{"binding":{"chosen_proof_sha256":"08f7647cfd9ab05023d94bc2fcca4631870d5f71d7f24de348f6c58aa5102a1d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"49e29b7526fb817a02fc6d72904a3c42721a221b956feae8f16c4d2d99722e34","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9b67a823dcea17b7fc9288545c47193c34fe1b90d1bd9cf3ef9be1964ef22f5d","source_sha256":"bf073edff8092ae47149eb64a7d94002d4cd84fb4941057816c6cf0f37fc36aa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isPosSemidef_def, Matrix.posSemidef_iff_dotProduct_mulVec]\n  apply and_congr (B.isSymm_iff_isHermitian_toMatrix b)\n  rw [isNonneg_def]\n  refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩\n  · rw [star_dotProduct_toMatrix₂_mulVec]\n    exact h _\n  · rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]\n    exact h _","hard_negative":false,"metrics":{"chosen_tokens":58,"rejected_tokens":63,"token_jaccard":0.864865,"token_length_ratio":1.086207},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"006a588191b537987ce94f10d10cf124332130be53a18cec7aeaffc2d310f19f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Matrix.PosDef\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Etienne Marion. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Etienne Marion\n-/\n/-!\n# Sesquilinear forms over a star ring\n\nThis file provides some properties about sesquilinear forms `M →ₗ⋆[R] M →ₗ[R] R` when `R` is a\n`StarRing`.\n-/\n\npublic section\n\nopen Module LinearMap\n\nvariable {R M n : Type*} [CommSemiring R] [StarRing R] [AddCommMonoid M] [Module R M]\n  [Fintype n] [DecidableEq n]\n  {B : M →ₗ⋆[R] M →ₗ[R] R} (b : Basis n R M)\n\nlemma LinearMap.isSymm_iff_basis {ι : Type*} (b : Basis ι R M) :\n    IsSymm B ↔ ∀ i j, star (B (b i) (b j)) = B (b j) (b i) where\n  mp h i j := h.eq _ _\n  mpr := by\n    refine fun h ↦ ⟨fun x y ↦ ?_⟩\n    obtain ⟨fx, tx, ix, -, hx⟩ := Submodule.mem_span_iff_exists_finset_subset.1\n      (by simp : x ∈ Submodule.span R (Set.range b))\n    obtain ⟨fy, ty, iy, -, hy⟩ := Submodule.mem_span_iff_exists_finset_subset.1\n      (by simp : y ∈ Submodule.span R (Set.range b))\n    rw [← hx, ← hy]\n    simp only [map_sum, LinearMap.map_smulₛₗ, starRingEnd_apply, map_smul, coe_sum,\n      Finset.sum_apply, smul_apply, smul_eq_mul, Finset.mul_sum, map_mul, star_star]\n    rw [Finset.sum_comm]\n    refine Finset.sum_congr rfl (fun b₁ h₁ ↦ Finset.sum_congr rfl fun b₂ h₂ ↦ ?_)\n    rw [mul_left_comm]\n    obtain ⟨i, rfl⟩ := ix h₁\n    obtain ⟨j, rfl⟩ := iy h₂\n    rw [h]\n\nlemma LinearMap.isSymm_iff_isHermitian_toMatrix : B.IsSymm ↔ (toMatrix₂ b b B).IsHermitian := by\n  rw [isSymm_iff_basis b, Matrix.IsHermitian.ext_iff, forall_comm]\n  simp [Eq.comm]\n\nlemma star_dotProduct_toMatrix₂_mulVec (x y : n → R) :\n    star x ⬝ᵥ (toMatrix₂ b b B).mulVec y = B (b.equivFun.symm x) (b.equivFun.symm y) :=\n  dotProduct_toMatrix₂_mulVec b b B x y\n\nlemma apply_eq_star_dotProduct_toMatrix₂_mulVec (x y : M) :\n    B x y = star (b.repr x) ⬝ᵥ (toMatrix₂ b b B).mulVec (b.repr y) :=\n  apply_eq_dotProduct_toMatrix₂_mulVec b b B x y\n\nvariable {R : Type*} [CommRing R] [StarRing R] [PartialOrder R] [Module R M]\n  {B : M →ₗ⋆[R] M →ₗ[R] R} (b : Basis n R M)\n\nTarget:\nlemma LinearMap.isPosSemidef_iff_posSemidef_toMatrix :\n    B.IsPosSemidef ↔ (toMatrix₂ b b B).PosSemidef :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  rw [isPosSemidef_def, Matrix.posSemidef_iff_dotProduct_mulVec]\n  apply and_congr (B.isSymm_iff_isHermitian_toMatrix b)\n  rw [isNonneg_def]\n  refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩\n  · rw [star_dotProduct_toMatrix₂_mulVec]\n    exact h _\n  · rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]\n    exact h _","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/SesquilinearForm","family_id":"linearmap","file_id":"mathlib/Mathlib/LinearAlgebra/SesquilinearForm/Star.lean","sample_id":"9b67a823dcea17b7fc9288545c47193c34fe1b90d1bd9cf3ef9be1964ef22f5d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"343a166fea2e90fed4c3043781c3d8f443aa011a6cdb85e131c437406e24ed78","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1f5f28afd315b861373a4cc9f83e458915950feee7818959152461dd28920518","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f98433b4daad566e797079ce130b71338bc97d14ae011a049aadb8ae222adbed","source_sha256":"92db59c0cca19f9c28529fd17474c1f344e67bd83f08b4344e7abe51502b9b08","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : C a • D f = D (C a * f) := by simp\n  rw [this, C_mul', D.map_smul]","hard_negative":false,"metrics":{"chosen_tokens":29,"rejected_tokens":36,"token_jaccard":0.916667,"token_length_ratio":1.241379},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"008161ec5aa1fb2766571828af325a0c8509ce89105a1ad79eadcdffbbbdc6f5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.AlgebraMap\npublic import Mathlib.Algebra.Polynomial.Derivative\npublic import Mathlib.Algebra.Polynomial.Module.AEval\npublic import Mathlib.RingTheory.Adjoin.Polynomial.Basic\npublic import Mathlib.RingTheory.Derivation.Basic\n\nNamespace:\nPolynomial\n\nLocal context:\n/-\nCopyright (c) 2023 Kevin Buzzard. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Richard M. Hill\n-/\n/-!\n# Derivations of univariate polynomials\n\nIn this file we prove that an `R`-derivation of `Polynomial R` is determined by its value on `X`.\nWe also provide a constructor `Polynomial.mkDerivation` that\nbuilds a derivation from its value on `X`, and a linear equivalence\n`Polynomial.mkDerivationEquiv` between `A` and `Derivation (Polynomial R) A`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nnamespace Polynomial\n\nsection CommSemiring\n\nvariable {R A : Type*} [CommSemiring R]\n\n/-- `Polynomial.derivative` as a derivation. -/\n@[simps]\ndef derivative' : Derivation R R[X] R[X] where\n  toFun := derivative\n  map_add' _ _ := derivative_add\n  map_smul' := derivative_smul\n  map_one_eq_zero' := derivative_one\n  leibniz' f g := by simp [mul_comm, add_comm, derivative_mul]\n\nvariable [AddCommMonoid A] [Module R A] [Module (Polynomial R) A]\n\n@[simp]\ntheorem derivation_C (D : Derivation R R[X] A) (a : R) : D (C a) = 0 :=\n  D.map_algebraMap a\n\n@[simp]\n\nTarget:\ntheorem C_smul_derivation_apply (D : Derivation R R[X] A) (a : R) (f : R[X]) :\n    C a • D f = a • D f :=\n\nProof body:\n","rejected":"```lean\nby\n  have : C a • D f = D (C a * f) := by simp\n  rw [this, C_mul', D.map_smul]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Polynomial","family_id":"c_smul_derivation_apply","file_id":"mathlib/Mathlib/Algebra/Polynomial/Derivation.lean","sample_id":"f98433b4daad566e797079ce130b71338bc97d14ae011a049aadb8ae222adbed"},"split":"train"}
{"binding":{"chosen_proof_sha256":"667083b2ccb953eab7042f8125e1495f0c41208219db3ae4328356bea107af18","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e486eab1ccf683613dfe0b4c7f4076b42faa49c71bab6abb88be2134dec9198b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"76ce3bf080d160d098641f9edded107effa5e468a4e154be6bb90faf62e605f0","source_sha256":"3b1ad71a44a0ba95fe172fff4adb35c1c3d0ab3dacc0cacca37d6024c51fd12b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [div_def, ← @Rat.round_cast ℝ _ _, ← @Rat.round_cast ℝ _ _]\n  simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, add_mul]","hard_negative":false,"metrics":{"chosen_tokens":36,"rejected_tokens":40,"token_jaccard":0.857143,"token_length_ratio":1.111111},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"00b4d46a98db70243ec6d7266484f8f7eb851885d9aaa9d16dbcf198f8f67cd7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Archimedean.Real.Basic\npublic import Mathlib.Data.Complex.Basic\npublic import Mathlib.Data.Nat.Prime.Basic\npublic import Mathlib.NumberTheory.Zsqrtd.Basic\n\nNamespace:\nGaussianInt\n\nLocal context:\n/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Gaussian integers\n\nThe Gaussian integers are complex integer, complex numbers whose real and imaginary parts are both\nintegers.\n\n## Main definitions\n\nThe Euclidean domain structure on `ℤ[i]` is defined in this file.\n\nThe homomorphism `GaussianInt.toComplex` into the complex numbers is also defined in this file.\n\n## See also\n\nSee `NumberTheory.Zsqrtd.QuadraticReciprocity` for:\n* `prime_iff_mod_four_eq_three_of_nat_prime`:\n  A prime natural number is prime in `ℤ[i]` if and only if it is `3` mod `4`\n\n## Notation\n\nThis file uses the local notation `ℤ[i]` for `GaussianInt`\n\n## Implementation notes\n\nGaussian integers are implemented using the more general definition `Zsqrtd`, the type of integers\nadjoined a square root of `d`, in this case `-1`. The definition is reducible, so that properties\nand definitions about `Zsqrtd` can easily be used.\n-/\n\n@[expose] public section\n\n\nopen Zsqrtd Complex\n\nopen scoped ComplexConjugate\n\n/-- The Gaussian integers, defined as `ℤ√(-1)`. -/\nabbrev GaussianInt : Type :=\n  Zsqrtd (-1)\n\nlocal notation \"ℤ[i]\" => GaussianInt\n\nnamespace GaussianInt\n\ninstance : Repr ℤ[i] :=\n  ⟨fun x _ => \"⟨\" ++ repr x.re ++ \", \" ++ repr x.im ++ \"⟩\"⟩\n\ninstance instCommRing : CommRing ℤ[i] :=\n  Zsqrtd.commRing\n\nsection\n\nattribute [-instance] Complex.instField -- Avoid making things noncomputable unnecessarily.\n\n/-- The embedding of the Gaussian integers into the complex numbers, as a ring homomorphism. -/\ndef toComplex : ℤ[i] →+* ℂ :=\n  Zsqrtd.lift ⟨I, by simp⟩\n\nend\n\ninstance : Coe ℤ[i] ℂ :=\n  ⟨toComplex⟩\n\ntheorem toComplex_def (x : ℤ[i]) : (x : ℂ) = x.re + x.im * I :=\n  rfl\n\ntheorem toComplex_def' (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ) = x + y * I := by simp [toComplex_def]\n\ntheorem toComplex_def₂ (x : ℤ[i]) : (x : ℂ) = ⟨x.re, x.im⟩ := by\n  apply Complex.ext <;> simp [toComplex_def]\n\n@[simp]\ntheorem intCast_re (x : ℤ[i]) : ((x.re : ℤ) : ℝ) = (x : ℂ).re := by simp [toComplex_def]\n\n@[simp]\ntheorem intCast_im (x : ℤ[i]) : ((x.im : ℤ) : ℝ) = (x : ℂ).im := by simp [toComplex_def]\n\n@[simp]\ntheorem re_toComplex (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).re = x := by simp [toComplex_def]\n\n@[simp]\ntheorem im_toComplex (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).im = y := by simp [toComplex_def]\n\ntheorem toComplex_add (x y : ℤ[i]) : ((x + y : ℤ[i]) : ℂ) = x + y :=\n  toComplex.map_add _ _\n\ntheorem toComplex_mul (x y : ℤ[i]) : ((x * y : ℤ[i]) : ℂ) = x * y :=\n  toComplex.map_mul _ _\n\ntheorem toComplex_one : ((1 : ℤ[i]) : ℂ) = 1 :=\n  toComplex.map_one\n\ntheorem toComplex_zero : ((0 : ℤ[i]) : ℂ) = 0 :=\n  toComplex.map_zero\n\ntheorem toComplex_neg (x : ℤ[i]) : ((-x : ℤ[i]) : ℂ) = -x :=\n  toComplex.map_neg _\n\ntheorem toComplex_sub (x y : ℤ[i]) : ((x - y : ℤ[i]) : ℂ) = x - y :=\n  toComplex.map_sub _ _\n\n@[simp]\ntheorem toComplex_star (x : ℤ[i]) : ((star x : ℤ[i]) : ℂ) = conj (x : ℂ) := by\n  rw [toComplex_def₂, toComplex_def₂]\n  exact congr_arg₂ _ rfl (Int.cast_neg _)\n\n@[simp]\ntheorem toComplex_inj {x y : ℤ[i]} : (x : ℂ) = y ↔ x = y := by\n  cases x; cases y; simp [toComplex_def₂]\n\nlemma toComplex_injective : Function.Injective GaussianInt.toComplex :=\n  fun ⦃_ _⦄ ↦ toComplex_inj.mp\n\n@[simp]\ntheorem toComplex_eq_zero {x : ℤ[i]} : (x : ℂ) = 0 ↔ x = 0 := by\n  rw [← toComplex_zero, toComplex_inj]\n\n@[simp]\ntheorem intCast_real_norm (x : ℤ[i]) : (x.norm : ℝ) = Complex.normSq (x : ℂ) := by\n  rw [Zsqrtd.norm, normSq]; simp\n\n@[simp]\ntheorem intCast_complex_norm (x : ℤ[i]) : (x.norm : ℂ) = Complex.normSq (x : ℂ) := by\n  cases x; rw [Zsqrtd.norm, normSq]; simp\n\ntheorem norm_nonneg (x : ℤ[i]) : 0 ≤ norm x :=\n  Zsqrtd.norm_nonneg (by simp) _\n\n@[simp]\ntheorem norm_eq_zero {x : ℤ[i]} : norm x = 0 ↔ x = 0 := by rw [← @Int.cast_inj ℝ _ _ _]; simp\n\ntheorem norm_pos {x : ℤ[i]} : 0 < norm x ↔ x ≠ 0 := by\n  rw [lt_iff_le_and_ne, Ne, eq_comm, norm_eq_zero]; simp [norm_nonneg]\n\ntheorem abs_natCast_norm (x : ℤ[i]) : (x.norm.natAbs : ℤ) = x.norm :=\n  Int.natAbs_of_nonneg (norm_nonneg _)\n\ntheorem natCast_natAbs_norm {α : Type*} [AddGroupWithOne α] (x : ℤ[i]) :\n    (x.norm.natAbs : α) = x.norm := by\n  simp\n\ntheorem natAbs_norm_eq (x : ℤ[i]) :\n    x.norm.natAbs = x.re.natAbs * x.re.natAbs + x.im.natAbs * x.im.natAbs := by\n  zify\n  rw [abs_norm (by simp)]\n  simp [Zsqrtd.norm]\n\ninstance : Div ℤ[i] :=\n  ⟨fun x y =>\n    let n := (norm y : ℚ)⁻¹\n    let c := star y\n    ⟨round ((x * c).re * n : ℚ), round ((x * c).im * n : ℚ)⟩⟩\n\ntheorem div_def (x y : ℤ[i]) :\n    x / y = ⟨round ((x * star y).re / norm y : ℚ), round ((x * star y).im / norm y : ℚ)⟩ :=\n  show Zsqrtd.mk _ _ = _ by simp [div_eq_mul_inv]\n\ntheorem toComplex_re_div (x y : ℤ[i]) : ((x / y : ℤ[i]) : ℂ).re = round (x / y : ℂ).re := by\n  rw [div_def, ← @Rat.round_cast ℝ _ _]\n  simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, add_mul]\n\nTarget:\ntheorem toComplex_im_div (x y : ℤ[i]) : ((x / y : ℤ[i]) : ℂ).im = round (x / y : ℂ).im :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [div_def, ← @Rat.round_cast ℝ _ _, ← @Rat.round_cast ℝ _ _]\n  simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, add_mul]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Zsqrtd","family_id":"tocomplex_im_div","file_id":"mathlib/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean","sample_id":"76ce3bf080d160d098641f9edded107effa5e468a4e154be6bb90faf62e605f0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8f3a85b33714251a1a29f92568064949152f5e84c72c953dd09287cb0b6f1455","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2a102d88cbe43376559a5015393d625cba3403a19843dca5452c2e82c2e44526","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"309747378fad33151e6310726ff0174aa0d05877efaa42bd524d8d5cf49ca2f9","source_sha256":"d9780449d2613fd42914cf0df8a7f4380debc01adc843be01bb83642da361938","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [NormNum.IsNat.to_eq h rfl]\n  exact Nat.cast_nonneg' n","hard_negative":false,"metrics":{"chosen_tokens":16,"rejected_tokens":21,"token_jaccard":0.736842,"token_length_ratio":1.3125},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"018ece668935a62be4abb3cf4d8d6f329c266da880680cdb88ac456c40f56ad6","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Invertible\npublic import Mathlib.Algebra.Order.Ring.Cast\npublic import Mathlib.Tactic.HaveI\npublic import Mathlib.Tactic.NormNum.Core\n\nNamespace:\nMathlib.Meta.Positivity\n\nLocal context:\n/-\nCopyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Heather Macbeth, Yaël Dillies\n-/\npublic meta import Mathlib.Control.Basic\n\n/-!\n## `positivity` core functionality\n\nThis file sets up the `positivity` tactic and the `@[positivity]` attribute,\nwhich allow for plugging in new positivity functionality around a positivity-based driver.\nThe actual behavior is in `@[positivity]`-tagged definitions in `Tactic.Positivity.Basic`\nand elsewhere.\n-/\n\npublic meta section\n\nopen Lean\nopen Lean.Meta Qq Lean.Elab Term\n\n/-- A definition of type `PositivityExt` tagged `@[positivity t]` extends the `positivity` tactic.\nThe term (with underscores) `t` indicates which expressions this extension accepts.\nAn extension will be given an expression `e : α`, together with hypotheses\n`[Zero α] [PartialOrder α]` and attempts to prove `e > 0`, `e ≥ 0`, or `e ≠ 0`.\n\nWhen `Positivity.core` calls this extension on an expression `e`, it does not guarantee that `e`\nmatches `t` perfectly: validate the form of the expression (using e.g.\n`match_expr (← withReducible (whnf e))`) before building a proof. See also the\n`let .app ... ← withReducible (whnf e) | throwError ...` lines in the example below.\n\nAn extension can call `Mathlib.Meta.Positivity.core` to recursively solve subgoals.\n\nExample:\n```lean\n@[positivity ite _ _ _] def evalIte : PositivityExt where eval {u α} zα pα e := do\n  let .app (.app (.app (.app f (p : Q(Prop))) (_ : Q(Decidable $p))) (a : Q($α))) (b : Q($α))\n    ← withReducible (whnf e) | throwError \"not ite\"\n  haveI' : $e =Q ite $p $a $b := ⟨⟩\n  guard <| ← withDefault <| withNewMCtxDepth <| isDefEq f q(ite (α := $α))\n  let ra ← core zα pα a; let rb ← core zα pα b\n  ...\n```\n-/\nsyntax (name := positivity) \"positivity \" term,+ : attr\n\nlemma ne_of_ne_of_eq' {α : Sort*} {a c b : α} (hab : (a : α) ≠ c) (hbc : a = b) : b ≠ c := hbc ▸ hab\n\nnamespace Mathlib.Meta.Positivity\n\nvariable {u : Level} {α : Q(Type u)} (zα : Q(Zero $α))\n\n/-- The result of `positivity` running on an expression `e` of type `α`. -/\ninductive Strictness (e : Q($α)) : Option Q(PartialOrder $α) → Type where\n  | positive {pα : Q(PartialOrder $α)} (pf : Q(0 < $e)) : Strictness e pα\n  | nonnegative {pα : Q(PartialOrder $α)} (pf : Q(0 ≤ $e)) : Strictness e pα\n  | nonzero {pα?} (pf : Q($e ≠ 0)) : Strictness e pα?\n  | none {pα?} : Strictness e pα?\n\n/-- Gives a generic description of the `positivity` result. -/\ndef Strictness.toString {e pα?} : Strictness zα e pα? → String\n  | positive _ => \"positive\"\n  | nonnegative _ => \"nonnegative\"\n  | nonzero _ => \"nonzero\"\n  | none => \"none\"\n\n/-- Extract a proof that `e` is positive, if possible, from `Strictness` information about `e`. -/\ndef Strictness.toPositive {e pα} : Strictness zα e (some pα) → Option Q(0 < $e)\n  | .positive pf => some pf\n  | _ => .none\n\n/-- Extract a proof that `e` is nonnegative, if possible, from `Strictness` information about `e`.\n-/\ndef Strictness.toNonneg {e pα} : Strictness zα e (some pα) → Option Q(0 ≤ $e)\n  | .positive pf => some q(le_of_lt $pf)\n  | .nonnegative pf => some pf\n  | _ => .none\n\n/-- Extract a proof that `e` is nonzero, if possible, from `Strictness` information about `e`. -/\ndef Strictness.toNonzero {e pα?} : Strictness zα e pα? → Option Q($e ≠ 0)\n  | .positive pf => some q(ne_of_gt $pf)\n  | .nonzero pf => some pf\n  | _ => .none\n\n/-- An extension for `positivity`. -/\nstructure PositivityExt where\n  /-- Attempts to prove an expression `e : α` is `>0`, `≥0`, or `≠0`. -/\n  eval {u : Level} {α : Q(Type u)} (zα : Q(Zero $α)) (pα? : Option Q(PartialOrder $α)) (e : Q($α)) :\n    MetaM (Strictness zα e pα?)\n\n/-- Read a `positivity` extension from a declaration of the right type. -/\ndef mkPositivityExt (n : Name) : ImportM PositivityExt := do\n  let { env, opts, .. } ← read\n  IO.ofExcept <| unsafe env.evalConstCheck PositivityExt opts ``PositivityExt n\n\n/-- Each `positivity` extension is labelled with a collection of patterns\nwhich determine the expressions to which it should be applied. -/\nabbrev Entry := Array (Array DiscrTree.Key) × Name\n\n/-- Environment extensions for `positivity` declarations -/\ninitialize positivityExt : PersistentEnvExtension Entry (Entry × PositivityExt)\n    (List Entry × DiscrTree PositivityExt) ←\n  -- we only need this to deduplicate entries in the DiscrTree\n  have : BEq PositivityExt := ⟨fun _ _ => false⟩\n  let insert kss v dt := kss.foldl (fun dt ks => dt.insertKeyValue ks v) dt\n  registerPersistentEnvExtension {\n    mkInitial := pure ([], {})\n    addImportedFn := fun s => do\n      let dt ← s.foldlM (init := {}) fun dt s => s.foldlM (init := dt) fun dt (kss, n) => do\n        pure (insert kss (← mkPositivityExt n) dt)\n      pure ([], dt)\n    addEntryFn := fun (entries, s) ((kss, n), ext) => ((kss, n) :: entries, insert kss ext s)\n    exportEntriesFn := fun s => s.1.reverse.toArray\n  }\n\ninitialize registerBuiltinAttribute {\n  name := `positivity\n  descr := \"adds a positivity extension\"\n  applicationTime := .afterCompilation\n  add := fun declName stx kind => match stx with\n    | `(attr| positivity $es,*) => do\n      ensureAttrDeclIsMeta `positivity declName kind\n      unless kind == AttributeKind.global do\n        throwError \"invalid attribute 'positivity', must be global\"\n      let env ← getEnv\n      unless (env.getModuleIdxFor? declName).isNone do\n        throwError \"invalid attribute 'positivity', declaration is in an imported module\"\n      if (IR.getSorryDep env declName).isSome then return -- ignore in progress definitions\n      let ext ← mkPositivityExt declName\n      let keys ← MetaM.run' <| es.getElems.mapM fun stx => do\n        let e ← TermElabM.run' <| withSaveInfoContext <| withAutoBoundImplicit <|\n          withReader ({ · with ignoreTCFailures := true }) do\n            let e ← elabTerm stx none\n            let (_, _, e) ← lambdaMetaTelescope (← mkLambdaFVars (← getLCtx).getFVars e)\n            return e\n        DiscrTree.mkPath e\n      setEnv <| positivityExt.addEntry env ((keys, declName), ext)\n      -- TODO: track what `[positivity]` decls are actually used at use sites\n      recordExtraRevUseOfCurrentModule\n    | _ => throwUnsupportedSyntax\n}\n\nvariable {A : Type*} {e : A}\n\nlemma pos_of_isNat {n : ℕ} [Semiring A] [PartialOrder A] [IsOrderedRing A] [Nontrivial A]\n    (h : NormNum.IsNat e n) (w : Nat.ble 1 n = true) : 0 < (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  apply Nat.cast_pos.2\n  simpa using! w\n\nlemma pos_of_isNat' {n : ℕ}\n    [AddMonoidWithOne A] [PartialOrder A] [AddLeftMono A] [ZeroLEOneClass A] [h'' : NeZero (1 : A)]\n    (h : NormNum.IsNat e n) (w : Nat.ble 1 n = true) : 0 < (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  apply Nat.cast_pos'.2\n  simpa using! w\n\nlemma nonneg_of_isNat {n : ℕ} [Semiring A] [PartialOrder A] [IsOrderedRing A]\n    (h : NormNum.IsNat e n) : 0 ≤ (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  exact Nat.cast_nonneg n\n\nTarget:\nlemma nonneg_of_isNat' {n : ℕ}\n    [AddMonoidWithOne A] [PartialOrder A] [AddLeftMono A] [ZeroLEOneClass A]\n    (h : NormNum.IsNat e n) : 0 ≤ (e : A) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  rw [NormNum.IsNat.to_eq h rfl]\n  exact Nat.cast_nonneg' n","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Positivity","family_id":"nonneg_of_isnat","file_id":"mathlib/Mathlib/Tactic/Positivity/Core.lean","sample_id":"309747378fad33151e6310726ff0174aa0d05877efaa42bd524d8d5cf49ca2f9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3d1a7e2cbfbf665394764afad32089ad753a926b64cde1fb10b4f271be3fc87d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ac00d8bac021edc80086045826c5231fb9c8b98b4c9ec6afaac414206bf5e6ea","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0f5b96fc655dafea499598a2e613044abf5991cec77163c042656873851c8051","source_sha256":"5e905b178f0e9eddf55a6c8ae9c3e3ec7191d0314931708fc81e73ffffb986ad","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain (hs | hs) := hs\n  · exact hs.2\n  · exact hs.1.isCompact","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":25,"token_jaccard":0.8125,"token_length_ratio":1.190476},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"01e20b3186c068c1f21ac0d461eb91dad6eacf8557baadfe969cd4871e4b9f7b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Spectral.Basic\npublic import Mathlib.Topology.WithTopology\npublic import Mathlib.Topology.JacobsonSpace\npublic import Mathlib.Data.Set.Card\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Jiedong Jiang, Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Jiedong Jiang, Christian Merten\n-/\n/-!\n# Constructible topology\n\nIn this file we define the constructible topology on a topological space. This is the topology\ngenerated by compact open subsets and their complements.\n\n## Main definitions and results\n\n- `WithConstructibleTopology`: `X` equipped with its constructible topology.\n- `compactSpace_withConstructibleTopology`: If `X` is quasi-separated, quasi-sober,\n  prespectral and quasi-compact, then `X` is still quasi-compact in the constructible topology.\n\n## TODOs\n\n- If `X` is a spectral space, show that `X` is T2 and totally disconnected (@chrisflav).\n-/\n\n@[expose] public section\n\nvariable {X : Type*} [TopologicalSpace X]\n\n/-- The subbasis of the constructible topology on a topological space `X`: It consists\nof the open and compact sets of `X` and their complements. -/\ndef constructibleTopologySubbasis (X : Type*) [TopologicalSpace X] : Set (Set X) :=\n  { s | IsOpen s ∧ IsCompact s } ∪ { s | IsClosed s ∧ IsCompact sᶜ }\n\n/-- The constructible topology on a topological space `X` has as a subbasis\nthe open and compact sets of `X` and their complements. -/\n@[implicit_reducible]\ndef constructibleTopology (X : Type*) [TopologicalSpace X] : TopologicalSpace X :=\n  .generateFrom (constructibleTopologySubbasis X)\n\n/-- A type synonym for `X` that is equipped with the constructible topology of `X`. -/\n@[nolint unusedArguments]\nabbrev WithConstructibleTopology (X : Type*) [TopologicalSpace X] : Type _ :=\n  WithTopology X (constructibleTopology X)\n\nopen Topology\n\nlemma WithConstructibleTopology.isOpen_iff {s : Set (WithConstructibleTopology X)} :\n    IsOpen s ↔ IsOpen[constructibleTopology X] (WithTopology.toTopology _ ⁻¹' s) :=\n  WithTopology.isOpen_iff _\n\nlemma WithConstructibleTopology.isClosed_iff {s : Set (WithConstructibleTopology X)} :\n    IsClosed s ↔ IsClosed[constructibleTopology X] (WithTopology.toTopology _ ⁻¹' s) :=\n  WithTopology.isClosed_iff _\n\nlemma IsCompact.isOpen_constructibleTopology_of_isOpen {s : Set X}\n    (hs : IsCompact s) (ho : IsOpen s) : IsOpen[constructibleTopology X] s := by\n  apply TopologicalSpace.isOpen_generateFrom_of_mem\n  simp [constructibleTopologySubbasis, ho, hs]\n\nlemma IsCompact.isOpen_constructibleTopology_of_isClosed {s : Set X}\n    (hs : IsCompact sᶜ) (ho : IsClosed s) : IsOpen[constructibleTopology X] s := by\n  apply TopologicalSpace.isOpen_generateFrom_of_mem\n  simp [constructibleTopologySubbasis, ho, hs]\n\n@[simp]\nlemma compl_mem_constructibleTopologySubbasis_iff {s : Set X} :\n    sᶜ ∈ constructibleTopologySubbasis X ↔ s ∈ constructibleTopologySubbasis X := by\n  grind [constructibleTopologySubbasis, isClosed_compl_iff, compl_compl]\n\nTarget:\nlemma isCompact_of_mem_constructibleTopologySubbasis [CompactSpace X] {s : Set X}\n    (hs : s ∈ constructibleTopologySubbasis X) : IsCompact s :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  obtain (hs | hs) := hs\n  · exact hs.2\n  · exact hs.1.isCompact","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Spectral","family_id":"iscompact_of_mem_constructibletopologysubbasis","file_id":"mathlib/Mathlib/Topology/Spectral/ConstructibleTopology.lean","sample_id":"0f5b96fc655dafea499598a2e613044abf5991cec77163c042656873851c8051"},"split":"train"}
{"binding":{"chosen_proof_sha256":"255334aceee88256ccde5808751019138f069b260876b485bc11ea482a1aed8d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e0497951fa64a3a8bf8993417b48930a17afd784b7a57ac8d34e2b837d3c4c49","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"04f65d5b50a300cc6227733efdcd3f39e044ddd8767d10f8047fdbe408fc2df7","source_sha256":"98d8a988714b8411039a52ca8cb94b19a051f27bd0e995b1332bddd6c23da9ea","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro b hb\n  have hg' := hg b hb\n  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hg'⟩ := hg'\n  refine ⟨c₁, hc₁_mem, c₂, hc₂_mem, ?_⟩\n  filter_upwards [hg', (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg, hfg]\n    with x hx₁ hx₂ hx₃\n  intro u hu\n  rw [hx₂ u hu.1, hx₃]\n  exact hx₁ u hu","hard_negative":true,"metrics":{"chosen_tokens":95,"rejected_tokens":3,"token_jaccard":0.05,"token_length_ratio":0.031579},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"01f45cf449314a1f396201cad53d979f4eb7eb8729d86df2e208459d7c12a39e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Pow.Real\npublic import Mathlib.Algebra.Order.ToIntervalMod\npublic import Mathlib.Analysis.SpecialFunctions.Log.Base\nimport Mathlib.Algebra.Order.Interval.Set.Group\n\nNamespace:\nAkraBazziRecurrence.GrowsPolynomially\n\nLocal context:\n/-\nCopyright (c) 2023 Frédéric Dupuis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Frédéric Dupuis\n-/\n/-!\n# Akra-Bazzi theorem: the polynomial growth condition\n\nThis file defines and develops an API for the polynomial growth condition that appears in the\nstatement of the Akra-Bazzi theorem: for the theorem to hold, the function `g` must\nsatisfy the condition that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b*n` and `n` for any\nconstant `b ∈ (0,1)`.\n\n## Implementation notes\n\nOur definition requires that the condition hold for any `b ∈ (0,1)`. This is equivalent to requiring\nit only for `b = 1 / 2` (or any other particular value in `(0, 1)`). While this could, in principle,\nmake it harder to prove that a particular function grows polynomially, this issue does not seem to\narise in practice.\n\n-/\n\n@[expose] public section\n\nopen Finset Real Filter Asymptotics\nopen scoped Topology\n\nnamespace AkraBazziRecurrence\n\n/-- The growth condition that the function `g` must satisfy for the Akra-Bazzi theorem to apply.\nIt roughly states that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b * n` and `n`, for any\nconstant `b ∈ (0, 1)`. -/\ndef GrowsPolynomially (f : ℝ → ℝ) : Prop :=\n  ∀ b ∈ Set.Ioo 0 1, ∃ c₁ > 0, ∃ c₂ > 0,\n    ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * (f x)) (c₂ * f x)\n\nnamespace GrowsPolynomially\n\nTarget:\nlemma congr_of_eventuallyEq {f g : ℝ → ℝ} (hfg : f =ᶠ[atTop] g) (hg : GrowsPolynomially g) :\n    GrowsPolynomially f :=\n\nProof body:\n","rejected":"by\n  exact congr_of_eventuallyEq","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"5e5c1b87bdcc6b043dfed46c9a7c541bf3d9a6766666dc2a8e16b19fdb4ccfca","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability/AkraBazzi","family_id":"congr_of_eventuallyeq","file_id":"mathlib/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean","sample_id":"04f65d5b50a300cc6227733efdcd3f39e044ddd8767d10f8047fdbe408fc2df7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"61dd4a4b69745b7e0fa68c4103cf026764280161c8ce4459f05b5e5caa88e47f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ff92774dd6570cff0844a95accc32cba20c68bc0bb5d35a0778f354de4405f0e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7de96c1ef7802d716a78d8f34f82e51e820ae7e56b3d2af666ad2f6428d02fa0","source_sha256":"15e2b4cca4919764d55bc958e94a932ae425f5e9b6e7dc6036a891b4490f43b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [univLE_iff_cardinal_le]; apply le_total","hard_negative":false,"metrics":{"chosen_tokens":8,"rejected_tokens":13,"token_jaccard":0.615385,"token_length_ratio":1.625},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"0208076cd7af211c6929eeda1e2a5d13d9881f1f09cbf03935e09d4a89f55e1b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Logic.UnivLE\npublic import Mathlib.SetTheory.Ordinal.Univ\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu\n-/\n/-!\n# UnivLE and cardinals\n-/\n\npublic section\n\nnoncomputable section\n\nuniverse u v\n\nopen Cardinal\n\ntheorem univLE_iff_cardinal_le : UnivLE.{u, v} ↔ univ.{u, v + 1} ≤ univ.{v, u + 1} := by\n  simp_rw [univLE_iff, small_iff_lift_mk_lt_univ]\n  contrapose!\n  -- strange: simp_rw [univ_umax.{v,u}] doesn't work\n  refine ⟨fun ⟨α, le⟩ ↦ ?_, fun h ↦ ?_⟩\n  · rw [univ_umax.{v, u}, ← lift_le.{u + 1}, lift_univ, lift_lift] at le\n    exact le.trans_lt (lift_lt_univ'.{u, v + 1} #α)\n  · obtain ⟨⟨α⟩, h⟩ := lt_univ'.mp h; use α\n    rw [univ_umax.{v, u}, ← lift_le.{u + 1}, lift_univ, lift_lift]\n    exact h.le\n\ntheorem univLE_iff_exists_embedding : UnivLE.{u, v} ↔ Nonempty (Ordinal.{u} ↪ Ordinal.{v}) := by\n  rw [univLE_iff_cardinal_le]\n  exact lift_mk_le'\n\ntheorem Ordinal.univLE_of_injective {f : Ordinal.{u} → Ordinal.{v}} (h : f.Injective) :\n    UnivLE.{u, v} :=\n  univLE_iff_exists_embedding.2 ⟨f, h⟩\n\n/-- Together with transitivity, this shows `UnivLE` is a total preorder. -/\n\nTarget:\ntheorem univLE_total : UnivLE.{u, v} ∨ UnivLE.{v, u} :=\n\nProof body:\n","rejected":"by\n  simp_rw [univLE_iff_cardinal_le]; apply le_total\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Cardinal","family_id":"univle_total","file_id":"mathlib/Mathlib/SetTheory/Cardinal/UnivLE.lean","sample_id":"7de96c1ef7802d716a78d8f34f82e51e820ae7e56b3d2af666ad2f6428d02fa0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1c6409609dd76697d622025294c8bc5145bcfe414ffbcf9cd053bd8d9c3b4785","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"de181494cf6df2f46264b7108f8b5fe8ade01b1f3c5c162ddf30528ffc055308","source_sha256":"e262008d04b45974f36575d7d297716d0a028e3928a5aa24f21cf62385c66b81","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let w'' : V₂.op ⋙ eB.op.functor ≅ eT.op.functor ⋙ V₃.op := NatIso.op w'\n  have : (w ≫ₕ w'.hom).op = (w.op ≫ᵥ w''.hom) := by ext; simp [w'']\n  rw [← guitartExact_op_iff, ← guitartExact_op_iff w,\n    ← vComp_iff_of_equivalences _ _ _ w'', this]\n  rfl","hard_negative":false,"metrics":{"chosen_tokens":80,"rejected_tokens":2,"token_jaccard":0.025,"token_length_ratio":0.025},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"0258a2a508961b1612849636bcad3885eb05b24688e91001df56fe74d3e18880","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.GuitartExact.Opposite\n\nNamespace:\nCategoryTheory.TwoSquare.GuitartExact\n\nLocal context:\n/-\nCopyright (c) 2026 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Horizontal composition of Guitart exact squares\n\nIn this file, we show that the horizontal composition of Guitart exact squares\nis Guitart exact.\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Category\n\nvariable {C₁ C₂ C₃ D₁ D₂ D₃ : Type*} [Category* C₁] [Category* C₂] [Category* C₃]\n  [Category* D₁] [Category* D₂] [Category* D₃]\n\nnamespace TwoSquare\n\nsection WhiskerHorizontal\n\nvariable {T : C₁ ⥤ D₁} {L : C₁ ⥤ C₂} {R : D₁ ⥤ D₂} {B : C₂ ⥤ D₂} (w : TwoSquare T L R B)\n  {T' : C₁ ⥤ D₁} {B' : C₂ ⥤ D₂}\n\n/-- Given `w : TwoSquare T L R B`, one may obtain a 2-square `TwoSquare T' L R B'` if we\nprovide natural transformations `α : T ⟶ T'` and `β : B' ⟶ B`. -/\n@[simps!]\ndef whiskerHorizontal (α : T' ⟶ T) (β : B ⟶ B') :\n    TwoSquare T' L R B' :=\n  (w.whiskerTop α).whiskerBottom β\n\nnamespace GuitartExact\n\nset_option backward.defeqAttrib.useBackward true in\n/-- A 2-square stays Guitart exact if we replace the top and bottom functors\nby isomorphic functors. See also `whiskerHorizontal_iff`. -/\nlemma whiskerHorizontal [w.GuitartExact] (α : T ≅ T') (β : B ≅ B') :\n    (w.whiskerHorizontal α.inv β.hom).GuitartExact := by\n  rw [guitartExact_iff_final]\n  intro X₂\n  let e : costructuredArrowRightwards (w.whiskerHorizontal α.inv β.hom) X₂ ≅\n      w.costructuredArrowRightwards X₂ ⋙ (CostructuredArrow.mapIso (β.app X₂)).functor :=\n    NatIso.ofComponents (fun f ↦ CostructuredArrow.isoMk (α.symm.app f.left))\n  rw [Functor.final_natIso_iff e]\n  infer_instance\n\n/-- A 2-square is Guitart exact iff it is so after replacing the top and bottom functors by\nisomorphic functors. -/\n@[simp]\nlemma whiskerHorizontal_iff (α : T ≅ T') (β : B ≅ B') :\n    (w.whiskerHorizontal α.inv β.hom).GuitartExact ↔ w.GuitartExact := by\n  rw [← guitartExact_op_iff, ← w.guitartExact_op_iff,\n    ← whiskerVertical_iff w.op (NatIso.op α.symm) (NatIso.op β.symm)]\n  rfl\n\ninstance [w.GuitartExact] (α : T' ⟶ T) (β : B ⟶ B')\n    [IsIso α] [IsIso β] : (w.whiskerHorizontal α β).GuitartExact :=\n  whiskerHorizontal w (asIso α).symm (asIso β)\n\nend GuitartExact\n\nend WhiskerHorizontal\n\nsection HorizontalComposition\n\nvariable {V₁ : C₁ ⥤ D₁} {T₁ : C₁ ⥤ C₂} {B₁ : D₁ ⥤ D₂} {V₂ : C₂ ⥤ D₂}\n  (w : TwoSquare T₁ V₁ V₂ B₁)\n  {T₂ : C₂ ⥤ C₃} {B₂ : D₂ ⥤ D₃} {V₃ : C₃ ⥤ D₃}\n  (w' : TwoSquare T₂ V₂ V₃ B₂)\n\n/-- The horizontal composition of 2-squares. (Variant where we allow the replacement of\nthe horizontal compositions by isomorphic functors.) -/\n@[simps!]\ndef hComp' {T₁₂ : C₁ ⥤ C₃} {B₁₂ : D₁ ⥤ D₃} (eT : T₁ ⋙ T₂ ≅ T₁₂) (eB : B₁ ⋙ B₂ ≅ B₁₂) :\n    TwoSquare T₁₂ V₁ V₃ B₁₂ :=\n  (w ≫ₕ w').whiskerHorizontal eT.inv eB.hom\n\nnamespace GuitartExact\n\nset_option backward.defeqAttrib.useBackward true in\ninstance hComp [w.GuitartExact] [w'.GuitartExact] :\n    (w ≫ₕ w').GuitartExact := by\n  rw [← guitartExact_op_iff]\n  have : (w ≫ₕ w').op = w.op ≫ᵥ w'.op := by ext; simp\n  rw [this]\n  exact inferInstanceAs (w.op ≫ᵥ w'.op).GuitartExact\n\ninstance hComp' {T₁₂ : C₁ ⥤ C₃} {B₁₂ : D₁ ⥤ D₃} (eT : T₁ ⋙ T₂ ≅ T₁₂) (eB : B₁ ⋙ B₂ ≅ B₁₂)\n    [w.GuitartExact] [w'.GuitartExact] :\n    (w.hComp' w' eT eB).GuitartExact := by\n  dsimp only [TwoSquare.hComp']\n  infer_instance\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The canonical isomorphism between\n`w.costructuredArrowRightwards Y₁ ⋙ w'.costructuredArrowRightwards (B₁.obj Y₁)` and\n`(w ≫ₕ w').costructuredArrowRightwards Y₁`. -/\ndef costructuredArrowRightwardsComp (Y₁ : D₁) :\n    w.costructuredArrowRightwards Y₁ ⋙ w'.costructuredArrowRightwards (B₁.obj Y₁) ≅\n      (w ≫ₕ w').costructuredArrowRightwards Y₁ :=\n  NatIso.ofComponents (fun _ => CostructuredArrow.isoMk (Iso.refl _))\n\nlemma of_hComp [B₁.EssSurj] [w.GuitartExact] [(w ≫ₕ w').GuitartExact] :\n    w'.GuitartExact := by\n  rw [guitartExact_iff_final]\n  intro Y₂\n  rw [costructuredArrowRightwards_final_iff_of_iso _ (B₁.objObjPreimageIso Y₂).symm]\n  have : (w.costructuredArrowRightwards (B₁.objPreimage Y₂) ⋙\n      w'.costructuredArrowRightwards (B₁.obj (B₁.objPreimage Y₂))).Final :=\n    (Functor.final_of_natIso (costructuredArrowRightwardsComp w w' _).symm :)\n  exact Functor.final_of_final_comp (w.costructuredArrowRightwards (B₁.objPreimage Y₂)) _\n\nlemma of_hComp' {T₁₂ : C₁ ⥤ C₃} {B₁₂ : D₁ ⥤ D₃} (eT : T₁ ⋙ T₂ ≅ T₁₂) (eB : B₁ ⋙ B₂ ≅ B₁₂)\n    [B₁.EssSurj] [w.GuitartExact] [h : (w.hComp' w' eT eB).GuitartExact] :\n    w'.GuitartExact := by\n  dsimp [TwoSquare.hComp'] at h\n  rw [whiskerHorizontal_iff] at h\n  exact of_hComp w w'\n\nlemma hComp_iff_of_essSurj [B₁.EssSurj] [w.GuitartExact] :\n    (w ≫ₕ w').GuitartExact ↔ w'.GuitartExact :=\n  ⟨fun _ ↦ of_hComp w w', fun _ ↦ inferInstance⟩\n\nlemma hComp'_iff_of_essSurj\n    {T₁₂ : C₁ ⥤ C₃} {B₁₂ : D₁ ⥤ D₃} (eT : T₁ ⋙ T₂ ≅ T₁₂) (eB : B₁ ⋙ B₂ ≅ B₁₂)\n    [B₁.EssSurj] [w.GuitartExact] :\n    (w.hComp' w' eT eB).GuitartExact ↔ w'.GuitartExact :=\n  ⟨fun _ ↦ of_hComp' w w' eT eB, fun _ ↦ inferInstance⟩\n\nset_option backward.defeqAttrib.useBackward true in\n\nTarget:\nlemma hComp_iff_of_equivalences (eT : C₂ ≌ C₃) (eB : D₂ ≌ D₃)\n    (w' : eT.functor ⋙ V₃ ≅ V₂ ⋙ eB.functor) :\n    (w ≫ₕ w'.hom).GuitartExact ↔ w.GuitartExact :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/GuitartExact","family_id":"hcomp_iff_of_equivalences","file_id":"mathlib/Mathlib/CategoryTheory/GuitartExact/HorizontalComposition.lean","sample_id":"de181494cf6df2f46264b7108f8b5fe8ade01b1f3c5c162ddf30528ffc055308"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2933327708396b2202fe5681ecb1091bcb221fad36d86c22e4ed4f768e81a87c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"77f925e6d7fb9f855b12d8cba8f5cb36cbebbc68e7684132b1a6af65dd6fe312","source_sha256":"fcd4a8db6b3cd3d3eac0ed5f4bd02b4de008d2f8c57c99f188ee276c869b4813","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext <;> simp","hard_negative":true,"metrics":{"chosen_tokens":6,"rejected_tokens":8,"token_jaccard":0.083333,"token_length_ratio":1.333333},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"02667132161f29e3e1dba5dd662419fad71580543d17984d1f9c0b446c0abaec","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Bicategory.LocallyDiscrete\npublic import Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo\n\nNamespace:\nCategoryTheory.Pseudofunctor.Grothendieck\n\nLocal context:\n/-\nCopyright (c) 2024 Calle Sönne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Calle Sönne, Joseph Hua\n-/\n/-!\n# The Grothendieck and CoGrothendieck constructions\n\n## The Grothendieck construction\n\nGiven a category `𝒮` and any pseudofunctor `F` from `𝒮` to `Cat`, we associate to it a category\n`∫ F`, defined as follows:\n* Objects: pairs `(S, a)` where `S` is an object of the base category and `a` is an object of the\n  category `F(S)`.\n* Morphisms: morphisms `(R, b) ⟶ (S, a)` are defined as pairs `(f, h)` where `f : R ⟶ S` is a\n  morphism in `𝒮` and `h : F(f)(a) ⟶ b`\n\nThe category `∫ F` is equipped with a projection functor `∫ F ⥤ 𝒮`,\ngiven by projecting to the first factors, i.e.\n* On objects, it sends `(S, a)` to `S`\n* On morphisms, it sends `(f, h)` to `f`\n\n## The CoGrothendieck construction\n\nGiven a category `𝒮` and any pseudofunctor `F` from `𝒮ᵒᵖ` to `Cat`,\nwe associate to it a category `∫ᶜ F`, defined as follows:\n* Objects: pairs `(S, a)` where `S` is an object of the base category and `a` is an object of the\n  category `F(S)`.\n* Morphisms: morphisms `(R, b) ⟶ (S, a)` are defined as pairs `(f, h)` where `f : R ⟶ S` is a\n  morphism in `𝒮` and `h : b ⟶ F(f)(a)`\n\nThe category `∫ᶜ F` is equipped with a functor `∫ᶜ F ⥤ 𝒮`,\ngiven by projecting to the first factors, i.e.\n* On objects, it sends `(S, a)` to `S`\n* On morphisms, it sends `(f, h)` to `f`\n\n## Naming conventions\n\nThe name `Grothendieck` is reserved for the construction on covariant pseudofunctors from `𝒮` to\n`Cat`, whereas the word `CoGrothendieck` is used for the contravariant construction.\nThis is consistent with the convention for the Grothendieck construction on 1-functors\n`CategoryTheory.Grothendieck`.\n\n## Future work / TODO\n\n1. Once the bicategory of pseudofunctors has been defined, show that this construction forms a\n   pseudofunctor from `LocallyDiscrete 𝒮 ⥤ᵖ Catᵒᵖ` to `Cat`.\n2. Deduce the results in `CategoryTheory.Grothendieck` as a specialization of\n   `Pseudofunctor.Grothendieck`.\n\n## References\n[Vistoli2008] \"Notes on Grothendieck Topologies, Fibered Categories and Descent Theory\" by\nAngelo Vistoli\n\n-/\n\nset_option backward.defeqAttrib.useBackward true\n\n@[expose] public section\n\nnamespace CategoryTheory.Pseudofunctor\n\nuniverse w v₁ v₂ v₃ u₁ u₂ u₃\n\nopen Functor Category Opposite Discrete Bicategory StrongTrans\n\nvariable {𝒮 : Type u₁} [Category.{v₁} 𝒮]\n\n/-- The type of objects in the fibered category associated to a pseudofunctor from a\n1-category to Cat. -/\n@[ext]\nstructure Grothendieck (F : LocallyDiscrete 𝒮 ⥤ᵖ Cat.{v₂, u₂}) where\n  /-- The underlying object in the base category. -/\n  base : 𝒮\n  /-- The object in the fiber of the base object. -/\n  fiber : F.obj ⟨base⟩\n\nnamespace Grothendieck\n\nvariable {F : LocallyDiscrete 𝒮 ⥤ᵖ Cat.{v₂, u₂}}\n\n/-- Notation for the Grothendieck category associated to a pseudofunctor `F`. -/\nscoped prefix:75 \"∫ \" => Grothendieck\n\n/-- A morphism in the Grothendieck construction `∫ F` between two points `X Y : ∫ F` consists of\na morphism in the base category `base : X.base ⟶ Y.base` and\na morphism in a fiber `f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber`. -/\nstructure Hom (X Y : ∫ F) where\n  /-- The morphism between base objects. -/\n  base : X.base ⟶ Y.base\n  /-- The morphism in the fiber over the domain. -/\n  fiber : (F.map base.toLoc).toFunctor.obj X.fiber ⟶ Y.fiber\n\n@[simps! id_base id_fiber comp_base comp_fiber]\ninstance categoryStruct : CategoryStruct (∫ F) where\n  Hom X Y := Hom X Y\n  id X := {\n    base := 𝟙 X.base\n    fiber := (F.mapId ⟨X.base⟩).hom.toNatTrans.app X.fiber }\n  comp {X _ _} f g := {\n    base := f.base ≫ g.base\n    fiber := (F.mapComp f.base.toLoc g.base.toLoc).hom.toNatTrans.app X.fiber ≫\n      (F.map g.base.toLoc).toFunctor.map f.fiber ≫ g.fiber }\n\ninstance (X : ∫ F) : Inhabited (Hom X X) :=\n  ⟨𝟙 X⟩\n\nsection\n\nvariable {a b : ∫ F}\n\n@[ext (iff := false)]\nlemma Hom.ext (f g : a ⟶ b) (hfg₁ : f.base = g.base)\n    (hfg₂ : eqToHom (hfg₁ ▸ rfl) ≫ f.fiber = g.fiber) : f = g := by\n  cases f; cases g\n  dsimp at hfg₁ hfg₂\n  cat_disch\n\nlemma Hom.ext_iff (f g : a ⟶ b) :\n    f = g ↔ ∃ (hfg : f.base = g.base), eqToHom (hfg ▸ rfl) ≫ f.fiber = g.fiber where\n  mp hfg := by subst hfg; simp\n  mpr := fun ⟨hfg₁, hfg₂⟩ => Hom.ext f g hfg₁ hfg₂\n\nlemma Hom.congr {a b : ∫ F} {f g : a ⟶ b} (h : f = g) :\n    f.fiber = eqToHom (h ▸ rfl) ≫ g.fiber := by\n  subst h\n  simp\n\nend\n\nset_option backward.isDefEq.respectTransparency false in\nattribute [local simp] PrelaxFunctor.map₂_eqToHom in\n/-- The category structure on `∫ F`. -/\ninstance category : Category (∫ F) where\n  toCategoryStruct := Pseudofunctor.Grothendieck.categoryStruct\n  id_comp {a b} f := by\n    ext\n    · simp\n    · simp [F.mapComp_id_left_hom_app, Strict.leftUnitor_eqToIso, ← Functor.map_comp_assoc,\n        ← Cat.Hom₂.comp_app]\n  comp_id {a b} f := by\n    ext\n    · simp\n    · simp [F.mapComp_id_right_hom_app, Strict.rightUnitor_eqToIso, ← reassoc_of% Cat.Hom₂.comp_app]\n  assoc f g h := by\n    ext\n    · simp\n    · simp [mapComp_assoc_right_hom_app_assoc, Strict.associator_eqToIso]\n\nvariable (F)\n\n/-- The projection `∫ F ⥤ 𝒮` given by projecting both objects and homs to the first factor. -/\n@[simps]\ndef forget (F : Pseudofunctor (LocallyDiscrete 𝒮) Cat.{v₂, u₂}) : ∫ F ⥤ 𝒮 where\n  obj X := X.base\n  map f := f.base\n\nsection\n\nattribute [local simp]\n  Strict.leftUnitor_eqToIso Strict.rightUnitor_eqToIso Strict.associator_eqToIso\n\nvariable {F} {G : Pseudofunctor (LocallyDiscrete 𝒮) Cat.{v₂, u₂}}\n  {H : Pseudofunctor (LocallyDiscrete 𝒮) Cat.{v₂, u₂}}\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The Grothendieck construction is functorial: a strong natural transformation `α : F ⟶ G`\ninduces a functor `Grothendieck.map : ∫ F ⥤ ∫ G`. -/\n@[simps!]\ndef map (α : F ⟶ G) : ∫ F ⥤ ∫ G where\n  obj a := {\n    base := a.base\n    fiber := (α.app ⟨a.base⟩).toFunctor.obj a.fiber }\n  map {a b} f := {\n    base := f.1\n    fiber := (α.naturality f.1.toLoc).inv.toNatTrans.app a.fiber ≫\n      (α.app ⟨b.base⟩).toFunctor.map f.2 }\n  map_id a := by\n    ext\n    · dsimp\n    · simp [StrongTrans.naturality_id_inv_app, ← map_comp, ← Cat.Hom₂.comp_app]\n  map_comp {a b c} f g := by\n    ext\n    · dsimp\n    · simp only [Cat.Hom.comp_toFunctor, comp_obj, categoryStruct_comp_base, Quiver.Hom.comp_toLoc,\n        categoryStruct_comp_fiber, eqToHom_refl, map_comp, ← Cat.Hom.comp_map, assoc,\n        NatTrans.naturality_assoc]\n      simp [naturality_comp_inv_app, ← Functor.map_comp, ← reassoc_of% Cat.Hom₂.comp_app]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\n\nTarget:\nlemma map_id_map {x y : ∫ F} (f : x ⟶ y) : (map (𝟙 F)).map f = f :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"3a85eef59907097f2969ba69d7196a93b4304624baf94c3312cb153590fd8dc3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Bicategory","family_id":"map_id_map","file_id":"mathlib/Mathlib/CategoryTheory/Bicategory/Grothendieck.lean","sample_id":"77f925e6d7fb9f855b12d8cba8f5cb36cbebbc68e7684132b1a6af65dd6fe312"},"split":"train"}
{"binding":{"chosen_proof_sha256":"306e987910ff023d43690d85687bf5ae9a7b00372cbbe2300a8443541f4fbbe4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3b8de9776eeb7f86a13700a0710fcf62496749d80a3452e3d1fdaf5a9f42d96d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"20b48f91a7f4ad693944c040d924608c454aa3ef8bafb869719f264b357fd194","source_sha256":"42683ce9ec3a96f77aab204680a294682617adafa7d756b1f3dbef8d091d30cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro Y f hf\n  simp only [Presheaf.χ_app, Opposite.op_unop] at hf ⊢\n  choose a ha using fun Z (g : Z ⟶ Y) (hg : (Sieve.pullback f ((χ m).app X x)).arrows g) => hg\n  refine ⟨(hF _ hf).amalgamate a ?_, ?_⟩\n  · introv Y₁ h\n    apply (mono_iff_injective (m.app (.op Z))).mp inferInstance\n    simp_rw [NatTrans.naturality_apply, ← ha, ← Functor.map_comp_apply, ← op_comp,\n      reassoc_of% h]\n  · refine (hG _ hf).ext fun Z f' hf' => ?_\n    rw [← NatTrans.naturality_apply, (hF _ hf).valid_glue _ _ hf', ← (ha _ _ _),\n      op_comp, Functor.map_comp_apply]","hard_negative":false,"metrics":{"chosen_tokens":165,"rejected_tokens":169,"token_jaccard":0.958333,"token_length_ratio":1.024242},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"02b4094cf6a650f96ac5a6e3e983121316efa3c3c4fd9d7c36f69d7a3987b7e3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Sites.Closed\npublic import Mathlib.CategoryTheory.Sites.Equivalence\npublic import Mathlib.CategoryTheory.Subobject.Classifier.Defs\npublic import Mathlib.CategoryTheory.Subfunctor.Image\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Edward van de Meent. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Edward van de Meent\n-/\n/-!\n\n# (Elementary) Sheaf Topos\n\nWe define a subobject classifier for categories of sheaves of (large enough) types.\n\n## Main definitions\n\nLet `C` refer to a category with (when relevant) Grothendieck topology `J`.\n\n* `Presheaf.classifier C` is a construction of a subobject classifier in `Cᵒᵖ ⥤ Type (max u v)`.\n* `Sheaf.classifier J` is a construction of a subobject classifier in `Sheaf J (Type (max u v))`.\n* `inferInstance : HasClassifier (Cᵒᵖ ⥤ Type w)` says that `Cᵒᵖ ⥤ Type w` has a subobject\n  classifier if `C` is `w`-essentially small.\n* `inferInstance : HasClassifier (Sheaf J (Type w))` says that `Sheaf J (Type w)` has a\n  subobject classifier if `C` is `w`-essentially small.\n\n## Main results\n\n* Any category of sheaves of types has a subobject classifier if the site is essentially small.\n* As a consequence, (because categories of sheaves are cartesian monoidal and have finite limits,)\n  such categories are Elementary Topoi.\n\n## TODOS:\n\n* generalize `Presheaf.isClosed_χ_app_apply_of` to only assuming `G` is separated\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nnamespace CategoryTheory\n\nvariable {C : Type u} [Category.{v} C]\nopen Limits\n\nsection presheaf\n\nvariable (C) in\n/-- The truth morphism in the category of presheaves. At each component `X : C`, it is the constant\nmap returning `⊤ : Sieve X`. -/\n@[simps]\ndef Presheaf.truth : (Functor.const _).obj PUnit ⟶ Functor.sieves C where\n  app X := ↾fun _ ↦ (⊤ : Sieve X.unop)\n\nvariable {F G : Cᵒᵖ ⥤ Type (max u v)}\n\nset_option backward.defeqAttrib.useBackward true in\n/--\nThe characteristic map of an inclusion of presheaves.\nGiven a monomorphism of sheaves `m : F ⟶ G`, an object X of the site, map an element `x : G(X)`\nto the (closed) sieve on X where `f : Y → X` is in the sieve iff\n  `∃ a ∈ F(Y), G(f)(x) = m_Y(a)`\n-/\n@[simps app]\ndef Presheaf.χ (m : F ⟶ G) : G ⟶ Functor.sieves C where\n  app X := ↾fun x ↦ ⟨fun Y f => ∃ a, G.map f.op x = m.app (.op Y) a, by\n    intro Y Z f ⟨a, ha⟩ g\n    use F.map g.op a\n    simp [ha, NatTrans.naturality_apply]⟩\n\nlemma Presheaf.comp_χ_eq (m : F ⟶ G) : m ≫ Presheaf.χ m =\n    (Functor.isTerminalConst _ Types.isTerminalPUnit).from F ≫ Presheaf.truth C := by\n  ext\n  apply Sieve.ext\n  simp [← NatTrans.naturality_apply]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma Presheaf.isPullback_χ_truth (m : F ⟶ G) [Mono m] :\n    IsPullback m ((Functor.isTerminalConst _ Types.isTerminalPUnit).from F) (χ m) (truth C) := by\n  refine IsPullback.of_forall_isPullback_app fun X => ?_\n  rw [Types.isPullback_iff]\n  refine ⟨congr(($(comp_χ_eq m)).app X), ?_, ?_⟩\n  · simpa using! (mono_iff_injective (m.app X)).mp (inferInstance)\n  · simp only [Functor.const_obj_obj, Functor.sieves_obj, χ_app, Opposite.op_unop,\n      TypeCat.hom_ofHom, TypeCat.Fun.coe_mk, truth_app, Functor.isTerminalConst_from_app,\n      Types.isTerminalPUnit_from_apply, and_true, forall_const]\n    intro p hp\n    simpa [eq_comm] using! congr($(hp).arrows (𝟙 _))\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma Presheaf.χ_unique (m : F ⟶ G) (χ' : G ⟶ Functor.sieves C)\n    (hχ' : IsPullback m ((Functor.isTerminalConst _ Types.isTerminalPUnit).from _) χ' (truth C)) :\n    χ' = χ m := by\n  ext X x\n  simp only [IsPullback.iff_app, Functor.const_obj_obj, Functor.sieves_obj,\n    Functor.isTerminalConst_from_app, truth_app, Types.isPullback_iff,\n    Types.isTerminalPUnit_from_apply, and_true, TypeCat.hom_ofHom, TypeCat.Fun.coe_mk,\n    forall_const, forall_and] at hχ'\n  obtain ⟨h₁, h₂, h₃⟩ := hχ'\n  refine Sieve.ext fun Y f => ?_\n  simp only [χ_app, Opposite.op_unop]\n  rw [Sieve.mem_iff_pullback_eq_top, ← Quiver.Hom.unop_op f]\n  dsimp\n  have := ConcreteCategory.congr_hom (Functor.sieves_map C (f.op)) (χ'.app X x)\n  rw [← dsimp% this, ← dsimp% NatTrans.naturality_apply χ' f.op x]\n  constructor\n  · intro h\n    obtain ⟨z, hz⟩ := h₃ _ _ h\n    use z, hz.symm\n  · rintro ⟨a, h⟩\n    rw [h]\n    simpa using congr($(h₁ (.op Y)) a)\n\nvariable (C) in\n/-- A construction of a subject classifier in a category of presheaves. -/\n@[simps! Ω truth Ω₀ χ χ₀]\ndef Presheaf.classifier : Subobject.Classifier (Cᵒᵖ ⥤ Type (max u v)) :=\n  .mkOfTerminalΩ₀ ((Functor.const Cᵒᵖ).obj PUnit)\n    (Functor.isTerminalConst _ (Types.isTerminalPUnit)) (Functor.sieves C) (Presheaf.truth C)\n    (Presheaf.χ ·) Presheaf.isPullback_χ_truth (Presheaf.χ_unique ·)\n\n/-- Presheaf categories on an essentially small domain have a subobject classifier. -/\ninstance [EssentiallySmall.{w} C] : HasSubobjectClassifier (Cᵒᵖ ⥤ Type w) where\n  exists_classifier := ⟨(Presheaf.classifier (SmallModel C)).ofEquivalence\n    (Equivalence.congrLeft (E := Type w) (equivSmallModel C).op).symm⟩\n\nend presheaf\n\nvariable {J : GrothendieckTopology C}\n\nopen Presheaf in\n\nTarget:\nlemma GrothendieckTopology.isClosed_χ_app_apply_of_isSheaf_of_isSeparated\n    {F G : Cᵒᵖ ⥤ Type (max u v)} (m : F ⟶ G) [Mono m] (hF : Presieve.IsSheaf J F)\n    (hG : Presieve.IsSeparated J G) (X : Cᵒᵖ) (x : G.obj X) :\n    J.IsClosed ((Presheaf.χ m).app X x) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  intro Y f hf\n  simp only [Presheaf.χ_app, Opposite.op_unop] at hf ⊢\n  choose a ha using fun Z (g : Z ⟶ Y) (hg : (Sieve.pullback f ((χ m).app X x)).arrows g) => hg\n  refine ⟨(hF _ hf).amalgamate a ?_, ?_⟩\n  · introv Y₁ h\n    apply (mono_iff_injective (m.app (.op Z))).mp inferInstance\n    simp_rw [NatTrans.naturality_apply, ← ha, ← Functor.map_comp_apply, ← op_comp,\n      reassoc_of% h]\n  · refine (hG _ hf).ext fun Z f' hf' => ?_\n    rw [← NatTrans.naturality_apply, (hF _ hf).valid_glue _ _ hf', ← (ha _ _ _),\n      op_comp, Functor.map_comp_apply]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Topos","family_id":"grothendiecktopology","file_id":"mathlib/Mathlib/CategoryTheory/Topos/Sheaf.lean","sample_id":"20b48f91a7f4ad693944c040d924608c454aa3ef8bafb869719f264b357fd194"},"split":"train"}
{"binding":{"chosen_proof_sha256":"665f9b24ed35095c9258c237fa3353b33a4af46379e1ca75f0d890346defc187","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f402ded8ec1b3fd49c322fa61f9e9685ec1e2964a86fb413b54058727b02c395","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"df00ef8b543a3838a8dcfd15337dcf5374c33e276a9693b751546ead28ce4bfb","source_sha256":"351582fbc976816e329c33b0cb83e6e30d58c72551427bea563b172b7814b639","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply EqualizerCondition.mk\n  intro Z B π _ _\n  refine ⟨fun a b h ↦ ?_, fun ⟨a, ha⟩ ↦ ?_⟩\n  · simp only [yonedaPresheaf, comp, Quiver.Hom.unop_op, TypeCat.Fun.coe_mk,\n      Set.coe_setOf, mapToEqualizer, Set.mem_setOf_eq, ConcreteCategory.hom_ofHom, Subtype.mk.injEq,\n      mk.injEq] at h\n    simp only [yonedaPresheaf, unop_op]\n    ext x\n    obtain ⟨y, hy⟩ := (hq Z B π).surjective x\n    rw [← hy]\n    exact congr_fun h y\n  · simp only [yonedaPresheaf, comp, Quiver.Hom.unop_op, ConcreteCategory.hom_ofHom,\n      TypeCat.Fun.coe_mk, mk.injEq, Set.mem_setOf_eq] at ha\n    simp only [yonedaPresheaf, comp, Quiver.Hom.unop_op, TypeCat.Fun.coe_mk,\n      Set.coe_setOf, mapToEqualizer, Set.mem_setOf_eq, ConcreteCategory.hom_ofHom, Subtype.mk.injEq]\n    simp only [yonedaPresheaf, unop_op] at a\n    refine ⟨(hq Z B π).lift a (factorsThrough_of_pullbackCondition G X ha), ?_⟩\n    congr 1\n    exact DFunLike.ext'_iff.mp ((hq Z B π).lift_comp a (factorsThrough_of_pullbackCondition G X ha))\n\n/--\nIf `G` preserves finite coproducts (which is the case when `C` is `CompHaus`, `Profinite` or\n`Stonean`), then `yonedaPresheaf` preserves finite products, which is required to be a sheaf for\nthe extensive topology.\n-/\nnoncomputable instance [PreservesFiniteCoproducts G] :\n    PreservesFiniteProducts (yonedaPresheaf G X) :=\n  have := preservesFiniteProducts_op G\n  ⟨fun _ ↦ comp_preservesLimitsOfShape G.op (yonedaPresheaf' X)⟩","hard_negative":true,"metrics":{"chosen_tokens":325,"rejected_tokens":2,"token_jaccard":0.009615,"token_length_ratio":0.006154},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"031569d80dcce1eeb2a0ab012dc89afd319979620e1bf1a8d9b4bcce58098c02","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Preserves.Opposites\npublic import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison\npublic import Mathlib.Condensed.Basic\npublic import Mathlib.Topology.Category.TopCat.Yoneda\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# The functor from topological spaces to condensed sets\n\nThis file builds on the API from the file `TopCat.Yoneda`. If the forgetful functor to `TopCat` has\nnice properties, like preserving pullbacks and finite coproducts, then this Yoneda presheaf\nsatisfies the sheaf condition for the regular and extensive topologies respectively.\n\nWe apply this API to `CompHaus` and define the functor\n`topCatToCondensedSet : TopCat.{u + 1} ⥤ CondensedSet.{u}`.\n\n-/\n\n@[expose] public section\n\nuniverse w w' v u\n\nopen CategoryTheory Opposite Limits regularTopology ContinuousMap Topology\n\nvariable {C : Type u} [Category.{v} C] (G : C ⥤ TopCat.{w})\n  (X : Type w') [TopologicalSpace X]\n\n/--\nAn auxiliary lemma to that allows us to use `IsQuotientMap.lift` in the proof of\n`equalizerCondition_yonedaPresheaf`.\n-/\ntheorem factorsThrough_of_pullbackCondition {Z B : C} {π : Z ⟶ B} [HasPullback π π]\n    [PreservesLimit (cospan π π) G]\n    {a : C(G.obj Z, X)}\n    (ha : a ∘ (G.map (pullback.fst _ _)) = a ∘ (G.map (pullback.snd π π))) :\n    Function.FactorsThrough a (G.map π) := by\n  intro x y hxy\n  let xy : G.obj (pullback π π) := (PreservesPullback.iso G π π).inv <|\n    (TopCat.pullbackIsoProdSubtype (G.map π) (G.map π)).inv ⟨(x, y), hxy⟩\n  have ha' := congr_fun ha xy\n  dsimp at ha'\n  have h₁ : ∀ y, G.map (pullback.fst _ _) ((PreservesPullback.iso G π π).inv y) =\n      pullback.fst (G.map π) (G.map π) y := by\n    simp only [← PreservesPullback.iso_inv_fst]; intro y; rfl\n  have h₂ : ∀ y, G.map (pullback.snd _ _) ((PreservesPullback.iso G π π).inv y) =\n      pullback.snd (G.map π) (G.map π) y := by\n    simp only [← PreservesPullback.iso_inv_snd]; intro y; rfl\n  rw [h₁, h₂, TopCat.pullbackIsoProdSubtype_inv_fst_apply,\n    TopCat.pullbackIsoProdSubtype_inv_snd_apply] at ha'\n  simpa using ha'\n\nset_option backward.isDefEq.respectTransparency false in\n/--\nIf `G` preserves the relevant pullbacks and every effective epi in `C` is a quotient map (which is\nthe case when `C` is `CompHaus` or `Profinite`), then `yonedaPresheaf` satisfies the equalizer\ncondition which is required to be a sheaf for the regular topology.\n-/\n\nTarget:\ntheorem equalizerCondition_yonedaPresheaf\n    [∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π], PreservesLimit (cospan π π) G]\n    (hq : ∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π], IsQuotientMap (G.map π)) :\n      EqualizerCondition (yonedaPresheaf G X) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_df00ef8b543a","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"7e4c508849f8e6007a94799fa3f1a0eb791e8921b5c410b8fd282a7b2b82ab3e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed","family_id":"equalizercondition_yonedapresheaf","file_id":"mathlib/Mathlib/Condensed/TopComparison.lean","sample_id":"df00ef8b543a3838a8dcfd15337dcf5374c33e276a9693b751546ead28ce4bfb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a7995287f4a56e7a4f65ad1427f3f2d4f2887e143a6beb10ba5ae796bad00236","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cc3728dc6e72e50fd255371dff3e5f651d3ce2d9c638da326516235746fae11d","source_sha256":"02c39b43c769989fbb2fff40a558624b26a5d6a7cd12a1509d9e9ee7ee0339ba","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← subsingleton_iff_zero_eq_one, OreLocalization.one_def,\n    OreLocalization.zero_def, oreDiv_eq_iff]\n  simp","hard_negative":false,"metrics":{"chosen_tokens":17,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.176471},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"0339380785d74e10a0a891ffeb27765d4a0522dac924f45152effc53d4080fc6","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Submonoid.DistribMulAction\npublic import Mathlib.GroupTheory.OreLocalization.Basic\npublic import Mathlib.Algebra.GroupWithZero.Defs\n\nNamespace:\nOreLocalization\n\nLocal context:\n/-\nCopyright (c) 2022 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer, Kevin Klinge, Andrew Yang\n-/\n/-!\n\n# Localization over left Ore sets.\n\nThis file proves results on the localization of rings (monoids with zeros) over a left Ore set.\n\n## References\n\n* <https://ncatlab.org/nlab/show/Ore+localization>\n* [Zoran Škoda, *Noncommutative localization in noncommutative geometry*][skoda2006]\n\n\n## Tags\nlocalization, Ore, non-commutative\n\n-/\n\n@[expose] public section\n\nassert_not_exists RelIso\n\nuniverse u\n\nnamespace OreLocalization\n\nsection MonoidWithZero\n\nvariable {R : Type*} [MonoidWithZero R] {S : Submonoid R} [OreSet S]\n\n@[simp]\ntheorem zero_oreDiv' (s : S) : (0 : R) /ₒ s = 0 := by\n  rw [OreLocalization.zero_def, oreDiv_eq_iff]\n  exact ⟨s, 1, by simp [Submonoid.smul_def]⟩\n\ninstance : MonoidWithZero R[S⁻¹] where\n  zero_mul x := by\n    induction x using OreLocalization.ind with | _ r s\n    rw [OreLocalization.zero_def, oreDiv_mul_char 0 r 1 s 0 1 (by simp), zero_mul, one_mul]\n  mul_zero x := by\n    induction x using OreLocalization.ind with | _ r s\n    rw [OreLocalization.zero_def, mul_div_one, mul_zero, zero_oreDiv', zero_oreDiv']\n\nTarget:\ntheorem subsingleton_iff :\n    Subsingleton R[S⁻¹] ↔ 0 ∈ S :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/OreLocalization","family_id":"subsingleton_iff","file_id":"mathlib/Mathlib/RingTheory/OreLocalization/Basic.lean","sample_id":"cc3728dc6e72e50fd255371dff3e5f651d3ce2d9c638da326516235746fae11d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4525ed279745c1a825a2d966ade6667dde0e9859d5d810c453aa56b8f6012282","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4ae32efd86a43fc03720d3484c6c7ad3df7ae093e40ae7dd44e806625970e54c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"06f153f627e2ad13220f5061c7800c30d1bdd6c1da9c7ac7f9963af2c8d4a748","source_sha256":"522c0739e8f7535bf0f954aff339b473dbd0f9e903c0b5a9a4fff87d391d66b3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only using! congr_fun (f.isCompat' φ) x","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":5,"token_jaccard":0.125,"token_length_ratio":0.384615},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"03461c44c88cb9a1380cb0dfcdd1fdd3b1fa120bf5139ca5cd6765379579bf3e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorProduct.RightExactness\npublic import Mathlib.RingTheory.Congruence.Hom\npublic import Mathlib.RingTheory.FiniteType\npublic import Mathlib.RingTheory.TensorProduct.DirectLimitFG\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir & María-Inés de Frutos-Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir & María-Inés de Frutos-Fernández\n-/\n/-! # Polynomial laws on modules\n\nLet `M` and `N` be a modules over a commutative ring `R`.\nA polynomial law `f : PolynomialLaw R M N`, with notation `f : M →ₚₗₗ[R] N`,\nis a “law” that assigns a natural map `PolynomialLaw.toFun' f S : S ⊗[R] M → S ⊗[R] N`\nfor every `R`-algebra `S`.\n\nFor type-theoretic reasons, if `R : Type u`, then the definition of the polynomial map `f`\nis restricted to `R`-algebras `S` such that `S : Type u`.\nUsing the fact that a module is the direct limit of its finitely generated submodules, that a\nfinitely generated subalgebra is a quotient of a polynomial ring in the universe `u`, plus\nthe commutation of tensor products with direct limits, we extend the functor\nto all `R`-algebras.\n\nThe two fields involving the definition of `PolynomialLaw`,\n`PolynomialLaw.toFun'` and `PolynomialLaw.isCompat'` are primed.\nThey are superseded by their universe-polymorphic counterparts,\nthe definition `PolynomialLaw.toFun` and the lemma `PolynomialLaw.isCompat`\nwhich should be used once the theory is properly stated.\n\nFor constructions of general definitions of `PolynomialLaw`\nat a universe-polymorphic level, one needs to lift\nelements in a tensor product to smaller universes.\nFor this, one can make use of\n`PolynomialLaw.exists_lift` or `PolynomialLaw.exists_lift'`,\nor establish appropriate generalizations.\n\n## Main definitions/lemmas\n\n* Instance : `Module R (M →ₚₗ[R] N)` shows that polynomial laws form an `R`-module.\n\n* `PolynomialLaw.ground f` is the map `M → N` corresponding to `PolynomialLaw.toFun' f R` under\n  the isomorphisms `R ⊗[R] M ≃ₗ[R] M`, and similarly for `N`.\n\nIn further works, we construct the coefficients of a polynomial law and show the relation with\npolynomials (when the module `M` is free and finite).\n\n## Implementation notes\n\nIn the literature, the theory is written for commutative rings, but this implementation\nonly assumes `R` is a commutative semiring.\n\n## References\n\n* [Roby, Norbert. 1963. «Lois polynomes et lois formelles en théorie des modules».\n  Annales scientifiques de l’École Normale Supérieure 80 (3): 213‑348](Roby-1963)\n\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nnoncomputable section PolynomialLaw\n\nopen scoped TensorProduct\n\nopen LinearMap TensorProduct AlgHom RingCon\n\n/-- A polynomial law `M →ₚₗ[R] N` between `R`-modules is a functorial family of maps\n`S ⊗[R] M → S ⊗[R] N`, for all `R`-algebras `S`.\n\nFor universe reasons, `S` has to be restricted to the same universe as `R`. -/\n@[ext]\nstructure PolynomialLaw (R : Type u) [CommSemiring R]\n    (M : Type*) [AddCommMonoid M] [Module R M] (N : Type*) [AddCommMonoid N] [Module R N] where\n  /-- The functions `S ⊗[R] M → S ⊗[R] N` underlying a polynomial law -/\n  toFun' (S : Type u) [CommSemiring S] [Algebra R S] : S ⊗[R] M → S ⊗[R] N\n  /-- The compatibility relations between the functions underlying a polynomial law -/\n  isCompat' {S : Type u} [CommSemiring S] [Algebra R S]\n    {S' : Type u} [CommSemiring S'] [Algebra R S'] (φ : S →ₐ[R] S') :\n    φ.toLinearMap.rTensor N ∘ toFun' S = toFun' S' ∘ φ.toLinearMap.rTensor M := by aesop\n\n/-- `M →ₚₗ[R] N` is the type of `R`-polynomial laws from `M` to `N`. -/\nnotation:25 M \" →ₚₗ[\" R:25 \"] \" N:0 => PolynomialLaw R M N\n\n@[local simp]\n\nTarget:\ntheorem PolynomialLaw.isCompat_apply'\n    {R : Type u} [CommSemiring R] {M : Type*} [AddCommMonoid M] [Module R M]\n    {N : Type*} [AddCommMonoid N] [Module R N] {f : M →ₚₗ[R] N}\n    {S : Type u} [CommSemiring S] [Algebra R S] {S' : Type u} [CommSemiring S'] [Algebra R S']\n    (φ : S →ₐ[R] S') (x : S ⊗[R] M) :\n    (φ.toLinearMap.rTensor N) ((f.toFun' S) x) = (f.toFun' S') (φ.toLinearMap.rTensor M x) :=\n\nProof body:\n","rejected":"by\n  exact PolynomialLaw.isCompat_apply","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"09b14f7b49ae29a5a60e049189e78be13343a428b834c70791d0b5537b9de052","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/PolynomialLaw","family_id":"polynomiallaw","file_id":"mathlib/Mathlib/RingTheory/PolynomialLaw/Basic.lean","sample_id":"06f153f627e2ad13220f5061c7800c30d1bdd6c1da9c7ac7f9963af2c8d4a748"},"split":"train"}
{"binding":{"chosen_proof_sha256":"faf07d4b9f81d896a4dedf3d8bef12a774f393144e0c5f30d0f0b0458195b2b5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"79e65930fe73a626b39d9d7e987131fdfa46bda1bbf7a03aa7b7123b1240dda2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"332d57c10825bff92a5f9c100471ccaa87a97a30c29a674c9e1339e0c80d29bd","source_sha256":"74b6571c8b4ffa69041159c6a44858bd2f705cc61f45fce1f1c46bf3161b1e36","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  introv W' hf hg hh hi w\n  simpa only [IsPushout.flip_iff, IsPullback.flip_iff, and_comm] using\n    H' g' f' i' h' αW αY αX αZ hg hf hi hh w.flip","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":46,"token_jaccard":0.909091,"token_length_ratio":1.095238},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"03669718c3d3faaf071366f6cd0ec55acbcead621fe0c78cd84310a1c4f84c3c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Extensive\npublic import Mathlib.CategoryTheory.Limits.Shapes.KernelPair\npublic import Mathlib.CategoryTheory.Limits.Constructions.EpiMono\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang, Jack McKoen\n-/\n/-!\n\n# Adhesive categories\n\n## Main definitions\n- `CategoryTheory.IsPushout.IsVanKampen`: A convenience formulation for a pushout being\n  a van Kampen colimit.\n- `CategoryTheory.Adhesive`: A category is adhesive if it has pushouts and pullbacks along\n  monomorphisms, and such pushouts are van Kampen.\n\n## Main Results\n- `CategoryTheory.Type.adhesive`: The category of `Type` is adhesive.\n- `CategoryTheory.Adhesive.isPullback_of_isPushout_of_mono_left`: In adhesive categories,\n  pushouts along monomorphisms are pullbacks.\n- `CategoryTheory.Adhesive.mono_of_isPushout_of_mono_left`: In adhesive categories,\n  monomorphisms are stable under pushouts.\n- `CategoryTheory.Adhesive.toRegularMonoCategory`: Monomorphisms in adhesive categories are\n  regular (this implies that adhesive categories are balanced).\n- `CategoryTheory.adhesive_functor`: The category `C ⥤ D` is adhesive if `D`\n  has all pullbacks and all pushouts and is adhesive\n\n## References\n- https://ncatlab.org/nlab/show/adhesive+category\n- [Stephen Lack and Paweł Sobociński, Adhesive Categories][adhesive2004]\n\n-/\n\n@[expose] public section\n\n\nnamespace CategoryTheory\n\nopen Limits\n\nuniverse v' u' v u\n\nvariable {J : Type v'} [Category.{u'} J] {C : Type u} [Category.{v} C]\nvariable {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z}\n\n-- This only makes sense when the original diagram is a pushout.\n/-- A convenient formulation for a pushout being a van Kampen colimit. For any commutative cube of\nwhich a van Kampen pushout forms the bottom face and the back faces are pullbacks, the front faces\nare pullbacks if and only if the top face is a pushout. See `IsPushout.isVanKampen_iff` below. -/\n@[nolint unusedArguments]\ndef IsPushout.IsVanKampen (_ : IsPushout f g h i) : Prop :=\n  ∀ ⦃W' X' Y' Z' : C⦄ (f' : W' ⟶ X') (g' : W' ⟶ Y') (h' : X' ⟶ Z') (i' : Y' ⟶ Z') (αW : W' ⟶ W)\n    (αX : X' ⟶ X) (αY : Y' ⟶ Y) (αZ : Z' ⟶ Z) (_ : IsPullback f' αW αX f)\n    (_ : IsPullback g' αW αY g) (_ : CommSq h' αX αZ h) (_ : CommSq i' αY αZ i)\n    (_ : CommSq f' g' h' i'), IsPushout f' g' h' i' ↔ IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i\n\n/-- If a van Kampen pushout forms the bottom face of a commutative \"half-cube\" whose front faces\nare pullbacks, then there exist two back faces which are pullbacks and a top face which is a\npushout. -/\nlemma IsPushout.IsVanKampen.exists_cube_filling {H : IsPushout f g h i} (H' : H.IsVanKampen)\n    {X' Y' Z' : C} {h' : X' ⟶ Z'} {i' : Y' ⟶ Z'} {αX : X' ⟶ X} {αY : Y' ⟶ Y} {αZ : Z' ⟶ Z}\n    [HasPullback αX f] (hh : IsPullback h' αX αZ h) (hi : IsPullback i' αY αZ i) :\n    ∃ (W' : C) (f' : W' ⟶ X') (g' : W' ⟶ Y') (αW : W' ⟶ W),\n      IsPullback f' αW αX f ∧ IsPullback g' αW αY g ∧ IsPushout f' g' h' i' := by\n  let l := hi.lift ((pullback.fst αX f) ≫ h') ((pullback.snd αX f) ≫ g)\n    (by simp only [Category.assoc, hh.toCommSq.w, pullback.condition_assoc, ← H.w])\n  use (pullback αX f), (pullback.fst αX f), l, (pullback.snd αX f)\n  refine ⟨IsPullback.of_hasPullback αX f, ?_, ?_⟩\n  · refine IsPullback.of_right' ?_ hi\n    rw [← H.w]\n    exact IsPullback.paste_horiz (IsPullback.of_hasPullback αX f) hh\n  · refine (H' (pullback.fst αX f) l h' i' (pullback.snd αX f) αX αY αZ\n      (IsPullback.of_hasPullback αX f) ?_\n        hh.toCommSq hi.toCommSq ⟨by simp only [IsPullback.lift_fst, l]⟩).2 ⟨hh, hi⟩\n    · refine IsPullback.of_right' ?_ hi\n      rw [← H.w]\n      exact IsPullback.paste_horiz (IsPullback.of_hasPullback αX f) hh\n\nTarget:\ntheorem IsPushout.IsVanKampen.flip {H : IsPushout f g h i} (H' : H.IsVanKampen) :\n    H.flip.IsVanKampen :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  introv W' hf hg hh hi w\n  simpa only [IsPushout.flip_iff, IsPullback.flip_iff, and_comm] using\n    H' g' f' i' h' αW αY αX αZ hg hf hi hh w.flip","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Adhesive","family_id":"ispushout","file_id":"mathlib/Mathlib/CategoryTheory/Adhesive/Basic.lean","sample_id":"332d57c10825bff92a5f9c100471ccaa87a97a30c29a674c9e1339e0c80d29bd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8f9fde22f4f5e3c6a342c774b51f2e9a3888b3631f985c3a4bfd65b729742429","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9f5d681bdf15caa79e692f23201793dd5e7cf30262fbd1bb9bad727c7ed9cb17","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bbc6b49baeedf3e55b05cb3f8d67c144875e1981fa10f0ed9325586b2eeede01","source_sha256":"92db59c0cca19f9c28529fd17474c1f344e67bd83f08b4344e7abe51502b9b08","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext : 1\n  simp [derivative']","hard_negative":true,"metrics":{"chosen_tokens":8,"rejected_tokens":3,"token_jaccard":0.1,"token_length_ratio":0.375},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"03988e09076bf188ebdad8f32cae975a6b519a883bea2ca78ba354b3ef45f06b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.AlgebraMap\npublic import Mathlib.Algebra.Polynomial.Derivative\npublic import Mathlib.Algebra.Polynomial.Module.AEval\npublic import Mathlib.RingTheory.Adjoin.Polynomial.Basic\npublic import Mathlib.RingTheory.Derivation.Basic\n\nNamespace:\nPolynomial\n\nLocal context:\n/-\nCopyright (c) 2023 Kevin Buzzard. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Richard M. Hill\n-/\n/-!\n# Derivations of univariate polynomials\n\nIn this file we prove that an `R`-derivation of `Polynomial R` is determined by its value on `X`.\nWe also provide a constructor `Polynomial.mkDerivation` that\nbuilds a derivation from its value on `X`, and a linear equivalence\n`Polynomial.mkDerivationEquiv` between `A` and `Derivation (Polynomial R) A`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nnamespace Polynomial\n\nsection CommSemiring\n\nvariable {R A : Type*} [CommSemiring R]\n\n/-- `Polynomial.derivative` as a derivation. -/\n@[simps]\ndef derivative' : Derivation R R[X] R[X] where\n  toFun := derivative\n  map_add' _ _ := derivative_add\n  map_smul' := derivative_smul\n  map_one_eq_zero' := derivative_one\n  leibniz' f g := by simp [mul_comm, add_comm, derivative_mul]\n\nvariable [AddCommMonoid A] [Module R A] [Module (Polynomial R) A]\n\n@[simp]\ntheorem derivation_C (D : Derivation R R[X] A) (a : R) : D (C a) = 0 :=\n  D.map_algebraMap a\n\n@[simp]\ntheorem C_smul_derivation_apply (D : Derivation R R[X] A) (a : R) (f : R[X]) :\n    C a • D f = a • D f := by\n  have : C a • D f = D (C a * f) := by simp\n  rw [this, C_mul', D.map_smul]\n\n@[ext]\ntheorem derivation_ext {D₁ D₂ : Derivation R R[X] A} (h : D₁ X = D₂ X) : D₁ = D₂ :=\n  Derivation.ext fun f => Derivation.eqOn_adjoin (Set.eqOn_singleton.2 h) <| by\n    simp only [adjoin_X, Algebra.coe_top, Set.mem_univ]\n\nvariable [IsScalarTower R (Polynomial R) A]\nvariable (R)\n\n/-- The derivation on `R[X]` that takes the value `a` on `X`. -/\ndef mkDerivation : A →ₗ[R] Derivation R R[X] A where\n  toFun := fun a ↦ (LinearMap.toSpanSingleton R[X] A a).compDer derivative'\n  map_add' := fun a b ↦ by ext; simp\n  map_smul' := fun t a ↦ by ext; simp\n\nlemma mkDerivation_apply (a : A) (f : R[X]) :\n    mkDerivation R a f = derivative f • a := by\n  rfl\n\n@[simp]\ntheorem mkDerivation_X (a : A) : mkDerivation R a X = a := by simp [mkDerivation_apply]\n\nTarget:\nlemma mkDerivation_one_eq_derivative' : mkDerivation R (1 : R[X]) = derivative' :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_bbc6b49baeed","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1bd067824a37314782da7518e377e1fd33ad11304939633f5b47140ea3e90fd6","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Polynomial","family_id":"mkderivation_one_eq_derivative","file_id":"mathlib/Mathlib/Algebra/Polynomial/Derivation.lean","sample_id":"bbc6b49baeedf3e55b05cb3f8d67c144875e1981fa10f0ed9325586b2eeede01"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5416e2b8428ca9ff8cc9626edea3896864ca77c250a86dc31c56a1ba22b8daf5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a2c1aaa8d0f8e5dae27717b6de1757cb5fb5083bc7f2860877dc52f001fa4d09","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"91fa9e01efb72d4bb46c66a3c3321fad75748aa4aba9170a103de81feb5b067d","source_sha256":"c9b2b17bec68389df88ab608b3808c19ca8eaaffb1ef5c857eb1b5ade046a67e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold Measure.mconv\n  rw [integral_map (by fun_prop) hf.1, integral_prod]\n  exact (integrable_map_measure hf.1 (by fun_prop)).mp hf","hard_negative":true,"metrics":{"chosen_tokens":32,"rejected_tokens":2,"token_jaccard":0.05,"token_length_ratio":0.0625},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"03a2139af4db8c546781b6d035205f2757bd776fd1019d5707999d001d400ac5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Group.Convolution\npublic import Mathlib.MeasureTheory.Integral.Prod\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n-/\n/-!\n# Bochner integrals of convolutions\n\nThis file contains results about the Bochner integrals of convolutions of measures.\n\nThese results are not placed in the main convolution file because we don't want to import Bochner\nintegrals over there.\n\n## Main statements\n\n* `integrable_conv_iff`: A function is integrable with respect to the convolution `μ ∗ ν` iff\n  the function `y ↦ f (x + y)` is integrable with respect to `ν` for `μ`-almost every `x` and\n  the function `x ↦ ∫ y, ‖f (x + y)‖ ∂ν` is integrable with respect to `μ`.\n* `integral_conv`: if `f` is integrable with respect to the convolution `μ ∗ ν`, then\n  `∫ x, f x ∂(μ ∗ₘ ν) = ∫ x, ∫ y, f (x + y) ∂ν ∂μ`.\n-/\n\npublic section\n\nnamespace MeasureTheory\n\nvariable {M F : Type*} [Monoid M] {mM : MeasurableSpace M} [MeasurableMul₂ M]\n  [NormedAddCommGroup F] {μ ν : Measure M} {f : M → F}\n\n@[to_additive]\nlemma integrable_mconv_iff [SFinite ν] (hf : AEStronglyMeasurable f (μ ∗ₘ ν)) :\n    Integrable f (μ ∗ₘ ν)\n      ↔ (∀ᵐ x ∂μ, Integrable (fun y ↦ f (x * y)) ν)\n        ∧ Integrable (fun x ↦ ∫ y, ‖f (x * y)‖ ∂ν) μ := by\n  simp [Measure.mconv, integrable_map_measure hf (by fun_prop),\n    integrable_prod_iff (hf.comp_measurable (by fun_prop))]\n\n@[to_additive]\n\nTarget:\nlemma integral_mconv [NormedSpace ℝ F] [SFinite μ] [SFinite ν] (hf : Integrable f (μ ∗ₘ ν)) :\n    ∫ x, f x ∂(μ ∗ₘ ν) = ∫ x, ∫ y, f (x * y) ∂ν ∂μ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_91fa9e01efb7","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"a7719d524d4f92f6ff0164abd392f193e5fa01840c627a209ed9ff60798ea72a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Group","family_id":"integral_mconv","file_id":"mathlib/Mathlib/MeasureTheory/Group/IntegralConvolution.lean","sample_id":"91fa9e01efb72d4bb46c66a3c3321fad75748aa4aba9170a103de81feb5b067d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"91dab1cd2e2e53fc5f8d17a5c93b18cccb419f8155621f1a414c77fee50249a1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9f0aba5e8c958a1721ba7a16d006099909f296bcaa86d09dd7b80102ebf2e9d4","source_sha256":"0d1f9dee5bc403406539e90adb439a195f88865e97423c979d76639acfd9231c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : x < 0 := lt_of_tendsto_atBot 0 hx\n  intro n\n  simpa [abs_of_neg this, lt_neg] using! lt_of_tendsto_atBot (-n) hx","hard_negative":true,"metrics":{"chosen_tokens":27,"rejected_tokens":8,"token_jaccard":0.153846,"token_length_ratio":0.296296},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"03b4b63db8e58f06409315b64b5d42c967e16e1402ac2260952d2a9b06f48669","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Ring.StandardPart\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.Order.Filter.FilterProduct\n\nNamespace:\nHyperreal\n\nLocal context:\n/-\nCopyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Abhimanyu Pallavi Sudhir, Violeta Hernández Palacios\n-/\n/-!\n# Construction of the hyperreal numbers as an ultraproduct of real sequences\n\nWe define the `Hyperreal` numbers as quotients of sequences `ℕ → ℝ` by an ultrafilter. These form\na field, and we prove some of their basic properties.\n\nNote that most of the machinery that is usually defined for the specific purpose of non-standard\nanalysis (infinitesimal and infinite elements, standard parts) has been generalized to other\nnon-archimedean fields. In particular:\n\n- `ArchimedeanClass` can be used to measure whether an element is infinitesimal (`0 < mk x`) or\n  infinite (`mk x < 0`).\n- `ArchimedeanClass.stdPart` generalizes the standard part function to a general ordered field.\n\n## Todo\n\nUse Łoś's Theorem `FirstOrder.Language.Ultraproduct.sentence_realize` to formalize the transfer\nprinciple on `Hyperreal`.\n-/\n\n@[expose] public section\n\nopen ArchimedeanClass Filter Germ Topology\n\nnoncomputable section\n\n/-- Hyperreal numbers on the ultrafilter extending the cofinite filter. -/\ndef Hyperreal : Type :=\n  Germ (hyperfilter ℕ : Filter ℕ) ℝ\nderiving Inhabited\n\nnamespace Hyperreal\n\n@[inherit_doc] notation \"ℝ*\" => Hyperreal\n\ninstance : Field ℝ* :=\n  inferInstanceAs (Field (Germ _ _))\n\ninstance : LinearOrder ℝ* :=\n  inferInstanceAs (LinearOrder (Germ _ _))\n\ninstance : IsStrictOrderedRing ℝ* :=\n  inferInstanceAs (IsStrictOrderedRing (Germ _ _))\n\n/-- Natural embedding `ℝ → ℝ*`. -/\n@[coe] def ofReal : ℝ → ℝ* := const\n\ninstance : CoeTC ℝ ℝ* := ⟨ofReal⟩\n\n@[simp, norm_cast]\ntheorem coe_eq_coe {x y : ℝ} : (x : ℝ*) = y ↔ x = y :=\n  Germ.const_inj\n\ntheorem coe_ne_coe {x y : ℝ} : (x : ℝ*) ≠ y ↔ x ≠ y :=\n  coe_eq_coe.not\n\n@[simp, norm_cast]\ntheorem coe_eq_zero {x : ℝ} : (x : ℝ*) = 0 ↔ x = 0 :=\n  coe_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_eq_one {x : ℝ} : (x : ℝ*) = 1 ↔ x = 1 :=\n  coe_eq_coe\n\n@[norm_cast]\ntheorem coe_ne_zero {x : ℝ} : (x : ℝ*) ≠ 0 ↔ x ≠ 0 :=\n  coe_ne_coe\n\n@[norm_cast]\ntheorem coe_ne_one {x : ℝ} : (x : ℝ*) ≠ 1 ↔ x ≠ 1 :=\n  coe_ne_coe\n\n@[simp, norm_cast]\ntheorem coe_one : ↑(1 : ℝ) = (1 : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_zero : ↑(0 : ℝ) = (0 : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_inv (x : ℝ) : ↑x⁻¹ = (x⁻¹ : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_neg (x : ℝ) : ↑(-x) = (-x : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_add (x y : ℝ) : ↑(x + y) = (x + y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_ofNat (n : ℕ) [n.AtLeastTwo] :\n    ((ofNat(n) : ℝ) : ℝ*) = OfNat.ofNat n :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_mul (x y : ℝ) : ↑(x * y) = (x * y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_div (x y : ℝ) : ↑(x / y) = (x / y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sub (x y : ℝ) : ↑(x - y) = (x - y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_le_coe {x y : ℝ} : (x : ℝ*) ≤ y ↔ x ≤ y :=\n  Germ.const_le_iff\n\n@[simp, norm_cast]\ntheorem coe_lt_coe {x y : ℝ} : (x : ℝ*) < y ↔ x < y :=\n  Germ.const_lt_iff\n\n@[simp, norm_cast]\ntheorem coe_nonneg {x : ℝ} : 0 ≤ (x : ℝ*) ↔ 0 ≤ x :=\n  coe_le_coe\n\n@[simp, norm_cast]\ntheorem coe_pos {x : ℝ} : 0 < (x : ℝ*) ↔ 0 < x :=\n  coe_lt_coe\n\n@[simp, norm_cast]\ntheorem coe_abs (x : ℝ) : ((|x| : ℝ) : ℝ*) = |↑x| :=\n  const_abs x\n\n@[simp, norm_cast]\ntheorem coe_max (x y : ℝ) : ((max x y : ℝ) : ℝ*) = max ↑x ↑y :=\n  Germ.const_max _ _\n\n@[simp, norm_cast]\ntheorem coe_min (x y : ℝ) : ((min x y : ℝ) : ℝ*) = min ↑x ↑y :=\n  Germ.const_min _ _\n\n/-- The canonical map `ℝ → ℝ*` as an `OrderRingHom`. -/\n@[simps]\ndef coeRingHom : ℝ →+*o ℝ* where\n  toFun x := x\n  map_zero' := rfl\n  map_one' := rfl\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  monotone' _ _ := coe_le_coe.2\n\n@[simp]\ntheorem archimedeanClassMk_coe_nonneg (x : ℝ) : 0 ≤ mk (x : ℝ*) :=\n  mk_map_nonneg_of_archimedean coeRingHom x\n\n@[simp]\ntheorem archimdeanClassMk_coe {x : ℝ} (hx : x ≠ 0) : mk (x : ℝ*) = 0 :=\n  mk_map_of_archimedean' coeRingHom hx\n\n@[simp]\ntheorem stdPart_coe (x : ℝ) : stdPart (x : ℝ*) = x :=\n  stdPart_map_real coeRingHom x\n\n/-! ### Basic constants -/\n\n/-- Construct a hyperreal number from a sequence of real numbers. -/\ndef ofSeq (f : ℕ → ℝ) : ℝ* := (↑f : Germ (hyperfilter ℕ : Filter ℕ) ℝ)\n\ntheorem ofSeq_surjective : Function.Surjective ofSeq := Quot.exists_rep\n\ntheorem ofSeq_lt_ofSeq {f g : ℕ → ℝ} : ofSeq f < ofSeq g ↔ ∀ᶠ n in hyperfilter ℕ, f n < g n :=\n  Germ.coe_lt\n\ntheorem ofSeq_le_ofSeq {f g : ℕ → ℝ} : ofSeq f ≤ ofSeq g ↔ ∀ᶠ n in hyperfilter ℕ, f n ≤ g n :=\n  Germ.coe_le\n\n/-! #### ω -/\n\n/-- A sample infinite hyperreal ω = ⟦(0, 1, 2, 3, ⋯)⟧. -/\ndef omega : ℝ* := ofSeq Nat.cast\n\n@[inherit_doc] scoped notation \"ω\" => Hyperreal.omega\nrecommended_spelling \"omega\" for \"ω\" in [omega, «termω»]\n\ntheorem coe_lt_omega (r : ℝ) : r < ω := by\n  apply ofSeq_lt_ofSeq.2 <| Filter.Eventually.filter_mono Nat.hyperfilter_le_atTop _\n  obtain ⟨n, hn⟩ := exists_nat_gt r\n  rw [eventually_atTop]\n  exact ⟨n, fun m hm ↦ hn.trans_le (mod_cast hm)⟩\n\ntheorem omega_pos : 0 < ω :=\n  coe_lt_omega 0\n\n@[simp]\ntheorem omega_ne_zero : ω ≠ 0 :=\n  omega_pos.ne'\n\n@[simp]\ntheorem abs_omega : |ω| = ω :=\n  abs_of_pos omega_pos\n\n@[simp]\ntheorem archimedeanClassMk_omega_neg : mk ω < 0 :=\n  fun n ↦ by simpa using! coe_lt_omega n\n\n@[simp]\ntheorem stdPart_omega : stdPart ω = 0 := by\n  rw [stdPart_eq_zero]\n  exact archimedeanClassMk_omega_neg.ne\n\n/-! #### ε -/\n\n/-- A sample infinitesimal hyperreal ε = ⟦(0, 1, 1/2, 1/3, ⋯)⟧. -/\ndef epsilon : ℝ* :=\n  ofSeq fun n => n⁻¹\n\n@[inherit_doc] scoped notation \"ε\" => Hyperreal.epsilon\nrecommended_spelling \"epsilon\" for \"ε\" in [epsilon, «termε»]\n\n@[simp]\ntheorem inv_omega : ω⁻¹ = ε :=\n  rfl\n\n@[simp]\ntheorem inv_epsilon : ε⁻¹ = ω :=\n  @inv_inv _ _ ω\n\n@[simp]\ntheorem epsilon_pos : 0 < ε :=\n  inv_pos_of_pos omega_pos\n\n@[simp]\ntheorem epsilon_ne_zero : ε ≠ 0 :=\n  epsilon_pos.ne'\n\n@[simp]\ntheorem epsilon_mul_omega : ε * ω = 1 :=\n  @inv_mul_cancel₀ _ _ ω omega_ne_zero\n\n@[simp]\ntheorem archimedeanClassMk_epsilon_pos : 0 < mk ε := by\n  simp [← inv_omega]\n\n/-!\n### Some facts about `Tendsto`\n-/\n\n@[simp]\ntheorem tendsto_ofSeq {f : ℕ → ℝ} {lb : Filter ℝ} :\n    (ofSeq f).Tendsto lb ↔ Tendsto f (hyperfilter ℕ) lb :=\n  .rfl\n\ntheorem stdPart_map {x : ℝ*} {r : ℝ} {f : ℝ → ℝ} (hf : ContinuousAt f r)\n    (hxr : x.Tendsto (𝓝 r)) : (x.map f).Tendsto (𝓝 (f r)) := by\n  rcases ofSeq_surjective x with ⟨g, rfl⟩\n  exact hf.tendsto.comp hxr\n\ntheorem stdPart_map₂ {x y : ℝ*} {r s : ℝ} {f : ℝ → ℝ → ℝ}\n    (hxr : x.Tendsto (𝓝 r)) (hys : y.Tendsto (𝓝 s))\n    (hf : ContinuousAt (Function.uncurry f) (r, s)) : (x.map₂ f y).Tendsto (𝓝 (f r s)) := by\n  rcases ofSeq_surjective x with ⟨x, rfl⟩\n  rcases ofSeq_surjective y with ⟨y, rfl⟩\n  exact hf.tendsto.comp (hxr.prodMk_nhds hys)\n\ntheorem tendsto_iff_forall {x : ℝ*} {r : ℝ} :\n    x.Tendsto (𝓝 r) ↔ (∀ s < r, s ≤ x) ∧ (∀ s > r, x ≤ s) := by\n  rcases ofSeq_surjective x with ⟨f, rfl⟩\n  rw [tendsto_ofSeq, (nhds_basis_Ioo _).tendsto_right_iff]\n  simp_rw [Set.mem_Ioo, eventually_and, ← ofSeq_lt_ofSeq]\n  refine ⟨fun H ↦ ⟨fun s hs ↦ ?_, fun s hs ↦ ?_⟩, fun H ⟨s, t⟩ ⟨hs, ht⟩ ↦ ⟨?_, ?_⟩⟩\n  · obtain ⟨t, ht⟩ := exists_gt r\n    exact (H ⟨s, t⟩ ⟨hs, ht⟩).1.le\n  · obtain ⟨t, ht⟩ := exists_lt r\n    exact (H ⟨t, s⟩ ⟨ht, hs⟩).2.le\n  · obtain ⟨u, hu, hu'⟩ := exists_between hs\n    exact (coe_lt_coe.2 hu).trans_le (H.1 _ hu')\n  · obtain ⟨u, hu, hu'⟩ := exists_between ht\n    exact (H.2 _ hu).trans_lt (coe_lt_coe.2 hu')\n\ntheorem archimedeanClassMk_nonneg_of_tendsto {x : ℝ*} {r : ℝ} (hx : x.Tendsto (𝓝 r)) :\n    0 ≤ mk x := by\n  rw [tendsto_iff_forall] at hx\n  obtain ⟨s, hs⟩ := exists_lt r\n  obtain ⟨t, ht⟩ := exists_gt r\n  exact mk_nonneg_of_le_of_le_of_archimedean coeRingHom (hx.1 s hs) (hx.2 t ht)\n\ntheorem stdPart_of_tendsto {x : ℝ*} {r : ℝ} (hx : x.Tendsto (𝓝 r)) : stdPart x = r := by\n  rw [tendsto_iff_forall] at hx\n  exact stdPart_eq coeRingHom hx.1 hx.2\n\ntheorem archimedeanClassMk_pos_of_tendsto {x : ℝ*} (hx : x.Tendsto (𝓝 0)) : 0 < mk x := by\n  apply (archimedeanClassMk_nonneg_of_tendsto hx).lt_of_ne'\n  rw [← stdPart_eq_zero, stdPart_of_tendsto hx]\n\n@[simp]\ntheorem stdPart_epsilon : stdPart ε = 0 :=\n  stdPart_eq_zero.2 <| archimedeanClassMk_epsilon_pos.ne'\n\ntheorem epsilon_lt_of_pos {r : ℝ} : 0 < r → ε < r :=\n  lt_of_pos_of_archimedean coeRingHom archimedeanClassMk_epsilon_pos\n\ntheorem epsilon_lt_of_neg {r : ℝ} : r < 0 → r < ε :=\n  lt_of_neg_of_archimedean coeRingHom archimedeanClassMk_epsilon_pos\n\n@[deprecated (since := \"2026-01-05\")]\nalias epsilon_lt_pos := epsilon_lt_of_pos\n\n@[deprecated archimedeanClassMk_pos_of_tendsto (since := \"2026-01-05\")]\ntheorem lt_of_tendsto_zero_of_pos {f : ℕ → ℝ} (hf : Tendsto f atTop (𝓝 0)) :\n    ∀ {r : ℝ}, 0 < r → ofSeq f < (r : ℝ*) := fun hr ↦\n  ofSeq_lt_ofSeq.2 <| (hf.eventually <| gt_mem_nhds hr).filter_mono Nat.hyperfilter_le_atTop\n\n@[deprecated archimedeanClassMk_pos_of_tendsto (since := \"2026-01-05\")]\ntheorem neg_lt_of_tendsto_zero_of_pos {f : ℕ → ℝ} (hf : Tendsto f atTop (𝓝 0)) :\n    ∀ {r : ℝ}, 0 < r → (-r : ℝ*) < ofSeq f := fun hr =>\n  have hg := hf.neg\n  neg_lt_of_neg_lt (by rw [neg_zero] at hg; exact lt_of_tendsto_zero_of_pos hg hr)\n\n@[deprecated archimedeanClassMk_pos_of_tendsto (since := \"2026-01-05\")]\ntheorem gt_of_tendsto_zero_of_neg {f : ℕ → ℝ} (hf : Tendsto f atTop (𝓝 0)) :\n    ∀ {r : ℝ}, r < 0 → (r : ℝ*) < ofSeq f := fun {r} hr => by\n  rw [← neg_neg r, coe_neg]; exact neg_lt_of_tendsto_zero_of_pos hf (neg_pos.mpr hr)\n\ntheorem lt_of_tendsto_atTop {x : ℝ*} (r : ℝ) (hx : x.Tendsto atTop) : r < x := by\n  rcases ofSeq_surjective x with ⟨f, rfl⟩\n  rw [tendsto_ofSeq] at hx\n  exact ofSeq_lt_ofSeq.2 <| hx.eventually_mem (Ioi_mem_atTop r)\n\ntheorem lt_of_tendsto_atBot {x : ℝ*} (r : ℝ) (hx : x.Tendsto atBot) : x < r := by\n  rcases ofSeq_surjective x with ⟨f, rfl⟩\n  rw [tendsto_ofSeq] at hx\n  exact ofSeq_lt_ofSeq.2 <| hx.eventually_mem (Iio_mem_atBot r)\n\ntheorem archimedeanClassMk_neg_of_tendsto_atTop {x : ℝ*} (hx : x.Tendsto atTop) : mk x < 0 := by\n  have : 0 < x := lt_of_tendsto_atTop 0 hx\n  intro n\n  simpa [abs_of_pos this] using! lt_of_tendsto_atTop n hx\n\nTarget:\ntheorem archimedeanClassMk_neg_of_tendsto_atBot {x : ℝ*} (hx : x.Tendsto atBot) : mk x < 0 :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"39839f0bf7bb15e2c15464e4b0aebb811f80512edef7ba8a27da7a2779825d38","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Real","family_id":"archimedeanclassmk_neg_of_tendsto_atbot","file_id":"mathlib/Mathlib/Analysis/Real/Hyperreal.lean","sample_id":"9f0aba5e8c958a1721ba7a16d006099909f296bcaa86d09dd7b80102ebf2e9d4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2f6287791c3459636988da88fa63c514ca770a07a4d1537e256b0a039a8246d2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f6e07b9557819a58269647c6b5189639d71a69d808a07410892e1fb416291719","source_sha256":"29ef0dde93e398484e13c344e681af6a10869ecf4559604c0c8bb7026329d317","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases l.exists_antitone_basis with ⟨u, hu⟩\n  rcases (Filter.hasBasis_self.mpr hl'.exists_measurable_subset).exists_antitone_subbasis with\n    ⟨v, v_meas, hv⟩\n  simp only [hu.tendsto_iff hv.toHasBasis, true_imp_iff, true_and, setOf_forall, setOf_exists]\n  exact .iInter fun n ↦ .iUnion fun _ ↦ .biInter (to_countable _) fun i _ ↦\n    (v_meas n).2.preimage (hf i)","hard_negative":false,"metrics":{"chosen_tokens":83,"rejected_tokens":2,"token_jaccard":0.021739,"token_length_ratio":0.024096},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"03c4c28dae193b43ef87179c737549426d04fa775fcbfd7ace25e33dfdabf23e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.MeasurableSpace.Constructions\npublic import Mathlib.Order.Filter.AtTopBot.CompleteLattice\npublic import Mathlib.Order.Filter.AtTopBot.CountablyGenerated\npublic import Mathlib.Order.Filter.SmallSets\npublic import Mathlib.Order.LiminfLimsup\npublic import Mathlib.Tactic.FinCases\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro\n-/\n/-!\n# Measurably generated filters\n\nWe say that a filter `f` is measurably generated if every set `s ∈ f` includes a measurable\nset `t ∈ f`. This property is useful, e.g., to extract a measurable witness of `Filter.Eventually`.\n-/\n\npublic section\n\nopen Set Filter\n\nuniverse uι\n\nvariable {α β γ δ : Type*} {ι : Sort uι}\n\nnamespace MeasurableSpace\n\n/-- The sigma-algebra generated by a single set `s` is `{∅, s, sᶜ, univ}`. -/\n@[simp] theorem generateFrom_singleton (s : Set α) :\n    generateFrom {s} = MeasurableSpace.comap (· ∈ s) ⊤ := by\n  classical\n  letI : MeasurableSpace α := generateFrom {s}\n  refine le_antisymm (generateFrom_le fun t ht => ⟨{True}, trivial, by simp [ht.symm]⟩) ?_\n  rintro _ ⟨u, -, rfl⟩\n  exact (show MeasurableSet s from GenerateMeasurable.basic _ <| mem_singleton s).mem trivial\n\nlemma generateFrom_singleton_le {m : MeasurableSpace α} {s : Set α} (hs : MeasurableSet s) :\n    MeasurableSpace.generateFrom {s} ≤ m :=\n  generateFrom_le (fun _ ht ↦ mem_singleton_iff.1 ht ▸ hs)\n\nlemma comap_indicator_const_le_generateFrom_singleton {M : Type*} [Zero M] [MeasurableSpace M]\n    (s : Set α) (c : M) :\n    MeasurableSpace.comap (s.indicator (fun _ ↦ c)) inferInstance ≤\n      MeasurableSpace.generateFrom {s} :=\n  (measurable_const.indicator (measurableSet_generateFrom (by simp))).comap_le\n\nend MeasurableSpace\n\nnamespace MeasureTheory\n\ntheorem measurableSet_generateFrom_singleton_iff {s t : Set α} :\n    MeasurableSet[MeasurableSpace.generateFrom {s}] t ↔ t = ∅ ∨ t = s ∨ t = sᶜ ∨ t = univ := by\n  simp_rw +instances [MeasurableSpace.generateFrom_singleton]\n  unfold MeasurableSet MeasurableSpace.MeasurableSet' MeasurableSpace.comap\n  simp_rw [MeasurableSpace.measurableSet_top, true_and]\n  constructor\n  · rintro ⟨x, rfl⟩\n    by_cases hT : True ∈ x\n    · by_cases hF : False ∈ x\n      · suffices x = univ by grind\n        grind [univ_eq_true_false]\n      · grind\n    · by_cases hF : False ∈ x\n      · grind\n      · suffices x ⊆ ∅ by grind\n        intro p hp\n        fin_cases p <;> contradiction\n  · rintro (rfl | rfl | rfl | rfl)\n    on_goal 1 => use ∅\n    on_goal 2 => use {True}\n    on_goal 3 => use {False}\n    on_goal 4 => use Set.univ\n    all_goals\n      simp [compl_def]\n\nend MeasureTheory\n\nnamespace Filter\n\nvariable [MeasurableSpace α]\n\n/-- A filter `f` is measurably generated if each `s ∈ f` includes a measurable `t ∈ f`. -/\nclass IsMeasurablyGenerated (f : Filter α) : Prop where\n  exists_measurable_subset : ∀ ⦃s⦄, s ∈ f → ∃ t ∈ f, MeasurableSet t ∧ t ⊆ s\n\ninstance isMeasurablyGenerated_bot : IsMeasurablyGenerated (⊥ : Filter α) :=\n  ⟨fun _ _ => ⟨∅, mem_bot, MeasurableSet.empty, empty_subset _⟩⟩\n\ninstance isMeasurablyGenerated_top : IsMeasurablyGenerated (⊤ : Filter α) :=\n  ⟨fun _s hs => ⟨univ, univ_mem, MeasurableSet.univ, fun x _ => hs x⟩⟩\n\ntheorem Eventually.exists_measurable_mem {f : Filter α} [IsMeasurablyGenerated f] {p : α → Prop}\n    (h : ∀ᶠ x in f, p x) : ∃ s ∈ f, MeasurableSet s ∧ ∀ x ∈ s, p x :=\n  IsMeasurablyGenerated.exists_measurable_subset h\n\ntheorem Eventually.exists_measurable_mem_of_smallSets {f : Filter α} [IsMeasurablyGenerated f]\n    {p : Set α → Prop} (h : ∀ᶠ s in f.smallSets, p s) : ∃ s ∈ f, MeasurableSet s ∧ p s :=\n  let ⟨_s, hsf, hs⟩ := eventually_smallSets.1 h\n  let ⟨t, htf, htm, hts⟩ := IsMeasurablyGenerated.exists_measurable_subset hsf\n  ⟨t, htf, htm, hs t hts⟩\n\ninstance inf_isMeasurablyGenerated (f g : Filter α) [IsMeasurablyGenerated f]\n    [IsMeasurablyGenerated g] : IsMeasurablyGenerated (f ⊓ g) := by\n  constructor\n  rintro t ⟨sf, hsf, sg, hsg, rfl⟩\n  rcases IsMeasurablyGenerated.exists_measurable_subset hsf with ⟨s'f, hs'f, hmf, hs'sf⟩\n  rcases IsMeasurablyGenerated.exists_measurable_subset hsg with ⟨s'g, hs'g, hmg, hs'sg⟩\n  refine ⟨s'f ∩ s'g, inter_mem_inf hs'f hs'g, hmf.inter hmg, ?_⟩\n  exact inter_subset_inter hs'sf hs'sg\n\ntheorem principal_isMeasurablyGenerated_iff {s : Set α} :\n    IsMeasurablyGenerated (𝓟 s) ↔ MeasurableSet s := by\n  refine ⟨?_, fun hs => ⟨fun t ht => ⟨s, mem_principal_self s, hs, ht⟩⟩⟩\n  rintro ⟨hs⟩\n  rcases hs (mem_principal_self s) with ⟨t, ht, htm, hts⟩\n  have : t = s := hts.antisymm ht\n  rwa [← this]\n\nalias ⟨_, _root_.MeasurableSet.principal_isMeasurablyGenerated⟩ :=\n  principal_isMeasurablyGenerated_iff\n\ninstance iInf_isMeasurablyGenerated {f : ι → Filter α} [∀ i, IsMeasurablyGenerated (f i)] :\n    IsMeasurablyGenerated (⨅ i, f i) := by\n  refine ⟨fun s hs => ?_⟩\n  rw [← Equiv.plift.surjective.iInf_comp, mem_iInf] at hs\n  rcases hs with ⟨t, ht, ⟨V, hVf, rfl⟩⟩\n  choose U hUf hU using fun i => IsMeasurablyGenerated.exists_measurable_subset (hVf i)\n  refine ⟨⋂ i : t, U i, ?_, ?_, ?_⟩\n  · rw [← Equiv.plift.surjective.iInf_comp, mem_iInf]\n    exact ⟨t, ht, U, hUf, rfl⟩\n  · haveI := ht.countable.toEncodable.countable\n    exact MeasurableSet.iInter fun i => (hU i).1\n  · exact iInter_mono fun i => (hU i).2\n\nend Filter\n\n/-- The set of points for which a sequence of measurable functions converges to a given value\nis measurable. -/\n@[measurability]\n\nTarget:\nlemma measurableSet_tendsto {_ : MeasurableSpace β} [MeasurableSpace γ]\n    [Countable δ] {l : Filter δ} [l.IsCountablyGenerated]\n    (l' : Filter γ) [l'.IsCountablyGenerated] [hl' : l'.IsMeasurablyGenerated]\n    {f : δ → β → γ} (hf : ∀ i, Measurable (f i)) :\n    MeasurableSet { x | Tendsto (fun n ↦ f n x) l l' } :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/MeasurableSpace","family_id":"measurableset_tendsto","file_id":"mathlib/Mathlib/MeasureTheory/MeasurableSpace/MeasurablyGenerated.lean","sample_id":"f6e07b9557819a58269647c6b5189639d71a69d808a07410892e1fb416291719"},"split":"train"}
{"binding":{"chosen_proof_sha256":"318c2cb9db795a2dd46d3f5a27d1f76a3d76756692909ad6bddc78fa03561671","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7ad57aa84d793cc3401f5f4a8dc65aac7c2a625b38a0756094d767468b9097fb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7da1d9db371103784bbc6daf0c80df8acd3ad5e9a823b6012d8ef842a94b24f4","source_sha256":"e3bd64c53d80fc2f3ac9f455d14462bc0198848b67a51278e36a13f2aa735979","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  rw [← trans_dualMap_symm_flip e]\n  exact LinearEquiv.bijective _","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":3,"token_jaccard":0.142857,"token_length_ratio":0.230769},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"03e83ffab3b24511547a5c1a3e2a00b485a60092b86eeaa712e710a6b54d8fe7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Dual.Lemmas\n\nNamespace:\nLinearEquiv\n\nLocal context:\n/-\nCopyright (c) 2023 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash, Yaël Dillies\n-/\n/-!\n# Perfect pairings\n\nThis file defines perfect pairings of modules.\n\nA perfect pairing of two (left) modules may be defined either as:\n1. A bilinear map `M × N → R` such that the induced maps `M → Dual R N` and `N → Dual R M` are both\n  bijective. It follows from this that both `M` and `N` are reflexive modules.\n2. A linear equivalence `N ≃ Dual R M` for which `M` is reflexive. (It then follows that `N` is\n  reflexive.)\n\nIn this file we provide a definition `IsPerfPair` corresponding to 1 above, together with logic\nto connect 1 and 2.\n-/\n\n@[expose] public section\n\nopen Function Module\n\nnamespace LinearMap\nvariable {R K M M' N N' : Type*} [AddCommGroup M] [AddCommGroup N] [AddCommGroup M']\n  [AddCommGroup N']\n\nsection CommRing\nvariable [CommRing R] [Module R M] [Module R M'] [Module R N] [Module R N']\n  {p : M →ₗ[R] N →ₗ[R] R} {x : M} {y : N}\n\n/-- For a ring `R` and two modules `M` and `N`, a perfect pairing is a bilinear map `M × N → R`\nthat is bijective in both arguments. -/\n@[ext]\nclass IsPerfPair (p : M →ₗ[R] N →ₗ[R] R) where\n  bijective_left (p) : Bijective p\n  bijective_right (p) : Bijective p.flip\n\n/-- Given a perfect pairing between `M` and `N`, we may interchange the roles of `M` and `N`. -/\nprotected lemma IsPerfPair.flip (hp : p.IsPerfPair) : p.flip.IsPerfPair where\n  bijective_left := IsPerfPair.bijective_right p\n  bijective_right := IsPerfPair.bijective_left p\n\nvariable [p.IsPerfPair]\n\n/-- Given a perfect pairing between `M` and `N`, we may interchange the roles of `M` and `N`. -/\ninstance flip.instIsPerfPair : p.flip.IsPerfPair := .flip ‹_›\n\nvariable (p)\n\n/-- Turn a perfect pairing between `M` and `N` into an isomorphism between `M` and the dual of `N`.\n-/\nnoncomputable def toPerfPair : M ≃ₗ[R] Dual R N :=\n  .ofBijective { toFun := _, map_add' x y := by simp, map_smul' r x := by simp } <|\n    IsPerfPair.bijective_left p\n\n@[simp] lemma toLinearMap_toPerfPair (x : M) : p.toPerfPair x = p x := rfl\n@[simp] lemma toPerfPair_apply (x : M) (y : N) : p.toPerfPair x y = p x y := rfl\n\n@[simp] lemma apply_symm_toPerfPair_self (f : Dual R N) : p (p.toPerfPair.symm f) = f :=\n  p.toPerfPair.apply_symm_apply f\n\n@[simp] lemma apply_toPerfPair_flip (f : Dual R M) (x : M) : p x (p.flip.toPerfPair.symm f) = f x :=\n  congr($(p.flip.apply_symm_toPerfPair_self ..) x)\n\ninclude p in\nlemma _root_.Module.IsReflexive.of_isPerfPair : IsReflexive R M where\n  bijective_dual_eval' := by\n    convert! (p.toPerfPair.trans p.flip.toPerfPair.dualMap.symm).bijective\n    ext x f\n    simp\n\ninclude p in\nlemma _root_.Module.finrank_of_isPerfPair [Module.Finite R M] [Module.Free R M] :\n    finrank R M = finrank R N :=\n  ((Module.Free.chooseBasis R M).toDualEquiv.trans p.flip.toPerfPair.symm).finrank_eq\n\n/-- A reflexive module has a perfect pairing with its dual. -/\nprotected instance IsPerfPair.id [IsReflexive R M] : IsPerfPair (.id (R := R) (M := Dual R M)) where\n  bijective_left := bijective_id\n  bijective_right := bijective_dual_eval R M\n\n/-- A reflexive module has a perfect pairing with its dual. -/\ninstance IsPerfPair.dualEval [IsReflexive R M] : IsPerfPair (Dual.eval R M) := .flip .id\n\ninstance IsPerfPair.compl₁₂ (eM : M' ≃ₗ[R] M) (eN : N' ≃ₗ[R] N) :\n    (p.compl₁₂ eM eN : M' →ₗ[R] N' →ₗ[R] R).IsPerfPair :=\n  ⟨((LinearEquiv.congrLeft R R eN).symm.bijective.comp\n    (IsPerfPair.bijective_left p)).comp eM.bijective,\n    ((LinearEquiv.congrLeft R R eM).symm.bijective.comp\n    (IsPerfPair.bijective_right p)).comp eN.bijective⟩\n\nlemma IsPerfPair.congr (eM : M' ≃ₗ[R] M) (eN : N' ≃ₗ[R] N) (q : M' →ₗ[R] N' →ₗ[R] R)\n    (H : q.compl₁₂ eM.symm eN.symm = p) : q.IsPerfPair := by\n  obtain rfl : q = p.compl₁₂ eM eN := by subst H; ext; simp\n  infer_instance\n\nlemma IsPerfPair.of_bijective (p : M →ₗ[R] N →ₗ[R] R) [IsReflexive R N] (h : Bijective p) :\n    IsPerfPair p :=\n  inferInstanceAs ((LinearMap.id (R := R) (M := Dual R N)).compl₁₂\n    (LinearEquiv.ofBijective p h : M →ₗ[R] N →ₗ[R] R)\n    (LinearEquiv.refl R N : N →ₗ[R] N)).IsPerfPair\n\nend CommRing\n\nsection Field\nvariable [Field K] [Module K M] [Module K N] {p : M →ₗ[K] N →ₗ[K] K} {x : M} {y : N}\n\n/-- If the coefficients are a field, and one of the spaces is finite-dimensional, it is sufficient\nto check only injectivity instead of bijectivity of the bilinear pairing. -/\nlemma IsPerfPair.of_injective [FiniteDimensional K M] (h : Injective p) (h' : Injective p.flip) :\n    p.IsPerfPair where\n  bijective_left := ⟨h, by rwa [← p.flip_injective_iff₁]⟩\n  bijective_right := ⟨h', by\n    have : FiniteDimensional K N := FiniteDimensional.of_injective p.flip h'\n    rwa [← p.flip.flip_injective_iff₁, LinearMap.flip_flip]⟩\n\n/-- If the coefficients are a field, and one of the spaces is finite-dimensional, it is sufficient\nto check only injectivity instead of bijectivity of the bilinear pairing. -/\nlemma IsPerfPair.of_injective' [FiniteDimensional K N] (h : Injective p) (h' : Injective p.flip) :\n    p.IsPerfPair := .flip <| .of_injective h' h\n\nend Field\nend LinearMap\n\nnoncomputable section\n\nvariable {R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]\n\nnamespace LinearMap\nvariable {p : M →ₗ[R] N →ₗ[R] R} [p.IsPerfPair]\n\nvariable (p) in\n/-- Given a perfect pairing `p` between `M` and `N`, we say a pair of submodules `U` in `M` and\n`V` in `N` are perfectly complementary w.r.t. `p` if their dual annihilators are complementary,\nusing `p` to identify `M` and `N` with dual spaces. -/\nstructure IsPerfectCompl (U : Submodule R M) (V : Submodule R N) : Prop where\n  isCompl_left : IsCompl U (V.dualAnnihilator.map (p.toPerfPair.symm : Dual R N →ₗ[R] M))\n  isCompl_right : IsCompl V (U.dualAnnihilator.map (p.flip.toPerfPair.symm : Dual R M →ₗ[R] N))\n\nnamespace IsPerfectCompl\nvariable {U : Submodule R M} {V : Submodule R N}\n\nprotected lemma flip (h : p.IsPerfectCompl U V) :\n    p.flip.IsPerfectCompl V U where\n  isCompl_left := h.isCompl_right\n  isCompl_right := h.isCompl_left\n\n@[simp]\nprotected lemma flip_iff :\n    p.flip.IsPerfectCompl V U ↔ p.IsPerfectCompl U V :=\n  ⟨fun h ↦ h.flip, fun h ↦ h.flip⟩\n\n@[simp]\nlemma left_top_iff :\n    p.IsPerfectCompl ⊤ V ↔ V = ⊤ := by\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · exact eq_top_of_isCompl_bot <| by simpa using h.isCompl_right\n  · rw [h]\n    exact\n      { isCompl_left := by simpa using isCompl_top_bot\n        isCompl_right := by simpa using isCompl_top_bot }\n\n@[simp]\nlemma right_top_iff :\n    p.IsPerfectCompl U ⊤ ↔ U = ⊤ := by\n  rw [← IsPerfectCompl.flip_iff]\n  exact left_top_iff\n\nend IsPerfectCompl\n\nend LinearMap\n\nvariable [IsReflexive R M]\n\nvariable (e : N ≃ₗ[R] Dual R M)\n\nnamespace LinearEquiv\n\n/-- For a reflexive module `M`, an equivalence `N ≃ₗ[R] Dual R M` naturally yields an equivalence\n`M ≃ₗ[R] Dual R N`. Such equivalences are known as perfect pairings. -/\ndef flip : M ≃ₗ[R] Dual R N :=\n  (evalEquiv R M).trans e.dualMap\n\n@[simp] lemma coe_toLinearMap_flip : e.flip = (↑e : N →ₗ[R] Dual R M).flip := rfl\n\n@[simp] lemma flip_apply (m : M) (n : N) : e.flip m n = e n m := rfl\n\nlemma symm_flip : e.flip.symm = e.symm.dualMap.trans (evalEquiv R M).symm := rfl\n\nlemma trans_dualMap_symm_flip : e.trans e.flip.symm.dualMap = Dual.eval R N := by\n  ext; simp [symm_flip]\n\ninclude e in\n/-- If `N` is in perfect pairing with `M`, then it is reflexive. -/\n\nTarget:\nlemma isReflexive_of_equiv_dual_of_isReflexive : IsReflexive R N :=\n\nProof body:\n","rejected":"by\n  exact isReflexive_of_equiv_dual_of_isReflexive","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"cdb778f32decca19a855e9a5065b72a08874ee34b39c0ee27b4c189e9421d036","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/PerfectPairing","family_id":"isreflexive_of_equiv_dual_of_isreflexive","file_id":"mathlib/Mathlib/LinearAlgebra/PerfectPairing/Basic.lean","sample_id":"7da1d9db371103784bbc6daf0c80df8acd3ad5e9a823b6012d8ef842a94b24f4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7361ae6d5ff40c196b32b028010a5c56393ce19a055fbd12b76df3365e97ad61","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6729f70f1e32fd9d52ce0ff9275d6acf57200f41a2cf0189cdeb6707f92d3fff","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"108e499b9132e898676c8d66a09fbbcb3ff6a762c243e0f235bc6414a3079513","source_sha256":"d7743ebc410d9ee31a12263ae8d894bf6aecd1d52cd1a367eccb2f5de9ab8de8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h ↦ eq_univ_iff_forall.mpr h.mem_periodicPts, fun h ↦ ?_⟩\n  rw [Finite.injective_iff_surjective, ← range_eq_univ, ← univ_subset_iff, ← h]\n  apply periodicPts_subset_range","hard_negative":true,"metrics":{"chosen_tokens":36,"rejected_tokens":3,"token_jaccard":0.038462,"token_length_ratio":0.083333},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"049191310a0dddf7afcd055460d88623cc382f2ac12ebb46054d2e2d91633a24","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GCDMonoid.Finset\npublic import Mathlib.Data.Nat.Prime.Basic\npublic import Mathlib.Data.PNat.Basic\npublic import Mathlib.Dynamics.PeriodicPts.Defs\npublic import Mathlib.Order.Lattice.Nat\n\nNamespace:\nFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Extra lemmas about periodic points\n-/\n\npublic section\n\nopen Nat Set\n\nnamespace Function\nvariable {α : Type*} {f : α → α} {x y : α}\n\nopen Function (Commute)\n\ntheorem directed_ptsOfPeriod_pnat (f : α → α) : Directed (· ⊆ ·) fun n : ℕ+ => ptsOfPeriod f n :=\n  fun m n => ⟨m * n, fun _ hx => hx.mul_const n, fun _ hx => hx.const_mul m⟩\n\nvariable (f) in\ntheorem bijOn_periodicPts : BijOn f (periodicPts f) (periodicPts f) :=\n  iUnion_pnat_ptsOfPeriod f ▸\n    bijOn_iUnion_of_directed (directed_ptsOfPeriod_pnat f) fun i => bijOn_ptsOfPeriod f i.pos\n\ntheorem minimalPeriod_eq_prime_iff {p : ℕ} [hp : Fact p.Prime] :\n    minimalPeriod f x = p ↔ IsPeriodicPt f p x ∧ ¬IsFixedPt f x := by\n  rw [Function.isPeriodicPt_iff_minimalPeriod_dvd, Nat.dvd_prime hp.out,\n    ← minimalPeriod_eq_one_iff_isFixedPt.not, or_and_right, and_not_self_iff, false_or,\n    iff_self_and]\n  exact fun h ↦ ne_of_eq_of_ne h hp.out.ne_one\n\ntheorem minimalPeriod_eq_sInf_n_pos_IsPeriodicPt :\n    minimalPeriod f x = sInf { n > 0 | IsPeriodicPt f n x } := by\n  dsimp +instances [minimalPeriod, periodicPts, sInf]\n  grind\n\n/-- The backward direction of `minimalPeriod_eq_prime_iff`. -/\ntheorem minimalPeriod_eq_prime {p : ℕ} [hp : Fact p.Prime] (hper : IsPeriodicPt f p x)\n    (hfix : ¬IsFixedPt f x) : minimalPeriod f x = p :=\n  minimalPeriod_eq_prime_iff.mpr ⟨hper, hfix⟩\n\ntheorem minimalPeriod_eq_prime_pow {p k : ℕ} [hp : Fact p.Prime] (hk : ¬IsPeriodicPt f (p ^ k) x)\n    (hk1 : IsPeriodicPt f (p ^ (k + 1)) x) : minimalPeriod f x = p ^ (k + 1) := by\n  apply Nat.eq_prime_pow_of_dvd_least_prime_pow hp.out <;>\n    rwa [← isPeriodicPt_iff_minimalPeriod_dvd]\n\ntheorem Commute.minimalPeriod_of_comp_dvd_mul {g : α → α} (h : Commute f g) :\n    minimalPeriod (f ∘ g) x ∣ minimalPeriod f x * minimalPeriod g x :=\n  dvd_trans h.minimalPeriod_of_comp_dvd_lcm (Nat.lcm_dvd_mul _ _)\n\ntheorem Commute.minimalPeriod_of_comp_eq_mul_of_coprime {g : α → α} (h : Commute f g)\n    (hco : Coprime (minimalPeriod f x) (minimalPeriod g x)) :\n    minimalPeriod (f ∘ g) x = minimalPeriod f x * minimalPeriod g x := by\n  apply h.minimalPeriod_of_comp_dvd_mul.antisymm\n  suffices ∀ {f g : α → α},\n      Commute f g →\n        Coprime (minimalPeriod f x) (minimalPeriod g x) →\n          minimalPeriod f x ∣ minimalPeriod (f ∘ g) x from\n    hco.mul_dvd_of_dvd_of_dvd (this h hco) (h.comp_eq.symm ▸ this h.symm hco.symm)\n  intro f g h hco\n  refine hco.dvd_of_dvd_mul_left (IsPeriodicPt.left_of_comp h ?_ ?_).minimalPeriod_dvd\n  · exact (isPeriodicPt_minimalPeriod _ _).const_mul _\n  · exact (isPeriodicPt_minimalPeriod _ _).mul_const _\n\nsection Fintype\n\nopen Fintype\n\ntheorem minimalPeriod_le_card [Fintype α] : minimalPeriod f x ≤ card α := by\n  rw [← periodicOrbit_length]\n  exact List.Nodup.length_le_card nodup_periodicOrbit\n\ntheorem isPeriodicPt_factorial_card_of_mem_periodicPts [Fintype α] (h : x ∈ periodicPts f) :\n    IsPeriodicPt f (card α)! x :=\n  isPeriodicPt_iff_minimalPeriod_dvd.mpr\n    (Nat.dvd_factorial (minimalPeriod_pos_of_mem_periodicPts h) minimalPeriod_le_card)\n\ntheorem mem_periodicPts_iff_isPeriodicPt_factorial_card [Fintype α] :\n    x ∈ periodicPts f ↔ IsPeriodicPt f (card α)! x where\n  mp := isPeriodicPt_factorial_card_of_mem_periodicPts\n  mpr h := minimalPeriod_pos_iff_mem_periodicPts.mp\n    (IsPeriodicPt.minimalPeriod_pos (Nat.factorial_pos _) h)\n\ntheorem Injective.mem_periodicPts [Finite α] (h : Injective f) (x : α) : x ∈ periodicPts f := by\n  obtain ⟨m, n, heq, hne⟩ : ∃ m n, f^[m] x = f^[n] x ∧ m ≠ n := by\n    simpa [Injective] using not_injective_infinite_finite (f^[·] x)\n  rcases lt_or_gt_of_ne hne with hlt | hlt\n  · exact mk_mem_periodicPts (by lia) (iterate_cancel h heq.symm)\n  · exact mk_mem_periodicPts (by lia) (iterate_cancel h heq)\n\nTarget:\ntheorem injective_iff_periodicPts_eq_univ [Finite α] : Injective f ↔ periodicPts f = univ :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_108e499b9132","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"e9369201a8ae13c9e08dbb556769b0fb5ac5f4aa659c3b39e30c09ae5f57502a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/PeriodicPts","family_id":"injective_iff_periodicpts_eq_univ","file_id":"mathlib/Mathlib/Dynamics/PeriodicPts/Lemmas.lean","sample_id":"108e499b9132e898676c8d66a09fbbcb3ff6a762c243e0f235bc6414a3079513"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c66eb3df5899011ef7909b5b28a8dbc7ead7f1cd7984dec7b597a92d174e6203","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e4be22c9b9981e432ef9024a9996296ed850a5052a0e1ee68ff3cc82dce6a051","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bf0246d237d6d9933a85d317c9082d92e62a7779e9ca287e29ec9e8f9ef938b0","source_sha256":"696827f59141a7e073b768692576d4aa28b090a0ebc3937f46b2b10aa2472ecd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [SModEq.sub_mem] at hxy ⊢\n  rw [← smul_sub]\n  exact smul_mem_smul hc hxy","hard_negative":true,"metrics":{"chosen_tokens":19,"rejected_tokens":3,"token_jaccard":0.125,"token_length_ratio":0.157895},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"0497dfba73125893b1e63c80e2a829a70b784bd03d43948f3bedeea907ff7248","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Operations\npublic import Mathlib.LinearAlgebra.SModEq.Basic\n\nNamespace:\nSModEq\n\nLocal context:\n/-\nCopyright (c) 2025 Jiedong Jiang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jiedong Jiang\n-/\n/-!\n# Pointwise lemmas for modular equivalence\n\nIn this file, we record more lemmas about `SModEq` on elements\nof modules or rings.\n-/\n\npublic section\n\nopen Submodule\n\nopen Polynomial\n\nvariable {R : Type*} [Ring R] {I : Ideal R}\nvariable {M : Type*} [AddCommGroup M] [Module R M] {U : Submodule R M}\nvariable {x y : M}\n\nnamespace SModEq\n\n/--\nA variant of `SModEq.smul`, where the scalar belongs to an ideal.\n-/\n\nTarget:\ntheorem smul' (hxy : x ≡ y [SMOD U])\n    {c : R} (hc : c ∈ I) : c • x ≡ c • y [SMOD (I • U)] :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_bf0246d237d6","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"998eeff469aca5126fc003e01a64a753056863204f1986fc278ccae4f1bdc251","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/SModEq","family_id":"smul","file_id":"mathlib/Mathlib/LinearAlgebra/SModEq/Pointwise.lean","sample_id":"bf0246d237d6d9933a85d317c9082d92e62a7779e9ca287e29ec9e8f9ef938b0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a0b20474523f493304d2e5c3c4796a60a88d5f5165ad276749d342179c8ed04e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0f8900d3e899ee934779d7ba7219c40f2683efc29f54c88c95ae34478c5fd6b3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6f6ae3b60a22d3ee8598639d07edd15dc214dc93b32db436d37ba0364ed7ec2a","source_sha256":"3e26e2de40fc89c98c0734f65f6d65c61b9e098984ba72f67bb093182d517ead","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  rw [separatingLeft_iff_ker_eq_bot, ker_eq_bot]\n  let e := LinearEquiv.prodComm R _ _ ≪≫ₗ Module.dualProdDualEquivDual R (Module.Dual R M) M\n  let h_d := e.symm.toLinearMap.comp (dualProd R M)\n  refine (Function.Injective.of_comp_iff e.symm.injective\n    (dualProd R M)).symm.trans ?_\n  rw [← LinearEquiv.coe_toLinearMap, ← coe_comp]\n  change Function.Injective h_d ↔ _\n  have : h_d = prodMap id (Module.Dual.eval R M) := by\n    refine ext fun x => Prod.ext ?_ ?_\n    · ext\n      dsimp [e, h_d, Module.Dual.eval, LinearEquiv.prodComm]\n      simp\n    · ext\n      dsimp [e, h_d, Module.Dual.eval, LinearEquiv.prodComm]\n      simp\n  rw [this, coe_prodMap]\n  refine Prod.map_injective.trans ?_\n  exact and_iff_right Function.injective_id","hard_negative":true,"metrics":{"chosen_tokens":174,"rejected_tokens":5,"token_jaccard":0.046875,"token_length_ratio":0.028736},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"04a1f03b51301ea5b5c4fc1791ce61e223b3c20ca1450cf564e127aa61be91fa","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Dual.Lemmas\npublic import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv\npublic import Mathlib.LinearAlgebra.QuadraticForm.Prod\n\nNamespace:\nLinearMap\n\nLocal context:\n/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Quadratic form structures related to `Module.Dual`\n\n## Main definitions\n\n* `LinearMap.dualProd R M`, the bilinear form on `(f, x) : Module.Dual R M × M` defined as\n  `f x`.\n* `QuadraticForm.dualProd R M`, the quadratic form on `(f, x) : Module.Dual R M × M` defined as\n  `f x`.\n* `QuadraticForm.toDualProd : (Q.prod <| -Q) →qᵢ QuadraticForm.dualProd R M` a form-preserving map\n  from `(Q.prod <| -Q)` to `QuadraticForm.dualProd R M`.\n\n-/\n\n@[expose] public section\n\nvariable (R M N : Type*)\n\nnamespace LinearMap\n\nsection Semiring\n\nvariable [CommSemiring R] [AddCommMonoid M] [Module R M]\n\n/-- The symmetric bilinear form on `Module.Dual R M × M` defined as\n`B (f, x) (g, y) = f y + g x`. -/\n@[simps!]\ndef dualProd : LinearMap.BilinForm R (Module.Dual R M × M) :=\n    (applyₗ.comp (snd R (Module.Dual R M) M)).compl₂ (fst R (Module.Dual R M) M) +\n      ((applyₗ.comp (snd R (Module.Dual R M) M)).compl₂ (fst R (Module.Dual R M) M)).flip\n\ntheorem isSymm_dualProd : (dualProd R M).IsSymm := ⟨fun _x _y => add_comm _ _⟩\n\nend Semiring\n\nsection Ring\n\nvariable [CommRing R] [AddCommGroup M] [Module R M]\n\nTarget:\ntheorem separatingLeft_dualProd :\n    (dualProd R M).SeparatingLeft ↔ Function.Injective (Module.Dual.eval R M) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_6f6ae3b60a22","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"a1cd792e41f8264e0873890de7e6e0c0c697cec886773af634497dc1fb85445c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/QuadraticForm","family_id":"separatingleft_dualprod","file_id":"mathlib/Mathlib/LinearAlgebra/QuadraticForm/Dual.lean","sample_id":"6f6ae3b60a22d3ee8598639d07edd15dc214dc93b32db436d37ba0364ed7ec2a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5fc5fcc6ed6270b255f9132aa985cf7070d173e497c862b80ffbe80e51562de8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"36e6a6a064db8ebbe5935e08c6a7f9aae4b4d3115561d63fc6249d7b668a9771","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0f82b3064f5d92a16f748b7b934596ecec0fdc7677d78faf22eb29829f183996","source_sha256":"dbc79381fe985a1a450b97d09f056dd0638cc629a12da566daa44070340e962c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [Set.Sized, Set.mem_iUnion, forall_exists_index]\n  exact forall_comm\n\n-- `simp` normal form is `sized_iUnion`.","hard_negative":true,"metrics":{"chosen_tokens":27,"rejected_tokens":2,"token_jaccard":0.05,"token_length_ratio":0.074074},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"04a5fd50bb702262dca90065fecf02aad0d4b646d83b5ae4421cb8cb17361903","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Order.Interval.Finset.Nat\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nNamespace:\nSet\n\nLocal context:\n/-\nCopyright (c) 2021 Bhavik Mehta, Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Alena Gusakov, Yaël Dillies\n-/\n/-!\n# `r`-sets and slice\n\nThis file defines the `r`-th slice of a set family and provides a way to say that a set family is\nmade of `r`-sets.\n\nAn `r`-set is a finset of cardinality `r` (aka of *size* `r`). The `r`-th slice of a set family is\nthe set family made of its `r`-sets.\n\n## Main declarations\n\n* `Set.Sized`: `A.Sized r` means that `A` only contains `r`-sets.\n* `Finset.slice`: `A.slice r` is the set of `r`-sets in `A`.\n\n## Notation\n\n`A # r` is notation for `A.slice r` in scope `finset_family`.\n-/\n\n@[expose] public section\n\n\nopen Finset Nat\n\nvariable {α : Type*} {ι : Sort*} {κ : ι → Sort*}\n\nnamespace Set\n\nvariable {A B : Set (Finset α)} {s : Finset α} {r : ℕ}\n\n/-! ### Families of `r`-sets -/\n\n\n/-- `Sized r A` means that every Finset in `A` has size `r`. -/\ndef Sized (r : ℕ) (A : Set (Finset α)) : Prop := ∀ ⦃x⦄, x ∈ A → #x = r\n\ntheorem Sized.mono (h : A ⊆ B) (hB : B.Sized r) : A.Sized r := fun _x hx => hB <| h hx\n\n@[simp] lemma sized_empty : (∅ : Set (Finset α)).Sized r := by simp [Sized]\n@[simp] lemma sized_singleton : ({s} : Set (Finset α)).Sized r ↔ #s = r := by simp [Sized]\n\ntheorem sized_union : (A ∪ B).Sized r ↔ A.Sized r ∧ B.Sized r :=\n  ⟨fun hA => ⟨hA.mono subset_union_left, hA.mono subset_union_right⟩, fun hA _x hx =>\n    hx.elim (fun h => hA.1 h) fun h => hA.2 h⟩\n\nalias ⟨_, sized.union⟩ := sized_union\n\n--TODO: A `forall_iUnion` lemma would be handy here.\n@[simp]\n\nTarget:\ntheorem sized_iUnion {f : ι → Set (Finset α)} : (⋃ i, f i).Sized r ↔ ∀ i, (f i).Sized r :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_0f82b3064f5d","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"ccbccaa3b15516808c6e7c4993eb990ce90eaaad1e5262b3004b3cb74c374b7e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finset","family_id":"sized_iunion","file_id":"mathlib/Mathlib/Data/Finset/Slice.lean","sample_id":"0f82b3064f5d92a16f748b7b934596ecec0fdc7677d78faf22eb29829f183996"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e79d2c183534cfffc99b615577f1db04cdc59929d66d45cf7905520be00385d4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2a949889ffca8542db6fbc433458938646cdc565445f5f1b5c344477b7981aab","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5627650f4b1175b07bdab889615a32ef5fc6133ef66c020ba65b75f9c09ed833","source_sha256":"de50c3991d383f79a1e56820d274c7129a795179a5130040268b4911014d87d8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  obtain ⟨F₁, hF₁, hAF₁⟩ := hA.sq_covBySMul\n  obtain ⟨F₂, hF₂, hBF₂⟩ := hB.sq_covBySMul\n  have := hA.one_le\n  choose f hf using exists_smul_inter_smul_subset_smul_inv_mul_inter_inv_mul A B\n  refine ⟨.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1)), ?_, ?_⟩\n  · calc\n      (#(.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1))) : ℝ)\n      _ ≤ #(F₁ ^ (m - 1)) * #(F₂ ^ (n - 1)) := mod_cast Finset.card_image₂_le ..\n      _ ≤ #F₁ ^ (m - 1) * #F₂ ^ (n - 1) := by gcongr <;> exact mod_cast Finset.card_pow_le\n      _ ≤ K ^ (m - 1) * L ^ (n - 1) := by gcongr\n  · calc\n      A ^ m ∩ B ^ n ⊆ (F₁ ^ (m - 1) * A) ∩ (F₂ ^ (n - 1) * B) := by\n        gcongr <;> apply pow_subset_pow_mul_of_sq_subset_mul <;> norm_cast <;> lia\n      _ = ⋃ (a ∈ F₁ ^ (m - 1)) (b ∈ F₂ ^ (n - 1)), a • A ∩ b • B := by\n        simp_rw [← smul_eq_mul, ← iUnion_smul_set, iUnion₂_inter_iUnion₂]; norm_cast\n      _ ⊆ ⋃ (a ∈ F₁ ^ (m - 1)) (b ∈ F₂ ^ (n - 1)), f a b • (A⁻¹ * A ∩ (B⁻¹ * B)) := by\n        gcongr; exact hf ..\n      _ = (Finset.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1))) * (A ^ 2 ∩ B ^ 2) := by\n        simp_rw [hA.inv_eq_self, hB.inv_eq_self, ← sq]\n        rw [Finset.coe_image₂, ← smul_eq_mul, ← iUnion_smul_set, biUnion_image2]\n        simp_rw [Finset.mem_coe]","hard_negative":false,"metrics":{"chosen_tokens":429,"rejected_tokens":436,"token_jaccard":0.977011,"token_length_ratio":1.016317},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"04eaba62f4e2be0fad56c2419db1bd1553812e384969072ab892f566a8dcf757","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Subgroup.Pointwise\npublic import Mathlib.Combinatorics.Additive.CovBySMul\npublic import Mathlib.Combinatorics.Additive.RuzsaCovering\npublic import Mathlib.Combinatorics.Additive.SmallTripling\n\nNamespace:\nIsApproximateSubgroup\n\nLocal context:\n/-\nCopyright (c) 2024 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Approximate subgroups\n\nThis file defines approximate subgroups of a group, namely symmetric sets `A` such that `A * A` can\nbe covered by a small number of translates of `A`.\n\n## Main results\n\nApproximate subgroups are a central concept in additive combinatorics, as a natural weakening and\nflexible substitute of genuine subgroups. As such, they share numerous properties with subgroups:\n* `IsApproximateSubgroup.image`: Group homomorphisms send approximate subgroups to approximate\n  subgroups\n* `IsApproximateSubgroup.pow_inter_pow`: The intersection of (non-trivial powers of) two approximate\n  subgroups is an approximate subgroup. Warning: The intersection of two approximate subgroups isn't\n  an approximate subgroup in general.\n\nApproximate subgroups are close qualitatively and quantitatively to other concepts in additive\ncombinatorics:\n* `IsApproximateSubgroup.card_pow_le`: An approximate subgroup has small powers.\n* `IsApproximateSubgroup.of_small_tripling`: A set of small tripling can be made an approximate\n  subgroup by squaring.\n\nIt can be readily confirmed that approximate subgroups are a weakening of subgroups:\n* `isApproximateSubgroup_one`: A 1-approximate subgroup is the same thing as a subgroup.\n-/\n\npublic section\n\nopen scoped Finset Pointwise\n\nvariable {G : Type*} [Group G] {A B : Set G} {K L : ℝ} {m n : ℕ}\n\n/--\nAn approximate subgroup in a group is a symmetric set `A` containing the identity and such that\n`A + A` can be covered by a small number of translates of `A`.\n\nIn practice, we will take `K` fixed and `A` large but finite.\n-/\nstructure IsApproximateAddSubgroup {G : Type*} [AddGroup G] (K : ℝ) (A : Set G) : Prop where\n  zero_mem : 0 ∈ A\n  neg_eq_self : -A = A\n  two_nsmul_covByVAdd : CovByVAdd G K (2 • A) A\n\n/--\nAn approximate subgroup in a group is a symmetric set `A` containing the identity and such that\n`A * A` can be covered by a small number of translates of `A`.\n\nIn practice, we will take `K` fixed and `A` large but finite.\n-/\n@[to_additive]\nstructure IsApproximateSubgroup (K : ℝ) (A : Set G) : Prop where\n  one_mem : 1 ∈ A\n  inv_eq_self : A⁻¹ = A\n  sq_covBySMul : CovBySMul G K (A ^ 2) A\n\nnamespace IsApproximateSubgroup\n\n@[to_additive] lemma nonempty (hA : IsApproximateSubgroup K A) : A.Nonempty := ⟨1, hA.one_mem⟩\n\n@[to_additive one_le]\nlemma one_le (hA : IsApproximateSubgroup K A) : 1 ≤ K := by\n  obtain ⟨F, hF, hSF⟩ := hA.sq_covBySMul\n  grw [← hF]\n  have : F.Nonempty := by by_contra! rfl; simp [hA.nonempty.ne_empty] at hSF\n  simpa\n\n@[to_additive]\nlemma mono (hKL : K ≤ L) (hA : IsApproximateSubgroup K A) : IsApproximateSubgroup L A where\n  one_mem := hA.one_mem\n  inv_eq_self := hA.inv_eq_self\n  sq_covBySMul := hA.sq_covBySMul.mono hKL\n\n@[to_additive]\nlemma card_pow_le [DecidableEq G] {A : Finset G} (hA : IsApproximateSubgroup K (A : Set G)) :\n    ∀ {n}, #(A ^ n) ≤ K ^ (n - 1) * #A\n  | 0 => by simpa using hA.nonempty\n  | 1 => by simp\n  | n + 2 => by\n    obtain ⟨F, hF, hSF⟩ := hA.sq_covBySMul\n    calc\n      (#(A ^ (n + 2)) : ℝ) ≤ #(F ^ (n + 1) * A) := by\n        gcongr; exact mod_cast Set.pow_subset_pow_mul_of_sq_subset_mul hSF (by lia)\n      _ ≤ #(F ^ (n + 1)) * #A := mod_cast Finset.card_mul_le\n      _ ≤ #F ^ (n + 1) * #A := by gcongr; exact mod_cast Finset.card_pow_le\n      _ ≤ K ^ (n + 1) * #A := by gcongr\n\n@[to_additive]\nlemma card_mul_self_le [DecidableEq G] {A : Finset G} (hA : IsApproximateSubgroup K (A : Set G)) :\n    #(A * A) ≤ K * #A := by simpa [sq] using hA.card_pow_le (n := 2)\n\n@[to_additive]\nlemma image {F H : Type*} [Group H] [FunLike F G H] [MonoidHomClass F G H] (f : F)\n    (hA : IsApproximateSubgroup K A) : IsApproximateSubgroup K (f '' A) where\n  one_mem := ⟨1, hA.one_mem, map_one _⟩\n  inv_eq_self := by simp [← Set.image_inv, hA.inv_eq_self]\n  sq_covBySMul := by\n    classical\n    obtain ⟨F, hF, hAF⟩ := hA.sq_covBySMul\n    refine ⟨F.image f, ?_, ?_⟩\n    · calc\n        (#(F.image f) : ℝ) ≤ #F := mod_cast F.card_image_le\n        _ ≤ K := hF\n    · simp only [← Set.image_pow, Finset.coe_image, ← Set.image_mul, smul_eq_mul] at hAF ⊢\n      gcongr\n\n@[to_additive]\nlemma subgroup {S : Type*} [SetLike S G] [SubgroupClass S G] {H : S} :\n    IsApproximateSubgroup 1 (H : Set G) where\n  one_mem := OneMemClass.one_mem H\n  inv_eq_self := inv_coe_set\n  sq_covBySMul := ⟨{1}, by simp⟩\n\nopen Finset in\n@[to_additive]\nlemma of_small_tripling [DecidableEq G] {A : Finset G} (hA₁ : 1 ∈ A) (hAsymm : A⁻¹ = A)\n    (hA : #(A ^ 3) ≤ K * #A) : IsApproximateSubgroup (K ^ 3) (A ^ 2 : Set G) where\n  one_mem := by rw [sq, ← one_mul 1]; exact Set.mul_mem_mul hA₁ hA₁\n  inv_eq_self := by simp [← inv_pow, hAsymm, ← coe_inv]\n  sq_covBySMul := by\n    replace hA := calc (#(A ^ 4 * A) : ℝ)\n      _ = #(A ^ 5) := by rw [← pow_succ]\n      _ ≤ K ^ 3 * #A := small_pow_of_small_tripling (by lia) hA hAsymm\n    have hA₀ : A.Nonempty := ⟨1, hA₁⟩\n    obtain ⟨F, -, hF, hAF⟩ := ruzsa_covering_mul hA₀ hA\n    exact ⟨F, hF, by norm_cast; simpa [div_eq_mul_inv, pow_succ, mul_assoc, hAsymm] using hAF⟩\n\nopen Set in\n@[to_additive]\n\nTarget:\nlemma pow_inter_pow_covBySMul_sq_inter_sq\n    (hA : IsApproximateSubgroup K A) (hB : IsApproximateSubgroup L B) (hm : 2 ≤ m) (hn : 2 ≤ n) :\n    CovBySMul G (K ^ (m - 1) * L ^ (n - 1)) (A ^ m ∩ B ^ n) (A ^ 2 ∩ B ^ 2) :=\n\nProof body:\n","rejected":"```lean\nby\n  classical\n  obtain ⟨F₁, hF₁, hAF₁⟩ := hA.sq_covBySMul\n  obtain ⟨F₂, hF₂, hBF₂⟩ := hB.sq_covBySMul\n  have := hA.one_le\n  choose f hf using exists_smul_inter_smul_subset_smul_inv_mul_inter_inv_mul A B\n  refine ⟨.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1)), ?_, ?_⟩\n  · calc\n      (#(.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1))) : ℝ)\n      _ ≤ #(F₁ ^ (m - 1)) * #(F₂ ^ (n - 1)) := mod_cast Finset.card_image₂_le ..\n      _ ≤ #F₁ ^ (m - 1) * #F₂ ^ (n - 1) := by gcongr <;> exact mod_cast Finset.card_pow_le\n      _ ≤ K ^ (m - 1) * L ^ (n - 1) := by gcongr\n  · calc\n      A ^ m ∩ B ^ n ⊆ (F₁ ^ (m - 1) * A) ∩ (F₂ ^ (n - 1) * B) := by\n        gcongr <;> apply pow_subset_pow_mul_of_sq_subset_mul <;> norm_cast <;> lia\n      _ = ⋃ (a ∈ F₁ ^ (m - 1)) (b ∈ F₂ ^ (n - 1)), a • A ∩ b • B := by\n        simp_rw [← smul_eq_mul, ← iUnion_smul_set, iUnion₂_inter_iUnion₂]; norm_cast\n      _ ⊆ ⋃ (a ∈ F₁ ^ (m - 1)) (b ∈ F₂ ^ (n - 1)), f a b • (A⁻¹ * A ∩ (B⁻¹ * B)) := by\n        gcongr; exact hf ..\n      _ = (Finset.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1))) * (A ^ 2 ∩ B ^ 2) := by\n        simp_rw [hA.inv_eq_self, hB.inv_eq_self, ← sq]\n        rw [Finset.coe_image₂, ← smul_eq_mul, ← iUnion_smul_set, biUnion_image2]\n        simp_rw [Finset.mem_coe]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Additive","family_id":"pow_inter_pow_covbysmul_sq_inter_sq","file_id":"mathlib/Mathlib/Combinatorics/Additive/ApproximateSubgroup.lean","sample_id":"5627650f4b1175b07bdab889615a32ef5fc6133ef66c020ba65b75f9c09ed833"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9ea2d48ebe00bb73f1eea97bd737a2f6b8f66a4067141833800162c91765c532","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dc8404380687b2e37f6ac350d18b83560147a6545b4e3c6396ca1a3bc04d0633","source_sha256":"5a251c5b9da944aa52b3b3c01ae0c840fa23c2c1f9c9e5ace9139527d3f0e3e9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [GradedMonoid.GOne.one, TensorPower.toTensorAlgebra_tprod]","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":8,"token_jaccard":0.058824,"token_length_ratio":0.615385},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"05239f75c87635d84490583f3f9a8c44b6f1a90bcc8585dd96bf1b926ab86034","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorAlgebra.Basic\npublic import Mathlib.LinearAlgebra.TensorPower.Basic\n\nNamespace:\nTensorPower\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Tensor algebras as direct sums of tensor powers\n\nIn this file we show that `TensorAlgebra R M` is isomorphic to a direct sum of tensor powers, as\n`TensorAlgebra.equivDirectSum`.\n-/\n\n@[expose] public section\n\nopen scoped DirectSum TensorProduct\n\nvariable {R M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M]\n\nnamespace TensorPower\n\n/-- The canonical embedding from a tensor power to the tensor algebra -/\ndef toTensorAlgebra {n} : ⨂[R]^n M →ₗ[R] TensorAlgebra R M :=\n  PiTensorProduct.lift (TensorAlgebra.tprod R M n)\n\n@[simp]\ntheorem toTensorAlgebra_tprod {n} (x : Fin n → M) :\n    TensorPower.toTensorAlgebra (PiTensorProduct.tprod R x) = TensorAlgebra.tprod R M n x :=\n  PiTensorProduct.lift.tprod _\n\n@[simp]\n\nTarget:\ntheorem toTensorAlgebra_gOne :\n    TensorPower.toTensorAlgebra (@GradedMonoid.GOne.one _ (fun n => ⨂[R]^n M) _ _) = 1 :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"98aefb3cadd5f1aade6e34aae778184241b0dff9f78ebac25e4dfc51ce971434","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/TensorAlgebra","family_id":"totensoralgebra_gone","file_id":"mathlib/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean","sample_id":"dc8404380687b2e37f6ac350d18b83560147a6545b4e3c6396ca1a3bc04d0633"},"split":"train"}
{"binding":{"chosen_proof_sha256":"389a8937a3f2af4740351819b93e71736e399c56154cd81ab2e2e646a0bfda28","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b0b58afdb4cef1d362a08061526d43713fac33d2fd2c151db38d446b2e4f58cd","source_sha256":"36cdfa8e1740ff8584851a3963e63d4eb571008501c17fdeb244e2a7e9d81b8d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [eq_comm, prod_eq_prod_of_nonempty h, eq_comm, @eq_comm _ eY']","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.2},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"052eb700b25f8527bd38890cecc9319b535d5dc2c6e56d3ff48a7b865dd7a646","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.OpenPartialHomeomorph.Composition\n\nNamespace:\nOpenPartialHomeomorph\n\nLocal context:\n/-\nCopyright (c) 2019 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Constructions of new partial homeomorphisms from old\n\n## Main definitions\n\n* `OpenPartialHomeomorph.const`: an open partial homeomorphism which is a constant map,\n  whose source and target are necessarily singleton sets\n* `OpenPartialHomeomorph.subtypeRestr`: restriction to a subtype\n* `OpenPartialHomeomorph.prod`: the product of two open partial homeomorphisms,\n  as an open partial homeomorphism on the product space\n* `OpenPartialHomeomorph.pi`: the product of a finite family of open partial homeomorphisms\n* `OpenPartialHomeomorph.disjointUnion`: combine two open partial homeomorphisms with disjoint\n  sources and disjoint targets\n* `OpenPartialHomeomorph.lift_openEmbedding`: extend an open partial homeomorphism `X → Y`\n  under an open embedding `X → X'`, to an open partial homeomorphism `X' → Z`.\n  (This is used to define the disjoint union of charted spaces.)\n-/\n\n@[expose] public section\n\nopen Function Set Filter Topology\n\nvariable {X X' : Type*} {Y Y' : Type*} {Z Z' : Type*}\n  [TopologicalSpace X] [TopologicalSpace X'] [TopologicalSpace Y] [TopologicalSpace Y']\n  [TopologicalSpace Z] [TopologicalSpace Z']\n\nnamespace OpenPartialHomeomorph\n\nvariable (e : OpenPartialHomeomorph X Y)\n\n/-!\n## Constants\n\n`PartialEquiv.const` as an open partial homeomorphism\n-/\nsection const\n\nvariable {a : X} {b : Y}\n\n/--\nThis is `PartialEquiv.single` as an open partial homeomorphism: a constant map,\nwhose source and target are necessarily singleton sets.\n-/\ndef const (ha : IsOpen {a}) (hb : IsOpen {b}) : OpenPartialHomeomorph X Y where\n  toPartialEquiv := PartialEquiv.single a b\n  open_source := ha\n  open_target := hb\n  continuousOn_toFun := by simp\n  continuousOn_invFun := by simp\n\n@[simp, mfld_simps]\nlemma const_apply (ha : IsOpen {a}) (hb : IsOpen {b}) (x : X) : (const ha hb) x = b := rfl\n\n@[simp, mfld_simps]\nlemma const_source (ha : IsOpen {a}) (hb : IsOpen {b}) : (const ha hb).source = {a} := rfl\n\n@[simp, mfld_simps]\nlemma const_target (ha : IsOpen {a}) (hb : IsOpen {b}) : (const ha hb).target = {b} := rfl\n\nend const\n\n/-!\n## Products\n\nProduct of two open partial homeomorphisms\n-/\nsection Prod\n\n/-- The product of two open partial homeomorphisms, as an open partial homeomorphism on the product\nspace. -/\n@[simps! (attr := mfld_simps) -fullyApplied toPartialHomeomorph apply,\n  simps! -isSimp source target symm_apply]\ndef prod (eX : OpenPartialHomeomorph X X') (eY : OpenPartialHomeomorph Y Y') :\n    OpenPartialHomeomorph (X × Y) (X' × Y') where\n  open_source := eX.open_source.prod eY.open_source\n  open_target := eX.open_target.prod eY.open_target\n  continuousOn_toFun := eX.continuousOn.prodMap eY.continuousOn\n  continuousOn_invFun := eX.continuousOn_symm.prodMap eY.continuousOn_symm\n  toPartialEquiv := eX.toPartialEquiv.prod eY.toPartialEquiv\n\n@[simp, mfld_simps]\ntheorem prod_symm (eX : OpenPartialHomeomorph X X') (eY : OpenPartialHomeomorph Y Y') :\n    (eX.prod eY).symm = eX.symm.prod eY.symm :=\n  rfl\n\n@[simp]\ntheorem refl_prod_refl : (OpenPartialHomeomorph.refl X).prod (OpenPartialHomeomorph.refl Y) =\n    OpenPartialHomeomorph.refl (X × Y) :=\n  OpenPartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) univ_prod_univ\n\n@[simp, mfld_simps]\ntheorem prod_trans (e : OpenPartialHomeomorph X Y) (f : OpenPartialHomeomorph Y Z)\n    (e' : OpenPartialHomeomorph X' Y') (f' : OpenPartialHomeomorph Y' Z') :\n    (e.prod e').trans (f.prod f') = (e.trans f).prod (e'.trans f') :=\n  toPartialEquiv_injective <| e.1.prod_trans ..\n\ntheorem prod_eq_prod_of_nonempty {eX eX' : OpenPartialHomeomorph X X'}\n    {eY eY' : OpenPartialHomeomorph Y Y'} (h : (eX.prod eY).source.Nonempty) :\n    eX.prod eY = eX'.prod eY' ↔ eX = eX' ∧ eY = eY' := by\n  obtain ⟨⟨x, y⟩, -⟩ := id h\n  haveI : Nonempty X := ⟨x⟩\n  haveI : Nonempty X' := ⟨eX x⟩\n  haveI : Nonempty Y := ⟨y⟩\n  haveI : Nonempty Y' := ⟨eY y⟩\n  simp_rw [OpenPartialHomeomorph.ext_iff, prod_apply, prod_symm_apply, prod_source, Prod.ext_iff,\n    Set.prod_eq_prod_iff_of_nonempty h, forall_and, Prod.forall, forall_const,\n    and_assoc, and_left_comm]\n\nTarget:\ntheorem prod_eq_prod_of_nonempty'\n    {eX eX' : OpenPartialHomeomorph X X'} {eY eY' : OpenPartialHomeomorph Y Y'}\n    (h : (eX'.prod eY').source.Nonempty) : eX.prod eY = eX'.prod eY' ↔ eX = eX' ∧ eY = eY' :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/OpenPartialHomeomorph","family_id":"prod_eq_prod_of_nonempty","file_id":"mathlib/Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean","sample_id":"b0b58afdb4cef1d362a08061526d43713fac33d2fd2c151db38d446b2e4f58cd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0e11e10bbaaaed2dc873b84e165eba0bc5466c4e31acb825aeb49b92a5fe94d8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0996acbca200c0dccec01a46d4e81205ed46c98fd950cb9f91af86182dcfbe7d","source_sha256":"5630df9717d5ebdb1ddaef0ed587220f1abeb4c5c8759131e52bd9025a112e08","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  -- Let `Z` be a minimal set of prime ideals such that their product is contained in `J`.\n  obtain ⟨Z₀, hZ₀⟩ := exists_primeSpectrum_prod_le_and_ne_bot_of_domain hNF hI0\n  obtain ⟨Z, ⟨hZI, hprodZ⟩, h_eraseZ⟩ :=\n    wellFounded_lt.has_min\n      {Z | (Z.map asIdeal).prod ≤ I ∧ (Z.map asIdeal).prod ≠ ⊥}\n      ⟨Z₀, hZ₀.1, hZ₀.2⟩\n  obtain ⟨_, hPZ', hPM⟩ := hM.isPrime.multiset_prod_le.mp (hZI.trans hIM)\n  -- Then in fact there is a `P ∈ Z` with `P ≤ M`.\n  obtain ⟨P, hPZ, rfl⟩ := Multiset.mem_map.mp hPZ'\n  classical\n    have := Multiset.map_erase asIdeal (fun _ _ => PrimeSpectrum.ext) P Z\n    obtain ⟨hP0, hZP0⟩ : P.asIdeal ≠ ⊥ ∧ ((Z.erase P).map asIdeal).prod ≠ ⊥ := by\n      rwa [Ne, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←\n        this] at hprodZ\n    -- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.\n    have hPM' := (P.isPrime.isMaximal hP0).eq_of_le hM.ne_top hPM\n    subst hPM'\n    -- By minimality of `Z`, erasing `P` from `Z` is exactly what we need.\n    refine ⟨Z.erase P, ?_, ?_⟩\n    · convert! hZI\n      rw [this, Multiset.cons_erase hPZ']\n    · refine fun h => h_eraseZ (Z.erase P) ⟨h, ?_⟩ (Multiset.erase_lt.mpr hPZ)\n      exact hZP0","hard_negative":false,"metrics":{"chosen_tokens":346,"rejected_tokens":2,"token_jaccard":0.008197,"token_length_ratio":0.00578},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"05511de79a89488a1357faf2a592ef25671df129fd76d0f508cfe5bc6eeb30fa","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Subalgebra.Pointwise\npublic import Mathlib.RingTheory.DedekindDomain.Basic\npublic import Mathlib.RingTheory.FractionalIdeal.Inverse\npublic import Mathlib.RingTheory.Spectrum.Prime.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Kenji Nakagawa. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio\n-/\n/-!\n# Dedekind domains and invertible ideals\n\nIn this file, we show a ring is a Dedekind domain iff all fractional ideals are invertible,\nand prove instances such as the unique factorization of ideals.\nFurther results on the structure of ideals in a Dedekind domain are found in\n`Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean`.\n\n## Main definitions\n\n- `IsDedekindDomainInv` alternatively defines a Dedekind domain as an integral domain where\n  every nonzero fractional ideal is invertible.\n- `isDedekindDomainInv_iff` shows that this does not depend on the choice of field of\n  fractions.\n\n## Main results:\n\n- `isDedekindDomain_iff_isDedekindDomainInv`\n- `Ideal.uniqueFactorizationMonoid`\n\n## Implementation notes\n\nThe definitions that involve a field of fractions choose a canonical field of fractions,\nbut are independent of that choice. The `..._iff` lemmas express this independence.\n\nOften, definitions assume that Dedekind domains are not fields. We found it more practical\nto add a `(h : ¬ IsField A)` assumption whenever this is explicitly needed.\n\n## References\n\n* [D. Marcus, *Number Fields*][marcus1977number]\n* [J.W.S. Cassels, A. Fröhlich, *Algebraic Number Theory*][cassels1967algebraic]\n* [J. Neukirch, *Algebraic Number Theory*][Neukirch1992]\n\n## Tags\n\ndedekind domain, dedekind ring\n-/\n\n@[expose] public section\n\nvariable (R A K : Type*) [CommRing R] [CommRing A] [Field K]\n\nopen scoped nonZeroDivisors Polynomial\n\nsection Inverse\n\nsection IsDedekindDomainInv\n\nvariable [IsDomain A]\n/-- A Dedekind domain is an integral domain such that every fractional ideal has an inverse.\n\nThis is equivalent to `IsDedekindDomain`.\nIn particular we provide a `CommGroupWithZero` instance,\nassuming `IsDedekindDomain A`, which implies `IsDedekindDomainInv`. For **integral** domain,\n`IsDedekindDomain`(`Inv`) implies only `Ideal.isCancelMulZero`.\n-/\ndef IsDedekindDomainInv : Prop :=\n  ∀ I ≠ (⊥ : FractionalIdeal A⁰ (FractionRing A)), I * I⁻¹ = 1\n\nopen FractionalIdeal\n\nvariable {R A K}\n\ntheorem isDedekindDomainInv_iff [Algebra A K] [IsFractionRing A K] :\n    IsDedekindDomainInv A ↔ ∀ I ≠ (⊥ : FractionalIdeal A⁰ K), I * I⁻¹ = 1 := by\n  let h : FractionalIdeal A⁰ (FractionRing A) ≃+* FractionalIdeal A⁰ K :=\n    FractionalIdeal.mapEquiv (FractionRing.algEquiv A K)\n  refine h.toEquiv.forall_congr (fun {x} => ?_)\n  rw [← h.toEquiv.apply_eq_iff_eq]\n  simp [h]\n\ntheorem FractionalIdeal.adjoinIntegral_eq_one_of_isUnit [Algebra A K] [IsFractionRing A K] (x : K)\n    (hx : IsIntegral A x) (hI : IsUnit (adjoinIntegral A⁰ x hx)) : adjoinIntegral A⁰ x hx = 1 := by\n  set I := adjoinIntegral A⁰ x hx\n  have mul_self : IsIdempotentElem I := by\n    apply coeToSubmodule_injective\n    simp only [coe_mul, adjoinIntegral_coe, I]\n    rw [(Algebra.adjoin A {x}).isIdempotentElem_toSubmodule]\n  convert! congr_arg (· * I⁻¹) mul_self <;>\n    simp only [(mul_inv_cancel_iff_isUnit K).mpr hI, mul_assoc, mul_one]\n\nnamespace IsDedekindDomainInv\n\nvariable [Algebra A K] [IsFractionRing A K] (h : IsDedekindDomainInv A) {I J : FractionalIdeal A⁰ K}\ninclude h\n\n/-- `IsDedekindDomainInv A` implies that fractional ideals over it form a commutative group with\nzero. -/\nnoncomputable abbrev commGroupWithZero : CommGroupWithZero (FractionalIdeal A⁰ K) where\n  inv_zero := inv_zero' _\n  mul_inv_cancel := isDedekindDomainInv_iff.mp h\n  div_eq_mul_inv I J := by\n    obtain rfl | hJ := eq_or_ne J 0\n    · simp [inv_zero']\n    refine le_antisymm ?_ ((FractionalIdeal.le_div_iff_mul_le hJ).2 ?_)\n    · suffices I / J * J ≤ I by\n        simpa [mul_assoc, isDedekindDomainInv_iff.mp h _ hJ] using mul_left_mono (a := J⁻¹) this\n      simp [FractionalIdeal.mul_le, mem_div_iff_of_ne_zero hJ]\n    · rw [mul_assoc, mul_comm _ J, isDedekindDomainInv_iff.mp h _ hJ, mul_one]\n\ntheorem isNoetherianRing : IsNoetherianRing A := by\n  let := h.commGroupWithZero (K := FractionRing A)\n  refine isNoetherianRing_iff.mpr ⟨fun I : Ideal A => ?_⟩\n  by_cases hI : I = ⊥\n  · rw [hI]; apply Submodule.fg_bot\n  have hI : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI\n  exact I.fg_of_isUnit (IsFractionRing.injective A (FractionRing A)) hI.isUnit\n\ntheorem integrallyClosed : IsIntegrallyClosed A := by\n  let := h.commGroupWithZero (K := FractionRing A)\n  -- It suffices to show that for integral `x`,\n  -- `A[x]` (which is a fractional ideal) is in fact equal to `A`.\n  refine (isIntegrallyClosed_iff (FractionRing A)).mpr (fun {x hx} => ?_)\n  rw [← Set.mem_range, ← Algebra.mem_bot, ← Subalgebra.mem_toSubmodule, Algebra.toSubmodule_bot,\n    Submodule.one_eq_span, ← coe_spanSingleton A⁰ (1 : FractionRing A), spanSingleton_one, ←\n    FractionalIdeal.adjoinIntegral_eq_one_of_isUnit x hx (Ne.isUnit _)]\n  · exact mem_adjoinIntegral_self A⁰ x hx\n  · exact fun h => one_ne_zero (eq_zero_iff.mp h 1 (Algebra.adjoin A {x}).one_mem)\n\nopen Ring\n\ntheorem dimensionLEOne : DimensionLEOne A := by\n  -- We're going to show that `P` is maximal because any (maximal) ideal `M`\n  -- that is strictly larger would be `⊤`.\n  let := h.commGroupWithZero (K := FractionRing A)\n  constructor\n  rintro P P_ne hP\n  refine Ideal.isMaximal_def.mpr ⟨hP.ne_top, fun M hM => ?_⟩\n  -- We may assume `P` and `M` (as fractional ideals) are nonzero.\n  have P'_ne : (P : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr P_ne\n  have M'_ne : (M : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hM.ne_bot\n  -- In particular, we'll show `M⁻¹ * P ≤ P`\n  suffices (M⁻¹ : FractionalIdeal A⁰ (FractionRing A)) * P ≤ P by\n    rw [eq_top_iff, ← coeIdeal_le_coeIdeal (FractionRing A), coeIdeal_top]\n    calc\n      (1 : FractionalIdeal A⁰ (FractionRing A)) = (↑M)⁻¹ * P * ((↑P)⁻¹ * M) := by\n        simp [mul_assoc, *]\n      _ ≤ P * ((↑P)⁻¹ * M) := by gcongr\n      _ = M := by simp [*]\n  -- Suppose we have `x ∈ M⁻¹ * P`, then in fact `x = algebraMap _ _ y` for some `y`.\n  intro x hx\n  have le_one : (M⁻¹ : FractionalIdeal A⁰ (FractionRing A)) * P ≤ 1 := by\n    rw [← inv_mul_cancel₀ M'_ne]; gcongr\n  obtain ⟨y, _hy, rfl⟩ := (mem_coeIdeal _).mp (le_one hx)\n  -- Since `M` is strictly greater than `P`, let `z ∈ M \\ P`.\n  obtain ⟨z, hzM, hzp⟩ := SetLike.exists_of_lt hM\n  -- We have `z * y ∈ M * (M⁻¹ * P) = P`.\n  have zy_mem := mul_mem_mul (mem_coeIdeal_of_mem A⁰ hzM) hx\n  rw [← map_mul, ← mul_assoc, mul_inv_cancel₀ M'_ne, one_mul] at zy_mem\n  obtain ⟨zy, hzy, zy_eq⟩ := (mem_coeIdeal A⁰).mp zy_mem\n  rw [IsFractionRing.injective A (FractionRing A) zy_eq] at hzy\n  -- But `P` is a prime ideal, so `z ∉ P` implies `y ∈ P`, as desired.\n  exact mem_coeIdeal_of_mem A⁰ (Or.resolve_left (hP.mem_or_mem hzy) hzp)\n\n/-- Showing one side of the equivalence between the definitions\n`IsDedekindDomainInv` and `IsDedekindDomain` of Dedekind domains. -/\ntheorem isDedekindDomain : IsDedekindDomain A :=\n  { h.isNoetherianRing, h.dimensionLEOne, h.integrallyClosed with }\n\nend IsDedekindDomainInv\n\nend IsDedekindDomainInv\n\nvariable [Algebra A K] [IsFractionRing A K]\n\nvariable {A K}\n\ntheorem FractionalIdeal.one_mem_inv_coe_ideal [IsDomain A] {I : Ideal A} (hI : I ≠ ⊥) :\n    (1 : K) ∈ (I : FractionalIdeal A⁰ K)⁻¹ := by\n  rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI)]\n  intro y hy\n  rw [one_mul]\n  exact coeIdeal_le_one hy\n\n@[deprecated (since := \"2026-04-16\")]\nalias one_mem_inv_coe_ideal := FractionalIdeal.one_mem_inv_coe_ideal\n\n/-- Specialization of `exists_primeSpectrum_prod_le_and_ne_bot_of_domain` to Dedekind domains:\nLet `I : Ideal A` be a nonzero ideal, where `A` is a Dedekind domain that is not a field.\nThen `exists_primeSpectrum_prod_le_and_ne_bot_of_domain` states we can find a product of prime\nideals that is contained within `I`. This lemma extends that result by making the product minimal:\nlet `M` be a maximal ideal that contains `I`, then the product including `M` is contained within `I`\nand the product excluding `M` is not contained within `I`. -/\n\nTarget:\ntheorem PrimeSpectrum.exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A]\n    (hNF : ¬IsField A) {I M : Ideal A} (hI0 : I ≠ ⊥) (hIM : I ≤ M) [hM : M.IsMaximal] :\n    ∃ Z : Multiset (PrimeSpectrum A),\n      (M ::ₘ Z.map asIdeal).prod ≤ I ∧\n        ¬Multiset.prod (Z.map asIdeal) ≤ I :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/DedekindDomain","family_id":"primespectrum","file_id":"mathlib/Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean","sample_id":"0996acbca200c0dccec01a46d4e81205ed46c98fd950cb9f91af86182dcfbe7d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"843286c15dd0e5c3687da59e28ddbf32542077397522d384d533e4f852e034a9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"96285e98a90c2e00636f856577b756dddec1260552689de71259778e263dc959","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2c90f2e32be3da6fb260df3bb94bf0c15d7de35d2f1513e19789e1136caf26c6","source_sha256":"fe6f78e60ae83f5b6cc4fe056a4280594015e555feafeac8a4ec205bff00b8f8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [map_comp]\n  exact hg.trans <| map_mono _ hf","hard_negative":true,"metrics":{"chosen_tokens":14,"rejected_tokens":3,"token_jaccard":0.133333,"token_length_ratio":0.214286},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"05578208c6008563b61054e814ca53c19b5bccf2e619f3de237f20f838328eea","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal\npublic import Mathlib.RingTheory.GradedAlgebra.RingHom\n\nNamespace:\nHomogeneousIdeal\n\nLocal context:\n/-\nCopyright (c) 2025 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n/-!\n# Maps on homogeneous ideals\n\nIn this file we define `HomogeneousIdeal.map` and `HomogeneousIdeal.comap`.\n-/\n\n@[expose] public section\n\nnamespace HomogeneousIdeal\n\nsection arbitrary_grading\n\nvariable {A B C σ τ ω ι F G : Type*}\n  [Semiring A] [Semiring B] [Semiring C]\n  [SetLike σ A] [SetLike τ B] [SetLike ω C]\n  [AddSubmonoidClass σ A] [AddSubmonoidClass τ B] [AddSubmonoidClass ω C]\n  [DecidableEq ι] [AddMonoid ι]\n  {𝒜 : ι → σ} {ℬ : ι → τ} {𝒞 : ι → ω}\n  [GradedRing 𝒜] [GradedRing ℬ] [GradedRing 𝒞]\n  (f : 𝒜 →+*ᵍ ℬ) (g : ℬ →+*ᵍ 𝒞)\n\n/-- Map a homogeneous ideal along a graded ring homomorphism. The underlying ideal is\n(definitionally) equal to `Ideal.map`. -/\ndef map (I : HomogeneousIdeal 𝒜) : HomogeneousIdeal ℬ where\n  __ := I.toIdeal.map f\n  is_homogeneous' i b hb := by\n    rw [Ideal.map] at hb\n    induction hb using Submodule.span_induction generalizing i with\n    | zero => simp\n    | add => simp [*, Ideal.add_mem]\n    | mem a ha =>\n      obtain ⟨a, ha, rfl⟩ := ha\n      rw [← f.map_directSumDecompose]\n      exact Ideal.mem_map_of_mem _ (I.2 _ ha)\n    | smul a₁ a₂ ha₂ ih =>\n      classical rw [smul_eq_mul, DirectSum.decompose_mul, DirectSum.coe_mul_apply]\n      exact sum_mem fun ij hij ↦ Ideal.mul_mem_left _ _ <| ih _\n\n/-- Pull back a homogeneous ideal along a graded ring homomorphism.\nThe underlying ideal is (definitionally) equal to `Ideal.comap`, whose underlying set is\ndefinitionally equal to the preimage. -/\ndef comap (I : HomogeneousIdeal ℬ) : HomogeneousIdeal 𝒜 where\n  __ := I.toIdeal.comap f\n  is_homogeneous' n a ha := by\n    rw [Ideal.mem_comap, HomogeneousIdeal.mem_iff, f.map_directSumDecompose]\n    exact I.2 _ ha\n\nvariable {I I₁ I₂ I₃ : HomogeneousIdeal 𝒜} {J J₁ J₂ J₃ : HomogeneousIdeal ℬ}\n  {K : HomogeneousIdeal 𝒞}\n\nlemma map_le_iff_le_comap : I.map f ≤ J ↔ I ≤ J.comap f := Ideal.map_le_iff_le_comap\n\nalias ⟨le_comap_of_map_le, map_le_of_le_comap⟩ := map_le_iff_le_comap\n\ntheorem gc_map_comap : GaloisConnection (map f) (comap f) := fun _ _ ↦\n  map_le_iff_le_comap f\n\n@[mono, aesop safe apply] lemma map_mono : Monotone (map f) := (gc_map_comap f).monotone_l\n\n@[gcongr, mono] lemma comap_mono : Monotone (comap f) := (gc_map_comap f).monotone_u\n\n@[simp] lemma toIdeal_comap : (J.comap f).toIdeal = J.toIdeal.comap f := rfl\n\n@[simp] lemma coe_comap : J.comap f = f ⁻¹' J := rfl\n\n@[simp] lemma toIdeal_map : (I.map f).toIdeal = I.toIdeal.map f := rfl\n\ninstance isPrime_comap [J.toIdeal.IsPrime] : (J.comap f).toIdeal.IsPrime :=\n  inferInstanceAs (J.toIdeal.comap f).IsPrime -- this shows that the simpNF already has the instance\n\n@[simp] lemma map_id : I.map (GradedRingHom.id 𝒜) = I := ext <| Ideal.map_id _\n\nlemma map_map : (I.map f).map g = I.map (g.comp f) := ext <| Ideal.map_map _ _\n\nlemma map_comp : I.map (g.comp f) = (I.map f).map g := (map_map f g).symm\n\n@[simp] lemma comap_id : I.comap (GradedRingHom.id 𝒜) = I := rfl\n\nlemma comap_comap : (K.comap g).comap f = K.comap (g.comp f) := rfl\n\nend arbitrary_grading\n\nsection canonical_grading\n\nvariable {A B C σ τ ω ι F G : Type*}\n  [Semiring A] [Semiring B] [Semiring C]\n  [SetLike σ A] [SetLike τ B] [SetLike ω C]\n  [AddSubmonoidClass σ A] [AddSubmonoidClass τ B] [AddSubmonoidClass ω C]\n  [DecidableEq ι] [AddCommMonoid ι] [PartialOrder ι] [CanonicallyOrderedAdd ι]\n  {𝒜 : ι → σ} {ℬ : ι → τ} {𝒞 : ι → ω}\n  [GradedRing 𝒜] [GradedRing ℬ] [GradedRing 𝒞]\n  {f : 𝒜 →+*ᵍ ℬ} {g : ℬ →+*ᵍ 𝒞}\n\nTarget:\ntheorem irrelevant_le_map_comp\n    (hf : ℬ₊ ≤ 𝒜₊.map f) (hg : 𝒞₊ ≤ ℬ₊.map g) : 𝒞₊ ≤ 𝒜₊.map (g.comp f) :=\n\nProof body:\n","rejected":"by\n  exact irrelevant_le_map_comp","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"fa9e40f9ac772bc5aa5a2780dca6e94d3723a408946d5514dcb8cd1bae663a85","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/GradedAlgebra","family_id":"irrelevant_le_map_comp","file_id":"mathlib/Mathlib/RingTheory/GradedAlgebra/Homogeneous/Maps.lean","sample_id":"2c90f2e32be3da6fb260df3bb94bf0c15d7de35d2f1513e19789e1136caf26c6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"de67f9b6451b5db3ba017889f07c8f6879cb98f3fc2736a86445d62c14d1c664","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3d470e7ad773073099cd7eb08ccd0ead6e5c8e489dd732b9e3bfbbb2b003c60c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ba8a1ee7158309c944c89fb3fff36b668ca7628b981cba7184536316677b8d1f","source_sha256":"94021f194f9dbe2b3ff10c0fe9e60d21f920f7ec47e18de0732170ff2d7ff47e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Squarefree, not_forall]\n  exact ⟨0, by simp⟩","hard_negative":false,"metrics":{"chosen_tokens":14,"rejected_tokens":18,"token_jaccard":0.75,"token_length_ratio":1.285714},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"05ff9d4452c635482cdef1b657e4ec716ce9b8154a9b10402ed87294aeedbfe7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Coprime.Lemmas\npublic import Mathlib.RingTheory.Nilpotent.Basic\npublic import Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid\npublic import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Squarefree elements of monoids\nAn element of a monoid is squarefree when it is not divisible by any squares\nexcept the squares of units.\n\nResults about squarefree natural numbers are proved in `Data.Nat.Squarefree`.\n\n## Main Definitions\n- `Squarefree r` indicates that `r` is only divisible by `x * x` if `x` is a unit.\n\n## Main Results\n- `multiplicity.squarefree_iff_emultiplicity_le_one`: `x` is `Squarefree` iff for every `y`, either\n  `emultiplicity y x ≤ 1` or `IsUnit y`.\n- `UniqueFactorizationMonoid.squarefree_iff_nodup_factors`: A nonzero element `x` of a unique\n  factorization monoid is squarefree iff `factors x` has no duplicate factors.\n\n## Tags\nsquarefree, multiplicity\n\n-/\n\n@[expose] public section\n\n\nvariable {R : Type*}\n\n/-- An element of a monoid is squarefree if the only squares that\n  divide it are the squares of units. -/\ndef Squarefree [Monoid R] (r : R) : Prop :=\n  ∀ x : R, x * x ∣ r → IsUnit x\n\ntheorem IsRelPrime.of_squarefree_mul [CommMonoid R] {m n : R} (h : Squarefree (m * n)) :\n    IsRelPrime m n := fun c hca hcb ↦ h c (mul_dvd_mul hca hcb)\n\n@[simp]\ntheorem IsUnit.squarefree [CommMonoid R] {x : R} (h : IsUnit x) : Squarefree x := fun _ hdvd =>\n  isUnit_of_mul_isUnit_left (isUnit_of_dvd_unit hdvd h)\n\ntheorem squarefree_one [CommMonoid R] : Squarefree (1 : R) :=\n  isUnit_one.squarefree\n\n@[simp]\n\nTarget:\ntheorem not_squarefree_zero [MonoidWithZero R] [Nontrivial R] : ¬Squarefree (0 : R) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [Squarefree, not_forall]\n  exact ⟨0, by simp⟩","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Squarefree","family_id":"not_squarefree_zero","file_id":"mathlib/Mathlib/Algebra/Squarefree/Basic.lean","sample_id":"ba8a1ee7158309c944c89fb3fff36b668ca7628b981cba7184536316677b8d1f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dfb0cfa54bf916f968be75edf3a87f502ebb8bc73b72d17a24c14c29c6f3e807","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"dfa2375d52b2512c773d6c8fddc24e78daa2ad33a4c48eccb3a13f32d9a43be5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5718bd96cf9f59ca4c83944f21497324dfdb689e13608aa103fc1d5ac63a1635","source_sha256":"3827253dc949c32959719d5a285bec59e92b58138d7cf11fd2dc87768305b52a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  convert! sup_mem_subalgebra_closure A f g\n  apply SetLike.ext'\n  simp","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":17,"token_jaccard":0.705882,"token_length_ratio":1.416667},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"060d74d0d090822121affa5c6d6fbf46b7f7ce76bc162e83c6a261b674b0c44d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Subalgebra.Tower\npublic import Mathlib.Analysis.RCLike.Basic\npublic import Mathlib.Topology.Algebra.Star.Real\npublic import Mathlib.Topology.Algebra.StarSubalgebra\npublic import Mathlib.Topology.Algebra.NonUnitalStarAlgebra\npublic import Mathlib.Topology.ContinuousMap.ContinuousMapZero\npublic import Mathlib.Topology.ContinuousMap.Lattice\npublic import Mathlib.Topology.ContinuousMap.Weierstrass\npublic import Mathlib.Algebra.Order.Module.Basic\n\nNamespace:\nContinuousMap\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison, Heather Macbeth\n-/\n/-!\n# The Stone-Weierstrass theorem\n\nIf a subalgebra `A` of `C(X, ℝ)`, where `X` is a compact topological space,\nseparates points, then it is dense.\n\nWe argue as follows.\n\n* In any subalgebra `A` of `C(X, ℝ)`, if `f ∈ A`, then `abs f ∈ A.topologicalClosure`.\n  This follows from the Weierstrass approximation theorem on `[-‖f‖, ‖f‖]` by\n  approximating `abs` uniformly thereon by polynomials.\n* This ensures that `A.topologicalClosure` is actually a sublattice:\n  if it contains `f` and `g`, then it contains the pointwise supremum `f ⊔ g`\n  and the pointwise infimum `f ⊓ g`.\n* Any nonempty sublattice `L` of `C(X, ℝ)` which separates points is dense,\n  by a nice argument approximating a given `f` above and below using separating functions.\n  For each `x y : X`, we pick a function `g x y ∈ L` so `g x y x = f x` and `g x y y = f y`.\n  By continuity these functions remain close to `f` on small patches around `x` and `y`.\n  We use compactness to identify a certain finitely indexed infimum of finitely indexed supremums\n  which is then close to `f` everywhere, obtaining the desired approximation.\n* Finally we put these pieces together. `L = A.topologicalClosure` is a nonempty sublattice\n  which separates points since `A` does, and so is dense (in fact equal to `⊤`).\n\nWe then prove the complex version for star subalgebras `A`, by separately approximating\nthe real and imaginary parts using the real subalgebra of real-valued functions in `A`\n(which still separates points, by taking the norm-square of a separating function).\n\n## Future work\n\nExtend to cover the case of subalgebras of the continuous functions vanishing at infinity,\non non-compact spaces.\n\n-/\n\n@[expose] public section\n\nassert_not_exists Unitization\n\nnoncomputable section\n\nnamespace ContinuousMap\n\nvariable {X : Type*} [TopologicalSpace X] [CompactSpace X]\n\nopen scoped Polynomial\n\n/-- Turn a function `f : C(X, ℝ)` into a continuous map into `Set.Icc (-‖f‖) (‖f‖)`,\nthereby explicitly attaching bounds.\n-/\ndef attachBound (f : C(X, ℝ)) : C(X, Set.Icc (-‖f‖) ‖f‖) where\n  toFun x := ⟨f x, ⟨neg_norm_le_apply f x, apply_le_norm f x⟩⟩\n\n@[simp]\ntheorem attachBound_apply_coe (f : C(X, ℝ)) (x : X) : ((attachBound f) x : ℝ) = f x :=\n  rfl\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\ntheorem polynomial_comp_attachBound (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :\n    (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound =\n      Polynomial.aeval f g := by\n  ext\n  simp only [Polynomial.aeval_subalgebra_coe, Polynomial.aeval_continuousMap_apply]\n  simp\n\n/-- Given a continuous function `f` in a subalgebra of `C(X, ℝ)`, postcomposing by a polynomial\ngives another function in `A`.\n\nThis lemma proves something slightly more subtle than this:\nwe take `f`, and think of it as a function into the restricted target `Set.Icc (-‖f‖) ‖f‖)`,\nand then postcompose with a polynomial function on that interval.\nThis is in fact the same situation as above, and so also gives a function in `A`.\n-/\ntheorem polynomial_comp_attachBound_mem (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :\n    (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound ∈ A := by\n  rw [polynomial_comp_attachBound]\n  apply SetLike.coe_mem\n\ntheorem comp_attachBound_mem_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A)\n    (p : C(Set.Icc (-‖f‖) ‖f‖, ℝ)) : p.comp (attachBound (f : C(X, ℝ))) ∈ A.topologicalClosure := by\n  -- `p` itself is in the closure of polynomials, by the Weierstrass theorem,\n  have mem_closure : p ∈ (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)).topologicalClosure :=\n    continuousMap_mem_polynomialFunctions_closure _ _ p\n  -- and so there are polynomials arbitrarily close.\n  have frequently_mem_polynomials := mem_closure_iff_frequently.mp mem_closure\n  -- To prove `p.comp (attachBound f)` is in the closure of `A`,\n  -- we show there are elements of `A` arbitrarily close.\n  apply mem_closure_iff_frequently.mpr\n  -- To show that, we pull back the polynomials close to `p`,\n  refine\n    ((compRightContinuousMap ℝ (attachBound (f : C(X, ℝ)))).continuousAt\n            p).tendsto.frequently_map\n      _ ?_ frequently_mem_polynomials\n  -- but need to show that those pullbacks are actually in `A`.\n  rintro _ ⟨g, ⟨-, rfl⟩⟩\n  simp only [SetLike.mem_coe, AlgHom.coe_toRingHom, compRightContinuousMap_apply,\n    Polynomial.toContinuousMapOnAlgHom_apply]\n  apply polynomial_comp_attachBound_mem\n\ntheorem abs_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A) :\n    |(f : C(X, ℝ))| ∈ A.topologicalClosure := by\n  let f' := attachBound (f : C(X, ℝ))\n  let abs : C(Set.Icc (-‖f‖) ‖f‖, ℝ) := { toFun := fun x : Set.Icc (-‖f‖) ‖f‖ => |(x : ℝ)| }\n  change abs.comp f' ∈ A.topologicalClosure\n  apply comp_attachBound_mem_closure\n\ntheorem inf_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :\n    (f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A.topologicalClosure := by\n  rw [inf_eq_half_smul_add_sub_abs_sub' ℝ]\n  refine\n    A.topologicalClosure.smul_mem\n      (A.topologicalClosure.sub_mem\n        (A.topologicalClosure.add_mem (A.le_topologicalClosure f.property)\n          (A.le_topologicalClosure g.property))\n        ?_)\n      _\n  exact mod_cast abs_mem_subalgebra_closure A _\n\ntheorem inf_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))\n    (f g : A) : (f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A := by\n  convert! inf_mem_subalgebra_closure A f g\n  apply SetLike.ext'\n  symm\n  rw [Subalgebra.topologicalClosure_coe, closure_eq_iff_isClosed]\n  exact h\n\ntheorem sup_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :\n    (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A.topologicalClosure := by\n  rw [sup_eq_half_smul_add_add_abs_sub' ℝ]\n  refine\n    A.topologicalClosure.smul_mem\n      (A.topologicalClosure.add_mem\n        (A.topologicalClosure.add_mem (A.le_topologicalClosure f.property)\n          (A.le_topologicalClosure g.property))\n        ?_)\n      _\n  exact mod_cast abs_mem_subalgebra_closure A _\n\nTarget:\ntheorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))\n    (f g : A) : (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  convert! sup_mem_subalgebra_closure A f g\n  apply SetLike.ext'\n  simp","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/ContinuousMap","family_id":"sup_mem_closed_subalgebra","file_id":"mathlib/Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean","sample_id":"5718bd96cf9f59ca4c83944f21497324dfdb689e13608aa103fc1d5ac63a1635"},"split":"train"}
{"binding":{"chosen_proof_sha256":"30015a6154bd2aef8d0848639c98385965744ef4b2fb839692a5ae2632c8bacb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"61faa742462a962beb76f3f1545ae04bd1db702ab9662b4c763eedf05c97a9ff","source_sha256":"ebbd09c0a09c3969b7a62f3c013274751c70d4ab23cc8901bc28ba4a69c7b881","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← List.filterMap_eq_filter]\n  apply listFilterMap .id\n  simp only [Primrec₂, Option.guard, decide_eq_true_eq]\n  exact ite (hf.comp snd) (option_some_iff.mpr snd) (const none)","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":8,"token_jaccard":0.027778,"token_length_ratio":0.190476},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"061085a06360faa3c0ed20fb10bd3809c69af39fc97327ee2a3024bc727d74da","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Computability.Primrec.Basic\npublic import Mathlib.Logic.Encodable.Pi\n\nNamespace:\nPrimrec\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Primitive recursive functions on Lists\n\nThe primitive recursive functions are defined in `Mathlib.Computability.Primrec.Basic`.\nThis file contains definitions and theorems about primitive recursive functions that\nrelate to operation on lists.\n\n## References\n\n* [Mario Carneiro, *Formalizing computability theory via partial recursive functions*][carneiro2019]\n-/\n\n@[expose] public section\n\nopen List (Vector)\nopen Denumerable Encodable Function\n\n\nsection\n\nvariable {α : Type*} {β : Type*} {σ : Type*}\nvariable [Primcodable α] [Primcodable β] [Primcodable σ]\nvariable (H : Nat.Primrec fun n => Encodable.encode (@decode (List β) _ n))\n\nopen Primrec\n\nset_option backward.privateInPublic true in\n@[implicit_reducible]\nprivate def prim : Primcodable (List β) := ⟨H⟩\n\nprivate theorem list_casesOn' {f : α → List β} {g : α → σ} {h : α → β × List β → σ}\n    (hf : haveI := prim H; Primrec f) (hg : Primrec g) (hh : haveI := prim H; Primrec₂ h) :\n    @Primrec _ σ _ _ fun a => List.casesOn (f a) (g a) fun b l => h a (b, l) :=\n  letI := prim H\n  have :\n    @Primrec _ (Option σ) _ _ fun a =>\n      (@decode (Option (β × List β)) _ (encode (f a))).map fun o => Option.casesOn o (g a) (h a) :=\n    ((@map_decode_iff _ (Option (β × List β)) _ _ _ _ _).2 <|\n      to₂ <|\n        option_casesOn snd (hg.comp fst) (hh.comp₂ (fst.comp₂ Primrec₂.left) Primrec₂.right)).comp\n      .id (encode_iff.2 hf)\n  option_some_iff.1 <| this.of_eq fun a => by rcases f a with - | ⟨b, l⟩ <;> simp [encodek]\n\nset_option backward.privateInPublic true in\nprivate theorem list_foldl' {f : α → List β} {g : α → σ} {h : α → σ × β → σ}\n    (hf : haveI := prim H; Primrec f) (hg : Primrec g) (hh : haveI := prim H; Primrec₂ h) :\n    Primrec fun a => (f a).foldl (fun s b => h a (s, b)) (g a) := by\n  letI := prim H\n  let G (a : α) (IH : σ × List β) : σ × List β := List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\n  have hG : Primrec₂ G := list_casesOn' H (snd.comp snd) snd <|\n    to₂ <|\n    pair (hh.comp (fst.comp fst) <| pair ((fst.comp snd).comp fst) (fst.comp snd))\n      (snd.comp snd)\n  let F := fun (a : α) (n : ℕ) => (G a)^[n] (g a, f a)\n  have hF : Primrec fun a => (F a (encode (f a))).1 :=\n    (fst.comp <|\n      nat_iterate (encode_iff.2 hf) (pair hg hf) <|\n      hG)\n  suffices ∀ a n, F a n = (((f a).take n).foldl (fun s b => h a (s, b)) (g a), (f a).drop n) by\n    refine hF.of_eq fun a => ?_\n    rw [this, List.take_of_length_le (length_le_encode _)]\n  introv\n  dsimp only [F]\n  generalize f a = l\n  generalize g a = x\n  induction n generalizing l x with\n  | zero => rfl\n  | succ n IH =>\n    simp only [iterate_succ, comp_apply]\n    rcases l with - | ⟨b, l⟩ <;> simp [G, IH]\n\nset_option backward.privateInPublic true in\nprivate theorem list_cons' : (haveI := prim H; Primrec₂ (@List.cons β)) :=\n  letI := prim H\n  encode_iff.1 (succ.comp <| Primrec₂.natPair.comp (encode_iff.2 fst) (encode_iff.2 snd))\n\nset_option backward.privateInPublic true in\nprivate theorem list_reverse' :\n    haveI := prim H\n    Primrec (@List.reverse β) :=\n  letI := prim H\n  (list_foldl' H .id (const []) <| to₂ <| ((list_cons' H).comp snd fst).comp snd).of_eq\n    (suffices ∀ l r, List.foldl (fun (s : List β) (b : β) => b :: s) r l = List.reverseAux l r from\n      fun l => this l []\n    fun l => by induction l <;> simp [*, List.reverseAux])\n\nend\n\nnamespace Primcodable\n\nvariable {α : Type*} {β : Type*}\nvariable [Primcodable α] [Primcodable β]\n\nopen Primrec\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nset_option linter.flexible false in -- TODO: revisit this after #13791 is merged\ninstance list : Primcodable (List α) :=\n  ⟨letI H := Primcodable.prim (List ℕ)\n    have : Primrec₂ fun (a : α) (o : Option (List ℕ)) => o.map (List.cons (encode a)) :=\n      option_map snd <| (list_cons' H).comp ((@Primrec.encode α _).comp (fst.comp fst)) snd\n    have :\n      Primrec fun n =>\n        (ofNat (List ℕ) n).reverse.foldl\n          (fun o m => (@decode α _ m).bind fun a => o.map (List.cons (encode a))) (some []) :=\n      list_foldl' H ((list_reverse' H).comp (.ofNat (List ℕ))) (const (some []))\n        (Primrec.comp₂ (bind_decode_iff.2 <| .swap this) Primrec₂.right)\n    nat_iff.1 <|\n      (encode_iff.2 this).of_eq fun n => by\n        rw [List.foldl_reverse]\n        apply Nat.case_strong_induction_on n; · simp\n        intro n IH; simp\n        rcases @decode α _ n.unpair.1 with - | a; · rfl\n        simp only [Option.bind_some, Option.map_some]\n        suffices ∀ (o : Option (List ℕ)) (p), encode o = encode p →\n            encode (Option.map (List.cons (encode a)) o) = encode (Option.map (List.cons a) p) from\n          this _ _ (IH _ (Nat.unpair_right_le n))\n        intro o p IH\n        cases o <;> cases p\n        · rfl\n        · injection IH\n        · injection IH\n        · exact congr_arg (fun k => (Nat.pair (encode a) k).succ.succ) (Nat.succ.inj IH)⟩\nend Primcodable\n\nnamespace Primrec\n\nvariable {α : Type*} {β : Type*} {γ : Type*} {σ : Type*}\nvariable [Primcodable α] [Primcodable β] [Primcodable γ] [Primcodable σ]\n\ntheorem list_cons : Primrec₂ (@List.cons α) :=\n  list_cons' (Primcodable.prim _)\n\ntheorem list_casesOn {f : α → List β} {g : α → σ} {h : α → β × List β → σ} :\n    Primrec f →\n      Primrec g →\n        Primrec₂ h → @Primrec _ σ _ _ fun a => List.casesOn (f a) (g a) fun b l => h a (b, l) :=\n  list_casesOn' (Primcodable.prim _)\n\ntheorem list_foldl {f : α → List β} {g : α → σ} {h : α → σ × β → σ} :\n    Primrec f →\n      Primrec g → Primrec₂ h → Primrec fun a => (f a).foldl (fun s b => h a (s, b)) (g a) :=\n  list_foldl' (Primcodable.prim _)\n\ntheorem list_reverse : Primrec (@List.reverse α) :=\n  list_reverse' (Primcodable.prim _)\n\ntheorem list_foldr {f : α → List β} {g : α → σ} {h : α → β × σ → σ} (hf : Primrec f)\n    (hg : Primrec g) (hh : Primrec₂ h) :\n    Primrec fun a => (f a).foldr (fun b s => h a (b, s)) (g a) :=\n  (list_foldl (list_reverse.comp hf) hg <| to₂ <| hh.comp fst <| (pair snd fst).comp snd).of_eq\n    fun a => by simp [List.foldl_reverse]\n\ntheorem list_head? : Primrec (@List.head? α) :=\n  (list_casesOn .id (const none) (option_some_iff.2 <| fst.comp snd).to₂).of_eq fun l => by\n    cases l <;> rfl\n\ntheorem list_headI [Inhabited α] : Primrec (@List.headI α _) :=\n  (option_getD_default.comp list_head?).of_eq fun l => l.head!_eq_head?_getD.symm\n\ntheorem list_tail : Primrec (@List.tail α) :=\n  (list_casesOn .id (const []) (snd.comp snd).to₂).of_eq fun l => by cases l <;> rfl\n\ntheorem list_rec {f : α → List β} {g : α → σ} {h : α → β × List β × σ → σ} (hf : Primrec f)\n    (hg : Primrec g) (hh : Primrec₂ h) :\n    @Primrec _ σ _ _ fun a => List.recOn (f a) (g a) fun b l IH => h a (b, l, IH) :=\n  let F (a : α) := (f a).foldr (fun (b : β) (s : List β × σ) => (b :: s.1, h a (b, s))) ([], g a)\n  have : Primrec F :=\n    list_foldr hf (pair (const []) hg) <|\n      to₂ <| pair ((list_cons.comp fst (fst.comp snd)).comp snd) hh\n  (snd.comp this).of_eq fun a => by\n    suffices F a = (f a, List.recOn (f a) (g a) fun b l IH => h a (b, l, IH)) by rw [this]\n    dsimp [F]\n    induction f a <;> simp [*]\n\ntheorem list_getElem? : Primrec₂ ((·[·]? : List α → ℕ → Option α)) :=\n  let F (l : List α) (n : ℕ) :=\n    l.foldl\n      (fun (s : ℕ ⊕ α) (a : α) =>\n        Sum.casesOn s (@Nat.casesOn (fun _ => ℕ ⊕ α) · (Sum.inr a) Sum.inl) Sum.inr)\n      (Sum.inl n)\n  have hF : Primrec₂ F :=\n    (list_foldl fst (sumInl.comp snd)\n      ((sumCasesOn fst (nat_casesOn snd (sumInr.comp <| snd.comp fst) (sumInl.comp snd).to₂).to₂\n              (sumInr.comp snd).to₂).comp\n          snd).to₂).to₂\n  have :\n    @Primrec _ (Option α) _ _ fun p : List α × ℕ => Sum.casesOn (F p.1 p.2) (fun _ => none) some :=\n    sumCasesOn hF (const none).to₂ (option_some.comp snd).to₂\n  this.to₂.of_eq fun l n => by\n    dsimp; symm\n    induction l generalizing n with\n    | nil => rfl\n    | cons a l IH =>\n      rcases n with - | n\n      · dsimp [F]\n        clear IH\n        induction l <;> simp_all\n      · simpa using! IH ..\n\ntheorem list_getD (d : α) : Primrec₂ fun l n => List.getD l n d := by\n  simp only [List.getD_eq_getElem?_getD]\n  exact option_getD.comp₂ list_getElem? (const _)\n\ntheorem list_getI [Inhabited α] : Primrec₂ (@List.getI α _) :=\n  list_getD _\n\ntheorem list_append : Primrec₂ ((· ++ ·) : List α → List α → List α) :=\n  (list_foldr fst snd <| to₂ <| comp (@list_cons α _) snd).to₂.of_eq fun l₁ l₂ => by\n    induction l₁ <;> simp [*]\n\ntheorem list_concat : Primrec₂ fun l (a : α) => l ++ [a] :=\n  list_append.comp fst (list_cons.comp snd (const []))\n\ntheorem list_map {f : α → List β} {g : α → β → σ} (hf : Primrec f) (hg : Primrec₂ g) :\n    Primrec fun a => (f a).map (g a) :=\n  (list_foldr hf (const []) <|\n        to₂ <| list_cons.comp (hg.comp fst (fst.comp snd)) (snd.comp snd)).of_eq\n    fun a => by induction f a <;> simp [*]\n\ntheorem list_range : Primrec List.range :=\n  (nat_rec' .id (const []) ((list_concat.comp snd fst).comp snd).to₂).of_eq fun n => by\n    simp; induction n <;> simp [*, List.range_succ]\n\ntheorem list_flatten : Primrec (@List.flatten α) :=\n  (list_foldr .id (const []) <| to₂ <| comp (@list_append α _) snd).of_eq fun l => by\n    dsimp; induction l <;> simp [*]\n\ntheorem list_flatMap {f : α → List β} {g : α → β → List σ} (hf : Primrec f) (hg : Primrec₂ g) :\n    Primrec (fun a => (f a).flatMap (g a)) := list_flatten.comp (list_map hf hg)\n\ntheorem optionToList : Primrec (Option.toList : Option α → List α) :=\n  (option_casesOn Primrec.id (const [])\n    ((list_cons.comp Primrec.id (const [])).comp₂ Primrec₂.right)).of_eq\n  (fun o => by rcases o <;> simp)\n\ntheorem listFilterMap {f : α → List β} {g : α → β → Option σ}\n    (hf : Primrec f) (hg : Primrec₂ g) : Primrec fun a => (f a).filterMap (g a) :=\n  (list_flatMap hf (comp₂ optionToList hg)).of_eq\n    fun _ ↦ Eq.symm <| List.filterMap_eq_flatMap_toList _ _\n\nvariable {p : α → Prop} [DecidablePred p]\n\ntheorem list_length : Primrec (@List.length α) :=\n  (list_foldr (@Primrec.id (List α) _) (const 0) <| to₂ <| (succ.comp <| snd.comp snd).to₂).of_eq\n    fun l => by dsimp; induction l <;> simp [*]\n\n/-- Filtering a list for elements that satisfy a decidable predicate is primitive recursive. -/\n\nTarget:\ntheorem listFilter (hf : PrimrecPred p) : Primrec fun L ↦ List.filter (p ·) L :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"754536d407ca07e61abeb3df0dafa3d8e2b6dafeec4230ddccf28b66656c9c91","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability/Primrec","family_id":"listfilter","file_id":"mathlib/Mathlib/Computability/Primrec/List.lean","sample_id":"61faa742462a962beb76f3f1545ae04bd1db702ab9662b4c763eedf05c97a9ff"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6bde054e8dd9e2e89dc5431f020cd1a2610cdbc6535abeb8845dde9b2fdd1677","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e31719f32ba1c30d133bd4fc148ed4aecac3d763ec2d36cf51797fc59dfc0b02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6c18e3a6e36fcd3e1caadefef8b6d28d63c691769c6b3ffbae3c86bf10ef7481","source_sha256":"8134c926e471796d362206cf36f0ce764965a63064ad2f4ac11aa45b1280dd60","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  change\n    Quotient.map (@funMap L M ps.toStructure n f) Prestructure.fun_equiv (Quotient.finChoice _) =\n      _\n  rw [Quotient.finChoice_eq, Quotient.map_mk]","hard_negative":true,"metrics":{"chosen_tokens":37,"rejected_tokens":3,"token_jaccard":0.035714,"token_length_ratio":0.081081},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"0618a971244095a66f24ab982d7243a5dab1c42ea87f6f3b0a8a261ecd3336bb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Quotient\npublic import Mathlib.ModelTheory.Semantics\n\nNamespace:\nFirstOrder.Language\n\nLocal context:\n/-\nCopyright (c) 2022 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Quotients of First-Order Structures\n\nThis file defines prestructures and quotients of first-order structures.\n\n## Main Definitions\n\n- If `s` is a setoid (equivalence relation) on `M`, a `FirstOrder.Language.Prestructure s` is the\n  data for a first-order structure on `M` that will still be a structure when modded out by `s`.\n- The structure `FirstOrder.Language.quotientStructure s` is the resulting structure on\n  `Quotient s`.\n-/\n\npublic section\n\n\nnamespace FirstOrder\n\nnamespace Language\n\nvariable (L : Language) {M : Type*}\n\nopen FirstOrder\n\nopen Structure\n\n/-- A prestructure is a first-order structure with a `Setoid` equivalence relation on it,\n  such that quotienting by that equivalence relation is still a structure. -/\nclass Prestructure (s : Setoid M) where\n  /-- The underlying first-order structure -/\n  toStructure : L.Structure M\n  fun_equiv : ∀ {n} {f : L.Functions n} (x y : Fin n → M), x ≈ y → funMap f x ≈ funMap f y\n  rel_equiv : ∀ {n} {r : L.Relations n} (x y : Fin n → M) (_ : x ≈ y), RelMap r x = RelMap r y\n\nvariable {L} {s : Setoid M}\nvariable [ps : L.Prestructure s]\n\ninstance quotientStructure : L.Structure (Quotient s) where\n  funMap {n} f x :=\n    Quotient.map (@funMap L M ps.toStructure n f) Prestructure.fun_equiv (Quotient.finChoice x)\n  RelMap {n} r x :=\n    Quotient.lift (@RelMap L M ps.toStructure n r) Prestructure.rel_equiv (Quotient.finChoice x)\n\nvariable (s)\n\nTarget:\ntheorem funMap_quotient_mk' {n : ℕ} (f : L.Functions n) (x : Fin n → M) :\n    (funMap f fun i => (⟦x i⟧ : Quotient s)) = ⟦@funMap _ _ ps.toStructure _ f x⟧ :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_6c18e3a6e36f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f3a9f2a24581b149d0f9b691dc031fc933d078bb9d82d773d53581f517ce5bd8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"ModelTheory","family_id":"funmap_quotient_mk","file_id":"mathlib/Mathlib/ModelTheory/Quotients.lean","sample_id":"6c18e3a6e36fcd3e1caadefef8b6d28d63c691769c6b3ffbae3c86bf10ef7481"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9863212e81e0182629bd15d927cf3f594339d8438f634ee6f333a33c3de5d6eb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b3ad2032874ba6a0b6f50397e2b035a5757e486348575d6dc25170f7fefc69ec","source_sha256":"5f13df63ef1a33fb7d6444e769a7e9b9cf3ab09860a26a91c473157d6239794f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  conv_rhs => simp [← h.commute_iff, Commute.symm_iff (a := e), commute_iff_eq, ← mem_commutant_iff]","hard_negative":false,"metrics":{"chosen_tokens":24,"rejected_tokens":3,"token_jaccard":0.045455,"token_length_ratio":0.125},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"0666a5e08b451e9ce6f99af41733fd943daa44e5ac2b70d7428f09ae8d10e4d8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.CStarAlgebra.Classes\npublic import Mathlib.Analysis.InnerProductSpace.Adjoint\n\nNamespace:\nVonNeumannAlgebra\n\nLocal context:\n/-\nCopyright (c) 2022 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Von Neumann algebras\n\nWe give the \"abstract\" and \"concrete\" definitions of a von Neumann algebra.\nWe still have a major project ahead of us to show the equivalence between these definitions!\n\nAn abstract von Neumann algebra `WStarAlgebra M` is a C⋆ algebra with a Banach space predual,\nper Sakai (1971).\n\nA concrete von Neumann algebra `VonNeumannAlgebra H` (where `H` is a Hilbert space)\nis a \\*-closed subalgebra of bounded operators on `H` which is equal to its double commutant.\n\nWe'll also need to prove the von Neumann double commutant theorem,\nthat the concrete definition is equivalent to a \\*-closed subalgebra which is weakly closed.\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\n/-- Sakai's definition of a von Neumann algebra as a C⋆ algebra with a Banach space predual.\n\nSo that we can unambiguously talk about these \"abstract\" von Neumann algebras\nin parallel with the \"concrete\" ones (weakly closed \\*-subalgebras of B(H)),\nwe name this definition `WStarAlgebra`.\n\nNote that for now we only assert the mere existence of predual, rather than picking one.\nThis may later prove problematic, and need to be revisited.\nPicking one may cause problems with definitional unification of different instances.\nOne the other hand, not picking one means that the weak-\\* topology\n(which depends on a choice of predual) must be defined using the choice,\nand we may be unhappy with the resulting opaqueness of the definition.\n-/\nclass WStarAlgebra (M : Type u) [CStarAlgebra M] : Prop where\n  /-- There is a Banach space `X` whose dual is isometrically (conjugate-linearly) isomorphic\n  to the `WStarAlgebra`. -/\n  exists_predual :\n    ∃ (X : Type u) (_ : NormedAddCommGroup X) (_ : NormedSpace ℂ X) (_ : CompleteSpace X),\n      Nonempty (StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M)\n\n-- TODO: Without this, `VonNeumannAlgebra` times out. Why?\n/-- The double commutant definition of a von Neumann algebra,\nas a \\*-closed subalgebra of bounded operators on a Hilbert space,\nwhich is equal to its double commutant.\n\nNote that this definition is parameterised by the Hilbert space\non which the algebra faithfully acts, as is standard in the literature.\nSee `WStarAlgebra` for the abstract notion (a C⋆-algebra with Banach space predual).\n\nNote this is a bundled structure, parameterised by the Hilbert space `H`,\nrather than a typeclass on the type of elements.\nThus we can't say that the bounded operators `H →L[ℂ] H` form a `VonNeumannAlgebra`\n(although we will later construct the instance `WStarAlgebra (H →L[ℂ] H)`),\nand instead will use `⊤ : VonNeumannAlgebra H`.\n-/\nstructure VonNeumannAlgebra (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H]\n    [CompleteSpace H] extends StarSubalgebra ℂ (H →L[ℂ] H) where\n  /-- The double commutant (a.k.a. centralizer) of a `VonNeumannAlgebra` is itself. -/\n  centralizer_centralizer' : Set.centralizer (Set.centralizer carrier) = carrier\n\n/-- Consider a von Neumann algebra acting on a Hilbert space `H` as a \\*-subalgebra of `H →L[ℂ] H`.\n(That is, we forget that it is equal to its double commutant\nor equivalently that it is closed in the weak and strong operator topologies.)\n-/\nadd_decl_doc VonNeumannAlgebra.toStarSubalgebra\n\nnamespace VonNeumannAlgebra\n\nvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n\ninstance instSetLike : SetLike (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  coe S := S.carrier\n  coe_injective S T h := by obtain ⟨⟨⟨⟨⟨⟨_, _⟩, _⟩, _⟩, _⟩, _⟩, _⟩ := S; cases T; congr\n\ninstance : PartialOrder (VonNeumannAlgebra H) := .ofSetLike (VonNeumannAlgebra H) (H →L[ℂ] H)\n\nnoncomputable instance instStarMemClass : StarMemClass (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  star_mem {s} := s.star_mem'\n\ninstance instSubringClass : SubringClass (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  one_mem {s} := s.one_mem'\n  zero_mem {s} := s.zero_mem'\n  neg_mem {s} a ha := show -a ∈ s.toStarSubalgebra from neg_mem ha\n\n@[simp]\ntheorem mem_carrier {S : VonNeumannAlgebra H} {x : H →L[ℂ] H} :\n    x ∈ S.toStarSubalgebra ↔ x ∈ (S : Set (H →L[ℂ] H)) :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toStarSubalgebra (S : VonNeumannAlgebra H) :\n    (S.toStarSubalgebra : Set (H →L[ℂ] H)) = S :=\n  rfl\n\n@[simp]\ntheorem coe_mk (S : StarSubalgebra ℂ (H →L[ℂ] H)) (h) :\n    ((⟨S, h⟩ : VonNeumannAlgebra H) : Set (H →L[ℂ] H)) = S :=\n  rfl\n\n@[ext]\ntheorem ext {S T : VonNeumannAlgebra H} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem centralizer_centralizer (S : VonNeumannAlgebra H) :\n    Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S :=\n  S.centralizer_centralizer'\n\n/-- The centralizer of a `VonNeumannAlgebra`, as a `VonNeumannAlgebra`. -/\nnoncomputable def commutant (S : VonNeumannAlgebra H) : VonNeumannAlgebra H where\n  toStarSubalgebra := StarSubalgebra.centralizer ℂ (S : Set (H →L[ℂ] H))\n  centralizer_centralizer' := by simp\n\n@[simp]\ntheorem coe_commutant (S : VonNeumannAlgebra H) :\n    ↑S.commutant = Set.centralizer (S : Set (H →L[ℂ] H)) := by\n  simp [commutant]\n\n@[simp]\ntheorem mem_commutant_iff {S : VonNeumannAlgebra H} {z : H →L[ℂ] H} :\n    z ∈ S.commutant ↔ ∀ g ∈ S, g * z = z * g := by\n  rw [← SetLike.mem_coe, coe_commutant]\n  rfl\n\n@[simp]\ntheorem commutant_commutant (S : VonNeumannAlgebra H) : S.commutant.commutant = S :=\n  SetLike.coe_injective <| by simp\n\nopen ContinuousLinearMap in\n/-- An idempotent is an element in a von Neumann algebra if and only if\nits range and kernel are invariant under the commutant. -/\n\nTarget:\ntheorem IsIdempotentElem.mem_iff {e : H →L[ℂ] H} (h : IsIdempotentElem e)\n    (S : VonNeumannAlgebra H) :\n    e ∈ S ↔ ∀ y ∈ S.commutant,\n      e.range ∈ Module.End.invtSubmodule y ∧ e.ker ∈ Module.End.invtSubmodule y :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/VonNeumannAlgebra","family_id":"isidempotentelem","file_id":"mathlib/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean","sample_id":"b3ad2032874ba6a0b6f50397e2b035a5757e486348575d6dc25170f7fefc69ec"},"split":"train"}
{"binding":{"chosen_proof_sha256":"03e21e37518786f0e8f480f42a07a2927cdb9237ca44f9d215146d88cd8ce9cd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1721659186d255075a58ab2c96c00145a28645dc3bdd670ebe46b5feb79e7a6a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7f63bf8b080e808117284c406a2db8ef2f14a5c00152c31fb40b5d94668e2b76","source_sha256":"ccfcbde3f10bc565628c0a834c762e9d5e446934b2c86ddc567af141b0f9967f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n    obtain ⟨s, hs⟩ := hI\n    rw [← adjoin_monomial_eq_reesAlgebra, ← hs]\n    use s.image (monomial 1)\n    rw [Finset.coe_image]\n    change\n      _ =\n        Algebra.adjoin R\n          (Submodule.map (monomial 1 : R →ₗ[R] R[X]) (Submodule.span R ↑s) : Set R[X])\n    rw [Submodule.map_span, Algebra.adjoin_span]","hard_negative":false,"metrics":{"chosen_tokens":83,"rejected_tokens":88,"token_jaccard":0.911111,"token_length_ratio":1.060241},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"06dc353dcc2bade935863497c4cfbb148bf7871adc6cd4ad051486cab904ed46","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Ideal.BigOperators\npublic import Mathlib.RingTheory.FiniteType\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Rees algebra\n\nThe Rees algebra of an ideal `I` is the subalgebra `R[It]` of `R[t]` defined as `R[It] = ⨁ₙ Iⁿ tⁿ`.\nThis is used to prove the Artin-Rees lemma, and will potentially enable us to calculate some\nblowup in the future.\n\n## Main definition\n\n- `reesAlgebra` : The Rees algebra of an ideal `I`, defined as a subalgebra of `R[X]`.\n- `adjoin_monomial_eq_reesAlgebra` : The Rees algebra is generated by the degree one elements.\n- `reesAlgebra.fg` : The Rees algebra of a f.g. ideal is of finite type. In particular, this\n  implies that the rees algebra over a Noetherian ring is still Noetherian.\n\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\nvariable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)\n\nopen Polynomial\n\n/-- The Rees algebra of an ideal `I`, defined as the subalgebra of `R[X]` whose `i`-th coefficient\nfalls in `I ^ i`. -/\ndef reesAlgebra : Subalgebra R R[X] where\n  carrier := { f | ∀ i, f.coeff i ∈ I ^ i }\n  mul_mem' hf hg i := by\n    rw [coeff_mul]\n    apply Ideal.sum_mem\n    rintro ⟨j, k⟩ e\n    rw [← Finset.mem_antidiagonal.mp e, pow_add]\n    exact Ideal.mul_mem_mul (hf j) (hg k)\n  one_mem' i := by\n    rw [coeff_one]\n    split_ifs with h\n    · subst h\n      simp\n    · simp\n  add_mem' hf hg i := by\n    rw [coeff_add]\n    exact Ideal.add_mem _ (hf i) (hg i)\n  zero_mem' _ := Ideal.zero_mem _\n  algebraMap_mem' r i := by\n    rw [algebraMap_apply, coeff_C]\n    split_ifs with h\n    · subst h\n      simp\n    · simp\n\ntheorem mem_reesAlgebra_iff (f : R[X]) : f ∈ reesAlgebra I ↔ ∀ i, f.coeff i ∈ I ^ i :=\n  Iff.rfl\n\ntheorem mem_reesAlgebra_iff_support (f : R[X]) :\n    f ∈ reesAlgebra I ↔ ∀ i ∈ f.support, f.coeff i ∈ I ^ i := by\n  apply forall_congr'\n  intro a\n  rw [mem_support_iff, Iff.comm, Classical.imp_iff_right_iff, Ne, ← imp_iff_not_or]\n  exact fun e => e.symm ▸ (I ^ a).zero_mem\n\ntheorem reesAlgebra.monomial_mem {I : Ideal R} {i : ℕ} {r : R} :\n    monomial i r ∈ reesAlgebra I ↔ r ∈ I ^ i := by\n  simp +contextual [mem_reesAlgebra_iff_support, coeff_monomial, ←\n    imp_iff_not_or]\n\ntheorem monomial_mem_adjoin_monomial {I : Ideal R} {n : ℕ} {r : R} (hr : r ∈ I ^ n) :\n    monomial n r ∈ Algebra.adjoin R (Submodule.map (monomial 1 : R →ₗ[R] R[X]) I : Set R[X]) := by\n  induction n generalizing r with\n  | zero => exact Subalgebra.algebraMap_mem _ _\n  | succ n hn =>\n    rw [pow_succ'] at hr\n    refine Submodule.smul_induction_on hr ?_ ?_\n    · intro r hr s hs\n      rw [add_comm n 1, smul_eq_mul, ← monomial_mul_monomial]\n      exact Subalgebra.mul_mem _ (Algebra.subset_adjoin (Set.mem_image_of_mem _ hr)) (hn hs)\n    · intro x y hx hy\n      rw [map_add]\n      exact Subalgebra.add_mem _ hx hy\n\ntheorem adjoin_monomial_eq_reesAlgebra :\n    Algebra.adjoin R (Submodule.map (monomial 1 : R →ₗ[R] R[X]) I : Set R[X]) = reesAlgebra I := by\n  apply le_antisymm\n  · apply Algebra.adjoin_le _\n    rintro _ ⟨r, hr, rfl⟩\n    exact reesAlgebra.monomial_mem.mpr (by rwa [pow_one])\n  · intro p hp\n    rw [p.as_sum_support]\n    apply Subalgebra.sum_mem _ _\n    rintro i -\n    exact monomial_mem_adjoin_monomial (hp i)\n\nvariable {I}\n\nTarget:\ntheorem reesAlgebra.fg (hI : I.FG) : (reesAlgebra I).FG :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  classical\n    obtain ⟨s, hs⟩ := hI\n    rw [← adjoin_monomial_eq_reesAlgebra, ← hs]\n    use s.image (monomial 1)\n    rw [Finset.coe_image]\n    change\n      _ =\n        Algebra.adjoin R\n          (Submodule.map (monomial 1 : R →ₗ[R] R[X]) (Submodule.span R ↑s) : Set R[X])\n    rw [Submodule.map_span, Algebra.adjoin_span]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory","family_id":"reesalgebra","file_id":"mathlib/Mathlib/RingTheory/ReesAlgebra.lean","sample_id":"7f63bf8b080e808117284c406a2db8ef2f14a5c00152c31fb40b5d94668e2b76"},"split":"train"}
{"binding":{"chosen_proof_sha256":"66e860b3b879ffe015e76af344d6b2852af86ab51c1939b65912c04b3badbf97","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6c05ada6fb867e067b57945bee363d3116da5202db3c69d06f6b508b9b741204","source_sha256":"d43040b92adbb8ebcddab07470e26da05f34a625604831787d2b69fe79c9281e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h => ?_, fun h => h.symm ▸ mulSingle_one i⟩\n  rw [← mulSingle_eq_same i x, h, one_apply]","hard_negative":false,"metrics":{"chosen_tokens":30,"rejected_tokens":3,"token_jaccard":0.041667,"token_length_ratio":0.1},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"06f4b1a5a05acacd143cae3af399d7ba0f61702a697dbafab2a9718292c4e543","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Pi.Defs\npublic import Mathlib.Logic.Function.Basic\n\nNamespace:\nPi\n\nLocal context:\n/-\nCopyright (c) 2020 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot, Eric Wieser\n-/\n/-!\n# Very basic algebraic operations on pi types\n\nThis file provides very basic algebraic operations on functions.\n-/\n\n@[expose] public section\n\nassert_not_exists Monoid Preorder\n\nopen Function\n\nvariable {ι ι' α β : Type*} {G M N O : ι → Type*}\n\nnamespace Pi\nvariable [∀ i, One (M i)] [∀ i, One (N i)] [∀ i, One (O i)] [DecidableEq ι] {i : ι} {x : M i}\n\n/-- The function supported at `i`, with value `x` there, and `1` elsewhere. -/\n@[to_additive /-- The function supported at `i`, with value `x` there, and `0` elsewhere. -/]\ndef mulSingle (i : ι) (x : M i) : ∀ j, M j := Function.update 1 i x\n\n@[to_additive (attr := simp)]\nlemma mulSingle_eq_same (i : ι) (x : M i) : mulSingle i x i = x := Function.update_self i x _\n\n@[to_additive (attr := simp)]\nlemma mulSingle_eq_of_ne {i i' : ι} (h : i' ≠ i) (x : M i) : mulSingle i x i' = 1 :=\n  Function.update_of_ne h x _\n\n/-- Abbreviation for `mulSingle_eq_of_ne h.symm`, for ease of use by `simp`. -/\n@[to_additive (attr := simp)\n  /-- Abbreviation for `single_eq_of_ne h.symm`, for ease of use by `simp`. -/]\nlemma mulSingle_eq_of_ne' {i i' : ι} (h : i ≠ i') (x : M i) : mulSingle i x i' = 1 :=\n  mulSingle_eq_of_ne h.symm x\n\n@[to_additive (attr := simp)]\nlemma mulSingle_one (i : ι) : mulSingle i (1 : M i) = 1 := Function.update_eq_self _ _\n\n@[to_additive (attr := simp)]\n\nTarget:\nlemma mulSingle_eq_one_iff : mulSingle i x = 1 ↔ x = 1 :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Notation","family_id":"mulsingle_eq_one_iff","file_id":"mathlib/Mathlib/Algebra/Notation/Pi/Basic.lean","sample_id":"6c05ada6fb867e067b57945bee363d3116da5202db3c69d06f6b508b9b741204"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0c7a4b44bba5c20da7810399c16de1e098a5c39bdee940dd765e6735e6e9d672","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"130996c934f7e3dc9a1f69e7437ed491cdeda3ef578a265295d518eee014bbc5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"af27a105e43a072189a2b49287038416eed37dc52c7226788734b79f2baf138e","source_sha256":"6b6fd17d69ce9cda1c5ca51d87b6241ac80bfb6aa3b060b832bdfe6d51236491","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [IsSelfAdjoint.star_eq, ← sub_eq_add_neg, (CFC.sqrt_nonneg (1 - a ^ 2 : A)).isSelfAdjoint]","hard_negative":true,"metrics":{"chosen_tokens":27,"rejected_tokens":5,"token_jaccard":0.08,"token_length_ratio":0.185185},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"0703b97fe34fc512203f8faaeb1e06c6a252bbb272b6c832827dc51a489746cc","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order\npublic import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary\npublic import Mathlib.Analysis.Normed.Module.Normalize\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\n/-! # Unitary elements span C⋆-algebras\n\n## Main results\n\n+ `CStarAlgebra.exists_sum_four_unitary`: every element `x` in a unital C⋆-algebra is a linear\n  combination of four unitary elements, and the norm of each coefficient does not exceed `‖x‖ / 2`.\n+ `CStarAlgebra.span_unitary`: a unital C⋆-algebra is spanned by its unitary elements.\n-/\n\n@[expose] public section\n\nvariable {A : Type*} [CStarAlgebra A]\n\nopen scoped ComplexStarModule\nopen Complex\n\nsection Ordered\n\nvariable [PartialOrder A] [StarOrderedRing A]\n\n/-- If `a : A` is a selfadjoint element in a C⋆-algebra with `‖a‖ ≤ 1`,\nthen `a + I • CFC.sqrt (1 - a ^ 2)` is unitary.\n\nThis is the key tool to show that a C⋆-algebra is spanned by its unitary elements. -/\nlemma IsSelfAdjoint.self_add_I_smul_cfcSqrt_sub_sq_mem_unitary (a : A) (ha : IsSelfAdjoint a)\n    (ha_norm : ‖a‖ ≤ 1) : a + I • CFC.sqrt (1 - a ^ 2) ∈ unitary A := by\n  obtain (_ | _) := subsingleton_or_nontrivial A\n  · simp [Subsingleton.elim (a + I • CFC.sqrt (1 - a ^ 2)) 1, one_mem (unitary A)]\n  have key : a + I • CFC.sqrt (1 - a ^ 2) = cfc (fun x : ℂ ↦ x.re + I * √(1 - x.re ^ 2)) a := by\n    rw [CFC.sqrt_eq_real_sqrt (1 - a ^ 2) ?nonneg]\n    case nonneg =>\n      rwa [sub_nonneg, ← CStarAlgebra.norm_le_one_iff_of_nonneg (a ^ 2), sq, ha.norm_mul_self,\n        sq_le_one_iff₀ (by positivity)]\n    rw [cfc_add .., cfc_const_mul .., ← cfc_real_eq_complex (fun x ↦ x) ha, cfc_id' ℝ a,\n      ← cfc_real_eq_complex (fun x ↦ √(1 - x ^ 2)) ha, cfcₙ_eq_cfc, cfc_comp' (√·) (1 - · ^ 2) a,\n      cfc_sub .., cfc_pow .., cfc_const_one .., cfc_id' ..]\n  rw [key, cfc_unitary_iff ..]\n  intro x hx\n  rw [← starRingEnd_apply, ← Complex.normSq_eq_conj_mul_self,\n    Complex.normSq_ofReal_add_I_mul_sqrt_one_sub, Complex.ofReal_one]\n  exact spectrum.norm_le_norm_of_mem (ha.spectrumRestricts.apply_mem hx) |>.trans ha_norm\n\n/-- For `a` selfadjoint with `‖a‖ ≤ 1`, this is the unitary `a + I • √(1 - a ^ 2)`. -/\n@[simps]\nnoncomputable def selfAdjoint.unitarySelfAddISMul (a : selfAdjoint A) (ha_norm : ‖a‖ ≤ 1) :\n    unitary A :=\n  ⟨(a : A) + I • CFC.sqrt (1 - a ^ 2 : A), a.2.self_add_I_smul_cfcSqrt_sub_sq_mem_unitary _ ha_norm⟩\n\nTarget:\nlemma selfAdjoint.star_coe_unitarySelfAddISMul (a : selfAdjoint A) (ha_norm : ‖a‖ ≤ 1) :\n    (star (unitarySelfAddISMul a ha_norm) : A) = a - I • CFC.sqrt (1 - a ^ 2 : A) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_af27a105e43a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"910d805da3f33b7ad1594723f71f7d85d99ffb7daf7416900d5275e49dd9c46f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/CStarAlgebra","family_id":"selfadjoint","file_id":"mathlib/Mathlib/Analysis/CStarAlgebra/Unitary/Span.lean","sample_id":"af27a105e43a072189a2b49287038416eed37dc52c7226788734b79f2baf138e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d20eb1ad8941b8e34a808b1966083034fd1512741d71702f610e71a800689ea8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ac967f3faf553306a5c24145a1c9caf628fb00eae9e83bd29d882cbc48ac201d","source_sha256":"5596724266063c95bd7c3b79e894544045556a2fdd0d7b0136189582e7829a9b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← single_zero_one, orderTop_single one_ne_zero, WithTop.coe_eq_zero]","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":8,"token_jaccard":0.055556,"token_length_ratio":0.615385},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"07c76a63455edc8852677c642af5e6de4077c2e679b530966ab8dae1e572e2d7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Subalgebra.Lattice\npublic import Mathlib.Algebra.GroupWithZero.Regular\npublic import Mathlib.Algebra.Module.BigOperators\npublic import Mathlib.Data.Finset.MulAntidiagonal\npublic import Mathlib.Data.Finset.SMulAntidiagonal\npublic import Mathlib.GroupTheory.GroupAction.Ring\npublic import Mathlib.RingTheory.HahnSeries.Addition\n\nNamespace:\nHahnSeries\n\nLocal context:\n/-\nCopyright (c) 2021 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson, Scott Carnahan\n-/\n/-!\n# Multiplicative properties of Hahn series\nIf `Γ` is ordered and `R` has zero, then `R⟦Γ⟧` consists of formal series over `Γ` with\ncoefficients in `R`, whose supports are partially well-ordered. This module introduces\nmultiplication and scalar multiplication on Hahn series. If `Γ` is an ordered cancellative\ncommutative additive monoid and `R` is a semiring, then we get a semiring structure on\n`R⟦Γ⟧`. If `Γ` has an ordered vector-addition on `Γ'` and `R` has a scalar multiplication\non `V`, we define `HahnModule Γ' R V` as a type alias for `V⟦Γ'⟧` that admits a scalar\nmultiplication from `R⟦Γ⟧`. The scalar action of `R` on `R⟦Γ⟧` is compatible\nwith the action of `R⟦Γ⟧` on `HahnModule Γ' R V`.\n\n## Main Definitions\n* `HahnModule` is a type alias for `HahnSeries`, which we use for defining scalar multiplication\n  of `R⟦Γ⟧` on `HahnModule Γ' R V` for an `R`-module `V`, where `Γ'` admits an ordered\n  cancellative vector addition operation from `Γ`. The type alias allows us to avoid a potential\n  instance diamond.\n* `HahnModule.of` is the isomorphism from `V⟦Γ⟧` to `HahnModule Γ R V`.\n* `HahnSeries.C` is the `constant term` ring homomorphism `R →+* R⟦Γ⟧`.\n* `HahnSeries.embDomainRingHom` is the ring homomorphism `R⟦Γ⟧ →+* R⟦Γ'⟧`\n  induced by an order embedding `Γ ↪o Γ'`.\n* `HahnSeries.orderTopSubOnePos` is the group of invertible Hahn series close to 1, i.e., those\n  series such that subtracting one yields a series with strictly positive `orderTop`.\n\n## Main results\n* If `R` is a (commutative) (semi-)ring, then so is `R⟦Γ⟧`.\n* If `V` is an `R`-module, then `HahnModule Γ' R V` is a `R⟦Γ⟧`-module.\n\n## TODO\nThe following may be useful for composing vertex operators, but they seem to take time.\n* rightTensorMap: `HahnModule Γ' R U ⊗[R] V →ₗ[R] HahnModule Γ' R (U ⊗[R] V)`\n* leftTensorMap: `U ⊗[R] HahnModule Γ' R V →ₗ[R] HahnModule Γ' R (U ⊗[R] V)`\n\n## References\n- [J. van der Hoeven, *Operators on Generalized Power Series*][van_der_hoeven]\n-/\n\n@[expose] public section\n\nopen Finset Function HahnSeries Pointwise\n\nnoncomputable section\n\nvariable {Γ Γ' R S V : Type*}\n\nnamespace HahnSeries\n\nvariable [Zero Γ] [PartialOrder Γ]\n\ninstance [Zero R] [One R] : One R⟦Γ⟧ where one := single 0 1\ninstance [Zero R] [NatCast R] : NatCast R⟦Γ⟧ where natCast n := single 0 n\ninstance [Zero R] [IntCast R] : IntCast R⟦Γ⟧ where intCast z := single 0 z\ninstance [Zero R] [NNRatCast R] : NNRatCast R⟦Γ⟧ where nnratCast q := single 0 q\ninstance [Zero R] [RatCast R] : RatCast R⟦Γ⟧ where ratCast q := single 0 q\n\nopen Classical in\n@[simp]\ntheorem coeff_one [Zero R] [One R] {a : Γ} : (1 : R⟦Γ⟧).coeff a = if a = 0 then 1 else 0 :=\n  coeff_single\n\n@[simp] theorem single_zero_one [Zero R] [One R] : single (0 : Γ) (1 : R) = 1 := rfl\ntheorem single_zero_natCast [Zero R] [NatCast R] (n : ℕ) : single (0 : Γ) (n : R) = n := rfl\ntheorem single_zero_intCast [Zero R] [IntCast R] (z : ℤ) : single (0 : Γ) (z : R) = z := rfl\ntheorem single_zero_nnratCast [Zero R] [NNRatCast R] (q : ℚ≥0) : single (0 : Γ) (q : R) = q := rfl\ntheorem single_zero_ratCast [Zero R] [RatCast R] (q : ℚ) : single (0 : Γ) (q : R) = q := rfl\n\ntheorem single_zero_ofNat [Zero R] [NatCast R] (n : ℕ) [n.AtLeastTwo] :\n    single (0 : Γ) (ofNat(n) : R) = ofNat(n) := rfl\n\n@[simp]\ntheorem support_one [MulZeroOneClass R] [Nontrivial R] : support (1 : R⟦Γ⟧) = {0} :=\n  support_single_of_ne one_ne_zero\n\n@[simp]\n\nTarget:\ntheorem orderTop_one [Zero R] [One R] [NeZero (1 : R)] : orderTop (1 : R⟦Γ⟧) = 0 :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b4e59b5d70d3037f13266d6f3c34244679154076146ca872ca947e693cb6e8a2","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/HahnSeries","family_id":"ordertop_one","file_id":"mathlib/Mathlib/RingTheory/HahnSeries/Multiplication.lean","sample_id":"ac967f3faf553306a5c24145a1c9caf628fb00eae9e83bd29d882cbc48ac201d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"930122cf7815506e3d0570b55467812e4caec3bf1c598928f36877dea55601fa","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f6f3ce4e022ec29b5bf2237824915ea6274dba67f177197d204b8daccb2930a5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"636a9fcc3e8a95e1374f41b3672c8c890f7576b9c9fa7d5f0caae9002e2a7ade","source_sha256":"d6fe9940fe5118171b24b843fb7690a27e16198b1d58f5f56373a58891dc6904","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only [h.comp_eq] using bijOn_fixedPoints_comp f g","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.25},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"07f520b641c5b18e3b4e90eab8fb5dab450d77fb31707558fe45ff66c8fb599f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.End\npublic import Mathlib.Data.Set.Function\npublic import Mathlib.Dynamics.FixedPoints.Defs\n\nNamespace:\nFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Fixed points of a self-map\n\nWe prove some simple lemmas about `IsFixedPt` and `∘`, `iterate`, and `Semiconj`.\n\n## Tags\n\nfixed point\n-/\n\npublic section\n\nopen Equiv\n\nuniverse u v\n\nvariable {α β : Type*} {f fa g : α → α} {x : α} {fb : β → β} {e : Perm α}\n\nnamespace Function\n\nopen Function (Commute)\n\nnamespace IsFixedPt\n\n/-- If `x` is a fixed point of `f` and `g`, then it is a fixed point of `f ∘ g`. -/\nprotected theorem comp (hf : IsFixedPt f x) (hg : IsFixedPt g x) : IsFixedPt (f ∘ g) x :=\n  calc\n    f (g x) = f x := congr_arg f hg\n    _ = x := hf\n\n/-- If `x` is a fixed point of `f`, then it is a fixed point of `f^[n]`. -/\nprotected theorem iterate (hf : IsFixedPt f x) (n : ℕ) : IsFixedPt f^[n] x :=\n  iterate_fixed hf n\n\n/-- If `x` is a fixed point of `f ∘ g` and `g`, then it is a fixed point of `f`. -/\ntheorem left_of_comp (hfg : IsFixedPt (f ∘ g) x) (hg : IsFixedPt g x) : IsFixedPt f x :=\n  calc\n    f x = f (g x) := congr_arg f hg.symm\n    _ = x := hfg\n\n/-- If `x` is a fixed point of `f` and `g` is a left inverse of `f`, then `x` is a fixed\npoint of `g`. -/\ntheorem to_leftInverse (hf : IsFixedPt f x) (h : LeftInverse g f) : IsFixedPt g x :=\n  calc\n    g x = g (f x) := congr_arg g hf.symm\n    _ = x := h x\n\n/-- If `g` (semi)conjugates `fa` to `fb`, then it sends fixed points of `fa` to fixed points\nof `fb`. -/\nprotected theorem map {x : α} (hx : IsFixedPt fa x) {g : α → β} (h : Semiconj g fa fb) :\n    IsFixedPt fb (g x) :=\n  calc\n    fb (g x) = g (fa x) := (h.eq x).symm\n    _ = g x := congr_arg g hx\n\nprotected theorem apply {x : α} (hx : IsFixedPt f x) : IsFixedPt f (f x) := by convert! hx\n\ntheorem preimage_iterate {s : Set α} (h : IsFixedPt (Set.preimage f) s) (n : ℕ) :\n    IsFixedPt (Set.preimage f^[n]) s := by\n  rw [Set.preimage_iterate_eq]\n  exact h.iterate n\n\nlemma image_iterate {s : Set α} (h : IsFixedPt (Set.image f) s) (n : ℕ) :\n    IsFixedPt (Set.image f^[n]) s :=\n  Set.image_iterate_eq ▸ h.iterate n\n\nprotected theorem equiv_symm (h : IsFixedPt e x) : IsFixedPt e.symm x :=\n  h.to_leftInverse e.leftInverse_symm\n\n@[simp]\ntheorem equiv_symm_iff : IsFixedPt e.symm x ↔ IsFixedPt e x :=\n  ⟨fun h ↦ e.symm_symm ▸ h.equiv_symm, .equiv_symm⟩\n\nprotected theorem perm_inv (h : IsFixedPt e x) : IsFixedPt (⇑e⁻¹) x :=\n  h.equiv_symm\n\nprotected theorem perm_pow (h : IsFixedPt e x) (n : ℕ) : IsFixedPt (⇑(e ^ n)) x := h.iterate _\n\nprotected theorem perm_zpow (h : IsFixedPt e x) : ∀ n : ℤ, IsFixedPt (⇑(e ^ n)) x\n  | Int.ofNat _ => h.perm_pow _\n  | Int.negSucc n => (h.perm_pow <| n + 1).perm_inv\n\nend IsFixedPt\n\n@[simp]\ntheorem fixedPoints_symm : fixedPoints e.symm = fixedPoints e := by\n  simp [Set.ext_iff]\n\n@[simp]\ntheorem Injective.isFixedPt_apply_iff (hf : Injective f) {x : α} :\n    IsFixedPt f (f x) ↔ IsFixedPt f x :=\n  ⟨fun h => hf h.eq, IsFixedPt.apply⟩\n\n/-- If `g` semiconjugates `fa` to `fb`, then it sends fixed points of `fa` to fixed points\nof `fb`. -/\ntheorem Semiconj.mapsTo_fixedPoints {g : α → β} (h : Semiconj g fa fb) :\n    Set.MapsTo g (fixedPoints fa) (fixedPoints fb) := fun _ hx => hx.map h\n\n/-- Any two maps `f : α → β` and `g : β → α` are inverse of each other on the sets of fixed points\nof `f ∘ g` and `g ∘ f`, respectively. -/\ntheorem invOn_fixedPoints_comp (f : α → β) (g : β → α) :\n    Set.InvOn f g (fixedPoints <| f ∘ g) (fixedPoints <| g ∘ f) :=\n  ⟨fun _ => id, fun _ => id⟩\n\n/-- Any map `f` sends fixed points of `g ∘ f` to fixed points of `f ∘ g`. -/\ntheorem mapsTo_fixedPoints_comp (f : α → β) (g : β → α) :\n    Set.MapsTo f (fixedPoints <| g ∘ f) (fixedPoints <| f ∘ g) := fun _ hx => hx.map fun _ => rfl\n\n/-- Given two maps `f : α → β` and `g : β → α`, `g` is a bijective map between the fixed points\nof `f ∘ g` and the fixed points of `g ∘ f`. The inverse map is `f`, see `invOn_fixedPoints_comp`. -/\ntheorem bijOn_fixedPoints_comp (f : α → β) (g : β → α) :\n    Set.BijOn g (fixedPoints <| f ∘ g) (fixedPoints <| g ∘ f) :=\n  (invOn_fixedPoints_comp f g).bijOn (mapsTo_fixedPoints_comp g f) (mapsTo_fixedPoints_comp f g)\n\n/-- If self-maps `f` and `g` commute, then they are inverse of each other on the set of fixed points\nof `f ∘ g`. This is a particular case of `Function.invOn_fixedPoints_comp`. -/\ntheorem Commute.invOn_fixedPoints_comp (h : Commute f g) :\n    Set.InvOn f g (fixedPoints <| f ∘ g) (fixedPoints <| f ∘ g) := by\n  simpa only [h.comp_eq] using Function.invOn_fixedPoints_comp f g\n\n/-- If self-maps `f` and `g` commute, then `f` is bijective on the set of fixed points of `f ∘ g`.\nThis is a particular case of `Function.bijOn_fixedPoints_comp`. -/\ntheorem Commute.left_bijOn_fixedPoints_comp (h : Commute f g) :\n    Set.BijOn f (fixedPoints <| f ∘ g) (fixedPoints <| f ∘ g) := by\n  simpa only [h.comp_eq] using bijOn_fixedPoints_comp g f\n\n/-- If self-maps `f` and `g` commute, then `g` is bijective on the set of fixed points of `f ∘ g`.\nThis is a particular case of `Function.bijOn_fixedPoints_comp`. -/\n\nTarget:\ntheorem Commute.right_bijOn_fixedPoints_comp (h : Commute f g) :\n    Set.BijOn g (fixedPoints <| f ∘ g) (fixedPoints <| f ∘ g) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_636a9fcc3e8a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"702594ef4837066088b7ae06ce9809d9afcf03ff374ea8d6139efd4bd4f8ba5c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/FixedPoints","family_id":"commute","file_id":"mathlib/Mathlib/Dynamics/FixedPoints/Basic.lean","sample_id":"636a9fcc3e8a95e1374f41b3672c8c890f7576b9c9fa7d5f0caae9002e2a7ade"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2c01520ab51bc25fbe7980ca7811c6f4fd2614f49e05e36aaa18fc8557eeaddd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"71ba16669017a6fd8a1b9efc9f308604b881f14de29d88f4bf40d45c3fc655da","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f57b40da55b42c3f0a2ab8a4512907a91408e6b8713cf272d9f3041b94ceb710","source_sha256":"21c0805d79514be5c2fec0b605f66875bb5bf5631a7a75dc4887fad5491c11a0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [modEq_comm]\n  simp_rw [modEq_iff_eq_add_zsmul, QuotientAddGroup.eq_iff_sub_mem, AddSubgroup.mem_zmultiples_iff,\n    eq_sub_iff_add_eq', eq_comm]","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":2,"token_jaccard":0.0625,"token_length_ratio":0.095238},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"08e596b83bd67565d7054379b9b8267b62ec6f2c76149ecb9585b1ecbe650c43","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.ModEq\npublic import Mathlib.GroupTheory.QuotientGroup.Defs\npublic import Mathlib.Algebra.Group.Subgroup.ZPowers.Basic\n\nNamespace:\nAddCommGroup\n\nLocal context:\n/-\nCopyright (c) 2023 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n\n/-!\n# Congruence modulo multiples and congruence modulo `AddSubgroup.zmultiples _`\n\nIn this file we show that in an additive commutative group, the congruence relation `a ≡ b [PMOD p]`\nis equivalent to the coercions of `a` and `b` to `G ⧸ AddSubgroup.zmultiples p` being equal.\n-/\n\npublic section\n\nnamespace AddCommGroup\n\nvariable {G : Type*} [AddCommGroup G] {a b p : G}\n\nTarget:\ntheorem modEq_iff_eq_mod_zmultiples : a ≡ b [PMOD p] ↔ (a : G ⧸ AddSubgroup.zmultiples p) = b :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_f57b40da55b4","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"2f57cca57a4dcb8b94248dc8a844a6b0ce98685f58004564fb111d1f4119b100","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/QuotientGroup","family_id":"modeq_iff_eq_mod_zmultiples","file_id":"mathlib/Mathlib/GroupTheory/QuotientGroup/ModEq.lean","sample_id":"f57b40da55b42c3f0a2ab8a4512907a91408e6b8713cf272d9f3041b94ceb710"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b82467ef621f5a21aaae8eaa62dc55df65f51c9c9619a918a2bac8a08b7ed281","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1979aaf06c49a077cffd0cbacc14afdba7fdea1957b85022a0c5fbe376b51129","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d0d0f38b9e3e428b2ae38db46c5384471141fd719dfd1fd904da355d5cb234fc","source_sha256":"b5b8fef319876fc584704ec0902863ab8016fe0ba4dee5dd358f86f4688b13cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  delta Irrational\n  contrapose!\n  rintro ⟨⟨rx, rfl⟩, ⟨ry, rfl⟩⟩\n  exact ⟨rx + ry, cast_add rx ry⟩","hard_negative":false,"metrics":{"chosen_tokens":29,"rejected_tokens":34,"token_jaccard":0.75,"token_length_ratio":1.172414},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"09f879999edc81c7f7930cb98b2b558018a3026a99ebfe4af6f059f43c356d46","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Rat\npublic import Mathlib.Data.Nat.Prime.Int\npublic import Mathlib.Data.Rat.Sqrt\npublic import Mathlib.Analysis.Real.Sqrt\npublic import Mathlib.RingTheory.Algebraic.Basic\npublic import Mathlib.Tactic.CrossRefAttribute\npublic import Mathlib.Tactic.IntervalCases\n\nNamespace:\nIrrational\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov\n-/\n/-!\n# Irrational real numbers\n\nIn this file we define a predicate `Irrational` on `ℝ`, prove that the `n`-th root of an integer\nnumber is irrational if it is not integer, and that `√(q : ℚ)` is irrational if and only if\n`¬IsSquare q ∧ 0 ≤ q`.\n\nWe also provide dot-style constructors like `Irrational.add_ratCast`, `Irrational.ratCast_sub` etc.\n\nWith the `Decidable` instances in this file, is possible to prove `Irrational √n` using `decide`,\nwhen `n` is a numeric literal or cast;\nbut this only works if you `unseal Nat.sqrt.iter in` before the theorem where you use this proof.\n-/\n\n@[expose] public section\n\n\nopen Rat Real\n\n/-- A real number is irrational if it is not equal to any rational number. -/\n@[wikidata Q607728]\ndef Irrational (x : ℝ) :=\n  x ∉ Set.range ((↑) : ℚ → ℝ)\n\ntheorem irrational_iff_ne_rational (x : ℝ) : Irrational x ↔ ∀ a b : ℤ, b ≠ 0 → x ≠ a / b := by\n  simp [Irrational, Rat.forall, eq_comm]\n\ntheorem Irrational.ne_rational {x : ℝ} (hx : Irrational x) (a b : ℤ) : x ≠ a / b := by\n  rintro rfl; exact hx ⟨a / b, by simp⟩\n\ntheorem exists_rat_of_not_irrational {x : ℝ} (hx : ¬ Irrational x) : ∃ (q : ℚ), x = q := by\n  grind [Irrational]\n\n/-- A transcendental real number is irrational. -/\ntheorem Transcendental.irrational {r : ℝ} (tr : Transcendental ℚ r) : Irrational r := by\n  rintro ⟨a, rfl⟩\n  exact tr (isAlgebraic_algebraMap a)\n\n/-!\n### Irrationality of roots of integer and rational numbers\n-/\n\n\n/-- If `x^n`, `n > 0`, is integer and is not the `n`-th power of an integer, then\n`x` is irrational. -/\ntheorem irrational_nrt_of_notint_nrt {x : ℝ} (n : ℕ) (m : ℤ) (hxr : x ^ n = m)\n    (hv : ¬∃ y : ℤ, x = y) (hnpos : 0 < n) : Irrational x := by\n  rintro ⟨⟨N, D, P, C⟩, rfl⟩\n  rw [← cast_pow] at hxr\n  have c1 : ((D : ℤ) : ℝ) ≠ 0 := by\n    rw [Int.cast_ne_zero, Int.natCast_ne_zero]\n    exact P\n  have c2 : ((D : ℤ) : ℝ) ^ n ≠ 0 := pow_ne_zero _ c1\n  rw [mk_eq_divInt, cast_pow, cast_divInt, div_pow, div_eq_iff_mul_eq c2, ← Int.cast_pow,\n    ← Int.cast_pow, ← Int.cast_mul, Int.cast_inj] at hxr\n  have hdivn : (D : ℤ) ^ n ∣ N ^ n := Dvd.intro_left m hxr\n  rw [← Int.dvd_natAbs, ← Int.natCast_pow, Int.natCast_dvd_natCast, Int.natAbs_pow,\n    Nat.pow_dvd_pow_iff hnpos.ne'] at hdivn\n  obtain rfl : D = 1 := by rw [← Nat.gcd_eq_right hdivn, C.gcd_eq_one]\n  refine hv ⟨N, ?_⟩\n  rw [mk_eq_divInt, Int.ofNat_one, divInt_one, cast_intCast]\n\n/-- If `x^n = m` is an integer and `n` does not divide the `multiplicity p m`, then `x`\nis irrational. -/\ntheorem irrational_nrt_of_n_not_dvd_multiplicity {x : ℝ} (n : ℕ) {m : ℤ} (hm : m ≠ 0) (p : ℕ)\n    [hp : Fact p.Prime] (hxr : x ^ n = m)\n    (hv : multiplicity (p : ℤ) m % n ≠ 0) :\n    Irrational x := by\n  rcases Nat.eq_zero_or_pos n with (rfl | hnpos)\n  · rw [eq_comm, pow_zero, ← Int.cast_one, Int.cast_inj] at hxr\n    simp [hxr, multiplicity_of_one_right (mt isUnit_iff_dvd_one.1\n      (mt Int.natCast_dvd_natCast.1 hp.1.not_dvd_one))] at hv\n  refine irrational_nrt_of_notint_nrt _ _ hxr ?_ hnpos\n  rintro ⟨y, rfl⟩\n  rw [← Int.cast_pow, Int.cast_inj] at hxr\n  subst m\n  have : y ≠ 0 := by rintro rfl; rw [zero_pow hnpos.ne'] at hm; exact hm rfl\n  rw [(Int.finiteMultiplicity_iff.2 ⟨by simp [hp.1.ne_one], this⟩).multiplicity_pow\n    (Nat.prime_iff_prime_int.1 hp.1), Nat.mul_mod_right] at hv\n  exact hv rfl\n\ntheorem irrational_sqrt_of_multiplicity_odd (m : ℤ) (hm : 0 < m) (p : ℕ) [hp : Fact p.Prime]\n    (Hpv : multiplicity (p : ℤ) m % 2 = 1) :\n    Irrational (√m) :=\n  @irrational_nrt_of_n_not_dvd_multiplicity _ 2 _ (Ne.symm (ne_of_lt hm)) p hp\n    (sq_sqrt (Int.cast_nonneg hm.le)) (by rw [Hpv]; exact one_ne_zero)\n\n@[simp] theorem not_irrational_zero : ¬Irrational 0 := not_not_intro ⟨0, Rat.cast_zero⟩\n@[simp] theorem not_irrational_one : ¬Irrational 1 := not_not_intro ⟨1, Rat.cast_one⟩\n\ntheorem irrational_sqrt_ratCast_iff_of_nonneg {q : ℚ} (hq : 0 ≤ q) :\n    Irrational (√q) ↔ ¬IsSquare q := by\n  refine Iff.not (?_ : Exists _ ↔ Exists _)\n  constructor\n  · rintro ⟨y, hy⟩\n    refine ⟨y, Rat.cast_injective (α := ℝ) ?_⟩\n    rw [Rat.cast_mul, hy, mul_self_sqrt (Rat.cast_nonneg.2 hq)]\n  · rintro ⟨q', rfl⟩\n    exact ⟨|q'|, mod_cast (sqrt_mul_self_eq_abs q').symm⟩\n\ntheorem irrational_sqrt_ratCast_iff {q : ℚ} :\n    Irrational (√q) ↔ ¬IsSquare q ∧ 0 ≤ q := by\n  obtain hq | hq := le_or_gt 0 q\n  · simp_rw [irrational_sqrt_ratCast_iff_of_nonneg hq, and_iff_left hq]\n  · rw [sqrt_eq_zero_of_nonpos (Rat.cast_nonpos.2 hq.le)]\n    simp_rw [not_irrational_zero, false_iff, not_and, not_le, hq, implies_true]\n\ntheorem irrational_sqrt_intCast_iff_of_nonneg {z : ℤ} (hz : 0 ≤ z) :\n    Irrational (√z) ↔ ¬IsSquare z := by\n  rw [← Rat.isSquare_intCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg (mod_cast hz),\n    Rat.cast_intCast]\n\ntheorem irrational_sqrt_intCast_iff {z : ℤ} :\n    Irrational (√z) ↔ ¬IsSquare z ∧ 0 ≤ z := by\n  rw [← Rat.cast_intCast, irrational_sqrt_ratCast_iff, Rat.isSquare_intCast_iff,\n    Int.cast_nonneg_iff]\n\ntheorem irrational_sqrt_natCast_iff {n : ℕ} : Irrational (√n) ↔ ¬IsSquare n := by\n  rw [← Rat.isSquare_natCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg n.cast_nonneg,\n    Rat.cast_natCast]\n\ntheorem irrational_sqrt_ofNat_iff {n : ℕ} [n.AtLeastTwo] :\n    Irrational √(ofNat(n)) ↔ ¬IsSquare ofNat(n) :=\n  irrational_sqrt_natCast_iff\n\ntheorem Nat.Prime.irrational_sqrt {p : ℕ} (hp : Nat.Prime p) : Irrational (√p) :=\n  irrational_sqrt_natCast_iff.mpr hp.not_isSquare\n\n/-- **Irrationality of the Square Root of 2** -/\ntheorem irrational_sqrt_two : Irrational (√2) := by\n  simpa using Nat.prime_two.irrational_sqrt\n\n/--\nThis can be used as\n```lean\nunseal Nat.sqrt.iter in\nexample : Irrational √24 := by decide\n```\n-/\ninstance {n : ℕ} [n.AtLeastTwo] : Decidable (Irrational √(ofNat(n))) :=\n  decidable_of_iff' _ irrational_sqrt_ofNat_iff\n\ninstance (n : ℕ) : Decidable (Irrational (√n)) :=\n  decidable_of_iff' _ irrational_sqrt_natCast_iff\n\ninstance (z : ℤ) : Decidable (Irrational (√z)) :=\n  decidable_of_iff' _ irrational_sqrt_intCast_iff\n\ninstance (q : ℚ) : Decidable (Irrational (√q)) :=\n  decidable_of_iff' _ irrational_sqrt_ratCast_iff\n\n/-!\n### Dot-style operations on `Irrational`\n\n#### Coercion of a rational/integer/natural number is not irrational\n-/\n\n\nnamespace Irrational\n\nvariable {x : ℝ}\n\n/-!\n#### Irrational number is not equal to a rational/integer/natural number\n-/\n\n\ntheorem ne_rat (h : Irrational x) (q : ℚ) : x ≠ q := fun hq => h ⟨q, hq.symm⟩\n\ntheorem ne_int (h : Irrational x) (m : ℤ) : x ≠ m := by\n  rw [← Rat.cast_intCast]\n  exact h.ne_rat _\n\ntheorem ne_nat (h : Irrational x) (m : ℕ) : x ≠ m :=\n  h.ne_int m\n\ntheorem ne_zero (h : Irrational x) : x ≠ 0 := mod_cast h.ne_nat 0\n\ntheorem ne_one (h : Irrational x) : x ≠ 1 := by simpa only [Nat.cast_one] using h.ne_nat 1\n\n@[simp] theorem ne_ofNat (h : Irrational x) (n : ℕ) [n.AtLeastTwo] : x ≠ ofNat(n) :=\n  h.ne_nat n\n\nend Irrational\n\n@[simp]\ntheorem Rat.not_irrational (q : ℚ) : ¬Irrational q := fun h => h ⟨q, rfl⟩\n\n@[simp]\ntheorem Int.not_irrational (m : ℤ) : ¬Irrational m := fun h => h.ne_int m rfl\n\n@[simp]\ntheorem Nat.not_irrational (m : ℕ) : ¬Irrational m := fun h => h.ne_nat m rfl\n\n@[simp] theorem not_irrational_ofNat (n : ℕ) [n.AtLeastTwo] : ¬Irrational ofNat(n) :=\n  n.not_irrational\nnamespace Irrational\n\nvariable (q : ℚ) {x y : ℝ}\n\n/-!\n#### Addition of rational/integer/natural numbers\n-/\n\n\n/-- If `x + y` is irrational, then at least one of `x` and `y` is irrational. -/\n\nTarget:\ntheorem add_cases : Irrational (x + y) → Irrational x ∨ Irrational y :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  delta Irrational\n  contrapose!\n  rintro ⟨⟨rx, rfl⟩, ⟨ry, rfl⟩⟩\n  exact ⟨rx + ry, cast_add rx ry⟩","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Real","family_id":"add_cases","file_id":"mathlib/Mathlib/NumberTheory/Real/Irrational.lean","sample_id":"d0d0f38b9e3e428b2ae38db46c5384471141fd719dfd1fd904da355d5cb234fc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e0cdaf3a4b43a4f8fc2adc6c6eee964e632fc1aa70e6d8ebca268efd1191ebba","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"56b98b81c35c609877ff6bd3d13a65d0676190defb990145e8ea12fdb67e7fa4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3c58404f57528d0f09544fa0d6422c11989243f1a83384b09f5595af7aef1431","source_sha256":"59e5e41190d4901dfcf9839eb63e1593f695e949ec1d2d7f0996228faa2acac0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Presieve.EffectiveEpimorphic.iff_forall_isSheafFor_yoneda] at hR\n  rw [← isSheafFor_comp_uliftFunctor_iff]\n  refine Presieve.isSheafFor_iso (F ⋙ uliftFunctor.{v}).uliftYonedaReprXIso ?_\n  dsimp only [uliftYoneda, Functor.comp_obj, Functor.whiskeringRight_obj_obj]\n  rw [isSheafFor_comp_uliftFunctor_iff]\n  exact hR _","hard_negative":true,"metrics":{"chosen_tokens":53,"rejected_tokens":7,"token_jaccard":0.147059,"token_length_ratio":0.132075},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"0a52e07b7cf84d666234eba2ae7ee1a32165323848b5c641186745b70b5e6d89","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Sites.SheafOfTypes\npublic import Mathlib.CategoryTheory.EffectiveEpi.Basic\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2023 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\n/-!\n\n# Effective epimorphic sieves\n\nWe define the notion of effective epimorphic (pre)sieves and provide some API for relating the\nnotion with the notions of effective epimorphism and effective epimorphic family.\n\nMore precisely, if `f` is a morphism, then `f` is an effective epi if and only if the sieve\nit generates is effective epimorphic; see `CategoryTheory.Sieve.effectiveEpimorphic_singleton`.\nThe analogous statement for a family of morphisms is in the theorem\n`CategoryTheory.Sieve.effectiveEpimorphic_family`.\n\n-/\n\nuniverse w v u\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Limits\n\nvariable {C : Type u} [Category.{v} C]\n\n/-- A sieve is effective epimorphic if the associated cocone is a colimit cocone. -/\ndef Sieve.EffectiveEpimorphic {X : C} (S : Sieve X) : Prop :=\n  Nonempty (IsColimit (S : Presieve X).cocone)\n\n/-- A presieve is effective epimorphic if the cocone associated to the sieve it generates\nis a colimit cocone. -/\nabbrev Presieve.EffectiveEpimorphic {X : C} (S : Presieve X) : Prop :=\n  (Sieve.generate S).EffectiveEpimorphic\n\n/--\nThe sieve of morphisms which factor through a given morphism `f`.\nThis is equal to `Sieve.generate (Presieve.singleton f)`, but has\nmore convenient definitional properties.\n-/\ndef Sieve.generateSingleton {X Y : C} (f : Y ⟶ X) : Sieve X where\n  arrows Z g := ∃ (e : Z ⟶ Y), e ≫ f = g\n  downward_closed := by\n    rintro W Z g ⟨e, rfl⟩ q\n    exact ⟨q ≫ e, by simp⟩\n\nlemma Sieve.generateSingleton_eq {X Y : C} (f : Y ⟶ X) :\n    Sieve.generate (Presieve.singleton f) = Sieve.generateSingleton f := by\n  ext Z g\n  constructor\n  · rintro ⟨W, i, p, ⟨⟩, rfl⟩\n    exact ⟨i, rfl⟩\n  · rintro ⟨g, h⟩\n    exact ⟨Y, g, f, ⟨⟩, h⟩\n\nlemma Sieve.EffectiveEpimorphic.iff_forall_isSheafFor_yoneda {X : C} (S : Sieve X) :\n    S.EffectiveEpimorphic ↔ ∀ Y, S.arrows.IsSheafFor (yoneda.obj Y) :=\n  S.forallYonedaIsSheaf_iff_colimit.symm\n\nlemma Presieve.EffectiveEpimorphic.iff_forall_isSheafFor_yoneda {X : C} (R : Presieve X) :\n    R.EffectiveEpimorphic ↔ ∀ Y, R.IsSheafFor (yoneda.obj Y) := by\n  simp_rw [Presieve.isSheafFor_iff_generate R,\n    Presieve.EffectiveEpimorphic, Sieve.EffectiveEpimorphic.iff_forall_isSheafFor_yoneda]\n\nset_option backward.defeqAttrib.useBackward true in\n\nTarget:\nlemma Presieve.EffectiveEpimorphic.isSheafFor_of_isRepresentable {X : C} {R : Presieve X}\n    (hR : R.EffectiveEpimorphic) (F : Cᵒᵖ ⥤ Type w) [F.IsRepresentable] :\n    R.IsSheafFor F :=\n\nProof body:\n","rejected":"by\n  exact Presieve.EffectiveEpimorphic.isSheafFor_of_isRepresentable","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"b171306b2273d77e4ffbabf0661f638c1a433e1edb980c40120f8d5ac80db39f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Sites","family_id":"presieve","file_id":"mathlib/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean","sample_id":"3c58404f57528d0f09544fa0d6422c11989243f1a83384b09f5595af7aef1431"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5b4476fdc1c0c0f51888b9b0f5b1dfa03fd37018bf6998a42fadc18efa75285e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f2d3f78b3cc280c53094018699cfb4d8122ad93d85fbb1e3b31a5ac4877cc8b7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3681767d9aba4e16656d0211cddaff6efe2e454378f2ebc6b91e0ff3858dab3f","source_sha256":"7e0891f362b30145c7c3c4e0ec7d7143a1d3e97fd040572f98d49bd047596de1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [quasicategory_iff_hasLiftingProperty.{u} _ terminalIsTerminal, innerFibration_iff]\n  exact ⟨fun h _ _ _ ⟨i, h0, hn⟩ ↦ h h0 hn,\n    fun h _ _ h0 hn ↦ h _ (horn_ι_mem_innerHornInclusions h0 hn)⟩","hard_negative":false,"metrics":{"chosen_tokens":49,"rejected_tokens":53,"token_jaccard":0.9,"token_length_ratio":1.081633},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"0a5ee35651cecb0719c3fb631c61bfd0f3b663e500e036ec908c65f8c59a54df","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.Quasicategory.Basic\n\nNamespace:\nSSet\n\nLocal context:\n/-\nCopyright (c) 2026 Jack McKoen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jack McKoen\n-/\n/-!\n# Inner fibrations\n\nInner fibrations of simplicial sets are the morphisms in `SSet` which have the right lifting\nproperty with respect to all inner horn inclusions.\n\nBasic consequences of inner fibrations with respect to the definition of quasi-categories are\nformalized.\n\n-/\n\npublic section\n\nopen CategoryTheory MorphismProperty Simplicial Limits\n\nuniverse u\n\nnamespace SSet\n\n/-- The family of morphisms in `SSet` which consists of inner horn inclusions\n`Λ[n, i].ι : Λ[n, i] ⟶ Δ[n]` (for `0 < i < n`). -/\ninductive innerHornInclusions : MorphismProperty SSet.{u} where\n  | intro {n : ℕ} (i : Fin (n + 3)) (h0 : 0 < i) (hn : i < Fin.last (n + 2)) :\n    innerHornInclusions Λ[n + 2, i].ι\n\nlemma horn_ι_mem_innerHornInclusions {n : ℕ} {i : Fin (n + 1)}\n    (h0 : 0 < i) (hn : i < Fin.last n) : innerHornInclusions (horn.{u} n i).ι := by\n  obtain _ | _ | k := n\n  · grind\n  · grind\n  · exact ⟨i, h0, hn⟩\n\nlemma innerHornInclusions_eq_iSup :\n    innerHornInclusions.{u} =\n    ⨆ n, .ofHoms (fun p : {p : Fin (n + 3) // 0 < p ∧ p < Fin.last (n + 2)} ↦ Λ[n + 2, p].ι) := by\n  ext\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · obtain @⟨n, i, h0, hn⟩ := h\n    simp only [iSup_iff, ofHoms_iff, Subtype.exists, exists_prop]\n    use n, i\n  · simp only [iSup_iff, ofHoms_iff] at h\n    obtain ⟨n, ⟨i, h0, hn⟩, _, _⟩ := h\n    exact horn_ι_mem_innerHornInclusions h0 hn\n\nlemma innerHornInclusions_le_J : innerHornInclusions.{u} ≤ modelCategoryQuillen.J :=\n  fun _ _ _ ⟨_, _, _⟩ ↦ modelCategoryQuillen.horn_ι_mem_J ..\n\nlemma innerHornInclusions_le_monomorphisms :\n    innerHornInclusions.{u} ≤ monomorphisms SSet :=\n  innerHornInclusions_le_J.trans modelCategoryQuillen.J_le_monomorphisms\n\n/-- The inner fibrations are the morphisms which have the right lifting property\nwith respect to inner horn inclusions. -/\n@[expose, kerodon 01BA]\ndef innerFibrations : MorphismProperty SSet.{u} := innerHornInclusions.rlp\nderiving IsMultiplicative, RespectsIso, IsStableUnderBaseChange,\n  IsStableUnderRetracts\n\n/-- A morphism `q` satisfies `[InnerFibration q]` if it belongs to `innerFibrations`. -/\n@[mk_iff]\nclass InnerFibration {X Y : SSet} (q : X ⟶ Y) : Prop where\n  mem : innerFibrations q\n\nlemma mem_innerFibrations {X Y : SSet} (q : X ⟶ Y) [InnerFibration q] : innerFibrations q :=\n  InnerFibration.mem\n\nTarget:\nlemma quasicategory_iff_innerFibration (X : SSet.{u}) :\n    Quasicategory X ↔ InnerFibration (terminal.from X) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [quasicategory_iff_hasLiftingProperty.{u} _ terminalIsTerminal, innerFibration_iff]\n  exact ⟨fun h _ _ _ ⟨i, h0, hn⟩ ↦ h h0 hn,\n    fun h _ _ h0 hn ↦ h _ (horn_ι_mem_innerHornInclusions h0 hn)⟩","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/Quasicategory","family_id":"quasicategory_iff_innerfibration","file_id":"mathlib/Mathlib/AlgebraicTopology/Quasicategory/InnerFibration.lean","sample_id":"3681767d9aba4e16656d0211cddaff6efe2e454378f2ebc6b91e0ff3858dab3f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"db5676c2a4baaaa8601683c727e173d0b3b0fe87aa92feb60828f3a50296fa53","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2679434db9e07f43c58fe299cdd22e963388d3f2fb60235d26d3f8376cc70086","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ba002fff735a1eceaabf1a8e0dbcc41b8470b3783bbd4a79e63a536436ef24f3","source_sha256":"cb81d90c8dbabbe33b1b9939fcf99c692948a640de2613d83a1e798cde8653e5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hp : ((f.support : Set G).smulAntidiagonal (Function.support x) p).Finite :=\n    Set.SMulAntidiagonal.finite_of_finite_fst f.support.finite_toSet x.support p\n  set s : Set (G × P) := ↑(Finset.SMulAntidiagonal p hp)\n  have h₁ : s.InjOn Prod.fst := fun _ h₁ _ h₂ h ↦ by\n    rw [Finset.mem_coe, mem_smulAntidiagonal_of_group] at h₁ h₂\n    aesop\n  have h₂ : s.MapsTo Prod.fst ↑f.support := fun g hg ↦ by aesop\n  have h₃ (g : G) (hg : g ∈ f.support) (hgn : g ∉ Prod.fst '' s) : f g • x (g⁻¹ • p) = 0 := by\n    obtain (h | h) : f g = 0 ∨ ∀ q, ¬ x q = 0 → ¬g • q = p := by aesop\n    · simp [h]\n    · have := h (g⁻¹ • p)\n      aesop\n  rw [smul_eq, Finset.sum_of_injOn Prod.fst h₁ h₂ h₃]\n  aesop","hard_negative":true,"metrics":{"chosen_tokens":224,"rejected_tokens":3,"token_jaccard":0.014493,"token_length_ratio":0.013393},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"0a6e443d072d5bc8ede2abaa1c80c0f4fafce1fd08c414a38f1bc5be8fb249ab","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MonoidAlgebra.Defs\npublic import Mathlib.Data.Finset.SMulAntidiagonal\n\nNamespace:\nMonoidAlgebra\n\nLocal context:\n/-\nCopyright (c) 2025 Scott Carnahan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Carnahan\n-/\n/-!\n# Scalar multiplication by (additive) monoid rings on formal functions.\nGiven sets `G` and `P`, with a left-cancellative scalar-multiplication (or vector-addition) of `G`\non `P`, together with a module `V` over a semiring `R`, we define a convolution action of the monoid\nalgebra `R[G]` on the set of functions `P → V`.\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nvariable {G P R V : Type*}\n\nnamespace MonoidAlgebra\n\n@[to_additive]\ntheorem mem_smulAntidiagonal_of_group [Group G] [MulAction G P] [Semiring R] [Zero V]\n    (f : R[G]) (x : P → V) (p : P) (gh : G × P) :\n    gh ∈ Finset.SMulAntidiagonal p\n      (Set.SMulAntidiagonal.finite_of_finite_fst f.support.finite_toSet x.support p) ↔\n      f gh.1 ≠ 0 ∧ x gh.2 ≠ 0 ∧ gh.2 = gh.1⁻¹ • p := by\n  rw [Finset.mem_smulAntidiagonal, eq_inv_smul_iff, Function.mem_support, Finset.mem_coe,\n    Finsupp.mem_support_iff]\n\n/-- A convolution-type scalar multiplication of the monoid algebra on the set of formal\nfunctions. -/\n@[to_additive (dont_translate := R) /-- A convolution-type scalar multiplication of the additive\nmonoid algebra on the set of formal functions. -/]\nscoped instance [SMul G P] [IsLeftCancelSMul G P] [Semiring R] [AddCommMonoid V]\n    [SMulWithZero R V] :\n    SMul (R[G]) (P → V) where\n  smul f x p := ∑ gh ∈ Finset.SMulAntidiagonal p\n    (Set.SMulAntidiagonal.finite_of_finite_fst f.support.finite_toSet x.support p), f gh.1 • x gh.2\n\n@[to_additive (dont_translate := R) smul_eq]\ntheorem smul_eq [SMul G P] [IsLeftCancelSMul G P] [Semiring R] [AddCommMonoid V] [SMulWithZero R V]\n    (f : R[G]) (x : P → V) (p : P)\n    (hp : ((f.support : Set G).smulAntidiagonal (Function.support x) p).Finite :=\n      Set.SMulAntidiagonal.finite_of_finite_fst f.support.finite_toSet x.support p) :\n    (f • x) p = ∑ gh ∈ Finset.SMulAntidiagonal p hp, f gh.1 • x gh.2 :=\n  rfl\n\n@[to_additive (dont_translate := R) smul_apply_addAction]\n\nTarget:\ntheorem smul_apply_mulAction [Group G] [MulAction G P] [Semiring R] [AddCommMonoid V]\n    [SMulWithZero R V] (f : MonoidAlgebra R G) (x : P → V) (p : P) :\n    (f • x) p = ∑ i ∈ f.support, (f i) • x (i⁻¹ • p) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_ba002fff735a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"4b27115d1268c1c427d90643d07e532a568da181ed0905bcd4e8b2dd2373cc44","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/MonoidAlgebra","family_id":"smul_apply_mulaction","file_id":"mathlib/Mathlib/Algebra/MonoidAlgebra/PointwiseSMul.lean","sample_id":"ba002fff735a1eceaabf1a8e0dbcc41b8470b3783bbd4a79e63a536436ef24f3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8a67a77843a75ba583cb43ac0e535963aaf96ee59db334d6516f871f6e4da8bd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"09f07bfd57e55788ddf24145640ae69665e081f5b1afc8e5c77d86db298399ca","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9f9a70e78a68c5afe844cbabb734e8a42046f9373491df56a598c7ff77b730eb","source_sha256":"bdcd2022da453373ffbdfe5bdbc273db0a9656383ede2bdc6575e68ee03c9aa0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← @isBounded_univ, isBounded_induced, image_univ]","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.272727},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"0a78cfc513926591aed12ca62d4a6439f8a4e598a136a02008a0952b537d8ca8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.TypeTags.Basic\npublic import Mathlib.Topology.Bornology.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Bornology structure on products and subtypes\n\nIn this file we define `Bornology` and `BoundedSpace` instances on `α × β`, `Π i, X i`, and\n`{x // p x}`. We also prove basic lemmas about `Bornology.cobounded` and `Bornology.IsBounded`\non these types.\n-/\n\npublic section\n\n\nopen Set Filter Bornology Function\n\nopen Filter\n\nvariable {α β ι : Type*} {X : ι → Type*} [Bornology α] [Bornology β]\n  [∀ i, Bornology (X i)]\n\ninstance Prod.instBornology : Bornology (α × β) where\n  cobounded := (cobounded α).coprod (cobounded β)\n  le_cofinite :=\n    @coprod_cofinite α β ▸ coprod_mono ‹Bornology α›.le_cofinite ‹Bornology β›.le_cofinite\n\ninstance Pi.instBornology : Bornology (∀ i, X i) where\n  cobounded := Filter.coprodᵢ fun i => cobounded (X i)\n  le_cofinite := iSup_le fun _ ↦ (comap_mono (Bornology.le_cofinite _)).trans (comap_cofinite_le _)\n\n/-- Inverse image of a bornology. -/\nabbrev Bornology.induced {α β : Type*} [Bornology β] (f : α → β) : Bornology α where\n  cobounded := comap f (cobounded β)\n  le_cofinite := (comap_mono (Bornology.le_cofinite β)).trans (comap_cofinite_le _)\n\ninstance {p : α → Prop} : Bornology (Subtype p) :=\n  Bornology.induced (Subtype.val : Subtype p → α)\n\nnamespace Bornology\n\n/-!\n### Bounded sets in `α × β`\n-/\n\n\ntheorem cobounded_prod : cobounded (α × β) = (cobounded α).coprod (cobounded β) :=\n  rfl\n\ntheorem isBounded_image_fst_and_snd {s : Set (α × β)} :\n    IsBounded (Prod.fst '' s) ∧ IsBounded (Prod.snd '' s) ↔ IsBounded s :=\n  compl_mem_coprod.symm\n\nlemma IsBounded.image_fst {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.fst '' s) :=\n  (isBounded_image_fst_and_snd.2 hs).1\n\nlemma IsBounded.image_snd {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.snd '' s) :=\n  (isBounded_image_fst_and_snd.2 hs).2\n\nvariable {s : Set α} {t : Set β} {S : ∀ i, Set (X i)}\n\ntheorem IsBounded.fst_of_prod (h : IsBounded (s ×ˢ t)) (ht : t.Nonempty) : IsBounded s :=\n  fst_image_prod s ht ▸ h.image_fst\n\ntheorem IsBounded.snd_of_prod (h : IsBounded (s ×ˢ t)) (hs : s.Nonempty) : IsBounded t :=\n  snd_image_prod hs t ▸ h.image_snd\n\ntheorem IsBounded.prod (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s ×ˢ t) :=\n  isBounded_image_fst_and_snd.1\n    ⟨hs.subset <| fst_image_prod_subset _ _, ht.subset <| snd_image_prod_subset _ _⟩\n\ntheorem isBounded_prod_of_nonempty (hne : Set.Nonempty (s ×ˢ t)) :\n    IsBounded (s ×ˢ t) ↔ IsBounded s ∧ IsBounded t :=\n  ⟨fun h => ⟨h.fst_of_prod hne.snd, h.snd_of_prod hne.fst⟩, fun h => h.1.prod h.2⟩\n\ntheorem isBounded_prod : IsBounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ IsBounded s ∧ IsBounded t := by\n  rcases s.eq_empty_or_nonempty with (rfl | hs); · simp\n  rcases t.eq_empty_or_nonempty with (rfl | ht); · simp\n  simp only [hs.ne_empty, ht.ne_empty, isBounded_prod_of_nonempty (hs.prod ht), false_or]\n\ntheorem isBounded_prod_self : IsBounded (s ×ˢ s) ↔ IsBounded s := by\n  rcases s.eq_empty_or_nonempty with (rfl | hs); · simp\n  exact (isBounded_prod_of_nonempty (hs.prod hs)).trans and_self_iff\n\n/-!\n### Bounded sets in `Π i, X i`\n-/\n\n\ntheorem cobounded_pi : cobounded (∀ i, X i) = Filter.coprodᵢ fun i => cobounded (X i) :=\n  rfl\n\ntheorem forall_isBounded_image_eval_iff {s : Set (∀ i, X i)} :\n    (∀ i, IsBounded (eval i '' s)) ↔ IsBounded s :=\n  compl_mem_coprodᵢ.symm\n\nlemma IsBounded.image_eval {s : Set (∀ i, X i)} (hs : IsBounded s) (i : ι) :\n    IsBounded (eval i '' s) :=\n  forall_isBounded_image_eval_iff.2 hs i\n\ntheorem IsBounded.pi (h : ∀ i, IsBounded (S i)) : IsBounded (pi univ S) :=\n  forall_isBounded_image_eval_iff.1 fun i => (h i).subset eval_image_univ_pi_subset\n\ntheorem isBounded_pi_of_nonempty (hne : (pi univ S).Nonempty) :\n    IsBounded (pi univ S) ↔ ∀ i, IsBounded (S i) :=\n  ⟨fun H i => @eval_image_univ_pi _ _ _ i hne ▸ forall_isBounded_image_eval_iff.2 H i, IsBounded.pi⟩\n\ntheorem isBounded_pi : IsBounded (pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ i, IsBounded (S i) := by\n  by_cases hne : ∃ i, S i = ∅\n  · simp [hne, univ_pi_eq_empty_iff.2 hne]\n  · simp only [hne, false_or]\n    simp only [not_exists, ← nonempty_iff_ne_empty, ← univ_pi_nonempty_iff] at hne\n    exact isBounded_pi_of_nonempty hne\n\n/-!\n### Bounded sets in `{x // p x}`\n-/\n\n\ntheorem isBounded_induced {α β : Type*} [Bornology β] {f : α → β} {s : Set α} :\n    @IsBounded α (Bornology.induced f) s ↔ IsBounded (f '' s) :=\n  compl_mem_comap\n\ntheorem isBounded_image_subtype_val {p : α → Prop} {s : Set { x // p x }} :\n    IsBounded (Subtype.val '' s) ↔ IsBounded s :=\n  isBounded_induced.symm\n\nend Bornology\n\n/-!\n### Bounded spaces\n-/\n\n\nopen Bornology\n\ninstance [BoundedSpace α] [BoundedSpace β] : BoundedSpace (α × β) := by\n  simp [← cobounded_eq_bot_iff, cobounded_prod]\n\ninstance [∀ i, BoundedSpace (X i)] : BoundedSpace (∀ i, X i) := by\n  simp [← cobounded_eq_bot_iff, cobounded_pi]\n\nTarget:\ntheorem boundedSpace_induced_iff {α β : Type*} [Bornology β] {f : α → β} :\n    @BoundedSpace α (Bornology.induced f) ↔ IsBounded (range f) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_9f9a70e78a68","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"796a0d5b85da265f6eb593b01f9e4e208d0fada9915216503476680da4badb66","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Bornology","family_id":"boundedspace_induced_iff","file_id":"mathlib/Mathlib/Topology/Bornology/Constructions.lean","sample_id":"9f9a70e78a68c5afe844cbabb734e8a42046f9373491df56a598c7ff77b730eb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c8b268100380170538174b78ca7f2045ab5992f487a935451a91e1cbc0d99b37","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"81a0159374a335acfc3a3be57440b1fb6c4f215c8d52062ffeefc8554afc5a68","source_sha256":"da1f22be161c5bc7cbd676252bea6b38bb4aa0497872bcdf52a178b9d4951829","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h₁ : a i < supOfSeq a :=\n    Nat.lt_add_one_iff.mpr (le_max_of_le_right <| Finset.le_sup (by simp))\n  have h₂ : supOfSeq a ≤ (i + 1) * (supOfSeq a)! + 1 :=\n    le_trans (self_le_factorial _) (le_trans (Nat.le_mul_of_pos_left (supOfSeq a)! (succ_pos i))\n      (le_add_right _ _))\n  simpa only [coprimes] using lt_of_lt_of_le h₁ h₂","hard_negative":false,"metrics":{"chosen_tokens":87,"rejected_tokens":2,"token_jaccard":0.02439,"token_length_ratio":0.022989},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"0a8487e37be145dbfc31c1579c657fe78361881899b9b1eedfba5163aa857af1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.ModEq\npublic import Mathlib.Data.Nat.ChineseRemainder\npublic import Mathlib.Data.Nat.Prime.Defs\npublic import Mathlib.Data.Nat.Pairing\npublic import Mathlib.Order.Fin.Basic\npublic import Mathlib.Data.Finset.Lattice.Fold\npublic import Mathlib.Data.Fintype.Basic\npublic import Mathlib.Data.Nat.Factorial.Basic\n\nNamespace:\nNat\n\nLocal context:\n/-\nCopyright (c) 2023 Shogo Saito. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Shogo Saito. Adapted for mathlib by Hunter Monroe\n-/\n/-!\n# Gödel's Beta Function Lemma\n\nThis file proves Gödel's Beta Function Lemma, used to prove the First Incompleteness Theorem. It\npermits quantification over finite sequences of natural numbers in formal theories of arithmetic.\nThis Beta Function has no connection with the unrelated Beta Function defined in analysis. Note\nthat `Nat.beta` and `Nat.unbeta` provide similar functionality to `Encodable.encodeList` and\n`Encodable.decodeList`. We define these separately, because it is easier to prove that `Nat.beta`\nand `Nat.unbeta` are arithmetically definable, and this is hard to prove that for\n`Encodable.encodeList` and `Encodable.decodeList` directly. The arithmetic\ndefinability is needed for the proof of the First Incompleteness Theorem.\n\n## Main result\n\n- `beta_unbeta_coe`: Gödel's Beta Function Lemma.\n\n## Implementation note\n\nThis code is a step towards eventually including a proof of Gödel's First Incompleteness Theorem\nand other key results from the repository https://github.com/iehality/lean4-logic.\n\n## References\n\n* [R. Kaye, *Models of Peano arithmetic*][kaye1991]\n* <https://en.wikipedia.org/wiki/G%C3%B6del%27s_%CE%B2_function>\n\n## Tags\n\nGödel, beta function\n-/\n\n@[expose] public section\n\nnamespace Nat\n\nlemma coprime_mul_succ {n m a} (ha : m - n ∣ a) : Coprime (n * a + 1) (m * a + 1) :=\n  Nat.coprime_of_dvd fun p pp hn hm => by\n    have : p ∣ (m - n) * a := by\n      simpa [Nat.succ_sub_succ, ← Nat.mul_sub_right_distrib] using\n        Nat.dvd_sub hm hn\n    have : p ∣ a := by\n      rcases (Nat.Prime.dvd_mul pp).mp this with (hp | hp)\n      · exact Nat.dvd_trans hp ha\n      · exact hp\n    apply pp.ne_one\n    simpa [Nat.add_sub_cancel_left] using Nat.dvd_sub hn (this.mul_left n)\n\nvariable {m : ℕ}\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nprivate def supOfSeq (a : Fin m → ℕ) : ℕ := max m (Finset.sup .univ a) + 1\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nprivate def coprimes (a : Fin m → ℕ) : Fin m → ℕ := fun i => (i + 1) * (supOfSeq a)! + 1\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n\nTarget:\nlemma coprimes_lt (a : Fin m → ℕ) (i) : a i < coprimes a i :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Godel","family_id":"coprimes_lt","file_id":"mathlib/Mathlib/Logic/Godel/GodelBetaFunction.lean","sample_id":"81a0159374a335acfc3a3be57440b1fb6c4f215c8d52062ffeefc8554afc5a68"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a329996be1763b313924738bf153d1074388b636f52e211888a4d4f3a26dea07","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a0d831455a2a6ecae7a67822f37fbbc5fcdca386afc2a04183d261185470b574","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1f97d0501e044f00fde760711297ef657cb86ce42d441db370fdf596ff19fb0c","source_sha256":"155010ab3058b3fe6da475eb20a319e6c48a44ac24aa3149ecf584265954195c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using w\n\n-- It would be nice to mark this as `aesop cases` if\n-- https://github.com/leanprover-community/aesop/issues/59\n-- is resolved.","hard_negative":true,"metrics":{"chosen_tokens":44,"rejected_tokens":2,"token_jaccard":0.032258,"token_length_ratio":0.045455},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"0a8e903b47393fd7ef91b9f2c671e4129acdd81baabebf6f649eae768f838692","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Logic.Equiv.Defs\npublic import Mathlib.Tactic.MkIffOfInductiveProp\npublic import Mathlib.Tactic.PPWithUniv\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Small types\n\nA type is `w`-small if there exists an equivalence to some `S : Type w`.\n\nWe provide a noncomputable model `Shrink α : Type w`, and `equivShrink α : α ≃ Shrink α`.\n\nA subsingleton type is `w`-small for any `w`.\n\nIf `α ≃ β`, then `Small.{w} α ↔ Small.{w} β`.\n\nSee `Mathlib/Logic/Small/Basic.lean` for further instances and theorems.\n-/\n\n@[expose] public section\n\nuniverse u w v v'\n\n/-- A type is `Small.{w}` if there exists an equivalence to some `S : Type w`.\n-/\n-- After https://github.com/leanprover/lean4/pull/12286 and\n-- https://github.com/leanprover/lean4/pull/12423: `v` is a true output (determined by `α`),\n-- but we need the attribute to prevent `w` from also being treated as output.\n-- See Note [universe output parameters and typeclass caching].\n@[univ_out_params v, mk_iff, pp_with_univ]\nclass Small (α : Type v) : Prop where\n  /-- If a type is `Small.{w}`, then there exists an equivalence with some `S : Type w` -/\n  equiv_small : ∃ S : Type w, Nonempty (α ≃ S)\n\n/-- Constructor for `Small α` from an explicit witness type and equivalence.\n-/\ntheorem Small.mk' {α : Type v} {S : Type w} (e : α ≃ S) : Small.{w} α :=\n  ⟨⟨S, ⟨e⟩⟩⟩\n\n/-- An arbitrarily chosen model in `Type w` for a `w`-small type.\n-/\n@[pp_with_univ, no_expose]\ndef Shrink (α : Type v) [Small.{w} α] : Type w :=\n  Classical.choose (@Small.equiv_small α _)\n\n/-- The noncomputable equivalence between a `w`-small type and a model.\n-/\n@[no_expose]\nnoncomputable def equivShrink (α : Type v) [Small.{w} α] : α ≃ Shrink α :=\n  Nonempty.some (Classical.choose_spec (@Small.equiv_small α _))\n\n@[ext]\n\nTarget:\ntheorem Shrink.ext {α : Type v} [Small.{w} α] {x y : Shrink α}\n    (w : (equivShrink _).symm x = (equivShrink _).symm y) : x = y :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_1f97d0501e04","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"74a9dff03c20ebc22e4a21a286c8f7662a7f74a8f96225fa1f95e61c1c8d2fb6","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Small","family_id":"shrink","file_id":"mathlib/Mathlib/Logic/Small/Defs.lean","sample_id":"1f97d0501e044f00fde760711297ef657cb86ce42d441db370fdf596ff19fb0c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"49bdfcf01bbb1a8c4c2fb91482006ee7e27f5a1dbdbae3ae54b463c762ee66e9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"65e0c46ab80804e8a5cd071bf84cbfc579f19ac01cda5e58e506b75975df165e","source_sha256":"35550c7de96df6629396d8c6f9793bd06aaa1489dd49abd1f5374fad7d4fd60d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext X; simp [Equivalence.unit]","hard_negative":false,"metrics":{"chosen_tokens":10,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.3},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"0ac5e9a770e587ea1d3b7656c21bc964e951f5fcf37004453b8fc03ae50b47ef","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Adjunction.Basic\npublic import Mathlib.CategoryTheory.Yoneda\npublic import Mathlib.CategoryTheory.Opposites\n\nNamespace:\nCategoryTheory.Adjunction\n\nLocal context:\n/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Thomas Read, Andrew Yang\n-/\n/-!\n# Opposite adjunctions\n\nThis file contains constructions to relate adjunctions of functors to adjunctions of their\nopposites.\n\n## Tags\nadjunction, opposite, uniqueness\n-/\n\n@[expose] public section\n\n\nopen CategoryTheory\n\nuniverse v₁ v₂ u₁ u₂\n\n-- morphism levels before object levels. See note [category theory universes].\nvariable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]\n\nnamespace CategoryTheory.Adjunction\n\nattribute [local simp] homEquiv_unit homEquiv_counit\n\n/-- If `G` is adjoint to `F` then `F.unop` is adjoint to `G.unop`. -/\n@[simps]\ndef unop {F : Cᵒᵖ ⥤ Dᵒᵖ} {G : Dᵒᵖ ⥤ Cᵒᵖ} (h : G ⊣ F) : F.unop ⊣ G.unop where\n  unit := NatTrans.unop h.counit\n  counit := NatTrans.unop h.unit\n  left_triangle_components _ := Quiver.Hom.op_inj (h.right_triangle_components _)\n  right_triangle_components _ := Quiver.Hom.op_inj (h.left_triangle_components _)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- If `G` is adjoint to `F` then `F.op` is adjoint to `G.op`. -/\n@[simps]\ndef op {F : C ⥤ D} {G : D ⥤ C} (h : G ⊣ F) : F.op ⊣ G.op where\n  unit := NatTrans.op h.counit\n  counit := NatTrans.op h.unit\n  left_triangle_components _ := Quiver.Hom.unop_inj (by simp)\n  right_triangle_components _ := Quiver.Hom.unop_inj (by simp)\n\n/-- If `F` is adjoint to `G.leftOp` then `G` is adjoint to `F.leftOp`. -/\n@[simps]\ndef leftOp {F : C ⥤ Dᵒᵖ} {G : D ⥤ Cᵒᵖ} (a : F ⊣ G.leftOp) : G ⊣ F.leftOp where\n  unit := NatTrans.unop a.counit\n  counit := NatTrans.op a.unit\n  left_triangle_components X := congr($(a.right_triangle_components (.op X)).op)\n  right_triangle_components X := congr($(a.left_triangle_components X.unop).unop)\n\n/-- If `F.rightOp` is adjoint to `G` then `G.rightOp` is adjoint to `F`. -/\n@[simps]\ndef rightOp {F : Cᵒᵖ ⥤ D} {G : Dᵒᵖ ⥤ C} (a : F.rightOp ⊣ G) : G.rightOp ⊣ F where\n  unit := NatTrans.unop a.counit\n  counit := NatTrans.op a.unit\n  left_triangle_components X := congr($(a.right_triangle_components (.op X)).op)\n  right_triangle_components X := congr($(a.left_triangle_components X.unop).unop)\n\nset_option backward.defeqAttrib.useBackward true in\nlemma leftOp_eq {F : C ⥤ Dᵒᵖ} {G : D ⥤ Cᵒᵖ} (a : F ⊣ G.leftOp) :\n    a.leftOp = (opOpEquivalence D).symm.toAdjunction.comp a.op := by\n  ext X; simp [Equivalence.unit]\n\nset_option backward.defeqAttrib.useBackward true in\n\nTarget:\nlemma rightOp_eq {F : Cᵒᵖ ⥤ D} {G : Dᵒᵖ ⥤ C} (a : F.rightOp ⊣ G) :\n    a.rightOp = (opOpEquivalence D).symm.toAdjunction.comp a.op :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Adjunction","family_id":"rightop_eq","file_id":"mathlib/Mathlib/CategoryTheory/Adjunction/Opposites.lean","sample_id":"65e0c46ab80804e8a5cd071bf84cbfc579f19ac01cda5e58e506b75975df165e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4bc1dfb496047d2c2f53f0ca55577975a296dcb639bf7ace11b6e910a7a97f44","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5d913d38e4c8561f809e8527a3507548e336a321ed8ff63d6ecacdea3d5cd929","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"452212369db5a32584e53a4244413eb62217af8a259514599237694d17941ea9","source_sha256":"5f13df63ef1a33fb7d6444e769a7e9b9cf3ab09860a26a91c473157d6239794f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← SetLike.mem_coe, coe_commutant]\n  rfl","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.272727},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"0b553713d409ec94f1445d12ac0178f733a55ace1591583196d71be797f241a7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.CStarAlgebra.Classes\npublic import Mathlib.Analysis.InnerProductSpace.Adjoint\n\nNamespace:\nVonNeumannAlgebra\n\nLocal context:\n/-\nCopyright (c) 2022 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Von Neumann algebras\n\nWe give the \"abstract\" and \"concrete\" definitions of a von Neumann algebra.\nWe still have a major project ahead of us to show the equivalence between these definitions!\n\nAn abstract von Neumann algebra `WStarAlgebra M` is a C⋆ algebra with a Banach space predual,\nper Sakai (1971).\n\nA concrete von Neumann algebra `VonNeumannAlgebra H` (where `H` is a Hilbert space)\nis a \\*-closed subalgebra of bounded operators on `H` which is equal to its double commutant.\n\nWe'll also need to prove the von Neumann double commutant theorem,\nthat the concrete definition is equivalent to a \\*-closed subalgebra which is weakly closed.\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\n/-- Sakai's definition of a von Neumann algebra as a C⋆ algebra with a Banach space predual.\n\nSo that we can unambiguously talk about these \"abstract\" von Neumann algebras\nin parallel with the \"concrete\" ones (weakly closed \\*-subalgebras of B(H)),\nwe name this definition `WStarAlgebra`.\n\nNote that for now we only assert the mere existence of predual, rather than picking one.\nThis may later prove problematic, and need to be revisited.\nPicking one may cause problems with definitional unification of different instances.\nOne the other hand, not picking one means that the weak-\\* topology\n(which depends on a choice of predual) must be defined using the choice,\nand we may be unhappy with the resulting opaqueness of the definition.\n-/\nclass WStarAlgebra (M : Type u) [CStarAlgebra M] : Prop where\n  /-- There is a Banach space `X` whose dual is isometrically (conjugate-linearly) isomorphic\n  to the `WStarAlgebra`. -/\n  exists_predual :\n    ∃ (X : Type u) (_ : NormedAddCommGroup X) (_ : NormedSpace ℂ X) (_ : CompleteSpace X),\n      Nonempty (StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M)\n\n-- TODO: Without this, `VonNeumannAlgebra` times out. Why?\n/-- The double commutant definition of a von Neumann algebra,\nas a \\*-closed subalgebra of bounded operators on a Hilbert space,\nwhich is equal to its double commutant.\n\nNote that this definition is parameterised by the Hilbert space\non which the algebra faithfully acts, as is standard in the literature.\nSee `WStarAlgebra` for the abstract notion (a C⋆-algebra with Banach space predual).\n\nNote this is a bundled structure, parameterised by the Hilbert space `H`,\nrather than a typeclass on the type of elements.\nThus we can't say that the bounded operators `H →L[ℂ] H` form a `VonNeumannAlgebra`\n(although we will later construct the instance `WStarAlgebra (H →L[ℂ] H)`),\nand instead will use `⊤ : VonNeumannAlgebra H`.\n-/\nstructure VonNeumannAlgebra (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H]\n    [CompleteSpace H] extends StarSubalgebra ℂ (H →L[ℂ] H) where\n  /-- The double commutant (a.k.a. centralizer) of a `VonNeumannAlgebra` is itself. -/\n  centralizer_centralizer' : Set.centralizer (Set.centralizer carrier) = carrier\n\n/-- Consider a von Neumann algebra acting on a Hilbert space `H` as a \\*-subalgebra of `H →L[ℂ] H`.\n(That is, we forget that it is equal to its double commutant\nor equivalently that it is closed in the weak and strong operator topologies.)\n-/\nadd_decl_doc VonNeumannAlgebra.toStarSubalgebra\n\nnamespace VonNeumannAlgebra\n\nvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n\ninstance instSetLike : SetLike (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  coe S := S.carrier\n  coe_injective S T h := by obtain ⟨⟨⟨⟨⟨⟨_, _⟩, _⟩, _⟩, _⟩, _⟩, _⟩ := S; cases T; congr\n\ninstance : PartialOrder (VonNeumannAlgebra H) := .ofSetLike (VonNeumannAlgebra H) (H →L[ℂ] H)\n\nnoncomputable instance instStarMemClass : StarMemClass (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  star_mem {s} := s.star_mem'\n\ninstance instSubringClass : SubringClass (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  one_mem {s} := s.one_mem'\n  zero_mem {s} := s.zero_mem'\n  neg_mem {s} a ha := show -a ∈ s.toStarSubalgebra from neg_mem ha\n\n@[simp]\ntheorem mem_carrier {S : VonNeumannAlgebra H} {x : H →L[ℂ] H} :\n    x ∈ S.toStarSubalgebra ↔ x ∈ (S : Set (H →L[ℂ] H)) :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toStarSubalgebra (S : VonNeumannAlgebra H) :\n    (S.toStarSubalgebra : Set (H →L[ℂ] H)) = S :=\n  rfl\n\n@[simp]\ntheorem coe_mk (S : StarSubalgebra ℂ (H →L[ℂ] H)) (h) :\n    ((⟨S, h⟩ : VonNeumannAlgebra H) : Set (H →L[ℂ] H)) = S :=\n  rfl\n\n@[ext]\ntheorem ext {S T : VonNeumannAlgebra H} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem centralizer_centralizer (S : VonNeumannAlgebra H) :\n    Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S :=\n  S.centralizer_centralizer'\n\n/-- The centralizer of a `VonNeumannAlgebra`, as a `VonNeumannAlgebra`. -/\nnoncomputable def commutant (S : VonNeumannAlgebra H) : VonNeumannAlgebra H where\n  toStarSubalgebra := StarSubalgebra.centralizer ℂ (S : Set (H →L[ℂ] H))\n  centralizer_centralizer' := by simp\n\n@[simp]\ntheorem coe_commutant (S : VonNeumannAlgebra H) :\n    ↑S.commutant = Set.centralizer (S : Set (H →L[ℂ] H)) := by\n  simp [commutant]\n\n@[simp]\n\nTarget:\ntheorem mem_commutant_iff {S : VonNeumannAlgebra H} {z : H →L[ℂ] H} :\n    z ∈ S.commutant ↔ ∀ g ∈ S, g * z = z * g :=\n\nProof body:\n","rejected":"by\n  exact mem_commutant_iff","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"54af599effc2a1b7c5096aef5feeec82537207f5daca0259a03dd76057e88671","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/VonNeumannAlgebra","family_id":"mem_commutant_iff","file_id":"mathlib/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean","sample_id":"452212369db5a32584e53a4244413eb62217af8a259514599237694d17941ea9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e5473258af32213de8f9e9baf793bc2d791875a7c5122f73158f25e7055ff270","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bff817808a5cc69dcd74325b621cd3b889cc91e9bdb2f5fd6f80c7083b9ec6d9","source_sha256":"f589366b76253f55a5f00af110f0088b3f55c67577d56940b18639c5f1a2a3b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only [sub_eq_add_neg] using!\n    (f.commute_add_nat n).inverses_right (Equiv.addRight _).right_inv (Equiv.addRight _).left_inv","hard_negative":false,"metrics":{"chosen_tokens":32,"rejected_tokens":5,"token_jaccard":0.086957,"token_length_ratio":0.15625},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"0be81a42db189b271d14e206ad1626fd3414d8540d79efd35868f814691f6636","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.Order.SemiconjSup\n\nNamespace:\nCircleDeg1Lift\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Translation number of a monotone real map that commutes with `x ↦ x + 1`\n\nLet `f : ℝ → ℝ` be a monotone map such that `f (x + 1) = f x + 1` for all `x`. Then the limit\n$$\n  \\tau(f)=\\lim_{n\\to\\infty}{f^n(x)-x}{n}\n$$\nexists and does not depend on `x`. This number is called the *translation number* of `f`.\nDifferent authors use different notation for this number: `τ`, `ρ`, `rot`, etc\n\nIn this file we define a structure `CircleDeg1Lift` for bundled maps with these properties, define\ntranslation number of `f : CircleDeg1Lift`, prove some estimates relating `f^n(x)-x` to `τ(f)`. In\ncase of a continuous map `f` we also prove that `f` admits a point `x` such that `f^n(x)=x+m` if and\nonly if `τ(f)=m/n`.\n\nMaps of this type naturally appear as lifts of orientation-preserving circle homeomorphisms. More\nprecisely, let `f` be an orientation-preserving homeomorphism of the circle $S^1=ℝ/ℤ$, and\nconsider a real number `a` such that\n`⟦a⟧ = f 0`, where `⟦⟧` means the natural projection `ℝ → ℝ/ℤ`. Then there exists a unique\ncontinuous function `F : ℝ → ℝ` such that `F 0 = a` and `⟦F x⟧ = f ⟦x⟧` for all `x` (this fact is\nnot formalized yet). This function is strictly monotone, continuous, and satisfies\n`F (x + 1) = F x + 1`. The number `⟦τ F⟧ : ℝ / ℤ` is called the *rotation number* of `f`.\nIt does not depend on the choice of `a`.\n\n## Main definitions\n\n* `CircleDeg1Lift`: a monotone map `f : ℝ → ℝ` such that `f (x + 1) = f x + 1` for all `x`;\n  the type `CircleDeg1Lift` is equipped with `Lattice` and `Monoid` structures; the\n  multiplication is given by composition: `(f * g) x = f (g x)`.\n* `CircleDeg1Lift.translationNumber`: translation number of `f : CircleDeg1Lift`.\n\n## Main statements\n\nWe prove the following properties of `CircleDeg1Lift.translationNumber`.\n\n* `CircleDeg1Lift.translationNumber_eq_of_dist_bounded`: if the distance between `(f^n) 0`\n  and `(g^n) 0` is bounded from above uniformly in `n : ℕ`, then `f` and `g` have equal\n  translation numbers.\n\n* `CircleDeg1Lift.translationNumber_eq_of_semiconjBy`: if two `CircleDeg1Lift` maps `f`, `g`\n  are semiconjugate by a `CircleDeg1Lift` map, then `τ f = τ g`.\n\n* `CircleDeg1Lift.translationNumber_units_inv`: if `f` is an invertible `CircleDeg1Lift` map\n  (equivalently, `f` is a lift of an orientation-preserving circle homeomorphism), then\n  the translation number of `f⁻¹` is the negative of the translation number of `f`.\n\n* `CircleDeg1Lift.translationNumber_mul_of_commute`: if `f` and `g` commute, then\n  `τ (f * g) = τ f + τ g`.\n\n* `CircleDeg1Lift.translationNumber_eq_rat_iff`: the translation number of `f` is equal to\n  a rational number `m / n` if and only if `(f^n) x = x + m` for some `x`.\n\n* `CircleDeg1Lift.semiconj_of_bijective_of_translationNumber_eq`: if `f` and `g` are two\n  bijective `CircleDeg1Lift` maps and their translation numbers are equal, then these\n  maps are semiconjugate to each other.\n\n* `CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq`: let `f₁` and `f₂` be\n  two actions of a group `G` on the circle by degree 1 maps (formally, `f₁` and `f₂` are two\n  homomorphisms from `G →* CircleDeg1Lift`). If the translation numbers of `f₁ g` and `f₂ g` are\n  equal to each other for all `g : G`, then these two actions are semiconjugate by some\n  `F : CircleDeg1Lift`. This is a version of Proposition 5.4 from [Étienne Ghys, Groupes\n  d'homéomorphismes du cercle et cohomologie bornée][ghys87:groupes].\n\n## Notation\n\nWe use a local notation `τ` for the translation number of `f : CircleDeg1Lift`.\n\n## Implementation notes\n\nWe define the translation number of `f : CircleDeg1Lift` to be the limit of the sequence\n`(f ^ (2 ^ n)) 0 / (2 ^ n)`, then prove that `((f ^ n) x - x) / n` tends to this number for any `x`.\nThis way it is much easier to prove that the limit exists and basic properties of the limit.\n\nWe define translation number for a wider class of maps `f : ℝ → ℝ` instead of lifts of orientation\npreserving circle homeomorphisms for two reasons:\n\n* non-strictly monotone circle self-maps with discontinuities naturally appear as Poincaré maps\n  for some flows on the two-torus (e.g., one can take a constant flow and glue in a few Cherry\n  cells);\n* definition and some basic properties still work for this class.\n\n## References\n\n* [Étienne Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée][ghys87:groupes]\n\n## TODO\n\nHere are some short-term goals.\n\n* Introduce a structure or a typeclass for lifts of circle homeomorphisms. We use\n  `Units CircleDeg1Lift` for now, but it's better to have a dedicated type (or a typeclass?).\n\n* Prove that the `SemiconjBy` relation on circle homeomorphisms is an equivalence relation.\n\n* Introduce `ConditionallyCompleteLattice` structure, use it in the proof of\n  `CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq`.\n\n* Prove that the orbits of the irrational rotation are dense in the circle. Deduce that a\n  homeomorphism with an irrational rotation is semiconjugate to the corresponding irrational\n  translation by a continuous `CircleDeg1Lift`.\n\n## Tags\n\ncircle homeomorphism, rotation number\n-/\n\n@[expose] public section\n\nopen Filter Set Int Topology\nopen Function hiding Commute\n\n/-!\n### Definition and monoid structure\n-/\n\n/-- A lift of a monotone degree one map `S¹ → S¹`. -/\nstructure CircleDeg1Lift : Type extends ℝ →o ℝ where\n  map_add_one' : ∀ x, toFun (x + 1) = toFun x + 1\n\nnamespace CircleDeg1Lift\n\ninstance : FunLike CircleDeg1Lift ℝ ℝ where\n  coe f := f.toFun\n  coe_injective | ⟨⟨_, _⟩, _⟩, ⟨⟨_, _⟩, _⟩, rfl => rfl\n\ninstance : OrderHomClass CircleDeg1Lift ℝ ℝ where\n  map_rel f _ _ h := f.monotone' h\n\n@[simp] theorem coe_mk (f h) : ⇑(mk f h) = f := rfl\n\nvariable (f g : CircleDeg1Lift)\n\n@[simp] theorem coe_toOrderHom : ⇑f.toOrderHom = f := rfl\n\nprotected theorem monotone : Monotone f := f.monotone'\n\n@[gcongr, mono] theorem mono {x y} (h : x ≤ y) : f x ≤ f y := f.monotone h\n\ntheorem strictMono_iff_injective : StrictMono f ↔ Injective f :=\n  f.monotone.strictMono_iff_injective\n\n@[simp]\ntheorem map_add_one : ∀ x, f (x + 1) = f x + 1 :=\n  f.map_add_one'\n\n@[simp]\ntheorem map_one_add (x : ℝ) : f (1 + x) = 1 + f x := by rw [add_comm, map_add_one, add_comm 1]\n\n@[ext]\ntheorem ext ⦃f g : CircleDeg1Lift⦄ (h : ∀ x, f x = g x) : f = g :=\n  DFunLike.ext f g h\n\ninstance : Monoid CircleDeg1Lift where\n  mul f g :=\n    { toOrderHom := f.1.comp g.1\n      map_add_one' := fun x => by simp [map_add_one] }\n  one := ⟨.id, fun _ => rfl⟩\n  mul_one _ := rfl\n  one_mul _ := rfl\n  mul_assoc _ _ _ := DFunLike.coe_injective rfl\n\ninstance : Inhabited CircleDeg1Lift := ⟨1⟩\n\n@[simp]\ntheorem coe_mul : ⇑(f * g) = f ∘ g :=\n  rfl\n\ntheorem mul_apply (x) : (f * g) x = f (g x) :=\n  rfl\n\n@[simp]\ntheorem coe_one : ⇑(1 : CircleDeg1Lift) = id :=\n  rfl\n\ninstance unitsHasCoeToFun : CoeFun CircleDeg1Liftˣ fun _ => ℝ → ℝ :=\n  ⟨fun f => ⇑(f : CircleDeg1Lift)⟩\n\n@[simp]\ntheorem units_inv_apply_apply (f : CircleDeg1Liftˣ) (x : ℝ) :\n    (f⁻¹ : CircleDeg1Liftˣ) (f x) = x := by simp only [← mul_apply, f.inv_mul, coe_one, id]\n\n@[simp]\ntheorem units_apply_inv_apply (f : CircleDeg1Liftˣ) (x : ℝ) :\n    f ((f⁻¹ : CircleDeg1Liftˣ) x) = x := by simp only [← mul_apply, f.mul_inv, coe_one, id]\n\n/-- If a lift of a circle map is bijective, then it is an order automorphism of the line. -/\ndef toOrderIso : CircleDeg1Liftˣ →* ℝ ≃o ℝ where\n  toFun f :=\n    { toFun := f\n      invFun := ⇑f⁻¹\n      left_inv := units_inv_apply_apply f\n      right_inv := units_apply_inv_apply f\n      map_rel_iff' := ⟨fun h => by simpa using mono (↑f⁻¹) h, mono f⟩ }\n  map_one' := rfl\n  map_mul' _ _ := rfl\n\n@[simp]\ntheorem coe_toOrderIso (f : CircleDeg1Liftˣ) : ⇑(toOrderIso f) = f :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderIso_symm (f : CircleDeg1Liftˣ) :\n    ⇑(toOrderIso f).symm = (f⁻¹ : CircleDeg1Liftˣ) :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderIso_inv (f : CircleDeg1Liftˣ) : ⇑(toOrderIso f)⁻¹ = (f⁻¹ : CircleDeg1Liftˣ) :=\n  rfl\n\ntheorem isUnit_iff_bijective {f : CircleDeg1Lift} : IsUnit f ↔ Bijective f :=\n  ⟨fun ⟨u, h⟩ => h ▸ (toOrderIso u).bijective, fun h =>\n    Units.isUnit\n      { val := f\n        inv :=\n          { toFun := (Equiv.ofBijective f h).symm\n            monotone' := fun x y hxy =>\n              (f.strictMono_iff_injective.2 h.1).le_iff_le.1\n                (by simp only [Equiv.ofBijective_apply_symm_apply f h, hxy])\n            map_add_one' := fun x =>\n              h.1 <| by simp only [Equiv.ofBijective_apply_symm_apply f, f.map_add_one] }\n        val_inv := ext <| Equiv.ofBijective_apply_symm_apply f h\n        inv_val := ext <| Equiv.ofBijective_symm_apply_apply f h }⟩\n\ntheorem coe_pow : ∀ n : ℕ, ⇑(f ^ n) = f^[n]\n  | 0 => rfl\n  | n + 1 => by\n    simp [coe_pow n, pow_succ]\n\ntheorem semiconjBy_iff_semiconj {f g₁ g₂ : CircleDeg1Lift} :\n    SemiconjBy f g₁ g₂ ↔ Semiconj f g₁ g₂ :=\n  CircleDeg1Lift.ext_iff\n\ntheorem commute_iff_commute {f g : CircleDeg1Lift} : Commute f g ↔ Function.Commute f g :=\n  CircleDeg1Lift.ext_iff\n\n/-!\n### Translate by a constant\n-/\n\n\n/-- The map `y ↦ x + y` as a `CircleDeg1Lift`. More precisely, we define a homomorphism from\n`Multiplicative ℝ` to `CircleDeg1Liftˣ`, so the translation by `x` is\n`translation (Multiplicative.ofAdd x)`. -/\ndef translate : Multiplicative ℝ →* CircleDeg1Liftˣ := MonoidHom.toHomUnits <|\n  { toFun x := ⟨⟨fun y => x.toAdd + y, add_right_mono⟩, fun _ => (add_assoc ..).symm⟩\n    map_one' := ext zero_add\n    map_mul' _ _ := ext <| add_assoc _ _ }\n\n@[simp]\ntheorem translate_apply (x y : ℝ) : translate (Multiplicative.ofAdd x) y = x + y :=\n  rfl\n\n@[simp]\ntheorem translate_inv_apply (x y : ℝ) : (translate <| Multiplicative.ofAdd x)⁻¹ y = -x + y :=\n  rfl\n\n@[simp]\ntheorem translate_zpow (x : ℝ) (n : ℤ) :\n    translate (Multiplicative.ofAdd x) ^ n = translate (Multiplicative.ofAdd <| ↑n * x) := by\n  simp only [← zsmul_eq_mul, ofAdd_zsmul, map_zpow]\n\n@[simp]\ntheorem translate_pow (x : ℝ) (n : ℕ) :\n    translate (Multiplicative.ofAdd x) ^ n = translate (Multiplicative.ofAdd <| ↑n * x) :=\n  translate_zpow x n\n\n@[simp]\ntheorem translate_iterate (x : ℝ) (n : ℕ) :\n    (translate (Multiplicative.ofAdd x))^[n] = translate (Multiplicative.ofAdd <| ↑n * x) := by\n  rw [← coe_pow, ← Units.val_pow_eq_pow_val, translate_pow]\n\n/-!\n### Commutativity with integer translations\n\nIn this section we prove that `f` commutes with translations by an integer number.\nFirst we formulate these statements (for a natural or an integer number,\naddition on the left or on the right, addition or subtraction) using `Function.Commute`,\nthen reformulate as `simp` lemmas `map_int_add` etc.\n-/\n\ntheorem commute_nat_add (n : ℕ) : Function.Commute f (n + ·) := by\n  simpa only [nsmul_one, add_left_iterate] using Function.Commute.iterate_right f.map_one_add n\n\ntheorem commute_add_nat (n : ℕ) : Function.Commute f (· + n) := by\n  simp only [add_comm _ (n : ℝ), f.commute_nat_add n]\n\nTarget:\ntheorem commute_sub_nat (n : ℕ) : Function.Commute f (· - n) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/Circle","family_id":"commute_sub_nat","file_id":"mathlib/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean","sample_id":"bff817808a5cc69dcd74325b621cd3b889cc91e9bdb2f5fd6f80c7083b9ec6d9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"73aa3a7c73d07e3e5e0d2c9a0dc0fab61b3371c9b2e729b99cd088422d654f85","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"dc3039cff6d4052f266ec1fe01fc169549d07b6d752b31189ee8df8095c3340c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ba99dd8f6392afa104bfc6ac9136b281cb75f223a9c829f9749723664ecb6a1c","source_sha256":"135835936222dd833aca95a55105deb9bdc869adf3601290cb91b6f312b8a295","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases Finite.exists_equiv_fin β with ⟨n, ⟨e⟩⟩\n  classical exact .of_equiv (Set.range (e ∘ f)) (Equiv.ofInjective _ (e.injective.comp H)).symm\n\n-- see Note [lower instance priority]","hard_negative":true,"metrics":{"chosen_tokens":53,"rejected_tokens":5,"token_jaccard":0.078947,"token_length_ratio":0.09434},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"0c04056bc242789491cf4239f93203ab8a94034db86100bb40bfbb56c8706dd1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Card\npublic import Mathlib.Data.List.NodupEquivFin\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Equivalences between `Fintype`, `Fin` and `Finite`\n\nThis file defines the bijection between a `Fintype α` and `Fin (Fintype.card α)`, and uses this to\nrelate `Fintype` with `Finite`. From that we can derive properties of `Finite` and `Infinite`,\nand show some instances of `Infinite`.\n\n## Main declarations\n\n* `Fintype.truncEquivFin`: A fintype `α` is computably equivalent to `Fin (card α)`. The\n  `Trunc`-free, noncomputable version is `Fintype.equivFin`.\n* `Fintype.truncEquivOfCardEq` `Fintype.equivOfCardEq`: Two fintypes of same cardinality are\n  equivalent. See above.\n* `Fin.equiv_iff_eq`: `Fin m ≃ Fin n` iff `m = n`.\n* `Infinite.natEmbedding`: An embedding of `ℕ` into an infinite type.\n\nTypes which have an injection from/a surjection to an `Infinite` type are themselves `Infinite`.\nSee `Infinite.of_injective` and `Infinite.of_surjective`.\n\n## Instances\n\nWe provide `Infinite` instances for\n* specific types: `ℕ`, `ℤ`, `String`\n* type constructors: `Multiset α`, `List α`\n\n-/\n\n@[expose] public section\n\nassert_not_exists Monoid\n\nopen Function\n\nuniverse u v\n\nvariable {α β γ : Type*}\n\nopen Finset\n\nnamespace Fintype\n\n/-- There is (computably) an equivalence between `α` and `Fin (card α)`.\n\nSince it is not unique and depends on which permutation\nof the universe list is used, the equivalence is wrapped in `Trunc` to\npreserve computability.\n\nSee `Fintype.equivFin` for the noncomputable version,\nand `Fintype.truncEquivFinOfCardEq` and `Fintype.equivFinOfCardEq`\nfor an equiv `α ≃ Fin n` given `Fintype.card α = n`.\n\nSee `Fintype.truncFinBijection` for a version without `[DecidableEq α]`.\n-/\ndef truncEquivFin (α) [DecidableEq α] [Fintype α] : Trunc (α ≃ Fin (card α)) := by\n  unfold card Finset.card\n  exact\n    Quot.recOnSubsingleton\n      (motive := fun s : Multiset α =>\n        (∀ x : α, x ∈ s) → s.Nodup → Trunc (α ≃ Fin (Multiset.card s)))\n      univ.val\n      (fun l (h : ∀ x : α, x ∈ l) (nd : l.Nodup) => Trunc.mk (nd.getEquivOfForallMemList _ h).symm)\n      mem_univ_val univ.2\n\n/-- There is (noncomputably) an equivalence between `α` and `Fin (card α)`.\n\nSee `Fintype.truncEquivFin` for the computable version,\nand `Fintype.truncEquivFinOfCardEq` and `Fintype.equivFinOfCardEq`\nfor an equiv `α ≃ Fin n` given `Fintype.card α = n`.\n-/\nnoncomputable def equivFin (α) [Fintype α] : α ≃ Fin (card α) :=\n  letI := Classical.decEq α\n  (truncEquivFin α).out\n\n/-- There is (computably) a bijection between `Fin (card α)` and `α`.\n\nSince it is not unique and depends on which permutation\nof the universe list is used, the bijection is wrapped in `Trunc` to\npreserve computability.\n\nSee `Fintype.truncEquivFin` for a version that gives an equivalence\ngiven `[DecidableEq α]`.\n-/\ndef truncFinBijection (α) [Fintype α] : Trunc { f : Fin (card α) → α // Bijective f } := by\n  unfold card Finset.card\n  refine\n    Quot.recOnSubsingleton\n      (motive := fun s : Multiset α =>\n        (∀ x : α, x ∈ s) → s.Nodup → Trunc {f : Fin (Multiset.card s) → α // Bijective f})\n      univ.val\n      (fun l (h : ∀ x : α, x ∈ l) (nd : l.Nodup) => Trunc.mk (nd.getBijectionOfForallMemList _ h))\n      mem_univ_val univ.2\n\nend Fintype\n\nnamespace Fintype\n\nsection\n\nvariable [Fintype α] [Fintype β]\n\n/-- If the cardinality of `α` is `n`, there is computably a bijection between `α` and `Fin n`.\n\nSee `Fintype.equivFinOfCardEq` for the noncomputable definition,\nand `Fintype.truncEquivFin` and `Fintype.equivFin` for the bijection `α ≃ Fin (card α)`.\n-/\ndef truncEquivFinOfCardEq [DecidableEq α] {n : ℕ} (h : Fintype.card α = n) : Trunc (α ≃ Fin n) :=\n  (truncEquivFin α).map fun e => e.trans (finCongr h)\n\n/-- If the cardinality of `α` is `n`, there is noncomputably a bijection between `α` and `Fin n`.\n\nSee `Fintype.truncEquivFinOfCardEq` for the computable definition,\nand `Fintype.truncEquivFin` and `Fintype.equivFin` for the bijection `α ≃ Fin (card α)`.\n-/\nnoncomputable def equivFinOfCardEq {n : ℕ} (h : Fintype.card α = n) : α ≃ Fin n :=\n  letI := Classical.decEq α\n  (truncEquivFinOfCardEq h).out\n\n/-- Two `Fintype`s with the same cardinality are (computably) in bijection.\n\nSee `Fintype.equivOfCardEq` for the noncomputable version,\nand `Fintype.truncEquivFinOfCardEq` and `Fintype.equivFinOfCardEq` for\nthe specialization to `Fin`.\n-/\ndef truncEquivOfCardEq [DecidableEq α] [DecidableEq β] (h : card α = card β) : Trunc (α ≃ β) :=\n  (truncEquivFinOfCardEq h).bind fun e => (truncEquivFin β).map fun e' => e.trans e'.symm\n\n/-- Two `Fintype`s with the same cardinality are (noncomputably) in bijection.\n\nSee `Fintype.truncEquivOfCardEq` for the computable version,\nand `Fintype.truncEquivFinOfCardEq` and `Fintype.equivFinOfCardEq` for\nthe specialization to `Fin`.\n-/\nnoncomputable def equivOfCardEq (h : card α = card β) : α ≃ β := by\n  letI := Classical.decEq α\n  letI := Classical.decEq β\n  exact (truncEquivOfCardEq h).out\n\nend\n\ntheorem card_eq {α β} [_F : Fintype α] [_G : Fintype β] : card α = card β ↔ Nonempty (α ≃ β) :=\n  ⟨fun h =>\n    haveI := Classical.propDecidable\n    (truncEquivOfCardEq h).nonempty,\n    fun ⟨f⟩ => card_congr f⟩\n\nend Fintype\n\n/-!\n### Relation to `Finite`\n\nIn this section we prove that `α : Type*` is `Finite` if and only if `Fintype α` is nonempty.\n-/\n\nprotected theorem Fintype.finite {α : Type*} (_inst : Fintype α) : Finite α :=\n  ⟨Fintype.equivFin α⟩\n\nset_option linter.unusedFintypeInType false in\n/-- For efficiency reasons, we want `Finite` instances to have higher\npriority than ones coming from `Fintype` instances. -/\ninstance (priority := 900) Finite.of_fintype (α : Type*) [Fintype α] : Finite α :=\n  Fintype.finite ‹_›\n\ntheorem finite_iff_nonempty_fintype (α : Type*) : Finite α ↔ Nonempty (Fintype α) :=\n  ⟨fun _ => nonempty_fintype α, fun ⟨_⟩ => inferInstance⟩\n\n/-- Noncomputably get a `Fintype` instance from a `Finite` instance. This is not an\ninstance because we want `Fintype` instances to be useful for computations. -/\n@[instance_reducible]\nnoncomputable def Fintype.ofFinite (α : Type*) [Finite α] : Fintype α :=\n  (nonempty_fintype α).some\n\nTarget:\ntheorem Finite.of_injective {α β : Sort*} [Finite β] (f : α → β) (H : Injective f) : Finite α :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_ba99dd8f6392","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"fcde561416e510a9e2b0cad44babb3c67f4c02e586f2f88c2dc1275f420c0434","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Fintype","family_id":"finite","file_id":"mathlib/Mathlib/Data/Fintype/EquivFin.lean","sample_id":"ba99dd8f6392afa104bfc6ac9136b281cb75f223a9c829f9749723664ecb6a1c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8b5027e289eea500a3bcfb155a81791b1a51bc31d201ba8ef228cee84d9c2f28","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d2e80bfd09f7c8ffd630b3233e0baf00abf1e879c638b96895f9ef4ded614180","source_sha256":"d4d995f1a37fa9568f07e686f3f287ad8080507cb2490d13806922b7a824453c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [IsIdempotentElem, ← star_mul]\n\nalias ⟨_, IsIdempotentElem.star⟩ := IsIdempotentElem.star_iff","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":2,"token_jaccard":0.058824,"token_length_ratio":0.1},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"0c6b1d877bcc3fc76fa790db5be247d036677f9b6071fc297fb6d3808318556f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Star.SelfAdjoint\npublic import Mathlib.Algebra.Module.Basic\npublic import Mathlib.Algebra.Module.Equiv.Defs\npublic import Mathlib.Algebra.Module.LinearMap.Star\npublic import Mathlib.Algebra.Module.Rat\npublic import Mathlib.LinearAlgebra.Prod\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser, Frédéric Dupuis\n-/\n/-!\n# The star operation, bundled as a star-linear equiv\n\nWe define `starLinearEquiv`, which is the star operation bundled as a star-linear map.\nIt is defined on a star algebra `A` over the base ring `R`.\n\nThis file also provides some lemmas that need `Algebra.Module.Basic` imported to prove.\n\n## TODO\n\n- Define `starLinearEquiv` for noncommutative `R`. We only the commutative case for now since,\n  in the noncommutative case, the ring hom needs to reverse the order of multiplication. This\n  requires a ring hom of type `R →+* Rᵐᵒᵖ`, which is very undesirable in the commutative case.\n  One way out would be to define a new typeclass `IsOp R S` and have an instance `IsOp R R`\n  for commutative `R`.\n- Also note that such a definition involving `Rᵐᵒᵖ` or `is_op R S` would require adding\n  the appropriate `RingHomInvPair` instances to be able to define the semilinear\n  equivalence.\n-/\n\n@[expose] public section\n\n\nsection SMulLemmas\n\nvariable {R M : Type*}\n\n@[simp]\ntheorem star_natCast_smul [Semiring R] [AddCommMonoid M] [Module R M] [StarAddMonoid M] (n : ℕ)\n    (x : M) : star ((n : R) • x) = (n : R) • star x :=\n  map_natCast_smul (starAddEquiv : M ≃+ M) R R n x\n\n@[simp]\ntheorem star_intCast_smul [Ring R] [AddCommGroup M] [Module R M] [StarAddMonoid M] (n : ℤ)\n    (x : M) : star ((n : R) • x) = (n : R) • star x :=\n  map_intCast_smul (starAddEquiv : M ≃+ M) R R n x\n\n@[simp]\ntheorem star_inv_natCast_smul [DivisionSemiring R] [AddCommMonoid M] [Module R M] [StarAddMonoid M]\n    (n : ℕ) (x : M) : star ((n⁻¹ : R) • x) = (n⁻¹ : R) • star x :=\n  map_inv_natCast_smul (starAddEquiv : M ≃+ M) R R n x\n\n@[simp]\ntheorem star_inv_intCast_smul [DivisionRing R] [AddCommGroup M] [Module R M] [StarAddMonoid M]\n    (n : ℤ) (x : M) : star ((n⁻¹ : R) • x) = (n⁻¹ : R) • star x :=\n  map_inv_intCast_smul (starAddEquiv : M ≃+ M) R R n x\n\n@[simp]\ntheorem star_ratCast_smul [DivisionRing R] [AddCommGroup M] [Module R M] [StarAddMonoid M] (n : ℚ)\n    (x : M) : star ((n : R) • x) = (n : R) • star x :=\n  map_ratCast_smul (starAddEquiv : M ≃+ M) _ _ _ x\n\n/-!\nPer the naming convention, these two lemmas call `(q • ·)` `nnrat_smul` and `rat_smul` respectively,\nrather than `nnqsmul` and `qsmul` because the latter are reserved to the actions coming from\n`DivisionSemiring` and `DivisionRing`. We provide aliases with `nnqsmul` and `qsmul` for\ndiscoverability.\n-/\n\n/-- Note that this lemma holds for an arbitrary `ℚ≥0`-action, rather than merely one coming from a\n`DivisionSemiring`. We keep both the `nnqsmul` and `nnrat_smul` naming conventions for\ndiscoverability. See `star_nnqsmul`. -/\n@[simp high]\nlemma star_nnrat_smul [AddCommMonoid R] [StarAddMonoid R] [Module ℚ≥0 R] (q : ℚ≥0) (x : R) :\n    star (q • x) = q • star x := map_nnrat_smul (starAddEquiv : R ≃+ R) _ _\n\n/-- Note that this lemma holds for an arbitrary `ℚ`-action, rather than merely one coming from a\n`DivisionRing`. We keep both the `qsmul` and `rat_smul` naming conventions for discoverability.\nSee `star_qsmul`. -/\n@[simp high] lemma star_rat_smul [AddCommGroup R] [StarAddMonoid R] [Module ℚ R] (q : ℚ) (x : R) :\n    star (q • x) = q • star x :=\n  map_rat_smul (starAddEquiv : R ≃+ R) _ _\n\n/-- Note that this lemma holds for an arbitrary `ℚ≥0`-action, rather than merely one coming from a\n`DivisionSemiring`. We keep both the `nnqsmul` and `nnrat_smul` naming conventions for\ndiscoverability. See `star_nnrat_smul`. -/\nalias star_nnqsmul := star_nnrat_smul\n\n/-- Note that this lemma holds for an arbitrary `ℚ`-action, rather than merely one coming from a\n`DivisionRing`. We keep both the `qsmul` and `rat_smul` naming conventions for\ndiscoverability. See `star_rat_smul`. -/\nalias star_qsmul := star_rat_smul\n\ninstance StarAddMonoid.toStarModuleNNRat [AddCommMonoid R] [Module ℚ≥0 R] [StarAddMonoid R] :\n    StarModule ℚ≥0 R where star_smul := star_nnrat_smul\n\ninstance StarAddMonoid.toStarModuleRat [AddCommGroup R] [Module ℚ R] [StarAddMonoid R] :\n    StarModule ℚ R where star_smul := star_rat_smul\n\nend SMulLemmas\n\nsection starLinearEquiv\n\nvariable (R : Type*) {A : Type*}\n  [CommSemiring R] [StarRing R] [AddCommMonoid A] [StarAddMonoid A] [Module R A] [StarModule R A]\n\n/-- If `A` is a module over a commutative `R` with compatible actions,\nthen `star` is a semilinear equivalence. -/\n@[simps! apply]\ndef starLinearEquiv : A ≃ₗ⋆[R] A where\n  __ := starAddEquiv\n  map_smul' := star_smul\n\n@[simp]\ntheorem toAddEquiv_starLinearEquiv :\n    (starLinearEquiv R : A ≃ₗ⋆[R] A).toAddEquiv = starAddEquiv :=\n  rfl\n\n@[simp]\ntheorem symm_starLinearEquiv : (starLinearEquiv R : A ≃ₗ⋆[R] A).symm = starLinearEquiv R :=\n  rfl\n\n@[deprecated \"Use `symm_starLinearEquiv` and `starLinearEquiv_apply` instead\"\n  (since := \"2026-06-03\")]\ntheorem starLinearEquiv_symm_apply (x : A) :\n    (starLinearEquiv R).symm x = starAddEquiv.invFun x := by\n  simp\n\nend starLinearEquiv\n\nsection SelfSkewAdjoint\n\nvariable (R : Type*) (A : Type*) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A]\n  [Module R A] [StarAddMonoid A] [StarModule R A]\n\n/-- The self-adjoint elements of a star module, as a submodule. -/\ndef selfAdjoint.submodule : Submodule R A :=\n  { selfAdjoint A with smul_mem' := fun _ _ => (IsSelfAdjoint.all _).smul }\n\n/-- The skew-adjoint elements of a star module, as a submodule. -/\ndef skewAdjoint.submodule : Submodule R A :=\n  { skewAdjoint A with smul_mem' := skewAdjoint.smul_mem }\n\nvariable {A} [Invertible (2 : R)]\n\n/-- The self-adjoint part of an element of a star module, as a linear map. -/\n@[simps]\ndef selfAdjointPart : A →ₗ[R] selfAdjoint A where\n  toFun x :=\n    ⟨(⅟2 : R) • (x + star x), by\n      rw [selfAdjoint.mem_iff, star_smul, star_trivial, star_add, star_star, add_comm]⟩\n  map_add' x y := by\n    ext\n    simp [add_add_add_comm]\n  map_smul' r x := by\n    ext\n    simp [← mul_smul, show ⅟2 * r = r * ⅟2 from Commute.invOf_left <| (2 : ℕ).cast_commute r]\n\n/-- The skew-adjoint part of an element of a star module, as a linear map. -/\n@[simps]\ndef skewAdjointPart : A →ₗ[R] skewAdjoint A where\n  toFun x :=\n    ⟨(⅟2 : R) • (x - star x), by\n      simp only [skewAdjoint.mem_iff, star_smul, star_sub, star_star, star_trivial, ← smul_neg,\n        neg_sub]⟩\n  map_add' x y := by\n    ext\n    simp only [sub_add, ← smul_add, sub_sub_eq_add_sub, star_add, AddSubgroup.coe_add]\n  map_smul' r x := by\n    ext\n    simp [← mul_smul, ← smul_sub,\n      show r * ⅟2 = ⅟2 * r from Commute.invOf_right <| (2 : ℕ).commute_cast r]\n\ntheorem StarModule.selfAdjointPart_add_skewAdjointPart (x : A) :\n    (selfAdjointPart R x : A) + skewAdjointPart R x = x := by\n  simp only [smul_sub, selfAdjointPart_apply_coe, smul_add, skewAdjointPart_apply_coe,\n    add_add_sub_cancel, invOf_two_smul_add_invOf_two_smul]\n\ntheorem IsSelfAdjoint.coe_selfAdjointPart_apply {x : A} (hx : IsSelfAdjoint x) :\n    (selfAdjointPart R x : A) = x := by\n  rw [selfAdjointPart_apply_coe, hx.star_eq, smul_add, invOf_two_smul_add_invOf_two_smul]\n\ntheorem IsSelfAdjoint.selfAdjointPart_apply {x : A} (hx : IsSelfAdjoint x) :\n    selfAdjointPart R x = ⟨x, hx⟩ :=\n  Subtype.ext (hx.coe_selfAdjointPart_apply R)\n\n@[simp]\ntheorem selfAdjointPart_comp_subtype_selfAdjoint :\n    (selfAdjointPart R).comp (selfAdjoint.submodule R A).subtype = .id :=\n  LinearMap.ext fun x ↦ x.2.selfAdjointPart_apply R\n\ntheorem IsSelfAdjoint.skewAdjointPart_apply {x : A} (hx : IsSelfAdjoint x) :\n    skewAdjointPart R x = 0 := Subtype.ext <| by\n  rw [skewAdjointPart_apply_coe, hx.star_eq, sub_self, smul_zero, ZeroMemClass.coe_zero]\n\n@[simp]\ntheorem skewAdjointPart_comp_subtype_selfAdjoint :\n    (skewAdjointPart R).comp (selfAdjoint.submodule R A).subtype = 0 :=\n  LinearMap.ext fun x ↦ x.2.skewAdjointPart_apply R\n\n@[simp]\ntheorem selfAdjointPart_comp_subtype_skewAdjoint :\n    (selfAdjointPart R).comp (skewAdjoint.submodule R A).subtype = 0 :=\n  LinearMap.ext fun ⟨x, (hx : _ = _)⟩ ↦ Subtype.ext <| by simp [hx]\n\n@[simp]\ntheorem skewAdjointPart_comp_subtype_skewAdjoint :\n    (skewAdjointPart R).comp (skewAdjoint.submodule R A).subtype = .id :=\n  LinearMap.ext fun ⟨x, (hx : _ = _)⟩ ↦ Subtype.ext <| by\n    simp only [LinearMap.comp_apply, Submodule.subtype_apply, skewAdjointPart_apply_coe, hx,\n      sub_neg_eq_add, smul_add, invOf_two_smul_add_invOf_two_smul]; rfl\n\nvariable (A)\n\n/-- The decomposition of elements of a star module into their self- and skew-adjoint parts,\nas a linear equivalence. -/\n@[simps!]\ndef StarModule.decomposeProdAdjoint : A ≃ₗ[R] selfAdjoint A × skewAdjoint A := by\n  refine LinearEquiv.ofLinear ((selfAdjointPart R).prod (skewAdjointPart R))\n    (LinearMap.coprod ((selfAdjoint.submodule R A).subtype) (skewAdjoint.submodule R A).subtype)\n    ?_ (LinearMap.ext <| StarModule.selfAdjointPart_add_skewAdjointPart R)\n  -- Note: with https://github.com/leanprover-community/mathlib4/pull/6965 `Submodule.coe_subtype` doesn't fire in `dsimp` or `simp`\n  ext x <;> dsimp <;> erw [Submodule.coe_subtype, Submodule.coe_subtype] <;> simp\n\nend SelfSkewAdjoint\n\nsection algebraMap\n\nvariable {R A : Type*} [CommSemiring R] [StarRing R] [Semiring A]\nvariable [StarMul A] [Algebra R A] [StarModule R A]\n\n@[simp]\ntheorem algebraMap_star_comm (r : R) : algebraMap R A (star r) = star (algebraMap R A r) := by\n  simp only [Algebra.algebraMap_eq_smul_one, star_smul, star_one]\n\nvariable (A) in\nprotected lemma IsSelfAdjoint.algebraMap {r : R} (hr : IsSelfAdjoint r) :\n    IsSelfAdjoint (algebraMap R A r) := by\n  simpa using! congr(algebraMap R A $(hr.star_eq))\n\nlemma isSelfAdjoint_algebraMap_iff {r : R} (h : Function.Injective (algebraMap R A)) :\n    IsSelfAdjoint (algebraMap R A r) ↔ IsSelfAdjoint r :=\n  ⟨fun hr ↦ h <| algebraMap_star_comm r (A := A) ▸ hr.star_eq, IsSelfAdjoint.algebraMap A⟩\n\nend algebraMap\n\nTarget:\ntheorem IsIdempotentElem.star_iff {R : Type*} [Mul R] [StarMul R] {a : R} :\n    IsIdempotentElem (star a) ↔ IsIdempotentElem a :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Star","family_id":"isidempotentelem","file_id":"mathlib/Mathlib/Algebra/Star/Module.lean","sample_id":"d2e80bfd09f7c8ffd630b3233e0baf00abf1e879c638b96895f9ef4ded614180"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c55c96a734207c6a61a412e0cab744c2c07b34209605ac6a4b05c1a6bd545ce9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f264b4d3af83d4231d10d5975f60cb548c0335d6d69af32d290c04593446c5ca","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4d49f41524ca78215a8298981fc8795e5f250086d26cc6a75dc11a9339f59a8c","source_sha256":"5a251c5b9da944aa52b3b3c01ae0c840fa23c2c1f9c9e5ace9139527d3f0e3e9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [tprod_apply, map_list_prod, List.map_ofFn]\n  simp_rw [Function.comp_def, toDirectSum_ι]\n  rw [DirectSum.list_prod_ofFn_of_eq_dProd]\n  apply DirectSum.of_eq_of_gradedMonoid_eq\n  rw [GradedMonoid.mk_list_dProd]\n  rw [TensorPower.list_prod_gradedMonoid_mk_single]","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":47,"token_jaccard":0.851852,"token_length_ratio":1.119048},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"0caff8188aeaebb78e5afc923133859529113710cced5a8a8bae09f050d8f218","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorAlgebra.Basic\npublic import Mathlib.LinearAlgebra.TensorPower.Basic\n\nNamespace:\nTensorAlgebra\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Tensor algebras as direct sums of tensor powers\n\nIn this file we show that `TensorAlgebra R M` is isomorphic to a direct sum of tensor powers, as\n`TensorAlgebra.equivDirectSum`.\n-/\n\n@[expose] public section\n\nopen scoped DirectSum TensorProduct\n\nvariable {R M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M]\n\nnamespace TensorPower\n\n/-- The canonical embedding from a tensor power to the tensor algebra -/\ndef toTensorAlgebra {n} : ⨂[R]^n M →ₗ[R] TensorAlgebra R M :=\n  PiTensorProduct.lift (TensorAlgebra.tprod R M n)\n\n@[simp]\ntheorem toTensorAlgebra_tprod {n} (x : Fin n → M) :\n    TensorPower.toTensorAlgebra (PiTensorProduct.tprod R x) = TensorAlgebra.tprod R M n x :=\n  PiTensorProduct.lift.tprod _\n\n@[simp]\ntheorem toTensorAlgebra_gOne :\n    TensorPower.toTensorAlgebra (@GradedMonoid.GOne.one _ (fun n => ⨂[R]^n M) _ _) = 1 := by\n  simp [GradedMonoid.GOne.one, TensorPower.toTensorAlgebra_tprod]\n\n@[simp]\ntheorem toTensorAlgebra_gMul {i j} (a : (⨂[R]^i) M) (b : (⨂[R]^j) M) :\n    TensorPower.toTensorAlgebra (@GradedMonoid.GMul.mul _ (fun n => ⨂[R]^n M) _ _ _ _ a b) =\n      TensorPower.toTensorAlgebra a * TensorPower.toTensorAlgebra b := by\n  -- change `a` and `b` to `tprod R a` and `tprod R b`\n  rw [TensorPower.gMul_eq_coe_linearMap, ← LinearMap.compr₂_apply, ← @LinearMap.mul_apply' R, ←\n    LinearMap.compl₂_apply, ← LinearMap.comp_apply]\n  refine LinearMap.congr_fun (LinearMap.congr_fun ?_ a) b\n  clear! a b\n  ext (a b)\n  simp only [LinearMap.compMultilinearMap_apply, LinearMap.compr₂_apply, ← gMul_def,\n    TensorProduct.mk_apply, LinearEquiv.coe_coe, tprod_mul_tprod, toTensorAlgebra_tprod,\n    TensorAlgebra.tprod_apply, LinearMap.comp_apply, LinearMap.compl₂_apply]\n  refine Eq.trans ?_ List.prod_append\n  congr\n  rw [List.ofFn_comp' _ (TensorAlgebra.ι R), List.ofFn_comp' _ (TensorAlgebra.ι R),\n    List.ofFn_comp' _ (TensorAlgebra.ι R), ← List.map_append, List.ofFn_fin_append]\n\n@[simp]\ntheorem toTensorAlgebra_galgebra_toFun (r : R) :\n    TensorPower.toTensorAlgebra (DirectSum.GAlgebra.toFun (R := R) (A := fun n => ⨂[R]^n M) r) =\n      algebraMap _ _ r := by\n  rw [TensorPower.galgebra_toFun_def, TensorPower.algebraMap₀_eq_smul_one, map_smul,\n    TensorPower.toTensorAlgebra_gOne, Algebra.algebraMap_eq_smul_one]\n\nend TensorPower\n\nnamespace TensorAlgebra\n\n/-- The canonical map from a direct sum of tensor powers to the tensor algebra. -/\ndef ofDirectSum : (⨁ n, ⨂[R]^n M) →ₐ[R] TensorAlgebra R M :=\n  DirectSum.toAlgebra _ _ (fun _ => TensorPower.toTensorAlgebra) TensorPower.toTensorAlgebra_gOne\n    (fun {_ _} => TensorPower.toTensorAlgebra_gMul)\n\n@[simp]\ntheorem ofDirectSum_of_tprod {n} (x : Fin n → M) :\n    ofDirectSum (DirectSum.of _ n (PiTensorProduct.tprod R x)) = tprod R M n x :=\n  (DirectSum.toAddMonoid_of\n    (fun _ ↦ LinearMap.toAddMonoidHom TensorPower.toTensorAlgebra) _ _).trans\n  (TensorPower.toTensorAlgebra_tprod _)\n\n/-- The canonical map from the tensor algebra to a direct sum of tensor powers. -/\ndef toDirectSum : TensorAlgebra R M →ₐ[R] ⨁ n, ⨂[R]^n M :=\n  TensorAlgebra.lift R <|\n    DirectSum.lof R ℕ (fun n => ⨂[R]^n M) _ ∘ₗ\n      (LinearEquiv.symm <| PiTensorProduct.subsingletonEquiv (0 : Fin 1) : M ≃ₗ[R] _).toLinearMap\n\n@[simp]\ntheorem toDirectSum_ι (x : M) :\n    toDirectSum (ι R x) =\n      DirectSum.of (fun n => ⨂[R]^n M) _ (PiTensorProduct.tprod R fun _ : Fin 1 => x) := by\n  simp [toDirectSum, TensorAlgebra.lift_ι_apply, DirectSum.lof_eq_of]\n\ntheorem ofDirectSum_comp_toDirectSum :\n    ofDirectSum.comp toDirectSum = AlgHom.id R (TensorAlgebra R M) := by\n  ext\n  simp [tprod_apply]\n\n@[simp]\ntheorem ofDirectSum_toDirectSum (x : TensorAlgebra R M) :\n    ofDirectSum (TensorAlgebra.toDirectSum x) = x :=\n  AlgHom.congr_fun ofDirectSum_comp_toDirectSum x\n\n@[simp]\ntheorem mk_reindex_cast {n m : ℕ} (h : n = m) (x : ⨂[R]^n M) :\n    GradedMonoid.mk (A := fun i => (⨂[R]^i) M) m\n    (PiTensorProduct.reindex R (fun _ ↦ M) (Equiv.cast <| congr_arg Fin h) x) =\n    GradedMonoid.mk n x :=\n  Eq.symm (PiTensorProduct.gradedMonoid_eq_of_reindex_cast h rfl)\n\n@[simp]\ntheorem mk_reindex_fin_cast {n m : ℕ} (h : n = m) (x : ⨂[R]^n M) :\n    GradedMonoid.mk (A := fun i => (⨂[R]^i) M) m\n    (PiTensorProduct.reindex R (fun _ ↦ M) (finCongr h) x) = GradedMonoid.mk n x := by\n  rw [finCongr_eq_equivCast, mk_reindex_cast h]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The product of tensor products made of a single vector is the same as a single product of\nall the vectors. -/\ntheorem _root_.TensorPower.list_prod_gradedMonoid_mk_single (n : ℕ) (x : Fin n → M) :\n    ((List.finRange n).map fun a =>\n          (GradedMonoid.mk _ (PiTensorProduct.tprod R fun _ : Fin 1 => x a) :\n            GradedMonoid fun n => ⨂[R]^n M)).prod =\n      GradedMonoid.mk n (PiTensorProduct.tprod R x) := by\n  refine Fin.consInduction ?_ ?_ x <;> clear x\n  · rw [List.finRange_zero, List.map_nil, List.prod_nil]\n    rfl\n  · intro n x₀ x ih\n    rw [List.finRange_succ, List.map_cons, List.prod_cons, List.map_map]\n    simp_rw [Function.comp_def, Fin.cons_zero, Fin.cons_succ]\n    rw [ih, GradedMonoid.mk_mul_mk, TensorPower.tprod_mul_tprod]\n    refine TensorPower.gradedMonoid_eq_of_cast (add_comm _ _) ?_\n    dsimp only [GradedMonoid.mk]\n    rw [TensorPower.cast_tprod]\n    simp_rw [Fin.append_left_eq_cons, Function.comp_def]\n    congr 1 with i\n\nTarget:\ntheorem toDirectSum_tensorPower_tprod {n} (x : Fin n → M) :\n    toDirectSum (tprod R M n x) = DirectSum.of _ n (PiTensorProduct.tprod R x) :=\n\nProof body:\n","rejected":"by\n  rw [tprod_apply, map_list_prod, List.map_ofFn]\n  simp_rw [Function.comp_def, toDirectSum_ι]\n  rw [DirectSum.list_prod_ofFn_of_eq_dProd]\n  apply DirectSum.of_eq_of_gradedMonoid_eq\n  rw [GradedMonoid.mk_list_dProd]\n  rw [TensorPower.list_prod_gradedMonoid_mk_single]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/TensorAlgebra","family_id":"todirectsum_tensorpower_tprod","file_id":"mathlib/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean","sample_id":"4d49f41524ca78215a8298981fc8795e5f250086d26cc6a75dc11a9339f59a8c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"153be3b3963b7b4e295fef95384346f85509c50a1d50e691a207cccee178ce99","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"95e884eecf0883ef717211cc0723527aed42480bad4f0b0ce44ee9602eb2c4e8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fffbd570a9bc968d3091c7ef16d1e7aecf0edf42a3c5fce61027b019b7b6b538","source_sha256":"16dd8d8cb25a5fc3397ffc119122019b2cf6ddea68dae9b29de54500250b117b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · rw [Presheaf.isLocallySurjective_iff_whisker_forget (J := extensiveTopology C)]\n    exact fun h _ ↦\n      surjective_of_isLocallySurjective_sheaf_of_types (Functor.whiskerRight f (forget D)) h\n  · exact fun a ↦\n      Presheaf.isLocallySurjective_of_surjective _ _ (fun _ ↦ a _)","hard_negative":false,"metrics":{"chosen_tokens":49,"rejected_tokens":56,"token_jaccard":0.933333,"token_length_ratio":1.142857},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"0d1fd8531ff775156e1d6e9cb157b3759f5bb0b9fbdf51afe9265aa6b6e8a259","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology\npublic import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison\npublic import Mathlib.CategoryTheory.Sites.LocallySurjective\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# Locally surjective morphisms of coherent sheaves\n\nThis file characterises locally surjective morphisms of presheaves for the coherent, regular\nand extensive topologies.\n\n## Main results\n\n* `regularTopology.isLocallySurjective_iff` A morphism of presheaves `f : F ⟶ G` is locally\n  surjective for the regular topology iff for every object `X` of `C`, and every `y : G(X)`, there\n  is an effective epimorphism `φ : X' ⟶ X` and an `x : F(X)` such that `f_{X'}(x) = G(φ)(y)`.\n\n* `coherentTopology.isLocallySurjective_iff` a morphism of sheaves for the coherent topology on a\n  preregular finitary extensive category is locally surjective if and only if it is\n  locally surjective for the regular topology.\n\n* `extensiveTopology.isLocallySurjective_iff` a morphism of sheaves for the extensive topology on a\n  finitary extensive category is locally surjective iff it is objectwise surjective.\n-/\n\npublic section\n\nuniverse w\n\nopen CategoryTheory Sheaf Limits Opposite\n\nnamespace CategoryTheory\n\nvariable {C : Type*} (D : Type*) [Category* C] [Category* D] {FD : D → D → Type*} {CD : D → Type w}\n  [∀ X Y, FunLike (FD X Y) (CD X) (CD Y)] [ConcreteCategory.{w} D FD]\n\nlemma regularTopology.isLocallySurjective_iff [Preregular C] {F G : Cᵒᵖ ⥤ D} (f : F ⟶ G) :\n    Presheaf.IsLocallySurjective (regularTopology C) f ↔\n      ∀ (X : C) (y : ToType (G.obj ⟨X⟩)), (∃ (X' : C) (φ : X' ⟶ X) (_ : EffectiveEpi φ)\n        (x : ToType (F.obj ⟨X'⟩)),\n        f.app ⟨X'⟩ x = G.map ⟨φ⟩ y) := by\n  constructor\n  · intro ⟨h⟩ X y\n    specialize h y\n    rw [regularTopology.mem_sieves_iff_hasEffectiveEpi] at h\n    obtain ⟨X', π, h, h'⟩ := h\n    exact ⟨X', π, h, h'⟩\n  · intro h\n    refine ⟨fun y ↦ ?_⟩\n    obtain ⟨X', π, h, h'⟩ := h _ y\n    rw [regularTopology.mem_sieves_iff_hasEffectiveEpi]\n    exact ⟨X', π, h, h'⟩\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma extensiveTopology.surjective_of_isLocallySurjective_sheaf_of_types [FinitaryPreExtensive C]\n    {F G : Cᵒᵖ ⥤ Type w} (f : F ⟶ G) [PreservesFiniteProducts F] [PreservesFiniteProducts G]\n      (h : Presheaf.IsLocallySurjective (extensiveTopology C) f) {X : C} :\n        Function.Surjective (f.app (op X)) := by\n  intro x\n  replace h := h.1 x\n  rw [mem_sieves_iff_contains_colimit_cofan] at h\n  obtain ⟨α, _, Y, π, h, h'⟩ := h\n  let y : (a : α) → (F.obj ⟨Y a⟩) := fun a ↦ (h' a).choose\n  let _ : Fintype α := Fintype.ofFinite _\n  let ht := (Types.productLimitCone (fun a ↦ F.obj ⟨Y a⟩)).isLimit\n  let ht' := (Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse\n    (Cocone.op (Cofan.mk X π))).symm h.some.op\n  let i : ((a : α) → (F.obj ⟨Y a⟩)) ≅ (F.obj ⟨X⟩) :=\n    ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht')\n      (Discrete.natIso (fun _ ↦ (Iso.refl (F.obj ⟨_⟩))))\n  refine ⟨i.hom y, ?_⟩\n  apply Concrete.isLimit_ext _ (isLimitOfPreserves G ht')\n  intro ⟨a⟩\n  simp only [Functor.comp_obj, Discrete.opposite_inverse_obj, Functor.op_obj, Discrete.functor_obj,\n    Functor.mapCone_pt, Cone.whisker_pt, Cocone.op_pt, Cofan.mk_pt, Functor.const_obj_obj,\n    Functor.mapCone_π_app, Cone.whisker_π, Cocone.op_π, Functor.whiskerLeft_app, NatTrans.op_app,\n    Cofan.mk_ι_app]\n  rw [← (h' a).choose_spec, ← NatTrans.naturality_apply (φ := f)]\n  simp only [IsLimit.conePointsIsoOfNatIso_hom, ← comp_apply, i]\n  erw [IsLimit.map_π]\n  rfl\n\nTarget:\nlemma extensiveTopology.presheafIsLocallySurjective_iff [FinitaryPreExtensive C] {F G : Cᵒᵖ ⥤ D}\n    (f : F ⟶ G) [PreservesFiniteProducts F] [PreservesFiniteProducts G]\n      [PreservesFiniteProducts (forget D)] : Presheaf.IsLocallySurjective (extensiveTopology C) f ↔\n        ∀ (X : C), Function.Surjective (f.app (op X)) :=\n\nProof body:\n","rejected":"```lean\nby\n  constructor\n  · rw [Presheaf.isLocallySurjective_iff_whisker_forget (J := extensiveTopology C)]\n    exact fun h _ ↦\n      surjective_of_isLocallySurjective_sheaf_of_types (Functor.whiskerRight f (forget D)) h\n  · exact fun a ↦\n      Presheaf.isLocallySurjective_of_surjective _ _ (fun _ ↦ a _)\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Sites","family_id":"extensivetopology","file_id":"mathlib/Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean","sample_id":"fffbd570a9bc968d3091c7ef16d1e7aecf0edf42a3c5fce61027b019b7b6b538"},"split":"train"}
{"binding":{"chosen_proof_sha256":"741b44ba22accbbd70bb07bc169d65e562975536f8496b4d492a4c4b786fdd83","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7efc732133ef1a6bdb2e645313e4807fc50596a44444a2a39484ea3f4ee979c1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6c6463defa0ee8680fe4b3b439ae453214eb49f285173f4fb331b4b3b6e3c669","source_sha256":"f5ba420ebee6ce69c4b05f94e3188f18366a5272d50259681e169766e1289584","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  lift N to {N : Submodule R P // N.FG} using hN\n  apply_fun (Module.fgSystem.equiv R P).symm.toLinearMap.rTensor M at eq\n  apply_fun directLimitLeft _ _ at eq\n  simp_rw [← LinearMap.rTensor_comp_apply, ← (LinearEquiv.eq_toLinearMap_symm_comp _ _).mpr\n    (Module.fgSystem.equiv_comp_of N), directLimitLeft_rTensor_of] at eq\n  have ⟨N', le, eq⟩ := Module.DirectLimit.exists_eq_of_of_eq eq\n  exact ⟨_, N'.2, le, eq⟩","hard_negative":false,"metrics":{"chosen_tokens":101,"rejected_tokens":106,"token_jaccard":0.928571,"token_length_ratio":1.049505},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"0d9684eeb4b924392c1cf04413710a5e4332c159efdbd6a9769a41d394954fcb","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Colimit.Finiteness\npublic import Mathlib.LinearAlgebra.TensorProduct.DirectLimit\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu\n-/\n/-!\n# Tensor product with direct limit of finitely generated submodules\n\nWe show that if `M` and `P` are arbitrary modules and `N` is a finitely generated submodule\nof a module `P`, then two elements of `N ⊗ M` have the same image in `P ⊗ M` if and only if\nthey already have the same image in `N' ⊗ M` for some finitely generated submodule `N' ≥ N`.\nThis is the theorem `Submodule.FG.exists_rTensor_fg_inclusion_eq`. The key facts used are\nthat every module is the direct limit of its finitely generated submodules and that tensor\nproduct preserves colimits.\n-/\n\npublic section\n\nopen TensorProduct\n\nvariable {R M P : Type*} [CommSemiring R]\nvariable [AddCommMonoid M] [Module R M] [AddCommMonoid P] [Module R P]\n\nTarget:\ntheorem Submodule.FG.exists_rTensor_fg_inclusion_eq {N : Submodule R P} (hN : N.FG)\n    {x y : N ⊗[R] M} (eq : N.subtype.rTensor M x = N.subtype.rTensor M y) :\n    ∃ N', N'.FG ∧ ∃ h : N ≤ N', (N.inclusion h).rTensor M x = (N.inclusion h).rTensor M y :=\n\nProof body:\n","rejected":"by\n  classical\n  lift N to {N : Submodule R P // N.FG} using hN\n  apply_fun (Module.fgSystem.equiv R P).symm.toLinearMap.rTensor M at eq\n  apply_fun directLimitLeft _ _ at eq\n  simp_rw [← LinearMap.rTensor_comp_apply, ← (LinearEquiv.eq_toLinearMap_symm_comp _ _).mpr\n    (Module.fgSystem.equiv_comp_of N), directLimitLeft_rTensor_of] at eq\n  have ⟨N', le, eq⟩ := Module.DirectLimit.exists_eq_of_of_eq eq\n  exact ⟨_, N'.2, le, eq⟩\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Colimit","family_id":"submodule","file_id":"mathlib/Mathlib/Algebra/Colimit/TensorProduct.lean","sample_id":"6c6463defa0ee8680fe4b3b439ae453214eb49f285173f4fb331b4b3b6e3c669"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8563be0743a9370b287fc6eea19b728e9532ff758a843c12bb7a89f00678f9fd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"23133e4cdbd3c18af59ef41f856739915fb4a88cdab3bfe7a080ab4e9717b499","source_sha256":"7fb2a027e17dd48df4c5d05af7f579e28e12ec18c52dd1aaeae9c6854e93da06","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← cons_comp_cycleRange, map_perm, Matrix.vecCons]\n  simp [Units.smul_def]","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":8,"token_jaccard":0.05,"token_length_ratio":0.444444},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"0da39cfbc840edc20c3c262711a5f19d80075ff165598bd1d86f47a4d5a05a1a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Alternating.Curry\npublic import Mathlib.GroupTheory.Perm.Fin\npublic import Mathlib.Data.Fin.Parity\n\nNamespace:\nAlternatingMap\n\nLocal context:\n/-\nCopyright (c) 2025 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Uncurrying alternating maps\n\nGiven a function `f` which is linear in the first argument\nand is alternating form in the other `n` arguments,\nthis file defines an alternating form `AlternatingMap.alternatizeUncurryFin f` in `n + 1` arguments.\n\nThis function is given by\n```\nAlternatingMap.alternatizeUncurryFin f v =\n  ∑ i : Fin (n + 1), (-1) ^ (i : ℕ) • f (v i) (removeNth i v)\n```\n\nGiven an alternating map `f` of `n + 1` arguments,\neach term in the sum above written for `f.curryLeft` equals the original map,\nthus `f.curryLeft.alternatizeUncurryFin = (n + 1) • f`.\n\nWe do not multiply the result of `alternatizeUncurryFin` by `(n + 1)⁻¹`\nso that the construction works for `R`-multilinear maps over any commutative ring `R`,\nnot only a field of characteristic zero.\n\n## Main results\n\n- `AlternatingMap.alternatizeUncurryFin_curryLeft`:\n  the round-trip formula for currying/uncurrying, see above.\n\n- `AlternatingMap.alternatizeUncurryFin_alternatizeUncurryFinLM_comp_of_symmetric`:\n  If `f` is a symmetric bilinear map taking values in the space of alternating maps,\n  then the twice uncurried `f` is zero.\n\nA version of the latter theorem for continuous alternating maps\nwill be used to prove that the second exterior derivative of a differential form is zero.\n-/\n\n@[expose] public section\n\nopen Fin Function\n\nnamespace AlternatingMap\n\nvariable {R : Type*} {M M₂ N N₂ : Type*} [CommRing R] [AddCommGroup M]\n  [AddCommGroup M₂] [AddCommGroup N] [AddCommGroup N₂] [Module R M] [Module R M₂]\n  [Module R N] [Module R N₂] {n : ℕ}\n\n/-- If `f` is a `(n + 1)`-multilinear alternating map, `x` is an element of the domain,\nand `v` is an `n`-vector, then the value of `f` at `v` with `x` inserted at the `p`th place\nequals `(-1) ^ p` times the value of `f` at `v` with `x` prepended. -/\n\nTarget:\ntheorem map_insertNth (f : M [⋀^Fin (n + 1)]→ₗ[R] N) (p : Fin (n + 1)) (x : M) (v : Fin n → M) :\n    f (p.insertNth x v) = (-1) ^ (p : ℕ) • f (Matrix.vecCons x v) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"414ffd38143406d29403ac97d39105771ea951770a10afa1be9dcbaecdef4717","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Alternating","family_id":"map_insertnth","file_id":"mathlib/Mathlib/LinearAlgebra/Alternating/Uncurry/Fin.lean","sample_id":"23133e4cdbd3c18af59ef41f856739915fb4a88cdab3bfe7a080ab4e9717b499"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7aefead92a5663164517806ecfcb8b153d186d51071288da4cb411af6d57db41","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e479d953b87af298df6f7cefb47da40ed2d5b92620080f3988416adfc6a4f973","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c860402497f1b4dc3bdc93021a1e3e3c1ea1267241369ebbfdacfe5570fa2d4c","source_sha256":"7fb2a027e17dd48df4c5d05af7f579e28e12ec18c52dd1aaeae9c6854e93da06","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext v\n  simp [alternatizeUncurryFin_apply, ← map_insertNth]\n\nvariable {S : Type*} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N]","hard_negative":true,"metrics":{"chosen_tokens":32,"rejected_tokens":5,"token_jaccard":0.038462,"token_length_ratio":0.15625},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"0db1c93e6172899c3e5cca6d143d76e4e9f9a0d2546c4cc028be9853ab6fc386","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Alternating.Curry\npublic import Mathlib.GroupTheory.Perm.Fin\npublic import Mathlib.Data.Fin.Parity\n\nNamespace:\nAlternatingMap\n\nLocal context:\n/-\nCopyright (c) 2025 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Uncurrying alternating maps\n\nGiven a function `f` which is linear in the first argument\nand is alternating form in the other `n` arguments,\nthis file defines an alternating form `AlternatingMap.alternatizeUncurryFin f` in `n + 1` arguments.\n\nThis function is given by\n```\nAlternatingMap.alternatizeUncurryFin f v =\n  ∑ i : Fin (n + 1), (-1) ^ (i : ℕ) • f (v i) (removeNth i v)\n```\n\nGiven an alternating map `f` of `n + 1` arguments,\neach term in the sum above written for `f.curryLeft` equals the original map,\nthus `f.curryLeft.alternatizeUncurryFin = (n + 1) • f`.\n\nWe do not multiply the result of `alternatizeUncurryFin` by `(n + 1)⁻¹`\nso that the construction works for `R`-multilinear maps over any commutative ring `R`,\nnot only a field of characteristic zero.\n\n## Main results\n\n- `AlternatingMap.alternatizeUncurryFin_curryLeft`:\n  the round-trip formula for currying/uncurrying, see above.\n\n- `AlternatingMap.alternatizeUncurryFin_alternatizeUncurryFinLM_comp_of_symmetric`:\n  If `f` is a symmetric bilinear map taking values in the space of alternating maps,\n  then the twice uncurried `f` is zero.\n\nA version of the latter theorem for continuous alternating maps\nwill be used to prove that the second exterior derivative of a differential form is zero.\n-/\n\n@[expose] public section\n\nopen Fin Function\n\nnamespace AlternatingMap\n\nvariable {R : Type*} {M M₂ N N₂ : Type*} [CommRing R] [AddCommGroup M]\n  [AddCommGroup M₂] [AddCommGroup N] [AddCommGroup N₂] [Module R M] [Module R M₂]\n  [Module R N] [Module R N₂] {n : ℕ}\n\n/-- If `f` is a `(n + 1)`-multilinear alternating map, `x` is an element of the domain,\nand `v` is an `n`-vector, then the value of `f` at `v` with `x` inserted at the `p`th place\nequals `(-1) ^ p` times the value of `f` at `v` with `x` prepended. -/\ntheorem map_insertNth (f : M [⋀^Fin (n + 1)]→ₗ[R] N) (p : Fin (n + 1)) (x : M) (v : Fin n → M) :\n    f (p.insertNth x v) = (-1) ^ (p : ℕ) • f (Matrix.vecCons x v) := by\n  rw [← cons_comp_cycleRange, map_perm, Matrix.vecCons]\n  simp [Units.smul_def]\n\ntheorem neg_one_pow_smul_map_insertNth (f : M [⋀^Fin (n + 1)]→ₗ[R] N) (p : Fin (n + 1)) (x : M)\n    (v : Fin n → M) :\n    (-1) ^ (p : ℕ) • f (p.insertNth x v) = f (Matrix.vecCons x v) := by\n  rw [map_insertNth, smul_smul, ← pow_add, Even.neg_one_pow, one_smul]\n  use p\n\n/-- Let `v` be an `(n + 1)`-tuple with two equal elements `v i = v j`, `i ≠ j`.\nLet `w i` (resp., `w j`) be the vector `v` with `i`th (resp., `j`th) element removed.\nThen `(-1) ^ i • f (w i) + (-1) ^ j • f (w j) = 0`.\nThis follows from the fact that these two vectors differ by a permutation of sign `(-1) ^ (i + j)`.\n\nThese are the only two nonzero terms in the proof of `map_eq_zero_of_eq`\nin the definition of `alternatizeUncurryFin` below. -/\ntheorem neg_one_pow_smul_map_removeNth_add_eq_zero_of_eq (f : M [⋀^Fin n]→ₗ[R] N)\n    {v : Fin (n + 1) → M} {i j : Fin (n + 1)} (hvij : v i = v j) (hij : i ≠ j) :\n    (-1) ^ (i : ℕ) • f (i.removeNth v) + (-1) ^ (j : ℕ) • f (j.removeNth v) = 0 := by\n  rcases exists_succAbove_eq hij with ⟨i, rfl⟩\n  obtain ⟨m, rfl⟩ : ∃ m, m + 1 = n := by simp [i.pos]\n  rw [← (i.predAbove j).insertNth_self_removeNth (removeNth _ _), ← removeNth_removeNth_eq_swap,\n    removeNth, succAbove_succAbove_predAbove, map_insertNth, ← neg_one_pow_smul_map_insertNth,\n    insertNth_removeNth, update_eq_self_iff.2, smul_smul, ← pow_add,\n    neg_one_pow_succAbove_add_predAbove, neg_smul, pow_add, mul_smul,\n    smul_smul (_ ^ i.val), ← sq, ← pow_mul, pow_mul', neg_one_pow_two, one_pow, one_smul,\n    neg_add_cancel]\n  exact hvij.symm\n\n/-- Given a function which is linear in the first argument\nand is alternating in the other `n` arguments,\nbuild an alternating form in `n + 1` arguments.\n\nThe function is given by\n```\nalternatizeUncurryFin f v = ∑ i : Fin (n + 1), (-1) ^ (i : ℕ) • f (v i) (removeNth i v)\n```\n\nNote that the round-trip with `curryFin` multiplies the form by `n + 1`,\nsince we want to avoid division in this definition. -/\ndef alternatizeUncurryFin (f : M →ₗ[R] M [⋀^Fin n]→ₗ[R] N) :\n    M [⋀^Fin (n + 1)]→ₗ[R] N where\n  toMultilinearMap :=\n    ∑ p : Fin (n + 1), (-1) ^ (p : ℕ) • LinearMap.uncurryMid p (toMultilinearMapLM ∘ₗ f)\n  map_eq_zero_of_eq' := by\n    intro v i j hvij hij\n    suffices ∑ k : Fin (n + 1), (-1) ^ (k : ℕ) • f (v k) (k.removeNth v) = 0 by simpa\n    calc\n      _ = (-1) ^ (i : ℕ) • f (v i) (i.removeNth v) + (-1) ^ (j : ℕ) • f (v j) (j.removeNth v) := by\n        refine Fintype.sum_eq_add _ _ hij fun k ⟨hki, hkj⟩ ↦ ?_\n        rcases exists_succAbove_eq hki.symm with ⟨i, rfl⟩\n        rcases exists_succAbove_eq hkj.symm with ⟨j, rfl⟩\n        rw [(f (v k)).map_eq_zero_of_eq _ hvij (ne_of_apply_ne _ hij), smul_zero]\n      _ = 0 := by\n        rw [hvij, neg_one_pow_smul_map_removeNth_add_eq_zero_of_eq] <;> assumption\n\ntheorem alternatizeUncurryFin_apply (f : M →ₗ[R] M [⋀^Fin n]→ₗ[R] N) (v : Fin (n + 1) → M) :\n    alternatizeUncurryFin f v = ∑ i : Fin (n + 1), (-1) ^ (i : ℕ) • f (v i) (removeNth i v) := by\n  simp [alternatizeUncurryFin]\n\n@[simp]\ntheorem alternatizeUncurryFin_add (f g : M →ₗ[R] M [⋀^Fin n]→ₗ[R] N) :\n    alternatizeUncurryFin (f + g) = alternatizeUncurryFin f + alternatizeUncurryFin g := by\n  ext\n  simp [alternatizeUncurryFin_apply, Finset.sum_add_distrib]\n\n@[simp]\n\nTarget:\nlemma alternatizeUncurryFin_curryLeft (f : M [⋀^Fin (n + 1)]→ₗ[R] N) :\n    alternatizeUncurryFin (curryLeft f) = (n + 1) • f :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_c860402497f1","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"7a492ef398182a1fda647614b5a804310589449f64085d07b3becbc74516bbc2","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Alternating","family_id":"alternatizeuncurryfin_curryleft","file_id":"mathlib/Mathlib/LinearAlgebra/Alternating/Uncurry/Fin.lean","sample_id":"c860402497f1b4dc3bdc93021a1e3e3c1ea1267241369ebbfdacfe5570fa2d4c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7b7262505c003dbad9c3aa96a2764e75ade60fa912085a50b5740b06b8e012c5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a8d869c19b5e1d1d76d3c28a6dc188be2ec2a1a13841d22f5d203f9a92b23e39","source_sha256":"93096837382e572977abeb7e5338f6f449e78fd61b5b1c8bf94ffe55e531a689","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro H\n  refine fermatLastTheoremThree_of_three_dvd_only_c (fun a b c hc ha hb ⟨x, hx⟩ hcoprime h ↦ ?_)\n  refine H a b c 1 (by simp [hc]) (fun hdvd ↦ ha ?_) (fun hdvd ↦ hb ?_) ?_ ?_ ?_\n  · rwa [← Ideal.norm_dvd_iff (hζ.prime_norm_toInteger_sub_one_of_prime_ne_two' (by decide)),\n      hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide)] at hdvd\n  · rwa [← Ideal.norm_dvd_iff (hζ.prime_norm_toInteger_sub_one_of_prime_ne_two' (by decide)),\n      hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide)] at hdvd\n  · exact dvd_trans hζ.toInteger_sub_one_dvd_prime' ⟨x, by simp [hx]⟩\n  · exact IsCoprime.intCast hcoprime\n  · simpa using mod_cast h","hard_negative":false,"metrics":{"chosen_tokens":144,"rejected_tokens":3,"token_jaccard":0.0625,"token_length_ratio":0.020833},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"0db391223a64e0ebea4439977f721c7b19ac386e18bcef7133e4c6935c3471d3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.FLT.Basic\npublic import Mathlib.NumberTheory.NumberField.Cyclotomic.PID\npublic import Mathlib.NumberTheory.NumberField.Cyclotomic.Three\npublic import Mathlib.Algebra.Ring.Divisibility.Lemmas\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Riccardo Brasca. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Riccardo Brasca, Sanyam Gupta, Omar Haddad, David Lowry-Duda,\n  Lorenzo Luccioli, Pietro Monticone, Alexis Saurin, Florent Schaffhauser\n-/\n/-!\n# Fermat Last Theorem in the case `n = 3`\nThe goal of this file is to prove Fermat's Last Theorem in the case `n = 3`.\n\n## Main results\n* `fermatLastTheoremThree`: Fermat's Last Theorem for `n = 3`: if `a b c : ℕ` are all non-zero then\n  `a ^ 3 + b ^ 3 ≠ c ^ 3`.\n\n## Implementation details\nWe follow the proof in <https://webusers.imj-prg.fr/~marc.hindry/Cours-arith.pdf>, page 43.\n\nThe strategy is the following:\n* The so-called \"Case 1\", when `3 ∣ a * b * c` is completely elementary and is proved using\n  congruences modulo `9`.\n* To prove case 2, we consider the generalized equation `a ^ 3 + b ^ 3 = u * c ^ 3`, where `a`, `b`,\n  and `c` are in the cyclotomic ring `ℤ[ζ₃]` (where `ζ₃` is a primitive cube root of unity) and `u`\n  is a unit of `ℤ[ζ₃]`. `FermatLastTheoremForThree_of_FermatLastTheoremThreeGen` (whose proof is\n  rather elementary on paper) says that to prove Fermat's last theorem for exponent `3`, it is\n  enough to prove that this equation has no solutions such that `c ≠ 0`, `¬ λ ∣ a`, `¬ λ ∣ b`,\n  `λ ∣ c` and `IsCoprime a b` (where we set `λ := ζ₃ - 1`). We call such a tuple a `Solution'`.\n  A `Solution` is the same as a `Solution'` with the additional assumption that `λ ^ 2 ∣ a + b`.\n  We then prove that, given `S' : Solution'`, there is `S : Solution` such that the multiplicity of\n  `λ = ζ₃ - 1` in `c` is the same in `S'` and `S` (see `exists_Solution_of_Solution'`).\n  In particular it is enough to prove that no `Solution` exists. The key point is a descent argument\n  on the multiplicity of `λ` in `c`: starting with `S : Solution` we can find `S₁ : Solution` with\n  multiplicity strictly smaller (see `exists_Solution_multiplicity_lt`) and this finishes the proof.\n  To construct `S₁` we go through a `Solution'` and then back to a `Solution`. More importantly, we\n  cannot control the unit `u`, and this is the reason why we need to consider the generalized\n  equation `a ^ 3 + b ^ 3 = u * c ^ 3`. The construction is completely explicit, but it depends\n  crucially on `IsCyclotomicExtension.Rat.Three.eq_one_or_neg_one_of_unit_of_congruent`, a special\n  case of Kummer's lemma.\n* Note that we don't prove Case 1 for the generalized equation (in particular we don't prove that\n  the generalized equation has no nontrivial solutions). This is because the proof, even if\n  elementary on paper, would be quite annoying to formalize: indeed it involves a lot of explicit\n  computations in `ℤ[ζ₃] / (λ)`: this ring is isomorphic to `ℤ / 9ℤ`, but of course, even if we\n  construct such an isomorphism, tactics like `decide` would not work.\n\n-/\n\nsection case1\n\nopen ZMod\n\nlemma cube_of_castHom_ne_zero {n : ZMod 9} :\n    castHom (show 3 ∣ 9 by simp) (ZMod 3) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8 := by\n  revert n; decide\n\nlemma cube_of_not_dvd {n : ℤ} (h : ¬ 3 ∣ n) :\n    (n : ZMod 9) ^ 3 = 1 ∨ (n : ZMod 9) ^ 3 = 8 := by\n  apply cube_of_castHom_ne_zero\n  rwa [map_intCast, Ne, ZMod.intCast_zmod_eq_zero_iff_dvd]\n\n/-- If `a b c : ℤ` are such that `¬ 3 ∣ a * b * c`, then `a ^ 3 + b ^ 3 ≠ c ^ 3`. -/\ntheorem fermatLastTheoremThree_case_1 {a b c : ℤ} (hdvd : ¬ 3 ∣ a * b * c) :\n    a ^ 3 + b ^ 3 ≠ c ^ 3 := by\n  simp_rw [Int.prime_three.dvd_mul, not_or] at hdvd\n  apply mt (congrArg (Int.cast : ℤ → ZMod 9))\n  simp_rw [Int.cast_add, Int.cast_pow]\n  rcases cube_of_not_dvd hdvd.1.1 with ha | ha <;>\n  rcases cube_of_not_dvd hdvd.1.2 with hb | hb <;>\n  rcases cube_of_not_dvd hdvd.2 with hc | hc <;>\n  rw [ha, hb, hc] <;> decide\n\nend case1\n\nsection case2\n\nlemma three_dvd_b_of_dvd_a_of_gcd_eq_one_of_case2 {a b c : ℤ} (ha : a ≠ 0)\n    (Hgcd : Finset.gcd {a, b, c} id = 1) (h3a : 3 ∣ a) (HF : a ^ 3 + b ^ 3 + c ^ 3 = 0)\n    (H : ∀ a b c : ℤ, c ≠ 0 → ¬ 3 ∣ a → ¬ 3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3) :\n    3 ∣ b := by\n  have hbc : IsCoprime (-b) (-c) := by\n    refine IsCoprime.neg_neg ?_\n    rw [add_comm (a ^ 3), add_assoc, add_comm (a ^ 3), ← add_assoc] at HF\n    refine isCoprime_of_gcd_eq_one_of_FLT ?_ HF\n    convert! Hgcd using 2\n    rw [Finset.pair_comm, Finset.insert_comm]\n  by_contra! h3b\n  by_cases h3c : 3 ∣ c\n  · apply h3b\n    rw [add_assoc, add_comm (b ^ 3), ← add_assoc] at HF\n    exact dvd_c_of_prime_of_dvd_a_of_dvd_b_of_FLT Int.prime_three h3a h3c HF\n  · refine H (-b) (-c) a ha (by simp [h3b]) (by simp [h3c]) h3a hbc ?_\n    rw [add_eq_zero_iff_eq_neg, ← (show Odd 3 by decide).neg_pow] at HF\n    rw [← HF]\n    ring\n\nopen Finset in\nlemma fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 {a b c : ℤ} (ha : a ≠ 0)\n    (h3a : 3 ∣ a) (Hgcd : Finset.gcd {a, b, c} id = 1)\n    (H : ∀ a b c : ℤ, c ≠ 0 → ¬ 3 ∣ a → ¬ 3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3) :\n    a ^ 3 + b ^ 3 + c ^ 3 ≠ 0 := by\n  intro HF\n  apply (show ¬(3 ∣ (1 : ℤ)) by decide)\n  rw [← Hgcd]\n  refine dvd_gcd (fun x hx ↦ ?_)\n  simp only [mem_insert, mem_singleton] at hx\n  have h3b : 3 ∣ b := by\n    refine three_dvd_b_of_dvd_a_of_gcd_eq_one_of_case2 ha ?_ h3a HF H\n    simp only [← Hgcd, gcd_insert, gcd_singleton, id_eq, ← Int.abs_eq_normalize]\n  rcases hx with hx | hx | hx\n  · exact hx ▸ h3a\n  · exact hx ▸ h3b\n  · simpa [hx] using dvd_c_of_prime_of_dvd_a_of_dvd_b_of_FLT Int.prime_three h3a h3b HF\n\nopen Finset Int in\n/--\nTo prove Fermat's Last Theorem for `n = 3`, it is enough to show that for all `a`, `b`, `c`\nin `ℤ` such that `c ≠ 0`, `¬ 3 ∣ a`, `¬ 3 ∣ b`, `a` and `b` are coprime and `3 ∣ c`, we have\n`a ^ 3 + b ^ 3 ≠ c ^ 3`.\n-/\ntheorem fermatLastTheoremThree_of_three_dvd_only_c\n    (H : ∀ a b c : ℤ, c ≠ 0 → ¬ 3 ∣ a → ¬ 3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3) :\n    FermatLastTheoremFor 3 := by\n  rw [fermatLastTheoremFor_iff_int]\n  refine fermatLastTheoremWith_of_fermatLastTheoremWith_coprime (fun a b c ha hb hc Hgcd hF ↦ ?_)\n  by_cases h1 : 3 ∣ a * b * c\n  swap\n  · exact fermatLastTheoremThree_case_1 h1 hF\n  rw [prime_three.dvd_mul, prime_three.dvd_mul] at h1\n  rw [← sub_eq_zero, sub_eq_add_neg, ← (show Odd 3 by decide).neg_pow] at hF\n  rcases h1 with (h3a | h3b) | h3c\n  · refine fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 ha h3a ?_ H hF\n    simp only [← Hgcd, gcd_insert, gcd_singleton, id_eq, ← abs_eq_normalize, abs_neg]\n  · rw [add_comm (a ^ 3)] at hF\n    refine fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 hb h3b ?_ H hF\n    simp only [← Hgcd, insert_comm, gcd_insert, gcd_singleton, id_eq, ← abs_eq_normalize, abs_neg]\n  · rw [add_comm _ ((-c) ^ 3), ← add_assoc] at hF\n    refine fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 (neg_ne_zero.2 hc) (by simp [h3c])\n      ?_ H hF\n    rw [Finset.insert_comm (-c), Finset.pair_comm (-c) b]\n    simp only [← Hgcd, gcd_insert, gcd_singleton, id_eq, ← abs_eq_normalize, abs_neg]\n\nsection eisenstein\n\nopen NumberField IsCyclotomicExtension.Rat.Three\n\nvariable {K : Type*} [Field K]\nvariable {ζ : K} (hζ : IsPrimitiveRoot ζ 3)\n\nlocal notation3 \"η\" => (IsPrimitiveRoot.isUnit (hζ.toInteger_isPrimitiveRoot) (by decide)).unit\nlocal notation3 \"λ\" => hζ.toInteger - 1\n\n/-- `FermatLastTheoremForThreeGen` is the statement that `a ^ 3 + b ^ 3 = u * c ^ 3` has no\nnontrivial solutions in `𝓞 K` for all `u : (𝓞 K)ˣ` such that `¬ λ ∣ a`, `¬ λ ∣ b` and `λ ∣ c`.\nThe reason to consider `FermatLastTheoremForThreeGen` is to make a descent argument working. -/\ndef FermatLastTheoremForThreeGen : Prop :=\n  ∀ a b c : 𝓞 K, ∀ u : (𝓞 K)ˣ, c ≠ 0 → ¬ λ ∣ a → ¬ λ ∣ b → λ ∣ c → IsCoprime a b →\n    a ^ 3 + b ^ 3 ≠ u * c ^ 3\n\n/-- To prove `FermatLastTheoremFor 3`, it is enough to prove `FermatLastTheoremForThreeGen`. -/\n\nTarget:\nlemma FermatLastTheoremForThree_of_FermatLastTheoremThreeGen\n    [NumberField K] [IsCyclotomicExtension {3} ℚ K] :\n    FermatLastTheoremForThreeGen hζ → FermatLastTheoremFor 3 :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/FLT","family_id":"fermatlasttheoremforthree_of_fermatlasttheoremthreegen","file_id":"mathlib/Mathlib/NumberTheory/FLT/Three.lean","sample_id":"a8d869c19b5e1d1d76d3c28a6dc188be2ec2a1a13841d22f5d203f9a92b23e39"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5710d934b0159d30d30972bfc5b7ddfd3eb8cd27b8c73a16ac7a346cdcff98c4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cd30468868389860cfba3fec091f109642c834aceaa42da567fd878f6c1965fc","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"519e1718b1b4803bcd063f65500a31317c48e550e0d65b418a9de8968712bef5","source_sha256":"c2ae48c6bd398849186d6b191b0d4c349dab84b9865f7c6ccfbd7ab5b2b3ca24","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rfl\n\nvariable (φ : A →ₐc[R] B)","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":20,"token_jaccard":0.75,"token_length_ratio":1.333333},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"0dd952312495f24c6b1343e1a91489e0d6647c9097cd6a24460b0899e08ab85f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Coalgebra.Hom\npublic import Mathlib.RingTheory.Bialgebra.Basic\n\nNamespace:\nBialgHom\n\nLocal context:\n/-\nCopyright (c) 2024 Amelia Livingston. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov, Amelia Livingston\n-/\n/-!\n# Homomorphisms of `R`-bialgebras\n\nThis file defines bundled homomorphisms of `R`-bialgebras. We simply mimic\n`Mathlib/Algebra/Algebra/Hom.lean`.\n\n## Main definitions\n\n* `BialgHom R A B`: the type of `R`-bialgebra morphisms from `A` to `B`.\n* `Bialgebra.counitBialgHom R A : A →ₐc[R] R`: the counit of a bialgebra as a bialgebra\n  homomorphism.\n\n## Notation\n\n* `A →ₐc[R] B` : `R`-bialgebra homomorphism from `A` to `B`.\n\n-/\n\n@[expose] public section\n\nopen TensorProduct Bialgebra Coalgebra Function\n\nuniverse u v w\n\n/-- Given `R`-algebras `A, B` with comultiplication maps `Δ_A, Δ_B` and counit maps\n`ε_A, ε_B`, an `R`-bialgebra homomorphism `A →ₐc[R] B` is an `R`-algebra map `f` such that\n`ε_B ∘ f = ε_A` and `(f ⊗ f) ∘ Δ_A = Δ_B ∘ f`. -/\nstructure BialgHom (R A B : Type*) [CommSemiring R]\n    [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]\n    [CoalgebraStruct R A] [CoalgebraStruct R B] extends A →ₗc[R] B, A →* B\n\n/-- Reinterpret a `BialgHom` as a `MonoidHom` -/\nadd_decl_doc BialgHom.toMonoidHom\n\n@[inherit_doc BialgHom]\ninfixr:25 \" →ₐc \" => BialgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐc[\" R \"] \" B => BialgHom R A B\n\n/-- `BialgHomClass F R A B` asserts `F` is a type of bundled bialgebra homomorphisms\nfrom `A` to `B`. -/\nclass BialgHomClass (F : Type*) (R A B : outParam Type*)\n    [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]\n    [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] : Prop\n    extends CoalgHomClass F R A B, MonoidHomClass F A B\n\nnamespace BialgHomClass\n\nvariable {R A B F : Type*}\n\nsection\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]\n  [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B]\n  [BialgHomClass F R A B]\n\ninstance (priority := 100) toAlgHomClass : AlgHomClass F R A B where\n  map_mul := map_mul\n  map_one := map_one\n  map_add := map_add\n  map_zero := map_zero\n  commutes := fun c r => by\n    simp only [Algebra.algebraMap_eq_smul_one, map_smul, map_one]\n\n/-- Turn an element of a type `F` satisfying `BialgHomClass F R A B` into an actual\n`BialgHom`. This is declared as the default coercion from `F` to `A →ₐc[R] B`. -/\n@[coe]\ndef toBialgHom (f : F) : A →ₐc[R] B :=\n  { CoalgHomClass.toCoalgHom f, AlgHomClass.toAlgHom f with\n    toFun := f }\n\ninstance instCoeToBialgHom :\n    CoeHead F (A →ₐc[R] B) :=\n  ⟨BialgHomClass.toBialgHom⟩\n\nend\nsection\nvariable [CommSemiring R] [Semiring A] [Bialgebra R A] [Semiring B] [Bialgebra R B]\n  [FunLike F A B] [BialgHomClass F R A B]\n\n@[simp]\ntheorem counitAlgHom_comp (f : F) :\n    (counitAlgHom R B).comp (AlgHomClass.toAlgHom f) = counitAlgHom R A :=\n  AlgHom.toLinearMap_injective (CoalgHomClass.counit_comp f)\n\n@[simp]\ntheorem map_comp_comulAlgHom (f : F) :\n    (Algebra.TensorProduct.map (AlgHomClass.toAlgHom f) (AlgHomClass.toAlgHom f)).comp\n      (comulAlgHom R A) = (comulAlgHom R B).comp (AlgHomClass.toAlgHom f) :=\n  AlgHom.toLinearMap_injective (CoalgHomClass.map_comp_comul f)\n\nend\nend BialgHomClass\n\nnamespace BialgHom\n\nvariable {R A B C D : Type*} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nsection AlgebraCoalgebra\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n  [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] [CoalgebraStruct R D]\n\ninstance funLike : FunLike (A →ₐc[R] B) A B where\n  coe f := f.toFun\n  coe_injective f g h := by\n    rcases f with ⟨_, _⟩\n    rcases g with ⟨_, _⟩\n    simp_all\n\ninstance bialgHomClass : BialgHomClass (A →ₐc[R] B) R A B where\n  map_add := fun f => f.map_add'\n  map_smulₛₗ := fun f => f.map_smul'\n  counit_comp := fun f => f.counit_comp\n  map_comp_comul := fun f => f.map_comp_comul\n  map_mul := fun f => f.map_mul'\n  map_one := fun f => f.map_one'\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R α β : Type*} [CommSemiring R]\n    [Semiring α] [Algebra R α] [Semiring β]\n    [Algebra R β] [CoalgebraStruct R α] [CoalgebraStruct R β]\n    (f : α →ₐc[R] β) : α → β := f\n\ninitialize_simps_projections BialgHom (toFun → apply, as_prefix toCoalgHom)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [FunLike F A B] [BialgHomClass F R A B] (f : F) :\n    ⇑(f : A →ₐc[R] B) = f :=\n  rfl\n\n@[simp]\ntheorem coe_mk {f : A →ₗc[R] B} (h h₁) : ((⟨f, h, h₁⟩ : A →ₐc[R] B) : A → B) = f :=\n  rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₀ h₁ h₂ h₃ h₄ h₅) :\n    ⇑(⟨⟨⟨⟨f, h₀⟩, h₁⟩, h₂, h₃⟩, h₄, h₅⟩ : A →ₐc[R] B) = f :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_coalgHom_mk {f : A →ₗc[R] B} (h h₁) :\n    ((⟨f, h, h₁⟩ : A →ₐc[R] B) : A →ₗc[R] B) = f := by\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_toCoalgHom (f : A →ₐc[R] B) : ⇑(f : A →ₗc[R] B) = f :=\n  rfl\n\nlemma toCoalgHom_apply (f : A →ₐc[R] B) (a : A) : f.toCoalgHom a = f a := rfl\n\n@[simp, norm_cast]\ntheorem coe_toLinearMap (f : A →ₐc[R] B) : ⇑(f : A →ₗ[R] B) = f :=\n  rfl\n\n/-- Turn a bialgebra homomorphism into an algebra homomorphism. -/\n@[coe]\ndef toAlgHom (f : A →ₐc[R] B) : A →ₐ[R] B where\n  __ := f\n  map_zero' := f.map_zero\n  commutes' := by\n    simp [Algebra.algebraMap_eq_smul_one, toCoalgHom_apply]\n\ninstance : Coe (A →ₐc[R] B) (A →ₐ[R] B) := ⟨toAlgHom⟩\n\n@[simp, norm_cast]\ntheorem coe_toAlgHom (f : A →ₐc[R] B) : ⇑(f : A →ₐ[R] B) = f :=\n  rfl\n\nTarget:\ntheorem toAlgHom_toLinearMap (f : A →ₐc[R] B) :\n    ((f : A →ₐ[R] B) : A →ₗ[R] B) = f :=\n\nProof body:\n","rejected":"by\n  rfl\n\nvariable (φ : A →ₐc[R] B)\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Bialgebra","family_id":"toalghom_tolinearmap","file_id":"mathlib/Mathlib/RingTheory/Bialgebra/Hom.lean","sample_id":"519e1718b1b4803bcd063f65500a31317c48e550e0d65b418a9de8968712bef5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f77254043cd1457d306845776290c0242267d27c173b195a06e4b8478f5ec277","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3d805ac3a07f605169cd17d09107c30e012f4d0f06e8571fe5f44b26cb0ba912","source_sha256":"1ee321a81f228358638453570a83aa9bc5ea6966d6e7864b764bf262f805e0a4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Set.eq_univ_iff_forall]\n  intro x\n  rw [Set.mem_iUnion]\n  exact 𝒰.exists_eq x","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":8,"token_jaccard":0.052632,"token_length_ratio":0.4},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"0ddbd7c1276c002874d5a8f0a44004d9e5a50f9c034b2c486e6c8933ddcaac9b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Sites.MorphismProperty\npublic import Mathlib.CategoryTheory.MorphismProperty.Limits\n\nNamespace:\nAlgebraicGeometry.Scheme\n\nLocal context:\n/-\nCopyright (c) 2024 Christian Merten, Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten, Andrew Yang\n-/\n/-!\n# Covers of schemes\n\nThis file provides the basic API for covers of schemes. A cover of a scheme `X` with respect to\na morphism property `P` is a jointly surjective indexed family of scheme morphisms with\ntarget `X` all satisfying `P`.\n\n## Implementation details\n\nThe definition on the pullback of a cover along a morphism depends on results that\nare developed later in the import tree. Hence in this file, they have additional assumptions\nthat will be automatically satisfied in later files. The motivation here is that we already\nknow that these assumptions are satisfied for open immersions and hence the cover API for open\nimmersions can be used to deduce these assumptions in the general case.\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen TopologicalSpace CategoryTheory Opposite CategoryTheory.Limits\n\nuniverse v v₁ v₂ u\n\nnamespace AlgebraicGeometry\n\nnamespace Scheme\n\nvariable (K : Precoverage Scheme.{u})\n\n/-- A coverage `K` on `Scheme` is called jointly surjective if every covering family in `K`\nis jointly surjective. -/\nclass JointlySurjective (K : Precoverage Scheme.{u}) : Prop where\n  exists_eq {X : Scheme.{u}} (S : Presieve X) (hS : S ∈ K X) (x : X) :\n    ∃ (Y : Scheme.{u}) (g : Y ⟶ X), S g ∧ x ∈ Set.range g\n\n/-- A cover of `X` in the coverage `K` is a `0`-hypercover for `K`. -/\nabbrev Cover (K : Precoverage Scheme.{u}) := Precoverage.ZeroHypercover.{v} K\n\nvariable {K}\n\nvariable {X Y Z : Scheme.{u}} (𝒰 : X.Cover K) (f : X ⟶ Z) (g : Y ⟶ Z)\nvariable [∀ x, HasPullback (𝒰.f x ≫ f) g]\n\nlemma Cover.exists_eq [JointlySurjective K] (𝒰 : X.Cover K) (x : X) :\n    ∃ i y, 𝒰.f i y = x := by\n  obtain ⟨Y, g, ⟨i⟩, y, hy⟩ := JointlySurjective.exists_eq 𝒰.presieve₀ 𝒰.mem₀ x\n  use i, y\n\n/-- A choice of an index `i` such that `x` is in the range of `𝒰.f i`. -/\ndef Cover.idx [JointlySurjective K] (𝒰 : X.Cover K) (x : X) : 𝒰.I₀ :=\n  (𝒰.exists_eq x).choose\n\nlemma Cover.covers [JointlySurjective K] (𝒰 : X.Cover K) (x : X) :\n    x ∈ Set.range (𝒰.f (𝒰.idx x)) :=\n  (𝒰.exists_eq x).choose_spec\n\nTarget:\ntheorem Cover.iUnion_range [JointlySurjective K] {X : Scheme.{u}} (𝒰 : X.Cover K) :\n    ⋃ i, Set.range (𝒰.f i) = Set.univ :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"8b989a86843ca5ae3710db04a49f1153f8ddd365f27f0d927511a31e0a169f4c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Cover","family_id":"cover","file_id":"mathlib/Mathlib/AlgebraicGeometry/Cover/MorphismProperty.lean","sample_id":"3d805ac3a07f605169cd17d09107c30e012f4d0f06e8571fe5f44b26cb0ba912"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f1c76c9c21cec19c82ec1c29aff36596a09a140cc79532bb60d64518174b61b0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6f9d3bcbb3b89d6ec55e4c066a99b476820b4a256f60b7c9eb9ba07f8b3b75d4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2f46fd8127f7a4707cce4dc2e25df1a6780c736d06d1699ff4e93a369a16eb74","source_sha256":"a6234c28d344c069679c6c490f57e0cad723a087d9f05f1fbee4f95afdc863a6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  convert! LinearMap.iSupIndep_map _ hv (iSupIndep_range_lsingle ι R R)\n  ext; simp [mem_span_singleton]","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":5,"token_jaccard":0.090909,"token_length_ratio":0.25},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"0df90959052c85b7da770b80286a359296eb4c0dc02f03b95af008fccb619e05","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.Tuple.Reflection\npublic import Mathlib.LinearAlgebra.Dual.Defs\npublic import Mathlib.LinearAlgebra.Finsupp.SumProd\npublic import Mathlib.LinearAlgebra.LinearIndependent.Basic\npublic import Mathlib.LinearAlgebra.Pi\npublic import Mathlib.Logic.Equiv.Fin.Rotate\npublic import Mathlib.Tactic.FinCases\npublic import Mathlib.Tactic.Module\npublic import Mathlib.Tactic.Abel\npublic import Mathlib.Tactic.NormNum.Ineq\nimport Mathlib.Algebra.Module.Torsion.Field\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen\n-/\n/-!\n# Linear independence\n\nThis file collects consequences of linear (in)dependence and includes specialized tests for\nspecific families of vectors, requiring more theory to state.\n\n## Main statements\n\nWe prove several specialized tests for linear independence of families of vectors and of sets of\nvectors.\n\n* `linearIndependent_option`, `linearIndependent_finCons`,\n  `linearIndependent_finSucc`, `linearIndependent_finSnoc`: type-specific tests for linear\n  independence of families of vector fields;\n* `linearIndependent_insert`, `linearIndependent_pair`: linear independence tests for set operations\n\nIn many cases we additionally provide dot-style operations (e.g., `LinearIndependent.union`) to\nmake the linear independence tests usable as `hv.insert ha` etc.\n\nWe also prove that, when working over a division ring,\nany family of vectors includes a linear independent subfamily spanning the same subspace.\n\n## TODO\n\nRework proofs to hold in semirings, by avoiding the path through\n`ker (Finsupp.linearCombination R v) = ⊥`.\n\n## Tags\n\nlinearly dependent, linear dependence, linearly independent, linear independence\n\n-/\n\n@[expose] public section\n\n\nassert_not_exists Cardinal\n\nnoncomputable section\n\nopen Function Module Set Submodule\n\nuniverse u' u\n\nvariable {ι : Type u'} {ι' : Type*} {R : Type*} {K : Type*} {s : Set ι}\nvariable {M : Type*} {M' : Type*} {V : Type u}\n\nsection Semiring\n\n\nvariable {v : ι → M}\nvariable [Semiring R] [AddCommMonoid M] [AddCommMonoid M']\nvariable [Module R M] [Module R M']\nvariable (R) (v)\n\nvariable {R v}\n\n/-- A finite family of vectors `v i` is linear independent iff the linear map that sends\n`c : ι → R` to `∑ i, c i • v i` is injective. -/\ntheorem Fintype.linearIndependent_iff'ₛ [Fintype ι] [DecidableEq ι] :\n    LinearIndependent R v ↔\n      Injective (LinearMap.lsum R (fun _ ↦ R) ℕ fun i ↦ LinearMap.id.smulRight (v i)) := by\n  simp [Fintype.linearIndependent_iffₛ, Injective, funext_iff]\n\nlemma LinearIndependent.pair_iffₛ {x y : M} :\n    LinearIndependent R ![x, y] ↔\n      ∀ (s t s' t' : R), s • x + t • y = s' • x + t' • y → s = s' ∧ t = t' := by\n  simp [Fintype.linearIndependent_iffₛ, Fin.forall_fin_two, ← FinVec.forall_iff]; rfl\n\nlemma LinearIndependent.eq_of_pair {x y : M} (h : LinearIndependent R ![x, y])\n    {s t s' t' : R} (h' : s • x + t • y = s' • x + t' • y) : s = s' ∧ t = t' :=\n  pair_iffₛ.mp h _ _ _ _ h'\n\nlemma LinearIndependent.eq_zero_of_pair' {x y : M} (h : LinearIndependent R ![x, y])\n    {s t : R} (h' : s • x = t • y) : s = 0 ∧ t = 0 := by\n  suffices H : s = 0 ∧ 0 = t from ⟨H.1, H.2.symm⟩\n  exact h.eq_of_pair (by simpa using h')\n\nlemma LinearIndependent.eq_zero_of_pair {x y : M} (h : LinearIndependent R ![x, y])\n    {s t : R} (h' : s • x + t • y = 0) : s = 0 ∧ t = 0 := by\n  replace h := @h (.single 0 s + .single 1 t) 0 ?_\n  · exact ⟨by simpa using congr($h 0), by simpa using congr($h 1)⟩\n  simpa\n\nsection Indexed\n\ntheorem linearIndepOn_iUnion_of_directed {η : Type*} {s : η → Set ι} (hs : Directed (· ⊆ ·) s)\n    (h : ∀ i, LinearIndepOn R v (s i)) : LinearIndepOn R v (⋃ i, s i) := by\n  by_cases hη : Nonempty η\n  · refine linearIndepOn_of_finite (⋃ i, s i) fun t ht ft => ?_\n    rcases finite_subset_iUnion ft ht with ⟨I, fi, hI⟩\n    rcases hs.finset_le fi.toFinset with ⟨i, hi⟩\n    exact (h i).mono (Subset.trans hI <| iUnion₂_subset fun j hj => hi j (fi.mem_toFinset.2 hj))\n  · refine (linearIndepOn_empty R v).mono (t := iUnion (s ·)) ?_\n    rintro _ ⟨_, ⟨i, _⟩, _⟩\n    exact hη ⟨i⟩\n\ntheorem linearIndepOn_sUnion_of_directed {s : Set (Set ι)} (hs : DirectedOn (· ⊆ ·) s)\n    (h : ∀ a ∈ s, LinearIndepOn R v a) : LinearIndepOn R v (⋃₀ s) := by\n  rw [sUnion_eq_iUnion]\n  exact linearIndepOn_iUnion_of_directed hs.directed_val (by simpa using h)\n\ntheorem linearIndepOn_biUnion_of_directed {η} {s : Set η} {t : η → Set ι}\n    (hs : DirectedOn (t ⁻¹'o (· ⊆ ·)) s) (h : ∀ a ∈ s, LinearIndepOn R v (t a)) :\n    LinearIndepOn R v (⋃ a ∈ s, t a) := by\n  rw [biUnion_eq_iUnion]\n  exact linearIndepOn_iUnion_of_directed (directed_comp.2 <| hs.directed_val) (by simpa using h)\n\nend Indexed\n\nsection repr\n\nvariable (ι R M) in\ntheorem iSupIndep_range_lsingle :\n    iSupIndep fun i : ι ↦ LinearMap.range (Finsupp.lsingle (R := R) (M := M) i) := by\n  refine fun i ↦ disjoint_iff_inf_le.mpr ?_\n  rintro x ⟨⟨m, rfl⟩, hm⟩\n  suffices ⨆ j ≠ i, LinearMap.range (Finsupp.lsingle j) ≤ Finsupp.supported M R {i}ᶜ by\n    have := (Finsupp.mem_supported ..).mp (this hm); simp_all\n  refine iSup₂_le fun j ne ↦ ?_\n  rintro _ ⟨m, rfl⟩\n  simp [Finsupp.mem_supported, ne]\n\ntheorem LinearMap.iSupIndep_map (f : M →ₗ[R] M') (inj : Injective f) {m : ι → Submodule R M}\n    (ind : iSupIndep m) : iSupIndep fun i ↦ (m i).map f := by\n  simp_rw [iSupIndep, disjoint_iff_inf_le] at ind ⊢\n  rintro i _ ⟨⟨x, hxi, rfl⟩, hx⟩\n  rw [ind i ⟨hxi, _⟩]; · simp\n  simp_rw [← Submodule.map_iSup] at hx\n  have ⟨y, hy, eq⟩ := hx\n  simpa [← inj eq]\n\nvariable (hv : LinearIndependent R v)\n\n/-- See also `iSupIndep_iff_linearIndependent_of_ne_zero`. -/\n\nTarget:\ntheorem LinearIndependent.iSupIndep_span_singleton (hv : LinearIndependent R v) :\n    iSupIndep fun i => R ∙ v i :=\n\nProof body:\n","rejected":"by\n  exact LinearIndependent.iSupIndep_span_singleton","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"246bb52515751dbe99ec85d48ee54869e6232628148a728aeddd74895968305a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/LinearIndependent","family_id":"linearindependent","file_id":"mathlib/Mathlib/LinearAlgebra/LinearIndependent/Lemmas.lean","sample_id":"2f46fd8127f7a4707cce4dc2e25df1a6780c736d06d1699ff4e93a369a16eb74"},"split":"train"}
{"binding":{"chosen_proof_sha256":"426d456c7ecb6c1bdb729eb16ba1456459d7dc0d59c4a4d027a77d78ce78493f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"56f2b093801a2a41875bf30e118429a5cc73ba8976a1adb97741fabeeedbed7b","source_sha256":"c86276a84562d270f98c4177786987e3596479c9ba6b4dd02bd972b6ae1c24e5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [← Nat.not_odd_iff_even]\n  constructor <;> intro h\n  · intro _ w ho\n    have := (w.three_le_chromaticNumber_of_odd_loop ho).trans h.chromaticNumber_le\n    norm_cast\n  · apply colorable_iff_forall_connectedComponents.2\n    intro c\n    obtain ⟨_, hv⟩ := c.nonempty_supp\n    use fun a ↦ Fin.ofNat 2 (c.connected_toSimpleGraph ⟨_, hv⟩ a).some.length\n    intro a b hab he\n    apply h _ <| (((c.connected_toSimpleGraph ⟨_, hv⟩ a).some.concat hab).append\n                 (c.connected_toSimpleGraph ⟨_, hv⟩ b).some.reverse).map c.toSimpleGraph_hom\n    rw [length_map, length_append, length_concat, length_reverse, add_right_comm]\n    have : ((Nonempty.some (c.connected_toSimpleGraph ⟨_, hv⟩ a)).length) % 2 =\n        (Nonempty.some (c.connected_toSimpleGraph ⟨_, hv⟩ b)).length % 2 := by\n      simp_rw [← Fin.val_natCast, ← Fin.ofNat_eq_cast, he]\n    exact (Nat.even_iff.mpr (by lia)).add_one","hard_negative":false,"metrics":{"chosen_tokens":212,"rejected_tokens":2,"token_jaccard":0.013889,"token_length_ratio":0.009434},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"0dfdab81ce49e67057fbe7cd8872948ee26472252bc15c8d7db5de09649af203","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.SimpleGraph.Bipartite\npublic import Mathlib.Combinatorics.SimpleGraph.Circulant\npublic import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex\npublic import Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite\npublic import Mathlib.Combinatorics.SimpleGraph.Hasse\npublic import Mathlib.Data.Fin.Parity\n\nNamespace:\nSimpleGraph\n\nLocal context:\n/-\nCopyright (c) 2023 Iván Renison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Iván Renison\n-/\n/-!\n# Concrete colorings of common graphs\n\nThis file defines colorings for some common graphs.\n\n## Main declarations\n\n* `SimpleGraph.pathGraph.bicoloring`: Bicoloring of a path graph.\n\n-/\n\n@[expose] public section\n\nassert_not_exists Field\n\nnamespace SimpleGraph\n\n/-- Bicoloring of a path graph -/\ndef pathGraph.bicoloring (n : ℕ) :\n    Coloring (pathGraph n) Bool :=\n  Coloring.mk (fun u ↦ u.val % 2 = 0) <| by\n    intro u v\n    rw [pathGraph_adj]\n    rintro (h | h) <;> simp [← h, not_iff, Nat.succ_mod_two_eq_zero_iff]\n\n/-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/\ndef pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where\n  toFun v := ⟨v, trans v.2 h⟩\n  inj' := by\n    rintro v w\n    rw [Fin.mk.injEq]\n    exact Fin.ext\n  map_rel_iff' := by simp [pathGraph]\n\ntheorem chromaticNumber_pathGraph (n : ℕ) (h : 2 ≤ n) :\n    (pathGraph n).chromaticNumber = 2 := by\n  have hc := (pathGraph.bicoloring n).colorable\n  apply le_antisymm\n  · exact hc.chromaticNumber_le\n  · have hadj : (pathGraph n).Adj ⟨0, Nat.zero_lt_of_lt h⟩ ⟨1, h⟩ := by simp [pathGraph_adj]\n    exact two_le_chromaticNumber_of_adj hadj\n\ntheorem Coloring.even_length_iff_congr {α} {G : SimpleGraph α}\n    (c : G.Coloring Bool) {u v : α} (p : G.Walk u v) :\n    Even p.length ↔ (c u ↔ c v) := by\n  induction p with\n  | nil => simp\n  | @cons u v w h p ih =>\n    simp only [Walk.length_cons, Nat.even_add_one]\n    have : ¬ c u = true ↔ c v = true := by\n      rw [← not_iff, ← Bool.eq_iff_iff]\n      exact c.valid h\n    tauto\n\ntheorem Coloring.odd_length_iff_not_congr {α} {G : SimpleGraph α}\n    (c : G.Coloring Bool) {u v : α} (p : G.Walk u v) :\n    Odd p.length ↔ (¬c u ↔ c v) := by\n  rw [← Nat.not_even_iff_odd, c.even_length_iff_congr p]\n  tauto\n\ntheorem Walk.three_le_chromaticNumber_of_odd_loop {α} {G : SimpleGraph α} {u : α} (p : G.Walk u u)\n    (hOdd : Odd p.length) : 3 ≤ G.chromaticNumber := Classical.by_contradiction <| by\n  intro h\n  have h' : G.chromaticNumber ≤ 2 := Order.le_of_lt_add_one <| not_le.mp h\n  let c : G.Coloring (Fin 2) := (chromaticNumber_le_iff_colorable.mp h').some\n  let c' : G.Coloring Bool := recolorOfEquiv G finTwoEquiv c\n  have : ¬c' u ↔ c' u := (c'.odd_length_iff_not_congr p).mp hOdd\n  simp_all\n\n/-- Bicoloring of a cycle graph of even size -/\ndef cycleGraph.bicoloring_of_even (n : ℕ) (h : Even n) : Coloring (cycleGraph n) Bool :=\n  Coloring.mk (fun u ↦ u.val % 2 = 0) <| by\n    intro u v hadj\n    match n with\n    | 0 => exact u.elim0\n    | 1 => simp at h\n    | n + 2 =>\n      simp only [ne_eq, decide_eq_decide]\n      simp only [cycleGraph_adj] at hadj\n      cases hadj with\n      | inl huv | inr huv =>\n        rw [← add_eq_of_eq_sub' huv.symm, ← Fin.even_iff_mod_of_even h,\n          ← Fin.even_iff_mod_of_even h, Fin.even_add_one_iff_odd]\n        apply Classical.not_iff.mpr\n        simp [Fin.not_odd_iff_even_of_even h, Fin.not_even_iff_odd_of_even h]\n\ntheorem chromaticNumber_cycleGraph_of_even (n : ℕ) (h : 2 ≤ n) (hEven : Even n) :\n    (cycleGraph n).chromaticNumber = 2 := by\n  have hc := (cycleGraph.bicoloring_of_even n hEven).colorable\n  apply le_antisymm\n  · apply hc.chromaticNumber_le\n  · have hadj : (cycleGraph n).Adj ⟨0, Nat.zero_lt_of_lt h⟩ ⟨1, h⟩ := by\n      simp [cycleGraph_adj', Fin.sub_val_of_le]\n    exact two_le_chromaticNumber_of_adj hadj\n\n/-- Tricoloring of a cycle graph -/\ndef cycleGraph.tricoloring (n : ℕ) (h : 2 ≤ n) : Coloring (cycleGraph n)\n    (Fin 3) := Coloring.mk (fun u ↦ if u.val = n - 1 then 2 else ⟨u.val % 2, by lia⟩) <| by\n    intro u v hadj\n    match n with\n    | 0 => exact u.elim0\n    | 1 => simp at h\n    | n + 2 =>\n      simp only [cycleGraph_adj] at hadj\n      split_ifs with hu hv\n      · simp [Fin.eq_mk_iff_val_eq.mpr hu, Fin.eq_mk_iff_val_eq.mpr hv] at hadj\n      · refine (Fin.ne_of_lt (Fin.mk_lt_of_lt_val (?_))).symm\n        exact v.val.mod_lt Nat.zero_lt_two\n      · refine (Fin.ne_of_lt (Fin.mk_lt_of_lt_val ?_))\n        exact u.val.mod_lt Nat.zero_lt_two\n      · simp only [ne_eq, Fin.ext_iff]\n        have hu' : u.val + (1 : Fin (n + 2)) < n + 2 := by fin_omega\n        have hv' : v.val + (1 : Fin (n + 2)) < n + 2 := by fin_omega\n        cases hadj with\n        | inl huv | inr huv =>\n          rw [← add_eq_of_eq_sub' huv.symm]\n          simp only [Fin.val_add_eq_of_add_lt hv', Fin.val_add_eq_of_add_lt hu', Fin.val_one]\n          rw [show ∀ x y : ℕ, x % 2 = y % 2 ↔ (Even x ↔ Even y) by simp [Nat.even_iff]; lia,\n            Nat.even_add]\n          simp only [Nat.not_even_one, iff_false, not_iff_self, iff_not_self]\n          exact id\n\ntheorem chromaticNumber_cycleGraph_of_odd (n : ℕ) (h : 2 ≤ n) (hOdd : Odd n) :\n    (cycleGraph n).chromaticNumber = 3 := by\n  have hc := (cycleGraph.tricoloring n h).colorable\n  apply le_antisymm\n  · apply hc.chromaticNumber_le\n  · have hn3 : n - 3 + 3 = n := by\n      refine Nat.sub_add_cancel (Nat.succ_le_of_lt (Nat.lt_of_le_of_ne h ?_))\n      intro h2\n      rw [← h2] at hOdd\n      exact (Nat.not_odd_iff.mpr rfl) hOdd\n    let w : (cycleGraph (n - 3 + 3)).Walk 0 0 := cycleGraph.cycle (n - 3)\n    have hOdd' : Odd w.length := by\n      rw [cycleGraph.length_cycle, hn3]\n      exact hOdd\n    rw [← hn3]\n    exact Walk.three_le_chromaticNumber_of_odd_loop w hOdd'\n\nsection CompleteEquipartiteGraph\n\nvariable {r t : ℕ}\n\n/-- The injection `(x₁, x₂) ↦ x₁` is always an `r`-coloring of a `completeEquipartiteGraph r ·`. -/\ndef Coloring.completeEquipartiteGraph :\n  (completeEquipartiteGraph r t).Coloring (Fin r) := ⟨Prod.fst, id⟩\n\n/-- The `completeEquipartiteGraph r t` is always `r`-colorable. -/\ntheorem completeEquipartiteGraph_colorable :\n  (completeEquipartiteGraph r t).Colorable r := ⟨Coloring.completeEquipartiteGraph⟩\n\nend CompleteEquipartiteGraph\n\nopen Walk\n\nTarget:\nlemma two_colorable_iff_forall_loop_even {α : Type*} {G : SimpleGraph α} :\n    G.Colorable 2 ↔ ∀ u, ∀ (w : G.Walk u u), Even w.length :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/SimpleGraph","family_id":"two_colorable_iff_forall_loop_even","file_id":"mathlib/Mathlib/Combinatorics/SimpleGraph/Coloring/Constructions.lean","sample_id":"56f2b093801a2a41875bf30e118429a5cc73ba8976a1adb97741fabeeedbed7b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9fa7af5d48ddb6db2529f448df0e30aa81c953ad207f9980f8c0f3df88f4e48e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"421e4acb4fbfc76c1e4546e0b6c98c0caf799dafceed449d8ee7acde11a16c4a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"66542818e479c89882f725bfc7d9605b37b594439204873e38030237ce086893","source_sha256":"c1ecc80ead99a17d01db2984c5f06df1499354702fc192601863dee22fbd7b8f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro V\n  have : ∀ x : V.1, ∃ (i : ι) (r : Γ(X, V.1)) (rU : Γ(X, U i)),\n      X.basicOpen r = X.basicOpen rU ∧ x.1 ∈ X.basicOpen r := by\n    intro ⟨x, hxV⟩\n    obtain ⟨i, hi⟩ := TopologicalSpace.Opens.mem_iSup.mp (hU.ge (Set.mem_univ x))\n    exact ⟨i, exists_basicOpen_le_affine_inter V.2 (U i).2 _ ⟨hxV, hi⟩⟩\n  choose i r rU e hxr using this\n  have : Ideal.span (Set.range r) = ⊤ := by\n    rw [← V.2.self_le_iSup_basicOpen_iff]\n    exact fun x hxV ↦ TopologicalSpace.Opens.mem_iSup.mpr ⟨⟨_, _, rfl⟩, hxr ⟨x, hxV⟩⟩\n  have inst := V.2.isLocalization_basicOpen\n  refine Submodule.le_of_isLocalized_span _ this (fun i ↦ Γ(X, X.basicOpen i.1))\n    (fun i ↦ Algebra.linearMap Γ(X, V.1) Γ(X, X.basicOpen i.1)) ?_\n  rintro ⟨_, j, rfl⟩\n  dsimp\n  simp +instances only [← Submodule.restrictScalars_localized' Γ(X, X.basicOpen (r j)),\n    Ideal.localized'_eq_map, RingHom.algebraMap_toAlgebra]\n  erw [I.map_ideal (U := ⟨_, V.2.basicOpen _⟩) (X.basicOpen_le (r j)),\n    J.map_ideal (U := ⟨_, V.2.basicOpen _⟩) (X.basicOpen_le (r j))]\n  delta algebra_section_section_basicOpen\n  rw! [e]\n  rw [← I.map_ideal (V := (U _)) (X.basicOpen_le _), ← J.map_ideal (V := (U _)) (X.basicOpen_le _)]\n  exact Ideal.map_mono (f := (X.presheaf.map (homOfLE (X.basicOpen_le (rU j))).op).hom) (H (i j))","hard_negative":true,"metrics":{"chosen_tokens":413,"rejected_tokens":3,"token_jaccard":0.020833,"token_length_ratio":0.007264},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"0e6113fde0cb11fe0b8f319c0a645ee92621c85bf22ef1c0c1383732c04168fa","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact\npublic import Mathlib.AlgebraicGeometry.Properties\npublic import Mathlib.Tactic.DepRewrite\n\nNamespace:\nAlgebraicGeometry.Scheme.IdealSheafData\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Ideal sheaves on schemes\n\nWe define ideal sheaves of schemes and provide various constructors for it.\n\n## Main definition\n* `AlgebraicGeometry.Scheme.IdealSheafData`: A structure that contains the data to uniquely define\n  an ideal sheaf, consisting of\n  1. an ideal `I(U) ≤ Γ(X, U)` for every affine open `U`\n  2. a proof that `I(D(f)) = I(U)_f` for every affine open `U` and every section `f : Γ(X, U)`.\n* `AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals`:\n  The largest ideal sheaf contained in a family of ideals.\n* `AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine`:\n  Over affine schemes, ideal sheaves are in bijection with ideals of the global sections.\n* `AlgebraicGeometry.Scheme.IdealSheafData.support`: The support of an ideal sheaf.\n* `AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal`: The vanishing ideal of a set.\n* `AlgebraicGeometry.Scheme.Hom.ker`: The kernel of a morphism.\n\n## Main results\n* `AlgebraicGeometry.Scheme.IdealSheafData.gc`:\n  `support` and `vanishingIdeal` forms a Galois connection.\n* `AlgebraicGeometry.Scheme.Hom.support_ker`: The support of a kernel of a quasi-compact morphism\n  is the closure of the range.\n\n## Implementation detail\n\nIdeal sheaves are not yet defined in this file as actual subsheaves of `𝒪ₓ`.\nInstead, for the ease of development and application,\nwe define the structure `IdealSheafData` containing all necessary data to uniquely define an\nideal sheaf. This should be refactored as a constructor for ideal sheaves once they are introduced\ninto mathlib.\n\n-/\n\n@[expose] public section\n\nopen CategoryTheory TopologicalSpace\n\nuniverse u\n\nnamespace AlgebraicGeometry.Scheme\n\nvariable {X : Scheme.{u}}\n\n/--\nA structure that contains the data to uniquely define an ideal sheaf, consisting of\n1. an ideal `I(U) ≤ Γ(X, U)` for every affine open `U`\n2. a proof that `I(D(f)) = I(U)_f` for every affine open `U` and every section `f : Γ(X, U)`\n3. a subset of `X` equal to the support.\n\nAlso see `Scheme.IdealSheafData.mkOfMemSupportIff` for a constructor with the condition on the\nsupport being (usually) easier to prove.\n-/\nstructure IdealSheafData (X : Scheme.{u}) : Type u where\n  /-- The component of an ideal sheaf at an affine open. -/\n  ideal : ∀ U : X.affineOpens, Ideal Γ(X, U)\n  /-- Also see `AlgebraicGeometry.Scheme.IdealSheafData.map_ideal` -/\n  map_ideal_basicOpen : ∀ (U : X.affineOpens) (f : Γ(X, U)),\n    (ideal U).map (X.presheaf.map (homOfLE <| X.basicOpen_le f).op).hom =\n      ideal (X.affineBasicOpen f)\n  /-- The support of an ideal sheaf. Use `IdealSheafData.support` instead for most occasions. -/\n  supportSet : Set X := ⋂ U, X.zeroLocus (U := U.1) (ideal U)\n  supportSet_eq_iInter_zeroLocus : supportSet = ⋂ U, X.zeroLocus (U := U.1) (ideal U) := by rfl\n\nnamespace IdealSheafData\n\n@[ext]\nprotected lemma ext {I J : X.IdealSheafData} (h : I.ideal = J.ideal) : I = J := by\n  obtain ⟨i, _, s, hs⟩ := I\n  obtain ⟨j, _, t, ht⟩ := J\n  subst h\n  congr\n  rw [hs, ht]\n\nsection Order\n\ninstance : PartialOrder (IdealSheafData X) := PartialOrder.lift ideal fun _ _ ↦ IdealSheafData.ext\n\nlemma le_def {I J : IdealSheafData X} : I ≤ J ↔ ∀ U, I.ideal U ≤ J.ideal U := .rfl\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : CompleteSemilatticeSup (IdealSheafData X) where\n  sSup s :=\n  { ideal := sSup (ideal '' s),\n    map_ideal_basicOpen := by\n      have : sSup (ideal '' s) = ⨆ i : s, ideal i.1 := by\n        conv_lhs => rw [← Subtype.range_val (s := s), ← Set.range_comp]\n        rfl\n      simp only [this, iSup_apply, Ideal.map_iSup, map_ideal_basicOpen, implies_true] }\n  isLUB_sSup _ := .of_image (f := ideal) le_def (isLUB_sSup _)\n\n/-- The largest ideal sheaf contained in a family of ideals. -/\ndef ofIdeals (I : ∀ U : X.affineOpens, Ideal Γ(X, U)) : IdealSheafData X :=\n  sSup { J : IdealSheafData X | J.ideal ≤ I }\n\nlemma ideal_ofIdeals_le (I : ∀ U : X.affineOpens, Ideal Γ(X, U)) :\n    (ofIdeals I).ideal ≤ I :=\n  sSup_le (Set.forall_mem_image.mpr fun _ ↦ id)\n\n/-- The Galois coinsertion between ideal sheaves and arbitrary families of ideals. -/\nprotected def gci : GaloisCoinsertion ideal (ofIdeals (X := X)) where\n  choice I hI :=\n  { ideal := I\n    map_ideal_basicOpen U f :=\n      (ideal_ofIdeals_le I).antisymm hI ▸ (ofIdeals I).map_ideal_basicOpen U f }\n  gc _ _ := ⟨(le_sSup ·), (le_trans · (ideal_ofIdeals_le _))⟩\n  u_l_le _ := sSup_le fun _ ↦ id\n  choice_eq I hI := IdealSheafData.ext (hI.antisymm (ideal_ofIdeals_le I))\n\nlemma strictMono_ideal : StrictMono (ideal (X := X)) := IdealSheafData.gci.strictMono_l\nlemma ideal_mono : Monotone (ideal (X := X)) := strictMono_ideal.monotone\nlemma ofIdeals_mono : Monotone (ofIdeals (X := X)) := IdealSheafData.gci.gc.monotone_u\nlemma ofIdeals_ideal (I : IdealSheafData X) : ofIdeals I.ideal = I := IdealSheafData.gci.u_l_eq _\nlemma le_ofIdeals_iff {I : IdealSheafData X} {J} : I ≤ ofIdeals J ↔ I.ideal ≤ J :=\n  IdealSheafData.gci.gc.le_iff_le.symm\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : OrderTop (IdealSheafData X) where\n  top.ideal := ⊤\n  top.map_ideal_basicOpen := by simp [Ideal.map_top]\n  top.supportSet := ⊥\n  top.supportSet_eq_iInter_zeroLocus := by\n    ext x\n    simpa using X.isBasis_affineOpens.exists_subset_of_mem_open (Set.mem_univ x) isOpen_univ\n  le_top I U := le_top\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : OrderBot (IdealSheafData X) where\n  bot.ideal := ⊥\n  bot.map_ideal_basicOpen := by simp\n  bot.supportSet := ⊤\n  bot.supportSet_eq_iInter_zeroLocus := by ext; simp\n  bot_le I U := bot_le\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : SemilatticeInf (IdealSheafData X) where\n  inf I J :=\n  { ideal := I.ideal ⊓ J.ideal\n    map_ideal_basicOpen U f := by\n      dsimp\n      have : (X.presheaf.map (homOfLE (X.basicOpen_le f)).op).hom = algebraMap _ _ := rfl\n      have inst := U.2.isLocalization_basicOpen f\n      rw [← I.map_ideal_basicOpen U f, ← J.map_ideal_basicOpen U f, this]\n      ext x\n      obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq (.powers f) x\n      simp only [IsLocalization.mk'_mem_map_algebraMap_iff, Submonoid.mem_powers_iff, Ideal.mem_inf,\n        exists_exists_eq_and]\n      refine ⟨fun ⟨n, h₁, h₂⟩ ↦ ⟨⟨n, h₁⟩, ⟨n, h₂⟩⟩, ?_⟩\n      rintro ⟨⟨n₁, h₁⟩, ⟨n₂, h₂⟩⟩\n      refine ⟨n₁ + n₂, ?_, ?_⟩\n      · rw [add_comm, pow_add, mul_assoc]; exact Ideal.mul_mem_left _ _ h₁\n      · rw [pow_add, mul_assoc]; exact Ideal.mul_mem_left _ _ h₂ }\n  inf_le_left I J U := inf_le_left\n  inf_le_right I J U := inf_le_right\n  le_inf I J K hIJ hIK U := le_inf (hIJ U) (hIK U)\n\ninstance : CompleteLattice (IdealSheafData X) where\n  __ := (inferInstance : OrderTop (IdealSheafData X))\n  __ := (inferInstance : OrderBot (IdealSheafData X))\n  __ := (inferInstance : SemilatticeInf (IdealSheafData X))\n  __ := (inferInstance : CompleteSemilatticeSup (IdealSheafData X))\n  __ := IdealSheafData.gci.liftCompleteLattice\n\n@[simp]\nlemma ideal_top : ideal (X := X) ⊤ = ⊤ := rfl\n\n@[simp]\nlemma ideal_bot : ideal (X := X) ⊥ = ⊥ := rfl\n\n@[simp]\nlemma ideal_sup {I J : IdealSheafData X} : (I ⊔ J).ideal = I.ideal ⊔ J.ideal := rfl\n\n@[simp]\nlemma ideal_sSup {I : Set (IdealSheafData X)} : (sSup I).ideal = sSup (ideal '' I) := rfl\n\n@[simp]\nlemma ideal_iSup {ι : Type*} {I : ι → IdealSheafData X} : (iSup I).ideal = ⨆ i, (I i).ideal := by\n  rw [← sSup_range, ← sSup_range, ideal_sSup, ← Set.range_comp, Function.comp_def]\n\n@[simp]\nlemma ideal_inf {I J : IdealSheafData X} : (I ⊓ J).ideal = I.ideal ⊓ J.ideal := rfl\n\n@[simp]\nlemma ideal_biInf {ι : Type*} (I : ι → IdealSheafData X) {s : Set ι} (hs : s.Finite) :\n    (⨅ i ∈ s, I i).ideal = ⨅ i ∈ s, (I i).ideal := by\n  refine hs.induction_on _ (by simp) fun {i s} his hs e ↦ ?_\n  simp only [iInf_insert, e, ideal_inf]\n\n@[simp]\nlemma ideal_iInf {ι : Type*} (I : ι → IdealSheafData X) [Finite ι] :\n    (⨅ i, I i).ideal = ⨅ i, (I i).ideal := by\n  simpa using ideal_biInf I Set.finite_univ\n\nend Order\n\nvariable (I : IdealSheafData X)\n\nsection map_ideal\n\n/-- subsumed by `IdealSheafData.map_ideal` below. -/\nprivate lemma map_ideal_basicOpen_of_eq\n    {U V : X.affineOpens} (f : Γ(X, U)) (hV : V = X.affineBasicOpen f) :\n    (I.ideal U).map (X.presheaf.map\n        (homOfLE (X := X.Opens) (hV.trans_le (X.affineBasicOpen_le f))).op).hom =\n      I.ideal V := by\n  subst hV; exact I.map_ideal_basicOpen _ _\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma map_ideal {U V : X.affineOpens} (h : U ≤ V) :\n    (I.ideal V).map (X.presheaf.map (homOfLE h).op).hom = I.ideal U := by\n  rw [U.2.ideal_ext_iff]\n  intro x hxU\n  obtain ⟨f, g, hfg, hxf⟩ := exists_basicOpen_le_affine_inter U.2 V.2 x ⟨hxU, h hxU⟩\n  have := I.map_ideal_basicOpen_of_eq (V := X.affineBasicOpen g) f (Subtype.ext hfg.symm)\n  rw [← I.map_ideal_basicOpen] at this\n  apply_fun Ideal.map (X.presheaf.germ (X.basicOpen g) x (hfg ▸ hxf)).hom at this\n  simp only [Ideal.map_map, ← CommRingCat.hom_comp, affineBasicOpen_coe, X.presheaf.germ_res]\n    at this ⊢\n  simp only [homOfLE_leOfHom, TopCat.Presheaf.germ_res', this]\n\n/-- A form of `map_ideal` that is easier to rewrite with. -/\nlemma map_ideal' {U V : X.affineOpens} (h : Opposite.op V.1 ⟶ .op U.1) :\n    (I.ideal V).map (X.presheaf.map h).hom = I.ideal U :=\n  map_ideal _ _\n\nlemma ideal_le_comap_ideal {U V : X.affineOpens} (h : U ≤ V) :\n    I.ideal V ≤ (I.ideal U).comap (X.presheaf.map (homOfLE h).op).hom := by\n  rw [← Ideal.map_le_iff_le_comap, ← I.map_ideal h]\n\nTarget:\nlemma le_of_iSup_eq_top {I J : X.IdealSheafData} {ι : Type*}\n    (U : ι → X.affineOpens) (hU : ⨆ i, (U i).1 = ⊤) (H : ∀ i, I.ideal (U i) ≤ J.ideal (U i)) :\n    I ≤ J :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_66542818e479","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1df6a5ea50ccab30bd25357159721a0a1062edd2e2b068cbc46c48a875124846","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/IdealSheaf","family_id":"le_of_isup_eq_top","file_id":"mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/Basic.lean","sample_id":"66542818e479c89882f725bfc7d9605b37b594439204873e38030237ce086893"},"split":"train"}
{"binding":{"chosen_proof_sha256":"be2b6932dda4e0b3c56c94860ddf8d756e2f7715633872f9e4a77499f7decf69","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c0c4b5e544c8ef5b36acb2b62220ff6fea80826bd9cad1193261c98202dffd52","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0700ede7788b015baae01a8e4aa3cd674212af76354c06e79abb4f1dca42ce86","source_sha256":"7fb54be1edb74454dd74d15ac42b7739b2a4aa95bc5649191f339f2298801887","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [cmpUsing, Ordering.ite_eq_gt_distrib, if_false_right, and_true, if_false_left,\n    and_iff_right_iff_imp, reduceCtorEq]\n  exact fun hba hab ↦ (irrefl a) (_root_.trans hab hba)","hard_negative":true,"metrics":{"chosen_tokens":36,"rejected_tokens":3,"token_jaccard":0.074074,"token_length_ratio":0.083333},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"0f302e6100c5e60c2551c6013829b2a049adbd1cfccb2ef6ca9406a84d706fcb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Ordering.Basic\npublic import Mathlib.Order.Defs.Unbundled\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Some `Ordering` lemmas\n-/\n\npublic section\n\nuniverse u\n\nnamespace Ordering\n\n@[simp]\ntheorem ite_eq_lt_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.lt) = if c then a = Ordering.lt else b = Ordering.lt := by\n  by_cases c <;> simp [*]\n\n@[simp]\ntheorem ite_eq_eq_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.eq) = if c then a = Ordering.eq else b = Ordering.eq := by\n  by_cases c <;> simp [*]\n\n@[simp]\ntheorem ite_eq_gt_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.gt) = if c then a = Ordering.gt else b = Ordering.gt := by\n  by_cases c <;> simp [*]\n\n@[simp]\nlemma dthen_eq_then (o₁ o₂ : Ordering) : o₁.dthen (fun _ => o₂) = o₁.then o₂ := by\n  cases o₁ <;> rfl\n\nend Ordering\n\nsection\n\nvariable {α : Type u} {lt : α → α → Prop} [DecidableRel lt]\n\nattribute [local simp] cmpUsing\n\n@[simp]\ntheorem cmpUsing_eq_lt (a b : α) : (cmpUsing lt a b = Ordering.lt) = lt a b := by\n  simp only [cmpUsing, Ordering.ite_eq_lt_distrib, ite_self, if_false_right, and_true, reduceCtorEq]\n\n@[simp]\n\nTarget:\ntheorem cmpUsing_eq_gt [IsStrictOrder α lt] (a b : α) : cmpUsing lt a b = Ordering.gt ↔ lt b a :=\n\nProof body:\n","rejected":"by\n  exact cmpUsing_eq_gt","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"8e309914c85f0f9214cffcb1369d67da64de4623628add3a1611ca74f64b7744","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Ordering","family_id":"cmpusing_eq_gt","file_id":"mathlib/Mathlib/Data/Ordering/Lemmas.lean","sample_id":"0700ede7788b015baae01a8e4aa3cd674212af76354c06e79abb4f1dca42ce86"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e934aed5577882b0e1846e82a16eff1d5d3efe027757ba1c1868dea83273eb47","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2d684b3c442a8ef210bbe9338a5474c025b33cb1271057595a26bab18822d713","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a0442559c44ff5e1900294674d91b30edd8758751f47d7b1327b13d6af184bd6","source_sha256":"e2952ea887763387582fd966bcaa7890973b527717726e02bfae4aafd3204443","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨T, hT₁, hT₂⟩ := has_injective_coseparator (op G) ((isSeparator_op_iff _).2 hG)\n  exact ⟨unop T, inferInstance, (isSeparator_unop_iff _).2 hT₂⟩","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":46,"token_jaccard":0.888889,"token_length_ratio":1.095238},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"0f3c8391133da095103e258e525a42d307bc35201e9b0693aac86734df32de84","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Abelian.Subobject\npublic import Mathlib.CategoryTheory.Limits.EssentiallySmall\npublic import Mathlib.CategoryTheory.Preadditive.Injective.Basic\npublic import Mathlib.CategoryTheory.Generator.Preadditive\npublic import Mathlib.CategoryTheory.Abelian.Opposite\n\nNamespace:\nCategoryTheory.Abelian\n\nLocal context:\n/-\nCopyright (c) 2022 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\n/-!\n# A complete abelian category with enough injectives and a separator has an injective coseparator\n\n## Future work\n* Once we know that Grothendieck categories have enough injectives, we can use this to conclude\n  that Grothendieck categories have an injective coseparator.\n\n## References\n* [Peter J Freyd, *Abelian Categories* (Theorem 3.37)][freyd1964abelian]\n\n-/\n\npublic section\n\n\nopen CategoryTheory CategoryTheory.Limits Opposite\n\nuniverse v u\n\nnamespace CategoryTheory.Abelian\n\nvariable {C : Type u} [Category.{v} C] [Abelian C]\n\ntheorem has_injective_coseparator [HasLimits C] [EnoughInjectives C] (G : C) (hG : IsSeparator G) :\n    ∃ G : C, Injective G ∧ IsCoseparator G := by\n  haveI : WellPowered.{v} C := wellPowered_of_isDetector G hG.isDetector\n  haveI : HasProductsOfShape (Subobject (op G)) C := hasProductsOfShape_of_small.{v} _ _\n  let T : C := Injective.under (piObj fun P : Subobject (op G) => unop P)\n  refine ⟨T, inferInstance, (Preadditive.isCoseparator_iff _).2 fun X Y f hf => ?_⟩\n  refine (Preadditive.isSeparator_iff _).1 hG _ fun h => ?_\n  suffices hh : factorThruImage (h ≫ f) = 0 by\n    rw [← Limits.image.fac (h ≫ f), hh, zero_comp]\n  let R := Subobject.mk (factorThruImage (h ≫ f)).op\n  let q₁ : image (h ≫ f) ⟶ unop R :=\n    (Subobject.underlyingIso (factorThruImage (h ≫ f)).op).unop.hom\n  let q₂ : unop (R : Cᵒᵖ) ⟶ piObj fun P : Subobject (op G) => unop P :=\n    section_ (Pi.π (fun P : Subobject (op G) => (unop P : C)) R)\n  let q : image (h ≫ f) ⟶ T := q₁ ≫ q₂ ≫ Injective.ι _\n  exact zero_of_comp_mono q\n    (by rw [← Injective.comp_factorThru q (Limits.image.ι (h ≫ f)), Limits.image.fac_assoc,\n      Category.assoc, hf, comp_zero])\n\nTarget:\ntheorem has_projective_separator [HasColimits C] [EnoughProjectives C] (G : C)\n    (hG : IsCoseparator G) : ∃ G : C, Projective G ∧ IsSeparator G :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  obtain ⟨T, hT₁, hT₂⟩ := has_injective_coseparator (op G) ((isSeparator_op_iff _).2 hG)\n  exact ⟨unop T, inferInstance, (isSeparator_unop_iff _).2 hT₂⟩","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Generator","family_id":"has_projective_separator","file_id":"mathlib/Mathlib/CategoryTheory/Generator/Abelian.lean","sample_id":"a0442559c44ff5e1900294674d91b30edd8758751f47d7b1327b13d6af184bd6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"002c7d89e4368c0249a9cb4cfc7e30d967572004732ef0400a272686c72ce645","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"af3b95b0732be5f55a3324fb2e3c5ad5c5f888a12a687a186421ed96ab233b01","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8c1abff1d102ab8663f817566b5f1e0679aded535e3b61cc03bd6dcfd4532f76","source_sha256":"b8e096f7469de9f8271b2d043d71be202bf790e32cee2afb83b7783c6d94bbb7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa [basis_repr_apply] using ιMultiDual_apply_nondiag R n b t s hst.symm","hard_negative":true,"metrics":{"chosen_tokens":16,"rejected_tokens":3,"token_jaccard":0.055556,"token_length_ratio":0.1875},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"0ff750a82ba5bd53fc54a12e80f647294feaafce2008f9ddcd471768014fccbd","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.ExteriorPower.Basic\npublic import Mathlib.LinearAlgebra.ExteriorPower.Pairing\npublic import Mathlib.RingTheory.Finiteness.Subalgebra\npublic import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition\n\nNamespace:\nexteriorPower\n\nLocal context:\n/-\nCopyright (c) 2025 Daniel Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sophie Morel, Daniel Morrison\n-/\n/-!\n# Constructs a basis for exterior powers\n-/\n\n@[expose] public section\n\nvariable {R K M E : Type*} {n : ℕ}\n  [CommRing R] [Field K] [AddCommGroup M] [Module R M] [AddCommGroup E] [Module K E]\n\nnamespace exteriorPower\n\n/-! Finiteness of the exterior power. -/\n\n/-- The `n`th exterior power of a finite module is a finite module. -/\ninstance instFinite [Module.Finite R M] : Module.Finite R (⋀[R]^n M) := by\n  rw [Module.Finite.iff_fg, ExteriorAlgebra.exteriorPower, LinearMap.range_eq_map]\n  exact Submodule.FG.pow (Submodule.FG.map _ Module.Finite.fg_top) n\n\n/-! We construct a basis of `⋀[R]^n M` from a basis of `M`. -/\n\nopen Module Set Set.powersetCard\n\nvariable (R n)\n\n/-- If `b` is a basis of `M` indexed by a linearly ordered type `I` and `s` is a finset of\n`I` of cardinality `n`, then we get a linear form on the `n`th exterior power of `M` by\napplying the `exteriorPower.linearForm` construction to the family of linear forms\ngiven by the coordinates of `b` indexed by elements of `s` (ordered using the given order on\n`I`). -/\nnoncomputable def ιMultiDual {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s : powersetCard I n) : Module.Dual R (⋀[R]^n M) :=\n  pairingDual R M n (ιMulti_family R n b.coord s)\n\n@[simp]\nlemma ιMultiDual_apply_ιMulti {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s : powersetCard I n) (v : Fin n → M) :\n    ιMultiDual R n b s (ιMulti R n v) =\n    (Matrix.of fun i j => b.coord (powersetCard.ofFinEmbEquiv.symm s j) (v i)).det := by\n  simp [ιMultiDual, ιMulti_family, pairingDual_ιMulti_ιMulti]\n\n/-- Let `b` be a basis of `M` indexed by a linearly ordered type `I` and `s` be a finset of `I`\nof cardinality `n`. If we apply the linear form on `⋀[R]^n M` defined by `b` and `s`\nto the exterior product of the `b i` for `i ∈ s`, then we get `1`. -/\nlemma ιMultiDual_apply_diag {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s : powersetCard I n) :\n    ιMultiDual R n b s (ιMulti_family R n b s) = 1 := by\n  rw [ιMulti_family, ιMultiDual_apply_ιMulti]\n  suffices Matrix.of (fun i j => b.coord (powersetCard.ofFinEmbEquiv.symm s j)\n    (b (powersetCard.ofFinEmbEquiv.symm s i))) = 1 by\n    simp_rw [Function.comp_apply, this, Matrix.det_one]\n  ext\n  simp [Matrix.one_apply, Finsupp.single_apply]\n\n/-- Let `b` be a basis of `M` indexed by a linearly ordered type `I` and `s` be a finset of `I`\nof cardinality `n`. Let `t` be a finset of `I` of cardinality `n` such that `s ≠ t`. If we apply\nthe linear form on `⋀[R]^n M` defined by `b` and `s` to the exterior product of the\n`b i` for `i ∈ t`, then we get `0`. -/\nlemma ιMultiDual_apply_nondiag {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s t : powersetCard I n) (hst : s ≠ t) :\n    ιMultiDual R n b s (ιMulti_family R n b t) = 0 := by\n  rw [ιMulti_family, ιMultiDual_apply_ιMulti]\n  obtain ⟨i, his, hit⟩ := (exists_mem_notMem_iff_ne s t).mp hst\n  obtain ⟨k, rfl⟩ := (mem_range_ofFinEmbEquiv_symm_iff_mem s i).mpr his\n  apply Matrix.det_eq_zero_of_column_eq_zero k\n  simp_rw [Matrix.of_apply, Basis.coord_apply, Function.comp_apply, Basis.repr_self]\n  intro j\n  apply Finsupp.single_eq_of_ne\n  by_contra! h\n  apply hit\n  rw [h, powersetCard.ofFinEmbEquiv_symm_apply, ← powersetCard.mem_coe_iff]\n  exact Finset.orderEmbOfFin_mem t.val t.prop j\n\n/-- If `b` is a basis of `M` (indexed by a linearly ordered type), then the family\n`exteriorPower.ιMulti R n b` of the `n`-fold exterior products of its elements is linearly\nindependent in the `n`th exterior power of `M`. -/\nlemma ιMulti_family_linearIndependent_ofBasis {I : Type*} [LinearOrder I] (b : Basis I R M) :\n    LinearIndependent R (ιMulti_family R n b) :=\n  LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun s ↦ ιMultiDual R n b s)\n    (fun _ _ h => ιMultiDual_apply_nondiag R n b _ _ h)\n    (fun _ => ιMultiDual_apply_diag _ _ _ _)\n\nvariable {R} in\n/-- If `b` is a basis of `M` (indexed by a linearly ordered type), the basis of the `n`th\nexterior power of `M` formed by the `n`-fold exterior products of elements of `b`. -/\nnoncomputable def _root_.Module.Basis.exteriorPower {I : Type*} [LinearOrder I] (b : Basis I R M) :\n    Basis (powersetCard I n) R (⋀[R]^n M) :=\n  Basis.mk (ιMulti_family_linearIndependent_ofBasis _ _ _)\n    (eq_top_iff.mp <| ιMulti_family_span_of_span R b.span_eq)\n\n@[simp]\nlemma coe_basis {I : Type*} [LinearOrder I] (b : Basis I R M) :\n    DFunLike.coe (b.exteriorPower n) = ιMulti_family R n b :=\n  Basis.coe_mk _ _\n\nlemma basis_apply {I : Type*} [LinearOrder I] (b : Basis I R M) (s : powersetCard I n) :\n    b.exteriorPower n s = ιMulti_family R n b s := by\n  rw [coe_basis]\n\n/-- If `b` is a basis of `M` indexed by a linearly ordered type `I` and `B` is the corresponding\nbasis of the `n`th exterior power of `M`, indexed by the set of finsets `s` of `I` of cardinality\n`n`, then the coordinate function of `B` at `s` is the linear form on the `n`th exterior power\ndefined by `b` and `s` in `exteriorPower.ιMultiDual`. -/\nlemma basis_coord {I : Type*} [LinearOrder I] (b : Basis I R M) (s : powersetCard I n) :\n    Basis.coord (b.exteriorPower n) s = ιMultiDual R n b s := by\n  apply LinearMap.ext_on (ιMulti_family_span_of_span R (Basis.span_eq b))\n  rintro x ⟨t, rfl⟩\n  rw [Basis.coord_apply]\n  by_cases! hst : s = t\n  · rw [hst, ιMultiDual_apply_diag, ← basis_apply, Basis.repr_self, Finsupp.single_eq_same]\n  · rw [ιMultiDual_apply_nondiag R n b s t hst, ← basis_apply, Basis.repr_self,\n      Finsupp.single_eq_of_ne hst]\n\nlemma basis_repr_apply {I : Type*} [LinearOrder I] (b : Basis I R M) (x : ⋀[R]^n M)\n    (s : powersetCard I n) :\n    Basis.repr (b.exteriorPower n) x s = ιMultiDual R n b s x := by\n  simpa [← Basis.coord_apply] using LinearMap.congr_fun (basis_coord R n b s) x\n\n@[simp]\nlemma basis_repr_self {I : Type*} [LinearOrder I] (b : Basis I R M) (s : powersetCard I n) :\n    Basis.repr (b.exteriorPower n) (ιMulti_family R n b s) s = 1 := by\n  simpa [basis_repr_apply] using ιMultiDual_apply_diag R n b s\n\n@[simp]\n\nTarget:\nlemma basis_repr_ne {I : Type*} [LinearOrder I] (b : Basis I R M)\n    {s t : powersetCard I n} (hst : s ≠ t) :\n    Basis.repr (b.exteriorPower n) (ιMulti_family R n b s) t = 0 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_8c1abff1d102","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d0800bfe131a7b35e86e7a8d89ea4dcb0eb72c8916d94f85532caed9306d0fc7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/ExteriorPower","family_id":"basis_repr_ne","file_id":"mathlib/Mathlib/LinearAlgebra/ExteriorPower/Basis.lean","sample_id":"8c1abff1d102ab8663f817566b5f1e0679aded535e3b61cc03bd6dcfd4532f76"},"split":"train"}
{"binding":{"chosen_proof_sha256":"05a30ac01794d34e26abf0bb89eb49ff18aa58a6481d8190121651504b0314f8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"956162824dd70572aa71bea9f0c8aee184b663e35c297fa1f233790107d5caec","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b28f2d047f3f02792ce315aca654c65c6b846562c70579c8ee832efe4c79dcf3","source_sha256":"b5f291a07b438990d3572a453453e2d3f7dd16b400e7d0a6bac2c42be601d0ed","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases eq_or_ne n 0 with rfl | hn₀; · simp\n  rcases eq_or_ne p 1 with rfl | hp₁; · simp\n  simpa [*] using pow_dvd_iff_le_padicValNat (k := 1) hp₁ hn₀ |>.symm |>.not","hard_negative":true,"metrics":{"chosen_tokens":48,"rejected_tokens":5,"token_jaccard":0.058824,"token_length_ratio":0.104167},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"101b9f31af38d3bb25030b452279db35540d353bee016b62e03a68ff0fb175e3","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.MaxPowDiv\npublic import Mathlib.RingTheory.Multiplicity\npublic import Mathlib.Data.Nat.Factors\n\nNamespace:\npadicValNat\n\nLocal context:\n/-\nCopyright (c) 2018 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis, Matthew Robert Ballard\n-/\n/-!\n# `p`-adic Valuation\n\nThis file defines the `p`-adic valuation on `ℕ`, `ℤ`, and `ℚ`.\n\nThe `p`-adic valuation on `ℚ` is the difference of the multiplicities of `p` in the numerator and\ndenominator of `q`. This function obeys the standard properties of a valuation, with the appropriate\nassumptions on `p`. The `p`-adic valuations on `ℕ` and `ℤ` agree with that on `ℚ`.\n\nThe valuation induces a norm on `ℚ`. This norm is defined in\n`Mathlib/NumberTheory/Padics/PadicNorm.lean`.\n-/\n\n@[expose] public section\n\nassert_not_exists Field\n\nuniverse u\n\nopen Nat\n\nvariable {p : ℕ}\n\ntheorem padicValNat_eq_emultiplicity_of_ne_one (hp : p ≠ 1) {n : ℕ} (hn : n ≠ 0) :\n    padicValNat p n = emultiplicity p n := by\n  rw [eq_comm, emultiplicity_eq_coe, pow_dvd_iff_le_padicValNat hp hn,\n    pow_dvd_iff_le_padicValNat hp hn]\n  simp\n\n@[simp]\ntheorem Nat.toNat_emultiplicity (p n : ℕ) : (emultiplicity p n).toNat = padicValNat p n := by\n  rcases eq_or_ne p 1 with rfl | hp\n  · simp\n  · rcases eq_or_ne n 0 with rfl | hn\n    · simp\n    · simp [← padicValNat_eq_emultiplicity_of_ne_one, *]\n\ntheorem padicValNat_def' {n : ℕ} (hp : p ≠ 1) (hn : n ≠ 0) :\n    padicValNat p n = multiplicity p n :=\n  .symm <| multiplicity_eq_of_emultiplicity_eq_some <| .symm <|\n    padicValNat_eq_emultiplicity_of_ne_one hp hn\n\n/-- A simplification of `padicValNat` when one input is prime, by analogy with\n`padicValRat_def`. -/\ntheorem padicValNat_def [hp : Fact p.Prime] {n : ℕ} (hn : n ≠ 0) :\n    padicValNat p n = multiplicity p n :=\n  padicValNat_def' hp.out.ne_one hn\n\n/-- A simplification of `padicValNat` when one input is prime, by analogy with\n`padicValRat_def`. -/\ntheorem padicValNat_eq_emultiplicity [hp : Fact p.Prime] {n : ℕ} (hn : n ≠ 0) :\n    padicValNat p n = emultiplicity p n :=\n  padicValNat_eq_emultiplicity_of_ne_one hp.out.ne_one hn\n\nnamespace padicValNat\n\n@[deprecated (since := \"2026-03-15\")]\nalias maxPowDiv_eq_emultiplicity := padicValNat_eq_emultiplicity\n\n@[deprecated (since := \"2026-03-15\")]\nalias maxPowDiv_eq_multiplicity := padicValNat_def'\n\n@[deprecated padicValNat_zero_right (since := \"2026-03-15\")]\nprotected theorem zero : padicValNat p 0 = 0 := padicValNat_zero_right p\n\n@[deprecated padicValNat_one_right (since := \"2026-03-15\")]\nprotected theorem one : padicValNat p 1 = 0 := padicValNat_one_right p\n\n@[simp]\n\nTarget:\ntheorem eq_zero_iff {n : ℕ} : padicValNat p n = 0 ↔ p = 1 ∨ n = 0 ∨ ¬p ∣ n :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_b28f2d047f3f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1c902c86f1e0b64c65fd4be7eca2c3e35b5812cc9a34f1b5e6a751b13e0c94d3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Padics","family_id":"eq_zero_iff","file_id":"mathlib/Mathlib/NumberTheory/Padics/PadicVal/Defs.lean","sample_id":"b28f2d047f3f02792ce315aca654c65c6b846562c70579c8ee832efe4c79dcf3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"335297b55a581bb6fe6dcb38f704182637c77f8f85b5c081f025c29aaa5ebb61","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c092795e5fd0e22d540c502fd1b4fd017f05beade7817be0597a69f2c004e7d6","source_sha256":"c0e3fa9405ddf4fdd99cd3aa6c9ff7258178f89496e31a48d0f8f121f656258a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Option.some_inj]\n  simp [finSuccAboveOrderIso, finSuccAboveEquiv, OrderIso.symm]","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":8,"token_jaccard":0.05,"token_length_ratio":0.444444},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"102f4f29c2e5e7606343e16a8f2e16caeb3bda5ce52f3a2743cde60da4663a62","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.VecNotation\npublic import Mathlib.Logic.Equiv.Fin.Basic\npublic import Mathlib.Order.Fin.Basic\npublic import Mathlib.Order.PiLex\npublic import Mathlib.Order.Interval.Set.Defs\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes\n-/\n/-!\n# Order properties on tuples\n-/\n\n@[expose] public section\n\nassert_not_exists Monoid\n\nopen Function Set\n\nnamespace Fin\nvariable {m n : ℕ} {α : Fin (n + 1) → Type*} (x : α 0) (q : ∀ i, α i) (p : ∀ i : Fin n, α i.succ)\n  (i : Fin n) (y : α i.succ) (z : α 0)\n\nlemma pi_lex_lt_cons_cons {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ}\n    (s : ∀ {i : Fin n.succ}, α i → α i → Prop) :\n    Pi.Lex (· < ·) (@s) (Fin.cons x₀ x) (Fin.cons y₀ y) ↔\n      s x₀ y₀ ∨ x₀ = y₀ ∧ Pi.Lex (· < ·) (@fun i : Fin n ↦ @s i.succ) x y := by\n  simp_rw [Pi.Lex, Fin.exists_fin_succ, Fin.cons_succ, Fin.cons_zero, Fin.forall_iff_succ]\n  simp [and_assoc, exists_and_left]\n\nvariable [∀ i, Preorder (α i)]\n\nlemma insertNth_mem_Icc {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)}\n    {q₁ q₂ : ∀ j, α j} :\n    i.insertNth x p ∈ Icc q₁ q₂ ↔\n      x ∈ Icc (q₁ i) (q₂ i) ∧ p ∈ Icc (fun j ↦ q₁ (i.succAbove j)) fun j ↦ q₂ (i.succAbove j) := by\n  simp only [mem_Icc, insertNth_le_iff, le_insertNth_iff, and_assoc, @and_left_comm (x ≤ q₂ i)]\n\nlemma preimage_insertNth_Icc_of_mem {i : Fin (n + 1)} {x : α i} {q₁ q₂ : ∀ j, α j}\n    (hx : x ∈ Icc (q₁ i) (q₂ i)) :\n    i.insertNth x ⁻¹' Icc q₁ q₂ = Icc (fun j ↦ q₁ (i.succAbove j)) fun j ↦ q₂ (i.succAbove j) :=\n  Set.ext fun p ↦ by simp only [mem_preimage, insertNth_mem_Icc, hx, true_and]\n\nlemma preimage_insertNth_Icc_of_notMem {i : Fin (n + 1)} {x : α i} {q₁ q₂ : ∀ j, α j}\n    (hx : x ∉ Icc (q₁ i) (q₂ i)) : i.insertNth x ⁻¹' Icc q₁ q₂ = ∅ :=\n  Set.ext fun p ↦ by\n    simp only [mem_preimage, insertNth_mem_Icc, hx, false_and, mem_empty_iff_false]\n\nend Fin\n\nopen Fin Matrix\n\nvariable {α : Type*}\n\nopen scoped Relator in\nlemma liftFun_vecCons {n : ℕ} (r : α → α → Prop) [IsTrans α r] {f : Fin (n + 1) → α} {a : α} :\n    ((· < ·) ⇒ r) (vecCons a f) (vecCons a f) ↔ r a (f 0) ∧ ((· < ·) ⇒ r) f f := by\n  simp only [liftFun_iff_succ r, forall_iff_succ, cons_val_succ, cons_val_zero, ← succ_castSucc,\n    castSucc_zero]\n\nvariable [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α}\n\n@[simp] lemma strictMono_vecCons : StrictMono (vecCons a f) ↔ a < f 0 ∧ StrictMono f :=\n  liftFun_vecCons (· < ·)\n\n@[simp]\nlemma monotone_vecCons : Monotone (vecCons a f) ↔ a ≤ f 0 ∧ Monotone f := by\n  simpa only [monotone_iff_forall_lt] using! @liftFun_vecCons α n (· ≤ ·) _ f a\n\n@[simp] lemma monotone_vecEmpty : Monotone ![a]\n  | ⟨0, _⟩, ⟨0, _⟩, _ => le_refl _\n\n@[simp] lemma strictMono_vecEmpty : StrictMono ![a]\n  | ⟨0, _⟩, ⟨0, _⟩, h => (irrefl _ h).elim\n\n@[simp] lemma strictAnti_vecCons : StrictAnti (vecCons a f) ↔ f 0 < a ∧ StrictAnti f :=\n  liftFun_vecCons (· > ·)\n\n@[simp] lemma antitone_vecCons : Antitone (vecCons a f) ↔ f 0 ≤ a ∧ Antitone f :=\n  monotone_vecCons (α := αᵒᵈ)\n\n@[simp] lemma antitone_vecEmpty : Antitone (vecCons a vecEmpty)\n  | ⟨0, _⟩, ⟨0, _⟩, _ => le_rfl\n\n@[simp] lemma strictAnti_vecEmpty : StrictAnti (vecCons a vecEmpty)\n  | ⟨0, _⟩, ⟨0, _⟩, h => (irrefl _ h).elim\n\nlemma StrictMono.vecCons (hf : StrictMono f) (ha : a < f 0) : StrictMono (vecCons a f) :=\n  strictMono_vecCons.2 ⟨ha, hf⟩\n\nlemma StrictAnti.vecCons (hf : StrictAnti f) (ha : f 0 < a) : StrictAnti (vecCons a f) :=\n  strictAnti_vecCons.2 ⟨ha, hf⟩\n\nlemma Monotone.vecCons (hf : Monotone f) (ha : a ≤ f 0) : Monotone (vecCons a f) :=\n  monotone_vecCons.2 ⟨ha, hf⟩\n\nlemma Antitone.vecCons (hf : Antitone f) (ha : f 0 ≤ a) : Antitone (vecCons a f) :=\n  antitone_vecCons.2 ⟨ha, hf⟩\n\nexample : Monotone ![1, 2, 2, 3] := by decide\n\n\nvariable {n : ℕ}\n\n/-- `Π i : Fin 2, α i` is order equivalent to `α 0 × α 1`. See also `OrderIso.finTwoArrowEquiv`\nfor a non-dependent version. -/\ndef OrderIso.piFinTwoIso (α : Fin 2 → Type*) [∀ i, Preorder (α i)] : (∀ i, α i) ≃o α 0 × α 1 where\n  toEquiv := piFinTwoEquiv α\n  map_rel_iff' := Iff.symm Fin.forall_fin_two\n\n/-- The space of functions `Fin 2 → α` is order equivalent to `α × α`. See also\n`OrderIso.piFinTwoIso`. -/\ndef OrderIso.finTwoArrowIso (α : Type*) [Preorder α] : (Fin 2 → α) ≃o α × α :=\n  { OrderIso.piFinTwoIso fun _ => α with toEquiv := finTwoArrowEquiv α }\n\nnamespace Fin\n\n/-- Order isomorphism between tuples of length `n + 1` and pairs of an element and a tuple of length\n`n` given by separating out the first element of the tuple.\n\nThis is `Fin.cons` as an `OrderIso`. -/\n@[simps!, simps toEquiv]\ndef consOrderIso (α : Fin (n + 1) → Type*) [∀ i, LE (α i)] :\n    α 0 × (∀ i, α (succ i)) ≃o ∀ i, α i where\n  toEquiv := consEquiv α\n  map_rel_iff' := forall_iff_succ\n\n/-- Order isomorphism between tuples of length `n + 1` and pairs of an element and a tuple of length\n`n` given by separating out the last element of the tuple.\n\nThis is `Fin.snoc` as an `OrderIso`. -/\n@[simps!, simps toEquiv]\ndef snocOrderIso (α : Fin (n + 1) → Type*) [∀ i, LE (α i)] :\n    α (last n) × (∀ i, α (castSucc i)) ≃o ∀ i, α i where\n  toEquiv := snocEquiv α\n  map_rel_iff' := by simp [Pi.le_def, Prod.le_def, forall_iff_castSucc]\n\n/-- Order isomorphism between tuples of length `n + 1` and pairs of an element and a tuple of length\n`n` given by separating out the `p`-th element of the tuple.\n\nThis is `Fin.insertNth` as an `OrderIso`. -/\n@[simps!, simps toEquiv]\ndef insertNthOrderIso (α : Fin (n + 1) → Type*) [∀ i, LE (α i)] (p : Fin (n + 1)) :\n    α p × (∀ i, α (p.succAbove i)) ≃o ∀ i, α i where\n  toEquiv := insertNthEquiv α p\n  map_rel_iff' := by simp [Pi.le_def, Prod.le_def, p.forall_iff_succAbove]\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp] lemma insertNthOrderIso_zero (α : Fin (n + 1) → Type*) [∀ i, LE (α i)] :\n    insertNthOrderIso α 0 = consOrderIso α := by ext; simp [insertNthOrderIso]\n\n/-- Note this lemma can only be written about non-dependent tuples as `insertNth (last n) = snoc` is\nnot a definitional equality. -/\n@[simp] lemma insertNthOrderIso_last (n : ℕ) (α : Type*) [LE α] :\n    insertNthOrderIso (fun _ ↦ α) (last n) = snocOrderIso (fun _ ↦ α) := by ext; simp\n\nend Fin\n\n/-- `Fin.succAbove` as an order isomorphism between `Fin n` and `{x : Fin (n + 1) // x ≠ p}`. -/\ndef finSuccAboveOrderIso (p : Fin (n + 1)) : Fin n ≃o { x : Fin (n + 1) // x ≠ p } where\n  __ := finSuccAboveEquiv p\n  map_rel_iff' := p.succAboveOrderEmb.map_rel_iff'\n\nlemma finSuccAboveOrderIso_apply (p : Fin (n + 1)) (i : Fin n) :\n    finSuccAboveOrderIso p i = ⟨p.succAbove i, p.succAbove_ne i⟩ := rfl\n\nTarget:\nlemma finSuccAboveOrderIso_symm_apply_last (x : { x : Fin (n + 1) // x ≠ Fin.last n }) :\n    (finSuccAboveOrderIso (Fin.last n)).symm x = Fin.castLT x.1 (Fin.val_lt_last x.2) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"26272c5b978e7bc20d7fbe034534b0d95cfe66a7458381f8324928d807d8d750","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Fin","family_id":"finsuccaboveorderiso_symm_apply_last","file_id":"mathlib/Mathlib/Order/Fin/Tuple.lean","sample_id":"c092795e5fd0e22d540c502fd1b4fd017f05beade7817be0597a69f2c004e7d6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"60c914c7101fd19d31a3d057e5fece308a5e2edc08549f3280fae32cf5fabb05","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c3ebc26cd68e8f84cca3c5c39ecb58bd11bd37f0e946789ac34e3ea71722700a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cf2066a6aac720a3b5d609e8b3fe1b26d84bc8deb53c0ec1b5795944dd568539","source_sha256":"16ec77c2931a23c29f0fb086af6455ba8ec16f170f0fd8c2178cb81f79494219","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp only [mem_setOf_eq]\n  rw [← Submodule.map_top, ← iSup_range_single, Submodule.map_iSup]\n  congr; funext i\n  rw [← LinearMap.range_comp, diagonal_comp_single, ← range_smul']","hard_negative":true,"metrics":{"chosen_tokens":36,"rejected_tokens":5,"token_jaccard":0.08,"token_length_ratio":0.138889},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"103c790c5979157196aa0b654513e4ee65a14337f977ea4a4938c113b6f058c1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Dimension.LinearMap\npublic import Mathlib.LinearAlgebra.Matrix.ToLin\n\nNamespace:\nMatrix\n\nLocal context:\n/-\nCopyright (c) 2019 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen\n-/\n/-!\n# Diagonal matrices\n\nThis file contains some results on the linear map corresponding to a\ndiagonal matrix (`range`, `ker` and `rank`).\n\n## Tags\n\nmatrix, diagonal, linear map\n-/\n\npublic section\n\n\nnoncomputable section\n\nopen LinearMap Matrix Set Submodule Matrix\n\nuniverse u v w\n\nnamespace Matrix\n\nsection CommSemiring\n\nvariable {n : Type*} [Fintype n] [DecidableEq n] {R : Type v} [CommSemiring R]\n\ntheorem proj_diagonal (i : n) (w : n → R) : (proj i).comp (toLin' (diagonal w)) = w i • proj i :=\n  LinearMap.ext fun _ => mulVec_diagonal _ _ _\n\ntheorem diagonal_comp_single (w : n → R) (i : n) :\n    (diagonal w).toLin'.comp (LinearMap.single R (fun _ : n => R) i) =\n      w i • LinearMap.single R (fun _ : n => R) i :=\n  LinearMap.ext fun x => (diagonal_mulVec_single w _ _).trans (Pi.single_smul' i (w i) x)\n\ntheorem diagonal_toLin' (w : n → R) :\n    toLin' (diagonal w) = LinearMap.pi fun i => w i • LinearMap.proj i :=\n  LinearMap.ext fun _ => funext fun _ => mulVec_diagonal _ _ _\n\nend CommSemiring\n\nsection Semifield\n\nvariable {m : Type*} [Fintype m] {K : Type u} [Semifield K]\n\n-- maybe try to relax the universe constraint\ntheorem ker_diagonal_toLin' [DecidableEq m] (w : m → K) :\n    ker (toLin' (diagonal w)) =\n      ⨆ i ∈ { i | w i = 0 }, LinearMap.range (LinearMap.single K (fun _ => K) i) := by\n  rw [← comap_bot, ← iInf_ker_proj, comap_iInf]\n  have := fun i : m => ker_comp (toLin' (diagonal w)) (proj i)\n  simp only [← this, proj_diagonal, ker_smul']\n  have : univ ⊆ { i : m | w i = 0 } ∪ { i : m | w i = 0 }ᶜ := by rw [Set.union_compl_self]\n  exact (iSup_range_single_eq_iInf_ker_proj K (fun _ : m => K) disjoint_compl_right this\n    (Set.toFinite _)).symm\n\nTarget:\ntheorem range_diagonal [DecidableEq m] (w : m → K) :\n    LinearMap.range (toLin' (diagonal w)) =\n      ⨆ i ∈ { i | w i ≠ 0 }, LinearMap.range (LinearMap.single K (fun _ => K) i) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_cf2066a6aac7","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"e9673935d7b2cea4000578bcc71a46f2a8ae67be458f335b8f76d526c1ddb947","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Matrix","family_id":"range_diagonal","file_id":"mathlib/Mathlib/LinearAlgebra/Matrix/Diagonal.lean","sample_id":"cf2066a6aac720a3b5d609e8b3fe1b26d84bc8deb53c0ec1b5795944dd568539"},"split":"train"}
{"binding":{"chosen_proof_sha256":"41ba997d60350b8410456fab7fd86644a15e80166bdc8dbee3ebfacac2920661","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ab793cebc21f139d2eab524640ed552866c6f6bd28b0f8c110edadda18cfb9c9","source_sha256":"2fda79aa400532989a3a948709c014243f7a3167b1476a14416ee8c8dd7c852b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction l with\n  | nil => simp only [List.not_mem_nil, false_and, exists_false] at hl\n  | cons x l ih =>\n    simp only [List.mem_cons, exists_eq_or_imp] at hl\n    rcases hl with h | hal\n    · simpa only [List.map_cons, List.prod_cons] using I.mul_mem_right _ _ h\n    · simpa only [List.map_cons, List.prod_cons] using I.mul_mem_left _ _ <| ih hal","hard_negative":true,"metrics":{"chosen_tokens":84,"rejected_tokens":8,"token_jaccard":0.023256,"token_length_ratio":0.095238},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"104721a19b73f9cbc1177310e6714afccb50a3e37cc85f233cec7bd70fc5bc15","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Congruence.BigOperators\npublic import Mathlib.RingTheory.TwoSidedIdeal.Basic\n\nNamespace:\nTwoSidedIdeal\n\nLocal context:\n/-\nCopyright (c) 2024 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang\n-/\n/-!\n# Interactions between `∑, ∏` and two-sided ideals\n\n-/\n\npublic section\n\nnamespace TwoSidedIdeal\n\nsection sum\n\nvariable {R : Type*} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R)\n\nlemma listSum_mem {ι : Type*} (l : List ι) (f : ι → R) (hl : ∀ x ∈ l, f x ∈ I) :\n    (l.map f).sum ∈ I := by\n  rw [mem_iff, ← List.sum_map_zero]\n  exact I.ringCon.listSum l hl\n\nlemma multisetSum_mem {ι : Type*} (s : Multiset ι) (f : ι → R) (hs : ∀ x ∈ s, f x ∈ I) :\n    (s.map f).sum ∈ I := by\n  rw [mem_iff, ← Multiset.sum_map_zero]\n  exact I.ringCon.multisetSum s hs\n\nlemma finsetSum_mem {ι : Type*} (s : Finset ι) (f : ι → R) (hs : ∀ x ∈ s, f x ∈ I) :\n    s.sum f ∈ I := by\n  rw [mem_iff, ← Finset.sum_const_zero]\n  exact I.ringCon.finsetSum s hs\n\nlemma finsuppSum_mem {ι : Type*} {β : Type*} [Zero β]\n    {f : ι →₀ β} (g : ι → β → R) (h : ∀ i ∈ f.support, g i (f i) ∈ I) :\n    f.sum g ∈ I :=\n  finsetSum_mem _ _ _ h\n\nlemma dfinsuppSum_mem {ι : Type*} {β : ι → Type*}\n    [DecidableEq ι] [∀ i, Zero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)]\n    {f : Π₀ i, β i} (g : (i : ι) → β i → R) (h : ∀ i ∈ f.support, g i (f i) ∈ I) :\n    f.sum g ∈ I :=\n  finsetSum_mem _ _ _ h\n\nend sum\n\nsection prod\n\nsection ring\n\nvariable {R : Type*} [Ring R] (I : TwoSidedIdeal R)\n\nTarget:\nlemma listProd_mem {ι : Type*} (l : List ι) (f : ι → R) (hl : ∃ x ∈ l, f x ∈ I) :\n    (l.map f).prod ∈ I :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"713e8111a6705ca467e58514aca0c28acd47aab4443715540f78cb003c041f78","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/TwoSidedIdeal","family_id":"listprod_mem","file_id":"mathlib/Mathlib/RingTheory/TwoSidedIdeal/BigOperators.lean","sample_id":"ab793cebc21f139d2eab524640ed552866c6f6bd28b0f8c110edadda18cfb9c9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"741b44ba22accbbd70bb07bc169d65e562975536f8496b4d492a4c4b786fdd83","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"92d3d3be6de007f629396e1b5b63d1e6068a147e8c6a1b838362d488459f36f2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6c6463defa0ee8680fe4b3b439ae453214eb49f285173f4fb331b4b3b6e3c669","source_sha256":"f5ba420ebee6ce69c4b05f94e3188f18366a5272d50259681e169766e1289584","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  lift N to {N : Submodule R P // N.FG} using hN\n  apply_fun (Module.fgSystem.equiv R P).symm.toLinearMap.rTensor M at eq\n  apply_fun directLimitLeft _ _ at eq\n  simp_rw [← LinearMap.rTensor_comp_apply, ← (LinearEquiv.eq_toLinearMap_symm_comp _ _).mpr\n    (Module.fgSystem.equiv_comp_of N), directLimitLeft_rTensor_of] at eq\n  have ⟨N', le, eq⟩ := Module.DirectLimit.exists_eq_of_of_eq eq\n  exact ⟨_, N'.2, le, eq⟩","hard_negative":true,"metrics":{"chosen_tokens":101,"rejected_tokens":5,"token_jaccard":0.055556,"token_length_ratio":0.049505},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"106dcfa1524001859755232ec9a2c321a174a4198acec8fe8f17060acc300c8b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Colimit.Finiteness\npublic import Mathlib.LinearAlgebra.TensorProduct.DirectLimit\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu\n-/\n/-!\n# Tensor product with direct limit of finitely generated submodules\n\nWe show that if `M` and `P` are arbitrary modules and `N` is a finitely generated submodule\nof a module `P`, then two elements of `N ⊗ M` have the same image in `P ⊗ M` if and only if\nthey already have the same image in `N' ⊗ M` for some finitely generated submodule `N' ≥ N`.\nThis is the theorem `Submodule.FG.exists_rTensor_fg_inclusion_eq`. The key facts used are\nthat every module is the direct limit of its finitely generated submodules and that tensor\nproduct preserves colimits.\n-/\n\npublic section\n\nopen TensorProduct\n\nvariable {R M P : Type*} [CommSemiring R]\nvariable [AddCommMonoid M] [Module R M] [AddCommMonoid P] [Module R P]\n\nTarget:\ntheorem Submodule.FG.exists_rTensor_fg_inclusion_eq {N : Submodule R P} (hN : N.FG)\n    {x y : N ⊗[R] M} (eq : N.subtype.rTensor M x = N.subtype.rTensor M y) :\n    ∃ N', N'.FG ∧ ∃ h : N ≤ N', (N.inclusion h).rTensor M x = (N.inclusion h).rTensor M y :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_6c6463defa0e","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d2feac7966fa052974ec08df95436627672438dcdc40f075ce1a5862c3d0834a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Colimit","family_id":"submodule","file_id":"mathlib/Mathlib/Algebra/Colimit/TensorProduct.lean","sample_id":"6c6463defa0ee8680fe4b3b439ae453214eb49f285173f4fb331b4b3b6e3c669"},"split":"train"}
{"binding":{"chosen_proof_sha256":"238c36aa3d5b83b0f47d94bdb4d2f15b92abc10f86ad4c7efd31a50fddc87c83","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f5318c3795c3dc2787b18f8ea59670daa71cdbfea157a0978c2a730bf24e4d84","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d594da1416489b60a7466e1d8f39b122bd55afcaa513107653ab63deea58dfe7","source_sha256":"e0af099a1dfedc907a6ae420e8f9c06d8630aad18426a643f2d72445c5f1b960","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [IsTranscendenceBasis, maximal_iff, Subtype.range_val]; rfl","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.230769},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"10e4df5eafaf1e015c2551b512e073b7473e6bced07b93f99c50cb165d59c14d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.CommRing\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Algebraic Independence\n\nThis file defines algebraic independence of a family of elements of an `R` algebra.\n\n## Main definitions\n\n* `AlgebraicIndependent` - `AlgebraicIndependent R x` states the family of elements `x`\n  is algebraically independent over `R`, meaning that the canonical map out of the multivariable\n  polynomial ring is injective.\n\n* `AlgebraicIndependent.aevalEquiv` - The canonical isomorphism from the polynomial ring to the\n  subalgebra generated by an algebraic independent family.\n\n* `AlgebraicIndependent.repr` - The canonical map from the subalgebra generated by an\n  algebraic independent family into the polynomial ring. It is the inverse of\n  `AlgebraicIndependent.aevalEquiv`.\n\n* `IsTranscendenceBasis R x` - a family `x` is a transcendence basis over `R` if it is a maximal\n  algebraically independent subset.\n\n## Main results\n\nWe show that algebraic independence is preserved under injective maps of the indices.\n\n## References\n\n* [Stacks: Transcendence](https://stacks.math.columbia.edu/tag/030D)\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen Function Set Subalgebra MvPolynomial Algebra\n\nvariable {ι ι' : Type*} (R : Type*) {K A A' : Type*} (x : ι → A)\nvariable [CommRing R] [CommRing A] [CommRing A'] [Algebra R A] [Algebra R A']\n\n/-- `AlgebraicIndependent R x` states the family of elements `x`\n  is algebraically independent over `R`, meaning that the canonical\n  map out of the multivariable polynomial ring is injective. -/\n@[stacks 030E \"(1)\"] def AlgebraicIndependent : Prop :=\n  Injective (MvPolynomial.aeval x : MvPolynomial ι R →ₐ[R] A)\n\n/-- `AlgebraicIndepOn R v s` states that the elements in the family `v` that are indexed by the\nelements of `s` are algebraically independent over `R`. -/\nabbrev AlgebraicIndepOn (s : Set ι) : Prop := AlgebraicIndependent R fun i : s ↦ x i\n\nvariable {R} {x}\n\ntheorem algebraicIndependent_iff :\n    AlgebraicIndependent R x ↔\n      ∀ p : MvPolynomial ι R, MvPolynomial.aeval (x : ι → A) p = 0 → p = 0 :=\n  injective_iff_map_eq_zero _\n\ntheorem AlgebraicIndependent.eq_zero_of_aeval_eq_zero (h : AlgebraicIndependent R x) :\n    ∀ p : MvPolynomial ι R, MvPolynomial.aeval (x : ι → A) p = 0 → p = 0 :=\n  algebraicIndependent_iff.1 h\n\ntheorem algebraicIndependent_iff_injective_aeval :\n    AlgebraicIndependent R x ↔ Injective (MvPolynomial.aeval x : MvPolynomial ι R →ₐ[R] A) :=\n  Iff.rfl\n\nnamespace AlgebraicIndependent\n\ntheorem of_comp (f : A →ₐ[R] A') (hfv : AlgebraicIndependent R (f ∘ x)) :\n    AlgebraicIndependent R x := by\n  have : aeval (f ∘ x) = f.comp (aeval x) := by ext; simp\n  rw [AlgebraicIndependent, this, AlgHom.coe_comp] at hfv\n  exact hfv.of_comp\n\nvariable (hx : AlgebraicIndependent R x)\ninclude hx\n\ntheorem comp (f : ι' → ι) (hf : Function.Injective f) : AlgebraicIndependent R (x ∘ f) := by\n  intro p q\n  simpa [aeval_rename, (rename_injective f hf).eq_iff] using @hx (rename f p) (rename f q)\n\ntheorem coe_range : AlgebraicIndependent R ((↑) : range x → A) := by\n  simpa using hx.comp _ (rangeSplitting_injective x)\n\nend AlgebraicIndependent\n\nopen AlgebraicIndependent\n\ntheorem algebraicIndependent_equiv (e : ι ≃ ι') {f : ι' → A} :\n    AlgebraicIndependent R (f ∘ e) ↔ AlgebraicIndependent R f :=\n  ⟨fun h => Function.comp_id f ▸ e.self_comp_symm ▸ h.comp _ e.symm.injective,\n    fun h => h.comp _ e.injective⟩\n\ntheorem algebraicIndependent_equiv' (e : ι ≃ ι') {f : ι' → A} {g : ι → A} (h : f ∘ e = g) :\n    AlgebraicIndependent R g ↔ AlgebraicIndependent R f :=\n  h ▸ algebraicIndependent_equiv e\n\ntheorem algebraicIndependent_subtype_range {ι} {f : ι → A} (hf : Injective f) :\n    AlgebraicIndependent R ((↑) : range f → A) ↔ AlgebraicIndependent R f :=\n  Iff.symm <| algebraicIndependent_equiv' (Equiv.ofInjective f hf) rfl\n\nalias ⟨AlgebraicIndependent.of_subtype_range, _⟩ := algebraicIndependent_subtype_range\n\ntheorem algebraicIndependent_image {ι} {s : Set ι} {f : ι → A} (hf : Set.InjOn f s) :\n    (AlgebraicIndependent R fun x : s => f x) ↔ AlgebraicIndependent R fun x : f '' s => (x : A) :=\n  algebraicIndependent_equiv' (Equiv.Set.imageOfInjOn _ _ hf) rfl\n\nlemma AlgebraicIndepOn.mono {s t : Set ι} (H : AlgebraicIndepOn R x t) (hst : s ⊆ t) :\n    AlgebraicIndepOn R x s := by\n  simpa [Function.comp] using! H.comp (Set.inclusion hst) (Set.inclusion_injective hst)\n\n@[simp]\nlemma AlgebraicIndepOn.univ : AlgebraicIndepOn R x .univ ↔ AlgebraicIndependent R x :=\n  algebraicIndependent_equiv (Equiv.Set.univ ι)\n\nnamespace AlgebraicIndependent\n\ntheorem mono {t s : Set A} (h : t ⊆ s)\n    (hx : AlgebraicIndependent R ((↑) : s → A)) : AlgebraicIndependent R ((↑) : t → A) :=\n  AlgebraicIndepOn.mono (x := id) hx h\n\nsection repr\n\nvariable (hx : AlgebraicIndependent R x)\ninclude hx\n\n/-- Canonical isomorphism between polynomials and the subalgebra generated by\n  algebraically independent elements. -/\n@[simps! apply_coe]\ndef aevalEquiv : MvPolynomial ι R ≃ₐ[R] Algebra.adjoin R (range x) :=\n  (AlgEquiv.ofInjective (aeval x) (algebraicIndependent_iff_injective_aeval.1 hx)).trans\n    (Subalgebra.equivOfEq _ _ (Algebra.adjoin_range_eq_range_aeval R x).symm)\n\ntheorem algebraMap_aevalEquiv (p : MvPolynomial ι R) :\n    algebraMap (Algebra.adjoin R (range x)) A (hx.aevalEquiv p) = aeval x p :=\n  rfl\n\n/-- The canonical map from the subalgebra generated by an algebraic independent family\n  into the polynomial ring. -/\ndef repr : Algebra.adjoin R (range x) →ₐ[R] MvPolynomial ι R :=\n  hx.aevalEquiv.symm\n\n@[simp]\ntheorem aeval_repr (p) : aeval x (hx.repr p) = p :=\n  Subtype.ext_iff.1 (AlgEquiv.apply_symm_apply hx.aevalEquiv p)\n\ntheorem aeval_comp_repr : (aeval x).comp hx.repr = Subalgebra.val _ :=\n  AlgHom.ext hx.aeval_repr\n\n/-- Canonical isomorphism between polynomials and the subalgebra generated by\n  algebraically independent elements.\n  `AlgebraicIndepOn` version of `AlgebraicIndependent.aevalEquiv`. -/\ndef _root_.AlgebraicIndepOn.aevalEquiv {s : Set ι} (hx : AlgebraicIndepOn R x s) :\n    MvPolynomial s R ≃ₐ[R] Algebra.adjoin R (x '' s) :=\n  (AlgebraicIndependent.aevalEquiv hx).trans\n    (Subalgebra.equivOfEq _ _ congr(Algebra.adjoin _ $(by aesop)))\n\nend repr\n\nend AlgebraicIndependent\n\nvariable (R) in\n/-- A family is a transcendence basis if it is a maximal algebraically independent subset. -/\n@[stacks 030E \"(4)\"] def IsTranscendenceBasis (x : ι → A) : Prop :=\n  AlgebraicIndependent R x ∧\n    ∀ (s : Set A) (_ : AlgebraicIndepOn R id s) (_ : range x ⊆ s), range x = s\n\nTarget:\ntheorem isTranscendenceBasis_iff_maximal {s : Set A} :\n    IsTranscendenceBasis R ((↑) : s → A) ↔ Maximal (AlgebraicIndepOn R id) s :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_d594da141648","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"741ac458cf0fb72b5cdcc8beb0e6c5f9a9cf275a0f754b54b2d5a91028025efa","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/AlgebraicIndependent","family_id":"istranscendencebasis_iff_maximal","file_id":"mathlib/Mathlib/RingTheory/AlgebraicIndependent/Defs.lean","sample_id":"d594da1416489b60a7466e1d8f39b122bd55afcaa513107653ab63deea58dfe7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"054bd373a1fa3d5f15bd0fe1ba1c34023be8febf1ced40b818a6af0feb340b77","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6e36103246bee59d4bc6e678583d099025644f446037302a958480771b365f9c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1f3bfea624fb86d6c8b755b05ec1c6fe8d68cd178710d19c21a20e4947546990","source_sha256":"ced30280014822262039ce197c9e4ae821d789623b770f4b0e6844d4852109a3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [dp_def]\n  rw [mkAlgHom_rel (A := MvPolynomial (ℕ × M) R) R Rel.add, map_sum,\n    Finset.sum_congr rfl (fun k _ ↦ by rw [_root_.map_mul])]","hard_negative":false,"metrics":{"chosen_tokens":45,"rejected_tokens":50,"token_jaccard":0.861111,"token_length_ratio":1.111111},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"10f7cd4dd7f69120be1fc6fe09923db551ae345f65e7adbac643f35c79b467ef","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.Eval\npublic import Mathlib.Algebra.RingQuot\npublic import Mathlib.RingTheory.DividedPowers.Basic\n\nNamespace:\nDividedPowerAlgebra\n\nLocal context:\n/-\nCopyright (c) 2026 Antoine Chambert-Loir & María Inés de Frutos—Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir, María Inés de Frutos—Fernández\n-/\n/-!\n# The universal divided power algebra\n\nLet `R` be a (commutative) semiring and `M` be an `R`-module. In this file we define `Γ_R(M)`,\nthe universal divided power algebra of `M`, as the ring quotient of the polynomial ring\nin the variables `ℕ × M` by the relation `DividedPowerAlgebra.Rel`.\n\n`DividedPowerAlgebra R M` satisfies a weak universal property for morphisms to rings with\ndivided powers (`DividedPowerAlgebra.lift`).\n\n## Main definitions\n\n* `DividedPowerAlgebra.Rel`: the type coding the basic relations that will give rise to the\n  divided power algebra.\n\n* `DividedPowerAlgebra R M`: the universal divided power algebra of the `R`-module `M`,\n  defined as `RingQuot` of `DividedPowerAlgebra.Rel R M`.\n\n* `DividedPowerAlgebra.dp R n m`: for `n : ℕ` and `m : M`, this is the equivalence class of\n  `MvPolynomial.X (⟨n, m⟩)` in `DividedPowerAlgebra R M`.\n\n  When that algebra is endowed with its canonical divided power structure (to be defined),\n  the image of `MvPolynomial.X (n, m)`, for any `n : ℕ` and `m : M`, is equal to\n  the `n`th divided power of the image of `m`.\n\n  The API will be setup so that it is never (never say never…) necessary to lift to `MvPolynomial`.\n\n* `DividedPowerAlgebra.lift`: the weak universal property of `DividedPowerAlgebra R M`.\n\n* `DividedPowerAlgebra.map`: the functoriality map between divided power algebras\n  associated with a linear map of the underlying modules.\n  Given an `R`-algebra `S`, an `S`-module `N` and an `R`-linear map `f : M →ₗ[R] N`,\n  this is the map `DividedPowerAlgebra R M →ₐ[R] DividedPowerAlgebra S N`\n  sending `dp R n m` to `dp S n (f m)`.\n\n## References\n\n* [P. Berthelot (1974), *Cohomologie cristalline des schémas de\n  caractéristique $p$ > 0*][Berthelot-1974]\n\n* [P. Berthelot and A. Ogus (1978), *Notes on crystalline\n  cohomology*][BerthelotOgus-1978]\n\n* [N. Roby (1963), *Lois polynomes et lois formelles en théorie des\n  modules*][Roby-1963]\n\n* [N. Roby (1965), *Les algèbres à puissances dividées*][Roby-1965]\n\n## TODO\n\n* Show in upcoming files that `DividedPowerAlgebra R M` has divided powers.\n\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Finset Ideal MvPolynomial RingQuot\n\nvariable (R M : Type*) [CommSemiring R] [AddCommMonoid M] [Module R M]\n\nnamespace DividedPowerAlgebra\n\n/-- The type coding the basic relations that will give rise to the divided power algebra.\n  The class of `MvPolynomial.X (n, a)` will be equal to `dpow n a`, for `a ∈ M`. -/\ninductive Rel : MvPolynomial (ℕ × M) R → MvPolynomial (ℕ × M) R → Prop\n  | rfl_zero : Rel 0 0 -- Needed for technical reasons.\n  | zero {a : M} : Rel (X (0, a)) 1\n  | smul {r : R} {n : ℕ} {a : M} : Rel (X (n, r • a)) (r ^ n • X (n, a))\n  | mul {m n : ℕ} {a : M} : Rel (X (m, a) * X (n, a)) (Nat.choose (m + n) m • X (m + n, a))\n  | add {n : ℕ} {a b : M} :\n    Rel (X (n, a + b)) ((Finset.antidiagonal n).sum fun k ↦ X (k.1, a) * X (k.2, b))\n\n/-- The ideal of `MvPolynomial (ℕ × M) R` generated by `Rel`. -/\ndef RelI : Ideal (MvPolynomial (ℕ × M) R) := ofRel (DividedPowerAlgebra.Rel R M)\n\nend DividedPowerAlgebra\n\n/-- The divided power algebra of a module M is defined as the ring quotient of the polynomial ring\n  in the variables `ℕ × M` by the ring relation defined by `DividedPowerAlgebra.Rel`.\n  We will later show that that `DividedPowerAlgebra R M` has divided powers.\n  It satisfies a weak universal property for morphisms to rings with divided powers. -/\nabbrev DividedPowerAlgebra := RingQuot (DividedPowerAlgebra.Rel R M)\n\nnamespace DividedPowerAlgebra\n\nopen MvPolynomial\n\nvariable {R M}\n\nlemma mkAlgHom_surjective : Function.Surjective (mkAlgHom R (Rel R M)) :=\n  RingQuot.mkAlgHom_surjective _ _\n\nlemma mkAlgHom_C (a : R) :\n    mkAlgHom R (Rel R M) (C a) = algebraMap R (DividedPowerAlgebra R M) a := by\n  rw [← MvPolynomial.algebraMap_eq, AlgHom.commutes]\n\nlemma mkRingHom_C (a : R) :\n    mkRingHom (Rel R M) (C a) = algebraMap R (DividedPowerAlgebra R M) a := by\n  rw [← mkAlgHom_C, mkAlgHom, AlgHom.coe_mk]\n\nvariable (R) in\n/-- `dp R n m` is the equivalence class of `X (⟨n, m⟩)` in `DividedPowerAlgebra R M`. -/\ndef dp (n : ℕ) (m : M) : DividedPowerAlgebra R M := mkAlgHom R (Rel R M) (X ⟨n, m⟩)\n\ntheorem dp_def (n : ℕ) (m : M) :\n  dp R n m = mkAlgHom R (Rel R M) (X ⟨n, m⟩) := rfl\n\nprotected theorem induction_on' {P : DividedPowerAlgebra R M → Prop} (f : DividedPowerAlgebra R M)\n    (h_C : ∀ a, P (mkAlgHom R (Rel R M) (C a))) (h_add : ∀ f g, P f → P g → P (f + g))\n    (h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)) : P f := by\n  obtain ⟨F, hf⟩ := RingQuot.mkRingHom_surjective (DividedPowerAlgebra.Rel R M) f\n  rw [← hf]\n  induction F using MvPolynomial.induction_on generalizing f with\n  | C a =>\n      convert! h_C a using 1\n      rw [mkAlgHom, AlgHom.coe_mk]\n  | add g1 g2 hg1 hg2 =>\n      rw [map_add]\n      exact h_add _ _ (hg1 ((mkRingHom (Rel R M)) g1) rfl) (hg2 ((mkRingHom (Rel R M)) g2) rfl)\n  | mul_X g nm h =>\n      have h' : (mkRingHom (Rel R M)) (X nm) = dp R nm.1 nm.2 := by\n        simp only [dp_def, Prod.mk.eta, mkAlgHom, AlgHom.coe_mk]\n      rw [_root_.map_mul, h']\n      exact h_dp _ _ _ (h (mkRingHom (Rel R M) g) rfl)\n\n@[elab_as_elim]\nprotected theorem induction_on {P : DividedPowerAlgebra R M → Prop} (f : DividedPowerAlgebra R M)\n    (C : ∀ a, P (algebraMap R _ a)) (add : ∀ f g, P f → P g → P (f + g))\n    (dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)) : P f :=\n  DividedPowerAlgebra.induction_on' f (fun a ↦ by rw [mkAlgHom_C]; exact C a) add dp\n\ntheorem dp_eq_mkRingHom (n : ℕ) (m : M) :\n    dp R n m = mkRingHom (Rel R M) (X (⟨n, m⟩)) := by\n  simp [dp, mkRingHom, mkAlgHom]\n\ntheorem dp_zero {m : M} : dp R 0 m = 1 := by\n  rw [dp_def, ← map_one (mkAlgHom R (Rel R M))]\n  exact RingQuot.mkAlgHom_rel R Rel.zero\n\ntheorem dp_smul {r : R} {n : ℕ} {m : M} : dp R n (r • m) = r ^ n • dp R n m := by\n  rw [dp_def, dp_def, ← map_smul]\n  exact mkAlgHom_rel R Rel.smul\n\ntheorem dp_null {n : ℕ} : dp R n (0 : M) = if n = 0 then 1 else 0 := by\n  cases Nat.eq_zero_or_pos n with\n  | inl hn =>\n    rw [if_pos hn, hn, dp_zero]\n  | inr hn =>\n    rw [if_neg (ne_of_gt hn), ← zero_smul R (0 : M), dp_smul]\n    rw [zero_pow (Nat.pos_iff_ne_zero.mp hn), zero_smul]\n\ntheorem dp_null_of_ne_zero {n : ℕ} (hn : n ≠ 0) : dp R n (0 : M) = 0 := by\n  rw [dp_null, if_neg hn]\n\ntheorem dp_mul {n p : ℕ} {m : M} :\n    dp R n m * dp R p m = (n + p).choose n • dp R (n + p) m := by\n  simp only [dp_def, ← _root_.map_mul, ← map_nsmul]\n  exact mkAlgHom_rel R Rel.mul\n\nTarget:\ntheorem dp_add {n : ℕ} {x y : M} :\n    dp R n (x + y) = (antidiagonal n).sum fun k ↦ dp R k.1 x * dp R k.2 y :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  simp only [dp_def]\n  rw [mkAlgHom_rel (A := MvPolynomial (ℕ × M) R) R Rel.add, map_sum,\n    Finset.sum_congr rfl (fun k _ ↦ by rw [_root_.map_mul])]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/DividedPowerAlgebra","family_id":"dp_add","file_id":"mathlib/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean","sample_id":"1f3bfea624fb86d6c8b755b05ec1c6fe8d68cd178710d19c21a20e4947546990"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ff1ae62e27377baaefea54f50e782d5e8647a507a59b2f78b75ee566cf2b6bd5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0e4d7ccbd372ec11fb03813b285ac23f53686f0fc73527400c7defe75cdef26d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5c3559b3cf2ee9a241bb81b7e23213b68dd9f8307cc2252bb99a753e0c04c076","source_sha256":"73ebc2e72f7678b20eb20bcb8222b112e9c86c209dc6aa3492fcf8a265aa9fa7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [iff_comp_injective]\n  intro B _ _ I hI f₁ f₂ e\n  ext x\n  have : f₁ x - f₂ x ∈ I := by\n    simpa [Ideal.Quotient.mk_eq_mk_iff_sub_mem] using AlgHom.congr_fun e x\n  have := Polynomial.eval_add_of_sq_eq_zero ((minpoly K x).map (algebraMap K B)) (f₂ x)\n    (f₁ x - f₂ x) (show (f₁ x - f₂ x) ^ 2 ∈ ⊥ from hI ▸ Ideal.pow_mem_pow this 2)\n  simp only [add_sub_cancel, eval_map_algebraMap, aeval_algHom_apply, minpoly.aeval, map_zero,\n    derivative_map, zero_add] at this\n  rwa [eq_comm, ((isUnit_iff_ne_zero.mpr\n    ((Algebra.IsSeparable.isSeparable K x).aeval_derivative_ne_zero\n      (minpoly.aeval K x))).map f₂).mul_right_eq_zero, sub_eq_zero] at this\n\nvariable [FormallyUnramified K A] [EssFiniteType K A]\nvariable [FormallyUnramified K L] [EssFiniteType K L]","hard_negative":true,"metrics":{"chosen_tokens":187,"rejected_tokens":5,"token_jaccard":0.027027,"token_length_ratio":0.026738},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"11153394b2fd3826377a26f1f7235603afc3f98ca5e0f5c7b795565ee3da6084","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.PurelyInseparable.Basic\npublic import Mathlib.RingTheory.Artinian.Ring\npublic import Mathlib.RingTheory.Unramified.Finite\npublic import Mathlib.RingTheory.Unramified.Locus\n\nNamespace:\nAlgebra.FormallyUnramified\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Unramified algebras over fields\n\n## Main results\n\nLet `K` be a field, `A` be a `K`-algebra and `L` be a field extension of `K`.\n\n- `Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_isLocalRing`:\n    If `A` is `K`-unramified and `K` is alg-closed, then `K = A`.\n- `Algebra.FormallyUnramified.isReduced_of_field`:\n    If `A` is `K`-unramified then `A` is reduced.\n- `Algebra.FormallyUnramified.iff_isSeparable`:\n    `L` is unramified over `K` iff `L` is separable over `K`.\n\n## References\n\n- [B. Iversen, *Generic Local Structure of the Morphisms in Commutative Algebra*][iversen]\n\n-/\n\npublic section\n\nopen Algebra Module Polynomial\nopen scoped TensorProduct\n\nuniverse u\n\nvariable (K A L : Type*) [Field K] [Field L] [CommRing A] [Algebra K A] [Algebra K L]\n\nnamespace Algebra.FormallyUnramified\n\nTarget:\ntheorem of_isSeparable [Algebra.IsSeparable K L] : FormallyUnramified K L :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_5c3559b3cf2e","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"eabd019117069e68452051f85a3dd05fc138657a1ad98f95e1f18b007985ebb6","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Unramified","family_id":"of_isseparable","file_id":"mathlib/Mathlib/RingTheory/Unramified/Field.lean","sample_id":"5c3559b3cf2ee9a241bb81b7e23213b68dd9f8307cc2252bb99a753e0c04c076"},"split":"train"}
{"binding":{"chosen_proof_sha256":"47f42a3f39b92164ba24c6342d141209f44e0c15b9d67ad24b48bf7df68f4b85","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"71edde365de57e77fa976ba82cd9d031cd577508e86b1eefe9de90e3f9e321a9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2293cf123c21972c1d2af5e35604c06efbc61aa089ba31c3f82c50d912dfbf35","source_sha256":"50e4a4d798ba2dd2872d163b6a5aeb55d8cf128d153e35dcc8c630f4eadcbabe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := degree_lt_wf.min_mem _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)\n  simp_all [uniformizingPolynomial]","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":20,"token_jaccard":0.789474,"token_length_ratio":1.333333},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"111f39b0ab4fcde985d09b101e8807657290d9354b21627c95d755f6e40945d4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.Finite.Valuation\npublic import Mathlib.NumberTheory.FunctionField\npublic import Mathlib.RingTheory.Valuation.Discrete.Basic\n\nNamespace:\nRatFunc\n\nLocal context:\n/-\nCopyright (c) 2025 María Inés de Frutos-Fernández & Xavier Généreux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: María Inés de Frutos-Fernández, Xavier Généreux\n-/\n/-!\n# Ostrowski's theorem for `K(X)`\n\nThis file proves Ostrowski's theorem for the field of rational functions `K(X)`, where `K` is any\nfield: if `v` is a discrete valuation on `K(X)` which is trivial on elements of `K`, then `v` is\nequivalent to either the `I`-adic valuation for some `I : HeightOneSpectrum K[X]`, or to the\nvaluation at infinity `FunctionField.inftyValuation K`.\n\n## Main results\n- `RatFunc.valuation_isEquiv_infty_or_adic`: Ostrowski's theorem for `K(X)`.\n-/\n\n@[expose] public noncomputable section\n\n\nopen Multiplicative WithZero\n\nvariable {K Γ : Type*} [Field K] [LinearOrderedCommGroupWithZero Γ] {v : Valuation (RatFunc K) Γ}\n\nnamespace RatFunc\n\nsection Infinity\n\nopen Polynomial Valuation\n\nlemma valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X {f : RatFunc K}\n    [v.IsTrivialOn K] (hlt : 1 < v X) (hf : f ≠ 0) : v f = v RatFunc.X ^ f.intDegree := by\n  induction f using RatFunc.induction_on with\n  | f p q hq =>\n    rw [intDegree_div (by grind only) (by grind only), v.map_div, zpow_sub₀ (ne_zero_of_lt hlt)]\n    simp_rw [intDegree_polynomial, zpow_natCast, ← coePolynomial_eq_algebraMap]\n    have hp : p ≠ 0 := by contrapose hf; simp [hf]\n    rw [valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X _ hlt hp,\n      valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X _ hlt hq]\n\nvariable [DecidableEq (RatFunc K)]\n\nlemma valuation_isEquiv_inftyValuation_of_one_lt_valuation_X [v.IsTrivialOn K] (hlt : 1 < v X) :\n    v.IsEquiv (inftyValuation K) := by\n  refine isEquiv_iff_val_lt_one.mpr fun {f} ↦ ?_\n  rcases eq_or_ne f 0 with rfl | hf\n  · simp\n  · have hlt' : 1 < inftyValuation K X := by simp [← exp_zero]\n    rw [valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt hf,\n      valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt' hf]\n    grind [one_le_zpow_iff_right₀]\n\nend Infinity\n\nopen IsDedekindDomain HeightOneSpectrum Set Valuation Polynomial\n\nlemma setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty [v.IsNontrivial] [v.IsTrivialOn K]\n    (hle : v RatFunc.X ≤ 1) : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty := by\n  obtain ⟨w, h0, h1⟩ := IsNontrivial.exists_lt_one (v := v)\n  induction w using RatFunc.induction_on with\n  | f p q =>\n    simp only [ne_eq, _root_.div_eq_zero_iff, FaithfulSMul.algebraMap_eq_zero_iff, not_or,\n      map_div₀] at *\n    have hor : ¬v ↑p = 1 ∨ ¬v ↑q = 1 := by rw [← not_and_or]; aesop\n    suffices ∀ r : K[X], v (↑r) ≠ 1 → r ≠ 0 → {p : K[X] | v ↑p < 1 ∧ ¬p = 0}.Nonempty by\n      exact Or.elim hor (fun hp ↦ this p hp h0.1) (fun hq ↦ this q hq h0.2)\n    exact fun r hr hr0 ↦ ⟨r, lt_iff_le_and_ne.mpr\n      ⟨Polynomial.valuation_le_one_of_valuation_X_le_one _ hle r, hr⟩, hr0⟩\n\nprivate lemma one_le_valuation_factor (hne : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty) {a b : K[X]}\n    (hab : v ↑(a * b) < 1 ∧ a ≠ 0 ∧ b ≠ 0) (hπᵥ : degree_lt_wf.min _ hne = a * b)\n    (hb : ¬IsUnit b) : 1 ≤ v ↑a := by\n  set πᵥ := degree_lt_wf.min _ hne\n  have hda : a.degree < πᵥ.degree := by\n    have hbpos := degree_pos_of_ne_zero_of_nonunit hab.2.2 hb\n    simp_rw [hπᵥ, degree_mul, degree_eq_natDegree hab.2.1, degree_eq_natDegree hab.2.2] at hbpos ⊢\n    norm_cast\n    simpa using hbpos\n  have hlea := imp_not_comm.mp (degree_lt_wf.not_lt_min _) hda\n  grind\n\nlemma irreducible_min_polynomial_valuation_lt_one_and_ne_zero [v.IsTrivialOn K]\n    (hne : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty) :\n    Irreducible (degree_lt_wf.min {p : K[X] | v p < 1 ∧ p ≠ 0} hne) := by\n  set πᵥ := degree_lt_wf.min _ hne\n  have hπᵥ : v πᵥ < 1 ∧ πᵥ ≠ 0 := degree_lt_wf.min_mem _ hne\n  refine irreducible_iff.mpr ⟨?_, fun a b hab ↦ ?_⟩\n  · simp only [Polynomial.isUnit_iff, isUnit_iff_ne_zero]\n    intro ⟨a, ha0, ha⟩\n    rw [← ha, coePolynomial, algebraMap_C, ← algebraMap_eq_C] at hπᵥ\n    grind\n  · by_contra! H\n    simp only [hab, ne_eq, mul_eq_zero, not_or] at hπᵥ\n    have hva := one_le_valuation_factor hne hπᵥ hab H.2\n    simp only [mul_comm a b, @and_comm (¬a = 0)] at hπᵥ hab\n    have := Right.one_le_mul (one_le_valuation_factor hne hπᵥ hab H.1) hva\n    simp only [coePolynomial_eq_algebraMap, map_mul] at hπᵥ this\n    grind\n\nsection valuation_X_le_one\n\nvariable [v.IsNontrivial] [v.IsTrivialOn K] (hle : v RatFunc.X ≤ 1)\n\n/-- A uniformizing element for the valuation `v`, as a polynomial in `K[X]`. -/\nabbrev uniformizingPolynomial : K[X] :=\n  WellFounded.min degree_lt_wf _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)\n\n@[inherit_doc]\nlocal notation \"πᵥ\" => uniformizingPolynomial hle\n\nTarget:\nlemma uniformizingPolynomial_ne_zero : πᵥ ≠ 0 :=\n\nProof body:\n","rejected":"by\n  have := degree_lt_wf.min_mem _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)\n  simp_all [uniformizingPolynomial]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/RatFunc","family_id":"uniformizingpolynomial_ne_zero","file_id":"mathlib/Mathlib/NumberTheory/RatFunc/Ostrowski.lean","sample_id":"2293cf123c21972c1d2af5e35604c06efbc61aa089ba31c3f82c50d912dfbf35"},"split":"train"}
{"binding":{"chosen_proof_sha256":"82d206dfe82328407b70971415868cac0e9eb31b2da76433329c285b45739547","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f2d217efd8dc67e5b8a5a62c60104ed7db1fe692e856a2299afd32cbd082efda","source_sha256":"20a211529741be444705515bbef40327474b8d60b75b7d4d28a459e31d8d7bbe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply Linear.comp_smul","hard_negative":true,"metrics":{"chosen_tokens":5,"rejected_tokens":8,"token_jaccard":0.090909,"token_length_ratio":1.6},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"113bf5fcc78d83b283b1abd59164a20a5b7d6f38b4351b823bdf98f6140cf016","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Defs\npublic import Mathlib.Algebra.Module.TransferInstance\npublic import Mathlib.Algebra.Group.Invertible.Defs\npublic import Mathlib.Algebra.Module.Equiv.Defs\npublic import Mathlib.CategoryTheory.Preadditive.Basic\n\nNamespace:\nCategoryTheory.Linear\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Linear categories\n\nAn `R`-linear category is a category in which `X ⟶ Y` is an `R`-module in such a way that\ncomposition of morphisms is `R`-linear in both variables.\n\nNote that sometimes in the literature a \"linear category\" is further required to be abelian.\n\n## Implementation\n\nCorresponding to the fact that we need to have an `AddCommGroup X` structure in place\nto talk about a `Module R X` structure,\nwe need `Preadditive C` as a prerequisite typeclass for `Linear R C`.\nThis makes for longer signatures than would be ideal.\n\n## Future work\n\nIt would be nice to have a usable framework of enriched categories in which this would just be\na category enriched in `Module R`.\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nopen CategoryTheory.Limits\n\nopen LinearMap\n\nnamespace CategoryTheory\n\n/-- A category is called `R`-linear if `P ⟶ Q` is an `R`-module such that composition is\n`R`-linear in both variables. -/\nclass Linear (R : Type w) [Semiring R] (C : Type u) [Category.{v} C] [Preadditive C] where\n  homModule : ∀ X Y : C, Module R (X ⟶ Y) := by infer_instance\n  /-- compatibility of the scalar multiplication with the post-composition -/\n  smul_comp : ∀ (X Y Z : C) (r : R) (f : X ⟶ Y) (g : Y ⟶ Z), (r • f) ≫ g = r • f ≫ g := by\n    cat_disch\n  /-- compatibility of the scalar multiplication with the pre-composition -/\n  comp_smul : ∀ (X Y Z : C) (f : X ⟶ Y) (r : R) (g : Y ⟶ Z), f ≫ (r • g) = r • f ≫ g := by\n    cat_disch\n\nattribute [instance_reducible, instance] Linear.homModule\n\nattribute [simp] Linear.smul_comp Linear.comp_smul\n\n-- (the linter doesn't like `simp` on the `_assoc` lemma)\nend CategoryTheory\n\nopen CategoryTheory\n\nnamespace CategoryTheory.Linear\n\nvariable {C : Type u} [Category.{v} C] [Preadditive C]\n\ninstance preadditiveNatLinear : Linear ℕ C where\n  smul_comp X _Y _Z r f g := by exact (Preadditive.rightComp X g).map_nsmul r f\n  comp_smul _X _Y Z f r g := by exact (Preadditive.leftComp Z f).map_nsmul r g\n\ninstance preadditiveIntLinear : Linear ℤ C where\n  smul_comp X _Y _Z r f g := by exact (Preadditive.rightComp X g).map_zsmul r f\n  comp_smul _X _Y Z f r g := by exact (Preadditive.leftComp Z f).map_zsmul r g\n\nsection End\n\nvariable {R : Type w}\n\ninstance [Semiring R] [Linear R C] (X : C) : Module R (End X) :=\n  inferInstanceAs <| Module R (X ⟶ X)\n\ninstance [CommSemiring R] [Linear R C] (X : C) : Algebra R (End X) :=\n  Algebra.ofModule (fun _ _ _ => comp_smul _ _ _ _ _ _) fun _ _ _ => smul_comp _ _ _ _ _ _\n\nend End\n\nsection\n\nvariable {R : Type w} [Semiring R] [Linear R C]\n\nsection InducedCategory\n\nuniverse u'\n\nvariable {D : Type u'} (F : D → C)\n\ninstance inducedCategory : Linear.{w, v} R (InducedCategory C F) where\n  homModule X Y := Equiv.module _ InducedCategory.homEquiv\n  smul_comp _ _ _ _ _ _ := by ext; apply smul_comp\n  comp_smul _ _ _ _ _ _ := by ext; apply comp_smul\n\nvariable {F} in\n/-- The linear equivalence `(X ⟶ Y) ≃+ (F X ⟶ F Y)` when `F : D → C` and\n`C` is a `R`-linear category. -/\n@[simps!]\ndef _root_.CategoryTheory.InducedCategory.homLinearEquiv\n    {X Y : InducedCategory C F} :\n    (X ⟶ Y) ≃ₗ[R] (F X ⟶ F Y) where\n  toAddEquiv := InducedCategory.homAddEquiv\n  map_smul' := by cat_disch\n\nend InducedCategory\n\ninstance fullSubcategory (Z : ObjectProperty C) : Linear.{w, v} R Z.FullSubcategory :=\n  inducedCategory _\n\nvariable (R)\n\n/-- Composition by a fixed left argument as an `R`-linear map. -/\n@[simps]\ndef leftComp {X Y : C} (Z : C) (f : X ⟶ Y) : (Y ⟶ Z) →ₗ[R] X ⟶ Z where\n  toFun g := f ≫ g\n  map_add' := by simp\n  map_smul' := by simp\n\n/-- Composition by a fixed right argument as an `R`-linear map. -/\n@[simps]\ndef rightComp (X : C) {Y Z : C} (g : Y ⟶ Z) : (X ⟶ Y) →ₗ[R] X ⟶ Z where\n  toFun f := f ≫ g\n  map_add' := by simp\n  map_smul' := by simp\n\ninstance {X Y : C} (f : X ⟶ Y) [Epi f] (r : R) [Invertible r] : Epi (r • f) :=\n  ⟨fun g g' H => by\n    rw [smul_comp, smul_comp, ← comp_smul, ← comp_smul, cancel_epi] at H\n    simpa [smul_smul] using congr_arg (fun f => ⅟r • f) H⟩\n\ninstance {X Y : C} (f : X ⟶ Y) [Mono f] (r : R) [Invertible r] : Mono (r • f) :=\n  ⟨fun g g' H => by\n    rw [comp_smul, comp_smul, ← smul_comp, ← smul_comp, cancel_mono] at H\n    simpa [smul_smul] using congr_arg (fun f => ⅟r • f) H⟩\n\n/-- Given isomorphic objects `X ≅ Y, W ≅ Z` in a `k`-linear category, we have a `k`-linear\nisomorphism between `Hom(X, W)` and `Hom(Y, Z).` -/\ndef homCongr (k : Type*) {C : Type*} [Category* C] [Semiring k] [Preadditive C] [Linear k C]\n    {X Y W Z : C} (f₁ : X ≅ Y) (f₂ : W ≅ Z) : (X ⟶ W) ≃ₗ[k] Y ⟶ Z :=\n  {\n    (rightComp k Y f₂.hom).comp\n      (leftComp k W\n        f₁.symm.hom) with\n    invFun := (leftComp k W f₁.hom).comp (rightComp k Y f₂.symm.hom)\n    left_inv := fun x => by\n      simp only [Iso.symm_hom, LinearMap.toFun_eq_coe, LinearMap.coe_comp, Function.comp_apply,\n        leftComp_apply, rightComp_apply, Category.assoc, Iso.hom_inv_id, Category.comp_id,\n        Iso.hom_inv_id_assoc]\n    right_inv := fun x => by\n      simp only [Iso.symm_hom, LinearMap.coe_comp, Function.comp_apply, rightComp_apply,\n        leftComp_apply, LinearMap.toFun_eq_coe, Iso.inv_hom_id_assoc, Category.assoc,\n        Iso.inv_hom_id, Category.comp_id] }\n\ntheorem homCongr_apply (k : Type*) {C : Type*} [Category* C] [Semiring k] [Preadditive C]\n    [Linear k C] {X Y W Z : C} (f₁ : X ≅ Y) (f₂ : W ≅ Z) (f : X ⟶ W) :\n    homCongr k f₁ f₂ f = (f₁.inv ≫ f) ≫ f₂.hom :=\n  rfl\n\ntheorem homCongr_symm_apply (k : Type*) {C : Type*} [Category* C] [Semiring k] [Preadditive C]\n    [Linear k C] {X Y W Z : C} (f₁ : X ≅ Y) (f₂ : W ≅ Z) (f : Y ⟶ Z) :\n    (homCongr k f₁ f₂).symm f = f₁.hom ≫ f ≫ f₂.inv :=\n  rfl\n\nvariable {R}\n\n@[simp]\nlemma units_smul_comp {X Y Z : C} (r : Rˣ) (f : X ⟶ Y) (g : Y ⟶ Z) :\n    (r • f) ≫ g = r • f ≫ g := by\n  apply Linear.smul_comp\n\n@[simp]\n\nTarget:\nlemma comp_units_smul {X Y Z : C} (f : X ⟶ Y) (r : Rˣ) (g : Y ⟶ Z) :\n    f ≫ (r • g) = r • f ≫ g :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"cf1dfd0c153f9c26912a120a41a2059d682480ccdab5e991e58d2753dc498894","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Linear","family_id":"comp_units_smul","file_id":"mathlib/Mathlib/CategoryTheory/Linear/Basic.lean","sample_id":"f2d217efd8dc67e5b8a5a62c60104ed7db1fe692e856a2299afd32cbd082efda"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0be2d8fff9f4e0668c3d36339336ba1af35cd1dd0eb1d09d433511d22d232ffa","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"43e80cf9725299057626e124ada0c7fa7af7a66a7d45d1cc2cc1eef0f94a344c","source_sha256":"7077627c568c3d5d4d50c5ab694c1de5a4aa92663f9998f759f2dc7e9ceb07d6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [← h₀.ordConnected_iff_of_bdd (OrderBot.bddBelow I) h₂, ordConnected_iff_disjoint_Ioo_empty]","hard_negative":true,"metrics":{"chosen_tokens":19,"rejected_tokens":8,"token_jaccard":0.043478,"token_length_ratio":0.421053},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"11d9a304193cb818fdea7f52225abf8e7b39b0f3f55e11ef2325d44d8810d343","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Int.ConditionallyCompleteOrder\npublic import Mathlib.Data.Int.Interval\npublic import Mathlib.Data.Int.SuccPred\npublic import Mathlib.Order.Lattice.Nat\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Bhavik Mehta, Oliver Nash\n-/\n/-!\n# Order-connected subsets of linear orders\n\nIn this file we provide some results about order-connected subsets of linear orders, together with\nsome convenience lemmas for characterising closed intervals in certain concrete types such as `ℤ`,\n`ℕ`, and `Fin n`.\n\n## Main results:\n* `Set.ordConnected_iff_disjoint_Ioo_empty`: a characterisation of `Set.OrdConnected` for\n  locally-finite linear orders.\n* `Set.Nonempty.ordConnected_iff_of_bdd`: a characterisation of closed intervals for locally-finite\n  conditionally complete linear orders.\n* `Set.Nonempty.ordConnected_iff_of_bdd'`: a characterisation of closed intervals for\n  locally-finite complete linear orders (convenient for `Fin n`).\n* `Set.Nonempty.eq_Icc_iff_nat`: characterisation of closed intervals for `ℕ`.\n* `Set.Nonempty.eq_Icc_iff_int`: characterisation of closed intervals for `ℤ`.\n-/\n\npublic section\n\nvariable {α : Type*} {I : Set α}\n\nlemma Set.Nonempty.ordConnected_iff_of_bdd\n    [ConditionallyCompleteLinearOrder α] [LocallyFiniteOrder α]\n    (h₀ : I.Nonempty) (h₁ : BddBelow I) (h₂ : BddAbove I) :\n    I.OrdConnected ↔ I = Icc (sInf I) (sSup I) :=\n  have h₄ : I.Finite := h₁.finite_of_bddAbove h₂\n  ⟨fun _ ↦ le_antisymm (subset_Icc_csInf_csSup h₁ h₂)\n    (I.Icc_subset (h₀.csInf_mem h₄) (h₀.csSup_mem h₄)), fun h₃ ↦ h₃ ▸ ordConnected_Icc⟩\n\n/-- A version of `Set.Nonempty.ordConnected_iff_of_bdd` for complete linear orders, such as `Fin n`,\nin which the explicit boundedness hypotheses are not necessary. -/\nlemma Set.Nonempty.ordConnected_iff_of_bdd' [ConditionallyCompleteLinearOrder α]\n    [OrderTop α] [OrderBot α] [LocallyFiniteOrder α]\n    (h₀ : I.Nonempty) :\n    I.OrdConnected ↔ I = Icc (sInf I) (sSup I) :=\n  h₀.ordConnected_iff_of_bdd (OrderBot.bddBelow I) (OrderTop.bddAbove I)\n\n/- TODO The `LocallyFiniteOrder` assumption here is probably too strong (e.g., it rules out `ℝ`\nfor which this result holds). However at the time of writing it is not clear what weaker\nassumption(s) should replace it. -/\nlemma Set.ordConnected_iff_disjoint_Ioo_empty [LinearOrder α] [LocallyFiniteOrder α] :\n    I.OrdConnected ↔ ∀ᵉ (x ∈ I) (y ∈ I), Disjoint (Ioo x y) I → Ioo x y = ∅ := by\n  simp_rw [← Set.subset_compl_iff_disjoint_right]\n  refine ⟨fun h' x hx y hy hxy ↦ ?_, fun h' ↦ ordConnected_of_Ioo fun x hx y hy hxy z hz ↦ ?_⟩\n  · suffices ∀ z, x < z → y ≤ z by ext z; simpa using this z\n    intro z hz\n    suffices z ∉ Ioo x y by simp_all\n    exact fun contra ↦ hxy contra <| h'.out hx hy <| mem_Icc_of_Ioo contra\n  · by_contra hz'\n    obtain ⟨x', hx', hx''⟩ :=\n      ((finite_Icc x z).inter_of_right I).exists_le_maximal ⟨hx, le_refl _, hz.1.le⟩\n    have hxz : x' < z := lt_of_le_of_ne hx''.1.2.2 (ne_of_mem_of_not_mem hx''.1.1 hz')\n    obtain ⟨y', hy', hy''⟩ :=\n      ((finite_Icc z y).inter_of_right I).exists_le_minimal ⟨hy, hz.2.le, le_refl _⟩\n    have hzy : z < y' := lt_of_le_of_ne' hy''.1.2.1 (ne_of_mem_of_not_mem hy''.1.1 hz')\n    have h₃ : Ioc x' z ⊆ Iᶜ := fun t ht ht' ↦ hx''.not_gt (⟨ht', le_trans hx' ht.1.le, ht.2⟩) ht.1\n    have h₄ : Ico z y' ⊆ Iᶜ := fun t ht ht' ↦ hy''.not_lt (⟨ht', ht.1, le_trans ht.2.le hy'⟩) ht.2\n    have h₅ : Ioo x' y' ⊆ Iᶜ := by\n      simp only [← Ioc_union_Ico_eq_Ioo hxz hzy, union_subset_iff, and_true, h₃, h₄]\n    exact eq_empty_iff_forall_notMem.1 (h' x' hx''.prop.1 y' hy''.prop.1 h₅) z ⟨hxz, hzy⟩\n\nTarget:\nlemma Set.Nonempty.eq_Icc_iff_nat {I : Set ℕ}\n    (h₀ : I.Nonempty) (h₂ : BddAbove I) :\n    I = Icc (sInf I) (sSup I) ↔ ∀ᵉ (x ∈ I) (y ∈ I), Disjoint (Ioo x y) I → y ≤ x + 1 :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"227e47caac3491472fbc0a591a4991f7bc68b507907afe0ed6bc45240bc4a7d5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Interval","family_id":"set","file_id":"mathlib/Mathlib/Order/Interval/Set/OrdConnectedLinear.lean","sample_id":"43e80cf9725299057626e124ada0c7fa7af7a66a7d45d1cc2cc1eef0f94a344c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e9b602b11908c749a9de1394766214188bf5ab4658bf596bbcd37cd55e75b8a2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e2bfab7a5a1b1912f82cb1a9e10696dbb413a79fc583a7da1b4cad05b058c943","source_sha256":"c0e006eb7064864ca13410a03d1b3208f2b5aa17e594e64bbe3404463f2e5f7f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← S.toNatSubmodule_toAddSubmonoid, ← Submodule.fg_iff_addSubmonoid_fg,\n    Submodule.fg_iff_exists_fin_linearMap]\n  exact exists_congr fun n => ⟨fun ⟨f, hf⟩ => ⟨f, hf ▸ LinearMap.range_toAddSubmonoid _⟩,\n    fun ⟨f, hf⟩ => ⟨f.toNatLinearMap, Submodule.toAddSubmonoid_inj.mp <|\n      hf ▸ LinearMap.range_toAddSubmonoid _⟩⟩","hard_negative":false,"metrics":{"chosen_tokens":68,"rejected_tokens":2,"token_jaccard":0.032258,"token_length_ratio":0.029412},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"11e9b3779fc382981c845ea6beaa9b846ec218c67277ebbf1f72ccb81e2f4499","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Congruence.Defs\npublic import Mathlib.LinearAlgebra.Basis.Cardinality\npublic import Mathlib.LinearAlgebra.DFinsupp\npublic import Mathlib.LinearAlgebra.Isomorphisms\npublic import Mathlib.LinearAlgebra.StdBasis\npublic import Mathlib.RingTheory.Finiteness.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\n/-!\n# Finite modules and types with finitely many elements\n\nThis file relates `Module.Finite` and `_root_.Finite`.\n\n-/\n\n@[expose] public section\n\nopen Function (Surjective)\nopen Finsupp\n\nsection ModuleAndAlgebra\n\nvariable (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]\n\nopen Module in\ntheorem Submodule.fg_iff_exists_fin_linearMap {N : Submodule R M} :\n    N.FG ↔ ∃ (n : ℕ) (f : (Fin n → R) →ₗ[R] M), LinearMap.range f = N := by\n  simp_rw [fg_iff_exists_fin_generating_family, ← ((Pi.basisFun R _).constr ℕ).exists_congr_right]\n  simp [Basis.constr_range]\n\nTarget:\ntheorem AddSubmonoid.fg_iff_exists_fin_addMonoidHom {M : Type*} [AddCommMonoid M]\n    {S : AddSubmonoid M} : S.FG ↔ ∃ (n : ℕ) (f : (Fin n → ℕ) →+ M), AddMonoidHom.mrange f = S :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Finiteness","family_id":"addsubmonoid","file_id":"mathlib/Mathlib/RingTheory/Finiteness/Cardinality.lean","sample_id":"e2bfab7a5a1b1912f82cb1a9e10696dbb413a79fc583a7da1b4cad05b058c943"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cf1663d691843ffb2d1c18b62f9a504b9533198687278010c9c58b7251f1dcfb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"52db5e0abeb3da4c111971ea0930014c39ce9e4ef49c8fcec292db20ddcc2dc7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8b1fcb5d375f7d0c76ac8b5dc5be9785fa4e3630fe7326a206b1b90b1962024a","source_sha256":"5efd5852b126afb746be3b592d500b1af5de2649a35029fd57ec7bbb27770fcb","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp only [LiftR, LiftR']\n  apply exists_iff_exists_of_mono F (f' rr _ _) (g' rr _ _)\n  · ext i ⟨x, _⟩ : 2\n    cases i <;> rfl\n  · intros\n    simp +unfoldPartialApp only [map_map, TypeVec.comp]\n    apply iff_of_eq -- Switch to `eq` so we can use `ext`\n    congr <;> ext i ⟨x, _⟩ <;> cases i <;> rfl","hard_negative":true,"metrics":{"chosen_tokens":88,"rejected_tokens":3,"token_jaccard":0.019608,"token_length_ratio":0.034091},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"123694f69ca9d7aa457964871adf8f032cb3113d42db59793c500fd1e7849745","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.Fin2\npublic import Mathlib.Data.TypeVec\npublic import Mathlib.Logic.Equiv.Defs\n\nNamespace:\nMvFunctor\n\nLocal context:\n/-\nCopyright (c) 2018 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Mario Carneiro, Simon Hudon\n-/\n/-!\n\n# Functors between the category of tuples of types, and the category Type\n\nFeatures:\n\n* `MvFunctor n` : the type class of multivariate functors\n* `f <$$> x`    : notation for map\n\n-/\n\n@[expose] public section\n\n\nuniverse u v w\n\nopen MvFunctor\n\n/-- Multivariate functors, i.e. functor between the category of type vectors\nand the category of Type -/\nclass MvFunctor {n : ℕ} (F : TypeVec n → Type*) where\n  /-- Multivariate map, if `f : α ⟹ β` and `x : F α` then `f <$$> x : F β`. -/\n  map : ∀ {α β : TypeVec n}, α ⟹ β → F α → F β\n\n/-- Multivariate map, if `f : α ⟹ β` and `x : F α` then `f <$$> x : F β` -/\nscoped[MvFunctor] infixr:100 \" <$$> \" => MvFunctor.map\n\nvariable {n : ℕ}\n\nnamespace MvFunctor\n\nvariable {α β : TypeVec.{u} n} {F : TypeVec.{u} n → Type v} [MvFunctor F]\n\n/-- predicate lifting over multivariate functors -/\ndef LiftP {α : TypeVec n} (P : ∀ i, α i → Prop) (x : F α) : Prop :=\n  ∃ u : F (fun i => Subtype (P i)), (fun i => @Subtype.val _ (P i)) <$$> u = x\n\n/-- relational lifting over multivariate functors -/\ndef LiftR {α : TypeVec n} (R : ∀ ⦃i⦄, α i → α i → Prop) (x y : F α) : Prop :=\n  ∃ u : F (fun i => { p : α i × α i // R p.fst p.snd }),\n    (fun i (t : { p : α i × α i // R p.fst p.snd }) => t.val.fst) <$$> u = x ∧\n      (fun i (t : { p : α i × α i // R p.fst p.snd }) => t.val.snd) <$$> u = y\n\n/-- given `x : F α` and a projection `i` of type vector `α`, `supp x i` is the set\nof `α.i` contained in `x` -/\ndef supp {α : TypeVec n} (x : F α) (i : Fin2 n) : Set (α i) :=\n  { y : α i | ∀ ⦃P⦄, LiftP P x → P i y }\n\ntheorem of_mem_supp {α : TypeVec n} {x : F α} {P : ∀ ⦃i⦄, α i → Prop} (h : LiftP P x) (i : Fin2 n) :\n    ∀ y ∈ supp x i, P y := fun _y hy => hy h\n\nend MvFunctor\n\n\n\n/-- laws for `MvFunctor` -/\nclass LawfulMvFunctor {n : ℕ} (F : TypeVec n → Type*) [MvFunctor F] : Prop where\n  /-- `map` preserved identities, i.e., maps identity on `α` to identity on `F α` -/\n  id_map : ∀ {α : TypeVec n} (x : F α), TypeVec.id <$$> x = x\n  /-- `map` preserves compositions -/\n  comp_map :\n    ∀ {α β γ : TypeVec n} (g : α ⟹ β) (h : β ⟹ γ) (x : F α), (h ⊚ g) <$$> x = h <$$> g <$$> x\n\nopen Nat TypeVec\n\nnamespace MvFunctor\n\nexport LawfulMvFunctor (comp_map)\n\nopen LawfulMvFunctor\n\nvariable {α β γ : TypeVec.{u} n}\nvariable {F : TypeVec.{u} n → Type v} [MvFunctor F]\nvariable (P : α ⟹ «repeat» n Prop) (R : α ⊗ α ⟹ «repeat» n Prop)\n\n/-- adapt `MvFunctor.LiftP` to accept predicates as arrows -/\ndef LiftP' : F α → Prop :=\n  MvFunctor.LiftP fun i x => ofRepeat <| P i x\n\n\n/-- adapt `MvFunctor.LiftR` to accept relations as arrows -/\ndef LiftR' : F α → F α → Prop :=\n  MvFunctor.LiftR @fun i x y => ofRepeat <| R i <| TypeVec.prod.mk _ x y\n\nvariable [LawfulMvFunctor F]\n\n@[simp]\ntheorem id_map (x : F α) : TypeVec.id <$$> x = x :=\n  LawfulMvFunctor.id_map x\n\n@[simp]\ntheorem id_map' (x : F α) : (fun _i a => a) <$$> x = x :=\n  id_map x\n\ntheorem map_map (g : α ⟹ β) (h : β ⟹ γ) (x : F α) : h <$$> g <$$> x = (h ⊚ g) <$$> x :=\n  Eq.symm <| comp_map _ _ _\n\nsection LiftP'\n\nvariable (F) in\ntheorem exists_iff_exists_of_mono {P : F α → Prop} {q : F β → Prop}\n    (f : α ⟹ β) (g : β ⟹ α)\n    (h₀ : f ⊚ g = TypeVec.id)\n    (h₁ : ∀ u : F α, P u ↔ q (f <$$> u)) :\n    (∃ u : F α, P u) ↔ ∃ u : F β, q u := by\n  constructor <;> rintro ⟨u, h₂⟩\n  · refine ⟨f <$$> u, ?_⟩\n    apply (h₁ u).mp h₂\n  · refine ⟨g <$$> u, ?_⟩\n    rw [h₁]\n    simp only [MvFunctor.map_map, h₀, LawfulMvFunctor.id_map, h₂]\n\ntheorem LiftP_def (x : F α) : LiftP' P x ↔ ∃ u : F (Subtype_ P), subtypeVal P <$$> u = x :=\n  exists_iff_exists_of_mono F _ _ (toSubtype_of_subtype P) (by simp [MvFunctor.map_map])\n\ntheorem LiftR_def (x y : F α) :\n    LiftR' R x y ↔\n      ∃ u : F (Subtype_ R),\n        (TypeVec.prod.fst ⊚ subtypeVal R) <$$> u = x ∧\n          (TypeVec.prod.snd ⊚ subtypeVal R) <$$> u = y :=\n  exists_iff_exists_of_mono _ _ _ (toSubtype'_of_subtype' R) (by\n    simp only [map_map, comp_assoc, subtypeVal_toSubtype']\n    simp +unfoldPartialApp [comp])\n\nend LiftP'\n\nend MvFunctor\n\nnamespace MvFunctor\n\nsection LiftPLastPredIff\n\nvariable {F : TypeVec.{u} (n + 1) → Type*} [MvFunctor F] [LawfulMvFunctor F] {α : TypeVec.{u} n}\n\nvariable {β : Type u}\nvariable (pp : β → Prop)\n\nprivate def f :\n    ∀ n α,\n      (fun i : Fin2 (n + 1) => { p_1 // ofRepeat (PredLast' α pp i p_1) }) ⟹ fun i : Fin2 (n + 1) =>\n        { p_1 : (α ::: β) i // PredLast α pp p_1 }\n  | _, α, Fin2.fs i, x =>\n    ⟨x.val, cast (by grind [PredLast]) x.property⟩\n  | _, _, Fin2.fz, x => ⟨x.val, x.property⟩\n\nprivate def g :\n    ∀ n α,\n      (fun i : Fin2 (n + 1) => { p_1 : (α ::: β) i // PredLast α pp p_1 }) ⟹ fun i : Fin2 (n + 1) =>\n        { p_1 // ofRepeat (PredLast' α pp i p_1) }\n  | _, α, Fin2.fs i, x =>\n    ⟨x.val, cast (by simp only [PredLast]; erw [const_iff_true]) x.property⟩\n  | _, _, Fin2.fz, x => ⟨x.val, x.property⟩\n\ntheorem LiftP_PredLast_iff {β} (P : β → Prop) (x : F (α ::: β)) :\n    LiftP' (PredLast' _ P) x ↔ LiftP (PredLast _ P) x := by\n  dsimp only [LiftP, LiftP']\n  apply exists_iff_exists_of_mono F (f _ n α) (g _ n α)\n  · ext i ⟨x, _⟩\n    cases i <;> rfl\n  · intros\n    rw [MvFunctor.map_map]\n    dsimp +unfoldPartialApp [(· ⊚ ·)]\n    suffices (fun i => Subtype.val) = (fun i x => (MvFunctor.f P n α i x).val) by rw [this]\n    ext i ⟨x, _⟩\n    cases i <;> rfl\n\nvariable (rr : β → β → Prop)\n\nset_option backward.isDefEq.respectTransparency false in\nprivate def f' :\n    ∀ n α,\n      (fun i : Fin2 (n + 1) =>\n          { p_1 : _ × _ // ofRepeat (RelLast' α rr i (TypeVec.prod.mk _ p_1.fst p_1.snd)) }) ⟹\n        fun i : Fin2 (n + 1) => { p_1 : (α ::: β) i × _ // RelLast α rr p_1.fst p_1.snd }\n  | _, α, Fin2.fs i, x =>\n    ⟨x.val, cast (by simp only [RelLast]; erw [repeatEq_iff_eq]) x.property⟩\n  | _, _, Fin2.fz, x => ⟨x.val, x.property⟩\n\nset_option backward.isDefEq.respectTransparency false in\nprivate def g' :\n    ∀ n α,\n      (fun i : Fin2 (n + 1) => { p_1 : (α ::: β) i × _ // RelLast α rr p_1.fst p_1.snd }) ⟹\n        fun i : Fin2 (n + 1) =>\n        { p_1 : _ × _ // ofRepeat (RelLast' α rr i (TypeVec.prod.mk _ p_1.1 p_1.2)) }\n  | _, α, Fin2.fs i, x =>\n    ⟨x.val, cast (by simp only [RelLast]; erw [repeatEq_iff_eq]) x.property⟩\n  | _, _, Fin2.fz, x => ⟨x.val, x.property⟩\n\nTarget:\ntheorem LiftR_RelLast_iff (x y : F (α ::: β)) :\n    LiftR' (RelLast' _ rr) x y ↔ LiftR (RelLast _ rr) x y :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_8b1fcb5d375f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"760005ab7ad9dcc50777246fcb558f4d3a74328ebf40f905fea69392731e06fc","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control/Functor","family_id":"liftr_rellast_iff","file_id":"mathlib/Mathlib/Control/Functor/Multivariate.lean","sample_id":"8b1fcb5d375f7d0c76ac8b5dc5be9785fa4e3630fe7326a206b1b90b1962024a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"732ed3f932374a610e819e8e805afcfdf95b627321f8ae3cc0670b9127888807","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5b7bc29de3804ac7bf45079e461f320c398aef01effbb7aab20e4f7fde575a9c","source_sha256":"5262ad7ee8fe0f47e01721cbd60f82b52bb0748fdb1cf1263478016dd35847f5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨n, rfl⟩ | ⟨n, rfl⟩ := Int.even_or_odd n\n  · simp [zpow_add, ← mul_zpow, Int.negOnePow_even _ (Even.add_self n)]\n  · rw [Int.negOnePow_odd _ (by exact odd_two_mul_add_one n)]\n    simp [Units.smul_def, zpow_add, Int.two_mul, ← mul_zpow]","hard_negative":false,"metrics":{"chosen_tokens":66,"rejected_tokens":3,"token_jaccard":0.060606,"token_length_ratio":0.045455},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"1238e57de9bec4706c122e43ebc803cddbb30dcae8333b00b62191a945d561b2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Center.Preadditive\npublic import Mathlib.Algebra.Ring.NegOnePow\n\nNamespace:\nCategoryTheory.CatCenter\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Powers of `-1` in the center of a preadditive category\n\n-/\n\npublic section\n\nuniverse v u\n\nnamespace CategoryTheory.CatCenter\n\nvariable {C : Type u} [Category.{v} C] [Preadditive C]\n\nopen scoped IsMulCommutative in\n@[simp]\n\nTarget:\nlemma app_neg_one_zpow (n : ℤ) (X : C) :\n    ((-1) ^ n : (CatCenter C)ˣ).val.app X = n.negOnePow • 𝟙 X :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Center","family_id":"app_neg_one_zpow","file_id":"mathlib/Mathlib/CategoryTheory/Center/NegOnePow.lean","sample_id":"5b7bc29de3804ac7bf45079e461f320c398aef01effbb7aab20e4f7fde575a9c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"36011e8dfa494a26101903f706aa37335eb673289a71da56c05014c7180fb544","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e1e83a15a2543ab2c7734d45b0801536ee9f8edb36b69899defb9d3f1b0cfbea","source_sha256":"837c21dde83dde778dcfbbb9710a388801c6131a30965944eb186a17738be704","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction n generalizing g with\n  | zero =>\n    funext (x : Fin 1)\n    simp [diagonalSuccIsoTensorTrivial, diagonalOneIsoLeftRegular, Subsingleton.elim x 0,\n      ofMulAction_V, types_tensorObj_def, types_tensorUnit_def]\n  | succ n hn =>\n    funext x\n    induction x using Fin.cases with\n    | zero => simp; rfl\n    | succ i =>\n      simpa [diagonalSuccIsoTensorTrivial, types_tensorObj_def, mul_assoc, Fin.partialProd_succ',\n        ofMulAction_V] using! congrFun (hn (g * f 0) (Fin.tail f)) i","hard_negative":false,"metrics":{"chosen_tokens":91,"rejected_tokens":2,"token_jaccard":0.022222,"token_length_ratio":0.021978},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"12ad3b5272bffd369a46fe99108f471e7d23d4015468668e32a31487e34afd36","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Fin\npublic import Mathlib.CategoryTheory.Monoidal.Linear\npublic import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory\npublic import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence\npublic import Mathlib.CategoryTheory.Monoidal.Transport\npublic import Mathlib.CategoryTheory.Monoidal.Types.Basic\npublic import Mathlib.CategoryTheory.Action.Concrete\npublic import Mathlib.CategoryTheory.Action.Limits\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Induced monoidal structure on `Action V G`\n\nWe show:\n\n* When `V` is monoidal, braided, or symmetric, so is `Action V G`.\n* When `V` is rigid and `G` is a group, `Action V G` is also rigid.\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen CategoryTheory Limits MonoidalCategory\n\nvariable {V : Type*} [Category* V] {G : Type*} [Monoid G]\n\nnamespace Action\n\nsection Monoidal\n\nopen MonoidalCategory\n\nvariable [MonoidalCategory V]\n\n@[simps! tensorUnit_V tensorObj_V tensorHom_hom whiskerLeft_hom whiskerRight_hom\n  associator_hom_hom associator_inv_hom leftUnitor_hom_hom leftUnitor_inv_hom\n  rightUnitor_hom_hom rightUnitor_inv_hom, reducible]\ninstance instMonoidalCategory : MonoidalCategory (Action V G) where\n  tensorObj X Y := Action.mk (X.V ⊗ Y.V) _\n  tensorUnit := Action.mk (𝟙_ _) _\n  __ := Monoidal.transport (Action.functorCategoryEquivalence _ _).symm\n\n@[simp]\ntheorem tensorUnit_ρ {g : G} :\n    @DFunLike.coe (G →* End (𝟙_ V)) _ _ _ (𝟙_ (Action V G)).ρ g = 𝟙 (𝟙_ V) :=\n  rfl\n\n@[simp]\ntheorem tensor_ρ {X Y : Action V G} {g : G} :\n    @DFunLike.coe (G →* End (X.V ⊗ Y.V)) _ _ _ (X ⊗ Y).ρ g = X.ρ g ⊗ₘ Y.ρ g :=\n  rfl\n\n/-- Given an object `X` isomorphic to the tensor unit of `V`, `X` equipped with the trivial action\nis isomorphic to the tensor unit of `Action V G`. -/\ndef tensorUnitIso {X : V} (f : 𝟙_ V ≅ X) : 𝟙_ (Action V G) ≅ Action.mk X 1 :=\n  Action.mkIso f\n\nvariable (V G)\n\nset_option backward.defeqAttrib.useBackward true in\ninstance : (Action.forget V G).Monoidal :=\n  Functor.CoreMonoidal.toMonoidal\n    { εIso := Iso.refl _\n      μIso := fun _ _ ↦ Iso.refl _ }\n\nopen Functor.LaxMonoidal Functor.OplaxMonoidal\n\n@[simp] lemma forget_ε : ε (Action.forget V G) = 𝟙 _ := rfl\n@[simp] lemma forget_η : η (Action.forget V G) = 𝟙 _ := rfl\n\nvariable {V G}\n\n@[simp] lemma forget_μ (X Y : Action V G) : μ (Action.forget V G) X Y = 𝟙 _ := rfl\n@[simp] lemma forget_δ (X Y : Action V G) : δ (Action.forget V G) X Y = 𝟙 _ := rfl\n\nvariable (V G)\n\nsection\n\nvariable [BraidedCategory V]\n\nset_option backward.defeqAttrib.useBackward true in\ninstance : BraidedCategory (Action V G) :=\n  .ofFaithful (Action.forget V G) fun X Y ↦ mkIso (β_ _ _) fun g ↦ by simp\n\n@[simp]\ntheorem β_hom_hom {X Y : Action V G} : (β_ X Y).hom.hom = (β_ X.V Y.V).hom := rfl\n\n@[simp]\ntheorem β_inv_hom {X Y : Action V G} : (β_ X Y).inv.hom = (β_ X.V Y.V).inv := rfl\n\nset_option backward.defeqAttrib.useBackward true in\n/-- When `V` is braided the forgetful functor `Action V G` to `V` is braided. -/\ninstance : (Action.forget V G).Braided where\n\nend\n\ninstance [SymmetricCategory V] : SymmetricCategory (Action V G) :=\n  .ofFaithful (Action.forget V G)\n\nsection\n\nvariable [Preadditive V] [MonoidalPreadditive V]\n\nattribute [local simp] MonoidalPreadditive.whiskerLeft_add MonoidalPreadditive.add_whiskerRight\n\ninstance : MonoidalPreadditive (Action V G) where\n\nvariable {R : Type*} [Semiring R] [Linear R V] [MonoidalLinear R V]\n\ninstance : MonoidalLinear R (Action V G) where\n\nend\n\nnoncomputable section\n\n/-- Upgrading the functor `Action V G ⥤ (SingleObj G ⥤ V)` to a monoidal functor. -/\ninstance FunctorCategoryEquivalence.functorMonoidal :\n    (FunctorCategoryEquivalence.functor (V := V) (G := G)).Monoidal :=\n  inferInstanceAs (Monoidal.equivalenceTransported\n    (Action.functorCategoryEquivalence V G).symm).inverse.Monoidal\n\ninstance functorCategoryEquivalenceFunctorMonoidal :\n    (functorCategoryEquivalence V G).functor.Monoidal :=\n  inferInstanceAs FunctorCategoryEquivalence.functor.Monoidal\n\n/-- Upgrading the functor `(SingleObj G ⥤ V) ⥤ Action V G` to a monoidal functor. -/\ninstance FunctorCategoryEquivalence.inverseMonoidal :\n    (FunctorCategoryEquivalence.inverse (V := V) (G := G)).Monoidal :=\n  inferInstanceAs (Monoidal.equivalenceTransported\n    (Action.functorCategoryEquivalence V G).symm).functor.Monoidal\n\ninstance functorCategoryEquivalenceInverseMonoidal :\n    (functorCategoryEquivalence V G).inverse.Monoidal :=\n  inferInstanceAs FunctorCategoryEquivalence.inverse.Monoidal\n\n@[simp]\nlemma FunctorCategoryEquivalence.functor_ε :\n    ε (FunctorCategoryEquivalence.functor (V := V) (G := G)) = 𝟙 _ := rfl\n\n@[simp]\nlemma FunctorCategoryEquivalence.functor_η :\n    η (FunctorCategoryEquivalence.functor (V := V) (G := G)) = 𝟙 _ := rfl\n\n@[simp]\nlemma FunctorCategoryEquivalence.functor_μ (A B : Action V G) :\n    μ FunctorCategoryEquivalence.functor A B = 𝟙 _ := rfl\n\n@[simp]\nlemma FunctorCategoryEquivalence.functor_δ (A B : Action V G) :\n    δ FunctorCategoryEquivalence.functor A B = 𝟙 _ := rfl\n\n\nvariable (H : Type*) [Group H]\n\ninstance [RightRigidCategory V] : RightRigidCategory (SingleObj H ⥤ V) := by\n  infer_instance\n\n/-- If `V` is right rigid, so is `Action V G`. -/\ninstance [RightRigidCategory V] : RightRigidCategory (Action V H) :=\n  rightRigidCategoryOfEquivalence\n    (functorCategoryEquivalence V H).toAdjunction\n\ninstance [LeftRigidCategory V] : LeftRigidCategory (SingleObj H ⥤ V) := by\n  infer_instance\n\n/-- If `V` is left rigid, so is `Action V G`. -/\ninstance [LeftRigidCategory V] : LeftRigidCategory (Action V H) :=\n  leftRigidCategoryOfEquivalence (functorCategoryEquivalence V H).toAdjunction\n\ninstance [RigidCategory V] : RigidCategory (SingleObj H ⥤ V) := by\n  infer_instance\n\n/-- If `V` is rigid, so is `Action V G`. -/\ninstance [RigidCategory V] : RigidCategory (Action V H) :=\n  rigidCategoryOfEquivalence (functorCategoryEquivalence V H).toAdjunction\n\nvariable {V H}\nvariable (X : Action V H)\n\n@[simp]\ntheorem rightDual_v [RightRigidCategory V] : Xᘁ.V = X.Vᘁ :=\n  rfl\n\n@[simp]\ntheorem leftDual_v [LeftRigidCategory V] : (ᘁX).V = ᘁX.V :=\n  rfl\n\ntheorem rightDual_ρ [RightRigidCategory V] (h : H) : Xᘁ.ρ h = (X.ρ (h⁻¹ : H))ᘁ := by\n  rw [← SingleObj.inv_as_inv]; rfl\n\ntheorem leftDual_ρ [LeftRigidCategory V] (h : H) : (ᘁX).ρ h = ᘁX.ρ (h⁻¹ : H) := by\n  rw [← SingleObj.inv_as_inv]; rfl\n\nend\n\nend Monoidal\n\nsection\n\nopen MonoidalCategory\n\nvariable (G : Type u)\n\n/-- The natural isomorphism of `G`-sets `Gⁿ⁺¹ ≅ G × Gⁿ`, where `G` acts by left multiplication on\neach factor. -/\n@[simps! hom_hom inv_hom]\nnoncomputable def diagonalSuccIsoTensorDiagonal [Monoid G] (n : ℕ) :\n    diagonal G (n + 1) ≅ leftRegular G ⊗ diagonal G n :=\n  mkIso (Fin.consEquiv _).symm.toIso fun _ => rfl\n\nvariable [Group G]\n\n/-- Given `X : Action (Type u) G` for `G` a group, then `G × X` (with `G` acting as left\nmultiplication on the first factor and by `X.ρ` on the second) is isomorphic as a `G`-set to\n`G × X` (with `G` acting as left multiplication on the first factor and trivially on the second).\nThe isomorphism is given by `(g, x) ↦ (g, g⁻¹ • x)`. -/\n@[simps! hom_hom inv_hom]\nnoncomputable def leftRegularTensorIso (X : Action (Type u) G) :\n    leftRegular G ⊗ X ≅ leftRegular G ⊗ trivial G X.V :=\n  mkIso (Equiv.toIso {\n    toFun g := ⟨g.1, (X.ρ (g.1⁻¹ : G) g.2 : X.V)⟩\n    invFun g := ⟨g.1, X.ρ g.1 g.2⟩\n    left_inv _ := Prod.ext rfl <| by simp\n    right_inv _ := Prod.ext rfl <| by simp }) <| fun _ => by\n      ext _\n      simp only [tensorObj_V, tensor_ρ]\n      simp [types_tensorObj_def]\n      rfl\n\n\n/-- An isomorphism of `G`-sets `Gⁿ⁺¹ ≅ G × Gⁿ`, where `G` acts by left multiplication on `Gⁿ⁺¹` and\n`G` but trivially on `Gⁿ`. The map sends `(g₀, ..., gₙ) ↦ (g₀, (g₀⁻¹g₁, g₁⁻¹g₂, ..., gₙ₋₁⁻¹gₙ))`,\nand the inverse is `(g₀, (g₁, ..., gₙ)) ↦ (g₀, g₀g₁, g₀g₁g₂, ..., g₀g₁...gₙ).` -/\nnoncomputable def diagonalSuccIsoTensorTrivial :\n    ∀ n : ℕ, diagonal G (n + 1) ≅ leftRegular G ⊗ trivial G (Fin n → G)\n  | 0 =>\n    diagonalOneIsoLeftRegular G ≪≫\n      (ρ_ _).symm ≪≫ tensorIso (Iso.refl _) (tensorUnitIso (Equiv.ofUnique PUnit _).toIso)\n  | n + 1 =>\n    diagonalSuccIsoTensorDiagonal _ _ ≪≫\n      tensorIso (Iso.refl _) (diagonalSuccIsoTensorTrivial n) ≪≫\n        leftRegularTensorIso _ _ ≪≫\n          tensorIso (Iso.refl _)\n            (mkIso (Fin.insertNthEquiv (fun _ => G) 0).toIso fun _ => rfl)\n\nvariable {G}\n\n@[simp]\ntheorem diagonalSuccIsoTensorTrivial_hom_hom_apply {n : ℕ} (f : Fin (n + 1) → G) :\n    dsimp% (diagonalSuccIsoTensorTrivial G n).hom.hom f =\n      (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f i.succ) := by\n  induction n with\n  | zero => exact Prod.ext rfl (funext fun x => Fin.elim0 x)\n  | succ n hn =>\n    refine Prod.ext rfl (funext fun x => ?_)\n    induction x using Fin.cases\n    <;> simp_all [diagonalSuccIsoTensorTrivial, types_tensorObj_def]\n    <;> rfl\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\n\nTarget:\ntheorem diagonalSuccIsoTensorTrivial_inv_hom_apply {n : ℕ} (g : G) (f : Fin n → G) :\n    dsimp% (diagonalSuccIsoTensorTrivial G n).inv.hom (g, f) =\n      (g • Fin.partialProd f : Fin (n + 1) → G) :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Action","family_id":"diagonalsuccisotensortrivial_inv_hom_apply","file_id":"mathlib/Mathlib/CategoryTheory/Action/Monoidal.lean","sample_id":"e1e83a15a2543ab2c7734d45b0801536ee9f8edb36b69899defb9d3f1b0cfbea"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f3a8fc7a4995882f5a643c316d6ec43e868e48edcba54745dfaa0649321d1a67","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"87307a21217d56836522ac81f517e1c61d9d870f235550b0f053c57d27e92497","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"28eb45ede6284ff2c969d0ba34870b558934c40303d056e5740787b98d7dcd28","source_sha256":"287a704d503b237a43f85c1d8d085e3a9c8170677a7fe6bc6d74659961e522de","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hb' : b.Dom := Part.dom_iff_mem.mpr ⟨a.get ha, hb⟩\n  rwa [← eq_get_iff_mem hb', eq_iff_of_dom ha hb'] at hb","hard_negative":false,"metrics":{"chosen_tokens":33,"rejected_tokens":38,"token_jaccard":0.862069,"token_length_ratio":1.151515},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"12b5ce0e040d5ad4af11f2224d32439feaac81dee4904790968fd1ebb58a4cfa","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Defs\npublic import Mathlib.Data.Set.Subsingleton\npublic import Mathlib.Logic.Equiv.Defs\n\nNamespace:\nPart\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Jeremy Avigad, Simon Hudon\n-/\n/-!\n# Partial values of a type\n\nThis file defines `Part α`, the partial values of a type.\n`o : Part α` carries a proposition `o.Dom`, its domain, along with a function `get : o.Dom → α`, its\nvalue. The rule is then that every partial value has a value but, to access it, you need to provide\na proof of the domain.\n`Part α` behaves the same as `Option α` except that `o : Option α` is decidably `none` or `some a`\nfor some `a : α`, while the domain of `o : Part α` doesn't have to be decidable. That means you can\ntranslate back and forth between a partial value with a decidable domain and an option, and\n`Option α` and `Part α` are classically equivalent. In general, `Part α` is bigger than `Option α`.\n\n## Main declarations\n`Option`-like declarations:\n* `Part.none`: The partial value whose domain is `False`.\n* `Part.some a`: The partial value whose domain is `True` and whose value is `a`.\n* `Part.ofOption`: Converts an `Option α` to a `Part α` by sending `none` to `none` and `some a` to\n  `some a`.\n* `Part.toOption`: Converts a `Part α` with a decidable domain to an `Option α`.\n* `Part.equivOption`: Classical equivalence between `Part α` and `Option α`.\n\nMonadic structure:\n* `Part.bind`: `o.bind f` has value `(f (o.get _)).get _` (`f o` morally) and is defined when `o`\n  and `f (o.get _)` are defined.\n* `Part.map`: Maps the value and keeps the same domain.\n\nOther:\n* `Part.restrict`: `Part.restrict p o` replaces the domain of `o : Part α` by `p : Prop` so long as\n  `p → o.Dom`.\n* `Part.assert`: `assert p f` appends `p` to the domains of the values of a partial function.\n* `Part.unwrap`: Gets the value of a partial value regardless of its domain. Unsound.\n\n## Notation\nFor `a : α`, `o : Part α`, `a ∈ o` means that `o` is defined and equal to `a`. Formally, it means\n`o.Dom` and `o.get _ = a`.\n-/\n\n@[expose] public section\n\nassert_not_exists RelIso\n\nopen Function\n\n/-- `Part α` is the type of \"partial values\" of type `α`. It\n  is similar to `Option α` except the domain condition can be an\n  arbitrary proposition, not necessarily decidable. -/\nstructure Part.{u} (α : Type u) : Type u where\n  /-- The domain of a partial value -/\n  Dom : Prop\n  /-- Extract a value from a partial value given a proof of `Dom` -/\n  get : Dom → α\n\nnamespace Part\n\nvariable {α : Type*} {β : Type*} {γ : Type*}\n\n/-- Convert a `Part α` with a decidable domain to an option -/\ndef toOption (o : Part α) [Decidable o.Dom] : Option α :=\n  if h : Dom o then some (o.get h) else none\n\n@[simp] lemma toOption_isSome (o : Part α) [Decidable o.Dom] : o.toOption.isSome ↔ o.Dom := by\n  by_cases h : o.Dom <;> simp [h, toOption]\n\n@[simp] lemma toOption_eq_none (o : Part α) [Decidable o.Dom] : o.toOption = none ↔ ¬o.Dom := by\n  by_cases h : o.Dom <;> simp [h, toOption]\n\n/-- `Part` extensionality -/\ntheorem ext' : ∀ {o p : Part α}, (o.Dom ↔ p.Dom) → (∀ h₁ h₂, o.get h₁ = p.get h₂) → o = p\n  | ⟨od, o⟩, ⟨pd, p⟩, H1, H2 => by\n    have t : od = pd := propext H1\n    cases t; rw [show o = p from funext fun p => H2 p p]\n\n/-- `Part` eta expansion -/\n@[simp]\ntheorem eta : ∀ o : Part α, (⟨o.Dom, fun h => o.get h⟩ : Part α) = o\n  | ⟨_, _⟩ => rfl\n\n/-- `a ∈ o` means that `o` is defined and equal to `a` -/\nprotected def Mem (o : Part α) (a : α) : Prop :=\n  ∃ h, o.get h = a\n\ninstance : Membership α (Part α) :=\n  ⟨Part.Mem⟩\n\ntheorem mem_eq (a : α) (o : Part α) : (a ∈ o) = ∃ h, o.get h = a :=\n  rfl\n\ntheorem dom_iff_mem : ∀ {o : Part α}, o.Dom ↔ ∃ y, y ∈ o\n  | ⟨_, f⟩ => ⟨fun h => ⟨f h, h, rfl⟩, fun ⟨_, h, rfl⟩ => h⟩\n\ntheorem get_mem {o : Part α} (h) : get o h ∈ o :=\n  ⟨_, rfl⟩\n\n@[simp]\ntheorem mem_mk_iff {p : Prop} {o : p → α} {a : α} : a ∈ Part.mk p o ↔ ∃ h, o h = a :=\n  Iff.rfl\n\n/-- `Part` extensionality -/\n@[ext]\ntheorem ext {o p : Part α} (H : ∀ a, a ∈ o ↔ a ∈ p) : o = p :=\n  (ext' ⟨fun h => ((H _).1 ⟨h, rfl⟩).fst, fun h => ((H _).2 ⟨h, rfl⟩).fst⟩) fun _ _ =>\n    ((H _).2 ⟨_, rfl⟩).snd\n\n/-- The `none` value in `Part` has a `False` domain and an empty function. -/\ndef none : Part α :=\n  ⟨False, False.rec⟩\n\ninstance : Inhabited (Part α) :=\n  ⟨none⟩\n\n@[simp]\ntheorem notMem_none (a : α) : a ∉ @none α := fun h => h.fst\n\n/-- The `some a` value in `Part` has a `True` domain and the\n  function returns `a`. -/\ndef some (a : α) : Part α :=\n  ⟨True, fun _ => a⟩\n\n@[simp]\ntheorem some_dom (a : α) : (some a).Dom :=\n  trivial\n\ntheorem mem_unique : ∀ {a b : α} {o : Part α}, a ∈ o → b ∈ o → a = b\n  | _, _, ⟨_, _⟩, ⟨_, rfl⟩, ⟨_, rfl⟩ => rfl\n\ntheorem mem_right_unique : ∀ {a : α} {o p : Part α}, a ∈ o → a ∈ p → o = p\n  | _, _, _, ⟨ho, _⟩, ⟨hp, _⟩ => ext' (iff_of_true ho hp) (by simp [*])\n\ntheorem Mem.left_unique : Relator.LeftUnique ((· ∈ ·) : α → Part α → Prop) := fun _ _ _ =>\n  mem_unique\n\ntheorem Mem.right_unique : Relator.RightUnique ((· ∈ ·) : α → Part α → Prop) := fun _ _ _ =>\n  mem_right_unique\n\ntheorem get_eq_of_mem {o : Part α} {a} (h : a ∈ o) (h') : get o h' = a :=\n  mem_unique ⟨_, rfl⟩ h\n\nprotected theorem subsingleton (o : Part α) : Set.Subsingleton { a | a ∈ o } := fun _ ha _ hb =>\n  mem_unique ha hb\n\n@[simp]\ntheorem get_some {a : α} (ha : (some a).Dom) : get (some a) ha = a :=\n  rfl\n\ntheorem mem_some (a : α) : a ∈ some a :=\n  ⟨trivial, rfl⟩\n\n@[simp]\ntheorem mem_some_iff {a b} : b ∈ (some a : Part α) ↔ b = a :=\n  ⟨fun ⟨_, e⟩ => e.symm, fun e => ⟨trivial, e.symm⟩⟩\n\ntheorem eq_some_iff {a : α} {o : Part α} : o = some a ↔ a ∈ o :=\n  ⟨fun e => e.symm ▸ mem_some _, fun ⟨h, e⟩ => e ▸ ext' (iff_true_intro h) fun _ _ => rfl⟩\n\ntheorem eq_none_iff {o : Part α} : o = none ↔ ∀ a, a ∉ o :=\n  ⟨fun e => e.symm ▸ notMem_none, fun h => ext (by simpa)⟩\n\ntheorem eq_none_iff' {o : Part α} : o = none ↔ ¬o.Dom :=\n  ⟨fun e => e.symm ▸ id, fun h => eq_none_iff.2 fun _ h' => h h'.fst⟩\n\n@[simp]\ntheorem not_none_dom : ¬(none : Part α).Dom :=\n  id\n\n@[simp]\ntheorem some_ne_none (x : α) : some x ≠ none := by\n  intro h\n  exact true_ne_false (congr_arg Dom h)\n\n@[simp]\ntheorem none_ne_some (x : α) : none ≠ some x :=\n  (some_ne_none x).symm\n\ntheorem ne_none_iff {o : Part α} : o ≠ none ↔ ∃ x, o = some x := by\n  constructor\n  · rw [Ne, eq_none_iff', not_not]\n    exact fun h => ⟨o.get h, eq_some_iff.2 (get_mem h)⟩\n  · rintro ⟨x, rfl⟩\n    apply some_ne_none\n\ntheorem eq_none_or_eq_some (o : Part α) : o = none ∨ ∃ x, o = some x :=\n  or_iff_not_imp_left.2 ne_none_iff.1\n\ntheorem some_injective : Injective (@Part.some α) := fun _ _ h =>\n  congr_fun (eq_of_heq (Part.mk.inj h).2) trivial\n\n@[simp]\ntheorem some_inj {a b : α} : Part.some a = some b ↔ a = b :=\n  some_injective.eq_iff\n\n@[simp]\ntheorem some_get {a : Part α} (ha : a.Dom) : Part.some (Part.get a ha) = a :=\n  Eq.symm (eq_some_iff.2 ⟨ha, rfl⟩)\n\ntheorem get_eq_iff_eq_some {a : Part α} {ha : a.Dom} {b : α} : a.get ha = b ↔ a = some b :=\n  ⟨fun h => by simp [h.symm], fun h => by simp [h]⟩\n\ntheorem get_eq_get_of_eq (a : Part α) (ha : a.Dom) {b : Part α} (h : a = b) :\n    a.get ha = b.get (h ▸ ha) := by\n  congr\n\ntheorem get_eq_iff_mem {o : Part α} {a : α} (h : o.Dom) : o.get h = a ↔ a ∈ o :=\n  ⟨fun H => ⟨h, H⟩, fun ⟨_, H⟩ => H⟩\n\ntheorem eq_get_iff_mem {o : Part α} {a : α} (h : o.Dom) : a = o.get h ↔ a ∈ o :=\n  eq_comm.trans (get_eq_iff_mem h)\n\ntheorem eq_of_get_eq_get {a b : Part α} (ha : a.Dom) (hb : b.Dom) (hab : a.get ha = b.get hb) :\n    a = b :=\n  ext' (iff_of_true ha hb) fun _ _ => hab\n\ntheorem eq_iff_of_dom {a b : Part α} (ha : a.Dom) (hb : b.Dom) : a.get ha = b.get hb ↔ a = b :=\n  ⟨eq_of_get_eq_get ha hb, get_eq_get_of_eq a ha⟩\n\nTarget:\ntheorem eq_of_mem {a b : Part α} (ha : a.Dom) (hb : a.get ha ∈ b) : a = b :=\n\nProof body:\n","rejected":"by\n  have hb' : b.Dom := Part.dom_iff_mem.mpr ⟨a.get ha, hb⟩\n  rwa [← eq_get_iff_mem hb', eq_iff_of_dom ha hb'] at hb\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data","family_id":"eq_of_mem","file_id":"mathlib/Mathlib/Data/Part.lean","sample_id":"28eb45ede6284ff2c969d0ba34870b558934c40303d056e5740787b98d7dcd28"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5bd7c7167db7f215623bed49eb0f0085e5528c13934e7559ad6540e992b06a8a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b950a5480671beab0f3b98bc2ebc1a58c3277b092f1e6675f0749476ef524dec","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2e71596933c22369bea1ce7d19358deda1acb57a1b511e60283fae72b0ed1051","source_sha256":"8d66dfc09aa4bf19e9637e6b2cc5cd204dddcf3c4b5353ddb78dc856eda79a64","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  letI := h𝕜.rclike\n  exact hs.mono_field","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":5,"token_jaccard":0.25,"token_length_ratio":0.454545},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"12da3d0165d4f2605052966862ff9ecba9709b056f9679cc74d75845448b03ca","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.RCLike.Basic\npublic import Mathlib.Topology.Instances.RealVectorSpace\npublic import Mathlib.Analysis.Calculus.TangentCone.Real\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-! # Relationships between unique differentiability over `ℝ` and `ℂ`\n\nA set of unique differentiability for `ℝ` is also a set of unique differentiability for `ℂ`\n(or for a general field satisfying `IsRCLikeNormedField 𝕜`).\n-/\n\npublic section\n\nvariable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [h𝕜 : IsRCLikeNormedField 𝕜]\n  {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace ℝ E]\n  {s : Set E} {x : E}\n\ntheorem tangentConeAt_real_subset_isRCLikeNormedField :\n    tangentConeAt ℝ s x ⊆ tangentConeAt 𝕜 s x := by\n  letI := h𝕜.rclike\n  exact tangentConeAt_mono_field\n\nTarget:\ntheorem UniqueDiffWithinAt.of_real (hs : UniqueDiffWithinAt ℝ s x) :\n    UniqueDiffWithinAt 𝕜 s x :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_2e71596933c2","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d43b23fba541f580915d0cbf2955ba12b24bd443cf53c79df4cb87516f00df92","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/RCLike","family_id":"uniquediffwithinat","file_id":"mathlib/Mathlib/Analysis/RCLike/TangentCone.lean","sample_id":"2e71596933c22369bea1ce7d19358deda1acb57a1b511e60283fae72b0ed1051"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5b8c55401ccfdd0326db1a35194aa3affc02ada3be7e2075a756a2a2bba77cb4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"499fbc700dadc344361ad4af367c82f43393dc7aa3a2dfdb704f625e9fa4c334","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fad9f1ea8d6654621705b02879012779787dbbf5d7900dc540f712fce41505c9","source_sha256":"51daf98e481b660f2839034f65b84ec1148fc7fc2de859367b6b6e85ce12e57c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isCoskeletal_iff]\n  exact isRightKanExtension_iff_isIso ((coskAdj n).unit.app X)\n    ((coskAdj n).counit.app _) (𝟙 _) ((coskAdj n).left_triangle_components X)","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":5,"token_jaccard":0.142857,"token_length_ratio":0.119048},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"134f85183e27480fb30c0967287e04534edbed39b6600d743b5aa5e4f98543ae","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.SimplicialObject.Basic\npublic import Mathlib.CategoryTheory.Functor.KanExtension.Adjunction\npublic import Mathlib.CategoryTheory.Functor.KanExtension.Basic\n\nNamespace:\nCategoryTheory.SimplicialObject\n\nLocal context:\n/-\nCopyright (c) 2024 Emily Riehl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Emily Riehl, Joël Riou\n-/\n/-!\n# Coskeletal simplicial objects\n\nThe identity natural transformation exhibits a simplicial object `X` as a right extension of its\nrestriction along `(Truncated.inclusion n).op` recorded by `rightExtensionInclusion X n`.\n\nThe simplicial object `X` is *n-coskeletal* if `rightExtensionInclusion X n` is a right Kan\nextension.\n\nWhen the ambient category admits right Kan extensions along `(Truncated.inclusion n).op`,\nthen when `X` is `n`-coskeletal, the unit of `coskAdj n` defines an isomorphism:\n`isoCoskOfIsCoskeletal : X ≅ (cosk n).obj X`.\n\nTODO: Prove that `X` is `n`-coskeletal whenever a certain canonical cone is a limit cone.\n-/\n\n@[expose] public section\n\nopen Opposite\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits CategoryTheory.Functor SimplexCategory\n\nuniverse v u v' u'\n\nnamespace CategoryTheory\n\nnamespace SimplicialObject\nvariable {C : Type u} [Category.{v} C]\nvariable (X : SimplicialObject C) (n : ℕ)\n\nnamespace Truncated\n\n/-- The identity natural transformation exhibits a simplicial set as a right extension of its\nrestriction along `(Truncated.inclusion n).op`. -/\n@[simps!]\ndef rightExtensionInclusion :\n    RightExtension (Truncated.inclusion n).op\n      ((Truncated.inclusion n).op ⋙ X) := RightExtension.mk _ (𝟙 _)\n\nend Truncated\n\nopen Truncated\n\n/-- A simplicial object `X` is `n`-coskeletal when it is the right Kan extension of its restriction\nalong `(Truncated.inclusion n).op` via the identity natural transformation. -/\n@[mk_iff]\nclass IsCoskeletal : Prop where\n  isRightKanExtension : IsRightKanExtension X (𝟙 ((Truncated.inclusion n).op ⋙ X))\n\nattribute [instance] IsCoskeletal.isRightKanExtension\n\nsection\n\nvariable [∀ (F : (SimplexCategory.Truncated n)ᵒᵖ ⥤ C),\n    (SimplexCategory.Truncated.inclusion n).op.HasRightKanExtension F]\n\n/-- If `X` is `n`-coskeletal, then `Truncated.rightExtensionInclusion X n` is a terminal object in\nthe category `RightExtension (Truncated.inclusion n).op (Truncated.inclusion.op ⋙ X)`. -/\nnoncomputable def IsCoskeletal.isUniversalOfIsRightKanExtension [X.IsCoskeletal n] :\n    (rightExtensionInclusion X n).IsUniversal := by\n  apply Functor.isUniversalOfIsRightKanExtension\n\nTarget:\ntheorem isCoskeletal_iff_isIso : X.IsCoskeletal n ↔ IsIso ((coskAdj n).unit.app X) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_fad9f1ea8d66","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"ef01acce0d3b29fbce563bc5624051edb1cac56e5b580d086e3141e806c29aa6","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplicialObject","family_id":"iscoskeletal_iff_isiso","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplicialObject/Coskeletal.lean","sample_id":"fad9f1ea8d6654621705b02879012779787dbbf5d7900dc540f712fce41505c9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"054bd373a1fa3d5f15bd0fe1ba1c34023be8febf1ced40b818a6af0feb340b77","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1f3bfea624fb86d6c8b755b05ec1c6fe8d68cd178710d19c21a20e4947546990","source_sha256":"ced30280014822262039ce197c9e4ae821d789623b770f4b0e6844d4852109a3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [dp_def]\n  rw [mkAlgHom_rel (A := MvPolynomial (ℕ × M) R) R Rel.add, map_sum,\n    Finset.sum_congr rfl (fun k _ ↦ by rw [_root_.map_mul])]","hard_negative":true,"metrics":{"chosen_tokens":45,"rejected_tokens":8,"token_jaccard":0.055556,"token_length_ratio":0.177778},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"136a283650b81dc17a88ee08c13f7cf55dc2feffafa957322ff8bbbe501b8b07","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.Eval\npublic import Mathlib.Algebra.RingQuot\npublic import Mathlib.RingTheory.DividedPowers.Basic\n\nNamespace:\nDividedPowerAlgebra\n\nLocal context:\n/-\nCopyright (c) 2026 Antoine Chambert-Loir & María Inés de Frutos—Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir, María Inés de Frutos—Fernández\n-/\n/-!\n# The universal divided power algebra\n\nLet `R` be a (commutative) semiring and `M` be an `R`-module. In this file we define `Γ_R(M)`,\nthe universal divided power algebra of `M`, as the ring quotient of the polynomial ring\nin the variables `ℕ × M` by the relation `DividedPowerAlgebra.Rel`.\n\n`DividedPowerAlgebra R M` satisfies a weak universal property for morphisms to rings with\ndivided powers (`DividedPowerAlgebra.lift`).\n\n## Main definitions\n\n* `DividedPowerAlgebra.Rel`: the type coding the basic relations that will give rise to the\n  divided power algebra.\n\n* `DividedPowerAlgebra R M`: the universal divided power algebra of the `R`-module `M`,\n  defined as `RingQuot` of `DividedPowerAlgebra.Rel R M`.\n\n* `DividedPowerAlgebra.dp R n m`: for `n : ℕ` and `m : M`, this is the equivalence class of\n  `MvPolynomial.X (⟨n, m⟩)` in `DividedPowerAlgebra R M`.\n\n  When that algebra is endowed with its canonical divided power structure (to be defined),\n  the image of `MvPolynomial.X (n, m)`, for any `n : ℕ` and `m : M`, is equal to\n  the `n`th divided power of the image of `m`.\n\n  The API will be setup so that it is never (never say never…) necessary to lift to `MvPolynomial`.\n\n* `DividedPowerAlgebra.lift`: the weak universal property of `DividedPowerAlgebra R M`.\n\n* `DividedPowerAlgebra.map`: the functoriality map between divided power algebras\n  associated with a linear map of the underlying modules.\n  Given an `R`-algebra `S`, an `S`-module `N` and an `R`-linear map `f : M →ₗ[R] N`,\n  this is the map `DividedPowerAlgebra R M →ₐ[R] DividedPowerAlgebra S N`\n  sending `dp R n m` to `dp S n (f m)`.\n\n## References\n\n* [P. Berthelot (1974), *Cohomologie cristalline des schémas de\n  caractéristique $p$ > 0*][Berthelot-1974]\n\n* [P. Berthelot and A. Ogus (1978), *Notes on crystalline\n  cohomology*][BerthelotOgus-1978]\n\n* [N. Roby (1963), *Lois polynomes et lois formelles en théorie des\n  modules*][Roby-1963]\n\n* [N. Roby (1965), *Les algèbres à puissances dividées*][Roby-1965]\n\n## TODO\n\n* Show in upcoming files that `DividedPowerAlgebra R M` has divided powers.\n\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Finset Ideal MvPolynomial RingQuot\n\nvariable (R M : Type*) [CommSemiring R] [AddCommMonoid M] [Module R M]\n\nnamespace DividedPowerAlgebra\n\n/-- The type coding the basic relations that will give rise to the divided power algebra.\n  The class of `MvPolynomial.X (n, a)` will be equal to `dpow n a`, for `a ∈ M`. -/\ninductive Rel : MvPolynomial (ℕ × M) R → MvPolynomial (ℕ × M) R → Prop\n  | rfl_zero : Rel 0 0 -- Needed for technical reasons.\n  | zero {a : M} : Rel (X (0, a)) 1\n  | smul {r : R} {n : ℕ} {a : M} : Rel (X (n, r • a)) (r ^ n • X (n, a))\n  | mul {m n : ℕ} {a : M} : Rel (X (m, a) * X (n, a)) (Nat.choose (m + n) m • X (m + n, a))\n  | add {n : ℕ} {a b : M} :\n    Rel (X (n, a + b)) ((Finset.antidiagonal n).sum fun k ↦ X (k.1, a) * X (k.2, b))\n\n/-- The ideal of `MvPolynomial (ℕ × M) R` generated by `Rel`. -/\ndef RelI : Ideal (MvPolynomial (ℕ × M) R) := ofRel (DividedPowerAlgebra.Rel R M)\n\nend DividedPowerAlgebra\n\n/-- The divided power algebra of a module M is defined as the ring quotient of the polynomial ring\n  in the variables `ℕ × M` by the ring relation defined by `DividedPowerAlgebra.Rel`.\n  We will later show that that `DividedPowerAlgebra R M` has divided powers.\n  It satisfies a weak universal property for morphisms to rings with divided powers. -/\nabbrev DividedPowerAlgebra := RingQuot (DividedPowerAlgebra.Rel R M)\n\nnamespace DividedPowerAlgebra\n\nopen MvPolynomial\n\nvariable {R M}\n\nlemma mkAlgHom_surjective : Function.Surjective (mkAlgHom R (Rel R M)) :=\n  RingQuot.mkAlgHom_surjective _ _\n\nlemma mkAlgHom_C (a : R) :\n    mkAlgHom R (Rel R M) (C a) = algebraMap R (DividedPowerAlgebra R M) a := by\n  rw [← MvPolynomial.algebraMap_eq, AlgHom.commutes]\n\nlemma mkRingHom_C (a : R) :\n    mkRingHom (Rel R M) (C a) = algebraMap R (DividedPowerAlgebra R M) a := by\n  rw [← mkAlgHom_C, mkAlgHom, AlgHom.coe_mk]\n\nvariable (R) in\n/-- `dp R n m` is the equivalence class of `X (⟨n, m⟩)` in `DividedPowerAlgebra R M`. -/\ndef dp (n : ℕ) (m : M) : DividedPowerAlgebra R M := mkAlgHom R (Rel R M) (X ⟨n, m⟩)\n\ntheorem dp_def (n : ℕ) (m : M) :\n  dp R n m = mkAlgHom R (Rel R M) (X ⟨n, m⟩) := rfl\n\nprotected theorem induction_on' {P : DividedPowerAlgebra R M → Prop} (f : DividedPowerAlgebra R M)\n    (h_C : ∀ a, P (mkAlgHom R (Rel R M) (C a))) (h_add : ∀ f g, P f → P g → P (f + g))\n    (h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)) : P f := by\n  obtain ⟨F, hf⟩ := RingQuot.mkRingHom_surjective (DividedPowerAlgebra.Rel R M) f\n  rw [← hf]\n  induction F using MvPolynomial.induction_on generalizing f with\n  | C a =>\n      convert! h_C a using 1\n      rw [mkAlgHom, AlgHom.coe_mk]\n  | add g1 g2 hg1 hg2 =>\n      rw [map_add]\n      exact h_add _ _ (hg1 ((mkRingHom (Rel R M)) g1) rfl) (hg2 ((mkRingHom (Rel R M)) g2) rfl)\n  | mul_X g nm h =>\n      have h' : (mkRingHom (Rel R M)) (X nm) = dp R nm.1 nm.2 := by\n        simp only [dp_def, Prod.mk.eta, mkAlgHom, AlgHom.coe_mk]\n      rw [_root_.map_mul, h']\n      exact h_dp _ _ _ (h (mkRingHom (Rel R M) g) rfl)\n\n@[elab_as_elim]\nprotected theorem induction_on {P : DividedPowerAlgebra R M → Prop} (f : DividedPowerAlgebra R M)\n    (C : ∀ a, P (algebraMap R _ a)) (add : ∀ f g, P f → P g → P (f + g))\n    (dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)) : P f :=\n  DividedPowerAlgebra.induction_on' f (fun a ↦ by rw [mkAlgHom_C]; exact C a) add dp\n\ntheorem dp_eq_mkRingHom (n : ℕ) (m : M) :\n    dp R n m = mkRingHom (Rel R M) (X (⟨n, m⟩)) := by\n  simp [dp, mkRingHom, mkAlgHom]\n\ntheorem dp_zero {m : M} : dp R 0 m = 1 := by\n  rw [dp_def, ← map_one (mkAlgHom R (Rel R M))]\n  exact RingQuot.mkAlgHom_rel R Rel.zero\n\ntheorem dp_smul {r : R} {n : ℕ} {m : M} : dp R n (r • m) = r ^ n • dp R n m := by\n  rw [dp_def, dp_def, ← map_smul]\n  exact mkAlgHom_rel R Rel.smul\n\ntheorem dp_null {n : ℕ} : dp R n (0 : M) = if n = 0 then 1 else 0 := by\n  cases Nat.eq_zero_or_pos n with\n  | inl hn =>\n    rw [if_pos hn, hn, dp_zero]\n  | inr hn =>\n    rw [if_neg (ne_of_gt hn), ← zero_smul R (0 : M), dp_smul]\n    rw [zero_pow (Nat.pos_iff_ne_zero.mp hn), zero_smul]\n\ntheorem dp_null_of_ne_zero {n : ℕ} (hn : n ≠ 0) : dp R n (0 : M) = 0 := by\n  rw [dp_null, if_neg hn]\n\ntheorem dp_mul {n p : ℕ} {m : M} :\n    dp R n m * dp R p m = (n + p).choose n • dp R (n + p) m := by\n  simp only [dp_def, ← _root_.map_mul, ← map_nsmul]\n  exact mkAlgHom_rel R Rel.mul\n\nTarget:\ntheorem dp_add {n : ℕ} {x y : M} :\n    dp R n (x + y) = (antidiagonal n).sum fun k ↦ dp R k.1 x * dp R k.2 y :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"8411547c3325da7e440e5445c9a3eeb534ad0c850ae122ee86bc61793e60967f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/DividedPowerAlgebra","family_id":"dp_add","file_id":"mathlib/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean","sample_id":"1f3bfea624fb86d6c8b755b05ec1c6fe8d68cd178710d19c21a20e4947546990"},"split":"train"}
{"binding":{"chosen_proof_sha256":"58196917b1a24b2be60c95c2b5d60e57c3e3c717c1a775b2f741e63bbfde64ca","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"97ab846b1a668a1e9d99cd8860bda5a20a029ec0c9fd4329151337202d3b3abe","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"41dfd38896021be841aea499e4a561d7890d2e4973f5f2dd9b0701af439b41dd","source_sha256":"931482f0b3c940b38e80debe6ca13abb163dc5dd9b614292ad2b1da12781602e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h := lintegral_condCDF ((dirac Unit.unit).prod μ) x\n  simpa only [fst_prod, prod_prod, measure_univ, one_mul, lintegral_dirac] using! h","hard_negative":true,"metrics":{"chosen_tokens":33,"rejected_tokens":2,"token_jaccard":0.037037,"token_length_ratio":0.060606},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"13986a6c4fe41e20089c2362cfe5dc3dd9de737a83594672fa21f71ee4ef1fe8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Kernel.Disintegration.CondCDF\npublic import Mathlib.Tactic.CrossRefAttribute\n\nNamespace:\nProbabilityTheory\n\nLocal context:\n/-\nCopyright (c) 2023 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n-/\n/-!\n# Cumulative distribution function of a real probability measure\n\nThe cumulative distribution function (cdf) of a probability measure over `ℝ` is a monotone, right\ncontinuous function with limit 0 at -∞ and 1 at +∞, such that `cdf μ x = μ (Iic x)` for all `x : ℝ`.\nTwo probability measures are equal if and only if they have the same cdf.\n\n## Main definitions\n\n* `ProbabilityTheory.cdf μ`: cumulative distribution function of `μ : Measure ℝ`, defined as the\n  conditional cdf (`ProbabilityTheory.condCDF`) of the product measure\n  `(Measure.dirac Unit.unit).prod μ` evaluated at `Unit.unit`.\n\nThe definition could be replaced by the more elementary `cdf μ x = μ.real (Iic x)`, but using\n`condCDF` gives us access to its API, from which most properties of the cdf follow directly.\n\n## Main statements\n\n* `ProbabilityTheory.ofReal_cdf`: for a probability measure `μ` and `x : ℝ`,\n  `ENNReal.ofReal (cdf μ x) = μ (Iic x)`.\n* `MeasureTheory.Measure.ext_of_cdf`: two probability measures are equal if and only if they have\n  the same cdf.\n\n## TODO\n\nThe definition could be extended to a finite measure by rescaling `condCDF`, but it would be nice\nto have more structure on Stieltjes functions first. Right now, if `f` is a Stieltjes function,\n`2 • f` makes no sense. We could define Stieltjes functions as a submodule.\n\nThe definition could be extended to `ℝⁿ`, either by extending the definition of `condCDF`, or by\nusing another construction here.\n-/\n\n@[expose] public section\n\nopen MeasureTheory Measure Set Filter\n\nopen scoped Topology\n\nnamespace ProbabilityTheory\n\n/-- Cumulative distribution function of a real measure. The definition currently makes sense only\nfor probability measures. In that case, it satisfies `cdf μ x = μ.real (Iic x)` (see\n`ProbabilityTheory.cdf_eq_real`). -/\n@[wikidata Q386228]\nnoncomputable\ndef cdf (μ : Measure ℝ) : StieltjesFunction ℝ :=\n  condCDF ((dirac Unit.unit).prod μ) Unit.unit\n\nsection ExplicitMeasureArg\nvariable (μ : Measure ℝ)\n\n/-- The cdf is non-negative. -/\nlemma cdf_nonneg (x : ℝ) : 0 ≤ cdf μ x := condCDF_nonneg _ _ _\n\n/-- The cdf is lower or equal to 1. -/\nlemma cdf_le_one (x : ℝ) : cdf μ x ≤ 1 := condCDF_le_one _ _ _\n\n/-- The cdf is monotone. -/\nlemma monotone_cdf : Monotone (cdf μ) := (condCDF _ _).mono\n\n/-- The cdf tends to 0 at -∞. -/\nlemma tendsto_cdf_atBot : Tendsto (cdf μ) atBot (𝓝 0) := tendsto_condCDF_atBot _ _\n\n/-- The cdf tends to 1 at +∞. -/\nlemma tendsto_cdf_atTop : Tendsto (cdf μ) atTop (𝓝 1) := tendsto_condCDF_atTop _ _\n\nTarget:\nlemma ofReal_cdf [IsProbabilityMeasure μ] (x : ℝ) : ENNReal.ofReal (cdf μ x) = μ (Iic x) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_41dfd3889602","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"a610e84eb9224551be0aa8be53bc3bee6efd049bdfc7e68b15ca1fe7c4347e26","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability","family_id":"ofreal_cdf","file_id":"mathlib/Mathlib/Probability/CDF.lean","sample_id":"41dfd38896021be841aea499e4a561d7890d2e4973f5f2dd9b0701af439b41dd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d0f64a8fe631e03ccedfddafbaffe13122e5547990701f69b60f4ed018241b86","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3c4a724f529b48895bc878479987d955226ae8930ccc5208fd2ff1e6255b3bca","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fbfa2ca2104fb09021e16e7f3542762a6ce83fb6ca653dc82a4df2bcb75e0375","source_sha256":"f3daaba8f2638bcf797b824f67c87ade5dbc78d42b30d85f5c7e88fd1a4627c0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext f : 1\n  apply (Matrix.stdBasis _ _ _).ext\n  intro ⟨i, j⟩\n  simp only [LinearMap.coe_comp, LinearMap.coe_mk, AddHom.coe_mk, Function.comp_apply, map_sum,\n    map_smul, stdBasis_eq_single, LinearMap.coe_sum, Finset.sum_apply,\n    LinearMap.smul_apply, LinearMap.id_coe, id_eq]\n  ext k l\n  simp [sum_apply, Matrix.mul_apply, single, Fintype.sum_prod_type, ite_and]","hard_negative":true,"metrics":{"chosen_tokens":84,"rejected_tokens":5,"token_jaccard":0.042553,"token_length_ratio":0.059524},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"13e26e708e6845ee7441fb127715c21882228aa2b8dcba9ce2e6ab0b196599c3","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Azumaya.Defs\npublic import Mathlib.LinearAlgebra.FreeModule.Finite.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Yunzhou Xie. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yunzhou Xie, Jujian Zhang\n-/\n/-!\n# Matrix algebra is an Azumaya algebra over R\n\nIn this file we prove that finite-dimensional matrix algebra `Matrix n n R` over `R`\nis an Azumaya algebra where `R` is a commutative ring.\n\n## Main Results\n\n- `IsAzumaya.Matrix`: Finite-dimensional matrix algebra over `R` is Azumaya.\n\n-/\n\npublic section\nopen scoped TensorProduct\n\nvariable (R n : Type*) [CommSemiring R] [Fintype n] [DecidableEq n]\n\nnoncomputable section\n\nopen Matrix MulOpposite\n\n/-- `AlgHom.mulLeftRight` for matrix algebra sends basis Eᵢⱼ⊗Eₖₗ to\n  the map `f : Eₛₜ ↦ Eᵢⱼ * Eₛₜ * Eₖₗ = δⱼₛδₜₖEᵢₗ`, therefore we construct the inverse\n  by sending `f` to `∑ᵢₗₛₜ f(Eₛₜ)ᵢₗ • Eᵢₛ⊗Eₜₗ`. -/\nabbrev AlgHom.mulLeftRightMatrix_inv :\n    Module.End R (Matrix n n R) →ₗ[R] Matrix n n R ⊗[R] (Matrix n n R)ᵐᵒᵖ where\n  toFun f := ∑ ⟨⟨i, j⟩, k, l⟩ : (n × n) × n × n,\n    f (single j k 1) i l • (single i j 1) ⊗ₜ[R] op (single k l 1)\n  map_add' f1 f2 := by simp [add_smul, Finset.sum_add_distrib]\n  map_smul' r f := by simp [mul_smul, Finset.smul_sum]\n\nlemma AlgHom.mulLeftRightMatrix.inv_comp :\n    (AlgHom.mulLeftRightMatrix_inv R n).comp\n    (AlgHom.mulLeftRight R (Matrix n n R)).toLinearMap = .id :=\n  ((Matrix.stdBasis _ _ _).tensorProduct ((Matrix.stdBasis _ _ _).map (opLinearEquiv ..))).ext\n  fun ⟨⟨i0, j0⟩, k0, l0⟩ ↦ by\n    simp [stdBasis_eq_single, ite_and, Fintype.sum_prod_type,\n      mulLeftRight_apply, single, Matrix.mul_apply]\n\nTarget:\nlemma AlgHom.mulLeftRightMatrix.comp_inv :\n    (AlgHom.mulLeftRight R (Matrix n n R)).toLinearMap.comp\n    (AlgHom.mulLeftRightMatrix_inv R n) = .id :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_fbfa2ca2104f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"72a57f4ee1b877a803a09a0288ff6deccbb80cf8b538686339ee535cd1b804a7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Azumaya","family_id":"alghom","file_id":"mathlib/Mathlib/Algebra/Azumaya/Matrix.lean","sample_id":"fbfa2ca2104fb09021e16e7f3542762a6ce83fb6ca653dc82a4df2bcb75e0375"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bdd21497f8131a5f5a6c2c2e48cfeb6b69583b6be9858ce150b32d0a5afe6448","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2f1e37c134e39703c11252b834c1e076724924538d8af83dbd0bc1c8dab12e37","source_sha256":"7f41b4373f1155c76bd92e338de143f67cd5443f9a140d26207473ba9237819c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Representation.IntertwiningMap.toLinearMap_apply, d_eq]; rfl","hard_negative":false,"metrics":{"chosen_tokens":14,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.214286},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"14a196523e8eee2c5a254a4d0ea875d01e109b19f7641888f9fac3859f033a2c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Category.ModuleCat.Adjunctions\npublic import Mathlib.AlgebraicTopology.ExtraDegeneracy\npublic import Mathlib.CategoryTheory.Abelian.Ext\npublic import Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced\npublic import Mathlib.RepresentationTheory.Rep.Iso\n\nNamespace:\nRep.standardComplex\n\nLocal context:\n/-\nCopyright (c) 2022 Amelia Livingston. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Amelia Livingston\n-/\n/-!\n# The standard and bar resolutions of `k` as a trivial `k`-linear `G`-representation\n\nGiven a commutative ring `k` and a group `G`, this file defines two projective resolutions of `k`\nas a trivial `k`-linear `G`-representation.\n\nThe first one, the standard resolution, has objects `k[Gⁿ⁺¹]` equipped with the diagonal\nrepresentation, and differential defined by `(g₀, ..., gₙ) ↦ ∑ (-1)ⁱ • (g₀, ..., ĝᵢ, ..., gₙ)`.\n\nWe define this as the alternating face map complex associated to an appropriate simplicial\n`k`-linear `G`-representation. This simplicial object is the `linearization` of the simplicial\n`G`-set given by the universal cover of the classifying space of `G`, `EG`. We prove this\nsimplicial `G`-set `EG` is isomorphic to the Čech nerve of the natural arrow of `G`-sets\n`G ⟶ {pt}`.\n\nWe then use this isomorphism to deduce that as a complex of `k`-modules, the standard resolution\nof `k` as a trivial `G`-representation is homotopy equivalent to the complex with `k` at 0 and 0\nelsewhere.\n\nPutting this material together allows us to define `Rep.standardResolution`, the\nstandard projective resolution of `k` as a trivial `k`-linear `G`-representation.\n\nWe then construct the bar resolution. The `n`th object in this complex is the representation on\n`Gⁿ →₀ k[G]` defined pointwise by the left regular representation on `k[G]`. The differentials are\ndefined by sending `(g₀, ..., gₙ)` to\n`g₀·(g₁, ..., gₙ) + ∑ (-1)ʲ⁺¹·(g₀, ..., gⱼgⱼ₊₁, ..., gₙ) + (-1)ⁿ⁺¹·(g₀, ..., gₙ₋₁)` for\n`j = 0, ..., n - 1`.\n\nIn `RepresentationTheory.Rep` we define an isomorphism `Rep.diagonalSuccIsoFree` between\n`k[Gⁿ⁺¹] ≅ (Gⁿ →₀ k[G])` sending `(g₀, ..., gₙ) ↦ g₀·(g₀⁻¹g₁, ..., gₙ₋₁⁻¹gₙ)`.\nWe show that this isomorphism defines a commutative square with the bar resolution differential and\nthe standard resolution differential, and thus conclude that the bar resolution differential\nsquares to zero and that `Rep.diagonalSuccIsoFree` defines an isomorphism between the two\ncomplexes. We carry the exactness properties across this isomorphism to conclude the bar resolution\nis a projective resolution too, in `Rep.barResolution`.\n\nIn `Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean` and\n`Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean`, we then use\n`Rep.barResolution` to define the inhomogeneous (co)chains of a representation, useful for\ncomputing group (co)homology.\n\n## Main definitions\n\n * `groupCohomology.resolution.ofMulActionBasis`\n * `classifyingSpaceUniversalCover`\n * `Rep.standardComplex.forget₂ToModuleCatHomotopyEquiv`\n * `Rep.standardResolution`\n\nTODO: There's bad DefEq abuses in `Action` and the way we do `Rep.standardComplex` should be\n  unified with continuous cohomology, therefore we should remove the use of `Action` in `Rep` which\n  would remove all the unification hints in this file.\n-/\n\n@[expose] public noncomputable section\n\nsuppress_compilation\n\nopen CategoryTheory Finsupp\nopen scoped MonoidAlgebra\n\nuniverse u v w\n\nvariable {k G : Type u} [CommRing k] {n : ℕ}\n\nlocal notation \"Gⁿ\" => Fin n → G\n\nset_option quotPrecheck false\nlocal notation \"Gⁿ⁺¹\" => Fin (n + 1) → G\n\nvariable (G)\n\n/-- The simplicial `G`-set sending `[n]` to `Gⁿ⁺¹` equipped with the diagonal action of `G`. -/\n@[simps obj map]\ndef classifyingSpaceUniversalCover [Monoid G] :\n    SimplicialObject (Action (Type u) G) where\n  obj n := Action.ofMulAction G (Fin (n.unop.len + 1) → G)\n  map f :=\n    { hom := ↾fun x => x ∘ f.unop.toOrderHom\n      comm := fun _ => rfl }\n  map_id _ := rfl\n  map_comp _ _ := rfl\n\nnamespace classifyingSpaceUniversalCover\n\nopen CategoryTheory.Limits\n\nvariable [Monoid G]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- When the category is `G`-Set, `cechNerveTerminalFrom` of `G` with the left regular action is\nisomorphic to `EG`, the universal cover of the classifying space of `G` as a simplicial `G`-set. -/\ndef cechNerveTerminalFromIso : cechNerveTerminalFrom (Action.ofMulAction G (G)) ≅\n    classifyingSpaceUniversalCover G :=\n  NatIso.ofComponents (fun _ => limit.isoLimitCone (Action.ofMulActionLimitCone _ _)) fun f => by\n    refine IsLimit.hom_ext (Action.ofMulActionLimitCone.{u, 0} G fun _ => G).2 fun j => ?_\n    dsimp only [cechNerveTerminalFrom, Pi.lift]\n    rw [Category.assoc, limit.isoLimitCone_hom_π, limit.lift_π, Category.assoc]\n    exact (limit.isoLimitCone_hom_π _ _).symm\n\n/-- As a simplicial set, `cechNerveTerminalFrom` of a monoid `G` is isomorphic to the universal\ncover of the classifying space of `G` as a simplicial set. -/\ndef cechNerveTerminalFromIsoCompForget :\n    cechNerveTerminalFrom G ≅ classifyingSpaceUniversalCover G ⋙ forget _ := by\n  refine NatIso.ofComponents (fun _ => Types.productIso _) fun _ => ?_\n  ext : 2\n  exact Matrix.ext fun _ _ => Pi.lift_π_apply (f := fun _ ↦ G) _ _ _\n\nvariable (k)\n\nopen AlgebraicTopology SimplicialObject.Augmented SimplicialObject CategoryTheory.Arrow\n\n/-- The universal cover of the classifying space of `G` as a simplicial set, augmented by the map\nfrom `Fin 1 → G` to the terminal object in `Type u`. -/\ndef compForgetAugmented : SimplicialObject.Augmented (Type u) :=\n  SimplicialObject.augment (classifyingSpaceUniversalCover G ⋙ forget _) (terminal _)\n    (terminal.from _) fun _ _ _ => Subsingleton.elim _ _\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The augmented Čech nerve of the map from `Fin 1 → G` to the terminal object in `Type u` has an\nextra degeneracy. -/\ndef extraDegeneracyAugmentedCechNerve :\n    ExtraDegeneracy (Arrow.mk <| terminal.from G).augmentedCechNerve :=\n  AugmentedCechNerve.extraDegeneracy (Arrow.mk <| terminal.from G)\n    ⟨↾fun _ => (1 : G), by cat_disch⟩\n\n/-- The universal cover of the classifying space of `G` as a simplicial set, augmented by the map\nfrom `Fin 1 → G` to the terminal object in `Type u`, has an extra degeneracy. -/\ndef extraDegeneracyCompForgetAugmented : ExtraDegeneracy (compForgetAugmented G) := by\n  refine\n    ExtraDegeneracy.ofIso (?_ : (Arrow.mk <| terminal.from G).augmentedCechNerve ≅ _)\n      (extraDegeneracyAugmentedCechNerve G)\n  exact\n    Comma.isoMk (CechNerveTerminalFrom.iso G ≪≫ cechNerveTerminalFromIsoCompForget G)\n      (Iso.refl _) (by ext : 1; exact IsTerminal.hom_ext terminalIsTerminal _ _)\n\n/-- The free functor `Type u ⥤ ModuleCat.{u} k` applied to the universal cover of the classifying\nspace of `G` as a simplicial set, augmented by the map from `Fin 1 → G` to the terminal object\nin `Type u`. -/\ndef compForgetAugmented.toModule : SimplicialObject.Augmented (ModuleCat.{u} k) :=\n  ((SimplicialObject.Augmented.whiskering _ _).obj (ModuleCat.free k)).obj (compForgetAugmented G)\n\n/-- If we augment the universal cover of the classifying space of `G` as a simplicial set by the\nmap from `Fin 1 → G` to the terminal object in `Type u`, then apply the free functor\n`Type u ⥤ ModuleCat.{u} k`, the resulting augmented simplicial `k`-module has an extra\ndegeneracy. -/\ndef extraDegeneracyCompForgetAugmentedToModule :\n    ExtraDegeneracy (compForgetAugmented.toModule k G) :=\n  ExtraDegeneracy.map (extraDegeneracyCompForgetAugmented G) (ModuleCat.free k)\n\nend classifyingSpaceUniversalCover\n\nvariable (k)\n\n/-- The standard resolution of `k` as a trivial representation, defined as the alternating\nface map complex of a simplicial `k`-linear `G`-representation. -/\ndef Rep.standardComplex [Monoid G] :=\n  (AlgebraicTopology.alternatingFaceMapComplex (Rep k G)).obj\n    (classifyingSpaceUniversalCover G ⋙ linearization k G)\n\nnamespace Rep.standardComplex\n\nopen classifyingSpaceUniversalCover AlgebraicTopology CategoryTheory.Limits\n\n/-- The `k`-linear map underlying the differential in the standard resolution of `k` as a trivial\n`k`-linear `G`-representation. It sends `(g₀, ..., gₙ) ↦ ∑ (-1)ⁱ • (g₀, ..., ĝᵢ, ..., gₙ)`. -/\ndef d (G : Type u) (n : ℕ) : ((Fin (n + 1) → G) →₀ k) →ₗ[k] (Fin n → G) →₀ k :=\n  Finsupp.lift ((Fin n → G) →₀ k) k (Fin (n + 1) → G) fun g =>\n    (@Finset.univ (Fin (n + 1)) _).sum fun p =>\n      Finsupp.single (g ∘ p.succAbove) ((-1 : k) ^ (p : ℕ))\n\nvariable {k G}\n\n@[simp]\ntheorem d_of {n : ℕ} (c : Fin (n + 1) → G) :\n    d k G n (Finsupp.single c 1) =\n      Finset.univ.sum fun p : Fin (n + 1) =>\n        Finsupp.single (c ∘ p.succAbove) ((-1 : k) ^ (p : ℕ)) := by\n  simp [d]\n\nlemma d_single {n : ℕ} (c : Fin (n + 1) → G) (r : k) :\n    d k G n (Finsupp.single c r) =\n      Finset.univ.sum fun p : Fin (n + 1) =>\n        Finsupp.single (c ∘ p.succAbove) (r * (-1 : k) ^ (p : ℕ)) := by\n  rw [← mul_one r, ← smul_eq_mul, ← smul_single, map_smul, d_of]\n  simp [Finset.smul_sum]\n\nvariable (k G) [Monoid G]\n\n/-- The `n`th object of the standard resolution of `k` is definitionally isomorphic to `k[Gⁿ⁺¹]`\nequipped with the representation induced by the diagonal action of `G`. -/\ndef xIso (n : ℕ) : (standardComplex k G).X n ≅ Rep.ofMulAction k G (Fin (n + 1) → G) :=\n  Iso.refl _\n\ninstance x_projective (G : Type u) [Group G] (n : ℕ) :\n    Projective ((standardComplex k G).X n) := by\n  classical exact inferInstanceAs <| Projective (Rep.diagonal k G (n + 1))\n\nset_option backward.defeqAttrib.useBackward true in\nunif_hint where ⊢ Action.V (Action.ofMulAction G (Fin (n + 1) → G)) ≟ Fin (n + 1) → G in\nset_option backward.isDefEq.respectTransparency false in\n/-- Simpler expression for the differential in the standard resolution of `k` as a\n`G`-representation. It sends `(g₀, ..., gₙ₊₁) ↦ ∑ (-1)ⁱ • (g₀, ..., ĝᵢ, ..., gₙ₊₁)`. -/\ntheorem d_eq (n : ℕ) : ((standardComplex k G).d (n + 1) n).hom.toLinearMap =\n    d k G (n + 1) := by\n  refine Finsupp.lhom_ext' fun (x : Fin (n + 2) → G) => LinearMap.ext_ring ?_\n  simp [standardComplex, Action.ofMulAction_V, SimplicialObject.δ, SimplexCategory.δ,\n    Fin.succAboveOrderEmb, ← Int.cast_smul_eq_zsmul k ((-1) ^ _ : ℤ), ← ofHom_smul, ← ofHom_sum,\n    Representation.IntertwiningMap.coe_toLinearMap, Representation.IntertwiningMap.sum_apply,\n    Representation.IntertwiningMap.smul_apply, (Representation.linearizeMap_single), smul_single,\n    smul_eq_mul, mul_one]\n\nTarget:\nlemma d_apply {n : ℕ} (f : (Fin (n + 1 + 1) → G) →₀ k) :\n    ((standardComplex k G).d (n + 1) n).hom f = d k G (n + 1) f :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RepresentationTheory/Homological","family_id":"d_apply","file_id":"mathlib/Mathlib/RepresentationTheory/Homological/Resolution.lean","sample_id":"2f1e37c134e39703c11252b834c1e076724924538d8af83dbd0bc1c8dab12e37"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7e80ec87feb47441a23ea0489b04ea9d8954472b614c08ce067659d50bc27e69","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"38130104b915c0d587ef920e495ec4fb8346201ebb2169cc467b266a759136af","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ec0e8191257eb538c96b568fc34a540eaf43a4e996b7fbafb697f19a5d8137e8","source_sha256":"8edd066203db5b8ec30e5a0f3cf792a4af8c9625cc6277b004728c090fd82f43","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [hasBasis_self, (hasBasis_nhdsSet _).forall_iff (by grind)]\n  intro U ⟨hU, h_KU⟩\n  obtain ⟨L, hL, hKL, hLU⟩ := exists_compact_between hK hU h_KU\n  exact ⟨L, by rwa [← subset_interior_iff_mem_nhdsSet], hL, hLU⟩","hard_negative":true,"metrics":{"chosen_tokens":53,"rejected_tokens":5,"token_jaccard":0.090909,"token_length_ratio":0.09434},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"1557b918bebf25efc27a3f1b3820e53c93492fdf3d0e3a4430dd2097015b629e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Compactness.Compact\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\n/-!\n# Locally compact spaces\n\nThis file contains basic results about locally compact spaces.\n-/\n\npublic section\n\nopen Set Filter Topology TopologicalSpace\n\nvariable {X : Type*} {Y : Type*} {ι : Type*}\nvariable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}\n\ninstance [WeaklyLocallyCompactSpace X] [WeaklyLocallyCompactSpace Y] :\n    WeaklyLocallyCompactSpace (X × Y) where\n  exists_compact_mem_nhds x :=\n    let ⟨s₁, hc₁, h₁⟩ := exists_compact_mem_nhds x.1\n    let ⟨s₂, hc₂, h₂⟩ := exists_compact_mem_nhds x.2\n    ⟨s₁ ×ˢ s₂, hc₁.prod hc₂, prod_mem_nhds h₁ h₂⟩\n\ninstance {ι : Type*} [Finite ι] {X : ι → Type*} [(i : ι) → TopologicalSpace (X i)]\n    [(i : ι) → WeaklyLocallyCompactSpace (X i)] :\n    WeaklyLocallyCompactSpace ((i : ι) → X i) where\n  exists_compact_mem_nhds f := by\n    choose s hsc hs using fun i ↦ exists_compact_mem_nhds (f i)\n    exact ⟨pi univ s, isCompact_univ_pi hsc, set_pi_mem_nhds univ.toFinite fun i _ ↦ hs i⟩\n\ninstance (priority := 100) [CompactSpace X] : WeaklyLocallyCompactSpace X where\n  exists_compact_mem_nhds _ := ⟨univ, isCompact_univ, univ_mem⟩\n\nprotected theorem Topology.IsClosedEmbedding.weaklyLocallyCompactSpace [WeaklyLocallyCompactSpace Y]\n    {f : X → Y} (hf : IsClosedEmbedding f) : WeaklyLocallyCompactSpace X where\n  exists_compact_mem_nhds x :=\n    let ⟨K, hK, hKx⟩ := exists_compact_mem_nhds (f x)\n    ⟨f ⁻¹' K, hf.isCompact_preimage hK, hf.continuous.continuousAt hKx⟩\n\nprotected theorem IsClosed.weaklyLocallyCompactSpace [WeaklyLocallyCompactSpace X]\n    {s : Set X} (hs : IsClosed s) : WeaklyLocallyCompactSpace s :=\n  hs.isClosedEmbedding_subtypeVal.weaklyLocallyCompactSpace\n\ntheorem IsOpenQuotientMap.weaklyLocallyCompactSpace [WeaklyLocallyCompactSpace X]\n    {f : X → Y} (hf : IsOpenQuotientMap f) : WeaklyLocallyCompactSpace Y where\n  exists_compact_mem_nhds := by\n    refine hf.surjective.forall.2 fun x ↦ ?_\n    rcases exists_compact_mem_nhds x with ⟨K, hKc, hKx⟩\n    exact ⟨f '' K, hKc.image hf.continuous, hf.isOpenMap.image_mem_nhds hKx⟩\n\n/-- In a weakly locally compact space,\nevery compact set is contained in the interior of a compact set. -/\ntheorem exists_compact_superset [WeaklyLocallyCompactSpace X] {K : Set X} (hK : IsCompact K) :\n    ∃ K', IsCompact K' ∧ K ⊆ interior K' := by\n  choose s hc hmem using fun x : X ↦ exists_compact_mem_nhds x\n  rcases hK.elim_nhds_subcover _ fun x _ ↦ interior_mem_nhds.2 (hmem x) with ⟨I, -, hIK⟩\n  refine ⟨⋃ x ∈ I, s x, I.isCompact_biUnion fun _ _ ↦ hc _, hIK.trans ?_⟩\n  exact iUnion₂_subset fun x hx ↦ interior_mono <| subset_iUnion₂ (s := fun x _ ↦ s x) x hx\n\n/-- In a weakly locally compact space,\nthe filters `𝓝 x` and `cocompact X` are disjoint for all `X`. -/\ntheorem disjoint_nhds_cocompact [WeaklyLocallyCompactSpace X] (x : X) :\n    Disjoint (𝓝 x) (cocompact X) :=\n  let ⟨_, hc, hx⟩ := exists_compact_mem_nhds x\n  disjoint_of_disjoint_of_mem disjoint_compl_right hx hc.compl_mem_cocompact\n\ntheorem compact_basis_nhds [LocallyCompactSpace X] (x : X) :\n    (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsCompact s) fun s => s :=\n  hasBasis_self.2 <| by simpa only [and_comm] using LocallyCompactSpace.local_compact_nhds x\n\ntheorem local_compact_nhds [LocallyCompactSpace X] {x : X} {n : Set X} (h : n ∈ 𝓝 x) :\n    ∃ s ∈ 𝓝 x, s ⊆ n ∧ IsCompact s :=\n  LocallyCompactSpace.local_compact_nhds _ _ h\n\ntheorem LocallyCompactSpace.of_hasBasis {ι : X → Type*} {p : ∀ x, ι x → Prop}\n    {s : ∀ x, ι x → Set X} (h : ∀ x, (𝓝 x).HasBasis (p x) (s x))\n    (hc : ∀ x i, p x i → IsCompact (s x i)) : LocallyCompactSpace X :=\n  ⟨fun x _t ht =>\n    let ⟨i, hp, ht⟩ := (h x).mem_iff.1 ht\n    ⟨s x i, (h x).mem_of_mem hp, ht, hc x i hp⟩⟩\n\ninstance Prod.locallyCompactSpace (X : Type*) (Y : Type*) [TopologicalSpace X]\n    [TopologicalSpace Y] [LocallyCompactSpace X] [LocallyCompactSpace Y] :\n    LocallyCompactSpace (X × Y) :=\n  have := fun x : X × Y => (compact_basis_nhds x.1).prod_nhds' (compact_basis_nhds x.2)\n  .of_hasBasis this fun _ _ ⟨⟨_, h₁⟩, _, h₂⟩ => h₁.prod h₂\n\nsection Pi\n\nvariable {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i, LocallyCompactSpace (X i)]\n\n/-- In general it suffices that all but finitely many of the spaces are compact,\n  but that's not straightforward to state and use. -/\ninstance Pi.locallyCompactSpace_of_finite [Finite ι] : LocallyCompactSpace (∀ i, X i) :=\n  ⟨fun t n hn => by\n    rw [nhds_pi, Filter.mem_pi] at hn\n    obtain ⟨s, -, n', hn', hsub⟩ := hn\n    choose n'' hn'' hsub' hc using fun i =>\n      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)\n    refine ⟨(Set.univ : Set ι).pi n'', ?_, subset_trans (fun _ h => ?_) hsub, isCompact_univ_pi hc⟩\n    · exact (set_pi_mem_nhds_iff (@Set.finite_univ ι _) _).mpr fun i _ => hn'' i\n    · exact fun i _ => hsub' i (h i trivial)⟩\n\n/-- For spaces that are not Hausdorff. -/\ninstance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=\n  ⟨fun t n hn => by\n    rw [nhds_pi, Filter.mem_pi] at hn\n    obtain ⟨s, hs, n', hn', hsub⟩ := hn\n    choose n'' hn'' hsub' hc using fun i =>\n      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)\n    refine ⟨s.pi n'', ?_, subset_trans (fun _ => ?_) hsub, ?_⟩\n    · exact (set_pi_mem_nhds_iff hs _).mpr fun i _ => hn'' i\n    · exact forall₂_imp fun i _ hi' => hsub' i hi'\n    · classical\n      rw [← Set.univ_pi_ite]\n      refine isCompact_univ_pi fun i => ?_\n      by_cases h : i ∈ s\n      · rw [if_pos h]\n        exact hc i\n      · rw [if_neg h]\n        exact CompactSpace.isCompact_univ⟩\n\ninstance Function.locallyCompactSpace_of_finite [Finite ι] [LocallyCompactSpace Y] :\n    LocallyCompactSpace (ι → Y) :=\n  Pi.locallyCompactSpace_of_finite\n\ninstance Function.locallyCompactSpace [LocallyCompactSpace Y] [CompactSpace Y] :\n    LocallyCompactSpace (ι → Y) :=\n  Pi.locallyCompactSpace\n\nend Pi\n\ninstance (priority := 900) [LocallyCompactSpace X] : LocallyCompactPair X Y where\n  exists_mem_nhds_isCompact_mapsTo hf hs :=\n    let ⟨K, hKx, hKs, hKc⟩ := local_compact_nhds (hf.continuousAt hs); ⟨K, hKx, hKc, hKs⟩\n\ninstance (priority := 100) [LocallyCompactSpace X] : WeaklyLocallyCompactSpace X where\n  exists_compact_mem_nhds (x : X) :=\n    let ⟨K, hx, _, hKc⟩ := local_compact_nhds (x := x) univ_mem; ⟨K, hKc, hx⟩\n\n/-- A reformulation of the definition of locally compact space: In a locally compact space,\n  every open set containing `x` has a compact subset containing `x` in its interior. -/\ntheorem exists_compact_subset [LocallyCompactSpace X] {x : X} {U : Set X} (hU : IsOpen U)\n    (hx : x ∈ U) : ∃ K : Set X, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U := by\n  rcases LocallyCompactSpace.local_compact_nhds x U (hU.mem_nhds hx) with ⟨K, h1K, h2K, h3K⟩\n  exact ⟨K, h3K, mem_interior_iff_mem_nhds.2 h1K, h2K⟩\n\n/-- If `f : X → Y` is a continuous map in a locally compact pair of topological spaces,\n`K : set X` is a compact set, and `U` is an open neighbourhood of `f '' K`,\nthen there exists a compact neighbourhood `L` of `K` such that `f` maps `L` to `U`.\n\nThis is a generalization of `exists_mem_nhds_isCompact_mapsTo`. -/\nlemma exists_mem_nhdsSet_isCompact_mapsTo [LocallyCompactPair X Y] {f : X → Y} {K : Set X}\n    {U : Set Y} (hf : Continuous f) (hK : IsCompact K) (hU : IsOpen U) (hKU : MapsTo f K U) :\n    ∃ L ∈ 𝓝ˢ K, IsCompact L ∧ MapsTo f L U := by\n  choose! V hxV hVc hVU using fun x (hx : x ∈ K) ↦\n    exists_mem_nhds_isCompact_mapsTo hf (hU.mem_nhds (hKU hx))\n  rcases hK.elim_nhds_subcover_nhdsSet hxV with ⟨s, hsK, hKs⟩\n  exact ⟨_, hKs, s.isCompact_biUnion fun x hx ↦ hVc x (hsK x hx), mapsTo_iUnion₂.2 fun x hx ↦\n    hVU x (hsK x hx)⟩\n\n/-- In a locally compact space, for every containment `K ⊆ U` of a compact set `K` in an open\n  set `U`, there is a compact neighborhood `L` such that `K ⊆ L ⊆ U`: equivalently, there is a\n  compact `L` such that `K ⊆ interior L` and `L ⊆ U`.\n  See also `exists_compact_closed_between`, in which one guarantees additionally that `L` is closed\n  if the space is regular. -/\ntheorem exists_compact_between [LocallyCompactSpace X] {K U : Set X} (hK : IsCompact K)\n    (hU : IsOpen U) (h_KU : K ⊆ U) : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U :=\n  let ⟨L, hKL, hL, hLU⟩ := exists_mem_nhdsSet_isCompact_mapsTo continuous_id hK hU h_KU\n  ⟨L, hL, subset_interior_iff_mem_nhdsSet.2 hKL, hLU⟩\n\n/-- In a (possibly non-Hausdorff) locally compact space, for every compact set `K`,\n`𝓝ˢ K` has a basis consisting of compact sets. -/\n\nTarget:\ntheorem IsCompact.nhdsSet_basis_isCompact [LocallyCompactSpace X] {K : Set X} (hK : IsCompact K) :\n    (𝓝ˢ K).HasBasis (fun L ↦ L ∈ 𝓝ˢ K ∧ IsCompact L) id :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_ec0e8191257e","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f776b64abd4ab228e7bc3fa509451301acd66378d871d6391ea3729236523b9d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Compactness","family_id":"iscompact","file_id":"mathlib/Mathlib/Topology/Compactness/LocallyCompact.lean","sample_id":"ec0e8191257eb538c96b568fc34a540eaf43a4e996b7fbafb697f19a5d8137e8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fda78651fadb9c3fc00301a4f0a7e0c199693b04a98a2e407d36dcf722f12255","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"50baa089f2e40a96f16f849fe21aaf0cb1a2bbb9df440662791a1a5773aa0950","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fc015e197e7e22d0acabad316890661898d1a5fa61e903e26832dbe043d9fa2c","source_sha256":"d2c15a0760158d76d516fc83add1bd6dc5918d2a6efe03e65d28acdb282a684f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let e := Fintype.equivFin ι\n  obtain ⟨i₀, i₁, ne, h⟩ := h.exists_approx_aux (Fintype.card ι) hε hb fun x y ↦ A x (e.symm y)\n  refine ⟨i₀, i₁, ne, fun k ↦ ?_⟩\n  convert! h (e k) <;> simp only [e.symm_apply_apply]","hard_negative":true,"metrics":{"chosen_tokens":78,"rejected_tokens":2,"token_jaccard":0.022727,"token_length_ratio":0.025641},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"15a0c8eda23c1afbc71fd926b46f04b76f2ac80f3afffc3668c0a8d69ca39f2e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Real.Basic\npublic import Mathlib.Combinatorics.Pigeonhole\npublic import Mathlib.Algebra.Order.AbsoluteValue.Euclidean\n\nNamespace:\nAbsoluteValue.IsAdmissible\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Admissible absolute values\nThis file defines a structure `AbsoluteValue.IsAdmissible` which we use to show the class number\nof the ring of integers of a global field is finite.\n\n## Main definitions\n\n* `AbsoluteValue.IsAdmissible abv` states the absolute value `abv : R → ℤ`\n  respects the Euclidean domain structure on `R`, and that a large enough set\n  of elements of `R^n` contains a pair of elements whose remainders are\n  pointwise close together.\n\n## Main results\n\n* `AbsoluteValue.absIsAdmissible` shows the \"standard\" absolute value on `ℤ`,\n  mapping negative `x` to `-x`, is admissible.\n* `Polynomial.cardPowDegreeIsAdmissible` shows `cardPowDegree`,\n  mapping `p : Polynomial 𝔽_q` to `q ^ degree p`, is admissible\n-/\n\npublic section\n\nlocal infixl:50 \" ≺ \" => EuclideanDomain.r\n\nnamespace AbsoluteValue\n\nvariable {R : Type*} [EuclideanDomain R]\nvariable (abv : AbsoluteValue R ℤ)\n\n/-- An absolute value `R → ℤ` is admissible if it respects the Euclidean domain\nstructure and a large enough set of elements in `R^n` will contain a pair of\nelements whose remainders are pointwise close together. -/\nstructure IsAdmissible extends IsEuclidean abv where\n  /-- The cardinality required for a given `ε`. -/\n  protected card : ℝ → ℕ\n  /-- For all `ε > 0` and finite families `A`, we can partition the remainders of `A` mod `b`\n  into `abv.card ε` sets, such that all elements in each part of remainders are close together. -/\n  exists_partition' :\n    ∀ (n : ℕ) {ε : ℝ} (_ : 0 < ε) {b : R} (_ : b ≠ 0) (A : Fin n → R),\n      ∃ t : Fin n → Fin (card ε), ∀ i₀ i₁, t i₀ = t i₁ → (abv (A i₁ % b - A i₀ % b) : ℝ) < abv b • ε\n\nnamespace IsAdmissible\n\nvariable {abv}\n\n/-- For all `ε > 0` and finite families `A`, we can partition the remainders of `A` mod `b`\ninto `abv.card ε` sets, such that all elements in each part of remainders are close together. -/\ntheorem exists_partition {ι : Type*} [Finite ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0)\n    (A : ι → R) (h : abv.IsAdmissible) : ∃ t : ι → Fin (h.card ε),\n      ∀ i₀ i₁, t i₀ = t i₁ → (abv (A i₁ % b - A i₀ % b) : ℝ) < abv b • ε := by\n  rcases Finite.exists_equiv_fin ι with ⟨n, ⟨e⟩⟩\n  obtain ⟨t, ht⟩ := h.exists_partition' n hε hb (A ∘ e.symm)\n  refine ⟨t ∘ e, fun i₀ i₁ h ↦ ?_⟩\n  convert! (config := { transparency := .default }) ht (e i₀) (e i₁) h <;>\n    simp only [e.symm_apply_apply]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Any large enough family of vectors in `R^n` has a pair of elements\nwhose remainders are close together, pointwise. -/\ntheorem exists_approx_aux (n : ℕ) (h : abv.IsAdmissible) :\n    ∀ {ε : ℝ} (_hε : 0 < ε) {b : R} (_hb : b ≠ 0) (A : Fin (h.card ε ^ n).succ → Fin n → R),\n      ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ k, (abv (A i₁ k % b - A i₀ k % b) : ℝ) < abv b • ε := by\n  haveI := Classical.decEq R\n  induction n with\n  | zero =>\n    intro ε _hε b _hb A\n    refine ⟨0, 1, ?_, ?_⟩\n    · simp\n    rintro ⟨i, ⟨⟩⟩\n  | succ n ih =>\n  intro ε hε b hb A\n  let M := h.card ε\n  -- By the \"nicer\" pigeonhole principle, we can find a collection `s`\n  -- of more than `M ^ n` remainders where the first components lie close together:\n  obtain ⟨s, s_inj, hs⟩ :\n    ∃ s : Fin (M ^ n).succ → Fin (M ^ n.succ).succ,\n      Function.Injective s ∧ ∀ i₀ i₁, (abv (A (s i₁) 0 % b - A (s i₀) 0 % b) : ℝ) < abv b • ε := by\n    -- We can partition the `A`s into `M` subsets where\n    -- the first components lie close together:\n    obtain ⟨t, ht⟩ :\n      ∃ t : Fin (M ^ n.succ).succ → Fin M,\n        ∀ i₀ i₁, t i₀ = t i₁ → (abv (A i₁ 0 % b - A i₀ 0 % b) : ℝ) < abv b • ε :=\n      h.exists_partition hε hb fun x ↦ A x 0\n    -- Since the `M` subsets contain more than `M * M^n` elements total,\n    -- there must be a subset that contains more than `M^n` elements.\n    obtain ⟨s, hs⟩ :=\n      Fintype.exists_lt_card_fiber_of_mul_lt_card (f := t)\n        (by simpa only [Fintype.card_fin, pow_succ'] using Nat.lt_succ_self (M ^ n.succ))\n    have : (M ^ n).succ ≤ (Finset.toList {x | t x = s}).length := by\n      rwa [Finset.length_toList]\n    refine ⟨fun i ↦ (Finset.toList {x | t x = s})[i.castLE this], fun i j h ↦ ?_,\n      fun i₀ i₁ ↦ ht _ _ ?_⟩\n    · simpa [(Finset.nodup_toList _).getElem_inj_iff, Fin.val_inj] using h\n    · have (i : Fin (M ^ n).succ) : t (Finset.toList {x | t x = s})[i.castLE this] = s :=\n        (Finset.mem_filter.mp ((Finset.mem_toList (s := {x | t x = s})).mp (List.getElem_mem _))).2\n      simp_rw [this]\n  -- Since `s` is large enough, there are two elements of `A ∘ s`\n  -- where the second components lie close together.\n  obtain ⟨k₀, k₁, hk, h⟩ := ih hε hb fun x ↦ Fin.tail (A (s x))\n  refine ⟨s k₀, s k₁, fun h ↦ hk (s_inj h), fun i ↦ Fin.cases ?_ (fun i ↦ ?_) i⟩\n  · exact hs k₀ k₁\n  · exact h i\n\n/-- Any large enough family of vectors in `R^ι` has a pair of elements\nwhose remainders are close together, pointwise. -/\n\nTarget:\ntheorem exists_approx {ι : Type*} [Fintype ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0)\n    (h : abv.IsAdmissible) (A : Fin (h.card ε ^ Fintype.card ι).succ → ι → R) :\n    ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ k, (abv (A i₁ k % b - A i₀ k % b) : ℝ) < abv b • ε :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_fc015e197e7e","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e45111bb0b93cea36e589f696c3c74e8b459c24eba96686ecf4c7310d6898d34","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/ClassNumber","family_id":"exists_approx","file_id":"mathlib/Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean","sample_id":"fc015e197e7e22d0acabad316890661898d1a5fa61e903e26832dbe043d9fa2c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7f711a73112f71461efde233fe8c7cca840501a653c0391902a7be231d8cd075","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f0926250948255762b99b2b1a18aec1ff03af8c440c7ae762e999efd06557812","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6a93f461afc68c4564b026d64e62d3228a07b26e1dd6066974bf51a8514702ac","source_sha256":"96ffeb6042bed2ccd5d2880386938c633598ffa0d830f64affcb9938dbdcf908","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction l with\n  | nil => simp\n  | cons head tail ih => simp [← ih, smul_mul_smul_comm, pow_succ']","hard_negative":true,"metrics":{"chosen_tokens":23,"rejected_tokens":3,"token_jaccard":0.05,"token_length_ratio":0.130435},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"15adc09976b9d60d2ce9fd6d471b2cbd8ead9b311b77ef0e696213326487a4fa","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Finprod\npublic import Mathlib.Algebra.GroupWithZero.Action.Defs\npublic import Mathlib.Algebra.Order.Group.Multiset\npublic import Mathlib.Data.Finset.Basic\npublic import Mathlib.Algebra.Group.Action.Basic\npublic import Mathlib.Algebra.Group.Units.Equiv\n\nNamespace:\nList\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Lemmas about group actions on big operators\n\nThis file contains results about two kinds of actions:\n\n* sums over `DistribSMul`: `r • ∑ x ∈ s, f x = ∑ x ∈ s, r • f x`\n* products over `MulDistribMulAction` (with primed name): `r • ∏ x ∈ s, f x = ∏ x ∈ s, r • f x`\n* products over `SMulCommClass` (with unprimed name):\n  `b ^ s.card • ∏ x ∈ s, f x = ∏ x ∈ s, b • f x`\n\nNote that analogous lemmas for `Module`s like `Finset.sum_smul` appear in other files.\n-/\n\npublic section\n\n\nvariable {M N γ : Type*}\n\nsection\n\nvariable [AddMonoid N] [DistribSMul M N]\n\ntheorem List.smul_sum {r : M} {l : List N} : r • l.sum = (l.map (r • ·)).sum :=\n  map_list_sum (DistribSMul.toAddMonoidHom N r) l\n\nend\n\nsection\n\nvariable [Monoid M] [Monoid N] [MulDistribMulAction M N]\n\ntheorem List.smul_prod' {r : M} {l : List N} : r • l.prod = (l.map (r • ·)).prod :=\n  map_list_prod (MulDistribMulAction.toMonoidHom N r) l\n\nend\n\nsection\n\nvariable [AddCommMonoid N] [DistribSMul M N] {r : M}\n\ntheorem Multiset.smul_sum {s : Multiset N} : r • s.sum = (s.map (r • ·)).sum :=\n  (DistribSMul.toAddMonoidHom N r).map_multiset_sum s\n\ntheorem Finset.smul_sum {f : γ → N} {s : Finset γ} :\n    (r • ∑ x ∈ s, f x) = ∑ x ∈ s, r • f x :=\n  map_sum (DistribSMul.toAddMonoidHom N r) f s\n\ntheorem smul_finsum_mem {f : γ → N} {s : Set γ} (hs : s.Finite) :\n    r • ∑ᶠ x ∈ s, f x = ∑ᶠ x ∈ s, r • f x :=\n  (DistribSMul.toAddMonoidHom N r).map_finsum_mem f hs\n\nend\n\nsection\n\nvariable [Monoid M] [CommMonoid N] [MulDistribMulAction M N]\n\ntheorem Multiset.smul_prod' {r : M} {s : Multiset N} : r • s.prod = (s.map (r • ·)).prod :=\n  (MulDistribMulAction.toMonoidHom N r).map_multiset_prod s\n\ntheorem Finset.smul_prod' {r : M} {f : γ → N} {s : Finset γ} :\n    (r • ∏ x ∈ s, f x) = ∏ x ∈ s, r • f x :=\n  map_prod (MulDistribMulAction.toMonoidHom N r) f s\n\ntheorem smul_finprod' {ι : Sort*} [Finite ι] {f : ι → N} (r : M) :\n    r • ∏ᶠ x : ι, f x = ∏ᶠ x : ι, r • (f x) := by\n  cases nonempty_fintype (PLift ι)\n  simp only [finprod_eq_prod_plift_of_mulSupport_subset (s := Finset.univ) (by simp),\n    Finset.smul_prod']\n\nvariable {G : Type*} [Group G] [MulDistribMulAction G N]\n\ntheorem Finset.smul_prod_perm [Fintype G] (b : N) (g : G) :\n    (g • ∏ h : G, h • b) = ∏ h : G, h • b := by\n  simp only [smul_prod', smul_smul]\n  exact Finset.prod_bijective (g * ·) (Group.mulLeft_bijective g) (by simp) (fun _ _ ↦ rfl)\n\ntheorem smul_finprod_perm [Finite G] (b : N) (g : G) :\n    (g • ∏ᶠ h : G, h • b) = ∏ᶠ h : G, h • b := by\n  cases nonempty_fintype G\n  simp only [finprod_eq_prod_of_fintype, Finset.smul_prod_perm]\n\nend\n\nnamespace List\n\n@[to_additive]\n\nTarget:\ntheorem smul_prod [Monoid M] [MulOneClass N] [MulAction M N] [IsScalarTower M N N]\n    [SMulCommClass M N N] (l : List N) (m : M) :\n    m ^ l.length • l.prod = (l.map (m • ·)).prod :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_6a93f461afc6","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"70f686ce2128aff900747b363091f223e238069ba1df9cbcf34186c6033e3bea","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/BigOperators","family_id":"smul_prod","file_id":"mathlib/Mathlib/Algebra/BigOperators/GroupWithZero/Action.lean","sample_id":"6a93f461afc68c4564b026d64e62d3228a07b26e1dd6066974bf51a8514702ac"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0a479f8dae478b3bb9603d60f313d74028b2da15cce1a50538dcbe628c6b865b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2b9e2ca58e134ad4578da49d74b391a4ef05b73b213f399f3d3257947965fc7d","source_sha256":"cbd935405b8857ab21a8766b03dbdc75523269b01677822430d1eb4f04caf0ba","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine (QuadraticMap.associated_rightInverse R).injective ?_\n  ext m₁ m₂ m₁' m₂' m₁'' m₂''\n  simp only [associated_tmul, QuadraticMap.associated_comp]\n  exact mul_assoc _ _ _","hard_negative":false,"metrics":{"chosen_tokens":45,"rejected_tokens":3,"token_jaccard":0.12,"token_length_ratio":0.066667},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"15d4793de2886ba8abb09fbae0ec61951e6f8040568683fe4191bd5e3aeddb43","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct\npublic import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv\n\nNamespace:\nQuadraticForm\n\nLocal context:\n/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Linear equivalences of tensor products as isometries\n\nThese results are separate from the definition of `QuadraticForm.tmul` as that file is very slow.\n\n## Main definitions\n\n* `QuadraticForm.Isometry.tmul`: `TensorProduct.map` as a `QuadraticForm.Isometry`\n* `QuadraticForm.tensorComm`: `TensorProduct.comm` as a `QuadraticForm.IsometryEquiv`\n* `QuadraticForm.tensorAssoc`: `TensorProduct.assoc` as a `QuadraticForm.IsometryEquiv`\n* `QuadraticForm.tensorRId`: `TensorProduct.rid` as a `QuadraticForm.IsometryEquiv`\n* `QuadraticForm.tensorLId`: `TensorProduct.lid` as a `QuadraticForm.IsometryEquiv`\n-/\n\n@[expose] public section\n\nuniverse uR uM₁ uM₂ uM₃ uM₄\nvariable {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} {M₄ : Type uM₄}\n\nopen scoped TensorProduct\n\nopen QuadraticMap\n\nnamespace QuadraticForm\n\nvariable [CommRing R]\nvariable [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M₄]\nvariable [Module R M₁] [Module R M₂] [Module R M₃] [Module R M₄] [Invertible (2 : R)]\n\n@[simp]\ntheorem tmul_comp_tensorMap\n    {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂}\n    {Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄}\n    (f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) :\n    (Q₂.tmul Q₄).comp (TensorProduct.map f.toLinearMap g.toLinearMap) = Q₁.tmul Q₃ := by\n  have h₁ : Q₁ = Q₂.comp f.toLinearMap := QuadraticMap.ext fun x => (f.map_app x).symm\n  have h₃ : Q₃ = Q₄.comp g.toLinearMap := QuadraticMap.ext fun x => (g.map_app x).symm\n  refine (QuadraticMap.associated_rightInverse R).injective ?_\n  ext m₁ m₃ m₁' m₃'\n  simp [h₁, h₃, associated_tmul]\n\n@[simp]\ntheorem tmul_tensorMap_apply\n    {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂}\n    {Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄}\n    (f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) (x : M₁ ⊗[R] M₃) :\n    Q₂.tmul Q₄ (TensorProduct.map f.toLinearMap g.toLinearMap x) = Q₁.tmul Q₃ x :=\n  DFunLike.congr_fun (tmul_comp_tensorMap f g) x\n\nnamespace Isometry\n\n/-- `TensorProduct.map` for `QuadraticForm.Isometry`s -/\ndef _root_.QuadraticMap.Isometry.tmul\n    {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂}\n    {Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄}\n    (f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) : (Q₁.tmul Q₃) →qᵢ (Q₂.tmul Q₄) where\n  toLinearMap := TensorProduct.map f.toLinearMap g.toLinearMap\n  map_app' := tmul_tensorMap_apply f g\n\n@[simp]\ntheorem _root_.QuadraticMap.Isometry.tmul_apply\n    {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂}\n    {Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄}\n    (f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) (x : M₁ ⊗[R] M₃) :\n    f.tmul g x = TensorProduct.map f.toLinearMap g.toLinearMap x :=\n  rfl\n\nend Isometry\n\nsection tensorComm\n\n@[simp]\ntheorem tmul_comp_tensorComm (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) :\n    (Q₂.tmul Q₁).comp (TensorProduct.comm R M₁ M₂) = Q₁.tmul Q₂ := by\n  refine (QuadraticMap.associated_rightInverse R).injective ?_\n  ext m₁ m₂ m₁' m₂'\n  simp only [associated_tmul, QuadraticMap.associated_comp]\n  exact mul_comm _ _\n\n@[simp]\ntheorem tmul_tensorComm_apply\n    (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (x : M₁ ⊗[R] M₂) :\n    Q₂.tmul Q₁ (TensorProduct.comm R M₁ M₂ x) = Q₁.tmul Q₂ x :=\n  DFunLike.congr_fun (tmul_comp_tensorComm Q₁ Q₂) x\n\n/-- `TensorProduct.comm` preserves tensor products of quadratic forms. -/\n@[simps toLinearEquiv]\ndef tensorComm (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) :\n    (Q₁.tmul Q₂).IsometryEquiv (Q₂.tmul Q₁) where\n  toLinearEquiv := TensorProduct.comm R M₁ M₂\n  map_app' := tmul_tensorComm_apply Q₁ Q₂\n\n@[simp] lemma tensorComm_apply (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂)\n    (x : M₁ ⊗[R] M₂) :\n    tensorComm Q₁ Q₂ x = TensorProduct.comm R M₁ M₂ x :=\n  rfl\n\n@[simp] lemma tensorComm_symm (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) :\n    (tensorComm Q₁ Q₂).symm = tensorComm Q₂ Q₁ :=\n  rfl\n\nend tensorComm\n\nsection tensorAssoc\n\n@[simp]\n\nTarget:\ntheorem tmul_comp_tensorAssoc\n    (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (Q₃ : QuadraticForm R M₃) :\n    (Q₁.tmul (Q₂.tmul Q₃)).comp (TensorProduct.assoc R M₁ M₂ M₃) = (Q₁.tmul Q₂).tmul Q₃ :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/QuadraticForm","family_id":"tmul_comp_tensorassoc","file_id":"mathlib/Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean","sample_id":"2b9e2ca58e134ad4578da49d74b391a4ef05b73b213f399f3d3257947965fc7d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"79e94f24c2f6775d53dff748682f833bc2540c933d9b7a7fd1509ddae3499adc","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6397f9eeed335e5c4d5285b3d24ecfded6dd566f396ae49dc9f12108ccdd3ee3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0fc65d8000935f5344eab88fab4e2c3713d4a97efe4538dc3faf285b863f943a","source_sha256":"95ccb03aba37ba4c0bbb12dee38bb2450c829862e49cf64d5a888d7fa6218e0d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← ofSupportFinite_coe (f := a) (hf := Set.toFinite _),\n    prod_congr (fun r _ ↦ rfl), ← coeff_linearCombination_X_pow]\n  simp [linearCombination_apply, sum_of_support_subset (s := Finset.univ)]","hard_negative":true,"metrics":{"chosen_tokens":44,"rejected_tokens":5,"token_jaccard":0.0625,"token_length_ratio":0.113636},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"161bf6cc6a74be59df96303ddd5bf9baac644f0485c1e3a30d9221fb2bc03556","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.Basic\npublic import Mathlib.Data.Nat.Choose.Multinomial\n\nNamespace:\nMvPolynomial\n\nLocal context:\n/-\nCopyright (c) 2026 Antoine Chambert-Loir, María Inés de Frutos-Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir, María Inés de Frutos-Fernández\n-/\n/-!\n# Formulas for coefficients of multivariate polynomials\n\n## Main Results\n\n* `MvPolynomial.coeff_add_pow`: the formula for the `d`th coefficient of `(X 0 + X 1) ^ n`.\n\n-/\n\npublic section\n\nnoncomputable section\n\nnamespace MvPolynomial\n\nopen Finsupp\n\nvariable {R σ : Type*} [CommSemiring R] {s : σ →₀ ℕ}\n\nprivate lemma coeff_linearCombination_X_pow_of_eq (a : σ →₀ R) {n : ℕ}\n    (hs : s.sum (fun _ m ↦ m) = n) :\n    coeff s (((a.linearCombination R X : MvPolynomial σ R)) ^ n) =\n      s.multinomial * s.prod (fun r m ↦ a r ^ m) := by\n  classical\n  simp only [sum, linearCombination_apply, Finset.sum_pow_eq_sum_piAntidiag, coeff_sum,\n    ← C_eq_coe_nat, coeff_C_mul, smul_eq_C_mul, mul_pow, Finset.prod_mul_distrib, ← map_pow,\n    ← map_prod, coeff_prod_X_pow, mul_ite, mul_one, mul_zero]\n  rw [Finset.sum_eq_single (s : σ → ℕ)]\n  · simp_rw [eq_indicator_self_iff]\n    split_ifs with hs'\n    · rw [prod_of_support_subset _ hs' _ (by simp), Finsupp.multinomial_of_support_subset hs']\n    · rw [Finset.subset_iff] at hs'\n      simp only [Finsupp.mem_support_iff, ne_eq, not_forall, Decidable.not_not] at hs'\n      obtain ⟨i, hsi, hai⟩ := hs'\n      rw [← mul_prod_erase _ i _ (by simpa), hai, zero_pow hsi, zero_mul, mul_zero]\n  · simp only [Finset.mem_piAntidiag, ne_eq, Finsupp.mem_support_iff, ite_eq_right_iff, and_imp]\n    intro _ _ _ _ hed\n    simp [Finsupp.ext_iff] at hed\n    grind\n  · simp_rw [ite_eq_right_iff]\n    intro hs' hs''\n    rw [eq_indicator_self_iff] at hs''\n    exfalso\n    rw [Finset.mem_piAntidiag, not_and_or] at hs'\n    rcases hs' with hs' | hs'\n    · apply hs'\n      rw [← hs, sum_of_support_subset _ hs'' _ (by simp)]\n    · grind\n\nprivate lemma coeff_linearCombination_X_pow_of_ne (a : σ →₀ R) {n : ℕ}\n    (hs : s.sum (fun _ m ↦ m) ≠ n) :\n    coeff s (((a.linearCombination R X : MvPolynomial σ R)) ^ n) = 0 := by\n  classical\n  simp only [sum, linearCombination_apply, Finset.sum_pow_eq_sum_piAntidiag, coeff_sum, ← map_pow,\n    ← C_eq_coe_nat, coeff_C_mul, smul_eq_C_mul, mul_pow, Finset.prod_mul_distrib, ← map_prod,\n    coeff_prod_X_pow, mul_ite, mul_one, mul_zero]\n  apply Finset.sum_eq_zero (fun x hx ↦ ?_)\n  rw [if_neg]\n  rintro ⟨rfl⟩\n  apply hs\n  simp only [Finset.mem_piAntidiag] at hx\n  rw [sum_of_support_subset _ (support_indicator_subset a.support _) _ (by simp), ← hx.1]\n  congr\n  ext i\n  by_cases hi : i ∈ a.support\n  · simp [Finsupp.indicator_of_mem hi]\n  · grind [Finsupp.indicator_of_notMem hi]\n\nlemma coeff_linearCombination_X_pow (a : σ →₀ R) (s : σ →₀ ℕ) (n : ℕ) :\n    coeff s (((a.linearCombination R X : MvPolynomial σ R)) ^ n) =\n      if s.sum (fun _ m ↦ m) = n then s.multinomial * s.prod (fun r m ↦ a r ^ m) else 0 := by\n  split_ifs with hs\n  · exact coeff_linearCombination_X_pow_of_eq a hs\n  · exact coeff_linearCombination_X_pow_of_ne a hs\n\nTarget:\nlemma coeff_linearCombination_X_pow_of_fintype [Fintype σ] (a : σ → R) (s : σ →₀ ℕ) (n : ℕ) :\n    coeff s (((∑ i, a i • X i : MvPolynomial σ R)) ^ n) =\n      if s.sum (fun _ m ↦ m) = n then s.multinomial * s.prod (fun r m ↦ a r ^ m) else 0 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_0fc65d800093","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"5c13b1a19a8f2289b412aabb782fb06dae214f3342ccbdeef2bf8f64b0ce43bf","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/MvPolynomial","family_id":"coeff_linearcombination_x_pow_of_fintype","file_id":"mathlib/Mathlib/Algebra/MvPolynomial/Coeff.lean","sample_id":"0fc65d8000935f5344eab88fab4e2c3713d4a97efe4538dc3faf285b863f943a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5e809a8a5b6dc52f86458c7ec1ba0de189d9ba286d3abe6189f2eb23181c4aa7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7f909b761cf539f37fcd85869fee376cc556800c01c64b800ef8cc533771a533","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"94279022a062f44341d707de6cd3452c8f04400967480dbebc54e8c3a85912a9","source_sha256":"320ae2b6adb110a4f3c6cc9d28358d174f6e3d360ad3ba6f7f42c745ee840c5c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine Set.ext fun R ↦ ⟨fun ⟨g, hg⟩ ↦ hg ▸ ⟨hQ.smul g, hP.smul g⟩, fun h ↦ ?_⟩\n  have : R.IsPrime := h.1\n  obtain ⟨g, hg⟩ := exists_smul_of_under_eq A B G Q R (hP.over.symm.trans h.2.over)\n  exact ⟨g, hg.symm⟩","hard_negative":true,"metrics":{"chosen_tokens":80,"rejected_tokens":3,"token_jaccard":0.051282,"token_length_ratio":0.0375},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"1645b26b14e94d51a6518bacd842c0ae67a8c4ef1d36e8dcb9fea9c07faaf1bb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Invariant.Defs\npublic import Mathlib.RingTheory.IntegralClosure.IntegralRestrict\npublic import Mathlib.RingTheory.LocalRing.ResidueField.Ideal\n\nNamespace:\nAlgebra.IsInvariant\n\nLocal context:\n/-\nCopyright (c) 2024 Thomas Browning. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Thomas Browning\n-/\n/-!\n# Invariant Extensions of Rings\n\nGiven an extension of rings `B/A` and an action of `G` on `B`, we introduce a predicate\n`Algebra.IsInvariant A B G` which states that every fixed point of `B` lies in the image of `A`.\n\nThe main application is in algebraic number theory, where `G := Gal(L/K)` is the Galois group\nof some finite Galois extension of number fields, and `A := 𝓞K` and `B := 𝓞L` are their ring of\nintegers. This main result in this file implies the existence of Frobenius elements in this setting.\nSee `Mathlib/RingTheory/Frobenius.lean`.\n\n## Main statements\n\nLet `G` be a finite group acting on a commutative ring `B` satisfying `Algebra.IsInvariant A B G`.\n\n* `Algebra.IsInvariant.isIntegral`: `B/A` is an integral extension.\n* `Algebra.IsInvariant.exists_smul_of_under_eq`: `G` acts transitivity on the prime ideals of `B`\n  lying above a given prime ideal of `A`.\n\nIf `Q` is a prime ideal of `B` lying over a prime ideal `P` of `A`, then\n\n* `IsFractionRing.stabilizerHom_surjective`:\n  The stabilizer subgroup of `Q` surjects onto `Aut(Frac(B/Q)/Frac(A/P))`.\n* `Ideal.Quotient.stabilizerHom_surjective`:\n  The stabilizer subgroup of `Q` surjects onto `Aut((B/Q)/(A/P))`.\n* `Ideal.Quotient.exists_algEquiv_fixedPoint_quotient_under`:\n  If `k` is a domain containing `B/Q`, then any `A/P`-algebra automorphism of `k` restricts to\n  an automorphism of `B/Q`.\n-/\n\n@[expose] public section\n\nopen scoped Pointwise\n\nsection Galois\n\nvariable (A K L B : Type*) [CommRing A] [CommRing B] [Field K] [Field L]\n  [Algebra A K] [Algebra B L] [IsFractionRing A K] [IsFractionRing B L]\n  [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L]\n  [IsIntegrallyClosed A] [IsIntegralClosure B A L]\n\n/-- In the AKLB setup, the Galois group of `L/K` acts on `B`. -/\n@[implicit_reducible]\nnoncomputable def IsIntegralClosure.MulSemiringAction [Algebra.IsAlgebraic K L] :\n    MulSemiringAction Gal(L/K) B :=\n  MulSemiringAction.compHom B (galRestrict A K L B).toMonoidHom\n\ninstance [Algebra.IsAlgebraic K L] : let := IsIntegralClosure.MulSemiringAction A K L B\n    SMulDistribClass Gal(L/K) B L :=\n  let := IsIntegralClosure.MulSemiringAction A K L B\n  ⟨fun g b l ↦ by\n    simp only [Algebra.smul_def, smul_mul', mul_eq_mul_right_iff]\n    exact Or.inl (algebraMap_galRestrictHom_apply A K L B g b).symm⟩\n\n/-- In the AKLB setup, every fixed point of `B` lies in the image of `A`. -/\ntheorem Algebra.isInvariant_of_isGalois [FiniteDimensional K L] [h : IsGalois K L] :\n    letI := IsIntegralClosure.MulSemiringAction A K L B\n    Algebra.IsInvariant A B Gal(L/K) := by\n  replace h := ((IsGalois.tfae (F := K) (E := L)).out 0 1).mp h\n  letI := IsIntegralClosure.MulSemiringAction A K L B\n  refine ⟨fun b hb ↦ ?_⟩\n  replace hb : algebraMap B L b ∈ IntermediateField.fixedField (⊤ : Subgroup Gal(L/K)) := by\n    rintro ⟨g, -⟩\n    exact (algebraMap_galRestrict_apply A g b).symm.trans (congrArg (algebraMap B L) (hb g))\n  rw [h, IntermediateField.mem_bot] at hb\n  obtain ⟨k, hk⟩ := hb\n  have hb : IsIntegral A b := IsIntegralClosure.isIntegral A L b\n  rw [← isIntegral_algebraMap_iff (FaithfulSMul.algebraMap_injective B L), ← hk,\n    isIntegral_algebraMap_iff (FaithfulSMul.algebraMap_injective K L)] at hb\n  obtain ⟨a, rfl⟩ := IsIntegrallyClosed.algebraMap_eq_of_integral hb\n  rw [← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply A B L,\n    (FaithfulSMul.algebraMap_injective B L).eq_iff] at hk\n  exact ⟨a, hk⟩\n\n/-- A variant of `Algebra.isInvariant_of_isGalois`, replacing `Gal(L/K)` by `Aut(B/A)`. -/\ntheorem Algebra.isInvariant_of_isGalois' [FiniteDimensional K L] [IsGalois K L] :\n    Algebra.IsInvariant A B (B ≃ₐ[A] B) :=\n  ⟨fun b h ↦ (isInvariant_of_isGalois A K L B).1 b (fun g ↦ h (galRestrict A K L B g))⟩\n\nend Galois\n\nsection Quotient\n\nvariable {A B : Type*} [CommRing A] [CommRing B] [Algebra A B]\nvariable {G : Type*} [Group G] [MulSemiringAction G B] [SMulCommClass G A B]\n\ninstance (H : Subgroup G) [H.Normal] :\n    MulSemiringAction (G ⧸ H) (FixedPoints.subring B H) where\n  smul := Quotient.lift (fun g x ↦ ⟨g • x, fun h ↦ by\n    simpa [mul_smul] using! congr(g • $(x.2 ⟨_, ‹H.Normal›.conj_mem' _ h.2 g⟩))⟩) (by\n    rintro _ a ⟨⟨⟨b⟩, hb⟩, rfl⟩\n    ext c\n    simpa [mul_smul] using! congr(a • $(c.2 ⟨b, hb⟩)))\n  one_smul b := Subtype.ext (one_smul G b.1)\n  mul_smul := Quotient.ind₂ fun _ _ _ ↦ Subtype.ext (mul_smul _ _ _)\n  smul_zero := Quotient.ind fun _ ↦ Subtype.ext (smul_zero _)\n  smul_add := Quotient.ind fun _ _ _ ↦ Subtype.ext (smul_add _ _ _)\n  smul_one := Quotient.ind fun _ ↦ Subtype.ext (smul_one _)\n  smul_mul := Quotient.ind fun _ _ _ ↦ Subtype.ext (MulSemiringAction.smul_mul _ _ _)\n\ninstance (H : Subgroup G) [H.Normal] :\n    MulSemiringAction (G ⧸ H) (FixedPoints.subalgebra A B H) :=\n  inferInstanceAs (MulSemiringAction (G ⧸ H) (FixedPoints.subring B H))\n\ninstance (H : Subgroup G) [H.Normal] :\n    SMulCommClass (G ⧸ H) A (FixedPoints.subalgebra A B H) where\n  smul_comm := Quotient.ind fun g r h ↦ Subtype.ext (smul_comm g r h.1)\n\ninstance (H : Subgroup G) [H.Normal] [Algebra.IsInvariant A B G] :\n    Algebra.IsInvariant A (FixedPoints.subalgebra A B H) (G ⧸ H) where\n  isInvariant x hx := by\n    obtain ⟨y, hy⟩ := Algebra.IsInvariant.isInvariant (A := A) (G := G) x.1\n      (fun g ↦ congr_arg Subtype.val (hx g))\n    exact ⟨y, Subtype.ext hy⟩\n\nend Quotient\n\nsection transitivity\n\nvariable (A B G : Type*) [CommRing A] [CommRing B] [Algebra A B] [Group G] [MulSemiringAction G B]\n\nnamespace MulSemiringAction\n\nopen Polynomial\n\nvariable {B} [Fintype G]\n\n/-- Characteristic polynomial of a finite group action on a ring. -/\nnoncomputable def charpoly (b : B) : B[X] := ∏ g : G, (X - C (g • b))\n\ntheorem charpoly_eq (b : B) : charpoly G b = ∏ g : G, (X - C (g • b)) := rfl\n\ntheorem charpoly_eq_prod_smul (b : B) : charpoly G b = ∏ g : G, g • (X - C b) := by\n  simp only [smul_sub, smul_C, smul_X, charpoly_eq]\n\ntheorem monic_charpoly (b : B) : (charpoly G b).Monic :=\n  monic_prod_of_monic _ _ (fun _ _ ↦ monic_X_sub_C _)\n\ntheorem splits_charpoly (b : B) : (charpoly G b).Splits :=\n  .prod fun g _ ↦ .X_sub_C (g • b)\n\ntheorem eval_charpoly (b : B) : (charpoly G b).eval b = 0 := by\n  rw [charpoly_eq, eval_prod]\n  apply Finset.prod_eq_zero (Finset.mem_univ (1 : G))\n  rw [one_smul, eval_sub, eval_C, eval_X, sub_self]\n\nvariable {G}\n\ntheorem smul_charpoly (b : B) (g : G) : g • (charpoly G b) = charpoly G b := by\n  rw [charpoly_eq_prod_smul, Finset.smul_prod_perm]\n\ntheorem smul_coeff_charpoly (b : B) (n : ℕ) (g : G) :\n    g • (charpoly G b).coeff n = (charpoly G b).coeff n := by\n  rw [← coeff_smul, smul_charpoly]\n\nend MulSemiringAction\n\nnamespace Algebra.IsInvariant\n\nopen MulSemiringAction Polynomial\n\nvariable [IsInvariant A B G]\n\ntheorem charpoly_mem_lifts [Fintype G] (b : B) :\n    charpoly G b ∈ Polynomial.lifts (algebraMap A B) :=\n  (charpoly G b).lifts_iff_coeff_lifts.mpr fun n ↦ isInvariant _ (smul_coeff_charpoly b n)\n\ntheorem isIntegral [Finite G] : Algebra.IsIntegral A B := by\n  cases nonempty_fintype G\n  refine ⟨fun b ↦ ?_⟩\n  obtain ⟨p, hp1, -, hp2⟩ := Polynomial.lifts_and_natDegree_eq_and_monic\n    (charpoly_mem_lifts A B G b) (monic_charpoly G b)\n  exact ⟨p, hp2, by rw [← eval_map, hp1, eval_charpoly]⟩\n\n/-- `G` acts transitively on the prime ideals of `B` above a given prime ideal of `A`. -/\ntheorem exists_smul_of_under_eq [Finite G] [SMulCommClass G A B]\n    (P Q : Ideal B) [hP : P.IsPrime] [hQ : Q.IsPrime]\n    (hPQ : P.under A = Q.under A) :\n    ∃ g : G, Q = g • P := by\n  cases nonempty_fintype G\n  have : ∀ (P Q : Ideal B) [P.IsPrime] [Q.IsPrime], P.under A = Q.under A →\n      ∃ g ∈ (⊤ : Finset G), Q ≤ g • P := by\n    intro P Q hP hQ hPQ\n    rw [← Ideal.subset_union_prime 1 1 (fun _ _ _ _ ↦ hP.smul _)]\n    intro b hb\n    suffices h : ∃ g ∈ Finset.univ, g • b ∈ P by\n      obtain ⟨g, -, hg⟩ := h\n      apply Set.mem_biUnion (Finset.mem_univ g⁻¹) (Ideal.mem_inv_pointwise_smul_iff.mpr hg)\n    obtain ⟨a, ha⟩ := isInvariant (A := A) (∏ g : G, g • b) (Finset.smul_prod_perm b)\n    rw [← hP.prod_mem_iff, ← ha, ← P.mem_comap, ← P.under_def A,\n      hPQ, Q.mem_comap, ha, hQ.prod_mem_iff]\n    exact ⟨1, Finset.mem_univ 1, (one_smul G b).symm ▸ hb⟩\n  obtain ⟨g, -, hg⟩ := this P Q hPQ\n  obtain ⟨g', -, hg'⟩ := this Q (g • P) ((P.under_smul A g).trans hPQ).symm\n  exact ⟨g, le_antisymm hg (smul_eq_of_le_smul (hg.trans hg') ▸ hg')⟩\n\nTarget:\ntheorem orbit_eq_primesOver [Finite G] [SMulCommClass G A B] (P : Ideal A) (Q : Ideal B)\n    [hP : Q.LiesOver P] [hQ : Q.IsPrime] : MulAction.orbit G Q = P.primesOver B :=\n\nProof body:\n","rejected":"by\n  exact orbit_eq_primesOver","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"106f678c679aa863ad967512be9c635f35b53268a547623947663476ac597609","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Invariant","family_id":"orbit_eq_primesover","file_id":"mathlib/Mathlib/RingTheory/Invariant/Basic.lean","sample_id":"94279022a062f44341d707de6cd3452c8f04400967480dbebc54e8c3a85912a9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a2f7906395a8fceee2188be53f90d4ed089166e4d7b933b3ba61e10fb185f1d5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2518e676d38db0b863404107b061157ad05645cfc444d4b1dd91915ce9184a85","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"aa6161864a0c5f5bc6339fe001dfcdf0b66c0b5bdb3cf1678dcb5f82be308aa0","source_sha256":"3e56b0bca06aab56ed706da176a00f9bb677f8dddf1dc55819d7e75bf8119e45","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  split_ifs with h\n  exacts [hs h, ht h]","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":5,"token_jaccard":0.071429,"token_length_ratio":0.416667},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"166d368d8d70810ca361a38fb07d0a15efaf31c4eca46313f823dae9fb24e3ba","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Countable\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\npublic import Mathlib.Tactic.CrossRefAttribute\npublic import Mathlib.Tactic.FunProp.Attr\npublic import Mathlib.Tactic.Measurability\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro\n-/\n/-!\n# Measurable spaces and measurable functions\n\nThis file defines measurable spaces and measurable functions.\n\nA measurable space is a set equipped with a σ-algebra, a collection of\nsubsets closed under complementation and countable union. A function\nbetween measurable spaces is measurable if the preimage of each\nmeasurable subset is measurable.\n\nσ-algebras on a fixed set `α` form a complete lattice. Here we order\nσ-algebras by writing `m₁ ≤ m₂` if every set which is `m₁`-measurable is\nalso `m₂`-measurable (that is, `m₁` is a subset of `m₂`). In particular, any\ncollection of subsets of `α` generates a smallest σ-algebra which\ncontains all of them.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Measurable_space>\n* <https://en.wikipedia.org/wiki/Sigma-algebra>\n* <https://en.wikipedia.org/wiki/Dynkin_system>\n\n## Tags\n\nmeasurable space, σ-algebra, measurable function\n-/\n\n@[expose] public section\n\nassert_not_exists Covariant MonoidWithZero\n\nopen Set Encodable Function Equiv\n\nvariable {α β γ δ δ' : Type*} {ι : Sort*} {s t u : Set α}\n\n/-- A measurable space is a space equipped with a σ-algebra. -/\n@[class] structure MeasurableSpace (α : Type*) where\n  /-- Predicate saying that a given set is measurable. Use `MeasurableSet` in the root namespace\n  instead. -/\n  MeasurableSet' : Set α → Prop\n  /-- The empty set is a measurable set. Use `MeasurableSet.empty` instead. -/\n  measurableSet_empty : MeasurableSet' ∅\n  /-- The complement of a measurable set is a measurable set. Use `MeasurableSet.compl` instead. -/\n  measurableSet_compl : ∀ s, MeasurableSet' s → MeasurableSet' sᶜ\n  /-- The union of a sequence of measurable sets is a measurable set. Use a more general\n  `MeasurableSet.iUnion` instead. -/\n  measurableSet_iUnion : ∀ f : ℕ → Set α, (∀ i, MeasurableSet' (f i)) → MeasurableSet' (⋃ i, f i)\n\ninstance [h : MeasurableSpace α] : MeasurableSpace αᵒᵈ := h\n\n/-- `MeasurableSet s` means that `s` is measurable (in the ambient measure space on `α`) -/\ndef MeasurableSet [MeasurableSpace α] (s : Set α) : Prop :=\n  ‹MeasurableSpace α›.MeasurableSet' s\n\n/-- Notation for `MeasurableSet` with respect to a non-standard σ-algebra. -/\nscoped[MeasureTheory] notation \"MeasurableSet[\" m \"]\" => @MeasurableSet _ m\n\nopen MeasureTheory\n\nsection\n\nopen scoped symmDiff\n\n@[simp, measurability]\ntheorem MeasurableSet.empty [MeasurableSpace α] : MeasurableSet (∅ : Set α) :=\n  MeasurableSpace.measurableSet_empty _\n\nvariable {m : MeasurableSpace α}\n\n@[measurability]\nprotected theorem MeasurableSet.compl : MeasurableSet s → MeasurableSet sᶜ :=\n  MeasurableSpace.measurableSet_compl _ s\n\nprotected theorem MeasurableSet.of_compl (h : MeasurableSet sᶜ) : MeasurableSet s :=\n  compl_compl s ▸ h.compl\n\n@[simp]\ntheorem MeasurableSet.compl_iff : MeasurableSet sᶜ ↔ MeasurableSet s :=\n  ⟨.of_compl, .compl⟩\n\n@[simp, measurability]\nprotected theorem MeasurableSet.univ : MeasurableSet (univ : Set α) :=\n  .of_compl <| by simp\n\n@[nontriviality, measurability]\ntheorem Subsingleton.measurableSet [Subsingleton α] {s : Set α} : MeasurableSet s :=\n  Subsingleton.set_cases MeasurableSet.empty MeasurableSet.univ s\n\ntheorem MeasurableSet.congr {s t : Set α} (hs : MeasurableSet s) (h : s = t) : MeasurableSet t := by\n  rwa [← h]\n\n@[measurability]\nprotected theorem MeasurableSet.iUnion [Countable ι] ⦃f : ι → Set α⦄\n    (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋃ b, f b) := by\n  cases isEmpty_or_nonempty ι\n  · simp\n  · rcases exists_surjective_nat ι with ⟨e, he⟩\n    rw [← iUnion_congr_of_surjective _ he (fun _ => rfl)]\n    exact m.measurableSet_iUnion _ fun _ => h _\n\nprotected theorem MeasurableSet.biUnion {f : β → Set α} {s : Set β} (hs : s.Countable)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) := by\n  rw [biUnion_eq_iUnion]\n  have := hs.to_subtype\n  exact MeasurableSet.iUnion (by simpa using h)\n\ntheorem Set.Finite.measurableSet_biUnion {f : β → Set α} {s : Set β} (hs : s.Finite)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) :=\n  .biUnion hs.countable h\n\ntheorem Finset.measurableSet_biUnion {f : β → Set α} (s : Finset β)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) :=\n  s.finite_toSet.measurableSet_biUnion h\n\nprotected theorem MeasurableSet.sUnion {s : Set (Set α)} (hs : s.Countable)\n    (h : ∀ t ∈ s, MeasurableSet t) : MeasurableSet (⋃₀ s) := by\n  rw [sUnion_eq_biUnion]\n  exact .biUnion hs h\n\ntheorem Set.Finite.measurableSet_sUnion {s : Set (Set α)} (hs : s.Finite)\n    (h : ∀ t ∈ s, MeasurableSet t) : MeasurableSet (⋃₀ s) :=\n  MeasurableSet.sUnion hs.countable h\n\n@[measurability]\ntheorem MeasurableSet.iInter [Countable ι] {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) :\n    MeasurableSet (⋂ b, f b) :=\n  .of_compl <| by rw [compl_iInter]; exact .iUnion fun b => (h b).compl\n\ntheorem MeasurableSet.biInter {f : β → Set α} {s : Set β} (hs : s.Countable)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) :=\n  .of_compl <| by rw [compl_iInter₂]; exact .biUnion hs fun b hb => (h b hb).compl\n\ntheorem Set.Finite.measurableSet_biInter {f : β → Set α} {s : Set β} (hs : s.Finite)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) :=\n  .biInter hs.countable h\n\ntheorem Finset.measurableSet_biInter {f : β → Set α} (s : Finset β)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) :=\n  s.finite_toSet.measurableSet_biInter h\n\ntheorem MeasurableSet.sInter {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s, MeasurableSet t) :\n    MeasurableSet (⋂₀ s) := by\n  rw [sInter_eq_biInter]\n  exact MeasurableSet.biInter hs h\n\ntheorem Set.Finite.measurableSet_sInter {s : Set (Set α)} (hs : s.Finite)\n    (h : ∀ t ∈ s, MeasurableSet t) : MeasurableSet (⋂₀ s) :=\n  MeasurableSet.sInter hs.countable h\n\n@[simp, measurability]\nprotected theorem MeasurableSet.union {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)\n    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ ∪ s₂) := by\n  rw [union_eq_iUnion]\n  exact .iUnion (Bool.forall_bool.2 ⟨h₂, h₁⟩)\n\n@[simp, measurability]\nprotected theorem MeasurableSet.inter {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)\n    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ ∩ s₂) := by\n  rw [inter_eq_compl_compl_union_compl]\n  exact (h₁.compl.union h₂.compl).compl\n\n@[simp, measurability]\nprotected theorem MeasurableSet.diff {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)\n    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ \\ s₂) :=\n  h₁.inter h₂.compl\n\n@[simp, measurability]\nprotected lemma MeasurableSet.himp {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂) :\n    MeasurableSet (s₁ ⇨ s₂) := by rw [himp_eq]; exact h₂.union h₁.compl\n\n@[simp, measurability]\nprotected theorem MeasurableSet.symmDiff {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)\n    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ ∆ s₂) :=\n  (h₁.diff h₂).union (h₂.diff h₁)\n\n@[simp, measurability]\nprotected lemma MeasurableSet.bihimp {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)\n    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ ⇔ s₂) := (h₂.himp h₁).inter (h₁.himp h₂)\n\n@[simp, measurability]\nprotected theorem MeasurableSet.ite {t s₁ s₂ : Set α} (ht : MeasurableSet t)\n    (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂) : MeasurableSet (t.ite s₁ s₂) :=\n  (h₁.inter ht).union (h₂.diff ht)\n\nopen Classical in\n\nTarget:\ntheorem MeasurableSet.ite' {s t : Set α} {p : Prop} (hs : p → MeasurableSet s)\n    (ht : ¬p → MeasurableSet t) : MeasurableSet (ite p s t) :=\n\nProof body:\n","rejected":"by\n  exact MeasurableSet.ite","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"7633c5feba12763231f58ac18145c871d09aea68d2e42caf7b72e6733a566bdf","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/MeasurableSpace","family_id":"measurableset","file_id":"mathlib/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","sample_id":"aa6161864a0c5f5bc6339fe001dfcdf0b66c0b5bdb3cf1678dcb5f82be308aa0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"586f1604059baac7ae4d22c8ed3b6c941cba1beaead4cb818f84462ea38efbce","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f99fb33905c3e834807b289faba5aa4e5d1d620c06c7bc55e1f4ca0533d6c2bb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dce4fa550c062aaa1e0e5d299822fa3b4591439eeb6e8fb744a8c9d9b1a2391e","source_sha256":"ba493885a2be935359e1e232e1bbaeb68ad80f382975839ead7d3190a1fce95f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [nonempty_iff_ne_empty, Ne, sumLexLift_eq_empty, not_and_or, exists_prop, not_forall]","hard_negative":true,"metrics":{"chosen_tokens":16,"rejected_tokens":5,"token_jaccard":0.0625,"token_length_ratio":0.3125},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"168419567e71cd2ce3231e9a917cf68f6a7546013ba74a7f1f96962eb22381fd","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Sum\npublic import Mathlib.Data.Sum.Order\npublic import Mathlib.Order.Interval.Finset.Defs\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Finite intervals in a disjoint union\n\nThis file provides the `LocallyFiniteOrder` instance for the disjoint sum and linear sum of two\norders and calculates the cardinality of their finite intervals.\n-/\n\n@[expose] public section\n\n\nopen Function Sum\n\nnamespace Finset\n\nvariable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}\n\nsection SumLift₂\n\nvariable (f f₁ g₁ : α₁ → β₁ → Finset γ₁) (g f₂ g₂ : α₂ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁` and `α₂ → β₂ → Finset γ₂` to a map\n`α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to `Alternative` functors if we can\nmake sure to keep computability and universe polymorphism. -/\n@[simp]\ndef sumLift₂ : ∀ (_ : α₁ ⊕ α₂) (_ : β₁ ⊕ β₂), Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f a b).map Embedding.inl\n  | inl _, inr _ => ∅\n  | inr _, inl _ => ∅\n  | inr a, inr b => (g a b).map Embedding.inr\n\nvariable {f f₁ g₁ g f₂ g₂} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\ntheorem mem_sumLift₂ :\n    c ∈ sumLift₂ f g a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨\n        ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g a₂ b₂ := by\n  constructor\n  · rcases a with a | a <;> rcases b with b | b\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr ⟨a, b, c, rfl, rfl, rfl, hc⟩\n  · rintro (⟨a, b, c, rfl, rfl, rfl, h⟩ | ⟨a, b, c, rfl, rfl, rfl, h⟩) <;> exact mem_map_of_mem _ h\n\ntheorem inl_mem_sumLift₂ {c₁ : γ₁} :\n    inl c₁ ∈ sumLift₂ f g a b ↔ ∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f a₁ b₁ := by\n  rw [mem_sumLift₂, or_iff_left]\n  · simp only [inl.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inl_ne_inr h\n\ntheorem inr_mem_sumLift₂ {c₂ : γ₂} :\n    inr c₂ ∈ sumLift₂ f g a b ↔ ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ g a₂ b₂ := by\n  rw [mem_sumLift₂, or_iff_right]\n  · simp only [inr.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inr_ne_inl h\n\ntheorem sumLift₂_eq_empty :\n    sumLift₂ f g a b = ∅ ↔\n      (∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f a₁ b₁ = ∅) ∧\n        ∀ a₂ b₂, a = inr a₂ → b = inr b₂ → g a₂ b₂ = ∅ := by\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · constructor <;>\n    · rintro a b rfl rfl\n      exact map_eq_empty.1 h\n  cases a <;> cases b\n  · exact map_eq_empty.2 (h.1 _ _ rfl rfl)\n  · rfl\n  · rfl\n  · exact map_eq_empty.2 (h.2 _ _ rfl rfl)\n\ntheorem sumLift₂_nonempty :\n    (sumLift₂ f g a b).Nonempty ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ (f a₁ b₁).Nonempty) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ (g a₂ b₂).Nonempty := by\n  simp only [nonempty_iff_ne_empty, Ne, sumLift₂_eq_empty, not_and_or, not_forall, exists_prop]\n\ntheorem sumLift₂_mono (h₁ : ∀ a b, f₁ a b ⊆ g₁ a b) (h₂ : ∀ a b, f₂ a b ⊆ g₂ a b) :\n    ∀ a b, sumLift₂ f₁ f₂ a b ⊆ sumLift₂ g₁ g₂ a b\n  | inl _, inl _ => map_subset_map.2 (h₁ _ _)\n  | inl _, inr _ => Subset.rfl\n  | inr _, inl _ => Subset.rfl\n  | inr _, inr _ => map_subset_map.2 (h₂ _ _)\n\nend SumLift₂\n\nsection SumLexLift\nvariable (f₁ f₁' : α₁ → β₁ → Finset γ₁) (f₂ f₂' : α₂ → β₂ → Finset γ₂)\n  (g₁ g₁' : α₁ → β₂ → Finset γ₁) (g₂ g₂' : α₁ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁`, `α₂ → β₂ → Finset γ₂`, `α₁ → β₂ → Finset γ₁`,\n`α₂ → β₂ → Finset γ₂`  to a map `α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to\nalternative monads if we can make sure to keep computability and universe polymorphism. -/\ndef sumLexLift : α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f₁ a b).map Embedding.inl\n  | inl a, inr b => (g₁ a b).disjSum (g₂ a b)\n  | inr _, inl _ => ∅\n  | inr a, inr b => (f₂ a b).map ⟨_, inr_injective⟩\n\n@[simp]\nlemma sumLexLift_inl_inl (a : α₁) (b : β₁) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inl b) = (f₁ a b).map Embedding.inl := rfl\n\n@[simp]\nlemma sumLexLift_inl_inr (a : α₁) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b) = (g₁ a b).disjSum (g₂ a b) := rfl\n\n@[simp]\nlemma sumLexLift_inr_inl (a : α₂) (b : β₁) : sumLexLift f₁ f₂ g₁ g₂ (inr a) (inl b) = ∅ := rfl\n\n@[simp]\nlemma sumLexLift_inr_inr (a : α₂) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inr a) (inr b) = (f₂ a b).map ⟨_, inr_injective⟩ := rfl\n\nvariable {f₁ g₁ f₂ g₂ f₁' g₁' f₂' g₂'} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\nlemma mem_sumLexLift :\n    c ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        (∃ a₁ b₂ c₁, a = inl a₁ ∧ b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨\n          (∃ a₁ b₂ c₂, a = inl a₁ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n            ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂ := by\n  constructor\n  · obtain a | a := a <;> obtain b | b := b\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (mem_disjSum.1 h).elim ?_ ?_\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inr <| Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr (Or.inr <| Or.inr <| ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n  · rintro (⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩ |\n      ⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact mem_map_of_mem _ hc\n    · exact inl_mem_disjSum.2 hc\n    · exact inr_mem_disjSum.2 hc\n    · exact mem_map_of_mem _ hc\n\nlemma inl_mem_sumLexLift {c₁ : γ₁} :\n    inl c₁ ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        ∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ c₁ ∈ g₁ a₁ b₂ := by\n  simp [mem_sumLexLift]\n\nlemma inr_mem_sumLexLift {c₂ : γ₂} :\n    inr c₂ ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ f₂ a₂ b₂ := by\n  simp [mem_sumLexLift]\n\nlemma sumLexLift_mono (hf₁ : ∀ a b, f₁ a b ⊆ f₁' a b) (hf₂ : ∀ a b, f₂ a b ⊆ f₂' a b)\n    (hg₁ : ∀ a b, g₁ a b ⊆ g₁' a b) (hg₂ : ∀ a b, g₂ a b ⊆ g₂' a b) (a : α₁ ⊕ α₂)\n    (b : β₁ ⊕ β₂) : sumLexLift f₁ f₂ g₁ g₂ a b ⊆ sumLexLift f₁' f₂' g₁' g₂' a b := by\n  cases a <;> cases b\n  exacts [map_subset_map.2 (hf₁ _ _), disjSum_mono (hg₁ _ _) (hg₂ _ _), Subset.rfl,\n    map_subset_map.2 (hf₂ _ _)]\n\nlemma sumLexLift_eq_empty :\n    sumLexLift f₁ f₂ g₁ g₂ a b = ∅ ↔\n      (∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f₁ a₁ b₁ = ∅) ∧\n        (∀ a₁ b₂, a = inl a₁ → b = inr b₂ → g₁ a₁ b₂ = ∅ ∧ g₂ a₁ b₂ = ∅) ∧\n          ∀ a₂ b₂, a = inr a₂ → b = inr b₂ → f₂ a₂ b₂ = ∅ := by\n  refine ⟨fun h ↦ ⟨?_, ?_, ?_⟩, fun h ↦ ?_⟩\n  any_goals rintro a b rfl rfl; exact map_eq_empty.1 h\n  · rintro a b rfl rfl; exact disjSum_eq_empty.1 h\n  cases a <;> cases b\n  · exact map_eq_empty.2 (h.1 _ _ rfl rfl)\n  · simp [h.2.1 _ _ rfl rfl]\n  · rfl\n  · exact map_eq_empty.2 (h.2.2 _ _ rfl rfl)\n\nTarget:\nlemma sumLexLift_nonempty :\n    (sumLexLift f₁ f₂ g₁ g₂ a b).Nonempty ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ (f₁ a₁ b₁).Nonempty) ∨\n        (∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ ((g₁ a₁ b₂).Nonempty ∨ (g₂ a₁ b₂).Nonempty)) ∨\n          ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ (f₂ a₂ b₂).Nonempty :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_dce4fa550c06","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"5d88000b3f5335cf008faace2e19656483624637da8c8d1c74d5eb157d098efb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sum","family_id":"sumlexlift_nonempty","file_id":"mathlib/Mathlib/Data/Sum/Interval.lean","sample_id":"dce4fa550c062aaa1e0e5d299822fa3b4591439eeb6e8fb744a8c9d9b1a2391e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ace5857b04711e3f6e5ec35f7db5255eacbf0110e349e471daaf3e18ca693ed4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"21693a2d3819a5c289e31cd4b3eeb3089492e9b06db09c51d168640b64281356","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"22290c022264f0f12d52570adb279eb6ca130b8e7ce84838db022df1840fc31a","source_sha256":"f47eb7e9445cb948d9f1cba9a87c2cadc067f856113f1f8e2a417357cd103064","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases! hn : n = 0\n  · rw [pow_eq_top_iff.mpr (Or.inr hn)]\n    exact isUnit_of_subsingleton _\n  exact (isUnit_mk_pow_iff_notMem I hn).mpr hx","hard_negative":true,"metrics":{"chosen_tokens":33,"rejected_tokens":7,"token_jaccard":0.107143,"token_length_ratio":0.212121},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"1701f83daf9a8ccb237893bb8c124eb99499fe170657b2a99d0d067ab5fddc54","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Ideal.Quotient.Operations\npublic import Mathlib.RingTheory.Nilpotent.Lemmas\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Nilpotent elements in quotient rings\n-/\n\npublic section\n\ntheorem Ideal.isRadical_iff_quotient_reduced {R : Type*} [CommRing R] (I : Ideal R) :\n    I.IsRadical ↔ IsReduced (R ⧸ I) := by\n  conv_lhs => rw [← @Ideal.mk_ker R _ I]\n  exact RingHom.ker_isRadical_iff_reduced_of_surjective Quotient.mk_surjective\n\nvariable {S : Type*} [CommRing S] (I : Ideal S)\n\n/-- Let `P` be a property on ideals. If `P` holds for square-zero ideals, and if\n  `P I → P (J ⧸ I) → P J`, then `P` holds for all nilpotent ideals. -/\ntheorem Ideal.IsNilpotent.induction_on (hI : IsNilpotent I)\n    {P : ∀ ⦃S : Type _⦄ [CommRing S], Ideal S → Prop}\n    (h₁ : ∀ ⦃S : Type _⦄ [CommRing S], ∀ I : Ideal S, I ^ 2 = ⊥ → P I)\n    (h₂ : ∀ ⦃S : Type _⦄ [CommRing S], ∀ I J : Ideal S, I ≤ J → P I →\n      P (J.map (Ideal.Quotient.mk I)) → P J) :\n    P I := by\n  obtain ⟨n, hI : I ^ n = ⊥⟩ := hI\n  induction n using Nat.strong_induction_on generalizing S with | _ n H\n  by_cases hI' : I = ⊥\n  · subst hI'\n    apply h₁\n    rw [← Ideal.zero_eq_bot, zero_pow two_ne_zero]\n  rcases n with - | n\n  · rw [pow_zero, Ideal.one_eq_top] at hI\n    haveI := subsingleton_of_bot_eq_top hI.symm\n    exact (hI' (Subsingleton.elim _ _)).elim\n  rcases n with - | n\n  · rw [pow_one] at hI\n    exact (hI' hI).elim\n  apply h₂ (I ^ 2) _ (Ideal.pow_le_self two_ne_zero)\n  · apply H n.succ _ (I ^ 2)\n    · rw [← pow_mul, eq_bot_iff, ← hI, Nat.succ_eq_add_one]\n      apply Ideal.pow_le_pow_right (by lia)\n    · exact n.succ.lt_succ_self\n  · apply h₁\n    rw [← Ideal.map_pow, Ideal.map_quotient_self]\n\ntheorem IsNilpotent.isUnit_quotient_mk_iff {R : Type*} [CommRing R] {I : Ideal R}\n    (hI : IsNilpotent I) {x : R} : IsUnit (Ideal.Quotient.mk I x) ↔ IsUnit x := by\n  refine ⟨?_, fun h => h.map <| Ideal.Quotient.mk I⟩\n  revert x\n  apply Ideal.IsNilpotent.induction_on (S := R) I hI <;> clear hI I\n  swap\n  · introv e h₁ h₂ h₃\n    apply h₁\n    apply h₂\n    exact\n      h₃.map\n        ((DoubleQuot.quotQuotEquivQuotSup I J).trans\n              (Ideal.quotEquivOfEq (sup_eq_right.mpr e))).symm.toRingHom\n  · introv e H\n    obtain ⟨y, hy⟩ := Ideal.Quotient.mk_surjective (↑H.unit⁻¹ : S ⧸ I)\n    have : Ideal.Quotient.mk I (x * y) = Ideal.Quotient.mk I 1 := by\n      rw [map_one, map_mul, hy, IsUnit.mul_val_inv]\n    rw [Ideal.Quotient.eq] at this\n    have : (x * y - 1) ^ 2 = 0 := by\n      rw [← Ideal.mem_bot, ← e]\n      exact Ideal.pow_mem_pow this _\n    have : x * (y * (2 - x * y)) = 1 := by\n      rw [eq_comm, ← sub_eq_zero, ← this]\n      ring\n    exact .of_mul_eq_one _ this\n\ntheorem Ideal.Quotient.isUnit_mk_pow_iff_isUnit_mk {x : S} {n : ℕ} (hn : n ≠ 0) :\n    IsUnit (Ideal.Quotient.mk (I ^ n) x) ↔ IsUnit (Ideal.Quotient.mk I x) := by\n  rw [← IsNilpotent.isUnit_quotient_mk_iff (I := Ideal.map (Ideal.Quotient.mk (I ^ n)) I)]\n  · rw [← isUnit_map_iff (DoubleQuot.quotQuotEquivQuotOfLE (Ideal.pow_le_self hn))]\n    rfl\n  · use n\n    simp [← Ideal.map_pow]\n\ntheorem Ideal.Quotient.isUnit_mk_pow_iff_notMem [I.IsMaximal] {n : ℕ} (hn : n ≠ 0) {x : S} :\n    IsUnit (mk (I ^ n) x) ↔ x ∉ I := by\n  let := Ideal.Quotient.field I\n  rw [isUnit_mk_pow_iff_isUnit_mk I hn, isUnit_iff_ne_zero]\n  exact Ideal.Quotient.eq_zero_iff_mem.not\n\nTarget:\ntheorem Ideal.Quotient.isUnit_mk_pow_of_notMem [I.IsMaximal] {n : ℕ} {x : S} (hx : x ∉ I) :\n    IsUnit (mk (I ^ n) x) :=\n\nProof body:\n","rejected":"by\n  exact Ideal.Quotient.isUnit_mk_pow_of_notMem","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"fd4afe14d32008c27559fa53b80c4501450857db78d8170a348af7e96bc29b7c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Ideal","family_id":"ideal","file_id":"mathlib/Mathlib/RingTheory/Ideal/Quotient/Nilpotent.lean","sample_id":"22290c022264f0f12d52570adb279eb6ca130b8e7ce84838db022df1840fc31a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ff5b125214a3c976ea99dae2474b0a2f292cf08626af894b2e62bc6e985e11b4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"519681ab2bbfd1d17a2a2e4441b416bdb43fb158bfd507e6a5c824c10ac4fe23","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2c8f4c46b465c3b61a9d9f12832774c7822ae516e1e560bc50ecc34f4e2ef998","source_sha256":"d2b257e0c7de516a23c8b73fe5ea40e7a7931660c3e6a7b037f3eda8ab6e5072","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply freeYonedaHomEquiv.injective\n  simp only [freeYonedaHomEquiv_comp, Equiv.apply_symm_apply]\n\nvariable (C)","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":5,"token_jaccard":0.1,"token_length_ratio":0.277778},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"173c12566e6c511e91d7e5214d20bdfe9ec289cde116dad6c7e888bb87b9700d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Generator.Basic\npublic import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic\n\nNamespace:\nCategoryTheory.Presheaf\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Generators in the category of presheaves\n\nIn this file, we show that if `A` is a category with zero morphisms that\nhas a separator (and suitable coproducts), then the category of\npresheaves `Cᵒᵖ ⥤ A` also has a separator.\n\n-/\n\n@[expose] public section\n\nuniverse w v' v u' u\n\nnamespace CategoryTheory\n\nopen Limits Opposite\n\nnamespace Presheaf\n\nvariable {C : Type u} [Category.{v} C] {A : Type u'} [Category.{v'} A]\n  [HasCoproducts.{v} A]\n\n/-- Given `X : C` and `M : A`, this is the presheaf `Cᵒᵖ ⥤ A` which sends\n`Y : Cᵒᵖ` to the coproduct of copies of `M` indexed by `Y.unop ⟶ X`. -/\n@[simps]\nnoncomputable def freeYoneda (X : C) (M : A) : Cᵒᵖ ⥤ A where\n  obj Y := ∐ (fun (i : (yoneda.obj X).obj Y) ↦ M)\n  map f := Sigma.map' ((yoneda.obj X).map f) (fun _ ↦ 𝟙 M)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The bijection `(Presheaf.freeYoneda X M ⟶ F) ≃ (M ⟶ F.obj (op X))`. -/\nnoncomputable def freeYonedaHomEquiv {X : C} {M : A} {F : Cᵒᵖ ⥤ A} :\n    (freeYoneda X M ⟶ F) ≃ (M ⟶ F.obj (op X)) where\n  toFun f := Sigma.ι (fun (i : (yoneda.obj X).obj _) ↦ M) (𝟙 _) ≫ f.app (op X)\n  invFun g :=\n    { app Y := Sigma.desc (fun φ ↦ g ≫ F.map φ.op)\n      naturality _ _ _ := Sigma.hom_ext _ _ (by simp) }\n  left_inv f := by\n    ext Y\n    refine Sigma.hom_ext _ _ (fun φ ↦ ?_)\n    simpa using (Sigma.ι _ (𝟙 _) ≫= f.naturality φ.op).symm\n  right_inv g := by simp\n\nset_option backward.defeqAttrib.useBackward true in\n@[reassoc]\nlemma freeYonedaHomEquiv_comp {X : C} {M : A} {F G : Cᵒᵖ ⥤ A}\n    (α : freeYoneda X M ⟶ F) (f : F ⟶ G) :\n    freeYonedaHomEquiv (α ≫ f) = freeYonedaHomEquiv α ≫ f.app (op X) := by\n  simp [freeYonedaHomEquiv]\n\n@[reassoc]\n\nTarget:\nlemma freeYonedaHomEquiv_symm_comp {X : C} {M : A} {F G : Cᵒᵖ ⥤ A} (α : M ⟶ F.obj (op X))\n    (f : F ⟶ G) :\n    freeYonedaHomEquiv.symm α ≫ f = freeYonedaHomEquiv.symm (α ≫ f.app (op X)) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_2c8f4c46b465","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d5b5f26625a749abdd13b6d8dd0df4e409a58b18a87268059c904774c3314726","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Generator","family_id":"freeyonedahomequiv_symm_comp","file_id":"mathlib/Mathlib/CategoryTheory/Generator/Presheaf.lean","sample_id":"2c8f4c46b465c3b61a9d9f12832774c7822ae516e1e560bc50ecc34f4e2ef998"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b2e750b3c5b8740fb444ea49e43055beae89260e7b9e80c29b949ec289033978","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d4463edd748beb11f3b39ffe113ee78d6baf11270fc163c33fc0896820ef4623","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f0978345095eddbdb23e72c7878138dd691ea3f6ff7d3034237e21db83bc2237","source_sha256":"155f9e8471ee4b30d28b3dbc51f2958ab69f6c864e7fa3081250ca89e1bfbb0f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [of_apply, ← lift_symm_apply, Equiv.symm_apply_apply]","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.166667},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"174bd753ceebdb7cc5ef3da50a9be98699ff246dd327bc4146ac464206699635","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.SkewMonoidAlgebra.Basic\npublic import Mathlib.Algebra.Module.BigOperators\npublic import Mathlib.Algebra.Algebra.Equiv\n\nNamespace:\nSkewMonoidAlgebra\n\nLocal context:\n/-\nCopyright (c) 2025 Xavier Généreux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: María Inés de Frutos Fernández, Xavier Généreux\n-/\n/-!\n# Lemmas about different kinds of \"lifts\" to `SkewMonoidAlgebra`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nnamespace SkewMonoidAlgebra\n\nvariable {k G H : Type*}\n\nsection lift\n\nvariable [CommSemiring k] [Monoid G] [Monoid H]\nvariable {A B : Type*} [Semiring A] [Algebra k A] [Semiring B] [Algebra k B]\n\n/-- `liftNCRingHom` as an `AlgHom`, for when `f` is an `AlgHom` -/\ndef liftNCAlgHom [MulSemiringAction G A] [SMulCommClass G k A] (f : A →ₐ[k] B)\n    (g : G →* B) (h_comm : ∀ {x y}, (f (y • x)) * g y = (g y) * (f x)) :\n    SkewMonoidAlgebra A G →ₐ[k] B where\n  __ := liftNCRingHom (f : A →+* B) g h_comm\n  commutes' := by simp [liftNCRingHom]\n\n/- Hypotheses needed for `k`-algebra homomorphism from `SkewMonoidAlgebra k G`-/\nvariable [MulSemiringAction G k] [SMulCommClass G k k]\n\nvariable (k G A)\n\n/-- Any monoid homomorphism `G →* A` can be lifted to an algebra homomorphism\n  `SkewMonoidAlgebra k G →ₐ[k] A`. -/\ndef lift : (G →* A) ≃ (AlgHom k (SkewMonoidAlgebra k G) A) where\n  invFun f := (f : SkewMonoidAlgebra k G →* A).comp (of k G)\n  toFun F := by\n    apply liftNCAlgHom (Algebra.ofId k A) F\n    simp_rw [show ∀ (g : G) (r : k), g • r = r by\n        exact fun _ _ ↦ smul_algebraMap _ (algebraMap k k _)]\n    exact Algebra.commutes _ _\n  left_inv f := by\n    ext\n    simp [liftNCAlgHom, liftNCRingHom]\n  right_inv F := by\n    ext\n    simp [liftNCAlgHom, liftNCRingHom]\n\nvariable {k G A}\n\ntheorem lift_apply' (F : G →* A) (f : SkewMonoidAlgebra k G) :\n    lift k G A F f = f.sum fun a b ↦ algebraMap k A b * F a := rfl\n\ntheorem lift_apply (F : G →* A) (f : SkewMonoidAlgebra k G) :\n    lift k G A F f = f.sum fun a b ↦ b • F a := by simp [lift_apply', Algebra.smul_def]\n\ntheorem lift_def (F : G →* A) : (lift k G A F : SkewMonoidAlgebra k G → A) =\n    liftNC ((algebraMap k A : k →+* A) : k →+ A) F := rfl\n\n@[simp]\ntheorem lift_symm_apply (F : AlgHom k (SkewMonoidAlgebra k G) A) (x : G) :\n    (lift k G A).symm F x = F (single x 1) := rfl\n\nTarget:\ntheorem lift_of (F : G →* A) (x) : lift k G A F (of k G x) = F x :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_f0978345095e","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"c16d93f2a95c3a2b1da59d485dd74fee64886ba975bdce9c6e9238e1ddb922a2","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/SkewMonoidAlgebra","family_id":"lift_of","file_id":"mathlib/Mathlib/Algebra/SkewMonoidAlgebra/Lift.lean","sample_id":"f0978345095eddbdb23e72c7878138dd691ea3f6ff7d3034237e21db83bc2237"},"split":"train"}
{"binding":{"chosen_proof_sha256":"187dbb6cbe73173dbea1f8f380afdbd3c2e979ea26a53fbec596e63a5f1c929e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"804a5cbe713c4c438d0140658a7f1c0cb5ba46f26459ab357963225a2bcc170e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9f7eb7bcc0d3153a05f5019eda72791728388ea6466ec87e305f6457e18d50ee","source_sha256":"72c080c49c4d252a9088a8037fd0035ad61560d6f34ce2947cf3bd3ceaf7e3f4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by by_cases a = 0 <;> grind [zero_le]","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.25},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"175439ef0c42bbbe1e8bac772f544e44cc3241661700538c492707e4cdfa2ee7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.Finite.Basic\npublic import Mathlib.RingTheory.Valuation.Basic\n\nNamespace:\nFiniteField\n\nLocal context:\n/-\nCopyright (c) 2026 María Inés de Frutos-Fernández, Xavier Généreux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: María Inés de Frutos-Fernández, Xavier Généreux\n-/\n/-!\n# Valuations on an algebra over a finite field.\n-/\n\npublic section\n\nnamespace FiniteField\n\nopen Valuation\n\nvariable {Fq A Γ : Type*} [Field Fq] [Finite Fq] [Ring A] [Algebra Fq A]\n  [LinearOrderedCommMonoidWithZero Γ] (v : Valuation A Γ)\n\n@[grind =>]\nlemma valuation_algebraMap_eq_one (a : Fq) (ha : a ≠ 0) : v (algebraMap Fq A a) = 1 := by\n  have : Fintype Fq := Fintype.ofFinite Fq\n  have hpow : (v (algebraMap Fq A a)) ^ (Fintype.card Fq - 1) = 1 := by\n    simp [← map_pow, FiniteField.pow_card_sub_one_eq_one a ha]\n  grind [pow_eq_one_iff, → IsPrimePow.two_le, FiniteField.isPrimePow_card]\n\nTarget:\nlemma valuation_algebraMap_le_one (v : Valuation A Γ) (a : Fq) :\n    v (algebraMap Fq A a) ≤ 1 :=\n\nProof body:\n","rejected":"by\n  exact valuation_algebraMap_le_one","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"4f13116cf6bbd60f2b97058bfd97798f3a29901b6ed86c01536641ccd5c882e9","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Finite","family_id":"valuation_algebramap_le_one","file_id":"mathlib/Mathlib/FieldTheory/Finite/Valuation.lean","sample_id":"9f7eb7bcc0d3153a05f5019eda72791728388ea6466ec87e305f6457e18d50ee"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8390e404c259cefd040f4b8484fc4ca8346f13b8985655df805e82d4eca600cf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"62dbb2c1f7ca5d01d29e0e066c6ef072aa7c2eb151ad094933b54216671e965f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"923ffa343160dc678018bf65a4585befc8d6632387419cd5ee47dffbd11b76e4","source_sha256":"0aa3275c10d58f2b53efa7a12fc422557c22614d050f0318487bd09968cf8e9f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x\n  induction x <;> rfl","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.555556},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"1771f005f27ba38ac9053aa45d62343058b7b4ca393d378927bcad73a781926b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Basic\npublic import Mathlib.Algebra.Group.Equiv.Defs\npublic import Mathlib.Algebra.Group.WithOne.Defs\n\nNamespace:\nWithOne\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Johan Commelin\n-/\n/-!\n# More operations on `WithOne` and `WithZero`\n\nThis file defines various bundled morphisms on `WithOne` and `WithZero`\nthat were not available in `Algebra/Group/WithOne/Defs`.\n\n## Main definitions\n\n* `WithOne.lift`, `WithZero.lift`\n* `WithOne.map`, `WithZero.map`\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero DenselyOrdered\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\nnamespace WithOne\n\n@[to_additive]\ninstance instInvolutiveInv [InvolutiveInv α] : InvolutiveInv (WithOne α) where\n  inv_inv a := (Option.map_map _ _ _).trans <| by simp_rw [inv_comp_inv, Option.map_id, id]\n\nsection\n\n/-- `WithOne.coe` as a bundled morphism -/\n@[to_additive (attr := simps apply) /-- `WithZero.coe` as a bundled morphism -/]\ndef coeMulHom [Mul α] : α →ₙ* WithOne α where\n  toFun := coe\n  map_mul' _ _ := rfl\n\nend\n\nsection lift\n\nvariable [Mul α] [MulOneClass β]\n\n/-- Lift a semigroup homomorphism `f` to a bundled monoid homomorphism. -/\n@[to_additive /--\nLift an additive semigroup homomorphism `f` to a bundled additive monoid homomorphism. -/]\ndef lift : (α →ₙ* β) ≃ (WithOne α →* β) where\n  toFun f :=\n    { toFun := WithOne.recOneCoe 1 f, map_one' := rfl,\n      map_mul' := fun x y => x.cases_on (by simp) (fun x => y.cases_on (by simp) (f.map_mul x)) }\n  invFun F := F.toMulHom.comp coeMulHom\n  right_inv F := MonoidHom.ext fun x => WithOne.cases_on x F.map_one.symm (fun _ => rfl)\n\nvariable (f : α →ₙ* β)\n\n@[to_additive (attr := simp)]\ntheorem lift_coe (x : α) : lift f x = f x :=\n  rfl\n\n@[to_additive (attr := simp)]\ntheorem lift_one : lift f 1 = 1 :=\n  rfl\n\n@[to_additive]\ntheorem lift_unique (f : WithOne α →* β) : f = lift (f.toMulHom.comp coeMulHom) :=\n  (lift.apply_symm_apply f).symm\n\n@[to_additive (attr := simp)]\ntheorem lift_symm_apply (f : WithOne α →* β) (x : α) : lift.symm f x = f x := rfl\n\n@[to_additive]\nlemma lift_symm_injective_of_injective {f : WithOne α →* β} (hf : Function.Injective f) :\n    Function.Injective (lift.symm f) :=\n  fun _ _ ↦ by simp [hf.eq_iff]\n\nend lift\n\nsection Map\n\nvariable [Mul α] [Mul β] [Mul γ]\n\n/-- Given a multiplicative map from `α → β` returns a monoid homomorphism\n  from `WithOne α` to `WithOne β` -/\n@[to_additive /-- Given an additive map from `α → β` returns an additive monoid homomorphism from\n`WithZero α` to `WithZero β` -/]\ndef mapMulHom (f : α →ₙ* β) : WithOne α →* WithOne β :=\n  lift (coeMulHom.comp f)\n\n@[to_additive (attr := simp)]\ntheorem mapMulHom_coe (f : α →ₙ* β) (a : α) : mapMulHom f (a : WithOne α) = f a :=\n  rfl\n\n@[to_additive (attr := simp)]\n\nTarget:\ntheorem mapMulHom_id : mapMulHom (MulHom.id α) = MonoidHom.id (WithOne α) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_923ffa343160","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"6210317329d985e44b4f599f05b36c7e58246f42db347bc8803db118dd7be3e6","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Group","family_id":"mapmulhom_id","file_id":"mathlib/Mathlib/Algebra/Group/WithOne/Basic.lean","sample_id":"923ffa343160dc678018bf65a4585befc8d6632387419cd5ee47dffbd11b76e4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1fd411efd0adcfd9a0a62aeb48dc422a81d7841cde6fbd3cbad674180785c839","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"54b19734bb7d230ea7d5c492620ce8dadeaac913d83d2ad2c5dfe1223b5a3aca","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bdf9d42bef05adc3f26afabef77136e641ea973a1c33f1f631c7b584f6c81787","source_sha256":"ef9864ce066b7696b62174e7a3b3a449882e3ab320bb2e19ef0e02cda6607291","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp +instances only [r_iff_exists, Subtype.exists, exists_prop, OreLocalization.oreEqv,\n    smul_eq_mul, Submonoid.mk_smul]\n  constructor\n  · rintro ⟨u, hu, e⟩\n    exact ⟨_, mul_mem hu x.2.2, u * y.2, by rw [mul_assoc, mul_assoc, ← e], mul_right_comm _ _ _⟩\n  · rintro ⟨u, hu, v, e₁, e₂⟩\n    exact ⟨u, hu, by rw [← mul_assoc, e₂, mul_right_comm, ← e₁, mul_assoc, mul_comm y.1]⟩","hard_negative":true,"metrics":{"chosen_tokens":112,"rejected_tokens":5,"token_jaccard":0.068182,"token_length_ratio":0.044643},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"1825f31b028bc4ce17a9f9a8bb9486506368df20e6716ed439d6960c238c6ead","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\npublic import Mathlib.Algebra.Group.Submonoid.Operations\npublic import Mathlib.Algebra.Regular.Basic\npublic import Mathlib.GroupTheory.Congruence.Hom\npublic import Mathlib.GroupTheory.OreLocalization.Basic\n\nNamespace:\nLocalization\n\nLocal context:\n/-\nCopyright (c) 2019 Amelia Livingston. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Amelia Livingston\n-/\n/-!\n# Localizations of commutative monoids\n\nLocalizing a commutative ring at one of its submonoids does not rely on the ring's addition, so\nwe can generalize localizations to commutative monoids.\n\nWe characterize the localization of a commutative monoid `M` at a submonoid `S` up to\nisomorphism; that is, a commutative monoid `N` is the localization of `M` at `S` iff we can find a\nmonoid homomorphism `f : M →* N` satisfying 3 properties:\n1. For all `y ∈ S`, `f y` is a unit;\n2. For all `z : N`, there exists `(x, y) : M × S` such that `z * f y = f x`;\n3. For all `x, y : M` such that `f x = f y`, there exists `c ∈ S` such that `x * c = y * c`.\n   (The converse is a consequence of 1.)\n\nGiven such a localization map `f : M →* N`, we can define the surjection\n`Submonoid.LocalizationMap.mk'` sending `(x, y) : M × S` to `f x * (f y)⁻¹`. Mapping properties\nof the localization (e.g. extending a map from `M → P` to `N` if the image of `S` is contained in\nthe units) are treated in a later file `Mathlib.GroupTheory.MonoidLocalization.Maps`.\n\nWe also define the quotient of `M × S` by the unique congruence relation (equivalence relation\npreserving a binary operation) `r` such that for any other congruence relation `s` on `M × S`\nsatisfying '`∀ y ∈ S`, `(1, 1) ∼ (y, y)` under `s`', we have that `(x₁, y₁) ∼ (x₂, y₂)` by `s`\nwhenever `(x₁, y₁) ∼ (x₂, y₂)` by `r`. We show this relation is equivalent to the standard\nlocalization relation.\nThis defines the localization as a quotient type, `Localization`, but the majority of\nsubsequent lemmas in the file are given in terms of localizations up to isomorphism, using maps\nwhich satisfy the characteristic predicate.\n\nThe Grothendieck group construction corresponds to localizing at the top submonoid, namely making\nevery element invertible.\n\n## Implementation notes\n\nIn maths it is natural to reason up to isomorphism, but in Lean we cannot naturally `rewrite` one\nstructure with an isomorphic one; one way around this is to isolate a predicate characterizing\na structure up to isomorphism, and reason about things that satisfy the predicate.\n\nThe infimum form of the localization congruence relation is chosen as 'canonical' here, since it\nshortens some proofs.\n\nTo reason about the localization as a quotient type, use `mk_eq_monoidOf_mk'` and associated\nlemmas. These show the quotient map `mk : M → S → Localization S` equals the\nsurjection `LocalizationMap.mk'` induced by the map\n`Localization.monoidOf : Submonoid.LocalizationMap S (Localization S)` (where `of` establishes the\nlocalization as a quotient type satisfies the characteristic predicate). The lemma\n`mk_eq_monoidOf_mk'` hence gives you access to the results in the rest of the file, which are about\nthe `LocalizationMap.mk'` induced by any localization map.\n\n## TODO\n\n* Show that the localization at the top monoid is a group.\n* Generalise to (nonempty) subsemigroups.\n* If we acquire more bundlings, we can make `Localization.mkOrderEmbedding` be an ordered monoid\n  embedding.\n\n## Tags\nlocalization, monoid localization, quotient monoid, congruence relation, characteristic predicate,\ncommutative monoid, grothendieck group\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero Ring\n\nopen Function\nnamespace AddSubmonoid\n\nvariable {M : Type*} [AddCommMonoid M] (S : AddSubmonoid M) (N : Type*) [AddCommMonoid N]\n\nvariable {N} in\n/-- A predicate characterizing homomorphisms between additive monoids `M` and `N` that form a\ncommutative triangle with the canonical map from `M` to its localization at `S` and\nsome isomorphism between `N` and the localization. -/\nstructure IsLocalizationMap (S : AddSubmonoid M) (f : M → N) where\n  map_addUnits (y : S) : IsAddUnit (f y)\n  surj (z : N) : ∃ x : M × S, z + f x.2 = f x.1\n  exists_of_eq {x y} : f x = f y → ∃ c : S, c + x = c + y\n\n/-- The type of AddMonoid homomorphisms satisfying the characteristic predicate: if `f : M →+ N`\nsatisfies this predicate, then `N` is isomorphic to the localization of `M` at `S`. -/\nstructure LocalizationMap extends M →ₙ+ N where\n  isLocalizationMap : IsLocalizationMap S toFun\n\n/-- The additive homomorphism underlying a `LocalizationMap` of `AddCommMonoid`s. -/\nadd_decl_doc LocalizationMap.toAddHom\n\nend AddSubmonoid\n\nsection CommMonoid\n\nvariable {M : Type*} [CommMonoid M] (S : Submonoid M) (N : Type*) [CommMonoid N] {P : Type*}\n  [CommMonoid P]\n\nnamespace Submonoid\n\nvariable {N} in\n/-- A predicate characterizing homomorphisms between monoids `M` and `N` that form a\ncommutative triangle with the canonical map from `M` to its localization at `S` and\nsome isomorphism between `N` and the localization. -/\n@[to_additive (attr := mk_iff)]\nstructure IsLocalizationMap (S : Submonoid M) (f : M → N) where\n  map_units (y : S) : IsUnit (f y)\n  surj (z : N) : ∃ x : M × S, z * f x.2 = f x.1\n  exists_of_eq {x y} : f x = f y → ∃ c : S, c * x = c * y\n\n/-- The type of monoid homomorphisms satisfying the characteristic predicate: if `f : M →* N`\nsatisfies this predicate, then `N` is isomorphic to the localization of `M` at `S`. -/\n@[to_additive] structure LocalizationMap extends M →ₙ* N where\n  isLocalizationMap : IsLocalizationMap S toFun\n\n/-- The multiplicative homomorphism underlying a `LocalizationMap`. -/\nadd_decl_doc LocalizationMap.toMulHom\n\nend Submonoid\n\nnamespace Localization\n\n/- Ensure that `@[to_additive]` uses the right namespace before the definition of `Localization`. -/\ninsert_to_additive_translation Localization AddLocalization\n\n/-- The congruence relation on `M × S`, `M` a `CommMonoid` and `S` a submonoid of `M`, whose\nquotient is the localization of `M` at `S`, defined as the unique congruence relation on\n`M × S` such that for any other congruence relation `s` on `M × S` where for all `y ∈ S`,\n`(1, 1) ∼ (y, y)` under `s`, we have that `(x₁, y₁) ∼ (x₂, y₂)` by `r` implies\n`(x₁, y₁) ∼ (x₂, y₂)` by `s`. -/\n@[to_additive\n/-- The congruence relation on `M × S`, `M` an `AddCommMonoid` and `S` an `AddSubmonoid` of `M`,\nwhose quotient is the localization of `M` at `S`, defined as the unique congruence relation on\n`M × S` such that for any other congruence relation `s` on `M × S` where for all `y ∈ S`,\n`(0, 0) ∼ (y, y)` under `s`, we have that `(x₁, y₁) ∼ (x₂, y₂)` by `r` implies\n`(x₁, y₁) ∼ (x₂, y₂)` by `s`. -/]\ndef r (S : Submonoid M) : Con (M × S) :=\n  sInf { c | ∀ y : S, c 1 (y, y) }\n\n/-- An alternate form of the congruence relation on `M × S`, `M` a `CommMonoid` and `S` a\nsubmonoid of `M`, whose quotient is the localization of `M` at `S`. -/\n@[to_additive\n/-- An alternate form of the congruence relation on `M × S`, `M` a `CommMonoid` and `S` a\nsubmonoid of `M`, whose quotient is the localization of `M` at `S`. -/]\ndef r' : Con (M × S) := by\n  -- note we multiply by `c` on the left so that we can later generalize to `•`\n  refine\n    { r := fun a b : M × S ↦ ∃ c : S, ↑c * (↑b.2 * a.1) = c * (a.2 * b.1)\n      iseqv := ⟨fun a ↦ ⟨1, rfl⟩, fun ⟨c, hc⟩ ↦ ⟨c, hc.symm⟩, ?_⟩\n      mul' := ?_ }\n  · rintro a b c ⟨t₁, ht₁⟩ ⟨t₂, ht₂⟩\n    use t₂ * t₁ * b.2\n    simp only [Submonoid.coe_mul]\n    calc\n      (t₂ * t₁ * b.2 : M) * (c.2 * a.1) = t₂ * c.2 * (t₁ * (b.2 * a.1)) := by ac_rfl\n      _ = t₁ * a.2 * (t₂ * (c.2 * b.1)) := by rw [ht₁]; ac_rfl\n      _ = t₂ * t₁ * b.2 * (a.2 * c.1) := by rw [ht₂]; ac_rfl\n  · rintro a b c d ⟨t₁, ht₁⟩ ⟨t₂, ht₂⟩\n    use t₂ * t₁\n    calc\n      (t₂ * t₁ : M) * (b.2 * d.2 * (a.1 * c.1)) = t₂ * (d.2 * c.1) * (t₁ * (b.2 * a.1)) := by ac_rfl\n      _ = (t₂ * t₁ : M) * (a.2 * c.2 * (b.1 * d.1)) := by rw [ht₁, ht₂]; ac_rfl\n\n/-- The congruence relation used to localize a `CommMonoid` at a submonoid can be expressed\nequivalently as an infimum (see `Localization.r`) or explicitly\n(see `Localization.r'`). -/\n@[to_additive\n/-- The additive congruence relation used to localize an `AddCommMonoid` at a submonoid can be\nexpressed equivalently as an infimum (see `AddLocalization.r`) or explicitly\n(see `AddLocalization.r'`). -/]\ntheorem r_eq_r' : r S = r' S :=\n  le_antisymm (sInf_le fun _ ↦ ⟨1, by simp⟩) <|\n    le_sInf fun b H ⟨p, q⟩ ⟨x, y⟩ ⟨t, ht⟩ ↦ by\n      rw [← one_mul (p, q), ← one_mul (x, y)]\n      refine b.trans (b.mul (H (t * y)) (b.refl _)) ?_\n      convert! b.symm (b.mul (H (t * q)) (b.refl (x, y))) using 1\n      dsimp only [Prod.mk_mul_mk, Submonoid.coe_mul] at ht ⊢\n      simp_rw [mul_assoc, ht, mul_comm y q]\n\nvariable {S}\n\n@[to_additive]\ntheorem r_iff_exists {x y : M × S} : r S x y ↔ ∃ c : S, ↑c * (↑y.2 * x.1) = c * (x.2 * y.1) := by\n  simp only [r_eq_r' S, r', Con.rel_mk]\n\n@[to_additive]\n\nTarget:\ntheorem r_iff_oreEqv_r {x y : M × S} : r S x y ↔ (OreLocalization.oreEqv S M).r x y :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_bdf9d42bef05","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1bcba9fb6cf91d3da91be0032bd1550ab9ea750edd24277ecc68262545d60fb9","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/MonoidLocalization","family_id":"r_iff_oreeqv_r","file_id":"mathlib/Mathlib/GroupTheory/MonoidLocalization/Basic.lean","sample_id":"bdf9d42bef05adc3f26afabef77136e641ea973a1c33f1f631c7b584f6c81787"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b654cbf08d5ade13b5e6184e32b41bf84d2fa139990565902f32727bb1db98ee","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0eb67f824175dac229b7646063b2648b75a17b30e9821e1a51528362c79353d3","source_sha256":"0722d600b193d93cb9e55becbab169fa607364d0057f53064083850c9da2e6cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun ⟨g, hg⟩ ↦ ?_, fun h ↦ (isCuspForm_iff f).mpr (isZeroAt_of_coeffZero_eq_zero f h)⟩\n  rw [← hg, qExpansion_coeff_zero one_pos\n    (ModularFormClass.analyticAt_cuspFunction_zero _ one_pos one_mem_strictPeriods_SL)\n    (periodic_comp_ofComplex _ one_mem_strictPeriods_SL)]\n  exact (CuspFormClass.zero_at_infty g).valueAtInfty_eq_zero","hard_negative":false,"metrics":{"chosen_tokens":58,"rejected_tokens":3,"token_jaccard":0.090909,"token_length_ratio":0.051724},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"1856d90552b32adf5a2c98910bdd2383fef974bed25e093a11cc17ae9d79696f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.ModularForms.QExpansion\npublic import Mathlib.NumberTheory.ModularForms.LevelOne.Basic\npublic import Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion\n\nNamespace:\nModularForm\n\nLocal context:\n/-\nCopyright (c) 2026 Chris Birkbeck. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Birkbeck\n-/\n/-!\n# Cusp form submodule and IsCuspForm predicate\n\nThis file defines the inclusion of cusp forms into modular forms as a linear map, the cusp form\nsubmodule of modular forms, and the `IsCuspForm` predicate. It also provides a direct constructor\n`ModularForm.toCuspForm` for building cusp forms from modular forms with vanishing constant\nq-expansion coefficient (for `𝒮ℒ`).\n\n## Main definitions\n\n* `CuspForm.toModularFormₗ`: the inclusion `CuspForm Γ k →ₗ[ℂ] ModularForm Γ k`.\n* `ModularForm.cuspFormSubmodule`: the submodule of `ModularForm Γ k` consisting of cusp forms.\n* `ModularForm.IsCuspForm`: predicate that a modular form lies in the cusp form submodule.\n* `ModularForm.toCuspForm`: builds a `CuspForm 𝒮ℒ k` from a `ModularForm` whose q-expansion\n  has vanishing constant term.\n\n## Main results\n\n* `CuspForm.toModularFormₗ_injective`: the inclusion is injective.\n* `CuspForm.equivCuspFormSubmodule`: `CuspForm Γ k ≃ₗ[ℂ] cuspFormSubmodule Γ k`.\n* `ModularForm.isCuspForm_iff_coeffZero_eq_zero`: for `𝒮ℒ`, `IsCuspForm` is equivalent to the\n  q-expansion having vanishing constant term.\n-/\n\n@[expose] public noncomputable section\n\nopen UpperHalfPlane ModularForm Complex SlashInvariantForm SlashInvariantFormClass\n  ModularFormClass MatrixGroups OnePoint Filter Topology\n\nvariable {Γ : Subgroup (GL (Fin 2) ℝ)} {k : ℤ}\n\nnamespace CuspForm\n\n/-- The inclusion of cusp forms into modular forms, as a ℂ-linear map. -/\ndef toModularFormₗ [Γ.HasDetOne] : CuspForm Γ k →ₗ[ℂ] ModularForm Γ k where\n  toFun := ModularFormClass.modularForm\n  map_add' _ _ := rfl\n  map_smul' _ _ := rfl\n\n@[simp]\nlemma toModularFormₗ_apply [Γ.HasDetOne] (f : CuspForm Γ k) (z : ℍ) :\n    (toModularFormₗ f) z = f z := rfl\n\nlemma toModularFormₗ_eq_coe [Γ.HasDetOne] (f : CuspForm Γ k) :\n    toModularFormₗ f = (f : ModularForm Γ k) := rfl\n\nlemma toModularFormₗ_injective [Γ.HasDetOne] :\n    Function.Injective (toModularFormₗ : CuspForm Γ k → ModularForm Γ k) :=\n  fun _ _ h ↦ DFunLike.ext _ _ fun z ↦ DFunLike.congr_fun h z\n\nend CuspForm\n\nnamespace ModularForm\n\n/-- The submodule of `ModularForm Γ k` consisting of cusp forms, defined as the range of\nthe inclusion `CuspForm.toModularFormₗ`. -/\ndef cuspFormSubmodule (Γ : Subgroup (GL (Fin 2) ℝ)) (k : ℤ) [Γ.HasDetOne] :\n    Submodule ℂ (ModularForm Γ k) :=\n  LinearMap.range CuspForm.toModularFormₗ\n\n/-- A modular form is a cusp form if it lies in the cusp form submodule. -/\ndef IsCuspForm [Γ.HasDetOne] (f : ModularForm Γ k) : Prop :=\n  f ∈ cuspFormSubmodule Γ k\n\n@[simp]\nlemma mem_cuspFormSubmodule_iff [Γ.HasDetOne] {f : ModularForm Γ k} :\n    f ∈ cuspFormSubmodule Γ k ↔ IsCuspForm f := Iff.rfl\n\n/-- The cusp form submodule is linearly equivalent to the type of cusp forms. -/\ndef CuspForm.equivCuspFormSubmodule (Γ : Subgroup (GL (Fin 2) ℝ)) (k : ℤ) [Γ.HasDetOne] :\n    CuspForm Γ k ≃ₗ[ℂ] cuspFormSubmodule Γ k :=\n  LinearEquiv.ofInjective CuspForm.toModularFormₗ CuspForm.toModularFormₗ_injective\n\n/-- The underlying modular form (via `toModularFormₗ`) of a `CuspForm` is itself a cusp form. -/\nlemma CuspForm.isCuspForm_toModularFormₗ {Γ : Subgroup (GL (Fin 2) ℝ)} [Γ.HasDetOne]\n    (f : CuspForm Γ k) : ModularForm.IsCuspForm f.toModularFormₗ := by\n  simp [← mem_cuspFormSubmodule_iff, ModularForm.cuspFormSubmodule]\n\n/-- A modular form is a cusp form if and only if it vanishes at every cusp. This is the\ngeneral characterization valid for any subgroup. -/\nlemma isCuspForm_iff [Γ.HasDetOne] (f : ModularForm Γ k) :\n    IsCuspForm f ↔ ∀ {c}, IsCusp c Γ → c.IsZeroAt f k :=\n  ⟨fun ⟨g, hg⟩ _ ↦ hg ▸ g.zero_at_cusps', fun h ↦ ⟨⟨f, f.holo', h⟩, rfl⟩⟩\n\n/-- A modular form with `valueAtInfty f = 0` is zero at infinity. -/\nlemma isZeroAtImInfty_of_valueAtInfty_eq_zero {F : Type*} [FunLike F ℍ ℂ]\n    [DiscreteTopology Γ] [Γ.HasDetPlusMinusOne] [Fact (IsCusp ∞ Γ)] [ModularFormClass F Γ k]\n    (f : F) (h : valueAtInfty f = 0) : IsZeroAtImInfty f := by\n  have hh : 0 < Γ.strictWidthInfty := Γ.strictWidthInfty_pos_iff.mpr Fact.out\n  have hΓ : Γ.strictWidthInfty ∈ Γ.strictPeriods := Γ.strictWidthInfty_mem_strictPeriods\n  have hanal := ModularFormClass.analyticAt_cuspFunction_zero f hh hΓ\n  have hper := periodic_comp_ofComplex f hΓ\n  simp_rw [IsZeroAtImInfty, ZeroAtFilter, ← h, ← cuspFunction_apply_zero hh hanal hper]\n  exact (hanal.continuousAt.tendsto.comp (qParam_tendsto_atImInfty hh)).congr\n    (fun τ ↦ SlashInvariantFormClass.eq_cuspFunction f τ hΓ hh.ne')\n\nsection SL2Z\n\nvariable {k : ℤ}\n\n/-- An `𝒮ℒ` modular form with vanishing q-expansion constant term vanishes at every cusp. -/\nlemma isZeroAt_of_coeffZero_eq_zero (f : ModularForm 𝒮ℒ k)\n    (h : (qExpansion 1 f).coeff 0 = 0) {c : OnePoint ℝ} (hc : IsCusp c 𝒮ℒ) :\n    c.IsZeroAt f k := by\n  rw [Subgroup.IsArithmetic.isCusp_iff_isCusp_SL2Z] at hc\n  rw [isZeroAt_iff_forall_SL2Z hc]\n  intro γ _\n  rw [show (⇑f ∣[k] γ) = ⇑f from f.slash_action_eq' _ ⟨γ, rfl⟩]\n  exact isZeroAtImInfty_of_valueAtInfty_eq_zero f <| by\n    rwa [← qExpansion_coeff_zero one_pos\n      (ModularFormClass.analyticAt_cuspFunction_zero f one_pos one_mem_strictPeriods_SL)\n      (periodic_comp_ofComplex f one_mem_strictPeriods_SL)]\n\n/-- Build a `CuspForm 𝒮ℒ k` from a `ModularForm 𝒮ℒ k` whose q-expansion has vanishing\nconstant term. The resulting cusp form has the same underlying function. -/\ndef toCuspForm (f : ModularForm 𝒮ℒ k) (h : (qExpansion 1 f).coeff 0 = 0) : CuspForm 𝒮ℒ k :=\n  { f with zero_at_cusps' := isZeroAt_of_coeffZero_eq_zero f h }\n\n@[simp]\nlemma toCuspForm_apply (f : ModularForm 𝒮ℒ k) (h : (qExpansion 1 f).coeff 0 = 0)\n    (z : ℍ) : (toCuspForm f h) z = f z := rfl\n\n/-- For `𝒮ℒ` modular forms, `IsCuspForm` is equivalent to the q-expansion having vanishing\nconstant term. -/\n\nTarget:\nlemma isCuspForm_iff_coeffZero_eq_zero (f : ModularForm 𝒮ℒ k) :\n    IsCuspForm f ↔ (qExpansion 1 f).coeff 0 = 0 :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/ModularForms","family_id":"iscuspform_iff_coeffzero_eq_zero","file_id":"mathlib/Mathlib/NumberTheory/ModularForms/CuspFormSubmodule.lean","sample_id":"0eb67f824175dac229b7646063b2648b75a17b30e9821e1a51528362c79353d3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"abdd5d1c7dac8f0ca69859030dcc532b57d2b6a28ac7caf0a9109fccea1058bd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"229b30587f745dbc42850d319f503873d65fa313ef55e24e07dade3f8ba24db5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5b70195082dfe681d1970751f7a4885c8fdbaa9cac549078cb4a1812112d4d46","source_sha256":"5ec4c4d2769bb0652aea247f81d7ff99e572b3159717c948f413b35f1ff454f9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]\n  apply sdiff_unique\n  · calc\n      x ⊓ y \\ z ⊔ (z ⊓ x ⊔ x \\ y) = (x ⊔ (z ⊓ x ⊔ x \\ y)) ⊓ (y \\ z ⊔ (z ⊓ x ⊔ x \\ y)) := by\n          rw [sup_inf_right]\n      _ = (x ⊔ x ⊓ z ⊔ x \\ y) ⊓ (y \\ z ⊔ (x ⊓ z ⊔ x \\ y)) := by ac_rfl\n      _ = x ⊓ (y \\ z ⊔ (x ⊓ z ⊔ x ⊓ y) ⊔ x \\ y) := by\n          rw [sup_inf_self, sup_sdiff_left, ← sup_assoc, sup_inf_left, sdiff_sup_self',\n            inf_sup_right, sup_comm y, inf_sdiff_sup_right, inf_sup_left x z y]\n      _ = x ⊓ (y \\ z ⊔ (x ⊓ z ⊔ (x ⊓ y ⊔ x \\ y))) := by ac_rfl\n      _ = x := by simp\n  · calc\n      x ⊓ y \\ z ⊓ (z ⊓ x ⊔ x \\ y) = x ⊓ y \\ z ⊓ (z ⊓ x) ⊔ x ⊓ y \\ z ⊓ x \\ y := by rw [inf_sup_left]\n      _ = x ⊓ (y \\ z ⊓ z ⊓ x) ⊔ x ⊓ y \\ z ⊓ x \\ y := by ac_rfl\n      _ = x ⊓ y \\ z ⊓ x \\ y := by rw [inf_sdiff_self_left, bot_inf_eq, inf_bot_eq, bot_sup_eq]\n      _ = x ⊓ (y \\ z ⊓ y) ⊓ x \\ y := by conv_lhs => rw [← inf_sdiff_left]\n      _ = x ⊓ (y \\ z ⊓ (y ⊓ x \\ y)) := by ac_rfl\n      _ = ⊥ := by rw [inf_sdiff_self_right, inf_bot_eq, inf_bot_eq]","hard_negative":true,"metrics":{"chosen_tokens":334,"rejected_tokens":2,"token_jaccard":0.021739,"token_length_ratio":0.005988},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"187a34dd603e3dfa90090f73bc875db7f57892ceaf4f46da61b226a333376914","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.BooleanAlgebra.Defs\npublic import Mathlib.Tactic.GRewrite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Bryan Gin-ge Chen\n-/\n/-!\n# Basic properties of Boolean algebras\n\nThis file provides some basic definitions, functions as well as lemmas for functions and type\nclasses related to Boolean algebras as defined in `Mathlib/Order/BooleanAlgebra/Defs.lean`.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>\n* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]\n* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]\n\n## Tags\n\ngeneralized Boolean algebras, Boolean algebras, lattices, sdiff, compl\n\n-/\n\npublic section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type*} {x y z : α}\n\n/-!\n### Generalized Boolean algebras\n\nSome of the lemmas in this section are from:\n\n* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]\n* <https://ncatlab.org/nlab/show/relative+complement>\n* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>\n\n-/\n\n-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,\n-- however we'd need another type class for lattices with bot, and all the API for that.\nsection GeneralizedBooleanAlgebra\n\nvariable [GeneralizedBooleanAlgebra α]\n\n@[simp]\ntheorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \\ y = x :=\n  GeneralizedBooleanAlgebra.sup_inf_sdiff _ _\n\n@[simp]\ntheorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \\ y = ⊥ :=\n  GeneralizedBooleanAlgebra.inf_inf_sdiff _ _\n\n@[simp]\ntheorem sup_sdiff_inf (x y : α) : x \\ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]\n\n@[simp]\ntheorem inf_sdiff_inf (x y : α) : x \\ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α where\n  __ := GeneralizedBooleanAlgebra.toBot\n  bot_le a := by\n    rw [← inf_inf_sdiff a a, inf_assoc]\n    exact inf_le_left\n\ntheorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \\ y) :=\n  disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le\n\n-- TODO: in distributive lattices, relative complements are unique when they exist\ntheorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \\ y = z := by\n  conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]\n  rw [sup_comm] at s\n  conv_rhs at i => rw [← inf_inf_sdiff x y, inf_comm]\n  rw [inf_comm] at i\n  exact (eq_of_inf_eq_sup_eq i s).symm\n\n-- Use `sdiff_le`\nprivate theorem sdiff_le' : x \\ y ≤ x :=\n  calc\n    x \\ y ≤ x ⊓ y ⊔ x \\ y := le_sup_right\n    _ = x := sup_inf_sdiff x y\n\nset_option backward.privateInPublic true in\n-- Use `sdiff_sup_self`\nprivate theorem sdiff_sup_self' : y \\ x ⊔ x = y ⊔ x :=\n  calc\n    y \\ x ⊔ x = y \\ x ⊔ (x ⊔ x ⊓ y) := by rw [sup_inf_self]\n    _ = y ⊓ x ⊔ y \\ x ⊔ x := by ac_rfl\n    _ = y ⊔ x := by rw [sup_inf_sdiff]\n\n@[simp]\ntheorem sdiff_inf_sdiff : x \\ y ⊓ y \\ x = ⊥ :=\n  Eq.symm <|\n    calc\n      ⊥ = x ⊓ (y ⊓ x ⊔ y \\ x) ⊓ x \\ y := by rw [← inf_inf_sdiff, sup_inf_sdiff]\n      _ = (x ⊓ (y ⊓ x) ⊔ x ⊓ y \\ x) ⊓ x \\ y := by rw [inf_sup_left]\n      _ = (y ⊓ (x ⊓ x) ⊔ x ⊓ y \\ x) ⊓ x \\ y := by ac_rfl\n      _ = x ⊓ y \\ x ⊓ x \\ y := by\n          rw [inf_idem, inf_sup_right, ← inf_comm x y, inf_inf_sdiff, bot_sup_eq]\n      _ = x ⊓ x \\ y ⊓ y \\ x := by ac_rfl\n      _ = x \\ y ⊓ y \\ x := by rw [inf_of_le_right sdiff_le']\n\ntheorem disjoint_sdiff_sdiff : Disjoint (x \\ y) (y \\ x) :=\n  disjoint_iff_inf_le.mpr sdiff_inf_sdiff.le\n\n@[simp]\ntheorem inf_sdiff_self_right : x ⊓ y \\ x = ⊥ :=\n  calc\n    x ⊓ y \\ x = (x ⊓ y ⊔ x \\ y) ⊓ y \\ x := by rw [sup_inf_sdiff]\n    _ = ⊥ := by rw [inf_sup_right, inf_comm x y, inf_inf_sdiff, sdiff_inf_sdiff, bot_sup_eq]\n\n@[simp]\ntheorem inf_sdiff_self_left : y \\ x ⊓ x = ⊥ := by rw [inf_comm, inf_sdiff_self_right]\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n-- see Note [lower instance priority]\ninstance (priority := 100) GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra :\n    GeneralizedCoheytingAlgebra α where\n  __ := ‹GeneralizedBooleanAlgebra α›\n  __ := GeneralizedBooleanAlgebra.toOrderBot\n  sdiff := (· \\ ·)\n  sdiff_le_iff y x z :=\n    ⟨fun h =>\n      le_of_inf_le_sup_le\n        (le_of_eq\n          (by grind [sdiff_le', inf_of_le_right, inf_eq_right, inf_sdiff_self_right, bot_sup_eq,\n            inf_sup_right]))\n        (calc\n          y ⊔ y \\ x ≤ y \\ x ⊔ x ⊔ z := by\n            grind [sup_of_le_left, sdiff_le', le_sup_left, sdiff_sup_self']\n          _ = x ⊔ z ⊔ y \\ x := by ac_rfl),\n      fun h => le_of_inf_le_sup_le (inf_sdiff_self_left.trans_le bot_le) (calc\n        y \\ x ⊔ x = y ⊔ x := sdiff_sup_self'\n        _ ≤ x ⊔ z ⊔ x := sup_le_sup_right h x\n        _ ≤ z ⊔ x := by rw [sup_assoc, sup_comm, sup_assoc, sup_idem])⟩\n\ntheorem disjoint_sdiff_self_left : Disjoint (y \\ x) x :=\n  disjoint_iff_inf_le.mpr inf_sdiff_self_left.le\n\ntheorem disjoint_sdiff_self_right : Disjoint x (y \\ x) :=\n  disjoint_iff_inf_le.mpr inf_sdiff_self_right.le\n\nlemma le_sdiff : x ≤ y \\ z ↔ x ≤ y ∧ Disjoint x z :=\n  ⟨fun h ↦ ⟨h.trans sdiff_le, disjoint_sdiff_self_left.mono_left h⟩, fun h ↦\n    by rw [← h.2.sdiff_eq_left]; exact sdiff_le_sdiff_right h.1⟩\n\n@[simp] lemma sdiff_eq_left : x \\ y = x ↔ Disjoint x y :=\n  ⟨fun h ↦ disjoint_sdiff_self_left.mono_left h.ge, Disjoint.sdiff_eq_left⟩\n\n/- TODO: we could make an alternative constructor for `GeneralizedBooleanAlgebra` using\n`Disjoint x (y \\ x)` and `x ⊔ (y \\ x) = y` as axioms. -/\ntheorem Disjoint.sdiff_eq_of_sup_eq (hi : Disjoint x z) (hs : x ⊔ z = y) : y \\ x = z :=\n  have h : y ⊓ x = x := inf_eq_right.2 <| le_sup_left.trans hs.le\n  sdiff_unique (by rw [h, hs]) (by rw [h, hi.eq_bot])\n\nprotected theorem Disjoint.sdiff_unique (hd : Disjoint x z) (hz : z ≤ y) (hs : y ≤ x ⊔ z) :\n    y \\ x = z :=\n  sdiff_unique\n    (by\n      rw [← inf_eq_right] at hs\n      rwa [sup_inf_right, inf_sup_right, sup_comm x, inf_sup_self, inf_comm, sup_comm z,\n        hs, sup_eq_left])\n    (by rw [inf_assoc, hd.eq_bot, inf_bot_eq])\n\n-- cf. `IsCompl.disjoint_left_iff` and `IsCompl.disjoint_right_iff`\ntheorem disjoint_sdiff_iff_le (hz : z ≤ y) (hx : x ≤ y) : Disjoint z (y \\ x) ↔ z ≤ x :=\n  ⟨fun H =>\n    le_of_inf_le_sup_le (le_trans H.le_bot bot_le)\n      (by\n        rw [sup_sdiff_cancel_right hx]\n        grw [sdiff_le]\n        rw [sup_eq_right.2 hz]),\n    fun H => disjoint_sdiff_self_right.mono_left H⟩\n\n-- cf. `IsCompl.le_left_iff` and `IsCompl.le_right_iff`\ntheorem le_iff_disjoint_sdiff (hz : z ≤ y) (hx : x ≤ y) : z ≤ x ↔ Disjoint z (y \\ x) :=\n  (disjoint_sdiff_iff_le hz hx).symm\n\n-- cf. `IsCompl.inf_left_eq_bot_iff` and `IsCompl.inf_right_eq_bot_iff`\ntheorem inf_sdiff_eq_bot_iff (hz : z ≤ y) (hx : x ≤ y) : z ⊓ y \\ x = ⊥ ↔ z ≤ x := by\n  rw [← disjoint_iff]\n  exact disjoint_sdiff_iff_le hz hx\n\n-- cf. `IsCompl.left_le_iff` and `IsCompl.right_le_iff`\ntheorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \\ x :=\n  ⟨fun H => (sup_sdiff_cancel' H hz).symm,\n    fun H => by\n    conv_lhs at H => rw [← sup_sdiff_cancel_right hx]\n    refine le_of_inf_le_sup_le ?_ H.le\n    rw [inf_sdiff_self_right]\n    exact bot_le⟩\n\n-- cf. `IsCompl.sup_inf`\ntheorem sdiff_sup : y \\ (x ⊔ z) = y \\ x ⊓ y \\ z :=\n  sdiff_unique\n    (calc\n      y ⊓ (x ⊔ z) ⊔ y \\ x ⊓ y \\ z = (y ⊓ x ⊔ y ⊓ z ⊔ y \\ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \\ z) := by\n          rw [sup_inf_left, inf_sup_left y]\n      _ = (y ⊓ z ⊔ (y ⊓ x ⊔ y \\ x)) ⊓ (y ⊓ x ⊔ (y ⊓ z ⊔ y \\ z)) := by ac_rfl\n      _ = (y ⊓ z ⊔ y) ⊓ (y ⊓ x ⊔ y) := by rw [sup_inf_sdiff, sup_inf_sdiff]\n      _ = (y ⊔ y ⊓ z) ⊓ (y ⊔ y ⊓ x) := by ac_rfl\n      _ = y := by rw [sup_inf_self, sup_inf_self, inf_idem])\n    (calc\n      y ⊓ (x ⊔ z) ⊓ (y \\ x ⊓ y \\ z) = y ⊓ x ⊓ (y \\ x ⊓ y \\ z) ⊔ y ⊓ z ⊓ (y \\ x ⊓ y \\ z) := by\n          rw [inf_sup_left, inf_sup_right]\n      _ = y ⊓ x ⊓ y \\ x ⊓ y \\ z ⊔ y \\ x ⊓ (y \\ z ⊓ (y ⊓ z)) := by ac_rfl\n      _ = ⊥ := by simp)\n\ntheorem sdiff_eq_sdiff_iff_inf_eq_inf : y \\ x = y \\ z ↔ y ⊓ x = y ⊓ z :=\n  ⟨fun h => eq_of_inf_eq_sup_eq (a := y \\ x) (by rw [inf_inf_sdiff, h, inf_inf_sdiff])\n    (by rw [sup_inf_sdiff, h, sup_inf_sdiff]),\n    fun h => by rw [← sdiff_inf_self_right, ← sdiff_inf_self_right z y, inf_comm, h, inf_comm]⟩\n\ntheorem sdiff_eq_self_iff_disjoint : x \\ y = x ↔ Disjoint y x := sdiff_eq_left.trans disjoint_comm\n\ntheorem sdiff_lt (hx : y ≤ x) (hy : y ≠ ⊥) : x \\ y < x := by\n  refine sdiff_le.lt_of_ne fun h => hy ?_\n  rw [sdiff_eq_left, disjoint_iff] at h\n  rw [← h, inf_eq_right.mpr hx]\n\ntheorem sdiff_lt_left : x \\ y < x ↔ ¬ Disjoint y x := by\n  rw [lt_iff_le_and_ne, Ne, sdiff_eq_self_iff_disjoint, and_iff_right sdiff_le]\n\n@[simp]\ntheorem le_sdiff_right : x ≤ y \\ x ↔ x = ⊥ :=\n  ⟨fun h => disjoint_self.1 (disjoint_sdiff_self_right.mono_right h), fun h => h.le.trans bot_le⟩\n\n@[simp] lemma sdiff_eq_right : x \\ y = y ↔ x = ⊥ ∧ y = ⊥ := by\n  rw [disjoint_sdiff_self_left.eq_iff]; simp_all\n\nlemma sdiff_ne_right : x \\ y ≠ y ↔ x ≠ ⊥ ∨ y ≠ ⊥ := sdiff_eq_right.not.trans not_and_or\n\ntheorem sdiff_lt_sdiff_right (h : x < y) (hz : z ≤ x) : x \\ z < y \\ z :=\n  (sdiff_le_sdiff_right h.le).lt_of_not_ge\n    fun h' => h.not_ge <| le_sdiff_sup.trans <| sup_le_of_le_sdiff_right h' hz\n\ntheorem sup_inf_inf_sdiff : x ⊓ y ⊓ z ⊔ y \\ z = x ⊓ y ⊔ y \\ z := by\n  rw [inf_assoc, sup_inf_right, sup_inf_sdiff, inf_sup_right, inf_sdiff_left]\n\nTarget:\ntheorem sdiff_sdiff_right : x \\ (y \\ z) = x \\ y ⊔ x ⊓ y ⊓ z :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_5b70195082df","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b3624a21daba327ae92c5e75facfbd8decf99068f26fa435e8d144b09e10d957","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/BooleanAlgebra","family_id":"sdiff_sdiff_right","file_id":"mathlib/Mathlib/Order/BooleanAlgebra/Basic.lean","sample_id":"5b70195082dfe681d1970751f7a4885c8fdbaa9cac549078cb4a1812112d4d46"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c30b62521a4c15bdccd0e80ec4d2a61314f23daa4911bb4ad6a467ce05c1c053","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"88c76701a2939d9e21c6541f82d98c2885e23683bed56dfcfe631a579dcb74b2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ff91d6fc0a99e05c49e2fa58e7af54a13ccacbe396be47e3ff0015e67bd86101","source_sha256":"0155ab8151a9daebd303562a6d33da3f5df4ece30ebde16cdbf90ade81f9770b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h I hIX ↦ ?_, fun h ↦ ext_indep rfl fun I (hI : I ⊆ X) ↦ ?_⟩\n  · rw [← and_iff_left (a := (M.Indep I)) hIX, ← and_iff_left (a := (M'.Indep I)) hIX,\n      ← restrict_indep_iff, h, restrict_indep_iff]\n  rw [restrict_indep_iff, and_iff_left hI, restrict_indep_iff, and_iff_left hI, h _ hI]","hard_negative":true,"metrics":{"chosen_tokens":83,"rejected_tokens":5,"token_jaccard":0.055556,"token_length_ratio":0.060241},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"18acb6283a9a8bc8c34b90b4872bfac179477d7368a2a006c3cbf8d32846b582","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Matroid.Dual\n\nNamespace:\nMatroid\n\nLocal context:\n/-\nCopyright (c) 2023 Peter Nelson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Peter Nelson\n-/\n/-!\n# Matroid Restriction\n\nGiven `M : Matroid α` and `R : Set α`, the independent sets of `M` that are contained in `R`\nare the independent sets of another matroid `M ↾ R` with ground set `R`,\ncalled the 'restriction' of `M` to `R`.\nFor `I ⊆ R ⊆ M.E`, `I` is a basis of `R` in `M` if and only if `I` is a base\nof the restriction `M ↾ R`, so this construction relates `Matroid.IsBasis` to `Matroid.IsBase`.\n\nIf `N M : Matroid α` satisfy `N = M ↾ R` for some `R ⊆ M.E`,\nthen we call `N` a 'restriction of `M`', and write `N ≤r M`. This is a partial order.\n\nThis file proves that the restriction is a matroid and that the `≤r` order is a partial order,\nand gives related API.\nIt also proves some `Matroid.IsBasis` analogues of `Matroid.IsBase` lemmas that,\nwhile they could be stated in `Data.Matroid.Basic`,\nare hard to prove without `Matroid.restrict` API.\n\n## Main Definitions\n\n* `M.restrict R`, written `M ↾ R`, is the restriction of `M : Matroid α` to `R : Set α`: i.e.\n  the matroid with ground set `R` whose independent sets are the `M`-independent subsets of `R`.\n\n* `Matroid.Restriction N M`, written `N ≤r M`, means that `N = M ↾ R` for some `R ⊆ M.E`.\n\n* `Matroid.IsStrictRestriction N M`, written `N <r M`, means that `N = M ↾ R` for some `R ⊂ M.E`.\n\n* `Matroidᵣ α` is a type synonym for `Matroid α`, equipped with the `PartialOrder` `≤r`.\n\n## Implementation Notes\n\nSince `R` and `M.E` are both terms in `Set α`, to define the restriction `M ↾ R`,\nwe need to either insist that `R ⊆ M.E`, or to say what happens when `R` contains the junk\noutside `M.E`.\n\nIt turns out that `R ⊆ M.E` is just an unnecessary hypothesis; if we say the restriction\n`M ↾ R` has ground set `R` and its independent sets are the `M`-independent subsets of `R`,\nwe always get a matroid, in which the elements of `R \\ M.E` aren't in any independent sets.\nWe could instead define this matroid to always be 'smaller' than `M` by setting\n`(M ↾ R).E := R ∩ M.E`, but this is worse definitionally, and more generally less convenient.\n\nThis makes it possible to actually restrict a matroid 'upwards'; for instance, if `M : Matroid α`\nsatisfies `M.E = ∅`, then `M ↾ Set.univ` is the matroid on `α` whose ground set is all of `α`,\nwhere the empty set is the only independent set.\n(In general, elements of `R \\ M.E` are all 'loops' of the matroid `M ↾ R`;\nsee `Matroid.loops` and `Matroid.restrict_loops_eq'` for a precise version of this statement.)\nThis is mathematically strange, but is useful for API building.\n\nThe cost of allowing a restriction of `M` to be 'bigger' than `M` itself is that\nthe statement `M ↾ R ≤r M` is only true with the hypothesis `R ⊆ M.E`\n(at least, if we want `≤r` to be a partial order).\nBut this isn't too inconvenient in practice. Indeed `(· ⊆ M.E)` proofs\ncan often be automatically provided by `aesop_mat`.\n\nWe define the restriction order `≤r` to give a `PartialOrder` instance on the type synonym\n`Matroidᵣ α` rather than `Matroid α` itself, because the `PartialOrder (Matroid α)` instance is\nreserved for the more mathematically important 'minor' order; see `Matroid.IsMinor`.\n-/\n\n@[expose] public section\n\nassert_not_exists Field\n\nopen Set\n\nnamespace Matroid\n\nvariable {α : Type*} {M : Matroid α} {R I X Y : Set α}\n\nsection restrict\n\n/-- The `IndepMatroid` whose independent sets are the independent subsets of `R`. -/\n@[simps] def restrictIndepMatroid (M : Matroid α) (R : Set α) : IndepMatroid α where\n  E := R\n  Indep I := M.Indep I ∧ I ⊆ R\n  indep_empty := ⟨M.empty_indep, empty_subset _⟩\n  indep_subset := fun _ _ h hIJ ↦ ⟨h.1.subset hIJ, hIJ.trans h.2⟩\n  indep_aug := by\n    rintro I I' ⟨hI, hIY⟩ (hIn : ¬ M.IsBasis' I R) (hI' : M.IsBasis' I' R)\n    rw [isBasis'_iff_isBasis_inter_ground] at hIn hI'\n    obtain ⟨B', hB', rfl⟩ := hI'.exists_isBase\n    obtain ⟨B, hB, hIB, hBIB'⟩ := hI.exists_isBase_subset_union_isBase hB'\n    rw [hB'.inter_isBasis_iff_compl_inter_isBasis_dual, sdiff_inter_sdiff] at hI'\n    have hss : M.E \\ (B' ∪ (R ∩ M.E)) ⊆ M.E \\ (B ∪ (R ∩ M.E)) := by\n      apply sdiff_subset_sdiff_right\n      rw [union_subset_iff, and_iff_left subset_union_right, union_comm]\n      exact hBIB'.trans (union_subset_union_left _ (subset_inter hIY hI.subset_ground))\n    have hi : M✶.Indep (M.E \\ (B ∪ (R ∩ M.E))) := by\n      rw [dual_indep_iff_exists]\n      exact ⟨B, hB, disjoint_of_subset_right subset_union_left disjoint_sdiff_left⟩\n    have h_eq := hI'.eq_of_subset_indep hi hss\n      (sdiff_subset_sdiff_right subset_union_right)\n    rw [h_eq, ← sdiff_inter_sdiff, ← hB.inter_isBasis_iff_compl_inter_isBasis_dual] at hI'\n    obtain ⟨J, hJ, hIJ⟩ := hI.subset_isBasis_of_subset\n      (subset_inter hIB (subset_inter hIY hI.subset_ground))\n    obtain rfl := hI'.indep.eq_of_isBasis hJ\n    have hIJ' : I ⊂ B ∩ (R ∩ M.E) := hIJ.ssubset_of_ne (fun he ↦ hIn (by rwa [he]))\n    obtain ⟨e, he⟩ := exists_of_ssubset hIJ'\n    exact ⟨e, ⟨⟨(hBIB' he.1.1).elim (fun h ↦ (he.2 h).elim) id,he.1.2⟩, he.2⟩,\n      hI'.indep.subset (insert_subset he.1 hIJ), insert_subset he.1.2.1 hIY⟩\n  indep_maximal := by\n    rintro A hAR I ⟨hI, _⟩ hIA\n    obtain ⟨J, hJ, hIJ⟩ := hI.subset_isBasis'_of_subset hIA\n    use J\n    simp only [hIJ, and_assoc, maximal_subset_iff, hJ.indep, hJ.subset, and_imp, true_and,\n      hJ.subset.trans hAR]\n    exact fun K hK _ hKA hJK ↦ hJ.eq_of_subset_indep hK hJK hKA\n  subset_ground _ := And.right\n\n/-- Change the ground set of a matroid to some `R : Set α`. The independent sets of the restriction\nare the independent subsets of the new ground set. Most commonly used when `R ⊆ M.E`,\nbut it is convenient not to require this. The elements of `R \\ M.E` become 'loops'. -/\ndef restrict (M : Matroid α) (R : Set α) : Matroid α := (M.restrictIndepMatroid R).matroid\n\n/-- `M ↾ R` means `M.restrict R`. -/\nscoped infixl:65 \" ↾ \" => Matroid.restrict\n\n@[simp] theorem restrict_indep_iff : (M ↾ R).Indep I ↔ M.Indep I ∧ I ⊆ R := Iff.rfl\n\ntheorem Indep.indep_restrict_of_subset (h : M.Indep I) (hIR : I ⊆ R) : (M ↾ R).Indep I :=\n  restrict_indep_iff.mpr ⟨h,hIR⟩\n\ntheorem Indep.of_restrict (hI : (M ↾ R).Indep I) : M.Indep I :=\n  (restrict_indep_iff.1 hI).1\n\n@[simp] theorem restrict_ground_eq : (M ↾ R).E = R := rfl\n\ntheorem restrict_finite {R : Set α} (hR : R.Finite) : (M ↾ R).Finite :=\n  ⟨hR⟩\n\n@[simp] theorem restrict_dep_iff : (M ↾ R).Dep X ↔ ¬ M.Indep X ∧ X ⊆ R := by\n  rw [Dep, restrict_indep_iff, restrict_ground_eq]; tauto\n\n@[simp] theorem restrict_ground_eq_self (M : Matroid α) : (M ↾ M.E) = M := by\n  refine ext_indep rfl ?_; simp_all\n\ntheorem restrict_restrict_eq {R₁ R₂ : Set α} (M : Matroid α) (hR : R₂ ⊆ R₁) :\n    (M ↾ R₁) ↾ R₂ = M ↾ R₂ := by\n  refine ext_indep rfl ?_\n  simp only [restrict_ground_eq, restrict_indep_iff, and_congr_left_iff, and_iff_left_iff_imp]\n  exact fun _ h _ _ ↦ h.trans hR\n\n@[simp] theorem restrict_idem (M : Matroid α) (R : Set α) : M ↾ R ↾ R = M ↾ R := by\n  rw [M.restrict_restrict_eq Subset.rfl]\n\n@[simp] theorem isBase_restrict_iff (hX : X ⊆ M.E := by aesop_mat) :\n    (M ↾ X).IsBase I ↔ M.IsBasis I X := by\n  simp_rw [isBase_iff_maximal_indep, IsBasis, and_iff_left hX, maximal_iff, restrict_indep_iff]\n\ntheorem isBase_restrict_iff' : (M ↾ X).IsBase I ↔ M.IsBasis' I X := by\n  simp_rw [isBase_iff_maximal_indep, IsBasis', maximal_iff, restrict_indep_iff]\n\ntheorem IsBasis'.isBase_restrict (hI : M.IsBasis' I X) : (M ↾ X).IsBase I :=\n  isBase_restrict_iff'.1 hI\n\ntheorem IsBasis.restrict_isBase (h : M.IsBasis I X) : (M ↾ X).IsBase I :=\n  (isBase_restrict_iff h.subset_ground).2 h\n\ninstance restrict_rankFinite [M.RankFinite] (R : Set α) : (M ↾ R).RankFinite :=\n  let ⟨_, hB⟩ := (M ↾ R).exists_isBase\n  hB.rankFinite_of_finite (hB.indep.of_restrict.finite)\n\ninstance restrict_finitary [Finitary M] (R : Set α) : Finitary (M ↾ R) := by\n  refine ⟨fun I hI ↦ ?_⟩\n  simp only [restrict_indep_iff] at *\n  rw [indep_iff_forall_finite_subset_indep]\n  exact ⟨fun J hJ hJfin ↦ (hI J hJ hJfin).1,\n    fun e heI ↦ singleton_subset_iff.1 (hI _ (by simpa) (toFinite _)).2⟩\n\n@[simp] theorem IsBasis.isBase_restrict (h : M.IsBasis I X) : (M ↾ X).IsBase I :=\n  (isBase_restrict_iff h.subset_ground).mpr h\n\ntheorem IsBasis.isBasis_restrict_of_subset (hI : M.IsBasis I X) (hXY : X ⊆ Y) :\n    (M ↾ Y).IsBasis I X := by\n  rwa [← isBase_restrict_iff, M.restrict_restrict_eq hXY, isBase_restrict_iff]\n\ntheorem isBasis'_restrict_iff : (M ↾ R).IsBasis' I X ↔ M.IsBasis' I (X ∩ R) ∧ I ⊆ R := by\n  simp_rw [IsBasis', maximal_iff, restrict_indep_iff, subset_inter_iff, and_imp]\n  tauto\n\ntheorem isBasis_restrict_iff' : (M ↾ R).IsBasis I X ↔ M.IsBasis I (X ∩ M.E) ∧ X ⊆ R := by\n  rw [isBasis_iff_isBasis'_subset_ground, isBasis'_restrict_iff, restrict_ground_eq,\n    and_congr_left_iff, ← isBasis'_iff_isBasis_inter_ground]\n  intro hXR\n  rw [inter_eq_self_of_subset_left hXR, and_iff_left_iff_imp]\n  exact fun h ↦ h.subset.trans hXR\n\ntheorem isBasis_restrict_iff (hR : R ⊆ M.E := by aesop_mat) :\n    (M ↾ R).IsBasis I X ↔ M.IsBasis I X ∧ X ⊆ R := by\n  rw [isBasis_restrict_iff', and_congr_left_iff]\n  intro hXR\n  rw [← isBasis'_iff_isBasis_inter_ground, isBasis'_iff_isBasis]\n\nlemma isBasis'_iff_isBasis_restrict_univ : M.IsBasis' I X ↔ (M ↾ univ).IsBasis I X := by\n  rw [isBasis_restrict_iff', isBasis'_iff_isBasis_inter_ground, and_iff_left (subset_univ _)]\n\nTarget:\ntheorem restrict_eq_restrict_iff (M M' : Matroid α) (X : Set α) :\n    M ↾ X = M' ↾ X ↔ ∀ I, I ⊆ X → (M.Indep I ↔ M'.Indep I) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_ff91d6fc0a99","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"98195a6ebe3fbaae305452dab65c75ec48d1f7928a5a8ac7678037b2b7f61276","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Matroid","family_id":"restrict_eq_restrict_iff","file_id":"mathlib/Mathlib/Combinatorics/Matroid/Minor/Restrict.lean","sample_id":"ff91d6fc0a99e05c49e2fa58e7af54a13ccacbe396be47e3ff0015e67bd86101"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a6d320117b4f0031ed5a5af663cf8c7397e183bff5fdfb69cb2aa2ba76b6c4c2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"40cd2f9260990c60dc526a39078b883362332866caba5970307c3bf8bf510d9b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a81f25208ce01c9c1e94e1a7e16ac4440ac59daf394e99a5ec3de20f35dc63a8","source_sha256":"22d84e531512fb453c5c22eb12c263f109619421771ae337215e1ec65fb0d46f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← mul_right_inj (x : M), mul_inv_cancel, mul_leftInvEquiv_symm]","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":2,"token_jaccard":0.066667,"token_length_ratio":0.133333},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"18f15646ec5716bd8bc6471ff21fea7d2c095cf9bb7bcec128646ec4dbd5cbb1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Submonoid.Pointwise\n\nNamespace:\nSubmonoid\n\nLocal context:\n/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Submonoid of inverses\n\nGiven a submonoid `N` of a monoid `M`, we define the submonoid `N.leftInv` as the submonoid of\nleft inverses of `N`. When `M` is commutative, we may define `fromCommLeftInv : N.leftInv →* N`\nsince the inverses are unique. When `N ≤ IsUnit.Submonoid M`, this is precisely\nthe pointwise inverse of `N`, and we may define `leftInvEquiv : S.leftInv ≃* S`.\n\nFor the pointwise inverse of submonoids of groups, please refer to the file\n`Mathlib/Algebra/Group/Submonoid/Pointwise.lean`.\n\n`N.leftInv` is distinct from `N.units`, which is the subgroup of `Mˣ` containing all units that are\nin `N`. See the implementation notes of `Mathlib/Algebra/Group/Submonoid/Units.lean` for more\ndetails on related constructions.\n\n## TODO\n\nDefine the submonoid of right inverses and two-sided inverses.\nSee the comments of https://github.com/leanprover-community/mathlib4/pull/10679 for a possible\nimplementation.\n-/\n\n@[expose] public section\n\n\nvariable {M : Type*}\n\nnamespace Submonoid\n\n@[to_additive]\nnoncomputable instance [Monoid M] : Group (IsUnit.submonoid M) :=\n  { (inferInstance : Monoid (IsUnit.submonoid M)) with\n    inv := fun x ↦ ⟨x.prop.unit⁻¹.val, x.prop.unit⁻¹.isUnit⟩\n    inv_mul_cancel := fun x ↦\n      Subtype.ext ((Units.val_mul x.prop.unit⁻¹ _).trans x.prop.unit.inv_val) }\n\n@[to_additive]\nnoncomputable instance [CommMonoid M] : CommGroup (IsUnit.submonoid M) :=\n  { (inferInstance : Group (IsUnit.submonoid M)) with\n    mul_comm := fun a b ↦ by convert! mul_comm a b }\n\n@[to_additive]\ntheorem _root_.IsUnit.submonoid.coe_inv [Monoid M] (x : IsUnit.submonoid M) :\n    ↑x⁻¹ = (↑x.prop.unit⁻¹ : M) :=\n  rfl\n\n@[deprecated (since := \"2026-05-24\")]\nalias _root_.AddSubmonoid.IsUnit.Submonoid.coe_neg := IsAddUnit.addSubmonoid.coe_neg\n@[to_additive existing, deprecated (since := \"2026-05-24\")]\nalias IsUnit.Submonoid.coe_inv := IsUnit.submonoid.coe_inv\n\nsection Monoid\n\nvariable [Monoid M] (S : Submonoid M)\n\n/-- `S.leftInv` is the submonoid containing all the left inverses of `S`. -/\n@[to_additive\n/-- `S.leftNeg` is the additive submonoid containing all the left additive inverses of `S`. -/]\ndef leftInv : Submonoid M where\n  carrier := { x : M | ∃ y : S, x * y = 1 }\n  one_mem' := ⟨1, mul_one 1⟩\n  mul_mem' := fun {a} _b ⟨a', ha⟩ ⟨b', hb⟩ ↦\n    ⟨b' * a', by simp only [coe_mul, ← mul_assoc, mul_assoc a, hb, mul_one, ha]⟩\n\n@[to_additive]\ntheorem leftInv_leftInv_le : S.leftInv.leftInv ≤ S := by\n  rintro x ⟨⟨y, z, h₁⟩, h₂ : x * y = 1⟩\n  convert! z.prop\n  rw [← mul_one x, ← h₁, ← mul_assoc, h₂, one_mul]\n\n@[to_additive]\ntheorem unit_mem_leftInv (x : Mˣ) (hx : (x : M) ∈ S) : ((x⁻¹ :) : M) ∈ S.leftInv :=\n  ⟨⟨x, hx⟩, x.inv_val⟩\n\n@[to_additive]\ntheorem leftInv_leftInv_eq (hS : S ≤ IsUnit.submonoid M) : S.leftInv.leftInv = S := by\n  refine le_antisymm S.leftInv_leftInv_le ?_\n  intro x hx\n  have : x = ((hS hx).unit⁻¹⁻¹ : Mˣ) := by\n    rw [inv_inv (hS hx).unit]\n    rfl\n  rw [this]\n  exact S.leftInv.unit_mem_leftInv _ (S.unit_mem_leftInv _ hx)\n\n/-- The function from `S.leftInv` to `S` sending an element to its right inverse in `S`.\nThis is a `MonoidHom` when `M` is commutative. -/\n@[to_additive\n/-- The function from `S.leftAdd` to `S` sending an element to its right additive\ninverse in `S`. This is an `AddMonoidHom` when `M` is commutative. -/]\nnoncomputable def fromLeftInv : S.leftInv → S := fun x ↦ x.prop.choose\n\n@[to_additive (attr := simp)]\ntheorem mul_fromLeftInv (x : S.leftInv) : (x : M) * S.fromLeftInv x = 1 :=\n  x.prop.choose_spec\n\n@[to_additive (attr := simp)]\ntheorem fromLeftInv_one : S.fromLeftInv 1 = 1 :=\n  (one_mul _).symm.trans (Subtype.ext <| S.mul_fromLeftInv 1)\n\nend Monoid\n\nsection CommMonoid\n\nvariable [CommMonoid M] (S : Submonoid M)\n\n@[to_additive (attr := simp)]\ntheorem fromLeftInv_mul (x : S.leftInv) : (S.fromLeftInv x : M) * x = 1 := by\n  rw [mul_comm, mul_fromLeftInv]\n\n@[to_additive]\ntheorem leftInv_le_isUnit : S.leftInv ≤ IsUnit.submonoid M := fun x ⟨y, hx⟩ ↦\n  ⟨⟨x, y, hx, mul_comm x y ▸ hx⟩, rfl⟩\n\n@[to_additive]\ntheorem fromLeftInv_eq_iff (a : S.leftInv) (b : M) :\n    (S.fromLeftInv a : M) = b ↔ (a : M) * b = 1 := by\n  rw [← IsUnit.mul_right_inj (leftInv_le_isUnit _ a.prop), S.mul_fromLeftInv, eq_comm]\n\n/-- The `MonoidHom` from `S.leftInv` to `S` sending an element to its right inverse in `S`. -/\n@[to_additive (attr := simps) /-- The `AddMonoidHom` from `S.leftNeg` to `S` sending an element to\nits right additive inverse in  `S`. -/]\nnoncomputable def fromCommLeftInv : S.leftInv →* S where\n  toFun := S.fromLeftInv\n  map_one' := S.fromLeftInv_one\n  map_mul' x y :=\n    Subtype.ext <| by\n      rw [fromLeftInv_eq_iff, mul_comm x, Submonoid.coe_mul, Submonoid.coe_mul, mul_assoc, ←\n        mul_assoc (x : M), mul_fromLeftInv, one_mul, mul_fromLeftInv]\n\nvariable (hS : S ≤ IsUnit.submonoid M)\n\n/-- The submonoid of pointwise inverse of `S` is `MulEquiv` to `S`. -/\n@[to_additive (attr := simps apply) /-- The additive submonoid of pointwise additive inverse of `S`\nis `AddEquiv` to `S`. -/]\nnoncomputable def leftInvEquiv : S.leftInv ≃* S :=\n  { S.fromCommLeftInv with\n    invFun := fun x ↦ ⟨↑(hS x.2).unit⁻¹, x, by simp⟩\n    left_inv := by\n      intro x\n      ext\n      simp [← Units.mul_eq_one_iff_inv_eq]\n    right_inv := by\n      rintro ⟨x, hx⟩\n      ext\n      simp [fromLeftInv_eq_iff] }\n\n@[to_additive (attr := simp)]\ntheorem fromLeftInv_leftInvEquiv_symm (x : S) : S.fromLeftInv ((S.leftInvEquiv hS).symm x) = x :=\n  (S.leftInvEquiv hS).right_inv x\n\n@[to_additive (attr := simp)]\ntheorem leftInvEquiv_symm_fromLeftInv (x : S.leftInv) :\n    (S.leftInvEquiv hS).symm (S.fromLeftInv x) = x :=\n  (S.leftInvEquiv hS).left_inv x\n\n@[to_additive]\ntheorem leftInvEquiv_mul (x : S.leftInv) : (S.leftInvEquiv hS x : M) * x = 1 := by\n  simpa only [leftInvEquiv_apply, fromCommLeftInv] using fromLeftInv_mul S x\n\n@[to_additive]\ntheorem mul_leftInvEquiv (x : S.leftInv) : (x : M) * S.leftInvEquiv hS x = 1 := by\n  simp only [leftInvEquiv_apply, fromCommLeftInv, mul_fromLeftInv]\n\n@[to_additive (attr := simp)]\ntheorem leftInvEquiv_symm_mul (x : S) : ((S.leftInvEquiv hS).symm x : M) * x = 1 := by\n  convert! S.mul_leftInvEquiv hS ((S.leftInvEquiv hS).symm x)\n  simp\n\n@[to_additive (attr := simp)]\ntheorem mul_leftInvEquiv_symm (x : S) : (x : M) * (S.leftInvEquiv hS).symm x = 1 := by\n  convert! S.leftInvEquiv_mul hS ((S.leftInvEquiv hS).symm x)\n  simp\n\nend CommMonoid\n\nsection Group\n\nvariable [Group M] (S : Submonoid M)\n\nopen scoped Pointwise\n\n@[to_additive]\ntheorem leftInv_eq_inv : S.leftInv = S⁻¹ :=\n  Submonoid.ext fun _ ↦\n    ⟨fun h ↦ Submonoid.mem_inv.mpr ((inv_eq_of_mul_eq_one_right h.choose_spec).symm ▸\n      h.choose.prop),\n      fun h ↦ ⟨⟨_, h⟩, mul_inv_cancel _⟩⟩\n\n@[to_additive (attr := simp)]\ntheorem fromLeftInv_eq_inv (x : S.leftInv) : (S.fromLeftInv x : M) = (x : M)⁻¹ := by\n  rw [← mul_right_inj (x : M), mul_inv_cancel, mul_fromLeftInv]\n\nend Group\n\nsection CommGroup\n\nvariable [CommGroup M] (S : Submonoid M) (hS : S ≤ IsUnit.submonoid M)\n\n@[to_additive (attr := simp)]\n\nTarget:\ntheorem leftInvEquiv_symm_eq_inv (x : S) : ((S.leftInvEquiv hS).symm x : M) = (x : M)⁻¹ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_a81f25208ce0","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"8814970482573410bd42fe91c8cefa95b69a972141faa1ac27902b4878e1303f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Submonoid","family_id":"leftinvequiv_symm_eq_inv","file_id":"mathlib/Mathlib/GroupTheory/Submonoid/Inverses.lean","sample_id":"a81f25208ce01c9c1e94e1a7e16ac4440ac59daf394e99a5ec3de20f35dc63a8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"da9b69d9b6dc7fdb8d3935d0065bc697214d7335fb75b8a7df981601b0c50a86","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"928089061b1d0efe8c64d1c8e4d6ab36c630bf94f5147818228af3cfdc500a8b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c850dd5f680796ef8939e552d2ed07dbbf81b4eaedf7784baebc1c5d3cb8013c","source_sha256":"ebea26ad83d186da19f8d5aec5a4532b5e10776580381e96c50d87a2ed0ca5b4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  simpa using (eq_bot_or_eq_top _).symm","hard_negative":true,"metrics":{"chosen_tokens":10,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.3},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"1913f79605220401903e5448744ec4bd7b57561660b48894b3a1378d624127a8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Lattice\npublic import Mathlib.Order.Atoms\npublic import Mathlib.Order.Interval.Finset.Defs\npublic import Mathlib.Order.Preorder.Finite\n\nNamespace:\nFintype.IsSimpleOrder\n\nLocal context:\n/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Atoms, Coatoms, Simple Lattices, and Finiteness\n\nThis module contains some results on atoms and simple lattices in the finite context.\n\n## Main results\n* `Finite.to_isAtomic`, `Finite.to_isCoatomic`: Finite partial orders with bottom resp. top\n  are atomic resp. coatomic.\n\n-/\n\npublic section\n\n\nvariable {α β : Type*}\n\nnamespace IsSimpleOrder\n\nvariable [LE α] [BoundedOrder α] [IsSimpleOrder α]\n\nsection DecidableEq\n\n/-- It is important that `IsSimpleOrder` is the last type-class argument of this instance,\nso that type-class inference fails quickly if it doesn't apply.\n\nNote that as of 2025-08-13, this is false. Could someone investigate? -/\nscoped instance (priority := 200) [DecidableEq α] : Fintype α :=\n  Fintype.ofEquiv Bool equivBool.symm\n\nend DecidableEq\n\nscoped instance (priority := 200) : Finite α := by classical infer_instance\n\nend IsSimpleOrder\n\nnamespace Fintype\n\nnamespace IsSimpleOrder\n\nopen scoped _root_.IsSimpleOrder\n\nvariable [LE α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α]\n\nTarget:\ntheorem univ : (Finset.univ : Finset α) = {⊤, ⊥} :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_c850dd5f6807","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1dc8d8ed96cfa63fdcbe0c2735eef4b06488d2d6f83e501b2dc2c2dd8ae49e05","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Atoms","family_id":"univ","file_id":"mathlib/Mathlib/Order/Atoms/Finite.lean","sample_id":"c850dd5f680796ef8939e552d2ed07dbbf81b4eaedf7784baebc1c5d3cb8013c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"882f9cd65c324f45860677174f52bfb0dee89c309337831b30132904e328a1fb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"955d5266a22901504a1cd758ee8e4fc19d0cd521386eb837724b543ccf57240a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5ad3f4c1074336dbfd17b6d6041c87d9308d36a6123c6ad702c726184bd0c5fe","source_sha256":"aac9c42310c3c68ba48257d401225da11673aa0693f3a499d296961d7c43ce42","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  lift s to Finset α using hs\n  classical\n  rw [stabilizer_coe_finset, mem_stabilizer_finset_iff_smul_finset_subset, ← Finset.coe_smul_finset,\n    Finset.coe_subset]","hard_negative":true,"metrics":{"chosen_tokens":24,"rejected_tokens":3,"token_jaccard":0.047619,"token_length_ratio":0.125},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"1927b176490d7255104eb1720cdfa2193cabb331aaa8cf3aee8932929e1a32f0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Pointwise.Finset\npublic import Mathlib.GroupTheory.QuotientGroup.Defs\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\n\nNamespace:\nMulAction\n\nLocal context:\n/-\nCopyright (c) 2023 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Stabilizer of a set under a pointwise action\n\nThis file characterises the stabilizer of a set/finset under the pointwise action of a group.\n-/\n\npublic section\n\nopen Function MulOpposite Set\nopen scoped Pointwise\n\nnamespace MulAction\nvariable {G H α : Type*}\n\n/-! ### Stabilizer of a set -/\n\nsection Set\nsection Group\nvariable [Group G] [Group H] [MulAction G α] {a : G} {s t : Set α}\n\n@[to_additive (attr := simp)]\nlemma stabilizer_empty : stabilizer G (∅ : Set α) = ⊤ :=\n  Subgroup.coe_eq_univ.1 <| eq_univ_of_forall fun _a ↦ smul_set_empty\n\n@[to_additive (attr := simp)]\nlemma stabilizer_univ : stabilizer G (Set.univ : Set α) = ⊤ := by\n  ext\n  simp\n\n@[to_additive (attr := simp)]\nlemma stabilizer_singleton (b : α) : stabilizer G ({b} : Set α) = stabilizer G b := by ext; simp\n\n@[to_additive]\nlemma mem_stabilizer_set {s : Set α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s := by\n  refine mem_stabilizer_iff.trans ⟨fun h b ↦ ?_, fun h ↦ ?_⟩\n  · rw [← (smul_mem_smul_set_iff : a • b ∈ _ ↔ _), h]\n  simp_rw [Set.ext_iff, mem_smul_set_iff_inv_smul_mem]\n  exact ((MulAction.toPerm a).forall_congr' <| by simp [Iff.comm]).1 h\n\n@[to_additive]\nlemma map_stabilizer_le (f : G →* H) (s : Set G) :\n    (stabilizer G s).map f ≤ stabilizer H (f '' s) := by\n  rintro a\n  simp only [Subgroup.mem_map, mem_stabilizer_iff, forall_exists_index, and_imp]\n  rintro a ha rfl\n  rw [← image_smul_distrib, ha]\n\n@[to_additive (attr := simp)]\nlemma stabilizer_mul_self (s : Set G) : (stabilizer G s : Set G) * s = s := by\n  ext\n  refine ⟨?_, fun h ↦ ⟨_, (stabilizer G s).one_mem, _, h, one_mul _⟩⟩\n  rintro ⟨a, ha, b, hb, rfl⟩\n  rw [← mem_stabilizer_iff.1 ha]\n  exact smul_mem_smul_set hb\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_apply₂ {f : Set α → Set α → Set α}\n    (hf : ∀ a : G, a • f s t = f (a • s) (a • t)) :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (f s t) := by aesop (add simp [SetLike.le_def])\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_union :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s ∪ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_union\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_inter :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s ∩ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_inter\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_sdiff :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s \\ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_sdiff\n\n@[to_additive]\nlemma stabilizer_union_eq_left (hdisj : Disjoint s t) (hstab : stabilizer G s ≤ stabilizer G t)\n    (hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G t) :\n    stabilizer G (s ∪ t) = stabilizer G s := by\n  refine le_antisymm ?_ ?_\n  · calc\n      stabilizer G (s ∪ t)\n        ≤ stabilizer G (s ∪ t) ⊓ stabilizer G t := by simpa\n      _ ≤ stabilizer G ((s ∪ t) \\ t) := stabilizer_inf_stabilizer_le_stabilizer_sdiff\n      _ = stabilizer G s := by rw [union_sdiff_cancel_right]; simpa [← disjoint_iff_inter_eq_empty]\n  · calc\n      stabilizer G s\n        ≤ stabilizer G s ⊓ stabilizer G t := by simpa\n      _ ≤ stabilizer G (s ∪ t) := stabilizer_inf_stabilizer_le_stabilizer_union\n\n@[to_additive]\nlemma stabilizer_union_eq_right (hdisj : Disjoint s t) (hstab : stabilizer G t ≤ stabilizer G s)\n    (hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G s) :\n    stabilizer G (s ∪ t) = stabilizer G t := by\n  rw [union_comm, stabilizer_union_eq_left hdisj.symm hstab (union_comm .. ▸ hstab_union)]\n\nvariable {s : Set G}\n\nopen scoped RightActions in\n@[to_additive]\nlemma op_smul_set_stabilizer_subset (ha : a ∈ s) : (stabilizer G s : Set G) <• a ⊆ s :=\n  smul_set_subset_iff.2 fun b hb ↦ by rw [← hb]; exact smul_mem_smul_set ha\n\n@[to_additive]\nlemma stabilizer_subset_div_right (ha : a ∈ s) : ↑(stabilizer G s) ⊆ s / {a} := fun b hb ↦\n  ⟨_, by rwa [← smul_eq_mul, mem_stabilizer_set.1 hb], _, mem_singleton _, mul_div_cancel_right _ _⟩\n\n@[to_additive]\nlemma stabilizer_finite (hs₀ : s.Nonempty) (hs : s.Finite) : (stabilizer G s : Set G).Finite := by\n  obtain ⟨a, ha⟩ := hs₀\n  exact (hs.div <| finite_singleton _).subset <| stabilizer_subset_div_right ha\n\nend Group\n\nsection CommGroup\nvariable [CommGroup G] {s t : Set G} {a : G}\n\n@[to_additive]\nlemma smul_set_stabilizer_subset (ha : a ∈ s) : a • (stabilizer G s : Set G) ⊆ s := by\n  simpa using op_smul_set_stabilizer_subset ha\n\nend CommGroup\nend Set\n\nvariable [Group G] [Group H] [MulAction G α] {a : G}\n\n/-! ### Stabilizer of a subgroup -/\n\nsection Subgroup\n\n-- TODO: Is there a lemma that could unify the following three very similar lemmas?\n\n@[to_additive (attr := simp)]\nlemma stabilizer_subgroup (s : Subgroup G) : stabilizer G (s : Set G) = s := by\n  simp_rw [SetLike.ext_iff, mem_stabilizer_set]\n  refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_left ha⟩\n  simpa only [smul_eq_mul, SetLike.mem_coe, mul_one] using (h 1).2 s.one_mem\n\n@[to_additive (attr := simp)]\nlemma stabilizer_op_subgroup (s : Subgroup G) : stabilizer Gᵐᵒᵖ (s : Set G) = s.op := by\n  simp_rw [SetLike.ext_iff, mem_stabilizer_set]\n  simp only [smul_eq_mul_unop, SetLike.mem_coe, Subgroup.mem_op, «forall», unop_op]\n  refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_right ha⟩\n  simpa only [op_smul_eq_mul, SetLike.mem_coe, one_mul] using (h 1).2 s.one_mem\n\n@[to_additive (attr := simp)]\nlemma stabilizer_subgroup_op (s : Subgroup Gᵐᵒᵖ) : stabilizer G (s : Set Gᵐᵒᵖ) = s.unop := by\n  simp_rw [SetLike.ext_iff, mem_stabilizer_set]\n  refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_right ha⟩\n  have : 1 * MulOpposite.op a ∈ s := (h 1).2 s.one_mem\n  simpa only [op_smul_eq_mul, SetLike.mem_coe, one_mul] using! this\n\nend Subgroup\n\n/-! ### Stabilizer of a finset -/\n\nsection Finset\nvariable [DecidableEq α]\n\n@[to_additive (attr := simp, norm_cast)]\nlemma stabilizer_coe_finset (s : Finset α) : stabilizer G (s : Set α) = stabilizer G s := by\n  ext; simp [← Finset.coe_inj]\n\n@[to_additive (attr := simp)]\nlemma stabilizer_finset_empty : stabilizer G (∅ : Finset α) = ⊤ :=\n  Subgroup.coe_eq_univ.1 <| eq_univ_of_forall Finset.smul_finset_empty\n\n@[to_additive (attr := simp)]\nlemma stabilizer_finset_univ [Fintype α] : stabilizer G (Finset.univ : Finset α) = ⊤ := by\n  ext\n  simp\n\n@[to_additive (attr := simp)]\nlemma stabilizer_finset_singleton (b : α) : stabilizer G ({b} : Finset α) = stabilizer G b := by\n  ext; simp\n\n@[to_additive]\nlemma mem_stabilizer_finset {s : Finset α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s := by\n  simp_rw [← stabilizer_coe_finset, mem_stabilizer_set, Finset.mem_coe]\n\n@[to_additive]\nlemma mem_stabilizer_finset_iff_subset_smul_finset {s : Finset α} :\n    a ∈ stabilizer G s ↔ s ⊆ a • s := by\n  rw [mem_stabilizer_iff, Finset.subset_iff_eq_of_card_le (Finset.card_smul_finset _ _).le, eq_comm]\n\n@[to_additive]\nlemma mem_stabilizer_finset_iff_smul_finset_subset {s : Finset α} :\n    a ∈ stabilizer G s ↔ a • s ⊆ s := by\n  rw [mem_stabilizer_iff, Finset.subset_iff_eq_of_card_le (Finset.card_smul_finset _ _).ge]\n\n@[to_additive]\nlemma mem_stabilizer_finset' {s : Finset α} : a ∈ stabilizer G s ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ s := by\n  rw [← Subgroup.inv_mem_iff, mem_stabilizer_finset_iff_subset_smul_finset]\n  simp_rw [← Finset.mem_inv_smul_finset_iff, Finset.subset_iff]\n\nend Finset\n\n/-! ### Stabilizer of a finite set -/\n\nvariable {s : Set α}\n\n@[to_additive]\nlemma mem_stabilizer_set_iff_subset_smul_set {s : Set α} (hs : s.Finite) :\n    a ∈ stabilizer G s ↔ s ⊆ a • s := by\n  lift s to Finset α using hs\n  classical\n  rw [stabilizer_coe_finset, mem_stabilizer_finset_iff_subset_smul_finset, ← Finset.coe_smul_finset,\n    Finset.coe_subset]\n\n@[to_additive]\n\nTarget:\nlemma mem_stabilizer_set_iff_smul_set_subset {s : Set α} (hs : s.Finite) :\n    a ∈ stabilizer G s ↔ a • s ⊆ s :=\n\nProof body:\n","rejected":"by\n  exact mem_stabilizer_set_iff_smul_set_subset","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"4944e8aa27a61f3e74b3aac570b34e0cbd605869d883587860b440c0b809b44e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Pointwise","family_id":"mem_stabilizer_set_iff_smul_set_subset","file_id":"mathlib/Mathlib/Algebra/Pointwise/Stabilizer.lean","sample_id":"5ad3f4c1074336dbfd17b6d6041c87d9308d36a6123c6ad702c726184bd0c5fe"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4644b39b47ec31a8d61ff06e763a66da7cadd0e1fb2b3309d9d1db8d535d144c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6d45385fed9b0ad68c7032cd158ad8e6ed294ff9a6d585f7ef6679ce0eea1fd4","source_sha256":"45bff0b3bef16dadf0bda36c7fa30586e758b281d4f3d18c6409891ebf942e27","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← sub_eq_zero]; exact hS.eq_zero_of_inner_left 𝕜 (by simpa [inner_sub_left, sub_eq_zero])","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":2,"token_jaccard":0.055556,"token_length_ratio":0.095238},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"1935bd5654919074f9c058e7ac6464f42c7ceabdee9597b2f5defc0f6e32db4f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.InnerProductSpace.Basic\npublic import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Zhouhang Zhou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis\n-/\n/-!\n# Continuity of inner product\n\nWe show that the inner product is continuous, `continuous_inner`.\n\n## Tags\n\ninner product space, Hilbert space, norm\n\n-/\n\npublic section\n\nnoncomputable section\n\nopen RCLike Real Filter Topology ComplexConjugate Finsupp\nopen LinearMap renaming BilinForm → BilinForm\n\nvariable {𝕜 E F : Type*} [RCLike 𝕜]\n\n\nsection Continuous\n\nvariable [SeminormedAddCommGroup E] [InnerProductSpace 𝕜 E]\n\nlocal notation \"⟪\" x \", \" y \"⟫\" => inner 𝕜 x y\n\n/-!\n### Continuity of the inner product\n-/\n\n/-- When an inner product space `E` over `𝕜` is considered as a real normed space, its inner\nproduct satisfies `IsBoundedBilinearMap`.\n\nIn order to state these results, we need a `NormedSpace ℝ E` instance. We will later establish\nsuch an instance by restriction-of-scalars, `InnerProductSpace.rclikeToReal 𝕜 E`, but this\ninstance may be not definitionally equal to some other “natural” instance. So, we assume\n`[NormedSpace ℝ E]`.\n-/\ntheorem _root_.isBoundedBilinearMap_inner [NormedSpace ℝ E] [IsScalarTower ℝ 𝕜 E] :\n    IsBoundedBilinearMap ℝ fun p : E × E => ⟪p.1, p.2⟫ :=\n  { add_left := inner_add_left\n    smul_left := fun r x y => by\n      simp only [← algebraMap_smul 𝕜 r x, algebraMap_eq_ofReal, inner_smul_real_left]\n    add_right := inner_add_right\n    smul_right := fun r x y => by\n      simp only [← algebraMap_smul 𝕜 r y, algebraMap_eq_ofReal, inner_smul_real_right]\n    bound :=\n      ⟨1, zero_lt_one, fun x y => by\n        rw [one_mul]\n        exact norm_inner_le_norm x y⟩ }\n\ntheorem continuous_inner : Continuous fun p : E × E => ⟪p.1, p.2⟫ :=\n  letI : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E\n  haveI := IsScalarTower.restrictScalars ℝ 𝕜 E\n  isBoundedBilinearMap_inner.continuous\n\nvariable {α : Type*}\n\ntheorem Filter.Tendsto.inner {f g : α → E} {l : Filter α} {x y : E} (hf : Tendsto f l (𝓝 x))\n    (hg : Tendsto g l (𝓝 y)) : Tendsto (fun t => ⟪f t, g t⟫) l (𝓝 ⟪x, y⟫) :=\n  (continuous_inner.tendsto _).comp (hf.prodMk_nhds hg)\n\nvariable [TopologicalSpace α] {f g : α → E} {x : α} {s : Set α}\n\ntheorem ContinuousWithinAt.inner (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :\n    ContinuousWithinAt (fun t => ⟪f t, g t⟫) s x :=\n  Filter.Tendsto.inner hf hg\n\n@[fun_prop]\ntheorem ContinuousAt.inner (hf : ContinuousAt f x) (hg : ContinuousAt g x) :\n    ContinuousAt (fun t => ⟪f t, g t⟫) x :=\n  Filter.Tendsto.inner hf hg\n\n@[fun_prop]\ntheorem ContinuousOn.inner (hf : ContinuousOn f s) (hg : ContinuousOn g s) :\n    ContinuousOn (fun t => ⟪f t, g t⟫) s := fun x hx => (hf x hx).inner (hg x hx)\n\n@[continuity, fun_prop]\ntheorem Continuous.inner (hf : Continuous f) (hg : Continuous g) : Continuous fun t => ⟪f t, g t⟫ :=\n  continuous_iff_continuousAt.2 fun _x => by fun_prop\n\nend Continuous\n\nopen Submodule\n\nvariable {E F ι : Type*}\nvariable (𝕜 : Type*) [RCLike 𝕜]\nvariable [NormedAddCommGroup E] [NormedAddCommGroup F]\nvariable [InnerProductSpace 𝕜 E] [InnerProductSpace ℝ F]\nvariable {x y : E} {S : Set E} {f : ι → E}\n\nlocal notation \"⟪\" x \", \" y \"⟫\" => inner 𝕜 x y\n\ntheorem Dense.eq_zero_of_inner_left (hS : Dense S) (h : ∀ v ∈ S, ⟪x, v⟫ = 0) : x = 0 := by\n  let K := span 𝕜 S\n  have hK : Dense (K : Set E) := hS.mono subset_span\n  have : (⟪x, ·⟫) = 0 := (continuous_const.inner continuous_id).ext_on\n    hK continuous_const fun v ↦ Submodule.span_induction h (by simp)\n      (by simp +contextual [inner_add_right]) (by simp +contextual [inner_smul_right])\n  simpa using congr_fun this x\n\ntheorem Dense.eq_zero_of_inner_right (hS : Dense S) (h : ∀ v ∈ S, ⟪v, x⟫ = 0) : x = 0 :=\n  hS.eq_zero_of_inner_left 𝕜 fun v hv ↦ by rw! [← inner_conj_symm]; simp [-inner_conj_symm, h, hv]\n\nTarget:\ntheorem Dense.eq_of_inner_left (hS : Dense S) (h : ∀ v ∈ S, ⟪x, v⟫ = ⟪y, v⟫) : x = y :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/InnerProductSpace","family_id":"dense","file_id":"mathlib/Mathlib/Analysis/InnerProductSpace/Continuous.lean","sample_id":"6d45385fed9b0ad68c7032cd158ad8e6ed294ff9a6d585f7ef6679ce0eea1fd4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8563be0743a9370b287fc6eea19b728e9532ff758a843c12bb7a89f00678f9fd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"23133e4cdbd3c18af59ef41f856739915fb4a88cdab3bfe7a080ab4e9717b499","source_sha256":"7fb2a027e17dd48df4c5d05af7f579e28e12ec18c52dd1aaeae9c6854e93da06","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← cons_comp_cycleRange, map_perm, Matrix.vecCons]\n  simp [Units.smul_def]","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":5,"token_jaccard":0.055556,"token_length_ratio":0.277778},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"195c39d1d06cd99336255d53d3b3b2299904add93819aa5be185104bdd40c3c5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Alternating.Curry\npublic import Mathlib.GroupTheory.Perm.Fin\npublic import Mathlib.Data.Fin.Parity\n\nNamespace:\nAlternatingMap\n\nLocal context:\n/-\nCopyright (c) 2025 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Uncurrying alternating maps\n\nGiven a function `f` which is linear in the first argument\nand is alternating form in the other `n` arguments,\nthis file defines an alternating form `AlternatingMap.alternatizeUncurryFin f` in `n + 1` arguments.\n\nThis function is given by\n```\nAlternatingMap.alternatizeUncurryFin f v =\n  ∑ i : Fin (n + 1), (-1) ^ (i : ℕ) • f (v i) (removeNth i v)\n```\n\nGiven an alternating map `f` of `n + 1` arguments,\neach term in the sum above written for `f.curryLeft` equals the original map,\nthus `f.curryLeft.alternatizeUncurryFin = (n + 1) • f`.\n\nWe do not multiply the result of `alternatizeUncurryFin` by `(n + 1)⁻¹`\nso that the construction works for `R`-multilinear maps over any commutative ring `R`,\nnot only a field of characteristic zero.\n\n## Main results\n\n- `AlternatingMap.alternatizeUncurryFin_curryLeft`:\n  the round-trip formula for currying/uncurrying, see above.\n\n- `AlternatingMap.alternatizeUncurryFin_alternatizeUncurryFinLM_comp_of_symmetric`:\n  If `f` is a symmetric bilinear map taking values in the space of alternating maps,\n  then the twice uncurried `f` is zero.\n\nA version of the latter theorem for continuous alternating maps\nwill be used to prove that the second exterior derivative of a differential form is zero.\n-/\n\n@[expose] public section\n\nopen Fin Function\n\nnamespace AlternatingMap\n\nvariable {R : Type*} {M M₂ N N₂ : Type*} [CommRing R] [AddCommGroup M]\n  [AddCommGroup M₂] [AddCommGroup N] [AddCommGroup N₂] [Module R M] [Module R M₂]\n  [Module R N] [Module R N₂] {n : ℕ}\n\n/-- If `f` is a `(n + 1)`-multilinear alternating map, `x` is an element of the domain,\nand `v` is an `n`-vector, then the value of `f` at `v` with `x` inserted at the `p`th place\nequals `(-1) ^ p` times the value of `f` at `v` with `x` prepended. -/\n\nTarget:\ntheorem map_insertNth (f : M [⋀^Fin (n + 1)]→ₗ[R] N) (p : Fin (n + 1)) (x : M) (v : Fin n → M) :\n    f (p.insertNth x v) = (-1) ^ (p : ℕ) • f (Matrix.vecCons x v) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Alternating","family_id":"map_insertnth","file_id":"mathlib/Mathlib/LinearAlgebra/Alternating/Uncurry/Fin.lean","sample_id":"23133e4cdbd3c18af59ef41f856739915fb4a88cdab3bfe7a080ab4e9717b499"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1917f8da7209b00ed86e3ac8e7571325a8372a2d3f50420181edba9e230a5abb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"12e052d74731a6e1b56e362de439a6012d603050ad7ea0afe32d18e575e9a7d6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c126f860a1889f0eed701887869d359f44f2980da9414820754ce91fa0c2db26","source_sha256":"6d14a53c89ceedf8fdc211fa474370698532792d974cd7aecfc721711170cc1d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_all [SProd.sprod, product, mem_flatMap, mem_map, Prod.ext_iff, and_left_comm]","hard_negative":true,"metrics":{"chosen_tokens":19,"rejected_tokens":3,"token_jaccard":0.0625,"token_length_ratio":0.157895},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"19bfe2c4bc3f9e9a72874f8d7b6fdefa80280a2eaee7103131920ace1ccb936f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.List.Defs\npublic import Mathlib.Tactic.Attr.Core\npublic import Mathlib.Tactic.Common\n\nNamespace:\nList\n\nLocal context:\n/-\nCopyright (c) 2015 Leonardo de Moura. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura, Mario Carneiro\n-/\n/-!\n# Lists in product and sigma types\n\nThis file proves basic properties of `List.product` and `List.sigma`, which are list constructions\nliving in `Prod` and `Sigma` types respectively. Their definitions can be found in\n[`Data.List.Defs`](./defs). Beware, this is not about `List.prod`, the multiplicative product.\n-/\n\npublic section\n\n\nvariable {α β : Type*}\n\nnamespace List\n\n/-! ### product -/\n\n\n@[simp]\ntheorem nil_product (l : List β) : (@nil α) ×ˢ l = [] :=\n  rfl\n\n@[simp]\ntheorem product_cons (a : α) (l₁ : List α) (l₂ : List β) :\n    (a :: l₁) ×ˢ l₂ = map (fun b => (a, b)) l₂ ++ (l₁ ×ˢ l₂) :=\n  rfl\n\n@[simp]\ntheorem product_nil : ∀ l : List α, l ×ˢ (@nil β) = []\n  | [] => rfl\n  | _ :: l => by simp [product_cons, product_nil l]\n\n@[simp]\n\nTarget:\ntheorem mem_product {l₁ : List α} {l₂ : List β} {a : α} {b : β} :\n    (a, b) ∈ l₁ ×ˢ l₂ ↔ a ∈ l₁ ∧ b ∈ l₂ :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_c126f860a188","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"9a8f321056d918cc559b72d365bfb30014ad9588a4f1ee9ba90fdbe7d61c2180","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/List","family_id":"mem_product","file_id":"mathlib/Mathlib/Data/List/ProdSigma.lean","sample_id":"c126f860a1889f0eed701887869d359f44f2980da9414820754ce91fa0c2db26"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a40d38439b063352a0a57202db0bbc83f3e167b29ffee4ace96a68c95b1c7096","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9aff7c65eeea9c95f2ed37bd598fa3d4c3d36b657db023ad927a8423779dff9f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"74e41a4b5bb950621cded82f802859d90e618c5ffbeaff846b1d0a22fbcf641d","source_sha256":"158ce289cc94776d7ace3f5f8788d7841da0114f64a958ffadba75342e4312aa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply Subtype.ext; funext n\n  rcases h : s.val n; · rfl\n  refine absurd ?_ H; exact ⟨⟨_, _, h.symm⟩⟩","hard_negative":true,"metrics":{"chosen_tokens":36,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.083333},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"19d7dfdccc79acb63f890787c64cf1a0ede3e92acc699603daf4b52337a904c2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.Find\npublic import Mathlib.Data.Stream.Init\npublic import Mathlib.Logic.Relator\npublic import Mathlib.Tactic.Common\npublic import Batteries.Tactic.Lint.Simp\n\nNamespace:\nComputation\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Coinductive formalization of unbounded computations.\n\nThis file provides a `Computation` type where `Computation α` is the type of\nunbounded computations returning `α`.\n-/\n\n@[expose] public section\n\nopen Function\n\nuniverse u v w\n\n/-\ncoinductive Computation (α : Type u) : Type u\n| pure : α → Computation α\n| think : Computation α → Computation α\n-/\n/-- `Computation α` is the type of unbounded computations returning `α`.\n  An element of `Computation α` is an infinite sequence of `Option α` such\n  that if `f n = some a` for some `n` then it is constantly `some a` after that. -/\ndef Computation (α : Type u) : Type u :=\n  { f : Stream' (Option α) // ∀ ⦃n a⦄, f n = some a → f (n + 1) = some a }\n\nnamespace Computation\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n-- constructors\n/-- `pure a` is the computation that immediately terminates with result `a`. -/\ndef pure (a : α) : Computation α :=\n  ⟨Stream'.const (some a), fun _ _ => id⟩\n\ninstance : CoeTC α (Computation α) :=\n  ⟨pure⟩\n\n-- note [use has_coe_t]\n/-- `think c` is the computation that delays for one \"tick\" and then performs\n  computation `c`. -/\ndef think (c : Computation α) : Computation α :=\n  ⟨Stream'.cons none c.1, fun n a h => by\n    rcases n with - | n\n    · contradiction\n    · exact c.2 h⟩\n\n/-- `thinkN c n` is the computation that delays for `n` ticks and then performs\n  computation `c`. -/\ndef thinkN (c : Computation α) : ℕ → Computation α\n  | 0 => c\n  | n + 1 => think (thinkN c n)\n\n-- check for immediate result\n/-- `head c` is the first step of computation, either `some a` if `c = pure a`\n  or `none` if `c = think c'`. -/\ndef head (c : Computation α) : Option α :=\n  c.1.head\n\n-- one step of computation\n/-- `tail c` is the remainder of computation, either `c` if `c = pure a`\n  or `c'` if `c = think c'`. -/\ndef tail (c : Computation α) : Computation α :=\n  ⟨c.1.tail, fun _ _ h => c.2 h⟩\n\n/-- `empty α` is the computation that never returns, an infinite sequence of\n  `think`s. -/\ndef empty (α) : Computation α :=\n  ⟨Stream'.const none, fun _ _ => id⟩\n\ninstance : Inhabited (Computation α) :=\n  ⟨empty _⟩\n\n/-- `runFor c n` evaluates `c` for `n` steps and returns the result, or `none`\n  if it did not terminate after `n` steps. -/\ndef runFor : Computation α → ℕ → Option α :=\n  Subtype.val\n\n/-- `destruct c` is the destructor for `Computation α` as a coinductive type.\n  It returns `inl a` if `c = pure a` and `inr c'` if `c = think c'`. -/\ndef destruct (c : Computation α) : α ⊕ (Computation α) :=\n  match c.1 0 with\n  | none => Sum.inr (tail c)\n  | some a => Sum.inl a\n\n/-- `run c` is an unsound meta function that runs `c` to completion, possibly\n  resulting in an infinite loop in the VM. -/\nunsafe def run : Computation α → α\n  | c =>\n    match destruct c with\n    | Sum.inl a => a\n    | Sum.inr ca => run ca\n\ntheorem destruct_eq_pure {s : Computation α} {a : α} : destruct s = Sum.inl a → s = pure a := by\n  dsimp [destruct]\n  cases f0 : s.1 0 <;> intro h\n  · contradiction\n  · apply Subtype.ext\n    funext n\n    induction n with\n    | zero => injection h with h'; rwa [h'] at f0\n    | succ n IH => exact s.2 IH\n\ntheorem destruct_eq_think {s : Computation α} {s'} : destruct s = Sum.inr s' → s = think s' := by\n  dsimp [destruct]\n  rcases f0 : s.1 0 with - | a' <;> intro h\n  · injection h with h'\n    rw [← h']\n    obtain ⟨f, al⟩ := s\n    apply Subtype.ext\n    dsimp [think, tail]\n    rw [← f0]\n    exact (Stream'.eta f).symm\n  · contradiction\n\n@[simp]\ntheorem destruct_pure (a : α) : destruct (pure a) = Sum.inl a :=\n  rfl\n\n@[simp]\ntheorem destruct_think : ∀ s : Computation α, destruct (think s) = Sum.inr s\n  | ⟨_, _⟩ => rfl\n\n@[simp]\ntheorem destruct_empty : destruct (empty α) = Sum.inr (empty α) :=\n  rfl\n\n@[simp]\ntheorem head_pure (a : α) : head (pure a) = some a :=\n  rfl\n\n@[simp]\ntheorem head_think (s : Computation α) : head (think s) = none :=\n  rfl\n\n@[simp]\ntheorem head_empty : head (empty α) = none :=\n  rfl\n\n@[simp]\ntheorem tail_pure (a : α) : tail (pure a) = pure a :=\n  rfl\n\n@[simp]\ntheorem tail_think (s : Computation α) : tail (think s) = s := rfl\n\n@[simp]\ntheorem tail_empty : tail (empty α) = empty α :=\n  rfl\n\ntheorem think_empty : empty α = think (empty α) :=\n  destruct_eq_think destruct_empty\n\n/-- Recursion principle for computations, compare with `List.recOn`. -/\n@[elab_as_elim]\ndef recOn {motive : Computation α → Sort v} (s : Computation α) (pure : ∀ a, motive (pure a))\n    (think : ∀ s, motive (think s)) : motive s :=\n  match H : destruct s with\n  | Sum.inl v => by\n    rw [destruct_eq_pure H]\n    apply pure\n  | Sum.inr v => match v with\n    | ⟨a, s'⟩ => by\n      rw [destruct_eq_think H]\n      apply think\n\n/-- Corecursor constructor for `corec` -/\ndef Corec.f (f : β → α ⊕ β) : α ⊕ β → Option α × (α ⊕ β)\n  | Sum.inl a => (some a, Sum.inl a)\n  | Sum.inr b =>\n    (match f b with\n      | Sum.inl a => some a\n      | Sum.inr _ => none,\n      f b)\n\n/-- `corec f b` is the corecursor for `Computation α` as a coinductive type.\n  If `f b = inl a` then `corec f b = pure a`, and if `f b = inl b'` then\n  `corec f b = think (corec f b')`. -/\ndef corec (f : β → α ⊕ β) (b : β) : Computation α := by\n  refine ⟨Stream'.corec' (Corec.f f) (Sum.inr b), fun n a' h => ?_⟩\n  rw [Stream'.corec'_eq]\n  change Stream'.corec' (Corec.f f) (Corec.f f (Sum.inr b)).2 n = some a'\n  revert h; generalize Sum.inr b = o\n  induction n generalizing o with\n  | zero =>\n    change (Corec.f f o).1 = some a' → (Corec.f f (Corec.f f o).2).1 = some a'\n    rcases o with _ | b <;> intro h\n    · exact h\n    unfold Corec.f at *; split <;> simp_all\n  | succ n IH =>\n    rw [Stream'.corec'_eq (Corec.f f) (Corec.f f o).2, Stream'.corec'_eq (Corec.f f) o]\n    exact IH (Corec.f f o).2\n\n/-- left map of `⊕` -/\ndef lmap (f : α → β) : α ⊕ γ → β ⊕ γ\n  | Sum.inl a => Sum.inl (f a)\n  | Sum.inr b => Sum.inr b\n\n/-- right map of `⊕` -/\ndef rmap (f : β → γ) : α ⊕ β → α ⊕ γ\n  | Sum.inl a => Sum.inl a\n  | Sum.inr b => Sum.inr (f b)\n\nattribute [simp] lmap rmap\n\n@[simp]\ntheorem corec_eq (f : β → α ⊕ β) (b : β) : destruct (corec f b) = rmap (corec f) (f b) := by\n  dsimp [corec, destruct]\n  rw [show Stream'.corec' (Corec.f f) (Sum.inr b) 0 =\n    Sum.rec Option.some (fun _ ↦ none) (f b) by\n    dsimp [Corec.f, Stream'.corec', Stream'.corec, Stream'.map, Stream'.get, Stream'.iterate]\n    match (f b) with\n    | Sum.inl x => rfl\n    | Sum.inr x => rfl]\n  rcases h : f b with a | b'; · rfl\n  dsimp [Corec.f, destruct]\n  apply congr_arg; apply Subtype.ext\n  dsimp [corec, tail]\n  rw [Stream'.corec'_eq, Stream'.tail_cons]\n  dsimp [Corec.f]; rw [h]\n\nsection Bisim\n\nvariable (R : Computation α → Computation α → Prop)\n\n/-- bisimilarity relation -/\nlocal infixl:50 \" ~ \" => R\n\n/-- Bisimilarity over a sum of `Computation`s -/\ndef BisimO : α ⊕ (Computation α) → α ⊕ (Computation α) → Prop\n  | Sum.inl a, Sum.inl a' => a = a'\n  | Sum.inr s, Sum.inr s' => R s s'\n  | _, _ => False\n\nattribute [simp] BisimO\nattribute [nolint simpNF] BisimO.eq_3\n\n/-- Attribute expressing bisimilarity over two `Computation`s -/\ndef IsBisimulation :=\n  ∀ ⦃s₁ s₂⦄, s₁ ~ s₂ → BisimO R (destruct s₁) (destruct s₂)\n\n-- If two computations are bisimilar, then they are equal\ntheorem eq_of_bisim (bisim : IsBisimulation R) {s₁ s₂} (r : s₁ ~ s₂) : s₁ = s₂ := by\n  apply Subtype.ext\n  apply Stream'.eq_of_bisim fun x y => ∃ s s' : Computation α, s.1 = x ∧ s'.1 = y ∧ R s s'\n  · dsimp [Stream'.IsBisimulation]\n    intro t₁ t₂ e\n    match t₁, t₂, e with\n    | _, _, ⟨s, s', rfl, rfl, r⟩ =>\n      suffices head s = head s' ∧ R (tail s) (tail s') from\n        And.imp id (fun r => ⟨tail s, tail s', by cases s; rfl, by cases s'; rfl, r⟩) this\n      have h := bisim r; revert r h\n      refine recOn s ?_ ?_ <;> intro r' <;> refine recOn s' ?_ ?_ <;> intro a' r h\n      · constructor <;> dsimp at h\n        · rw [h]\n        · rw [h] at r\n          rw [tail_pure, tail_pure, h]\n          assumption\n      · rw [destruct_pure, destruct_think] at h\n        exact False.elim h\n      · rw [destruct_pure, destruct_think] at h\n        exact False.elim h\n      · simp_all\n  · exact ⟨s₁, s₂, rfl, rfl, r⟩\n\nend Bisim\n\n-- It's more of a stretch to use ∈ for this relation, but it\n-- asserts that the computation limits to the given value.\n/-- Assertion that a `Computation` limits to a given value -/\nprotected def Mem (s : Computation α) (a : α) :=\n  some a ∈ s.1\n\ninstance : Membership α (Computation α) :=\n  ⟨Computation.Mem⟩\n\ntheorem le_stable (s : Computation α) {a m n} (h : m ≤ n) : s.1 m = some a → s.1 n = some a := by\n  obtain ⟨f, al⟩ := s\n  induction h with\n  | refl => exact id\n  | step _ IH => exact fun h2 ↦ al (IH h2)\n\ntheorem mem_unique {s : Computation α} {a b : α} : a ∈ s → b ∈ s → a = b\n  | ⟨m, ha⟩, ⟨n, hb⟩ => by\n    injection\n      (le_stable s (le_max_left m n) ha.symm).symm.trans (le_stable s (le_max_right m n) hb.symm)\n\ntheorem Mem.left_unique : Relator.LeftUnique ((· ∈ ·) : α → Computation α → Prop) := fun _ _ _ =>\n  mem_unique\n\n/-- `Terminates s` asserts that the computation `s` eventually terminates with some value. -/\nclass Terminates (s : Computation α) : Prop where\n  /-- assertion that there is some term `a` such that the `Computation` terminates -/\n  term : ∃ a, a ∈ s\n\ntheorem terminates_iff (s : Computation α) : Terminates s ↔ ∃ a, a ∈ s :=\n  ⟨fun h => h.1, Terminates.mk⟩\n\ntheorem terminates_of_mem {s : Computation α} {a : α} (h : a ∈ s) : Terminates s :=\n  ⟨⟨a, h⟩⟩\n\ntheorem terminates_def (s : Computation α) : Terminates s ↔ ∃ n, (s.1 n).isSome :=\n  ⟨fun ⟨⟨a, n, h⟩⟩ =>\n    ⟨n, by\n      dsimp [Stream'.get] at h\n      rw [← h]\n      exact rfl⟩,\n    fun ⟨n, h⟩ => ⟨⟨Option.get _ h, n, (Option.eq_some_of_isSome h).symm⟩⟩⟩\n\ntheorem ret_mem (a : α) : a ∈ pure a :=\n  Exists.intro 0 rfl\n\ntheorem eq_of_pure_mem {a a' : α} (h : a' ∈ pure a) : a' = a :=\n  mem_unique h (ret_mem _)\n\n@[simp]\ntheorem mem_pure_iff (a b : α) : a ∈ pure b ↔ a = b :=\n  ⟨eq_of_pure_mem, fun h => h ▸ ret_mem _⟩\n\ninstance ret_terminates (a : α) : Terminates (pure a) :=\n  terminates_of_mem (ret_mem _)\n\ntheorem think_mem {s : Computation α} {a} : a ∈ s → a ∈ think s\n  | ⟨n, h⟩ => ⟨n + 1, h⟩\n\ninstance think_terminates (s : Computation α) : ∀ [Terminates s], Terminates (think s)\n  | ⟨⟨a, n, h⟩⟩ => ⟨⟨a, n + 1, h⟩⟩\n\ntheorem of_think_mem {s : Computation α} {a} : a ∈ think s → a ∈ s\n  | ⟨n, h⟩ => by\n    rcases n with - | n'\n    · contradiction\n    · exact ⟨n', h⟩\n\ntheorem of_think_terminates {s : Computation α} : Terminates (think s) → Terminates s\n  | ⟨⟨a, h⟩⟩ => ⟨⟨a, of_think_mem h⟩⟩\n\ntheorem notMem_empty (a : α) : a ∉ empty α := fun ⟨n, h⟩ => by contradiction\n\ntheorem not_terminates_empty : ¬Terminates (empty α) := fun ⟨⟨a, h⟩⟩ => notMem_empty a h\n\nTarget:\ntheorem eq_empty_of_not_terminates {s} (H : ¬Terminates s) : s = empty α :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_74e41a4b5bb9","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d277a78876e97e303acf8528c188b44052546aaaf8a3b321d1763a66902a4060","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Seq","family_id":"eq_empty_of_not_terminates","file_id":"mathlib/Mathlib/Data/Seq/Computation.lean","sample_id":"74e41a4b5bb950621cded82f802859d90e618c5ffbeaff846b1d0a22fbcf641d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"47f42a3f39b92164ba24c6342d141209f44e0c15b9d67ad24b48bf7df68f4b85","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cca863f431b487d24cfdcc63d33eb97176150aa66a1f7cf715b74e4dcfb0a563","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2293cf123c21972c1d2af5e35604c06efbc61aa089ba31c3f82c50d912dfbf35","source_sha256":"50e4a4d798ba2dd2872d163b6a5aeb55d8cf128d153e35dcc8c630f4eadcbabe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := degree_lt_wf.min_mem _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)\n  simp_all [uniformizingPolynomial]","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":5,"token_jaccard":0.111111,"token_length_ratio":0.333333},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"1a057fc2644995df354dfa488219ee91dbd732f7e95299ccb4f8634d258295be","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.Finite.Valuation\npublic import Mathlib.NumberTheory.FunctionField\npublic import Mathlib.RingTheory.Valuation.Discrete.Basic\n\nNamespace:\nRatFunc\n\nLocal context:\n/-\nCopyright (c) 2025 María Inés de Frutos-Fernández & Xavier Généreux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: María Inés de Frutos-Fernández, Xavier Généreux\n-/\n/-!\n# Ostrowski's theorem for `K(X)`\n\nThis file proves Ostrowski's theorem for the field of rational functions `K(X)`, where `K` is any\nfield: if `v` is a discrete valuation on `K(X)` which is trivial on elements of `K`, then `v` is\nequivalent to either the `I`-adic valuation for some `I : HeightOneSpectrum K[X]`, or to the\nvaluation at infinity `FunctionField.inftyValuation K`.\n\n## Main results\n- `RatFunc.valuation_isEquiv_infty_or_adic`: Ostrowski's theorem for `K(X)`.\n-/\n\n@[expose] public noncomputable section\n\n\nopen Multiplicative WithZero\n\nvariable {K Γ : Type*} [Field K] [LinearOrderedCommGroupWithZero Γ] {v : Valuation (RatFunc K) Γ}\n\nnamespace RatFunc\n\nsection Infinity\n\nopen Polynomial Valuation\n\nlemma valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X {f : RatFunc K}\n    [v.IsTrivialOn K] (hlt : 1 < v X) (hf : f ≠ 0) : v f = v RatFunc.X ^ f.intDegree := by\n  induction f using RatFunc.induction_on with\n  | f p q hq =>\n    rw [intDegree_div (by grind only) (by grind only), v.map_div, zpow_sub₀ (ne_zero_of_lt hlt)]\n    simp_rw [intDegree_polynomial, zpow_natCast, ← coePolynomial_eq_algebraMap]\n    have hp : p ≠ 0 := by contrapose hf; simp [hf]\n    rw [valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X _ hlt hp,\n      valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X _ hlt hq]\n\nvariable [DecidableEq (RatFunc K)]\n\nlemma valuation_isEquiv_inftyValuation_of_one_lt_valuation_X [v.IsTrivialOn K] (hlt : 1 < v X) :\n    v.IsEquiv (inftyValuation K) := by\n  refine isEquiv_iff_val_lt_one.mpr fun {f} ↦ ?_\n  rcases eq_or_ne f 0 with rfl | hf\n  · simp\n  · have hlt' : 1 < inftyValuation K X := by simp [← exp_zero]\n    rw [valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt hf,\n      valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt' hf]\n    grind [one_le_zpow_iff_right₀]\n\nend Infinity\n\nopen IsDedekindDomain HeightOneSpectrum Set Valuation Polynomial\n\nlemma setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty [v.IsNontrivial] [v.IsTrivialOn K]\n    (hle : v RatFunc.X ≤ 1) : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty := by\n  obtain ⟨w, h0, h1⟩ := IsNontrivial.exists_lt_one (v := v)\n  induction w using RatFunc.induction_on with\n  | f p q =>\n    simp only [ne_eq, _root_.div_eq_zero_iff, FaithfulSMul.algebraMap_eq_zero_iff, not_or,\n      map_div₀] at *\n    have hor : ¬v ↑p = 1 ∨ ¬v ↑q = 1 := by rw [← not_and_or]; aesop\n    suffices ∀ r : K[X], v (↑r) ≠ 1 → r ≠ 0 → {p : K[X] | v ↑p < 1 ∧ ¬p = 0}.Nonempty by\n      exact Or.elim hor (fun hp ↦ this p hp h0.1) (fun hq ↦ this q hq h0.2)\n    exact fun r hr hr0 ↦ ⟨r, lt_iff_le_and_ne.mpr\n      ⟨Polynomial.valuation_le_one_of_valuation_X_le_one _ hle r, hr⟩, hr0⟩\n\nprivate lemma one_le_valuation_factor (hne : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty) {a b : K[X]}\n    (hab : v ↑(a * b) < 1 ∧ a ≠ 0 ∧ b ≠ 0) (hπᵥ : degree_lt_wf.min _ hne = a * b)\n    (hb : ¬IsUnit b) : 1 ≤ v ↑a := by\n  set πᵥ := degree_lt_wf.min _ hne\n  have hda : a.degree < πᵥ.degree := by\n    have hbpos := degree_pos_of_ne_zero_of_nonunit hab.2.2 hb\n    simp_rw [hπᵥ, degree_mul, degree_eq_natDegree hab.2.1, degree_eq_natDegree hab.2.2] at hbpos ⊢\n    norm_cast\n    simpa using hbpos\n  have hlea := imp_not_comm.mp (degree_lt_wf.not_lt_min _) hda\n  grind\n\nlemma irreducible_min_polynomial_valuation_lt_one_and_ne_zero [v.IsTrivialOn K]\n    (hne : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty) :\n    Irreducible (degree_lt_wf.min {p : K[X] | v p < 1 ∧ p ≠ 0} hne) := by\n  set πᵥ := degree_lt_wf.min _ hne\n  have hπᵥ : v πᵥ < 1 ∧ πᵥ ≠ 0 := degree_lt_wf.min_mem _ hne\n  refine irreducible_iff.mpr ⟨?_, fun a b hab ↦ ?_⟩\n  · simp only [Polynomial.isUnit_iff, isUnit_iff_ne_zero]\n    intro ⟨a, ha0, ha⟩\n    rw [← ha, coePolynomial, algebraMap_C, ← algebraMap_eq_C] at hπᵥ\n    grind\n  · by_contra! H\n    simp only [hab, ne_eq, mul_eq_zero, not_or] at hπᵥ\n    have hva := one_le_valuation_factor hne hπᵥ hab H.2\n    simp only [mul_comm a b, @and_comm (¬a = 0)] at hπᵥ hab\n    have := Right.one_le_mul (one_le_valuation_factor hne hπᵥ hab H.1) hva\n    simp only [coePolynomial_eq_algebraMap, map_mul] at hπᵥ this\n    grind\n\nsection valuation_X_le_one\n\nvariable [v.IsNontrivial] [v.IsTrivialOn K] (hle : v RatFunc.X ≤ 1)\n\n/-- A uniformizing element for the valuation `v`, as a polynomial in `K[X]`. -/\nabbrev uniformizingPolynomial : K[X] :=\n  WellFounded.min degree_lt_wf _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)\n\n@[inherit_doc]\nlocal notation \"πᵥ\" => uniformizingPolynomial hle\n\nTarget:\nlemma uniformizingPolynomial_ne_zero : πᵥ ≠ 0 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_2293cf123c21","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"83c06b073bcf3e0618a7ac8f675b562f071866e8873afde5c632a1cc8fe80d75","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/RatFunc","family_id":"uniformizingpolynomial_ne_zero","file_id":"mathlib/Mathlib/NumberTheory/RatFunc/Ostrowski.lean","sample_id":"2293cf123c21972c1d2af5e35604c06efbc61aa089ba31c3f82c50d912dfbf35"},"split":"train"}
{"binding":{"chosen_proof_sha256":"297365a1d8f1a47429f28c814575b3f64d65dc37c4e0b829e52928a4af8f384b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7f59f02f1adc338374d83c01438102e2efac30fffd4f3810fcf815de0d1655df","source_sha256":"aac9c42310c3c68ba48257d401225da11673aa0693f3a499d296961d7c43ce42","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine le_antisymm ?_ ?_\n  · calc\n      stabilizer G (s ∪ t)\n        ≤ stabilizer G (s ∪ t) ⊓ stabilizer G t := by simpa\n      _ ≤ stabilizer G ((s ∪ t) \\ t) := stabilizer_inf_stabilizer_le_stabilizer_sdiff\n      _ = stabilizer G s := by rw [union_sdiff_cancel_right]; simpa [← disjoint_iff_inter_eq_empty]\n  · calc\n      stabilizer G s\n        ≤ stabilizer G s ⊓ stabilizer G t := by simpa\n      _ ≤ stabilizer G (s ∪ t) := stabilizer_inf_stabilizer_le_stabilizer_union","hard_negative":false,"metrics":{"chosen_tokens":90,"rejected_tokens":3,"token_jaccard":0.032258,"token_length_ratio":0.033333},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"1aff229bc63d479ebebb6ab307970995ae1afb5b3944fe76aef5de353a235217","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Pointwise.Finset\npublic import Mathlib.GroupTheory.QuotientGroup.Defs\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\n\nNamespace:\nMulAction\n\nLocal context:\n/-\nCopyright (c) 2023 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Stabilizer of a set under a pointwise action\n\nThis file characterises the stabilizer of a set/finset under the pointwise action of a group.\n-/\n\npublic section\n\nopen Function MulOpposite Set\nopen scoped Pointwise\n\nnamespace MulAction\nvariable {G H α : Type*}\n\n/-! ### Stabilizer of a set -/\n\nsection Set\nsection Group\nvariable [Group G] [Group H] [MulAction G α] {a : G} {s t : Set α}\n\n@[to_additive (attr := simp)]\nlemma stabilizer_empty : stabilizer G (∅ : Set α) = ⊤ :=\n  Subgroup.coe_eq_univ.1 <| eq_univ_of_forall fun _a ↦ smul_set_empty\n\n@[to_additive (attr := simp)]\nlemma stabilizer_univ : stabilizer G (Set.univ : Set α) = ⊤ := by\n  ext\n  simp\n\n@[to_additive (attr := simp)]\nlemma stabilizer_singleton (b : α) : stabilizer G ({b} : Set α) = stabilizer G b := by ext; simp\n\n@[to_additive]\nlemma mem_stabilizer_set {s : Set α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s := by\n  refine mem_stabilizer_iff.trans ⟨fun h b ↦ ?_, fun h ↦ ?_⟩\n  · rw [← (smul_mem_smul_set_iff : a • b ∈ _ ↔ _), h]\n  simp_rw [Set.ext_iff, mem_smul_set_iff_inv_smul_mem]\n  exact ((MulAction.toPerm a).forall_congr' <| by simp [Iff.comm]).1 h\n\n@[to_additive]\nlemma map_stabilizer_le (f : G →* H) (s : Set G) :\n    (stabilizer G s).map f ≤ stabilizer H (f '' s) := by\n  rintro a\n  simp only [Subgroup.mem_map, mem_stabilizer_iff, forall_exists_index, and_imp]\n  rintro a ha rfl\n  rw [← image_smul_distrib, ha]\n\n@[to_additive (attr := simp)]\nlemma stabilizer_mul_self (s : Set G) : (stabilizer G s : Set G) * s = s := by\n  ext\n  refine ⟨?_, fun h ↦ ⟨_, (stabilizer G s).one_mem, _, h, one_mul _⟩⟩\n  rintro ⟨a, ha, b, hb, rfl⟩\n  rw [← mem_stabilizer_iff.1 ha]\n  exact smul_mem_smul_set hb\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_apply₂ {f : Set α → Set α → Set α}\n    (hf : ∀ a : G, a • f s t = f (a • s) (a • t)) :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (f s t) := by aesop (add simp [SetLike.le_def])\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_union :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s ∪ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_union\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_inter :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s ∩ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_inter\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_sdiff :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s \\ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_sdiff\n\n@[to_additive]\n\nTarget:\nlemma stabilizer_union_eq_left (hdisj : Disjoint s t) (hstab : stabilizer G s ≤ stabilizer G t)\n    (hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G t) :\n    stabilizer G (s ∪ t) = stabilizer G s :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Pointwise","family_id":"stabilizer_union_eq_left","file_id":"mathlib/Mathlib/Algebra/Pointwise/Stabilizer.lean","sample_id":"7f59f02f1adc338374d83c01438102e2efac30fffd4f3810fcf815de0d1655df"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e0b9bed3a6acbad85454a981928799bf6c6adb0fb6d0013326ad4cea47a0b244","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"194f0f4367719ea1e6824ec86d27af7e10162a98d826fa13df412b7a41e3d577","source_sha256":"ef9864ce066b7696b62174e7a3b3a449882e3ab320bb2e19ef0e02cda6607291","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [r_eq_r' S, r', Con.rel_mk]","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":8,"token_jaccard":0.055556,"token_length_ratio":0.615385},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"1b2701093ed9e2bc39f5a1c97f74b26c79bfcf69fcce92f5a8854ec382843f96","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\npublic import Mathlib.Algebra.Group.Submonoid.Operations\npublic import Mathlib.Algebra.Regular.Basic\npublic import Mathlib.GroupTheory.Congruence.Hom\npublic import Mathlib.GroupTheory.OreLocalization.Basic\n\nNamespace:\nLocalization\n\nLocal context:\n/-\nCopyright (c) 2019 Amelia Livingston. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Amelia Livingston\n-/\n/-!\n# Localizations of commutative monoids\n\nLocalizing a commutative ring at one of its submonoids does not rely on the ring's addition, so\nwe can generalize localizations to commutative monoids.\n\nWe characterize the localization of a commutative monoid `M` at a submonoid `S` up to\nisomorphism; that is, a commutative monoid `N` is the localization of `M` at `S` iff we can find a\nmonoid homomorphism `f : M →* N` satisfying 3 properties:\n1. For all `y ∈ S`, `f y` is a unit;\n2. For all `z : N`, there exists `(x, y) : M × S` such that `z * f y = f x`;\n3. For all `x, y : M` such that `f x = f y`, there exists `c ∈ S` such that `x * c = y * c`.\n   (The converse is a consequence of 1.)\n\nGiven such a localization map `f : M →* N`, we can define the surjection\n`Submonoid.LocalizationMap.mk'` sending `(x, y) : M × S` to `f x * (f y)⁻¹`. Mapping properties\nof the localization (e.g. extending a map from `M → P` to `N` if the image of `S` is contained in\nthe units) are treated in a later file `Mathlib.GroupTheory.MonoidLocalization.Maps`.\n\nWe also define the quotient of `M × S` by the unique congruence relation (equivalence relation\npreserving a binary operation) `r` such that for any other congruence relation `s` on `M × S`\nsatisfying '`∀ y ∈ S`, `(1, 1) ∼ (y, y)` under `s`', we have that `(x₁, y₁) ∼ (x₂, y₂)` by `s`\nwhenever `(x₁, y₁) ∼ (x₂, y₂)` by `r`. We show this relation is equivalent to the standard\nlocalization relation.\nThis defines the localization as a quotient type, `Localization`, but the majority of\nsubsequent lemmas in the file are given in terms of localizations up to isomorphism, using maps\nwhich satisfy the characteristic predicate.\n\nThe Grothendieck group construction corresponds to localizing at the top submonoid, namely making\nevery element invertible.\n\n## Implementation notes\n\nIn maths it is natural to reason up to isomorphism, but in Lean we cannot naturally `rewrite` one\nstructure with an isomorphic one; one way around this is to isolate a predicate characterizing\na structure up to isomorphism, and reason about things that satisfy the predicate.\n\nThe infimum form of the localization congruence relation is chosen as 'canonical' here, since it\nshortens some proofs.\n\nTo reason about the localization as a quotient type, use `mk_eq_monoidOf_mk'` and associated\nlemmas. These show the quotient map `mk : M → S → Localization S` equals the\nsurjection `LocalizationMap.mk'` induced by the map\n`Localization.monoidOf : Submonoid.LocalizationMap S (Localization S)` (where `of` establishes the\nlocalization as a quotient type satisfies the characteristic predicate). The lemma\n`mk_eq_monoidOf_mk'` hence gives you access to the results in the rest of the file, which are about\nthe `LocalizationMap.mk'` induced by any localization map.\n\n## TODO\n\n* Show that the localization at the top monoid is a group.\n* Generalise to (nonempty) subsemigroups.\n* If we acquire more bundlings, we can make `Localization.mkOrderEmbedding` be an ordered monoid\n  embedding.\n\n## Tags\nlocalization, monoid localization, quotient monoid, congruence relation, characteristic predicate,\ncommutative monoid, grothendieck group\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero Ring\n\nopen Function\nnamespace AddSubmonoid\n\nvariable {M : Type*} [AddCommMonoid M] (S : AddSubmonoid M) (N : Type*) [AddCommMonoid N]\n\nvariable {N} in\n/-- A predicate characterizing homomorphisms between additive monoids `M` and `N` that form a\ncommutative triangle with the canonical map from `M` to its localization at `S` and\nsome isomorphism between `N` and the localization. -/\nstructure IsLocalizationMap (S : AddSubmonoid M) (f : M → N) where\n  map_addUnits (y : S) : IsAddUnit (f y)\n  surj (z : N) : ∃ x : M × S, z + f x.2 = f x.1\n  exists_of_eq {x y} : f x = f y → ∃ c : S, c + x = c + y\n\n/-- The type of AddMonoid homomorphisms satisfying the characteristic predicate: if `f : M →+ N`\nsatisfies this predicate, then `N` is isomorphic to the localization of `M` at `S`. -/\nstructure LocalizationMap extends M →ₙ+ N where\n  isLocalizationMap : IsLocalizationMap S toFun\n\n/-- The additive homomorphism underlying a `LocalizationMap` of `AddCommMonoid`s. -/\nadd_decl_doc LocalizationMap.toAddHom\n\nend AddSubmonoid\n\nsection CommMonoid\n\nvariable {M : Type*} [CommMonoid M] (S : Submonoid M) (N : Type*) [CommMonoid N] {P : Type*}\n  [CommMonoid P]\n\nnamespace Submonoid\n\nvariable {N} in\n/-- A predicate characterizing homomorphisms between monoids `M` and `N` that form a\ncommutative triangle with the canonical map from `M` to its localization at `S` and\nsome isomorphism between `N` and the localization. -/\n@[to_additive (attr := mk_iff)]\nstructure IsLocalizationMap (S : Submonoid M) (f : M → N) where\n  map_units (y : S) : IsUnit (f y)\n  surj (z : N) : ∃ x : M × S, z * f x.2 = f x.1\n  exists_of_eq {x y} : f x = f y → ∃ c : S, c * x = c * y\n\n/-- The type of monoid homomorphisms satisfying the characteristic predicate: if `f : M →* N`\nsatisfies this predicate, then `N` is isomorphic to the localization of `M` at `S`. -/\n@[to_additive] structure LocalizationMap extends M →ₙ* N where\n  isLocalizationMap : IsLocalizationMap S toFun\n\n/-- The multiplicative homomorphism underlying a `LocalizationMap`. -/\nadd_decl_doc LocalizationMap.toMulHom\n\nend Submonoid\n\nnamespace Localization\n\n/- Ensure that `@[to_additive]` uses the right namespace before the definition of `Localization`. -/\ninsert_to_additive_translation Localization AddLocalization\n\n/-- The congruence relation on `M × S`, `M` a `CommMonoid` and `S` a submonoid of `M`, whose\nquotient is the localization of `M` at `S`, defined as the unique congruence relation on\n`M × S` such that for any other congruence relation `s` on `M × S` where for all `y ∈ S`,\n`(1, 1) ∼ (y, y)` under `s`, we have that `(x₁, y₁) ∼ (x₂, y₂)` by `r` implies\n`(x₁, y₁) ∼ (x₂, y₂)` by `s`. -/\n@[to_additive\n/-- The congruence relation on `M × S`, `M` an `AddCommMonoid` and `S` an `AddSubmonoid` of `M`,\nwhose quotient is the localization of `M` at `S`, defined as the unique congruence relation on\n`M × S` such that for any other congruence relation `s` on `M × S` where for all `y ∈ S`,\n`(0, 0) ∼ (y, y)` under `s`, we have that `(x₁, y₁) ∼ (x₂, y₂)` by `r` implies\n`(x₁, y₁) ∼ (x₂, y₂)` by `s`. -/]\ndef r (S : Submonoid M) : Con (M × S) :=\n  sInf { c | ∀ y : S, c 1 (y, y) }\n\n/-- An alternate form of the congruence relation on `M × S`, `M` a `CommMonoid` and `S` a\nsubmonoid of `M`, whose quotient is the localization of `M` at `S`. -/\n@[to_additive\n/-- An alternate form of the congruence relation on `M × S`, `M` a `CommMonoid` and `S` a\nsubmonoid of `M`, whose quotient is the localization of `M` at `S`. -/]\ndef r' : Con (M × S) := by\n  -- note we multiply by `c` on the left so that we can later generalize to `•`\n  refine\n    { r := fun a b : M × S ↦ ∃ c : S, ↑c * (↑b.2 * a.1) = c * (a.2 * b.1)\n      iseqv := ⟨fun a ↦ ⟨1, rfl⟩, fun ⟨c, hc⟩ ↦ ⟨c, hc.symm⟩, ?_⟩\n      mul' := ?_ }\n  · rintro a b c ⟨t₁, ht₁⟩ ⟨t₂, ht₂⟩\n    use t₂ * t₁ * b.2\n    simp only [Submonoid.coe_mul]\n    calc\n      (t₂ * t₁ * b.2 : M) * (c.2 * a.1) = t₂ * c.2 * (t₁ * (b.2 * a.1)) := by ac_rfl\n      _ = t₁ * a.2 * (t₂ * (c.2 * b.1)) := by rw [ht₁]; ac_rfl\n      _ = t₂ * t₁ * b.2 * (a.2 * c.1) := by rw [ht₂]; ac_rfl\n  · rintro a b c d ⟨t₁, ht₁⟩ ⟨t₂, ht₂⟩\n    use t₂ * t₁\n    calc\n      (t₂ * t₁ : M) * (b.2 * d.2 * (a.1 * c.1)) = t₂ * (d.2 * c.1) * (t₁ * (b.2 * a.1)) := by ac_rfl\n      _ = (t₂ * t₁ : M) * (a.2 * c.2 * (b.1 * d.1)) := by rw [ht₁, ht₂]; ac_rfl\n\n/-- The congruence relation used to localize a `CommMonoid` at a submonoid can be expressed\nequivalently as an infimum (see `Localization.r`) or explicitly\n(see `Localization.r'`). -/\n@[to_additive\n/-- The additive congruence relation used to localize an `AddCommMonoid` at a submonoid can be\nexpressed equivalently as an infimum (see `AddLocalization.r`) or explicitly\n(see `AddLocalization.r'`). -/]\ntheorem r_eq_r' : r S = r' S :=\n  le_antisymm (sInf_le fun _ ↦ ⟨1, by simp⟩) <|\n    le_sInf fun b H ⟨p, q⟩ ⟨x, y⟩ ⟨t, ht⟩ ↦ by\n      rw [← one_mul (p, q), ← one_mul (x, y)]\n      refine b.trans (b.mul (H (t * y)) (b.refl _)) ?_\n      convert! b.symm (b.mul (H (t * q)) (b.refl (x, y))) using 1\n      dsimp only [Prod.mk_mul_mk, Submonoid.coe_mul] at ht ⊢\n      simp_rw [mul_assoc, ht, mul_comm y q]\n\nvariable {S}\n\n@[to_additive]\n\nTarget:\ntheorem r_iff_exists {x y : M × S} : r S x y ↔ ∃ c : S, ↑c * (↑y.2 * x.1) = c * (x.2 * y.1) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"94e24a42d1169d605949dda928a07c4171eb9f911df0f2c03338dc723c0cf0ce","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/MonoidLocalization","family_id":"r_iff_exists","file_id":"mathlib/Mathlib/GroupTheory/MonoidLocalization/Basic.lean","sample_id":"194f0f4367719ea1e6824ec86d27af7e10162a98d826fa13df412b7a41e3d577"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6296f9ab27c166fefa518a334c1d06c276278c6742ae2b92e22ac06c24c52565","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6775aa7a9f8509a5e61608d136b1fa3addef965d34c87b8593996c354ac35167","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c88d0c7a08db8e0ee629366fac952f34c781e69b50a4902cc54b66d0c91187ae","source_sha256":"20106a17e82d1783ea6f9971b872fe8ba3aa3b18f09bbcdf6c093b13edf6c381","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  conv_rhs => rw [← sum_single a, ← sum_single b]\n  simp_rw [mul_def, map_finsuppSum, liftNC_single, Finsupp.sum_mul, Finsupp.mul_sum]\n  refine Finset.sum_congr rfl fun y hy => Finset.sum_congr rfl fun x _hx => ?_\n  simp [mul_assoc, (h_comm hy).left_comm]","hard_negative":true,"metrics":{"chosen_tokens":59,"rejected_tokens":5,"token_jaccard":0.051282,"token_length_ratio":0.084746},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"1b800e55e58c2bf813d8b2e51ce54465c3efa3cb636b13eb467e653adb70fd2e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MonoidAlgebra.Defs\n\nNamespace:\nMonoidAlgebra\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Yury Kudryashov, Kim Morrison\n-/\n/-!\n# Lifting monoid algebras\n\nThis file defines `liftNC`. For the definition of `MonoidAlgebra.lift`, see\n`Mathlib/Algebra/MonoidAlgebra/Basic.lean`.\n\n## Main results\n* `MonoidAlgebra.liftNC`, `AddMonoidAlgebra.liftNC`: lift a homomorphism `f : k →+ R` and a\n  function `g : G → R` to a homomorphism `k[G] →+ R`.\n-/\n\n@[expose] public section\n\nassert_not_exists NonUnitalAlgHom AlgEquiv\n\nnoncomputable section\n\nopen Finsupp hiding single\n\nuniverse u₁ u₂ u₃ u₄\n\nvariable (k : Type u₁) (G : Type u₂) (H : Type*) {R S T M : Type*}\n\n/-! ### Multiplicative monoids -/\n\nnamespace MonoidAlgebra\n\nvariable {k G}\n\nsection\n\nvariable [Semiring k] [NonUnitalNonAssocSemiring R]\n\n/-- A non-commutative version of `MonoidAlgebra.lift`: given an additive homomorphism `f : k →+ R`\nand a homomorphism `g : G → R`, returns the additive homomorphism from\n`k[G]` such that `liftNC f g (single a b) = f b * g a`. If `f` is a ring homomorphism\nand the range of either `f` or `g` is in center of `R`, then the result is a ring homomorphism.  If\n`R` is a `k`-algebra and `f = algebraMap k R`, then the result is an algebra homomorphism called\n`MonoidAlgebra.lift`. -/\ndef liftNC (f : k →+ R) (g : G → R) : k[G] →+ R :=\n  liftAddHom fun x : G => (AddMonoidHom.mulRight (g x)).comp f\n\n@[simp]\ntheorem liftNC_single (f : k →+ R) (g : G → R) (a : G) (b : k) :\n    liftNC f g (single a b) = f b * g a :=\n  liftAddHom_apply_single _ _ _\n\nend\n\nsection Mul\n\nvariable [Semiring k] [Mul G] [Semiring R]\n\nTarget:\ntheorem liftNC_mul {g_hom : Type*} [FunLike g_hom G R] [MulHomClass g_hom G R]\n    (f : k →+* R) (g : g_hom) (a b : k[G])\n    (h_comm : ∀ {x y}, y ∈ a.support → Commute (f (b x)) (g y)) :\n    liftNC (f : k →+ R) g (a * b) = liftNC (f : k →+ R) g a * liftNC (f : k →+ R) g b :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_c88d0c7a08db","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"2b155782d8d28f9712302bdd48a7ab8ff39d800118237fa6fc0bc0dd2ef8f3f0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/MonoidAlgebra","family_id":"liftnc_mul","file_id":"mathlib/Mathlib/Algebra/MonoidAlgebra/Lift.lean","sample_id":"c88d0c7a08db8e0ee629366fac952f34c781e69b50a4902cc54b66d0c91187ae"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5cffe96cad332b72c0ae2797275016448548d2c1b107d88e2f58540117e63993","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f6946cb3c4bfb0278075b43218a3f062725f7b92d4a1e74a22d268efa20acb64","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"501b3d3640125c7dca7fdd0dca6da79f794dca628efb68f4fea9ebb8533db64a","source_sha256":"f4774116e10dc728faae25beafedcc16dba2b4511270606e92d6263e552fccf8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    obtain ⟨a', m⟩ := w.mem_essImage α\n    obtain ⟨m'⟩ := m\n    exact ⟨a', ⟨(Iso.toEquiv m').symm.trans Equiv.ulift⟩⟩","hard_negative":true,"metrics":{"chosen_tokens":38,"rejected_tokens":5,"token_jaccard":0.125,"token_length_ratio":0.131579},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"1b9dc9f4c97d0b6a8ffbac9f793b15465283301d55cefc78c179fa3545231cde","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.EssentialImage\npublic import Mathlib.CategoryTheory.Types.Basic\npublic import Mathlib.Logic.UnivLE\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Universe inequalities and essential surjectivity of `uliftFunctor`.\n\nWe show `UnivLE.{max u v, v} ↔ EssSurj (uliftFunctor.{u, v} : Type v ⥤ Type max u v)`.\n-/\n\n@[expose] public section\n\nopen CategoryTheory\n\nuniverse u v\n\nnoncomputable section\n\nTarget:\ntheorem UnivLE.ofEssSurj (w : (uliftFunctor.{u, v} : Type v ⥤ Type max u v).EssSurj) :\n    UnivLE.{max u v, v} where\n  small α :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_501b3d364012","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"0ab2687159bd543ea36b52c22feaf5c779f21a802821f63d591475b8f15e0bb8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory","family_id":"univle","file_id":"mathlib/Mathlib/CategoryTheory/UnivLE.lean","sample_id":"501b3d3640125c7dca7fdd0dca6da79f794dca628efb68f4fea9ebb8533db64a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5e2bc200513a76ad4fb8bd2871f6348a8c6776f4fb4e2960dcaa904707705fee","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6a8e5453b7a4db469efd31b432f7b75acf3e7eec129c2d2a6b546cf031d8a71a","source_sha256":"80fda85bf9297b0a26c4ff540728683da76211a2d7626ecf60d2ca016feea2f7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  -- Using `have` not `let` since it is only existence of `NormedSpace` structure that we need.\n  have : NormedSpace ℝ E := .restrictScalars ℝ K E\n  have : FiniteDimensional ℝ E := FiniteDimensional.trans ℝ K E\n  infer_instance\n\nvariable {E}","hard_negative":false,"metrics":{"chosen_tokens":53,"rejected_tokens":3,"token_jaccard":0.030303,"token_length_ratio":0.056604},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"1bb09380576cee7409ea4c91452dc6959355926ed229f9c9de9be57da2e8dbda","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Module.FiniteDimension\npublic import Mathlib.Analysis.RCLike.Basic\npublic import Mathlib.Topology.Instances.RealVectorSpace\n\nNamespace:\nFiniteDimensional\n\nLocal context:\n/-\nCopyright (c) 2020 Frédéric Dupuis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Frédéric Dupuis\n-/\n/-! # Further lemmas about `RCLike` -/\n\npublic section\n\nopen scoped Finset\n\nvariable {K E : Type*} [RCLike K]\n\nopen ComplexOrder RCLike in\nlemma convex_RCLike_iff_convex_real [AddCommMonoid E] [Module K E] [Module ℝ E]\n    [IsScalarTower ℝ K E] {s : Set E} : Convex K s ↔ Convex ℝ s :=\n  ⟨Convex.lift ℝ,\n  fun hs => convex_of_nonneg_surjective_algebraMap _ (fun _ => nonneg_iff_exists_ofReal.mp) hs⟩\n\nnamespace Polynomial\n\ntheorem ofReal_eval (p : ℝ[X]) (x : ℝ) : (↑(p.eval x) : K) = aeval (↑x) p :=\n  (@aeval_algebraMap_apply_eq_algebraMap_eval ℝ K _ _ _ x p).symm\n\nend Polynomial\n\nvariable (K) in\nlemma RCLike.span_one_I : Submodule.span ℝ (M := K) {1, I} = ⊤ := by\n  suffices ∀ x : K, ∃ a b : ℝ, a • 1 + b • I = x by\n    simpa [Submodule.eq_top_iff', Submodule.mem_span_pair]\n  exact fun x ↦ ⟨re x, im x, by simp [real_smul_eq_coe_mul]⟩\n\nvariable (K) in\nlemma RCLike.rank_le_two : Module.rank ℝ K ≤ 2 :=\n  calc\n    _ = Module.rank ℝ ↥(Submodule.span ℝ ({1, I} : Set K)) := by rw [span_one_I]; simp\n    _ ≤ #({1, I} : Finset K) := by\n      -- TODO: `simp` doesn't rewrite inside the type argument to `Module.rank`, but `rw` does.\n      -- We should introduce `Submodule.rank` to fix this.\n      have := rank_span_finset_le (R := ℝ) (M := K) {1, I}\n      rw [Finset.coe_pair] at this\n      simpa [span_one_I] using this\n    _ ≤ 2 := mod_cast Finset.card_le_two\n\nvariable (K) in\nlemma RCLike.finrank_le_two : Module.finrank ℝ K ≤ 2 :=\n  Module.finrank_le_of_rank_le <| rank_le_two _\n\nnamespace FiniteDimensional\n\nopen RCLike\n\nlibrary_note «RCLike instance» /--\nThis instance generates a type-class problem with a metavariable `?m` that should satisfy\n`RCLike ?m`. Since this can only be satisfied by `ℝ` or `ℂ`, this does not cause problems. -/\n\n/-- An `RCLike` field is finite-dimensional over `ℝ`, since it is spanned by `{1, I}`. -/\ninstance rclike_to_real : FiniteDimensional ℝ K := ⟨{1, I}, by simp [span_one_I]⟩\n\nvariable (K E)\nvariable [NormedAddCommGroup E] [NormedSpace K E]\n\n/-- A finite-dimensional vector space over an `RCLike` is a proper metric space.\n\nThis is not an instance because it would cause a search for `FiniteDimensional ?x E` before\n`RCLike ?x`. -/\n\nTarget:\ntheorem proper_rclike [FiniteDimensional K E] : ProperSpace E :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/RCLike","family_id":"proper_rclike","file_id":"mathlib/Mathlib/Analysis/RCLike/Lemmas.lean","sample_id":"6a8e5453b7a4db469efd31b432f7b75acf3e7eec129c2d2a6b546cf031d8a71a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"230ad1a22636c034f298a25415dfe821b738da09682e6993796fa2ce84eb0a0a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"16bdb0569806200171e13b3e0d9b0d935a38c6f276941a0946af8ff38e8cee10","source_sha256":"25ee7ccac39d94bf37419f1553f1c5ec483939108b5bb4af754da5f173eef853","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← nsmul_one, D.map_smul_of_tower n, map_one_eq_zero, smul_zero]","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":2,"token_jaccard":0.071429,"token_length_ratio":0.133333},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"1bc307d1b4b2666b956ef06221e5269c0b98aed952e53dcd53d11700c0aa253a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.AlgebraMap\npublic import Mathlib.Algebra.Polynomial.Derivative\n\nNamespace:\nDerivation\n\nLocal context:\n/-\nCopyright (c) 2020 Nicolò Cavalleri. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Nicolò Cavalleri, Andrew Yang\n-/\n/-!\n# Derivations\n\nThis file defines derivation. A derivation `D` from the `R`-algebra `A` to the `A`-module `M` is an\n`R`-linear map that satisfy the Leibniz rule `D (a * b) = a * D b + D a * b`.\n\n## Main results\n\n- `Derivation`: The type of `R`-derivations from `A` to `M`. This has an `A`-module structure.\n- `Derivation.llcomp`: We may compose linear maps and derivations to obtain a derivation,\n  and the composition is bilinear.\n\nSee `Mathlib/RingTheory/Derivation/Lie.lean` for\n- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.\n\nand `Mathlib/RingTheory/Derivation/ToSquareZero.lean` for\n- `derivationToSquareZeroEquivLift`: The `R`-derivations from `A` into a square-zero ideal `I`\n  of `B` corresponds to the lifts `A →ₐ[R] B` of the map `A →ₐ[R] B ⧸ I`.\n\n## Future project\n\n- Generalize derivations into bimodules.\n\n-/\n\n@[expose] public section\n\nopen Algebra\n\n/-- `D : Derivation R A M` is an `R`-linear map from `A` to `M` that satisfies the `leibniz`\nequality. We also require that `D 1 = 0`. See `Derivation.mk'` for a constructor that deduces this\nassumption from the Leibniz rule when `M` is cancellative.\n\nTODO: update this when bimodules are defined. -/\nstructure Derivation (R : Type*) (A : Type*) (M : Type*)\n    [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M]\n    extends A →ₗ[R] M where\n  protected map_one_eq_zero' : toLinearMap 1 = 0\n  protected leibniz' (a b : A) : toLinearMap (a * b) = a • toLinearMap b + b • toLinearMap a\n\n/-- The `LinearMap` underlying a `Derivation`. -/\nadd_decl_doc Derivation.toLinearMap\n\nnamespace Derivation\n\nsection\n\nvariable {R : Type*} {A : Type*} {B : Type*} {M : Type*}\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B] [AddCommMonoid M]\nvariable [Algebra R A] [Algebra R B]\nvariable [Module A M] [Module B M] [Module R M]\n\n\nvariable (D : Derivation R A M) {D1 D2 : Derivation R A M} (r : R) (a b : A)\n\ninstance : FunLike (Derivation R A M) A M where\n  coe D := D.toFun\n  coe_injective D1 D2 h := by cases D1; cases D2; congr; exact DFunLike.coe_injective h\n\ninstance : AddMonoidHomClass (Derivation R A M) A M where\n  map_add D := D.toLinearMap.map_add'\n  map_zero D := D.toLinearMap.map_zero\n\n-- Not a simp lemma because it can be proved via `coeFn_coe` + `toLinearMap_eq_coe`\ntheorem toFun_eq_coe : D.toFun = ⇑D :=\n  rfl\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply (D : Derivation R A M) : A → M := D\n\ninitialize_simps_projections Derivation (toFun → apply)\n\nattribute [coe] toLinearMap\n\ninstance hasCoeToLinearMap : Coe (Derivation R A M) (A →ₗ[R] M) :=\n  ⟨fun D => D.toLinearMap⟩\n\n@[simp]\ntheorem mk_coe (f : A →ₗ[R] M) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : Derivation R A M) : A → M) = f :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coeFn_coe (f : Derivation R A M) : ⇑(f : A →ₗ[R] M) = f :=\n  rfl\n\ntheorem coe_injective : @Function.Injective (Derivation R A M) (A → M) DFunLike.coe :=\n  DFunLike.coe_injective\n\n@[ext]\ntheorem ext (H : ∀ a, D1 a = D2 a) : D1 = D2 :=\n  DFunLike.ext _ _ H\n\ntheorem congr_fun (h : D1 = D2) (a : A) : D1 a = D2 a :=\n  DFunLike.congr_fun h a\n\nprotected theorem map_add : D (a + b) = D a + D b :=\n  map_add D a b\n\nprotected theorem map_zero : D 0 = 0 :=\n  map_zero D\n\n@[simp]\ntheorem map_smul : D (r • a) = r • D a :=\n  D.toLinearMap.map_smul r a\n\n@[simp]\ntheorem leibniz : D (a * b) = a • D b + b • D a :=\n  D.leibniz' _ _\n\n@[simp]\ntheorem map_smul_of_tower {S : Type*} [SMul S A] [SMul S M] [LinearMap.CompatibleSMul A M S R]\n    (D : Derivation R A M) (r : S) (a : A) : D (r • a) = r • D a :=\n  D.toLinearMap.map_smul_of_tower r a\n\n@[simp]\ntheorem map_one_eq_zero : D 1 = 0 :=\n  D.map_one_eq_zero'\n\n@[simp]\ntheorem map_algebraMap : D (algebraMap R A r) = 0 := by\n  rw [← mul_one r, map_mul, map_one, ← smul_def, map_smul, map_one_eq_zero, smul_zero]\n\n@[simp]\n\nTarget:\ntheorem map_natCast (n : ℕ) : D (n : A) = 0 :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Derivation","family_id":"map_natcast","file_id":"mathlib/Mathlib/RingTheory/Derivation/Basic.lean","sample_id":"16bdb0569806200171e13b3e0d9b0d935a38c6f276941a0946af8ff38e8cee10"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4d337297249649c3a21f0b564e2b0bbdb408fd0e071471a465068a7173e4de87","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cfc88893dfc4400feb7142404abe76629a26bbb69437adaf877cb8c74ee10580","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"91bf84bd60b26a35f7dabcc39cf740009b92e0c19b956edd8d544cfa8e467dbb","source_sha256":"bf073edff8092ae47149eb64a7d94002d4cd84fb4941057816c6cf0f37fc36aa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isSymm_iff_basis b, Matrix.IsHermitian.ext_iff, forall_comm]\n  simp [Eq.comm]","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":3,"token_jaccard":0.058824,"token_length_ratio":0.15},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"1bd5510092a909f6adfdd0a74c9a302aaba11f78e08015a0b3c9ad2dc39c0db8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Matrix.PosDef\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Etienne Marion. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Etienne Marion\n-/\n/-!\n# Sesquilinear forms over a star ring\n\nThis file provides some properties about sesquilinear forms `M →ₗ⋆[R] M →ₗ[R] R` when `R` is a\n`StarRing`.\n-/\n\npublic section\n\nopen Module LinearMap\n\nvariable {R M n : Type*} [CommSemiring R] [StarRing R] [AddCommMonoid M] [Module R M]\n  [Fintype n] [DecidableEq n]\n  {B : M →ₗ⋆[R] M →ₗ[R] R} (b : Basis n R M)\n\nlemma LinearMap.isSymm_iff_basis {ι : Type*} (b : Basis ι R M) :\n    IsSymm B ↔ ∀ i j, star (B (b i) (b j)) = B (b j) (b i) where\n  mp h i j := h.eq _ _\n  mpr := by\n    refine fun h ↦ ⟨fun x y ↦ ?_⟩\n    obtain ⟨fx, tx, ix, -, hx⟩ := Submodule.mem_span_iff_exists_finset_subset.1\n      (by simp : x ∈ Submodule.span R (Set.range b))\n    obtain ⟨fy, ty, iy, -, hy⟩ := Submodule.mem_span_iff_exists_finset_subset.1\n      (by simp : y ∈ Submodule.span R (Set.range b))\n    rw [← hx, ← hy]\n    simp only [map_sum, LinearMap.map_smulₛₗ, starRingEnd_apply, map_smul, coe_sum,\n      Finset.sum_apply, smul_apply, smul_eq_mul, Finset.mul_sum, map_mul, star_star]\n    rw [Finset.sum_comm]\n    refine Finset.sum_congr rfl (fun b₁ h₁ ↦ Finset.sum_congr rfl fun b₂ h₂ ↦ ?_)\n    rw [mul_left_comm]\n    obtain ⟨i, rfl⟩ := ix h₁\n    obtain ⟨j, rfl⟩ := iy h₂\n    rw [h]\n\nTarget:\nlemma LinearMap.isSymm_iff_isHermitian_toMatrix : B.IsSymm ↔ (toMatrix₂ b b B).IsHermitian :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_91bf84bd60b2","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"a9d3f6a1c0980cf019f6eb771bf669545173be1302c5d6721278e94b70605a5d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/SesquilinearForm","family_id":"linearmap","file_id":"mathlib/Mathlib/LinearAlgebra/SesquilinearForm/Star.lean","sample_id":"91bf84bd60b26a35f7dabcc39cf740009b92e0c19b956edd8d544cfa8e467dbb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5110e356531b5aca6a5cc8c637eddfe770b9e7e43c44858448f93b443e747ce9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6c64e74b2c20ff26c2afb932dbf9dff923649fb90ef50d86ead94e0006609bba","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"948388cf0ecec93fe348cbe9fadcd98478b5894a8ffcc1cd67d7ba5e880077f1","source_sha256":"76bf26908fe5bb3f22e0b1de59c408f5480271bc0d25bef1d614882cce4d1085","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun ⟨H⟩ ↦ ⟨?_⟩, fun ⟨H⟩ ↦ ⟨?_⟩⟩ <;>\n    simp only [isCardinalFiltered_aleph0_iff] at * <;>\n    exact H","hard_negative":false,"metrics":{"chosen_tokens":38,"rejected_tokens":43,"token_jaccard":0.807692,"token_length_ratio":1.131579},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"1bd8a7dfa24371e52b3607102934209e9286447ca70cf379b655b95928f2895a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Filtered\npublic import Mathlib.CategoryTheory.Limits.Preserves.Filtered\npublic import Mathlib.CategoryTheory.Limits.Types.Filtered\npublic import Mathlib.CategoryTheory.MorphismProperty.Basic\npublic import Mathlib.CategoryTheory.Presentable.Basic\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Finitely Presentable Objects\n\nWe define finitely presentable objects as a synonym for `ℵ₀`-presentable objects,\nand link this definition with the preservation of filtered colimits.\n\n-/\n\n@[expose] public section\n\n\nuniverse w v' v u' u\n\nnamespace CategoryTheory\n\nopen Limits Opposite Cardinal\n\nvariable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]\n\nattribute [local instance] fact_isRegular_aleph0\n\n/-- A functor `F : C ⥤ D` is finitely accessible if it is `ℵ₀`-accessible.\nEquivalently, it preserves all filtered colimits.\nSee `CategoryTheory.Functor.IsFinitelyAccessible_iff_preservesFilteredColimits`. -/\nabbrev Functor.IsFinitelyAccessible (F : C ⥤ D) : Prop := IsCardinalAccessible.{w} F ℵ₀\n\nlemma Functor.IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize {F : C ⥤ D} :\n    IsFinitelyAccessible.{w} F ↔ PreservesFilteredColimitsOfSize.{w, w} F := by\n  refine ⟨fun ⟨H⟩ ↦ ⟨?_⟩, fun ⟨H⟩ ↦ ⟨?_⟩⟩ <;>\n    simp only [isCardinalFiltered_aleph0_iff] at * <;>\n    exact H\n\nlemma Functor.isFinitelyAccessible_iff_preservesFilteredColimits {F : C ⥤ D} :\n    IsFinitelyAccessible.{v'} F ↔ PreservesFilteredColimits F :=\n  IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize\n\n/-- An object `X` is finitely presentable if `Hom(X, -)` preserves all filtered colimits. -/\nabbrev IsFinitelyPresentable (X : C) : Prop :=\n  IsCardinalPresentable.{w} X ℵ₀\n\nvariable (C) in\n/-- `IsFinitelyPresentable` as an `ObjectProperty` on `C`. This is sometimes called \"compact\". -/\ndef ObjectProperty.isFinitelyPresentable : ObjectProperty C := fun X ↦ IsFinitelyPresentable.{w} X\n\nlemma ObjectProperty.isFinitelyPresentable_eq_isCardinalPresentable :\n    isFinitelyPresentable.{w} C = isCardinalPresentable.{w} C ℵ₀ :=\n  rfl\n\nvariable (C) in\n/-- A morphism `f : X ⟶ Y` is finitely presentable if it is so as an object of `Under X`. -/\ndef MorphismProperty.isFinitelyPresentable : MorphismProperty C :=\n  fun _ _ f ↦ ObjectProperty.isFinitelyPresentable.{w} _ (CategoryTheory.Under.mk f)\n\nlemma isFinitelyPresentable_iff_preservesFilteredColimitsOfSize {X : C} :\n    IsFinitelyPresentable.{w} X ↔ PreservesFilteredColimitsOfSize.{w, w} (coyoneda.obj (op X)) :=\n  Functor.IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize\n\nlemma isFinitelyPresentable_iff_preservesFilteredColimits {X : C} :\n    IsFinitelyPresentable.{v} X ↔ PreservesFilteredColimits (coyoneda.obj (op X)) :=\n  Functor.IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize\n\ninstance (X : C) [IsFinitelyPresentable.{w} X] :\n    PreservesFilteredColimitsOfSize.{w, w} (coyoneda.obj (op X)) := by\n  rw [← isFinitelyPresentable_iff_preservesFilteredColimitsOfSize]\n  infer_instance\n\ninstance (X : (ObjectProperty.isFinitelyPresentable.{w} C).FullSubcategory) :\n    IsFinitelyPresentable.{w} ((ObjectProperty.isFinitelyPresentable.{w} C).ι.obj X) :=\n  X.property\n\nlemma IsFinitelyPresentable.exists_hom_of_isColimit {J : Type w} [SmallCategory J] [IsFiltered J]\n    {D : J ⥤ C} {c : Cocone D} (hc : IsColimit c) {X : C} [IsFinitelyPresentable.{w} X]\n    (f : X ⟶ c.pt) :\n    ∃ (j : J) (p : X ⟶ D.obj j), p ≫ c.ι.app j = f :=\n  Types.jointly_surjective_of_isColimit (isColimitOfPreserves (coyoneda.obj (op X)) hc) f\n\nlemma IsFinitelyPresentable.exists_eq_of_isColimit {J : Type w} [SmallCategory J] [IsFiltered J]\n    {D : J ⥤ C} {c : Cocone D} (hc : IsColimit c) {X : C} [IsFinitelyPresentable.{w} X]\n    {i j : J} (f : X ⟶ D.obj i) (g : X ⟶ D.obj j) (h : f ≫ c.ι.app i = g ≫ c.ι.app j) :\n    ∃ (k : J) (u : i ⟶ k) (v : j ⟶ k), f ≫ D.map u = g ≫ D.map v :=\n  (Types.FilteredColimit.isColimit_eq_iff _ (isColimitOfPreserves (coyoneda.obj (op X)) hc)).mp h\n\nlemma IsFinitelyPresentable.exists_hom_of_isColimit_under\n    {J : Type w} [SmallCategory J] [IsFiltered J] {D : J ⥤ C} {c : Cocone D} (hc : IsColimit c)\n    {X A : C} (p : X ⟶ A) (s : (Functor.const J).obj X ⟶ D)\n    [IsFinitelyPresentable.{w} (Under.mk p)]\n    (f : A ⟶ c.pt) (h : ∀ (j : J), s.app j ≫ c.ι.app j = p ≫ f) :\n    ∃ (j : J) (q : A ⟶ D.obj j), p ≫ q = s.app j ∧ q ≫ c.ι.app j = f := by\n  have : Nonempty J := IsFiltered.nonempty\n  let hc' := Under.isColimitLiftCocone D s c (p ≫ f) h hc\n  obtain ⟨j, q, hq⟩ := exists_hom_of_isColimit (X := Under.mk p) hc' (Under.homMk f rfl)\n  use j, q.right, Under.w q, congr($(hq).right)\n\nTarget:\nlemma HasCardinalFilteredColimits_iff_hasFilteredColimitsOfSize :\n    HasCardinalFilteredColimits.{w} C ℵ₀ ↔ HasFilteredColimitsOfSize.{w, w} C :=\n\nProof body:\n","rejected":"by\n  refine ⟨fun ⟨H⟩ ↦ ⟨?_⟩, fun ⟨H⟩ ↦ ⟨?_⟩⟩ <;>\n    simp only [isCardinalFiltered_aleph0_iff] at * <;>\n    exact H\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Presentable","family_id":"hascardinalfilteredcolimits_iff_hasfilteredcolimitsofsize","file_id":"mathlib/Mathlib/CategoryTheory/Presentable/Finite.lean","sample_id":"948388cf0ecec93fe348cbe9fadcd98478b5894a8ffcc1cd67d7ba5e880077f1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"60c914c7101fd19d31a3d057e5fece308a5e2edc08549f3280fae32cf5fabb05","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3924443c538f044c69cb7756f1f4c2d5ba5a0970c695f80abe0a59f67dc9ea46","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cf2066a6aac720a3b5d609e8b3fe1b26d84bc8deb53c0ec1b5795944dd568539","source_sha256":"16ec77c2931a23c29f0fb086af6455ba8ec16f170f0fd8c2178cb81f79494219","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp only [mem_setOf_eq]\n  rw [← Submodule.map_top, ← iSup_range_single, Submodule.map_iSup]\n  congr; funext i\n  rw [← LinearMap.range_comp, diagonal_comp_single, ← range_smul']","hard_negative":false,"metrics":{"chosen_tokens":36,"rejected_tokens":41,"token_jaccard":0.846154,"token_length_ratio":1.138889},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"1c0192c27d304c374cc3035fb149cdc4ebbd070fb58a3eac86060b37f98bb029","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Dimension.LinearMap\npublic import Mathlib.LinearAlgebra.Matrix.ToLin\n\nNamespace:\nMatrix\n\nLocal context:\n/-\nCopyright (c) 2019 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen\n-/\n/-!\n# Diagonal matrices\n\nThis file contains some results on the linear map corresponding to a\ndiagonal matrix (`range`, `ker` and `rank`).\n\n## Tags\n\nmatrix, diagonal, linear map\n-/\n\npublic section\n\n\nnoncomputable section\n\nopen LinearMap Matrix Set Submodule Matrix\n\nuniverse u v w\n\nnamespace Matrix\n\nsection CommSemiring\n\nvariable {n : Type*} [Fintype n] [DecidableEq n] {R : Type v} [CommSemiring R]\n\ntheorem proj_diagonal (i : n) (w : n → R) : (proj i).comp (toLin' (diagonal w)) = w i • proj i :=\n  LinearMap.ext fun _ => mulVec_diagonal _ _ _\n\ntheorem diagonal_comp_single (w : n → R) (i : n) :\n    (diagonal w).toLin'.comp (LinearMap.single R (fun _ : n => R) i) =\n      w i • LinearMap.single R (fun _ : n => R) i :=\n  LinearMap.ext fun x => (diagonal_mulVec_single w _ _).trans (Pi.single_smul' i (w i) x)\n\ntheorem diagonal_toLin' (w : n → R) :\n    toLin' (diagonal w) = LinearMap.pi fun i => w i • LinearMap.proj i :=\n  LinearMap.ext fun _ => funext fun _ => mulVec_diagonal _ _ _\n\nend CommSemiring\n\nsection Semifield\n\nvariable {m : Type*} [Fintype m] {K : Type u} [Semifield K]\n\n-- maybe try to relax the universe constraint\ntheorem ker_diagonal_toLin' [DecidableEq m] (w : m → K) :\n    ker (toLin' (diagonal w)) =\n      ⨆ i ∈ { i | w i = 0 }, LinearMap.range (LinearMap.single K (fun _ => K) i) := by\n  rw [← comap_bot, ← iInf_ker_proj, comap_iInf]\n  have := fun i : m => ker_comp (toLin' (diagonal w)) (proj i)\n  simp only [← this, proj_diagonal, ker_smul']\n  have : univ ⊆ { i : m | w i = 0 } ∪ { i : m | w i = 0 }ᶜ := by rw [Set.union_compl_self]\n  exact (iSup_range_single_eq_iInf_ker_proj K (fun _ : m => K) disjoint_compl_right this\n    (Set.toFinite _)).symm\n\nTarget:\ntheorem range_diagonal [DecidableEq m] (w : m → K) :\n    LinearMap.range (toLin' (diagonal w)) =\n      ⨆ i ∈ { i | w i ≠ 0 }, LinearMap.range (LinearMap.single K (fun _ => K) i) :=\n\nProof body:\n","rejected":"by\n  dsimp only [mem_setOf_eq]\n  rw [← Submodule.map_top, ← iSup_range_single, Submodule.map_iSup]\n  congr; funext i\n  rw [← LinearMap.range_comp, diagonal_comp_single, ← range_smul']\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Matrix","family_id":"range_diagonal","file_id":"mathlib/Mathlib/LinearAlgebra/Matrix/Diagonal.lean","sample_id":"cf2066a6aac720a3b5d609e8b3fe1b26d84bc8deb53c0ec1b5795944dd568539"},"split":"train"}
{"binding":{"chosen_proof_sha256":"76978dc753664885955d4019ab183b19aab997594734243a4e6b99373e308b2b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"30fea064c64881073cd6086113f489a6588683e24e95a47f14c696b86deaffd3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4ea77ade5bcaf82f263f63d2e35e31706e2c9a6c7ae719c3b1558eff0efaaa66","source_sha256":"f83ecd09e0c1dc0009269016250874e7bd19aed3be89d47a83b8fee4b9eb7cc4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [isCompactElement_iff_exists_le_sSup_of_le_sSup]\n  refine ⟨fun h k s hs => ?_, fun h s => ?_⟩\n  · obtain ⟨t, ⟨hts, htsup⟩⟩ := h s\n    use t, hts\n    rwa [← htsup]\n  · obtain ⟨t, ⟨hts, htsup⟩⟩ := h (sSup s) s (by rfl)\n    have : sSup s = t.sup id := by\n      suffices t.sup id ≤ sSup s by apply le_antisymm <;> assumption\n      simp only [id, Finset.sup_le_iff]\n      intro x hx\n      exact le_sSup (hts hx)\n    exact ⟨t, hts, this⟩","hard_negative":true,"metrics":{"chosen_tokens":120,"rejected_tokens":5,"token_jaccard":0.053571,"token_length_ratio":0.041667},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"1c14d8095a58066d8e93466c454c9725166b376161f25f01d0da053085d5ff09","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Atoms\npublic import Mathlib.Order.OrderIsoNat\npublic import Mathlib.Order.RelIso.Set\npublic import Mathlib.Order.SupClosed\npublic import Mathlib.Order.SupIndep\npublic import Mathlib.Order.Zorn\npublic import Mathlib.Data.Finset.Order\npublic import Mathlib.Order.Interval.Set.OrderIso\npublic import Mathlib.Data.Finite.Set\npublic import Mathlib.Tactic.TFAE\n\nNamespace:\nCompleteLattice\n\nLocal context:\n/-\nCopyright (c) 2021 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Compactness properties for complete lattices\n\nFor complete lattices, there are numerous equivalent ways to express the fact that the relation `>`\nis well-founded. In this file we define three especially-useful characterisations and provide\nproofs that they are indeed equivalent to well-foundedness.\n\n## Main definitions\n* `CompleteLattice.IsSupClosedCompact`\n* `CompleteLattice.IsSupFiniteCompact`\n* `IsCompactElement`\n* `IsCompactlyGenerated`\n\n## Main results\nThe main result is that the following four conditions are equivalent for a complete lattice:\n* `well_founded (>)`\n* `CompleteLattice.IsSupClosedCompact`\n* `CompleteLattice.IsSupFiniteCompact`\n* `∀ k, IsCompactElement k`\n\nThis is demonstrated by means of the following four lemmas:\n* `CompleteLattice.WellFounded.isSupFiniteCompact`\n* `CompleteLattice.IsSupFiniteCompact.isSupClosedCompact`\n* `CompleteLattice.IsSupClosedCompact.wellFounded`\n* `CompleteLattice.isSupFiniteCompact_iff_all_elements_compact`\n\nWe also show well-founded lattices are compactly generated\n(`CompleteLattice.isCompactlyGenerated_of_wellFounded`).\n\n## References\n- [G. Călugăreanu, *Lattice Concepts of Module Theory*][calugareanu]\n\n## Tags\n\ncomplete lattice, well-founded, compact\n-/\n\n@[expose] public section\n\nopen Set\n/-- An element `k` is compact if any directed set with `LUB` (least upper bound) above\n`k` has already got above `k` at some point in the set.\nSuch an element is also called \"finite\" or \"S-compact\". -/\ndef IsCompactElement {α : Type*} [PartialOrder α] (k : α) :=\n  ∀ (s : Set α) (u : α),\n    s.Nonempty →\n    DirectedOn (· ≤ ·) s →\n    IsLUB s u →\n    k ≤ u →\n    ∃ x ∈ s, k ≤ x\n\nvariable {ι : Sort*} {α : Type*} [CompleteLattice α] {f : ι → α}\n\nnamespace CompleteLattice\n\nvariable (α)\n\n/-- A compactness property for a complete lattice is that any `sup`-closed non-empty subset\ncontains its `sSup`. -/\ndef IsSupClosedCompact : Prop :=\n  ∀ (s : Set α) (_ : s.Nonempty), SupClosed s → sSup s ∈ s\n\n/-- A compactness property for a complete lattice is that any subset has a finite subset with the\nsame `sSup`. -/\ndef IsSupFiniteCompact : Prop :=\n  ∀ s : Set α, ∃ t : Finset α, ↑t ⊆ s ∧ sSup s = t.sup id\n\n/-- An element `k` is compact if and only if any directed set with `sSup` above\n`k` already got above `k` at some point in the set. -/\ntheorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :\n    IsCompactElement k ↔\n      ∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x := by\n  constructor\n  · intro hk s hs hs' h_le\n    exact hk s (sSup s) hs hs' (isLUB_sSup s) h_le\n  · intro h s u hs hs' hu h_le\n    rw [isLUB_iff_sSup_eq] at hu\n    rw [← hu] at h_le\n    exact h s hs hs' h_le\n\n/-- An element `k` of is compact if any set with `sSup`\nabove `k` has a finite subset with `sSup` above `k`. -/\ntheorem isCompactElement_iff_exists_le_sSup_of_le_sSup (k : α) :\n    IsCompactElement k ↔ ∀ s : Set α, k ≤ sSup s → ∃ t : Finset α, ↑t ⊆ s ∧ k ≤ t.sup id := by\n  classical\n    rw [isCompactElement_iff_le_of_directed_sSup_le]\n    constructor\n    · intro hk s hsup\n      -- Consider the set of finite joins of elements of the (plain) set s.\n      let S : Set α := { x | ∃ t : Finset α, ↑t ⊆ s ∧ x = t.sup id }\n      -- S is directed, nonempty, and still has sup above k.\n      have dir_US : DirectedOn (· ≤ ·) S := by\n        rintro x ⟨c, hc⟩ y ⟨d, hd⟩\n        use x ⊔ y\n        constructor\n        · use c ∪ d\n          constructor\n          · simp only [hc.left, hd.left, Set.union_subset_iff, Finset.coe_union, and_self_iff]\n          · simp only [hc.right, hd.right, Finset.sup_union]\n        simp only [and_self_iff, le_sup_left, le_sup_right]\n      have sup_S : sSup s ≤ sSup S := by\n        apply sSup_le_sSup\n        intro x hx\n        use {x}\n        simpa only [and_true, id, Finset.coe_singleton, eq_self_iff_true,\n          Finset.sup_singleton, Set.singleton_subset_iff]\n      have Sne : S.Nonempty := by\n        suffices ⊥ ∈ S from Set.nonempty_of_mem this\n        use ∅\n        simp\n      -- Now apply the defn of compact and finish.\n      obtain ⟨j, ⟨hjS, hjk⟩⟩ := hk S Sne dir_US (le_trans hsup sup_S)\n      obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS\n      use t\n      exact ⟨htS, by rwa [← htsup]⟩\n    · intro hk s hne hdir hsup\n      obtain ⟨t, ht⟩ := hk s hsup\n      -- certainly every element of t is below something in s, since ↑t ⊆ s.\n      have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y := fun x hxt => ⟨x, ht.left hxt, le_rfl⟩\n      obtain ⟨x, ⟨hxs, hsupx⟩⟩ := Finset.sup_le_of_le_directed s hne hdir t t_below_s\n      exact ⟨x, ⟨hxs, le_trans ht.right hsupx⟩⟩\n\ntheorem isCompactElement_iff_exists_le_iSup_of_le_iSup.{u} {α : Type u} [CompleteLattice α]\n    (k : α) : IsCompactElement k ↔\n      ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s := by\n  classical\n    rw [isCompactElement_iff_exists_le_sSup_of_le_sSup]\n    constructor\n    · intro H ι s hs\n      obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs\n      have : ∀ x : t, ∃ i, s i = x := fun x => ht x.prop\n      choose f hf using this\n      refine ⟨Finset.univ.image f, ht'.trans ?_⟩\n      rw [Finset.sup_le_iff]\n      intro b hb\n      rw [← show s (f ⟨b, hb⟩) = id b from hf _]\n      exact Finset.le_sup (Finset.mem_image_of_mem f <| Finset.mem_univ (Subtype.mk b hb))\n    · intro H s hs\n      obtain ⟨t, ht⟩ :=\n        H s Subtype.val\n          (by\n            delta iSup\n            rwa [Subtype.range_coe])\n      refine ⟨t.image Subtype.val, by simp, ht.trans ?_⟩\n      rw [Finset.sup_le_iff]\n      exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem Subtype.val hx)\n\ntheorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type*}\n    (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by\n  classical\n    rw [isCompactElement_iff_le_of_directed_sSup_le] at hk\n    let g : Finset ι → α := fun s => ⨆ i ∈ s, f i\n    have h1 : DirectedOn (· ≤ ·) (Set.range g) := by\n      rintro - ⟨s, rfl⟩ - ⟨t, rfl⟩\n      exact\n        ⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, iSup_le_iSup_of_subset Finset.subset_union_left,\n          iSup_le_iSup_of_subset Finset.subset_union_right⟩\n    have h2 : k ≤ sSup (Set.range g) :=\n      h.trans\n        (iSup_le fun i =>\n          le_sSup_of_le ⟨{i}, rfl⟩\n            (le_iSup_of_le i (le_iSup_of_le (Finset.mem_singleton_self i) le_rfl)))\n    obtain ⟨-, ⟨s, rfl⟩, hs⟩ := hk (Set.range g) (Set.range_nonempty g) h1 h2\n    exact ⟨s, hs⟩\n\n/-- A compact element `k` has the property that any directed set lying strictly below `k` has\nits `sSup` strictly below `k`. -/\ntheorem IsCompactElement.directed_sSup_lt_of_lt {α : Type*} [CompleteLattice α] {k : α}\n    (hk : IsCompactElement k) {s : Set α} (hemp : s.Nonempty) (hdir : DirectedOn (· ≤ ·) s)\n    (hbelow : ∀ x ∈ s, x < k) : sSup s < k := by\n  rw [isCompactElement_iff_le_of_directed_sSup_le] at hk\n  by_contra h\n  have sSup' : sSup s ≤ k := sSup_le fun s hs => (hbelow s hs).le\n  replace sSup : sSup s = k := eq_iff_le_not_lt.mpr ⟨sSup', h⟩\n  obtain ⟨x, hxs, hkx⟩ := hk s hemp hdir sSup.symm.le\n  obtain hxk := hbelow x hxs\n  exact hxk.ne (hxk.le.antisymm hkx)\n\ntheorem isCompactElement_finsetSup {α β : Type*} [CompleteLattice α] {f : β → α} (s : Finset β)\n    (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by\n  classical\n    simp_rw [isCompactElement_iff_le_of_directed_sSup_le] at ⊢ h\n    intro d hemp hdir hsup\n    rw [← Function.id_comp f]\n    rw [← Finset.sup_image]\n    apply Finset.sup_le_of_le_directed d hemp hdir\n    rintro x hx\n    obtain ⟨p, ⟨hps, rfl⟩⟩ := Finset.mem_image.mp hx\n    specialize h p hps\n    specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)\n    simpa only [exists_prop]\n\ntheorem WellFoundedGT.isSupFiniteCompact [WellFoundedGT α] :\n    IsSupFiniteCompact α := fun s => by\n  let S := { x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x }\n  obtain ⟨m, ⟨t, ⟨ht₁, rfl⟩⟩, hm⟩ := wellFounded_gt.has_min S ⟨⊥, ∅, by simp⟩\n  refine ⟨t, ht₁, (sSup_le fun y hy => ?_).antisymm ?_⟩\n  · classical\n    rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))\n        (hm _ ⟨insert y t, by simp [Set.insert_subset_iff, hy, ht₁]⟩)]\n    simp\n  · rw [Finset.sup_id_eq_sSup]\n    exact sSup_le_sSup ht₁\n\ntheorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) :\n    IsSupClosedCompact α := by\n  intro s hne hsc; obtain ⟨t, ht₁, ht₂⟩ := h s; clear h\n  rcases t.eq_empty_or_nonempty with rfl | h\n  · rw [Finset.sup_empty] at ht₂\n    rw [ht₂]\n    simp [eq_singleton_bot_of_sSup_eq_bot_of_nonempty ht₂ hne]\n  · rw [ht₂]\n    exact hsc.finsetSup_mem h ht₁\n\ntheorem IsSupClosedCompact.wellFoundedGT (h : IsSupClosedCompact α) :\n    WellFoundedGT α where\n  wf := by\n    refine RelEmbedding.wellFounded_iff_isEmpty.mpr ⟨fun a => ?_⟩\n    suffices sSup (Set.range a) ∈ Set.range a by\n      obtain ⟨n, hn⟩ := Set.mem_range.mp this\n      have h' : sSup (Set.range a) < a (n + 1) := by\n        change _ > _\n        simp [← hn, a.map_rel_iff]\n      apply lt_irrefl (a (n + 1))\n      apply lt_of_le_of_lt _ h'\n      apply le_sSup\n      apply Set.mem_range_self\n    apply h (Set.range a)\n    · use a 37\n      apply Set.mem_range_self\n    · rintro x ⟨m, hm⟩ y ⟨n, hn⟩\n      use m ⊔ n\n      rw [← hm, ← hn]\n      apply RelHomClass.map_sup a\n\nTarget:\ntheorem isSupFiniteCompact_iff_all_elements_compact :\n    IsSupFiniteCompact α ↔ ∀ k : α, IsCompactElement k :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_4ea77ade5bca","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"7a14d292732b6fa82657755401f0f91bf894beabc44c4466e3f22df9195eeed5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/CompactlyGenerated","family_id":"issupfinitecompact_iff_all_elements_compact","file_id":"mathlib/Mathlib/Order/CompactlyGenerated/Basic.lean","sample_id":"4ea77ade5bcaf82f263f63d2e35e31706e2c9a6c7ae719c3b1558eff0efaaa66"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9ea2d48ebe00bb73f1eea97bd737a2f6b8f66a4067141833800162c91765c532","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"90d3570afa5eb87a779588a4326f9118d8f86493d7e087dc02225d9a416adcbe","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dc8404380687b2e37f6ac350d18b83560147a6545b4e3c6396ca1a3bc04d0633","source_sha256":"5a251c5b9da944aa52b3b3c01ae0c840fa23c2c1f9c9e5ace9139527d3f0e3e9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [GradedMonoid.GOne.one, TensorPower.toTensorAlgebra_tprod]","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":18,"token_jaccard":0.6875,"token_length_ratio":1.384615},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"1c2700cf986b307f1856ac9930109e2840b0dabcdd028abb973d3051ccbf595b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorAlgebra.Basic\npublic import Mathlib.LinearAlgebra.TensorPower.Basic\n\nNamespace:\nTensorPower\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Tensor algebras as direct sums of tensor powers\n\nIn this file we show that `TensorAlgebra R M` is isomorphic to a direct sum of tensor powers, as\n`TensorAlgebra.equivDirectSum`.\n-/\n\n@[expose] public section\n\nopen scoped DirectSum TensorProduct\n\nvariable {R M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M]\n\nnamespace TensorPower\n\n/-- The canonical embedding from a tensor power to the tensor algebra -/\ndef toTensorAlgebra {n} : ⨂[R]^n M →ₗ[R] TensorAlgebra R M :=\n  PiTensorProduct.lift (TensorAlgebra.tprod R M n)\n\n@[simp]\ntheorem toTensorAlgebra_tprod {n} (x : Fin n → M) :\n    TensorPower.toTensorAlgebra (PiTensorProduct.tprod R x) = TensorAlgebra.tprod R M n x :=\n  PiTensorProduct.lift.tprod _\n\n@[simp]\n\nTarget:\ntheorem toTensorAlgebra_gOne :\n    TensorPower.toTensorAlgebra (@GradedMonoid.GOne.one _ (fun n => ⨂[R]^n M) _ _) = 1 :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  simp [GradedMonoid.GOne.one, TensorPower.toTensorAlgebra_tprod]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/TensorAlgebra","family_id":"totensoralgebra_gone","file_id":"mathlib/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean","sample_id":"dc8404380687b2e37f6ac350d18b83560147a6545b4e3c6396ca1a3bc04d0633"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dc1780cb3fb0b15d1b902233abc1a076e6c3e1b7b64ffb6ddfc98b9761298fce","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1a6f779a26d6933a1ab93cb5daf06274698653e526a8b75426906873d50dc3c8","source_sha256":"36095e70549b743e6d6e18e94107fa315e72325dc59944c049976dea39a27dfe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine .fromProd (fun a d hd₁ hd₂ c hdc ↦ ?_) (fun b d hd₁ hd₂ c hdc ↦ ?_)\n  · have e1 : (fun b ↦ (f a, g b)) '' d = {f a} ×ˢ (g '' d) := by aesop\n    simp_rw [Prod.map_apply, e1]\n    exact .prod (singleton_nonempty _) (hd₁.image _) isLUB_singleton (hg hd₁ hd₂ hdc)\n  · have e2 : ((fun a ↦ (f a, g b)) '' d) = (f '' d) ×ˢ {g b} := by aesop\n    simp_rw [Prod.map_apply, e2]\n    exact .prod (hd₁.image _) (singleton_nonempty _) (hf hd₁ hd₂ hdc) isLUB_singleton","hard_negative":true,"metrics":{"chosen_tokens":166,"rejected_tokens":8,"token_jaccard":0.081633,"token_length_ratio":0.048193},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"1c3d1a5f7d948f10649ad19223cfeb5d4332e76e441403983b9667aee959f26e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ScottContinuity\npublic import Mathlib.Order.Bounds.Lattice\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Christopher Hoskin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christopher Hoskin\n-/\n/-!\n# Scott continuity on product spaces\n\n## Main result\n\n- `ScottContinuous_prod_of_ScottContinuous`: A function is Scott continuous on a product space if it\n  is Scott continuous in each variable.\n- `ScottContinuousOn.inf₂`: For complete linear orders, the meet operation is Scott continuous.\n\n-/\n\npublic section\n\nopen Set\n\nvariable {α β γ : Type*}\n\n/-- If  is Scott continuous on a product space if it is Scott continuous and monotone in each\nvariable -/\nlemma ScottContinuousOn.fromProd [Preorder α] [Preorder β] [Preorder γ]\n    {f : α × β → γ} {D : Set (Set (α × β))}\n    (h₁ : ∀ a, ScottContinuousOn ((fun d => Prod.snd '' d) '' D) (fun b => f (a, b)))\n    (h₂ : ∀ b, ScottContinuousOn ((fun d => Prod.fst '' d) '' D) (fun a => f (a, b)))\n    (h₁' : ∀ a, Monotone (fun b => f (a, b))) (h₂' : ∀ b, Monotone (fun a => f (a, b))) :\n    ScottContinuousOn D f := fun d hX hd₁ hd₂ ⟨p1, p2⟩ hdp => by\n  rw [isLUB_congr ((monotone_prod_iff.mpr ⟨h₁', h₂'⟩).upperBounds_image_of_directedOn_prod hd₂),\n    ← iUnion_of_singleton_coe (Prod.fst '' d), iUnion_prod_const, image_iUnion,\n    ← isLUB_iUnion_iff_of_isLUB (fun a => by\n      rw [singleton_prod, image_image f (fun b ↦ (a, b))]\n      exact h₁ _ (mem_image_of_mem (fun d ↦ Prod.snd '' d) hX) (Nonempty.image Prod.snd hd₁)\n        (DirectedOn.snd hd₂) (isLUB_prod.mp hdp).2) _, Set.range]\n  convert!\n    (h₂ _ (mem_image_of_mem (fun d ↦ Prod.fst '' d) hX) (Nonempty.image Prod.fst hd₁)\n      (DirectedOn.fst hd₂) (isLUB_prod.mp hdp).1)\n  ext : 1\n  simp_all only [Subtype.exists, mem_image, Prod.exists,\n    exists_and_right, exists_eq_right, exists_prop, mem_setOf_eq]\n\nlemma ScottContinuous.fromProd {γ : Type*} [Preorder α] [Preorder β] [Preorder γ]\n    {f : α × β → γ} (h₁ : ∀ a, ScottContinuous (fun b => f (a, b)))\n    (h₂ : ∀ b, ScottContinuous (fun a => f (a, b))) : ScottContinuous f := by\n  simp_rw [← scottContinuousOn_univ] at ⊢\n  exact .fromProd (fun a ↦ (h₁ a).scottContinuousOn) (fun b ↦ (h₂ b).scottContinuousOn)\n    (fun a ↦ (h₁ a).monotone) (fun b ↦ (h₂ b).monotone)\n\nTarget:\nlemma ScottContinuous.prod {α' β' : Type*} [Preorder α] [Preorder β] [Preorder α'] [Preorder β']\n    {f : α → α'} {g : β → β'} (hf : ScottContinuous f) (hg : ScottContinuous g) :\n    ScottContinuous (Prod.map f g) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"89a1205b433729915e2a2c0e75dbddfa06b4e7e46a17bf69c69831147495e55f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/ScottContinuity","family_id":"scottcontinuous","file_id":"mathlib/Mathlib/Order/ScottContinuity/Prod.lean","sample_id":"1a6f779a26d6933a1ab93cb5daf06274698653e526a8b75426906873d50dc3c8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"228b521f0401eea33d29434cfc850bb07b080dd681e51957a62ad89f2e88ea24","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"200254cdeea524e0fb3be35346d663e1d47538d7e3d287bc5c967ddfaadb9e33","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3edd16c3ca0318c2e5d045f7be510639e5b1ebe6db412620dbe457e127e3ccb4","source_sha256":"d55fd8bd8c5069614d92f45b31915666a5cb1c81ecb8657cfb3d4cc03acc7e68","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rwa [sign_apply, if_neg <| asymm ha, if_pos]","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.230769},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"1c3d33173424041454c9eab467c7619157401f50447ef5bbcb2a735caffa19c2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GroupWithZero.Defs\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Algebra.Order.Ring.Defs\npublic import Mathlib.Tactic.DeriveFintype  -- shake: keep (deriving handlers not tracked yet)\npublic import Mathlib.Data.Multiset.Defs\npublic import Mathlib.Data.Fintype.Defs\npublic import Mathlib.Algebra.Group.Equiv.Defs\npublic import Mathlib.Algebra.Group.Int.Defs\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Eric Rodriguez. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Rodriguez\n-/\n/-!\n# Sign type\n\nThis file defines the type of signs $\\{-1, 0, 1\\}$ and its basic arithmetic instances.\n-/\n\n@[expose] public section\n\n-- Don't generate unnecessary `sizeOf_spec` lemmas which the `simpNF` linter will complain about.\nset_option genSizeOfSpec false in\n/-- The type of signs. -/\ninductive SignType\n  | zero\n  | neg\n  | pos\n  deriving DecidableEq, Inhabited, Fintype\n\nnamespace SignType\n\ninstance : Zero SignType :=\n  ⟨zero⟩\n\ninstance : One SignType :=\n  ⟨pos⟩\n\ninstance : Neg SignType :=\n  ⟨fun s =>\n    match s with\n    | neg => pos\n    | zero => zero\n    | pos => neg⟩\n\n@[simp]\ntheorem zero_eq_zero : zero = 0 :=\n  rfl\n\n@[simp]\ntheorem neg_eq_neg_one : neg = -1 :=\n  rfl\n\n@[simp]\ntheorem pos_eq_one : pos = 1 :=\n  rfl\n\ntheorem trichotomy (a : SignType) : a = -1 ∨ a = 0 ∨ a = 1 := by\n  cases a <;> simp\n\ninstance : Mul SignType :=\n  ⟨fun x y =>\n    match x with\n    | neg => -y\n    | zero => zero\n    | pos => y⟩\n\n/-- The less-than-or-equal relation on signs. -/\nprotected inductive LE : SignType → SignType → Prop\n  | of_neg (a) : SignType.LE neg a\n  | zero : SignType.LE zero zero\n  | of_pos (a) : SignType.LE a pos\n\ninstance : LE SignType :=\n  ⟨SignType.LE⟩\n\ninstance : DecidableLE SignType := fun a b => by\n  cases a <;> cases b <;> first | exact isTrue (by constructor) | exact isFalse (by rintro ⟨_⟩)\n\n/-- We can define a `Field` instance on `SignType`, but it's not mathematically sensible,\nso we only define the `CommGroupWithZero`. -/\ninstance : CommGroupWithZero SignType where\n  inv := id\n  mul_zero a := by cases a <;> rfl\n  zero_mul a := by cases a <;> rfl\n  mul_one a := by cases a <;> rfl\n  one_mul a := by cases a <;> rfl\n  mul_inv_cancel a ha := by cases a <;> trivial\n  mul_comm := by decide\n  mul_assoc := by decide\n  exists_pair_ne := ⟨0, 1, by rintro ⟨_⟩⟩\n  inv_zero := rfl\n\ninstance : LinearOrder SignType where\n  le_refl a := by cases a <;> constructor\n  le_total := by decide\n  le_antisymm := by decide\n  le_trans := by decide\n  toDecidableLE := instDecidableLE\n\ninstance : BoundedOrder SignType where\n  top := 1\n  le_top := LE.of_pos\n  bot := -1\n  bot_le :=\n    #adaptation_note /-- https://github.com/leanprover/lean4/pull/6053\n    Added `by exact`, but don't understand why it was needed. -/\n    by exact LE.of_neg\n\ninstance : HasDistribNeg SignType where\n  neg_neg := by rintro ⟨_⟩ <;> rfl\n  neg_mul := by rintro ⟨_⟩ ⟨_⟩ <;> rfl\n  mul_neg := by rintro ⟨_⟩ ⟨_⟩ <;> rfl\n\n/-- `SignType` is equivalent to `Fin 3`. -/\ndef fin3Equiv : SignType ≃* Fin 3 where\n  toFun a :=\n    match a with\n    | 0 => ⟨0, by simp⟩\n    | 1 => ⟨1, by simp⟩\n    | -1 => ⟨2, by simp⟩\n  invFun a :=\n    match a with\n    | ⟨0, _⟩ => 0\n    | ⟨1, _⟩ => 1\n    | ⟨2, _⟩ => -1\n  left_inv a := by cases a <;> rfl\n  right_inv a :=\n    match a with\n    | ⟨0, _⟩ => by simp\n    | ⟨1, _⟩ => by simp\n    | ⟨2, _⟩ => by simp\n  map_mul' a b := by\n    cases a <;> cases b <;> rfl\n\nsection CaseBashing\n\ntheorem nonneg_iff {a : SignType} : 0 ≤ a ↔ a = 0 ∨ a = 1 := by decide +revert\n\ntheorem nonneg_iff_ne_neg_one {a : SignType} : 0 ≤ a ↔ a ≠ -1 := by decide +revert\n\ntheorem neg_one_lt_iff {a : SignType} : -1 < a ↔ 0 ≤ a := by decide +revert\n\ntheorem nonpos_iff {a : SignType} : a ≤ 0 ↔ a = -1 ∨ a = 0 := by decide +revert\n\ntheorem nonpos_iff_ne_one {a : SignType} : a ≤ 0 ↔ a ≠ 1 := by decide +revert\n\ntheorem lt_one_iff {a : SignType} : a < 1 ↔ a ≤ 0 := by decide +revert\n\n@[simp]\ntheorem neg_iff {a : SignType} : a < 0 ↔ a = -1 := by decide +revert\n\n@[simp]\ntheorem le_neg_one_iff {a : SignType} : a ≤ -1 ↔ a = -1 :=\n  le_bot_iff\n\n@[simp]\ntheorem pos_iff {a : SignType} : 0 < a ↔ a = 1 := by decide +revert\n\n@[simp]\ntheorem one_le_iff {a : SignType} : 1 ≤ a ↔ a = 1 :=\n  top_le_iff\n\n@[simp]\ntheorem neg_one_le (a : SignType) : -1 ≤ a :=\n  bot_le\n\n@[simp]\ntheorem le_one (a : SignType) : a ≤ 1 :=\n  le_top\n\n@[simp]\ntheorem not_lt_neg_one (a : SignType) : ¬a < -1 :=\n  not_lt_bot\n\n@[simp]\ntheorem not_one_lt (a : SignType) : ¬1 < a :=\n  not_top_lt\n\n@[simp]\ntheorem self_eq_neg_iff {a : SignType} : a = -a ↔ a = 0 := by decide +revert\n\n@[simp]\ntheorem neg_eq_self_iff {a : SignType} : -a = a ↔ a = 0 := by decide +revert\n\n@[simp]\ntheorem neg_eq_zero_iff {a : SignType} : -a = 0 ↔ a = 0 := by decide +revert\n\n@[simp]\ntheorem neg_one_lt_one : (-1 : SignType) < 1 :=\n  bot_lt_top\n\n@[simp]\nprotected theorem neg_le_neg_iff {a b : SignType} : -a ≤ -b ↔ b ≤ a := by decide +revert\n\n@[simp]\nprotected theorem neg_lt_neg_iff {a b : SignType} : -a < -b ↔ b < a := by decide +revert\n\nend CaseBashing\n\nsection cast\n\nvariable {α : Type*} [Zero α] [One α] [Neg α]\n\n/-- Turn a `SignType` into zero, one, or minus one. This is a coercion instance. -/\n@[coe]\ndef cast : SignType → α\n  | zero => 0\n  | pos => 1\n  | neg => -1\n\n/--\nThis can't be a `CoeTail` or `Coe` instance because we don't want it to fire when `SignType` isn't\ninvolved in the coercion (or `CoeHead` or `CoeOut` because of `outParam`s). The only other\nuser-exposed option is `CoeDep` then, which allows us to match on both given and expected type.\n-/\ninstance (s : SignType) : CoeDep SignType s α :=\n  ⟨cast s⟩\n\n/-- Casting out of `SignType` respects composition with functions preserving `0, 1, -1`. -/\nlemma map_cast' {β : Type*} [One β] [Neg β] [Zero β]\n    (f : α → β) (h₁ : f 1 = 1) (h₂ : f 0 = 0) (h₃ : f (-1) = -1) (s : SignType) :\n    f s = s := by\n  cases s <;> simp only [SignType.cast, h₁, h₂, h₃]\n\n/-- Casting out of `SignType` respects composition with suitable bundled homomorphism types. -/\nlemma map_cast {α β F : Type*} [AddGroupWithOne α] [One β] [SubtractionMonoid β]\n    [FunLike F α β] [AddMonoidHomClass F α β] [OneHomClass F α β] (f : F) (s : SignType) :\n    f s = s := by\n  apply map_cast' <;> simp\n\n@[simp]\ntheorem coe_zero : ↑(0 : SignType) = (0 : α) :=\n  rfl\n\n@[simp]\ntheorem coe_one : ↑(1 : SignType) = (1 : α) :=\n  rfl\n\n@[simp]\ntheorem coe_neg_one : ↑(-1 : SignType) = (-1 : α) :=\n  rfl\n\n@[simp, norm_cast]\nlemma coe_neg {α : Type*} [One α] [SubtractionMonoid α] (s : SignType) :\n    (↑(-s) : α) = -↑s := by\n  cases s <;> simp\n\nend cast\n\nend SignType\n\nvariable {α : Type*}\n\nopen SignType\n\nsection Preorder\n\nvariable [Zero α] [Preorder α] [DecidableLT α] {a : α}\n\n/-- The sign of an element is 1 if it's positive, -1 if negative, 0 otherwise. -/\ndef SignType.sign : α →o SignType :=\n  ⟨fun a => if 0 < a then 1 else if a < 0 then -1 else 0, fun a b h => by\n    dsimp\n    split_ifs with h₁ h₂ h₃ h₄ _ _ h₂ h₃ <;> try constructor\n    · cases lt_irrefl 0 (h₁.trans <| h.trans_lt h₃)\n    · cases h₂ (h₁.trans_le h)\n    · cases h₄ (h.trans_lt h₃)⟩\n\ntheorem sign_apply : sign a = ite (0 < a) 1 (ite (a < 0) (-1) 0) :=\n  rfl\n\n@[simp]\ntheorem sign_zero : sign (0 : α) = 0 := by simp [sign_apply]\n\n@[simp]\ntheorem sign_pos (ha : 0 < a) : sign a = 1 := by rwa [sign_apply, if_pos]\n\n@[simp]\n\nTarget:\ntheorem sign_neg (ha : a < 0) : sign a = -1 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_3edd16c3ca03","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"a4d454f625fb5d30fc0fc504aad6499b7d5099f59be59599b7ef7aa51349853a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sign","family_id":"sign_neg","file_id":"mathlib/Mathlib/Data/Sign/Defs.lean","sample_id":"3edd16c3ca0318c2e5d045f7be510639e5b1ebe6db412620dbe457e127e3ccb4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"efb528cee40199b8a7f10c4af81ecc5b602aa05a4991b3b60a5968591ef7ad2c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2c31379e40cbecc4b70db87b9684b86d81f7fb6fcff5b69f3c8ba705b4e20f03","source_sha256":"7aa6ff736d259829acf4509d7de6faf2fc5cbce6e43d3b6e8a128a0408a44be4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [section_]\n  rw [assoc, dsimp% X.hom.naturality, comp_id]\n  exact ed.s'_comp_ε","hard_negative":false,"metrics":{"chosen_tokens":24,"rejected_tokens":5,"token_jaccard":0.095238,"token_length_ratio":0.208333},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"1c5e479d9829761933af88ce30d0ec454f6ea70f8735f1685c4b9d9a4639bff5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.Homotopy\npublic import Mathlib.AlgebraicTopology.AlternatingFaceMapComplex\npublic import Mathlib.AlgebraicTopology.CechNerve\npublic import Mathlib.AlgebraicTopology.SimplicialObject.DeltaZeroIter\npublic import Mathlib.AlgebraicTopology.SimplicialObject.Homotopy\npublic import Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex\n\nNamespace:\nCategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy\n\nLocal context:\n/-\nCopyright (c) 2022 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n\n# Augmented simplicial objects with an extra degeneracy\n\nIn simplicial homotopy theory, in order to prove that the connected components\nof a simplicial set `X` are contractible, it suffices to construct an extra\ndegeneracy as it is defined in *Simplicial Homotopy Theory* by Goerss-Jardine p. 190.\nIt consists of a series of maps `π₀ X → X _⦋0⦌` and `X _⦋n⦌ → X _⦋n+1⦌` which\nbehave formally like an extra degeneracy `σ (-1)`. It can be thought as a datum\nassociated to the augmented simplicial set `X → π₀ X`.\n\nIn this file, we adapt this definition to the case of augmented\nsimplicial objects in any category.\n\n## Main definitions\n\n- the structure `ExtraDegeneracy X` for any `X : SimplicialObject.Augmented C`\n- `ExtraDegeneracy.map`: extra degeneracies are preserved by the application of any\n  functor `C ⥤ D`\n- `SSet.Augmented.StandardSimplex.extraDegeneracy`: the standard `n`-simplex has\n  an extra degeneracy\n- `Arrow.AugmentedCechNerve.extraDegeneracy`: the Čech nerve of a split\n  epimorphism has an extra degeneracy\n- `ExtraDegeneracy.homotopyEquiv`: in the case the category `C` is preadditive,\n  if we have an extra degeneracy on `X : SimplicialObject.Augmented C`, then\n  the augmentation on the alternating face map complex of `X` is a homotopy\n  equivalence.\n- `ExtraDegeneracy.homotopy`: if we have an extra degeneracy `ed` on\n  `X : SimplicialObject.Augmented C` (for any category `C`), then\n  the morphism `X.hom ≫ ed.section_` is homotopic to `𝟙 X.left`.\n\n## References\n* [Paul G. Goerss, John F. Jardine, *Simplicial Homotopy Theory*][goerss-jardine-2009]\n* [M. Barr, J. Kennison, J. and R. Robert,\n  *Contractible simplicial objects*][barr-kennison-robert-2019]\n\n-/\n\nset_option backward.defeqAttrib.useBackward true\n\n@[expose] public section\n\n\nopen CategoryTheory Category SimplicialObject.Augmented Opposite Simplicial\n\nnamespace CategoryTheory\n\nnamespace SimplicialObject\n\nnamespace Augmented\n\nvariable {C : Type*} [Category* C]\n\n/-- The datum of an extra degeneracy is a technical condition on\naugmented simplicial objects. The morphisms `s'` and `s n` of the\nstructure formally behave like extra degeneracies `σ (-1)`. -/\n@[ext]\nstructure ExtraDegeneracy (X : SimplicialObject.Augmented C) where\n  /-- a section of the augmentation in dimension `0` -/\n  s' : X.right ⟶ X.left _⦋0⦌\n  /-- the extra degeneracy -/\n  s : ∀ n : ℕ, X.left _⦋n⦌ ⟶ X.left _⦋n + 1⦌\n  s'_comp_ε : dsimp% s' ≫ X.hom.app (op ⦋0⦌) = 𝟙 X.right := by cat_disch\n  s₀_comp_δ₁ : dsimp% s 0 ≫ X.left.δ 1 = X.hom.app (op ⦋0⦌) ≫ s' := by cat_disch\n  s_comp_δ₀ : ∀ n : ℕ, s n ≫ X.left.δ 0 = 𝟙 _ := by cat_disch\n  s_comp_δ :\n    ∀ (n : ℕ) (i : Fin (n + 2)), s (n + 1) ≫ X.left.δ i.succ = X.left.δ i ≫ s n := by cat_disch\n  s_comp_σ :\n    ∀ (n : ℕ) (i : Fin (n + 1)), s n ≫ X.left.σ i.succ = X.left.σ i ≫ s (n + 1) := by cat_disch\n\nnamespace ExtraDegeneracy\n\nattribute [reassoc] s₀_comp_δ₁ s_comp_δ s_comp_σ\nattribute [reassoc (attr := simp)] s'_comp_ε s_comp_δ₀\n\nattribute [local simp←] Functor.map_comp in\nattribute [local simp] s₀_comp_δ₁ s_comp_δ s_comp_σ in\n/-- If `ed` is an extra degeneracy for `X : SimplicialObject.Augmented C` and\n`F : C ⥤ D` is a functor, then `ed.map F` is an extra degeneracy for the\naugmented simplicial object in `D` obtained by applying `F` to `X`. -/\ndef map {D : Type*} [Category* D] {X : SimplicialObject.Augmented C} (ed : ExtraDegeneracy X)\n    (F : C ⥤ D) : ExtraDegeneracy (((whiskering _ _).obj F).obj X) where\n  s' := F.map ed.s'\n  s n := F.map (ed.s n)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- If `X` and `Y` are isomorphic augmented simplicial objects, then an extra\ndegeneracy for `X` gives also an extra degeneracy for `Y` -/\ndef ofIso {X Y : SimplicialObject.Augmented C} (e : X ≅ Y) (ed : ExtraDegeneracy X) :\n    ExtraDegeneracy Y where\n  s' := (point.mapIso e).inv ≫ ed.s' ≫ (drop.mapIso e).hom.app (op ⦋0⦌)\n  s n := (drop.mapIso e).inv.app (op ⦋n⦌) ≫ ed.s n ≫ (drop.mapIso e).hom.app (op ⦋n + 1⦌)\n  s'_comp_ε := by\n    simpa [dsimp% w₀] using dsimp% (point.mapIso e).inv_hom_id\n  s₀_comp_δ₁ := by\n    simp [← SimplicialObject.δ_naturality, s₀_comp_δ₁_assoc, dsimp% w₀_assoc]\n  s_comp_δ₀ n := by\n    simpa [← SimplicialObject.δ_naturality] using\n      congr_app (drop.mapIso e).inv_hom_id (op ⦋n⦌)\n  s_comp_δ n i := by\n    simp [← SimplicialObject.δ_naturality, s_comp_δ_assoc,\n      ← SimplicialObject.δ_naturality_assoc]\n  s_comp_σ n i := by\n    simp [← SimplicialObject.σ_naturality, s_comp_σ_assoc,\n      ← SimplicialObject.σ_naturality_assoc]\n\nvariable {X : SimplicialObject.Augmented C} (ed : ExtraDegeneracy X)\n\nattribute [local simp←] Functor.map_comp in\n/-- The section of the augmentation that is induced by the extradegeneracy. -/\ndef section_ : (SimplicialObject.const C).obj X.right ⟶ X.left where\n  app n := ed.s' ≫ X.left.map (SimplexCategory.isTerminalZero.from _).op\n\n@[simp]\nlemma section_app_op_mk_zero :\n    ed.section_.app (op ⦋0⦌) = ed.s' := by\n  simp [section_]\n\n@[reassoc (attr := simp)]\n\nTarget:\nlemma section_app_comp_hom_app (n : SimplexCategoryᵒᵖ) :\n    dsimp% ed.section_.app n ≫ X.hom.app n = 𝟙 _ :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology","family_id":"section_app_comp_hom_app","file_id":"mathlib/Mathlib/AlgebraicTopology/ExtraDegeneracy.lean","sample_id":"2c31379e40cbecc4b70db87b9684b86d81f7fb6fcff5b69f3c8ba705b4e20f03"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f4dbb4c53a6e3a2d5b935a93c9fc29cc9b719e218918f120b3b408ba8d371698","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5a2ece2a22fcf936fd41bf7ec3c1d0ec84ea941a229b765b720040afa6959a7e","source_sha256":"695353e91eba40e45d9ecd0e58f8b00a56d575b25190f682a1689e29c7316a51","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold logDeriv\n  simp [field, Derivation.leibniz_div]","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.272727},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"1c70f55398586dcf9a6bd21af6ef17a54558ada3de20a56346dc50824d01493c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Derivation.MapCoeffs\npublic import Mathlib.FieldTheory.PrimitiveElement\n\nNamespace:\nDifferential\n\nLocal context:\n/-\nCopyright (c) 2024 Daniel Weber. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Weber\n-/\n/-!\n# Differential Fields\n\nThis file defines the logarithmic derivative `Differential.logDeriv` and proves properties of it.\nThis is defined algebraically, compared to `logDeriv` which is analytical.\n-/\n\n@[expose] public section\n\nnamespace Differential\n\nopen algebraMap Polynomial IntermediateField\n\nvariable {R : Type*} [Field R] [Differential R] (a b : R)\n\n/--\nThe logarithmic derivative of a is a′ / a.\n-/\ndef logDeriv : R := a′ / a\n\n@[simp]\nlemma logDeriv_zero : logDeriv (0 : R) = 0 := by\n  simp [logDeriv]\n\n@[simp]\nlemma logDeriv_one : logDeriv (1 : R) = 0 := by\n  simp [logDeriv]\n\nlemma logDeriv_mul (ha : a ≠ 0) (hb : b ≠ 0) : logDeriv (a * b) = logDeriv a + logDeriv b := by\n  unfold logDeriv\n  simp [field]\n  ring\n\nTarget:\nlemma logDeriv_div (ha : a ≠ 0) (hb : b ≠ 0) : logDeriv (a / b) = logDeriv a - logDeriv b :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Differential","family_id":"logderiv_div","file_id":"mathlib/Mathlib/FieldTheory/Differential/Basic.lean","sample_id":"5a2ece2a22fcf936fd41bf7ec3c1d0ec84ea941a229b765b720040afa6959a7e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c91bd3015c099e4be465fbcb7168b44d0092e9266e1434d35ffc1f51e248e636","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"befab83d2930909beac8cfad141326e33a75bf337f03f5f78fa5a152c0c13634","source_sha256":"f8317928b1cbc96407d5af706b485b7c6547cb67a94dbd8e0bac5c444710ec85","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  obtain ⟨I, hIC, hI, hst⟩ := hC.sdiff_eq_sUnion' s hs t ht\n  refine ⟨.ofErase I (supIndep_iff_pairwiseDisjoint.mpr hI) ?_, ?_⟩\n  · rw [sup_id_eq_sSup, sSup_eq_sUnion, hst]\n  · grw [Finpartition.ofErase_parts, Finset.erase_subset, hIC]","hard_negative":false,"metrics":{"chosen_tokens":59,"rejected_tokens":3,"token_jaccard":0.052632,"token_length_ratio":0.050847},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"1c76e5ec838c937b1ff12063d0bdf455a4d7d9ee1d7fb0474b8503c48ae44e4e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Pairwise.Lattice\npublic import Mathlib.MeasureTheory.PiSystem\npublic import Mathlib.Order.Lattice.Nat\npublic import Mathlib.Order.Partition.Finpartition\npublic import Mathlib.Order.SetAccumulate\npublic import Mathlib.Order.SupClosed\n\nNamespace:\nMeasureTheory.IsSetSemiring\n\nLocal context:\n/-\nCopyright (c) 2023 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne, Peter Pfaffelhuber\n-/\n/-! # Semirings and rings of sets\n\nA semi-ring of sets `C` (in the sense of measure theory) is a family of sets containing `∅`,\nstable by intersection and such that for all `s, t ∈ C`, `t \\ s` is equal to a disjoint union of\nfinitely many sets in `C`. Note that a semi-ring of sets may not contain unions.\n\nAn important example of a semi-ring of sets is intervals in `ℝ`. The intersection of two intervals\nis an interval (possibly empty). The union of two intervals may not be an interval.\nThe set difference of two intervals may not be an interval, but it will be a disjoint union of\ntwo intervals.\n\nA ring of sets is a set of sets containing `∅`, stable by union, set difference and intersection.\n\n## Main definitions\n\n* `MeasureTheory.IsSetSemiring C`: property of being a semi-ring of sets.\n* `MeasureTheory.IsSetSemiring.disjointOfDiff hs ht`: for `s, t` in a semi-ring `C`\n  (with `hC : IsSetSemiring C`) with `hs : s ∈ C`, `ht : t ∈ C`, this is a `Finset` of\n  pairwise disjoint sets such that `s \\ t = ⋃₀ hC.disjointOfDiff hs ht`.\n* `MeasureTheory.IsSetSemiring.disjointOfDiffUnion hs hI`: for `hs : s ∈ C` and a finset\n  `I` of sets in `C` (with `hI : ↑I ⊆ C`), this is a `Finset` of pairwise disjoint sets such that\n  `s \\ ⋃₀ I = ⋃₀ hC.disjointOfDiffUnion hs hI`.\n* `MeasureTheory.IsSetSemiring.disjointOfUnion hJ`: for `hJ ⊆ C`, this is a\n  `Finset` of pairwise disjoint sets such that `⋃₀ J = ⋃₀ hC.disjointOfUnion hJ`.\n\n* `MeasureTheory.IsSetRing`: property of being a ring of sets.\n\n## Main statements\n\n* `MeasureTheory.IsSetSemiring.exists_disjoint_finset_sdiff_eq`: the existence of the `Finset` given\n  by the definition `IsSetSemiring.disjointOfDiffUnion` (see above).\n* `MeasureTheory.IsSetSemiring.disjointOfUnion_props`: In a `hC : IsSetSemiring C`,\n  for a `J : Finset (Set α)` with `J ⊆ C`, there is\n  for every `x in J` some `K x ⊆ C` finite, such that\n  * `⋃ x ∈ J, K x` are pairwise disjoint and do not contain ∅,\n  * `⋃ s ∈ K x, s ⊆ x`,\n  * `⋃ x ∈ J, x = ⋃ x ∈ J, ⋃ s ∈ K x, s`.\n\n-/\n\n@[expose] public section\n\nopen Finset Set\n\nnamespace MeasureTheory\n\nvariable {α : Type*} {C : Set (Set α)} {s t : Set α}\n\n/-- A semi-ring of sets `C` is a family of sets containing `∅`, stable by intersection and such that\nfor all `s, t ∈ C`, `s \\ t` is equal to a disjoint union of finitely many sets in `C`. -/\nstructure IsSetSemiring (C : Set (Set α)) : Prop where\n  empty_mem : ∅ ∈ C\n  inter_mem : ∀ s ∈ C, ∀ t ∈ C, s ∩ t ∈ C\n  sdiff_eq_sUnion' : ∀ s ∈ C, ∀ t ∈ C,\n    ∃ I : Finset (Set α), ↑I ⊆ C ∧ PairwiseDisjoint (I : Set (Set α)) id ∧ s \\ t = ⋃₀ I\n\n/-- A ring of sets `C` is a family of sets containing `∅`, stable by union and set difference.\nIt is then also stable by intersection (see `IsSetRing.inter_mem`). -/\nstructure IsSetRing (C : Set (Set α)) : Prop where\n  empty_mem : ∅ ∈ C\n  union_mem ⦃s t : Set α⦄ : s ∈ C → t ∈ C → s ∪ t ∈ C\n  sdiff_mem ⦃s t : Set α⦄ : s ∈ C → t ∈ C → s \\ t ∈ C\n\nnamespace IsSetSemiring\n\nlemma isPiSystem (hC : IsSetSemiring C) : IsPiSystem C := fun s hs t ht _ ↦ hC.inter_mem s hs t ht\n\nTarget:\ntheorem exists_finpartition_sdiff (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :\n    ∃ P : Finpartition (s \\ t), ↑P.parts ⊆ C :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory","family_id":"exists_finpartition_sdiff","file_id":"mathlib/Mathlib/MeasureTheory/SetSemiring.lean","sample_id":"befab83d2930909beac8cfad141326e33a75bf337f03f5f78fa5a152c0c13634"},"split":"train"}
{"binding":{"chosen_proof_sha256":"59b2a49c3792a452986527f1fd02d5e769f33e00ec8381ee8f5c72d68ad2ecd4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5f8fd4810614a2a60bbe889c763a731558149e6e22da170e586da7e6c50b4cc4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"99f1e37e3ec119044c219636cfcc8ffdd76a19fdac55cff796f9c655385230aa","source_sha256":"6b463b31746b56f6d89e4d588151f63717a66001d29dfed4ab7ccf9fc8571299","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [le_antisymm_iff, h x y, h y x]\n\nvariable [IsDomain S] [Nontrivial R]","hard_negative":true,"metrics":{"chosen_tokens":22,"rejected_tokens":2,"token_jaccard":0.066667,"token_length_ratio":0.090909},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"1c81b906bbab5d4877e330b3d16d118fdf8abef713ba75b6c3319ea41f0cf60e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Field.WithAbs\npublic import Mathlib.Analysis.SpecialFunctions.Pow.Real\n\nNamespace:\nAbsoluteValue\n\nLocal context:\n/-\nCopyright (c) 2025 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Equivalence of real-valued absolute values\n\nTwo absolute values `v₁, v₂ : AbsoluteValue R ℝ` are *equivalent* if there exists a\npositive real number `c` such that `v₁ x ^ c = v₂ x` for all `x : R`.\n-/\n\n@[expose] public section\n\nnamespace AbsoluteValue\n\nsection OrderedSemiring\n\nvariable {R : Type*} [Semiring R] {S : Type*} [Semiring S] [PartialOrder S]\n  (v w : AbsoluteValue R S)\n\n/-- Two absolute values `v` and `w` are *equivalent* if `v x ≤ v y` precisely when\n`w x ≤ w y`.\n\nNote that for real absolute values this condition is equivalent to the existence of a positive\nreal number `c` such that `v x ^ c = w x` for all `x`. See\n`AbsoluteValue.isEquiv_iff_exists_rpow_eq`. -/\ndef IsEquiv : Prop := ∀ x y, v x ≤ v y ↔ w x ≤ w y\n\ntheorem IsEquiv.refl : v.IsEquiv v := fun _ _ ↦ .rfl\n\nvariable {v w}\n\ntheorem IsEquiv.rfl : v.IsEquiv v := fun _ _ ↦ .rfl\n\ntheorem IsEquiv.symm (h : v.IsEquiv w) : w.IsEquiv v := fun _ _ ↦ (h _ _).symm\n\ntheorem IsEquiv.trans {u : AbsoluteValue R S} (h₁ : v.IsEquiv w)\n    (h₂ : w.IsEquiv u) : v.IsEquiv u := fun _ _ ↦ (h₁ _ _).trans (h₂ _ _)\n\ninstance : Setoid (AbsoluteValue R S) where\n  r := IsEquiv\n  iseqv := {\n    refl := .refl\n    symm := .symm\n    trans := .trans\n  }\n\ntheorem IsEquiv.le_iff_le (h : v.IsEquiv w) {x y : R} : v x ≤ v y ↔ w x ≤ w y := h ..\n\ntheorem IsEquiv.lt_iff_lt (h : v.IsEquiv w) {x y : R} : v x < v y ↔ w x < w y :=\n  lt_iff_lt_of_le_iff_le' (h y x) (h x y)\n\nTarget:\ntheorem IsEquiv.eq_iff_eq (h : v.IsEquiv w) {x y : R} : v x = v y ↔ w x = w y :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_99f1e37e3ec1","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"3ea7a7afc7fec1b41400f7ee608ccf41f6101bd37537d05f2e6e1bff003060a7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/AbsoluteValue","family_id":"isequiv","file_id":"mathlib/Mathlib/Analysis/AbsoluteValue/Equivalence.lean","sample_id":"99f1e37e3ec119044c219636cfcc8ffdd76a19fdac55cff796f9c655385230aa"},"split":"train"}
{"binding":{"chosen_proof_sha256":"55394e3b94e4002c95f3216c1764d8bfdefb9347ecc72f11001d5f75ebad2dcb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2401688d1603099d05da8c9eb9c2a484d628ba1e3b53286d63c216455fb01f8b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"65f5358ecacae8480f414a0ec307a37a8131f8d46fb796e1817fa646fac75533","source_sha256":"b302dc459101a0b475ce68391aafc25fa9a801436e4b136046216248b891e1f7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rintro x ⟨hf, hg⟩\n  rw [mem_fixedPoints_iff] at *\n  rw [comp_apply, hg, hf]","hard_negative":false,"metrics":{"chosen_tokens":22,"rejected_tokens":26,"token_jaccard":0.789474,"token_length_ratio":1.181818},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"1cc62332447619be8ebbf06d08c8d77cd5753ad76027f45bd1d852ad80f89bce","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Operations\n\nNamespace:\nFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Fixed points of a self-map\n\nIn this file we define the set `Function.fixedPoints` of fixed points of a function `f : α → α`.\nThe related predicate `IsFixedPt` is defined in `Mathlib.Logic.Function.Defs`.\n\n## Tags\n\nfixed point\n-/\n\n@[expose] public section\n\nnamespace Function\n\nvariable {α : Type*} {x : α} {f g : α → α}\n\n/-- The set of fixed points of a map `f : α → α`. -/\ndef fixedPoints (f : α → α) : Set α :=\n  { x : α | IsFixedPt f x }\n\ninstance fixedPoints.decidable [DecidableEq α] (f : α → α) (x : α) :\n    Decidable (x ∈ fixedPoints f) :=\n  IsFixedPt.decidable\n\n@[simp]\ntheorem mem_fixedPoints : x ∈ fixedPoints f ↔ IsFixedPt f x :=\n  .rfl\n\ntheorem mem_fixedPoints_iff {α : Type*} {f : α → α} {x : α} : x ∈ fixedPoints f ↔ f x = x :=\n  .rfl\n\n@[simp]\ntheorem fixedPoints_id : fixedPoints (@id α) = Set.univ :=\n  Set.ext fun _ => by simpa using isFixedPt_id _\n\n@[simp]\n\nTarget:\ntheorem inter_subset_fixedPoints_comp : fixedPoints f ∩ fixedPoints g ⊆ fixedPoints (f ∘ g) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rintro x ⟨hf, hg⟩\n  rw [mem_fixedPoints_iff] at *\n  rw [comp_apply, hg, hf]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/FixedPoints","family_id":"inter_subset_fixedpoints_comp","file_id":"mathlib/Mathlib/Dynamics/FixedPoints/Defs.lean","sample_id":"65f5358ecacae8480f414a0ec307a37a8131f8d46fb796e1817fa646fac75533"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c601c0ce6ee21cbe086a05718f7a601ae3b0678e21055794ba74900b0b12e072","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"451ace8d95c1b634db051c4f50d43d08f0484bdd00fcb364c7b3114561147163","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"15c500daf4b584078896ed64949b949708a161dcfff41d36df74a29d2e07383f","source_sha256":"e2e260981e2cd985149de5ebb4a157aa3216ee7c359bb9bfe83a133b3f869574","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [IsPeriodicPt, hco.comp_iterate] at hfg\n  exact hfg.left_of_comp hg","hard_negative":false,"metrics":{"chosen_tokens":16,"rejected_tokens":23,"token_jaccard":0.875,"token_length_ratio":1.4375},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"1cd7d344796e466ec0c7769727f23526cf3aa9c6bd4272e56a276fa4af1e9592","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Defs\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.Sub.Basic\npublic import Mathlib.Data.List.Cycle\npublic import Mathlib.Data.PNat.Notation\npublic import Mathlib.Dynamics.FixedPoints.Basic\n\nNamespace:\nFunction.IsPeriodicPt\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Periodic points\n\nA point `x : α` is a periodic point of `f : α → α` of period `n` if `f^[n] x = x`.\n\n## Main definitions\n\n* `IsPeriodicPt f n x` : `x` is a periodic point of `f` of period `n`, i.e. `f^[n] x = x`.\n  We do not require `n > 0` in the definition.\n* `ptsOfPeriod f n` : the set `{x | IsPeriodicPt f n x}`. Note that `n` is not required to\n  be the minimal period of `x`.\n* `periodicPts f` : the set of all periodic points of `f`.\n* `minimalPeriod f x` : the minimal period of a point `x` under an endomorphism `f` or zero\n  if `x` is not a periodic point of `f`.\n* `orbit f x`: the cycle `[x, f x, f (f x), ...]` for a periodic point.\n* `MulAction.period g x` : the minimal period of a point `x` under the multiplicative action of `g`;\n  an equivalent `AddAction.period g x` is defined for additive actions.\n\n## Main statements\n\nWe provide “dot syntax”-style operations on terms of the form `h : IsPeriodicPt f n x` including\narithmetic operations on `n` and `h.map (hg : SemiconjBy g f f')`. We also prove that `f`\nis bijective on each set `ptsOfPeriod f n` and on `periodicPts f`. Finally, we prove that `x`\nis a periodic point of `f` of period `n` if and only if `minimalPeriod f x | n`.\n\n## References\n\n* https://en.wikipedia.org/wiki/Periodic_point\n\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero\n\n\nopen Set\n\nnamespace Function\n\nopen Function (Commute)\n\nvariable {α : Type*} {β : Type*} {f fa : α → α} {fb : β → β} {x y : α} {m n : ℕ}\n\n/-- A point `x` is a periodic point of `f : α → α` of period `n` if `f^[n] x = x`.\nNote that we do not require `0 < n` in this definition. Many theorems about periodic points\nneed this assumption. -/\ndef IsPeriodicPt (f : α → α) (n : ℕ) (x : α) :=\n  IsFixedPt f^[n] x\n\n/-- A fixed point of `f` is a periodic point of `f` of any prescribed period. -/\ntheorem IsFixedPt.isPeriodicPt (hf : IsFixedPt f x) (n : ℕ) : IsPeriodicPt f n x :=\n  hf.iterate n\n\n/-- For the identity map, all points are periodic. -/\ntheorem is_periodic_id (n : ℕ) (x : α) : IsPeriodicPt id n x :=\n  (isFixedPt_id x).isPeriodicPt n\n\n/-- Any point is a periodic point of period `0`. -/\ntheorem isPeriodicPt_zero (f : α → α) (x : α) : IsPeriodicPt f 0 x :=\n  isFixedPt_id x\n\nnamespace IsPeriodicPt\n\n@[nontriviality]\ntheorem of_subsingleton [Subsingleton α] (f : α → α) (n : ℕ) (x : α) : IsPeriodicPt f n x :=\n  IsFixedPt.of_subsingleton _ _\n\ninstance [DecidableEq α] {f : α → α} {n : ℕ} {x : α} : Decidable (IsPeriodicPt f n x) :=\n  IsFixedPt.decidable\n\nprotected theorem isFixedPt (hf : IsPeriodicPt f n x) : IsFixedPt f^[n] x :=\n  hf\n\nprotected theorem map (hx : IsPeriodicPt fa n x) {g : α → β} (hg : Semiconj g fa fb) :\n    IsPeriodicPt fb n (g x) :=\n  IsFixedPt.map hx (hg.iterate_right n)\n\ntheorem apply_iterate (hx : IsPeriodicPt f n x) (m : ℕ) : IsPeriodicPt f n (f^[m] x) :=\n  hx.map <| Commute.iterate_self f m\n\nprotected theorem apply (hx : IsPeriodicPt f n x) : IsPeriodicPt f n (f x) :=\n  hx.apply_iterate 1\n\nprotected theorem add (hn : IsPeriodicPt f n x) (hm : IsPeriodicPt f m x) :\n    IsPeriodicPt f (n + m) x := by\n  rw [IsPeriodicPt, iterate_add]\n  exact hn.comp hm\n\ntheorem left_of_add (hn : IsPeriodicPt f (n + m) x) (hm : IsPeriodicPt f m x) :\n    IsPeriodicPt f n x := by\n  rw [IsPeriodicPt, iterate_add] at hn\n  exact hn.left_of_comp hm\n\ntheorem right_of_add (hn : IsPeriodicPt f (n + m) x) (hm : IsPeriodicPt f n x) :\n    IsPeriodicPt f m x := by\n  rw [add_comm] at hn\n  exact hn.left_of_add hm\n\nprotected theorem sub (hm : IsPeriodicPt f m x) (hn : IsPeriodicPt f n x) :\n    IsPeriodicPt f (m - n) x := by\n  rcases le_total n m with h | h\n  · refine left_of_add ?_ hn\n    rwa [tsub_add_cancel_of_le h]\n  · rw [tsub_eq_zero_iff_le.mpr h]\n    apply isPeriodicPt_zero\n\nprotected theorem mul_const (hm : IsPeriodicPt f m x) (n : ℕ) : IsPeriodicPt f (m * n) x := by\n  simp only [IsPeriodicPt, iterate_mul, hm.isFixedPt.iterate n]\n\nprotected theorem const_mul (hm : IsPeriodicPt f m x) (n : ℕ) : IsPeriodicPt f (n * m) x := by\n  simp only [mul_comm n, hm.mul_const n]\n\ntheorem trans_dvd (hm : IsPeriodicPt f m x) {n : ℕ} (hn : m ∣ n) : IsPeriodicPt f n x :=\n  let ⟨k, hk⟩ := hn\n  hk.symm ▸ hm.mul_const k\n\nprotected theorem iterate (hf : IsPeriodicPt f n x) (m : ℕ) : IsPeriodicPt f^[m] n x := by\n  rw [IsPeriodicPt, ← iterate_mul, mul_comm, iterate_mul]\n  exact hf.isFixedPt.iterate m\n\ntheorem comp {g : α → α} (hco : Commute f g) (hf : IsPeriodicPt f n x) (hg : IsPeriodicPt g n x) :\n    IsPeriodicPt (f ∘ g) n x := by\n  rw [IsPeriodicPt, hco.comp_iterate]\n  exact IsFixedPt.comp hf hg\n\ntheorem comp_lcm {g : α → α} (hco : Commute f g) (hf : IsPeriodicPt f m x)\n    (hg : IsPeriodicPt g n x) : IsPeriodicPt (f ∘ g) (Nat.lcm m n) x :=\n  (hf.trans_dvd <| Nat.dvd_lcm_left _ _).comp hco (hg.trans_dvd <| Nat.dvd_lcm_right _ _)\n\nTarget:\ntheorem left_of_comp {g : α → α} (hco : Commute f g) (hfg : IsPeriodicPt (f ∘ g) n x)\n    (hg : IsPeriodicPt g n x) : IsPeriodicPt f n x :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [IsPeriodicPt, hco.comp_iterate] at hfg\n  exact hfg.left_of_comp hg\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/PeriodicPts","family_id":"left_of_comp","file_id":"mathlib/Mathlib/Dynamics/PeriodicPts/Defs.lean","sample_id":"15c500daf4b584078896ed64949b949708a161dcfff41d36df74a29d2e07383f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b62df895e5ecbe83fea44bae35972a3514a09e499e02633c6e24648777cc00bf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9a39643b19053974b943a97a575958ebd9f8fed79c371c4523680775abf39743","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5e07b58c14b8fda2d13e55aab1c1520c2c2eed27aece7742846d0bae8fae4ed6","source_sha256":"5d8f7f707a69d48e10d7817e517c71568b55a688a7dc3473a0fa55b05f45f810","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro contr\n  obtain ⟨a, b, hab, mcfab⟩ := mcf\n  specialize contr ![![a, b], ![b, a]]\n  rw [Fin.sum_univ_two', ← mcfab.left, ← two_nsmul] at contr\n  have sharp :\n    2 • ((ω.tt ![![a, b], ![b, a]]).map (fun _ => f ![a, b])).sum <\n    2 • ((ω.tt ![![a, b], ![b, a]]).map f).sum := by\n    have half_sharp :\n      ((ω.tt ![![a, b], ![b, a]]).map (fun _ => f ![a, b])).sum <\n      ((ω.tt ![![a, b], ![b, a]]).map f).sum := by\n      apply Multiset.sum_lt_sum\n      · intro r rin\n        exact le_of_lt (mcfab.rows_lt_aux hab symmega rin)\n      · obtain ⟨g, _⟩ := valid.contains\n        have : (fun i => g ((Function.swap ![![a, b], ![b, a]]) i)) ∈ ω.tt ![![a, b], ![b, a]] := by\n          simp only [FractionalOperation.tt, Multiset.mem_map]\n          use g\n        exact ⟨_, this, mcfab.rows_lt_aux hab symmega this⟩\n    rw [two_nsmul, two_nsmul]\n    exact add_lt_add half_sharp half_sharp\n  have impos : 2 • (ω.map (fun _ => f ![a, b])).sum < ω.size • 2 • f ![a, b] := by\n    convert! lt_of_lt_of_le sharp contr\n    simp [FractionalOperation.tt, Multiset.map_map]\n  have rhs_swap : ω.size • 2 • f ![a, b] = 2 • ω.size • f ![a, b] := nsmul_left_comm ..\n  have distrib : (ω.map (fun _ => f ![a, b])).sum = ω.size • f ![a, b] := by simp\n  rw [rhs_swap, distrib] at impos\n  exact ne_of_lt impos rfl","hard_negative":true,"metrics":{"chosen_tokens":456,"rejected_tokens":2,"token_jaccard":0.012821,"token_length_ratio":0.004386},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"1ce64bcd27abe28dc787819952db5b0292664c4448ec32b5954cb9cd13a0ba7c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Fin\npublic import Mathlib.Algebra.Order.BigOperators.Group.Multiset\npublic import Mathlib.Data.Fin.VecNotation\npublic import Mathlib.LinearAlgebra.Matrix.Notation\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Martin Dvorak. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Martin Dvorak\n-/\n/-!\n\n# General-Valued Constraint Satisfaction Problems\n\nGeneral-Valued CSP is a very broad class of problems in discrete optimization.\nGeneral-Valued CSP subsumes Min-Cost-Hom (including 3-SAT for example) and Finite-Valued CSP.\n\n## Main definitions\n* `ValuedCSP`: A VCSP template; fixes a domain, a codomain, and allowed cost functions.\n* `ValuedCSP.Term`: One summand in a VCSP instance; calls a concrete function from given template.\n* `ValuedCSP.Term.evalSolution`: An evaluation of the VCSP term for given solution.\n* `ValuedCSP.Instance`: An instance of a VCSP problem over given template.\n* `ValuedCSP.Instance.evalSolution`: An evaluation of the VCSP instance for given solution.\n* `ValuedCSP.Instance.IsOptimumSolution`: Is given solution a minimum of the VCSP instance?\n* `Function.HasMaxCutProperty`: Can given binary function express the Max-Cut problem?\n* `FractionalOperation`: Multiset of operations on given domain of the same arity.\n* `FractionalOperation.IsSymmetricFractionalPolymorphismFor`: Is given fractional operation a\n  symmetric fractional polymorphism for given VCSP template?\n\n## References\n* [D. A. Cohen, M. C. Cooper, P. Creed, P. G. Jeavons, S. Živný,\n  *An Algebraic Theory of Complexity for Discrete Optimisation*][cohen2012]\n\n-/\n\n@[expose] public section\n\n/-- A template for a valued CSP problem over a domain `D` with costs in `C`.\nRegarding `C` we want to support `Bool`, `Nat`, `ENat`, `Int`, `Rat`, `NNRat`,\n`Real`, `NNReal`, `EReal`, `ENNReal`, and tuples made of any of those types. -/\n@[nolint unusedArguments]\nabbrev ValuedCSP (D C : Type*) [AddCommMonoid C] [PartialOrder C] [IsOrderedAddMonoid C] :=\n  Set (Σ (n : ℕ), (Fin n → D) → C) -- Cost functions `D^n → C` for any `n`\n\nvariable {D C : Type*} [AddCommMonoid C] [PartialOrder C] [IsOrderedAddMonoid C]\n\n/-- A term in a valued CSP instance over the template `Γ`. -/\nstructure ValuedCSP.Term (Γ : ValuedCSP D C) (ι : Type*) where\n  /-- Arity of the function -/\n  n : ℕ\n  /-- Which cost function is instantiated -/\n  f : (Fin n → D) → C\n  /-- The cost function comes from the template -/\n  inΓ : ⟨n, f⟩ ∈ Γ\n  /-- Which variables are plugged as arguments to the cost function -/\n  app : Fin n → ι\n\n/-- Evaluation of a `Γ` term `t` for given solution `x`. -/\ndef ValuedCSP.Term.evalSolution {Γ : ValuedCSP D C} {ι : Type*}\n    (t : Γ.Term ι) (x : ι → D) : C :=\n  t.f (x ∘ t.app)\n\n/-- A valued CSP instance over the template `Γ` with variables indexed by `ι`. -/\nabbrev ValuedCSP.Instance (Γ : ValuedCSP D C) (ι : Type*) : Type _ :=\n  Multiset (Γ.Term ι)\n\n/-- Evaluation of a `Γ` instance `I` for given solution `x`. -/\ndef ValuedCSP.Instance.evalSolution {Γ : ValuedCSP D C} {ι : Type*}\n    (I : Γ.Instance ι) (x : ι → D) : C :=\n  (I.map (·.evalSolution x)).sum\n\n/-- Condition for `x` being an optimum solution (min) to given `Γ` instance `I`. -/\ndef ValuedCSP.Instance.IsOptimumSolution {Γ : ValuedCSP D C} {ι : Type*}\n    (I : Γ.Instance ι) (x : ι → D) : Prop :=\n  ∀ y : ι → D, I.evalSolution x ≤ I.evalSolution y\n\n/-- Function `f` has Max-Cut property at labels `a` and `b` when `argmin f` is exactly\n`{ ![a, b], ![b, a] }`. -/\ndef Function.HasMaxCutPropertyAt (f : (Fin 2 → D) → C) (a b : D) : Prop :=\n  f ![a, b] = f ![b, a] ∧\n    ∀ x y : D, f ![a, b] ≤ f ![x, y] ∧ (f ![a, b] = f ![x, y] → a = x ∧ b = y ∨ a = y ∧ b = x)\n\n/-- Function `f` has Max-Cut property at some two non-identical labels. -/\ndef Function.HasMaxCutProperty (f : (Fin 2 → D) → C) : Prop :=\n  ∃ a b : D, a ≠ b ∧ f.HasMaxCutPropertyAt a b\n\n/-- Fractional operation is a finite unordered collection of D^m → D possibly with duplicates. -/\nabbrev FractionalOperation (D : Type*) (m : ℕ) : Type _ :=\n  Multiset ((Fin m → D) → D)\n\nvariable {m : ℕ}\n\n/-- Arity of the \"output\" of the fractional operation. -/\n@[simp]\ndef FractionalOperation.size (ω : FractionalOperation D m) : ℕ := ω.card\n\n/-- Fractional operation is valid iff nonempty. -/\ndef FractionalOperation.IsValid (ω : FractionalOperation D m) : Prop :=\n  ω ≠ ∅\n\n/-- Valid fractional operation contains an operation. -/\nlemma FractionalOperation.IsValid.contains {ω : FractionalOperation D m} (valid : ω.IsValid) :\n    ∃ g : (Fin m → D) → D, g ∈ ω :=\n  Multiset.exists_mem_of_ne_zero valid\n\n/-- Fractional operation applied to a transposed table of values. -/\ndef FractionalOperation.tt {ι : Type*} (ω : FractionalOperation D m) (x : Fin m → ι → D) :\n    Multiset (ι → D) :=\n  ω.map (fun (g : (Fin m → D) → D) (i : ι) => g ((Function.swap x) i))\n\n/-- Cost function admits given fractional operation, i.e., `ω` improves `f` in the `≤` sense. -/\ndef Function.AdmitsFractional {n : ℕ} (f : (Fin n → D) → C) (ω : FractionalOperation D m) : Prop :=\n  ∀ x : (Fin m → (Fin n → D)),\n    m • ((ω.tt x).map f).sum ≤ ω.size • Finset.univ.sum (fun i => f (x i))\n\n/-- Fractional operation is a fractional polymorphism for given VCSP template. -/\ndef FractionalOperation.IsFractionalPolymorphismFor\n    (ω : FractionalOperation D m) (Γ : ValuedCSP D C) : Prop :=\n  ∀ f ∈ Γ, f.snd.AdmitsFractional ω\n\n/-- Fractional operation is symmetric. -/\ndef FractionalOperation.IsSymmetric (ω : FractionalOperation D m) : Prop :=\n  ∀ x y : (Fin m → D), List.Perm (List.ofFn x) (List.ofFn y) → ∀ g ∈ ω, g x = g y\n\n/-- Fractional operation is a symmetric fractional polymorphism for given VCSP template. -/\ndef FractionalOperation.IsSymmetricFractionalPolymorphismFor\n    (ω : FractionalOperation D m) (Γ : ValuedCSP D C) : Prop :=\n  ω.IsFractionalPolymorphismFor Γ ∧ ω.IsSymmetric\n\nlemma Function.HasMaxCutPropertyAt.rows_lt_aux {C : Type*} [PartialOrder C]\n    {f : (Fin 2 → D) → C} {a b : D} (mcf : f.HasMaxCutPropertyAt a b) (hab : a ≠ b)\n    {ω : FractionalOperation D 2} (symmega : ω.IsSymmetric)\n    {r : Fin 2 → D} (rin : r ∈ (ω.tt ![![a, b], ![b, a]])) :\n    f ![a, b] < f r := by\n  rw [FractionalOperation.tt, Multiset.mem_map] at rin\n  rw [show r = ![r 0, r 1] by simp [← List.ofFn_inj]]\n  apply lt_of_le_of_ne (mcf.right (r 0) (r 1)).left\n  intro equ\n  have asymm : r 0 ≠ r 1 := by\n    rcases (mcf.right (r 0) (r 1)).right equ with ⟨ha0, hb1⟩ | ⟨ha1, hb0⟩\n    · rw [ha0, hb1] at hab\n      exact hab\n    · rw [ha1, hb0] at hab\n      exact hab.symm\n  apply asymm\n  obtain ⟨o, in_omega, rfl⟩ := rin\n  change o (fun j => ![![a, b], ![b, a]] j 0) = o (fun j => ![![a, b], ![b, a]] j 1)\n  convert! symmega ![a, b] ![b, a] (by simp [List.Perm.swap]) o in_omega using 2 <;>\n    simp [Matrix.const_fin1_eq]\n\nvariable {C : Type*} [AddCommMonoid C] [PartialOrder C] [IsOrderedCancelAddMonoid C]\n\nTarget:\nlemma Function.HasMaxCutProperty.forbids_commutativeFractionalPolymorphism\n    {f : (Fin 2 → D) → C} (mcf : f.HasMaxCutProperty)\n    {ω : FractionalOperation D 2} (valid : ω.IsValid) (symmega : ω.IsSymmetric) :\n    ¬ f.AdmitsFractional ω :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_5e07b58c14b8","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"cf29f216a84d9eebfb60bc8f39a6318a9a8f0b5acda098716b846e43d86446f1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Optimization","family_id":"function","file_id":"mathlib/Mathlib/Combinatorics/Optimization/ValuedCSP.lean","sample_id":"5e07b58c14b8fda2d13e55aab1c1520c2c2eed27aece7742846d0bae8fae4ed6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3db3fcbd4e98076f3d0533d22872c1a4a27f5605675d9e65936bcaf6671c864f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5f10fa5c3e8b99e44987fd6f1820e919de6c733d5031ca673f738ff243d9f672","source_sha256":"36095e70549b743e6d6e18e94107fa315e72325dc59944c049976dea39a27dfe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [← scottContinuousOn_univ] at ⊢\n  exact .fromProd (fun a ↦ (h₁ a).scottContinuousOn) (fun b ↦ (h₂ b).scottContinuousOn)\n    (fun a ↦ (h₁ a).monotone) (fun b ↦ (h₂ b).monotone)","hard_negative":false,"metrics":{"chosen_tokens":59,"rejected_tokens":3,"token_jaccard":0.086957,"token_length_ratio":0.050847},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"1d03a2cb305d0db293e1462157048bc889c4d80c250b9e1954c678bad30b1ec9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ScottContinuity\npublic import Mathlib.Order.Bounds.Lattice\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Christopher Hoskin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christopher Hoskin\n-/\n/-!\n# Scott continuity on product spaces\n\n## Main result\n\n- `ScottContinuous_prod_of_ScottContinuous`: A function is Scott continuous on a product space if it\n  is Scott continuous in each variable.\n- `ScottContinuousOn.inf₂`: For complete linear orders, the meet operation is Scott continuous.\n\n-/\n\npublic section\n\nopen Set\n\nvariable {α β γ : Type*}\n\n/-- If  is Scott continuous on a product space if it is Scott continuous and monotone in each\nvariable -/\nlemma ScottContinuousOn.fromProd [Preorder α] [Preorder β] [Preorder γ]\n    {f : α × β → γ} {D : Set (Set (α × β))}\n    (h₁ : ∀ a, ScottContinuousOn ((fun d => Prod.snd '' d) '' D) (fun b => f (a, b)))\n    (h₂ : ∀ b, ScottContinuousOn ((fun d => Prod.fst '' d) '' D) (fun a => f (a, b)))\n    (h₁' : ∀ a, Monotone (fun b => f (a, b))) (h₂' : ∀ b, Monotone (fun a => f (a, b))) :\n    ScottContinuousOn D f := fun d hX hd₁ hd₂ ⟨p1, p2⟩ hdp => by\n  rw [isLUB_congr ((monotone_prod_iff.mpr ⟨h₁', h₂'⟩).upperBounds_image_of_directedOn_prod hd₂),\n    ← iUnion_of_singleton_coe (Prod.fst '' d), iUnion_prod_const, image_iUnion,\n    ← isLUB_iUnion_iff_of_isLUB (fun a => by\n      rw [singleton_prod, image_image f (fun b ↦ (a, b))]\n      exact h₁ _ (mem_image_of_mem (fun d ↦ Prod.snd '' d) hX) (Nonempty.image Prod.snd hd₁)\n        (DirectedOn.snd hd₂) (isLUB_prod.mp hdp).2) _, Set.range]\n  convert!\n    (h₂ _ (mem_image_of_mem (fun d ↦ Prod.fst '' d) hX) (Nonempty.image Prod.fst hd₁)\n      (DirectedOn.fst hd₂) (isLUB_prod.mp hdp).1)\n  ext : 1\n  simp_all only [Subtype.exists, mem_image, Prod.exists,\n    exists_and_right, exists_eq_right, exists_prop, mem_setOf_eq]\n\nTarget:\nlemma ScottContinuous.fromProd {γ : Type*} [Preorder α] [Preorder β] [Preorder γ]\n    {f : α × β → γ} (h₁ : ∀ a, ScottContinuous (fun b => f (a, b)))\n    (h₂ : ∀ b, ScottContinuous (fun a => f (a, b))) : ScottContinuous f :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/ScottContinuity","family_id":"scottcontinuous","file_id":"mathlib/Mathlib/Order/ScottContinuity/Prod.lean","sample_id":"5f10fa5c3e8b99e44987fd6f1820e919de6c733d5031ca673f738ff243d9f672"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4d337297249649c3a21f0b564e2b0bbdb408fd0e071471a465068a7173e4de87","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"91bf84bd60b26a35f7dabcc39cf740009b92e0c19b956edd8d544cfa8e467dbb","source_sha256":"bf073edff8092ae47149eb64a7d94002d4cd84fb4941057816c6cf0f37fc36aa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isSymm_iff_basis b, Matrix.IsHermitian.ext_iff, forall_comm]\n  simp [Eq.comm]","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":3,"token_jaccard":0.058824,"token_length_ratio":0.15},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"1d375f9665e8577d89c6d3f9584fa76b875d53793acc232d7804d58a067049ac","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Matrix.PosDef\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Etienne Marion. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Etienne Marion\n-/\n/-!\n# Sesquilinear forms over a star ring\n\nThis file provides some properties about sesquilinear forms `M →ₗ⋆[R] M →ₗ[R] R` when `R` is a\n`StarRing`.\n-/\n\npublic section\n\nopen Module LinearMap\n\nvariable {R M n : Type*} [CommSemiring R] [StarRing R] [AddCommMonoid M] [Module R M]\n  [Fintype n] [DecidableEq n]\n  {B : M →ₗ⋆[R] M →ₗ[R] R} (b : Basis n R M)\n\nlemma LinearMap.isSymm_iff_basis {ι : Type*} (b : Basis ι R M) :\n    IsSymm B ↔ ∀ i j, star (B (b i) (b j)) = B (b j) (b i) where\n  mp h i j := h.eq _ _\n  mpr := by\n    refine fun h ↦ ⟨fun x y ↦ ?_⟩\n    obtain ⟨fx, tx, ix, -, hx⟩ := Submodule.mem_span_iff_exists_finset_subset.1\n      (by simp : x ∈ Submodule.span R (Set.range b))\n    obtain ⟨fy, ty, iy, -, hy⟩ := Submodule.mem_span_iff_exists_finset_subset.1\n      (by simp : y ∈ Submodule.span R (Set.range b))\n    rw [← hx, ← hy]\n    simp only [map_sum, LinearMap.map_smulₛₗ, starRingEnd_apply, map_smul, coe_sum,\n      Finset.sum_apply, smul_apply, smul_eq_mul, Finset.mul_sum, map_mul, star_star]\n    rw [Finset.sum_comm]\n    refine Finset.sum_congr rfl (fun b₁ h₁ ↦ Finset.sum_congr rfl fun b₂ h₂ ↦ ?_)\n    rw [mul_left_comm]\n    obtain ⟨i, rfl⟩ := ix h₁\n    obtain ⟨j, rfl⟩ := iy h₂\n    rw [h]\n\nTarget:\nlemma LinearMap.isSymm_iff_isHermitian_toMatrix : B.IsSymm ↔ (toMatrix₂ b b B).IsHermitian :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/SesquilinearForm","family_id":"linearmap","file_id":"mathlib/Mathlib/LinearAlgebra/SesquilinearForm/Star.lean","sample_id":"91bf84bd60b26a35f7dabcc39cf740009b92e0c19b956edd8d544cfa8e467dbb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0276c875b5de26026fe85dbf0c34475b3acdef44fb07206bfeb84e06366c9607","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1ca3e2d479b890c0c8ec12f1bf45f7c09e78c813a278da66b2270ecf4fe4ac81","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"16839e522d1756068dfaf97c745b531d7271caa8d21dfa9ef1a00b09a41091af","source_sha256":"f31e0ea5c5dcaaef4fe7a595be40e4dcc670dcec12bcae3dfce625de497aea8b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  funext n\n  induction n with\n  | zero => rfl\n  | succ n ih =>\n    unfold map iterate get\n    rw [map, get] at ih\n    rw [iterate]\n    exact congrArg f ih","hard_negative":false,"metrics":{"chosen_tokens":35,"rejected_tokens":42,"token_jaccard":0.92,"token_length_ratio":1.2},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"1d3c20ecf7ff84fe6effc00abb1d04b58ba0b4d32e01c964b880c81e429d2a20","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Stream.Defs\npublic import Mathlib.Logic.Function.Basic\npublic import Mathlib.Data.Nat.Basic\npublic import Mathlib.Tactic.Common\n\nNamespace:\nStream'\n\nLocal context:\n/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Streams a.k.a. infinite lists a.k.a. infinite sequences\n-/\n\n@[expose] public section\n\nopen Nat Function Option\n\nnamespace Stream'\n\nuniverse u v w\nvariable {α : Type u} {β : Type v} {δ : Type w}\nvariable (m n : ℕ) (x y : List α) (a b : Stream' α)\n\ninstance [Inhabited α] : Inhabited (Stream' α) :=\n  ⟨Stream'.const default⟩\n\n@[simp] protected theorem eta (s : Stream' α) : head s :: tail s = s :=\n  funext fun i => by cases i <;> rfl\n\n/-- Alias for `Stream'.eta` to match `List` API. -/\nalias cons_head_tail := Stream'.eta\n\n@[ext]\nprotected theorem ext {s₁ s₂ : Stream' α} : (∀ n, get s₁ n = get s₂ n) → s₁ = s₂ :=\n  fun h => funext h\n\n@[simp]\ntheorem get_zero_cons (a : α) (s : Stream' α) : get (a::s) 0 = a :=\n  rfl\n\n@[simp]\ntheorem head_cons (a : α) (s : Stream' α) : head (a::s) = a :=\n  rfl\n\n@[simp]\ntheorem tail_cons (a : α) (s : Stream' α) : tail (a::s) = s :=\n  rfl\n\n@[simp]\ntheorem get_drop (n m : ℕ) (s : Stream' α) : get (drop m s) n = get s (m + n) := by\n  rw [Nat.add_comm]\n  rfl\n\ntheorem tail_eq_drop (s : Stream' α) : tail s = drop 1 s :=\n  rfl\n\n@[simp]\ntheorem drop_drop (n m : ℕ) (s : Stream' α) : drop n (drop m s) = drop (m + n) s := by\n  ext; simp [Nat.add_assoc]\n\n@[simp] theorem get_tail {n : ℕ} {s : Stream' α} : s.tail.get n = s.get (n + 1) := rfl\n\n@[simp] theorem tail_drop' {i : ℕ} {s : Stream' α} : tail (drop i s) = s.drop (i + 1) := by\n  ext; simp [Nat.add_comm, Nat.add_left_comm]\n\n@[simp] theorem drop_tail' {i : ℕ} {s : Stream' α} : drop i (tail s) = s.drop (i + 1) := rfl\n\ntheorem tail_drop (n : ℕ) (s : Stream' α) : tail (drop n s) = drop n (tail s) := by simp\n\ntheorem get_succ (n : ℕ) (s : Stream' α) : get s (succ n) = get (tail s) n :=\n  rfl\n\n@[simp]\ntheorem get_succ_cons (n : ℕ) (s : Stream' α) (x : α) : get (x :: s) n.succ = get s n :=\n  rfl\n\n@[simp] lemma get_cons_append_zero {a : α} {x : List α} {s : Stream' α} :\n    (a :: x ++ₛ s).get 0 = a := rfl\n\n@[simp] lemma append_eq_cons {a : α} {as : Stream' α} : [a] ++ₛ as = a :: as := rfl\n\n@[simp] theorem drop_zero {s : Stream' α} : s.drop 0 = s := rfl\n\ntheorem drop_succ (n : ℕ) (s : Stream' α) : drop (succ n) s = drop n (tail s) :=\n  rfl\n\ntheorem head_drop (a : Stream' α) (n : ℕ) : (a.drop n).head = a.get n := by simp\n\ntheorem cons_injective2 : Function.Injective2 (cons : α → Stream' α → Stream' α) := fun x y s t h =>\n  ⟨by rw [← get_zero_cons x s, h, get_zero_cons],\n    Stream'.ext fun n => by rw [← get_succ_cons n _ x, h, get_succ_cons]⟩\n\ntheorem cons_injective_left (s : Stream' α) : Function.Injective fun x => cons x s :=\n  cons_injective2.left _\n\ntheorem cons_injective_right (x : α) : Function.Injective (cons x) :=\n  cons_injective2.right _\n\ntheorem all_def (p : α → Prop) (s : Stream' α) : All p s = ∀ n, p (get s n) :=\n  rfl\n\ntheorem any_def (p : α → Prop) (s : Stream' α) : Any p s = ∃ n, p (get s n) :=\n  rfl\n\n@[simp]\ntheorem mem_cons (a : α) (s : Stream' α) : a ∈ a::s :=\n  Exists.intro 0 rfl\n\ntheorem mem_cons_of_mem {a : α} {s : Stream' α} (b : α) : a ∈ s → a ∈ b::s := fun ⟨n, h⟩ =>\n  Exists.intro (succ n) (by rw [get_succ, tail_cons, h])\n\ntheorem eq_or_mem_of_mem_cons {a b : α} {s : Stream' α} : (a ∈ b::s) → a = b ∨ a ∈ s :=\n    fun ⟨n, h⟩ => by\n  rcases n with - | n'\n  · left\n    exact h\n  · right\n    rw [get_succ, tail_cons] at h\n    exact ⟨n', h⟩\n\ntheorem mem_of_get_eq {n : ℕ} {s : Stream' α} {a : α} : a = get s n → a ∈ s := fun h =>\n  Exists.intro n h\n\ntheorem mem_iff_exists_get_eq {s : Stream' α} {a : α} : a ∈ s ↔ ∃ n, a = s.get n where\n  mp := by simp [Membership.mem, any_def]\n  mpr h := mem_of_get_eq h.choose_spec\n\nsection Map\n\nvariable (f : α → β)\n\ntheorem drop_map (n : ℕ) (s : Stream' α) : drop n (map f s) = map f (drop n s) :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem get_map (n : ℕ) (s : Stream' α) : get (map f s) n = f (get s n) :=\n  rfl\n\ntheorem tail_map (s : Stream' α) : tail (map f s) = map f (tail s) := rfl\n\n@[simp]\ntheorem head_map (s : Stream' α) : head (map f s) = f (head s) :=\n  rfl\n\ntheorem map_eq (s : Stream' α) : map f s = f (head s)::map f (tail s) := by\n  rw [← Stream'.eta (map f s), tail_map, head_map]\n\ntheorem map_cons (a : α) (s : Stream' α) : map f (a::s) = f a::map f s := by\n  rw [← Stream'.eta (map f (a::s)), map_eq]; rfl\n\n@[simp]\ntheorem map_id (s : Stream' α) : map id s = s :=\n  rfl\n\n@[simp]\ntheorem map_map (g : β → δ) (f : α → β) (s : Stream' α) : map g (map f s) = map (g ∘ f) s :=\n  rfl\n\n@[simp]\ntheorem map_tail (s : Stream' α) : map f (tail s) = tail (map f s) :=\n  rfl\n\ntheorem mem_map {a : α} {s : Stream' α} : a ∈ s → f a ∈ map f s := fun ⟨n, h⟩ =>\n  Exists.intro n (by rw [get_map, h])\n\ntheorem exists_of_mem_map {f} {b : β} {s : Stream' α} : b ∈ map f s → ∃ a, a ∈ s ∧ f a = b :=\n  fun ⟨n, h⟩ => ⟨get s n, ⟨n, rfl⟩, h.symm⟩\n\nend Map\n\nsection Zip\n\nvariable (f : α → β → δ)\n\ntheorem drop_zip (n : ℕ) (s₁ : Stream' α) (s₂ : Stream' β) :\n    drop n (zip f s₁ s₂) = zip f (drop n s₁) (drop n s₂) :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem get_zip (n : ℕ) (s₁ : Stream' α) (s₂ : Stream' β) :\n    get (zip f s₁ s₂) n = f (get s₁ n) (get s₂ n) :=\n  rfl\n\ntheorem head_zip (s₁ : Stream' α) (s₂ : Stream' β) : head (zip f s₁ s₂) = f (head s₁) (head s₂) :=\n  rfl\n\ntheorem tail_zip (s₁ : Stream' α) (s₂ : Stream' β) :\n    tail (zip f s₁ s₂) = zip f (tail s₁) (tail s₂) :=\n  rfl\n\ntheorem zip_eq (s₁ : Stream' α) (s₂ : Stream' β) :\n    zip f s₁ s₂ = f (head s₁) (head s₂)::zip f (tail s₁) (tail s₂) := by\n  rw [← Stream'.eta (zip f s₁ s₂)]; rfl\n\n@[simp]\ntheorem get_enum (s : Stream' α) (n : ℕ) : get (enum s) n = (n, s.get n) :=\n  rfl\n\ntheorem enum_eq_zip (s : Stream' α) : enum s = zip Prod.mk nats s :=\n  rfl\n\nend Zip\n\n@[simp]\ntheorem mem_const (a : α) : a ∈ const a :=\n  Exists.intro 0 rfl\n\ntheorem const_eq (a : α) : const a = a::const a := by\n  apply Stream'.ext; intro n\n  cases n <;> rfl\n\n@[simp]\ntheorem tail_const (a : α) : tail (const a) = const a :=\n  suffices tail (a::const a) = const a by rwa [← const_eq] at this\n  rfl\n\n@[simp]\ntheorem map_const (f : α → β) (a : α) : map f (const a) = const (f a) :=\n  rfl\n\n@[simp]\ntheorem get_const (n : ℕ) (a : α) : get (const a) n = a :=\n  rfl\n\n@[simp]\ntheorem drop_const (n : ℕ) (a : α) : drop n (const a) = const a :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem head_iterate (f : α → α) (a : α) : head (iterate f a) = a :=\n  rfl\n\ntheorem get_succ_iterate' (n : ℕ) (f : α → α) (a : α) :\n    get (iterate f a) (succ n) = f (get (iterate f a) n) := rfl\n\ntheorem tail_iterate (f : α → α) (a : α) : tail (iterate f a) = iterate f (f a) := by\n  ext n\n  rw [get_tail]\n  induction n with\n  | zero => rfl\n  | succ n ih => rw [get_succ_iterate', ih, get_succ_iterate']\n\ntheorem iterate_eq (f : α → α) (a : α) : iterate f a = a::iterate f (f a) := by\n  rw [← Stream'.eta (iterate f a)]\n  rw [tail_iterate]; rfl\n\n@[simp]\ntheorem get_zero_iterate (f : α → α) (a : α) : get (iterate f a) 0 = a :=\n  rfl\n\ntheorem get_succ_iterate (n : ℕ) (f : α → α) (a : α) :\n    get (iterate f a) (succ n) = get (iterate f (f a)) n := by rw [get_succ, tail_iterate]\n\nsection Bisim\n\nvariable (R : Stream' α → Stream' α → Prop)\n\n/-- equivalence relation -/\nlocal infixl:50 \" ~ \" => R\n\n/-- Streams `s₁` and `s₂` are defined to be bisimulations if\ntheir heads are equal and tails are bisimulations. -/\ndef IsBisimulation :=\n  ∀ ⦃s₁ s₂⦄, s₁ ~ s₂ →\n      head s₁ = head s₂ ∧ tail s₁ ~ tail s₂\n\ntheorem get_of_bisim (bisim : IsBisimulation R) {s₁ s₂} :\n    ∀ n, s₁ ~ s₂ → get s₁ n = get s₂ n ∧ drop (n + 1) s₁ ~ drop (n + 1) s₂\n  | 0, h => bisim h\n  | n + 1, h =>\n    match bisim h with\n    | ⟨_, trel⟩ => get_of_bisim bisim n trel\n\n-- If two streams are bisimilar, then they are equal\ntheorem eq_of_bisim (bisim : IsBisimulation R) {s₁ s₂} : s₁ ~ s₂ → s₁ = s₂ := fun r =>\n  Stream'.ext fun n => And.left (get_of_bisim R bisim n r)\n\nend Bisim\n\ntheorem bisim_simple (s₁ s₂ : Stream' α) :\n    head s₁ = head s₂ → s₁ = tail s₁ → s₂ = tail s₂ → s₁ = s₂ := fun hh ht₁ ht₂ =>\n  eq_of_bisim (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂)\n    (fun s₁ s₂ ⟨h₁, h₂, h₃⟩ => by grind)\n    (And.intro hh (And.intro ht₁ ht₂))\n\ntheorem coinduction {s₁ s₂ : Stream' α} :\n    head s₁ = head s₂ →\n      (∀ (β : Type u) (fr : Stream' α → β),\n      fr s₁ = fr s₂ → fr (tail s₁) = fr (tail s₂)) → s₁ = s₂ :=\n  fun hh ht =>\n  eq_of_bisim\n    (fun s₁ s₂ =>\n      head s₁ = head s₂ ∧\n        ∀ (β : Type u) (fr : Stream' α → β), fr s₁ = fr s₂ → fr (tail s₁) = fr (tail s₂))\n    (fun s₁ s₂ h =>\n      have h₁ : head s₁ = head s₂ := And.left h\n      have h₂ : head (tail s₁) = head (tail s₂) := And.right h α (@head α) h₁\n      have h₃ :\n        ∀ (β : Type u) (fr : Stream' α → β),\n          fr (tail s₁) = fr (tail s₂) → fr (tail (tail s₁)) = fr (tail (tail s₂)) :=\n        fun β fr => And.right h β fun s => fr (tail s)\n      And.intro h₁ (And.intro h₂ h₃))\n    (And.intro hh ht)\n\n@[simp]\ntheorem iterate_id (a : α) : iterate id a = const a :=\n  coinduction rfl fun β fr ch => by rw [tail_iterate, tail_const]; exact ch\n\nTarget:\ntheorem map_iterate (f : α → α) (a : α) : iterate f (f a) = map f (iterate f a) :=\n\nProof body:\n","rejected":"```lean\nby\n  funext n\n  induction n with\n  | zero => rfl\n  | succ n ih =>\n    unfold map iterate get\n    rw [map, get] at ih\n    rw [iterate]\n    exact congrArg f ih\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Stream","family_id":"map_iterate","file_id":"mathlib/Mathlib/Data/Stream/Init.lean","sample_id":"16839e522d1756068dfaf97c745b531d7271caa8d21dfa9ef1a00b09a41091af"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9ee4d424ebdd38c7fae4d49084f0c22e4992fdbfa58721d1092ad5e07b192bfb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"05563c96d23b122c116d8c9bb388bbb8f9750c86c78006cd8d30cd42ca3fd710","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ea7f7ba297977c69513c6abcc1bfadf80c88bd3e61590c3a43e8307b939f76d2","source_sha256":"e2ead2d6b414091f83a91e8561b014394407eaedab6a9801bc467d9fa54fc95c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  exact div_nonneg (Finset.sum_nonneg fun _ _ => sq_nonneg _) <| sq_nonneg _","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":3,"token_jaccard":0.133333,"token_length_ratio":0.166667},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"1d4cd11e885f5f84e9f0fa61052e1c14c92153f8e1e846bca89df83e6367bb43","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.NatInt\npublic import Mathlib.Algebra.Order.BigOperators.Group.Finset\npublic import Mathlib.Combinatorics.SimpleGraph.Density\npublic import Mathlib.Data.Rat.BigOperators\n\nNamespace:\nFinpartition\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Energy of a partition\n\nThis file defines the energy of a partition.\n\nThe energy is the auxiliary quantity that drives the induction process in the proof of Szemerédi's\nRegularity Lemma. As long as we do not have a suitable equipartition, we will find a new one that\nhas an energy greater than the previous one plus some fixed constant.\n\n## References\n\n[Yaël Dillies, Bhavik Mehta, *Formalising Szemerédi’s Regularity Lemma in Lean*][srl_itp]\n-/\n\n@[expose] public section\n\n\nopen Finset\n\nvariable {α : Type*} [DecidableEq α] {s : Finset α} (P : Finpartition s) (G : SimpleGraph α)\n  [DecidableRel G.Adj]\n\nnamespace Finpartition\n\n/-- The energy of a partition, also known as index. Auxiliary quantity for Szemerédi's regularity\nlemma. -/\ndef energy : ℚ :=\n  ((∑ uv ∈ P.parts.offDiag, G.edgeDensity uv.1 uv.2 ^ 2) : ℚ) / (#P.parts : ℚ) ^ 2\n\nTarget:\ntheorem energy_nonneg : 0 ≤ P.energy G :=\n\nProof body:\n","rejected":"by\n  exact energy_nonneg","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"7813bf8c7517122bcb478d6a0820bb9199f373c14cb7ccdeefd936aac55eb6ae","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/SimpleGraph","family_id":"energy_nonneg","file_id":"mathlib/Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean","sample_id":"ea7f7ba297977c69513c6abcc1bfadf80c88bd3e61590c3a43e8307b939f76d2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"65d184d09b9f9abbbd2d10bfaea51c9a0bf698460f9d72b6fbec0c7ce1a8ad5d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f2679e48bf9685f04472782b58d5cd7288a2a2ea83cf96ca512c6701884e5398","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2b26d3e25a280904d8577d75d632bf5a13763ff21493b59168968dcb1e0a5669","source_sha256":"f94f55c4ba2322bc4c471438f5b24643adb8c39a2c5e74b34a6830fdd6741d84","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction n with\n  | zero => rfl\n  | succ n ih =>\n    rw [replicate_succ]\n    conv => rhs; rw [replicate_succ]\n    rw [snoc_cons, ih]","hard_negative":true,"metrics":{"chosen_tokens":31,"rejected_tokens":3,"token_jaccard":0.047619,"token_length_ratio":0.096774},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"1dc74bea0ea98e8b2f52ddeedd30f3539cf880cbe2bc01a624dfd8c402c88335","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Vector.Basic\n\nNamespace:\nList.Vector\n\nLocal context:\n/-\nCopyright (c) 2023 Alex Keizer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alex Keizer\n-/\n/-!\n  This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single\n  element to the back of a vector.\n\n  It provides a collection of lemmas that show how different `Vector` operations reduce when their\n  argument is `snoc xs x`.\n\n  Also, an alternative, reverse, induction principle is added, that breaks down a vector into\n  `snoc xs x` for its inductive case. Effectively doing induction from right-to-left\n-/\n\n@[expose] public section\n\nnamespace List\n\nnamespace Vector\n\nvariable {α β σ φ : Type*} {n : ℕ} {x : α} {s : σ} (xs : Vector α n)\n\n/-- Append a single element to the end of a vector -/\ndef snoc : Vector α n → α → Vector α (n + 1) :=\n  fun xs x => xs ++ x ::ᵥ Vector.nil\n\n/-! ## Simplification lemmas -/\n\nsection Simp\n\nvariable {y : α}\n\n@[simp]\ntheorem snoc_cons : (x ::ᵥ xs).snoc y = x ::ᵥ (xs.snoc y) :=\n  rfl\n\n@[simp]\ntheorem snoc_nil : (nil.snoc x) = x ::ᵥ nil :=\n  rfl\n\n@[simp]\ntheorem reverse_cons : reverse (x ::ᵥ xs) = (reverse xs).snoc x := by\n  cases xs\n  simp only [reverse, cons, toList_mk, List.reverse_cons, snoc]\n  congr\n\n@[simp]\ntheorem reverse_snoc : reverse (xs.snoc x) = x ::ᵥ (reverse xs) := by\n  cases xs\n  simp only [reverse, snoc, cons, toList_mk]\n  congr\n  simp [toList, append_def]\n\nTarget:\ntheorem replicate_succ_to_snoc (val : α) :\n    replicate (n + 1) val = (replicate n val).snoc val :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_2b26d3e25a28","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"5043bf761356b5236c4a9b754def8b442bede0cf7f829e3e9ebd957a2a93827f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Vector","family_id":"replicate_succ_to_snoc","file_id":"mathlib/Mathlib/Data/Vector/Snoc.lean","sample_id":"2b26d3e25a280904d8577d75d632bf5a13763ff21493b59168968dcb1e0a5669"},"split":"train"}
{"binding":{"chosen_proof_sha256":"68a591d0520d49abb38f3df228ef3dd797bb020e537249311751475dbc1d52d8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"44ece7bf253c5f603c55e81415554957d2378036c79d5b76487dc473ce633518","source_sha256":"837dba1e44eb02eb7837e8e7b72b59c277e85899454f45d3be66de44ecd98632","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [max_def]; split_ifs with h\n  exacts [hb h, ha (le_of_not_ge h)]","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":5,"token_jaccard":0.05,"token_length_ratio":0.25},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"1dcaa35be0155baefef7d63fc386f0a95aeee7283837dfdd74f859ebfcf99410","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Classes.Order\npublic import Batteries.Tactic.Trans\npublic import Mathlib.Data.Ordering.Basic\npublic import Mathlib.Tactic.ExtendDoc\npublic import Mathlib.Tactic.Push.Attr\npublic import Mathlib.Tactic.Simps.Basic\npublic import Mathlib.Tactic.SplitIfs\npublic import Mathlib.Order.Defs.PartialOrder\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Orders\n\nDefines classes for linear orders and proves some basic lemmas about them.\n\nWe intentionally avoid using `grind` in this fundamental file to keep the proofs understandable,\nrather than hiding the reasoning behind automation.\n-/\n\n@[expose] public section\n\nvariable {α : Type*}\n\nsection LinearOrder\n\n/-!\n### Definition of `LinearOrder` and lemmas about types with a linear order\n-/\n\n/-- Default definition of `max`. -/\ndef maxDefault [LE α] [DecidableLE α] (a b : α) :=\n  if a ≤ b then b else a\n\n/-- Default definition of `min`. -/\ndef minDefault [LE α] [DecidableLE α] (a b : α) :=\n  if a ≤ b then a else b\n\n/-- This attempts to prove that a given instance of `compare` is equal to `compareOfLessAndEq` by\nintroducing the arguments and trying the following approaches in order:\n\n1. seeing if `rfl` works\n2. seeing if the `compare` at hand is nonetheless essentially `compareOfLessAndEq`, but, because of\n   implicit arguments, requires us to unfold the defs and split the `if`s in the definition of\n   `compareOfLessAndEq`\n3. seeing if we can split by cases on the arguments, then see if the defs work themselves out\n   (useful when `compare` is defined via a `match` statement, as it is for `Bool`)\n-/\nmacro \"compareOfLessAndEq_rfl\" : tactic =>\n  `(tactic| (intro a b; first | rfl |\n    (simp only [compare, compareOfLessAndEq]; split_ifs <;> rfl) |\n    (induction a <;> induction b <;> simp +decide only)))\n\n/-- A linear order is reflexive, transitive, antisymmetric and total relation `≤`.\nWe assume that every linear ordered type has decidable `(≤)`, `(<)`, and `(=)`. -/\nclass LinearOrder (α : Type*) extends PartialOrder α, Min α, Max α, Ord α where\n  /-- A linear order is total. -/\n  protected le_total (a b : α) : a ≤ b ∨ b ≤ a\n  /-- In a linearly ordered type, we assume the order relations are all decidable. -/\n  toDecidableLE : DecidableLE α\n  /-- In a linearly ordered type, we assume the order relations are all decidable. -/\n  toDecidableEq : DecidableEq α := @decidableEqOfDecidableLE _ _ toDecidableLE\n  /-- In a linearly ordered type, we assume the order relations are all decidable. -/\n  toDecidableLT : DecidableLT α := @decidableLTOfDecidableLE _ _ toDecidableLE\n  min := fun a b => if a ≤ b then a else b\n  max := fun a b => if a ≤ b then b else a\n  /-- The minimum function is equivalent to the one you get from `minOfLe`. -/\n  protected min_def : ∀ a b, min a b = if a ≤ b then a else b := by intros; rfl\n  /-- The minimum function is equivalent to the one you get from `maxOfLe`. -/\n  protected max_def : ∀ a b, max a b = if a ≤ b then b else a := by intros; rfl\n  compare a b := compareOfLessAndEq a b\n  /-- Comparison via `compare` is equal to the canonical comparison given decidable `<` and `=`. -/\n  compare_eq_compareOfLessAndEq : ∀ a b, compare a b = compareOfLessAndEq a b := by\n    compareOfLessAndEq_rfl\n\nattribute [to_dual existing] LinearOrder.toMax\n\nvariable [LinearOrder α] {a b c : α}\n\nattribute [instance_reducible, instance 900] LinearOrder.toDecidableLT\nattribute [instance_reducible, instance 900] LinearOrder.toDecidableLE\nattribute [instance_reducible, instance 900] LinearOrder.toDecidableEq\n\ninstance : Std.IsLinearOrder α where\n  le_total := LinearOrder.le_total\n\n@[to_dual self] lemma le_total : ∀ a b : α, a ≤ b ∨ b ≤ a := LinearOrder.le_total\n\n@[to_dual self] lemma le_of_not_ge : ¬a ≤ b → b ≤ a := (le_total a b).resolve_left\n@[to_dual self] lemma lt_of_not_ge (h : ¬b ≤ a) : a < b := lt_of_le_not_ge (le_of_not_ge h) h\n\n@[to_dual self] lemma lt_or_ge (a b : α) : a < b ∨ b ≤ a :=\n  if hba : b ≤ a then Or.inr hba else Or.inl <| lt_of_not_ge hba\n\n@[to_dual self] lemma le_or_gt (a b : α) : a ≤ b ∨ b < a := (lt_or_ge b a).symm\n\n@[to_dual gt_trichotomy]\nlemma lt_trichotomy (a b : α) : a < b ∨ a = b ∨ b < a :=\n  (lt_or_ge a b).imp_right (fun h ↦ (Decidable.lt_or_eq_of_le' h).symm)\n\n@[to_dual self]\nlemma le_of_not_gt (h : ¬b < a) : a ≤ b := (le_or_gt a b).resolve_right h\n\n@[to_dual gt_or_lt_of_ne]\nlemma lt_or_gt_of_ne (h : a ≠ b) : a < b ∨ b < a :=\n  (lt_trichotomy a b).imp_right (fun h' ↦ h'.resolve_left h)\n\n@[to_dual ne_iff_gt_or_lt]\nlemma ne_iff_lt_or_gt : a ≠ b ↔ a < b ∨ b < a := ⟨lt_or_gt_of_ne, (Or.elim · ne_of_lt ne_of_gt)⟩\n\n@[to_dual self] lemma lt_iff_not_ge : a < b ↔ ¬b ≤ a := ⟨not_le_of_gt, lt_of_not_ge⟩\n\n@[simp, push, to_dual self] lemma not_lt : ¬a < b ↔ b ≤ a := ⟨le_of_not_gt, not_lt_of_ge⟩\n@[simp, push, to_dual self] lemma not_le : ¬a ≤ b ↔ b < a := lt_iff_not_ge.symm\n\n@[to_dual eq_or_lt_of_not_gt]\nlemma eq_or_gt_of_not_lt (h : ¬a < b) : a = b ∨ b < a :=\n  if h₁ : a = b then Or.inl h₁ else Or.inr (lt_of_not_ge fun hge => h (lt_of_le_of_ne hge h₁))\n\n@[to_dual self]\ntheorem le_imp_le_of_lt_imp_lt {α β} [Preorder α] [LinearOrder β] {a b : α} {c d : β}\n    (H : d < c → b < a) (h : a ≤ b) : c ≤ d :=\n  le_of_not_gt fun h' => not_le_of_gt (H h') h\n\n@[grind =]\nlemma min_def (a b : α) : min a b = if a ≤ b then a else b := LinearOrder.min_def a b\n@[grind =]\nlemma max_def (a b : α) : max a b = if a ≤ b then b else a := LinearOrder.max_def a b\n\ntheorem min_ind {motive : α → Prop} (ha : a ≤ b → motive a) (hb : b ≤ a → motive b) :\n    motive (min a b) := by\n  rw [min_def]; split_ifs with h\n  exacts [ha h, hb (le_of_not_ge h)]\n\n@[to_dual existing (attr := elab_as_elim)]\n\nTarget:\ntheorem max_ind {motive : α → Prop} (ha : b ≤ a → motive a) (hb : a ≤ b → motive b) :\n    motive (max a b) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Defs","family_id":"max_ind","file_id":"mathlib/Mathlib/Order/Defs/LinearOrder.lean","sample_id":"44ece7bf253c5f603c55e81415554957d2378036c79d5b76487dc473ce633518"},"split":"train"}
{"binding":{"chosen_proof_sha256":"362e79b2088809d4572239c27914a8ae94ced25dc58902293b9373aadce248ad","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"215bc10c17214a80cdf028f691cd9e0cb3d40b6838bce81a23b452ba150e84be","source_sha256":"1b477641415d2c99d052691b438f4aaca539c9f1e08663583ff663a2f6ef7d8c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI : Fintype ι := Fintype.ofFinite ι\n  rw [fromDirectSumEquiv, ← fromDFinsuppEquiv_single]\n  convert! rfl","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":2,"token_jaccard":0.052632,"token_length_ratio":0.1},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"1e0a4619db9fd7836d4cadc3d561a082f6b82b443e5747c3cf361aa81f6b1176","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.DirectSum.Module\npublic import Mathlib.LinearAlgebra.Multilinear.DFinsupp\n\nNamespace:\nMultilinearMap\n\nLocal context:\n/-\nCopyright (c) 2024 Sophie Morel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sophie Morel\n-/\n/-!\n# Multilinear maps from direct sums\n\nThis file describes multilinear maps on direct sums.\n\n## Main results\n\n* `MultilinearMap.fromDirectSumEquiv` : If `ι` is a `Fintype`, `κ i` is a family of types\n  indexed by `ι` and we are given an `R`-module `M i j` for every `i : ι` and `j : κ i`, this is\n  the linear equivalence between `Π p : (i : ι) → κ i, MultilinearMap R (fun i ↦ M i (p i)) M'` and\n  `MultilinearMap R (fun i ↦ ⨁ j : κ i, M i j) M'`.\n-/\n\n@[expose] public section\n\nnamespace MultilinearMap\n\nopen DirectSum\n\nvariable {R ι M' : Type*} {κ : ι → Type*} {M : (i : ι) → κ i → Type*}\nvariable [CommSemiring R]\nvariable [∀ i j, AddCommMonoid (M i j)] [∀ i j, Module R (M i j)] [AddCommMonoid M'] [Module R M']\n\n/-- Two multilinear maps from direct sums are equal if they agree on the generators. -/\n@[ext]\ntheorem directSum_ext [Finite ι] [(i : ι) → DecidableEq (κ i)]\n    ⦃f g : MultilinearMap R (fun i ↦ ⨁ j : κ i, M i j) M'⦄\n    (h : ∀ p : (i : ι) → κ i,\n      f.compLinearMap (fun i => DirectSum.lof _ _ _ (p i)) =\n      g.compLinearMap (fun i => DirectSum.lof _ _ _ (p i))) : f = g :=\n  dfinsupp_ext h\n\nvariable [DecidableEq ι]\n\n/-- The linear equivalence between families indexed by `p : Π i : ι, κ i` of multilinear maps\non the `fun i ↦ M i (p i)` and the space of multilinear map on `fun i ↦ ⨁ j : κ i, M i j`. -/\nnoncomputable def fromDirectSumEquiv [Finite ι] :\n    ((p : (i : ι) → κ i) → MultilinearMap R (fun i ↦ M i (p i)) M') ≃ₗ[R]\n    MultilinearMap R (fun i ↦ ⨁ j : κ i, M i j) M' :=\n  haveI : Fintype ι := Fintype.ofFinite ι\n  haveI : (i : ι) → DecidableEq (κ i) := fun i ↦ Classical.typeDecidableEq (κ i)\n  fromDFinsuppEquiv _ _\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\n\nTarget:\ntheorem fromDirectSumEquiv_lof [Finite ι] [(i : ι) → DecidableEq (κ i)]\n    (f : (p : (i : ι) → κ i) → MultilinearMap R (fun i ↦ M i (p i)) M')\n    (p : (i : ι) → κ i) (x : (i : ι) → M i (p i)) :\n    fromDirectSumEquiv f (fun i => lof R _ _ _ (x i)) = f p x :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Multilinear","family_id":"fromdirectsumequiv_lof","file_id":"mathlib/Mathlib/LinearAlgebra/Multilinear/DirectSum.lean","sample_id":"215bc10c17214a80cdf028f691cd9e0cb3d40b6838bce81a23b452ba150e84be"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c46ea331c2028c22d1cfe25ee9b2c63dfdbb45feb5dd1bf4ebf33f7cd796b96a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"00521cab1e8d5624effd7f9080f88a2a7c2a46ecd1ff08ed8fe772073ff48814","source_sha256":"bd4b2b01d950a3b9c8fd78b82f11b816abe9ec16bf6d55a504ac2e051e9cf9fe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h z => ?_, fun h n => by simpa only [zpow_natCast] using! h (n : ℤ)⟩\n  by_cases! hz : 0 ≤ z\n  · unfold Tendsto\n    lift z to ℕ using hz\n    simpa using! h z\n  · have : Tendsto (fun a => k a ^ z) l (𝓝 0) :=\n      Tendsto.comp (tendsto_zpow_atTop_zero hz) hk\n    have h : Tendsto f l (𝓝 0) := by simpa using! h 0\n    exact zero_mul (0 : β) ▸ this.mul h\n\nvariable {f}","hard_negative":true,"metrics":{"chosen_tokens":110,"rejected_tokens":8,"token_jaccard":0.070175,"token_length_ratio":0.072727},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"1e2a719b0fb45f2cd1076752524941609195fe373f3a27b5c43fa1bd4d6d026c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Eval.Defs\npublic import Mathlib.Analysis.Asymptotics.Lemmas\n\nNamespace:\nAsymptotics\n\nLocal context:\n/-\nCopyright (c) 2021 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\n/-!\n# Super-Polynomial Function Decay\n\nThis file defines a predicate `Asymptotics.SuperpolynomialDecay f` for a function satisfying\none of the following equivalent definitions (the definition is in terms of the first condition):\n\n* `x ^ n * f` tends to `𝓝 0` for all (or sufficiently large) naturals `n`\n* `|x ^ n * f|` tends to `𝓝 0` for all naturals `n` (`superpolynomialDecay_iff_abs_tendsto_zero`)\n* `|x ^ n * f|` is bounded for all naturals `n` (`superpolynomialDecay_iff_abs_isBoundedUnder`)\n* `f` is `o(x ^ c)` for all integers `c` (`superpolynomialDecay_iff_isLittleO`)\n* `f` is `O(x ^ c)` for all integers `c` (`superpolynomialDecay_iff_isBigO`)\n\nThese conditions are all equivalent to conditions in terms of polynomials, replacing `x ^ c` with\n  `p(x)` or `p(x)⁻¹` as appropriate, since asymptotically `p(x)` behaves like `X ^ p.natDegree`.\nThese further equivalences are not proven in mathlib but would be good future projects.\n\nThe definition of superpolynomial decay for `f : α → β` is relative to a parameter `k : α → β`.\nSuper-polynomial decay then means `f x` decays faster than `(k x) ^ c` for all integers `c`.\nEquivalently `f x` decays faster than `p.eval (k x)` for all polynomials `p : β[X]`.\nThe definition is also relative to a filter `l : Filter α` where the decay rate is compared.\n\nWhen the map `k` is given by `n ↦ ↑n : ℕ → ℝ` this defines negligible functions:\nhttps://en.wikipedia.org/wiki/Negligible_function\n\nWhen the map `k` is given by `(r₁,...,rₙ) ↦ r₁*...*rₙ : ℝⁿ → ℝ` this is equivalent\n  to the definition of rapidly decreasing functions given here:\nhttps://ncatlab.org/nlab/show/rapidly+decreasing+function\n\n## Main statements\n\n* `SuperpolynomialDecay.polynomial_mul` says that if `f(x)` is negligible,\n    then so is `p(x) * f(x)` for any polynomial `p`.\n* `superpolynomialDecay_iff_zpow_tendsto_zero` gives an equivalence between definitions in terms\n    of decaying faster than `k(x) ^ n` for all naturals `n` or `k(x) ^ c` for all integer `c`.\n-/\n\n@[expose] public section\n\n\nnamespace Asymptotics\n\nopen Topology Polynomial\n\nopen Filter\n\n/-- `f` has superpolynomial decay in parameter `k` along filter `l` if\n  `k ^ n * f` tends to zero at `l` for all naturals `n` -/\ndef SuperpolynomialDecay {α β : Type*} [TopologicalSpace β] [CommSemiring β] (l : Filter α)\n    (k : α → β) (f : α → β) :=\n  ∀ n : ℕ, Tendsto (fun a : α => k a ^ n * f a) l (𝓝 0)\n\nvariable {α β : Type*} {l : Filter α} {k : α → β} {f g g' : α → β}\n\nsection CommSemiring\n\nvariable [TopologicalSpace β] [CommSemiring β]\n\ntheorem SuperpolynomialDecay.congr' (hf : SuperpolynomialDecay l k f) (hfg : f =ᶠ[l] g) :\n    SuperpolynomialDecay l k g := fun z =>\n  (hf z).congr' (EventuallyEq.mul (EventuallyEq.refl l _) hfg)\n\ntheorem SuperpolynomialDecay.congr (hf : SuperpolynomialDecay l k f) (hfg : ∀ x, f x = g x) :\n    SuperpolynomialDecay l k g := fun z =>\n  (hf z).congr fun x => (congr_arg fun a => k x ^ z * a) <| hfg x\n\n@[simp]\ntheorem superpolynomialDecay_zero (l : Filter α) (k : α → β) : SuperpolynomialDecay l k 0 :=\n  fun z => by simpa only [Pi.zero_apply, mul_zero] using tendsto_const_nhds\n\ntheorem SuperpolynomialDecay.add [ContinuousAdd β] (hf : SuperpolynomialDecay l k f)\n    (hg : SuperpolynomialDecay l k g) : SuperpolynomialDecay l k (f + g) := fun z => by\n  simpa only [mul_add, add_zero, Pi.add_apply] using (hf z).add (hg z)\n\ntheorem SuperpolynomialDecay.mul [ContinuousMul β] (hf : SuperpolynomialDecay l k f)\n    (hg : SuperpolynomialDecay l k g) : SuperpolynomialDecay l k (f * g) := fun z => by\n  simpa only [mul_assoc, one_mul, mul_zero, pow_zero] using! (hf z).mul (hg 0)\n\ntheorem SuperpolynomialDecay.mul_const [ContinuousMul β] (hf : SuperpolynomialDecay l k f) (c : β) :\n    SuperpolynomialDecay l k fun n => f n * c := fun z => by\n  simpa only [← mul_assoc, zero_mul] using Tendsto.mul_const c (hf z)\n\ntheorem SuperpolynomialDecay.const_mul [ContinuousMul β] (hf : SuperpolynomialDecay l k f) (c : β) :\n    SuperpolynomialDecay l k fun n => c * f n :=\n  (hf.mul_const c).congr fun _ => mul_comm _ _\n\ntheorem SuperpolynomialDecay.param_mul (hf : SuperpolynomialDecay l k f) :\n    SuperpolynomialDecay l k (k * f) := fun z =>\n  tendsto_nhds.2 fun s hs hs0 =>\n    l.sets_of_superset ((tendsto_nhds.1 (hf <| z + 1)) s hs hs0) fun x hx => by\n      simpa only [Set.mem_preimage, Pi.mul_apply, ← mul_assoc, ← pow_succ] using hx\n\ntheorem SuperpolynomialDecay.mul_param (hf : SuperpolynomialDecay l k f) :\n    SuperpolynomialDecay l k (f * k) :=\n  hf.param_mul.congr fun _ => mul_comm _ _\n\ntheorem SuperpolynomialDecay.param_pow_mul (hf : SuperpolynomialDecay l k f) (n : ℕ) :\n    SuperpolynomialDecay l k (k ^ n * f) := by\n  induction n with\n  | zero => simpa only [one_mul, pow_zero] using hf\n  | succ n hn => simpa only [pow_succ', mul_assoc] using hn.param_mul\n\ntheorem SuperpolynomialDecay.mul_param_pow (hf : SuperpolynomialDecay l k f) (n : ℕ) :\n    SuperpolynomialDecay l k (f * k ^ n) :=\n  (hf.param_pow_mul n).congr fun _ => mul_comm _ _\n\ntheorem SuperpolynomialDecay.polynomial_mul [ContinuousAdd β] [ContinuousMul β]\n    (hf : SuperpolynomialDecay l k f) (p : β[X]) :\n    SuperpolynomialDecay l k fun x => (p.eval <| k x) * f x :=\n  Polynomial.induction_on' p (fun p q hp hq => by simpa [add_mul] using! hp.add hq) fun n c => by\n    simpa [mul_assoc] using! (hf.param_pow_mul n).const_mul c\n\ntheorem SuperpolynomialDecay.mul_polynomial [ContinuousAdd β] [ContinuousMul β]\n    (hf : SuperpolynomialDecay l k f) (p : β[X]) :\n    SuperpolynomialDecay l k fun x => f x * (p.eval <| k x) :=\n  (hf.polynomial_mul p).congr fun _ => mul_comm _ _\n\nend CommSemiring\n\nsection OrderedCommSemiring\n\nvariable [TopologicalSpace β] [CommSemiring β] [PartialOrder β] [IsOrderedRing β] [OrderTopology β]\n\ntheorem SuperpolynomialDecay.trans_eventuallyLE (hk : 0 ≤ᶠ[l] k) (hg : SuperpolynomialDecay l k g)\n    (hg' : SuperpolynomialDecay l k g') (hfg : g ≤ᶠ[l] f) (hfg' : f ≤ᶠ[l] g') :\n    SuperpolynomialDecay l k f := fun z =>\n  tendsto_of_tendsto_of_tendsto_of_le_of_le' (hg z) (hg' z)\n    (by filter_upwards [hfg, hk] with x hx (hx' : 0 ≤ k x) using by gcongr)\n    (by filter_upwards [hfg', hk] with x hx (hx' : 0 ≤ k x) using by gcongr)\n\nend OrderedCommSemiring\n\nsection LinearOrderedCommRing\n\nvariable [TopologicalSpace β] [CommRing β] [LinearOrder β] [IsStrictOrderedRing β] [OrderTopology β]\nvariable (l k f)\n\ntheorem superpolynomialDecay_iff_abs_tendsto_zero :\n    SuperpolynomialDecay l k f ↔ ∀ n : ℕ, Tendsto (fun a : α => |k a ^ n * f a|) l (𝓝 0) :=\n  ⟨fun h z => (tendsto_zero_iff_abs_tendsto_zero _).1 (h z), fun h z =>\n    (tendsto_zero_iff_abs_tendsto_zero _).2 (h z)⟩\n\ntheorem superpolynomialDecay_iff_superpolynomialDecay_abs :\n    SuperpolynomialDecay l k f ↔ SuperpolynomialDecay l (fun a => |k a|) fun a => |f a| :=\n  (superpolynomialDecay_iff_abs_tendsto_zero l k f).trans\n    (by simp_rw [SuperpolynomialDecay, abs_mul, abs_pow])\n\nvariable {l k f}\n\ntheorem SuperpolynomialDecay.trans_eventually_abs_le (hf : SuperpolynomialDecay l k f)\n    (hfg : abs ∘ g ≤ᶠ[l] abs ∘ f) : SuperpolynomialDecay l k g := by\n  rw [superpolynomialDecay_iff_abs_tendsto_zero] at hf ⊢\n  refine fun z =>\n    tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds (hf z)\n      (Eventually.of_forall fun x => abs_nonneg _) (hfg.mono fun x hx => ?_)\n  calc\n    |k x ^ z * g x| = |k x ^ z| * |g x| := abs_mul (k x ^ z) (g x)\n    _ ≤ |k x ^ z| * |f x| := by gcongr _ * ?_; exact hx\n    _ = |k x ^ z * f x| := (abs_mul (k x ^ z) (f x)).symm\n\ntheorem SuperpolynomialDecay.trans_abs_le (hf : SuperpolynomialDecay l k f)\n    (hfg : ∀ x, |g x| ≤ |f x|) : SuperpolynomialDecay l k g :=\n  hf.trans_eventually_abs_le (Eventually.of_forall hfg)\n\nend LinearOrderedCommRing\n\nsection Field\n\nvariable [TopologicalSpace β] [Field β] (l k f)\n\ntheorem superpolynomialDecay_mul_const_iff [ContinuousMul β] {c : β} (hc0 : c ≠ 0) :\n    (SuperpolynomialDecay l k fun n => f n * c) ↔ SuperpolynomialDecay l k f :=\n  ⟨fun h => (h.mul_const c⁻¹).congr fun x => by simp [mul_assoc, mul_inv_cancel₀ hc0], fun h =>\n    h.mul_const c⟩\n\ntheorem superpolynomialDecay_const_mul_iff [ContinuousMul β] {c : β} (hc0 : c ≠ 0) :\n    (SuperpolynomialDecay l k fun n => c * f n) ↔ SuperpolynomialDecay l k f :=\n  ⟨fun h => (h.const_mul c⁻¹).congr fun x => by simp [← mul_assoc, inv_mul_cancel₀ hc0], fun h =>\n    h.const_mul c⟩\n\nend Field\n\nsection LinearOrderedField\n\nvariable [TopologicalSpace β] [Field β] [LinearOrder β] [IsStrictOrderedRing β] [OrderTopology β]\nvariable (f)\n\ntheorem superpolynomialDecay_iff_abs_isBoundedUnder (hk : Tendsto k l atTop) :\n    SuperpolynomialDecay l k f ↔\n    ∀ z : ℕ, IsBoundedUnder (· ≤ ·) l fun a : α => |k a ^ z * f a| := by\n  refine\n    ⟨fun h z => Tendsto.isBoundedUnder_le (Tendsto.abs (h z)), fun h =>\n      (superpolynomialDecay_iff_abs_tendsto_zero l k f).2 fun z => ?_⟩\n  obtain ⟨m, hm⟩ := h (z + 1)\n  have h1 : Tendsto (fun _ : α => (0 : β)) l (𝓝 0) := tendsto_const_nhds\n  have h2 : Tendsto (fun a : α => |(k a)⁻¹| * m) l (𝓝 0) :=\n    zero_mul m ▸\n      Tendsto.mul_const m ((tendsto_zero_iff_abs_tendsto_zero _).1 hk.inv_tendsto_atTop)\n  refine\n    tendsto_of_tendsto_of_tendsto_of_le_of_le' h1 h2 (Eventually.of_forall fun x => abs_nonneg _)\n      ((eventually_map.1 hm).mp ?_)\n  refine (hk.eventually_ne_atTop 0).mono fun x hk0 hx => ?_\n  refine Eq.trans_le ?_ (mul_le_mul_of_nonneg_left hx <| abs_nonneg (k x)⁻¹)\n  rw [← abs_mul, ← mul_assoc, pow_succ', ← mul_assoc, inv_mul_cancel₀ hk0, one_mul]\n\nTarget:\ntheorem superpolynomialDecay_iff_zpow_tendsto_zero (hk : Tendsto k l atTop) :\n    SuperpolynomialDecay l k f ↔ ∀ z : ℤ, Tendsto (fun a : α => k a ^ z * f a) l (𝓝 0) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"06e7f55fce5af1b527b2a553dd0083585710e63c5f85144bcb171b8a1acee043","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Asymptotics","family_id":"superpolynomialdecay_iff_zpow_tendsto_zero","file_id":"mathlib/Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean","sample_id":"00521cab1e8d5624effd7f9080f88a2a7c2a46ecd1ff08ed8fe772073ff48814"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c8dca49869410f2bacfcd9d441c978765c739555c35836f4fba20bb9b5e2825f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"34c67d0984704f562f600f9107ee26cdff7c7034435d16dae2284506bc3e8668","source_sha256":"837dba1e44eb02eb7837e8e7b72b59c277e85899454f45d3be66de44ecd98632","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases h : compare a b\n  · simpa using le_of_lt <| compare_lt_iff_lt.1 h\n  · simpa using le_of_eq <| compare_eq_iff_eq.1 h\n  · simpa using compare_gt_iff_gt.1 h","hard_negative":false,"metrics":{"chosen_tokens":34,"rejected_tokens":2,"token_jaccard":0.05,"token_length_ratio":0.058824},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"1e59313f3ff1b4f456767c5d0490ded72de17d8073d50c54083166b6b35bdcc9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Classes.Order\npublic import Batteries.Tactic.Trans\npublic import Mathlib.Data.Ordering.Basic\npublic import Mathlib.Tactic.ExtendDoc\npublic import Mathlib.Tactic.Push.Attr\npublic import Mathlib.Tactic.Simps.Basic\npublic import Mathlib.Tactic.SplitIfs\npublic import Mathlib.Order.Defs.PartialOrder\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Orders\n\nDefines classes for linear orders and proves some basic lemmas about them.\n\nWe intentionally avoid using `grind` in this fundamental file to keep the proofs understandable,\nrather than hiding the reasoning behind automation.\n-/\n\n@[expose] public section\n\nvariable {α : Type*}\n\nsection LinearOrder\n\n/-!\n### Definition of `LinearOrder` and lemmas about types with a linear order\n-/\n\n/-- Default definition of `max`. -/\ndef maxDefault [LE α] [DecidableLE α] (a b : α) :=\n  if a ≤ b then b else a\n\n/-- Default definition of `min`. -/\ndef minDefault [LE α] [DecidableLE α] (a b : α) :=\n  if a ≤ b then a else b\n\n/-- This attempts to prove that a given instance of `compare` is equal to `compareOfLessAndEq` by\nintroducing the arguments and trying the following approaches in order:\n\n1. seeing if `rfl` works\n2. seeing if the `compare` at hand is nonetheless essentially `compareOfLessAndEq`, but, because of\n   implicit arguments, requires us to unfold the defs and split the `if`s in the definition of\n   `compareOfLessAndEq`\n3. seeing if we can split by cases on the arguments, then see if the defs work themselves out\n   (useful when `compare` is defined via a `match` statement, as it is for `Bool`)\n-/\nmacro \"compareOfLessAndEq_rfl\" : tactic =>\n  `(tactic| (intro a b; first | rfl |\n    (simp only [compare, compareOfLessAndEq]; split_ifs <;> rfl) |\n    (induction a <;> induction b <;> simp +decide only)))\n\n/-- A linear order is reflexive, transitive, antisymmetric and total relation `≤`.\nWe assume that every linear ordered type has decidable `(≤)`, `(<)`, and `(=)`. -/\nclass LinearOrder (α : Type*) extends PartialOrder α, Min α, Max α, Ord α where\n  /-- A linear order is total. -/\n  protected le_total (a b : α) : a ≤ b ∨ b ≤ a\n  /-- In a linearly ordered type, we assume the order relations are all decidable. -/\n  toDecidableLE : DecidableLE α\n  /-- In a linearly ordered type, we assume the order relations are all decidable. -/\n  toDecidableEq : DecidableEq α := @decidableEqOfDecidableLE _ _ toDecidableLE\n  /-- In a linearly ordered type, we assume the order relations are all decidable. -/\n  toDecidableLT : DecidableLT α := @decidableLTOfDecidableLE _ _ toDecidableLE\n  min := fun a b => if a ≤ b then a else b\n  max := fun a b => if a ≤ b then b else a\n  /-- The minimum function is equivalent to the one you get from `minOfLe`. -/\n  protected min_def : ∀ a b, min a b = if a ≤ b then a else b := by intros; rfl\n  /-- The minimum function is equivalent to the one you get from `maxOfLe`. -/\n  protected max_def : ∀ a b, max a b = if a ≤ b then b else a := by intros; rfl\n  compare a b := compareOfLessAndEq a b\n  /-- Comparison via `compare` is equal to the canonical comparison given decidable `<` and `=`. -/\n  compare_eq_compareOfLessAndEq : ∀ a b, compare a b = compareOfLessAndEq a b := by\n    compareOfLessAndEq_rfl\n\nattribute [to_dual existing] LinearOrder.toMax\n\nvariable [LinearOrder α] {a b c : α}\n\nattribute [instance_reducible, instance 900] LinearOrder.toDecidableLT\nattribute [instance_reducible, instance 900] LinearOrder.toDecidableLE\nattribute [instance_reducible, instance 900] LinearOrder.toDecidableEq\n\ninstance : Std.IsLinearOrder α where\n  le_total := LinearOrder.le_total\n\n@[to_dual self] lemma le_total : ∀ a b : α, a ≤ b ∨ b ≤ a := LinearOrder.le_total\n\n@[to_dual self] lemma le_of_not_ge : ¬a ≤ b → b ≤ a := (le_total a b).resolve_left\n@[to_dual self] lemma lt_of_not_ge (h : ¬b ≤ a) : a < b := lt_of_le_not_ge (le_of_not_ge h) h\n\n@[to_dual self] lemma lt_or_ge (a b : α) : a < b ∨ b ≤ a :=\n  if hba : b ≤ a then Or.inr hba else Or.inl <| lt_of_not_ge hba\n\n@[to_dual self] lemma le_or_gt (a b : α) : a ≤ b ∨ b < a := (lt_or_ge b a).symm\n\n@[to_dual gt_trichotomy]\nlemma lt_trichotomy (a b : α) : a < b ∨ a = b ∨ b < a :=\n  (lt_or_ge a b).imp_right (fun h ↦ (Decidable.lt_or_eq_of_le' h).symm)\n\n@[to_dual self]\nlemma le_of_not_gt (h : ¬b < a) : a ≤ b := (le_or_gt a b).resolve_right h\n\n@[to_dual gt_or_lt_of_ne]\nlemma lt_or_gt_of_ne (h : a ≠ b) : a < b ∨ b < a :=\n  (lt_trichotomy a b).imp_right (fun h' ↦ h'.resolve_left h)\n\n@[to_dual ne_iff_gt_or_lt]\nlemma ne_iff_lt_or_gt : a ≠ b ↔ a < b ∨ b < a := ⟨lt_or_gt_of_ne, (Or.elim · ne_of_lt ne_of_gt)⟩\n\n@[to_dual self] lemma lt_iff_not_ge : a < b ↔ ¬b ≤ a := ⟨not_le_of_gt, lt_of_not_ge⟩\n\n@[simp, push, to_dual self] lemma not_lt : ¬a < b ↔ b ≤ a := ⟨le_of_not_gt, not_lt_of_ge⟩\n@[simp, push, to_dual self] lemma not_le : ¬a ≤ b ↔ b < a := lt_iff_not_ge.symm\n\n@[to_dual eq_or_lt_of_not_gt]\nlemma eq_or_gt_of_not_lt (h : ¬a < b) : a = b ∨ b < a :=\n  if h₁ : a = b then Or.inl h₁ else Or.inr (lt_of_not_ge fun hge => h (lt_of_le_of_ne hge h₁))\n\n@[to_dual self]\ntheorem le_imp_le_of_lt_imp_lt {α β} [Preorder α] [LinearOrder β] {a b : α} {c d : β}\n    (H : d < c → b < a) (h : a ≤ b) : c ≤ d :=\n  le_of_not_gt fun h' => not_le_of_gt (H h') h\n\n@[grind =]\nlemma min_def (a b : α) : min a b = if a ≤ b then a else b := LinearOrder.min_def a b\n@[grind =]\nlemma max_def (a b : α) : max a b = if a ≤ b then b else a := LinearOrder.max_def a b\n\ntheorem min_ind {motive : α → Prop} (ha : a ≤ b → motive a) (hb : b ≤ a → motive b) :\n    motive (min a b) := by\n  rw [min_def]; split_ifs with h\n  exacts [ha h, hb (le_of_not_ge h)]\n\n@[to_dual existing (attr := elab_as_elim)]\ntheorem max_ind {motive : α → Prop} (ha : b ≤ a → motive a) (hb : a ≤ b → motive b) :\n    motive (max a b) := by\n  rw [max_def]; split_ifs with h\n  exacts [hb h, ha (le_of_not_ge h)]\n\n@[to_dual existing max_def]\ntheorem min_def' (a b : α) : min a b = if b ≤ a then b else a := by\n  obtain h | h | h := lt_trichotomy a b <;> simp [le_of_lt, not_le_of_gt, h, min_def]\n\n@[to_dual existing min_def]\ntheorem max_def' (a b : α) : max a b = if b ≤ a then a else b := by\n  obtain h | h | h := lt_trichotomy a b <;> simp [le_of_lt, not_le_of_gt, h, max_def]\n\n@[to_dual le_max_left]\nlemma min_le_left (a b : α) : min a b ≤ a := by\n  rw [min_def]\n  split_ifs with h <;> simp [h, le_of_not_ge]\n\n@[to_dual le_max_right]\nlemma min_le_right (a b : α) : min a b ≤ b := by\n  rw [min_def]\n  split_ifs with h <;> simp [h]\n\n@[to_dual max_le]\nlemma le_min (h₁ : c ≤ a) (h₂ : c ≤ b) : c ≤ min a b := by\n  rw [min_def]\n  split_ifs <;> assumption\n\n@[to_dual]\nlemma eq_min (h₁ : c ≤ a) (h₂ : c ≤ b) (h₃ : ∀ {d}, d ≤ a → d ≤ b → d ≤ c) : c = min a b :=\n  le_antisymm (le_min h₁ h₂) (h₃ (min_le_left a b) (min_le_right a b))\n\n@[to_dual]\nlemma min_comm (a b : α) : min a b = min b a :=\n  eq_min (min_le_right a b) (min_le_left a b) fun h₁ h₂ => le_min h₂ h₁\n\n@[to_dual]\nlemma min_assoc (a b c : α) : min (min a b) c = min a (min b c) :=\n  eq_min\n    (le_trans (min_le_left ..) (min_le_left ..))\n    (le_min (le_trans (min_le_left ..) (min_le_right ..)) (min_le_right ..))\n    (fun h₁ h₂ ↦\n      le_min (le_min h₁ (le_trans h₂ (min_le_left ..))) (le_trans h₂ (min_le_right ..)))\n\n@[to_dual]\nlemma min_left_comm (a b c : α) : min a (min b c) = min b (min a c) := by\n  rw [← min_assoc, min_comm a, min_assoc]\n\n@[to_dual (attr := simp)] lemma min_self (a : α) : min a a = a := by rw [min_def, ite_id]\n\n@[to_dual]\nlemma min_eq_left (h : a ≤ b) : min a b = a := (eq_min le_rfl h (fun h _ ↦ h)).symm\n\n@[to_dual]\nlemma min_eq_right (h : b ≤ a) : min a b = b := min_comm b a ▸ min_eq_left h\n\n@[to_dual] lemma min_eq_left_of_lt (h : a < b) : min a b = a := min_eq_left (le_of_lt h)\n@[to_dual] lemma min_eq_right_of_lt (h : b < a) : min a b = b := min_eq_right (le_of_lt h)\n\n@[to_dual max_lt]\nlemma lt_min (h₁ : a < b) (h₂ : a < c) : a < min b c := by\n  cases le_total b c <;> simp [min_eq_left, min_eq_right, *]\n\nsection Ord\n\nlemma compare_lt_iff_lt : compare a b = .lt ↔ a < b := by\n  rw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq_eq_lt]\n\nlemma compare_eq_iff_eq : compare a b = .eq ↔ a = b := by\n  rw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq_eq_eq le_refl not_le]\n\nlemma compare_gt_iff_gt : compare a b = .gt ↔ b < a := by\n  rw [LinearOrder.compare_eq_compareOfLessAndEq,\n    compareOfLessAndEq_eq_gt le_antisymm le_total not_le]\n\nTarget:\nlemma compare_le_iff_le : compare a b ≠ .gt ↔ a ≤ b :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Defs","family_id":"compare_le_iff_le","file_id":"mathlib/Mathlib/Order/Defs/LinearOrder.lean","sample_id":"34c67d0984704f562f600f9107ee26cdff7c7034435d16dae2284506bc3e8668"},"split":"train"}
{"binding":{"chosen_proof_sha256":"38de9a26cb91dd01aaa9b7d35dd028817eb092a585a923cb7e7b911ff5d759bb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"573f1aa7ff0f0816637422072f791f0e7b8be3fe3f296b4756b97cbe25ce6448","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"836ef9c3cd55ae2e3f293b4d27ae834545f003ae5a97617e40dffc59a7b9fdae","source_sha256":"98d8a988714b8411039a52ca8cb94b19a051f27bd0e995b1332bddd6c23da9ea","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine fun _ _ => ⟨1, by norm_num, 1, by norm_num, ?_⟩\n  filter_upwards [] with x\n  simp","hard_negative":false,"metrics":{"chosen_tokens":26,"rejected_tokens":31,"token_jaccard":0.772727,"token_length_ratio":1.192308},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"1e73e4e7d2dec99b3134a197be1fba0088f8938bdd15892f4bc9f7a2e8e75495","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Pow.Real\npublic import Mathlib.Algebra.Order.ToIntervalMod\npublic import Mathlib.Analysis.SpecialFunctions.Log.Base\nimport Mathlib.Algebra.Order.Interval.Set.Group\n\nNamespace:\nAkraBazziRecurrence\n\nLocal context:\n/-\nCopyright (c) 2023 Frédéric Dupuis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Frédéric Dupuis\n-/\n/-!\n# Akra-Bazzi theorem: the polynomial growth condition\n\nThis file defines and develops an API for the polynomial growth condition that appears in the\nstatement of the Akra-Bazzi theorem: for the theorem to hold, the function `g` must\nsatisfy the condition that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b*n` and `n` for any\nconstant `b ∈ (0,1)`.\n\n## Implementation notes\n\nOur definition requires that the condition hold for any `b ∈ (0,1)`. This is equivalent to requiring\nit only for `b = 1 / 2` (or any other particular value in `(0, 1)`). While this could, in principle,\nmake it harder to prove that a particular function grows polynomially, this issue does not seem to\narise in practice.\n\n-/\n\n@[expose] public section\n\nopen Finset Real Filter Asymptotics\nopen scoped Topology\n\nnamespace AkraBazziRecurrence\n\n/-- The growth condition that the function `g` must satisfy for the Akra-Bazzi theorem to apply.\nIt roughly states that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b * n` and `n`, for any\nconstant `b ∈ (0, 1)`. -/\ndef GrowsPolynomially (f : ℝ → ℝ) : Prop :=\n  ∀ b ∈ Set.Ioo 0 1, ∃ c₁ > 0, ∃ c₂ > 0,\n    ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * (f x)) (c₂ * f x)\n\nnamespace GrowsPolynomially\n\nlemma congr_of_eventuallyEq {f g : ℝ → ℝ} (hfg : f =ᶠ[atTop] g) (hg : GrowsPolynomially g) :\n    GrowsPolynomially f := by\n  intro b hb\n  have hg' := hg b hb\n  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hg'⟩ := hg'\n  refine ⟨c₁, hc₁_mem, c₂, hc₂_mem, ?_⟩\n  filter_upwards [hg', (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg, hfg]\n    with x hx₁ hx₂ hx₃\n  intro u hu\n  rw [hx₂ u hu.1, hx₃]\n  exact hx₁ u hu\n\nlemma iff_eventuallyEq {f g : ℝ → ℝ} (h : f =ᶠ[atTop] g) :\n    GrowsPolynomially f ↔ GrowsPolynomially g :=\n  ⟨fun hf => congr_of_eventuallyEq h.symm hf, fun hg => congr_of_eventuallyEq h hg⟩\n\nvariable {f : ℝ → ℝ}\n\nlemma eventually_atTop_le {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ≤ c * f x := by\n  obtain ⟨c₁, _, c₂, hc₂, h⟩ := hf b hb\n  refine ⟨c₂, hc₂, ?_⟩\n  filter_upwards [h]\n  exact fun _ H u hu => (H u hu).2\n\nlemma eventually_atTop_le_nat {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (b * n) n, f u ≤ c * f n := by\n  obtain ⟨c, hc_mem, hc⟩ := hf.eventually_atTop_le hb\n  exact ⟨c, hc_mem, hc.natCast_atTop⟩\n\nlemma eventually_atTop_ge {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, c * f x ≤ f u := by\n  obtain ⟨c₁, hc₁, c₂, _, h⟩ := hf b hb\n  refine ⟨c₁, hc₁, ?_⟩\n  filter_upwards [h]\n  exact fun _ H u hu => (H u hu).1\n\nlemma eventually_atTop_ge_nat {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (b * n) n, c * f n ≤ f u := by\n  obtain ⟨c, hc_mem, hc⟩ := hf.eventually_atTop_ge hb\n  exact ⟨c, hc_mem, hc.natCast_atTop⟩\n\nlemma eventually_zero_of_frequently_zero (hf : GrowsPolynomially f) (hf' : ∃ᶠ x in atTop, f x = 0) :\n    ∀ᶠ x in atTop, f x = 0 := by\n  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf (1 / 2) (by norm_num)\n  rw [frequently_atTop] at hf'\n  filter_upwards [eventually_forall_ge_atTop.mpr hf, eventually_gt_atTop 0] with x hx hx_pos\n  obtain ⟨x₀, hx₀_ge, hx₀⟩ := hf' (max x 1)\n  have x₀_pos := calc\n    0 < 1 := by norm_num\n    _ ≤ x₀ := le_of_max_le_right hx₀_ge\n  have hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z →\n      z ∈ Set.Icc ((2 : ℝ) ^ (-(m : ℤ) - 1) * x₀) ((2 : ℝ) ^ (-(m : ℤ)) * x₀) → f z = 0 := by\n    intro m\n    induction m with\n    | zero =>\n      simp only [CharP.cast_eq_zero, neg_zero, zero_sub, zpow_zero, one_mul] at *\n      specialize hx x₀ (le_of_max_le_left hx₀_ge)\n      simp only [hx₀, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx\n      refine fun z _ hz => hx _ ?_\n      simp only [zpow_neg, zpow_one] at hz\n      simp only [one_div, hz]\n    | succ k ih =>\n      intro z hxz hz\n      simp only [Nat.cast_add, Nat.cast_one] at *\n      have hx' : x ≤ (2 : ℝ) ^ (-(k : ℤ) - 1) * x₀ := by\n        calc x ≤ z := hxz\n          _ ≤ _ := by simp only [neg_add, ← sub_eq_add_neg] at hz; exact hz.2\n      specialize hx ((2 : ℝ) ^ (-(k : ℤ) - 1) * x₀) hx' z\n      specialize ih ((2 : ℝ) ^ (-(k : ℤ) - 1) * x₀) hx' ?ineq\n      case ineq =>\n        rw [Set.left_mem_Icc]\n        gcongr\n        · norm_num\n        · lia\n      simp only [ih, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx\n      refine hx ⟨?lb₁, ?ub₁⟩\n      case lb₁ =>\n        rw [one_div, ← zpow_neg_one, ← mul_assoc, ← zpow_add₀ (by norm_num)]\n        have h₁ : (-1 : ℤ) + (-k - 1) = -k - 2 := by ring\n        have h₂ : -(k + (1 : ℤ)) - 1 = -k - 2 := by ring\n        rw [h₁]\n        rw [h₂] at hz\n        exact hz.1\n      case ub₁ =>\n        have := hz.2\n        simp only [neg_add, ← sub_eq_add_neg] at this\n        exact this\n  refine hmain ⌊-logb 2 (x / x₀)⌋₊ x le_rfl ⟨?lb, ?ub⟩\n  case lb =>\n    rw [← le_div_iff₀ x₀_pos]\n    refine (logb_le_logb (b := 2) (by norm_num) (zpow_pos (by norm_num) _)\n      (by positivity)).mp ?_\n    rw [← rpow_intCast, logb_rpow (by norm_num) (by norm_num), ← neg_le_neg_iff]\n    simp only [Int.cast_sub, Int.cast_neg, Int.cast_natCast, Int.cast_one, neg_sub, sub_neg_eq_add]\n    calc -logb 2 (x / x₀) ≤ ⌈-logb 2 (x / x₀)⌉₊ := Nat.le_ceil (-logb 2 (x / x₀))\n         _ ≤ _ := by rw [add_comm]; exact_mod_cast Nat.ceil_le_floor_add_one _\n  case ub =>\n    rw [← div_le_iff₀ x₀_pos]\n    refine (logb_le_logb (b := 2) (by norm_num) (by positivity)\n      (zpow_pos (by norm_num) _)).mp ?_\n    rw [← rpow_intCast, logb_rpow (by norm_num) (by norm_num), ← neg_le_neg_iff]\n    simp only [Int.cast_neg, Int.cast_natCast, neg_neg]\n    have : 0 ≤ -logb 2 (x / x₀) := by\n      rw [neg_nonneg]\n      refine logb_nonpos (by norm_num) (by positivity) ?_\n      rw [div_le_one x₀_pos]\n      exact le_of_max_le_left hx₀_ge\n    exact_mod_cast Nat.floor_le this\n\nlemma eventually_atTop_nonneg_or_nonpos (hf : GrowsPolynomially f) :\n    (∀ᶠ x in atTop, 0 ≤ f x) ∨ (∀ᶠ x in atTop, f x ≤ 0) := by\n  obtain ⟨c₁, _, c₂, _, h⟩ := hf (1 / 2) (by norm_num)\n  match lt_trichotomy c₁ c₂ with\n  | .inl hlt => -- c₁ < c₂\n    left\n    filter_upwards [h, eventually_ge_atTop 0] with x hx hx_nonneg\n    have h' : 3 / 4 * x ∈ Set.Icc (1 / 2 * x) x := by\n      rw [Set.mem_Icc]\n      exact ⟨by gcongr ?_ * x; norm_num, by linarith⟩\n    have hu := hx (3 / 4 * x) h'\n    have hu := Set.nonempty_of_mem hu\n    rw [Set.nonempty_Icc] at hu\n    have hu' : 0 ≤ (c₂ - c₁) * f x := by linarith\n    exact nonneg_of_mul_nonneg_right hu' (by linarith)\n  | .inr (.inr hgt) => -- c₂ < c₁\n    right\n    filter_upwards [h, eventually_ge_atTop 0] with x hx hx_nonneg\n    have h' : 3 / 4 * x ∈ Set.Icc (1 / 2 * x) x := by\n      rw [Set.mem_Icc]\n      exact ⟨by gcongr ?_ * x; norm_num, by linarith⟩\n    have hu := hx (3 / 4 * x) h'\n    have hu := Set.nonempty_of_mem hu\n    rw [Set.nonempty_Icc] at hu\n    have hu' : (c₁ - c₂) * f x ≤ 0 := by linarith\n    exact nonpos_of_mul_nonpos_right hu' (by linarith)\n  | .inr (.inl heq) => -- c₁ = c₂\n    have hmain : ∃ c, ∀ᶠ x in atTop, f x = c := by\n      simp only [heq, Set.Icc_self, Set.mem_singleton_iff] at h\n      rw [eventually_atTop] at h\n      obtain ⟨n₀, hn₀⟩ := h\n      refine ⟨f (max n₀ 2), ?_⟩\n      rw [eventually_atTop]\n      refine ⟨max n₀ 2, ?_⟩\n      refine Real.induction_Ico_mul _ 2 (by norm_num) (by positivity) ?base ?step\n      case base => grind\n      case step =>\n        intro n _ _ z _\n        have le_2n : max n₀ 2 ≤ (2 : ℝ) ^ n * max n₀ 2 := by\n          simp [one_le_pow₀ (show (1 : ℝ) ≤ 2 by norm_num1)]\n        have half_z_to_base : f (1 / 2 * z) = f (max n₀ 2) := by\n          grind [mul_assoc]\n        grind\n    obtain ⟨c, hc⟩ := hmain\n    cases le_or_gt 0 c with\n    | inl hpos =>\n      exact Or.inl <| by filter_upwards [hc] with _ hc; simpa only [hc]\n    | inr hneg =>\n      right\n      filter_upwards [hc] with x hc\n      exact le_of_lt <| by simpa only [hc]\n\nlemma eventually_atTop_zero_or_pos_or_neg (hf : GrowsPolynomially f) :\n    (∀ᶠ x in atTop, f x = 0) ∨ (∀ᶠ x in atTop, 0 < f x) ∨ (∀ᶠ x in atTop, f x < 0) := by\n  by_cases! h : ∃ᶠ x in atTop, f x = 0\n  · exact Or.inl <| eventually_zero_of_frequently_zero hf h\n  · cases eventually_atTop_nonneg_or_nonpos hf with\n    | inl h' =>\n      refine Or.inr (Or.inl ?_)\n      simp only [lt_iff_le_and_ne]\n      rw [eventually_and]\n      exact ⟨h', by filter_upwards [h] with x hx; exact hx.symm⟩\n    | inr h' =>\n      refine Or.inr (Or.inr ?_)\n      simp only [lt_iff_le_and_ne]\n      rw [eventually_and]\n      exact ⟨h', h⟩\n\nprotected lemma neg {f : ℝ → ℝ} (hf : GrowsPolynomially f) : GrowsPolynomially (-f) := by\n  intro b hb\n  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf b hb\n  refine ⟨c₂, hc₂_mem, c₁, hc₁_mem, ?_⟩\n  filter_upwards [hf] with x hx\n  intro u hu\n  simp only [Pi.neg_apply, Set.neg_mem_Icc_iff, neg_mul_eq_mul_neg, neg_neg]\n  exact hx u hu\n\nprotected lemma neg_iff {f : ℝ → ℝ} : GrowsPolynomially f ↔ GrowsPolynomially (-f) :=\n  ⟨fun hf => hf.neg, fun hf => by rw [← neg_neg f]; exact hf.neg⟩\n\nprotected lemma abs (hf : GrowsPolynomially f) : GrowsPolynomially (fun x => |f x|) := by\n  cases eventually_atTop_nonneg_or_nonpos hf with\n  | inl hf' =>\n    have hmain : f =ᶠ[atTop] fun x => |f x| := by\n      filter_upwards [hf'] with x hx\n      rw [abs_of_nonneg hx]\n    rw [← iff_eventuallyEq hmain]\n    exact hf\n  | inr hf' =>\n    have hmain : -f =ᶠ[atTop] fun x => |f x| := by\n      filter_upwards [hf'] with x hx\n      simp only [Pi.neg_apply, abs_of_nonpos hx]\n    rw [← iff_eventuallyEq hmain]\n    exact hf.neg\n\nprotected lemma norm (hf : GrowsPolynomially f) : GrowsPolynomially (fun x => ‖f x‖) := by\n  simp only [norm_eq_abs]\n  exact hf.abs\n\nend GrowsPolynomially\n\nvariable {f : ℝ → ℝ}\n\nTarget:\nlemma growsPolynomially_const {c : ℝ} : GrowsPolynomially (fun _ => c) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  refine fun _ _ => ⟨1, by norm_num, 1, by norm_num, ?_⟩\n  filter_upwards [] with x\n  simp","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability/AkraBazzi","family_id":"growspolynomially_const","file_id":"mathlib/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean","sample_id":"836ef9c3cd55ae2e3f293b4d27ae834545f003ae5a97617e40dffc59a7b9fdae"},"split":"train"}
{"binding":{"chosen_proof_sha256":"958795982b90c7f479780dff2a7eb96698a38b055a8fc856a14fd71baa43ed6a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ef64a7044dcbaec85a782fa2c5e1d2c9e957e4c92134e4db0c06011461f3684d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7885120c783d836f7907adab08dd0290a06a8a8cf35af5f387b7e22d740bb408","source_sha256":"4068661203cf4c8a8500ce96a89d4eba2dfb134d6a130d9d8f6fe5c686e7101d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rw [IsLUB.sSup_eq (h d₁ d₂ (isLUB_iff_sSup_eq.mpr rfl))]\n  mpr h _ d₁ d₂ _ hda := by rw [isLUB_iff_sSup_eq, ← (h d₁ d₂), IsLUB.sSup_eq hda]\n\nalias ⟨ScottContinuous.map_sSup, ScottContinuous.of_map_sSup⟩ :=\n  scottContinuous_iff_map_sSup","hard_negative":false,"metrics":{"chosen_tokens":61,"rejected_tokens":68,"token_jaccard":0.933333,"token_length_ratio":1.114754},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"1eaccc5fe0295642d0577697157bdc7b3afbcb0e1338c4617610da8dd3771e37","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ScottContinuity.Prod\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Christopher Hoskin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christopher Hoskin\n-/\n/-!\n\n# Scott continuity on complete lattices\n\n## Main results\n\n- `scottContinuous_iff_map_sSup`: A function is Scott continuous if and only if it commutes with\n  `sSup` on directed sets.\n\n-/\n\npublic section\n\nvariable {α β : Type*}\n\nsection CompleteLattice\n\nvariable [CompleteLattice α] [CompleteLattice β]\n\n/- `f` is Scott continuous if and only if it commutes with `sSup` on directed sets -/\n\nTarget:\nlemma scottContinuous_iff_map_sSup {f : α → β} :\n    ScottContinuous f ↔\n      ∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (· ≤ ·) d → f (sSup d) = sSup (f '' d) where\n  mp h _ d₁ d₂ :=\n\nProof body:\n","rejected":"```lean\nby rw [IsLUB.sSup_eq (h d₁ d₂ (isLUB_iff_sSup_eq.mpr rfl))]\n  mpr h _ d₁ d₂ _ hda := by rw [isLUB_iff_sSup_eq, ← (h d₁ d₂), IsLUB.sSup_eq hda]\n\nalias ⟨ScottContinuous.map_sSup, ScottContinuous.of_map_sSup⟩ :=\n  scottContinuous_iff_map_sSup\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/ScottContinuity","family_id":"scottcontinuous_iff_map_ssup","file_id":"mathlib/Mathlib/Order/ScottContinuity/Complete.lean","sample_id":"7885120c783d836f7907adab08dd0290a06a8a8cf35af5f387b7e22d740bb408"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3f65d3f4792abc1e42ea24793b4621de9bedda45327f9e9eb8d4aee686a05437","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9eff011adcb66f66cc2b152b3ad39ee8a1b4fc446edb5657de3f0ac1212971ec","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"45bf8f219b5d1572a7b7bfc82220b21da6d9c0dadf354c8db833eeedf98c284d","source_sha256":"5f15c2a4e2cafa011e3dad0d984da70295687e57a0186d3fe7fe49952e8b2ae6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← LinearOrderedCommGroupWithZero.wellFoundedOn_setOf_ge_gt_iff_nonempty_discrete_of_ne_zero\n    one_ne_zero, ← Set.wellFoundedOn_range]\n  classical\n  refine ⟨fun h ↦ (h.mapsTo Subtype.val ?_).mono' (by simp), fun h ↦ (h.mapsTo ?_ ?_).mono' ?_⟩\n  · rintro ⟨_, x, rfl⟩\n    simp only [← Subtype.coe_le_coe, OneMemClass.coe_one, Set.mem_setOf_eq, Set.mem_range,\n      Function.comp_apply]\n    intro hx\n    obtain ⟨y, rfl⟩ := hv.exists_of_le_one hx\n    exact ⟨y, by simp⟩\n  · exact fun x ↦ if hx : x ∈ MonoidHom.mrange v then ⟨x, hx⟩ else 1\n  · intro\n    simp only [Set.mem_range, Function.comp_apply, MonoidHom.mem_mrange, Set.mem_setOf_eq,\n      forall_exists_index]\n    rintro x rfl\n    simp [← Subtype.coe_le_coe, hv.map_le_one]\n  · simp [Function.onFun]","hard_negative":true,"metrics":{"chosen_tokens":173,"rejected_tokens":3,"token_jaccard":0.032258,"token_length_ratio":0.017341},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"1ec96dfc2868a805008ab528b4fca80b826a4802121f5b52e24b34e5451f7e0d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.ArchimedeanDensely\npublic import Mathlib.RingTheory.Valuation.ValuationRing\n\nNamespace:\nValuation.Integers\n\nLocal context:\n/-\nCopyright (c) 2024 Yakov Pechersky. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yakov Pechersky\n-/\n/-!\n# Ring of integers under a given valuation in a multiplicatively archimedean codomain\n\n-/\n\npublic section\n\nsection Field\n\nvariable {F Γ₀ O : Type*} [Field F] [LinearOrderedCommGroupWithZero Γ₀]\n  [CommRing O] [Algebra O F] {v : Valuation F Γ₀}\n\ninstance MonoidWithZeroHom.instLinearOrderedCommGroupWithZeroMrange (v : F →*₀ Γ₀) :\n    LinearOrderedCommGroupWithZero (MonoidHom.mrange v) where\n  bot := ⟨⊥, by simp [bot_eq_zero]⟩\n  bot_le a := by simp [bot_eq_zero, ← Subtype.coe_le_coe]\n  isBot_zero a := by simp [← Subtype.coe_le_coe]\n  mul_lt_mul_of_pos_left := by\n    simp only [← Subtype.coe_lt_coe, val_mrange_zero, Submonoid.coe_mul, Subtype.forall,\n      MonoidHom.mem_mrange, forall_exists_index, forall_apply_eq_imp_iff]\n    rintro a ha b c hbc\n    gcongr\n\ninstance Valuation.instLinearOrderedCommGroupWithZeroMrange :\n    LinearOrderedCommGroupWithZero (MonoidHom.mrange v) :=\n  inferInstanceAs (LinearOrderedCommGroupWithZero (MonoidHom.mrange (.ofClass v : F →*₀ Γ₀)))\n\nnamespace Valuation.Integers\n\nopen scoped Function in\nlemma wfDvdMonoid_iff_wellFounded_gt_on_v (hv : Integers v O) :\n    WfDvdMonoid O ↔ WellFounded ((· > ·) on (v ∘ algebraMap O F)) := by\n  refine ⟨fun _ ↦ wellFounded_dvdNotUnit.mono ?_, fun h ↦ ⟨h.mono ?_⟩⟩ <;>\n  simp [Function.onFun, hv.dvdNotUnit_iff_lt]\n\nopen scoped Function WithZero in\n\nTarget:\nlemma wellFounded_gt_on_v_iff_discrete_mrange [Nontrivial (MonoidHom.mrange v)ˣ]\n    (hv : Integers v O) :\n    WellFounded ((· > ·) on (v ∘ algebraMap O F)) ↔\n      Nonempty (MonoidHom.mrange v ≃*o ℤᵐ⁰) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_45bf8f219b5d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"b0015ebf930c50bb357b947c5ab7810c6341b7b3049c08cf0ec977dd527f9716","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Valuation","family_id":"wellfounded_gt_on_v_iff_discrete_mrange","file_id":"mathlib/Mathlib/RingTheory/Valuation/Archimedean.lean","sample_id":"45bf8f219b5d1572a7b7bfc82220b21da6d9c0dadf354c8db833eeedf98c284d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6217ca4bcfee77224af82f13005a67bfc164dca4064e34175a675cb4920a7928","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"afe49b555821b24ebf64bf6c1b67f3d1c075f97fc87866f6aaf38fc1d4f60a18","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c422540f094e4df2b71b39ea0c0e9f6334865968fac379ba2de62eb381a0a862","source_sha256":"ebbd09c0a09c3969b7a62f3c013274751c70d4ab23cc8901bc28ba4a69c7b881","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [List.getD_eq_getElem?_getD]\n  exact option_getD.comp₂ list_getElem? (const _)","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":26,"token_jaccard":0.826087,"token_length_ratio":1.238095},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"1f13755f9fac8b85082b7bf833e02c7718221274488e29bf8af25ff2ef8411be","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Computability.Primrec.Basic\npublic import Mathlib.Logic.Encodable.Pi\n\nNamespace:\nPrimrec\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Primitive recursive functions on Lists\n\nThe primitive recursive functions are defined in `Mathlib.Computability.Primrec.Basic`.\nThis file contains definitions and theorems about primitive recursive functions that\nrelate to operation on lists.\n\n## References\n\n* [Mario Carneiro, *Formalizing computability theory via partial recursive functions*][carneiro2019]\n-/\n\n@[expose] public section\n\nopen List (Vector)\nopen Denumerable Encodable Function\n\n\nsection\n\nvariable {α : Type*} {β : Type*} {σ : Type*}\nvariable [Primcodable α] [Primcodable β] [Primcodable σ]\nvariable (H : Nat.Primrec fun n => Encodable.encode (@decode (List β) _ n))\n\nopen Primrec\n\nset_option backward.privateInPublic true in\n@[implicit_reducible]\nprivate def prim : Primcodable (List β) := ⟨H⟩\n\nprivate theorem list_casesOn' {f : α → List β} {g : α → σ} {h : α → β × List β → σ}\n    (hf : haveI := prim H; Primrec f) (hg : Primrec g) (hh : haveI := prim H; Primrec₂ h) :\n    @Primrec _ σ _ _ fun a => List.casesOn (f a) (g a) fun b l => h a (b, l) :=\n  letI := prim H\n  have :\n    @Primrec _ (Option σ) _ _ fun a =>\n      (@decode (Option (β × List β)) _ (encode (f a))).map fun o => Option.casesOn o (g a) (h a) :=\n    ((@map_decode_iff _ (Option (β × List β)) _ _ _ _ _).2 <|\n      to₂ <|\n        option_casesOn snd (hg.comp fst) (hh.comp₂ (fst.comp₂ Primrec₂.left) Primrec₂.right)).comp\n      .id (encode_iff.2 hf)\n  option_some_iff.1 <| this.of_eq fun a => by rcases f a with - | ⟨b, l⟩ <;> simp [encodek]\n\nset_option backward.privateInPublic true in\nprivate theorem list_foldl' {f : α → List β} {g : α → σ} {h : α → σ × β → σ}\n    (hf : haveI := prim H; Primrec f) (hg : Primrec g) (hh : haveI := prim H; Primrec₂ h) :\n    Primrec fun a => (f a).foldl (fun s b => h a (s, b)) (g a) := by\n  letI := prim H\n  let G (a : α) (IH : σ × List β) : σ × List β := List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\n  have hG : Primrec₂ G := list_casesOn' H (snd.comp snd) snd <|\n    to₂ <|\n    pair (hh.comp (fst.comp fst) <| pair ((fst.comp snd).comp fst) (fst.comp snd))\n      (snd.comp snd)\n  let F := fun (a : α) (n : ℕ) => (G a)^[n] (g a, f a)\n  have hF : Primrec fun a => (F a (encode (f a))).1 :=\n    (fst.comp <|\n      nat_iterate (encode_iff.2 hf) (pair hg hf) <|\n      hG)\n  suffices ∀ a n, F a n = (((f a).take n).foldl (fun s b => h a (s, b)) (g a), (f a).drop n) by\n    refine hF.of_eq fun a => ?_\n    rw [this, List.take_of_length_le (length_le_encode _)]\n  introv\n  dsimp only [F]\n  generalize f a = l\n  generalize g a = x\n  induction n generalizing l x with\n  | zero => rfl\n  | succ n IH =>\n    simp only [iterate_succ, comp_apply]\n    rcases l with - | ⟨b, l⟩ <;> simp [G, IH]\n\nset_option backward.privateInPublic true in\nprivate theorem list_cons' : (haveI := prim H; Primrec₂ (@List.cons β)) :=\n  letI := prim H\n  encode_iff.1 (succ.comp <| Primrec₂.natPair.comp (encode_iff.2 fst) (encode_iff.2 snd))\n\nset_option backward.privateInPublic true in\nprivate theorem list_reverse' :\n    haveI := prim H\n    Primrec (@List.reverse β) :=\n  letI := prim H\n  (list_foldl' H .id (const []) <| to₂ <| ((list_cons' H).comp snd fst).comp snd).of_eq\n    (suffices ∀ l r, List.foldl (fun (s : List β) (b : β) => b :: s) r l = List.reverseAux l r from\n      fun l => this l []\n    fun l => by induction l <;> simp [*, List.reverseAux])\n\nend\n\nnamespace Primcodable\n\nvariable {α : Type*} {β : Type*}\nvariable [Primcodable α] [Primcodable β]\n\nopen Primrec\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nset_option linter.flexible false in -- TODO: revisit this after #13791 is merged\ninstance list : Primcodable (List α) :=\n  ⟨letI H := Primcodable.prim (List ℕ)\n    have : Primrec₂ fun (a : α) (o : Option (List ℕ)) => o.map (List.cons (encode a)) :=\n      option_map snd <| (list_cons' H).comp ((@Primrec.encode α _).comp (fst.comp fst)) snd\n    have :\n      Primrec fun n =>\n        (ofNat (List ℕ) n).reverse.foldl\n          (fun o m => (@decode α _ m).bind fun a => o.map (List.cons (encode a))) (some []) :=\n      list_foldl' H ((list_reverse' H).comp (.ofNat (List ℕ))) (const (some []))\n        (Primrec.comp₂ (bind_decode_iff.2 <| .swap this) Primrec₂.right)\n    nat_iff.1 <|\n      (encode_iff.2 this).of_eq fun n => by\n        rw [List.foldl_reverse]\n        apply Nat.case_strong_induction_on n; · simp\n        intro n IH; simp\n        rcases @decode α _ n.unpair.1 with - | a; · rfl\n        simp only [Option.bind_some, Option.map_some]\n        suffices ∀ (o : Option (List ℕ)) (p), encode o = encode p →\n            encode (Option.map (List.cons (encode a)) o) = encode (Option.map (List.cons a) p) from\n          this _ _ (IH _ (Nat.unpair_right_le n))\n        intro o p IH\n        cases o <;> cases p\n        · rfl\n        · injection IH\n        · injection IH\n        · exact congr_arg (fun k => (Nat.pair (encode a) k).succ.succ) (Nat.succ.inj IH)⟩\nend Primcodable\n\nnamespace Primrec\n\nvariable {α : Type*} {β : Type*} {γ : Type*} {σ : Type*}\nvariable [Primcodable α] [Primcodable β] [Primcodable γ] [Primcodable σ]\n\ntheorem list_cons : Primrec₂ (@List.cons α) :=\n  list_cons' (Primcodable.prim _)\n\ntheorem list_casesOn {f : α → List β} {g : α → σ} {h : α → β × List β → σ} :\n    Primrec f →\n      Primrec g →\n        Primrec₂ h → @Primrec _ σ _ _ fun a => List.casesOn (f a) (g a) fun b l => h a (b, l) :=\n  list_casesOn' (Primcodable.prim _)\n\ntheorem list_foldl {f : α → List β} {g : α → σ} {h : α → σ × β → σ} :\n    Primrec f →\n      Primrec g → Primrec₂ h → Primrec fun a => (f a).foldl (fun s b => h a (s, b)) (g a) :=\n  list_foldl' (Primcodable.prim _)\n\ntheorem list_reverse : Primrec (@List.reverse α) :=\n  list_reverse' (Primcodable.prim _)\n\ntheorem list_foldr {f : α → List β} {g : α → σ} {h : α → β × σ → σ} (hf : Primrec f)\n    (hg : Primrec g) (hh : Primrec₂ h) :\n    Primrec fun a => (f a).foldr (fun b s => h a (b, s)) (g a) :=\n  (list_foldl (list_reverse.comp hf) hg <| to₂ <| hh.comp fst <| (pair snd fst).comp snd).of_eq\n    fun a => by simp [List.foldl_reverse]\n\ntheorem list_head? : Primrec (@List.head? α) :=\n  (list_casesOn .id (const none) (option_some_iff.2 <| fst.comp snd).to₂).of_eq fun l => by\n    cases l <;> rfl\n\ntheorem list_headI [Inhabited α] : Primrec (@List.headI α _) :=\n  (option_getD_default.comp list_head?).of_eq fun l => l.head!_eq_head?_getD.symm\n\ntheorem list_tail : Primrec (@List.tail α) :=\n  (list_casesOn .id (const []) (snd.comp snd).to₂).of_eq fun l => by cases l <;> rfl\n\ntheorem list_rec {f : α → List β} {g : α → σ} {h : α → β × List β × σ → σ} (hf : Primrec f)\n    (hg : Primrec g) (hh : Primrec₂ h) :\n    @Primrec _ σ _ _ fun a => List.recOn (f a) (g a) fun b l IH => h a (b, l, IH) :=\n  let F (a : α) := (f a).foldr (fun (b : β) (s : List β × σ) => (b :: s.1, h a (b, s))) ([], g a)\n  have : Primrec F :=\n    list_foldr hf (pair (const []) hg) <|\n      to₂ <| pair ((list_cons.comp fst (fst.comp snd)).comp snd) hh\n  (snd.comp this).of_eq fun a => by\n    suffices F a = (f a, List.recOn (f a) (g a) fun b l IH => h a (b, l, IH)) by rw [this]\n    dsimp [F]\n    induction f a <;> simp [*]\n\ntheorem list_getElem? : Primrec₂ ((·[·]? : List α → ℕ → Option α)) :=\n  let F (l : List α) (n : ℕ) :=\n    l.foldl\n      (fun (s : ℕ ⊕ α) (a : α) =>\n        Sum.casesOn s (@Nat.casesOn (fun _ => ℕ ⊕ α) · (Sum.inr a) Sum.inl) Sum.inr)\n      (Sum.inl n)\n  have hF : Primrec₂ F :=\n    (list_foldl fst (sumInl.comp snd)\n      ((sumCasesOn fst (nat_casesOn snd (sumInr.comp <| snd.comp fst) (sumInl.comp snd).to₂).to₂\n              (sumInr.comp snd).to₂).comp\n          snd).to₂).to₂\n  have :\n    @Primrec _ (Option α) _ _ fun p : List α × ℕ => Sum.casesOn (F p.1 p.2) (fun _ => none) some :=\n    sumCasesOn hF (const none).to₂ (option_some.comp snd).to₂\n  this.to₂.of_eq fun l n => by\n    dsimp; symm\n    induction l generalizing n with\n    | nil => rfl\n    | cons a l IH =>\n      rcases n with - | n\n      · dsimp [F]\n        clear IH\n        induction l <;> simp_all\n      · simpa using! IH ..\n\nTarget:\ntheorem list_getD (d : α) : Primrec₂ fun l n => List.getD l n d :=\n\nProof body:\n","rejected":"by\n  simp only [List.getD_eq_getElem?_getD]\n  exact option_getD.comp₂ list_getElem? (const _)\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability/Primrec","family_id":"list_getd","file_id":"mathlib/Mathlib/Computability/Primrec/List.lean","sample_id":"c422540f094e4df2b71b39ea0c0e9f6334865968fac379ba2de62eb381a0a862"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4824a182947d6de23921d345ff1615bb9443926fe4268ae7e79b135cafd2e987","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4b36ce42ac987a500d5fe55c650474a5f3ef5a7b0b4174df7e06c353194c96bf","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f97b91f107eb7b4f9d321215e0984f13daac29b819bde8fd09eea122778dc081","source_sha256":"7e5df4bf710966952754657aad6fa1ad4d76931701275f754a0493a614aef912","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    rintro X ⟨hX⟩\n    have := hX.prop_diag_obj\n    simp only [isCardinalPresentable_iff] at this ⊢\n    exact isCardinalPresentable_of_isColimit _ hX.isColimit κ hJ","hard_negative":true,"metrics":{"chosen_tokens":27,"rejected_tokens":3,"token_jaccard":0.08,"token_length_ratio":0.111111},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"1f38ce0889a5bdc99d3d42c5b2b55ceab67814f70f03d2a9bde4d97b6b753ef4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Types.Filtered\npublic import Mathlib.CategoryTheory.Limits.Yoneda\npublic import Mathlib.CategoryTheory.Presentable.Basic\npublic import Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Colimits of presentable objects\n\nIn this file, we show that `κ`-accessible functors (to the category of types)\nare stable under limits indexed by a category `K` such that\n`HasCardinalLT (Arrow K) κ`.\nIn particular, `κ`-presentable objects are stable by colimits indexed\nby a category `K` such that `HasCardinalLT (Arrow K) κ`.\n\n-/\n\n@[expose] public section\n\nuniverse w w' v' v u' u\n\nnamespace CategoryTheory\n\nopen Opposite Limits\n\nvariable {C : Type u} [Category.{v} C]\n\nnamespace Functor\n\nnamespace Accessible\n\nnamespace Limits\n\nsection\n\nvariable {K : Type u'} [Category.{v'} K] {F : K ⥤ C ⥤ Type w'}\n  (c : Cone F) (hc : ∀ (Y : C), IsLimit (((evaluation _ _).obj Y).mapCone c))\n  (κ : Cardinal.{w}) [Fact κ.IsRegular]\n  (hK : HasCardinalLT (Arrow K) κ)\n  {J : Type w} [SmallCategory J] [IsCardinalFiltered J κ]\n  {X : J ⥤ C} (cX : Cocone X)\n  (hF : ∀ (k : K), IsColimit ((F.obj k).mapCocone cX))\n\nnamespace isColimitMapCocone\n\ninclude hc hF hK\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma surjective (x : c.pt.obj cX.pt) :\n    ∃ (j : J) (x' : c.pt.obj (X.obj j)), x = (c.pt.mapCocone cX).ι.app j x' := by\n  have := isFiltered_of_isCardinalFiltered J κ\n  obtain ⟨y, hy⟩ := (Types.isLimitEquivSections (hc cX.pt)).symm.surjective x\n  obtain ⟨j₀, z, hz⟩ : ∃ (j₀ : J) (z : (k : K) → (F.obj k).obj (X.obj j₀)),\n      ∀ (k : K), y.1 k = (F.obj k).map (cX.ι.app j₀) (z k) := by\n    have H (k : K) := Types.jointly_surjective_of_isColimit (hF k) (y.1 k)\n    let j (k : K) : J := (H k).choose\n    let z (k : K) : (F.obj k).obj (X.obj (j k)) := (H k).choose_spec.choose\n    have hz (k : K) : (F.obj k).map (cX.ι.app (j k)) (z k) = y.1 k :=\n      (H k).choose_spec.choose_spec\n    exact ⟨IsCardinalFiltered.max j (hasCardinalLT_of_hasCardinalLT_arrow hK),\n      fun k ↦ (F.obj k).map (X.map (IsCardinalFiltered.toMax j _ k)) (z k),\n        fun k ↦ by rw [← hz, ← comp_apply, ← Functor.map_comp, cX.w]; rfl⟩\n  obtain ⟨j₁, α, hα⟩ : ∃ (j₁ : J) (α : j₀ ⟶ j₁), ∀ ⦃k k' : K⦄ (φ : k ⟶ k'),\n      (F.obj k').map (X.map α) ((F.map φ).app _ (z k)) =\n        (F.obj k').map (X.map α) (z k') := by\n    have H {k k' : K} (φ : k ⟶ k') :=\n      (Types.FilteredColimit.isColimit_eq_iff' (ht := hF k')\n        (x := (F.map φ).app _ (z k)) (y := z k')).1 (by\n          dsimp at hz ⊢\n          simpa only [← NatTrans.naturality_apply, ← hz] using! y.2 φ)\n    let j {k k' : K} (φ : k ⟶ k') : J := (H φ).choose\n    let g {k k' : K} (φ : k ⟶ k') : j₀ ⟶ j φ := (H φ).choose_spec.choose\n    have hg {k k' : K} (φ : k ⟶ k') :\n        (F.obj k').map (X.map (g φ)) ((F.map φ).app _ (z k)) =\n          (F.obj k').map (X.map (g φ)) (z k') := (H φ).choose_spec.choose_spec\n    obtain ⟨j₁, α, β, hα⟩ : ∃ (j₁ : J) (α : j₀ ⟶ j₁)\n        (β : ∀ ⦃k k' : K⦄ (φ : k ⟶ k'), j φ ⟶ j₁),\n        ∀ ⦃k k' : K⦄ (φ : k ⟶ k'), α = g φ ≫ β φ := by\n      let j'' (f : Arrow K) : J := j f.hom\n      let ψ (f : Arrow K) : j₀ ⟶ IsCardinalFiltered.max j'' hK :=\n        g f.hom ≫ IsCardinalFiltered.toMax j'' hK f\n      refine ⟨IsCardinalFiltered.coeq ψ hK, IsCardinalFiltered.toCoeq ψ hK,\n        fun k k' φ ↦ IsCardinalFiltered.toMax j'' hK φ ≫ IsCardinalFiltered.coeqHom ψ hK,\n        fun k k' φ ↦ ?_⟩\n      simpa [ψ] using! (IsCardinalFiltered.coeq_condition ψ hK (Arrow.mk φ)).symm\n    exact ⟨j₁, α, fun k k' φ ↦ by simp [hα φ, hg]⟩\n  let s : (F ⋙ (evaluation C (Type w')).obj (X.obj j₁)).sections :=\n    { val k := (F.obj k).map (X.map α) (z k)\n      property {k k'} φ := by\n        dsimp\n        rw [NatTrans.naturality_apply, ← hα φ] }\n  refine ⟨j₁, (Types.isLimitEquivSections (hc (X.obj j₁))).symm s, ?_⟩\n  apply (Types.isLimitEquivSections (hc cX.pt)).injective\n  rw [← hy, Equiv.apply_symm_apply]\n  ext k\n  have h₁ := Types.isLimitEquivSections_apply (hc cX.pt) k\n    (c.pt.map (cX.ι.app j₁) ((Types.isLimitEquivSections (hc (X.obj j₁))).symm s))\n  have h₂ := Types.isLimitEquivSections_symm_apply (hc (X.obj j₁)) s k\n  dsimp at h₁ h₂ ⊢\n  rw [h₁, hz, NatTrans.naturality_apply, h₂, ← comp_apply, ← Functor.map_comp, cX.w]\n  rfl\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma injective (j : J) (x₁ x₂ : c.pt.obj (X.obj j))\n    (h : c.pt.map (cX.ι.app j) x₁ = c.pt.map (cX.ι.app j) x₂) :\n    ∃ (j' : J) (α : j ⟶ j'),\n    c.pt.map (X.map α) x₁ = c.pt.map (X.map α) x₂ := by\n  have := isFiltered_of_isCardinalFiltered J κ\n  let y₁ := Types.isLimitEquivSections (hc (X.obj j)) x₁\n  let y₂ := Types.isLimitEquivSections (hc (X.obj j)) x₂\n  have hy₁ : (Types.isLimitEquivSections (hc (X.obj j))).symm y₁ = x₁ := by simp [y₁]\n  have hy₂ : (Types.isLimitEquivSections (hc (X.obj j))).symm y₂ = x₂ := by simp [y₂]\n  have H (k : K) := (Types.FilteredColimit.isColimit_eq_iff' (ht := hF k)\n    (x := y₁.1 k) (y := y₂.1 k)).1 (by\n      simp only [y₁, y₂, Types.isLimitEquivSections_apply]\n      dsimp at h ⊢\n      simp only [← NatTrans.naturality_apply, h])\n  let j₁ (k : K) : J := (H k).choose\n  let f (k : K) : j ⟶ j₁ k := (H k).choose_spec.choose\n  have hf (k : K) : (F.obj k).map (X.map (f k)) (y₁.1 k) =\n      (F.obj k).map (X.map (f k)) (y₂.1 k) :=\n    (H k).choose_spec.choose_spec\n  have hK' := hasCardinalLT_of_hasCardinalLT_arrow hK\n  let ψ (k : K) : j ⟶ IsCardinalFiltered.max j₁ hK' :=\n    f k ≫ IsCardinalFiltered.toMax j₁ hK' k\n  refine ⟨IsCardinalFiltered.coeq ψ hK', IsCardinalFiltered.toCoeq ψ hK', ?_⟩\n  apply (Types.isLimitEquivSections (hc _)).injective\n  ext k\n  simp only [Types.isLimitEquivSections_apply, ← hy₁, ← hy₂]\n  have h₁ := Types.isLimitEquivSections_symm_apply (hc (X.obj j)) y₁ k\n  have h₂ := Types.isLimitEquivSections_symm_apply (hc (X.obj j)) y₂ k\n  dsimp at h₁ h₂ ⊢\n  simp [h₁, h₂, ← IsCardinalFiltered.coeq_condition ψ hK' k, ψ, hf]\n\nend isColimitMapCocone\n\n/-- Auxiliary definition for `isCardinalAccessible_of_isLimit`. -/\nnoncomputable def isColimitMapCocone : IsColimit (c.pt.mapCocone cX) := by\n  have := isFiltered_of_isCardinalFiltered J κ\n  apply Types.FilteredColimit.isColimitOf'\n  · exact isColimitMapCocone.surjective c hc κ hK cX hF\n  · exact isColimitMapCocone.injective c hc κ hK cX hF\n\nend\n\nend Limits\n\nend Accessible\n\nlemma isCardinalAccessible_of_isLimit {K : Type u'} [Category.{v'} K] {F : K ⥤ C ⥤ Type w'}\n    (c : Cone F) (hc : IsLimit c) (κ : Cardinal.{w}) [Fact κ.IsRegular]\n    [HasLimitsOfShape K (Type w')] (hK : HasCardinalLT (Arrow K) κ)\n    [∀ k, (F.obj k).IsCardinalAccessible κ] :\n    c.pt.IsCardinalAccessible κ where\n  preservesColimitOfShape {J _ _} := ⟨fun {X} ↦ ⟨fun {cX} hcX ↦ by\n    have := fun k ↦ preservesColimitsOfShape_of_isCardinalAccessible (F.obj k) κ J\n    exact ⟨Accessible.Limits.isColimitMapCocone c\n      (fun Y ↦ isLimitOfPreserves ((evaluation C (Type w')).obj Y) hc) κ hK cX\n      (fun k ↦ isColimitOfPreserves (F.obj k) hcX)⟩⟩⟩\n\nend Functor\n\nset_option backward.defeqAttrib.useBackward true in\n/-- In case `C` is locally `w`-small, use `isCardinalPresentable_of_isColimit`. -/\nlemma isCardinalPresentable_of_isColimit'\n    {K : Type u'} [Category.{v'} K] {Y : K ⥤ C}\n    (c : Cocone Y) (hc : IsColimit c) (κ : Cardinal.{w}) [Fact κ.IsRegular]\n    [HasLimitsOfShape Kᵒᵖ (Type v)] (hK : HasCardinalLT (Arrow K) κ)\n    [∀ k, IsCardinalPresentable (Y.obj k) κ] :\n    IsCardinalPresentable c.pt κ := by\n  have (k : Kᵒᵖ) : ((Y.op ⋙ coyoneda).obj k).IsCardinalAccessible κ := by\n    dsimp; infer_instance\n  exact Functor.isCardinalAccessible_of_isLimit\n    (coyoneda.mapCone c.op) (isLimitOfPreserves _ hc.op) κ (by simpa)\n\nset_option backward.defeqAttrib.useBackward true in\nlemma isCardinalPresentable_of_isColimit [LocallySmall.{w} C]\n    {K : Type u'} [Category.{v'} K] [HasLimitsOfShape Kᵒᵖ (Type w)] {Y : K ⥤ C}\n    (c : Cocone Y) (hc : IsColimit c) (κ : Cardinal.{w}) [Fact κ.IsRegular]\n    (hK : HasCardinalLT (Arrow K) κ)\n    [∀ k, IsCardinalPresentable (Y.obj k) κ] :\n    IsCardinalPresentable c.pt κ := by\n  let e := ShrinkHoms.equivalence.{w} C\n  have (k : K) : IsCardinalPresentable ((Y ⋙ e.functor).obj k) κ := by\n    dsimp; infer_instance\n  rw [← isCardinalPresentable_iff_of_isEquivalence c.pt κ e.functor]\n  exact isCardinalPresentable_of_isColimit' _\n    (isColimitOfPreserves e.functor hc) κ hK\n\nvariable (C) in\n\nTarget:\nlemma isClosedUnderColimitsOfShape_isCardinalPresentable [LocallySmall.{w} C]\n    {κ : Cardinal.{w}} [Fact κ.IsRegular]\n    {J : Type u'} [Category.{v'} J] [HasLimitsOfShape Jᵒᵖ (Type w)]\n    (hJ : HasCardinalLT (Arrow J) κ) :\n    (isCardinalPresentable C κ).IsClosedUnderColimitsOfShape J where\n  colimitsOfShape_le :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_f97b91f107eb","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"3c80c5bff675cb62613aab6c8932c9203aa877d0968e76ea1b36500aeae22287","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Presentable","family_id":"isclosedundercolimitsofshape_iscardinalpresentable","file_id":"mathlib/Mathlib/CategoryTheory/Presentable/Limits.lean","sample_id":"f97b91f107eb7b4f9d321215e0984f13daac29b819bde8fd09eea122778dc081"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1e7bb9669789dabc1d0b4a44d7bd38a7c58136fdfc801571fd7b5e8b4e1c7606","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4c02a391b7ac277a192dd766770fc91e8893ee8cde8243ce667618594d9c0e76","source_sha256":"81278dc64ee13852107171790942ce238560fcc410268d3130067bd338a7e15e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  conv_lhs => rw [← (mateEquiv adj₁ adj₂).right_inv α]\n  exact (mateEquiv_counit adj₁ adj₂ ((mateEquiv adj₁ adj₂).symm α) d)","hard_negative":false,"metrics":{"chosen_tokens":38,"rejected_tokens":2,"token_jaccard":0.047619,"token_length_ratio":0.052632},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"1f3ae663cf661ed6e010b892bb65315befc9ab321fc23dc63f8133c1190a666b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Adjunction.Basic\npublic import Mathlib.CategoryTheory.Functor.TwoSquare\npublic import Mathlib.CategoryTheory.HomCongr\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Emily Riehl, Joël Riou\n-/\n/-!\n# Mate of natural transformations\n\nThis file establishes the bijection between the 2-cells\n\n```\n         L₁                  R₁\n      C --→ D             C ←-- D\n    G ↓  ↗  ↓ H         G ↓  ↘  ↓ H\n      E --→ F             E ←-- F\n         L₂                  R₂\n```\n\nwhere `L₁ ⊣ R₁` and `L₂ ⊣ R₂`. The corresponding natural transformations are called mates.\n\nThis bijection includes a number of interesting cases as specializations. For instance, in the\nspecial case where `G,H` are identity functors then the bijection preserves and reflects\nisomorphisms (i.e. we have bijections `(L₂ ⟶ L₁) ≃ (R₁ ⟶ R₂)`, and if either side is an iso then the\nother side is as well). This demonstrates that adjoints to a given functor are unique up to\nisomorphism (since if `L₁ ≅ L₂` then we deduce `R₁ ≅ R₂`).\n\nAnother example arises from considering the square representing that a functor `H` preserves\nproducts, in particular the morphism `H A ⨯ H- ⟶ H (A ⨯ -)`. Then provided `(A ⨯ -)` and `H A ⨯ -`\nhave left adjoints (for instance if the relevant categories are Cartesian closed), the transferred\nnatural transformation is the exponential comparison morphism: `H (A ^ -) ⟶ H A ^ H-`.\nFurthermore if `H` has a left adjoint `L`, this morphism is an isomorphism iff its mate\n`L (H A ⨯ -) ⟶ A ⨯ L-` is an isomorphism, see\nhttps://ncatlab.org/nlab/show/Frobenius+reciprocity#InCategoryTheory.\nThis also relates to Grothendieck's yoga of six operations, though this is not spelled out in\nmathlib: https://ncatlab.org/nlab/show/six+operations.\n-/\n\nset_option backward.defeqAttrib.useBackward true\n\n@[expose] public section\n\nuniverse v₁ v₂ v₃ v₄ v₅ v₆ v₇ v₈ v₉ u₁ u₂ u₃ u₄ u₅ u₆ u₇ u₈ u₉\nnamespace CategoryTheory\n\nopen Category Functor Adjunction NatTrans TwoSquare\n\nsection mateEquiv\n\nvariable {C : Type u₁} {D : Type u₂} {E : Type u₃} {F : Type u₄}\nvariable [Category.{v₁} C] [Category.{v₂} D] [Category.{v₃} E] [Category.{v₄} F]\nvariable {G : C ⥤ E} {H : D ⥤ F} {L₁ : C ⥤ D} {R₁ : D ⥤ C} {L₂ : E ⥤ F} {R₂ : F ⥤ E}\nvariable (adj₁ : L₁ ⊣ R₁) (adj₂ : L₂ ⊣ R₂)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- Suppose we have a square of functors (where the top and bottom are adjunctions `L₁ ⊣ R₁`\nand `L₂ ⊣ R₂` respectively).\n\n```\n      C ↔ D\n    G ↓   ↓ H\n      E ↔ F\n```\n\nThen we have a bijection between natural transformations `G ⋙ L₂ ⟶ L₁ ⋙ H` and\n`R₁ ⋙ G ⟶ H ⋙ R₂`. This can be seen as a bijection of the 2-cells:\n\n```\n         L₁                  R₁\n      C --→ D             C ←-- D\n    G ↓  ↗  ↓ H         G ↓  ↘  ↓ H\n      E --→ F             E ←-- F\n         L₂                  R₂\n```\n\nNote that if one of the transformations is an iso, it does not imply the other is an iso.\n-/\n@[simps]\ndef mateEquiv : TwoSquare G L₁ L₂ H ≃ TwoSquare R₁ H G R₂ where\n  toFun α := .mk _ _ _ _ <|\n    (rightUnitor _).inv ≫\n    whiskerLeft (R₁ ⋙ G) adj₂.unit ≫\n    (associator _ _ _).hom ≫ whiskerLeft _ (associator _ _ _).inv ≫\n    whiskerLeft R₁ (whiskerRight α.natTrans R₂) ≫\n    whiskerLeft _ (associator _ _ _).hom ≫ (associator _ _ _).inv ≫\n    whiskerRight adj₁.counit (H ⋙ R₂) ≫\n    (leftUnitor _).hom\n  invFun β := .mk _ _ _ _ <|\n    (leftUnitor _).inv ≫\n    whiskerRight adj₁.unit (G ⋙ L₂) ≫\n    (associator _ _ _).inv ≫ whiskerRight (associator _ _ _).hom _ ≫\n    whiskerRight (whiskerLeft L₁ β.natTrans) L₂ ≫\n    whiskerRight (associator _ _ _).inv _ ≫ (associator _ _ _).hom ≫\n    whiskerLeft (L₁ ⋙ H) adj₂.counit ≫\n    (rightUnitor _).hom\n  left_inv α := by\n    ext\n    simp only [comp_obj, whiskerLeft_comp, whiskerLeft_twice, assoc, Iso.hom_inv_id_assoc,\n      whiskerRight_comp, comp_app, id_obj, leftUnitor_inv_app, Functor.whiskerRight_app,\n      Functor.comp_map, associator_inv_app, associator_hom_app, map_id, Functor.whiskerLeft_app,\n      rightUnitor_inv_app, leftUnitor_hom_app, rightUnitor_hom_app, comp_id, id_comp,\n      counit_naturality, counit_naturality_assoc, left_triangle_components_assoc]\n    rw [← assoc, ← Functor.comp_map, α.natTrans.naturality, Functor.comp_map, assoc, ← H.map_comp,\n      left_triangle_components, map_id]\n    simp only [comp_obj, comp_id]\n  right_inv β := by\n    ext\n    simp only [comp_obj, whiskerRight_comp, whiskerRight_twice, assoc, Iso.inv_hom_id_assoc,\n      whiskerLeft_comp, comp_app, id_obj, rightUnitor_inv_app, Functor.whiskerLeft_app,\n      associator_hom_app, associator_inv_app, Functor.whiskerRight_app, leftUnitor_inv_app, map_id,\n      Functor.comp_map, rightUnitor_hom_app, leftUnitor_hom_app, comp_id, id_comp,\n      unit_naturality_assoc, right_triangle_components_assoc]\n    rw [← assoc, ← Functor.comp_map, assoc, ← β.natTrans.naturality, ← assoc, Functor.comp_map,\n      ← G.map_comp, right_triangle_components, map_id, id_comp]\n\nset_option backward.defeqAttrib.useBackward true in\n/-- A component of a transposed version of the mates correspondence. -/\ntheorem mateEquiv_counit (α : TwoSquare G L₁ L₂ H) (d : D) :\n    L₂.map ((mateEquiv adj₁ adj₂ α).app _) ≫ adj₂.counit.app _ =\n      α.app _ ≫ H.map (adj₁.counit.app d) := by simp\n\n/-- A component of a transposed version of the inverse mates correspondence. -/\n\nTarget:\ntheorem mateEquiv_counit_symm (α : TwoSquare R₁ H G R₂) (d : D) :\n    L₂.map (α.app _) ≫ adj₂.counit.app _ =\n      ((mateEquiv adj₁ adj₂).symm α).app _ ≫ H.map (adj₁.counit.app d) :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Adjunction","family_id":"mateequiv_counit_symm","file_id":"mathlib/Mathlib/CategoryTheory/Adjunction/Mates.lean","sample_id":"4c02a391b7ac277a192dd766770fc91e8893ee8cde8243ce667618594d9c0e76"},"split":"train"}
{"binding":{"chosen_proof_sha256":"aa83c7abdbb25c29bbdc8a05c929751eafeeab66442c5c16368b86eecdc282c9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2abedb34a57507348c7123447df9943ccb023fa39d74eb1e15a4dbda90a697ca","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"423b546f69a419e2fd6b478d8c8a356a73c2e1fde067e823b879f61db25761a8","source_sha256":"087c4fdfa1fe64dbc38821c1fa8aea61c46d7fa4bef6ee724aa05ce31b21dbc2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← image_singleton, preimage_image_mk_eq_mul]\n  simp","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":5,"token_jaccard":0.076923,"token_length_ratio":0.555556},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"1f41ea2a813c362b4242c16209efe3475213597187c51d3590d30ac759880597","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Quotient\npublic import Mathlib.Algebra.Group.Action.Opposite\npublic import Mathlib.Algebra.Group.Subgroup.MulOpposite\npublic import Mathlib.GroupTheory.GroupAction.Defs\npublic import Mathlib.Algebra.Group.Pointwise.Set.Basic\n\nNamespace:\nQuotientGroup\n\nLocal context:\n/-\nCopyright (c) 2018 Mitchell Rowett. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mitchell Rowett, Kim Morrison\n-/\n/-!\n# Cosets\n\nThis file develops the basic theory of left and right cosets.\n\nWhen `G` is a group and `a : G`, `s : Set G`, with  `open scoped Pointwise` we can write:\n* the left coset of `s` by `a` as `a • s`\n* the right coset of `s` by `a` as `MulOpposite.op a • s` (or `op a • s` with `open MulOpposite`,\n  or `s <• a` with `open scoped Pointwise RightActions`)\n\nIf instead `G` is an additive group, we can write (with  `open scoped Pointwise` still)\n* the left coset of `s` by `a` as `a +ᵥ s`\n* the right coset of `s` by `a` as `AddOpposite.op a +ᵥ s` (or `op a +ᵥ s` with `open AddOpposite`,\n  or `s <+ᵥ a` with `open scoped Pointwise RightActions`)\n\n## Main definitions\n\n* `QuotientGroup.quotient s`: the quotient type representing the left cosets with respect to a\n  subgroup `s`, for an `AddGroup` this is `QuotientAddGroup.quotient s`.\n* `QuotientGroup.mk`: the canonical map from `α` to `α/s` for a subgroup `s` of `α`, for an\n  `AddGroup` this is `QuotientAddGroup.mk`.\n\n## Notation\n\n* `G ⧸ H` is the quotient of the (additive) group `G` by the (additive) subgroup `H`\n\n## TODO\n\nProperly merge with pointwise actions on sets, by renaming and deduplicating lemmas as appropriate.\n-/\n\n@[expose] public section\n\nassert_not_exists Cardinal\n\nopen Function Set\nopen scoped Pointwise\n\nvariable {α : Type*}\n\n/- Ensure that `@[to_additive]` uses the right namespace. -/\ninsert_to_additive_translation QuotientGroup QuotientAddGroup\n\nnamespace QuotientGroup\n\nvariable [Group α] (s : Subgroup α)\n\n/-- The equivalence relation corresponding to the partition of a group by left cosets\nof a subgroup. -/\n@[to_additive (attr := implicit_reducible)\n  /-- The equivalence relation corresponding to the partition of a group by left cosets\nof a subgroup. -/]\ndef leftRel : Setoid α :=\n  MulAction.orbitRel s.op α\n\nvariable {s} in\n@[to_additive]\ntheorem leftRel_apply {x y : α} : leftRel s x y ↔ x⁻¹ * y ∈ s :=\n  calc\n    (∃ a : s.op, y * MulOpposite.unop a = x) ↔ ∃ a : s, y * a = x :=\n      s.equivOp.symm.exists_congr_left\n    _ ↔ ∃ a : s, x⁻¹ * y = a⁻¹ := by\n      simp only [inv_mul_eq_iff_eq_mul, Subgroup.coe_inv, eq_mul_inv_iff_mul_eq]\n    _ ↔ x⁻¹ * y ∈ s := by simp [exists_inv_mem_iff_exists_mem]\n\n@[to_additive]\ntheorem leftRel_eq : ⇑(leftRel s) = fun x y => x⁻¹ * y ∈ s :=\n  funext₂ <| by\n    simp only [eq_iff_iff]\n    apply leftRel_apply\n\n@[to_additive]\ninstance leftRelDecidable [DecidablePred (· ∈ s)] : DecidableRel (leftRel s).r := fun x y => by\n  rw [leftRel_eq]\n  exact ‹DecidablePred (· ∈ s)› _\n\n/-- `α ⧸ s` is the quotient type representing the left cosets of `s`. If `s` is a normal subgroup,\n`α ⧸ s` is a group -/\n@[to_additive /-- `α ⧸ s` is the quotient type representing the left cosets of `s`. If `s` is a\nnormal subgroup, `α ⧸ s` is a group -/]\ninstance instHasQuotientSubgroup : HasQuotient α (Subgroup α) :=\n  ⟨fun s => Quotient (leftRel s)⟩\n\n@[to_additive]\ninstance [DecidablePred (· ∈ s)] : DecidableEq (α ⧸ s) :=\n  @Quotient.decidableEq _ _ (leftRelDecidable _)\n\n/-- The equivalence relation corresponding to the partition of a group by right cosets of a\nsubgroup. -/\n@[to_additive (attr := implicit_reducible)\n  /-- The equivalence relation corresponding to the partition of a group by right cosets\n  of a subgroup. -/]\ndef rightRel : Setoid α :=\n  MulAction.orbitRel s α\n\nvariable {s} in\n@[to_additive]\ntheorem rightRel_apply {x y : α} : rightRel s x y ↔ y * x⁻¹ ∈ s :=\n  calc\n    (∃ a : s, (a : α) * y = x) ↔ ∃ a : s, y * x⁻¹ = a⁻¹ := by\n      simp only [mul_inv_eq_iff_eq_mul, Subgroup.coe_inv, eq_inv_mul_iff_mul_eq]\n    _ ↔ y * x⁻¹ ∈ s := by simp [exists_inv_mem_iff_exists_mem]\n\n@[to_additive]\ntheorem rightRel_eq : ⇑(rightRel s) = fun x y => y * x⁻¹ ∈ s :=\n  funext₂ <| by\n    simp only [eq_iff_iff]\n    apply rightRel_apply\n\n@[to_additive]\ninstance rightRelDecidable [DecidablePred (· ∈ s)] : DecidableRel (rightRel s).r := fun x y => by\n  rw [rightRel_eq]\n  exact ‹DecidablePred (· ∈ s)› _\n\n/-- Right cosets are in bijection with left cosets. -/\n@[to_additive /-- Right cosets are in bijection with left cosets. -/]\ndef quotientRightRelEquivQuotientLeftRel : Quotient (QuotientGroup.rightRel s) ≃ α ⧸ s where\n  toFun :=\n    Quotient.map' (fun g => g⁻¹) fun a b => by\n      rw [leftRel_apply, rightRel_apply]\n      exact fun h => (congr_arg (· ∈ s) (by simp)).mp (s.inv_mem h)\n  invFun :=\n    Quotient.map' (fun g => g⁻¹) fun a b => by\n      rw [leftRel_apply, rightRel_apply]\n      exact fun h => (congr_arg (· ∈ s) (by simp)).mp (s.inv_mem h)\n  left_inv g :=\n    Quotient.inductionOn' g fun g =>\n      Quotient.sound'\n        (by\n          simp only [inv_inv]\n          exact Quotient.exact' rfl)\n  right_inv g :=\n    Quotient.inductionOn' g fun g =>\n      Quotient.sound'\n        (by\n          simp only [inv_inv]\n          exact Quotient.exact' rfl)\n\nend QuotientGroup\n\nnamespace QuotientGroup\n\nvariable [Group α] {s : Subgroup α}\n\n/-- The canonical map from a group `α` to the quotient `α ⧸ s`. -/\n@[to_additive (attr := coe)\n/-- The canonical map from an `AddGroup` `α` to the quotient `α ⧸ s`. -/]\nabbrev mk (a : α) : α ⧸ s :=\n  Quotient.mk'' a\n\n@[to_additive]\ntheorem mk_surjective : Function.Surjective <| @mk _ _ s :=\n  Quotient.mk''_surjective\n\n@[to_additive (attr := simp)]\nlemma range_mk : range (QuotientGroup.mk (s := s)) = univ := range_eq_univ.mpr mk_surjective\n\n@[to_additive (attr := elab_as_elim)]\ntheorem induction_on {C : α ⧸ s → Prop} (x : α ⧸ s) (H : ∀ z, C (QuotientGroup.mk z)) : C x :=\n  Quotient.inductionOn' x H\n\n@[to_additive]\ninstance : Coe α (α ⧸ s) :=\n  ⟨mk⟩\n\n@[to_additive] alias induction_on' := induction_on\n\n@[to_additive (attr := simp)]\ntheorem quotient_liftOn_mk {β} (f : α → β) (h) (x : α) : Quotient.liftOn' (x : α ⧸ s) f h = f x :=\n  rfl\n\n@[to_additive]\ntheorem forall_mk {C : α ⧸ s → Prop} : (∀ x : α ⧸ s, C x) ↔ ∀ x : α, C x :=\n  mk_surjective.forall\n\n@[to_additive]\ntheorem exists_mk {C : α ⧸ s → Prop} : (∃ x : α ⧸ s, C x) ↔ ∃ x : α, C x :=\n  mk_surjective.exists\n\n@[to_additive]\ninstance (s : Subgroup α) : Inhabited (α ⧸ s) :=\n  ⟨((1 : α) : α ⧸ s)⟩\n\n@[to_additive]\nprotected theorem eq {a b : α} : (a : α ⧸ s) = b ↔ a⁻¹ * b ∈ s :=\n  calc\n    _ ↔ leftRel s a b := Quotient.eq''\n    _ ↔ _ := by rw [leftRel_apply]\n\n@[to_additive]\ntheorem out_eq' (a : α ⧸ s) : mk a.out = a :=\n  Quotient.out_eq' a\n\nvariable (s)\n\n/-- It can be useful to write `obtain ⟨h, H⟩ := mk_out_eq_mul ...`, and then `rw [H]` or\n`simp_rw [H]` or `simp only [H]`. In order for `simp_rw` and `simp only` to work, this lemma is\nstated in terms of an arbitrary `h : s`, rather than the specific `h = g⁻¹ * (mk g).out`. -/\n@[to_additive QuotientAddGroup.mk_out_eq_mul]\ntheorem mk_out_eq_mul (g : α) : ∃ h : s, (mk g : α ⧸ s).out = g * h :=\n  ⟨⟨g⁻¹ * (mk g).out, QuotientGroup.eq.mp (mk g).out_eq'.symm⟩, by rw [mul_inv_cancel_left]⟩\n\nvariable {s} {a b : α}\n\n@[to_additive (attr := simp)]\ntheorem mk_mul_of_mem (a : α) (hb : b ∈ s) : (mk (a * b) : α ⧸ s) = mk a := by\n  rwa [QuotientGroup.eq, mul_inv_rev, inv_mul_cancel_right, s.inv_mem_iff]\n\n@[to_additive]\ntheorem preimage_image_mk (N : Subgroup α) (s : Set α) :\n    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = ⋃ x : N, (· * (x : α)) ⁻¹' s := by\n  ext x\n  simp only [QuotientGroup.eq, SetLike.exists, exists_prop, Set.mem_preimage, Set.mem_iUnion,\n    Set.mem_image]\n  exact\n    ⟨fun ⟨y, hs, hN⟩ => ⟨_, N.inv_mem hN, by simpa using hs⟩, fun ⟨z, hz, hxz⟩ =>\n      ⟨x * z, hxz, by simpa using hz⟩⟩\n\n@[to_additive]\ntheorem preimage_image_mk_eq_iUnion_image (N : Subgroup α) (s : Set α) :\n    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = ⋃ x : N, (· * (x : α)) '' s := by\n  rw [preimage_image_mk, iUnion_congr_of_surjective (·⁻¹) inv_surjective]\n  exact fun x ↦ image_mul_right'\n\n@[to_additive]\ntheorem preimage_image_mk_eq_mul (N : Subgroup α) (s : Set α) :\n    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = s * N := by\n  rw [preimage_image_mk_eq_iUnion_image, iUnion_subtype, ← image2_mul, ← iUnion_image_right]\n  simp only [SetLike.mem_coe]\n\n@[to_additive]\n\nTarget:\ntheorem preimage_mk_one (N : Subgroup α) :\n    mk ⁻¹' {(mk : α → α ⧸ N) 1} = N :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_423b546f69a4","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"cd1f9ec868d11cffd7a9f773bdf2cabea557d39df2809266924c65586cafea5c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Coset","family_id":"preimage_mk_one","file_id":"mathlib/Mathlib/GroupTheory/Coset/Defs.lean","sample_id":"423b546f69a419e2fd6b478d8c8a356a73c2e1fde067e823b879f61db25761a8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2038792152bb22c0a5ae377d16d2f16e94b117cceae2db86d9ede3e4d520885b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7437481b96e36485900fd3de7576c86e052cc66a933bd5a9f55150ba401a7490","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"59fa1cfe920c44354539a08a73d5bce71392689b4372837276159b664aba9fb8","source_sha256":"5a8a601bded85c9c17f5a633b68ceab704a71b9b3614d423fc312dd608794122","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← cancel_epi ((shiftFunctorZero _ A).inv.app X), Iso.inv_hom_id_app,\n    pullbackShiftFunctorZero_inv_app, assoc, Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app]\n  rfl","hard_negative":true,"metrics":{"chosen_tokens":35,"rejected_tokens":3,"token_jaccard":0.041667,"token_length_ratio":0.085714},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"1f9a6b2a01e16f0b64bd55904d72ea4772ab017d64f8062d26f7c5cc9958ef95","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Shift.Adjunction\npublic import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2023 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The pullback of a shift by a monoid morphism\n\nGiven a shift by a monoid `B` on a category `C` and a monoid morphism `φ : A →+ B`,\nwe define a shift by `A` on a category `PullbackShift C φ` which is a type synonym for `C`.\n\nIf `F : C ⥤ D` is a functor between categories equipped with shifts by `B`, we define\na type synonym `PullbackShift.functor F φ` for `F`. When `F` has a `CommShift` structure\nby `B`, we define a pulled back `CommShift` structure by `A` on `PullbackShift.functor F φ`.\n\nSimilarly, if `τ` is a natural transformation between functors `F,G : C ⥤ D`, we define\na type synonym\n`PullbackShift.natTrans τ φ : PullbackShift.functor F φ ⟶ PullbackShift.functor G φ`.\nWhen `τ` has a `CommShift` structure by `B` (i.e. is compatible with `CommShift` structures\non `F` and `G`), we define a pulled back `CommShift` structure by `A` on\n`PullbackShift.natTrans τ φ`.\n\nFinally, if we have an adjunction `F ⊣ G` (with `G : D ⥤ C`), we define a type synonym\n`PullbackShift.adjunction adj φ : PullbackShift.functor F φ ⊣ PullbackShift.functor G φ`\nand we show that, if `adj` is compatible with `CommShift` structures\non `F` and `G`, then `PullbackShift.adjunction adj φ` is also compatible with the pulled back\n`CommShift` structures.\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Limits Category\n\nvariable (C : Type*) [Category* C] {A B : Type*} [AddMonoid A] [AddMonoid B]\n\n/-- The category `PullbackShift C φ` is equipped with a shift such that for all `a`,\nthe shift functor by `a` is `shiftFunctor C (φ a)`. -/\n@[nolint unusedArguments]\ndef PullbackShift [HasShift C B] (_ : A →+ B) := C\nderiving Category\n\nattribute [local instance] endofunctorMonoidalCategory\n\nvariable [HasShift C B] (φ : A →+ B)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The shift on `PullbackShift C φ` is obtained by precomposing the shift on `C` with\nthe monoidal functor `Discrete.addMonoidalFunctor φ : Discrete A ⥤ Discrete B`. -/\ninstance : HasShift (PullbackShift C φ) A where\n  shift := Discrete.addMonoidalFunctor φ ⋙ shiftMonoidalFunctor C B\n\ninstance [HasZeroObject C] : HasZeroObject (PullbackShift C φ) :=\n  inferInstanceAs <| HasZeroObject C\n\ninstance [Preadditive C] : Preadditive (PullbackShift C φ) :=\n  inferInstanceAs <| Preadditive C\n\ninstance [Preadditive C] (a : A) [(shiftFunctor C (φ a)).Additive] :\n    (shiftFunctor (PullbackShift C φ) a).Additive :=\n  inferInstanceAs (shiftFunctor C (φ a)).Additive\n\n/-- When `b = φ a`, this is the canonical\nisomorphism `shiftFunctor (PullbackShift C φ) a ≅ shiftFunctor C b`. -/\ndef pullbackShiftIso (a : A) (b : B) (h : b = φ a) :\n    shiftFunctor (PullbackShift C φ) a ≅ shiftFunctor C b := eqToIso (by subst h; rfl)\n\nvariable {C}\nvariable (X : PullbackShift C φ) (a₁ a₂ a₃ : A) (h : a₁ + a₂ = a₃) (b₁ b₂ b₃ : B)\n  (h₁ : b₁ = φ a₁) (h₂ : b₂ = φ a₂) (h₃ : b₃ = φ a₃)\n\nset_option backward.defeqAttrib.useBackward true in\nlemma pullbackShiftFunctorZero_inv_app :\n    (shiftFunctorZero _ A).inv.app X =\n      (shiftFunctorZero C B).inv.app X ≫ (pullbackShiftIso C φ 0 0 (by simp)).inv.app X := by\n  change (shiftFunctorZero C B).inv.app X ≫ _ = _\n  dsimp [Discrete.eqToHom, Discrete.addMonoidalFunctor_ε]\n  congr 2\n  apply eqToHom_map\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma pullbackShiftFunctorZero_hom_app :\n    (shiftFunctorZero _ A).hom.app X =\n      (pullbackShiftIso C φ 0 0 (by simp)).hom.app X ≫ (shiftFunctorZero C B).hom.app X :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_59fa1cfe920c","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f3cb0b77a3862415cb35d2403d03bcc5001d152d5ea51afc3f6b83026e1ec580","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Shift","family_id":"pullbackshiftfunctorzero_hom_app","file_id":"mathlib/Mathlib/CategoryTheory/Shift/Pullback.lean","sample_id":"59fa1cfe920c44354539a08a73d5bce71392689b4372837276159b664aba9fb8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7098d4ede92516c6303541ad23377423cb7f62e04005055a6e4b359a31869d3c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2b9acd22df8b3e0b46d29d9433ec7adfa97c3ca8da662c2868b5f6dac8fb61d2","source_sha256":"1c0dcf6a3ce55bec7a88672314ea6ae6741a6599903854de40abb691f935e16b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    intro ⟨h1, h2⟩; use y.drop x.length, h2\n    nth_rw 1 [← List.take_append_drop x.length y]\n    simpa [-List.take_append_drop, List.prefix_iff_eq_take, hl] using h1\n  mpr := by simp +contextual [pullSub]","hard_negative":false,"metrics":{"chosen_tokens":53,"rejected_tokens":5,"token_jaccard":0.027778,"token_length_ratio":0.09434},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"1fe42d65f919c227f84d3279c6a8e793a998ceb09e4dffe3c82b2cc424b86b0a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.CompleteLattice.SetLike\n\nNamespace:\nDescriptive.Tree\n\nLocal context:\n/-\nCopyright (c) 2024 Sven Manthe. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sven Manthe\n-/\n/-!\n# Trees in the sense of descriptive set theory\n\nThis file defines trees of depth `ω` in the sense of descriptive set theory as sets of finite\nsequences that are stable under taking prefixes.\n\n## Main declarations\n\n* `tree A`: a (possibly infinite) tree of depth at most `ω` with nodes in `A`\n-/\n\n@[expose] public section\n\nnamespace Descriptive\n\n/-- A tree is a set of finite sequences, implemented as `List A`, that is stable under\n  taking prefixes. For the definition we use the equivalent property `x ++ [a] ∈ T → x ∈ T`,\n  which is more convenient to check. We define `tree A` as a complete sublattice of\n  `Set (List A)`, which coerces to the type of trees on `A`. -/\ndef tree (A : Type*) : CompleteSublattice (Set (List A)) :=\n  CompleteSublattice.mk' {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T}\n    (by rintro S hS x a ⟨t, ht, hx⟩; use t, ht, hS ht hx)\n    (by rintro S hS x a h T hT; exact hS hT <| h T hT)\n\n@[simps!] instance (A : Type*) : SetLike (tree A) (List A) := SetLike.instSubtypeSet\n\nexample (A : Type*) : PartialOrder (tree A) := inferInstance\n\nnamespace Tree\nvariable {A : Type*} {S T : tree A}\n\nlemma mem_of_append {x y : List A} (h : x ++ y ∈ T) : x ∈ T := by\n  induction y generalizing x with\n  | nil => simpa using h\n  | cons y ys ih => exact T.prop (ih (by simpa))\n\nlemma mem_of_prefix {x y : List A} (h' : x <+: y) (h : y ∈ T) : x ∈ T := by\n  obtain ⟨_, rfl⟩ := h'; exact mem_of_append h\n\ninstance : Trans List.IsPrefix (fun x (T : tree A) ↦ x ∈ T) (fun x T ↦ x ∈ T) where\n  trans := mem_of_prefix\n\nlemma singleton_mem (T : tree A) {a : A} {x : List A} (h : a :: x ∈ T) : [a] ∈ T :=\n  mem_of_prefix ⟨x, rfl⟩ h\n\n@[simp] lemma tree_eq_bot : T = ⊥ ↔ [] ∉ T where\n  mp := by rintro rfl; simp\n  mpr h := by ext x; simpa using fun h' ↦ h <| mem_of_prefix x.nil_prefix h'\n\nlemma take_mem {n : ℕ} (x : T) : x.val.take n ∈ T :=\n  mem_of_prefix (x.val.take_prefix n) x.prop\n\n/-- A variant of `List.take` internally to a tree -/\n@[simps] def take (n : ℕ) (x : T) : T := ⟨x.val.take n, take_mem x⟩\n\n@[simp] lemma take_take (m n : ℕ) (x : T) : take m (take n x) = take (m ⊓ n) x := by\n  simp [Subtype.ext_iff, List.take_take]\n\n@[simp] lemma take_eq_take {x : T} {m n : ℕ} :\n    take m x = take n x ↔ m ⊓ x.val.length = n ⊓ x.val.length := by simp [Subtype.ext_iff]\n\n-- ### `subAt`\n\nvariable (T) (x y : List A)\n\n/-- The residual tree obtained by regarding the node x as new root -/\ndef subAt : tree A :=\n  ⟨(x ++ ·)⁻¹' T, fun _ a _ ↦ mem_of_append (y := [a]) (by rwa [List.append_assoc])⟩\n\n@[simp] lemma mem_subAt : y ∈ subAt T x ↔ x ++ y ∈ T := Iff.rfl\n\n@[simp] lemma subAt_nil : subAt T [] = T := rfl\n\n@[simp] lemma subAt_append : subAt (subAt T x) y = subAt T (x ++ y) := by ext; simp\n\n@[gcongr] lemma subAt_mono (h : S ≤ T) : subAt S x ≤ subAt T x :=\n  Set.preimage_mono h\n\n/-- A variant of `List.drop` that takes values in `subAt` -/\n@[simps] def drop (n : ℕ) (x : T) : subAt T (Tree.take n x).val :=\n  ⟨x.val.drop n, by simp⟩\n\n-- ### `pullSub`\n\n/-- Adjoint of `subAt`, given by pasting x before the root of T. Explicitly,\n  elements are prefixes of x or x with an element of T appended -/\ndef pullSub : tree A where\n  val := { y | y.take x.length <+: x ∧ y.drop x.length ∈ T }\n  property := fun y a ⟨h1, h2⟩ ↦\n    ⟨((y.prefix_append [a]).take x.length).trans h1,\n    mem_of_prefix ((y.prefix_append [a]).drop x.length) h2⟩\n\nvariable {T x y}\n\nlemma mem_pullSub_short (hl : y.length ≤ x.length) : y ∈ pullSub T x ↔ y <+: x ∧ [] ∈ T := by\n  simp [pullSub, List.take_of_length_le hl, List.drop_eq_nil_iff.mpr hl]\n\nTarget:\nlemma mem_pullSub_long (hl : x.length ≤ y.length) : y ∈ pullSub T x ↔ ∃ z ∈ T, y = x ++ z where\n  mp :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Descriptive","family_id":"mem_pullsub_long","file_id":"mathlib/Mathlib/SetTheory/Descriptive/Tree.lean","sample_id":"2b9acd22df8b3e0b46d29d9433ec7adfa97c3ca8da662c2868b5f6dac8fb61d2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9d69af709cb1a99e4edeafcd19d4f6a19e8a85efb124f30d847acee7dc29b2e3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b0369a5a54c56e81b58aafefa0901fd2dbc75fdc809ee1103b64d4a18103c12d","source_sha256":"47c01594d211b3c54399310f0cb3b9f706ed5bf9e27bc1387c290b2a49698c96","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  suffices ∀ x : FreeMonoid α, (xs.map FreeMonoid.of).foldl (· * ·) x = x * ofList xs by\n    simpa [← List.foldl_map, lift_mk_eq_foldl, toFreeMonoid, lift] using this (FreeMonoid.of x)\n  induction xs with grind [ofList_nil, ofList_cons]","hard_negative":false,"metrics":{"chosen_tokens":60,"rejected_tokens":2,"token_jaccard":0.027778,"token_length_ratio":0.033333},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"20375be39ef088f14861660f5ae3c3b62767d1c461e58188c9048f680527ffed","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.FreeMonoid.Basic\npublic import Mathlib.Algebra.Free\npublic import Mathlib.Algebra.Group.WithOne.Basic\npublic import Mathlib.Algebra.Group.Units.Basic\npublic import Mathlib.Data.Set.Operations\nimport Mathlib.Data.Set.Insert\n\nNamespace:\nFreeSemigroup\n\nLocal context:\n/-\nCopyright (c) 2026 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\n/-!\n# Relation between the free semigroup and the free monoid\n\nWe provide some constructions relating the free semigroup and the free monoid on the same type.\n\n## Main definitions\n* `FreeSemigroup.toFreeMonoid`: the natural embedding of the free semigroup into the free monoid.\n* `FreeMonoid.equivWithOneFreeSemigroup`: the free monoid is isomorphic to the free semigroup\n  with a `1` added.\n-/\n\npublic section\n\nvariable {α : Type*}\n\nnamespace FreeSemigroup\n\nopen FreeMonoid\n\n/--\nThe natural embedding of the free semigroup into the free monoid.\nThis is injective (`FreeSemigroup.toFreeMonoid_injective`), and its image\nconsists of all non-`1` elements of the free monoid (`FreeSemigroup.eq_one_or_toFreeMonoid`).\n-/\n@[expose, to_additive /-- The natural embedding of the free additive semigroup into the\nfree additive monoid. This is injective (`FreeAddSemigroup.toFreeAddMonoid_injective`), and its\nimage consists of all non-`0` elements of the free additive monoid\n(`FreeAddSemigroup.eq_zero_or_toFreeAddMonoid`). -/]\ndef toFreeMonoid : FreeSemigroup α →ₙ* FreeMonoid α :=\n  lift FreeMonoid.of\n\n@[to_additive (attr := simp, grind =)]\nlemma toFreeMonoid_of (x : α) : toFreeMonoid (.of x) = .of x := rfl\n\n@[to_additive]\n\nTarget:\nlemma toFreeMonoid_mk_eq_cons (x : α) (xs : List α) :\n    toFreeMonoid ⟨x, xs⟩ = FreeMonoid.ofList (x :: xs) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/FreeMonoid","family_id":"tofreemonoid_mk_eq_cons","file_id":"mathlib/Mathlib/Algebra/FreeMonoid/FreeSemigroup.lean","sample_id":"b0369a5a54c56e81b58aafefa0901fd2dbc75fdc809ee1103b64d4a18103c12d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ab45fc24fe8d81ce6bc79891c129b27df70dce3fefc1109e3aecbeb52c81794e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0ccbfff2e32bc19e9550d2514637413d97b726466dde45498d6c8f3060c8d074","source_sha256":"18fda008e7b40ec7b81fa0fd9f7c55475c44e029c9af90e5639e54fa98674982","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext; simp [transvection.apply]","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":5,"token_jaccard":0.076923,"token_length_ratio":0.555556},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"2043a201ba3d1f11258029b2d9f795706ce9c63283af85f00f7795c6c0b1babb","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup\npublic import Mathlib.LinearAlgebra.Charpoly.BaseChange\npublic import Mathlib.LinearAlgebra.Dual.BaseChange\npublic import Mathlib.LinearAlgebra.Dual.Lemmas\npublic import Mathlib.LinearAlgebra.FixedSubmodule\n\nNamespace:\nLinearMap.transvection\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir\n-/\n\n/-!\n# Transvections in a module\n\n* When `f : Module.Dual R V` and `v : V`,\n  `LinearMap.transvection f v` is the linear map given by `x ↦ x + f x • v`,\n\n* `LinearMap.transvection.det` shows that the determinant of\n  `LinearMap.transvection f v` is equal to `1 + f v`.\n\n* If, moreover, `f v = 0`, then `LinearEquiv.transvection` shows that it is\n  a linear equivalence.\n\n* `LinearMap.transvections R V`: the set of transvections.\n\n* `LinearEquiv.dilatransvections R V`: the set of linear equivalences\n  whose associated linear map is of the form `LinearMap.transvection f v`.\n\n* `LinearEquiv.transvection.det` shows that it has determinant `1`.\n\n## Note on terminology\n\nIn the mathematical literature, linear maps of the form `LinearMap.transvection f v`\nare only called “transvections” when `f v = 0`. Otherwise, they are sometimes\ncalled “dilations” (especially if `f v ≠ -1`).\n\nThe definition is almost the same as that of `Module.preReflection f v`,\nup to a sign change, which are interesting when `f v = 2`, because they give “reflections”.\n\n-/\n\n@[expose] public section\n\nnamespace LinearMap\n\nopen Module\n\nvariable {R V : Type*} [Semiring R] [AddCommMonoid V] [Module R V]\n\n/-- The transvection associated with a linear form `f` and a vector `v`.\n\nNB. In mathematics, these linear maps are only called “transvections” when `f v = 0`.\nSee also `Module.preReflection` for a similar definition, up to a sign. -/\ndef transvection (f : Dual R V) (v : V) : V →ₗ[R] V where\n  toFun x := x + f x • v\n  map_add' x y := by simp [add_add_add_comm, add_smul]\n  map_smul' r x := by simp [smul_eq_mul, smul_add, mul_smul]\n\nnamespace transvection\n\nopen Submodule LinearMap\n\ntheorem apply (f : Dual R V) (v x : V) :\n    transvection f v x = x + f x • v :=\n  rfl\n\ntheorem comp_of_left_eq_apply {f : Dual R V} {v w : V} {x : V} (hw : f w = 0) :\n    transvection f v (transvection f w x) = transvection f (v + w) x := by\n  simp [transvection, map_add, hw, add_assoc]\n\ntheorem comp_of_left_eq {f : Dual R V} {v w : V} (hw : f w = 0) :\n    (transvection f v) ∘ₗ (transvection f w) = transvection f (v + w) := by\n  ext; simp [comp_of_left_eq_apply hw]\n\ntheorem comp_of_right_eq_apply {f g : Dual R V} {v : V} {x : V} (hf : f v = 0) :\n    (transvection f v) (transvection g v x) = transvection (f + g) v x := by\n  simp [transvection, map_add, hf, add_smul, add_assoc]\n\ntheorem comp_of_right_eq {f g : Dual R V} {v : V} (hf : f v = 0) :\n    (transvection f v) ∘ₗ (transvection g v) = transvection (f + g) v := by\n  ext; simp [comp_of_right_eq_apply hf]\n\n@[simp]\ntheorem of_left_eq_zero (v : V) :\n    transvection (0 : Dual R V) v = id := by\n  ext\n  simp [transvection]\n\n@[simp]\ntheorem of_right_eq_zero (f : Dual R V) :\n    transvection f 0 = id := by\n  ext\n  simp [transvection]\n\ntheorem comp_smul_smul {f : Dual R V} {v : V} {r s : R} :\n    transvection f (r • v) ∘ₗ transvection f (s • v) =\n      transvection f ((r + s + s * f v * r) • v) := by\n  ext x\n  simp only [LinearMap.comp_apply, apply, map_add, map_smul, add_assoc]\n  simp only [smul_add, ← mul_smul, ← add_smul, ← mul_add (f x), mul_assoc]\n\ntheorem eq_id_of_finrank_le_one\n    {R V : Type*} [CommSemiring R] [AddCommMonoid V] [Module R V]\n    [Free R V] [Module.Finite R V] [StrongRankCondition R]\n    {f : Dual R V} {v : V} (hfv : f v = 0) (h1 : finrank R V ≤ 1) :\n    transvection f v = id := by\n  interval_cases h : finrank R V\n  · have : Subsingleton V := (finrank_eq_zero_iff_of_free R V).mp h\n    simp [Subsingleton.eq_zero v]\n  · let b := finBasis R V\n    ext x\n    suffices f x • v = 0 by\n      simp [apply, this]\n    let i : Fin (finrank R V) := ⟨0, by simp [h]⟩\n    suffices ∀ x, x = b.repr x i • (b i) by\n      rw [this v, map_smul, smul_eq_mul, mul_comm] at hfv\n      rw [this x, this v, map_smul, smul_eq_mul, ← mul_smul, mul_assoc, hfv, mul_zero, zero_smul]\n    intro x\n    have : x = ∑ i, b.repr x i • b i := (b.sum_equivFun x).symm\n    rwa [Finset.sum_eq_single_of_mem i (Finset.mem_univ i) (by grind)] at this\n\nTarget:\ntheorem congr {W : Type*} [AddCommMonoid W] [Module R W]\n    (f : Dual R V) (v : V) (e : V ≃ₗ[R] W) :\n    e ∘ₗ (transvection f v) ∘ₗ e.symm = transvection (f ∘ₗ e.symm) (e v) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Transvection","family_id":"congr","file_id":"mathlib/Mathlib/LinearAlgebra/Transvection/Basic.lean","sample_id":"0ccbfff2e32bc19e9550d2514637413d97b726466dde45498d6c8f3060c8d074"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a24cc2dfc6b516924da62f552de8528961f8853b68b3464a392b74c6594e45ef","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"33c5a04717df2d635bf7df2da69f293f07603674ff1d481440ce1211cda1adae","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2f3292c8af1fa98f0dfb39f23e5916bd9e2de4337e8d92b7857cea3205fe336c","source_sha256":"6a1fd2aa47744142955f801b29fb06cc46e0d439ea80c9cccba97a19a60fbbe9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x\n  apply (IsIntegralClosure.equiv A (integralClosure A L₃) L₃ B₃).symm.injective\n  ext\n  simp [galRestrict', Subalgebra.algebraMap_eq]","hard_negative":true,"metrics":{"chosen_tokens":33,"rejected_tokens":5,"token_jaccard":0.076923,"token_length_ratio":0.151515},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"2058139325b159e410505b33a24c651c8d2222d2a7f153f2d9afde493a2c6027","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.IntegralClosure\npublic import Mathlib.RingTheory.RingHom.Finite\npublic import Mathlib.RingTheory.Localization.LocalizationLocalization\npublic import Mathlib.RingTheory.Localization.NormTrace\npublic import Mathlib.RingTheory.Norm.Transitivity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Restriction of various maps between fields to integrally closed subrings.\n\nIn this file, we assume `A` is an integrally closed domain; `K` is the fraction ring of `A`;\n`L` is a finite extension of `K`; `B` is the integral closure of `A` in `L`.\nWe call this the AKLB setup.\n\n## Main definitions\n- `galRestrict`: The restriction `Aut(L/K) → Aut(B/A)` as an `MulEquiv` in an AKLB setup.\n- `Algebra.intTrace`: The trace map of a finite extension of integrally closed domains `B/A` is\n  defined to be the restriction of the trace map of `Frac(B)/Frac(A)`.\n- `Algebra.intNorm`: The norm map of a finite extension of integrally closed domains `B/A` is\n  defined to be the restriction of the norm map of `Frac(B)/Frac(A)`.\n\n-/\n\n@[expose] public section\n\nopen Module nonZeroDivisors\n\nvariable (A K L L₂ L₃ B B₂ B₃ : Type*)\nvariable [CommRing A] [CommRing B] [CommRing B₂] [CommRing B₃]\nvariable [Algebra A B] [Algebra A B₂] [Algebra A B₃]\nvariable [Field K] [Field L] [Field L₂] [Field L₃]\nvariable [Algebra A K] [IsFractionRing A K]\nvariable [Algebra K L] [Algebra A L] [IsScalarTower A K L]\nvariable [Algebra K L₂] [Algebra A L₂] [IsScalarTower A K L₂]\nvariable [Algebra K L₃] [Algebra A L₃] [IsScalarTower A K L₃]\nvariable [Algebra B L] [IsScalarTower A B L] [IsIntegralClosure B A L]\nvariable [Algebra B₂ L₂] [IsScalarTower A B₂ L₂] [IsIntegralClosure B₂ A L₂]\nvariable [Algebra B₃ L₃] [IsScalarTower A B₃ L₃] [IsIntegralClosure B₃ A L₃]\n\nsection galois\n\nsection galRestrict'\nvariable {K L L₂ L₃}\nomit [IsFractionRing A K]\n\n/-- A generalization of `galRestrictHom` beyond endomorphisms. -/\nnoncomputable\ndef galRestrict' (f : L →ₐ[K] L₂) : (B →ₐ[A] B₂) :=\n  (IsIntegralClosure.equiv A (integralClosure A L₂) L₂ B₂).toAlgHom.comp\n      (((f.restrictScalars A).comp (IsScalarTower.toAlgHom A B L)).codRestrict\n        (integralClosure A L₂) (fun x ↦ IsIntegral.map _ (IsIntegralClosure.isIntegral A L x)))\n\n@[simp]\nlemma algebraMap_galRestrict'_apply (σ : L →ₐ[K] L₂) (x : B) :\n    algebraMap B₂ L₂ (galRestrict' A B B₂ σ x) = σ (algebraMap B L x) := by\n  simp [galRestrict', galRestrict', Subalgebra.algebraMap_eq]\n\n@[simp]\ntheorem galRestrict'_id : galRestrict' A B B (.id K L) = .id A B := by\n  ext\n  apply IsIntegralClosure.algebraMap_injective B A L\n  simp\n\nTarget:\ntheorem galRestrict'_comp (σ : L →ₐ[K] L₂) (σ' : L₂ →ₐ[K] L₃) :\n    galRestrict' A B B₃ (σ'.comp σ) = (galRestrict' A B₂ B₃ σ').comp (galRestrict' A B B₂ σ) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_2f3292c8af1f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"06db84e7e262dcb197086ffa15bf778fe4416ab799bf00209254707991705d98","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/IntegralClosure","family_id":"galrestrict'_comp","file_id":"mathlib/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean","sample_id":"2f3292c8af1fa98f0dfb39f23e5916bd9e2de4337e8d92b7857cea3205fe336c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2d27cea8f3ca8caf10e9d8775b5b6885d300093e2302453f512d2957b166201a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4e9ba5114733e3b96be27c5b4d01de8367929d2bf280e4296d00c4e5e9fb55bb","source_sha256":"73ebc2e72f7678b20eb20bcb8222b112e9c86c209dc6aa3492fcf8a265aa9fa7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let : Field (A ⧸ p) := Ideal.Quotient.field p\n  rw [Ideal.isRadical_iff_quotient_reduced]\n  exact Algebra.FormallyUnramified.isReduced_of_field (A ⧸ p) (B ⧸ p.map (algebraMap A B))","hard_negative":false,"metrics":{"chosen_tokens":45,"rejected_tokens":5,"token_jaccard":0.071429,"token_length_ratio":0.111111},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"206751964446c8c6e0b9dbc6e1fa0ecb1f3304df04fa5d0296f19d8754ed6347","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.PurelyInseparable.Basic\npublic import Mathlib.RingTheory.Artinian.Ring\npublic import Mathlib.RingTheory.Unramified.Finite\npublic import Mathlib.RingTheory.Unramified.Locus\n\nNamespace:\nAlgebra.FormallyUnramified\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Unramified algebras over fields\n\n## Main results\n\nLet `K` be a field, `A` be a `K`-algebra and `L` be a field extension of `K`.\n\n- `Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_isLocalRing`:\n    If `A` is `K`-unramified and `K` is alg-closed, then `K = A`.\n- `Algebra.FormallyUnramified.isReduced_of_field`:\n    If `A` is `K`-unramified then `A` is reduced.\n- `Algebra.FormallyUnramified.iff_isSeparable`:\n    `L` is unramified over `K` iff `L` is separable over `K`.\n\n## References\n\n- [B. Iversen, *Generic Local Structure of the Morphisms in Commutative Algebra*][iversen]\n\n-/\n\npublic section\n\nopen Algebra Module Polynomial\nopen scoped TensorProduct\n\nuniverse u\n\nvariable (K A L : Type*) [Field K] [Field L] [CommRing A] [Algebra K A] [Algebra K L]\n\nnamespace Algebra.FormallyUnramified\n\ntheorem of_isSeparable [Algebra.IsSeparable K L] : FormallyUnramified K L := by\n  rw [iff_comp_injective]\n  intro B _ _ I hI f₁ f₂ e\n  ext x\n  have : f₁ x - f₂ x ∈ I := by\n    simpa [Ideal.Quotient.mk_eq_mk_iff_sub_mem] using AlgHom.congr_fun e x\n  have := Polynomial.eval_add_of_sq_eq_zero ((minpoly K x).map (algebraMap K B)) (f₂ x)\n    (f₁ x - f₂ x) (show (f₁ x - f₂ x) ^ 2 ∈ ⊥ from hI ▸ Ideal.pow_mem_pow this 2)\n  simp only [add_sub_cancel, eval_map_algebraMap, aeval_algHom_apply, minpoly.aeval, map_zero,\n    derivative_map, zero_add] at this\n  rwa [eq_comm, ((isUnit_iff_ne_zero.mpr\n    ((Algebra.IsSeparable.isSeparable K x).aeval_derivative_ne_zero\n      (minpoly.aeval K x))).map f₂).mul_right_eq_zero, sub_eq_zero] at this\n\nvariable [FormallyUnramified K A] [EssFiniteType K A]\nvariable [FormallyUnramified K L] [EssFiniteType K L]\n\ntheorem bijective_of_isAlgClosed_of_isLocalRing\n    [IsAlgClosed K] [IsLocalRing A] :\n    Function.Bijective (algebraMap K A) := by\n  have := finite_of_free (R := K) (S := A)\n  have : IsArtinianRing A := isArtinian_of_tower K inferInstance\n  have hA : IsNilpotent (IsLocalRing.maximalIdeal A) := by\n    rw [← IsLocalRing.jacobson_eq_maximalIdeal ⊥]\n    · exact IsArtinianRing.isNilpotent_jacobson_bot\n    · exact bot_ne_top\n  let e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A := {\n    __ := Algebra.ofId K (A ⧸ IsLocalRing.maximalIdeal A)\n    __ := Equiv.ofBijective _ IsAlgClosed.algebraMap_bijective_of_isIntegral }\n  let e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=\n    (Algebra.TensorProduct.congr AlgEquiv.refl e.symm).trans (Algebra.TensorProduct.rid K A A)\n  let f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap.comp (sec K A _)\n  have hf : (Algebra.ofId _ _).toLinearMap ∘ₗ f = LinearMap.id := by\n    dsimp [f]\n    rw [← LinearMap.comp_assoc, ← comp_sec K A]\n    congr 1\n    apply LinearMap.restrictScalars_injective K\n    apply _root_.TensorProduct.ext'\n    intro r s\n    obtain ⟨s, rfl⟩ := e.surjective s\n    suffices s • (Ideal.Quotient.mk (IsLocalRing.maximalIdeal A)) r = r • e s by\n      simpa [ofId, e']\n    simp [Algebra.smul_def, e, ofId, mul_comm]\n  have hf₁ : f 1 • (1 : A ⧸ IsLocalRing.maximalIdeal A) = 1 := by\n    rw [← algebraMap_eq_smul_one]\n    exact LinearMap.congr_fun hf 1\n  have hf₂ : 1 - f 1 ∈ IsLocalRing.maximalIdeal A := by\n    rw [← Ideal.Quotient.eq_zero_iff_mem, map_sub, map_one, ← Ideal.Quotient.algebraMap_eq,\n     algebraMap_eq_smul_one, hf₁, sub_self]\n  have hf₃ : IsIdempotentElem (1 - f 1) := by\n    apply IsIdempotentElem.one_sub\n    rw [IsIdempotentElem, ← smul_eq_mul, ← map_smul, hf₁]\n  have hf₄ : f 1 = 1 := by\n    obtain ⟨n, hn⟩ := hA\n    have : (1 - f 1) ^ n = 0 := by\n      rw [← Ideal.mem_bot, ← Ideal.zero_eq_bot, ← hn]\n      exact Ideal.pow_mem_pow hf₂ n\n    rw [eq_comm, ← sub_eq_zero, ← hf₃.pow_succ_eq n, pow_succ, this, zero_mul]\n  refine Equiv.bijective ⟨algebraMap K A, ⇑e.symm ∘ ⇑(algebraMap A _), fun x ↦ by simp, fun x ↦ ?_⟩\n  have : ⇑(algebraMap K A) = ⇑f ∘ ⇑e := by\n    ext k\n    conv_rhs => rw [← mul_one k, ← smul_eq_mul, Function.comp_apply, map_smul,\n      LinearMap.map_smul_of_tower, map_one, hf₄, ← algebraMap_eq_smul_one]\n  rw [this]\n  simp only [Function.comp_apply, AlgEquiv.apply_symm_apply, algebraMap_eq_smul_one,\n    map_smul, hf₄, smul_eq_mul, mul_one]\n\ntheorem isField_of_isAlgClosed_of_isLocalRing\n    [IsAlgClosed K] [IsLocalRing A] : IsField A := by\n  rw [IsLocalRing.isField_iff_maximalIdeal_eq, eq_bot_iff]\n  intro x hx\n  obtain ⟨x, rfl⟩ := (bijective_of_isAlgClosed_of_isLocalRing K A).surjective x\n  change _ = 0\n  rw [← (algebraMap K A).map_zero]\n  by_contra hx'\n  exact hx ((isUnit_iff_ne_zero.mpr\n    (fun e ↦ hx' ((algebraMap K A).congr_arg e))).map (algebraMap K A))\n\ninclude K in\ntheorem isReduced_of_field :\n    IsReduced A := by\n  constructor\n  intro x hx\n  let f := (Algebra.TensorProduct.includeRight (R := K) (A := AlgebraicClosure K) (B := A))\n  have : Function.Injective f := by\n    have : ⇑f = (LinearMap.rTensor A (Algebra.ofId K (AlgebraicClosure K)).toLinearMap).comp\n        (Algebra.TensorProduct.lid K A).symm.toLinearMap := by\n      ext x; simp [f]\n    rw [this]\n    suffices Function.Injective\n        (LinearMap.rTensor A (Algebra.ofId K (AlgebraicClosure K)).toLinearMap) by\n      exact this.comp (Algebra.TensorProduct.lid K A).symm.injective\n    apply Module.Flat.rTensor_preserves_injective_linearMap\n    exact (algebraMap K _).injective\n  apply this\n  rw [map_zero]\n  apply eq_zero_of_localization\n  intro M hM\n  have hy := (hx.map f).map (algebraMap _ (Localization.AtPrime M))\n  generalize algebraMap _ (Localization.AtPrime M) (f x) = y at *\n  have := EssFiniteType.of_isLocalization (Localization.AtPrime M) M.primeCompl\n  have := of_isLocalization (Rₘ := Localization.AtPrime M) M.primeCompl\n  have := EssFiniteType.comp (AlgebraicClosure K) (AlgebraicClosure K ⊗[K] A)\n    (Localization.AtPrime M)\n  have := comp (AlgebraicClosure K) (AlgebraicClosure K ⊗[K] A)\n    (Localization.AtPrime M)\n  letI := (isField_of_isAlgClosed_of_isLocalRing (AlgebraicClosure K)\n    (A := Localization.AtPrime M)).toField\n  exact hy.eq_zero\n\nTarget:\ntheorem isRadical_map_isMaximal (B : Type*) [CommRing B] [Algebra A B]\n    [Algebra.EssFiniteType A B] [Algebra.FormallyUnramified A B] (p : Ideal A) [p.IsMaximal] :\n    (p.map (algebraMap A B)).IsRadical :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Unramified","family_id":"isradical_map_ismaximal","file_id":"mathlib/Mathlib/RingTheory/Unramified/Field.lean","sample_id":"4e9ba5114733e3b96be27c5b4d01de8367929d2bf280e4296d00c4e5e9fb55bb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ba5868d82bf97e5ff1d70a97d543dc36082af3a6a7d0e0038bdd9326c14ca5d6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c5320067570b9bfc10d7ea602085cbc08d55ba3d758c561b8444485698919964","source_sha256":"05b71bb08d5e0d5da2d7d8beb37172741e5772665eaf54d30c9d72516996fb96","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [CompleteLattice.isCompactElement_iff_exists_le_iSup_of_le_iSup,\n    Finset.sup_eq_iSup] at h ⊢\n  intro ι s hb\n  replace hb : (b : α) ≤ iSup ((↑) ∘ s) := le_trans hb <| (coe_iSup s) ▸ le_refl _\n  obtain ⟨t, ht⟩ := h ι ((↑) ∘ s) hb\n  exact ⟨t, (by simpa using ht : (b : α) ≤ _)⟩","hard_negative":false,"metrics":{"chosen_tokens":84,"rejected_tokens":5,"token_jaccard":0.065217,"token_length_ratio":0.059524},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"2099a0bcb20f7c4581b35ce061769025425b83b33f4b376cfcc46ae6810157d8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.CompleteLatticeIntervals\npublic import Mathlib.Order.CompactlyGenerated.Basic\n\nNamespace:\nSet.Iic\n\nLocal context:\n/-\nCopyright (c) 2024 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Results about compactness properties for intervals in complete lattices\n-/\n\npublic section\n\nvariable {ι α : Type*} [CompleteLattice α]\n\nnamespace Set.Iic\n\nTarget:\ntheorem isCompactElement {a : α} {b : Iic a} (h : IsCompactElement (b : α)) :\n    IsCompactElement b :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/CompactlyGenerated","family_id":"iscompactelement","file_id":"mathlib/Mathlib/Order/CompactlyGenerated/Intervals.lean","sample_id":"c5320067570b9bfc10d7ea602085cbc08d55ba3d758c561b8444485698919964"},"split":"train"}
{"binding":{"chosen_proof_sha256":"32e319c4110fd1b6339d57505a0712e65f79dbff575ad63d342ca445e0185468","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"828fcf5f665efc678113e458ba63e19ba96a40a56bbfc3f5735582f797454ccb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5f583e779a0f586168eb801c8f41ed46b5920677578a5ac3171bab04bd27faca","source_sha256":"b76902cf8d4ceba8892c478a05df06c4c7f74a8ad6562beb851b28c9078eec8d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [← HomologicalComplex.comp_f]\n\n/-- The homotopy equivalence from `X.chainComplex R` to `X.normalizedChainComplex R`. -/\nnoncomputable def homotopyEquivNormalizedChainComplex :\n    HomotopyEquiv (X.chainComplex R) (X.normalizedChainComplex R) :=\n  SimplicialObject.Splitting.homotopyEquivNondegComplex _","hard_negative":true,"metrics":{"chosen_tokens":55,"rejected_tokens":5,"token_jaccard":0.057143,"token_length_ratio":0.090909},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"20a91c87f68c4d8fc3929aba655bcd39d928258f55b95422c2e4fe0be42f4460","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.QuasiIso\npublic import Mathlib.AlgebraicTopology.SimplicialSet.Homology.Basic\npublic import Mathlib.AlgebraicTopology.SimplicialSet.Splitting\npublic import Mathlib.AlgebraicTopology.SimplicialSet.Dimension\npublic import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject\npublic import Mathlib.CategoryTheory.Limits.Preserves.SigmaConst\n\nNamespace:\nSSet\n\nLocal context:\n/-\nCopyright (c) 2026 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Computing homology using nondegenerate simplices\n\nIn this file, we introduce the normalized chain complex `X.normalizedChainComplex R`\nof a simplicial set `X` with coefficients in `R` (where `R` is an object of a\npreadditive category `C` with coproducts). The `n`-chains of this complex\nidentify to the coproduct of copies of `R` indexed by the nondegenerate\n`n`-simplices of `X`. In particular, we deduce that the homology is zero in degree `≥ d`\nwhen `X` has dimension `< d`.\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nopen CategoryTheory Limits HomologicalComplex Simplicial\n  AlgebraicTopology.DoldKan\n\nnamespace SSet\n\nvariable {C : Type u} [Category.{v} C] [HasCoproducts.{w} C] [Preadditive C]\n  (X Y : SSet.{w}) (f : X ⟶ Y) (R : C)\n\n/-- The normalized chain complex of a simplicial set `X` with coefficients in `R`.\nIn degree `n`, it consists of a coproduct of copies of `R` indexed by the\nnondegenerate `n`-simplices of `X`. -/\nnoncomputable def normalizedChainComplex : ChainComplex C ℕ :=\n  (X.splitting.map (sigmaConst.obj R)).nondegComplex\n\n/-- The split epi `X.chainComplex R ⟶ X.normalizedChainComplex R`. -/\nnoncomputable def toNormalizedChainComplex : X.chainComplex R ⟶ X.normalizedChainComplex R :=\n  (X.splitting.map (sigmaConst.obj R)).toNondegComplex\n\n/-- The split mono `X.normalizedChainComplex R ⟶ X.chainComplex R`. -/\nnoncomputable def fromNormalizedChainComplex : X.normalizedChainComplex R ⟶ X.chainComplex R :=\n  (X.splitting.map (sigmaConst.obj R)).fromNondegComplex\n\n@[reassoc (attr := simp)]\nlemma PInfty_toNormalizedChainComplex :\n    PInfty ≫ X.toNormalizedChainComplex R = X.toNormalizedChainComplex R :=\n  SimplicialObject.Splitting.PInfty_toNondegComplex _\n\ninstance : IsSplitEpi (X.toNormalizedChainComplex R) :=\n  SimplicialObject.Splitting.isSplitEpi_toNondegComplex _\n\ninstance : IsSplitMono (X.fromNormalizedChainComplex R) :=\n  SimplicialObject.Splitting.isSplitMono_fromNondegComplex _\n\n@[reassoc (attr := simp)]\nlemma fromNormalizedChainComplex_toNormalizedChainComplex :\n    X.fromNormalizedChainComplex R ≫ X.toNormalizedChainComplex R = 𝟙 _ :=\n  SimplicialObject.Splitting.fromNondegComplex_toNondegComplex _\n\n@[reassoc (attr := simp)]\nlemma fromNormalizedChainComplex_f_toNormalizedChainComplex_f (n : ℕ) :\n    (X.fromNormalizedChainComplex R).f n ≫ (X.toNormalizedChainComplex R).f n = 𝟙 _ := by\n  simp [← HomologicalComplex.comp_f]\n\n@[reassoc (attr := simp)]\nlemma toNormalizedChainComplex_fromNormalizedChainComplex :\n    X.toNormalizedChainComplex R ≫ X.fromNormalizedChainComplex R = PInfty :=\n  SimplicialObject.Splitting.toNondegComplex_fromNondegComplex _\n\n@[reassoc (attr := simp)]\n\nTarget:\nlemma toNormalizedChainComplex_f_fromNormalizedChainComplex_f (n : ℕ) :\n    (X.toNormalizedChainComplex R).f n ≫ (X.fromNormalizedChainComplex R).f n = PInfty.f n :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_5f583e779a0f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"e5a05192a5d6bf57cb49b99f7154208699208d23a0b9f2f05dcc74314645821f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplicialSet","family_id":"tonormalizedchaincomplex_f_fromnormalizedchaincomplex_f","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplicialSet/Homology/Nondegenerate.lean","sample_id":"5f583e779a0f586168eb801c8f41ed46b5920677578a5ac3171bab04bd27faca"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d39e8a584cb9cadcbcf3f438cba249cdb73b9b66b3d2c3c40bd6185edf5ea6c9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"af7cda194f325da0ba382a80db632aee5a45876e1050994aa06518ce0c53a98d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"75d34468da2034a80fca64c4e5869a12c798eeacfe03a8618a187bb63ae913dc","source_sha256":"2f00d04aba5c305cbf490ef92448e4a2acde5744e8e9969803e8bbd6f318fa29","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h : f = 𝟙 j := by cat_disch\n  rw [h]\n  simp","hard_negative":true,"metrics":{"chosen_tokens":16,"rejected_tokens":3,"token_jaccard":0.0625,"token_length_ratio":0.1875},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"20e11165a19c0a6466b2cc097857d16b2a8587d4d5e3f4ea88b300d2325e2b2f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Pi.Basic\n\nNamespace:\nCategoryTheory.Discrete\n\nLocal context:\n/-\nCopyright (c) 2017 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Kim Morrison, Floris van Doorn\n-/\n/-!\n# Discrete categories\n\nWe define `Discrete α` as a structure containing a term `a : α` for any type `α`,\nand use this type alias to provide a `SmallCategory` instance\nwhose only morphisms are the identities.\n\nThere is an annoying technical difficulty that it has turned out to be inconvenient\nto allow categories with morphisms living in `Prop`,\nso instead of defining `X ⟶ Y` in `Discrete α` as `X = Y`,\none might define it as `PLift (X = Y)`.\nIn fact, to allow `Discrete α` to be a `SmallCategory`\n(i.e. with morphisms in the same universe as the objects),\nwe actually define the hom type `X ⟶ Y` as `ULift (PLift (X = Y))`.\n\n`Discrete.functor` promotes a function `f : I → C` (for any category `C`) to a functor\n`Discrete.functor f : Discrete I ⥤ C`.\n\nSimilarly, `Discrete.natTrans` and `Discrete.natIso` promote `I`-indexed families of morphisms,\nor `I`-indexed families of isomorphisms to natural transformations or natural isomorphism.\n\nWe show equivalences of types are the same as (categorical) equivalences of the corresponding\ndiscrete categories.\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\n-- morphism levels before object levels. See note [category theory universes].\nuniverse v₁ v₂ v₃ u₁ u₁' u₂ u₃\n\n-- This is intentionally a structure rather than a type synonym\n-- to enforce using `DiscreteEquiv` (or `Discrete.mk` and `Discrete.as`) to move between\n-- `Discrete α` and `α`. Otherwise there is too much API leakage.\n/-- A wrapper for promoting any type to a category,\nwith the only morphisms being equalities.\n-/\n@[ext, aesop safe cases (rule_sets := [CategoryTheory])]\nstructure Discrete (α : Type u₁) where\n  /-- A wrapper for promoting any type to a category,\n  with the only morphisms being equalities. -/\n  as : α\n\n@[simp]\ntheorem Discrete.mk_as {α : Type u₁} (X : Discrete α) : Discrete.mk X.as = X :=\n  rfl\n\n/-- `Discrete α` is equivalent to the original type `α`. -/\n@[simps]\ndef discreteEquiv {α : Type u₁} : Discrete α ≃ α where\n  toFun := Discrete.as\n  invFun := Discrete.mk\n  left_inv := by cat_disch\n  right_inv := by cat_disch\n\ninstance {α : Type u₁} [DecidableEq α] : DecidableEq (Discrete α) :=\n  discreteEquiv.decidableEq\n\n/-- The \"Discrete\" category on a type, whose morphisms are equalities.\n\nBecause we do not allow morphisms in `Prop` (only in `Type`),\nsomewhat annoyingly we have to define `X ⟶ Y` as `ULift (PLift (X = Y))`. -/\n@[stacks 001A]\ninstance discreteCategory (α : Type u₁) : SmallCategory (Discrete α) where\n  Hom X Y := ULift (PLift (X.as = Y.as))\n  id _ := ULift.up (PLift.up rfl)\n  comp {X Y Z} g f := by\n    cases X\n    cases Y\n    cases Z\n    rcases f with ⟨⟨⟨⟩⟩⟩\n    exact g\n\nnamespace Discrete\n\nvariable {α : Type u₁}\n\ninstance [Inhabited α] : Inhabited (Discrete α) :=\n  ⟨⟨default⟩⟩\n\ninstance [Subsingleton α] : Subsingleton (Discrete α) :=\n  ⟨by cat_disch⟩\n\ninstance instSubsingletonDiscreteHom (X Y : Discrete α) : Subsingleton (X ⟶ Y) :=\n  show Subsingleton (ULift (PLift _)) from inferInstance\n\n/-- A simple tactic to run `cases` on any `Discrete α` hypotheses. -/\nmacro \"discrete_cases\" : tactic =>\n  `(tactic| fail_if_no_progress casesm* Discrete _, (_ : Discrete _) ⟶ (_ : Discrete _), PLift _)\n\nopen Lean Elab Tactic in\n/--\nUse:\n```\nattribute [local aesop safe tactic (rule_sets := [CategoryTheory])]\n  CategoryTheory.Discrete.discreteCases\n```\nto locally give `cat_disch` the ability to call `cases` on\n`Discrete` and `(_ : Discrete _) ⟶ (_ : Discrete _)` hypotheses.\n-/\nmeta def discreteCases : TacticM Unit := do\n  evalTactic (← `(tactic| discrete_cases))\n\n-- TODO: investigate turning on either\n-- `attribute [aesop safe cases (rule_sets := [CategoryTheory])] Discrete`\n-- or\n-- `attribute [aesop safe tactic (rule_sets := [CategoryTheory])] discreteCases`\n-- globally.\n\ninstance [Unique α] : Unique (Discrete α) :=\n  Unique.mk' (Discrete α)\n\n/-- Extract the equation from a morphism in a discrete category. -/\ntheorem eq_of_hom {X Y : Discrete α} (i : X ⟶ Y) : X.as = Y.as :=\n  i.down.down\n\n/-- Promote an equation between the wrapped terms in `X Y : Discrete α` to a morphism `X ⟶ Y`\nin the discrete category. -/\nprotected abbrev eqToHom {X Y : Discrete α} (h : X.as = Y.as) : X ⟶ Y :=\n  eqToHom (by cat_disch)\n\n/-- Promote an equation between the wrapped terms in `X Y : Discrete α` to an isomorphism `X ≅ Y`\nin the discrete category. -/\nprotected abbrev eqToIso {X Y : Discrete α} (h : X.as = Y.as) : X ≅ Y :=\n  eqToIso (by cat_disch)\n\n/-- A variant of `eqToHom` that lifts terms to the discrete category. -/\nabbrev eqToHom' {a b : α} (h : a = b) : Discrete.mk a ⟶ Discrete.mk b :=\n  Discrete.eqToHom h\n\n/-- A variant of `eqToIso` that lifts terms to the discrete category. -/\nabbrev eqToIso' {a b : α} (h : a = b) : Discrete.mk a ≅ Discrete.mk b :=\n  Discrete.eqToIso h\n\n@[simp]\ntheorem id_def (X : Discrete α) : ULift.up (PLift.up (Eq.refl X.as)) = 𝟙 X :=\n  rfl\n\n@[simp]\ntheorem id_def' (X : α) : ULift.up (PLift.up (Eq.refl X)) = 𝟙 (⟨X⟩ : Discrete α) :=\n  rfl\n\nvariable {C : Type u₂} [Category.{v₂} C]\n\ninstance {I : Type u₁} {i j : Discrete I} (f : i ⟶ j) : IsIso f :=\n  ⟨⟨Discrete.eqToHom (eq_of_hom f).symm, by cat_disch⟩⟩\n\nattribute [local aesop safe tactic (rule_sets := [CategoryTheory])]\n  CategoryTheory.Discrete.discreteCases\n\n/-- Any function `I → C` gives a functor `Discrete I ⥤ C`. -/\ndef functor {I : Type u₁} (F : I → C) : Discrete I ⥤ C where\n  obj := F ∘ Discrete.as\n  map {X Y} f := by\n    dsimp\n    rcases f with ⟨⟨h⟩⟩\n    exact eqToHom (congrArg _ h)\n\n@[simp]\ntheorem functor_obj {I : Type u₁} (F : I → C) (i : I) :\n    (Discrete.functor F).obj (Discrete.mk i) = F i :=\n  rfl\n\ntheorem functor_map {I : Type u₁} (F : I → C) {i : Discrete I} (f : i ⟶ i) :\n    (Discrete.functor F).map f = 𝟙 (F i.as) := by cat_disch\n\n@[simp]\ntheorem functor_obj_eq_as {I : Type u₁} (F : I → C) (X : Discrete I) :\n    (Discrete.functor F).obj X = F X.as :=\n  rfl\n\n@[ext]\nlemma functor_ext {I : Type u₁} {G F : Discrete I ⥤ C} (h : (i : I) → G.obj ⟨i⟩ = F.obj ⟨i⟩) :\n    G = F := by\n  fapply Functor.ext\n  · intro I; rw [h]\n  · intro ⟨X⟩ ⟨Y⟩ ⟨⟨p⟩⟩; simp only at p; induction p; simp\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The discrete functor induced by a composition of maps can be written as a\ncomposition of two discrete functors.\n-/\n@[simps!]\ndef functorComp {I : Type u₁} {J : Type u₁'} (f : J → C) (g : I → J) :\n    Discrete.functor (f ∘ g) ≅ Discrete.functor (Discrete.mk ∘ g) ⋙ Discrete.functor f :=\n  NatIso.ofComponents fun _ => Iso.refl _\n\n/-- For functors out of a discrete category,\na natural transformation is just a collection of maps,\nas the naturality squares are trivial.\n-/\n@[simps]\ndef natTrans {I : Type u₁} {F G : Discrete I ⥤ C} (f : ∀ i : Discrete I, F.obj i ⟶ G.obj i) :\n    F ⟶ G where\n  app := f\n  naturality := fun {X Y} ⟨⟨g⟩⟩ => by\n    discrete_cases\n    rcases g\n    change F.map (𝟙 _) ≫ _ = _ ≫ G.map (𝟙 _)\n    simp\n\n/-- For functors out of a discrete category,\na natural isomorphism is just a collection of isomorphisms,\nas the naturality squares are trivial.\n-/\n@[simps!]\ndef natIso {I : Type u₁} {F G : Discrete I ⥤ C} (f : ∀ i : Discrete I, F.obj i ≅ G.obj i) :\n    F ≅ G :=\n  NatIso.ofComponents f fun ⟨⟨g⟩⟩ => by\n    discrete_cases\n    rcases g\n    change F.map (𝟙 _) ≫ _ = _ ≫ G.map (𝟙 _)\n    simp\n\ninstance {I : Type*} {F G : Discrete I ⥤ C} (f : ∀ i, F.obj i ⟶ G.obj i) [∀ i, IsIso (f i)] :\n    IsIso (Discrete.natTrans f) := by\n  change IsIso (Discrete.natIso (fun i => asIso (f i))).hom\n  infer_instance\n\n@[simp]\ntheorem natIso_app {I : Type u₁} {F G : Discrete I ⥤ C} (f : ∀ i : Discrete I, F.obj i ≅ G.obj i)\n    (i : Discrete I) : (Discrete.natIso f).app i = f i := by cat_disch\n\n/-- Every functor `F` from a discrete category is naturally isomorphic (actually, equal) to\n  `Discrete.functor (F.obj)`. -/\n@[simps!]\ndef natIsoFunctor {I : Type u₁} {F : Discrete I ⥤ C} : F ≅ Discrete.functor (F.obj ∘ Discrete.mk) :=\n  natIso fun _ => Iso.refl _\n\n/-- Composing `Discrete.functor F` with another functor `G` amounts to composing `F` with `G.obj` -/\n@[simps!]\ndef compNatIsoDiscrete {I : Type u₁} {D : Type u₃} [Category.{v₃} D] (F : I → C) (G : C ⥤ D) :\n    Discrete.functor F ⋙ G ≅ Discrete.functor (G.obj ∘ F) :=\n  natIso fun _ => Iso.refl _\n\n/-- We can promote a type-level `Equiv` to\nan equivalence between the corresponding `discrete` categories.\n-/\n@[simps]\ndef equivalence {I : Type u₁} {J : Type u₂} (e : I ≃ J) : Discrete I ≌ Discrete J where\n  functor := Discrete.functor (Discrete.mk ∘ (e : I → J))\n  inverse := Discrete.functor (Discrete.mk ∘ (e.symm : J → I))\n  unitIso :=\n    Discrete.natIso fun i => eqToIso (by simp)\n  counitIso :=\n    Discrete.natIso fun j => eqToIso (by simp)\n\n/-- We can convert an equivalence of `discrete` categories to a type-level `Equiv`. -/\n@[simps]\ndef equivOfEquivalence {α : Type u₁} {β : Type u₂} (h : Discrete α ≌ Discrete β) : α ≃ β where\n  toFun := Discrete.as ∘ h.functor.obj ∘ Discrete.mk\n  invFun := Discrete.as ∘ h.inverse.obj ∘ Discrete.mk\n  left_inv a := by simpa using eq_of_hom (h.unitIso.app (Discrete.mk a)).2\n  right_inv a := by simpa using eq_of_hom (h.counitIso.app (Discrete.mk a)).1\n\nend Discrete\n\nnamespace Discrete\n\nvariable {J : Type v₁}\n\nopen Opposite\n\n/-- A discrete category is equivalent to its opposite category. -/\n@[simps! functor_obj_as inverse_obj]\nprotected def opposite (α : Type u₁) : (Discrete α)ᵒᵖ ≌ Discrete α :=\n  let F : Discrete α ⥤ (Discrete α)ᵒᵖ := Discrete.functor fun x => op (Discrete.mk x)\n  { functor := F.leftOp\n    inverse := F\n    unitIso := NatIso.ofComponents fun ⟨_⟩ => Iso.refl _\n    counitIso := Discrete.natIso fun ⟨_⟩ => Iso.refl _ }\n\nvariable {C : Type u₂} [Category.{v₂} C]\n\n@[simp]\n\nTarget:\ntheorem functor_map_id (F : Discrete J ⥤ C) {j : Discrete J} (f : j ⟶ j) :\n    F.map f = 𝟙 (F.obj j) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_75d34468da20","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"ac48af2a61ec57ee92967b946c9b267c9f73e5b417dbfe008e0417dd7ec6cb65","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Discrete","family_id":"functor_map_id","file_id":"mathlib/Mathlib/CategoryTheory/Discrete/Basic.lean","sample_id":"75d34468da2034a80fca64c4e5869a12c798eeacfe03a8618a187bb63ae913dc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"96180be036a5aa5c1b27a5d6a0c89b53acf0cf1ee820da9c2ba106ab55ce193e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1daa9f465ccaa4038e45b7b45c48af664510fd77f6ef72028805211ad957a3be","source_sha256":"089de13c5349089f72463c4ad215460456301c821effdb48ac9998fc0a8a4b16","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold traverse\n  simp only [Functor.map_map, fmap_def, map_comp_coe_apply, Multiset.fmap_def, ←\n    Multiset.map_traverse]","hard_negative":false,"metrics":{"chosen_tokens":23,"rejected_tokens":3,"token_jaccard":0.055556,"token_length_ratio":0.130435},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"20ef6d184262595939cc8ae8b399edad7ce94caebe09ccf683b11902780870ff","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Control.AlternativeMonad\npublic import Mathlib.Data.Finset.Lattice.Union\npublic import Mathlib.Data.Finset.NAry\npublic import Mathlib.Data.Multiset.Functor\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2021 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Kim Morrison\n-/\n/-!\n# Functoriality of `Finset`\n\nThis file defines the functor structure of `Finset`.\n\n## TODO\n\nCurrently, all instances are classical because the functor classes want to run over all types. If\ninstead we could state that a functor is lawful/applicative/traversable... between two given types,\nthen we could provide the instances for types with decidable equality.\n-/\n\n@[expose] public section\n\n\nuniverse u\n\nopen Function\n\nnamespace Finset\n\n/-! ### Functor -/\n\nsection Functor\n\nvariable {α β : Type u} [∀ P, Decidable P]\n\n/-- Because `Finset.image` requires a `DecidableEq` instance for the target type, we can only\nconstruct `Functor Finset` when working classically. -/\nprotected instance functor : Functor Finset where map f s := s.image f\n\ninstance lawfulFunctor : LawfulFunctor Finset where\n  id_map _ := image_id\n  comp_map _ _ _ := image_image.symm\n  map_const {α} {β} := by simp only [Functor.mapConst, Functor.map]\n\n@[simp]\ntheorem fmap_def {s : Finset α} (f : α → β) : f <$> s = s.image f := rfl\n\nend Functor\n\n/-! ### Pure -/\n\n\nprotected instance pure : Pure Finset :=\n  ⟨fun x => {x}⟩\n\n@[simp]\ntheorem pure_def {α} : (pure : α → Finset α) = singleton := rfl\n\n/-! ### Applicative functor -/\n\n\nsection Applicative\n\nvariable {α β : Type u} [∀ P, Decidable P]\n\nprotected instance applicative : Applicative Finset :=\n  { Finset.functor, Finset.pure with\n    seq := fun t s => t.sup fun f => (s ()).image f\n    seqLeft := fun s t => if t () = ∅ then ∅ else s\n    seqRight := fun s t => if s = ∅ then ∅ else t () }\n\n@[simp]\ntheorem seq_def (s : Finset α) (t : Finset (α → β)) : t <*> s = t.sup fun f => s.image f :=\n  rfl\n\n@[simp]\ntheorem seqLeft_def (s : Finset α) (t : Finset β) : s <* t = if t = ∅ then ∅ else s :=\n  rfl\n\n@[simp]\ntheorem seqRight_def (s : Finset α) (t : Finset β) : s *> t = if s = ∅ then ∅ else t :=\n  rfl\n\n/-- `Finset.image₂` in terms of monadic operations. Note that this can't be taken as the definition\nbecause of the lack of universe polymorphism. -/\ntheorem image₂_def {α β γ : Type u} (f : α → β → γ) (s : Finset α) (t : Finset β) :\n    image₂ f s t = f <$> s <*> t := by\n  ext\n  simp [mem_sup]\n\ninstance lawfulApplicative : LawfulApplicative Finset :=\n  { Finset.lawfulFunctor with\n    seqLeft_eq := fun s t => by\n      rw [seq_def, fmap_def, seqLeft_def]\n      obtain rfl | ht := t.eq_empty_or_nonempty\n      · simp_rw [image_empty, if_true]\n        exact (sup_bot _).symm\n      · ext a\n        rw [if_neg ht.ne_empty, mem_sup]\n        refine ⟨fun ha => ⟨const _ a, mem_image_of_mem _ ha, mem_image_const_self.2 ht⟩, ?_⟩\n        rintro ⟨f, hf, ha⟩\n        rw [mem_image] at hf ha\n        obtain ⟨b, hb, rfl⟩ := hf\n        obtain ⟨_, _, rfl⟩ := ha\n        exact hb\n    seqRight_eq := fun s t => by\n      rw [seq_def, fmap_def, seqRight_def]\n      obtain rfl | hs := s.eq_empty_or_nonempty\n      · rw [if_pos rfl, image_empty, sup_empty, bot_eq_empty]\n      · ext a\n        rw [if_neg hs.ne_empty, mem_sup]\n        refine ⟨fun ha => ⟨id, mem_image_const_self.2 hs, by rwa [image_id]⟩, ?_⟩\n        rintro ⟨f, hf, ha⟩\n        rw [mem_image] at hf ha\n        obtain ⟨b, hb, rfl⟩ := ha\n        obtain ⟨_, _, rfl⟩ := hf\n        exact hb\n    pure_seq := fun f s => by simp only [pure_def, seq_def, sup_singleton, fmap_def]\n    map_pure := fun _ _ => image_singleton _ _\n    seq_pure := fun _ _ => sup_singleton_apply _ _\n    seq_assoc := fun s t u => by\n      ext a\n      simp_rw [seq_def, fmap_def]\n      simp only [mem_sup, mem_image]\n      constructor\n      · rintro ⟨g, hg, b, ⟨f, hf, a, ha, rfl⟩, rfl⟩\n        exact ⟨g ∘ f, ⟨comp g, ⟨g, hg, rfl⟩, f, hf, rfl⟩, a, ha, rfl⟩\n      · rintro ⟨c, ⟨_, ⟨g, hg, rfl⟩, f, hf, rfl⟩, a, ha, rfl⟩\n        exact ⟨g, hg, f a, ⟨f, hf, a, ha, rfl⟩, rfl⟩ }\n\ninstance commApplicative : CommApplicative Finset :=\n  { Finset.lawfulApplicative with\n    commutative_prod := fun s t => by\n      simp_rw [seq_def, fmap_def, sup_image, sup_eq_biUnion]\n      change (s.biUnion fun a => t.image fun b => (a, b))\n        = t.biUnion fun b => s.image fun a => (a, b)\n      trans s ×ˢ t <;> [rw [product_eq_biUnion]; rw [product_eq_biUnion_right]] }\n\nend Applicative\n\n/-! ### Monad -/\n\n\nsection Monad\n\nvariable [∀ P, Decidable P]\n\ninstance : Monad Finset :=\n  { Finset.applicative with bind := sup }\n\n@[simp]\ntheorem bind_def {α β} : (· >>= ·) = sup (α := Finset α) (β := β) :=\n  rfl\n\ninstance : LawfulMonad Finset :=\n  { Finset.lawfulApplicative with\n    bind_pure_comp := fun _ _ => sup_singleton_apply _ _\n    bind_map := fun _ _ => rfl\n    pure_bind := fun _ _ => sup_singleton\n    bind_assoc := fun s f g => by simp only [bind, sup_eq_biUnion, biUnion_biUnion] }\n\nend Monad\n\n/-! ### Alternative functor -/\n\n\nsection Alternative\n\nvariable [∀ P, Decidable P]\n\ninstance : AlternativeMonad Finset where\n  orElse s t := s ∪ t ()\n  failure := ∅\n\ninstance : LawfulAlternative Finset where\n  map_failure _ := Finset.image_empty _\n  failure_seq _ := Finset.sup_empty\n  orElse_failure _ := Finset.union_empty _\n  failure_orElse _ := Finset.empty_union _\n  orElse_assoc _ _ _ := Finset.union_assoc _ _ _ |>.symm\n  map_orElse _ _ _ := Finset.image_union _ _\n\nend Alternative\n\n/-! ### Traversable functor -/\n\n\nsection Traversable\n\nvariable {α β γ : Type u} {F G : Type u → Type u} [Applicative F] [Applicative G]\n  [CommApplicative F] [CommApplicative G]\n\n/-- Traverse function for `Finset`. -/\ndef traverse [DecidableEq β] (f : α → F β) (s : Finset α) : F (Finset β) :=\n  Multiset.toFinset <$> Multiset.traverse f s.1\n\n@[simp]\ntheorem id_traverse [DecidableEq α] (s : Finset α) : traverse (pure : α → Id α) s = pure s := by\n  rw [traverse, Multiset.id_traverse]\n  exact s.val_toFinset\n\nopen scoped Classical in\n@[simp]\ntheorem map_comp_coe (h : α → β) :\n    Functor.map h ∘ Multiset.toFinset = Multiset.toFinset ∘ Functor.map h :=\n  funext fun _ => image_toFinset\n\nopen scoped Classical in\n@[simp]\ntheorem map_comp_coe_apply (h : α → β) (s : Multiset α) :\n    s.toFinset.image h = (h <$> s).toFinset :=\n  congrFun (map_comp_coe h) s\n\nopen scoped Classical in\n\nTarget:\ntheorem map_traverse (g : α → G β) (h : β → γ) (s : Finset α) :\n    Functor.map h <$> traverse g s = traverse (Functor.map h ∘ g) s :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finset","family_id":"map_traverse","file_id":"mathlib/Mathlib/Data/Finset/Functor.lean","sample_id":"1daa9f465ccaa4038e45b7b45c48af664510fd77f6ef72028805211ad957a3be"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8b8c47751a3cd16893a65f53a169d429e22504bc42c9299e12e8db816e1165e6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"eabbb9ba32c0200e9777fe721b3dc25c57d7503803b9652f7e2ea66508c0be60","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a559373815454d0166c5dab8fbbe7db64a117d1cd999728a36144f94e03e6450","source_sha256":"e05f0de1b38fa7ce1307d0699fed1c367aa4ac0c4de0f78f8d97c1a3bc8f1b0d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  induction s using Finset.induction generalizing j with\n  | empty => simp [coeff_one, apply_ite, isIntegral_zero, isIntegral_one]\n  | insert a s has IH =>\n    rw [Finset.prod_insert has, coeff_mul]\n    exact IsIntegral.sum _ fun i hi ↦ .mul (H _ (by simp) _) (IH (fun _ _ ↦ H _ (by aesop)) _)","hard_negative":true,"metrics":{"chosen_tokens":76,"rejected_tokens":3,"token_jaccard":0.047619,"token_length_ratio":0.039474},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"210fee043b3d7eee34bcd396fde23a67c62b1147e37b172c11ed187622d8a575","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Multiset.Fintype\npublic import Mathlib.RingTheory.AdjoinRoot\npublic import Mathlib.RingTheory.Polynomial.RationalRoot\npublic import Mathlib.RingTheory.IntegralClosure.IsIntegral.AlmostIntegral\n\nNamespace:\nPolynomial\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Results about coefficients of polynomials being integral\n\n## Main results\n- `Polynomial.isIntegral_coeff_of_dvd`: If a monic polynomial `p` divides another monic polynomial\n  with integral coefficients, then the coefficients of `p` are themselves integral.\n- `Polynomial.isIntegral_iff_isIntegral_coeff`:\n  `p : S[X]` is integral over `R[X]` iff the coefficients of `p` are integral over `R`.\n- `MvPolynomial.isIntegral_iff_isIntegral_coeff`: `p : MvPolynomial σ S` is integral over\n  `MvPolynomial σ R` iff the coefficients of `p` are integral over `R`.\n- We also provide the instance `[IsIntegrallyClosed R] : IsIntegrallyClosed R[X]`.\n\n-/\n\npublic section\n\nvariable {R S ι : Type*} [CommRing R] [CommRing S] [Algebra R S]\n\nnamespace Polynomial\n\nTarget:\nlemma isIntegral_coeff_prod\n    (s : Finset ι) (p : ι → S[X]) (H : ∀ i ∈ s, ∀ j, IsIntegral R ((p i).coeff j)) (j : ℕ) :\n    IsIntegral R ((s.prod p).coeff j) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_a55937381545","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"a1e17c2cc733a9cc8252fd4f77b1d99a6ea1797b79d584c683d338809894d0d7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Polynomial","family_id":"isintegral_coeff_prod","file_id":"mathlib/Mathlib/RingTheory/Polynomial/IsIntegral.lean","sample_id":"a559373815454d0166c5dab8fbbe7db64a117d1cd999728a36144f94e03e6450"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ddb596461bd0c4bd75d4fb0cc2c7a82f17820be10b8f87f0c46c723e2b555f0f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fe21963d2aeeba6cb61599867ca841b158d2155ae815d66b4331eaa8e5ce4e3b","source_sha256":"018e0555cbbfa42b8d2c4f98114f5ffea5e65210561d826e089516f6ee1b9864","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hg : DifferentiableAt ℂ (fun z ↦ log (1 - z)) 0 := by\n    have : 1 - 0 ∈ slitPlane := (sub_zero (1 : ℂ)).symm ▸ one_mem_slitPlane\n    fun_prop\n  have : (fun z ↦ log (1 - z)) =O[𝓝 0] id := by\n    simpa only [sub_zero, log_one] using! hg.isBigO_sub\n  exact this.comp_summable hsum","hard_negative":true,"metrics":{"chosen_tokens":80,"rejected_tokens":8,"token_jaccard":0.090909,"token_length_ratio":0.1},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"2170087dc9172d1aca007e714c75aa73bcb5efa102fecaf611bb1fac2c7800f4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Module.FiniteDimension\npublic import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv\npublic import Mathlib.LinearAlgebra.Complex.FiniteDimensional\npublic import Mathlib.NumberTheory.EulerProduct.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Logarithms of Euler Products\n\nWe consider `f : ℕ →*₀ ℂ` and show that `exp (∑ p in Primes, log (1 - f p)⁻¹) = ∑ n : ℕ, f n`\nunder suitable conditions on `f`. This can be seen as a logarithmic version of the\nEuler product for `f`.\n-/\n\npublic section\n\nopen Complex\n\nopen Topology in\n/-- If `f : α → ℂ` is summable, then so is `n ↦ log (1 - f n)`. -/\n\nTarget:\nlemma Summable.clog_one_sub {α : Type*} {f : α → ℂ} (hsum : Summable f) :\n    Summable fun n ↦ log (1 - f n) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"87d3fb4739a3ef97859cdbda2fe4d2d5a0363863e1ebc7bd39b18affa66c29bb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/EulerProduct","family_id":"summable","file_id":"mathlib/Mathlib/NumberTheory/EulerProduct/ExpLog.lean","sample_id":"fe21963d2aeeba6cb61599867ca841b158d2155ae815d66b4331eaa8e5ce4e3b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ccb01e73a5d93776c5ed6830bc49ca880d5f04d2137609fc522a9ded57d8bf1c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"be074731ee1e86ad6f0c254a1e86d8fd22d9ad9d3fb311c4cb117c2e21e27796","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6c405ac245686730f2e5f7fda35bbf73c516cea158bccf92ad5f323956571fcb","source_sha256":"e095ac1997df0a4695785cbe0250e8d2d4a54f54912b6bc94870e67de3b0352a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine Multifork.IsLimit.hom_ext hc fun a ↦ ?_\n  have heq := d.condition ⟨⟨(f.s₀ a), (g.s₀ a)⟩, H.H a⟩\n  simp only [multicospanIndex_right, multicospanShape_fst, multicospanIndex_left,\n    multicospanIndex_fst, multicospanShape_snd, multicospanIndex_snd] at heq\n  simp [-Homotopy.wl, -Homotopy.wr, ← H.wl, ← H.wr, reassoc_of% heq]","hard_negative":true,"metrics":{"chosen_tokens":86,"rejected_tokens":3,"token_jaccard":0.021277,"token_length_ratio":0.034884},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"2188c73f5289f0ffaf8f47240824366172c03ed1b754ed46e6eb1cc578e0f200","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Quotient\npublic import Mathlib.CategoryTheory.Sites.Hypercover.One\npublic import Mathlib.CategoryTheory.Filtered.Basic\n\nNamespace:\nCategoryTheory.PreOneHypercover\n\nLocal context:\n/-\nCopyright (c) 2025 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten\n-/\n/-!\n# The category of `1`-hypercovers up to homotopy\n\nIn this file we define the category of `1`-hypercovers up to homotopy. This is the category of\n`1`-hypercovers, but where morphisms are considered up to existence of a homotopy.\n\n## Main definitions\n\n- `CategoryTheory.PreOneHypercover.Homotopy`: A homotopy of refinements `E ⟶ F` is a family of\n  morphisms `Xᵢ ⟶ Yₐ` where `Yₐ` is a component of the cover of `X_{f(i)} ×[S] X_{g(i)}`.\n- `CategoryTheory.GrothendieckTopology.HOneHypercover`: The category of `1`-hypercovers\n  with respect to a Grothendieck topology and morphisms up to homotopy.\n\n## Main results\n\n- `CategoryTheory.GrothendieckTopology.HOneHypercover.isCofiltered_of_hasPullbacks`: The\n  category of `1`-hypercovers up to homotopy is cofiltered if `C` has pullbacks.\n-/\n\n@[expose] public section\n\nuniverse w'' w' w v u\n\nnamespace CategoryTheory\n\nopen Limits\n\nvariable {C : Type u} [Category.{v} C]\n\nnamespace PreOneHypercover\n\nvariable {S : C} {E : PreOneHypercover.{w} S} {F : PreOneHypercover.{w'} S}\n\n/-- A homotopy of refinements `E ⟶ F` is a family of morphisms `Xᵢ ⟶ Yₐ` where\n`Yₐ` is a component of the cover of `X_{f(i)} ×[S] X_{g(i)}`. -/\nstructure Homotopy (f g : E.Hom F) where\n  /-- The index map sending `i : E.I₀` to `a` above `(f(i), g(i))`. -/\n  H (i : E.I₀) : F.I₁ (f.s₀ i) (g.s₀ i)\n  /-- The morphism `Xᵢ ⟶ Yₐ`. -/\n  a (i : E.I₀) : E.X i ⟶ F.Y (H i)\n  wl (i : E.I₀) : a i ≫ F.p₁ (H i) = f.h₀ i\n  wr (i : E.I₀) : a i ≫ F.p₂ (H i) = g.h₀ i\n\nattribute [reassoc (attr := simp)] Homotopy.wl Homotopy.wr\n\nsection\n\nvariable {A : Type*} [Category* A]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- Homotopic refinements induce the same map on multiequalizers. -/\n\nTarget:\nlemma Homotopy.mapMultiforkOfIsLimit_eq\n    {E F : PreOneHypercover.{w} S} {f g : E.Hom F} (H : Homotopy f g)\n    (P : Cᵒᵖ ⥤ A) {c : Multifork (E.multicospanIndex P)} (hc : IsLimit c)\n    (d : Multifork (F.multicospanIndex P)) :\n    f.mapMultiforkOfIsLimit P hc d = g.mapMultiforkOfIsLimit P hc d :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_6c405ac24568","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"82567c60cf45e02cc4f6dc664bdb48a87f306a767686ce7592ffd5e9512f9343","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Sites","family_id":"homotopy","file_id":"mathlib/Mathlib/CategoryTheory/Sites/Hypercover/Homotopy.lean","sample_id":"6c405ac245686730f2e5f7fda35bbf73c516cea158bccf92ad5f323956571fcb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"543d94c4b8beec8d8219800c1a531bf78be218b6d235d8e934ed1722e2e75902","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4561b215385a78d721dfe36030198c0459e29f4db8031b7c27e613ae90c4a078","source_sha256":"9204017e76fd98a1b3e65ba16535a42898573646b8f44e5554ab4317200de227","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine measure_mono_null (fun x hx ↦ ?_)\n    (Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet _\n      (isClosed_closure (s := s)).measurableSet)\n  by_cases h : x ∈ closure s <;>\n    simp only [mem_compl_iff, mem_setOf, h, not_false_eq_true, indicator_of_notMem,\n      indicator_of_mem, Pi.one_apply]\n  · refine aux₁ fun _ ↦ hs.compl.exists_subset_ball <| frontier_subset_closure ?_\n    rwa [frontier_compl]\n  · exact aux₀ fun _ ↦ hs.exists_subset_ball <| frontier_subset_closure hx","hard_negative":true,"metrics":{"chosen_tokens":91,"rejected_tokens":8,"token_jaccard":0.055556,"token_length_ratio":0.087912},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"218bc6adf15ff3c66ce3f2201e3d3d27307ece96351373cc78eb1849b1aa5b29","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Order.UpperLower\npublic import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace\npublic import Mathlib.Topology.Order.DenselyOrdered\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies, Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Kexing Ying\n-/\n/-!\n# Order-connected sets are null-measurable\n\nThis file proves that order-connected sets in `ℝⁿ` under the pointwise order are null-measurable.\nRecall that `x ≤ y` iff `∀ i, x i ≤ y i`, and `s` is order-connected iff\n`∀ x y ∈ s, ∀ z, x ≤ z → z ≤ y → z ∈ s`.\n\n## Main declarations\n\n* `Set.OrdConnected.null_frontier`: The frontier of an order-connected set in `ℝⁿ` has measure `0`.\n\n## Notes\n\nWe prove null-measurability in `ℝⁿ` with the `∞`-metric, but this transfers directly to `ℝⁿ` with\nthe Euclidean metric because they have the same measurable sets.\n\nNull-measurability can't be strengthened to measurability because any antichain (and in particular\nany subset of the antidiagonal `{(x, y) | x + y = 0}`) is order-connected.\n\n## Sketch proof\n\n1. To show an order-connected set is null-measurable, it is enough to show it has null frontier.\n2. Since an order-connected set is the intersection of its upper and lower closure, it's enough to\n  show that upper and lower sets have null frontier.\n3. WLOG let's prove it for an upper set `s`.\n4. By the Lebesgue density theorem, it is enough to show that any frontier point `x` of `s` is not a\n  Lebesgue point, namely we want the density of `s` over small balls centered at `x` to not tend to\n  either `0` or `1`.\n5. This is true, since by the upper setness of `s` we can intercalate a ball of radius `δ / 4` in\n  `s` intersected with the upper quadrant of the ball of radius `δ` centered at `x` (recall that the\n  balls are taken in the ∞-norm, so they are cubes), and another ball of radius `δ / 4` in `sᶜ` and\n  the lower quadrant of the ball of radius `δ` centered at `x`.\n\n## TODO\n\nGeneralize so that it also applies to `ℝ × ℝ`, for example.\n-/\n\npublic section\n\nopen Filter MeasureTheory Metric Set\nopen scoped Topology\n\nvariable {ι : Type*} [Fintype ι] {s : Set (ι → ℝ)} {x : ι → ℝ}\n\n/-- If we can fit a small ball inside a set `s` intersected with any neighborhood of `x`, then the\ndensity of `s` near `x` is not `0`.\n\nAlong with `aux₁`, this proves that `x` is not a Lebesgue point of `s`. This will be used to prove\nthat the frontier of an order-connected set is null. -/\nprivate lemma aux₀\n    (h : ∀ δ, 0 < δ →\n      ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior s) :\n    ¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)\n        (𝓝 0) := by\n  choose f hf₀ hf₁ using h\n  intro H\n  obtain ⟨ε, -, hε', hε₀⟩ := exists_seq_strictAnti_tendsto_nhdsWithin (0 : ℝ)\n  refine not_eventually.2\n    (Frequently.of_forall fun _ ↦ lt_irrefl <| ENNReal.ofReal <| 4⁻¹ ^ Fintype.card ι)\n    ((Filter.Tendsto.eventually_lt (H.comp hε₀) tendsto_const_nhds ?_).mono fun n ↦\n      lt_of_le_of_lt ?_)\n  on_goal 2 =>\n    calc\n      ENNReal.ofReal (4⁻¹ ^ Fintype.card ι)\n        = volume (closedBall (f (ε n) (hε' n)) (ε n / 4)) / volume (closedBall x (ε n)) := ?_\n      _ ≤ volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n)) := by\n        gcongr\n        exact subset_inter ((hf₁ _ <| hε' n).trans interior_subset_closure) <| hf₀ _ <| hε' n\n    have := hε' n\n    rw [Real.volume_pi_closedBall, Real.volume_pi_closedBall, ← ENNReal.ofReal_div_of_pos,\n      ← div_pow, mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]\n  all_goals positivity\n\n/-- If we can fit a small ball inside a set `sᶜ` intersected with any neighborhood of `x`, then the\ndensity of `s` near `x` is not `1`.\n\nAlong with `aux₀`, this proves that `x` is not a Lebesgue point of `s`. This will be used to prove\nthat the frontier of an order-connected set is null. -/\nprivate lemma aux₁\n    (h : ∀ δ, 0 < δ →\n      ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior sᶜ) :\n    ¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)\n        (𝓝 1) := by\n  choose f hf₀ hf₁ using h\n  intro H\n  obtain ⟨ε, -, hε', hε₀⟩ := exists_seq_strictAnti_tendsto_nhdsWithin (0 : ℝ)\n  refine not_eventually.2\n      (Frequently.of_forall fun _ ↦ lt_irrefl <| 1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι))\n      ((Filter.Tendsto.eventually_lt tendsto_const_nhds (H.comp hε₀) <|\n            ENNReal.sub_lt_self ENNReal.one_ne_top one_ne_zero ?_).mono\n        fun n ↦ lt_of_le_of_lt' ?_)\n  on_goal 2 =>\n    calc\n      volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n))\n        ≤ volume (closedBall x (ε n) \\ closedBall (f (ε n) <| hε' n) (ε n / 4)) /\n          volume (closedBall x (ε n)) := by\n        gcongr\n        rw [sdiff_eq_compl_inter]\n        refine inter_subset_inter_left _ ?_\n        rw [subset_compl_comm, ← interior_compl]\n        exact hf₁ _ _\n      _ = 1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι) := ?_\n    have := hε' n\n    rw [measure_sdiff (hf₀ _ _) _ ((Real.volume_pi_closedBall _ _).trans_ne ENNReal.ofReal_ne_top),\n      Real.volume_pi_closedBall, Real.volume_pi_closedBall, ENNReal.sub_div fun _ _ ↦ _,\n      ENNReal.div_self _ ENNReal.ofReal_ne_top, ← ENNReal.ofReal_div_of_pos, ← div_pow,\n      mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]\n  all_goals try positivity\n  · simp_all\n  · exact measurableSet_closedBall.nullMeasurableSet\n\nTarget:\ntheorem IsUpperSet.null_frontier (hs : IsUpperSet s) : volume (frontier s) = 0 :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e3c55b818770a3ddaef15bbc762355387e1adddb238521610a9a0ca65bcf1929","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Order","family_id":"isupperset","file_id":"mathlib/Mathlib/MeasureTheory/Order/UpperLower.lean","sample_id":"4561b215385a78d721dfe36030198c0459e29f4db8031b7c27e613ae90c4a078"},"split":"train"}
{"binding":{"chosen_proof_sha256":"576771f2ca0b7fae2b561a23269a331367199c4b2800aa18415fec083cc832ae","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"be9b8d29bc8008d9f2006f7c0e2e488c12668f4a1e6f2ba4ffcf6937f3ace739","source_sha256":"d0eafab6c9f23d1000e4c23ec8fbc2844b38b0956624818eec3c9dbb37162d5b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [liftOfDerivationToSquareZero_apply, map_add, Ideal.Quotient.eq_zero_iff_mem.mpr (d x).prop,\n    zero_add, Ideal.Quotient.mk_algebraMap]","hard_negative":false,"metrics":{"chosen_tokens":29,"rejected_tokens":3,"token_jaccard":0.047619,"token_length_ratio":0.103448},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"2194b225115fd1e4dd66bcc6f9d049a994339361c2e4aa4cd0d920fec25041c0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Derivation.Basic\npublic import Mathlib.RingTheory.Ideal.Quotient.Operations\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Nicolò Cavalleri. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Nicolò Cavalleri, Andrew Yang\n-/\n/-!\n# Derivations into Square-Zero Ideals\n\n## Main statements\n\n- `derivationToSquareZeroOfLift`: The `R`-derivations from `A` into a square-zero ideal `I`\n  of `B` corresponds to the lifts `A →ₐ[R] B` of the map `A →ₐ[R] B ⧸ I`.\n\n-/\n\n@[expose] public section\n\n\nsection ToSquareZero\n\nuniverse u v w\n\nvariable {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommRing B]\nvariable [Algebra R A] [Algebra R B] (I : Ideal B)\n\n/-- If `f₁ f₂ : A →ₐ[R] B` are two lifts of the same `A →ₐ[R] B ⧸ I`,\n  we may define a map `f₁ - f₂ : A →ₗ[R] I`. -/\ndef diffToIdealOfQuotientCompEq (f₁ f₂ : A →ₐ[R] B)\n    (e : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂) : A →ₗ[R] I :=\n  LinearMap.codRestrict (I.restrictScalars _) (f₁.toLinearMap - f₂.toLinearMap)\n    (fun x => by simpa [Ideal.Quotient.eq] using congr($e x))\n\n@[simp]\ntheorem diffToIdealOfQuotientCompEq_apply (f₁ f₂ : A →ₐ[R] B)\n    (e : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂) (x : A) :\n    ((diffToIdealOfQuotientCompEq I f₁ f₂ e) x : B) = f₁ x - f₂ x :=\n  rfl\n\nvariable [Algebra A B]\n\n/-- Given a tower of algebras `R → A → B`, and a square-zero `I : Ideal B`, each lift `A →ₐ[R] B`\nof the canonical map `A →ₐ[R] B ⧸ I` corresponds to an `R`-derivation from `A` to `I`. -/\ndef derivationToSquareZeroOfLift [IsScalarTower R A B] (hI : I ^ 2 = ⊥) (f : A →ₐ[R] B)\n    (e : (Ideal.Quotient.mkₐ R I).comp f = IsScalarTower.toAlgHom R A (B ⧸ I)) :\n    Derivation R A I := by\n  refine\n    { diffToIdealOfQuotientCompEq I f (IsScalarTower.toAlgHom R A B) ?_ with\n      map_one_eq_zero' := ?_\n      leibniz' := ?_ }\n  · ext; simp [e]\n  · ext; simp\n  · intro x y\n    let F := diffToIdealOfQuotientCompEq I f (IsScalarTower.toAlgHom R A B) (by rw [e]; ext; rfl)\n    have : (f x - algebraMap A B x) * (f y - algebraMap A B y) = 0 := by\n      rw [← Ideal.mem_bot, ← hI, pow_two]\n      convert! Ideal.mul_mem_mul (F x).2 (F y).2 using 1\n    ext\n    dsimp only [Submodule.coe_add, Submodule.coe_mk, LinearMap.coe_mk,\n      diffToIdealOfQuotientCompEq_apply, Submodule.coe_smul_of_tower, IsScalarTower.coe_toAlgHom',\n      LinearMap.toFun_eq_coe]\n    simp only [map_mul, sub_mul, mul_sub, Algebra.smul_def] at this ⊢\n    rw [sub_eq_iff_eq_add, sub_eq_iff_eq_add] at this\n    simp only [this]\n    ring\n\nvariable (hI : I ^ 2 = ⊥)\n\ntheorem derivationToSquareZeroOfLift_apply [IsScalarTower R A B] (f : A →ₐ[R] B)\n    (e : (Ideal.Quotient.mkₐ R I).comp f = IsScalarTower.toAlgHom R A (B ⧸ I)) (x : A) :\n    (derivationToSquareZeroOfLift I hI f e x : B) = f x - algebraMap A B x :=\n  rfl\n\n/-- Given a tower of algebras `R → A → B`, and a square-zero `I : Ideal B`, each `R`-derivation\nfrom `A` to `I` corresponds to a lift `A →ₐ[R] B` of the canonical map `A →ₐ[R] B ⧸ I`. -/\n@[simps -isSimp]\ndef liftOfDerivationToSquareZero [IsScalarTower R A B] (hI : I ^ 2 = ⊥) (f : Derivation R A I) :\n    A →ₐ[R] B :=\n  { ((I.restrictScalars R).subtype.comp f.toLinearMap + (IsScalarTower.toAlgHom R A B).toLinearMap :\n      A →ₗ[R] B) with\n    toFun := fun x => f x + algebraMap A B x\n    map_one' := by\n      rw [map_one (algebraMap _ _), f.map_one_eq_zero, Submodule.coe_zero, zero_add]\n    map_mul' := fun x y => by\n      have : (f x : B) * f y = 0 := by\n        rw [← Ideal.mem_bot, ← hI, pow_two]\n        convert! Ideal.mul_mem_mul (f x).2 (f y).2 using 1\n      simp only [map_mul, f.leibniz, add_mul, mul_add, Submodule.coe_add,\n        Submodule.coe_smul_of_tower, Algebra.smul_def, this]\n      ring\n    commutes' := fun r => by\n      simp only [Derivation.map_algebraMap, zero_add, Submodule.coe_zero, ←\n        IsScalarTower.algebraMap_apply R A B r]\n    map_zero' := ((I.restrictScalars R).subtype.comp f.toLinearMap +\n      (IsScalarTower.toAlgHom R A B).toLinearMap).map_zero }\n\n-- simp normal form is `liftOfDerivationToSquareZero_mk_apply'`\n\nTarget:\ntheorem liftOfDerivationToSquareZero_mk_apply [IsScalarTower R A B] (d : Derivation R A I) (x : A) :\n    Ideal.Quotient.mk I (liftOfDerivationToSquareZero I hI d x) = algebraMap A (B ⧸ I) x :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Derivation","family_id":"liftofderivationtosquarezero_mk_apply","file_id":"mathlib/Mathlib/RingTheory/Derivation/ToSquareZero.lean","sample_id":"be9b8d29bc8008d9f2006f7c0e2e488c12668f4a1e6f2ba4ffcf6937f3ace739"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5e09f885c765d1667ba798f4a88bd6a2385014bfb40b892ea398481ff81d2e8d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3ad3f81ff3635c43ae0eded3174baf2b8d3e1f824ea5c939c33ca67b0a90e7c7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"abc072079305390f0bef511c2c60314512d25bfb1b8f705b8e28581cf1707079","source_sha256":"c7fff7c9f2f71a64bbae1f040362fa5f16b7992ea23b8c9d1ff27baa1835cd99","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  -- The key is that φ preserves labels, so edges labelled 's' stay labelled 's'\n  let e : v ⟶ ι s • v := ⟨s, rfl⟩\n  -- φ.map e is an edge from φ.obj v, and its label is preserved\n  have h := (φ.map e).property\n  -- This says: `ι (φ.map e).val • φ.obj v = φ.obj (ι s • v)`\n  -- We need to show `(φ.map e).val = s`\n  have label_eq : (φ.map e).val = s := by\n    -- `φm` says `φ ⋙q labelling = labelling`\n    -- So `(φ ⋙q labelling).map e = labelling.map e`\n    have : (φ ⋙q labelling W ι).map e = (labelling V ι).map e := by\n      rw [φm]\n    simp only [Prefunctor.comp_map, labelling_map] at this\n    exact this\n  rw [label_eq] at h\n  exact h.symm","hard_negative":true,"metrics":{"chosen_tokens":216,"rejected_tokens":5,"token_jaccard":0.040541,"token_length_ratio":0.023148},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"22033c151df62556097d54d858c083343cdfdcd18fd0202c69ec560ff57021bd","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Defs\npublic import Mathlib.Algebra.Group.Basic\npublic import Mathlib.Combinatorics.Quiver.Covering\npublic import Mathlib.Combinatorics.Quiver.SingleObj\n\nNamespace:\nQuiver.SchreierGraph\n\nLocal context:\n/-\nCopyright (c) 2026 Runtian Zhou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Runtian Zhou\n-/\n/-!\n# Schreier Graphs\n\nThis module defines Schreier graphs as quivers with labelled edges.\n\nGiven a monoid `M` acting on a type `V` and a map `ι : S → M`, the Schreier graph has\nvertices `V` and a directed edge `x → ι(s) • x` for each `x : V` and `s : S`.\n\n## Main definitions\n\n* `SchreierGraph V ι` - The Schreier graph of an action, with vertices of type `V` and edges\n  labelled by elements of `S` via `ι : S → M`.\n* `SchreierGraph.labelling` - The prefunctor from a Schreier graph to `SingleObj S` that\n  extracts edge labels.\n\n## Main results\n\n* `SchreierGraph.labelling_isCovering` - The labelling prefunctor is a covering when we have\n  a group action.\n\n## Examples\n\n* The (left) **Cayley graph** of a group `M` with generators `ι : S → M` is the Schreier graph\n  where `V = M` and the action is left multiplication.\n\n## Implementation notes\n\nAlthough referred to informally as graphs, Schreier graphs have multiple, directed, labelled\nedges between nodes and so are implemented here as quivers.\n\n## References\n\n* [Y. Vorobets, *Notes on the Schreier graphs of the Grigorchuk group*][Vorobets2012]\n-/\n\n@[expose] public section\n\nnamespace Quiver\n\n/-- A Schreier graph for a monoid `M` acting on `V` with generators `ι : S → M`.\nVertices are elements of `V`, and there is an edge from `x` to `y` for each `s : S`\nsuch that `ι s • x = y`. -/\n@[nolint unusedArguments, ext]\nstructure SchreierGraph (V : Type*) {M : Type*} [SMul M V] {S : Type*} (_ι : S → M) where\n  /-- Wraps a vertex of the acted-upon type into the Schreier graph. -/\n  ofVertex ::\n  /-- The underlying vertex. -/\n  toVertex : V\n\nnamespace SchreierGraph\n\nsection Basic\n\nvariable (V : Type*) {M : Type*} [SMul M V] {S : Type*} (ι : S → M)\n\n/-- Equivalence between the original vertex type and the Schreier graph type. -/\n@[simps]\ndef equiv : V ≃ SchreierGraph V ι where\n  toFun := SchreierGraph.ofVertex\n  invFun := SchreierGraph.toVertex\n  left_inv _ := rfl\n  right_inv _ := rfl\n\n/-- Transport the scalar multiplication to the Schreier graph vertices. -/\ninstance schreierGraphSMul : SMul M (SchreierGraph V ι) where\n  smul x y := ⟨x • y.toVertex⟩\n\n/-- The quiver structure on a Schreier graph. An arrow from `x` to `y` exists when\nthere is an `s : S` such that `(ι s) • x = y`. -/\ninstance schreierGraphQuiver : Quiver (SchreierGraph V ι) where\n  Hom x y := { s : S // (ι s) • x = y }\n\n/-- The labelling of arrows in a Schreier graph by elements of `S`.\nThis is encoded as a prefunctor to `SingleObj S`. -/\n@[simps]\ndef labelling : SchreierGraph V ι ⥤q SingleObj S where\n  obj _ := SingleObj.star S\n  map e := e.val\n\nend Basic\n\nsection MulAction\n\nvariable (V : Type*) {M : Type*} [Monoid M] [MulAction M V] {S : Type*} (ι : S → M)\n\n/-- The monoid acts on the vertices of the Schreier graph. -/\ninstance : MulAction M (SchreierGraph V ι) where\n  one_smul x := by\n    ext\n    exact one_smul M x.toVertex\n  mul_smul a b x := by\n    ext\n    exact mul_smul a b x.toVertex\n\nend MulAction\n\nsection GroupAction\n\n/-!\n### Schreier graphs for group actions\n\nWhen we have a group action, the labelling becomes a covering.\n-/\n\nvariable {V : Type*} {M : Type*} [Group M] [MulAction M V] {S : Type*} (ι : S → M)\n\n/-- The star map of the labelling prefunctor as an equivalence. -/\n@[simps]\ndef labellingStarEquiv (x : SchreierGraph V ι) :\n    Quiver.Star x ≃ Quiver.Star (SingleObj.star S) where\n  toFun := (labelling V ι).star x\n  invFun := fun ⟨_, s⟩ => ⟨ι s • x, s, rfl⟩\n  left_inv := fun ⟨_, _, rfl⟩ => rfl\n  right_inv := fun ⟨_, _⟩ => rfl\n\n/-- The costar map of the labelling prefunctor as an equivalence. -/\n@[simps]\ndef labellingCostarEquiv (x : SchreierGraph V ι) :\n    Quiver.Costar x ≃ Quiver.Costar (SingleObj.star S) where\n  toFun := (labelling V ι).costar x\n  invFun := fun ⟨_, s⟩ => ⟨(ι s)⁻¹ • x, s, by simp⟩\n  left_inv := by\n    rintro ⟨v, s, hs⟩\n    simp only [Prefunctor.costar_apply, labelling_map]\n    have : (ι s)⁻¹ • x = v := by rw [← hs, inv_smul_smul]\n    subst this; rfl\n  right_inv := fun ⟨_, _⟩ => rfl\n\n/-- The labelling prefunctor is a covering for Schreier graphs with group actions. -/\ntheorem labelling_isCovering : (labelling V ι).IsCovering where\n  star_bijective u := (labellingStarEquiv ι u).bijective\n  costar_bijective u := (labellingCostarEquiv ι u).bijective\n\n/-- If a prefunctor between Schreier graphs commutes with the labelling (i.e., labels are\npreserved), then it commutes with the group action. In other words, morphisms that preserve edge\nlabels also preserve the group structure. -/\n\nTarget:\nlemma map_smul_of_comp_labelling_eq {W : Type*} [MulAction M W]\n    (φ : SchreierGraph V ι ⥤q SchreierGraph W ι) (φm : φ ⋙q labelling W ι = labelling V ι)\n    (v : SchreierGraph V ι) (s : S) :\n    φ.obj (ι s • v) = ι s • (φ.obj v) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_abc072079305","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"41d517f223c4bfd5fec91ffe89df47ad54a9cee072c5ce2e9c3a8d291d4b2aec","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Quiver","family_id":"map_smul_of_comp_labelling_eq","file_id":"mathlib/Mathlib/Combinatorics/Quiver/Schreier.lean","sample_id":"abc072079305390f0bef511c2c60314512d25bfb1b8f705b8e28581cf1707079"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4f2754f3c1709f7213213c711ba2777cb527cb9fa5e9c5f811685e4d0a510864","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"aaa0680a89a9aa6b80dccb48330d64d6e725c16a4a09fd3e03a441abb32d0e21","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2ba6bee601213da996a0d324b595d9dabc6794f2b60ae27ff16595b74c0086ad","source_sha256":"1cf64ac04a5f1f9a7c585c7ed1d6a4cf445109aff4a46ab4c7d2a0b61c1da479","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← IsLocalization.mk'_one (M := K[X]⁰) (FractionRing K[X]) p, ← mk_coe_def, Submonoid.coe_one]","hard_negative":false,"metrics":{"chosen_tokens":32,"rejected_tokens":37,"token_jaccard":0.84,"token_length_ratio":1.15625},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"22ae332d660e412f06421002ab5b39535de5d5f6a3c80a61fa6812141c61f8b4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Basic\npublic import Mathlib.Algebra.Ring.NonZeroDivisors\npublic import Mathlib.RingTheory.Localization.FractionRing\n\nNamespace:\nRatFunc\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# The field of rational functions\n\nFiles in this folder define the field `K⟮X⟯` of rational functions over a field `K`, show it\nis the field of fractions of `K[X]` and provide the main results concerning it. This file contains\nthe basic definition.\n\nFor connections with Laurent Series, see `Mathlib/RingTheory/LaurentSeries.lean`.\n\n## Main definitions\nWe provide a set of recursion and induction principles:\n- `RatFunc.liftOn`: define a function by mapping a fraction of polynomials `p/q` to `f p q`,\n  if `f` is well-defined in the sense that `p/q = p'/q' → f p q = f p' q'`.\n- `RatFunc.liftOn'`: define a function by mapping a fraction of polynomials `p/q` to `f p q`,\n  if `f` is well-defined in the sense that `f (a * p) (a * q) = f p' q'`.\n- `RatFunc.induction_on`: if `P` holds on `p / q` for all polynomials `p q`, then `P` holds on all\n  rational functions\n\n## Implementation notes\n\nTo provide good API encapsulation and speed up unification problems,\n`RatFunc` is defined as a structure, and all operations are `@[irreducible] def`s\n\nWe need a couple of maps to set up the `Field` and `IsFractionRing` structure,\nnamely `RatFunc.ofFractionRing`, `RatFunc.toFractionRing`, `RatFunc.mk` and\n`RatFunc.toFractionRingRingEquiv`.\nAll these maps get `simp`ed to bundled morphisms like `algebraMap K[X] K⟮X⟯`\nand `IsLocalization.algEquiv`.\n\nThere are separate lifts and maps of homomorphisms, to provide routes of lifting even when\nthe codomain is not a field or even an integral domain.\n\n## References\n\n* [Kleiman, *Misconceptions about $K_X$*][kleiman1979]\n* https://freedommathdance.blogspot.com/2012/11/misconceptions-about-kx.html\n* https://stacks.math.columbia.edu/tag/01X1\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen scoped nonZeroDivisors Polynomial\n\nuniverse u v\n\nvariable (K : Type u)\n\n/-- `RatFunc K` is `K(X)`, the field of rational functions over `K`.\n\nThe inclusion of polynomials into `RatFunc` is `algebraMap K[X] K⟮X⟯`,\nthe maps between `K⟮X⟯` and another field of fractions of `K[X]`,\nespecially `FractionRing K[X]`, are given by `IsLocalization.algEquiv`.\n-/\nstructure RatFunc [CommRing K] : Type u where ofFractionRing ::\n/-- the coercion to the fraction ring of the polynomial ring -/\n  toFractionRing : FractionRing K[X]\n\n@[inherit_doc] scoped[RatFunc] notation:9000 R \"⟮X⟯\" => RatFunc R\n\nnamespace RatFunc\n\nsection CommRing\n\nvariable {K}\nvariable [CommRing K]\n\nsection Rec\n\n/-! ### Constructing `RatFunc`s and their induction principles -/\n\ntheorem ofFractionRing_injective : Function.Injective (ofFractionRing : _ → K⟮X⟯) :=\n  fun _ _ => ofFractionRing.inj\n\ntheorem toFractionRing_injective : Function.Injective (toFractionRing : _ → FractionRing K[X])\n  | ⟨x⟩, ⟨y⟩, xy => by subst xy; rfl\n\n@[simp] lemma toFractionRing_inj {x y : K⟮X⟯} :\n    toFractionRing x = toFractionRing y ↔ x = y :=\n  toFractionRing_injective.eq_iff\n\n/-- Non-dependent recursion principle for `K⟮X⟯`:\nTo construct a term of `P : Sort*` out of `x : K⟮X⟯`,\nit suffices to provide a constructor `f : Π (p q : K[X]), P`\nand a proof that `f p q = f p' q'` for all `p q p' q'` such that `q' * p = q * p'` where\nboth `q` and `q'` are not zero divisors, stated as `q ∉ K[X]⁰`, `q' ∉ K[X]⁰`.\n\nIf considering `K` as an integral domain, this is the same as saying that\nwe construct a value of `P` for such elements of `K⟮X⟯` by setting\n`liftOn (p / q) f _ = f p q`.\n\nWhen `[IsDomain K]`, one can use `RatFunc.liftOn'`, which has the stronger requirement\nof `∀ {p q a : K[X]} (hq : q ≠ 0) (ha : a ≠ 0), f (a * p) (a * q) = f p q)`.\n-/\nprotected irreducible_def liftOn {P : Sort v} (x : K⟮X⟯) (f : K[X] → K[X] → P)\n    (H : ∀ {p q p' q'} (_hq : q ∈ K[X]⁰) (_hq' : q' ∈ K[X]⁰), q' * p = q * p' → f p q = f p' q') :\n    P :=\n  Localization.liftOn (toFractionRing x) (fun p q => f p q) fun {_ _ q q'} h =>\n    H q.2 q'.2 (let ⟨⟨_, _⟩, mul_eq⟩ := Localization.r_iff_exists.mp h\n      mul_cancel_left_coe_nonZeroDivisors.mp mul_eq)\n\ntheorem liftOn_ofFractionRing_mk {P : Sort v} (n : K[X]) (d : K[X]⁰) (f : K[X] → K[X] → P)\n    (H : ∀ {p q p' q'} (_hq : q ∈ K[X]⁰) (_hq' : q' ∈ K[X]⁰), q' * p = q * p' → f p q = f p' q') :\n    RatFunc.liftOn (ofFractionRing (Localization.mk n d)) f @H = f n d := by\n  rw [RatFunc.liftOn]\n  exact Localization.liftOn_mk _ _ _ _\n\ntheorem liftOn_condition_of_liftOn'_condition {P : Sort v} {f : K[X] → K[X] → P}\n    (H : ∀ {p q a} (_ : q ≠ 0) (_ha : a ≠ 0), f (a * p) (a * q) = f p q) ⦃p q p' q' : K[X]⦄\n    (hq : q ≠ 0) (hq' : q' ≠ 0) (h : q' * p = q * p') : f p q = f p' q' :=\n  calc\n    f p q = f (q' * p) (q' * q) := (H hq hq').symm\n    _ = f (q * p') (q * q') := by rw [h, mul_comm q']\n    _ = f p' q' := H hq' hq\n\nsection IsDomain\n\nvariable [IsDomain K]\n\n/-- `RatFunc.mk (p q : K[X])` is `p / q` as a rational function.\n\nIf `q = 0`, then `mk` returns 0.\n\nThis is an auxiliary definition used to define an `Algebra` structure on `RatFunc`;\nthe `simp` normal form of `mk p q` is `algebraMap _ _ p / algebraMap _ _ q`.\n-/\nprotected irreducible_def mk (p q : K[X]) : K⟮X⟯ :=\n  ofFractionRing (algebraMap _ _ p / algebraMap _ _ q)\n\ntheorem mk_eq_div' (p q : K[X]) :\n    RatFunc.mk p q = ofFractionRing (algebraMap _ _ p / algebraMap _ _ q) := by rw [RatFunc.mk]\n\ntheorem mk_zero (p : K[X]) : RatFunc.mk p 0 = ofFractionRing (0 : FractionRing K[X]) := by\n  rw [mk_eq_div', map_zero, div_zero]\n\ntheorem mk_coe_def (p : K[X]) (q : K[X]⁰) :\n    RatFunc.mk p q = ofFractionRing (IsLocalization.mk' _ p q) := by\n  simp only [mk_eq_div', ← Localization.mk_eq_mk', FractionRing.mk_eq_div]\n\ntheorem mk_def_of_mem (p : K[X]) {q} (hq : q ∈ K[X]⁰) :\n    RatFunc.mk p q = ofFractionRing (IsLocalization.mk' (FractionRing K[X]) p ⟨q, hq⟩) := by\n  simp only [← mk_coe_def]\n\ntheorem mk_def_of_ne (p : K[X]) {q : K[X]} (hq : q ≠ 0) :\n    RatFunc.mk p q =\n      ofFractionRing (IsLocalization.mk' (FractionRing K[X]) p\n        ⟨q, mem_nonZeroDivisors_iff_ne_zero.mpr hq⟩) :=\n  mk_def_of_mem p _\n\ntheorem mk_eq_localization_mk (p : K[X]) {q : K[X]} (hq : q ≠ 0) :\n    RatFunc.mk p q =\n      ofFractionRing (Localization.mk p ⟨q, mem_nonZeroDivisors_iff_ne_zero.mpr hq⟩) := by\n  rw [mk_def_of_ne _ hq, Localization.mk_eq_mk']\n\nTarget:\ntheorem mk_one' (p : K[X]) :\n    RatFunc.mk p 1 = ofFractionRing (algebraMap _ _ p) :=\n\nProof body:\n","rejected":"by\n  rw [← IsLocalization.mk'_one (M := K[X]⁰) (FractionRing K[X]) p, ← mk_coe_def, Submonoid.coe_one]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/RatFunc","family_id":"mk_one","file_id":"mathlib/Mathlib/FieldTheory/RatFunc/Defs.lean","sample_id":"2ba6bee601213da996a0d324b595d9dabc6794f2b60ae27ff16595b74c0086ad"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6dfab6bad543a2d9c97fdd7bad870518b9b6feb74bbecb3fcb1c2ddbf6b41bd4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"44eed7b983007fa4c1d9c73cb7fb1872722dad2bf7f08f9f9ca305f717904a0a","source_sha256":"e3bd64c53d80fc2f3ac9f455d14462bc0198848b67a51278e36a13f2aa735979","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : IsReflexive R N := e.isReflexive_of_equiv_dual_of_isReflexive\n  suffices\n      (p.map (e.symm : Dual R M →ₗ[R] N)).dualAnnihilator.map (e.flip.symm : Dual R N →ₗ[R] M) =\n        (p.dualCoannihilator.map (e.flip : M →ₗ[R] Dual R N)).map (e.flip.symm : Dual R N →ₗ[R] M)\n    from (Submodule.map_injective_of_injective e.flip.symm.injective this).symm\n  rw [← dualCoannihilator_map_linearEquiv_flip, ← LinearEquiv.coe_toLinearMap_flip,\n    LinearEquiv.flip_flip, ← map_comp, ← map_comp]\n  simp [-coe_toLinearMap_flip]","hard_negative":false,"metrics":{"chosen_tokens":135,"rejected_tokens":2,"token_jaccard":0.02381,"token_length_ratio":0.014815},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"22e313d13272ce019dfc3eb44b147564aa4726759d0b7ea0dac62e5bd66d7058","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Dual.Lemmas\n\nNamespace:\nSubmodule\n\nLocal context:\n/-\nCopyright (c) 2023 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash, Yaël Dillies\n-/\n/-!\n# Perfect pairings\n\nThis file defines perfect pairings of modules.\n\nA perfect pairing of two (left) modules may be defined either as:\n1. A bilinear map `M × N → R` such that the induced maps `M → Dual R N` and `N → Dual R M` are both\n  bijective. It follows from this that both `M` and `N` are reflexive modules.\n2. A linear equivalence `N ≃ Dual R M` for which `M` is reflexive. (It then follows that `N` is\n  reflexive.)\n\nIn this file we provide a definition `IsPerfPair` corresponding to 1 above, together with logic\nto connect 1 and 2.\n-/\n\n@[expose] public section\n\nopen Function Module\n\nnamespace LinearMap\nvariable {R K M M' N N' : Type*} [AddCommGroup M] [AddCommGroup N] [AddCommGroup M']\n  [AddCommGroup N']\n\nsection CommRing\nvariable [CommRing R] [Module R M] [Module R M'] [Module R N] [Module R N']\n  {p : M →ₗ[R] N →ₗ[R] R} {x : M} {y : N}\n\n/-- For a ring `R` and two modules `M` and `N`, a perfect pairing is a bilinear map `M × N → R`\nthat is bijective in both arguments. -/\n@[ext]\nclass IsPerfPair (p : M →ₗ[R] N →ₗ[R] R) where\n  bijective_left (p) : Bijective p\n  bijective_right (p) : Bijective p.flip\n\n/-- Given a perfect pairing between `M` and `N`, we may interchange the roles of `M` and `N`. -/\nprotected lemma IsPerfPair.flip (hp : p.IsPerfPair) : p.flip.IsPerfPair where\n  bijective_left := IsPerfPair.bijective_right p\n  bijective_right := IsPerfPair.bijective_left p\n\nvariable [p.IsPerfPair]\n\n/-- Given a perfect pairing between `M` and `N`, we may interchange the roles of `M` and `N`. -/\ninstance flip.instIsPerfPair : p.flip.IsPerfPair := .flip ‹_›\n\nvariable (p)\n\n/-- Turn a perfect pairing between `M` and `N` into an isomorphism between `M` and the dual of `N`.\n-/\nnoncomputable def toPerfPair : M ≃ₗ[R] Dual R N :=\n  .ofBijective { toFun := _, map_add' x y := by simp, map_smul' r x := by simp } <|\n    IsPerfPair.bijective_left p\n\n@[simp] lemma toLinearMap_toPerfPair (x : M) : p.toPerfPair x = p x := rfl\n@[simp] lemma toPerfPair_apply (x : M) (y : N) : p.toPerfPair x y = p x y := rfl\n\n@[simp] lemma apply_symm_toPerfPair_self (f : Dual R N) : p (p.toPerfPair.symm f) = f :=\n  p.toPerfPair.apply_symm_apply f\n\n@[simp] lemma apply_toPerfPair_flip (f : Dual R M) (x : M) : p x (p.flip.toPerfPair.symm f) = f x :=\n  congr($(p.flip.apply_symm_toPerfPair_self ..) x)\n\ninclude p in\nlemma _root_.Module.IsReflexive.of_isPerfPair : IsReflexive R M where\n  bijective_dual_eval' := by\n    convert! (p.toPerfPair.trans p.flip.toPerfPair.dualMap.symm).bijective\n    ext x f\n    simp\n\ninclude p in\nlemma _root_.Module.finrank_of_isPerfPair [Module.Finite R M] [Module.Free R M] :\n    finrank R M = finrank R N :=\n  ((Module.Free.chooseBasis R M).toDualEquiv.trans p.flip.toPerfPair.symm).finrank_eq\n\n/-- A reflexive module has a perfect pairing with its dual. -/\nprotected instance IsPerfPair.id [IsReflexive R M] : IsPerfPair (.id (R := R) (M := Dual R M)) where\n  bijective_left := bijective_id\n  bijective_right := bijective_dual_eval R M\n\n/-- A reflexive module has a perfect pairing with its dual. -/\ninstance IsPerfPair.dualEval [IsReflexive R M] : IsPerfPair (Dual.eval R M) := .flip .id\n\ninstance IsPerfPair.compl₁₂ (eM : M' ≃ₗ[R] M) (eN : N' ≃ₗ[R] N) :\n    (p.compl₁₂ eM eN : M' →ₗ[R] N' →ₗ[R] R).IsPerfPair :=\n  ⟨((LinearEquiv.congrLeft R R eN).symm.bijective.comp\n    (IsPerfPair.bijective_left p)).comp eM.bijective,\n    ((LinearEquiv.congrLeft R R eM).symm.bijective.comp\n    (IsPerfPair.bijective_right p)).comp eN.bijective⟩\n\nlemma IsPerfPair.congr (eM : M' ≃ₗ[R] M) (eN : N' ≃ₗ[R] N) (q : M' →ₗ[R] N' →ₗ[R] R)\n    (H : q.compl₁₂ eM.symm eN.symm = p) : q.IsPerfPair := by\n  obtain rfl : q = p.compl₁₂ eM eN := by subst H; ext; simp\n  infer_instance\n\nlemma IsPerfPair.of_bijective (p : M →ₗ[R] N →ₗ[R] R) [IsReflexive R N] (h : Bijective p) :\n    IsPerfPair p :=\n  inferInstanceAs ((LinearMap.id (R := R) (M := Dual R N)).compl₁₂\n    (LinearEquiv.ofBijective p h : M →ₗ[R] N →ₗ[R] R)\n    (LinearEquiv.refl R N : N →ₗ[R] N)).IsPerfPair\n\nend CommRing\n\nsection Field\nvariable [Field K] [Module K M] [Module K N] {p : M →ₗ[K] N →ₗ[K] K} {x : M} {y : N}\n\n/-- If the coefficients are a field, and one of the spaces is finite-dimensional, it is sufficient\nto check only injectivity instead of bijectivity of the bilinear pairing. -/\nlemma IsPerfPair.of_injective [FiniteDimensional K M] (h : Injective p) (h' : Injective p.flip) :\n    p.IsPerfPair where\n  bijective_left := ⟨h, by rwa [← p.flip_injective_iff₁]⟩\n  bijective_right := ⟨h', by\n    have : FiniteDimensional K N := FiniteDimensional.of_injective p.flip h'\n    rwa [← p.flip.flip_injective_iff₁, LinearMap.flip_flip]⟩\n\n/-- If the coefficients are a field, and one of the spaces is finite-dimensional, it is sufficient\nto check only injectivity instead of bijectivity of the bilinear pairing. -/\nlemma IsPerfPair.of_injective' [FiniteDimensional K N] (h : Injective p) (h' : Injective p.flip) :\n    p.IsPerfPair := .flip <| .of_injective h' h\n\nend Field\nend LinearMap\n\nnoncomputable section\n\nvariable {R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]\n\nnamespace LinearMap\nvariable {p : M →ₗ[R] N →ₗ[R] R} [p.IsPerfPair]\n\nvariable (p) in\n/-- Given a perfect pairing `p` between `M` and `N`, we say a pair of submodules `U` in `M` and\n`V` in `N` are perfectly complementary w.r.t. `p` if their dual annihilators are complementary,\nusing `p` to identify `M` and `N` with dual spaces. -/\nstructure IsPerfectCompl (U : Submodule R M) (V : Submodule R N) : Prop where\n  isCompl_left : IsCompl U (V.dualAnnihilator.map (p.toPerfPair.symm : Dual R N →ₗ[R] M))\n  isCompl_right : IsCompl V (U.dualAnnihilator.map (p.flip.toPerfPair.symm : Dual R M →ₗ[R] N))\n\nnamespace IsPerfectCompl\nvariable {U : Submodule R M} {V : Submodule R N}\n\nprotected lemma flip (h : p.IsPerfectCompl U V) :\n    p.flip.IsPerfectCompl V U where\n  isCompl_left := h.isCompl_right\n  isCompl_right := h.isCompl_left\n\n@[simp]\nprotected lemma flip_iff :\n    p.flip.IsPerfectCompl V U ↔ p.IsPerfectCompl U V :=\n  ⟨fun h ↦ h.flip, fun h ↦ h.flip⟩\n\n@[simp]\nlemma left_top_iff :\n    p.IsPerfectCompl ⊤ V ↔ V = ⊤ := by\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · exact eq_top_of_isCompl_bot <| by simpa using h.isCompl_right\n  · rw [h]\n    exact\n      { isCompl_left := by simpa using isCompl_top_bot\n        isCompl_right := by simpa using isCompl_top_bot }\n\n@[simp]\nlemma right_top_iff :\n    p.IsPerfectCompl U ⊤ ↔ U = ⊤ := by\n  rw [← IsPerfectCompl.flip_iff]\n  exact left_top_iff\n\nend IsPerfectCompl\n\nend LinearMap\n\nvariable [IsReflexive R M]\n\nvariable (e : N ≃ₗ[R] Dual R M)\n\nnamespace LinearEquiv\n\n/-- For a reflexive module `M`, an equivalence `N ≃ₗ[R] Dual R M` naturally yields an equivalence\n`M ≃ₗ[R] Dual R N`. Such equivalences are known as perfect pairings. -/\ndef flip : M ≃ₗ[R] Dual R N :=\n  (evalEquiv R M).trans e.dualMap\n\n@[simp] lemma coe_toLinearMap_flip : e.flip = (↑e : N →ₗ[R] Dual R M).flip := rfl\n\n@[simp] lemma flip_apply (m : M) (n : N) : e.flip m n = e n m := rfl\n\nlemma symm_flip : e.flip.symm = e.symm.dualMap.trans (evalEquiv R M).symm := rfl\n\nlemma trans_dualMap_symm_flip : e.trans e.flip.symm.dualMap = Dual.eval R N := by\n  ext; simp [symm_flip]\n\ninclude e in\n/-- If `N` is in perfect pairing with `M`, then it is reflexive. -/\nlemma isReflexive_of_equiv_dual_of_isReflexive : IsReflexive R N := by\n  constructor\n  rw [← trans_dualMap_symm_flip e]\n  exact LinearEquiv.bijective _\n\n@[simp] lemma flip_flip (h : IsReflexive R N := isReflexive_of_equiv_dual_of_isReflexive e) :\n    e.flip.flip = e := by\n  ext; rfl\n\ninstance : e.toLinearMap.IsPerfPair where\n  bijective_left := e.bijective\n  bijective_right := e.flip.bijective\n\nend LinearEquiv\n\nnamespace Submodule\n\nopen LinearEquiv\n\nomit [IsReflexive R M] in\n@[simp]\nlemma dualCoannihilator_map_linearEquiv_flip (p : Submodule R M) :\n    (p.map e.toLinearMap.flip).dualCoannihilator =\n      p.dualAnnihilator.map (e.symm : Dual R M →ₗ[R] N) := by\n  ext; simp\n\n@[simp]\nlemma map_dualAnnihilator_linearEquiv_flip_symm (p : Submodule R N) :\n    p.dualAnnihilator.map (e.flip.symm : Dual R N →ₗ[R] M) =\n      (p.map (e : N →ₗ[R] Dual R M)).dualCoannihilator := by\n  have : IsReflexive R N := e.isReflexive_of_equiv_dual_of_isReflexive\n  rw [← dualCoannihilator_map_linearEquiv_flip, ← LinearEquiv.coe_toLinearMap_flip,\n    LinearEquiv.flip_flip]\n\n@[simp]\n\nTarget:\nlemma map_dualCoannihilator_linearEquiv_flip (p : Submodule R (Dual R M)) :\n    p.dualCoannihilator.map e.toLinearMap.flip =\n      (p.map (e.symm : Dual R M →ₗ[R] N)).dualAnnihilator :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/PerfectPairing","family_id":"map_dualcoannihilator_linearequiv_flip","file_id":"mathlib/Mathlib/LinearAlgebra/PerfectPairing/Basic.lean","sample_id":"44eed7b983007fa4c1d9c73cb7fb1872722dad2bf7f08f9f9ca305f717904a0a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"255334aceee88256ccde5808751019138f069b260876b485bc11ea482a1aed8d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8a92de4cf09f54437fffe352359b2fe17de76c67cc1f4f0cdf36c80b9799f8f6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"04f65d5b50a300cc6227733efdcd3f39e044ddd8767d10f8047fdbe408fc2df7","source_sha256":"98d8a988714b8411039a52ca8cb94b19a051f27bd0e995b1332bddd6c23da9ea","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro b hb\n  have hg' := hg b hb\n  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hg'⟩ := hg'\n  refine ⟨c₁, hc₁_mem, c₂, hc₂_mem, ?_⟩\n  filter_upwards [hg', (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg, hfg]\n    with x hx₁ hx₂ hx₃\n  intro u hu\n  rw [hx₂ u hu.1, hx₃]\n  exact hx₁ u hu","hard_negative":false,"metrics":{"chosen_tokens":95,"rejected_tokens":99,"token_jaccard":0.928571,"token_length_ratio":1.042105},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"2335bedc644e23a0e929bce581ada7640866b3e1884652288ea617cc065a13f8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Pow.Real\npublic import Mathlib.Algebra.Order.ToIntervalMod\npublic import Mathlib.Analysis.SpecialFunctions.Log.Base\nimport Mathlib.Algebra.Order.Interval.Set.Group\n\nNamespace:\nAkraBazziRecurrence.GrowsPolynomially\n\nLocal context:\n/-\nCopyright (c) 2023 Frédéric Dupuis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Frédéric Dupuis\n-/\n/-!\n# Akra-Bazzi theorem: the polynomial growth condition\n\nThis file defines and develops an API for the polynomial growth condition that appears in the\nstatement of the Akra-Bazzi theorem: for the theorem to hold, the function `g` must\nsatisfy the condition that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b*n` and `n` for any\nconstant `b ∈ (0,1)`.\n\n## Implementation notes\n\nOur definition requires that the condition hold for any `b ∈ (0,1)`. This is equivalent to requiring\nit only for `b = 1 / 2` (or any other particular value in `(0, 1)`). While this could, in principle,\nmake it harder to prove that a particular function grows polynomially, this issue does not seem to\narise in practice.\n\n-/\n\n@[expose] public section\n\nopen Finset Real Filter Asymptotics\nopen scoped Topology\n\nnamespace AkraBazziRecurrence\n\n/-- The growth condition that the function `g` must satisfy for the Akra-Bazzi theorem to apply.\nIt roughly states that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b * n` and `n`, for any\nconstant `b ∈ (0, 1)`. -/\ndef GrowsPolynomially (f : ℝ → ℝ) : Prop :=\n  ∀ b ∈ Set.Ioo 0 1, ∃ c₁ > 0, ∃ c₂ > 0,\n    ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * (f x)) (c₂ * f x)\n\nnamespace GrowsPolynomially\n\nTarget:\nlemma congr_of_eventuallyEq {f g : ℝ → ℝ} (hfg : f =ᶠ[atTop] g) (hg : GrowsPolynomially g) :\n    GrowsPolynomially f :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  intro b hb\n  have hg' := hg b hb\n  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hg'⟩ := hg'\n  refine ⟨c₁, hc₁_mem, c₂, hc₂_mem, ?_⟩\n  filter_upwards [hg', (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg, hfg]\n    with x hx₁ hx₂ hx₃\n  intro u hu\n  rw [hx₂ u hu.1, hx₃]\n  exact hx₁ u hu","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability/AkraBazzi","family_id":"congr_of_eventuallyeq","file_id":"mathlib/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean","sample_id":"04f65d5b50a300cc6227733efdcd3f39e044ddd8767d10f8047fdbe408fc2df7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"46daf5b2e7bf4d90e3013b1a193808b8c73aa5defa9de5956e0736e8e4908309","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b1b68db34cb48a6039b6507dde691821271f6dc0755eada87fdde3b2fa4932ab","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9439016a6f4e6a57c85536640298da69f3a589358dd7d886fc77db2d3daea612","source_sha256":"18b2ea13d158ef44c4da5d71e0384f4a95c10d686d0e4d8e534d2cf89fc900ba","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply of_span_eq_top_target _ hs\n  rintro - ⟨i, rfl⟩\n  exact .equiv <| (IsLocalization.algEquiv (.powers <| s i) _ (T i)).symm |>.restrictScalars R","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":47,"token_jaccard":0.848485,"token_length_ratio":1.119048},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"239a92bc3dca9067c2fa51e48aac57dbb7a02d1c7d477729df4282490bd80097","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Finiteness.FiniteTypeLocal\npublic import Mathlib.RingTheory.Localization.Away.AdjoinRoot\n\nNamespace:\nAlgebra.FinitePresentation\n\nLocal context:\n/-\nCopyright (c) 2024 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten\n-/\n/-!\n\n# `Algebra.FinitePresentation` is local\n\nIn this file we show that being a finitely presented algebra is local.\n\n## Main results\n\n- `Algebra.FinitePresentation.of_span_eq_top_target`: finite presentation is local on the\n  (algebraic) target\n\n-/\n\npublic section\n\nopen scoped Pointwise TensorProduct\n\nnamespace Algebra.FinitePresentation\n\nvariable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]\n\n/--\nIf `S` is an `R`-algebra with a surjection from a finitely-presented `R`-algebra `A`, such that\nlocalized at a spanning set `{ r }` of elements of `A`, `Sᵣ` is finitely-presented, then\n`S` is finitely presented.\nThis is almost `finitePresentation_ofLocalizationSpanTarget`. The difference is,\nthat here the set `t` generates the unit ideal of `A`, while in the general version,\nit only generates a quotient of `A`.\n-/\nlemma of_span_eq_top_target_aux {A : Type*} [CommRing A] [Algebra R A]\n    [Algebra.FinitePresentation R A] (f : A →ₐ[R] S) (hf : Function.Surjective f)\n    (t : Finset A) (ht : Ideal.span (t : Set A) = ⊤)\n    (H : ∀ g : t, Algebra.FinitePresentation R (Localization.Away (f g))) :\n    Algebra.FinitePresentation R S := by\n  apply Algebra.FinitePresentation.of_surjective hf\n  apply RingHom.ker_fg_of_localizationSpan t ht\n  intro g\n  let f' : Localization.Away g.val →ₐ[R] Localization.Away (f g) :=\n    Localization.awayMapₐ f g.val\n  have (g : t) : Algebra.FinitePresentation R (Localization.Away g.val) :=\n    haveI : Algebra.FinitePresentation A (Localization.Away g.val) :=\n      IsLocalization.Away.finitePresentation g.val\n    Algebra.FinitePresentation.trans R A (Localization.Away g.val)\n  apply Algebra.FinitePresentation.ker_fG_of_surjective f'\n  exact IsLocalization.Away.mapₐ_surjective_of_surjective _ hf\n\nuniverse u\n\n/-- Finite-presentation can be checked on a standard covering of the target. -/\nlemma of_span_eq_top_target (s : Set S) (hs : Ideal.span (s : Set S) = ⊤)\n    (h : ∀ i ∈ s, Algebra.FinitePresentation R (Localization.Away i)) :\n    Algebra.FinitePresentation R S := by\n  obtain ⟨s, h₁, hs⟩ := (Ideal.span_eq_top_iff_finite s).mp hs\n  replace h (i : s) : Algebra.FinitePresentation R (Localization.Away i.val) := h i (h₁ i.property)\n  classical\n  /-\n  We already know that `S` is of finite type over `R`, so we have a surjection\n  `MvPolynomial (Fin n) R →ₐ[R] S`. To reason about the kernel, we want to check it on the stalks\n  of preimages of `s`. But the preimages do not necessarily span `MvPolynomial (Fin n) R`, so\n  we quotient out by an ideal and apply `finitePresentation_ofLocalizationSpanTarget_aux`.\n  -/\n  have hfintype : Algebra.FiniteType R S := by\n    apply Algebra.FiniteType.of_span_eq_top_target s hs\n    intro x hx\n    have := h ⟨x, hx⟩\n    infer_instance\n  obtain ⟨n, f, hf⟩ := Algebra.FiniteType.iff_quotient_mvPolynomial''.mp hfintype\n  obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_linearCombination S (s : Set S) 1).mp\n      (show (1 : S) ∈ Ideal.span (s : Set S) by rw [hs]; trivial)\n  choose g' hg' using (fun g : s ↦ hf g)\n  choose h' hh' using (fun g : s ↦ hf (l g))\n  let I : Ideal (MvPolynomial (Fin n) R) := Ideal.span { ∑ g : s, g' g * h' g - 1 }\n  let A := MvPolynomial (Fin n) R ⧸ I\n  have hfI : ∀ a ∈ I, f a = 0 := by\n    intro p hp\n    simp only [Finset.univ_eq_attach, I, Ideal.mem_span_singleton] at hp\n    obtain ⟨q, rfl⟩ := hp\n    simp only [map_mul, map_sub, map_sum, map_one, hg', hh']\n    rw [Finsupp.linearCombination_apply_of_mem_supported (α := (s : Set S)) S (s := s.attach)] at hl\n    · rw [← hl]\n      simp only [Finset.coe_sort_coe, smul_eq_mul, mul_comm, sub_self, zero_mul]\n    · rintro a -\n      simp\n  let f' : A →ₐ[R] S := Ideal.Quotient.liftₐ I f hfI\n  have hf' : Function.Surjective f' :=\n    Ideal.Quotient.lift_surjective_of_surjective I hfI hf\n  let t : Finset A := Finset.image (fun g ↦ g' g) Finset.univ\n  have ht : Ideal.span (t : Set A) = ⊤ := by\n    rw [Ideal.eq_top_iff_one]\n    have : ∑ g : { x // x ∈ s }, g' g * h' g = (1 : A) := by\n      apply eq_of_sub_eq_zero\n      rw [← map_one (Ideal.Quotient.mk I), ← map_sub, Ideal.Quotient.eq_zero_iff_mem]\n      apply Ideal.subset_span\n      simp\n    simp_rw [← this, Finset.univ_eq_attach, map_sum, map_mul]\n    refine Ideal.sum_mem _ (fun g _ ↦ Ideal.mul_mem_right _ _ <| Ideal.subset_span ?_)\n    simp [t]\n  have : Algebra.FinitePresentation R A := by\n    apply Algebra.FinitePresentation.quotient\n    simp only [Finset.univ_eq_attach, I]\n    exact ⟨{∑ g ∈ s.attach, g' g * h' g - 1}, by simp⟩\n  have Ht (g : t) : Algebra.FinitePresentation R (Localization.Away (f' g)) := by\n    have : ∃ (a : S) (hb : a ∈ s), (Ideal.Quotient.mk I) (g' ⟨a, hb⟩) = g.val := by\n      obtain ⟨g, hg⟩ := g\n      convert! hg\n      simp [A, t]\n    obtain ⟨r, hr, hrr⟩ := this\n    simp only [f']\n    rw [← hrr, Ideal.Quotient.liftₐ_apply, Ideal.Quotient.lift_mk]\n    simp_rw +instances [RingHom.coe_coe]\n    rw [hg']\n    apply h\n  exact of_span_eq_top_target_aux f' hf' t ht Ht\n\n/-- Finite-presentation can be checked on a standard covering of the target. -/\n\nTarget:\nlemma of_span_eq_top_target_of_isLocalizationAway {ι : Type*} (s : ι → S)\n    (hs : Ideal.span (Set.range s) = ⊤) (T : ι → Type*) [∀ i, CommRing (T i)] [∀ i, Algebra R (T i)]\n    [∀ i, Algebra S (T i)] [∀ i, IsScalarTower R S (T i)] [∀ i, IsLocalization.Away (s i) (T i)]\n    [∀ i, Algebra.FinitePresentation R (T i)] :\n    Algebra.FinitePresentation R S :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  apply of_span_eq_top_target _ hs\n  rintro - ⟨i, rfl⟩\n  exact .equiv <| (IsLocalization.algEquiv (.powers <| s i) _ (T i)).symm |>.restrictScalars R","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Finiteness","family_id":"of_span_eq_top_target_of_islocalizationaway","file_id":"mathlib/Mathlib/RingTheory/Finiteness/FinitePresentationLocal.lean","sample_id":"9439016a6f4e6a57c85536640298da69f3a589358dd7d886fc77db2d3daea612"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2933327708396b2202fe5681ecb1091bcb221fad36d86c22e4ed4f768e81a87c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d58f1e0ee25611d5386590a468f4a791724666cd5a828ff119f2caeb8a944a6e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"77f925e6d7fb9f855b12d8cba8f5cb36cbebbc68e7684132b1a6af65dd6fe312","source_sha256":"fcd4a8db6b3cd3d3eac0ed5f4bd02b4de008d2f8c57c99f188ee276c869b4813","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext <;> simp","hard_negative":false,"metrics":{"chosen_tokens":6,"rejected_tokens":11,"token_jaccard":0.545455,"token_length_ratio":1.833333},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"24208e0adf1f8ddd0ac8dac34ac4834a4ae66d40044f9c3eee4e3696e49f4a73","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Bicategory.LocallyDiscrete\npublic import Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo\n\nNamespace:\nCategoryTheory.Pseudofunctor.Grothendieck\n\nLocal context:\n/-\nCopyright (c) 2024 Calle Sönne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Calle Sönne, Joseph Hua\n-/\n/-!\n# The Grothendieck and CoGrothendieck constructions\n\n## The Grothendieck construction\n\nGiven a category `𝒮` and any pseudofunctor `F` from `𝒮` to `Cat`, we associate to it a category\n`∫ F`, defined as follows:\n* Objects: pairs `(S, a)` where `S` is an object of the base category and `a` is an object of the\n  category `F(S)`.\n* Morphisms: morphisms `(R, b) ⟶ (S, a)` are defined as pairs `(f, h)` where `f : R ⟶ S` is a\n  morphism in `𝒮` and `h : F(f)(a) ⟶ b`\n\nThe category `∫ F` is equipped with a projection functor `∫ F ⥤ 𝒮`,\ngiven by projecting to the first factors, i.e.\n* On objects, it sends `(S, a)` to `S`\n* On morphisms, it sends `(f, h)` to `f`\n\n## The CoGrothendieck construction\n\nGiven a category `𝒮` and any pseudofunctor `F` from `𝒮ᵒᵖ` to `Cat`,\nwe associate to it a category `∫ᶜ F`, defined as follows:\n* Objects: pairs `(S, a)` where `S` is an object of the base category and `a` is an object of the\n  category `F(S)`.\n* Morphisms: morphisms `(R, b) ⟶ (S, a)` are defined as pairs `(f, h)` where `f : R ⟶ S` is a\n  morphism in `𝒮` and `h : b ⟶ F(f)(a)`\n\nThe category `∫ᶜ F` is equipped with a functor `∫ᶜ F ⥤ 𝒮`,\ngiven by projecting to the first factors, i.e.\n* On objects, it sends `(S, a)` to `S`\n* On morphisms, it sends `(f, h)` to `f`\n\n## Naming conventions\n\nThe name `Grothendieck` is reserved for the construction on covariant pseudofunctors from `𝒮` to\n`Cat`, whereas the word `CoGrothendieck` is used for the contravariant construction.\nThis is consistent with the convention for the Grothendieck construction on 1-functors\n`CategoryTheory.Grothendieck`.\n\n## Future work / TODO\n\n1. Once the bicategory of pseudofunctors has been defined, show that this construction forms a\n   pseudofunctor from `LocallyDiscrete 𝒮 ⥤ᵖ Catᵒᵖ` to `Cat`.\n2. Deduce the results in `CategoryTheory.Grothendieck` as a specialization of\n   `Pseudofunctor.Grothendieck`.\n\n## References\n[Vistoli2008] \"Notes on Grothendieck Topologies, Fibered Categories and Descent Theory\" by\nAngelo Vistoli\n\n-/\n\nset_option backward.defeqAttrib.useBackward true\n\n@[expose] public section\n\nnamespace CategoryTheory.Pseudofunctor\n\nuniverse w v₁ v₂ v₃ u₁ u₂ u₃\n\nopen Functor Category Opposite Discrete Bicategory StrongTrans\n\nvariable {𝒮 : Type u₁} [Category.{v₁} 𝒮]\n\n/-- The type of objects in the fibered category associated to a pseudofunctor from a\n1-category to Cat. -/\n@[ext]\nstructure Grothendieck (F : LocallyDiscrete 𝒮 ⥤ᵖ Cat.{v₂, u₂}) where\n  /-- The underlying object in the base category. -/\n  base : 𝒮\n  /-- The object in the fiber of the base object. -/\n  fiber : F.obj ⟨base⟩\n\nnamespace Grothendieck\n\nvariable {F : LocallyDiscrete 𝒮 ⥤ᵖ Cat.{v₂, u₂}}\n\n/-- Notation for the Grothendieck category associated to a pseudofunctor `F`. -/\nscoped prefix:75 \"∫ \" => Grothendieck\n\n/-- A morphism in the Grothendieck construction `∫ F` between two points `X Y : ∫ F` consists of\na morphism in the base category `base : X.base ⟶ Y.base` and\na morphism in a fiber `f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber`. -/\nstructure Hom (X Y : ∫ F) where\n  /-- The morphism between base objects. -/\n  base : X.base ⟶ Y.base\n  /-- The morphism in the fiber over the domain. -/\n  fiber : (F.map base.toLoc).toFunctor.obj X.fiber ⟶ Y.fiber\n\n@[simps! id_base id_fiber comp_base comp_fiber]\ninstance categoryStruct : CategoryStruct (∫ F) where\n  Hom X Y := Hom X Y\n  id X := {\n    base := 𝟙 X.base\n    fiber := (F.mapId ⟨X.base⟩).hom.toNatTrans.app X.fiber }\n  comp {X _ _} f g := {\n    base := f.base ≫ g.base\n    fiber := (F.mapComp f.base.toLoc g.base.toLoc).hom.toNatTrans.app X.fiber ≫\n      (F.map g.base.toLoc).toFunctor.map f.fiber ≫ g.fiber }\n\ninstance (X : ∫ F) : Inhabited (Hom X X) :=\n  ⟨𝟙 X⟩\n\nsection\n\nvariable {a b : ∫ F}\n\n@[ext (iff := false)]\nlemma Hom.ext (f g : a ⟶ b) (hfg₁ : f.base = g.base)\n    (hfg₂ : eqToHom (hfg₁ ▸ rfl) ≫ f.fiber = g.fiber) : f = g := by\n  cases f; cases g\n  dsimp at hfg₁ hfg₂\n  cat_disch\n\nlemma Hom.ext_iff (f g : a ⟶ b) :\n    f = g ↔ ∃ (hfg : f.base = g.base), eqToHom (hfg ▸ rfl) ≫ f.fiber = g.fiber where\n  mp hfg := by subst hfg; simp\n  mpr := fun ⟨hfg₁, hfg₂⟩ => Hom.ext f g hfg₁ hfg₂\n\nlemma Hom.congr {a b : ∫ F} {f g : a ⟶ b} (h : f = g) :\n    f.fiber = eqToHom (h ▸ rfl) ≫ g.fiber := by\n  subst h\n  simp\n\nend\n\nset_option backward.isDefEq.respectTransparency false in\nattribute [local simp] PrelaxFunctor.map₂_eqToHom in\n/-- The category structure on `∫ F`. -/\ninstance category : Category (∫ F) where\n  toCategoryStruct := Pseudofunctor.Grothendieck.categoryStruct\n  id_comp {a b} f := by\n    ext\n    · simp\n    · simp [F.mapComp_id_left_hom_app, Strict.leftUnitor_eqToIso, ← Functor.map_comp_assoc,\n        ← Cat.Hom₂.comp_app]\n  comp_id {a b} f := by\n    ext\n    · simp\n    · simp [F.mapComp_id_right_hom_app, Strict.rightUnitor_eqToIso, ← reassoc_of% Cat.Hom₂.comp_app]\n  assoc f g h := by\n    ext\n    · simp\n    · simp [mapComp_assoc_right_hom_app_assoc, Strict.associator_eqToIso]\n\nvariable (F)\n\n/-- The projection `∫ F ⥤ 𝒮` given by projecting both objects and homs to the first factor. -/\n@[simps]\ndef forget (F : Pseudofunctor (LocallyDiscrete 𝒮) Cat.{v₂, u₂}) : ∫ F ⥤ 𝒮 where\n  obj X := X.base\n  map f := f.base\n\nsection\n\nattribute [local simp]\n  Strict.leftUnitor_eqToIso Strict.rightUnitor_eqToIso Strict.associator_eqToIso\n\nvariable {F} {G : Pseudofunctor (LocallyDiscrete 𝒮) Cat.{v₂, u₂}}\n  {H : Pseudofunctor (LocallyDiscrete 𝒮) Cat.{v₂, u₂}}\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The Grothendieck construction is functorial: a strong natural transformation `α : F ⟶ G`\ninduces a functor `Grothendieck.map : ∫ F ⥤ ∫ G`. -/\n@[simps!]\ndef map (α : F ⟶ G) : ∫ F ⥤ ∫ G where\n  obj a := {\n    base := a.base\n    fiber := (α.app ⟨a.base⟩).toFunctor.obj a.fiber }\n  map {a b} f := {\n    base := f.1\n    fiber := (α.naturality f.1.toLoc).inv.toNatTrans.app a.fiber ≫\n      (α.app ⟨b.base⟩).toFunctor.map f.2 }\n  map_id a := by\n    ext\n    · dsimp\n    · simp [StrongTrans.naturality_id_inv_app, ← map_comp, ← Cat.Hom₂.comp_app]\n  map_comp {a b c} f g := by\n    ext\n    · dsimp\n    · simp only [Cat.Hom.comp_toFunctor, comp_obj, categoryStruct_comp_base, Quiver.Hom.comp_toLoc,\n        categoryStruct_comp_fiber, eqToHom_refl, map_comp, ← Cat.Hom.comp_map, assoc,\n        NatTrans.naturality_assoc]\n      simp [naturality_comp_inv_app, ← Functor.map_comp, ← reassoc_of% Cat.Hom₂.comp_app]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\n\nTarget:\nlemma map_id_map {x y : ∫ F} (f : x ⟶ y) : (map (𝟙 F)).map f = f :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  ext <;> simp","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Bicategory","family_id":"map_id_map","file_id":"mathlib/Mathlib/CategoryTheory/Bicategory/Grothendieck.lean","sample_id":"77f925e6d7fb9f855b12d8cba8f5cb36cbebbc68e7684132b1a6af65dd6fe312"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f9aa1f2b0c3ec5667a46cb932e6917f3da85ddecbe4e4201b9fff6f973d74358","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ca4d4f9887ee3a892e6506dc6023e4d2417fcaf8ede1273a6e1bb19013900ef6","source_sha256":"575fbe654cd51b260a28c1b494f58357a3030804869742bdef7a7dc326096822","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨a, b, hij⟩ | ⟨a, b, hab⟩ := h\n  · exact Lex.left _ _ (hr _ _ hij)\n  · exact Lex.right _ _ (hs _ _ _ hab)","hard_negative":false,"metrics":{"chosen_tokens":46,"rejected_tokens":5,"token_jaccard":0.12,"token_length_ratio":0.108696},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"242bc8e63e5959bf2431ff03014f6c585fb6037bb738514c6a4bc1d0bb00d08a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Logic.Function.Defs\npublic import Mathlib.Order.Defs.Unbundled\npublic import Batteries.Logic\n\nNamespace:\nSigma\n\nLocal context:\n/-\nCopyright (c) 2021 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Lexicographic order on a sigma type\n\nThis defines the lexicographical order of two arbitrary relations on a sigma type and proves some\nlemmas about `PSigma.Lex`, which is defined in core Lean.\n\nGiven a relation in the index type and a relation on each summand, the lexicographical order on the\nsigma type relates `a` and `b` if their summands are related or they are in the same summand and\nrelated by the summand's relation.\n\n## See also\n\nRelated files are:\n* `Combinatorics.CoLex`: Colexicographic order on finite sets.\n* `Data.List.Lex`: Lexicographic order on lists.\n* `Data.Sigma.Order`: Lexicographic order on `Σ i, α i` per say.\n* `Data.PSigma.Order`: Lexicographic order on `Σ' i, α i`.\n* `Data.Prod.Lex`: Lexicographic order on `α × β`. Can be thought of as the special case of\n  `Sigma.Lex` where all summands are the same\n-/\n\npublic section\n\n\nnamespace Sigma\n\nvariable {ι : Type*} {α : ι → Type*} {r r₁ r₂ : ι → ι → Prop} {s s₁ s₂ : ∀ i, α i → α i → Prop}\n  {a b : Σ i, α i}\n\n/-- The lexicographical order on a sigma type. It takes in a relation on the index type and a\nrelation for each summand. `a` is related to `b` iff their summands are related or they are in the\nsame summand and are related through the summand's relation. -/\ninductive Lex (r : ι → ι → Prop) (s : ∀ i, α i → α i → Prop) : ∀ _ _ : Σ i, α i, Prop\n  | left {i j : ι} (a : α i) (b : α j) : r i j → Lex r s ⟨i, a⟩ ⟨j, b⟩\n  | right {i : ι} (a b : α i) : s i a b → Lex r s ⟨i, a⟩ ⟨i, b⟩\n\ntheorem lex_iff : Lex r s a b ↔ r a.1 b.1 ∨ ∃ h : a.1 = b.1, s b.1 (h.rec a.2) b.2 := by\n  constructor\n  · rintro (⟨a, b, hij⟩ | ⟨a, b, hab⟩)\n    · exact Or.inl hij\n    · exact Or.inr ⟨rfl, hab⟩\n  · obtain ⟨i, a⟩ := a\n    dsimp only\n    rintro (h | ⟨rfl, h⟩)\n    · exact Lex.left _ _ h\n    · exact Lex.right _ _ h\n\ninstance Lex.decidable (r : ι → ι → Prop) (s : ∀ i, α i → α i → Prop) [DecidableEq ι]\n    [DecidableRel r] [∀ i, DecidableRel (s i)] : DecidableRel (Lex r s) := fun _ _ =>\n  decidable_of_decidable_of_iff lex_iff.symm\n\nTarget:\ntheorem Lex.mono (hr : ∀ a b, r₁ a b → r₂ a b) (hs : ∀ i a b, s₁ i a b → s₂ i a b) {a b : Σ i, α i}\n    (h : Lex r₁ s₁ a b) : Lex r₂ s₂ a b :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sigma","family_id":"lex","file_id":"mathlib/Mathlib/Data/Sigma/Lex.lean","sample_id":"ca4d4f9887ee3a892e6506dc6023e4d2417fcaf8ede1273a6e1bb19013900ef6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"da9b69d9b6dc7fdb8d3935d0065bc697214d7335fb75b8a7df981601b0c50a86","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c850dd5f680796ef8939e552d2ed07dbbf81b4eaedf7784baebc1c5d3cb8013c","source_sha256":"ebea26ad83d186da19f8d5aec5a4532b5e10776580381e96c50d87a2ed0ca5b4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  simpa using (eq_bot_or_eq_top _).symm","hard_negative":false,"metrics":{"chosen_tokens":10,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.3},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"244021c0c967282e0f564e4ca4fc44557fc485ce1fe12759dc70e9ea9bec6cb5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Lattice\npublic import Mathlib.Order.Atoms\npublic import Mathlib.Order.Interval.Finset.Defs\npublic import Mathlib.Order.Preorder.Finite\n\nNamespace:\nFintype.IsSimpleOrder\n\nLocal context:\n/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Atoms, Coatoms, Simple Lattices, and Finiteness\n\nThis module contains some results on atoms and simple lattices in the finite context.\n\n## Main results\n* `Finite.to_isAtomic`, `Finite.to_isCoatomic`: Finite partial orders with bottom resp. top\n  are atomic resp. coatomic.\n\n-/\n\npublic section\n\n\nvariable {α β : Type*}\n\nnamespace IsSimpleOrder\n\nvariable [LE α] [BoundedOrder α] [IsSimpleOrder α]\n\nsection DecidableEq\n\n/-- It is important that `IsSimpleOrder` is the last type-class argument of this instance,\nso that type-class inference fails quickly if it doesn't apply.\n\nNote that as of 2025-08-13, this is false. Could someone investigate? -/\nscoped instance (priority := 200) [DecidableEq α] : Fintype α :=\n  Fintype.ofEquiv Bool equivBool.symm\n\nend DecidableEq\n\nscoped instance (priority := 200) : Finite α := by classical infer_instance\n\nend IsSimpleOrder\n\nnamespace Fintype\n\nnamespace IsSimpleOrder\n\nopen scoped _root_.IsSimpleOrder\n\nvariable [LE α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α]\n\nTarget:\ntheorem univ : (Finset.univ : Finset α) = {⊤, ⊥} :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Atoms","family_id":"univ","file_id":"mathlib/Mathlib/Order/Atoms/Finite.lean","sample_id":"c850dd5f680796ef8939e552d2ed07dbbf81b4eaedf7784baebc1c5d3cb8013c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"447b3ac2404a7fdae48d1d41ba080e9c5434ee1fd77cdc32e3d7e6b39eb7aa1d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b38b14c96b38afbba7c3259cf4c4e7b4c145e1b2ac6c153a770feeac3a7076d5","source_sha256":"ffc9175aacac138c9dfb496e1bcda0b6ab6a54d2ace871f4d5aff79b2c6d448a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h ↦ ?_, fun h ↦ h.units_smul w⟩\n  convert! h.units_smul (fun i ↦ (w i)⁻¹)\n  simp [funext_iff]","hard_negative":true,"metrics":{"chosen_tokens":37,"rejected_tokens":8,"token_jaccard":0.033333,"token_length_ratio":0.216216},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"245ea563a30fcf5dc95495367db27739788352c8c6bc6e7f5623f0a4a17a6a43","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Fin\npublic import Mathlib.LinearAlgebra.LinearIndependent.Defs\npublic import Mathlib.Logic.Equiv.Fin.Rotate\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen\n-/\n/-!\n# Linear independence\n\nThis file collects basic consequences of linear (in)dependence and includes specialized tests for\nspecific families of vectors.\n\n## Main statements\n\nWe prove several specialized tests for linear independence of families of vectors and of sets of\nvectors.\n\n* `linearIndependent_empty_type`: a family indexed by an empty type is linearly independent;\n* `linearIndependent_unique_iff`: if `ι` is a singleton, then `LinearIndependent K v` is\n  equivalent to `v default ≠ 0`;\n* `linearIndependent_sum`: type-specific test for linear independence of families of vector\n  fields;\n* `linearIndependent_singleton`: linear independence tests for set operations.\n\nIn many cases we additionally provide dot-style operations (e.g., `LinearIndependent.union`) to\nmake the linear independence tests usable as `hv.insert ha` etc.\n\n## TODO\n\nRework proofs to hold in semirings, by avoiding the path through\n`ker (Finsupp.linearCombination R v) = ⊥`.\n\n## Tags\n\nlinearly dependent, linear dependence, linearly independent, linear independence\n\n-/\n\npublic section\n\nassert_not_exists Cardinal\n\nnoncomputable section\n\nopen Function Set Submodule\n\nuniverse u' u\n\nvariable {ι : Type u'} {ι' : Type*} {R : Type*} {K : Type*} {s : Set ι}\nvariable {M : Type*} {M' : Type*} {V : Type u}\n\nsection Semiring\n\n\nvariable {v : ι → M}\nvariable [Semiring R] [AddCommMonoid M] [AddCommMonoid M']\nvariable [Module R M] [Module R M']\nvariable (R) (v)\n\nvariable {R v}\n\n/-- A set of linearly independent vectors in a module `M` over a semiring `K` is also linearly\nindependent over a subring `R` of `K`.\n\nSee also `LinearIndependent.restrict_scalars'` for a version with more convenient typeclass\nassumptions.\n\nTODO : `LinearIndepOn` version. -/\ntheorem LinearIndependent.restrict_scalars [Semiring K] [SMulWithZero R K] [Module K M]\n    [IsScalarTower R K M] (hinj : Injective fun r : R ↦ r • (1 : K))\n    (li : LinearIndependent K v) : LinearIndependent R v := by\n  intro x y hxy\n  let f := fun r : R => r • (1 : K)\n  have := @li (x.mapRange f (by simp [f])) (y.mapRange f (by simp [f])) ?_\n  · ext i\n    exact hinj congr($this i)\n  simpa [Finsupp.linearCombination, f, Finsupp.sum_mapRange_index]\n\nvariable (R) in\ntheorem LinearIndependent.restrict_scalars' [Semiring K] [SMulWithZero R K] [Module K M]\n    [IsScalarTower R K M] [FaithfulSMul R K] [IsScalarTower R K K] {v : ι → M}\n    (li : LinearIndependent K v) : LinearIndependent R v :=\n  restrict_scalars ((faithfulSMul_iff_injective_smul_one R K).mp inferInstance) li\n\n/-- If `v` is an injective family of vectors such that `f ∘ v` is linearly independent, then `v`\n    spans a submodule disjoint from the kernel of `f`.\nTODO : `LinearIndepOn` version. -/\ntheorem Submodule.range_ker_disjoint {f : M →ₗ[R] M'}\n    (hv : LinearIndependent R (f ∘ v)) :\n    Disjoint (span R (range v)) (LinearMap.ker f) := by\n  rw [LinearIndependent, Finsupp.linearCombination_linear_comp] at hv\n  rw [disjoint_iff_inf_le, ← Set.image_univ, Finsupp.span_image_eq_map_linearCombination,\n    map_inf_eq_map_inf_comap, (LinearMap.ker_comp _ _).symm.trans\n      (LinearMap.ker_eq_bot_of_injective hv), inf_bot_eq, map_bot]\n\n/-- If `M / R` and `M' / R'` are modules, `i : R' → R` is a map, `j : M →+ M'` is a monoid map,\nsuch that they are both injective, and compatible with the scalar\nmultiplications on `M` and `M'`, then `j` sends linearly independent families of vectors to\nlinearly independent families of vectors. As a special case, taking `R = R'`\nit is `LinearIndependent.map_injOn`.\nTODO : `LinearIndepOn` version. -/\ntheorem LinearIndependent.map_of_injective_injectiveₛ {R' M' : Type*}\n    [Semiring R'] [AddCommMonoid M'] [Module R' M'] (hv : LinearIndependent R v)\n    (i : R' → R) (j : M →+ M') (hi : Injective i) (hj : Injective j)\n    (hc : ∀ (r : R') (m : M), j (i r • m) = r • j m) : LinearIndependent R' (j ∘ v) := by\n  rw [linearIndependent_iff'ₛ] at hv ⊢\n  intro S r₁ r₂ H s hs\n  simp_rw [comp_apply, ← hc, ← map_sum] at H\n  exact hi <| hv _ _ _ (hj H) s hs\n\n/-- If `M / R` and `M' / R'` are modules, `i : R → R'` is a surjective map,\nand `j : M →+ M'` is an injective monoid map, such that the scalar multiplications\non `M` and `M'` are compatible, then `j` sends linearly independent families\nof vectors to linearly independent families of vectors. As a special case, taking `R = R'`\nit is `LinearIndependent.map_injOn`.\nTODO : `LinearIndepOn` version. -/\ntheorem LinearIndependent.map_of_surjective_injectiveₛ {R' M' : Type*}\n    [Semiring R'] [AddCommMonoid M'] [Module R' M'] (hv : LinearIndependent R v)\n    (i : R → R') (j : M →+ M') (hi : Surjective i) (hj : Injective j)\n    (hc : ∀ (r : R) (m : M), j (r • m) = i r • j m) : LinearIndependent R' (j ∘ v) := by\n  obtain ⟨i', hi'⟩ := hi.hasRightInverse\n  refine hv.map_of_injective_injectiveₛ i' j (fun _ _ h ↦ ?_) hj fun r m ↦ ?_\n  · apply_fun i at h\n    rwa [hi', hi'] at h\n  rw [hc (i' r) m, hi']\n\n/-- If a linear map is injective on the span of a family of linearly independent vectors, then\nthe family stays linearly independent after composing with the linear map.\nSee `LinearIndependent.map` for the version with `Set.InjOn` replaced by `Disjoint`\nwhen working over a ring. -/\ntheorem LinearIndependent.map_injOn (hv : LinearIndependent R v) (f : M →ₗ[R] M')\n    (hf_inj : Set.InjOn f (span R (Set.range v))) : LinearIndependent R (f ∘ v) :=\n  (f.linearIndependent_iff_of_injOn hf_inj).mpr hv\n\ntheorem LinearIndepOn.map_injOn (hv : LinearIndepOn R v s) (f : M →ₗ[R] M')\n    (hf_inj : Set.InjOn f (span R (v '' s))) : LinearIndepOn R (f ∘ v) s :=\n  (f.linearIndepOn_iff_of_injOn hf_inj).mpr hv\n\ntheorem LinearIndepOn.comp_of_image {s : Set ι'} {f : ι' → ι} (h : LinearIndepOn R v (f '' s))\n    (hf : InjOn f s) : LinearIndepOn R (v ∘ f) s :=\n  LinearIndependent.comp h _ (Equiv.Set.imageOfInjOn _ _ hf).injective\n\ntheorem LinearIndepOn.image_of_comp (f : ι → ι') (g : ι' → M) (hs : LinearIndepOn R (g ∘ f) s) :\n    LinearIndepOn R g (f '' s) := by\n  nontriviality R\n  have : InjOn f s := injOn_iff_injective.2 hs.injective.of_comp\n  exact (linearIndependent_equiv' (Equiv.Set.imageOfInjOn f s this) rfl).1 hs\n\ntheorem LinearIndepOn.id_image (hs : LinearIndepOn R v s) : LinearIndepOn R id (v '' s) :=\n  LinearIndepOn.image_of_comp v id hs\n\ntheorem LinearIndepOn_iff_linearIndepOn_image_injOn [Nontrivial R] :\n    LinearIndepOn R v s ↔ LinearIndepOn R id (v '' s) ∧ InjOn v s :=\n  ⟨fun h ↦ ⟨h.id_image, h.injOn⟩, fun h ↦ (linearIndepOn_iff_image h.2).2 h.1⟩\n\ntheorem linearIndepOn_congr {w : ι → M} (h : EqOn v w s) :\n    LinearIndepOn R v s ↔ LinearIndepOn R w s := by\n  rw [LinearIndepOn, LinearIndepOn]\n  convert! Iff.rfl using 2\n  ext x\n  exact h.symm x.2\n\ntheorem LinearIndepOn.congr {w : ι → M} (hli : LinearIndepOn R v s) (h : EqOn v w s) :\n    LinearIndepOn R w s :=\n  (linearIndepOn_congr h).1 hli\n\ntheorem LinearIndependent.group_smul {G : Type*} [hG : Group G] [MulAction G R]\n    [SMul G M] [IsScalarTower G R M] [SMulCommClass G R M] {v : ι → M}\n    (hv : LinearIndependent R v) (w : ι → G) : LinearIndependent R (w • v) := by\n  rw [linearIndependent_iff''ₛ] at hv ⊢\n  intro s g₁ g₂ hgs hsum i\n  refine (Group.isUnit (w i)).smul_left_cancel.mp ?_\n  refine hv s (fun i ↦ w i • g₁ i) (fun i ↦ w i • g₂ i) (fun i hi ↦ ?_) ?_ i\n  · simp_rw [hgs i hi]\n  · simpa only [smul_assoc, smul_comm] using! hsum\n\n@[simp]\ntheorem LinearIndependent.group_smul_iff {G : Type*} [hG : Group G] [MulAction G R]\n    [MulAction G M] [IsScalarTower G R M] [SMulCommClass G R M] (v : ι → M) (w : ι → G) :\n    LinearIndependent R (w • v) ↔ LinearIndependent R v := by\n  refine ⟨fun h ↦ ?_, fun h ↦ h.group_smul w⟩\n  convert! h.group_smul (fun i ↦ (w i)⁻¹)\n  simp [funext_iff]\n\n-- This lemma cannot be proved with `LinearIndependent.group_smul` since the action of\n-- `Rˣ` on `R` is not commutative.\ntheorem LinearIndependent.units_smul {v : ι → M} (hv : LinearIndependent R v) (w : ι → Rˣ) :\n    LinearIndependent R (w • v) := by\n  rw [linearIndependent_iff''ₛ] at hv ⊢\n  intro s g₁ g₂ hgs hsum i\n  rw [← (w i).mul_left_inj]\n  refine hv s (fun i ↦ g₁ i • w i) (fun i ↦ g₂ i • w i) (fun i hi ↦ ?_) ?_ i\n  · simp_rw [hgs i hi]\n  · simpa only [smul_eq_mul, mul_smul, Pi.smul_apply'] using! hsum\n\n@[simp]\n\nTarget:\ntheorem LinearIndependent.units_smul_iff (v : ι → M) (w : ι → Rˣ) :\n    LinearIndependent R (w • v) ↔ LinearIndependent R v :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"fb2973245bb56815e051c21d24450521ae551b5d9f39883d1ff1e55acbd6736f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/LinearIndependent","family_id":"linearindependent","file_id":"mathlib/Mathlib/LinearAlgebra/LinearIndependent/Basic.lean","sample_id":"b38b14c96b38afbba7c3259cf4c4e7b4c145e1b2ac6c153a770feeac3a7076d5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8da08d0db4cb5d343e5a5a407377b94360b60c72eeb504f2655147b8545bacfb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"24a5ad421e4a7eee0e5feabb06c49c3c686c8f596032a3cb50682135e2ce5773","source_sha256":"f2a66bdccdec789c239da17203e2b07ced0f35d264cc0fb10abbfb178b2db08a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases isEmpty_or_nonempty α <;> simp [FreeCommRing]","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":8,"token_jaccard":0.058824,"token_length_ratio":0.727273},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"2503ee875e5ead31a29d5772dd8a0d1b9662b0bc149536c48b10d5b8c012d855","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.FreeAbelianGroup.Finsupp\npublic import Mathlib.Algebra.Ring.TransferInstance\npublic import Mathlib.Data.Finsupp.Fintype\npublic import Mathlib.Data.ZMod.Defs\npublic import Mathlib.GroupTheory.FreeGroup.Reduce\npublic import Mathlib.RingTheory.FreeCommRing\npublic import Mathlib.SetTheory.Cardinal.Arithmetic\npublic import Mathlib.SetTheory.Cardinal.Finsupp\n\nNamespace:\nCardinal\n\nLocal context:\n/-\nCopyright (c) 2024 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser, Daniel Weber\n-/\n/-!\n# Cardinalities of free constructions\n\nThis file shows that all the free constructions over `α` have cardinality `max #α ℵ₀`,\nand are thus infinite, and specifically countable over countable generators.\n\nCombined with the ring `Fin n` for the finite cases, this lets us show that there is a `CommRing` of\nany cardinality.\n-/\n\npublic section\n\nuniverse u\nvariable (α : Type u)\n\nsection Infinite\n\n@[to_additive]\ninstance [Nonempty α] : Infinite (FreeMonoid α) := inferInstanceAs <| Infinite (List α)\n\n@[to_additive]\ninstance [Nonempty α] : Infinite (FreeGroup α) := by\n  classical\n  exact Infinite.of_surjective FreeGroup.norm FreeGroup.norm_surjective\n\ninstance [Nonempty α] : Infinite (FreeAbelianGroup α) :=\n  (FreeAbelianGroup.equivFinsupp α).toEquiv.infinite_iff.2 inferInstance\n\nderiving instance Infinite for FreeRing, FreeCommRing\n\nend Infinite\n\nsection Countable\n\nvariable [Countable α]\n\n@[to_additive]\ninstance : Countable (FreeMonoid α) := inferInstanceAs <| Countable (List α)\n\n@[to_additive]\ninstance : Countable (FreeGroup α) := inferInstanceAs <| Countable (Quot _)\n\ninstance : Countable (FreeAbelianGroup α) := inferInstanceAs <| Countable (Quot _)\n\ninstance : Countable (FreeRing α) := inferInstanceAs <| Countable (Quot _)\n\ninstance : Countable (FreeCommRing α) :=\n  inferInstanceAs <| Countable (FreeAbelianGroup (Multiset α))\n\nend Countable\n\nnamespace Cardinal\n\ntheorem mk_abelianization_le (G : Type u) [Group G] :\n    #(Abelianization G) ≤ #G := Cardinal.mk_le_of_surjective Quotient.mk_surjective\n\n@[to_additive (attr := simp)]\ntheorem mk_freeMonoid [Nonempty α] : #(FreeMonoid α) = max #α ℵ₀ :=\n    Cardinal.mk_list_eq_max_mk_aleph0 _\n\n@[to_additive (attr := simp)]\ntheorem mk_freeGroup [Nonempty α] : #(FreeGroup α) = max #α ℵ₀ := by\n  classical\n  apply le_antisymm\n  · apply (mk_le_of_injective (FreeGroup.toWord_injective (α := α))).trans_eq\n    simp only [mk_list_eq_max_mk_aleph0, mk_prod, lift_uzero, mk_fintype, Fintype.card_bool,\n      Nat.cast_ofNat, lift_ofNat]\n    obtain hα | hα := lt_or_ge #α ℵ₀\n    · simp only [hα.le, max_eq_right, max_eq_right_iff]\n      exact (mul_lt_aleph0 hα natCast_lt_aleph0).le\n    · rw [max_eq_left hα, max_eq_left (hα.trans <| Cardinal.le_mul_right two_ne_zero),\n        Cardinal.mul_eq_left hα _ (by simp)]\n      exact natCast_le_aleph0.trans hα\n  · apply max_le\n    · exact mk_le_of_injective FreeGroup.of_injective\n    · simp\n\n@[simp]\ntheorem mk_freeAbelianGroup [Nonempty α] : #(FreeAbelianGroup α) = max #α ℵ₀ := by\n  rw [Cardinal.mk_congr (FreeAbelianGroup.equivFinsupp α).toEquiv]\n  simp\n\n@[simp]\ntheorem mk_freeRing : #(FreeRing α) = max #α ℵ₀ := by\n  cases isEmpty_or_nonempty α <;> simp [FreeRing]\n\n@[simp]\n\nTarget:\ntheorem mk_freeCommRing : #(FreeCommRing α) = max #α ℵ₀ :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"89889556b4425adfa9c730e30d09ec974755f2bba8657580a1f9b42b33e524ec","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Cardinal","family_id":"mk_freecommring","file_id":"mathlib/Mathlib/SetTheory/Cardinal/Free.lean","sample_id":"24a5ad421e4a7eee0e5feabb06c49c3c686c8f596032a3cb50682135e2ce5773"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bb5b4eccf3c9035f221ecef92ea8d7966625741540da04e726a173d5b6907beb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f1e96e086d107cbb628f9333e3a0d90e5f01c368a40a55b24a768acd38a2bd9b","source_sha256":"69a9767d2da3b234c9bb0ca0a2aec7e4918b4afa1e5dd9c10eea2ad03e813fc9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply Functor.ext; rotate_left\n  · intro X\n    ext i\n    specialize h i\n    have := congr_obj h X\n    simpa\n  · intro X Y g\n    funext i\n    specialize h i\n    have := congr_hom h g\n    simpa","hard_negative":false,"metrics":{"chosen_tokens":37,"rejected_tokens":2,"token_jaccard":0.045455,"token_length_ratio":0.054054},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"253126cef318b090d400d45200ee703aa9a04b71caffb4c24bab1dc6eae042b8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.EqToHom\npublic import Mathlib.CategoryTheory.NatIso\npublic import Mathlib.CategoryTheory.Products.Basic\n\nNamespace:\nFunctor\n\nLocal context:\n/-\nCopyright (c) 2020 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Kim Morrison\n-/\n/-!\n# Categories of indexed families of objects.\n\nWe define the pointwise category structure on indexed families of objects in a category\n(and also the dependent generalization).\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Functor\n\nuniverse w₀ w₁ w₂ v₁ v₂ v₃ u₁ u₂ u₃\n\nvariable {I : Type w₀} {J : Type w₁} (C : I → Type u₁) [∀ i, Category.{v₁} (C i)]\n\n\n/-- `pi C` gives the Cartesian product of an indexed family of categories.\n-/\ninstance pi : Category.{max w₀ v₁} (∀ i, C i) where\n  Hom X Y := ∀ i, X i ⟶ Y i\n  id X i := 𝟙 (X i)\n  comp f g i := f i ≫ g i\n\nnamespace Pi\n\n@[simp]\ntheorem id_apply (X : ∀ i, C i) (i) : (𝟙 X : ∀ i, X i ⟶ X i) i = 𝟙 (X i) :=\n  rfl\n\n@[simp]\ntheorem comp_apply {X Y Z : ∀ i, C i} (f : X ⟶ Y) (g : Y ⟶ Z) (i) :\n    (f ≫ g : ∀ i, X i ⟶ Z i) i = f i ≫ g i :=\n  rfl\n\n@[ext]\nlemma ext {X Y : ∀ i, C i} {f g : X ⟶ Y} (w : ∀ i, f i = g i) : f = g :=\n  funext (w ·)\n\n/--\nThe evaluation functor at `i : I`, sending an `I`-indexed family of objects to the object over `i`.\n-/\n@[simps]\ndef eval (i : I) : (∀ i, C i) ⥤ C i where\n  obj f := f i\n  map α := α i\n\nsection\n\nvariable {J : Type w₁}\n\ninstance (f : J → I) : (j : J) → Category ((C ∘ f) j) :=\n  inferInstanceAs <| (j : J) → Category (C (f j))\n\n/-- Pull back an `I`-indexed family of objects to a `J`-indexed family, along a function `J → I`.\n-/\n@[simps]\ndef comap (h : J → I) : (∀ i, C i) ⥤ (∀ j, C (h j)) where\n  obj f i := f (h i)\n  map α i := α (h i)\n\nvariable (I)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The natural isomorphism between\npulling back a grading along the identity function,\nand the identity functor. -/\n@[simps]\ndef comapId : comap C (id : I → I) ≅ 𝟭 (∀ i, C i) where\n  hom := { app := fun X => 𝟙 X }\n  inv := { app := fun X => 𝟙 X }\n\nexample (g : J → I) : (j : J) → Category (C (g j)) := by infer_instance\n\nvariable {I}\nvariable {K : Type w₂}\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The natural isomorphism comparing between\npulling back along two successive functions, and\npulling back along their composition\n-/\n@[simps!]\ndef comapComp (f : K → J) (g : J → I) : comap C g ⋙ comap (C ∘ g) f ≅ comap C (g ∘ f) where\n  hom :=\n  { app := fun X b => 𝟙 (X (g (f b)))\n    naturality := fun X Y f' => by simp only [comap, Function.comp]; funext; simp }\n  inv :=\n  { app := fun X b => 𝟙 (X (g (f b)))\n    naturality := fun X Y f' => by simp only [comap, Function.comp]; funext; simp }\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The natural isomorphism between pulling back then evaluating, and just evaluating. -/\n@[simps!]\ndef comapEvalIsoEval (h : J → I) (j : J) : comap C h ⋙ eval (C ∘ h) j ≅ eval C (h j) :=\n  NatIso.ofComponents (fun _ => Iso.refl _) (by simp)\n\nend\n\nsection\n\nvariable {J : Type w₀} {D : J → Type u₁} [∀ j, Category.{v₁} (D j)]\n\ninstance sumElimCategory : ∀ s : I ⊕ J, Category.{v₁} (Sum.elim C D s)\n  | Sum.inl i => inferInstanceAs <| Category (C i)\n  | Sum.inr j => inferInstanceAs <| Category (D j)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The bifunctor combining an `I`-indexed family of objects with a `J`-indexed family of objects\nto obtain an `I ⊕ J`-indexed family of objects.\n-/\n@[simps]\ndef sum : (∀ i, C i) ⥤ (∀ j, D j) ⥤ ∀ s : I ⊕ J, Sum.elim C D s where\n  obj X :=\n    { obj := fun Y s =>\n        match s with\n        | .inl i => X i\n        | .inr j => Y j\n      map := fun {_} {_} f s =>\n        match s with\n        | .inl i => 𝟙 (X i)\n        | .inr j => f j }\n  map {X} {X'} f :=\n    { app := fun Y s =>\n        match s with\n        | .inl i => f i\n        | .inr j => 𝟙 (Y j) }\n\nend\n\nvariable {C}\n\n/-- A family of isomorphisms gives rise to an isomorphism of families. -/\n@[simps]\ndef isoMk {X Y : ∀ i, C i} (iso : ∀ i, X i ≅ Y i) :\n    X ≅ Y where\n  hom := fun i => (iso i).hom\n  inv := fun i => (iso i).inv\n\n/-- An isomorphism between `I`-indexed objects gives an isomorphism between each\npair of corresponding components. -/\n@[simps]\ndef isoApp {X Y : ∀ i, C i} (f : X ≅ Y) (i : I) : X i ≅ Y i :=\n  ⟨f.hom i, f.inv i,\n    by rw [← comp_apply, Iso.hom_inv_id, id_apply], by rw [← comp_apply, Iso.inv_hom_id, id_apply]⟩\n\n@[simp]\ntheorem isoApp_refl (X : ∀ i, C i) (i : I) : isoApp (Iso.refl X) i = Iso.refl (X i) :=\n  rfl\n\n@[simp]\ntheorem isoApp_symm {X Y : ∀ i, C i} (f : X ≅ Y) (i : I) : isoApp f.symm i = (isoApp f i).symm :=\n  rfl\n\n@[simp]\ntheorem isoApp_trans {X Y Z : ∀ i, C i} (f : X ≅ Y) (g : Y ≅ Z) (i : I) :\n    isoApp (f ≪≫ g) i = isoApp f i ≪≫ isoApp g i :=\n  rfl\n\nend Pi\n\nnamespace Functor\n\nvariable {C}\nvariable {D : I → Type u₂} [∀ i, Category.{v₂} (D i)] {A : Type u₃} [Category.{v₃} A]\n\n/-- Assemble an `I`-indexed family of functors into a functor between the pi types.\n-/\n@[simps]\ndef pi (F : ∀ i, C i ⥤ D i) : (∀ i, C i) ⥤ ∀ i, D i where\n  obj f i := (F i).obj (f i)\n  map α i := (F i).map (α i)\n\n/-- Similar to `pi`, but all functors come from the same category `A`\n-/\n@[simps]\ndef pi' (f : ∀ i, A ⥤ C i) : A ⥤ ∀ i, C i where\n  obj a i := (f i).obj a\n  map h i := (f i).map h\n\n/-- The projections of `Functor.pi' F` are isomorphic to the functors of the family `F` -/\n@[simps!]\ndef pi'CompEval {A : Type*} [Category* A] (F : ∀ i, A ⥤ C i) (i : I) :\n    pi' F ⋙ Pi.eval C i ≅ F i :=\n  Iso.refl _\n\nsection EqToHom\n\n@[simp]\ntheorem eqToHom_proj {x x' : ∀ i, C i} (h : x = x') (i : I) :\n    (eqToHom h : x ⟶ x') i = eqToHom (funext_iff.mp h i) := by\n  subst h\n  rfl\n\nend EqToHom\n\n-- One could add some natural isomorphisms showing\n-- how `Functor.pi` commutes with `Pi.eval` and `Pi.comap`.\n@[simp]\ntheorem pi'_eval (f : ∀ i, A ⥤ C i) (i : I) : pi' f ⋙ Pi.eval C i = f i :=\n  rfl\n\n/-- Two functors to a product category are equal iff they agree on every coordinate. -/\n\nTarget:\ntheorem pi_ext (f f' : A ⥤ ∀ i, C i) (h : ∀ i, f ⋙ (Pi.eval C i) = f' ⋙ (Pi.eval C i)) :\n    f = f' :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Pi","family_id":"pi_ext","file_id":"mathlib/Mathlib/CategoryTheory/Pi/Basic.lean","sample_id":"f1e96e086d107cbb628f9333e3a0d90e5f01c368a40a55b24a768acd38a2bd9b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7b7262505c003dbad9c3aa96a2764e75ade60fa912085a50b5740b06b8e012c5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b01e50f28aa56d2ac7f58b9ce851e208ce0a6ff3a1d25fba2c8975fa9fbc30bc","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a8d869c19b5e1d1d76d3c28a6dc188be2ec2a1a13841d22f5d203f9a92b23e39","source_sha256":"93096837382e572977abeb7e5338f6f449e78fd61b5b1c8bf94ffe55e531a689","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro H\n  refine fermatLastTheoremThree_of_three_dvd_only_c (fun a b c hc ha hb ⟨x, hx⟩ hcoprime h ↦ ?_)\n  refine H a b c 1 (by simp [hc]) (fun hdvd ↦ ha ?_) (fun hdvd ↦ hb ?_) ?_ ?_ ?_\n  · rwa [← Ideal.norm_dvd_iff (hζ.prime_norm_toInteger_sub_one_of_prime_ne_two' (by decide)),\n      hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide)] at hdvd\n  · rwa [← Ideal.norm_dvd_iff (hζ.prime_norm_toInteger_sub_one_of_prime_ne_two' (by decide)),\n      hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide)] at hdvd\n  · exact dvd_trans hζ.toInteger_sub_one_dvd_prime' ⟨x, by simp [hx]⟩\n  · exact IsCoprime.intCast hcoprime\n  · simpa using mod_cast h","hard_negative":true,"metrics":{"chosen_tokens":144,"rejected_tokens":3,"token_jaccard":0.040816,"token_length_ratio":0.020833},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"25337fbdcf5bb30be3c010180143c4b5c9a5d4b6e4b4017cb8464eb4d36b235d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.FLT.Basic\npublic import Mathlib.NumberTheory.NumberField.Cyclotomic.PID\npublic import Mathlib.NumberTheory.NumberField.Cyclotomic.Three\npublic import Mathlib.Algebra.Ring.Divisibility.Lemmas\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Riccardo Brasca. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Riccardo Brasca, Sanyam Gupta, Omar Haddad, David Lowry-Duda,\n  Lorenzo Luccioli, Pietro Monticone, Alexis Saurin, Florent Schaffhauser\n-/\n/-!\n# Fermat Last Theorem in the case `n = 3`\nThe goal of this file is to prove Fermat's Last Theorem in the case `n = 3`.\n\n## Main results\n* `fermatLastTheoremThree`: Fermat's Last Theorem for `n = 3`: if `a b c : ℕ` are all non-zero then\n  `a ^ 3 + b ^ 3 ≠ c ^ 3`.\n\n## Implementation details\nWe follow the proof in <https://webusers.imj-prg.fr/~marc.hindry/Cours-arith.pdf>, page 43.\n\nThe strategy is the following:\n* The so-called \"Case 1\", when `3 ∣ a * b * c` is completely elementary and is proved using\n  congruences modulo `9`.\n* To prove case 2, we consider the generalized equation `a ^ 3 + b ^ 3 = u * c ^ 3`, where `a`, `b`,\n  and `c` are in the cyclotomic ring `ℤ[ζ₃]` (where `ζ₃` is a primitive cube root of unity) and `u`\n  is a unit of `ℤ[ζ₃]`. `FermatLastTheoremForThree_of_FermatLastTheoremThreeGen` (whose proof is\n  rather elementary on paper) says that to prove Fermat's last theorem for exponent `3`, it is\n  enough to prove that this equation has no solutions such that `c ≠ 0`, `¬ λ ∣ a`, `¬ λ ∣ b`,\n  `λ ∣ c` and `IsCoprime a b` (where we set `λ := ζ₃ - 1`). We call such a tuple a `Solution'`.\n  A `Solution` is the same as a `Solution'` with the additional assumption that `λ ^ 2 ∣ a + b`.\n  We then prove that, given `S' : Solution'`, there is `S : Solution` such that the multiplicity of\n  `λ = ζ₃ - 1` in `c` is the same in `S'` and `S` (see `exists_Solution_of_Solution'`).\n  In particular it is enough to prove that no `Solution` exists. The key point is a descent argument\n  on the multiplicity of `λ` in `c`: starting with `S : Solution` we can find `S₁ : Solution` with\n  multiplicity strictly smaller (see `exists_Solution_multiplicity_lt`) and this finishes the proof.\n  To construct `S₁` we go through a `Solution'` and then back to a `Solution`. More importantly, we\n  cannot control the unit `u`, and this is the reason why we need to consider the generalized\n  equation `a ^ 3 + b ^ 3 = u * c ^ 3`. The construction is completely explicit, but it depends\n  crucially on `IsCyclotomicExtension.Rat.Three.eq_one_or_neg_one_of_unit_of_congruent`, a special\n  case of Kummer's lemma.\n* Note that we don't prove Case 1 for the generalized equation (in particular we don't prove that\n  the generalized equation has no nontrivial solutions). This is because the proof, even if\n  elementary on paper, would be quite annoying to formalize: indeed it involves a lot of explicit\n  computations in `ℤ[ζ₃] / (λ)`: this ring is isomorphic to `ℤ / 9ℤ`, but of course, even if we\n  construct such an isomorphism, tactics like `decide` would not work.\n\n-/\n\nsection case1\n\nopen ZMod\n\nlemma cube_of_castHom_ne_zero {n : ZMod 9} :\n    castHom (show 3 ∣ 9 by simp) (ZMod 3) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8 := by\n  revert n; decide\n\nlemma cube_of_not_dvd {n : ℤ} (h : ¬ 3 ∣ n) :\n    (n : ZMod 9) ^ 3 = 1 ∨ (n : ZMod 9) ^ 3 = 8 := by\n  apply cube_of_castHom_ne_zero\n  rwa [map_intCast, Ne, ZMod.intCast_zmod_eq_zero_iff_dvd]\n\n/-- If `a b c : ℤ` are such that `¬ 3 ∣ a * b * c`, then `a ^ 3 + b ^ 3 ≠ c ^ 3`. -/\ntheorem fermatLastTheoremThree_case_1 {a b c : ℤ} (hdvd : ¬ 3 ∣ a * b * c) :\n    a ^ 3 + b ^ 3 ≠ c ^ 3 := by\n  simp_rw [Int.prime_three.dvd_mul, not_or] at hdvd\n  apply mt (congrArg (Int.cast : ℤ → ZMod 9))\n  simp_rw [Int.cast_add, Int.cast_pow]\n  rcases cube_of_not_dvd hdvd.1.1 with ha | ha <;>\n  rcases cube_of_not_dvd hdvd.1.2 with hb | hb <;>\n  rcases cube_of_not_dvd hdvd.2 with hc | hc <;>\n  rw [ha, hb, hc] <;> decide\n\nend case1\n\nsection case2\n\nlemma three_dvd_b_of_dvd_a_of_gcd_eq_one_of_case2 {a b c : ℤ} (ha : a ≠ 0)\n    (Hgcd : Finset.gcd {a, b, c} id = 1) (h3a : 3 ∣ a) (HF : a ^ 3 + b ^ 3 + c ^ 3 = 0)\n    (H : ∀ a b c : ℤ, c ≠ 0 → ¬ 3 ∣ a → ¬ 3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3) :\n    3 ∣ b := by\n  have hbc : IsCoprime (-b) (-c) := by\n    refine IsCoprime.neg_neg ?_\n    rw [add_comm (a ^ 3), add_assoc, add_comm (a ^ 3), ← add_assoc] at HF\n    refine isCoprime_of_gcd_eq_one_of_FLT ?_ HF\n    convert! Hgcd using 2\n    rw [Finset.pair_comm, Finset.insert_comm]\n  by_contra! h3b\n  by_cases h3c : 3 ∣ c\n  · apply h3b\n    rw [add_assoc, add_comm (b ^ 3), ← add_assoc] at HF\n    exact dvd_c_of_prime_of_dvd_a_of_dvd_b_of_FLT Int.prime_three h3a h3c HF\n  · refine H (-b) (-c) a ha (by simp [h3b]) (by simp [h3c]) h3a hbc ?_\n    rw [add_eq_zero_iff_eq_neg, ← (show Odd 3 by decide).neg_pow] at HF\n    rw [← HF]\n    ring\n\nopen Finset in\nlemma fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 {a b c : ℤ} (ha : a ≠ 0)\n    (h3a : 3 ∣ a) (Hgcd : Finset.gcd {a, b, c} id = 1)\n    (H : ∀ a b c : ℤ, c ≠ 0 → ¬ 3 ∣ a → ¬ 3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3) :\n    a ^ 3 + b ^ 3 + c ^ 3 ≠ 0 := by\n  intro HF\n  apply (show ¬(3 ∣ (1 : ℤ)) by decide)\n  rw [← Hgcd]\n  refine dvd_gcd (fun x hx ↦ ?_)\n  simp only [mem_insert, mem_singleton] at hx\n  have h3b : 3 ∣ b := by\n    refine three_dvd_b_of_dvd_a_of_gcd_eq_one_of_case2 ha ?_ h3a HF H\n    simp only [← Hgcd, gcd_insert, gcd_singleton, id_eq, ← Int.abs_eq_normalize]\n  rcases hx with hx | hx | hx\n  · exact hx ▸ h3a\n  · exact hx ▸ h3b\n  · simpa [hx] using dvd_c_of_prime_of_dvd_a_of_dvd_b_of_FLT Int.prime_three h3a h3b HF\n\nopen Finset Int in\n/--\nTo prove Fermat's Last Theorem for `n = 3`, it is enough to show that for all `a`, `b`, `c`\nin `ℤ` such that `c ≠ 0`, `¬ 3 ∣ a`, `¬ 3 ∣ b`, `a` and `b` are coprime and `3 ∣ c`, we have\n`a ^ 3 + b ^ 3 ≠ c ^ 3`.\n-/\ntheorem fermatLastTheoremThree_of_three_dvd_only_c\n    (H : ∀ a b c : ℤ, c ≠ 0 → ¬ 3 ∣ a → ¬ 3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3) :\n    FermatLastTheoremFor 3 := by\n  rw [fermatLastTheoremFor_iff_int]\n  refine fermatLastTheoremWith_of_fermatLastTheoremWith_coprime (fun a b c ha hb hc Hgcd hF ↦ ?_)\n  by_cases h1 : 3 ∣ a * b * c\n  swap\n  · exact fermatLastTheoremThree_case_1 h1 hF\n  rw [prime_three.dvd_mul, prime_three.dvd_mul] at h1\n  rw [← sub_eq_zero, sub_eq_add_neg, ← (show Odd 3 by decide).neg_pow] at hF\n  rcases h1 with (h3a | h3b) | h3c\n  · refine fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 ha h3a ?_ H hF\n    simp only [← Hgcd, gcd_insert, gcd_singleton, id_eq, ← abs_eq_normalize, abs_neg]\n  · rw [add_comm (a ^ 3)] at hF\n    refine fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 hb h3b ?_ H hF\n    simp only [← Hgcd, insert_comm, gcd_insert, gcd_singleton, id_eq, ← abs_eq_normalize, abs_neg]\n  · rw [add_comm _ ((-c) ^ 3), ← add_assoc] at hF\n    refine fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_of_case2 (neg_ne_zero.2 hc) (by simp [h3c])\n      ?_ H hF\n    rw [Finset.insert_comm (-c), Finset.pair_comm (-c) b]\n    simp only [← Hgcd, gcd_insert, gcd_singleton, id_eq, ← abs_eq_normalize, abs_neg]\n\nsection eisenstein\n\nopen NumberField IsCyclotomicExtension.Rat.Three\n\nvariable {K : Type*} [Field K]\nvariable {ζ : K} (hζ : IsPrimitiveRoot ζ 3)\n\nlocal notation3 \"η\" => (IsPrimitiveRoot.isUnit (hζ.toInteger_isPrimitiveRoot) (by decide)).unit\nlocal notation3 \"λ\" => hζ.toInteger - 1\n\n/-- `FermatLastTheoremForThreeGen` is the statement that `a ^ 3 + b ^ 3 = u * c ^ 3` has no\nnontrivial solutions in `𝓞 K` for all `u : (𝓞 K)ˣ` such that `¬ λ ∣ a`, `¬ λ ∣ b` and `λ ∣ c`.\nThe reason to consider `FermatLastTheoremForThreeGen` is to make a descent argument working. -/\ndef FermatLastTheoremForThreeGen : Prop :=\n  ∀ a b c : 𝓞 K, ∀ u : (𝓞 K)ˣ, c ≠ 0 → ¬ λ ∣ a → ¬ λ ∣ b → λ ∣ c → IsCoprime a b →\n    a ^ 3 + b ^ 3 ≠ u * c ^ 3\n\n/-- To prove `FermatLastTheoremFor 3`, it is enough to prove `FermatLastTheoremForThreeGen`. -/\n\nTarget:\nlemma FermatLastTheoremForThree_of_FermatLastTheoremThreeGen\n    [NumberField K] [IsCyclotomicExtension {3} ℚ K] :\n    FermatLastTheoremForThreeGen hζ → FermatLastTheoremFor 3 :=\n\nProof body:\n","rejected":"by\n  exact FermatLastTheoremForThree_of_FermatLastTheoremThreeGen","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"c60aba24f3a207097e7a17518d4dfb7e37cc24baba00072a5990f1e9cd88fef8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/FLT","family_id":"fermatlasttheoremforthree_of_fermatlasttheoremthreegen","file_id":"mathlib/Mathlib/NumberTheory/FLT/Three.lean","sample_id":"a8d869c19b5e1d1d76d3c28a6dc188be2ec2a1a13841d22f5d203f9a92b23e39"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5ca120de9531c196478bb85a2839de944f04348088dc99c356e217a11b64504f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0e4bac0eeddb51f4e6c61f8570f09dd4bde909a0e782357c90c7a237447aef47","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e2cd3c1a8076b3af47b98c32f101f8a48aa6a72e4acab38020fc2356c4738a6b","source_sha256":"5fbd6e0c9aed4ae0af15e49b0a292eb4e252a42e876e277383b0144f2f430dad","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction xs generalizing ys i with\n  | nil => cases h₂\n  | cons x' xs xs_ih =>\n    cases i\n    · simp only [length_cons, lt_add_iff_pos_left, add_pos_iff, Nat.lt_add_one, or_true,\n        getElem?_eq_getElem, getElem_cons_zero, Option.some.injEq] at h₂\n      subst h₂\n      cases ys\n      · cases h₁\n      · simp\n    · cases ys\n      · cases h₁\n      · obtain - | ⟨h₀, h₁⟩ := h₀\n        simp only [getElem?_cons_succ, zip_cons_cons, applyId_cons] at h₂ ⊢\n        rw [if_neg]\n        · apply xs_ih <;> solve_by_elim [Nat.succ.inj]\n        · apply h₀; apply List.mem_of_getElem? h₂","hard_negative":true,"metrics":{"chosen_tokens":129,"rejected_tokens":5,"token_jaccard":0.048387,"token_length_ratio":0.03876},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"269750fb8edad59304aa811ce6a0dbbd33afc5b6c48f9ad4f1b728523d361996","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.MLList.Basic\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.ZeroLEOne\npublic import Mathlib.Data.DFinsupp.Defs\npublic import Mathlib.Data.Finsupp.Defs\npublic import Mathlib.Data.Int.Range\npublic import Mathlib.Data.List.Sigma\npublic import Mathlib.Data.PNat.Notation\npublic import Mathlib.Tactic.Bound.Init\n\nNamespace:\nPlausible.InjectiveFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\npublic meta import Mathlib.Data.Finsupp.ToDFinsupp\npublic meta import Mathlib.Data.Int.Range\npublic meta import Mathlib.Data.List.Sigma\npublic meta import Plausible.Functions\n\n/-!\n## `Plausible`: generators for functions\n\nThis file defines `Sampleable` instances for `ℤ → ℤ` injective functions.\n\nInjective functions are generated by creating a list of numbers and\na permutation of that list. The permutation insures that every input\nis mapped to a unique output. When an input is not found in the list\nthe input itself is used as an output.\n\nInjective functions `f : α → α` could be generated easily instead of\n`ℤ → ℤ` by generating a `List α`, removing duplicates and creating a\npermutation. One has to be careful when generating the domain to make\nit vast enough that, when generating arguments to apply `f` to,\nthey argument should be likely to lie in the domain of `f`. This is\nthe reason that injective functions `f : ℤ → ℤ` are generated by\nfixing the domain to the range `[-2*size .. 2*size]`, with `size`\nthe size parameter of the `gen` monad.\n\nMuch of the machinery provided in this file is applicable to generate\ninjective functions of type `α → α` and new instances should be easy\nto define.\n\nOther classes of functions such as monotone functions can generated using\nsimilar techniques. For monotone functions, generating two lists, sorting them\nand matching them should suffice, with appropriate default values.\nSome care must be taken for shrinking such functions to make sure\ntheir defining property is invariant through shrinking. Injective\nfunctions are an example of how complicated it can get.\n-/\n\n@[expose] public meta section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type v}\n\nnamespace Plausible\n\nnamespace TotalFunction\n\nsection Finsupp\n\nvariable [DecidableEq α]\n\n/--\nThis theorem exists because plausible does not have access to dlookup but\nmathlib has all the theory for it and wants to use it. We probably want to\nbring these two together at some point.\n-/\nprivate theorem apply_eq_dlookup (m : List (Σ _ : α, β)) (y : β) (x : α) :\n     (withDefault m y).apply x = (m.dlookup x).getD y := by\n  dsimp only [apply]\n  congr 1\n  induction m with\n  | nil => simp\n  | cons p m ih =>\n    rcases p with ⟨fst, snd⟩\n    by_cases heq : fst = x\n    · simp [heq]\n    · rw [List.dlookup_cons_ne]\n      · simp [heq, ih]\n      · symm\n        simp [heq]\n\nvariable [Zero β] [DecidableEq β]\n\n/-- Map a `TotalFunction` to one whose default value is zero so that it represents a `Finsupp`. -/\n@[simp]\ndef zeroDefault : TotalFunction α β → TotalFunction α β\n  | .withDefault A _ => .withDefault A 0\n\n/-- The support of a zero default `TotalFunction`. -/\ndef zeroDefaultSupp : TotalFunction α β → Finset α\n  | .withDefault A _ =>\n    List.toFinset <| (A.dedupKeys.filter fun ab => Sigma.snd ab ≠ 0).map Sigma.fst\n\n/-- Create a finitely supported function from a total function by taking the default value to\nzero. -/\ndef applyFinsupp (tf : TotalFunction α β) : α →₀ β where\n  support := zeroDefaultSupp tf\n  toFun := tf.zeroDefault.apply\n  mem_support_toFun := by\n    intro a\n    rcases tf with ⟨A, y⟩\n    simp only [zeroDefaultSupp, List.mem_map, List.mem_filter, exists_and_right,\n      List.mem_toFinset, exists_eq_right, Sigma.exists, Ne, zeroDefault]\n    rw [apply_eq_dlookup]\n    constructor\n    · rintro ⟨od, hval, hod⟩\n      have := List.mem_dlookup (List.nodupKeys_dedupKeys A) hval\n      rw [(_ : List.dlookup a A = od)]\n      · simpa using hod\n      · simpa [List.dlookup_dedupKeys]\n    · intro h\n      use (A.dlookup a).getD (0 : β)\n      rw [← List.dlookup_dedupKeys] at h ⊢\n      simp only [h, ← List.mem_dlookup_iff A.nodupKeys_dedupKeys, not_false_iff, Option.mem_def]\n      cases haA : List.dlookup a A.dedupKeys\n      · simp [haA] at h\n      · simp\n\nvariable [SampleableExt α] [SampleableExt β] [Repr α]\n\ninstance Finsupp.sampleableExt : SampleableExt (α →₀ β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\n-- TODO: support a non-constant codomain type\ninstance DFinsupp.sampleableExt : SampleableExt (Π₀ _ : α, β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp.toDFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\nend Finsupp\nend TotalFunction\n\nopen _root_.List\n\n/-- Data structure specifying a total function using a list of pairs\nand a default value returned when the input is not in the domain of\nthe partial function.\n\n`mapToSelf f` encodes `x ↦ f x` when `x ∈ f` and `x ↦ x`,\ni.e. `x` to itself, otherwise.\n\nWe use `Σ` to encode mappings instead of `×` because we\nrely on the association list API defined in `Mathlib/Data/List/Sigma.lean`.\n-/\ninductive InjectiveFunction (α : Type u) : Type u\n  | mapToSelf (xs : List (Σ _ : α, α)) :\n      xs.map Sigma.fst ~ xs.map Sigma.snd → List.Nodup (xs.map Sigma.snd) → InjectiveFunction α\n\ninstance : Inhabited (InjectiveFunction α) :=\n  ⟨⟨[], List.Perm.nil, List.nodup_nil⟩⟩\n\nnamespace InjectiveFunction\n\n/-- Apply a total function to an argument. -/\ndef apply [DecidableEq α] : InjectiveFunction α → α → α\n  | InjectiveFunction.mapToSelf m _ _, x => (m.dlookup x).getD x\n\n/-- Produce a string for a given `InjectiveFunction`.\nThe output is of the form `[x₀ ↦ f x₀, .. xₙ ↦ f xₙ, x ↦ x]`.\nUnlike for `TotalFunction`, the default value is not a constant\nbut the identity function.\n-/\nprotected def repr [Repr α] : InjectiveFunction α → String\n  | InjectiveFunction.mapToSelf m _ _ => s! \"[{TotalFunction.reprAux m}x ↦ x]\"\n\ninstance (α : Type u) [Repr α] : Repr (InjectiveFunction α) where\n  reprPrec f _p := InjectiveFunction.repr f\n\n/-- Interpret a list of pairs as a total function, defaulting to\nthe identity function when no entries are found for a given function -/\ndef List.applyId [DecidableEq α] (xs : List (α × α)) (x : α) : α :=\n  ((xs.map Prod.toSigma).dlookup x).getD x\n\n@[simp]\ntheorem List.applyId_cons [DecidableEq α] (xs : List (α × α)) (x y z : α) :\n    List.applyId ((y, z)::xs) x = if y = x then z else List.applyId xs x := by\n  simp only [List.applyId, List.dlookup, eq_rec_constant, Prod.toSigma, List.map]\n  split_ifs <;> rfl\n\nopen Function\nopen List\n\nopen Nat\n\nTarget:\ntheorem List.applyId_zip_eq [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs)\n    (h₁ : xs.length = ys.length) (x y : α) (i : ℕ) (h₂ : xs[i]? = some x) :\n    List.applyId.{u} (xs.zip ys) x = y ↔ ys[i]? = some y :=\n\nProof body:\n","rejected":"by\n  exact List.applyId_zip_eq","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"79ffa88d073191f8ba5409d33573682fd570e65b0c7a9d766971052451f60a0c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Testing/Plausible","family_id":"list","file_id":"mathlib/Mathlib/Testing/Plausible/Functions.lean","sample_id":"e2cd3c1a8076b3af47b98c32f101f8a48aa6a72e4acab38020fc2356c4738a6b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4fe5d002e1432ce371569b334eb570afebb4011df3c1ab6e1eb29382c5b0d1c6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ee5d0df3c7f8bc043a92caea4286a93e5e0c94dddc909c0df2918e11365aefae","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5d67e5e7b9950556ea8bd8cdfa3767dbe10a6562717cb86050afc39cdaf6a4b7","source_sha256":"85d2f87939059f796ffb510eeb3d6af5b97a8cdc2c5223d679b28800dce394e2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI := Nonempty.map f ‹_›\n  haveI := Nonempty.map g ‹_›\n  exact (map_injective.and map_surjective).trans and_and_and_comm","hard_negative":false,"metrics":{"chosen_tokens":29,"rejected_tokens":34,"token_jaccard":0.826087,"token_length_ratio":1.172414},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"26a9f6314cb5e3df27f4d4e9cb0a3d7933c47d635727e22a495a91e9b104fb51","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Lean.PrettyPrinter.Delaborator.Builtins\npublic import Mathlib.Logic.Function.Defs\npublic import Mathlib.Logic.Function.Iterate\npublic import Mathlib.Tactic.Inhabit\npublic import Batteries.Tactic.Trans\nimport Mathlib.Tactic.Attr.Register\n\nNamespace:\nProd\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Extra facts about `Prod`\n\nThis file proves various simple lemmas about `Prod`.\nIt also defines better delaborators for product projections.\n-/\n\n@[expose] public section\n\nvariable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}\n\nnamespace Prod\n\nlemma swap_eq_iff_eq_swap {x : α × β} {y : β × α} : x.swap = y ↔ x = y.swap := by grind\n\ndef mk.injArrow {x₁ : α} {y₁ : β} {x₂ : α} {y₂ : β} :\n    (x₁, y₁) = (x₂, y₂) → ∀ ⦃P : Sort*⦄, (x₁ = x₂ → y₁ = y₂ → P) → P := by\n  intros h P w\n  cases h\n  exact w rfl rfl\n\n@[simp]\ntheorem mk.eta : ∀ {p : α × β}, (p.1, p.2) = p\n  | (_, _) => rfl\n\ntheorem forall' {p : α → β → Prop} : (∀ x : α × β, p x.1 x.2) ↔ ∀ a b, p a b :=\n  Prod.forall\n\ntheorem exists' {p : α → β → Prop} : (∃ x : α × β, p x.1 x.2) ↔ ∃ a b, p a b :=\n  Prod.exists\n\n@[simp]\ntheorem snd_comp_mk (x : α) : Prod.snd ∘ (Prod.mk x : β → α × β) = id :=\n  rfl\n\n@[simp]\ntheorem fst_comp_mk (x : α) : Prod.fst ∘ (Prod.mk x : β → α × β) = Function.const β x :=\n  rfl\n\nattribute [mfld_simps] map_apply\n\n-- This was previously a `simp` lemma, but no longer is on the basis that it destructures the pair.\n--  See `map_apply`, `map_fst`, and `map_snd` for slightly weaker lemmas in the `simp` set.\ntheorem map_apply' (f : α → γ) (g : β → δ) (p : α × β) : map f g p = (f p.1, g p.2) :=\n  rfl\n\ntheorem map_fst' (f : α → γ) (g : β → δ) : Prod.fst ∘ map f g = f ∘ Prod.fst :=\n  funext <| map_fst f g\n\ntheorem map_snd' (f : α → γ) (g : β → δ) : Prod.snd ∘ map f g = g ∘ Prod.snd :=\n  funext <| map_snd f g\n\ntheorem mk_inj {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) = (a₂, b₂) ↔ a₁ = a₂ ∧ b₁ = b₂ := by simp\n\ntheorem mk_right_injective {α β : Type*} (a : α) : (mk a : β → α × β).Injective := by\n  intro b₁ b₂ h\n  simpa only [true_and, Prod.mk_inj, eq_self_iff_true] using h\n\ntheorem mk_left_injective {α β : Type*} (b : β) : (fun a ↦ mk a b : α → α × β).Injective := by\n  intro b₁ b₂ h\n  simpa only [and_true, eq_self_iff_true, mk_inj] using h\n\nlemma mk_right_inj {a : α} {b₁ b₂ : β} : (a, b₁) = (a, b₂) ↔ b₁ = b₂ :=\n    (mk_right_injective _).eq_iff\n\nlemma mk_left_inj {a₁ a₂ : α} {b : β} : (a₁, b) = (a₂, b) ↔ a₁ = a₂ := (mk_left_injective _).eq_iff\n\ntheorem map_def {f : α → γ} {g : β → δ} : Prod.map f g = fun p : α × β ↦ (f p.1, g p.2) :=\n  funext fun p ↦ Prod.ext (map_fst f g p) (map_snd f g p)\n\ntheorem id_prod : (fun p : α × β ↦ (p.1, p.2)) = id :=\n  rfl\n\n@[simp]\ntheorem map_iterate (f : α → α) (g : β → β) (n : ℕ) :\n    (Prod.map f g)^[n] = Prod.map f^[n] g^[n] := by induction n <;> simp [*, Prod.map_comp_map]\n\ntheorem fst_surjective [h : Nonempty β] : Function.Surjective (@fst α β) :=\n  fun x ↦ h.elim fun y ↦ ⟨⟨x, y⟩, rfl⟩\n\ntheorem snd_surjective [h : Nonempty α] : Function.Surjective (@snd α β) :=\n  fun y ↦ h.elim fun x ↦ ⟨⟨x, y⟩, rfl⟩\n\ntheorem fst_injective [Subsingleton β] : Function.Injective (@fst α β) :=\n  fun _ _ h ↦ Prod.ext h (Subsingleton.elim _ _)\n\ntheorem snd_injective [Subsingleton α] : Function.Injective (@snd α β) :=\n  fun _ _ h ↦ Prod.ext (Subsingleton.elim _ _) h\n\n@[simp]\ntheorem swap_leftInverse : Function.LeftInverse (@swap α β) swap :=\n  swap_swap\n\n@[simp]\ntheorem swap_rightInverse : Function.RightInverse (@swap α β) swap :=\n  swap_swap\n\ntheorem swap_injective : Function.Injective (@swap α β) :=\n  swap_leftInverse.injective\n\ntheorem swap_surjective : Function.Surjective (@swap α β) :=\n  swap_leftInverse.surjective\n\ntheorem swap_bijective : Function.Bijective (@swap α β) :=\n  ⟨swap_injective, swap_surjective⟩\n\ntheorem _root_.Function.Semiconj.swap_map (f : α → α) (g : β → β) :\n    Function.Semiconj swap (map f g) (map g f) :=\n  Function.semiconj_iff_comp_eq.2 (map_comp_swap g f).symm\n\ntheorem eq_iff_fst_eq_snd_eq : ∀ {p q : α × β}, p = q ↔ p.1 = q.1 ∧ p.2 = q.2\n  | ⟨p₁, p₂⟩, ⟨q₁, q₂⟩ => by simp\n\ntheorem fst_eq_iff : ∀ {p : α × β} {x : α}, p.1 = x ↔ p = (x, p.2)\n  | ⟨a, b⟩, x => by simp\n\ntheorem snd_eq_iff : ∀ {p : α × β} {x : β}, p.2 = x ↔ p = (p.1, x)\n  | ⟨a, b⟩, x => by simp\n\nvariable {r : α → α → Prop} {s : β → β → Prop} {x y : α × β}\n\nlemma lex_iff : Prod.Lex r s x y ↔ r x.1 y.1 ∨ x.1 = y.1 ∧ s x.2 y.2 := lex_def\n\ninstance Lex.decidable [DecidableEq α]\n    (r : α → α → Prop) (s : β → β → Prop) [DecidableRel r] [DecidableRel s] :\n    DecidableRel (Prod.Lex r s) :=\n  fun _ _ ↦ decidable_of_decidable_of_iff lex_def.symm\n\n@[refl]\ntheorem Lex.refl_left (r : α → α → Prop) (s : β → β → Prop) [Std.Refl r] : ∀ x, Prod.Lex r s x x\n  | (_, _) => Lex.left _ _ (refl _)\n\ninstance {r : α → α → Prop} {s : β → β → Prop} [Std.Refl r] : Std.Refl (Prod.Lex r s) :=\n  ⟨Lex.refl_left _ _⟩\n\n@[refl]\ntheorem Lex.refl_right (r : α → α → Prop) (s : β → β → Prop) [Std.Refl s] : ∀ x, Prod.Lex r s x x\n  | (_, _) => Lex.right _ (refl _)\n\ninstance {r : α → α → Prop} {s : β → β → Prop} [Std.Refl s] : Std.Refl (Prod.Lex r s) :=\n  ⟨Lex.refl_right _ _⟩\n\ninstance [Std.Irrefl r] [Std.Irrefl s] : Std.Irrefl (Prod.Lex r s) :=\n  ⟨by rintro ⟨i, a⟩ (⟨_, _, h⟩ | ⟨_, h⟩) <;> exact irrefl _ h⟩\n\nset_option linter.style.whitespace false in -- manual alignment is not recognised\n@[trans]\ntheorem Lex.trans {r : α → α → Prop} {s : β → β → Prop} [IsTrans α r] [IsTrans β s] :\n    ∀ {x y z : α × β}, Prod.Lex r s x y → Prod.Lex r s y z → Prod.Lex r s x z\n  | (_, _), (_, _), (_, _), left  _ _ hxy₁, left  _ _ hyz₁ => left  _ _ (_root_.trans hxy₁ hyz₁)\n  | (_, _), (_, _), (_, _), left  _ _ hxy₁, right _ _      => left  _ _ hxy₁\n  | (_, _), (_, _), (_, _), right _ _,      left  _ _ hyz₁ => left  _ _ hyz₁\n  | (_, _), (_, _), (_, _), right _ hxy₂,   right _ hyz₂   => right _ (_root_.trans hxy₂ hyz₂)\n\ninstance {r : α → α → Prop} {s : β → β → Prop} [IsTrans α r] [IsTrans β s] :\n    IsTrans (α × β) (Prod.Lex r s) :=\n  ⟨fun _ _ _ ↦ Lex.trans⟩\n\ninstance {r : α → α → Prop} {s : β → β → Prop} [IsStrictOrder α r] [Std.Antisymm s] :\n    Std.Antisymm (Prod.Lex r s) :=\n  ⟨fun x₁ x₂ h₁₂ h₂₁ ↦\n    match x₁, x₂, h₁₂, h₂₁ with\n    | (a, _), (_, _), .left  _ _ hr₁, .left  _ _ hr₂ => (irrefl a (_root_.trans hr₁ hr₂)).elim\n    | (_, _), (_, _), .left  _ _ hr₁, .right _ _     => (irrefl _ hr₁).elim\n    | (_, _), (_, _), .right _ _,     .left  _ _ hr₂ => (irrefl _ hr₂).elim\n    | (_, _), (_, _), .right _ hs₁,   .right _ hs₂   => antisymm hs₁ hs₂ ▸ rfl⟩\n\ninstance total_left {r : α → α → Prop} {s : β → β → Prop} [Std.Total r] :\n    Std.Total (Prod.Lex r s) :=\n  ⟨fun ⟨a₁, _⟩ ⟨a₂, _⟩ ↦ (Std.Total.total a₁ a₂).imp (Lex.left _ _) (Lex.left _ _)⟩\n\ninstance total_right {r : α → α → Prop} {s : β → β → Prop} [Std.Trichotomous r] [Std.Total s] :\n    Std.Total (Prod.Lex r s) :=\n  ⟨fun ⟨i, a⟩ ⟨j, b⟩ ↦ by\n    obtain hij | rfl | hji := trichotomous_of r i j\n    · exact Or.inl (.left _ _ hij)\n    · exact (total_of s a b).imp (.right _) (.right _)\n    · exact Or.inr (.left _ _ hji) ⟩\n\ninstance trichotomous [Std.Trichotomous r] [Std.Trichotomous s] :\n    Std.Trichotomous (Prod.Lex r s) :=\n  Std.trichotomous_of_rel_or_eq_or_rel_swap <| by\n    intro ⟨i, a⟩ ⟨j, b⟩\n    obtain hij | rfl | hji := trichotomous_of r i j\n    { exact Or.inl (Lex.left _ _ hij) }\n    { exact (trichotomous_of (s) a b).imp3 (Lex.right _) (congr_arg _) (Lex.right _) }\n    { exact Or.inr (Or.inr <| Lex.left _ _ hji) }\n\ninstance [Std.Asymm r] [Std.Asymm s] :\n    Std.Asymm (Prod.Lex r s) where\n  asymm\n  | (_a₁, _a₂), (_b₁, _b₂), .left _ _ h₁, .left _ _ h₂ => Std.Asymm.asymm _ _ h₂ h₁\n  | (_a₁, _a₂), (_, _b₂), .left _ _ h₁, .right _ _ => Std.Asymm.asymm _ _ h₁ h₁\n  | (_a₁, _a₂), (_, _b₂), .right _ _, .left _ _ h₂ => Std.Asymm.asymm _ _ h₂ h₂\n  | (_a₁, _a₂), (_, _b₂), .right _ h₁, .right _ h₂ => Std.Asymm.asymm _ _ h₁ h₂\n\nend Prod\n\nopen Prod\n\nnamespace Function\n\nvariable {f : α → γ} {g : β → δ} {f₁ : α → β} {g₁ : γ → δ} {f₂ : β → α} {g₂ : δ → γ}\n\ntheorem Injective.prodMap (hf : Injective f) (hg : Injective g) : Injective (map f g) :=\n  fun _ _ h ↦ Prod.ext (hf <| congr_arg Prod.fst h) (hg <| congr_arg Prod.snd h)\n\ntheorem Surjective.prodMap (hf : Surjective f) (hg : Surjective g) : Surjective (map f g) :=\n  fun p ↦\n  let ⟨x, hx⟩ := hf p.1\n  let ⟨y, hy⟩ := hg p.2\n  ⟨(x, y), Prod.ext hx hy⟩\n\ntheorem Bijective.prodMap (hf : Bijective f) (hg : Bijective g) : Bijective (map f g) :=\n  ⟨hf.1.prodMap hg.1, hf.2.prodMap hg.2⟩\n\ntheorem LeftInverse.prodMap (hf : LeftInverse f₁ f₂) (hg : LeftInverse g₁ g₂) :\n    LeftInverse (map f₁ g₁) (map f₂ g₂) :=\n  fun a ↦ by rw [Prod.map_map, hf.comp_eq_id, hg.comp_eq_id, map_id, id]\n\ntheorem RightInverse.prodMap :\n    RightInverse f₁ f₂ → RightInverse g₁ g₂ → RightInverse (map f₁ g₁) (map f₂ g₂) :=\n  LeftInverse.prodMap\n\ntheorem Involutive.prodMap {f : α → α} {g : β → β} :\n    Involutive f → Involutive g → Involutive (map f g) :=\n  LeftInverse.prodMap\n\nend Function\n\nnamespace Prod\n\nopen Function\n\n@[simp]\ntheorem map_injective [Nonempty α] [Nonempty β] {f : α → γ} {g : β → δ} :\n    Injective (map f g) ↔ Injective f ∧ Injective g :=\n  ⟨fun h =>\n    ⟨fun a₁ a₂ ha => by\n      inhabit β\n      injection\n        @h (a₁, default) (a₂, default) (congr_arg (fun c : γ => Prod.mk c (g default)) ha :),\n      fun b₁ b₂ hb => by\n      inhabit α\n      injection @h (default, b₁) (default, b₂) (congr_arg (Prod.mk (f default)) hb :)⟩,\n    fun h => h.1.prodMap h.2⟩\n\n@[simp]\ntheorem map_surjective [Nonempty γ] [Nonempty δ] {f : α → γ} {g : β → δ} :\n    Surjective (map f g) ↔ Surjective f ∧ Surjective g :=\n  ⟨fun h =>\n    ⟨fun c => by\n      inhabit δ\n      obtain ⟨⟨a, b⟩, h⟩ := h (c, default)\n      exact ⟨a, congr_arg Prod.fst h⟩,\n      fun d => by\n      inhabit γ\n      obtain ⟨⟨a, b⟩, h⟩ := h (default, d)\n      exact ⟨b, congr_arg Prod.snd h⟩⟩,\n    fun h => h.1.prodMap h.2⟩\n\n@[simp]\n\nTarget:\ntheorem map_bijective [Nonempty α] [Nonempty β] {f : α → γ} {g : β → δ} :\n    Bijective (map f g) ↔ Bijective f ∧ Bijective g :=\n\nProof body:\n","rejected":"by\n  haveI := Nonempty.map f ‹_›\n  haveI := Nonempty.map g ‹_›\n  exact (map_injective.and map_surjective).trans and_and_and_comm\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Prod","family_id":"map_bijective","file_id":"mathlib/Mathlib/Data/Prod/Basic.lean","sample_id":"5d67e5e7b9950556ea8bd8cdfa3767dbe10a6562717cb86050afc39cdaf6a4b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4e24363fb7ba779b1d49eb8030dc7b2f8941ee6210796f291a186ebbe93a5464","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"bffda7eb658c234a9b6c2ed0b066a47fa2bff85af6a53b9d9b38a9ace4fa3206","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"31d3eaad908c26db9fa8cd7a18a465f80cc33c5d4402214fcc63790030c8fb99","source_sha256":"05b71bb08d5e0d5da2d7d8beb37172741e5772665eaf54d30c9d72516996fb96","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply complementedLattice_of_sSup_atoms_eq_top\n  have : ∀ i ∈ s, ∃ t : Set α, f i = sSup t ∧ ∀ a ∈ t, IsAtom a := fun i hi ↦ by\n    replace h := complementedLattice_iff_isAtomistic.mp (h i hi)\n    obtain ⟨u, hu, hu'⟩ := eq_sSup_atoms (⊤ : Iic (f i))\n    refine ⟨(↑) '' u, ?_, ?_⟩\n    · replace hu : f i = ↑(sSup u) := Subtype.ext_iff.mp hu\n      simp_rw [hu, Iic.coe_sSup]\n    · rintro b ⟨⟨a, ha'⟩, ha, rfl⟩\n      exact IsAtom.of_isAtom_coe_Iic (hu' _ ha)\n  choose t ht ht' using this\n  let u : Set α := ⋃ i, ⋃ hi : i ∈ s, t i hi\n  have hu₁ : u ⊆ {a | IsAtom a} := by\n    rintro a ⟨-, ⟨i, rfl⟩, ⟨-, ⟨hi, rfl⟩, ha : a ∈ t i hi⟩⟩\n    exact ht' i hi a ha\n  have hu₂ : sSup u = ⨆ i ∈ s, f i := by simp_rw [u, sSup_iUnion, biSup_congr' ht]\n  rw [eq_top_iff, ← h', ← hu₂]\n  exact sSup_le_sSup hu₁","hard_negative":true,"metrics":{"chosen_tokens":244,"rejected_tokens":3,"token_jaccard":0.02439,"token_length_ratio":0.012295},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"26c97243878f536132b113f5de48c4a6850fadfc42945681a20d71f40d507235","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.CompleteLatticeIntervals\npublic import Mathlib.Order.CompactlyGenerated.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Results about compactness properties for intervals in complete lattices\n-/\n\npublic section\n\nvariable {ι α : Type*} [CompleteLattice α]\n\nnamespace Set.Iic\n\ntheorem isCompactElement {a : α} {b : Iic a} (h : IsCompactElement (b : α)) :\n    IsCompactElement b := by\n  simp only [CompleteLattice.isCompactElement_iff_exists_le_iSup_of_le_iSup,\n    Finset.sup_eq_iSup] at h ⊢\n  intro ι s hb\n  replace hb : (b : α) ≤ iSup ((↑) ∘ s) := le_trans hb <| (coe_iSup s) ▸ le_refl _\n  obtain ⟨t, ht⟩ := h ι ((↑) ∘ s) hb\n  exact ⟨t, (by simpa using ht : (b : α) ≤ _)⟩\n\ninstance instIsCompactlyGenerated [IsCompactlyGenerated α] {a : α} :\n    IsCompactlyGenerated (Iic a) := by\n  refine ⟨fun ⟨x, (hx : x ≤ a)⟩ ↦ ?_⟩\n  obtain ⟨s, hs, rfl⟩ := IsCompactlyGenerated.exists_sSup_eq x\n  rw [sSup_le_iff] at hx\n  let f : s → Iic a := fun y ↦ ⟨y, hx _ y.property⟩\n  refine ⟨range f, ?_, ?_⟩\n  · rintro - ⟨⟨y, hy⟩, hy', rfl⟩\n    exact isCompactElement (hs _ hy)\n  · rw [Subtype.ext_iff]\n    change sSup (((↑) : Iic a → α) '' (range f)) = sSup s\n    congr\n    ext b\n    simpa [f] using hx b\n\nend Set.Iic\n\nopen Set (Iic)\n\nTarget:\ntheorem complementedLattice_of_complementedLattice_Iic\n    [IsModularLattice α] [IsCompactlyGenerated α]\n    {s : Set ι} {f : ι → α}\n    (h : ∀ i ∈ s, ComplementedLattice <| Iic (f i))\n    (h' : ⨆ i ∈ s, f i = ⊤) :\n    ComplementedLattice α :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_31d3eaad908c","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"fcef7b034e506b6423a928364b9961e54d0c1d5f7a0d6df53b0d61e0af74f6e1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/CompactlyGenerated","family_id":"complementedlattice_of_complementedlattice_iic","file_id":"mathlib/Mathlib/Order/CompactlyGenerated/Intervals.lean","sample_id":"31d3eaad908c26db9fa8cd7a18a465f80cc33c5d4402214fcc63790030c8fb99"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f4a24f2bb7a6f0211b6b657b7e7dccc3a2dcf7c7a859099b25ae3a21d107e6f6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"956561a8c5c16a4f630b857e9ae4145c51a548f34056583aef5e42d25cb0ec66","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"955e34b4be7edd56dadecaa66b57be2059d1e36951adf123ec19009bee070aac","source_sha256":"7d247632b790bfc237d79d3c040fb7f21d9cff00d190a67e120ccb73056460a6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only [← hammingDist_zero_right, hf]\n    using! hammingDist_comp_le_hammingDist f (y := fun _ ↦ 0)","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":3,"token_jaccard":0.043478,"token_length_ratio":0.142857},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"26dfce741a44216871b84c3e29ec976fc17785b97e7ca021f994dae056a5ae6d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Group.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Wrenna Robson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Wrenna Robson\n-/\n/-!\n# Hamming spaces\n\nThe Hamming metric counts the number of places two members of a (finite) Pi type\ndiffer. The Hamming norm is the same as the Hamming metric over additive groups, and\ncounts the number of places a member of a (finite) Pi type differs from zero.\n\nThis is a useful notion in various applications, but in particular it is relevant\nin coding theory, in which it is fundamental for defining the minimum distance of a\ncode.\n\n## Main definitions\n* `hammingDist x y`: the Hamming distance between `x` and `y`, the number of entries which differ.\n* `hammingNorm x`: the Hamming norm of `x`, the number of non-zero entries.\n* `Hamming β`: a type synonym for `Π i, β i` with `dist` and `norm` provided by the above.\n* `Hamming.toHamming`, `Hamming.ofHamming`: functions for casting between `Hamming β` and\n  `Π i, β i`.\n* the Hamming norm forms a normed group on `Hamming β`.\n-/\n\n@[expose] public section\n\n\nsection HammingDistNorm\n\nopen Finset Function\n\nvariable {α ι : Type*} {β : ι → Type*} [Fintype ι] [∀ i, DecidableEq (β i)]\nvariable {γ : ι → Type*} [∀ i, DecidableEq (γ i)]\n\n/-- The Hamming distance function to the naturals. -/\ndef hammingDist (x y : ∀ i, β i) : ℕ := #{i | x i ≠ y i}\n\n/-- Corresponds to `dist_self`. -/\n@[simp]\ntheorem hammingDist_self (x : ∀ i, β i) : hammingDist x x = 0 := by\n  rw [hammingDist, card_eq_zero, filter_eq_empty_iff]\n  exact fun _ _ H => H rfl\n\n-- TODO: this seems unnecessary.\n/-- Corresponds to `dist_nonneg`. -/\ntheorem hammingDist_nonneg {x y : ∀ i, β i} : 0 ≤ hammingDist x y :=\n  zero_le\n\n/-- Corresponds to `dist_comm`. -/\ntheorem hammingDist_comm (x y : ∀ i, β i) : hammingDist x y = hammingDist y x := by\n  simp_rw [hammingDist, ne_comm]\n\n/-- Corresponds to `dist_triangle`. -/\ntheorem hammingDist_triangle (x y z : ∀ i, β i) :\n    hammingDist x z ≤ hammingDist x y + hammingDist y z := by\n  classical\n    unfold hammingDist\n    refine le_trans (card_mono ?_) (card_union_le _ _)\n    rw [← filter_or]\n    exact monotone_filter_right _ fun i _ h ↦ (h.ne_or_ne _).imp_right Ne.symm\n\n/-- Corresponds to `dist_triangle_left`. -/\ntheorem hammingDist_triangle_left (x y z : ∀ i, β i) :\n    hammingDist x y ≤ hammingDist z x + hammingDist z y := by\n  rw [hammingDist_comm z]\n  exact hammingDist_triangle _ _ _\n\n/-- Corresponds to `dist_triangle_right`. -/\ntheorem hammingDist_triangle_right (x y z : ∀ i, β i) :\n    hammingDist x y ≤ hammingDist x z + hammingDist y z := by\n  rw [hammingDist_comm y]\n  exact hammingDist_triangle _ _ _\n\n/-- Corresponds to `swap_dist`. -/\ntheorem swap_hammingDist : swap (@hammingDist _ β _ _) = hammingDist := by\n  funext x y\n  exact hammingDist_comm _ _\n\n/-- Corresponds to `eq_of_dist_eq_zero`. -/\ntheorem eq_of_hammingDist_eq_zero {x y : ∀ i, β i} : hammingDist x y = 0 → x = y := by\n  simp_rw [hammingDist, card_eq_zero, filter_eq_empty_iff, Classical.not_not, funext_iff, mem_univ,\n    forall_true_left, imp_self]\n\n/-- Corresponds to `dist_eq_zero`. -/\n@[simp]\ntheorem hammingDist_eq_zero {x y : ∀ i, β i} : hammingDist x y = 0 ↔ x = y :=\n  ⟨eq_of_hammingDist_eq_zero, fun H => by\n    rw [H]\n    exact hammingDist_self _⟩\n\n/-- Corresponds to `zero_eq_dist`. -/\n@[simp]\ntheorem hamming_zero_eq_dist {x y : ∀ i, β i} : 0 = hammingDist x y ↔ x = y := by\n  rw [eq_comm, hammingDist_eq_zero]\n\n/-- Corresponds to `dist_ne_zero`. -/\ntheorem hammingDist_ne_zero {x y : ∀ i, β i} : hammingDist x y ≠ 0 ↔ x ≠ y :=\n  hammingDist_eq_zero.not\n\n/-- Corresponds to `dist_pos`. -/\n@[simp]\ntheorem hammingDist_pos {x y : ∀ i, β i} : 0 < hammingDist x y ↔ x ≠ y := by\n  rw [← hammingDist_ne_zero, iff_not_comm, not_lt, Nat.le_zero]\n\ntheorem hammingDist_lt_one {x y : ∀ i, β i} : hammingDist x y < 1 ↔ x = y := by\n  rw [Nat.lt_one_iff, hammingDist_eq_zero]\n\ntheorem hammingDist_le_card_fintype {x y : ∀ i, β i} : hammingDist x y ≤ Fintype.card ι :=\n  card_le_univ _\n\ntheorem hammingDist_comp_le_hammingDist (f : ∀ i, γ i → β i) {x y : ∀ i, γ i} :\n    hammingDist (fun i => f i (x i)) (fun i => f i (y i)) ≤ hammingDist x y := by\n  dsimp [hammingDist]; gcongr; simp +contextual\n\ntheorem hammingDist_comp (f : ∀ i, γ i → β i) {x y : ∀ i, γ i} (hf : ∀ i, Injective (f i)) :\n    hammingDist (fun i => f i (x i)) (fun i => f i (y i)) = hammingDist x y :=\n  le_antisymm (hammingDist_comp_le_hammingDist _) <| by dsimp [hammingDist]; gcongr; exact @hf _ _ _\n\ntheorem hammingDist_smul_le_hammingDist [∀ i, SMul α (β i)] {k : α} {x y : ∀ i, β i} :\n    hammingDist (k • x) (k • y) ≤ hammingDist x y :=\n  hammingDist_comp_le_hammingDist fun i => (k • · : β i → β i)\n\n/-- Corresponds to `dist_smul` with the discrete norm on `α`. -/\ntheorem hammingDist_smul [∀ i, SMul α (β i)] {k : α} {x y : ∀ i, β i}\n    (hk : ∀ i, IsSMulRegular (β i) k) : hammingDist (k • x) (k • y) = hammingDist x y :=\n  hammingDist_comp (fun i => (k • · : β i → β i)) hk\n\nsection Zero\n\nvariable [∀ i, Zero (β i)] [∀ i, Zero (γ i)]\n\n/-- The Hamming weight function to the naturals. -/\ndef hammingNorm (x : ∀ i, β i) : ℕ := #{i | x i ≠ 0}\n\n/-- Corresponds to `dist_zero_right`. -/\n@[simp]\ntheorem hammingDist_zero_right (x : ∀ i, β i) : hammingDist x 0 = hammingNorm x :=\n  rfl\n\n/-- Corresponds to `dist_zero_left`. -/\n@[simp]\ntheorem hammingDist_zero_left : hammingDist (0 : ∀ i, β i) = hammingNorm :=\n  funext fun x => by rw [hammingDist_comm, hammingDist_zero_right]\n\n-- TODO: this seems unnecessary.\n/-- Corresponds to `norm_nonneg`. -/\ntheorem hammingNorm_nonneg {x : ∀ i, β i} : 0 ≤ hammingNorm x :=\n  zero_le\n\n/-- Corresponds to `norm_zero`. -/\n@[simp]\ntheorem hammingNorm_zero : hammingNorm (0 : ∀ i, β i) = 0 :=\n  hammingDist_self _\n\n/-- Corresponds to `norm_eq_zero`. -/\n@[simp]\ntheorem hammingNorm_eq_zero {x : ∀ i, β i} : hammingNorm x = 0 ↔ x = 0 :=\n  hammingDist_eq_zero\n\n/-- Corresponds to `norm_ne_zero_iff`. -/\ntheorem hammingNorm_ne_zero_iff {x : ∀ i, β i} : hammingNorm x ≠ 0 ↔ x ≠ 0 :=\n  hammingNorm_eq_zero.not\n\n/-- Corresponds to `norm_pos_iff`. -/\n@[simp]\ntheorem hammingNorm_pos_iff {x : ∀ i, β i} : 0 < hammingNorm x ↔ x ≠ 0 :=\n  hammingDist_pos\n\ntheorem hammingNorm_lt_one {x : ∀ i, β i} : hammingNorm x < 1 ↔ x = 0 :=\n  hammingDist_lt_one\n\ntheorem hammingNorm_le_card_fintype {x : ∀ i, β i} : hammingNorm x ≤ Fintype.card ι :=\n  hammingDist_le_card_fintype\n\nTarget:\ntheorem hammingNorm_comp_le_hammingNorm (f : ∀ i, γ i → β i) {x : ∀ i, γ i} (hf : ∀ i, f i 0 = 0) :\n    (hammingNorm fun i => f i (x i)) ≤ hammingNorm x :=\n\nProof body:\n","rejected":"by\n  exact hammingNorm_comp_le_hammingNorm","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"ac917cfe9c75c6e989bb977b96027860dd8dde27ca91cc532283cf877b46aa19","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"InformationTheory","family_id":"hammingnorm_comp_le_hammingnorm","file_id":"mathlib/Mathlib/InformationTheory/Hamming.lean","sample_id":"955e34b4be7edd56dadecaa66b57be2059d1e36951adf123ec19009bee070aac"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5e809a8a5b6dc52f86458c7ec1ba0de189d9ba286d3abe6189f2eb23181c4aa7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"72e24772baf3562fa19ff8d7c94a207bf1314dd59ae9ee733427021020caba70","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"94279022a062f44341d707de6cd3452c8f04400967480dbebc54e8c3a85912a9","source_sha256":"320ae2b6adb110a4f3c6cc9d28358d174f6e3d360ad3ba6f7f42c745ee840c5c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine Set.ext fun R ↦ ⟨fun ⟨g, hg⟩ ↦ hg ▸ ⟨hQ.smul g, hP.smul g⟩, fun h ↦ ?_⟩\n  have : R.IsPrime := h.1\n  obtain ⟨g, hg⟩ := exists_smul_of_under_eq A B G Q R (hP.over.symm.trans h.2.over)\n  exact ⟨g, hg.symm⟩","hard_negative":false,"metrics":{"chosen_tokens":80,"rejected_tokens":84,"token_jaccard":0.926829,"token_length_ratio":1.05},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"26ee24b7c91c431d3213ef463003a93c54a9723cb87d5a4868e3d1751b40c22d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Invariant.Defs\npublic import Mathlib.RingTheory.IntegralClosure.IntegralRestrict\npublic import Mathlib.RingTheory.LocalRing.ResidueField.Ideal\n\nNamespace:\nAlgebra.IsInvariant\n\nLocal context:\n/-\nCopyright (c) 2024 Thomas Browning. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Thomas Browning\n-/\n/-!\n# Invariant Extensions of Rings\n\nGiven an extension of rings `B/A` and an action of `G` on `B`, we introduce a predicate\n`Algebra.IsInvariant A B G` which states that every fixed point of `B` lies in the image of `A`.\n\nThe main application is in algebraic number theory, where `G := Gal(L/K)` is the Galois group\nof some finite Galois extension of number fields, and `A := 𝓞K` and `B := 𝓞L` are their ring of\nintegers. This main result in this file implies the existence of Frobenius elements in this setting.\nSee `Mathlib/RingTheory/Frobenius.lean`.\n\n## Main statements\n\nLet `G` be a finite group acting on a commutative ring `B` satisfying `Algebra.IsInvariant A B G`.\n\n* `Algebra.IsInvariant.isIntegral`: `B/A` is an integral extension.\n* `Algebra.IsInvariant.exists_smul_of_under_eq`: `G` acts transitivity on the prime ideals of `B`\n  lying above a given prime ideal of `A`.\n\nIf `Q` is a prime ideal of `B` lying over a prime ideal `P` of `A`, then\n\n* `IsFractionRing.stabilizerHom_surjective`:\n  The stabilizer subgroup of `Q` surjects onto `Aut(Frac(B/Q)/Frac(A/P))`.\n* `Ideal.Quotient.stabilizerHom_surjective`:\n  The stabilizer subgroup of `Q` surjects onto `Aut((B/Q)/(A/P))`.\n* `Ideal.Quotient.exists_algEquiv_fixedPoint_quotient_under`:\n  If `k` is a domain containing `B/Q`, then any `A/P`-algebra automorphism of `k` restricts to\n  an automorphism of `B/Q`.\n-/\n\n@[expose] public section\n\nopen scoped Pointwise\n\nsection Galois\n\nvariable (A K L B : Type*) [CommRing A] [CommRing B] [Field K] [Field L]\n  [Algebra A K] [Algebra B L] [IsFractionRing A K] [IsFractionRing B L]\n  [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L]\n  [IsIntegrallyClosed A] [IsIntegralClosure B A L]\n\n/-- In the AKLB setup, the Galois group of `L/K` acts on `B`. -/\n@[implicit_reducible]\nnoncomputable def IsIntegralClosure.MulSemiringAction [Algebra.IsAlgebraic K L] :\n    MulSemiringAction Gal(L/K) B :=\n  MulSemiringAction.compHom B (galRestrict A K L B).toMonoidHom\n\ninstance [Algebra.IsAlgebraic K L] : let := IsIntegralClosure.MulSemiringAction A K L B\n    SMulDistribClass Gal(L/K) B L :=\n  let := IsIntegralClosure.MulSemiringAction A K L B\n  ⟨fun g b l ↦ by\n    simp only [Algebra.smul_def, smul_mul', mul_eq_mul_right_iff]\n    exact Or.inl (algebraMap_galRestrictHom_apply A K L B g b).symm⟩\n\n/-- In the AKLB setup, every fixed point of `B` lies in the image of `A`. -/\ntheorem Algebra.isInvariant_of_isGalois [FiniteDimensional K L] [h : IsGalois K L] :\n    letI := IsIntegralClosure.MulSemiringAction A K L B\n    Algebra.IsInvariant A B Gal(L/K) := by\n  replace h := ((IsGalois.tfae (F := K) (E := L)).out 0 1).mp h\n  letI := IsIntegralClosure.MulSemiringAction A K L B\n  refine ⟨fun b hb ↦ ?_⟩\n  replace hb : algebraMap B L b ∈ IntermediateField.fixedField (⊤ : Subgroup Gal(L/K)) := by\n    rintro ⟨g, -⟩\n    exact (algebraMap_galRestrict_apply A g b).symm.trans (congrArg (algebraMap B L) (hb g))\n  rw [h, IntermediateField.mem_bot] at hb\n  obtain ⟨k, hk⟩ := hb\n  have hb : IsIntegral A b := IsIntegralClosure.isIntegral A L b\n  rw [← isIntegral_algebraMap_iff (FaithfulSMul.algebraMap_injective B L), ← hk,\n    isIntegral_algebraMap_iff (FaithfulSMul.algebraMap_injective K L)] at hb\n  obtain ⟨a, rfl⟩ := IsIntegrallyClosed.algebraMap_eq_of_integral hb\n  rw [← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply A B L,\n    (FaithfulSMul.algebraMap_injective B L).eq_iff] at hk\n  exact ⟨a, hk⟩\n\n/-- A variant of `Algebra.isInvariant_of_isGalois`, replacing `Gal(L/K)` by `Aut(B/A)`. -/\ntheorem Algebra.isInvariant_of_isGalois' [FiniteDimensional K L] [IsGalois K L] :\n    Algebra.IsInvariant A B (B ≃ₐ[A] B) :=\n  ⟨fun b h ↦ (isInvariant_of_isGalois A K L B).1 b (fun g ↦ h (galRestrict A K L B g))⟩\n\nend Galois\n\nsection Quotient\n\nvariable {A B : Type*} [CommRing A] [CommRing B] [Algebra A B]\nvariable {G : Type*} [Group G] [MulSemiringAction G B] [SMulCommClass G A B]\n\ninstance (H : Subgroup G) [H.Normal] :\n    MulSemiringAction (G ⧸ H) (FixedPoints.subring B H) where\n  smul := Quotient.lift (fun g x ↦ ⟨g • x, fun h ↦ by\n    simpa [mul_smul] using! congr(g • $(x.2 ⟨_, ‹H.Normal›.conj_mem' _ h.2 g⟩))⟩) (by\n    rintro _ a ⟨⟨⟨b⟩, hb⟩, rfl⟩\n    ext c\n    simpa [mul_smul] using! congr(a • $(c.2 ⟨b, hb⟩)))\n  one_smul b := Subtype.ext (one_smul G b.1)\n  mul_smul := Quotient.ind₂ fun _ _ _ ↦ Subtype.ext (mul_smul _ _ _)\n  smul_zero := Quotient.ind fun _ ↦ Subtype.ext (smul_zero _)\n  smul_add := Quotient.ind fun _ _ _ ↦ Subtype.ext (smul_add _ _ _)\n  smul_one := Quotient.ind fun _ ↦ Subtype.ext (smul_one _)\n  smul_mul := Quotient.ind fun _ _ _ ↦ Subtype.ext (MulSemiringAction.smul_mul _ _ _)\n\ninstance (H : Subgroup G) [H.Normal] :\n    MulSemiringAction (G ⧸ H) (FixedPoints.subalgebra A B H) :=\n  inferInstanceAs (MulSemiringAction (G ⧸ H) (FixedPoints.subring B H))\n\ninstance (H : Subgroup G) [H.Normal] :\n    SMulCommClass (G ⧸ H) A (FixedPoints.subalgebra A B H) where\n  smul_comm := Quotient.ind fun g r h ↦ Subtype.ext (smul_comm g r h.1)\n\ninstance (H : Subgroup G) [H.Normal] [Algebra.IsInvariant A B G] :\n    Algebra.IsInvariant A (FixedPoints.subalgebra A B H) (G ⧸ H) where\n  isInvariant x hx := by\n    obtain ⟨y, hy⟩ := Algebra.IsInvariant.isInvariant (A := A) (G := G) x.1\n      (fun g ↦ congr_arg Subtype.val (hx g))\n    exact ⟨y, Subtype.ext hy⟩\n\nend Quotient\n\nsection transitivity\n\nvariable (A B G : Type*) [CommRing A] [CommRing B] [Algebra A B] [Group G] [MulSemiringAction G B]\n\nnamespace MulSemiringAction\n\nopen Polynomial\n\nvariable {B} [Fintype G]\n\n/-- Characteristic polynomial of a finite group action on a ring. -/\nnoncomputable def charpoly (b : B) : B[X] := ∏ g : G, (X - C (g • b))\n\ntheorem charpoly_eq (b : B) : charpoly G b = ∏ g : G, (X - C (g • b)) := rfl\n\ntheorem charpoly_eq_prod_smul (b : B) : charpoly G b = ∏ g : G, g • (X - C b) := by\n  simp only [smul_sub, smul_C, smul_X, charpoly_eq]\n\ntheorem monic_charpoly (b : B) : (charpoly G b).Monic :=\n  monic_prod_of_monic _ _ (fun _ _ ↦ monic_X_sub_C _)\n\ntheorem splits_charpoly (b : B) : (charpoly G b).Splits :=\n  .prod fun g _ ↦ .X_sub_C (g • b)\n\ntheorem eval_charpoly (b : B) : (charpoly G b).eval b = 0 := by\n  rw [charpoly_eq, eval_prod]\n  apply Finset.prod_eq_zero (Finset.mem_univ (1 : G))\n  rw [one_smul, eval_sub, eval_C, eval_X, sub_self]\n\nvariable {G}\n\ntheorem smul_charpoly (b : B) (g : G) : g • (charpoly G b) = charpoly G b := by\n  rw [charpoly_eq_prod_smul, Finset.smul_prod_perm]\n\ntheorem smul_coeff_charpoly (b : B) (n : ℕ) (g : G) :\n    g • (charpoly G b).coeff n = (charpoly G b).coeff n := by\n  rw [← coeff_smul, smul_charpoly]\n\nend MulSemiringAction\n\nnamespace Algebra.IsInvariant\n\nopen MulSemiringAction Polynomial\n\nvariable [IsInvariant A B G]\n\ntheorem charpoly_mem_lifts [Fintype G] (b : B) :\n    charpoly G b ∈ Polynomial.lifts (algebraMap A B) :=\n  (charpoly G b).lifts_iff_coeff_lifts.mpr fun n ↦ isInvariant _ (smul_coeff_charpoly b n)\n\ntheorem isIntegral [Finite G] : Algebra.IsIntegral A B := by\n  cases nonempty_fintype G\n  refine ⟨fun b ↦ ?_⟩\n  obtain ⟨p, hp1, -, hp2⟩ := Polynomial.lifts_and_natDegree_eq_and_monic\n    (charpoly_mem_lifts A B G b) (monic_charpoly G b)\n  exact ⟨p, hp2, by rw [← eval_map, hp1, eval_charpoly]⟩\n\n/-- `G` acts transitively on the prime ideals of `B` above a given prime ideal of `A`. -/\ntheorem exists_smul_of_under_eq [Finite G] [SMulCommClass G A B]\n    (P Q : Ideal B) [hP : P.IsPrime] [hQ : Q.IsPrime]\n    (hPQ : P.under A = Q.under A) :\n    ∃ g : G, Q = g • P := by\n  cases nonempty_fintype G\n  have : ∀ (P Q : Ideal B) [P.IsPrime] [Q.IsPrime], P.under A = Q.under A →\n      ∃ g ∈ (⊤ : Finset G), Q ≤ g • P := by\n    intro P Q hP hQ hPQ\n    rw [← Ideal.subset_union_prime 1 1 (fun _ _ _ _ ↦ hP.smul _)]\n    intro b hb\n    suffices h : ∃ g ∈ Finset.univ, g • b ∈ P by\n      obtain ⟨g, -, hg⟩ := h\n      apply Set.mem_biUnion (Finset.mem_univ g⁻¹) (Ideal.mem_inv_pointwise_smul_iff.mpr hg)\n    obtain ⟨a, ha⟩ := isInvariant (A := A) (∏ g : G, g • b) (Finset.smul_prod_perm b)\n    rw [← hP.prod_mem_iff, ← ha, ← P.mem_comap, ← P.under_def A,\n      hPQ, Q.mem_comap, ha, hQ.prod_mem_iff]\n    exact ⟨1, Finset.mem_univ 1, (one_smul G b).symm ▸ hb⟩\n  obtain ⟨g, -, hg⟩ := this P Q hPQ\n  obtain ⟨g', -, hg'⟩ := this Q (g • P) ((P.under_smul A g).trans hPQ).symm\n  exact ⟨g, le_antisymm hg (smul_eq_of_le_smul (hg.trans hg') ▸ hg')⟩\n\nTarget:\ntheorem orbit_eq_primesOver [Finite G] [SMulCommClass G A B] (P : Ideal A) (Q : Ideal B)\n    [hP : Q.LiesOver P] [hQ : Q.IsPrime] : MulAction.orbit G Q = P.primesOver B :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  refine Set.ext fun R ↦ ⟨fun ⟨g, hg⟩ ↦ hg ▸ ⟨hQ.smul g, hP.smul g⟩, fun h ↦ ?_⟩\n  have : R.IsPrime := h.1\n  obtain ⟨g, hg⟩ := exists_smul_of_under_eq A B G Q R (hP.over.symm.trans h.2.over)\n  exact ⟨g, hg.symm⟩","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Invariant","family_id":"orbit_eq_primesover","file_id":"mathlib/Mathlib/RingTheory/Invariant/Basic.lean","sample_id":"94279022a062f44341d707de6cd3452c8f04400967480dbebc54e8c3a85912a9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"75232f0c801e33fc6a2f2b42c9fbde78aef8e1d36674a56a7714ae3434228c02","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8129b6b03a4962538cfee55f286a700211b4dd9a42f851c5da701ccd34c2d104","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"419897952885489eef42100d7198d985d0ed456c5f3c123051bf15c5b6fac476","source_sha256":"5fd82395c755ee110eacac536435f2600396b3d8488a7cfc9d34b3e8c0d5cde4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [(v.filterAt_basis_closedBall x).mem_iff, ← and_imp, subset_def, mem_setOf]","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":25,"token_jaccard":0.809524,"token_length_ratio":1.25},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"27dcd539c7528a16bab0e42c10877179eb044a1b2367e5446fd31b51e65116e0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Measure.AbsolutelyContinuous\n\nNamespace:\nVitaliFamily\n\nLocal context:\n/-\nCopyright (c) 2021 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Vitali families\n\nOn a metric space `X` with a measure `μ`, consider for each `x : X` a family of measurable sets with\nnonempty interiors, called `setsAt x`. This family is a Vitali family if it satisfies the following\nproperty: consider a (possibly non-measurable) set `s`, and for any `x` in `s` a\nsubfamily `f x` of `setsAt x` containing sets of arbitrarily small diameter. Then one can extract\na disjoint subfamily covering almost all `s`.\n\nVitali families are provided by covering theorems such as the Besicovitch covering theorem or the\nVitali covering theorem. They make it possible to formulate general versions of theorems on\ndifferentiations of measure that apply in both contexts.\n\nThis file gives the basic definition of Vitali families. More interesting developments of this\nnotion are deferred to other files:\n* constructions of specific Vitali families are provided by the Besicovitch covering theorem, in\n  `Besicovitch.vitaliFamily`, and by the Vitali covering theorem, in `Vitali.vitaliFamily`.\n* The main theorem on differentiation of measures along a Vitali family is proved in\n  `VitaliFamily.ae_tendsto_rnDeriv`.\n\n## Main definitions\n\n* `VitaliFamily μ` is a structure made, for each `x : X`, of a family of sets around `x`, such that\n  one can extract an almost everywhere disjoint covering from any subfamily containing sets of\n  arbitrarily small diameters.\n\nLet `v` be such a Vitali family.\n* `v.FineSubfamilyOn` describes the subfamilies of `v` from which one can extract almost\n  everywhere disjoint coverings. This property, called\n  `v.FineSubfamilyOn.exists_disjoint_covering_ae`, is essentially a restatement of the definition\n  of a Vitali family. We also provide an API to use efficiently such a disjoint covering.\n* `v.filterAt x` is a filter on sets of `X`, such that convergence with respect to this filter\n  means convergence when sets in the Vitali family shrink towards `x`.\n\n## References\n\n* [Herbert Federer, Geometric Measure Theory, Chapter 2.8][Federer1996]\n  (Vitali families are called Vitali relations there)\n-/\n\n@[expose] public section\n\n\nopen MeasureTheory Metric Set Filter TopologicalSpace MeasureTheory.Measure\nopen scoped Topology\n\nvariable {X : Type*} [PseudoMetricSpace X]\n\n/-- On a metric space `X` with a measure `μ`, consider for each `x : X` a family of measurable sets\nwith nonempty interiors, called `setsAt x`. This family is a Vitali family if it satisfies the\nfollowing property: consider a (possibly non-measurable) set `s`, and for any `x` in `s` a\nsubfamily `f x` of `setsAt x` containing sets of arbitrarily small diameter. Then one can extract\na disjoint subfamily covering almost all `s`.\n\nVitali families are provided by covering theorems such as the Besicovitch covering theorem or the\nVitali covering theorem. They make it possible to formulate general versions of theorems on\ndifferentiations of measure that apply in both contexts.\n-/\nstructure VitaliFamily {m : MeasurableSpace X} (μ : Measure X) where\n  /-- Sets of the family \"centered\" at a given point. -/\n  setsAt : X → Set (Set X)\n  /-- All sets of the family are measurable. -/\n  measurableSet : ∀ x : X, ∀ s ∈ setsAt x, MeasurableSet s\n  /-- All sets of the family have nonempty interior. -/\n  nonempty_interior : ∀ x : X, ∀ s ∈ setsAt x, (interior s).Nonempty\n  /-- For any closed ball around `x`, there exists a set of the family contained in this ball. -/\n  nontrivial : ∀ (x : X), ∀ ε > (0 : ℝ), ∃ s ∈ setsAt x, s ⊆ closedBall x ε\n  /-- Consider a (possibly non-measurable) set `s`,\n  and for any `x` in `s` a subfamily `f x` of `setsAt x`\n  containing sets of arbitrarily small diameter.\n  Then one can extract a disjoint subfamily covering almost all `s`. -/\n  covering : ∀ (s : Set X) (f : X → Set (Set X)),\n    (∀ x ∈ s, f x ⊆ setsAt x) → (∀ x ∈ s, ∀ ε > (0 : ℝ), ∃ t ∈ f x, t ⊆ closedBall x ε) →\n    ∃ t : Set (X × Set X), (∀ p ∈ t, p.1 ∈ s) ∧ (t.PairwiseDisjoint fun p ↦ p.2) ∧\n      (∀ p ∈ t, p.2 ∈ f p.1) ∧ μ (s \\ ⋃ p ∈ t, p.2) = 0\n\nnamespace VitaliFamily\n\nvariable {m0 : MeasurableSpace X} {μ : Measure X}\n\n/-- A Vitali family for a measure `μ` is also a Vitali family for any measure absolutely continuous\nwith respect to `μ`. -/\ndef mono (v : VitaliFamily μ) (ν : Measure X) (hν : ν ≪ μ) : VitaliFamily ν where\n  __ := v\n  covering s f h h' :=\n    let ⟨t, ts, disj, mem_f, hμ⟩ := v.covering s f h h'\n    ⟨t, ts, disj, mem_f, hν hμ⟩\n\n/-- Given a Vitali family `v` for a measure `μ`, a family `f` is a fine subfamily on a set `s` if\nevery point `x` in `s` belongs to arbitrarily small sets in `v.setsAt x ∩ f x`. This is precisely\nthe subfamilies for which the Vitali family definition ensures that one can extract a disjoint\ncovering of almost all `s`. -/\ndef FineSubfamilyOn (v : VitaliFamily μ) (f : X → Set (Set X)) (s : Set X) : Prop :=\n  ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε\n\nnamespace FineSubfamilyOn\n\nvariable {v : VitaliFamily μ} {f : X → Set (Set X)} {s : Set X} (h : v.FineSubfamilyOn f s)\ninclude h\n\ntheorem exists_disjoint_covering_ae :\n    ∃ t : Set (X × Set X),\n      (∀ p : X × Set X, p ∈ t → p.1 ∈ s) ∧\n      (t.PairwiseDisjoint fun p => p.2) ∧\n      (∀ p : X × Set X, p ∈ t → p.2 ∈ v.setsAt p.1 ∩ f p.1) ∧\n      μ (s \\ ⋃ (p : X × Set X) (_ : p ∈ t), p.2) = 0 :=\n  v.covering s (fun x => v.setsAt x ∩ f x) (fun _ _ => inter_subset_left) h\n\n/-- Given `h : v.FineSubfamilyOn f s`, then `h.index` is a set parametrizing a disjoint\ncovering of almost every `s`. -/\nprotected def index : Set (X × Set X) :=\n  h.exists_disjoint_covering_ae.choose\n\n/-- Given `h : v.FineSubfamilyOn f s`, then `h.covering p` is a set in the family,\nfor `p ∈ h.index`, such that these sets form a disjoint covering of almost every `s`. -/\n@[nolint unusedArguments]\nprotected def covering (_h : FineSubfamilyOn v f s) : X × Set X → Set X :=\n  fun p => p.2\n\ntheorem index_subset : ∀ p : X × Set X, p ∈ h.index → p.1 ∈ s :=\n  h.exists_disjoint_covering_ae.choose_spec.1\n\ntheorem covering_disjoint : h.index.PairwiseDisjoint h.covering :=\n  h.exists_disjoint_covering_ae.choose_spec.2.1\n\nopen scoped Function in -- required for scoped `on` notation\ntheorem covering_disjoint_subtype : Pairwise (Disjoint on fun x : h.index => h.covering x) :=\n  (pairwise_subtype_iff_pairwise_set _ _).2 h.covering_disjoint\n\ntheorem covering_mem {p : X × Set X} (hp : p ∈ h.index) : h.covering p ∈ f p.1 :=\n  (h.exists_disjoint_covering_ae.choose_spec.2.2.1 p hp).2\n\ntheorem covering_mem_family {p : X × Set X} (hp : p ∈ h.index) : h.covering p ∈ v.setsAt p.1 :=\n  (h.exists_disjoint_covering_ae.choose_spec.2.2.1 p hp).1\n\ntheorem measure_sdiff_biUnion : μ (s \\ ⋃ p ∈ h.index, h.covering p) = 0 :=\n  h.exists_disjoint_covering_ae.choose_spec.2.2.2\n\n@[deprecated (since := \"2026-06-03\")] alias measure_diff_biUnion := measure_sdiff_biUnion\n\ntheorem index_countable [SecondCountableTopology X] : h.index.Countable :=\n  h.covering_disjoint.countable_of_nonempty_interior fun _ hx =>\n    v.nonempty_interior _ _ (h.covering_mem_family hx)\n\nprotected theorem measurableSet_u {p : X × Set X} (hp : p ∈ h.index) :\n    MeasurableSet (h.covering p) :=\n  v.measurableSet p.1 _ (h.covering_mem_family hp)\n\ntheorem measure_le_tsum_of_absolutelyContinuous [SecondCountableTopology X] {ρ : Measure X}\n    (hρ : ρ ≪ μ) : ρ s ≤ ∑' p : h.index, ρ (h.covering p) :=\n  calc\n    ρ s ≤ ρ ((s \\ ⋃ p ∈ h.index, h.covering p) ∪ ⋃ p ∈ h.index, h.covering p) :=\n      measure_mono (by simp only [subset_union_left, sdiff_union_self])\n    _ ≤ ρ (s \\ ⋃ p ∈ h.index, h.covering p) + ρ (⋃ p ∈ h.index, h.covering p) :=\n      (measure_union_le _ _)\n    _ = ∑' p : h.index, ρ (h.covering p) := by\n      rw [hρ h.measure_sdiff_biUnion, zero_add,\n        measure_biUnion h.index_countable h.covering_disjoint fun x hx => h.measurableSet_u hx]\n\ntheorem measure_le_tsum [SecondCountableTopology X] : μ s ≤ ∑' x : h.index, μ (h.covering x) :=\n  h.measure_le_tsum_of_absolutelyContinuous Measure.AbsolutelyContinuous.rfl\n\nend FineSubfamilyOn\n\n/-- One can enlarge a Vitali family by adding to the sets `f x` at `x` all sets which are not\ncontained in a `δ`-neighborhood on `x`. This does not change the local filter at a point, but it\ncan be convenient to get a nicer global behavior. -/\ndef enlarge (v : VitaliFamily μ) (δ : ℝ) (δpos : 0 < δ) : VitaliFamily μ where\n  setsAt x := v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}\n  measurableSet := by\n    rintro x s (hs | hs)\n    exacts [v.measurableSet _ _ hs, hs.1]\n  nonempty_interior := by\n    rintro x s (hs | hs)\n    exacts [v.nonempty_interior _ _ hs, hs.2.1]\n  nontrivial := by\n    intro x ε εpos\n    rcases v.nontrivial x ε εpos with ⟨s, hs, h's⟩\n    exact ⟨s, mem_union_left _ hs, h's⟩\n  covering := by\n    intro s f fset ffine\n    let g : X → Set (Set X) := fun x => f x ∩ v.setsAt x\n    have : ∀ x ∈ s, ∀ ε : ℝ, ε > 0 → ∃ t ∈ g x, t ⊆ closedBall x ε := by\n      intro x hx ε εpos\n      obtain ⟨t, tf, ht⟩ : ∃ t ∈ f x, t ⊆ closedBall x (min ε δ) :=\n        ffine x hx (min ε δ) (lt_min εpos δpos)\n      rcases fset x hx tf with (h't | h't)\n      · exact ⟨t, ⟨tf, h't⟩, ht.trans (closedBall_subset_closedBall (min_le_left _ _))⟩\n      · refine False.elim (h't.2.2 ?_)\n        exact ht.trans (closedBall_subset_closedBall (min_le_right _ _))\n    rcases v.covering s g (fun x _ => inter_subset_right) this with ⟨t, ts, tdisj, tg, μt⟩\n    exact ⟨t, ts, tdisj, fun p hp => (tg p hp).1, μt⟩\n\nvariable (v : VitaliFamily μ)\n\n/-- Given a vitali family `v`, then `v.filterAt x` is the filter on `Set X` made of those families\nthat contain all sets of `v.setsAt x` of a sufficiently small diameter. This filter makes it\npossible to express limiting behavior when sets in `v.setsAt x` shrink to `x`. -/\ndef filterAt (x : X) : Filter (Set X) := (𝓝 x).smallSets ⊓ 𝓟 (v.setsAt x)\n\ntheorem _root_.Filter.HasBasis.vitaliFamily {ι : Sort*} {p : ι → Prop} {s : ι → Set X} {x : X}\n    (h : (𝓝 x).HasBasis p s) : (v.filterAt x).HasBasis p (fun i ↦ {t ∈ v.setsAt x | t ⊆ s i}) := by\n  simpa only [← Set.setOf_inter_eq_sep] using! h.smallSets.inf_principal _\n\ntheorem filterAt_basis_closedBall (x : X) :\n    (v.filterAt x).HasBasis (0 < ·) ({t ∈ v.setsAt x | t ⊆ closedBall x ·}) :=\n  nhds_basis_closedBall.vitaliFamily v\n\nTarget:\ntheorem mem_filterAt_iff {x : X} {s : Set (Set X)} :\n    s ∈ v.filterAt x ↔ ∃ ε > (0 : ℝ), ∀ t ∈ v.setsAt x, t ⊆ closedBall x ε → t ∈ s :=\n\nProof body:\n","rejected":"by\n  simp only [(v.filterAt_basis_closedBall x).mem_iff, ← and_imp, subset_def, mem_setOf]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Covering","family_id":"mem_filterat_iff","file_id":"mathlib/Mathlib/MeasureTheory/Covering/VitaliFamily.lean","sample_id":"419897952885489eef42100d7198d985d0ed456c5f3c123051bf15c5b6fac476"},"split":"train"}
{"binding":{"chosen_proof_sha256":"95de6bb494dcefbe243626eb340736b29bf3b291e155dac01267e899f1bb69c2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b30174f1e4c5cb8ee3644e42068a798768b42f432347c73b4edbba1ec70ef473","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c031c9432732d05cd237126cbcac6a473645115eab86e0eb5195fb77c4512d50","source_sha256":"afb2352b0e8eda21bd88c47cda5f9b0a4c162f3f78eedf36a33613392805b346","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [of_eq, Prefunctor.comp_assoc, Prefunctor.comp_assoc, Functor.toPrefunctor_comp]\n  dsimp [lift]\n  rw [Quotient.lift_spec, Paths.lift_spec, Quiver.Symmetrify.lift_spec]","hard_negative":true,"metrics":{"chosen_tokens":37,"rejected_tokens":3,"token_jaccard":0.05,"token_length_ratio":0.081081},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"27fb21e5b556ddac7ba755431ec6bda6f95a299bda9187fccb1e81541d6bde26","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Groupoid\npublic import Mathlib.CategoryTheory.PathCategory.Basic\n\nNamespace:\nQuiver.FreeGroupoid\n\nLocal context:\n/-\nCopyright (c) 2022 Rémi Bottinelli. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémi Bottinelli\n-/\n/-!\n# Free groupoid on a quiver\n\nThis file defines the free groupoid on a quiver, the lifting of a prefunctor to its unique\nextension as a functor from the free groupoid, and proves uniqueness of this extension.\n\n## Main results\n\nGiven the type `V` and a quiver instance on `V`:\n\n- `Quiver.FreeGroupoid V`: a type synonym for `V`.\n- `Quiver.FreeGroupoid.instGroupoid`: the `Groupoid` instance on `Quiver.FreeGroupoid V`.\n- `lift`: the lifting of a prefunctor from `V` to `V'` where `V'` is a groupoid, to a functor.\n  `Quiver.FreeGroupoid V ⥤ V'`.\n- `lift_spec` and `lift_unique`: the proofs that, respectively, `lift` indeed is a lifting\n  and is the unique one.\n\n## Implementation notes\n\nThe free groupoid is first defined by symmetrifying the quiver, taking the induced path category\nand finally quotienting by the reducibility relation.\n\n-/\n\n@[expose] public section\n\nopen Set Function\n\nnamespace Quiver\n\nopen CategoryTheory\n\nuniverse u v u' v' u'' v''\n\nvariable {V : Type u} [Quiver.{v} V]\n\n/-- Shorthand for the \"forward\" arrow corresponding to `f` in `paths <| symmetrify V` -/\nabbrev Hom.toPosPath {X Y : V} (f : X ⟶ Y) :\n    (CategoryTheory.Paths.categoryPaths <| Quiver.Symmetrify V).Hom X Y :=\n  f.toPos.toPath\n\n/-- Shorthand for the \"forward\" arrow corresponding to `f` in `paths <| symmetrify V` -/\nabbrev Hom.toNegPath {X Y : V} (f : X ⟶ Y) :\n    (CategoryTheory.Paths.categoryPaths <| Quiver.Symmetrify V).Hom Y X :=\n  f.toNeg.toPath\n\n/-- The \"reduction\" relation -/\ninductive FreeGroupoid.redStep : HomRel (Paths (Quiver.Symmetrify V))\n  | step (X Z : Quiver.Symmetrify V) (f : X ⟶ Z) :\n    redStep (𝟙 ((Paths.of (Quiver.Symmetrify V)).obj X)) (f.toPath ≫ (Quiver.reverse f).toPath)\n\n/-- The underlying vertices of the free groupoid -/\nprotected def FreeGroupoid (V) [Q : Quiver V] :=\n  CategoryTheory.Quotient (@FreeGroupoid.redStep V Q)\n\nnamespace FreeGroupoid\n\nopen Quiver\n\ninstance {V} [Quiver V] [Nonempty V] : Nonempty (Quiver.FreeGroupoid V) := by\n  inhabit V; exact ⟨⟨@default V _⟩⟩\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem congr_reverse {X Y : Paths <| Quiver.Symmetrify V} (p q : X ⟶ Y) :\n    HomRel.CompClosure redStep p q → HomRel.CompClosure redStep p.reverse q.reverse := by\n  rintro ⟨_, _, XW, _, _, WY, _, _, f⟩\n  have : HomRel.CompClosure redStep (WY.reverse ≫ 𝟙 _ ≫ XW.reverse)\n      (WY.reverse ≫ (f.toPath ≫ (Quiver.reverse f).toPath) ≫ XW.reverse) := by\n    constructor\n    constructor\n  simpa only [CategoryStruct.comp, CategoryStruct.id, Quiver.Path.reverse, Quiver.Path.nil_comp,\n    Quiver.Path.reverse_comp, Quiver.reverse_reverse, Quiver.Path.reverse_toPath,\n    Quiver.Path.comp_assoc] using this\n\nopen Relation in\ntheorem congr_comp_reverse {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :\n    Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (p ≫ p.reverse) =\n      Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (𝟙 X) := by\n  apply Quot.eqvGen_sound\n  induction p with\n  | nil => apply EqvGen.refl\n  | cons q f ih =>\n    simp only [Quiver.Path.reverse]\n    fapply EqvGen.trans\n    -- Porting note: dot notation for `Quiver.Path.*` and `Quiver.Hom.*` not working\n    · exact q ≫ Quiver.Path.reverse q\n    · apply EqvGen.symm\n      apply EqvGen.rel\n      have : HomRel.CompClosure redStep (q ≫ 𝟙 _ ≫ Quiver.Path.reverse q)\n          (q ≫ (Quiver.Hom.toPath f ≫ Quiver.Hom.toPath (Quiver.reverse f)) ≫\n            Quiver.Path.reverse q) := by\n        apply HomRel.CompClosure.intro\n        apply redStep.step\n      simp only [Category.assoc, Category.id_comp] at this ⊢\n      -- Porting note: `simp` cannot see how `Quiver.Path.comp_assoc` is relevant, so change to\n      -- category notation\n      change HomRel.CompClosure redStep (q ≫ Quiver.Path.reverse q)\n        (Quiver.Path.cons q f ≫ (Quiver.Hom.toPath (Quiver.reverse f)) ≫ (Quiver.Path.reverse q))\n      simp only [← Category.assoc] at this ⊢\n      exact this\n    · exact ih\n\ntheorem congr_reverse_comp {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :\n    Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (p.reverse ≫ p) =\n      Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (𝟙 Y) := by\n  nth_rw 2 [← Quiver.Path.reverse_reverse p]\n  apply congr_comp_reverse\n\ninstance : Category (Quiver.FreeGroupoid V) :=\n  Quotient.category redStep\n\n/-- The inverse of an arrow in the free groupoid -/\ndef quotInv {X Y : Quiver.FreeGroupoid V} (f : X ⟶ Y) : Y ⟶ X :=\n  Quot.liftOn f (fun pp => Quot.mk _ <| pp.reverse) fun pp qq con =>\n    Quot.sound <| congr_reverse pp qq con\n\ninstance instGroupoid : Groupoid (Quiver.FreeGroupoid V) where\n  inv := quotInv\n  inv_comp p := Quot.inductionOn p fun pp => congr_reverse_comp pp\n  comp_inv p := Quot.inductionOn p fun pp => congr_comp_reverse pp\n\n/-- The inclusion of the quiver on `V` to the underlying quiver on `FreeGroupoid V` -/\ndef of (V) [Quiver V] : V ⥤q Quiver.FreeGroupoid V where\n  obj X := ⟨X⟩\n  map f := Quot.mk _ f.toPosPath\n\ntheorem of_eq :\n    of V = (Quiver.Symmetrify.of ⋙q (Paths.of (Quiver.Symmetrify V))).comp\n      (Quotient.functor <| @redStep V _).toPrefunctor := rfl\n\nsection UniversalProperty\n\nvariable {V' : Type u'} [Groupoid V']\n\n/-- The lift of a prefunctor to a groupoid, to a functor from `FreeGroupoid V` -/\ndef lift (φ : V ⥤q V') : Quiver.FreeGroupoid V ⥤ V' :=\n  CategoryTheory.Quotient.lift _ (Paths.lift <| Quiver.Symmetrify.lift φ) <| by\n    rintro _ _ _ _ ⟨X, Y, f⟩\n    -- Porting note: `simp` does not work, so manually `rewrite`\n    erw [Paths.lift_nil, Paths.lift_cons, Quiver.Path.comp_nil, Paths.lift_toPath,\n      Quiver.Symmetrify.lift_reverse]\n    symm\n    apply Groupoid.comp_inv\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem lift_spec (φ : V ⥤q V') : of V ⋙q (lift φ).toPrefunctor = φ :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_c031c9432732","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"450640791efd04eabba4f685a05e93e504e4ea3c544156655c5431e6c20d8721","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Groupoid","family_id":"lift_spec","file_id":"mathlib/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean","sample_id":"c031c9432732d05cd237126cbcac6a473645115eab86e0eb5195fb77c4512d50"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eb3d60d437f15cfc69d1d37a56f9a9b73ef1279b8f1ffb7d0327dabbc53ce4fd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b5eed30bb36819b24fa9a164628ee2a2c0338152109146fc3f9ca3ebec531d4e","source_sha256":"9cb41eaef9fea958cbbd1f98ac6fcd22922a81e7fe77437a7ce0c4aae8ecfaf2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  simp only [alternatingMapLinearEquiv_comp_ιMulti, LinearMap.compAlternatingMap_apply,\n    LinearMap.coe_comp, comp_apply, alternatingMapLinearEquiv_apply_ιMulti]\n\n/-! Functoriality of the exterior powers. -/\n\nvariable (n) in\n/-- The linear map between `n`th exterior powers induced by a linear map between the modules. -/\nnoncomputable def map (f : M →ₗ[R] N) : ⋀[R]^n M →ₗ[R] ⋀[R]^n N :=\n  alternatingMapLinearEquiv ((ιMulti R n).compLinearMap f)","hard_negative":false,"metrics":{"chosen_tokens":110,"rejected_tokens":2,"token_jaccard":0.018868,"token_length_ratio":0.018182},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"27fd89cbbb71a9ef083392658abdb1c67642492f7c6ce3190a2e6b1491328683","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Presentation.Basic\npublic import Mathlib.LinearAlgebra.ExteriorAlgebra.OfAlternating\n\nNamespace:\nexteriorPower\n\nLocal context:\n/-\nCopyright (c) 2024 Sophie Morel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sophie Morel, Joël Riou\n-/\n/-!\n# Exterior powers\n\nWe study the exterior powers of a module `M` over a commutative ring `R`.\n\n## Definitions\n\n* `exteriorPower.ιMulti` is the canonical alternating map on `M` with values in `⋀[R]^n M`.\n\n* `exteriorPower.presentation R n M` is the standard presentation of the `R`-module `⋀[R]^n M`.\n\n* `exteriorPower.map n f : ⋀[R]^n M →ₗ[R] ⋀[R]^n N` is the linear map on `nth` exterior powers\n  induced by a linear map `f : M →ₗ[R] N`. (See the file\n  `Mathlib/Algebra/Category/ModuleCat/ExteriorPower.lean` for the corresponding functor\n  `ModuleCat R ⥤ ModuleCat R`.)\n\n## Theorems\n* `exteriorPower.ιMulti_span`: The image of `exteriorPower.ιMulti` spans `⋀[R]^n M`.\n\n* We construct `exteriorPower.alternatingMapLinearEquiv` which\n  expresses the universal property of the exterior power as a\n  linear equivalence `(M [⋀^Fin n]→ₗ[R] N) ≃ₗ[R] ⋀[R]^n M →ₗ[R] N` between\n  alternating maps and linear maps from the exterior power.\n\n-/\n\n@[expose] public section\n\nopen scoped TensorProduct\n\nuniverse u\n\nvariable (R : Type u) [CommRing R] (n : ℕ) {M N N' : Type*}\n  [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]\n  [AddCommGroup N'] [Module R N']\n\nnamespace exteriorPower\n\nopen Function Set Set.powersetCard\n\n/-! The canonical alternating map from `Fin n → M` to `⋀[R]^n M`. -/\n\n/-- `exteriorAlgebra.ιMulti` is the alternating map from `Fin n → M` to `⋀[r]^n M`\ninduced by `exteriorAlgebra.ιMulti`, i.e. sending a family of vectors `m : Fin n → M` to the\nproduct of its entries. -/\ndef ιMulti : M [⋀^Fin n]→ₗ[R] (⋀[R]^n M) :=\n  (ExteriorAlgebra.ιMulti R n).codRestrict (⋀[R]^n M) fun _ =>\n    ExteriorAlgebra.ιMulti_range R n <| Set.mem_range_self _\n\n@[simp] lemma ιMulti_apply_coe (a : Fin n → M) : ιMulti R n a = ExteriorAlgebra.ιMulti R n a := rfl\n\n/-- Given a linearly ordered family `v` of vectors of `M` and a natural number `n`, produce the\nfamily of `n`fold exterior products of elements of `v`, seen as members of the\n`n`th exterior power. -/\nnoncomputable def ιMulti_family {I : Type*} [LinearOrder I] (v : I → M)\n    (s : powersetCard I n) : ⋀[R]^n M :=\n  ιMulti R n (v ∘ (ofFinEmbEquiv.symm s))\n\nlemma ιMulti_family_eq_coe_comp {I : Type*} [LinearOrder I] (v : I → M) :\n    ExteriorAlgebra.ιMulti_family R n v = (↑) ∘ ιMulti_family R n v :=\n  rfl\n\n@[simp] lemma ιMulti_family_apply_coe {I : Type*} [LinearOrder I] (v : I → M)\n    (s : powersetCard I n) :\n    ιMulti_family R n v s = ExteriorAlgebra.ιMulti_family R n v s := rfl\n\nvariable (M)\n/-- The image of `ExteriorAlgebra.ιMulti R n` spans the `n`th exterior power. Variant of\n`ExteriorAlgebra.ιMulti_span_fixedDegree`, useful in rewrites. -/\nlemma ιMulti_span_fixedDegree :\n    Submodule.span R (Set.range (ExteriorAlgebra.ιMulti R n)) = ⋀[R]^n M :=\n  ExteriorAlgebra.ιMulti_span_fixedDegree R n\n\nopen Set Submodule in\n/-- If a set `s` spans the module `M`, then the set of all elements of the form `x₁ ∧ ⋯ ∧ xₙ`\nwhere `xᵢ ∈ s` spans `⋀ⁿ M`. -/\nlemma ιMulti_span_fixedDegree_of_span_eq_top {s : Set M} (hs : span R s = ⊤) :\n    span R (ExteriorAlgebra.ιMulti R n '' {a | range a ⊆ s}) = ⋀[R]^n M := by\n  apply le_antisymm\n  · rw [span_le]\n    rintro - ⟨y, ⟨y_mem, rfl⟩⟩\n    apply ExteriorAlgebra.ιMulti_range R n\n    simp\n  · rw [ExteriorAlgebra.exteriorPower, LinearMap.range_eq_map, ← hs, map_span, span_pow, span_le]\n    rintro x hx\n    obtain ⟨f, rfl⟩ := Set.mem_pow.mp hx\n    refine mem_span_of_mem ⟨ExteriorAlgebra.ιInv ∘ Subtype.val ∘ f, ?_, ?_⟩\n    · rw [Set.mem_setOf_eq, Set.range_comp, Set.image_subset_iff]\n      apply Subset.trans ?_ (s.image_subset_preimage_of_inverse ExteriorAlgebra.ι_leftInverse)\n      grind\n    · rw [ExteriorAlgebra.ιMulti_apply]\n      apply congrArg (List.prod ∘ List.ofFn)\n      ext i\n      obtain ⟨m, -, hm⟩ := (Set.mem_image _ _ _).mp (f i).2\n      rw [Function.comp_apply, Function.comp_apply, ← hm, ExteriorAlgebra.ι_leftInverse]\n\n/-- The image of `exteriorPower.ιMulti` spans `⋀[R]^n M`. -/\nlemma ιMulti_span :\n    Submodule.span R (Set.range (ιMulti R n)) = (⊤ : Submodule R (⋀[R]^n M)) := by\n  apply LinearMap.map_injective (Submodule.ker_subtype (⋀[R]^n M))\n  rw [LinearMap.map_span, ← Set.image_univ, Set.image_image]\n  simp only [Submodule.coe_subtype, ιMulti_apply_coe, Set.image_univ, Submodule.map_top,\n    Submodule.range_subtype]\n  exact ExteriorAlgebra.ιMulti_span_fixedDegree R n\n\nopen Set Submodule in\n/-- A version of `ιMulti_span_fixedDegree_of_span_eq_top` that works in the exterior power. -/\nlemma ιMulti_span_of_span {s : Set M} (hs : span R s = ⊤) :\n    span R (ιMulti R n '' {a | range a ⊆ s}) = ⊤ := by\n  apply LinearMap.map_injective (ker_subtype (⋀[R]^n M))\n  simpa [LinearMap.map_span, Set.image_image] using ιMulti_span_fixedDegree_of_span_eq_top R n M hs\n\nnamespace presentation\n\n/-- The index type for the relations in the standard presentation of `⋀[R]^n M`,\nin the particular case `ι` is `Fin n`. -/\ninductive Rels (ι : Type*) (M : Type*)\n  | add (m : ι → M) (i : ι) (x y : M)\n  | smul (m : ι → M) (i : ι) (r : R) (x : M)\n  | alt (m : ι → M) (i j : ι) (hm : m i = m j) (hij : i ≠ j)\n\n/-- The relations in the standard presentation of `⋀[R]^n M` with generators and relations. -/\n@[simps]\nnoncomputable def relations (ι : Type*) [DecidableEq ι] (M : Type*)\n    [AddCommGroup M] [Module R M] :\n    Module.Relations R where\n  G := ι → M\n  R := Rels R ι M\n  relation\n    | .add m i x y => Finsupp.single (update m i x) 1 +\n        Finsupp.single (update m i y) 1 -\n        Finsupp.single (update m i (x + y)) 1\n    | .smul m i r x => Finsupp.single (update m i (r • x)) 1 -\n        r • Finsupp.single (update m i x) 1\n    | .alt m _ _ _ _ => Finsupp.single m 1\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nvariable {R} in\n/-- The solutions in a module `N` to the linear equations\ngiven by `exteriorPower.relations R ι M` identify to alternating maps to `N`. -/\n@[simps!]\nnoncomputable def relationsSolutionEquiv {ι : Type*} [DecidableEq ι] {M : Type*}\n    [AddCommGroup M] [Module R M] :\n    (relations R ι M).Solution N ≃ AlternatingMap R M N ι where\n  toFun s :=\n    { toFun := fun m ↦ s.var m\n      map_update_add' := fun m i x y ↦ by\n        have := s.linearCombination_var_relation (.add m i x y)\n        dsimp at this ⊢\n        rw [map_sub, map_add, Finsupp.linearCombination_single, one_smul,\n          Finsupp.linearCombination_single, one_smul,\n          Finsupp.linearCombination_single, one_smul, sub_eq_zero] at this\n        convert! this.symm -- `convert` is necessary due to the implementation of `MultilinearMap`\n      map_update_smul' := fun m i r x ↦ by\n        have := s.linearCombination_var_relation (.smul m i r x)\n        dsimp at this ⊢\n        rw [Finsupp.smul_single, smul_eq_mul, mul_one, map_sub,\n          Finsupp.linearCombination_single, one_smul,\n          Finsupp.linearCombination_single, sub_eq_zero] at this\n        convert! this\n      map_eq_zero_of_eq' := fun v i j hm hij ↦\n        by simpa using s.linearCombination_var_relation (.alt v i j hm hij) }\n  invFun f :=\n    { var := fun m ↦ f m\n      linearCombination_var_relation := by\n        rintro (⟨m, i, x, y⟩ | ⟨m, i, r, x⟩ | ⟨v, i, j, hm, hij⟩)\n        · simp\n        · simp\n        · simpa using f.map_eq_zero_of_eq v hm hij }\n\n/-- The universal property of the exterior power. -/\nnoncomputable def isPresentationCore :\n    (relationsSolutionEquiv.symm (ιMulti R n (M := M))).IsPresentationCore where\n  desc s := LinearMap.comp (ExteriorAlgebra.liftAlternating\n      (Function.update 0 n (relationsSolutionEquiv s))) (Submodule.subtype _)\n  postcomp_desc s := by aesop\n  postcomp_injective {N _ _ f f' h} := by\n    rw [Submodule.linearMap_eq_iff_of_span_eq_top _ _ (ιMulti_span R n M)]\n    rintro ⟨_, ⟨f, rfl⟩⟩\n    exact Module.Relations.Solution.congr_var h f\n\nend presentation\n\n/-- The standard presentation of the `R`-module `⋀[R]^n M`. -/\n@[simps! G R relation var]\nnoncomputable def presentation : Module.Presentation R (⋀[R]^n M) :=\n  .ofIsPresentation (presentation.isPresentationCore R n M).isPresentation\n\nvariable {R M n}\n\n/-- Two linear maps on `⋀[R]^n M` that agree on the image of `exteriorPower.ιMulti`\nare equal. -/\n@[ext]\nlemma linearMap_ext {f : ⋀[R]^n M →ₗ[R] N} {g : ⋀[R]^n M →ₗ[R] N}\n    (heq : f.compAlternatingMap (ιMulti R n) = g.compAlternatingMap (ιMulti R n)) : f = g :=\n  (presentation R n M).postcomp_injective (by ext f; apply DFunLike.congr_fun heq)\n\n/-- The linear equivalence between `n`-fold alternating maps from `M` to `N` and linear maps from\n`⋀[R]^n M` to `N`: this is the universal property of the `n`th exterior power of `M`. -/\nnoncomputable def alternatingMapLinearEquiv : (M [⋀^Fin n]→ₗ[R] N) ≃ₗ[R] ⋀[R]^n M →ₗ[R] N :=\n  LinearEquiv.symm\n    (Equiv.toLinearEquiv\n      ((presentation R n M).linearMapEquiv.trans presentation.relationsSolutionEquiv)\n      { map_add := fun _ _ => rfl\n        map_smul := fun _ _ => rfl })\n\n@[simp]\nlemma alternatingMapLinearEquiv_comp_ιMulti (f : M [⋀^Fin n]→ₗ[R] N) :\n    (alternatingMapLinearEquiv f).compAlternatingMap (ιMulti R n) = f := by\n  obtain ⟨φ, rfl⟩ := alternatingMapLinearEquiv.symm.surjective f\n  dsimp [alternatingMapLinearEquiv]\n  simp only [LinearEquiv.symm_apply_apply]\n  rfl\n\n@[simp]\nlemma alternatingMapLinearEquiv_apply_ιMulti (f : M [⋀^Fin n]→ₗ[R] N) (a : Fin n → M) :\n    alternatingMapLinearEquiv f (ιMulti R n a) = f a :=\n  DFunLike.congr_fun (alternatingMapLinearEquiv_comp_ιMulti f) a\n\n@[simp]\nlemma alternatingMapLinearEquiv_symm_apply (F : ⋀[R]^n M →ₗ[R] N) (m : Fin n → M) :\n    alternatingMapLinearEquiv.symm F m = F.compAlternatingMap (ιMulti R n) m := by\n  obtain ⟨f, rfl⟩ := alternatingMapLinearEquiv.surjective F\n  simp only [LinearEquiv.symm_apply_apply, alternatingMapLinearEquiv_comp_ιMulti]\n\n@[simp]\nlemma alternatingMapLinearEquiv_ιMulti :\n    alternatingMapLinearEquiv (ιMulti R n (M := M)) = LinearMap.id := by\n  ext\n  simp only [alternatingMapLinearEquiv_comp_ιMulti, ιMulti_apply_coe,\n    LinearMap.compAlternatingMap_apply, LinearMap.id_coe, id_eq]\n\n/-- If `f` is an alternating map from `M` to `N`,\n`alternatingMapLinearEquiv f` is the corresponding linear map from `⋀[R]^n M` to `N`,\nand if `g` is a linear map from `N` to `N'`, then\nthe alternating map `g.compAlternatingMap f` from `M` to `N'` corresponds to the linear\nmap `g.comp (alternatingMapLinearEquiv f)` on `⋀[R]^n M`. -/\n\nTarget:\nlemma alternatingMapLinearEquiv_comp (g : N →ₗ[R] N') (f : M [⋀^Fin n]→ₗ[R] N) :\n    alternatingMapLinearEquiv (g.compAlternatingMap f) = g.comp (alternatingMapLinearEquiv f) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/ExteriorPower","family_id":"alternatingmaplinearequiv_comp","file_id":"mathlib/Mathlib/LinearAlgebra/ExteriorPower/Basic.lean","sample_id":"b5eed30bb36819b24fa9a164628ee2a2c0338152109146fc3f9ca3ebec531d4e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eeea18d1418cb641b169bfeb8bce930a2742ccdb2c216d63523e2c130f059c37","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cbe42b39bb30027dd1ac1ec3aabbd6e4868650ccfa4b29e6d7ea3da9ff9c5957","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"88630b1e90f48b536f58e1419d7c32cb382c083b0e23e75073e0b1e0fdf80749","source_sha256":"44a01634b6653e0d9e33dfb14b6ec5ffa4aa5d523005601762c92770961f9cd1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨χ, hχ⟩ := Finset.rado_selection_subtype (β := β) (fun s ↦ g s s.finite_toSet)\n  refine ⟨χ, fun s hs ↦ ?_⟩\n  obtain ⟨t, ht, hst⟩ := hχ hs.toFinset\n  simp only [Set.Finite.toFinset_subset] at ht\n  exact ⟨t, by simp_all⟩","hard_negative":false,"metrics":{"chosen_tokens":70,"rejected_tokens":75,"token_jaccard":0.902439,"token_length_ratio":1.071429},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"281dbc449cbb0b535f74a149deb0c06cde610f3d57c008e7bef30087774b47f1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\nimport Mathlib.Topology.Compactness.Compact\npublic import Mathlib.Data.Set.Finite.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\n/-!\n# Combinatorial compactness and the Rado selection lemma\n\nThis file contains compactness arguments for constructing infinite objects from finite\napproximations. The main result is a formalization of Rado's selection principle, as an application\nof compactness to combinatorics.\n\nWe give four versions, depending on whether the \"partial\" functions are defined locally or globally,\nand whether we use `Finset` or `Set.Finite`. The precise formulation of the lemma is therefore\n`Finset.rado_selection_subtype` or `Set.Finite.rado_selection_subtype`, but the versions avoiding\nsubtypes are easier to prove and often easier to apply, so they are provided too.\n\n## Main results\n\n* `Finset.rado_selection`: Given functions `g : Finset α → α → β` where `β` is finite,\n  there exists a single function `χ : α → β` which is constructed out of `g`.\n  More precisely, for each finite set `s`, there exists a larger set `t ⊇ s` such that\n  `χ` and `g t` agree on `s`.\n  In fact, we can more generally allow each `g s` to be a dependent function, as `(a : α) → β a`, so\n  the type of `g` will be `Finset α → (a : α) → β a`.\n\n* `Finset.rado_selection_subtype`: A variant where `g` takes elements in the subtype.\n\n* `Set.Finite.rado_selection`: A variant using `Set.Finite`.\n\n* `Set.Finite.rado_selection`: A variant using `Set.Finite` and where `g` takes elements in the\n  subtype.\n\n## Implementation notes\n\nThe proof uses the fact that the product of finite discrete spaces is compact\n(by Tychonoff's theorem). The closed sets corresponding to \"agreeing with `g s` on `s`\"\nhave the finite intersection property, so their intersection is nonempty.\n\n## References\n\n* de Bruijn, N. G.; Erdős, P. (1951). \"A colour problem for infinite graphs and a problem\n  in the theory of relations\".\n* Rado, R. (1949). \"Axiomatic treatment of rank in infinite sets\".\n\n-/\n\npublic section\n\nvariable {α : Type*} {β : α → Type*} [∀ a, Finite (β a)]\n\n/--\nGiven a (dependent) function `g s : (a : α) → β a` for each finset `s` of `α`, provided that\neach `β a` is finite, we can find another function `χ : (a : α) → β a` such that on every `s`,\nthere is some larger `t` such that `χ` agrees with `g t` on `s`.\nInformally, we are stitching together the local functions `g s` into a global `χ` such that on\neach `s`, `χ` can be expressed in terms of one of the `g`.\n-/\ntheorem Finset.rado_selection (g : Finset α → (a : α) → β a) :\n    ∃ χ : (a : α) → β a, ∀ s : Finset α, ∃ t : Finset α, s ⊆ t ∧ ∀ x ∈ s, χ x = g t x := by\n  classical\n  let instTop (a : α) : TopologicalSpace (β a) := ⊥\n  have instDiscr (a : α) : DiscreteTopology (β a) := discreteTopology_bot _\n  let e (s : Finset α) : Set ((a : α) → β a) := {f | ∃ t, s ⊆ t ∧ ∀ x ∈ s, f x = g t x}\n  have (s : Finset α) : s.restrict ⁻¹' {f | ∃ t, s ⊆ t ∧ ∀ x, f x = g t x} = e s := by simp [e]\n  have he' (s : Finset α) : IsClosed (e s) := by\n    rw [← this]\n    exact (isClosed_discrete _).preimage (by fun_prop)\n  have he'' (B : Finset (Finset α)) : (⋂ i ∈ B, e i).Nonempty := by\n    refine ⟨g (B.biUnion id), ?_⟩\n    simp only [Set.mem_iInter, Set.mem_setOf_eq, e]\n    intro i hi\n    exact ⟨_, subset_biUnion_of_mem id hi, by simp⟩\n  simpa using! CompactSpace.iInter_nonempty he' he''\n\n/--\nGiven a (dependent) function `g s : (a : s) → β a` for each finset `s` of `α`, provided that\neach `β a` is finite, we can find another function `χ : (a : α) → β a` such that on every `s`,\nthere is some larger `t` such that `χ` agrees with `g t` on `s`.\nInformally, we are stitching together the local functions `g s` into a global `χ` such that on\neach `s`, `χ` can be expressed in terms of one of the `g`.\n-/\ntheorem Finset.rado_selection_subtype (g : (s : Finset α) → (a : s) → β a) :\n    ∃ χ : (a : α) → β a, ∀ s : Finset α,\n      ∃ (t : Finset α) (hst : s ⊆ t), ∀ x : s, χ x = g t (Set.inclusion hst x) := by\n  classical\n  have (a : α) : Nonempty (β a) := ⟨g {a} ⟨a, by simp⟩⟩\n  let g' (s) (a : α) : β a := if ha : a ∈ s then g s ⟨a, ha⟩ else Classical.arbitrary (β a)\n  have hg (s : Finset α) (x : s) : g s x = g' s x := by simp [g']\n  simpa [hg] using Finset.rado_selection g'\n\n/--\nGiven a (dependent) function `g s : (a : α) → β a` for each finite set `s` of `α`, provided that\neach `β a` is finite, we can find another function `χ : (a : α) → β a` such that on every `s`,\nthere is some larger `t` such that `χ` agrees with `g t` on `s`.\nInformally, we are stitching together the local functions `g s` into a global `χ` such that on\neach `s`, `χ` can be expressed in terms of one of the `g`.\n-/\ntheorem Set.Finite.rado_selection (g : (s : Set α) → s.Finite → (a : α) → β a) :\n    ∃ χ : (a : α) → β a, ∀ s : Set α, s.Finite →\n      ∃ (t : Set α) (ht : t.Finite), s ⊆ t ∧ ∀ x ∈ s, χ x = g t ht x := by\n  obtain ⟨χ, hχ⟩ := Finset.rado_selection (fun s ↦ g s s.finite_toSet)\n  refine ⟨χ, fun s hs ↦ ?_⟩\n  obtain ⟨t, ht, ht'⟩ := hχ hs.toFinset\n  exact ⟨t, by simp_all⟩\n\n/--\nGiven a (dependent) function `g s : (a : s) → β a` for each finite set `s` of `α`, provided that\neach `β a` is finite, we can find another function `χ : (a : α) → β a` such that on every `s`,\nthere is some larger `t` such that `χ` agrees with `g t` on `s`.\nInformally, we are stitching together the local functions `g s` into a global `χ` such that on\neach `s`, `χ` can be expressed in terms of one of the `g`.\n-/\n\nTarget:\ntheorem Set.Finite.rado_selection_subtype (g : (s : Set α) → s.Finite → (a : s) → β a) :\n    ∃ χ : (a : α) → β a, ∀ s : Set α, s.Finite →\n      ∃ (t : Set α) (ht : t.Finite) (hst : s ⊆ t), ∀ x : s, χ x = g t ht (Set.inclusion hst x) :=\n\nProof body:\n","rejected":"by\n  obtain ⟨χ, hχ⟩ := Finset.rado_selection_subtype (β := β) (fun s ↦ g s s.finite_toSet)\n  refine ⟨χ, fun s hs ↦ ?_⟩\n  obtain ⟨t, ht, hst⟩ := hχ hs.toFinset\n  simp only [Set.Finite.toFinset_subset] at ht\n  exact ⟨t, by simp_all⟩\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics","family_id":"set","file_id":"mathlib/Mathlib/Combinatorics/Compactness.lean","sample_id":"88630b1e90f48b536f58e1419d7c32cb382c083b0e23e75073e0b1e0fdf80749"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0276c875b5de26026fe85dbf0c34475b3acdef44fb07206bfeb84e06366c9607","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"36e832bb8cc9ec1ccec45966728b66a2c4051379529ccd6772318576d494b034","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"16839e522d1756068dfaf97c745b531d7271caa8d21dfa9ef1a00b09a41091af","source_sha256":"f31e0ea5c5dcaaef4fe7a595be40e4dcc670dcec12bcae3dfce625de497aea8b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  funext n\n  induction n with\n  | zero => rfl\n  | succ n ih =>\n    unfold map iterate get\n    rw [map, get] at ih\n    rw [iterate]\n    exact congrArg f ih","hard_negative":true,"metrics":{"chosen_tokens":35,"rejected_tokens":2,"token_jaccard":0.041667,"token_length_ratio":0.057143},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"2821bb255314f9ee038db5ea697ede39bca49b53562eaa0159e2eba953c39b59","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Stream.Defs\npublic import Mathlib.Logic.Function.Basic\npublic import Mathlib.Data.Nat.Basic\npublic import Mathlib.Tactic.Common\n\nNamespace:\nStream'\n\nLocal context:\n/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Streams a.k.a. infinite lists a.k.a. infinite sequences\n-/\n\n@[expose] public section\n\nopen Nat Function Option\n\nnamespace Stream'\n\nuniverse u v w\nvariable {α : Type u} {β : Type v} {δ : Type w}\nvariable (m n : ℕ) (x y : List α) (a b : Stream' α)\n\ninstance [Inhabited α] : Inhabited (Stream' α) :=\n  ⟨Stream'.const default⟩\n\n@[simp] protected theorem eta (s : Stream' α) : head s :: tail s = s :=\n  funext fun i => by cases i <;> rfl\n\n/-- Alias for `Stream'.eta` to match `List` API. -/\nalias cons_head_tail := Stream'.eta\n\n@[ext]\nprotected theorem ext {s₁ s₂ : Stream' α} : (∀ n, get s₁ n = get s₂ n) → s₁ = s₂ :=\n  fun h => funext h\n\n@[simp]\ntheorem get_zero_cons (a : α) (s : Stream' α) : get (a::s) 0 = a :=\n  rfl\n\n@[simp]\ntheorem head_cons (a : α) (s : Stream' α) : head (a::s) = a :=\n  rfl\n\n@[simp]\ntheorem tail_cons (a : α) (s : Stream' α) : tail (a::s) = s :=\n  rfl\n\n@[simp]\ntheorem get_drop (n m : ℕ) (s : Stream' α) : get (drop m s) n = get s (m + n) := by\n  rw [Nat.add_comm]\n  rfl\n\ntheorem tail_eq_drop (s : Stream' α) : tail s = drop 1 s :=\n  rfl\n\n@[simp]\ntheorem drop_drop (n m : ℕ) (s : Stream' α) : drop n (drop m s) = drop (m + n) s := by\n  ext; simp [Nat.add_assoc]\n\n@[simp] theorem get_tail {n : ℕ} {s : Stream' α} : s.tail.get n = s.get (n + 1) := rfl\n\n@[simp] theorem tail_drop' {i : ℕ} {s : Stream' α} : tail (drop i s) = s.drop (i + 1) := by\n  ext; simp [Nat.add_comm, Nat.add_left_comm]\n\n@[simp] theorem drop_tail' {i : ℕ} {s : Stream' α} : drop i (tail s) = s.drop (i + 1) := rfl\n\ntheorem tail_drop (n : ℕ) (s : Stream' α) : tail (drop n s) = drop n (tail s) := by simp\n\ntheorem get_succ (n : ℕ) (s : Stream' α) : get s (succ n) = get (tail s) n :=\n  rfl\n\n@[simp]\ntheorem get_succ_cons (n : ℕ) (s : Stream' α) (x : α) : get (x :: s) n.succ = get s n :=\n  rfl\n\n@[simp] lemma get_cons_append_zero {a : α} {x : List α} {s : Stream' α} :\n    (a :: x ++ₛ s).get 0 = a := rfl\n\n@[simp] lemma append_eq_cons {a : α} {as : Stream' α} : [a] ++ₛ as = a :: as := rfl\n\n@[simp] theorem drop_zero {s : Stream' α} : s.drop 0 = s := rfl\n\ntheorem drop_succ (n : ℕ) (s : Stream' α) : drop (succ n) s = drop n (tail s) :=\n  rfl\n\ntheorem head_drop (a : Stream' α) (n : ℕ) : (a.drop n).head = a.get n := by simp\n\ntheorem cons_injective2 : Function.Injective2 (cons : α → Stream' α → Stream' α) := fun x y s t h =>\n  ⟨by rw [← get_zero_cons x s, h, get_zero_cons],\n    Stream'.ext fun n => by rw [← get_succ_cons n _ x, h, get_succ_cons]⟩\n\ntheorem cons_injective_left (s : Stream' α) : Function.Injective fun x => cons x s :=\n  cons_injective2.left _\n\ntheorem cons_injective_right (x : α) : Function.Injective (cons x) :=\n  cons_injective2.right _\n\ntheorem all_def (p : α → Prop) (s : Stream' α) : All p s = ∀ n, p (get s n) :=\n  rfl\n\ntheorem any_def (p : α → Prop) (s : Stream' α) : Any p s = ∃ n, p (get s n) :=\n  rfl\n\n@[simp]\ntheorem mem_cons (a : α) (s : Stream' α) : a ∈ a::s :=\n  Exists.intro 0 rfl\n\ntheorem mem_cons_of_mem {a : α} {s : Stream' α} (b : α) : a ∈ s → a ∈ b::s := fun ⟨n, h⟩ =>\n  Exists.intro (succ n) (by rw [get_succ, tail_cons, h])\n\ntheorem eq_or_mem_of_mem_cons {a b : α} {s : Stream' α} : (a ∈ b::s) → a = b ∨ a ∈ s :=\n    fun ⟨n, h⟩ => by\n  rcases n with - | n'\n  · left\n    exact h\n  · right\n    rw [get_succ, tail_cons] at h\n    exact ⟨n', h⟩\n\ntheorem mem_of_get_eq {n : ℕ} {s : Stream' α} {a : α} : a = get s n → a ∈ s := fun h =>\n  Exists.intro n h\n\ntheorem mem_iff_exists_get_eq {s : Stream' α} {a : α} : a ∈ s ↔ ∃ n, a = s.get n where\n  mp := by simp [Membership.mem, any_def]\n  mpr h := mem_of_get_eq h.choose_spec\n\nsection Map\n\nvariable (f : α → β)\n\ntheorem drop_map (n : ℕ) (s : Stream' α) : drop n (map f s) = map f (drop n s) :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem get_map (n : ℕ) (s : Stream' α) : get (map f s) n = f (get s n) :=\n  rfl\n\ntheorem tail_map (s : Stream' α) : tail (map f s) = map f (tail s) := rfl\n\n@[simp]\ntheorem head_map (s : Stream' α) : head (map f s) = f (head s) :=\n  rfl\n\ntheorem map_eq (s : Stream' α) : map f s = f (head s)::map f (tail s) := by\n  rw [← Stream'.eta (map f s), tail_map, head_map]\n\ntheorem map_cons (a : α) (s : Stream' α) : map f (a::s) = f a::map f s := by\n  rw [← Stream'.eta (map f (a::s)), map_eq]; rfl\n\n@[simp]\ntheorem map_id (s : Stream' α) : map id s = s :=\n  rfl\n\n@[simp]\ntheorem map_map (g : β → δ) (f : α → β) (s : Stream' α) : map g (map f s) = map (g ∘ f) s :=\n  rfl\n\n@[simp]\ntheorem map_tail (s : Stream' α) : map f (tail s) = tail (map f s) :=\n  rfl\n\ntheorem mem_map {a : α} {s : Stream' α} : a ∈ s → f a ∈ map f s := fun ⟨n, h⟩ =>\n  Exists.intro n (by rw [get_map, h])\n\ntheorem exists_of_mem_map {f} {b : β} {s : Stream' α} : b ∈ map f s → ∃ a, a ∈ s ∧ f a = b :=\n  fun ⟨n, h⟩ => ⟨get s n, ⟨n, rfl⟩, h.symm⟩\n\nend Map\n\nsection Zip\n\nvariable (f : α → β → δ)\n\ntheorem drop_zip (n : ℕ) (s₁ : Stream' α) (s₂ : Stream' β) :\n    drop n (zip f s₁ s₂) = zip f (drop n s₁) (drop n s₂) :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem get_zip (n : ℕ) (s₁ : Stream' α) (s₂ : Stream' β) :\n    get (zip f s₁ s₂) n = f (get s₁ n) (get s₂ n) :=\n  rfl\n\ntheorem head_zip (s₁ : Stream' α) (s₂ : Stream' β) : head (zip f s₁ s₂) = f (head s₁) (head s₂) :=\n  rfl\n\ntheorem tail_zip (s₁ : Stream' α) (s₂ : Stream' β) :\n    tail (zip f s₁ s₂) = zip f (tail s₁) (tail s₂) :=\n  rfl\n\ntheorem zip_eq (s₁ : Stream' α) (s₂ : Stream' β) :\n    zip f s₁ s₂ = f (head s₁) (head s₂)::zip f (tail s₁) (tail s₂) := by\n  rw [← Stream'.eta (zip f s₁ s₂)]; rfl\n\n@[simp]\ntheorem get_enum (s : Stream' α) (n : ℕ) : get (enum s) n = (n, s.get n) :=\n  rfl\n\ntheorem enum_eq_zip (s : Stream' α) : enum s = zip Prod.mk nats s :=\n  rfl\n\nend Zip\n\n@[simp]\ntheorem mem_const (a : α) : a ∈ const a :=\n  Exists.intro 0 rfl\n\ntheorem const_eq (a : α) : const a = a::const a := by\n  apply Stream'.ext; intro n\n  cases n <;> rfl\n\n@[simp]\ntheorem tail_const (a : α) : tail (const a) = const a :=\n  suffices tail (a::const a) = const a by rwa [← const_eq] at this\n  rfl\n\n@[simp]\ntheorem map_const (f : α → β) (a : α) : map f (const a) = const (f a) :=\n  rfl\n\n@[simp]\ntheorem get_const (n : ℕ) (a : α) : get (const a) n = a :=\n  rfl\n\n@[simp]\ntheorem drop_const (n : ℕ) (a : α) : drop n (const a) = const a :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem head_iterate (f : α → α) (a : α) : head (iterate f a) = a :=\n  rfl\n\ntheorem get_succ_iterate' (n : ℕ) (f : α → α) (a : α) :\n    get (iterate f a) (succ n) = f (get (iterate f a) n) := rfl\n\ntheorem tail_iterate (f : α → α) (a : α) : tail (iterate f a) = iterate f (f a) := by\n  ext n\n  rw [get_tail]\n  induction n with\n  | zero => rfl\n  | succ n ih => rw [get_succ_iterate', ih, get_succ_iterate']\n\ntheorem iterate_eq (f : α → α) (a : α) : iterate f a = a::iterate f (f a) := by\n  rw [← Stream'.eta (iterate f a)]\n  rw [tail_iterate]; rfl\n\n@[simp]\ntheorem get_zero_iterate (f : α → α) (a : α) : get (iterate f a) 0 = a :=\n  rfl\n\ntheorem get_succ_iterate (n : ℕ) (f : α → α) (a : α) :\n    get (iterate f a) (succ n) = get (iterate f (f a)) n := by rw [get_succ, tail_iterate]\n\nsection Bisim\n\nvariable (R : Stream' α → Stream' α → Prop)\n\n/-- equivalence relation -/\nlocal infixl:50 \" ~ \" => R\n\n/-- Streams `s₁` and `s₂` are defined to be bisimulations if\ntheir heads are equal and tails are bisimulations. -/\ndef IsBisimulation :=\n  ∀ ⦃s₁ s₂⦄, s₁ ~ s₂ →\n      head s₁ = head s₂ ∧ tail s₁ ~ tail s₂\n\ntheorem get_of_bisim (bisim : IsBisimulation R) {s₁ s₂} :\n    ∀ n, s₁ ~ s₂ → get s₁ n = get s₂ n ∧ drop (n + 1) s₁ ~ drop (n + 1) s₂\n  | 0, h => bisim h\n  | n + 1, h =>\n    match bisim h with\n    | ⟨_, trel⟩ => get_of_bisim bisim n trel\n\n-- If two streams are bisimilar, then they are equal\ntheorem eq_of_bisim (bisim : IsBisimulation R) {s₁ s₂} : s₁ ~ s₂ → s₁ = s₂ := fun r =>\n  Stream'.ext fun n => And.left (get_of_bisim R bisim n r)\n\nend Bisim\n\ntheorem bisim_simple (s₁ s₂ : Stream' α) :\n    head s₁ = head s₂ → s₁ = tail s₁ → s₂ = tail s₂ → s₁ = s₂ := fun hh ht₁ ht₂ =>\n  eq_of_bisim (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂)\n    (fun s₁ s₂ ⟨h₁, h₂, h₃⟩ => by grind)\n    (And.intro hh (And.intro ht₁ ht₂))\n\ntheorem coinduction {s₁ s₂ : Stream' α} :\n    head s₁ = head s₂ →\n      (∀ (β : Type u) (fr : Stream' α → β),\n      fr s₁ = fr s₂ → fr (tail s₁) = fr (tail s₂)) → s₁ = s₂ :=\n  fun hh ht =>\n  eq_of_bisim\n    (fun s₁ s₂ =>\n      head s₁ = head s₂ ∧\n        ∀ (β : Type u) (fr : Stream' α → β), fr s₁ = fr s₂ → fr (tail s₁) = fr (tail s₂))\n    (fun s₁ s₂ h =>\n      have h₁ : head s₁ = head s₂ := And.left h\n      have h₂ : head (tail s₁) = head (tail s₂) := And.right h α (@head α) h₁\n      have h₃ :\n        ∀ (β : Type u) (fr : Stream' α → β),\n          fr (tail s₁) = fr (tail s₂) → fr (tail (tail s₁)) = fr (tail (tail s₂)) :=\n        fun β fr => And.right h β fun s => fr (tail s)\n      And.intro h₁ (And.intro h₂ h₃))\n    (And.intro hh ht)\n\n@[simp]\ntheorem iterate_id (a : α) : iterate id a = const a :=\n  coinduction rfl fun β fr ch => by rw [tail_iterate, tail_const]; exact ch\n\nTarget:\ntheorem map_iterate (f : α → α) (a : α) : iterate f (f a) = map f (iterate f a) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_16839e522d17","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"f8b9f546e80bddd9ebbe91538ecae622fc54e7250a98abf6d6d8156c75444397","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Stream","family_id":"map_iterate","file_id":"mathlib/Mathlib/Data/Stream/Init.lean","sample_id":"16839e522d1756068dfaf97c745b531d7271caa8d21dfa9ef1a00b09a41091af"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3dd577e7ce2be81d92ba051dd678c134f2a6bcae0a357014a7b7228cadb929c6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"872abf858e904265f85b85d943b70513c747a0ef833b4e67caab136109df4748","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1a64a2331eda0da0fbaef78bea3f8bc0d6cc844d45d3bd42a1481a8c40fbd0be","source_sha256":"76ff543811de607fc5619ca4ca6c9c5878d83cf85372b02e848f8ebf54663e8c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine strictMonoOn_of_lt_add_one ⟨?_⟩ fun n _ _ hn ↦\n    pow_ssubset_pow_succ_of_pow_ne_closure hX₁ hX hn\n  rintro - - n hn m ⟨-, hmn⟩ hm\n  apply hn\n  obtain rfl | hm₀ := m.eq_zero_or_pos\n  · simp [eq_comm (a := (1 : Set _)), coe_set_eq_one, -Set.subset_singleton_iff,\n      hX.coe.not_subset_singleton] at hm\n  · calc (X : Set G) ^ (n - 1)\n    _ = X ^ (n - m) * X ^ (m - 1) := by rw [← pow_add]; congr 1; lia\n    _ = closure (X : Set G) := by rw [hm, Set.pow_mul_subgroupClosure hX.nonempty.to_set]","hard_negative":true,"metrics":{"chosen_tokens":138,"rejected_tokens":3,"token_jaccard":0.016129,"token_length_ratio":0.021739},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"2828126f12b3797d60c2acd2a09b7f61c43d10d82ed21da7d2021cf5e54597ce","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Pointwise.Finset\npublic import Mathlib.Algebra.Group.Subgroup.Pointwise\npublic import Mathlib.Data.Nat.SuccPred\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2024 Yaël Dillies, Patrick Luo, Eric Rodriguez. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Patrick Luo, Eric Rodriguez\n-/\n/-!\n# Linear lower bound on the growth of a generating set\n\nThis file proves that the growth of a set generating an infinite group is at least linear.\n-/\n\npublic section\n\nopen Subgroup\nopen scoped Pointwise\n\nnamespace Finset\nvariable {G : Type*} [Group G] [DecidableEq G] {X : Finset G} {n : ℕ}\n\n@[to_additive]\nlemma pow_ssubset_pow_succ_of_pow_ne_closure (hX₁ : (1 : G) ∈ X) (hX : X.Nontrivial)\n    (hXclosure : (X ^ n : Set G) ≠ closure (X : Set G)) : X ^ n ⊂ X ^ (n + 1) := by\n  obtain rfl | hn := eq_or_ne n 0\n  · simpa [ssubset_iff_subset_not_subset, hX₁, -Finset.subset_singleton_iff]\n      using! hX.not_subset_singleton\n  refine (pow_subset_pow_right hX₁ <| n.le_add_right _).ssubset_of_ne ?_\n  contrapose hXclosure with hXn\n  rw [← closure_pow (mod_cast hX₁) hn]\n  wlog hn₁ : n = 1\n  · simp +contextual only [pow_one] at this\n    replace hXn d : X ^ (n + d) = X ^ n := by\n      induction d with\n      | zero => rw [add_zero]\n      | succ d hd =>\n        rw [pow_add, pow_one] at hXn\n        rw [← add_assoc, pow_add, pow_one, hd, ← hXn]\n    exact mod_cast this (one_mem_pow hX₁) (hX.pow hn) one_ne_zero\n      (by simp [hXn, ← pow_mul, mul_two]) (by simp)\n  subst hn₁\n  simp only [ne_eq, one_ne_zero, not_false_eq_true, Nat.reduceAdd, pow_one] at *\n  let Xgp : Subgroup G :=\n  { carrier := X\n    mul_mem' := fun {x y} hx hy ↦ by\n      norm_cast at *\n      simpa [← hXn, ← sq] using! mul_mem_mul hx hy\n    one_mem' := hX₁\n    inv_mem' := fun {x} hx ↦ by\n      norm_cast at *\n      have : x • X ⊆ X := by\n        simpa [← hXn, add_assoc, ← sq] using! smul_finset_subset_mul (t := X) hx\n      have : x • X = X := eq_of_subset_of_card_le this (card_smul_finset ..).ge\n      rw [← eq_inv_smul_iff] at this\n      rw [this]\n      simpa [mem_inv_smul_finset_iff] }\n  exact subset_closure.antisymm <| (closure_le Xgp).2 subset_rfl\n\n@[to_additive]\n\nTarget:\nlemma pow_right_strictMonoOn (hX₁ : 1 ∈ X) (hX : X.Nontrivial) :\n    StrictMonoOn (fun n ↦ X ^ n) {n | (X ^ (n - 1) : Set G) ≠ closure (X : Set G)} :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_1a64a2331eda","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"51bb6274b82ac391cf9ea275c995259c64685cf2d84285860d04e4f8c5ab170d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/Group","family_id":"pow_right_strictmonoon","file_id":"mathlib/Mathlib/Geometry/Group/Growth/LinearLowerBound.lean","sample_id":"1a64a2331eda0da0fbaef78bea3f8bc0d6cc844d45d3bd42a1481a8c40fbd0be"},"split":"train"}
{"binding":{"chosen_proof_sha256":"76e3cedf98327795879092160501512043429427fee2f5b0e827ab58125ee1b5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f4641d98f5a96bd0f077eb90f92ed674d374bbac275dc02e77eaac0b91c10057","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"856d27b4c3ec825fbdd58bf9dd767cc218a09754dd1106fc2031d94545f79fef","source_sha256":"a532f8917e86d9e21f6417bd7287cabd2ab46e6ff2388a4cee138b7afaddc6d0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [IsSimplyConnected, simply_connected_iff_loops_nullhomotopic,\n    ← isPathConnected_iff_pathConnectedSpace]\n  refine .and .rfl ⟨fun h x p hp ↦ ?_, fun h x p ↦ ?_⟩\n  · lift x to s using by simpa using hp 0\n    rcases h x {\n      toFun := fun t ↦ ⟨p t, hp t⟩\n      source' := by simp\n      target' := by simp\n    } with ⟨F⟩\n    exact ⟨F.map (.restrict s (.id _)), fun t ↦ (F t).2⟩\n  · rcases h x (p.map continuous_subtype_val) (fun t ↦ (p t).2) with ⟨F, hF⟩\n    exact ⟨{\n      toFun t := ⟨F t, hF t⟩\n      map_zero_left := by simp\n      map_one_left := by simp\n      prop' := by simp\n    }⟩","hard_negative":false,"metrics":{"chosen_tokens":153,"rejected_tokens":158,"token_jaccard":0.929825,"token_length_ratio":1.03268},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"28bfaea17662d50921c524d94deb2540779e3255131b3c10b4164a04d7348200","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps\npublic import Mathlib.Topology.Homotopy.Contractible\npublic import Mathlib.CategoryTheory.PUnit\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.PUnit\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Praneeth Kolichala. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Praneeth Kolichala\n-/\n/-!\n# Simply connected spaces\nThis file defines simply connected spaces.\nA topological space is simply connected if its fundamental groupoid is equivalent to `Unit`.\n\nWe also define the corresponding predicate for sets.\n\n## Main theorems\n  - `simply_connected_iff_unique_homotopic` - A space is simply connected if and only if it is\n    nonempty and there is a unique path up to homotopy between any two points\n\n  - `SimplyConnectedSpace.ofContractible` - A contractible space is simply connected\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen CategoryTheory\nopen scoped ContinuousMap\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n\n/-- A simply connected space is one whose fundamental groupoid is equivalent to `Discrete Unit` -/\n@[mk_iff]\nclass SimplyConnectedSpace (X : Type*) [TopologicalSpace X] : Prop where\n  equiv_unit : Nonempty (FundamentalGroupoid X ≌ Discrete Unit)\n\n@[deprecated (since := \"2026-01-08\")]\nalias simply_connected_def := simplyConnectedSpace_iff\n\ntheorem simply_connected_iff_unique_homotopic (X : Type*) [TopologicalSpace X] :\n    SimplyConnectedSpace X ↔\n      Nonempty X ∧ ∀ x y : X, Nonempty (Unique (Path.Homotopic.Quotient x y)) := by\n  simp only [simplyConnectedSpace_iff, equiv_punit_iff_unique,\n    FundamentalGroupoid.nonempty_iff X, and_congr_right_iff, Nonempty.forall]\n  intros\n  exact ⟨fun h _ _ => h _ _, fun h _ _ => h _ _⟩\n\ntheorem ContinuousMap.HomotopyEquiv.simplyConnectedSpace [hY : SimplyConnectedSpace Y]\n    (e : X ≃ₕ Y) : SimplyConnectedSpace X :=\n  ⟨hY.1.map (FundamentalGroupoidFunctor.equivOfHomotopyEquiv e).trans⟩\n\ntheorem ContinuousMap.HomotopyEquiv.simplyConnectedSpace_iff (e : X ≃ₕ Y) :\n    SimplyConnectedSpace X ↔ SimplyConnectedSpace Y :=\n  ⟨fun _ ↦ e.symm.simplyConnectedSpace, fun _ ↦ e.simplyConnectedSpace⟩\n\nnamespace SimplyConnectedSpace\n\nvariable {X : Type*} [TopologicalSpace X] [SimplyConnectedSpace X]\n\ninstance (x y : X) : Subsingleton (Path.Homotopic.Quotient x y) :=\n  @Unique.instSubsingleton _ (Nonempty.some (by\n    rw [simply_connected_iff_unique_homotopic] at *; tauto))\n\ninstance (x : X) : Subsingleton (FundamentalGroup X x) :=\n  show Subsingleton (Path.Homotopic.Quotient x x) from inferInstance\n\ninstance (priority := 100) : PathConnectedSpace X :=\n  let unique_homotopic := (simply_connected_iff_unique_homotopic X).mp inferInstance\n  { nonempty := unique_homotopic.1\n    joined := fun x y => ⟨(unique_homotopic.2 x y).some.default.out⟩ }\n\n/-- In a simply connected space, any two paths are homotopic -/\ntheorem paths_homotopic {x y : X} (p₁ p₂ : Path x y) : Path.Homotopic p₁ p₂ :=\n  Quotient.eq.mp (@Subsingleton.elim (Path.Homotopic.Quotient x y) _ _ _)\n\ninstance (priority := 100) ofContractible (Y : Type*) [TopologicalSpace Y] [ContractibleSpace Y] :\n    SimplyConnectedSpace Y :=\n  haveI : SimplyConnectedSpace Unit := ⟨⟨FundamentalGroupoid.punitEquivDiscretePUnit⟩⟩\n  (ContractibleSpace.hequiv Y Unit).some.simplyConnectedSpace\n\nend SimplyConnectedSpace\n\n/-- A space is simply connected iff it is path connected, and there is at most one path\n  up to homotopy between any two points. -/\ntheorem simply_connected_iff_paths_homotopic :\n    SimplyConnectedSpace Y ↔\n      PathConnectedSpace Y ∧ ∀ x y : Y, Subsingleton (Path.Homotopic.Quotient x y) :=\n  ⟨by intro; constructor <;> infer_instance, fun h => by\n    cases h; rw [simply_connected_iff_unique_homotopic]\n    exact ⟨inferInstance, fun x y => ⟨uniqueOfSubsingleton ⟦PathConnectedSpace.somePath x y⟧⟩⟩⟩\n\n/-- Another version of `simply_connected_iff_paths_homotopic` -/\ntheorem simply_connected_iff_paths_homotopic' :\n    SimplyConnectedSpace Y ↔\n      PathConnectedSpace Y ∧ ∀ {x y : Y} (p₁ p₂ : Path x y), Path.Homotopic p₁ p₂ := by\n  convert! simply_connected_iff_paths_homotopic (Y := Y)\n  simp [Path.Homotopic.Quotient, Setoid.eq_top_iff]; rfl\n\nset_option backward.isDefEq.respectTransparency false in\nopen Path.Homotopic.Quotient in\n/-- A space is simply connected if and only if it is path-connected and every loop\n    at any basepoint is null-homotopic (i.e., homotopic to the constant loop). -/\ntheorem simply_connected_iff_loops_nullhomotopic :\n    SimplyConnectedSpace Y ↔\n      PathConnectedSpace Y ∧ ∀ (x : Y) (γ : Path x x), Path.Homotopic γ (Path.refl x) := by\n  rw [simply_connected_iff_paths_homotopic']\n  constructor\n  · -- Forward: all paths homotopic implies all loops null-homotopic\n    intro ⟨hpc, hall⟩\n    exact ⟨hpc, fun x γ => hall γ (Path.refl x)⟩\n  · -- Backward: all loops null-homotopic implies all paths homotopic\n    intro ⟨hpc, hloops⟩\n    refine ⟨hpc, fun {x y} p₁ p₂ => ?_⟩\n    -- Work in the quotient where structural steps can be done by simp\n    rw [← eq]\n    replace hloops : ∀ (x : Y) (γ : Path x x),\n        (⟦γ⟧ : Path.Homotopic.Quotient x x) = ⟦Path.refl x⟧ :=\n      fun x γ => Quotient.sound (hloops x γ)\n    have h : trans ⟦p₁⟧ (symm ⟦p₂⟧) = refl x := by\n      simpa using hloops x (p₁.trans p₂.symm)\n    calc ⟦p₁⟧\n      _ = trans (trans ⟦p₁⟧ (symm ⟦p₂⟧)) ⟦p₂⟧ := by simp\n      _ = ⟦p₂⟧ := by grind\n\n/-!\n### Simply connected sets\n-/\n\n/-- We say that a set is simply connected if it's a simply connected topological space\nin the induced topology. -/\ndef IsSimplyConnected (s : Set X) : Prop := SimplyConnectedSpace s\n\ntheorem IsSimplyConnected.simplyConnectedSpace {s : Set X} (hs : IsSimplyConnected s) :\n    SimplyConnectedSpace s := hs\n\ntheorem IsSimplyConnected.isPathConnected {s : Set X} (hs : IsSimplyConnected s) :\n    IsPathConnected s :=\n  have := hs.simplyConnectedSpace\n  isPathConnected_iff_pathConnectedSpace.mpr inferInstance\n\ntheorem IsSimplyConnected.nonempty {s : Set X} (hs : IsSimplyConnected s) : s.Nonempty :=\n  hs.isPathConnected.nonempty\n\ntheorem Topology.IsEmbedding.isSimplyConnected_image {f : X → Y} (hf : Topology.IsEmbedding f)\n    {s : Set X} :\n    IsSimplyConnected (f '' s) ↔ IsSimplyConnected s :=\n  hf.homeomorphImage s |>.toHomotopyEquiv |>.simplyConnectedSpace_iff |>.symm\n\n@[simp]\ntheorem Homeomorph.isSimplyConnected_image (f : X ≃ₜ Y) {s : Set X} :\n    IsSimplyConnected (f '' s) ↔ IsSimplyConnected s :=\n  f.isEmbedding.isSimplyConnected_image\n\n@[simp]\ntheorem Homeomorph.isSimplyConnected_preimage (f : X ≃ₜ Y) {s : Set Y} :\n    IsSimplyConnected (f ⁻¹' s) ↔ IsSimplyConnected s := by\n  rw [← image_symm, isSimplyConnected_image]\n\n/-- A set is simply connected iff it's path connected\nand any loop is homotopic to the constant path within `s`. -/\n\nTarget:\ntheorem isSimplyConnected_iff_exists_homotopy_refl_forall_mem {s : Set X} :\n    IsSimplyConnected s ↔ IsPathConnected s ∧ ∀ x, ∀ p : Path x x, (∀ t, p t ∈ s) →\n      ∃ F : p.Homotopy (.refl x), ∀ t, F t ∈ s :=\n\nProof body:\n","rejected":"by\n  rw [IsSimplyConnected, simply_connected_iff_loops_nullhomotopic,\n    ← isPathConnected_iff_pathConnectedSpace]\n  refine .and .rfl ⟨fun h x p hp ↦ ?_, fun h x p ↦ ?_⟩\n  · lift x to s using by simpa using hp 0\n    rcases h x {\n      toFun := fun t ↦ ⟨p t, hp t⟩\n      source' := by simp\n      target' := by simp\n    } with ⟨F⟩\n    exact ⟨F.map (.restrict s (.id _)), fun t ↦ (F t).2⟩\n  · rcases h x (p.map continuous_subtype_val) (fun t ↦ (p t).2) with ⟨F, hF⟩\n    exact ⟨{\n      toFun t := ⟨F t, hF t⟩\n      map_zero_left := by simp\n      map_one_left := by simp\n      prop' := by simp\n    }⟩\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/FundamentalGroupoid","family_id":"issimplyconnected_iff_exists_homotopy_refl_forall_mem","file_id":"mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean","sample_id":"856d27b4c3ec825fbdd58bf9dd767cc218a09754dd1106fc2031d94545f79fef"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c337966156f7fa0b44c661383f2baec35f6094ce1c36181a113cc67c17fd2ac2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4771507a2edaa490165ad6dbbb8bbee190b6099c2dfd923f1e5c8b47e63ba2b7","source_sha256":"7fb54be1edb74454dd74d15ac42b7739b2a4aa95bc5649191f339f2298801887","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases c <;> simp [*]","hard_negative":false,"metrics":{"chosen_tokens":10,"rejected_tokens":2,"token_jaccard":0.090909,"token_length_ratio":0.2},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"28c74d95d1f9cd44be4bb9036ba080299b3ffef4bcab6f7adbf1713af2c1f491","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Ordering.Basic\npublic import Mathlib.Order.Defs.Unbundled\n\nNamespace:\nOrdering\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Some `Ordering` lemmas\n-/\n\npublic section\n\nuniverse u\n\nnamespace Ordering\n\n@[simp]\ntheorem ite_eq_lt_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.lt) = if c then a = Ordering.lt else b = Ordering.lt := by\n  by_cases c <;> simp [*]\n\n@[simp]\ntheorem ite_eq_eq_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.eq) = if c then a = Ordering.eq else b = Ordering.eq := by\n  by_cases c <;> simp [*]\n\n@[simp]\n\nTarget:\ntheorem ite_eq_gt_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.gt) = if c then a = Ordering.gt else b = Ordering.gt :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Ordering","family_id":"ite_eq_gt_distrib","file_id":"mathlib/Mathlib/Data/Ordering/Lemmas.lean","sample_id":"4771507a2edaa490165ad6dbbb8bbee190b6099c2dfd923f1e5c8b47e63ba2b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"340281fbf0a45d34c3a34d2f4a61a24c355dc2f03e29d91afa10e3fb584bf2fd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3fd6053780e5eb00bf5e0f3cec4bde111bdf2750e7bc0579c6d7cb68f7c6a582","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3b594b8efd7138c1edf9cf4beb8848fba944bf82946b75971c431aea36b7d8df","source_sha256":"34c15bf85e4926f0e35f0d5bc6422a5c85336bdcbed113f50f3b65102f3e6cc1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro a b h\n  simp only [LT.lt, Finsupp.lex_def]\n  simp only [ofLex_toLex, Nat.lt_eq]\n  use a\n  constructor\n  · intro d hd\n    simp only [Finsupp.single_eq_of_ne hd.ne, Finsupp.single_eq_of_ne (hd.trans h).ne]\n  · simp [h.ne']","hard_negative":true,"metrics":{"chosen_tokens":61,"rejected_tokens":3,"token_jaccard":0.032258,"token_length_ratio":0.04918},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"28fd13829bb27468a9ae9f5e9535fe3acb144b75ec6ad7302e14a04ef6c06872","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finsupp.Order\npublic import Mathlib.Data.DFinsupp.Lex\npublic import Mathlib.Data.Finsupp.ToDFinsupp\n\nNamespace:\nFinsupp\n\nLocal context:\n/-\nCopyright (c) 2022 Damiano Testa. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Damiano Testa\n-/\n/-!\n# Lexicographic order on finitely supported functions\n\nThis file defines the lexicographic order on `Finsupp`.\n-/\n\n@[expose] public section\n\n\nvariable {α N : Type*}\n\nnamespace Finsupp\n\nsection NHasZero\n\nvariable [Zero N]\n\n/-- `Finsupp.Lex r s` is the lexicographic relation on `α →₀ N`, where `α` is ordered by `r`,\nand `N` is ordered by `s`.\n\nThe type synonym `Lex (α →₀ N)` has an order given by `Finsupp.Lex (· < ·) (· < ·)`.\n-/\nprotected def Lex (r : α → α → Prop) (s : N → N → Prop) (x y : α →₀ N) : Prop :=\n  Pi.Lex r s x y\n\ntheorem _root_.Pi.lex_eq_finsupp_lex {r : α → α → Prop} {s : N → N → Prop} (a b : α →₀ N) :\n    Pi.Lex r s a b = Finsupp.Lex r s a b :=\n  rfl\n\ntheorem lex_def {r : α → α → Prop} {s : N → N → Prop} {a b : α →₀ N} :\n    Finsupp.Lex r s a b ↔ ∃ j, (∀ d, r d j → a d = b d) ∧ s (a j) (b j) :=\n  .rfl\n\ntheorem lex_eq_invImage_dfinsupp_lex (r : α → α → Prop) (s : N → N → Prop) :\n    Finsupp.Lex r s = InvImage (DFinsupp.Lex r fun _ ↦ s) toDFinsupp :=\n  rfl\n\ninstance [LT α] [LT N] : LT (Lex (α →₀ N)) :=\n  ⟨fun f g ↦ Finsupp.Lex (· < ·) (· < ·) (ofLex f) (ofLex g)⟩\n\ninstance [LT α] [LT N] : LT (Colex (α →₀ N)) :=\n  ⟨fun f g ↦ Finsupp.Lex (· > ·) (· < ·) (ofColex f) (ofColex g)⟩\n\ntheorem Lex.lt_iff [LT α] [LT N] {a b : Lex (α →₀ N)} :\n    a < b ↔ ∃ i, (∀ j, j < i → a j = b j) ∧ a i < b i :=\n  .rfl\n\n@[deprecated (since := \"2025-11-29\")]\nalias lex_lt_iff := Lex.lt_iff\n\ntheorem Colex.lt_iff [LT α] [LT N] {a b : Colex (α →₀ N)} :\n    a < b ↔ ∃ i, (∀ j, i < j → a j = b j) ∧ a i < b i :=\n  .rfl\n\ntheorem lex_lt_of_lt_of_preorder [Preorder N] (r) [IsStrictOrder α r] {x y : α →₀ N} (hlt : x < y) :\n    ∃ i, (∀ j, r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i :=\n  DFinsupp.lex_lt_of_lt_of_preorder r (id hlt : x.toDFinsupp < y.toDFinsupp)\n\ntheorem lex_lt_of_lt [PartialOrder N] (r) [IsStrictOrder α r] {x y : α →₀ N} (hlt : x < y) :\n    Pi.Lex r (· < ·) x y :=\n  DFinsupp.lex_lt_of_lt r (id hlt : x.toDFinsupp < y.toDFinsupp)\n\ntheorem lex_iff_of_unique [Unique α] [LT N] {r} [Std.Irrefl r] {x y : α →₀ N} :\n    Finsupp.Lex r (· < ·) x y ↔ x default < y default :=\n  Pi.lex_iff_of_unique\n\ntheorem Lex.lt_iff_of_unique [Unique α] [LT N] [Preorder α] {x y : Lex (α →₀ N)} :\n    x < y ↔ x default < y default :=\n  lex_iff_of_unique\n\n@[deprecated (since := \"2025-11-29\")]\nalias lex_lt_iff_of_unique := Lex.lt_iff_of_unique\n\ntheorem Colex.lt_iff_of_unique [Unique α] [LT N] [Preorder α] {x y : Colex (α →₀ N)} :\n    x < y ↔ x default < y default :=\n  Lex.lt_iff_of_unique (α := αᵒᵈ)\n\nvariable [LinearOrder α]\n\ninstance Lex.isStrictOrder [PartialOrder N] : IsStrictOrder (Lex (α →₀ N)) (· < ·) where\n  irrefl _ := lt_irrefl (α := Lex (α → N)) _\n  trans _ _ _ := lt_trans (α := Lex (α → N))\n\ninstance Colex.isStrictOrder [PartialOrder N] : IsStrictOrder (Colex (α →₀ N)) (· < ·) :=\n  Lex.isStrictOrder (α := αᵒᵈ)\n\n/-- The partial order on `Finsupp`s obtained by the lexicographic ordering.\nSee `Finsupp.Lex.linearOrder` for a proof that this partial order is in fact linear. -/\ninstance Lex.partialOrder [PartialOrder N] : PartialOrder (Lex (α →₀ N)) where\n  lt := (· < ·)\n  le x y := ⇑(ofLex x) = ⇑(ofLex y) ∨ x < y\n  __ := PartialOrder.lift (fun x : Lex (α →₀ N) ↦ toLex (⇑(ofLex x)))\n    (DFunLike.coe_injective (F := Finsupp α N))\n\n/-- The partial order on `Finsupp`s obtained by the colexicographic ordering.\nSee `Finsupp.Colex.linearOrder` for a proof that this partial order is in fact linear. -/\ninstance Colex.partialOrder [PartialOrder N] : PartialOrder (Colex (α →₀ N)) where\n  lt := (· < ·)\n  le x y := ⇑(ofColex x) = ⇑(ofColex y) ∨ x < y\n  __ := PartialOrder.lift (fun x : Colex (α →₀ N) ↦ toColex (⇑(ofColex x)))\n    (DFunLike.coe_injective (F := Finsupp α N))\n\n/-- The linear order on `Finsupp`s obtained by the lexicographic ordering. -/\ninstance Lex.linearOrder [LinearOrder N] : LinearOrder (Lex (α →₀ N)) where\n  __ := Lex.partialOrder\n  __ := LinearOrder.lift' (toLex ∘ toDFinsupp ∘ ofLex) finsuppEquivDFinsupp.injective\n\n/-- The linear order on `Finsupp`s obtained by the colexicographic ordering. -/\ninstance Colex.linearOrder [LinearOrder N] : LinearOrder (Colex (α →₀ N)) where\n  lt := (· < ·)\n  le := (· ≤ ·)\n  __ := LinearOrder.lift' (toColex ∘ toDFinsupp ∘ ofColex) finsuppEquivDFinsupp.injective\n\ntheorem Lex.le_iff_of_unique [Unique α] [PartialOrder N] {x y : Lex (α →₀ N)} :\n    x ≤ y ↔ x default ≤ y default :=\n  Pi.lex_le_iff_of_unique\n\n@[deprecated (since := \"2025-11-29\")]\nalias lex_le_iff_of_unique := Lex.le_iff_of_unique\n\ntheorem Colex.le_iff_of_unique [Unique α] [PartialOrder N] {x y : Colex (α →₀ N)} :\n    x ≤ y ↔ x default ≤ y default :=\n  Lex.le_iff_of_unique (α := αᵒᵈ)\n\nTarget:\ntheorem Lex.single_strictAnti : StrictAnti fun (a : α) ↦ toLex (single a 1) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_3b594b8efd71","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"0e47a2a545dfed109ecf5e553bc0319f688e317d34d6d13da4ec88a015a1e011","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finsupp","family_id":"lex","file_id":"mathlib/Mathlib/Data/Finsupp/Lex.lean","sample_id":"3b594b8efd7138c1edf9cf4beb8848fba944bf82946b75971c431aea36b7d8df"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9c5374a4a76445b21ceb5288576943f3dfabe49605e7c1eedae05176a4a561c7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cb787666449dd4a52c1a3d0c14107bfed4df4b3d94d031fd1a98bc977a411bf3","source_sha256":"be2af13256fec9c1b55ab43e0b0388a9f91c486ebeddde5998dfc4d97729095b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← isLocallySurjective_iff_epi']\n  exact LightCondensed.isLocallySurjective_iff_locallySurjective_on_lightProfinite _ f","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":2,"token_jaccard":0.076923,"token_length_ratio":0.166667},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"293d1c19c91b7f2d834e7f66976ec97948e88a4fc64b98b947b9485f2d7cfdb3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Shapes.SequentialProduct\npublic import Mathlib.CategoryTheory.Sites.Coherent.SequentialLimit\npublic import Mathlib.Condensed.Light.Functors\npublic import Mathlib.Condensed.Light.Limits\n\nNamespace:\nLightCondSet\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# Epimorphisms of light condensed objects\n\nThis file characterises epimorphisms in light condensed sets and modules as the locally surjective\nmorphisms. Here, the condition of locally surjective is phrased in terms of continuous surjections\nof light profinite sets.\n\nFurther, we prove that the functor `lim : Discrete ℕ ⥤ LightCondMod R` preserves epimorphisms.\n-/\n\npublic section\n\nuniverse v u w u' v'\n\nopen CategoryTheory Sheaf Limits GrothendieckTopology\n\nnamespace LightCondensed\n\nvariable (A : Type u') [Category.{v'} A] {FA : A → A → Type*} {CA : A → Type w}\nvariable [∀ X Y, FunLike (FA X Y) (CA X) (CA Y)] [ConcreteCategory.{w} A FA]\n  [PreservesFiniteProducts (CategoryTheory.forget A)]\n\nvariable {X Y : LightCondensed.{u} A} (f : X ⟶ Y)\n\nlemma isLocallySurjective_iff_locallySurjective_on_lightProfinite : IsLocallySurjective f ↔\n    ∀ (S : LightProfinite) (y : ToType (Y.obj.obj ⟨S⟩)),\n      (∃ (S' : LightProfinite) (φ : S' ⟶ S) (_ : Function.Surjective φ)\n        (x : ToType (X.obj.obj ⟨S'⟩)),\n        f.hom.app ⟨S'⟩ x = Y.obj.map ⟨φ⟩ y) := by\n  rw [coherentTopology.isLocallySurjective_iff,\n    regularTopology.isLocallySurjective_iff]\n  simp_rw [LightProfinite.effectiveEpi_iff_surjective]\n\nend LightCondensed\n\nnamespace LightCondSet\n\nvariable {X Y : LightCondSet.{u}} (f : X ⟶ Y)\n\nTarget:\nlemma epi_iff_locallySurjective_on_lightProfinite : Epi f ↔\n    ∀ (S : LightProfinite) (y : Y.obj.obj ⟨S⟩),\n      (∃ (S' : LightProfinite) (φ : S' ⟶ S) (_ : Function.Surjective φ) (x : X.obj.obj ⟨S'⟩),\n        f.hom.app ⟨S'⟩ x = Y.obj.map ⟨φ⟩ y) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed/Light","family_id":"epi_iff_locallysurjective_on_lightprofinite","file_id":"mathlib/Mathlib/Condensed/Light/Epi.lean","sample_id":"cb787666449dd4a52c1a3d0c14107bfed4df4b3d94d031fd1a98bc977a411bf3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ce223990efe19a3952beb8c48c08eec52484b0eabebe4090119bef4f0e984746","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ebf2287be0dff80d8527d98a13ffae0d07d8ff69284070c38aa592036a9015a5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"210dd76ae67ac01eff7a888fdcf07f7b44a5b4586ca30f9a119d28a997965242","source_sha256":"b1dde0e82be30e8a9ca87b23c46d04afc92be61818622ba6d4603daa4d4a9f70","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by simp [mk, ← ofCauchy_add]","hard_negative":true,"metrics":{"chosen_tokens":8,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.625},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"2955e439973cf0e54f62fbb48c3368e4917d01fa3c8b1ca0c7212e8a25f36fc7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.CauSeq.Completion\npublic import Mathlib.Algebra.Order.Ring.Rat\npublic import Mathlib.Data.Rat.Cast.Defs\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Floris van Doorn\n-/\n/-!\n# Real numbers from Cauchy sequences\n\nThis file defines `ℝ` as the type of equivalence classes of Cauchy sequences of rational numbers.\nThis choice is motivated by how easy it is to prove that `ℝ` is a commutative ring, by simply\nlifting everything to `ℚ`.\n\nThe facts that the real numbers are an Archimedean floor ring,\nand a conditionally complete linear order,\nhave been deferred to the file `Mathlib/Data/Real/Archimedean.lean`,\nin order to keep the imports here simple.\n\nThe fact that the real numbers are a (trivial) \\*-ring has similarly been deferred to\n`Mathlib/Data/Real/Star.lean`.\n-/\n\n@[expose] public section\n\n\nassert_not_exists Finset Module Submonoid FloorRing\n\n/-- The type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational\nnumbers. -/\n@[wikidata Q12916, wikidata Q2584477]\nstructure Real where ofCauchy ::\n  /-- The underlying Cauchy completion -/\n  cauchy : CauSeq.Completion.Cauchy (abs : ℚ → ℚ)\n\n@[inherit_doc]\nnotation \"ℝ\" => Real\n\nnamespace CauSeq.Completion\n\n-- this can't go in `Data.Real.CauSeqCompletion` as the structure on `ℚ` isn't available\n@[simp]\ntheorem ofRat_rat {abv : ℚ → ℚ} [IsAbsoluteValue abv] (q : ℚ) :\n    ofRat (q : ℚ) = (q : Cauchy abv) :=\n  rfl\n\nend CauSeq.Completion\n\nnamespace Real\n\nopen CauSeq CauSeq.Completion\n\nvariable {x : ℝ}\n\ntheorem ext_cauchy_iff : ∀ {x y : Real}, x = y ↔ x.cauchy = y.cauchy\n  | ⟨a⟩, ⟨b⟩ => by rw [ofCauchy.injEq]\n\ntheorem ext_cauchy {x y : Real} : x.cauchy = y.cauchy → x = y :=\n  ext_cauchy_iff.2\n\n/-- The real numbers are isomorphic to the quotient of Cauchy sequences on the rationals. -/\ndef equivCauchy : ℝ ≃ CauSeq.Completion.Cauchy (abs : ℚ → ℚ) :=\n  ⟨Real.cauchy, Real.ofCauchy, fun ⟨_⟩ => rfl, fun _ => rfl⟩\n\nset_option backward.privateInPublic true in\n-- irreducible doesn't work for instances: https://github.com/leanprover-community/lean/issues/511\nprivate irreducible_def zero : ℝ :=\n  ⟨0⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def one : ℝ :=\n  ⟨1⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def add : ℝ → ℝ → ℝ\n  | ⟨a⟩, ⟨b⟩ => ⟨a + b⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def neg : ℝ → ℝ\n  | ⟨a⟩ => ⟨-a⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def mul : ℝ → ℝ → ℝ\n  | ⟨a⟩, ⟨b⟩ => ⟨a * b⟩\n\nset_option backward.privateInPublic true in\nprivate noncomputable irreducible_def inv' : ℝ → ℝ\n  | ⟨a⟩ => ⟨a⁻¹⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Zero ℝ :=\n  ⟨zero⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : One ℝ :=\n  ⟨one⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Add ℝ :=\n  ⟨add⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Neg ℝ :=\n  ⟨neg⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Mul ℝ :=\n  ⟨mul⟩\n\ninstance : Sub ℝ :=\n  ⟨fun a b => a + -b⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nnoncomputable instance : Inv ℝ :=\n  ⟨inv'⟩\n\ntheorem ofCauchy_zero : (⟨0⟩ : ℝ) = 0 :=\n  zero_def.symm\n\ntheorem ofCauchy_one : (⟨1⟩ : ℝ) = 1 :=\n  one_def.symm\n\ntheorem ofCauchy_add (a b) : (⟨a + b⟩ : ℝ) = ⟨a⟩ + ⟨b⟩ :=\n  (add_def _ _).symm\n\ntheorem ofCauchy_neg (a) : (⟨-a⟩ : ℝ) = -⟨a⟩ :=\n  (neg_def _).symm\n\ntheorem ofCauchy_sub (a b) : (⟨a - b⟩ : ℝ) = ⟨a⟩ - ⟨b⟩ := by\n  rw [sub_eq_add_neg, ofCauchy_add, ofCauchy_neg]\n  rfl\n\ntheorem ofCauchy_mul (a b) : (⟨a * b⟩ : ℝ) = ⟨a⟩ * ⟨b⟩ :=\n  (mul_def _ _).symm\n\ntheorem ofCauchy_inv {f} : (⟨f⁻¹⟩ : ℝ) = ⟨f⟩⁻¹ :=\n  show _ = inv' _ by rw [inv']\n\ntheorem cauchy_zero : (0 : ℝ).cauchy = 0 :=\n  show zero.cauchy = 0 by rw [zero_def]\n\ntheorem cauchy_one : (1 : ℝ).cauchy = 1 :=\n  show one.cauchy = 1 by rw [one_def]\n\ntheorem cauchy_add : ∀ a b, (a + b : ℝ).cauchy = a.cauchy + b.cauchy\n  | ⟨a⟩, ⟨b⟩ => show (add _ _).cauchy = _ by rw [add_def]\n\ntheorem cauchy_neg : ∀ a, (-a : ℝ).cauchy = -a.cauchy\n  | ⟨a⟩ => show (neg _).cauchy = _ by rw [neg_def]\n\ntheorem cauchy_mul : ∀ a b, (a * b : ℝ).cauchy = a.cauchy * b.cauchy\n  | ⟨a⟩, ⟨b⟩ => show (mul _ _).cauchy = _ by rw [mul_def]\n\ntheorem cauchy_sub : ∀ a b, (a - b : ℝ).cauchy = a.cauchy - b.cauchy\n  | ⟨a⟩, ⟨b⟩ => by\n    rw [sub_eq_add_neg, ← cauchy_neg, ← cauchy_add]\n    rfl\n\ntheorem cauchy_inv : ∀ f, (f⁻¹ : ℝ).cauchy = f.cauchy⁻¹\n  | ⟨f⟩ => show (inv' _).cauchy = _ by rw [inv']\n\ninstance instNatCast : NatCast ℝ where natCast n := ⟨n⟩\ninstance instIntCast : IntCast ℝ where intCast z := ⟨z⟩\ninstance instNNRatCast : NNRatCast ℝ where nnratCast q := ⟨q⟩\ninstance instRatCast : RatCast ℝ where ratCast q := ⟨q⟩\n\nlemma ofCauchy_natCast (n : ℕ) : (⟨n⟩ : ℝ) = n := rfl\nlemma ofCauchy_intCast (z : ℤ) : (⟨z⟩ : ℝ) = z := rfl\nlemma ofCauchy_nnratCast (q : ℚ≥0) : (⟨q⟩ : ℝ) = q := rfl\nlemma ofCauchy_ratCast (q : ℚ) : (⟨q⟩ : ℝ) = q := rfl\n\nlemma cauchy_natCast (n : ℕ) : (n : ℝ).cauchy = n := rfl\nlemma cauchy_intCast (z : ℤ) : (z : ℝ).cauchy = z := rfl\nlemma cauchy_nnratCast (q : ℚ≥0) : (q : ℝ).cauchy = q := rfl\nlemma cauchy_ratCast (q : ℚ) : (q : ℝ).cauchy = q := rfl\n\ninstance commRing : CommRing ℝ where\n  natCast n := ⟨n⟩\n  intCast z := ⟨z⟩\n  npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩\n  nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩\n  zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩ (@nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩)\n  add_zero a := by apply ext_cauchy; simp [cauchy_add, cauchy_zero]\n  zero_add a := by apply ext_cauchy; simp [cauchy_add, cauchy_zero]\n  add_comm a b := by apply ext_cauchy; simp only [cauchy_add, add_comm]\n  add_assoc a b c := by apply ext_cauchy; simp only [cauchy_add, add_assoc]\n  mul_zero a := by apply ext_cauchy; simp [cauchy_mul, cauchy_zero]\n  zero_mul a := by apply ext_cauchy; simp [cauchy_mul, cauchy_zero]\n  mul_one a := by apply ext_cauchy; simp [cauchy_mul, cauchy_one]\n  one_mul a := by apply ext_cauchy; simp [cauchy_mul, cauchy_one]\n  mul_comm a b := by apply ext_cauchy; simp only [cauchy_mul, mul_comm]\n  mul_assoc a b c := by apply ext_cauchy; simp only [cauchy_mul, mul_assoc]\n  left_distrib a b c := by apply ext_cauchy; simp only [cauchy_add, cauchy_mul, mul_add]\n  right_distrib a b c := by apply ext_cauchy; simp only [cauchy_add, cauchy_mul, add_mul]\n  neg_add_cancel a := by apply ext_cauchy; simp [cauchy_add, cauchy_neg, cauchy_zero]\n  natCast_zero := by apply ext_cauchy; simp [cauchy_zero]\n  natCast_succ n := by apply ext_cauchy; simp [cauchy_one, cauchy_add]\n  intCast_negSucc z := by apply ext_cauchy; simp [cauchy_neg, cauchy_natCast]\n\n/-- `Real.equivCauchy` as a ring equivalence. -/\n@[simps]\ndef ringEquivCauchy : ℝ ≃+* CauSeq.Completion.Cauchy (abs : ℚ → ℚ) :=\n  { equivCauchy with\n    toFun := cauchy\n    invFun := ofCauchy\n    map_add' := cauchy_add\n    map_mul' := cauchy_mul }\n\n/-! Extra instances to short-circuit type class resolution.\n\nThese short-circuits have an additional property of ensuring that a computable path is found; if\n`Field ℝ` is found first, then decaying it to these typeclasses would result in a `noncomputable`\nversion of them. -/\n\ninstance instRing : Ring ℝ := by infer_instance\n\ninstance : CommSemiring ℝ := by infer_instance\n\ninstance semiring : Semiring ℝ := by infer_instance\n\ninstance : CommMonoidWithZero ℝ := by infer_instance\n\ninstance : MonoidWithZero ℝ := by infer_instance\n\ninstance : AddCommGroup ℝ := by infer_instance\n\ninstance : AddGroup ℝ := by infer_instance\n\ninstance : AddCommMonoid ℝ := by infer_instance\n\ninstance : AddMonoid ℝ := by infer_instance\n\ninstance : AddLeftCancelSemigroup ℝ := by infer_instance\n\ninstance : AddRightCancelSemigroup ℝ := by infer_instance\n\ninstance : AddCommSemigroup ℝ := by infer_instance\n\ninstance : AddSemigroup ℝ := by infer_instance\n\ninstance : CommMonoid ℝ := by infer_instance\n\ninstance : Monoid ℝ := by infer_instance\n\ninstance : CommSemigroup ℝ := by infer_instance\n\ninstance : Semigroup ℝ := by infer_instance\n\ninstance : Inhabited ℝ :=\n  ⟨0⟩\n\n/-- Make a real number from a Cauchy sequence of rationals (by taking the equivalence class). -/\ndef mk (x : CauSeq ℚ abs) : ℝ :=\n  ⟨CauSeq.Completion.mk x⟩\n\ntheorem mk_eq {f g : CauSeq ℚ abs} : mk f = mk g ↔ f ≈ g :=\n  ext_cauchy_iff.trans CauSeq.Completion.mk_eq\n\nset_option backward.privateInPublic true in\nprivate irreducible_def lt : ℝ → ℝ → Prop\n  | ⟨x⟩, ⟨y⟩ =>\n    (Quotient.liftOn₂ x y (· < ·)) fun _ _ _ _ hf hg =>\n      propext <|\n        ⟨fun h => lt_of_eq_of_lt (Setoid.symm hf) (lt_of_lt_of_eq h hg), fun h =>\n          lt_of_eq_of_lt hf (lt_of_lt_of_eq h (Setoid.symm hg))⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : LT ℝ :=\n  ⟨lt⟩\n\ntheorem lt_cauchy {f g} : (⟨⟦f⟧⟩ : ℝ) < ⟨⟦g⟧⟩ ↔ f < g :=\n  show lt _ _ ↔ _ by rw [lt_def]; rfl\n\n@[simp]\ntheorem mk_lt {f g : CauSeq ℚ abs} : mk f < mk g ↔ f < g :=\n  lt_cauchy\n\ntheorem mk_zero : mk 0 = 0 := by rw [← ofCauchy_zero]; rfl\n\ntheorem mk_one : mk 1 = 1 := by rw [← ofCauchy_one]; rfl\n\nTarget:\ntheorem mk_add {f g : CauSeq ℚ abs} : mk (f + g) = mk f + mk g :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_210dd76ae67a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1361893137bf51dcbae79e0dd23223aed20c8d91ae57d717422c61a8df3efe13","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Real","family_id":"mk_add","file_id":"mathlib/Mathlib/Data/Real/Basic.lean","sample_id":"210dd76ae67ac01eff7a888fdcf07f7b44a5b4586ca30f9a119d28a997965242"},"split":"train"}
{"binding":{"chosen_proof_sha256":"de0e9cf3147279392578300fd27bc7777015480616482bce0bced9bd64ff6f36","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7c2cc263cb08cd0ae7b3b6f1e5d4cbec38b9a07797c8b210ba27cbac169d95dc","source_sha256":"67ccbb07cd4eb0a572ef6c3e31cad126033c1f36f3721cbbe9e484682e7571e0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [NatTrans.ext_iff]\n  simp [le_def, Set.subset_def, ConcreteCategory.hom_ext_iff, funext_iff]","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":5,"token_jaccard":0.052632,"token_length_ratio":0.238095},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"295ae110ea111122ef18bd440b3ae2b287b254623a0ac2b5d21783ab2ecebf6f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Shapes.Equalizers\npublic import Mathlib.CategoryTheory.Subfunctor.Image\n\nNamespace:\nCategoryTheory.Subfunctor\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The equalizer of two morphisms of functors, as a subfunctor\n\nIf `F₁` and `F₂` are type-valued functors, `A : Subfunctor F₁`, and\n`f` and `g` are two morphisms `A.toFunctor ⟶ F₂`, we introduce\n`Subcomplex.equalizer f g`, which is the subfunctor of `F₁` contained in `A`\nwhere `f` and `g` coincide.\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nnamespace CategoryTheory\n\nvariable {C : Type u} [Category.{v} C] {F₁ F₂ : C ⥤ Type w} {A : Subfunctor F₁}\n  (f g : A.toFunctor ⟶ F₂)\n\nnamespace Subfunctor\n\n/-- The equalizer of two morphisms of type-valued functors of types of the form\n`A.toFunctor ⟶ F₂` with `A : Subfunctor F₁`, as a subcomplex of `F₁`. -/\n@[simps -isSimp]\nprotected def equalizer : Subfunctor F₁ where\n  obj U := setOf (fun x ↦ ∃ (hx : x ∈ A.obj _), f.app _ ⟨x, hx⟩ = g.app _ ⟨x, hx⟩)\n  map φ x := by\n    rintro ⟨hx, h⟩\n    exact ⟨A.map _ hx,\n      (NatTrans.naturality_apply f φ ⟨x, hx⟩).trans (Eq.trans (by rw [h])\n        (NatTrans.naturality_apply g φ ⟨x, hx⟩).symm)⟩\n\nattribute [local simp] equalizer_obj\n\nlemma equalizer_le : Subfunctor.equalizer f g ≤ A :=\n  fun _ _ h ↦ h.1\n\n@[simp]\nlemma equalizer_self : Subfunctor.equalizer f f = A := by aesop\n\nlemma mem_equalizer_iff {i : C} (x : A.toFunctor.obj i) :\n    x.1 ∈ (Subfunctor.equalizer f g).obj i ↔ f.app i x = g.app i x := by\n  simp\n\nTarget:\nlemma range_le_equalizer_iff {G : C ⥤ Type w} (φ : G ⟶ A.toFunctor) :\n    range (φ ≫ A.ι) ≤ Subfunctor.equalizer f g ↔ φ ≫ f = φ ≫ g :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Subfunctor","family_id":"range_le_equalizer_iff","file_id":"mathlib/Mathlib/CategoryTheory/Subfunctor/Equalizer.lean","sample_id":"7c2cc263cb08cd0ae7b3b6f1e5d4cbec38b9a07797c8b210ba27cbac169d95dc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f6f5bbf3230230f3a0428b8083f535c36ff227e3eca17b3d7dbf4a73cb1bc39d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ea51963a627d887e53897ff9660cd0137a6ca3fccf9a879fe29c6140902dfe65","source_sha256":"f1d0cf668c2807493d79a9d4c8a96aaed3d5b4da33fe2397162a57706cbf448f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [weaklyEtale_iff, weaklyEtale_iff, Module.Flat.ulift_left_iff, Module.Flat.ulift_right_iff]\n  congr!\n  conv_rhs => rw [← RingHom.Flat.ulift_iff.{u₁, u₂}]\n  rw [TensorProduct.lmul'_ulift, AlgHom.toRingHom_eq_coe, AlgHom.comp_toRingHom]\n  exact RingHom.Flat.comp_iff_of_bijective_right (Equiv.bijective _)","hard_negative":true,"metrics":{"chosen_tokens":66,"rejected_tokens":8,"token_jaccard":0.02439,"token_length_ratio":0.121212},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"298cd7b25b0ffcbd5188ba9b9b27f37f11d09a876331551754ccfa878d7978ab","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.RingHom.Flat\npublic import Mathlib.RingTheory.Etale.Basic\npublic import Mathlib.RingTheory.Smooth.Flat\n\nNamespace:\nAlgebra.WeaklyEtale\n\nLocal context:\n/-\nCopyright (c) 2026 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten\n-/\n/-!\n# Weakly étale algebras\n\nIn this file we define weakly étale algebras. An `R`-algebra `S` is weakly étale if\n`S` is `R`-flat and the multiplication map `S ⊗[R] S → S` is flat.\n\n## TODOs\n\n- Show that a weakly étale algebra is formally unramified and in particular that\n  a weakly étale algebra of finite presentation is étale (@chrisflav).\n-/\n\npublic section\n\nuniverse u u₁ u₂ u₃\n\nopen TensorProduct\n\nnamespace Algebra\n\nvariable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]\n\n/-- `S` is a weakly-étale `R`-algebra if both `R → S` and `S ⊗[R] S → R` are flat.\nThis is also called absolutely flat. -/\n@[stacks 092B, mk_iff]\nclass WeaklyEtale (R S : Type*) [CommRing R] [CommRing S] [Algebra R S] where\n  flat : Module.Flat R S := by infer_instance\n  flat_lmul' (R S) : (Algebra.TensorProduct.lmul' R (S := S)).Flat\n\nattribute [instance] WeaklyEtale.flat\n\nnamespace WeaklyEtale\n\nattribute [local instance] ULift.algebra' in\n\nTarget:\nlemma ulift_iff : WeaklyEtale (ULift.{u₁} R) (ULift.{u₂} S) ↔ WeaklyEtale R S :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"21e84bd0a80657de1d30789cdb6932b85962cc35f335ed13c74e1d1a7824e0c7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Etale","family_id":"ulift_iff","file_id":"mathlib/Mathlib/RingTheory/Etale/Weakly.lean","sample_id":"ea51963a627d887e53897ff9660cd0137a6ca3fccf9a879fe29c6140902dfe65"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0d95bb02e9e9a9b98bdd4fbdf7abc16ec6edaa269b3beda79e93adb84be4b936","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5dd2d45a5990b3cf1553e9a7690924d8ccbc6a84d5460b755d61cad4687003ff","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"32b115c85276c93388c62a93ae6422efd911c7d0d06efd966d995ffe632b53e9","source_sha256":"53d8534063fc3b6cb81173ac579623e7fbacb797af1fdf83974b842dd64042c1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases h : k = 0\n  · simp [h]\n  exact (isPrimitiveRoot_exp k h).card_primitiveRoots","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":27,"token_jaccard":0.894737,"token_length_ratio":1.35},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"29ddd831492637e0a029b97532214282686237c74cad53e65a4e8dad36b4a2f0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Complex.Log\npublic import Mathlib.RingTheory.Int.Basic\npublic import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots\npublic import Mathlib.Tactic.Rify\npublic import Mathlib.Tactic.Qify\n\nNamespace:\nComplex\n\nLocal context:\n/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Snir Broshi\n-/\n/-!\n# Complex roots of unity\n\nIn this file we show that the `n`-th complex roots of unity\nare exactly the complex numbers `exp (2 * π * I * (i / n))` for `i ∈ Finset.range n`.\n\n## Main declarations\n\n* `Complex.mem_rootsOfUnity`: the complex `n`-th roots of unity are exactly the\n  complex numbers of the form `exp (2 * π * I * (i / n))` for some `i < n`.\n* `Complex.card_rootsOfUnity`: the number of `n`-th roots of unity is exactly `n`.\n* `Complex.norm_rootOfUnity_eq_one`: A complex root of unity has norm `1`.\n\n-/\n\npublic section\n\n\nnamespace Complex\n\nopen Polynomial Real\n\nopen scoped Nat Real\n\ntheorem isPrimitiveRoot_exp_of_isCoprime (i : ℤ) (n : ℕ) (h0 : n ≠ 0) (hi : IsCoprime i n) :\n    IsPrimitiveRoot (exp (2 * π * I * (i / n))) n := by\n  rw [IsPrimitiveRoot.iff_def]\n  simp only [← exp_nat_mul, exp_eq_one_iff]\n  constructor\n  · use i\n    simp (discharger := norm_cast) [field]\n  · simp only [forall_exists_index]\n    have hn0 : (n : ℂ) ≠ 0 := mod_cast h0\n    rintro l k hk\n    field_simp at hk\n    norm_cast at hk\n    exact Int.natCast_dvd_natCast.mp <| hi.symm.dvd_of_dvd_mul_right <| hk ▸ dvd_mul_right ..\n\ntheorem isPrimitiveRoot_exp_of_coprime (i n : ℕ) (h0 : n ≠ 0) (hi : i.Coprime n) :\n    IsPrimitiveRoot (exp (2 * π * I * (i / n))) n :=\n  isPrimitiveRoot_exp_of_isCoprime _ _ h0 hi.isCoprime\n\ntheorem isPrimitiveRoot_exp_rat (q : ℚ) : IsPrimitiveRoot (exp (2 * π * I * q)) q.den := by\n  convert!\n    isPrimitiveRoot_exp_of_isCoprime _ _ q.den_nz <| Int.isCoprime_iff_nat_coprime.mpr q.reduced\n  nth_rw 1 [← Rat.num_div_den q]\n  simp\n\ntheorem isPrimitiveRoot_exp_rat_of_even_num (q : ℚ) (h : Even q.num) :\n    IsPrimitiveRoot (exp (π * I * q)) q.den := by\n  have ⟨n, hn⟩ := even_iff_exists_two_nsmul _ |>.mp h\n  convert! isPrimitiveRoot_exp_rat (n / q.den) using 1\n  · nth_rw 1 [← q.num_div_den, hn, Int.nsmul_eq_mul]\n    push_cast\n    ring_nf\n  · rw [← Int.cast_natCast, ← Rat.divInt_eq_div, ← Rat.mk_eq_divInt (nz := by simp)]\n    apply Nat.Coprime.coprime_mul_left (k := 2)\n    convert! q.reduced\n    grind\n\ntheorem isPrimitiveRoot_exp_rat_of_odd_num (q : ℚ) (h : Odd q.num) :\n    IsPrimitiveRoot (exp (π * I * q)) (2 * q.den) := by\n  convert! isPrimitiveRoot_exp_rat (q / 2) using 1\n  · push_cast\n    ring_nf\n  · nth_rw 2 [← q.num_div_den]\n    rw [mul_comm, div_div, ← Int.cast_ofNat, ← Int.cast_natCast, ← Int.cast_mul,\n      ← Rat.divInt_eq_div, ← Nat.cast_ofNat (R := ℤ), ← Nat.cast_mul,\n      ← Rat.mk_eq_divInt (nz := by simp)\n        (c := Nat.Coprime.mul_right q.reduced h.natAbs.coprime_two_right)]\n\ntheorem isPrimitiveRoot_exp (n : ℕ) (h0 : n ≠ 0) : IsPrimitiveRoot (exp (2 * π * I / n)) n := by\n  simpa only [Nat.cast_one, one_div] using!\n    isPrimitiveRoot_exp_of_coprime 1 n h0 n.coprime_one_left\n\ntheorem isPrimitiveRoot_iff (ζ : ℂ) (n : ℕ) (hn : n ≠ 0) :\n    IsPrimitiveRoot ζ n ↔ ∃ i < n, ∃ _ : i.Coprime n, exp (2 * π * I * (i / n)) = ζ := by\n  have hn0 : (n : ℂ) ≠ 0 := mod_cast hn\n  constructor; swap\n  · rintro ⟨i, -, hi, rfl⟩; exact isPrimitiveRoot_exp_of_coprime i n hn hi\n  intro h\n  have : NeZero n := ⟨hn⟩\n  obtain ⟨i, hi, rfl⟩ :=\n    (isPrimitiveRoot_exp n hn).eq_pow_of_pow_eq_one h.pow_eq_one\n  refine ⟨i, hi, ((isPrimitiveRoot_exp n hn).pow_iff_coprime (Nat.pos_of_ne_zero hn) i).mp h, ?_⟩\n  rw [← exp_nat_mul]\n  congr 1\n  ring\n\n/-- The complex `n`-th roots of unity are exactly the\ncomplex numbers of the form `exp (2 * Real.pi * Complex.I * (i / n))` for some `i < n`. -/\nnonrec theorem mem_rootsOfUnity (n : ℕ) [NeZero n] (x : Units ℂ) :\n    x ∈ rootsOfUnity n ℂ ↔ ∃ i < n, exp (2 * π * I * (i / n)) = x := by\n  rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one]\n  have hn0 : (n : ℂ) ≠ 0 := mod_cast NeZero.out\n  constructor\n  · intro h\n    obtain ⟨i, hi, H⟩ : ∃ i < (n : ℕ), exp (2 * π * I / n) ^ i = x := by\n      simpa only using (isPrimitiveRoot_exp n NeZero.out).eq_pow_of_pow_eq_one h\n    refine ⟨i, hi, ?_⟩\n    rw [← H, ← exp_nat_mul]\n    congr 1\n    ring\n  · rintro ⟨i, _, H⟩\n    rw [← H, ← exp_nat_mul, exp_eq_one_iff]\n    use i\n    simp [field]\n\ntheorem card_rootsOfUnity (n : ℕ) [NeZero n] : Fintype.card (rootsOfUnity n ℂ) = n :=\n  (isPrimitiveRoot_exp n NeZero.out).card_rootsOfUnity\n\nTarget:\ntheorem card_primitiveRoots (k : ℕ) : (primitiveRoots k ℂ).card = φ k :=\n\nProof body:\n","rejected":"```lean\nby\n  by_cases h : k = 0\n  · simp [h]\n  exact (isPrimitiveRoot_exp k h).card_primitiveRoots\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/RootsOfUnity","family_id":"card_primitiveroots","file_id":"mathlib/Mathlib/RingTheory/RootsOfUnity/Complex.lean","sample_id":"32b115c85276c93388c62a93ae6422efd911c7d0d06efd966d995ffe632b53e9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"475307141c3a1cbf5e416ce7878a4f3fd1a2e5cb9833e295c816f2bfea6746a8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8718a7560479506834cd432f1d1bbbcdc14349fe8b10b51da3d79c17a3f32f95","source_sha256":"f0e2b6563fb6bf505db894994fffc54566c37c1de08ac8bba92d16e8dd3bcc5d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [Fintype.subtype_card, Finset.card_univ]","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":2,"token_jaccard":0.090909,"token_length_ratio":0.181818},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"29fa3bda623fc371960080f054aee6783eafa62672d775562bf40f579f6b0950","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Powerset\npublic import Mathlib.Data.Fintype.EquivFin\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# fintype instance for `Set α`, when `α` is a fintype\n-/\n\npublic section\n\n\nvariable {α : Type*}\n\nopen Finset\n\ninstance Finset.fintype [Fintype α] : Fintype (Finset α) :=\n  ⟨univ.powerset, fun _ => Finset.mem_powerset.2 (Finset.subset_univ _)⟩\n\n@[simp]\ntheorem Fintype.card_finset [Fintype α] : Fintype.card (Finset α) = 2 ^ Fintype.card α :=\n  Finset.card_powerset Finset.univ\n\nnamespace Finset\nvariable [Fintype α] {s : Finset α} {k : ℕ}\n\n@[simp] lemma powerset_univ : (univ : Finset α).powerset = univ :=\n  coe_injective <| by simp [-coe_eq_univ]\n\nlemma filter_subset_univ [DecidableEq α] (s : Finset α) :\n    ({t | t ⊆ s} : Finset _) = powerset s := by ext; simp\n\n@[simp] lemma powerset_eq_univ : s.powerset = univ ↔ s = univ := by\n  rw [← Finset.powerset_univ, powerset_inj]\n\nlemma mem_powersetCard_univ : s ∈ powersetCard k (univ : Finset α) ↔ #s = k :=\n  mem_powersetCard.trans <| and_iff_right <| subset_univ _\n\nvariable (α)\n\n@[simp] lemma univ_filter_card_eq (k : ℕ) :\n    ({s | #s = k} : Finset (Finset α)) = univ.powersetCard k := by ext; simp\n\nend Finset\n\n@[simp]\n\nTarget:\ntheorem Fintype.card_finset_len [Fintype α] (k : ℕ) :\n    Fintype.card { s : Finset α // #s = k } = Nat.choose (Fintype.card α) k :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Fintype","family_id":"fintype","file_id":"mathlib/Mathlib/Data/Fintype/Powerset.lean","sample_id":"8718a7560479506834cd432f1d1bbbcdc14349fe8b10b51da3d79c17a3f32f95"},"split":"train"}
{"binding":{"chosen_proof_sha256":"83fc6e2c4e62e7f25a2d9e2ffddabb7d5875dc78574f118790a50383327ea178","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"30d6aae9bfaa67acea75dfbb91ae0fa520b66296064e76c798d32752d9029399","source_sha256":"d2377d09b51f8862ecd1e9604b616865bc29923d42ea22ca4ee787791f41d5e7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨n, hn : ∀ m, n ≤ m → LinearMap.ker (f ^ n) = LinearMap.ker (f ^ m)⟩ :=\n    monotone_stabilizes_iff_noetherian.mpr inferInstance f.iterateKer\n  refine eventually_atTop.mpr ⟨n, fun m hm ↦ disjoint_iff.mpr ?_⟩\n  rw [← hn _ hm, Submodule.eq_bot_iff]\n  rintro - ⟨hx, ⟨x, rfl⟩⟩\n  apply pow_map_zero_of_le hm\n  replace hx : x ∈ LinearMap.ker (f ^ (n + m)) := by\n    simpa [f.pow_apply n, f.pow_apply m, ← f.pow_apply (n + m), ← iterate_add_apply] using hx\n  rwa [← hn _ (n.le_add_right m)] at hx","hard_negative":false,"metrics":{"chosen_tokens":142,"rejected_tokens":5,"token_jaccard":0.033898,"token_length_ratio":0.035211},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"2a3bb5be608940175f348a1e168fe6c221ca05e0c1856d0932fde4e42006f85e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Filter.AtTopBot.Basic\npublic import Mathlib.RingTheory.Finiteness.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Kevin Buzzard\n-/\n/-!\n# Noetherian rings and modules\n\nThe following are equivalent for a module M over a ring R:\n1. Every increasing chain of submodules M₁ ⊆ M₂ ⊆ M₃ ⊆ ⋯ eventually stabilises.\n2. Every submodule is finitely generated.\n\nA module satisfying these equivalent conditions is said to be a *Noetherian* R-module.\nA ring is a *Noetherian ring* if it is Noetherian as a module over itself.\n\n(Note that we do not assume yet that our rings are commutative,\nso perhaps this should be called \"left-Noetherian\".\nTo avoid cumbersome names once we specialize to the commutative case,\nwe don't make this explicit in the declaration names.)\n\n## Main definitions\n\nLet `R` be a ring and let `M` and `P` be `R`-modules. Let `N` be an `R`-submodule of `M`.\n\n* `IsNoetherian R M` is the proposition that `M` is a Noetherian `R`-module. It is a class,\n  implemented as the predicate that all `R`-submodules of `M` are finitely generated.\n\n## Main statements\n\n* `isNoetherian_iff` is the theorem that an R-module M is Noetherian iff `>` is well-founded on\n  `Submodule R M`.\n\nNote that the Hilbert basis theorem, that if a commutative ring R is Noetherian then so is R[X],\nis proved in `RingTheory.Polynomial`.\n\n## References\n\n* [M. F. Atiyah and I. G. Macdonald, *Introduction to commutative algebra*][atiyah-macdonald]\n* [P. Samuel, *Algebraic Theory of Numbers*][samuel1967]\n\n## Tags\n\nNoetherian, noetherian, Noetherian ring, Noetherian module, noetherian ring, noetherian module\n\n-/\n\n@[expose] public section\n\nassert_not_exists Finsupp.linearCombination Matrix Pi.basis\n\nopen Set Pointwise\n\n/-- `IsNoetherian R M` is the proposition that `M` is a Noetherian `R`-module,\nimplemented as the predicate that all `R`-submodules of `M` are finitely generated.\n-/\n-- TODO: should this be renamed to `Noetherian`?\nclass IsNoetherian (R M) [Semiring R] [AddCommMonoid M] [Module R M] : Prop where\n  noetherian : ∀ s : Submodule R M, s.FG\n\nattribute [inherit_doc IsNoetherian] IsNoetherian.noetherian\n\nsection\n\nvariable {R : Type*} {M : Type*} {P : Type*}\nvariable [Semiring R] [AddCommMonoid M] [AddCommMonoid P]\nvariable [Module R M] [Module R P]\n\nopen IsNoetherian\n\n/-- An R-module is Noetherian iff all its submodules are finitely-generated. -/\ntheorem isNoetherian_def : IsNoetherian R M ↔ ∀ s : Submodule R M, s.FG :=\n  ⟨fun h => h.noetherian, IsNoetherian.mk⟩\n\ntheorem isNoetherian_submodule {N : Submodule R M} :\n    IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG := by\n  refine ⟨fun ⟨hn⟩ => fun s hs =>\n    have : s ≤ LinearMap.range N.subtype := N.range_subtype.symm ▸ hs\n    Submodule.map_comap_eq_self this ▸ (hn _).map _,\n    fun h => ⟨fun s => ?_⟩⟩\n  specialize h (s.map N.subtype) (Submodule.map_subtype_le N s)\n  exact Submodule.fg_of_fg_map_injective N.subtype Subtype.val_injective h\n\ntheorem isNoetherian_submodule_left {N : Submodule R M} :\n    IsNoetherian R N ↔ ∀ s : Submodule R M, (N ⊓ s).FG :=\n  isNoetherian_submodule.trans ⟨fun H _ => H _ inf_le_left, fun H _ hs => inf_of_le_right hs ▸ H _⟩\n\ntheorem isNoetherian_submodule_right {N : Submodule R M} :\n    IsNoetherian R N ↔ ∀ s : Submodule R M, (s ⊓ N).FG :=\n  isNoetherian_submodule.trans ⟨fun H _ => H _ inf_le_right, fun H _ hs => inf_of_le_left hs ▸ H _⟩\n\ninstance isNoetherian_submodule' [IsNoetherian R M] (N : Submodule R M) : IsNoetherian R N :=\n  isNoetherian_submodule.2 fun _ _ => IsNoetherian.noetherian _\n\ntheorem isNoetherian_of_le {s t : Submodule R M} [ht : IsNoetherian R t] (h : s ≤ t) :\n    IsNoetherian R s :=\n  isNoetherian_submodule.mpr fun _ hs' => isNoetherian_submodule.mp ht _ (le_trans hs' h)\n\nend\n\nopen IsNoetherian Submodule Function\n\nsection\n\nuniverse w\n\nvariable {R M P : Type*} {N : Type w} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N]\n  [Module R N] [AddCommMonoid P] [Module R P]\n\ntheorem isNoetherian_iff' : IsNoetherian R M ↔ WellFoundedGT (Submodule R M) := by\n  refine .trans ?_ ((CompleteLattice.wellFoundedGT_characterisations <| Submodule R M).out 0 3).symm\n  exact\n    ⟨fun ⟨h⟩ => fun k => (fg_iff_compact k).mp (h k), fun h =>\n      ⟨fun k => (fg_iff_compact k).mpr (h k)⟩⟩\n\ntheorem isNoetherian_iff :\n    IsNoetherian R M ↔ WellFounded ((· > ·) : Submodule R M → Submodule R M → Prop) := by\n  rw [isNoetherian_iff', ← isWellFounded_iff]\n\nalias ⟨IsNoetherian.wf, _⟩ := isNoetherian_iff\n\nalias ⟨IsNoetherian.wellFoundedGT, isNoetherian_mk⟩ := isNoetherian_iff'\n\ninstance wellFoundedGT [h : IsNoetherian R M] : WellFoundedGT (Submodule R M) :=\n  h.wellFoundedGT\n\ntheorem isNoetherian_iff_fg_wellFounded :\n    IsNoetherian R M ↔ WellFoundedGT { N : Submodule R M // N.FG } := by\n  let α := { N : Submodule R M // N.FG }\n  constructor\n  · intro H\n    let f : α ↪o Submodule R M := OrderEmbedding.subtype _\n    exact OrderEmbedding.wellFoundedLT f.dual\n  · intro H\n    constructor\n    intro N\n    obtain ⟨⟨N₀, h₁⟩, e : N₀ ≤ N, h₂⟩ :=\n      WellFounded.has_min H.wf { N' : α | N'.1 ≤ N } ⟨⟨⊥, Submodule.fg_bot⟩, @bot_le _ _ _ N⟩\n    convert! h₁\n    refine (e.antisymm ?_).symm\n    by_contra h₃\n    obtain ⟨x, hx₁ : x ∈ N, hx₂ : x ∉ N₀⟩ := Set.not_subset.mp h₃\n    apply hx₂\n    rw [eq_of_le_of_not_lt (le_sup_right : N₀ ≤ _) (h₂\n      ⟨_, Submodule.FG.sup ⟨{x}, by rw [Finset.coe_singleton]⟩ h₁⟩ <|\n      sup_le ((Submodule.span_singleton_le_iff_mem _ _).mpr hx₁) e)]\n    exact (le_sup_left : R ∙ x ≤ _) (Submodule.mem_span_singleton_self _)\n\n/-- A module is Noetherian iff every nonempty set of submodules has a maximal submodule among them.\n-/\ntheorem set_has_maximal_iff_noetherian :\n    (∀ a : Set <| Submodule R M, a.Nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬M' < I) ↔ IsNoetherian R M := by\n  rw [isNoetherian_iff, WellFounded.wellFounded_iff_has_min]\n\n/-- A module is Noetherian iff every increasing chain of submodules stabilizes. -/\ntheorem monotone_stabilizes_iff_noetherian :\n    (∀ f : ℕ →o Submodule R M, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsNoetherian R M := by\n  rw [isNoetherian_iff', wellFoundedGT_iff_monotone_chain_condition]\n\nvariable [IsNoetherian R M]\n\nopen Filter\n/-- For an endomorphism of a Noetherian module, any sufficiently large iterate has disjoint kernel\nand range. -/\n\nTarget:\ntheorem Module.End.eventually_disjoint_ker_pow_range_pow (f : End R M) :\n    ∀ᶠ n in atTop, Disjoint (LinearMap.ker (f ^ n)) (LinearMap.range (f ^ n)) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Noetherian","family_id":"module","file_id":"mathlib/Mathlib/RingTheory/Noetherian/Defs.lean","sample_id":"30d6aae9bfaa67acea75dfbb91ae0fa520b66296064e76c798d32752d9029399"},"split":"train"}
{"binding":{"chosen_proof_sha256":"00de7867ae9086f2cf9c529fdfa5d87816a8332564eca883373884a7662e8cc7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"774c673a4deeecd41f98949f24cac76430a29b067534196cc6ad50eb5f15d0e9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7d66ee4f35ab6035f8dacc7a5cf8575522e26c3c67988759a6c4ebffd4318014","source_sha256":"d494c9de8c64f10c69db3a3e7c2cc9feffbd36585bacb5b41e8c3dcb64729d9b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [rowLen, Nat.lt_find_iff]\n  push Not\n  exact ⟨fun h _ hmj => μ.up_left_mem (by rfl) hmj h, fun h => h _ (by rfl)⟩","hard_negative":true,"metrics":{"chosen_tokens":38,"rejected_tokens":2,"token_jaccard":0.04,"token_length_ratio":0.052632},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"2a59de97015c1f1b510526fc5ab69a98b769bd65ede3133fcecd438a6e365b5c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Preimage\npublic import Mathlib.Data.Finset.Prod\npublic import Mathlib.Data.SetLike.Basic\npublic import Mathlib.Order.UpperLower.Basic\n\nNamespace:\nYoungDiagram\n\nLocal context:\n/-\nCopyright (c) 2022 Jake Levinson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jake Levinson\n-/\n/-!\n# Young diagrams\n\nA Young diagram is a finite set of up-left justified boxes:\n\n```text\n□□□□□\n□□□\n□□□\n□\n```\nThis Young diagram corresponds to the [5, 3, 3, 1] partition of 12.\n\nWe represent it as a lower set in `ℕ × ℕ` in the product partial order. We write `(i, j) ∈ μ`\nto say that `(i, j)` (in matrix coordinates) is in the Young diagram `μ`.\n\n## Main definitions\n\n- `YoungDiagram` : Young diagrams\n- `YoungDiagram.card` : the number of cells in a Young diagram (its *cardinality*)\n- `YoungDiagram.instDistribLatticeYoungDiagram` : a distributive lattice instance for Young diagrams\n  ordered by containment, with `(⊥ : YoungDiagram)` the empty diagram.\n- `YoungDiagram.row` and `YoungDiagram.rowLen`: rows of a Young diagram and their lengths\n- `YoungDiagram.col` and `YoungDiagram.colLen`: columns of a Young diagram and their lengths\n\n## Notation\n\nIn \"English notation\", a Young diagram is drawn so that (i1, j1) ≤ (i2, j2)\nmeans (i1, j1) is weakly up-and-left of (i2, j2). This terminology is used\nbelow, e.g. in `YoungDiagram.up_left_mem`.\n\n## Tags\n\nYoung diagram\n\n## References\n\n<https://en.wikipedia.org/wiki/Young_tableau>\n\n-/\n\n@[expose] public section\n\n\nopen Function\n\n/-- A Young diagram is a finite collection of cells on the `ℕ × ℕ` grid such that whenever\na cell is present, so are all the ones above and to the left of it. Like matrices, an `(i, j)` cell\nis a cell in row `i` and column `j`, where rows are enumerated downward and columns rightward.\n\nYoung diagrams are modeled as finite sets in `ℕ × ℕ` that are lower sets with respect to the\nstandard order on products. -/\n@[ext]\nstructure YoungDiagram where\n  /-- A finite set which represents a finite collection of cells on the `ℕ × ℕ` grid. -/\n  cells : Finset (ℕ × ℕ)\n  /-- Cells are up-left justified, witnessed by the fact that `cells` is a lower set in `ℕ × ℕ`. -/\n  isLowerSet : IsLowerSet (cells : Set (ℕ × ℕ))\n\nnamespace YoungDiagram\n\ninstance : SetLike YoungDiagram (ℕ × ℕ) where\n  coe y := y.cells\n  coe_injective μ ν h := by rwa [YoungDiagram.ext_iff, ← Finset.coe_inj]\n\ninstance : PartialOrder YoungDiagram := .ofSetLike YoungDiagram (ℕ × ℕ)\n\n@[simp]\ntheorem mem_cells {μ : YoungDiagram} (c : ℕ × ℕ) : c ∈ μ.cells ↔ c ∈ μ :=\n  Iff.rfl\n\n@[simp]\ntheorem mem_mk (c : ℕ × ℕ) (cells) (isLowerSet) :\n    c ∈ YoungDiagram.mk cells isLowerSet ↔ c ∈ cells :=\n  Iff.rfl\n\ninstance decidableMem (μ : YoungDiagram) : DecidablePred (· ∈ μ) :=\n  inferInstanceAs (DecidablePred (· ∈ μ.cells))\n\n/-- In \"English notation\", a Young diagram is drawn so that (i1, j1) ≤ (i2, j2)\nmeans (i1, j1) is weakly up-and-left of (i2, j2). -/\ntheorem up_left_mem (μ : YoungDiagram) {i1 i2 j1 j2 : ℕ} (hi : i1 ≤ i2) (hj : j1 ≤ j2)\n    (hcell : (i2, j2) ∈ μ) : (i1, j1) ∈ μ :=\n  μ.isLowerSet (Prod.mk_le_mk.mpr ⟨hi, hj⟩) hcell\n\nsection DistribLattice\n\n@[simp]\ntheorem cells_subset_iff {μ ν : YoungDiagram} : μ.cells ⊆ ν.cells ↔ μ ≤ ν :=\n  Iff.rfl\n\n@[simp]\ntheorem cells_ssubset_iff {μ ν : YoungDiagram} : μ.cells ⊂ ν.cells ↔ μ < ν :=\n  Iff.rfl\n\ninstance : Max YoungDiagram where\n  max μ ν :=\n    { cells := μ.cells ∪ ν.cells\n      isLowerSet := by\n        rw [Finset.coe_union]\n        exact μ.isLowerSet.union ν.isLowerSet }\n\n@[simp]\ntheorem cells_sup (μ ν : YoungDiagram) : (μ ⊔ ν).cells = μ.cells ∪ ν.cells :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sup (μ ν : YoungDiagram) : ↑(μ ⊔ ν) = (μ ∪ ν : Set (ℕ × ℕ)) :=\n  Finset.coe_union _ _\n\n@[simp]\ntheorem mem_sup {μ ν : YoungDiagram} {x : ℕ × ℕ} : x ∈ μ ⊔ ν ↔ x ∈ μ ∨ x ∈ ν :=\n  Finset.mem_union\n\ninstance : Min YoungDiagram where\n  min μ ν :=\n    { cells := μ.cells ∩ ν.cells\n      isLowerSet := by\n        rw [Finset.coe_inter]\n        exact μ.isLowerSet.inter ν.isLowerSet }\n\n@[simp]\ntheorem cells_inf (μ ν : YoungDiagram) : (μ ⊓ ν).cells = μ.cells ∩ ν.cells :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_inf (μ ν : YoungDiagram) : ↑(μ ⊓ ν) = (μ ∩ ν : Set (ℕ × ℕ)) :=\n  Finset.coe_inter _ _\n\n@[simp]\ntheorem mem_inf {μ ν : YoungDiagram} {x : ℕ × ℕ} : x ∈ μ ⊓ ν ↔ x ∈ μ ∧ x ∈ ν :=\n  Finset.mem_inter\n\n/-- The empty Young diagram is `(⊥ : YoungDiagram)`. -/\ninstance : OrderBot YoungDiagram where\n  bot :=\n    { cells := ∅\n      isLowerSet := by\n        intro a b _ h\n        simp only [Finset.coe_empty, Set.mem_empty_iff_false]\n        simp only [Finset.coe_empty, Set.mem_empty_iff_false] at h }\n  bot_le _ _ := by\n    intro y\n    simp only [mem_mk, Finset.notMem_empty] at y\n\n@[simp]\ntheorem cells_bot : (⊥ : YoungDiagram).cells = ∅ :=\n  rfl\n\n@[simp]\ntheorem notMem_bot (x : ℕ × ℕ) : x ∉ (⊥ : YoungDiagram) :=\n  Finset.notMem_empty x\n\n@[norm_cast]\ntheorem coe_bot : (⊥ : YoungDiagram) = (∅ : Set (ℕ × ℕ)) := by\n  ext; simp\n\ninstance : Inhabited YoungDiagram :=\n  ⟨⊥⟩\n\ninstance : DistribLattice YoungDiagram :=\n  Function.Injective.distribLattice YoungDiagram.cells (fun μ ν h ↦ by rwa [YoungDiagram.ext_iff])\n    .rfl .rfl (fun _ _ ↦ rfl) fun _ _ ↦ rfl\n\nend DistribLattice\n\n/-- Cardinality of a Young diagram -/\nprotected abbrev card (μ : YoungDiagram) : ℕ :=\n  μ.cells.card\n\nsection Transpose\n\n/-- The `transpose` of a Young diagram is obtained by swapping i's with j's. -/\ndef transpose (μ : YoungDiagram) : YoungDiagram where\n  cells := (Equiv.prodComm _ _).finsetCongr μ.cells\n  isLowerSet _ _ h := by\n    simp only [Finset.mem_coe, Equiv.finsetCongr_apply, Finset.mem_map_equiv]\n    intro hcell\n    apply μ.isLowerSet _ hcell\n    simp [h]\n\n@[simp]\ntheorem mem_transpose {μ : YoungDiagram} {c : ℕ × ℕ} : c ∈ μ.transpose ↔ c.swap ∈ μ := by\n  simp [transpose]\n\n@[simp]\ntheorem transpose_transpose (μ : YoungDiagram) : μ.transpose.transpose = μ := by\n  ext x\n  simp\n\ntheorem transpose_eq_iff_eq_transpose {μ ν : YoungDiagram} : μ.transpose = ν ↔ μ = ν.transpose := by\n  constructor <;>\n    · rintro rfl\n      simp\n\n@[simp]\ntheorem transpose_eq_iff {μ ν : YoungDiagram} : μ.transpose = ν.transpose ↔ μ = ν := by\n  rw [transpose_eq_iff_eq_transpose]\n  simp\n\n-- This is effectively both directions of `transpose_le_iff` below.\nprotected theorem le_of_transpose_le {μ ν : YoungDiagram} (h_le : μ.transpose ≤ ν) :\n    μ ≤ ν.transpose := fun c hc => by\n  simp only [mem_transpose]\n  apply h_le\n  simpa\n\n@[simp]\ntheorem transpose_le_iff {μ ν : YoungDiagram} : μ.transpose ≤ ν.transpose ↔ μ ≤ ν :=\n  ⟨fun h => by\n    convert! YoungDiagram.le_of_transpose_le h\n    simp, fun h => by\n    rw [← transpose_transpose μ] at h\n    exact YoungDiagram.le_of_transpose_le h ⟩\n\n@[gcongr, mono]\nprotected theorem transpose_mono {μ ν : YoungDiagram} (h_le : μ ≤ ν) : μ.transpose ≤ ν.transpose :=\n  transpose_le_iff.mpr h_le\n\n/-- Transposing Young diagrams is an `OrderIso`. -/\n@[simps]\ndef transposeOrderIso : YoungDiagram ≃o YoungDiagram :=\n  ⟨⟨transpose, transpose, fun _ => by simp, fun _ => by simp⟩, by simp⟩\n\nend Transpose\n\nsection Rows\n\n/-! ### Rows and row lengths of Young diagrams.\n\nThis section defines `μ.row` and `μ.rowLen`, with the following API:\n      1.  `(i, j) ∈ μ ↔ j < μ.rowLen i`\n      2.  `μ.row i = {i} ×ˢ (Finset.range (μ.rowLen i))`\n      3.  `μ.rowLen i = (μ.row i).card`\n      4.  `∀ {i1 i2}, i1 ≤ i2 → μ.rowLen i2 ≤ μ.rowLen i1`\n\nNote: #3 is not convenient for defining `μ.rowLen`; instead, `μ.rowLen` is defined\nas the smallest `j` such that `(i, j) ∉ μ`. -/\n\n\n/-- The `i`-th row of a Young diagram consists of the cells whose first coordinate is `i`. -/\ndef row (μ : YoungDiagram) (i : ℕ) : Finset (ℕ × ℕ) :=\n  μ.cells.filter fun c => c.fst = i\n\ntheorem mem_row_iff {μ : YoungDiagram} {i : ℕ} {c : ℕ × ℕ} : c ∈ μ.row i ↔ c ∈ μ ∧ c.fst = i := by\n  simp [row]\n\ntheorem mk_mem_row_iff {μ : YoungDiagram} {i j : ℕ} : (i, j) ∈ μ.row i ↔ (i, j) ∈ μ := by simp [row]\n\nprotected theorem exists_notMem_row (μ : YoungDiagram) (i : ℕ) : ∃ j, (i, j) ∉ μ := by\n  obtain ⟨j, hj⟩ :=\n    Infinite.exists_notMem_finset\n      (μ.cells.preimage (Prod.mk i) fun _ _ _ _ h => by\n        cases h\n        rfl)\n  rw [Finset.mem_preimage] at hj\n  exact ⟨j, hj⟩\n\n/-- Length of a row of a Young diagram -/\ndef rowLen (μ : YoungDiagram) (i : ℕ) : ℕ :=\n  Nat.find <| μ.exists_notMem_row i\n\nTarget:\ntheorem mem_iff_lt_rowLen {μ : YoungDiagram} {i j : ℕ} : (i, j) ∈ μ ↔ j < μ.rowLen i :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_7d66ee4f35ab","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"14f5edffae279320cebdad57bda0f496e1ad8d439457a621bc0f327193d068cd","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Young","family_id":"mem_iff_lt_rowlen","file_id":"mathlib/Mathlib/Combinatorics/Young/YoungDiagram.lean","sample_id":"7d66ee4f35ab6035f8dacc7a5cf8575522e26c3c67988759a6c4ebffd4318014"},"split":"train"}
{"binding":{"chosen_proof_sha256":"90db882f63b60882a9cc2eecf39977196f2e1c4d889d00ccb138dc84cbbbe60a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4ed39d9450f827e035b841d5a8626fbdbc747601015b7cb760148514ffb7149a","source_sha256":"9a532fef3a07582dfc854d1c03bc13ebe1db0f3d339ef6b3416a8baedc6e0ea2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [trop_add_def, trop_eq_iff_eq_untrop]\n  simp [min_eq_iff]","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":5,"token_jaccard":0.076923,"token_length_ratio":0.454545},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"2a85e4f910706c9db96e2f403364c74758f2445b5ca5e43864246d349f960c76","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.AddGroupWithTop\npublic import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax\npublic import Mathlib.Algebra.Order.Monoid.Unbundled.Pow\npublic import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Order.Hom.Basic\n\nNamespace:\nTropical\n\nLocal context:\n/-\nCopyright (c) 2021 Yakov Pechersky. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yakov Pechersky\n-/\n/-!\n\n# Tropical algebraic structures\n\nThis file defines algebraic structures of the (min-)tropical numbers, up to the tropical semiring.\nSome basic lemmas about conversion from the base type `R` to `Tropical R` are provided, as\nwell as the expected implementations of tropical addition and tropical multiplication.\n\n## Main declarations\n\n* `Tropical R`: The type synonym of the tropical interpretation of `R`.\n    If `[LinearOrder R]`, then addition on `R` is via `min`.\n* `Semiring (Tropical R)`: A `LinearOrderedAddCommMonoidWithTop R`\n    induces a `Semiring (Tropical R)`. If one solely has `[LinearOrderedAddCommMonoid R]`,\n    then the \"tropicalization of `R`\" would be `Tropical (WithTop R)`.\n\n## Implementation notes\n\nThe tropical structure relies on `Top` and `min`. For the max-tropical numbers, use\n`OrderDual R`.\n\nInspiration was drawn from the implementation of `Additive`/`Multiplicative`/`Opposite`,\nwhere a type synonym is created with some barebones API, and quickly made irreducible.\n\nAlgebraic structures are provided with as few typeclass assumptions as possible, even though\nmost references rely on `Semiring (Tropical R)` for building up the whole theory.\n\n## References followed\n\n* https://arxiv.org/pdf/math/0408099.pdf\n* https://www.mathenjeans.fr/sites/default/files/sujets/tropical_geometry_-_casagrande.pdf\n\n-/\n\n@[expose] public section\n\nassert_not_exists Nat.instMulOneClass\n\nuniverse u v\n\nvariable (R : Type u)\n\n/-- The tropicalization of a type `R`. -/\ndef Tropical : Type u :=\n  R\n\nvariable {R}\n\nnamespace Tropical\n\n/-- Reinterpret `x : R` as an element of `Tropical R`.\nSee `Tropical.tropEquiv` for the equivalence.\n-/\ndef trop : R → Tropical R :=\n  id\n\n/-- Reinterpret `x : Tropical R` as an element of `R`.\nSee `Tropical.tropEquiv` for the equivalence. -/\n@[pp_nodot]\ndef untrop : Tropical R → R :=\n  id\n\ntheorem trop_injective : Function.Injective (trop : R → Tropical R) := fun _ _ => id\n\ntheorem untrop_injective : Function.Injective (untrop : Tropical R → R) := fun _ _ => id\n\n@[simp]\ntheorem trop_inj_iff (x y : R) : trop x = trop y ↔ x = y :=\n  Iff.rfl\n\n@[simp]\ntheorem untrop_inj_iff (x y : Tropical R) : untrop x = untrop y ↔ x = y :=\n  Iff.rfl\n\n@[simp]\ntheorem trop_untrop (x : Tropical R) : trop (untrop x) = x :=\n  rfl\n\n@[simp]\ntheorem untrop_trop (x : R) : untrop (trop x) = x :=\n  rfl\n\nattribute [irreducible] Tropical\n\ntheorem leftInverse_trop : Function.LeftInverse (trop : R → Tropical R) untrop :=\n  trop_untrop\n\ntheorem rightInverse_trop : Function.RightInverse (trop : R → Tropical R) untrop :=\n  untrop_trop\n\n/-- Reinterpret `x : R` as an element of `Tropical R`.\nSee `Tropical.tropOrderIso` for the order-preserving equivalence. -/\ndef tropEquiv : R ≃ Tropical R where\n  toFun := trop\n  invFun := untrop\n  left_inv := untrop_trop\n  right_inv := trop_untrop\n\n@[simp]\ntheorem tropEquiv_coe_fn : (tropEquiv : R → Tropical R) = trop :=\n  rfl\n\n@[simp]\ntheorem tropEquiv_symm_coe_fn : (tropEquiv.symm : Tropical R → R) = untrop :=\n  rfl\n\ntheorem trop_eq_iff_eq_untrop {x : R} {y} : trop x = y ↔ x = untrop y :=\n  tropEquiv.apply_eq_iff_eq_symm_apply\n\ntheorem untrop_eq_iff_eq_trop {x} {y : R} : untrop x = y ↔ x = trop y :=\n  tropEquiv.symm.apply_eq_iff_eq_symm_apply\n\ntheorem injective_trop : Function.Injective (trop : R → Tropical R) :=\n  tropEquiv.injective\n\ntheorem injective_untrop : Function.Injective (untrop : Tropical R → R) :=\n  tropEquiv.symm.injective\n\ntheorem surjective_trop : Function.Surjective (trop : R → Tropical R) :=\n  tropEquiv.surjective\n\ntheorem surjective_untrop : Function.Surjective (untrop : Tropical R → R) :=\n  tropEquiv.symm.surjective\n\ninstance [Inhabited R] : Inhabited (Tropical R) :=\n  ⟨trop default⟩\n\n/-- Recursing on an `x' : Tropical R` is the same as recursing on an `x : R` reinterpreted\nas a term of `Tropical R` via `trop x`. -/\n@[simp]\ndef tropRec {F : Tropical R → Sort v} (h : ∀ X, F (trop X)) : ∀ X, F X := fun X => h (untrop X)\n\ninstance [DecidableEq R] : DecidableEq (Tropical R) := fun _ _ =>\n  decidable_of_iff _ injective_untrop.eq_iff\n\nsection Order\n\ninstance instLETropical [LE R] : LE (Tropical R) where le x y := untrop x ≤ untrop y\n\n@[simp]\ntheorem untrop_le_iff [LE R] {x y : Tropical R} : untrop x ≤ untrop y ↔ x ≤ y :=\n  Iff.rfl\n\ninstance decidableLE [LE R] [DecidableLE R] : DecidableLE (Tropical R) := fun x y =>\n  ‹DecidableLE R› (untrop x) (untrop y)\n\ninstance instLTTropical [LT R] : LT (Tropical R) where lt x y := untrop x < untrop y\n\n@[simp]\ntheorem untrop_lt_iff [LT R] {x y : Tropical R} : untrop x < untrop y ↔ x < y :=\n  Iff.rfl\n\ninstance decidableLT [LT R] [DecidableLT R] : DecidableLT (Tropical R) := fun x y =>\n  ‹DecidableLT R› (untrop x) (untrop y)\n\ninstance instPreorderTropical [Preorder R] : Preorder (Tropical R) :=\n  { instLETropical, instLTTropical with\n    le_refl := fun x => le_refl (untrop x)\n    le_trans := fun _ _ _ h h' => le_trans (α := R) h h'\n    lt_iff_le_not_ge := fun _ _ => lt_iff_le_not_ge (α := R) }\n\n/-- Reinterpret `x : R` as an element of `Tropical R`, preserving the order. -/\ndef tropOrderIso [Preorder R] : R ≃o Tropical R :=\n  { tropEquiv with map_rel_iff' := untrop_le_iff }\n\n@[simp]\ntheorem tropOrderIso_coe_fn [Preorder R] : (tropOrderIso : R → Tropical R) = trop :=\n  rfl\n\n@[simp]\ntheorem tropOrderIso_symm_coe_fn [Preorder R] : (tropOrderIso.symm : Tropical R → R) = untrop :=\n  rfl\n\ntheorem trop_monotone [Preorder R] : Monotone (trop : R → Tropical R) := fun _ _ => id\n\ntheorem untrop_monotone [Preorder R] : Monotone (untrop : Tropical R → R) := fun _ _ => id\n\ninstance instPartialOrderTropical [PartialOrder R] : PartialOrder (Tropical R) :=\n  { instPreorderTropical with le_antisymm := fun _ _ h h' => untrop_injective (le_antisymm h h') }\n\ninstance instZeroTropical [Top R] : Zero (Tropical R) :=\n  ⟨trop ⊤⟩\n\ninstance instTopTropical [Top R] : Top (Tropical R) :=\n  ⟨0⟩\n\n@[simp]\ntheorem untrop_zero [Top R] : untrop (0 : Tropical R) = ⊤ :=\n  rfl\n\n@[simp]\ntheorem trop_top [Top R] : trop (⊤ : R) = 0 :=\n  rfl\n\n@[simp]\ntheorem trop_coe_ne_zero (x : R) : trop (x : WithTop R) ≠ 0 :=\n  nofun\n\n@[simp]\ntheorem zero_ne_trop_coe (x : R) : 0 ≠ (trop x : Tropical (WithTop R)) :=\n  nofun\n\n@[simp]\ntheorem le_zero [LE R] [OrderTop R] (x : Tropical R) : x ≤ 0 :=\n  le_top (α := R)\n\ninstance [LE R] [OrderTop R] : OrderTop (Tropical R) :=\n  { instTopTropical with le_top := fun _ => le_top (α := R) }\n\nvariable [LinearOrder R]\n\n/-- Tropical addition is the minimum of two underlying elements of `R`. -/\ninstance : Add (Tropical R) :=\n  ⟨fun x y => trop (min (untrop x) (untrop y))⟩\n\ninstance instAddCommSemigroupTropical : AddCommSemigroup (Tropical R) where\n  add_assoc _ _ _ := untrop_injective (min_assoc _ _ _)\n  add_comm _ _ := untrop_injective (min_comm _ _)\n\n@[simp]\ntheorem untrop_add (x y : Tropical R) : untrop (x + y) = min (untrop x) (untrop y) :=\n  rfl\n\n@[simp]\ntheorem trop_min (x y : R) : trop (min x y) = trop x + trop y :=\n  rfl\n\n@[simp]\ntheorem trop_inf (x y : R) : trop (x ⊓ y) = trop x + trop y :=\n  rfl\n\ntheorem trop_add_def (x y : Tropical R) : x + y = trop (min (untrop x) (untrop y)) :=\n  rfl\n\ninstance instLinearOrderTropical : LinearOrder (Tropical R) :=\n  { instPartialOrderTropical with\n    le_total := fun a b => le_total (untrop a) (untrop b)\n    toDecidableLE := Tropical.decidableLE\n    toDecidableEq := Tropical.instDecidableEq\n    toDecidableLT := Tropical.decidableLT\n    max := fun a b => trop (max (untrop a) (untrop b))\n    max_def := fun a b => untrop_injective (by\n      simp only [max_def, untrop_le_iff, untrop_trop]; split_ifs <;> simp)\n    min := (· + ·)\n    min_def := fun a b => untrop_injective (by\n      simp only [untrop_add, min_def, untrop_le_iff]; split_ifs <;> simp) }\n\n@[simp]\ntheorem untrop_sup (x y : Tropical R) : untrop (x ⊔ y) = untrop x ⊔ untrop y :=\n  rfl\n\n@[simp]\ntheorem untrop_max (x y : Tropical R) : untrop (max x y) = max (untrop x) (untrop y) :=\n  rfl\n\n@[simp]\ntheorem min_eq_add : (min : Tropical R → Tropical R → Tropical R) = (· + ·) :=\n  rfl\n\n@[simp]\ntheorem inf_eq_add : ((· ⊓ ·) : Tropical R → Tropical R → Tropical R) = (· + ·) :=\n  rfl\n\ntheorem trop_max_def (x y : Tropical R) : max x y = trop (max (untrop x) (untrop y)) :=\n  rfl\n\ntheorem trop_sup_def (x y : Tropical R) : x ⊔ y = trop (untrop x ⊔ untrop y) :=\n  rfl\n\n@[simp]\ntheorem add_eq_left ⦃x y : Tropical R⦄ (h : x ≤ y) : x + y = x :=\n  untrop_injective (by simpa using h)\n\n@[simp]\ntheorem add_eq_right ⦃x y : Tropical R⦄ (h : y ≤ x) : x + y = y :=\n  untrop_injective (by simpa using h)\n\ntheorem add_eq_left_iff {x y : Tropical R} : x + y = x ↔ x ≤ y := by\n  rw [trop_add_def, trop_eq_iff_eq_untrop, ← untrop_le_iff, min_eq_left_iff]\n\ntheorem add_eq_right_iff {x y : Tropical R} : x + y = y ↔ y ≤ x := by\n  rw [trop_add_def, trop_eq_iff_eq_untrop, ← untrop_le_iff, min_eq_right_iff]\n\ntheorem add_self (x : Tropical R) : x + x = x :=\n  untrop_injective (min_eq_right le_rfl)\n\nTarget:\ntheorem add_eq_iff {x y z : Tropical R} : x + y = z ↔ x = z ∧ x ≤ y ∨ y = z ∧ y ≤ x :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Tropical","family_id":"add_eq_iff","file_id":"mathlib/Mathlib/Algebra/Tropical/Basic.lean","sample_id":"4ed39d9450f827e035b841d5a8626fbdbc747601015b7cb760148514ffb7149a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9a5efd93ec86ca2b78ac7e681d48ea70e75ceb2212f655294f2322b5b525cad8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"461385a2074d603323a793c95b869b667bfdb9c71b9da493fdf5249f006d209b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"aad3a3baeb6dacb034a541f0080f4a6bbc685cc200a408a1414f15d0a2cae0b7","source_sha256":"129f2f3aa776146d58955f965fc598f57556bb308d25e687562d21d0c89b2ff2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply isCyclic_of_card_dvd_prime (p := 2)\n  simpa [card_perm] using factorial_dvd_factorial hα","hard_negative":true,"metrics":{"chosen_tokens":16,"rejected_tokens":3,"token_jaccard":0.055556,"token_length_ratio":0.1875},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"2ab73f6131c86d470dc932cd10d46dfeadc70036eec08a87d447e6256ec412f8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Perm\npublic import Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic\npublic import Mathlib.SetTheory.Cardinal.Finite\n\nNamespace:\nEquiv.Perm\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir\n-/\n/-! # Properties of `Equiv.Perm` on `Finite` types\n\nLet `α` be a `Finite` type.\n\n* `Nat.card_perm`: cardinality of `Equiv.Perm α`.\n\n* `Equiv.Perm.isCyclic_of_card_le_two`: if `Nat.card α ≤ 2`,\n  then `Equiv.Perm α` is cyclic.\n\n* `Equiv.Perm.isCyclic_iff_card_le_two`: `Equiv.Perm α` is cyclic iff `Nat.card α ≤ 2`.\n\n* `Equiv.Perm.isMulCommutative_iff_card_le_two`: `Equiv.Perm α` is commutative iff `Nat.card α ≤ 2`.\n\n-/\n\npublic section\n\nassert_not_exists Field\n\nopen Equiv Nat\n\nvariable {α : Type*} [Finite α]\n\nnamespace Nat\n\ntheorem card_perm : Nat.card (Perm α) = (Nat.card α)! := by\n  classical\n  have := Fintype.ofFinite α\n  rw [card_eq_fintype_card, card_eq_fintype_card, Fintype.card_perm]\n\nend Nat\n\nnamespace Equiv.Perm\n\nTarget:\ntheorem isCyclic_of_card_le_two (hα : Nat.card α ≤ 2) :\n    IsCyclic (Perm α) :=\n\nProof body:\n","rejected":"by\n  exact isCyclic_of_card_le_two","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"9155e4475d081d0fb00927bdf2189664c909f701deba7ff97f1e82d130d74d32","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finite","family_id":"iscyclic_of_card_le_two","file_id":"mathlib/Mathlib/Data/Finite/Perm.lean","sample_id":"aad3a3baeb6dacb034a541f0080f4a6bbc685cc200a408a1414f15d0a2cae0b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3fc07c2c4acb95dfe246db6ffb6277493987fabf1c2b630f887c08fb23242d47","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1ab3415365954ac703a89b50238d1e7749b875697a54dc974228c428e68d72cc","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"879654c433cff299cfe89ef259733d4899e296266c7758cbde162533660dcf70","source_sha256":"d6accbe9b37eb545d619e89f4675da5baa0eb79327269932bbf60850434ef205","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.toShHom\n    g.toShHom)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]\n  erw [← TopCat.coe_comp]\n  rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]\n  dsimp only [SheafedSpace.forget]\n  rw [imageBasicOpen_image_preimage]\n  exact (imageBasicOpen f g U s).2","hard_negative":true,"metrics":{"chosen_tokens":75,"rejected_tokens":3,"token_jaccard":0.054054,"token_length_ratio":0.04},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"2ac05a5727ef4c514e115bbf3c8edae488f8e3e0c0319f738144097a4be32887","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Geometry.RingedSpace.LocallyRingedSpace\npublic import Mathlib.Algebra.Category.Ring.Constructions\npublic import Mathlib.Geometry.RingedSpace.OpenImmersion\npublic import Mathlib.CategoryTheory.Limits.Types.Coequalizers\npublic import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers\n\nNamespace:\nAlgebraicGeometry.LocallyRingedSpace.HasCoequalizer\n\nLocal context:\n/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Colimits of LocallyRingedSpace\n\nWe construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.\nIt then follows that `LocallyRingedSpace` has all colimits, and\n`forgetToSheafedSpace` preserves them.\n\n-/\n\n@[expose] public section\n\n\nnamespace AlgebraicGeometry\n\nuniverse w' w v u\n\nopen CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace\n\nattribute [local instance] Opposite.small\n\nnamespace SheafedSpace\n\nvariable {C : Type u} [Category.{v} C]\nvariable {J : Type w} [Category.{w'} J] [Small.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)\n\ntheorem isColimit_exists_rep [HasLimitsOfShape Jᵒᵖ C] {c : Cocone F} (hc : IsColimit c) (x : c.pt) :\n    ∃ (i : J) (y : F.obj i), (c.ι.app i).hom.base y = x :=\n  Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x\n\n-- Porting note: argument `C` of colimit need to be made explicit, otherwise we get universe issues\ntheorem colimit_exists_rep [HasLimitsOfShape Jᵒᵖ C] (x : colimit (C := SheafedSpace C) F) :\n    ∃ (i : J) (y : F.obj i), (colimit.ι F i).hom.base y = x :=\n  Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)\n    (isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x\n\nset_option backward.isDefEq.respectTransparency false in\ninstance [HasLimits C] {X Y : SheafedSpace C} (f g : X ⟶ Y) :\n    Epi (coequalizer.π f g).hom.base := by\n  rw [← show _ = (coequalizer.π f g).hom.base from\n      ι_comp_coequalizerComparison f g (SheafedSpace.forget C),\n      ← PreservesCoequalizer.iso_hom]\n  apply epi_comp\n\nend SheafedSpace\n\nnamespace LocallyRingedSpace\n\nsection HasCoproducts\n\nvariable {ι : Type v} [Small.{u} ι] (F : Discrete ι ⥤ LocallyRingedSpace.{u})\n\n/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/\nnoncomputable def coproduct : LocallyRingedSpace where\n  toSheafedSpace := colimit (C := SheafedSpace.{u + 1, u, u} CommRingCat.{u})\n    (F ⋙ forgetToSheafedSpace)\n  isLocalRing x := by\n    obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x\n    haveI : IsLocalRing (((F ⋙ forgetToSheafedSpace).obj i).presheaf.stalk y) :=\n      (F.obj i).isLocalRing _\n    exact\n      (asIso ((colimit.ι (C := SheafedSpace.{u + 1, u, u} CommRingCat.{u})\n        (F ⋙ forgetToSheafedSpace) i :).hom.stalkMap y)).symm.commRingCatIsoToRingEquiv.isLocalRing\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/\nnoncomputable def coproductCofan : Cocone F where\n  pt := coproduct F\n  ι :=\n    { app j := LocallyRingedSpace.homMk (colimit.ι (F ⋙ forgetToSheafedSpace) j)\n      naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; simp }\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The explicit coproduct cofan constructed in `coproductCofan` is indeed a colimit. -/\nnoncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where\n  desc s :=\n    LocallyRingedSpace.homMk (colimit.desc\n      (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) (by\n        intro x\n        obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x\n        have := PresheafedSpace.stalkMap.comp\n          (colimit.ι (F ⋙ forgetToSheafedSpace) i).hom\n          (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)).hom y\n        simp only [← IsIso.comp_inv_eq,\n          ← InducedCategory.comp_hom,\n          PresheafedSpace.stalkMap.congr_hom _ _\n            (congr_arg (InducedCategory.Hom.hom) (colimit.ι_desc\n              (forgetToSheafedSpace.mapCocone s) i))] at this\n        rw [← this]\n        dsimp\n        infer_instance)\n  fac _ _ :=\n    LocallyRingedSpace.forgetToSheafedSpace.map_injective\n     (colimit.ι_desc (C := SheafedSpace _) _ _)\n  uniq s f h :=\n    LocallyRingedSpace.forgetToSheafedSpace.map_injective\n      (IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.toShHom fun j =>\n        congr_arg LocallyRingedSpace.Hom.toShHom (h j))\n\ninstance : HasColimitsOfShape (Discrete ι) LocallyRingedSpace.{u} :=\n  ⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩\n\nnoncomputable instance : PreservesColimitsOfShape (Discrete.{v} ι) forgetToSheafedSpace.{u} :=\n  ⟨fun {G} =>\n    preservesColimit_of_preserves_colimit_cocone (coproductCofanIsColimit G)\n      ((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCat.{u}) _).ofIsoColimit\n        (Cocone.ext (Iso.refl _) fun _ => Category.comp_id _))⟩\n\nend HasCoproducts\n\nsection HasCoequalizer\n\nvariable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)\n\nnamespace HasCoequalizer\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[instance]\ntheorem coequalizer_π_app_isLocalHom\n    (U : TopologicalSpace.Opens (coequalizer f.toShHom g.toShHom).carrier) :\n    IsLocalHom ((coequalizer.π f.toShHom g.toShHom :).hom.c.app (op U)).hom := by\n  have := ι_comp_coequalizerComparison f.toShHom g.toShHom SheafedSpace.forgetToPresheafedSpace\n  dsimp at this\n  rw [← PreservesCoequalizer.iso_hom] at this\n  rw [← this, PresheafedSpace.comp_c_app,\n    ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]\n  -- Porting note (https://github.com/leanprover-community/mathlib4/issues/10754): this instance has to be manually added\n  haveI : IsIso (PreservesCoequalizer.iso\n      SheafedSpace.forgetToPresheafedSpace f.toShHom g.toShHom).hom.c :=\n    inferInstance\n  apply +allowSynthFailures RingHom.isLocalHom_comp\n  · apply +allowSynthFailures RingHom.isLocalHom_comp\n    · apply CommRingCat.equalizer_ι_isLocalHom'\n    · apply isLocalHom_of_isIso\n  · apply isLocalHom_of_isIso\n\n/-!\nWe roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms\nof locally ringed spaces, we want to show that the stalk map of\n`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that\n`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of\nlocally ringed space.\n\nGiven a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show\nthat `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the\nrestriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the\nbasic open set of `π⋆x`.\n\nSince `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have\n`π ⁻¹' π '' V = V` (as the underlying set map is merely the set-theoretic coequalizer).\nThis shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`\nare local ring homs.\n-/\n\n\nvariable (U : Opens (coequalizer f.toShHom g.toShHom).carrier)\nvariable (s : (coequalizer f.toShHom g.toShHom).presheaf.obj (op U))\n\n/-- (Implementation). The basic open set of the section `π꙳ s`. -/\nnoncomputable def imageBasicOpen : Opens Y :=\n  Y.toRingedSpace.basicOpen\n    (show Y.presheaf.obj (op (unop _)) from\n      ((coequalizer.π f.toShHom g.toShHom).hom.c.app (op U)) s)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\ntheorem imageBasicOpen_image_preimage :\n    (coequalizer.π f.toShHom g.toShHom).hom.base ⁻¹'\n      ((coequalizer.π f.toShHom g.toShHom).hom.base ''\n        (imageBasicOpen f g U s).1) = (imageBasicOpen f g U s).1 := by\n  fapply Types.coequalizer_preimage_image_eq_of_preimage_eq (↾f.base)\n    (↾g.base) (↾(coequalizer.π f.toShHom g.toShHom).hom.base)\n  · ext\n    simp only [TypeCat.Fun.toFun_apply, comp_apply, ConcreteCategory.hom_ofHom,\n      TypeCat.Fun.coe_mk, ← TopCat.comp_app, ← PresheafedSpace.comp_base]\n    congr 3\n    exact SheafedSpace.forgetToPresheafedSpace.congr_map\n      (coequalizer.condition f.toShHom g.toShHom)\n  · exact isColimitCoforkMapOfIsColimit (forget TopCat) _\n      (isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)\n      _ (coequalizerIsCoequalizer f.toShHom g.toShHom))\n  · suffices\n      (TopologicalSpace.Opens.map f.base).obj (imageBasicOpen f g U s) =\n        (TopologicalSpace.Opens.map g.base).obj (imageBasicOpen f g U s)\n      by injection this\n    delta imageBasicOpen\n    rw [preimage_basicOpen f, preimage_basicOpen g]\n    dsimp\n    rw [← ConcreteCategory.comp_apply, ← PresheafedSpace.comp_c_app,\n      ← CommRingCat.comp_apply, ← PresheafedSpace.comp_c_app]\n    -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11224): change `rw` to `erw`\n    erw [SheafedSpace.congr_hom_app (coequalizer.condition f.toShHom g.toShHom),\n      CommRingCat.comp_apply, X.toRingedSpace.basicOpen_res]\n    apply inf_eq_right.mpr\n    refine (RingedSpace.basicOpen_le _ _).trans ?_\n    rw [coequalizer.condition f.toShHom g.toShHom]\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem imageBasicOpen_image_open :\n    IsOpen ((coequalizer.π f.toShHom g.toShHom).hom.base '' (imageBasicOpen f g U s).1) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_879654c433cf","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"ea7254f8bddf0769d3a937fa65e7658170ea2f8ac3e535c08e68a10c753d8f82","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/RingedSpace","family_id":"imagebasicopen_image_open","file_id":"mathlib/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean","sample_id":"879654c433cff299cfe89ef259733d4899e296266c7758cbde162533660dcf70"},"split":"train"}
{"binding":{"chosen_proof_sha256":"16b3be2598502346de2754703a4649aa758c506146fa34fc091635ed0bf9b555","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"41091d79c58a55895294fdd103ee3f3bbfd8236b3858691c14a8a3dfe9a1b2a2","source_sha256":"04666e1a54002d0f7b310bb68c623ecf47eda2066cc741dda3dda2f5fc6f4c64","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases isEmpty_or_nonempty X with h | h\n  · exact ⟨Classical.arbitrary Y, funext <| h.elim⟩\n  · exact ⟨f (Classical.arbitrary X), hf.eq_const _⟩","hard_negative":true,"metrics":{"chosen_tokens":39,"rejected_tokens":8,"token_jaccard":0.032258,"token_length_ratio":0.205128},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"2ac185c77e31a997a73526df4ae9948823965acf12cb5c4f42000afeadb30542","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Indicator\npublic import Mathlib.Topology.Connected.LocallyConnected\npublic import Mathlib.Topology.Sets.Closeds\n\nNamespace:\nIsLocallyConstant\n\nLocal context:\n/-\nCopyright (c) 2021 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\n/-!\n# Locally constant functions\n\nThis file sets up the theory of locally constant function from a topological space to a type.\n\n## Main definitions and constructions\n\n* `IsLocallyConstant f` : a map `f : X → Y` where `X` is a topological space is locally\n                            constant if every set in `Y` has an open preimage.\n* `LocallyConstant X Y` : the type of locally constant maps from `X` to `Y`\n* `LocallyConstant.map` : push-forward of locally constant maps\n* `LocallyConstant.comap` : pull-back of locally constant maps\n-/\n\n@[expose] public section\n\nvariable {X Y Z α : Type*} [TopologicalSpace X]\n\nopen Set Filter\nopen scoped Topology\n\n/-- A function between topological spaces is locally constant if the preimage of any set is open. -/\ndef IsLocallyConstant (f : X → Y) : Prop :=\n  ∀ s : Set Y, IsOpen (f ⁻¹' s)\n\nnamespace IsLocallyConstant\n\nopen List in\nprotected theorem tfae (f : X → Y) :\n    TFAE [IsLocallyConstant f,\n      ∀ x, ∀ᶠ x' in 𝓝 x, f x' = f x,\n      ∀ x, IsOpen { x' | f x' = f x },\n      ∀ y, IsOpen (f ⁻¹' {y}),\n      ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x] := by\n  tfae_have 1 → 4 := fun h y => h {y}\n  tfae_have 4 → 3 := fun h x => h (f x)\n  tfae_have 3 → 2 := fun h x => IsOpen.mem_nhds (h x) rfl\n  tfae_have 2 → 5\n  | h, x => by\n    rcases mem_nhds_iff.1 (h x) with ⟨U, eq, hU, hx⟩\n    exact ⟨U, hU, hx, eq⟩\n  tfae_have 5 → 1\n  | h, s => by\n    refine isOpen_iff_forall_mem_open.2 fun x hx ↦ ?_\n    rcases h x with ⟨U, hU, hxU, eq⟩\n    exact ⟨U, fun x' hx' => mem_preimage.2 <| (eq x' hx').symm ▸ hx, hU, hxU⟩\n  tfae_finish\n\n@[nontriviality]\ntheorem of_discrete [DiscreteTopology X] (f : X → Y) : IsLocallyConstant f := fun _ =>\n  isOpen_discrete _\n\ntheorem isOpen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsOpen { x | f x = y } :=\n  hf {y}\n\ntheorem isClosed_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClosed { x | f x = y } :=\n  ⟨hf {y}ᶜ⟩\n\ntheorem isClopen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClopen { x | f x = y } :=\n  ⟨isClosed_fiber hf _, isOpen_fiber hf _⟩\n\ntheorem iff_exists_open (f : X → Y) :\n    IsLocallyConstant f ↔ ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x :=\n  (IsLocallyConstant.tfae f).out 0 4\n\ntheorem iff_eventually_eq (f : X → Y) : IsLocallyConstant f ↔ ∀ x, ∀ᶠ y in 𝓝 x, f y = f x :=\n  (IsLocallyConstant.tfae f).out 0 1\n\ntheorem exists_open {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x :=\n  (iff_exists_open f).1 hf x\n\nprotected theorem eventually_eq {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    ∀ᶠ y in 𝓝 x, f y = f x :=\n  (iff_eventually_eq f).1 hf x\n\ntheorem iff_isOpen_fiber_apply {f : X → Y} : IsLocallyConstant f ↔ ∀ x, IsOpen (f ⁻¹' {f x}) :=\n  (IsLocallyConstant.tfae f).out 0 2\n\ntheorem iff_isOpen_fiber {f : X → Y} : IsLocallyConstant f ↔ ∀ y, IsOpen (f ⁻¹' {y}) :=\n  (IsLocallyConstant.tfae f).out 0 3\n\nprotected theorem continuous [TopologicalSpace Y] {f : X → Y} (hf : IsLocallyConstant f) :\n    Continuous f :=\n  ⟨fun _ _ => hf _⟩\n\ntheorem iff_continuous {_ : TopologicalSpace Y} [DiscreteTopology Y] (f : X → Y) :\n    IsLocallyConstant f ↔ Continuous f :=\n  ⟨IsLocallyConstant.continuous, fun h s => h.isOpen_preimage s (isOpen_discrete _)⟩\n\ntheorem of_constant (f : X → Y) (h : ∀ x y, f x = f y) : IsLocallyConstant f :=\n  (iff_eventually_eq f).2 fun _ => Eventually.of_forall fun _ => h _ _\n\nprotected theorem const (y : Y) : IsLocallyConstant (Function.const X y) :=\n  of_constant _ fun _ _ => rfl\n\nprotected theorem comp {f : X → Y} (hf : IsLocallyConstant f) (g : Y → Z) :\n    IsLocallyConstant (g ∘ f) := fun s => by\n  rw [Set.preimage_comp]\n  exact hf _\n\ntheorem prodMk {Y'} {f : X → Y} {f' : X → Y'} (hf : IsLocallyConstant f)\n    (hf' : IsLocallyConstant f') : IsLocallyConstant fun x => (f x, f' x) :=\n  (iff_eventually_eq _).2 fun x =>\n    (hf.eventually_eq x).mp <| (hf'.eventually_eq x).mono fun _ hf' hf => Prod.ext hf hf'\n\ntheorem comp₂ {Y₁ Y₂ Z : Type*} {f : X → Y₁} {g : X → Y₂} (hf : IsLocallyConstant f)\n    (hg : IsLocallyConstant g) (h : Y₁ → Y₂ → Z) : IsLocallyConstant fun x => h (f x) (g x) :=\n  (hf.prodMk hg).comp fun x : Y₁ × Y₂ => h x.1 x.2\n\ntheorem comp_continuous [TopologicalSpace Y] {g : Y → Z} {f : X → Y} (hg : IsLocallyConstant g)\n    (hf : Continuous f) : IsLocallyConstant (g ∘ f) := fun s => by\n  rw [Set.preimage_comp]\n  exact hf.isOpen_preimage _ (hg _)\n\n/-- A locally constant function is constant on any preconnected set. -/\ntheorem apply_eq_of_isPreconnected {f : X → Y} (hf : IsLocallyConstant f) {s : Set X}\n    (hs : IsPreconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by\n  let U := f ⁻¹' {f y}\n  suffices x ∉ Uᶜ from Classical.not_not.1 this\n  intro hxV\n  specialize hs U Uᶜ (hf {f y}) (hf {f y}ᶜ) _ ⟨y, ⟨hy, rfl⟩⟩ ⟨x, ⟨hx, hxV⟩⟩\n  · simp only [union_compl_self, subset_univ]\n  · simp only [inter_empty, Set.not_nonempty_empty, inter_compl_self] at hs\n\ntheorem apply_eq_of_preconnectedSpace [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f)\n    (x y : X) : f x = f y :=\n  hf.apply_eq_of_isPreconnected isPreconnected_univ trivial trivial\n\ntheorem eq_const [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    f = Function.const X (f x) :=\n  funext fun y => hf.apply_eq_of_preconnectedSpace y x\n\nTarget:\ntheorem exists_eq_const [PreconnectedSpace X] [Nonempty Y] {f : X → Y} (hf : IsLocallyConstant f) :\n    ∃ y, f = Function.const X y :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"407636fda83c5b7a83adb732b1a29fb090cebf292fcc87156769bda23304b774","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/LocallyConstant","family_id":"exists_eq_const","file_id":"mathlib/Mathlib/Topology/LocallyConstant/Basic.lean","sample_id":"41091d79c58a55895294fdd103ee3f3bbfd8236b3858691c14a8a3dfe9a1b2a2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b656fc39071ddeaac02c29e4861f9ce7d1311a7b76fad806354a82acd6bcd29b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"785e48ee70dd3a551ef74441e20aad6ceed1e81f37833bb7ee09495a58b561be","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"66a9d446584881e6ae181bf60c62b6c4435815bff91f04bcf763c4007fc9aaae","source_sha256":"7bb1b48aebf21c1532e493ca79c87d1bb8ee632b2dfff76d51a712868ddd9efa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases isEmpty_or_nonempty ι\n  · rw [iSup_of_empty', iSup_of_empty']\n  · exact hdf.ciSup_le fun x ↦ hdg.le_ciSup_of_le B x (H x)","hard_negative":true,"metrics":{"chosen_tokens":28,"rejected_tokens":5,"token_jaccard":0.12,"token_length_ratio":0.178571},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"2ac81cb114284df4816758415945e26e9668a422c87ff18678889dd26f684ec4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ConditionallyCompletePartialOrder.Basic\npublic import Mathlib.Order.GaloisConnection.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2026 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\n/-!\n# Indexed sup / inf in conditionally complete lattices\n\nThis file proves lemmas about `iSup` and `iInf` for functions valued in a conditionally complete\npartial order, as opposed to a conditionally complete lattice.\n\n## TODO\n\n+ Use `@[to_dual]` in the `GaloisConnection` and `OrderIso` sections.\n\n-/\n\npublic section\n\n-- Guard against import creep\nassert_not_exists Multiset\n\nopen Function OrderDual Set\n\nvariable {α β γ : Type*} {ι : Sort*}\n\nsection ConditionallyCompletePartialOrderSup\n\nvariable [ConditionallyCompletePartialOrderSup α] {a b : α}\n\n@[to_dual]\ntheorem Directed.isLUB_ciSup [Nonempty ι] {f : ι → α} (hd : Directed (· ≤ ·) f)\n    (H : BddAbove (range f)) : IsLUB (range f) (⨆ i, f i) :=\n  hd.directedOn_range.isLUB_csSup (range_nonempty f) H\n\n@[to_dual]\ntheorem DirectedOn.isLUB_ciSup_set {f : β → α} {s : Set β} (hd : DirectedOn (· ≤ ·) (f '' s))\n    (H : BddAbove (f '' s)) (Hne : s.Nonempty) :\n    IsLUB (f '' s) (⨆ i : s, f i) := by\n  rw [← sSup_image']\n  exact hd.isLUB_csSup (Hne.image _) H\n\n@[to_dual Directed.le_ciInf_iff]\ntheorem Directed.ciSup_le_iff [Nonempty ι] {f : ι → α} {a : α}\n    (hd : Directed (· ≤ ·) f) (hf : BddAbove (range f)) :\n    iSup f ≤ a ↔ ∀ i, f i ≤ a :=\n  (isLUB_le_iff <| hd.isLUB_ciSup hf).trans forall_mem_range\n\n@[to_dual DirectedOn.le_ciInf_set_iff]\ntheorem DirectedOn.ciSup_set_le_iff {ι : Type*} {s : Set ι} {f : ι → α} {a : α} (hs : s.Nonempty)\n    (hd : DirectedOn (· ≤ ·) (f '' s)) (hf : BddAbove (f '' s)) :\n    ⨆ i : s, f i ≤ a ↔ ∀ i ∈ s, f i ≤ a :=\n  (isLUB_le_iff <| hd.isLUB_ciSup_set hf hs).trans forall_mem_image\n\n@[to_dual Directed.ciInf_le_of_le]\ntheorem Directed.le_ciSup_of_le {f : ι → α} (hd : Directed (· ≤ ·) f)\n    (H : BddAbove (range f)) (c : ι) (h : a ≤ f c) : a ≤ iSup f :=\n  le_trans h (hd.le_ciSup H c)\n\n/-- The indexed suprema of two functions are comparable if the functions are pointwise comparable -/\n@[to_dual (attr := gcongr low)\n/-- The indexed infimum of two functions are comparable if the functions are pointwise\ncomparable -/]\n\nTarget:\ntheorem Directed.ciSup_mono {f g : ι → α} (hdf : Directed (· ≤ ·) f)\n    (hdg : Directed (· ≤ ·) g) (B : BddAbove (range g)) (H : ∀ x, f x ≤ g x) :\n    iSup f ≤ iSup g :=\n\nProof body:\n","rejected":"by\n  exact Directed.ciSup_mono","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"3a82591f2c4c79b5346fbecdbd75c2479a35e84e068139b0084e9adc6b23192b","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/ConditionallyCompletePartialOrder","family_id":"directed","file_id":"mathlib/Mathlib/Order/ConditionallyCompletePartialOrder/Indexed.lean","sample_id":"66a9d446584881e6ae181bf60c62b6c4435815bff91f04bcf763c4007fc9aaae"},"split":"train"}
{"binding":{"chosen_proof_sha256":"91037b5a5d4107cfc6e47498527b6da8e785a774a3e33005feebe3831fc4a2a4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"357a0e4c1d470dcb98d6df000adeb3562e47d246bd50625bb6f9d70fa5388f2b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2031f82d6bd5b2086402ba4895d772d5452bfbf03175b7504ea5ff92425b4486","source_sha256":"113aaf52a8970f3d5731b3bef38b5eb29be69585d68fdd5fc22f4868fa9f35e6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← tendsto_norm_atTop_iff_cobounded]\n  simpa [norm_intCast_eq_abs_mul_norm_one] using tendsto_intCast_atTop_atTop\n    |>.comp (tendsto_abs_atBot_atTop.sup tendsto_abs_atTop_atTop)\n    |>.atTop_mul_const (norm_pos_iff.mpr one_ne_zero)","hard_negative":true,"metrics":{"chosen_tokens":32,"rejected_tokens":3,"token_jaccard":0.04,"token_length_ratio":0.09375},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"2b08faf0a8bb69affd853206e48d69c1aeab22e23f2004d393261561415dc3c6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Module\npublic import Mathlib.Algebra.Order.Field.Power\npublic import Mathlib.Algebra.Polynomial.Monic\npublic import Mathlib.Analysis.Asymptotics.Lemmas\npublic import Mathlib.Analysis.Normed.Ring.InfiniteSum\npublic import Mathlib.Analysis.Normed.Module.Basic\npublic import Mathlib.Analysis.Normed.Order.Lattice\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.Data.List.TFAE\npublic import Mathlib.Data.Nat.Choose.Bounds\npublic import Mathlib.Order.Filter.AtTopBot.ModEq\npublic import Mathlib.RingTheory.Polynomial.Pochhammer\npublic import Mathlib.Tactic.NoncommRing\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie, Patrick Massot\n-/\n/-!\n# A collection of specific limit computations\n\nThis file contains important specific limit computations in (semi-)normed groups/rings/spaces, as\nwell as such computations in `ℝ` when the natural proof passes through a fact about normed spaces.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Set Function Filter Finset Metric Module Asymptotics Topology Nat NNReal ENNReal\nopen scoped Ring\n\nvariable {α : Type*}\n\ntheorem tendsto_natCast_atTop_cobounded\n    [NormedRing α] [NormSMulClass ℤ α] [Nontrivial α] :\n    Tendsto Nat.cast atTop (Bornology.cobounded α) := by\n  rw [← tendsto_norm_atTop_iff_cobounded]\n  simpa [norm_natCast_eq_mul_norm_one] using tendsto_natCast_atTop_atTop\n    |>.atTop_mul_const (norm_pos_iff.mpr one_ne_zero)\n\nTarget:\ntheorem tendsto_intCast_atBot_sup_atTop_cobounded\n    [NormedRing α] [NormSMulClass ℤ α] [Nontrivial α] :\n    Tendsto Int.cast (atBot ⊔ atTop) (Bornology.cobounded α) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_2031f82d6bd5","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"e076ec07edeaefcdb4036d705e99faf54d5ddc24438a39deca0be7e543107511","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/SpecificLimits","family_id":"tendsto_intcast_atbot_sup_attop_cobounded","file_id":"mathlib/Mathlib/Analysis/SpecificLimits/Normed.lean","sample_id":"2031f82d6bd5b2086402ba4895d772d5452bfbf03175b7504ea5ff92425b4486"},"split":"train"}
{"binding":{"chosen_proof_sha256":"56b4e98d034de1994cb22dfefff34d5e35c25783ae7eef0d3fe3f45cbf3ff033","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e0b2d88f43c7c482b1479e13f5c9bc4c3c7292675c09d5ada6dc20e2083f376e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"27226b4cef484b3c92a34e5c76b5413440a1f9efdabd65d7d5bb3fbb80618898","source_sha256":"d5cffaaa7c7b308256b47b38d296e80bf7d53217db8c0a040e827e6ffc87b03a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using mul_vle_mul_left ((vle_total 0 1).resolve_right not_vle_one_zero) x","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":2,"token_jaccard":0.071429,"token_length_ratio":0.133333},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"2b0bbbc67eee158c9ea0dec75cec4dd96bb7a2bb679e29d1bd2ae5a4d729aae1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Valuation.Basic\npublic import Mathlib.Data.NNReal.Defs\npublic import Mathlib.Topology.Defs.Filter\npublic import Mathlib.Order.Filter.Bases.Basic\n\nNamespace:\nValuativeRel\n\nLocal context:\n/-\nCopyright (c) 2025 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Liu, Adam Topaz\n-/\n/-!\n\n# Valuative Relations\n\nIn this file we introduce a class called `ValuativeRel R` for a ring `R`.\nThis bundles a relation `vle : R → R → Prop` on `R` which mimics a\npreorder on `R` arising from a valuation.\nWe introduce the notation `x ≤ᵥ y` for this relation.\n\nRecall that the equivalence class of a valuation is *completely* characterized by\nsuch a preorder. Thus, we can think of `ValuativeRel R` as a way of\nsaying that `R` is endowed with an equivalence class of valuations.\n\n## Main Definitions\n\n- `ValuativeRel R` endows a semiring `R` with a relation \"arising from a valuation\". When `R` is a\n  ring, this is equivalent to fixing an equivalence class of valuations on `R`.\n  Use the notation `x ≤ᵥ y` for this relation.\n- `ValuativeRel.valuation R` is the \"canonical\" valuation associated to `ValuativeRel R`,\n  taking values in `ValuativeRel.ValueGroupWithZero R`.\n- Given a valuation `v` on `R` and an instance `[ValuativeRel R]`, writing `[v.Compatible]`\n  ensures that the relation `x ≤ᵥ y` is equivalent to `v x ≤ v y`. Note that\n  it is possible to have `[v.Compatible]` and `[w.Compatible]` for two different valuations on `R`.\n- Given `[ValuativeRel A]`, `[ValuativeRel B]` and `[Algebra A B]`, the class\n  `[ValuativeExtension A B]` ensures that the algebra map `A → B` is compatible with the valuations\n  on `A` and `B`. For example, this can be used to talk about extensions of valued fields.\n\n\n## Remark\n\nThe last two axioms in `ValuativeRel`, namely `vle_mul_cancel` and `not_vle_one_zero`, are\nused to ensure that we have a well-behaved valuation taking values in a *value group* (with zero).\nIn principle, it should be possible to drop these two axioms and obtain a value monoid,\nhowever, such a value monoid would not necessarily embed into an ordered abelian group with zero.\nSimilarly, without these axioms, the support of the valuation need not be a prime ideal.\nWe have thus opted to include these two axioms and obtain a `ValueGroupWithZero` associated to\na `ValuativeRel` in order to best align with the literature about valuations on commutative rings.\n\nFuture work could refactor `ValuativeRel` by dropping the `vle_mul_cancel` and `not_vle_one_zero`\naxioms, opting to make these mixins instead.\n\n## Projects\n\nThe `ValuativeRel` class should eventually replace the existing `Valued` typeclass.\nOnce such a refactor happens, `ValuativeRel` could be renamed to `Valued`.\n\n## TODO\nSplit this file. For instance, the universal properties of `ValueGroupWithZero` and definition of\n`IsRankLeOne` could be separated out.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\n/-- The class `[ValuativeRel R]` class introduces an operator `x ≤ᵥ y : Prop` for `x y : R`\nwhich is the natural relation arising from (the equivalence class of) a valuation on `R` when `R`\nis a ring. More precisely, if `v` is a valuation on `R` then the associated relation is\n`x ≤ᵥ y ↔ v x ≤ v y`. Use this class to talk about the case where `R` is equipped\nwith an equivalence class of valuations. -/\n@[ext]\nclass ValuativeRel (R : Type*) [Semiring R] where\n  /-- The valuation less-equal operator arising from `ValuativeRel`. -/\n  vle : R → R → Prop\n  vle_total (x y) : vle x y ∨ vle y x\n  vle_trans {z y x} : vle x y → vle y z → vle x z\n  vle_add {x y z} : vle x z → vle y z → vle (x + y) z\n  mul_vle_mul_left {x y} (h : vle x y) (z) : vle (x * z) (y * z)\n  vle_mul_cancel {x y z} : ¬ vle z 0 → vle (x * z) (y * z) → vle x y\n  not_vle_one_zero : ¬ vle 1 0\n  vle_mul_comm {x y} : vle (x * y) (y * x)\n\n@[inherit_doc] infix:50 \" ≤ᵥ \" => ValuativeRel.vle\n\nmacro_rules | `($a ≤ᵥ $b) => `(binrel% ValuativeRel.vle $a $b)\n\nnamespace Valuation\n\nvariable {R Γ : Type*} [Ring R] [LinearOrderedCommMonoidWithZero Γ]\n  (v : Valuation R Γ)\n\n/-- We say that a valuation `v` is `Compatible` if the relation `x ≤ᵥ y`\nis equivalent to `v x ≤ v y`. -/\nclass Compatible [ValuativeRel R] where\n  vle_iff_le (x y : R) : x ≤ᵥ y ↔ v x ≤ v y\n\nend Valuation\n\n/-- A preorder on a ring is said to be \"valuative\" if it agrees with the\nvaluative relation. -/\nclass ValuativePreorder (R : Type*) [Semiring R] [ValuativeRel R] [Preorder R] where\n  vle_iff_le (x y : R) : x ≤ᵥ y ↔ x ≤ y\n\nnamespace ValuativeRel\n\n@[deprecated (since := \"2025-12-20\")] alias Rel := vle\n@[deprecated (since := \"2025-12-20\")] alias rel_total := vle_total\n@[deprecated (since := \"2025-12-20\")] alias rel_trans := vle_trans\n@[deprecated (since := \"2025-12-20\")] alias rel_add := vle_add\n@[deprecated (since := \"2025-12-20\")] alias rel_mul_right := mul_vle_mul_left\n@[deprecated (since := \"2025-12-20\")] alias rel_mul_cancel := vle_mul_cancel\n@[deprecated (since := \"2025-12-20\")] alias not_rel_one_zero := not_vle_one_zero\n\nvariable {R : Type*} [Semiring R] [ValuativeRel R] {x x' y y' z : R}\n\n/-- The valuation less-than relation, defined as `x <ᵥ y ↔ ¬ y ≤ᵥ x`. -/\ndef vlt (x y : R) : Prop := ¬ y ≤ᵥ x\n\n@[deprecated (since := \"2025-12-20\")] alias SRel := vlt\n\n@[inherit_doc] infix:50 \" <ᵥ \" => ValuativeRel.vlt\n\nmacro_rules | `($a <ᵥ $b) => `(binrel% ValuativeRel.vlt $a $b)\n\n/-- The valuation equals relation, defined as `x =ᵥ y ↔ x ≤ᵥ y ∧ y ≤ᵥ x`. -/\ndef veq : R → R → Prop := AntisymmRel (· ≤ᵥ ·)\n\n@[inherit_doc] infix:50 \" =ᵥ \" => ValuativeRel.veq\n\nlemma veq_mul_comm (x y : R) : x * y =ᵥ y * x := ⟨vle_mul_comm, vle_mul_comm⟩\n\nmacro_rules | `($a =ᵥ $b) => `(binrel% ValuativeRel.veq $a $b)\n\n@[simp, grind =] lemma not_vle : ¬ x ≤ᵥ y ↔ y <ᵥ x := .rfl\n@[simp, grind =] lemma not_vlt : ¬ x <ᵥ y ↔ y ≤ᵥ x := not_vle.not_left\nlemma veq_def : x =ᵥ y ↔ x ≤ᵥ y ∧ y ≤ᵥ x := .rfl\n\n@[deprecated not_vle (since := \"2025-12-20\")]\nlemma srel_iff : x <ᵥ y ↔ ¬ y ≤ᵥ x := Iff.rfl\n\n@[deprecated (since := \"2025-12-20\")] alias not_srel_iff := not_vlt\n\nprotected alias ⟨_, vle.not_vlt⟩ := not_vlt\nprotected alias ⟨_, vlt.not_vle⟩ := not_vle\n\nlemma veq_comm : x =ᵥ y ↔ y =ᵥ x := antisymmRel_comm\n@[symm] protected alias ⟨veq.symm, _⟩ := veq_comm\n\ninstance : @Std.Symm R (· =ᵥ ·) where\n  symm _ _ := veq.symm\n\nlemma vle_of_veq (h : x =ᵥ y) : x ≤ᵥ y := h.1\nlemma vge_of_veq (h : x =ᵥ y) : y ≤ᵥ x := h.2\n\nprotected alias veq.vle := vle_of_veq\nprotected alias veq.vge := vge_of_veq\n\nlemma not_vlt_of_veq (h : x =ᵥ y) : ¬ x <ᵥ y := h.vge.not_vlt\nlemma not_vgt_of_veq (h : x =ᵥ y) : ¬ y <ᵥ x := h.vle.not_vlt\n\nprotected alias veq.not_vlt := not_vlt_of_veq\nprotected alias veq.not_vgt := not_vgt_of_veq\n\n@[simp, refl] lemma vle_refl (x : R) : x ≤ᵥ x := or_self_iff.1 <| vle_total x x\nlemma vle_rfl : x ≤ᵥ x := vle_refl x\n\n@[deprecated (since := \"2025-12-20\")] alias rel_refl := vle_refl\n@[deprecated (since := \"2025-12-20\")] alias rel_rfl := vle_rfl\n\nprotected alias vle.refl := vle_refl\nprotected alias vle.rfl := vle_rfl\n\ninstance : @Std.Refl R (· ≤ᵥ ·) where\n  refl _ := vle_rfl\n\n@[deprecated (since := \"2025-12-20\")] protected alias Rel.refl := vle.refl\n@[deprecated (since := \"2025-12-20\")] protected alias Rel.rfl := vle.rfl\n\n@[simp, refl] lemma veq_refl (x : R) : x =ᵥ x := AntisymmRel.rfl\nlemma veq_rfl : x =ᵥ x := veq_refl x\n\nprotected alias veq.refl := veq_refl\nprotected alias veq.rfl := veq_rfl\n\ninstance : @Std.Refl R (· =ᵥ ·) where\n  refl _ := veq_rfl\n\n@[simp]\n\nTarget:\ntheorem zero_vle (x : R) : 0 ≤ᵥ x :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_27226b4cef48","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"39c17893e48efecb814760c686e396dc4aeab84b04d2553f82ef8a7025bc94fc","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Valuation","family_id":"zero_vle","file_id":"mathlib/Mathlib/RingTheory/Valuation/ValuativeRel/Basic.lean","sample_id":"27226b4cef484b3c92a34e5c76b5413440a1f9efdabd65d7d5bb3fbb80618898"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cde0cb2c54480a9a9ba9dd0ed55105a1ff6f413d177d544464139dccab437478","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d4e1b22d41edfa6c165c88bc1ca936a47f961d4ff80fb80f370b5f07e28c1664","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dfc0baea2eabd164611e1d1c93cf31cf2ae01a14f1445c0b6a91023c66e00ae4","source_sha256":"5d27fa86937e3368de02fb4ec1d31ef3389387a61446fa516bf2a6e60873be5c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  simp_rw [← AlgHom.coe_toRingHom, ← coeff_map, subProdXSubC, Polynomial.map_sub,\n    Polynomial.map_prod, Polynomial.map_sub, map_X, map_C, toSplittingField,\n    AlgHom.coe_toRingHom, MvPolynomial.aeval_X, dif_pos h,\n    ← (finEquivRoots (Monics.splits_finsetProd h)).symm.prod_comp, Equiv.apply_symm_apply]\n  rw [Finset.prod_coe_sort (f := fun x : _ × ℕ ↦ X - C x.1), (Multiset.toEnumFinset _)\n    |>.prod_eq_multiset_prod, ← Function.comp_def (X - C ·) Prod.fst, ← Multiset.map_map,\n    Multiset.map_toEnumFinset_fst, map_map, AlgHom.comp_algebraMap]\n  conv in map _ _ => rw [Splits.eq_prod_roots (Monics.splits_finsetProd h)]\n  rw [leadingCoeff_map, f.2, map_one, C_1, one_mul, sub_self, coeff_zero]\n\nvariable (k)","hard_negative":true,"metrics":{"chosen_tokens":165,"rejected_tokens":3,"token_jaccard":0.012987,"token_length_ratio":0.018182},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"2b0ec5528b7a125de6b2e375a9b81c38254226fb1feaece9d8bc3df50c354a21","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharP.Algebra\npublic import Mathlib.Data.Multiset.Fintype\npublic import Mathlib.FieldTheory.IsAlgClosed.Basic\npublic import Mathlib.FieldTheory.SplittingField.Construction\n\nNamespace:\nAlgebraicClosure\n\nLocal context:\n/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n/-!\n# Algebraic Closure\n\nIn this file we construct the algebraic closure of a field\n\n## Main Definitions\n\n- `AlgebraicClosure k` is an algebraic closure of `k` (in the same universe).\n  It is constructed by taking the polynomial ring generated by indeterminates\n  $X_{f,1}, \\dots, X_{f,\\deg f}$ corresponding to roots of monic irreducible\n  polynomials `f` with coefficients in `k`, and quotienting out by a maximal\n  ideal containing every $f - \\prod_i (X - X_{f,i})$. The proof follows\n  https://kconrad.math.uconn.edu/blurbs/galoistheory/algclosureshorter.pdf.\n\n## Tags\n\nalgebraic closure, algebraically closed\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nnoncomputable section\n\nopen Polynomial\n\nvariable (k : Type u) [Field k]\n\nnamespace AlgebraicClosure\n\n/-- The subtype of monic polynomials. -/\ndef Monics : Type u := {f : k[X] // f.Monic}\n\n/-- `Vars k` provides `n` variables $X_{f,1}, \\dots, X_{f,n}$ for each monic polynomial\n`f : k[X]` of degree `n`. -/\ndef Vars : Type u := Σ f : Monics k, Fin f.1.natDegree\n\nvariable {k} in\n/-- Given a monic polynomial `f : k[X]`,\n`subProdXSubC f` is the polynomial $f - \\prod_i (X - X_{f,i})$. -/\ndef subProdXSubC (f : Monics k) : (MvPolynomial (Vars k) k)[X] :=\n  f.1.map (algebraMap _ _) - ∏ i : Fin f.1.natDegree, (X - C (MvPolynomial.X ⟨f, i⟩))\n\n/-- The span of all coefficients of `subProdXSubC f` as `f` ranges all polynomials in `k[X]`. -/\ndef spanCoeffs : Ideal (MvPolynomial (Vars k) k) :=\n  Ideal.span <| Set.range fun fn : Monics k × ℕ ↦ (subProdXSubC fn.1).coeff fn.2\n\nvariable {k}\n\n/-- If a monic polynomial `f : k[X]` splits in `K`,\nthen it has as many roots (counting multiplicity) as its degree. -/\ndef finEquivRoots {K} [Field K] [DecidableEq K] {i : k →+* K} {f : Monics k}\n    (hf : (f.1.map i).Splits) : Fin f.1.natDegree ≃ (f.1.map i).roots.toEnumFinset :=\n  .symm <| Finset.equivFinOfCardEq <| by\n    rwa [splits_iff_card_roots,\n      ← Multiset.card_toEnumFinset, f.2.natDegree_map] at hf\n\nlemma Monics.splits_finsetProd {s : Finset (Monics k)} {f : Monics k} (hf : f ∈ s) :\n    (f.1.map (algebraMap k (SplittingField (∏ f ∈ s, f.1)))).Splits :=\n  (splits_prod_iff fun j _ ↦ map_ne_zero j.2.ne_zero).mp\n    (by simpa [Polynomial.map_prod] using SplittingField.splits (∏ f ∈ s, f.1)) f hf\n\nopen Classical in\n/-- Given a finite set of monic polynomials, construct an algebra homomorphism\nto the splitting field of the product of the polynomials\nsending indeterminates $X_{f_i}$ to the distinct roots of `f`. -/\ndef toSplittingField (s : Finset (Monics k)) :\n    MvPolynomial (Vars k) k →ₐ[k] SplittingField (∏ f ∈ s, f.1) :=\n  MvPolynomial.aeval fun fi ↦\n    if hf : fi.1 ∈ s then (finEquivRoots (Monics.splits_finsetProd hf) fi.2).1.1 else 37\n\nTarget:\ntheorem toSplittingField_coeff {s : Finset (Monics k)} {f} (h : f ∈ s) (n) :\n    toSplittingField s ((subProdXSubC f).coeff n) = 0 :=\n\nProof body:\n","rejected":"by\n  exact toSplittingField_coeff","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"6dce43230cd4fb166894bdbd8b35b5ceb1af7ef9d3745ca0a76e68ef20ba9c47","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/IsAlgClosed","family_id":"tosplittingfield_coeff","file_id":"mathlib/Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean","sample_id":"dfc0baea2eabd164611e1d1c93cf31cf2ae01a14f1445c0b6a91023c66e00ae4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"055208fdff31df9d8fb76bf05bb6a68b4de72a41035c9aa9cc8d72c2b3b25d67","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"db95c7c59b1a744d161d6d13a5a1ed32694edc168ef81e065f31791b05479dc7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"029e00c3f6a53264f7ab33ca9285b495e3986d3f7052bd853ac23e6d759b17de","source_sha256":"38b81cd46858f243db931b63ae095bb873edc08d47b08e77c1c254a8ae6a5653","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [Algebra.smul_def, ← nat_rawCast_2]","hard_negative":true,"metrics":{"chosen_tokens":10,"rejected_tokens":5,"token_jaccard":0.153846,"token_length_ratio":0.5},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"2b163f1426e145f24aa67ae233bcfc05c164db8ac921f9658ca906355d474461","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Defs\npublic import Mathlib.Algebra.Algebra.Basic\npublic import Mathlib.Tactic.Ring.RingNF\n\nNamespace:\nMathlib.Tactic.Algebra\n\nLocal context:\n/-\nCopyright (c) 2025 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Arend Mellendijk\n-/\n/-! # Lemmas for the `algebra` tactic.\n-/\n\n@[expose] public section\n\nopen Mathlib.Meta.NormNum\n\nnamespace Mathlib.Tactic.Algebra\n\nsection ring\n\nvariable {R A : Type*} [CommRing R] [CommRing A] [Algebra R A]\n\n/- evalCast -/\ntheorem isInt_negOfNat_eq {a : A} {lit : ℕ} (h : IsInt a (Int.negOfNat lit)) :\n    a = algebraMap R A (Int.rawCast (Int.negOfNat lit) + 0 : R) + 0 := by\n  simp [h.out]\n\nend ring\n\nsection semifield\n\nvariable {R A : Type*} [Semifield R] [Semifield A] [Algebra R A]\n\n/- evalCast -/\ntheorem isNNRat_eq_rawCast {a : A} {n d : ℕ} (h : IsNNRat a n d) :\n    a = algebraMap R A (NNRat.rawCast n d + 0 : R) + 0 := by\n  simp [Mathlib.Tactic.Ring.cast_nnrat h]\n\nend semifield\n\nsection field\n\nvariable {R A : Type*} [Field R] [Field A] [Algebra R A]\n\n/- evalCast -/\ntheorem isRat_eq_rawCast {a : A} {n d : ℕ} (h : IsRat a (.negOfNat n) d) :\n    a = algebraMap R A (Rat.rawCast (.negOfNat n) d + 0 : R) + 0 := by\n  simp [Mathlib.Tactic.Ring.cast_rat h]\n\nend field\n\nvariable {R A : Type*} [sR : CommSemiring R] [sA : CommSemiring A] [sAlg : Algebra R A]\n\n/- evalCast -/\ntheorem isNat_zero_eq {a : A} (h : IsNat a 0) : a = 0 := by\n  have := h.out\n  simp [this]\n\n/- evalCast -/\ntheorem isNat_eq_rawCast {a : A} {lit : ℕ} (h : IsNat a lit) :\n    a = algebraMap R A (lit + 0 : R) + 0 := by\n  simp [h.out]\n\nsection cleanup\n\nvariable {n d : ℕ}\n\nsection cleanupSMul\n\ntheorem add_assoc_rev (a b c : R) : a + (b + c) = a + b + c := (add_assoc ..).symm\ntheorem mul_assoc_rev (a b c : R) : a * (b * c) = a * b * c := (mul_assoc ..).symm\ntheorem mul_neg {R} [Ring R] (a b : R) : a * -b = -(a * b) := by simp\ntheorem add_neg {R} [Ring R] (a b : R) : a + -b = a - b := (sub_eq_add_neg ..).symm\ntheorem nat_rawCast_0 : (Nat.rawCast 0 : R) = 0 := by simp\ntheorem nat_rawCast_1 : (Nat.rawCast 1 : R) = 1 := by simp\ntheorem nat_rawCast_2 [Nat.AtLeastTwo n] : (Nat.rawCast n : R) = OfNat.ofNat n := rfl\ntheorem int_rawCast_neg {R} [Ring R] : (Int.rawCast (.negOfNat n) : R) = -Nat.rawCast n := by simp\ntheorem nnrat_rawCast {R} [DivisionSemiring R] :\n    (NNRat.rawCast n d : R) = Nat.rawCast n / Nat.rawCast d := by simp\ntheorem rat_rawCast_neg {R} [DivisionRing R] :\n    (Rat.rawCast (.negOfNat n) d : R) = Int.rawCast (.negOfNat n) / Nat.rawCast d := by simp\n\nend cleanupSMul\nsection cleanupConsts\n\ntheorem ofNat_smul {R A} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    [n.AtLeastTwo] {a : A} :\n    (ofNat(n) : R) • a = ofNat(n) * a := by\n  simp_rw [← nat_rawCast_2]\n  simp [Nat.cast_smul_eq_nsmul]\n\ntheorem neg_ofNat_smul {R A} [CommRing R] [CommRing A] [Algebra R A] {a : A} [n.AtLeastTwo] :\n    (- ofNat(n) : R) • a = - (ofNat(n)) * a := by\n  simpa [← nat_rawCast_2] using! ofNat_smul\n\ntheorem neg_1_smul {R A} [CommRing R] [CommRing A] [Algebra R A] {a : A} :\n    (-1 : R) • a = - a := by\n  simp\n\ntheorem nnRat_ofNat_smul_1 {R A} [Semifield R] [Semifield A] [Algebra R A] {a : A}\n    [d.AtLeastTwo] :\n    (1 / ofNat(d) : R) • a = (1 / ofNat(d)) * a := by\n  simp [Algebra.smul_def, ← nat_rawCast_2]\n\ntheorem nnRat_ofNat_smul_2 {R A} [Semifield R] [Semifield A] [Algebra R A] {a : A}\n    [n.AtLeastTwo] [d.AtLeastTwo] :\n    (ofNat(n) / ofNat(d) : R) • a = (ofNat(n) / ofNat(d)) * a := by\n  simp [Algebra.smul_def, ← nat_rawCast_2]\n\nTarget:\ntheorem rat_ofNat_smul_1 {R A} [Field R] [Field A] [Algebra R A] {a : A}\n    [d.AtLeastTwo] :\n    ((- 1) / ofNat(d) : R) • a = ((- 1) / ofNat(d)) * a :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_029e00c3f6a5","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1603f346fdf1ead97ce3a40f1add2bd5044bc756b503ecc10dd47aef6f566edd","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Algebra","family_id":"rat_ofnat_smul","file_id":"mathlib/Mathlib/Tactic/Algebra/Lemmas.lean","sample_id":"029e00c3f6a53264f7ab33ca9285b495e3986d3f7052bd853ac23e6d759b17de"},"split":"train"}
{"binding":{"chosen_proof_sha256":"93aa954ac038910f3cc0d692da80205ae5a7230c0f479de23b1a6ec2e526a44a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4158bc4533e6e85b1b86e67b31a3d6eca02e3c8f701cb75f73a81c4775a53885","source_sha256":"5d8f7f707a69d48e10d7817e517c71568b55a688a7dc3473a0fa55b05f45f810","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [FractionalOperation.tt, Multiset.mem_map] at rin\n  rw [show r = ![r 0, r 1] by simp [← List.ofFn_inj]]\n  apply lt_of_le_of_ne (mcf.right (r 0) (r 1)).left\n  intro equ\n  have asymm : r 0 ≠ r 1 := by\n    rcases (mcf.right (r 0) (r 1)).right equ with ⟨ha0, hb1⟩ | ⟨ha1, hb0⟩\n    · rw [ha0, hb1] at hab\n      exact hab\n    · rw [ha1, hb0] at hab\n      exact hab.symm\n  apply asymm\n  obtain ⟨o, in_omega, rfl⟩ := rin\n  change o (fun j => ![![a, b], ![b, a]] j 0) = o (fun j => ![![a, b], ![b, a]] j 1)\n  convert! symmega ![a, b] ![b, a] (by simp [List.Perm.swap]) o in_omega using 2 <;>\n    simp [Matrix.const_fin1_eq]\n\nvariable {C : Type*} [AddCommMonoid C] [PartialOrder C] [IsOrderedCancelAddMonoid C]","hard_negative":false,"metrics":{"chosen_tokens":237,"rejected_tokens":3,"token_jaccard":0.025,"token_length_ratio":0.012658},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"2b2ae743c980fc73f4778bfd3781b160415ea67c8f7e1bb7df768553e0447e1b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Fin\npublic import Mathlib.Algebra.Order.BigOperators.Group.Multiset\npublic import Mathlib.Data.Fin.VecNotation\npublic import Mathlib.LinearAlgebra.Matrix.Notation\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Martin Dvorak. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Martin Dvorak\n-/\n/-!\n\n# General-Valued Constraint Satisfaction Problems\n\nGeneral-Valued CSP is a very broad class of problems in discrete optimization.\nGeneral-Valued CSP subsumes Min-Cost-Hom (including 3-SAT for example) and Finite-Valued CSP.\n\n## Main definitions\n* `ValuedCSP`: A VCSP template; fixes a domain, a codomain, and allowed cost functions.\n* `ValuedCSP.Term`: One summand in a VCSP instance; calls a concrete function from given template.\n* `ValuedCSP.Term.evalSolution`: An evaluation of the VCSP term for given solution.\n* `ValuedCSP.Instance`: An instance of a VCSP problem over given template.\n* `ValuedCSP.Instance.evalSolution`: An evaluation of the VCSP instance for given solution.\n* `ValuedCSP.Instance.IsOptimumSolution`: Is given solution a minimum of the VCSP instance?\n* `Function.HasMaxCutProperty`: Can given binary function express the Max-Cut problem?\n* `FractionalOperation`: Multiset of operations on given domain of the same arity.\n* `FractionalOperation.IsSymmetricFractionalPolymorphismFor`: Is given fractional operation a\n  symmetric fractional polymorphism for given VCSP template?\n\n## References\n* [D. A. Cohen, M. C. Cooper, P. Creed, P. G. Jeavons, S. Živný,\n  *An Algebraic Theory of Complexity for Discrete Optimisation*][cohen2012]\n\n-/\n\n@[expose] public section\n\n/-- A template for a valued CSP problem over a domain `D` with costs in `C`.\nRegarding `C` we want to support `Bool`, `Nat`, `ENat`, `Int`, `Rat`, `NNRat`,\n`Real`, `NNReal`, `EReal`, `ENNReal`, and tuples made of any of those types. -/\n@[nolint unusedArguments]\nabbrev ValuedCSP (D C : Type*) [AddCommMonoid C] [PartialOrder C] [IsOrderedAddMonoid C] :=\n  Set (Σ (n : ℕ), (Fin n → D) → C) -- Cost functions `D^n → C` for any `n`\n\nvariable {D C : Type*} [AddCommMonoid C] [PartialOrder C] [IsOrderedAddMonoid C]\n\n/-- A term in a valued CSP instance over the template `Γ`. -/\nstructure ValuedCSP.Term (Γ : ValuedCSP D C) (ι : Type*) where\n  /-- Arity of the function -/\n  n : ℕ\n  /-- Which cost function is instantiated -/\n  f : (Fin n → D) → C\n  /-- The cost function comes from the template -/\n  inΓ : ⟨n, f⟩ ∈ Γ\n  /-- Which variables are plugged as arguments to the cost function -/\n  app : Fin n → ι\n\n/-- Evaluation of a `Γ` term `t` for given solution `x`. -/\ndef ValuedCSP.Term.evalSolution {Γ : ValuedCSP D C} {ι : Type*}\n    (t : Γ.Term ι) (x : ι → D) : C :=\n  t.f (x ∘ t.app)\n\n/-- A valued CSP instance over the template `Γ` with variables indexed by `ι`. -/\nabbrev ValuedCSP.Instance (Γ : ValuedCSP D C) (ι : Type*) : Type _ :=\n  Multiset (Γ.Term ι)\n\n/-- Evaluation of a `Γ` instance `I` for given solution `x`. -/\ndef ValuedCSP.Instance.evalSolution {Γ : ValuedCSP D C} {ι : Type*}\n    (I : Γ.Instance ι) (x : ι → D) : C :=\n  (I.map (·.evalSolution x)).sum\n\n/-- Condition for `x` being an optimum solution (min) to given `Γ` instance `I`. -/\ndef ValuedCSP.Instance.IsOptimumSolution {Γ : ValuedCSP D C} {ι : Type*}\n    (I : Γ.Instance ι) (x : ι → D) : Prop :=\n  ∀ y : ι → D, I.evalSolution x ≤ I.evalSolution y\n\n/-- Function `f` has Max-Cut property at labels `a` and `b` when `argmin f` is exactly\n`{ ![a, b], ![b, a] }`. -/\ndef Function.HasMaxCutPropertyAt (f : (Fin 2 → D) → C) (a b : D) : Prop :=\n  f ![a, b] = f ![b, a] ∧\n    ∀ x y : D, f ![a, b] ≤ f ![x, y] ∧ (f ![a, b] = f ![x, y] → a = x ∧ b = y ∨ a = y ∧ b = x)\n\n/-- Function `f` has Max-Cut property at some two non-identical labels. -/\ndef Function.HasMaxCutProperty (f : (Fin 2 → D) → C) : Prop :=\n  ∃ a b : D, a ≠ b ∧ f.HasMaxCutPropertyAt a b\n\n/-- Fractional operation is a finite unordered collection of D^m → D possibly with duplicates. -/\nabbrev FractionalOperation (D : Type*) (m : ℕ) : Type _ :=\n  Multiset ((Fin m → D) → D)\n\nvariable {m : ℕ}\n\n/-- Arity of the \"output\" of the fractional operation. -/\n@[simp]\ndef FractionalOperation.size (ω : FractionalOperation D m) : ℕ := ω.card\n\n/-- Fractional operation is valid iff nonempty. -/\ndef FractionalOperation.IsValid (ω : FractionalOperation D m) : Prop :=\n  ω ≠ ∅\n\n/-- Valid fractional operation contains an operation. -/\nlemma FractionalOperation.IsValid.contains {ω : FractionalOperation D m} (valid : ω.IsValid) :\n    ∃ g : (Fin m → D) → D, g ∈ ω :=\n  Multiset.exists_mem_of_ne_zero valid\n\n/-- Fractional operation applied to a transposed table of values. -/\ndef FractionalOperation.tt {ι : Type*} (ω : FractionalOperation D m) (x : Fin m → ι → D) :\n    Multiset (ι → D) :=\n  ω.map (fun (g : (Fin m → D) → D) (i : ι) => g ((Function.swap x) i))\n\n/-- Cost function admits given fractional operation, i.e., `ω` improves `f` in the `≤` sense. -/\ndef Function.AdmitsFractional {n : ℕ} (f : (Fin n → D) → C) (ω : FractionalOperation D m) : Prop :=\n  ∀ x : (Fin m → (Fin n → D)),\n    m • ((ω.tt x).map f).sum ≤ ω.size • Finset.univ.sum (fun i => f (x i))\n\n/-- Fractional operation is a fractional polymorphism for given VCSP template. -/\ndef FractionalOperation.IsFractionalPolymorphismFor\n    (ω : FractionalOperation D m) (Γ : ValuedCSP D C) : Prop :=\n  ∀ f ∈ Γ, f.snd.AdmitsFractional ω\n\n/-- Fractional operation is symmetric. -/\ndef FractionalOperation.IsSymmetric (ω : FractionalOperation D m) : Prop :=\n  ∀ x y : (Fin m → D), List.Perm (List.ofFn x) (List.ofFn y) → ∀ g ∈ ω, g x = g y\n\n/-- Fractional operation is a symmetric fractional polymorphism for given VCSP template. -/\ndef FractionalOperation.IsSymmetricFractionalPolymorphismFor\n    (ω : FractionalOperation D m) (Γ : ValuedCSP D C) : Prop :=\n  ω.IsFractionalPolymorphismFor Γ ∧ ω.IsSymmetric\n\nTarget:\nlemma Function.HasMaxCutPropertyAt.rows_lt_aux {C : Type*} [PartialOrder C]\n    {f : (Fin 2 → D) → C} {a b : D} (mcf : f.HasMaxCutPropertyAt a b) (hab : a ≠ b)\n    {ω : FractionalOperation D 2} (symmega : ω.IsSymmetric)\n    {r : Fin 2 → D} (rin : r ∈ (ω.tt ![![a, b], ![b, a]])) :\n    f ![a, b] < f r :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Optimization","family_id":"function","file_id":"mathlib/Mathlib/Combinatorics/Optimization/ValuedCSP.lean","sample_id":"4158bc4533e6e85b1b86e67b31a3d6eca02e3c8f701cb75f73a81c4775a53885"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4bbd86e73cbf3bb6938d81696c0fac5d64455a44efb1861863f6e712d6a46845","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"286b97964583ec10d54aee23d52750899f4ec1e260fb348ac7eb9f7b0bf1c2f4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8a9a5d45eecf81f80cc6ac9acad10ad3865265d34cf144e568617dadd24602fc","source_sha256":"e073d0499bdb4f94f221cf1b385cb626e3018e60f9a1a6981e11696230ed0c39","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Finset.disjoint_left]\n  rintro x hx hxc\n  obtain ⟨x, hx', rfl⟩ := mem_map.mp hxc\n  exact hs.compl_notMem hx' hx","hard_negative":true,"metrics":{"chosen_tokens":30,"rejected_tokens":5,"token_jaccard":0.12,"token_length_ratio":0.166667},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"2b3a8d0a0da7d24222fe2a0e13bf55a4eafca860be427bfd992c20f73fb0e309","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Card\npublic import Mathlib.Order.UpperLower.Basic\n\nNamespace:\nSet\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Intersecting families\n\nThis file defines intersecting families and proves their basic properties.\n\n## Main declarations\n\n* `Set.Intersecting`: Predicate for a set of elements in a generalized Boolean algebra to be an\n  intersecting family.\n* `Set.Intersecting.card_le`: An intersecting family can only take up to half the elements, because\n  `a` and `aᶜ` cannot simultaneously be in it.\n* `Set.Intersecting.is_max_iff_card_eq`: Any maximal intersecting family takes up half the elements.\n* `Set.IsIntersectingOf`: Predicate stating that a family `𝒜` of finsets is `L`-intersecting, i.e.,\n  meaning the intersection size of every pair of distinct members of `𝒜` belongs to `L ⊆ ℕ`.\n\n## References\n\n* [D. J. Kleitman, *Families of non-disjoint subsets*][kleitman1966]\n-/\n\n@[expose] public section\n\nassert_not_exists Monoid\n\nopen Finset\n\nnamespace Set\n\nsection SemilatticeInf\n\nvariable {α : Type*}\n\nvariable [SemilatticeInf α] [OrderBot α] {s t : Set α} {a b c : α}\n\n/-- A set family is intersecting if every pair of elements is non-disjoint. -/\ndef Intersecting (s : Set α) : Prop :=\n  ∀ ⦃a⦄, a ∈ s → ∀ ⦃b⦄, b ∈ s → ¬Disjoint a b\n\n@[gcongr, mono]\ntheorem Intersecting.mono (h : t ⊆ s) (hs : s.Intersecting) : t.Intersecting := fun _a ha _b hb =>\n  hs (h ha) (h hb)\n\ntheorem Intersecting.bot_notMem (hs : s.Intersecting) : ⊥ ∉ s := fun h => hs h h disjoint_bot_left\n\ntheorem Intersecting.ne_bot (hs : s.Intersecting) (ha : a ∈ s) : a ≠ ⊥ :=\n  ne_of_mem_of_not_mem ha hs.bot_notMem\n\ntheorem intersecting_empty : (∅ : Set α).Intersecting := fun _ => False.elim\n\n@[simp]\ntheorem intersecting_singleton : ({a} : Set α).Intersecting ↔ a ≠ ⊥ := by simp [Intersecting]\n\nprotected theorem Intersecting.insert (hs : s.Intersecting) (ha : a ≠ ⊥)\n    (h : ∀ b ∈ s, ¬Disjoint a b) : (insert a s).Intersecting := by\n  rintro b (rfl | hb) c (rfl | hc)\n  · rwa [disjoint_self]\n  · exact h _ hc\n  · exact fun H => h _ hb H.symm\n  · exact hs hb hc\n\ntheorem intersecting_insert :\n    (insert a s).Intersecting ↔ s.Intersecting ∧ a ≠ ⊥ ∧ ∀ b ∈ s, ¬Disjoint a b :=\n  ⟨fun h =>\n    ⟨h.mono <| subset_insert _ _, h.ne_bot <| mem_insert _ _, fun _b hb =>\n      h (mem_insert _ _) <| mem_insert_of_mem _ hb⟩,\n    fun h => h.1.insert h.2.1 h.2.2⟩\n\ntheorem intersecting_iff_pairwise_not_disjoint :\n    s.Intersecting ↔ (s.Pairwise fun a b => ¬Disjoint a b) ∧ s ≠ {⊥} := by\n  refine ⟨fun h => ⟨fun a ha b hb _ => h ha hb, ?_⟩, fun h a ha b hb hab => ?_⟩\n  · rintro rfl\n    exact intersecting_singleton.1 h rfl\n  have := h.1.eq ha hb (Classical.not_not.2 hab)\n  rw [this, disjoint_self] at hab\n  rw [hab] at hb\n  exact\n    h.2\n      (eq_singleton_iff_unique_mem.2\n        ⟨hb, fun c hc => not_ne_iff.1 fun H => h.1 hb hc H.symm disjoint_bot_left⟩)\n\nprotected theorem Subsingleton.intersecting (hs : s.Subsingleton) : s.Intersecting ↔ s ≠ {⊥} :=\n  intersecting_iff_pairwise_not_disjoint.trans <| and_iff_right <| hs.pairwise _\n\ntheorem intersecting_iff_eq_empty_of_subsingleton [Subsingleton α] (s : Set α) :\n    s.Intersecting ↔ s = ∅ := by\n  refine\n    subsingleton_of_subsingleton.intersecting.trans\n      ⟨not_imp_comm.2 fun h => subsingleton_of_subsingleton.eq_singleton_of_mem ?_, ?_⟩\n  · obtain ⟨a, ha⟩ := nonempty_iff_ne_empty.2 h\n    rwa [Subsingleton.elim ⊥ a]\n  · rintro rfl\n    exact (Set.singleton_nonempty _).ne_empty.symm\n\n/-- Maximal intersecting families are upper sets. -/\nprotected theorem Intersecting.isUpperSet (hs : s.Intersecting)\n    (h : ∀ t : Set α, t.Intersecting → s ⊆ t → s = t) : IsUpperSet s := by\n  classical\n    rintro a b hab ha\n    rw [h (Insert.insert b s) _ (subset_insert _ _)]\n    · exact mem_insert _ _\n    exact\n      hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab\n\n/-- Maximal intersecting families are upper sets. Finset version. -/\ntheorem Intersecting.isUpperSet' {s : Finset α} (hs : (s : Set α).Intersecting)\n    (h : ∀ t : Finset α, (t : Set α).Intersecting → s ⊆ t → s = t) : IsUpperSet (s : Set α) := by\n  classical\n    rintro a b hab ha\n    rw [h (Insert.insert b s) _ (Finset.subset_insert _ _)]\n    · exact mem_insert_self _ _\n    rw [coe_insert]\n    exact\n      hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab\n\nend SemilatticeInf\n\nsection\n\nvariable {α : Type*}\n\ntheorem Intersecting.exists_mem_set {𝒜 : Set (Set α)} (h𝒜 : 𝒜.Intersecting) {s t : Set α}\n    (hs : s ∈ 𝒜) (ht : t ∈ 𝒜) : ∃ a, a ∈ s ∧ a ∈ t :=\n  not_disjoint_iff.1 <| h𝒜 hs ht\n\ntheorem Intersecting.exists_mem_finset [DecidableEq α] {𝒜 : Set (Finset α)} (h𝒜 : 𝒜.Intersecting)\n    {s t : Finset α} (hs : s ∈ 𝒜) (ht : t ∈ 𝒜) : ∃ a, a ∈ s ∧ a ∈ t :=\n  not_disjoint_iff.1 <| disjoint_coe.not.2 <| h𝒜 hs ht\n\nvariable [BooleanAlgebra α]\n\ntheorem Intersecting.compl_notMem {s : Set α} (hs : s.Intersecting) {a : α} (ha : a ∈ s) :\n    aᶜ ∉ s := fun h => hs ha h disjoint_compl_right\n\ntheorem Intersecting.notMem {s : Set α} (hs : s.Intersecting) {a : α} (ha : aᶜ ∈ s) : a ∉ s :=\n  fun h => hs ha h disjoint_compl_left\n\nTarget:\ntheorem Intersecting.disjoint_map_compl {s : Finset α} (hs : (s : Set α).Intersecting) :\n    Disjoint s (s.map ⟨compl, compl_injective⟩) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_8a9a5d45eecf","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"51d63d3c47d50c05930532233183dd1da11a31ab9f6720d72c6457d77a829717","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/SetFamily","family_id":"intersecting","file_id":"mathlib/Mathlib/Combinatorics/SetFamily/Intersecting.lean","sample_id":"8a9a5d45eecf81f80cc6ac9acad10ad3865265d34cf144e568617dadd24602fc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"12279e85ffc4d0d8cf16eae5dea8e0e2916e98350ea6156d422f2d5856bb9f1e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"15e4f134fc4396fd42f4d434d7605aca7a909498c0d0844c2e35d252b2adb530","source_sha256":"7bb1b48aebf21c1532e493ca79c87d1bb8ee632b2dfff76d51a712868ddd9efa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← sSup_image']\n  exact hd.isLUB_csSup (Hne.image _) H","hard_negative":false,"metrics":{"chosen_tokens":17,"rejected_tokens":2,"token_jaccard":0.058824,"token_length_ratio":0.117647},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"2b5503a356257d696db89f9343f09e137f376d747ad538d568a11847ac6bafbd","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ConditionallyCompletePartialOrder.Basic\npublic import Mathlib.Order.GaloisConnection.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2026 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\n/-!\n# Indexed sup / inf in conditionally complete lattices\n\nThis file proves lemmas about `iSup` and `iInf` for functions valued in a conditionally complete\npartial order, as opposed to a conditionally complete lattice.\n\n## TODO\n\n+ Use `@[to_dual]` in the `GaloisConnection` and `OrderIso` sections.\n\n-/\n\npublic section\n\n-- Guard against import creep\nassert_not_exists Multiset\n\nopen Function OrderDual Set\n\nvariable {α β γ : Type*} {ι : Sort*}\n\nsection ConditionallyCompletePartialOrderSup\n\nvariable [ConditionallyCompletePartialOrderSup α] {a b : α}\n\n@[to_dual]\ntheorem Directed.isLUB_ciSup [Nonempty ι] {f : ι → α} (hd : Directed (· ≤ ·) f)\n    (H : BddAbove (range f)) : IsLUB (range f) (⨆ i, f i) :=\n  hd.directedOn_range.isLUB_csSup (range_nonempty f) H\n\n@[to_dual]\n\nTarget:\ntheorem DirectedOn.isLUB_ciSup_set {f : β → α} {s : Set β} (hd : DirectedOn (· ≤ ·) (f '' s))\n    (H : BddAbove (f '' s)) (Hne : s.Nonempty) :\n    IsLUB (f '' s) (⨆ i : s, f i) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/ConditionallyCompletePartialOrder","family_id":"directedon","file_id":"mathlib/Mathlib/Order/ConditionallyCompletePartialOrder/Indexed.lean","sample_id":"15e4f134fc4396fd42f4d434d7605aca7a909498c0d0844c2e35d252b2adb530"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6ed251e98bcc09fc840097a1704c71dc97a2385a1bbef8d0c694a48f689a68b3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ae377ebe4e81614b721fd67079ff2abc3a5bfe01ce66b84d4f05d9eb4b4813fb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0d7956b6ebdcf8c47b68d95a35dc56b9f93a664feaf120a85ef061f559b47042","source_sha256":"1513994cab9129a1b43fa5d8ae4d7aeabdb9cc958d905bbce24e5e0da36801ec","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [OneSkeletonGraph_isLink, closedCell_zero_eq_singleton, closedCell_zero_eq_singleton,\n    singleton_union]","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.1,"token_length_ratio":0.272727},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"2ba6d52516f26a891f5a039837178ed0f3e0848640e5a25f848e4b0ab49b3026","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.CWComplex.Classical.Finite\npublic import Mathlib.Combinatorics.Graph.Basic\n\nNamespace:\nTopology.CWComplex.OneSkeletonGraph\n\nLocal context:\n/-\nCopyright (c) 2026 Jun Kwon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jun Kwon\n-/\n/-!\n# 1-skeletons of CW complexes as graphs\n\nIn this file we define the 1-skeleton of a CW complex as a graph.\n\n## Main definitions\n* `CWComplex.OneSkeletonGraph`: the 1-skeleton of a CW complex as a graph.\n\n-/\n\npublic section\n\nopen Metric Set Graph\n\nnamespace Topology\n\nvariable {X : Type*} [TopologicalSpace X]\n\n/-- The 1-skeleton of a CW complex as a graph. -/\n@[expose, simps]\ndef CWComplex.OneSkeletonGraph (C : Set X) [CWComplex C] : Graph (cell C 0) (cell C 1) where\n  vertexSet := univ\n  edgeSet := univ\n  IsLink e x y := cellFrontier 1 e = closedCell 0 x ∪ closedCell 0 y\n  isLink_symm := by grind [symm_def]\n  eq_or_eq_of_isLink_of_isLink e x y z w h1 h2 := by\n    simp_rw [closedCell_zero_eq_singleton] at h1 h2\n    rw [h1] at h2\n    simp only [(RelCWComplex.injective_map_zero C).eq_iff, union_singleton, pair_eq_pair_iff] at h2\n    tauto\n  left_mem_of_isLink _ _ _ _ := mem_univ _\n  edge_mem_iff_exists_isLink e := by\n    simp only [mem_univ, true_iff]\n    exact exists_cellFrontier_one_eq e\n\nnamespace CWComplex.OneSkeletonGraph\n\nvariable {C : Set X} [CWComplex C]\n\nTarget:\nlemma isLink_iff_pair (e : cell C 1) (x y : cell C 0) :\n    (OneSkeletonGraph C).IsLink e x y ↔ cellFrontier 1 e = {map 0 x ![], map 0 y ![]} :=\n\nProof body:\n","rejected":"by\n  exact isLink_iff_pair","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"47140aea32669f13703c8388ccbf1fc063152f5b01aafafc70d92b3035f0ed07","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/CWComplex","family_id":"islink_iff_pair","file_id":"mathlib/Mathlib/Topology/CWComplex/Classical/Graph.lean","sample_id":"0d7956b6ebdcf8c47b68d95a35dc56b9f93a664feaf120a85ef061f559b47042"},"split":"train"}
{"binding":{"chosen_proof_sha256":"61743a744bcd9aa71d7e4c51326303887dfd1d0a7a128d08960e283447fbb42c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"739d2e77b2892cb2da9b9ac9b16f4da929592ced9d530dbf87b4be7c879143a4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"06643b8b06bfa39bedc6e6457247fd95883e76318998fda10cf95f90e40967df","source_sha256":"65877b034edf8cebcde3630a91726f90cf5b7369055e4e4a4603c386cc7b9be8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [numLeaves_eq_numNodes_succ]\n  exact x.numNodes.zero_lt_succ","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":15,"token_jaccard":0.769231,"token_length_ratio":1.363636},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"2bb4ef543f7c09e9d62a0a4b621121477d8345b9e84f3f526d24e1a86336bd93","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.Notation\npublic import Mathlib.Util.CompileInductive\nimport Batteries.Tactic.Alias\n\nNamespace:\nBinaryTree\n\nLocal context:\n/-\nCopyright (c) 2019 mathlib community. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Wojciech Nawrocki\n-/\n/-!\n# Binary tree\n\nProvides binary tree storage for values of any type, with O(lg n) retrieval.\nSee also `Lean.Data.RBTree` for red-black trees - this version allows more operations\nto be defined and is better suited for in-kernel computation.\n\nWe also specialize for `BinaryTree Unit`, which is a binary tree without any\nadditional data. We provide the notation `a △ b` for making a `BinaryTree Unit` with children\n`a` and `b`.\n\n## References\n\n<https://leanprover-community.github.io/archive/stream/113488-general/topic/tactic.20question.html>\n-/\n\n@[expose] public section\n\n\n/-- A binary tree with values stored in non-leaf nodes. -/\ninductive BinaryTree.{u} (α : Type u) : Type u\n  | nil : BinaryTree α\n  | node (value : α) (left : BinaryTree α) (right : BinaryTree α) : BinaryTree α\n  deriving DecidableEq, Repr\ncompile_inductive% BinaryTree\n\n@[deprecated (since := \"2026-06-07\"), reducible]\nalias Tree := BinaryTree\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.nil`. -/\n@[deprecated BinaryTree.nil (since := \"2026-06-07\")]\nabbrev Tree.nil.{u} {α : Type u} : Tree α := BinaryTree.nil\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.node`. -/\n@[deprecated BinaryTree.node (since := \"2026-06-07\")]\nabbrev Tree.node.{u} {α : Type u}\n    (value : α) (left : Tree α) (right : Tree α) : Tree α :=\n  BinaryTree.node value left right\n\nnamespace BinaryTree\n\nuniverse u\n\nvariable {α : Type u}\n\ninstance : Inhabited (BinaryTree α) :=\n  ⟨nil⟩\n\n/--\nDo an action for every node of the tree.\nActions are taken in node -> left subtree -> right subtree recursive order.\nThis function is the `traverse` function for the `Traversable BinaryTree` instance.\n-/\ndef traverse\n    {m : Type* → Type*} [Applicative m] {α β} (f : α → m β) :\n    BinaryTree α → m (BinaryTree β)\n  | .nil => pure nil\n  | .node a l r => .node <$> f a <*> traverse f l <*> traverse f r\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.traverse`. -/\n@[deprecated BinaryTree.traverse (since := \"2026-06-07\")]\nabbrev _root_.Tree.traverse {m : Type* → Type*} [Applicative m] {α β} (f : α → m β)\n(t : Tree α) : m (Tree β) :=\n  BinaryTree.traverse f t\n\n/-- Apply a function to each value in the BinaryTree.\nThis is the `map` function for the `BinaryTree` functor.\n-/\ndef map {β} (f : α → β) : BinaryTree α → BinaryTree β\n  | nil => nil\n  | node a l r => node (f a) (map f l) (map f r)\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.map`. -/\n@[deprecated BinaryTree.map (since := \"2026-06-07\")]\nabbrev _root_.Tree.map {α β} (f : α → β) (t : Tree α) : Tree β := BinaryTree.map f t\n\ntheorem id_map (t : BinaryTree α) : t.map id = t := by\n  induction t with\n  | nil => rw [map]\n  | node v l r hl hr => rw [map, hl, hr, id_eq]\n\ntheorem comp_map {β γ : Type*} (f : α → β) (g : β → γ) (t : BinaryTree α) :\n    t.map (g ∘ f) = (t.map f).map g := by\n  induction t with\n  | nil => rw [map, map, map]\n  | node v l r hl hr => rw [map, map, map, hl, hr, Function.comp_apply]\n\ntheorem traverse_pure (t : BinaryTree α) {m : Type u → Type*}\n    [Applicative m] [LawfulApplicative m] :\n    t.traverse (pure : α → m α) = pure t := by\n  induction t with\n  | nil => rw [traverse]\n  | node v l r hl hr =>\n    rw [traverse, hl, hr, map_pure, pure_seq, seq_pure, map_pure, map_pure]\n\n/-- The number of internal nodes (i.e. not including leaves) of a binary tree -/\n@[simp]\ndef numNodes : BinaryTree α → ℕ\n  | nil => 0\n  | node _ a b => a.numNodes + b.numNodes + 1\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.numNodes`. -/\n@[deprecated BinaryTree.numNodes (since := \"2026-06-07\")]\nabbrev _root_.Tree.numNodes {α} (t : Tree α) : ℕ := BinaryTree.numNodes t\n\n/-- The number of leaves of a binary tree -/\n@[simp]\ndef numLeaves : BinaryTree α → ℕ\n  | nil => 1\n  | node _ a b => a.numLeaves + b.numLeaves\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.numLeaves`. -/\n@[deprecated BinaryTree.numLeaves (since := \"2026-06-07\")]\nabbrev _root_.Tree.numLeaves {α} (t : Tree α) : ℕ := BinaryTree.numLeaves t\n\n/-- The height - length of the longest path from the root - of a binary tree -/\n@[simp]\ndef height : BinaryTree α → ℕ\n  | nil => 0\n  | node _ a b => max a.height b.height + 1\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.height`. -/\n@[deprecated BinaryTree.height (since := \"2026-06-07\")]\nabbrev _root_.Tree.height {α} (t : Tree α) : ℕ := BinaryTree.height t\n\ntheorem numLeaves_eq_numNodes_succ (x : BinaryTree α) : x.numLeaves = x.numNodes + 1 := by\n  induction x <;> simp [*, Nat.add_comm, Nat.add_assoc, Nat.add_left_comm]\n\nTarget:\ntheorem numLeaves_pos (x : BinaryTree α) : 0 < x.numLeaves :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [numLeaves_eq_numNodes_succ]\n  exact x.numNodes.zero_lt_succ","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Tree","family_id":"numleaves_pos","file_id":"mathlib/Mathlib/Data/Tree/Basic.lean","sample_id":"06643b8b06bfa39bedc6e6457247fd95883e76318998fda10cf95f90e40967df"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fd3b7a2ce076a9542b33ce25f06638c50f65a2500253da07e8afbc3c349b995c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"59c573b6651dd9610dd6079b8c99d6d2fda95dbe61cad8eef27d531ec73203de","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f2813c1994d0014dcc0cbbe6ef4d234abd05fcb0aaba7d50c3b4450586130664","source_sha256":"c60b7992efa7a8848883c62ac5e7382d125b9535f551900578b8595483f08374","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction m with\n  | zero => simpa\n  | succ m ih =>\n    simp_rw [pow_succ _ m, pow_mul]\n    exact ih.pow_add_one hpI m.succ_ne_zero","hard_negative":false,"metrics":{"chosen_tokens":29,"rejected_tokens":36,"token_jaccard":0.916667,"token_length_ratio":1.241379},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"2bde5acd05e04ef6ef73cb14e5642881118639397ea895ddf44e4035b176129f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharP.Lemmas\npublic import Mathlib.LinearAlgebra.SModEq.Basic\npublic import Mathlib.RingTheory.Ideal.Operations\n\nNamespace:\nSModEq\n\nLocal context:\n/-\nCopyright (c) 2026 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n\n/-! # Lemmas about SModEq related to powers -/\n\npublic section\n\nnamespace SModEq\nvariable {R : Type*} [CommRing R] {I J : Ideal R} {p : ℕ} (hpI : (p : R) ∈ I)\ninclude hpI\n\ntheorem pow_mul_of_le {x y : R} (h : x ≡ y [SMOD J]) (hJI : J ≤ I) :\n    x ^ p ≡ y ^ p [SMOD J * I] := by\n  have h₁ := idealQuotientMk.mp <| h.mono hJI\n  rw [SModEq.sub_mem] at h ⊢\n  rw [← Commute.mul_neg_geom_sum₂ (.all _ _)]\n  refine Ideal.mul_mem_mul h ?_\n  have h₂ : (p : R ⧸ I) = 0 := by simpa using Ideal.Quotient.eq_zero_iff_mem.mpr hpI\n  simp only [← Ideal.Quotient.eq_zero_iff_mem, map_sum, map_mul, map_pow, h₁, ← pow_add]\n  trans ∑ x ∈ Finset.range p, Ideal.Quotient.mk I y ^ (p - 1)\n  · exact Finset.sum_congr rfl fun _ _ ↦ by grind\n  simp [h₂]\n\ntheorem pow_add_one {x y : R} {m : ℕ} (hm : m ≠ 0) (h : x ≡ y [SMOD I ^ m]) :\n    x ^ p ≡ y ^ p [SMOD I ^ (m + 1)] := h.pow_mul_of_le hpI <| I.pow_le_self hm\n\nTarget:\ntheorem pow_pow_add_one {x y : R} (h : x ≡ y [SMOD I]) (m : ℕ) :\n    x ^ p ^ m ≡ y ^ p ^ m [SMOD I ^ (m + 1)] :=\n\nProof body:\n","rejected":"```lean\nby\n  induction m with\n  | zero => simpa\n  | succ m ih =>\n    simp_rw [pow_succ _ m, pow_mul]\n    exact ih.pow_add_one hpI m.succ_ne_zero\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/SModEq","family_id":"pow_pow_add_one","file_id":"mathlib/Mathlib/LinearAlgebra/SModEq/Pow.lean","sample_id":"f2813c1994d0014dcc0cbbe6ef4d234abd05fcb0aaba7d50c3b4450586130664"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6db23e1003596f7373b150a65cdc1843f9cceb615009a5e13fae5265b46dd136","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3dbc97992882d9b356771d88365a2201f668dc9f8e7a14b9314d75b55360da6b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"70ee751da2d1168fe9d7a11059342b3a3f15c210604e29d540ce2dd98fb9afcf","source_sha256":"1d77f6aaa753df530e96edf4819026a453d32ae6fc473ac307ea722c26a6c807","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro L₁ L₂ hflat\n  apply Subtype.ext\n  exact h L₁.val L₂.val L₁.prop L₂.prop hflat","hard_negative":true,"metrics":{"chosen_tokens":30,"rejected_tokens":5,"token_jaccard":0.1875,"token_length_ratio":0.166667},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"2be29a7419cf004992680d994be85b01b44fe0aac71190ea7afe03feb9cc8d97","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Basic\n\nNamespace:\nInformationTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Elazar Gershuni. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Elazar Gershuni\n-/\n/-!\n# Uniquely Decodable Codes\n\nThis file defines uniquely decodable codes and proves basic properties.\n\n## Main definitions\n\n* `UniquelyDecodable`: A set of codewords is uniquely decodable if distinct concatenations\n  of codewords yield distinct strings.\n\n## Main results\n\n* `UniquelyDecodable.epsilon_not_mem`: Uniquely decodable codes cannot contain the empty\n  string.\n* `UniquelyDecodable.flatten_injective`: The flatten function is injective on lists of\n  codewords from a uniquely decodable code.\n-/\n\n@[expose] public section\n\nnamespace InformationTheory\n\nvariable {α : Type*}\n\n/-- A set of lists is uniquely decodable if distinct concatenations yield distinct strings. -/\ndef UniquelyDecodable (S : Set (List α)) : Prop :=\n  ∀ (L₁ L₂ : List (List α)),\n    (∀ w ∈ L₁, w ∈ S) → (∀ w ∈ L₂, w ∈ S) →\n    L₁.flatten = L₂.flatten → L₁ = L₂\n\nvariable {S : Set (List α)}\n\n/-- If a code is uniquely decodable, it does not contain the empty string.\n\nThe empty string can be \"decoded\" as either zero or two copies of itself,\nviolating unique decodability. -/\nlemma UniquelyDecodable.epsilon_not_mem\n    (h : UniquelyDecodable S) :\n    [] ∉ S := by\n  simpa using h [[]] [[], []]\n\nTarget:\nlemma UniquelyDecodable.flatten_injective (h : UniquelyDecodable S) :\n    Function.Injective (fun (L : {L : List (List α) // ∀ x ∈ L, x ∈ S}) => L.val.flatten) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_70ee751da2d1","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1c3163e847cc85bf955bf7271f36d6b7faea8ea4f36440e64281cf0552daec97","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"InformationTheory/Coding","family_id":"uniquelydecodable","file_id":"mathlib/Mathlib/InformationTheory/Coding/UniquelyDecodable.lean","sample_id":"70ee751da2d1168fe9d7a11059342b3a3f15c210604e29d540ce2dd98fb9afcf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ea6d6b98ce6bb5df330d0c3825d495329a7a811e1188283f31f2efe51dfb0168","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c37348c1479d7ebb1f5927cb69b83756ac21232b863e187128fe0df7669b5624","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4ea2d4d2d7f00624026e8f321443e14dc6b69d2bf90adb74245fe472882d93fb","source_sha256":"b6022f061f06753f0797fa4fa7ab43cc6fb4c7d61abaa7b4bb4707d4dc0299c4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← LinearMap.comp_assoc, ← LinearMap.comp_assoc, TensorProduct.map_map_comp_assoc_eq]","hard_negative":true,"metrics":{"chosen_tokens":17,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.117647},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"2be6f4c30e76734dcb0bfc3143e3ff5890d28146213a5d70f454f00cd40a5df0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorProduct.Tower\npublic import Mathlib.RingTheory.Coalgebra.Basic\nimport Mathlib.Tactic.Attr.Register\n\nNamespace:\nCoassocSimps\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang, Yaël Dillies\n-/\n/-!\n# Tactic to reassociate comultiplication in a coalgebra\n\n`coassoc_simps` is a simp set useful to prove tautologies on coalgebras.\n\nThe general algorithm it follows is to push the associators `TensorProduct.assoc` and\ncommutators `TensorProduct.comm` inwards (to the right) until they cancel against\nco-multiplications.\n\nThe simp set makes the following choice of normal form\n* It regards `TensorProduct.map`, `TensorProduct.assoc`, `TensorProduct.comm` as the primitive\n  constructions and rewrites everything else such as `lTensor`, `leftComm` using them.\n* It rewrites both sides into a right associated composition of linear maps.\n  In particular `LinearMap.comp_assoc` and `LinearEquiv.coe_trans` are tagged.\n* It rewrites `(f₂ ⊗ g₂) ∘ (f₁ ⊗ g₁)` into `(f₂ ∘ f₁) ⊗ (g₂ ∘ g₁)`.\n\n## Notes\n\n- It is not confluent with `(ε ⊗ₘ id) ∘ₗ δ = λ⁻¹`.\n  It is often useful to `trans` (or `calc`) with a term containing\n  `(ε ⊗ₘ _) ∘ₗ δ` or `(_ ⊗ₘ ε) ∘ₗ δ`,\n  and use one of `map_counit_comp_comul_left` `map_counit_comp_comul_right`\n  `map_counit_comp_comul_left_assoc` `map_counit_comp_comul_right_assoc` to continue.\n\n- Some lemmas (e.g. `lid_comp_map : λ ∘ₗ (f ⊗ₘ g) = g ∘ₗ λ ∘ₗ (f ⊗ₘ id)`) loops when tagged as simp,\n  so we wrap it inside a rudimentary simproc that only fires when `g ≠ id`.\n-/\n\n@[expose] public section\n\nopen TensorProduct\n\nopen LinearMap (id)\nopen Coalgebra\n\nopen Qq\nnamespace CoassocSimps\n\nvariable {R A M N P M' N' P' Q Q' M₁ M₂ M₃ N₁ N₂ N₃ : Type*}\n    [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A]\n    [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P]\n    [AddCommMonoid M'] [Module R M'] [AddCommMonoid N'] [Module R N']\n    [AddCommMonoid P'] [Module R P'] [AddCommMonoid Q] [Module R Q] [AddCommMonoid Q'] [Module R Q']\n    [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃]\n    [AddCommMonoid N₁] [AddCommMonoid N₂] [AddCommMonoid N₃]\n    [Module R M₁] [Module R M₂] [Module R M₃] [Module R N₁] [Module R N₂] [Module R N₃]\n\nlocal notation3 \"α\" => (TensorProduct.assoc R _ _ _).toLinearMap\nlocal notation3 \"α⁻¹\" => (TensorProduct.assoc R _ _ _).symm.toLinearMap\nlocal notation3 \"λ\" => (TensorProduct.lid R _).toLinearMap\nlocal notation3 \"λ⁻¹\" => (TensorProduct.lid R _).symm.toLinearMap\nlocal notation3 \"ρ\" => (TensorProduct.rid R _).toLinearMap\nlocal notation3 \"ρ⁻¹\" => (TensorProduct.rid R _).symm.toLinearMap\nlocal notation3 \"β\" => (TensorProduct.comm R _ _).toLinearMap\nlocal infix:90 \" ⊗ₘ \" => TensorProduct.map\nlocal notation3 \"δ\" => comul (R := R)\nlocal notation3 \"ε\" => counit (R := R)\n\nattribute [coassoc_simps] LinearMap.comp_id LinearMap.id_comp TensorProduct.map_id\n  LinearMap.lTensor_def LinearMap.rTensor_def LinearMap.comp_assoc\n  LinearEquiv.coe_trans LinearEquiv.trans_symm\n  LinearEquiv.refl_toLinearMap TensorProduct.toLinearMap_congr\n  LinearEquiv.comp_symm LinearEquiv.symm_comp LinearEquiv.symm_symm\n  LinearEquiv.coe_lTensor LinearEquiv.coe_lTensor_symm\n  LinearEquiv.coe_rTensor LinearEquiv.coe_rTensor_symm\n  IsCocomm.comm_comp_comul TensorProduct.AlgebraTensorModule.map_eq\n  TensorProduct.AlgebraTensorModule.assoc_eq TensorProduct.AlgebraTensorModule.rightComm_eq\n  TensorProduct.tensorTensorTensorComm TensorProduct.AlgebraTensorModule.tensorTensorTensorComm\n  TensorProduct.AlgebraTensorModule.congr_eq LinearEquiv.comp_symm_assoc\n  LinearEquiv.symm_comp_assoc TensorProduct.rightComm_def TensorProduct.leftComm_def\n  TensorProduct.comm_symm TensorProduct.comm_comp_comm TensorProduct.comm_comp_comm_assoc\n\nattribute [coassoc_simps← ] TensorProduct.map_comp TensorProduct.map_map_comp_assoc_eq\n  TensorProduct.map_map_comp_assoc_symm_eq\n\n@[coassoc_simps]\nlemma TensorProduct.map_comp_assoc\n    (f : M →ₗ[R] N) (g : N →ₗ[R] P) (f' : M' →ₗ[R] N') (g' : N' →ₗ[R] P') (φ : M₁ →ₗ[R] M ⊗[R] M') :\n    map g g' ∘ₗ map f f' ∘ₗ φ = map (g ∘ₗ f) (g' ∘ₗ f') ∘ₛₗ φ := by\n  rw [← LinearMap.comp_assoc, TensorProduct.map_comp]\n\n@[coassoc_simps← ]\n\nTarget:\nlemma TensorProduct.map_map_comp_assoc_eq_assoc\n    (f₁ : M₁ →ₗ[R] N₁) (f₂ : M₂ →ₗ[R] N₂) (f₃ : M₃ →ₗ[R] N₃) (f : M →ₗ[R] M₁ ⊗[R] M₂ ⊗[R] M₃) :\n    f₁ ⊗ₘ (f₂ ⊗ₘ f₃) ∘ₗ α ∘ₗ f = α ∘ₗ ((f₁ ⊗ₘ f₂) ⊗ₘ f₃) ∘ₗ f :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_4ea2d4d2d7f0","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"493e92c2ee8c8655b1b45bbcce545a2eaa2f55240dcbe5fd9fd93d2eb9e42712","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Coalgebra","family_id":"tensorproduct","file_id":"mathlib/Mathlib/RingTheory/Coalgebra/CoassocSimps.lean","sample_id":"4ea2d4d2d7f00624026e8f321443e14dc6b69d2bf90adb74245fe472882d93fb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a18d962dbf46cb1bc44d7afa74d89d68066f49e5784c2cb2792cd31f2ea27cd6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1047399b822c6a1bbaabd56b32edd0709c973672b47f8d7183452371a30af927","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5b6616e952970d213dc2af95acd62437ab9f4ee924f0692e7518320229a74eaa","source_sha256":"f440a69580513b2b91fa2379cabfe0bde1204ddec625489d040b307defdad141","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [H.apply_zero_path, H.apply_one_path, H.evalAt_eq]\n  erw [H.evalAt_eq]\n  dsimp only [prodToProdTopI]\n  constructor\n  · slice_lhs 2 4 => rw [eqToHom_trans, eqToHom_refl] -- Porting note: this ↓ `simp` didn't do this\n    slice_lhs 2 4 => simp [← CategoryTheory.Functor.map_comp]\n    rfl\n  · slice_lhs 2 4 => rw [eqToHom_trans, eqToHom_refl] -- Porting note: this ↓ `simp` didn't do this\n    slice_lhs 2 4 => simp [← CategoryTheory.Functor.map_comp]\n    rfl","hard_negative":true,"metrics":{"chosen_tokens":103,"rejected_tokens":3,"token_jaccard":0.025641,"token_length_ratio":0.029126},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"2bf277499374f1270da7f25a593a292876be0b7dbfdf9bfaa696619678754124","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Homotopy.Equiv\npublic import Mathlib.CategoryTheory.Equivalence\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.Product\n\nNamespace:\nContinuousMap.Homotopy\n\nLocal context:\n/-\nCopyright (c) 2022 Praneeth Kolichala. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Praneeth Kolichala, Yury Kudryashov\n-/\n/-!\n# Homotopic maps induce naturally isomorphic functors\n\n## Main definitions\n\n- `FundamentalGroupoidFunctor.homotopicMapsNatIso H` The natural isomorphism\n  between the induced functors `f : π(X) ⥤ π(Y)` and `g : π(X) ⥤ π(Y)`, given a homotopy\n  `H : f ∼ g`\n\n- `FundamentalGroupoidFunctor.equivOfHomotopyEquiv hequiv` The equivalence of the categories\n  `π(X)` and `π(Y)` given a homotopy equivalence `hequiv : X ≃ₕ Y` between them.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nuniverse u v\n\nopen FundamentalGroupoid CategoryTheory FundamentalGroupoidFunctor\nopen scoped FundamentalGroupoid unitInterval\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Let `F` be a homotopy between two continuous maps `f g : C(X, Y)`.\nGiven a path `p : Path x₁ x₂` in the domain, consider the following two paths in the codomain.\nOne path goes along the image of `p` under `f`, then along the trajectory of `x₂` under `F`.\nThe other path goes along the trajectory of `x₁` under `F`, then along the image of `p` under `g`.\n\nThese two paths are homotopic. -/\ntheorem Path.Homotopic.map_trans_evalAt {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n    {f g : C(X, Y)} (F : f.Homotopy g) {x₁ x₂ : X} (p : Path x₁ x₂) :\n    ((p.map (map_continuous f)).trans (F.evalAt x₂)).Homotopic\n      ((F.evalAt x₁).trans (p.map (map_continuous g))) := by\n  /- Let `G` be the continuous map on the unit square sending `(t, s)` to `F(t, p(s))`.\n  Then our homotopy is the image under `G` of a homotopy\n  between the two paths from `(0, 0)` to `(1, 1)` along the sides of the square. -/\n  set G : C(I × I, Y) := F.toContinuousMap.comp (.prodMap (.id _) p)\n  set p₁ : Path ((0, 0) : I × I) (1, 1) := .prod (.trans (.refl _) .id) (.trans .id (.refl _))\n  set p₂ : Path ((0, 0) : I × I) (1, 1) := .prod (.trans .id (.refl _)) (.trans (.refl _) .id)\n  set Fsq : p₁.Homotopy p₂ :=\n    Path.Homotopic.prodHomotopy (.trans (.reflTrans _) (.symm <| .transRefl _))\n      (.trans (.transRefl _) (.symm <| .reflTrans _))\n  refine ⟨((Fsq.map G).pathCast ?H0 ?H1).cast ?hp ?hq⟩\n  all_goals aesop (add simp Path.trans_apply)\n\nnamespace FundamentalGroupoidFunctor\n\nopen CategoryTheory\nopen scoped FundamentalGroupoid ContinuousMap\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n  {f g : C(X, Y)}\n\nset_option backward.isDefEq.respectTransparency false in\nset_option pp.proofs.withType true in\n/-- Given a homotopy H : f ∼ g, we have an associated natural isomorphism between the induced\nfunctors `map f` and `map g` on fundamental groupoids. -/\ndef homotopicMapsNatIso (H : ContinuousMap.Homotopy f g) : map f ⟶ map g where\n  app x := ⟦H.evalAt x.as⟧\n  naturality := by\n    rintro ⟨x⟩ ⟨y⟩ p\n    rcases Path.Homotopic.Quotient.mk_surjective p with ⟨p, rfl⟩\n    simp only [map_map, Path.Homotopic.Quotient.mk''_eq_mk, comp_eq,\n      ← Path.Homotopic.Quotient.mk_map, ← Path.Homotopic.Quotient.mk_trans]\n    rw [Path.Homotopic.Quotient.eq]\n    exact .map_trans_evalAt _ _\n\ninstance (H : ContinuousMap.Homotopy f g) : IsIso (homotopicMapsNatIso H) :=\n  NatIso.isIso_of_isIso_app _\n\nopen scoped ContinuousMap\n\n/-- Homotopy equivalent topological spaces have equivalent fundamental groupoids. -/\ndef equivOfHomotopyEquiv {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] (hequiv : X ≃ₕ Y) :\n    πₓ (.of X) ≌ πₓ (.of Y) := by\n  apply CategoryTheory.Equivalence.mk (map hequiv.toFun) (map hequiv.invFun)\n  · simpa only [FundamentalGroupoid.map_id, FundamentalGroupoid.map_comp]\n      using (asIso (homotopicMapsNatIso hequiv.left_inv.some)).symm\n  · simpa only [FundamentalGroupoid.map_id, FundamentalGroupoid.map_comp]\n      using asIso (homotopicMapsNatIso hequiv.right_inv.some)\n\nend FundamentalGroupoidFunctor\n\n/-!\n### Old proof\n\nThe rest of the file contains definitions and theorems required to write the same proof\nin a slightly different manner.\n\nThe proof was rewritten in 2025 for two reasons:\n\n- the new proof is much more straightforward;\n- the new proof is fully universe polymorphic.\n\nTODO: review which of these definitions and theorems are useful for other reasons,\nthen deprecate the rest of them.\n-/\n\nnamespace unitInterval\n\n/-- The path 0 ⟶ 1 in `I` -/\ndef path01 : Path (0 : I) 1 where\n  toFun := id\n  source' := rfl\n  target' := rfl\n\n/-- The path 0 ⟶ 1 in `ULift I` -/\ndef upath01 : Path (ULift.up 0 : ULift.{u} I) (ULift.up 1) where\n  toFun := ULift.up\n  source' := rfl\n  target' := rfl\n\n/-- The homotopy path class of 0 → 1 in `ULift I` -/\ndef uhpath01 : @fromTop (TopCat.of <| ULift.{u} I) (ULift.up (0 : I)) ⟶ fromTop (ULift.up 1) :=\n  ⟦upath01⟧\n\nend unitInterval\n\nnamespace ContinuousMap.Homotopy\n\nopen unitInterval (uhpath01)\n\nsection Casts\n\n/-- Abbreviation for `eqToHom` that accepts points in a topological space -/\nabbrev hcast {X : TopCat.{u}} {x₀ x₁ : X} (hx : x₀ = x₁) : fromTop x₀ ⟶ fromTop x₁ :=\n  eqToHom <| FundamentalGroupoid.ext hx\n\n@[simp]\ntheorem hcast_def {X : TopCat.{u}} {x₀ x₁ : X} (hx₀ : x₀ = x₁) :\n    hcast hx₀ = eqToHom (FundamentalGroupoid.ext hx₀) :=\n  rfl\n\nvariable {X₁ X₂ Y : TopCat.{u}} {f : C(X₁, Y)} {g : C(X₂, Y)} {x₀ x₁ : X₁} {x₂ x₃ : X₂}\n  {p : Path x₀ x₁} {q : Path x₂ x₃} (hfg : ∀ t, f (p t) = g (q t))\ninclude hfg\n\n/-- If `f(p(t) = g(q(t))` for two paths `p` and `q`, then the induced path homotopy classes\n`f(p)` and `g(p)` are the same as well, despite having a priori different types -/\ntheorem heq_path_of_eq_image :\n    (πₘ (TopCat.ofHom f)).map ⟦p⟧ ≍ (πₘ (TopCat.ofHom g)).map ⟦q⟧ := by\n  simp only [map_eq]\n  apply Path.Homotopic.hpath_hext\n  exact hfg\n\nset_option backward.privateInPublic true in\nprivate theorem start_path : f x₀ = g x₂ := by convert! hfg 0 <;> simp only [Path.source]\n\nset_option backward.privateInPublic true in\nprivate theorem end_path : f x₁ = g x₃ := by convert! hfg 1 <;> simp only [Path.target]\n\nset_option backward.isDefEq.respectTransparency false in\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ntheorem eq_path_of_eq_image :\n    (πₘ (TopCat.ofHom f)).map ⟦p⟧ =\n        hcast (start_path hfg) ≫ (πₘ (TopCat.ofHom g)).map ⟦q⟧ ≫ hcast (end_path hfg).symm := by\n  rw [conj_eqToHom_iff_heq\n    ((πₘ (TopCat.ofHom f)).map ⟦p⟧) ((πₘ (TopCat.ofHom g)).map ⟦q⟧)\n    (FundamentalGroupoid.ext <| start_path hfg)\n    (FundamentalGroupoid.ext <| end_path hfg)]\n  exact heq_path_of_eq_image hfg\n\nend Casts\n\n-- We let `X` and `Y` be spaces, and `f` and `g` be homotopic maps between them\nvariable {X Y : TopCat.{u}} {f g : C(X, Y)} (H : ContinuousMap.Homotopy f g) {x₀ x₁ : X}\n  (p : fromTop x₀ ⟶ fromTop x₁)\n\n/-!\nThese definitions set up the following diagram, for each path `p`:\n\n```\n            f(p)\n        *--------*\n        | \\      |\n    H₀  |   \\ d  |  H₁\n        |     \\  |\n        *--------*\n            g(p)\n```\n\nHere, `H₀ = H.evalAt x₀` is the path from `f(x₀)` to `g(x₀)`,\nand similarly for `H₁`. Similarly, `f(p)` denotes the\npath in Y that the induced map `f` takes `p`, and similarly for `g(p)`.\n\nFinally, `d`, the diagonal path, is H(0 ⟶ 1, p), the result of the induced `H` on\n`Path.Homotopic.prod (0 ⟶ 1) p`, where `(0 ⟶ 1)` denotes the path from `0` to `1` in `I`.\n\nIt is clear that the diagram commutes (`H₀ ≫ g(p) = d = f(p) ≫ H₁`), but unfortunately,\nmany of the paths do not have defeq starting/ending points, so we end up needing some casting.\n-/\n\n\n/-- Interpret a homotopy `H : C(I × X, Y)` as a map `C(ULift I × X, Y)` -/\ndef uliftMap : C(TopCat.of (ULift.{u} I × X), Y) :=\n  ⟨fun x => H (x.1.down, x.2),\n    H.continuous.comp ((continuous_uliftDown.comp continuous_fst).prodMk continuous_snd)⟩\n\ntheorem ulift_apply (i : ULift.{u} I) (x : X) : H.uliftMap (i, x) = H (i.down, x) :=\n  rfl\n\n/-- An abbreviation for `prodToProdTop`, with some types already in place to help the\ntypechecker. In particular, the first path should be on the ulifted unit interval. -/\nabbrev prodToProdTopI {a₁ a₂ : TopCat.of (ULift I)} {b₁ b₂ : X} (p₁ : fromTop a₁ ⟶ fromTop a₂)\n    (p₂ : fromTop b₁ ⟶ fromTop b₂) :=\n  (prodToProdTop (TopCat.of <| ULift I) X).map (X := (⟨a₁⟩, ⟨b₁⟩)) (Y := (⟨a₂⟩, ⟨b₂⟩)) (p₁, p₂)\n\n/-- The diagonal path `d` of a homotopy `H` on a path `p` -/\ndef diagonalPath : fromTop (H (0, x₀)) ⟶ fromTop (H (1, x₁)) :=\n  (πₘ (TopCat.ofHom H.uliftMap)).map (prodToProdTopI uhpath01 p)\n\n/-- The diagonal path, but starting from `f x₀` and going to `g x₁` -/\ndef diagonalPath' : fromTop (f x₀) ⟶ fromTop (g x₁) :=\n  hcast (H.apply_zero x₀).symm ≫ H.diagonalPath p ≫ hcast (H.apply_one x₁)\n\n/-- Proof that `f(p) = H(0 ⟶ 0, p)`, with the appropriate casts -/\ntheorem apply_zero_path : (πₘ (TopCat.ofHom f)).map p = hcast (H.apply_zero x₀).symm ≫\n    (πₘ (TopCat.ofHom H.uliftMap)).map\n      (prodToProdTopI (𝟙 (@fromTop (TopCat.of _) (ULift.up 0))) p) ≫\n    hcast (H.apply_zero x₁) :=\n  Quotient.inductionOn p fun p' => by\n    apply @eq_path_of_eq_image _ _ _ _ H.uliftMap _ _ _ _ _ ((Path.refl (ULift.up _)).prod p')\n    intros\n    rw [Path.prod_coe, ulift_apply H]\n    simp\n\n/-- Proof that `g(p) = H(1 ⟶ 1, p)`, with the appropriate casts -/\ntheorem apply_one_path : (πₘ (TopCat.ofHom g)).map p = hcast (H.apply_one x₀).symm ≫\n    (πₘ (TopCat.ofHom H.uliftMap)).map\n      (prodToProdTopI (𝟙 (@fromTop (TopCat.of _) (ULift.up 1))) p) ≫\n    hcast (H.apply_one x₁) :=\n  Quotient.inductionOn p fun p' => by\n    apply @eq_path_of_eq_image _ _ _ _ H.uliftMap _ _ _ _ _ ((Path.refl (ULift.up _)).prod p')\n    intros\n    rw [Path.prod_coe, ulift_apply H]\n    simp\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Proof that `H.evalAt x = H(0 ⟶ 1, x ⟶ x)`, with the appropriate casts -/\ntheorem evalAt_eq (x : X) : ⟦H.evalAt x⟧ = hcast (H.apply_zero x).symm ≫\n    (πₘ (TopCat.ofHom H.uliftMap)).map (prodToProdTopI uhpath01 (𝟙 (fromTop x))) ≫\n      hcast (H.apply_one x).symm.symm := by\n  dsimp only [prodToProdTopI, uhpath01, hcast]\n  refine (@conj_eqToHom_iff_heq (πₓ Y) _ _ _ _ _ _ _ _\n    (FundamentalGroupoid.ext <| H.apply_one x).symm).mpr ?_\n  simp only [map_eq]\n  apply Path.Homotopic.hpath_hext; intro; rfl\n\nset_option backward.isDefEq.respectTransparency false in\n-- Finally, we show `d = f(p) ≫ H₁ = H₀ ≫ g(p)`\n\nTarget:\ntheorem eq_diag_path : (πₘ (TopCat.ofHom f)).map p ≫ ⟦H.evalAt x₁⟧ = H.diagonalPath' p ∧\n    (⟦H.evalAt x₀⟧ ≫ (πₘ (TopCat.ofHom g)).map p :\n    fromTop (f x₀) ⟶ fromTop (g x₁)) = H.diagonalPath' p :=\n\nProof body:\n","rejected":"by\n  exact eq_diag_path","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"fc9bf722f12c4f26646a03c9a39810208625371d727a6886a920b9d029784b0c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/FundamentalGroupoid","family_id":"eq_diag_path","file_id":"mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean","sample_id":"5b6616e952970d213dc2af95acd62437ab9f4ee924f0692e7518320229a74eaa"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c46ea331c2028c22d1cfe25ee9b2c63dfdbb45feb5dd1bf4ebf33f7cd796b96a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6f67bba633372c0ed9e66b5d0fc499a3e73b2bb0435be558c4947409e967051e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"00521cab1e8d5624effd7f9080f88a2a7c2a46ecd1ff08ed8fe772073ff48814","source_sha256":"bd4b2b01d950a3b9c8fd78b82f11b816abe9ec16bf6d55a504ac2e051e9cf9fe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h z => ?_, fun h n => by simpa only [zpow_natCast] using! h (n : ℤ)⟩\n  by_cases! hz : 0 ≤ z\n  · unfold Tendsto\n    lift z to ℕ using hz\n    simpa using! h z\n  · have : Tendsto (fun a => k a ^ z) l (𝓝 0) :=\n      Tendsto.comp (tendsto_zpow_atTop_zero hz) hk\n    have h : Tendsto f l (𝓝 0) := by simpa using! h 0\n    exact zero_mul (0 : β) ▸ this.mul h\n\nvariable {f}","hard_negative":false,"metrics":{"chosen_tokens":110,"rejected_tokens":115,"token_jaccard":0.931034,"token_length_ratio":1.045455},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"2c1978589a299ee8a8fda14b5dfaa6fcfd4b87f815e623eafe7ee1da5e3220b9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Eval.Defs\npublic import Mathlib.Analysis.Asymptotics.Lemmas\n\nNamespace:\nAsymptotics\n\nLocal context:\n/-\nCopyright (c) 2021 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\n/-!\n# Super-Polynomial Function Decay\n\nThis file defines a predicate `Asymptotics.SuperpolynomialDecay f` for a function satisfying\none of the following equivalent definitions (the definition is in terms of the first condition):\n\n* `x ^ n * f` tends to `𝓝 0` for all (or sufficiently large) naturals `n`\n* `|x ^ n * f|` tends to `𝓝 0` for all naturals `n` (`superpolynomialDecay_iff_abs_tendsto_zero`)\n* `|x ^ n * f|` is bounded for all naturals `n` (`superpolynomialDecay_iff_abs_isBoundedUnder`)\n* `f` is `o(x ^ c)` for all integers `c` (`superpolynomialDecay_iff_isLittleO`)\n* `f` is `O(x ^ c)` for all integers `c` (`superpolynomialDecay_iff_isBigO`)\n\nThese conditions are all equivalent to conditions in terms of polynomials, replacing `x ^ c` with\n  `p(x)` or `p(x)⁻¹` as appropriate, since asymptotically `p(x)` behaves like `X ^ p.natDegree`.\nThese further equivalences are not proven in mathlib but would be good future projects.\n\nThe definition of superpolynomial decay for `f : α → β` is relative to a parameter `k : α → β`.\nSuper-polynomial decay then means `f x` decays faster than `(k x) ^ c` for all integers `c`.\nEquivalently `f x` decays faster than `p.eval (k x)` for all polynomials `p : β[X]`.\nThe definition is also relative to a filter `l : Filter α` where the decay rate is compared.\n\nWhen the map `k` is given by `n ↦ ↑n : ℕ → ℝ` this defines negligible functions:\nhttps://en.wikipedia.org/wiki/Negligible_function\n\nWhen the map `k` is given by `(r₁,...,rₙ) ↦ r₁*...*rₙ : ℝⁿ → ℝ` this is equivalent\n  to the definition of rapidly decreasing functions given here:\nhttps://ncatlab.org/nlab/show/rapidly+decreasing+function\n\n## Main statements\n\n* `SuperpolynomialDecay.polynomial_mul` says that if `f(x)` is negligible,\n    then so is `p(x) * f(x)` for any polynomial `p`.\n* `superpolynomialDecay_iff_zpow_tendsto_zero` gives an equivalence between definitions in terms\n    of decaying faster than `k(x) ^ n` for all naturals `n` or `k(x) ^ c` for all integer `c`.\n-/\n\n@[expose] public section\n\n\nnamespace Asymptotics\n\nopen Topology Polynomial\n\nopen Filter\n\n/-- `f` has superpolynomial decay in parameter `k` along filter `l` if\n  `k ^ n * f` tends to zero at `l` for all naturals `n` -/\ndef SuperpolynomialDecay {α β : Type*} [TopologicalSpace β] [CommSemiring β] (l : Filter α)\n    (k : α → β) (f : α → β) :=\n  ∀ n : ℕ, Tendsto (fun a : α => k a ^ n * f a) l (𝓝 0)\n\nvariable {α β : Type*} {l : Filter α} {k : α → β} {f g g' : α → β}\n\nsection CommSemiring\n\nvariable [TopologicalSpace β] [CommSemiring β]\n\ntheorem SuperpolynomialDecay.congr' (hf : SuperpolynomialDecay l k f) (hfg : f =ᶠ[l] g) :\n    SuperpolynomialDecay l k g := fun z =>\n  (hf z).congr' (EventuallyEq.mul (EventuallyEq.refl l _) hfg)\n\ntheorem SuperpolynomialDecay.congr (hf : SuperpolynomialDecay l k f) (hfg : ∀ x, f x = g x) :\n    SuperpolynomialDecay l k g := fun z =>\n  (hf z).congr fun x => (congr_arg fun a => k x ^ z * a) <| hfg x\n\n@[simp]\ntheorem superpolynomialDecay_zero (l : Filter α) (k : α → β) : SuperpolynomialDecay l k 0 :=\n  fun z => by simpa only [Pi.zero_apply, mul_zero] using tendsto_const_nhds\n\ntheorem SuperpolynomialDecay.add [ContinuousAdd β] (hf : SuperpolynomialDecay l k f)\n    (hg : SuperpolynomialDecay l k g) : SuperpolynomialDecay l k (f + g) := fun z => by\n  simpa only [mul_add, add_zero, Pi.add_apply] using (hf z).add (hg z)\n\ntheorem SuperpolynomialDecay.mul [ContinuousMul β] (hf : SuperpolynomialDecay l k f)\n    (hg : SuperpolynomialDecay l k g) : SuperpolynomialDecay l k (f * g) := fun z => by\n  simpa only [mul_assoc, one_mul, mul_zero, pow_zero] using! (hf z).mul (hg 0)\n\ntheorem SuperpolynomialDecay.mul_const [ContinuousMul β] (hf : SuperpolynomialDecay l k f) (c : β) :\n    SuperpolynomialDecay l k fun n => f n * c := fun z => by\n  simpa only [← mul_assoc, zero_mul] using Tendsto.mul_const c (hf z)\n\ntheorem SuperpolynomialDecay.const_mul [ContinuousMul β] (hf : SuperpolynomialDecay l k f) (c : β) :\n    SuperpolynomialDecay l k fun n => c * f n :=\n  (hf.mul_const c).congr fun _ => mul_comm _ _\n\ntheorem SuperpolynomialDecay.param_mul (hf : SuperpolynomialDecay l k f) :\n    SuperpolynomialDecay l k (k * f) := fun z =>\n  tendsto_nhds.2 fun s hs hs0 =>\n    l.sets_of_superset ((tendsto_nhds.1 (hf <| z + 1)) s hs hs0) fun x hx => by\n      simpa only [Set.mem_preimage, Pi.mul_apply, ← mul_assoc, ← pow_succ] using hx\n\ntheorem SuperpolynomialDecay.mul_param (hf : SuperpolynomialDecay l k f) :\n    SuperpolynomialDecay l k (f * k) :=\n  hf.param_mul.congr fun _ => mul_comm _ _\n\ntheorem SuperpolynomialDecay.param_pow_mul (hf : SuperpolynomialDecay l k f) (n : ℕ) :\n    SuperpolynomialDecay l k (k ^ n * f) := by\n  induction n with\n  | zero => simpa only [one_mul, pow_zero] using hf\n  | succ n hn => simpa only [pow_succ', mul_assoc] using hn.param_mul\n\ntheorem SuperpolynomialDecay.mul_param_pow (hf : SuperpolynomialDecay l k f) (n : ℕ) :\n    SuperpolynomialDecay l k (f * k ^ n) :=\n  (hf.param_pow_mul n).congr fun _ => mul_comm _ _\n\ntheorem SuperpolynomialDecay.polynomial_mul [ContinuousAdd β] [ContinuousMul β]\n    (hf : SuperpolynomialDecay l k f) (p : β[X]) :\n    SuperpolynomialDecay l k fun x => (p.eval <| k x) * f x :=\n  Polynomial.induction_on' p (fun p q hp hq => by simpa [add_mul] using! hp.add hq) fun n c => by\n    simpa [mul_assoc] using! (hf.param_pow_mul n).const_mul c\n\ntheorem SuperpolynomialDecay.mul_polynomial [ContinuousAdd β] [ContinuousMul β]\n    (hf : SuperpolynomialDecay l k f) (p : β[X]) :\n    SuperpolynomialDecay l k fun x => f x * (p.eval <| k x) :=\n  (hf.polynomial_mul p).congr fun _ => mul_comm _ _\n\nend CommSemiring\n\nsection OrderedCommSemiring\n\nvariable [TopologicalSpace β] [CommSemiring β] [PartialOrder β] [IsOrderedRing β] [OrderTopology β]\n\ntheorem SuperpolynomialDecay.trans_eventuallyLE (hk : 0 ≤ᶠ[l] k) (hg : SuperpolynomialDecay l k g)\n    (hg' : SuperpolynomialDecay l k g') (hfg : g ≤ᶠ[l] f) (hfg' : f ≤ᶠ[l] g') :\n    SuperpolynomialDecay l k f := fun z =>\n  tendsto_of_tendsto_of_tendsto_of_le_of_le' (hg z) (hg' z)\n    (by filter_upwards [hfg, hk] with x hx (hx' : 0 ≤ k x) using by gcongr)\n    (by filter_upwards [hfg', hk] with x hx (hx' : 0 ≤ k x) using by gcongr)\n\nend OrderedCommSemiring\n\nsection LinearOrderedCommRing\n\nvariable [TopologicalSpace β] [CommRing β] [LinearOrder β] [IsStrictOrderedRing β] [OrderTopology β]\nvariable (l k f)\n\ntheorem superpolynomialDecay_iff_abs_tendsto_zero :\n    SuperpolynomialDecay l k f ↔ ∀ n : ℕ, Tendsto (fun a : α => |k a ^ n * f a|) l (𝓝 0) :=\n  ⟨fun h z => (tendsto_zero_iff_abs_tendsto_zero _).1 (h z), fun h z =>\n    (tendsto_zero_iff_abs_tendsto_zero _).2 (h z)⟩\n\ntheorem superpolynomialDecay_iff_superpolynomialDecay_abs :\n    SuperpolynomialDecay l k f ↔ SuperpolynomialDecay l (fun a => |k a|) fun a => |f a| :=\n  (superpolynomialDecay_iff_abs_tendsto_zero l k f).trans\n    (by simp_rw [SuperpolynomialDecay, abs_mul, abs_pow])\n\nvariable {l k f}\n\ntheorem SuperpolynomialDecay.trans_eventually_abs_le (hf : SuperpolynomialDecay l k f)\n    (hfg : abs ∘ g ≤ᶠ[l] abs ∘ f) : SuperpolynomialDecay l k g := by\n  rw [superpolynomialDecay_iff_abs_tendsto_zero] at hf ⊢\n  refine fun z =>\n    tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds (hf z)\n      (Eventually.of_forall fun x => abs_nonneg _) (hfg.mono fun x hx => ?_)\n  calc\n    |k x ^ z * g x| = |k x ^ z| * |g x| := abs_mul (k x ^ z) (g x)\n    _ ≤ |k x ^ z| * |f x| := by gcongr _ * ?_; exact hx\n    _ = |k x ^ z * f x| := (abs_mul (k x ^ z) (f x)).symm\n\ntheorem SuperpolynomialDecay.trans_abs_le (hf : SuperpolynomialDecay l k f)\n    (hfg : ∀ x, |g x| ≤ |f x|) : SuperpolynomialDecay l k g :=\n  hf.trans_eventually_abs_le (Eventually.of_forall hfg)\n\nend LinearOrderedCommRing\n\nsection Field\n\nvariable [TopologicalSpace β] [Field β] (l k f)\n\ntheorem superpolynomialDecay_mul_const_iff [ContinuousMul β] {c : β} (hc0 : c ≠ 0) :\n    (SuperpolynomialDecay l k fun n => f n * c) ↔ SuperpolynomialDecay l k f :=\n  ⟨fun h => (h.mul_const c⁻¹).congr fun x => by simp [mul_assoc, mul_inv_cancel₀ hc0], fun h =>\n    h.mul_const c⟩\n\ntheorem superpolynomialDecay_const_mul_iff [ContinuousMul β] {c : β} (hc0 : c ≠ 0) :\n    (SuperpolynomialDecay l k fun n => c * f n) ↔ SuperpolynomialDecay l k f :=\n  ⟨fun h => (h.const_mul c⁻¹).congr fun x => by simp [← mul_assoc, inv_mul_cancel₀ hc0], fun h =>\n    h.const_mul c⟩\n\nend Field\n\nsection LinearOrderedField\n\nvariable [TopologicalSpace β] [Field β] [LinearOrder β] [IsStrictOrderedRing β] [OrderTopology β]\nvariable (f)\n\ntheorem superpolynomialDecay_iff_abs_isBoundedUnder (hk : Tendsto k l atTop) :\n    SuperpolynomialDecay l k f ↔\n    ∀ z : ℕ, IsBoundedUnder (· ≤ ·) l fun a : α => |k a ^ z * f a| := by\n  refine\n    ⟨fun h z => Tendsto.isBoundedUnder_le (Tendsto.abs (h z)), fun h =>\n      (superpolynomialDecay_iff_abs_tendsto_zero l k f).2 fun z => ?_⟩\n  obtain ⟨m, hm⟩ := h (z + 1)\n  have h1 : Tendsto (fun _ : α => (0 : β)) l (𝓝 0) := tendsto_const_nhds\n  have h2 : Tendsto (fun a : α => |(k a)⁻¹| * m) l (𝓝 0) :=\n    zero_mul m ▸\n      Tendsto.mul_const m ((tendsto_zero_iff_abs_tendsto_zero _).1 hk.inv_tendsto_atTop)\n  refine\n    tendsto_of_tendsto_of_tendsto_of_le_of_le' h1 h2 (Eventually.of_forall fun x => abs_nonneg _)\n      ((eventually_map.1 hm).mp ?_)\n  refine (hk.eventually_ne_atTop 0).mono fun x hk0 hx => ?_\n  refine Eq.trans_le ?_ (mul_le_mul_of_nonneg_left hx <| abs_nonneg (k x)⁻¹)\n  rw [← abs_mul, ← mul_assoc, pow_succ', ← mul_assoc, inv_mul_cancel₀ hk0, one_mul]\n\nTarget:\ntheorem superpolynomialDecay_iff_zpow_tendsto_zero (hk : Tendsto k l atTop) :\n    SuperpolynomialDecay l k f ↔ ∀ z : ℤ, Tendsto (fun a : α => k a ^ z * f a) l (𝓝 0) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  refine ⟨fun h z => ?_, fun h n => by simpa only [zpow_natCast] using! h (n : ℤ)⟩\n  by_cases! hz : 0 ≤ z\n  · unfold Tendsto\n    lift z to ℕ using hz\n    simpa using! h z\n  · have : Tendsto (fun a => k a ^ z) l (𝓝 0) :=\n      Tendsto.comp (tendsto_zpow_atTop_zero hz) hk\n    have h : Tendsto f l (𝓝 0) := by simpa using! h 0\n    exact zero_mul (0 : β) ▸ this.mul h\n\nvariable {f}","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Asymptotics","family_id":"superpolynomialdecay_iff_zpow_tendsto_zero","file_id":"mathlib/Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean","sample_id":"00521cab1e8d5624effd7f9080f88a2a7c2a46ecd1ff08ed8fe772073ff48814"},"split":"train"}
{"binding":{"chosen_proof_sha256":"62c5d9390a895014879439c4bd46fd00c9783796de9ef2a06a744f2c74871aea","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d69b8dbc815ff0dff2149b097f513181f220bdc5a96303ab1a9ff390558d832e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0f39df25f86b79636ec8efd74a8a03a810910ef6427851948542bfbbe91d6d31","source_sha256":"0e2737adb7e8846513bd14b3c89ab969bcc977f3e683f90a3402f41d55ebbfc2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply_fun ((Sigma.desc fun a ↦ pullback.fst g₁ (Sigma.ι X a)) ≫ ·) using\n    (fun a b ↦ (cancel_epi _).mp)\n  ext a\n  simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_ι_app]\n  rw [← Category.assoc, pullback.condition]\n  simp only [Category.assoc, colimit.ι_desc, Cofan.mk_ι_app]\n  apply_fun ((Sigma.desc fun a ↦ pullback.fst (pullback.fst _ _ ≫ g₂) (Sigma.ι X a)) ≫ ·)\n    using (fun a b ↦ (cancel_epi _).mp)\n  ext b\n  simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_ι_app]\n  simp only [← Category.assoc]\n  rw [(Category.assoc _ _ g₂), pullback.condition]\n  simp only [Category.assoc, colimit.ι_desc, Cofan.mk_ι_app]\n  rw [← Category.assoc]\n  apply h\n  apply_fun (pullback.fst g₁ (Sigma.ι X a) ≫ ·) at hg\n  rw [← Category.assoc, pullback.condition] at hg\n  simp only [Category.assoc, colimit.ι_desc, Cofan.mk_ι_app] at hg\n  apply_fun ((Sigma.ι (fun a ↦ pullback _ _) b) ≫ (Sigma.desc fun a ↦\n    pullback.fst (pullback.fst _ _ ≫ g₂) (Sigma.ι X a)) ≫ ·) at hg\n  simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_ι_app] at hg\n  simp only [← Category.assoc] at hg\n  rw [(Category.assoc _ _ g₂), pullback.condition] at hg\n  simpa using hg","hard_negative":false,"metrics":{"chosen_tokens":358,"rejected_tokens":363,"token_jaccard":0.90566,"token_length_ratio":1.013966},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"2c19b09382c1b789c100ae239eeab201d35a70c3b8be81f5c6e217c8b9ddd7dd","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.EffectiveEpi.Basic\npublic import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback\npublic import Mathlib.Tactic.ApplyFun\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2023 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# Effective epimorphic families and coproducts\n\nThis file proves that an effective epimorphic family induces an effective epi from the coproduct if\nthe coproduct exists, and the converse under some more conditions on the coproduct (that it\ninteracts well with pullbacks).\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Limits\n\nvariable {C : Type*} [Category* C]\n\nset_option backward.isDefEq.respectTransparency false in\n/--\nGiven an `EffectiveEpiFamily X π` and a corresponding coproduct cocone, the family descends to an\n`EffectiveEpi` from the coproduct.\n-/\nnoncomputable\ndef effectiveEpiStructIsColimitDescOfEffectiveEpiFamily {B : C} {α : Type*} (X : α → C)\n    (c : Cofan X) (hc : IsColimit c) (π : (a : α) → (X a ⟶ B)) [EffectiveEpiFamily X π] :\n    EffectiveEpiStruct (hc.desc (Cofan.mk B π)) where\n  desc e h := EffectiveEpiFamily.desc X π (fun a ↦ c.ι.app ⟨a⟩ ≫ e) (fun a₁ a₂ g₁ g₂ hg ↦ by\n    simp only [← Category.assoc]\n    exact h (g₁ ≫ c.ι.app ⟨a₁⟩) (g₂ ≫ c.ι.app ⟨a₂⟩) (by simpa))\n  fac e h := hc.hom_ext (fun ⟨j⟩ ↦ (by simp))\n  uniq e _ m hm := EffectiveEpiFamily.uniq X π (fun a ↦ c.ι.app ⟨a⟩ ≫ e)\n      (fun _ _ _ _ hg ↦ (by simp [← hm, reassoc_of% hg])) m (fun _ ↦ (by simp [← hm]))\n\nset_option backward.defeqAttrib.useBackward true in\ninstance {B : C} {α : Type*} (X : α → C) (π : (a : α) → (X a ⟶ B)) [HasCoproduct X]\n    [EffectiveEpiFamily X π] : EffectiveEpi (Sigma.desc π) := by\n  let e := effectiveEpiStructIsColimitDescOfEffectiveEpiFamily X _ (coproductIsCoproduct _) π\n  simp only [Cofan.mk_pt, coproductIsCoproduct, colimit.cocone_x, IsColimit.ofIsoColimit_desc,\n    Cocone.ext_inv_hom, Iso.refl_inv, colimit.isColimit_desc, Category.id_comp] at e\n  exact ⟨⟨e⟩⟩\n\nexample {B : C} {α : Type*} (X : α → C) (π : (a : α) → (X a ⟶ B)) [EffectiveEpiFamily X π]\n    [HasCoproduct X] : Epi (Sigma.desc π) := inferInstance\n\nset_option backward.isDefEq.respectTransparency false in\n/--\nThis is an auxiliary lemma used twice in the definition of  `EffectiveEpiFamilyOfEffectiveEpiDesc`.\nIt is the `h` hypothesis of `EffectiveEpi.desc` and `EffectiveEpi.fac`.\n-/\n\nTarget:\ntheorem effectiveEpiFamilyStructOfEffectiveEpiDesc_aux {B : C} {α : Type*} {X : α → C}\n    {π : (a : α) → X a ⟶ B} [HasCoproduct X]\n    [∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)]\n    [∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a ↦ pullback g (Sigma.ι X a)]\n    [∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a ↦ pullback.fst g (Sigma.ι X a))]\n    {W : C} {e : (a : α) → X a ⟶ W} (h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂),\n      g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂) {Z : C}\n    {g₁ g₂ : Z ⟶ ∐ fun b ↦ X b} (hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π) :\n    g₁ ≫ Sigma.desc e = g₂ ≫ Sigma.desc e :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  apply_fun ((Sigma.desc fun a ↦ pullback.fst g₁ (Sigma.ι X a)) ≫ ·) using\n    (fun a b ↦ (cancel_epi _).mp)\n  ext a\n  simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_ι_app]\n  rw [← Category.assoc, pullback.condition]\n  simp only [Category.assoc, colimit.ι_desc, Cofan.mk_ι_app]\n  apply_fun ((Sigma.desc fun a ↦ pullback.fst (pullback.fst _ _ ≫ g₂) (Sigma.ι X a)) ≫ ·)\n    using (fun a b ↦ (cancel_epi _).mp)\n  ext b\n  simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_ι_app]\n  simp only [← Category.assoc]\n  rw [(Category.assoc _ _ g₂), pullback.condition]\n  simp only [Category.assoc, colimit.ι_desc, Cofan.mk_ι_app]\n  rw [← Category.assoc]\n  apply h\n  apply_fun (pullback.fst g₁ (Sigma.ι X a) ≫ ·) at hg\n  rw [← Category.assoc, pullback.condition] at hg\n  simp only [Category.assoc, colimit.ι_desc, Cofan.mk_ι_app] at hg\n  apply_fun ((Sigma.ι (fun a ↦ pullback _ _) b) ≫ (Sigma.desc fun a ↦\n    pullback.fst (pullback.fst _ _ ≫ g₂) (Sigma.ι X a)) ≫ ·) at hg\n  simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_ι_app] at hg\n  simp only [← Category.assoc] at hg\n  rw [(Category.assoc _ _ g₂), pullback.condition] at hg\n  simpa using hg","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/EffectiveEpi","family_id":"effectiveepifamilystructofeffectiveepidesc_aux","file_id":"mathlib/Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean","sample_id":"0f39df25f86b79636ec8efd74a8a03a810910ef6427851948542bfbbe91d6d31"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1e710a9d7e1e6f7c130e5c280474bf460f6d3246c3fb6e99473017f60576e601","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8646e1a8a8dc5383d0680a35efcb4d004ebffec2fea8e92c53930b15808a4955","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8451c1001183081b0718334cf931fa3560f5cf1ca44e66d4b164528ee6a8ec57","source_sha256":"da03aef6be316c0f1e4d46eb4e8f0617ba6def9cf6bbead1497df151ed7bf988","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by cases x; rfl\n\nnonrec def StateT.callCC {σ} [MonadCont m] {α β : Type _}\n    (f : Label α (StateT σ m) β → StateT σ m α) : StateT σ m α :=\n  StateT.mk (fun r => callCC fun f' => (f <| StateT.mkLabel f').run r)","hard_negative":false,"metrics":{"chosen_tokens":71,"rejected_tokens":76,"token_jaccard":0.902439,"token_length_ratio":1.070423},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"2c267673684fc6654f8ab8891c4792a44bc9f8e00802b54d61113ee8b0c0ef66","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Control.Monad.Basic\npublic import Mathlib.Control.Monad.Writer\npublic import Mathlib.Control.Lawful\npublic import Batteries.Tactic.Congr\npublic import Batteries.Lean.Except\nimport all Init.Control.Option  -- for unfolding `Option.lift`\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\n/-!\n# Continuation Monad\n\nMonad encapsulating continuation passing programming style, similar to\nHaskell's `Cont`, `ContT` and `MonadCont`:\n<https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Cont.html>\n<https://hackage.haskell.org/package/transformers-0.6.2.0/docs/Control-Monad-Trans-Cont.html>\n-/\n\n@[expose] public section\n\nuniverse u v w u₀ u₁ v₀ v₁\n\nstructure MonadCont.Label (α : Type w) (m : Type u → Type v) (β : Type u) where\n  apply : α → m β\n\nabbrev MonadCont.goto {α β} {m : Type u → Type v} (f : MonadCont.Label α m β) (x : α) :=\n  f.apply x\n\nclass MonadCont (m : Type u → Type v) where\n  callCC : ∀ {α β}, (MonadCont.Label α m β → m α) → m α\n\nopen MonadCont\n\nclass LawfulMonadCont (m : Type u → Type v) [Monad m] [MonadCont m] : Prop\n    extends LawfulMonad m where\n  callCC_bind_right {α ω γ} (cmd : m α) (next : Label ω m γ → α → m ω) :\n    (callCC fun f => cmd >>= next f) = cmd >>= fun x => callCC fun f => next f x\n  callCC_bind_left {α} (β) (x : α) (dead : Label α m β → β → m α) :\n    (callCC fun f : Label α m β => goto f x >>= dead f) = pure x\n  callCC_dummy {α β} (dummy : m α) : (callCC fun _ : Label α m β => dummy) = dummy\n\nexport LawfulMonadCont (callCC_bind_right callCC_bind_left callCC_dummy)\n\ndef ContT (r : Type u) (m : Type u → Type v) (α : Type w) :=\n  (α → m r) → m r\n\nabbrev Cont (r : Type u) (α : Type w) :=\n  ContT r Id α\n\nnamespace ContT\n\nexport MonadCont (Label goto)\n\nvariable {r : Type u} {m : Type u → Type v} {α β : Type w}\n\n/-- Build a `ContT` from a function taking a continuation callback. -/\ndef mk (f : (α → m r) → m r) : ContT r m α := f\n\n/-- Run a `ContT` with a provided callback. -/\ndef run (x : ContT r m α) : (α → m r) → m r := x\n\ndef map (f : m r → m r) (x : ContT r m α) : ContT r m α :=\n  f ∘ x\n\ntheorem run_contT_map_contT (f : m r → m r) (x : ContT r m α) : run (map f x) = f ∘ run x :=\n  rfl\n\ndef withContT (f : (β → m r) → α → m r) (x : ContT r m α) : ContT r m β := fun g => x <| f g\n\ntheorem run_withContT (f : (β → m r) → α → m r) (x : ContT r m α) :\n    run (withContT f x) = run x ∘ f :=\n  rfl\n\n@[ext]\nprotected theorem ext {x y : ContT r m α} (h : ∀ f, x.run f = y.run f) : x = y := by\n  unfold ContT; ext; apply h\n\ninstance : Monad (ContT r m) where\n  pure x f := f x\n  bind x f g := x fun i => f i g\n\n@[simp]\ntheorem run_mk (f : (α → m r) → m r) (k : α → m r) : (.mk f : ContT r m α).run k = f k := rfl\n\n@[simp]\ntheorem run_pure (a : α) (k : α → m r) : (pure a : ContT r m α).run k = k a := rfl\n\n@[simp]\ntheorem run_bind (x : ContT r m α) (f : α → ContT r m β) (k : β → m r) :\n    (x >>= f).run k = x.run fun x => (f x).run k := rfl\n\n@[simp]\ntheorem run_map (f : α → β) (x : ContT r m α) (k : β → m r) :\n    (f <$> x).run k = x.run (k ∘ f) := rfl\n\n@[simp]\ntheorem run_seq (f : ContT r m (α → β)) (x : ContT r m α) (k : β → m r) :\n    (f <*> x).run k = f.run fun f => x.run (k ∘ f) := rfl\n\n@[simp]\ntheorem run_seqLeft (x : ContT r m α) (y : ContT r m β) (k : α → m r) :\n    (x <* y).run k = x.run fun x => y.run fun _ => k x := rfl\n\n@[simp]\ntheorem run_seqRight (x : ContT r m α) (y : ContT r m β) (k : β → m r) :\n    (x *> y).run k = x.run fun _ => y.run k := rfl\n\ninstance : LawfulMonad (ContT r m) := LawfulMonad.mk'\n  (id_map := by intros; rfl)\n  (pure_bind := by intros; ext; rfl)\n  (bind_assoc := by intros; ext; rfl)\n\ninstance [Monad m] : MonadLift m (ContT r m) where\n  monadLift x := .mk fun k => x >>= k\n\n@[simp]\ntheorem run_monadLift [Monad m] {α} (x : m α) (k : α → m r) :\n    (monadLift x : ContT r m α).run k = x >>= k := rfl\n\ntheorem monadLift_bind [Monad m] [LawfulMonad m] {α β} (x : m α) (f : α → m β) :\n    (monadLift (x >>= f) : ContT r m β) = monadLift x >>= monadLift ∘ f := by\n  ext\n  simp only [bind_assoc, run_bind, run_monadLift, Function.comp_apply]\n\ninstance : MonadCont (ContT r m) where\n  callCC f := .mk fun k => f ⟨fun x => .mk fun _ => k x⟩ k\n\n@[simp]\ntheorem run_callCC (f : Label α (ContT r m) β → ContT r m α) (k : α → m r) :\n    (callCC f).run k = (f ⟨fun x => .mk fun _ => k x⟩).run k := rfl\n\ninstance : LawfulMonadCont (ContT r m) where\n  callCC_bind_right := by intros; ext; rfl\n  callCC_bind_left := by intros; ext; rfl\n  callCC_dummy := by intros; ext; rfl\n\n/-- Note that `tryCatch` does not have correct behavior in this monad:\n```\ndef foo : ContT Bool (Except String) Bool := do\n  let x ← try\n    pure true\n  catch _ =>\n    return false\n  throw s!\"oh no {x}\"\n#eval foo.run pure\n-- `Except.ok false`, no error\n```\nHere, the `throwError` is being run inside the `try`.\nSee [Zulip](https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/MonadExcept.20in.20the.20ContT.20monad/near/375341221)\nfor further discussion.\n-/\ninstance (ε) [MonadExceptOf ε m] : MonadExceptOf ε (ContT r m) where\n  throw e := .mk fun _ => throw e\n  tryCatch act h := .mk fun k => tryCatch (act.run k) fun e => (h e).run k\n\n@[simp]\ntheorem run_throw {ε} [MonadExceptOf ε m]\n    (e : ε) (f : α → m r) :\n    (throw e : ContT r m α).run f = throw e := rfl\n\n@[simp]\ntheorem run_tryCatch {ε} [MonadExceptOf ε m]\n    (act : ContT r m α) (h : ε → ContT r m α) (f : α → m r) :\n    (tryCatch act h : ContT r m α).run f = tryCatch (act.run f) fun e => (h e).run f := rfl\n\nend ContT\n\nvariable {m : Type u → Type v}\n\nsection\nvariable [Monad m]\n\ndef ExceptT.mkLabel {α β ε} : Label (Except.{u, u} ε α) m β → Label α (ExceptT ε m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (Except.ok a)⟩\n\ntheorem ExceptT.goto_mkLabel {α β ε : Type _} (x : Label (Except.{u, u} ε α) m β) (i : α) :\n    goto (ExceptT.mkLabel x) i = ExceptT.mk (Except.ok <$> goto x (Except.ok i)) := by\n  cases x; rfl\n\nnonrec def ExceptT.callCC {ε} [MonadCont m] {α β : Type _}\n    (f : Label α (ExceptT ε m) β → ExceptT ε m α) : ExceptT ε m α :=\n  ExceptT.mk (callCC fun x : Label _ m β => ExceptT.run <| f (ExceptT.mkLabel x))\n\ninstance {ε} [MonadCont m] : MonadCont (ExceptT ε m) where\n  callCC := ExceptT.callCC\n\ninstance {ε} [MonadCont m] [LawfulMonadCont m] : LawfulMonadCont (ExceptT ε m) where\n  callCC_bind_right := by\n    intros; simp only [callCC, ExceptT.callCC, ExceptT.run_bind, callCC_bind_right]; ext\n    dsimp\n    congr with ⟨⟩ <;> simp [@callCC_dummy m _]\n  callCC_bind_left := by\n    intros\n    simp only [callCC, ExceptT.callCC, ExceptT.goto_mkLabel, map_eq_bind_pure_comp, Function.comp,\n      ExceptT.run_bind, ExceptT.run_mk, bind_assoc, pure_bind, @callCC_bind_left m _]\n    ext; rfl\n  callCC_dummy := by intros; simp only [callCC, ExceptT.callCC, @callCC_dummy m _]; ext; rfl\n\ndef OptionT.mkLabel {α β} : Label (Option.{u} α) m β → Label α (OptionT m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (some a)⟩\n\ntheorem OptionT.goto_mkLabel {α β : Type _} (x : Label (Option.{u} α) m β) (i : α) :\n    goto (OptionT.mkLabel x) i = OptionT.mk (goto x (some i) >>= fun a => pure (some a)) :=\n  (rfl)\n\nnonrec def OptionT.callCC [MonadCont m] {α β : Type _} (f : Label α (OptionT m) β → OptionT m α) :\n    OptionT m α :=\n  OptionT.mk (callCC fun x : Label _ m β => OptionT.run <| f (OptionT.mkLabel x) : m (Option α))\n\n@[simp]\nlemma run_callCC [MonadCont m] {α β : Type _} (f : Label α (OptionT m) β → OptionT m α) :\n    (OptionT.callCC f).run = (callCC fun x => OptionT.run <| f (OptionT.mkLabel x)) := (rfl)\n\ninstance [MonadCont m] : MonadCont (OptionT m) where\n  callCC := OptionT.callCC\n\ninstance [MonadCont m] [LawfulMonadCont m] : LawfulMonadCont (OptionT m) where\n  callCC_bind_right := by\n    refine fun _ _ => OptionT.ext ?_\n    simpa [callCC, Option.elimM, callCC_bind_right] using\n      bind_congr fun | some _ => rfl | none => by simp [@callCC_dummy m _]\n  callCC_bind_left := by\n    intros\n    ext\n    simp [callCC, OptionT.goto_mkLabel, @callCC_bind_left m _]\n  callCC_dummy := by intros; ext; simp [callCC, OptionT.callCC, @callCC_dummy m _]\n\ndef WriterT.mkLabel {α β ω} [EmptyCollection ω] : Label (α × ω) m β → Label α (WriterT ω m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (a, ∅)⟩\n\ndef WriterT.mkLabel' {α β ω} [Monoid ω] : Label (α × ω) m β → Label α (WriterT ω m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (a, 1)⟩\n\ntheorem WriterT.goto_mkLabel {α β ω : Type _} [EmptyCollection ω] (x : Label (α × ω) m β) (i : α) :\n    goto (WriterT.mkLabel x) i = monadLift (goto x (i, ∅)) := by cases x; rfl\n\ntheorem WriterT.goto_mkLabel' {α β ω : Type _} [Monoid ω] (x : Label (α × ω) m β) (i : α) :\n    goto (WriterT.mkLabel' x) i = monadLift (goto x (i, 1)) := by cases x; rfl\n\nnonrec def WriterT.callCC [MonadCont m] {α β ω : Type _} [EmptyCollection ω]\n    (f : Label α (WriterT ω m) β → WriterT ω m α) : WriterT ω m α :=\n  WriterT.mk <| callCC (WriterT.run ∘ f ∘ WriterT.mkLabel : Label (α × ω) m β → m (α × ω))\n\ndef WriterT.callCC' [MonadCont m] {α β ω : Type _} [Monoid ω]\n    (f : Label α (WriterT ω m) β → WriterT ω m α) : WriterT ω m α :=\n  WriterT.mk <|\n    MonadCont.callCC (WriterT.run ∘ f ∘ WriterT.mkLabel' : Label (α × ω) m β → m (α × ω))\n\nend\n\ninstance (ω) [Monad m] [EmptyCollection ω] [MonadCont m] : MonadCont (WriterT ω m) where\n  callCC := WriterT.callCC\n\ninstance (ω) [Monad m] [Monoid ω] [MonadCont m] : MonadCont (WriterT ω m) where\n  callCC := WriterT.callCC'\n\ndef StateT.mkLabel {α β σ : Type u} : Label (α × σ) m (β × σ) → Label α (StateT σ m) β\n  | ⟨f⟩ => ⟨fun a => StateT.mk (fun s => f (a, s))⟩\n\nTarget:\ntheorem StateT.goto_mkLabel {α β σ : Type u} (x : Label (α × σ) m (β × σ)) (i : α) :\n    goto (StateT.mkLabel x) i = StateT.mk (fun s => goto x (i, s)) :=\n\nProof body:\n","rejected":"Here is the proof:\nby cases x; rfl\n\nnonrec def StateT.callCC {σ} [MonadCont m] {α β : Type _}\n    (f : Label α (StateT σ m) β → StateT σ m α) : StateT σ m α :=\n  StateT.mk (fun r => callCC fun f' => (f <| StateT.mkLabel f').run r)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control/Monad","family_id":"statet","file_id":"mathlib/Mathlib/Control/Monad/Cont.lean","sample_id":"8451c1001183081b0718334cf931fa3560f5cf1ca44e66d4b164528ee6a8ec57"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fda78651fadb9c3fc00301a4f0a7e0c199693b04a98a2e407d36dcf722f12255","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fc015e197e7e22d0acabad316890661898d1a5fa61e903e26832dbe043d9fa2c","source_sha256":"d2c15a0760158d76d516fc83add1bd6dc5918d2a6efe03e65d28acdb282a684f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let e := Fintype.equivFin ι\n  obtain ⟨i₀, i₁, ne, h⟩ := h.exists_approx_aux (Fintype.card ι) hε hb fun x y ↦ A x (e.symm y)\n  refine ⟨i₀, i₁, ne, fun k ↦ ?_⟩\n  convert! h (e k) <;> simp only [e.symm_apply_apply]","hard_negative":false,"metrics":{"chosen_tokens":78,"rejected_tokens":2,"token_jaccard":0.022727,"token_length_ratio":0.025641},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"2c50b277bbfb256310683bc7b0b15cb2b90ce78f3d82f90c5c5c3e5391c889d6","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Real.Basic\npublic import Mathlib.Combinatorics.Pigeonhole\npublic import Mathlib.Algebra.Order.AbsoluteValue.Euclidean\n\nNamespace:\nAbsoluteValue.IsAdmissible\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Admissible absolute values\nThis file defines a structure `AbsoluteValue.IsAdmissible` which we use to show the class number\nof the ring of integers of a global field is finite.\n\n## Main definitions\n\n* `AbsoluteValue.IsAdmissible abv` states the absolute value `abv : R → ℤ`\n  respects the Euclidean domain structure on `R`, and that a large enough set\n  of elements of `R^n` contains a pair of elements whose remainders are\n  pointwise close together.\n\n## Main results\n\n* `AbsoluteValue.absIsAdmissible` shows the \"standard\" absolute value on `ℤ`,\n  mapping negative `x` to `-x`, is admissible.\n* `Polynomial.cardPowDegreeIsAdmissible` shows `cardPowDegree`,\n  mapping `p : Polynomial 𝔽_q` to `q ^ degree p`, is admissible\n-/\n\npublic section\n\nlocal infixl:50 \" ≺ \" => EuclideanDomain.r\n\nnamespace AbsoluteValue\n\nvariable {R : Type*} [EuclideanDomain R]\nvariable (abv : AbsoluteValue R ℤ)\n\n/-- An absolute value `R → ℤ` is admissible if it respects the Euclidean domain\nstructure and a large enough set of elements in `R^n` will contain a pair of\nelements whose remainders are pointwise close together. -/\nstructure IsAdmissible extends IsEuclidean abv where\n  /-- The cardinality required for a given `ε`. -/\n  protected card : ℝ → ℕ\n  /-- For all `ε > 0` and finite families `A`, we can partition the remainders of `A` mod `b`\n  into `abv.card ε` sets, such that all elements in each part of remainders are close together. -/\n  exists_partition' :\n    ∀ (n : ℕ) {ε : ℝ} (_ : 0 < ε) {b : R} (_ : b ≠ 0) (A : Fin n → R),\n      ∃ t : Fin n → Fin (card ε), ∀ i₀ i₁, t i₀ = t i₁ → (abv (A i₁ % b - A i₀ % b) : ℝ) < abv b • ε\n\nnamespace IsAdmissible\n\nvariable {abv}\n\n/-- For all `ε > 0` and finite families `A`, we can partition the remainders of `A` mod `b`\ninto `abv.card ε` sets, such that all elements in each part of remainders are close together. -/\ntheorem exists_partition {ι : Type*} [Finite ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0)\n    (A : ι → R) (h : abv.IsAdmissible) : ∃ t : ι → Fin (h.card ε),\n      ∀ i₀ i₁, t i₀ = t i₁ → (abv (A i₁ % b - A i₀ % b) : ℝ) < abv b • ε := by\n  rcases Finite.exists_equiv_fin ι with ⟨n, ⟨e⟩⟩\n  obtain ⟨t, ht⟩ := h.exists_partition' n hε hb (A ∘ e.symm)\n  refine ⟨t ∘ e, fun i₀ i₁ h ↦ ?_⟩\n  convert! (config := { transparency := .default }) ht (e i₀) (e i₁) h <;>\n    simp only [e.symm_apply_apply]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Any large enough family of vectors in `R^n` has a pair of elements\nwhose remainders are close together, pointwise. -/\ntheorem exists_approx_aux (n : ℕ) (h : abv.IsAdmissible) :\n    ∀ {ε : ℝ} (_hε : 0 < ε) {b : R} (_hb : b ≠ 0) (A : Fin (h.card ε ^ n).succ → Fin n → R),\n      ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ k, (abv (A i₁ k % b - A i₀ k % b) : ℝ) < abv b • ε := by\n  haveI := Classical.decEq R\n  induction n with\n  | zero =>\n    intro ε _hε b _hb A\n    refine ⟨0, 1, ?_, ?_⟩\n    · simp\n    rintro ⟨i, ⟨⟩⟩\n  | succ n ih =>\n  intro ε hε b hb A\n  let M := h.card ε\n  -- By the \"nicer\" pigeonhole principle, we can find a collection `s`\n  -- of more than `M ^ n` remainders where the first components lie close together:\n  obtain ⟨s, s_inj, hs⟩ :\n    ∃ s : Fin (M ^ n).succ → Fin (M ^ n.succ).succ,\n      Function.Injective s ∧ ∀ i₀ i₁, (abv (A (s i₁) 0 % b - A (s i₀) 0 % b) : ℝ) < abv b • ε := by\n    -- We can partition the `A`s into `M` subsets where\n    -- the first components lie close together:\n    obtain ⟨t, ht⟩ :\n      ∃ t : Fin (M ^ n.succ).succ → Fin M,\n        ∀ i₀ i₁, t i₀ = t i₁ → (abv (A i₁ 0 % b - A i₀ 0 % b) : ℝ) < abv b • ε :=\n      h.exists_partition hε hb fun x ↦ A x 0\n    -- Since the `M` subsets contain more than `M * M^n` elements total,\n    -- there must be a subset that contains more than `M^n` elements.\n    obtain ⟨s, hs⟩ :=\n      Fintype.exists_lt_card_fiber_of_mul_lt_card (f := t)\n        (by simpa only [Fintype.card_fin, pow_succ'] using Nat.lt_succ_self (M ^ n.succ))\n    have : (M ^ n).succ ≤ (Finset.toList {x | t x = s}).length := by\n      rwa [Finset.length_toList]\n    refine ⟨fun i ↦ (Finset.toList {x | t x = s})[i.castLE this], fun i j h ↦ ?_,\n      fun i₀ i₁ ↦ ht _ _ ?_⟩\n    · simpa [(Finset.nodup_toList _).getElem_inj_iff, Fin.val_inj] using h\n    · have (i : Fin (M ^ n).succ) : t (Finset.toList {x | t x = s})[i.castLE this] = s :=\n        (Finset.mem_filter.mp ((Finset.mem_toList (s := {x | t x = s})).mp (List.getElem_mem _))).2\n      simp_rw [this]\n  -- Since `s` is large enough, there are two elements of `A ∘ s`\n  -- where the second components lie close together.\n  obtain ⟨k₀, k₁, hk, h⟩ := ih hε hb fun x ↦ Fin.tail (A (s x))\n  refine ⟨s k₀, s k₁, fun h ↦ hk (s_inj h), fun i ↦ Fin.cases ?_ (fun i ↦ ?_) i⟩\n  · exact hs k₀ k₁\n  · exact h i\n\n/-- Any large enough family of vectors in `R^ι` has a pair of elements\nwhose remainders are close together, pointwise. -/\n\nTarget:\ntheorem exists_approx {ι : Type*} [Fintype ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0)\n    (h : abv.IsAdmissible) (A : Fin (h.card ε ^ Fintype.card ι).succ → ι → R) :\n    ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ k, (abv (A i₁ k % b - A i₀ k % b) : ℝ) < abv b • ε :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/ClassNumber","family_id":"exists_approx","file_id":"mathlib/Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean","sample_id":"fc015e197e7e22d0acabad316890661898d1a5fa61e903e26832dbe043d9fa2c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9d077355c8aafccdf2f765766657b2aef7013299b24845a61f82fd2eda23f5d5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"16e52315f7d70bd4a1b86cbcbc09e475d0eddeeace4a355cb687391707f5e085","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"471ab45e244c5046c4d6db5db33762cae30b42d81f33c3c388fbdb7ee1366b37","source_sha256":"bdc962e9d41e4ed417e034e5bc54ca34f64e30878dd13535c17ebf9d0ca4959d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  zify\n  convert! Rat.num_mul_num_eq_num_mul_gcd q₁ q₂ <;> norm_cast","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":2,"token_jaccard":0.066667,"token_length_ratio":0.133333},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"2c8ecd531757390c20fcd24074ef03d0d120c0b4ca2517dab488f82066625f6c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Rat\npublic import Mathlib.Algebra.Group.Indicator\npublic import Mathlib.Algebra.Order.Field.Rat\npublic import Mathlib.Data.Rat.Lemmas\npublic import Mathlib.Tactic.Zify\n\nNamespace:\nNNRat\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Field and action structures on the nonnegative rationals\n\nThis file provides additional results about `NNRat` that cannot live in earlier files due to import\ncycles.\n-/\n\n@[expose] public section\n\nopen Function\nopen scoped NNRat\n\nnamespace NNRat\nvariable {α : Type*} {q : ℚ≥0}\n\n@[simp, norm_cast]\nlemma coe_indicator (s : Set α) (f : α → ℚ≥0) (a : α) :\n    ((s.indicator f a : ℚ≥0) : ℚ) = s.indicator (fun x ↦ ↑(f x)) a :=\n  map_indicator coeHom _ _ _\n\nend NNRat\n\nopen NNRat\n\nnamespace Rat\n\nvariable {p q : ℚ}\n\nlemma toNNRat_inv (q : ℚ) : toNNRat q⁻¹ = (toNNRat q)⁻¹ := by\n  obtain hq | hq := le_total q 0\n  · rw [toNNRat_eq_zero.mpr hq, inv_zero, toNNRat_eq_zero.mpr (inv_nonpos.mpr hq)]\n  · nth_rw 1 [← Rat.coe_toNNRat q hq]\n    rw [← coe_inv, toNNRat_coe]\n\nlemma toNNRat_div (hp : 0 ≤ p) : toNNRat (p / q) = toNNRat p / toNNRat q := by\n  rw [div_eq_mul_inv, div_eq_mul_inv, ← toNNRat_inv, ← toNNRat_mul hp]\n\nlemma toNNRat_div' (hq : 0 ≤ q) : toNNRat (p / q) = toNNRat p / toNNRat q := by\n  rw [div_eq_inv_mul, div_eq_inv_mul, toNNRat_mul (inv_nonneg.2 hq), toNNRat_inv]\n\nend Rat\n\n/-! ### Numerator and denominator -/\n\nnamespace NNRat\n\nvariable {q : ℚ≥0}\n\n/-- A recursor for nonnegative rationals in terms of numerators and denominators. -/\nprotected def rec {α : ℚ≥0 → Sort*} (h : ∀ m n : ℕ, α (m / n)) (q : ℚ≥0) : α q := by\n  rw [← num_div_den q]; apply h\n\ntheorem mul_num (q₁ q₂ : ℚ≥0) :\n    (q₁ * q₂).num = q₁.num * q₂.num / Nat.gcd (q₁.num * q₂.num) (q₁.den * q₂.den) := by\n  zify\n  convert! Rat.mul_num q₁ q₂ <;> norm_cast\n\ntheorem mul_den (q₁ q₂ : ℚ≥0) :\n    (q₁ * q₂).den = q₁.den * q₂.den / Nat.gcd (q₁.num * q₂.num) (q₁.den * q₂.den) := by\n  convert! Rat.mul_den q₁ q₂\n  norm_cast\n\n/-- A version of `NNRat.mul_den` without division. -/\ntheorem den_mul_den_eq_den_mul_gcd (q₁ q₂ : ℚ≥0) :\n    q₁.den * q₂.den = (q₁ * q₂).den * ((q₁.num * q₂.num).gcd (q₁.den * q₂.den)) := by\n  convert! Rat.den_mul_den_eq_den_mul_gcd q₁ q₂\n  norm_cast\n\n/-- A version of `NNRat.mul_num` without division. -/\n\nTarget:\ntheorem num_mul_num_eq_num_mul_gcd (q₁ q₂ : ℚ≥0) :\n    q₁.num * q₂.num = (q₁ * q₂).num * ((q₁.num * q₂.num).gcd (q₁.den * q₂.den)) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_471ab45e244c","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"d1751be3c73d22e7f3a85e4a351cc12fa071e04c5f752395529289df9e5ff240","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/NNRat","family_id":"num_mul_num_eq_num_mul_gcd","file_id":"mathlib/Mathlib/Data/NNRat/Lemmas.lean","sample_id":"471ab45e244c5046c4d6db5db33762cae30b42d81f33c3c388fbdb7ee1366b37"},"split":"train"}
{"binding":{"chosen_proof_sha256":"da2d9310157ed9ceff8c58c2de745bed1ed2329391fedb4f8010b3557e4b56ed","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1782058706e57bee7d4d9918ffda568fb09fac99c8f6bf7aa18b2cb7f7640f7f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9b4e3e0d6a7653b54818bb9ece9503e507bf5b7abbd8d2d4b99515af39aba54b","source_sha256":"5719c6ab1b79cb92451b632f8631c4fdd8e58d363dc71c3722b36068c3f61537","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h : ∃ n, x ≤ t n := by\n    simp only [tendsto_atTop_atTop_iff_of_monotone ht_mono.monotone] at ht_tendsto\n    exact ht_tendsto x\n  have h' m := Nat.find_min h (m := m)\n  simp only [not_le] at h'\n  exact ⟨Nat.find h - 1, h' _ (by simp [hx]), by simp [Nat.find_spec h, hx]⟩","hard_negative":true,"metrics":{"chosen_tokens":76,"rejected_tokens":5,"token_jaccard":0.076923,"token_length_ratio":0.065789},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"2c9d193228e2c597c7245b340979c4e46077c99e7b447c9d227b2930b87d88a1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecificLimits.ArithmeticGeometric\npublic import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap\npublic import Mathlib.MeasureTheory.Function.L1Space.Integrable\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n-/\n/-!\n# Fernique's theorem for rotation-invariant measures\n\nLet `μ` be a finite measure on a second-countable normed space `E` such that the product measure\n`μ.prod μ` on `E × E` is invariant by rotation of angle `-π/4`.\nThen there exists a constant `C > 0` such that the function `x ↦ exp (C * ‖x‖ ^ 2)` is integrable\nwith respect to `μ`.\n\n## Sketch of the proof\n\nThe main case of the proof is for `μ` a probability measure such that there exists a positive\n`a : ℝ` such that `2⁻¹ < μ {x | ‖x‖ ≤ a} < 1`. If `μ` is a probability measure and `a` does not\nexist then we can show that there is a ball with finite radius of measure 1, and the result is true\nfor `C = 1` (for example), since `x ↦ exp (‖x‖ ^ 2)` is almost surely bounded.\nWe then choose such an `a`.\n\nIn order to show the existence of `C` such that `x ↦ exp (C * ‖x‖ ^ 2)` is integrable, we prove as\nintermediate result that for `a, c` with `2⁻¹ < c ≤ μ {x | ‖x‖ ≤ a}`,\nthe integral `∫⁻ x, exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2) ∂μ` is bounded by a finite quantity\n(`logRatio c` is a multiple of `log (c / (1 - c))`). We can then take `C = logRatio c * a⁻¹ ^ 2`.\n\nWe now turn to the proof of the intermediate result.\n\nFirst in `measure_le_mul_measure_gt_le_of_map_rotation_eq_self` we prove that if a measure `μ` is\nsuch that `μ.prod μ` is invariant by rotation of angle `-π/4` then\n`μ {x | ‖x‖ ≤ a} * μ {x | b < ‖x‖} ≤ μ {x | (b - a) / √2 < ‖x‖} ^ 2`.\nThe rotation invariance is used only through that inequality.\n\nWe define a sequence of thresholds `t n` inductively by `t 0 = a` and `t (n + 1) = √2 * t n + a`.\nThey are chosen such that the invariance by rotation gives\n`μ {x | ‖x‖ ≤ a} * μ {x | t (n + 1) < ‖x‖} ≤ μ {x | t n < ‖x‖} ^ 2`.\nThanks to that inequality we can show that `μ {x | t n < ‖x‖}` decreases fast with `n`:\nfor `mₐ = μ {x | ‖x‖ ≤ a}`, `μ {x | t n < ‖x‖} ≤ mₐ * exp (- log (mₐ / (1 - mₐ)) * 2 ^ n)`.\n\nWe cut the space into annuli `{x | t n < ‖x‖ ≤ t n + 1}` and bound the integral separately on\neach annulus. On that set the function `exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2)` is bounded by\n`exp (logRatio c * a⁻¹ ^ 2 * t (n + 1) ^ 2)`, which is in turn less than\n`exp (2⁻¹ * log (c / (1 - c)) * 2 ^ n)` (from the definition of the threshold `t` and `logRatio c`).\nThe measure of the annulus is bounded by `μ {x | t n < ‖x‖}`, for which we derived an upper bound\nabove. The function gets exponentially large, but `μ {x | t n < ‖x‖}` decreases even faster, so the\nintegral is bounded by a quantity of the form `exp (- u * 2 ^ n)` for `u>0`.\nSumming over all annuli (over `n`) gives a finite value for the integral.\n\n## Main statements\n\n* `lintegral_exp_mul_sq_norm_le_of_map_rotation_eq_self`: for `μ` a probability measure\n  whose product with itself is invariant by rotation and for `a, c` with\n  `2⁻¹ < c ≤ μ {x | ‖x‖ ≤ a}`, the integral `∫⁻ x, exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2) ∂μ`\n  is bounded by a quantity that does not depend on `a`.\n* `exists_integrable_exp_sq_of_map_rotation_eq_self`: Fernique's theorem for finite measures\n  whose product is invariant by rotation.\n\n## References\n\n* [Xavier Fernique, *Intégrabilité des vecteurs gaussiens*][fernique1970integrabilite]\n* [Martin Hairer, *An introduction to stochastic PDEs*][hairer2009introduction]\n\n## TODO\n\nFrom the intermediate result `lintegral_exp_mul_sq_norm_le_of_map_rotation_eq_self`,\nwe can deduce bounds on all the moments of the measure `μ` as function of powers of\nthe first moment.\n\n-/\n\n@[expose] public section\n\nopen MeasureTheory ProbabilityTheory Complex NormedSpace Filter\nopen scoped ENNReal NNReal Real Topology\n\nsection Aux\n\nTarget:\nlemma StrictMono.exists_between_of_tendsto_atTop {β : Type*} [LinearOrder β] {t : ℕ → β}\n    (ht_mono : StrictMono t) (ht_tendsto : Tendsto t atTop atTop) {x : β} (hx : t 0 < x) :\n    ∃ n, t n < x ∧ x ≤ t (n + 1) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_9b4e3e0d6a76","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"33208c466f736312792a71e1140ed400042f42855ffd04e3f5648d58acd3b384","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Distributions","family_id":"strictmono","file_id":"mathlib/Mathlib/Probability/Distributions/Fernique.lean","sample_id":"9b4e3e0d6a7653b54818bb9ece9503e507bf5b7abbd8d2d4b99515af39aba54b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fd4f7273c1abc3ee605f4507e24cb13680aff1584126fd18e76d89a937f7044b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"50332d0f41236ffc25367bcc287b5a5f97bce4b58e2df5be3cb1c6d4b44bee1a","source_sha256":"6495ae02a429a2011b5873f559e0c6908f832b9f2a026ddab4af530b07660425","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  exact Decidable.List.Lex.ne_iff H","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":5,"token_jaccard":0.166667,"token_length_ratio":0.454545},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"2ce04ac8ec1a20af89e935349fe3b4a32d2127f97b4da240c993eda6fcc8455e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.List.Basic\npublic import Mathlib.Data.Nat.Basic\npublic import Mathlib.Order.RelClasses\n\nNamespace:\nList.Lex\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Lexicographic ordering of lists.\n\nThe lexicographic order on `List α` is defined by `L < M` iff\n* `[] < (a :: L)` for any `a` and `L`,\n* `(a :: L) < (b :: M)` where `a < b`, or\n* `(a :: L) < (a :: M)` where `L < M`.\n\n## See also\n\nRelated files are:\n* `Mathlib/Combinatorics/Colex.lean`: Colexicographic order on finite sets.\n* `Mathlib/Data/PSigma/Order.lean`: Lexicographic order on `Σ' i, α i`.\n* `Mathlib/Order/PiLex.lean`: Lexicographic order on `Πₗ i, α i`.\n* `Mathlib/Data/Sigma/Order.lean`: Lexicographic order on `Σ i, α i`.\n* `Mathlib/Data/Prod/Lex.lean`: Lexicographic order on `α × β`.\n-/\n\npublic section\n\n\nnamespace List\n\nopen Nat\n\nuniverse u\n\nvariable {α : Type u}\n\n/-! ### lexicographic ordering -/\n\ntheorem lex_cons_iff {r : α → α → Prop} [Std.Irrefl r] {a l₁ l₂} :\n    Lex r (a :: l₁) (a :: l₂) ↔ Lex r l₁ l₂ :=\n  ⟨fun h => by obtain - | h | h := h; exacts [(irrefl_of r a h).elim, h], Lex.cons⟩\n\ntheorem lex_nil_or_eq_nil {r : α → α → Prop} (l : List α) : List.Lex r [] l ∨ l = [] :=\n  match l with\n  | [] => Or.inr rfl\n  | _ :: _ => .inl .nil\n\n@[simp]\ntheorem lex_singleton_iff {r : α → α → Prop} (a b : α) : List.Lex r [a] [b] ↔ r a b :=\n  ⟨fun | .rel h => h, .rel⟩\n\nnamespace Lex\n\ninstance isOrderConnected (r : α → α → Prop) [IsOrderConnected α r] [Std.Trichotomous r] :\n    IsOrderConnected (List α) (Lex r) where\n  conn := aux where\n    aux\n    | _, [], _ :: _, nil => Or.inr nil\n    | _, [], _ :: _, rel _ => Or.inr nil\n    | _, [], _ :: _, cons _ => Or.inr nil\n    | _, _ :: _, _ :: _, nil => Or.inl nil\n    | _ :: _, b :: _, _ :: _, rel h => (IsOrderConnected.conn _ b _ h).imp rel rel\n    | a :: l₁, b :: l₂, _ :: l₃, cons h => by\n      rcases trichotomous_of r a b with (ab | rfl | ab)\n      · exact Or.inl (rel ab)\n      · exact (aux _ l₂ _ h).imp cons cons\n      · exact Or.inr (rel ab)\n\ninstance trichotomous (r : α → α → Prop) [Std.Trichotomous r] : Std.Trichotomous (Lex r) where\n  trichotomous := aux where\n    aux\n    | [], [], _, _ => rfl\n    | [], _ :: _, hab, _ => hab nil |>.elim\n    | _ :: _, [], _, hba => hba nil |>.elim\n    | a :: l₁, b :: l₂, hab, hba => by\n      obtain rfl := Std.Trichotomous.trichotomous a b (mt rel hab) (mt rel hba)\n      rw [aux l₁ l₂ (mt cons hab) (mt cons hba)]\n\ninstance asymm (r : α → α → Prop) [Std.Asymm r] : Std.Asymm (Lex r) where\n  asymm := aux where\n    aux\n    | _, _, Lex.rel h₁, Lex.rel h₂ => _root_.asymm h₁ h₂\n    | _, _, Lex.rel h₁, Lex.cons _ => _root_.asymm h₁ h₁\n    | _, _, Lex.cons _, Lex.rel h₂ => _root_.asymm h₂ h₂\n    | _, _, Lex.cons h₁, Lex.cons h₂ => aux _ _ h₁ h₂\n\ninstance decidableRel [DecidableEq α] (r : α → α → Prop) [DecidableRel r] : DecidableRel (Lex r)\n  | l₁, [] => isFalse fun h => by cases h\n  | [], _ :: _ => isTrue Lex.nil\n  | a :: l₁, b :: l₂ => by\n    haveI := decidableRel r l₁ l₂\n    refine decidable_of_iff (r a b ∨ a = b ∧ Lex r l₁ l₂) ⟨fun h => ?_, fun h => ?_⟩\n    · rcases h with (h | ⟨rfl, h⟩)\n      · exact Lex.rel h\n      · exact Lex.cons h\n    · rcases h with (_ | h | h)\n      · exact Or.inl h\n      · exact Or.inr ⟨rfl, h⟩\n\ntheorem append_right (r : α → α → Prop) : ∀ {s₁ s₂} (t), Lex r s₁ s₂ → Lex r s₁ (s₂ ++ t)\n  | _, _, _, nil => nil\n  | _, _, _, cons h => cons (append_right r _ h)\n  | _, _, _, rel r => rel r\n\ntheorem append_left (R : α → α → Prop) {t₁ t₂} (h : Lex R t₁ t₂) : ∀ s, Lex R (s ++ t₁) (s ++ t₂)\n  | [] => h\n  | _ :: l => cons (append_left R h l)\n\ntheorem imp {r s : α → α → Prop} (H : ∀ a b, r a b → s a b) : ∀ l₁ l₂, Lex r l₁ l₂ → Lex s l₁ l₂\n  | _, _, nil => nil\n  | _, _, cons h => cons (imp H _ _ h)\n  | _, _, rel r => rel (H _ _ r)\n\ntheorem to_ne : ∀ {l₁ l₂ : List α}, Lex (· ≠ ·) l₁ l₂ → l₁ ≠ l₂\n  | _, _, cons h, e => to_ne h (List.cons.inj e).2\n  | _, _, rel r, e => r (List.cons.inj e).1\n\ntheorem _root_.Decidable.List.Lex.ne_iff [DecidableEq α] {l₁ l₂ : List α}\n    (H : length l₁ ≤ length l₂) : Lex (· ≠ ·) l₁ l₂ ↔ l₁ ≠ l₂ :=\n  ⟨to_ne, fun h => by\n    induction l₁ generalizing l₂ <;> rcases l₂ with - | ⟨b, l₂⟩\n    · contradiction\n    · apply nil\n    · exact (not_lt_of_ge H).elim (succ_pos _)\n    case cons.cons a l₁ IH =>\n      by_cases ab : a = b\n      · subst b\n        exact .cons <| IH (le_of_succ_le_succ H) (mt (congr_arg _) h)\n      · exact .rel ab ⟩\n\nTarget:\ntheorem ne_iff {l₁ l₂ : List α} (H : length l₁ ≤ length l₂) : Lex (· ≠ ·) l₁ l₂ ↔ l₁ ≠ l₂ :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/List","family_id":"ne_iff","file_id":"mathlib/Mathlib/Data/List/Lex.lean","sample_id":"50332d0f41236ffc25367bcc287b5a5f97bce4b58e2df5be3cb1c6d4b44bee1a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0ec8ef0b04cc9e360fe54d83ec9b445dee17ea05e7abf3a9b4757fa21a5fb78e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"71f1b32ec46df06988407bec9fec4c171851cda268d1134ad7fa0191f2142682","source_sha256":"b5b8fef319876fc584704ec0902863ab8016fe0ba4dee5dd358f86f4688b13cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [Irrational, Rat.forall, eq_comm]","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.272727},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"2d5d4073c53729387264afb981a4782a108c48dc9ec4ae0530c3ea2db1ba33f9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Rat\npublic import Mathlib.Data.Nat.Prime.Int\npublic import Mathlib.Data.Rat.Sqrt\npublic import Mathlib.Analysis.Real.Sqrt\npublic import Mathlib.RingTheory.Algebraic.Basic\npublic import Mathlib.Tactic.CrossRefAttribute\npublic import Mathlib.Tactic.IntervalCases\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov\n-/\n/-!\n# Irrational real numbers\n\nIn this file we define a predicate `Irrational` on `ℝ`, prove that the `n`-th root of an integer\nnumber is irrational if it is not integer, and that `√(q : ℚ)` is irrational if and only if\n`¬IsSquare q ∧ 0 ≤ q`.\n\nWe also provide dot-style constructors like `Irrational.add_ratCast`, `Irrational.ratCast_sub` etc.\n\nWith the `Decidable` instances in this file, is possible to prove `Irrational √n` using `decide`,\nwhen `n` is a numeric literal or cast;\nbut this only works if you `unseal Nat.sqrt.iter in` before the theorem where you use this proof.\n-/\n\n@[expose] public section\n\n\nopen Rat Real\n\n/-- A real number is irrational if it is not equal to any rational number. -/\n@[wikidata Q607728]\ndef Irrational (x : ℝ) :=\n  x ∉ Set.range ((↑) : ℚ → ℝ)\n\nTarget:\ntheorem irrational_iff_ne_rational (x : ℝ) : Irrational x ↔ ∀ a b : ℤ, b ≠ 0 → x ≠ a / b :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Real","family_id":"irrational_iff_ne_rational","file_id":"mathlib/Mathlib/NumberTheory/Real/Irrational.lean","sample_id":"71f1b32ec46df06988407bec9fec4c171851cda268d1134ad7fa0191f2142682"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b2349a1019905ca8cc28d0ef2d79317628410a89c7793fafcbce2e7a084a9bdd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"16580c7b1066ea68f7db2b4cc23ac07bcdd4aa4217971b759537bed8271b6fb6","source_sha256":"b8e3cefa8cfc385c8eb0b597e4d568bbff6c04129d45f91f63ded9f92406e88b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Ioi_union_left, ← Ioi_union_left]\n  exact\n    (surjOn_Ioi_of_monotone_surjective h_mono h_surj a).union_union\n      (@image_singleton _ _ f a ▸ surjOn_image _ _)","hard_negative":false,"metrics":{"chosen_tokens":30,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.066667},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"2db597b4238ff7e5c8ee68b5a011a6356f58ebff573573bd0ca7ebbc5c66e79e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Function\npublic import Mathlib.Order.Interval.Set.LinearOrder\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Heather Macbeth. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Heather Macbeth\n-/\n/-!\n# Monotone surjective functions are surjective on intervals\n\nA monotone surjective function sends any interval in the domain onto the interval with corresponding\nendpoints in the range.  This is expressed in this file using `Set.surjOn`, and provided for all\npermutations of interval endpoints.\n-/\n\npublic section\n\n\nvariable {α : Type*} {β : Type*} [LinearOrder α] [PartialOrder β] {f : α → β}\n\nopen Set Function\n\nopen OrderDual (toDual)\n\ntheorem surjOn_Ioo_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    (a b : α) : SurjOn f (Ioo a b) (Ioo (f a) (f b)) := by\n  intro p hp\n  rcases h_surj p with ⟨x, rfl⟩\n  refine ⟨x, mem_Ioo.2 ?_, rfl⟩\n  contrapose! hp\n  exact fun h => h.2.not_ge (h_mono <| hp <| h_mono.reflect_lt h.1)\n\ntheorem surjOn_Ico_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    (a b : α) : SurjOn f (Ico a b) (Ico (f a) (f b)) := by\n  obtain hab | hab := lt_or_ge a b\n  · intro p hp\n    rcases eq_left_or_mem_Ioo_of_mem_Ico hp with (rfl | hp')\n    · exact mem_image_of_mem f (left_mem_Ico.mpr hab)\n    · exact image_mono Ioo_subset_Ico_self <|\n        surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'\n  · rw [Ico_eq_empty (h_mono hab).not_gt]\n    exact surjOn_empty f _\n\ntheorem surjOn_Ioc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    (a b : α) : SurjOn f (Ioc a b) (Ioc (f a) (f b)) := by\n  simpa using! surjOn_Ico_of_monotone_surjective h_mono.dual h_surj (toDual b) (toDual a)\n\n-- to see that the hypothesis `a ≤ b` is necessary, consider a constant function\ntheorem surjOn_Icc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    {a b : α} (hab : a ≤ b) : SurjOn f (Icc a b) (Icc (f a) (f b)) := by\n  intro p hp\n  rcases eq_endpoints_or_mem_Ioo_of_mem_Icc hp with (rfl | rfl | hp')\n  · exact ⟨a, left_mem_Icc.mpr hab, rfl⟩\n  · exact ⟨b, right_mem_Icc.mpr hab, rfl⟩\n  · exact image_mono Ioo_subset_Icc_self <|\n      surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'\n\ntheorem surjOn_Ioi_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    (a : α) : SurjOn f (Ioi a) (Ioi (f a)) := by\n  rw [← compl_Iic, ← compl_compl (Ioi (f a))]\n  refine MapsTo.surjOn_compl ?_ h_surj\n  exact fun x hx => (h_mono hx).not_gt\n\ntheorem surjOn_Iio_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    (a : α) : SurjOn f (Iio a) (Iio (f a)) :=\n  @surjOn_Ioi_of_monotone_surjective _ _ _ _ _ h_mono.dual h_surj a\n\nTarget:\ntheorem surjOn_Ici_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    (a : α) : SurjOn f (Ici a) (Ici (f a)) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Interval","family_id":"surjon_ici_of_monotone_surjective","file_id":"mathlib/Mathlib/Order/Interval/Set/SurjOn.lean","sample_id":"16580c7b1066ea68f7db2b4cc23ac07bcdd4aa4217971b759537bed8271b6fb6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"badc607969163b460b5422c403e504297c9199325e39a0ffb23b23e6ce679da2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e3c482c18dafac02bf8a0fd6128bd4ffa9ee2fc18b435651f9c2883c55628c8a","source_sha256":"006922c150073d26eccce5bef3353d7015b01e87320ecec635275017a27177d4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [ncoeff_apply, neg_sub, Int.sub_neg, add_sub_cancel_left]","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":8,"token_jaccard":0.058824,"token_length_ratio":0.615385},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"2e30d261dca200bd42a30493920abddffedf67d94fa7edf2c803f48d58654f1f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Vertex.HVertexOperator\npublic import Mathlib.Data.Int.Interval\n\nNamespace:\nVertexOperator\n\nLocal context:\n/-\nCopyright (c) 2024 Scott Carnahan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Carnahan\n-/\n/-!\n# Vertex operators\nIn this file we introduce vertex operators as linear maps to Laurent series.\n\n## Definitions\n* `VertexOperator` is an `R`-linear map from an `R`-module `V` to `LaurentSeries V`.\n* `VertexOperator.ncoeff` is the coefficient of a vertex operator under normalized indexing.\n\n## TODO\n* `HasseDerivative` : A divided-power derivative.\n* `Locality` : A weak form of commutativity.\n* `Residue products` : A family of products on `VertexOperator R V` parametrized by integers.\n\n## References\n* [G. Mason, *Vertex rings and Pierce bundles*][mason2017]\n* [A. Matsuo, K. Nagatomo, *On axioms for a vertex algebra and locality of quantum\n  fields*][matsuo1997]\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nvariable {R V : Type*} [CommRing R] [AddCommGroup V] [Module R V]\n\n/-- A vertex operator over a commutative ring `R` is an `R`-linear map from an `R`-module `V` to\nLaurent series with coefficients in `V`.  We write this as a specialization of the heterogeneous\ncase. -/\nabbrev VertexOperator (R : Type*) (V : Type*) [CommRing R] [AddCommGroup V]\n    [Module R V] := HVertexOperator ℤ R V V\n\nnamespace VertexOperator\n\nopen HVertexOperator\n\n@[ext]\ntheorem ext (A B : VertexOperator R V) (h : ∀ v : V, A v = B v) :\n    A = B := LinearMap.ext h\n\n/-- The coefficient of a vertex operator under normalized indexing. -/\ndef ncoeff : VertexOperator R V →ₗ[R] ℤ → Module.End R V where\n  toFun A n := HVertexOperator.coeff A (-n - 1)\n  map_add' _ _ := by ext; simp\n  map_smul' _ _ := by ext; simp\n\ntheorem ncoeff_apply (A : VertexOperator R V) (n : ℤ) : ncoeff A n = coeff A (-n - 1) :=\n  rfl\n\n/-- In the literature, the `n`th normalized coefficient of a vertex operator `A` is written as\neither `Aₙ` or `A(n)`. -/\nscoped[VertexOperator] notation A \"[[\" n \"]]\" => ncoeff A n\n\n@[simp]\n\nTarget:\ntheorem coeff_eq_ncoeff (A : VertexOperator R V)\n    (n : ℤ) : HVertexOperator.coeff A n = A[[-n - 1]] :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"392a0fed2d32d7800f087ecfc18a0b4604bd4bbae3bc30b59f83e0a0e224af67","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Vertex","family_id":"coeff_eq_ncoeff","file_id":"mathlib/Mathlib/Algebra/Vertex/VertexOperator.lean","sample_id":"e3c482c18dafac02bf8a0fd6128bd4ffa9ee2fc18b435651f9c2883c55628c8a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c860446eb6ebd47f56bc80449291da30bdf73e713b21c6c028e07a7ce9d17e8f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"624cb99c0cbd1997bbdc0b1f1c47bd9e2e1caf360a81585dc2e8556116e02b79","source_sha256":"6f89eec31c4c30c0e54882861b32ee0325842f53b6b6cd8f4cb71a0f6148c838","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n      rw [Iso.conj_apply, Iso.eq_inv_comp]; simp [f.hom.comm]","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":3,"token_jaccard":0.0625,"token_length_ratio":0.15},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"2e7fec6cb7e34286c34ca3fdc95848ade499dfded5ed2034aa00d7c7218752a5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Category.Grp.Basic\npublic import Mathlib.Algebra.Ring.PUnit\npublic import Mathlib.CategoryTheory.Adjunction.Limits\npublic import Mathlib.CategoryTheory.Conj\npublic import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic\npublic import Mathlib.CategoryTheory.Limits.Preserves.Basic\npublic import Mathlib.CategoryTheory.SingleObj\npublic import Mathlib.Tactic.ApplyFun\n\nNamespace:\nAction\n\nLocal context:\n/-\nCopyright (c) 2020 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# `Action V G`, the category of actions of a monoid `G` inside some category `V`.\n\nThe prototypical example is `V = ModuleCat R`,\nwhere `Action (ModuleCat R) G` is the category of `R`-linear representations of `G`.\n\nWe check `Action V G ≌ (CategoryTheory.SingleObj G ⥤ V)`,\nand construct the restriction functors\n`res {G H} [Monoid G] [Monoid H] (f : G →* H) : Action V H ⥤ Action V G`.\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\nopen CategoryTheory Limits\n\nvariable (V : Type*) [Category* V]\n\n-- Note: this is _not_ a categorical action of `G` on `V`.\n/-- An `Action V G` represents a bundled action of\nthe monoid `G` on an object of some category `V`.\n\nAs an example, when `V = ModuleCat R`, this is an `R`-linear representation of `G`,\nwhile when `V = Type` this is a `G`-action.\n-/\nstructure Action (G : Type*) [Monoid G] where\n  /-- The object this action acts on -/\n  V : V\n  /-- The underlying monoid homomorphism of this action -/\n  ρ : G →* End V\n\nnamespace Action\n\nvariable {V}\n\ntheorem ρ_one {G : Type*} [Monoid G] (A : Action V G) : A.ρ 1 = 𝟙 A.V := by simp\n\n/-- When a group acts, we can lift the action to the group of automorphisms. -/\n@[simps]\ndef ρAut {G : Type*} [Group G] (A : Action V G) : G →* Aut A.V where\n  toFun g :=\n    { hom := A.ρ g\n      inv := A.ρ (g⁻¹ : G)\n      hom_inv_id := (A.ρ.map_mul (g⁻¹ : G) g).symm.trans (by rw [inv_mul_cancel, ρ_one])\n      inv_hom_id := (A.ρ.map_mul g (g⁻¹ : G)).symm.trans (by rw [mul_inv_cancel, ρ_one]) }\n  map_one' := Aut.ext A.ρ.map_one\n  map_mul' x y := Aut.ext (A.ρ.map_mul x y)\n\nvariable (G : Type*) [Monoid G]\n\nsection\n\n/-- The action defined by sending every monoid element to the identity. -/\n@[simps]\ndef trivial (X : V) : Action V G := { V := X, ρ := 1 }\n\ninstance inhabited' : Inhabited (Action Type* G) :=\n  ⟨⟨PUnit, 1⟩⟩\n\ninstance : Inhabited (Action AddCommGrpCat G) :=\n  ⟨trivial G <| AddCommGrpCat.of PUnit⟩\n\nend\n\nvariable {G}\n\n/-- A homomorphism of `Action V G`s is a morphism between the underlying objects,\ncommuting with the action of `G`.\n-/\n@[ext]\nstructure Hom (M N : Action V G) where\n  /-- The morphism between the underlying objects of this action -/\n  hom : M.V ⟶ N.V\n  comm : ∀ g : G, M.ρ g ≫ hom = hom ≫ N.ρ g := by cat_disch\n\nnamespace Hom\n\nattribute [reassoc] comm\nattribute [local simp] comm comm_assoc\n\n/-- The identity morphism on an `Action V G`. -/\n@[simps]\ndef id (M : Action V G) : Action.Hom M M where hom := 𝟙 M.V\n\ninstance (M : Action V G) : Inhabited (Action.Hom M M) :=\n  ⟨id M⟩\n\n/-- The composition of two `Action V G` homomorphisms is the composition of the underlying maps.\n-/\n@[simps]\ndef comp {M N K : Action V G} (p : Action.Hom M N) (q : Action.Hom N K) : Action.Hom M K where\n  hom := p.hom ≫ q.hom\n\nend Hom\n\ninstance : Category (Action V G) where\n  Hom M N := Hom M N\n  id M := Hom.id M\n  comp f g := Hom.comp f g\n\nlemma hom_injective {M N : Action V G} : Function.Injective (Hom.hom : (M ⟶ N) → (M.V ⟶ N.V)) :=\n  fun _ _ ↦ Hom.ext\n\n@[ext]\nlemma hom_ext {M N : Action V G} (φ₁ φ₂ : M ⟶ N) (h : φ₁.hom = φ₂.hom) : φ₁ = φ₂ :=\n  Hom.ext h\n\n@[simp]\ntheorem id_hom (M : Action V G) : (𝟙 M : Hom M M).hom = 𝟙 M.V :=\n  rfl\n\n@[simp, reassoc]\ntheorem comp_hom {M N K : Action V G} (f : M ⟶ N) (g : N ⟶ K) :\n    (f ≫ g : Hom M K).hom = f.hom ≫ g.hom :=\n  rfl\n\n@[reassoc (attr := simp)]\ntheorem hom_inv_hom {M N : Action V G} (f : M ≅ N) :\n    f.hom.hom ≫ f.inv.hom = 𝟙 M.V := by\n  rw [← comp_hom, Iso.hom_inv_id, id_hom]\n\n@[reassoc (attr := simp)]\ntheorem inv_hom_hom {M N : Action V G} (f : M ≅ N) :\n    f.inv.hom ≫ f.hom.hom = 𝟙 N.V := by\n  rw [← comp_hom, Iso.inv_hom_id, id_hom]\n\n/-- Construct an isomorphism of `G` actions/representations\nfrom an isomorphism of the underlying objects,\nwhere the forward direction commutes with the group action. -/\n@[simps]\ndef mkIso {M N : Action V G} (f : M.V ≅ N.V)\n    (comm : ∀ g : G, M.ρ g ≫ f.hom = f.hom ≫ N.ρ g := by cat_disch) : M ≅ N where\n  hom :=\n    { hom := f.hom\n      comm := comm }\n  inv :=\n    { hom := f.inv\n      comm := fun g => by have w := comm g =≫ f.inv; simp at w; simp [w] }\n\ninstance (priority := 100) isIso_of_hom_isIso {M N : Action V G} (f : M ⟶ N) [IsIso f.hom] :\n    IsIso f := (mkIso (asIso f.hom) f.comm).isIso_hom\n\ninstance isIso_hom_mk {M N : Action V G} (f : M.V ⟶ N.V) [IsIso f] (w) :\n    @IsIso _ _ M N (Hom.mk f w) :=\n  (mkIso (asIso f) w).isIso_hom\n\ninstance {M N : Action V G} (f : M ≅ N) : IsIso f.hom.hom where\n  out := ⟨f.inv.hom, by simp⟩\n\ninstance {M N : Action V G} (f : M ≅ N) : IsIso f.inv.hom where\n  out := ⟨f.hom.hom, by simp⟩\n\nnamespace FunctorCategoryEquivalence\n\n/-- Auxiliary definition for `functorCategoryEquivalence`. -/\n@[simps]\ndef functor : Action V G ⥤ SingleObj G ⥤ V where\n  obj M :=\n    { obj := fun _ => M.V\n      map := fun g => M.ρ g\n      map_id := fun _ => M.ρ.map_one\n      map_comp := fun g h => M.ρ.map_mul h g }\n  map f :=\n    { app := fun _ => f.hom\n      naturality := fun _ _ g => f.comm g }\n\n/-- Auxiliary definition for `functorCategoryEquivalence`. -/\n@[simps]\ndef inverse : (SingleObj G ⥤ V) ⥤ Action V G where\n  obj F :=\n    { V := F.obj PUnit.unit\n      ρ :=\n        { toFun := fun g => F.map g\n          map_one' := F.map_id PUnit.unit\n          map_mul' := fun g h => F.map_comp h g } }\n  map f :=\n    { hom := f.app PUnit.unit\n      comm := fun g => f.naturality g }\n\nset_option backward.defeqAttrib.useBackward true in\n/-- Auxiliary definition for `functorCategoryEquivalence`. -/\n@[simps!]\ndef unitIso : 𝟭 (Action V G) ≅ functor ⋙ inverse :=\n  NatIso.ofComponents fun M => mkIso (Iso.refl _)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Auxiliary definition for `functorCategoryEquivalence`. -/\n@[simps!]\ndef counitIso : inverse ⋙ functor ≅ 𝟭 (SingleObj G ⥤ V) :=\n  NatIso.ofComponents fun M => NatIso.ofComponents fun _ => Iso.refl _\n\nend FunctorCategoryEquivalence\n\nsection\n\nopen FunctorCategoryEquivalence\n\nvariable (V G)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The category of actions of `G` in the category `V`\nis equivalent to the functor category `SingleObj G ⥤ V`.\n-/\n@[simps]\ndef functorCategoryEquivalence : Action V G ≌ SingleObj G ⥤ V where\n  functor := functor\n  inverse := inverse\n  unitIso := unitIso\n  counitIso := counitIso\n\ninstance : (FunctorCategoryEquivalence.functor (V := V) (G := G)).IsEquivalence :=\n  (functorCategoryEquivalence V G).isEquivalence_functor\n\ninstance : (FunctorCategoryEquivalence.inverse (V := V) (G := G)).IsEquivalence :=\n  (functorCategoryEquivalence V G).isEquivalence_inverse\n\nend\n\nsection Forget\n\nvariable (V G)\n\n/-- (implementation) The forgetful functor from bundled actions to the underlying objects.\n\nUse the `CategoryTheory.forget` API provided by the `ConcreteCategory` instance below,\nrather than using this directly.\n-/\n@[simps]\ndef forget : Action V G ⥤ V where\n  obj M := M.V\n  map f := f.hom\n\ninstance : (forget V G).Faithful where map_injective w := Hom.ext w\n\n/-- The type of `V`-morphisms that can be lifted back to morphisms in the category `Action`. -/\nabbrev HomSubtype {FV : V → V → Type*} {CV : V → Type*} [∀ X Y, FunLike (FV X Y) (CV X) (CV Y)]\n    [ConcreteCategory V FV] (M N : Action V G) :=\n  { f : FV M.V N.V // ∀ g : G,\n      f ∘ ConcreteCategory.hom (M.ρ g) = ConcreteCategory.hom (N.ρ g) ∘ f }\n\ninstance {FV : V → V → Type*} {CV : V → Type*} [∀ X Y, FunLike (FV X Y) (CV X) (CV Y)]\n    [ConcreteCategory V FV] (M N : Action V G) :\n    FunLike (HomSubtype V G M N) (CV M.V) (CV N.V) where\n  coe f := f.1\n  coe_injective _ _ h := Subtype.ext (DFunLike.coe_injective h)\n\ninstance {FV : V → V → Type*} {CV : V → Type*} [∀ X Y, FunLike (FV X Y) (CV X) (CV Y)]\n    [ConcreteCategory V FV] : ConcreteCategory (Action V G) (HomSubtype V G) where\n  hom f := ⟨ConcreteCategory.hom (C := V) f.1, fun g => by\n    ext\n    simpa using CategoryTheory.congr_fun (f.2 g) _⟩\n  ofHom f := ⟨ConcreteCategory.ofHom (C := V) f, fun g => ConcreteCategory.ext_apply fun x => by\n    simpa [ConcreteCategory.hom_ofHom] using congr_fun (f.2 g) x⟩\n  hom_ofHom _ := by dsimp; ext; simp [ConcreteCategory.hom_ofHom]\n  ofHom_hom _ := by ext; simp [ConcreteCategory.ofHom_hom]\n  id_apply := ConcreteCategory.id_apply (C := V)\n  comp_apply _ _ := ConcreteCategory.comp_apply (C := V) _ _\n\ninstance hasForgetToV {FV : V → V → Type*} {CV : V → Type*} [∀ X Y, FunLike (FV X Y) (CV X) (CV Y)]\n    [ConcreteCategory V FV] : HasForget₂ (Action V G) V where forget₂ := forget V G\n\n/-- The forgetful functor is intertwined by `functorCategoryEquivalence` with\nevaluation at `PUnit.star`. -/\ndef functorCategoryEquivalenceCompEvaluation :\n    (functorCategoryEquivalence V G).functor ⋙ (evaluation _ _).obj PUnit.unit ≅ forget V G :=\n  Iso.refl _\n\nnoncomputable instance preservesLimits_forget [HasLimits V] :\n    PreservesLimits (forget V G) :=\n  Limits.preservesLimits_of_natIso (Action.functorCategoryEquivalenceCompEvaluation V G)\n\nnoncomputable instance preservesColimits_forget [HasColimits V] :\n    PreservesColimits (forget V G) :=\n  preservesColimits_of_natIso (Action.functorCategoryEquivalenceCompEvaluation V G)\n\n-- TODO construct categorical images?\nend Forget\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem Iso.conj_ρ {M N : Action V G} (f : M ≅ N) (g : G) :\n    N.ρ g = ((forget V G).mapIso f).conj (M.ρ g) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Action","family_id":"iso","file_id":"mathlib/Mathlib/CategoryTheory/Action/Basic.lean","sample_id":"624cb99c0cbd1997bbdc0b1f1c47bd9e2e1caf360a81585dc2e8556116e02b79"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ffc956ff392a03eea39dc40bd447e1d225bcdcdd1b88caef9ee391f6b7a9878c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1bdf66f02f800f9f6bc8f802f4eb7010531ea0fed644e6fb319377ffa1bd244a","source_sha256":"250e0bf39b2fceb88cc44db25e786e52b9f1629e4d745089c27db7ba4ae4e60e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [card_of_finrank k V h]","hard_negative":false,"metrics":{"chosen_tokens":8,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.625},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"2e8bc6afe474cf2e998dc2d18cf565d3c2b279a2df6330a0e1b96d0d8beee73b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.GeomSum\npublic import Mathlib.Algebra.GroupWithZero.Units.Fintype\npublic import Mathlib.Data.Finite.Sum\npublic import Mathlib.GroupTheory.GroupAction.Quotient\npublic import Mathlib.LinearAlgebra.Projectivization.Basic\n\nNamespace:\nProjectivization\n\nLocal context:\n/-\nCopyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Judith Ludwig, Christian Merten\n-/\n/-!\n# Cardinality of projective spaces\n\nWe compute the cardinality of `ℙ k V` if `k` is a finite field.\n\n-/\n\n@[expose] public section\n\nnamespace Projectivization\n\nopen scoped LinearAlgebra.Projectivization\n\nsection\n\nvariable (k V : Type*) [DivisionRing k] [AddCommGroup V] [Module k V]\n\n/-- `ℙ k V` is equivalent to the quotient of the non-zero elements of `V` by `kˣ`. -/\ndef equivQuotientOrbitRel : ℙ k V ≃ Quotient (MulAction.orbitRel kˣ { v : V // v ≠ 0 }) :=\n  Quotient.congr (Equiv.refl _) (fun x y ↦ (Units.orbitRel_nonZero_iff k V x y).symm)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The non-zero elements of `V` are equivalent to the product of `ℙ k V` with the units of `k`. -/\nnoncomputable def nonZeroEquivProjectivizationProdUnits : { v : V // v ≠ 0 } ≃ ℙ k V × kˣ :=\n  let e := MulAction.selfEquivOrbitsQuotientProd <| fun b ↦ by\n    rw [(Units.nonZeroSubMul k V).stabilizer_of_subMul,\n      Module.stabilizer_units_eq_bot_of_ne_zero k b.property]\n  e.trans (Equiv.prodCongrLeft (fun _ ↦ (equivQuotientOrbitRel k V).symm))\n\ninstance isEmpty_of_subsingleton [Subsingleton V] : IsEmpty (ℙ k V) := by\n  have : IsEmpty { v : V // v ≠ 0 } := ⟨fun v ↦ v.2 (Subsingleton.elim v.1 0)⟩\n  simpa using (nonZeroEquivProjectivizationProdUnits k V).symm.isEmpty\n\n/-- If `V` is a finite `k`-module and `k` is finite, `ℙ k V` is finite. -/\ninstance finite_of_finite [Finite V] : Finite (ℙ k V) :=\n  have : Finite (ℙ k V × kˣ) := Finite.of_equiv _ (nonZeroEquivProjectivizationProdUnits k V)\n  Finite.prod_left kˣ\n\nlemma finite_iff_of_finite [Finite k] : Finite (ℙ k V) ↔ Finite V := by\n  classical\n  refine ⟨fun h ↦ ?_, fun h ↦ inferInstance⟩\n  let e := nonZeroEquivProjectivizationProdUnits k V\n  have : Finite { v : V // v ≠ 0 } := Finite.of_equiv _ e.symm\n  let eq : { v : V // v ≠ 0 } ⊕ Unit ≃ V :=\n    ⟨(Sum.elim Subtype.val (fun _ ↦ 0)), fun v ↦ if h : v = 0 then Sum.inr () else Sum.inl ⟨v, h⟩,\n      by intro x; aesop, by intro x; aesop⟩\n  exact Finite.of_equiv _ eq\n\n/-- Fraction free cardinality formula for the points of `ℙ k V` if `k` and `V` are finite\n(for silly reasons the formula also holds when `k` and `V` are infinite).\nSee `Projectivization.card'` and `Projectivization.card''` for other spellings of the formula. -/\nlemma card : Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1) := by\n  nontriviality V\n  cases finite_or_infinite k with\n  | inr h =>\n    have : Infinite V := Module.Free.infinite k V\n    simp\n  | inl h =>\n  cases finite_or_infinite V with\n  | inr h =>\n    have := not_iff_not.mpr (finite_iff_of_finite k V)\n    push Not at this\n    have : Infinite (ℙ k V) := by rwa [this]\n    simp\n  | inl h =>\n  classical\n  haveI : Fintype V := Fintype.ofFinite V\n  haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)\n  haveI : Fintype k := Fintype.ofFinite k\n  have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp\n  simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]\n  rw [Fintype.card_congr (nonZeroEquivProjectivizationProdUnits k V), Fintype.card_prod]\n\n/-- Cardinality formula for the points of `ℙ k V` if `k` and `V` are finite with less\nnatural subtraction. -/\nlemma card' [Finite V] : Nat.card V = Nat.card (ℙ k V) * (Nat.card k - 1) + 1 := by\n  rw [← card k V]\n  have : Nat.card V > 0 := Nat.card_pos\n  lia\n\nend\n\nvariable (k V : Type*) [Field k] [AddCommGroup V] [Module k V]\n\n/-- Cardinality formula for the points of `ℙ k V` if `k` and `V` are finite expressed\nas a fraction. -/\nlemma card'' [Finite k] : Nat.card (ℙ k V) = (Nat.card V - 1) / (Nat.card k - 1) := by\n  have : 1 < Nat.card k := Finite.one_lt_card\n  rw [card k, Nat.mul_div_cancel]\n  lia\n\nlemma card_of_finrank [Finite k] {n : ℕ} (h : Module.finrank k V = n) :\n    Nat.card (ℙ k V) = ∑ i ∈ Finset.range n, Nat.card k ^ i := by\n  wlog hf : Finite V\n  · have : Infinite (ℙ k V) := by\n      contrapose! hf\n      rwa [finite_iff_of_finite] at hf\n    have : n = 0 := by\n      rw [← h]\n      apply Module.finrank_of_not_finite\n      contrapose hf\n      simpa using Module.finite_of_finite k\n    simp [this]\n  have : 1 < Nat.card k := Finite.one_lt_card\n  refine Nat.mul_right_cancel (m := Nat.card k - 1) (by lia) ?_\n  let e : V ≃ₗ[k] (Fin n → k) := LinearEquiv.ofFinrankEq _ _ (by simpa)\n  have hc : Nat.card V = Nat.card k ^ n := by simp [Nat.card_congr e.toEquiv, Nat.card_fun]\n  zify\n  conv_rhs => rw [Int.natCast_sub this.le, Int.natCast_one, geom_sum_mul]\n  rw [← Int.natCast_mul, ← card k V, hc]\n  simp\n\nTarget:\nlemma card_of_finrank_two [Finite k] (h : Module.finrank k V = 2) :\n    Nat.card (ℙ k V) = Nat.card k + 1 :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Projectivization","family_id":"card_of_finrank_two","file_id":"mathlib/Mathlib/LinearAlgebra/Projectivization/Cardinality.lean","sample_id":"1bdf66f02f800f9f6bc8f802f4eb7010531ea0fed644e6fb319377ffa1bd244a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5d0edc0cc6cc1668b437d5481146309b64fe078c9ece4c0df2b3120bce3ca534","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"80671f5e23cc162f2c4d734bcae6d4b38501c2182a9dd22e1a1e199c28f65d78","source_sha256":"139083b5f0b4f5b6fe528c0c9bf16219e4a814c177511d56fdc66704924ae51e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine TopologicalSpace.ext_nhds fun x ↦ ?_\n  ext s\n  rw [mem_nhds_toTopsp, H]","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":3,"token_jaccard":0.05,"token_length_ratio":0.166667},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"2ea70940f63d237d36781e67c4a60feca5dccf1aef9c3e4568c636040b9cbb3b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Analysis.Filter\npublic import Mathlib.Topology.Bases\npublic import Mathlib.Topology.LocallyFinite\n\nNamespace:\nCtop.Realizer\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Computational realization of topological spaces (experimental)\n\nThis file provides infrastructure to compute with topological spaces.\n\n## Main declarations\n\n* `Ctop`: Realization of a topology basis.\n* `Ctop.Realizer`: Realization of a topological space. `Ctop` that generates the given topology.\n* `LocallyFinite.Realizer`: Realization of the local finiteness of an indexed family of sets.\n* `Compact.Realizer`: Realization of the compactness of a set.\n-/\n\n@[expose] public section\n\n\nopen Set\n\nopen Filter hiding Realizer\n\nopen Topology\n\n/-- A `Ctop α σ` is a realization of a topology (basis) on `α`,\n  represented by a type `σ` together with operations for the top element and\n  the intersection operation. -/\nstructure Ctop (α σ : Type*) where\n  f : σ → Set α\n  top : α → σ\n  top_mem : ∀ x : α, x ∈ f (top x)\n  inter : ∀ (a b) (x : α), x ∈ f a ∩ f b → σ\n  inter_mem : ∀ a b x h, x ∈ f (inter a b x h)\n  inter_sub : ∀ a b x h, f (inter a b x h) ⊆ f a ∩ f b\n\nvariable {α : Type*} {β : Type*} {σ : Type*} {τ : Type*}\n\ninstance : Inhabited (Ctop α (Set α)) :=\n  ⟨{  f := id\n      top := singleton\n      top_mem := mem_singleton\n      inter := fun s t _ _ ↦ s ∩ t\n      inter_mem := fun _s _t _a ↦ id\n      inter_sub := fun _s _t _a _ha ↦ Subset.rfl }⟩\n\nnamespace Ctop\n\nsection\n\nvariable (F : Ctop α σ)\n\ninstance : CoeFun (Ctop α σ) fun _ ↦ σ → Set α :=\n  ⟨Ctop.f⟩\n\ntheorem coe_mk (f T h₁ I h₂ h₃ a) : (@Ctop.mk α σ f T h₁ I h₂ h₃) a = f a := rfl\n\n/-- Map a Ctop to an equivalent representation type. -/\ndef ofEquiv (E : σ ≃ τ) : Ctop α σ → Ctop α τ\n  | ⟨f, T, h₁, I, h₂, h₃⟩ =>\n    { f := fun a ↦ f (E.symm a)\n      top := fun x ↦ E (T x)\n      top_mem := fun x ↦ by simpa using h₁ x\n      inter := fun a b x h ↦ E (I (E.symm a) (E.symm b) x h)\n      inter_mem := fun a b x h ↦ by simpa using h₂ (E.symm a) (E.symm b) x h\n      inter_sub := fun a b x h ↦ by simpa using h₃ (E.symm a) (E.symm b) x h }\n\n@[simp]\ntheorem ofEquiv_val (E : σ ≃ τ) (F : Ctop α σ) (a : τ) : F.ofEquiv E a = F (E.symm a) := by\n  cases F; rfl\n\nend\n\n/-- Every `Ctop` is a topological space. -/\n@[implicit_reducible]\ndef toTopsp (F : Ctop α σ) : TopologicalSpace α := TopologicalSpace.generateFrom (Set.range F.f)\n\ntheorem toTopsp_isTopologicalBasis (F : Ctop α σ) :\n    @TopologicalSpace.IsTopologicalBasis _ F.toTopsp (Set.range F.f) :=\n  letI := F.toTopsp\n  ⟨fun _u ⟨a, e₁⟩ _v ⟨b, e₂⟩ ↦\n    e₁ ▸ e₂ ▸ fun x h ↦ ⟨_, ⟨_, rfl⟩, F.inter_mem a b x h, F.inter_sub a b x h⟩,\n    eq_univ_iff_forall.2 fun x ↦ ⟨_, ⟨_, rfl⟩, F.top_mem x⟩, rfl⟩\n\n@[simp]\ntheorem mem_nhds_toTopsp (F : Ctop α σ) {s : Set α} {a : α} :\n    s ∈ @nhds _ F.toTopsp a ↔ ∃ b, a ∈ F b ∧ F b ⊆ s :=\n  (@TopologicalSpace.IsTopologicalBasis.mem_nhds_iff _ F.toTopsp _ _ _\n        F.toTopsp_isTopologicalBasis).trans <|\n    ⟨fun ⟨_, ⟨x, rfl⟩, h⟩ ↦ ⟨x, h⟩, fun ⟨x, h⟩ ↦ ⟨_, ⟨x, rfl⟩, h⟩⟩\n\nend Ctop\n\n/-- A `Ctop` realizer for the topological space `T` is a `Ctop`\n  which generates `T`. -/\nstructure Ctop.Realizer (α) [T : TopologicalSpace α] where\n  σ : Type*\n  F : Ctop α σ\n  eq : F.toTopsp = T\n\nopen Ctop\n\n/-- A `Ctop` realizes the topological space it generates. -/\nprotected def Ctop.toRealizer (F : Ctop α σ) : @Ctop.Realizer _ F.toTopsp :=\n  @Ctop.Realizer.mk _ F.toTopsp σ F rfl\n\ninstance (F : Ctop α σ) : Inhabited (@Ctop.Realizer _ F.toTopsp) :=\n  ⟨F.toRealizer⟩\n\nnamespace Ctop.Realizer\n\nprotected theorem is_basis [T : TopologicalSpace α] (F : Realizer α) :\n    TopologicalSpace.IsTopologicalBasis (Set.range F.F.f) := by\n  have := toTopsp_isTopologicalBasis F.F; rwa [F.eq] at this\n\nprotected theorem mem_nhds [T : TopologicalSpace α] (F : Realizer α) {s : Set α} {a : α} :\n    s ∈ 𝓝 a ↔ ∃ b, a ∈ F.F b ∧ F.F b ⊆ s := by\n  have := @mem_nhds_toTopsp _ _ F.F s a; rwa [F.eq] at this\n\ntheorem isOpen_iff [TopologicalSpace α] (F : Realizer α) {s : Set α} :\n    IsOpen s ↔ ∀ a ∈ s, ∃ b, a ∈ F.F b ∧ F.F b ⊆ s :=\n  isOpen_iff_mem_nhds.trans <| forall₂_congr fun _a _h ↦ F.mem_nhds\n\ntheorem isClosed_iff [TopologicalSpace α] (F : Realizer α) {s : Set α} :\n    IsClosed s ↔ ∀ a, (∀ b, a ∈ F.F b → ∃ z, z ∈ F.F b ∩ s) → a ∈ s :=\n  isOpen_compl_iff.symm.trans <|\n    F.isOpen_iff.trans <|\n      forall_congr' fun a ↦\n        show (a ∉ s → ∃ b : F.σ, a ∈ F.F b ∧ ∀ z ∈ F.F b, z ∉ s) ↔ _ by\n          haveI := Classical.propDecidable; rw [not_imp_comm]\n          simp [not_exists, not_and, not_forall, and_comm]\n\ntheorem mem_interior_iff [TopologicalSpace α] (F : Realizer α) {s : Set α} {a : α} :\n    a ∈ interior s ↔ ∃ b, a ∈ F.F b ∧ F.F b ⊆ s :=\n  mem_interior_iff_mem_nhds.trans F.mem_nhds\n\nprotected theorem isOpen [TopologicalSpace α] (F : Realizer α) (s : F.σ) : IsOpen (F.F s) :=\n  isOpen_iff_nhds.2 fun a m ↦ by simpa using F.mem_nhds.2 ⟨s, m, Subset.refl _⟩\n\nTarget:\ntheorem ext' [T : TopologicalSpace α] {σ : Type*} {F : Ctop α σ}\n    (H : ∀ a s, s ∈ 𝓝 a ↔ ∃ b, a ∈ F b ∧ F b ⊆ s) : F.toTopsp = T :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Analysis","family_id":"ext","file_id":"mathlib/Mathlib/Data/Analysis/Topology.lean","sample_id":"80671f5e23cc162f2c4d734bcae6d4b38501c2182a9dd22e1a1e199c28f65d78"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9a8be05ccc00ea06c742ae40a8ee2f89b7edc8fc907070b2e2dab340e52a4333","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7f88709361e31e7cfe4a2d2240e22f16a88fdfa93df42a71071e525d8a5017f9","source_sha256":"99fe4affc634c570de70587e929805d16958622f49de95b17da87beb4716efa3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction t with\n  | nil => rw [traverse, traverse, η.preserves_pure]\n  | node v l r hl hr =>\n    rw [traverse, traverse, η.preserves_seq, η.preserves_seq, η.preserves_map, hl, hr,\n      Function.comp_apply]","hard_negative":true,"metrics":{"chosen_tokens":51,"rejected_tokens":8,"token_jaccard":0.032258,"token_length_ratio":0.156863},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"2ee7c8362d034ee756e03dca70c3cb5f2fc9f825f7cd2a70499dd0e3ff28860b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Tree.Basic\npublic import Mathlib.Control.Applicative\npublic import Mathlib.Control.Traversable.Basic\n\nNamespace:\nBinaryTree\n\nLocal context:\n/-\nCopyright (c) 2025 Edward van de Meent. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Edward van de Meent\n-/\n/-!\n# Traversable Binary Tree\n\nProvides a `Traversable` instance for the `Tree` type.\n-/\n\npublic section\n\nuniverse u v w\n\nnamespace BinaryTree\nsection Traverse\nvariable {α β : Type*}\n\ninstance : Traversable BinaryTree where\n  map := map\n  traverse := traverse\n\nlemma comp_traverse\n    {F : Type u → Type v} {G : Type v → Type w} [Applicative F] [Applicative G]\n    [LawfulApplicative G] {β : Type v} {γ : Type u} (f : β → F γ) (g : α → G β)\n    (t : BinaryTree α) : t.traverse (Functor.Comp.mk ∘ (f <$> ·) ∘ g) =\n      Functor.Comp.mk ((·.traverse f) <$> (t.traverse g)) := by\n  induction t with\n  | nil => rw [traverse, traverse, map_pure, traverse]; rfl\n  | node v l r hl hr =>\n    rw [traverse, hl, hr, traverse]\n    simp only [Function.comp_def, Function.comp_apply, Functor.Comp.map_mk, Functor.map_map,\n      Comp.seq_mk, seq_map_assoc, map_seq]\n    rfl\n\nlemma traverse_eq_map_id (f : α → β) (t : BinaryTree α) :\n    t.traverse ((pure : β → Id β) ∘ f) = pure (t.map f) := by\n  induction t with\n  | nil => rw [traverse, map]\n  | node v l r hl hr =>\n    rw [traverse, map, hl, hr, Function.comp_apply, map_pure, pure_seq, map_pure, pure_seq,\n      map_pure]\n\nTarget:\nlemma naturality {F G : Type u → Type*} [Applicative F] [Applicative G] [LawfulApplicative F]\n    [LawfulApplicative G] (η : ApplicativeTransformation F G) {β : Type u} (f : α → F β)\n    (t : BinaryTree α) : η (t.traverse f) = t.traverse (η.app β ∘ f : α → G β) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"99fc4937c1beeeedc14df853db7b4153aba39418f6309bf9998692ea9d3c7cca","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Tree","family_id":"naturality","file_id":"mathlib/Mathlib/Data/Tree/Traversable.lean","sample_id":"7f88709361e31e7cfe4a2d2240e22f16a88fdfa93df42a71071e525d8a5017f9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2b5a7d59d6ab27179c1c1bc56a6c116c83ca6fccc9d560e5875bd86a9aa70480","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"32ce66990d1ffba436ebb7dc106eea97303764f97b8b35040bc36958f838afdb","source_sha256":"2245dbaf366fe900a8a3490e27a54ff47c641f9f75713ae9f1dd06aaebe571c5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [List.mem_iff_get, Fin.exists_iff, eq_comm (b := b)]","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":2,"token_jaccard":0.0625,"token_length_ratio":0.111111},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"2ef2e1b1e567c7c4fddabd87a0a5934b768ccac7449678de69778525daeec2b3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Data.List.Defs\n\nNamespace:\nList\n\nLocal context:\n/-\nCopyright (c) 2024 Miyahara Kō. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Miyahara Kō\n-/\n/-!\n# iterate\n\nProves various lemmas about `List.iterate`.\n-/\n\npublic section\n\nvariable {α : Type*}\n\nnamespace List\n\n@[simp]\ntheorem length_iterate (f : α → α) (a : α) (n : ℕ) : length (iterate f a n) = n := by\n  induction n generalizing a <;> simp [*]\n\n@[simp]\ntheorem iterate_eq_nil {f : α → α} {a : α} {n : ℕ} : iterate f a n = [] ↔ n = 0 := by\n  rw [← length_eq_zero_iff, length_iterate]\n\ntheorem getElem?_iterate (f : α → α) (a : α) :\n    ∀ (n i : ℕ), i < n → (iterate f a n)[i]? = f^[i] a\n  | n + 1, 0, _ => by simp\n  | n + 1, i + 1, h => by simp [getElem?_iterate f (f a) n i (by simpa using h)]\n\n@[simp]\ntheorem getElem_iterate (f : α → α) (a : α) (n : ℕ) (i : Nat) (h : i < (iterate f a n).length) :\n    (iterate f a n)[i] = f^[i] a :=\n  (getElem_eq_iff _).2 <| getElem?_iterate _ _ _ _ <| by rwa [length_iterate] at h\n\n@[simp]\n\nTarget:\ntheorem mem_iterate {f : α → α} {a : α} {n : ℕ} {b : α} :\n    b ∈ iterate f a n ↔ ∃ m < n, b = f^[m] a :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/List","family_id":"mem_iterate","file_id":"mathlib/Mathlib/Data/List/Iterate.lean","sample_id":"32ce66990d1ffba436ebb7dc106eea97303764f97b8b35040bc36958f838afdb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b4230b54afe23a3c3ef25af1bbb191756411be86b06eb1ac6af58052e937afcd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c6c084351715c62d7f25bc79f783809f58cea66c38bcf07ed142593c77732c13","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d0d0056d6763aa2dc8a9996175ddeb4b3a077afcfff88c7bba8165c2216d8b23","source_sha256":"ea74f644091594213d0fb29edabe43ef72df0c1170b50dfeb2e61c30c872f245","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [PresheafedSpace.congr_app (F.map_id j)]","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":17,"token_jaccard":0.8,"token_length_ratio":1.307692},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"2f45dcf012036fe300d4224395396903214ef8dcc7c1e240e4cbe8144c6a0a95","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Geometry.RingedSpace.PresheafedSpace\npublic import Mathlib.Topology.Category.TopCat.Limits.Basic\npublic import Mathlib.Topology.Sheaves.Limits\npublic import Mathlib.CategoryTheory.ConcreteCategory.Elementwise\n\nNamespace:\nAlgebraicGeometry.PresheafedSpace\n\nLocal context:\n/-\nCopyright (c) 2020 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# `PresheafedSpace C` has colimits.\n\nIf `C` has limits, then the category `PresheafedSpace C` has colimits,\nand the forgetful functor to `TopCat` preserves these colimits.\n\nWhen restricted to a diagram where the underlying continuous maps are open embeddings,\nthis says that we can glue presheafed spaces.\n\nGiven a diagram `F : J ⥤ PresheafedSpace C`,\nwe first build the colimit of the underlying topological spaces,\nas `colimit (F ⋙ PresheafedSpace.forget C)`. Call that colimit space `X`.\n\nOur strategy is to push each of the presheaves `F.obj j`\nforward along the continuous map `colimit.ι (F ⋙ PresheafedSpace.forget C) j` to `X`.\nSince pushforward is functorial, we obtain a diagram `J ⥤ (presheaf C X)ᵒᵖ`\nof presheaves on a single space `X`.\n(Note that the arrows now point the other direction,\nbecause this is the way `PresheafedSpace C` is set up.)\n\nThe limit of this diagram then constitutes the colimit presheaf.\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nuniverse v' u' v u\n\nopen CategoryTheory Opposite CategoryTheory.Category CategoryTheory.Functor CategoryTheory.Limits\n  TopCat TopCat.Presheaf TopologicalSpace\n\nvariable {J : Type u'} [Category.{v'} J] {C : Type u} [Category.{v} C]\n\nnamespace AlgebraicGeometry\n\nnamespace PresheafedSpace\n\nattribute [local simp] eqToHom_map\n\n-- We could enable the following attribute:\n-- attribute [local aesop safe cases (rule_sets := [CategoryTheory])] Opens\n-- although it doesn't appear to help in this file, in any case.\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp]\n\nTarget:\ntheorem map_id_c_app (F : J ⥤ PresheafedSpace.{_, _, v} C) (j) (U) :\n    (F.map (𝟙 j)).c.app U =\n      (Pushforward.id (F.obj j).presheaf).inv.app U ≫\n        (pushforwardEq (by simp) (F.obj j).presheaf).hom.app U :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  simp [PresheafedSpace.congr_app (F.map_id j)]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/RingedSpace","family_id":"map_id_c_app","file_id":"mathlib/Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean","sample_id":"d0d0056d6763aa2dc8a9996175ddeb4b3a077afcfff88c7bba8165c2216d8b23"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dae1c5b0ed2fe337b0ef71d64dc3ef20222db1f81ac15117533aa09e94ca0f93","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f8bf5611d48800247cbe0df58105517cb729c0618719c6fcdd68a62dbe1a0152","source_sha256":"913b3712895896b9c1defc42d8c9355d3cd43a62b5cc81738e459895eb085302","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨u, u_cov, u_props⟩ := sc_st\n  exact\n    ⟨u,\n     s_sub.trans u_cov,\n     fun n ↦\n       ⟨(u_props n).1,\n        disjoint_of_subset (fun ⦃_⦄ a ↦ a) t_sub (u_props n).2⟩⟩","hard_negative":false,"metrics":{"chosen_tokens":50,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.06},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"2f7048e42002074dbdc755ed94fe25ad80d123afa4fad662ec0e4d7ada593fa5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Continuous\npublic import Mathlib.Topology.NhdsSet\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro\n-/\n/-!\n# Separated neighbourhoods\n\nThis file defines the predicates `SeparatedNhds` and `HasSeparatingCover`, which are used in\nformulating separation axioms for topological spaces.\n\n## Main definitions\n\n* `SeparatedNhds`: Two `Set`s are separated by neighbourhoods if they are contained in disjoint\n  open sets.\n* `HasSeparatingCover`: A set has a countable cover that can be used with\n  `hasSeparatingCovers_iff_separatedNhds` to witness when two `Set`s have `SeparatedNhds`.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Separation_axiom>\n* [Willard's *General Topology*][zbMATH02107988]\n-/\n\n@[expose] public section\n\nopen Function Set Filter Topology TopologicalSpace\n\nuniverse u v\n\nvariable {X : Type*} {Y : Type*} [TopologicalSpace X]\n\nsection Separation\n\n/--\n`SeparatedNhds` is a predicate on pairs of sub`Set`s of a topological space.  It holds if the two\nsub`Set`s are contained in disjoint open sets.\n-/\ndef SeparatedNhds : Set X → Set X → Prop := fun s t : Set X =>\n  ∃ U V : Set X, IsOpen U ∧ IsOpen V ∧ s ⊆ U ∧ t ⊆ V ∧ Disjoint U V\n\ntheorem separatedNhds_iff_disjoint {s t : Set X} : SeparatedNhds s t ↔ Disjoint (𝓝ˢ s) (𝓝ˢ t) := by\n  simp only [(hasBasis_nhdsSet s).disjoint_iff (hasBasis_nhdsSet t), SeparatedNhds, ←\n    exists_and_left, and_assoc, and_comm, and_left_comm]\n\nalias ⟨SeparatedNhds.disjoint_nhdsSet, _⟩ := separatedNhds_iff_disjoint\n\n/-- `HasSeparatingCover`s can be useful witnesses for `SeparatedNhds`. -/\ndef HasSeparatingCover : Set X → Set X → Prop := fun s t ↦\n  ∃ u : ℕ → Set X, s ⊆ ⋃ n, u n ∧ ∀ n, IsOpen (u n) ∧ Disjoint (closure (u n)) t\n\n/-- Used to prove that a regular topological space with Lindelöf topology is a normal space,\nand a perfectly normal space is a completely normal space. -/\ntheorem hasSeparatingCovers_iff_separatedNhds {s t : Set X} :\n    HasSeparatingCover s t ∧ HasSeparatingCover t s ↔ SeparatedNhds s t := by\n  constructor\n  · rintro ⟨⟨u, u_cov, u_props⟩, ⟨v, v_cov, v_props⟩⟩\n    have open_lemma : ∀ (u₀ a : ℕ → Set X), (∀ n, IsOpen (u₀ n)) →\n      IsOpen (⋃ n, u₀ n \\ closure (a n)) := fun _ _ u₀i_open ↦\n        isOpen_iUnion fun i ↦ (u₀i_open i).sdiff isClosed_closure\n    have cover_lemma : ∀ (h₀ : Set X) (u₀ v₀ : ℕ → Set X),\n        (h₀ ⊆ ⋃ n, u₀ n) → (∀ n, Disjoint (closure (v₀ n)) h₀) →\n        (h₀ ⊆ ⋃ n, u₀ n \\ closure (⋃ m ≤ n, v₀ m)) :=\n        fun h₀ u₀ v₀ h₀_cov dis x xinh ↦ by\n      rcases h₀_cov xinh with ⟨un, ⟨n, rfl⟩, xinun⟩\n      simp only [mem_iUnion]\n      refine ⟨n, xinun, ?_⟩\n      simp_all only [closure_iUnion₂_le_nat, disjoint_right, mem_iUnion,\n        exists_false, not_false_eq_true]\n    refine\n      ⟨⋃ n : ℕ, u n \\ (closure (⋃ m ≤ n, v m)),\n       ⋃ n : ℕ, v n \\ (closure (⋃ m ≤ n, u m)),\n       open_lemma u (fun n ↦ ⋃ m ≤ n, v m) (fun n ↦ (u_props n).1),\n       open_lemma v (fun n ↦ ⋃ m ≤ n, u m) (fun n ↦ (v_props n).1),\n       cover_lemma s u v u_cov (fun n ↦ (v_props n).2),\n       cover_lemma t v u v_cov (fun n ↦ (u_props n).2),\n       ?_⟩\n    rw [Set.disjoint_left]\n    rintro x ⟨un, ⟨n, rfl⟩, xinun⟩\n    suffices ∀ (m : ℕ), x ∈ v m → x ∈ closure (⋃ m' ∈ {m' | m' ≤ m}, u m') by simpa\n    intro m xinvm\n    have n_le_m : n ≤ m := by\n      by_contra m_gt_n\n      exact xinun.2 (subset_closure (mem_biUnion (le_of_lt (not_le.mp m_gt_n)) xinvm))\n    exact subset_closure (mem_biUnion n_le_m xinun.1)\n  · rintro ⟨U, V, U_open, V_open, h_sub_U, k_sub_V, UV_dis⟩\n    exact\n      ⟨⟨fun _ ↦ U,\n        h_sub_U.trans (iUnion_const U).symm.subset,\n        fun _ ↦\n          ⟨U_open, disjoint_of_subset (fun ⦃a⦄ a ↦ a) k_sub_V (UV_dis.closure_left V_open)⟩⟩,\n       ⟨fun _ ↦ V,\n        k_sub_V.trans (iUnion_const V).symm.subset,\n        fun _ ↦\n          ⟨V_open, disjoint_of_subset (fun ⦃a⦄ a ↦ a) h_sub_U (UV_dis.closure_right U_open).symm⟩⟩⟩\n\ntheorem Set.hasSeparatingCover_empty_left (s : Set X) : HasSeparatingCover ∅ s :=\n  ⟨fun _ ↦ ∅, empty_subset (⋃ _, ∅),\n   fun _ ↦ ⟨isOpen_empty, by simp only [closure_empty, empty_disjoint]⟩⟩\n\ntheorem Set.hasSeparatingCover_empty_right (s : Set X) : HasSeparatingCover s ∅ :=\n  ⟨fun _ ↦ univ, (subset_univ s).trans univ.iUnion_const.symm.subset,\n   fun _ ↦ ⟨isOpen_univ, by apply disjoint_empty⟩⟩\n\nTarget:\ntheorem HasSeparatingCover.mono {s₁ s₂ t₁ t₂ : Set X} (sc_st : HasSeparatingCover s₂ t₂)\n    (s_sub : s₁ ⊆ s₂) (t_sub : t₁ ⊆ t₂) : HasSeparatingCover s₁ t₁ :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Separation","family_id":"hasseparatingcover","file_id":"mathlib/Mathlib/Topology/Separation/SeparatedNhds.lean","sample_id":"f8bf5611d48800247cbe0df58105517cb729c0618719c6fcdd68a62dbe1a0152"},"split":"train"}
{"binding":{"chosen_proof_sha256":"318c2cb9db795a2dd46d3f5a27d1f76a3d76756692909ad6bddc78fa03561671","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7da1d9db371103784bbc6daf0c80df8acd3ad5e9a823b6012d8ef842a94b24f4","source_sha256":"e3bd64c53d80fc2f3ac9f455d14462bc0198848b67a51278e36a13f2aa735979","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  rw [← trans_dualMap_symm_flip e]\n  exact LinearEquiv.bijective _","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":3,"token_jaccard":0.142857,"token_length_ratio":0.230769},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"2fb98968e4727ea518c1bbf91f4fa71862e5e2dddc615ab5f111d4bde8d3d508","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Dual.Lemmas\n\nNamespace:\nLinearEquiv\n\nLocal context:\n/-\nCopyright (c) 2023 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash, Yaël Dillies\n-/\n/-!\n# Perfect pairings\n\nThis file defines perfect pairings of modules.\n\nA perfect pairing of two (left) modules may be defined either as:\n1. A bilinear map `M × N → R` such that the induced maps `M → Dual R N` and `N → Dual R M` are both\n  bijective. It follows from this that both `M` and `N` are reflexive modules.\n2. A linear equivalence `N ≃ Dual R M` for which `M` is reflexive. (It then follows that `N` is\n  reflexive.)\n\nIn this file we provide a definition `IsPerfPair` corresponding to 1 above, together with logic\nto connect 1 and 2.\n-/\n\n@[expose] public section\n\nopen Function Module\n\nnamespace LinearMap\nvariable {R K M M' N N' : Type*} [AddCommGroup M] [AddCommGroup N] [AddCommGroup M']\n  [AddCommGroup N']\n\nsection CommRing\nvariable [CommRing R] [Module R M] [Module R M'] [Module R N] [Module R N']\n  {p : M →ₗ[R] N →ₗ[R] R} {x : M} {y : N}\n\n/-- For a ring `R` and two modules `M` and `N`, a perfect pairing is a bilinear map `M × N → R`\nthat is bijective in both arguments. -/\n@[ext]\nclass IsPerfPair (p : M →ₗ[R] N →ₗ[R] R) where\n  bijective_left (p) : Bijective p\n  bijective_right (p) : Bijective p.flip\n\n/-- Given a perfect pairing between `M` and `N`, we may interchange the roles of `M` and `N`. -/\nprotected lemma IsPerfPair.flip (hp : p.IsPerfPair) : p.flip.IsPerfPair where\n  bijective_left := IsPerfPair.bijective_right p\n  bijective_right := IsPerfPair.bijective_left p\n\nvariable [p.IsPerfPair]\n\n/-- Given a perfect pairing between `M` and `N`, we may interchange the roles of `M` and `N`. -/\ninstance flip.instIsPerfPair : p.flip.IsPerfPair := .flip ‹_›\n\nvariable (p)\n\n/-- Turn a perfect pairing between `M` and `N` into an isomorphism between `M` and the dual of `N`.\n-/\nnoncomputable def toPerfPair : M ≃ₗ[R] Dual R N :=\n  .ofBijective { toFun := _, map_add' x y := by simp, map_smul' r x := by simp } <|\n    IsPerfPair.bijective_left p\n\n@[simp] lemma toLinearMap_toPerfPair (x : M) : p.toPerfPair x = p x := rfl\n@[simp] lemma toPerfPair_apply (x : M) (y : N) : p.toPerfPair x y = p x y := rfl\n\n@[simp] lemma apply_symm_toPerfPair_self (f : Dual R N) : p (p.toPerfPair.symm f) = f :=\n  p.toPerfPair.apply_symm_apply f\n\n@[simp] lemma apply_toPerfPair_flip (f : Dual R M) (x : M) : p x (p.flip.toPerfPair.symm f) = f x :=\n  congr($(p.flip.apply_symm_toPerfPair_self ..) x)\n\ninclude p in\nlemma _root_.Module.IsReflexive.of_isPerfPair : IsReflexive R M where\n  bijective_dual_eval' := by\n    convert! (p.toPerfPair.trans p.flip.toPerfPair.dualMap.symm).bijective\n    ext x f\n    simp\n\ninclude p in\nlemma _root_.Module.finrank_of_isPerfPair [Module.Finite R M] [Module.Free R M] :\n    finrank R M = finrank R N :=\n  ((Module.Free.chooseBasis R M).toDualEquiv.trans p.flip.toPerfPair.symm).finrank_eq\n\n/-- A reflexive module has a perfect pairing with its dual. -/\nprotected instance IsPerfPair.id [IsReflexive R M] : IsPerfPair (.id (R := R) (M := Dual R M)) where\n  bijective_left := bijective_id\n  bijective_right := bijective_dual_eval R M\n\n/-- A reflexive module has a perfect pairing with its dual. -/\ninstance IsPerfPair.dualEval [IsReflexive R M] : IsPerfPair (Dual.eval R M) := .flip .id\n\ninstance IsPerfPair.compl₁₂ (eM : M' ≃ₗ[R] M) (eN : N' ≃ₗ[R] N) :\n    (p.compl₁₂ eM eN : M' →ₗ[R] N' →ₗ[R] R).IsPerfPair :=\n  ⟨((LinearEquiv.congrLeft R R eN).symm.bijective.comp\n    (IsPerfPair.bijective_left p)).comp eM.bijective,\n    ((LinearEquiv.congrLeft R R eM).symm.bijective.comp\n    (IsPerfPair.bijective_right p)).comp eN.bijective⟩\n\nlemma IsPerfPair.congr (eM : M' ≃ₗ[R] M) (eN : N' ≃ₗ[R] N) (q : M' →ₗ[R] N' →ₗ[R] R)\n    (H : q.compl₁₂ eM.symm eN.symm = p) : q.IsPerfPair := by\n  obtain rfl : q = p.compl₁₂ eM eN := by subst H; ext; simp\n  infer_instance\n\nlemma IsPerfPair.of_bijective (p : M →ₗ[R] N →ₗ[R] R) [IsReflexive R N] (h : Bijective p) :\n    IsPerfPair p :=\n  inferInstanceAs ((LinearMap.id (R := R) (M := Dual R N)).compl₁₂\n    (LinearEquiv.ofBijective p h : M →ₗ[R] N →ₗ[R] R)\n    (LinearEquiv.refl R N : N →ₗ[R] N)).IsPerfPair\n\nend CommRing\n\nsection Field\nvariable [Field K] [Module K M] [Module K N] {p : M →ₗ[K] N →ₗ[K] K} {x : M} {y : N}\n\n/-- If the coefficients are a field, and one of the spaces is finite-dimensional, it is sufficient\nto check only injectivity instead of bijectivity of the bilinear pairing. -/\nlemma IsPerfPair.of_injective [FiniteDimensional K M] (h : Injective p) (h' : Injective p.flip) :\n    p.IsPerfPair where\n  bijective_left := ⟨h, by rwa [← p.flip_injective_iff₁]⟩\n  bijective_right := ⟨h', by\n    have : FiniteDimensional K N := FiniteDimensional.of_injective p.flip h'\n    rwa [← p.flip.flip_injective_iff₁, LinearMap.flip_flip]⟩\n\n/-- If the coefficients are a field, and one of the spaces is finite-dimensional, it is sufficient\nto check only injectivity instead of bijectivity of the bilinear pairing. -/\nlemma IsPerfPair.of_injective' [FiniteDimensional K N] (h : Injective p) (h' : Injective p.flip) :\n    p.IsPerfPair := .flip <| .of_injective h' h\n\nend Field\nend LinearMap\n\nnoncomputable section\n\nvariable {R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]\n\nnamespace LinearMap\nvariable {p : M →ₗ[R] N →ₗ[R] R} [p.IsPerfPair]\n\nvariable (p) in\n/-- Given a perfect pairing `p` between `M` and `N`, we say a pair of submodules `U` in `M` and\n`V` in `N` are perfectly complementary w.r.t. `p` if their dual annihilators are complementary,\nusing `p` to identify `M` and `N` with dual spaces. -/\nstructure IsPerfectCompl (U : Submodule R M) (V : Submodule R N) : Prop where\n  isCompl_left : IsCompl U (V.dualAnnihilator.map (p.toPerfPair.symm : Dual R N →ₗ[R] M))\n  isCompl_right : IsCompl V (U.dualAnnihilator.map (p.flip.toPerfPair.symm : Dual R M →ₗ[R] N))\n\nnamespace IsPerfectCompl\nvariable {U : Submodule R M} {V : Submodule R N}\n\nprotected lemma flip (h : p.IsPerfectCompl U V) :\n    p.flip.IsPerfectCompl V U where\n  isCompl_left := h.isCompl_right\n  isCompl_right := h.isCompl_left\n\n@[simp]\nprotected lemma flip_iff :\n    p.flip.IsPerfectCompl V U ↔ p.IsPerfectCompl U V :=\n  ⟨fun h ↦ h.flip, fun h ↦ h.flip⟩\n\n@[simp]\nlemma left_top_iff :\n    p.IsPerfectCompl ⊤ V ↔ V = ⊤ := by\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · exact eq_top_of_isCompl_bot <| by simpa using h.isCompl_right\n  · rw [h]\n    exact\n      { isCompl_left := by simpa using isCompl_top_bot\n        isCompl_right := by simpa using isCompl_top_bot }\n\n@[simp]\nlemma right_top_iff :\n    p.IsPerfectCompl U ⊤ ↔ U = ⊤ := by\n  rw [← IsPerfectCompl.flip_iff]\n  exact left_top_iff\n\nend IsPerfectCompl\n\nend LinearMap\n\nvariable [IsReflexive R M]\n\nvariable (e : N ≃ₗ[R] Dual R M)\n\nnamespace LinearEquiv\n\n/-- For a reflexive module `M`, an equivalence `N ≃ₗ[R] Dual R M` naturally yields an equivalence\n`M ≃ₗ[R] Dual R N`. Such equivalences are known as perfect pairings. -/\ndef flip : M ≃ₗ[R] Dual R N :=\n  (evalEquiv R M).trans e.dualMap\n\n@[simp] lemma coe_toLinearMap_flip : e.flip = (↑e : N →ₗ[R] Dual R M).flip := rfl\n\n@[simp] lemma flip_apply (m : M) (n : N) : e.flip m n = e n m := rfl\n\nlemma symm_flip : e.flip.symm = e.symm.dualMap.trans (evalEquiv R M).symm := rfl\n\nlemma trans_dualMap_symm_flip : e.trans e.flip.symm.dualMap = Dual.eval R N := by\n  ext; simp [symm_flip]\n\ninclude e in\n/-- If `N` is in perfect pairing with `M`, then it is reflexive. -/\n\nTarget:\nlemma isReflexive_of_equiv_dual_of_isReflexive : IsReflexive R N :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/PerfectPairing","family_id":"isreflexive_of_equiv_dual_of_isreflexive","file_id":"mathlib/Mathlib/LinearAlgebra/PerfectPairing/Basic.lean","sample_id":"7da1d9db371103784bbc6daf0c80df8acd3ad5e9a823b6012d8ef842a94b24f4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"38369068946cc9341941259052323d13d916cefa46cf4a0009f06b6990667aa7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0c0e4150f1cfb1e9439885a29fd7c58e255cd130ea43fb4bb8cf77ae83937d2b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dfb013882af3dab10b8a5b0e2b5f55bfe7bbceeec4f48cee972a6e94a5340486","source_sha256":"3ea4a54ee767d2f1814d8b90accb092fa9ec4a1fd88519664221a22cfd80fa82","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [basis_apply_powersetCard, exteriorPower.basis_apply, ιMulti_family_apply_coe]","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.166667},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"2fbd283df7a1a90cb9ef6ac980f77033db10ae8177874c32cf5b3a7d53423731","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.ExteriorAlgebra.Grading\npublic import Mathlib.LinearAlgebra.ExteriorPower.Basis\n\nNamespace:\nExteriorAlgebra\n\nLocal context:\n/-\nCopyright (c) 2026 Daniel Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Morrison\n-/\n/-!\n# Basis for `ExteriorAlgebra`\n-/\n\n@[expose] public section\n\nnamespace ExteriorAlgebra\n\nopen Module Set Set.powersetCard exteriorPower\n\nvariable {R M : Type*} {m n : ℕ} {I : Type*} [LinearOrder I] [CommRing R]\n  [AddCommGroup M] [Module R M] (b : Module.Basis I R M)\n\n/-- The direct sum decomposition of the exterior algebra from the graded algebra structure. -/\ninstance : DirectSum.Decomposition (fun n ↦ ⋀[R]^n M) :=\n  GradedRing.toDecomposition (self := ExteriorAlgebra.gradedAlgebra R M)\n\n/-- If `b` is a basis of `M` (indexed by a linearly ordered type), the basis of the exterior\nalgebra of `M` formed by the `n`-fold exterior products of elements of `b` for each `n`. -/\nnoncomputable def _root_.Module.Basis.ExteriorAlgebra : Basis (Finset I) R (ExteriorAlgebra R M) :=\n  .reindex\n    ((DirectSum.Decomposition.isInternal (fun n => ⋀[R]^n M)).collectedBasis b.exteriorPower)\n    Set.powersetCard.prodEquiv\n\nlemma basis_apply (s : Finset I) :\n    b.ExteriorAlgebra s = ιMulti_family R s.card b (prodEquiv.symm s).2 := by\n  simp [Basis.ExteriorAlgebra]\n\nlemma basis_apply_ofCard {s : Finset I} (s_card : s.card = n) :\n    b.ExteriorAlgebra s = ιMulti_family R n b (ofCard s_card) := by\n  subst s_card\n  simp [basis_apply]\n\nvariable (s : powersetCard I m) (t : powersetCard I n)\n\nlemma basis_apply_powersetCard :\n    b.ExteriorAlgebra s = ιMulti_family R m b s := by\n  simp [basis_apply_ofCard]\n\nTarget:\nlemma basis_eq_coe_basis :\n    b.ExteriorAlgebra s = (b.exteriorPower m s : ExteriorAlgebra R M) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_dfb013882af3","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"30b6b073d01cbc1db05c58ce24281a7140d17b1d4c820d26467f35fe4d819d36","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/ExteriorAlgebra","family_id":"basis_eq_coe_basis","file_id":"mathlib/Mathlib/LinearAlgebra/ExteriorAlgebra/Basis.lean","sample_id":"dfb013882af3dab10b8a5b0e2b5f55bfe7bbceeec4f48cee972a6e94a5340486"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7c1a95bd387fa99509421bd9a6c435e484eea5c2a9222b6585afb56f5e4cfe03","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fb8d70162809714734e39e1803545fb5adac8b0c13fe3dded4a257dc63c28f16","source_sha256":"79526dcfa55da9c5f7685aed6cead7ebb3b44c830f8cd589df3db791b7c151e7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply tendsto_nhdsNE_of_tendsto_atTop _ _ h_neg\n  convert! h_pos","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":8,"token_jaccard":0.071429,"token_length_ratio":0.888889},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"300e31c61a6838a2a0812f94718e734eb2606ade1af500f2636490adeca83f74","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Algebra.Order.Field\npublic import Mathlib.Topology.Maps.Basic\npublic import Mathlib.Analysis.Asymptotics.Lemmas\n\nNamespace:\nTactic.ComputeAsymptotics\n\nLocal context:\n/-\nCopyright (c) 2026 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\n/-!\n# Conversion lemmas\n\nThe main procedure of the `compute_asymptotics` tactic is able to compute limits of functions at\n`atTop` filter. This file contains lemmas we use to reduce other asymptotic goals to\nthe case `Tendsto f atTop l`.\n\n## Main theorems\n\nThis file contains the following lemmas:\n* `tendsto_nhdsGT_of_tendsto_atTop` for `Tendsto f (𝓝[>] c) l`\n* `tendsto_nhdsLT_of_tendsto_atTop` for `Tendsto f (𝓝[<] c) l`\n* `tendsto_nhdsNE_of_tendsto_atTop` for `Tendsto f (𝓝[≠] c) l`\n* `isBigO_of_div_tendsto_atTop` and `isBigO_of_div_tendsto_atBot` for `f =O[l] g`\n\nWe also use lemmas from other files:\n* `tendsto_comp_neg_atTop_iff` for `Tendsto f atBot l`\n* `IsLittleO.of_tendsto_div_atBot` and `IsLittleO.of_tendsto_div_atTop` for `f =o[l] g`\n* `isEquivalent_of_tendsto_one` for `f ∼ g`\n-/\n\npublic section\n\nopen Filter Topology Asymptotics\n\nnamespace Tactic.ComputeAsymptotics\n\nvariable {α 𝕜 : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [TopologicalSpace 𝕜]\n  [OrderTopology 𝕜] {l : Filter α} (f : 𝕜 → α) (c : 𝕜)\n\ntheorem tendsto_nhdsGT_of_tendsto_atTop (h : Tendsto (fun x ↦ f (c + x⁻¹)) atTop l) :\n    Tendsto f (𝓝[>] c) l := by\n  simpa [← Function.comp_def, Tendsto, ← Filter.map_map] using h\n\ntheorem tendsto_nhdsLT_of_tendsto_atTop (h : Tendsto (fun x ↦ f (c - x⁻¹)) atTop l) :\n    Tendsto f (𝓝[<] c) l := by\n  convert_to Tendsto (f ∘ (fun x ↦ c + x) ∘ Neg.neg ∘ Inv.inv) atTop l at h\n  · ext\n    simp [AddGroupWithOne.sub_eq_add_neg]\n  simpa [Tendsto, ← Filter.map_map] using h\n\ntheorem tendsto_nhdsNE_of_tendsto_atTop (h_neg : Tendsto (fun x ↦ f (c - x⁻¹)) atTop l)\n    (h_pos : Tendsto (fun x ↦ f (c + x⁻¹)) atTop l) :\n    Tendsto f (𝓝[≠] c) l := by\n  simpa [Tendsto, ← nhdsLT_sup_nhdsGT] using\n    ⟨tendsto_nhdsLT_of_tendsto_atTop _ _ h_neg, tendsto_nhdsGT_of_tendsto_atTop _ _ h_pos⟩\n\nTarget:\ntheorem tendsto_nhdsNE_of_tendsto_atTop_nhds_of_eq [TopologicalSpace α] {a b : α}\n    (h_neg : Tendsto (fun x ↦ f (c - x⁻¹)) atTop (𝓝 a))\n    (h_pos : Tendsto (fun x ↦ f (c + x⁻¹)) atTop (𝓝 b)) (h_eq : a = b) :\n    Tendsto f (𝓝[≠] c) (𝓝 a) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"5269fac6894e732c3c4e11054d038a307612f2e1d93c820690bd80916eba8453","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/ComputeAsymptotics","family_id":"tendsto_nhdsne_of_tendsto_attop_nhds_of_eq","file_id":"mathlib/Mathlib/Tactic/ComputeAsymptotics/Lemmas.lean","sample_id":"fb8d70162809714734e39e1803545fb5adac8b0c13fe3dded4a257dc63c28f16"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ba260adf630a9d5dff8237b3bea2f33b1eff285cacd7e89777abb74bff12da31","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3ac2f1d971950cbc9a18df02d16cfd27138b3550acd788b6fdc93fcd40fa14a9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"eb0438819632308cf7faaf08d1c110925f195fcd644c416f40abd06f69c1666f","source_sha256":"da03aef6be316c0f1e4d46eb4e8f0617ba6def9cf6bbead1497df151ed7bf988","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases x; rfl\n\nnonrec def ExceptT.callCC {ε} [MonadCont m] {α β : Type _}\n    (f : Label α (ExceptT ε m) β → ExceptT ε m α) : ExceptT ε m α :=\n  ExceptT.mk (callCC fun x : Label _ m β => ExceptT.run <| f (ExceptT.mkLabel x))","hard_negative":true,"metrics":{"chosen_tokens":73,"rejected_tokens":5,"token_jaccard":0.052632,"token_length_ratio":0.068493},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"30180d62520f234cfdf1320f58d1965db02564bfffdb00c5116629fe1647481c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Control.Monad.Basic\npublic import Mathlib.Control.Monad.Writer\npublic import Mathlib.Control.Lawful\npublic import Batteries.Tactic.Congr\npublic import Batteries.Lean.Except\nimport all Init.Control.Option  -- for unfolding `Option.lift`\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\n/-!\n# Continuation Monad\n\nMonad encapsulating continuation passing programming style, similar to\nHaskell's `Cont`, `ContT` and `MonadCont`:\n<https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Cont.html>\n<https://hackage.haskell.org/package/transformers-0.6.2.0/docs/Control-Monad-Trans-Cont.html>\n-/\n\n@[expose] public section\n\nuniverse u v w u₀ u₁ v₀ v₁\n\nstructure MonadCont.Label (α : Type w) (m : Type u → Type v) (β : Type u) where\n  apply : α → m β\n\nabbrev MonadCont.goto {α β} {m : Type u → Type v} (f : MonadCont.Label α m β) (x : α) :=\n  f.apply x\n\nclass MonadCont (m : Type u → Type v) where\n  callCC : ∀ {α β}, (MonadCont.Label α m β → m α) → m α\n\nopen MonadCont\n\nclass LawfulMonadCont (m : Type u → Type v) [Monad m] [MonadCont m] : Prop\n    extends LawfulMonad m where\n  callCC_bind_right {α ω γ} (cmd : m α) (next : Label ω m γ → α → m ω) :\n    (callCC fun f => cmd >>= next f) = cmd >>= fun x => callCC fun f => next f x\n  callCC_bind_left {α} (β) (x : α) (dead : Label α m β → β → m α) :\n    (callCC fun f : Label α m β => goto f x >>= dead f) = pure x\n  callCC_dummy {α β} (dummy : m α) : (callCC fun _ : Label α m β => dummy) = dummy\n\nexport LawfulMonadCont (callCC_bind_right callCC_bind_left callCC_dummy)\n\ndef ContT (r : Type u) (m : Type u → Type v) (α : Type w) :=\n  (α → m r) → m r\n\nabbrev Cont (r : Type u) (α : Type w) :=\n  ContT r Id α\n\nnamespace ContT\n\nexport MonadCont (Label goto)\n\nvariable {r : Type u} {m : Type u → Type v} {α β : Type w}\n\n/-- Build a `ContT` from a function taking a continuation callback. -/\ndef mk (f : (α → m r) → m r) : ContT r m α := f\n\n/-- Run a `ContT` with a provided callback. -/\ndef run (x : ContT r m α) : (α → m r) → m r := x\n\ndef map (f : m r → m r) (x : ContT r m α) : ContT r m α :=\n  f ∘ x\n\ntheorem run_contT_map_contT (f : m r → m r) (x : ContT r m α) : run (map f x) = f ∘ run x :=\n  rfl\n\ndef withContT (f : (β → m r) → α → m r) (x : ContT r m α) : ContT r m β := fun g => x <| f g\n\ntheorem run_withContT (f : (β → m r) → α → m r) (x : ContT r m α) :\n    run (withContT f x) = run x ∘ f :=\n  rfl\n\n@[ext]\nprotected theorem ext {x y : ContT r m α} (h : ∀ f, x.run f = y.run f) : x = y := by\n  unfold ContT; ext; apply h\n\ninstance : Monad (ContT r m) where\n  pure x f := f x\n  bind x f g := x fun i => f i g\n\n@[simp]\ntheorem run_mk (f : (α → m r) → m r) (k : α → m r) : (.mk f : ContT r m α).run k = f k := rfl\n\n@[simp]\ntheorem run_pure (a : α) (k : α → m r) : (pure a : ContT r m α).run k = k a := rfl\n\n@[simp]\ntheorem run_bind (x : ContT r m α) (f : α → ContT r m β) (k : β → m r) :\n    (x >>= f).run k = x.run fun x => (f x).run k := rfl\n\n@[simp]\ntheorem run_map (f : α → β) (x : ContT r m α) (k : β → m r) :\n    (f <$> x).run k = x.run (k ∘ f) := rfl\n\n@[simp]\ntheorem run_seq (f : ContT r m (α → β)) (x : ContT r m α) (k : β → m r) :\n    (f <*> x).run k = f.run fun f => x.run (k ∘ f) := rfl\n\n@[simp]\ntheorem run_seqLeft (x : ContT r m α) (y : ContT r m β) (k : α → m r) :\n    (x <* y).run k = x.run fun x => y.run fun _ => k x := rfl\n\n@[simp]\ntheorem run_seqRight (x : ContT r m α) (y : ContT r m β) (k : β → m r) :\n    (x *> y).run k = x.run fun _ => y.run k := rfl\n\ninstance : LawfulMonad (ContT r m) := LawfulMonad.mk'\n  (id_map := by intros; rfl)\n  (pure_bind := by intros; ext; rfl)\n  (bind_assoc := by intros; ext; rfl)\n\ninstance [Monad m] : MonadLift m (ContT r m) where\n  monadLift x := .mk fun k => x >>= k\n\n@[simp]\ntheorem run_monadLift [Monad m] {α} (x : m α) (k : α → m r) :\n    (monadLift x : ContT r m α).run k = x >>= k := rfl\n\ntheorem monadLift_bind [Monad m] [LawfulMonad m] {α β} (x : m α) (f : α → m β) :\n    (monadLift (x >>= f) : ContT r m β) = monadLift x >>= monadLift ∘ f := by\n  ext\n  simp only [bind_assoc, run_bind, run_monadLift, Function.comp_apply]\n\ninstance : MonadCont (ContT r m) where\n  callCC f := .mk fun k => f ⟨fun x => .mk fun _ => k x⟩ k\n\n@[simp]\ntheorem run_callCC (f : Label α (ContT r m) β → ContT r m α) (k : α → m r) :\n    (callCC f).run k = (f ⟨fun x => .mk fun _ => k x⟩).run k := rfl\n\ninstance : LawfulMonadCont (ContT r m) where\n  callCC_bind_right := by intros; ext; rfl\n  callCC_bind_left := by intros; ext; rfl\n  callCC_dummy := by intros; ext; rfl\n\n/-- Note that `tryCatch` does not have correct behavior in this monad:\n```\ndef foo : ContT Bool (Except String) Bool := do\n  let x ← try\n    pure true\n  catch _ =>\n    return false\n  throw s!\"oh no {x}\"\n#eval foo.run pure\n-- `Except.ok false`, no error\n```\nHere, the `throwError` is being run inside the `try`.\nSee [Zulip](https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/MonadExcept.20in.20the.20ContT.20monad/near/375341221)\nfor further discussion.\n-/\ninstance (ε) [MonadExceptOf ε m] : MonadExceptOf ε (ContT r m) where\n  throw e := .mk fun _ => throw e\n  tryCatch act h := .mk fun k => tryCatch (act.run k) fun e => (h e).run k\n\n@[simp]\ntheorem run_throw {ε} [MonadExceptOf ε m]\n    (e : ε) (f : α → m r) :\n    (throw e : ContT r m α).run f = throw e := rfl\n\n@[simp]\ntheorem run_tryCatch {ε} [MonadExceptOf ε m]\n    (act : ContT r m α) (h : ε → ContT r m α) (f : α → m r) :\n    (tryCatch act h : ContT r m α).run f = tryCatch (act.run f) fun e => (h e).run f := rfl\n\nend ContT\n\nvariable {m : Type u → Type v}\n\nsection\nvariable [Monad m]\n\ndef ExceptT.mkLabel {α β ε} : Label (Except.{u, u} ε α) m β → Label α (ExceptT ε m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (Except.ok a)⟩\n\nTarget:\ntheorem ExceptT.goto_mkLabel {α β ε : Type _} (x : Label (Except.{u, u} ε α) m β) (i : α) :\n    goto (ExceptT.mkLabel x) i = ExceptT.mk (Except.ok <$> goto x (Except.ok i)) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_eb0438819632","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"e5260bb4f8305a400825b4f750820f978d378cb8029ca6187dc7f52884515ddc","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control/Monad","family_id":"exceptt","file_id":"mathlib/Mathlib/Control/Monad/Cont.lean","sample_id":"eb0438819632308cf7faaf08d1c110925f195fcd644c416f40abd06f69c1666f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a24cc2dfc6b516924da62f552de8528961f8853b68b3464a392b74c6594e45ef","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0f0633797d29bc379a07723d24c5fedc5dea9d12be64f1efa311a7d0b83f3df4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2f3292c8af1fa98f0dfb39f23e5916bd9e2de4337e8d92b7857cea3205fe336c","source_sha256":"6a1fd2aa47744142955f801b29fb06cc46e0d439ea80c9cccba97a19a60fbbe9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x\n  apply (IsIntegralClosure.equiv A (integralClosure A L₃) L₃ B₃).symm.injective\n  ext\n  simp [galRestrict', Subalgebra.algebraMap_eq]","hard_negative":false,"metrics":{"chosen_tokens":33,"rejected_tokens":38,"token_jaccard":0.851852,"token_length_ratio":1.151515},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"30699fbdeb86fbc729a3410b7942ed38f363a49a822fb7ad86928efaeddc739e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.IntegralClosure\npublic import Mathlib.RingTheory.RingHom.Finite\npublic import Mathlib.RingTheory.Localization.LocalizationLocalization\npublic import Mathlib.RingTheory.Localization.NormTrace\npublic import Mathlib.RingTheory.Norm.Transitivity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Restriction of various maps between fields to integrally closed subrings.\n\nIn this file, we assume `A` is an integrally closed domain; `K` is the fraction ring of `A`;\n`L` is a finite extension of `K`; `B` is the integral closure of `A` in `L`.\nWe call this the AKLB setup.\n\n## Main definitions\n- `galRestrict`: The restriction `Aut(L/K) → Aut(B/A)` as an `MulEquiv` in an AKLB setup.\n- `Algebra.intTrace`: The trace map of a finite extension of integrally closed domains `B/A` is\n  defined to be the restriction of the trace map of `Frac(B)/Frac(A)`.\n- `Algebra.intNorm`: The norm map of a finite extension of integrally closed domains `B/A` is\n  defined to be the restriction of the norm map of `Frac(B)/Frac(A)`.\n\n-/\n\n@[expose] public section\n\nopen Module nonZeroDivisors\n\nvariable (A K L L₂ L₃ B B₂ B₃ : Type*)\nvariable [CommRing A] [CommRing B] [CommRing B₂] [CommRing B₃]\nvariable [Algebra A B] [Algebra A B₂] [Algebra A B₃]\nvariable [Field K] [Field L] [Field L₂] [Field L₃]\nvariable [Algebra A K] [IsFractionRing A K]\nvariable [Algebra K L] [Algebra A L] [IsScalarTower A K L]\nvariable [Algebra K L₂] [Algebra A L₂] [IsScalarTower A K L₂]\nvariable [Algebra K L₃] [Algebra A L₃] [IsScalarTower A K L₃]\nvariable [Algebra B L] [IsScalarTower A B L] [IsIntegralClosure B A L]\nvariable [Algebra B₂ L₂] [IsScalarTower A B₂ L₂] [IsIntegralClosure B₂ A L₂]\nvariable [Algebra B₃ L₃] [IsScalarTower A B₃ L₃] [IsIntegralClosure B₃ A L₃]\n\nsection galois\n\nsection galRestrict'\nvariable {K L L₂ L₃}\nomit [IsFractionRing A K]\n\n/-- A generalization of `galRestrictHom` beyond endomorphisms. -/\nnoncomputable\ndef galRestrict' (f : L →ₐ[K] L₂) : (B →ₐ[A] B₂) :=\n  (IsIntegralClosure.equiv A (integralClosure A L₂) L₂ B₂).toAlgHom.comp\n      (((f.restrictScalars A).comp (IsScalarTower.toAlgHom A B L)).codRestrict\n        (integralClosure A L₂) (fun x ↦ IsIntegral.map _ (IsIntegralClosure.isIntegral A L x)))\n\n@[simp]\nlemma algebraMap_galRestrict'_apply (σ : L →ₐ[K] L₂) (x : B) :\n    algebraMap B₂ L₂ (galRestrict' A B B₂ σ x) = σ (algebraMap B L x) := by\n  simp [galRestrict', galRestrict', Subalgebra.algebraMap_eq]\n\n@[simp]\ntheorem galRestrict'_id : galRestrict' A B B (.id K L) = .id A B := by\n  ext\n  apply IsIntegralClosure.algebraMap_injective B A L\n  simp\n\nTarget:\ntheorem galRestrict'_comp (σ : L →ₐ[K] L₂) (σ' : L₂ →ₐ[K] L₃) :\n    galRestrict' A B B₃ (σ'.comp σ) = (galRestrict' A B₂ B₃ σ').comp (galRestrict' A B B₂ σ) :=\n\nProof body:\n","rejected":"by\n  ext x\n  apply (IsIntegralClosure.equiv A (integralClosure A L₃) L₃ B₃).symm.injective\n  ext\n  simp [galRestrict', Subalgebra.algebraMap_eq]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/IntegralClosure","family_id":"galrestrict'_comp","file_id":"mathlib/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean","sample_id":"2f3292c8af1fa98f0dfb39f23e5916bd9e2de4337e8d92b7857cea3205fe336c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a8bd42ecc90333f6177ba81b0d401c4a0d5b772d591ccefb5c358cae0906bf4e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"389d6b6e4066fc2cabec7aefc51cac6e21581a5a3e730cc7243010a6f145a966","source_sha256":"8656d251697095f7fe7e214833a10429be9a10229d106380fe6a23432c3fbd73","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let e : (L ⊗[R] S) ≃ₐ[K] L ⊗[K] (K ⊗[R] S) :=\n    (Algebra.TensorProduct.cancelBaseChange R K K L S).symm\n  let e2 : L ⊗[K] (K ⊗[R] S) ≃ₐ[K] (K ⊗[R] S) ⊗[K] L := Algebra.TensorProduct.comm ..\n  have heq : Algebra.TensorProduct.map (Algebra.ofId K L) (AlgHom.id R S) =\n      (e.symm.toAlgHom.comp e2.symm.toAlgHom).comp\n        (IsScalarTower.toAlgHom K (K ⊗[R] S) ((K ⊗[R] S) ⊗[K] L)) := by\n    ext; simp [e, e2]\n  rw [heq]\n  simp only [AlgHom.toRingHom_eq_coe, AlgHom.comp_toRingHom,\n    AlgEquiv.toAlgHom_toRingHom, IsScalarTower.coe_toAlgHom, comap_comp]\n  exact (isHomeomorph_comap_of_isPurelyInseparable K L (K ⊗[R] S)).comp <|\n    (isHomeomorph_comap_of_bijective e2.symm.bijective).comp <|\n    isHomeomorph_comap_of_bijective e.symm.bijective","hard_negative":false,"metrics":{"chosen_tokens":226,"rejected_tokens":3,"token_jaccard":0.038462,"token_length_ratio":0.013274},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"3072adf4bc9854c82e8b39c351a72a959821193e70443d707921a045cdea1bd8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.PurelyInseparable.Basic\npublic import Mathlib.RingTheory.Flat.Basic\npublic import Mathlib.RingTheory.Spectrum.Prime.Topology\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten, Junyan Xu\n-/\n/-!\n# Purely inseparable extensions are universal homeomorphisms\n\nIf `K` is a purely inseparable extension of `k`, the induced map `Spec K ⟶ Spec k` is a universal\nhomeomorphism, i.e. it stays a homeomorphism after arbitrary base change.\n\n## Main results\n\n- `PrimeSpectrum.isHomeomorph_comap`: if `f : R →+* S` is a ring map with locally nilpotent kernel\n  such that for every `x : S`, there exists `n > 0` such that `x ^ n` is in the image of `f`,\n  `Spec f` is a homeomorphism.\n- `PrimeSpectrum.isHomeomorph_comap_of_isPurelyInseparable`: `Spec K ⟶ Spec k` is a universal\n  homeomorphism for a purely inseparable field extension `K` over `k`.\n-/\n\npublic section\n\nopen TensorProduct\n\nvariable (k K R S : Type*) [Field k] [Field K] [Algebra k K] [CommRing R] [Algebra k R] [CommRing S]\n\nvariable {R S} in\n/-- If the kernel of `f : R →+* S` consists of nilpotent elements and for every `x : S`,\nthere exists `n > 0` such that `x ^ n` is in the range of `f`, then `Spec f` is a homeomorphism.\nNote: This does not hold for semirings, because `ℕ →+* ℤ` satisfies these conditions, but\n`Spec ℕ` has one more point than `Spec ℤ`. -/\n@[stacks 0BR8 \"Homeomorphism part\"]\nlemma PrimeSpectrum.isHomeomorph_comap (f : R →+* S) (H : ∀ (x : S), ∃ n > 0, x ^ n ∈ f.range)\n    (hker : RingHom.ker f ≤ nilradical R) : IsHomeomorph (comap f) := by\n  have h1 : Function.Injective (comap f) := by\n    intro q q' hqq'\n    ext x\n    obtain ⟨n, hn, y, hy⟩ := H x\n    rw [← q.2.pow_mem_iff_mem _ hn, ← q'.2.pow_mem_iff_mem _ hn, ← hy]\n    rw [PrimeSpectrum.ext_iff, SetLike.ext_iff] at hqq'\n    apply hqq'\n  have hint : f.kerLift.IsIntegral := fun x ↦\n    have ⟨n, hn, y, hy⟩ := H x\n    let _ := f.kerLift.toAlgebra\n    IsIntegral.of_pow hn (hy ▸ f.kerLift.isIntegralElem_map (x := ⟦y⟧))\n  have hbij : Function.Bijective (comap f) :=\n    ⟨h1, (comap_quotientMk_bijective_of_le_nilradical hker).2.comp <|\n      hint.comap_surjective f.kerLift_injective⟩\n  refine ⟨continuous_comap f, ?_, h1, hbij.2⟩\n  rw [isTopologicalBasis_basic_opens.isOpenMap_iff]\n  rintro - ⟨s, rfl⟩\n  obtain ⟨n, hn, r, hr⟩ := H s\n  have : (comap f) '' (basicOpen s) = basicOpen r :=\n    (Set.eq_preimage_iff_image_eq hbij).mp <| by rw [← basicOpen_pow _ n hn, ← hr]; rfl\n  exact this ▸ isOpen_basicOpen\n\n/-- Purely inseparable field extensions are universal homeomorphisms. -/\n@[stacks 0BRA \"Special case for purely inseparable field extensions\"]\nlemma PrimeSpectrum.isHomeomorph_comap_of_isPurelyInseparable [IsPurelyInseparable k K] :\n    IsHomeomorph (comap <| algebraMap R (R ⊗[k] K)) := by\n  let q := ringExpChar k\n  refine isHomeomorph_comap _ (IsPurelyInseparable.exists_pow_mem_range_tensorProduct) ?_\n  convert! bot_le\n  rw [← RingHom.injective_iff_ker_eq_bot]\n  exact Algebra.TensorProduct.includeLeft_injective (S := R) (algebraMap k K).injective\n\n/-- If `L` is a purely inseparable extension of `K` over `R` and `S` is an `R`-algebra,\nthe induced map `Spec (L ⊗[R] S) ⟶ Spec (K ⊗[R] S)` is a homeomorphism. -/\n\nTarget:\nlemma PrimeSpectrum.isHomeomorph_comap_tensorProductMap_of_isPurelyInseparable [Algebra R K]\n    [Algebra R S] (L : Type*) [Field L] [Algebra R L] [Algebra K L] [IsScalarTower R K L]\n    [IsPurelyInseparable K L] :\n    IsHomeomorph (comap (Algebra.TensorProduct.map (Algebra.ofId K L) (.id R S)).toRingHom) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Spectrum","family_id":"primespectrum","file_id":"mathlib/Mathlib/RingTheory/Spectrum/Prime/Homeomorph.lean","sample_id":"389d6b6e4066fc2cabec7aefc51cac6e21581a5a3e730cc7243010a6f145a966"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2851538f3a7c40f88dfdee204d38577a3a636017ca196f0ed156206cf7eba4b2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"06f9768559f2fe8088fc3798603424c8e8c34edf39453ccd8c0d887cb067503b","source_sha256":"06f9836a8093746b062f55af45a77b834aa17f6fb321985aabfd940c6ffdaecf","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction hx using span_induction with\n  | mem x h => obtain ⟨i, rfl⟩ := h; cases j <;> simp [h]\n  | zero => simp\n  | add u v _ _ hu hv => simp [hu, hv]\n  | smul t u _ hu => simp [hu]","hard_negative":false,"metrics":{"chosen_tokens":59,"rejected_tokens":2,"token_jaccard":0.027778,"token_length_ratio":0.033898},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"307f86bc62b7fe9e54b9cf9748a187bc89efe04481ed5a2adff9a58a8305b32f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Lie.Matrix\npublic import Mathlib.Algebra.Lie.OfAssociative\npublic import Mathlib.Algebra.Lie.Weights.Basic\npublic import Mathlib.LinearAlgebra.Eigenspace.Matrix\npublic import Mathlib.LinearAlgebra.LinearIndependent.BaseChange\npublic import Mathlib.LinearAlgebra.RootSystem.CartanMatrix\n\nNamespace:\nRootPairing.GeckConstruction\n\nLocal context:\n/-\nCopyright (c) 2025 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Geck's construction of a Lie algebra associated to a root system\n\nThis file contains an implementation of Geck's construction of a semisimple Lie algebra from a\nreduced crystallographic root system. It follows [Geck](Geck2017) quite closely.\n\n## Main definitions:\n* `RootPairing.GeckConstruction.lieAlgebra`: the Geck construction of the Lie algebra associated to\n  a root system with distinguished base.\n* `RootPairing.GeckConstruction.cartanSubalgebra`: a distinguished subalgebra corresponding to a\n  Cartan subalgebra of the Geck construction.\n* `RootPairing.GeckConstruction.cartanSubalgebra_le_lieAlgebra`: the distinguished subalgebra is\n  contained in the Geck construction.\n\n## Alternative approaches\n\nThere are at least three ways to construct a Lie algebra from a root system:\n1. As a quotient of a free Lie algebra, using the Serre relations\n2. Directly defining the Lie bracket on $H ⊕ K^∣Φ|$\n3. The Geck construction\n\nWe comment on these as follows:\n1. This construction takes just a matrix as input. It yields a semisimple Lie algebra iff the\n   matrix is a Cartan matrix but it is quite a lot of work to prove this. On the other hand, it also\n   allows construction of Kac-Moody Lie algebras. It has been implemented as `Matrix.ToLieAlgebra`\n   but as of May 2025, almost nothing has been proved about it in Mathlib.\n2. This construction takes a root system with base as input, together with sufficient additional\n   data to determine a collection of extraspecial pairs of roots. The additional data for the\n   extraspecial pairs is required to pin down certain signs when defining the Lie bracket. (These\n   signs can be interpreted as a set-theoretic splitting of Tits's extension of the Weyl group by\n   an elementary 2-group of order $2^l$ where $l$ is the rank.)\n3. This construction takes a root system with base as input and is implemented here.\n\nThere seems to be no known construction of a Lie algebra from a root system without first choosing\na base: https://mathoverflow.net/questions/495434/\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Function Set Submodule\nopen scoped Matrix\n\nattribute [local simp] Matrix.mul_apply Matrix.one_apply Matrix.diagonal_apply\n\nnamespace RootPairing.GeckConstruction\n\nvariable {ι R M N : Type*} [CommRing R]\n  [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]\n  {P : RootPairing ι R M N} [P.IsCrystallographic] {b : P.Base}\n\n/-- Part of an `sl₂` triple used in Geck's construction of a Lie algebra from a root system. -/\ndef h (i : b.support) :\n    Matrix (b.support ⊕ ι) (b.support ⊕ ι) R :=\n  open scoped Classical in\n  .fromBlocks 0 0 0 (.diagonal (P.pairingIn ℤ · i))\n\nlemma h_def [DecidableEq ι] (i : b.support) :\n    h i = .fromBlocks 0 0 0 (.diagonal (P.pairingIn ℤ · i)) := by\n  ext (j | j) (k | k) <;> simp [h, Matrix.diagonal_apply]\n\nlemma h_eq_diagonal [DecidableEq ι] (i : b.support) :\n    h i = .diagonal (Sum.elim 0 (P.pairingIn ℤ · i)) := by\n  ext (j | j) (k | k) <;> simp [h, Matrix.diagonal_apply]\n\nvariable (b) in\nlemma linearIndependent_h [Finite ι] [CharZero R] [IsDomain R] [P.IsRootSystem] :\n    LinearIndependent R (h (b := b)) := by\n  classical\n  have : Matrix.diagLinearMap (b.support ⊕ ι) R R ∘ h =\n      Sum.elimZeroLeft ∘ fun i : b.support ↦ algebraMap ℤ R ∘ (P.pairingIn ℤ · i) := by\n    ext; rw [comp_apply, h_def]; aesop\n  apply LinearIndependent.of_comp (Matrix.diagLinearMap _ _ _)\n  rw [this, LinearMap.linearIndependent_iff_of_injOn _ Sum.elim_injective'.injOn,\n    linearIndependent_algebraMap_comp_iff]\n  suffices LinearIndependent ℤ (fun i j : b.support ↦ P.pairingIn ℤ j i) from\n    this.of_linearIndependent_subset b.support\n  apply b.cartanMatrix.transpose.linearIndependent_rows_of_det_ne_zero\n  rw [Matrix.det_transpose, ← Matrix.nondegenerate_iff_det_ne_zero]\n  exact b.cartanMatrix_nondegenerate\n\nlemma span_range_h_le_range_diagonal [DecidableEq ι] :\n    span R (range h) ≤ LinearMap.range (Matrix.diagonalLinearMap (b.support ⊕ ι) R R) := by\n  rw [span_le]\n  rintro - ⟨i, rfl⟩\n  rw [h_eq_diagonal]\n  exact LinearMap.mem_range_self _ _\n\nopen Matrix in\n@[simp] lemma diagonal_elim_mem_span_h_iff [DecidableEq ι] {d : ι → R} :\n    diagonal (Sum.elim 0 d) ∈ span R (range <| h (b := b)) ↔\n      d ∈ span R (range <| fun (i : b.support) j ↦ (P.pairingIn ℤ j i : R)) := by\n  let g : Matrix ι ι R →ₗ[R] Matrix (b.support ⊕ ι) (b.support ⊕ ι) R :=\n    { toFun := .fromBlocks 0 0 0\n      map_add' x y := by ext (i | i) (j | j) <;> simp\n      map_smul' t x := by ext (i | i) (j | j) <;> simp }\n  have h₀ : Injective (g ∘ diagonalLinearMap ι R R) := fun _ _ hd ↦ funext <| by simpa [g] using hd\n  have h₁ {d : ι → R} : diagonal (Sum.elim 0 d) = g (diagonalLinearMap ι R R d) := by\n    ext (i | i) (j | j) <;> simp [g]\n  have h₂ : range h = g '' (diagonalLinearMap ι R R ''\n    (range <| fun (i : b.support) j ↦ (P.pairingIn ℤ j i : R))) := by ext; simp [g, h_def]\n  simp_rw [h₁, h₂, span_image, ← map_comp, ← comp_apply (f := g), mem_map, LinearMap.coe_comp,\n    h₀.eq_iff, exists_eq_right]\n\nTarget:\nlemma apply_sum_inl_eq_zero_of_mem_span_h\n    (i : b.support) (j : b.support ⊕ ι) {x : Matrix (b.support ⊕ ι) (b.support ⊕ ι) R}\n    (hx : x ∈ span R (range h)) :\n    x j (Sum.inl i) = 0 :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/RootSystem","family_id":"apply_sum_inl_eq_zero_of_mem_span_h","file_id":"mathlib/Mathlib/LinearAlgebra/RootSystem/GeckConstruction/Basic.lean","sample_id":"06f9768559f2fe8088fc3798603424c8e8c34edf39453ccd8c0d887cb067503b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f6142b8f4680c491cc9a84d46312116399269b0b324f299edcee5c0f95b61ddc","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"58002da4ccd4d62f1b8ce3942439ad0ce33ed3890c7de24f9d70e2a2bdeae379","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ebebed96f1f59661bd31143f834b43b5b9941049affa74d994da26eb1385a91c","source_sha256":"edb52479240700af7f6c572a1b6c456a3b7b18b85aa47a4adb83e507587b6553","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro x y hxy\n  have := M.isUnit_algebraMap_end_of_le_basicOpen _ hle\n  rw [Module.End.isUnit_iff] at this\n  exact this.injective hxy","hard_negative":true,"metrics":{"chosen_tokens":27,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.074074},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"309ea5cc2b5b5afbc0a979b160d9dde441a4ddf2a815fb47ce97993a254db68c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Category.ModuleCat.Localization\npublic import Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent\npublic import Mathlib.AlgebraicGeometry.AffineScheme\npublic import Mathlib.AlgebraicGeometry.Modules.Sheaf\n\nNamespace:\nAlgebraicGeometry.Scheme.Modules\n\nLocal context:\n/-\nCopyright (c) 2024 Weihong Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Johan Commelin, Amelia Livingston, Sophie Morel, Jujian Zhang, Weihong Xu,\n  Andrew Yang, Brian Nugent\n-/\n/-!\n\n# Construction of M^~\n\nGiven any commutative ring `R` and `R`-module `M`, we construct the sheaf `M^~` of `𝒪_SpecR`-modules\nsuch that `M^~(U)` is the set of dependent functions that are locally fractions.\n\n## Main definitions\n* `AlgebraicGeometry.tilde` : `M^~` as a sheaf of `𝒪_{Spec R}`-modules.\n* `AlgebraicGeometry.tilde.adjunction` : `~` is left adjoint to taking global sections.\n\n-/\n\n@[expose] public noncomputable section\n\nuniverse u\n\nopen TopCat AlgebraicGeometry TopologicalSpace CategoryTheory Opposite\n\nvariable {R : CommRingCat.{u}} (M : ModuleCat.{u} R)\n\nnamespace AlgebraicGeometry\n\nopen _root_.PrimeSpectrum\n\n/-- The forgetful functor from `𝒪_{Spec R}` modules to sheaves of `R`-modules. -/\ndef modulesSpecToSheaf :\n    (Spec R).Modules ⥤ TopCat.Sheaf (ModuleCat R) (Spec R) :=\n  SheafOfModules.forgetToSheafModuleCat (Spec R).ringCatSheaf (.op ⊤)\n    (Limits.initialOpOfTerminal Limits.isTerminalTop) ⋙\n  sheafCompose _ (ModuleCat.restrictScalars (Scheme.ΓSpecIso R).inv.hom)\n\n/-- The global section functor for `𝒪_{Spec R}` modules -/\nnoncomputable\ndef moduleSpecΓFunctor : (Spec (.of R)).Modules ⥤ ModuleCat R :=\n  modulesSpecToSheaf ⋙ TopCat.Sheaf.forget _ _ ⋙ (evaluation _ _).obj (.op ⊤)\n\nset_option backward.isDefEq.respectTransparency false in\nopen PrimeSpectrum in\n/-- The forgetful functor from `𝒪_{Spec R}` modules to sheaves of `R`-modules is fully faithful. -/\ndef SpecModulesToSheafFullyFaithful : (modulesSpecToSheaf (R := R)).FullyFaithful where\n  preimage {M N} f := ⟨fun U ↦ ModuleCat.ofHom ⟨(f.1.app U).hom.toAddHom, by\n    intro t m\n    apply TopCat.Presheaf.IsSheaf.section_ext (modulesSpecToSheaf.obj N).2\n    intro x hxU\n    obtain ⟨a, ⟨_, ⟨r, rfl⟩, rfl⟩, hxr, hrU : basicOpen _ ≤ _⟩ :=\n      PrimeSpectrum.isBasis_basic_opens.exists_subset_of_mem_open hxU U.unop.2\n    refine ⟨_, hrU, hxr, ?_⟩\n    refine Eq.trans ?_ (N.val.map_smul (homOfLE hrU).op t _).symm\n    change N.1.map (homOfLE hrU).op (f.1.app _ _) = _ • N.1.map (homOfLE hrU).op (f.1.app _ _)\n    have (x : _) :\n        f.1.app _ (M.1.map (homOfLE hrU).op _) = N.1.map (homOfLE hrU).op (f.1.app _ x) :=\n      congr($(f.1.naturality (homOfLE hrU).op).hom x)\n    rw [← this, ← this, M.val.map_smul]\n    generalize (Spec R).ringCatSheaf.obj.map (homOfLE hrU).op t = t\n    letI := Module.compHom (R := Γ(Spec R, basicOpen r)) Γ(M, basicOpen r)\n      (algebraMap R Γ(Spec R, basicOpen r))\n    haveI : IsScalarTower R Γ(Spec R, basicOpen r) Γ(M, basicOpen r) :=\n      .of_algebraMap_smul fun _ _ ↦ rfl\n    letI := Module.compHom Γ(N, basicOpen r) (algebraMap R Γ(Spec R, basicOpen r))\n    haveI : IsScalarTower R Γ(Spec R, basicOpen r) Γ(N, basicOpen r) :=\n      .of_algebraMap_smul fun _ _ ↦ rfl\n    exact (IsLocalization.linearMap_compatibleSMul (.powers (M := R) r)\n      Γ(Spec R, basicOpen r) Γ(M, basicOpen r) Γ(N, basicOpen r)).map_smul\n      (f.hom.app _).hom _ _⟩, fun i ↦ by ext x; exact congr($(f.1.naturality i).hom x)⟩\n  map_preimage f := rfl\n  preimage_map f := rfl\n\ninstance : (modulesSpecToSheaf (R := R)).Faithful := SpecModulesToSheafFullyFaithful.faithful\n\ninstance : (modulesSpecToSheaf (R := R)).Full := SpecModulesToSheafFullyFaithful.full\n\nnamespace Scheme.Modules\n\nvariable {M : (Spec R).Modules} {U V : (Spec R).Opens}\n\ninstance : Module R Γ(M, U) :=\n  inferInstanceAs <| Module R ((modulesSpecToSheaf.obj M).obj.obj (.op U))\n\ninstance : IsScalarTower R Γ(Spec R, U) Γ(M, U) :=\n  IsScalarTower.of_compHom R Γ(Spec R, U) Γ(M, U)\n\nlemma smul_Spec_def (r : R) (x : Γ(M, U)) :\n    r • x = ((Spec R).presheaf.map U.leTop.op) ((Scheme.ΓSpecIso R).inv r) • x :=\n  rfl\n\n@[simp]\nlemma map_smul_Spec (hUV : .op V ⟶ .op U) (f : R) (x : Γ(M, V)) :\n    dsimp% M.presheaf.map hUV (f • x) = f • M.presheaf.map hUV x :=\n  ((modulesSpecToSheaf.obj M).obj.map hUV).hom.map_smul f x\n\nlemma isUnit_algebraMap_end_of_le_basicOpen (f : R) (hf : U ≤ PrimeSpectrum.basicOpen f) :\n    IsUnit (algebraMap R (Module.End R Γ(M, U)) f) := by\n  rw [Module.End.isUnit_iff]\n  have : ⇑((algebraMap R (Module.End ↑R ↑Γ(M, U))) f) =\n      algebraMap (Γ(Spec R, U)) (Module.End Γ(Spec R, U) Γ(M, U))\n        (((Spec R).presheaf.map (homOfLE hf).op) <| algebraMap R _ f) :=\n    rfl\n  rw [this, ← Module.End.isUnit_iff]\n  exact ((IsLocalization.Away.algebraMap_isUnit _).map _).map _\n\nTarget:\nlemma isSMulRegular_of_le_basicOpen {f : R} (hle : U ≤ PrimeSpectrum.basicOpen f) :\n    IsSMulRegular Γ(M, U) f :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_ebebed96f1f5","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"4a0292f291caead584e429a3c452856bf0e9bee08474cfd2ff7b88d99fd75dc3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Modules","family_id":"issmulregular_of_le_basicopen","file_id":"mathlib/Mathlib/AlgebraicGeometry/Modules/Tilde.lean","sample_id":"ebebed96f1f59661bd31143f834b43b5b9941049affa74d994da26eb1385a91c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6932312c5f36cec3a14fb1d6fd3d6a1837663ad65e9f1ee82ae3d44d58f2d03c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a5c039edafb94f9c2f4f534603a685dad7c654461a72b2106c4226b561dbae5d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"062acd11f87739fb15d069f3d86b913dd7cb13fa854b155bc79d97daea43f761","source_sha256":"c3981253356b7afc328e1095904e6a5b84d7b6867b9673136639d9cfe765100e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let _ : Nontrivial N := by simp [← Module.supportDim_ne_bot_iff_nontrivial R, dim]\n  rw [PrimeSpectrum.zeroLocus_eq_singleton]\n  apply le_antisymm\n  · intro p hp\n    by_contra nmem\n    push _ ∈ _ at nmem\n    have : p < ⟨maximalIdeal R, IsMaximal.isPrime' (maximalIdeal R)⟩ :=\n      lt_of_le_of_ne (IsLocalRing.le_maximalIdeal IsPrime.ne_top') nmem\n    have : Module.supportDim R N > 0 := by\n      simp only [Module.supportDim, gt_iff_lt, Order.krullDim_pos_iff, Subtype.exists,\n        Subtype.mk_lt_mk, exists_prop]\n      use p\n      simpa [hp] using! ⟨_, IsLocalRing.closedPoint_mem_support R N, this⟩\n    exact (ne_of_lt this) dim.symm\n  · simpa using! IsLocalRing.closedPoint_mem_support R N","hard_negative":true,"metrics":{"chosen_tokens":135,"rejected_tokens":2,"token_jaccard":0.015152,"token_length_ratio":0.014815},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"30e3cb0dfeec61431e5d6d5638ebcd422d7bddeaa842b3c83989a3620e4f922b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.KrullDimension.NonZeroDivisors\npublic import Mathlib.RingTheory.Spectrum.Prime.Module\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Nailin Guan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Nailin Guan\n-/\n/-!\n\n# Krull Dimension of Module\n\nIn this file we define `Module.supportDim R M` for an `R`-module `M` as\nthe krull dimension of its support. It is equal to the krull dimension of `R / Ann M` when\n`M` is finitely generated.\n\n-/\n\n@[expose] public section\n\nvariable (R : Type*) [CommRing R]\n\nvariable (M : Type*) [AddCommGroup M] [Module R M] (N : Type*) [AddCommGroup N] [Module R N]\n\nnamespace Module\n\nopen Order\n\n/-- The krull dimension of module, defined as `krullDim` of its support. -/\nnoncomputable def supportDim : WithBot ℕ∞ :=\n  krullDim (Module.support R M)\n\n@[nontriviality]\nlemma supportDim_eq_bot_of_subsingleton [Subsingleton M] : supportDim R M = ⊥ := by\n  simpa [supportDim, support_eq_empty_iff]\n\nlemma supportDim_ne_bot_of_nontrivial [Nontrivial M] : supportDim R M ≠ ⊥ := by\n  have : Nonempty (Module.support R M) := nonempty_support_of_nontrivial.to_subtype\n  simp [supportDim]\n\nlemma supportDim_eq_bot_iff_subsingleton : supportDim R M = ⊥ ↔ Subsingleton M := by\n  simp [supportDim, krullDim_eq_bot_iff, support_eq_empty_iff]\n\nlemma supportDim_ne_bot_iff_nontrivial : supportDim R M ≠ ⊥ ↔ Nontrivial M := by\n  simp [supportDim, krullDim_eq_bot_iff, support_eq_empty_iff, not_subsingleton_iff_nontrivial]\n\nlemma supportDim_eq_ringKrullDim_quotient_annihilator [Module.Finite R M] :\n    supportDim R M = ringKrullDim (R ⧸ annihilator R M) := by\n  simp only [supportDim]\n  rw [support_eq_zeroLocus, ringKrullDim_quotient]\n\nlemma supportDim_self_eq_ringKrullDim : supportDim R R = ringKrullDim R := by\n  have : annihilator R R = ⊥ :=\n    annihilator_eq_bot.mpr ((faithfulSMul_iff_algebraMap_injective R R).mpr fun {a₁ a₂} a ↦ a)\n  rw [supportDim_eq_ringKrullDim_quotient_annihilator, this]\n  exact (RingEquiv.ringKrullDim (RingEquiv.quotientBot R))\n\nlemma supportDim_le_ringKrullDim : supportDim R M ≤ ringKrullDim R :=\n  krullDim_le_of_strictMono (fun a ↦ a) fun {_ _} lt ↦ lt\n\nvariable {R M N}\n\nlemma supportDim_quotient_eq_ringKrullDim (I : Ideal R) :\n    supportDim R (R ⧸ I) = ringKrullDim (R ⧸ I) := by\n  rw [supportDim_eq_ringKrullDim_quotient_annihilator, Ideal.annihilator_quotient]\n\nlemma supportDim_le_of_injective (f : M →ₗ[R] N) (h : Function.Injective f) :\n    supportDim R M ≤ supportDim R N :=\n  krullDim_le_of_strictMono (fun a ↦ ⟨a.1, Module.support_subset_of_injective f h a.2⟩)\n    (fun {_ _} lt ↦ lt)\n\nlemma supportDim_le_of_surjective (f : M →ₗ[R] N) (h : Function.Surjective f) :\n    supportDim R N ≤ supportDim R M :=\n  krullDim_le_of_strictMono (fun a ↦ ⟨a.1, Module.support_subset_of_surjective f h a.2⟩)\n    (fun {_ _} lt ↦ lt)\n\nlemma supportDim_eq_of_equiv (e : M ≃ₗ[R] N) :\n    supportDim R M = supportDim R N :=\n  le_antisymm (supportDim_le_of_injective e e.injective)\n    (supportDim_le_of_surjective e e.surjective)\n\nend Module\n\nopen Ideal IsLocalRing\n\nTarget:\nlemma support_of_supportDim_eq_zero [IsLocalRing R]\n    (dim : Module.supportDim R N = 0) :\n    Module.support R N = PrimeSpectrum.zeroLocus (maximalIdeal R) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_062acd11f877","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b57e9138f752240bdd5cbb3bcb4852ad3078ba6a706fe4fc7a597da81e17fcd9","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/KrullDimension","family_id":"support_of_supportdim_eq_zero","file_id":"mathlib/Mathlib/RingTheory/KrullDimension/Module.lean","sample_id":"062acd11f87739fb15d069f3d86b913dd7cb13fa854b155bc79d97daea43f761"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7492371e89f22fb3632f66da0f33c643e10177499de4f2d7febebab11d42e4f8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c0a7c5068f13964705a2f5697590139cfa7263b7caacbab7db184ea5f0bc0d60","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"31295720883c04924c41b3b7a4b8a52765977942c19e8e921c01332c5cadf5db","source_sha256":"11459aad782565ea7bf0f74d44b1f758119860f533b2e2fde38a8a35d278f13d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  symm\n  calc\n    0 = ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆ := by\n      rw [(commute_lmul_lmul_sq (a + b + c)).lie_eq]\n    _ = _ := by rw [aux0, aux1, aux2, aux3, nsmul_add, nsmul_add]","hard_negative":false,"metrics":{"chosen_tokens":70,"rejected_tokens":77,"token_jaccard":0.935484,"token_length_ratio":1.1},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"31178cdf59865953c167821ec1d317a24be1a43656405ecf995a397d9de1fc38","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Lie.OfAssociative\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Christopher Hoskin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christopher Hoskin\n-/\n/-!\n# Jordan rings\n\nLet `A` be a non-unital, non-associative ring. Then `A` is said to be a (commutative, linear) Jordan\nring if the multiplication is commutative and satisfies a weak associativity law known as the\nJordan Identity: for all `a` and `b` in `A`,\n```\n(a * b) * a^2 = a * (b * a^2)\n```\ni.e. the operators of multiplication by `a` and `a^2` commute.\n\nA more general concept of a (non-commutative) Jordan ring can also be defined, as a\n(non-commutative, non-associative) ring `A` where, for each `a` in `A`, the operators of left and\nright multiplication by `a` and `a^2` commute.\n\nEvery associative algebra can be equipped with a symmetrized multiplication (characterized by\n`SymAlg.sym_mul_sym`) making it into a commutative Jordan algebra (`IsCommJordan`).\nJordan algebras arising this way are said to be special.\n\nA real Jordan algebra `A` can be introduced by\n```lean\nvariable {A : Type*} [NonUnitalNonAssocCommRing A] [Module ℝ A] [SMulCommClass ℝ A A]\n  [IsScalarTower ℝ A A] [IsCommJordan A]\n```\n\n## Main results\n\n- `two_nsmul_lie_lmul_lmul_add_add_eq_zero` : Linearisation of the commutative Jordan axiom\n\n## Implementation notes\n\nWe shall primarily be interested in linear Jordan algebras (i.e. over rings of characteristic not\ntwo) leaving quadratic algebras to those better versed in that theory.\n\nThe conventional way to linearise the Jordan axiom is to equate coefficients (more formally, assume\nthat the axiom holds in all field extensions). For simplicity we use brute force algebraic expansion\nand substitution instead.\n\n## Motivation\n\nEvery Jordan algebra `A` has a triple product defined, for `a` `b` and `c` in `A` by\n$$\n{a\\,b\\,c} = (a * b) * c - (a * c) * b + a * (b * c).\n$$\nVia this triple product Jordan algebras are related to a number of other mathematical structures:\nJordan triples, partial Jordan triples, Jordan pairs and quadratic Jordan algebras. In addition to\ntheir considerable algebraic interest ([mccrimmon2004]) these structures have been shown to have\ndeep connections to mathematical physics, functional analysis and differential geometry. For more\ninformation about these connections the interested reader is referred to [alfsenshultz2003],\n[chu2012], [friedmanscarr2005], [iordanescu2003] and [upmeier1987].\n\nThere are also exceptional Jordan algebras which can be shown not to be the symmetrization of any\nassociative algebra. The 3x3 matrices of octonions is the canonical example.\n\nNon-commutative Jordan algebras have connections to the Vidav-Palmer theorem\n[cabreragarciarodriguezpalacios2014].\n\n## References\n\n* [Cabrera García and Rodríguez Palacios, Non-associative normed algebras. Volume 1]\n  [cabreragarciarodriguezpalacios2014]\n* [Hanche-Olsen and Størmer, Jordan Operator Algebras][hancheolsenstormer1984]\n* [McCrimmon, A taste of Jordan algebras][mccrimmon2004]\n\n-/\n\npublic section\n\n\nvariable (A : Type*)\n\n/-- A (non-commutative) Jordan multiplication. -/\nclass IsJordan [Mul A] : Prop where\n  lmul_comm_rmul : ∀ a b : A, a * b * a = a * (b * a)\n  lmul_lmul_comm_lmul : ∀ a b : A, a * a * (a * b) = a * (a * a * b)\n  lmul_lmul_comm_rmul : ∀ a b : A, a * a * (b * a) = a * a * b * a\n  lmul_comm_rmul_rmul : ∀ a b : A, a * b * (a * a) = a * (b * (a * a))\n  rmul_comm_rmul_rmul : ∀ a b : A, b * a * (a * a) = b * (a * a) * a\n\n/-- A commutative Jordan multiplication -/\nclass IsCommJordan [CommMagma A] : Prop where\n  lmul_comm_rmul_rmul : ∀ a b : A, a * b * (a * a) = a * (b * (a * a))\n\n-- see Note [lower instance priority]\n/-- A (commutative) Jordan multiplication is also a Jordan multiplication -/\ninstance (priority := 100) IsCommJordan.toIsJordan [CommMagma A] [IsCommJordan A] : IsJordan A where\n  lmul_comm_rmul a b := by rw [mul_comm, mul_comm a b]\n  lmul_lmul_comm_lmul a b := by\n    rw [mul_comm (a * a) (a * b), IsCommJordan.lmul_comm_rmul_rmul,\n      mul_comm b (a * a)]\n  lmul_comm_rmul_rmul := IsCommJordan.lmul_comm_rmul_rmul\n  lmul_lmul_comm_rmul a b := by\n    rw [mul_comm (a * a) (b * a), mul_comm b a,\n      IsCommJordan.lmul_comm_rmul_rmul, mul_comm, mul_comm b (a * a)]\n  rmul_comm_rmul_rmul a b := by\n    rw [mul_comm b a, IsCommJordan.lmul_comm_rmul_rmul, mul_comm]\n\n-- see Note [lower instance priority]\n/-- Semigroup multiplication satisfies the (non-commutative) Jordan axioms -/\ninstance (priority := 100) Semigroup.isJordan [Semigroup A] : IsJordan A where\n  lmul_comm_rmul a b := by rw [mul_assoc]\n  lmul_lmul_comm_lmul a b := by rw [mul_assoc, mul_assoc]\n  lmul_comm_rmul_rmul a b := by rw [mul_assoc]\n  lmul_lmul_comm_rmul a b := by rw [← mul_assoc]\n  rmul_comm_rmul_rmul a b := by rw [← mul_assoc, ← mul_assoc]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) CommSemigroup.isCommJordan [CommSemigroup A] : IsCommJordan A where\n  lmul_comm_rmul_rmul _ _ := mul_assoc _ _ _\n\nlocal notation \"L\" => AddMonoid.End.mulLeft\n\nlocal notation \"R\" => AddMonoid.End.mulRight\n\n/-!\nThe Jordan axioms can be expressed in terms of commuting multiplication operators.\n-/\n\n\nsection Commute\n\nvariable {A} [NonUnitalNonAssocRing A] [IsJordan A]\n\n@[simp]\ntheorem commute_lmul_rmul (a : A) : Commute (L a) (R a) :=\n  AddMonoidHom.ext fun _ => (IsJordan.lmul_comm_rmul _ _).symm\n\n@[simp]\ntheorem commute_lmul_lmul_sq (a : A) : Commute (L a) (L (a * a)) :=\n  AddMonoidHom.ext fun _ => (IsJordan.lmul_lmul_comm_lmul _ _).symm\n\n@[simp]\ntheorem commute_lmul_rmul_sq (a : A) : Commute (L a) (R (a * a)) :=\n  AddMonoidHom.ext fun _ => (IsJordan.lmul_comm_rmul_rmul _ _).symm\n\n@[simp]\ntheorem commute_lmul_sq_rmul (a : A) : Commute (L (a * a)) (R a) :=\n  AddMonoidHom.ext fun _ => IsJordan.lmul_lmul_comm_rmul _ _\n\n@[simp]\ntheorem commute_rmul_rmul_sq (a : A) : Commute (R a) (R (a * a)) :=\n  AddMonoidHom.ext fun _ => (IsJordan.rmul_comm_rmul_rmul _ _).symm\n\nend Commute\n\nvariable {A} [NonUnitalNonAssocCommRing A]\n\nattribute [local instance 100] LieRing.ofAssociativeRing\n\n/-!\nThe endomorphisms on an additive monoid `AddMonoid.End` form a `Ring`, and this may be equipped\nwith a Lie Bracket via `Ring.bracket`.\n-/\n\ntheorem two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add [IsCommJordan A] (a b : A) :\n    2 • (⁅L a, L (a * b)⁆ + ⁅L b, L (b * a)⁆) = ⁅L (a * a), L b⁆ + ⁅L (b * b), L a⁆ := by\n  suffices 2 • ⁅L a, L (a * b)⁆ + 2 • ⁅L b, L (b * a)⁆ + ⁅L b, L (a * a)⁆ + ⁅L a, L (b * b)⁆ = 0 by\n    rwa [← sub_eq_zero, ← sub_sub, sub_eq_add_neg, sub_eq_add_neg, lie_skew, lie_skew, nsmul_add]\n  convert (commute_lmul_lmul_sq (a + b)).lie_eq\n  simp only [add_mul, mul_add, map_add, lie_add, add_lie, mul_comm b a,\n    (commute_lmul_lmul_sq a).lie_eq, (commute_lmul_lmul_sq b).lie_eq, zero_add, add_zero, two_smul]\n  abel\n\n-- Porting note: the monolithic `calc`-based proof of `two_nsmul_lie_lmul_lmul_add_add_eq_zero`\n-- has had four auxiliary parts `aux{0,1,2,3}` split off from it.\nprivate theorem aux0 {a b c : A} : ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆ =\n    ⁅L a + L b + L c, L (a * a) + L (b * b) + L (c * c) +\n    2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ := by\n  rw [add_mul, add_mul]\n  iterate 6 rw [mul_add]\n  iterate 10 rw [map_add]\n  rw [mul_comm b a, mul_comm c a, mul_comm c b]\n  iterate 3 rw [two_smul]\n  simp only [add_lie]\n  abel_nf\n\nprivate theorem aux1 {a b c : A} :\n    ⁅L a + L b + L c, L (a * a) + L (b * b) + L (c * c) +\n    2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆\n    =\n    ⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ +\n    ⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ + ⁅L a, 2 • L (b * c)⁆ +\n    (⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅L b, L (c * c)⁆ +\n    ⁅L b, 2 • L (a * b)⁆ + ⁅L b, 2 • L (c * a)⁆ + ⁅L b, 2 • L (b * c)⁆) +\n    (⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, L (c * c)⁆ +\n    ⁅L c, 2 • L (a * b)⁆ + ⁅L c, 2 • L (c * a)⁆ + ⁅L c, 2 • L (b * c)⁆) := by\n  rw [add_lie, add_lie]\n  iterate 15 rw [lie_add]\n\nvariable [IsCommJordan A]\n\nprivate theorem aux2 {a b c : A} :\n    ⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ +\n    ⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ + ⁅L a, 2 • L (b * c)⁆ +\n    (⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅L b, L (c * c)⁆ +\n    ⁅L b, 2 • L (a * b)⁆ + ⁅L b, 2 • L (c * a)⁆ + ⁅L b, 2 • L (b * c)⁆) +\n    (⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, L (c * c)⁆ +\n    ⁅L c, 2 • L (a * b)⁆ + ⁅L c, 2 • L (c * a)⁆ + ⁅L c, 2 • L (b * c)⁆)\n    =\n    ⁅L a, L (b * b)⁆ + ⁅L b, L (a * a)⁆ + 2 • (⁅L a, L (a * b)⁆ + ⁅L b, L (a * b)⁆) +\n    (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + 2 • (⁅L a, L (c * a)⁆ + ⁅L c, L (c * a)⁆)) +\n    (⁅L b, L (c * c)⁆ + ⁅L c, L (b * b)⁆ + 2 • (⁅L b, L (b * c)⁆ + ⁅L c, L (b * c)⁆)) +\n    (2 • ⁅L a, L (b * c)⁆ + 2 • ⁅L b, L (c * a)⁆ + 2 • ⁅L c, L (a * b)⁆) := by\n  rw [(commute_lmul_lmul_sq a).lie_eq, (commute_lmul_lmul_sq b).lie_eq,\n    (commute_lmul_lmul_sq c).lie_eq, zero_add, add_zero, add_zero]\n  simp only [lie_nsmul]\n  abel\n\nprivate theorem aux3 {a b c : A} :\n    ⁅L a, L (b * b)⁆ + ⁅L b, L (a * a)⁆ + 2 • (⁅L a, L (a * b)⁆ + ⁅L b, L (a * b)⁆) +\n    (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + 2 • (⁅L a, L (c * a)⁆ + ⁅L c, L (c * a)⁆)) +\n    (⁅L b, L (c * c)⁆ + ⁅L c, L (b * b)⁆ + 2 • (⁅L b, L (b * c)⁆ + ⁅L c, L (b * c)⁆)) +\n    (2 • ⁅L a, L (b * c)⁆ + 2 • ⁅L b, L (c * a)⁆ + 2 • ⁅L c, L (a * b)⁆)\n    =\n    2 • ⁅L a, L (b * c)⁆ + 2 • ⁅L b, L (c * a)⁆ + 2 • ⁅L c, L (a * b)⁆ := by\n  rw [add_eq_right]\n  nth_rw 2 [mul_comm a b]\n  nth_rw 1 [mul_comm c a]\n  nth_rw 2 [mul_comm b c]\n  iterate 3 rw [two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add]\n  iterate 2 rw [← lie_skew (L (a * a)), ← lie_skew (L (b * b)), ← lie_skew (L (c * c))]\n  abel\n\nTarget:\ntheorem two_nsmul_lie_lmul_lmul_add_add_eq_zero (a b c : A) :\n    2 • (⁅L a, L (b * c)⁆ + ⁅L b, L (c * a)⁆ + ⁅L c, L (a * b)⁆) = 0 :=\n\nProof body:\n","rejected":"```lean\nby\n  symm\n  calc\n    0 = ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆ := by\n      rw [(commute_lmul_lmul_sq (a + b + c)).lie_eq]\n    _ = _ := by rw [aux0, aux1, aux2, aux3, nsmul_add, nsmul_add]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Jordan","family_id":"two_nsmul_lie_lmul_lmul_add_add_eq_zero","file_id":"mathlib/Mathlib/Algebra/Jordan/Basic.lean","sample_id":"31295720883c04924c41b3b7a4b8a52765977942c19e8e921c01332c5cadf5db"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4824a182947d6de23921d345ff1615bb9443926fe4268ae7e79b135cafd2e987","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f97b91f107eb7b4f9d321215e0984f13daac29b819bde8fd09eea122778dc081","source_sha256":"7e5df4bf710966952754657aad6fa1ad4d76931701275f754a0493a614aef912","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    rintro X ⟨hX⟩\n    have := hX.prop_diag_obj\n    simp only [isCardinalPresentable_iff] at this ⊢\n    exact isCardinalPresentable_of_isColimit _ hX.isColimit κ hJ","hard_negative":false,"metrics":{"chosen_tokens":27,"rejected_tokens":2,"token_jaccard":0.04,"token_length_ratio":0.074074},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"311a07b10842cb222361fceadf3b61186e88f0fdf43ea5a2b1d6f5d623829169","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Types.Filtered\npublic import Mathlib.CategoryTheory.Limits.Yoneda\npublic import Mathlib.CategoryTheory.Presentable.Basic\npublic import Mathlib.CategoryTheory.ObjectProperty.ColimitsOfShape\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Colimits of presentable objects\n\nIn this file, we show that `κ`-accessible functors (to the category of types)\nare stable under limits indexed by a category `K` such that\n`HasCardinalLT (Arrow K) κ`.\nIn particular, `κ`-presentable objects are stable by colimits indexed\nby a category `K` such that `HasCardinalLT (Arrow K) κ`.\n\n-/\n\n@[expose] public section\n\nuniverse w w' v' v u' u\n\nnamespace CategoryTheory\n\nopen Opposite Limits\n\nvariable {C : Type u} [Category.{v} C]\n\nnamespace Functor\n\nnamespace Accessible\n\nnamespace Limits\n\nsection\n\nvariable {K : Type u'} [Category.{v'} K] {F : K ⥤ C ⥤ Type w'}\n  (c : Cone F) (hc : ∀ (Y : C), IsLimit (((evaluation _ _).obj Y).mapCone c))\n  (κ : Cardinal.{w}) [Fact κ.IsRegular]\n  (hK : HasCardinalLT (Arrow K) κ)\n  {J : Type w} [SmallCategory J] [IsCardinalFiltered J κ]\n  {X : J ⥤ C} (cX : Cocone X)\n  (hF : ∀ (k : K), IsColimit ((F.obj k).mapCocone cX))\n\nnamespace isColimitMapCocone\n\ninclude hc hF hK\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma surjective (x : c.pt.obj cX.pt) :\n    ∃ (j : J) (x' : c.pt.obj (X.obj j)), x = (c.pt.mapCocone cX).ι.app j x' := by\n  have := isFiltered_of_isCardinalFiltered J κ\n  obtain ⟨y, hy⟩ := (Types.isLimitEquivSections (hc cX.pt)).symm.surjective x\n  obtain ⟨j₀, z, hz⟩ : ∃ (j₀ : J) (z : (k : K) → (F.obj k).obj (X.obj j₀)),\n      ∀ (k : K), y.1 k = (F.obj k).map (cX.ι.app j₀) (z k) := by\n    have H (k : K) := Types.jointly_surjective_of_isColimit (hF k) (y.1 k)\n    let j (k : K) : J := (H k).choose\n    let z (k : K) : (F.obj k).obj (X.obj (j k)) := (H k).choose_spec.choose\n    have hz (k : K) : (F.obj k).map (cX.ι.app (j k)) (z k) = y.1 k :=\n      (H k).choose_spec.choose_spec\n    exact ⟨IsCardinalFiltered.max j (hasCardinalLT_of_hasCardinalLT_arrow hK),\n      fun k ↦ (F.obj k).map (X.map (IsCardinalFiltered.toMax j _ k)) (z k),\n        fun k ↦ by rw [← hz, ← comp_apply, ← Functor.map_comp, cX.w]; rfl⟩\n  obtain ⟨j₁, α, hα⟩ : ∃ (j₁ : J) (α : j₀ ⟶ j₁), ∀ ⦃k k' : K⦄ (φ : k ⟶ k'),\n      (F.obj k').map (X.map α) ((F.map φ).app _ (z k)) =\n        (F.obj k').map (X.map α) (z k') := by\n    have H {k k' : K} (φ : k ⟶ k') :=\n      (Types.FilteredColimit.isColimit_eq_iff' (ht := hF k')\n        (x := (F.map φ).app _ (z k)) (y := z k')).1 (by\n          dsimp at hz ⊢\n          simpa only [← NatTrans.naturality_apply, ← hz] using! y.2 φ)\n    let j {k k' : K} (φ : k ⟶ k') : J := (H φ).choose\n    let g {k k' : K} (φ : k ⟶ k') : j₀ ⟶ j φ := (H φ).choose_spec.choose\n    have hg {k k' : K} (φ : k ⟶ k') :\n        (F.obj k').map (X.map (g φ)) ((F.map φ).app _ (z k)) =\n          (F.obj k').map (X.map (g φ)) (z k') := (H φ).choose_spec.choose_spec\n    obtain ⟨j₁, α, β, hα⟩ : ∃ (j₁ : J) (α : j₀ ⟶ j₁)\n        (β : ∀ ⦃k k' : K⦄ (φ : k ⟶ k'), j φ ⟶ j₁),\n        ∀ ⦃k k' : K⦄ (φ : k ⟶ k'), α = g φ ≫ β φ := by\n      let j'' (f : Arrow K) : J := j f.hom\n      let ψ (f : Arrow K) : j₀ ⟶ IsCardinalFiltered.max j'' hK :=\n        g f.hom ≫ IsCardinalFiltered.toMax j'' hK f\n      refine ⟨IsCardinalFiltered.coeq ψ hK, IsCardinalFiltered.toCoeq ψ hK,\n        fun k k' φ ↦ IsCardinalFiltered.toMax j'' hK φ ≫ IsCardinalFiltered.coeqHom ψ hK,\n        fun k k' φ ↦ ?_⟩\n      simpa [ψ] using! (IsCardinalFiltered.coeq_condition ψ hK (Arrow.mk φ)).symm\n    exact ⟨j₁, α, fun k k' φ ↦ by simp [hα φ, hg]⟩\n  let s : (F ⋙ (evaluation C (Type w')).obj (X.obj j₁)).sections :=\n    { val k := (F.obj k).map (X.map α) (z k)\n      property {k k'} φ := by\n        dsimp\n        rw [NatTrans.naturality_apply, ← hα φ] }\n  refine ⟨j₁, (Types.isLimitEquivSections (hc (X.obj j₁))).symm s, ?_⟩\n  apply (Types.isLimitEquivSections (hc cX.pt)).injective\n  rw [← hy, Equiv.apply_symm_apply]\n  ext k\n  have h₁ := Types.isLimitEquivSections_apply (hc cX.pt) k\n    (c.pt.map (cX.ι.app j₁) ((Types.isLimitEquivSections (hc (X.obj j₁))).symm s))\n  have h₂ := Types.isLimitEquivSections_symm_apply (hc (X.obj j₁)) s k\n  dsimp at h₁ h₂ ⊢\n  rw [h₁, hz, NatTrans.naturality_apply, h₂, ← comp_apply, ← Functor.map_comp, cX.w]\n  rfl\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma injective (j : J) (x₁ x₂ : c.pt.obj (X.obj j))\n    (h : c.pt.map (cX.ι.app j) x₁ = c.pt.map (cX.ι.app j) x₂) :\n    ∃ (j' : J) (α : j ⟶ j'),\n    c.pt.map (X.map α) x₁ = c.pt.map (X.map α) x₂ := by\n  have := isFiltered_of_isCardinalFiltered J κ\n  let y₁ := Types.isLimitEquivSections (hc (X.obj j)) x₁\n  let y₂ := Types.isLimitEquivSections (hc (X.obj j)) x₂\n  have hy₁ : (Types.isLimitEquivSections (hc (X.obj j))).symm y₁ = x₁ := by simp [y₁]\n  have hy₂ : (Types.isLimitEquivSections (hc (X.obj j))).symm y₂ = x₂ := by simp [y₂]\n  have H (k : K) := (Types.FilteredColimit.isColimit_eq_iff' (ht := hF k)\n    (x := y₁.1 k) (y := y₂.1 k)).1 (by\n      simp only [y₁, y₂, Types.isLimitEquivSections_apply]\n      dsimp at h ⊢\n      simp only [← NatTrans.naturality_apply, h])\n  let j₁ (k : K) : J := (H k).choose\n  let f (k : K) : j ⟶ j₁ k := (H k).choose_spec.choose\n  have hf (k : K) : (F.obj k).map (X.map (f k)) (y₁.1 k) =\n      (F.obj k).map (X.map (f k)) (y₂.1 k) :=\n    (H k).choose_spec.choose_spec\n  have hK' := hasCardinalLT_of_hasCardinalLT_arrow hK\n  let ψ (k : K) : j ⟶ IsCardinalFiltered.max j₁ hK' :=\n    f k ≫ IsCardinalFiltered.toMax j₁ hK' k\n  refine ⟨IsCardinalFiltered.coeq ψ hK', IsCardinalFiltered.toCoeq ψ hK', ?_⟩\n  apply (Types.isLimitEquivSections (hc _)).injective\n  ext k\n  simp only [Types.isLimitEquivSections_apply, ← hy₁, ← hy₂]\n  have h₁ := Types.isLimitEquivSections_symm_apply (hc (X.obj j)) y₁ k\n  have h₂ := Types.isLimitEquivSections_symm_apply (hc (X.obj j)) y₂ k\n  dsimp at h₁ h₂ ⊢\n  simp [h₁, h₂, ← IsCardinalFiltered.coeq_condition ψ hK' k, ψ, hf]\n\nend isColimitMapCocone\n\n/-- Auxiliary definition for `isCardinalAccessible_of_isLimit`. -/\nnoncomputable def isColimitMapCocone : IsColimit (c.pt.mapCocone cX) := by\n  have := isFiltered_of_isCardinalFiltered J κ\n  apply Types.FilteredColimit.isColimitOf'\n  · exact isColimitMapCocone.surjective c hc κ hK cX hF\n  · exact isColimitMapCocone.injective c hc κ hK cX hF\n\nend\n\nend Limits\n\nend Accessible\n\nlemma isCardinalAccessible_of_isLimit {K : Type u'} [Category.{v'} K] {F : K ⥤ C ⥤ Type w'}\n    (c : Cone F) (hc : IsLimit c) (κ : Cardinal.{w}) [Fact κ.IsRegular]\n    [HasLimitsOfShape K (Type w')] (hK : HasCardinalLT (Arrow K) κ)\n    [∀ k, (F.obj k).IsCardinalAccessible κ] :\n    c.pt.IsCardinalAccessible κ where\n  preservesColimitOfShape {J _ _} := ⟨fun {X} ↦ ⟨fun {cX} hcX ↦ by\n    have := fun k ↦ preservesColimitsOfShape_of_isCardinalAccessible (F.obj k) κ J\n    exact ⟨Accessible.Limits.isColimitMapCocone c\n      (fun Y ↦ isLimitOfPreserves ((evaluation C (Type w')).obj Y) hc) κ hK cX\n      (fun k ↦ isColimitOfPreserves (F.obj k) hcX)⟩⟩⟩\n\nend Functor\n\nset_option backward.defeqAttrib.useBackward true in\n/-- In case `C` is locally `w`-small, use `isCardinalPresentable_of_isColimit`. -/\nlemma isCardinalPresentable_of_isColimit'\n    {K : Type u'} [Category.{v'} K] {Y : K ⥤ C}\n    (c : Cocone Y) (hc : IsColimit c) (κ : Cardinal.{w}) [Fact κ.IsRegular]\n    [HasLimitsOfShape Kᵒᵖ (Type v)] (hK : HasCardinalLT (Arrow K) κ)\n    [∀ k, IsCardinalPresentable (Y.obj k) κ] :\n    IsCardinalPresentable c.pt κ := by\n  have (k : Kᵒᵖ) : ((Y.op ⋙ coyoneda).obj k).IsCardinalAccessible κ := by\n    dsimp; infer_instance\n  exact Functor.isCardinalAccessible_of_isLimit\n    (coyoneda.mapCone c.op) (isLimitOfPreserves _ hc.op) κ (by simpa)\n\nset_option backward.defeqAttrib.useBackward true in\nlemma isCardinalPresentable_of_isColimit [LocallySmall.{w} C]\n    {K : Type u'} [Category.{v'} K] [HasLimitsOfShape Kᵒᵖ (Type w)] {Y : K ⥤ C}\n    (c : Cocone Y) (hc : IsColimit c) (κ : Cardinal.{w}) [Fact κ.IsRegular]\n    (hK : HasCardinalLT (Arrow K) κ)\n    [∀ k, IsCardinalPresentable (Y.obj k) κ] :\n    IsCardinalPresentable c.pt κ := by\n  let e := ShrinkHoms.equivalence.{w} C\n  have (k : K) : IsCardinalPresentable ((Y ⋙ e.functor).obj k) κ := by\n    dsimp; infer_instance\n  rw [← isCardinalPresentable_iff_of_isEquivalence c.pt κ e.functor]\n  exact isCardinalPresentable_of_isColimit' _\n    (isColimitOfPreserves e.functor hc) κ hK\n\nvariable (C) in\n\nTarget:\nlemma isClosedUnderColimitsOfShape_isCardinalPresentable [LocallySmall.{w} C]\n    {κ : Cardinal.{w}} [Fact κ.IsRegular]\n    {J : Type u'} [Category.{v'} J] [HasLimitsOfShape Jᵒᵖ (Type w)]\n    (hJ : HasCardinalLT (Arrow J) κ) :\n    (isCardinalPresentable C κ).IsClosedUnderColimitsOfShape J where\n  colimitsOfShape_le :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Presentable","family_id":"isclosedundercolimitsofshape_iscardinalpresentable","file_id":"mathlib/Mathlib/CategoryTheory/Presentable/Limits.lean","sample_id":"f97b91f107eb7b4f9d321215e0984f13daac29b819bde8fd09eea122778dc081"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0b0b9e8a744c1d7e4020f039217d12486b9369dcf62ab6b9ae3cf10e259b4ee9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f90fc72aaab263f72c2cf73ac9538f2e1ea027c3390cce04ffdf4abd7892e4b0","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"aeb71f8e68a66dc07c62978a48769bca8b7ac587e71fccaeb6caa13c22e6a3db","source_sha256":"68dce35114cf22d288a51cfbefe877f5725b4487e27909680b1aae1c2b647510","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [← factorPowSucc_comp_fontaineThetaModPPow n]","hard_negative":true,"metrics":{"chosen_tokens":7,"rejected_tokens":3,"token_jaccard":0.111111,"token_length_ratio":0.428571},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"3144e61f0b57a6d0d1a4eafa252cf17bd4a6b4dece679e96abc46487a19cf89d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.AdicCompletion.Functoriality\npublic import Mathlib.RingTheory.AdicCompletion.RingHom\npublic import Mathlib.RingTheory.Perfectoid.Untilt\npublic import Mathlib.RingTheory.WittVector.TeichmullerSeries\n\nNamespace:\nWittVector\n\nLocal context:\n/-\nCopyright (c) 2025 Jiedong Jiang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jiedong Jiang\n-/\n/-!\n# Fontaine's θ map\nIn this file, we define Fontaine's `θ` map, which is a ring\nhomomorphism from the Witt vector `𝕎 R♭` of the tilt of a perfectoid ring `R`\nto `R` itself. Our definition of `θ` does not require that `R` is perfectoid in the first place.\nWe only need `R` to be `p`-adically complete.\n\n## Main Definitions\n* `fontaineTheta` : Fontaine's θ map, which is a ring homomorphism from `𝕎 R♭` to `R`.\n\n## Main Theorems\n* `fontaineTheta_teichmuller` : `θ([x])` is the untilt of `x`.\n* `fontaineTheta_surjective` : Fontaine's θ map is surjective.\n\n## TODO\nEstablish that our definition (explicit construction of `θ mod p ^ n`) agrees with the\ndeformation-theoretic approach via the cotangent complex, as in\n[Bhatt, *Lecture notes for a class on perfectoid spaces*.\nRemark 6.1.7](https://www.math.ias.edu/~bhatt/teaching/mat679w17/lectures.pdf).\n\n## Tags\nFontaine's theta map, perfectoid theory, p-adic Hodge theory\n\n## Reference\n\n* [Fontaine, *Sur Certains Types de Représentations p-Adiques du Groupe de Galois d'un Corps Local;\n  Construction d'un Anneau de Barsotti-Tate*][fontaine1982certains]\n* [Fontaine, *Le corps des périodes p-adiques*][fontaine1994corps]\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen Ideal Quotient PreTilt WittVector\n\nnoncomputable section\n\nvariable {R : Type u} [CommRing R] {p : ℕ} [Fact p.Prime]\n\nlocal notation \"𝕎 \" A:100 => WittVector p A\nlocal notation A \"♭\" => PreTilt A p\nlocal notation3 \"𝔭\" => span {(p : R)}\n\nnamespace WittVector\n\n/-!\n## θ as a ring homomorphism\nLet `𝔭` denote the ideal of `R` generated by the prime number `p`. In this section, we first\ndefine the ring homomorphism `fontaineThetaModPPow : 𝕎 R♭ →+* R ⧸ 𝔭 ^ (n + 1)`.\nThen we show they are compatible with each other and lift to a\nring homomorphism `fontaineTheta : 𝕎 R♭ →+* R`.\n\nTo prove this, we define `fontaineThetaModPPow` as a composition of the following ring\nhomomorphisms.\n\n`𝕎 R♭ --𝕎(Frob^-n)-> 𝕎 R♭ --𝕎(coeff 0)-> 𝕎(R/𝔭) --gh_n-> R/𝔭^(n+1)`\n\nHere, the ring map `gh_n` fits in the following diagram.\n\n```\n𝕎(R)  --ghost_n->   R\n|                   |\nv                   v\n𝕎(R/𝔭) --gh_n-> R/𝔭^(n+1)\n```\n-/\n\ntheorem ker_map_le_ker_mk_comp_ghostComponent (n : ℕ) :\n    RingHom.ker (WittVector.map (Ideal.Quotient.mk 𝔭)) ≤\n    RingHom.ker (((Ideal.Quotient.mk (𝔭 ^ (n + 1)))).comp\n    (WittVector.ghostComponent (p := p) n)) := by\n  intro x\n  simp only [RingHom.mem_ker, map_eq_zero_iff, RingHom.comp_apply]\n  intro h\n  simp only [ghostComponent]\n  apply_fun Ideal.quotEquivOfEq (Ideal.span_singleton_pow _ (n + 1))\n  simp only [RingHom.coe_comp, Function.comp_apply, Pi.evalRingHom_apply, ghostMap_apply,\n    quotEquivOfEq_mk, map_zero]\n  simp only [eq_zero_iff_dvd] at h ⊢\n  exact pow_dvd_ghostComponent_of_dvd_coeff (fun _ _ ↦ h _)\n\n/--\nThe lift ring map `gh_n : 𝕎(R/𝔭) →+* R/𝔭^(n+1)` of the `n`-th ghost component\n`𝕎(R) →+* R` along the surjective ring map `𝕎(R) →+* 𝕎(R/𝔭)`.\n-/\ndef ghostComponentModPPow (n : ℕ) : 𝕎 (R ⧸ 𝔭) →+* R ⧸ 𝔭 ^ (n + 1) :=\n  RingHom.liftOfSurjective (WittVector.map (Ideal.Quotient.mk 𝔭))\n    (map_surjective _ Ideal.Quotient.mk_surjective) ⟨((Ideal.Quotient.mk (𝔭 ^ (n + 1)))).comp\n      (WittVector.ghostComponent n), ker_map_le_ker_mk_comp_ghostComponent n⟩\n\n@[simp]\ntheorem ghostComponentModPPow_map_mk (n : ℕ) (x : 𝕎 R) :\n    ghostComponentModPPow n (WittVector.map (Ideal.Quotient.mk 𝔭) x) =\n    WittVector.ghostComponent n x :=\n  RingHom.liftOfSurjective_comp_apply ..\n\n@[simp]\ntheorem quotEquivOfEq_ghostComponentModPPow (x : 𝕎 (R ⧸ 𝔭)) (h : 𝔭 ^ (0 + 1) = 𝔭) :\n    quotEquivOfEq h (ghostComponentModPPow 0 x) = ghostComponent 0 x := by\n  obtain ⟨y, hy⟩ := map_surjective _ Ideal.Quotient.mk_surjective x\n  simp [← hy, ghostComponent_apply]\n\nvariable [Fact ¬IsUnit (p : R)] [IsAdicComplete (span {(p : R)}) R]\n-- local notation 𝔭 does not work in [IsAdicComplete (span {(p : R)}) R]\n\n@[simp]\ntheorem ghostComponentModPPow_teichmuller_coeff (n : ℕ) (x : R♭) :\n    ghostComponentModPPow n (teichmuller p (PreTilt.coeff n x)) =\n    Ideal.Quotient.mk (𝔭 ^ (n + 1)) x.untilt := by\n  simpa using ghostComponentModPPow_map_mk n\n    (teichmuller p ((((_root_.frobeniusEquiv _ p).symm ^ n) x).untilt))\n\nvariable (R p) in\n/--\nThe Fontaine's theta map modulo `p^(n+1)`.\nIt is the composition of the following ring homomorphisms.\n`𝕎 R♭ --𝕎(Frob^-n)-> 𝕎 R♭ --𝕎(coeff 0)-> 𝕎(R/p) --gh_n-> R/p^(n+1)`\n-/\ndef fontaineThetaModPPow (n : ℕ) : 𝕎 R♭ →+* R ⧸ 𝔭 ^ (n + 1) :=\n  (ghostComponentModPPow n).comp (((WittVector.map (PreTilt.coeff 0))).comp\n    (WittVector.map ((_root_.frobeniusEquiv (R♭) p).symm ^ n : R♭ →+* R♭)))\n\n@[simp]\ntheorem fontaineThetaModPPow_teichmuller (n : ℕ) (x : R♭) :\n    fontaineThetaModPPow R p n (teichmuller p x) = Ideal.Quotient.mk _ x.untilt := by\n  simp [fontaineThetaModPPow]\n\ntheorem factorPowSucc_comp_fontaineThetaModPPow (n : ℕ) :\n    (factorPowSucc _ _).comp (fontaineThetaModPPow R p (n + 1)) = fontaineThetaModPPow R p n := by\n  apply eq_of_apply_teichmuller_eq ((factorPowSucc _ _).comp (fontaineThetaModPPow R p (n + 1)))\n    (fontaineThetaModPPow R p n)\n  · use n + 1\n    have : p = Ideal.Quotient.mk (𝔭 ^ (n + 1)) p := by\n      simp [map_natCast]\n    rw [this, ← map_pow, Ideal.Quotient.eq_zero_iff_mem]\n    exact Ideal.pow_mem_pow (mem_span_singleton_self _) _\n  simp [fontaineThetaModPPow]\n\nTarget:\ntheorem factorPowSucc_fontaineThetaModPPow_eq (n : ℕ) (x : 𝕎 R♭) :\n    factorPowSucc _ _ ((fontaineThetaModPPow R p (n + 1)) x) = fontaineThetaModPPow R p n x :=\n\nProof body:\n","rejected":"by\n  exact factorPowSucc_fontaineThetaModPPow_eq","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"47390c68f04a5c4852b50a19d0e6f80a4c3c0085c36161e1fef819687422d39f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Perfectoid","family_id":"factorpowsucc_fontainethetamodppow_eq","file_id":"mathlib/Mathlib/RingTheory/Perfectoid/FontaineTheta.lean","sample_id":"aeb71f8e68a66dc07c62978a48769bca8b7ac587e71fccaeb6caa13c22e6a3db"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8855b3aa1fcebcd1bfc8f7c993730be08249feb4e40c04eec6356b1306bf4c78","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ced9a8be21b100f92f5cf1865633692067c84047c80974cbc15b46518f499a9c","source_sha256":"7fb54be1edb74454dd74d15ac42b7739b2a4aa95bc5649191f339f2298801887","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases o₁ <;> rfl","hard_negative":true,"metrics":{"chosen_tokens":8,"rejected_tokens":8,"token_jaccard":0.071429,"token_length_ratio":1.0},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"31df3b675bb684e4f22e4199c419eb2e934c7c06b9bc949812b1d50a52940e3a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Ordering.Basic\npublic import Mathlib.Order.Defs.Unbundled\n\nNamespace:\nOrdering\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Some `Ordering` lemmas\n-/\n\npublic section\n\nuniverse u\n\nnamespace Ordering\n\n@[simp]\ntheorem ite_eq_lt_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.lt) = if c then a = Ordering.lt else b = Ordering.lt := by\n  by_cases c <;> simp [*]\n\n@[simp]\ntheorem ite_eq_eq_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.eq) = if c then a = Ordering.eq else b = Ordering.eq := by\n  by_cases c <;> simp [*]\n\n@[simp]\ntheorem ite_eq_gt_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.gt) = if c then a = Ordering.gt else b = Ordering.gt := by\n  by_cases c <;> simp [*]\n\n@[simp]\n\nTarget:\nlemma dthen_eq_then (o₁ o₂ : Ordering) : o₁.dthen (fun _ => o₂) = o₁.then o₂ :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"d97c118a51f4d94cbaa94d739d9fc739c65058be6992d3b1f3a09e87fcba875c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Ordering","family_id":"dthen_eq_then","file_id":"mathlib/Mathlib/Data/Ordering/Lemmas.lean","sample_id":"ced9a8be21b100f92f5cf1865633692067c84047c80974cbc15b46518f499a9c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f5148887f1a7dfc8799d4c5143dcee9d63a0a776aa7b9be04cad5dfcce2d500a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b53835cfef00f711bf8e92a6529c08ab65e2ebdbf08f272d243eb3a0265fab0b","source_sha256":"ce72ff57b0bb6ff2d5bfc5f2bf6fafec557312ecbed5c6bd326819562f53269d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun {U} hU ↦ ?_⟩\n  exact .of_comp (a := f.s₀) (𝒱.f ·) (f.h₀ ·)\n    (fun _ ↦ Scheme.Hom.continuous _) (fun i ↦ funext <| by simp [← Scheme.Hom.comp_apply])\n    (fun _ ↦ Scheme.Hom.continuous _) U.2 (hU.isCompactOpenCovered 𝒱)","hard_negative":false,"metrics":{"chosen_tokens":85,"rejected_tokens":2,"token_jaccard":0.025,"token_length_ratio":0.023529},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"31efb9aff66a5f087738cd23eea037915a00f12d873ecc4c6a98f9cb124b8488","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.Affine\npublic import Mathlib.AlgebraicGeometry.Properties\npublic import Mathlib.AlgebraicGeometry.PullbackCarrier\npublic import Mathlib.Topology.Sets.CompactOpenCovered\n\nNamespace:\nAlgebraicGeometry.QuasiCompactCover\n\nLocal context:\n/-\nCopyright (c) 2025 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten\n-/\n/-!\n# Quasi-compact covers\n\nA cover of a scheme is quasi-compact if every affine open of the base can be covered\nby a finite union of images of quasi-compact opens of the components.\n\nThis is used to define the fpqc (faithfully flat, quasi-compact) topology, where covers are given by\nflat covers that are quasi-compact.\n-/\n\n@[expose] public section\n\nuniverse w' w u v\n\nopen CategoryTheory Limits MorphismProperty TopologicalSpace.Opens AlgebraicGeometry\n\nnamespace AlgebraicGeometry\n\nvariable {S : Scheme.{u}}\n\n/--\nA cover of a scheme is quasi-compact if every affine open of the base can be covered\nby a finite union of images of quasi-compact opens of the components.\n-/\n@[stacks 022B, mk_iff]\nclass QuasiCompactCover (𝒰 : PreZeroHypercover.{v} S) : Prop where\n  isCompactOpenCovered_of_isAffineOpen {U : S.Opens} (hU : IsAffineOpen U) :\n    IsCompactOpenCovered (𝒰.f ·) (U : Set S)\n\nvariable (𝒰 : PreZeroHypercover.{v} S)\n\nlemma IsAffineOpen.isCompactOpenCovered [QuasiCompactCover 𝒰] {U : S.Opens} (hU : IsAffineOpen U) :\n    IsCompactOpenCovered (𝒰.f ·) (U : Set S) :=\n  QuasiCompactCover.isCompactOpenCovered_of_isAffineOpen hU\n\nnamespace QuasiCompactCover\n\nlemma isCompactOpenCovered_of_isCompact [QuasiCompactCover 𝒰]\n    {U : S.Opens} (hU : IsCompact (U : Set S)) :\n    IsCompactOpenCovered (𝒰.f ·) (U : Set S) := by\n  obtain ⟨Us, hUs, hUf, hUc⟩ := S.isBasis_affineOpens.exists_finite_of_isCompact hU\n  refine .of_biUnion_eq_of_finite (SetLike.coe '' Us) (by simp_all) (hUf.image _) ?_\n  simpa using fun t ht ↦ IsAffineOpen.isCompactOpenCovered 𝒰 (hUs ht)\n\nvariable {𝒰 : PreZeroHypercover.{v} S} {K : Precoverage Scheme.{u}}\n\nvariable (𝒰) in\nlemma exists_isAffineOpen_of_isCompact [QuasiCompactCover 𝒰] {U : S.Opens}\n    (hU : IsCompact (U : Set S)) :\n    ∃ (n : ℕ) (f : Fin n → 𝒰.I₀) (V : ∀ i, (𝒰.X (f i)).Opens),\n      (∀ i, IsAffineOpen (V i)) ∧\n      ⋃ i, 𝒰.f (f i) '' (V i) = U := by\n  obtain ⟨n, a, V, ha, heq⟩ := (isCompactOpenCovered_of_isCompact 𝒰 hU).exists_mem_of_isBasis\n    (fun i ↦ (𝒰.X i).isBasis_affineOpens) (fun _ _ h ↦ h.isCompact)\n  exact ⟨n, a, V, ha, heq⟩\n\n/-- If the component maps of `𝒰` are open, `𝒰` is quasi-compact. This in particular\napplies if `K` is the fppf topology (i.e., flat and of finite presentation) and hence in\nparticular for étale and Zariski covers. -/\n@[stacks 022C]\nlemma of_isOpenMap {𝒰 : S.Cover K} [Scheme.JointlySurjective K] (h : ∀ i, IsOpenMap (𝒰.f i)) :\n    QuasiCompactCover 𝒰.toPreZeroHypercover where\n  isCompactOpenCovered_of_isAffineOpen {U} hU := .of_isOpenMap\n    (fun i ↦ (𝒰.f i).continuous) h (fun x _ ↦ ⟨𝒰.idx x, 𝒰.covers x⟩) U.2 hU.isCompact\n\n/-- Any open cover is quasi-compact. -/\ninstance (𝒰 : S.OpenCover) : QuasiCompactCover 𝒰.toPreZeroHypercover :=\n  of_isOpenMap fun i ↦ (𝒰.f i).isOpenEmbedding.isOpenMap\n\n/-- If `𝒱` is a refinement of `𝒰` such that `𝒱` is quasicompact, also `𝒰` is quasicompact. -/\n@[stacks 03L8]\n\nTarget:\nlemma of_hom {𝒱 : PreZeroHypercover.{w'} S} (f : 𝒱.Hom 𝒰) [QuasiCompactCover 𝒱] :\n    QuasiCompactCover 𝒰 :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Cover","family_id":"of_hom","file_id":"mathlib/Mathlib/AlgebraicGeometry/Cover/QuasiCompact.lean","sample_id":"b53835cfef00f711bf8e92a6529c08ab65e2ebdbf08f272d243eb3a0265fab0b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"59b2a49c3792a452986527f1fd02d5e769f33e00ec8381ee8f5c72d68ad2ecd4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"99f1e37e3ec119044c219636cfcc8ffdd76a19fdac55cff796f9c655385230aa","source_sha256":"6b463b31746b56f6d89e4d588151f63717a66001d29dfed4ab7ccf9fc8571299","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [le_antisymm_iff, h x y, h y x]\n\nvariable [IsDomain S] [Nontrivial R]","hard_negative":false,"metrics":{"chosen_tokens":22,"rejected_tokens":2,"token_jaccard":0.066667,"token_length_ratio":0.090909},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"321f6b67827ad4d0c42945f305ba192341f6ba75030b8682c4ecdd08245d4199","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Field.WithAbs\npublic import Mathlib.Analysis.SpecialFunctions.Pow.Real\n\nNamespace:\nAbsoluteValue\n\nLocal context:\n/-\nCopyright (c) 2025 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Equivalence of real-valued absolute values\n\nTwo absolute values `v₁, v₂ : AbsoluteValue R ℝ` are *equivalent* if there exists a\npositive real number `c` such that `v₁ x ^ c = v₂ x` for all `x : R`.\n-/\n\n@[expose] public section\n\nnamespace AbsoluteValue\n\nsection OrderedSemiring\n\nvariable {R : Type*} [Semiring R] {S : Type*} [Semiring S] [PartialOrder S]\n  (v w : AbsoluteValue R S)\n\n/-- Two absolute values `v` and `w` are *equivalent* if `v x ≤ v y` precisely when\n`w x ≤ w y`.\n\nNote that for real absolute values this condition is equivalent to the existence of a positive\nreal number `c` such that `v x ^ c = w x` for all `x`. See\n`AbsoluteValue.isEquiv_iff_exists_rpow_eq`. -/\ndef IsEquiv : Prop := ∀ x y, v x ≤ v y ↔ w x ≤ w y\n\ntheorem IsEquiv.refl : v.IsEquiv v := fun _ _ ↦ .rfl\n\nvariable {v w}\n\ntheorem IsEquiv.rfl : v.IsEquiv v := fun _ _ ↦ .rfl\n\ntheorem IsEquiv.symm (h : v.IsEquiv w) : w.IsEquiv v := fun _ _ ↦ (h _ _).symm\n\ntheorem IsEquiv.trans {u : AbsoluteValue R S} (h₁ : v.IsEquiv w)\n    (h₂ : w.IsEquiv u) : v.IsEquiv u := fun _ _ ↦ (h₁ _ _).trans (h₂ _ _)\n\ninstance : Setoid (AbsoluteValue R S) where\n  r := IsEquiv\n  iseqv := {\n    refl := .refl\n    symm := .symm\n    trans := .trans\n  }\n\ntheorem IsEquiv.le_iff_le (h : v.IsEquiv w) {x y : R} : v x ≤ v y ↔ w x ≤ w y := h ..\n\ntheorem IsEquiv.lt_iff_lt (h : v.IsEquiv w) {x y : R} : v x < v y ↔ w x < w y :=\n  lt_iff_lt_of_le_iff_le' (h y x) (h x y)\n\nTarget:\ntheorem IsEquiv.eq_iff_eq (h : v.IsEquiv w) {x y : R} : v x = v y ↔ w x = w y :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/AbsoluteValue","family_id":"isequiv","file_id":"mathlib/Mathlib/Analysis/AbsoluteValue/Equivalence.lean","sample_id":"99f1e37e3ec119044c219636cfcc8ffdd76a19fdac55cff796f9c655385230aa"},"split":"train"}
{"binding":{"chosen_proof_sha256":"59cc5684ec22bcbe1a47c40b86884f4ebd0253cacb27e5dc09b1539a4441b454","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ca6497bd2a5f61079765826c69fc7379d1387ee73473e363675e959db77191cc","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dbf6ffda00f48d6763a755f6b1baea6be9639bc7a19fe67ae570fed274326119","source_sha256":"f3eaf728d7cb8f06fa2b67e9feb8de9fb95bffa5aec7e165f2a4546b07059ca7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa [ChainComplex.of.d] using ((inhomogeneousChains A).d_comp_d (n + 2) (n + 1) n)","hard_negative":true,"metrics":{"chosen_tokens":29,"rejected_tokens":3,"token_jaccard":0.05,"token_length_ratio":0.103448},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"322b38c853613ab119291a5ddb59093328d02f365a45e8dfea6c238874a66e71","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.ConcreteCategory\npublic import Mathlib.RepresentationTheory.Coinvariants\npublic import Mathlib.RepresentationTheory.Homological.Resolution\npublic import Mathlib.Tactic.CategoryTheory.Slice\npublic import Mathlib.CategoryTheory.Abelian.LeftDerived\n\nNamespace:\ngroupHomology\n\nLocal context:\n/-\nCopyright (c) 2025 Amelia Livingston. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Amelia Livingston\n-/\n/-!\n# The group homology of a `k`-linear `G`-representation\n\nLet `k` be a commutative ring and `G` a group. This file defines the group homology of\n`A : Rep k G` to be the homology of the complex\n$$\\dots \\to \\bigoplus_{G^2} A \\to \\bigoplus_{G^1} A \\to \\bigoplus_{G^0} A$$\nwith differential $d_n$ sending $a\\cdot (g_0, \\dots, g_n)$ to\n$$\\rho(g_0^{-1})(a)\\cdot (g_1, \\dots, g_n)$$\n$$+ \\sum_{i = 0}^{n - 1}(-1)^{i + 1}a\\cdot (g_0, \\dots, g_ig_{i + 1}, \\dots, g_n)$$\n$$+ (-1)^{n + 1}a\\cdot (g_0, \\dots, g_{n - 1})$$ (where `ρ` is the representation attached to `A`).\n\nWe have a `k`-linear isomorphism\n$\\bigoplus_{G^n} A \\cong (A \\otimes_k \\left(\\bigoplus_{G^n} k[G]\\right))_G$ given by\n`Rep.coinvariantsTensorFreeLEquiv`. If we conjugate the $n$th differential in $(A \\otimes_k P)_G$\nby this isomorphism, where `P` is the bar resolution of `k` as a trivial `k`-linear\n`G`-representation, then the resulting map agrees with the differential $d_n$ defined\nabove, a fact we prove.\n\nHence our $d_n$ squares to zero, and we get\n$\\mathrm{H}_n(G, A) \\cong \\mathrm{Tor}_n(A, k),$ where $\\mathrm{Tor}$ is defined by deriving the\nsecond argument of the functor $(A, B) \\mapsto (A \\otimes_k B)_G.$\n\nTo talk about homology in low degree, the file\n`Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean` provides API specialized to\n`H₀`, `H₁`, `H₂`.\n\n## Main definitions\n\n* `Rep.Tor k G n`: the left-derived functors given by deriving the second argument of\n  $(A, B) \\mapsto (A \\otimes_k B)_G$.\n* `groupHomology.inhomogeneousChains A`: a complex whose objects are\n  $\\bigoplus_{G^n} A$ and whose homology is the group homology $\\mathrm{H}_n(G, A).$\n* `groupHomology.inhomogeneousChainsIso A`: an isomorphism between the above two complexes.\n* `groupHomology A n`: this is $\\mathrm{H}_n(G, A),$ defined as the $n$th homology of the\n  second complex, `inhomogeneousChains A`.\n* `groupHomologyIsoTor A n`: an isomorphism $\\mathrm{H}_n(G, A) \\cong \\mathrm{Tor}_n(A, k)$\n  induced by `inhomogeneousChainsIso A`.\n\n## Implementation notes\n\nGroup homology is typically stated for `G`-modules, or equivalently modules over the group ring\n`ℤ[G].` However, `ℤ` can be generalized to any commutative ring `k`, which is what we use.\nMoreover, we express `k[G]`-module structures on a module `k`-module `A` using the `Rep` definition.\nWe avoid using instances `Module k[G] A` so that we do not run into possible scalar action diamonds.\n\nNote that the existing definition of `Tor` in `Mathlib.CategoryTheory.Monoidal.Tor` is for monoidal\ncategories, and the bifunctor we need to derive here maps to `ModuleCat k`. Hence we define\n`Rep.Tor k G n` by instead left-deriving the second argument of `Rep.coinvariantsTensor k G`:\n$(A, B) \\mapsto (A \\otimes_k B)_G$. The functor `Rep.coinvariantsTensor k G` is naturally\nisomorphic to the functor sending `A, B` to `A ⊗[k[G]] B`, where we give `A` the `k[G]ᵐᵒᵖ`-module\nstructure defined by `g • a := A.ρ g⁻¹ a`, but currently mathlib's `TensorProduct` is only defined\nfor commutative rings.\n\n## TODO\n\n* Upgrading `groupHomologyIsoTor` to an isomorphism of derived functors.\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nuniverse u v w\n\nopen CategoryTheory CategoryTheory.Limits\n\nvariable (k G : Type u) [CommRing k] [Group G]\n\nopen MonoidalCategory Representation Finsupp\n\nsection Tor\n\nvariable {k G} in\n/-- Given `A : Rep k G` and a chain complex `P` in `Rep k G`, this is the chain complex whose\n`n`th object is `(A ⊗ Pₙ)_G`. -/\nabbrev HomologicalComplex.coinvariantsTensorObj {α : Type*} [AddRightCancelSemigroup α] [One α]\n    (A : Rep k G) (P : ChainComplex (Rep k G) α) :\n    ChainComplex (ModuleCat k) α :=\n  (((Rep.coinvariantsTensor k G).obj A).mapHomologicalComplex _).obj P\n\nnamespace Rep\n\n/-- The left-derived functors given by deriving the second argument of `A, B ↦ (A ⊗[k] B)_G`. -/\n@[simps]\ndef Tor (n : ℕ) : Rep k G ⥤ Rep k G ⥤ ModuleCat k where\n  obj X := Functor.leftDerived ((coinvariantsTensor k G).obj X) n\n  map f := NatTrans.leftDerived ((coinvariantsTensor k G).map f) n\n\nvariable {k G} (A : Rep.{w} k G)\n\n/-- `Tor` can be computed using a projective resolution. -/\nabbrev torIso (A : Rep k G) {B : Rep k G} (P : ProjectiveResolution B) (n : ℕ) :\n    ((Rep.Tor k G n).obj A).obj B ≅ (P.complex.coinvariantsTensorObj A).homology n :=\n  P.isoLeftDerivedObj _ n\n\n/-- The higher `Tor` groups for `X` and `Y` are zero if `Y` is projective. -/\nlemma isZero_Tor_succ_of_projective (X Y : Rep k G) [Projective Y] (n : ℕ) :\n    IsZero (((Tor k G (n + 1)).obj X).obj Y) :=\n  Functor.isZero_leftDerived_obj_projective_succ ..\n\nend Rep\nend Tor\n\nnamespace groupHomology\n\nopen Rep Finsupp\n\nvariable {k G : Type u} [CommRing k] [Group G] (A : Rep.{u} k G) (n : ℕ)\n\nnamespace inhomogeneousChains\n\n/-- The differential in the complex of inhomogeneous chains used to calculate group homology. -/\ndef d : ModuleCat.of k ((Fin (n + 1) → G) →₀ A) ⟶ ModuleCat.of k ((Fin n → G) →₀ A) :=\n  ModuleCat.ofHom <| lsum (R := k) k fun g => lsingle (fun i => g i.succ) ∘ₗ A.ρ (g 0)⁻¹ +\n    Finset.univ.sum fun j : Fin (n + 1) =>\n      (-1 : k) ^ ((j : ℕ) + 1) • lsingle (Fin.contractNth j (· * ·) g)\n\nvariable {A n} in\n@[simp]\ntheorem d_single (n : ℕ) (g : Fin (n + 1) → G) (a : A) :\n    d A n (single g a) = single (fun i => g i.succ) (A.ρ (g 0)⁻¹ a) +\n      Finset.univ.sum fun j : Fin (n + 1) =>\n        (-1 : k) ^ ((j : ℕ) + 1) • single (Fin.contractNth j (· * ·) g) a := by\n  simp [d]\n\nopen ModuleCat.MonoidalCategory\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\ntheorem d_eq [DecidableEq G] :\n    d A n = (coinvariantsTensorFreeLEquiv A (Fin (n + 1) → G)).toModuleIso.inv ≫\n      ((barComplex k G).coinvariantsTensorObj A).d (n + 1) n ≫\n      (coinvariantsTensorFreeLEquiv A (Fin n → G)).toModuleIso.hom := by\n  ext : 3\n  simp [d_single (k := k), TensorProduct.tmul_add, TensorProduct.tmul_sum,\n    barComplex.d_single (k := k)]\n\nend inhomogeneousChains\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Given a `k`-linear `G`-representation `A`, this is the complex of inhomogeneous chains\n$$\\dots \\to \\bigoplus_{G^1} A \\to \\bigoplus_{G^0} A \\to 0$$\nwhich calculates the group homology of `A`. -/\nnoncomputable abbrev inhomogeneousChains :\n    ChainComplex (ModuleCat k) ℕ :=\n  ChainComplex.of (fun n => ModuleCat.of k ((Fin n → G) →₀ A))\n    (fun n => inhomogeneousChains.d A n) fun n => by\n    classical\n    rw [inhomogeneousChains.d_eq, inhomogeneousChains.d_eq]\n    slice_lhs 3 4 => rw [Iso.hom_inv_id]\n    slice_lhs 2 4 => rw [Category.id_comp, ((barComplex k G).coinvariantsTensorObj A).d_comp_d]\n    simp\n\nopen inhomogeneousChains\n\nvariable {A n} in\n@[ext]\ntheorem inhomogeneousChains.ext {M : ModuleCat k} {x y : (inhomogeneousChains A).X n ⟶ M}\n    (h : ∀ g, ModuleCat.ofHom (lsingle g) ≫ x = ModuleCat.ofHom (lsingle g) ≫ y) :\n    x = y := ModuleCat.hom_ext <| lhom_ext' fun g => ModuleCat.hom_ext_iff.1 (h g)\n\ntheorem inhomogeneousChains.d_def (n : ℕ) :\n    (inhomogeneousChains A).d (n + 1) n = d A n := by\n  simp [inhomogeneousChains]\n\nset_option backward.defeqAttrib.useBackward true in\n\nTarget:\ntheorem inhomogeneousChains.d_comp_d :\n    d A (n + 1) ≫ d A n = 0 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_dbf6ffda00f4","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"8e5eabca450ae3df49b2e77cdc620c2a9abbb398837513dd51c3ec8828389c1f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RepresentationTheory/Homological","family_id":"inhomogeneouschains","file_id":"mathlib/Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean","sample_id":"dbf6ffda00f48d6763a755f6b1baea6be9639bc7a19fe67ae570fed274326119"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9c3ded6353915dd072b1066cf2ce119a7132edcb22b959bc51e04f3927cf0477","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5adb15e5706aefda52cadd04c16407f0f1f64e18c9972fb156247c44ba666c48","source_sha256":"69f324bcc2121803a5bee9f9f572821f81fe33ac326f7f940f63d1e7e35d90b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h1 (y : ℝ) : 𝐞 y = fourier 1 (y : UnitAddCircle) := by\n    rw [fourierChar_apply, fourier_coe_apply]\n    push_cast\n    ring_nf\n  simpa only [h1, Int.cast_one, ofReal_one, div_one, mul_one] using hasDerivAt_fourier 1 1 x","hard_negative":false,"metrics":{"chosen_tokens":49,"rejected_tokens":2,"token_jaccard":0.029412,"token_length_ratio":0.040816},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"3263c8990b33b625e5485e8967a750f3aa5d55dc718bd0214523bdf17345bcd0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Calculus.FDeriv.Analytic\npublic import Mathlib.Analysis.Fourier.AddCircle\npublic import Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Calculus.ContDiff.Bounds\nimport Mathlib.Analysis.Calculus.ContDiff.CPolynomial\nimport Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts\nimport Mathlib.Analysis.Calculus.ParametricIntegral\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2024 David Loeffler. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alex Kontorovich, David Loeffler, Heather Macbeth, Sébastien Gouëzel\n-/\n/-!\n# Derivatives of the Fourier transform\n\nIn this file we compute the Fréchet derivative of the Fourier transform of `f`, where `f` is a\nfunction such that both `f` and `v ↦ ‖v‖ * ‖f v‖` are integrable. Here the Fourier transform is\nunderstood as an operator `(V → E) → (W → E)`, where `V` and `W` are normed `ℝ`-vector spaces\nand the Fourier transform is taken with respect to a continuous `ℝ`-bilinear\npairing `L : V × W → ℝ` and a given reference measure `μ`.\n\nWe also investigate higher derivatives: Assuming that `‖v‖^n * ‖f v‖` is integrable, we show\nthat the Fourier transform of `f` is `C^n`.\n\nWe also study in a parallel way the Fourier transform of the derivative, which is obtained by\ntensoring the Fourier transform of the original function with the bilinear form. We also get\nresults for iterated derivatives.\n\nA consequence of these results is that, if a function is smooth and all its derivatives are\nintegrable when multiplied by `‖v‖^k`, then the same goes for its Fourier transform, with\nexplicit bounds.\n\nWe give specialized versions of these results on inner product spaces (where `L` is the scalar\nproduct) and on the real line, where we express the one-dimensional derivative in more concrete\nterms, as the Fourier transform of `-2πI x * f x` (or `(-2πI x)^n * f x` for higher derivatives).\n\n## Main definitions and results\n\nWe introduce two convenience definitions:\n\n* `VectorFourier.fourierSMulRight L f`: given `f : V → E` and `L` a bilinear pairing\n  between `V` and `W`, then this is the function `fun v ↦ -(2 * π * I) (L v ⬝) • f v`,\n  from `V` to `Hom (W, E)`.\n  This is essentially `ContinuousLinearMap.smulRight`, up to the factor `- 2πI` designed to make\n  sure that the Fourier integral of `fourierSMulRight L f` is the derivative of the Fourier\n  integral of `f`.\n* `VectorFourier.fourierPowSMulRight` is the higher-order analogue for higher derivatives:\n  `fourierPowSMulRight L f v n` is informally `(-(2 * π * I))^n (L v ⬝)^n • f v`, in\n  the space of continuous multilinear maps `W [×n]→L[ℝ] E`.\n\nWith these definitions, the statements read as follows, first in a general context\n(arbitrary `L` and `μ`):\n\n* `VectorFourier.hasFDerivAt_fourierIntegral`: the Fourier integral of `f` is differentiable, with\n    derivative the Fourier integral of `fourierSMulRight L f`.\n* `VectorFourier.differentiable_fourierIntegral`: the Fourier integral of `f` is differentiable.\n* `VectorFourier.fderiv_fourierIntegral`: formula for the derivative of the Fourier integral of `f`.\n* `VectorFourier.fourierIntegral_fderiv`: formula for the Fourier integral of the derivative of `f`.\n* `VectorFourier.hasFTaylorSeriesUpTo_fourierIntegral`: under suitable integrability conditions,\n  the Fourier integral of `f` has an explicit Taylor series up to order `N`, given by the Fourier\n  integrals of `fun v ↦ fourierPowSMulRight L f v n`.\n* `VectorFourier.contDiff_fourierIntegral`: under suitable integrability conditions,\n  the Fourier integral of `f` is `C^n`.\n* `VectorFourier.iteratedFDeriv_fourierIntegral`: under suitable integrability conditions,\n  explicit formula for the `n`-th derivative of the Fourier integral of `f`, as the Fourier\n  integral of `fun v ↦ fourierPowSMulRight L f v n`.\n* `VectorFourier.pow_mul_norm_iteratedFDeriv_fourierIntegral_le`: explicit bounds for the `n`-th\n  derivative of the Fourier integral, multiplied by a power function, in terms of corresponding\n  integrals for the original function.\n\nThese statements are then specialized to the case of the usual Fourier transform on\nfinite-dimensional inner product spaces with their canonical Lebesgue measure (covering in\nparticular the case of the real line), replacing the namespace `VectorFourier` by\nthe namespace `Real` in the above statements.\n\nWe also give specialized versions of the one-dimensional real derivative (and iterated derivative)\nin `Real.deriv_fourierIntegral` and `Real.iteratedDeriv_fourierIntegral`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Real Complex MeasureTheory Filter TopologicalSpace\n\nopen scoped FourierTransform Topology ContDiff\n\n-- without this local instance, Lean tries first the instance\n-- `secondCountableTopologyEither_of_right` (whose priority is 100) and takes a very long time to\n-- fail. Since we only use the left instance in this file, we make sure it is tried first.\nattribute [local instance 101] secondCountableTopologyEither_of_left\n\nnamespace Real\n\nTarget:\nlemma hasDerivAt_fourierChar (x : ℝ) : HasDerivAt (𝐞 · : ℝ → ℂ) (2 * π * I * 𝐞 x) x :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Fourier","family_id":"hasderivat_fourierchar","file_id":"mathlib/Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","sample_id":"5adb15e5706aefda52cadd04c16407f0f1f64e18c9972fb156247c44ba666c48"},"split":"train"}
{"binding":{"chosen_proof_sha256":"894118c14dc5e14117bef012c677ec8f7a21a4cdd6ea74ed8d6415838166eb42","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0b5bf10ac555cb918a9c33514f288ff7c5d467ae47c35f515470efcd6f103f3e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a4e28351bd9a0b41f4d5ba3c08dc4215d63a0d8b4b1907903fe2be3c09e51fcd","source_sha256":"82faa569879adae1105c3c846784886561d45f7b1b0c67c1d5374fc07267c247","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases hE : CompleteSpace E; swap\n  · simp [integral, hE, hasSum_zero]\n  rw [integral_tsum (fun i ↦ (hF_int i).1)]\n  · exact (hF_sum.of_norm_bounded fun i ↦ norm_integral_le_integral_norm _).hasSum\n  have (i : ι) : ∫⁻ a, ‖F i a‖ₑ ∂μ = ‖∫ a, ‖F i a‖ ∂μ‖ₑ := by\n    dsimp [enorm]\n    rw [lintegral_coe_eq_integral _ (hF_int i).norm, coe_nnreal_eq, coe_nnnorm,\n      Real.norm_of_nonneg (integral_nonneg (fun a ↦ norm_nonneg (F i a)))]\n    simp only [coe_nnnorm]\n  rw [funext this]\n  exact ENNReal.tsum_coe_ne_top_iff_summable.2 <| NNReal.summable_coe.1 hF_sum.abs","hard_negative":true,"metrics":{"chosen_tokens":144,"rejected_tokens":5,"token_jaccard":0.045455,"token_length_ratio":0.034722},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"32b077fbda3a35ba2c9f49fb35b9c85123357eebdfeba69388e9ae7b391ae6a5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable\npublic import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic\nimport Mathlib.Topology.Algebra.IsUniformGroup.Order\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2019 Zhouhang Zhou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Zhouhang Zhou, Yury Kudryashov, Patrick Massot, Louis (Yiyang) Liu\n-/\n/-!\n# The dominated convergence theorem\n\nThis file collects various results related to the Lebesgue dominated convergence theorem\nfor the Bochner integral.\n\n## Main results\n- `MeasureTheory.tendsto_integral_of_dominated_convergence`:\n  the Lebesgue dominated convergence theorem for the Bochner integral\n- `MeasureTheory.hasSum_integral_of_dominated_convergence`:\n  the Lebesgue dominated convergence theorem for series\n- `MeasureTheory.integral_tsum`, `MeasureTheory.integral_tsum_of_summable_integral_norm`:\n  the integral and `tsum`s commute, if the norms of the functions form a summable series\n- `intervalIntegral.hasSum_integral_of_dominated_convergence`: the Lebesgue dominated convergence\n  theorem for parametric interval integrals\n- `intervalIntegral.continuous_of_dominated_interval`: continuity of the interval integral\n  w.r.t. a parameter\n- `intervalIntegral.continuous_primitive` and friends: primitives of interval integrable\n  measurable functions are continuous\n\n-/\n\npublic section\n\nopen MeasureTheory Metric\n\n/-!\n## The Lebesgue dominated convergence theorem for the Bochner integral\n-/\nsection DominatedConvergenceTheorem\n\nopen Set Filter TopologicalSpace ENNReal\nopen scoped Topology Interval\n\nnamespace MeasureTheory\n\nvariable {α E G : Type*}\n  [NormedAddCommGroup E] [NormedSpace ℝ E]\n  [NormedAddCommGroup G] [NormedSpace ℝ G]\n  {m : MeasurableSpace α} {μ : Measure α}\n\n/-- **Lebesgue dominated convergence theorem** provides sufficient conditions under which almost\n  everywhere convergence of a sequence of functions implies the convergence of their integrals.\n  We could weaken the condition `bound_integrable` to require `HasFiniteIntegral bound μ` instead\n  (i.e. not requiring that `bound` is measurable), but in all applications proving integrability\n  is easier. -/\ntheorem tendsto_integral_of_dominated_convergence {F : ℕ → α → G} {f : α → G} (bound : α → ℝ)\n    (F_measurable : ∀ n, AEStronglyMeasurable (F n) μ) (bound_integrable : Integrable bound μ)\n    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)\n    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :\n    Tendsto (fun n => ∫ a, F n a ∂μ) atTop (𝓝 <| ∫ a, f a ∂μ) := by\n  simp only [integral_eq_setToFun]\n  exact tendsto_setToFun_of_dominated_convergence (dominatedFinMeasAdditive_weightedSMul μ)\n    bound F_measurable bound_integrable h_bound h_lim\n\n/-- Lebesgue dominated convergence theorem for filters with a countable basis -/\ntheorem tendsto_integral_filter_of_dominated_convergence {ι} {l : Filter ι} [l.IsCountablyGenerated]\n    {F : ι → α → G} {f : α → G} (bound : α → ℝ) (hF_meas : ∀ᶠ n in l, AEStronglyMeasurable (F n) μ)\n    (h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (bound_integrable : Integrable bound μ)\n    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a))) :\n    Tendsto (fun n => ∫ a, F n a ∂μ) l (𝓝 <| ∫ a, f a ∂μ) := by\n  simp only [integral_eq_setToFun]\n  exact tendsto_setToFun_filter_of_dominated_convergence (dominatedFinMeasAdditive_weightedSMul μ)\n    bound hF_meas h_bound bound_integrable h_lim\n\n/-- Lebesgue dominated convergence theorem for series. -/\ntheorem hasSum_integral_of_dominated_convergence {ι} [Countable ι] {F : ι → α → G} {f : α → G}\n    (bound : ι → α → ℝ) (hF_meas : ∀ n, AEStronglyMeasurable (F n) μ)\n    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound n a)\n    (bound_summable : ∀ᵐ a ∂μ, Summable fun n => bound n a)\n    (bound_integrable : Integrable (fun a => ∑' n, bound n a) μ)\n    (h_lim : ∀ᵐ a ∂μ, HasSum (fun n => F n a) (f a)) :\n    HasSum (fun n => ∫ a, F n a ∂μ) (∫ a, f a ∂μ) := by\n  simp only [integral_eq_setToFun]\n  exact hasSum_setToFun_of_dominated_convergence _ bound hF_meas h_bound bound_summable\n    bound_integrable h_lim\n\ntheorem integral_tsum {ι} [Countable ι] {f : ι → α → G} (hf : ∀ i, AEStronglyMeasurable (f i) μ)\n    (hf' : ∑' i, ∫⁻ a : α, ‖f i a‖ₑ ∂μ ≠ ∞) :\n    ∫ a, ∑' i, f i a ∂μ = ∑' i, ∫ a, f i a ∂μ := by\n  by_cases hG : CompleteSpace G; swap\n  · simp [integral, hG]\n  simp only [integral_eq_setToFun]\n  exact setToFun_tsum _ hf hf'\n\nTarget:\nlemma hasSum_integral_of_summable_integral_norm {ι} [Countable ι] {F : ι → α → E}\n    (hF_int : ∀ i : ι, Integrable (F i) μ) (hF_sum : Summable fun i ↦ ∫ a, ‖F i a‖ ∂μ) :\n    HasSum (∫ a, F · a ∂μ) (∫ a, (∑' i, F i a) ∂μ) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_a4e28351bd9a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"9c93dea653893b9177b06b4187286709785db390a7ecf150278e352d356341cb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Integral","family_id":"hassum_integral_of_summable_integral_norm","file_id":"mathlib/Mathlib/MeasureTheory/Integral/DominatedConvergence.lean","sample_id":"a4e28351bd9a0b41f4d5ba3c08dc4215d63a0d8b4b1907903fe2be3c09e51fcd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a27f59db44cea34ebdac75d20e4acc76ed165c06147e346a2eafa82d1ac01d9a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5b2ad4056a63f71ffd19b6edff0a4e71e89bed5e4d41dbbbaadf0a9e6a420870","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3236ac1bb6f48e9d26b20fb63c3f2eb332af2bf7a5e2f318caa01e95dce53a89","source_sha256":"3ebef0ce1ff4e8804552a350539c21f4e3856ca570ca14d4062fa9367e8ccafd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨β, _, sgn, g, hg, hβ, hf⟩ := exists_signed_sum s f\n  refine\n    ⟨β ⊕ (Fin (n - ∑ i ∈ s, (f i).natAbs)), inferInstance, Sum.elim sgn 0,\n      Sum.elim g (Classical.arbitrary (Fin (n - Finset.sum s fun i => Int.natAbs (f i)) → α)),\n        ?_, by simp [hβ, h], fun a ha => by simp [hf _ ha]⟩\n  rintro (b | b) hb\n  · cases hb (hg _)\n  · rfl","hard_negative":true,"metrics":{"chosen_tokens":126,"rejected_tokens":3,"token_jaccard":0.036364,"token_length_ratio":0.02381},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"333846360b4e1d90250b7e95fdc5a68aafd1196164bdce2babb307a2306b4945","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GroupWithZero.Units.Lemmas\npublic import Mathlib.Algebra.Order.BigOperators.Group.Finset\npublic import Mathlib.Algebra.Order.Ring.Cast\npublic import Mathlib.Data.Fintype.BigOperators\npublic import Mathlib.Data.Sign.Defs\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Eric Rodriguez. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Rodriguez\n-/\n/-!\n# Sign function\n\nThis file defines the sign function for types with zero and a decidable less-than relation, and\nproves some basic theorems about it.\n-/\n\n@[expose] public section\n\nuniverse u\nvariable {α : Type u}\n\nnamespace SignType\n\n/-- Casting `SignType → ℤ → α` is the same as casting directly `SignType → α`. -/\n@[simp, norm_cast]\nlemma intCast_cast {α : Type*} [AddGroupWithOne α] (s : SignType) : ((s : ℤ) : α) = s :=\n  map_cast' _ Int.cast_one Int.cast_zero (@Int.cast_one α _ ▸ Int.cast_neg 1) _\n\ntheorem pow_odd (s : SignType) {n : ℕ} (hn : Odd n) : s ^ n = s := by\n  obtain ⟨k, rfl⟩ := hn\n  rw [pow_add, pow_one, pow_mul, sq]\n  cases s <;> simp\n\ntheorem zpow_odd (s : SignType) {z : ℤ} (hz : Odd z) : s ^ z = s := by\n  obtain rfl | hs := eq_or_ne s 0\n  · rw [zero_zpow]\n    rintro rfl\n    simp at hz\n  obtain ⟨k, rfl⟩ := hz\n  rw [zpow_add₀ hs, zpow_one, zpow_mul, zpow_two]\n  cases s <;> simp\n\nlemma pow_even (s : SignType) {n : ℕ} (hn : Even n) (hs : s ≠ 0) :\n    s ^ n = 1 := by\n  cases s <;> simp_all\n\nlemma zpow_even (s : SignType) {z : ℤ} (hz : Even z) (hs : s ≠ 0) :\n    s ^ z = 1 := by\n  cases s <;> simp_all [Even.neg_one_zpow]\n\n/-- `SignType.cast` as a `MulWithZeroHom`. -/\n@[simps]\ndef castHom {α} [MulZeroOneClass α] [HasDistribNeg α] : SignType →*₀ α where\n  toFun := cast\n  map_zero' := rfl\n  map_one' := rfl\n  map_mul' x y := by cases x <;> cases y <;> simp [zero_eq_zero, pos_eq_one, neg_eq_neg_one]\n\ntheorem univ_eq : (Finset.univ : Finset SignType) = {0, -1, 1} := by\n  decide\n\ntheorem range_eq {α} (f : SignType → α) : Set.range f = {f zero, f neg, f pos} := by\n  classical rw [← Fintype.coe_image_univ, univ_eq]\n  classical simp [Finset.coe_insert]\n\n@[simp, norm_cast] lemma coe_mul {α} [MulZeroOneClass α] [HasDistribNeg α] (a b : SignType) :\n    ↑(a * b) = (a : α) * b :=\n  map_mul SignType.castHom _ _\n\n@[simp, norm_cast] lemma coe_pow {α} [MonoidWithZero α] [HasDistribNeg α] (a : SignType) (k : ℕ) :\n    ↑(a ^ k) = (a : α) ^ k :=\n  map_pow SignType.castHom _ _\n\n@[simp, norm_cast] lemma coe_zpow {α} [GroupWithZero α] [HasDistribNeg α] (a : SignType) (k : ℤ) :\n    ↑(a ^ k) = (a : α) ^ k :=\n  map_zpow₀ SignType.castHom _ _\n\nend SignType\n\nopen SignType\n\nsection OrderedRing\n\n@[simp]\nlemma sign_intCast {α : Type*} [Ring α] [PartialOrder α] [IsOrderedRing α]\n    [Nontrivial α] [DecidableLT α] (n : ℤ) :\n    sign (n : α) = sign n := by\n  simp only [sign_apply, Int.cast_pos, Int.cast_lt_zero]\n\nend OrderedRing\n\nsection LinearOrderedRing\n\nvariable [Ring α] [LinearOrder α] [IsStrictOrderedRing α]\n\ntheorem sign_mul (x y : α) : sign (x * y) = sign x * sign y := by\n  rcases lt_trichotomy x 0 with (hx | hx | hx) <;> rcases lt_trichotomy y 0 with (hy | hy | hy) <;>\n    simp [hx, hy, mul_pos_of_neg_of_neg, mul_neg_of_neg_of_pos, mul_neg_of_pos_of_neg]\n\n@[simp] theorem sign_mul_abs (x : α) : (sign x * |x| : α) = x := by\n  rcases lt_trichotomy x 0 with hx | rfl | hx <;> simp [*, abs_of_pos, abs_of_neg]\n\n@[simp] theorem abs_mul_sign (x : α) : (|x| * sign x : α) = x := by\n  rcases lt_trichotomy x 0 with hx | rfl | hx <;> simp [*, abs_of_pos, abs_of_neg]\n\n@[simp]\ntheorem sign_mul_self (x : α) : sign x * x = |x| := by\n  rcases lt_trichotomy x 0 with hx | rfl | hx <;> simp [*, abs_of_pos, abs_of_neg]\n\n@[simp]\ntheorem self_mul_sign (x : α) : x * sign x = |x| := by\n  rcases lt_trichotomy x 0 with hx | rfl | hx <;> simp [*, abs_of_pos, abs_of_neg]\n\n/-- `SignType.sign` as a `MonoidWithZeroHom` for a nontrivial ordered semiring. Note that linearity\nis required; consider ℂ with the order `z ≤ w` iff they have the same imaginary part and\n`z - w ≤ 0` in the reals; then `1 + I` and `1 - I` are incomparable to zero, and thus we have:\n`0 * 0 = SignType.sign (1 + I) * SignType.sign (1 - I) ≠ SignType.sign 2 = 1`.\n(`Complex.orderedCommRing`) -/\n@[simps -fullyApplied]\ndef signHom : α →*₀ SignType where\n  toFun := sign\n  map_zero' := sign_zero\n  map_one' := sign_one\n  map_mul' := sign_mul\n\ntheorem sign_pow (x : α) (n : ℕ) : sign (x ^ n) = sign x ^ n := map_pow signHom x n\n\nend LinearOrderedRing\n\nsection LinearOrderedAddCommGroup\n\nvariable [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α]\n\ntheorem sign_sum {ι : Type*} {s : Finset ι} {f : ι → α} (hs : s.Nonempty) (t : SignType)\n    (h : ∀ i ∈ s, sign (f i) = t) : sign (∑ i ∈ s, f i) = t := by\n  cases t\n  · simp_rw [zero_eq_zero, sign_eq_zero_iff] at h ⊢\n    exact Finset.sum_eq_zero h\n  · simp_rw [neg_eq_neg_one, sign_eq_neg_one_iff] at h ⊢\n    exact Finset.sum_neg h hs\n  · simp_rw [pos_eq_one, sign_eq_one_iff] at h ⊢\n    exact Finset.sum_pos h hs\n\nend LinearOrderedAddCommGroup\n\nopen Finset Nat\n\nsection exists_signed_sum\n\n/-!\nIn this section we explicitly handle universe variables,\nbecause Lean creates a fresh universe variable for the type whose existence is asserted.\nBut we want the type to live in the same universe as the input type.\n-/\n\nprivate theorem exists_signed_sum_aux [DecidableEq α] (s : Finset α) (f : α → ℤ) :\n    ∃ (β : Type u) (t : Finset β) (sgn : β → SignType) (g : β → α),\n      (∀ b, g b ∈ s) ∧\n        (#t = ∑ a ∈ s, (f a).natAbs) ∧\n          ∀ a ∈ s, (∑ b ∈ t, if g b = a then (sgn b : ℤ) else 0) = f a := by\n  refine\n    ⟨(Σ _ : { x // x ∈ s }, ℕ), Finset.univ.sigma fun a => range (f a).natAbs,\n      fun a => sign (f a.1), fun a => a.1, fun a => a.1.2, ?_, ?_⟩\n  · simp [sum_attach (f := fun a => (f a).natAbs)]\n  · intro x hx\n    simp [sum_sigma, hx, ← Int.sign_eq_sign, Int.sign_mul_abs, mul_comm |f _|,\n      sum_attach (s := s) (f := fun y => if y = x then f y else 0)]\n\n/-- We can decompose a sum of absolute value `n` into a sum of `n` signs. -/\ntheorem exists_signed_sum [DecidableEq α] (s : Finset α) (f : α → ℤ) :\n    ∃ (β : Type u) (_ : Fintype β) (sgn : β → SignType) (g : β → α),\n      (∀ b, g b ∈ s) ∧\n        (Fintype.card β = ∑ a ∈ s, (f a).natAbs) ∧\n          ∀ a ∈ s, (∑ b, if g b = a then (sgn b : ℤ) else 0) = f a :=\n  let ⟨β, t, sgn, g, hg, ht, hf⟩ := exists_signed_sum_aux s f\n  ⟨t, inferInstance, fun b => sgn b, fun b => g b, fun b => hg b, by simp [ht], fun a ha =>\n    (sum_attach t fun b ↦ ite (g b = a) (sgn b : ℤ) 0).trans <| hf _ ha⟩\n\n/-- We can decompose a sum of absolute value less than `n` into a sum of at most `n` signs. -/\n\nTarget:\ntheorem exists_signed_sum' [Nonempty α] [DecidableEq α] (s : Finset α) (f : α → ℤ)\n    (n : ℕ) (h : (∑ i ∈ s, (f i).natAbs) ≤ n) :\n    ∃ (β : Type u) (_ : Fintype β) (sgn : β → SignType) (g : β → α),\n      (∀ b, g b ∉ s → sgn b = 0) ∧\n        Fintype.card β = n ∧ ∀ a ∈ s, (∑ i, if g i = a then (sgn i : ℤ) else 0) = f a :=\n\nProof body:\n","rejected":"by\n  exact exists_signed_sum","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"bf4d9a66f1b62856ef84e463cf148a2ae0f481a680d5220ce0fa63cce2ed698d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sign","family_id":"exists_signed_sum","file_id":"mathlib/Mathlib/Data/Sign/Basic.lean","sample_id":"3236ac1bb6f48e9d26b20fb63c3f2eb332af2bf7a5e2f318caa01e95dce53a89"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f18be9dd1c59e0ebc0b61eaff66514bafe594968c74bd37a4cddbaa1b0ea3c89","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"197f653d4b36fd6849b8b6ebccef410407e7a70fbd810f7a0a75770cb14b42e2","source_sha256":"bf4a6ea9ec041102f48b300b957ea9d15b34197752c294a77f2238db87be2629","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rintro _ (rfl | rfl | rfl) _ (rfl | rfl | rfl) h <;>\n  simp only\n    [cast_neg, cast_one, cast_zero, neg_eq_zero, one_ne_zero, zero_ne_one, zero_eq_neg] at h ⊢\n  · exact ((Ring.neg_one_ne_one_of_char_ne_two hR).symm h).elim\n  · exact ((Ring.neg_one_ne_one_of_char_ne_two hR) h).elim","hard_negative":false,"metrics":{"chosen_tokens":70,"rejected_tokens":3,"token_jaccard":0.058824,"token_length_ratio":0.042857},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"334182da61cfd887c965db718c91e2d37de833a6e42f5519a9ba4e93686c0b21","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharP.Defs\npublic import Mathlib.Algebra.Group.Fin.Basic\npublic import Mathlib.Algebra.Ring.ULift\npublic import Mathlib.Algebra.Ring.Opposite\npublic import Mathlib.Data.Int.ModEq\npublic import Mathlib.Data.Nat.Cast.Prod\npublic import Mathlib.Data.ULift\npublic import Mathlib.Order.Interval.Set.Defs\npublic import Mathlib.Algebra.Ring.GrindInstances\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Joey van Langen, Casper Putz\n-/\n/-!\n# Characteristic of semirings\n\nThis file collects some fundamental results on the characteristic of rings that don't need the extra\nimports of `Mathlib/Algebra/CharP/Lemmas.lean`.\n\nAs such, we can probably reorganize and find a better home for most of these lemmas.\n-/\n\npublic section\n\nassert_not_exists Finset TwoSidedIdeal\n\nopen Set\n\nvariable (R : Type*)\n\nnamespace CharP\nsection AddMonoidWithOne\nvariable [AddMonoidWithOne R] (p : ℕ)\n\nvariable [CharP R p] {a b : ℕ}\n\nlemma natCast_eq_natCast' (h : a ≡ b [MOD p]) : (a : R) = b := by\n  wlog hle : a ≤ b\n  · exact (this R p h.symm (le_of_not_ge hle)).symm\n  rw [Nat.modEq_iff_dvd' hle] at h\n  rw [← Nat.sub_add_cancel hle, Nat.cast_add, (cast_eq_zero_iff R p _).mpr h, zero_add]\n\nlemma natCast_eq_natCast_mod (a : ℕ) : (a : R) = a % p :=\n  natCast_eq_natCast' R p (Nat.mod_modEq a p).symm\n\nvariable [IsRightCancelAdd R]\n\nlemma natCast_eq_natCast : (a : R) = b ↔ a ≡ b [MOD p] := by\n  wlog hle : a ≤ b\n  · rw [eq_comm, this R p (le_of_not_ge hle), Nat.ModEq.comm]\n  rw [Nat.modEq_iff_dvd' hle, ← cast_eq_zero_iff R p (b - a),\n    ← add_right_cancel_iff (G := R) (a := a) (b := b - a), zero_add, ← Nat.cast_add,\n    Nat.sub_add_cancel hle, eq_comm]\n\nlemma natCast_injOn_Iio : (Set.Iio p).InjOn ((↑) : ℕ → R) :=\n  fun _a ha _b hb hab ↦ ((natCast_eq_natCast _ _).1 hab).eq_of_lt_of_lt ha hb\n\nend AddMonoidWithOne\n\nsection AddGroupWithOne\nvariable [AddGroupWithOne R] (p : ℕ) [CharP R p] {a b : ℤ}\n\nlemma intCast_eq_intCast : (a : R) = b ↔ a ≡ b [ZMOD p] := by\n  rw [eq_comm, ← sub_eq_zero, ← Int.cast_sub, CharP.intCast_eq_zero_iff R p, Int.modEq_iff_dvd]\n\nlemma intCast_eq_intCast_mod : (a : R) = a % (p : ℤ) :=\n  (CharP.intCast_eq_intCast R p).mpr (Int.mod_modEq a p).symm\n\nlemma intCast_injOn_Ico [IsRightCancelAdd R] : InjOn (Int.cast : ℤ → R) (Ico 0 p) := by\n  rintro a ⟨ha₀, ha⟩ b ⟨hb₀, hb⟩ hab\n  lift a to ℕ using ha₀\n  lift b to ℕ using hb₀\n  norm_cast at *\n  exact natCast_injOn_Iio _ _ ha hb hab\n\nend AddGroupWithOne\nend CharP\n\nnamespace CharP\n\nsection NonAssocSemiring\n\nvariable {R} [NonAssocSemiring R]\n\nvariable (R) in\n/-- If a ring `R` is of characteristic `p`, then for any prime number `q` different from `p`,\nit is not zero in `R`. -/\nlemma cast_ne_zero_of_ne_of_prime [Nontrivial R]\n    {p q : ℕ} [CharP R p] (hq : q.Prime) (hneq : p ≠ q) : (q : R) ≠ 0 := fun h ↦ by\n  rw [cast_eq_zero_iff R p q] at h\n  rcases hq.eq_one_or_self_of_dvd _ h with rfl | h\n  · exact false_of_nontrivial_of_char_one (R := R)\n  · exact hneq h\n\nlemma ringChar_of_prime_eq_zero [Nontrivial R] {p : ℕ} (hprime : Nat.Prime p)\n    (hp0 : (p : R) = 0) : ringChar R = p :=\n  Or.resolve_left ((Nat.dvd_prime hprime).1 (ringChar.dvd hp0)) ringChar_ne_one\n\nlemma charP_iff_prime_eq_zero [Nontrivial R] {p : ℕ} (hp : p.Prime) :\n    CharP R p ↔ (p : R) = 0 :=\n  ⟨fun _ => cast_eq_zero R p,\n   fun hp0 => (ringChar_of_prime_eq_zero hp hp0) ▸ inferInstance⟩\n\nend NonAssocSemiring\nend CharP\n\nsection\n\n/-- We have `2 ≠ 0` in a nontrivial ring whose characteristic is not `2`. -/\nprotected lemma Ring.two_ne_zero {R : Type*} [NonAssocSemiring R] [Nontrivial R]\n    (hR : ringChar R ≠ 2) : (2 : R) ≠ 0 := by\n  rw [Ne, (by norm_cast : (2 : R) = (2 : ℕ)), ringChar.spec, Nat.dvd_prime Nat.prime_two]\n  exact mt (or_iff_left hR).mp CharP.ringChar_ne_one\n\n-- We have `CharP.neg_one_ne_one`, which assumes `[Ring R] (p : ℕ) [CharP R p] [Fact (2 < p)]`.\n-- This is a version using `ringChar` instead.\n/-- Characteristic `≠ 2` and nontrivial implies that `-1 ≠ 1`. -/\nlemma Ring.neg_one_ne_one_of_char_ne_two {R : Type*} [NonAssocRing R] [Nontrivial R]\n    (hR : ringChar R ≠ 2) : (-1 : R) ≠ 1 := fun h =>\n  Ring.two_ne_zero hR (one_add_one_eq_two (R := R) ▸ neg_eq_iff_add_eq_zero.mp h)\n\n/-- Characteristic `≠ 2` in a domain implies that `-a = a` iff `a = 0`. -/\nlemma Ring.eq_self_iff_eq_zero_of_char_ne_two {R : Type*} [NonAssocRing R] [Nontrivial R]\n    [NoZeroDivisors R] (hR : ringChar R ≠ 2) {a : R} : -a = a ↔ a = 0 :=\n  ⟨fun h =>\n    (mul_eq_zero.mp <| (two_mul a).trans <| neg_eq_iff_add_eq_zero.mp h).resolve_left\n      (Ring.two_ne_zero hR),\n    fun h => ((congr_arg (fun x => -x) h).trans neg_zero).trans h.symm⟩\n\nend\n\nsection Prod\nvariable (S : Type*) [AddMonoidWithOne R] [AddMonoidWithOne S] (p q : ℕ) [CharP R p]\n\n/-- The characteristic of the product of rings is the least common multiple of the\ncharacteristics of the two rings. -/\ninstance Nat.lcm.charP [CharP S q] : CharP (R × S) (Nat.lcm p q) where\n  cast_eq_zero_iff := by\n    simp [Prod.ext_iff, CharP.cast_eq_zero_iff R p, CharP.cast_eq_zero_iff S q, Nat.lcm_dvd_iff]\n\n/-- The characteristic of the product of two rings of the same characteristic\n  is the same as the characteristic of the rings -/\ninstance Prod.charP [CharP S p] : CharP (R × S) p := by\n  convert! Nat.lcm.charP R S p p; simp\n\ninstance Prod.charZero_of_left [CharZero R] : CharZero (R × S) where\n  cast_injective _ _ h := CharZero.cast_injective congr(Prod.fst $h)\n\ninstance Prod.charZero_of_right [CharZero S] : CharZero (R × S) where\n  cast_injective _ _ h := CharZero.cast_injective congr(Prod.snd $h)\n\nend Prod\n\ninstance ULift.charP [AddMonoidWithOne R] (p : ℕ) [CharP R p] : CharP (ULift R) p where\n  cast_eq_zero_iff n := Iff.trans ULift.ext_iff <| CharP.cast_eq_zero_iff R p n\n\ninstance MulOpposite.charP [AddMonoidWithOne R] (p : ℕ) [CharP R p] : CharP Rᵐᵒᵖ p where\n  cast_eq_zero_iff n := MulOpposite.unop_inj.symm.trans <| CharP.cast_eq_zero_iff R p n\n\nsection\n\n/-- If two integers from `{0, 1, -1}` result in equal elements in a ring `R`\nthat is nontrivial and of characteristic not `2`, then they are equal. -/\n\nTarget:\nlemma Int.cast_injOn_of_ringChar_ne_two {R : Type*} [NonAssocRing R] [Nontrivial R]\n    (hR : ringChar R ≠ 2) : ({0, 1, -1} : Set ℤ).InjOn ((↑) : ℤ → R) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/CharP","family_id":"int","file_id":"mathlib/Mathlib/Algebra/CharP/Basic.lean","sample_id":"197f653d4b36fd6849b8b6ebccef410407e7a70fbd810f7a0a75770cb14b42e2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"87c6abbbe5679e2db362b7fc92a4107a665977a26b58a1424f12d41fb84a8bc6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"03a76a6ea9df8634d131435398f0347bcd73e900da0d8f9eb52c6f6448d3c6cf","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2e2771fc6ad14eff3a9b98719630e14505321f0c2d2dbcd9d1df71f5b966db93","source_sha256":"695353e91eba40e45d9ecd0e58f8b00a56d575b25190f682a1689e29c7316a51","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold logDeriv\n  simp [field]\n  ring","hard_negative":true,"metrics":{"chosen_tokens":8,"rejected_tokens":2,"token_jaccard":0.111111,"token_length_ratio":0.25},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"33517cf1c18f84492de57e053ec10f966db8785be8441cfe995877851451c62f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Derivation.MapCoeffs\npublic import Mathlib.FieldTheory.PrimitiveElement\n\nNamespace:\nDifferential\n\nLocal context:\n/-\nCopyright (c) 2024 Daniel Weber. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Weber\n-/\n/-!\n# Differential Fields\n\nThis file defines the logarithmic derivative `Differential.logDeriv` and proves properties of it.\nThis is defined algebraically, compared to `logDeriv` which is analytical.\n-/\n\n@[expose] public section\n\nnamespace Differential\n\nopen algebraMap Polynomial IntermediateField\n\nvariable {R : Type*} [Field R] [Differential R] (a b : R)\n\n/--\nThe logarithmic derivative of a is a′ / a.\n-/\ndef logDeriv : R := a′ / a\n\n@[simp]\nlemma logDeriv_zero : logDeriv (0 : R) = 0 := by\n  simp [logDeriv]\n\n@[simp]\nlemma logDeriv_one : logDeriv (1 : R) = 0 := by\n  simp [logDeriv]\n\nTarget:\nlemma logDeriv_mul (ha : a ≠ 0) (hb : b ≠ 0) : logDeriv (a * b) = logDeriv a + logDeriv b :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_2e2771fc6ad1","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"57eb9d50936b1cdf31f783b24a390476808b1b3f484884b80173a10db9d6f88d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Differential","family_id":"logderiv_mul","file_id":"mathlib/Mathlib/FieldTheory/Differential/Basic.lean","sample_id":"2e2771fc6ad14eff3a9b98719630e14505321f0c2d2dbcd9d1df71f5b966db93"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a73fac1f1d110d78557015f3b6ec0716b6ddeed8c82d3ea7e29e1289e23f50bb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8a0eee3ca3d7371d45983d225651a1f869b4128c95c979ad95cb7c08b51b4942","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8b9a52eec598c7910eb23c645754ae85116421ba3d320b896639db65e6cbf421","source_sha256":"087c4fdfa1fe64dbc38821c1fa8aea61c46d7fa4bef6ee724aa05ce31b21dbc2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [preimage_image_mk_eq_iUnion_image, iUnion_subtype, ← image2_mul, ← iUnion_image_right]\n  simp only [SetLike.mem_coe]","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":2,"token_jaccard":0.0625,"token_length_ratio":0.1},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"33a2ca26c5e926a0375358bcd717f7c3c324828494536a2a01ed6939665bfa63","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Quotient\npublic import Mathlib.Algebra.Group.Action.Opposite\npublic import Mathlib.Algebra.Group.Subgroup.MulOpposite\npublic import Mathlib.GroupTheory.GroupAction.Defs\npublic import Mathlib.Algebra.Group.Pointwise.Set.Basic\n\nNamespace:\nQuotientGroup\n\nLocal context:\n/-\nCopyright (c) 2018 Mitchell Rowett. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mitchell Rowett, Kim Morrison\n-/\n/-!\n# Cosets\n\nThis file develops the basic theory of left and right cosets.\n\nWhen `G` is a group and `a : G`, `s : Set G`, with  `open scoped Pointwise` we can write:\n* the left coset of `s` by `a` as `a • s`\n* the right coset of `s` by `a` as `MulOpposite.op a • s` (or `op a • s` with `open MulOpposite`,\n  or `s <• a` with `open scoped Pointwise RightActions`)\n\nIf instead `G` is an additive group, we can write (with  `open scoped Pointwise` still)\n* the left coset of `s` by `a` as `a +ᵥ s`\n* the right coset of `s` by `a` as `AddOpposite.op a +ᵥ s` (or `op a +ᵥ s` with `open AddOpposite`,\n  or `s <+ᵥ a` with `open scoped Pointwise RightActions`)\n\n## Main definitions\n\n* `QuotientGroup.quotient s`: the quotient type representing the left cosets with respect to a\n  subgroup `s`, for an `AddGroup` this is `QuotientAddGroup.quotient s`.\n* `QuotientGroup.mk`: the canonical map from `α` to `α/s` for a subgroup `s` of `α`, for an\n  `AddGroup` this is `QuotientAddGroup.mk`.\n\n## Notation\n\n* `G ⧸ H` is the quotient of the (additive) group `G` by the (additive) subgroup `H`\n\n## TODO\n\nProperly merge with pointwise actions on sets, by renaming and deduplicating lemmas as appropriate.\n-/\n\n@[expose] public section\n\nassert_not_exists Cardinal\n\nopen Function Set\nopen scoped Pointwise\n\nvariable {α : Type*}\n\n/- Ensure that `@[to_additive]` uses the right namespace. -/\ninsert_to_additive_translation QuotientGroup QuotientAddGroup\n\nnamespace QuotientGroup\n\nvariable [Group α] (s : Subgroup α)\n\n/-- The equivalence relation corresponding to the partition of a group by left cosets\nof a subgroup. -/\n@[to_additive (attr := implicit_reducible)\n  /-- The equivalence relation corresponding to the partition of a group by left cosets\nof a subgroup. -/]\ndef leftRel : Setoid α :=\n  MulAction.orbitRel s.op α\n\nvariable {s} in\n@[to_additive]\ntheorem leftRel_apply {x y : α} : leftRel s x y ↔ x⁻¹ * y ∈ s :=\n  calc\n    (∃ a : s.op, y * MulOpposite.unop a = x) ↔ ∃ a : s, y * a = x :=\n      s.equivOp.symm.exists_congr_left\n    _ ↔ ∃ a : s, x⁻¹ * y = a⁻¹ := by\n      simp only [inv_mul_eq_iff_eq_mul, Subgroup.coe_inv, eq_mul_inv_iff_mul_eq]\n    _ ↔ x⁻¹ * y ∈ s := by simp [exists_inv_mem_iff_exists_mem]\n\n@[to_additive]\ntheorem leftRel_eq : ⇑(leftRel s) = fun x y => x⁻¹ * y ∈ s :=\n  funext₂ <| by\n    simp only [eq_iff_iff]\n    apply leftRel_apply\n\n@[to_additive]\ninstance leftRelDecidable [DecidablePred (· ∈ s)] : DecidableRel (leftRel s).r := fun x y => by\n  rw [leftRel_eq]\n  exact ‹DecidablePred (· ∈ s)› _\n\n/-- `α ⧸ s` is the quotient type representing the left cosets of `s`. If `s` is a normal subgroup,\n`α ⧸ s` is a group -/\n@[to_additive /-- `α ⧸ s` is the quotient type representing the left cosets of `s`. If `s` is a\nnormal subgroup, `α ⧸ s` is a group -/]\ninstance instHasQuotientSubgroup : HasQuotient α (Subgroup α) :=\n  ⟨fun s => Quotient (leftRel s)⟩\n\n@[to_additive]\ninstance [DecidablePred (· ∈ s)] : DecidableEq (α ⧸ s) :=\n  @Quotient.decidableEq _ _ (leftRelDecidable _)\n\n/-- The equivalence relation corresponding to the partition of a group by right cosets of a\nsubgroup. -/\n@[to_additive (attr := implicit_reducible)\n  /-- The equivalence relation corresponding to the partition of a group by right cosets\n  of a subgroup. -/]\ndef rightRel : Setoid α :=\n  MulAction.orbitRel s α\n\nvariable {s} in\n@[to_additive]\ntheorem rightRel_apply {x y : α} : rightRel s x y ↔ y * x⁻¹ ∈ s :=\n  calc\n    (∃ a : s, (a : α) * y = x) ↔ ∃ a : s, y * x⁻¹ = a⁻¹ := by\n      simp only [mul_inv_eq_iff_eq_mul, Subgroup.coe_inv, eq_inv_mul_iff_mul_eq]\n    _ ↔ y * x⁻¹ ∈ s := by simp [exists_inv_mem_iff_exists_mem]\n\n@[to_additive]\ntheorem rightRel_eq : ⇑(rightRel s) = fun x y => y * x⁻¹ ∈ s :=\n  funext₂ <| by\n    simp only [eq_iff_iff]\n    apply rightRel_apply\n\n@[to_additive]\ninstance rightRelDecidable [DecidablePred (· ∈ s)] : DecidableRel (rightRel s).r := fun x y => by\n  rw [rightRel_eq]\n  exact ‹DecidablePred (· ∈ s)› _\n\n/-- Right cosets are in bijection with left cosets. -/\n@[to_additive /-- Right cosets are in bijection with left cosets. -/]\ndef quotientRightRelEquivQuotientLeftRel : Quotient (QuotientGroup.rightRel s) ≃ α ⧸ s where\n  toFun :=\n    Quotient.map' (fun g => g⁻¹) fun a b => by\n      rw [leftRel_apply, rightRel_apply]\n      exact fun h => (congr_arg (· ∈ s) (by simp)).mp (s.inv_mem h)\n  invFun :=\n    Quotient.map' (fun g => g⁻¹) fun a b => by\n      rw [leftRel_apply, rightRel_apply]\n      exact fun h => (congr_arg (· ∈ s) (by simp)).mp (s.inv_mem h)\n  left_inv g :=\n    Quotient.inductionOn' g fun g =>\n      Quotient.sound'\n        (by\n          simp only [inv_inv]\n          exact Quotient.exact' rfl)\n  right_inv g :=\n    Quotient.inductionOn' g fun g =>\n      Quotient.sound'\n        (by\n          simp only [inv_inv]\n          exact Quotient.exact' rfl)\n\nend QuotientGroup\n\nnamespace QuotientGroup\n\nvariable [Group α] {s : Subgroup α}\n\n/-- The canonical map from a group `α` to the quotient `α ⧸ s`. -/\n@[to_additive (attr := coe)\n/-- The canonical map from an `AddGroup` `α` to the quotient `α ⧸ s`. -/]\nabbrev mk (a : α) : α ⧸ s :=\n  Quotient.mk'' a\n\n@[to_additive]\ntheorem mk_surjective : Function.Surjective <| @mk _ _ s :=\n  Quotient.mk''_surjective\n\n@[to_additive (attr := simp)]\nlemma range_mk : range (QuotientGroup.mk (s := s)) = univ := range_eq_univ.mpr mk_surjective\n\n@[to_additive (attr := elab_as_elim)]\ntheorem induction_on {C : α ⧸ s → Prop} (x : α ⧸ s) (H : ∀ z, C (QuotientGroup.mk z)) : C x :=\n  Quotient.inductionOn' x H\n\n@[to_additive]\ninstance : Coe α (α ⧸ s) :=\n  ⟨mk⟩\n\n@[to_additive] alias induction_on' := induction_on\n\n@[to_additive (attr := simp)]\ntheorem quotient_liftOn_mk {β} (f : α → β) (h) (x : α) : Quotient.liftOn' (x : α ⧸ s) f h = f x :=\n  rfl\n\n@[to_additive]\ntheorem forall_mk {C : α ⧸ s → Prop} : (∀ x : α ⧸ s, C x) ↔ ∀ x : α, C x :=\n  mk_surjective.forall\n\n@[to_additive]\ntheorem exists_mk {C : α ⧸ s → Prop} : (∃ x : α ⧸ s, C x) ↔ ∃ x : α, C x :=\n  mk_surjective.exists\n\n@[to_additive]\ninstance (s : Subgroup α) : Inhabited (α ⧸ s) :=\n  ⟨((1 : α) : α ⧸ s)⟩\n\n@[to_additive]\nprotected theorem eq {a b : α} : (a : α ⧸ s) = b ↔ a⁻¹ * b ∈ s :=\n  calc\n    _ ↔ leftRel s a b := Quotient.eq''\n    _ ↔ _ := by rw [leftRel_apply]\n\n@[to_additive]\ntheorem out_eq' (a : α ⧸ s) : mk a.out = a :=\n  Quotient.out_eq' a\n\nvariable (s)\n\n/-- It can be useful to write `obtain ⟨h, H⟩ := mk_out_eq_mul ...`, and then `rw [H]` or\n`simp_rw [H]` or `simp only [H]`. In order for `simp_rw` and `simp only` to work, this lemma is\nstated in terms of an arbitrary `h : s`, rather than the specific `h = g⁻¹ * (mk g).out`. -/\n@[to_additive QuotientAddGroup.mk_out_eq_mul]\ntheorem mk_out_eq_mul (g : α) : ∃ h : s, (mk g : α ⧸ s).out = g * h :=\n  ⟨⟨g⁻¹ * (mk g).out, QuotientGroup.eq.mp (mk g).out_eq'.symm⟩, by rw [mul_inv_cancel_left]⟩\n\nvariable {s} {a b : α}\n\n@[to_additive (attr := simp)]\ntheorem mk_mul_of_mem (a : α) (hb : b ∈ s) : (mk (a * b) : α ⧸ s) = mk a := by\n  rwa [QuotientGroup.eq, mul_inv_rev, inv_mul_cancel_right, s.inv_mem_iff]\n\n@[to_additive]\ntheorem preimage_image_mk (N : Subgroup α) (s : Set α) :\n    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = ⋃ x : N, (· * (x : α)) ⁻¹' s := by\n  ext x\n  simp only [QuotientGroup.eq, SetLike.exists, exists_prop, Set.mem_preimage, Set.mem_iUnion,\n    Set.mem_image]\n  exact\n    ⟨fun ⟨y, hs, hN⟩ => ⟨_, N.inv_mem hN, by simpa using hs⟩, fun ⟨z, hz, hxz⟩ =>\n      ⟨x * z, hxz, by simpa using hz⟩⟩\n\n@[to_additive]\ntheorem preimage_image_mk_eq_iUnion_image (N : Subgroup α) (s : Set α) :\n    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = ⋃ x : N, (· * (x : α)) '' s := by\n  rw [preimage_image_mk, iUnion_congr_of_surjective (·⁻¹) inv_surjective]\n  exact fun x ↦ image_mul_right'\n\n@[to_additive]\n\nTarget:\ntheorem preimage_image_mk_eq_mul (N : Subgroup α) (s : Set α) :\n    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = s * N :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_8b9a52eec598","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"55b65fefefb991fd9bea405d330a6807b930a6aa429baf916c4d98b474149706","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Coset","family_id":"preimage_image_mk_eq_mul","file_id":"mathlib/Mathlib/GroupTheory/Coset/Defs.lean","sample_id":"8b9a52eec598c7910eb23c645754ae85116421ba3d320b896639db65e6cbf421"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4e24363fb7ba779b1d49eb8030dc7b2f8941ee6210796f291a186ebbe93a5464","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"31d3eaad908c26db9fa8cd7a18a465f80cc33c5d4402214fcc63790030c8fb99","source_sha256":"05b71bb08d5e0d5da2d7d8beb37172741e5772665eaf54d30c9d72516996fb96","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply complementedLattice_of_sSup_atoms_eq_top\n  have : ∀ i ∈ s, ∃ t : Set α, f i = sSup t ∧ ∀ a ∈ t, IsAtom a := fun i hi ↦ by\n    replace h := complementedLattice_iff_isAtomistic.mp (h i hi)\n    obtain ⟨u, hu, hu'⟩ := eq_sSup_atoms (⊤ : Iic (f i))\n    refine ⟨(↑) '' u, ?_, ?_⟩\n    · replace hu : f i = ↑(sSup u) := Subtype.ext_iff.mp hu\n      simp_rw [hu, Iic.coe_sSup]\n    · rintro b ⟨⟨a, ha'⟩, ha, rfl⟩\n      exact IsAtom.of_isAtom_coe_Iic (hu' _ ha)\n  choose t ht ht' using this\n  let u : Set α := ⋃ i, ⋃ hi : i ∈ s, t i hi\n  have hu₁ : u ⊆ {a | IsAtom a} := by\n    rintro a ⟨-, ⟨i, rfl⟩, ⟨-, ⟨hi, rfl⟩, ha : a ∈ t i hi⟩⟩\n    exact ht' i hi a ha\n  have hu₂ : sSup u = ⨆ i ∈ s, f i := by simp_rw [u, sSup_iUnion, biSup_congr' ht]\n  rw [eq_top_iff, ← h', ← hu₂]\n  exact sSup_le_sSup hu₁","hard_negative":false,"metrics":{"chosen_tokens":244,"rejected_tokens":2,"token_jaccard":0.012195,"token_length_ratio":0.008197},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"33d10c34098b6d9abba0d8d2da91231f5f747dc540405e82e5b6307d724ff69e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.CompleteLatticeIntervals\npublic import Mathlib.Order.CompactlyGenerated.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Results about compactness properties for intervals in complete lattices\n-/\n\npublic section\n\nvariable {ι α : Type*} [CompleteLattice α]\n\nnamespace Set.Iic\n\ntheorem isCompactElement {a : α} {b : Iic a} (h : IsCompactElement (b : α)) :\n    IsCompactElement b := by\n  simp only [CompleteLattice.isCompactElement_iff_exists_le_iSup_of_le_iSup,\n    Finset.sup_eq_iSup] at h ⊢\n  intro ι s hb\n  replace hb : (b : α) ≤ iSup ((↑) ∘ s) := le_trans hb <| (coe_iSup s) ▸ le_refl _\n  obtain ⟨t, ht⟩ := h ι ((↑) ∘ s) hb\n  exact ⟨t, (by simpa using ht : (b : α) ≤ _)⟩\n\ninstance instIsCompactlyGenerated [IsCompactlyGenerated α] {a : α} :\n    IsCompactlyGenerated (Iic a) := by\n  refine ⟨fun ⟨x, (hx : x ≤ a)⟩ ↦ ?_⟩\n  obtain ⟨s, hs, rfl⟩ := IsCompactlyGenerated.exists_sSup_eq x\n  rw [sSup_le_iff] at hx\n  let f : s → Iic a := fun y ↦ ⟨y, hx _ y.property⟩\n  refine ⟨range f, ?_, ?_⟩\n  · rintro - ⟨⟨y, hy⟩, hy', rfl⟩\n    exact isCompactElement (hs _ hy)\n  · rw [Subtype.ext_iff]\n    change sSup (((↑) : Iic a → α) '' (range f)) = sSup s\n    congr\n    ext b\n    simpa [f] using hx b\n\nend Set.Iic\n\nopen Set (Iic)\n\nTarget:\ntheorem complementedLattice_of_complementedLattice_Iic\n    [IsModularLattice α] [IsCompactlyGenerated α]\n    {s : Set ι} {f : ι → α}\n    (h : ∀ i ∈ s, ComplementedLattice <| Iic (f i))\n    (h' : ⨆ i ∈ s, f i = ⊤) :\n    ComplementedLattice α :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/CompactlyGenerated","family_id":"complementedlattice_of_complementedlattice_iic","file_id":"mathlib/Mathlib/Order/CompactlyGenerated/Intervals.lean","sample_id":"31d3eaad908c26db9fa8cd7a18a465f80cc33c5d4402214fcc63790030c8fb99"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f0fe75d1087a41898b65fcdcdb5116eac360500295f3e4e2a2c616668dd8129d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5ae786a0f2c5c57019f4c1e4edc7524aa65574003fae2570dc839740f449cb98","source_sha256":"afb2352b0e8eda21bd88c47cda5f9b0a4c162f3f78eedf36a33613392805b346","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  nth_rw 2 [← Quiver.Path.reverse_reverse p]\n  apply congr_comp_reverse","hard_negative":true,"metrics":{"chosen_tokens":14,"rejected_tokens":8,"token_jaccard":0.052632,"token_length_ratio":0.571429},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"33d42a290b722fdf6160b5687ab7e5668fd8df731d7a97c74ce50246f0133dcd","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Groupoid\npublic import Mathlib.CategoryTheory.PathCategory.Basic\n\nNamespace:\nQuiver.FreeGroupoid\n\nLocal context:\n/-\nCopyright (c) 2022 Rémi Bottinelli. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémi Bottinelli\n-/\n/-!\n# Free groupoid on a quiver\n\nThis file defines the free groupoid on a quiver, the lifting of a prefunctor to its unique\nextension as a functor from the free groupoid, and proves uniqueness of this extension.\n\n## Main results\n\nGiven the type `V` and a quiver instance on `V`:\n\n- `Quiver.FreeGroupoid V`: a type synonym for `V`.\n- `Quiver.FreeGroupoid.instGroupoid`: the `Groupoid` instance on `Quiver.FreeGroupoid V`.\n- `lift`: the lifting of a prefunctor from `V` to `V'` where `V'` is a groupoid, to a functor.\n  `Quiver.FreeGroupoid V ⥤ V'`.\n- `lift_spec` and `lift_unique`: the proofs that, respectively, `lift` indeed is a lifting\n  and is the unique one.\n\n## Implementation notes\n\nThe free groupoid is first defined by symmetrifying the quiver, taking the induced path category\nand finally quotienting by the reducibility relation.\n\n-/\n\n@[expose] public section\n\nopen Set Function\n\nnamespace Quiver\n\nopen CategoryTheory\n\nuniverse u v u' v' u'' v''\n\nvariable {V : Type u} [Quiver.{v} V]\n\n/-- Shorthand for the \"forward\" arrow corresponding to `f` in `paths <| symmetrify V` -/\nabbrev Hom.toPosPath {X Y : V} (f : X ⟶ Y) :\n    (CategoryTheory.Paths.categoryPaths <| Quiver.Symmetrify V).Hom X Y :=\n  f.toPos.toPath\n\n/-- Shorthand for the \"forward\" arrow corresponding to `f` in `paths <| symmetrify V` -/\nabbrev Hom.toNegPath {X Y : V} (f : X ⟶ Y) :\n    (CategoryTheory.Paths.categoryPaths <| Quiver.Symmetrify V).Hom Y X :=\n  f.toNeg.toPath\n\n/-- The \"reduction\" relation -/\ninductive FreeGroupoid.redStep : HomRel (Paths (Quiver.Symmetrify V))\n  | step (X Z : Quiver.Symmetrify V) (f : X ⟶ Z) :\n    redStep (𝟙 ((Paths.of (Quiver.Symmetrify V)).obj X)) (f.toPath ≫ (Quiver.reverse f).toPath)\n\n/-- The underlying vertices of the free groupoid -/\nprotected def FreeGroupoid (V) [Q : Quiver V] :=\n  CategoryTheory.Quotient (@FreeGroupoid.redStep V Q)\n\nnamespace FreeGroupoid\n\nopen Quiver\n\ninstance {V} [Quiver V] [Nonempty V] : Nonempty (Quiver.FreeGroupoid V) := by\n  inhabit V; exact ⟨⟨@default V _⟩⟩\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem congr_reverse {X Y : Paths <| Quiver.Symmetrify V} (p q : X ⟶ Y) :\n    HomRel.CompClosure redStep p q → HomRel.CompClosure redStep p.reverse q.reverse := by\n  rintro ⟨_, _, XW, _, _, WY, _, _, f⟩\n  have : HomRel.CompClosure redStep (WY.reverse ≫ 𝟙 _ ≫ XW.reverse)\n      (WY.reverse ≫ (f.toPath ≫ (Quiver.reverse f).toPath) ≫ XW.reverse) := by\n    constructor\n    constructor\n  simpa only [CategoryStruct.comp, CategoryStruct.id, Quiver.Path.reverse, Quiver.Path.nil_comp,\n    Quiver.Path.reverse_comp, Quiver.reverse_reverse, Quiver.Path.reverse_toPath,\n    Quiver.Path.comp_assoc] using this\n\nopen Relation in\ntheorem congr_comp_reverse {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :\n    Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (p ≫ p.reverse) =\n      Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (𝟙 X) := by\n  apply Quot.eqvGen_sound\n  induction p with\n  | nil => apply EqvGen.refl\n  | cons q f ih =>\n    simp only [Quiver.Path.reverse]\n    fapply EqvGen.trans\n    -- Porting note: dot notation for `Quiver.Path.*` and `Quiver.Hom.*` not working\n    · exact q ≫ Quiver.Path.reverse q\n    · apply EqvGen.symm\n      apply EqvGen.rel\n      have : HomRel.CompClosure redStep (q ≫ 𝟙 _ ≫ Quiver.Path.reverse q)\n          (q ≫ (Quiver.Hom.toPath f ≫ Quiver.Hom.toPath (Quiver.reverse f)) ≫\n            Quiver.Path.reverse q) := by\n        apply HomRel.CompClosure.intro\n        apply redStep.step\n      simp only [Category.assoc, Category.id_comp] at this ⊢\n      -- Porting note: `simp` cannot see how `Quiver.Path.comp_assoc` is relevant, so change to\n      -- category notation\n      change HomRel.CompClosure redStep (q ≫ Quiver.Path.reverse q)\n        (Quiver.Path.cons q f ≫ (Quiver.Hom.toPath (Quiver.reverse f)) ≫ (Quiver.Path.reverse q))\n      simp only [← Category.assoc] at this ⊢\n      exact this\n    · exact ih\n\nTarget:\ntheorem congr_reverse_comp {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :\n    Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (p.reverse ≫ p) =\n      Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (𝟙 Y) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"955da69216b933be4050ed3544213ab8dfce6fdcaa40ceacb0a2eab48bc8af13","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Groupoid","family_id":"congr_reverse_comp","file_id":"mathlib/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean","sample_id":"5ae786a0f2c5c57019f4c1e4edc7524aa65574003fae2570dc839740f449cb98"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5e2bc200513a76ad4fb8bd2871f6348a8c6776f4fb4e2960dcaa904707705fee","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cf6ae4c397cb6879927e93c99b97bb183866348c7185907ec01a438554d86622","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6a8e5453b7a4db469efd31b432f7b75acf3e7eec129c2d2a6b546cf031d8a71a","source_sha256":"80fda85bf9297b0a26c4ff540728683da76211a2d7626ecf60d2ca016feea2f7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  -- Using `have` not `let` since it is only existence of `NormedSpace` structure that we need.\n  have : NormedSpace ℝ E := .restrictScalars ℝ K E\n  have : FiniteDimensional ℝ E := FiniteDimensional.trans ℝ K E\n  infer_instance\n\nvariable {E}","hard_negative":true,"metrics":{"chosen_tokens":53,"rejected_tokens":3,"token_jaccard":0.030303,"token_length_ratio":0.056604},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"3404b06a9978715f843fe94f9488cdb567cb9c52972f0346db4807a6460f66aa","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Module.FiniteDimension\npublic import Mathlib.Analysis.RCLike.Basic\npublic import Mathlib.Topology.Instances.RealVectorSpace\n\nNamespace:\nFiniteDimensional\n\nLocal context:\n/-\nCopyright (c) 2020 Frédéric Dupuis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Frédéric Dupuis\n-/\n/-! # Further lemmas about `RCLike` -/\n\npublic section\n\nopen scoped Finset\n\nvariable {K E : Type*} [RCLike K]\n\nopen ComplexOrder RCLike in\nlemma convex_RCLike_iff_convex_real [AddCommMonoid E] [Module K E] [Module ℝ E]\n    [IsScalarTower ℝ K E] {s : Set E} : Convex K s ↔ Convex ℝ s :=\n  ⟨Convex.lift ℝ,\n  fun hs => convex_of_nonneg_surjective_algebraMap _ (fun _ => nonneg_iff_exists_ofReal.mp) hs⟩\n\nnamespace Polynomial\n\ntheorem ofReal_eval (p : ℝ[X]) (x : ℝ) : (↑(p.eval x) : K) = aeval (↑x) p :=\n  (@aeval_algebraMap_apply_eq_algebraMap_eval ℝ K _ _ _ x p).symm\n\nend Polynomial\n\nvariable (K) in\nlemma RCLike.span_one_I : Submodule.span ℝ (M := K) {1, I} = ⊤ := by\n  suffices ∀ x : K, ∃ a b : ℝ, a • 1 + b • I = x by\n    simpa [Submodule.eq_top_iff', Submodule.mem_span_pair]\n  exact fun x ↦ ⟨re x, im x, by simp [real_smul_eq_coe_mul]⟩\n\nvariable (K) in\nlemma RCLike.rank_le_two : Module.rank ℝ K ≤ 2 :=\n  calc\n    _ = Module.rank ℝ ↥(Submodule.span ℝ ({1, I} : Set K)) := by rw [span_one_I]; simp\n    _ ≤ #({1, I} : Finset K) := by\n      -- TODO: `simp` doesn't rewrite inside the type argument to `Module.rank`, but `rw` does.\n      -- We should introduce `Submodule.rank` to fix this.\n      have := rank_span_finset_le (R := ℝ) (M := K) {1, I}\n      rw [Finset.coe_pair] at this\n      simpa [span_one_I] using this\n    _ ≤ 2 := mod_cast Finset.card_le_two\n\nvariable (K) in\nlemma RCLike.finrank_le_two : Module.finrank ℝ K ≤ 2 :=\n  Module.finrank_le_of_rank_le <| rank_le_two _\n\nnamespace FiniteDimensional\n\nopen RCLike\n\nlibrary_note «RCLike instance» /--\nThis instance generates a type-class problem with a metavariable `?m` that should satisfy\n`RCLike ?m`. Since this can only be satisfied by `ℝ` or `ℂ`, this does not cause problems. -/\n\n/-- An `RCLike` field is finite-dimensional over `ℝ`, since it is spanned by `{1, I}`. -/\ninstance rclike_to_real : FiniteDimensional ℝ K := ⟨{1, I}, by simp [span_one_I]⟩\n\nvariable (K E)\nvariable [NormedAddCommGroup E] [NormedSpace K E]\n\n/-- A finite-dimensional vector space over an `RCLike` is a proper metric space.\n\nThis is not an instance because it would cause a search for `FiniteDimensional ?x E` before\n`RCLike ?x`. -/\n\nTarget:\ntheorem proper_rclike [FiniteDimensional K E] : ProperSpace E :=\n\nProof body:\n","rejected":"by\n  exact proper_rclike","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"7e7636e7a5f2a1767bb5ef73f1ed194bf27ea95bdd052084d26bc4ec894a94ed","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/RCLike","family_id":"proper_rclike","file_id":"mathlib/Mathlib/Analysis/RCLike/Lemmas.lean","sample_id":"6a8e5453b7a4db469efd31b432f7b75acf3e7eec129c2d2a6b546cf031d8a71a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"851de4796bc656b1357eef23d637285576683be0e867f23ab28b2999ef31f864","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5348f324874c43ec8d969e75ffd6ef39a0f98072430e686d7ccb4cf76d0c6077","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a95312207d0271d9516315b3dd214e2726915b26938f24e5ee114b92566a3ab3","source_sha256":"aa35c22bbec7297b71adc1698506626520835012db568036d1b21d7492380df0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  letI := (Sᵀ.toNormedAddCommGroup hS.transpose)\n  letI := (Sᵀ.toInnerProductSpace hS.transpose.posSemidef)\n  ext i j\n  rw [LDL.lowerInv, Basis.coePiBasisFun.toMatrix_eq_transpose, coe_gramSchmidtBasis]\n  rfl\n\nnoncomputable instance LDL.invertibleLowerInv : Invertible (LDL.lowerInv hS) := by\n  rw [LDL.lowerInv_eq_gramSchmidtBasis]\n  haveI :=\n    Basis.invertibleToMatrix (Pi.basisFun 𝕜 n)\n      (@gramSchmidtBasis 𝕜 (n → 𝕜) _ (Sᵀ.toNormedAddCommGroup hS.transpose)\n        (Sᵀ.toInnerProductSpace hS.transpose.posSemidef) n _ _ _ (Pi.basisFun 𝕜 n))\n  infer_instance","hard_negative":true,"metrics":{"chosen_tokens":119,"rejected_tokens":2,"token_jaccard":0.022222,"token_length_ratio":0.016807},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"3428710be41746cf04a0b99b80948cd0a8912b7e114c3e49afac2cf74b8265b5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho\npublic import Mathlib.Analysis.Matrix.PosDef\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Alexander Bentkamp. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alexander Bentkamp\n-/\n/-! # LDL decomposition\n\nThis file proves the LDL-decomposition of matrices: Any positive definite matrix `S` can be\ndecomposed as `S = LDLᴴ` where `L` is a lower-triangular matrix and `D` is a diagonal matrix.\n\n## Main definitions\n\n* `LDL.lower` is the lower triangular matrix `L`.\n* `LDL.lowerInv` is the inverse of the lower triangular matrix `L`.\n* `LDL.diag` is the diagonal matrix `D`.\n\n## Main result\n\n* `LDL.lower_conj_diag` states that any positive definite matrix can be decomposed as `LDLᴴ`.\n\n## TODO\n\n* Prove that `LDL.lower` is lower triangular from `LDL.lowerInv_triangular`.\n\n-/\n\n@[expose] public section\n\nopen Module\n\nvariable {𝕜 : Type*} [RCLike 𝕜]\nvariable {n : Type*} [LinearOrder n] [WellFoundedLT n] [LocallyFiniteOrderBot n]\n\nsection set_options\n\nset_option quotPrecheck false\nlocal notation \"⟪\" x \", \" y \"⟫ₑ\" => inner 𝕜 (WithLp.toLp 2 x) (WithLp.toLp 2 y)\n\nopen Matrix InnerProductSpace\n\nopen scoped ComplexOrder\n\nvariable {S : Matrix n n 𝕜} [Fintype n] (hS : S.PosDef)\n\n/-- The inverse of the lower triangular matrix `L` of the LDL-decomposition. It is obtained by\napplying Gram-Schmidt-Orthogonalization w.r.t. the inner product induced by `Sᵀ` on the standard\nbasis vectors `Pi.basisFun`. -/\nnoncomputable def LDL.lowerInv : Matrix n n 𝕜 :=\n  @gramSchmidt 𝕜 (n → 𝕜) _ (Sᵀ.toNormedAddCommGroup hS.transpose)\n    (Sᵀ.toInnerProductSpace hS.transpose.posSemidef) n _ _ _ (Pi.basisFun 𝕜 n)\n\nTarget:\ntheorem LDL.lowerInv_eq_gramSchmidtBasis :\n    LDL.lowerInv hS =\n      ((Pi.basisFun 𝕜 n).toMatrix\n          (@gramSchmidtBasis 𝕜 (n → 𝕜) _ (Sᵀ.toNormedAddCommGroup hS.transpose)\n            (Sᵀ.toInnerProductSpace hS.transpose.posSemidef) n _ _ _ (Pi.basisFun 𝕜 n)))ᵀ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_a95312207d02","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"eefca2e10e37cf2ff32fc78d18a0bacc23e8be541e733c2e09a93a61fc0765c4","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Matrix","family_id":"ldl","file_id":"mathlib/Mathlib/Analysis/Matrix/LDL.lean","sample_id":"a95312207d0271d9516315b3dd214e2726915b26938f24e5ee114b92566a3ab3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b2fc7f29216c9249b22ab675eb45a0b56030097a0051e019b9740cd584e92cb3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"34e8994f5ec220f3f31f1cf7f6b1b16bc75863bde20089fcb254304fbec368fa","source_sha256":"c0e3fa9405ddf4fdd99cd3aa6c9ff7258178f89496e31a48d0f8f121f656258a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [liftFun_iff_succ r, forall_iff_succ, cons_val_succ, cons_val_zero, ← succ_castSucc,\n    castSucc_zero]\n\nvariable [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α}","hard_negative":false,"metrics":{"chosen_tokens":45,"rejected_tokens":2,"token_jaccard":0.032258,"token_length_ratio":0.044444},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"34546ee3d7f2f4b3bc277e66aa167c125e73ba6842052a6263f51a0e9da915e7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.VecNotation\npublic import Mathlib.Logic.Equiv.Fin.Basic\npublic import Mathlib.Order.Fin.Basic\npublic import Mathlib.Order.PiLex\npublic import Mathlib.Order.Interval.Set.Defs\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes\n-/\n/-!\n# Order properties on tuples\n-/\n\n@[expose] public section\n\nassert_not_exists Monoid\n\nopen Function Set\n\nnamespace Fin\nvariable {m n : ℕ} {α : Fin (n + 1) → Type*} (x : α 0) (q : ∀ i, α i) (p : ∀ i : Fin n, α i.succ)\n  (i : Fin n) (y : α i.succ) (z : α 0)\n\nlemma pi_lex_lt_cons_cons {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ}\n    (s : ∀ {i : Fin n.succ}, α i → α i → Prop) :\n    Pi.Lex (· < ·) (@s) (Fin.cons x₀ x) (Fin.cons y₀ y) ↔\n      s x₀ y₀ ∨ x₀ = y₀ ∧ Pi.Lex (· < ·) (@fun i : Fin n ↦ @s i.succ) x y := by\n  simp_rw [Pi.Lex, Fin.exists_fin_succ, Fin.cons_succ, Fin.cons_zero, Fin.forall_iff_succ]\n  simp [and_assoc, exists_and_left]\n\nvariable [∀ i, Preorder (α i)]\n\nlemma insertNth_mem_Icc {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)}\n    {q₁ q₂ : ∀ j, α j} :\n    i.insertNth x p ∈ Icc q₁ q₂ ↔\n      x ∈ Icc (q₁ i) (q₂ i) ∧ p ∈ Icc (fun j ↦ q₁ (i.succAbove j)) fun j ↦ q₂ (i.succAbove j) := by\n  simp only [mem_Icc, insertNth_le_iff, le_insertNth_iff, and_assoc, @and_left_comm (x ≤ q₂ i)]\n\nlemma preimage_insertNth_Icc_of_mem {i : Fin (n + 1)} {x : α i} {q₁ q₂ : ∀ j, α j}\n    (hx : x ∈ Icc (q₁ i) (q₂ i)) :\n    i.insertNth x ⁻¹' Icc q₁ q₂ = Icc (fun j ↦ q₁ (i.succAbove j)) fun j ↦ q₂ (i.succAbove j) :=\n  Set.ext fun p ↦ by simp only [mem_preimage, insertNth_mem_Icc, hx, true_and]\n\nlemma preimage_insertNth_Icc_of_notMem {i : Fin (n + 1)} {x : α i} {q₁ q₂ : ∀ j, α j}\n    (hx : x ∉ Icc (q₁ i) (q₂ i)) : i.insertNth x ⁻¹' Icc q₁ q₂ = ∅ :=\n  Set.ext fun p ↦ by\n    simp only [mem_preimage, insertNth_mem_Icc, hx, false_and, mem_empty_iff_false]\n\nend Fin\n\nopen Fin Matrix\n\nvariable {α : Type*}\n\nopen scoped Relator in\n\nTarget:\nlemma liftFun_vecCons {n : ℕ} (r : α → α → Prop) [IsTrans α r] {f : Fin (n + 1) → α} {a : α} :\n    ((· < ·) ⇒ r) (vecCons a f) (vecCons a f) ↔ r a (f 0) ∧ ((· < ·) ⇒ r) f f :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Fin","family_id":"liftfun_veccons","file_id":"mathlib/Mathlib/Order/Fin/Tuple.lean","sample_id":"34e8994f5ec220f3f31f1cf7f6b1b16bc75863bde20089fcb254304fbec368fa"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d4687275514802a11968138ca35e457b8f90609ca53ed1eef831fff19e4d31f5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7b2f0763bd6dff91efe83580b9c6d31cae9870b9a7c3375d08365c053b25571d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fd18d30e8b2b216d023535e089915b87e05c113a9e9d53b5e82b2353ec29f830","source_sha256":"586377b5167eecc243d7b879009165834566ed22063836fb061931738feab6a2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Nat.card_prod]; exact Nat.card_congr Subgroup.groupEquivQuotientProdSubgroup","hard_negative":false,"metrics":{"chosen_tokens":16,"rejected_tokens":20,"token_jaccard":0.8125,"token_length_ratio":1.25},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"35035681bfee8134ac926abdc2b2d203fca61f4e5038b938b6a2cf480d2f0d06","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.Coset.Basic\npublic import Mathlib.SetTheory.Cardinal.Finite\n\nNamespace:\nSubgroup\n\nLocal context:\n/-\nCopyright (c) 2018 Mitchell Rowett. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mitchell Rowett, Kim Morrison\n-/\n/-!\n# Lagrange's theorem: the order of a subgroup divides the order of the group.\n\n* `Subgroup.card_subgroup_dvd_card`: Lagrange's theorem (for multiplicative groups);\n  there is an analogous version for additive groups\n\n-/\n\npublic section\n\nassert_not_exists Field\n\nopen scoped Pointwise\n\nvariable {α : Type*} [Group α] {s : Subgroup α}\n\nnamespace QuotientGroup\n\n@[to_additive]\ninstance fintype [Fintype α] (s : Subgroup α) [DecidableRel (leftRel s).r] : Fintype (α ⧸ s) :=\n  Quotient.fintype (leftRel s)\n\n@[to_additive]\ninstance (priority := 100) finite [Finite α] : Finite (α ⧸ s) :=\n  Quotient.finite _\n\n@[to_additive]\ninstance fintypeQuotientRightRel [Fintype (α ⧸ s)] :\n    Fintype (Quotient (QuotientGroup.rightRel s)) :=\n  .ofEquiv (α ⧸ s) (QuotientGroup.quotientRightRelEquivQuotientLeftRel s).symm\n\nvariable (s) in\n@[to_additive]\nlemma card_quotient_rightRel [Fintype (α ⧸ s)] :\n    Fintype.card (Quotient (QuotientGroup.rightRel s)) = Fintype.card (α ⧸ s) :=\n  Fintype.ofEquiv_card (QuotientGroup.quotientRightRelEquivQuotientLeftRel s).symm\n\nend QuotientGroup\n\nnamespace Subgroup\n\n@[to_additive AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup]\n\nTarget:\ntheorem card_eq_card_quotient_mul_card_subgroup (s : Subgroup α) :\n    Nat.card α = Nat.card (α ⧸ s) * Nat.card s :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [← Nat.card_prod]; exact Nat.card_congr Subgroup.groupEquivQuotientProdSubgroup","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Coset","family_id":"card_eq_card_quotient_mul_card_subgroup","file_id":"mathlib/Mathlib/GroupTheory/Coset/Card.lean","sample_id":"fd18d30e8b2b216d023535e089915b87e05c113a9e9d53b5e82b2353ec29f830"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2c1ae7a6d731fe18987c9a16e015e76772289e20b04a8b5d9fc3cf3d2c6b5c61","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"048b9be495a268385d167a57cca1fa78137f791b21974ba0172ed051eac3eda5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6eee5d2ff91023d609bd493224edd9b87b2fa60a603a305b0c65f8936a71c40f","source_sha256":"bc1107b51b023f15ba6b116e921d72825e5fd28811d110becc324cfa9ebc2fb7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold mkFinSnoc\n  exact coe_mk (v := Fin.snoc (N.subtype ∘ b) y) _ _\n\n/-- Let `b` be a basis for a submodule `N ≤ O`. If `y ∈ O` is linear independent of `N`\nand `y` and `N` together span the whole of `O`, then there is a basis for `O`\nwhose basis vectors are given by `Fin.snoc b y`. -/\nnoncomputable def mkFinSnocOfLE {n : ℕ} {N O : Submodule R M} (b : Basis (Fin n) R N)\n    (hNO : N ≤ O) (y : M) (yO : y ∈ O) (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0)\n    (hsp : ∀ z ∈ O, ∃ c : R, z + c • y ∈ N) : Basis (Fin (n + 1)) R O :=\n  mkFinSnoc (b.map (Submodule.comapSubtypeEquivOfLe hNO).symm) ⟨y, yO⟩\n    (fun c x hc hx => hli c x (Submodule.mem_comap.mp hc) (congr_arg ((↑) : O → M) hx))\n    fun z => hsp z z.2","hard_negative":true,"metrics":{"chosen_tokens":262,"rejected_tokens":5,"token_jaccard":0.033333,"token_length_ratio":0.019084},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"354b70acb2054097e8e076bbd89f081d20d6cc427827733badaf7c8b4d73e3bd","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Basis.Basic\npublic import Mathlib.LinearAlgebra.Pi\n\nNamespace:\nModule.Basis\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Kevin H. Wilson\n-/\n/-!\n# Bases indexed by `Fin`\n-/\n\n@[expose] public section\n\nassert_not_exists Ordinal\n\nnoncomputable section\n\nuniverse u\n\nopen Function Set Submodule Finsupp\n\nvariable {ι : Type*} {ι' : Type*} {R : Type*} {R₂ : Type*} {M : Type*} {M' : Type*}\n\nnamespace Module\n\nopen LinearMap\n\nvariable {v : ι → M}\nvariable [Ring R] [CommRing R₂] [AddCommGroup M]\nvariable [Module R M] [Module R₂ M]\nvariable {x y : M}\nvariable (b : Basis ι R M)\n\nnamespace Basis\n\nsection Fin\n\n/-- Let `b` be a basis for a submodule `N` of `M`. If `y : M` is linear independent of `N`\nand `y` and `N` together span the whole of `M`, then there is a basis for `M`\nwhose basis vectors are given by `Fin.cons y b`. -/\nnoncomputable def mkFinCons {n : ℕ} {N : Submodule R M} (y : M) (b : Basis (Fin n) R N)\n    (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :\n    Basis (Fin (n + 1)) R M :=\n  have span_b : N = Submodule.span R (Set.range (N.subtype ∘ b)) := by\n    rw [Set.range_comp, Submodule.span_image, b.span_eq, Submodule.map_subtype_top]\n  Basis.mk (v := Fin.cons y (N.subtype ∘ b))\n    ((b.linearIndependent.map' N.subtype (Submodule.ker_subtype _)).finCons' _ _\n      (by\n        intro c x hx hc\n        rw [← span_b] at hx\n        exact hli c x hx hc))\n    fun x _ => by simpa [Submodule.mem_span_insert', span_b] using hsp x\n\n@[simp]\ntheorem coe_mkFinCons {n : ℕ} {N : Submodule R M} (y : M) (b : Basis (Fin n) R N)\n    (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :\n    (mkFinCons y b hli hsp : Fin (n + 1) → M) = Fin.cons y ((↑) ∘ b) := by\n  unfold mkFinCons\n  exact coe_mk (v := Fin.cons y (N.subtype ∘ b)) _ _\n\n/-- Let `b` be a basis for a submodule `N ≤ O`. If `y ∈ O` is linear independent of `N`\nand `y` and `N` together span the whole of `O`, then there is a basis for `O`\nwhose basis vectors are given by `Fin.cons y b`. -/\nnoncomputable def mkFinConsOfLE {n : ℕ} {N O : Submodule R M} (y : M) (yO : y ∈ O)\n    (b : Basis (Fin n) R N) (hNO : N ≤ O) (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0)\n    (hsp : ∀ z ∈ O, ∃ c : R, z + c • y ∈ N) : Basis (Fin (n + 1)) R O :=\n  mkFinCons ⟨y, yO⟩ (b.map (Submodule.comapSubtypeEquivOfLe hNO).symm)\n    (fun c x hc hx => hli c x (Submodule.mem_comap.mp hc) (congr_arg ((↑) : O → M) hx))\n    fun z => hsp z z.2\n\n@[simp]\ntheorem coe_mkFinConsOfLE {n : ℕ} {N O : Submodule R M} (y : M) (yO : y ∈ O) (b : Basis (Fin n) R N)\n    (hNO : N ≤ O) (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0)\n    (hsp : ∀ z ∈ O, ∃ c : R, z + c • y ∈ N) :\n    (mkFinConsOfLE y yO b hNO hli hsp : Fin (n + 1) → O) =\n      Fin.cons ⟨y, yO⟩ (Submodule.inclusion hNO ∘ b) :=\n  coe_mkFinCons _ _ _ _\n\n/-- Let `b` be a basis for a submodule `N` of `M`. If `y : M` is linear independent of `N`\nand `y` and `N` together span the whole of `M`, then there is a basis for `M`\nwhose basis vectors are given by `Fin.snoc b y`. -/\nnoncomputable def mkFinSnoc {n : ℕ} {N : Submodule R M} (b : Basis (Fin n) R N) (y : M)\n    (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :\n    Basis (Fin (n + 1)) R M :=\n  have span_b : N = Submodule.span R (Set.range (N.subtype ∘ b)) := by\n    rw [Set.range_comp, Submodule.span_image, b.span_eq, Submodule.map_subtype_top]\n  Basis.mk (v := Fin.snoc (N.subtype ∘ b) y)\n    ((b.linearIndependent.map' N.subtype (Submodule.ker_subtype _)).finSnoc' _ _\n      (by\n        intro c x hx hc\n        rw [← span_b] at hx\n        exact hli c x hx hc))\n    fun x _ ↦ by simpa [Submodule.mem_span_insert', span_b] using hsp x\n\n@[simp]\n\nTarget:\ntheorem coe_mkFinSnoc {n : ℕ} {N : Submodule R M} (b : Basis (Fin n) R N) (y : M)\n    (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :\n    (mkFinSnoc b y hli hsp : Fin (n + 1) → M) = Fin.snoc ((↑) ∘ b) y :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_6eee5d2ff910","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"ba4e06754d8abe64f86af9d038e888dd3053fdf0e4425f235921d49f852880e1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Basis","family_id":"coe_mkfinsnoc","file_id":"mathlib/Mathlib/LinearAlgebra/Basis/Fin.lean","sample_id":"6eee5d2ff91023d609bd493224edd9b87b2fa60a603a305b0c65f8936a71c40f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"10363aa97aac5a8e4a6ed13d0ebf67594ed188c89c81a730878f4693d1b889fa","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"19c9230dd9532a6df572a3d4cf4ed8cfd2fc18850e231e933699018c9fe60632","source_sha256":"18fda008e7b40ec7b81fa0fd9f7c55475c44e029c9af90e5639e54fa98674982","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x\n  simp only [LinearMap.comp_apply, apply, map_add, map_smul, add_assoc]\n  simp only [smul_add, ← mul_smul, ← add_smul, ← mul_add (f x), mul_assoc]","hard_negative":false,"metrics":{"chosen_tokens":38,"rejected_tokens":2,"token_jaccard":0.04,"token_length_ratio":0.052632},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"354c58e32a655125a2d3160becc057859b10fbad5073f127e89904e27b6c879b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup\npublic import Mathlib.LinearAlgebra.Charpoly.BaseChange\npublic import Mathlib.LinearAlgebra.Dual.BaseChange\npublic import Mathlib.LinearAlgebra.Dual.Lemmas\npublic import Mathlib.LinearAlgebra.FixedSubmodule\n\nNamespace:\nLinearMap.transvection\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir\n-/\n\n/-!\n# Transvections in a module\n\n* When `f : Module.Dual R V` and `v : V`,\n  `LinearMap.transvection f v` is the linear map given by `x ↦ x + f x • v`,\n\n* `LinearMap.transvection.det` shows that the determinant of\n  `LinearMap.transvection f v` is equal to `1 + f v`.\n\n* If, moreover, `f v = 0`, then `LinearEquiv.transvection` shows that it is\n  a linear equivalence.\n\n* `LinearMap.transvections R V`: the set of transvections.\n\n* `LinearEquiv.dilatransvections R V`: the set of linear equivalences\n  whose associated linear map is of the form `LinearMap.transvection f v`.\n\n* `LinearEquiv.transvection.det` shows that it has determinant `1`.\n\n## Note on terminology\n\nIn the mathematical literature, linear maps of the form `LinearMap.transvection f v`\nare only called “transvections” when `f v = 0`. Otherwise, they are sometimes\ncalled “dilations” (especially if `f v ≠ -1`).\n\nThe definition is almost the same as that of `Module.preReflection f v`,\nup to a sign change, which are interesting when `f v = 2`, because they give “reflections”.\n\n-/\n\n@[expose] public section\n\nnamespace LinearMap\n\nopen Module\n\nvariable {R V : Type*} [Semiring R] [AddCommMonoid V] [Module R V]\n\n/-- The transvection associated with a linear form `f` and a vector `v`.\n\nNB. In mathematics, these linear maps are only called “transvections” when `f v = 0`.\nSee also `Module.preReflection` for a similar definition, up to a sign. -/\ndef transvection (f : Dual R V) (v : V) : V →ₗ[R] V where\n  toFun x := x + f x • v\n  map_add' x y := by simp [add_add_add_comm, add_smul]\n  map_smul' r x := by simp [smul_eq_mul, smul_add, mul_smul]\n\nnamespace transvection\n\nopen Submodule LinearMap\n\ntheorem apply (f : Dual R V) (v x : V) :\n    transvection f v x = x + f x • v :=\n  rfl\n\ntheorem comp_of_left_eq_apply {f : Dual R V} {v w : V} {x : V} (hw : f w = 0) :\n    transvection f v (transvection f w x) = transvection f (v + w) x := by\n  simp [transvection, map_add, hw, add_assoc]\n\ntheorem comp_of_left_eq {f : Dual R V} {v w : V} (hw : f w = 0) :\n    (transvection f v) ∘ₗ (transvection f w) = transvection f (v + w) := by\n  ext; simp [comp_of_left_eq_apply hw]\n\ntheorem comp_of_right_eq_apply {f g : Dual R V} {v : V} {x : V} (hf : f v = 0) :\n    (transvection f v) (transvection g v x) = transvection (f + g) v x := by\n  simp [transvection, map_add, hf, add_smul, add_assoc]\n\ntheorem comp_of_right_eq {f g : Dual R V} {v : V} (hf : f v = 0) :\n    (transvection f v) ∘ₗ (transvection g v) = transvection (f + g) v := by\n  ext; simp [comp_of_right_eq_apply hf]\n\n@[simp]\ntheorem of_left_eq_zero (v : V) :\n    transvection (0 : Dual R V) v = id := by\n  ext\n  simp [transvection]\n\n@[simp]\ntheorem of_right_eq_zero (f : Dual R V) :\n    transvection f 0 = id := by\n  ext\n  simp [transvection]\n\nTarget:\ntheorem comp_smul_smul {f : Dual R V} {v : V} {r s : R} :\n    transvection f (r • v) ∘ₗ transvection f (s • v) =\n      transvection f ((r + s + s * f v * r) • v) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Transvection","family_id":"comp_smul_smul","file_id":"mathlib/Mathlib/LinearAlgebra/Transvection/Basic.lean","sample_id":"19c9230dd9532a6df572a3d4cf4ed8cfd2fc18850e231e933699018c9fe60632"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eb4847d4ec18aeaffa781e4a9bdcbca4898991cdee673fe5996e2fd2c98cb1bf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b5c69aaa1bed68b17e4f32a5e7a2a5a68a9b885f214b93cc029b4d1067559363","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cb8c4de23317146f19f32df2485e1f047d44025176348b35088a93dfd2c0286c","source_sha256":"e4f7eab8ce40520bcf25316e25287e36bf06b7a538fd15f40c2755c88337db7b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Function.Injective] at hv\n  push Not at hv\n  rcases hv with ⟨i₁, i₂, heq, hne⟩\n  exact f.map_eq_zero_of_eq v heq hne\n\n/-!\n### Algebraic structure inherited from `MultilinearMap`\n\n`AlternatingMap` carries the same `AddCommMonoid`, `AddCommGroup`, and `Module` structure\nas `MultilinearMap`\n-/","hard_negative":true,"metrics":{"chosen_tokens":72,"rejected_tokens":2,"token_jaccard":0.022222,"token_length_ratio":0.027778},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"354ef4ba11542a9a99316901bb57688217619e2a7828de2ec5ac511b146c0d26","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.Perm.Sign\npublic import Mathlib.LinearAlgebra.LinearIndependent.Defs\npublic import Mathlib.LinearAlgebra.Multilinear.Basis\n\nNamespace:\nAlternatingMap\n\nLocal context:\n/-\nCopyright (c) 2020 Zhangir Azerbayev. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser, Zhangir Azerbayev\n-/\n/-!\n# Alternating Maps\n\nWe construct the bundled function `AlternatingMap`, which extends `MultilinearMap` with all the\narguments of the same type.\n\n## Main definitions\n* `AlternatingMap R M N ι` is the space of `R`-linear alternating maps from `ι → M` to `N`.\n* `f.map_eq_zero_of_eq` expresses that `f` is zero when two inputs are equal.\n* `f.map_swap` expresses that `f` is negated when two inputs are swapped.\n* `f.map_perm` expresses how `f` varies by a sign change under a permutation of its inputs.\n* An `AddCommMonoid`, `AddCommGroup`, and `Module` structure over `AlternatingMap`s that\n  matches the definitions over `MultilinearMap`s.\n* `AlternatingMap.domDomCongr`, for permuting the elements within a family.\n* `MultilinearMap.alternatization`, which makes an alternating map out of a non-alternating one.\n* `AlternatingMap.curryLeft`, for binding the leftmost argument of an alternating map indexed\n  by `Fin n.succ`.\n\n## Implementation notes\n`AlternatingMap` is defined in terms of `map_eq_zero_of_eq`, as this is easier to work with than\nusing `map_swap` as a definition, and does not require `Neg N`.\n\n`AlternatingMap`s are provided with a coercion to `MultilinearMap`, along with a set of\n`norm_cast` lemmas that act on the algebraic structure:\n\n* `AlternatingMap.coe_add`\n* `AlternatingMap.coe_zero`\n* `AlternatingMap.coe_sub`\n* `AlternatingMap.coe_neg`\n* `AlternatingMap.coe_smul`\n-/\n\n@[expose] public section\n\nopen Module\n\n-- semiring / add_comm_monoid\n\nvariable {R : Type*} [Semiring R]\nvariable {M : Type*} [AddCommMonoid M] [Module R M]\nvariable {N : Type*} [AddCommMonoid N] [Module R N]\nvariable {P : Type*} [AddCommMonoid P] [Module R P]\n\n-- semiring / add_comm_group\n\nvariable {M' : Type*} [AddCommGroup M'] [Module R M']\nvariable {N' : Type*} [AddCommGroup N'] [Module R N']\nvariable {ι ι' ι'' : Type*}\n\nsection\n\nvariable (R M N ι)\n\n/-- An alternating map from `ι → M` to `N`, denoted `M [⋀^ι]→ₗ[R] N`,\nis a multilinear map that vanishes when two of its arguments are equal. -/\nstructure AlternatingMap extends MultilinearMap R (fun _ : ι => M) N where\n  /-- The map is alternating: if `v` has two equal coordinates, then `f v = 0`. -/\n  map_eq_zero_of_eq' : ∀ (v : ι → M) (i j : ι), v i = v j → i ≠ j → toFun v = 0\n\n@[inherit_doc]\nnotation M \" [⋀^\" ι \"]→ₗ[\" R \"] \" N:100 => AlternatingMap R M N ι\n\nend\n\n/-- The multilinear map associated to an alternating map -/\nadd_decl_doc AlternatingMap.toMultilinearMap\n\nnamespace AlternatingMap\n\nvariable (f f' : M [⋀^ι]→ₗ[R] N)\nvariable (g g₂ : M [⋀^ι]→ₗ[R] N')\nvariable (g' : M' [⋀^ι]→ₗ[R] N')\nvariable (v : ι → M) (v' : ι → M')\n\nopen Function\n\n/-! Basic coercion simp lemmas, largely copied from `RingHom` and `MultilinearMap` -/\n\n\nsection Coercions\n\ninstance instFunLike : FunLike (M [⋀^ι]→ₗ[R] N) (ι → M) N where\n  coe f := f.toFun\n  coe_injective f g h := by\n    rcases f with ⟨⟨_, _, _⟩, _⟩\n    rcases g with ⟨⟨_, _, _⟩, _⟩\n    congr\n\ninitialize_simps_projections AlternatingMap (toFun → apply)\n\n@[simp]\ntheorem toFun_eq_coe : f.toFun = f :=\n  rfl\n\n@[simp]\ntheorem coe_mk (f : MultilinearMap R (fun _ : ι => M) N) (h) :\n    ⇑(⟨f, h⟩ : M [⋀^ι]→ₗ[R] N) = f :=\n  rfl\n\nprotected theorem congr_fun {f g : M [⋀^ι]→ₗ[R] N} (h : f = g) (x : ι → M) : f x = g x :=\n  congr_arg (fun h : M [⋀^ι]→ₗ[R] N => h x) h\n\nprotected theorem congr_arg (f : M [⋀^ι]→ₗ[R] N) {x y : ι → M} (h : x = y) : f x = f y :=\n  congr_arg (fun x : ι → M => f x) h\n\ntheorem coe_injective : Injective ((↑) : M [⋀^ι]→ₗ[R] N → (ι → M) → N) :=\n  DFunLike.coe_injective\n\n@[norm_cast]\ntheorem coe_inj {f g : M [⋀^ι]→ₗ[R] N} : (f : (ι → M) → N) = g ↔ f = g :=\n  coe_injective.eq_iff\n\n@[ext]\ntheorem ext {f f' : M [⋀^ι]→ₗ[R] N} (H : ∀ x, f x = f' x) : f = f' :=\n  DFunLike.ext _ _ H\n\nattribute [coe] AlternatingMap.toMultilinearMap\n\ninstance instCoe : Coe (M [⋀^ι]→ₗ[R] N) (MultilinearMap R (fun _ : ι => M) N) :=\n  ⟨fun x => x.toMultilinearMap⟩\n\n@[simp, norm_cast]\ntheorem coe_multilinearMap : ⇑(f : MultilinearMap R (fun _ : ι => M) N) = f :=\n  rfl\n\ntheorem coe_multilinearMap_injective :\n    Function.Injective ((↑) : M [⋀^ι]→ₗ[R] N → MultilinearMap R (fun _ : ι => M) N) :=\n  fun _ _ h => ext <| MultilinearMap.congr_fun h\n\ntheorem coe_multilinearMap_mk (f : (ι → M) → N) (h₁ h₂ h₃) :\n    ((⟨⟨f, h₁, h₂⟩, h₃⟩ : M [⋀^ι]→ₗ[R] N) : MultilinearMap R (fun _ : ι => M) N) =\n      ⟨f, @h₁, @h₂⟩ := by\n  simp\n\nend Coercions\n\n/-!\n### Simp-normal forms of the structure fields\n\nThese are expressed in terms of `⇑f` instead of `f.toFun`.\n-/\n\n\n@[simp]\ntheorem map_update_add [DecidableEq ι] (i : ι) (x y : M) :\n    f (update v i (x + y)) = f (update v i x) + f (update v i y) :=\n  f.map_update_add' v i x y\n\n@[simp]\ntheorem map_update_sub [DecidableEq ι] (i : ι) (x y : M') :\n    g' (update v' i (x - y)) = g' (update v' i x) - g' (update v' i y) :=\n  g'.toMultilinearMap.map_update_sub v' i x y\n\n@[simp]\ntheorem map_update_neg [DecidableEq ι] (i : ι) (x : M') :\n    g' (update v' i (-x)) = -g' (update v' i x) :=\n  g'.toMultilinearMap.map_update_neg v' i x\n\n@[simp]\ntheorem map_update_smul [DecidableEq ι] (i : ι) (r : R) (x : M) :\n    f (update v i (r • x)) = r • f (update v i x) :=\n  f.map_update_smul' v i r x\n\n-- Cannot be @[simp] because `i` and `j` cannot be inferred by `simp`.\ntheorem map_eq_zero_of_eq (v : ι → M) {i j : ι} (h : v i = v j) (hij : i ≠ j) : f v = 0 :=\n  f.map_eq_zero_of_eq' v i j h hij\n\ntheorem map_coord_zero {m : ι → M} (i : ι) (h : m i = 0) : f m = 0 :=\n  f.toMultilinearMap.map_coord_zero i h\n\n@[simp]\ntheorem map_update_zero [DecidableEq ι] (m : ι → M) (i : ι) : f (update m i 0) = 0 :=\n  f.toMultilinearMap.map_update_zero m i\n\n@[simp]\ntheorem map_zero [Nonempty ι] : f 0 = 0 :=\n  f.toMultilinearMap.map_zero\n\nTarget:\ntheorem map_eq_zero_of_not_injective (v : ι → M) (hv : ¬Function.Injective v) : f v = 0 :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_cb8c4de23317","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"04ff01998f8740cf4c1e0c58f8f437b3dd163e599eed72c322726a88eac06b4a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Alternating","family_id":"map_eq_zero_of_not_injective","file_id":"mathlib/Mathlib/LinearAlgebra/Alternating/Basic.lean","sample_id":"cb8c4de23317146f19f32df2485e1f047d44025176348b35088a93dfd2c0286c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fd406c6900b20879ac4c2cbd6762020da2098ae8a94437f8bd328ff516fb751d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ed1245fdebd56f8a9139f5759dda744d8fb6856b2b39689345dc4ef3a2c36e99","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9b1bbd8b0b0a1747cccd2009098a56d3ba588503f823228d885fa9ccd0f30c49","source_sha256":"2330b04bbf9f5616db3c6c30d466b364ee306b290c33249fea3b2f667c7433f7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := Adjunction.rightAdjointUniq_hom_counit (mapPullbackAdj (𝟙 X)) (id X).mapPullbackAdj\n  rw [pullbackId, Adjunction.rightAdjointUniq_hom_counit]","hard_negative":true,"metrics":{"chosen_tokens":27,"rejected_tokens":3,"token_jaccard":0.052632,"token_length_ratio":0.111111},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"3550c406481c2c9774a40659175f7ec68fe597fd70b5c89aa70e32bcefe9f7f0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Comma.Over.Pullback\npublic import Mathlib.CategoryTheory.Monoidal.Cartesian.Basic\npublic import Mathlib.CategoryTheory.Adjunction.Unique\npublic import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Defs\n\nNamespace:\nCategoryTheory.ChosenPullbacksAlong\n\nLocal context:\n/-\nCopyright (c) 2025 Sina Hazratpour. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sina Hazratpour\n-/\n/-!\n# Chosen pullbacks along a morphism\n\n## Main declarations\n\n- `ChosenPullbacksAlong` : For a morphism `f : Y ⟶ X` in `C`, the type class\n  `ChosenPullbacksAlong f` provides the data of a pullback functor `Over X ⥤ Over Y`\n  as a right adjoint to `Over.map f`.\n\n## Main results\n\n- We prove that `ChosenPullbacksAlong` has good closure properties: isos have chosen pullbacks,\n  and composition of morphisms with chosen pullbacks have chosen pullbacks.\n\n- We prove that chosen pullbacks yield usual pullbacks: `ChosenPullbacksAlong.isPullback`\n  proves that for morphisms `f` and `g` with the same codomain, the object\n  `ChosenPullbacksAlong.pullbackObj f g` together with morphisms\n  `ChosenPullbacksAlong.fst f g` and `ChosenPullbacksAlong.snd f g` form a pullback square\n  over `f` and `g`.\n\n- We prove that in cartesian monoidal categories, morphisms to the terminal tensor unit and\n  the product projections have chosen pullbacks.\n\n-/\n\n@[expose] public section\n\nuniverse v₁ v₂ u₁ u₂\n\nnamespace CategoryTheory\n\nopen Category Limits CartesianMonoidalCategory MonoidalCategory Over\n\nvariable {C : Type u₁} [Category.{v₁} C]\n\n/-- A functorial choice of pullbacks along a morphism `f : Y ⟶ X` in `C` given by a functor\n`Over X ⥤ Over Y` which is a right adjoint to the functor `Over.map f`. -/\nclass ChosenPullbacksAlong {Y X : C} (f : Y ⟶ X) where\n  /-- The pullback functor along `f`. -/\n  pullback : Over X ⥤ Over Y\n  /-- The adjunction between `Over.map f` and `pullback f`. -/\n  mapPullbackAdj (f) : Over.map f ⊣ pullback\n\nvariable (C) in\n/-- A category has chosen pullbacks if every morphism has a chosen pullback. -/\nabbrev ChosenPullbacks := Π {X Y : C} (f : Y ⟶ X), ChosenPullbacksAlong f\n\nnamespace ChosenPullbacksAlong\n\n/-- Relating the existing noncomputable `HasPullbacksAlong` typeclass to `ChosenPullbacksAlong`. -/\n@[simps, implicit_reducible]\nnoncomputable def ofHasPullbacksAlong {Y X : C} (f : Y ⟶ X) [HasPullbacksAlong f] :\n    ChosenPullbacksAlong f where\n  pullback := Over.pullback f\n  mapPullbackAdj := Over.mapPullbackAdj f\n\n/-- The identity morphism has a functorial choice of pullbacks. -/\n@[implicit_reducible]\ndef id (X : C) : ChosenPullbacksAlong (𝟙 X) where\n  pullback := 𝟭 _\n  mapPullbackAdj := (Adjunction.id).ofNatIsoLeft (Over.mapId _).symm\n\n/-- Any chosen pullback functor of the identity morphism is naturally isomorphic to the identity\nfunctor. -/\ndef pullbackId (X : C) [ChosenPullbacksAlong (𝟙 X)] :\n    pullback (𝟙 X) ≅ 𝟭 (Over X) :=\n  (mapPullbackAdj (𝟙 X)).rightAdjointUniq (id X).mapPullbackAdj\n\n@[reassoc (attr := simp)]\ntheorem unit_pullbackId_hom_app (X : C) [ChosenPullbacksAlong (𝟙 X)] (Y : Over X) :\n    (mapPullbackAdj (𝟙 X)).unit.app Y ≫ (pullbackId X).hom.app ((Over.map (𝟙 X)).obj Y) =\n      (id X).mapPullbackAdj.unit.app Y := by\n  rw [pullbackId, Adjunction.unit_rightAdjointUniq_hom_app]\n\n@[reassoc (attr := simp)]\ntheorem unit_pullbackId_hom (X : C) [ChosenPullbacksAlong (𝟙 X)] :\n    (mapPullbackAdj (𝟙 X)).unit ≫ (Over.map (𝟙 X)).whiskerLeft (pullbackId X).hom =\n      (id X).mapPullbackAdj.unit := by\n  rw [pullbackId, Adjunction.unit_rightAdjointUniq_hom]\n\n@[reassoc (attr := simp)]\n\nTarget:\ntheorem pullbackId_hom_counit (X : C) [ChosenPullbacksAlong (𝟙 X)] :\n    Functor.whiskerRight (pullbackId X).hom (Over.map (𝟙 X)) ≫ (id X).mapPullbackAdj.counit =\n      (mapPullbackAdj (𝟙 X)).counit :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_9b1bbd8b0b0a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"9b11e0022818b96a686a3e3f7c92484bae136688ad924ac338ecf6673e03b745","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/LocallyCartesianClosed","family_id":"pullbackid_hom_counit","file_id":"mathlib/Mathlib/CategoryTheory/LocallyCartesianClosed/ChosenPullbacksAlong.lean","sample_id":"9b1bbd8b0b0a1747cccd2009098a56d3ba588503f823228d885fa9ccd0f30c49"},"split":"train"}
{"binding":{"chosen_proof_sha256":"aedbd85f784d006a3b8e0a1c284ce353d4c9554e1c007c5246c151cde67cde07","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3d5cbd94823c1da9fd124543545cda0f81180d6bc6e05f5f4ff0b354b5e02f71","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a3caeca3aca5af45210b187259a35456fb858a420f6704c94186f1b14ec00a25","source_sha256":"898a445ef5a6164c4919dd80610d649e8321c8a06f3edd7c9d6402f5554964c9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  exact ⟨fun h ↦ ⟨⟨@effectiveEpiFamilyStructOfEffectiveEpiDesc _ _ _ _ X π _ h _ _ (fun g ↦\n    (FinitaryPreExtensive.isIso_sigmaDesc_fst (fun a ↦ Sigma.ι X a) g inferInstance).epi_of_iso)⟩⟩,\n    fun _ ↦ inferInstance⟩\n\nvariable {D : Type*} [Category* D] [FinitaryPreExtensive D]\nvariable (F : C ⥤ D) [PreservesFiniteCoproducts F]","hard_negative":true,"metrics":{"chosen_tokens":80,"rejected_tokens":5,"token_jaccard":0.075,"token_length_ratio":0.0625},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"355b1460fee3cbc6766e60cf44b6e229c86c0d18a4919fb0f51dd99e412010b5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.EffectiveEpi.Preserves\npublic import Mathlib.CategoryTheory.EffectiveEpi.Coproduct\npublic import Mathlib.CategoryTheory.Extensive\npublic import Mathlib.CategoryTheory.Limits.Preserves.Finite\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# Preserving and reflecting effective epis on extensive categories\n\nWe prove that a functor between `FinitaryPreExtensive` categories preserves (resp. reflects) finite\neffective epi families if it preserves (resp. reflects) effective epis.\n-/\n\npublic section\n\nnamespace CategoryTheory\n\nopen Limits\n\nvariable {C : Type*} [Category* C] [FinitaryPreExtensive C]\n\nTarget:\ntheorem effectiveEpi_desc_iff_effectiveEpiFamily {α : Type} [Finite α]\n    {B : C} (X : α → C) (π : (a : α) → X a ⟶ B) :\n    EffectiveEpi (Sigma.desc π) ↔ EffectiveEpiFamily X π :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_a3caeca3aca5","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f7987da539a33e873744d2af1896ff01b2868af55242809ccae7e41468d7fdb0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/EffectiveEpi","family_id":"effectiveepi_desc_iff_effectiveepifamily","file_id":"mathlib/Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean","sample_id":"a3caeca3aca5af45210b187259a35456fb858a420f6704c94186f1b14ec00a25"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7bfddf9976579c9b33ef1adb03a12367ce2324df54f03888b73e637b017ab565","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6ef6ae9c0f9613c7c3706f6b6b19cc3ed2c14562e6ea2548c5e346b1a3fa96cb","source_sha256":"5ec4c4d2769bb0652aea247f81d7ff99e572b3159717c948f413b35f1ff454f9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← disjoint_iff]\n  exact disjoint_sdiff_iff_le hz hx\n\n-- cf. `IsCompl.left_le_iff` and `IsCompl.right_le_iff`","hard_negative":true,"metrics":{"chosen_tokens":25,"rejected_tokens":8,"token_jaccard":0.041667,"token_length_ratio":0.32},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"3572f35d3a1f468f7ac6c404447b178fd422425b30981250e1fcc32346c4a46a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.BooleanAlgebra.Defs\npublic import Mathlib.Tactic.GRewrite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Bryan Gin-ge Chen\n-/\n/-!\n# Basic properties of Boolean algebras\n\nThis file provides some basic definitions, functions as well as lemmas for functions and type\nclasses related to Boolean algebras as defined in `Mathlib/Order/BooleanAlgebra/Defs.lean`.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>\n* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]\n* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]\n\n## Tags\n\ngeneralized Boolean algebras, Boolean algebras, lattices, sdiff, compl\n\n-/\n\npublic section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type*} {x y z : α}\n\n/-!\n### Generalized Boolean algebras\n\nSome of the lemmas in this section are from:\n\n* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]\n* <https://ncatlab.org/nlab/show/relative+complement>\n* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>\n\n-/\n\n-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,\n-- however we'd need another type class for lattices with bot, and all the API for that.\nsection GeneralizedBooleanAlgebra\n\nvariable [GeneralizedBooleanAlgebra α]\n\n@[simp]\ntheorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \\ y = x :=\n  GeneralizedBooleanAlgebra.sup_inf_sdiff _ _\n\n@[simp]\ntheorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \\ y = ⊥ :=\n  GeneralizedBooleanAlgebra.inf_inf_sdiff _ _\n\n@[simp]\ntheorem sup_sdiff_inf (x y : α) : x \\ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]\n\n@[simp]\ntheorem inf_sdiff_inf (x y : α) : x \\ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α where\n  __ := GeneralizedBooleanAlgebra.toBot\n  bot_le a := by\n    rw [← inf_inf_sdiff a a, inf_assoc]\n    exact inf_le_left\n\ntheorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \\ y) :=\n  disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le\n\n-- TODO: in distributive lattices, relative complements are unique when they exist\ntheorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \\ y = z := by\n  conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]\n  rw [sup_comm] at s\n  conv_rhs at i => rw [← inf_inf_sdiff x y, inf_comm]\n  rw [inf_comm] at i\n  exact (eq_of_inf_eq_sup_eq i s).symm\n\n-- Use `sdiff_le`\nprivate theorem sdiff_le' : x \\ y ≤ x :=\n  calc\n    x \\ y ≤ x ⊓ y ⊔ x \\ y := le_sup_right\n    _ = x := sup_inf_sdiff x y\n\nset_option backward.privateInPublic true in\n-- Use `sdiff_sup_self`\nprivate theorem sdiff_sup_self' : y \\ x ⊔ x = y ⊔ x :=\n  calc\n    y \\ x ⊔ x = y \\ x ⊔ (x ⊔ x ⊓ y) := by rw [sup_inf_self]\n    _ = y ⊓ x ⊔ y \\ x ⊔ x := by ac_rfl\n    _ = y ⊔ x := by rw [sup_inf_sdiff]\n\n@[simp]\ntheorem sdiff_inf_sdiff : x \\ y ⊓ y \\ x = ⊥ :=\n  Eq.symm <|\n    calc\n      ⊥ = x ⊓ (y ⊓ x ⊔ y \\ x) ⊓ x \\ y := by rw [← inf_inf_sdiff, sup_inf_sdiff]\n      _ = (x ⊓ (y ⊓ x) ⊔ x ⊓ y \\ x) ⊓ x \\ y := by rw [inf_sup_left]\n      _ = (y ⊓ (x ⊓ x) ⊔ x ⊓ y \\ x) ⊓ x \\ y := by ac_rfl\n      _ = x ⊓ y \\ x ⊓ x \\ y := by\n          rw [inf_idem, inf_sup_right, ← inf_comm x y, inf_inf_sdiff, bot_sup_eq]\n      _ = x ⊓ x \\ y ⊓ y \\ x := by ac_rfl\n      _ = x \\ y ⊓ y \\ x := by rw [inf_of_le_right sdiff_le']\n\ntheorem disjoint_sdiff_sdiff : Disjoint (x \\ y) (y \\ x) :=\n  disjoint_iff_inf_le.mpr sdiff_inf_sdiff.le\n\n@[simp]\ntheorem inf_sdiff_self_right : x ⊓ y \\ x = ⊥ :=\n  calc\n    x ⊓ y \\ x = (x ⊓ y ⊔ x \\ y) ⊓ y \\ x := by rw [sup_inf_sdiff]\n    _ = ⊥ := by rw [inf_sup_right, inf_comm x y, inf_inf_sdiff, sdiff_inf_sdiff, bot_sup_eq]\n\n@[simp]\ntheorem inf_sdiff_self_left : y \\ x ⊓ x = ⊥ := by rw [inf_comm, inf_sdiff_self_right]\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n-- see Note [lower instance priority]\ninstance (priority := 100) GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra :\n    GeneralizedCoheytingAlgebra α where\n  __ := ‹GeneralizedBooleanAlgebra α›\n  __ := GeneralizedBooleanAlgebra.toOrderBot\n  sdiff := (· \\ ·)\n  sdiff_le_iff y x z :=\n    ⟨fun h =>\n      le_of_inf_le_sup_le\n        (le_of_eq\n          (by grind [sdiff_le', inf_of_le_right, inf_eq_right, inf_sdiff_self_right, bot_sup_eq,\n            inf_sup_right]))\n        (calc\n          y ⊔ y \\ x ≤ y \\ x ⊔ x ⊔ z := by\n            grind [sup_of_le_left, sdiff_le', le_sup_left, sdiff_sup_self']\n          _ = x ⊔ z ⊔ y \\ x := by ac_rfl),\n      fun h => le_of_inf_le_sup_le (inf_sdiff_self_left.trans_le bot_le) (calc\n        y \\ x ⊔ x = y ⊔ x := sdiff_sup_self'\n        _ ≤ x ⊔ z ⊔ x := sup_le_sup_right h x\n        _ ≤ z ⊔ x := by rw [sup_assoc, sup_comm, sup_assoc, sup_idem])⟩\n\ntheorem disjoint_sdiff_self_left : Disjoint (y \\ x) x :=\n  disjoint_iff_inf_le.mpr inf_sdiff_self_left.le\n\ntheorem disjoint_sdiff_self_right : Disjoint x (y \\ x) :=\n  disjoint_iff_inf_le.mpr inf_sdiff_self_right.le\n\nlemma le_sdiff : x ≤ y \\ z ↔ x ≤ y ∧ Disjoint x z :=\n  ⟨fun h ↦ ⟨h.trans sdiff_le, disjoint_sdiff_self_left.mono_left h⟩, fun h ↦\n    by rw [← h.2.sdiff_eq_left]; exact sdiff_le_sdiff_right h.1⟩\n\n@[simp] lemma sdiff_eq_left : x \\ y = x ↔ Disjoint x y :=\n  ⟨fun h ↦ disjoint_sdiff_self_left.mono_left h.ge, Disjoint.sdiff_eq_left⟩\n\n/- TODO: we could make an alternative constructor for `GeneralizedBooleanAlgebra` using\n`Disjoint x (y \\ x)` and `x ⊔ (y \\ x) = y` as axioms. -/\ntheorem Disjoint.sdiff_eq_of_sup_eq (hi : Disjoint x z) (hs : x ⊔ z = y) : y \\ x = z :=\n  have h : y ⊓ x = x := inf_eq_right.2 <| le_sup_left.trans hs.le\n  sdiff_unique (by rw [h, hs]) (by rw [h, hi.eq_bot])\n\nprotected theorem Disjoint.sdiff_unique (hd : Disjoint x z) (hz : z ≤ y) (hs : y ≤ x ⊔ z) :\n    y \\ x = z :=\n  sdiff_unique\n    (by\n      rw [← inf_eq_right] at hs\n      rwa [sup_inf_right, inf_sup_right, sup_comm x, inf_sup_self, inf_comm, sup_comm z,\n        hs, sup_eq_left])\n    (by rw [inf_assoc, hd.eq_bot, inf_bot_eq])\n\n-- cf. `IsCompl.disjoint_left_iff` and `IsCompl.disjoint_right_iff`\ntheorem disjoint_sdiff_iff_le (hz : z ≤ y) (hx : x ≤ y) : Disjoint z (y \\ x) ↔ z ≤ x :=\n  ⟨fun H =>\n    le_of_inf_le_sup_le (le_trans H.le_bot bot_le)\n      (by\n        rw [sup_sdiff_cancel_right hx]\n        grw [sdiff_le]\n        rw [sup_eq_right.2 hz]),\n    fun H => disjoint_sdiff_self_right.mono_left H⟩\n\n-- cf. `IsCompl.le_left_iff` and `IsCompl.le_right_iff`\ntheorem le_iff_disjoint_sdiff (hz : z ≤ y) (hx : x ≤ y) : z ≤ x ↔ Disjoint z (y \\ x) :=\n  (disjoint_sdiff_iff_le hz hx).symm\n\n-- cf. `IsCompl.inf_left_eq_bot_iff` and `IsCompl.inf_right_eq_bot_iff`\n\nTarget:\ntheorem inf_sdiff_eq_bot_iff (hz : z ≤ y) (hx : x ≤ y) : z ⊓ y \\ x = ⊥ ↔ z ≤ x :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"4199460063626449aa34c8e36e46a0a6b0d0f3c678ba42e730f3296f9faee3a0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/BooleanAlgebra","family_id":"inf_sdiff_eq_bot_iff","file_id":"mathlib/Mathlib/Order/BooleanAlgebra/Basic.lean","sample_id":"6ef6ae9c0f9613c7c3706f6b6b19cc3ed2c14562e6ea2548c5e346b1a3fa96cb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"29e959508ac284606b1ea79d1f9c44fa587eb78cdc7fa0d4ba1b2b14a0490f11","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"13de1f24cd4205e4f85cb45f304f4be6801f8a44e1ba2f59e38a94d0c0c66525","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e5f7ea238c907be9131815ee500db761a2036d4f0c717eb8bb8747df40a1767e","source_sha256":"155010ab3058b3fe6da475eb20a319e6c48a44ac24aa3149ecf584265954195c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [Shrink.rec, eqRec_eq_cast, cast_eq_iff_heq]\n  rw [Equiv.symm_apply_apply]","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":5,"token_jaccard":0.117647,"token_length_ratio":0.277778},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"359ff4cd179e8ed7a2eba7d84f983d6230213a7bd8ecc1b1f10a2cc40c07580f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Logic.Equiv.Defs\npublic import Mathlib.Tactic.MkIffOfInductiveProp\npublic import Mathlib.Tactic.PPWithUniv\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Small types\n\nA type is `w`-small if there exists an equivalence to some `S : Type w`.\n\nWe provide a noncomputable model `Shrink α : Type w`, and `equivShrink α : α ≃ Shrink α`.\n\nA subsingleton type is `w`-small for any `w`.\n\nIf `α ≃ β`, then `Small.{w} α ↔ Small.{w} β`.\n\nSee `Mathlib/Logic/Small/Basic.lean` for further instances and theorems.\n-/\n\n@[expose] public section\n\nuniverse u w v v'\n\n/-- A type is `Small.{w}` if there exists an equivalence to some `S : Type w`.\n-/\n-- After https://github.com/leanprover/lean4/pull/12286 and\n-- https://github.com/leanprover/lean4/pull/12423: `v` is a true output (determined by `α`),\n-- but we need the attribute to prevent `w` from also being treated as output.\n-- See Note [universe output parameters and typeclass caching].\n@[univ_out_params v, mk_iff, pp_with_univ]\nclass Small (α : Type v) : Prop where\n  /-- If a type is `Small.{w}`, then there exists an equivalence with some `S : Type w` -/\n  equiv_small : ∃ S : Type w, Nonempty (α ≃ S)\n\n/-- Constructor for `Small α` from an explicit witness type and equivalence.\n-/\ntheorem Small.mk' {α : Type v} {S : Type w} (e : α ≃ S) : Small.{w} α :=\n  ⟨⟨S, ⟨e⟩⟩⟩\n\n/-- An arbitrarily chosen model in `Type w` for a `w`-small type.\n-/\n@[pp_with_univ, no_expose]\ndef Shrink (α : Type v) [Small.{w} α] : Type w :=\n  Classical.choose (@Small.equiv_small α _)\n\n/-- The noncomputable equivalence between a `w`-small type and a model.\n-/\n@[no_expose]\nnoncomputable def equivShrink (α : Type v) [Small.{w} α] : α ≃ Shrink α :=\n  Nonempty.some (Classical.choose_spec (@Small.equiv_small α _))\n\n@[ext]\ntheorem Shrink.ext {α : Type v} [Small.{w} α] {x y : Shrink α}\n    (w : (equivShrink _).symm x = (equivShrink _).symm y) : x = y := by\n  simpa using w\n\n-- It would be nice to mark this as `aesop cases` if\n-- https://github.com/leanprover-community/aesop/issues/59\n-- is resolved.\n@[induction_eliminator]\nprotected noncomputable def Shrink.rec {α : Type*} [Small.{w} α] {F : Shrink α → Sort v}\n    (h : ∀ X, F (equivShrink _ X)) : ∀ X, F X :=\n  fun X => ((equivShrink _).apply_symm_apply X) ▸ (h _)\n\n@[simp]\n\nTarget:\nlemma Shrink.rec_equivShrink {α : Type*} [Small.{w} α] {F : Shrink α → Sort v}\n    {f : (a : α) → F (equivShrink α a)} (a : α) : Shrink.rec f (equivShrink _ a) = f a :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_e5f7ea238c90","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c8583cdd28a0406dbadd3db22a26137d6515acf344009f797c96ca71ce54ab19","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Small","family_id":"shrink","file_id":"mathlib/Mathlib/Logic/Small/Defs.lean","sample_id":"e5f7ea238c907be9131815ee500db761a2036d4f0c717eb8bb8747df40a1767e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ee1a0ff1d116d0272d387712ee44ccfa46821cca224349b1934d9628b210bb5f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f97002eb6f5349cba0b0cba353ad0fc3306516a27d9c160813ec6a889f48cb83","source_sha256":"31a5542d1ac83b668eb669f80f15da1c85ba792b90d38c379ac11ec99cc62c14","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases isEmpty_or_nonempty ι\n  · simp\n  rcases isEmpty_or_nonempty ι'\n  · simp\n  refine ciSup_le fun j ↦ ?_\n  grw [hv.apply_sum_le]\n  simp only [map_mul]\n  have (f : ι → ℝ) : ⨆ i : ↥Finset.univ, f i.val = ⨆ i, f i :=\n    Function.Surjective.iSup_comp (fun i ↦ ⟨⟨i, Finset.mem_univ i⟩, rfl⟩) f\n  rw [this fun i ↦ v (A (j, i)) * v (x i)]\n  refine ciSup_le fun i ↦ ?_\n  gcongr\n  · exact Real.iSup_nonneg_of_nonnegHomClass v _\n  · exact Finite.le_ciSup_of_le (j, i) le_rfl\n  · exact Finite.le_ciSup_of_le i le_rfl","hard_negative":false,"metrics":{"chosen_tokens":136,"rejected_tokens":5,"token_jaccard":0.05,"token_length_ratio":0.036765},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"35a5a247925ca0a7a9efa829477d70646ee300ff31aed3d1c3659abc86188e50","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Homogenize\npublic import Mathlib.NumberTheory.Height.Basic\nimport Mathlib.Algebra.Order.Ring.IsNonarchimedean\nimport Mathlib.Data.Fintype.Order\nimport all Mathlib.NumberTheory.Height.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2026 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Height bounds for linear and polynomial maps\n\nWe prove an upper bound for the height of the image of a tuple under a linear map.\n\nWe also prove upper and lower bounds for the height of `fun i ↦ eval P i x`, where `P` is a family\nof homogeneous polynomials over the field `K` of the same degree `N` and `x : ι → K`\nwith `ι` finite.\n-/\n\npublic section\n\nsection aux\n\nprivate lemma Height.iSup_fun_eq_max (f : Fin 2 → ℝ) : iSup f = max (f 0) (f 1) := by\n  rw [show f = ![f 0, f 1] from List.ofFn_inj.mp rfl]\n  exact (max_eq_iSup ..).symm\n\nnamespace IsNonarchimedean\n\nvariable {R α β F : Type*} [CommRing R] [AddCommMonoid β] [FunLike F β ℝ] [NonnegHomClass F β ℝ]\n    [ZeroHomClass F β ℝ] {v : F} {l : α → β}\n\n-- NOTE: The following cannot be moved to Mathlib.Algebra.Order.Ring.IsNonarchimedean,\n--       because it needs the target to be the reals (to have the default value zero\n--       for empty iSups), which are not known there.\n/-- The ultrametric triangle inequality for finite sums. -/\nlemma apply_sum_le (hv : IsNonarchimedean v) {s : Finset α} :\n    v (∑ i ∈ s, l i) ≤ ⨆ i : s, v (l i) := by\n  classical\n  induction s using Finset.induction with\n  | empty => simp\n  | insert a s ha ih =>\n    rw [Finset.sum_insert ha]\n    grw [hv .., ih]\n    refine max_le ?_ ?_\n    · exact Finite.le_ciSup_of_le ⟨_, s.mem_insert_self a⟩ le_rfl\n    · rcases isEmpty_or_nonempty s with hs | hs\n      · simpa using Real.iSup_nonneg_of_nonnegHomClass v _\n      exact ciSup_le fun i ↦ Finite.le_ciSup_of_le (⟨i.val, Finset.mem_insert_of_mem i.prop⟩) le_rfl\n\n/-- The ultrametric triangle inequality for finite sums. -/\nlemma apply_sum_univ_le [Fintype α] (hv : IsNonarchimedean v) :\n    v (∑ i, l i) ≤ ⨆ i, v (l i) := by\n  grw [hv.apply_sum_le, ← cbiSup_eq_of_forall (by grind)]\n  simp\n\nend IsNonarchimedean\n\nend aux\n\n/-!\n### Upper bound for the height of the image under a linear map\n-/\n\nvariable {K : Type*} [Field K] {ι ι' : Type*} [Fintype ι] [Finite ι']\n\n-- The \"local\" version of the bound for (archimedean) absolute values.\nlemma AbsoluteValue.iSup_abv_linearMap_apply_le (v : AbsoluteValue K ℝ) (A : ι' × ι → K)\n    (x : ι → K) :\n    ⨆ j, v (∑ i, A (j, i) * x i) ≤ Nat.card ι * (⨆ ji, v (A ji)) * ⨆ i, v (x i) := by\n  rcases isEmpty_or_nonempty ι'\n  · simp\n  refine ciSup_le fun j ↦ ?_\n  grw [v.sum_le]\n  simp only [map_mul]\n  grw [Finset.sum_le_sum (g := fun _ ↦ (⨆ ji, v (A ji)) * ⨆ i, v (x i)) fun i _ ↦ ?h]\n  case h =>\n    gcongr\n    · exact Real.iSup_nonneg_of_nonnegHomClass v _\n    · exact Finite.le_ciSup_of_le (j, i) le_rfl\n    · exact Finite.le_ciSup_of_le i le_rfl\n  rw [Finset.sum_const, nsmul_eq_mul, mul_assoc, Finset.card_univ, Nat.card_eq_fintype_card]\n\n-- The \"local\" version of the bound for nonarchimedean absolute values.\n\nTarget:\nlemma IsNonarchimedean.iSup_abv_linearMap_apply_le {v : AbsoluteValue K ℝ} (hv : IsNonarchimedean v)\n    (A : ι' × ι → K) (x : ι → K) :\n    ⨆ j, v (∑ i, A (j, i) * x i) ≤ (⨆ ji, v (A ji)) * ⨆ i, v (x i) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Height","family_id":"isnonarchimedean","file_id":"mathlib/Mathlib/NumberTheory/Height/MvPolynomial.lean","sample_id":"f97002eb6f5349cba0b0cba353ad0fc3306516a27d9c160813ec6a889f48cb83"},"split":"train"}
{"binding":{"chosen_proof_sha256":"20d35803748fd7d6978af67d3ca14fe96ffdbfdea01cb8bdd6440cb3a20e70bb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"83f4c0050682df846c30f760830154dbb64df53f5da882fea6802894613db41a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"87698f733f1f8800ecece55747b606be522d5d9ae2965cfc2fd1ae226fa9c49c","source_sha256":"24ffbf256f6f6f7a2617901323c2d532e2d7871c826a8b11f0580b283e994302","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let nop := @neg_one_pow_div_two_of_three_mod_four\n  rw [quadratic_reciprocity', pow_mul, nop hp, nop hq, neg_one_mul] <;>\n  rwa [← Prime.mod_two_eq_one_iff_ne_two Fact.out, odd_of_mod_four_eq_three]","hard_negative":false,"metrics":{"chosen_tokens":35,"rejected_tokens":39,"token_jaccard":0.896552,"token_length_ratio":1.114286},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"35b8f9d00ad0947d1f95405b553de862111d1980aa5db086e1eff558b750bd4e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.LegendreSymbol.Basic\npublic import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum\n\nNamespace:\nlegendreSym\n\nLocal context:\n/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Michael Stoll\n-/\n/-!\n# Quadratic reciprocity.\n\n## Main results\n\nWe prove the law of quadratic reciprocity, see `legendreSym.quadratic_reciprocity` and\n`legendreSym.quadratic_reciprocity'`, as well as the\ninterpretations in terms of existence of square roots depending on the congruence mod 4,\n`ZMod.exists_sq_eq_prime_iff_of_mod_four_eq_one` and\n`ZMod.exists_sq_eq_prime_iff_of_mod_four_eq_three`.\n\nWe also prove the supplementary laws that give conditions for when `2` or `-2`\nis a square modulo a prime `p`:\n`legendreSym.at_two` and `ZMod.exists_sq_eq_two_iff` for `2` and\n`legendreSym.at_neg_two` and `ZMod.exists_sq_eq_neg_two_iff` for `-2`.\n\n## Implementation notes\n\nThe proofs use results for quadratic characters on arbitrary finite fields\nfrom `NumberTheory.LegendreSymbol.QuadraticChar.GaussSum`, which in turn are based on\nproperties of quadratic Gauss sums as provided by `NumberTheory.LegendreSymbol.GaussSum`.\n\n## Tags\n\nquadratic residue, quadratic nonresidue, Legendre symbol, quadratic reciprocity\n-/\n\npublic section\n\n\nopen Nat\n\nsection Values\n\nvariable {p : ℕ} [Fact p.Prime]\n\nopen ZMod\n\n/-!\n### The value of the Legendre symbol at `2` and `-2`\n\nSee `jacobiSym.at_two` and `jacobiSym.at_neg_two` for the corresponding statements\nfor the Jacobi symbol.\n-/\n\n\nnamespace legendreSym\n\n/-- `legendreSym p 2` is given by `χ₈ p`. -/\ntheorem at_two (hp : p ≠ 2) : legendreSym p 2 = χ₈ p := by\n  have : (2 : ZMod p) = (2 : ℤ) := by norm_cast\n  rw [legendreSym, ← this, quadraticChar_two ((ringChar_zmod_n p).substr hp), card p]\n\n/-- `legendreSym p (-2)` is given by `χ₈' p`. -/\ntheorem at_neg_two (hp : p ≠ 2) : legendreSym p (-2) = χ₈' p := by\n  have : (-2 : ZMod p) = (-2 : ℤ) := by norm_cast\n  rw [legendreSym, ← this, quadraticChar_neg_two ((ringChar_zmod_n p).substr hp), card p]\n\nend legendreSym\n\nnamespace ZMod\n\n/-- `2` is a square modulo an odd prime `p` iff `p` is congruent to `1` or `7` mod `8`. -/\ntheorem exists_sq_eq_two_iff (hp : p ≠ 2) : IsSquare (2 : ZMod p) ↔ p % 8 = 1 ∨ p % 8 = 7 := by\n  rw [FiniteField.isSquare_two_iff, card p]\n  have h₁ := (Prime.mod_two_eq_one_iff_ne_two Fact.out).mpr hp\n  lia\n\n/-- `-2` is a square modulo an odd prime `p` iff `p` is congruent to `1` or `3` mod `8`. -/\ntheorem exists_sq_eq_neg_two_iff (hp : p ≠ 2) : IsSquare (-2 : ZMod p) ↔ p % 8 = 1 ∨ p % 8 = 3 := by\n  rw [FiniteField.isSquare_neg_two_iff, card p]\n  have h₁ := (Prime.mod_two_eq_one_iff_ne_two Fact.out).mpr hp\n  lia\n\nend ZMod\n\nend Values\n\nsection Reciprocity\n\n/-!\n### The Law of Quadratic Reciprocity\n\nSee `jacobiSym.quadratic_reciprocity` and variants for a version of Quadratic Reciprocity\nfor the Jacobi symbol.\n-/\n\n\nvariable {p q : ℕ} [Fact p.Prime] [Fact q.Prime]\n\nnamespace legendreSym\n\nopen ZMod\n\n/-- **The Law of Quadratic Reciprocity**: if `p` and `q` are distinct odd primes, then\n`(q / p) * (p / q) = (-1)^((p-1)(q-1)/4)`. -/\ntheorem quadratic_reciprocity (hp : p ≠ 2) (hq : q ≠ 2) (hpq : p ≠ q) :\n    legendreSym q p * legendreSym p q = (-1) ^ (p / 2 * (q / 2)) := by\n  have hp₁ := (Prime.eq_two_or_odd <| @Fact.out p.Prime _).resolve_left hp\n  have hq₁ := (Prime.eq_two_or_odd <| @Fact.out q.Prime _).resolve_left hq\n  have hq₂ : ringChar (ZMod q) ≠ 2 := (ringChar_zmod_n q).substr hq\n  have h :=\n    quadraticChar_odd_prime ((ringChar_zmod_n p).substr hp) hq ((ringChar_zmod_n p).substr hpq)\n  rw [card p] at h\n  have nc : ∀ n r : ℕ, ((n : ℤ) : ZMod r) = n := fun n r => by norm_cast\n  have nc' : (((-1) ^ (p / 2) : ℤ) : ZMod q) = (-1) ^ (p / 2) := by norm_cast\n  rw [legendreSym, legendreSym, nc, nc, h, map_mul, mul_rotate', mul_comm (p / 2), ← pow_two,\n    quadraticChar_sq_one (prime_ne_zero q p hpq.symm), mul_one, pow_mul, χ₄_eq_neg_one_pow hp₁, nc',\n    map_pow, quadraticChar_neg_one hq₂, card q, χ₄_eq_neg_one_pow hq₁]\n\n/-- The Law of Quadratic Reciprocity: if `p` and `q` are odd primes, then\n`(q / p) = (-1)^((p-1)(q-1)/4) * (p / q)`. -/\ntheorem quadratic_reciprocity' (hp : p ≠ 2) (hq : q ≠ 2) :\n    legendreSym q p = (-1) ^ (p / 2 * (q / 2)) * legendreSym p q := by\n  rcases eq_or_ne p q with rfl | h\n  · rw [(eq_zero_iff p p).mpr (mod_cast natCast_self p), mul_zero]\n  · have qr := congr_arg (· * legendreSym p q) (quadratic_reciprocity hp hq h)\n    have : ((q : ℤ) : ZMod p) ≠ 0 := mod_cast prime_ne_zero p q h\n    simpa only [mul_assoc, ← pow_two, sq_one p this, mul_one] using qr\n\n/-- The Law of Quadratic Reciprocity: if `p` and `q` are odd primes and `p % 4 = 1`,\nthen `(q / p) = (p / q)`. -/\ntheorem quadratic_reciprocity_one_mod_four (hp : p % 4 = 1) (hq : q ≠ 2) :\n    legendreSym q p = legendreSym p q := by\n  rw [quadratic_reciprocity'\n      ((Prime.mod_two_eq_one_iff_ne_two Fact.out).mp (odd_of_mod_four_eq_one hp)) hq,\n    pow_mul, neg_one_pow_div_two_of_one_mod_four hp, one_pow, one_mul]\n\n/-- The Law of Quadratic Reciprocity: if `p` and `q` are primes that are both congruent\nto `3` mod `4`, then `(q / p) = -(p / q)`. -/\n\nTarget:\ntheorem quadratic_reciprocity_three_mod_four (hp : p % 4 = 3) (hq : q % 4 = 3) :\n    legendreSym q p = -legendreSym p q :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  let nop := @neg_one_pow_div_two_of_three_mod_four\n  rw [quadratic_reciprocity', pow_mul, nop hp, nop hq, neg_one_mul] <;>\n  rwa [← Prime.mod_two_eq_one_iff_ne_two Fact.out, odd_of_mod_four_eq_three]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/LegendreSymbol","family_id":"quadratic_reciprocity_three_mod_four","file_id":"mathlib/Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean","sample_id":"87698f733f1f8800ecece55747b606be522d5d9ae2965cfc2fd1ae226fa9c49c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7f6de06b7c519dc56f72cfe0e561560f9ccb6c57f8383a711dc7fbaa5be562ae","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c6fd5b3bbb6c2272c694e26c6809305221670a614d7e0915a43ee429e5ad389a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"04175e8f69832579c88e53e7defc7abb906d1749bebae6932b35a3cc3ac66b47","source_sha256":"5dcbaf49b7e0422efc9e44781278719441afae74990c650101eb7518f775293b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [absNorm_eq' 1 I₀ (by rw [one_smul]; rfl), OneMemClass.coe_one, map_one, abs_one,\n    Int.cast_one, _root_.div_one]","hard_negative":true,"metrics":{"chosen_tokens":33,"rejected_tokens":3,"token_jaccard":0.04,"token_length_ratio":0.090909},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"35d3ef8b373df58615d0d549084fb23dd8d55bd87d4de30b31d920652036aea4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.FractionalIdeal.Basic\npublic import Mathlib.RingTheory.Ideal.Norm.AbsNorm\npublic import Mathlib.RingTheory.Localization.NormTrace\n\nNamespace:\nFractionalIdeal\n\nLocal context:\n/-\nCopyright (c) 2024 Xavier Roblot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Xavier Roblot\n-/\n/-!\n\n# Fractional ideal norms\n\nThis file defines the absolute ideal norm of a fractional ideal `I : FractionalIdeal R⁰ K` where\n`K` is a fraction field of `R`. The norm is defined by\n`FractionalIdeal.absNorm I = Ideal.absNorm I.num / |Algebra.norm ℤ I.den|` where `I.num` is an\nideal of `R` and `I.den` an element of `R⁰` such that `I.den • I = I.num`.\n\n## Main definitions and results\n\n* `FractionalIdeal.absNorm`: the norm as a zero-preserving morphism with values in `ℚ`.\n* `FractionalIdeal.absNorm_eq'`: the value of the norm does not depend on the choice of\n  `I.num` and `I.den`.\n* `FractionalIdeal.abs_det_basis_change`: the norm is given by the determinant\n  of the basis change matrix.\n* `FractionalIdeal.absNorm_span_singleton`: the norm of a principal fractional ideal is the\n  norm of its generator\n-/\n\n@[expose] public section\n\nopen Module\nopen scoped Pointwise nonZeroDivisors\n\nnamespace FractionalIdeal\nvariable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free ℤ R] [Module.Finite ℤ R]\nvariable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]\n\ntheorem absNorm_div_norm_eq_absNorm_div_norm {I : FractionalIdeal R⁰ K} (a : R⁰) (I₀ : Ideal R)\n    (h : a • (I : Submodule R K) = Submodule.map (Algebra.linearMap R K) I₀) :\n    (Ideal.absNorm I.num : ℚ) / |Algebra.norm ℤ (I.den : R)| =\n      (Ideal.absNorm I₀ : ℚ) / |Algebra.norm ℤ (a : R)| := by\n  rw [div_eq_div_iff]\n  · replace h := congr_arg (I.den • ·) h\n    have h' := congr_arg (a • ·) (den_mul_self_eq_num I)\n    rw [smul_comm] at h\n    rw [h, Submonoid.smul_def, Submonoid.smul_def, ← Submodule.ideal_span_singleton_smul,\n      ← Submodule.ideal_span_singleton_smul, ← Submodule.map_smul'', ← Submodule.map_smul'',\n      (LinearMap.map_injective ?_).eq_iff, smul_eq_mul, smul_eq_mul] at h'\n    · simp_rw [← Nat.cast_natAbs, ← Nat.cast_mul, ← Ideal.absNorm_span_singleton]\n      rw [← map_mul, ← map_mul, mul_comm, ← h', mul_comm]\n    · exact LinearMap.ker_eq_bot.mpr (IsFractionRing.injective R K)\n  all_goals simp [Algebra.norm_eq_zero_iff]\n\n/-- The absolute norm of the fractional ideal `I` extending by multiplicativity the absolute norm\non (integral) ideals. -/\nnoncomputable def absNorm : FractionalIdeal R⁰ K →*₀ ℚ where\n  toFun I := (Ideal.absNorm I.num : ℚ) / |Algebra.norm ℤ (I.den : R)|\n  map_zero' := by\n    rw [num_zero_eq, Submodule.zero_eq_bot, Ideal.absNorm_bot, Nat.cast_zero, zero_div]\n    exact IsFractionRing.injective R K\n  map_one' := by\n    rw [absNorm_div_norm_eq_absNorm_div_norm 1 ⊤ (by simp [Submodule.one_eq_range]),\n      Ideal.absNorm_top, Nat.cast_one, OneMemClass.coe_one, map_one, abs_one,\n      Int.cast_one,\n      one_div_one]\n  map_mul' I J := by\n    rw [absNorm_div_norm_eq_absNorm_div_norm (I.den * J.den) (I.num * J.num) (by\n        have : Algebra.linearMap R K = (IsScalarTower.toAlgHom R R K).toLinearMap := rfl\n        rw [coe_mul, this, Submodule.map_mul, ← this, ← den_mul_self_eq_num, ← den_mul_self_eq_num]\n        exact Submodule.mul_smul_mul_eq_smul_mul_smul _ _ _ _),\n      Submonoid.coe_mul, map_mul, map_mul, Nat.cast_mul, div_mul_div_comm,\n      Int.cast_abs, Int.cast_abs, Int.cast_abs, ← abs_mul, Int.cast_mul]\n\ntheorem absNorm_eq (I : FractionalIdeal R⁰ K) :\n    absNorm I = (Ideal.absNorm I.num : ℚ) / |Algebra.norm ℤ (I.den : R)| := rfl\n\ntheorem absNorm_eq' {I : FractionalIdeal R⁰ K} (a : R⁰) (I₀ : Ideal R)\n    (h : a • (I : Submodule R K) = Submodule.map (Algebra.linearMap R K) I₀) :\n    absNorm I = (Ideal.absNorm I₀ : ℚ) / |Algebra.norm ℤ (a : R)| := by\n  rw [absNorm, ← absNorm_div_norm_eq_absNorm_div_norm a I₀ h, MonoidWithZeroHom.coe_mk,\n    ZeroHom.coe_mk]\n\ntheorem absNorm_nonneg (I : FractionalIdeal R⁰ K) : 0 ≤ absNorm I := by dsimp [absNorm]; positivity\n\ntheorem absNorm_bot : absNorm (⊥ : FractionalIdeal R⁰ K) = 0 := absNorm.map_zero'\n\ntheorem absNorm_one : absNorm (1 : FractionalIdeal R⁰ K) = 1 := by convert! absNorm.map_one'\n\ntheorem absNorm_eq_zero_iff [IsDomain K] {I : FractionalIdeal R⁰ K} :\n    absNorm I = 0 ↔ I = 0 := by\n  refine ⟨fun h ↦ zero_of_num_eq_bot zero_notMem_nonZeroDivisors ?_, fun h ↦ h ▸ absNorm_bot⟩\n  rw [absNorm_eq, div_eq_zero_iff] at h\n  refine Ideal.absNorm_eq_zero_iff.mp <| Nat.cast_eq_zero.mp <| h.resolve_right ?_\n  simp [Algebra.norm_eq_zero_iff]\n\nTarget:\ntheorem coeIdeal_absNorm (I₀ : Ideal R) :\n    absNorm (I₀ : FractionalIdeal R⁰ K) = Ideal.absNorm I₀ :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_04175e8f6983","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"0869978178e7c3420128e96c6381d97c5637f090367ea97d3c137f1faa732ccd","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/FractionalIdeal","family_id":"coeideal_absnorm","file_id":"mathlib/Mathlib/RingTheory/FractionalIdeal/Norm.lean","sample_id":"04175e8f69832579c88e53e7defc7abb906d1749bebae6932b35a3cc3ac66b47"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bc537854a12e226bc8123bdf85ce0d801d68aa77f9f581d84fafad195b8a7d94","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ca1391130a23e3d68ef153258875ebd71c4479710a9861a526aad29dd1d8ea0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7f4e05d647d8234808511f2d65c328365b205678685e9a78a56123a55f2d6582","source_sha256":"e4f58af6d4a9683ac7a3c2d77d21907a98d376b8c60bad21d7bf0c07390bc700","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨n, f, hf, P, rfl⟩ := P.exists_fin_rename\n  rw [IsNilpotent.map_iff (rename_injective _ hf), MvPolynomial.isNilpotent_iff_of_fintype]\n  lift f to Fin n ↪ σ using hf\n  refine ⟨fun H i ↦ ?_, fun H i ↦ by simpa using H (i.embDomain f)⟩\n  by_cases H : i ∈ Set.range (Finsupp.embDomain f)\n  · aesop\n  · rw [coeff_rename_eq_zero] <;> aesop (add simp Finsupp.embDomain_eq_mapDomain)","hard_negative":true,"metrics":{"chosen_tokens":97,"rejected_tokens":2,"token_jaccard":0.018519,"token_length_ratio":0.020619},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"361932539e02e5f0fb82ab0899499dd4514051803753a869c6c3ebcff7834d9c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.MvPolynomial.Homogeneous\npublic import Mathlib.RingTheory.Polynomial.Nilpotent\n\nNamespace:\nMvPolynomial\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Nilpotents and units in multivariate polynomial rings\n\nWe prove that\n- `MvPolynomial.isNilpotent_iff`:\n  A multivariate polynomial is nilpotent iff all its coefficients are.\n- `MvPolynomial.isUnit_iff`:\n  A multivariate polynomial is invertible iff its constant term is invertible\n  and its other coefficients are nilpotent.\n-/\n\npublic section\n\nnamespace MvPolynomial\n\nvariable {σ R : Type*} [CommRing R] {P : MvPolynomial σ R}\n\n-- Subsumed by `isNilpotent_iff` below.\nprivate theorem isNilpotent_iff_of_fintype [Finite σ] :\n    IsNilpotent P ↔ ∀ i, IsNilpotent (P.coeff i) := by\n  classical\n  -- Note: including `Fintype.ofFinite σ` in the entire context interferes with the `rw` below.\n  refine have := Fintype.ofFinite σ; Fintype.induction_empty_option ?_ ?_ ?_ σ P\n  · intro α β _ e h₁ P\n    rw [← IsNilpotent.map_iff (rename_injective _ e.symm.injective), h₁,\n      (Finsupp.equivCongrLeft e).forall_congr_left]\n    simp [Finsupp.equivMapDomain_eq_mapDomain, coeff_rename_mapDomain _ e.symm.injective]\n  · simp [Unique.forall_iff, ← IsNilpotent.map_iff (isEmptyRingEquiv R PEmpty).injective,\n      -isEmptyRingEquiv_apply, isEmptyRingEquiv_eq_coeff_zero]\n  · intro α _ H P\n    obtain ⟨P, rfl⟩ := (optionEquivLeft _ _).symm.surjective P\n    simp [IsNilpotent.map_iff (optionEquivLeft _ _).symm.injective,\n      Polynomial.isNilpotent_iff, H, Finsupp.optionEquiv.forall_congr_left,\n      ← optionEquivLeft_coeff_some_coeff_none, Finsupp.coe_update]\n\nTarget:\ntheorem isNilpotent_iff : IsNilpotent P ↔ ∀ i, IsNilpotent (P.coeff i) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_7f4e05d647d8","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"7513c53ea4da56aaee9f30770e9f196004a07f06d2f45b58ff91eac6762a8b80","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/MvPolynomial","family_id":"isnilpotent_iff","file_id":"mathlib/Mathlib/Algebra/MvPolynomial/Nilpotent.lean","sample_id":"7f4e05d647d8234808511f2d65c328365b205678685e9a78a56123a55f2d6582"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cfefd25aa03c8c5b75a26d1fb3bac847d72b8789c719fd94bddb3d949b0e73ab","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"fa88b07de990b63c90a15ac530da9aeac1ae116f8f2bab2532225d3022d2d910","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3872ad2e2e46d2587f8e296dc30252bf359d8b645c5d7c7851970aa025ddb54e","source_sha256":"d3c748be9a560274e8be6246cdd5ae4eb345ae9279176c7be340f873d2e38ac6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun hn x ↦ lt_of_le_of_lt (norm_coe_le_norm f x) hn, ?_⟩\n  · obtain (he | he) := isEmpty_or_nonempty α\n    · simpa\n    · obtain ⟨x, hx⟩ := exist_norm_eq h\n      exact fun h ↦ hx ▸ h x","hard_negative":true,"metrics":{"chosen_tokens":48,"rejected_tokens":2,"token_jaccard":0.033333,"token_length_ratio":0.041667},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"3633032d32bfd4e2dd33ff66dfccaf834d2efa205c6ae97b7e8f2f7d2788b9c6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.ContinuousMap.Bounded.Normed\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Yoh Tanimoto. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yoh Tanimoto\n-/\n/-!\n# Compactly supported bounded continuous functions\n\nThe two-sided ideal of compactly supported bounded continuous functions taking values in a metric\nspace, with the uniform distance.\n-/\n\n@[expose] public section\n\nopen Set BoundedContinuousFunction\n\nsection CompactlySupported\n\n/-- The two-sided ideal of compactly supported functions. -/\nnoncomputable def compactlySupported (α γ : Type*) [TopologicalSpace α] [NonUnitalNormedRing γ] :\n    TwoSidedIdeal (α →ᵇ γ) :=\n  .mk' {z | HasCompactSupport z} .zero .add .neg .mul_left .mul_right\n\nvariable {α γ : Type*} [TopologicalSpace α] [NonUnitalNormedRing γ]\n\n@[inherit_doc]\nscoped[BoundedContinuousFunction] notation\n  \"C_cb(\" α \", \" γ \")\" => compactlySupported α γ\n\nlemma mem_compactlySupported {f : α →ᵇ γ} :\n    f ∈ C_cb(α, γ) ↔ HasCompactSupport f :=\n  TwoSidedIdeal.mem_mk' {z : α →ᵇ γ | HasCompactSupport z} .zero .add .neg .mul_left .mul_right f\n\nlemma exist_norm_eq [c : Nonempty α] {f : α →ᵇ γ} (h : f ∈ C_cb(α, γ)) : ∃ (x : α),\n    ‖f x‖ = ‖f‖ := by\n  by_cases hs : (tsupport f).Nonempty\n  · obtain ⟨x, _, hmax⟩ := mem_compactlySupported.mp h |>.exists_isMaxOn hs <|\n      (map_continuous f).norm.continuousOn\n    refine ⟨x, le_antisymm (norm_coe_le_norm f x) (norm_le (norm_nonneg _) |>.mpr fun y ↦ ?_)⟩\n    by_cases hy : y ∈ tsupport f\n    · exact hmax hy\n    · simp [image_eq_zero_of_notMem_tsupport hy]\n  · suffices f = 0 by simp [this]\n    rwa [not_nonempty_iff_eq_empty, tsupport_eq_empty_iff, ← coe_zero, ← DFunLike.ext'_iff] at hs\n\nTarget:\ntheorem norm_lt_iff_of_compactlySupported {f : α →ᵇ γ} (h : f ∈ C_cb(α, γ)) {M : ℝ}\n    (M0 : 0 < M) : ‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_3872ad2e2e46","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"f381f1a70338f1234b5e7244fde4200b9bdd28c8a9ee1e3032b21c1552b0fdb0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/ContinuousMap","family_id":"norm_lt_iff_of_compactlysupported","file_id":"mathlib/Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean","sample_id":"3872ad2e2e46d2587f8e296dc30252bf359d8b645c5d7c7851970aa025ddb54e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eba6e30ed08f1c8872a40c988023ca98c14ec7e63590064e2ee0d7b74b74430f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"382d9c391f850ba4635e3d83c4c92384a1fa17b247df5a67f6099368f1582fb9","source_sha256":"bc35074d1a6058ec18619c6ed537b3d817e9e8ac77cd437323327cafd172da85","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine\n    Computation.eq_of_bisim\n      (fun c1 c2 =>\n        ∃ (l : List α) (s : WSeq α),\n          c1 = Computation.corec (fun ⟨n, s⟩ =>\n            match Seq.destruct s with\n            | none => Sum.inl n\n            | some (none, s') => Sum.inr (n, s')\n            | some (some _, s') => Sum.inr (n + 1, s')) (l.length, s) ∧\n            c2 = Computation.map List.length (Computation.corec (fun ⟨l, s⟩ =>\n              match Seq.destruct s with\n              | none => Sum.inl l.reverse\n              | some (none, s') => Sum.inr (l, s')\n              | some (some a, s') => Sum.inr (a::l, s')) (l, s)))\n      ?_ ⟨[], s, rfl, rfl⟩\n  intro s1 s2 h; rcases h with ⟨l, s, h⟩; rw [h.left, h.right]\n  induction s using WSeq.recOn with\n  | nil => simp [nil]\n  | cons a s => simpa using ⟨a::l, s, by simp, by simp⟩\n  | think s => simpa using ⟨l, s, by simp, by simp⟩","hard_negative":true,"metrics":{"chosen_tokens":263,"rejected_tokens":8,"token_jaccard":0.028986,"token_length_ratio":0.030418},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"367aaa573632060435fc2f1fbc6b6b90fc7718af178f2ea2857903c102b08e18","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.DList.Basic\npublic import Mathlib.Data.WSeq.Basic\n\nNamespace:\nStream'.WSeq\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Miscellaneous definitions concerning weak sequences\n\nThese definitions, as well as those in `Mathlib/Data/WSeq/Productive.lean`, are not needed for the\ndevelopment of `Mathlib/Data/Seq/Parallel.lean`.\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nnamespace Stream'.WSeq\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\nopen Function\n\n/-- Get the length of `s` (if it is finite and completes in finite time). -/\ndef length (s : WSeq α) : Computation ℕ :=\n  @Computation.corec ℕ (ℕ × WSeq α)\n    (fun ⟨n, s⟩ =>\n      match Seq.destruct s with\n      | none => Sum.inl n\n      | some (none, s') => Sum.inr (n, s')\n      | some (some _, s') => Sum.inr (n + 1, s'))\n    (0, s)\n\n/-- A weak sequence is finite if `toList s` terminates. Equivalently,\n  it is a finite number of `think` and `cons` applied to `nil`. -/\nclass IsFinite (s : WSeq α) : Prop where\n  out : (toList s).Terminates\n\ninstance toList_terminates (s : WSeq α) [h : IsFinite s] : (toList s).Terminates :=\n  h.out\n\n/-- Get the list corresponding to a finite weak sequence. -/\ndef get (s : WSeq α) [IsFinite s] : List α :=\n  (toList s).get\n\n/-- Replace the `n`th element of `s` with `a`. -/\ndef updateNth (s : WSeq α) (n : ℕ) (a : α) : WSeq α :=\n  @Seq.corec (Option α) (ℕ × WSeq α)\n    (fun ⟨n, s⟩ =>\n      match Seq.destruct s, n with\n      | none, _ => none\n      | some (none, s'), n => some (none, n, s')\n      | some (some a', s'), 0 => some (some a', 0, s')\n      | some (some _, s'), 1 => some (some a, 0, s')\n      | some (some a', s'), n + 2 => some (some a', n + 1, s'))\n    (n + 1, s)\n\n/-- Remove the `n`th element of `s`. -/\ndef removeNth (s : WSeq α) (n : ℕ) : WSeq α :=\n  @Seq.corec (Option α) (ℕ × WSeq α)\n    (fun ⟨n, s⟩ =>\n      match Seq.destruct s, n with\n      | none, _ => none\n      | some (none, s'), n => some (none, n, s')\n      | some (some a', s'), 0 => some (some a', 0, s')\n      | some (some _, s'), 1 => some (none, 0, s')\n      | some (some a', s'), n + 2 => some (some a', n + 1, s'))\n    (n + 1, s)\n\n/-- Map the elements of `s` over `f`, removing any values that yield `none`. -/\ndef filterMap (f : α → Option β) : WSeq α → WSeq β :=\n  Seq.corec fun s =>\n    match Seq.destruct s with\n    | none => none\n    | some (none, s') => some (none, s')\n    | some (some a, s') => some (f a, s')\n\n/-- Select the elements of `s` that satisfy `p`. -/\ndef filter (p : α → Prop) [DecidablePred p] : WSeq α → WSeq α :=\n  filterMap fun a => if p a then some a else none\n\n-- example of infinite list manipulations\n/-- Get the first element of `s` satisfying `p`. -/\ndef find (p : α → Prop) [DecidablePred p] (s : WSeq α) : Computation (Option α) :=\n  head <| filter p s\n\n/-- Zip a function over two weak sequences -/\ndef zipWith (f : α → β → γ) (s1 : WSeq α) (s2 : WSeq β) : WSeq γ :=\n  @Seq.corec (Option γ) (WSeq α × WSeq β)\n    (fun ⟨s1, s2⟩ =>\n      match Seq.destruct s1, Seq.destruct s2 with\n      | some (none, s1'), some (none, s2') => some (none, s1', s2')\n      | some (some _, _), some (none, s2') => some (none, s1, s2')\n      | some (none, s1'), some (some _, _) => some (none, s1', s2)\n      | some (some a1, s1'), some (some a2, s2') => some (some (f a1 a2), s1', s2')\n      | _, _ => none)\n    (s1, s2)\n\n/-- Zip two weak sequences into a single sequence of pairs -/\ndef zip : WSeq α → WSeq β → WSeq (α × β) :=\n  zipWith Prod.mk\n\n/-- Get the list of indexes of elements of `s` satisfying `p` -/\ndef findIndexes (p : α → Prop) [DecidablePred p] (s : WSeq α) : WSeq ℕ :=\n  (zip s (Stream'.nats : WSeq ℕ)).filterMap fun ⟨a, n⟩ => if p a then some n else none\n\n/-- Get the index of the first element of `s` satisfying `p` -/\ndef findIndex (p : α → Prop) [DecidablePred p] (s : WSeq α) : Computation ℕ :=\n  (fun o => Option.getD o 0) <$> head (findIndexes p s)\n\n/-- Get the index of the first occurrence of `a` in `s` -/\ndef indexOf [DecidableEq α] (a : α) : WSeq α → Computation ℕ :=\n  findIndex (Eq a)\n\n/-- Get the indexes of occurrences of `a` in `s` -/\ndef indexesOf [DecidableEq α] (a : α) : WSeq α → WSeq ℕ :=\n  findIndexes (Eq a)\n\n/-- `union s1 s2` is a weak sequence which interleaves `s1` and `s2` in\n  some order (nondeterministically). -/\ndef union (s1 s2 : WSeq α) : WSeq α :=\n  @Seq.corec (Option α) (WSeq α × WSeq α)\n    (fun ⟨s1, s2⟩ =>\n      match Seq.destruct s1, Seq.destruct s2 with\n      | none, none => none\n      | some (a1, s1'), none => some (a1, s1', nil)\n      | none, some (a2, s2') => some (a2, nil, s2')\n      | some (none, s1'), some (none, s2') => some (none, s1', s2')\n      | some (some a1, s1'), some (none, s2') => some (some a1, s1', s2')\n      | some (none, s1'), some (some a2, s2') => some (some a2, s1', s2')\n      | some (some a1, s1'), some (some a2, s2') => some (some a1, cons a2 s1', s2'))\n    (s1, s2)\n\n/-- Returns `true` if `s` is `nil` and `false` if `s` has an element -/\ndef isEmpty (s : WSeq α) : Computation Bool :=\n  Computation.map Option.isNone <| head s\n\n/-- Calculate one step of computation -/\ndef compute (s : WSeq α) : WSeq α :=\n  match Seq.destruct s with\n  | some (none, s') => s'\n  | _ => s\n\n/-- Get the first `n` elements of a weak sequence -/\ndef take (s : WSeq α) (n : ℕ) : WSeq α :=\n  @Seq.corec (Option α) (ℕ × WSeq α)\n    (fun ⟨n, s⟩ =>\n      match n, Seq.destruct s with\n      | 0, _ => none\n      | _ + 1, none => none\n      | m + 1, some (none, s') => some (none, m + 1, s')\n      | m + 1, some (some a, s') => some (some a, m, s'))\n    (n, s)\n\n/-- Split the sequence at position `n` into a finite initial segment\n  and the weak sequence tail -/\ndef splitAt (s : WSeq α) (n : ℕ) : Computation (List α × WSeq α) :=\n  @Computation.corec (List α × WSeq α) (ℕ × List α × WSeq α)\n    (fun ⟨n, l, s⟩ =>\n      match n, Seq.destruct s with\n      | 0, _ => Sum.inl (l.reverse, s)\n      | _ + 1, none => Sum.inl (l.reverse, s)\n      | _ + 1, some (none, s') => Sum.inr (n, l, s')\n      | m + 1, some (some a, s') => Sum.inr (m, a::l, s'))\n    (n, [], s)\n\n/-- Returns `true` if any element of `s` satisfies `p` -/\ndef any (s : WSeq α) (p : α → Bool) : Computation Bool :=\n  Computation.corec\n    (fun s : WSeq α =>\n      match Seq.destruct s with\n      | none => Sum.inl false\n      | some (none, s') => Sum.inr s'\n      | some (some a, s') => if p a then Sum.inl true else Sum.inr s')\n    s\n\n/-- Returns `true` if every element of `s` satisfies `p` -/\ndef all (s : WSeq α) (p : α → Bool) : Computation Bool :=\n  Computation.corec\n    (fun s : WSeq α =>\n      match Seq.destruct s with\n      | none => Sum.inl true\n      | some (none, s') => Sum.inr s'\n      | some (some a, s') => if p a then Sum.inr s' else Sum.inl false)\n    s\n\n/-- Apply a function to the elements of the sequence to produce a sequence\n  of partial results. (There is no `scanr` because this would require\n  working from the end of the sequence, which may not exist.) -/\ndef scanl (f : α → β → α) (a : α) (s : WSeq β) : WSeq α :=\n  cons a <|\n    @Seq.corec (Option α) (α × WSeq β)\n      (fun ⟨a, s⟩ =>\n        match Seq.destruct s with\n        | none => none\n        | some (none, s') => some (none, a, s')\n        | some (some b, s') =>\n          let a' := f a b\n          some (some a', a', s'))\n      (a, s)\n\n/-- Get the weak sequence of initial segments of the input sequence -/\ndef inits (s : WSeq α) : WSeq (List α) :=\n  cons [] <|\n    @Seq.corec (Option (List α)) (Batteries.DList α × WSeq α)\n      (fun ⟨l, s⟩ =>\n        match Seq.destruct s with\n        | none => none\n        | some (none, s') => some (none, l, s')\n        | some (some a, s') =>\n          let l' := l.push a\n          some (some l'.toList, l', s'))\n      (Batteries.DList.empty, s)\n\n/-- Like take, but does not wait for a result. Calculates `n` steps of\n  computation and returns the sequence computed so far -/\ndef collect (s : WSeq α) (n : ℕ) : List α :=\n  (Seq.take n s).filterMap id\n\nTarget:\ntheorem length_eq_map (s : WSeq α) : length s = Computation.map List.length (toList s) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"8168c5fc8d4ce8751edbf96033ee0bbc2c6a6cb93799b867bddf32abe0cbab1d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/WSeq","family_id":"length_eq_map","file_id":"mathlib/Mathlib/Data/WSeq/Defs.lean","sample_id":"382d9c391f850ba4635e3d83c4c92384a1fa17b247df5a67f6099368f1582fb9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"feace96534351d67a98372e9c910b5669a41f603a705ba51814a94d00a6c6ea8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cb73e3847e70012451e31d1a504621a0a7c3b09a69dfc4dc16d9adb5f791622f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5bb17713869ded8211fecb07cd342899842c304348ff9eddfbbe96f8cb169aa9","source_sha256":"91f7a9e43cd159c4e5d0d1764bbe2e984a1eb627973fe04a06238fa29e76fd73","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro _ ha _ hb hne\n  have := hrc ha hb (fun h ↦ hne (hf h))\n  grind","hard_negative":true,"metrics":{"chosen_tokens":23,"rejected_tokens":3,"token_jaccard":0.055556,"token_length_ratio":0.130435},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"36b7e16998f9fa3db57dd194d1fd49b44db17c969d46126a46fddbb6c1b7c888","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Notation\npublic import Mathlib.Data.Set.Pairwise.Basic\npublic import Mathlib.Data.SetLike.Basic\npublic import Mathlib.Order.Directed\npublic import Mathlib.Order.Hom.Set\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Chains and flags\n\nThis file defines chains for an arbitrary relation and flags for an order.\n\n## Main declarations\n\n* `IsChain s`: A chain `s` is a set of comparable elements.\n* `Flag`: The type of flags, aka maximal chains, of an order.\n\n## Notes\n\nOriginally ported from Isabelle/HOL. The\n[original file](https://isabelle.in.tum.de/dist/library/HOL/HOL/Zorn.html) was written by Jacques D.\nFleuriot, Tobias Nipkow, Christian Sternagel.\n-/\n\n@[expose] public section\n\nassert_not_exists CompleteLattice\n\nopen Set Set.Notation\n\nvariable {α β F : Type*}\n\n/-! ### Chains -/\n\n\nsection Chain\n\nvariable (r : α → α → Prop)\n\n/-- In this file, we use `≺` as a local notation for any relation `r`. -/\nlocal infixl:50 \" ≺ \" => r\n\n/-- A chain is a set `s` satisfying `x ≺ y ∨ x = y ∨ y ≺ x` for all `x y ∈ s`. -/\ndef IsChain (s : Set α) : Prop :=\n  s.Pairwise fun x y => x ≺ y ∨ y ≺ x\n\n/-- `SuperChain s t` means that `t` is a chain that strictly includes `s`. -/\ndef SuperChain (s t : Set α) : Prop :=\n  IsChain r t ∧ s ⊂ t\n\n/-- A chain `s` is a maximal chain if there does not exists a chain strictly including `s`. -/\ndef IsMaxChain (s : Set α) : Prop :=\n  IsChain r s ∧ ∀ ⦃t⦄, IsChain r t → s ⊆ t → s = t\n\nvariable {r} {c c₁ c₂ s t : Set α} {a b x y : α}\n\n@[simp] lemma IsChain.empty : IsChain r ∅ := pairwise_empty _\n@[simp] lemma IsChain.singleton : IsChain r {a} := pairwise_singleton ..\n\ntheorem Set.Subsingleton.isChain (hs : s.Subsingleton) : IsChain r s :=\n  hs.pairwise _\n\ntheorem IsChain.mono : s ⊆ t → IsChain r t → IsChain r s :=\n  Set.Pairwise.mono\n\ntheorem IsChain.mono_rel {r' : α → α → Prop} (h : IsChain r s) (h_imp : ∀ x y, r x y → r' x y) :\n    IsChain r' s :=\n  h.mono' fun x y => Or.imp (h_imp x y) (h_imp y x)\n\n/-- This can be used to turn `IsChain (≥)` into `IsChain (≤)` and vice-versa. -/\ntheorem IsChain.symm (h : IsChain r s) : IsChain (flip r) s :=\n  h.mono' fun _ _ => Or.symm\n\ntheorem isChain_of_trichotomous [Std.Trichotomous r] (s : Set α) : IsChain r s :=\n  fun a _ b _ hab => (trichotomous_of r a b).imp_right fun h => h.resolve_left hab\n\nprotected theorem IsChain.insert (hs : IsChain r s) (ha : ∀ b ∈ s, a ≠ b → a ≺ b ∨ b ≺ a) :\n    IsChain r (insert a s) :=\n  have : Std.Symm fun a b ↦ a ≺ b ∨ b ≺ a := { symm _ _ := Or.symm }\n  hs.insert_of_symm ha\n\nlemma IsChain.pair (h : r a b) : IsChain r {a, b} :=\n  IsChain.singleton.insert fun _ hb _ ↦ .inl <| (eq_of_mem_singleton hb).symm.recOn ‹_›\n\ntheorem isChain_univ_iff : IsChain r (univ : Set α) ↔ Std.Trichotomous r := by\n  refine ⟨fun h => ⟨fun a b => ?_⟩, fun h => @isChain_of_trichotomous _ _ h univ⟩\n  have : a ≠ b → (r a b ∨ r b a) := h trivial trivial\n  grind\n\ntheorem IsChain.image_of_map_rel (r : α → α → Prop) (s : β → β → Prop) (f : α → β)\n    (h : ∀ x y, r x y → s (f x) (f y)) {c : Set α} (hrc : IsChain r c) : IsChain s (f '' c) :=\n  fun _ ⟨_, ha₁, ha₂⟩ _ ⟨_, hb₁, hb₂⟩ =>\n  ha₂ ▸ hb₂ ▸ fun hxy => (hrc ha₁ hb₁ <| ne_of_apply_ne f hxy).imp (h _ _) (h _ _)\n\nTarget:\ntheorem IsChain.preimage (r : α → α → Prop) (s : β → β → Prop) (f : α → β)\n    (hf : Function.Injective f) (h : ∀ x y, s (f x) (f y) → r x y) {c : Set β} (hrc : IsChain s c) :\n    IsChain r (f ⁻¹' c) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_5bb17713869d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d66fbae3cb8bd0f12fa4601d9020527b7f02aa02daf44dc2259329e18605600e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Preorder","family_id":"ischain","file_id":"mathlib/Mathlib/Order/Preorder/Chain.lean","sample_id":"5bb17713869ded8211fecb07cd342899842c304348ff9eddfbbe96f8cb169aa9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f187198a893dd0403a98bdeb7c5bab4c0cada1bb485a8d70e6ce244cdd6c05b3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"10754817056ac7fe2728b7d3c57dd409b9884a07365c423b530430a8a51d4d7e","source_sha256":"42a1a9a755579f25de77a56470f5e17c3dfcdee299a3d64bf9907f705df67f49","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Icc_union_Ioc_eq_Icc le_rfl h, VectorMeasure.of_union (by simp)\n    measurableSet_Icc measurableSet_Ioc, Icc_self, hf.vectorMeasure_singleton,\n    hf.vectorMeasure_Ioc h]\n  simp","hard_negative":false,"metrics":{"chosen_tokens":30,"rejected_tokens":5,"token_jaccard":0.04,"token_length_ratio":0.166667},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"36dda3f7ca76f7f356892aac32d90c779d07bf95015f573ced8a2680847a87f2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Group.Defs\npublic import Mathlib.MeasureTheory.Measure.Stieltjes\npublic import Mathlib.MeasureTheory.VectorMeasure.Basic\npublic import Mathlib.Topology.EMetricSpace.BoundedVariation\nimport Mathlib.MeasureTheory.VectorMeasure.AddContent\n\nNamespace:\nBoundedVariationOn\n\nLocal context:\n/-\nCopyright (c) 2026 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Vector valued Stieltjes measure associated to a bounded variation function\n\nLet `α` be a dense linear order with compact segments (e.g. `ℝ` or `ℝ≥0`), and `f : α → E` a\nbounded variation function taking values in a complete additive normed group.\nWe associate to `f` a vector measure, called `BoundedVariationOn.vectorMeasure`. It gives\nmass `f.rightLim b - f.leftLim a` to the interval `[a, b]` (with similar formulas for\nother types of intervals).\n\nFor the construction, we define first an additive content on the set semiring of open-closed\nintervals `(a, b]`, mapping this interval to `f.rightLim b - f.rightLim a`. To extend this content\nto the whole sigma-algebra, by general extension theorems, it is enough to show that it is\ndominated by a finite measure. For this, we can use the Stieltjes measure associated to the\nvariation of `f.rightLim`. The extension we get is not exactly the desired vector measure, as we\nneed to tweak things if there is a bot element `a`: the previous vector measure gives to `{a}` the\nmass `0` instead of the desired `f.rightLim a - f a`, so we add a Dirac mass to correct this defect.\n-/\n\n@[expose] public section\n\nopen Filter Set MeasureTheory MeasurableSpace MeasureTheory\nopen scoped symmDiff Topology NNReal ENNReal\n\nvariable {α : Type*} [LinearOrder α] [DenselyOrdered α] [TopologicalSpace α] [OrderTopology α]\n  [SecondCountableTopology α] [CompactIccSpace α] [hα : MeasurableSpace α] [BorelSpace α]\n  {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]\n  {f : α → E} {a b : α}\n\nnamespace BoundedVariationOn\n\n/-- The Stieltjes function associated to a bounded variation function. It is given by\nthe variation of the function `f.rightLim` from a fixed base point.\nUsing right limits ensures the right continuity, which is used to construct Stieltjes measures. -/\n@[simps] noncomputable def stieltjesFunctionRightLim\n    (hf : BoundedVariationOn f univ) (x₀ : α) : StieltjesFunction α where\n  toFun x := variationOnFromTo f.rightLim univ x₀ x\n  mono' := by\n    rw [← monotoneOn_univ]\n    exact variationOnFromTo.monotoneOn hf.rightLim.locallyBoundedVariationOn (mem_univ _)\n  right_continuous' x := hf.continuousWithinAt_variationOnFromTo_rightLim_Ici\n\nopen scoped Classical in\n/-- Auxiliary measure used to construct the vector measure associated to a bounded variation\nfunction. This is *not* the total variation of this measure in general, as we need to adjust things\nwhen there is a bot element by adding a Dirac mass there. -/\nprivate noncomputable def measureAux (hf : BoundedVariationOn f univ) : Measure α :=\n  if h : Nonempty α then (hf.stieltjesFunctionRightLim h.some).measure else 0\n\nprivate instance (hf : BoundedVariationOn f univ) : IsFiniteMeasure hf.measureAux := by\n  by_cases h : Nonempty α; swap\n  · simp only [BoundedVariationOn.measureAux, h, ↓reduceDIte]\n    infer_instance\n  simp only [BoundedVariationOn.measureAux, h, ↓reduceDIte]\n  apply StieltjesFunction.isFiniteMeasure_of_forall_abs_le\n    (C := (eVariationOn f.rightLim univ).toReal) _ (fun x ↦ ?_)\n  exact variationOnFromTo.abs_le_eVariationOn hf.rightLim\n\n/-- Given a bounded variation function `f`, we can construct a vector measure giving\nmass `f.rightLim v - f.rightLim a` to each open-closed interval `(a, b]`. This is *not* the\nmeasure associated to `f` in general, as we may need to adjust things at the bot element if\nthere is one. -/\nprivate lemma exists_vectorMeasure_le_measureAux (hf : BoundedVariationOn f univ) :\n    ∃ m : VectorMeasure α E, (∀ u v, u ≤ v → m (Set.Ioc u v) = f.rightLim v - f.rightLim u) ∧\n      m botSet = 0 ∧ ∀ s, ‖m s‖ₑ ≤ hf.measureAux s := by\n  /- We will apply the general extension theorem\n  `VectorMeasure.exists_extension_of_isSetSemiring_of_le_measure_of_generateFrom`. For this, we\n  need to check that the additive content is bounded by the measure `measureAux`. -/\n  rcases isEmpty_or_nonempty α with h'α | h'α\n  · exact ⟨0, by simp⟩\n  let m := AddContent.onIoc f.rightLim\n  have A : ∀ s ∈ {s | ∃ u v, u ≤ v ∧ s = Ioc u v}, ‖m s‖ₑ ≤ hf.measureAux s := by\n    rintro s ⟨u, v, huv, rfl⟩\n    rw [AddContent.onIoc_apply huv]\n    simp only [BoundedVariationOn.measureAux, h'α, ↓reduceDIte, StieltjesFunction.measure_Ioc,\n      BoundedVariationOn.stieltjesFunctionRightLim_apply]\n    rw [← variationOnFromTo.add hf.rightLim.locallyBoundedVariationOn\n      (mem_univ h'α.some) (mem_univ u) (mem_univ v)]\n    simp only [add_sub_cancel_left, variationOnFromTo, huv, ↓reduceIte, univ_inter]\n    rw [ENNReal.ofReal_toReal]; swap\n    · exact ((eVariationOn.mono _ (subset_univ _)).trans_lt hf.rightLim.lt_top).ne\n    rw [← edist_eq_enorm_sub]\n    exact eVariationOn.edist_le _ (by grind) (by grind)\n  have B : hα = generateFrom {s | ∃ u v, u ≤ v ∧ s = Ioc u v} := by\n    borelize α\n    convert! borel_eq_generateFrom_Ioc_le α using 2\n    grind only\n  rcases VectorMeasure.exists_extension_of_isSetSemiring_of_le_measure_of_generateFrom\n    IsSetSemiring.Ioc A B with ⟨m', hm', h'm'⟩\n  refine ⟨m', fun u v huv ↦ ?_, ?_, h'm'⟩\n  · rw [hm']\n    · exact AddContent.onIoc_apply huv\n    · exact ⟨u, v, huv, rfl⟩\n  · apply enorm_eq_zero.1\n    apply le_bot_iff.1\n    exact (h'm' _).trans (by simp [measureAux, h'α])\n\nopen scoped Classical in\n/-- The vector measure associated to a bounded variation function `f`, giving mass\n`f.rightLim b - f.leftLim a` to closed intervals `[a, b]`, and similarly for other intervals. -/\n@[no_expose] noncomputable def vectorMeasure (hf : BoundedVariationOn f univ) : VectorMeasure α E :=\n  hf.exists_vectorMeasure_le_measureAux.choose +\n  (if h : ∃ x, IsBot x then VectorMeasure.dirac h.choose (f.rightLim h.choose - f h.choose) else 0)\n\nlemma vectorMeasure_Ioc (hf : BoundedVariationOn f univ) (h : a ≤ b) :\n    hf.vectorMeasure (Ioc a b) = f.rightLim b - f.rightLim a := by\n  classical\n  have A : hf.exists_vectorMeasure_le_measureAux.choose (Ioc a b) =\n      f.rightLim b - f.rightLim a :=\n    hf.exists_vectorMeasure_le_measureAux.choose_spec.1 a b h\n  have B : (if hx : ∃ (x : α), IsBot x then VectorMeasure.dirac hx.choose\n      (f.rightLim hx.choose - f hx.choose) else 0) (Ioc a b) = 0 := by\n    by_cases hx : ∃ (x : α), IsBot x\n    · simp only [hx, ↓reduceDIte]\n      rw [VectorMeasure.dirac_apply_of_notMem]\n      simp only [mem_Ioc, not_and_or, not_lt, not_le]\n      exact Or.inl (hx.choose_spec _)\n    · simp [hx]\n  simp [vectorMeasure, A, B]\n\nlemma vectorMeasure_singleton (hf : BoundedVariationOn f univ) :\n    hf.vectorMeasure {a} = f.rightLim a - f.leftLim a := by\n  by_cases ha : IsBot a\n  · have h : ∃ x, IsBot x := ⟨a, ha⟩\n    have heqa : h.choose = a := subsingleton_isBot _ h.choose_spec ha\n    have A : hf.exists_vectorMeasure_le_measureAux.choose {a} = 0 := by\n      rw [← botSet_eq_singleton_of_isBot ha]\n      exact hf.exists_vectorMeasure_le_measureAux.choose_spec.2.1\n    simp only [vectorMeasure, h, ↓reduceDIte, add_apply, A, zero_add]\n    rw [VectorMeasure.dirac_apply_of_mem (MeasurableSet.singleton a)]\n    · simpa only [heqa, sub_right_inj] using (leftLim_eq_of_isBot ha).symm\n    · simp [heqa]\n  obtain ⟨b, hb⟩ : ∃ b, b < a := by simpa only [IsBot, not_forall, not_le] using ha\n  obtain ⟨u, u_mono, u_lt_a, u_lim⟩ :\n      ∃ u : ℕ → α, StrictMono u ∧ (∀ n : ℕ, u n ∈ Ioo b a) ∧ Tendsto u atTop (𝓝 a) :=\n    exists_seq_strictMono_tendsto' hb\n  replace u_lt_a n : u n < a := (u_lt_a n).2\n  have A : {a} = ⋂ n, Ioc (u n) a := by\n    refine Subset.antisymm (fun x hx => by simp [mem_singleton_iff.1 hx, u_lt_a]) fun x hx => ?_\n    replace hx : ∀ (i : ℕ), u i < x ∧ x ≤ a := by simpa using hx\n    have : a ≤ x := le_of_tendsto' u_lim fun n => (hx n).1.le\n    simp [le_antisymm this (hx 0).2]\n  have L1 : Tendsto (fun n ↦ hf.vectorMeasure (Ioc (u n) a)) atTop (𝓝 (hf.vectorMeasure {a})) := by\n    rw [A]\n    apply VectorMeasure.tendsto_vectorMeasure_iInter_atTop_nat ?_ (fun n ↦ measurableSet_Ioc)\n    exact fun m n hmn ↦ Ioc_subset_Ioc_left (u_mono.monotone hmn)\n  have L2 : Tendsto (fun n ↦ hf.vectorMeasure (Ioc (u n) a)) atTop\n      (𝓝 (f.rightLim a - f.leftLim a)) := by\n    simp_rw [hf.vectorMeasure_Ioc (u_lt_a _).le]\n    apply tendsto_const_nhds.sub\n    have : Tendsto u atTop (𝓝[<] a) := tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _\n      u_lim (Eventually.of_forall u_lt_a)\n    convert! (hf.rightLim.tendsto_leftLim a).comp this using 2\n    have : (𝓝[<] a).NeBot := by\n      rw [← mem_closure_iff_nhdsWithin_neBot, closure_Iio' ⟨b, hb⟩]\n      exact self_mem_Iic\n    exact (leftLim_rightLim (hf.tendsto_leftLim _)).symm\n  exact tendsto_nhds_unique L1 L2\n\nTarget:\nlemma vectorMeasure_Icc (hf : BoundedVariationOn f univ) (h : a ≤ b) :\n    hf.vectorMeasure (Icc a b) = f.rightLim b - f.leftLim a :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/VectorMeasure","family_id":"vectormeasure_icc","file_id":"mathlib/Mathlib/MeasureTheory/VectorMeasure/BoundedVariation.lean","sample_id":"10754817056ac7fe2728b7d3c57dd409b9884a07365c423b530430a8a51d4d7e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"37b0177ca46b549523222694ca3ff53bf01def907bba860633ebbd21d6d3186d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d7dad36c4dd4b782ac303e0c16fb450acdb401810d5ca2a1d43aa67f80f20eeb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6e5dde6af19e66e06a4ae4317b8eb4e8eb4862f32bca8c9663051c5849e4b141","source_sha256":"bda612cd6d1ad1cbdd6e02bafd2e7e77c11e8cfa412747033a52bbe8062964f2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x\n  obtain ⟨I, rfl⟩ := ClassGroup.mk0_surjective x\n  rw [extendedHom_mk0, MonoidHom.one_apply]\n  exact (ClassGroup.mk0_eq_one_iff (extendedIdeal A B I).2).mpr (by simpa using h I)","hard_negative":true,"metrics":{"chosen_tokens":45,"rejected_tokens":5,"token_jaccard":0.090909,"token_length_ratio":0.111111},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"376259a361952449d7f02ec7e995816b2f078c596e2245e23effd7db12ad61c0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.FractionalIdeal.Extended\npublic import Mathlib.RingTheory.ClassGroup.Basic\n\nNamespace:\nClassGroup\n\nLocal context:\n/-\nCopyright (c) 2026 Riccardo Brasca. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Birkbeck, Riccardo Brasca\n-/\n/-!\n# Class group map induced by an extension of domains\n\nFor an injective extension `A → B` of commutative domains (equivalently `Module.IsTorsionFree A B`),\nwe construct the group homomorphism `ClassGroup.extendedHom : ClassGroup A →* ClassGroup B` given by\npushing fractional ideals forward along the algebra map.\n\n## Main definitions\n\n- `ClassGroup.extendedHom A B`: the induced map between the class groups.\n- `ClassGroup.extendedIdeal A B`: the extension of a nonzero integral ideal.\n\n## Main results\n\n- `ClassGroup.extendedHom_mk`: compatibility with representatives as fractional ideals.\n- `ClassGroup.extendedHom_mk0`: compatibility with representatives as nonzero integral ideals.\n- `ClassGroup.extendedHom_comp`: compatibility of extension in a tower `A → B → C`.\n- `ClassGroup.extendedHom_eq_one_of_forall_isPrincipal`: if the extension of every ideal is\n  principal, then `ClassGroup.extendedHom A B` is trivial.\n-/\n\npublic section\n\nopen scoped nonZeroDivisors\n\nvariable (A B : Type*) [CommRing A] [CommRing B] [Algebra A B]\n  [Module.IsTorsionFree A B]\n\nnamespace ClassGroup\n\nsection CommRing\n\nvariable [IsDomain A] [IsDomain B]\n\n/-- The monoid homomorphism `ClassGroup A → ClassGroup B` induced by an\ninjective extension of domains `A → B`. -/\nnoncomputable def extendedHom : ClassGroup A →* ClassGroup B :=\n  QuotientGroup.map _ _\n    (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom)\n    (by\n      rintro _ ⟨α, rfl⟩\n      refine ⟨Units.mk0 (IsFractionRing.map (j := algebraMap A B)\n        (FaithfulSMul.algebraMap_injective _ _) (α : FractionRing A))\n        (by simp [α.ne_zero]), ?_⟩\n      simpa [coe_toPrincipalIdeal, Units.coe_map, Units.val_mk0] using!\n        (FractionalIdeal.extendedHom_spanSingleton (FractionRing B) B _).symm)\n\n@[simp]\nlemma extendedHom_quotientMk (α : (FractionalIdeal A⁰ (FractionRing A))ˣ) :\n    extendedHom A B (QuotientGroup.mk α) = QuotientGroup.mk\n      (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom α) := by\n  rfl\n\n@[simp]\ntheorem extendedHom_mk (I : (FractionalIdeal A⁰ (FractionRing A))ˣ) :\n    extendedHom A B (ClassGroup.mk _ I) = ClassGroup.mk _\n        (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom I) := by\n  rw [← ClassGroup.Quot_mk_eq_mk, ← ClassGroup.Quot_mk_eq_mk]\n  exact extendedHom_quotientMk A B I\n\n/-- The extension of a nonzero integral ideal along an injective extension of domains. -/\nabbrev extendedIdeal (I : (Ideal A)⁰) : (Ideal B)⁰ :=\n  ⟨I.1.map (algebraMap A B), mem_nonZeroDivisors_iff_ne_zero.mpr <|\n    (Ideal.map_eq_bot_iff_of_injective (FaithfulSMul.algebraMap_injective A B)).not.mpr\n      (mem_nonZeroDivisors_iff_ne_zero.mp I.2)⟩\n\n@[simp]\ntheorem extendedIdeal_extendedIdeal (C : Type*) [CommRing C] [IsDomain C] [Algebra B C]\n    [Algebra A C] [IsScalarTower A B C] [Module.IsTorsionFree B C]\n    [Module.IsTorsionFree A C] (I : (Ideal A)⁰) :\n    extendedIdeal B C (extendedIdeal A B I) = extendedIdeal A C I := by\n  simp [Ideal.map_map, IsScalarTower.algebraMap_eq A B C]\n\nend CommRing\n\nsection DedekindDomain\n\nvariable [IsDedekindDomain A] (C : Type*) [CommRing C] [Algebra B C] [Algebra A C]\n  [IsScalarTower A B C] [Module.IsTorsionFree B C] [Module.IsTorsionFree A C]\n  [IsDedekindDomain C]\n\ntheorem extendedHom_mk0' [IsDomain B] (I : (Ideal A)⁰) :\n    extendedHom A B (ClassGroup.mk0 I) =\n      ClassGroup.mk _ (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom\n      (FractionalIdeal.mk0 (FractionRing A) I)) := by\n  rw [← ClassGroup.mk_mk0 (FractionRing A), extendedHom_mk]\n\nvariable [IsDedekindDomain B]\n\nlemma extendedHom_mk0 (I : (Ideal A)⁰) :\n    extendedHom A B (ClassGroup.mk0 I) = ClassGroup.mk0 (extendedIdeal A B I) := by\n  rw [mk0_eq_quotientMk, mk0_eq_quotientMk, extendedHom_quotientMk]\n  congr; ext : 1\n  exact FractionalIdeal.extendedHom_coeIdeal_eq_map (L := FractionRing B) (B := B) _\n\n\n@[simp]\ntheorem extendedHom_comp_apply (x : ClassGroup A) :\n      extendedHom B C (extendedHom A B x) = extendedHom A C x := by\n  obtain ⟨I, rfl⟩ := ClassGroup.mk0_surjective x\n  rw [extendedHom_mk0 A B I, extendedHom_mk0 B C (extendedIdeal A B I),\n    extendedHom_mk0 A C I, extendedIdeal_extendedIdeal]\n\ntheorem extendedHom_comp : (extendedHom B C).comp (extendedHom A B) = extendedHom A C := by\n  ext x\n  exact extendedHom_comp_apply A B C x\n\nTarget:\ntheorem extendedHom_eq_one_of_forall_isPrincipal\n    (h : ∀ I : (Ideal A), (I.map (algebraMap A B)).IsPrincipal) : extendedHom A B = 1 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_6e5dde6af19e","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"e996cae37284de7f25034b2334ab08e50ba7d6655f0a2622813c221c50635d65","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/ClassGroup","family_id":"extendedhom_eq_one_of_forall_isprincipal","file_id":"mathlib/Mathlib/RingTheory/ClassGroup/ExtendedHom.lean","sample_id":"6e5dde6af19e66e06a4ae4317b8eb4e8eb4862f32bca8c9663051c5849e4b141"},"split":"train"}
{"binding":{"chosen_proof_sha256":"471c71a8aa6e81210c7b7ea449c4bb32d4af4dfb79ecb5a4425fcd21336ff9fe","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"acf989bf4bb4722560bd130b7474c5acc042118d5f211afffa376e73cfffd6c3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a0a40b53d14535de5a165c9e14107454e6cc6b4afceef9775bdaaf0c31600cd0","source_sha256":"006922c150073d26eccce5bef3353d7015b01e87320ecec635275017a27177d4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [ncoeff, HVertexOperator.coeff, LinearMap.coe_mk, AddHom.coe_mk]\n  exact HahnSeries.coeff_eq_zero_of_lt_order h","hard_negative":true,"metrics":{"chosen_tokens":23,"rejected_tokens":5,"token_jaccard":0.157895,"token_length_ratio":0.217391},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"3858d7ef88881d29bc2475ddbae59ecdf6871d9d2c1f75f977b1629cd8491902","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Vertex.HVertexOperator\npublic import Mathlib.Data.Int.Interval\n\nNamespace:\nVertexOperator\n\nLocal context:\n/-\nCopyright (c) 2024 Scott Carnahan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Carnahan\n-/\n/-!\n# Vertex operators\nIn this file we introduce vertex operators as linear maps to Laurent series.\n\n## Definitions\n* `VertexOperator` is an `R`-linear map from an `R`-module `V` to `LaurentSeries V`.\n* `VertexOperator.ncoeff` is the coefficient of a vertex operator under normalized indexing.\n\n## TODO\n* `HasseDerivative` : A divided-power derivative.\n* `Locality` : A weak form of commutativity.\n* `Residue products` : A family of products on `VertexOperator R V` parametrized by integers.\n\n## References\n* [G. Mason, *Vertex rings and Pierce bundles*][mason2017]\n* [A. Matsuo, K. Nagatomo, *On axioms for a vertex algebra and locality of quantum\n  fields*][matsuo1997]\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nvariable {R V : Type*} [CommRing R] [AddCommGroup V] [Module R V]\n\n/-- A vertex operator over a commutative ring `R` is an `R`-linear map from an `R`-module `V` to\nLaurent series with coefficients in `V`.  We write this as a specialization of the heterogeneous\ncase. -/\nabbrev VertexOperator (R : Type*) (V : Type*) [CommRing R] [AddCommGroup V]\n    [Module R V] := HVertexOperator ℤ R V V\n\nnamespace VertexOperator\n\nopen HVertexOperator\n\n@[ext]\ntheorem ext (A B : VertexOperator R V) (h : ∀ v : V, A v = B v) :\n    A = B := LinearMap.ext h\n\n/-- The coefficient of a vertex operator under normalized indexing. -/\ndef ncoeff : VertexOperator R V →ₗ[R] ℤ → Module.End R V where\n  toFun A n := HVertexOperator.coeff A (-n - 1)\n  map_add' _ _ := by ext; simp\n  map_smul' _ _ := by ext; simp\n\ntheorem ncoeff_apply (A : VertexOperator R V) (n : ℤ) : ncoeff A n = coeff A (-n - 1) :=\n  rfl\n\n/-- In the literature, the `n`th normalized coefficient of a vertex operator `A` is written as\neither `Aₙ` or `A(n)`. -/\nscoped[VertexOperator] notation A \"[[\" n \"]]\" => ncoeff A n\n\n@[simp]\ntheorem coeff_eq_ncoeff (A : VertexOperator R V)\n    (n : ℤ) : HVertexOperator.coeff A n = A[[-n - 1]] := by\n  rw [ncoeff_apply, neg_sub, Int.sub_neg, add_sub_cancel_left]\n\nTarget:\ntheorem ncoeff_eq_zero_of_lt_order (A : VertexOperator R V) (n : ℤ) (x : V)\n    (h : -n - 1 < HahnSeries.order ((HahnModule.of R).symm (A x))) : (A[[n]]) x = 0 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_a0a40b53d145","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"48301007c1b8d53a738af50f821372f8bb24ae9827b904c9e8c0084f2eacbef4","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Vertex","family_id":"ncoeff_eq_zero_of_lt_order","file_id":"mathlib/Mathlib/Algebra/Vertex/VertexOperator.lean","sample_id":"a0a40b53d14535de5a165c9e14107454e6cc6b4afceef9775bdaaf0c31600cd0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9c3779b7663f368283b7a9eee3adc6092b2e62b6c24d85d914985cdd33c9ca16","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b7454ed33b7067c1c405e0b58237396fb00e395f6ff19576d9916ffdee3bc4d9","source_sha256":"4d4d0117550f0a508e4a45c091460a207bd2706a42ac9b83d9699e528b57c020","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun H => H.van_kampen' c₀ hc₀, fun H => ?_⟩\n  constructor\n  simp_rw [BinaryCofan.isVanKampen_iff] at H ⊢\n  intro X Y c hc X' Y' c' αX αY f hX hY\n  obtain ⟨d, hd, hd'⟩ :=\n    Limits.BinaryCofan.IsColimit.desc' hc (HT.from _ ≫ c₀.inl) (HT.from _ ≫ c₀.inr)\n  rw [H c' (αX ≫ HT.from _) (αY ≫ HT.from _) (f ≫ d) (by rw [← reassoc_of% hX, hd, Category.assoc])\n      (by rw [← reassoc_of% hY, hd', Category.assoc])]\n  obtain ⟨hl, hr⟩ := (H c (HT.from _) (HT.from _) d hd.symm hd'.symm).mp ⟨hc⟩\n  rw [hl.paste_vert_iff hX.symm, hr.paste_vert_iff hY.symm]","hard_negative":false,"metrics":{"chosen_tokens":195,"rejected_tokens":2,"token_jaccard":0.016667,"token_length_ratio":0.010256},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"3871df2da27da115e888c4cbd6568e5e7e1e0973f029f278eee1d39b110d90af","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Types.Coproducts\npublic import Mathlib.CategoryTheory.Limits.Types.Products\npublic import Mathlib.CategoryTheory.Limits.Types.Pullbacks\npublic import Mathlib.Topology.Category.TopCat.Limits.Pullbacks\npublic import Mathlib.CategoryTheory.Limits.VanKampen\npublic import Mathlib.CategoryTheory.Limits.MonoCoprod\npublic import Mathlib.CategoryTheory.Limits.Shapes.DisjointCoproduct\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Extensive categories\n\n## Main definitions\n- `CategoryTheory.FinitaryExtensive`: A category is (finitary) extensive if it has finite\n  coproducts, and binary coproducts are van Kampen.\n\n## Main Results\n- `CategoryTheory.hasStrictInitialObjects_of_finitaryExtensive`: The initial object\n  in extensive categories is strict.\n- `CategoryTheory.FinitaryExtensive.mono_inr_of_isColimit`: Coproduct injections are monic in\n  extensive categories.\n- `CategoryTheory.BinaryCofan.isPullback_initial_to_of_isVanKampen`: In extensive categories,\n  sums are disjoint, i.e. the pullback of `X ⟶ X ⨿ Y` and `Y ⟶ X ⨿ Y` is the initial object.\n- `CategoryTheory.types.finitaryExtensive`: The category of types is extensive.\n- `CategoryTheory.FinitaryExtensive_TopCat`:\n  The category `Top` is extensive.\n- `CategoryTheory.FinitaryExtensive_functor`: The category `C ⥤ D` is extensive if `D`\n  has all pullbacks and is extensive.\n- `CategoryTheory.FinitaryExtensive.isVanKampen_finiteCoproducts`: Finite coproducts in a\n  finitary extensive category are van Kampen.\n\n## References\n- https://ncatlab.org/nlab/show/extensive+category\n- [Carboni et al, Introduction to extensive and distributive categories][CARBONI1993145]\n\n-/\n\n@[expose] public section\n\nopen CategoryTheory.Limits Topology\n\nnamespace CategoryTheory\n\nuniverse v' u' v u v'' u''\n\nvariable {J : Type v'} [Category.{u'} J] {C : Type u} [Category.{v} C]\nvariable {D : Type u''} [Category.{v''} D]\n\nsection Extensive\n\nvariable {X Y : C}\n\n/-- A category has pullback of inclusions if it has all pullbacks along coproduct injections. -/\nclass HasPullbacksOfInclusions (C : Type u) [Category.{v} C] [HasBinaryCoproducts C] : Prop where\n  [hasPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), HasPullback coprod.inl f]\n\nattribute [instance] HasPullbacksOfInclusions.hasPullbackInl\n\n/--\nA functor preserves pullback of inclusions if it preserves all pullbacks along coproduct injections.\n-/\nclass PreservesPullbacksOfInclusions {C : Type*} [Category* C] {D : Type*} [Category* D]\n    (F : C ⥤ D) [HasBinaryCoproducts C] where\n  [preservesPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), PreservesLimit (cospan coprod.inl f) F]\n\nattribute [instance] PreservesPullbacksOfInclusions.preservesPullbackInl\n\n/-- A category is (finitary) pre-extensive if it has finite coproducts,\nand binary coproducts are universal. -/\nclass FinitaryPreExtensive (C : Type u) [Category.{v} C] : Prop where\n  [hasFiniteCoproducts : HasFiniteCoproducts C]\n  [hasPullbacksOfInclusions : HasPullbacksOfInclusions C]\n  /-- In a finitary extensive category, all coproducts are van Kampen -/\n  universal' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsUniversalColimit c\n\nattribute [instance] FinitaryPreExtensive.hasFiniteCoproducts\nattribute [instance] FinitaryPreExtensive.hasPullbacksOfInclusions\n\n/-- A category is (finitary) extensive if it has finite coproducts,\nand binary coproducts are van Kampen. -/\nclass FinitaryExtensive (C : Type u) [Category.{v} C] : Prop where\n  [hasFiniteCoproducts : HasFiniteCoproducts C]\n  [hasPullbacksOfInclusions : HasPullbacksOfInclusions C]\n  /-- In a finitary extensive category, all coproducts are van Kampen -/\n  van_kampen' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsVanKampenColimit c\n\nattribute [instance] FinitaryExtensive.hasFiniteCoproducts\nattribute [instance] FinitaryExtensive.hasPullbacksOfInclusions\n\ntheorem FinitaryExtensive.vanKampen [FinitaryExtensive C] {F : Discrete WalkingPair ⥤ C}\n    (c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c := by\n  let X := F.obj ⟨WalkingPair.left⟩\n  let Y := F.obj ⟨WalkingPair.right⟩\n  have : F = pair X Y := by\n    apply Functor.hext\n    · rintro ⟨⟨⟩⟩ <;> rfl\n    · rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simp [X, Y]\n  clear_value X Y\n  subst this\n  exact FinitaryExtensive.van_kampen' c hc\n\nnamespace HasPullbacksOfInclusions\n\ninstance (priority := 100) [HasBinaryCoproducts C] [HasPullbacks C] :\n    HasPullbacksOfInclusions C := ⟨⟩\n\nvariable [HasBinaryCoproducts C] [HasPullbacksOfInclusions C] {X Y Z : C} (f : Z ⟶ X ⨿ Y)\n\ninstance preservesPullbackInl' :\n    HasPullback f coprod.inl :=\n  hasPullback_symmetry _ _\n\nset_option backward.isDefEq.respectTransparency false in\ninstance hasPullbackInr' :\n    HasPullback f coprod.inr := by\n  have : IsPullback (𝟙 _) (f ≫ (coprod.braiding X Y).hom) f (coprod.braiding Y X).hom :=\n    IsPullback.of_horiz_isIso ⟨by simp⟩\n  have := (IsPullback.of_hasPullback (f ≫ (coprod.braiding X Y).hom) coprod.inl).paste_horiz this\n  simp only [coprod.braiding_hom, Category.comp_id, colimit.ι_desc,\n    BinaryCofan.ι_app_left, BinaryCofan.mk_inl] at this\n  exact ⟨⟨⟨_, this.isLimit⟩⟩⟩\n\ninstance hasPullbackInr :\n    HasPullback coprod.inr f :=\n  hasPullback_symmetry _ _\n\nend HasPullbacksOfInclusions\n\nnamespace PreservesPullbacksOfInclusions\n\nvariable {D : Type*} [Category* D] [HasBinaryCoproducts C] (F : C ⥤ D)\n\nnoncomputable\ninstance (priority := 100) [PreservesLimitsOfShape WalkingCospan F] :\n    PreservesPullbacksOfInclusions F := ⟨⟩\n\nvariable [PreservesPullbacksOfInclusions F] {X Y Z : C} (f : Z ⟶ X ⨿ Y)\n\nnoncomputable\ninstance preservesPullbackInl' :\n    PreservesLimit (cospan f coprod.inl) F :=\n  preservesPullback_symmetry _ _ _\n\nset_option backward.isDefEq.respectTransparency false in\nnoncomputable\ninstance preservesPullbackInr' :\n    PreservesLimit (cospan f coprod.inr) F := by\n  apply preservesLimit_of_iso_diagram (K₁ := cospan (f ≫ (coprod.braiding X Y).hom) coprod.inl)\n  apply cospanExt (Iso.refl _) (Iso.refl _) (coprod.braiding X Y).symm <;> simp\n\nnoncomputable\ninstance preservesPullbackInr :\n    PreservesLimit (cospan coprod.inr f) F :=\n  preservesPullback_symmetry _ _ _\n\nend PreservesPullbacksOfInclusions\n\ninstance (priority := 100) FinitaryExtensive.toFinitaryPreExtensive [FinitaryExtensive C] :\n    FinitaryPreExtensive C :=\n  ⟨fun c hc ↦ (FinitaryExtensive.van_kampen' c hc).isUniversal⟩\n\ntheorem FinitaryExtensive.mono_inr_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}\n    (hc : IsColimit c) : Mono c.inr :=\n  BinaryCofan.mono_inr_of_isVanKampen (FinitaryExtensive.vanKampen c hc)\n\ntheorem FinitaryExtensive.mono_inl_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}\n    (hc : IsColimit c) : Mono c.inl :=\n  FinitaryExtensive.mono_inr_of_isColimit (BinaryCofan.isColimitFlip hc)\n\ninstance (priority := low) [FinitaryExtensive C] : MonoCoprod C where\n  binaryCofan_inl _ _ _ hc := BinaryCofan.mono_inr_of_isVanKampen\n    (FinitaryExtensive.vanKampen _ (BinaryCofan.isColimitFlip hc))\n\ntheorem FinitaryExtensive.isPullback_initial_to_binaryCofan [FinitaryExtensive C]\n    {c : BinaryCofan X Y} (hc : IsColimit c) :\n    IsPullback (initial.to _) (initial.to _) c.inl c.inr :=\n  BinaryCofan.isPullback_initial_to_of_isVanKampen (FinitaryExtensive.vanKampen c hc)\n\nset_option backward.defeqAttrib.useBackward true in\ninstance (priority := 100) hasStrictInitialObjects_of_finitaryPreExtensive\n    [FinitaryPreExtensive C] : HasStrictInitialObjects C :=\n  hasStrictInitial_of_isUniversal (FinitaryPreExtensive.universal' _\n    ((BinaryCofan.isColimit_iff_isIso_inr initialIsInitial _).mpr (by\n      dsimp\n      infer_instance)).some)\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem finitaryExtensive_iff_of_isTerminal (C : Type u) [Category.{v} C] [HasFiniteCoproducts C]\n    [HasPullbacksOfInclusions C]\n    (T : C) (HT : IsTerminal T) (c₀ : BinaryCofan T T) (hc₀ : IsColimit c₀) :\n    FinitaryExtensive C ↔ IsVanKampenColimit c₀ :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory","family_id":"finitaryextensive_iff_of_isterminal","file_id":"mathlib/Mathlib/CategoryTheory/Extensive.lean","sample_id":"b7454ed33b7067c1c405e0b58237396fb00e395f6ff19576d9916ffdee3bc4d9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f994609f95dbe1be796aaafe12c37132a7b266348fd2421e6b7d6339c7e0ee00","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"92a52bbb327e83cdf5240725d5efdd23b48813429f9bb0e08c129150def1dc99","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"184e16fbe9e5eec732502dd0a1ed9c46036bcaecb5ded625d98d24b0ac54477f","source_sha256":"ff10564e9930cd55392330030cf618a47034f00b752e4ff3cc347c8e7c823f58","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using! (H.toChainHomotopy).homologyMap_eq n","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.25},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"38743134bb4a1adde6fff4899380853797dc96253099eb40e9181f50887d9237","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.SimplicialObject.Homotopy\npublic import Mathlib.AlgebraicTopology.AlternatingFaceMapComplex\npublic import Mathlib.Algebra.Homology.Homotopy\n\nNamespace:\nCategoryTheory.SimplicialObject.Homotopy\n\nLocal context:\n/-\nCopyright (c) 2025 Fabian Odermatt. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Fabian Odermatt\n-/\n/-!\n# Simplicial homotopies induce chain homotopies\n\nGiven a simplicial homotopy between morphisms of simplicial objects in a preadditive category,\nwe construct a chain homotopy between the induced morphisms on the alternating face map complexes.\n\nConcretely, if `H : Homotopy f g` gives maps\n`H.h i : X _⦋n⦌ ⟶ Y _⦋n+1⦌` indexed by `i : Fin (n + 1)`, we define the degree-`n` component\nof the chain homotopy as the opposite of alternating sum `∑ i, (-1)^i • H.h i`.\n-/\n\n@[expose] public section\n\nuniverse v u\n\nopen CategoryTheory CategoryTheory.SimplicialObject\nopen SimplexCategory Simplicial Opposite AlgebraicTopology\n\nnamespace CategoryTheory.SimplicialObject.Homotopy\n\nvariable {C : Type u} [Category.{v} C] [Preadditive C]\nvariable {X Y : SimplicialObject C} {f g : X ⟶ Y}\nvariable (H : Homotopy f g)\n\nnamespace ToChainHomotopy\n\n/-- The family of components of the induced chain homotopy -/\nnoncomputable def hom (p q : ℕ) : X _⦋p⦌ ⟶ Y _⦋q⦌ :=\n  if h : p + 1 = q then\n    -∑ k : Fin (p + 1), ((-1 : ℤ) ^ (k : ℕ)) • H.h k ≫ eqToHom (by simp [h])\n  else 0\n\n@[simp]\nlemma hom_eq (p : ℕ) :\n    hom H p (p + 1) = -∑ k : Fin (p + 1), ((-1 : ℤ) ^ (k : ℕ)) • H.h k := by\n  simp [hom]\n\n@[simp]\nlemma hom_eq_zero (p q : ℕ) (hpq : p + 1 ≠ q) :\n    hom H p q = 0 :=\n  dif_neg hpq\n\nprivate lemma comm_zero :\n    letI d : Y _⦋1⦌ ⟶ Y _⦋0⦌ := ((alternatingFaceMapComplex C).obj Y).d 1 0\n    f.app (op ⦋0⦌) = hom H 0 1 ≫ d + g.app (op ⦋0⦌) := by\n  simp [← H.h_last_comp_δ_last 0]\n\nprivate lemma comm_succ (n : ℕ) :\n    letI α : X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌ :=\n      ((alternatingFaceMapComplex C).obj X).d (n + 1) n ≫ ToChainHomotopy.hom H n (n + 1)\n    letI β : X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌ := hom H (n + 1) (n + 2) ≫\n      ((alternatingFaceMapComplex C).obj Y).d (n + 2) (n + 1)\n    f.app (op ⦋n + 1⦌) = α + β + g.app (op ⦋n + 1⦌) := by\n  rw [← H.h_zero_comp_δ_zero, ← H.h_last_comp_δ_last]\n  dsimp\n  simp only [alternatingFaceMapComplex_obj_d, AlternatingFaceMapComplex.objD, hom_eq,\n    Preadditive.comp_neg, Preadditive.neg_comp, Preadditive.comp_sum,\n    Preadditive.sum_comp, Preadditive.comp_zsmul, Preadditive.zsmul_comp,\n    smul_neg, Finset.sum_neg_distrib, ← Finset.sum_zsmul, smul_smul, ← pow_add]\n  let α (x : Fin (n + 1) × Fin (n + 2)) := (-1) ^ ((x.1 + x.2 : ℕ)) • X.δ x.2 ≫ H.h x.1\n  let β (x : Fin (n + 3) × Fin (n + 2)) := (-1) ^ ((x.1 + x.2 : ℕ)) • H.h x.2 ≫ Y.δ x.1\n  have h₂ (x : Fin (n + 1) × Fin (n + 2)) (hx : x.1.castSucc < x.2) :\n      α x = -β ⟨x.2.succ, x.1.castSucc⟩ := by\n    dsimp [α, β]\n    simp only [← H.h_castSucc_comp_δ_succ_of_lt x.2 x.1 hx,\n      pow_add, pow_one, mul_neg, mul_one, neg_mul, neg_smul, neg_neg]\n    rw [mul_comm]\n  rw [← Finset.sum_product .univ .univ α, ← Finset.sum_product .univ .univ β,\n    Finset.univ_product_univ, Finset.univ_product_univ]\n  let S : Finset (Fin (n + 1) × Fin (n + 2)) := { x | x.1.castSucc < x.2 }\n  let γ₁ (x : Fin (n + 1) × Fin (n + 2)) := (x.2.castSucc, x.1.succ)\n  let γ₂ (x : Fin (n + 1) × Fin (n + 2)) := (x.2.succ, x.1.castSucc)\n  let γ₃ (i : Fin (n + 1)) := (i.castSucc.succ, i.succ)\n  let γ₄ (i : Fin (n + 1)) := (i.castSucc.succ, i.castSucc)\n  have hγ₁ : Function.Injective γ₁ := fun _ _ ↦ by aesop\n  have hγ₂ : Function.Injective γ₂ := fun _ _ ↦ by aesop\n  have hγ₃ : Function.Injective γ₃ := fun _ _ ↦ by aesop\n  have hγ₄ : Function.Injective γ₄ := fun _ _ ↦ by aesop\n  have eq₁ : H.h 0 ≫ Y.δ 0 = β ⟨0, 0⟩ := by simp [β]\n  have eq₂ : H.h (Fin.last _) ≫ Y.δ (Fin.last _) = - β ⟨Fin.last _, Fin.last _⟩ := by\n    dsimp [β]\n    simp only [pow_add, even_two, Even.neg_pow, one_pow, mul_one,\n      pow_one, mul_neg, neg_smul, neg_neg]\n    rw [← pow_add, (Even.add_self n).neg_one_pow, one_smul]\n  have eq₃ : ∑ x ∈ Sᶜ, α x = - ∑ y ∈ Finset.image γ₁ Sᶜ, β y := by\n    rw [← Finset.sum_neg_distrib, Finset.sum_image hγ₁.injOn]\n    refine Finset.sum_congr rfl (fun x hx ↦ ?_)\n    dsimp [α, β, γ₁]\n    simp only [← H.h_succ_comp_δ_castSucc_of_lt x.2 x.1 (by simpa [S] using hx),\n      pow_add, pow_one, mul_neg, mul_one, neg_smul, neg_neg]\n    rw [mul_comm]\n  have eq₄ : ∑ x ∈ S, α x = - ∑ y ∈ Finset.image γ₂ S, β y := by\n    rw [← Finset.sum_neg_distrib, Finset.sum_image hγ₂.injOn]\n    refine Finset.sum_congr rfl (fun x hx ↦ ?_)\n    dsimp [α, β, γ₂]\n    simp only [← H.h_castSucc_comp_δ_succ_of_lt x.2 x.1 (by simpa [S] using hx),\n      pow_add, pow_one, mul_neg, mul_one, neg_mul, neg_smul, neg_neg]\n    rw [mul_comm]\n  have eq₅ : ∑ x, β (γ₄ x) = - ∑ x, β (γ₃ x) := by\n    rw [← Finset.sum_neg_distrib]\n    exact Finset.sum_congr rfl (fun x hx ↦ by simp [h_succ_comp_δ_castSucc_succ, β, γ₃, γ₄])\n  have h₁ : Disjoint (Finset.image γ₁ Sᶜ) (Finset.image γ₂ S) := by\n    rw [Finset.disjoint_iff_ne]\n    grind [Finset.mem_compl]\n  have h₂ : Disjoint (Finset.image γ₃ .univ) (Finset.image γ₄ .univ) := by\n    rw [Finset.disjoint_iff_ne]\n    grind\n  have h₃ : Disjoint (Finset.disjUnion _ _ h₂) {(0, 0), (Fin.last _, Fin.last _)} := by\n    rw [Finset.disjoint_iff_ne]\n    simp only [Finset.mem_insert, forall_eq_or_imp, Prod.forall]\n    rintro ⟨a, _⟩ ⟨b, _⟩\n    simp\n    grind\n  have h₄ : Disjoint (Finset.disjUnion _ _ h₁) (Finset.disjUnion _ _ h₃) := by\n    rw [Finset.disjoint_iff_ne]\n    simp only [Finset.compl_filter, not_lt, Finset.disjUnion_eq_union, Finset.mem_union,\n      Finset.mem_image, Finset.mem_filter, Finset.mem_univ, true_and, Prod.exists, ne_eq,\n      Finset.mem_insert, Finset.mem_singleton, Prod.forall, Prod.mk.injEq, not_and,\n      S, γ₁, γ₂, γ₃, γ₄]\n    rintro ⟨a, _⟩ ⟨b, _⟩ (⟨⟨j, _⟩, ⟨k, _⟩, h₁, h₂, h₃⟩ | ⟨⟨j, _⟩, ⟨k, _⟩, h₁, h₂, h₃⟩) _ _\n      ((⟨⟨i, _⟩, h₄, h₅⟩ | ⟨⟨i, _⟩, h₄, h₅⟩) | (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)) <;>\n        simp [Fin.ext_iff] at h₁ h₂ h₃ ⊢ <;> grind\n  have H : (Finset.disjUnion _ _ h₁)ᶜ = Finset.disjUnion _ _ h₃ :=\n    Finset.compl_eq_of_disjoint_of_card_add_eq h₄ (by\n      rw [Finset.card_disjUnion, Finset.card_disjUnion, Finset.card_disjUnion,\n        Finset.card_image_of_injective _ hγ₁, Finset.card_image_of_injective _ hγ₂,\n        Finset.card_image_of_injective _ hγ₃, Finset.card_image_of_injective _ hγ₄]\n      simp\n      lia)\n  rw [eq₁, eq₂, ← S.sum_add_sum_compl, eq₃, eq₄,\n    neg_add_rev, neg_neg, neg_neg, ← Finset.sum_disjUnion h₁,\n    ← (Finset.disjUnion _ _ h₁).sum_add_sum_compl, neg_add,\n    ← add_assoc, add_neg_cancel, zero_add, H,\n    Finset.sum_disjUnion, Finset.sum_disjUnion,\n    Finset.sum_image hγ₃.injOn, Finset.sum_image hγ₄.injOn,\n    Finset.sum_insert (by simp), Finset.sum_singleton,\n    neg_add_rev, neg_add_rev, neg_add_rev, eq₅]\n  simp\n\nend ToChainHomotopy\n\nset_option backward.isDefEq.respectTransparency false in\n/-- A simplicial homotopy between `f` and `g` induces a chain homotopy\nbetween the induced morphisms on the alternating face map complexes. -/\nnoncomputable def toChainHomotopy (H : Homotopy f g) :\n    _root_.Homotopy\n      ((alternatingFaceMapComplex C).map f)\n      ((alternatingFaceMapComplex C).map g) where\n  hom := ToChainHomotopy.hom H\n  zero i j hij := ToChainHomotopy.hom_eq_zero _ _ _ hij\n  comm n := by\n    cases n with\n    | zero =>\n      rw [prevD_eq (j' := 1) (w := by simp), dNext_eq_zero _ _ (by simp), zero_add]\n      simp [ToChainHomotopy.comm_zero H]\n    | succ n =>\n      rw [dNext_eq (i' := n) (w := by simp), prevD_eq (j' := n + 2) (w := by simp)]\n      simp [ToChainHomotopy.comm_succ H]\n\nTarget:\ntheorem map_homology_eq [CategoryWithHomology C] (H : Homotopy f g) (n : ℕ) :\n    (HomologicalComplex.homologyFunctor C _ n).map ((alternatingFaceMapComplex C).map f) =\n    (HomologicalComplex.homologyFunctor C _ n).map ((alternatingFaceMapComplex C).map g) :=\n\nProof body:\n","rejected":"by\n  exact map_homology_eq","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"2bfb5d937c1385a985e75f8506d07e9796deb0b660381a52ac172e0f2beb8a92","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplicialObject","family_id":"map_homology_eq","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplicialObject/ChainHomotopy.lean","sample_id":"184e16fbe9e5eec732502dd0a1ed9c46036bcaecb5ded625d98d24b0ac54477f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"19091c655ecaa51563cbf7438e93b5ad161e207efc740ddfb20c17af665fcd74","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"028d6d8509b98611e3f56cc2c5f547656c14ba62d311582bead44f78610f5634","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1d731b6f1a0aa28042e1c006480d5c7f9dc42d3f0a8f8f5b52bbb265d67e141d","source_sha256":"5fbd6e0c9aed4ae0af15e49b0a292eb4e252a42e876e277383b0144f2f430dad","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro x y h\n  by_cases hx : x ∈ xs <;> by_cases hy : y ∈ xs\n  · rw [List.mem_iff_getElem?] at hx hy\n    obtain ⟨i, hx⟩ := hx\n    obtain ⟨j, hy⟩ := hy\n    suffices some x = some y by injection this\n    have h₂ := h₁.length_eq\n    rw [List.applyId_zip_eq h₀ h₂ _ _ _ hx] at h\n    rw [← hx, ← hy]; congr\n    apply (List.getElem?_inj _ (h₁.nodup_iff.1 h₀)).mp\n    · symm; rw [h]\n      rw [← List.applyId_zip_eq] <;> assumption\n    · rw [← h₁.length_eq]\n      rw [List.getElem?_eq_some_iff] at hx\n      obtain ⟨hx, hx'⟩ := hx\n      exact hx\n  · rw [← applyId_mem_iff h₀ h₁] at hx hy\n    rw [h] at hx\n    contradiction\n  · rw [← applyId_mem_iff h₀ h₁] at hx hy\n    rw [h] at hx\n    contradiction\n  · rwa [List.applyId_eq_self, List.applyId_eq_self] at h <;> assumption","hard_negative":false,"metrics":{"chosen_tokens":215,"rejected_tokens":220,"token_jaccard":0.938462,"token_length_ratio":1.023256},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"387f03c2b097f6e2d3e2ea34e628249cba148a80923c64e0a21ac9ad1e4a1a0b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.MLList.Basic\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.ZeroLEOne\npublic import Mathlib.Data.DFinsupp.Defs\npublic import Mathlib.Data.Finsupp.Defs\npublic import Mathlib.Data.Int.Range\npublic import Mathlib.Data.List.Sigma\npublic import Mathlib.Data.PNat.Notation\npublic import Mathlib.Tactic.Bound.Init\n\nNamespace:\nPlausible.InjectiveFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\npublic meta import Mathlib.Data.Finsupp.ToDFinsupp\npublic meta import Mathlib.Data.Int.Range\npublic meta import Mathlib.Data.List.Sigma\npublic meta import Plausible.Functions\n\n/-!\n## `Plausible`: generators for functions\n\nThis file defines `Sampleable` instances for `ℤ → ℤ` injective functions.\n\nInjective functions are generated by creating a list of numbers and\na permutation of that list. The permutation insures that every input\nis mapped to a unique output. When an input is not found in the list\nthe input itself is used as an output.\n\nInjective functions `f : α → α` could be generated easily instead of\n`ℤ → ℤ` by generating a `List α`, removing duplicates and creating a\npermutation. One has to be careful when generating the domain to make\nit vast enough that, when generating arguments to apply `f` to,\nthey argument should be likely to lie in the domain of `f`. This is\nthe reason that injective functions `f : ℤ → ℤ` are generated by\nfixing the domain to the range `[-2*size .. 2*size]`, with `size`\nthe size parameter of the `gen` monad.\n\nMuch of the machinery provided in this file is applicable to generate\ninjective functions of type `α → α` and new instances should be easy\nto define.\n\nOther classes of functions such as monotone functions can generated using\nsimilar techniques. For monotone functions, generating two lists, sorting them\nand matching them should suffice, with appropriate default values.\nSome care must be taken for shrinking such functions to make sure\ntheir defining property is invariant through shrinking. Injective\nfunctions are an example of how complicated it can get.\n-/\n\n@[expose] public meta section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type v}\n\nnamespace Plausible\n\nnamespace TotalFunction\n\nsection Finsupp\n\nvariable [DecidableEq α]\n\n/--\nThis theorem exists because plausible does not have access to dlookup but\nmathlib has all the theory for it and wants to use it. We probably want to\nbring these two together at some point.\n-/\nprivate theorem apply_eq_dlookup (m : List (Σ _ : α, β)) (y : β) (x : α) :\n     (withDefault m y).apply x = (m.dlookup x).getD y := by\n  dsimp only [apply]\n  congr 1\n  induction m with\n  | nil => simp\n  | cons p m ih =>\n    rcases p with ⟨fst, snd⟩\n    by_cases heq : fst = x\n    · simp [heq]\n    · rw [List.dlookup_cons_ne]\n      · simp [heq, ih]\n      · symm\n        simp [heq]\n\nvariable [Zero β] [DecidableEq β]\n\n/-- Map a `TotalFunction` to one whose default value is zero so that it represents a `Finsupp`. -/\n@[simp]\ndef zeroDefault : TotalFunction α β → TotalFunction α β\n  | .withDefault A _ => .withDefault A 0\n\n/-- The support of a zero default `TotalFunction`. -/\ndef zeroDefaultSupp : TotalFunction α β → Finset α\n  | .withDefault A _ =>\n    List.toFinset <| (A.dedupKeys.filter fun ab => Sigma.snd ab ≠ 0).map Sigma.fst\n\n/-- Create a finitely supported function from a total function by taking the default value to\nzero. -/\ndef applyFinsupp (tf : TotalFunction α β) : α →₀ β where\n  support := zeroDefaultSupp tf\n  toFun := tf.zeroDefault.apply\n  mem_support_toFun := by\n    intro a\n    rcases tf with ⟨A, y⟩\n    simp only [zeroDefaultSupp, List.mem_map, List.mem_filter, exists_and_right,\n      List.mem_toFinset, exists_eq_right, Sigma.exists, Ne, zeroDefault]\n    rw [apply_eq_dlookup]\n    constructor\n    · rintro ⟨od, hval, hod⟩\n      have := List.mem_dlookup (List.nodupKeys_dedupKeys A) hval\n      rw [(_ : List.dlookup a A = od)]\n      · simpa using hod\n      · simpa [List.dlookup_dedupKeys]\n    · intro h\n      use (A.dlookup a).getD (0 : β)\n      rw [← List.dlookup_dedupKeys] at h ⊢\n      simp only [h, ← List.mem_dlookup_iff A.nodupKeys_dedupKeys, not_false_iff, Option.mem_def]\n      cases haA : List.dlookup a A.dedupKeys\n      · simp [haA] at h\n      · simp\n\nvariable [SampleableExt α] [SampleableExt β] [Repr α]\n\ninstance Finsupp.sampleableExt : SampleableExt (α →₀ β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\n-- TODO: support a non-constant codomain type\ninstance DFinsupp.sampleableExt : SampleableExt (Π₀ _ : α, β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp.toDFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\nend Finsupp\nend TotalFunction\n\nopen _root_.List\n\n/-- Data structure specifying a total function using a list of pairs\nand a default value returned when the input is not in the domain of\nthe partial function.\n\n`mapToSelf f` encodes `x ↦ f x` when `x ∈ f` and `x ↦ x`,\ni.e. `x` to itself, otherwise.\n\nWe use `Σ` to encode mappings instead of `×` because we\nrely on the association list API defined in `Mathlib/Data/List/Sigma.lean`.\n-/\ninductive InjectiveFunction (α : Type u) : Type u\n  | mapToSelf (xs : List (Σ _ : α, α)) :\n      xs.map Sigma.fst ~ xs.map Sigma.snd → List.Nodup (xs.map Sigma.snd) → InjectiveFunction α\n\ninstance : Inhabited (InjectiveFunction α) :=\n  ⟨⟨[], List.Perm.nil, List.nodup_nil⟩⟩\n\nnamespace InjectiveFunction\n\n/-- Apply a total function to an argument. -/\ndef apply [DecidableEq α] : InjectiveFunction α → α → α\n  | InjectiveFunction.mapToSelf m _ _, x => (m.dlookup x).getD x\n\n/-- Produce a string for a given `InjectiveFunction`.\nThe output is of the form `[x₀ ↦ f x₀, .. xₙ ↦ f xₙ, x ↦ x]`.\nUnlike for `TotalFunction`, the default value is not a constant\nbut the identity function.\n-/\nprotected def repr [Repr α] : InjectiveFunction α → String\n  | InjectiveFunction.mapToSelf m _ _ => s! \"[{TotalFunction.reprAux m}x ↦ x]\"\n\ninstance (α : Type u) [Repr α] : Repr (InjectiveFunction α) where\n  reprPrec f _p := InjectiveFunction.repr f\n\n/-- Interpret a list of pairs as a total function, defaulting to\nthe identity function when no entries are found for a given function -/\ndef List.applyId [DecidableEq α] (xs : List (α × α)) (x : α) : α :=\n  ((xs.map Prod.toSigma).dlookup x).getD x\n\n@[simp]\ntheorem List.applyId_cons [DecidableEq α] (xs : List (α × α)) (x y z : α) :\n    List.applyId ((y, z)::xs) x = if y = x then z else List.applyId xs x := by\n  simp only [List.applyId, List.dlookup, eq_rec_constant, Prod.toSigma, List.map]\n  split_ifs <;> rfl\n\nopen Function\nopen List\n\nopen Nat\n\ntheorem List.applyId_zip_eq [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs)\n    (h₁ : xs.length = ys.length) (x y : α) (i : ℕ) (h₂ : xs[i]? = some x) :\n    List.applyId.{u} (xs.zip ys) x = y ↔ ys[i]? = some y := by\n  induction xs generalizing ys i with\n  | nil => cases h₂\n  | cons x' xs xs_ih =>\n    cases i\n    · simp only [length_cons, lt_add_iff_pos_left, add_pos_iff, Nat.lt_add_one, or_true,\n        getElem?_eq_getElem, getElem_cons_zero, Option.some.injEq] at h₂\n      subst h₂\n      cases ys\n      · cases h₁\n      · simp\n    · cases ys\n      · cases h₁\n      · obtain - | ⟨h₀, h₁⟩ := h₀\n        simp only [getElem?_cons_succ, zip_cons_cons, applyId_cons] at h₂ ⊢\n        rw [if_neg]\n        · apply xs_ih <;> solve_by_elim [Nat.succ.inj]\n        · apply h₀; apply List.mem_of_getElem? h₂\n\ntheorem applyId_mem_iff [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs) (h₁ : xs ~ ys)\n    (x : α) : List.applyId.{u} (xs.zip ys) x ∈ ys ↔ x ∈ xs := by\n  simp only [List.applyId]\n  cases h₃ : List.dlookup x (List.map Prod.toSigma (xs.zip ys)) with\n  | none =>\n    dsimp [Option.getD]\n    rw [h₁.mem_iff]\n  | some val =>\n    have h₂ : ys.Nodup := h₁.nodup_iff.1 h₀\n    replace h₁ : xs.length = ys.length := h₁.length_eq\n    dsimp\n    induction xs generalizing ys with\n    | nil => contradiction\n    | cons x' xs xs_ih =>\n      rcases ys with - | ⟨y, ys⟩\n      · cases h₃\n      dsimp [List.dlookup] at h₃; split_ifs at h₃ with h\n      · rw [Option.some_inj] at h₃\n        subst x'; subst val\n        simp only [List.mem_cons, true_or]\n      · obtain - | ⟨h₀, h₅⟩ := h₀\n        obtain - | ⟨h₂, h₄⟩ := h₂\n        have h₆ := Nat.succ.inj h₁\n        specialize xs_ih h₅ h₃ h₄ h₆\n        simp only [Ne.symm h, xs_ih, List.mem_cons]\n        suffices val ∈ ys by tauto\n        rw [← Option.mem_def, List.mem_dlookup_iff] at h₃\n        · simp only [Prod.toSigma, List.mem_map, Prod.exists] at h₃\n          rcases h₃ with ⟨a, b, h₃, h₄, h₅⟩\n          apply (List.of_mem_zip h₃).2\n        simp only [List.NodupKeys, List.keys, comp_def, Prod.fst_toSigma, List.map_map]\n        rwa [List.map_fst_zip (le_of_eq h₆)]\n\ntheorem List.applyId_eq_self [DecidableEq α] {xs ys : List α} (x : α) :\n    x ∉ xs → List.applyId.{u} (xs.zip ys) x = x := by\n  intro h\n  dsimp [List.applyId]\n  rw [List.dlookup_eq_none.2]\n  · rfl\n  simp only [List.keys, not_exists, Prod.toSigma, exists_and_right, exists_eq_right, List.mem_map,\n    Function.comp_apply, List.map_map, Prod.exists]\n  intro y hy\n  exact h (List.of_mem_zip hy).1\n\nTarget:\ntheorem applyId_injective [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs) (h₁ : xs ~ ys) :\n    Injective.{u + 1, u + 1} (List.applyId (xs.zip ys)) :=\n\nProof body:\n","rejected":"by\n  intro x y h\n  by_cases hx : x ∈ xs <;> by_cases hy : y ∈ xs\n  · rw [List.mem_iff_getElem?] at hx hy\n    obtain ⟨i, hx⟩ := hx\n    obtain ⟨j, hy⟩ := hy\n    suffices some x = some y by injection this\n    have h₂ := h₁.length_eq\n    rw [List.applyId_zip_eq h₀ h₂ _ _ _ hx] at h\n    rw [← hx, ← hy]; congr\n    apply (List.getElem?_inj _ (h₁.nodup_iff.1 h₀)).mp\n    · symm; rw [h]\n      rw [← List.applyId_zip_eq] <;> assumption\n    · rw [← h₁.length_eq]\n      rw [List.getElem?_eq_some_iff] at hx\n      obtain ⟨hx, hx'⟩ := hx\n      exact hx\n  · rw [← applyId_mem_iff h₀ h₁] at hx hy\n    rw [h] at hx\n    contradiction\n  · rw [← applyId_mem_iff h₀ h₁] at hx hy\n    rw [h] at hx\n    contradiction\n  · rwa [List.applyId_eq_self, List.applyId_eq_self] at h <;> assumption\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Testing/Plausible","family_id":"applyid_injective","file_id":"mathlib/Mathlib/Testing/Plausible/Functions.lean","sample_id":"1d731b6f1a0aa28042e1c006480d5c7f9dc42d3f0a8f8f5b52bbb265d67e141d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e0b9bed3a6acbad85454a981928799bf6c6adb0fb6d0013326ad4cea47a0b244","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"af7418511b50f18257ec80fc628fa03b92c7fc67e34f32b29b3a4343d3d288e3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"194f0f4367719ea1e6824ec86d27af7e10162a98d826fa13df412b7a41e3d577","source_sha256":"ef9864ce066b7696b62174e7a3b3a449882e3ab320bb2e19ef0e02cda6607291","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [r_eq_r' S, r', Con.rel_mk]","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":18,"token_jaccard":0.705882,"token_length_ratio":1.384615},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"38825fc80997bced68503ed46f379384a0dff70237207ddf81f3043faa691812","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\npublic import Mathlib.Algebra.Group.Submonoid.Operations\npublic import Mathlib.Algebra.Regular.Basic\npublic import Mathlib.GroupTheory.Congruence.Hom\npublic import Mathlib.GroupTheory.OreLocalization.Basic\n\nNamespace:\nLocalization\n\nLocal context:\n/-\nCopyright (c) 2019 Amelia Livingston. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Amelia Livingston\n-/\n/-!\n# Localizations of commutative monoids\n\nLocalizing a commutative ring at one of its submonoids does not rely on the ring's addition, so\nwe can generalize localizations to commutative monoids.\n\nWe characterize the localization of a commutative monoid `M` at a submonoid `S` up to\nisomorphism; that is, a commutative monoid `N` is the localization of `M` at `S` iff we can find a\nmonoid homomorphism `f : M →* N` satisfying 3 properties:\n1. For all `y ∈ S`, `f y` is a unit;\n2. For all `z : N`, there exists `(x, y) : M × S` such that `z * f y = f x`;\n3. For all `x, y : M` such that `f x = f y`, there exists `c ∈ S` such that `x * c = y * c`.\n   (The converse is a consequence of 1.)\n\nGiven such a localization map `f : M →* N`, we can define the surjection\n`Submonoid.LocalizationMap.mk'` sending `(x, y) : M × S` to `f x * (f y)⁻¹`. Mapping properties\nof the localization (e.g. extending a map from `M → P` to `N` if the image of `S` is contained in\nthe units) are treated in a later file `Mathlib.GroupTheory.MonoidLocalization.Maps`.\n\nWe also define the quotient of `M × S` by the unique congruence relation (equivalence relation\npreserving a binary operation) `r` such that for any other congruence relation `s` on `M × S`\nsatisfying '`∀ y ∈ S`, `(1, 1) ∼ (y, y)` under `s`', we have that `(x₁, y₁) ∼ (x₂, y₂)` by `s`\nwhenever `(x₁, y₁) ∼ (x₂, y₂)` by `r`. We show this relation is equivalent to the standard\nlocalization relation.\nThis defines the localization as a quotient type, `Localization`, but the majority of\nsubsequent lemmas in the file are given in terms of localizations up to isomorphism, using maps\nwhich satisfy the characteristic predicate.\n\nThe Grothendieck group construction corresponds to localizing at the top submonoid, namely making\nevery element invertible.\n\n## Implementation notes\n\nIn maths it is natural to reason up to isomorphism, but in Lean we cannot naturally `rewrite` one\nstructure with an isomorphic one; one way around this is to isolate a predicate characterizing\na structure up to isomorphism, and reason about things that satisfy the predicate.\n\nThe infimum form of the localization congruence relation is chosen as 'canonical' here, since it\nshortens some proofs.\n\nTo reason about the localization as a quotient type, use `mk_eq_monoidOf_mk'` and associated\nlemmas. These show the quotient map `mk : M → S → Localization S` equals the\nsurjection `LocalizationMap.mk'` induced by the map\n`Localization.monoidOf : Submonoid.LocalizationMap S (Localization S)` (where `of` establishes the\nlocalization as a quotient type satisfies the characteristic predicate). The lemma\n`mk_eq_monoidOf_mk'` hence gives you access to the results in the rest of the file, which are about\nthe `LocalizationMap.mk'` induced by any localization map.\n\n## TODO\n\n* Show that the localization at the top monoid is a group.\n* Generalise to (nonempty) subsemigroups.\n* If we acquire more bundlings, we can make `Localization.mkOrderEmbedding` be an ordered monoid\n  embedding.\n\n## Tags\nlocalization, monoid localization, quotient monoid, congruence relation, characteristic predicate,\ncommutative monoid, grothendieck group\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero Ring\n\nopen Function\nnamespace AddSubmonoid\n\nvariable {M : Type*} [AddCommMonoid M] (S : AddSubmonoid M) (N : Type*) [AddCommMonoid N]\n\nvariable {N} in\n/-- A predicate characterizing homomorphisms between additive monoids `M` and `N` that form a\ncommutative triangle with the canonical map from `M` to its localization at `S` and\nsome isomorphism between `N` and the localization. -/\nstructure IsLocalizationMap (S : AddSubmonoid M) (f : M → N) where\n  map_addUnits (y : S) : IsAddUnit (f y)\n  surj (z : N) : ∃ x : M × S, z + f x.2 = f x.1\n  exists_of_eq {x y} : f x = f y → ∃ c : S, c + x = c + y\n\n/-- The type of AddMonoid homomorphisms satisfying the characteristic predicate: if `f : M →+ N`\nsatisfies this predicate, then `N` is isomorphic to the localization of `M` at `S`. -/\nstructure LocalizationMap extends M →ₙ+ N where\n  isLocalizationMap : IsLocalizationMap S toFun\n\n/-- The additive homomorphism underlying a `LocalizationMap` of `AddCommMonoid`s. -/\nadd_decl_doc LocalizationMap.toAddHom\n\nend AddSubmonoid\n\nsection CommMonoid\n\nvariable {M : Type*} [CommMonoid M] (S : Submonoid M) (N : Type*) [CommMonoid N] {P : Type*}\n  [CommMonoid P]\n\nnamespace Submonoid\n\nvariable {N} in\n/-- A predicate characterizing homomorphisms between monoids `M` and `N` that form a\ncommutative triangle with the canonical map from `M` to its localization at `S` and\nsome isomorphism between `N` and the localization. -/\n@[to_additive (attr := mk_iff)]\nstructure IsLocalizationMap (S : Submonoid M) (f : M → N) where\n  map_units (y : S) : IsUnit (f y)\n  surj (z : N) : ∃ x : M × S, z * f x.2 = f x.1\n  exists_of_eq {x y} : f x = f y → ∃ c : S, c * x = c * y\n\n/-- The type of monoid homomorphisms satisfying the characteristic predicate: if `f : M →* N`\nsatisfies this predicate, then `N` is isomorphic to the localization of `M` at `S`. -/\n@[to_additive] structure LocalizationMap extends M →ₙ* N where\n  isLocalizationMap : IsLocalizationMap S toFun\n\n/-- The multiplicative homomorphism underlying a `LocalizationMap`. -/\nadd_decl_doc LocalizationMap.toMulHom\n\nend Submonoid\n\nnamespace Localization\n\n/- Ensure that `@[to_additive]` uses the right namespace before the definition of `Localization`. -/\ninsert_to_additive_translation Localization AddLocalization\n\n/-- The congruence relation on `M × S`, `M` a `CommMonoid` and `S` a submonoid of `M`, whose\nquotient is the localization of `M` at `S`, defined as the unique congruence relation on\n`M × S` such that for any other congruence relation `s` on `M × S` where for all `y ∈ S`,\n`(1, 1) ∼ (y, y)` under `s`, we have that `(x₁, y₁) ∼ (x₂, y₂)` by `r` implies\n`(x₁, y₁) ∼ (x₂, y₂)` by `s`. -/\n@[to_additive\n/-- The congruence relation on `M × S`, `M` an `AddCommMonoid` and `S` an `AddSubmonoid` of `M`,\nwhose quotient is the localization of `M` at `S`, defined as the unique congruence relation on\n`M × S` such that for any other congruence relation `s` on `M × S` where for all `y ∈ S`,\n`(0, 0) ∼ (y, y)` under `s`, we have that `(x₁, y₁) ∼ (x₂, y₂)` by `r` implies\n`(x₁, y₁) ∼ (x₂, y₂)` by `s`. -/]\ndef r (S : Submonoid M) : Con (M × S) :=\n  sInf { c | ∀ y : S, c 1 (y, y) }\n\n/-- An alternate form of the congruence relation on `M × S`, `M` a `CommMonoid` and `S` a\nsubmonoid of `M`, whose quotient is the localization of `M` at `S`. -/\n@[to_additive\n/-- An alternate form of the congruence relation on `M × S`, `M` a `CommMonoid` and `S` a\nsubmonoid of `M`, whose quotient is the localization of `M` at `S`. -/]\ndef r' : Con (M × S) := by\n  -- note we multiply by `c` on the left so that we can later generalize to `•`\n  refine\n    { r := fun a b : M × S ↦ ∃ c : S, ↑c * (↑b.2 * a.1) = c * (a.2 * b.1)\n      iseqv := ⟨fun a ↦ ⟨1, rfl⟩, fun ⟨c, hc⟩ ↦ ⟨c, hc.symm⟩, ?_⟩\n      mul' := ?_ }\n  · rintro a b c ⟨t₁, ht₁⟩ ⟨t₂, ht₂⟩\n    use t₂ * t₁ * b.2\n    simp only [Submonoid.coe_mul]\n    calc\n      (t₂ * t₁ * b.2 : M) * (c.2 * a.1) = t₂ * c.2 * (t₁ * (b.2 * a.1)) := by ac_rfl\n      _ = t₁ * a.2 * (t₂ * (c.2 * b.1)) := by rw [ht₁]; ac_rfl\n      _ = t₂ * t₁ * b.2 * (a.2 * c.1) := by rw [ht₂]; ac_rfl\n  · rintro a b c d ⟨t₁, ht₁⟩ ⟨t₂, ht₂⟩\n    use t₂ * t₁\n    calc\n      (t₂ * t₁ : M) * (b.2 * d.2 * (a.1 * c.1)) = t₂ * (d.2 * c.1) * (t₁ * (b.2 * a.1)) := by ac_rfl\n      _ = (t₂ * t₁ : M) * (a.2 * c.2 * (b.1 * d.1)) := by rw [ht₁, ht₂]; ac_rfl\n\n/-- The congruence relation used to localize a `CommMonoid` at a submonoid can be expressed\nequivalently as an infimum (see `Localization.r`) or explicitly\n(see `Localization.r'`). -/\n@[to_additive\n/-- The additive congruence relation used to localize an `AddCommMonoid` at a submonoid can be\nexpressed equivalently as an infimum (see `AddLocalization.r`) or explicitly\n(see `AddLocalization.r'`). -/]\ntheorem r_eq_r' : r S = r' S :=\n  le_antisymm (sInf_le fun _ ↦ ⟨1, by simp⟩) <|\n    le_sInf fun b H ⟨p, q⟩ ⟨x, y⟩ ⟨t, ht⟩ ↦ by\n      rw [← one_mul (p, q), ← one_mul (x, y)]\n      refine b.trans (b.mul (H (t * y)) (b.refl _)) ?_\n      convert! b.symm (b.mul (H (t * q)) (b.refl (x, y))) using 1\n      dsimp only [Prod.mk_mul_mk, Submonoid.coe_mul] at ht ⊢\n      simp_rw [mul_assoc, ht, mul_comm y q]\n\nvariable {S}\n\n@[to_additive]\n\nTarget:\ntheorem r_iff_exists {x y : M × S} : r S x y ↔ ∃ c : S, ↑c * (↑y.2 * x.1) = c * (x.2 * y.1) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  simp only [r_eq_r' S, r', Con.rel_mk]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/MonoidLocalization","family_id":"r_iff_exists","file_id":"mathlib/Mathlib/GroupTheory/MonoidLocalization/Basic.lean","sample_id":"194f0f4367719ea1e6824ec86d27af7e10162a98d826fa13df412b7a41e3d577"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dd729798355787e2f50246686d3074ecb0b708500ef8ae38cb56d8d7dbde9c7f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"613d32e5c711b87b51ff7b2dba0bc6c8e15dd81cc10f27f546b3f4c85e9fbf7e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"724a8437e52e6581c6973db3a8c052c428831eeb06b3711d37815c86ff48be5a","source_sha256":"77767c60d3dd433d634008abee2943f40d23ac4eab051267efb892f5f5bd8ab6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  induction s using Finset.induction with\n  | empty => simp\n  | insert _ _ _ _ => simp_all [lcm_eq_nat_lcm, Nat.factorization_lcm]","hard_negative":true,"metrics":{"chosen_tokens":28,"rejected_tokens":2,"token_jaccard":0.045455,"token_length_ratio":0.071429},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"388bdb13cb4f1eaf25410a3dc865794ac0eeaf68456dbdaf1f6e3e6060f6df69","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GCDMonoid.Finset\npublic import Mathlib.Algebra.GCDMonoid.Nat\npublic import Mathlib.Data.Nat.GCD.Basic\npublic import Mathlib.RingTheory.Coprime.Lemmas\npublic import Mathlib.Data.Nat.Factorization.Basic\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2025 Snir Broshi. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Snir Broshi\n-/\n/-!\n# `Finset.lcm` lemmas\n\n## Tags\n\nfinset, lcm, prod, coprime, Rat.den\n-/\n\npublic section\n\nnamespace Finset\n\nvariable {ι α : Type*} [CommMonoidWithZero α] [NormalizedGCDMonoid α]\n\ntheorem lcm_dvd_prod (s : Finset ι) (f : ι → α) : s.lcm f ∣ s.prod f :=\n  lcm_dvd fun _ ↦ dvd_prod_of_mem _\n\ntheorem associated_lcm_prod {s : Finset ι} {f : ι → α} (h : Set.Pairwise s <| IsRelPrime.onFun f) :\n    Associated (s.lcm f) (s.prod f) :=\n  associated_of_dvd_dvd (s.lcm_dvd_prod f) (s.prod_dvd_of_isRelPrime h fun _ ↦ dvd_lcm)\n\ntheorem lcm_eq_prod {s : Finset ι} {f : ι → ℕ} (h : Set.Pairwise s <| Nat.Coprime.onFun f) :\n    s.lcm f = s.prod f := by\n  rw [show Nat.Coprime = IsRelPrime by ext; exact Nat.coprime_iff_isRelPrime] at h\n  exact associated_lcm_prod h |>.eq_of_normalized (normalize_eq _) (normalize_eq _)\n\n/-- An analogue of `Nat.factorization_lcm` for `Finset.lcm`. -/\n\nTarget:\ntheorem factorization_lcm {f : ι → ℕ} {s : Finset ι} (hf : ∀ k ∈ s, f k ≠ 0) (p : ℕ) :\n    (s.lcm f).factorization p = s.sup fun a ↦ (f a).factorization p :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_724a8437e52e","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"d92808b2d035bf16b2922a4c326d86cd275cb144aaa64c2aab393b1613255977","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/GCDMonoid","family_id":"factorization_lcm","file_id":"mathlib/Mathlib/Algebra/GCDMonoid/FinsetLemmas.lean","sample_id":"724a8437e52e6581c6973db3a8c052c428831eeb06b3711d37815c86ff48be5a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e1fced99958acc84dd8fa2ad1ee6184c3708ee54a7e8d9105ea3938d1bcfadce","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8ccb2339aba41c643ce9ab6237c697b1faecc53decbb2a2025ef71895c9c0e9f","source_sha256":"cd00025c9f1c99662e498ca3e267a5f89784b2f84527a76f8e7a37bf2514f3a2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  refine ⟨fun h₂ ↦ by simp [isBigO_of_le' (c := 0), h₂], ?_⟩\n  contrapose\n  intro h₁\n  obtain ⟨e, he⟩ := exists_single_le_pos (lt_of_le_of_ne h (h₁ ·.symm))\n  rw [isBigO_iff'']\n  push Not\n  intro a ha\n  simp only [Pi.one_apply, norm_eq_abs, frequently_atTop, abs_one]\n  intro b\n  obtain ⟨c, hc⟩ := eventually_atTop.1\n    (isLittleO_iff.1 (one_isLittleO_logCounting_single (e := e)) ha)\n  let ℓ := 1 + max ‖e‖ (max |b| |c|)\n  have h₁ℓ : c ≤ ℓ := by grind\n  have h₂ℓ : 1 ≤ ℓ := by simp [ℓ]\n  use 1 + ℓ, (show b ≤ 1 + ℓ by grind)\n  calc 1\n    _ ≤ (a * |logCounting (single e 1) ℓ|) := by simpa [h₁ℓ] using hc ℓ\n    _ ≤ (a * |logCounting D ℓ|) := by\n      gcongr\n      · apply logCounting_nonneg (single_pos.2 Int.one_pos).le h₂ℓ\n      · apply logCounting_le he h₂ℓ\n    _ < a * |logCounting D (1 + ℓ)| := by\n      gcongr 2\n      rw [abs_of_nonneg (logCounting_nonneg h h₂ℓ),\n        abs_of_nonneg (logCounting_nonneg h (by grind))]\n      apply logCounting_strictMono he <;> grind","hard_negative":false,"metrics":{"chosen_tokens":264,"rejected_tokens":3,"token_jaccard":0.02381,"token_length_ratio":0.011364},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"389a5dfeb9a6b6a183afa9e93ee57bbe820fdd85a3dfc495faed3ab952c2f26e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic\n\nNamespace:\nFunction.locallyFinsuppWithin\n\nLocal context:\n/-\nCopyright (c) 2026 Stefan Kebekus. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stefan Kebekus\n-/\n/-!\n# Asymptotic Behavior of the Logarithmic Counting Function\n\nIf `f` is meromorphic over a field `𝕜`, we show that the logarithmic counting function for the\npoles of `f` is asymptotically bounded if and only if `f` has only removable singularities.  See\nPage 170f of [Lang, *Introduction to Complex Hyperbolic Spaces*][MR886677] for a detailed\ndiscussion.\n\n## Implementation Notes\n\nWe establish the result first for the logarithmic counting function for functions with locally\nfinite support on `𝕜` and then specialize to the setting where the function with locally finite\nsupport is the pole or zero-divisor of a meromorphic function.\n\n## TODO\n\nEstablish the analogous characterization of meromorphic functions with finite set of poles, as\nfunctions whose logarithmic counting function is big-O of `log`.\n-/\n\npublic section\n\nopen Asymptotics Filter Function Real Set\n\nnamespace Function.locallyFinsuppWithin\n\nvariable\n  {E : Type*} [NormedAddCommGroup E]\n\n/-!\n## Logarithmic Counting Functions for Functions with Locally Finite Support\n-/\n\n/--\nQualitative consequence of `logCounting_single_eq_log_sub_const`. The constant function `1 : ℝ → ℝ`\nis little o of the logarithmic counting function attached to `single e`.\n-/\nlemma one_isLittleO_logCounting_single [DecidableEq E] [ProperSpace E] {e : E} :\n    (1 : ℝ → ℝ) =o[atTop] logCounting (single e 1) := by\n  have hΘ : (fun r ↦ log r - log ‖e‖) =Θ[atTop] log :=\n    (IsEquivalent.sub_isLittleO IsEquivalent.refl isLittleO_const_log_atTop).isTheta\n  have h₁ : (1 : ℝ → ℝ) =o[atTop] fun r ↦ log r - log ‖e‖ :=\n    (hΘ.isLittleO_congr_right).2 isLittleO_const_log_atTop\n  refine h₁.congr' EventuallyEq.rfl ?_\n  filter_upwards [eventually_ge_atTop ‖e‖] with r hr\n  simp [logCounting_single_eq_log_sub_const hr]\n\n/--\nA non-negative function with locally finite support is zero if and only if its logarithmic counting\nfunctions is asymptotically bounded.\n-/\n\nTarget:\nlemma zero_iff_logCounting_bounded [ProperSpace E]\n    {D : locallyFinsuppWithin (univ : Set E) ℤ} (h : 0 ≤ D) :\n    D = 0 ↔ logCounting D =O[atTop] (1 : ℝ → ℝ) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Complex","family_id":"zero_iff_logcounting_bounded","file_id":"mathlib/Mathlib/Analysis/Complex/ValueDistribution/LogCounting/Asymptotic.lean","sample_id":"8ccb2339aba41c643ce9ab6237c697b1faecc53decbb2a2025ef71895c9c0e9f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dab19792c72b6e7680af848aa1b358000086832bd7758b76eb7e5a5172d70aec","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"486f6d77bda3f3ad1fcdbdbc2929bf3ae5b28224eec869b29f4c0aa90253527b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"21c2606d6758e396ea3dedfc301c790b246ae8c817f34e167e84ed4e781597b5","source_sha256":"7745c09f53b57dd9e1369fa3be9c7884e924965f772e3b47328f6b6cb7e756a6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  grw [Sum.dist, ← le_add_of_nonneg_right dist_nonneg, ← le_add_of_nonneg_left dist_nonneg]","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":5,"token_jaccard":0.133333,"token_length_ratio":0.333333},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"38abc8c992b37abff93e816b2dffb12764189361fa90f853fa18bdb1118100ac","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ConditionallyCompleteLattice.Group\npublic import Mathlib.Topology.MetricSpace.Isometry\n\nNamespace:\nMetric\n\nLocal context:\n/-\nCopyright (c) 2019 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Metric space gluing\n\nGluing two metric spaces along a common subset. Formally, we are given\n\n```\n     Φ\n  Z ---> X\n  |\n  |Ψ\n  v\n  Y\n```\nwhere `hΦ : Isometry Φ` and `hΨ : Isometry Ψ`.\nWe want to complete the square by a space `GlueSpace hΦ hΨ` and two isometries\n`toGlueL hΦ hΨ` and `toGlueR hΦ hΨ` that make the square commute.\nWe start by defining a predistance on the disjoint union `X ⊕ Y`, for which\npoints `Φ p` and `Ψ p` are at distance 0. The (quotient) metric space associated\nto this predistance is the desired space.\n\nThis is an instance of a more general construction, where `Φ` and `Ψ` do not have to be isometries,\nbut the distances in the image almost coincide, up to `2ε` say. Then one can almost glue the two\nspaces so that the images of a point under `Φ` and `Ψ` are `ε`-close. If `ε > 0`, this yields a\nmetric space structure on `X ⊕ Y`, without the need to take a quotient. In particular,\nthis gives a natural metric space structure on `X ⊕ Y`, where the basepoints\nare at distance 1, say, and the distances between other points are obtained by going through the two\nbasepoints.\n(We also register the same metric space structure on a general disjoint union `Σ i, E i`).\n\nWe also define the inductive limit of metric spaces. Given\n```\n     f 0        f 1        f 2        f 3\nX 0 -----> X 1 -----> X 2 -----> X 3 -----> ...\n```\nwhere the `X n` are metric spaces and `f n` isometric embeddings, we define the inductive\nlimit of the `X n`, also known as the increasing union of the `X n` in this context, if we\nidentify `X n` and `X (n+1)` through `f n`. This is a metric space in which all `X n` embed\nisometrically and in a way compatible with `f n`.\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nuniverse u v w\n\nopen Function Set Uniformity Topology\n\nnamespace Metric\n\nsection ApproxGluing\n\nvariable {X : Type u} {Y : Type v} {Z : Type w}\nvariable [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} {ε : ℝ}\n\n/-- Define a predistance on `X ⊕ Y`, for which `Φ p` and `Ψ p` are at distance `ε` -/\ndef glueDist (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) : X ⊕ Y → X ⊕ Y → ℝ\n  | .inl x, .inl y => dist x y\n  | .inr x, .inr y => dist x y\n  | .inl x, .inr y => (⨅ p, dist x (Φ p) + dist y (Ψ p)) + ε\n  | .inr x, .inl y => (⨅ p, dist y (Φ p) + dist x (Ψ p)) + ε\n\nset_option backward.privateInPublic true in\nprivate theorem glueDist_self (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) : ∀ x, glueDist Φ Ψ ε x x = 0\n  | .inl _ => dist_self _\n  | .inr _ => dist_self _\n\ntheorem glueDist_glued_points [Nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (p : Z) :\n    glueDist Φ Ψ ε (.inl (Φ p)) (.inr (Ψ p)) = ε := by\n  have : ⨅ q, dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q) = 0 := by\n    have A : ∀ q, 0 ≤ dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q) := fun _ => by positivity\n    refine le_antisymm ?_ (le_ciInf A)\n    have : 0 = dist (Φ p) (Φ p) + dist (Ψ p) (Ψ p) := by simp\n    rw [this]\n    exact ciInf_le ⟨0, forall_mem_range.2 A⟩ p\n  simp only [glueDist, this, zero_add]\n\nset_option backward.privateInPublic true in\nprivate theorem glueDist_comm (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) :\n    ∀ x y, glueDist Φ Ψ ε x y = glueDist Φ Ψ ε y x\n  | .inl _, .inl _ => dist_comm _ _\n  | .inr _, .inr _ => dist_comm _ _\n  | .inl _, .inr _ => rfl\n  | .inr _, .inl _ => rfl\n\ntheorem glueDist_swap (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) :\n    ∀ x y, glueDist Ψ Φ ε x.swap y.swap = glueDist Φ Ψ ε x y\n  | .inl _, .inl _ => rfl\n  | .inr _, .inr _ => rfl\n  | .inl _, .inr _ => by simp only [glueDist, Sum.swap_inl, Sum.swap_inr, add_comm]\n  | .inr _, .inl _ => by simp only [glueDist, Sum.swap_inl, Sum.swap_inr, add_comm]\n\ntheorem le_glueDist_inl_inr (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x y) :\n    ε ≤ glueDist Φ Ψ ε (.inl x) (.inr y) :=\n  le_add_of_nonneg_left <| Real.iInf_nonneg fun _ => by positivity\n\ntheorem le_glueDist_inr_inl (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x y) :\n    ε ≤ glueDist Φ Ψ ε (.inr x) (.inl y) := by\n  rw [glueDist_comm]; apply le_glueDist_inl_inr\n\nsection\nvariable [Nonempty Z]\n\nprivate theorem glueDist_triangle_inl_inr_inr (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x : X) (y z : Y) :\n    glueDist Φ Ψ ε (.inl x) (.inr z) ≤\n      glueDist Φ Ψ ε (.inl x) (.inr y) + glueDist Φ Ψ ε (.inr y) (.inr z) := by\n  simp only [glueDist]\n  rw [add_right_comm, add_le_add_iff_right]\n  refine le_ciInf_add fun p => ciInf_le_of_le ⟨0, ?_⟩ p ?_\n  · exact forall_mem_range.2 fun _ => by positivity\n  · linarith [dist_triangle_left z (Ψ p) y]\n\nprivate theorem glueDist_triangle_inl_inr_inl (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ)\n    (H : ∀ p q, |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) (x : X) (y : Y) (z : X) :\n    glueDist Φ Ψ ε (.inl x) (.inl z) ≤\n      glueDist Φ Ψ ε (.inl x) (.inr y) + glueDist Φ Ψ ε (.inr y) (.inl z) := by\n  simp_rw [glueDist, add_add_add_comm _ ε, add_assoc]\n  refine le_ciInf_add fun p => ?_\n  rw [add_left_comm, add_assoc, ← two_mul]\n  refine le_ciInf_add fun q => ?_\n  rw [dist_comm z]\n  linarith [dist_triangle4 x (Φ p) (Φ q) z, dist_triangle_left (Ψ p) (Ψ q) y, (abs_le.1 (H p q)).2]\n\nset_option backward.privateInPublic true in\nprivate theorem glueDist_triangle (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ)\n    (H : ∀ p q, |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) :\n    ∀ x y z, glueDist Φ Ψ ε x z ≤ glueDist Φ Ψ ε x y + glueDist Φ Ψ ε y z\n  | .inl _, .inl _, .inl _ => dist_triangle _ _ _\n  | .inr _, .inr _, .inr _ => dist_triangle _ _ _\n  | .inr x, .inl y, .inl z => by\n    simp only [← glueDist_swap Φ]\n    apply glueDist_triangle_inl_inr_inr\n  | .inr x, .inr y, .inl z => by\n    simpa only [glueDist_comm, add_comm] using glueDist_triangle_inl_inr_inr _ _ _ z y x\n  | .inl x, .inl y, .inr z => by\n    simpa only [← glueDist_swap Φ, glueDist_comm, add_comm, Sum.swap_inl, Sum.swap_inr]\n      using glueDist_triangle_inl_inr_inr Ψ Φ ε z y x\n  | .inl _, .inr _, .inr _ => glueDist_triangle_inl_inr_inr ..\n  | .inl x, .inr y, .inl z => glueDist_triangle_inl_inr_inl Φ Ψ ε H x y z\n  | .inr x, .inl y, .inr z => by\n    simp only [← glueDist_swap Φ]\n    apply glueDist_triangle_inl_inr_inl\n    simpa only [abs_sub_comm]\n\nend\n\nset_option backward.privateInPublic true in\nprivate theorem eq_of_glueDist_eq_zero (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (ε0 : 0 < ε) :\n    ∀ p q : X ⊕ Y, glueDist Φ Ψ ε p q = 0 → p = q\n  | .inl x, .inl y, h => by rw [eq_of_dist_eq_zero h]\n  | .inl x, .inr y, h => by exfalso; linarith [le_glueDist_inl_inr Φ Ψ ε x y]\n  | .inr x, .inl y, h => by exfalso; linarith [le_glueDist_inr_inl Φ Ψ ε x y]\n  | .inr x, .inr y, h => by rw [eq_of_dist_eq_zero h]\n\ntheorem Sum.mem_uniformity_iff_glueDist (hε : 0 < ε) (s : Set ((X ⊕ Y) × (X ⊕ Y))) :\n    s ∈ 𝓤 (X ⊕ Y) ↔ ∃ δ > 0, ∀ a b, glueDist Φ Ψ ε a b < δ → (a, b) ∈ s := by\n  simp only [Sum.uniformity, Filter.mem_sup, Filter.mem_map, mem_uniformity_dist, mem_preimage]\n  constructor\n  · rintro ⟨⟨δX, δX0, hX⟩, δY, δY0, hY⟩\n    refine ⟨min (min δX δY) ε, lt_min (lt_min δX0 δY0) hε, ?_⟩\n    rintro (a | a) (b | b) h <;> simp only [lt_min_iff] at h\n    · exact hX h.1.1\n    · exact absurd h.2 (le_glueDist_inl_inr _ _ _ _ _).not_gt\n    · exact absurd h.2 (le_glueDist_inr_inl _ _ _ _ _).not_gt\n    · exact hY h.1.2\n  · rintro ⟨ε, ε0, H⟩\n    constructor <;> exact ⟨ε, ε0, fun _ _ h => H _ _ h⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Given two maps `Φ` and `Ψ` intro metric spaces `X` and `Y` such that the distances between\n`Φ p` and `Φ q`, and between `Ψ p` and `Ψ q`, coincide up to `2 ε` where `ε > 0`, one can almost\nglue the two spaces `X` and `Y` along the images of `Φ` and `Ψ`, so that `Φ p` and `Ψ p` are\nat distance `ε`. -/\n@[implicit_reducible]\ndef glueMetricApprox [Nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (ε0 : 0 < ε)\n    (H : ∀ p q, |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) : MetricSpace (X ⊕ Y) where\n  dist := glueDist Φ Ψ ε\n  dist_self := glueDist_self Φ Ψ ε\n  dist_comm := glueDist_comm Φ Ψ ε\n  dist_triangle := glueDist_triangle Φ Ψ ε H\n  eq_of_dist_eq_zero := eq_of_glueDist_eq_zero Φ Ψ ε ε0 _ _\n  toUniformSpace := Sum.instUniformSpace\n  uniformity_dist := uniformity_dist_of_mem_uniformity _ _ <| Sum.mem_uniformity_iff_glueDist ε0\n\nend ApproxGluing\n\nsection Sum\n\n/-!\n### Metric on `X ⊕ Y`\n\nA particular case of the previous construction is when one uses basepoints in `X` and `Y` and one\nglues only along the basepoints, putting them at distance 1. We give a direct definition of\nthe distance, without `iInf`, as it is easier to use in applications, and show that it is equal to\nthe gluing distance defined above to take advantage of the lemmas we have already proved.\n-/\nvariable {X : Type u} {Y : Type v} {Z : Type w}\nvariable [MetricSpace X] [MetricSpace Y]\n\n/-- Distance on a disjoint union. There are many (noncanonical) ways to put a distance compatible\nwith each factor.\nIf the two spaces are bounded, one can say for instance that each point in the first is at distance\n`diam X + diam Y + 1` of each point in the second.\nInstead, we choose a construction that works for unbounded spaces, but requires basepoints,\nchosen arbitrarily.\nWe embed isometrically each factor, set the basepoints at distance 1,\narbitrarily, and say that the distance from `a` to `b` is the sum of the distances of `a` and `b` to\ntheir respective basepoints, plus the distance 1 between the basepoints.\nSince there is an arbitrary choice in this construction, it is not an instance by default. -/\nprotected def Sum.dist : X ⊕ Y → X ⊕ Y → ℝ\n  | .inl a, .inl a' => dist a a'\n  | .inr b, .inr b' => dist b b'\n  | .inl a, .inr b => dist a (Nonempty.some ⟨a⟩) + 1 + dist (Nonempty.some ⟨b⟩) b\n  | .inr b, .inl a => dist b (Nonempty.some ⟨b⟩) + 1 + dist (Nonempty.some ⟨a⟩) a\n\ntheorem Sum.dist_eq_glueDist {p q : X ⊕ Y} (x : X) (y : Y) :\n    Sum.dist p q =\n      glueDist (fun _ : Unit => Nonempty.some ⟨x⟩) (fun _ : Unit => Nonempty.some ⟨y⟩) 1 p q := by\n  cases p <;> cases q <;> first | rfl | simp [Sum.dist, glueDist, dist_comm, add_comm,\n    add_left_comm, add_assoc]\n\nset_option backward.privateInPublic true in\nprivate theorem Sum.dist_comm (x y : X ⊕ Y) : Sum.dist x y = Sum.dist y x := by\n  cases x <;> cases y <;> simp [Sum.dist, _root_.dist_comm, add_comm, add_left_comm]\n\nTarget:\ntheorem Sum.one_le_dist_inl_inr {x : X} {y : Y} : 1 ≤ Sum.dist (.inl x) (.inr y) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_21c2606d6758","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"b15f3432909b95de75e4231a4ad93e083d4d8ae9d515f5f3f73c93226805f347","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/MetricSpace","family_id":"sum","file_id":"mathlib/Mathlib/Topology/MetricSpace/Gluing.lean","sample_id":"21c2606d6758e396ea3dedfc301c790b246ae8c817f34e167e84ed4e781597b5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1c8bfaba84928466d869f23a6dc4de48708f2d6673fcc79573e01b4de03f6a29","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4f39866c14e5ed6348379b32458fb4fbb16c3c70f88dda8d8800b80be3694787","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"30e2dcca73b76516afafa57bf58fa1f528e0a923278fa6441a9972d75d07dd33","source_sha256":"5446f542892910a2397a84cd0f024393cbca50d14ecc1fa829e9751db035abd3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [normal_iff] at h ⊢\n  intro x; specialize h (f.symm x)\n  rw [← f.apply_symm_apply x, minpoly.algEquiv_eq, ← f.toAlgHom.comp_algebraMap, ← map_map]\n  exact ⟨h.1.map f, h.2.map _⟩","hard_negative":false,"metrics":{"chosen_tokens":57,"rejected_tokens":61,"token_jaccard":0.914286,"token_length_ratio":1.070175},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"38d18d4dbc52298f7b0484bf3499a8e4168b74b0e74aac0fae48edc87eea15ac","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Splits\npublic import Mathlib.FieldTheory.Galois.Notation\npublic import Mathlib.FieldTheory.IntermediateField.Basic\npublic import Mathlib.FieldTheory.Minpoly.Field\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Thomas Browning, Patrick Lutz\n-/\n/-!\n# Normal field extensions\n\nIn this file we define normal field extensions.\n\n## Main Definitions\n\n- `Normal F K` where `K` is a field extension of `F`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Polynomial IsScalarTower\n\nvariable (F K : Type*) [Field F] [Field K] [Algebra F K]\n\n/-- Typeclass for normal field extensions: an algebraic extension of fields `K/F` is *normal*\nif the minimal polynomial of every element `x` in `K` splits in `K`, i.e. every `F`-conjugate\nof `x` is in `K`. -/\n@[stacks 09HM]\nclass Normal : Prop extends Algebra.IsAlgebraic F K where\n  splits' (x : K) : Splits ((minpoly F x).map (algebraMap F K))\n\nvariable {F K}\n\ntheorem Normal.isIntegral (_ : Normal F K) (x : K) : IsIntegral F x :=\n  Algebra.IsIntegral.isIntegral x\n\ntheorem Normal.splits (_ : Normal F K) (x : K) : Splits ((minpoly F x).map (algebraMap F K)) :=\n  Normal.splits' x\n\ntheorem normal_iff :\n    Normal F K ↔ ∀ x : K, IsIntegral F x ∧ Splits ((minpoly F x).map (algebraMap F K)) :=\n  ⟨fun h x => ⟨h.isIntegral x, h.splits x⟩, fun h =>\n    { isAlgebraic := fun x => (h x).1.isAlgebraic\n      splits' := fun x => (h x).2 }⟩\n\ntheorem Normal.out :\n    Normal F K → ∀ x : K, IsIntegral F x ∧ Splits ((minpoly F x).map (algebraMap F K)) :=\n  normal_iff.1\n\nvariable (F K)\n\ninstance normal_self : Normal F F where\n  isAlgebraic := fun _ => isIntegral_algebraMap.isAlgebraic\n  splits' := fun x => (minpoly.eq_X_sub_C' x).symm ▸ by simp\n\nsection NormalTower\n\nvariable (E : Type*) [Field E] [Algebra F E] [Algebra K E] [IsScalarTower F K E]\n\n@[stacks 09HN]\ntheorem Normal.tower_top_of_normal [h : Normal F E] : Normal K E :=\n  normal_iff.2 fun x => by\n    obtain ⟨hx, hhx⟩ := h.out x\n    rw [algebraMap_eq F K E, ← map_map] at hhx\n    exact ⟨hx.tower_top, hhx.of_dvd (map_ne_zero (map_ne_zero (minpoly.ne_zero hx)))\n      ((map_dvd_map' _).mpr (minpoly.dvd_map_of_isScalarTower F K x))⟩\n\ninstance IntermediateField.normal (K : IntermediateField F E) [Normal F E] : Normal K E :=\n  Normal.tower_top_of_normal F K E\n\ntheorem AlgHom.normal_bijective [h : Normal F E] (ϕ : E →ₐ[F] K) : Function.Bijective ϕ :=\n  h.toIsAlgebraic.bijective_of_isScalarTower' ϕ\n\nvariable {E F}\nvariable {E' : Type*} [Field E'] [Algebra F E']\n\nTarget:\ntheorem Normal.of_algEquiv [h : Normal F E] (f : E ≃ₐ[F] E') : Normal F E' :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [normal_iff] at h ⊢\n  intro x; specialize h (f.symm x)\n  rw [← f.apply_symm_apply x, minpoly.algEquiv_eq, ← f.toAlgHom.comp_algebraMap, ← map_map]\n  exact ⟨h.1.map f, h.2.map _⟩","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Normal","family_id":"normal","file_id":"mathlib/Mathlib/FieldTheory/Normal/Defs.lean","sample_id":"30e2dcca73b76516afafa57bf58fa1f528e0a923278fa6441a9972d75d07dd33"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5a49665e0c5d2151828a184678146c90b5369415677dd342dff1e4392eda48b8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"fdd6631a5a7abaea2efb2699b9a4116dd2cd36f2dc787b03bb1091f02f93db54","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"83ec59d8b8358fb8a455e8f01681be42384d6a03b957fc77c05e2d58d2ad540d","source_sha256":"c729c093a684b2a7b5066e0c93a4ca1c797418af44bab051680547cf260ff85c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [*]","hard_negative":true,"metrics":{"chosen_tokens":5,"rejected_tokens":3,"token_jaccard":0.142857,"token_length_ratio":0.6},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"38e72cf81d72d0b042e17ff6e770311e3ad2a4bb67bff51ae2c9b5c3344b4acc","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Ineq\npublic import Mathlib.Data.Nat.Cast.Order.Ring\n\nNamespace:\nMathlib.Tactic.Linarith\n\nLocal context:\n/-\nCopyright (c) 2020 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n-/\npublic meta import Batteries.Tactic.Lint.Basic\npublic meta import Mathlib.Data.Ineq\npublic meta import Mathlib.Tactic.ToAdditive\n\n/-!\n# Lemmas for `linarith`.\n\nThose in the `Linarith` namespace should stay here.\n\nThose outside the `Linarith` namespace may be deleted as they are ported to mathlib4.\n-/\n\npublic meta section\n\nnamespace Mathlib.Tactic.Linarith\n\nuniverse u\ntheorem lt_irrefl {α : Type u} [Preorder α] {a : α} : ¬a < a := _root_.lt_irrefl a\n\nTarget:\ntheorem eq_of_eq_of_eq {α} [Semiring α] {a b : α} (ha : a = 0) (hb : b = 0) : a + b = 0 :=\n\nProof body:\n","rejected":"by\n  exact eq_of_eq_of_eq","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"d15d5276ebe1989abbe921486f8971b3b92fbe64adcb62db6da4da2163362988","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Linarith","family_id":"eq_of_eq_of_eq","file_id":"mathlib/Mathlib/Tactic/Linarith/Lemmas.lean","sample_id":"83ec59d8b8358fb8a455e8f01681be42384d6a03b957fc77c05e2d58d2ad540d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8bb9f6292b5b7cdafeb1af9846ff9f4bfcbbf84b87fb669f9431c4e7c51bed33","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0532fa68927b4e9d297bcdb5210b2b8108229a0d71484a51358724860afda8a3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8b95081cd7f651acb1a47e7a53ca218a713ae886e12ac7935b03bee7470a950f","source_sha256":"3542347af6e76d4ad663bef58f83e85c4df560564e9c17485c7828e4299be8bb","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [iSup, hf.map_sSup', ← range_comp]\n  rfl","hard_negative":false,"metrics":{"chosen_tokens":14,"rejected_tokens":19,"token_jaccard":0.722222,"token_length_ratio":1.357143},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"38f7e4af5d4868840f3c0d810a6d8539234efb15d720ce203c92f010d9e827cb","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\npublic import Mathlib.Order.RelIso.Basic\n\nNamespace:\nLeftOrdContinuous\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Johannes Hölzl\n-/\n/-!\n# Order continuity\n\nWe say that a function is *left order continuous* if it sends all least upper bounds\nto least upper bounds. The order dual notion is called *right order continuity*.\n\nFor monotone functions `ℝ → ℝ` these notions correspond to the usual left and right continuity.\n\nWe prove some basic lemmas (`map_sup`, `map_sSup` etc) and prove that a `RelIso` is both left\nand right order continuous.\n-/\n\n@[expose] public section\n\n\nuniverse u v w x\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}\n\nopen Function OrderDual Set\n\n/-!\n### Definitions\n-/\n\n\n/-- A function `f` between preorders is left order continuous if it preserves all suprema.  We\ndefine it using `IsLUB` instead of `sSup` so that the proof works both for complete lattices and\nconditionally complete lattices. -/\n@[to_dual\n/-- A function `f` between preorders is right order continuous if it preserves all infima.  We\ndefine it using `IsGLB` instead of `sInf` so that the proof works both for complete lattices and\nconditionally complete lattices. -/]\ndef LeftOrdContinuous [Preorder α] [Preorder β] (f : α → β) :=\n  ∀ ⦃s : Set α⦄ ⦃x⦄, IsLUB s x → IsLUB (f '' s) (f x)\n\nnamespace LeftOrdContinuous\n\nsection Preorder\n\nvariable (α) [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β}\n\n@[to_dual]\nprotected theorem id : LeftOrdContinuous (id : α → α) := fun s x h => by\n  simpa only [image_id] using! h\n\nvariable {α}\n\n@[to_dual]\nprotected theorem dual :\n    LeftOrdContinuous f → RightOrdContinuous (toDual ∘ f ∘ ofDual) :=\n  id\n\n@[deprecated (since := \"2026-04-08\")] alias rightOrdContinuous_dual := LeftOrdContinuous.dual\n\n@[deprecated (since := \"2026-04-08\")] alias _root_.RightOrdContinuous.orderDual :=\n  RightOrdContinuous.dual\n\n@[to_dual]\ntheorem map_isGreatest (hf : LeftOrdContinuous f) {s : Set α} {x : α} (h : IsGreatest s x) :\n    IsGreatest (f '' s) (f x) :=\n  ⟨mem_image_of_mem f h.1, (hf h.isLUB).1⟩\n\n@[to_dual]\ntheorem mono (hf : LeftOrdContinuous f) : Monotone f := fun a₁ a₂ h =>\n  have : IsGreatest {a₁, a₂} a₂ := ⟨Or.inr rfl, by simp [*]⟩\n  (hf.map_isGreatest this).2 <| mem_image_of_mem _ (Or.inl rfl)\n\n@[to_dual]\ntheorem comp (hg : LeftOrdContinuous g) (hf : LeftOrdContinuous f) : LeftOrdContinuous (g ∘ f) :=\n  fun s x h => by simpa only [image_image] using! hg (hf h)\n\n@[to_dual]\nprotected theorem iterate {f : α → α} (hf : LeftOrdContinuous f) (n : ℕ) :\n    LeftOrdContinuous f^[n] :=\n  match n with\n  | 0 => LeftOrdContinuous.id α\n  | (n + 1) => (LeftOrdContinuous.iterate hf n).comp hf\n\nend Preorder\n\nsection SemilatticeSup\n\nvariable [SemilatticeSup α] [SemilatticeSup β] {f : α → β}\n\n@[to_dual]\ntheorem map_sup (hf : LeftOrdContinuous f) (x y : α) : f (x ⊔ y) = f x ⊔ f y :=\n  (hf isLUB_pair).unique <| by simp only [image_pair, isLUB_pair]\n\n@[to_dual]\ntheorem le_iff (hf : LeftOrdContinuous f) (h : Injective f) {x y} : f x ≤ f y ↔ x ≤ y := by\n  simp only [← sup_eq_right, ← hf.map_sup, h.eq_iff]\n\n@[to_dual]\ntheorem lt_iff (hf : LeftOrdContinuous f) (h : Injective f) {x y} : f x < f y ↔ x < y := by\n  simp only [lt_iff_le_not_ge, hf.le_iff h]\n\nvariable (f)\n\n/-- Convert an injective left order continuous function to an order embedding. -/\n@[to_dual\n/-- Convert an injective right order continuous function to an order embedding. -/]\ndef toOrderEmbedding (hf : LeftOrdContinuous f) (h : Injective f) : α ↪o β :=\n  ⟨⟨f, h⟩, hf.le_iff h⟩\n\nvariable {f}\n\n@[to_dual (attr := simp)]\ntheorem coe_toOrderEmbedding (hf : LeftOrdContinuous f) (h : Injective f) :\n    ⇑(hf.toOrderEmbedding f h) = f :=\n  rfl\n\nend SemilatticeSup\n\nsection CompleteLattice\n\nvariable [CompleteLattice α] [CompleteLattice β] {f : α → β}\n\n@[to_dual]\ntheorem map_sSup' (hf : LeftOrdContinuous f) (s : Set α) : f (sSup s) = sSup (f '' s) :=\n  (hf <| isLUB_sSup s).sSup_eq.symm\n\n@[to_dual]\ntheorem map_sSup (hf : LeftOrdContinuous f) (s : Set α) : f (sSup s) = ⨆ x ∈ s, f x := by\n  rw [hf.map_sSup', sSup_image]\n\n@[to_dual]\n\nTarget:\ntheorem map_iSup (hf : LeftOrdContinuous f) (g : ι → α) : f (⨆ i, g i) = ⨆ i, f (g i) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  simp only [iSup, hf.map_sSup', ← range_comp]\n  rfl","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order","family_id":"map_isup","file_id":"mathlib/Mathlib/Order/OrdContinuous.lean","sample_id":"8b95081cd7f651acb1a47e7a53ca218a713ae886e12ac7935b03bee7470a950f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e934aed5577882b0e1846e82a16eff1d5d3efe027757ba1c1868dea83273eb47","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2a1c4d5761edc1d34d895e3b69bd6ea2c79fcdd869d5c1a4c773391f846b1568","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a0442559c44ff5e1900294674d91b30edd8758751f47d7b1327b13d6af184bd6","source_sha256":"e2952ea887763387582fd966bcaa7890973b527717726e02bfae4aafd3204443","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨T, hT₁, hT₂⟩ := has_injective_coseparator (op G) ((isSeparator_op_iff _).2 hG)\n  exact ⟨unop T, inferInstance, (isSeparator_unop_iff _).2 hT₂⟩","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":3,"token_jaccard":0.08,"token_length_ratio":0.071429},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"397a2eb4305972dbfa1c04e564ffcecf32d4b57ad5d2d4665ea2e74e6789cc48","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Abelian.Subobject\npublic import Mathlib.CategoryTheory.Limits.EssentiallySmall\npublic import Mathlib.CategoryTheory.Preadditive.Injective.Basic\npublic import Mathlib.CategoryTheory.Generator.Preadditive\npublic import Mathlib.CategoryTheory.Abelian.Opposite\n\nNamespace:\nCategoryTheory.Abelian\n\nLocal context:\n/-\nCopyright (c) 2022 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\n/-!\n# A complete abelian category with enough injectives and a separator has an injective coseparator\n\n## Future work\n* Once we know that Grothendieck categories have enough injectives, we can use this to conclude\n  that Grothendieck categories have an injective coseparator.\n\n## References\n* [Peter J Freyd, *Abelian Categories* (Theorem 3.37)][freyd1964abelian]\n\n-/\n\npublic section\n\n\nopen CategoryTheory CategoryTheory.Limits Opposite\n\nuniverse v u\n\nnamespace CategoryTheory.Abelian\n\nvariable {C : Type u} [Category.{v} C] [Abelian C]\n\ntheorem has_injective_coseparator [HasLimits C] [EnoughInjectives C] (G : C) (hG : IsSeparator G) :\n    ∃ G : C, Injective G ∧ IsCoseparator G := by\n  haveI : WellPowered.{v} C := wellPowered_of_isDetector G hG.isDetector\n  haveI : HasProductsOfShape (Subobject (op G)) C := hasProductsOfShape_of_small.{v} _ _\n  let T : C := Injective.under (piObj fun P : Subobject (op G) => unop P)\n  refine ⟨T, inferInstance, (Preadditive.isCoseparator_iff _).2 fun X Y f hf => ?_⟩\n  refine (Preadditive.isSeparator_iff _).1 hG _ fun h => ?_\n  suffices hh : factorThruImage (h ≫ f) = 0 by\n    rw [← Limits.image.fac (h ≫ f), hh, zero_comp]\n  let R := Subobject.mk (factorThruImage (h ≫ f)).op\n  let q₁ : image (h ≫ f) ⟶ unop R :=\n    (Subobject.underlyingIso (factorThruImage (h ≫ f)).op).unop.hom\n  let q₂ : unop (R : Cᵒᵖ) ⟶ piObj fun P : Subobject (op G) => unop P :=\n    section_ (Pi.π (fun P : Subobject (op G) => (unop P : C)) R)\n  let q : image (h ≫ f) ⟶ T := q₁ ≫ q₂ ≫ Injective.ι _\n  exact zero_of_comp_mono q\n    (by rw [← Injective.comp_factorThru q (Limits.image.ι (h ≫ f)), Limits.image.fac_assoc,\n      Category.assoc, hf, comp_zero])\n\nTarget:\ntheorem has_projective_separator [HasColimits C] [EnoughProjectives C] (G : C)\n    (hG : IsCoseparator G) : ∃ G : C, Projective G ∧ IsSeparator G :=\n\nProof body:\n","rejected":"by\n  exact has_projective_separator","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"f54969bc658347d968513a8f995eae649f443a7b61aa863a63cef6e5c8684dd2","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Generator","family_id":"has_projective_separator","file_id":"mathlib/Mathlib/CategoryTheory/Generator/Abelian.lean","sample_id":"a0442559c44ff5e1900294674d91b30edd8758751f47d7b1327b13d6af184bd6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c198f493b5c0a079bb4ea7d468b89dbbd0202be048c196e631696690e8762509","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f83dd1b97c9a129dc210da5c52474019b307010c9a4d0c19de7a0ae4adc2ce13","source_sha256":"5fcf0f4a5da250f9e6a71f1db0e6fc4a861d3fb51b0fa6cda384195f89e3653e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain rfl | hμ := eq_zero_or_isProbabilityMeasure μ\n  · simp\n  simpa [Pi.mul_def, integral_of_ae_eq_zero_or_one, integral_one_sub_of_ae_eq_zero_or_one, mul_comm,\n    *] using condVar_of_ae_eq_zero_or_one bot_le hXmeas hX","hard_negative":true,"metrics":{"chosen_tokens":30,"rejected_tokens":8,"token_jaccard":0.064516,"token_length_ratio":0.266667},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"398bc786a96037f71cecf272ce5b090bfcbf3eff13b1ea92d677b68cfe449367","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Integral.Bochner.Basic\npublic import Mathlib.Probability.CondVar\nimport Mathlib.Probability.Notation\n\nNamespace:\nProbabilityTheory\n\nLocal context:\n/-\nCopyright (c) 2024 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Distributions on two values\n\nThis file proves a few lemmas about random variables that take at most two values.\n-/\n\npublic section\n\nopen MeasureTheory\nopen scoped ProbabilityTheory\n\nnamespace MeasureTheory\nvariable {Ω : Type*} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : Measure Ω}\n\n/-- If an `AEMeasurable` function is ae equal to `0` or `1`, then its integral is equal to the\nmeasure of the set where it equals `1`. -/\nlemma integral_of_ae_eq_zero_or_one (hXmeas : AEMeasurable X μ) (hX : ∀ᵐ ω ∂μ, X ω = 0 ∨ X ω = 1) :\n    μ[X] = μ.real {ω | X ω = 1} := by\n  refine (integral_map (f := id) hXmeas <| by fun_prop).symm.trans ?_\n  rw [(Measure.ae_eq_or_eq_iff_map_eq_dirac_add_dirac hXmeas zero_ne_one).1 hX]\n  by_cases h : μ {ω | X ω = 1} = ⊤\n  · simp [h, Measure.real, Set.preimage, integral_undef, Integrable, HasFiniteIntegral]\n  rw [integral_add_measure ⟨by fun_prop, by simp [HasFiniteIntegral]⟩ <|\n    .smul_measure (by simp [integrable_dirac]) h]\n  simp [Measure.real, Set.preimage]\n\n/-- If a random variable is ae equal to `0` or `1`, then one minus its expectation is equal to the\nprobability that it equals `0`. -/\nlemma integral_one_sub_of_ae_eq_zero_or_one (hXmeas : AEMeasurable X μ)\n    (hX : ∀ᵐ ω ∂μ, X ω = 0 ∨ X ω = 1) : ∫ ω, 1 - X ω ∂μ = μ.real {ω | X ω = 0} := by\n  calc\n    _ = μ.real {ω | 1 - X ω = 1} :=\n      integral_of_ae_eq_zero_or_one (aemeasurable_const (b := 1).sub hXmeas)\n        (by simpa [sub_eq_zero, or_comm, eq_comm (a := (1 : ℝ))] using hX)\n    _ = μ.real {ω | X ω = 0} := by simp\n\nend MeasureTheory\n\n\nnamespace ProbabilityTheory\nvariable {Ω : Type*} {m : MeasurableSpace Ω} {X Y : Ω → ℝ} {μ : Measure ℝ} {P : Measure Ω}\n\n/-- If a random variable is ae equal to `0` or `1`, then its conditional variance is the product of\nthe conditional probabilities that it's equal to `0` and that it's equal to `1`. -/\nlemma condVar_of_ae_eq_zero_or_one {m₀ : MeasurableSpace Ω} (hm : m ≤ m₀) {μ : Measure[m₀] Ω}\n    [IsFiniteMeasure μ] (hXmeas : AEMeasurable[m₀] X μ) (hX : ∀ᵐ ω ∂μ, X ω = 0 ∨ X ω = 1) :\n    Var[X; μ | m] =ᵐ[μ] μ[X | m] * μ[1 - X | m] := by\n  wlog hXmeas : Measurable[m₀] X\n  · obtain ⟨Y, hYmeas, hXY⟩ := ‹AEMeasurable[m₀] X μ›\n    calc\n      Var[X; μ | m]\n      _ =ᵐ[μ] Var[Y; μ | m] := condVar_congr_ae hXY\n      _ =ᵐ[μ] μ[Y | m] * μ[1 - Y | m] := by\n        refine this hm hYmeas.aemeasurable ?_ hYmeas\n        filter_upwards [hX, hXY] with ω hXω hXYω\n        simp [hXω, ← hXYω]\n      _ =ᵐ[μ] μ[X | m] * μ[1 - X | m] := by\n        refine .mul ?_ ?_ <;>\n          exact condExp_congr_ae <| by filter_upwards [hXY] with ω hω; simp [hω]\n  calc\n    _ =ᵐ[μ] μ[X ^ 2 | m] - μ[X | m] ^ 2 :=\n      condVar_ae_eq_condExp_sq_sub_sq_condExp hm <| .of_bound hXmeas.aestronglyMeasurable 1 <| by\n        filter_upwards [hX]; rintro ω (hω | hω) <;> simp [hω]\n    _ =ᵐ[μ] μ[X | m] - μ[X | m] ^ 2 := by\n      refine .sub ?_ ae_eq_rfl\n      exact condExp_congr_ae <| by filter_upwards [hX]; rintro ω (hω | hω) <;> simp [hω]\n    _ =ᵐ[μ] μ[X | m] * μ[1 - X | m] := by\n      rw [sq, ← one_sub_mul, mul_comm]\n      refine .mul ae_eq_rfl ?_\n      calc\n        1 - μ[X | m]\n        _ = μ[1 | m] - μ[X | m] := by simp [Pi.one_def, hm]\n        _ =ᵐ[μ] μ[1 - X | m] := by\n          refine (condExp_sub (integrable_const _)\n            (.of_bound (C := 1) hXmeas.aestronglyMeasurable ?_) _).symm\n          filter_upwards [hX]\n          rintro ω (hω | hω) <;> simp [hω]\n\n/-- If a random variable is ae equal to `0` or `1`, then its variance is the product of\nthe probabilities that it's equal to `0` and that it's equal to `1`. -/\n\nTarget:\nlemma variance_of_ae_eq_zero_or_one {μ : Measure Ω} [IsZeroOrProbabilityMeasure μ]\n    (hXmeas : AEMeasurable X μ) (hX : ∀ᵐ ω ∂μ, X ω = 0 ∨ X ω = 1) :\n    Var[X; μ] = μ.real {ω | X ω = 0} * μ.real {ω | X ω = 1} :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"2639225f5522d206f6e0dbe67fcf8efda531c80cbb5887e690e8a3255d373117","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Distributions","family_id":"variance_of_ae_eq_zero_or_one","file_id":"mathlib/Mathlib/Probability/Distributions/TwoValued.lean","sample_id":"f83dd1b97c9a129dc210da5c52474019b307010c9a4d0c19de7a0ae4adc2ce13"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3bdeb5d69c9d68024e63a03b42978b4ed5db36efc7c46e5dd1c026762b0bff70","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0499f9ddadeadf4353cd9529cac7c5ef8cb81587f40f09d0d81cd202b497e41b","source_sha256":"7e0891f362b30145c7c3c4e0ec7d7143a1d3e97fd040572f98d49bd047596de1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · obtain @⟨n, i, h0, hn⟩ := h\n    simp only [iSup_iff, ofHoms_iff, Subtype.exists, exists_prop]\n    use n, i\n  · simp only [iSup_iff, ofHoms_iff] at h\n    obtain ⟨n, ⟨i, h0, hn⟩, _, _⟩ := h\n    exact horn_ι_mem_innerHornInclusions h0 hn","hard_negative":false,"metrics":{"chosen_tokens":81,"rejected_tokens":2,"token_jaccard":0.027778,"token_length_ratio":0.024691},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"39bb1153b9c3ca38d62c94322fa2f5ad691a7dd49dd29b13a83a2be7090f6e3a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.Quasicategory.Basic\n\nNamespace:\nSSet\n\nLocal context:\n/-\nCopyright (c) 2026 Jack McKoen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jack McKoen\n-/\n/-!\n# Inner fibrations\n\nInner fibrations of simplicial sets are the morphisms in `SSet` which have the right lifting\nproperty with respect to all inner horn inclusions.\n\nBasic consequences of inner fibrations with respect to the definition of quasi-categories are\nformalized.\n\n-/\n\npublic section\n\nopen CategoryTheory MorphismProperty Simplicial Limits\n\nuniverse u\n\nnamespace SSet\n\n/-- The family of morphisms in `SSet` which consists of inner horn inclusions\n`Λ[n, i].ι : Λ[n, i] ⟶ Δ[n]` (for `0 < i < n`). -/\ninductive innerHornInclusions : MorphismProperty SSet.{u} where\n  | intro {n : ℕ} (i : Fin (n + 3)) (h0 : 0 < i) (hn : i < Fin.last (n + 2)) :\n    innerHornInclusions Λ[n + 2, i].ι\n\nlemma horn_ι_mem_innerHornInclusions {n : ℕ} {i : Fin (n + 1)}\n    (h0 : 0 < i) (hn : i < Fin.last n) : innerHornInclusions (horn.{u} n i).ι := by\n  obtain _ | _ | k := n\n  · grind\n  · grind\n  · exact ⟨i, h0, hn⟩\n\nTarget:\nlemma innerHornInclusions_eq_iSup :\n    innerHornInclusions.{u} =\n    ⨆ n, .ofHoms (fun p : {p : Fin (n + 3) // 0 < p ∧ p < Fin.last (n + 2)} ↦ Λ[n + 2, p].ι) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/Quasicategory","family_id":"innerhorninclusions_eq_isup","file_id":"mathlib/Mathlib/AlgebraicTopology/Quasicategory/InnerFibration.lean","sample_id":"0499f9ddadeadf4353cd9529cac7c5ef8cb81587f40f09d0d81cd202b497e41b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"68b7446878a3dfdb301fca861ee4b86aba230785df09d24821618a061945374b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0cf725ee07a3eb842b12fe329854e9d4ad59f496cf122c50f3f3510995f88d74","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a8f96640d4ba327fc7e9e6bd4ae65eab83d12cf3037d5abef2ad29e199cb314c","source_sha256":"f9d3644b75382b6a32e9f16c61996173c9614e3a85719811c47a399dfcb0f697","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n    let ⟨f⟩ := quotient_prod_linearEquiv p\n    exact rank_prod'.symm.trans f.rank_eq","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":5,"token_jaccard":0.176471,"token_length_ratio":0.277778},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"39d221fe035b23dabb00f03e2add60e036cd183cc5a454fd3597b92d4a0b90e7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Basis.VectorSpace\npublic import Mathlib.LinearAlgebra.Dimension.Finite\npublic import Mathlib.LinearAlgebra.Dimension.RankNullity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Johannes Hölzl, Sander Dahmen,\nKim Morrison, Chris Hughes, Anne Baanen, Junyan Xu\n-/\n/-!\n# Dimension of vector spaces\n\nIn this file we provide results about `Module.rank` and `Module.finrank` of vector spaces\nover division rings.\n\n## Main statements\n\nFor vector spaces (i.e. modules over a field), we have\n\n* `rank_quotient_add_rank_of_divisionRing`: if `V₁` is a submodule of `V`, then\n  `Module.rank (V/V₁) + Module.rank V₁ = Module.rank V`.\n* `rank_range_add_rank_ker`: the rank-nullity theorem.\n\nSee also `Mathlib/LinearAlgebra/Dimension/ErdosKaplansky.lean` for the Erdős-Kaplansky theorem.\n\n-/\n\npublic section\n\n\nnoncomputable section\n\nuniverse u₀ u v v' v'' u₁' w w'\n\nvariable {K : Type u} {V V₁ V₂ V₃ : Type v}\nvariable {ι : Type w}\n\nopen Cardinal Basis Submodule Function Set\n\nsection Module\n\nsection DivisionRing\n\nvariable [DivisionRing K]\nvariable [AddCommGroup V] [Module K V]\nvariable [AddCommGroup V₁] [Module K V₁]\n\n/-- If a vector space has a finite dimension, the index set of `Basis.ofVectorSpace` is finite. -/\ntheorem Module.Basis.finite_ofVectorSpaceIndex_of_rank_lt_aleph0 (h : Module.rank K V < ℵ₀) :\n    (Basis.ofVectorSpaceIndex K V).Finite :=\n  Set.finite_def.2 <| (Basis.ofVectorSpace K V).nonempty_fintype_index_of_rank_lt_aleph0 h\n\n/-- Also see `rank_quotient_add_rank`. -/\n\nTarget:\ntheorem rank_quotient_add_rank_of_divisionRing (p : Submodule K V) :\n    Module.rank K (V ⧸ p) + Module.rank K p = Module.rank K V :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_a8f96640d4ba","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"9cacb6448c156cbdc75eca2929b9bdd3b402b57e83aa832a50fcce2c960ba969","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Dimension","family_id":"rank_quotient_add_rank_of_divisionring","file_id":"mathlib/Mathlib/LinearAlgebra/Dimension/DivisionRing.lean","sample_id":"a8f96640d4ba327fc7e9e6bd4ae65eab83d12cf3037d5abef2ad29e199cb314c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7492371e89f22fb3632f66da0f33c643e10177499de4f2d7febebab11d42e4f8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4fa103ab04285b9702895808bf833a23ffe13a8d87d9c70cc82fd21a7bc26b3d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"31295720883c04924c41b3b7a4b8a52765977942c19e8e921c01332c5cadf5db","source_sha256":"11459aad782565ea7bf0f74d44b1f758119860f533b2e2fde38a8a35d278f13d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  symm\n  calc\n    0 = ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆ := by\n      rw [(commute_lmul_lmul_sq (a + b + c)).lie_eq]\n    _ = _ := by rw [aux0, aux1, aux2, aux3, nsmul_add, nsmul_add]","hard_negative":true,"metrics":{"chosen_tokens":70,"rejected_tokens":2,"token_jaccard":0.033333,"token_length_ratio":0.028571},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"39d6e22a1b8531dc38257f0d792244995aea973f34fa11533a96fa01b0f74248","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Lie.OfAssociative\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Christopher Hoskin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christopher Hoskin\n-/\n/-!\n# Jordan rings\n\nLet `A` be a non-unital, non-associative ring. Then `A` is said to be a (commutative, linear) Jordan\nring if the multiplication is commutative and satisfies a weak associativity law known as the\nJordan Identity: for all `a` and `b` in `A`,\n```\n(a * b) * a^2 = a * (b * a^2)\n```\ni.e. the operators of multiplication by `a` and `a^2` commute.\n\nA more general concept of a (non-commutative) Jordan ring can also be defined, as a\n(non-commutative, non-associative) ring `A` where, for each `a` in `A`, the operators of left and\nright multiplication by `a` and `a^2` commute.\n\nEvery associative algebra can be equipped with a symmetrized multiplication (characterized by\n`SymAlg.sym_mul_sym`) making it into a commutative Jordan algebra (`IsCommJordan`).\nJordan algebras arising this way are said to be special.\n\nA real Jordan algebra `A` can be introduced by\n```lean\nvariable {A : Type*} [NonUnitalNonAssocCommRing A] [Module ℝ A] [SMulCommClass ℝ A A]\n  [IsScalarTower ℝ A A] [IsCommJordan A]\n```\n\n## Main results\n\n- `two_nsmul_lie_lmul_lmul_add_add_eq_zero` : Linearisation of the commutative Jordan axiom\n\n## Implementation notes\n\nWe shall primarily be interested in linear Jordan algebras (i.e. over rings of characteristic not\ntwo) leaving quadratic algebras to those better versed in that theory.\n\nThe conventional way to linearise the Jordan axiom is to equate coefficients (more formally, assume\nthat the axiom holds in all field extensions). For simplicity we use brute force algebraic expansion\nand substitution instead.\n\n## Motivation\n\nEvery Jordan algebra `A` has a triple product defined, for `a` `b` and `c` in `A` by\n$$\n{a\\,b\\,c} = (a * b) * c - (a * c) * b + a * (b * c).\n$$\nVia this triple product Jordan algebras are related to a number of other mathematical structures:\nJordan triples, partial Jordan triples, Jordan pairs and quadratic Jordan algebras. In addition to\ntheir considerable algebraic interest ([mccrimmon2004]) these structures have been shown to have\ndeep connections to mathematical physics, functional analysis and differential geometry. For more\ninformation about these connections the interested reader is referred to [alfsenshultz2003],\n[chu2012], [friedmanscarr2005], [iordanescu2003] and [upmeier1987].\n\nThere are also exceptional Jordan algebras which can be shown not to be the symmetrization of any\nassociative algebra. The 3x3 matrices of octonions is the canonical example.\n\nNon-commutative Jordan algebras have connections to the Vidav-Palmer theorem\n[cabreragarciarodriguezpalacios2014].\n\n## References\n\n* [Cabrera García and Rodríguez Palacios, Non-associative normed algebras. Volume 1]\n  [cabreragarciarodriguezpalacios2014]\n* [Hanche-Olsen and Størmer, Jordan Operator Algebras][hancheolsenstormer1984]\n* [McCrimmon, A taste of Jordan algebras][mccrimmon2004]\n\n-/\n\npublic section\n\n\nvariable (A : Type*)\n\n/-- A (non-commutative) Jordan multiplication. -/\nclass IsJordan [Mul A] : Prop where\n  lmul_comm_rmul : ∀ a b : A, a * b * a = a * (b * a)\n  lmul_lmul_comm_lmul : ∀ a b : A, a * a * (a * b) = a * (a * a * b)\n  lmul_lmul_comm_rmul : ∀ a b : A, a * a * (b * a) = a * a * b * a\n  lmul_comm_rmul_rmul : ∀ a b : A, a * b * (a * a) = a * (b * (a * a))\n  rmul_comm_rmul_rmul : ∀ a b : A, b * a * (a * a) = b * (a * a) * a\n\n/-- A commutative Jordan multiplication -/\nclass IsCommJordan [CommMagma A] : Prop where\n  lmul_comm_rmul_rmul : ∀ a b : A, a * b * (a * a) = a * (b * (a * a))\n\n-- see Note [lower instance priority]\n/-- A (commutative) Jordan multiplication is also a Jordan multiplication -/\ninstance (priority := 100) IsCommJordan.toIsJordan [CommMagma A] [IsCommJordan A] : IsJordan A where\n  lmul_comm_rmul a b := by rw [mul_comm, mul_comm a b]\n  lmul_lmul_comm_lmul a b := by\n    rw [mul_comm (a * a) (a * b), IsCommJordan.lmul_comm_rmul_rmul,\n      mul_comm b (a * a)]\n  lmul_comm_rmul_rmul := IsCommJordan.lmul_comm_rmul_rmul\n  lmul_lmul_comm_rmul a b := by\n    rw [mul_comm (a * a) (b * a), mul_comm b a,\n      IsCommJordan.lmul_comm_rmul_rmul, mul_comm, mul_comm b (a * a)]\n  rmul_comm_rmul_rmul a b := by\n    rw [mul_comm b a, IsCommJordan.lmul_comm_rmul_rmul, mul_comm]\n\n-- see Note [lower instance priority]\n/-- Semigroup multiplication satisfies the (non-commutative) Jordan axioms -/\ninstance (priority := 100) Semigroup.isJordan [Semigroup A] : IsJordan A where\n  lmul_comm_rmul a b := by rw [mul_assoc]\n  lmul_lmul_comm_lmul a b := by rw [mul_assoc, mul_assoc]\n  lmul_comm_rmul_rmul a b := by rw [mul_assoc]\n  lmul_lmul_comm_rmul a b := by rw [← mul_assoc]\n  rmul_comm_rmul_rmul a b := by rw [← mul_assoc, ← mul_assoc]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) CommSemigroup.isCommJordan [CommSemigroup A] : IsCommJordan A where\n  lmul_comm_rmul_rmul _ _ := mul_assoc _ _ _\n\nlocal notation \"L\" => AddMonoid.End.mulLeft\n\nlocal notation \"R\" => AddMonoid.End.mulRight\n\n/-!\nThe Jordan axioms can be expressed in terms of commuting multiplication operators.\n-/\n\n\nsection Commute\n\nvariable {A} [NonUnitalNonAssocRing A] [IsJordan A]\n\n@[simp]\ntheorem commute_lmul_rmul (a : A) : Commute (L a) (R a) :=\n  AddMonoidHom.ext fun _ => (IsJordan.lmul_comm_rmul _ _).symm\n\n@[simp]\ntheorem commute_lmul_lmul_sq (a : A) : Commute (L a) (L (a * a)) :=\n  AddMonoidHom.ext fun _ => (IsJordan.lmul_lmul_comm_lmul _ _).symm\n\n@[simp]\ntheorem commute_lmul_rmul_sq (a : A) : Commute (L a) (R (a * a)) :=\n  AddMonoidHom.ext fun _ => (IsJordan.lmul_comm_rmul_rmul _ _).symm\n\n@[simp]\ntheorem commute_lmul_sq_rmul (a : A) : Commute (L (a * a)) (R a) :=\n  AddMonoidHom.ext fun _ => IsJordan.lmul_lmul_comm_rmul _ _\n\n@[simp]\ntheorem commute_rmul_rmul_sq (a : A) : Commute (R a) (R (a * a)) :=\n  AddMonoidHom.ext fun _ => (IsJordan.rmul_comm_rmul_rmul _ _).symm\n\nend Commute\n\nvariable {A} [NonUnitalNonAssocCommRing A]\n\nattribute [local instance 100] LieRing.ofAssociativeRing\n\n/-!\nThe endomorphisms on an additive monoid `AddMonoid.End` form a `Ring`, and this may be equipped\nwith a Lie Bracket via `Ring.bracket`.\n-/\n\ntheorem two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add [IsCommJordan A] (a b : A) :\n    2 • (⁅L a, L (a * b)⁆ + ⁅L b, L (b * a)⁆) = ⁅L (a * a), L b⁆ + ⁅L (b * b), L a⁆ := by\n  suffices 2 • ⁅L a, L (a * b)⁆ + 2 • ⁅L b, L (b * a)⁆ + ⁅L b, L (a * a)⁆ + ⁅L a, L (b * b)⁆ = 0 by\n    rwa [← sub_eq_zero, ← sub_sub, sub_eq_add_neg, sub_eq_add_neg, lie_skew, lie_skew, nsmul_add]\n  convert (commute_lmul_lmul_sq (a + b)).lie_eq\n  simp only [add_mul, mul_add, map_add, lie_add, add_lie, mul_comm b a,\n    (commute_lmul_lmul_sq a).lie_eq, (commute_lmul_lmul_sq b).lie_eq, zero_add, add_zero, two_smul]\n  abel\n\n-- Porting note: the monolithic `calc`-based proof of `two_nsmul_lie_lmul_lmul_add_add_eq_zero`\n-- has had four auxiliary parts `aux{0,1,2,3}` split off from it.\nprivate theorem aux0 {a b c : A} : ⁅L (a + b + c), L ((a + b + c) * (a + b + c))⁆ =\n    ⁅L a + L b + L c, L (a * a) + L (b * b) + L (c * c) +\n    2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆ := by\n  rw [add_mul, add_mul]\n  iterate 6 rw [mul_add]\n  iterate 10 rw [map_add]\n  rw [mul_comm b a, mul_comm c a, mul_comm c b]\n  iterate 3 rw [two_smul]\n  simp only [add_lie]\n  abel_nf\n\nprivate theorem aux1 {a b c : A} :\n    ⁅L a + L b + L c, L (a * a) + L (b * b) + L (c * c) +\n    2 • L (a * b) + 2 • L (c * a) + 2 • L (b * c)⁆\n    =\n    ⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ +\n    ⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ + ⁅L a, 2 • L (b * c)⁆ +\n    (⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅L b, L (c * c)⁆ +\n    ⁅L b, 2 • L (a * b)⁆ + ⁅L b, 2 • L (c * a)⁆ + ⁅L b, 2 • L (b * c)⁆) +\n    (⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, L (c * c)⁆ +\n    ⁅L c, 2 • L (a * b)⁆ + ⁅L c, 2 • L (c * a)⁆ + ⁅L c, 2 • L (b * c)⁆) := by\n  rw [add_lie, add_lie]\n  iterate 15 rw [lie_add]\n\nvariable [IsCommJordan A]\n\nprivate theorem aux2 {a b c : A} :\n    ⁅L a, L (a * a)⁆ + ⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ +\n    ⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ + ⁅L a, 2 • L (b * c)⁆ +\n    (⁅L b, L (a * a)⁆ + ⁅L b, L (b * b)⁆ + ⁅L b, L (c * c)⁆ +\n    ⁅L b, 2 • L (a * b)⁆ + ⁅L b, 2 • L (c * a)⁆ + ⁅L b, 2 • L (b * c)⁆) +\n    (⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, L (c * c)⁆ +\n    ⁅L c, 2 • L (a * b)⁆ + ⁅L c, 2 • L (c * a)⁆ + ⁅L c, 2 • L (b * c)⁆)\n    =\n    ⁅L a, L (b * b)⁆ + ⁅L b, L (a * a)⁆ + 2 • (⁅L a, L (a * b)⁆ + ⁅L b, L (a * b)⁆) +\n    (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + 2 • (⁅L a, L (c * a)⁆ + ⁅L c, L (c * a)⁆)) +\n    (⁅L b, L (c * c)⁆ + ⁅L c, L (b * b)⁆ + 2 • (⁅L b, L (b * c)⁆ + ⁅L c, L (b * c)⁆)) +\n    (2 • ⁅L a, L (b * c)⁆ + 2 • ⁅L b, L (c * a)⁆ + 2 • ⁅L c, L (a * b)⁆) := by\n  rw [(commute_lmul_lmul_sq a).lie_eq, (commute_lmul_lmul_sq b).lie_eq,\n    (commute_lmul_lmul_sq c).lie_eq, zero_add, add_zero, add_zero]\n  simp only [lie_nsmul]\n  abel\n\nprivate theorem aux3 {a b c : A} :\n    ⁅L a, L (b * b)⁆ + ⁅L b, L (a * a)⁆ + 2 • (⁅L a, L (a * b)⁆ + ⁅L b, L (a * b)⁆) +\n    (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + 2 • (⁅L a, L (c * a)⁆ + ⁅L c, L (c * a)⁆)) +\n    (⁅L b, L (c * c)⁆ + ⁅L c, L (b * b)⁆ + 2 • (⁅L b, L (b * c)⁆ + ⁅L c, L (b * c)⁆)) +\n    (2 • ⁅L a, L (b * c)⁆ + 2 • ⁅L b, L (c * a)⁆ + 2 • ⁅L c, L (a * b)⁆)\n    =\n    2 • ⁅L a, L (b * c)⁆ + 2 • ⁅L b, L (c * a)⁆ + 2 • ⁅L c, L (a * b)⁆ := by\n  rw [add_eq_right]\n  nth_rw 2 [mul_comm a b]\n  nth_rw 1 [mul_comm c a]\n  nth_rw 2 [mul_comm b c]\n  iterate 3 rw [two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add]\n  iterate 2 rw [← lie_skew (L (a * a)), ← lie_skew (L (b * b)), ← lie_skew (L (c * c))]\n  abel\n\nTarget:\ntheorem two_nsmul_lie_lmul_lmul_add_add_eq_zero (a b c : A) :\n    2 • (⁅L a, L (b * c)⁆ + ⁅L b, L (c * a)⁆ + ⁅L c, L (a * b)⁆) = 0 :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_31295720883c","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"c81f4ca586ac14043b7a60e123d96d0230ead2144925f26f43d82eaa5b335743","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Jordan","family_id":"two_nsmul_lie_lmul_lmul_add_add_eq_zero","file_id":"mathlib/Mathlib/Algebra/Jordan/Basic.lean","sample_id":"31295720883c04924c41b3b7a4b8a52765977942c19e8e921c01332c5cadf5db"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b8dd807b486dcb43a0cfd46155c68422d627f5cb03d289ff5154b4c959da2515","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"70091202abe0a1d76505ded2eda1922a6e55f006910256559adb12050fa4d2a2","source_sha256":"49729ab7657bf5b96e3883707dabc942ba77c2ab9a085e0f2d0959061ea37dde","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa [← coe_subset_coe] using! Set.not_subset","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.181818},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"3a06c731d9c2783a1be222ebacbcd02a3d1c160019702b6a851e9760fc69f86e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Tactic.Monotonicity.Attr\npublic import Mathlib.Tactic.SetLike\npublic import Mathlib.Data.Set.Basic\n\nNamespace:\nSetLike\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Typeclass for types with a set-like extensionality property\n\nThe `Membership` typeclass is used to let terms of a type have elements.\nMany instances of `Membership` have a set-like extensionality property:\nthings are equal iff they have the same elements.  The `SetLike`\ntypeclass provides a unified interface to define a `Membership` that is\nextensional in this way.\n\nThe main use of `SetLike` is for algebraic subobjects (such as\n`Submonoid` and `Submodule`), whose non-proof data consists only of a\ncarrier set.  In such a situation, the projection to the carrier set\nis injective.\n\nIn general, a type `A` is `SetLike` with elements of type `B` if it\nhas an injective map to `Set B`.  This module provides standard\nboilerplate for every `SetLike`: a `coe_sort`, a `coe` to set,\nand various extensionality and simp lemmas. The order induced by set inclusion is\ncalled `PartialOrder.ofSetlike`: this is not an instance for flexibility in choosing orders.\nThe class `IsConcreteLE` abstractly states the order is equal to that induced by set inclusion;\nan instance is automatically available when defining a `PartialOrder` as\n`.ofSetLike (MySubobject X) X`.\n\nA typical subobject should be declared as:\n```\nstructure MySubobject (X : Type*) [ObjectTypeclass X] where\n  (carrier : Set X)\n  (op_mem' : ∀ {x : X}, x ∈ carrier → sorry ∈ carrier)\n\nnamespace MySubobject\n\nvariable {X : Type*} [ObjectTypeclass X] {x : X}\n\ninstance : SetLike (MySubobject X) X :=\n  ⟨MySubobject.carrier, fun p q h => by cases p; cases q; congr!⟩\n\ninstance : PartialOrder (MySubobject X) := .ofSetLike (MySubobject X) X\n\n@[simp] lemma mem_carrier {p : MySubobject X} : x ∈ p.carrier ↔ x ∈ (p : Set X) := Iff.rfl\n\n@[ext] theorem ext {p q : MySubobject X} (h : ∀ x, x ∈ p ↔ x ∈ q) : p = q := SetLike.ext h\n\n/-- Copy of a `MySubobject` with a new `carrier` equal to the old one. Useful to fix definitional\nequalities. See Note [range copy pattern]. -/\nprotected def copy (p : MySubobject X) (s : Set X) (hs : s = ↑p) : MySubobject X :=\n  { carrier := s\n    op_mem' := hs.symm ▸ p.op_mem' }\n\n@[simp] lemma coe_copy (p : MySubobject X) (s : Set X) (hs : s = ↑p) :\n  (p.copy s hs : Set X) = s := rfl\n\nlemma copy_eq (p : MySubobject X) (s : Set X) (hs : s = ↑p) : p.copy s hs = p :=\n  SetLike.coe_injective hs\n\nend MySubobject\n```\n\nAn alternative to `SetLike` could have been an extensional `Membership` typeclass:\n```\nclass ExtMembership (α : out_param <| Type u) (β : Type v) extends Membership α β where\n  (ext_iff : ∀ {s t : β}, s = t ↔ ∀ (x : α), x ∈ s ↔ x ∈ t)\n```\nWhile this is equivalent, `SetLike` conveniently uses a carrier set projection directly.\n\n## Tags\n\nsubobjects\n-/\n\n@[expose] public section\n\nassert_not_exists RelIso\n\n/-- A class to indicate that there is a canonical injection between `A` and `Set B`.\n\nThis has the effect of giving terms of `A` elements of type `B` (through a `Membership`\ninstance) and a compatible coercion to `Type*` as a subtype.\n\nNote: if `SetLike.coe` is a projection, implementers should create a simp lemma such as\n```\n@[simp] lemma mem_carrier {p : MySubobject X} : x ∈ p.carrier ↔ x ∈ (p : Set X) := Iff.rfl\n```\nto normalize terms.\n\nIf you declare an unbundled subclass of `SetLike`, for example:\n```\nclass MulMemClass (S : Type*) (M : Type*) [Mul M] [SetLike S M] where\n  ...\n```\nThen you should *not* repeat the `outParam` declaration so `SetLike` will supply the value instead.\nThis ensures your subclass will not have issues with synthesis of the `[Mul M]` parameter starting\nbefore the value of `M` is known.\n-/\n@[notation_class* carrier Simps.findCoercionArgs]\nclass SetLike (A : Type*) (B : outParam Type*) where\n  /-- The coercion from a term of a `SetLike` to its corresponding `Set`. -/\n  protected coe : A → Set B\n  /-- The coercion from a term of a `SetLike` to its corresponding `Set` is injective. -/\n  coe_injective : Function.Injective coe\n\nattribute [coe] SetLike.coe\n\nnamespace SetLike\n\nvariable {A : Type*} {B : Type*} [i : SetLike A B]\n\n@[deprecated (since := \"2026-06-04\")] alias coe_injective' := coe_injective\n\ninstance : CoeTC A (Set B) where coe := SetLike.coe\n\ninstance (priority := 100) instMembership : Membership B A :=\n  ⟨fun p x => x ∈ (p : Set B)⟩\n\ninstance (priority := 100) : CoeSort A (Type _) :=\n  ⟨fun p => { x : B // x ∈ p }⟩\n\nsection Delab\nopen Lean PrettyPrinter.Delaborator SubExpr\n\n/-- For terms that match the `CoeSort` instance's body, pretty print as `↥S`\nrather than as `{ x // x ∈ S }`. The discriminating feature is that membership\nuses the `SetLike.instMembership` instance. -/\n@[app_delab Subtype]\nmeta def delabSubtypeSetLike : Delab := whenPPOption getPPNotation do\n  let #[_, .lam n _ body _] := (← getExpr).getAppArgs | failure\n  guard <| body.isAppOf ``Membership.mem\n  let #[_, _, inst, _, .bvar 0] := body.getAppArgs | failure\n  guard <| inst.isAppOfArity ``instMembership 3\n  let S ← withAppArg <| withBindingBody n <| withNaryArg 3 delab\n  `(↥$S)\n\nend Delab\n\nvariable (p q : A)\n\n@[simp, norm_cast]\ntheorem coe_sort_coe : ((p : Set B) : Type _) = p :=\n  rfl\n\nvariable {p q}\n\nprotected theorem «exists» {q : p → Prop} : (∃ x, q x) ↔ ∃ (x : B) (h : x ∈ p), q ⟨x, ‹_›⟩ :=\n  SetCoe.exists\n\nprotected theorem «forall» {q : p → Prop} : (∀ x, q x) ↔ ∀ (x : B) (h : x ∈ p), q ⟨x, ‹_›⟩ :=\n  SetCoe.forall\n\n@[simp, norm_cast]\ntheorem coe_set_eq : (p : Set B) = q ↔ p = q :=\n  coe_injective.eq_iff\n\n@[norm_cast] lemma coe_ne_coe : (p : Set B) ≠ q ↔ p ≠ q := coe_injective.ne_iff\n\ntheorem ext' (h : (p : Set B) = q) : p = q :=\n  coe_injective h\n\ntheorem ext'_iff : p = q ↔ (p : Set B) = q :=\n  coe_set_eq.symm\n\n/-- Note: implementers of `SetLike` must copy this lemma in order to tag it with `@[ext]`. -/\ntheorem ext (h : ∀ x, x ∈ p ↔ x ∈ q) : p = q :=\n  coe_injective <| Set.ext h\n\ntheorem ext_iff : p = q ↔ ∀ x, x ∈ p ↔ x ∈ q :=\n  coe_injective.eq_iff.symm.trans Set.ext_iff\n\n@[simp, push]\ntheorem mem_coe {x : B} : x ∈ (p : Set B) ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_eq_coe {x y : p} : (x : B) = y ↔ x = y :=\n  Subtype.ext_iff.symm\n\n@[simp]\ntheorem coe_mem (x : p) : (x : B) ∈ p :=\n  x.2\n\n@[aesop 5% (rule_sets := [SetLike!])]\nlemma mem_of_subset {s : Set B} (hp : s ⊆ p) {x : B} (hx : x ∈ s) : x ∈ p := hp hx\n\n@[simp]\nprotected theorem eta (x : p) (hx : (x : B) ∈ p) : (⟨x, hx⟩ : p) = x := rfl\n\n@[simp] lemma setOf_mem_eq (a : A) : {b | b ∈ a} = a := rfl\n\n@[nontriviality]\nlemma mem_of_subsingleton [Subsingleton B] (S : A) [h : Nonempty S] {b : B} : b ∈ S := by\n  obtain ⟨s, hs⟩ := nonempty_subtype.mp h\n  simpa [Subsingleton.elim b s]\n\n/-- If `s` is a proper element of a `SetLike` structure (i.e., `s ≠ ⊤`) and the top element\ncoerces to the universal set, then there exists an element not in `s`. -/\nlemma exists_not_mem_of_ne_top [LE A] [OrderTop A] (s : A) (hs : s ≠ ⊤)\n    (h_top : ((⊤ : A) : Set B) = Set.univ := by simp) :\n    ∃ b : B, b ∉ s := by\n  simpa [-SetLike.coe_set_eq, SetLike.ext'_iff, h_top, Set.ne_univ_iff_exists_notMem] using hs\n\nend SetLike\n\n/-- A class to indicate that the canonical injection between `A` and `Set B` is order-preserving.\n\nAn instance of this class is automatically available on any partial order defined as\n`PartialOrder.ofSetLike`.\n-/\nclass IsConcreteLE (A : Type*) (B : outParam Type*) [SetLike A B] [LE A] where\n  /-- The coercion from a `SetLike` type preserves the ordering. -/\n  protected coe_subset_coe' {S T : A} : SetLike.coe S ⊆ SetLike.coe T ↔ S ≤ T\n\nsection default\n\nvariable (A B : Type*) [SetLike A B]\n\n/-- The order induced from a `SetLike` instance by inclusion. -/\n@[reducible] def LE.ofSetLike : LE A where\n  le := fun H K => ∀ ⦃x⦄, x ∈ H → x ∈ K\n\n/-- The partial order induced from a `SetLike` instance by inclusion.\n\nA partial order defined as `.ofSetLike` will automatically make available an instance\nof `IsConcreteLE`.\n-/\n@[reducible] def PartialOrder.ofSetLike : PartialOrder A where\n  __ := LE.ofSetLike A B\n  __ := PartialOrder.lift (SetLike.coe : A → Set B) SetLike.coe_injective\n\ninstance : letI := PartialOrder.ofSetLike A B; IsConcreteLE A B :=\n  letI := PartialOrder.ofSetLike A B; { coe_subset_coe' := Iff.rfl }\n\nend default\n\nnamespace SetLike\n\nvariable {A B : Type*} [SetLike A B]\n\nsection LE\n\nvariable [LE A] [IsConcreteLE A B] {p q : A}\n\n@[simp, norm_cast, gcongr] lemma coe_subset_coe {S T : A} : (S : Set B) ⊆ T ↔ S ≤ T :=\n  IsConcreteLE.coe_subset_coe'\n\ntheorem le_def {S T : A} : S ≤ T ↔ ∀ ⦃x : B⦄, x ∈ S → x ∈ T := by\n  simp [← coe_subset_coe, Set.subset_def]\n\n@[gcongr low] -- lower priority than `Set.mem_of_subset_of_mem`\nalias ⟨_root_.mem_of_le_of_mem, _⟩ := le_def\n\n@[deprecated (since := \"2026-01-07\")] alias GCongr.mem_of_le_of_mem := _root_.mem_of_le_of_mem\n\nTarget:\ntheorem not_le_iff_exists : ¬p ≤ q ↔ ∃ x ∈ p, x ∉ q :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/SetLike","family_id":"not_le_iff_exists","file_id":"mathlib/Mathlib/Data/SetLike/Basic.lean","sample_id":"70091202abe0a1d76505ded2eda1922a6e55f006910256559adb12050fa4d2a2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"73ba32450f7eaa32ac31b9569cb16af401e9b0fc7b4daddcdd761838e6b70efb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8138a062ed7aa5e0b5f610f4aeff3a70100ca75f6fdd0cf8c052c2cf58487e47","source_sha256":"04666e1a54002d0f7b310bb68c623ecf47eda2066cc741dda3dda2f5fc6f4c64","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  letI : TopologicalSpace Y := ⊥; haveI := discreteTopology_bot Y\n  exact (isCompact_range hf.continuous).finite_of_discrete","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":2,"token_jaccard":0.052632,"token_length_ratio":0.095238},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"3a07fbb92b7fcdf273d60be8f269cfe6b472d207b4f840aaa78c2528d2cb7bb9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Indicator\npublic import Mathlib.Topology.Connected.LocallyConnected\npublic import Mathlib.Topology.Sets.Closeds\n\nNamespace:\nIsLocallyConstant\n\nLocal context:\n/-\nCopyright (c) 2021 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\n/-!\n# Locally constant functions\n\nThis file sets up the theory of locally constant function from a topological space to a type.\n\n## Main definitions and constructions\n\n* `IsLocallyConstant f` : a map `f : X → Y` where `X` is a topological space is locally\n                            constant if every set in `Y` has an open preimage.\n* `LocallyConstant X Y` : the type of locally constant maps from `X` to `Y`\n* `LocallyConstant.map` : push-forward of locally constant maps\n* `LocallyConstant.comap` : pull-back of locally constant maps\n-/\n\n@[expose] public section\n\nvariable {X Y Z α : Type*} [TopologicalSpace X]\n\nopen Set Filter\nopen scoped Topology\n\n/-- A function between topological spaces is locally constant if the preimage of any set is open. -/\ndef IsLocallyConstant (f : X → Y) : Prop :=\n  ∀ s : Set Y, IsOpen (f ⁻¹' s)\n\nnamespace IsLocallyConstant\n\nopen List in\nprotected theorem tfae (f : X → Y) :\n    TFAE [IsLocallyConstant f,\n      ∀ x, ∀ᶠ x' in 𝓝 x, f x' = f x,\n      ∀ x, IsOpen { x' | f x' = f x },\n      ∀ y, IsOpen (f ⁻¹' {y}),\n      ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x] := by\n  tfae_have 1 → 4 := fun h y => h {y}\n  tfae_have 4 → 3 := fun h x => h (f x)\n  tfae_have 3 → 2 := fun h x => IsOpen.mem_nhds (h x) rfl\n  tfae_have 2 → 5\n  | h, x => by\n    rcases mem_nhds_iff.1 (h x) with ⟨U, eq, hU, hx⟩\n    exact ⟨U, hU, hx, eq⟩\n  tfae_have 5 → 1\n  | h, s => by\n    refine isOpen_iff_forall_mem_open.2 fun x hx ↦ ?_\n    rcases h x with ⟨U, hU, hxU, eq⟩\n    exact ⟨U, fun x' hx' => mem_preimage.2 <| (eq x' hx').symm ▸ hx, hU, hxU⟩\n  tfae_finish\n\n@[nontriviality]\ntheorem of_discrete [DiscreteTopology X] (f : X → Y) : IsLocallyConstant f := fun _ =>\n  isOpen_discrete _\n\ntheorem isOpen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsOpen { x | f x = y } :=\n  hf {y}\n\ntheorem isClosed_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClosed { x | f x = y } :=\n  ⟨hf {y}ᶜ⟩\n\ntheorem isClopen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClopen { x | f x = y } :=\n  ⟨isClosed_fiber hf _, isOpen_fiber hf _⟩\n\ntheorem iff_exists_open (f : X → Y) :\n    IsLocallyConstant f ↔ ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x :=\n  (IsLocallyConstant.tfae f).out 0 4\n\ntheorem iff_eventually_eq (f : X → Y) : IsLocallyConstant f ↔ ∀ x, ∀ᶠ y in 𝓝 x, f y = f x :=\n  (IsLocallyConstant.tfae f).out 0 1\n\ntheorem exists_open {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x :=\n  (iff_exists_open f).1 hf x\n\nprotected theorem eventually_eq {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    ∀ᶠ y in 𝓝 x, f y = f x :=\n  (iff_eventually_eq f).1 hf x\n\ntheorem iff_isOpen_fiber_apply {f : X → Y} : IsLocallyConstant f ↔ ∀ x, IsOpen (f ⁻¹' {f x}) :=\n  (IsLocallyConstant.tfae f).out 0 2\n\ntheorem iff_isOpen_fiber {f : X → Y} : IsLocallyConstant f ↔ ∀ y, IsOpen (f ⁻¹' {y}) :=\n  (IsLocallyConstant.tfae f).out 0 3\n\nprotected theorem continuous [TopologicalSpace Y] {f : X → Y} (hf : IsLocallyConstant f) :\n    Continuous f :=\n  ⟨fun _ _ => hf _⟩\n\ntheorem iff_continuous {_ : TopologicalSpace Y} [DiscreteTopology Y] (f : X → Y) :\n    IsLocallyConstant f ↔ Continuous f :=\n  ⟨IsLocallyConstant.continuous, fun h s => h.isOpen_preimage s (isOpen_discrete _)⟩\n\ntheorem of_constant (f : X → Y) (h : ∀ x y, f x = f y) : IsLocallyConstant f :=\n  (iff_eventually_eq f).2 fun _ => Eventually.of_forall fun _ => h _ _\n\nprotected theorem const (y : Y) : IsLocallyConstant (Function.const X y) :=\n  of_constant _ fun _ _ => rfl\n\nprotected theorem comp {f : X → Y} (hf : IsLocallyConstant f) (g : Y → Z) :\n    IsLocallyConstant (g ∘ f) := fun s => by\n  rw [Set.preimage_comp]\n  exact hf _\n\ntheorem prodMk {Y'} {f : X → Y} {f' : X → Y'} (hf : IsLocallyConstant f)\n    (hf' : IsLocallyConstant f') : IsLocallyConstant fun x => (f x, f' x) :=\n  (iff_eventually_eq _).2 fun x =>\n    (hf.eventually_eq x).mp <| (hf'.eventually_eq x).mono fun _ hf' hf => Prod.ext hf hf'\n\ntheorem comp₂ {Y₁ Y₂ Z : Type*} {f : X → Y₁} {g : X → Y₂} (hf : IsLocallyConstant f)\n    (hg : IsLocallyConstant g) (h : Y₁ → Y₂ → Z) : IsLocallyConstant fun x => h (f x) (g x) :=\n  (hf.prodMk hg).comp fun x : Y₁ × Y₂ => h x.1 x.2\n\ntheorem comp_continuous [TopologicalSpace Y] {g : Y → Z} {f : X → Y} (hg : IsLocallyConstant g)\n    (hf : Continuous f) : IsLocallyConstant (g ∘ f) := fun s => by\n  rw [Set.preimage_comp]\n  exact hf.isOpen_preimage _ (hg _)\n\n/-- A locally constant function is constant on any preconnected set. -/\ntheorem apply_eq_of_isPreconnected {f : X → Y} (hf : IsLocallyConstant f) {s : Set X}\n    (hs : IsPreconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by\n  let U := f ⁻¹' {f y}\n  suffices x ∉ Uᶜ from Classical.not_not.1 this\n  intro hxV\n  specialize hs U Uᶜ (hf {f y}) (hf {f y}ᶜ) _ ⟨y, ⟨hy, rfl⟩⟩ ⟨x, ⟨hx, hxV⟩⟩\n  · simp only [union_compl_self, subset_univ]\n  · simp only [inter_empty, Set.not_nonempty_empty, inter_compl_self] at hs\n\ntheorem apply_eq_of_preconnectedSpace [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f)\n    (x y : X) : f x = f y :=\n  hf.apply_eq_of_isPreconnected isPreconnected_univ trivial trivial\n\ntheorem eq_const [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    f = Function.const X (f x) :=\n  funext fun y => hf.apply_eq_of_preconnectedSpace y x\n\ntheorem exists_eq_const [PreconnectedSpace X] [Nonempty Y] {f : X → Y} (hf : IsLocallyConstant f) :\n    ∃ y, f = Function.const X y := by\n  rcases isEmpty_or_nonempty X with h | h\n  · exact ⟨Classical.arbitrary Y, funext <| h.elim⟩\n  · exact ⟨f (Classical.arbitrary X), hf.eq_const _⟩\n\ntheorem iff_is_const [PreconnectedSpace X] {f : X → Y} : IsLocallyConstant f ↔ ∀ x y, f x = f y :=\n  ⟨fun h _ _ => h.apply_eq_of_isPreconnected isPreconnected_univ trivial trivial, of_constant _⟩\n\nTarget:\ntheorem range_finite [CompactSpace X] {f : X → Y} (hf : IsLocallyConstant f) :\n    (Set.range f).Finite :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/LocallyConstant","family_id":"range_finite","file_id":"mathlib/Mathlib/Topology/LocallyConstant/Basic.lean","sample_id":"8138a062ed7aa5e0b5f610f4aeff3a70100ca75f6fdd0cf8c052c2cf58487e47"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fc005df75a037e9e437ea57ce191d036c65586027b4016497e6ed6b62c40e9f3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"05a262d7393581017bce712b907edb438417440eb57a4689f9e11c3881a2d526","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9f7fcfe93c14fb87931e79553027be989fd1cf1216f6e4592f76a5cfd75311b1","source_sha256":"5efd5852b126afb746be3b592d500b1af5de2649a35029fd57ec7bbb27770fcb","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp only [LiftP, LiftP']\n  apply exists_iff_exists_of_mono F (f _ n α) (g _ n α)\n  · ext i ⟨x, _⟩\n    cases i <;> rfl\n  · intros\n    rw [MvFunctor.map_map]\n    dsimp +unfoldPartialApp [(· ⊚ ·)]\n    suffices (fun i => Subtype.val) = (fun i x => (MvFunctor.f P n α i x).val) by rw [this]\n    ext i ⟨x, _⟩\n    cases i <;> rfl\n\nvariable (rr : β → β → Prop)","hard_negative":false,"metrics":{"chosen_tokens":111,"rejected_tokens":116,"token_jaccard":0.927273,"token_length_ratio":1.045045},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"3a4a4a438ba5b6ac6006e54e5014ce6261a493e9e2fb0bb07c6668ba380ae03a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.Fin2\npublic import Mathlib.Data.TypeVec\npublic import Mathlib.Logic.Equiv.Defs\n\nNamespace:\nMvFunctor\n\nLocal context:\n/-\nCopyright (c) 2018 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Mario Carneiro, Simon Hudon\n-/\n/-!\n\n# Functors between the category of tuples of types, and the category Type\n\nFeatures:\n\n* `MvFunctor n` : the type class of multivariate functors\n* `f <$$> x`    : notation for map\n\n-/\n\n@[expose] public section\n\n\nuniverse u v w\n\nopen MvFunctor\n\n/-- Multivariate functors, i.e. functor between the category of type vectors\nand the category of Type -/\nclass MvFunctor {n : ℕ} (F : TypeVec n → Type*) where\n  /-- Multivariate map, if `f : α ⟹ β` and `x : F α` then `f <$$> x : F β`. -/\n  map : ∀ {α β : TypeVec n}, α ⟹ β → F α → F β\n\n/-- Multivariate map, if `f : α ⟹ β` and `x : F α` then `f <$$> x : F β` -/\nscoped[MvFunctor] infixr:100 \" <$$> \" => MvFunctor.map\n\nvariable {n : ℕ}\n\nnamespace MvFunctor\n\nvariable {α β : TypeVec.{u} n} {F : TypeVec.{u} n → Type v} [MvFunctor F]\n\n/-- predicate lifting over multivariate functors -/\ndef LiftP {α : TypeVec n} (P : ∀ i, α i → Prop) (x : F α) : Prop :=\n  ∃ u : F (fun i => Subtype (P i)), (fun i => @Subtype.val _ (P i)) <$$> u = x\n\n/-- relational lifting over multivariate functors -/\ndef LiftR {α : TypeVec n} (R : ∀ ⦃i⦄, α i → α i → Prop) (x y : F α) : Prop :=\n  ∃ u : F (fun i => { p : α i × α i // R p.fst p.snd }),\n    (fun i (t : { p : α i × α i // R p.fst p.snd }) => t.val.fst) <$$> u = x ∧\n      (fun i (t : { p : α i × α i // R p.fst p.snd }) => t.val.snd) <$$> u = y\n\n/-- given `x : F α` and a projection `i` of type vector `α`, `supp x i` is the set\nof `α.i` contained in `x` -/\ndef supp {α : TypeVec n} (x : F α) (i : Fin2 n) : Set (α i) :=\n  { y : α i | ∀ ⦃P⦄, LiftP P x → P i y }\n\ntheorem of_mem_supp {α : TypeVec n} {x : F α} {P : ∀ ⦃i⦄, α i → Prop} (h : LiftP P x) (i : Fin2 n) :\n    ∀ y ∈ supp x i, P y := fun _y hy => hy h\n\nend MvFunctor\n\n\n\n/-- laws for `MvFunctor` -/\nclass LawfulMvFunctor {n : ℕ} (F : TypeVec n → Type*) [MvFunctor F] : Prop where\n  /-- `map` preserved identities, i.e., maps identity on `α` to identity on `F α` -/\n  id_map : ∀ {α : TypeVec n} (x : F α), TypeVec.id <$$> x = x\n  /-- `map` preserves compositions -/\n  comp_map :\n    ∀ {α β γ : TypeVec n} (g : α ⟹ β) (h : β ⟹ γ) (x : F α), (h ⊚ g) <$$> x = h <$$> g <$$> x\n\nopen Nat TypeVec\n\nnamespace MvFunctor\n\nexport LawfulMvFunctor (comp_map)\n\nopen LawfulMvFunctor\n\nvariable {α β γ : TypeVec.{u} n}\nvariable {F : TypeVec.{u} n → Type v} [MvFunctor F]\nvariable (P : α ⟹ «repeat» n Prop) (R : α ⊗ α ⟹ «repeat» n Prop)\n\n/-- adapt `MvFunctor.LiftP` to accept predicates as arrows -/\ndef LiftP' : F α → Prop :=\n  MvFunctor.LiftP fun i x => ofRepeat <| P i x\n\n\n/-- adapt `MvFunctor.LiftR` to accept relations as arrows -/\ndef LiftR' : F α → F α → Prop :=\n  MvFunctor.LiftR @fun i x y => ofRepeat <| R i <| TypeVec.prod.mk _ x y\n\nvariable [LawfulMvFunctor F]\n\n@[simp]\ntheorem id_map (x : F α) : TypeVec.id <$$> x = x :=\n  LawfulMvFunctor.id_map x\n\n@[simp]\ntheorem id_map' (x : F α) : (fun _i a => a) <$$> x = x :=\n  id_map x\n\ntheorem map_map (g : α ⟹ β) (h : β ⟹ γ) (x : F α) : h <$$> g <$$> x = (h ⊚ g) <$$> x :=\n  Eq.symm <| comp_map _ _ _\n\nsection LiftP'\n\nvariable (F) in\ntheorem exists_iff_exists_of_mono {P : F α → Prop} {q : F β → Prop}\n    (f : α ⟹ β) (g : β ⟹ α)\n    (h₀ : f ⊚ g = TypeVec.id)\n    (h₁ : ∀ u : F α, P u ↔ q (f <$$> u)) :\n    (∃ u : F α, P u) ↔ ∃ u : F β, q u := by\n  constructor <;> rintro ⟨u, h₂⟩\n  · refine ⟨f <$$> u, ?_⟩\n    apply (h₁ u).mp h₂\n  · refine ⟨g <$$> u, ?_⟩\n    rw [h₁]\n    simp only [MvFunctor.map_map, h₀, LawfulMvFunctor.id_map, h₂]\n\ntheorem LiftP_def (x : F α) : LiftP' P x ↔ ∃ u : F (Subtype_ P), subtypeVal P <$$> u = x :=\n  exists_iff_exists_of_mono F _ _ (toSubtype_of_subtype P) (by simp [MvFunctor.map_map])\n\ntheorem LiftR_def (x y : F α) :\n    LiftR' R x y ↔\n      ∃ u : F (Subtype_ R),\n        (TypeVec.prod.fst ⊚ subtypeVal R) <$$> u = x ∧\n          (TypeVec.prod.snd ⊚ subtypeVal R) <$$> u = y :=\n  exists_iff_exists_of_mono _ _ _ (toSubtype'_of_subtype' R) (by\n    simp only [map_map, comp_assoc, subtypeVal_toSubtype']\n    simp +unfoldPartialApp [comp])\n\nend LiftP'\n\nend MvFunctor\n\nnamespace MvFunctor\n\nsection LiftPLastPredIff\n\nvariable {F : TypeVec.{u} (n + 1) → Type*} [MvFunctor F] [LawfulMvFunctor F] {α : TypeVec.{u} n}\n\nvariable {β : Type u}\nvariable (pp : β → Prop)\n\nprivate def f :\n    ∀ n α,\n      (fun i : Fin2 (n + 1) => { p_1 // ofRepeat (PredLast' α pp i p_1) }) ⟹ fun i : Fin2 (n + 1) =>\n        { p_1 : (α ::: β) i // PredLast α pp p_1 }\n  | _, α, Fin2.fs i, x =>\n    ⟨x.val, cast (by grind [PredLast]) x.property⟩\n  | _, _, Fin2.fz, x => ⟨x.val, x.property⟩\n\nprivate def g :\n    ∀ n α,\n      (fun i : Fin2 (n + 1) => { p_1 : (α ::: β) i // PredLast α pp p_1 }) ⟹ fun i : Fin2 (n + 1) =>\n        { p_1 // ofRepeat (PredLast' α pp i p_1) }\n  | _, α, Fin2.fs i, x =>\n    ⟨x.val, cast (by simp only [PredLast]; erw [const_iff_true]) x.property⟩\n  | _, _, Fin2.fz, x => ⟨x.val, x.property⟩\n\nTarget:\ntheorem LiftP_PredLast_iff {β} (P : β → Prop) (x : F (α ::: β)) :\n    LiftP' (PredLast' _ P) x ↔ LiftP (PredLast _ P) x :=\n\nProof body:\n","rejected":"by\n  dsimp only [LiftP, LiftP']\n  apply exists_iff_exists_of_mono F (f _ n α) (g _ n α)\n  · ext i ⟨x, _⟩\n    cases i <;> rfl\n  · intros\n    rw [MvFunctor.map_map]\n    dsimp +unfoldPartialApp [(· ⊚ ·)]\n    suffices (fun i => Subtype.val) = (fun i x => (MvFunctor.f P n α i x).val) by rw [this]\n    ext i ⟨x, _⟩\n    cases i <;> rfl\n\nvariable (rr : β → β → Prop)\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control/Functor","family_id":"liftp_predlast_iff","file_id":"mathlib/Mathlib/Control/Functor/Multivariate.lean","sample_id":"9f7fcfe93c14fb87931e79553027be989fd1cf1216f6e4592f76a5cfd75311b1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8dd1af80a36345fc761ec289040691b52374d545c9e7b00957a6cfdf28435160","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1a652632662ffd2c69af3248efeb0f3501cac7bf09cd6225eb81a59c2f6b3b05","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"48750c221d7b3b3a511e093001b0678d1d19b0dff2bd9c9bd5031532b6945ec8","source_sha256":"f92ad0e7d471b925eba21d3dc40708d34e9a126b63fb834c30a1fe2061fbbcfe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [intDegree, add_sub, sub_add, sub_sub_eq_add_sub, sub_sub, sub_eq_sub_iff_add_eq_add]\n  norm_cast\n  rw [← Polynomial.natDegree_mul x.denom_ne_zero y.denom_ne_zero, ←\n    Polynomial.natDegree_mul (RatFunc.num_ne_zero (mul_ne_zero hx hy))\n      (mul_ne_zero x.denom_ne_zero y.denom_ne_zero),\n    ← Polynomial.natDegree_mul (RatFunc.num_ne_zero hx) (RatFunc.num_ne_zero hy), ←\n    Polynomial.natDegree_mul (mul_ne_zero (RatFunc.num_ne_zero hx) (RatFunc.num_ne_zero hy))\n      (x * y).denom_ne_zero,\n    RatFunc.num_denom_mul]","hard_negative":true,"metrics":{"chosen_tokens":102,"rejected_tokens":2,"token_jaccard":0.032258,"token_length_ratio":0.019608},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"3a5a101eec821896472f0b33c586b7bf9380083e1b65d17d00d49c592d32e2a5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.RatFunc.AsPolynomial\npublic import Mathlib.RingTheory.EuclideanDomain\npublic import Mathlib.RingTheory.Localization.FractionRing\npublic import Mathlib.RingTheory.Polynomial.Content\n\nNamespace:\nRatFunc\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# The degree of rational functions\n\n## Main definitions\nWe define the degree of a rational function, with values in `ℤ`:\n- `intDegree` is the degree of a rational function, defined as the difference between the\n  `natDegree` of its numerator and the `natDegree` of its denominator. In particular,\n  `intDegree 0 = 0`.\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nuniverse u\n\nvariable {K : Type u}\n\nnamespace RatFunc\n\nsection IntDegree\n\nopen Polynomial\n\nvariable [Field K]\n\n/-- `intDegree x` is the degree of the rational function `x`, defined as the difference between\nthe `natDegree` of its numerator and the `natDegree` of its denominator. In particular,\n`intDegree 0 = 0`. -/\ndef intDegree (x : K⟮X⟯) : ℤ :=\n  natDegree x.num - natDegree x.denom\n\n@[simp]\ntheorem intDegree_zero : intDegree (0 : K⟮X⟯) = 0 := by\n  rw [intDegree, num_zero, natDegree_zero, denom_zero, natDegree_one, sub_self]\n\n@[simp]\ntheorem intDegree_one : intDegree (1 : K⟮X⟯) = 0 := by\n  rw [intDegree, num_one, denom_one, sub_self]\n\n@[simp]\ntheorem intDegree_C (k : K) : intDegree (C k) = 0 := by\n  rw [intDegree, num_C, natDegree_C, denom_C, natDegree_one, sub_self]\n\n@[simp]\ntheorem intDegree_X : intDegree (X : K⟮X⟯) = 1 := by\n  rw [intDegree, num_X, Polynomial.natDegree_X, denom_X, Polynomial.natDegree_one,\n    Int.ofNat_one, Int.ofNat_zero, sub_zero]\n\n@[simp]\ntheorem intDegree_polynomial {p : K[X]} :\n    intDegree (algebraMap K[X] K⟮X⟯ p) = natDegree p := by\n  rw [intDegree, RatFunc.num_algebraMap, RatFunc.denom_algebraMap, Polynomial.natDegree_one,\n    Int.ofNat_zero, sub_zero]\n\nTarget:\ntheorem intDegree_mul {x y : K⟮X⟯} (hx : x ≠ 0) (hy : y ≠ 0) :\n    intDegree (x * y) = intDegree x + intDegree y :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_48750c221d7b","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"3b576b6b84b139a92e301c39735c8b03b468a4037c2b3acd023ba7b45d98b88c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/RatFunc","family_id":"intdegree_mul","file_id":"mathlib/Mathlib/FieldTheory/RatFunc/Degree.lean","sample_id":"48750c221d7b3b3a511e093001b0678d1d19b0dff2bd9c9bd5031532b6945ec8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e849d95b5db02f9c868ba8b0e417961eeea08fc774f074fe05a938dadf630d6e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d15e4ade9e9b79b5272408bd0853223ef464e6a201f0499f57810cff4cb0f0e7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bf6d1403eb4460ab7069e5f743251c32cc0683a5c043812b0396ebf8e760f9d6","source_sha256":"af794759eaa35d9b988712c78093779e987eaf10b925193a6e559d007441b0b2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_contra!\n  exact hA (hP.subobject_eq_top this)","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.166667,"token_length_ratio":0.272727},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"3af692074f3d66c71d252186b387fa6e55bdf72b605049d20700002c9de61aff","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.ExtremalEpi\npublic import Mathlib.CategoryTheory.Generator.Basic\npublic import Mathlib.CategoryTheory.Limits.Presentation\n\nNamespace:\nCategoryTheory.ObjectProperty.IsStrongGenerator\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Strong generators\n\nIf `P : ObjectProperty C`, we say that `P` is a strong generator if it is a\ngenerator (in the sense that `IsSeparating P` holds) such that for any\nproper subobject `A ⊂ X`, there exists a morphism `G ⟶ X` which does not factor\nthrough `A` from an object satisfying `P`.\n\nThe main result is the lemma `isStrongGenerator_iff_exists_extremalEpi` which\nsays that if `P` is `w`-small, `C` is locally `w`-small and\nhas coproducts of size `w`, then `P` is a strong generator iff any\nobject of `C` is the target of an extremal epimorphism from a coproduct of\nobjects satisfying `P`.\n\nWe also show that if any object in `C` is a colimit of objects in `S`,\nthen `S` is a strong generator.\n\n## References\n* [Adámek, J. and Rosický, J., *Locally presentable and accessible categories*][Adamek_Rosicky_1994]\n\n-/\n\n@[expose] public section\n\nuniverse w' w v u\n\nnamespace CategoryTheory\n\nopen Limits\n\n\nnamespace ObjectProperty\n\nvariable {C : Type u} [Category.{v} C] (P : ObjectProperty C)\n\n/-- A property `P : ObjectProperty C` is a strong generator\nif it is separating and for any proper subobject `A ⊂ X`, there exists\na morphism `G ⟶ X` which does not factor through `A` from an object\nsuch that `P G` holds. -/\ndef IsStrongGenerator : Prop :=\n  P.IsSeparating ∧ ∀ ⦃X : C⦄ (A : Subobject X),\n    (∀ (G : C) (_ : P G) (f : G ⟶ X), Subobject.Factors A f) → A = ⊤\n\nvariable {P}\n\nlemma isStrongGenerator_iff :\n    P.IsStrongGenerator ↔ P.IsSeparating ∧\n      ∀ ⦃X Y : C⦄ (i : X ⟶ Y) [Mono i],\n        (∀ (G : C) (_ : P G), Function.Surjective (fun (f : G ⟶ X) ↦ f ≫ i)) → IsIso i := by\n  refine ⟨fun ⟨hS₁, hS₂⟩ ↦ ⟨hS₁, fun X Y i _ h ↦ ?_⟩,\n    fun ⟨hS₁, hS₂⟩ ↦ ⟨hS₁, fun X A hA ↦ ?_⟩⟩\n  · rw [Subobject.isIso_iff_mk_eq_top]\n    refine hS₂ _ (fun G hG g ↦ ?_)\n    rw [Subobject.mk_factors_iff]\n    exact h G hG g\n  · rw [← Subobject.isIso_arrow_iff_eq_top]\n    exact hS₂ A.arrow (fun G hG g ↦ ⟨_, Subobject.factorThru_arrow _ _ (hA G hG g)⟩)\n\nnamespace IsStrongGenerator\n\nsection\n\nvariable (hP : P.IsStrongGenerator)\n\ninclude hP\n\nlemma isSeparating : P.IsSeparating := hP.1\n\nlemma subobject_eq_top {X : C} {A : Subobject X}\n    (hA : ∀ (G : C) (_ : P G) (f : G ⟶ X), Subobject.Factors A f) :\n    A = ⊤ :=\n  hP.2 _ hA\n\nlemma isIso_of_mono ⦃X Y : C⦄ (i : X ⟶ Y) [Mono i]\n    (hi : ∀ (G : C) (_ : P G), Function.Surjective (fun (f : G ⟶ X) ↦ f ≫ i)) : IsIso i :=\n  (isStrongGenerator_iff.1 hP).2 i hi\n\nTarget:\nlemma exists_of_subobject_ne_top {X : C} {A : Subobject X} (hA : A ≠ ⊤) :\n    ∃ (G : C) (_ : P G) (f : G ⟶ X), ¬ Subobject.Factors A f :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_bf6d1403eb44","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"12fd2466f71c4f41864681a45b9aed05fb29563b40aa89f30ef2163c0b7d6374","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Generator","family_id":"exists_of_subobject_ne_top","file_id":"mathlib/Mathlib/CategoryTheory/Generator/StrongGenerator.lean","sample_id":"bf6d1403eb4460ab7069e5f743251c32cc0683a5c043812b0396ebf8e760f9d6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c785440b2e793b1a2afd030ccba707286c555ec39b3feca32e7d614cb913da7e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7734458b70658c76e232f731e838552dc282d9587d72ef9af109c7d77dc4c28f","source_sha256":"56d11af83a40725621595d46ae5793c5903e2fe6ca0b04b01b11f907274c0c49","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let ⟨z, hz⟩ := (F.map (algebraMap R ℤ_[p])).evalSubFactor x y\n  simp only [Polynomial.eval_map_algebraMap] at hz\n  calc\n    ‖F.aeval x - F.aeval y‖ = ‖z‖ * ‖x - y‖ := by simp [hz]\n    _ ≤ 1 * ‖x - y‖ := by gcongr; apply PadicInt.norm_le_one\n    _ = ‖x - y‖ := by simp","hard_negative":false,"metrics":{"chosen_tokens":90,"rejected_tokens":2,"token_jaccard":0.02381,"token_length_ratio":0.022222},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"3b38342a69e6bae52493abf42ed745b63648ef60756ce74e01bb424663256091","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Identities\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.NumberTheory.Padics.PadicIntegers\npublic import Mathlib.Topology.Algebra.Polynomial\npublic import Mathlib.Topology.MetricSpace.CauSeqFilter\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n-/\n/-!\n# Hensel's lemma on `ℤ_p`\n\nThis file proves Hensel's lemma on `ℤ_p`, roughly following Keith Conrad's writeup:\n<http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/hensel.pdf>\n\nHensel's lemma gives a simple condition for the existence of a root of a polynomial.\n\nThe proof and motivation are described in the paper\n[R. Y. Lewis, *A formal proof of Hensel's lemma over the p-adic integers*][lewis2019].\n\n## References\n\n* <http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/hensel.pdf>\n* [R. Y. Lewis, *A formal proof of Hensel's lemma over the p-adic integers*][lewis2019]\n* <https://en.wikipedia.org/wiki/Hensel%27s_lemma>\n\n## Tags\n\np-adic, p adic, padic, p-adic integer\n-/\n\npublic section\n\n\nnoncomputable section\n\nopen Topology\n\n-- We begin with some general lemmas that are used below in the computation.\n\nTarget:\ntheorem padic_polynomial_dist {p : ℕ} [Fact p.Prime] {R : Type*} [CommSemiring R] [Algebra R ℤ_[p]]\n    (F : Polynomial R) (x y : ℤ_[p]) :\n    ‖F.aeval x - F.aeval y‖ ≤ ‖x - y‖ :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Padics","family_id":"padic_polynomial_dist","file_id":"mathlib/Mathlib/NumberTheory/Padics/Hensel.lean","sample_id":"7734458b70658c76e232f731e838552dc282d9587d72ef9af109c7d77dc4c28f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1ce00e07b29f18d90867caf068379ad134d68969aa30ca0e3ea5aac4b2537419","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0ec603376cff234c3462a3aa6c0a2d3215d8de77afbecfc23ef258a73225dd9f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9d858434b5715b6466fd7efd13eba53c42a1ae0530865db41d7c8c0a8af13979","source_sha256":"5a8a601bded85c9c17f5a633b68ceab704a71b9b3614d423fc312dd608794122","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [pullbackShiftFunctorZero_inv_app]\n  simp only [Functor.id_obj, pullbackShiftIso, eqToIso.inv, eqToHom_app, shiftFunctorZero',\n    Iso.trans_inv, NatTrans.comp_app, eqToIso_refl, Iso.refl_inv, NatTrans.id_app, assoc]\n  erw [comp_id]","hard_negative":true,"metrics":{"chosen_tokens":46,"rejected_tokens":3,"token_jaccard":0.035714,"token_length_ratio":0.065217},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"3b4d48befed62bf0f1226229b22f74380c118d044223f7ce9d875ed9d154902e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Shift.Adjunction\npublic import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2023 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The pullback of a shift by a monoid morphism\n\nGiven a shift by a monoid `B` on a category `C` and a monoid morphism `φ : A →+ B`,\nwe define a shift by `A` on a category `PullbackShift C φ` which is a type synonym for `C`.\n\nIf `F : C ⥤ D` is a functor between categories equipped with shifts by `B`, we define\na type synonym `PullbackShift.functor F φ` for `F`. When `F` has a `CommShift` structure\nby `B`, we define a pulled back `CommShift` structure by `A` on `PullbackShift.functor F φ`.\n\nSimilarly, if `τ` is a natural transformation between functors `F,G : C ⥤ D`, we define\na type synonym\n`PullbackShift.natTrans τ φ : PullbackShift.functor F φ ⟶ PullbackShift.functor G φ`.\nWhen `τ` has a `CommShift` structure by `B` (i.e. is compatible with `CommShift` structures\non `F` and `G`), we define a pulled back `CommShift` structure by `A` on\n`PullbackShift.natTrans τ φ`.\n\nFinally, if we have an adjunction `F ⊣ G` (with `G : D ⥤ C`), we define a type synonym\n`PullbackShift.adjunction adj φ : PullbackShift.functor F φ ⊣ PullbackShift.functor G φ`\nand we show that, if `adj` is compatible with `CommShift` structures\non `F` and `G`, then `PullbackShift.adjunction adj φ` is also compatible with the pulled back\n`CommShift` structures.\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Limits Category\n\nvariable (C : Type*) [Category* C] {A B : Type*} [AddMonoid A] [AddMonoid B]\n\n/-- The category `PullbackShift C φ` is equipped with a shift such that for all `a`,\nthe shift functor by `a` is `shiftFunctor C (φ a)`. -/\n@[nolint unusedArguments]\ndef PullbackShift [HasShift C B] (_ : A →+ B) := C\nderiving Category\n\nattribute [local instance] endofunctorMonoidalCategory\n\nvariable [HasShift C B] (φ : A →+ B)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The shift on `PullbackShift C φ` is obtained by precomposing the shift on `C` with\nthe monoidal functor `Discrete.addMonoidalFunctor φ : Discrete A ⥤ Discrete B`. -/\ninstance : HasShift (PullbackShift C φ) A where\n  shift := Discrete.addMonoidalFunctor φ ⋙ shiftMonoidalFunctor C B\n\ninstance [HasZeroObject C] : HasZeroObject (PullbackShift C φ) :=\n  inferInstanceAs <| HasZeroObject C\n\ninstance [Preadditive C] : Preadditive (PullbackShift C φ) :=\n  inferInstanceAs <| Preadditive C\n\ninstance [Preadditive C] (a : A) [(shiftFunctor C (φ a)).Additive] :\n    (shiftFunctor (PullbackShift C φ) a).Additive :=\n  inferInstanceAs (shiftFunctor C (φ a)).Additive\n\n/-- When `b = φ a`, this is the canonical\nisomorphism `shiftFunctor (PullbackShift C φ) a ≅ shiftFunctor C b`. -/\ndef pullbackShiftIso (a : A) (b : B) (h : b = φ a) :\n    shiftFunctor (PullbackShift C φ) a ≅ shiftFunctor C b := eqToIso (by subst h; rfl)\n\nvariable {C}\nvariable (X : PullbackShift C φ) (a₁ a₂ a₃ : A) (h : a₁ + a₂ = a₃) (b₁ b₂ b₃ : B)\n  (h₁ : b₁ = φ a₁) (h₂ : b₂ = φ a₂) (h₃ : b₃ = φ a₃)\n\nset_option backward.defeqAttrib.useBackward true in\nlemma pullbackShiftFunctorZero_inv_app :\n    (shiftFunctorZero _ A).inv.app X =\n      (shiftFunctorZero C B).inv.app X ≫ (pullbackShiftIso C φ 0 0 (by simp)).inv.app X := by\n  change (shiftFunctorZero C B).inv.app X ≫ _ = _\n  dsimp [Discrete.eqToHom, Discrete.addMonoidalFunctor_ε]\n  congr 2\n  apply eqToHom_map\n\nset_option backward.isDefEq.respectTransparency false in\nlemma pullbackShiftFunctorZero_hom_app :\n    (shiftFunctorZero _ A).hom.app X =\n      (pullbackShiftIso C φ 0 0 (by simp)).hom.app X ≫ (shiftFunctorZero C B).hom.app X := by\n  rw [← cancel_epi ((shiftFunctorZero _ A).inv.app X), Iso.inv_hom_id_app,\n    pullbackShiftFunctorZero_inv_app, assoc, Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app]\n  rfl\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma pullbackShiftFunctorZero'_inv_app :\n    (shiftFunctorZero _ A).inv.app X = (shiftFunctorZero' C (φ 0) (by rw [map_zero])).inv.app X ≫\n      (pullbackShiftIso C φ 0 (φ 0) rfl).inv.app X :=\n\nProof body:\n","rejected":"by\n  exact pullbackShiftFunctorZero'_inv_app","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"e62659ff4a83b8ae5b72ed688538e06a93578e7bade6a16c0ff9a87a6897ba1d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Shift","family_id":"pullbackshiftfunctorzero'_inv_app","file_id":"mathlib/Mathlib/CategoryTheory/Shift/Pullback.lean","sample_id":"9d858434b5715b6466fd7efd13eba53c42a1ae0530865db41d7c8c0a8af13979"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d409d608f76c2016efc5fdd42930cab75c91e811c1cdb05b5a33c8b551b1384f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"326f40f774f4b7c77d4400168187e2382705a1c9ec9556d6e046ae0cef8df589","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"63d4c6f7bdfaad78a31356f5390bcb2b6c5c4366f0f8ad12785fcd4685b1c7e7","source_sha256":"231219a77cbb3d5335e2c2c8fee21d9a8c54aa585e63a8a73c88281c1837f6a8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro i\n  have h_eq_iUnion : {ω | τ ω ≤ i} = ⋃ k ≤ i, {ω | τ ω = k} := by\n    ext ω\n    simp only [Set.mem_setOf_eq, Set.mem_iUnion, exists_prop]\n    cases τ ω with\n    | top => simp\n    | coe a => norm_cast; simp\n  rw [h_eq_iUnion]\n  refine MeasurableSet.biUnion (Set.to_countable _) fun k hk => ?_\n  exact f.mono hk _ (hτ k)","hard_negative":true,"metrics":{"chosen_tokens":91,"rejected_tokens":3,"token_jaccard":0.039216,"token_length_ratio":0.032967},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"3b68759fae54cb38b8d8ee08eb72bfc3f98facf04d55052ddbe64f3d9c882365","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Process.Adapted\npublic import Mathlib.MeasureTheory.Constructions.BorelSpace.WithTop\npublic import Mathlib.Data.ENat.Lattice\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2021 Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kexing Ying, Rémy Degenne\n-/\n/-!\n# Stopping times, stopped processes and stopped values\n\nDefinition and properties of stopping times.\n\n## Main definitions\n\n* `MeasureTheory.IsStoppingTime`: a stopping time with respect to some filtration `f` on a\n  measurable space `Ω` is a function `τ : Ω → WithTop ι` such that for all `i : ι`,\n  the preimage of `{j | j ≤ i}` along `τ` is `f i`-measurable\n* `MeasureTheory.IsStoppingTime.measurableSpace`: the σ-algebra associated with a stopping time\n\n## Main results\n\n* `IsStronglyProgressive.stoppedProcess`: the stopped process of a progressively measurable process\n  is progressively measurable.\n* `memLp_stoppedProcess`: if a process belongs to `ℒp` at every time in `ℕ`, then its stopped\n  process belongs to `ℒp` as well.\n\n## Implementation notes\n\nFor a filtration on a type `ι`, we define stopping times as functions from the measurable space `Ω`\nto `WithTop ι`, which allows stopping times that can take an infinite value, represented by\n`⊤ : WithTop ι`.\n\nThis means that if we have a process `X : ι → Ω → β` and a stopping time `τ : Ω → WithTop ι`, then\nto consider the value of `X` at the stopping time `τ ω`, we need to write `X (τ ω).untopA ω`,\nin which `(τ ω).untopA` is the value of `τ ω` in `ι` if `τ ω ≠ ⊤` and some arbitrary value if\n`τ ω = ⊤`.\n\nWhile indexing would be more convenient if we defined stopping times as functions from `Ω` to `ι`,\nthis would prevent us from using stopping times as in standard mathematical literature, where a\ntypical example of stopping time is the first time an event occurs, which may never happen.\nConsider for example the first time a coin lands heads when flipping it infinitely many times:\nthis is almost surely finite, but possibly infinite. We could also not use a function `Ω → ι` with\narbitrary value for the infinite case, because this would be incompatible with the stopping time\nproperty.\n\n## Tags\n\nstopping time, stochastic process\n\n-/\n\n@[expose] public section\n\nopen Filter Order TopologicalSpace WithTop\n\nopen scoped MeasureTheory NNReal ENNReal Topology\n\nnamespace MeasureTheory\n\nvariable {Ω β ι : Type*} {m : MeasurableSpace Ω}\n\n/-! ### Stopping times -/\n\n\n/-- A stopping time with respect to some filtration `f` is a function\n`τ` such that for all `i`, the preimage of `{j | j ≤ i}` along `τ` is measurable\nwith respect to `f i`.\n\nIntuitively, the stopping time `τ` describes some stopping rule such that at time\n`i`, we may determine it with the information we have at time `i`. -/\ndef IsStoppingTime [Preorder ι] (f : Filtration ι m) (τ : Ω → WithTop ι) :=\n  ∀ i : ι, MeasurableSet[f i] <| {ω | τ ω ≤ i}\n\ntheorem isStoppingTime_const [Preorder ι] (f : Filtration ι m) (i : ι) :\n    IsStoppingTime f fun _ => i := fun j => by simp only [MeasurableSet.const]\n\nsection MeasurableSet\n\nsection Preorder\n\nvariable [Preorder ι] {f : Filtration ι m} {τ : Ω → WithTop ι}\n\nprotected theorem IsStoppingTime.measurableSet_le (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω ≤ i} :=\n  hτ i\n\ntheorem IsStoppingTime.measurableSet_lt_of_pred [PredOrder ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω < i} := by\n  by_cases hi_min : IsMin i\n  · suffices {ω : Ω | τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i)\n    ext1 ω\n    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]\n    rw [isMin_iff_forall_not_lt] at hi_min\n    cases τ ω with\n    | top => simp\n    | coe t => exact mod_cast hi_min t\n  have : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iic (pred i : ι) := by\n    ext ω\n    push _ ∈ _\n    cases τ ω with\n    | top => simp\n    | coe t =>\n      simp only [coe_lt_coe, coe_le_coe]\n      rw [le_pred_iff_of_not_isMin hi_min]\n  rw [this]\n  exact f.mono (pred_le i) _ (hτ.measurableSet_le <| pred i)\n\nend Preorder\n\nsection CountableStoppingTime\n\nnamespace IsStoppingTime\n\nvariable [PartialOrder ι] {τ : Ω → WithTop ι} {f : Filtration ι m}\n\nprotected theorem measurableSet_eq_of_countable_range (hτ : IsStoppingTime f τ)\n    (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω | τ ω = i} := by\n  have : {ω | τ ω = i} = {ω | τ ω ≤ i} \\ ⋃ (j ∈ Set.range τ) (_ : j < i), {ω | τ ω ≤ j} := by\n    ext1 a\n    simp only [Set.mem_setOf_eq, Set.mem_range, Set.iUnion_exists, Set.iUnion_iUnion_eq',\n      Set.mem_sdiff, Set.mem_iUnion, exists_prop, not_exists, not_and]\n    constructor <;> intro h\n    · simp only [h, lt_iff_le_not_ge, le_refl, and_imp, imp_self, imp_true_iff, and_self_iff]\n    · exact h.1.eq_or_lt.resolve_right fun h_lt => h.2 a h_lt le_rfl\n  rw [this]\n  refine (hτ.measurableSet_le i).diff ?_\n  refine MeasurableSet.biUnion h_countable fun j _ => ?_\n  classical\n  rw [Set.iUnion_eq_if]\n  split_ifs with hji\n  · lift j to ι using (ne_top_of_lt hji)\n    exact f.mono (mod_cast hji.le) _ (hτ.measurableSet_le j)\n  · exact @MeasurableSet.empty _ (f i)\n\nprotected theorem measurableSet_eq_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω = i} :=\n  hτ.measurableSet_eq_of_countable_range (Set.to_countable _) i\n\nprotected theorem measurableSet_lt_of_countable_range (hτ : IsStoppingTime f τ)\n    (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω | τ ω < i} := by\n  have : {ω | τ ω < i} = {ω | τ ω ≤ i} \\ {ω | τ ω = i} := by ext1 ω; simp [lt_iff_le_and_ne]\n  rw [this]\n  exact (hτ.measurableSet_le i).diff (hτ.measurableSet_eq_of_countable_range h_countable i)\n\nprotected theorem measurableSet_lt_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω < i} :=\n  hτ.measurableSet_lt_of_countable_range (Set.to_countable _) i\n\nprotected theorem measurableSet_ge_of_countable_range {ι} [LinearOrder ι] {τ : Ω → WithTop ι}\n    {f : Filtration ι m} (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) :\n    MeasurableSet[f i] {ω | i ≤ τ ω} := by\n  have : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ := by\n    ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt]\n  rw [this]\n  exact (hτ.measurableSet_lt_of_countable_range h_countable i).compl\n\nprotected theorem measurableSet_ge_of_countable {ι} [LinearOrder ι] {τ : Ω → WithTop ι}\n    {f : Filtration ι m} [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | i ≤ τ ω} :=\n  hτ.measurableSet_ge_of_countable_range (Set.to_countable _) i\n\nend IsStoppingTime\n\nend CountableStoppingTime\n\nsection LinearOrder\n\nvariable [LinearOrder ι] {f : Filtration ι m} {τ : Ω → WithTop ι}\n\ntheorem IsStoppingTime.measurableSet_gt (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | i < τ ω} := by\n  have : {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ := by\n    ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_le]\n  rw [this]\n  exact (hτ.measurableSet_le i).compl\n\nsection TopologicalSpace\n\nvariable [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι]\n\n/-- Auxiliary lemma for `MeasureTheory.IsStoppingTime.measurableSet_lt`. -/\ntheorem IsStoppingTime.measurableSet_lt_of_isLUB (hτ : IsStoppingTime f τ) (i : ι)\n    (h_lub : IsLUB (Set.Iio i) i) : MeasurableSet[f i] {ω | τ ω < i} := by\n  by_cases hi_min : IsMin i\n  · suffices {ω | τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i)\n    ext1 ω\n    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]\n    cases τ ω with\n    | top => simp\n    | coe t => norm_cast; exact isMin_iff_forall_not_lt.mp hi_min t\n  obtain ⟨seq, -, -, h_tendsto, h_bound⟩ :\n      ∃ seq : ℕ → ι, Monotone seq ∧ (∀ j, seq j ≤ i) ∧ Tendsto seq atTop (𝓝 i) ∧ ∀ j, seq j < i :=\n    h_lub.exists_seq_monotone_tendsto (not_isMin_iff.mp hi_min)\n  have h_Iio_eq_Union : Set.Iio (i : WithTop ι) = ⋃ j, {k : WithTop ι | k ≤ seq j} := by\n    ext1 k\n    push _ ∈ _\n    refine ⟨fun hk_lt_i => ?_, fun h_exists_k_le_seq => ?_⟩\n    · rw [tendsto_atTop'] at h_tendsto\n      cases k with\n      | top => simp at hk_lt_i\n      | coe k =>\n        norm_cast at hk_lt_i ⊢\n        have h_nhds : Set.Ici k ∈ 𝓝 i :=\n          mem_nhds_iff.mpr ⟨Set.Ioi k, Set.Ioi_subset_Ici le_rfl, isOpen_Ioi, hk_lt_i⟩\n        obtain ⟨a, ha⟩ : ∃ a : ℕ, ∀ b : ℕ, b ≥ a → k ≤ seq b := h_tendsto (Set.Ici k) h_nhds\n        exact ⟨a, ha a le_rfl⟩\n    · obtain ⟨j, hk_seq_j⟩ := h_exists_k_le_seq\n      exact hk_seq_j.trans_lt (mod_cast h_bound j)\n  have h_lt_eq_preimage : {ω | τ ω < i} = τ ⁻¹' Set.Iio i := by\n    ext1 ω; push _ ∈ _; rfl\n  rw [h_lt_eq_preimage, h_Iio_eq_Union]\n  simp only [Set.preimage_iUnion, Set.preimage_setOf_eq]\n  exact MeasurableSet.iUnion fun n => f.mono (h_bound n).le _ (hτ.measurableSet_le (seq n))\n\ntheorem IsStoppingTime.measurableSet_lt (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω < i} := by\n  obtain ⟨i', hi'_lub⟩ : ∃ i', IsLUB (Set.Iio i) i' := exists_lub_Iio i\n  rcases lub_Iio_eq_self_or_Iio_eq_Iic i hi'_lub with hi'_eq_i | h_Iio_eq_Iic\n  · rw [← hi'_eq_i] at hi'_lub ⊢\n    exact hτ.measurableSet_lt_of_isLUB i' hi'_lub\n  · have h_lt_eq_preimage : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iio i := rfl\n    have h_Iio_eq_Iic' : Set.Iio (i : WithTop ι) = Set.Iic (i' : WithTop ι) := by\n      rw [← image_coe_Iio, ← image_coe_Iic, h_Iio_eq_Iic]\n    rw [h_lt_eq_preimage, h_Iio_eq_Iic']\n    exact f.mono (le_of_isLUB_Iio i hi'_lub) _ (hτ.measurableSet_le i')\n\ntheorem IsStoppingTime.measurableSet_ge (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | i ≤ τ ω} := by\n  have : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ := by\n    ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt]\n  rw [this]\n  exact (hτ.measurableSet_lt i).compl\n\ntheorem IsStoppingTime.measurableSet_eq (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω = i} := by\n  have : {ω | τ ω = i} = {ω | τ ω ≤ i} ∩ {ω | τ ω ≥ i} := by\n    ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_inter_iff, le_antisymm_iff]\n  rw [this]\n  exact (hτ.measurableSet_le i).inter (hτ.measurableSet_ge i)\n\ntheorem IsStoppingTime.measurableSet_eq_le (hτ : IsStoppingTime f τ) {i j : ι} (hle : i ≤ j) :\n    MeasurableSet[f j] {ω | τ ω = i} :=\n  f.mono hle _ <| hτ.measurableSet_eq i\n\ntheorem IsStoppingTime.measurableSet_lt_le (hτ : IsStoppingTime f τ) {i j : ι} (hle : i ≤ j) :\n    MeasurableSet[f j] {ω | τ ω < i} :=\n  f.mono hle _ <| hτ.measurableSet_lt i\n\nend TopologicalSpace\n\nend LinearOrder\n\nsection Countable\n\nTarget:\ntheorem isStoppingTime_of_measurableSet_eq [Preorder ι] [Countable ι] {f : Filtration ι m}\n    {τ : Ω → WithTop ι} (hτ : ∀ i, MeasurableSet[f i] {ω | τ ω = i}) : IsStoppingTime f τ :=\n\nProof body:\n","rejected":"by\n  exact isStoppingTime_of_measurableSet_eq","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"144990c5d793be30b9b90f94ecb488925dcf459e714ad2816a4dd8892b3f5831","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Process","family_id":"isstoppingtime_of_measurableset_eq","file_id":"mathlib/Mathlib/Probability/Process/Stopping.lean","sample_id":"63d4c6f7bdfaad78a31356f5390bcb2b6c5c4366f0f8ad12785fcd4685b1c7e7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"887cd45eb0ad0270c148f43d5c6fb5160031574c8c6c116689e7fb4bdff4ffd8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b26b39d9245fa55448b7c078ba618090e6feb2dd8d0cbd1bbea3ea947d8d976b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"00174d518458fef9a309864e49f69be9bd090110f25e5905b3ad9090d83909e4","source_sha256":"0558464e45d5ed116146fb004c344db832eef9939fa1268196e1a1a60376c0f3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases finite_or_infinite X with _ | _\n  · have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_right S X)\n    rwa [lift_umax.{v, u}, lift_id'] at this\n  have key (x : X) (s s' : S) (h : s • x = s' • x) (hc : Commute s s') : x /ₒ s = x /ₒ s' := by\n    rw [oreDiv_eq_iff]\n    refine ⟨s, s'.1, h, ?_⟩\n    · exact_mod_cast hc\n  let i (x : X × S) := x.1 /ₒ x.2\n  have hsurj : Surjective i := Quotient.mk''_surjective\n  have hi := rightInverse_surjInv hsurj\n  let j := (fun x : X × S ↦ (x.1, x.2 • x.1)) ∘ surjInv hsurj\n  suffices Injective j by\n    have := lift_mk_le_lift_mk_of_injective this\n    rwa [lift_umax.{v, u}, lift_id', mk_prod, lift_id, lift_mul, mul_eq_self (by simp)] at this\n  intro\n  grind","hard_negative":false,"metrics":{"chosen_tokens":190,"rejected_tokens":195,"token_jaccard":0.948052,"token_length_ratio":1.026316},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"3bd32c00b5c37bde1ffd4746746b5e3391cdbcf278af9ffbc3ede35edbc230f9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Pigeonhole\npublic import Mathlib.GroupTheory.OreLocalization.Basic\npublic import Mathlib.SetTheory.Cardinal.Arithmetic\n\nNamespace:\nOreLocalization\n\nLocal context:\n/-\nCopyright (c) 2024 Jz Pan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jz Pan\n-/\n/-!\n\n# Cardinality of Ore localizations\n\nThis file contains some results on cardinality of Ore localizations.\n\n## TODO\n\n- Prove or disprove `OreLocalization.cardinalMk_le_lift_cardinalMk_of_commute`\n  with `Commute` assumption removed.\n\n-/\n\npublic section\n\nuniverse u v\n\nopen Cardinal Function\n\nnamespace OreLocalization\n\nvariable {R : Type u} [Monoid R] (S : Submonoid R) [OreLocalization.OreSet S]\n  (X : Type v) [MulAction R X]\n\n@[to_additive]\ntheorem oreDiv_one_surjective_of_finite_left [Finite S] :\n    Surjective (fun x ↦ x /ₒ (1 : ↥S) : X → OreLocalization S X) := by\n  refine OreLocalization.ind fun x s ↦ ?_\n  obtain ⟨i, j, hne, heq⟩ := Finite.exists_ne_map_eq_of_infinite (α := ℕ) (s ^ ·)\n  wlog! hlt : j < i generalizing i j\n  · exact this j i hne.symm heq.symm (hne.lt_of_le hlt)\n  use s ^ (i - (j + 1)) • x\n  rw [oreDiv_eq_iff]\n  refine ⟨s ^ j, (s ^ (j + 1)).1, ?_, ?_⟩\n  · change s ^ j • x = s ^ (j + 1) • s ^ (i - (j + 1)) • x\n    rw [← mul_smul, ← pow_add, Nat.add_sub_cancel' hlt, heq]\n  · simp_rw [SubmonoidClass.coe_pow, OneMemClass.coe_one, mul_one, pow_succ]\n\n@[to_additive]\ntheorem oreDiv_one_surjective_of_finite_right [Finite X] :\n    Surjective (fun x ↦ x /ₒ (1 : ↥S) : X → OreLocalization S X) := by\n  refine OreLocalization.ind fun x s ↦ ?_\n  obtain ⟨i, j, hne, heq⟩ := Finite.exists_ne_map_eq_of_infinite (α := ℕ) (s ^ · • x)\n  wlog! hlt : j < i generalizing i j\n  · exact this j i hne.symm heq.symm (hne.lt_of_le hlt)\n  use s ^ (i - (j + 1)) • x\n  rw [oreDiv_eq_iff]\n  refine ⟨s ^ j, (s ^ (j + 1)).1, ?_, ?_⟩\n  · change s ^ j • x = s ^ (j + 1) • s ^ (i - (j + 1)) • x\n    rw [← mul_smul, ← pow_add, Nat.add_sub_cancel' hlt, heq]\n  · simp_rw [SubmonoidClass.coe_pow, OneMemClass.coe_one, mul_one, pow_succ]\n\n@[to_additive]\ntheorem numeratorHom_surjective_of_finite [Finite S] : Surjective (numeratorHom (S := S)) :=\n  oreDiv_one_surjective_of_finite_left S R\n\n@[to_additive]\ntheorem cardinalMk_le_max : #(OreLocalization S X) ≤ max (lift.{v} #S) (lift.{u} #X) := by\n  rcases finite_or_infinite X with _ | _\n  · have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_right S X)\n    rw [lift_umax.{v, u}, lift_id'] at this\n    exact le_max_of_le_right this\n  rcases finite_or_infinite S with _ | _\n  · have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_left S X)\n    rw [lift_umax.{v, u}, lift_id'] at this\n    exact le_max_of_le_right this\n  convert! ←\n    mk_le_of_surjective (show Surjective fun x : X × S ↦ x.1 /ₒ x.2 from Quotient.mk''_surjective)\n  rw [mk_prod, mul_comm]\n  refine mul_eq_max ?_ ?_ <;> simp\n\n@[to_additive]\ntheorem cardinalMk_le : #(OreLocalization S R) ≤ #R := by\n  convert! ← cardinalMk_le_max S R\n  simp_rw [lift_id, max_eq_right_iff, mk_subtype_le]\n\n-- TODO: remove the `Commute` assumption\n@[to_additive]\n\nTarget:\ntheorem cardinalMk_le_lift_cardinalMk_of_commute (hc : ∀ s s' : S, Commute s s') :\n    #(OreLocalization S X) ≤ lift.{u} #X :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  rcases finite_or_infinite X with _ | _\n  · have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_right S X)\n    rwa [lift_umax.{v, u}, lift_id'] at this\n  have key (x : X) (s s' : S) (h : s • x = s' • x) (hc : Commute s s') : x /ₒ s = x /ₒ s' := by\n    rw [oreDiv_eq_iff]\n    refine ⟨s, s'.1, h, ?_⟩\n    · exact_mod_cast hc\n  let i (x : X × S) := x.1 /ₒ x.2\n  have hsurj : Surjective i := Quotient.mk''_surjective\n  have hi := rightInverse_surjInv hsurj\n  let j := (fun x : X × S ↦ (x.1, x.2 • x.1)) ∘ surjInv hsurj\n  suffices Injective j by\n    have := lift_mk_le_lift_mk_of_injective this\n    rwa [lift_umax.{v, u}, lift_id', mk_prod, lift_id, lift_mul, mul_eq_self (by simp)] at this\n  intro\n  grind","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/OreLocalization","family_id":"cardinalmk_le_lift_cardinalmk_of_commute","file_id":"mathlib/Mathlib/GroupTheory/OreLocalization/Cardinality.lean","sample_id":"00174d518458fef9a309864e49f69be9bd090110f25e5905b3ad9090d83909e4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"212d1f84cff2b596f7c1f93804fd9c282408c326066650fb7feba4fd785cf41f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c49a0b60eabd3506186f70e85a5871422137df56b53553e203b445997183b81b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a53b7c1c6dc7380167b7eef2678e6391bfc432a0b3abcb53f2f3e41f881be729","source_sha256":"52cc50e04f911b3cb7014b8c73cd594d90bc999d3e9e3dfd7bf800868c589bb6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [max_def, val_le_val, ← apply_ite]\n  rfl","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":16,"token_jaccard":0.666667,"token_length_ratio":1.454545},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"3c047b19fcdba34445a6063b591186adf8ebf083cdb763379dca9c9e3743aadb","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Hom.Basic\npublic import Mathlib.Algebra.Group.Units.Defs\n\nNamespace:\nUnits\n\nLocal context:\n/-\nCopyright (c) 2016 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl\n-/\n/-!\n# Units in ordered monoids\n-/\n\n@[expose] public section\n\nnamespace Units\n\nvariable {α : Type*}\n\n@[to_additive]\ninstance [Monoid α] [Preorder α] : Preorder αˣ :=\n  Preorder.lift val\n\n@[to_additive (attr := simp, norm_cast)]\ntheorem val_le_val [Monoid α] [Preorder α] {a b : αˣ} : (a : α) ≤ b ↔ a ≤ b :=\n  Iff.rfl\n\n@[to_additive (attr := simp, norm_cast)]\ntheorem val_lt_val [Monoid α] [Preorder α] {a b : αˣ} : (a : α) < b ↔ a < b :=\n  Iff.rfl\n\n@[to_additive]\ninstance instPartialOrderUnits [Monoid α] [PartialOrder α] : PartialOrder αˣ :=\n  PartialOrder.lift val val_injective\n\n@[to_additive]\ninstance [Monoid α] [LinearOrder α] : Max αˣ where\n  max a b := if a ≤ b then b else a\n\n@[to_additive]\ninstance [Monoid α] [LinearOrder α] : Min αˣ where\n  min a b := if a ≤ b then a else b\n\n\n@[to_additive (attr := simp, norm_cast)]\n\nTarget:\ntheorem max_val [Monoid α] [LinearOrder α] (a b : αˣ) : (max a b).val = max a.val b.val :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  simp_rw [max_def, val_le_val, ← apply_ite]\n  rfl","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Order","family_id":"max_val","file_id":"mathlib/Mathlib/Algebra/Order/Monoid/Units.lean","sample_id":"a53b7c1c6dc7380167b7eef2678e6391bfc432a0b3abcb53f2f3e41f881be729"},"split":"train"}
{"binding":{"chosen_proof_sha256":"afd4fe6929e62277a458a32fdf80154f965674b8199603407ac172cf56934b1a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"feb0ac643b6d6e33049a45eaff6295a840fb3e4de3349a4fe12fdb819cc1ae9c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7b702b3f54a49b24809e03e0a017a431793ec768bf4e679fe9f430ed6393e5c4","source_sha256":"704b186061db0b8a3733d8cc4c73447096dc6713be40f982a89e46b27c980656","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine\n    le_trans ?_ <| ((Subobject.pullback (π(π₁, prod.map F.f (𝟙 _) ≫ G.F))).monotone F.le).trans <|\n    le_trans ?_ <| ((Subobject.pullback (prod.map F.f (𝟙 Z.tgt))).monotone G.le).trans ?_\n    <;> simp [← Subobject.pullback_comp]","hard_negative":true,"metrics":{"chosen_tokens":90,"rejected_tokens":3,"token_jaccard":0.027778,"token_length_ratio":0.033333},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"3c17ff7e5b9bfc7f99a0aef982097f9786d4b8bde18a9c5d705828b74dac9053","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Subobject.Basic\n\nNamespace:\nCategoryTheory.Dial\n\nLocal context:\n/-\nCopyright (c) 2024 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Dialectica category\n\nWe define the category `Dial` of the Dialectica interpretation, after [dialectica1989].\n\n## Background\n\nDialectica categories are important models of linear type theory. They satisfy most of the\ndistinctions that linear logic was meant to introduce and many models do not satisfy, like the\nindependence of constants. Many linear type theories are being used at the\nmoment--[nLab] describes some of them: for quantum systems, for effects in programming, for linear\ndependent types. In particular, dialectica categories are connected to polynomial functors, being a\nslightly more sophisticated version of polynomial types, as discussed, for instance, in Moss and\nvon Glehn's [*Dialectica models of type theory*]. As such they are related to the polynomial\nconstructions being [developed][Poly] by Awodey, Riehl, and Hazratpour. For the non-dependent\nversion developed here several applications are known to Petri Nets, small cardinals\nin Set Theory, state in imperative programming, and others, see [Dialectica Categories].\n\n## References\n\n* [Valeria de Paiva, The Dialectica Categories.][dialectica1989]\n  ([pdf](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-213.pdf))\n\n[nLab]: https://ncatlab.org/nlab/show/linear+type+theory\n[*Dialectica models of type theory*]: https://arxiv.org/abs/2105.00283\n[Poly]: https://github.com/sinhp/Poly\n[Dialectica Categories]: https://github.com/vcvpaiva/DialecticaCategories\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nnamespace CategoryTheory\n\nopen Limits\n\nuniverse v u\nvariable {C : Type u} [Category.{v} C] [HasFiniteProducts C] [HasPullbacks C]\n\nvariable (C) in\n/-- The Dialectica category. An object of the category is a triple `⟨U, X, α ⊆ U × X⟩`,\nand a morphism from `⟨U, X, α⟩` to `⟨V, Y, β⟩` is a pair `(f : U ⟶ V, F : U ⨯ Y ⟶ X)` such that\n`{(u,y) | α(u, F(u, y))} ⊆ {(u,y) | β(f(u), y)}`. The subset `α` is actually encoded as an element\nof `Subobject (U × X)`, and the above inequality is expressed using pullbacks. -/\nstructure Dial where\n  /-- The source object -/\n  src : C\n  /-- The target object -/\n  tgt : C\n  /-- A subobject of `src ⨯ tgt`, interpreted as a relation -/\n  rel : Subobject (src ⨯ tgt)\n\nnamespace Dial\n\nlocal notation \"π₁\" => prod.fst\nlocal notation \"π₂\" => prod.snd\nlocal notation \"π(\" a \", \" b \")\" => prod.lift a b\n\n/-- A morphism in the `Dial C` category from `⟨U, X, α⟩` to `⟨V, Y, β⟩` is a pair\n`(f : U ⟶ V, F : U ⨯ Y ⟶ X)` such that `{(u,y) | α(u, F(u, y))} ≤ {(u,y) | β(f(u), y)}`. -/\n@[ext] structure Hom (X Y : Dial C) where\n  /-- Maps the sources -/\n  f : X.src ⟶ Y.src\n  /-- Maps the targets (contravariantly) -/\n  F : X.src ⨯ Y.tgt ⟶ X.tgt\n  /-- This says `{(u, y) | α(u, F(u, y))} ⊆ {(u, y) | β(f(u), y)}` using subobject pullbacks -/\n  le :\n    (Subobject.pullback π(π₁, F)).obj X.rel ≤\n    (Subobject.pullback (prod.map f (𝟙 _))).obj Y.rel\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem comp_le_lemma {X Y Z : Dial C} (F : Dial.Hom X Y) (G : Dial.Hom Y Z) :\n    (Subobject.pullback π(π₁, π(π₁, prod.map F.f (𝟙 _) ≫ G.F) ≫ F.F)).obj X.rel ≤\n    (Subobject.pullback (prod.map (F.f ≫ G.f) (𝟙 Z.tgt))).obj Z.rel :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_7b702b3f54a4","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"4c2b0791f2aa28619c5d5f056ac2aed2e8ced281ff1d586aab6d5bcad2bfbbba","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Dialectica","family_id":"comp_le_lemma","file_id":"mathlib/Mathlib/CategoryTheory/Dialectica/Basic.lean","sample_id":"7b702b3f54a49b24809e03e0a017a431793ec768bf4e679fe9f430ed6393e5c4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3fa3b1641e3a223c1b1c2acb29844761f01c0429d5fdec1fe9c7e636fcc33dde","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e97cf30cee9ddef793712f354c924a5ad07437c6d2813b405d7ca3ad25f55874","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"717673c42ec9cf536eab52497d443d7e188d9ccddc0b2dc34242dbb48af892e4","source_sha256":"a29151d12db11777bb9d805b9535e76a5c2a8f2d2a791e6ae7bfa4c562e67f6b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  suffices h₁ : n ≤ sInf {m | p (m - n)} by\n    convert! sInf_add h₁\n    simp_rw [Nat.add_sub_cancel_right]\n  obtain ⟨m, hm⟩ := nonempty_of_pos_sInf h\n  refine\n    le_csInf ⟨m + n, ?_⟩ fun b hb ↦\n      le_of_not_gt fun hbn ↦\n        ne_of_mem_of_not_mem ?_ (notMem_of_lt_sInf h) (Nat.sub_eq_zero_of_le hbn.le)\n  · dsimp\n    rwa [Nat.add_sub_cancel_right]\n  · exact hb","hard_negative":true,"metrics":{"chosen_tokens":83,"rejected_tokens":3,"token_jaccard":0.038462,"token_length_ratio":0.036145},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"3c9e8ad53dd79d8ba619d1bbe1715a818d4944e9d3302209978a9f4fed85562b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ConditionallyCompleteLattice.Finset\npublic import Mathlib.Order.Interval.Finset.Nat\n\nNamespace:\nNat\n\nLocal context:\n/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Floris van Doorn, Gabriel Ebner, Yury Kudryashov\n-/\n/-!\n# Conditionally complete linear order structure on `ℕ`\n\nIn this file we\n\n* define a `ConditionallyCompleteLinearOrderBot` structure on `ℕ`;\n* prove a few lemmas about `iSup`/`iInf`/`Set.iUnion`/`Set.iInter` and natural numbers.\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero\n\nopen Set\n\nnamespace Nat\n\nopen scoped Classical in\nnoncomputable instance : InfSet ℕ :=\n  ⟨fun s ↦ if h : ∃ n, n ∈ s then @Nat.find (fun n ↦ n ∈ s) _ h else 0⟩\n\nopen scoped Classical in\nnoncomputable instance : SupSet ℕ :=\n  ⟨fun s ↦ if h : ∃ n, ∀ a ∈ s, a ≤ n then @Nat.find (fun n ↦ ∀ a ∈ s, a ≤ n) _ h else 0⟩\n\nopen scoped Classical in\ntheorem sInf_def {s : Set ℕ} (h : s.Nonempty) : sInf s = @Nat.find (fun n ↦ n ∈ s) _ h :=\n  dif_pos _\n\nopen scoped Classical in\ntheorem sSup_def {s : Set ℕ} (h : ∃ n, ∀ a ∈ s, a ≤ n) :\n    sSup s = @Nat.find (fun n ↦ ∀ a ∈ s, a ≤ n) _ h :=\n  dif_pos _\n\ntheorem _root_.Set.Infinite.Nat.sSup_eq_zero {s : Set ℕ} (h : s.Infinite) : sSup s = 0 :=\n  dif_neg fun ⟨n, hn⟩ ↦\n    let ⟨k, hks, hk⟩ := h.exists_gt n\n    (hn k hks).not_gt hk\n\ntheorem sSup_of_not_bddAbove {s : Set ℕ} (h : ¬BddAbove s) : sSup s = 0 :=\n  Set.Infinite.Nat.sSup_eq_zero <| Set.infinite_of_not_bddAbove h\n\nlemma iSup_of_not_bddAbove {ι : Sort*} {f : ι → ℕ} (h : ¬ BddAbove (Set.range f)) :\n    (⨆ i, f i : ℕ) = 0 := Nat.sSup_of_not_bddAbove h\n\n@[simp]\ntheorem sInf_eq_zero {s : Set ℕ} : sInf s = 0 ↔ 0 ∈ s ∨ s = ∅ := by\n  cases eq_empty_or_nonempty s with\n  | inl h => subst h\n             simp only [or_true, InfSet.sInf,\n                        mem_empty_iff_false, exists_false, dif_neg, not_false_iff]\n  | inr h => simp only [h.ne_empty, or_false, Nat.sInf_def, h, Nat.find_eq_zero]\n\n@[simp]\ntheorem sInf_empty : sInf ∅ = 0 := by\n  rw [sInf_eq_zero]\n  right\n  rfl\n\n@[simp]\ntheorem iInf_of_empty {ι : Sort*} [IsEmpty ι] (f : ι → ℕ) : iInf f = 0 := by\n  rw [iInf_of_isEmpty, sInf_empty]\n\n/-- This combines `Nat.iInf_of_empty` with `ciInf_const`. -/\n@[simp]\nlemma iInf_const_zero {ι : Sort*} : ⨅ _ : ι, 0 = 0 :=\n  (isEmpty_or_nonempty ι).elim (fun h ↦ by simp) fun h ↦ sInf_eq_zero.2 <| by simp\n\ntheorem sInf_mem {s : Set ℕ} (h : s.Nonempty) : sInf s ∈ s := by\n  classical\n  rw [Nat.sInf_def h]\n  exact Nat.find_spec h\n\ntheorem notMem_of_lt_sInf {s : Set ℕ} {m : ℕ} (hm : m < sInf s) : m ∉ s := by\n  classical\n  cases eq_empty_or_nonempty s with\n  | inl h => subst h; apply notMem_empty\n  | inr h => rw [Nat.sInf_def h] at hm; exact Nat.find_min h hm\n\nprotected theorem sInf_le {s : Set ℕ} {m : ℕ} (hm : m ∈ s) : sInf s ≤ m := by\n  classical\n  rw [Nat.sInf_def ⟨m, hm⟩]\n  exact Nat.find_min' ⟨m, hm⟩ hm\n\ntheorem nonempty_of_pos_sInf {s : Set ℕ} (h : 0 < sInf s) : s.Nonempty := by\n  by_contra contra\n  rw [Set.not_nonempty_iff_eq_empty] at contra\n  have h' : sInf s ≠ 0 := ne_of_gt h\n  apply h'\n  rw [Nat.sInf_eq_zero]\n  right\n  assumption\n\ntheorem nonempty_of_sInf_eq_succ {s : Set ℕ} {k : ℕ} (h : sInf s = k + 1) : s.Nonempty :=\n  nonempty_of_pos_sInf (h.symm ▸ succ_pos k : sInf s > 0)\n\ntheorem eq_Ici_of_nonempty_of_upward_closed {s : Set ℕ} (hs : s.Nonempty)\n    (hs' : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s) : s = Ici (sInf s) :=\n  ext fun n ↦ ⟨fun H ↦ Nat.sInf_le H, fun H ↦ hs' (sInf s) n H (sInf_mem hs)⟩\n\ntheorem sInf_upward_closed_eq_succ_iff {s : Set ℕ} (hs : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s)\n    (k : ℕ) : sInf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s := by\n  classical\n  constructor\n  · intro H\n    rw [eq_Ici_of_nonempty_of_upward_closed (nonempty_of_sInf_eq_succ _) hs, H, mem_Ici, mem_Ici]\n    · exact ⟨le_rfl, k.not_succ_le_self⟩\n    · exact k\n    · assumption\n  · rintro ⟨H, H'⟩\n    rw [sInf_def (⟨_, H⟩ : s.Nonempty), find_eq_iff]\n    exact ⟨H, fun n hnk hns ↦ H' <| hs n k (Nat.lt_succ_iff.mp hnk) hns⟩\n\n/-- This instance is necessary, otherwise the lattice operations would be derived via\n`ConditionallyCompleteLinearOrderBot` and marked as noncomputable. -/\ninstance : Lattice ℕ :=\n  LinearOrder.toLattice\n\nopen scoped Classical in\nnoncomputable instance : ConditionallyCompleteLinearOrderBot ℕ :=\n  { (inferInstance : OrderBot ℕ), (LinearOrder.toLattice : Lattice ℕ),\n    (inferInstance : LinearOrder ℕ) with\n    isLUB_csSup _ hn hb := sSup_def hb ▸ Nat.isLeast_find hb\n    isGLB_csInf _ hn hb := sInf_def hn ▸ (Nat.isLeast_find hn).isGLB\n    csSup_empty := by\n      simp only [sSup_def, Set.mem_empty_iff_false, forall_const, forall_prop_of_false,\n        not_false_iff, exists_const]\n      apply bot_unique (Nat.find_min' _ _)\n      trivial\n    csSup_of_not_bddAbove := by\n      intro s hs\n      simp only [sSup,\n        mem_empty_iff_false, IsEmpty.forall_iff, forall_const, exists_const, dite_true]\n      rw [dif_neg]\n      · exact le_antisymm (zero_le _) (find_le trivial)\n      · exact hs\n    csInf_of_not_bddBelow := fun s hs ↦ by simp at hs }\n\ntheorem sSup_mem {s : Set ℕ} (h₁ : s.Nonempty) (h₂ : BddAbove s) : sSup s ∈ s :=\n  let ⟨k, hk⟩ := h₂\n  h₁.csSup_mem ((finite_le_nat k).subset hk)\n\ntheorem sInf_add {n : ℕ} {p : ℕ → Prop} (hn : n ≤ sInf { m | p m }) :\n    sInf { m | p (m + n) } + n = sInf { m | p m } := by\n  classical\n  obtain h | ⟨m, hm⟩ := { m | p (m + n) }.eq_empty_or_nonempty\n  · rw [h, Nat.sInf_empty, zero_add]\n    obtain hnp | hnp := hn.eq_or_lt\n    · exact hnp\n    suffices hp : p (sInf { m | p m } - n + n) from (h.subset hp).elim\n    rw [Nat.sub_add_cancel hn]\n    exact csInf_mem (nonempty_of_pos_sInf <| n.zero_le.trans_lt hnp)\n  · have hp : ∃ n, n ∈ { m | p m } := ⟨_, hm⟩\n    rw [Nat.sInf_def ⟨m, hm⟩, Nat.sInf_def hp]\n    rw [Nat.sInf_def hp] at hn\n    exact find_add hn\n\nTarget:\ntheorem sInf_add' {n : ℕ} {p : ℕ → Prop} (h : 0 < sInf { m | p m }) :\n    sInf { m | p m } + n = sInf { m | p (m - n) } :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_717673c42ec9","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"65ae4901ffba7d5e92ccaaa270adc5cc0e665aa90f4c92ece65c19f6c6b6dbee","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Lattice","family_id":"sinf_add","file_id":"mathlib/Mathlib/Order/Lattice/Nat.lean","sample_id":"717673c42ec9cf536eab52497d443d7e188d9ccddc0b2dc34242dbb48af892e4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"801cfb0a412f7e9f521af786c135901e8a7681f3edb0eee3dd6e24550333a788","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"786cb7a4f6894264ed36b3120e49a200184d139c5431a2fefd4dcc4302b7110b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fcb3cb79819283ab318174f307cb21c4367decc586f83a6b3fb969ef48ad9494","source_sha256":"ba493885a2be935359e1e232e1bbaeb68ad80f382975839ead7d3190a1fce95f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · obtain a | a := a <;> obtain b | b := b\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (mem_disjSum.1 h).elim ?_ ?_\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inr <| Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr (Or.inr <| Or.inr <| ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n  · rintro (⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩ |\n      ⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact mem_map_of_mem _ hc\n    · exact inl_mem_disjSum.2 hc\n    · exact inr_mem_disjSum.2 hc\n    · exact mem_map_of_mem _ hc","hard_negative":false,"metrics":{"chosen_tokens":287,"rejected_tokens":291,"token_jaccard":0.93617,"token_length_ratio":1.013937},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"3cb5651472f70e5a2555b16bdf02f2a3ff2b28e2dd1e0d9c0e6422b5fd393164","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Sum\npublic import Mathlib.Data.Sum.Order\npublic import Mathlib.Order.Interval.Finset.Defs\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Finite intervals in a disjoint union\n\nThis file provides the `LocallyFiniteOrder` instance for the disjoint sum and linear sum of two\norders and calculates the cardinality of their finite intervals.\n-/\n\n@[expose] public section\n\n\nopen Function Sum\n\nnamespace Finset\n\nvariable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}\n\nsection SumLift₂\n\nvariable (f f₁ g₁ : α₁ → β₁ → Finset γ₁) (g f₂ g₂ : α₂ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁` and `α₂ → β₂ → Finset γ₂` to a map\n`α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to `Alternative` functors if we can\nmake sure to keep computability and universe polymorphism. -/\n@[simp]\ndef sumLift₂ : ∀ (_ : α₁ ⊕ α₂) (_ : β₁ ⊕ β₂), Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f a b).map Embedding.inl\n  | inl _, inr _ => ∅\n  | inr _, inl _ => ∅\n  | inr a, inr b => (g a b).map Embedding.inr\n\nvariable {f f₁ g₁ g f₂ g₂} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\ntheorem mem_sumLift₂ :\n    c ∈ sumLift₂ f g a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨\n        ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g a₂ b₂ := by\n  constructor\n  · rcases a with a | a <;> rcases b with b | b\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr ⟨a, b, c, rfl, rfl, rfl, hc⟩\n  · rintro (⟨a, b, c, rfl, rfl, rfl, h⟩ | ⟨a, b, c, rfl, rfl, rfl, h⟩) <;> exact mem_map_of_mem _ h\n\ntheorem inl_mem_sumLift₂ {c₁ : γ₁} :\n    inl c₁ ∈ sumLift₂ f g a b ↔ ∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f a₁ b₁ := by\n  rw [mem_sumLift₂, or_iff_left]\n  · simp only [inl.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inl_ne_inr h\n\ntheorem inr_mem_sumLift₂ {c₂ : γ₂} :\n    inr c₂ ∈ sumLift₂ f g a b ↔ ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ g a₂ b₂ := by\n  rw [mem_sumLift₂, or_iff_right]\n  · simp only [inr.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inr_ne_inl h\n\ntheorem sumLift₂_eq_empty :\n    sumLift₂ f g a b = ∅ ↔\n      (∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f a₁ b₁ = ∅) ∧\n        ∀ a₂ b₂, a = inr a₂ → b = inr b₂ → g a₂ b₂ = ∅ := by\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · constructor <;>\n    · rintro a b rfl rfl\n      exact map_eq_empty.1 h\n  cases a <;> cases b\n  · exact map_eq_empty.2 (h.1 _ _ rfl rfl)\n  · rfl\n  · rfl\n  · exact map_eq_empty.2 (h.2 _ _ rfl rfl)\n\ntheorem sumLift₂_nonempty :\n    (sumLift₂ f g a b).Nonempty ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ (f a₁ b₁).Nonempty) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ (g a₂ b₂).Nonempty := by\n  simp only [nonempty_iff_ne_empty, Ne, sumLift₂_eq_empty, not_and_or, not_forall, exists_prop]\n\ntheorem sumLift₂_mono (h₁ : ∀ a b, f₁ a b ⊆ g₁ a b) (h₂ : ∀ a b, f₂ a b ⊆ g₂ a b) :\n    ∀ a b, sumLift₂ f₁ f₂ a b ⊆ sumLift₂ g₁ g₂ a b\n  | inl _, inl _ => map_subset_map.2 (h₁ _ _)\n  | inl _, inr _ => Subset.rfl\n  | inr _, inl _ => Subset.rfl\n  | inr _, inr _ => map_subset_map.2 (h₂ _ _)\n\nend SumLift₂\n\nsection SumLexLift\nvariable (f₁ f₁' : α₁ → β₁ → Finset γ₁) (f₂ f₂' : α₂ → β₂ → Finset γ₂)\n  (g₁ g₁' : α₁ → β₂ → Finset γ₁) (g₂ g₂' : α₁ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁`, `α₂ → β₂ → Finset γ₂`, `α₁ → β₂ → Finset γ₁`,\n`α₂ → β₂ → Finset γ₂`  to a map `α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to\nalternative monads if we can make sure to keep computability and universe polymorphism. -/\ndef sumLexLift : α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f₁ a b).map Embedding.inl\n  | inl a, inr b => (g₁ a b).disjSum (g₂ a b)\n  | inr _, inl _ => ∅\n  | inr a, inr b => (f₂ a b).map ⟨_, inr_injective⟩\n\n@[simp]\nlemma sumLexLift_inl_inl (a : α₁) (b : β₁) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inl b) = (f₁ a b).map Embedding.inl := rfl\n\n@[simp]\nlemma sumLexLift_inl_inr (a : α₁) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b) = (g₁ a b).disjSum (g₂ a b) := rfl\n\n@[simp]\nlemma sumLexLift_inr_inl (a : α₂) (b : β₁) : sumLexLift f₁ f₂ g₁ g₂ (inr a) (inl b) = ∅ := rfl\n\n@[simp]\nlemma sumLexLift_inr_inr (a : α₂) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inr a) (inr b) = (f₂ a b).map ⟨_, inr_injective⟩ := rfl\n\nvariable {f₁ g₁ f₂ g₂ f₁' g₁' f₂' g₂'} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\nTarget:\nlemma mem_sumLexLift :\n    c ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        (∃ a₁ b₂ c₁, a = inl a₁ ∧ b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨\n          (∃ a₁ b₂ c₂, a = inl a₁ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n            ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂ :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  constructor\n  · obtain a | a := a <;> obtain b | b := b\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (mem_disjSum.1 h).elim ?_ ?_\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inr <| Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr (Or.inr <| Or.inr <| ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n  · rintro (⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩ |\n      ⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact mem_map_of_mem _ hc\n    · exact inl_mem_disjSum.2 hc\n    · exact inr_mem_disjSum.2 hc\n    · exact mem_map_of_mem _ hc","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sum","family_id":"mem_sumlexlift","file_id":"mathlib/Mathlib/Data/Sum/Interval.lean","sample_id":"fcb3cb79819283ab318174f307cb21c4367decc586f83a6b3fb969ef48ad9494"},"split":"train"}
{"binding":{"chosen_proof_sha256":"faf07d4b9f81d896a4dedf3d8bef12a774f393144e0c5f30d0f0b0458195b2b5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9aed89b367ed3619b282b85e2e0b0b35016d47c70362c6487609356cd5adeaf2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"332d57c10825bff92a5f9c100471ccaa87a97a30c29a674c9e1339e0c80d29bd","source_sha256":"74b6571c8b4ffa69041159c6a44858bd2f705cc61f45fce1f1c46bf3161b1e36","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  introv W' hf hg hh hi w\n  simpa only [IsPushout.flip_iff, IsPullback.flip_iff, and_comm] using\n    H' g' f' i' h' αW αY αX αZ hg hf hi hh w.flip","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":7,"token_jaccard":0.125,"token_length_ratio":0.166667},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"3cc62d4238d5417ca23d555d365d225f5c526a10344d2df48da1c03bcc8e2fd1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Extensive\npublic import Mathlib.CategoryTheory.Limits.Shapes.KernelPair\npublic import Mathlib.CategoryTheory.Limits.Constructions.EpiMono\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang, Jack McKoen\n-/\n/-!\n\n# Adhesive categories\n\n## Main definitions\n- `CategoryTheory.IsPushout.IsVanKampen`: A convenience formulation for a pushout being\n  a van Kampen colimit.\n- `CategoryTheory.Adhesive`: A category is adhesive if it has pushouts and pullbacks along\n  monomorphisms, and such pushouts are van Kampen.\n\n## Main Results\n- `CategoryTheory.Type.adhesive`: The category of `Type` is adhesive.\n- `CategoryTheory.Adhesive.isPullback_of_isPushout_of_mono_left`: In adhesive categories,\n  pushouts along monomorphisms are pullbacks.\n- `CategoryTheory.Adhesive.mono_of_isPushout_of_mono_left`: In adhesive categories,\n  monomorphisms are stable under pushouts.\n- `CategoryTheory.Adhesive.toRegularMonoCategory`: Monomorphisms in adhesive categories are\n  regular (this implies that adhesive categories are balanced).\n- `CategoryTheory.adhesive_functor`: The category `C ⥤ D` is adhesive if `D`\n  has all pullbacks and all pushouts and is adhesive\n\n## References\n- https://ncatlab.org/nlab/show/adhesive+category\n- [Stephen Lack and Paweł Sobociński, Adhesive Categories][adhesive2004]\n\n-/\n\n@[expose] public section\n\n\nnamespace CategoryTheory\n\nopen Limits\n\nuniverse v' u' v u\n\nvariable {J : Type v'} [Category.{u'} J] {C : Type u} [Category.{v} C]\nvariable {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z}\n\n-- This only makes sense when the original diagram is a pushout.\n/-- A convenient formulation for a pushout being a van Kampen colimit. For any commutative cube of\nwhich a van Kampen pushout forms the bottom face and the back faces are pullbacks, the front faces\nare pullbacks if and only if the top face is a pushout. See `IsPushout.isVanKampen_iff` below. -/\n@[nolint unusedArguments]\ndef IsPushout.IsVanKampen (_ : IsPushout f g h i) : Prop :=\n  ∀ ⦃W' X' Y' Z' : C⦄ (f' : W' ⟶ X') (g' : W' ⟶ Y') (h' : X' ⟶ Z') (i' : Y' ⟶ Z') (αW : W' ⟶ W)\n    (αX : X' ⟶ X) (αY : Y' ⟶ Y) (αZ : Z' ⟶ Z) (_ : IsPullback f' αW αX f)\n    (_ : IsPullback g' αW αY g) (_ : CommSq h' αX αZ h) (_ : CommSq i' αY αZ i)\n    (_ : CommSq f' g' h' i'), IsPushout f' g' h' i' ↔ IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i\n\n/-- If a van Kampen pushout forms the bottom face of a commutative \"half-cube\" whose front faces\nare pullbacks, then there exist two back faces which are pullbacks and a top face which is a\npushout. -/\nlemma IsPushout.IsVanKampen.exists_cube_filling {H : IsPushout f g h i} (H' : H.IsVanKampen)\n    {X' Y' Z' : C} {h' : X' ⟶ Z'} {i' : Y' ⟶ Z'} {αX : X' ⟶ X} {αY : Y' ⟶ Y} {αZ : Z' ⟶ Z}\n    [HasPullback αX f] (hh : IsPullback h' αX αZ h) (hi : IsPullback i' αY αZ i) :\n    ∃ (W' : C) (f' : W' ⟶ X') (g' : W' ⟶ Y') (αW : W' ⟶ W),\n      IsPullback f' αW αX f ∧ IsPullback g' αW αY g ∧ IsPushout f' g' h' i' := by\n  let l := hi.lift ((pullback.fst αX f) ≫ h') ((pullback.snd αX f) ≫ g)\n    (by simp only [Category.assoc, hh.toCommSq.w, pullback.condition_assoc, ← H.w])\n  use (pullback αX f), (pullback.fst αX f), l, (pullback.snd αX f)\n  refine ⟨IsPullback.of_hasPullback αX f, ?_, ?_⟩\n  · refine IsPullback.of_right' ?_ hi\n    rw [← H.w]\n    exact IsPullback.paste_horiz (IsPullback.of_hasPullback αX f) hh\n  · refine (H' (pullback.fst αX f) l h' i' (pullback.snd αX f) αX αY αZ\n      (IsPullback.of_hasPullback αX f) ?_\n        hh.toCommSq hi.toCommSq ⟨by simp only [IsPullback.lift_fst, l]⟩).2 ⟨hh, hi⟩\n    · refine IsPullback.of_right' ?_ hi\n      rw [← H.w]\n      exact IsPullback.paste_horiz (IsPullback.of_hasPullback αX f) hh\n\nTarget:\ntheorem IsPushout.IsVanKampen.flip {H : IsPushout f g h i} (H' : H.IsVanKampen) :\n    H.flip.IsVanKampen :=\n\nProof body:\n","rejected":"by\n  exact IsPushout.IsVanKampen.flip","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"b1d49eaa2ed29b14dcc02cf4b5722a5b80f1e2e4d2685c5083d7d8643a95c268","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Adhesive","family_id":"ispushout","file_id":"mathlib/Mathlib/CategoryTheory/Adhesive/Basic.lean","sample_id":"332d57c10825bff92a5f9c100471ccaa87a97a30c29a674c9e1339e0c80d29bd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"65460242af6486d44c1e55ce08a5b548dbca87ef42697a0dbcd7738054ac5b9c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0864def967194569451c916a4bd0e8926491ddb099f124674791947f7d0a7f93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8415a8306797307a1399dfa46e7bc4bffb6c158dd15399817b536ab5989fba40","source_sha256":"79526dcfa55da9c5f7685aed6cead7ebb3b44c830f8cd589df3db791b7c151e7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  convert_to Tendsto (f ∘ (fun x ↦ c + x) ∘ Neg.neg ∘ Inv.inv) atTop l at h\n  · ext\n    simp [AddGroupWithOne.sub_eq_add_neg]\n  simpa [Tendsto, ← Filter.map_map] using h","hard_negative":true,"metrics":{"chosen_tokens":46,"rejected_tokens":2,"token_jaccard":0.028571,"token_length_ratio":0.043478},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"3ce4b3fbf4b32ed258d9807f73f0b5fae91570a0605e4acb716eddecab76db59","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Algebra.Order.Field\npublic import Mathlib.Topology.Maps.Basic\npublic import Mathlib.Analysis.Asymptotics.Lemmas\n\nNamespace:\nTactic.ComputeAsymptotics\n\nLocal context:\n/-\nCopyright (c) 2026 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\n/-!\n# Conversion lemmas\n\nThe main procedure of the `compute_asymptotics` tactic is able to compute limits of functions at\n`atTop` filter. This file contains lemmas we use to reduce other asymptotic goals to\nthe case `Tendsto f atTop l`.\n\n## Main theorems\n\nThis file contains the following lemmas:\n* `tendsto_nhdsGT_of_tendsto_atTop` for `Tendsto f (𝓝[>] c) l`\n* `tendsto_nhdsLT_of_tendsto_atTop` for `Tendsto f (𝓝[<] c) l`\n* `tendsto_nhdsNE_of_tendsto_atTop` for `Tendsto f (𝓝[≠] c) l`\n* `isBigO_of_div_tendsto_atTop` and `isBigO_of_div_tendsto_atBot` for `f =O[l] g`\n\nWe also use lemmas from other files:\n* `tendsto_comp_neg_atTop_iff` for `Tendsto f atBot l`\n* `IsLittleO.of_tendsto_div_atBot` and `IsLittleO.of_tendsto_div_atTop` for `f =o[l] g`\n* `isEquivalent_of_tendsto_one` for `f ∼ g`\n-/\n\npublic section\n\nopen Filter Topology Asymptotics\n\nnamespace Tactic.ComputeAsymptotics\n\nvariable {α 𝕜 : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [TopologicalSpace 𝕜]\n  [OrderTopology 𝕜] {l : Filter α} (f : 𝕜 → α) (c : 𝕜)\n\ntheorem tendsto_nhdsGT_of_tendsto_atTop (h : Tendsto (fun x ↦ f (c + x⁻¹)) atTop l) :\n    Tendsto f (𝓝[>] c) l := by\n  simpa [← Function.comp_def, Tendsto, ← Filter.map_map] using h\n\nTarget:\ntheorem tendsto_nhdsLT_of_tendsto_atTop (h : Tendsto (fun x ↦ f (c - x⁻¹)) atTop l) :\n    Tendsto f (𝓝[<] c) l :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_8415a8306797","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"60380dc2b99cd141789ee90f0acaca100e1fd4773b7102e1287d29e151a4a27a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/ComputeAsymptotics","family_id":"tendsto_nhdslt_of_tendsto_attop","file_id":"mathlib/Mathlib/Tactic/ComputeAsymptotics/Lemmas.lean","sample_id":"8415a8306797307a1399dfa46e7bc4bffb6c158dd15399817b536ab5989fba40"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d4e8af4e9759204cb08f477f781bb80c363e8cafacffe36b1eb6bdb2061a0215","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a5523b6e9518ffe55d198ff8a834dd6444ded26e2b428ae0b590b1ebf7c42899","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4cb237f6549cf75bf0312f10927aa64c8d7dbadae8e86cca221ffb114034bffb","source_sha256":"5a984bf431d6ea343adf1846001af32b3b0dc44766966d2864ca4d0d545aa12b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨?_, Submodule.projectionOnto_comp_surjective_of_exact hfg _ hmap⟩\n  rwa [LinearMap.coe_comp, Set.InjOn.injective_iff ↑(LinearMap.range f) _ subset_rfl]\n  simpa [← LinearMap.disjoint_ker_iff_injOn, ← hfg.linearMap_ker_eq]","hard_negative":true,"metrics":{"chosen_tokens":46,"rejected_tokens":5,"token_jaccard":0.09375,"token_length_ratio":0.108696},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"3cf9f2e4907291de3b141b90b5c812818f902ddc22515d404dbebeeee9470552","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Exact.Basic\npublic import Mathlib.LinearAlgebra.Basis.Basic\npublic import Mathlib.LinearAlgebra.Projection\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten\n-/\n/-!\n# Basis from a split exact sequence\n\nLet `0 → K → M → P → 0` be a split exact sequence of `R`-modules, let `s : M → K` be a\nretraction of `f` and `v` be a basis of `M` indexed by `κ ⊕ σ`. Then\nif `s vᵢ = 0` for `i : κ` and `(s vⱼ)ⱼ` is linear independent for `j : σ`, then\nthe images of `vᵢ` for `i : κ` form a basis of `P`.\n\nWe treat linear independence and the span condition separately. For convenience this\nis stated not for `κ ⊕ σ`, but for an arbitrary type `ι` with two maps `κ → ι` and `σ → ι`.\n-/\n\n@[expose] public section\n\nvariable {R M K P : Type*} [Ring R] [AddCommGroup M] [AddCommGroup K] [AddCommGroup P]\nvariable [Module R M] [Module R K] [Module R P]\nvariable {f : K →ₗ[R] M} {g : M →ₗ[R] P} {s : M →ₗ[R] K}\nvariable (hs : s ∘ₗ f = LinearMap.id) (hfg : Function.Exact f g)\nvariable {ι κ σ : Type*} {v : ι → M} {a : κ → ι} {b : σ → ι}\n\nsection\ninclude hs hfg\n\nlemma LinearIndependent.linearIndependent_of_exact_of_retraction\n    (hainj : Function.Injective a) (hsa : ∀ i, s (v (a i)) = 0)\n    (hli : LinearIndependent R v) :\n    LinearIndependent R (g ∘ v ∘ a) := by\n  apply (LinearIndependent.comp hli a hainj).map\n  rw [Submodule.disjoint_def, hfg.linearMap_ker_eq]\n  rintro - hy ⟨y, rfl⟩\n  have hz : s (f y) = 0 := by\n    revert hy\n    generalize f y = x\n    intro hy\n    induction hy using Submodule.span_induction with\n    | mem m hm => obtain ⟨i, rfl⟩ := hm; apply hsa\n    | zero => simp_all\n    | add => simp_all\n    | smul => simp_all\n  replace hs := DFunLike.congr_fun hs y\n  simp only [LinearMap.coe_comp, Function.comp_apply, LinearMap.id_coe, id_eq] at hs\n  rw [← hs, hz, map_zero]\n\nprivate lemma top_le_span_of_aux (v : κ ⊕ σ → M)\n    (hg : Function.Surjective g) (hslzero : ∀ i, s (v (.inl i)) = 0)\n    (hli : LinearIndependent R (s ∘ v ∘ .inr)) (hsp : ⊤ ≤ Submodule.span R (Set.range v)) :\n    ⊤ ≤ Submodule.span R (Set.range <| g ∘ v ∘ .inl) := by\n  rintro p -\n  obtain ⟨m, rfl⟩ := hg p\n  wlog h : m ∈ LinearMap.ker s\n  · let x : M := f (s m)\n    rw [show g m = g (m - f (s m)) by simp [hfg.apply_apply_eq_zero]]\n    apply this hs hfg v hg hslzero hli hsp\n    replace hs := DFunLike.congr_fun hs (s m)\n    simp only [LinearMap.coe_comp, Function.comp_apply, LinearMap.id_coe, id_eq] at hs\n    simp [hs]\n  have : m ∈ Submodule.span R (Set.range v) := hsp Submodule.mem_top\n  obtain ⟨c, rfl⟩ := Finsupp.mem_span_range_iff_exists_finsupp.mp this\n  simp only [LinearMap.mem_ker, Finsupp.sum, map_sum, map_smul,\n    Finset.sum_sum_eq_sum_toLeft_add_sum_toRight, map_add, hslzero, smul_zero,\n    Finset.sum_const_zero, zero_add] at h\n  replace hli := (linearIndependent_iff'.mp hli) c.support.toRight (c ∘ .inr) h\n  simp only [Finset.mem_toRight, Finsupp.mem_support_iff, Function.comp_apply, not_imp_self] at hli\n  simp only [Finsupp.sum, Finset.sum_sum_eq_sum_toLeft_add_sum_toRight, hli, zero_smul,\n    Finset.sum_const_zero, add_zero, map_sum, map_smul]\n  exact Submodule.sum_mem _ (fun i hi ↦ Submodule.smul_mem _ _ <| Submodule.subset_span ⟨i, rfl⟩)\n\nlemma Submodule.top_le_span_of_exact_of_retraction (hg : Function.Surjective g)\n    (hsa : ∀ i, s (v (a i)) = 0) (hlib : LinearIndependent R (s ∘ v ∘ b))\n    (hab : Codisjoint (Set.range a) (Set.range b))\n    (hsp : ⊤ ≤ Submodule.span R (Set.range v)) :\n    ⊤ ≤ Submodule.span R (Set.range <| g ∘ v ∘ a) := by\n  apply top_le_span_of_aux hs hfg (Sum.elim (v ∘ a) (v ∘ b)) hg hsa hlib\n  simp only [codisjoint_iff, Set.sup_eq_union, Set.top_eq_univ] at hab\n  rwa [Set.Sum.elim_range, Set.range_comp, Set.range_comp, ← Set.image_union, hab, Set.image_univ]\n\n/-- Let `0 → K → M → P → 0` be a split exact sequence of `R`-modules, let `s : M → K` be a\nretraction of `f` and `v` be a basis of `M` indexed by `κ ⊕ σ`. Then\nif `s vᵢ = 0` for `i : κ` and `(s vⱼ)ⱼ` is linear independent for `j : σ`, then\nthe images of `vᵢ` for `i : κ` form a basis of `P`.\n\nFor convenience this is stated for an arbitrary type `ι` with two maps `κ → ι` and `σ → ι`. -/\nnoncomputable def Module.Basis.ofSplitExact (hg : Function.Surjective g) (v : Basis ι R M)\n    (hainj : Function.Injective a) (hsa : ∀ i, s (v (a i)) = 0)\n    (hlib : LinearIndependent R (s ∘ v ∘ b))\n    (hab : Codisjoint (Set.range a) (Set.range b)) :\n    Basis κ R P :=\n  .mk (v.linearIndependent.linearIndependent_of_exact_of_retraction hs hfg hainj hsa)\n    (Submodule.top_le_span_of_exact_of_retraction hs hfg hg hsa hlib hab (by rw [v.span_eq]))\n\n@[simp]\nlemma Module.Basis.ofSplitExact_apply (hg : Function.Surjective g) (v : Basis ι R M)\n    (hainj : Function.Injective a) (hsa : ∀ i, s (v (a i)) = 0)\n    (hlib : LinearIndependent R (s ∘ v ∘ b))\n    (hab : Codisjoint (Set.range a) (Set.range b)) (k : κ) :\n    ofSplitExact hs hfg hg v hainj hsa hlib hab k = g (v (a k)) := by\n  simp [ofSplitExact]\n\nend\n\nsection\ninclude hfg\n\nlemma Submodule.projectionOnto_comp_surjective_of_exact\n    {p q : Submodule R M} (hpq : IsCompl p q)\n    (hmap : Submodule.map g q = ⊤) :\n    Function.Surjective (Submodule.projectionOnto p q hpq ∘ₗ f) := by\n  rw [← Set.surjOn_univ, LinearMap.coe_comp, Set.surjOn_comp_iff, Set.image_univ]\n  rw [← LinearMap.coe_range, ← Submodule.top_coe (R := R), surjOn_iff_le_map,\n    ← hfg.linearMap_ker_eq]\n  intro x triv\n  obtain ⟨a, haq, ha⟩ : g x.val ∈ q.map g := by rwa [hmap]\n  exact ⟨x - a, by simp [← ha], by simpa⟩\n\n@[deprecated (since := \"2026-05-05\")] alias\n  Submodule.linearProjOfIsCompl_comp_surjective_of_exact :=\n  Submodule.projectionOnto_comp_surjective_of_exact\n\nTarget:\nlemma Submodule.projectionOnto_comp_bijective_of_exact\n    (hf : Function.Injective f) {p q : Submodule R M} (hpq : IsCompl p q)\n    (hker : Disjoint (LinearMap.ker g) q) (hmap : Submodule.map g q = ⊤) :\n    Function.Bijective (Submodule.projectionOnto p q hpq ∘ₗ f) :=\n\nProof body:\n","rejected":"by\n  exact Submodule.projectionOnto_comp_bijective_of_exact","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"2ef5cf11c17f87bf0a7179215c448c7be9f9c4cf08ba0f4505033356cb16a365","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Basis","family_id":"submodule","file_id":"mathlib/Mathlib/LinearAlgebra/Basis/Exact.lean","sample_id":"4cb237f6549cf75bf0312f10927aa64c8d7dbadae8e86cca221ffb114034bffb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9f76daa2d17794583c7f31dfaabd84b49dfce6e55e1f6c203ac803f9bd29220e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f212eda5800dde28c047e59ae432b8c938197bd68af0cd5502133a63cb13c83a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1e270d1bd1eeb14c5bf62361efc62c628e247ff0c8b0d9ddc26f734fc3b353a7","source_sha256":"1c0dcf6a3ce55bec7a88672314ea6ae6741a6599903854de40abb691f935e16b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rw [← subAt_pullSub S x]; gcongr\n  mpr _ := le_trans (by gcongr) (pullSub_subAt T x)","hard_negative":true,"metrics":{"chosen_tokens":23,"rejected_tokens":3,"token_jaccard":0.05,"token_length_ratio":0.130435},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"3cfc69799a4ef209d786f1ad1b775559aaae79ceced7a39346b2693969cd045f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.CompleteLattice.SetLike\n\nNamespace:\nDescriptive.Tree\n\nLocal context:\n/-\nCopyright (c) 2024 Sven Manthe. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sven Manthe\n-/\n/-!\n# Trees in the sense of descriptive set theory\n\nThis file defines trees of depth `ω` in the sense of descriptive set theory as sets of finite\nsequences that are stable under taking prefixes.\n\n## Main declarations\n\n* `tree A`: a (possibly infinite) tree of depth at most `ω` with nodes in `A`\n-/\n\n@[expose] public section\n\nnamespace Descriptive\n\n/-- A tree is a set of finite sequences, implemented as `List A`, that is stable under\n  taking prefixes. For the definition we use the equivalent property `x ++ [a] ∈ T → x ∈ T`,\n  which is more convenient to check. We define `tree A` as a complete sublattice of\n  `Set (List A)`, which coerces to the type of trees on `A`. -/\ndef tree (A : Type*) : CompleteSublattice (Set (List A)) :=\n  CompleteSublattice.mk' {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T}\n    (by rintro S hS x a ⟨t, ht, hx⟩; use t, ht, hS ht hx)\n    (by rintro S hS x a h T hT; exact hS hT <| h T hT)\n\n@[simps!] instance (A : Type*) : SetLike (tree A) (List A) := SetLike.instSubtypeSet\n\nexample (A : Type*) : PartialOrder (tree A) := inferInstance\n\nnamespace Tree\nvariable {A : Type*} {S T : tree A}\n\nlemma mem_of_append {x y : List A} (h : x ++ y ∈ T) : x ∈ T := by\n  induction y generalizing x with\n  | nil => simpa using h\n  | cons y ys ih => exact T.prop (ih (by simpa))\n\nlemma mem_of_prefix {x y : List A} (h' : x <+: y) (h : y ∈ T) : x ∈ T := by\n  obtain ⟨_, rfl⟩ := h'; exact mem_of_append h\n\ninstance : Trans List.IsPrefix (fun x (T : tree A) ↦ x ∈ T) (fun x T ↦ x ∈ T) where\n  trans := mem_of_prefix\n\nlemma singleton_mem (T : tree A) {a : A} {x : List A} (h : a :: x ∈ T) : [a] ∈ T :=\n  mem_of_prefix ⟨x, rfl⟩ h\n\n@[simp] lemma tree_eq_bot : T = ⊥ ↔ [] ∉ T where\n  mp := by rintro rfl; simp\n  mpr h := by ext x; simpa using fun h' ↦ h <| mem_of_prefix x.nil_prefix h'\n\nlemma take_mem {n : ℕ} (x : T) : x.val.take n ∈ T :=\n  mem_of_prefix (x.val.take_prefix n) x.prop\n\n/-- A variant of `List.take` internally to a tree -/\n@[simps] def take (n : ℕ) (x : T) : T := ⟨x.val.take n, take_mem x⟩\n\n@[simp] lemma take_take (m n : ℕ) (x : T) : take m (take n x) = take (m ⊓ n) x := by\n  simp [Subtype.ext_iff, List.take_take]\n\n@[simp] lemma take_eq_take {x : T} {m n : ℕ} :\n    take m x = take n x ↔ m ⊓ x.val.length = n ⊓ x.val.length := by simp [Subtype.ext_iff]\n\n-- ### `subAt`\n\nvariable (T) (x y : List A)\n\n/-- The residual tree obtained by regarding the node x as new root -/\ndef subAt : tree A :=\n  ⟨(x ++ ·)⁻¹' T, fun _ a _ ↦ mem_of_append (y := [a]) (by rwa [List.append_assoc])⟩\n\n@[simp] lemma mem_subAt : y ∈ subAt T x ↔ x ++ y ∈ T := Iff.rfl\n\n@[simp] lemma subAt_nil : subAt T [] = T := rfl\n\n@[simp] lemma subAt_append : subAt (subAt T x) y = subAt T (x ++ y) := by ext; simp\n\n@[gcongr] lemma subAt_mono (h : S ≤ T) : subAt S x ≤ subAt T x :=\n  Set.preimage_mono h\n\n/-- A variant of `List.drop` that takes values in `subAt` -/\n@[simps] def drop (n : ℕ) (x : T) : subAt T (Tree.take n x).val :=\n  ⟨x.val.drop n, by simp⟩\n\n-- ### `pullSub`\n\n/-- Adjoint of `subAt`, given by pasting x before the root of T. Explicitly,\n  elements are prefixes of x or x with an element of T appended -/\ndef pullSub : tree A where\n  val := { y | y.take x.length <+: x ∧ y.drop x.length ∈ T }\n  property := fun y a ⟨h1, h2⟩ ↦\n    ⟨((y.prefix_append [a]).take x.length).trans h1,\n    mem_of_prefix ((y.prefix_append [a]).drop x.length) h2⟩\n\nvariable {T x y}\n\nlemma mem_pullSub_short (hl : y.length ≤ x.length) : y ∈ pullSub T x ↔ y <+: x ∧ [] ∈ T := by\n  simp [pullSub, List.take_of_length_le hl, List.drop_eq_nil_iff.mpr hl]\n\nlemma mem_pullSub_long (hl : x.length ≤ y.length) : y ∈ pullSub T x ↔ ∃ z ∈ T, y = x ++ z where\n  mp := by\n    intro ⟨h1, h2⟩; use y.drop x.length, h2\n    nth_rw 1 [← List.take_append_drop x.length y]\n    simpa [-List.take_append_drop, List.prefix_iff_eq_take, hl] using h1\n  mpr := by simp +contextual [pullSub]\n\n@[simp] lemma mem_pullSub_append : x ++ y ∈ pullSub T x ↔ y ∈ T := by simp [mem_pullSub_long]\n\n@[simp] lemma mem_pullSub_self : x ∈ pullSub T x ↔ [] ∈ T := by\n  simpa using mem_pullSub_append (y := [])\n\n\nvariable (T x y)\n\nlemma pullSub_subAt : pullSub (subAt T x) x ≤ T := by\n  intro y (h : y ∈ pullSub _ x); rcases le_total y.length x.length with h' | h'\n  · rw [mem_pullSub_short h'] at h; exact mem_of_prefix h.1 (by simpa using h.2)\n  · rw [mem_pullSub_long h'] at h; obtain ⟨_, h, rfl⟩ := h; exact h\n\n@[simp] lemma subAt_pullSub : subAt (pullSub T x) x = T := by\n  ext y; simp\n\n@[gcongr] lemma pullSub_mono (h : S ≤ T) x : pullSub S x ≤ pullSub T x :=\n  fun _ ⟨h1, h2⟩ ↦ ⟨h1, h h2⟩\n\nTarget:\nlemma pullSub_adjunction (S T : tree A) (x : List A) : pullSub S x ≤ T ↔ S ≤ subAt T x where\n  mp _ :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_1e270d1bd1ee","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"bec847eafd109630d8de47c73aeee0c6d21cbb5a0f58c59988d31c6a33285397","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Descriptive","family_id":"pullsub_adjunction","file_id":"mathlib/Mathlib/SetTheory/Descriptive/Tree.lean","sample_id":"1e270d1bd1eeb14c5bf62361efc62c628e247ff0c8b0d9ddc26f734fc3b353a7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dfefda437f41ddea0a1ee24ef23b40384ea8acbbe4173109f7ed5eb612dda632","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f660384a053241b3494fd1356d2eb560d02440a2703e9c40c666787377d767ce","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c48cf6260a277b10c2e23f28a0f369c22b3db624af2c1f400c3f2ea77b4dd0c6","source_sha256":"b91134b0321c98fdbc750a31ba0314dea5693d2a0ca446a2ffe21e21866c8af5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let f := dirichletSummandHom χ <| ne_zero_of_one_lt_re hs\n  have h n : term ↗χ s n = f n := by\n    rcases eq_or_ne n 0 with rfl | hn\n    · simp only [term_zero, map_zero]\n    · simp only [ne_eq, hn, not_false_eq_true, term_of_ne_zero, div_eq_mul_inv,\n        dirichletSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, f]\n  simpa only [LSeries, h]\n    using! exp_tsum_primes_log_eq_tsum (f := f) <| summable_dirichletSummand χ hs","hard_negative":true,"metrics":{"chosen_tokens":88,"rejected_tokens":5,"token_jaccard":0.038462,"token_length_ratio":0.056818},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"3d16f71ff88635e8b8e9ec8ba3a19f836d157e2701d2d425ee7d80a8ab373bb8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.EulerProduct.ExpLog\npublic import Mathlib.NumberTheory.LSeries.Dirichlet\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# The Euler Product for the Riemann Zeta Function and Dirichlet L-Series\n\nThe first main result of this file is the Euler Product formula for the Riemann ζ function\n$$\\prod_p \\frac{1}{1 - p^{-s}}\n   = \\lim_{n \\to \\infty} \\prod_{p < n} \\frac{1}{1 - p^{-s}} = \\zeta(s)$$\nfor $s$ with real part $> 1$ ($p$ runs through the primes).\n`riemannZeta_eulerProduct` is the second equality above. There are versions\n`riemannZeta_eulerProduct_hasProd` and `riemannZeta_eulerProduct_tprod` in terms of `HasProd`\nand `tprod`, respectively.\n\nThe second result is `dirichletLSeries_eulerProduct` (with variants\n`dirichletLSeries_eulerProduct_hasProd` and `dirichletLSeries_eulerProduct_tprod`),\nwhich is the analogous statement for Dirichlet L-series.\n-/\n\n@[expose] public section\n\nopen Complex\n\nvariable {s : ℂ}\n\n/-- When `s ≠ 0`, the map `n ↦ n^(-s)` is completely multiplicative and vanishes at zero. -/\nnoncomputable\ndef riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where\n  toFun n := (n : ℂ) ^ (-s)\n  map_zero' := by simp [hs]\n  map_one' := by simp\n  map_mul' m n := by\n    simpa only [Nat.cast_mul, ofReal_natCast]\n      using mul_cpow_ofReal_nonneg m.cast_nonneg n.cast_nonneg _\n\n/-- When `χ` is a Dirichlet character and `s ≠ 0`, the map `n ↦ χ n * n^(-s)` is completely\nmultiplicative and vanishes at zero. -/\nnoncomputable\ndef dirichletSummandHom {n : ℕ} (χ : DirichletCharacter ℂ n) (hs : s ≠ 0) : ℕ →*₀ ℂ where\n  toFun n := χ n * (n : ℂ) ^ (-s)\n  map_zero' := by simp [hs]\n  map_one' := by simp\n  map_mul' m n := by\n    simp_rw [← ofReal_natCast]\n    simpa only [Nat.cast_mul, IsUnit.mul_iff, not_and, map_mul, ofReal_mul,\n      mul_cpow_ofReal_nonneg m.cast_nonneg n.cast_nonneg _]\n      using mul_mul_mul_comm ..\n\n/-- When `s.re > 1`, the map `n ↦ n^(-s)` is norm-summable. -/\nlemma summable_riemannZetaSummand (hs : 1 < s.re) :\n    Summable (fun n ↦ ‖riemannZetaSummandHom (ne_zero_of_one_lt_re hs) n‖) := by\n  simp only [riemannZetaSummandHom, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]\n  convert! Real.summable_nat_rpow_inv.mpr hs with n\n  rw [← ofReal_natCast,\n    norm_cpow_eq_rpow_re_of_nonneg (Nat.cast_nonneg n) <| re_neg_ne_zero_of_one_lt_re hs,\n    neg_re, Real.rpow_neg <| Nat.cast_nonneg n]\n\nlemma tsum_riemannZetaSummand (hs : 1 < s.re) :\n    ∑' (n : ℕ), riemannZetaSummandHom (ne_zero_of_one_lt_re hs) n = riemannZeta s := by\n  have hsum := summable_riemannZetaSummand hs\n  rw [zeta_eq_tsum_one_div_nat_add_one_cpow hs, hsum.of_norm.tsum_eq_zero_add, map_zero, zero_add]\n  simp only [riemannZetaSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk,\n    Nat.cast_add, Nat.cast_one, one_div]\n\n/-- When `s.re > 1`, the map `n ↦ χ(n) * n^(-s)` is norm-summable. -/\nlemma summable_dirichletSummand {N : ℕ} (χ : DirichletCharacter ℂ N) (hs : 1 < s.re) :\n    Summable (fun n ↦ ‖dirichletSummandHom χ (ne_zero_of_one_lt_re hs) n‖) := by\n  simp only [dirichletSummandHom, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, norm_mul]\n  exact (summable_riemannZetaSummand hs).of_nonneg_of_le (fun _ ↦ by positivity)\n    (fun n ↦ mul_le_of_le_one_left (norm_nonneg _) <| χ.norm_le_one n)\n\nopen scoped LSeries.notation in\nlemma tsum_dirichletSummand {N : ℕ} (χ : DirichletCharacter ℂ N) (hs : 1 < s.re) :\n    ∑' (n : ℕ), dirichletSummandHom χ (ne_zero_of_one_lt_re hs) n = L ↗χ s := by\n  simp only [dirichletSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, LSeries,\n    LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs), div_eq_mul_inv]\n\nopen Filter Nat Topology EulerProduct\n\n/-- The Euler product for the Riemann ζ function, valid for `s.re > 1`.\nThis version is stated in terms of `HasProd`. -/\ntheorem riemannZeta_eulerProduct_hasProd (hs : 1 < s.re) :\n    HasProd (fun p : Primes ↦ (1 - (p : ℂ) ^ (-s))⁻¹) (riemannZeta s) := by\n  rw [← tsum_riemannZetaSummand hs]\n  apply eulerProduct_completely_multiplicative_hasProd <| summable_riemannZetaSummand hs\n\n/-- The Euler product for the Riemann ζ function, valid for `s.re > 1`.\nThis version is stated in terms of `tprod`. -/\ntheorem riemannZeta_eulerProduct_tprod (hs : 1 < s.re) :\n    ∏' p : Primes, (1 - (p : ℂ) ^ (-s))⁻¹ = riemannZeta s :=\n  (riemannZeta_eulerProduct_hasProd hs).tprod_eq\n\n/-- The Euler product for the Riemann ζ function, valid for `s.re > 1`.\nThis version is stated in the form of convergence of finite partial products. -/\ntheorem riemannZeta_eulerProduct (hs : 1 < s.re) :\n    Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - (p : ℂ) ^ (-s))⁻¹) atTop\n      (𝓝 (riemannZeta s)) := by\n  rw [← tsum_riemannZetaSummand hs]\n  apply eulerProduct_completely_multiplicative <| summable_riemannZetaSummand hs\n\nopen scoped LSeries.notation\n\n/-- The Euler product for Dirichlet L-series, valid for `s.re > 1`.\nThis version is stated in terms of `HasProd`. -/\ntheorem DirichletCharacter.LSeries_eulerProduct_hasProd {N : ℕ} (χ : DirichletCharacter ℂ N)\n    (hs : 1 < s.re) :\n    HasProd (fun p : Primes ↦ (1 - χ p * (p : ℂ) ^ (-s))⁻¹) (L ↗χ s) := by\n  rw [← tsum_dirichletSummand χ hs]\n  convert! eulerProduct_completely_multiplicative_hasProd <| summable_dirichletSummand χ hs\n\n/-- The Euler product for Dirichlet L-series, valid for `s.re > 1`.\nThis version is stated in terms of `tprod`. -/\ntheorem DirichletCharacter.LSeries_eulerProduct_tprod {N : ℕ} (χ : DirichletCharacter ℂ N)\n    (hs : 1 < s.re) :\n    ∏' p : Primes, (1 - χ p * (p : ℂ) ^ (-s))⁻¹ = L ↗χ s :=\n  (DirichletCharacter.LSeries_eulerProduct_hasProd χ hs).tprod_eq\n\n/-- The Euler product for Dirichlet L-series, valid for `s.re > 1`.\nThis version is stated in the form of convergence of finite partial products. -/\ntheorem DirichletCharacter.LSeries_eulerProduct {N : ℕ} (χ : DirichletCharacter ℂ N)\n    (hs : 1 < s.re) :\n    Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - χ p * (p : ℂ) ^ (-s))⁻¹) atTop\n      (𝓝 (L ↗χ s)) := by\n  rw [← tsum_dirichletSummand χ hs]\n  apply eulerProduct_completely_multiplicative <| summable_dirichletSummand χ hs\n\nopen LSeries\n\n/-- A variant of the Euler product for Dirichlet L-series. -/\n\nTarget:\ntheorem DirichletCharacter.LSeries_eulerProduct_exp_log {N : ℕ} (χ : DirichletCharacter ℂ N)\n    {s : ℂ} (hs : 1 < s.re) :\n    exp (∑' p : Nat.Primes, -log (1 - χ p * p ^ (-s))) = L ↗χ s :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_c48cf6260a27","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c5b82e864493151862c2c4b00482a603f251d2f2075317263a7e011eed2d2a6a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/EulerProduct","family_id":"dirichletcharacter","file_id":"mathlib/Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean","sample_id":"c48cf6260a277b10c2e23f28a0f369c22b3db624af2c1f400c3f2ea77b4dd0c6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"228e4c456772f8dbd4f2347154e877aedbab995a4c162aadcd1eff84790d42d5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8b02f39ef9581487d59d58b35216989c805c41b002f7b469ea3f61d4f16336bd","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d61440398863b603b3185bd9be1a7f02c573d18ffa317ffa26f3c3cf5dacdb05","source_sha256":"9404d94c318d3af35f74e686f2ec04bdb0f4088fe90fc5c0cd1b8f6b94df8f36","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [mul_app']\n  exact x.naturality (y.app X)","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.133333,"token_length_ratio":0.2},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"3d343d1be63e7723c094f54ff312708399000705855cc60212de9fc8f2c83457","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Endomorphism\n\nNamespace:\nCategoryTheory.CatCenter\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The center of a category\n\nGiven a category `C`, we introduce an abbreviation `CatCenter C` for\nthe center of the category `C`, which is `End (𝟭 C)`, the\ntype of endomorphisms of the identity functor of `C`.\n\n## References\n* https://ncatlab.org/nlab/show/center+of+a+category\n\n-/\n\npublic section\nuniverse v u\n\nnamespace CategoryTheory\n\nopen Category\n\nvariable (C : Type u) [Category.{v} C]\n\n/-- The center of a category `C` is the type `End (𝟭 C)` of the endomorphisms\nof the identify functor of `C`. -/\nabbrev CatCenter := End (𝟭 C)\n\nnamespace CatCenter\n\nvariable {C}\n\n/-- The action of the center of a category on an object. (This is necessary as\n`NatTrans.app x X` is syntactically an endomorphism of `(𝟭 C).obj X`\nrather than of `X`.) -/\nabbrev app (x : CatCenter C) (X : C) : X ⟶ X := NatTrans.app x X\n\n@[ext]\nlemma ext (x y : CatCenter C) (h : ∀ (X : C), x.app X = y.app X) : x = y :=\n  NatTrans.ext (funext h)\n\n@[reassoc]\nlemma naturality (z : CatCenter C) {X Y : C} (f : X ⟶ Y) :\n    f ≫ z.app Y = z.app X ≫ f := NatTrans.naturality z f\n\n@[reassoc]\nlemma mul_app' (x y : CatCenter C) (X : C) : (x * y).app X = y.app X ≫ x.app X := rfl\n\n@[reassoc]\n\nTarget:\nlemma mul_app (x y : CatCenter C) (X : C) : (x * y).app X = x.app X ≫ y.app X :=\n\nProof body:\n","rejected":"by\n  exact mul_app","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"8a39fdab4c6da6f2fc1a88b934c7e550826c810b163fcfd72eedd824d61d5425","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Center","family_id":"mul_app","file_id":"mathlib/Mathlib/CategoryTheory/Center/Basic.lean","sample_id":"d61440398863b603b3185bd9be1a7f02c573d18ffa317ffa26f3c3cf5dacdb05"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7222ee9e22761482f65e7ecb6d7dbb6c39c2c6b75d4bfee6e89b7661c7ff185a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d95a8f0403ed836e86d315e2fe09240f96c711c7c3ecfd907c7c91bd90e81d53","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9a1583a6f100445b753a9ba00ceefda2c0d6b67968367089199c06ca3f1c14f0","source_sha256":"1232470fae437db186c8cbd5a8bd94565b5cb80ee4c956cf59b078515170c352","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine le_antisymm ?_ subset_span\n  let s' : Submodule R M :=\n    { carrier := s\n      add_mem' := add_mem\n      zero_mem' := zero_mem _\n      smul_mem' := SMulMemClass.smul_mem }\n  exact span_le (p := s') |>.mpr le_rfl","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.119048},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"3d507e3b61dfeedd980b5997b6e6a6820851cb57bd1ff64a04d56180e74e0c43","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Submodule.Lattice\npublic import Mathlib.Algebra.Group.Pointwise.Set.Basic\n\nNamespace:\nSubmodule\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,\n  Heather Macbeth\n-/\n/-!\n# The span of a set of vectors, as a submodule\n\n* `Submodule.span s` is defined to be the smallest submodule containing the set `s`.\n\n## Notation\n\n* We introduce the notation `R ∙ v` for the span of a singleton, `Submodule.span R {v}`.  This is\n  `\\span`, not the same as the scalar multiplication `•`/`\\bub`.\n\n-/\n\n@[expose] public section\n\nassert_not_exists Field\n\nvariable {R R₂ K M M₂ V S : Type*}\n\nnamespace Submodule\n\nopen Function Set\n\nopen scoped Pointwise\n\nsection AddCommMonoid\n\nvariable [Semiring R] [AddCommMonoid M] [Module R M]\nvariable {x : M} (p p' : Submodule R M)\nvariable [Semiring R₂] {σ₁₂ : R →+* R₂}\nvariable [AddCommMonoid M₂] [Module R₂ M₂]\nvariable {F : Type*} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂]\n\nsection\n\nvariable (R) in\n/-- The span of a set `s ⊆ M` is the smallest submodule of M that contains `s`. -/\ndef span (s : Set M) : Submodule R M :=\n  sInf { p | s ⊆ p }\n\n/-- An `R`-submodule of `M` is principal if it is generated by one element. -/\n@[mk_iff, wikidata Q44382]\nclass IsPrincipal (S : Submodule R M) : Prop where\n  principal (S) : ∃ a, S = span R {a}\n\ninstance (x : R) : (span R {x}).IsPrincipal := ⟨x, rfl⟩\n\nnamespace IsPrincipal\n\n/-- `generator I`, if `I` is a principal submodule, is an `x ∈ M` such that `span R {x} = I` -/\nnoncomputable def generator (S : Submodule R M) [S.IsPrincipal] : M :=\n  Classical.choose (principal S)\n\ntheorem span_singleton_generator (S : Submodule R M) [S.IsPrincipal] : span R {generator S} = S :=\n  (Classical.choose_spec (principal S)).symm\n\nend IsPrincipal\n\nend\n\nvariable {s t : Set M}\n\ntheorem mem_span : x ∈ span R s ↔ ∀ p : Submodule R M, s ⊆ p → x ∈ p :=\n  mem_iInter₂\n\n@[simp, aesop safe 20 (rule_sets := [SetLike])]\ntheorem subset_span : s ⊆ span R s := fun _ h => mem_span.2 fun _ hp => hp h\n\n@[aesop 80% (rule_sets := [SetLike])]\ntheorem mem_span_of_mem {s : Set M} {x : M} (hx : x ∈ s) : x ∈ span R s := subset_span hx\n\ntheorem span_le {p} : span R s ≤ p ↔ s ⊆ p :=\n  ⟨Subset.trans subset_span, fun ss _ h => mem_span.1 h _ ss⟩\n\n@[gcongr] theorem span_mono (h : s ⊆ t) : span R s ≤ span R t :=\n  span_le.2 <| Subset.trans h subset_span\n\ntheorem span_monotone : Monotone (span R : Set M → Submodule R M) := fun _ _ => span_mono\n\ntheorem span_eq_of_le (h₁ : s ⊆ p) (h₂ : p ≤ span R s) : span R s = p :=\n  le_antisymm (span_le.2 h₁) h₂\n\ntheorem span_eq : span R (p : Set M) = p :=\n  span_eq_of_le _ (Subset.refl _) subset_span\n\ntheorem span_eq_span (hs : s ⊆ span R t) (ht : t ⊆ span R s) : span R s = span R t :=\n  le_antisymm (span_le.2 hs) (span_le.2 ht)\n\n/-- A version of `Submodule.span_eq` for subobjects closed under addition and scalar multiplication\nand containing zero. In general, this should not be used directly, but can be used to quickly\ngenerate proofs for specific types of subobjects. -/\n\nTarget:\nlemma coe_span_eq_self [SetLike S M] [AddSubmonoidClass S M] [SMulMemClass S R M] (s : S) :\n    (span R (s : Set M) : Set M) = s :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_9a1583a6f100","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"29f15baa0da7df9b6f698b7e82b9606103e74eba5e8fb62b08622971e61073f8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Span","family_id":"coe_span_eq_self","file_id":"mathlib/Mathlib/LinearAlgebra/Span/Defs.lean","sample_id":"9a1583a6f100445b753a9ba00ceefda2c0d6b67968367089199c06ca3f1c14f0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b8f5229735b933ede0144ab9a48b8b3042b761eb7777cebb1d91a92fa7760495","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ad36c8eeea38b3617b2fa01b60e158adc224ea844614163c66de06f8b4d2fee4","source_sha256":"42f43c83c3b42913128aa30933a73363c485cf2bc69fc613214486004b504ca0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Subgroup.rank_congr (commutator_eq_closure G)]\n  apply Subgroup.rank_closure_finite_le_nat_card\n\nvariable [Group.FG G]","hard_negative":false,"metrics":{"chosen_tokens":22,"rejected_tokens":3,"token_jaccard":0.055556,"token_length_ratio":0.136364},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"3d9ff7536d3ada5080bdbad6467ec1a18a13b5c923dd730cfffbec5b02c63761","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Subgroup.Finite\npublic import Mathlib.GroupTheory.Commutator.Basic\npublic import Mathlib.GroupTheory.Rank\npublic import Mathlib.GroupTheory.Index\n\nNamespace:\nSubgroup\n\nLocal context:\n/-\nCopyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jordan Brown, Thomas Browning, Patrick Lutz\n-/\n/-!\nThe commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n\npublic section\n\nvariable {G : Type*} [Group G]\n\nnamespace Subgroup\n\n/-- The commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n@[to_additive /-- The commutator of a finite direct product is contained in the direct product of\nthe commutators. -/]\ntheorem commutator_pi_pi_of_finite {η : Type*} [Finite η] {Gs : η → Type*} [∀ i, Group (Gs i)]\n    (H K : ∀ i, Subgroup (Gs i)) : ⁅Subgroup.pi Set.univ H, Subgroup.pi Set.univ K⁆ =\n    Subgroup.pi Set.univ fun i => ⁅H i, K i⁆ := by\n  classical\n    apply le_antisymm (commutator_pi_pi_le H K)\n    rw [pi_le_iff]\n    intro i hi\n    rw [map_commutator]\n    apply commutator_mono <;>\n      · rw [le_pi_iff]\n        intro j _hj\n        rintro _ ⟨x, hx, rfl⟩\n        by_cases h : j = i\n        · subst h\n          simpa using hx\n        · simp [h, one_mem]\n\nvariable [Finite (commutatorSet G)]\n\ninstance : Group.FG (_root_.commutator G) := by\n  rw [commutator_eq_closure]; apply Group.closure_finite_fg\n\nvariable (G) in\n\nTarget:\nlemma rank_commutator_le_card : Group.rank (_root_.commutator G) ≤ Nat.card (commutatorSet G) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Commutator","family_id":"rank_commutator_le_card","file_id":"mathlib/Mathlib/GroupTheory/Commutator/Finite.lean","sample_id":"ad36c8eeea38b3617b2fa01b60e158adc224ea844614163c66de06f8b4d2fee4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8e1e8082a0a84f2600d4003971ff82d46d02db0f4b2396697813b11973e2972c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f5f31c487c9867affa52b7440d78c3121f50ce3816f050fa3e638b6c8edfd584","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"660e9e4b92da9e1464e29ce063157b6ff472ec59e69123eed2b8146bcadb1cb1","source_sha256":"74b0e866ba6855a98543cc59c9d908f25a74c6e6456b4e8b44974d8edb78dfd2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rintro ⟨p⟩\n  rw [← @WeaklyConnectedComponent.eq (Generators G), eq_comm, ← FreeGroup.of_injective.eq_iff, ←\n    mul_inv_eq_one]\n  let X := FreeGroup (WeaklyConnectedComponent <| Generators G)\n  let f : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\n  let F : G ⥤ CategoryTheory.SingleObj.{u} (X : Type u) := SingleObj.differenceFunctor f\n  change (F.map p) = ((@CategoryTheory.Functor.const G _ _ (SingleObj.category X)).obj ()).map p\n  congr; ext\n  rw [Functor.const_obj_map, id_as_one, differenceFunctor_map, @mul_inv_eq_one _ _ (f _)]\n  apply congr_arg FreeGroup.of\n  apply (WeaklyConnectedComponent.eq _ _).mpr\n  exact ⟨Hom.toPath (Sum.inr (by assumption))⟩","hard_negative":true,"metrics":{"chosen_tokens":168,"rejected_tokens":3,"token_jaccard":0.028986,"token_length_ratio":0.017857},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"3dbbe6c2a1df224f71f3ae2f345e3757b09334f0c34335e5cbaa5d541e4a89cb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Action\npublic import Mathlib.Combinatorics.Quiver.Arborescence\npublic import Mathlib.Combinatorics.Quiver.ConnectedComponent\npublic import Mathlib.GroupTheory.FreeGroup.IsFreeGroup\n\nNamespace:\nIsFreeGroupoid\n\nLocal context:\n/-\nCopyright (c) 2021 David Wärn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David Wärn\n-/\n/-!\n# The Nielsen-Schreier theorem\n\nThis file proves that a subgroup of a free group is itself free.\n\n## Main result\n\n- `subgroupIsFreeOfIsFree H`: an instance saying that a subgroup of a free group is free.\n\n## Proof overview\n\nThe proof is analogous to the proof using covering spaces and fundamental groups of graphs,\nbut we work directly with groupoids instead of topological spaces. Under this analogy,\n\n- `IsFreeGroupoid G` corresponds to saying that a space is a graph.\n- `endMulEquivSubgroup H` plays the role of replacing 'subgroup of fundamental group' with\n  'fundamental group of covering space'.\n- `actionGroupoidIsFree G A` corresponds to the fact that a covering of a (single-vertex)\n  graph is a graph.\n- `endIsFree T` corresponds to the fact that, given a spanning tree `T` of a\n  graph, its fundamental group is free (generated by loops from the complement of the tree).\n\n## Implementation notes\n\nOur definition of `IsFreeGroupoid` is nonstandard. Normally one would require that functors\n`G ⥤ X` to any _groupoid_ `X` are given by graph homomorphisms from the generators, but we only\nconsider _groups_ `X`. This simplifies the argument since functor equality is complicated in\ngeneral, but simple for functors to single object categories.\n\n## References\n\nhttps://ncatlab.org/nlab/show/Nielsen-Schreier+theorem\n\n## Tags\n\nfree group, free groupoid, Nielsen-Schreier\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nuniverse v u\n\nopen CategoryTheory CategoryTheory.ActionCategory CategoryTheory.SingleObj Quiver FreeGroup\n\n/-- `IsFreeGroupoid.Generators G` is a type synonym for `G`. We think of this as\nthe vertices of the generating quiver of `G` when `G` is free. We can't use `G` directly,\nsince `G` already has a quiver instance from being a groupoid. -/\n@[nolint unusedArguments]\ndef IsFreeGroupoid.Generators (G) [Groupoid G] :=\n  G\n\n/--\nA groupoid `G` is free when we have the following data:\n- a quiver on `IsFreeGroupoid.Generators G` (a type synonym for `G`)\n- a function `of` taking a generating arrow to a morphism in `G`\n- such that a functor from `G` to any group `X` is uniquely determined\n  by assigning labels in `X` to the generating arrows.\n\nThis definition is nonstandard. Normally one would require that functors `G ⥤ X`\nto any _groupoid_ `X` are given by graph homomorphisms from `generators`. -/\nclass IsFreeGroupoid (G) [Groupoid.{v} G] where\n  quiverGenerators : Quiver.{v} (IsFreeGroupoid.Generators G)\n  of : ∀ {a b : IsFreeGroupoid.Generators G}, (a ⟶ b) → ((show G from a) ⟶ b)\n  unique_lift :\n    ∀ {X : Type v} [Group X] (f : Labelling (IsFreeGroupoid.Generators G) X),\n      ∃! F : G ⥤ CategoryTheory.SingleObj X, ∀ (a b) (g : a ⟶ b), F.map (of g) = f g\n\nattribute [nolint docBlame] IsFreeGroupoid.of IsFreeGroupoid.unique_lift\n\nnamespace IsFreeGroupoid\n\nattribute [instance_reducible, instance] quiverGenerators\n\n/-- Two functors from a free groupoid to a group are equal when they agree on the generating\nquiver. -/\n@[ext]\ntheorem ext_functor {G} [Groupoid.{v} G] [IsFreeGroupoid G] {X : Type v} [Group X]\n    (f g : G ⥤ CategoryTheory.SingleObj X) (h : ∀ (a b) (e : a ⟶ b), f.map (of e) = g.map (of e)) :\n    f = g :=\n  let ⟨_, _, u⟩ := @unique_lift G _ _ X _ fun (a b : Generators G) (e : a ⟶ b) => g.map (of e)\n  _root_.trans (u _ h) (u _ fun _ _ _ => rfl).symm\n\n/-- An action groupoid over a free group is free. More generally, one could show that the groupoid\nof elements over a free groupoid is free, but this version is easier to prove and suffices for our\npurposes.\n\nAnalogous to the fact that a covering space of a graph is a graph. (A free groupoid is like a graph,\nand a groupoid of elements is like a covering space.) -/\ninstance actionGroupoidIsFree {G A : Type u} [Group G] [IsFreeGroup G] [MulAction G A] :\n    IsFreeGroupoid (ActionCategory G A) where\n  quiverGenerators :=\n    ⟨fun a b => { e : IsFreeGroup.Generators G // IsFreeGroup.of e • a.back = b.back }⟩\n  of := fun (e : Subtype _) => ⟨IsFreeGroup.of e, e.property⟩\n  unique_lift := by\n    intro X _ f\n    let f' : IsFreeGroup.Generators G → (A → X) ⋊[mulAutArrow] G := fun e =>\n      ⟨fun b => @f ⟨(), _⟩ ⟨(), b⟩ ⟨e, smul_inv_smul _ b⟩, IsFreeGroup.of e⟩\n    rcases IsFreeGroup.unique_lift f' with ⟨F', hF', uF'⟩\n    refine ⟨uncurry F' ?_, ?_, ?_⟩\n    · suffices SemidirectProduct.rightHom.comp F' = MonoidHom.id _ by\n        exact DFunLike.ext_iff.mp this\n      apply IsFreeGroup.ext_hom (fun x ↦ ?_)\n      rw [MonoidHom.comp_apply, hF']\n      rfl\n    · rintro ⟨⟨⟩, a : A⟩ ⟨⟨⟩, b⟩ ⟨e, h : IsFreeGroup.of e • a = b⟩\n      change (F' (IsFreeGroup.of _)).left _ = _\n      rw [hF']\n      cases inv_smul_eq_iff.mpr h.symm\n      rfl\n    · intro E hE\n      have : curry E = F' := by\n        apply uF'\n        intro e\n        ext\n        · convert! hE _ _ _\n          rfl\n        · rfl\n      apply Functor.hext\n      · intro\n        apply Unit.ext\n      · refine ActionCategory.cases ?_\n        intros\n        simp only [← this, uncurry_map, curry_apply_left, coe_back, homOfPair.val]\n        rfl\n\nnamespace SpanningTree\n\n/- In this section, we suppose we have a free groupoid with a spanning tree for its generating\nquiver. The goal is to prove that the vertex group at the root is free. A picture to have in mind\nis that we are 'pulling' the endpoints of all the edges of the quiver along the spanning tree to\nthe root. -/\nvariable {G : Type u} [Groupoid.{u} G] [IsFreeGroupoid G]\n  (T : WideSubquiver (Symmetrify <| Generators G)) [Arborescence T]\n\nset_option backward.privateInPublic true in\n/-- The root of `T`, except its type is `G` instead of the type synonym `T`. -/\nprivate def root' : G :=\n  show T from root T\n\n-- this has to be marked noncomputable, see issue https://github.com/leanprover-community/mathlib4/pull/451.\n-- It might be nicer to define this in terms of `composePath`\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- A path in the tree gives a hom, by composition. -/\ndef homOfPath : ∀ {a : G}, Path (root T) a → (root' T ⟶ a)\n  | _, Path.nil => 𝟙 _\n  | _, Path.cons p f => homOfPath p ≫ Sum.recOn f.val (fun e => of e) fun e => inv (of e)\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- For every vertex `a`, there is a canonical hom from the root, given by the path in the tree. -/\ndef treeHom (a : G) : root' T ⟶ a :=\n  homOfPath T default\n\n/-- Any path to `a` gives `treeHom T a`, since paths in the tree are unique. -/\ntheorem treeHom_eq {a : G} (p : Path (root T) a) : treeHom T a = homOfPath T p := by\n  rw [treeHom, Unique.default_eq]\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n@[simp]\ntheorem treeHom_root : treeHom T (root' T) = 𝟙 _ :=\n  -- this should just be `treeHom_eq T Path.nil`, but Lean treats `homOfPath` with suspicion.\n    _root_.trans\n    (treeHom_eq T Path.nil) rfl\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Any hom in `G` can be made into a loop, by conjugating with `treeHom`s. -/\ndef loopOfHom {a b : G} (p : a ⟶ b) : End (root' T) :=\n  treeHom T a ≫ p ≫ inv (treeHom T b)\n\nset_option backward.isDefEq.respectTransparency false in\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Turning an edge in the spanning tree into a loop gives the identity loop. -/\ntheorem loopOfHom_eq_id {a b : Generators G} (e) (H : e ∈ wideSubquiverSymmetrify T a b) :\n    loopOfHom T (of e) = 𝟙 (root' T) := by\n  rw [loopOfHom, ← Category.assoc, IsIso.comp_inv_eq, Category.id_comp]\n  rcases H with H | H\n  · rw [treeHom_eq T (Path.cons default ⟨Sum.inl e, H⟩), homOfPath]\n    rfl\n  · rw [treeHom_eq T (Path.cons default ⟨Sum.inr e, H⟩), homOfPath]\n    simp only [IsIso.inv_hom_id, Category.comp_id, Category.assoc, treeHom]\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Since a hom gives a loop, any homomorphism from the vertex group at the root\nextends to a functor on the whole groupoid. -/\n@[simps]\ndef functorOfMonoidHom {X} [Monoid X] (f : End (root' T) →* X) :\n    G ⥤ CategoryTheory.SingleObj X where\n  obj _ := ()\n  map p := f (loopOfHom T p)\n  map_id := by\n    intro a\n    dsimp only [loopOfHom]\n    rw [Category.id_comp, IsIso.hom_inv_id, ← End.one_def, f.map_one, id_as_one]\n  map_comp := by\n    intros\n    rw [comp_as_mul, ← f.map_mul]\n    simp only [IsIso.inv_hom_id_assoc, loopOfHom, End.mul_def, Category.assoc]\n\nset_option backward.isDefEq.respectTransparency false in\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nopen scoped Classical in\n/-- Given a free groupoid and an arborescence of its generating quiver, the vertex\ngroup at the root is freely generated by loops coming from generating arrows\nin the complement of the tree. -/\nlemma endIsFree : IsFreeGroup (End (root' T)) :=\n  IsFreeGroup.ofUniqueLift ((wideSubquiverEquivSetTotal <| wideSubquiverSymmetrify T)ᶜ : Set _)\n    (fun e => loopOfHom T (of e.val.hom))\n    (by\n      intro X _ f\n      let f' : Labelling (Generators G) X := fun a b e =>\n        if h : e ∈ wideSubquiverSymmetrify T a b then 1 else f ⟨⟨a, b, e⟩, h⟩\n      rcases unique_lift f' with ⟨F', hF', uF'⟩\n      refine ⟨F'.mapEnd _, ?_, ?_⟩\n      · suffices ∀ {x y} (q : x ⟶ y), F'.map (loopOfHom T q) = (F'.map q : X) by\n          rintro ⟨⟨a, b, e⟩, h⟩\n          simp only [Functor.mapEnd, DFunLike.coe, this, hF']\n          exact dif_neg h\n        intro x y q\n        suffices ∀ {a} (p : Path (root T) a), F'.map (homOfPath T p) = 1 by\n          simp only [this, treeHom, comp_as_mul, inv_as_inv, loopOfHom, inv_one, mul_one,\n            one_mul, Functor.map_inv, Functor.map_comp]\n        intro a p\n        induction p with\n        | nil => rw [homOfPath, F'.map_id, id_as_one]\n        | cons p e ih =>\n          rw [homOfPath, F'.map_comp, comp_as_mul, ih, mul_one]\n          rcases e with ⟨e | e, eT⟩\n          · rw [hF']\n            exact dif_pos (Or.inl eT)\n          · rw [F'.map_inv, inv_as_inv, inv_eq_one, hF']\n            exact dif_pos (Or.inr eT)\n      · intro E hE\n        ext x\n        suffices (functorOfMonoidHom T E).map x = F'.map x by\n          simpa only [loopOfHom, functorOfMonoidHom, IsIso.inv_id, treeHom_root,\n            Category.id_comp, Category.comp_id] using! this\n        congr\n        apply uF'\n        intro a b e\n        change E (loopOfHom T _) = dite _ _ _\n        split_ifs with h\n        · rw [loopOfHom_eq_id T e h, ← End.one_def, E.map_one]\n        · exact hE ⟨⟨a, b, e⟩, h⟩)\n\nend SpanningTree\n\nset_option backward.privateInPublic true in\n/-- Another name for the identity function `G → G`, to help type checking. -/\nprivate def symgen {G : Type u} [Groupoid.{v} G] [IsFreeGroupoid G] :\n    G → Symmetrify (Generators G) :=\n  id\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- If there exists a morphism `a → b` in a free groupoid, then there also exists a zigzag\nfrom `a` to `b` in the generating quiver. -/\n\nTarget:\ntheorem path_nonempty_of_hom {G} [Groupoid.{u, u} G] [IsFreeGroupoid G] {a b : G} :\n    Nonempty (a ⟶ b) → Nonempty (Path (symgen a) (symgen b)) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_660e9e4b92da","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f863721a1e1ac4e6f631465e0938a20c3367c4d7bfa58dad02fa7c5902448e37","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/FreeGroup","family_id":"path_nonempty_of_hom","file_id":"mathlib/Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean","sample_id":"660e9e4b92da9e1464e29ce063157b6ff472ec59e69123eed2b8146bcadb1cb1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dca549e925c3a05e8b8bc1baae8e33cdf145c6f1d35ee1dc1c15d213d724ea07","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"97f596d6d90eb6f474110e8621a17881009e62dfd1861b34761429e37cb4ef6d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6aa245c658f8d36bf9da386e1af0245093910d695a91f4bad772966128f1d7d5","source_sha256":"5a992f718439ad7f2b978ed86a7b3a95db162d52a4ab577079eb77f016af349f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by cat_disch) : f = g :=\n  Hom.ext w","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":19,"token_jaccard":0.857143,"token_length_ratio":1.583333},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"3dc38da595ad5a94746cd27ae2d50fc5f4d4fda06f20773ebac8e9ef94fd9d48","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Shapes.IsTerminal\npublic import Mathlib.CategoryTheory.Functor.EpiMono\n\nNamespace:\nCategoryTheory.Endofunctor.Coalgebra\n\nLocal context:\n/-\nCopyright (c) 2022 Joseph Hua. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison, Bhavik Mehta, Johan Commelin, Reid Barton, Robert Y. Lewis, Joseph Hua\n-/\n/-!\n\n# Algebras of endofunctors\n\nThis file defines (co)algebras of an endofunctor, and provides the category instance for them.\nIt also defines the forgetful functor from the category of (co)algebras. It is shown that the\nstructure map of the initial algebra of an endofunctor is an isomorphism. Furthermore, it is shown\nthat for an adjunction `F ⊣ G` the category of algebras over `F` is equivalent to the category of\ncoalgebras over `G`.\n\n## TODO\n\n* Prove that if the countable infinite product over the powers of the endofunctor exists, then\n  algebras over the endofunctor coincide with algebras over the free monad on the endofunctor.\n-/\n\n@[expose] public section\n\n\nuniverse v u\n\nnamespace CategoryTheory\n\nnamespace Endofunctor\n\nvariable {C : Type u} [Category.{v} C]\n\n/-- An algebra of an endofunctor; `str` stands for \"structure morphism\" -/\nstructure Algebra (F : C ⥤ C) where\n  /-- carrier of the algebra -/\n  a : C\n  /-- structure morphism of the algebra -/\n  str : F.obj a ⟶ a\n\ninstance [Inhabited C] : Inhabited (Algebra (𝟭 C)) :=\n  ⟨⟨default, 𝟙 _⟩⟩\n\nnamespace Algebra\n\nvariable {F : C ⥤ C} (A : Algebra F) {A₀ A₁ A₂ : Algebra F}\n\n/-\n```\n        str\n   F A₀ -----> A₀\n    |          |\nF f |          | f\n    V          V\n   F A₁ -----> A₁\n        str\n```\n-/\n/-- A morphism between algebras of endofunctor `F` -/\n@[ext]\nstructure Hom (A₀ A₁ : Algebra F) where\n  /-- underlying morphism between the carriers -/\n  f : A₀.1 ⟶ A₁.1\n  /-- compatibility condition -/\n  h : F.map f ≫ A₁.str = A₀.str ≫ f := by cat_disch\n\nattribute [reassoc (attr := simp)] Hom.h\n\nnamespace Hom\n\n/-- The identity morphism of an algebra of endofunctor `F` -/\ndef id : Hom A A where f := 𝟙 _\n\ninstance : Inhabited (Hom A A) :=\n  ⟨{ f := 𝟙 _ }⟩\n\n/-- The composition of morphisms between algebras of endofunctor `F` -/\ndef comp (f : Hom A₀ A₁) (g : Hom A₁ A₂) : Hom A₀ A₂ where f := f.1 ≫ g.1\n\nend Hom\n\ninstance (F : C ⥤ C) : CategoryStruct (Algebra F) where\n  Hom := Hom\n  id := Hom.id\n  comp := @Hom.comp _ _ _\n\n@[ext]\nlemma ext {A B : Algebra F} {f g : A ⟶ B} (w : f.f = g.f := by cat_disch) : f = g :=\n  Hom.ext w\n\n@[simp]\ntheorem id_eq_id : Algebra.Hom.id A = 𝟙 A :=\n  rfl\n\n@[simp]\ntheorem id_f : (𝟙 _ : A ⟶ A).1 = 𝟙 A.1 :=\n  rfl\n\nvariable (f : A₀ ⟶ A₁) (g : A₁ ⟶ A₂)\n\n@[simp]\ntheorem comp_eq_comp : Algebra.Hom.comp f g = f ≫ g :=\n  rfl\n\n@[simp]\ntheorem comp_f : (f ≫ g).1 = f.1 ≫ g.1 :=\n  rfl\n\n/-- Algebras of an endofunctor `F` form a category -/\ninstance (F : C ⥤ C) : Category (Algebra F) := { }\n\n/-- To construct an isomorphism of algebras, it suffices to give an isomorphism of the As which\ncommutes with the structure morphisms.\n-/\n@[simps!]\ndef isoMk (h : A₀.1 ≅ A₁.1) (w : F.map h.hom ≫ A₁.str = A₀.str ≫ h.hom := by cat_disch) :\n    A₀ ≅ A₁ where\n  hom := { f := h.hom }\n  inv :=\n    { f := h.inv\n      h := by\n        rw [h.eq_comp_inv, Category.assoc, ← w, ← Functor.map_comp_assoc]\n        simp }\n\n/-- The forgetful functor from the category of algebras, forgetting the algebraic structure. -/\n@[simps]\ndef forget (F : C ⥤ C) : Algebra F ⥤ C where\n  obj A := A.1\n  map := Hom.f\n\n/-- An algebra morphism with an underlying isomorphism hom in `C` is an algebra isomorphism. -/\ntheorem iso_of_iso (f : A₀ ⟶ A₁) [IsIso f.1] : IsIso f :=\n  ⟨⟨{ f := inv f.1\n      h := by simp }, by cat_disch, by cat_disch⟩⟩\n\ninstance forget_reflects_iso : (forget F).ReflectsIsomorphisms where reflects := iso_of_iso\n\ninstance forget_faithful : (forget F).Faithful := { }\n\n/-- An algebra morphism with an underlying epimorphism hom in `C` is an algebra epimorphism. -/\ntheorem epi_of_epi {X Y : Algebra F} (f : X ⟶ Y) [h : Epi f.1] : Epi f :=\n  (forget F).epi_of_epi_map h\n\n/-- An algebra morphism with an underlying monomorphism hom in `C` is an algebra monomorphism. -/\ntheorem mono_of_mono {X Y : Algebra F} (f : X ⟶ Y) [h : Mono f.1] : Mono f :=\n  (forget F).mono_of_mono_map h\n\n/-- From a natural transformation `α : G → F` we get a functor from\nalgebras of `F` to algebras of `G`.\n-/\n@[simps]\ndef functorOfNatTrans {F G : C ⥤ C} (α : G ⟶ F) : Algebra F ⥤ Algebra G where\n  obj A :=\n    { a := A.1\n      str := α.app _ ≫ A.str }\n  map f := { f := f.1 }\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The identity transformation induces the identity endofunctor on the category of algebras. -/\n@[simps!]\ndef functorOfNatTransId : functorOfNatTrans (𝟙 F) ≅ 𝟭 _ :=\n  NatIso.ofComponents fun X => isoMk (Iso.refl _)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- A composition of natural transformations gives the composition of corresponding functors. -/\n@[simps!]\ndef functorOfNatTransComp {F₀ F₁ F₂ : C ⥤ C} (α : F₀ ⟶ F₁) (β : F₁ ⟶ F₂) :\n    functorOfNatTrans (α ≫ β) ≅ functorOfNatTrans β ⋙ functorOfNatTrans α :=\n  NatIso.ofComponents fun X => isoMk (Iso.refl _)\n\n/--\nIf `α` and `β` are two equal natural transformations, then the functors of algebras induced by them\nare isomorphic.\nWe define it like this as opposed to using `eq_to_iso` so that the components are nicer to prove\nlemmas about.\n-/\n@[simps!]\ndef functorOfNatTransEq {F G : C ⥤ C} {α β : F ⟶ G} (h : α = β) :\n    functorOfNatTrans α ≅ functorOfNatTrans β :=\n  NatIso.ofComponents fun X => isoMk (Iso.refl _)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- Naturally isomorphic endofunctors give equivalent categories of algebras.\nFurthermore, they are equivalent as categories over `C`, that is,\nwe have `equiv_of_nat_iso h ⋙ forget = forget`.\n-/\n@[simps]\ndef equivOfNatIso {F G : C ⥤ C} (α : F ≅ G) : Algebra F ≌ Algebra G where\n  functor := functorOfNatTrans α.inv\n  inverse := functorOfNatTrans α.hom\n  unitIso := functorOfNatTransId.symm ≪≫ functorOfNatTransEq (by simp) ≪≫ functorOfNatTransComp _ _\n  counitIso :=\n    (functorOfNatTransComp _ _).symm ≪≫ functorOfNatTransEq (by simp) ≪≫ functorOfNatTransId\n\nnamespace Initial\n\nvariable {A : Algebra F} (h : Limits.IsInitial A)\n/-- The inverse of the structure map of an initial algebra -/\n@[simp]\ndef strInv : A.1 ⟶ F.obj A.1 :=\n  (h.to ⟨F.obj A.a, F.map A.str⟩).f\n\ntheorem left_inv' :\n    ⟨strInv h ≫ A.str, by rw [← Category.assoc, F.map_comp, strInv, ← Hom.h]⟩ = 𝟙 A :=\n  Limits.IsInitial.hom_ext h _ (𝟙 A)\n\ntheorem left_inv : strInv h ≫ A.str = 𝟙 _ :=\n  congr_arg Hom.f (left_inv' h)\n\ntheorem right_inv : A.str ≫ strInv h = 𝟙 _ := by\n  rw [strInv, ← (h.to ⟨F.obj A.1, F.map A.str⟩).h, ← F.map_id, ← F.map_comp]\n  congr\n  exact left_inv h\n\n/-- The structure map of the initial algebra is an isomorphism,\nhence endofunctors preserve their initial algebras\n-/\ntheorem str_isIso (h : Limits.IsInitial A) : IsIso A.str :=\n  { out := ⟨strInv h, right_inv _, left_inv _⟩ }\n\nend Initial\n\nend Algebra\n\n/-- A coalgebra of an endofunctor; `str` stands for \"structure morphism\" -/\nstructure Coalgebra (F : C ⥤ C) where\n  /-- carrier of the coalgebra -/\n  V : C\n  /-- structure morphism of the coalgebra -/\n  str : V ⟶ F.obj V\n\ninstance [Inhabited C] : Inhabited (Coalgebra (𝟭 C)) :=\n  ⟨⟨default, 𝟙 _⟩⟩\n\nnamespace Coalgebra\n\nvariable {F : C ⥤ C} (V : Coalgebra F) {V₀ V₁ V₂ : Coalgebra F}\n\n/-\n```\n        str\n    V₀ -----> F V₀\n    |          |\n  f |          | F f\n    V          V\n    V₁ -----> F V₁\n        str\n```\n-/\n/-- A morphism between coalgebras of an endofunctor `F` -/\n@[ext]\nstructure Hom (V₀ V₁ : Coalgebra F) where\n  /-- underlying morphism between two carriers -/\n  f : V₀.1 ⟶ V₁.1\n  /-- compatibility condition -/\n  h : V₀.str ≫ F.map f = f ≫ V₁.str := by cat_disch\n\nattribute [reassoc (attr := simp)] Hom.h\n\nnamespace Hom\n\n/-- The identity morphism of an algebra of endofunctor `F` -/\ndef id : Hom V V where f := 𝟙 _\n\ninstance : Inhabited (Hom V V) :=\n  ⟨{ f := 𝟙 _ }⟩\n\n/-- The composition of morphisms between algebras of endofunctor `F` -/\ndef comp (f : Hom V₀ V₁) (g : Hom V₁ V₂) : Hom V₀ V₂ where f := f.1 ≫ g.1\n\nend Hom\n\ninstance (F : C ⥤ C) : CategoryStruct (Coalgebra F) where\n  Hom := Hom\n  id := Hom.id\n  comp := @Hom.comp _ _ _\n\n@[ext]\n\nTarget:\nlemma ext {A B : Coalgebra F} {f g : A ⟶ B} (w : f.f = g.f :=\n\nProof body:\n","rejected":"```lean\nby cat_disch) : f = g :=\n  Hom.ext w\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Endofunctor","family_id":"ext","file_id":"mathlib/Mathlib/CategoryTheory/Endofunctor/Algebra.lean","sample_id":"6aa245c658f8d36bf9da386e1af0245093910d695a91f4bad772966128f1d7d5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dfefda437f41ddea0a1ee24ef23b40384ea8acbbe4173109f7ed5eb612dda632","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c48cf6260a277b10c2e23f28a0f369c22b3db624af2c1f400c3f2ea77b4dd0c6","source_sha256":"b91134b0321c98fdbc750a31ba0314dea5693d2a0ca446a2ffe21e21866c8af5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let f := dirichletSummandHom χ <| ne_zero_of_one_lt_re hs\n  have h n : term ↗χ s n = f n := by\n    rcases eq_or_ne n 0 with rfl | hn\n    · simp only [term_zero, map_zero]\n    · simp only [ne_eq, hn, not_false_eq_true, term_of_ne_zero, div_eq_mul_inv,\n        dirichletSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, f]\n  simpa only [LSeries, h]\n    using! exp_tsum_primes_log_eq_tsum (f := f) <| summable_dirichletSummand χ hs","hard_negative":false,"metrics":{"chosen_tokens":88,"rejected_tokens":2,"token_jaccard":0.02,"token_length_ratio":0.022727},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"3dccb6049738626e34d7b81ec8264639896af9e5d100b048afa43790435319d3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.EulerProduct.ExpLog\npublic import Mathlib.NumberTheory.LSeries.Dirichlet\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# The Euler Product for the Riemann Zeta Function and Dirichlet L-Series\n\nThe first main result of this file is the Euler Product formula for the Riemann ζ function\n$$\\prod_p \\frac{1}{1 - p^{-s}}\n   = \\lim_{n \\to \\infty} \\prod_{p < n} \\frac{1}{1 - p^{-s}} = \\zeta(s)$$\nfor $s$ with real part $> 1$ ($p$ runs through the primes).\n`riemannZeta_eulerProduct` is the second equality above. There are versions\n`riemannZeta_eulerProduct_hasProd` and `riemannZeta_eulerProduct_tprod` in terms of `HasProd`\nand `tprod`, respectively.\n\nThe second result is `dirichletLSeries_eulerProduct` (with variants\n`dirichletLSeries_eulerProduct_hasProd` and `dirichletLSeries_eulerProduct_tprod`),\nwhich is the analogous statement for Dirichlet L-series.\n-/\n\n@[expose] public section\n\nopen Complex\n\nvariable {s : ℂ}\n\n/-- When `s ≠ 0`, the map `n ↦ n^(-s)` is completely multiplicative and vanishes at zero. -/\nnoncomputable\ndef riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where\n  toFun n := (n : ℂ) ^ (-s)\n  map_zero' := by simp [hs]\n  map_one' := by simp\n  map_mul' m n := by\n    simpa only [Nat.cast_mul, ofReal_natCast]\n      using mul_cpow_ofReal_nonneg m.cast_nonneg n.cast_nonneg _\n\n/-- When `χ` is a Dirichlet character and `s ≠ 0`, the map `n ↦ χ n * n^(-s)` is completely\nmultiplicative and vanishes at zero. -/\nnoncomputable\ndef dirichletSummandHom {n : ℕ} (χ : DirichletCharacter ℂ n) (hs : s ≠ 0) : ℕ →*₀ ℂ where\n  toFun n := χ n * (n : ℂ) ^ (-s)\n  map_zero' := by simp [hs]\n  map_one' := by simp\n  map_mul' m n := by\n    simp_rw [← ofReal_natCast]\n    simpa only [Nat.cast_mul, IsUnit.mul_iff, not_and, map_mul, ofReal_mul,\n      mul_cpow_ofReal_nonneg m.cast_nonneg n.cast_nonneg _]\n      using mul_mul_mul_comm ..\n\n/-- When `s.re > 1`, the map `n ↦ n^(-s)` is norm-summable. -/\nlemma summable_riemannZetaSummand (hs : 1 < s.re) :\n    Summable (fun n ↦ ‖riemannZetaSummandHom (ne_zero_of_one_lt_re hs) n‖) := by\n  simp only [riemannZetaSummandHom, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]\n  convert! Real.summable_nat_rpow_inv.mpr hs with n\n  rw [← ofReal_natCast,\n    norm_cpow_eq_rpow_re_of_nonneg (Nat.cast_nonneg n) <| re_neg_ne_zero_of_one_lt_re hs,\n    neg_re, Real.rpow_neg <| Nat.cast_nonneg n]\n\nlemma tsum_riemannZetaSummand (hs : 1 < s.re) :\n    ∑' (n : ℕ), riemannZetaSummandHom (ne_zero_of_one_lt_re hs) n = riemannZeta s := by\n  have hsum := summable_riemannZetaSummand hs\n  rw [zeta_eq_tsum_one_div_nat_add_one_cpow hs, hsum.of_norm.tsum_eq_zero_add, map_zero, zero_add]\n  simp only [riemannZetaSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk,\n    Nat.cast_add, Nat.cast_one, one_div]\n\n/-- When `s.re > 1`, the map `n ↦ χ(n) * n^(-s)` is norm-summable. -/\nlemma summable_dirichletSummand {N : ℕ} (χ : DirichletCharacter ℂ N) (hs : 1 < s.re) :\n    Summable (fun n ↦ ‖dirichletSummandHom χ (ne_zero_of_one_lt_re hs) n‖) := by\n  simp only [dirichletSummandHom, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, norm_mul]\n  exact (summable_riemannZetaSummand hs).of_nonneg_of_le (fun _ ↦ by positivity)\n    (fun n ↦ mul_le_of_le_one_left (norm_nonneg _) <| χ.norm_le_one n)\n\nopen scoped LSeries.notation in\nlemma tsum_dirichletSummand {N : ℕ} (χ : DirichletCharacter ℂ N) (hs : 1 < s.re) :\n    ∑' (n : ℕ), dirichletSummandHom χ (ne_zero_of_one_lt_re hs) n = L ↗χ s := by\n  simp only [dirichletSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, LSeries,\n    LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs), div_eq_mul_inv]\n\nopen Filter Nat Topology EulerProduct\n\n/-- The Euler product for the Riemann ζ function, valid for `s.re > 1`.\nThis version is stated in terms of `HasProd`. -/\ntheorem riemannZeta_eulerProduct_hasProd (hs : 1 < s.re) :\n    HasProd (fun p : Primes ↦ (1 - (p : ℂ) ^ (-s))⁻¹) (riemannZeta s) := by\n  rw [← tsum_riemannZetaSummand hs]\n  apply eulerProduct_completely_multiplicative_hasProd <| summable_riemannZetaSummand hs\n\n/-- The Euler product for the Riemann ζ function, valid for `s.re > 1`.\nThis version is stated in terms of `tprod`. -/\ntheorem riemannZeta_eulerProduct_tprod (hs : 1 < s.re) :\n    ∏' p : Primes, (1 - (p : ℂ) ^ (-s))⁻¹ = riemannZeta s :=\n  (riemannZeta_eulerProduct_hasProd hs).tprod_eq\n\n/-- The Euler product for the Riemann ζ function, valid for `s.re > 1`.\nThis version is stated in the form of convergence of finite partial products. -/\ntheorem riemannZeta_eulerProduct (hs : 1 < s.re) :\n    Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - (p : ℂ) ^ (-s))⁻¹) atTop\n      (𝓝 (riemannZeta s)) := by\n  rw [← tsum_riemannZetaSummand hs]\n  apply eulerProduct_completely_multiplicative <| summable_riemannZetaSummand hs\n\nopen scoped LSeries.notation\n\n/-- The Euler product for Dirichlet L-series, valid for `s.re > 1`.\nThis version is stated in terms of `HasProd`. -/\ntheorem DirichletCharacter.LSeries_eulerProduct_hasProd {N : ℕ} (χ : DirichletCharacter ℂ N)\n    (hs : 1 < s.re) :\n    HasProd (fun p : Primes ↦ (1 - χ p * (p : ℂ) ^ (-s))⁻¹) (L ↗χ s) := by\n  rw [← tsum_dirichletSummand χ hs]\n  convert! eulerProduct_completely_multiplicative_hasProd <| summable_dirichletSummand χ hs\n\n/-- The Euler product for Dirichlet L-series, valid for `s.re > 1`.\nThis version is stated in terms of `tprod`. -/\ntheorem DirichletCharacter.LSeries_eulerProduct_tprod {N : ℕ} (χ : DirichletCharacter ℂ N)\n    (hs : 1 < s.re) :\n    ∏' p : Primes, (1 - χ p * (p : ℂ) ^ (-s))⁻¹ = L ↗χ s :=\n  (DirichletCharacter.LSeries_eulerProduct_hasProd χ hs).tprod_eq\n\n/-- The Euler product for Dirichlet L-series, valid for `s.re > 1`.\nThis version is stated in the form of convergence of finite partial products. -/\ntheorem DirichletCharacter.LSeries_eulerProduct {N : ℕ} (χ : DirichletCharacter ℂ N)\n    (hs : 1 < s.re) :\n    Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - χ p * (p : ℂ) ^ (-s))⁻¹) atTop\n      (𝓝 (L ↗χ s)) := by\n  rw [← tsum_dirichletSummand χ hs]\n  apply eulerProduct_completely_multiplicative <| summable_dirichletSummand χ hs\n\nopen LSeries\n\n/-- A variant of the Euler product for Dirichlet L-series. -/\n\nTarget:\ntheorem DirichletCharacter.LSeries_eulerProduct_exp_log {N : ℕ} (χ : DirichletCharacter ℂ N)\n    {s : ℂ} (hs : 1 < s.re) :\n    exp (∑' p : Nat.Primes, -log (1 - χ p * p ^ (-s))) = L ↗χ s :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/EulerProduct","family_id":"dirichletcharacter","file_id":"mathlib/Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean","sample_id":"c48cf6260a277b10c2e23f28a0f369c22b3db624af2c1f400c3f2ea77b4dd0c6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3939293b987d00efdc8989f961d8fe481cabdbec2fac5708f8344374642f9236","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"66509cfd3b611bcc3ba67f00ce88462a0218dee03d4503cf26ee350e187ae0c3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dd448d7bddd39539db63b6005ceef026bb709e5272ecd6e5386e656fa08db32f","source_sha256":"6ccf0444fc2138afeb704541116189c71bea73fd06b5fb4f13fd001240de6123","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [TopologicalSpace.nhds_generateFrom]\n  simp only [comap_iInf, comap_principal]\n  intro s hs\n  rw [← le_principal_iff]\n  refine iInf_le_of_le { u | s ∈ u } ?_\n  refine iInf_le_of_le ⟨hs, ⟨s, rfl⟩⟩ ?_\n  exact principal_mono.2 fun _ ↦ id","hard_negative":true,"metrics":{"chosen_tokens":54,"rejected_tokens":3,"token_jaccard":0.055556,"token_length_ratio":0.055556},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"3df02329fafbe9629ebc4a4224fe722ab657f0b0963f6322be4c1cc6a5af1846","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Bases\npublic import Mathlib.Topology.DenseEmbedding\npublic import Mathlib.Topology.Connected.TotallyDisconnected\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton\n-/\n/-! # Stone-Čech compactification\n\nConstruction of the Stone-Čech compactification using ultrafilters.\n\nFor any topological space `α`, we build a compact Hausdorff space `StoneCech α` and a continuous\nmap `stoneCechUnit : α → StoneCech α` which is minimal in the sense of the following universal\nproperty: for any compact Hausdorff space `β` and every map `f : α → β` such that\n`hf : Continuous f`, there is a unique map `stoneCechExtend hf : StoneCech α → β` such that\n`stoneCechExtend_extends : stoneCechExtend hf ∘ stoneCechUnit = f`.\nContinuity of this extension is asserted by `continuous_stoneCechExtend` and uniqueness by\n`stoneCech_hom_ext`.\n\nBeware that the terminology “extend” is slightly misleading since `stoneCechUnit` is not always\ninjective, so one cannot always think of `α` as being “inside” its compactification `StoneCech α`.\n\n## Implementation notes\n\nParts of the formalization are based on “Ultrafilters and Topology”\nby Marius Stekelenburg, particularly section 5. However the construction in the general\ncase is different because the equivalence relation on spaces of ultrafilters described\nby Stekelenburg causes issues with universes since it involves a condition\non all compact Hausdorff spaces. We replace it by a two steps construction.\nThe first step called `PreStoneCech` guarantees the expected universal property but\nnot the Hausdorff condition. We then define `StoneCech α` as `T2Quotient (PreStoneCech α)`.\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen Filter Set\n\nopen Topology\n\nuniverse u v\n\nsection Ultrafilter\n\n/- The set of ultrafilters on α carries a natural topology which makes\n  it the Stone-Čech compactification of α (viewed as a discrete space). -/\n/-- Basis for the topology on `Ultrafilter α`. -/\ndef ultrafilterBasis (α : Type u) : Set (Set (Ultrafilter α)) :=\n  range fun s : Set α ↦ { u | s ∈ u }\n\nvariable {α : Type u}\n\ninstance Ultrafilter.topologicalSpace : TopologicalSpace (Ultrafilter α) :=\n  TopologicalSpace.generateFrom (ultrafilterBasis α)\n\ntheorem ultrafilterBasis_is_basis : TopologicalSpace.IsTopologicalBasis (ultrafilterBasis α) :=\n  ⟨by\n    rintro _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ u ⟨ua, ub⟩\n    refine ⟨_, ⟨a ∩ b, rfl⟩, inter_mem ua ub, fun v hv ↦ ⟨?_, ?_⟩⟩ <;> apply mem_of_superset hv <;>\n      simp [inter_subset_right],\n    eq_univ_of_univ_subset <| subset_sUnion_of_mem <| ⟨univ, eq_univ_of_forall fun _ ↦ univ_mem⟩,\n    rfl⟩\n\n/-- The basic open sets for the topology on ultrafilters are open. -/\ntheorem ultrafilter_isOpen_basic (s : Set α) : IsOpen { u : Ultrafilter α | s ∈ u } :=\n  ultrafilterBasis_is_basis.isOpen ⟨s, rfl⟩\n\n/-- The basic open sets for the topology on ultrafilters are also closed. -/\ntheorem ultrafilter_isClosed_basic (s : Set α) : IsClosed { u : Ultrafilter α | s ∈ u } := by\n  rw [← isOpen_compl_iff]\n  convert! ultrafilter_isOpen_basic sᶜ using 1\n  ext u\n  exact Ultrafilter.compl_mem_iff_notMem.symm\n\n/-- Every ultrafilter `u` on `Ultrafilter α` converges to a unique\n  point of `Ultrafilter α`, namely `joinM u`. -/\ntheorem ultrafilter_converges_iff {u : Ultrafilter (Ultrafilter α)} {x : Ultrafilter α} :\n    ↑u ≤ 𝓝 x ↔ x = joinM u := by\n  rw [eq_comm, ← Ultrafilter.coe_le_coe]\n  change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, { v : Ultrafilter α | s ∈ v } ∈ u\n  simp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff,\n    mem_setOf_eq]\n  constructor\n  · intro h a ha\n    exact h _ ⟨ha, a, rfl⟩\n  · rintro h a ⟨xi, a, rfl⟩\n    exact h _ xi\n\ninstance ultrafilter_compact : CompactSpace (Ultrafilter α) :=\n  ⟨isCompact_iff_ultrafilter_le_nhds.mpr fun f _ ↦\n      ⟨joinM f, trivial, ultrafilter_converges_iff.mpr rfl⟩⟩\n\ninstance Ultrafilter.t2Space : T2Space (Ultrafilter α) :=\n  t2_iff_ultrafilter.mpr fun {x y} f fx fy ↦\n    have hx : x = joinM f := ultrafilter_converges_iff.mp fx\n    have hy : y = joinM f := ultrafilter_converges_iff.mp fy\n    hx.trans hy.symm\n\ninstance : TotallyDisconnectedSpace (Ultrafilter α) := by\n  rw [totallyDisconnectedSpace_iff_connectedComponent_singleton]\n  intro A\n  simp only [Set.eq_singleton_iff_unique_mem, mem_connectedComponent, true_and]\n  intro B hB\n  rw [← Ultrafilter.coe_le_coe]\n  intro s hs\n  rw [connectedComponent_eq_iInter_isClopen, Set.mem_iInter] at hB\n  let Z := { F : Ultrafilter α | s ∈ F }\n  have hZ : IsClopen Z := ⟨ultrafilter_isClosed_basic s, ultrafilter_isOpen_basic s⟩\n  exact hB ⟨Z, hZ, hs⟩\n\n@[simp] theorem Ultrafilter.tendsto_pure_self (b : Ultrafilter α) : Tendsto pure b (𝓝 b) := by\n  rw [Tendsto, ← coe_map, ultrafilter_converges_iff]\n  ext s\n  change s ∈ b ↔ {t | s ∈ t} ∈ map pure b\n  simp_rw [mem_map, preimage_setOf_eq, mem_pure, setOf_mem_eq]\n\nTarget:\ntheorem ultrafilter_comap_pure_nhds (b : Ultrafilter α) : comap pure (𝓝 b) ≤ b :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_dd448d7bddd3","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"e147a58bd4b4460aefa4b06edb8fef3525c30b474f4922fe385d9e3a3cdfdcdd","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Compactification","family_id":"ultrafilter_comap_pure_nhds","file_id":"mathlib/Mathlib/Topology/Compactification/StoneCech.lean","sample_id":"dd448d7bddd39539db63b6005ceef026bb709e5272ecd6e5386e656fa08db32f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4477bb12e5cc2a166f61746299c4ae29af03f6b7f73cd9a6d3ff697daf2ff447","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"865c2117838a5f9b6bdb0bfba027decbe6b709de52a194b8d1a351aa165f6f08","source_sha256":"42f43c83c3b42913128aa30933a73363c485cf2bc69fc613214486004b504ca0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G\n  rw [← hS1, ← Fintype.card_coe, ← Nat.card_eq_fintype_card, ← Finset.coe_sort_coe, ← Nat.card_fun]\n  exact Finite.card_le_of_embedding (quotientCenterEmbedding hS2)","hard_negative":false,"metrics":{"chosen_tokens":47,"rejected_tokens":5,"token_jaccard":0.060606,"token_length_ratio":0.106383},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"3e11b4af2d21ab27b87e43fe5d516e5e868ee72a8bb2899c772a35db1c4f772b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Subgroup.Finite\npublic import Mathlib.GroupTheory.Commutator.Basic\npublic import Mathlib.GroupTheory.Rank\npublic import Mathlib.GroupTheory.Index\n\nNamespace:\nSubgroup\n\nLocal context:\n/-\nCopyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jordan Brown, Thomas Browning, Patrick Lutz\n-/\n/-!\nThe commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n\npublic section\n\nvariable {G : Type*} [Group G]\n\nnamespace Subgroup\n\n/-- The commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n@[to_additive /-- The commutator of a finite direct product is contained in the direct product of\nthe commutators. -/]\ntheorem commutator_pi_pi_of_finite {η : Type*} [Finite η] {Gs : η → Type*} [∀ i, Group (Gs i)]\n    (H K : ∀ i, Subgroup (Gs i)) : ⁅Subgroup.pi Set.univ H, Subgroup.pi Set.univ K⁆ =\n    Subgroup.pi Set.univ fun i => ⁅H i, K i⁆ := by\n  classical\n    apply le_antisymm (commutator_pi_pi_le H K)\n    rw [pi_le_iff]\n    intro i hi\n    rw [map_commutator]\n    apply commutator_mono <;>\n      · rw [le_pi_iff]\n        intro j _hj\n        rintro _ ⟨x, hx, rfl⟩\n        by_cases h : j = i\n        · subst h\n          simpa using hx\n        · simp [h, one_mem]\n\nvariable [Finite (commutatorSet G)]\n\ninstance : Group.FG (_root_.commutator G) := by\n  rw [commutator_eq_closure]; apply Group.closure_finite_fg\n\nvariable (G) in\nlemma rank_commutator_le_card : Group.rank (_root_.commutator G) ≤ Nat.card (commutatorSet G) := by\n  rw [Subgroup.rank_congr (commutator_eq_closure G)]\n  apply Subgroup.rank_closure_finite_le_nat_card\n\nvariable [Group.FG G]\n\ninstance finiteIndex_center : FiniteIndex (center G) := by\n  obtain ⟨S, -, hS⟩ := Group.rank_spec G\n  exact ⟨mt (Finite.card_eq_zero_of_embedding (quotientCenterEmbedding hS)) Finite.card_pos.ne'⟩\n\nvariable (G) in\n\nTarget:\nlemma index_center_le_pow : (center G).index ≤ Nat.card (commutatorSet G) ^ Group.rank G :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Commutator","family_id":"index_center_le_pow","file_id":"mathlib/Mathlib/GroupTheory/Commutator/Finite.lean","sample_id":"865c2117838a5f9b6bdb0bfba027decbe6b709de52a194b8d1a351aa165f6f08"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a0b44731945f1ae77d84c174d4c4d596bfd46fdd862ded445d0f40690c121d90","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e5d640efe871327e6d7a85ba9e52dd3140e74fe8671f6a1d11428174bcd3a187","source_sha256":"88c666c6d936856962069aacdb5158224b504be63ef88b462fdb4de48e42b813","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := h.hasNondegenerateEdges.two_le\n  by_contra! hlt\n  interval_cases n\n  exact (h 0).injective.ne (by decide : (0 : Fin 3) ≠ 2) (by simp)","hard_negative":false,"metrics":{"chosen_tokens":39,"rejected_tokens":2,"token_jaccard":0.038462,"token_length_ratio":0.051282},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"3e1422ddee30d54e40877c67ae7a4a3d3b240bd8a0eddc3fd13783341c033ae8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Convex.Between\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Tactic.Continuity\n\nNamespace:\nPolygon\n\nLocal context:\n/-\nCopyright (c) 2026 A. M. Berns. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: A. M. Berns\n-/\n/-!\n# Polygons\n\nThis file defines polygons in affine spaces.\nFor the special case `n = 3`, an interconversion is provided with `Affine.Triangle`.\n\n## Main definitions\n\n* `Polygon P n`: A polygon with `n` vertices in a type `P`.\n\n-/\n\n@[expose] public section\n\nopen Set\n\n/-- A polygon with `n` vertices in a type `P`. -/\nstructure Polygon (P : Type*) (n : ℕ) where\n  /-- The vertices of the polygon, indexed by `Fin n`. -/\n  vertices : Fin n → P\n\nnamespace Polygon\n\nvariable {R V P : Type*} {n : ℕ}\n\n/-- A coercion to function so that vertices can\nbe written as `poly i` instead of `poly.vertices i` -/\ninstance : CoeFun (Polygon P n) (fun _ => Fin n → P) where\n  coe := Polygon.vertices\n\n/-- A polygon has nondegenerate edges if adjacent vertices are distinct. -/\ndef HasNondegenerateEdges (poly : Polygon P n) : Prop :=\n  ∀ i : Fin n, poly i ≠ poly (finRotate n i)\n\ntheorem HasNondegenerateEdges.two_le [NeZero n] {poly : Polygon P n}\n    (h : poly.HasNondegenerateEdges) : 2 ≤ n := by\n  by_contra! hlt\n  interval_cases n\n  · simp_all only [neZero_zero_iff_false]\n  · exact h 0 (by simp)\n\nvariable [Ring R] [AddCommGroup V] [Module R V] [AddTorsor V P]\n\nvariable (R) in\n/-- The `i`-th edge as an affine map `R →ᵃ[R] P`. -/\ndef edgePath (poly : Polygon P n) (i : Fin n) : R →ᵃ[R] P :=\n  AffineMap.lineMap (poly i) (poly (finRotate n i))\n\nvariable (R) in\n/-- The `i`-th edge as a set of points using an `affineSegment`. -/\ndef edgeSet [PartialOrder R] (poly : Polygon P n) (i : Fin n) : Set P :=\n  affineSegment R (poly i) (poly (finRotate n i))\n\nvariable (R) in\n/-- The `edgeSet` is equivalent to the image of the `edgePath`. -/\ntheorem edgeSet_eq_image_edgePath [PartialOrder R] (poly : Polygon P n) (i : Fin n) :\n    poly.edgeSet R i = poly.edgePath R i '' Icc (0 : R) 1 := rfl\n\nvariable (R) in\n/-- The boundary of the polygon is the union of all its edges. -/\ndef boundary [PartialOrder R] (poly : Polygon P n) : Set P :=\n  ⋃ i, poly.edgeSet R i\n\nvariable (R) in\n/-- A polygon has nondegenerate vertices if any three consecutive vertices\nare affinely independent. -/\ndef HasNondegenerateVertices [NeZero n] (poly : Polygon P n) : Prop :=\n  ∀ i : Fin n, AffineIndependent R ![poly i, poly (i + 1), poly (i + 2)]\n\n/-- Polygons with nondegenerate vertices also have nondegenerate edges. -/\ntheorem HasNondegenerateVertices.hasNondegenerateEdges [NeZero n] [Nontrivial R]\n    {poly : Polygon P n}\n    (h : poly.HasNondegenerateVertices R) : poly.HasNondegenerateEdges := by\n  obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (NeZero.ne n)\n  intro i\n  simpa using (h i).injective.ne (by decide : (0 : Fin 3) ≠ 1)\n\nTarget:\ntheorem HasNondegenerateVertices.three_le [NeZero n] [Nontrivial R] {poly : Polygon P n}\n    (h : poly.HasNondegenerateVertices R) : 3 ≤ n :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/Polygon","family_id":"hasnondegeneratevertices","file_id":"mathlib/Mathlib/Geometry/Polygon/Basic.lean","sample_id":"e5d640efe871327e6d7a85ba9e52dd3140e74fe8671f6a1d11428174bcd3a187"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fd4f7273c1abc3ee605f4507e24cb13680aff1584126fd18e76d89a937f7044b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"50332d0f41236ffc25367bcc287b5a5f97bce4b58e2df5be3cb1c6d4b44bee1a","source_sha256":"6495ae02a429a2011b5873f559e0c6908f832b9f2a026ddab4af530b07660425","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  exact Decidable.List.Lex.ne_iff H","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":8,"token_jaccard":0.066667,"token_length_ratio":0.727273},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"3e26ba83b0fa37c1a985506cdfa0bccd4e96811ef10705b18d2a058fceef37a9","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.List.Basic\npublic import Mathlib.Data.Nat.Basic\npublic import Mathlib.Order.RelClasses\n\nNamespace:\nList.Lex\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Lexicographic ordering of lists.\n\nThe lexicographic order on `List α` is defined by `L < M` iff\n* `[] < (a :: L)` for any `a` and `L`,\n* `(a :: L) < (b :: M)` where `a < b`, or\n* `(a :: L) < (a :: M)` where `L < M`.\n\n## See also\n\nRelated files are:\n* `Mathlib/Combinatorics/Colex.lean`: Colexicographic order on finite sets.\n* `Mathlib/Data/PSigma/Order.lean`: Lexicographic order on `Σ' i, α i`.\n* `Mathlib/Order/PiLex.lean`: Lexicographic order on `Πₗ i, α i`.\n* `Mathlib/Data/Sigma/Order.lean`: Lexicographic order on `Σ i, α i`.\n* `Mathlib/Data/Prod/Lex.lean`: Lexicographic order on `α × β`.\n-/\n\npublic section\n\n\nnamespace List\n\nopen Nat\n\nuniverse u\n\nvariable {α : Type u}\n\n/-! ### lexicographic ordering -/\n\ntheorem lex_cons_iff {r : α → α → Prop} [Std.Irrefl r] {a l₁ l₂} :\n    Lex r (a :: l₁) (a :: l₂) ↔ Lex r l₁ l₂ :=\n  ⟨fun h => by obtain - | h | h := h; exacts [(irrefl_of r a h).elim, h], Lex.cons⟩\n\ntheorem lex_nil_or_eq_nil {r : α → α → Prop} (l : List α) : List.Lex r [] l ∨ l = [] :=\n  match l with\n  | [] => Or.inr rfl\n  | _ :: _ => .inl .nil\n\n@[simp]\ntheorem lex_singleton_iff {r : α → α → Prop} (a b : α) : List.Lex r [a] [b] ↔ r a b :=\n  ⟨fun | .rel h => h, .rel⟩\n\nnamespace Lex\n\ninstance isOrderConnected (r : α → α → Prop) [IsOrderConnected α r] [Std.Trichotomous r] :\n    IsOrderConnected (List α) (Lex r) where\n  conn := aux where\n    aux\n    | _, [], _ :: _, nil => Or.inr nil\n    | _, [], _ :: _, rel _ => Or.inr nil\n    | _, [], _ :: _, cons _ => Or.inr nil\n    | _, _ :: _, _ :: _, nil => Or.inl nil\n    | _ :: _, b :: _, _ :: _, rel h => (IsOrderConnected.conn _ b _ h).imp rel rel\n    | a :: l₁, b :: l₂, _ :: l₃, cons h => by\n      rcases trichotomous_of r a b with (ab | rfl | ab)\n      · exact Or.inl (rel ab)\n      · exact (aux _ l₂ _ h).imp cons cons\n      · exact Or.inr (rel ab)\n\ninstance trichotomous (r : α → α → Prop) [Std.Trichotomous r] : Std.Trichotomous (Lex r) where\n  trichotomous := aux where\n    aux\n    | [], [], _, _ => rfl\n    | [], _ :: _, hab, _ => hab nil |>.elim\n    | _ :: _, [], _, hba => hba nil |>.elim\n    | a :: l₁, b :: l₂, hab, hba => by\n      obtain rfl := Std.Trichotomous.trichotomous a b (mt rel hab) (mt rel hba)\n      rw [aux l₁ l₂ (mt cons hab) (mt cons hba)]\n\ninstance asymm (r : α → α → Prop) [Std.Asymm r] : Std.Asymm (Lex r) where\n  asymm := aux where\n    aux\n    | _, _, Lex.rel h₁, Lex.rel h₂ => _root_.asymm h₁ h₂\n    | _, _, Lex.rel h₁, Lex.cons _ => _root_.asymm h₁ h₁\n    | _, _, Lex.cons _, Lex.rel h₂ => _root_.asymm h₂ h₂\n    | _, _, Lex.cons h₁, Lex.cons h₂ => aux _ _ h₁ h₂\n\ninstance decidableRel [DecidableEq α] (r : α → α → Prop) [DecidableRel r] : DecidableRel (Lex r)\n  | l₁, [] => isFalse fun h => by cases h\n  | [], _ :: _ => isTrue Lex.nil\n  | a :: l₁, b :: l₂ => by\n    haveI := decidableRel r l₁ l₂\n    refine decidable_of_iff (r a b ∨ a = b ∧ Lex r l₁ l₂) ⟨fun h => ?_, fun h => ?_⟩\n    · rcases h with (h | ⟨rfl, h⟩)\n      · exact Lex.rel h\n      · exact Lex.cons h\n    · rcases h with (_ | h | h)\n      · exact Or.inl h\n      · exact Or.inr ⟨rfl, h⟩\n\ntheorem append_right (r : α → α → Prop) : ∀ {s₁ s₂} (t), Lex r s₁ s₂ → Lex r s₁ (s₂ ++ t)\n  | _, _, _, nil => nil\n  | _, _, _, cons h => cons (append_right r _ h)\n  | _, _, _, rel r => rel r\n\ntheorem append_left (R : α → α → Prop) {t₁ t₂} (h : Lex R t₁ t₂) : ∀ s, Lex R (s ++ t₁) (s ++ t₂)\n  | [] => h\n  | _ :: l => cons (append_left R h l)\n\ntheorem imp {r s : α → α → Prop} (H : ∀ a b, r a b → s a b) : ∀ l₁ l₂, Lex r l₁ l₂ → Lex s l₁ l₂\n  | _, _, nil => nil\n  | _, _, cons h => cons (imp H _ _ h)\n  | _, _, rel r => rel (H _ _ r)\n\ntheorem to_ne : ∀ {l₁ l₂ : List α}, Lex (· ≠ ·) l₁ l₂ → l₁ ≠ l₂\n  | _, _, cons h, e => to_ne h (List.cons.inj e).2\n  | _, _, rel r, e => r (List.cons.inj e).1\n\ntheorem _root_.Decidable.List.Lex.ne_iff [DecidableEq α] {l₁ l₂ : List α}\n    (H : length l₁ ≤ length l₂) : Lex (· ≠ ·) l₁ l₂ ↔ l₁ ≠ l₂ :=\n  ⟨to_ne, fun h => by\n    induction l₁ generalizing l₂ <;> rcases l₂ with - | ⟨b, l₂⟩\n    · contradiction\n    · apply nil\n    · exact (not_lt_of_ge H).elim (succ_pos _)\n    case cons.cons a l₁ IH =>\n      by_cases ab : a = b\n      · subst b\n        exact .cons <| IH (le_of_succ_le_succ H) (mt (congr_arg _) h)\n      · exact .rel ab ⟩\n\nTarget:\ntheorem ne_iff {l₁ l₂ : List α} (H : length l₁ ≤ length l₂) : Lex (· ≠ ·) l₁ l₂ ↔ l₁ ≠ l₂ :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"f32c7f31abe05d24ffd62435c0aa53a68f663765217d99ecfe0ccf132431e40f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/List","family_id":"ne_iff","file_id":"mathlib/Mathlib/Data/List/Lex.lean","sample_id":"50332d0f41236ffc25367bcc287b5a5f97bce4b58e2df5be3cb1c6d4b44bee1a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"594293ce5d84984c59a5d88c876de5c8200038ce21c7d54ee21a2ba975b3d8b6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d7b302fa097a85ad40628a8309be13365dc90a026d8ed805296461111bd052b1","source_sha256":"ae9de95805258feb351386be6f3451ef917af14e0cd4a5d5b560efd102bade12","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rw [eq_bot_iff, sup_le_iff]; simp","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":2,"token_jaccard":0.1,"token_length_ratio":0.222222},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"3e4ac19298ec725a0f96e7f76e7052ba4a0ff43fa59cbbc95477d3c3d32ea310","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.BoundedOrder.Basic\npublic import Mathlib.Order.Lattice\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Bounded lattices\n\nThis file contains miscellaneous lemmas about lattices with top or bottom elements.\n\n## Common lattices\n\n* Distributive lattices with a bottom element. Notated by `[DistribLattice α] [OrderBot α]`.\n  It captures the properties of `Disjoint` that are common to `GeneralizedBooleanAlgebra` and\n  `DistribLattice` when `OrderBot`.\n* Bounded and distributive lattice. Notated by `[DistribLattice α] [BoundedOrder α]`.\n  Typical examples include `Prop` and `Set α`.\n-/\n\npublic section\n\nopen Function OrderDual\n\nvariable {α β : Type*}\n\n/-! ### Top, bottom element -/\n\nsection SemilatticeSupTop\n\nvariable [SemilatticeSup α] [OrderTop α]\n\n@[to_dual] theorem top_sup_eq (a : α) : ⊤ ⊔ a = ⊤ := sup_of_le_left le_top\n@[to_dual] theorem sup_top_eq (a : α) : a ⊔ ⊤ = ⊤ := sup_of_le_right le_top\n\nend SemilatticeSupTop\n\nsection SemilatticeSupBot\n\nvariable [SemilatticeSup α] [OrderBot α] {a b : α}\n\n@[to_dual] theorem bot_sup_eq (a : α) : ⊥ ⊔ a = a := sup_of_le_right bot_le\n@[to_dual] theorem sup_bot_eq (a : α) : a ⊔ ⊥ = a := sup_of_le_left bot_le\n\n@[to_dual (attr := simp, grind =)]\n\nTarget:\ntheorem sup_eq_bot_iff : a ⊔ b = ⊥ ↔ a = ⊥ ∧ b = ⊥ :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/BoundedOrder","family_id":"sup_eq_bot_iff","file_id":"mathlib/Mathlib/Order/BoundedOrder/Lattice.lean","sample_id":"d7b302fa097a85ad40628a8309be13365dc90a026d8ed805296461111bd052b1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"faeb8949323faf61ca5ff806cf60d2c823a2170b0852dfc5d54df3a865ffc0fd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"64df50078f89549dc86aa8460f006bfc2de2619575085c7a305917f01b031133","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"988fd8cb4d8e17e38f2a76981b98bfba51705b13e2c6dc4822f9d73a946a1e06","source_sha256":"8ca4ffc48e91301292a0b84aa8e139dd34c405da224b4628cd66cbb6fa0e45d8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [zariskisMainProperty_iff, ← top_le_iff, SetLike.le_def,\n    Submonoid.mem_powers_iff, mem_integralClosure_iff]","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":5,"token_jaccard":0.117647,"token_length_ratio":0.277778},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"3e570820a241bc081149e3a3850d889496c72e288a64e272af8263655b2f25ba","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Algebraic.StronglyTranscendental\npublic import Mathlib.RingTheory.Conductor\npublic import Mathlib.RingTheory.Ideal.Quotient.Nilpotent\npublic import Mathlib.RingTheory.IntegralClosure.GoingDown\npublic import Mathlib.RingTheory.Polynomial.IsIntegral\npublic import Mathlib.RingTheory.QuasiFinite.Polynomial\npublic import Mathlib.Algebra.Algebra.Shrink\n\nNamespace:\nAlgebra\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Algebraic Zariski's Main Theorem\n\nThe statement of Zariski's main theorem is the following:\nGiven a finite type `R`-algebra `S`, and `p` a prime of `S` such that `S` is quasi-finite at `R`,\nthen there exists a `f ∉ p` such that `S[1/f]` is isomorphic to `R'[1/f]` where `R'` is the integral\nclosure of `R` in `S`.\n\nWe follow https://stacks.math.columbia.edu/tag/00PI and proceed in the following steps\n\n1. `Algebra.ZariskisMainProperty.of_adjoin_eq_top`:\n  The case where `S = R[X]/I`.\n  The key is `Polynomial.not_ker_le_map_C_of_surjective_of_quasiFiniteAt`\n  which shows that there exists some `g ∈ I` such that some coefficient `gᵢ ∉ p`.\n  Then one basically takes `f = gᵢ` and `g` becomes monic in `R[1/gᵢ][X]` up to some minor technical\n  issues, and then `S[1/gᵢ]` is basically integral over `R[1/gᵢ]`.\n2. `Algebra.ZariskisMainProperty.of_algHom_polynomial`:\n  The case where `S` is finite over `R⟨x⟩` for some `x : S`.\n  The following key results are first established:\n  - `isStronglyTranscendental_mk_radical_conductor`:\n    Let `𝔣` be the conductor of `x` (i.e. the largest `S`-ideal in `R⟨x⟩`).\n    `x` as an element of `S/√𝔣` is strongly transcendental over `R`.\n  - `Algebra.not_quasiFiniteAt_of_stronglyTranscendental`:\n    If `S` is reduced, then `x : S` is not strongly transcendental over `R`.\n    One first reduces to when `R ⊆ S` are domains, and then to when `R` is integrally closed.\n    A going down theorem is now available, which could be applied to\n    `Polynomial.map_under_lt_comap_of_quasiFiniteAt`:`(p ∩ R)[X] < p ∩ R<x>` to get a contradiction.\n\n  The second result applied to `S/√𝔣` together with the first result implies that\n  `p` does not contain `𝔣`.\n  The claim then follows from `Localization.localRingHom_bijective_of_not_conductor_le`.\n3. `Algebra.ZariskisMainProperty.of_algHom_mvPolynomial`:\n  The case where `S` is finite over `R⟨x₁,...,xₙ⟩`. This is proved using induction on `n`.\n\n## Main definition and results\n- `Algebra.ZariskisMainProperty`:\n  We say that an `R` algebra `S` satisfies the Zariski's main property at a prime `p` of `S`\n  if there exists `r ∉ p` in the integral closure `S'` of `R` in `S`, such that `S'[1/r] = S[1/r]`.\n- `Algebra.ZariskisMainProperty.of_finiteType`:\n  If `S` is finite type over `R` and quasi-finite at `p`, then `ZariskisMainProperty` holds.\n- `Algebra.QuasiFiniteAt.exists_fg_and_exists_notMem_and_awayMap_bijective`:\n  If `S` is finite type over `R` and quasi-finite at `p`,\n  then there exists a subalgebra `S'` of `R` that is finitely generated as an `R`-module,\n  and some `r ∈ S'` such that `r ∉ p` and `S'[1/r] = S[1/r]`.\n-/\n\n@[expose] public section\n\nvariable {R S T : Type*} [CommRing R] [CommRing S] [Algebra R S] [CommRing T] [Algebra R T]\n\nopen scoped TensorProduct nonZeroDivisors\n\nopen Polynomial\n\nnamespace Algebra\n\nvariable (R) in\n/-- We say that an `R` algebra `S` satisfies the Zariski's main property at a prime `p` of `S`\nif there exists `r ∉ p` in the integral closure `S'` of `R` in `S`, such that `S'[1/r] = S[1/r]`. -/\ndef ZariskisMainProperty (p : Ideal S) : Prop :=\n  ∃ r : integralClosure R S, r.1 ∉ p ∧ Function.Bijective\n    (Localization.awayMap (integralClosure R S).val.toRingHom r)\n\nlemma zariskisMainProperty_iff {p : Ideal S} :\n    ZariskisMainProperty R p ↔ ∃ r ∉ p, IsIntegral R r ∧ ∀ x, ∃ m, IsIntegral R (r ^ m * x) := by\n  simp only [ZariskisMainProperty, Subtype.exists, ← exists_prop, @exists_comm (_ ∉ p)]\n  refine exists₃_congr fun r hr hrp ↦ ?_\n  rw [Function.Bijective, and_iff_right\n    (by exact IsLocalization.map_injective_of_injective _ _ _ Subtype.val_injective),\n    Localization.awayMap_surjective_iff]\n  simp [mem_integralClosure_iff]\n\nlemma zariskisMainProperty_iff' {p : Ideal S} :\n    ZariskisMainProperty R p ↔ ∃ r ∉ p, ∀ x, ∃ m, IsIntegral R (r ^ m * x) := by\n  refine zariskisMainProperty_iff.trans (exists_congr fun r ↦ and_congr_right fun hrp ↦\n    and_iff_right_of_imp fun H ↦ ?_)\n  obtain ⟨n, hn⟩ := H r\n  rw [← pow_succ] at hn\n  exact (IsIntegral.pow_iff (by simp)).mp hn\n\nTarget:\nlemma zariskisMainProperty_iff_exists_saturation_eq_top {p : Ideal S} :\n    ZariskisMainProperty R p ↔ ∃ r ∉ p, ∃ h : IsIntegral R r,\n      (integralClosure R S).saturation (.powers r) (by simpa [Submonoid.powers_le]) = ⊤ :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_988fd8cb4d8e","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"be396a1d6255c43c2383fef3588f29e2a2356a266cd92a0664681e1a6443fcb8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory","family_id":"zariskismainproperty_iff_exists_saturation_eq_top","file_id":"mathlib/Mathlib/RingTheory/ZariskisMainTheorem.lean","sample_id":"988fd8cb4d8e17e38f2a76981b98bfba51705b13e2c6dc4822f9d73a946a1e06"},"split":"train"}
{"binding":{"chosen_proof_sha256":"61743a744bcd9aa71d7e4c51326303887dfd1d0a7a128d08960e283447fbb42c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d4232d0429bbf109d599498b2662b7bb33bf4085569f416b9dac682235a68918","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"06643b8b06bfa39bedc6e6457247fd95883e76318998fda10cf95f90e40967df","source_sha256":"65877b034edf8cebcde3630a91726f90cf5b7369055e4e4a4603c386cc7b9be8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [numLeaves_eq_numNodes_succ]\n  exact x.numNodes.zero_lt_succ","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.181818,"token_length_ratio":0.272727},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"3ecedd4aaaf8c951c48fc9fbab0d112c6a6c1f72cd3a5f8976febb9bdfe7fb7f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.Notation\npublic import Mathlib.Util.CompileInductive\nimport Batteries.Tactic.Alias\n\nNamespace:\nBinaryTree\n\nLocal context:\n/-\nCopyright (c) 2019 mathlib community. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Wojciech Nawrocki\n-/\n/-!\n# Binary tree\n\nProvides binary tree storage for values of any type, with O(lg n) retrieval.\nSee also `Lean.Data.RBTree` for red-black trees - this version allows more operations\nto be defined and is better suited for in-kernel computation.\n\nWe also specialize for `BinaryTree Unit`, which is a binary tree without any\nadditional data. We provide the notation `a △ b` for making a `BinaryTree Unit` with children\n`a` and `b`.\n\n## References\n\n<https://leanprover-community.github.io/archive/stream/113488-general/topic/tactic.20question.html>\n-/\n\n@[expose] public section\n\n\n/-- A binary tree with values stored in non-leaf nodes. -/\ninductive BinaryTree.{u} (α : Type u) : Type u\n  | nil : BinaryTree α\n  | node (value : α) (left : BinaryTree α) (right : BinaryTree α) : BinaryTree α\n  deriving DecidableEq, Repr\ncompile_inductive% BinaryTree\n\n@[deprecated (since := \"2026-06-07\"), reducible]\nalias Tree := BinaryTree\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.nil`. -/\n@[deprecated BinaryTree.nil (since := \"2026-06-07\")]\nabbrev Tree.nil.{u} {α : Type u} : Tree α := BinaryTree.nil\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.node`. -/\n@[deprecated BinaryTree.node (since := \"2026-06-07\")]\nabbrev Tree.node.{u} {α : Type u}\n    (value : α) (left : Tree α) (right : Tree α) : Tree α :=\n  BinaryTree.node value left right\n\nnamespace BinaryTree\n\nuniverse u\n\nvariable {α : Type u}\n\ninstance : Inhabited (BinaryTree α) :=\n  ⟨nil⟩\n\n/--\nDo an action for every node of the tree.\nActions are taken in node -> left subtree -> right subtree recursive order.\nThis function is the `traverse` function for the `Traversable BinaryTree` instance.\n-/\ndef traverse\n    {m : Type* → Type*} [Applicative m] {α β} (f : α → m β) :\n    BinaryTree α → m (BinaryTree β)\n  | .nil => pure nil\n  | .node a l r => .node <$> f a <*> traverse f l <*> traverse f r\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.traverse`. -/\n@[deprecated BinaryTree.traverse (since := \"2026-06-07\")]\nabbrev _root_.Tree.traverse {m : Type* → Type*} [Applicative m] {α β} (f : α → m β)\n(t : Tree α) : m (Tree β) :=\n  BinaryTree.traverse f t\n\n/-- Apply a function to each value in the BinaryTree.\nThis is the `map` function for the `BinaryTree` functor.\n-/\ndef map {β} (f : α → β) : BinaryTree α → BinaryTree β\n  | nil => nil\n  | node a l r => node (f a) (map f l) (map f r)\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.map`. -/\n@[deprecated BinaryTree.map (since := \"2026-06-07\")]\nabbrev _root_.Tree.map {α β} (f : α → β) (t : Tree α) : Tree β := BinaryTree.map f t\n\ntheorem id_map (t : BinaryTree α) : t.map id = t := by\n  induction t with\n  | nil => rw [map]\n  | node v l r hl hr => rw [map, hl, hr, id_eq]\n\ntheorem comp_map {β γ : Type*} (f : α → β) (g : β → γ) (t : BinaryTree α) :\n    t.map (g ∘ f) = (t.map f).map g := by\n  induction t with\n  | nil => rw [map, map, map]\n  | node v l r hl hr => rw [map, map, map, hl, hr, Function.comp_apply]\n\ntheorem traverse_pure (t : BinaryTree α) {m : Type u → Type*}\n    [Applicative m] [LawfulApplicative m] :\n    t.traverse (pure : α → m α) = pure t := by\n  induction t with\n  | nil => rw [traverse]\n  | node v l r hl hr =>\n    rw [traverse, hl, hr, map_pure, pure_seq, seq_pure, map_pure, map_pure]\n\n/-- The number of internal nodes (i.e. not including leaves) of a binary tree -/\n@[simp]\ndef numNodes : BinaryTree α → ℕ\n  | nil => 0\n  | node _ a b => a.numNodes + b.numNodes + 1\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.numNodes`. -/\n@[deprecated BinaryTree.numNodes (since := \"2026-06-07\")]\nabbrev _root_.Tree.numNodes {α} (t : Tree α) : ℕ := BinaryTree.numNodes t\n\n/-- The number of leaves of a binary tree -/\n@[simp]\ndef numLeaves : BinaryTree α → ℕ\n  | nil => 1\n  | node _ a b => a.numLeaves + b.numLeaves\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.numLeaves`. -/\n@[deprecated BinaryTree.numLeaves (since := \"2026-06-07\")]\nabbrev _root_.Tree.numLeaves {α} (t : Tree α) : ℕ := BinaryTree.numLeaves t\n\n/-- The height - length of the longest path from the root - of a binary tree -/\n@[simp]\ndef height : BinaryTree α → ℕ\n  | nil => 0\n  | node _ a b => max a.height b.height + 1\n\nset_option linter.deprecated false in\n/-- **Alias** of `BinaryTree.height`. -/\n@[deprecated BinaryTree.height (since := \"2026-06-07\")]\nabbrev _root_.Tree.height {α} (t : Tree α) : ℕ := BinaryTree.height t\n\ntheorem numLeaves_eq_numNodes_succ (x : BinaryTree α) : x.numLeaves = x.numNodes + 1 := by\n  induction x <;> simp [*, Nat.add_comm, Nat.add_assoc, Nat.add_left_comm]\n\nTarget:\ntheorem numLeaves_pos (x : BinaryTree α) : 0 < x.numLeaves :=\n\nProof body:\n","rejected":"by\n  exact numLeaves_pos","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"008a525c4b8018985d5916fe71a5543656f1d1500759a1c71f193d194a3c3883","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Tree","family_id":"numleaves_pos","file_id":"mathlib/Mathlib/Data/Tree/Basic.lean","sample_id":"06643b8b06bfa39bedc6e6457247fd95883e76318998fda10cf95f90e40967df"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b8dd807b486dcb43a0cfd46155c68422d627f5cb03d289ff5154b4c959da2515","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6150e96d7c55abb9f7848262359606a5319554921ab6d8a96487846e3eb3b999","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"70091202abe0a1d76505ded2eda1922a6e55f006910256559adb12050fa4d2a2","source_sha256":"49729ab7657bf5b96e3883707dabc942ba77c2ab9a085e0f2d0959061ea37dde","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa [← coe_subset_coe] using! Set.not_subset","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.181818},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"3f072022e825bc41c97969f9ad9f2d7a4d4981c0e425b7518f21970bfce4a02a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Tactic.Monotonicity.Attr\npublic import Mathlib.Tactic.SetLike\npublic import Mathlib.Data.Set.Basic\n\nNamespace:\nSetLike\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Typeclass for types with a set-like extensionality property\n\nThe `Membership` typeclass is used to let terms of a type have elements.\nMany instances of `Membership` have a set-like extensionality property:\nthings are equal iff they have the same elements.  The `SetLike`\ntypeclass provides a unified interface to define a `Membership` that is\nextensional in this way.\n\nThe main use of `SetLike` is for algebraic subobjects (such as\n`Submonoid` and `Submodule`), whose non-proof data consists only of a\ncarrier set.  In such a situation, the projection to the carrier set\nis injective.\n\nIn general, a type `A` is `SetLike` with elements of type `B` if it\nhas an injective map to `Set B`.  This module provides standard\nboilerplate for every `SetLike`: a `coe_sort`, a `coe` to set,\nand various extensionality and simp lemmas. The order induced by set inclusion is\ncalled `PartialOrder.ofSetlike`: this is not an instance for flexibility in choosing orders.\nThe class `IsConcreteLE` abstractly states the order is equal to that induced by set inclusion;\nan instance is automatically available when defining a `PartialOrder` as\n`.ofSetLike (MySubobject X) X`.\n\nA typical subobject should be declared as:\n```\nstructure MySubobject (X : Type*) [ObjectTypeclass X] where\n  (carrier : Set X)\n  (op_mem' : ∀ {x : X}, x ∈ carrier → sorry ∈ carrier)\n\nnamespace MySubobject\n\nvariable {X : Type*} [ObjectTypeclass X] {x : X}\n\ninstance : SetLike (MySubobject X) X :=\n  ⟨MySubobject.carrier, fun p q h => by cases p; cases q; congr!⟩\n\ninstance : PartialOrder (MySubobject X) := .ofSetLike (MySubobject X) X\n\n@[simp] lemma mem_carrier {p : MySubobject X} : x ∈ p.carrier ↔ x ∈ (p : Set X) := Iff.rfl\n\n@[ext] theorem ext {p q : MySubobject X} (h : ∀ x, x ∈ p ↔ x ∈ q) : p = q := SetLike.ext h\n\n/-- Copy of a `MySubobject` with a new `carrier` equal to the old one. Useful to fix definitional\nequalities. See Note [range copy pattern]. -/\nprotected def copy (p : MySubobject X) (s : Set X) (hs : s = ↑p) : MySubobject X :=\n  { carrier := s\n    op_mem' := hs.symm ▸ p.op_mem' }\n\n@[simp] lemma coe_copy (p : MySubobject X) (s : Set X) (hs : s = ↑p) :\n  (p.copy s hs : Set X) = s := rfl\n\nlemma copy_eq (p : MySubobject X) (s : Set X) (hs : s = ↑p) : p.copy s hs = p :=\n  SetLike.coe_injective hs\n\nend MySubobject\n```\n\nAn alternative to `SetLike` could have been an extensional `Membership` typeclass:\n```\nclass ExtMembership (α : out_param <| Type u) (β : Type v) extends Membership α β where\n  (ext_iff : ∀ {s t : β}, s = t ↔ ∀ (x : α), x ∈ s ↔ x ∈ t)\n```\nWhile this is equivalent, `SetLike` conveniently uses a carrier set projection directly.\n\n## Tags\n\nsubobjects\n-/\n\n@[expose] public section\n\nassert_not_exists RelIso\n\n/-- A class to indicate that there is a canonical injection between `A` and `Set B`.\n\nThis has the effect of giving terms of `A` elements of type `B` (through a `Membership`\ninstance) and a compatible coercion to `Type*` as a subtype.\n\nNote: if `SetLike.coe` is a projection, implementers should create a simp lemma such as\n```\n@[simp] lemma mem_carrier {p : MySubobject X} : x ∈ p.carrier ↔ x ∈ (p : Set X) := Iff.rfl\n```\nto normalize terms.\n\nIf you declare an unbundled subclass of `SetLike`, for example:\n```\nclass MulMemClass (S : Type*) (M : Type*) [Mul M] [SetLike S M] where\n  ...\n```\nThen you should *not* repeat the `outParam` declaration so `SetLike` will supply the value instead.\nThis ensures your subclass will not have issues with synthesis of the `[Mul M]` parameter starting\nbefore the value of `M` is known.\n-/\n@[notation_class* carrier Simps.findCoercionArgs]\nclass SetLike (A : Type*) (B : outParam Type*) where\n  /-- The coercion from a term of a `SetLike` to its corresponding `Set`. -/\n  protected coe : A → Set B\n  /-- The coercion from a term of a `SetLike` to its corresponding `Set` is injective. -/\n  coe_injective : Function.Injective coe\n\nattribute [coe] SetLike.coe\n\nnamespace SetLike\n\nvariable {A : Type*} {B : Type*} [i : SetLike A B]\n\n@[deprecated (since := \"2026-06-04\")] alias coe_injective' := coe_injective\n\ninstance : CoeTC A (Set B) where coe := SetLike.coe\n\ninstance (priority := 100) instMembership : Membership B A :=\n  ⟨fun p x => x ∈ (p : Set B)⟩\n\ninstance (priority := 100) : CoeSort A (Type _) :=\n  ⟨fun p => { x : B // x ∈ p }⟩\n\nsection Delab\nopen Lean PrettyPrinter.Delaborator SubExpr\n\n/-- For terms that match the `CoeSort` instance's body, pretty print as `↥S`\nrather than as `{ x // x ∈ S }`. The discriminating feature is that membership\nuses the `SetLike.instMembership` instance. -/\n@[app_delab Subtype]\nmeta def delabSubtypeSetLike : Delab := whenPPOption getPPNotation do\n  let #[_, .lam n _ body _] := (← getExpr).getAppArgs | failure\n  guard <| body.isAppOf ``Membership.mem\n  let #[_, _, inst, _, .bvar 0] := body.getAppArgs | failure\n  guard <| inst.isAppOfArity ``instMembership 3\n  let S ← withAppArg <| withBindingBody n <| withNaryArg 3 delab\n  `(↥$S)\n\nend Delab\n\nvariable (p q : A)\n\n@[simp, norm_cast]\ntheorem coe_sort_coe : ((p : Set B) : Type _) = p :=\n  rfl\n\nvariable {p q}\n\nprotected theorem «exists» {q : p → Prop} : (∃ x, q x) ↔ ∃ (x : B) (h : x ∈ p), q ⟨x, ‹_›⟩ :=\n  SetCoe.exists\n\nprotected theorem «forall» {q : p → Prop} : (∀ x, q x) ↔ ∀ (x : B) (h : x ∈ p), q ⟨x, ‹_›⟩ :=\n  SetCoe.forall\n\n@[simp, norm_cast]\ntheorem coe_set_eq : (p : Set B) = q ↔ p = q :=\n  coe_injective.eq_iff\n\n@[norm_cast] lemma coe_ne_coe : (p : Set B) ≠ q ↔ p ≠ q := coe_injective.ne_iff\n\ntheorem ext' (h : (p : Set B) = q) : p = q :=\n  coe_injective h\n\ntheorem ext'_iff : p = q ↔ (p : Set B) = q :=\n  coe_set_eq.symm\n\n/-- Note: implementers of `SetLike` must copy this lemma in order to tag it with `@[ext]`. -/\ntheorem ext (h : ∀ x, x ∈ p ↔ x ∈ q) : p = q :=\n  coe_injective <| Set.ext h\n\ntheorem ext_iff : p = q ↔ ∀ x, x ∈ p ↔ x ∈ q :=\n  coe_injective.eq_iff.symm.trans Set.ext_iff\n\n@[simp, push]\ntheorem mem_coe {x : B} : x ∈ (p : Set B) ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_eq_coe {x y : p} : (x : B) = y ↔ x = y :=\n  Subtype.ext_iff.symm\n\n@[simp]\ntheorem coe_mem (x : p) : (x : B) ∈ p :=\n  x.2\n\n@[aesop 5% (rule_sets := [SetLike!])]\nlemma mem_of_subset {s : Set B} (hp : s ⊆ p) {x : B} (hx : x ∈ s) : x ∈ p := hp hx\n\n@[simp]\nprotected theorem eta (x : p) (hx : (x : B) ∈ p) : (⟨x, hx⟩ : p) = x := rfl\n\n@[simp] lemma setOf_mem_eq (a : A) : {b | b ∈ a} = a := rfl\n\n@[nontriviality]\nlemma mem_of_subsingleton [Subsingleton B] (S : A) [h : Nonempty S] {b : B} : b ∈ S := by\n  obtain ⟨s, hs⟩ := nonempty_subtype.mp h\n  simpa [Subsingleton.elim b s]\n\n/-- If `s` is a proper element of a `SetLike` structure (i.e., `s ≠ ⊤`) and the top element\ncoerces to the universal set, then there exists an element not in `s`. -/\nlemma exists_not_mem_of_ne_top [LE A] [OrderTop A] (s : A) (hs : s ≠ ⊤)\n    (h_top : ((⊤ : A) : Set B) = Set.univ := by simp) :\n    ∃ b : B, b ∉ s := by\n  simpa [-SetLike.coe_set_eq, SetLike.ext'_iff, h_top, Set.ne_univ_iff_exists_notMem] using hs\n\nend SetLike\n\n/-- A class to indicate that the canonical injection between `A` and `Set B` is order-preserving.\n\nAn instance of this class is automatically available on any partial order defined as\n`PartialOrder.ofSetLike`.\n-/\nclass IsConcreteLE (A : Type*) (B : outParam Type*) [SetLike A B] [LE A] where\n  /-- The coercion from a `SetLike` type preserves the ordering. -/\n  protected coe_subset_coe' {S T : A} : SetLike.coe S ⊆ SetLike.coe T ↔ S ≤ T\n\nsection default\n\nvariable (A B : Type*) [SetLike A B]\n\n/-- The order induced from a `SetLike` instance by inclusion. -/\n@[reducible] def LE.ofSetLike : LE A where\n  le := fun H K => ∀ ⦃x⦄, x ∈ H → x ∈ K\n\n/-- The partial order induced from a `SetLike` instance by inclusion.\n\nA partial order defined as `.ofSetLike` will automatically make available an instance\nof `IsConcreteLE`.\n-/\n@[reducible] def PartialOrder.ofSetLike : PartialOrder A where\n  __ := LE.ofSetLike A B\n  __ := PartialOrder.lift (SetLike.coe : A → Set B) SetLike.coe_injective\n\ninstance : letI := PartialOrder.ofSetLike A B; IsConcreteLE A B :=\n  letI := PartialOrder.ofSetLike A B; { coe_subset_coe' := Iff.rfl }\n\nend default\n\nnamespace SetLike\n\nvariable {A B : Type*} [SetLike A B]\n\nsection LE\n\nvariable [LE A] [IsConcreteLE A B] {p q : A}\n\n@[simp, norm_cast, gcongr] lemma coe_subset_coe {S T : A} : (S : Set B) ⊆ T ↔ S ≤ T :=\n  IsConcreteLE.coe_subset_coe'\n\ntheorem le_def {S T : A} : S ≤ T ↔ ∀ ⦃x : B⦄, x ∈ S → x ∈ T := by\n  simp [← coe_subset_coe, Set.subset_def]\n\n@[gcongr low] -- lower priority than `Set.mem_of_subset_of_mem`\nalias ⟨_root_.mem_of_le_of_mem, _⟩ := le_def\n\n@[deprecated (since := \"2026-01-07\")] alias GCongr.mem_of_le_of_mem := _root_.mem_of_le_of_mem\n\nTarget:\ntheorem not_le_iff_exists : ¬p ≤ q ↔ ∃ x ∈ p, x ∉ q :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_70091202abe0","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"1ca45044ccec6859f1ba9dd39aa479df5fc4b786c9682126f4cd39514593235c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/SetLike","family_id":"not_le_iff_exists","file_id":"mathlib/Mathlib/Data/SetLike/Basic.lean","sample_id":"70091202abe0a1d76505ded2eda1922a6e55f006910256559adb12050fa4d2a2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"53c62f34c0746917df7ca2e7b0151f3d1cc2f5a2e7dd8a4133dc4a6cf7d81680","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"42a4c36eb13ded6148878cebea2ece5533a6d03f906d1cecf0df378f10eb9caa","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"21df6f68f8686f6d7b4ab378fa3f7d2564d582425a9b504a6d726062d4362c74","source_sha256":"96d763a04d380b57c61dd8bcbe51e54cac3fe376ddcd3d5a1be83055f4846a12","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro f g hfg\n  have : f.toAlgHom = g.toAlgHom := by\n    apply (hμ.powerBasis K).algHom_ext\n    rw [AlgEquiv.coe_algHom, AlgEquiv.coe_algHom, powerBasis_gen,\n      ← autToPow_spec K hμ g, ← autToPow_spec K hμ f, hfg]\n  exact AlgEquiv.coe_algHom_injective this","hard_negative":true,"metrics":{"chosen_tokens":59,"rejected_tokens":2,"token_jaccard":0.03125,"token_length_ratio":0.033898},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"3f35029584b59c15e415f5d98bba7e4c2535a67565bed969284d86af78d34d23","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots\npublic import Mathlib.FieldTheory.PolynomialGaloisGroup\n\nNamespace:\nIsPrimitiveRoot\n\nLocal context:\n/-\nCopyright (c) 2022 Eric Rodriguez. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Rodriguez\n-/\n/-!\n# Galois group of cyclotomic extensions\n\nIn this file, we show the relationship between the Galois group of `K(ζₙ)` and `(ZMod n)ˣ`;\nit is always a subgroup, and if the `n`th cyclotomic polynomial is irreducible, they are isomorphic.\n\n## Main results\n\n* `IsPrimitiveRoot.autToPow_injective`: `IsPrimitiveRoot.autToPow` is injective\n  in the case that it's considered over a cyclotomic field extension.\n* `IsCyclotomicExtension.autEquivPow`: If the `n`th cyclotomic polynomial is irreducible in `K`,\n  then `IsPrimitiveRoot.autToPow` is a `MulEquiv` (for example, in `ℚ` and certain `𝔽ₚ`).\n* `galXPowEquivUnitsZMod`, `galCyclotomicEquivUnitsZMod`: Repackage\n  `IsCyclotomicExtension.autEquivPow` in terms of `Polynomial.Gal`.\n* `IsCyclotomicExtension.Aut.commGroup`: Cyclotomic extensions are abelian.\n\n## References\n\n* https://kconrad.math.uconn.edu/blurbs/galoistheory/cyclotomic.pdf\n\n## TODO\n\n* We currently can get away with the fact that the power of a primitive root is a primitive root,\n  but the correct long-term solution for computing other explicit Galois groups is creating\n  `PowerBasis.map_conjugate`; but figuring out the exact correct assumptions + proof for this is\n  mathematically nontrivial. (Current thoughts: the correct condition is that the annihilating\n  ideal of both elements is equal. This may not hold in an ID, and definitely holds in an ICD.)\n\n-/\n\n@[expose] public section\n\n\nvariable {n : ℕ} [NeZero n] (K : Type*) [Field K] {L : Type*} {μ : L}\n\nopen Polynomial IsCyclotomicExtension\n\nopen scoped Cyclotomic\n\nnamespace IsPrimitiveRoot\n\nvariable [CommRing L] [IsDomain L] (hμ : IsPrimitiveRoot μ n) [Algebra K L]\n  [IsCyclotomicExtension {n} K L]\n\n/-- `IsPrimitiveRoot.autToPow` is injective in the case that it's considered over a cyclotomic\nfield extension. -/\n\nTarget:\ntheorem autToPow_injective : Function.Injective <| hμ.autToPow K :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_21df6f68f868","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"94deba3e6bca2364d76a0900744d7e1cd329b86fc42df56259f2b88a68805cd8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Cyclotomic","family_id":"auttopow_injective","file_id":"mathlib/Mathlib/NumberTheory/Cyclotomic/Gal.lean","sample_id":"21df6f68f8686f6d7b4ab378fa3f7d2564d582425a9b504a6d726062d4362c74"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b7ecb19a982d835fa985827cc7886e0d4262e615155091b32c40d6c7877f5f38","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"42a9f1a907278d22812ff81dd94d8d168bcbffa80c21609d5e1957007c2f10c4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5bfec5ae03bfd55ef1bbbff853e84f4a25242edd9b4ad0bd7f154c9852ef86aa","source_sha256":"d7743ebc410d9ee31a12263ae8d894bf6aecd1d52cd1a367eccb2f5de9ab8de8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [minimalPeriod_piMap]\n  by_cases h : {n | 0 < n ∧ ∀ (i : ι), minimalPeriod (f i) (x i) ∣ n}.Nonempty\n  · exact (Nat.sInf_mem h).2 i\n  · simp [not_nonempty_iff_eq_empty.mp h]","hard_negative":false,"metrics":{"chosen_tokens":56,"rejected_tokens":61,"token_jaccard":0.897436,"token_length_ratio":1.089286},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"3f3af03442b8b49fc7366904d8c4e13b9cc84360b97f5c12bd2469121c57cc16","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GCDMonoid.Finset\npublic import Mathlib.Data.Nat.Prime.Basic\npublic import Mathlib.Data.PNat.Basic\npublic import Mathlib.Dynamics.PeriodicPts.Defs\npublic import Mathlib.Order.Lattice.Nat\n\nNamespace:\nFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Extra lemmas about periodic points\n-/\n\npublic section\n\nopen Nat Set\n\nnamespace Function\nvariable {α : Type*} {f : α → α} {x y : α}\n\nopen Function (Commute)\n\ntheorem directed_ptsOfPeriod_pnat (f : α → α) : Directed (· ⊆ ·) fun n : ℕ+ => ptsOfPeriod f n :=\n  fun m n => ⟨m * n, fun _ hx => hx.mul_const n, fun _ hx => hx.const_mul m⟩\n\nvariable (f) in\ntheorem bijOn_periodicPts : BijOn f (periodicPts f) (periodicPts f) :=\n  iUnion_pnat_ptsOfPeriod f ▸\n    bijOn_iUnion_of_directed (directed_ptsOfPeriod_pnat f) fun i => bijOn_ptsOfPeriod f i.pos\n\ntheorem minimalPeriod_eq_prime_iff {p : ℕ} [hp : Fact p.Prime] :\n    minimalPeriod f x = p ↔ IsPeriodicPt f p x ∧ ¬IsFixedPt f x := by\n  rw [Function.isPeriodicPt_iff_minimalPeriod_dvd, Nat.dvd_prime hp.out,\n    ← minimalPeriod_eq_one_iff_isFixedPt.not, or_and_right, and_not_self_iff, false_or,\n    iff_self_and]\n  exact fun h ↦ ne_of_eq_of_ne h hp.out.ne_one\n\ntheorem minimalPeriod_eq_sInf_n_pos_IsPeriodicPt :\n    minimalPeriod f x = sInf { n > 0 | IsPeriodicPt f n x } := by\n  dsimp +instances [minimalPeriod, periodicPts, sInf]\n  grind\n\n/-- The backward direction of `minimalPeriod_eq_prime_iff`. -/\ntheorem minimalPeriod_eq_prime {p : ℕ} [hp : Fact p.Prime] (hper : IsPeriodicPt f p x)\n    (hfix : ¬IsFixedPt f x) : minimalPeriod f x = p :=\n  minimalPeriod_eq_prime_iff.mpr ⟨hper, hfix⟩\n\ntheorem minimalPeriod_eq_prime_pow {p k : ℕ} [hp : Fact p.Prime] (hk : ¬IsPeriodicPt f (p ^ k) x)\n    (hk1 : IsPeriodicPt f (p ^ (k + 1)) x) : minimalPeriod f x = p ^ (k + 1) := by\n  apply Nat.eq_prime_pow_of_dvd_least_prime_pow hp.out <;>\n    rwa [← isPeriodicPt_iff_minimalPeriod_dvd]\n\ntheorem Commute.minimalPeriod_of_comp_dvd_mul {g : α → α} (h : Commute f g) :\n    minimalPeriod (f ∘ g) x ∣ minimalPeriod f x * minimalPeriod g x :=\n  dvd_trans h.minimalPeriod_of_comp_dvd_lcm (Nat.lcm_dvd_mul _ _)\n\ntheorem Commute.minimalPeriod_of_comp_eq_mul_of_coprime {g : α → α} (h : Commute f g)\n    (hco : Coprime (minimalPeriod f x) (minimalPeriod g x)) :\n    minimalPeriod (f ∘ g) x = minimalPeriod f x * minimalPeriod g x := by\n  apply h.minimalPeriod_of_comp_dvd_mul.antisymm\n  suffices ∀ {f g : α → α},\n      Commute f g →\n        Coprime (minimalPeriod f x) (minimalPeriod g x) →\n          minimalPeriod f x ∣ minimalPeriod (f ∘ g) x from\n    hco.mul_dvd_of_dvd_of_dvd (this h hco) (h.comp_eq.symm ▸ this h.symm hco.symm)\n  intro f g h hco\n  refine hco.dvd_of_dvd_mul_left (IsPeriodicPt.left_of_comp h ?_ ?_).minimalPeriod_dvd\n  · exact (isPeriodicPt_minimalPeriod _ _).const_mul _\n  · exact (isPeriodicPt_minimalPeriod _ _).mul_const _\n\nsection Fintype\n\nopen Fintype\n\ntheorem minimalPeriod_le_card [Fintype α] : minimalPeriod f x ≤ card α := by\n  rw [← periodicOrbit_length]\n  exact List.Nodup.length_le_card nodup_periodicOrbit\n\ntheorem isPeriodicPt_factorial_card_of_mem_periodicPts [Fintype α] (h : x ∈ periodicPts f) :\n    IsPeriodicPt f (card α)! x :=\n  isPeriodicPt_iff_minimalPeriod_dvd.mpr\n    (Nat.dvd_factorial (minimalPeriod_pos_of_mem_periodicPts h) minimalPeriod_le_card)\n\ntheorem mem_periodicPts_iff_isPeriodicPt_factorial_card [Fintype α] :\n    x ∈ periodicPts f ↔ IsPeriodicPt f (card α)! x where\n  mp := isPeriodicPt_factorial_card_of_mem_periodicPts\n  mpr h := minimalPeriod_pos_iff_mem_periodicPts.mp\n    (IsPeriodicPt.minimalPeriod_pos (Nat.factorial_pos _) h)\n\ntheorem Injective.mem_periodicPts [Finite α] (h : Injective f) (x : α) : x ∈ periodicPts f := by\n  obtain ⟨m, n, heq, hne⟩ : ∃ m n, f^[m] x = f^[n] x ∧ m ≠ n := by\n    simpa [Injective] using not_injective_infinite_finite (f^[·] x)\n  rcases lt_or_gt_of_ne hne with hlt | hlt\n  · exact mk_mem_periodicPts (by lia) (iterate_cancel h heq.symm)\n  · exact mk_mem_periodicPts (by lia) (iterate_cancel h heq)\n\ntheorem injective_iff_periodicPts_eq_univ [Finite α] : Injective f ↔ periodicPts f = univ := by\n  refine ⟨fun h ↦ eq_univ_iff_forall.mpr h.mem_periodicPts, fun h ↦ ?_⟩\n  rw [Finite.injective_iff_surjective, ← range_eq_univ, ← univ_subset_iff, ← h]\n  apply periodicPts_subset_range\n\ntheorem injective_iff_iterate_factorial_card_eq_id [Fintype α] :\n    Injective f ↔ f^[(card α)!] = id := by\n  simp only [injective_iff_periodicPts_eq_univ, mem_periodicPts_iff_isPeriodicPt_factorial_card,\n    funext_iff, eq_univ_iff_forall, IsPeriodicPt, id, IsFixedPt]\n\nend Fintype\n\nend Function\n\nnamespace Function\n\nsection Prod\n\nvariable {α β : Type*} {f : α → α} {g : β → β} {x : α × β} {a : α} {b : β} {m n : ℕ}\n\ntheorem minimalPeriod_prodMap (f : α → α) (g : β → β) (x : α × β) :\n    minimalPeriod (Prod.map f g) x = (minimalPeriod f x.1).lcm (minimalPeriod g x.2) :=\n  eq_of_forall_dvd <| by simp [← isPeriodicPt_iff_minimalPeriod_dvd, Nat.lcm_dvd_iff]\n\ntheorem minimalPeriod_fst_dvd : minimalPeriod f x.1 ∣ minimalPeriod (Prod.map f g) x := by\n  rw [minimalPeriod_prodMap]; exact Nat.dvd_lcm_left _ _\n\ntheorem minimalPeriod_snd_dvd : minimalPeriod g x.2 ∣ minimalPeriod (Prod.map f g) x := by\n  rw [minimalPeriod_prodMap]; exact Nat.dvd_lcm_right _ _\n\nend Prod\n\nsection Pi\n\nvariable {ι : Type*} {α : ι → Type*} {f : ∀ i, α i → α i} {x : ∀ i, α i}\n\n/-- This `sInf` can be regarded as a generalized version of LCM\nfor possibly infinite sets and types. -/\ntheorem minimalPeriod_piMap :\n    minimalPeriod (Pi.map f) x = sInf { n > 0 | ∀ i, minimalPeriod (f i) (x i) ∣ n } := by\n  conv_lhs => simp [minimalPeriod_eq_sInf_n_pos_IsPeriodicPt]\n  simp [← isPeriodicPt_iff_minimalPeriod_dvd]\n\ntheorem minimalPeriod_piMap_fintype [Fintype ι] :\n    minimalPeriod (Pi.map f) x = Finset.univ.lcm (fun i => minimalPeriod (f i) (x i)) :=\n  eq_of_forall_dvd <| by simp [← isPeriodicPt_iff_minimalPeriod_dvd]\n\nTarget:\ntheorem minimalPeriod_single_dvd_minimalPeriod_piMap (i : ι) :\n    minimalPeriod (f i) (x i) ∣ minimalPeriod (Pi.map f) x :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  simp only [minimalPeriod_piMap]\n  by_cases h : {n | 0 < n ∧ ∀ (i : ι), minimalPeriod (f i) (x i) ∣ n}.Nonempty\n  · exact (Nat.sInf_mem h).2 i\n  · simp [not_nonempty_iff_eq_empty.mp h]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/PeriodicPts","family_id":"minimalperiod_single_dvd_minimalperiod_pimap","file_id":"mathlib/Mathlib/Dynamics/PeriodicPts/Lemmas.lean","sample_id":"5bfec5ae03bfd55ef1bbbff853e84f4a25242edd9b4ad0bd7f154c9852ef86aa"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a8bd42ecc90333f6177ba81b0d401c4a0d5b772d591ccefb5c358cae0906bf4e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"66a10d4bc2e644ca79ec68399eade5487ca74804aa7aa3c3538e9529456e8bc5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"389d6b6e4066fc2cabec7aefc51cac6e21581a5a3e730cc7243010a6f145a966","source_sha256":"8656d251697095f7fe7e214833a10429be9a10229d106380fe6a23432c3fbd73","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let e : (L ⊗[R] S) ≃ₐ[K] L ⊗[K] (K ⊗[R] S) :=\n    (Algebra.TensorProduct.cancelBaseChange R K K L S).symm\n  let e2 : L ⊗[K] (K ⊗[R] S) ≃ₐ[K] (K ⊗[R] S) ⊗[K] L := Algebra.TensorProduct.comm ..\n  have heq : Algebra.TensorProduct.map (Algebra.ofId K L) (AlgHom.id R S) =\n      (e.symm.toAlgHom.comp e2.symm.toAlgHom).comp\n        (IsScalarTower.toAlgHom K (K ⊗[R] S) ((K ⊗[R] S) ⊗[K] L)) := by\n    ext; simp [e, e2]\n  rw [heq]\n  simp only [AlgHom.toRingHom_eq_coe, AlgHom.comp_toRingHom,\n    AlgEquiv.toAlgHom_toRingHom, IsScalarTower.coe_toAlgHom, comap_comp]\n  exact (isHomeomorph_comap_of_isPurelyInseparable K L (K ⊗[R] S)).comp <|\n    (isHomeomorph_comap_of_bijective e2.symm.bijective).comp <|\n    isHomeomorph_comap_of_bijective e.symm.bijective","hard_negative":true,"metrics":{"chosen_tokens":226,"rejected_tokens":3,"token_jaccard":0.038462,"token_length_ratio":0.013274},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"3f62b7423568eb97eade30c7f53ed16ddc5871078a750a11cf768939c6793698","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.PurelyInseparable.Basic\npublic import Mathlib.RingTheory.Flat.Basic\npublic import Mathlib.RingTheory.Spectrum.Prime.Topology\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten, Junyan Xu\n-/\n/-!\n# Purely inseparable extensions are universal homeomorphisms\n\nIf `K` is a purely inseparable extension of `k`, the induced map `Spec K ⟶ Spec k` is a universal\nhomeomorphism, i.e. it stays a homeomorphism after arbitrary base change.\n\n## Main results\n\n- `PrimeSpectrum.isHomeomorph_comap`: if `f : R →+* S` is a ring map with locally nilpotent kernel\n  such that for every `x : S`, there exists `n > 0` such that `x ^ n` is in the image of `f`,\n  `Spec f` is a homeomorphism.\n- `PrimeSpectrum.isHomeomorph_comap_of_isPurelyInseparable`: `Spec K ⟶ Spec k` is a universal\n  homeomorphism for a purely inseparable field extension `K` over `k`.\n-/\n\npublic section\n\nopen TensorProduct\n\nvariable (k K R S : Type*) [Field k] [Field K] [Algebra k K] [CommRing R] [Algebra k R] [CommRing S]\n\nvariable {R S} in\n/-- If the kernel of `f : R →+* S` consists of nilpotent elements and for every `x : S`,\nthere exists `n > 0` such that `x ^ n` is in the range of `f`, then `Spec f` is a homeomorphism.\nNote: This does not hold for semirings, because `ℕ →+* ℤ` satisfies these conditions, but\n`Spec ℕ` has one more point than `Spec ℤ`. -/\n@[stacks 0BR8 \"Homeomorphism part\"]\nlemma PrimeSpectrum.isHomeomorph_comap (f : R →+* S) (H : ∀ (x : S), ∃ n > 0, x ^ n ∈ f.range)\n    (hker : RingHom.ker f ≤ nilradical R) : IsHomeomorph (comap f) := by\n  have h1 : Function.Injective (comap f) := by\n    intro q q' hqq'\n    ext x\n    obtain ⟨n, hn, y, hy⟩ := H x\n    rw [← q.2.pow_mem_iff_mem _ hn, ← q'.2.pow_mem_iff_mem _ hn, ← hy]\n    rw [PrimeSpectrum.ext_iff, SetLike.ext_iff] at hqq'\n    apply hqq'\n  have hint : f.kerLift.IsIntegral := fun x ↦\n    have ⟨n, hn, y, hy⟩ := H x\n    let _ := f.kerLift.toAlgebra\n    IsIntegral.of_pow hn (hy ▸ f.kerLift.isIntegralElem_map (x := ⟦y⟧))\n  have hbij : Function.Bijective (comap f) :=\n    ⟨h1, (comap_quotientMk_bijective_of_le_nilradical hker).2.comp <|\n      hint.comap_surjective f.kerLift_injective⟩\n  refine ⟨continuous_comap f, ?_, h1, hbij.2⟩\n  rw [isTopologicalBasis_basic_opens.isOpenMap_iff]\n  rintro - ⟨s, rfl⟩\n  obtain ⟨n, hn, r, hr⟩ := H s\n  have : (comap f) '' (basicOpen s) = basicOpen r :=\n    (Set.eq_preimage_iff_image_eq hbij).mp <| by rw [← basicOpen_pow _ n hn, ← hr]; rfl\n  exact this ▸ isOpen_basicOpen\n\n/-- Purely inseparable field extensions are universal homeomorphisms. -/\n@[stacks 0BRA \"Special case for purely inseparable field extensions\"]\nlemma PrimeSpectrum.isHomeomorph_comap_of_isPurelyInseparable [IsPurelyInseparable k K] :\n    IsHomeomorph (comap <| algebraMap R (R ⊗[k] K)) := by\n  let q := ringExpChar k\n  refine isHomeomorph_comap _ (IsPurelyInseparable.exists_pow_mem_range_tensorProduct) ?_\n  convert! bot_le\n  rw [← RingHom.injective_iff_ker_eq_bot]\n  exact Algebra.TensorProduct.includeLeft_injective (S := R) (algebraMap k K).injective\n\n/-- If `L` is a purely inseparable extension of `K` over `R` and `S` is an `R`-algebra,\nthe induced map `Spec (L ⊗[R] S) ⟶ Spec (K ⊗[R] S)` is a homeomorphism. -/\n\nTarget:\nlemma PrimeSpectrum.isHomeomorph_comap_tensorProductMap_of_isPurelyInseparable [Algebra R K]\n    [Algebra R S] (L : Type*) [Field L] [Algebra R L] [Algebra K L] [IsScalarTower R K L]\n    [IsPurelyInseparable K L] :\n    IsHomeomorph (comap (Algebra.TensorProduct.map (Algebra.ofId K L) (.id R S)).toRingHom) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_389d6b6e4066","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"008529422e97546d74652288c6977e48f8e59d3a573d905e1e78b431e5bcd8ab","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Spectrum","family_id":"primespectrum","file_id":"mathlib/Mathlib/RingTheory/Spectrum/Prime/Homeomorph.lean","sample_id":"389d6b6e4066fc2cabec7aefc51cac6e21581a5a3e730cc7243010a6f145a966"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bcf496a337f71ad6a5e3548ca780931dad5f52b6127e92076ca6f453d0ce3736","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"444126766a6d3867417667c40dd37f00d68870fb70c1031516cd452dd27d2227","source_sha256":"3d16eac88c0b4f07fc5014be8b3e025e599c272f2809936f3c6e3dad4757fa05","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine le_trans (measure_iUnion_le _) (le_trans\n    (ENNReal.tsum_le_tsum <| notConvergentSeqLTIndex_spec (half_pos hε) hf hsm hs hfg) ?_)\n  simp_rw [ENNReal.ofReal_mul (half_pos hε).le]\n  rw [ENNReal.tsum_mul_left, ← ENNReal.ofReal_tsum_of_nonneg, inv_eq_one_div, tsum_geometric_two,\n    ← ENNReal.ofReal_mul (half_pos hε).le, div_mul_cancel₀ ε two_ne_zero]\n  · intro n; positivity\n  · rw [inv_eq_one_div]\n    exact summable_geometric_two","hard_negative":false,"metrics":{"chosen_tokens":86,"rejected_tokens":3,"token_jaccard":0.045455,"token_length_ratio":0.034884},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"3f9a3dab90c7c3bc1749dd82fc7754b2199ebea3f2baa34867ed9fd7884fce8f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic\n\nNamespace:\nMeasureTheory.Egorov\n\nLocal context:\n/-\nCopyright (c) 2022 Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kexing Ying\n-/\n/-!\n# Egorov theorem\n\nThis file contains the Egorov theorem which states that an almost everywhere convergent\nsequence on a finite measure space converges uniformly except on an arbitrarily small set.\nThis theorem is useful for the Vitali convergence theorem as well as theorems regarding\nconvergence in measure.\n\n## Main results\n\n* `MeasureTheory.tendstoUniformlyOn_of_ae_tendsto`: Egorov's theorem which shows that a sequence of\n  almost everywhere convergent functions converges uniformly except on an arbitrarily small set.\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen MeasureTheory NNReal ENNReal Topology\n\nnamespace MeasureTheory\n\nopen Set Filter TopologicalSpace\n\nvariable {α β ι : Type*} {m : MeasurableSpace α} [PseudoEMetricSpace β] {μ : Measure α}\n\nnamespace Egorov\n\n/-- Given a sequence of functions `f` and a function `g`, `notConvergentSeq f g n j` is the\nset of elements such that `f k x` and `g x` are separated by at least `1 / (n + 1)` for some\n`k ≥ j`.\n\nThis definition is useful for Egorov's theorem. -/\ndef notConvergentSeq [Preorder ι] (f : ι → α → β) (g : α → β) (n : ℕ) (j : ι) : Set α :=\n  ⋃ (k) (_ : j ≤ k), { x | (n : ℝ≥0∞)⁻¹ < edist (f k x) (g x) }\n\nvariable {n : ℕ} {j : ι} {s : Set α} {ε : ℝ} {f : ι → α → β} {g : α → β}\n\ntheorem mem_notConvergentSeq_iff [Preorder ι] {x : α} :\n    x ∈ notConvergentSeq f g n j ↔ ∃ k ≥ j, (n : ℝ≥0∞)⁻¹ < edist (f k x) (g x) := by\n  simp_rw [notConvergentSeq, Set.mem_iUnion, exists_prop, mem_setOf]\n\ntheorem notConvergentSeq_antitone [Preorder ι] : Antitone (notConvergentSeq f g n) :=\n  fun _ _ hjk => Set.iUnion₂_mono' fun l hl => ⟨l, le_trans hjk hl, Set.Subset.rfl⟩\n\ntheorem measure_inter_notConvergentSeq_eq_zero [SemilatticeSup ι] [Nonempty ι]\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :\n    μ (s ∩ ⋂ j, notConvergentSeq f g n j) = 0 := by\n  simp_rw [EMetric.tendsto_atTop, ae_iff] at hfg\n  rw [← nonpos_iff_eq_zero, ← hfg]\n  refine measure_mono fun x => ?_\n  simp only [Set.mem_inter_iff, Set.mem_iInter, mem_notConvergentSeq_iff]\n  push Not\n  rintro ⟨hmem, hx⟩\n  refine ⟨hmem, (n : ℝ≥0∞)⁻¹, by simp, fun N => ?_⟩\n  obtain ⟨n, hn₁, hn₂⟩ := hx N\n  exact ⟨n, hn₁, hn₂.le⟩\n\ntheorem notConvergentSeq_measurableSet [Preorder ι] [Countable ι]\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a))) :\n    MeasurableSet (notConvergentSeq f g n j) :=\n  MeasurableSet.iUnion fun k ↦ MeasurableSet.iUnion fun _ ↦\n      measurableSet_lt measurable_const <| hf k\n\ntheorem measure_notConvergentSeq_tendsto_zero [SemilatticeSup ι] [Countable ι]\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a))) (hsm : MeasurableSet s)\n    (hs : μ s ≠ ∞) (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :\n    Tendsto (fun j => μ (s ∩ notConvergentSeq f g n j)) atTop (𝓝 0) := by\n  rcases isEmpty_or_nonempty ι with h | h\n  · have : (fun j => μ (s ∩ notConvergentSeq f g n j)) = fun j => 0 := by\n      simp only [eq_iff_true_of_subsingleton]\n    rw [this]\n    exact tendsto_const_nhds\n  rw [← measure_inter_notConvergentSeq_eq_zero hfg n, Set.inter_iInter]\n  refine tendsto_measure_iInter_atTop\n    (fun n ↦ (hsm.inter <| notConvergentSeq_measurableSet hf).nullMeasurableSet)\n    (fun k l hkl => Set.inter_subset_inter_right _ <| notConvergentSeq_antitone hkl)\n    ⟨h.some, ne_top_of_le_ne_top hs (measure_mono Set.inter_subset_left)⟩\n\nvariable [SemilatticeSup ι] [Nonempty ι] [Countable ι]\n\ntheorem exists_notConvergentSeq_lt (hε : 0 < ε)\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a)))\n    (hsm : MeasurableSet s) (hs : μ s ≠ ∞)\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :\n    ∃ j : ι, μ (s ∩ notConvergentSeq f g n j) ≤ ENNReal.ofReal (ε * 2⁻¹ ^ n) := by\n  have ⟨N, hN⟩ := (ENNReal.tendsto_atTop ENNReal.zero_ne_top).1\n    (measure_notConvergentSeq_tendsto_zero hf hsm hs hfg n) (.ofReal (ε * 2⁻¹ ^ n))\n      (by positivity)\n  rw [zero_add] at hN\n  exact ⟨N, (hN N le_rfl).2⟩\n\n/-- Given some `ε > 0`, `notConvergentSeqLTIndex` provides the index such that\n`notConvergentSeq` (intersected with a set of finite measure) has measure less than\n`ε * 2⁻¹ ^ n`.\n\nThis definition is useful for Egorov's theorem. -/\ndef notConvergentSeqLTIndex (hε : 0 < ε)\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a)))\n    (hsm : MeasurableSet s) (hs : μ s ≠ ∞)\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) : ι :=\n  Classical.choose <| exists_notConvergentSeq_lt hε hf hsm hs hfg n\n\ntheorem notConvergentSeqLTIndex_spec (hε : 0 < ε)\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a)))\n    (hsm : MeasurableSet s) (hs : μ s ≠ ∞)\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :\n    μ (s ∩ notConvergentSeq f g n (notConvergentSeqLTIndex hε hf hsm hs hfg n)) ≤\n      ENNReal.ofReal (ε * 2⁻¹ ^ n) :=\n  Classical.choose_spec <| exists_notConvergentSeq_lt hε hf hsm hs hfg n\n\n/-- Given some `ε > 0`, `iUnionNotConvergentSeq` is the union of `notConvergentSeq` with\nspecific indices such that `iUnionNotConvergentSeq` has measure less equal than `ε`.\n\nThis definition is useful for Egorov's theorem. -/\ndef iUnionNotConvergentSeq (hε : 0 < ε)\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a)))\n    (hsm : MeasurableSet s) (hs : μ s ≠ ∞)\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) : Set α :=\n  ⋃ n, s ∩ notConvergentSeq f g n (notConvergentSeqLTIndex (half_pos hε) hf hsm hs hfg n)\n\ntheorem iUnionNotConvergentSeq_measurableSet (hε : 0 < ε)\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a)))\n    (hsm : MeasurableSet s) (hs : μ s ≠ ∞)\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) :\n    MeasurableSet <| iUnionNotConvergentSeq hε hf hsm hs hfg :=\n  MeasurableSet.iUnion fun _ ↦ hsm.inter <| notConvergentSeq_measurableSet hf\n\nTarget:\ntheorem measure_iUnionNotConvergentSeq (hε : 0 < ε)\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a)))\n    (hsm : MeasurableSet s) (hs : μ s ≠ ∞)\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) :\n    μ (iUnionNotConvergentSeq hε hf hsm hs hfg) ≤ ENNReal.ofReal ε :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Function","family_id":"measure_iunionnotconvergentseq","file_id":"mathlib/Mathlib/MeasureTheory/Function/Egorov.lean","sample_id":"444126766a6d3867417667c40dd37f00d68870fb70c1031516cd452dd27d2227"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3b000d981c87ecc371d9ed638c299bf8ec24374d90b8958428a050b3089d276d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6b8295f80bbfd1f35f38e61aa6f8084c84def0ddff64970bd5ce951ce909b0d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c8ff660e3cb075cd21a289a8fe0e908d06e3a660d3865e6edea15898913062b8","source_sha256":"0d1f9dee5bc403406539e90adb439a195f88865e97423c979d76639acfd9231c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases ofSeq_surjective x with ⟨f, rfl⟩\n  rw [tendsto_ofSeq, (nhds_basis_Ioo _).tendsto_right_iff]\n  simp_rw [Set.mem_Ioo, eventually_and, ← ofSeq_lt_ofSeq]\n  refine ⟨fun H ↦ ⟨fun s hs ↦ ?_, fun s hs ↦ ?_⟩, fun H ⟨s, t⟩ ⟨hs, ht⟩ ↦ ⟨?_, ?_⟩⟩\n  · obtain ⟨t, ht⟩ := exists_gt r\n    exact (H ⟨s, t⟩ ⟨hs, ht⟩).1.le\n  · obtain ⟨t, ht⟩ := exists_lt r\n    exact (H ⟨t, s⟩ ⟨ht, hs⟩).2.le\n  · obtain ⟨u, hu, hu'⟩ := exists_between hs\n    exact (coe_lt_coe.2 hu).trans_le (H.1 _ hu')\n  · obtain ⟨u, hu, hu'⟩ := exists_between ht\n    exact (H.2 _ hu).trans_lt (coe_lt_coe.2 hu')","hard_negative":true,"metrics":{"chosen_tokens":186,"rejected_tokens":2,"token_jaccard":0.018868,"token_length_ratio":0.010753},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"3faaf69e6802aa84354387cba4bbfb3bfcfdf27ee27ea284b5b5030113b56803","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Ring.StandardPart\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.Order.Filter.FilterProduct\n\nNamespace:\nHyperreal\n\nLocal context:\n/-\nCopyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Abhimanyu Pallavi Sudhir, Violeta Hernández Palacios\n-/\n/-!\n# Construction of the hyperreal numbers as an ultraproduct of real sequences\n\nWe define the `Hyperreal` numbers as quotients of sequences `ℕ → ℝ` by an ultrafilter. These form\na field, and we prove some of their basic properties.\n\nNote that most of the machinery that is usually defined for the specific purpose of non-standard\nanalysis (infinitesimal and infinite elements, standard parts) has been generalized to other\nnon-archimedean fields. In particular:\n\n- `ArchimedeanClass` can be used to measure whether an element is infinitesimal (`0 < mk x`) or\n  infinite (`mk x < 0`).\n- `ArchimedeanClass.stdPart` generalizes the standard part function to a general ordered field.\n\n## Todo\n\nUse Łoś's Theorem `FirstOrder.Language.Ultraproduct.sentence_realize` to formalize the transfer\nprinciple on `Hyperreal`.\n-/\n\n@[expose] public section\n\nopen ArchimedeanClass Filter Germ Topology\n\nnoncomputable section\n\n/-- Hyperreal numbers on the ultrafilter extending the cofinite filter. -/\ndef Hyperreal : Type :=\n  Germ (hyperfilter ℕ : Filter ℕ) ℝ\nderiving Inhabited\n\nnamespace Hyperreal\n\n@[inherit_doc] notation \"ℝ*\" => Hyperreal\n\ninstance : Field ℝ* :=\n  inferInstanceAs (Field (Germ _ _))\n\ninstance : LinearOrder ℝ* :=\n  inferInstanceAs (LinearOrder (Germ _ _))\n\ninstance : IsStrictOrderedRing ℝ* :=\n  inferInstanceAs (IsStrictOrderedRing (Germ _ _))\n\n/-- Natural embedding `ℝ → ℝ*`. -/\n@[coe] def ofReal : ℝ → ℝ* := const\n\ninstance : CoeTC ℝ ℝ* := ⟨ofReal⟩\n\n@[simp, norm_cast]\ntheorem coe_eq_coe {x y : ℝ} : (x : ℝ*) = y ↔ x = y :=\n  Germ.const_inj\n\ntheorem coe_ne_coe {x y : ℝ} : (x : ℝ*) ≠ y ↔ x ≠ y :=\n  coe_eq_coe.not\n\n@[simp, norm_cast]\ntheorem coe_eq_zero {x : ℝ} : (x : ℝ*) = 0 ↔ x = 0 :=\n  coe_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_eq_one {x : ℝ} : (x : ℝ*) = 1 ↔ x = 1 :=\n  coe_eq_coe\n\n@[norm_cast]\ntheorem coe_ne_zero {x : ℝ} : (x : ℝ*) ≠ 0 ↔ x ≠ 0 :=\n  coe_ne_coe\n\n@[norm_cast]\ntheorem coe_ne_one {x : ℝ} : (x : ℝ*) ≠ 1 ↔ x ≠ 1 :=\n  coe_ne_coe\n\n@[simp, norm_cast]\ntheorem coe_one : ↑(1 : ℝ) = (1 : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_zero : ↑(0 : ℝ) = (0 : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_inv (x : ℝ) : ↑x⁻¹ = (x⁻¹ : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_neg (x : ℝ) : ↑(-x) = (-x : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_add (x y : ℝ) : ↑(x + y) = (x + y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_ofNat (n : ℕ) [n.AtLeastTwo] :\n    ((ofNat(n) : ℝ) : ℝ*) = OfNat.ofNat n :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_mul (x y : ℝ) : ↑(x * y) = (x * y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_div (x y : ℝ) : ↑(x / y) = (x / y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sub (x y : ℝ) : ↑(x - y) = (x - y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_le_coe {x y : ℝ} : (x : ℝ*) ≤ y ↔ x ≤ y :=\n  Germ.const_le_iff\n\n@[simp, norm_cast]\ntheorem coe_lt_coe {x y : ℝ} : (x : ℝ*) < y ↔ x < y :=\n  Germ.const_lt_iff\n\n@[simp, norm_cast]\ntheorem coe_nonneg {x : ℝ} : 0 ≤ (x : ℝ*) ↔ 0 ≤ x :=\n  coe_le_coe\n\n@[simp, norm_cast]\ntheorem coe_pos {x : ℝ} : 0 < (x : ℝ*) ↔ 0 < x :=\n  coe_lt_coe\n\n@[simp, norm_cast]\ntheorem coe_abs (x : ℝ) : ((|x| : ℝ) : ℝ*) = |↑x| :=\n  const_abs x\n\n@[simp, norm_cast]\ntheorem coe_max (x y : ℝ) : ((max x y : ℝ) : ℝ*) = max ↑x ↑y :=\n  Germ.const_max _ _\n\n@[simp, norm_cast]\ntheorem coe_min (x y : ℝ) : ((min x y : ℝ) : ℝ*) = min ↑x ↑y :=\n  Germ.const_min _ _\n\n/-- The canonical map `ℝ → ℝ*` as an `OrderRingHom`. -/\n@[simps]\ndef coeRingHom : ℝ →+*o ℝ* where\n  toFun x := x\n  map_zero' := rfl\n  map_one' := rfl\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  monotone' _ _ := coe_le_coe.2\n\n@[simp]\ntheorem archimedeanClassMk_coe_nonneg (x : ℝ) : 0 ≤ mk (x : ℝ*) :=\n  mk_map_nonneg_of_archimedean coeRingHom x\n\n@[simp]\ntheorem archimdeanClassMk_coe {x : ℝ} (hx : x ≠ 0) : mk (x : ℝ*) = 0 :=\n  mk_map_of_archimedean' coeRingHom hx\n\n@[simp]\ntheorem stdPart_coe (x : ℝ) : stdPart (x : ℝ*) = x :=\n  stdPart_map_real coeRingHom x\n\n/-! ### Basic constants -/\n\n/-- Construct a hyperreal number from a sequence of real numbers. -/\ndef ofSeq (f : ℕ → ℝ) : ℝ* := (↑f : Germ (hyperfilter ℕ : Filter ℕ) ℝ)\n\ntheorem ofSeq_surjective : Function.Surjective ofSeq := Quot.exists_rep\n\ntheorem ofSeq_lt_ofSeq {f g : ℕ → ℝ} : ofSeq f < ofSeq g ↔ ∀ᶠ n in hyperfilter ℕ, f n < g n :=\n  Germ.coe_lt\n\ntheorem ofSeq_le_ofSeq {f g : ℕ → ℝ} : ofSeq f ≤ ofSeq g ↔ ∀ᶠ n in hyperfilter ℕ, f n ≤ g n :=\n  Germ.coe_le\n\n/-! #### ω -/\n\n/-- A sample infinite hyperreal ω = ⟦(0, 1, 2, 3, ⋯)⟧. -/\ndef omega : ℝ* := ofSeq Nat.cast\n\n@[inherit_doc] scoped notation \"ω\" => Hyperreal.omega\nrecommended_spelling \"omega\" for \"ω\" in [omega, «termω»]\n\ntheorem coe_lt_omega (r : ℝ) : r < ω := by\n  apply ofSeq_lt_ofSeq.2 <| Filter.Eventually.filter_mono Nat.hyperfilter_le_atTop _\n  obtain ⟨n, hn⟩ := exists_nat_gt r\n  rw [eventually_atTop]\n  exact ⟨n, fun m hm ↦ hn.trans_le (mod_cast hm)⟩\n\ntheorem omega_pos : 0 < ω :=\n  coe_lt_omega 0\n\n@[simp]\ntheorem omega_ne_zero : ω ≠ 0 :=\n  omega_pos.ne'\n\n@[simp]\ntheorem abs_omega : |ω| = ω :=\n  abs_of_pos omega_pos\n\n@[simp]\ntheorem archimedeanClassMk_omega_neg : mk ω < 0 :=\n  fun n ↦ by simpa using! coe_lt_omega n\n\n@[simp]\ntheorem stdPart_omega : stdPart ω = 0 := by\n  rw [stdPart_eq_zero]\n  exact archimedeanClassMk_omega_neg.ne\n\n/-! #### ε -/\n\n/-- A sample infinitesimal hyperreal ε = ⟦(0, 1, 1/2, 1/3, ⋯)⟧. -/\ndef epsilon : ℝ* :=\n  ofSeq fun n => n⁻¹\n\n@[inherit_doc] scoped notation \"ε\" => Hyperreal.epsilon\nrecommended_spelling \"epsilon\" for \"ε\" in [epsilon, «termε»]\n\n@[simp]\ntheorem inv_omega : ω⁻¹ = ε :=\n  rfl\n\n@[simp]\ntheorem inv_epsilon : ε⁻¹ = ω :=\n  @inv_inv _ _ ω\n\n@[simp]\ntheorem epsilon_pos : 0 < ε :=\n  inv_pos_of_pos omega_pos\n\n@[simp]\ntheorem epsilon_ne_zero : ε ≠ 0 :=\n  epsilon_pos.ne'\n\n@[simp]\ntheorem epsilon_mul_omega : ε * ω = 1 :=\n  @inv_mul_cancel₀ _ _ ω omega_ne_zero\n\n@[simp]\ntheorem archimedeanClassMk_epsilon_pos : 0 < mk ε := by\n  simp [← inv_omega]\n\n/-!\n### Some facts about `Tendsto`\n-/\n\n@[simp]\ntheorem tendsto_ofSeq {f : ℕ → ℝ} {lb : Filter ℝ} :\n    (ofSeq f).Tendsto lb ↔ Tendsto f (hyperfilter ℕ) lb :=\n  .rfl\n\ntheorem stdPart_map {x : ℝ*} {r : ℝ} {f : ℝ → ℝ} (hf : ContinuousAt f r)\n    (hxr : x.Tendsto (𝓝 r)) : (x.map f).Tendsto (𝓝 (f r)) := by\n  rcases ofSeq_surjective x with ⟨g, rfl⟩\n  exact hf.tendsto.comp hxr\n\ntheorem stdPart_map₂ {x y : ℝ*} {r s : ℝ} {f : ℝ → ℝ → ℝ}\n    (hxr : x.Tendsto (𝓝 r)) (hys : y.Tendsto (𝓝 s))\n    (hf : ContinuousAt (Function.uncurry f) (r, s)) : (x.map₂ f y).Tendsto (𝓝 (f r s)) := by\n  rcases ofSeq_surjective x with ⟨x, rfl⟩\n  rcases ofSeq_surjective y with ⟨y, rfl⟩\n  exact hf.tendsto.comp (hxr.prodMk_nhds hys)\n\nTarget:\ntheorem tendsto_iff_forall {x : ℝ*} {r : ℝ} :\n    x.Tendsto (𝓝 r) ↔ (∀ s < r, s ≤ x) ∧ (∀ s > r, x ≤ s) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_c8ff660e3cb0","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"21fdcf88254979f047f517e27c5dbbe277b2e92ef5713174fdfe6e774a5d6e04","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Real","family_id":"tendsto_iff_forall","file_id":"mathlib/Mathlib/Analysis/Real/Hyperreal.lean","sample_id":"c8ff660e3cb075cd21a289a8fe0e908d06e3a660d3865e6edea15898913062b8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a0b44731945f1ae77d84c174d4c4d596bfd46fdd862ded445d0f40690c121d90","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d9c98e23776b09be391189ac538d2208b80743aa22e7f2a2c50784d270cab064","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e5d640efe871327e6d7a85ba9e52dd3140e74fe8671f6a1d11428174bcd3a187","source_sha256":"88c666c6d936856962069aacdb5158224b504be63ef88b462fdb4de48e42b813","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := h.hasNondegenerateEdges.two_le\n  by_contra! hlt\n  interval_cases n\n  exact (h 0).injective.ne (by decide : (0 : Fin 3) ≠ 2) (by simp)","hard_negative":true,"metrics":{"chosen_tokens":39,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.076923},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"40850515c35b73afd9c6fddb4d13e762283825c2543e457b6a967d6d61cebde3","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Convex.Between\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Tactic.Continuity\n\nNamespace:\nPolygon\n\nLocal context:\n/-\nCopyright (c) 2026 A. M. Berns. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: A. M. Berns\n-/\n/-!\n# Polygons\n\nThis file defines polygons in affine spaces.\nFor the special case `n = 3`, an interconversion is provided with `Affine.Triangle`.\n\n## Main definitions\n\n* `Polygon P n`: A polygon with `n` vertices in a type `P`.\n\n-/\n\n@[expose] public section\n\nopen Set\n\n/-- A polygon with `n` vertices in a type `P`. -/\nstructure Polygon (P : Type*) (n : ℕ) where\n  /-- The vertices of the polygon, indexed by `Fin n`. -/\n  vertices : Fin n → P\n\nnamespace Polygon\n\nvariable {R V P : Type*} {n : ℕ}\n\n/-- A coercion to function so that vertices can\nbe written as `poly i` instead of `poly.vertices i` -/\ninstance : CoeFun (Polygon P n) (fun _ => Fin n → P) where\n  coe := Polygon.vertices\n\n/-- A polygon has nondegenerate edges if adjacent vertices are distinct. -/\ndef HasNondegenerateEdges (poly : Polygon P n) : Prop :=\n  ∀ i : Fin n, poly i ≠ poly (finRotate n i)\n\ntheorem HasNondegenerateEdges.two_le [NeZero n] {poly : Polygon P n}\n    (h : poly.HasNondegenerateEdges) : 2 ≤ n := by\n  by_contra! hlt\n  interval_cases n\n  · simp_all only [neZero_zero_iff_false]\n  · exact h 0 (by simp)\n\nvariable [Ring R] [AddCommGroup V] [Module R V] [AddTorsor V P]\n\nvariable (R) in\n/-- The `i`-th edge as an affine map `R →ᵃ[R] P`. -/\ndef edgePath (poly : Polygon P n) (i : Fin n) : R →ᵃ[R] P :=\n  AffineMap.lineMap (poly i) (poly (finRotate n i))\n\nvariable (R) in\n/-- The `i`-th edge as a set of points using an `affineSegment`. -/\ndef edgeSet [PartialOrder R] (poly : Polygon P n) (i : Fin n) : Set P :=\n  affineSegment R (poly i) (poly (finRotate n i))\n\nvariable (R) in\n/-- The `edgeSet` is equivalent to the image of the `edgePath`. -/\ntheorem edgeSet_eq_image_edgePath [PartialOrder R] (poly : Polygon P n) (i : Fin n) :\n    poly.edgeSet R i = poly.edgePath R i '' Icc (0 : R) 1 := rfl\n\nvariable (R) in\n/-- The boundary of the polygon is the union of all its edges. -/\ndef boundary [PartialOrder R] (poly : Polygon P n) : Set P :=\n  ⋃ i, poly.edgeSet R i\n\nvariable (R) in\n/-- A polygon has nondegenerate vertices if any three consecutive vertices\nare affinely independent. -/\ndef HasNondegenerateVertices [NeZero n] (poly : Polygon P n) : Prop :=\n  ∀ i : Fin n, AffineIndependent R ![poly i, poly (i + 1), poly (i + 2)]\n\n/-- Polygons with nondegenerate vertices also have nondegenerate edges. -/\ntheorem HasNondegenerateVertices.hasNondegenerateEdges [NeZero n] [Nontrivial R]\n    {poly : Polygon P n}\n    (h : poly.HasNondegenerateVertices R) : poly.HasNondegenerateEdges := by\n  obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (NeZero.ne n)\n  intro i\n  simpa using (h i).injective.ne (by decide : (0 : Fin 3) ≠ 1)\n\nTarget:\ntheorem HasNondegenerateVertices.three_le [NeZero n] [Nontrivial R] {poly : Polygon P n}\n    (h : poly.HasNondegenerateVertices R) : 3 ≤ n :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_e5d640efe871","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"028bed2aa68d24b3604bb8238e46d52e315ff4a220e82e0ed0e8b682a757ce7e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/Polygon","family_id":"hasnondegeneratevertices","file_id":"mathlib/Mathlib/Geometry/Polygon/Basic.lean","sample_id":"e5d640efe871327e6d7a85ba9e52dd3140e74fe8671f6a1d11428174bcd3a187"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d0424b22b522a25f022cebb1346824da37ff580ade98ae91173e442bd2f6ba06","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f78ddd49796f36d4b728f9170045f34f377db64dfe529812548eacc978bc1b8b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"093333b00fcbe2feec5384aaaba84c83717794caa26f2c6955f67a6005b04fa4","source_sha256":"6ab13085b4631b0c5e2a6d22147b2e528fc5eed37a5ef3189daeebd872d6998b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply IsPullback.of_right (t := (inf_isPullback ((«exists» f).obj A') B').flip)\n    (p := by simp [frobeniusMorphism])\n  simpa [frobeniusMorphism, IsPullback.lift_fst, ← imageFactorisation_F_m,\n    (isPullback f B').paste_horiz_iff] using\n    (inf_isPullback A' ((Subobject.pullback f).obj B')).flip","hard_negative":true,"metrics":{"chosen_tokens":72,"rejected_tokens":3,"token_jaccard":0.028571,"token_length_ratio":0.041667},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"409e23a7e9e1f849d416f1317cd9c65b096e53f63d93d4dc0c771f033fa9c906","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.ExtremalEpi\npublic import Mathlib.CategoryTheory.MorphismProperty.Limits\npublic import Mathlib.CategoryTheory.Sites.Coherent.Basic\n\nNamespace:\nCategoryTheory.Regular\n\nLocal context:\n/-\nCopyright (c) 2025 Fernando Chu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Fernando Chu\n-/\n/-!\n# Regular categories\n\nA regular category is a category with finite limits such that each kernel pair has a coequalizer\nand such that regular epimorphisms are stable under pullback.\n\nThese categories provide a good ground to develop the calculus of relations, as well as being the\nsemantics for regular logic.\n\n## Main results\n\n* We show that every regular category has strong epi-mono factorisations, following Theorem 1.11\n  in [Gran2021].\n* We show that every regular category satisfies Frobenius reciprocity. That is, that in their\n  internal language, we have `∃ x, (P(x) ⊓ Q)` iff `(∃ x, P(x)) ⊓ Q`, for a proposition `Q` not\n  depending on `x`.\n\n## Future work\n* Show that every topos is regular\n* Show that regular logic has an interpretation in regular categories\n\n## References\n* [Marino Gran, An Introduction to Regular Categories][Gran2021]\n* <https://ncatlab.org/nlab/show/regular+category>\n-/\n\n@[expose] public section\n\nopen CategoryTheory Limits\n\nuniverse u v\n\nnamespace CategoryTheory\n\nvariable (C : Type u) [Category.{v} C]\n\n/--\nA regular category is a category with finite limits, such that every kernel pair has a coequalizer,\nand such that regular epimorphisms are stable under base change.\n-/\nclass Regular extends HasFiniteLimits C where\n  hasCoequalizer_of_isKernelPair {X Y Z : C} {f : X ⟶ Y} {g₁ g₂ : Z ⟶ X} :\n    IsKernelPair f g₁ g₂ → HasCoequalizer g₁ g₂\n  regularEpiIsStableUnderBaseChange : MorphismProperty.IsStableUnderBaseChange (.regularEpi C)\n\nvariable {C} [Regular C]\n\ninstance {X Y B : C} (f : X ⟶ B) (g : Y ⟶ B) [HasPullback f g] [IsRegularEpi f] :\n    IsRegularEpi (pullback.snd f g) := by\n  apply Regular.regularEpiIsStableUnderBaseChange.of_isPullback (IsPullback.of_hasPullback f g)\n  dsimp [MorphismProperty.regularEpi]\n  infer_instance\n\ninstance {X Y B : C} (f : X ⟶ B) (g : Y ⟶ B) [HasPullback f g] [IsRegularEpi g] :\n    IsRegularEpi (pullback.fst f g) := by\n  apply Regular.regularEpiIsStableUnderBaseChange.of_isPullback (IsPullback.of_hasPullback f g).flip\n  dsimp [MorphismProperty.regularEpi]\n  infer_instance\n\ninstance : Preregular C where\n  exists_fac f g := ⟨_, pullback.snd g f, inferInstance, pullback.fst g f, pullback.condition⟩\n\nvariable {X Y : C} (f : X ⟶ Y)\n\nnamespace Regular\n\nsection StrongEpiMonoFactorisation\n\nlocal instance : HasCoequalizer (pullback.fst f f) (pullback.snd f f) :=\n  Regular.hasCoequalizer_of_isKernelPair <| IsKernelPair.of_hasPullback f\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : Mono (coequalizer.desc f pullback.condition) := by\n  -- It suffices to show that the two projections from the kernel pair are equal:\n  apply (IsKernelPair.of_hasPullback _).mono_of_eq_fst_snd\n  /- We fill in the kernel pair square of `f` as follows:\n  ```\n                  g₁                   fst\n  pullback f f------->pullback e k₁----------> X\n        |                 |                    |\n      g₂|                 |snd                 |e\n        v        fst      v            k₁      v\n  pullback k₂ e------>pullback m m---------->coeq\n        |                                      |\n     snd|                                      |m\n        v              e ≫ m = f               v\n        X------------------------------------->Y\n  ```\n  Where `m`, `e`, `k₁`, `k₂`, `g₁`, `g₂` are defined below, `fst` and `snd` denote the projections\n  in the pullbacks indicated as the source of those morphisms, and `coeq` is the coequalizer of the\n  two projections in from the kernel pair of `f`.\n  -/\n  let m := (coequalizer.desc f pullback.condition)\n  let e := coequalizer.π (pullback.fst f f) (pullback.snd f f)\n  let k₁ := pullback.fst m m\n  let k₂ := pullback.snd m m\n  let d : pullback f f ⟶ (pullback m m) :=\n    pullback.lift (pullback.fst f f ≫ e) (pullback.snd f f ≫ e) (by simp [m, e, pullback.condition])\n  let g₁ : pullback f f ⟶ (pullback e k₁) := pullback.lift (pullback.fst f f) d (by simp [d, k₁])\n  let g₂ : pullback f f ⟶ (pullback k₂ e) := pullback.lift d (pullback.snd f f) (by simp [d, k₂])\n  /-\n  Since the big square, the bottom square, and the top right square above are pullback squares,\n  the top left square is also a pullback square.\n  -/\n  have h : IsPullback g₁ g₂ (pullback.snd e k₁) (pullback.fst k₂ e) := by\n    refine .of_right ?_ (by simp [g₁, g₂]) (.of_hasPullback e k₁)\n    refine .of_bot ?_ ?_ (.paste_horiz (.of_hasPullback k₂ e) (.of_hasPullback m m))\n    · simpa [g₁, g₂, e, m, pullback.lift_fst, pullback.lift_snd] using .of_hasPullback f f\n    · simp [g₁, g₂, k₁, d]\n  /-\n  Since `g₁` is the base change of a regular epi (the map `fst` in the middle row of the diagram\n  above, which itself is a regular epi because it is a base change of the regular epi `e`),\n  it is a regular epi.\n  -/\n  have : IsRegularEpi g₁ := by\n    apply Regular.regularEpiIsStableUnderBaseChange.of_isPullback h.flip\n    dsimp [MorphismProperty.regularEpi]\n    infer_instance\n  -- We precompose with the epimorphism `g₁ ≫ pullback.snd e k₁`, and finish\n  rw [← cancel_epi (g₁ ≫ pullback.snd e k₁)]\n  convert! coequalizer.condition (pullback.fst f f) (pullback.snd f f) using 1\n  all_goals cat_disch\n\nset_option backward.isDefEq.respectTransparency false in\n/--\nIn a regular category, every morphism `f : X ⟶ Y` factors as `e ≫ m`, where `e` is the projection\nmap to the coequalizer of the kernel pair of `f`, and `m` is the canonical map from that\ncoequalizer to `Y`. In particular, `f` factors as a strong epimorphism followed by a monomorphism.\n-/\nnoncomputable def strongEpiMonoFactorisation : StrongEpiMonoFactorisation f where\n  I := coequalizer (pullback.fst f f) (pullback.snd f f)\n  m := coequalizer.desc f pullback.condition\n  e := coequalizer.π (pullback.fst f f) (pullback.snd f f)\n\ninstance : IsRegularEpi (strongEpiMonoFactorisation f).e := by\n  dsimp [strongEpiMonoFactorisation]\n  infer_instance\n\n/--\nIn a regular category, every morphism `f` factors as `e ≫ m`, with `e` a strong epimorphism\nand `m` a monomorphism.\n-/\ninstance hasStrongEpiMonoFactorisations : HasStrongEpiMonoFactorisations C where\n  has_fac f := ⟨strongEpiMonoFactorisation f⟩\n\nset_option backward.isDefEq.respectTransparency false in\n/-- In a regular category, every extremal epimorphism is a regular epimorphism. -/\nnoncomputable def regularEpiOfExtremalEpi [h : ExtremalEpi f] : RegularEpi f :=\n  have := h.isIso (strongEpiMonoFactorisation f).e (strongEpiMonoFactorisation f).m (by simp)\n  RegularEpi.ofArrowIso (Arrow.isoMk (f := .mk (strongEpiMonoFactorisation f).e) (Iso.refl _)\n    (asIso (strongEpiMonoFactorisation f).m)) (IsRegularEpi.getStruct _)\n\ninstance isRegularEpi_of_extremalEpi (f : X ⟶ Y) [ExtremalEpi f] : IsRegularEpi f :=\n  ⟨⟨regularEpiOfExtremalEpi f⟩⟩\n\nend StrongEpiMonoFactorisation\n\nsection Frobenius\n\nopen Subobject\n\nvariable {A B : C} (f : A ⟶ B) (A' : Subobject A) (B' : Subobject B)\n\nset_option backward.isDefEq.respectTransparency false in\n/--\nGiven a morphism `f : A ⟶ B` and subobjects `A' ⟶ A` and `B' ⟶ B`, we have a canonical morphism\n`(A' ⊓ (Subobject.pullback f).obj B') ⟶ ((«exists» f).obj A' ⊓ B')`.\nThis morphism is part of a `StrongEpiMonoFactorisation` of\n`(A' ⊓ (Subobject.pullback f).obj B').arrow ≫ f`, see `frobeniusStrongEpiMonoFactorisation`.\n-/\nnoncomputable def frobeniusMorphism :\n    underlying.obj (A' ⊓ (Subobject.pullback f).obj B') ⟶\n      underlying.obj ((«exists» f).obj A' ⊓ B') :=\n  (inf_isPullback ((«exists» f).obj A') B').flip.lift\n    ((ofLE _ _ (inf_le_right A' ((Subobject.pullback f).obj B'))) ≫ (pullbackπ _ _))\n    ((ofLE _ _ (inf_le_left A' ((Subobject.pullback f).obj B'))) ≫ (imageFactorisation f A').F.e)\n    (by simp [← imageFactorisation_F_m, (isPullback _ _).w])\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma frobeniusMorphism_isPullback :\n    IsPullback (frobeniusMorphism f A' B')\n      ((ofLE _ _ (inf_le_left A' ((Subobject.pullback f).obj B'))))\n      ((ofLE _ _ (inf_le_left ((«exists» f).obj A') B')))\n      (imageFactorisation _ _).F.e :=\n\nProof body:\n","rejected":"by\n  exact frobeniusMorphism_isPullback","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"95c11f5057ea83aae1bae672ce95f1b298fbcc3ceb1de9912c81f822227d2730","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/RegularCategory","family_id":"frobeniusmorphism_ispullback","file_id":"mathlib/Mathlib/CategoryTheory/RegularCategory/Basic.lean","sample_id":"093333b00fcbe2feec5384aaaba84c83717794caa26f2c6955f67a6005b04fa4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"81a9d05bf0ed54d4fa50cc7311fe12bd2332adf9f758867f731c0d0d23db9024","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cb0401825ad6d1e1b558ca9c25c51b4a8e4b1b05044ff31d12c8d300f731fff7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"15f253266f8e1f550df58301525f21097e11cc98ed777b7a946120509154235e","source_sha256":"e01a6cb1d107abe5a5bc2e366e62863bb223a3de261cf05b6f9895b4d0fc06e8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext i\n  by_cases h : i = i₀\n  · subst h\n    simp\n  · rw [subtypeNeLift_of_neq _ _ _ _ h]","hard_negative":false,"metrics":{"chosen_tokens":24,"rejected_tokens":28,"token_jaccard":0.8,"token_length_ratio":1.166667},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"40b71e0f45fedf8cbe1826c33ae76fdc78b2339d3989c264820ccc6f3613b8f6","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Logic.Equiv.Option\n\nNamespace:\nFunction\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Extending a function from the complement of a singleton\n\nIn this file, we define `Function.subtypeNeLift` which allows to\nextend a (dependent) function defined on the complement of a singleton.\n\n-/\n\n@[expose] public section\n\nnamespace Function\n\nvariable {ι : Type*} [DecidableEq ι] {M : ι → Type*} (i₀ : ι)\n  (f : ∀ (j : { i // i ≠ i₀ }), M j) (x : M i₀)\n\n/-- Given `i₀ : ι` and `x : M i₀`, this is the (dependent) map `(i : ι) → M i`\nwhose value at `i₀` is `x` and which extends a given map on the complement of `{i₀}`. -/\ndef subtypeNeLift (i : ι) : M i :=\n  if h : i = i₀ then by rw [h]; exact x else f ⟨i, h⟩\n\n@[simp]\nlemma subtypeNeLift_self : subtypeNeLift i₀ f x i₀ = x := dif_pos rfl\n\nlemma subtypeNeLift_of_neq (i : ι) (h : i ≠ i₀) :\n    subtypeNeLift i₀ f x i = f ⟨i, h⟩ := dif_neg h\n\n@[simp]\n\nTarget:\nlemma subtypeNeLift_restriction (φ : ∀ i, M i) (i₀ : ι) :\n    subtypeNeLift i₀ (fun i ↦ φ i) (φ i₀) = φ :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  ext i\n  by_cases h : i = i₀\n  · subst h\n    simp\n  · rw [subtypeNeLift_of_neq _ _ _ _ h]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data","family_id":"subtypenelift_restriction","file_id":"mathlib/Mathlib/Data/SubtypeNeLift.lean","sample_id":"15f253266f8e1f550df58301525f21097e11cc98ed777b7a946120509154235e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2cd47198bc2f9b64bb28556c6ab7308c7324a109d2a3891b6dff94f0ce2c3121","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9ba3ca3c063a13ab1bd3991a202b055b4a965f5ac7583f897a336971b812fcc","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2b6b462ef4cebc5cf1be9ae0f138ed0a50ba4a689e61f2c8626b10e8cc24c3b7","source_sha256":"a29151d12db11777bb9d805b9535e76a5c2a8f2d2a791e6ae7bfa4c562e67f6b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [Nat.lt_add_one_iff, biSup_le_eq_sup]","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":3,"token_jaccard":0.090909,"token_length_ratio":0.333333},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"412ec17cbcd87d1977e8216dc98818c7e4904c9cb4c879617067b58f2a88b3af","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ConditionallyCompleteLattice.Finset\npublic import Mathlib.Order.Interval.Finset.Nat\n\nNamespace:\nNat\n\nLocal context:\n/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Floris van Doorn, Gabriel Ebner, Yury Kudryashov\n-/\n/-!\n# Conditionally complete linear order structure on `ℕ`\n\nIn this file we\n\n* define a `ConditionallyCompleteLinearOrderBot` structure on `ℕ`;\n* prove a few lemmas about `iSup`/`iInf`/`Set.iUnion`/`Set.iInter` and natural numbers.\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero\n\nopen Set\n\nnamespace Nat\n\nopen scoped Classical in\nnoncomputable instance : InfSet ℕ :=\n  ⟨fun s ↦ if h : ∃ n, n ∈ s then @Nat.find (fun n ↦ n ∈ s) _ h else 0⟩\n\nopen scoped Classical in\nnoncomputable instance : SupSet ℕ :=\n  ⟨fun s ↦ if h : ∃ n, ∀ a ∈ s, a ≤ n then @Nat.find (fun n ↦ ∀ a ∈ s, a ≤ n) _ h else 0⟩\n\nopen scoped Classical in\ntheorem sInf_def {s : Set ℕ} (h : s.Nonempty) : sInf s = @Nat.find (fun n ↦ n ∈ s) _ h :=\n  dif_pos _\n\nopen scoped Classical in\ntheorem sSup_def {s : Set ℕ} (h : ∃ n, ∀ a ∈ s, a ≤ n) :\n    sSup s = @Nat.find (fun n ↦ ∀ a ∈ s, a ≤ n) _ h :=\n  dif_pos _\n\ntheorem _root_.Set.Infinite.Nat.sSup_eq_zero {s : Set ℕ} (h : s.Infinite) : sSup s = 0 :=\n  dif_neg fun ⟨n, hn⟩ ↦\n    let ⟨k, hks, hk⟩ := h.exists_gt n\n    (hn k hks).not_gt hk\n\ntheorem sSup_of_not_bddAbove {s : Set ℕ} (h : ¬BddAbove s) : sSup s = 0 :=\n  Set.Infinite.Nat.sSup_eq_zero <| Set.infinite_of_not_bddAbove h\n\nlemma iSup_of_not_bddAbove {ι : Sort*} {f : ι → ℕ} (h : ¬ BddAbove (Set.range f)) :\n    (⨆ i, f i : ℕ) = 0 := Nat.sSup_of_not_bddAbove h\n\n@[simp]\ntheorem sInf_eq_zero {s : Set ℕ} : sInf s = 0 ↔ 0 ∈ s ∨ s = ∅ := by\n  cases eq_empty_or_nonempty s with\n  | inl h => subst h\n             simp only [or_true, InfSet.sInf,\n                        mem_empty_iff_false, exists_false, dif_neg, not_false_iff]\n  | inr h => simp only [h.ne_empty, or_false, Nat.sInf_def, h, Nat.find_eq_zero]\n\n@[simp]\ntheorem sInf_empty : sInf ∅ = 0 := by\n  rw [sInf_eq_zero]\n  right\n  rfl\n\n@[simp]\ntheorem iInf_of_empty {ι : Sort*} [IsEmpty ι] (f : ι → ℕ) : iInf f = 0 := by\n  rw [iInf_of_isEmpty, sInf_empty]\n\n/-- This combines `Nat.iInf_of_empty` with `ciInf_const`. -/\n@[simp]\nlemma iInf_const_zero {ι : Sort*} : ⨅ _ : ι, 0 = 0 :=\n  (isEmpty_or_nonempty ι).elim (fun h ↦ by simp) fun h ↦ sInf_eq_zero.2 <| by simp\n\ntheorem sInf_mem {s : Set ℕ} (h : s.Nonempty) : sInf s ∈ s := by\n  classical\n  rw [Nat.sInf_def h]\n  exact Nat.find_spec h\n\ntheorem notMem_of_lt_sInf {s : Set ℕ} {m : ℕ} (hm : m < sInf s) : m ∉ s := by\n  classical\n  cases eq_empty_or_nonempty s with\n  | inl h => subst h; apply notMem_empty\n  | inr h => rw [Nat.sInf_def h] at hm; exact Nat.find_min h hm\n\nprotected theorem sInf_le {s : Set ℕ} {m : ℕ} (hm : m ∈ s) : sInf s ≤ m := by\n  classical\n  rw [Nat.sInf_def ⟨m, hm⟩]\n  exact Nat.find_min' ⟨m, hm⟩ hm\n\ntheorem nonempty_of_pos_sInf {s : Set ℕ} (h : 0 < sInf s) : s.Nonempty := by\n  by_contra contra\n  rw [Set.not_nonempty_iff_eq_empty] at contra\n  have h' : sInf s ≠ 0 := ne_of_gt h\n  apply h'\n  rw [Nat.sInf_eq_zero]\n  right\n  assumption\n\ntheorem nonempty_of_sInf_eq_succ {s : Set ℕ} {k : ℕ} (h : sInf s = k + 1) : s.Nonempty :=\n  nonempty_of_pos_sInf (h.symm ▸ succ_pos k : sInf s > 0)\n\ntheorem eq_Ici_of_nonempty_of_upward_closed {s : Set ℕ} (hs : s.Nonempty)\n    (hs' : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s) : s = Ici (sInf s) :=\n  ext fun n ↦ ⟨fun H ↦ Nat.sInf_le H, fun H ↦ hs' (sInf s) n H (sInf_mem hs)⟩\n\ntheorem sInf_upward_closed_eq_succ_iff {s : Set ℕ} (hs : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s)\n    (k : ℕ) : sInf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s := by\n  classical\n  constructor\n  · intro H\n    rw [eq_Ici_of_nonempty_of_upward_closed (nonempty_of_sInf_eq_succ _) hs, H, mem_Ici, mem_Ici]\n    · exact ⟨le_rfl, k.not_succ_le_self⟩\n    · exact k\n    · assumption\n  · rintro ⟨H, H'⟩\n    rw [sInf_def (⟨_, H⟩ : s.Nonempty), find_eq_iff]\n    exact ⟨H, fun n hnk hns ↦ H' <| hs n k (Nat.lt_succ_iff.mp hnk) hns⟩\n\n/-- This instance is necessary, otherwise the lattice operations would be derived via\n`ConditionallyCompleteLinearOrderBot` and marked as noncomputable. -/\ninstance : Lattice ℕ :=\n  LinearOrder.toLattice\n\nopen scoped Classical in\nnoncomputable instance : ConditionallyCompleteLinearOrderBot ℕ :=\n  { (inferInstance : OrderBot ℕ), (LinearOrder.toLattice : Lattice ℕ),\n    (inferInstance : LinearOrder ℕ) with\n    isLUB_csSup _ hn hb := sSup_def hb ▸ Nat.isLeast_find hb\n    isGLB_csInf _ hn hb := sInf_def hn ▸ (Nat.isLeast_find hn).isGLB\n    csSup_empty := by\n      simp only [sSup_def, Set.mem_empty_iff_false, forall_const, forall_prop_of_false,\n        not_false_iff, exists_const]\n      apply bot_unique (Nat.find_min' _ _)\n      trivial\n    csSup_of_not_bddAbove := by\n      intro s hs\n      simp only [sSup,\n        mem_empty_iff_false, IsEmpty.forall_iff, forall_const, exists_const, dite_true]\n      rw [dif_neg]\n      · exact le_antisymm (zero_le _) (find_le trivial)\n      · exact hs\n    csInf_of_not_bddBelow := fun s hs ↦ by simp at hs }\n\ntheorem sSup_mem {s : Set ℕ} (h₁ : s.Nonempty) (h₂ : BddAbove s) : sSup s ∈ s :=\n  let ⟨k, hk⟩ := h₂\n  h₁.csSup_mem ((finite_le_nat k).subset hk)\n\ntheorem sInf_add {n : ℕ} {p : ℕ → Prop} (hn : n ≤ sInf { m | p m }) :\n    sInf { m | p (m + n) } + n = sInf { m | p m } := by\n  classical\n  obtain h | ⟨m, hm⟩ := { m | p (m + n) }.eq_empty_or_nonempty\n  · rw [h, Nat.sInf_empty, zero_add]\n    obtain hnp | hnp := hn.eq_or_lt\n    · exact hnp\n    suffices hp : p (sInf { m | p m } - n + n) from (h.subset hp).elim\n    rw [Nat.sub_add_cancel hn]\n    exact csInf_mem (nonempty_of_pos_sInf <| n.zero_le.trans_lt hnp)\n  · have hp : ∃ n, n ∈ { m | p m } := ⟨_, hm⟩\n    rw [Nat.sInf_def ⟨m, hm⟩, Nat.sInf_def hp]\n    rw [Nat.sInf_def hp] at hn\n    exact find_add hn\n\ntheorem sInf_add' {n : ℕ} {p : ℕ → Prop} (h : 0 < sInf { m | p m }) :\n    sInf { m | p m } + n = sInf { m | p (m - n) } := by\n  suffices h₁ : n ≤ sInf {m | p (m - n)} by\n    convert! sInf_add h₁\n    simp_rw [Nat.add_sub_cancel_right]\n  obtain ⟨m, hm⟩ := nonempty_of_pos_sInf h\n  refine\n    le_csInf ⟨m + n, ?_⟩ fun b hb ↦\n      le_of_not_gt fun hbn ↦\n        ne_of_mem_of_not_mem ?_ (notMem_of_lt_sInf h) (Nat.sub_eq_zero_of_le hbn.le)\n  · dsimp\n    rwa [Nat.add_sub_cancel_right]\n  · exact hb\n\nsection\n\nvariable {α : Type*} [CompleteLattice α]\n\nTarget:\ntheorem iSup_lt_succ (u : ℕ → α) (n : ℕ) : ⨆ k < n + 1, u k = (⨆ k < n, u k) ⊔ u n :=\n\nProof body:\n","rejected":"by\n  exact iSup_lt_succ","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"b910b8c480ed9e9b033051ea554b60079013651fd3fe80596df977ae866fce57","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Lattice","family_id":"isup_lt_succ","file_id":"mathlib/Mathlib/Order/Lattice/Nat.lean","sample_id":"2b6b462ef4cebc5cf1be9ae0f138ed0a50ba4a689e61f2c8626b10e8cc24c3b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7749888414d06cd3f5ca86c4419d34b5759e4af27af9cf5a4232c6e45cfd9c6a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"36b8d32675fdb938bd83c305167a8b369ebab6dbb89b168f6a1a6366bf149799","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"099bd67010bd3bdf0391a625cbe08e92d2569c8b6c447860d4abbc5d4f43fc92","source_sha256":"f589366b76253f55a5f00af110f0088b3f55c67577d56940b18639c5f1a2a3b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only [add_comm _ (n : ℝ)] using f.commute_add_int n","hard_negative":true,"metrics":{"chosen_tokens":17,"rejected_tokens":2,"token_jaccard":0.058824,"token_length_ratio":0.117647},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"413625687de6e6eba503b6192443285c3af8347f5e29f50a245268adc30911ce","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.Order.SemiconjSup\n\nNamespace:\nCircleDeg1Lift\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Translation number of a monotone real map that commutes with `x ↦ x + 1`\n\nLet `f : ℝ → ℝ` be a monotone map such that `f (x + 1) = f x + 1` for all `x`. Then the limit\n$$\n  \\tau(f)=\\lim_{n\\to\\infty}{f^n(x)-x}{n}\n$$\nexists and does not depend on `x`. This number is called the *translation number* of `f`.\nDifferent authors use different notation for this number: `τ`, `ρ`, `rot`, etc\n\nIn this file we define a structure `CircleDeg1Lift` for bundled maps with these properties, define\ntranslation number of `f : CircleDeg1Lift`, prove some estimates relating `f^n(x)-x` to `τ(f)`. In\ncase of a continuous map `f` we also prove that `f` admits a point `x` such that `f^n(x)=x+m` if and\nonly if `τ(f)=m/n`.\n\nMaps of this type naturally appear as lifts of orientation-preserving circle homeomorphisms. More\nprecisely, let `f` be an orientation-preserving homeomorphism of the circle $S^1=ℝ/ℤ$, and\nconsider a real number `a` such that\n`⟦a⟧ = f 0`, where `⟦⟧` means the natural projection `ℝ → ℝ/ℤ`. Then there exists a unique\ncontinuous function `F : ℝ → ℝ` such that `F 0 = a` and `⟦F x⟧ = f ⟦x⟧` for all `x` (this fact is\nnot formalized yet). This function is strictly monotone, continuous, and satisfies\n`F (x + 1) = F x + 1`. The number `⟦τ F⟧ : ℝ / ℤ` is called the *rotation number* of `f`.\nIt does not depend on the choice of `a`.\n\n## Main definitions\n\n* `CircleDeg1Lift`: a monotone map `f : ℝ → ℝ` such that `f (x + 1) = f x + 1` for all `x`;\n  the type `CircleDeg1Lift` is equipped with `Lattice` and `Monoid` structures; the\n  multiplication is given by composition: `(f * g) x = f (g x)`.\n* `CircleDeg1Lift.translationNumber`: translation number of `f : CircleDeg1Lift`.\n\n## Main statements\n\nWe prove the following properties of `CircleDeg1Lift.translationNumber`.\n\n* `CircleDeg1Lift.translationNumber_eq_of_dist_bounded`: if the distance between `(f^n) 0`\n  and `(g^n) 0` is bounded from above uniformly in `n : ℕ`, then `f` and `g` have equal\n  translation numbers.\n\n* `CircleDeg1Lift.translationNumber_eq_of_semiconjBy`: if two `CircleDeg1Lift` maps `f`, `g`\n  are semiconjugate by a `CircleDeg1Lift` map, then `τ f = τ g`.\n\n* `CircleDeg1Lift.translationNumber_units_inv`: if `f` is an invertible `CircleDeg1Lift` map\n  (equivalently, `f` is a lift of an orientation-preserving circle homeomorphism), then\n  the translation number of `f⁻¹` is the negative of the translation number of `f`.\n\n* `CircleDeg1Lift.translationNumber_mul_of_commute`: if `f` and `g` commute, then\n  `τ (f * g) = τ f + τ g`.\n\n* `CircleDeg1Lift.translationNumber_eq_rat_iff`: the translation number of `f` is equal to\n  a rational number `m / n` if and only if `(f^n) x = x + m` for some `x`.\n\n* `CircleDeg1Lift.semiconj_of_bijective_of_translationNumber_eq`: if `f` and `g` are two\n  bijective `CircleDeg1Lift` maps and their translation numbers are equal, then these\n  maps are semiconjugate to each other.\n\n* `CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq`: let `f₁` and `f₂` be\n  two actions of a group `G` on the circle by degree 1 maps (formally, `f₁` and `f₂` are two\n  homomorphisms from `G →* CircleDeg1Lift`). If the translation numbers of `f₁ g` and `f₂ g` are\n  equal to each other for all `g : G`, then these two actions are semiconjugate by some\n  `F : CircleDeg1Lift`. This is a version of Proposition 5.4 from [Étienne Ghys, Groupes\n  d'homéomorphismes du cercle et cohomologie bornée][ghys87:groupes].\n\n## Notation\n\nWe use a local notation `τ` for the translation number of `f : CircleDeg1Lift`.\n\n## Implementation notes\n\nWe define the translation number of `f : CircleDeg1Lift` to be the limit of the sequence\n`(f ^ (2 ^ n)) 0 / (2 ^ n)`, then prove that `((f ^ n) x - x) / n` tends to this number for any `x`.\nThis way it is much easier to prove that the limit exists and basic properties of the limit.\n\nWe define translation number for a wider class of maps `f : ℝ → ℝ` instead of lifts of orientation\npreserving circle homeomorphisms for two reasons:\n\n* non-strictly monotone circle self-maps with discontinuities naturally appear as Poincaré maps\n  for some flows on the two-torus (e.g., one can take a constant flow and glue in a few Cherry\n  cells);\n* definition and some basic properties still work for this class.\n\n## References\n\n* [Étienne Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée][ghys87:groupes]\n\n## TODO\n\nHere are some short-term goals.\n\n* Introduce a structure or a typeclass for lifts of circle homeomorphisms. We use\n  `Units CircleDeg1Lift` for now, but it's better to have a dedicated type (or a typeclass?).\n\n* Prove that the `SemiconjBy` relation on circle homeomorphisms is an equivalence relation.\n\n* Introduce `ConditionallyCompleteLattice` structure, use it in the proof of\n  `CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq`.\n\n* Prove that the orbits of the irrational rotation are dense in the circle. Deduce that a\n  homeomorphism with an irrational rotation is semiconjugate to the corresponding irrational\n  translation by a continuous `CircleDeg1Lift`.\n\n## Tags\n\ncircle homeomorphism, rotation number\n-/\n\n@[expose] public section\n\nopen Filter Set Int Topology\nopen Function hiding Commute\n\n/-!\n### Definition and monoid structure\n-/\n\n/-- A lift of a monotone degree one map `S¹ → S¹`. -/\nstructure CircleDeg1Lift : Type extends ℝ →o ℝ where\n  map_add_one' : ∀ x, toFun (x + 1) = toFun x + 1\n\nnamespace CircleDeg1Lift\n\ninstance : FunLike CircleDeg1Lift ℝ ℝ where\n  coe f := f.toFun\n  coe_injective | ⟨⟨_, _⟩, _⟩, ⟨⟨_, _⟩, _⟩, rfl => rfl\n\ninstance : OrderHomClass CircleDeg1Lift ℝ ℝ where\n  map_rel f _ _ h := f.monotone' h\n\n@[simp] theorem coe_mk (f h) : ⇑(mk f h) = f := rfl\n\nvariable (f g : CircleDeg1Lift)\n\n@[simp] theorem coe_toOrderHom : ⇑f.toOrderHom = f := rfl\n\nprotected theorem monotone : Monotone f := f.monotone'\n\n@[gcongr, mono] theorem mono {x y} (h : x ≤ y) : f x ≤ f y := f.monotone h\n\ntheorem strictMono_iff_injective : StrictMono f ↔ Injective f :=\n  f.monotone.strictMono_iff_injective\n\n@[simp]\ntheorem map_add_one : ∀ x, f (x + 1) = f x + 1 :=\n  f.map_add_one'\n\n@[simp]\ntheorem map_one_add (x : ℝ) : f (1 + x) = 1 + f x := by rw [add_comm, map_add_one, add_comm 1]\n\n@[ext]\ntheorem ext ⦃f g : CircleDeg1Lift⦄ (h : ∀ x, f x = g x) : f = g :=\n  DFunLike.ext f g h\n\ninstance : Monoid CircleDeg1Lift where\n  mul f g :=\n    { toOrderHom := f.1.comp g.1\n      map_add_one' := fun x => by simp [map_add_one] }\n  one := ⟨.id, fun _ => rfl⟩\n  mul_one _ := rfl\n  one_mul _ := rfl\n  mul_assoc _ _ _ := DFunLike.coe_injective rfl\n\ninstance : Inhabited CircleDeg1Lift := ⟨1⟩\n\n@[simp]\ntheorem coe_mul : ⇑(f * g) = f ∘ g :=\n  rfl\n\ntheorem mul_apply (x) : (f * g) x = f (g x) :=\n  rfl\n\n@[simp]\ntheorem coe_one : ⇑(1 : CircleDeg1Lift) = id :=\n  rfl\n\ninstance unitsHasCoeToFun : CoeFun CircleDeg1Liftˣ fun _ => ℝ → ℝ :=\n  ⟨fun f => ⇑(f : CircleDeg1Lift)⟩\n\n@[simp]\ntheorem units_inv_apply_apply (f : CircleDeg1Liftˣ) (x : ℝ) :\n    (f⁻¹ : CircleDeg1Liftˣ) (f x) = x := by simp only [← mul_apply, f.inv_mul, coe_one, id]\n\n@[simp]\ntheorem units_apply_inv_apply (f : CircleDeg1Liftˣ) (x : ℝ) :\n    f ((f⁻¹ : CircleDeg1Liftˣ) x) = x := by simp only [← mul_apply, f.mul_inv, coe_one, id]\n\n/-- If a lift of a circle map is bijective, then it is an order automorphism of the line. -/\ndef toOrderIso : CircleDeg1Liftˣ →* ℝ ≃o ℝ where\n  toFun f :=\n    { toFun := f\n      invFun := ⇑f⁻¹\n      left_inv := units_inv_apply_apply f\n      right_inv := units_apply_inv_apply f\n      map_rel_iff' := ⟨fun h => by simpa using mono (↑f⁻¹) h, mono f⟩ }\n  map_one' := rfl\n  map_mul' _ _ := rfl\n\n@[simp]\ntheorem coe_toOrderIso (f : CircleDeg1Liftˣ) : ⇑(toOrderIso f) = f :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderIso_symm (f : CircleDeg1Liftˣ) :\n    ⇑(toOrderIso f).symm = (f⁻¹ : CircleDeg1Liftˣ) :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderIso_inv (f : CircleDeg1Liftˣ) : ⇑(toOrderIso f)⁻¹ = (f⁻¹ : CircleDeg1Liftˣ) :=\n  rfl\n\ntheorem isUnit_iff_bijective {f : CircleDeg1Lift} : IsUnit f ↔ Bijective f :=\n  ⟨fun ⟨u, h⟩ => h ▸ (toOrderIso u).bijective, fun h =>\n    Units.isUnit\n      { val := f\n        inv :=\n          { toFun := (Equiv.ofBijective f h).symm\n            monotone' := fun x y hxy =>\n              (f.strictMono_iff_injective.2 h.1).le_iff_le.1\n                (by simp only [Equiv.ofBijective_apply_symm_apply f h, hxy])\n            map_add_one' := fun x =>\n              h.1 <| by simp only [Equiv.ofBijective_apply_symm_apply f, f.map_add_one] }\n        val_inv := ext <| Equiv.ofBijective_apply_symm_apply f h\n        inv_val := ext <| Equiv.ofBijective_symm_apply_apply f h }⟩\n\ntheorem coe_pow : ∀ n : ℕ, ⇑(f ^ n) = f^[n]\n  | 0 => rfl\n  | n + 1 => by\n    simp [coe_pow n, pow_succ]\n\ntheorem semiconjBy_iff_semiconj {f g₁ g₂ : CircleDeg1Lift} :\n    SemiconjBy f g₁ g₂ ↔ Semiconj f g₁ g₂ :=\n  CircleDeg1Lift.ext_iff\n\ntheorem commute_iff_commute {f g : CircleDeg1Lift} : Commute f g ↔ Function.Commute f g :=\n  CircleDeg1Lift.ext_iff\n\n/-!\n### Translate by a constant\n-/\n\n\n/-- The map `y ↦ x + y` as a `CircleDeg1Lift`. More precisely, we define a homomorphism from\n`Multiplicative ℝ` to `CircleDeg1Liftˣ`, so the translation by `x` is\n`translation (Multiplicative.ofAdd x)`. -/\ndef translate : Multiplicative ℝ →* CircleDeg1Liftˣ := MonoidHom.toHomUnits <|\n  { toFun x := ⟨⟨fun y => x.toAdd + y, add_right_mono⟩, fun _ => (add_assoc ..).symm⟩\n    map_one' := ext zero_add\n    map_mul' _ _ := ext <| add_assoc _ _ }\n\n@[simp]\ntheorem translate_apply (x y : ℝ) : translate (Multiplicative.ofAdd x) y = x + y :=\n  rfl\n\n@[simp]\ntheorem translate_inv_apply (x y : ℝ) : (translate <| Multiplicative.ofAdd x)⁻¹ y = -x + y :=\n  rfl\n\n@[simp]\ntheorem translate_zpow (x : ℝ) (n : ℤ) :\n    translate (Multiplicative.ofAdd x) ^ n = translate (Multiplicative.ofAdd <| ↑n * x) := by\n  simp only [← zsmul_eq_mul, ofAdd_zsmul, map_zpow]\n\n@[simp]\ntheorem translate_pow (x : ℝ) (n : ℕ) :\n    translate (Multiplicative.ofAdd x) ^ n = translate (Multiplicative.ofAdd <| ↑n * x) :=\n  translate_zpow x n\n\n@[simp]\ntheorem translate_iterate (x : ℝ) (n : ℕ) :\n    (translate (Multiplicative.ofAdd x))^[n] = translate (Multiplicative.ofAdd <| ↑n * x) := by\n  rw [← coe_pow, ← Units.val_pow_eq_pow_val, translate_pow]\n\n/-!\n### Commutativity with integer translations\n\nIn this section we prove that `f` commutes with translations by an integer number.\nFirst we formulate these statements (for a natural or an integer number,\naddition on the left or on the right, addition or subtraction) using `Function.Commute`,\nthen reformulate as `simp` lemmas `map_int_add` etc.\n-/\n\ntheorem commute_nat_add (n : ℕ) : Function.Commute f (n + ·) := by\n  simpa only [nsmul_one, add_left_iterate] using Function.Commute.iterate_right f.map_one_add n\n\ntheorem commute_add_nat (n : ℕ) : Function.Commute f (· + n) := by\n  simp only [add_comm _ (n : ℝ), f.commute_nat_add n]\n\ntheorem commute_sub_nat (n : ℕ) : Function.Commute f (· - n) := by\n  simpa only [sub_eq_add_neg] using!\n    (f.commute_add_nat n).inverses_right (Equiv.addRight _).right_inv (Equiv.addRight _).left_inv\n\nTarget:\ntheorem commute_add_int : ∀ n : ℤ, Function.Commute f (· + n)\n  | (n : ℕ) => f.commute_add_nat n\n  | -[n+1] => by simpa [sub_eq_add_neg] using f.commute_sub_nat (n + 1)\n\ntheorem commute_int_add (n : ℤ) : Function.Commute f (n + ·) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_099bd67010bd","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"7233c5be510663c049a4b3b52e65f41e1840c00dfd8ba69e41c5a9871c5438aa","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/Circle","family_id":"commute_add_int","file_id":"mathlib/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean","sample_id":"099bd67010bd3bdf0391a625cbe08e92d2569c8b6c447860d4abbc5d4f43fc92"},"split":"train"}
{"binding":{"chosen_proof_sha256":"62c33a874b3140b3e0d71815632c5a3a67b39c9eeeb9eb02b51c7aa26e877fd4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"bf9223aaabfc90647618733bf83627a63533b7794e13d7246825be247173665a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b7d8bbdbc675ed1b504fd154e9b2c440cdd8d44f35c4f66773b08147245061fc","source_sha256":"e6c6110c32737b6ae741871eb4fd44ef0a8fec8abe5efc331215d91b55ffecbc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor <;> grind [IsZero]","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":14,"token_jaccard":0.642857,"token_length_ratio":1.555556},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"4151f0bfbe5d215e252eaaadc98de727d1b233371bf45ae4e22f1c36f6dab589","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs\n\nNamespace:\nTactic.ComputeAsymptotics.MultiseriesExpansion.IsZero\n\nLocal context:\n/-\nCopyright (c) 2026 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\n/-!\n# Trimming of multiseries\n\nA multiseries is *trimmed* when its leading coefficient (the head of its expansion) is itself\ntrimmed and non-zero. For a trimmed multiseries, the leading monomial captures the main\nasymptotic behavior of the approximated function.\n\n## Main definitions\n\n* `IsZero`: a multiseries represents the zero function — it is either the real number `0`\n  (for the empty basis) or has an empty underlying sequence (`.nil`).\n* `Trimmed` and `Multiseries.Trimmed`: a multiseries is trimmed in the sense above. The former\n  is defined inductively for `MultiseriesExpansion`, and the latter for `Multiseries` is\n  derived from it.\n\nWe also prove structural lemmas relating these predicates to `seq` and to the `cons`/`nil`\nconstructors.\n\n-/\n\n@[expose] public section\n\nnamespace Tactic.ComputeAsymptotics\n\nnamespace MultiseriesExpansion\n\nopen Filter Topology Stream'\n\n/-- A multiseries is zero if it is the real constant `0` or has an empty sequence. -/\ninductive IsZero : {basis : Basis} → MultiseriesExpansion basis → Prop\n| const {c : MultiseriesExpansion []} (hc : c.toReal = 0) : IsZero c\n| nil {basis_hd} {basis_tl} (f) : @IsZero (basis_hd :: basis_tl) (mk .nil f)\n\nnamespace IsZero\n\n@[simp]\n\nTarget:\ntheorem const_iff {c : MultiseriesExpansion []} : IsZero c ↔ c.toReal = 0 :=\n\nProof body:\n","rejected":"by\n  constructor <;> grind [IsZero]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/ComputeAsymptotics","family_id":"const_iff","file_id":"mathlib/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Trimming.lean","sample_id":"b7d8bbdbc675ed1b504fd154e9b2c440cdd8d44f35c4f66773b08147245061fc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"badd85084558f6da2abee0803d9cfe41d37ed7236c401bd880c419ea46047156","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7b470604d9e322074df2adc2ca1effcd921f6f9f24f85545a0b4aaa60460b18d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"73b36450d7beabc8b542d5bd6e54c7d1928d98c7ccc3d417ffbb153021170e98","source_sha256":"3ebf52bd4f56640ff82f3663b47b89c207fb7b51b21d5bfbbe3cfb7edd34865e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [lt_univ']\n  constructor\n  · rintro ⟨β, e⟩\n    exact ⟨#β, lift_mk_eq.{u, _, v + 1}.2 e⟩\n  · rintro ⟨c, hc⟩\n    exact ⟨⟨c.out, lift_mk_eq.{u, _, v + 1}.1 (hc.trans (congr rfl c.mk_out.symm))⟩⟩","hard_negative":true,"metrics":{"chosen_tokens":76,"rejected_tokens":5,"token_jaccard":0.081081,"token_length_ratio":0.065789},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"418d287f6bb67962af655be24a04191f0a4c55c72dc5859f9559bb4f93ed64ba","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.SetTheory.Ordinal.Basic\n\nNamespace:\nCardinal\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Floris van Doorn\n-/\n/-!\n# Universal ordinal and cardinal\n\n`Cardinal.univ` is the cardinality of the cardinals themselves. Likewise, `Ordinal.univ` is the\norder type of the ordinals. These are related via `Cardinal.univ.ord = Ordinal.univ` and\n`Ordinal.univ.card = Cardinal.univ`.\n\nThe cardinal `Cardinal.univ` is strongly inaccessible. This reflects the fact that in ZFC, the\ncardinals form a proper class. See `IsInaccessible.univ` for a proof.\n\n## Implementation notes\n\nWe actually define `Cardinal.univ` as the cardinality of `Ordinal`, rather than that of `Cardinal`.\nThis makes the basic API easier to set up. See `Cardinal.mk_cardinal` for a proof that\n`Cardinal.univ = #Cardinal`.\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nset_option linter.checkUnivs false in\nopen Ordinal in\n-- intended to be used with explicit universe parameters\n/-- The ordinal `univ.{u, v}` is the order type of `Ordinal.{u}` or `Cardinal.{u}`, as an element of\n`Ordinal.{v}` (when `u < v`). -/\n@[pp_with_univ, nolint checkUnivs]\ndef Ordinal.univ : Ordinal.{max (u + 1) v} :=\n  lift.{v, u + 1} (typeLT Ordinal)\n\nset_option linter.checkUnivs false in\nopen Cardinal in\n-- intended to be used with explicit universe parameters\n/-- The cardinal `univ.{u, v}` is the cardinality of `Ordinal.{u}` or `Cardinal.{u}`, as an element\nof `Cardinal.{v}` (when `u < v`). -/\n@[pp_with_univ, nolint checkUnivs]\ndef Cardinal.univ : Cardinal.{max (u + 1) v} :=\n  lift.{v, u + 1} #Ordinal\n\n/-! ### Universal ordinal -/\n\nnamespace Ordinal\n\n@[simp]\ntheorem type_lt_ordinal : typeLT Ordinal = univ.{u, u + 1} :=\n  (lift_id _).symm\n\n@[deprecated type_lt_ordinal (since := \"2026-03-20\")]\ntheorem univ_id : univ.{u, u + 1} = typeLT Ordinal :=\n  lift_id _\n\n@[simp]\ntheorem lift_univ : lift.{w} univ.{u, v} = univ.{u, max v w} :=\n  lift_lift _\n\ntheorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=\n  congr_fun lift_umax _\n\n/-- Principal segment version of the lift operation on ordinals, embedding `Ordinal.{u}` in\n`Ordinal.{v}` as a principal segment when `u < v`. -/\ndef liftPrincipalSeg : Ordinal.{u} <i Ordinal.{max (u + 1) v} :=\n  ⟨liftInitialSeg.{max (u + 1) v, u}, univ.{u, v}, by\n    refine fun b => inductionOn b ?_; intro β s _\n    rw [univ, ← lift_umax]; constructor <;> intro h\n    · obtain ⟨a, e⟩ := h\n      rw [← e]\n      refine inductionOn a ?_\n      intro α r _\n      exact lift_type_lt.{u, u + 1, max (u + 1) v}.2 ⟨typein r⟩\n    · rw [← lift_id (type s)] at h ⊢\n      obtain ⟨f⟩ := lift_type_lt.{_,_,v}.1 h\n      obtain ⟨f, a, hf⟩ := f\n      exists a\n      induction a using inductionOn with | type α r\n      refine lift_type_eq.{u, max (u + 1) v, max (u + 1) v}.2\n        ⟨(RelIso.ofSurjective (RelEmbedding.ofMonotone ?_ ?_) ?_).symm⟩\n      · exact fun b => enum r ⟨f b, (hf _).1 ⟨_, rfl⟩⟩\n      · refine fun a b h => (typein_lt_typein r).1 ?_\n        rw [typein_enum, typein_enum]\n        exact f.map_rel_iff.2 h\n      · intro a'\n        obtain ⟨b, e⟩ := (hf _).2 (typein_lt_type _ a')\n        exists b\n        simp only [RelEmbedding.ofMonotone_coe]\n        simp [e]⟩\n\n@[simp]\ntheorem liftPrincipalSeg_coe :\n    (liftPrincipalSeg.{u, v} : Ordinal → Ordinal) = lift.{max (u + 1) v} :=\n  rfl\n\n@[simp]\ntheorem liftPrincipalSeg_top : (liftPrincipalSeg.{u, v}).top = univ.{u, v} :=\n  rfl\n\n@[deprecated liftPrincipalSeg_top (since := \"2026-03-20\")]\ntheorem liftPrincipalSeg_top' : liftPrincipalSeg.{u, u + 1}.top = typeLT Ordinal := by\n  simp\n\n@[simp]\ntheorem card_univ : card univ.{u, v} = Cardinal.univ.{u, v} :=\n  rfl\n\nend Ordinal\n\n/-! ### Universal cardinal -/\n\nnamespace Cardinal\n\n@[simp]\ntheorem mk_ordinal : #Ordinal = univ.{u, u + 1} :=\n  (lift_id _).symm\n\n@[deprecated mk_ordinal (since := \"2026-04-22\")]\ntheorem univ_id : univ.{u, u + 1} = #Ordinal :=\n  lift_id _\n\n@[simp]\ntheorem lift_univ : lift.{w} univ.{u, v} = univ.{u, max v w} :=\n  lift_lift _\n\ntheorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=\n  congr_fun lift_umax _\n\ntheorem lift_lt_univ (c : Cardinal) : lift.{u + 1, u} c < univ.{u, u + 1} := by\n  simpa only [Ordinal.liftPrincipalSeg_coe, lift_ord, lift_succ, ord_le, Order.succ_le_iff] using!\n    le_of_lt (Ordinal.liftPrincipalSeg.{u, u + 1}.lt_top (Order.succ c).ord)\n\ntheorem lift_lt_univ' (c : Cardinal) : lift.{max (u + 1) v, u} c < univ.{u, v} := by\n  have := lift_lt.{_, max (u + 1) v}.2 (lift_lt_univ c)\n  rw [lift_lift, lift_univ, univ_umax.{u, v}] at this\n  exact this\n\n@[simp]\ntheorem aleph0_lt_univ : ℵ₀ < univ.{u, v} := by\n  simpa using lift_lt_univ' ℵ₀\n\n@[simp]\ntheorem nat_lt_univ (n : ℕ) : n < univ.{u, v} := natCast_lt_aleph0.trans aleph0_lt_univ\n\n@[simp]\ntheorem univ_pos : 0 < univ.{u, v} :=\n  aleph0_lt_univ.pos\n\n@[simp]\ntheorem univ_ne_zero : univ.{u, v} ≠ 0 :=\n  univ_pos.ne'\n\n@[simp]\ntheorem ord_univ : ord univ.{u, v} = Ordinal.univ.{u, v} := by\n  refine le_antisymm (ord_card_le _) <| le_of_forall_lt fun o h => lt_ord.2 ?_\n  have := Ordinal.liftPrincipalSeg.mem_range_of_rel_top (by simpa using h)\n  rcases this with ⟨o, h'⟩\n  rw [← h', Ordinal.liftPrincipalSeg_coe, ← Ordinal.lift_card]\n  apply lift_lt_univ'\n\ntheorem lt_univ {c} : c < univ.{u, u + 1} ↔ ∃ c', c = lift.{u + 1, u} c' :=\n  ⟨fun h => by\n    have := ord_lt_ord.2 h\n    rw [ord_univ] at this\n    obtain ⟨o, e⟩ := Ordinal.liftPrincipalSeg.mem_range_of_rel_top (by simpa)\n    have := card_ord c\n    rw [← e, Ordinal.liftPrincipalSeg_coe, ← Ordinal.lift_card] at this\n    exact ⟨_, this.symm⟩, fun ⟨_, e⟩ => e.symm ▸ lift_lt_univ _⟩\n\ntheorem lt_univ' {c} : c < univ.{u, v} ↔ ∃ c', c = lift.{max (u + 1) v, u} c' :=\n  ⟨fun h => by\n    let ⟨a, h', e⟩ := lt_lift_iff.1 h\n    rw [mk_ordinal] at h'\n    rcases lt_univ.{u}.1 h' with ⟨c', rfl⟩\n    exact ⟨c', by simp only [e.symm, lift_lift]⟩, fun ⟨_, e⟩ => e.symm ▸ lift_lt_univ' _⟩\n\ntheorem IsStrongLimit.univ : IsStrongLimit univ.{u, v} :=\n  ⟨univ_ne_zero, fun c h ↦ let ⟨w, h⟩ := lt_univ'.1 h; lt_univ'.2 ⟨2 ^ w, by simp [h]⟩⟩\n\nTarget:\ntheorem small_iff_lift_mk_lt_univ {α : Type u} :\n    Small.{v} α ↔ Cardinal.lift.{v + 1, _} #α < univ.{v, max u (v + 1)} :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_73b36450d7be","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"313bb6754713b0d9c9e719b9d2f04612099b39d13035f3fb89659e2872359669","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Ordinal","family_id":"small_iff_lift_mk_lt_univ","file_id":"mathlib/Mathlib/SetTheory/Ordinal/Univ.lean","sample_id":"73b36450d7beabc8b542d5bd6e54c7d1928d98c7ccc3d417ffbb153021170e98"},"split":"train"}
{"binding":{"chosen_proof_sha256":"363f3ee37f17de1a17c56efabc2dfd6a0b7c412da4a7733db969356b0041dbda","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"55e5c34623a2e778c9d9153c0ad047a58585be7ec2184015ff3484db7a201425","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a921660930bcd0f4c9bd0e614c1768bb72ffc642b6565ee11d368d38df7c680d","source_sha256":"a4e7292deef4515ad5e54a931a89dcffc81e82a037be0f261e9c4a92f324bcea","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply SetLike.coe_injective\n  simp only [coe_centralizer, Set.centralizer_centralizer_centralizer]\n\nvariable {M} in","hard_negative":true,"metrics":{"chosen_tokens":19,"rejected_tokens":5,"token_jaccard":0.095238,"token_length_ratio":0.263158},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"41905867a23c2ab967075a2663c218509f7d9bd078a0eb73b697e53b9a45cbff","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.Subsemigroup.Centralizer\npublic import Mathlib.GroupTheory.Submonoid.Center\n\nNamespace:\nSubmonoid\n\nLocal context:\n/-\nCopyright (c) 2021 Thomas Browning. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Thomas Browning\n-/\n/-!\n# Centralizers of magmas and monoids\n\n## Main definitions\n\n* `Submonoid.centralizer`: the centralizer of a subset of a monoid\n* `AddSubmonoid.centralizer`: the centralizer of a subset of an additive monoid\n\nWe provide `Subgroup.centralizer`, `AddSubgroup.centralizer` in other files.\n-/\n\n@[expose] public section\n\n-- Guard against import creep\nassert_not_exists Finset\n\nvariable {M : Type*} {S T : Set M}\n\nnamespace Submonoid\n\nsection\n\nvariable [Monoid M] (S)\n\n/-- The centralizer of a subset of a monoid `M`. -/\n@[to_additive /-- The centralizer of a subset of an additive monoid. -/]\ndef centralizer : Submonoid M where\n  carrier := S.centralizer\n  one_mem' := S.one_mem_centralizer\n  mul_mem' := Set.mul_mem_centralizer\n\n@[to_additive (attr := simp, norm_cast)]\ntheorem coe_centralizer : ↑(centralizer S) = S.centralizer :=\n  rfl\n\n@[to_additive AddSubmonoid.centralizer_toAddSubsemigroup]\ntheorem centralizer_toSubsemigroup : (centralizer S).toSubsemigroup = Subsemigroup.centralizer S :=\n  rfl\n\nvariable {S}\n\n@[to_additive]\ntheorem mem_centralizer_iff {z : M} : z ∈ centralizer S ↔ ∀ g ∈ S, g * z = z * g :=\n  Iff.rfl\n\n@[to_additive]\ntheorem center_le_centralizer (s) : center M ≤ centralizer s :=\n  s.center_subset_centralizer\n\n@[to_additive]\ninstance decidableMemCentralizer (a) [Decidable <| ∀ b ∈ S, b * a = a * b] :\n    Decidable (a ∈ centralizer S) :=\n  decidable_of_iff' _ mem_centralizer_iff\n\n@[to_additive]\ntheorem centralizer_le (h : S ⊆ T) : centralizer T ≤ centralizer S :=\n  Set.centralizer_subset h\n\n@[to_additive (attr := simp)]\ntheorem centralizer_eq_top_iff_subset {s : Set M} : centralizer s = ⊤ ↔ s ⊆ center M :=\n  SetLike.ext'_iff.trans Set.centralizer_eq_top_iff_subset\n\nvariable (M)\n\n@[to_additive (attr := simp)]\ntheorem centralizer_univ : centralizer Set.univ = center M :=\n  SetLike.ext' (Set.centralizer_univ M)\n\n@[to_additive]\nlemma le_centralizer_centralizer {s : Submonoid M} : s ≤ centralizer (centralizer (s : Set M)) :=\n  Set.subset_centralizer_centralizer\n\n@[to_additive (attr := simp)]\n\nTarget:\nlemma centralizer_centralizer_centralizer {s : Set M} :\n    centralizer s.centralizer.centralizer = centralizer s :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_a921660930bc","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"2ba0c7b76fc548bb992dcea9353fc73ddd4024956c677acbcd5ef72a0eb96706","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Submonoid","family_id":"centralizer_centralizer_centralizer","file_id":"mathlib/Mathlib/GroupTheory/Submonoid/Centralizer.lean","sample_id":"a921660930bcd0f4c9bd0e614c1768bb72ffc642b6565ee11d368d38df7c680d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b77e7e870da0ed2ed6f8b7ac1969e927be1488555d4e8afb15051655e1aaced2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"105c96e643efa1f749e66cefffdea4f338def0d05aec0167af11a6685aed3ac3","source_sha256":"85fcf92cc8640c9f278d7a2006b517c6ae93694387f3542b94cbe753427dd693","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [← Function.comp_def, Tendsto, ← map_map, map_inv_atTop]","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":2,"token_jaccard":0.076923,"token_length_ratio":0.133333},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"41aaa5590fb5c2f7a5763e39b7c2d516b3ff11baab056f0de0eff84cf2faec91","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.MinMax\npublic import Mathlib.Order.Filter.AtTopBot.Basic\npublic import Mathlib.Order.Filter.AtTopBot.Map\npublic import Mathlib.Order.Filter.AtTopBot.Monoid\n\nNamespace:\nFilter\n\nLocal context:\n/-\nCopyright (c) 2019 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Convergence to ±infinity in ordered commutative groups\n-/\n\npublic section\n\nvariable {α G : Type*}\nopen Set\n\nnamespace Filter\n\nsection OrderedCommGroup\n\nvariable [CommGroup G] [PartialOrder G] [IsOrderedMonoid G] (l : Filter α) {f g : α → G}\n\n@[to_additive]\ntheorem tendsto_atTop_mul_left_of_le' (C : G) (hf : ∀ᶠ x in l, C ≤ f x) (hg : Tendsto g l atTop) :\n    Tendsto (fun x => f x * g x) l atTop :=\n  .atTop_of_isBoundedUnder_le_mul (f := f⁻¹) ⟨C⁻¹, by simpa⟩ (by simpa)\n\n@[to_additive]\ntheorem tendsto_atBot_mul_left_of_ge' (C : G) (hf : ∀ᶠ x in l, f x ≤ C) (hg : Tendsto g l atBot) :\n    Tendsto (fun x => f x * g x) l atBot :=\n  tendsto_atTop_mul_left_of_le' (G := Gᵒᵈ) _ C hf hg\n\n@[to_additive]\ntheorem tendsto_atTop_mul_left_of_le (C : G) (hf : ∀ x, C ≤ f x) (hg : Tendsto g l atTop) :\n    Tendsto (fun x => f x * g x) l atTop :=\n  tendsto_atTop_mul_left_of_le' l C (univ_mem' hf) hg\n\n@[to_additive]\ntheorem tendsto_atBot_mul_left_of_ge (C : G) (hf : ∀ x, f x ≤ C) (hg : Tendsto g l atBot) :\n    Tendsto (fun x => f x * g x) l atBot :=\n  tendsto_atTop_mul_left_of_le (G := Gᵒᵈ) _ C hf hg\n\n@[to_additive]\ntheorem tendsto_atTop_mul_right_of_le' (C : G) (hf : Tendsto f l atTop) (hg : ∀ᶠ x in l, C ≤ g x) :\n    Tendsto (fun x => f x * g x) l atTop :=\n  .atTop_of_mul_isBoundedUnder_le (g := g⁻¹) ⟨C⁻¹, by simpa⟩ (by simpa)\n\n@[to_additive]\ntheorem tendsto_atBot_mul_right_of_ge' (C : G) (hf : Tendsto f l atBot) (hg : ∀ᶠ x in l, g x ≤ C) :\n    Tendsto (fun x => f x * g x) l atBot :=\n  tendsto_atTop_mul_right_of_le' (G := Gᵒᵈ) _ C hf hg\n\n@[to_additive]\ntheorem tendsto_atTop_mul_right_of_le (C : G) (hf : Tendsto f l atTop) (hg : ∀ x, C ≤ g x) :\n    Tendsto (fun x => f x * g x) l atTop :=\n  tendsto_atTop_mul_right_of_le' l C hf (univ_mem' hg)\n\n@[to_additive]\ntheorem tendsto_atBot_mul_right_of_ge (C : G) (hf : Tendsto f l atBot) (hg : ∀ x, g x ≤ C) :\n    Tendsto (fun x => f x * g x) l atBot :=\n  tendsto_atTop_mul_right_of_le (G := Gᵒᵈ) _ C hf hg\n\n@[to_additive]\ntheorem tendsto_atTop_mul_const_left (C : G) (hf : Tendsto f l atTop) :\n    Tendsto (fun x => C * f x) l atTop :=\n  tendsto_atTop_mul_left_of_le' l C (univ_mem' fun _ => le_refl C) hf\n\n@[to_additive]\ntheorem tendsto_atBot_mul_const_left (C : G) (hf : Tendsto f l atBot) :\n    Tendsto (fun x => C * f x) l atBot :=\n  tendsto_atTop_mul_const_left (G := Gᵒᵈ) _ C hf\n\n@[to_additive]\ntheorem tendsto_atTop_mul_const_right (C : G) (hf : Tendsto f l atTop) :\n    Tendsto (fun x => f x * C) l atTop :=\n  tendsto_atTop_mul_right_of_le' l C hf (univ_mem' fun _ => le_refl C)\n\n@[to_additive]\ntheorem tendsto_atBot_mul_const_right (C : G) (hf : Tendsto f l atBot) :\n    Tendsto (fun x => f x * C) l atBot :=\n  tendsto_atTop_mul_const_right (G := Gᵒᵈ) _ C hf\n\n@[to_additive]\ntheorem map_inv_atBot : map (Inv.inv : G → G) atBot = atTop :=\n  (OrderIso.inv G).map_atBot\n\n@[to_additive]\ntheorem map_inv_atTop : map (Inv.inv : G → G) atTop = atBot :=\n  (OrderIso.inv G).map_atTop\n\n@[to_additive]\ntheorem comap_inv_atBot : comap (Inv.inv : G → G) atBot = atTop :=\n  (OrderIso.inv G).comap_atTop\n\n@[to_additive]\ntheorem comap_inv_atTop : comap (Inv.inv : G → G) atTop = atBot :=\n  (OrderIso.inv G).comap_atBot\n\n@[to_additive]\ntheorem tendsto_inv_atTop_atBot : Tendsto (Inv.inv : G → G) atTop atBot :=\n  (OrderIso.inv G).tendsto_atTop\n\n@[to_additive]\ntheorem tendsto_inv_atBot_atTop : Tendsto (Inv.inv : G → G) atBot atTop :=\n  tendsto_inv_atTop_atBot (G := Gᵒᵈ)\n\nvariable {l}\n\n@[to_additive (attr := simp)]\ntheorem tendsto_inv_atTop_iff : Tendsto (fun x => (f x)⁻¹) l atTop ↔ Tendsto f l atBot :=\n  (OrderIso.inv G).tendsto_atBot_iff\n\n@[to_additive (attr := simp)]\ntheorem tendsto_inv_atBot_iff : Tendsto (fun x => (f x)⁻¹) l atBot ↔ Tendsto f l atTop :=\n  (OrderIso.inv G).tendsto_atTop_iff\n\n@[to_additive (attr := simp)]\n\nTarget:\ntheorem tendsto_comp_inv_atTop_iff {f : G → α} :\n    Tendsto (fun x ↦ f (x⁻¹)) atTop l ↔ Tendsto f atBot l :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Filter","family_id":"tendsto_comp_inv_attop_iff","file_id":"mathlib/Mathlib/Order/Filter/AtTopBot/Group.lean","sample_id":"105c96e643efa1f749e66cefffdea4f338def0d05aec0167af11a6685aed3ac3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d3371ad1ae3b9f3bec6391019a5fb1c62af2682915c2a167e06751160d75c560","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f5c3570decc7db710ef7c794c68d9c0b62e8e6967dc4c3bfe5c338cfbcf456b8","source_sha256":"f6bedc69ab0e67353ba4b185fe6aec08e361441cb856f5f803e100c5cd160384","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hb' : (0 : ℝ) < ↑(abs b) := Int.cast_pos.mpr (abs_pos.mpr hb)\n  have hbε : 0 < abs b • ε := by\n    rw [Algebra.smul_def]\n    exact mul_pos hb' hε\n  have hfloor : ∀ i, 0 ≤ floor ((A i % b : ℤ) / abs b • ε : ℝ) :=\n    fun _ ↦ floor_nonneg.mpr (div_nonneg (cast_nonneg (emod_nonneg _ hb)) hbε.le)\n  refine ⟨fun i ↦ ⟨natAbs (floor ((A i % b : ℤ) / abs b • ε : ℝ)), ?_⟩, ?_⟩\n  · rw [← ofNat_lt, natAbs_of_nonneg (hfloor i), floor_lt, Algebra.smul_def, eq_intCast, ← div_div]\n    apply lt_of_lt_of_le _ (Nat.le_ceil _)\n    gcongr\n    rw [div_lt_one hb', cast_lt]\n    exact Int.emod_lt_abs _ hb\n  intro i₀ i₁ hi\n  have hi : (⌊↑(A i₀ % b) / abs b • ε⌋.natAbs : ℤ) = ⌊↑(A i₁ % b) / abs b • ε⌋.natAbs :=\n    congr_arg ((↑) : ℕ → ℤ) (Fin.mk_eq_mk.mp hi)\n  rw [natAbs_of_nonneg (hfloor i₀), natAbs_of_nonneg (hfloor i₁)] at hi\n  have hi := abs_sub_lt_one_of_floor_eq_floor hi\n  rw [abs_sub_comm, ← sub_div, abs_div, abs_of_nonneg hbε.le, div_lt_iff₀ hbε, one_mul] at hi\n  rwa [Int.cast_abs, Int.cast_sub]\n\n/-- `abs : ℤ → ℤ` is an admissible absolute value. -/\nnoncomputable def absIsAdmissible : IsAdmissible AbsoluteValue.abs :=\n  { AbsoluteValue.abs_isEuclidean with\n    card := fun ε ↦ ⌈1 / ε⌉₊\n    exists_partition' := fun n _ hε _ hb ↦ exists_partition_int n hε hb }\n\nnoncomputable instance : Inhabited (IsAdmissible AbsoluteValue.abs) :=\n  ⟨absIsAdmissible⟩","hard_negative":true,"metrics":{"chosen_tokens":376,"rejected_tokens":8,"token_jaccard":0.033333,"token_length_ratio":0.021277},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"41bf18869ead390331de6d70aa9fe0df126ffb58db84de53e086454f37eda47b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Basic\npublic import Mathlib.Algebra.Order.Archimedean.Real.Basic\npublic import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue\n\nNamespace:\nAbsoluteValue\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Admissible absolute value on the integers\nThis file defines an admissible absolute value `AbsoluteValue.absIsAdmissible`\nwhich we use to show the class number of the ring of integers of a number field\nis finite.\n\n## Main results\n\n* `AbsoluteValue.absIsAdmissible` shows the \"standard\" absolute value on `ℤ`,\n  mapping negative `x` to `-x`, is admissible.\n-/\n\n@[expose] public section\n\n\nnamespace AbsoluteValue\n\nopen Int\n\n/-- We can partition a finite family into `partition_card ε` sets, such that the remainders\nin each set are close together. -/\n\nTarget:\ntheorem exists_partition_int (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : ℤ} (hb : b ≠ 0) (A : Fin n → ℤ) :\n    ∃ t : Fin n → Fin ⌈1 / ε⌉₊,\n    ∀ i₀ i₁, t i₀ = t i₁ → ↑(abs (A i₁ % b - A i₀ % b)) < abs b • ε :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"40599cf467faf304b330c65bb7be1aeda3b8acded303696c63cc6650ecda71a7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/ClassNumber","family_id":"exists_partition_int","file_id":"mathlib/Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean","sample_id":"f5c3570decc7db710ef7c794c68d9c0b62e8e6967dc4c3bfe5c338cfbcf456b8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0cde5a74deeec0335214ee6e0227e2c1e15f0001cffe80b1335cabf0968874c8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"80af364b54e4f9d5bc6e01d40bc9d559f947616a68e18a5600b526c2e866a061","source_sha256":"345dde00cb75d3f9bc3f5909b7969856d423edbbc6d0a61a7a141d42b1e88b8d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : (p.map C).LiesOver p := ⟨IsMaximal.eq_of_le inferInstance IsPrime.ne_top' le_comap_map⟩\n  simp [height_eq_height_add_of_liesOver_of_hasGoingDown p]","hard_negative":false,"metrics":{"chosen_tokens":28,"rejected_tokens":2,"token_jaccard":0.041667,"token_length_ratio":0.071429},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"421fee0bd506a99ef01333645913b8967970e234d8674a8be7167254f046f498","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.FieldDivision\npublic import Mathlib.RingTheory.KrullDimension.PID\npublic import Mathlib.RingTheory.LocalRing.ResidueField.Fiber\npublic import Mathlib.RingTheory.Ideal.KrullsHeightTheorem\npublic import Mathlib.RingTheory.KrullDimension.NonZeroDivisors\n\nNamespace:\nPolynomial\n\nLocal context:\n/-\nCopyright (c) 2025 Jingting Wang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jingting Wang, Sihan Su, Yi Song, Christian Merten\n-/\n/-!\n# Krull dimension of polynomial ring\n\nThis file proves properties of the Krull dimension of the polynomial ring over a commutative ring\n\n## Main results\n\n* `Polynomial.ringKrullDim_le`: the Krull dimension of the polynomial ring over a commutative ring\n  `R` is less than `2 * (ringKrullDim R) + 1`.\n\nFor noetherian rings:\n* `Polynomial.ringKrullDim_of_isNoetherianRing`: the Krull dimension of `R[X]` is `dim R + 1`.\n* `MvPolynomial.ringKrullDim_of_isNoetherianRing`: the Krull dimension of `R[X₁, ..., Xₙ]` is\n  `dim R + n`.\n-/\n\npublic section\n\ntheorem Polynomial.ringKrullDim_le {R : Type*} [CommRing R] :\n    ringKrullDim (Polynomial R) ≤ 2 * (ringKrullDim R) + 1 := by\n  rw [ringKrullDim, ringKrullDim]\n  apply Order.krullDim_le_of_krullDim_preimage_le' (PrimeSpectrum.comap C) ?_ (fun p ↦ ?_)\n  · exact fun {a b} h ↦ Ideal.comap_mono h\n  · rw [show C = (algebraMap R (Polynomial R)) from rfl, Order.krullDim_eq_of_orderIso\n      (PrimeSpectrum.preimageOrderIsoFiber R (Polynomial R) p), ← ringKrullDim,\n      ← ringKrullDim_eq_of_ringEquiv (polyEquivTensor R (p.asIdeal.ResidueField)).toRingEquiv,\n      ← Ring.krullDimLE_iff]\n    infer_instance\n\nvariable {R : Type*} [CommRing R] [IsNoetherianRing R]\n\nnamespace Polynomial\n\nopen Ideal IsLocalization\n\n/--\nLet `p` be a maximal ideal of `A`. If `P` is a maximal ideal of `A[X]` lying above `p`,\nthen `ht(P) = ht(p) + 1`.\nSee `Polynomial.height_eq_height_add_one` for the more general version that does not assume `p` is\nmaximal.\n-/\nprivate lemma height_eq_height_add_one_of_isMaximal (p : Ideal R) [p.IsMaximal] (P : Ideal R[X])\n    [P.IsMaximal] [P.LiesOver p] : P.height = p.height + 1 := by\n  let _ : Field (R ⧸ p) := Quotient.field p\n  suffices h : (P.map (Ideal.Quotient.mk (Ideal.map (algebraMap R R[X]) p))).height = 1 by\n    rw [height_eq_height_add_of_liesOver_of_hasGoingDown p, h]\n  let e : (R[X] ⧸ (p.map (algebraMap R R[X]))) ≃+* (R ⧸ p)[X] :=\n    (polynomialQuotientEquivQuotientPolynomial p).symm\n  let P' : Ideal (R ⧸ p)[X] := Ideal.map e <| P.map (Ideal.Quotient.mk <| p.map (algebraMap R R[X]))\n  have : (P.map (Ideal.Quotient.mk <| p.map (algebraMap R R[X]))).IsMaximal := by\n    refine .map_of_surjective_of_ker_le Quotient.mk_surjective ?_\n    rw [mk_ker, LiesOver.over (P := P) (p := p)]\n    exact map_comap_le\n  have : P'.IsMaximal := map_isMaximal_of_equiv e\n  have : P'.height = 1 := IsPrincipalIdealRing.height_eq_one_of_isMaximal P' polynomial_not_isField\n  rwa [← e.height_map <| P.map (Ideal.Quotient.mk <| p.map (algebraMap R R[X]))]\n\n/-- Let `p` be a maximal ideal of `R`. Then the height of `p[X]` equals the height of `p`. -/\n\nTarget:\nlemma height_map_C (p : Ideal R) [p.IsMaximal] : (p.map C).height = p.height :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/KrullDimension","family_id":"height_map_c","file_id":"mathlib/Mathlib/RingTheory/KrullDimension/Polynomial.lean","sample_id":"80af364b54e4f9d5bc6e01d40bc9d559f947616a68e18a5600b526c2e866a061"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e1906f3243bcdc4427e534120589c5393cc550f48188ef4c67cb950df5b01321","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2640774717b5cc5cf0106fb4ef5d3f69a7c6539d5084db036f9ef862ac01288e","source_sha256":"823d2b692e062acc3ec291c52e92559d9b37f8adc1d4d307badd142b4bd43de9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h : whiskerLeft F (add F s i a b).hom = _ :=\n    ((whiskeringLeft C D D).obj F).map_preimage _\n  exact (NatTrans.congr_app h X).trans (by simp)","hard_negative":false,"metrics":{"chosen_tokens":47,"rejected_tokens":2,"token_jaccard":0.033333,"token_length_ratio":0.042553},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"42244ad7dbff3768e8368a8e7395ddc7343cb9206d4f9c76434b5c059e1f2c87","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Shift.CommShift\n\nNamespace:\nCategoryTheory.HasShift.Induced\n\nLocal context:\n/-\nCopyright (c) 2023 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Shift induced from a category to another\n\nIn this file, we introduce a sufficient condition on a functor\n`F : C ⥤ D` so that a shift on `C` by a monoid `A` induces a shift on `D`.\nMore precisely, when the functor `(D ⥤ D) ⥤ C ⥤ D` given\nby the precomposition with `F` is fully faithful, and that\nall the shift functors on `C` can be lifted to functors `D ⥤ D`\n(i.e. we have functors `s a : D ⥤ D` for all `a : A`, and isomorphisms\n`F ⋙ s a ≅ shiftFunctor C a ⋙ F`), then these functors `s a` are\nthe shift functors of a term of type `HasShift D A`.\n\nAs this condition on the functor `F` is satisfied for quotient and localization\nfunctors, the main construction `HasShift.induced` in this file shall be\nused for both quotient and localized shifts.\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Functor\n\nvariable {C D : Type _} [Category* C] [Category* D]\n  (F : C ⥤ D) {A : Type _} [AddMonoid A] [HasShift C A]\n  (s : A → D ⥤ D) (i : ∀ a, F ⋙ s a ≅ shiftFunctor C a ⋙ F)\n  [((whiskeringLeft C D D).obj F).Full] [((whiskeringLeft C D D).obj F).Faithful]\n\nnamespace HasShift\n\nnamespace Induced\n\n/-- The `zero` field of the `ShiftMkCore` structure for the induced shift. -/\nnoncomputable def zero : s 0 ≅ 𝟭 D :=\n  ((whiskeringLeft C D D).obj F).preimageIso ((i 0) ≪≫\n    isoWhiskerRight (shiftFunctorZero C A) F ≪≫ F.leftUnitor ≪≫ F.rightUnitor.symm)\n\n/-- The `add` field of the `ShiftMkCore` structure for the induced shift. -/\nnoncomputable def add (a b : A) : s (a + b) ≅ s a ⋙ s b :=\n  ((whiskeringLeft C D D).obj F).preimageIso\n    (i (a + b) ≪≫ isoWhiskerRight (shiftFunctorAdd C a b) F ≪≫\n      Functor.associator _ _ _ ≪≫\n        isoWhiskerLeft _ (i b).symm ≪≫ (Functor.associator _ _ _).symm ≪≫\n        isoWhiskerRight (i a).symm _ ≪≫ Functor.associator _ _ _)\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp]\nlemma zero_hom_app_obj (X : C) :\n    (zero F s i).hom.app (F.obj X) =\n      (i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X) := by\n  have h : whiskerLeft F (zero F s i).hom = _ :=\n    ((whiskeringLeft C D D).obj F).map_preimage _\n  exact (NatTrans.congr_app h X).trans (by simp)\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp]\nlemma zero_inv_app_obj (X : C) :\n    (zero F s i).inv.app (F.obj X) =\n      F.map ((shiftFunctorZero C A).inv.app X) ≫ (i 0).inv.app X := by\n  have h : whiskerLeft F (zero F s i).inv = _ :=\n    ((whiskeringLeft C D D).obj F).map_preimage _\n  exact (NatTrans.congr_app h X).trans (by simp)\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp]\n\nTarget:\nlemma add_hom_app_obj (a b : A) (X : C) :\n    (add F s i a b).hom.app (F.obj X) =\n      (i (a + b)).hom.app X ≫ F.map ((shiftFunctorAdd C a b).hom.app X) ≫\n        (i b).inv.app ((shiftFunctor C a).obj X) ≫ (s b).map ((i a).inv.app X) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Shift","family_id":"add_hom_app_obj","file_id":"mathlib/Mathlib/CategoryTheory/Shift/Induced.lean","sample_id":"2640774717b5cc5cf0106fb4ef5d3f69a7c6539d5084db036f9ef862ac01288e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7cabac5c0510d840ec215a55b2c97b8bd4ff4bb3ba7ca644c381e4c621c7c4ab","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b76866134268ce8c1e930e844ff2aaa363a473acc169ff6b4db1a719899e0bd6","source_sha256":"354c54d863521e4c2e2cb8eb45baaec320a74608e472a8d10bc34f45771b2911","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction rs generalizing M N with\n  | nil => simp\n  | cons x _ ih =>\n    simp only [List.map_cons, isWeaklyRegular_cons_iff] at reg ⊢\n    have e := (QuotSMulTop.algebraMapTensorEquivTensorQuotSMulTop x M S).symm ≪≫ₗ\n      QuotSMulTop.congr ((algebraMap R S) x) hf.equiv\n    have hg : IsBaseChange S <|\n        e.toLinearMap.restrictScalars R ∘ₗ TensorProduct.mk R S (QuotSMulTop x M) 1 :=\n      IsBaseChange.of_equiv e (fun _ ↦ by simp)\n    exact ⟨reg.1.of_flat_of_isBaseChange hf, ih hg reg.2⟩","hard_negative":false,"metrics":{"chosen_tokens":112,"rejected_tokens":2,"token_jaccard":0.016129,"token_length_ratio":0.017857},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"4237a088b715916ec909bae34c81cafba67a8d66f249d8e1c436aa703a6fb56c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Flat.FaithfullyFlat.Basic\npublic import Mathlib.RingTheory.Flat.Localization\npublic import Mathlib.RingTheory.Regular.RegularSequence\n\nNamespace:\nRingTheory.Sequence\n\nLocal context:\n/-\nCopyright (c) 2025 Yongle Hu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yongle Hu, Nailin Guan\n-/\n/-!\n# `RingTheory.Sequence.IsWeaklyRegular` is stable under flat base change\n\n## Main results\n* `RingTheory.Sequence.IsWeaklyRegular.of_flat_of_isBaseChange`: Let `R` be a commutative ring,\n  `M` be an `R`-module, `S` be a flat `R`-algebra, `N` be the base change of `M` to `S`.\n  If `[r₁, …, rₙ]` is a weakly regular `M`-sequence, then its image in `N` is a weakly regular\n  `N`-sequence.\n-/\n\npublic section\n\nnamespace RingTheory.Sequence\n\nopen Module\n\nvariable {R S M N : Type*} [CommRing R] [CommRing S] [Algebra R S]\n  [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower R S N]\n\n/-- Let `R` be a commutative ring, `M` be an `R`-module, `S` be a flat `R`-algebra, `N` be the base\n  change of `M` to `S`. If `[r₁, …, rₙ]` is a weakly regular `M`-sequence, then its image in `N` is\n  a weakly regular `N`-sequence. -/\n\nTarget:\ntheorem IsWeaklyRegular.of_flat_of_isBaseChange [Flat R S] {f : M →ₗ[R] N} (hf : IsBaseChange S f)\n    {rs : List R} (reg : IsWeaklyRegular M rs) : IsWeaklyRegular N (rs.map (algebraMap R S)) :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Regular","family_id":"isweaklyregular","file_id":"mathlib/Mathlib/RingTheory/Regular/Flat.lean","sample_id":"b76866134268ce8c1e930e844ff2aaa363a473acc169ff6b4db1a719899e0bd6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9c3ded6353915dd072b1066cf2ce119a7132edcb22b959bc51e04f3927cf0477","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"def6a45549df833e35870bc3cedcd463aa656f27044d41b28de2a288d3116271","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5adb15e5706aefda52cadd04c16407f0f1f64e18c9972fb156247c44ba666c48","source_sha256":"69f324bcc2121803a5bee9f9f572821f81fe33ac326f7f940f63d1e7e35d90b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h1 (y : ℝ) : 𝐞 y = fourier 1 (y : UnitAddCircle) := by\n    rw [fourierChar_apply, fourier_coe_apply]\n    push_cast\n    ring_nf\n  simpa only [h1, Int.cast_one, ofReal_one, div_one, mul_one] using hasDerivAt_fourier 1 1 x","hard_negative":true,"metrics":{"chosen_tokens":49,"rejected_tokens":2,"token_jaccard":0.029412,"token_length_ratio":0.040816},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"427b404de34fe85f67a0576d38af56f0cab238b2cb99d4bcb670fd4d4899ba40","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Calculus.FDeriv.Analytic\npublic import Mathlib.Analysis.Fourier.AddCircle\npublic import Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Calculus.ContDiff.Bounds\nimport Mathlib.Analysis.Calculus.ContDiff.CPolynomial\nimport Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts\nimport Mathlib.Analysis.Calculus.ParametricIntegral\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2024 David Loeffler. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alex Kontorovich, David Loeffler, Heather Macbeth, Sébastien Gouëzel\n-/\n/-!\n# Derivatives of the Fourier transform\n\nIn this file we compute the Fréchet derivative of the Fourier transform of `f`, where `f` is a\nfunction such that both `f` and `v ↦ ‖v‖ * ‖f v‖` are integrable. Here the Fourier transform is\nunderstood as an operator `(V → E) → (W → E)`, where `V` and `W` are normed `ℝ`-vector spaces\nand the Fourier transform is taken with respect to a continuous `ℝ`-bilinear\npairing `L : V × W → ℝ` and a given reference measure `μ`.\n\nWe also investigate higher derivatives: Assuming that `‖v‖^n * ‖f v‖` is integrable, we show\nthat the Fourier transform of `f` is `C^n`.\n\nWe also study in a parallel way the Fourier transform of the derivative, which is obtained by\ntensoring the Fourier transform of the original function with the bilinear form. We also get\nresults for iterated derivatives.\n\nA consequence of these results is that, if a function is smooth and all its derivatives are\nintegrable when multiplied by `‖v‖^k`, then the same goes for its Fourier transform, with\nexplicit bounds.\n\nWe give specialized versions of these results on inner product spaces (where `L` is the scalar\nproduct) and on the real line, where we express the one-dimensional derivative in more concrete\nterms, as the Fourier transform of `-2πI x * f x` (or `(-2πI x)^n * f x` for higher derivatives).\n\n## Main definitions and results\n\nWe introduce two convenience definitions:\n\n* `VectorFourier.fourierSMulRight L f`: given `f : V → E` and `L` a bilinear pairing\n  between `V` and `W`, then this is the function `fun v ↦ -(2 * π * I) (L v ⬝) • f v`,\n  from `V` to `Hom (W, E)`.\n  This is essentially `ContinuousLinearMap.smulRight`, up to the factor `- 2πI` designed to make\n  sure that the Fourier integral of `fourierSMulRight L f` is the derivative of the Fourier\n  integral of `f`.\n* `VectorFourier.fourierPowSMulRight` is the higher-order analogue for higher derivatives:\n  `fourierPowSMulRight L f v n` is informally `(-(2 * π * I))^n (L v ⬝)^n • f v`, in\n  the space of continuous multilinear maps `W [×n]→L[ℝ] E`.\n\nWith these definitions, the statements read as follows, first in a general context\n(arbitrary `L` and `μ`):\n\n* `VectorFourier.hasFDerivAt_fourierIntegral`: the Fourier integral of `f` is differentiable, with\n    derivative the Fourier integral of `fourierSMulRight L f`.\n* `VectorFourier.differentiable_fourierIntegral`: the Fourier integral of `f` is differentiable.\n* `VectorFourier.fderiv_fourierIntegral`: formula for the derivative of the Fourier integral of `f`.\n* `VectorFourier.fourierIntegral_fderiv`: formula for the Fourier integral of the derivative of `f`.\n* `VectorFourier.hasFTaylorSeriesUpTo_fourierIntegral`: under suitable integrability conditions,\n  the Fourier integral of `f` has an explicit Taylor series up to order `N`, given by the Fourier\n  integrals of `fun v ↦ fourierPowSMulRight L f v n`.\n* `VectorFourier.contDiff_fourierIntegral`: under suitable integrability conditions,\n  the Fourier integral of `f` is `C^n`.\n* `VectorFourier.iteratedFDeriv_fourierIntegral`: under suitable integrability conditions,\n  explicit formula for the `n`-th derivative of the Fourier integral of `f`, as the Fourier\n  integral of `fun v ↦ fourierPowSMulRight L f v n`.\n* `VectorFourier.pow_mul_norm_iteratedFDeriv_fourierIntegral_le`: explicit bounds for the `n`-th\n  derivative of the Fourier integral, multiplied by a power function, in terms of corresponding\n  integrals for the original function.\n\nThese statements are then specialized to the case of the usual Fourier transform on\nfinite-dimensional inner product spaces with their canonical Lebesgue measure (covering in\nparticular the case of the real line), replacing the namespace `VectorFourier` by\nthe namespace `Real` in the above statements.\n\nWe also give specialized versions of the one-dimensional real derivative (and iterated derivative)\nin `Real.deriv_fourierIntegral` and `Real.iteratedDeriv_fourierIntegral`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Real Complex MeasureTheory Filter TopologicalSpace\n\nopen scoped FourierTransform Topology ContDiff\n\n-- without this local instance, Lean tries first the instance\n-- `secondCountableTopologyEither_of_right` (whose priority is 100) and takes a very long time to\n-- fail. Since we only use the left instance in this file, we make sure it is tried first.\nattribute [local instance 101] secondCountableTopologyEither_of_left\n\nnamespace Real\n\nTarget:\nlemma hasDerivAt_fourierChar (x : ℝ) : HasDerivAt (𝐞 · : ℝ → ℂ) (2 * π * I * 𝐞 x) x :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_5adb15e5706a","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"f62c287cad842a967ad265077e93d0382dde31fad821d5f07974688d59ec3726","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Fourier","family_id":"hasderivat_fourierchar","file_id":"mathlib/Mathlib/Analysis/Fourier/FourierTransformDeriv.lean","sample_id":"5adb15e5706aefda52cadd04c16407f0f1f64e18c9972fb156247c44ba666c48"},"split":"train"}
{"binding":{"chosen_proof_sha256":"443eea2a8757775ee8a442c763e06d326c2fd7f65ba73927f6537e2b3fa75567","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"63406a1df7aba3e4d515c54c2b6910fb073c857cebaad69808804490f6844a24","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"36013d3b030a4beac323ba03c7b15bdae8a250c801f46d872bcc359cdf27248b","source_sha256":"73244aa9467f7c855deb57de598d12f743787aaa6e228fe2bbfaffba906d38dc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [finite_iff_subgroup_quotient]; constructor <;> assumption","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":2,"token_jaccard":0.090909,"token_length_ratio":0.181818},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"42afa11a4574b933e082d57a47661c8b6e6329d27a320999fa48a1a5dfa34f9f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Subgroup.Finite\npublic import Mathlib.Data.Finite.Prod\npublic import Mathlib.GroupTheory.QuotientGroup.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Kevin Buzzard, Patrick Massot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Patrick Massot\n-/\n-- This file is to a certain extent based on `quotient_module.lean` by Johannes Hölzl.\n/-!\n# Deducing finiteness of a group.\n-/\n\n@[expose] public section\n\nopen Function QuotientGroup Subgroup\nopen scoped Pointwise\n\n\nvariable {F G H : Type*} [Group F] [Group G] [Group H] [Fintype F] [Fintype H]\nvariable (f : F →* G) (g : G →* H)\n\nnamespace Group\n\nopen scoped Classical in\n/-- If `F` and `H` are finite such that `ker(G →* H) ≤ im(F →* G)`, then `G` is finite. -/\n@[to_additive (attr := implicit_reducible)\n/-- If `F` and `H` are finite such that `ker(G →+ H) ≤ im(F →+ G)`, then `G` is finite. -/]\nnoncomputable def fintypeOfKerLeRange (h : g.ker ≤ f.range) : Fintype G :=\n  @Fintype.ofEquiv _ _\n    (@instFintypeProd _ _ (Fintype.ofInjective _ <| kerLift_injective g) <|\n      Fintype.ofInjective _ <| inclusion_injective h)\n    groupEquivQuotientProdSubgroup.symm\n\n/-- If `F` and `H` are finite such that `ker(G →* H) = im(F →* G)`, then `G` is finite. -/\n@[to_additive (attr := implicit_reducible)\n/-- If `F` and `H` are finite such that `ker(G →+ H) = im(F →+ G)`, then `G` is finite. -/]\nnoncomputable def fintypeOfKerEqRange (h : g.ker = f.range) : Fintype G :=\n  fintypeOfKerLeRange _ _ h.le\n\n/-- If `ker(G →* H)` and `H` are finite, then `G` is finite. -/\n@[to_additive (attr := implicit_reducible)\n  /-- If `ker(G →+ H)` and `H` are finite, then `G` is finite. -/]\nnoncomputable def fintypeOfKerOfCodom [Fintype g.ker] : Fintype G :=\n  fintypeOfKerLeRange ((topEquiv : _ ≃* G).toMonoidHom.comp <| inclusion le_top) g fun x hx =>\n    ⟨⟨x, hx⟩, rfl⟩\n\n/-- If `F` and `coker(F →* G)` are finite, then `G` is finite. -/\n@[to_additive (attr := implicit_reducible)\n  /-- If `F` and `coker(F →+ G)` are finite, then `G` is finite. -/]\nnoncomputable def fintypeOfDomOfCoker [Normal f.range] [Fintype <| G ⧸ f.range] : Fintype G :=\n  fintypeOfKerLeRange _ (mk' f.range) fun x => (eq_one_iff x).mp\n\nend Group\n\n@[to_additive]\nlemma finite_iff_subgroup_quotient (H : Subgroup G) : Finite G ↔ Finite H ∧ Finite (G ⧸ H) := by\n  rw [(groupEquivQuotientProdSubgroup (s := H)).finite_iff, Prod.finite_iff, and_comm]\n\n@[to_additive]\n\nTarget:\nlemma Finite.of_subgroup_quotient (H : Subgroup G) [Finite H] [Finite (G ⧸ H)] : Finite G :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_36013d3b030a","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"8d8d2b17aa2978443abf69aead49998c8c2a9378fc7c577f892442c1a20f3703","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/QuotientGroup","family_id":"finite","file_id":"mathlib/Mathlib/GroupTheory/QuotientGroup/Finite.lean","sample_id":"36013d3b030a4beac323ba03c7b15bdae8a250c801f46d872bcc359cdf27248b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b4230b54afe23a3c3ef25af1bbb191756411be86b06eb1ac6af58052e937afcd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5be88ee7294e42cf554af7ccb31b4028b464c56bb950879cfb0843670b5a0955","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d0d0056d6763aa2dc8a9996175ddeb4b3a077afcfff88c7bba8165c2216d8b23","source_sha256":"ea74f644091594213d0fb29edabe43ef72df0c1170b50dfeb2e61c30c872f245","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [PresheafedSpace.congr_app (F.map_id j)]","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.230769},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"42bc257cccf504a46028be6d493122c7bfe94ede9ad83935e960f2f8ff13591c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Geometry.RingedSpace.PresheafedSpace\npublic import Mathlib.Topology.Category.TopCat.Limits.Basic\npublic import Mathlib.Topology.Sheaves.Limits\npublic import Mathlib.CategoryTheory.ConcreteCategory.Elementwise\n\nNamespace:\nAlgebraicGeometry.PresheafedSpace\n\nLocal context:\n/-\nCopyright (c) 2020 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# `PresheafedSpace C` has colimits.\n\nIf `C` has limits, then the category `PresheafedSpace C` has colimits,\nand the forgetful functor to `TopCat` preserves these colimits.\n\nWhen restricted to a diagram where the underlying continuous maps are open embeddings,\nthis says that we can glue presheafed spaces.\n\nGiven a diagram `F : J ⥤ PresheafedSpace C`,\nwe first build the colimit of the underlying topological spaces,\nas `colimit (F ⋙ PresheafedSpace.forget C)`. Call that colimit space `X`.\n\nOur strategy is to push each of the presheaves `F.obj j`\nforward along the continuous map `colimit.ι (F ⋙ PresheafedSpace.forget C) j` to `X`.\nSince pushforward is functorial, we obtain a diagram `J ⥤ (presheaf C X)ᵒᵖ`\nof presheaves on a single space `X`.\n(Note that the arrows now point the other direction,\nbecause this is the way `PresheafedSpace C` is set up.)\n\nThe limit of this diagram then constitutes the colimit presheaf.\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nuniverse v' u' v u\n\nopen CategoryTheory Opposite CategoryTheory.Category CategoryTheory.Functor CategoryTheory.Limits\n  TopCat TopCat.Presheaf TopologicalSpace\n\nvariable {J : Type u'} [Category.{v'} J] {C : Type u} [Category.{v} C]\n\nnamespace AlgebraicGeometry\n\nnamespace PresheafedSpace\n\nattribute [local simp] eqToHom_map\n\n-- We could enable the following attribute:\n-- attribute [local aesop safe cases (rule_sets := [CategoryTheory])] Opens\n-- although it doesn't appear to help in this file, in any case.\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp]\n\nTarget:\ntheorem map_id_c_app (F : J ⥤ PresheafedSpace.{_, _, v} C) (j) (U) :\n    (F.map (𝟙 j)).c.app U =\n      (Pushforward.id (F.obj j).presheaf).inv.app U ≫\n        (pushforwardEq (by simp) (F.obj j).presheaf).hom.app U :=\n\nProof body:\n","rejected":"by\n  exact map_id_c_app","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"d6d3a55f8bf79e2314a6e5a78536172922b0e24dff67a01c324663d7634c0bfe","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/RingedSpace","family_id":"map_id_c_app","file_id":"mathlib/Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean","sample_id":"d0d0056d6763aa2dc8a9996175ddeb4b3a077afcfff88c7bba8165c2216d8b23"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c311ad4b581caa0a3b895b5e4dcbcd3cc30063d829252ea41d678e32db406435","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f3389f96c5e2aa369eaf83ff655a44ee24b26357bab6687ac815547af6b99bb0","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4edfa6f1dfc9284bddb186d5cd60c15b919148603ac348edfd11fc15fed97087","source_sha256":"f341bd031305e6d14f35c1e928ff55667e741d845f3190520037617945b7b186","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  tfae_have 1 ↔ 2 := Sheaf.isConstant_iff_isIso_counit_app _ _ _\n  tfae_have 1 ↔ 3 := ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩\n  tfae_have 1 ↔ 4 := Sheaf.isConstant_iff_mem_essImage _ LightProfinite.isTerminalPUnit adjunction X\n  tfae_have 1 ↔ 5 :=\n    have : functor.Faithful := inferInstance\n    have : functor.Full := inferInstance\n    -- These `have` statements above shouldn't be needed, but they are.\n    Sheaf.isConstant_iff_isIso_counit_app' _ LightProfinite.isTerminalPUnit adjunction X\n  tfae_have 6 → 4 := fun h ↦\n    mem_locallyConstant_essImage_of_isColimit_mapCocone X (fun S ↦ (h S).some)\n  tfae_have 4 → 6 := fun ⟨Y, ⟨i⟩⟩ S ↦\n    ⟨IsColimit.mapCoconeEquiv ((sheafToPresheaf _ _).mapIso i)\n      (LightCondensed.isColimitLocallyConstantPresheafDiagram Y S)⟩\n  tfae_finish","hard_negative":true,"metrics":{"chosen_tokens":149,"rejected_tokens":3,"token_jaccard":0.016667,"token_length_ratio":0.020134},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"42c981131762e6f0836ac9a3cd49e07a810490e54d3fb1bfdaf235a3228600c4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Condensed.Discrete.Colimit\npublic import Mathlib.Condensed.Discrete.Module\n\nNamespace:\nLightCondSet\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# Characterizing discrete condensed sets and `R`-modules.\n\nThis file proves a characterization of discrete condensed sets, discrete condensed `R`-modules over\na ring `R`, discrete light condensed sets, and discrete light condensed `R`-modules over a ring `R`.\nsee `CondensedSet.isDiscrete_tfae`, `CondensedMod.isDiscrete_tfae`, `LightCondSet.isDiscrete_tfae`,\nand `LightCondMod.isDiscrete_tfae`.\n\nInformally, we can say: The following conditions characterize a condensed set `X` as discrete\n(`CondensedSet.isDiscrete_tfae`):\n\n1. There exists a set `X'` and an isomorphism `X ≅ cst X'`, where `cst X'` denotes the constant\n   sheaf on `X'`.\n2. The counit induces an isomorphism `cst X(*) ⟶ X`.\n3. There exists a set `X'` and an isomorphism `X ≅ LocallyConstant · X'`.\n4. The counit induces an isomorphism `LocallyConstant · X(*) ⟶ X`.\n5. For every profinite set `S = limᵢSᵢ`, the canonical map `colimᵢX(Sᵢ) ⟶ X(S)` is an isomorphism.\n\nThe analogues for light condensed sets, condensed `R`-modules over any ring, and light\ncondensed `R`-modules are nearly identical (`CondensedMod.isDiscrete_tfae`,\n`LightCondSet.isDiscrete_tfae`, and `LightCondMod.isDiscrete_tfae`).\n-/\n\npublic section\n\nuniverse u\n\nopen CategoryTheory Limits Functor FintypeCat\n\nnamespace Condensed\n\nvariable {C : Type*} [Category* C] [HasWeakSheafify (coherentTopology CompHaus.{u}) C]\n\n/--\nA condensed object is *discrete* if it is constant as a sheaf, i.e. isomorphic to a constant sheaf.\n-/\nabbrev IsDiscrete (X : Condensed.{u} C) := X.IsConstant (coherentTopology CompHaus)\n\nend Condensed\n\nnamespace CondensedSet\n\nopen CompHausLike.LocallyConstant\n\nlemma mem_locallyConstant_essImage_of_isColimit_mapCocone (X : CondensedSet.{u})\n    (h : ∀ S : Profinite.{u}, IsColimit <|\n      (profiniteToCompHaus.op ⋙ X.obj).mapCocone S.asLimitCone.op) :\n    CondensedSet.LocallyConstant.functor.essImage X := by\n  let e : CondensedSet.{u} ≌ Sheaf (coherentTopology Profinite) _ :=\n    (Condensed.ProfiniteCompHaus.equivalence (Type (u + 1))).symm\n  let i : (e.functor.obj X).obj ≅ (e.functor.obj (LocallyConstant.functor.obj _)).obj :=\n    Condensed.isoLocallyConstantOfIsColimit _ h\n  exact ⟨_, ⟨e.functor.preimageIso ((sheafToPresheaf _ _).preimageIso i.symm)⟩⟩\n\n/--\n`CondensedSet.LocallyConstant.functor` is left adjoint to the forgetful functor from condensed\nsets to sets.\n-/\nnoncomputable abbrev LocallyConstant.adjunction :\n    CondensedSet.LocallyConstant.functor ⊣ Condensed.underlying (Type (u + 1)) :=\n  CompHausLike.LocallyConstant.adjunction _ _\n\nopen Condensed\n\nopen CondensedSet.LocallyConstant List in\ntheorem isDiscrete_tfae (X : CondensedSet.{u}) :\n    TFAE\n    [ X.IsDiscrete\n    , IsIso ((Condensed.discreteUnderlyingAdj _).counit.app X)\n    , (Condensed.discrete _).essImage X\n    , CondensedSet.LocallyConstant.functor.essImage X\n    , IsIso (CondensedSet.LocallyConstant.adjunction.counit.app X)\n    , Sheaf.IsConstant (coherentTopology Profinite)\n        ((Condensed.ProfiniteCompHaus.equivalence _).inverse.obj X)\n    , ∀ S : Profinite.{u}, Nonempty\n        (IsColimit <| (profiniteToCompHaus.op ⋙ X.obj).mapCocone S.asLimitCone.op)\n    ] := by\n  tfae_have 1 ↔ 2 := Sheaf.isConstant_iff_isIso_counit_app _ _ _\n  tfae_have 1 ↔ 3 := ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩\n  tfae_have 1 ↔ 4 := Sheaf.isConstant_iff_mem_essImage _ CompHaus.isTerminalPUnit adjunction _\n  tfae_have 1 ↔ 5 :=\n    have : functor.Faithful := inferInstance\n    have : functor.Full := inferInstance\n    -- These `have` statements above shouldn't be needed, but they are.\n    Sheaf.isConstant_iff_isIso_counit_app' _ CompHaus.isTerminalPUnit adjunction _\n  tfae_have 1 ↔ 6 :=\n    (Sheaf.isConstant_iff_of_equivalence (coherentTopology Profinite)\n      (coherentTopology CompHaus) profiniteToCompHaus Profinite.isTerminalPUnit\n      CompHaus.isTerminalPUnit _).symm\n  tfae_have 7 → 4 := fun h ↦\n    mem_locallyConstant_essImage_of_isColimit_mapCocone X (fun S ↦ (h S).some)\n  tfae_have 4 → 7 := fun ⟨Y, ⟨i⟩⟩ S ↦\n    ⟨IsColimit.mapCoconeEquiv (isoWhiskerLeft profiniteToCompHaus.op\n      ((sheafToPresheaf _ _).mapIso i))\n      (Condensed.isColimitLocallyConstantPresheafDiagram Y S)⟩\n  tfae_finish\n\nend CondensedSet\n\nnamespace CondensedMod\n\nvariable (R : Type (u + 1)) [Ring R]\n\nlemma isDiscrete_iff_isDiscrete_forget (M : CondensedMod R) :\n    M.IsDiscrete ↔ ((Condensed.forget R).obj M).IsDiscrete :=\n  Sheaf.isConstant_iff_forget (coherentTopology CompHaus)\n    (forget (ModuleCat R)) M CompHaus.isTerminalPUnit\n\ninstance : HasLimitsOfSize.{u, u + 1} (ModuleCat.{u + 1} R) :=\n  hasLimitsOfSizeShrink.{u, u + 1, u + 1, u + 1} _\n\nopen CondensedMod.LocallyConstant List in\ntheorem isDiscrete_tfae (M : CondensedMod.{u} R) :\n    TFAE\n    [ M.IsDiscrete\n    , IsIso ((Condensed.discreteUnderlyingAdj _).counit.app M)\n    , (Condensed.discrete _).essImage M\n    , (CondensedMod.LocallyConstant.functor R).essImage M\n    , IsIso ((CondensedMod.LocallyConstant.adjunction R).counit.app M)\n    , Sheaf.IsConstant (coherentTopology Profinite)\n        ((Condensed.ProfiniteCompHaus.equivalence _).inverse.obj M)\n    , ∀ S : Profinite.{u}, Nonempty\n        (IsColimit <| (profiniteToCompHaus.op ⋙ M.obj).mapCocone S.asLimitCone.op)\n    ] := by\n  tfae_have 1 ↔ 2 := Sheaf.isConstant_iff_isIso_counit_app _ _ _\n  tfae_have 1 ↔ 3 := ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩\n  tfae_have 1 ↔ 4 := Sheaf.isConstant_iff_mem_essImage _ CompHaus.isTerminalPUnit (adjunction R) _\n  tfae_have 1 ↔ 5 :=\n    have : (functor R).Faithful := inferInstance\n    have : (functor R).Full := inferInstance\n    -- These `have` statements above shouldn't be needed, but they are.\n    Sheaf.isConstant_iff_isIso_counit_app' _ CompHaus.isTerminalPUnit (adjunction R) _\n  tfae_have 1 ↔ 6 :=\n    (Sheaf.isConstant_iff_of_equivalence (coherentTopology Profinite)\n      (coherentTopology CompHaus) profiniteToCompHaus Profinite.isTerminalPUnit\n      CompHaus.isTerminalPUnit _).symm\n  tfae_have 7 → 1 := by\n    intro h\n    rw [isDiscrete_iff_isDiscrete_forget, ((CondensedSet.isDiscrete_tfae _).out 0 6 :)]\n    intro S\n    letI : PreservesFilteredColimitsOfSize.{u, u} (forget (ModuleCat R)) :=\n      preservesFilteredColimitsOfSize_shrink.{u, u + 1, u, u + 1} _\n    exact ⟨isColimitOfPreserves (forget (ModuleCat R)) (h S).some⟩\n  tfae_have 1 → 7 := by\n    intro h S\n    rw [isDiscrete_iff_isDiscrete_forget, ((CondensedSet.isDiscrete_tfae _).out 0 6 :)] at h\n    letI : ReflectsFilteredColimitsOfSize.{u, u} (forget (ModuleCat R)) :=\n      reflectsFilteredColimitsOfSize_shrink.{u, u + 1, u, u + 1} _\n    exact ⟨isColimitOfReflects (forget (ModuleCat R)) (h S).some⟩\n  tfae_finish\n\nend CondensedMod\n\nnamespace LightCondensed\n\nvariable {C : Type*} [Category* C] [HasWeakSheafify (coherentTopology LightProfinite.{u}) C]\n\n/--\nA light condensed object is *discrete* if it is constant as a sheaf, i.e. isomorphic to a constant\nsheaf.\n-/\nabbrev IsDiscrete (X : LightCondensed.{u} C) := X.IsConstant (coherentTopology LightProfinite)\n\nend LightCondensed\n\nnamespace LightCondSet\n\nlemma mem_locallyConstant_essImage_of_isColimit_mapCocone (X : LightCondSet.{u})\n    (h : ∀ S : LightProfinite.{u}, IsColimit <|\n      X.obj.mapCocone (coconeRightOpOfCone S.asLimitCone)) :\n    LightCondSet.LocallyConstant.functor.essImage X := by\n  let i : X.obj ≅ (LightCondSet.LocallyConstant.functor.obj _).obj :=\n    LightCondensed.isoLocallyConstantOfIsColimit _ h\n  exact ⟨_, ⟨((sheafToPresheaf _ _).preimageIso i.symm)⟩⟩\n\n/--\n`LightCondSet.LocallyConstant.functor` is left adjoint to the forgetful functor from light condensed\nsets to sets.\n-/\nnoncomputable abbrev LocallyConstant.adjunction :\n    LightCondSet.LocallyConstant.functor ⊣ LightCondensed.underlying (Type u) :=\n  CompHausLike.LocallyConstant.adjunction _ _\n\nopen LightCondSet.LocallyConstant List in\n\nTarget:\ntheorem isDiscrete_tfae (X : LightCondSet.{u}) :\n    TFAE\n    [ X.IsDiscrete\n    , IsIso ((LightCondensed.discreteUnderlyingAdj _).counit.app X)\n    , (LightCondensed.discrete _).essImage X\n    , LightCondSet.LocallyConstant.functor.essImage X\n    , IsIso (LightCondSet.LocallyConstant.adjunction.counit.app X)\n    , ∀ S : LightProfinite.{u}, Nonempty\n        (IsColimit <| X.obj.mapCocone (coconeRightOpOfCone S.asLimitCone))\n    ] :=\n\nProof body:\n","rejected":"by\n  exact isDiscrete_tfae","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"f0ce0420ae2db90c54c78b6cceacc46310171dc4aa362a5844cc160ba76fcf78","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed/Discrete","family_id":"isdiscrete_tfae","file_id":"mathlib/Mathlib/Condensed/Discrete/Characterization.lean","sample_id":"4edfa6f1dfc9284bddb186d5cd60c15b919148603ac348edfd11fc15fed97087"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6217ca4bcfee77224af82f13005a67bfc164dca4064e34175a675cb4920a7928","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5c22024a34a06e016e56b56461bf861c464ede134f7a2786ddced665088e6547","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c422540f094e4df2b71b39ea0c0e9f6334865968fac379ba2de62eb381a0a862","source_sha256":"ebbd09c0a09c3969b7a62f3c013274751c70d4ab23cc8901bc28ba4a69c7b881","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [List.getD_eq_getElem?_getD]\n  exact option_getD.comp₂ list_getElem? (const _)","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":5,"token_jaccard":0.142857,"token_length_ratio":0.238095},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"42ebd72c7dbadd77d1b508eac24ad71e68f693ee3708ac09b30b3525f6a6dec4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Computability.Primrec.Basic\npublic import Mathlib.Logic.Encodable.Pi\n\nNamespace:\nPrimrec\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Primitive recursive functions on Lists\n\nThe primitive recursive functions are defined in `Mathlib.Computability.Primrec.Basic`.\nThis file contains definitions and theorems about primitive recursive functions that\nrelate to operation on lists.\n\n## References\n\n* [Mario Carneiro, *Formalizing computability theory via partial recursive functions*][carneiro2019]\n-/\n\n@[expose] public section\n\nopen List (Vector)\nopen Denumerable Encodable Function\n\n\nsection\n\nvariable {α : Type*} {β : Type*} {σ : Type*}\nvariable [Primcodable α] [Primcodable β] [Primcodable σ]\nvariable (H : Nat.Primrec fun n => Encodable.encode (@decode (List β) _ n))\n\nopen Primrec\n\nset_option backward.privateInPublic true in\n@[implicit_reducible]\nprivate def prim : Primcodable (List β) := ⟨H⟩\n\nprivate theorem list_casesOn' {f : α → List β} {g : α → σ} {h : α → β × List β → σ}\n    (hf : haveI := prim H; Primrec f) (hg : Primrec g) (hh : haveI := prim H; Primrec₂ h) :\n    @Primrec _ σ _ _ fun a => List.casesOn (f a) (g a) fun b l => h a (b, l) :=\n  letI := prim H\n  have :\n    @Primrec _ (Option σ) _ _ fun a =>\n      (@decode (Option (β × List β)) _ (encode (f a))).map fun o => Option.casesOn o (g a) (h a) :=\n    ((@map_decode_iff _ (Option (β × List β)) _ _ _ _ _).2 <|\n      to₂ <|\n        option_casesOn snd (hg.comp fst) (hh.comp₂ (fst.comp₂ Primrec₂.left) Primrec₂.right)).comp\n      .id (encode_iff.2 hf)\n  option_some_iff.1 <| this.of_eq fun a => by rcases f a with - | ⟨b, l⟩ <;> simp [encodek]\n\nset_option backward.privateInPublic true in\nprivate theorem list_foldl' {f : α → List β} {g : α → σ} {h : α → σ × β → σ}\n    (hf : haveI := prim H; Primrec f) (hg : Primrec g) (hh : haveI := prim H; Primrec₂ h) :\n    Primrec fun a => (f a).foldl (fun s b => h a (s, b)) (g a) := by\n  letI := prim H\n  let G (a : α) (IH : σ × List β) : σ × List β := List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\n  have hG : Primrec₂ G := list_casesOn' H (snd.comp snd) snd <|\n    to₂ <|\n    pair (hh.comp (fst.comp fst) <| pair ((fst.comp snd).comp fst) (fst.comp snd))\n      (snd.comp snd)\n  let F := fun (a : α) (n : ℕ) => (G a)^[n] (g a, f a)\n  have hF : Primrec fun a => (F a (encode (f a))).1 :=\n    (fst.comp <|\n      nat_iterate (encode_iff.2 hf) (pair hg hf) <|\n      hG)\n  suffices ∀ a n, F a n = (((f a).take n).foldl (fun s b => h a (s, b)) (g a), (f a).drop n) by\n    refine hF.of_eq fun a => ?_\n    rw [this, List.take_of_length_le (length_le_encode _)]\n  introv\n  dsimp only [F]\n  generalize f a = l\n  generalize g a = x\n  induction n generalizing l x with\n  | zero => rfl\n  | succ n IH =>\n    simp only [iterate_succ, comp_apply]\n    rcases l with - | ⟨b, l⟩ <;> simp [G, IH]\n\nset_option backward.privateInPublic true in\nprivate theorem list_cons' : (haveI := prim H; Primrec₂ (@List.cons β)) :=\n  letI := prim H\n  encode_iff.1 (succ.comp <| Primrec₂.natPair.comp (encode_iff.2 fst) (encode_iff.2 snd))\n\nset_option backward.privateInPublic true in\nprivate theorem list_reverse' :\n    haveI := prim H\n    Primrec (@List.reverse β) :=\n  letI := prim H\n  (list_foldl' H .id (const []) <| to₂ <| ((list_cons' H).comp snd fst).comp snd).of_eq\n    (suffices ∀ l r, List.foldl (fun (s : List β) (b : β) => b :: s) r l = List.reverseAux l r from\n      fun l => this l []\n    fun l => by induction l <;> simp [*, List.reverseAux])\n\nend\n\nnamespace Primcodable\n\nvariable {α : Type*} {β : Type*}\nvariable [Primcodable α] [Primcodable β]\n\nopen Primrec\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nset_option linter.flexible false in -- TODO: revisit this after #13791 is merged\ninstance list : Primcodable (List α) :=\n  ⟨letI H := Primcodable.prim (List ℕ)\n    have : Primrec₂ fun (a : α) (o : Option (List ℕ)) => o.map (List.cons (encode a)) :=\n      option_map snd <| (list_cons' H).comp ((@Primrec.encode α _).comp (fst.comp fst)) snd\n    have :\n      Primrec fun n =>\n        (ofNat (List ℕ) n).reverse.foldl\n          (fun o m => (@decode α _ m).bind fun a => o.map (List.cons (encode a))) (some []) :=\n      list_foldl' H ((list_reverse' H).comp (.ofNat (List ℕ))) (const (some []))\n        (Primrec.comp₂ (bind_decode_iff.2 <| .swap this) Primrec₂.right)\n    nat_iff.1 <|\n      (encode_iff.2 this).of_eq fun n => by\n        rw [List.foldl_reverse]\n        apply Nat.case_strong_induction_on n; · simp\n        intro n IH; simp\n        rcases @decode α _ n.unpair.1 with - | a; · rfl\n        simp only [Option.bind_some, Option.map_some]\n        suffices ∀ (o : Option (List ℕ)) (p), encode o = encode p →\n            encode (Option.map (List.cons (encode a)) o) = encode (Option.map (List.cons a) p) from\n          this _ _ (IH _ (Nat.unpair_right_le n))\n        intro o p IH\n        cases o <;> cases p\n        · rfl\n        · injection IH\n        · injection IH\n        · exact congr_arg (fun k => (Nat.pair (encode a) k).succ.succ) (Nat.succ.inj IH)⟩\nend Primcodable\n\nnamespace Primrec\n\nvariable {α : Type*} {β : Type*} {γ : Type*} {σ : Type*}\nvariable [Primcodable α] [Primcodable β] [Primcodable γ] [Primcodable σ]\n\ntheorem list_cons : Primrec₂ (@List.cons α) :=\n  list_cons' (Primcodable.prim _)\n\ntheorem list_casesOn {f : α → List β} {g : α → σ} {h : α → β × List β → σ} :\n    Primrec f →\n      Primrec g →\n        Primrec₂ h → @Primrec _ σ _ _ fun a => List.casesOn (f a) (g a) fun b l => h a (b, l) :=\n  list_casesOn' (Primcodable.prim _)\n\ntheorem list_foldl {f : α → List β} {g : α → σ} {h : α → σ × β → σ} :\n    Primrec f →\n      Primrec g → Primrec₂ h → Primrec fun a => (f a).foldl (fun s b => h a (s, b)) (g a) :=\n  list_foldl' (Primcodable.prim _)\n\ntheorem list_reverse : Primrec (@List.reverse α) :=\n  list_reverse' (Primcodable.prim _)\n\ntheorem list_foldr {f : α → List β} {g : α → σ} {h : α → β × σ → σ} (hf : Primrec f)\n    (hg : Primrec g) (hh : Primrec₂ h) :\n    Primrec fun a => (f a).foldr (fun b s => h a (b, s)) (g a) :=\n  (list_foldl (list_reverse.comp hf) hg <| to₂ <| hh.comp fst <| (pair snd fst).comp snd).of_eq\n    fun a => by simp [List.foldl_reverse]\n\ntheorem list_head? : Primrec (@List.head? α) :=\n  (list_casesOn .id (const none) (option_some_iff.2 <| fst.comp snd).to₂).of_eq fun l => by\n    cases l <;> rfl\n\ntheorem list_headI [Inhabited α] : Primrec (@List.headI α _) :=\n  (option_getD_default.comp list_head?).of_eq fun l => l.head!_eq_head?_getD.symm\n\ntheorem list_tail : Primrec (@List.tail α) :=\n  (list_casesOn .id (const []) (snd.comp snd).to₂).of_eq fun l => by cases l <;> rfl\n\ntheorem list_rec {f : α → List β} {g : α → σ} {h : α → β × List β × σ → σ} (hf : Primrec f)\n    (hg : Primrec g) (hh : Primrec₂ h) :\n    @Primrec _ σ _ _ fun a => List.recOn (f a) (g a) fun b l IH => h a (b, l, IH) :=\n  let F (a : α) := (f a).foldr (fun (b : β) (s : List β × σ) => (b :: s.1, h a (b, s))) ([], g a)\n  have : Primrec F :=\n    list_foldr hf (pair (const []) hg) <|\n      to₂ <| pair ((list_cons.comp fst (fst.comp snd)).comp snd) hh\n  (snd.comp this).of_eq fun a => by\n    suffices F a = (f a, List.recOn (f a) (g a) fun b l IH => h a (b, l, IH)) by rw [this]\n    dsimp [F]\n    induction f a <;> simp [*]\n\ntheorem list_getElem? : Primrec₂ ((·[·]? : List α → ℕ → Option α)) :=\n  let F (l : List α) (n : ℕ) :=\n    l.foldl\n      (fun (s : ℕ ⊕ α) (a : α) =>\n        Sum.casesOn s (@Nat.casesOn (fun _ => ℕ ⊕ α) · (Sum.inr a) Sum.inl) Sum.inr)\n      (Sum.inl n)\n  have hF : Primrec₂ F :=\n    (list_foldl fst (sumInl.comp snd)\n      ((sumCasesOn fst (nat_casesOn snd (sumInr.comp <| snd.comp fst) (sumInl.comp snd).to₂).to₂\n              (sumInr.comp snd).to₂).comp\n          snd).to₂).to₂\n  have :\n    @Primrec _ (Option α) _ _ fun p : List α × ℕ => Sum.casesOn (F p.1 p.2) (fun _ => none) some :=\n    sumCasesOn hF (const none).to₂ (option_some.comp snd).to₂\n  this.to₂.of_eq fun l n => by\n    dsimp; symm\n    induction l generalizing n with\n    | nil => rfl\n    | cons a l IH =>\n      rcases n with - | n\n      · dsimp [F]\n        clear IH\n        induction l <;> simp_all\n      · simpa using! IH ..\n\nTarget:\ntheorem list_getD (d : α) : Primrec₂ fun l n => List.getD l n d :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_c422540f094e","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"6a1bb39b68ad5369fe7c1e60cc8c2b1eee6398a743be347d1335c1c38c8d2b48","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability/Primrec","family_id":"list_getd","file_id":"mathlib/Mathlib/Computability/Primrec/List.lean","sample_id":"c422540f094e4df2b71b39ea0c0e9f6334865968fac379ba2de62eb381a0a862"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6124388c54e3601f5b392168c48f6a150dee4519aaf6dbd646732ffc8061300b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"366f97549100ba64f200cb187b78f8d0bb7a865a03ffc134d0a9bf76055485b7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"84f48056d33f497341acedeaa75d75fead8d9c6cb4f1d4df07a45458a9009e00","source_sha256":"ab525a505d882c41e1965c3ef070c66d963cb0848742b2b2ba04dd9efd351a5a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x <;> simp","hard_negative":false,"metrics":{"chosen_tokens":7,"rejected_tokens":11,"token_jaccard":0.636364,"token_length_ratio":1.571429},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"42f73a2bc8d62a71682deae578c650538a5d9258fc7bd84cbbc9cf55b6d32972","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.RootSystem.Basic\npublic import Mathlib.LinearAlgebra.RootSystem.Defs\n\nNamespace:\nRootPairing.Hom\n\nLocal context:\n/-\nCopyright (c) 2024 Scott Carnahan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Carnahan\n-/\n/-!\n# Morphisms of root pairings\nThis file defines morphisms of root pairings, following the definition of morphisms of root data\ngiven in SGA III Exp. 21 Section 6.\n\n## Main definitions:\n* `Hom`: A morphism of root pairings is a linear map of weight spaces, its transverse on coweight\n  spaces, and a bijection on the set that indexes roots and coroots.\n* `Hom.id`: The identity morphism.\n* `Hom.comp`: The composite of two morphisms.\n* `End`: The endomorphism monoid of a root pairing.\n* `Hom.weightHom`: The homomorphism from the endomorphism monoid to linear endomorphisms on the\n  weight space.\n* `Hom.coweightHom`: The homomorphism from the endomorphism monoid to the opposite monoid of linear\n  endomorphisms on the coweight space.\n* `Equiv`: An equivalence of root pairings is a morphism for which the maps on weight spaces and\n  coweight spaces are bijective.\n* `Equiv.toHom`: The morphism underlying an equivalence.\n* `Equiv.weightEquiv`: The linear isomorphism on weight spaces given by an equivalence.\n* `Equiv.coweightEquiv`: The linear isomorphism on coweight spaces given by an equivalence.\n* `Equiv.id`: The identity equivalence.\n* `Equiv.comp`: The composite of two equivalences.\n* `Equiv.symm`: The inverse of an equivalence.\n* `Aut`: The automorphism group of a root pairing.\n* `Equiv.toEndUnit`: The group isomorphism between the automorphism group of a root pairing and the\n  group of invertible endomorphisms.\n* `Equiv.weightHom`: The homomorphism from the automorphism group to linear automorphisms on the\n  weight space.\n* `Equiv.coweightHom`: The homomorphism from the automorphism group to the opposite group of linear\n  automorphisms on the coweight space.\n* `Equiv.reflection`: The automorphism of a root pairing given by reflection in a root and\n  coreflection in the corresponding coroot.\n\n## TODO\n* Special types of morphisms: Isogenies, weight/coweight space embeddings\n* Weyl group reimplementation?\n\n-/\n\n@[expose] public section\n\nopen Set Function\n\nnoncomputable section\n\nvariable {ι R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]\n\nnamespace RootPairing\n\n/-- A morphism of root pairings is a pair of mutually transposed maps of weight and coweight spaces\nthat preserves roots and coroots.  We make the map of indexing sets explicit. -/\n@[ext]\nstructure Hom {ι₂ M₂ N₂ : Type*}\n    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) where\n  /-- A linear map on weight space. -/\n  weightMap : M →ₗ[R] M₂\n  /-- A contravariant linear map on coweight space. -/\n  coweightMap : N₂ →ₗ[R] N\n  /-- A bijection on index sets. -/\n  indexEquiv : ι ≃ ι₂\n  weight_coweight_transpose :\n    weightMap.dualMap ∘ₗ Q.flip.toPerfPair = P.flip.toPerfPair ∘ₗ coweightMap\n  root_weightMap : weightMap ∘ P.root = Q.root ∘ indexEquiv\n  coroot_coweightMap : coweightMap ∘ Q.coroot = P.coroot ∘ indexEquiv.symm\n\nnamespace Hom\n\nlemma weight_coweight_transpose_apply {ι₂ M₂ N₂ : Type*}\n    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (x : N₂) (f : Hom P Q) :\n    f.weightMap.dualMap (Q.flip.toPerfPair x) = P.flip.toPerfPair (f.coweightMap x) :=\n  Eq.mp (propext LinearMap.ext_iff) f.weight_coweight_transpose x\n\nlemma root_weightMap_apply {ι₂ M₂ N₂ : Type*}\n    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (i : ι) (f : Hom P Q) :\n    f.weightMap (P.root i) = Q.root (f.indexEquiv i) :=\n  Eq.mp (propext funext_iff) f.root_weightMap i\n\nlemma coroot_coweightMap_apply {ι₂ M₂ N₂ : Type*}\n    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (i : ι₂) (f : Hom P Q) :\n    f.coweightMap (Q.coroot i) = P.coroot (f.indexEquiv.symm i) :=\n  Eq.mp (propext funext_iff) f.coroot_coweightMap i\n\n/-- The identity morphism of a root pairing. -/\n@[simps!]\ndef id (P : RootPairing ι R M N) : Hom P P where\n  weightMap := LinearMap.id\n  coweightMap := LinearMap.id\n  indexEquiv := Equiv.refl ι\n  weight_coweight_transpose := by simp\n  root_weightMap := by simp\n  coroot_coweightMap := by simp\n\n/-- Composition of morphisms -/\n@[simps!]\ndef comp {ι₁ M₁ N₁ ι₂ M₂ N₂ : Type*} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁]\n    [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂}\n    (g : Hom P₁ P₂) (f : Hom P P₁) : Hom P P₂ where\n  weightMap := g.weightMap ∘ₗ f.weightMap\n  coweightMap := f.coweightMap ∘ₗ g.coweightMap\n  indexEquiv := f.indexEquiv.trans g.indexEquiv\n  weight_coweight_transpose := by\n    ext φ x\n    rw [← LinearMap.dualMap_comp_dualMap, ← LinearMap.comp_assoc _ f.coweightMap,\n      ← f.weight_coweight_transpose, LinearMap.comp_assoc g.coweightMap,\n      ← g.weight_coweight_transpose, ← LinearMap.comp_assoc]\n  root_weightMap := by\n    ext i\n    simp only [LinearMap.coe_comp, Equiv.coe_trans]\n    rw [comp_assoc, f.root_weightMap, ← comp_assoc, g.root_weightMap, comp_assoc]\n  coroot_coweightMap := by\n    ext i\n    simp only [LinearMap.coe_comp]\n    rw [comp_assoc, g.coroot_coweightMap, ← comp_assoc, f.coroot_coweightMap, comp_assoc]\n    simp\n\n@[simp]\nlemma id_comp {ι₂ M₂ N₂ : Type*}\n    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : Hom P Q) :\n    comp f (id P) = f := by\n  ext x <;> simp\n\n@[simp]\n\nTarget:\nlemma comp_id {ι₂ M₂ N₂ : Type*}\n    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : Hom P Q) :\n    comp (id Q) f = f :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  ext x <;> simp","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/RootSystem","family_id":"comp_id","file_id":"mathlib/Mathlib/LinearAlgebra/RootSystem/Hom.lean","sample_id":"84f48056d33f497341acedeaa75d75fead8d9c6cb4f1d4df07a45458a9009e00"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2bb1a47c41a1654399bf9970e1dd3f153a3e08560ed360131ac9536f9ce93d74","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d14998e2d236d5892c9ed492b6063a072a0ec193e356245fa60104974e67b50c","source_sha256":"88899366de27f8f7e91c2113d9cd607e3b4b1f4debab3b6b3a2046a9e9f7c31f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction s using Finset.cons_induction with\n  | empty => simp\n  | cons a s ha ihs =>\n    simp_rw [Finset.sum_cons, ihs]\n    refine iSup_add_iSup fun i j ↦ (hf i j).imp fun k hk ↦ ?_\n    gcongr\n    exacts [(hk a).1, (hk _).2]","hard_negative":true,"metrics":{"chosen_tokens":63,"rejected_tokens":8,"token_jaccard":0.022222,"token_length_ratio":0.126984},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"4314373bc9e76418578dbe986a91ec7ffc724e28a63a92118a4c7f20de99e3d5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Ring.Finset\npublic import Mathlib.Algebra.BigOperators.WithTop\npublic import Mathlib.Algebra.Order.BigOperators.Group.Finset\npublic import Mathlib.Data.ENat.Lattice\n\nNamespace:\nENat\n\nLocal context:\n/-\nCopyright (c) 2024 Joachim Breitner, Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joachim Breitner, Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Sum of suprema in `ENat`\n-/\n\npublic section\n\nassert_not_exists Field\n\nnamespace ENat\n\nvariable {a b c d : ℕ∞} {r p q : ℕ}\n\nsection OperationsAndInfty\n\nvariable {α : Type*}\n\n@[simp]\ntheorem toNat_prod {ι : Type*} {s : Finset ι} {f : ι → ℕ∞} :\n    (∏ i ∈ s, f i).toNat = ∏ i ∈ s, (f i).toNat :=\n  map_prod toNatHom _ _\n\ntheorem iInf_sum {ι α : Type*} {f : ι → α → ℕ∞} {s : Finset α} [Nonempty ι]\n    (h : ∀ (t : Finset α) (i j : ι), ∃ k, ∀ a ∈ t, f k a ≤ f i a ∧ f k a ≤ f j a) :\n    ⨅ i, ∑ a ∈ s, f i a = ∑ a ∈ s, ⨅ i, f i a := by\n  induction s using Finset.cons_induction_on with\n  | empty => simp only [Finset.sum_empty, ciInf_const]\n  | cons a s ha ih =>\n    simp only [Finset.sum_cons, ← ih]\n    refine (iInf_add_iInf fun i j => ?_).symm\n    refine (h (Finset.cons a s ha) i j).imp fun k hk => ?_\n    rw [Finset.forall_mem_cons] at hk\n    exact add_le_add hk.1.1 (Finset.sum_le_sum fun a ha => (hk.2 a ha).2)\n\nend OperationsAndInfty\n\nsection Sum\n\nopen Finset\n\nvariable {α : Type*} {s : Finset α} {f : α → ℕ∞}\n\n/-- A product of finite numbers is still finite. -/\nlemma prod_ne_top (h : ∀ a ∈ s, f a ≠ ⊤) : ∏ a ∈ s, f a ≠ ⊤ := WithTop.prod_ne_top h\n\n/-- A product of finite numbers is still finite. -/\nlemma prod_lt_top (h : ∀ a ∈ s, f a < ⊤) : ∏ a ∈ s, f a < ⊤ := WithTop.prod_lt_top h\n\n/-- A sum is infinite iff one of the summands is infinite. -/\n@[simp] lemma sum_eq_top : ∑ x ∈ s, f x = ⊤ ↔ ∃ a ∈ s, f a = ⊤ := WithTop.sum_eq_top\n\n/-- A sum is finite iff all summands are finite. -/\nlemma sum_ne_top : ∑ a ∈ s, f a ≠ ⊤ ↔ ∀ a ∈ s, f a ≠ ⊤ := WithTop.sum_ne_top\n\n/-- A sum is finite iff all summands are finite. -/\n@[simp] lemma sum_lt_top : ∑ a ∈ s, f a < ⊤ ↔ ∀ a ∈ s, f a < ⊤ := WithTop.sum_lt_top\n\ntheorem lt_top_of_sum_ne_top {s : Finset α} {f : α → ℕ∞} (h : ∑ x ∈ s, f x ≠ ⊤) {a : α}\n    (ha : a ∈ s) : f a < ⊤ :=\n  sum_lt_top.1 h.lt_top a ha\n\n/-- Seeing `ℕ∞` as `ℕ` does not change their sum, unless one of the `ℕ∞` is\ninfinity -/\ntheorem toNat_sum {s : Finset α} {f : α → ℕ∞} (hf : ∀ a ∈ s, f a ≠ ⊤) :\n    ENat.toNat (∑ a ∈ s, f a) = ∑ a ∈ s, ENat.toNat (f a) := by\n  rw [← coe_inj, coe_toNat (sum_ne_top.2 hf), Nat.cast_sum]\n  exact sum_congr rfl fun x hx => (coe_toNat (hf x hx)).symm\n\ntheorem sum_lt_sum_of_nonempty {s : Finset α} (hs : s.Nonempty) {f g : α → ℕ∞}\n    (Hlt : ∀ i ∈ s, f i < g i) : ∑ i ∈ s, f i < ∑ i ∈ s, g i := by\n  induction hs using Nonempty.cons_induction with\n  | singleton => simp [Hlt _ (mem_singleton_self _)]\n  | cons _ _ _ _ ih =>\n    simp only [sum_cons, forall_mem_cons] at Hlt ⊢\n    exact ENat.add_lt_add Hlt.1 (ih Hlt.2)\n\ntheorem exists_le_of_sum_le {s : Finset α} (hs : s.Nonempty) {f g : α → ℕ∞}\n    (Hle : ∑ i ∈ s, f i ≤ ∑ i ∈ s, g i) : ∃ i ∈ s, f i ≤ g i := by\n  contrapose! Hle\n  apply sum_lt_sum_of_nonempty hs Hle\n\nend Sum\n\nTarget:\nlemma sum_iSup {α ι : Type*} {s : Finset α} {f : α → ι → ℕ∞}\n    (hf : ∀ i j, ∃ k, ∀ a, f a i ≤ f a k ∧ f a j ≤ f a k) :\n    ∑ a ∈ s, ⨆ i, f a i = ⨆ i, ∑ a ∈ s, f a i :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"67d4a1fc4b3abafd7606f3d1ac1adeb2bc3c73b6f5c30d42313a15c003af2cd8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/ENat","family_id":"sum_isup","file_id":"mathlib/Mathlib/Data/ENat/BigOperators.lean","sample_id":"d14998e2d236d5892c9ed492b6063a072a0ec193e356245fa60104974e67b50c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d0709903e2751d33af1193290b412e76b1fe0c171376fb60a72f4abe9461ccc4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ea366b04c54fa2d802b07a4ed0e9ad19edd8f2e0e916dddae3f37ee74b249b0d","source_sha256":"b1dde0e82be30e8a9ca87b23c46d04afc92be61818622ba6d4603daa4d4a9f70","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [sub_eq_add_neg, ofCauchy_add, ofCauchy_neg]\n  rfl","hard_negative":false,"metrics":{"chosen_tokens":10,"rejected_tokens":3,"token_jaccard":0.090909,"token_length_ratio":0.3},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"43270389f6dff85cce58dcaae21c59c2e5ac18066570f242419a03d53633d632","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.CauSeq.Completion\npublic import Mathlib.Algebra.Order.Ring.Rat\npublic import Mathlib.Data.Rat.Cast.Defs\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Floris van Doorn\n-/\n/-!\n# Real numbers from Cauchy sequences\n\nThis file defines `ℝ` as the type of equivalence classes of Cauchy sequences of rational numbers.\nThis choice is motivated by how easy it is to prove that `ℝ` is a commutative ring, by simply\nlifting everything to `ℚ`.\n\nThe facts that the real numbers are an Archimedean floor ring,\nand a conditionally complete linear order,\nhave been deferred to the file `Mathlib/Data/Real/Archimedean.lean`,\nin order to keep the imports here simple.\n\nThe fact that the real numbers are a (trivial) \\*-ring has similarly been deferred to\n`Mathlib/Data/Real/Star.lean`.\n-/\n\n@[expose] public section\n\n\nassert_not_exists Finset Module Submonoid FloorRing\n\n/-- The type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational\nnumbers. -/\n@[wikidata Q12916, wikidata Q2584477]\nstructure Real where ofCauchy ::\n  /-- The underlying Cauchy completion -/\n  cauchy : CauSeq.Completion.Cauchy (abs : ℚ → ℚ)\n\n@[inherit_doc]\nnotation \"ℝ\" => Real\n\nnamespace CauSeq.Completion\n\n-- this can't go in `Data.Real.CauSeqCompletion` as the structure on `ℚ` isn't available\n@[simp]\ntheorem ofRat_rat {abv : ℚ → ℚ} [IsAbsoluteValue abv] (q : ℚ) :\n    ofRat (q : ℚ) = (q : Cauchy abv) :=\n  rfl\n\nend CauSeq.Completion\n\nnamespace Real\n\nopen CauSeq CauSeq.Completion\n\nvariable {x : ℝ}\n\ntheorem ext_cauchy_iff : ∀ {x y : Real}, x = y ↔ x.cauchy = y.cauchy\n  | ⟨a⟩, ⟨b⟩ => by rw [ofCauchy.injEq]\n\ntheorem ext_cauchy {x y : Real} : x.cauchy = y.cauchy → x = y :=\n  ext_cauchy_iff.2\n\n/-- The real numbers are isomorphic to the quotient of Cauchy sequences on the rationals. -/\ndef equivCauchy : ℝ ≃ CauSeq.Completion.Cauchy (abs : ℚ → ℚ) :=\n  ⟨Real.cauchy, Real.ofCauchy, fun ⟨_⟩ => rfl, fun _ => rfl⟩\n\nset_option backward.privateInPublic true in\n-- irreducible doesn't work for instances: https://github.com/leanprover-community/lean/issues/511\nprivate irreducible_def zero : ℝ :=\n  ⟨0⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def one : ℝ :=\n  ⟨1⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def add : ℝ → ℝ → ℝ\n  | ⟨a⟩, ⟨b⟩ => ⟨a + b⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def neg : ℝ → ℝ\n  | ⟨a⟩ => ⟨-a⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def mul : ℝ → ℝ → ℝ\n  | ⟨a⟩, ⟨b⟩ => ⟨a * b⟩\n\nset_option backward.privateInPublic true in\nprivate noncomputable irreducible_def inv' : ℝ → ℝ\n  | ⟨a⟩ => ⟨a⁻¹⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Zero ℝ :=\n  ⟨zero⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : One ℝ :=\n  ⟨one⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Add ℝ :=\n  ⟨add⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Neg ℝ :=\n  ⟨neg⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Mul ℝ :=\n  ⟨mul⟩\n\ninstance : Sub ℝ :=\n  ⟨fun a b => a + -b⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nnoncomputable instance : Inv ℝ :=\n  ⟨inv'⟩\n\ntheorem ofCauchy_zero : (⟨0⟩ : ℝ) = 0 :=\n  zero_def.symm\n\ntheorem ofCauchy_one : (⟨1⟩ : ℝ) = 1 :=\n  one_def.symm\n\ntheorem ofCauchy_add (a b) : (⟨a + b⟩ : ℝ) = ⟨a⟩ + ⟨b⟩ :=\n  (add_def _ _).symm\n\ntheorem ofCauchy_neg (a) : (⟨-a⟩ : ℝ) = -⟨a⟩ :=\n  (neg_def _).symm\n\nTarget:\ntheorem ofCauchy_sub (a b) : (⟨a - b⟩ : ℝ) = ⟨a⟩ - ⟨b⟩ :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Real","family_id":"ofcauchy_sub","file_id":"mathlib/Mathlib/Data/Real/Basic.lean","sample_id":"ea366b04c54fa2d802b07a4ed0e9ad19edd8f2e0e916dddae3f37ee74b249b0d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"79e94f24c2f6775d53dff748682f833bc2540c933d9b7a7fd1509ddae3499adc","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0fc65d8000935f5344eab88fab4e2c3713d4a97efe4538dc3faf285b863f943a","source_sha256":"95ccb03aba37ba4c0bbb12dee38bb2450c829862e49cf64d5a888d7fa6218e0d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← ofSupportFinite_coe (f := a) (hf := Set.toFinite _),\n    prod_congr (fun r _ ↦ rfl), ← coeff_linearCombination_X_pow]\n  simp [linearCombination_apply, sum_of_support_subset (s := Finset.univ)]","hard_negative":false,"metrics":{"chosen_tokens":44,"rejected_tokens":2,"token_jaccard":0.033333,"token_length_ratio":0.045455},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"43399dbcc53e05d762408b22db24f31f2fea58c81f98d6f6dd8244b5b1ca6f9a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.Basic\npublic import Mathlib.Data.Nat.Choose.Multinomial\n\nNamespace:\nMvPolynomial\n\nLocal context:\n/-\nCopyright (c) 2026 Antoine Chambert-Loir, María Inés de Frutos-Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir, María Inés de Frutos-Fernández\n-/\n/-!\n# Formulas for coefficients of multivariate polynomials\n\n## Main Results\n\n* `MvPolynomial.coeff_add_pow`: the formula for the `d`th coefficient of `(X 0 + X 1) ^ n`.\n\n-/\n\npublic section\n\nnoncomputable section\n\nnamespace MvPolynomial\n\nopen Finsupp\n\nvariable {R σ : Type*} [CommSemiring R] {s : σ →₀ ℕ}\n\nprivate lemma coeff_linearCombination_X_pow_of_eq (a : σ →₀ R) {n : ℕ}\n    (hs : s.sum (fun _ m ↦ m) = n) :\n    coeff s (((a.linearCombination R X : MvPolynomial σ R)) ^ n) =\n      s.multinomial * s.prod (fun r m ↦ a r ^ m) := by\n  classical\n  simp only [sum, linearCombination_apply, Finset.sum_pow_eq_sum_piAntidiag, coeff_sum,\n    ← C_eq_coe_nat, coeff_C_mul, smul_eq_C_mul, mul_pow, Finset.prod_mul_distrib, ← map_pow,\n    ← map_prod, coeff_prod_X_pow, mul_ite, mul_one, mul_zero]\n  rw [Finset.sum_eq_single (s : σ → ℕ)]\n  · simp_rw [eq_indicator_self_iff]\n    split_ifs with hs'\n    · rw [prod_of_support_subset _ hs' _ (by simp), Finsupp.multinomial_of_support_subset hs']\n    · rw [Finset.subset_iff] at hs'\n      simp only [Finsupp.mem_support_iff, ne_eq, not_forall, Decidable.not_not] at hs'\n      obtain ⟨i, hsi, hai⟩ := hs'\n      rw [← mul_prod_erase _ i _ (by simpa), hai, zero_pow hsi, zero_mul, mul_zero]\n  · simp only [Finset.mem_piAntidiag, ne_eq, Finsupp.mem_support_iff, ite_eq_right_iff, and_imp]\n    intro _ _ _ _ hed\n    simp [Finsupp.ext_iff] at hed\n    grind\n  · simp_rw [ite_eq_right_iff]\n    intro hs' hs''\n    rw [eq_indicator_self_iff] at hs''\n    exfalso\n    rw [Finset.mem_piAntidiag, not_and_or] at hs'\n    rcases hs' with hs' | hs'\n    · apply hs'\n      rw [← hs, sum_of_support_subset _ hs'' _ (by simp)]\n    · grind\n\nprivate lemma coeff_linearCombination_X_pow_of_ne (a : σ →₀ R) {n : ℕ}\n    (hs : s.sum (fun _ m ↦ m) ≠ n) :\n    coeff s (((a.linearCombination R X : MvPolynomial σ R)) ^ n) = 0 := by\n  classical\n  simp only [sum, linearCombination_apply, Finset.sum_pow_eq_sum_piAntidiag, coeff_sum, ← map_pow,\n    ← C_eq_coe_nat, coeff_C_mul, smul_eq_C_mul, mul_pow, Finset.prod_mul_distrib, ← map_prod,\n    coeff_prod_X_pow, mul_ite, mul_one, mul_zero]\n  apply Finset.sum_eq_zero (fun x hx ↦ ?_)\n  rw [if_neg]\n  rintro ⟨rfl⟩\n  apply hs\n  simp only [Finset.mem_piAntidiag] at hx\n  rw [sum_of_support_subset _ (support_indicator_subset a.support _) _ (by simp), ← hx.1]\n  congr\n  ext i\n  by_cases hi : i ∈ a.support\n  · simp [Finsupp.indicator_of_mem hi]\n  · grind [Finsupp.indicator_of_notMem hi]\n\nlemma coeff_linearCombination_X_pow (a : σ →₀ R) (s : σ →₀ ℕ) (n : ℕ) :\n    coeff s (((a.linearCombination R X : MvPolynomial σ R)) ^ n) =\n      if s.sum (fun _ m ↦ m) = n then s.multinomial * s.prod (fun r m ↦ a r ^ m) else 0 := by\n  split_ifs with hs\n  · exact coeff_linearCombination_X_pow_of_eq a hs\n  · exact coeff_linearCombination_X_pow_of_ne a hs\n\nTarget:\nlemma coeff_linearCombination_X_pow_of_fintype [Fintype σ] (a : σ → R) (s : σ →₀ ℕ) (n : ℕ) :\n    coeff s (((∑ i, a i • X i : MvPolynomial σ R)) ^ n) =\n      if s.sum (fun _ m ↦ m) = n then s.multinomial * s.prod (fun r m ↦ a r ^ m) else 0 :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/MvPolynomial","family_id":"coeff_linearcombination_x_pow_of_fintype","file_id":"mathlib/Mathlib/Algebra/MvPolynomial/Coeff.lean","sample_id":"0fc65d8000935f5344eab88fab4e2c3713d4a97efe4538dc3faf285b863f943a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d9b98cd109749b4d841c4e5ec47310732f89bc8df99f9f6e87e3aa11beb61e37","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"33493801268bbdfd455e186fa50a17ff9c38e950d8c9e9bcab0dd5bfd9827ec7","source_sha256":"7cfbfc972c134864031de87e3b5157be2d7af83e375a2c0e1cc9ff7db598678d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have ⟨hμν_ac, hκη_ac⟩ := Measure.absolutelyContinuous_compProd_iff.mp h_ac\n  rw [← integrable_rnDeriv_mul_log_iff h_ac] at h_int\n  replace h_int := convexOn_mul_log.integrable_apply_rnDeriv_of_integrable_compProd\n    continuous_mul_log.stronglyMeasurable continuous_mul_log.continuousWithinAt h_int hκη_ac\n  exact (integrable_rnDeriv_mul_log_iff hμν_ac).mp h_int","hard_negative":false,"metrics":{"chosen_tokens":56,"rejected_tokens":2,"token_jaccard":0.029412,"token_length_ratio":0.035714},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"43db951c40561c3a9b04b48cd002bec5689d669066453a8455597ff2c678829e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.InformationTheory.KullbackLeibler.Basic\npublic import Mathlib.Probability.Kernel.Composition.MeasureCompProd\npublic import Mathlib.Probability.Notation\nimport Mathlib.Probability.Kernel.Composition.IntegralCompProd\nimport Mathlib.Probability.Kernel.Composition.RadonNikodym\n\nNamespace:\nInformationTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne, Lorenzo Luccioli\n-/\n/-!\n# Chain rule for the Kullback-Leibler divergence\n\nSuppose that we have two finite joint measures on a product `𝓧 × 𝓨`, which can be decomposed as\n`μ ⊗ₘ κ` and `ν ⊗ₘ η`, where `μ` and `ν` are measures on `𝓧` and `κ` and `η` are Markov kernels\nfrom `𝓧` to `𝓨`. Then we can express the Kullback-Leibler divergence between these two joint\nmeasures as a sum of `klDiv μ ν` and the conditional Kullback-Leibler divergence between the kernels\n`κ` and `η`, averaged over `μ`. The resulting equality is most often written as\n`klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + μ[fun x ↦ klDiv (κ x) (η x)]`.\n\nHere we first prove the following version:\n`klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)`.\nThis version avoids the issue of measurability of the function `x ↦ klDiv (κ x) (η x)`, which is not\nalways guaranteed, and thus holds for all measurable spaces `𝓧` and `𝓨`, without any assumptions.\n\n## Main statements\n\n* `klDiv_compProd_eq_add`: `klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)`\n* `klDiv_compProd_left`: `klDiv (μ ⊗ₘ κ) (ν ⊗ₘ κ) = klDiv μ ν`\n\n## Proof\n\nThe main ingredient is the chain rule for Radon-Nikodym derivatives:\n`∂(μ ⊗ₘ κ)/∂(ν ⊗ₘ η) = ∂μ/∂ν * ∂(μ ⊗ₘ κ)/∂(μ ⊗ₘ η)`.\nThen, omitting edge cases, the Kullback-Leibler divergence is an integral of a logarithm of the\nderivative on the left, which decomposes into a sum of two integrals of logarithms.\nWe now give a more detailed outline of the proof.\n\nThe Kullback-Leibler divergence `klDiv μ ν` is defined with an if-then-else statement:\nif the measures are absolutely continuous (`μ ≪ ν`) and the log-likelihood ratio `llr μ ν` is\nintegrable, then it is defined as `∫ x, llr μ ν x ∂μ + ν.real univ - μ.real univ`, otherwise\nit is defined to be `∞`.\n\nWe first deal with the case in which absolute continuity does not hold. The main observation is\nthat `μ ⊗ₘ κ ≪ ν ⊗ₘ η ↔ μ ≪ ν ∧ μ ⊗ₘ κ ≪ μ ⊗ₘ η`, which means that if one of the two sides of the\nKL equality is infinite because of lack of absolute continuity, then the other side is also infinite\nfor the same reason.\n\nThen, we deal with the case in which absolute continuity holds but integrability does not. Again,\nwe can show a similar equivalence for integrability, which allows us to conclude that both sides\nare infinite.\n`Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)` is equivalent to\n`Integrable (llr μ ν) μ ∧ Integrable (llr (μ ⊗ₘ κ) (μ ⊗ₘ η)) (μ ⊗ₘ κ)`.\nThe proof of this equivalence relies on the convexity of the function `x ↦ x * log x`.\n\nFinally, we prove the equality in the case in which both absolute continuity and integrability hold.\nIn that case, `klDiv μ ν = ∫ x, llr μ ν x ∂μ + ν.real univ - μ.real univ` and similarly for\nthe other terms. It is easy to see that it suffices to prove the equality of the integrals parts.\nFinally, the computation for the integral of the log-likelihood ratio is as follows:\n```\n∫ p, llr (μ ⊗ₘ κ) (ν ⊗ₘ η) p ∂(μ ⊗ₘ κ)\n_ = ∫ p, ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal * log ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal ∂(ν ⊗ₘ η)\n_ = ∫ p, ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal *\n    (log ((∂μ/∂ν) p.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal) ∂(ν ⊗ₘ η)\n_ = ∫ p, (log ((∂μ/∂ν) p.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal) ∂(μ ⊗ₘ κ)\n_ = ∫ p, log ((∂μ/∂ν) p.1).toReal ∂(μ ⊗ₘ κ) + ∫ p, log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal ∂(μ ⊗ₘ κ)\n_ = ∫ a, llr μ ν a ∂μ + ∫ p, llr (μ ⊗ₘ κ) (μ ⊗ₘ η) p ∂(μ ⊗ₘ κ)\n```\n\n## TODO\n\nAdd a version of the chain rule for the integral form of the contional KL divergence, i.e.\n`μ[fun x ↦ klDiv (κ x) (η x)]`.\n\n-/\n\npublic section\n\nopen Real MeasureTheory Set ProbabilityTheory\nopen scoped ENNReal\n\nnamespace InformationTheory\n\nvariable {𝓧 𝓨 : Type*} {m𝓧 : MeasurableSpace 𝓧} {m𝓨 : MeasurableSpace 𝓨}\n  {μ ν : Measure 𝓧} {κ η : Kernel 𝓧 𝓨}\n  [IsFiniteMeasure μ] [IsFiniteMeasure ν] [IsMarkovKernel κ] [IsMarkovKernel η]\n\n/-- If the log-likelihood ration between two composition-products is integrable, then so is the\nlog-likelihood ratio between the two measures on the first space. -/\n\nTarget:\nlemma integrable_llr_of_integrable_llr_compProd\n    (h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η) (h_int : Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)) :\n    Integrable (llr μ ν) μ :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"InformationTheory/KullbackLeibler","family_id":"integrable_llr_of_integrable_llr_compprod","file_id":"mathlib/Mathlib/InformationTheory/KullbackLeibler/ChainRule.lean","sample_id":"33493801268bbdfd455e186fa50a17ff9c38e950d8c9e9bcab0dd5bfd9827ec7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9c6b8baf38f61be33d84aa02acad26a64e1f3aa8d253f05349f099a61ea5244e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"370d662bc0a91476deac5c89915d2f05cb999a5bdef941bdf4c4d32527ff8234","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ff5f243c31598fd2aadb8dc36b39e21f486eb66be282ba3b720cbff07478671d","source_sha256":"94021f194f9dbe2b3ff10c0fe9e60d21f920f7ec47e18de0732170ff2d7ff47e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h p hp hp' ↦ hp.not_isUnit (h p hp'), fun h d hd ↦ by_contra fun hdu ↦ ?_⟩\n  have hd₀ : d ≠ 0 := ne_zero_of_dvd_ne_zero (ne_zero_of_dvd_ne_zero hx₀ hd) (dvd_mul_left d d)\n  obtain ⟨p, irr, dvd⟩ := WfDvdMonoid.exists_irreducible_factor hdu hd₀\n  exact h p irr ((mul_dvd_mul dvd dvd).trans hd)","hard_negative":false,"metrics":{"chosen_tokens":79,"rejected_tokens":84,"token_jaccard":0.904762,"token_length_ratio":1.063291},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"43dee5eb4f9b1b70ac165d43236017772b80a86bf36a420bee2da4e81e6741f3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Coprime.Lemmas\npublic import Mathlib.RingTheory.Nilpotent.Basic\npublic import Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid\npublic import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Squarefree elements of monoids\nAn element of a monoid is squarefree when it is not divisible by any squares\nexcept the squares of units.\n\nResults about squarefree natural numbers are proved in `Data.Nat.Squarefree`.\n\n## Main Definitions\n- `Squarefree r` indicates that `r` is only divisible by `x * x` if `x` is a unit.\n\n## Main Results\n- `multiplicity.squarefree_iff_emultiplicity_le_one`: `x` is `Squarefree` iff for every `y`, either\n  `emultiplicity y x ≤ 1` or `IsUnit y`.\n- `UniqueFactorizationMonoid.squarefree_iff_nodup_factors`: A nonzero element `x` of a unique\n  factorization monoid is squarefree iff `factors x` has no duplicate factors.\n\n## Tags\nsquarefree, multiplicity\n\n-/\n\n@[expose] public section\n\n\nvariable {R : Type*}\n\n/-- An element of a monoid is squarefree if the only squares that\n  divide it are the squares of units. -/\ndef Squarefree [Monoid R] (r : R) : Prop :=\n  ∀ x : R, x * x ∣ r → IsUnit x\n\ntheorem IsRelPrime.of_squarefree_mul [CommMonoid R] {m n : R} (h : Squarefree (m * n)) :\n    IsRelPrime m n := fun c hca hcb ↦ h c (mul_dvd_mul hca hcb)\n\n@[simp]\ntheorem IsUnit.squarefree [CommMonoid R] {x : R} (h : IsUnit x) : Squarefree x := fun _ hdvd =>\n  isUnit_of_mul_isUnit_left (isUnit_of_dvd_unit hdvd h)\n\ntheorem squarefree_one [CommMonoid R] : Squarefree (1 : R) :=\n  isUnit_one.squarefree\n\n@[simp]\ntheorem not_squarefree_zero [MonoidWithZero R] [Nontrivial R] : ¬Squarefree (0 : R) := by\n  rw [Squarefree, not_forall]\n  exact ⟨0, by simp⟩\n\ntheorem Squarefree.ne_zero [MonoidWithZero R] [Nontrivial R] {m : R} (hm : Squarefree (m : R)) :\n    m ≠ 0 := by\n  rintro rfl\n  exact not_squarefree_zero hm\n\n@[simp]\ntheorem Irreducible.squarefree [CommMonoid R] {x : R} (h : Irreducible x) : Squarefree x := by\n  rintro y ⟨z, hz⟩\n  rw [mul_assoc] at hz\n  rcases h.isUnit_or_isUnit hz with (hu | hu)\n  · exact hu\n  · apply isUnit_of_mul_isUnit_left hu\n\n@[simp]\ntheorem Prime.squarefree [CommMonoidWithZero R] [IsCancelMulZero R] {x : R} (h : Prime x) :\n    Squarefree x :=\n  h.irreducible.squarefree\n\ntheorem Squarefree.of_mul_left [Monoid R] {m n : R} (hmn : Squarefree (m * n)) : Squarefree m :=\n  fun p hp => hmn p (dvd_mul_of_dvd_left hp n)\n\ntheorem Squarefree.of_mul_right [CommMonoid R] {m n : R} (hmn : Squarefree (m * n)) :\n    Squarefree n := fun p hp => hmn p (dvd_mul_of_dvd_right hp m)\n\ntheorem Squarefree.squarefree_of_dvd [Monoid R] {x y : R} (hdvd : x ∣ y) (hsq : Squarefree y) :\n    Squarefree x := fun _ h => hsq _ (h.trans hdvd)\n\ntheorem Associated.squarefree_iff [Monoid R] {x y : R} (h : Associated x y) :\n    Squarefree x ↔ Squarefree y :=\n  ⟨fun hx ↦ hx.squarefree_of_dvd h.dvd', fun hy ↦ hy.squarefree_of_dvd h.dvd⟩\n\ntheorem Squarefree.eq_zero_or_one_of_pow_of_not_isUnit [Monoid R] {x : R} {n : ℕ}\n    (h : Squarefree (x ^ n)) (h' : ¬ IsUnit x) :\n    n = 0 ∨ n = 1 := by\n  contrapose! h'\n  replace h' : 2 ≤ n := by lia\n  have : x * x ∣ x ^ n := by rw [← sq]; exact pow_dvd_pow x h'\n  exact h.squarefree_of_dvd this x (refl _)\n\ntheorem Squarefree.pow_dvd_of_pow_dvd [Monoid R] {x y : R} {n : ℕ}\n    (hx : Squarefree y) (h : x ^ n ∣ y) : x ^ n ∣ x := by\n  by_cases hu : IsUnit x\n  · exact (hu.pow n).dvd\n  · rcases (hx.squarefree_of_dvd h).eq_zero_or_one_of_pow_of_not_isUnit hu with rfl | rfl <;> simp\n\nsection SquarefreeGcdOfSquarefree\n\nvariable {α : Type*} [CommMonoidWithZero α] [GCDMonoid α]\n\ntheorem Squarefree.gcd_right (a : α) {b : α} (hb : Squarefree b) : Squarefree (gcd a b) :=\n  hb.squarefree_of_dvd (gcd_dvd_right _ _)\n\ntheorem Squarefree.gcd_left {a : α} (b : α) (ha : Squarefree a) : Squarefree (gcd a b) :=\n  ha.squarefree_of_dvd (gcd_dvd_left _ _)\n\nend SquarefreeGcdOfSquarefree\n\ntheorem squarefree_iff_emultiplicity_le_one [CommMonoid R] (r : R) :\n    Squarefree r ↔ ∀ x : R, emultiplicity x r ≤ 1 ∨ IsUnit x := by\n  refine forall_congr' fun a => ?_\n  rw [← sq, pow_dvd_iff_le_emultiplicity, or_iff_not_imp_left, not_le, imp_congr _ Iff.rfl]\n  norm_cast\n  rw [← one_add_one_eq_two]\n  exact Order.add_one_le_iff_of_not_isMax (by simp)\n\nsection Irreducible\n\nvariable [CommMonoidWithZero R] [WfDvdMonoid R]\n\nTarget:\ntheorem squarefree_iff_no_irreducibles {x : R} (hx₀ : x ≠ 0) :\n    Squarefree x ↔ ∀ p, Irreducible p → ¬ (p * p ∣ x) :=\n\nProof body:\n","rejected":"by\n  refine ⟨fun h p hp hp' ↦ hp.not_isUnit (h p hp'), fun h d hd ↦ by_contra fun hdu ↦ ?_⟩\n  have hd₀ : d ≠ 0 := ne_zero_of_dvd_ne_zero (ne_zero_of_dvd_ne_zero hx₀ hd) (dvd_mul_left d d)\n  obtain ⟨p, irr, dvd⟩ := WfDvdMonoid.exists_irreducible_factor hdu hd₀\n  exact h p irr ((mul_dvd_mul dvd dvd).trans hd)\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Squarefree","family_id":"squarefree_iff_no_irreducibles","file_id":"mathlib/Mathlib/Algebra/Squarefree/Basic.lean","sample_id":"ff5f243c31598fd2aadb8dc36b39e21f486eb66be282ba3b720cbff07478671d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"da195f4abae7ab0a2c1c536bf1f2974db5c532e362792146fd16a70b703b9aa9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f4c589a7f4818f438b23af657c2db224655a94001f25229b505dee753deb8fe1","source_sha256":"cc48552fd983736c871bbaff8e8b09090c080c9d6073d135463b6637bbb51f7b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨cf, rfl⟩ := Code.exists_code.1 hf\n  obtain ⟨cg, rfl⟩ := Code.exists_code.1 hg\n  have : Nat.Partrec fun n => Nat.rfindOpt fun k => cf.evaln k n <|> cg.evaln k n :=\n    Partrec.nat_iff.1\n      (Partrec.rfindOpt <|\n        Primrec.option_orElse.to_comp.comp\n          (Code.primrec_evaln.to_comp.comp <| (snd.pair (const cf)).pair fst)\n          (Code.primrec_evaln.to_comp.comp <| (snd.pair (const cg)).pair fst))\n  refine ⟨_, this, fun n => ?_⟩\n  have : ∀ x ∈ rfindOpt fun k ↦ Code.evaln k cf n <|> Code.evaln k cg n,\n      x ∈ Code.eval cf n ∨ x ∈ Code.eval cg n := by\n    intro x h\n    obtain ⟨k, e⟩ := Nat.rfindOpt_spec h\n    rw [Option.mem_def, Option.orElse_eq_some, ← Option.mem_def, ← Option.mem_def] at e\n    obtain e | ⟨-, e⟩ := e <;> simp [Code.evaln_sound e]\n  refine ⟨this, fun h ↦ (this _ ⟨h, rfl⟩).imp Exists.fst Exists.fst, fun h ↦ ?_⟩\n  rw [Nat.rfindOpt_dom]\n  simp only [dom_iff_mem, Code.evaln_complete, Option.mem_def] at h\n  obtain ⟨x, k, e⟩ | ⟨x, k, e⟩ := h\n  · exact ⟨k, x, by simp [e]⟩\n  · refine ⟨k, ?_⟩\n    rcases cf.evaln k n with - | y\n    · exact ⟨x, by simp [e]⟩\n    · exact ⟨y, by simp⟩","hard_negative":true,"metrics":{"chosen_tokens":350,"rejected_tokens":8,"token_jaccard":0.05,"token_length_ratio":0.022857},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"43fbb68d8cf74e083d6d85d1fec91975f1e3922051bad8e402c9c94ada4f4e48","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Computability.PartrecCode\n\nNamespace:\nNat.Partrec\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Computable and Recursively Enumerable Predicates\n\nThis file defines computable (`ComputablePred`) and recursively enumerable (`REPred`)\npredicates. It also provides basic closure properties and Post's theorem on the\nequivalence of recursive, r.e., and co-r.e. sets.\n-/\n\n@[expose] public section\n\nopen List (Vector)\nopen Encodable Denumerable\n\nnamespace Nat.Partrec\n\nopen Computable Part\n\nTarget:\ntheorem merge' {f g} (hf : Nat.Partrec f) (hg : Nat.Partrec g) :\n    ∃ h, Nat.Partrec h ∧\n      ∀ a, (∀ x ∈ h a, x ∈ f a ∨ x ∈ g a) ∧ ((h a).Dom ↔ (f a).Dom ∨ (g a).Dom) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"644436e402e5479e5e92ff08568e4adad83df762908082eda5f53e5632900b4e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability","family_id":"merge","file_id":"mathlib/Mathlib/Computability/RE.lean","sample_id":"f4c589a7f4818f438b23af657c2db224655a94001f25229b505dee753deb8fe1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"99a954ce5f0b66b4d53e9b41552229a51be60d1da33548a2621f7dcf8fed671d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8908242e67265bda7ad427e8c08cbaa7bc892a07689e108509f2c3eb2190a540","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4fa9f7ffaee479f67dcc3f60683ec20977b8a4ed853209875ba4ba0d88e8525f","source_sha256":"fb00166246598a433f1c5ccafeed443b215760ffb2773f535ee79e2155bf0bd1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  use 1, S.one_mem\n  intro b hb\n  rw [one_smul]\n  obtain ⟨b', b'_mem, rfl⟩ := mem_one.mp (h hb)\n  exact Set.mem_range_self b'","hard_negative":true,"metrics":{"chosen_tokens":35,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.085714},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"443b2062bc4ef7eebc3867958ff1b04b4b850bc2b88b5440b840137cb1448001","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Localization.Integer\npublic import Mathlib.RingTheory.Localization.Submodule\n\nNamespace:\nFractionalIdeal\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen, Filippo A. E. Nuccio\n-/\n/-!\n# Fractional ideals\n\nThis file defines fractional ideals of an integral domain and proves basic facts about them.\n\n## Main definitions\nLet `S` be a submonoid of an integral domain `R` and `P` the localization of `R` at `S`.\n* `IsFractional` defines which `R`-submodules of `P` are fractional ideals\n* `FractionalIdeal S P` is the type of fractional ideals in `P`\n* a coercion `coeIdeal : Ideal R → FractionalIdeal S P`\n* `CommSemiring (FractionalIdeal S P)` instance:\n  the typical ideal operations generalized to fractional ideals\n* `Lattice (FractionalIdeal S P)` instance\n\n## Main statements\n\n  * the `MulLeftMono` and `MulRightMono` instances state that ideal multiplication is monotone\n  * `mul_div_self_cancel_iff` states that `1 / I` is the inverse of `I` if one exists\n\n## Implementation notes\n\nFractional ideals are considered equal when they contain the same elements,\nindependent of the denominator `a : R` such that `a I ⊆ R`.\nThus, we define `FractionalIdeal` to be the subtype of the predicate `IsFractional`,\ninstead of having `FractionalIdeal` be a structure of which `a` is a field.\n\nMost definitions in this file specialize operations from submodules to fractional ideals,\nproving that the result of this operation is fractional if the input is fractional.\nExceptions to this rule are defining `(+) := (⊔)` and `⊥ := 0`,\nin order to reuse their respective proof terms.\nWe can still use `simp` to show `↑I + ↑J = ↑(I + J)` and `↑⊥ = ↑0`.\n\nMany results in fact do not need that `P` is a localization, only that `P` is an\n`R`-algebra. We omit the `IsLocalization` parameter whenever this is practical.\nSimilarly, we don't assume that the localization is a field until we need it to\ndefine ideal quotients. When this assumption is needed, we replace `S` with `R⁰`,\nmaking the localization a field.\n\n## References\n\n  * https://en.wikipedia.org/wiki/Fractional_ideal\n\n## Tags\n\nfractional ideal, fractional ideals, invertible ideal\n-/\n\n@[expose] public section\n\n\nopen IsLocalization Pointwise nonZeroDivisors\n\nsection Defs\n\nvariable {R : Type*} [CommRing R] {S : Submonoid R} {P : Type*} [CommRing P]\nvariable [Algebra R P]\nvariable (S)\n\n/-- A submodule `I` is a fractional ideal with respect to a submonoid `S`\nif `a I ⊆ R` for some `a ∈ S`. -/\ndef IsFractional (I : Submodule R P) :=\n  ∃ a ∈ S, ∀ b ∈ I, IsInteger R (a • b)\n\nvariable (P)\n\n/-- The fractional ideals of a domain `R` are ideals of `R` divided by some `a ∈ R`.\n\nMore precisely, let `P` be a localization of `R` at some submonoid `S`,\nthen a fractional ideal `I ⊆ P` is an `R`-submodule of `P`,\nsuch that there is an `a ∈ S` with `a I ⊆ R`.\n-/\n@[wikidata Q1497184]\ndef FractionalIdeal :=\n  { I : Submodule R P // IsFractional S I }\n\nend Defs\n\nnamespace FractionalIdeal\n\nopen Set Submodule\n\nvariable {R : Type*} [CommRing R] {S : Submonoid R} {P : Type*} [CommRing P]\nvariable [Algebra R P]\n\n/-- Map a fractional ideal `I` to a submodule by forgetting that `∃ a, a I ⊆ R`.\n\nThis implements the coercion `FractionalIdeal S P → Submodule R P`.\n-/\n@[coe]\ndef coeToSubmodule (I : FractionalIdeal S P) : Submodule R P :=\n  I.val\n\n/-- Map a fractional ideal `I` to a submodule by forgetting that `∃ a, a I ⊆ R`.\n\nThis coercion is typically called `coeToSubmodule` in lemma names\n(or `coe` when the coercion is clear from the context),\nnot to be confused with `IsLocalization.coeSubmodule : Ideal R → Submodule R P`\n(which we use to define `coe : Ideal R → FractionalIdeal S P`).\n-/\ninstance : CoeOut (FractionalIdeal S P) (Submodule R P) :=\n  ⟨coeToSubmodule⟩\n\nprotected theorem isFractional (I : FractionalIdeal S P) : IsFractional S (I : Submodule R P) :=\n  I.prop\n\n/-- An element of `S` such that `I.den • I = I.num`, see `FractionalIdeal.num` and\n`FractionalIdeal.den_mul_self_eq_num`. -/\nnoncomputable def den (I : FractionalIdeal S P) : S :=\n  ⟨I.2.choose, I.2.choose_spec.1⟩\n\n/-- An ideal of `R` such that `I.den • I = I.num`, see `FractionalIdeal.den` and\n`FractionalIdeal.den_mul_self_eq_num`. -/\nnoncomputable def num (I : FractionalIdeal S P) : Ideal R :=\n  (I.den • (I : Submodule R P)).comap (Algebra.linearMap R P)\n\ntheorem den_mul_self_eq_num (I : FractionalIdeal S P) :\n    I.den • (I : Submodule R P) = Submodule.map (Algebra.linearMap R P) I.num := by\n  rw [den, num, Submodule.map_comap_eq]\n  refine (inf_of_le_right ?_).symm\n  rintro _ ⟨a, ha, rfl⟩\n  exact I.2.choose_spec.2 a ha\n\n/-- The linear equivalence between the fractional ideal `I` and the integral ideal `I.num`\ndefined by mapping `x` to `I.den • x`, assuming scalar multiplication by `I.den` is injective. -/\nnoncomputable abbrev equivNumOfIsSMulRegular [FaithfulSMul R P] {I : FractionalIdeal S P}\n    (reg : IsSMulRegular P I.den) : I ≃ₗ[R] I.num := by\n  refine LinearEquiv.trans\n    (LinearEquiv.ofBijective ((DistribSMul.toLinearMap R P I.den).restrict fun _ hx ↦ ?_)\n      ⟨fun _ _ hxy ↦ ?_, fun ⟨y, hy⟩ ↦ ?_⟩)\n    (Submodule.equivMapOfInjective (Algebra.linearMap R P)\n      (FaithfulSMul.algebraMap_injective R P) (num I)).symm\n  · rw [← den_mul_self_eq_num]\n    exact Submodule.smul_mem_pointwise_smul _ _ _ hx\n  · simpa [LinearMap.restrict_apply, reg.eq_iff] using hxy\n  · rw [← den_mul_self_eq_num] at hy\n    obtain ⟨x, hx, hxy⟩ := hy\n    exact ⟨⟨x, hx⟩, by simp_rw [LinearMap.restrict_apply, Subtype.ext_iff, ← hxy]; rfl⟩\n\n/-- The linear equivalence between the fractional ideal `I` and the integral ideal `I.num`\ndefined by mapping `x` to `I.den • x`. -/\nnoncomputable def equivNum [IsDomain R] [Module.IsTorsionFree R P] [Nontrivial P]\n    {I : FractionalIdeal S P} (h_nz : (I.den : R) ≠ 0) : I ≃ₗ[R] I.num :=\n  equivNumOfIsSMulRegular (smul_right_injective P h_nz)\n\n/-- The linear equivalence between the fractional ideal `I` in a faithful localization\nand the integral ideal `I.num`. -/\nnoncomputable def equivNumOfIsLocalization [FaithfulSMul R P] [IsLocalization S P]\n    (I : FractionalIdeal S P) : I ≃ₗ[R] I.num :=\n  equivNumOfIsSMulRegular (smul_bijective ..).1\n\nsection SetLike\n\ninstance : SetLike (FractionalIdeal S P) P where\n  coe I := ↑(I : Submodule R P)\n  coe_injective := SetLike.coe_injective.comp Subtype.coe_injective\n\ninstance : PartialOrder (FractionalIdeal S P) := .ofSetLike (FractionalIdeal S P) P\n\n@[simp]\ntheorem mem_coe {I : FractionalIdeal S P} {x : P} : x ∈ (I : Submodule R P) ↔ x ∈ I :=\n  Iff.rfl\n\n/-- Partially-applied version of `FractionalIdeal.ext`. -/\ntheorem coe_ext {I J : FractionalIdeal S P} : (I : Submodule R P) = (J : Submodule R P) → I = J :=\n  Subtype.ext\n\n/-- Partially-applied version of `FractionalIdeal.ext_iff`. -/\ntheorem coe_ext_iff {I J : FractionalIdeal S P} :\n    I = J ↔ (I : Submodule R P) = (J : Submodule R P) :=\n  Subtype.ext_iff\n\n@[ext]\ntheorem ext {I J : FractionalIdeal S P} : (∀ x, x ∈ I ↔ x ∈ J) → I = J :=\n  SetLike.ext\n\n@[simp]\n theorem equivNum_apply [IsDomain R] [Module.IsTorsionFree R P] [Nontrivial P]\n    {I : FractionalIdeal S P} (h_nz : (I.den : R) ≠ 0) (x : I) :\n    algebraMap R P (equivNum h_nz x) = I.den • x := by\n  change Algebra.linearMap R P _ = _\n  rw [equivNum, LinearEquiv.trans_apply, LinearEquiv.ofBijective_apply, LinearMap.restrict_apply,\n    Submodule.map_equivMapOfInjective_symm_apply, Subtype.coe_mk,\n    DistribSMul.toLinearMap_apply]\n\n/-- Copy of a `FractionalIdeal` with a new underlying set equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : FractionalIdeal S P :=\n  ⟨Submodule.copy p s hs, by\n    convert! p.isFractional\n    ext\n    simp only [hs]\n    rfl⟩\n\n@[simp]\ntheorem coe_copy (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : ↑(p.copy s hs) = s :=\n  rfl\n\ntheorem coe_eq (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : p.copy s hs = p :=\n  SetLike.coe_injective hs\n\nend SetLike\n\nlemma zero_mem (I : FractionalIdeal S P) : 0 ∈ I := I.coeToSubmodule.zero_mem\n\n@[simp]\ntheorem val_eq_coe (I : FractionalIdeal S P) : I.val = I :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_mk (I : Submodule R P) (hI : IsFractional S I) :\n    coeToSubmodule ⟨I, hI⟩ = I :=\n  rfl\n\ntheorem coeToSet_coeToSubmodule (I : FractionalIdeal S P) :\n    ((I : Submodule R P) : Set P) = I :=\n  rfl\n\n/-! Transfer instances from `Submodule R P` to `FractionalIdeal S P`. -/\n\ninstance (I : FractionalIdeal S P) : Module R I :=\n  Submodule.module (I : Submodule R P)\n\ntheorem coeToSubmodule_injective :\n    Function.Injective (fun (I : FractionalIdeal S P) ↦ (I : Submodule R P)) :=\n  Subtype.coe_injective\n\ntheorem coeToSubmodule_inj {I J : FractionalIdeal S P} : (I : Submodule R P) = J ↔ I = J :=\n  coeToSubmodule_injective.eq_iff\n\nTarget:\ntheorem isFractional_of_le_one (I : Submodule R P) (h : I ≤ 1) : IsFractional S I :=\n\nProof body:\n","rejected":"by\n  exact isFractional_of_le_one","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"f23acf797a473e5c80bcb0cc47f71b4940f2238c84932eadb1e018f45336dfc5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/FractionalIdeal","family_id":"isfractional_of_le_one","file_id":"mathlib/Mathlib/RingTheory/FractionalIdeal/Basic.lean","sample_id":"4fa9f7ffaee479f67dcc3f60683ec20977b8a4ed853209875ba4ba0d88e8525f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8f3b01149f94ae9d113cd2afe1cda1c376344890d5f097114694fdbd6fd42eb6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"624824fed1f8ed60e18c6dc25bea0252c969203f2e3d4afffc1243ef67d1654d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c7abf66d2d061cad451c9a404bc80bcd09d739b8aa88cc477da70de900f24a00","source_sha256":"bdcd2022da453373ffbdfe5bdbc273db0a9656383ede2bdc6575e68ee03c9aa0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases s.eq_empty_or_nonempty with (rfl | hs); · simp\n  rcases t.eq_empty_or_nonempty with (rfl | ht); · simp\n  simp only [hs.ne_empty, ht.ne_empty, isBounded_prod_of_nonempty (hs.prod ht), false_or]","hard_negative":false,"metrics":{"chosen_tokens":48,"rejected_tokens":52,"token_jaccard":0.888889,"token_length_ratio":1.083333},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"4443a4d5840f418065b914a8e00090a38533f3915f9789fc4c54d9a17c70d9bb","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.TypeTags.Basic\npublic import Mathlib.Topology.Bornology.Basic\n\nNamespace:\nBornology\n\nLocal context:\n/-\nCopyright (c) 2022 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Bornology structure on products and subtypes\n\nIn this file we define `Bornology` and `BoundedSpace` instances on `α × β`, `Π i, X i`, and\n`{x // p x}`. We also prove basic lemmas about `Bornology.cobounded` and `Bornology.IsBounded`\non these types.\n-/\n\npublic section\n\n\nopen Set Filter Bornology Function\n\nopen Filter\n\nvariable {α β ι : Type*} {X : ι → Type*} [Bornology α] [Bornology β]\n  [∀ i, Bornology (X i)]\n\ninstance Prod.instBornology : Bornology (α × β) where\n  cobounded := (cobounded α).coprod (cobounded β)\n  le_cofinite :=\n    @coprod_cofinite α β ▸ coprod_mono ‹Bornology α›.le_cofinite ‹Bornology β›.le_cofinite\n\ninstance Pi.instBornology : Bornology (∀ i, X i) where\n  cobounded := Filter.coprodᵢ fun i => cobounded (X i)\n  le_cofinite := iSup_le fun _ ↦ (comap_mono (Bornology.le_cofinite _)).trans (comap_cofinite_le _)\n\n/-- Inverse image of a bornology. -/\nabbrev Bornology.induced {α β : Type*} [Bornology β] (f : α → β) : Bornology α where\n  cobounded := comap f (cobounded β)\n  le_cofinite := (comap_mono (Bornology.le_cofinite β)).trans (comap_cofinite_le _)\n\ninstance {p : α → Prop} : Bornology (Subtype p) :=\n  Bornology.induced (Subtype.val : Subtype p → α)\n\nnamespace Bornology\n\n/-!\n### Bounded sets in `α × β`\n-/\n\n\ntheorem cobounded_prod : cobounded (α × β) = (cobounded α).coprod (cobounded β) :=\n  rfl\n\ntheorem isBounded_image_fst_and_snd {s : Set (α × β)} :\n    IsBounded (Prod.fst '' s) ∧ IsBounded (Prod.snd '' s) ↔ IsBounded s :=\n  compl_mem_coprod.symm\n\nlemma IsBounded.image_fst {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.fst '' s) :=\n  (isBounded_image_fst_and_snd.2 hs).1\n\nlemma IsBounded.image_snd {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.snd '' s) :=\n  (isBounded_image_fst_and_snd.2 hs).2\n\nvariable {s : Set α} {t : Set β} {S : ∀ i, Set (X i)}\n\ntheorem IsBounded.fst_of_prod (h : IsBounded (s ×ˢ t)) (ht : t.Nonempty) : IsBounded s :=\n  fst_image_prod s ht ▸ h.image_fst\n\ntheorem IsBounded.snd_of_prod (h : IsBounded (s ×ˢ t)) (hs : s.Nonempty) : IsBounded t :=\n  snd_image_prod hs t ▸ h.image_snd\n\ntheorem IsBounded.prod (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s ×ˢ t) :=\n  isBounded_image_fst_and_snd.1\n    ⟨hs.subset <| fst_image_prod_subset _ _, ht.subset <| snd_image_prod_subset _ _⟩\n\ntheorem isBounded_prod_of_nonempty (hne : Set.Nonempty (s ×ˢ t)) :\n    IsBounded (s ×ˢ t) ↔ IsBounded s ∧ IsBounded t :=\n  ⟨fun h => ⟨h.fst_of_prod hne.snd, h.snd_of_prod hne.fst⟩, fun h => h.1.prod h.2⟩\n\nTarget:\ntheorem isBounded_prod : IsBounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ IsBounded s ∧ IsBounded t :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rcases s.eq_empty_or_nonempty with (rfl | hs); · simp\n  rcases t.eq_empty_or_nonempty with (rfl | ht); · simp\n  simp only [hs.ne_empty, ht.ne_empty, isBounded_prod_of_nonempty (hs.prod ht), false_or]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Bornology","family_id":"isbounded_prod","file_id":"mathlib/Mathlib/Topology/Bornology/Constructions.lean","sample_id":"c7abf66d2d061cad451c9a404bc80bcd09d739b8aa88cc477da70de900f24a00"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b232bd6f68b5950f3c5e2ed9ba68ca285d64e0f598a20d32ae0f8c799f5bfc41","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b14089476197b86942d51dd07351b5cf69d062ba1bdb2cb6a8c8c4b42ae306f4","source_sha256":"d1237e92863e296159e7253e7e582eee623003d46d97c2c45df765267113b5c1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let M' := algebraMapSubmonoid S M\n  have hM' : algebraMapSubmonoid S M ≤ S⁰ := algebraMapSubmonoid_le_nonZeroDivisors_of_faithfulSMul\n    _ hM\n  let f₁ : Rₘ →+* K := map _ (T := R⁰) (RingHom.id R) hM\n  let f₂ : Sₘ →+* L := map _ (T := S⁰) (RingHom.id S) hM'\n  algebraize [f₁, f₂]\n  have := localization_isScalarTower_of_submonoid_le Rₘ K _ _ hM\n  have := localization_isScalarTower_of_submonoid_le Sₘ L _ _ hM'\n  have := isFractionRing_of_isDomain_of_isLocalization M Rₘ K\n  have := isFractionRing_of_isDomain_of_isLocalization M' Sₘ L\n  have : IsDomain Rₘ := isDomain_of_le_nonZeroDivisors _ hM\n  apply Algebra.IsSeparable.of_equiv_equiv (FractionRing.algEquiv Rₘ K).symm.toRingEquiv\n    (FractionRing.algEquiv Sₘ L).symm.toRingEquiv\n  apply ringHom_ext R⁰\n  ext\n  simp only [RingHom.coe_comp,\n      RingHom.coe_coe, Function.comp_apply, ← algebraMap_apply]\n  rw [algebraMap_apply R Rₘ (FractionRing R), AlgEquiv.coe_ringEquiv, AlgEquiv.commutes,\n    algebraMap_apply R S L, algebraMap_apply S Sₘ L, AlgEquiv.coe_ringEquiv, AlgEquiv.commutes]\n  simp only [← algebraMap_apply]\n  rw [algebraMap_apply R Rₘ (FractionRing Rₘ), ← algebraMap_apply Rₘ, ← algebraMap_apply]","hard_negative":false,"metrics":{"chosen_tokens":238,"rejected_tokens":3,"token_jaccard":0.015625,"token_length_ratio":0.012605},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"44522ed7d3eecdd9540e4b48e25a0457b28b4dbf3d6a4a377bdb7e059762f911","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.PID\npublic import Mathlib.FieldTheory.Separable\npublic import Mathlib.RingTheory.RingHom.Finite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Riccardo Brasca. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Riccardo Brasca\n-/\n/-!\n# Instances for Dedekind domains\nThis file contains various instances to work with localization of a ring extension.\n\nA very common situation in number theory is to have an extension of (say) Dedekind domains `R` and\n`S`, and to prove a property of this extension it is useful to consider the localization `Rₚ` of `R`\nat `P`, a prime ideal of `R`. One also works with the corresponding localization `Sₚ` of `S` and the\nfraction fields `K` and `L` of `R` and `S`. In this situation there are many compatible algebra\nstructures and various properties of the rings involved. Another situation is when we have a\ntower extension `R ⊆ S ⊆ T` and thus we work with `Rₚ ⊆ Sₚ ⊆ Tₚ` where\n`Tₚ` is the localization of `T` at `P`. This file contains a collection of such instances.\n\n## Implementation details\nIn general one wants all the results below for any algebra satisfying `IsLocalization`, but those\ncannot be instances (since Lean has no way of guessing the submonoid). Having the instances in the\nspecial case of *the* localization at a prime ideal is useful in working with Dedekind domains.\n\n-/\n\npublic section\n\nopen nonZeroDivisors IsLocalization Algebra Module IsFractionRing IsScalarTower\n\nattribute [local instance] FractionRing.liftAlgebra\n\nvariable {R : Type*} (S : Type*) (T : Type*) [CommRing R] [CommRing S] [CommRing T] [IsDomain R]\n  [IsDomain S] [IsDomain T] [Algebra R S]\n\nlocal notation3 \"K\" => FractionRing R\nlocal notation3 \"L\" => FractionRing S\nlocal notation3 \"F\" => FractionRing T\n\nsection\n\ntheorem algebraMapSubmonoid_le_nonZeroDivisors_of_faithfulSMul {A : Type*} (B : Type*)\n    [CommSemiring A] [CommSemiring B] [Algebra A B] [NoZeroDivisors B] [FaithfulSMul A B]\n    {S : Submonoid A} (hS : S ≤ A⁰) : algebraMapSubmonoid B S ≤ B⁰ :=\n  map_le_nonZeroDivisors_of_injective _ (FaithfulSMul.algebraMap_injective A B) hS\n\nvariable (Rₘ Sₘ : Type*) [CommRing Rₘ] [CommRing Sₘ] [Algebra R Rₘ] [IsTorsionFree R S]\n    [Algebra.IsSeparable (FractionRing R) (FractionRing S)] {M : Submonoid R} [IsLocalization M Rₘ]\n    [Algebra Rₘ Sₘ] [Algebra S Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ]\n    [IsScalarTower R S Sₘ] [IsLocalization (algebraMapSubmonoid S M) Sₘ]\n    [Algebra (FractionRing Rₘ) (FractionRing Sₘ)]\n    [IsScalarTower Rₘ (FractionRing Rₘ) (FractionRing Sₘ)]\n\nset_option backward.isDefEq.respectTransparency false in\ninclude R S in\n\nTarget:\ntheorem FractionRing.isSeparable_of_isLocalization (hM : M ≤ R⁰) :\n    Algebra.IsSeparable (FractionRing Rₘ) (FractionRing Sₘ) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/DedekindDomain","family_id":"fractionring","file_id":"mathlib/Mathlib/RingTheory/DedekindDomain/Instances.lean","sample_id":"b14089476197b86942d51dd07351b5cf69d062ba1bdb2cb6a8c8c4b42ae306f4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3d1a7e2cbfbf665394764afad32089ad753a926b64cde1fb10b4f271be3fc87d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"29350a05453bd19d7ef84c08e0437d9c4f3708e80c1fb8e846de11c72c3e3a7f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0f5b96fc655dafea499598a2e613044abf5991cec77163c042656873851c8051","source_sha256":"5e905b178f0e9eddf55a6c8ae9c3e3ec7191d0314931708fc81e73ffffb986ad","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain (hs | hs) := hs\n  · exact hs.2\n  · exact hs.1.isCompact","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":3,"token_jaccard":0.142857,"token_length_ratio":0.142857},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"445cb03a8b8a3c4caca14fcdce043fb8b437bafdf262c1e0a51c7cd96ae6c7aa","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Spectral.Basic\npublic import Mathlib.Topology.WithTopology\npublic import Mathlib.Topology.JacobsonSpace\npublic import Mathlib.Data.Set.Card\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Jiedong Jiang, Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Jiedong Jiang, Christian Merten\n-/\n/-!\n# Constructible topology\n\nIn this file we define the constructible topology on a topological space. This is the topology\ngenerated by compact open subsets and their complements.\n\n## Main definitions and results\n\n- `WithConstructibleTopology`: `X` equipped with its constructible topology.\n- `compactSpace_withConstructibleTopology`: If `X` is quasi-separated, quasi-sober,\n  prespectral and quasi-compact, then `X` is still quasi-compact in the constructible topology.\n\n## TODOs\n\n- If `X` is a spectral space, show that `X` is T2 and totally disconnected (@chrisflav).\n-/\n\n@[expose] public section\n\nvariable {X : Type*} [TopologicalSpace X]\n\n/-- The subbasis of the constructible topology on a topological space `X`: It consists\nof the open and compact sets of `X` and their complements. -/\ndef constructibleTopologySubbasis (X : Type*) [TopologicalSpace X] : Set (Set X) :=\n  { s | IsOpen s ∧ IsCompact s } ∪ { s | IsClosed s ∧ IsCompact sᶜ }\n\n/-- The constructible topology on a topological space `X` has as a subbasis\nthe open and compact sets of `X` and their complements. -/\n@[implicit_reducible]\ndef constructibleTopology (X : Type*) [TopologicalSpace X] : TopologicalSpace X :=\n  .generateFrom (constructibleTopologySubbasis X)\n\n/-- A type synonym for `X` that is equipped with the constructible topology of `X`. -/\n@[nolint unusedArguments]\nabbrev WithConstructibleTopology (X : Type*) [TopologicalSpace X] : Type _ :=\n  WithTopology X (constructibleTopology X)\n\nopen Topology\n\nlemma WithConstructibleTopology.isOpen_iff {s : Set (WithConstructibleTopology X)} :\n    IsOpen s ↔ IsOpen[constructibleTopology X] (WithTopology.toTopology _ ⁻¹' s) :=\n  WithTopology.isOpen_iff _\n\nlemma WithConstructibleTopology.isClosed_iff {s : Set (WithConstructibleTopology X)} :\n    IsClosed s ↔ IsClosed[constructibleTopology X] (WithTopology.toTopology _ ⁻¹' s) :=\n  WithTopology.isClosed_iff _\n\nlemma IsCompact.isOpen_constructibleTopology_of_isOpen {s : Set X}\n    (hs : IsCompact s) (ho : IsOpen s) : IsOpen[constructibleTopology X] s := by\n  apply TopologicalSpace.isOpen_generateFrom_of_mem\n  simp [constructibleTopologySubbasis, ho, hs]\n\nlemma IsCompact.isOpen_constructibleTopology_of_isClosed {s : Set X}\n    (hs : IsCompact sᶜ) (ho : IsClosed s) : IsOpen[constructibleTopology X] s := by\n  apply TopologicalSpace.isOpen_generateFrom_of_mem\n  simp [constructibleTopologySubbasis, ho, hs]\n\n@[simp]\nlemma compl_mem_constructibleTopologySubbasis_iff {s : Set X} :\n    sᶜ ∈ constructibleTopologySubbasis X ↔ s ∈ constructibleTopologySubbasis X := by\n  grind [constructibleTopologySubbasis, isClosed_compl_iff, compl_compl]\n\nTarget:\nlemma isCompact_of_mem_constructibleTopologySubbasis [CompactSpace X] {s : Set X}\n    (hs : s ∈ constructibleTopologySubbasis X) : IsCompact s :=\n\nProof body:\n","rejected":"by\n  exact isCompact_of_mem_constructibleTopologySubbasis","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"cee7b61a688398a79f854a07bfc4284d79892216bef2800461145b9ca25b869a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Spectral","family_id":"iscompact_of_mem_constructibletopologysubbasis","file_id":"mathlib/Mathlib/Topology/Spectral/ConstructibleTopology.lean","sample_id":"0f5b96fc655dafea499598a2e613044abf5991cec77163c042656873851c8051"},"split":"train"}
{"binding":{"chosen_proof_sha256":"04ff734dba3e12dd755f41758fae7a61d3a513148d7d550f593f25ac71b54468","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"57c3ec1326de5e5cf945d22e65a425ed4806bdef05d031767a6473451f983e0e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9e767d0b834339320d21f2fe6faa7f7606cdac3e6fc203de41750386839ebf3b","source_sha256":"06b8f3152156917b43eb16aa25f88f706675fd088336840c57324ab27b671934","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  subst coeff_eq deg_eq_deg coeff_eq_deg\n  exact natDegree_eq_of_le_of_coeff_ne_zero ‹_› ‹_›","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":18,"token_jaccard":0.666667,"token_length_ratio":1.384615},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"44970deeb22ba5843fc7e05ffa095e2ca77a6dc89aff4296a4575f8daee5110b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Degree.Lemmas\n\nNamespace:\nMathlib.Tactic.ComputeDegree\n\nLocal context:\n/-\nCopyright (c) 2023 Damiano Testa. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Damiano Testa\n-/\n/-!\n\n# `compute_degree` and `monicity`: tactics for explicit polynomials\n\nThis file defines two related tactics: `compute_degree` and `monicity`.\n\nUsing `compute_degree` when the goal is of one of the seven forms\n* `natDegree f ≤ d` (or `<`),\n* `degree f ≤ d` (or `<`),\n* `natDegree f = d`,\n* `degree f = d`,\n* `coeff f d = r`, if `d` is the degree of `f`,\n\ntries to solve the goal.\nIt may leave side-goals, in case it is not entirely successful.\n\nUsing `monicity` when the goal is of the form `Monic f` tries to solve the goal.\nIt may leave side-goals, in case it is not entirely successful.\n\nBoth tactics admit a `!` modifier (`compute_degree!` and `monicity!`) instructing\nLean to try harder to close the goal.\n\nSee the doc-strings for more details.\n\n## Future work\n\n* Currently, `compute_degree` does not deal correctly with some edge cases.  For instance,\n  ```lean\n  example [Semiring R] : natDegree (C 0 : R[X]) = 0 := by\n    compute_degree\n  --  ⊢ 0 ≠ 0\n  ```\n  Still, it may not be worth to provide special support for `natDegree f = 0`.\n* Make sure that numerals in coefficients are treated correctly.\n* Make sure that `compute_degree` works with goals of the form `degree f ≤ ↑d`, with an\n  explicit coercion from `ℕ` on the RHS.\n* Add support for proving goals of the from `natDegree f ≠ 0` and `degree f ≠ 0`.\n* Make sure that `degree`, `natDegree` and `coeff` are equally supported.\n\n## Implementation details\n\nAssume that `f : R[X]` is a polynomial with coefficients in a semiring `R` and\n`d` is either in `ℕ` or in `WithBot ℕ`.\n\nIf the goal has the form `natDegree f < d`, then we convert it to two separate goals:\n* `natDegree f ≤ ?_`, on which we apply the following steps;\n* `?_ < d`;\n\nwhere `?_` is a metavariable that `compute_degree` computes in its process.\nWe proceed similarly for `degree f < d`.\n\nIf the goal has the form `natDegree f = d`, then we convert it to three separate goals:\n* `natDegree f ≤ d`;\n* `coeff f d = r`;\n* `r ≠ 0`.\n\nSimilarly, an initial goal of the form `degree f = d` gives rise to goals of the form\n* `degree f ≤ d`;\n* `coeff f d = r`;\n* `r ≠ 0`.\n\nNext, we apply successively lemmas whose side-goals all have the shape\n* `natDegree f ≤ d`;\n* `degree f ≤ d`;\n* `coeff f d = r`;\n\nplus possibly \"numerical\" identities and choices of elements in `ℕ`, `WithBot ℕ`, and `R`.\n\nRecursing into `f`, we break apart additions, multiplications, powers, subtractions,...\nThe leaves of the process are\n* numerals, `C a`, `X` and `monomial a n`, to which we assign degree `0`, `1` and `a` respectively;\n* `fvar`s `f`, to which we tautologically assign degree `natDegree f`.\n-/\n\npublic meta section\n\nopen Polynomial\n\nnamespace Mathlib.Tactic.ComputeDegree\n\nsection recursion_lemmas\n/-!\n### Simple lemmas about `natDegree`\n\nThe lemmas in this section all have the form `natDegree <some form of cast> ≤ 0`.\nTheir proofs are weakenings of the stronger lemmas `natDegree <same> = 0`.\nThese are the lemmas called by `compute_degree` on (almost) all the leaves of its recursion.\n-/\n\nvariable {R : Type*}\n\nsection semiring\nvariable [Semiring R]\n\ntheorem natDegree_C_le (a : R) : natDegree (C a) ≤ 0 := (natDegree_C a).le\n\ntheorem natDegree_natCast_le (n : ℕ) : natDegree (n : R[X]) ≤ 0 := (natDegree_natCast _).le\ntheorem natDegree_zero_le : natDegree (0 : R[X]) ≤ 0 := natDegree_zero.le\ntheorem natDegree_one_le : natDegree (1 : R[X]) ≤ 0 := natDegree_one.le\n\ntheorem coeff_add_of_eq {n : ℕ} {a b : R} {f g : R[X]}\n    (h_add_left : f.coeff n = a) (h_add_right : g.coeff n = b) :\n    (f + g).coeff n = a + b := by subst ‹_› ‹_›; apply coeff_add\n\ntheorem coeff_mul_add_of_le_natDegree_of_eq_ite {d df dg : ℕ} {a b : R} {f g : R[X]}\n    (h_mul_left : natDegree f ≤ df) (h_mul_right : natDegree g ≤ dg)\n    (h_mul_left : f.coeff df = a) (h_mul_right : g.coeff dg = b) (ddf : df + dg ≤ d) :\n    (f * g).coeff d = if d = df + dg then a * b else 0 := by\n  split_ifs with h\n  · subst h_mul_left h_mul_right h\n    exact coeff_mul_add_eq_of_natDegree_le ‹_› ‹_›\n  · apply coeff_eq_zero_of_natDegree_lt\n    apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ddf ?_)\n    · exact natDegree_mul_le_of_le ‹_› ‹_›\n    · exact ne_comm.mp h\n\ntheorem coeff_pow_of_natDegree_le_of_eq_ite' {m n o : ℕ} {a : R} {p : R[X]}\n    (h_pow : natDegree p ≤ n) (h_exp : m * n ≤ o) (h_pow_bas : coeff p n = a) :\n    coeff (p ^ m) o = if o = m * n then a ^ m else 0 := by\n  split_ifs with h\n  · subst h h_pow_bas\n    exact coeff_pow_of_natDegree_le ‹_›\n  · apply coeff_eq_zero_of_natDegree_lt\n    apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ‹_› ?_)\n    · exact natDegree_pow_le_of_le m ‹_›\n    · exact Iff.mp ne_comm h\n\nsection SMul\n\nvariable {S : Type*} [SMulZeroClass S R] {n : ℕ} {a : S} {f : R[X]}\n\ntheorem natDegree_smul_le_of_le (hf : natDegree f ≤ n) :\n    natDegree (a • f) ≤ n :=\n  (natDegree_smul_le a f).trans hf\n\ntheorem degree_smul_le_of_le (hf : degree f ≤ n) :\n    degree (a • f) ≤ n :=\n  (degree_smul_le a f).trans hf\n\ntheorem coeff_smul : (a • f).coeff n = a • f.coeff n := rfl\n\nend SMul\n\nsection congr_lemmas\n\n/-- The following two lemmas should be viewed as a hand-made \"congr\"-lemmas.\nThey achieve the following goals.\n* They introduce *two* fresh metavariables replacing the given one `deg`,\n  one for the `natDegree ≤` computation and one for the `coeff =` computation.\n  This helps `compute_degree`, since it does not \"pre-estimate\" the degree,\n  but it \"picks it up along the way\".\n* They split checking the inequality `coeff p n ≠ 0` into the task of\n  finding a value `c` for the `coeff` and then\n  proving that this value is non-zero by `coeff_ne_zero`.\n-/\n\nTarget:\ntheorem natDegree_eq_of_le_of_coeff_ne_zero' {deg m o : ℕ} {c : R} {p : R[X]}\n    (h_natDeg_le : natDegree p ≤ m) (coeff_eq : coeff p o = c)\n    (coeff_ne_zero : c ≠ 0) (deg_eq_deg : m = deg) (coeff_eq_deg : o = deg) :\n    natDegree p = deg :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  subst coeff_eq deg_eq_deg coeff_eq_deg\n  exact natDegree_eq_of_le_of_coeff_ne_zero ‹_› ‹_›","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic","family_id":"natdegree_eq_of_le_of_coeff_ne_zero","file_id":"mathlib/Mathlib/Tactic/ComputeDegree.lean","sample_id":"9e767d0b834339320d21f2fe6faa7f7606cdac3e6fc203de41750386839ebf3b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3f00e9105f1d5704619d01594b14b1a2c0b0c68fe25f9bbf22b6c6f972b81c9f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a26f956f637c764b2f59cef3d4d51fd447c50773705cdd84c865b39ccbebb619","source_sha256":"009168811e0aa7b7b2aa8bada71ce25e1da7960123a5f42f2e4c63ec9822c044","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [moduleCat_exact_iff_function_exact] at hC ⊢\n  exact rTensor_exact M hC","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.166667},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"449a1c68066140eb5949a1a19a0cf946b71ddb684ad734d9df343ea208d3a031","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Flat.Basic\npublic import Mathlib.Algebra.Homology.ShortComplex.ModuleCat\npublic import Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\nNamespace:\nModule.Flat\n\nLocal context:\n/-\nCopyright (c) 2024 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang\n-/\n/-!\n# Tensoring with a flat module is an exact functor\n\nIn this file we prove that tensoring with a flat module is an exact functor.\n\n## Main results\n\n- `Module.Flat.iff_lTensor_preserves_shortComplex_exact`: an `R`-module `M` is flat if and only if\n  for every exact sequence `A ⟶ B ⟶ C`, `M ⊗ A ⟶ M ⊗ B ⟶ M ⊗ C` is also exact.\n\n- `Module.Flat.iff_rTensor_preserves_shortComplex_exact`: an `R`-module `M` is flat if and only if\n  for every short exact sequence `A ⟶ B ⟶ C`, `A ⊗ M ⟶ B ⊗ M ⟶ C ⊗ M` is also exact.\n\n## TODO\n\n- Relate flatness with `Tor`\n\n-/\n\npublic section\n\nuniverse u\n\nopen CategoryTheory MonoidalCategory ShortComplex.ShortExact\n\nnamespace Module.Flat\n\nvariable {R : Type u} [CommRing R] (M : ModuleCat.{u} R)\n\nlemma lTensor_shortComplex_exact [Flat R M] (C : ShortComplex <| ModuleCat R) (hC : C.Exact) :\n    C.map (tensorLeft M) |>.Exact := by\n  rw [moduleCat_exact_iff_function_exact] at hC ⊢\n  exact lTensor_exact M hC\n\nTarget:\nlemma rTensor_shortComplex_exact [Flat R M] (C : ShortComplex <| ModuleCat R) (hC : C.Exact) :\n    C.map (tensorRight M) |>.Exact :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Flat","family_id":"rtensor_shortcomplex_exact","file_id":"mathlib/Mathlib/RingTheory/Flat/CategoryTheory.lean","sample_id":"a26f956f637c764b2f59cef3d4d51fd447c50773705cdd84c865b39ccbebb619"},"split":"train"}
{"binding":{"chosen_proof_sha256":"332fb89312684e7786baadfdf197587e5cdba892591555a524aedd43ed4113c6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6f14ad8f52f1b968fcf983bc79e14954c8b5b7be8afbe68ca15b647c761ebe7b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e31fb78612321e73c41c554f6e35840ea8a109f4f7b6f9fddcff3cf0f701ebac","source_sha256":"810b2eb230c1a433d88acadf7472080648a9bb39145bc2d8989cf9f5cfc8a13b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction n with\n  | zero =>\n    simp\n  | succ n IH =>\n    simp_rw [iteratedLineDerivOp_succ_left, IH, lineDerivOp_add]","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":25,"token_jaccard":0.8,"token_length_ratio":1.190476},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"44c57f9801266bf37e860450a49958c1a5a11070241ef83db3a933cf2b98c716","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Equiv.Defs\npublic import Mathlib.Data.Fin.Tuple.Basic\npublic import Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic\npublic import Mathlib.Analysis.InnerProductSpace.CanonicalTensor\n\nNamespace:\nLineDeriv\n\nLocal context:\n/-\nCopyright (c) 2025 Moritz Doll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Moritz Doll\n-/\n/-! # Type classes for derivatives and the Laplacian\n\nIn this file we define notation type classes for line derivatives, also known as partial\nderivatives, and for the Laplacian.\n\nMoreover, we provide type-classes that encode the linear structure.\nWe also define the iterated line derivative and prove elementary properties.\nWe define a Laplacian based on the sum of second derivatives formula and prove that the Laplacian\nthus defined is independent of the choice of basis.\n\nCurrently, this type class is only used by Schwartz functions. Future uses include derivatives on\ntest functions, distributions, tempered distributions, and Sobolev spaces (and other generalized\nfunction spaces).\n-/\n\n@[expose] public noncomputable section\n\nuniverse u' u v w\n\nvariable {ι ι' 𝕜 R V E F V₁ V₂ V₃ : Type*}\n\n/-! ## Line derivative -/\n\nopen Fin\n\n/--\nThe notation typeclass for the line derivative.\n-/\nclass LineDeriv (V : Type u) (E : Type v) (F : outParam (Type w)) where\n  /-- `∂_{v} f` is the line derivative of `f` in direction `v`. The meaning of this notation is\n  type-dependent. -/\n  lineDerivOp : V → E → F\n\nnamespace LineDeriv\n\n@[inherit_doc] scoped notation \"∂_{\" v \"}\" => LineDeriv.lineDerivOp v\n\nvariable {V E : Type*} [LineDeriv V E E]\n\n/-- `∂^{m} f` is the iterated line derivative of `f`, where `m` is a finite number of (different)\ndirections. -/\ndef iteratedLineDerivOp {n : ℕ} : (Fin n → V) → E → E :=\n  Nat.recOn n (fun _ ↦ id) (fun _ rec y ↦ LineDeriv.lineDerivOp (y 0) ∘ rec (tail y))\n\n@[inherit_doc] scoped notation \"∂^{\" v \"}\" => LineDeriv.iteratedLineDerivOp v\n\n@[simp]\ntheorem iteratedLineDerivOp_fin_zero (m : Fin 0 → V) (f : E) : ∂^{m} f = f :=\n  rfl\n\n@[simp]\ntheorem iteratedLineDerivOp_one (m : Fin 1 → V) (f : E) : ∂^{m} f = ∂_{m 0} f :=\n  rfl\n\ntheorem iteratedLineDerivOp_succ_left {n : ℕ} (m : Fin (n + 1) → V) (f : E) :\n    ∂^{m} f = ∂_{m 0} (∂^{tail m} f) :=\n  rfl\n\ntheorem iteratedLineDerivOp_succ_right {n : ℕ} (m : Fin (n + 1) → V) (f : E) :\n    ∂^{m} f = ∂^{init m} (∂_{m (last n)} f) := by\n  induction n with\n  | zero => rfl\n  -- The proof is `∂^{n + 2} = ∂ ∂^{n + 1} = ∂ ∂^n ∂ = ∂^{n+1} ∂`\n  | succ n IH =>\n    have hmzero : init m 0 = m 0 := by simp only [init_def, castSucc_zero]\n    have hmtail : tail m (last n) = m (last n.succ) := by\n      simp only [tail_def, succ_last]\n    calc\n      _ = ∂_{m 0} (∂^{tail m} f) := iteratedLineDerivOp_succ_left _ _\n      _ = ∂_{m 0} (∂^{init <| tail m} (∂_{tail m <| last n} f)) := by\n        congr 1\n        exact IH _\n      _ = _ := by\n        rw [hmtail, iteratedLineDerivOp_succ_left, hmzero, tail_init_eq_init_tail]\n\n@[simp]\ntheorem iteratedLineDerivOp_const_eq_iter_lineDerivOp (n : ℕ) (y : V) (f : E) :\n    ∂^{fun (_ : Fin n) ↦ y} f = ∂_{y}^[n] f := by\n  induction n with\n  | zero => rfl\n  | succ n IH =>\n    rw [iteratedLineDerivOp_succ_left, Function.iterate_succ_apply']\n    congr\n\nend LineDeriv\n\nopen LineDeriv\n\n/--\nThe line derivative is additive, `∂_{v} (x + y) = ∂_{v} x + ∂_{v} y` for all `x y : E`\nand `∂_{v + w} x = ∂_{v} x + ∂_{w} y` for all `v w : V`.\n\nNote that `lineDeriv` on functions is not additive.\n-/\nclass LineDerivAdd (V : Type u) (E : Type v) (F : outParam (Type w))\n    [AddCommGroup V] [AddCommGroup E] [AddCommGroup F] [LineDeriv V E F] where\n  lineDerivOp_add (v : V) (x y : E) : ∂_{v} (x + y) = ∂_{v} x + ∂_{v} y\n  lineDerivOp_left_add (v w : V) (x : E) : ∂_{v + w} x = ∂_{v} x + ∂_{w} x\n\n/--\nThe line derivative commutes with scalar multiplication, `∂_{v} (r • x) = r • ∂_{v} x` for all\n`r : R` and `x : E`.\n-/\nclass LineDerivSMul (R : Type*) (V : Type u) (E : Type v) (F : outParam (Type w))\n    [SMul R E] [SMul R F] [LineDeriv V E F] where\n  lineDerivOp_smul (v : V) (r : R) (x : E) : ∂_{v} (r • x) = r • ∂_{v} x\n\n/--\nThe line derivative commutes with scalar multiplication, `∂_{r • v} x = r • ∂_{v} x` for all\n`r : R` and `v : V`.\n-/\nclass LineDerivLeftSMul (R : Type*) (V : Type u) (E : Type v) (F : outParam (Type w))\n    [SMul R V] [SMul R F] [LineDeriv V E F] where\n  lineDerivOp_left_smul (r : R) (v : V) (x : E) : ∂_{r • v} x = r • ∂_{v} x\n\n/--\nThe line derivative is continuous.\n-/\nclass ContinuousLineDeriv (V : Type u) (E : Type v) (F : outParam (Type w))\n    [TopologicalSpace E] [TopologicalSpace F] [LineDeriv V E F] where\n  continuous_lineDerivOp (v : V) : Continuous (∂_{v} : E → F)\n\nattribute [fun_prop] ContinuousLineDeriv.continuous_lineDerivOp\n\nnamespace LineDeriv\n\nexport LineDerivAdd (lineDerivOp_add)\nexport LineDerivAdd (lineDerivOp_left_add)\nexport LineDerivSMul (lineDerivOp_smul)\nexport LineDerivLeftSMul (lineDerivOp_left_smul)\nexport ContinuousLineDeriv (continuous_lineDerivOp)\n\nsection lineDerivOp\n\nvariable [AddCommGroup V] [AddCommGroup E] [AddCommGroup F] [LineDeriv V E F] [LineDerivAdd V E F]\n\n@[simp]\ntheorem lineDerivOp_zero (v : V) : ∂_{v} (0 : E) = 0 :=\n  map_zero (AddMonoidHom.mk' ∂_{v} (lineDerivOp_add v))\n\n@[simp]\ntheorem lineDerivOp_neg (v : V) (x : E) : ∂_{v} (-x) = - ∂_{v} x :=\n  map_neg (AddMonoidHom.mk' ∂_{v} (lineDerivOp_add v)) x\n\n@[simp]\ntheorem lineDerivOp_sum (v : V) (f : ι → E) (s : Finset ι) :\n    ∂_{v} (∑ i ∈ s, f i) = ∑ i ∈ s, ∂_{v} (f i) :=\n  map_sum (AddMonoidHom.mk' ∂_{v} (lineDerivOp_add v)) f s\n\n@[simp]\ntheorem lineDerivOp_left_zero (x : E) : ∂_{(0 : V)} x = 0 :=\n  map_zero (AddMonoidHom.mk' (∂_{·} x) (lineDerivOp_left_add · · x))\n\n@[simp]\ntheorem lineDerivOp_left_neg (v : V) (x : E) : ∂_{-v} x = - ∂_{v} x :=\n  map_neg (AddMonoidHom.mk' (∂_{·} x) (lineDerivOp_left_add · · x)) v\n\n@[simp]\ntheorem lineDerivOp_left_sum (f : ι → V) (x : E) (s : Finset ι) :\n    ∂_{∑ i ∈ s, f i} x = ∑ i ∈ s, ∂_{f i} x :=\n  map_sum (AddMonoidHom.mk' (∂_{·} x) (lineDerivOp_left_add · · x)) f s\n\nend lineDerivOp\n\nsection lineDerivOpCLM\n\nvariable [Ring R] [AddCommGroup E] [Module R E] [AddCommGroup F] [Module R F]\n  [TopologicalSpace E] [TopologicalSpace F] [AddCommGroup V]\n  [LineDeriv V E F] [LineDerivAdd V E F] [LineDerivSMul R V E F] [ContinuousLineDeriv V E F]\n\nvariable (R E) in\n/-- The line derivative as a continuous linear map. -/\ndef lineDerivOpCLM (m : V) : E →L[R] F where\n  toFun := ∂_{m}\n  map_add' := lineDerivOp_add m\n  map_smul' := lineDerivOp_smul m\n\n@[simp]\ntheorem lineDerivOpCLM_apply (m : V) (x : E) :\n    lineDerivOpCLM R E m x = ∂_{m} x := rfl\n\nend lineDerivOpCLM\n\nsection iteratedLineDerivOp\n\nvariable [LineDeriv V E E]\nvariable {n : ℕ} (m : Fin n → V)\n\nsection add\n\nvariable [AddCommGroup V] [AddCommGroup E] [LineDerivAdd V E E]\n\nTarget:\ntheorem iteratedLineDerivOp_add (x y : E) :\n    ∂^{m} (x + y) = ∂^{m} x + ∂^{m} y :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  induction n with\n  | zero =>\n    simp\n  | succ n IH =>\n    simp_rw [iteratedLineDerivOp_succ_left, IH, lineDerivOp_add]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Distribution","family_id":"iteratedlinederivop_add","file_id":"mathlib/Mathlib/Analysis/Distribution/DerivNotation.lean","sample_id":"e31fb78612321e73c41c554f6e35840ea8a109f4f7b6f9fddcff3cf0f701ebac"},"split":"train"}
{"binding":{"chosen_proof_sha256":"98bbccba21050a97745f14c454bcd7300ae6caac0c9cdef2d7b7857a2d6f0324","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"28dab043652a1525adfd5ddc52e6243324bae2d9c4d94e5b8f18a41fa2a7d11f","source_sha256":"ca6f378e1485ca426ed934f409d878b070ba40b0fbd38c240a94cbf961d2f64f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← totallyDisconnectedSpace_subtype_iff]\n  have : Countable s := hs\n  infer_instance","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.230769},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"44d7ca051765c1b692857da001cd0bab371164b96f3e1dd1b046fd0f9a3cc100","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.GDelta.MetrizableSpace\npublic import Mathlib.Topology.Separation.CompletelyRegular\npublic import Mathlib.Topology.Separation.Profinite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Bryan Wang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Geoffrey Irving, Bryan Wang, Oliver Nash\n-/\n/-!\n# Further separation lemmas\n-/\n\npublic section\n\nvariable {X : Type*}\n\nnamespace CompletelyRegularSpace\n\nvariable [TopologicalSpace X] [T35Space X]\n\ntheorem totallySeparatedSpace_of_cardinalMk_lt_continuum (h : Cardinal.mk X < Cardinal.continuum) :\n    TotallySeparatedSpace X :=\n  totallySeparatedSpace_of_t0_of_basis_clopen <|\n    CompletelyRegularSpace.isTopologicalBasis_clopens_of_cardinalMk_lt_continuum h\n\ninstance [Countable X] : TotallySeparatedSpace X :=\n  totallySeparatedSpace_of_cardinalMk_lt_continuum <|\n    (Cardinal.mk_le_aleph0_iff.mpr inferInstance).trans_lt Cardinal.aleph0_lt_continuum\n\nprotected lemma _root_.Set.Countable.totallySeparatedSpace {s : Set X} (h : s.Countable) :\n    TotallySeparatedSpace s :=\n  have : _root_.Countable s := h\n  inferInstanceAs (TotallySeparatedSpace s)\n\nend CompletelyRegularSpace\n\n/-- Countable subsets of metric spaces are totally disconnected. -/\n\nTarget:\ntheorem Set.Countable.isTotallyDisconnected [MetricSpace X] {s : Set X} (hs : s.Countable) :\n    IsTotallyDisconnected s :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Separation","family_id":"set","file_id":"mathlib/Mathlib/Topology/Separation/Lemmas.lean","sample_id":"28dab043652a1525adfd5ddc52e6243324bae2d9c4d94e5b8f18a41fa2a7d11f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1a29d536dd839cb2ac804ef83ab5abb79ba3d4ec297a0fa4ca91c3477c2aaaa0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"539a7c9270ba302e32332d5783827731099600c5b6bb97d8bc9c71f4436296b8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a793e6cc882c9887527edba47d8bc663c406cd6c0a53d249021c37e71fed657c","source_sha256":"0558464e45d5ed116146fb004c344db832eef9939fa1268196e1a1a60376c0f3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine OreLocalization.ind fun x s ↦ ?_\n  obtain ⟨i, j, hne, heq⟩ := Finite.exists_ne_map_eq_of_infinite (α := ℕ) (s ^ · • x)\n  wlog! hlt : j < i generalizing i j\n  · exact this j i hne.symm heq.symm (hne.lt_of_le hlt)\n  use s ^ (i - (j + 1)) • x\n  rw [oreDiv_eq_iff]\n  refine ⟨s ^ j, (s ^ (j + 1)).1, ?_, ?_⟩\n  · change s ^ j • x = s ^ (j + 1) • s ^ (i - (j + 1)) • x\n    rw [← mul_smul, ← pow_add, Nat.add_sub_cancel' hlt, heq]\n  · simp_rw [SubmonoidClass.coe_pow, OneMemClass.coe_one, mul_one, pow_succ]","hard_negative":true,"metrics":{"chosen_tokens":165,"rejected_tokens":5,"token_jaccard":0.047619,"token_length_ratio":0.030303},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"4518950c919b80b9300c2dcb446a82150b72c276ff66f4ff2f0d3c618718526b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Pigeonhole\npublic import Mathlib.GroupTheory.OreLocalization.Basic\npublic import Mathlib.SetTheory.Cardinal.Arithmetic\n\nNamespace:\nOreLocalization\n\nLocal context:\n/-\nCopyright (c) 2024 Jz Pan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jz Pan\n-/\n/-!\n\n# Cardinality of Ore localizations\n\nThis file contains some results on cardinality of Ore localizations.\n\n## TODO\n\n- Prove or disprove `OreLocalization.cardinalMk_le_lift_cardinalMk_of_commute`\n  with `Commute` assumption removed.\n\n-/\n\npublic section\n\nuniverse u v\n\nopen Cardinal Function\n\nnamespace OreLocalization\n\nvariable {R : Type u} [Monoid R] (S : Submonoid R) [OreLocalization.OreSet S]\n  (X : Type v) [MulAction R X]\n\n@[to_additive]\ntheorem oreDiv_one_surjective_of_finite_left [Finite S] :\n    Surjective (fun x ↦ x /ₒ (1 : ↥S) : X → OreLocalization S X) := by\n  refine OreLocalization.ind fun x s ↦ ?_\n  obtain ⟨i, j, hne, heq⟩ := Finite.exists_ne_map_eq_of_infinite (α := ℕ) (s ^ ·)\n  wlog! hlt : j < i generalizing i j\n  · exact this j i hne.symm heq.symm (hne.lt_of_le hlt)\n  use s ^ (i - (j + 1)) • x\n  rw [oreDiv_eq_iff]\n  refine ⟨s ^ j, (s ^ (j + 1)).1, ?_, ?_⟩\n  · change s ^ j • x = s ^ (j + 1) • s ^ (i - (j + 1)) • x\n    rw [← mul_smul, ← pow_add, Nat.add_sub_cancel' hlt, heq]\n  · simp_rw [SubmonoidClass.coe_pow, OneMemClass.coe_one, mul_one, pow_succ]\n\n@[to_additive]\n\nTarget:\ntheorem oreDiv_one_surjective_of_finite_right [Finite X] :\n    Surjective (fun x ↦ x /ₒ (1 : ↥S) : X → OreLocalization S X) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_a793e6cc882c","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c91fee88ad1ed12eba6e009039b69dd7aee2e448ff9d91e392161ba9b48d9406","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/OreLocalization","family_id":"orediv_one_surjective_of_finite_right","file_id":"mathlib/Mathlib/GroupTheory/OreLocalization/Cardinality.lean","sample_id":"a793e6cc882c9887527edba47d8bc663c406cd6c0a53d249021c37e71fed657c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b7ecb19a982d835fa985827cc7886e0d4262e615155091b32c40d6c7877f5f38","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5bfec5ae03bfd55ef1bbbff853e84f4a25242edd9b4ad0bd7f154c9852ef86aa","source_sha256":"d7743ebc410d9ee31a12263ae8d894bf6aecd1d52cd1a367eccb2f5de9ab8de8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [minimalPeriod_piMap]\n  by_cases h : {n | 0 < n ∧ ∀ (i : ι), minimalPeriod (f i) (x i) ∣ n}.Nonempty\n  · exact (Nat.sInf_mem h).2 i\n  · simp [not_nonempty_iff_eq_empty.mp h]","hard_negative":true,"metrics":{"chosen_tokens":56,"rejected_tokens":8,"token_jaccard":0.05,"token_length_ratio":0.142857},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"4549190c469bc55b3512f4988ce585891baf93f6ba6eee53863200e8a1009e27","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GCDMonoid.Finset\npublic import Mathlib.Data.Nat.Prime.Basic\npublic import Mathlib.Data.PNat.Basic\npublic import Mathlib.Dynamics.PeriodicPts.Defs\npublic import Mathlib.Order.Lattice.Nat\n\nNamespace:\nFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Extra lemmas about periodic points\n-/\n\npublic section\n\nopen Nat Set\n\nnamespace Function\nvariable {α : Type*} {f : α → α} {x y : α}\n\nopen Function (Commute)\n\ntheorem directed_ptsOfPeriod_pnat (f : α → α) : Directed (· ⊆ ·) fun n : ℕ+ => ptsOfPeriod f n :=\n  fun m n => ⟨m * n, fun _ hx => hx.mul_const n, fun _ hx => hx.const_mul m⟩\n\nvariable (f) in\ntheorem bijOn_periodicPts : BijOn f (periodicPts f) (periodicPts f) :=\n  iUnion_pnat_ptsOfPeriod f ▸\n    bijOn_iUnion_of_directed (directed_ptsOfPeriod_pnat f) fun i => bijOn_ptsOfPeriod f i.pos\n\ntheorem minimalPeriod_eq_prime_iff {p : ℕ} [hp : Fact p.Prime] :\n    minimalPeriod f x = p ↔ IsPeriodicPt f p x ∧ ¬IsFixedPt f x := by\n  rw [Function.isPeriodicPt_iff_minimalPeriod_dvd, Nat.dvd_prime hp.out,\n    ← minimalPeriod_eq_one_iff_isFixedPt.not, or_and_right, and_not_self_iff, false_or,\n    iff_self_and]\n  exact fun h ↦ ne_of_eq_of_ne h hp.out.ne_one\n\ntheorem minimalPeriod_eq_sInf_n_pos_IsPeriodicPt :\n    minimalPeriod f x = sInf { n > 0 | IsPeriodicPt f n x } := by\n  dsimp +instances [minimalPeriod, periodicPts, sInf]\n  grind\n\n/-- The backward direction of `minimalPeriod_eq_prime_iff`. -/\ntheorem minimalPeriod_eq_prime {p : ℕ} [hp : Fact p.Prime] (hper : IsPeriodicPt f p x)\n    (hfix : ¬IsFixedPt f x) : minimalPeriod f x = p :=\n  minimalPeriod_eq_prime_iff.mpr ⟨hper, hfix⟩\n\ntheorem minimalPeriod_eq_prime_pow {p k : ℕ} [hp : Fact p.Prime] (hk : ¬IsPeriodicPt f (p ^ k) x)\n    (hk1 : IsPeriodicPt f (p ^ (k + 1)) x) : minimalPeriod f x = p ^ (k + 1) := by\n  apply Nat.eq_prime_pow_of_dvd_least_prime_pow hp.out <;>\n    rwa [← isPeriodicPt_iff_minimalPeriod_dvd]\n\ntheorem Commute.minimalPeriod_of_comp_dvd_mul {g : α → α} (h : Commute f g) :\n    minimalPeriod (f ∘ g) x ∣ minimalPeriod f x * minimalPeriod g x :=\n  dvd_trans h.minimalPeriod_of_comp_dvd_lcm (Nat.lcm_dvd_mul _ _)\n\ntheorem Commute.minimalPeriod_of_comp_eq_mul_of_coprime {g : α → α} (h : Commute f g)\n    (hco : Coprime (minimalPeriod f x) (minimalPeriod g x)) :\n    minimalPeriod (f ∘ g) x = minimalPeriod f x * minimalPeriod g x := by\n  apply h.minimalPeriod_of_comp_dvd_mul.antisymm\n  suffices ∀ {f g : α → α},\n      Commute f g →\n        Coprime (minimalPeriod f x) (minimalPeriod g x) →\n          minimalPeriod f x ∣ minimalPeriod (f ∘ g) x from\n    hco.mul_dvd_of_dvd_of_dvd (this h hco) (h.comp_eq.symm ▸ this h.symm hco.symm)\n  intro f g h hco\n  refine hco.dvd_of_dvd_mul_left (IsPeriodicPt.left_of_comp h ?_ ?_).minimalPeriod_dvd\n  · exact (isPeriodicPt_minimalPeriod _ _).const_mul _\n  · exact (isPeriodicPt_minimalPeriod _ _).mul_const _\n\nsection Fintype\n\nopen Fintype\n\ntheorem minimalPeriod_le_card [Fintype α] : minimalPeriod f x ≤ card α := by\n  rw [← periodicOrbit_length]\n  exact List.Nodup.length_le_card nodup_periodicOrbit\n\ntheorem isPeriodicPt_factorial_card_of_mem_periodicPts [Fintype α] (h : x ∈ periodicPts f) :\n    IsPeriodicPt f (card α)! x :=\n  isPeriodicPt_iff_minimalPeriod_dvd.mpr\n    (Nat.dvd_factorial (minimalPeriod_pos_of_mem_periodicPts h) minimalPeriod_le_card)\n\ntheorem mem_periodicPts_iff_isPeriodicPt_factorial_card [Fintype α] :\n    x ∈ periodicPts f ↔ IsPeriodicPt f (card α)! x where\n  mp := isPeriodicPt_factorial_card_of_mem_periodicPts\n  mpr h := minimalPeriod_pos_iff_mem_periodicPts.mp\n    (IsPeriodicPt.minimalPeriod_pos (Nat.factorial_pos _) h)\n\ntheorem Injective.mem_periodicPts [Finite α] (h : Injective f) (x : α) : x ∈ periodicPts f := by\n  obtain ⟨m, n, heq, hne⟩ : ∃ m n, f^[m] x = f^[n] x ∧ m ≠ n := by\n    simpa [Injective] using not_injective_infinite_finite (f^[·] x)\n  rcases lt_or_gt_of_ne hne with hlt | hlt\n  · exact mk_mem_periodicPts (by lia) (iterate_cancel h heq.symm)\n  · exact mk_mem_periodicPts (by lia) (iterate_cancel h heq)\n\ntheorem injective_iff_periodicPts_eq_univ [Finite α] : Injective f ↔ periodicPts f = univ := by\n  refine ⟨fun h ↦ eq_univ_iff_forall.mpr h.mem_periodicPts, fun h ↦ ?_⟩\n  rw [Finite.injective_iff_surjective, ← range_eq_univ, ← univ_subset_iff, ← h]\n  apply periodicPts_subset_range\n\ntheorem injective_iff_iterate_factorial_card_eq_id [Fintype α] :\n    Injective f ↔ f^[(card α)!] = id := by\n  simp only [injective_iff_periodicPts_eq_univ, mem_periodicPts_iff_isPeriodicPt_factorial_card,\n    funext_iff, eq_univ_iff_forall, IsPeriodicPt, id, IsFixedPt]\n\nend Fintype\n\nend Function\n\nnamespace Function\n\nsection Prod\n\nvariable {α β : Type*} {f : α → α} {g : β → β} {x : α × β} {a : α} {b : β} {m n : ℕ}\n\ntheorem minimalPeriod_prodMap (f : α → α) (g : β → β) (x : α × β) :\n    minimalPeriod (Prod.map f g) x = (minimalPeriod f x.1).lcm (minimalPeriod g x.2) :=\n  eq_of_forall_dvd <| by simp [← isPeriodicPt_iff_minimalPeriod_dvd, Nat.lcm_dvd_iff]\n\ntheorem minimalPeriod_fst_dvd : minimalPeriod f x.1 ∣ minimalPeriod (Prod.map f g) x := by\n  rw [minimalPeriod_prodMap]; exact Nat.dvd_lcm_left _ _\n\ntheorem minimalPeriod_snd_dvd : minimalPeriod g x.2 ∣ minimalPeriod (Prod.map f g) x := by\n  rw [minimalPeriod_prodMap]; exact Nat.dvd_lcm_right _ _\n\nend Prod\n\nsection Pi\n\nvariable {ι : Type*} {α : ι → Type*} {f : ∀ i, α i → α i} {x : ∀ i, α i}\n\n/-- This `sInf` can be regarded as a generalized version of LCM\nfor possibly infinite sets and types. -/\ntheorem minimalPeriod_piMap :\n    minimalPeriod (Pi.map f) x = sInf { n > 0 | ∀ i, minimalPeriod (f i) (x i) ∣ n } := by\n  conv_lhs => simp [minimalPeriod_eq_sInf_n_pos_IsPeriodicPt]\n  simp [← isPeriodicPt_iff_minimalPeriod_dvd]\n\ntheorem minimalPeriod_piMap_fintype [Fintype ι] :\n    minimalPeriod (Pi.map f) x = Finset.univ.lcm (fun i => minimalPeriod (f i) (x i)) :=\n  eq_of_forall_dvd <| by simp [← isPeriodicPt_iff_minimalPeriod_dvd]\n\nTarget:\ntheorem minimalPeriod_single_dvd_minimalPeriod_piMap (i : ι) :\n    minimalPeriod (f i) (x i) ∣ minimalPeriod (Pi.map f) x :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"59d633f642b6cb2164b6d1c43a7c438f0a2966b7435cb98112c3bf8684251005","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/PeriodicPts","family_id":"minimalperiod_single_dvd_minimalperiod_pimap","file_id":"mathlib/Mathlib/Dynamics/PeriodicPts/Lemmas.lean","sample_id":"5bfec5ae03bfd55ef1bbbff853e84f4a25242edd9b4ad0bd7f154c9852ef86aa"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3a259627f5a5b32fa98a969454c4436a807645c0fdfa0bcf717382246cd616fd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d2824e2c18d86e102371bf1923e5cd437621f7aa2725972b2459646211e58782","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"89ecc5dec26210b39010cde2e335b6d4cf0a4c1bf7eb4d3bd4ba691ff88b6aba","source_sha256":"f31e0ea5c5dcaaef4fe7a595be40e4dcc670dcec12bcae3dfce625de497aea8b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply Stream'.ext; intro n\n  cases n <;> rfl","hard_negative":false,"metrics":{"chosen_tokens":14,"rejected_tokens":19,"token_jaccard":0.705882,"token_length_ratio":1.357143},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"4608cd25c866a01d302c793bf0d5d65101cf3eb929904f973da70b8245e93bdd","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Stream.Defs\npublic import Mathlib.Logic.Function.Basic\npublic import Mathlib.Data.Nat.Basic\npublic import Mathlib.Tactic.Common\n\nNamespace:\nStream'\n\nLocal context:\n/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Streams a.k.a. infinite lists a.k.a. infinite sequences\n-/\n\n@[expose] public section\n\nopen Nat Function Option\n\nnamespace Stream'\n\nuniverse u v w\nvariable {α : Type u} {β : Type v} {δ : Type w}\nvariable (m n : ℕ) (x y : List α) (a b : Stream' α)\n\ninstance [Inhabited α] : Inhabited (Stream' α) :=\n  ⟨Stream'.const default⟩\n\n@[simp] protected theorem eta (s : Stream' α) : head s :: tail s = s :=\n  funext fun i => by cases i <;> rfl\n\n/-- Alias for `Stream'.eta` to match `List` API. -/\nalias cons_head_tail := Stream'.eta\n\n@[ext]\nprotected theorem ext {s₁ s₂ : Stream' α} : (∀ n, get s₁ n = get s₂ n) → s₁ = s₂ :=\n  fun h => funext h\n\n@[simp]\ntheorem get_zero_cons (a : α) (s : Stream' α) : get (a::s) 0 = a :=\n  rfl\n\n@[simp]\ntheorem head_cons (a : α) (s : Stream' α) : head (a::s) = a :=\n  rfl\n\n@[simp]\ntheorem tail_cons (a : α) (s : Stream' α) : tail (a::s) = s :=\n  rfl\n\n@[simp]\ntheorem get_drop (n m : ℕ) (s : Stream' α) : get (drop m s) n = get s (m + n) := by\n  rw [Nat.add_comm]\n  rfl\n\ntheorem tail_eq_drop (s : Stream' α) : tail s = drop 1 s :=\n  rfl\n\n@[simp]\ntheorem drop_drop (n m : ℕ) (s : Stream' α) : drop n (drop m s) = drop (m + n) s := by\n  ext; simp [Nat.add_assoc]\n\n@[simp] theorem get_tail {n : ℕ} {s : Stream' α} : s.tail.get n = s.get (n + 1) := rfl\n\n@[simp] theorem tail_drop' {i : ℕ} {s : Stream' α} : tail (drop i s) = s.drop (i + 1) := by\n  ext; simp [Nat.add_comm, Nat.add_left_comm]\n\n@[simp] theorem drop_tail' {i : ℕ} {s : Stream' α} : drop i (tail s) = s.drop (i + 1) := rfl\n\ntheorem tail_drop (n : ℕ) (s : Stream' α) : tail (drop n s) = drop n (tail s) := by simp\n\ntheorem get_succ (n : ℕ) (s : Stream' α) : get s (succ n) = get (tail s) n :=\n  rfl\n\n@[simp]\ntheorem get_succ_cons (n : ℕ) (s : Stream' α) (x : α) : get (x :: s) n.succ = get s n :=\n  rfl\n\n@[simp] lemma get_cons_append_zero {a : α} {x : List α} {s : Stream' α} :\n    (a :: x ++ₛ s).get 0 = a := rfl\n\n@[simp] lemma append_eq_cons {a : α} {as : Stream' α} : [a] ++ₛ as = a :: as := rfl\n\n@[simp] theorem drop_zero {s : Stream' α} : s.drop 0 = s := rfl\n\ntheorem drop_succ (n : ℕ) (s : Stream' α) : drop (succ n) s = drop n (tail s) :=\n  rfl\n\ntheorem head_drop (a : Stream' α) (n : ℕ) : (a.drop n).head = a.get n := by simp\n\ntheorem cons_injective2 : Function.Injective2 (cons : α → Stream' α → Stream' α) := fun x y s t h =>\n  ⟨by rw [← get_zero_cons x s, h, get_zero_cons],\n    Stream'.ext fun n => by rw [← get_succ_cons n _ x, h, get_succ_cons]⟩\n\ntheorem cons_injective_left (s : Stream' α) : Function.Injective fun x => cons x s :=\n  cons_injective2.left _\n\ntheorem cons_injective_right (x : α) : Function.Injective (cons x) :=\n  cons_injective2.right _\n\ntheorem all_def (p : α → Prop) (s : Stream' α) : All p s = ∀ n, p (get s n) :=\n  rfl\n\ntheorem any_def (p : α → Prop) (s : Stream' α) : Any p s = ∃ n, p (get s n) :=\n  rfl\n\n@[simp]\ntheorem mem_cons (a : α) (s : Stream' α) : a ∈ a::s :=\n  Exists.intro 0 rfl\n\ntheorem mem_cons_of_mem {a : α} {s : Stream' α} (b : α) : a ∈ s → a ∈ b::s := fun ⟨n, h⟩ =>\n  Exists.intro (succ n) (by rw [get_succ, tail_cons, h])\n\ntheorem eq_or_mem_of_mem_cons {a b : α} {s : Stream' α} : (a ∈ b::s) → a = b ∨ a ∈ s :=\n    fun ⟨n, h⟩ => by\n  rcases n with - | n'\n  · left\n    exact h\n  · right\n    rw [get_succ, tail_cons] at h\n    exact ⟨n', h⟩\n\ntheorem mem_of_get_eq {n : ℕ} {s : Stream' α} {a : α} : a = get s n → a ∈ s := fun h =>\n  Exists.intro n h\n\ntheorem mem_iff_exists_get_eq {s : Stream' α} {a : α} : a ∈ s ↔ ∃ n, a = s.get n where\n  mp := by simp [Membership.mem, any_def]\n  mpr h := mem_of_get_eq h.choose_spec\n\nsection Map\n\nvariable (f : α → β)\n\ntheorem drop_map (n : ℕ) (s : Stream' α) : drop n (map f s) = map f (drop n s) :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem get_map (n : ℕ) (s : Stream' α) : get (map f s) n = f (get s n) :=\n  rfl\n\ntheorem tail_map (s : Stream' α) : tail (map f s) = map f (tail s) := rfl\n\n@[simp]\ntheorem head_map (s : Stream' α) : head (map f s) = f (head s) :=\n  rfl\n\ntheorem map_eq (s : Stream' α) : map f s = f (head s)::map f (tail s) := by\n  rw [← Stream'.eta (map f s), tail_map, head_map]\n\ntheorem map_cons (a : α) (s : Stream' α) : map f (a::s) = f a::map f s := by\n  rw [← Stream'.eta (map f (a::s)), map_eq]; rfl\n\n@[simp]\ntheorem map_id (s : Stream' α) : map id s = s :=\n  rfl\n\n@[simp]\ntheorem map_map (g : β → δ) (f : α → β) (s : Stream' α) : map g (map f s) = map (g ∘ f) s :=\n  rfl\n\n@[simp]\ntheorem map_tail (s : Stream' α) : map f (tail s) = tail (map f s) :=\n  rfl\n\ntheorem mem_map {a : α} {s : Stream' α} : a ∈ s → f a ∈ map f s := fun ⟨n, h⟩ =>\n  Exists.intro n (by rw [get_map, h])\n\ntheorem exists_of_mem_map {f} {b : β} {s : Stream' α} : b ∈ map f s → ∃ a, a ∈ s ∧ f a = b :=\n  fun ⟨n, h⟩ => ⟨get s n, ⟨n, rfl⟩, h.symm⟩\n\nend Map\n\nsection Zip\n\nvariable (f : α → β → δ)\n\ntheorem drop_zip (n : ℕ) (s₁ : Stream' α) (s₂ : Stream' β) :\n    drop n (zip f s₁ s₂) = zip f (drop n s₁) (drop n s₂) :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem get_zip (n : ℕ) (s₁ : Stream' α) (s₂ : Stream' β) :\n    get (zip f s₁ s₂) n = f (get s₁ n) (get s₂ n) :=\n  rfl\n\ntheorem head_zip (s₁ : Stream' α) (s₂ : Stream' β) : head (zip f s₁ s₂) = f (head s₁) (head s₂) :=\n  rfl\n\ntheorem tail_zip (s₁ : Stream' α) (s₂ : Stream' β) :\n    tail (zip f s₁ s₂) = zip f (tail s₁) (tail s₂) :=\n  rfl\n\ntheorem zip_eq (s₁ : Stream' α) (s₂ : Stream' β) :\n    zip f s₁ s₂ = f (head s₁) (head s₂)::zip f (tail s₁) (tail s₂) := by\n  rw [← Stream'.eta (zip f s₁ s₂)]; rfl\n\n@[simp]\ntheorem get_enum (s : Stream' α) (n : ℕ) : get (enum s) n = (n, s.get n) :=\n  rfl\n\ntheorem enum_eq_zip (s : Stream' α) : enum s = zip Prod.mk nats s :=\n  rfl\n\nend Zip\n\n@[simp]\ntheorem mem_const (a : α) : a ∈ const a :=\n  Exists.intro 0 rfl\n\nTarget:\ntheorem const_eq (a : α) : const a = a::const a :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  apply Stream'.ext; intro n\n  cases n <;> rfl","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Stream","family_id":"const_eq","file_id":"mathlib/Mathlib/Data/Stream/Init.lean","sample_id":"89ecc5dec26210b39010cde2e335b6d4cf0a4c1bf7eb4d3bd4ba691ff88b6aba"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1f1ea0e13f9e5b99d77e0d90716326ac0d10ebd31e81aeaa4cd98f72f2f5c63c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5a66cefb33286413a5fb6e69c074a044e6555b7d91542cc621e7ce84e9aac9c7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"19249b12421b93ef9aca4c5c6690238801a0a01bd844a6075bfc5c80d4237b2d","source_sha256":"d0e82c6901f418bcc62ae1e69134f5914117f0eba42bc658092940528f8b500f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction n, hn using Nat.le_induction with\n  | base => simp\n  | succ k hk ih =>\n    have hck : 0 ≤ c k := hc k hk\n    have heq : c k * ∑ j ∈ Ico n₀ k, b j * ∏ i ∈ Ico (j + 1) k, c i + b k =\n        ∑ j ∈ Ico n₀ (k + 1), b j * ∏ i ∈ Ico (j + 1) (k + 1), c i := by\n      rw [sum_Ico_succ_top hk, mul_sum, Ico_self, prod_empty, mul_one]\n      refine congr_arg (· + b k) (sum_congr rfl fun j hj ↦ ?_)\n      rw [prod_Ico_succ_top (by have := mem_Ico.mp hj; omega)]; ring\n    calc u (k + 1)\n      _ ≤ c k * u k + b k := hu k hk\n      _ ≤ c k * (u n₀ * ∏ i ∈ Ico n₀ k, c i +\n            ∑ j ∈ Ico n₀ k, b j * ∏ i ∈ Ico (j + 1) k, c i) + b k := by gcongr\n      _ = u n₀ * ∏ i ∈ Ico n₀ (k + 1), c i +\n            ∑ j ∈ Ico n₀ (k + 1), b j * ∏ i ∈ Ico (j + 1) (k + 1), c i := by\n          rw [← heq, ← prod_Ico_mul_eq_prod_Ico_add_one hk]; ring","hard_negative":false,"metrics":{"chosen_tokens":286,"rejected_tokens":291,"token_jaccard":0.946667,"token_length_ratio":1.017483},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"462c157226fc5a39159b42cf3af0c16a56927ceac1aea7407a13612ed30bf77c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.ExpDeriv\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2026 Dennj Osele. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dennj Osele\n-/\n/-!\n# Discrete Grönwall inequality\n\nVarious forms of the discrete Grönwall inequality, bounding solutions to recurrence\ninequalities `u (n+1) ≤ c n * u n + b n` and `u (n+1) ≤ (1 + c n) * u n + b n`.\n\n## Main results\n\n* `discrete_gronwall_prod_general`: product form, over any ordered commutative semiring.\n* `discrete_gronwall`: classical exponential bound for the `(1 + c)` form, over `ℝ`.\n* `discrete_gronwall_Ico`: uniform bound over an interval, over `ℝ`.\n\n## References\n\n* [T. H. Grönwall, *Note on the derivatives with respect to a parameter of the solutions of a\n  system of differential equations*][Gronwall_1919]\n\n## See also\n\n* `Mathlib.Analysis.ODE.Gronwall` for the continuous Grönwall inequality for ODEs.\n-/\n\n@[expose] public section\n\nopen Real Finset\n\nsection General\n\n/-! ### Generalized product form -/\n\nvariable {R : Type*} [CommSemiring R] [PartialOrder R] [IsOrderedRing R] {u b c : ℕ → R}\n\n/-- Discrete Grönwall inequality, product form: if `u (n+1) ≤ c n * u n + b n` and `0 ≤ c n`\nthen `u n ≤ u n₀ * ∏ c i + ∑ b k * ∏ c i` over the appropriate ranges. -/\n\nTarget:\ntheorem discrete_gronwall_prod_general {n₀ : ℕ} (hu : ∀ n ≥ n₀, u (n + 1) ≤ c n * u n + b n)\n    (hc : ∀ n ≥ n₀, 0 ≤ c n) ⦃n : ℕ⦄ (hn : n₀ ≤ n) :\n    u n ≤ u n₀ * ∏ i ∈ Ico n₀ n, c i +\n      ∑ k ∈ Ico n₀ n, b k * ∏ i ∈ Ico (k + 1) n, c i :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  induction n, hn using Nat.le_induction with\n  | base => simp\n  | succ k hk ih =>\n    have hck : 0 ≤ c k := hc k hk\n    have heq : c k * ∑ j ∈ Ico n₀ k, b j * ∏ i ∈ Ico (j + 1) k, c i + b k =\n        ∑ j ∈ Ico n₀ (k + 1), b j * ∏ i ∈ Ico (j + 1) (k + 1), c i := by\n      rw [sum_Ico_succ_top hk, mul_sum, Ico_self, prod_empty, mul_one]\n      refine congr_arg (· + b k) (sum_congr rfl fun j hj ↦ ?_)\n      rw [prod_Ico_succ_top (by have := mem_Ico.mp hj; omega)]; ring\n    calc u (k + 1)\n      _ ≤ c k * u k + b k := hu k hk\n      _ ≤ c k * (u n₀ * ∏ i ∈ Ico n₀ k, c i +\n            ∑ j ∈ Ico n₀ k, b j * ∏ i ∈ Ico (j + 1) k, c i) + b k := by gcongr\n      _ = u n₀ * ∏ i ∈ Ico n₀ (k + 1), c i +\n            ∑ j ∈ Ico n₀ (k + 1), b j * ∏ i ∈ Ico (j + 1) (k + 1), c i := by\n          rw [← heq, ← prod_Ico_mul_eq_prod_Ico_add_one hk]; ring","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/ODE","family_id":"discrete_gronwall_prod_general","file_id":"mathlib/Mathlib/Analysis/ODE/DiscreteGronwall.lean","sample_id":"19249b12421b93ef9aca4c5c6690238801a0a01bd844a6075bfc5c80d4237b2d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ddb596461bd0c4bd75d4fb0cc2c7a82f17820be10b8f87f0c46c723e2b555f0f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fe21963d2aeeba6cb61599867ca841b158d2155ae815d66b4331eaa8e5ce4e3b","source_sha256":"018e0555cbbfa42b8d2c4f98114f5ffea5e65210561d826e089516f6ee1b9864","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hg : DifferentiableAt ℂ (fun z ↦ log (1 - z)) 0 := by\n    have : 1 - 0 ∈ slitPlane := (sub_zero (1 : ℂ)).symm ▸ one_mem_slitPlane\n    fun_prop\n  have : (fun z ↦ log (1 - z)) =O[𝓝 0] id := by\n    simpa only [sub_zero, log_one] using! hg.isBigO_sub\n  exact this.comp_summable hsum","hard_negative":false,"metrics":{"chosen_tokens":80,"rejected_tokens":5,"token_jaccard":0.045455,"token_length_ratio":0.0625},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"463c792303982d2516b86bdebd3033e45a5fb0ca38bc68da9b73b19dad89415a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Module.FiniteDimension\npublic import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv\npublic import Mathlib.LinearAlgebra.Complex.FiniteDimensional\npublic import Mathlib.NumberTheory.EulerProduct.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Logarithms of Euler Products\n\nWe consider `f : ℕ →*₀ ℂ` and show that `exp (∑ p in Primes, log (1 - f p)⁻¹) = ∑ n : ℕ, f n`\nunder suitable conditions on `f`. This can be seen as a logarithmic version of the\nEuler product for `f`.\n-/\n\npublic section\n\nopen Complex\n\nopen Topology in\n/-- If `f : α → ℂ` is summable, then so is `n ↦ log (1 - f n)`. -/\n\nTarget:\nlemma Summable.clog_one_sub {α : Type*} {f : α → ℂ} (hsum : Summable f) :\n    Summable fun n ↦ log (1 - f n) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/EulerProduct","family_id":"summable","file_id":"mathlib/Mathlib/NumberTheory/EulerProduct/ExpLog.lean","sample_id":"fe21963d2aeeba6cb61599867ca841b158d2155ae815d66b4331eaa8e5ce4e3b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0d4d6dccf2c84c2f8c5355bc4fd7dec43e2d86ae85215b172de94088a2f48bcb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f874782215c2982072ce26a83604cb1ed65939016dddaf29babaa9fd388a56ea","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dd8a4792fd8238e946d75aea51e934ebc1fab1139603667bd836f2fb5b0acf87","source_sha256":"62aebf63d41c64c52529eefd998601766828e0e4d42e1874f5417f5225e112d0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x\n  simp only [spectrum.mem_iff, isUnit_iff_forall_ne_zero, not_forall, sub_apply,\n    Classical.not_not, Set.mem_range,\n    sub_eq_zero, @eq_comm _ (x • 1 : R) _, Set.mem_preimage, Algebra.algebraMap_eq_smul_one,\n    smul_apply, one_apply]","hard_negative":false,"metrics":{"chosen_tokens":50,"rejected_tokens":55,"token_jaccard":0.891892,"token_length_ratio":1.1},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"466b04cf1d13f2e2ae9b5dd42fe730941d7c173793514d7917691485e8fa6fbf","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Ring.Units\npublic import Mathlib.Algebra.Algebra.Spectrum.Basic\npublic import Mathlib.Topology.ContinuousMap.Algebra\n\nNamespace:\nContinuousMap\n\nLocal context:\n/-\nCopyright (c) 2022 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\n/-!\n# Units of continuous functions\n\nThis file concerns itself with `C(X, M)ˣ` and `C(X, Mˣ)` when `X` is a topological space\nand `M` has some monoid structure compatible with its topology.\n-/\n\n@[expose] public section\n\n\nvariable {X M R 𝕜 : Type*} [TopologicalSpace X]\n\nnamespace ContinuousMap\n\nsection Monoid\n\nvariable [Monoid M] [TopologicalSpace M] [ContinuousMul M]\n\n/-- Equivalence between continuous maps into the units of a monoid with continuous multiplication\nand the units of the monoid of continuous maps. -/\n-- `simps` generates some lemmas here with LHS not in simp normal form,\n-- so we write them out manually below.\n@[to_additive (attr := simps apply_val_apply symm_apply_apply_val)\n/-- Equivalence between continuous maps into the additive units of an additive monoid with\ncontinuous addition and the additive units of the additive monoid of continuous maps. -/]\ndef unitsLift : C(X, Mˣ) ≃ C(X, M)ˣ where\n  toFun f :=\n    { val := ⟨fun x => f x, Units.continuous_val.comp f.continuous⟩\n      inv := ⟨fun x => ↑(f x)⁻¹, Units.continuous_val.comp (continuous_inv.comp f.continuous)⟩\n      val_inv := ext fun _ => Units.mul_inv _\n      inv_val := ext fun _ => Units.inv_mul _ }\n  invFun f :=\n    { toFun := fun x =>\n        ⟨(f : C(X, M)) x, (↑f⁻¹ : C(X, M)) x,\n          ContinuousMap.congr_fun f.mul_inv x, ContinuousMap.congr_fun f.inv_mul x⟩\n      continuous_toFun := continuous_induced_rng.2 <|\n        (f : C(X, M)).continuous.prodMk <|\n        MulOpposite.continuous_op.comp (↑f⁻¹ : C(X, M)).continuous }\n\n@[to_additive (attr := simp)]\nlemma unitsLift_apply_inv_apply (f : C(X, Mˣ)) (x : X) :\n    (↑(ContinuousMap.unitsLift f)⁻¹ : C(X, M)) x = (f x)⁻¹ :=\n  rfl\n\n@[to_additive (attr := simp)]\nlemma unitsLift_symm_apply_apply_inv' (f : C(X, M)ˣ) (x : X) :\n    (ContinuousMap.unitsLift.symm f x)⁻¹ = (↑f⁻¹ : C(X, M)) x := by\n  rfl\n\nend Monoid\n\nsection NormedRing\n\nvariable [NormedRing R] [CompleteSpace R]\n\ntheorem continuous_isUnit_unit {f : C(X, R)} (h : ∀ x, IsUnit (f x)) :\n    Continuous fun x => (h x).unit := by\n  refine\n    continuous_induced_rng.2\n      (Continuous.prodMk f.continuous\n        (MulOpposite.continuous_op.comp (continuous_iff_continuousAt.mpr fun x => ?_)))\n  have := NormedRing.inverse_continuousAt (h x).unit\n  simp only\n  simp only [← Ring.inverse_unit, IsUnit.unit_spec] at this ⊢\n  exact this.comp (f.continuousAt x)\n\n/-- Construct a continuous map into the group of units of a normed ring from a function into the\nnormed ring and a proof that every element of the range is a unit. -/\n@[simps]\nnoncomputable def unitsOfForallIsUnit {f : C(X, R)} (h : ∀ x, IsUnit (f x)) : C(X, Rˣ) where\n  toFun x := (h x).unit\n  continuous_toFun := continuous_isUnit_unit h\n\ninstance canLift :\n    CanLift C(X, R) C(X, Rˣ) (fun f => ⟨fun x => f x, Units.continuous_val.comp f.continuous⟩)\n      fun f => ∀ x, IsUnit (f x) where\n  prf f h := ⟨unitsOfForallIsUnit h, by ext; rfl⟩\n\ntheorem isUnit_iff_forall_isUnit (f : C(X, R)) : IsUnit f ↔ ∀ x, IsUnit (f x) :=\n  Iff.intro (fun h => fun x => ⟨unitsLift.symm h.unit x, rfl⟩) fun h =>\n    ⟨ContinuousMap.unitsLift (unitsOfForallIsUnit h), by ext; rfl⟩\n\nend NormedRing\n\nsection NormedField\n\nvariable [NormedField 𝕜] [NormedDivisionRing R] [Algebra 𝕜 R] [CompleteSpace R]\n\ntheorem isUnit_iff_forall_ne_zero (f : C(X, R)) : IsUnit f ↔ ∀ x, f x ≠ 0 := by\n  simp_rw [f.isUnit_iff_forall_isUnit, isUnit_iff_ne_zero]\n\nTarget:\ntheorem spectrum_eq_preimage_range (f : C(X, R)) :\n    spectrum 𝕜 f = algebraMap _ _ ⁻¹' Set.range f :=\n\nProof body:\n","rejected":"by\n  ext x\n  simp only [spectrum.mem_iff, isUnit_iff_forall_ne_zero, not_forall, sub_apply,\n    Classical.not_not, Set.mem_range,\n    sub_eq_zero, @eq_comm _ (x • 1 : R) _, Set.mem_preimage, Algebra.algebraMap_eq_smul_one,\n    smul_apply, one_apply]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/ContinuousMap","family_id":"spectrum_eq_preimage_range","file_id":"mathlib/Mathlib/Topology/ContinuousMap/Units.lean","sample_id":"dd8a4792fd8238e946d75aea51e934ebc1fab1139603667bd836f2fb5b0acf87"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fd3b7a2ce076a9542b33ce25f06638c50f65a2500253da07e8afbc3c349b995c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"39df060b78e1e695bb13dca8ed0e62daa81eb218aab7135ea5abb7ca1ce46c9c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f2813c1994d0014dcc0cbbe6ef4d234abd05fcb0aaba7d50c3b4450586130664","source_sha256":"c60b7992efa7a8848883c62ac5e7382d125b9535f551900578b8595483f08374","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction m with\n  | zero => simpa\n  | succ m ih =>\n    simp_rw [pow_succ _ m, pow_mul]\n    exact ih.pow_add_one hpI m.succ_ne_zero","hard_negative":true,"metrics":{"chosen_tokens":29,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.068966},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"468c36fbb6b109c4d1e4335a3a0a063208dbd6dfd028ef8a60550f5a0ceeb992","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharP.Lemmas\npublic import Mathlib.LinearAlgebra.SModEq.Basic\npublic import Mathlib.RingTheory.Ideal.Operations\n\nNamespace:\nSModEq\n\nLocal context:\n/-\nCopyright (c) 2026 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n\n/-! # Lemmas about SModEq related to powers -/\n\npublic section\n\nnamespace SModEq\nvariable {R : Type*} [CommRing R] {I J : Ideal R} {p : ℕ} (hpI : (p : R) ∈ I)\ninclude hpI\n\ntheorem pow_mul_of_le {x y : R} (h : x ≡ y [SMOD J]) (hJI : J ≤ I) :\n    x ^ p ≡ y ^ p [SMOD J * I] := by\n  have h₁ := idealQuotientMk.mp <| h.mono hJI\n  rw [SModEq.sub_mem] at h ⊢\n  rw [← Commute.mul_neg_geom_sum₂ (.all _ _)]\n  refine Ideal.mul_mem_mul h ?_\n  have h₂ : (p : R ⧸ I) = 0 := by simpa using Ideal.Quotient.eq_zero_iff_mem.mpr hpI\n  simp only [← Ideal.Quotient.eq_zero_iff_mem, map_sum, map_mul, map_pow, h₁, ← pow_add]\n  trans ∑ x ∈ Finset.range p, Ideal.Quotient.mk I y ^ (p - 1)\n  · exact Finset.sum_congr rfl fun _ _ ↦ by grind\n  simp [h₂]\n\ntheorem pow_add_one {x y : R} {m : ℕ} (hm : m ≠ 0) (h : x ≡ y [SMOD I ^ m]) :\n    x ^ p ≡ y ^ p [SMOD I ^ (m + 1)] := h.pow_mul_of_le hpI <| I.pow_le_self hm\n\nTarget:\ntheorem pow_pow_add_one {x y : R} (h : x ≡ y [SMOD I]) (m : ℕ) :\n    x ^ p ^ m ≡ y ^ p ^ m [SMOD I ^ (m + 1)] :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_f2813c1994d0","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"22f31c6c6cb622fcceff3b4a2289445422846e31fdcc6eca71ab63a9d7712d14","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/SModEq","family_id":"pow_pow_add_one","file_id":"mathlib/Mathlib/LinearAlgebra/SModEq/Pow.lean","sample_id":"f2813c1994d0014dcc0cbbe6ef4d234abd05fcb0aaba7d50c3b4450586130664"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4967cf04180cb0748d25ecdf64a662a7f6a1d6bdd998e61ce6f3d7d2c6f082a5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"756685a57c1c40f2d410134b5fc2580aacd3ec04b425aa2b56ef0fc77bf58bf8","source_sha256":"39dfdad0e4eff3ed2f02c59604c6ff92849183f69003baf79902622224068963","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← map_add]\n  exact m.toSyn_monotone le_self_add","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.181818},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"4699d4773f8622959c8e22375a66dcf62b79560afe2577aab0a229e17b6e9796","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finsupp.Lex\npublic import Mathlib.Data.Finsupp.WellFounded\npublic import Mathlib.Data.List.TFAE\n\nNamespace:\nMonomialOrder\n\nLocal context:\n/-\nCopyright (c) 2024 Antoine Chambert-Loir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir\n-/\n/-! # Monomial orders\n\n## Monomial orders\n\nA *monomial order* is well ordering relation on a type of the form `σ →₀ ℕ` which\nis compatible with addition and for which `0` is the smallest element.\nSince several monomial orders may have to be used simultaneously, one cannot\nget them as instances.\n\nIn this formalization, they are presented as a structure `MonomialOrder` which encapsulates\n`MonomialOrder.toSyn`, an additive and monotone isomorphism to a linearly ordered cancellative\nadditive commutative monoid.\nThe entry `MonomialOrder.wf` asserts that `MonomialOrder.syn` is well founded.\n\nThe terminology comes from commutative algebra and algebraic geometry, especially Gröbner bases,\nwhere `c : σ →₀ ℕ` are exponents of monomials.\n\nGiven a monomial order `m : MonomialOrder σ`, we provide the notation\n`c ≼[m] d` and `c ≺[m] d` to compare `c d : σ →₀ ℕ` with respect to `m`.\nIt is activated using `open scoped MonomialOrder`.\n\n## Examples\n\nCommutative algebra defines many monomial orders, with different usefulness ranges.\nIn this file, we provide the basic example of lexicographic ordering.\nFor the graded lexicographic ordering, see `Mathlib/Data/Finsupp/MonomialOrder/DegLex.lean`\n\n* `MonomialOrder.lex` : the lexicographic ordering on `σ →₀ ℕ`.\n  For this, `σ` needs to be embedded with an ordering relation which satisfies `WellFoundedGT σ`.\n  (This last property is automatic when `σ` is finite).\n\nThe type synonym is `Lex (σ →₀ ℕ)` and the two lemmas `MonomialOrder.lex_le_iff`\nand `MonomialOrder.lex_lt_iff` rewrite the ordering as comparisons in the type `Lex (σ →₀ ℕ)`.\n\n## References\n\n* [Cox, Little and O'Shea, *Ideals, varieties, and algorithms*][coxlittleoshea1997]\n* [Becker and Weispfenning, *Gröbner bases*][Becker-Weispfenning1993]\n\n## Note\n\nIn algebraic geometry, when the finitely many variables are indexed by integers,\nit is customary to order them using the opposite order : `MvPolynomial.X 0 > MvPolynomial.X 1 > … `\n\n-/\n\n@[expose] public section\n\n/-- Monomial orders : equivalence of `σ →₀ ℕ` with a well-ordered type -/\nstructure MonomialOrder (σ : Type*) where\n  /-- The synonym type -/\n  syn : Type*\n  /-- `syn` is an additive commutative monoid -/\n  acm : AddCommMonoid syn := by infer_instance\n  /-- `syn` is linearly ordered -/\n  lo : LinearOrder syn := by infer_instance\n  /-- `syn` is a linearly ordered cancellative additive commutative monoid -/\n  iocam : IsOrderedCancelAddMonoid syn := by infer_instance\n  /-- the additive equivalence from `σ →₀ ℕ` to `syn` -/\n  toSyn : (σ →₀ ℕ) ≃+ syn\n  /-- `toSyn` is monotone -/\n  toSyn_monotone : Monotone toSyn\n  /-- `syn` is a well ordering -/\n  wf : WellFoundedLT syn := by infer_instance\n\nattribute [instance] MonomialOrder.acm MonomialOrder.lo MonomialOrder.iocam MonomialOrder.wf\n\nnamespace MonomialOrder\n\nvariable {σ : Type*} (m : MonomialOrder σ)\n\nTarget:\nlemma le_add_right (a b : σ →₀ ℕ) :\n    m.toSyn a ≤ m.toSyn a + m.toSyn b :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finsupp","family_id":"le_add_right","file_id":"mathlib/Mathlib/Data/Finsupp/MonomialOrder.lean","sample_id":"756685a57c1c40f2d410134b5fc2580aacd3ec04b425aa2b56ef0fc77bf58bf8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f11fe3945a9851f302dfa7c95843f91ea71198e392581ee0ee3ba699adf230d7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6f236f3198461aabd72d37b97a5e2b5e82970cee4f352355afa3f62c491f1499","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9757f7a78bda71d8e7517609a4d26326ae1122e116d5aaf449c4d98b5651cdcf","source_sha256":"b7c2e561541c6c2f2a91b3c7b8f53597d9188114f47fe81a231097a061f20b60","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [toPrincipalIdeal]; exact Units.ext_iff","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.166667,"token_length_ratio":0.272727},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"47916be85ef61c5281f8a87f23e798534748b12a7ec195b21ae3ebb2aa6a9930","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.Ideal.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# The ideal class group\n\nThis file defines the ideal class group `ClassGroup R` of fractional ideals of `R`\ninside its field of fractions.\n\n## Main definitions\n\n- `toPrincipalIdeal` sends an invertible `x : K` to an invertible fractional ideal\n- `ClassGroup` is the quotient of invertible fractional ideals modulo `toPrincipalIdeal.range`\n- `ClassGroup.mk0` sends a nonzero integral ideal in a Dedekind domain to its class\n\n## Main results\n- `ClassGroup.mk0_eq_mk0_iff` shows the equivalence with the \"classical\" definition,\n  where `I ~ J` iff `x I = y J` for `x y ≠ (0 : R)`\n- `ClassGroup.mk0_eq_quotientMk` identifies `ClassGroup.mk0` with the canonical quotient\n  projection on `(FractionalIdeal R⁰ (FractionRing R))ˣ`\n\n## Implementation details\n\nThe definition of `ClassGroup R` involves `FractionRing R`. However, the API should be completely\nidentical no matter the choice of field of fractions for `R`.\n-/\n\n@[expose] public noncomputable section\n\n\nvariable {R K : Type*} [CommRing R] [Field K] [Algebra R K] [IsFractionRing R K]\n\nopen scoped nonZeroDivisors\n\nopen IsLocalization IsFractionRing FractionalIdeal Units\n\nsection\n\nvariable (R K)\n\n/-- `toPrincipalIdeal R K x` sends `x ≠ 0 : K` to the fractional `R`-ideal generated by `x` -/\nirreducible_def toPrincipalIdeal : Kˣ →* (FractionalIdeal R⁰ K)ˣ :=\n  { toFun := fun x =>\n      ⟨spanSingleton _ x, spanSingleton _ x⁻¹, by\n        simp only [spanSingleton_one, Units.mul_inv', spanSingleton_mul_spanSingleton], by\n        simp only [spanSingleton_one, Units.inv_mul', spanSingleton_mul_spanSingleton]⟩\n    map_mul' := fun x y =>\n      ext (by simp only [Units.val_mul, spanSingleton_mul_spanSingleton])\n    map_one' := ext (by simp only [spanSingleton_one, Units.val_one]) }\n\nvariable {R K}\n\n@[simp]\ntheorem coe_toPrincipalIdeal (x : Kˣ) :\n    (toPrincipalIdeal R K x : FractionalIdeal R⁰ K) = spanSingleton _ (x : K) := by\n  simp only [toPrincipalIdeal]; rfl\n\n@[simp]\n\nTarget:\ntheorem toPrincipalIdeal_eq_iff {I : (FractionalIdeal R⁰ K)ˣ} {x : Kˣ} :\n    toPrincipalIdeal R K x = I ↔ spanSingleton R⁰ (x : K) = I :=\n\nProof body:\n","rejected":"by\n  exact toPrincipalIdeal_eq_iff","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"61c619d88b56c6afe0decbceed7ed363ee114dda3ea91469b8f148ba3d4099fd","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/ClassGroup","family_id":"toprincipalideal_eq_iff","file_id":"mathlib/Mathlib/RingTheory/ClassGroup/Basic.lean","sample_id":"9757f7a78bda71d8e7517609a4d26326ae1122e116d5aaf449c4d98b5651cdcf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7605cac82936a34e51b347b493387fd67f6c60e4f590ba9df4da446fc48c38a4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"059486db01226e7fb2c9a1e273235fb9255a92436306e27b0e9d7e56bd1eefb6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1e9f94a5050c3f631d6ef4550fc4148994113fabe92f719d1a84b2cac8998b0b","source_sha256":"74b6571c8b4ffa69041159c6a44858bd2f705cc61f45fce1f1c46bf3161b1e36","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let l := hi.lift ((pullback.fst αX f) ≫ h') ((pullback.snd αX f) ≫ g)\n    (by simp only [Category.assoc, hh.toCommSq.w, pullback.condition_assoc, ← H.w])\n  use (pullback αX f), (pullback.fst αX f), l, (pullback.snd αX f)\n  refine ⟨IsPullback.of_hasPullback αX f, ?_, ?_⟩\n  · refine IsPullback.of_right' ?_ hi\n    rw [← H.w]\n    exact IsPullback.paste_horiz (IsPullback.of_hasPullback αX f) hh\n  · refine (H' (pullback.fst αX f) l h' i' (pullback.snd αX f) αX αY αZ\n      (IsPullback.of_hasPullback αX f) ?_\n        hh.toCommSq hi.toCommSq ⟨by simp only [IsPullback.lift_fst, l]⟩).2 ⟨hh, hi⟩\n    · refine IsPullback.of_right' ?_ hi\n      rw [← H.w]\n      exact IsPullback.paste_horiz (IsPullback.of_hasPullback αX f) hh","hard_negative":true,"metrics":{"chosen_tokens":219,"rejected_tokens":3,"token_jaccard":0.039216,"token_length_ratio":0.013699},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"47c8bd73ad0c3e992b5d632b5a1a0ca97d22bfec50099a1c647c645010c1f6b1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Extensive\npublic import Mathlib.CategoryTheory.Limits.Shapes.KernelPair\npublic import Mathlib.CategoryTheory.Limits.Constructions.EpiMono\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang, Jack McKoen\n-/\n/-!\n\n# Adhesive categories\n\n## Main definitions\n- `CategoryTheory.IsPushout.IsVanKampen`: A convenience formulation for a pushout being\n  a van Kampen colimit.\n- `CategoryTheory.Adhesive`: A category is adhesive if it has pushouts and pullbacks along\n  monomorphisms, and such pushouts are van Kampen.\n\n## Main Results\n- `CategoryTheory.Type.adhesive`: The category of `Type` is adhesive.\n- `CategoryTheory.Adhesive.isPullback_of_isPushout_of_mono_left`: In adhesive categories,\n  pushouts along monomorphisms are pullbacks.\n- `CategoryTheory.Adhesive.mono_of_isPushout_of_mono_left`: In adhesive categories,\n  monomorphisms are stable under pushouts.\n- `CategoryTheory.Adhesive.toRegularMonoCategory`: Monomorphisms in adhesive categories are\n  regular (this implies that adhesive categories are balanced).\n- `CategoryTheory.adhesive_functor`: The category `C ⥤ D` is adhesive if `D`\n  has all pullbacks and all pushouts and is adhesive\n\n## References\n- https://ncatlab.org/nlab/show/adhesive+category\n- [Stephen Lack and Paweł Sobociński, Adhesive Categories][adhesive2004]\n\n-/\n\n@[expose] public section\n\n\nnamespace CategoryTheory\n\nopen Limits\n\nuniverse v' u' v u\n\nvariable {J : Type v'} [Category.{u'} J] {C : Type u} [Category.{v} C]\nvariable {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z}\n\n-- This only makes sense when the original diagram is a pushout.\n/-- A convenient formulation for a pushout being a van Kampen colimit. For any commutative cube of\nwhich a van Kampen pushout forms the bottom face and the back faces are pullbacks, the front faces\nare pullbacks if and only if the top face is a pushout. See `IsPushout.isVanKampen_iff` below. -/\n@[nolint unusedArguments]\ndef IsPushout.IsVanKampen (_ : IsPushout f g h i) : Prop :=\n  ∀ ⦃W' X' Y' Z' : C⦄ (f' : W' ⟶ X') (g' : W' ⟶ Y') (h' : X' ⟶ Z') (i' : Y' ⟶ Z') (αW : W' ⟶ W)\n    (αX : X' ⟶ X) (αY : Y' ⟶ Y) (αZ : Z' ⟶ Z) (_ : IsPullback f' αW αX f)\n    (_ : IsPullback g' αW αY g) (_ : CommSq h' αX αZ h) (_ : CommSq i' αY αZ i)\n    (_ : CommSq f' g' h' i'), IsPushout f' g' h' i' ↔ IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i\n\n/-- If a van Kampen pushout forms the bottom face of a commutative \"half-cube\" whose front faces\nare pullbacks, then there exist two back faces which are pullbacks and a top face which is a\npushout. -/\n\nTarget:\nlemma IsPushout.IsVanKampen.exists_cube_filling {H : IsPushout f g h i} (H' : H.IsVanKampen)\n    {X' Y' Z' : C} {h' : X' ⟶ Z'} {i' : Y' ⟶ Z'} {αX : X' ⟶ X} {αY : Y' ⟶ Y} {αZ : Z' ⟶ Z}\n    [HasPullback αX f] (hh : IsPullback h' αX αZ h) (hi : IsPullback i' αY αZ i) :\n    ∃ (W' : C) (f' : W' ⟶ X') (g' : W' ⟶ Y') (αW : W' ⟶ W),\n      IsPullback f' αW αX f ∧ IsPullback g' αW αY g ∧ IsPushout f' g' h' i' :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_1e9f94a5050c","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"0b627c7602bf1ad0488ae8c863103641b650ddaf0c810931559c4a71ff85535d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Adhesive","family_id":"ispushout","file_id":"mathlib/Mathlib/CategoryTheory/Adhesive/Basic.lean","sample_id":"1e9f94a5050c3f631d6ef4550fc4148994113fabe92f719d1a84b2cac8998b0b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f4514d3004c77d49e14e5cb3065d22f4527708fcd18c9c1d909c571ef396f6f1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e96a9f2e6f2cd142ce46e8c5c7977a2fb22f0ea1c0daabf15e8cbf9d7c647459","source_sha256":"56c135165b3ce5c6f5239e9a81f21fe0754d5a166fa0d32c24c14c3224af6216","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨n, hn⟩ := hnp\n  obtain ⟨k, hk⟩ := IsNilpotent.nilpotent R I M\n  have hk' : I.lcs M k = ⊥ := by\n    simp only [← toSubmodule_inj, I.coe_lcs_eq, hk, bot_toSubmodule]\n  suffices ∀ l, lowerCentralSeries R L M (l * n) ≤ I.lcs M l by\n    rw [isNilpotent_iff R]\n    use k * n\n    simpa [hk'] using this k\n  intro l\n  induction l with\n  | zero => simp\n  | succ l ih => exact (l.succ_mul n).symm ▸ lcs_le_lcs_of_is_nilpotent_span_sup_eq_top hxI hn ih","hard_negative":false,"metrics":{"chosen_tokens":112,"rejected_tokens":3,"token_jaccard":0.032787,"token_length_ratio":0.026786},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"47fc992b83f050545243413d13ccb7226de7fc1e73009880c9e8461a7c0e2255","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Lie.AdjointAction.Basic\npublic import Mathlib.Algebra.Lie.Nilpotent\npublic import Mathlib.Algebra.Lie.Normalizer\n\nNamespace:\nLieSubmodule\n\nLocal context:\n/-\nCopyright (c) 2022 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Engel's theorem\n\nThis file contains a proof of Engel's theorem providing necessary and sufficient conditions for Lie\nalgebras and Lie modules to be nilpotent.\n\nThe key result `LieModule.isNilpotent_iff_forall` says that if `M` is a Lie module of a\nNoetherian Lie algebra `L`, then `M` is nilpotent iff the image of `L → End(M)` consists of\nnilpotent elements. In the special case that we have the adjoint representation `M = L`, this says\nthat a Lie algebra is nilpotent iff `ad x : End(L)` is nilpotent for all `x : L`.\n\nEngel's theorem is true for any coefficients (i.e., it is really a theorem about Lie rings) and so\nwe work with coefficients in any commutative ring `R` throughout.\n\nOn the other hand, Engel's theorem is not true for infinite-dimensional Lie algebras and so a\nfinite-dimensionality assumption is required. We prove the theorem subject to the assumption\nthat the Lie algebra is Noetherian as an `R`-module, though actually we only need the slightly\nweaker property that the relation `>` is well-founded on the complete lattice of Lie subalgebras.\n\n## Remarks about the proof\n\nEngel's theorem is usually proved in the special case that the coefficients are a field, and uses\nan inductive argument on the dimension of the Lie algebra. One begins by choosing either a maximal\nproper Lie subalgebra (in some proofs) or a maximal nilpotent Lie subalgebra (in other proofs, at\nthe cost of obtaining a weaker end result).\n\nSince we work with general coefficients, we cannot induct on dimension and an alternate approach\nmust be taken. The key ingredient is the concept of nilpotency, not just for Lie algebras, but for\nLie modules. Using this concept, we define an _Engelian Lie algebra_ `LieAlgebra.IsEngelian` to\nbe one for which a Lie module is nilpotent whenever the action consists of nilpotent endomorphisms.\nThe argument then proceeds by selecting a maximal Engelian Lie subalgebra and showing that it cannot\nbe proper.\n\nThe first part of the traditional statement of Engel's theorem consists of the statement that if `M`\nis a non-trivial `R`-module and `L ⊆ End(M)` is a finite-dimensional Lie subalgebra of nilpotent\nelements, then there exists a non-zero element `m : M` that is annihilated by every element of `L`.\nThis follows trivially from the result established here `LieModule.isNilpotent_iff_forall`, that\n`M` is a nilpotent Lie module over `L`, since the last non-zero term in the lower central series\nwill consist of such elements `m` (see: `LieModule.nontrivial_max_triv_of_isNilpotent`). It seems\nthat this result has not previously been established at this level of generality.\n\nThe second part of the traditional statement of Engel's theorem concerns nilpotency of the Lie\nalgebra and a proof of this for general coefficients appeared in the literature as long ago\n[as 1937](zorn1937). This also follows trivially from `LieModule.isNilpotent_iff_forall` simply by\ntaking `M = L`.\n\nIt is pleasing that the two parts of the traditional statements of Engel's theorem are thus unified\ninto a single statement about nilpotency of Lie modules. This is not usually emphasised.\n\n## Main definitions\n\n  * `LieAlgebra.IsEngelian`\n  * `LieAlgebra.isEngelian_of_isNoetherian`\n  * `LieModule.isNilpotent_iff_forall`\n  * `LieAlgebra.isNilpotent_iff_forall`\n\n-/\n\n@[expose] public section\n\n\nuniverse u₁ u₂ u₃ u₄\n\nvariable {R : Type u₁} {L : Type u₂} {L₂ : Type u₃} {M : Type u₄}\nvariable [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L₂] [LieAlgebra R L₂]\nvariable [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]\n\nnamespace LieSubmodule\n\nopen LieModule\n\nvariable {I : LieIdeal R L} {x : L} (hxI : R ∙ x ⊔ I = ⊤)\ninclude hxI\n\ntheorem exists_smul_add_of_span_sup_eq_top (y : L) : ∃ t : R, ∃ z ∈ I, y = t • x + z := by\n  have hy : y ∈ (⊤ : Submodule R L) := Submodule.mem_top\n  simp only [← hxI, Submodule.mem_sup, Submodule.mem_span_singleton] at hy\n  obtain ⟨-, ⟨t, rfl⟩, z, hz, rfl⟩ := hy\n  exact ⟨t, z, hz, rfl⟩\n\ntheorem lie_top_eq_of_span_sup_eq_top (N : LieSubmodule R L M) :\n    (↑⁅(⊤ : LieIdeal R L), N⁆ : Submodule R M) =\n      (N : Submodule R M).map (toEnd R L M x) ⊔ (↑⁅I, N⁆ : Submodule R M) := by\n  simp only [lieIdeal_oper_eq_linear_span', Submodule.sup_span, mem_top, true_and,\n    Submodule.map_coe, toEnd_apply_apply]\n  refine le_antisymm (Submodule.span_le.mpr ?_) (Submodule.span_mono fun z hz => ?_)\n  · rintro z ⟨y, n, hn : n ∈ N, rfl⟩\n    obtain ⟨t, z, hz, rfl⟩ := exists_smul_add_of_span_sup_eq_top hxI y\n    simp only [SetLike.mem_coe, Submodule.span_union, Submodule.mem_sup]\n    exact\n      ⟨t • ⁅x, n⁆, Submodule.subset_span ⟨t • n, N.smul_mem' t hn, lie_smul t x n⟩, ⁅z, n⁆,\n        Submodule.subset_span ⟨z, hz, n, hn, rfl⟩, by simp⟩\n  · rcases hz with (⟨m, hm, rfl⟩ | ⟨y, -, m, hm, rfl⟩)\n    exacts [⟨x, m, hm, rfl⟩, ⟨y, m, hm, rfl⟩]\n\ntheorem lcs_le_lcs_of_is_nilpotent_span_sup_eq_top {n i j : ℕ}\n    (hxn : toEnd R L M x ^ n = 0) (hIM : lowerCentralSeries R L M i ≤ I.lcs M j) :\n    lowerCentralSeries R L M (i + n) ≤ I.lcs M (j + 1) := by\n  suffices\n    ∀ l,\n      ((⊤ : LieIdeal R L).lcs M (i + l) : Submodule R M) ≤\n        (I.lcs M j : Submodule R M).map (toEnd R L M x ^ l) ⊔\n          (I.lcs M (j + 1) : Submodule R M)\n    by simpa only [bot_sup_eq, LieIdeal.incl_coe, Submodule.map_zero, hxn] using! this n\n  intro l\n  induction l with\n  | zero =>\n    simp only [add_zero, LieIdeal.lcs_succ, pow_zero, Module.End.one_eq_id,\n      Submodule.map_id]\n    exact le_sup_of_le_left hIM\n  | succ l ih =>\n    simp only [LieIdeal.lcs_succ, i.add_succ l, lie_top_eq_of_span_sup_eq_top hxI, sup_le_iff]\n    refine ⟨(Submodule.map_mono ih).trans ?_, le_sup_of_le_right ?_⟩\n    · rw [Submodule.map_sup, ← Submodule.map_comp, ← Module.End.mul_eq_comp, ← pow_succ', ←\n        I.lcs_succ]\n      grw [coe_map_toEnd_le]\n    · norm_cast\n      gcongr\n      exact le_trans (antitone_lowerCentralSeries R L M le_self_add) hIM\n\nTarget:\ntheorem isNilpotentOfIsNilpotentSpanSupEqTop (hnp : IsNilpotent <| toEnd R L M x)\n    (hIM : IsNilpotent I M) : IsNilpotent L M :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Lie","family_id":"isnilpotentofisnilpotentspansupeqtop","file_id":"mathlib/Mathlib/Algebra/Lie/Engel.lean","sample_id":"e96a9f2e6f2cd142ce46e8c5c7977a2fb22f0ea1c0daabf15e8cbf9d7c647459"},"split":"train"}
{"binding":{"chosen_proof_sha256":"31a79f6998a821c6e1751ba23882d5c421febc08424475fbe1cf68e517fa33e6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"75ec7c738aaa4368c89d8dfb1f412cc8678a88ad71476a93e594c94c2fe463da","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6c68bdabc3643e303bfd886e5b6ece2de0d41819d4c4fb2cea020fa9d7f7d7c8","source_sha256":"c611981e5313e82f67b996518241b02b4fd5776e6033546107f5407921b773b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases L <;> grind","hard_negative":true,"metrics":{"chosen_tokens":7,"rejected_tokens":3,"token_jaccard":0.111111,"token_length_ratio":0.428571},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"48082a975f8d67251c063815152b2f46689e0316ebcb5ec10b20066e5292be94","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono\n\nNamespace:\nSimplexCategoryGenRel\n\nLocal context:\n/-\nCopyright (c) 2025 Robin Carlier. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robin Carlier\n-/\n/-! # Normal forms for morphisms in `SimplexCategoryGenRel`.\n\nIn this file, we establish that `P_δ` and `P_σ` morphisms in `SimplexCategoryGenRel`\neach admits a normal form.\n\nIn both cases, the normal forms are encoded as an integer `m`, and a strictly increasing\nlist of integers `[i₀,…,iₙ]` such that `iₖ ≤ m + k` for all `k`. We define a predicate\n`isAdmissible m : List ℕ → Prop` encoding this property. And provide some lemmas to help\nwork with such lists.\n\nNormal forms for `P_σ` morphisms are encoded by `m`-admissible lists, in which case the list\n`[i₀,…,iₙ]` represents the morphism `σ iₙ ≫ ⋯ ≫ σ i₀ : .mk (m + n) ⟶ .mk n`.\n\nNormal forms for `P_δ` morphisms are encoded by `(m + 1)`-admissible lists, in which case the list\n`[i₀,…,iₙ]` represents the morphism `δ i₀ ≫ ⋯ ≫ δ iₙ : .mk n ⟶ .mk (m + n)`.\n\nThe results in this file are to be treated as implementation-only, and they only serve as stepping\nstones towards proving that the canonical functor\n`toSimplexCategory : SimplexCategoryGenRel ⥤ SimplexCategory` is an equivalence.\n\n## References:\n* [Kerodon Tag 04FQ](https://kerodon.net/tag/04FQ)\n* [Kerodon Tag 04FT](https://kerodon.net/tag/04FT)\n\n## TODOs:\n- Show that every `P_δ` admits a unique normal form.\n-/\n\n@[expose] public section\n\nnamespace SimplexCategoryGenRel\n\nopen CategoryTheory\n\nsection AdmissibleLists\n-- Impl. note: We are not bundling admissible lists as a subtype of `List ℕ` so that it remains\n-- easier to perform inductive constructions and proofs on such lists, and we instead bundle\n-- propositions asserting that various List constructions produce admissible lists.\n\nvariable (m : ℕ)\n/-- A list of natural numbers `[i₀, ⋯, iₙ]` is said to be `m`-admissible (for `m : ℕ`) if\n`i₀ < ⋯ < iₙ` and `iₖ ≤ m + k` for all `k`. This would suggest the definition\n`L.IsChain (· < ·) ∧ ∀ k, (h : k < L.length) → L[k] ≤ m + k`.\nHowever, we instead define `IsAdmissible` inductively and show, in\n`isAdmissible_iff_isChain_and_le`, that this is equivalent to the non-inductive definition.\n-/\n@[mk_iff]\ninductive IsAdmissible : (m : ℕ) → (L : List ℕ) → Prop\n  | nil (m : ℕ) : IsAdmissible m []\n  | singleton {m a} (ha : a ≤ m) : IsAdmissible m [a]\n  | cons_cons {m a b L'} (hab : a < b) (hbL : IsAdmissible (m + 1) (b :: L'))\n      (ha : a ≤ m) : IsAdmissible m (a :: b :: L')\n\nattribute [simp, grind ←] IsAdmissible.nil\nattribute [grind →] IsAdmissible.cons_cons\n\nsection IsAdmissible\n\nvariable {m a b : ℕ} {L : List ℕ}\n\n@[simp, grind =]\ntheorem isAdmissible_singleton_iff : IsAdmissible m [a] ↔ a ≤ m :=\n  ⟨fun | .singleton h => h, .singleton⟩\n\n@[simp, grind =]\ntheorem isAdmissible_cons_cons_iff : IsAdmissible m (a :: b :: L) ↔\n    a < b ∧ IsAdmissible (m + 1) (b :: L) ∧ a ≤ m :=\n  ⟨fun | .cons_cons hab hbL ha => ⟨hab, hbL, ha⟩, by grind⟩\n\nTarget:\ntheorem isAdmissible_cons_iff : IsAdmissible m (a :: L) ↔\n    a ≤ m ∧ ((_ : 0 < L.length) → a < L[0]) ∧ IsAdmissible (m + 1) L :=\n\nProof body:\n","rejected":"by\n  exact isAdmissible_cons_iff","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"69789abbe8338ba4f64b45438a4926152ece4153683f4c16b89a33a6da6cec97","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplexCategory","family_id":"isadmissible_cons_iff","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/NormalForms.lean","sample_id":"6c68bdabc3643e303bfd886e5b6ece2de0d41819d4c4fb2cea020fa9d7f7d7c8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9ed2efcc5bc8272d2a70d89cfb0ba3dc939ae6d8c6a39b0504302e5c2afc9b8d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b765d1892775d5e6e6a62ecd85051d3ac26c69fedc8cb19fe0955539acfca9f3","source_sha256":"d1ba22e22a26245161ec2703bf29aa4e6e9c0ade6d5458aee3e70435922bd387","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨x, v₁, v₂⟩ := x\n  obtain ⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩ := h\n  simp [Prod.toFun', Prod.invFun', h₁, h₂]","hard_negative":false,"metrics":{"chosen_tokens":53,"rejected_tokens":5,"token_jaccard":0.038462,"token_length_ratio":0.09434},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"481296d7ceb149b5ed19cb7a106ebed9c4f91f66c76a67967e99b87abc75347a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.FiberBundle.Basic\n\nNamespace:\nBundle.Trivialization\n\nLocal context:\n/-\nCopyright (c) 2022 Nicolò Cavalleri. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn\n-/\n/-!\n# Standard constructions on fiber bundles\n\nThis file contains several standard constructions on fiber bundles:\n\n* `Bundle.Trivial.fiberBundle 𝕜 B F`: the trivial fiber bundle with model fiber `F` over the base\n  `B`\n\n* `FiberBundle.prod`: for fiber bundles `E₁` and `E₂` over a common base, a fiber bundle structure\n  on their fiberwise product `E₁ ×ᵇ E₂` (the notation stands for `fun x ↦ E₁ x × E₂ x`).\n\n* `FiberBundle.pullback`: for a fiber bundle `E` over `B`, a fiber bundle structure on its\n  pullback `f *ᵖ E` by a map `f : B' → B` (the notation is a type synonym for `E ∘ f`).\n\n## Tags\n\nfiber bundle, fibre bundle, fiberwise product, pullback\n\n-/\n\n@[expose] public section\n\nopen Bundle Filter Set TopologicalSpace Topology\n\n/-! ### The trivial bundle -/\n\nnamespace Bundle\n\nnamespace Trivial\n\nvariable (B : Type*) (F : Type*)\n\n-- TODO: use `TotalSpace.toProd`\ninstance topologicalSpace [t₁ : TopologicalSpace B]\n    [t₂ : TopologicalSpace F] : TopologicalSpace (TotalSpace F (Trivial B F)) :=\n  induced TotalSpace.proj t₁ ⊓ induced (TotalSpace.trivialSnd B F) t₂\n\nvariable [TopologicalSpace B] [TopologicalSpace F]\n\ntheorem isInducing_toProd : IsInducing (TotalSpace.toProd B F) :=\n  ⟨by simp only [instTopologicalSpaceProd, induced_inf, induced_compose]; rfl⟩\n\n/-- Homeomorphism between the total space of the trivial bundle and the Cartesian product. -/\n@[simps!]\ndef homeomorphProd : TotalSpace F (Trivial B F) ≃ₜ B × F :=\n  (TotalSpace.toProd _ _).toHomeomorphOfIsInducing (isInducing_toProd B F)\n\n/-- Local trivialization for trivial bundle. -/\n@[simps!]\ndef trivialization : Trivialization F (π F (Bundle.Trivial B F)) where\n  toOpenPartialHomeomorph := (homeomorphProd B F).toOpenPartialHomeomorph\n  baseSet := univ\n  open_baseSet := isOpen_univ\n  source_eq := rfl\n  target_eq := univ_prod_univ.symm\n  proj_toFun _ _ := rfl\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp] lemma trivialization_symm_apply [Zero F] (b : B) (f : F) :\n    (trivialization B F).symm b f = f := by\n  simp [trivialization, homeomorphProd, TotalSpace.toProd, Trivialization.symm,\n    Pretrivialization.symm, Trivialization.toPretrivialization]\n\n@[simp] lemma toOpenPartialHomeomorph_trivialization_symm_apply (v : B × F) :\n    (trivialization B F).toOpenPartialHomeomorph.symm v = ⟨v.1, v.2⟩ := rfl\n\n/-- Fiber bundle instance on the trivial bundle. -/\n@[simps] instance fiberBundle : FiberBundle F (Bundle.Trivial B F) where\n  trivializationAtlas' := {trivialization B F}\n  trivializationAt' _ := trivialization B F\n  mem_baseSet_trivializationAt' := mem_univ\n  trivialization_mem_atlas' _ := mem_singleton _\n  totalSpaceMk_isInducing' _ := (homeomorphProd B F).symm.isInducing.comp\n    (isInducing_const_prod.2 .id)\n\ntheorem eq_trivialization (e : Trivialization F (π F (Bundle.Trivial B F)))\n    [i : MemTrivializationAtlas e] : e = trivialization B F := i.out\n\nend Trivial\n\nend Bundle\n\n/-! ### Fibrewise product of two bundles -/\n\n\nsection Prod\n\nvariable {B : Type*}\n\nsection Defs\n\nvariable (F₁ : Type*) (E₁ : B → Type*) (F₂ : Type*) (E₂ : B → Type*)\nvariable [TopologicalSpace (TotalSpace F₁ E₁)] [TopologicalSpace (TotalSpace F₂ E₂)]\n\n/-- Equip the total space of the fiberwise product of two fiber bundles `E₁`, `E₂` with\nthe induced topology from the diagonal embedding into `TotalSpace F₁ E₁ × TotalSpace F₂ E₂`. -/\ninstance FiberBundle.Prod.topologicalSpace : TopologicalSpace (TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂)) :=\n  TopologicalSpace.induced\n    (fun p ↦ ((⟨p.1, p.2.1⟩ : TotalSpace F₁ E₁), (⟨p.1, p.2.2⟩ : TotalSpace F₂ E₂)))\n    inferInstance\n\n/-- The diagonal map from the total space of the fiberwise product of two fiber bundles\n`E₁`, `E₂` into `TotalSpace F₁ E₁ × TotalSpace F₂ E₂` is an inducing map. -/\ntheorem FiberBundle.Prod.isInducing_diag :\n    IsInducing (fun p ↦ (⟨p.1, p.2.1⟩, ⟨p.1, p.2.2⟩) :\n      TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂) :=\n  ⟨rfl⟩\n\nend Defs\n\nopen FiberBundle\n\nvariable [TopologicalSpace B] (F₁ : Type*) [TopologicalSpace F₁] (E₁ : B → Type*)\n  [TopologicalSpace (TotalSpace F₁ E₁)] (F₂ : Type*) [TopologicalSpace F₂] (E₂ : B → Type*)\n  [TopologicalSpace (TotalSpace F₂ E₂)]\n\nnamespace Bundle.Trivialization\n\nvariable {F₁ E₁ F₂ E₂}\nvariable (e₁ : Trivialization F₁ (π F₁ E₁)) (e₂ : Trivialization F₂ (π F₂ E₂))\n\n/-- Given trivializations `e₁`, `e₂` for fiber bundles `E₁`, `E₂` over a base `B`, the forward\nfunction for the construction `Trivialization.prod`, the induced\ntrivialization for the fiberwise product of `E₁` and `E₂`. -/\ndef Prod.toFun' : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) → B × F₁ × F₂ :=\n  fun p ↦ ⟨p.1, (e₁ ⟨p.1, p.2.1⟩).2, (e₂ ⟨p.1, p.2.2⟩).2⟩\n\nvariable {e₁ e₂}\n\ntheorem Prod.continuous_to_fun : ContinuousOn (Prod.toFun' e₁ e₂)\n    (π (F₁ × F₂) (E₁ ×ᵇ E₂) ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) := by\n  let f₁ : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ :=\n    fun p ↦ ((⟨p.1, p.2.1⟩ : TotalSpace F₁ E₁), (⟨p.1, p.2.2⟩ : TotalSpace F₂ E₂))\n  let f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p ↦ ⟨e₁ p.1, e₂ p.2⟩\n  let f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p ↦ ⟨p.1.1, p.1.2, p.2.2⟩\n  have hf₁ : Continuous f₁ := (Prod.isInducing_diag F₁ E₁ F₂ E₂).continuous\n  have hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) :=\n    e₁.toOpenPartialHomeomorph.continuousOn.prodMap e₂.toOpenPartialHomeomorph.continuousOn\n  have hf₃ : Continuous f₃ := by fun_prop\n  refine ((hf₃.comp_continuousOn hf₂).comp hf₁.continuousOn ?_).congr ?_\n  · rw [e₁.source_eq, e₂.source_eq]\n    exact mapsTo_preimage _ _\n  rintro ⟨b, v₁, v₂⟩ ⟨hb₁, _⟩\n  simp only [f₁, f₂, f₃, Prod.toFun', Prod.mk_inj, Function.comp_apply, and_true]\n  rw [e₁.coe_fst]\n  rw [e₁.source_eq, mem_preimage]\n  exact hb₁\n\nvariable (e₁ e₂) [∀ x, Zero (E₁ x)] [∀ x, Zero (E₂ x)]\n\n/-- Given trivializations `e₁`, `e₂` for fiber bundles `E₁`, `E₂` over a base `B`, the inverse\nfunction for the construction `Trivialization.prod`, the induced\ntrivialization for the fiberwise product of `E₁` and `E₂`. -/\nnoncomputable def Prod.invFun' (p : B × F₁ × F₂) : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) :=\n  ⟨p.1, e₁.symm p.1 p.2.1, e₂.symm p.1 p.2.2⟩\n\nvariable {e₁ e₂}\n\nTarget:\ntheorem Prod.left_inv {x : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂)}\n    (h : x ∈ π (F₁ × F₂) (E₁ ×ᵇ E₂) ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) :\n    Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/FiberBundle","family_id":"prod","file_id":"mathlib/Mathlib/Topology/FiberBundle/Constructions.lean","sample_id":"b765d1892775d5e6e6a62ecd85051d3ac26c69fedc8cb19fe0955539acfca9f3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0ec8ef0b04cc9e360fe54d83ec9b445dee17ea05e7abf3a9b4757fa21a5fb78e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5cf02018572ea2a48c79b73191cb5822bde47b0ed84cae6ea2a427310937ee64","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"71f1b32ec46df06988407bec9fec4c171851cda268d1134ad7fa0191f2142682","source_sha256":"b5b8fef319876fc584704ec0902863ab8016fe0ba4dee5dd358f86f4688b13cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [Irrational, Rat.forall, eq_comm]","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.272727},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"482c56b5cd119129b12043dad116decd800531b148fd7edef8496e9739657eb1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Rat\npublic import Mathlib.Data.Nat.Prime.Int\npublic import Mathlib.Data.Rat.Sqrt\npublic import Mathlib.Analysis.Real.Sqrt\npublic import Mathlib.RingTheory.Algebraic.Basic\npublic import Mathlib.Tactic.CrossRefAttribute\npublic import Mathlib.Tactic.IntervalCases\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov\n-/\n/-!\n# Irrational real numbers\n\nIn this file we define a predicate `Irrational` on `ℝ`, prove that the `n`-th root of an integer\nnumber is irrational if it is not integer, and that `√(q : ℚ)` is irrational if and only if\n`¬IsSquare q ∧ 0 ≤ q`.\n\nWe also provide dot-style constructors like `Irrational.add_ratCast`, `Irrational.ratCast_sub` etc.\n\nWith the `Decidable` instances in this file, is possible to prove `Irrational √n` using `decide`,\nwhen `n` is a numeric literal or cast;\nbut this only works if you `unseal Nat.sqrt.iter in` before the theorem where you use this proof.\n-/\n\n@[expose] public section\n\n\nopen Rat Real\n\n/-- A real number is irrational if it is not equal to any rational number. -/\n@[wikidata Q607728]\ndef Irrational (x : ℝ) :=\n  x ∉ Set.range ((↑) : ℚ → ℝ)\n\nTarget:\ntheorem irrational_iff_ne_rational (x : ℝ) : Irrational x ↔ ∀ a b : ℤ, b ≠ 0 → x ≠ a / b :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_71f1b32ec46d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"74c9ec3867b26ad3dc254c1ba114e63a10ea00360df83b58e3eb2cbb3a745aad","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Real","family_id":"irrational_iff_ne_rational","file_id":"mathlib/Mathlib/NumberTheory/Real/Irrational.lean","sample_id":"71f1b32ec46df06988407bec9fec4c171851cda268d1134ad7fa0191f2142682"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6986b7de3b811139639af9103eba6dc70b8f13e73607506a5de302cee24db96a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1e74a857d727dcb19bacc65e1dd39ea3f06dc3533cc251ea326cf338c51e0a18","source_sha256":"6b438bdcb0b80057beec48c7a9a1f234b44b153474b61732effd69c42c628d3a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [BilinForm.toMatrixAux, Matrix.toBilin'Aux, toLinearMap₂'Aux_toMatrix₂Aux]","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":2,"token_jaccard":0.0625,"token_length_ratio":0.111111},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"482f8820fe9468f5ac36722732b4434fb6efe4e8e2b98e55873e1b71de28547d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.BilinearForm.Properties\npublic import Mathlib.LinearAlgebra.Matrix.SesquilinearForm\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen, Kexing Ying\n-/\n/-!\n# Bilinear form\n\nThis file defines the conversion between bilinear forms and matrices.\n\n## Main definitions\n\n* `Matrix.toBilin` given a basis define a bilinear form\n* `Matrix.toBilin'` define the bilinear form on `n → R`\n* `BilinForm.toMatrix`: calculate the matrix coefficients of a bilinear form\n* `BilinForm.toMatrix'`: calculate the matrix coefficients of a bilinear form on `n → R`\n\n## Notation\n\nIn this file we use the following type variables:\n- `M₁` is a module over the commutative semiring `R₁`,\n- `M₂` is a module over the commutative ring `R₂`.\n\n## Tags\n\nbilinear form, bilin form, BilinearForm, matrix, basis\n\n-/\n\n@[expose] public section\n\nopen LinearMap (BilinForm)\nopen Module\n\nvariable {R₁ : Type*} {M₁ : Type*} [CommSemiring R₁] [AddCommMonoid M₁] [Module R₁ M₁]\nvariable {R₂ : Type*} {M₂ : Type*} [CommRing R₂] [AddCommGroup M₂] [Module R₂ M₂]\n\nsection Matrix\n\nvariable {n o : Type*}\n\nopen Finset LinearMap Matrix\n\nopen Matrix\n\n/-- The map from `Matrix n n R` to bilinear forms on `n → R`.\n\nThis is an auxiliary definition for the equivalence `Matrix.toBilin'`. -/\ndef Matrix.toBilin'Aux [Fintype n] (M : Matrix n n R₁) : BilinForm R₁ (n → R₁) :=\n  Matrix.toLinearMap₂'Aux _ _ M\n\ntheorem Matrix.toBilin'Aux_single [Fintype n] [DecidableEq n] (M : Matrix n n R₁) (i j : n) :\n    M.toBilin'Aux (Pi.single i 1) (Pi.single j 1) = M i j :=\n  Matrix.toLinearMap₂'Aux_single _ _ _ _ _\n\n/-- The linear map from bilinear forms to `Matrix n n R` given an `n`-indexed basis.\n\nThis is an auxiliary definition for the equivalence `Matrix.toBilin'`. -/\ndef LinearMap.BilinForm.toMatrixAux (b : n → M₁) : BilinForm R₁ M₁ →ₗ[R₁] Matrix n n R₁ :=\n  LinearMap.toMatrix₂Aux R₁ b b\n\n@[deprecated (since := \"2026-01-16\")] alias BilinForm.toMatrixAux := LinearMap.BilinForm.toMatrixAux\n\n@[simp]\ntheorem LinearMap.BilinForm.toMatrixAux_apply (B : BilinForm R₁ M₁) (b : n → M₁) (i j : n) :\n    BilinForm.toMatrixAux b B i j = B (b i) (b j) :=\n  LinearMap.toMatrix₂Aux_apply R₁ B _ _ _ _\n\nvariable [Fintype n] [Fintype o]\n\nTarget:\ntheorem LinearMap.toBilin'Aux_toMatrixAux [DecidableEq n] (B₂ : BilinForm R₁ (n → R₁)) :\n    Matrix.toBilin'Aux (BilinForm.toMatrixAux (fun j => Pi.single j 1) B₂) = B₂ :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Matrix","family_id":"linearmap","file_id":"mathlib/Mathlib/LinearAlgebra/Matrix/BilinearForm.lean","sample_id":"1e74a857d727dcb19bacc65e1dd39ea3f06dc3533cc251ea326cf338c51e0a18"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0520ccfd6af7e55e37364ebbda9a263098edd7a0ec6ad9af960d91cce2f0e2d9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"541da4b081e2e34b78b074868e478c10ad66ab82360120d258e46aaa2f75d6bf","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b16015c2c8cb752b26e08d47468536e58a1c1a2c7cc95ef8b2185f3dd54359d8","source_sha256":"bda612cd6d1ad1cbdd6e02bafd2e7e77c11e8cfa412747033a52bbe8062964f2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← ClassGroup.mk_mk0 (FractionRing A), extendedHom_mk]\n\nvariable [IsDedekindDomain B]","hard_negative":true,"metrics":{"chosen_tokens":19,"rejected_tokens":3,"token_jaccard":0.052632,"token_length_ratio":0.157895},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"48517f2220c82cb31768b141753f55ef0959b070279bb311b8f0e291c1a23938","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.FractionalIdeal.Extended\npublic import Mathlib.RingTheory.ClassGroup.Basic\n\nNamespace:\nClassGroup\n\nLocal context:\n/-\nCopyright (c) 2026 Riccardo Brasca. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Birkbeck, Riccardo Brasca\n-/\n/-!\n# Class group map induced by an extension of domains\n\nFor an injective extension `A → B` of commutative domains (equivalently `Module.IsTorsionFree A B`),\nwe construct the group homomorphism `ClassGroup.extendedHom : ClassGroup A →* ClassGroup B` given by\npushing fractional ideals forward along the algebra map.\n\n## Main definitions\n\n- `ClassGroup.extendedHom A B`: the induced map between the class groups.\n- `ClassGroup.extendedIdeal A B`: the extension of a nonzero integral ideal.\n\n## Main results\n\n- `ClassGroup.extendedHom_mk`: compatibility with representatives as fractional ideals.\n- `ClassGroup.extendedHom_mk0`: compatibility with representatives as nonzero integral ideals.\n- `ClassGroup.extendedHom_comp`: compatibility of extension in a tower `A → B → C`.\n- `ClassGroup.extendedHom_eq_one_of_forall_isPrincipal`: if the extension of every ideal is\n  principal, then `ClassGroup.extendedHom A B` is trivial.\n-/\n\npublic section\n\nopen scoped nonZeroDivisors\n\nvariable (A B : Type*) [CommRing A] [CommRing B] [Algebra A B]\n  [Module.IsTorsionFree A B]\n\nnamespace ClassGroup\n\nsection CommRing\n\nvariable [IsDomain A] [IsDomain B]\n\n/-- The monoid homomorphism `ClassGroup A → ClassGroup B` induced by an\ninjective extension of domains `A → B`. -/\nnoncomputable def extendedHom : ClassGroup A →* ClassGroup B :=\n  QuotientGroup.map _ _\n    (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom)\n    (by\n      rintro _ ⟨α, rfl⟩\n      refine ⟨Units.mk0 (IsFractionRing.map (j := algebraMap A B)\n        (FaithfulSMul.algebraMap_injective _ _) (α : FractionRing A))\n        (by simp [α.ne_zero]), ?_⟩\n      simpa [coe_toPrincipalIdeal, Units.coe_map, Units.val_mk0] using!\n        (FractionalIdeal.extendedHom_spanSingleton (FractionRing B) B _).symm)\n\n@[simp]\nlemma extendedHom_quotientMk (α : (FractionalIdeal A⁰ (FractionRing A))ˣ) :\n    extendedHom A B (QuotientGroup.mk α) = QuotientGroup.mk\n      (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom α) := by\n  rfl\n\n@[simp]\ntheorem extendedHom_mk (I : (FractionalIdeal A⁰ (FractionRing A))ˣ) :\n    extendedHom A B (ClassGroup.mk _ I) = ClassGroup.mk _\n        (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom I) := by\n  rw [← ClassGroup.Quot_mk_eq_mk, ← ClassGroup.Quot_mk_eq_mk]\n  exact extendedHom_quotientMk A B I\n\n/-- The extension of a nonzero integral ideal along an injective extension of domains. -/\nabbrev extendedIdeal (I : (Ideal A)⁰) : (Ideal B)⁰ :=\n  ⟨I.1.map (algebraMap A B), mem_nonZeroDivisors_iff_ne_zero.mpr <|\n    (Ideal.map_eq_bot_iff_of_injective (FaithfulSMul.algebraMap_injective A B)).not.mpr\n      (mem_nonZeroDivisors_iff_ne_zero.mp I.2)⟩\n\n@[simp]\ntheorem extendedIdeal_extendedIdeal (C : Type*) [CommRing C] [IsDomain C] [Algebra B C]\n    [Algebra A C] [IsScalarTower A B C] [Module.IsTorsionFree B C]\n    [Module.IsTorsionFree A C] (I : (Ideal A)⁰) :\n    extendedIdeal B C (extendedIdeal A B I) = extendedIdeal A C I := by\n  simp [Ideal.map_map, IsScalarTower.algebraMap_eq A B C]\n\nend CommRing\n\nsection DedekindDomain\n\nvariable [IsDedekindDomain A] (C : Type*) [CommRing C] [Algebra B C] [Algebra A C]\n  [IsScalarTower A B C] [Module.IsTorsionFree B C] [Module.IsTorsionFree A C]\n  [IsDedekindDomain C]\n\nTarget:\ntheorem extendedHom_mk0' [IsDomain B] (I : (Ideal A)⁰) :\n    extendedHom A B (ClassGroup.mk0 I) =\n      ClassGroup.mk _ (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom\n      (FractionalIdeal.mk0 (FractionRing A) I)) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_b16015c2c8cb","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"89a6928a37b9590ad665943f05e8e25c828348e19982a1bbefaa574957ecb9d5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/ClassGroup","family_id":"extendedhom_mk","file_id":"mathlib/Mathlib/RingTheory/ClassGroup/ExtendedHom.lean","sample_id":"b16015c2c8cb752b26e08d47468536e58a1c1a2c7cc95ef8b2185f3dd54359d8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bd9e89042700066e34db2eae2a5bdca2cc788a55ff318687cbca751f9272e192","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"25e33e2ad73b99a239389fcab2e1bb17f4373e01742868a9ff04d6e9bc1274a4","source_sha256":"5fbd6e0c9aed4ae0af15e49b0a292eb4e252a42e876e277383b0144f2f430dad","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [List.applyId, List.dlookup, eq_rec_constant, Prod.toSigma, List.map]\n  split_ifs <;> rfl","hard_negative":true,"metrics":{"chosen_tokens":27,"rejected_tokens":8,"token_jaccard":0.04,"token_length_ratio":0.296296},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"48880fabbd2f921943ec9b8d75691d0088f5fc23d4308e5f8cf1f1e61e20ee3c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.MLList.Basic\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.ZeroLEOne\npublic import Mathlib.Data.DFinsupp.Defs\npublic import Mathlib.Data.Finsupp.Defs\npublic import Mathlib.Data.Int.Range\npublic import Mathlib.Data.List.Sigma\npublic import Mathlib.Data.PNat.Notation\npublic import Mathlib.Tactic.Bound.Init\n\nNamespace:\nPlausible.InjectiveFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\npublic meta import Mathlib.Data.Finsupp.ToDFinsupp\npublic meta import Mathlib.Data.Int.Range\npublic meta import Mathlib.Data.List.Sigma\npublic meta import Plausible.Functions\n\n/-!\n## `Plausible`: generators for functions\n\nThis file defines `Sampleable` instances for `ℤ → ℤ` injective functions.\n\nInjective functions are generated by creating a list of numbers and\na permutation of that list. The permutation insures that every input\nis mapped to a unique output. When an input is not found in the list\nthe input itself is used as an output.\n\nInjective functions `f : α → α` could be generated easily instead of\n`ℤ → ℤ` by generating a `List α`, removing duplicates and creating a\npermutation. One has to be careful when generating the domain to make\nit vast enough that, when generating arguments to apply `f` to,\nthey argument should be likely to lie in the domain of `f`. This is\nthe reason that injective functions `f : ℤ → ℤ` are generated by\nfixing the domain to the range `[-2*size .. 2*size]`, with `size`\nthe size parameter of the `gen` monad.\n\nMuch of the machinery provided in this file is applicable to generate\ninjective functions of type `α → α` and new instances should be easy\nto define.\n\nOther classes of functions such as monotone functions can generated using\nsimilar techniques. For monotone functions, generating two lists, sorting them\nand matching them should suffice, with appropriate default values.\nSome care must be taken for shrinking such functions to make sure\ntheir defining property is invariant through shrinking. Injective\nfunctions are an example of how complicated it can get.\n-/\n\n@[expose] public meta section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type v}\n\nnamespace Plausible\n\nnamespace TotalFunction\n\nsection Finsupp\n\nvariable [DecidableEq α]\n\n/--\nThis theorem exists because plausible does not have access to dlookup but\nmathlib has all the theory for it and wants to use it. We probably want to\nbring these two together at some point.\n-/\nprivate theorem apply_eq_dlookup (m : List (Σ _ : α, β)) (y : β) (x : α) :\n     (withDefault m y).apply x = (m.dlookup x).getD y := by\n  dsimp only [apply]\n  congr 1\n  induction m with\n  | nil => simp\n  | cons p m ih =>\n    rcases p with ⟨fst, snd⟩\n    by_cases heq : fst = x\n    · simp [heq]\n    · rw [List.dlookup_cons_ne]\n      · simp [heq, ih]\n      · symm\n        simp [heq]\n\nvariable [Zero β] [DecidableEq β]\n\n/-- Map a `TotalFunction` to one whose default value is zero so that it represents a `Finsupp`. -/\n@[simp]\ndef zeroDefault : TotalFunction α β → TotalFunction α β\n  | .withDefault A _ => .withDefault A 0\n\n/-- The support of a zero default `TotalFunction`. -/\ndef zeroDefaultSupp : TotalFunction α β → Finset α\n  | .withDefault A _ =>\n    List.toFinset <| (A.dedupKeys.filter fun ab => Sigma.snd ab ≠ 0).map Sigma.fst\n\n/-- Create a finitely supported function from a total function by taking the default value to\nzero. -/\ndef applyFinsupp (tf : TotalFunction α β) : α →₀ β where\n  support := zeroDefaultSupp tf\n  toFun := tf.zeroDefault.apply\n  mem_support_toFun := by\n    intro a\n    rcases tf with ⟨A, y⟩\n    simp only [zeroDefaultSupp, List.mem_map, List.mem_filter, exists_and_right,\n      List.mem_toFinset, exists_eq_right, Sigma.exists, Ne, zeroDefault]\n    rw [apply_eq_dlookup]\n    constructor\n    · rintro ⟨od, hval, hod⟩\n      have := List.mem_dlookup (List.nodupKeys_dedupKeys A) hval\n      rw [(_ : List.dlookup a A = od)]\n      · simpa using hod\n      · simpa [List.dlookup_dedupKeys]\n    · intro h\n      use (A.dlookup a).getD (0 : β)\n      rw [← List.dlookup_dedupKeys] at h ⊢\n      simp only [h, ← List.mem_dlookup_iff A.nodupKeys_dedupKeys, not_false_iff, Option.mem_def]\n      cases haA : List.dlookup a A.dedupKeys\n      · simp [haA] at h\n      · simp\n\nvariable [SampleableExt α] [SampleableExt β] [Repr α]\n\ninstance Finsupp.sampleableExt : SampleableExt (α →₀ β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\n-- TODO: support a non-constant codomain type\ninstance DFinsupp.sampleableExt : SampleableExt (Π₀ _ : α, β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp.toDFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\nend Finsupp\nend TotalFunction\n\nopen _root_.List\n\n/-- Data structure specifying a total function using a list of pairs\nand a default value returned when the input is not in the domain of\nthe partial function.\n\n`mapToSelf f` encodes `x ↦ f x` when `x ∈ f` and `x ↦ x`,\ni.e. `x` to itself, otherwise.\n\nWe use `Σ` to encode mappings instead of `×` because we\nrely on the association list API defined in `Mathlib/Data/List/Sigma.lean`.\n-/\ninductive InjectiveFunction (α : Type u) : Type u\n  | mapToSelf (xs : List (Σ _ : α, α)) :\n      xs.map Sigma.fst ~ xs.map Sigma.snd → List.Nodup (xs.map Sigma.snd) → InjectiveFunction α\n\ninstance : Inhabited (InjectiveFunction α) :=\n  ⟨⟨[], List.Perm.nil, List.nodup_nil⟩⟩\n\nnamespace InjectiveFunction\n\n/-- Apply a total function to an argument. -/\ndef apply [DecidableEq α] : InjectiveFunction α → α → α\n  | InjectiveFunction.mapToSelf m _ _, x => (m.dlookup x).getD x\n\n/-- Produce a string for a given `InjectiveFunction`.\nThe output is of the form `[x₀ ↦ f x₀, .. xₙ ↦ f xₙ, x ↦ x]`.\nUnlike for `TotalFunction`, the default value is not a constant\nbut the identity function.\n-/\nprotected def repr [Repr α] : InjectiveFunction α → String\n  | InjectiveFunction.mapToSelf m _ _ => s! \"[{TotalFunction.reprAux m}x ↦ x]\"\n\ninstance (α : Type u) [Repr α] : Repr (InjectiveFunction α) where\n  reprPrec f _p := InjectiveFunction.repr f\n\n/-- Interpret a list of pairs as a total function, defaulting to\nthe identity function when no entries are found for a given function -/\ndef List.applyId [DecidableEq α] (xs : List (α × α)) (x : α) : α :=\n  ((xs.map Prod.toSigma).dlookup x).getD x\n\n@[simp]\n\nTarget:\ntheorem List.applyId_cons [DecidableEq α] (xs : List (α × α)) (x y z : α) :\n    List.applyId ((y, z)::xs) x = if y = x then z else List.applyId xs x :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"072f58d4d9323e32ce9b0bd3261f93220e3af16282fde86ec58c222727cf35bc","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Testing/Plausible","family_id":"list","file_id":"mathlib/Mathlib/Testing/Plausible/Functions.lean","sample_id":"25e33e2ad73b99a239389fcab2e1bb17f4373e01742868a9ff04d6e9bc1274a4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8b4eba4a7d76ab6115b8794619b18b2fd26643e7d931d0fcde4744d29b34df3e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c5d49b24160a98ef5dde849af319186f4d5a71a4762042b5236c692256d7519c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"40d4ba4e788dd7fbc85ecb01598bd57d742b372ed1f0e1d36e21ca910229b975","source_sha256":"6ea826f0633d3930f7872110f571f16f76059ec3668e205ef02a1089b9b31873","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n    rw [← card_image_of_injective s' Subtype.coe_injective]\n    convert! ht (s'.image fun z => z.1) using 1\n    apply congr_arg\n    ext y\n    simp","hard_negative":false,"metrics":{"chosen_tokens":32,"rejected_tokens":36,"token_jaccard":0.9,"token_length_ratio":1.125},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"488f6387366706ce14512350ac0b3218a31bed8ca127ad6eb0e177a9ca559bb7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Basic\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Data.Set.Finite.Basic\n\nNamespace:\nHallMarriageTheorem\n\nLocal context:\n/-\nCopyright (c) 2021 Alena Gusakov, Bhavik Mehta, Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alena Gusakov, Bhavik Mehta, Kyle Miller\n-/\n/-!\n# Hall's Marriage Theorem for finite index types\n\nThis module proves the basic form of Hall's theorem.\nIn contrast to the theorem described in `Combinatorics.Hall.Basic`, this\nversion requires that the indexed family `t : ι → Finset α` have `ι` be finite.\nThe `Combinatorics.Hall.Basic` module applies a compactness argument to this version\nto remove the `Finite` constraint on `ι`.\n\nThe modules are split like this since the generalized statement\ndepends on the topology and category theory libraries, but the finite\ncase in this module has few dependencies.\n\nA description of this formalization is in [Gusakov2021].\n\n## Main statements\n\n* `Finset.all_card_le_biUnion_card_iff_existsInjective'` is Hall's theorem with\n  a finite index set.  This is elsewhere generalized to\n  `Finset.all_card_le_biUnion_card_iff_existsInjective`.\n\n## Tags\n\nHall's Marriage Theorem, indexed families\n-/\n\npublic section\n\n\nopen Finset\n\nuniverse u v\n\nnamespace HallMarriageTheorem\n\nvariable {ι : Type u} {α : Type v} [DecidableEq α] {t : ι → Finset α}\n\nsection Fintype\n\nvariable [Fintype ι]\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem hall_cond_of_erase {x : ι} (a : α)\n    (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t))\n    (s' : Finset { x' : ι | x' ≠ x }) : #s' ≤ #(s'.biUnion fun x' => (t x').erase a) := by\n  haveI := Classical.decEq ι\n  specialize ha (s'.image fun z => z.1)\n  rw [image_nonempty, Finset.card_image_of_injective s' Subtype.coe_injective] at ha\n  by_cases! he : s'.Nonempty\n  · have ha' : #s' < #(s'.biUnion fun x => t x) := by\n      convert! ha he fun h => by simpa [← h] using mem_univ x using 2\n      ext x\n      simp only [mem_image, mem_biUnion, SetCoe.exists, exists_and_right,\n        exists_eq_right]\n    rw [← erase_biUnion]\n    by_cases hb : a ∈ s'.biUnion fun x => t x\n    · rw [card_erase_of_mem hb]\n      exact Nat.le_sub_one_of_lt ha'\n    · rw [erase_eq_of_notMem hb]\n      exact Nat.le_of_lt ha'\n  · subst s'\n    simp\n\n/-- First case of the inductive step: assuming that\n`∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)`\nand that the statement of **Hall's Marriage Theorem** is true for all\n`ι'` of cardinality ≤ `n`, then it is true for `ι` of cardinality `n + 1`.\n-/\ntheorem hall_hard_inductive_step_A {n : ℕ} (hn : Fintype.card ι = n + 1)\n    (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))\n    (ih :\n      ∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),\n        Fintype.card ι' ≤ n →\n          (∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →\n            ∃ f : ι' → α, Function.Injective f ∧ ∀ x, f x ∈ t' x)\n    (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t)) :\n    ∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by\n  haveI : Nonempty ι := Fintype.card_pos_iff.mp (hn.symm ▸ Nat.succ_pos _)\n  haveI := Classical.decEq ι\n  -- Choose an arbitrary element `x : ι` and `y : t x`.\n  let x := Classical.arbitrary ι\n  have tx_ne : (t x).Nonempty := by\n    rw [← Finset.card_pos]\n    calc\n      0 < 1 := Nat.one_pos\n      _ ≤ #(.biUnion {x} t) := ht {x}\n      _ = (t x).card := by rw [Finset.singleton_biUnion]\n  choose y hy using tx_ne\n  -- Restrict to everything except `x` and `y`.\n  let ι' := { x' : ι | x' ≠ x }\n  let t' : ι' → Finset α := fun x' => (t x').erase y\n  have card_ι' : Fintype.card ι' = n :=\n    calc\n      Fintype.card ι' = Fintype.card ι - 1 := Set.card_ne_eq _\n      _ = n := by rw [hn, Nat.add_succ_sub_one, add_zero]\n  rcases ih t' card_ι'.le (hall_cond_of_erase y ha) with ⟨f', hfinj, hfr⟩\n  -- Extend the resulting function.\n  refine ⟨fun z => if h : z = x then y else f' ⟨z, h⟩, ?_, ?_⟩\n  · rintro z₁ z₂\n    have key : ∀ {x}, y ≠ f' x := by\n      intro x h\n      simpa [t', ← h] using hfr x\n    by_cases h₁ : z₁ = x <;> by_cases h₂ : z₂ = x <;>\n      simp [h₁, h₂, hfinj.eq_iff, key, key.symm]\n  · intro z\n    simp only\n    split_ifs with hz\n    · rwa [hz]\n    · specialize hfr ⟨z, hz⟩\n      rw [mem_erase] at hfr\n      exact hfr.2\n\nTarget:\ntheorem hall_cond_of_restrict {ι : Type u} {t : ι → Finset α} {s : Finset ι}\n    (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t)) (s' : Finset (s : Set ι)) :\n    #s' ≤ #(s'.biUnion fun a' => t a') :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  classical\n    rw [← card_image_of_injective s' Subtype.coe_injective]\n    convert! ht (s'.image fun z => z.1) using 1\n    apply congr_arg\n    ext y\n    simp","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Hall","family_id":"hall_cond_of_restrict","file_id":"mathlib/Mathlib/Combinatorics/Hall/Finite.lean","sample_id":"40d4ba4e788dd7fbc85ecb01598bd57d742b372ed1f0e1d36e21ca910229b975"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3c1f07b625c847d49cc43a89c05d3139e535c28f43f07276611fd8f7f1db78de","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0c4813e84ac03a598437ff70e6508e2117e5d23b23f80a998fcfa79efe8bb4aa","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cb9967aa5d54ae74d6b7c334cf672946e1ac267085a38bfbf8b9d07e6fea6e09","source_sha256":"5baad0a48c840bf6cedcfdfce26a9261ba521811fb4fc226af3afd123575d446","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by trunc) (hb : b.len ≤ n := by trunc)\n    (hc : c.len ≤ n := by trunc) :\n    tr (f ≫ g) = tr f ≫ tr g :=\n  rfl","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":47,"token_jaccard":0.833333,"token_length_ratio":1.119048},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"48ec5534fbd07f864eeeeb6740f31ec73673670c82f80e0e2b824f5f84d0f00e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Category.Preorder\npublic import Mathlib.CategoryTheory.Opposites\npublic import Mathlib.Order.Fin.Basic\npublic import Mathlib.Util.Superscript\n\nNamespace:\nSimplexCategory.Truncated\n\nLocal context:\n/-\nCopyright (c) 2020 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Kim Morrison, Adam Topaz\n-/\n/-! # The simplex category\n\nWe construct a skeletal model of the simplex category, with an object `⦋n⦌` for each `n : ℕ`, and\nmorphisms `⦋n⦌ ⟶ ⦋m⦌` identify to monotone maps from `Fin (n + 1)` to `Fin (m + 1)`.\n\nIn `Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean`, we show that this category\nis equivalent to `NonemptyFinLinOrd`.\n\n## Remarks\n\nWe provide the following functions to work with these objects:\n1. `SimplexCategory.mk` creates an object of `SimplexCategory` out of a natural number.\n  Use the notation `⦋n⦌` in the `Simplicial` locale.\n2. `SimplexCategory.len` gives the \"length\" of an object of `SimplexCategory`, as a natural.\n3. `SimplexCategory.Hom.mk` makes a morphism out of a monotone map between `Fin`'s.\n4. `SimplexCategory.Hom.toOrderHom` gives the underlying monotone map associated to a\n  term of `SimplexCategory.Hom`.\n\n## Notation\n\n* `⦋n⦌` denotes the `n`-dimensional simplex. This notation is available with\n  `open Simplicial`.\n* `⦋m⦌ₙ` denotes the `m`-dimensional simplex in the `n`-truncated simplex category.\n  The truncation proof `p : m ≤ n` can also be provided using the syntax `⦋m, p⦌ₙ`.\n  This notation is available with `open SimplexCategory.Truncated`.\n-/\n\n@[expose] public section\n\nuniverse v\n\nopen CategoryTheory\n\n/-- The simplex category:\n* for each `n : ℕ`, there is an object `⦋n⦌`;\n* morphisms `⦋n⦌ ⟶ ⦋m⦌` are monotone functions `Fin (n+1) → Fin (m+1)`\n-/\n@[ext]\nstructure SimplexCategory : Type where\n  /-- Constructor `ℕ → SimplexCategory`. -/\n  mk ::\n  /-- The length of an object in `SimplexCategory` -/\n  len : ℕ\n\nnamespace SimplexCategory\n\n/-- the `n`-dimensional simplex can be denoted `⦋n⦌` -/\nscoped[Simplicial] notation \"⦋\" n \"⦌\" => SimplexCategory.mk n\n\nopen Simplicial\n\ntheorem len_mk (n : ℕ) : ⦋n⦌.len = n := rfl\n\n@[simp]\ntheorem mk_len (n : SimplexCategory) : ⦋n.len⦌ = n :=\n  rfl\n\n/-- Morphisms in the `SimplexCategory`. -/\nprotected def Hom (a b : SimplexCategory) :=\n  Fin (a.len + 1) →o Fin (b.len + 1)\n\nnamespace Hom\n\n/-- Make a morphism in `SimplexCategory` from a monotone map of `Fin`'s. -/\ndef mk {a b : SimplexCategory} (f : Fin (a.len + 1) →o Fin (b.len + 1)) : SimplexCategory.Hom a b :=\n  f\n\n/-- Recover the monotone map from a morphism in the simplex category. -/\ndef toOrderHom {a b : SimplexCategory} (f : SimplexCategory.Hom a b) :\n    Fin (a.len + 1) →o Fin (b.len + 1) :=\n  f\n\ntheorem ext' {a b : SimplexCategory} (f g : SimplexCategory.Hom a b) :\n    f.toOrderHom = g.toOrderHom → f = g :=\n  id\n\n@[simp]\ntheorem mk_toOrderHom {a b : SimplexCategory} (f : SimplexCategory.Hom a b) : mk f.toOrderHom = f :=\n  rfl\n\n@[simp]\ntheorem toOrderHom_mk {a b : SimplexCategory} (f : Fin (a.len + 1) →o Fin (b.len + 1)) :\n    (mk f).toOrderHom = f :=\n  rfl\n\ntheorem mk_toOrderHom_apply {a b : SimplexCategory} (f : Fin (a.len + 1) →o Fin (b.len + 1))\n    (i : Fin (a.len + 1)) : (mk f).toOrderHom i = f i :=\n  rfl\n\n/-- Identity morphisms of `SimplexCategory`. -/\n@[simp]\ndef id (a : SimplexCategory) : SimplexCategory.Hom a a :=\n  mk OrderHom.id\n\n/-- Composition of morphisms of `SimplexCategory`. -/\n@[simp]\ndef comp {a b c : SimplexCategory} (f : SimplexCategory.Hom b c) (g : SimplexCategory.Hom a b) :\n    SimplexCategory.Hom a c :=\n  mk <| f.toOrderHom.comp g.toOrderHom\n\nend Hom\n\nattribute [irreducible] SimplexCategory.Hom\n\ninstance smallCategory : SmallCategory.{0} SimplexCategory where\n  Hom n m := SimplexCategory.Hom n m\n  id _ := SimplexCategory.Hom.id _\n  comp f g := SimplexCategory.Hom.comp g f\n\n@[simp]\nlemma id_toOrderHom (a : SimplexCategory) :\n    Hom.toOrderHom (𝟙 a) = OrderHom.id := rfl\n\n@[simp]\nlemma comp_toOrderHom {a b c : SimplexCategory} (f : a ⟶ b) (g : b ⟶ c) :\n    (f ≫ g).toOrderHom = g.toOrderHom.comp f.toOrderHom := rfl\n\n@[ext]\ntheorem Hom.ext {a b : SimplexCategory} (f g : a ⟶ b) :\n    f.toOrderHom = g.toOrderHom → f = g :=\n  Hom.ext' _ _\n\n/-- Homs in `SimplexCategory` are equivalent to order-preserving functions of finite linear\norders. -/\ndef homEquivOrderHom {a b : SimplexCategory} :\n    (a ⟶ b) ≃ (Fin (a.len + 1) →o Fin (b.len + 1)) where\n  toFun := Hom.toOrderHom\n  invFun := Hom.mk\n\n/-- Homs in `SimplexCategory` are equivalent to functors between finite linear orders. -/\ndef homEquivFunctor {a b : SimplexCategory} :\n    (a ⟶ b) ≃ (Fin (a.len + 1) ⥤ Fin (b.len + 1)) :=\n  SimplexCategory.homEquivOrderHom.trans OrderHom.equivFunctor\n\n/-- The truncated simplex category. -/\nabbrev Truncated (n : ℕ) :=\n  ObjectProperty.FullSubcategory fun a : SimplexCategory => a.len ≤ n\n\nnamespace Truncated\n\ninstance {n} : Inhabited (Truncated n) :=\n  ⟨⟨⦋0⦌, by simp⟩⟩\n\n/-- The fully faithful inclusion of the truncated simplex category into the usual\nsimplex category.\n-/\nabbrev inclusion (n : ℕ) : SimplexCategory.Truncated n ⥤ SimplexCategory :=\n  ObjectProperty.ι _\n\n/-- A proof that the full subcategory inclusion is fully faithful -/\nnoncomputable def inclusion.fullyFaithful (n : ℕ) :\n    (inclusion n : Truncated n ⥤ _).op.FullyFaithful :=\n  Functor.FullyFaithful.ofFullyFaithful _\n\n@[ext]\ntheorem Hom.ext {n} {a b : Truncated n} (f g : a ⟶ b)\n    (h : f.hom.toOrderHom = g.hom.toOrderHom) : f = g :=\n  ObjectProperty.hom_ext _ (SimplexCategory.Hom.ext _ _ h)\n\n/-- A quick attempt to prove that `⦋m⦌` is `n`-truncated (`⦋m⦌.len ≤ n`). -/\nscoped macro \"trunc\" : tactic =>\n  `(tactic| first | assumption | dsimp only [SimplexCategory.len_mk] <;> lia)\n\nopen Mathlib.Tactic (subscriptTerm) in\n/-- For `m ≤ n`, `⦋m⦌ₙ` is the `m`-dimensional simplex in `Truncated n`. The\nproof `p : m ≤ n` can also be provided using the syntax `⦋m, p⦌ₙ`. -/\nscoped syntax:max (name := mkNotation)\n  \"⦋\" term (\",\" term)? \"⦌\" noWs subscriptTerm : term\nscoped macro_rules\n  | `(⦋$m:term⦌$n:subscript) =>\n    `((⟨SimplexCategory.mk $m, by first | trunc |\n      fail \"Failed to prove truncation property. Try writing `⦋m, by ...⦌ₙ`.\"⟩ :\n      SimplexCategory.Truncated $n))\n  | `(⦋$m:term, $p:term⦌$n:subscript) =>\n    `((⟨SimplexCategory.mk $m, $p⟩ : SimplexCategory.Truncated $n))\n\n/-- Make a morphism in `Truncated n` from a morphism in `SimplexCategory`. This\nis equivalent to `@id (⦋a⦌ₙ ⟶ ⦋b⦌ₙ) f`. -/\nabbrev Hom.tr {n : ℕ} {a b : SimplexCategory} (f : a ⟶ b)\n    (ha : a.len ≤ n := by trunc) (hb : b.len ≤ n := by trunc) :\n    (⟨a, ha⟩ : Truncated n) ⟶ ⟨b, hb⟩ :=\n  ObjectProperty.homMk f\n\n@[simp]\nlemma Hom.tr_id {n : ℕ} (a : SimplexCategory) (ha : a.len ≤ n := by trunc) :\n    Hom.tr (𝟙 a) ha = 𝟙 _ := rfl\n\n@[reassoc]\n\nTarget:\nlemma Hom.tr_comp {n : ℕ} {a b c : SimplexCategory} (f : a ⟶ b) (g : b ⟶ c)\n    (ha : a.len ≤ n :=\n\nProof body:\n","rejected":"by trunc) (hb : b.len ≤ n := by trunc)\n    (hc : c.len ≤ n := by trunc) :\n    tr (f ≫ g) = tr f ≫ tr g :=\n  rfl\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplexCategory","family_id":"hom","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplexCategory/Defs.lean","sample_id":"cb9967aa5d54ae74d6b7c334cf672946e1ac267085a38bfbf8b9d07e6fea6e09"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6dfab6bad543a2d9c97fdd7bad870518b9b6feb74bbecb3fcb1c2ddbf6b41bd4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"99db565f5bf6a82c918f0e5129424afa9bc1b4ea5c09b335daef9165e292b4ec","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"44eed7b983007fa4c1d9c73cb7fb1872722dad2bf7f08f9f9ca305f717904a0a","source_sha256":"e3bd64c53d80fc2f3ac9f455d14462bc0198848b67a51278e36a13f2aa735979","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : IsReflexive R N := e.isReflexive_of_equiv_dual_of_isReflexive\n  suffices\n      (p.map (e.symm : Dual R M →ₗ[R] N)).dualAnnihilator.map (e.flip.symm : Dual R N →ₗ[R] M) =\n        (p.dualCoannihilator.map (e.flip : M →ₗ[R] Dual R N)).map (e.flip.symm : Dual R N →ₗ[R] M)\n    from (Submodule.map_injective_of_injective e.flip.symm.injective this).symm\n  rw [← dualCoannihilator_map_linearEquiv_flip, ← LinearEquiv.coe_toLinearMap_flip,\n    LinearEquiv.flip_flip, ← map_comp, ← map_comp]\n  simp [-coe_toLinearMap_flip]","hard_negative":true,"metrics":{"chosen_tokens":135,"rejected_tokens":5,"token_jaccard":0.045455,"token_length_ratio":0.037037},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"48f1969b216dbd95031c3b48ec4e85623403157065635f327862e043e91270eb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Dual.Lemmas\n\nNamespace:\nSubmodule\n\nLocal context:\n/-\nCopyright (c) 2023 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash, Yaël Dillies\n-/\n/-!\n# Perfect pairings\n\nThis file defines perfect pairings of modules.\n\nA perfect pairing of two (left) modules may be defined either as:\n1. A bilinear map `M × N → R` such that the induced maps `M → Dual R N` and `N → Dual R M` are both\n  bijective. It follows from this that both `M` and `N` are reflexive modules.\n2. A linear equivalence `N ≃ Dual R M` for which `M` is reflexive. (It then follows that `N` is\n  reflexive.)\n\nIn this file we provide a definition `IsPerfPair` corresponding to 1 above, together with logic\nto connect 1 and 2.\n-/\n\n@[expose] public section\n\nopen Function Module\n\nnamespace LinearMap\nvariable {R K M M' N N' : Type*} [AddCommGroup M] [AddCommGroup N] [AddCommGroup M']\n  [AddCommGroup N']\n\nsection CommRing\nvariable [CommRing R] [Module R M] [Module R M'] [Module R N] [Module R N']\n  {p : M →ₗ[R] N →ₗ[R] R} {x : M} {y : N}\n\n/-- For a ring `R` and two modules `M` and `N`, a perfect pairing is a bilinear map `M × N → R`\nthat is bijective in both arguments. -/\n@[ext]\nclass IsPerfPair (p : M →ₗ[R] N →ₗ[R] R) where\n  bijective_left (p) : Bijective p\n  bijective_right (p) : Bijective p.flip\n\n/-- Given a perfect pairing between `M` and `N`, we may interchange the roles of `M` and `N`. -/\nprotected lemma IsPerfPair.flip (hp : p.IsPerfPair) : p.flip.IsPerfPair where\n  bijective_left := IsPerfPair.bijective_right p\n  bijective_right := IsPerfPair.bijective_left p\n\nvariable [p.IsPerfPair]\n\n/-- Given a perfect pairing between `M` and `N`, we may interchange the roles of `M` and `N`. -/\ninstance flip.instIsPerfPair : p.flip.IsPerfPair := .flip ‹_›\n\nvariable (p)\n\n/-- Turn a perfect pairing between `M` and `N` into an isomorphism between `M` and the dual of `N`.\n-/\nnoncomputable def toPerfPair : M ≃ₗ[R] Dual R N :=\n  .ofBijective { toFun := _, map_add' x y := by simp, map_smul' r x := by simp } <|\n    IsPerfPair.bijective_left p\n\n@[simp] lemma toLinearMap_toPerfPair (x : M) : p.toPerfPair x = p x := rfl\n@[simp] lemma toPerfPair_apply (x : M) (y : N) : p.toPerfPair x y = p x y := rfl\n\n@[simp] lemma apply_symm_toPerfPair_self (f : Dual R N) : p (p.toPerfPair.symm f) = f :=\n  p.toPerfPair.apply_symm_apply f\n\n@[simp] lemma apply_toPerfPair_flip (f : Dual R M) (x : M) : p x (p.flip.toPerfPair.symm f) = f x :=\n  congr($(p.flip.apply_symm_toPerfPair_self ..) x)\n\ninclude p in\nlemma _root_.Module.IsReflexive.of_isPerfPair : IsReflexive R M where\n  bijective_dual_eval' := by\n    convert! (p.toPerfPair.trans p.flip.toPerfPair.dualMap.symm).bijective\n    ext x f\n    simp\n\ninclude p in\nlemma _root_.Module.finrank_of_isPerfPair [Module.Finite R M] [Module.Free R M] :\n    finrank R M = finrank R N :=\n  ((Module.Free.chooseBasis R M).toDualEquiv.trans p.flip.toPerfPair.symm).finrank_eq\n\n/-- A reflexive module has a perfect pairing with its dual. -/\nprotected instance IsPerfPair.id [IsReflexive R M] : IsPerfPair (.id (R := R) (M := Dual R M)) where\n  bijective_left := bijective_id\n  bijective_right := bijective_dual_eval R M\n\n/-- A reflexive module has a perfect pairing with its dual. -/\ninstance IsPerfPair.dualEval [IsReflexive R M] : IsPerfPair (Dual.eval R M) := .flip .id\n\ninstance IsPerfPair.compl₁₂ (eM : M' ≃ₗ[R] M) (eN : N' ≃ₗ[R] N) :\n    (p.compl₁₂ eM eN : M' →ₗ[R] N' →ₗ[R] R).IsPerfPair :=\n  ⟨((LinearEquiv.congrLeft R R eN).symm.bijective.comp\n    (IsPerfPair.bijective_left p)).comp eM.bijective,\n    ((LinearEquiv.congrLeft R R eM).symm.bijective.comp\n    (IsPerfPair.bijective_right p)).comp eN.bijective⟩\n\nlemma IsPerfPair.congr (eM : M' ≃ₗ[R] M) (eN : N' ≃ₗ[R] N) (q : M' →ₗ[R] N' →ₗ[R] R)\n    (H : q.compl₁₂ eM.symm eN.symm = p) : q.IsPerfPair := by\n  obtain rfl : q = p.compl₁₂ eM eN := by subst H; ext; simp\n  infer_instance\n\nlemma IsPerfPair.of_bijective (p : M →ₗ[R] N →ₗ[R] R) [IsReflexive R N] (h : Bijective p) :\n    IsPerfPair p :=\n  inferInstanceAs ((LinearMap.id (R := R) (M := Dual R N)).compl₁₂\n    (LinearEquiv.ofBijective p h : M →ₗ[R] N →ₗ[R] R)\n    (LinearEquiv.refl R N : N →ₗ[R] N)).IsPerfPair\n\nend CommRing\n\nsection Field\nvariable [Field K] [Module K M] [Module K N] {p : M →ₗ[K] N →ₗ[K] K} {x : M} {y : N}\n\n/-- If the coefficients are a field, and one of the spaces is finite-dimensional, it is sufficient\nto check only injectivity instead of bijectivity of the bilinear pairing. -/\nlemma IsPerfPair.of_injective [FiniteDimensional K M] (h : Injective p) (h' : Injective p.flip) :\n    p.IsPerfPair where\n  bijective_left := ⟨h, by rwa [← p.flip_injective_iff₁]⟩\n  bijective_right := ⟨h', by\n    have : FiniteDimensional K N := FiniteDimensional.of_injective p.flip h'\n    rwa [← p.flip.flip_injective_iff₁, LinearMap.flip_flip]⟩\n\n/-- If the coefficients are a field, and one of the spaces is finite-dimensional, it is sufficient\nto check only injectivity instead of bijectivity of the bilinear pairing. -/\nlemma IsPerfPair.of_injective' [FiniteDimensional K N] (h : Injective p) (h' : Injective p.flip) :\n    p.IsPerfPair := .flip <| .of_injective h' h\n\nend Field\nend LinearMap\n\nnoncomputable section\n\nvariable {R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]\n\nnamespace LinearMap\nvariable {p : M →ₗ[R] N →ₗ[R] R} [p.IsPerfPair]\n\nvariable (p) in\n/-- Given a perfect pairing `p` between `M` and `N`, we say a pair of submodules `U` in `M` and\n`V` in `N` are perfectly complementary w.r.t. `p` if their dual annihilators are complementary,\nusing `p` to identify `M` and `N` with dual spaces. -/\nstructure IsPerfectCompl (U : Submodule R M) (V : Submodule R N) : Prop where\n  isCompl_left : IsCompl U (V.dualAnnihilator.map (p.toPerfPair.symm : Dual R N →ₗ[R] M))\n  isCompl_right : IsCompl V (U.dualAnnihilator.map (p.flip.toPerfPair.symm : Dual R M →ₗ[R] N))\n\nnamespace IsPerfectCompl\nvariable {U : Submodule R M} {V : Submodule R N}\n\nprotected lemma flip (h : p.IsPerfectCompl U V) :\n    p.flip.IsPerfectCompl V U where\n  isCompl_left := h.isCompl_right\n  isCompl_right := h.isCompl_left\n\n@[simp]\nprotected lemma flip_iff :\n    p.flip.IsPerfectCompl V U ↔ p.IsPerfectCompl U V :=\n  ⟨fun h ↦ h.flip, fun h ↦ h.flip⟩\n\n@[simp]\nlemma left_top_iff :\n    p.IsPerfectCompl ⊤ V ↔ V = ⊤ := by\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · exact eq_top_of_isCompl_bot <| by simpa using h.isCompl_right\n  · rw [h]\n    exact\n      { isCompl_left := by simpa using isCompl_top_bot\n        isCompl_right := by simpa using isCompl_top_bot }\n\n@[simp]\nlemma right_top_iff :\n    p.IsPerfectCompl U ⊤ ↔ U = ⊤ := by\n  rw [← IsPerfectCompl.flip_iff]\n  exact left_top_iff\n\nend IsPerfectCompl\n\nend LinearMap\n\nvariable [IsReflexive R M]\n\nvariable (e : N ≃ₗ[R] Dual R M)\n\nnamespace LinearEquiv\n\n/-- For a reflexive module `M`, an equivalence `N ≃ₗ[R] Dual R M` naturally yields an equivalence\n`M ≃ₗ[R] Dual R N`. Such equivalences are known as perfect pairings. -/\ndef flip : M ≃ₗ[R] Dual R N :=\n  (evalEquiv R M).trans e.dualMap\n\n@[simp] lemma coe_toLinearMap_flip : e.flip = (↑e : N →ₗ[R] Dual R M).flip := rfl\n\n@[simp] lemma flip_apply (m : M) (n : N) : e.flip m n = e n m := rfl\n\nlemma symm_flip : e.flip.symm = e.symm.dualMap.trans (evalEquiv R M).symm := rfl\n\nlemma trans_dualMap_symm_flip : e.trans e.flip.symm.dualMap = Dual.eval R N := by\n  ext; simp [symm_flip]\n\ninclude e in\n/-- If `N` is in perfect pairing with `M`, then it is reflexive. -/\nlemma isReflexive_of_equiv_dual_of_isReflexive : IsReflexive R N := by\n  constructor\n  rw [← trans_dualMap_symm_flip e]\n  exact LinearEquiv.bijective _\n\n@[simp] lemma flip_flip (h : IsReflexive R N := isReflexive_of_equiv_dual_of_isReflexive e) :\n    e.flip.flip = e := by\n  ext; rfl\n\ninstance : e.toLinearMap.IsPerfPair where\n  bijective_left := e.bijective\n  bijective_right := e.flip.bijective\n\nend LinearEquiv\n\nnamespace Submodule\n\nopen LinearEquiv\n\nomit [IsReflexive R M] in\n@[simp]\nlemma dualCoannihilator_map_linearEquiv_flip (p : Submodule R M) :\n    (p.map e.toLinearMap.flip).dualCoannihilator =\n      p.dualAnnihilator.map (e.symm : Dual R M →ₗ[R] N) := by\n  ext; simp\n\n@[simp]\nlemma map_dualAnnihilator_linearEquiv_flip_symm (p : Submodule R N) :\n    p.dualAnnihilator.map (e.flip.symm : Dual R N →ₗ[R] M) =\n      (p.map (e : N →ₗ[R] Dual R M)).dualCoannihilator := by\n  have : IsReflexive R N := e.isReflexive_of_equiv_dual_of_isReflexive\n  rw [← dualCoannihilator_map_linearEquiv_flip, ← LinearEquiv.coe_toLinearMap_flip,\n    LinearEquiv.flip_flip]\n\n@[simp]\n\nTarget:\nlemma map_dualCoannihilator_linearEquiv_flip (p : Submodule R (Dual R M)) :\n    p.dualCoannihilator.map e.toLinearMap.flip =\n      (p.map (e.symm : Dual R M →ₗ[R] N)).dualAnnihilator :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_44eed7b98300","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"9d4013c29b476e968b384a48f875a1490053a0e3881ddb4af45ad8c5170c048a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/PerfectPairing","family_id":"map_dualcoannihilator_linearequiv_flip","file_id":"mathlib/Mathlib/LinearAlgebra/PerfectPairing/Basic.lean","sample_id":"44eed7b983007fa4c1d9c73cb7fb1872722dad2bf7f08f9f9ca305f717904a0a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b925b94bb1cfb3ff34ceacbe250abd23143fbe7540078964bfa526bb642e3c9f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f7bbb727f1ea351360c7052b952d4fc3113423ef399aac4de4fe2b1d680daf36","source_sha256":"31cc90ac880af63d7e71cce55cea648ad2ff660c87dd9690b4a90bd3c1b360e1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  simpa [dotProduct_comm] using! (isSymmetric_toEuclideanLin_iff.mpr hA).im_inner_self_apply _","hard_negative":true,"metrics":{"chosen_tokens":17,"rejected_tokens":8,"token_jaccard":0.045455,"token_length_ratio":0.470588},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"490ea10d3bc19ff6e1430cf39619d1a15a51e8866b65121442418c21bc5f4646","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.InnerProductSpace.PiL2\npublic import Mathlib.LinearAlgebra.Matrix.Hermitian\nimport Mathlib.Analysis.InnerProductSpace.Adjoint\n\nNamespace:\nMatrix\n\nLocal context:\n/-\nCopyright (c) 2022 Alexander Bentkamp. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alexander Bentkamp\n-/\n/-!\n# Hermitian matrices over ℝ and ℂ\n\nThis file proves that Hermitian matrices over ℝ and ℂ are exactly the ones whose corresponding\nlinear map is self-adjoint.\n\n## Tags\n\nself-adjoint matrix, hermitian matrix\n-/\n\npublic section\n\n-- TODO:\n-- assert_not_exists MonoidAlgebra\n\nopen RCLike\n\nnamespace Matrix\n\nvariable {𝕜 m n : Type*} {A : Matrix n n 𝕜} [RCLike 𝕜]\n\n/-- The diagonal elements of a complex Hermitian matrix are real. -/\nlemma IsHermitian.coe_re_apply_self (h : A.IsHermitian) (i : n) : (re (A i i) : 𝕜) = A i i := by\n  rw [← conj_eq_iff_re, ← star_def, ← conjTranspose_apply, h.eq]\n\n/-- The diagonal elements of a complex Hermitian matrix are real. -/\nlemma IsHermitian.coe_re_diag (h : A.IsHermitian) : (fun i => (re (A.diag i) : 𝕜)) = A.diag :=\n  funext h.coe_re_apply_self\n\n/-- A matrix is Hermitian iff the corresponding linear map with an orthonormal basis is\nsymmetric. -/\n@[simp]\nlemma isSymmetric_toLin_iff [Fintype n] [DecidableEq n] {E : Type*}\n    [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (b : OrthonormalBasis n 𝕜 E) :\n    (A.toLin b.toBasis b.toBasis).IsSymmetric ↔ A.IsHermitian := by\n  have : FiniteDimensional 𝕜 E := b.toBasis.finiteDimensional_of_finite\n  simp_rw [LinearMap.IsSymmetric, ← LinearMap.adjoint_inner_left, ← toLin_conjTranspose]\n  refine ⟨fun h ↦ ?_, fun h _ _ ↦ by rw [h.eq]⟩\n  simpa using! (LinearMap.ext fun x ↦ ext_inner_right _ (h x)).symm\n\n/-- A matrix is Hermitian iff the corresponding linear map on the Euclidean space is\nsymmetric. -/\n@[simp]\nlemma isSymmetric_toEuclideanLin_iff [Fintype n] [DecidableEq n] :\n    A.toEuclideanLin.IsSymmetric ↔ A.IsHermitian :=\n  isSymmetric_toLin_iff (EuclideanSpace.basisFun n 𝕜)\n\n@[deprecated isSymmetric_toEuclideanLin_iff \"use isSymmetric_toEuclideanLin_iff.symm\"\n  (since := \"2026-03-30\")]\nlemma isHermitian_iff_isSymmetric [Fintype n] [DecidableEq n] :\n    IsHermitian A ↔ A.toEuclideanLin.IsSymmetric := isSymmetric_toEuclideanLin_iff.symm\n\nTarget:\nlemma IsHermitian.im_star_dotProduct_mulVec_self [Fintype n] (hA : A.IsHermitian) (x : n → 𝕜) :\n     RCLike.im (star x ⬝ᵥ A *ᵥ x) = 0 :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"85ae0ff883f1bceaaf5429811a4454c8f50827d09e85c1e537a7723d25e5300e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Matrix","family_id":"ishermitian","file_id":"mathlib/Mathlib/Analysis/Matrix/Hermitian.lean","sample_id":"f7bbb727f1ea351360c7052b952d4fc3113423ef399aac4de4fe2b1d680daf36"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b33ee369bdfae6b8f7816eb5661ea638ec7dc5ec9125117e507e6aaa3923f7b0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1da3d2e9dd76dc2f56adcba04528e39b3dd32ce83ca140add0d49391d97b5c7c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cfce958e665e03e21c1db8b838eb32bd3a0604379292b7cebef2b66be097f2db","source_sha256":"dcc51bf41b9226e46d854f6feb1d45a4ec3794545cf9d910bab23bf813dc4f3f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro a b h\n  refine le_of_lt_or_eq (or_iff_not_imp_right.2 fun hne ↦ ?_)\n  classical\n  exact ⟨Finset.min' _ (nonempty_neLocus_iff.2 hne),\n    fun j hj ↦ notMem_neLocus.1 fun h ↦ (Finset.min'_le _ _ h).not_gt hj,\n    (h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)⟩","hard_negative":false,"metrics":{"chosen_tokens":72,"rejected_tokens":77,"token_jaccard":0.9,"token_length_ratio":1.069444},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"4928049a9146bf2342c8c2325ad661352892774ef229ee483ac53b80a3b1a328","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.PiLex\npublic import Mathlib.Data.DFinsupp.Order\npublic import Mathlib.Data.DFinsupp.NeLocus\npublic import Mathlib.Order.WellFoundedSet\n\nNamespace:\nDFinsupp\n\nLocal context:\n/-\nCopyright (c) 2022 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Damiano Testa, Junyan Xu\n-/\n/-!\n# Lexicographic order on finitely supported dependent functions\n\nThis file defines the lexicographic order on `DFinsupp`.\n-/\n\n@[expose] public section\n\n\nvariable {ι : Type*} {α : ι → Type*}\n\nnamespace DFinsupp\n\nsection Zero\n\nvariable [∀ i, Zero (α i)]\n\n/-- `DFinsupp.Lex r s` is the lexicographic relation on `Π₀ i, α i`, where `ι` is ordered by `r`,\nand `α i` is ordered by `s i`.\n\nThe type synonym `Lex (Π₀ i, α i)` has an order given by `DFinsupp.Lex (· < ·) (· < ·)`, whereas\n`Colex (Π₀ i, α i)` has an order given by `DFinsupp.Lex (· > ·) (· < ·)`.\n-/\nprotected def Lex (r : ι → ι → Prop) (s : ∀ i, α i → α i → Prop) (x y : Π₀ i, α i) : Prop :=\n  Pi.Lex r (s _) x y\n\ntheorem _root_.Pi.lex_eq_dfinsupp_lex {r : ι → ι → Prop} {s : ∀ i, α i → α i → Prop}\n    (a b : Π₀ i, α i) : Pi.Lex r (s _) (a : ∀ i, α i) b = DFinsupp.Lex r s a b :=\n  rfl\n\ntheorem lex_def {r : ι → ι → Prop} {s : ∀ i, α i → α i → Prop} {a b : Π₀ i, α i} :\n    DFinsupp.Lex r s a b ↔ ∃ j, (∀ d, r d j → a d = b d) ∧ s j (a j) (b j) :=\n  .rfl\n\ninstance [LT ι] [∀ i, LT (α i)] : LT (Lex (Π₀ i, α i)) :=\n  ⟨fun f g ↦ DFinsupp.Lex (· < ·) (fun _ ↦ (· < ·)) (ofLex f) (ofLex g)⟩\n\ninstance [LT ι] [∀ i, LT (α i)] : LT (Colex (Π₀ i, α i)) :=\n  ⟨fun f g ↦ DFinsupp.Lex (· > ·) (fun _ ↦ (· < ·)) (ofColex f) (ofColex g)⟩\n\ntheorem Lex.lt_iff [LT ι] [∀ i, LT (α i)] {a b : Lex (Π₀ i, α i)} :\n    a < b ↔ ∃ i, (∀ j, j < i → a j = b j) ∧ a i < b i :=\n  .rfl\n\n@[deprecated (since := \"2025-11-29\")]\nalias lex_lt_iff := Lex.lt_iff\n\ntheorem Colex.lt_iff [LT ι] [∀ i, LT (α i)] {a b : Colex (Π₀ i, α i)} :\n    a < b ↔ ∃ i, (∀ j, i < j → a j = b j) ∧ a i < b i :=\n  .rfl\n\ntheorem lex_lt_of_lt_of_preorder [∀ i, Preorder (α i)] (r) [IsStrictOrder ι r] {x y : Π₀ i, α i}\n    (hlt : x < y) : ∃ i, (∀ j, r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i := by\n  obtain ⟨hle, j, hlt⟩ := Pi.lt_def.1 hlt\n  classical\n  have : (x.neLocus y : Set ι).WellFoundedOn r := (x.neLocus y).finite_toSet.wellFoundedOn\n  obtain ⟨i, hi, hl⟩ := this.has_min { i | x i < y i } ⟨⟨j, mem_neLocus.2 hlt.ne⟩, hlt⟩\n  refine ⟨i, fun k hk ↦ ⟨hle k, ?_⟩, hi⟩\n  exact of_not_not fun h ↦ hl ⟨k, mem_neLocus.2 (ne_of_not_le h).symm⟩ ((hle k).lt_of_not_ge h) hk\n\ntheorem lex_lt_of_lt [∀ i, PartialOrder (α i)] (r) [IsStrictOrder ι r] {x y : Π₀ i, α i}\n    (hlt : x < y) : Pi.Lex r (· < ·) x y := by\n  simp_rw [Pi.Lex, le_antisymm_iff]\n  exact lex_lt_of_lt_of_preorder r hlt\n\ntheorem lex_iff_of_unique [Unique ι] [∀ i, LT (α i)] {r} [Std.Irrefl r] {x y : Π₀ i, α i} :\n    DFinsupp.Lex r (fun _ ↦ (· < ·)) x y ↔ x default < y default :=\n  Pi.lex_iff_of_unique\n\ntheorem Lex.lt_iff_of_unique [Unique ι] [∀ i, LT (α i)] [Preorder ι] {x y : Lex (Π₀ i, α i)} :\n    x < y ↔ x default < y default :=\n  lex_iff_of_unique\n\n@[deprecated (since := \"2025-11-29\")]\nalias lex_lt_iff_of_unique := Lex.lt_iff_of_unique\n\ntheorem colex_lt_iff_of_unique [Unique ι] [∀ i, LT (α i)] [Preorder ι] {x y : Colex (Π₀ i, α i)} :\n    x < y ↔ x default < y default :=\n  lex_iff_of_unique\n\nvariable [LinearOrder ι]\n\ninstance Lex.isStrictOrder [∀ i, PartialOrder (α i)] :\n    IsStrictOrder (Lex (Π₀ i, α i)) (· < ·) where\n  irrefl _ := lt_irrefl (α := Lex (∀ i, α i)) _\n  trans _ _ _ := lt_trans (α := Lex (∀ i, α i))\n\ninstance Colex.isStrictOrder [∀ i, PartialOrder (α i)] :\n    IsStrictOrder (Colex (Π₀ i, α i)) (· < ·) :=\n  Lex.isStrictOrder (ι := ιᵒᵈ)\n\n/-- The partial order on `DFinsupp`s obtained by the lexicographic ordering.\nSee `DFinsupp.Lex.linearOrder` for a proof that this partial order is in fact linear. -/\ninstance Lex.partialOrder [∀ i, PartialOrder (α i)] : PartialOrder (Lex (Π₀ i, α i)) where\n  le x y := ⇑(ofLex x) = ⇑(ofLex y) ∨ x < y\n  __ := PartialOrder.lift (fun x : Lex (Π₀ i, α i) ↦ toLex (⇑(ofLex x)))\n    (DFunLike.coe_injective (F := DFinsupp α))\n\n/-- The partial order on `DFinsupp`s obtained by the colexicographic ordering.\nSee `DFinsupp.Colex.linearOrder` for a proof that this partial order is in fact linear. -/\ninstance Colex.partialOrder [∀ i, PartialOrder (α i)] : PartialOrder (Colex (Π₀ i, α i)) where\n  le x y := ⇑(ofColex x) = ⇑(ofColex y) ∨ x < y\n  __ := PartialOrder.lift (fun x : Colex (Π₀ i, α i) ↦ toColex (⇑(ofColex x)))\n    (DFunLike.coe_injective (F := DFinsupp α))\n\ntheorem Lex.le_iff_of_unique [Unique ι] [∀ i, PartialOrder (α i)] {x y : Lex (Π₀ i, α i)} :\n    x ≤ y ↔ x default ≤ y default :=\n  Pi.lex_le_iff_of_unique\n\n@[deprecated (since := \"2025-11-29\")]\nalias lex_le_iff_of_unique := Lex.le_iff_of_unique\n\ntheorem Colex.le_iff_of_unique [Unique ι] [∀ i, PartialOrder (α i)] {x y : Colex (Π₀ i, α i)} :\n    x ≤ y ↔ x default ≤ y default :=\n  Lex.le_iff_of_unique (ι := ιᵒᵈ)\n\nsection LinearOrder\n\nvariable [∀ i, LinearOrder (α i)]\n\nset_option backward.privateInPublic true in\n/-- Auxiliary helper to case split computably. There is no need for this to be public, as it\ncan be written with `Or.by_cases` on `lt_trichotomy` once the instances below are constructed. -/\nprivate def lt_trichotomy_rec {P : Lex (Π₀ i, α i) → Lex (Π₀ i, α i) → Sort*}\n    (h_lt : ∀ {f g}, toLex f < toLex g → P (toLex f) (toLex g))\n    (h_eq : ∀ {f g}, toLex f = toLex g → P (toLex f) (toLex g))\n    (h_gt : ∀ {f g}, toLex g < toLex f → P (toLex f) (toLex g)) : ∀ f g, P f g :=\n  Lex.rec fun f ↦ Lex.rec fun g ↦ match (motive := ∀ y, (f.neLocus g).min = y → _) _, rfl with\n  | ⊤, h => h_eq (neLocus_eq_empty.mp <| Finset.min_eq_top.mp h)\n  | (wit : ι), h => by\n    apply (mem_neLocus.mp <| Finset.mem_of_min h).lt_or_gt.by_cases <;> intro hwit\n    · exact h_lt ⟨wit, fun j hj ↦ notMem_neLocus.mp (Finset.notMem_of_lt_min hj h), hwit⟩\n    · exact h_gt ⟨wit, fun j hj ↦\n        notMem_neLocus.mp (Finset.notMem_of_lt_min hj <| by rwa [neLocus_comm]), hwit⟩\n\ninstance Lex.total_le : @Std.Total (Lex (Π₀ i, α i)) (· ≤ ·) where\n  total := lt_trichotomy_rec (fun h ↦ Or.inl h.le) (fun h ↦ Or.inl h.le) fun h ↦ Or.inr h.le\n\ninstance Colex.total_le : @Std.Total (Colex (Π₀ i, α i)) (· ≤ ·) :=\n  Lex.total_le (ι := ιᵒᵈ)\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- The less-or-equal relation for the lexicographic ordering is decidable. -/\ninstance Lex.decidableLE : DecidableLE (Lex (Π₀ i, α i)) :=\n  lt_trichotomy_rec (fun h ↦ isTrue <| Or.inr h)\n    (fun h ↦ isTrue <| Or.inl <| congr_arg _ h)\n    fun h ↦ isFalse fun h' ↦ lt_irrefl _ (h.trans_le h')\n\n/-- The less-or-equal relation for the colexicographic ordering is decidable. -/\ninstance Colex.decidableLE : DecidableLE (Colex (Π₀ i, α i)) :=\n  Lex.decidableLE (ι := ιᵒᵈ)\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- The less-than relation for the lexicographic ordering is decidable. -/\ninstance Lex.decidableLT : DecidableLT (Lex (Π₀ i, α i)) :=\n  lt_trichotomy_rec (fun h ↦ isTrue h) (fun h ↦ isFalse h.not_lt) fun h ↦ isFalse h.asymm\n\n/-- The less-than relation for the colexicographic ordering is decidable. -/\ninstance Colex.decidableLT : DecidableLT (Colex (Π₀ i, α i)) :=\n  Lex.decidableLT (ι := ιᵒᵈ)\n\n/-- The linear order on `DFinsupp`s obtained by the lexicographic ordering. -/\ninstance Lex.linearOrder : LinearOrder (Lex (Π₀ i, α i)) where\n  __ := Lex.partialOrder\n  le_total := total_of _\n  toDecidableLT := decidableLT\n  toDecidableLE := decidableLE\n\n/-- The linear order on `DFinsupp`s obtained by the colexicographic ordering. -/\ninstance Colex.linearOrder : LinearOrder (Colex (Π₀ i, α i)) where\n  __ := Colex.partialOrder\n  le_total := total_of _\n  toDecidableLT := decidableLT\n  toDecidableLE := decidableLE\n\nend LinearOrder\n\nvariable [∀ i, PartialOrder (α i)]\n\nTarget:\ntheorem toLex_monotone : Monotone (@toLex (Π₀ i, α i)) :=\n\nProof body:\n","rejected":"by\n  intro a b h\n  refine le_of_lt_or_eq (or_iff_not_imp_right.2 fun hne ↦ ?_)\n  classical\n  exact ⟨Finset.min' _ (nonempty_neLocus_iff.2 hne),\n    fun j hj ↦ notMem_neLocus.1 fun h ↦ (Finset.min'_le _ _ h).not_gt hj,\n    (h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)⟩\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/DFinsupp","family_id":"tolex_monotone","file_id":"mathlib/Mathlib/Data/DFinsupp/Lex.lean","sample_id":"cfce958e665e03e21c1db8b838eb32bd3a0604379292b7cebef2b66be097f2db"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e79e7e0dc18cc849a7cc458c4f413b9fd00b3758b5b3041d108cd5fc7c5dd925","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"43a95204387fcb917327a44f5490f1b702dfabe466759d72dbcc1d90f4cdf8d9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"05a81ca678f684a0e635006515f905d4791271bb1386b472a3d4d7bf0d040e70","source_sha256":"94fce5005b3452fffd0935ed1c8380f811db0b64c02673ffec8176805d9749dd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun _ ↦ ?_, fun _ ↦ inferInstance⟩\n  let e : G.inv ⋙ G ⋙ F ≅ F := (associator _ _ _).symm ≪≫\n    isoWhiskerRight (G.asEquivalence.counitIso) _ ≪≫ F.leftUnitor\n  exact of_iso e","hard_negative":true,"metrics":{"chosen_tokens":54,"rejected_tokens":5,"token_jaccard":0.088235,"token_length_ratio":0.092593},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"493c3d055e550b3d98ba02927645c7b31c70375b6a0e3bf8514c1d058f370249","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Functor.KanExtension.DenseAt\npublic import Mathlib.CategoryTheory.Limits.Presheaf\npublic import Mathlib.CategoryTheory.Generator.StrongGenerator\n\nNamespace:\nCategoryTheory.Functor\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Dense functors\n\nA functor `F : C ⥤ D` is dense (`F.IsDense`) if `𝟭 D` is a pointwise\nleft Kan extension of `F` along itself, i.e. any `Y : D` is the\ncolimit of all `F.obj X` for all morphisms `F.obj X ⟶ Y` (which\nis the condition `F.DenseAt Y`).\nWhen `F` is full, we show that this\nis equivalent to saying that the restricted Yoneda functor\n`D ⥤ Cᵒᵖ ⥤ Type _` is fully faithful (see the lemma\n`Functor.isDense_iff_fullyFaithful_restrictedULiftYoneda`).\n\nWe also show that the range of a dense functor is a strong\ngenerator (see `Functor.isStrongGenerator_of_isDense`).\n\n## References\n\n* https://ncatlab.org/nlab/show/dense+subcategory\n\n-/\n\n@[expose] public section\n\nuniverse w v₁ v₂ v₃ u₁ u₂ u₃\n\nnamespace CategoryTheory\n\nopen Limits Opposite Presheaf ConcreteCategory\n\nvariable {C : Type u₁} {D : Type u₂} [Category.{v₁} C] [Category.{v₂} D]\n  {C' : Type u₃} [Category.{v₃} C']\n\nnamespace Functor\n\n/-- A functor `F : C ⥤ D` is dense if any `Y : D` is a canonical colimit\nrelatively to `F`. -/\nclass IsDense (F : C ⥤ D) : Prop where\n  isDenseAt (F) (Y : D) : F.isDenseAt Y\n\n/-- This is a choice of structure `F.DenseAt Y` when `F : C ⥤ D`\nis dense, and `Y : D`. -/\nnoncomputable def denseAt (F : C ⥤ D) [F.IsDense] (Y : D) : F.DenseAt Y :=\n  (IsDense.isDenseAt F Y).some\n\nlemma isDense_iff_nonempty_isPointwiseLeftKanExtension (F : C ⥤ D) :\n    F.IsDense ↔\n      Nonempty ((LeftExtension.mk _ (rightUnitor F).inv).IsPointwiseLeftKanExtension) :=\n  ⟨fun _ ↦ ⟨fun _ ↦ F.denseAt _⟩, fun ⟨h⟩ ↦ ⟨fun _ ↦ ⟨h _⟩⟩⟩\n\ninstance (F : C ⥤ D) [F.IsDense] : Functor.IsLeftKanExtension (𝟭 D) (Functor.rightUnitor F).inv :=\n  ((Functor.isDense_iff_nonempty_isPointwiseLeftKanExtension F).mp ‹_›).some.isLeftKanExtension\n\ninstance (F : C ⥤ D) [F.IsDense] : F.HasPointwiseLeftKanExtension F :=\n  fun X ↦ (Functor.IsDense.isDenseAt F X).some.hasPointwiseLeftKanExtensionAt\n\nlemma IsDense.of_iso {F G : C ⥤ D} (e : F ≅ G) [F.IsDense] :\n    G.IsDense where\n  isDenseAt Y := by\n    rw [← Functor.congr_isDenseAt e]\n    exact ⟨F.denseAt Y⟩\n\nlemma IsDense.iff_of_iso {F G : C ⥤ D} (e : F ≅ G) :\n    F.IsDense ↔ G.IsDense :=\n  ⟨fun _ ↦ of_iso e, fun _ ↦ of_iso e.symm⟩\n\nvariable (F : C ⥤ D)\n\ninstance (G : C' ⥤ C) [F.IsDense] [G.IsEquivalence] :\n    (G ⋙ F).IsDense where\n  isDenseAt Y := ⟨(F.denseAt Y).precompOfFinal G⟩\n\nTarget:\nlemma IsDense.comp_left_iff_of_isEquivalence (G : C' ⥤ C) [G.IsEquivalence] :\n    (G ⋙ F).IsDense ↔ F.IsDense :=\n\nProof body:\n","rejected":"by\n  exact IsDense.comp_left_iff_of_isEquivalence","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"01f19691fd40e213a02c079d707e728b6e329f4561624e61745877ccf57be5af","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Functor","family_id":"isdense","file_id":"mathlib/Mathlib/CategoryTheory/Functor/KanExtension/Dense.lean","sample_id":"05a81ca678f684a0e635006515f905d4791271bb1386b472a3d4d7bf0d040e70"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6dbb4280aff812e88179a089ca7e30391af8146bdc02e8489a28db8f88a2918d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f56b4dc966c61de3f4b4118e398dfcc1e44b0ddfc9b9e3ab8b9799975c8612d7","source_sha256":"482679c638fc474fa50b011d709adb8dcdc790257773e34a00e39190b4d36d68","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext a\n  apply Subtype.ext\n  ext ⟨i, hi⟩\n  obtain ⟨⟨n, a, ⟨b, hb'⟩, i, rfl : _ = b⟩, rfl⟩ := mk_surjective a\n  simp only [homOfLE_leOfHom, CommRingCat.hom_comp, RingHom.coe_comp, Function.comp_apply]\n  erw [ProjectiveSpectrum.Proj.awayToSection_apply]\n  rw [CommRingCat.hom_ofHom, val_awayMap_mk, Localization.mk_eq_mk', IsLocalization.map_mk',\n    ← Localization.mk_eq_mk']\n  refine Localization.mk_eq_mk_iff.mpr ?_\n  rw [Localization.r_iff_exists]\n  use 1\n  simp [hx]\n  ring","hard_negative":true,"metrics":{"chosen_tokens":107,"rejected_tokens":8,"token_jaccard":0.052632,"token_length_ratio":0.074766},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"497637912598f337141e1b86c2a8b5aa8303640a4d6ced13db40675c1e2642d6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme\npublic import Mathlib.AlgebraicGeometry.AffineScheme\npublic import Mathlib.AlgebraicGeometry.Gluing\n\nNamespace:\nAlgebraicGeometry.Proj\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Basic properties of the scheme `Proj A`\n\nThe scheme `Proj 𝒜` for a graded ring `𝒜` is constructed in\n`Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean`.\nIn this file we provide basic properties of the scheme.\n\n## Main results\n- `AlgebraicGeometry.Proj.toSpecZero`: The structure map `Proj A ⟶ Spec (A 0)`.\n- `AlgebraicGeometry.Proj.basicOpenIsoSpec`:\n  The canonical isomorphism `Proj A |_ D₊(f) ≅ Spec (A_f)₀`\n  when `f` is homogeneous of positive degree.\n- `AlgebraicGeometry.Proj.awayι`: The open immersion `Spec (A_f)₀ ⟶ Proj A`.\n- `AlgebraicGeometry.Proj.affineOpenCover`: The open cover of `Proj A` by `Spec (A_f)₀` for all\n  homogeneous `f` of positive degree.\n- `AlgebraicGeometry.Proj.stalkIso`:\n  The stalk of `Proj A` at `x` is the degree `0` part of the localization of `A` at `x`.\n- `AlgebraicGeometry.Proj.fromOfGlobalSections`:\n  Given a map `f : A →+* Γ(X, ⊤)` such that the image of the irrelevant ideal under `f`\n  generates the whole ring, we can construct a map `X ⟶ Proj 𝒜`.\n\n-/\n\n@[expose] public section\n\nnamespace AlgebraicGeometry.Proj\n\nopen HomogeneousLocalization CategoryTheory\n\nuniverse u\n\nvariable {σ : Type*} {A : Type u}\nvariable [CommRing A] [SetLike σ A] [AddSubgroupClass σ A]\nvariable (𝒜 : ℕ → σ)\nvariable [GradedRing 𝒜]\n\nsection basicOpen\n\nvariable (f g : A)\n\n/-- The basic open set `D₊(f)` associated to `f : A`. -/\ndef basicOpen : (Proj 𝒜).Opens :=\n  ProjectiveSpectrum.basicOpen 𝒜 f\n\n@[simp]\ntheorem mem_basicOpen (x : Proj 𝒜) :\n    x ∈ basicOpen 𝒜 f ↔ f ∉ x.asHomogeneousIdeal :=\n  Iff.rfl\n\n@[simp] theorem basicOpen_one : basicOpen 𝒜 1 = ⊤ := ProjectiveSpectrum.basicOpen_one ..\n\n@[simp] theorem basicOpen_zero : basicOpen 𝒜 0 = ⊥ := ProjectiveSpectrum.basicOpen_zero ..\n\n@[simp] theorem basicOpen_pow (n) (hn : 0 < n) : basicOpen 𝒜 (f ^ n) = basicOpen 𝒜 f :=\n  ProjectiveSpectrum.basicOpen_pow 𝒜 f n hn\n\ntheorem basicOpen_mul : basicOpen 𝒜 (f * g) = basicOpen 𝒜 f ⊓ basicOpen 𝒜 g :=\n  ProjectiveSpectrum.basicOpen_mul ..\n\ntheorem basicOpen_mono (hfg : f ∣ g) : basicOpen 𝒜 g ≤ basicOpen 𝒜 f :=\n  (hfg.choose_spec ▸ basicOpen_mul 𝒜 f _).trans_le inf_le_left\n\ntheorem basicOpen_eq_iSup_proj (f : A) :\n    basicOpen 𝒜 f = ⨆ i : ℕ, basicOpen 𝒜 (GradedRing.proj 𝒜 i f) :=\n  ProjectiveSpectrum.basicOpen_eq_union_of_projection ..\n\ntheorem isBasis_basicOpen :\n    TopologicalSpace.Opens.IsBasis (Set.range (basicOpen 𝒜)) := by\n  delta TopologicalSpace.Opens.IsBasis\n  convert! ProjectiveSpectrum.isTopologicalBasis_basic_opens 𝒜\n  exact (Set.range_comp _ _).symm\n\n/-- If `{ xᵢ }` spans the irrelevant ideal of `A`, then `D₊(xᵢ)` covers `Proj A`. -/\nlemma iSup_basicOpen_eq_top {ι : Type*} (f : ι → A)\n    (hf : (HomogeneousIdeal.irrelevant 𝒜).toIdeal ≤ Ideal.span (Set.range f)) :\n    ⨆ i, Proj.basicOpen 𝒜 (f i) = ⊤ := by\n  classical\n  refine top_le_iff.mp fun x hx ↦ TopologicalSpace.Opens.mem_iSup.mpr ?_\n  by_contra! H\n  simp only [mem_basicOpen, Decidable.not_not] at H\n  refine x.not_irrelevant_le (hf.trans ?_)\n  rwa [Ideal.span_le, Set.range_subset_iff]\n\n/-- If `{ xᵢ }` are homogeneous and span `A` as an `A₀` algebra, then `D₊(xᵢ)` covers `Proj A`. -/\nlemma iSup_basicOpen_eq_top' {ι : Type*} (f : ι → A)\n    (hfn : ∀ i, ∃ n, f i ∈ 𝒜 n)\n    (hf : Algebra.adjoin (𝒜 0) (Set.range f) = ⊤) :\n    ⨆ i, Proj.basicOpen 𝒜 (f i) = ⊤ := by\n  classical\n  apply Proj.iSup_basicOpen_eq_top\n  intro x hx\n  convert_to x - GradedRing.projZeroRingHom 𝒜 x ∈ _\n  · rw [GradedRing.projZeroRingHom_apply, ← GradedRing.proj_apply,\n      (HomogeneousIdeal.mem_irrelevant_iff _ _).mp hx, sub_zero]\n  clear hx\n  have := (eq_iff_iff.mp congr(x ∈ $hf)).mpr trivial\n  induction this using Algebra.adjoin_induction with\n  | mem x hx =>\n    obtain ⟨i, rfl⟩ := hx\n    obtain ⟨n, hn⟩ := hfn i\n    rw [GradedRing.projZeroRingHom_apply]\n    by_cases hn' : n = 0\n    · rw [DirectSum.decompose_of_mem_same 𝒜 (hn' ▸ hn), sub_self]\n      exact zero_mem _\n    · rw [DirectSum.decompose_of_mem_ne 𝒜 hn hn', sub_zero]\n      exact Ideal.subset_span ⟨_, rfl⟩\n  | algebraMap r =>\n    convert! zero_mem (Ideal.span _)\n    rw [sub_eq_zero]\n    exact (DirectSum.decompose_of_mem_same 𝒜 r.2).symm\n  | add x y hx hy _ _ =>\n    rw [map_add, add_sub_add_comm]\n    exact add_mem ‹_› ‹_›\n  | mul x y hx hy hx' hy' =>\n    convert!\n      add_mem (Ideal.mul_mem_left _ x hy')\n        (Ideal.mul_mem_right (GradedRing.projZeroRingHom 𝒜 y) _ hx') using 1\n    rw [map_mul]\n    ring\n\n/-- The canonical map `(A_f)₀ ⟶ Γ(Proj A, D₊(f))`.\nThis is an isomorphism when `f` is homogeneous of positive degree. See `basicOpenIsoAway` below. -/\ndef awayToSection : CommRingCat.of (Away 𝒜 f) ⟶ Γ(Proj 𝒜, basicOpen 𝒜 f) :=\n  ProjectiveSpectrum.Proj.awayToSection ..\n\n/-- The canonical map `Proj A |_ D₊(f) ⟶ Spec (A_f)₀`.\nThis is an isomorphism when `f` is homogeneous of positive degree. See `basicOpenIsoSpec` below. -/\nnoncomputable\ndef basicOpenToSpec : (basicOpen 𝒜 f).toScheme ⟶ Spec (.of <| Away 𝒜 f) :=\n  (basicOpen 𝒜 f).toSpecΓ ≫ Spec.map (awayToSection 𝒜 f)\n\nlemma basicOpenToSpec_app_top :\n    (basicOpenToSpec 𝒜 f).app ⊤ = (Scheme.ΓSpecIso _).hom ≫ awayToSection 𝒜 f ≫\n      (basicOpen 𝒜 f).topIso.inv := by\n  simp [basicOpenToSpec, Scheme.Opens.toSpecΓ_appTop]\n\n/-- The structure map `Proj A ⟶ Spec A₀`. -/\nnoncomputable\ndef toSpecZero : Proj 𝒜 ⟶ Spec (.of <| 𝒜 0) :=\n  (Scheme.topIso _).inv ≫ (Scheme.isoOfEq _ (basicOpen_one _)).inv ≫\n    basicOpenToSpec 𝒜 1 ≫ Spec.map (CommRingCat.ofHom (fromZeroRingHom 𝒜 _))\n\nvariable {m} (f_deg : f ∈ 𝒜 m) (hm : 0 < m)\n\n/-- The canonical isomorphism `Proj A |_ D₊(f) ≅ Spec (A_f)₀`\nwhen `f` is homogeneous of positive degree. -/\n@[simps! -isSimp hom]\nnoncomputable\ndef basicOpenIsoSpec : (basicOpen 𝒜 f).toScheme ≅ Spec (.of <| Away 𝒜 f) :=\n  have : IsIso (basicOpenToSpec 𝒜 f) := by\n    apply (isIso_iff_of_reflects_iso _ Scheme.forgetToLocallyRingedSpace).mp ?_\n    convert! ProjectiveSpectrum.Proj.isIso_toSpec 𝒜 f f_deg hm using 1\n    refine Eq.trans ?_ (ΓSpec.locallyRingedSpaceAdjunction.homEquiv_apply _ _ _).symm\n    dsimp [basicOpenToSpec, Scheme.Opens.toSpecΓ]\n    simp only [Category.assoc, ← Spec.map_comp]\n    rfl\n  asIso (basicOpenToSpec 𝒜 f)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The canonical isomorphism `(A_f)₀ ≅ Γ(Proj A, D₊(f))`\nwhen `f` is homogeneous of positive degree. -/\n@[simps! -isSimp hom]\nnoncomputable\ndef basicOpenIsoAway : CommRingCat.of (Away 𝒜 f) ≅ Γ(Proj 𝒜, basicOpen 𝒜 f) :=\n  have : IsIso (awayToSection 𝒜 f) := by\n    have := basicOpenToSpec_app_top 𝒜 f\n    rw [← Iso.inv_comp_eq, Iso.eq_comp_inv] at this\n    rw [← this, ← basicOpenIsoSpec_hom 𝒜 f f_deg hm]\n    infer_instance\n  asIso (awayToSection 𝒜 f)\n\n/-- The open immersion `Spec (A_f)₀ ⟶ Proj A`. -/\nnoncomputable\ndef awayι : Spec (.of <| Away 𝒜 f) ⟶ Proj 𝒜 :=\n  (basicOpenIsoSpec 𝒜 f f_deg hm).inv ≫ (Proj.basicOpen 𝒜 f).ι\n\n@[reassoc]\nlemma basicOpenIsoSpec_inv_ι :\n    (basicOpenIsoSpec 𝒜 f f_deg hm).inv ≫ (Proj.basicOpen 𝒜 f).ι = awayι 𝒜 f f_deg hm := rfl\n\ninstance : IsOpenImmersion (Proj.awayι 𝒜 f f_deg hm) :=\n  IsOpenImmersion.comp _ _\n\nlemma opensRange_awayι :\n    (Proj.awayι 𝒜 f f_deg hm).opensRange = Proj.basicOpen 𝒜 f :=\n  (Scheme.Hom.opensRange_comp_of_isIso _ _).trans (basicOpen 𝒜 f).opensRange_ι\n\ninclude f_deg hm in\nlemma isAffineOpen_basicOpen : IsAffineOpen (basicOpen 𝒜 f) := by\n  rw [← opensRange_awayι 𝒜 f f_deg hm]\n  exact isAffineOpen_opensRange (awayι _ _ _ _)\n\n@[reassoc]\nlemma awayι_toSpecZero : awayι 𝒜 f f_deg hm ≫ toSpecZero 𝒜 =\n    Spec.map (CommRingCat.ofHom (fromZeroRingHom 𝒜 _)) := by\n  rw [toSpecZero, basicOpenToSpec, awayι]\n  simp only [Category.assoc, Iso.inv_comp_eq, basicOpenIsoSpec_hom]\n  have (U) (e : U = ⊤) : (basicOpen 𝒜 f).ι ≫ (Scheme.topIso _).inv ≫ (Scheme.isoOfEq _ e).inv =\n      Scheme.homOfLE _ (le_top.trans_eq e.symm) := by\n    simp only [← Category.assoc, Iso.comp_inv_eq]\n    simp only [Scheme.topIso_hom, Category.assoc, Scheme.isoOfEq_hom_ι, Scheme.homOfLE_ι]\n  rw [reassoc_of% this, ← Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_assoc, basicOpenToSpec,\n    Category.assoc, ← Spec.map_comp, ← Spec.map_comp, ← Spec.map_comp]\n  rfl\n\nvariable {f}\nvariable {m' : ℕ} {g : A} (g_deg : g ∈ 𝒜 m') (hm' : 0 < m') {x : A} (hx : x = f * g)\n\n@[reassoc]\n\nTarget:\nlemma awayMap_awayToSection :\n    CommRingCat.ofHom (awayMap 𝒜 g_deg hx) ≫ awayToSection 𝒜 x =\n      awayToSection 𝒜 f ≫ (Proj 𝒜).presheaf.map (homOfLE (basicOpen_mono _ _ _ ⟨_, hx⟩)).op :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"c06c447ac10e48d8e2833722fad093807ac47177780c23119db9fe5c4f22ddc5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/ProjectiveSpectrum","family_id":"awaymap_awaytosection","file_id":"mathlib/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.lean","sample_id":"f56b4dc966c61de3f4b4118e398dfcc1e44b0ddfc9b9e3ab8b9799975c8612d7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f144c9feaed254a6570ff3fe42b4f3dd16c597e6f392545a7ebcf73899e7bd04","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a7cf948f3b6fdc8d09936e8976199a52527a423e8d8d93a75fa33891fbbd66f2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c5c582457298af65c0d36d7c0a352351e64ed55551e90cd77c5b767876e6c965","source_sha256":"cc48552fd983736c871bbaff8e8b09090c080c9d6073d135463b6637bbb51f7b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical exact computable_iff_re_compl_re","hard_negative":true,"metrics":{"chosen_tokens":4,"rejected_tokens":2,"token_jaccard":0.2,"token_length_ratio":0.5},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"49a3383574538cf7706fa8db894494a2c48dcd177c2ddc8b5313f4b99fadcc2c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Computability.PartrecCode\n\nNamespace:\nComputablePred\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Computable and Recursively Enumerable Predicates\n\nThis file defines computable (`ComputablePred`) and recursively enumerable (`REPred`)\npredicates. It also provides basic closure properties and Post's theorem on the\nequivalence of recursive, r.e., and co-r.e. sets.\n-/\n\n@[expose] public section\n\nopen List (Vector)\nopen Encodable Denumerable\n\nnamespace Nat.Partrec\n\nopen Computable Part\n\ntheorem merge' {f g} (hf : Nat.Partrec f) (hg : Nat.Partrec g) :\n    ∃ h, Nat.Partrec h ∧\n      ∀ a, (∀ x ∈ h a, x ∈ f a ∨ x ∈ g a) ∧ ((h a).Dom ↔ (f a).Dom ∨ (g a).Dom) := by\n  obtain ⟨cf, rfl⟩ := Code.exists_code.1 hf\n  obtain ⟨cg, rfl⟩ := Code.exists_code.1 hg\n  have : Nat.Partrec fun n => Nat.rfindOpt fun k => cf.evaln k n <|> cg.evaln k n :=\n    Partrec.nat_iff.1\n      (Partrec.rfindOpt <|\n        Primrec.option_orElse.to_comp.comp\n          (Code.primrec_evaln.to_comp.comp <| (snd.pair (const cf)).pair fst)\n          (Code.primrec_evaln.to_comp.comp <| (snd.pair (const cg)).pair fst))\n  refine ⟨_, this, fun n => ?_⟩\n  have : ∀ x ∈ rfindOpt fun k ↦ Code.evaln k cf n <|> Code.evaln k cg n,\n      x ∈ Code.eval cf n ∨ x ∈ Code.eval cg n := by\n    intro x h\n    obtain ⟨k, e⟩ := Nat.rfindOpt_spec h\n    rw [Option.mem_def, Option.orElse_eq_some, ← Option.mem_def, ← Option.mem_def] at e\n    obtain e | ⟨-, e⟩ := e <;> simp [Code.evaln_sound e]\n  refine ⟨this, fun h ↦ (this _ ⟨h, rfl⟩).imp Exists.fst Exists.fst, fun h ↦ ?_⟩\n  rw [Nat.rfindOpt_dom]\n  simp only [dom_iff_mem, Code.evaln_complete, Option.mem_def] at h\n  obtain ⟨x, k, e⟩ | ⟨x, k, e⟩ := h\n  · exact ⟨k, x, by simp [e]⟩\n  · refine ⟨k, ?_⟩\n    rcases cf.evaln k n with - | y\n    · exact ⟨x, by simp [e]⟩\n    · exact ⟨y, by simp⟩\n\nend Nat.Partrec\n\nnamespace Partrec\n\nvariable {α : Type*} {β : Type*} {γ : Type*} {σ : Type*}\nvariable [Primcodable α] [Primcodable β] [Primcodable γ] [Primcodable σ]\n\nopen Computable Part\n\nopen Nat.Partrec (Code)\n\nopen Nat.Partrec.Code\n\ntheorem merge' {f g : α →. σ} (hf : Partrec f) (hg : Partrec g) :\n    ∃ k : α →. σ,\n      Partrec k ∧ ∀ a, (∀ x ∈ k a, x ∈ f a ∨ x ∈ g a) ∧ ((k a).Dom ↔ (f a).Dom ∨ (g a).Dom) := by\n  let ⟨k, hk, H⟩ := Nat.Partrec.merge' (bind_decode₂_iff.1 hf) (bind_decode₂_iff.1 hg)\n  let k' (a : α) := (k (encode a)).bind fun n => (decode (α := σ) n : Part σ)\n  refine\n    ⟨k', ((nat_iff.2 hk).comp Computable.encode).bind (Computable.decode.ofOption.comp snd).to₂,\n      fun a => ?_⟩\n  have : ∀ x ∈ k' a, x ∈ f a ∨ x ∈ g a := by\n    intro x h'\n    simp only [k', mem_coe, mem_bind_iff, Option.mem_def] at h'\n    obtain ⟨n, hn, hx⟩ := h'\n    have := (H _).1 _ hn\n    simp only [decode₂_encode, coe_some, bind_some, mem_map_iff] at this\n    obtain ⟨a', ha, rfl⟩ | ⟨a', ha, rfl⟩ := this <;> simp only [encodek, Option.some_inj] at hx <;>\n      rw [hx] at ha\n    · exact Or.inl ha\n    · exact Or.inr ha\n  refine ⟨this, ⟨fun h => (this _ ⟨h, rfl⟩).imp Exists.fst Exists.fst, ?_⟩⟩\n  intro h\n  rw [bind_dom]\n  have hk : (k (encode a)).Dom :=\n    (H _).2.2 (by simpa only [encodek₂, bind_some, coe_some] using! h)\n  exists hk\n  simp only [mem_map_iff, mem_coe, mem_bind_iff, Option.mem_def] at H\n  obtain ⟨a', _, y, _, e⟩ | ⟨a', _, y, _, e⟩ := (H _).1 _ ⟨hk, rfl⟩ <;>\n    simp only [e.symm, encodek, coe_some, some_dom]\n\ntheorem merge {f g : α →. σ} (hf : Partrec f) (hg : Partrec g)\n    (H : ∀ (a), ∀ x ∈ f a, ∀ y ∈ g a, x = y) :\n    ∃ k : α →. σ, Partrec k ∧ ∀ a x, x ∈ k a ↔ x ∈ f a ∨ x ∈ g a :=\n  let ⟨k, hk, K⟩ := merge' hf hg\n  ⟨k, hk, fun a x =>\n    ⟨(K _).1 _, fun h => by\n      have : (k a).Dom := (K _).2.2 (h.imp Exists.fst Exists.fst)\n      refine ⟨this, ?_⟩\n      rcases h with h | h <;> rcases (K _).1 _ ⟨this, rfl⟩ with h' | h'\n      · exact mem_unique h' h\n      · exact (H _ _ h _ h').symm\n      · exact H _ _ h' _ h\n      · exact mem_unique h' h⟩⟩\n\ntheorem cond {c : α → Bool} {f : α →. σ} {g : α →. σ} (hc : Computable c) (hf : Partrec f)\n    (hg : Partrec g) : Partrec fun a => cond (c a) (f a) (g a) :=\n  let ⟨cf, ef⟩ := exists_code.1 hf\n  let ⟨cg, eg⟩ := exists_code.1 hg\n  ((eval_part.comp (Computable.cond hc (const cf) (const cg)) Computable.encode).bind\n    ((@Computable.decode σ _).comp snd).ofOption.to₂).of_eq\n    fun a => by cases c a <;> simp [ef, eg, encodek]\n\nnonrec theorem sumCasesOn {f : α → β ⊕ γ} {g : α → β →. σ} {h : α → γ →. σ} (hf : Computable f)\n    (hg : Partrec₂ g) (hh : Partrec₂ h) : @Partrec _ σ _ _ fun a => Sum.casesOn (f a) (g a) (h a) :=\n  option_some_iff.1 <|\n    (cond (sumCasesOn hf (const true).to₂ (const false).to₂)\n          (sumCasesOn_left hf (option_some_iff.2 hg).to₂ (const Option.none).to₂)\n          (sumCasesOn_right hf (const Option.none).to₂ (option_some_iff.2 hh).to₂)).of_eq\n      fun a => by cases f a <;> simp only [Bool.cond_true, Bool.cond_false]\n\nend Partrec\n\n/-- A computable predicate is one whose indicator function is computable. -/\ndef ComputablePred {α} [Primcodable α] (p : α → Prop) :=\n  ∃ (_ : DecidablePred p), Computable fun a => decide (p a)\n\nsection decide\n\nvariable {α} [Primcodable α]\n\nprotected lemma ComputablePred.decide {p : α → Prop} [DecidablePred p] (hp : ComputablePred p) :\n    Computable (fun a => decide (p a)) := by\n  convert! hp.choose_spec\n\nlemma Computable.computablePred {p : α → Prop} [DecidablePred p]\n    (hp : Computable (fun a => decide (p a))) : ComputablePred p :=\n  ⟨inferInstance, hp⟩\n\nlemma computablePred_iff_computable_decide {p : α → Prop} [DecidablePred p] :\n    ComputablePred p ↔ Computable (fun a => decide (p a)) where\n  mp := ComputablePred.decide\n  mpr := Computable.computablePred\n\nlemma PrimrecPred.computablePred {α} [Primcodable α] {p : α → Prop} :\n    (hp : PrimrecPred p) → ComputablePred p\n  | ⟨_, hp⟩ => hp.to_comp.computablePred\n\nend decide\n\n/-- A recursively enumerable predicate is one which is the domain of a computable partial function.\n-/\ndef REPred {α} [Primcodable α] (p : α → Prop) :=\n  Partrec fun a => Part.assert (p a) fun _ => Part.some ()\n\ntheorem REPred.of_eq {α} [Primcodable α] {p q : α → Prop} (hp : REPred p) (H : ∀ a, p a ↔ q a) :\n    REPred q :=\n  (funext fun a => propext (H a) : p = q) ▸ hp\n\ntheorem Partrec.dom_re {α β} [Primcodable α] [Primcodable β] {f : α →. β} (h : Partrec f) :\n    REPred fun a => (f a).Dom :=\n  (h.map (Computable.const ()).to₂).of_eq fun n => Part.ext fun _ => by simp [Part.dom_iff_mem]\n\ntheorem ComputablePred.of_eq {α} [Primcodable α] {p q : α → Prop} (hp : ComputablePred p)\n    (H : ∀ a, p a ↔ q a) : ComputablePred q :=\n  (funext fun a => propext (H a) : p = q) ▸ hp\n\nnamespace Computable\n\n/-- If `P` is computable, and if for every `x` there exists an `n` such that `P x n` holds,\nthen the function mapping `x` to the minimal such `n` (using `Nat.find`) is computable.\nThis formally bridges `Partrec.rfind` with total unbounded search. -/\nlemma find {α : Type*} [Primcodable α] {P : α → ℕ → Prop} [DecidableRel P]\n    (hP_comp : ComputablePred (fun p : α × ℕ => P p.1 p.2)) (hP_ex : ∀ x, ∃ n, P x n) :\n    Computable (fun x => Nat.find (hP_ex x)) := by\n  have h : Partrec (fun x ↦ Nat.rfind fun n => Part.some (decide (P x n))) :=\n    Partrec.rfind hP_comp.decide.partrec\n  refine h.of_eq_tot fun x ↦ ?_\n  simp +contextual [Nat.find_spec]\n\nend Computable\n\nnamespace ComputablePred\n\nvariable {α : Type*} [Primcodable α]\n\nopen Nat.Partrec (Code)\n\nopen Nat.Partrec.Code Computable\n\ntheorem computable_iff {p : α → Prop} :\n    ComputablePred p ↔ ∃ f : α → Bool, Computable f ∧ p = fun a => (f a : Prop) :=\n  ⟨fun ⟨_, h⟩ => ⟨_, h, funext fun _ => propext (Bool.decide_iff _).symm⟩, by\n    rintro ⟨f, h, rfl⟩; exact ⟨by infer_instance, by simpa using h⟩⟩\n\nprotected theorem not {p : α → Prop} :\n    (hp : ComputablePred p) → ComputablePred fun a => ¬p a\n  | ⟨_, hp⟩ => Computable.computablePred <| Primrec.not.to_comp.comp hp |>.of_eq <| by simp\n\n/-- The computable functions are closed under if-then-else definitions\nwith computable predicates. -/\ntheorem ite {f₁ f₂ : ℕ → ℕ} (hf₁ : Computable f₁) (hf₂ : Computable f₂)\n    {c : ℕ → Prop} [DecidablePred c] (hc : ComputablePred c) :\n    Computable fun k ↦ if c k then f₁ k else f₂ k := by\n  simpa [Bool.cond_decide] using hc.decide.cond hf₁ hf₂\n\ntheorem to_re {p : α → Prop} (hp : ComputablePred p) : REPred p := by\n  obtain ⟨f, hf, rfl⟩ := computable_iff.1 hp\n  unfold REPred\n  dsimp only []\n  refine\n    (Partrec.cond hf (Decidable.Partrec.const' (Part.some ())) Partrec.none).of_eq fun n =>\n      Part.ext fun a => ?_\n  cases a; cases f n <;> simp\n\n-- Post's theorem on the equivalence of r.e., co-r.e. sets and\n-- computable sets. The assumption that p is decidable is required\n-- unless we assume Markov's principle or LEM.\nset_option linter.unusedDecidableInType false in\ntheorem computable_iff_re_compl_re {p : α → Prop} [DecidablePred p] :\n    ComputablePred p ↔ REPred p ∧ REPred fun a => ¬p a :=\n  ⟨fun h => ⟨h.to_re, h.not.to_re⟩, fun ⟨h₁, h₂⟩ =>\n    ⟨‹_›, by\n      obtain ⟨k, pk, hk⟩ :=\n        Partrec.merge (h₁.map (Computable.const true).to₂) (h₂.map (Computable.const false).to₂)\n        (by\n          intro a x hx y hy\n          simp only [Part.mem_map_iff, Part.mem_assert_iff, Part.mem_some_iff, exists_prop,\n            and_true, exists_const] at hx hy\n          cases hy.1 hx.1)\n      refine Partrec.of_eq pk fun n => Part.eq_some_iff.2 ?_\n      rw [hk]\n      simp only [Part.mem_map_iff, Part.mem_assert_iff, Part.mem_some_iff, exists_prop, and_true,\n        true_eq_decide_iff, and_self, exists_const, false_eq_decide_iff]\n      apply Decidable.em⟩⟩\n\nTarget:\ntheorem computable_iff_re_compl_re' {p : α → Prop} :\n    ComputablePred p ↔ REPred p ∧ REPred fun a => ¬p a :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_c5c582457298","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"1afb3554effcd8d9a859a71ae2f4ec43ef495b60eb5b31f20811b66773e11dd3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability","family_id":"computable_iff_re_compl_re","file_id":"mathlib/Mathlib/Computability/RE.lean","sample_id":"c5c582457298af65c0d36d7c0a352351e64ed55551e90cd77c5b767876e6c965"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5bc8e347ea35f5e54b983043b84f723d4bb439695fcffd6e054bbe0bc3c13d4f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0ccc7aefbf796c9bc413bd34f90036d969463ddb4ebea0b7afe1f034f8ff048d","source_sha256":"fe20ac9adb6e069125c46c6cc0a23c1dfac91d49367534266b3b31063c05d603","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  constructor\n  · simp only [basis, TwoSidedIdeal.le_iff, TwoSidedIdeal.coe_mk', setOf_subset_setOf]\n    intro h\n    constructor\n    · intro x hx\n      have (d' : _) : coeff d' (C (σ := σ) x) ∈ J := by\n        rw [coeff_C]; split_ifs <;> [exact hx; exact J.zero_mem]\n      simpa using h (C x) (fun _ _ ↦ this _) _ zero_le\n    · by_contra h'\n      apply hK\n      rw [eq_top_iff]\n      intro x _\n      have (d') (hd'_le : d' ≤ d) : coeff d' (monomial e x) ∈ J := by\n        rw [coeff_monomial]\n        split_ifs with hd' <;> [exact (h' (hd' ▸ hd'_le)).elim; exact J.zero_mem]\n      simpa using h (monomial e x) this _ le_rfl\n  · rintro ⟨hJK, hed⟩\n    exact basis_le hJK hed\n\nvariable [TopologicalSpace R]\n\n-- We endow MvPowerSeries σ R with the product topology.","hard_negative":true,"metrics":{"chosen_tokens":185,"rejected_tokens":8,"token_jaccard":0.05,"token_length_ratio":0.043243},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"49f9dd29a62867eb5d79cefb4522701eb2f74d8f0400dcdff911ad022772b489","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finsupp.Interval\npublic import Mathlib.RingTheory.Ideal.Quotient.Defs\npublic import Mathlib.RingTheory.MvPowerSeries.PiTopology\npublic import Mathlib.Topology.Algebra.LinearTopology\npublic import Mathlib.RingTheory.TwoSidedIdeal.Operations\n\nNamespace:\nMvPowerSeries.LinearTopology\n\nLocal context:\n/-\nCopyright (c) 2024 Antoine Chambert-Loir, María Inés de Frutos-Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir, María Inés de Frutos-Fernández\n-/\n/-! # Linear topology on the ring of multivariate power series\n\n- `MvPowerSeries.LinearTopology.basis`: the ideals of the ring of multivariate power series\n  all coefficients the exponent of which is smaller than some bound vanish.\n\n- `MvPowerSeries.LinearTopology.hasBasis_nhds_zero` :\n  the two-sided ideals from `MvPowerSeries.LinearTopology.basis` form a basis\n  of neighborhoods of `0` if the topology of `R` is (left and right) linear.\n\n## Instances :\n\nIf `R` has a linear topology, then the product topology on `MvPowerSeries σ R`\nis a linear topology.\n\nThis applies in particular when `R` has the discrete topology.\n\n## Note\n\nIf we had an analogue of `PolynomialModule` for power series,\nmeaning that we could consider the `R⟦X⟧`-module `M⟦X⟧` when `M` is an `R`-module,\nthen one could prove that `M⟦X⟧` is linearly topologized over `R⟦X⟧`\nwhenever `M` is linearly topologized over `R`.\nTo recover the ring case, it would remain to show that the isomorphism between\n`Rᵐᵒᵖ⟦X⟧` and `R⟦X⟧ᵐᵒᵖ` identifies their respective actions on `R⟦X⟧`.\n(And likewise in the multivariate case.)\n\n-/\n\n@[expose] public section\n\nnamespace MvPowerSeries\n\nnamespace LinearTopology\n\nopen scoped Topology\n\nopen Set SetLike Filter\n\n/-- The underlying family for the basis of ideals in a multivariate power series ring. -/\nnoncomputable def basis (σ : Type*) (R : Type*) [Ring R] (Jd : TwoSidedIdeal R × (σ →₀ ℕ)) :\n    TwoSidedIdeal (MvPowerSeries σ R) :=\n  TwoSidedIdeal.mk' {f | ∀ e ≤ Jd.2, coeff e f ∈ Jd.1}\n    (by simp [coeff_zero])\n    (fun hf hg e he ↦ by rw [map_add]; exact add_mem (hf e he) (hg e he))\n    (fun {f} hf e he ↦ by simp only [map_neg, neg_mem, hf e he])\n    (fun {f g} hg e he ↦ by\n      classical\n      rw [coeff_mul]\n      apply sum_mem\n      rintro uv huv\n      exact TwoSidedIdeal.mul_mem_left _ _ _ (hg _ (le_trans (Finset.antidiagonal.snd_le huv) he)))\n    (fun {f g} hf e he ↦ by\n      classical\n      rw [coeff_mul]\n      apply sum_mem\n      rintro uv huv\n      exact TwoSidedIdeal.mul_mem_right _ _ _ (hf _ (le_trans (Finset.antidiagonal.fst_le huv) he)))\n\nvariable {σ : Type*} {R : Type*} [Ring R]\n\n/-- A power series `f` belongs to the two-sided ideal `basis σ R ⟨J, d⟩`\nif and only if `coeff e f ∈ J` for all `e ≤ d`. -/\ntheorem mem_basis_iff {f : MvPowerSeries σ R} {Jd : TwoSidedIdeal R × (σ →₀ ℕ)} :\n    f ∈ basis σ R Jd ↔ ∀ e ≤ Jd.2, coeff e f ∈ Jd.1 := by\n  simp [basis]\n\n/-- If `J ≤ K` and `e ≤ d`, then we have the inclusion of two-sided ideals\n`basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e,>`. -/\ntheorem basis_le {Jd Ke : TwoSidedIdeal R × (σ →₀ ℕ)} (hJK : Jd.1 ≤ Ke.1) (hed : Ke.2 ≤ Jd.2) :\n    basis σ R Jd ≤ basis σ R Ke :=\n  fun _ ↦ forall_imp (fun _ h hue ↦ hJK (h (le_trans hue hed)))\n\n/-- `basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e⟩` if and only if `J ≤ K` and `e ≤ d`. -/\n\nTarget:\ntheorem basis_le_iff {J K : TwoSidedIdeal R} {d e : σ →₀ ℕ} (hK : K ≠ ⊤) :\n    basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e⟩ ↔ J ≤ K ∧ e ≤ d :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"fb4dc1caa077054f40a0a4acf45c5cefbc8d0baf1ffff33e96cb1ce32b9d275e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/MvPowerSeries","family_id":"basis_le_iff","file_id":"mathlib/Mathlib/RingTheory/MvPowerSeries/LinearTopology.lean","sample_id":"0ccc7aefbf796c9bc413bd34f90036d969463ddb4ebea0b7afe1f034f8ff048d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b232bd6f68b5950f3c5e2ed9ba68ca285d64e0f598a20d32ae0f8c799f5bfc41","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"285ac08c42ed5e3ea6a0bc84afe24ee43953ddfec199138f1b03d43553c70fdb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b14089476197b86942d51dd07351b5cf69d062ba1bdb2cb6a8c8c4b42ae306f4","source_sha256":"d1237e92863e296159e7253e7e582eee623003d46d97c2c45df765267113b5c1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let M' := algebraMapSubmonoid S M\n  have hM' : algebraMapSubmonoid S M ≤ S⁰ := algebraMapSubmonoid_le_nonZeroDivisors_of_faithfulSMul\n    _ hM\n  let f₁ : Rₘ →+* K := map _ (T := R⁰) (RingHom.id R) hM\n  let f₂ : Sₘ →+* L := map _ (T := S⁰) (RingHom.id S) hM'\n  algebraize [f₁, f₂]\n  have := localization_isScalarTower_of_submonoid_le Rₘ K _ _ hM\n  have := localization_isScalarTower_of_submonoid_le Sₘ L _ _ hM'\n  have := isFractionRing_of_isDomain_of_isLocalization M Rₘ K\n  have := isFractionRing_of_isDomain_of_isLocalization M' Sₘ L\n  have : IsDomain Rₘ := isDomain_of_le_nonZeroDivisors _ hM\n  apply Algebra.IsSeparable.of_equiv_equiv (FractionRing.algEquiv Rₘ K).symm.toRingEquiv\n    (FractionRing.algEquiv Sₘ L).symm.toRingEquiv\n  apply ringHom_ext R⁰\n  ext\n  simp only [RingHom.coe_comp,\n      RingHom.coe_coe, Function.comp_apply, ← algebraMap_apply]\n  rw [algebraMap_apply R Rₘ (FractionRing R), AlgEquiv.coe_ringEquiv, AlgEquiv.commutes,\n    algebraMap_apply R S L, algebraMap_apply S Sₘ L, AlgEquiv.coe_ringEquiv, AlgEquiv.commutes]\n  simp only [← algebraMap_apply]\n  rw [algebraMap_apply R Rₘ (FractionRing Rₘ), ← algebraMap_apply Rₘ, ← algebraMap_apply]","hard_negative":true,"metrics":{"chosen_tokens":238,"rejected_tokens":5,"token_jaccard":0.046875,"token_length_ratio":0.021008},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"4a147518f54534069e68b3beddcdf163a8d59f0beb48758b885e2d4cccabdac4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.PID\npublic import Mathlib.FieldTheory.Separable\npublic import Mathlib.RingTheory.RingHom.Finite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Riccardo Brasca. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Riccardo Brasca\n-/\n/-!\n# Instances for Dedekind domains\nThis file contains various instances to work with localization of a ring extension.\n\nA very common situation in number theory is to have an extension of (say) Dedekind domains `R` and\n`S`, and to prove a property of this extension it is useful to consider the localization `Rₚ` of `R`\nat `P`, a prime ideal of `R`. One also works with the corresponding localization `Sₚ` of `S` and the\nfraction fields `K` and `L` of `R` and `S`. In this situation there are many compatible algebra\nstructures and various properties of the rings involved. Another situation is when we have a\ntower extension `R ⊆ S ⊆ T` and thus we work with `Rₚ ⊆ Sₚ ⊆ Tₚ` where\n`Tₚ` is the localization of `T` at `P`. This file contains a collection of such instances.\n\n## Implementation details\nIn general one wants all the results below for any algebra satisfying `IsLocalization`, but those\ncannot be instances (since Lean has no way of guessing the submonoid). Having the instances in the\nspecial case of *the* localization at a prime ideal is useful in working with Dedekind domains.\n\n-/\n\npublic section\n\nopen nonZeroDivisors IsLocalization Algebra Module IsFractionRing IsScalarTower\n\nattribute [local instance] FractionRing.liftAlgebra\n\nvariable {R : Type*} (S : Type*) (T : Type*) [CommRing R] [CommRing S] [CommRing T] [IsDomain R]\n  [IsDomain S] [IsDomain T] [Algebra R S]\n\nlocal notation3 \"K\" => FractionRing R\nlocal notation3 \"L\" => FractionRing S\nlocal notation3 \"F\" => FractionRing T\n\nsection\n\ntheorem algebraMapSubmonoid_le_nonZeroDivisors_of_faithfulSMul {A : Type*} (B : Type*)\n    [CommSemiring A] [CommSemiring B] [Algebra A B] [NoZeroDivisors B] [FaithfulSMul A B]\n    {S : Submonoid A} (hS : S ≤ A⁰) : algebraMapSubmonoid B S ≤ B⁰ :=\n  map_le_nonZeroDivisors_of_injective _ (FaithfulSMul.algebraMap_injective A B) hS\n\nvariable (Rₘ Sₘ : Type*) [CommRing Rₘ] [CommRing Sₘ] [Algebra R Rₘ] [IsTorsionFree R S]\n    [Algebra.IsSeparable (FractionRing R) (FractionRing S)] {M : Submonoid R} [IsLocalization M Rₘ]\n    [Algebra Rₘ Sₘ] [Algebra S Sₘ] [Algebra R Sₘ] [IsScalarTower R Rₘ Sₘ]\n    [IsScalarTower R S Sₘ] [IsLocalization (algebraMapSubmonoid S M) Sₘ]\n    [Algebra (FractionRing Rₘ) (FractionRing Sₘ)]\n    [IsScalarTower Rₘ (FractionRing Rₘ) (FractionRing Sₘ)]\n\nset_option backward.isDefEq.respectTransparency false in\ninclude R S in\n\nTarget:\ntheorem FractionRing.isSeparable_of_isLocalization (hM : M ≤ R⁰) :\n    Algebra.IsSeparable (FractionRing Rₘ) (FractionRing Sₘ) :=\n\nProof body:\n","rejected":"by\n  exact FractionRing.isSeparable_of_isLocalization","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"de136c05feed13899ff87bc574cd2851d498acde159eba003f0885cdc7826e17","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/DedekindDomain","family_id":"fractionring","file_id":"mathlib/Mathlib/RingTheory/DedekindDomain/Instances.lean","sample_id":"b14089476197b86942d51dd07351b5cf69d062ba1bdb2cb6a8c8c4b42ae306f4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7f711a73112f71461efde233fe8c7cca840501a653c0391902a7be231d8cd075","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6a93f461afc68c4564b026d64e62d3228a07b26e1dd6066974bf51a8514702ac","source_sha256":"96ffeb6042bed2ccd5d2880386938c633598ffa0d830f64affcb9938dbdcf908","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction l with\n  | nil => simp\n  | cons head tail ih => simp [← ih, smul_mul_smul_comm, pow_succ']","hard_negative":false,"metrics":{"chosen_tokens":23,"rejected_tokens":3,"token_jaccard":0.05,"token_length_ratio":0.130435},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"4a169661d08d0e1795e69ce39176cb889a7821c78ef58b4724918e9ea69351ec","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Finprod\npublic import Mathlib.Algebra.GroupWithZero.Action.Defs\npublic import Mathlib.Algebra.Order.Group.Multiset\npublic import Mathlib.Data.Finset.Basic\npublic import Mathlib.Algebra.Group.Action.Basic\npublic import Mathlib.Algebra.Group.Units.Equiv\n\nNamespace:\nList\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Lemmas about group actions on big operators\n\nThis file contains results about two kinds of actions:\n\n* sums over `DistribSMul`: `r • ∑ x ∈ s, f x = ∑ x ∈ s, r • f x`\n* products over `MulDistribMulAction` (with primed name): `r • ∏ x ∈ s, f x = ∏ x ∈ s, r • f x`\n* products over `SMulCommClass` (with unprimed name):\n  `b ^ s.card • ∏ x ∈ s, f x = ∏ x ∈ s, b • f x`\n\nNote that analogous lemmas for `Module`s like `Finset.sum_smul` appear in other files.\n-/\n\npublic section\n\n\nvariable {M N γ : Type*}\n\nsection\n\nvariable [AddMonoid N] [DistribSMul M N]\n\ntheorem List.smul_sum {r : M} {l : List N} : r • l.sum = (l.map (r • ·)).sum :=\n  map_list_sum (DistribSMul.toAddMonoidHom N r) l\n\nend\n\nsection\n\nvariable [Monoid M] [Monoid N] [MulDistribMulAction M N]\n\ntheorem List.smul_prod' {r : M} {l : List N} : r • l.prod = (l.map (r • ·)).prod :=\n  map_list_prod (MulDistribMulAction.toMonoidHom N r) l\n\nend\n\nsection\n\nvariable [AddCommMonoid N] [DistribSMul M N] {r : M}\n\ntheorem Multiset.smul_sum {s : Multiset N} : r • s.sum = (s.map (r • ·)).sum :=\n  (DistribSMul.toAddMonoidHom N r).map_multiset_sum s\n\ntheorem Finset.smul_sum {f : γ → N} {s : Finset γ} :\n    (r • ∑ x ∈ s, f x) = ∑ x ∈ s, r • f x :=\n  map_sum (DistribSMul.toAddMonoidHom N r) f s\n\ntheorem smul_finsum_mem {f : γ → N} {s : Set γ} (hs : s.Finite) :\n    r • ∑ᶠ x ∈ s, f x = ∑ᶠ x ∈ s, r • f x :=\n  (DistribSMul.toAddMonoidHom N r).map_finsum_mem f hs\n\nend\n\nsection\n\nvariable [Monoid M] [CommMonoid N] [MulDistribMulAction M N]\n\ntheorem Multiset.smul_prod' {r : M} {s : Multiset N} : r • s.prod = (s.map (r • ·)).prod :=\n  (MulDistribMulAction.toMonoidHom N r).map_multiset_prod s\n\ntheorem Finset.smul_prod' {r : M} {f : γ → N} {s : Finset γ} :\n    (r • ∏ x ∈ s, f x) = ∏ x ∈ s, r • f x :=\n  map_prod (MulDistribMulAction.toMonoidHom N r) f s\n\ntheorem smul_finprod' {ι : Sort*} [Finite ι] {f : ι → N} (r : M) :\n    r • ∏ᶠ x : ι, f x = ∏ᶠ x : ι, r • (f x) := by\n  cases nonempty_fintype (PLift ι)\n  simp only [finprod_eq_prod_plift_of_mulSupport_subset (s := Finset.univ) (by simp),\n    Finset.smul_prod']\n\nvariable {G : Type*} [Group G] [MulDistribMulAction G N]\n\ntheorem Finset.smul_prod_perm [Fintype G] (b : N) (g : G) :\n    (g • ∏ h : G, h • b) = ∏ h : G, h • b := by\n  simp only [smul_prod', smul_smul]\n  exact Finset.prod_bijective (g * ·) (Group.mulLeft_bijective g) (by simp) (fun _ _ ↦ rfl)\n\ntheorem smul_finprod_perm [Finite G] (b : N) (g : G) :\n    (g • ∏ᶠ h : G, h • b) = ∏ᶠ h : G, h • b := by\n  cases nonempty_fintype G\n  simp only [finprod_eq_prod_of_fintype, Finset.smul_prod_perm]\n\nend\n\nnamespace List\n\n@[to_additive]\n\nTarget:\ntheorem smul_prod [Monoid M] [MulOneClass N] [MulAction M N] [IsScalarTower M N N]\n    [SMulCommClass M N N] (l : List N) (m : M) :\n    m ^ l.length • l.prod = (l.map (m • ·)).prod :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/BigOperators","family_id":"smul_prod","file_id":"mathlib/Mathlib/Algebra/BigOperators/GroupWithZero/Action.lean","sample_id":"6a93f461afc68c4564b026d64e62d3228a07b26e1dd6066974bf51a8514702ac"},"split":"train"}
{"binding":{"chosen_proof_sha256":"893f9954189be14b6650b8e11183442e164c9e72d10c7e09ce63fc59ebc05f3e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d56eafe3c4252de8949772acbb02a5dc70b73280feb05a0bdb40abf43f9fe7b5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e5021bc99d39f6a7ba2ff9754e2b8736041ba7f4cf828fd50b3785d42ee868a1","source_sha256":"1ff5a6a6007c7fc57cd72503958e574b72763b74e50bd4d47517631475da6ea9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext; simp [equiv]","hard_negative":false,"metrics":{"chosen_tokens":7,"rejected_tokens":12,"token_jaccard":0.583333,"token_length_ratio":1.714286},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"4a2a14743e7cbac8ba974fd6f72cd9ae030cfd3cc412237d98d7ae97a4cc753a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Colimit.Module\npublic import Mathlib.RingTheory.Finiteness.Basic\n\nNamespace:\nModule.fgSystem\n\nLocal context:\n/-\nCopyright (c) 2024 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu\n-/\n/-!\n# Modules as direct limits of finitely generated submodules\n\nWe show that every module is the direct limit of its finitely generated submodules.\n\n## Main definitions\n\n* `Module.fgSystem`: the directed system of finitely generated submodules of a module.\n\n* `Module.fgSystem.equiv`: the isomorphism between a module and the direct limit of its\n  finitely generated submodules.\n-/\n\n@[expose] public section\n\nnamespace Module\n\nvariable (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]\n\n/-- The directed system of finitely generated submodules of a module. -/\ndef fgSystem (N₁ N₂ : {N : Submodule R M // N.FG}) (le : N₁ ≤ N₂) : N₁ →ₗ[R] N₂ :=\n  Submodule.inclusion le\n\nopen DirectLimit\n\nnamespace fgSystem\n\ninstance : IsDirectedOrder {N : Submodule R M // N.FG} where\n  directed N₁ N₂ :=\n    ⟨⟨_, N₁.2.sup N₂.2⟩, Subtype.coe_le_coe.mp le_sup_left, Subtype.coe_le_coe.mp le_sup_right⟩\n\ninstance : DirectedSystem _ (fgSystem R M · · · ·) where\n  map_self _ _ := rfl\n  map_map _ _ _ _ _ _ := rfl\n\nvariable [DecidableEq (Submodule R M)]\n\nopen Submodule in\n/-- Every module is the direct limit of its finitely generated submodules. -/\nnoncomputable def equiv : DirectLimit _ (fgSystem R M) ≃ₗ[R] M :=\n  .ofBijective (lift _ _ _ _ (fun _ ↦ Submodule.subtype _) fun _ _ _ _ ↦ rfl)\n    ⟨lift_injective _ _ fun _ ↦ Subtype.val_injective, fun x ↦\n      ⟨of _ _ _ _ ⟨_, fg_span_singleton x⟩ ⟨x, subset_span <| by rfl⟩, lift_of ..⟩⟩\n\nvariable {R M}\n\nTarget:\nlemma equiv_comp_of (N : {N : Submodule R M // N.FG}) :\n    (equiv R M).toLinearMap ∘ₗ of _ _ _ _ N = N.1.subtype :=\n\nProof body:\n","rejected":"by\n  ext; simp [equiv]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Colimit","family_id":"equiv_comp_of","file_id":"mathlib/Mathlib/Algebra/Colimit/Finiteness.lean","sample_id":"e5021bc99d39f6a7ba2ff9754e2b8736041ba7f4cf828fd50b3785d42ee868a1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7f54226990013dd5c5400459b377a357ecf590ee9b93a656a04179c0ea205d15","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"fe8a92a902f5ce0e14ed5c2fddfd6500672f57758772c16637af76973411d10c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4e7c68a0ae1706e7fbfa84a0b8e103da95b624a1edb0fcbc393efbb9763f79d8","source_sha256":"d54536321e31ab6ecb143f6ff47411119fb9708f05bec728ae60cd91a7b3224f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isUnit_iff_coprime]\n  exact (Prime.dvd_iff_not_coprime <| prime_of_mem_primeFactors h).mp <| dvd_of_mem_primeFactors h","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":26,"token_jaccard":0.772727,"token_length_ratio":1.238095},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"4a2e65127e4c3fb0d1d24d1f5ea4ac7eb5df9f7b0ebce25e57878b5fc082c2d8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Associated\npublic import Mathlib.Data.ZMod.Basic\npublic import Mathlib.RingTheory.Coprime.Lemmas\n\nNamespace:\nZMod\n\nLocal context:\n/-\nCopyright (c) 2023 Moritz Firsching. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Moritz Firsching, Ashvni Narayanan, Michael Stoll\n-/\n/-!\n# Lemmas about units in `ZMod`.\n-/\n\n@[expose] public section\n\nassert_not_exists TwoSidedIdeal\n\nnamespace ZMod\n\nvariable {n m : ℕ}\n/-- `unitsMap` is a group homomorphism that maps units of `ZMod m` to units of `ZMod n` when `n`\ndivides `m`. -/\ndef unitsMap (hm : n ∣ m) : (ZMod m)ˣ →* (ZMod n)ˣ := Units.map (castHom hm (ZMod n))\n\nlemma unitsMap_def (hm : n ∣ m) : unitsMap hm = Units.map (castHom hm (ZMod n)) := rfl\n\nlemma unitsMap_comp {d : ℕ} (hm : n ∣ m) (hd : m ∣ d) :\n    (unitsMap hm).comp (unitsMap hd) = unitsMap (dvd_trans hm hd) := by\n  simp only [unitsMap_def]\n  rw [← Units.map_comp]\n  exact congr_arg Units.map <| congr_arg RingHom.toMonoidHom <| castHom_comp hm hd\n\n@[simp]\nlemma unitsMap_self (n : ℕ) : unitsMap (dvd_refl n) = MonoidHom.id _ := by\n  simp [unitsMap, castHom_self]\n\n/-- `unitsMap_val` shows that coercing from `(ZMod m)ˣ` to `ZMod n` gives the same result\nwhen going via `(ZMod n)ˣ` and `ZMod m`. -/\nlemma unitsMap_val (h : n ∣ m) (a : (ZMod m)ˣ) :\n    ↑(unitsMap h a) = ((a : ZMod m).cast : ZMod n) := rfl\n\nlemma isUnit_cast_of_dvd (hm : n ∣ m) (a : Units (ZMod m)) : IsUnit (cast (a : ZMod m) : ZMod n) :=\n  Units.isUnit (unitsMap hm a)\ntheorem unitsMap_surjective [hm : NeZero m] (h : n ∣ m) :\n    Function.Surjective (unitsMap h) := by\n  suffices ∀ x : ℕ, x.Coprime n → ∃ k : ℕ, (x + k * n).Coprime m by\n    intro x\n    have ⟨k, hk⟩ := this x.val.val (val_coe_unit_coprime x)\n    refine ⟨unitOfCoprime _ hk, Units.ext ?_⟩\n    have : NeZero n := ⟨fun hn ↦ hm.out (eq_zero_of_zero_dvd (hn ▸ h))⟩\n    simp [unitsMap_def, -castHom_apply]\n  intro x hx\n  let ps : Finset ℕ := {p ∈ m.primeFactors | ¬p ∣ x}\n  use ps.prod id\n  apply Nat.coprime_of_dvd\n  intro p pp hp hpn\n  by_cases hpx : p ∣ x\n  · have h := Nat.dvd_sub hp hpx\n    rw [add_comm, Nat.add_sub_cancel] at h\n    rcases pp.dvd_mul.mp h with h | h\n    · have ⟨q, hq, hq'⟩ := (pp.prime.dvd_finsetProd_iff id).mp h\n      rw [Finset.mem_filter, Nat.mem_primeFactors,\n        ← (Nat.prime_dvd_prime_iff_eq pp hq.1.1).mp hq'] at hq\n      exact hq.2 hpx\n    · exact Nat.Prime.not_coprime_iff_dvd.mpr ⟨p, pp, hpx, h⟩ hx\n  · have pps : p ∈ ps := Finset.mem_filter.mpr ⟨Nat.mem_primeFactors.mpr ⟨pp, hpn, hm.out⟩, hpx⟩\n    have h := Nat.dvd_sub hp ((Finset.dvd_prod_of_mem id pps).mul_right n)\n    rw [Nat.add_sub_cancel] at h\n    contradiction\n\n-- This needs `Nat.primeFactors`, so cannot go into `Mathlib/Data/ZMod/Basic.lean`.\nopen Nat in\n\nTarget:\nlemma not_isUnit_of_mem_primeFactors {n p : ℕ} (h : p ∈ n.primeFactors) :\n    ¬ IsUnit (p : ZMod n) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  rw [isUnit_iff_coprime]\n  exact (Prime.dvd_iff_not_coprime <| prime_of_mem_primeFactors h).mp <| dvd_of_mem_primeFactors h","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/ZMod","family_id":"not_isunit_of_mem_primefactors","file_id":"mathlib/Mathlib/Data/ZMod/Units.lean","sample_id":"4e7c68a0ae1706e7fbfa84a0b8e103da95b624a1edb0fcbc393efbb9763f79d8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3fa4e6a02899a81962f514978e80ca0d79529b0b1a773bcbf2dd1c22e7f2b19e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6402417a187a534989eab3a2c52dac736091b114295a76d321088d940f0ef62a","source_sha256":"d30cbd6919379ea815bed52dd0199b189327488f1457353c63e7bcff05fa2d66","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Order.covBy_iff_add_one_eq, Order.covBy_iff_add_one_eq]\n  exact Int.natCast_inj","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":8,"token_jaccard":0.058824,"token_length_ratio":0.533333},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"4a7c478931aad4647762863e031a70f36b2c32df5133c08acf3b1024fd23df0e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Ring.Int\npublic import Mathlib.Data.Nat.SuccPred\n\nNamespace:\nInt\n\nLocal context:\n/-\nCopyright (c) 2021 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Successors and predecessors of integers\n\nIn this file, we show that `ℤ` is both an archimedean `SuccOrder` and an archimedean `PredOrder`.\n-/\n\npublic section\n\n\nopen Function Order\n\nnamespace Int\n\ninstance instSuccOrder : SuccOrder ℤ :=\n  { SuccOrder.ofSuccLeIff succ fun {_ _} => Iff.rfl with succ := succ }\n\ninstance instSuccAddOrder : SuccAddOrder ℤ := ⟨fun _ => rfl⟩\n\ninstance instPredOrder : PredOrder ℤ where\n  pred := pred\n  pred_le _ := (sub_one_lt_of_le le_rfl).le\n  min_of_le_pred ha := ((sub_one_lt_of_le le_rfl).not_ge ha).elim\n  le_pred_of_lt {_ _} := le_sub_one_of_lt\n\ninstance instPredSubOrder : PredSubOrder ℤ := ⟨fun _ => rfl⟩\n\n@[simp]\ntheorem succ_eq_succ : Order.succ = succ :=\n  rfl\n\n@[simp]\ntheorem pred_eq_pred : Order.pred = pred :=\n  rfl\n\ninstance : IsSuccArchimedean ℤ :=\n  ⟨fun {a b} h =>\n    ⟨(b - a).toNat, by rw [succ_iterate, toNat_sub_of_le h, ← add_sub_assoc, add_sub_cancel_left]⟩⟩\n\ninstance : IsPredArchimedean ℤ :=\n  ⟨fun {a b} h =>\n    ⟨(b - a).toNat, by rw [pred_iterate, toNat_sub_of_le h, sub_sub_cancel]⟩⟩\n\n/-! ### Covering relation -/\n\n\n@[simp, norm_cast]\n\nTarget:\ntheorem natCast_covBy {a b : ℕ} : (a : ℤ) ⋖ b ↔ a ⋖ b :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"c6c8a02e22119901ef4648f31170b8bcb81e8923b31c22280cd140bf1bd2b748","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Int","family_id":"natcast_covby","file_id":"mathlib/Mathlib/Data/Int/SuccPred.lean","sample_id":"6402417a187a534989eab3a2c52dac736091b114295a76d321088d940f0ef62a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a76aedca09d06687f23aabb71a976948a3f5261d4be2092b4354b20a250ef4d4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c124e4499748820a17df65d5a170a2889e6ded7f6f676ef140d015398e461d38","source_sha256":"38b81cd46858f243db931b63ae095bb873edc08d47b08e77c1c254a8ae6a5653","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [h]\n\n/- ExProd.equateZero -/","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.25},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"4aa4753f2d907c1750f160a2a0f3efc6fb9e73bfed28c5c83018eb27201bd56e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Defs\npublic import Mathlib.Algebra.Algebra.Basic\npublic import Mathlib.Tactic.Ring.RingNF\n\nNamespace:\nMathlib.Tactic.Algebra\n\nLocal context:\n/-\nCopyright (c) 2025 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Arend Mellendijk\n-/\n/-! # Lemmas for the `algebra` tactic.\n-/\n\n@[expose] public section\n\nopen Mathlib.Meta.NormNum\n\nnamespace Mathlib.Tactic.Algebra\n\nsection ring\n\nvariable {R A : Type*} [CommRing R] [CommRing A] [Algebra R A]\n\n/- evalCast -/\ntheorem isInt_negOfNat_eq {a : A} {lit : ℕ} (h : IsInt a (Int.negOfNat lit)) :\n    a = algebraMap R A (Int.rawCast (Int.negOfNat lit) + 0 : R) + 0 := by\n  simp [h.out]\n\nend ring\n\nsection semifield\n\nvariable {R A : Type*} [Semifield R] [Semifield A] [Algebra R A]\n\n/- evalCast -/\ntheorem isNNRat_eq_rawCast {a : A} {n d : ℕ} (h : IsNNRat a n d) :\n    a = algebraMap R A (NNRat.rawCast n d + 0 : R) + 0 := by\n  simp [Mathlib.Tactic.Ring.cast_nnrat h]\n\nend semifield\n\nsection field\n\nvariable {R A : Type*} [Field R] [Field A] [Algebra R A]\n\n/- evalCast -/\ntheorem isRat_eq_rawCast {a : A} {n d : ℕ} (h : IsRat a (.negOfNat n) d) :\n    a = algebraMap R A (Rat.rawCast (.negOfNat n) d + 0 : R) + 0 := by\n  simp [Mathlib.Tactic.Ring.cast_rat h]\n\nend field\n\nvariable {R A : Type*} [sR : CommSemiring R] [sA : CommSemiring A] [sAlg : Algebra R A]\n\n/- evalCast -/\ntheorem isNat_zero_eq {a : A} (h : IsNat a 0) : a = 0 := by\n  have := h.out\n  simp [this]\n\n/- evalCast -/\ntheorem isNat_eq_rawCast {a : A} {lit : ℕ} (h : IsNat a lit) :\n    a = algebraMap R A (lit + 0 : R) + 0 := by\n  simp [h.out]\n\nsection cleanup\n\nvariable {n d : ℕ}\n\nsection cleanupSMul\n\ntheorem add_assoc_rev (a b c : R) : a + (b + c) = a + b + c := (add_assoc ..).symm\ntheorem mul_assoc_rev (a b c : R) : a * (b * c) = a * b * c := (mul_assoc ..).symm\ntheorem mul_neg {R} [Ring R] (a b : R) : a * -b = -(a * b) := by simp\ntheorem add_neg {R} [Ring R] (a b : R) : a + -b = a - b := (sub_eq_add_neg ..).symm\ntheorem nat_rawCast_0 : (Nat.rawCast 0 : R) = 0 := by simp\ntheorem nat_rawCast_1 : (Nat.rawCast 1 : R) = 1 := by simp\ntheorem nat_rawCast_2 [Nat.AtLeastTwo n] : (Nat.rawCast n : R) = OfNat.ofNat n := rfl\ntheorem int_rawCast_neg {R} [Ring R] : (Int.rawCast (.negOfNat n) : R) = -Nat.rawCast n := by simp\ntheorem nnrat_rawCast {R} [DivisionSemiring R] :\n    (NNRat.rawCast n d : R) = Nat.rawCast n / Nat.rawCast d := by simp\ntheorem rat_rawCast_neg {R} [DivisionRing R] :\n    (Rat.rawCast (.negOfNat n) d : R) = Int.rawCast (.negOfNat n) / Nat.rawCast d := by simp\n\nend cleanupSMul\nsection cleanupConsts\n\ntheorem ofNat_smul {R A} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    [n.AtLeastTwo] {a : A} :\n    (ofNat(n) : R) • a = ofNat(n) * a := by\n  simp_rw [← nat_rawCast_2]\n  simp [Nat.cast_smul_eq_nsmul]\n\ntheorem neg_ofNat_smul {R A} [CommRing R] [CommRing A] [Algebra R A] {a : A} [n.AtLeastTwo] :\n    (- ofNat(n) : R) • a = - (ofNat(n)) * a := by\n  simpa [← nat_rawCast_2] using! ofNat_smul\n\ntheorem neg_1_smul {R A} [CommRing R] [CommRing A] [Algebra R A] {a : A} :\n    (-1 : R) • a = - a := by\n  simp\n\ntheorem nnRat_ofNat_smul_1 {R A} [Semifield R] [Semifield A] [Algebra R A] {a : A}\n    [d.AtLeastTwo] :\n    (1 / ofNat(d) : R) • a = (1 / ofNat(d)) * a := by\n  simp [Algebra.smul_def, ← nat_rawCast_2]\n\ntheorem nnRat_ofNat_smul_2 {R A} [Semifield R] [Semifield A] [Algebra R A] {a : A}\n    [n.AtLeastTwo] [d.AtLeastTwo] :\n    (ofNat(n) / ofNat(d) : R) • a = (ofNat(n) / ofNat(d)) * a := by\n  simp [Algebra.smul_def, ← nat_rawCast_2]\n\ntheorem rat_ofNat_smul_1 {R A} [Field R] [Field A] [Algebra R A] {a : A}\n    [d.AtLeastTwo] :\n    ((- 1) / ofNat(d) : R) • a = ((- 1) / ofNat(d)) * a := by\n  simp [Algebra.smul_def, ← nat_rawCast_2]\n\ntheorem rat_ofNat_smul_2 {R A} [Field R] [Field A] [Algebra R A] {a : A}\n    [n.AtLeastTwo] [d.AtLeastTwo] :\n    ((- ofNat(n)) / ofNat(d) : R) • a = ((- ofNat(n)) / ofNat(d)) * a := by\n  simp [Algebra.smul_def, ← nat_rawCast_2]\n\nend cleanupConsts\n\nend cleanup\n\nsection equateScalars\n\n/- ExProd.equateZero -/\n\nTarget:\ntheorem smul_one_eq_zero {r : R} (h : r = 0) :\n    r • (1 : A) = 0 :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Algebra","family_id":"smul_one_eq_zero","file_id":"mathlib/Mathlib/Tactic/Algebra/Lemmas.lean","sample_id":"c124e4499748820a17df65d5a170a2889e6ded7f6f676ef140d015398e461d38"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4de448a7d71906907793d1dc7af096bf74aa03a276d6cf774b8ce324536dc08e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"03c570aaf8f111c107d758da0bd5e98a1f4ca9d08005e7436d51a2a50c34c894","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e762d0b76da2ca7a3ba42e4fe5df7c426f0bb2153a670147e34342a186a3e955","source_sha256":"4db60ea328ce3789c20719560845d3c8bdc3720bf72ad08c23d3cd9a57d627b0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => ?_\n  rw [Set.indicator_of_notMem]\n  simp only [not_le, Set.mem_setOf_eq]\n  refine lt_of_le_of_lt (Nat.le_ceil _) ?_\n  refine lt_of_lt_of_le (lt_add_one _) ?_\n  norm_cast\n\nvariable {p : ℝ≥0∞}","hard_negative":false,"metrics":{"chosen_tokens":71,"rejected_tokens":78,"token_jaccard":0.959184,"token_length_ratio":1.098592},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"4abffdce2654b5dd3e238172fe02bc0a2bdbd1dff1ff272140cb26170e39fceb","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Function.ConvergenceInMeasure\npublic import Mathlib.MeasureTheory.Function.L1Space.Integrable\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2022 Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kexing Ying\n-/\n/-!\n# Uniform integrability\n\nThis file contains the definitions for uniform integrability (both in the measure theory sense\nas well as the probability theory sense). This file also contains the Vitali convergence theorem\nwhich establishes a relation between uniform integrability, convergence in measure and\nLp convergence.\n\nUniform integrability plays a vital role in the theory of martingales and most notably is used to\nformulate the martingale convergence theorem.\n\n## Main definitions\n\n* `MeasureTheory.UnifIntegrable`: uniform integrability in the measure theory sense.\n  In particular, a sequence of functions `f` is uniformly integrable if for all `ε > 0`, there\n  exists some `δ > 0` such that for all sets `s` of smaller measure than `δ`, the Lp-norm of\n  `f i` restricted to `s` is smaller than `ε` for all `i`.\n* `MeasureTheory.UniformIntegrable`: uniform integrability in the probability theory sense.\n  In particular, a sequence of measurable functions `f` is uniformly integrable in the\n  probability theory sense if it is uniformly integrable in the measure theory sense and\n  has uniformly bounded Lp-norm.\n\n## Main results\n\n* `MeasureTheory.unifIntegrable_finite`: a finite sequence of Lp functions is uniformly\n  integrable.\n* `MeasureTheory.tendsto_Lp_finite_of_tendsto_ae`: a sequence of Lp functions which is uniformly\n  integrable converges in Lp if they converge almost everywhere.\n* `MeasureTheory.tendstoInMeasure_iff_tendsto_Lp_finite`: Vitali convergence theorem:\n  a sequence of Lp functions converges in Lp if and only if it is uniformly integrable\n  and converges in measure.\n\n## Tags\nuniformly integrable, uniformly absolutely continuous integral, Vitali convergence theorem\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen scoped MeasureTheory NNReal ENNReal Topology\n\nnamespace MeasureTheory\n\nopen Set Filter TopologicalSpace\n\nvariable {α β ι : Type*} {m : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup β]\n\n/-- Uniform integrability in the measure theory sense.\n\nA sequence of functions `f` is said to be uniformly integrable if for all `ε > 0`, there exists\nsome `δ > 0` such that for all sets `s` with measure less than `δ`, the Lp-norm of `f i`\nrestricted to `s` is less than `ε`.\n\nUniform integrability is also known as uniformly absolutely continuous integrals. -/\ndef UnifIntegrable {_ : MeasurableSpace α} (f : ι → α → β) (p : ℝ≥0∞) (μ : Measure α) : Prop :=\n  ∀ ⦃ε : ℝ⦄ (_ : 0 < ε), ∃ (δ : ℝ) (_ : 0 < δ), ∀ i s,\n    MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε\n\n/-- In probability theory, a family of measurable functions is uniformly integrable if it is\nuniformly integrable in the measure theory sense and is uniformly bounded. -/\ndef UniformIntegrable {_ : MeasurableSpace α} (f : ι → α → β) (p : ℝ≥0∞) (μ : Measure α) : Prop :=\n  (∀ i, AEStronglyMeasurable (f i) μ) ∧ UnifIntegrable f p μ ∧ ∃ C : ℝ≥0, ∀ i, eLpNorm (f i) p μ ≤ C\n\nnamespace UniformIntegrable\n\nprotected theorem aestronglyMeasurable {f : ι → α → β} {p : ℝ≥0∞} (hf : UniformIntegrable f p μ)\n    (i : ι) : AEStronglyMeasurable (f i) μ :=\n  hf.1 i\n\nprotected theorem unifIntegrable {f : ι → α → β} {p : ℝ≥0∞} (hf : UniformIntegrable f p μ) :\n    UnifIntegrable f p μ :=\n  hf.2.1\n\nprotected theorem memLp {f : ι → α → β} {p : ℝ≥0∞} (hf : UniformIntegrable f p μ) (i : ι) :\n    MemLp (f i) p μ :=\n  ⟨hf.1 i,\n    let ⟨_, _, hC⟩ := hf.2\n    lt_of_le_of_lt (hC i) ENNReal.coe_lt_top⟩\n\nend UniformIntegrable\n\nsection UnifIntegrable\n\n/-! ### `UnifIntegrable`\n\nThis section deals with uniform integrability in the measure theory sense. -/\n\n\nnamespace UnifIntegrable\n\nvariable {f g : ι → α → β} {p : ℝ≥0∞}\n\nprotected theorem add (hf : UnifIntegrable f p μ) (hg : UnifIntegrable g p μ) (hp : 1 ≤ p)\n    (hf_meas : ∀ i, AEStronglyMeasurable (f i) μ) (hg_meas : ∀ i, AEStronglyMeasurable (g i) μ) :\n    UnifIntegrable (f + g) p μ := by\n  intro ε hε\n  have hε2 : 0 < ε / 2 := half_pos hε\n  obtain ⟨δ₁, hδ₁_pos, hfδ₁⟩ := hf hε2\n  obtain ⟨δ₂, hδ₂_pos, hgδ₂⟩ := hg hε2\n  refine ⟨min δ₁ δ₂, lt_min hδ₁_pos hδ₂_pos, fun i s hs hμs => ?_⟩\n  simp_rw [Pi.add_apply, Set.indicator_add']\n  refine (eLpNorm_add_le ((hf_meas i).indicator hs) ((hg_meas i).indicator hs) hp).trans ?_\n  have hε_halves : ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2) := by\n    rw [← ENNReal.ofReal_add hε2.le hε2.le, add_halves]\n  rw [hε_halves]\n  exact add_le_add (hfδ₁ i s hs (hμs.trans (ENNReal.ofReal_le_ofReal (min_le_left _ _))))\n    (hgδ₂ i s hs (hμs.trans (ENNReal.ofReal_le_ofReal (min_le_right _ _))))\n\nprotected theorem neg (hf : UnifIntegrable f p μ) : UnifIntegrable (-f) p μ := by\n  simp_rw [UnifIntegrable, Pi.neg_apply, Set.indicator_neg', eLpNorm_neg]\n  exact hf\n\nprotected theorem sub (hf : UnifIntegrable f p μ) (hg : UnifIntegrable g p μ) (hp : 1 ≤ p)\n    (hf_meas : ∀ i, AEStronglyMeasurable (f i) μ) (hg_meas : ∀ i, AEStronglyMeasurable (g i) μ) :\n    UnifIntegrable (f - g) p μ := by\n  rw [sub_eq_add_neg]\n  exact hf.add hg.neg hp hf_meas fun i => (hg_meas i).neg\n\nprotected theorem ae_eq (hf : UnifIntegrable f p μ) (hfg : ∀ n, f n =ᵐ[μ] g n) :\n    UnifIntegrable g p μ := by\n  classical\n  intro ε hε\n  obtain ⟨δ, hδ_pos, hfδ⟩ := hf hε\n  refine ⟨δ, hδ_pos, fun n s hs hμs => (le_of_eq <| eLpNorm_congr_ae ?_).trans (hfδ n s hs hμs)⟩\n  filter_upwards [hfg n] with x hx\n  simp_rw [Set.indicator_apply, hx]\n\n/-- Uniform integrability is preserved by restriction of the functions to a set. -/\nprotected theorem indicator (hf : UnifIntegrable f p μ) (E : Set α) :\n    UnifIntegrable (fun i => E.indicator (f i)) p μ := fun ε hε ↦ by\n  obtain ⟨δ, hδ_pos, hε⟩ := hf hε\n  refine ⟨δ, hδ_pos, fun i s hs hμs ↦ ?_⟩\n  calc\n    eLpNorm (s.indicator (E.indicator (f i))) p μ\n      = eLpNorm (E.indicator (s.indicator (f i))) p μ := by\n      simp only [indicator_indicator, inter_comm]\n    _ ≤ eLpNorm (s.indicator (f i)) p μ := eLpNorm_indicator_le _\n    _ ≤ ENNReal.ofReal ε := hε _ _ hs hμs\n\n/-- Uniform integrability is preserved by restriction of the measure to a set. -/\nprotected theorem restrict (hf : UnifIntegrable f p μ) (E : Set α) :\n    UnifIntegrable f p (μ.restrict E) := fun ε hε ↦ by\n  obtain ⟨δ, hδ_pos, hδε⟩ := hf hε\n  refine ⟨δ, hδ_pos, fun i s hs hμs ↦ ?_⟩\n  rw [μ.restrict_apply hs, ← measure_toMeasurable] at hμs\n  calc\n    eLpNorm (indicator s (f i)) p (μ.restrict E) = eLpNorm (f i) p (μ.restrict (s ∩ E)) := by\n      rw [eLpNorm_indicator_eq_eLpNorm_restrict hs, μ.restrict_restrict hs]\n    _ ≤ eLpNorm (f i) p (μ.restrict (toMeasurable μ (s ∩ E))) :=\n      eLpNorm_mono_measure _ <| Measure.restrict_mono (subset_toMeasurable _ _) le_rfl\n    _ = eLpNorm (indicator (toMeasurable μ (s ∩ E)) (f i)) p μ :=\n      (eLpNorm_indicator_eq_eLpNorm_restrict (measurableSet_toMeasurable _ _)).symm\n    _ ≤ ENNReal.ofReal ε := hδε i _ (measurableSet_toMeasurable _ _) hμs\n\nend UnifIntegrable\n\ntheorem unifIntegrable_zero_meas [MeasurableSpace α] {p : ℝ≥0∞} {f : ι → α → β} :\n    UnifIntegrable f p (0 : Measure α) :=\n  fun ε _ => ⟨1, one_pos, fun i s _ _ => by simp⟩\n\ntheorem unifIntegrable_congr_ae {p : ℝ≥0∞} {f g : ι → α → β} (hfg : ∀ n, f n =ᵐ[μ] g n) :\n    UnifIntegrable f p μ ↔ UnifIntegrable g p μ :=\n  ⟨fun hf => hf.ae_eq hfg, fun hg => hg.ae_eq fun n => (hfg n).symm⟩\n\nTarget:\ntheorem tendsto_indicator_ge (f : α → β) (x : α) :\n    Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=\n\nProof body:\n","rejected":"```lean\nby\n  refine tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => ?_\n  rw [Set.indicator_of_notMem]\n  simp only [not_le, Set.mem_setOf_eq]\n  refine lt_of_le_of_lt (Nat.le_ceil _) ?_\n  refine lt_of_lt_of_le (lt_add_one _) ?_\n  norm_cast\n\nvariable {p : ℝ≥0∞}\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Function","family_id":"tendsto_indicator_ge","file_id":"mathlib/Mathlib/MeasureTheory/Function/UniformIntegrable.lean","sample_id":"e762d0b76da2ca7a3ba42e4fe5df7c426f0bb2153a670147e34342a186a3e955"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2c1ae7a6d731fe18987c9a16e015e76772289e20b04a8b5d9fc3cf3d2c6b5c61","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6eee5d2ff91023d609bd493224edd9b87b2fa60a603a305b0c65f8936a71c40f","source_sha256":"bc1107b51b023f15ba6b116e921d72825e5fd28811d110becc324cfa9ebc2fb7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold mkFinSnoc\n  exact coe_mk (v := Fin.snoc (N.subtype ∘ b) y) _ _\n\n/-- Let `b` be a basis for a submodule `N ≤ O`. If `y ∈ O` is linear independent of `N`\nand `y` and `N` together span the whole of `O`, then there is a basis for `O`\nwhose basis vectors are given by `Fin.snoc b y`. -/\nnoncomputable def mkFinSnocOfLE {n : ℕ} {N O : Submodule R M} (b : Basis (Fin n) R N)\n    (hNO : N ≤ O) (y : M) (yO : y ∈ O) (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0)\n    (hsp : ∀ z ∈ O, ∃ c : R, z + c • y ∈ N) : Basis (Fin (n + 1)) R O :=\n  mkFinSnoc (b.map (Submodule.comapSubtypeEquivOfLe hNO).symm) ⟨y, yO⟩\n    (fun c x hc hx => hli c x (Submodule.mem_comap.mp hc) (congr_arg ((↑) : O → M) hx))\n    fun z => hsp z z.2","hard_negative":false,"metrics":{"chosen_tokens":262,"rejected_tokens":2,"token_jaccard":0.011236,"token_length_ratio":0.007634},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"4ae27756f716074d2045181f1258cbabe74c4011df5883ed418ce765c93ff95a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Basis.Basic\npublic import Mathlib.LinearAlgebra.Pi\n\nNamespace:\nModule.Basis\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Kevin H. Wilson\n-/\n/-!\n# Bases indexed by `Fin`\n-/\n\n@[expose] public section\n\nassert_not_exists Ordinal\n\nnoncomputable section\n\nuniverse u\n\nopen Function Set Submodule Finsupp\n\nvariable {ι : Type*} {ι' : Type*} {R : Type*} {R₂ : Type*} {M : Type*} {M' : Type*}\n\nnamespace Module\n\nopen LinearMap\n\nvariable {v : ι → M}\nvariable [Ring R] [CommRing R₂] [AddCommGroup M]\nvariable [Module R M] [Module R₂ M]\nvariable {x y : M}\nvariable (b : Basis ι R M)\n\nnamespace Basis\n\nsection Fin\n\n/-- Let `b` be a basis for a submodule `N` of `M`. If `y : M` is linear independent of `N`\nand `y` and `N` together span the whole of `M`, then there is a basis for `M`\nwhose basis vectors are given by `Fin.cons y b`. -/\nnoncomputable def mkFinCons {n : ℕ} {N : Submodule R M} (y : M) (b : Basis (Fin n) R N)\n    (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :\n    Basis (Fin (n + 1)) R M :=\n  have span_b : N = Submodule.span R (Set.range (N.subtype ∘ b)) := by\n    rw [Set.range_comp, Submodule.span_image, b.span_eq, Submodule.map_subtype_top]\n  Basis.mk (v := Fin.cons y (N.subtype ∘ b))\n    ((b.linearIndependent.map' N.subtype (Submodule.ker_subtype _)).finCons' _ _\n      (by\n        intro c x hx hc\n        rw [← span_b] at hx\n        exact hli c x hx hc))\n    fun x _ => by simpa [Submodule.mem_span_insert', span_b] using hsp x\n\n@[simp]\ntheorem coe_mkFinCons {n : ℕ} {N : Submodule R M} (y : M) (b : Basis (Fin n) R N)\n    (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :\n    (mkFinCons y b hli hsp : Fin (n + 1) → M) = Fin.cons y ((↑) ∘ b) := by\n  unfold mkFinCons\n  exact coe_mk (v := Fin.cons y (N.subtype ∘ b)) _ _\n\n/-- Let `b` be a basis for a submodule `N ≤ O`. If `y ∈ O` is linear independent of `N`\nand `y` and `N` together span the whole of `O`, then there is a basis for `O`\nwhose basis vectors are given by `Fin.cons y b`. -/\nnoncomputable def mkFinConsOfLE {n : ℕ} {N O : Submodule R M} (y : M) (yO : y ∈ O)\n    (b : Basis (Fin n) R N) (hNO : N ≤ O) (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0)\n    (hsp : ∀ z ∈ O, ∃ c : R, z + c • y ∈ N) : Basis (Fin (n + 1)) R O :=\n  mkFinCons ⟨y, yO⟩ (b.map (Submodule.comapSubtypeEquivOfLe hNO).symm)\n    (fun c x hc hx => hli c x (Submodule.mem_comap.mp hc) (congr_arg ((↑) : O → M) hx))\n    fun z => hsp z z.2\n\n@[simp]\ntheorem coe_mkFinConsOfLE {n : ℕ} {N O : Submodule R M} (y : M) (yO : y ∈ O) (b : Basis (Fin n) R N)\n    (hNO : N ≤ O) (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0)\n    (hsp : ∀ z ∈ O, ∃ c : R, z + c • y ∈ N) :\n    (mkFinConsOfLE y yO b hNO hli hsp : Fin (n + 1) → O) =\n      Fin.cons ⟨y, yO⟩ (Submodule.inclusion hNO ∘ b) :=\n  coe_mkFinCons _ _ _ _\n\n/-- Let `b` be a basis for a submodule `N` of `M`. If `y : M` is linear independent of `N`\nand `y` and `N` together span the whole of `M`, then there is a basis for `M`\nwhose basis vectors are given by `Fin.snoc b y`. -/\nnoncomputable def mkFinSnoc {n : ℕ} {N : Submodule R M} (b : Basis (Fin n) R N) (y : M)\n    (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :\n    Basis (Fin (n + 1)) R M :=\n  have span_b : N = Submodule.span R (Set.range (N.subtype ∘ b)) := by\n    rw [Set.range_comp, Submodule.span_image, b.span_eq, Submodule.map_subtype_top]\n  Basis.mk (v := Fin.snoc (N.subtype ∘ b) y)\n    ((b.linearIndependent.map' N.subtype (Submodule.ker_subtype _)).finSnoc' _ _\n      (by\n        intro c x hx hc\n        rw [← span_b] at hx\n        exact hli c x hx hc))\n    fun x _ ↦ by simpa [Submodule.mem_span_insert', span_b] using hsp x\n\n@[simp]\n\nTarget:\ntheorem coe_mkFinSnoc {n : ℕ} {N : Submodule R M} (b : Basis (Fin n) R N) (y : M)\n    (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :\n    (mkFinSnoc b y hli hsp : Fin (n + 1) → M) = Fin.snoc ((↑) ∘ b) y :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Basis","family_id":"coe_mkfinsnoc","file_id":"mathlib/Mathlib/LinearAlgebra/Basis/Fin.lean","sample_id":"6eee5d2ff91023d609bd493224edd9b87b2fa60a603a305b0c65f8936a71c40f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"002c7d89e4368c0249a9cb4cfc7e30d967572004732ef0400a272686c72ce645","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8c1abff1d102ab8663f817566b5f1e0679aded535e3b61cc03bd6dcfd4532f76","source_sha256":"b8e096f7469de9f8271b2d043d71be202bf790e32cee2afb83b7783c6d94bbb7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa [basis_repr_apply] using ιMultiDual_apply_nondiag R n b t s hst.symm","hard_negative":false,"metrics":{"chosen_tokens":16,"rejected_tokens":2,"token_jaccard":0.058824,"token_length_ratio":0.125},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"4b08ea0b081d1f430bdce7a8c28f1ac7ade56b4d2514b0e01e9f0534c864c834","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.ExteriorPower.Basic\npublic import Mathlib.LinearAlgebra.ExteriorPower.Pairing\npublic import Mathlib.RingTheory.Finiteness.Subalgebra\npublic import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition\n\nNamespace:\nexteriorPower\n\nLocal context:\n/-\nCopyright (c) 2025 Daniel Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sophie Morel, Daniel Morrison\n-/\n/-!\n# Constructs a basis for exterior powers\n-/\n\n@[expose] public section\n\nvariable {R K M E : Type*} {n : ℕ}\n  [CommRing R] [Field K] [AddCommGroup M] [Module R M] [AddCommGroup E] [Module K E]\n\nnamespace exteriorPower\n\n/-! Finiteness of the exterior power. -/\n\n/-- The `n`th exterior power of a finite module is a finite module. -/\ninstance instFinite [Module.Finite R M] : Module.Finite R (⋀[R]^n M) := by\n  rw [Module.Finite.iff_fg, ExteriorAlgebra.exteriorPower, LinearMap.range_eq_map]\n  exact Submodule.FG.pow (Submodule.FG.map _ Module.Finite.fg_top) n\n\n/-! We construct a basis of `⋀[R]^n M` from a basis of `M`. -/\n\nopen Module Set Set.powersetCard\n\nvariable (R n)\n\n/-- If `b` is a basis of `M` indexed by a linearly ordered type `I` and `s` is a finset of\n`I` of cardinality `n`, then we get a linear form on the `n`th exterior power of `M` by\napplying the `exteriorPower.linearForm` construction to the family of linear forms\ngiven by the coordinates of `b` indexed by elements of `s` (ordered using the given order on\n`I`). -/\nnoncomputable def ιMultiDual {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s : powersetCard I n) : Module.Dual R (⋀[R]^n M) :=\n  pairingDual R M n (ιMulti_family R n b.coord s)\n\n@[simp]\nlemma ιMultiDual_apply_ιMulti {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s : powersetCard I n) (v : Fin n → M) :\n    ιMultiDual R n b s (ιMulti R n v) =\n    (Matrix.of fun i j => b.coord (powersetCard.ofFinEmbEquiv.symm s j) (v i)).det := by\n  simp [ιMultiDual, ιMulti_family, pairingDual_ιMulti_ιMulti]\n\n/-- Let `b` be a basis of `M` indexed by a linearly ordered type `I` and `s` be a finset of `I`\nof cardinality `n`. If we apply the linear form on `⋀[R]^n M` defined by `b` and `s`\nto the exterior product of the `b i` for `i ∈ s`, then we get `1`. -/\nlemma ιMultiDual_apply_diag {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s : powersetCard I n) :\n    ιMultiDual R n b s (ιMulti_family R n b s) = 1 := by\n  rw [ιMulti_family, ιMultiDual_apply_ιMulti]\n  suffices Matrix.of (fun i j => b.coord (powersetCard.ofFinEmbEquiv.symm s j)\n    (b (powersetCard.ofFinEmbEquiv.symm s i))) = 1 by\n    simp_rw [Function.comp_apply, this, Matrix.det_one]\n  ext\n  simp [Matrix.one_apply, Finsupp.single_apply]\n\n/-- Let `b` be a basis of `M` indexed by a linearly ordered type `I` and `s` be a finset of `I`\nof cardinality `n`. Let `t` be a finset of `I` of cardinality `n` such that `s ≠ t`. If we apply\nthe linear form on `⋀[R]^n M` defined by `b` and `s` to the exterior product of the\n`b i` for `i ∈ t`, then we get `0`. -/\nlemma ιMultiDual_apply_nondiag {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s t : powersetCard I n) (hst : s ≠ t) :\n    ιMultiDual R n b s (ιMulti_family R n b t) = 0 := by\n  rw [ιMulti_family, ιMultiDual_apply_ιMulti]\n  obtain ⟨i, his, hit⟩ := (exists_mem_notMem_iff_ne s t).mp hst\n  obtain ⟨k, rfl⟩ := (mem_range_ofFinEmbEquiv_symm_iff_mem s i).mpr his\n  apply Matrix.det_eq_zero_of_column_eq_zero k\n  simp_rw [Matrix.of_apply, Basis.coord_apply, Function.comp_apply, Basis.repr_self]\n  intro j\n  apply Finsupp.single_eq_of_ne\n  by_contra! h\n  apply hit\n  rw [h, powersetCard.ofFinEmbEquiv_symm_apply, ← powersetCard.mem_coe_iff]\n  exact Finset.orderEmbOfFin_mem t.val t.prop j\n\n/-- If `b` is a basis of `M` (indexed by a linearly ordered type), then the family\n`exteriorPower.ιMulti R n b` of the `n`-fold exterior products of its elements is linearly\nindependent in the `n`th exterior power of `M`. -/\nlemma ιMulti_family_linearIndependent_ofBasis {I : Type*} [LinearOrder I] (b : Basis I R M) :\n    LinearIndependent R (ιMulti_family R n b) :=\n  LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun s ↦ ιMultiDual R n b s)\n    (fun _ _ h => ιMultiDual_apply_nondiag R n b _ _ h)\n    (fun _ => ιMultiDual_apply_diag _ _ _ _)\n\nvariable {R} in\n/-- If `b` is a basis of `M` (indexed by a linearly ordered type), the basis of the `n`th\nexterior power of `M` formed by the `n`-fold exterior products of elements of `b`. -/\nnoncomputable def _root_.Module.Basis.exteriorPower {I : Type*} [LinearOrder I] (b : Basis I R M) :\n    Basis (powersetCard I n) R (⋀[R]^n M) :=\n  Basis.mk (ιMulti_family_linearIndependent_ofBasis _ _ _)\n    (eq_top_iff.mp <| ιMulti_family_span_of_span R b.span_eq)\n\n@[simp]\nlemma coe_basis {I : Type*} [LinearOrder I] (b : Basis I R M) :\n    DFunLike.coe (b.exteriorPower n) = ιMulti_family R n b :=\n  Basis.coe_mk _ _\n\nlemma basis_apply {I : Type*} [LinearOrder I] (b : Basis I R M) (s : powersetCard I n) :\n    b.exteriorPower n s = ιMulti_family R n b s := by\n  rw [coe_basis]\n\n/-- If `b` is a basis of `M` indexed by a linearly ordered type `I` and `B` is the corresponding\nbasis of the `n`th exterior power of `M`, indexed by the set of finsets `s` of `I` of cardinality\n`n`, then the coordinate function of `B` at `s` is the linear form on the `n`th exterior power\ndefined by `b` and `s` in `exteriorPower.ιMultiDual`. -/\nlemma basis_coord {I : Type*} [LinearOrder I] (b : Basis I R M) (s : powersetCard I n) :\n    Basis.coord (b.exteriorPower n) s = ιMultiDual R n b s := by\n  apply LinearMap.ext_on (ιMulti_family_span_of_span R (Basis.span_eq b))\n  rintro x ⟨t, rfl⟩\n  rw [Basis.coord_apply]\n  by_cases! hst : s = t\n  · rw [hst, ιMultiDual_apply_diag, ← basis_apply, Basis.repr_self, Finsupp.single_eq_same]\n  · rw [ιMultiDual_apply_nondiag R n b s t hst, ← basis_apply, Basis.repr_self,\n      Finsupp.single_eq_of_ne hst]\n\nlemma basis_repr_apply {I : Type*} [LinearOrder I] (b : Basis I R M) (x : ⋀[R]^n M)\n    (s : powersetCard I n) :\n    Basis.repr (b.exteriorPower n) x s = ιMultiDual R n b s x := by\n  simpa [← Basis.coord_apply] using LinearMap.congr_fun (basis_coord R n b s) x\n\n@[simp]\nlemma basis_repr_self {I : Type*} [LinearOrder I] (b : Basis I R M) (s : powersetCard I n) :\n    Basis.repr (b.exteriorPower n) (ιMulti_family R n b s) s = 1 := by\n  simpa [basis_repr_apply] using ιMultiDual_apply_diag R n b s\n\n@[simp]\n\nTarget:\nlemma basis_repr_ne {I : Type*} [LinearOrder I] (b : Basis I R M)\n    {s t : powersetCard I n} (hst : s ≠ t) :\n    Basis.repr (b.exteriorPower n) (ιMulti_family R n b s) t = 0 :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/ExteriorPower","family_id":"basis_repr_ne","file_id":"mathlib/Mathlib/LinearAlgebra/ExteriorPower/Basis.lean","sample_id":"8c1abff1d102ab8663f817566b5f1e0679aded535e3b61cc03bd6dcfd4532f76"},"split":"train"}
{"binding":{"chosen_proof_sha256":"abdd5d1c7dac8f0ca69859030dcc532b57d2b6a28ac7caf0a9109fccea1058bd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5b70195082dfe681d1970751f7a4885c8fdbaa9cac549078cb4a1812112d4d46","source_sha256":"5ec4c4d2769bb0652aea247f81d7ff99e572b3159717c948f413b35f1ff454f9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [sup_comm, inf_comm, ← inf_assoc, sup_inf_inf_sdiff]\n  apply sdiff_unique\n  · calc\n      x ⊓ y \\ z ⊔ (z ⊓ x ⊔ x \\ y) = (x ⊔ (z ⊓ x ⊔ x \\ y)) ⊓ (y \\ z ⊔ (z ⊓ x ⊔ x \\ y)) := by\n          rw [sup_inf_right]\n      _ = (x ⊔ x ⊓ z ⊔ x \\ y) ⊓ (y \\ z ⊔ (x ⊓ z ⊔ x \\ y)) := by ac_rfl\n      _ = x ⊓ (y \\ z ⊔ (x ⊓ z ⊔ x ⊓ y) ⊔ x \\ y) := by\n          rw [sup_inf_self, sup_sdiff_left, ← sup_assoc, sup_inf_left, sdiff_sup_self',\n            inf_sup_right, sup_comm y, inf_sdiff_sup_right, inf_sup_left x z y]\n      _ = x ⊓ (y \\ z ⊔ (x ⊓ z ⊔ (x ⊓ y ⊔ x \\ y))) := by ac_rfl\n      _ = x := by simp\n  · calc\n      x ⊓ y \\ z ⊓ (z ⊓ x ⊔ x \\ y) = x ⊓ y \\ z ⊓ (z ⊓ x) ⊔ x ⊓ y \\ z ⊓ x \\ y := by rw [inf_sup_left]\n      _ = x ⊓ (y \\ z ⊓ z ⊓ x) ⊔ x ⊓ y \\ z ⊓ x \\ y := by ac_rfl\n      _ = x ⊓ y \\ z ⊓ x \\ y := by rw [inf_sdiff_self_left, bot_inf_eq, inf_bot_eq, bot_sup_eq]\n      _ = x ⊓ (y \\ z ⊓ y) ⊓ x \\ y := by conv_lhs => rw [← inf_sdiff_left]\n      _ = x ⊓ (y \\ z ⊓ (y ⊓ x \\ y)) := by ac_rfl\n      _ = ⊥ := by rw [inf_sdiff_self_right, inf_bot_eq, inf_bot_eq]","hard_negative":false,"metrics":{"chosen_tokens":334,"rejected_tokens":2,"token_jaccard":0.021739,"token_length_ratio":0.005988},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"4b31703d672509e84c008e56dee9469a7908f79dedcc86e97867842f79f70d8d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.BooleanAlgebra.Defs\npublic import Mathlib.Tactic.GRewrite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Bryan Gin-ge Chen\n-/\n/-!\n# Basic properties of Boolean algebras\n\nThis file provides some basic definitions, functions as well as lemmas for functions and type\nclasses related to Boolean algebras as defined in `Mathlib/Order/BooleanAlgebra/Defs.lean`.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>\n* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]\n* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]\n\n## Tags\n\ngeneralized Boolean algebras, Boolean algebras, lattices, sdiff, compl\n\n-/\n\npublic section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type*} {x y z : α}\n\n/-!\n### Generalized Boolean algebras\n\nSome of the lemmas in this section are from:\n\n* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]\n* <https://ncatlab.org/nlab/show/relative+complement>\n* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>\n\n-/\n\n-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,\n-- however we'd need another type class for lattices with bot, and all the API for that.\nsection GeneralizedBooleanAlgebra\n\nvariable [GeneralizedBooleanAlgebra α]\n\n@[simp]\ntheorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \\ y = x :=\n  GeneralizedBooleanAlgebra.sup_inf_sdiff _ _\n\n@[simp]\ntheorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \\ y = ⊥ :=\n  GeneralizedBooleanAlgebra.inf_inf_sdiff _ _\n\n@[simp]\ntheorem sup_sdiff_inf (x y : α) : x \\ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]\n\n@[simp]\ntheorem inf_sdiff_inf (x y : α) : x \\ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α where\n  __ := GeneralizedBooleanAlgebra.toBot\n  bot_le a := by\n    rw [← inf_inf_sdiff a a, inf_assoc]\n    exact inf_le_left\n\ntheorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \\ y) :=\n  disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le\n\n-- TODO: in distributive lattices, relative complements are unique when they exist\ntheorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \\ y = z := by\n  conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]\n  rw [sup_comm] at s\n  conv_rhs at i => rw [← inf_inf_sdiff x y, inf_comm]\n  rw [inf_comm] at i\n  exact (eq_of_inf_eq_sup_eq i s).symm\n\n-- Use `sdiff_le`\nprivate theorem sdiff_le' : x \\ y ≤ x :=\n  calc\n    x \\ y ≤ x ⊓ y ⊔ x \\ y := le_sup_right\n    _ = x := sup_inf_sdiff x y\n\nset_option backward.privateInPublic true in\n-- Use `sdiff_sup_self`\nprivate theorem sdiff_sup_self' : y \\ x ⊔ x = y ⊔ x :=\n  calc\n    y \\ x ⊔ x = y \\ x ⊔ (x ⊔ x ⊓ y) := by rw [sup_inf_self]\n    _ = y ⊓ x ⊔ y \\ x ⊔ x := by ac_rfl\n    _ = y ⊔ x := by rw [sup_inf_sdiff]\n\n@[simp]\ntheorem sdiff_inf_sdiff : x \\ y ⊓ y \\ x = ⊥ :=\n  Eq.symm <|\n    calc\n      ⊥ = x ⊓ (y ⊓ x ⊔ y \\ x) ⊓ x \\ y := by rw [← inf_inf_sdiff, sup_inf_sdiff]\n      _ = (x ⊓ (y ⊓ x) ⊔ x ⊓ y \\ x) ⊓ x \\ y := by rw [inf_sup_left]\n      _ = (y ⊓ (x ⊓ x) ⊔ x ⊓ y \\ x) ⊓ x \\ y := by ac_rfl\n      _ = x ⊓ y \\ x ⊓ x \\ y := by\n          rw [inf_idem, inf_sup_right, ← inf_comm x y, inf_inf_sdiff, bot_sup_eq]\n      _ = x ⊓ x \\ y ⊓ y \\ x := by ac_rfl\n      _ = x \\ y ⊓ y \\ x := by rw [inf_of_le_right sdiff_le']\n\ntheorem disjoint_sdiff_sdiff : Disjoint (x \\ y) (y \\ x) :=\n  disjoint_iff_inf_le.mpr sdiff_inf_sdiff.le\n\n@[simp]\ntheorem inf_sdiff_self_right : x ⊓ y \\ x = ⊥ :=\n  calc\n    x ⊓ y \\ x = (x ⊓ y ⊔ x \\ y) ⊓ y \\ x := by rw [sup_inf_sdiff]\n    _ = ⊥ := by rw [inf_sup_right, inf_comm x y, inf_inf_sdiff, sdiff_inf_sdiff, bot_sup_eq]\n\n@[simp]\ntheorem inf_sdiff_self_left : y \\ x ⊓ x = ⊥ := by rw [inf_comm, inf_sdiff_self_right]\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n-- see Note [lower instance priority]\ninstance (priority := 100) GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra :\n    GeneralizedCoheytingAlgebra α where\n  __ := ‹GeneralizedBooleanAlgebra α›\n  __ := GeneralizedBooleanAlgebra.toOrderBot\n  sdiff := (· \\ ·)\n  sdiff_le_iff y x z :=\n    ⟨fun h =>\n      le_of_inf_le_sup_le\n        (le_of_eq\n          (by grind [sdiff_le', inf_of_le_right, inf_eq_right, inf_sdiff_self_right, bot_sup_eq,\n            inf_sup_right]))\n        (calc\n          y ⊔ y \\ x ≤ y \\ x ⊔ x ⊔ z := by\n            grind [sup_of_le_left, sdiff_le', le_sup_left, sdiff_sup_self']\n          _ = x ⊔ z ⊔ y \\ x := by ac_rfl),\n      fun h => le_of_inf_le_sup_le (inf_sdiff_self_left.trans_le bot_le) (calc\n        y \\ x ⊔ x = y ⊔ x := sdiff_sup_self'\n        _ ≤ x ⊔ z ⊔ x := sup_le_sup_right h x\n        _ ≤ z ⊔ x := by rw [sup_assoc, sup_comm, sup_assoc, sup_idem])⟩\n\ntheorem disjoint_sdiff_self_left : Disjoint (y \\ x) x :=\n  disjoint_iff_inf_le.mpr inf_sdiff_self_left.le\n\ntheorem disjoint_sdiff_self_right : Disjoint x (y \\ x) :=\n  disjoint_iff_inf_le.mpr inf_sdiff_self_right.le\n\nlemma le_sdiff : x ≤ y \\ z ↔ x ≤ y ∧ Disjoint x z :=\n  ⟨fun h ↦ ⟨h.trans sdiff_le, disjoint_sdiff_self_left.mono_left h⟩, fun h ↦\n    by rw [← h.2.sdiff_eq_left]; exact sdiff_le_sdiff_right h.1⟩\n\n@[simp] lemma sdiff_eq_left : x \\ y = x ↔ Disjoint x y :=\n  ⟨fun h ↦ disjoint_sdiff_self_left.mono_left h.ge, Disjoint.sdiff_eq_left⟩\n\n/- TODO: we could make an alternative constructor for `GeneralizedBooleanAlgebra` using\n`Disjoint x (y \\ x)` and `x ⊔ (y \\ x) = y` as axioms. -/\ntheorem Disjoint.sdiff_eq_of_sup_eq (hi : Disjoint x z) (hs : x ⊔ z = y) : y \\ x = z :=\n  have h : y ⊓ x = x := inf_eq_right.2 <| le_sup_left.trans hs.le\n  sdiff_unique (by rw [h, hs]) (by rw [h, hi.eq_bot])\n\nprotected theorem Disjoint.sdiff_unique (hd : Disjoint x z) (hz : z ≤ y) (hs : y ≤ x ⊔ z) :\n    y \\ x = z :=\n  sdiff_unique\n    (by\n      rw [← inf_eq_right] at hs\n      rwa [sup_inf_right, inf_sup_right, sup_comm x, inf_sup_self, inf_comm, sup_comm z,\n        hs, sup_eq_left])\n    (by rw [inf_assoc, hd.eq_bot, inf_bot_eq])\n\n-- cf. `IsCompl.disjoint_left_iff` and `IsCompl.disjoint_right_iff`\ntheorem disjoint_sdiff_iff_le (hz : z ≤ y) (hx : x ≤ y) : Disjoint z (y \\ x) ↔ z ≤ x :=\n  ⟨fun H =>\n    le_of_inf_le_sup_le (le_trans H.le_bot bot_le)\n      (by\n        rw [sup_sdiff_cancel_right hx]\n        grw [sdiff_le]\n        rw [sup_eq_right.2 hz]),\n    fun H => disjoint_sdiff_self_right.mono_left H⟩\n\n-- cf. `IsCompl.le_left_iff` and `IsCompl.le_right_iff`\ntheorem le_iff_disjoint_sdiff (hz : z ≤ y) (hx : x ≤ y) : z ≤ x ↔ Disjoint z (y \\ x) :=\n  (disjoint_sdiff_iff_le hz hx).symm\n\n-- cf. `IsCompl.inf_left_eq_bot_iff` and `IsCompl.inf_right_eq_bot_iff`\ntheorem inf_sdiff_eq_bot_iff (hz : z ≤ y) (hx : x ≤ y) : z ⊓ y \\ x = ⊥ ↔ z ≤ x := by\n  rw [← disjoint_iff]\n  exact disjoint_sdiff_iff_le hz hx\n\n-- cf. `IsCompl.left_le_iff` and `IsCompl.right_le_iff`\ntheorem le_iff_eq_sup_sdiff (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \\ x :=\n  ⟨fun H => (sup_sdiff_cancel' H hz).symm,\n    fun H => by\n    conv_lhs at H => rw [← sup_sdiff_cancel_right hx]\n    refine le_of_inf_le_sup_le ?_ H.le\n    rw [inf_sdiff_self_right]\n    exact bot_le⟩\n\n-- cf. `IsCompl.sup_inf`\ntheorem sdiff_sup : y \\ (x ⊔ z) = y \\ x ⊓ y \\ z :=\n  sdiff_unique\n    (calc\n      y ⊓ (x ⊔ z) ⊔ y \\ x ⊓ y \\ z = (y ⊓ x ⊔ y ⊓ z ⊔ y \\ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \\ z) := by\n          rw [sup_inf_left, inf_sup_left y]\n      _ = (y ⊓ z ⊔ (y ⊓ x ⊔ y \\ x)) ⊓ (y ⊓ x ⊔ (y ⊓ z ⊔ y \\ z)) := by ac_rfl\n      _ = (y ⊓ z ⊔ y) ⊓ (y ⊓ x ⊔ y) := by rw [sup_inf_sdiff, sup_inf_sdiff]\n      _ = (y ⊔ y ⊓ z) ⊓ (y ⊔ y ⊓ x) := by ac_rfl\n      _ = y := by rw [sup_inf_self, sup_inf_self, inf_idem])\n    (calc\n      y ⊓ (x ⊔ z) ⊓ (y \\ x ⊓ y \\ z) = y ⊓ x ⊓ (y \\ x ⊓ y \\ z) ⊔ y ⊓ z ⊓ (y \\ x ⊓ y \\ z) := by\n          rw [inf_sup_left, inf_sup_right]\n      _ = y ⊓ x ⊓ y \\ x ⊓ y \\ z ⊔ y \\ x ⊓ (y \\ z ⊓ (y ⊓ z)) := by ac_rfl\n      _ = ⊥ := by simp)\n\ntheorem sdiff_eq_sdiff_iff_inf_eq_inf : y \\ x = y \\ z ↔ y ⊓ x = y ⊓ z :=\n  ⟨fun h => eq_of_inf_eq_sup_eq (a := y \\ x) (by rw [inf_inf_sdiff, h, inf_inf_sdiff])\n    (by rw [sup_inf_sdiff, h, sup_inf_sdiff]),\n    fun h => by rw [← sdiff_inf_self_right, ← sdiff_inf_self_right z y, inf_comm, h, inf_comm]⟩\n\ntheorem sdiff_eq_self_iff_disjoint : x \\ y = x ↔ Disjoint y x := sdiff_eq_left.trans disjoint_comm\n\ntheorem sdiff_lt (hx : y ≤ x) (hy : y ≠ ⊥) : x \\ y < x := by\n  refine sdiff_le.lt_of_ne fun h => hy ?_\n  rw [sdiff_eq_left, disjoint_iff] at h\n  rw [← h, inf_eq_right.mpr hx]\n\ntheorem sdiff_lt_left : x \\ y < x ↔ ¬ Disjoint y x := by\n  rw [lt_iff_le_and_ne, Ne, sdiff_eq_self_iff_disjoint, and_iff_right sdiff_le]\n\n@[simp]\ntheorem le_sdiff_right : x ≤ y \\ x ↔ x = ⊥ :=\n  ⟨fun h => disjoint_self.1 (disjoint_sdiff_self_right.mono_right h), fun h => h.le.trans bot_le⟩\n\n@[simp] lemma sdiff_eq_right : x \\ y = y ↔ x = ⊥ ∧ y = ⊥ := by\n  rw [disjoint_sdiff_self_left.eq_iff]; simp_all\n\nlemma sdiff_ne_right : x \\ y ≠ y ↔ x ≠ ⊥ ∨ y ≠ ⊥ := sdiff_eq_right.not.trans not_and_or\n\ntheorem sdiff_lt_sdiff_right (h : x < y) (hz : z ≤ x) : x \\ z < y \\ z :=\n  (sdiff_le_sdiff_right h.le).lt_of_not_ge\n    fun h' => h.not_ge <| le_sdiff_sup.trans <| sup_le_of_le_sdiff_right h' hz\n\ntheorem sup_inf_inf_sdiff : x ⊓ y ⊓ z ⊔ y \\ z = x ⊓ y ⊔ y \\ z := by\n  rw [inf_assoc, sup_inf_right, sup_inf_sdiff, inf_sup_right, inf_sdiff_left]\n\nTarget:\ntheorem sdiff_sdiff_right : x \\ (y \\ z) = x \\ y ⊔ x ⊓ y ⊓ z :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/BooleanAlgebra","family_id":"sdiff_sdiff_right","file_id":"mathlib/Mathlib/Order/BooleanAlgebra/Basic.lean","sample_id":"5b70195082dfe681d1970751f7a4885c8fdbaa9cac549078cb4a1812112d4d46"},"split":"train"}
{"binding":{"chosen_proof_sha256":"82e688aad755eac6f657be89199e8e6fa7e63dacb10bb688605e04b2c9309872","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d3b554a7490cfc938221d5d7fff9423a8ad834bad53b65c4db2638f83535b57e","source_sha256":"e2806df73ecfc81f91bdaba3be881cdfeef7117f485c371a23cf9b886b2f1661","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    suffices IsSquare (-x) → x = 0 by aesop\n    exact fun hc ↦ le_antisymm (nonpos_of_neg_nonneg (IsSquare.nonneg hc)) h\n  mpr := IsSquare.nonneg\n\nalias ⟨_root_.IsSquare.of_nonneg, _⟩ := nonneg_iff_isSquare","hard_negative":false,"metrics":{"chosen_tokens":45,"rejected_tokens":2,"token_jaccard":0.032258,"token_length_ratio":0.044444},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"4b5862fa895c466abf9019621fad19d0a32eb81493407242a73c5b5cb0f06e63","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Degree.Domain\npublic import Mathlib.Algebra.Polynomial.Eval.Defs\npublic import Mathlib.Algebra.Ring.Semireal.Defs\npublic import Mathlib.Tactic.LinearCombination\n\nNamespace:\nIsRealClosed\n\nLocal context:\n/-\nCopyright (c) 2025 Artie Khovanov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Artie Khovanov\n-/\n/-!\n# Real Closed Field\n\nA field `R` is real closed if all of the following hold:\n1. `R` is real (that is, `-1` is not a sum of squares in `R`).\n2. for every `x` in `R`, one of `x` or `-x` is a square.\n3. every odd-degree polynomial over `R` has a root in `R`.\n\nA real closed field is an algebraic generalisation of the real numbers.\n\nIn this file we define real closed fields and prove some of their properties.\n\nTODO (Artie Khovanov) : equivalent conditions for a real field to be real closed\nTODO (Artie Khovanov) : real numbers, real algebraic numbers, hyperreals form a real closed field\n\n## Main Definitions\n\n- `IsRealClosed R` is the typeclass saying `R` is a real closed field.\n\n## Tags\n\nreal closed, rcf\n\n-/\n\npublic section\n\nopen Polynomial\n\n/--\nA field `R` is real closed if all of the following hold:\n1. `R` is real (that is, `-1` is not a sum of squares in `R`).\n2. for every `x` in `R`, one of `x` or `-x` is a square.\n3. every odd-degree polynomial over `R` has a root in `R`.\n-/\nclass IsRealClosed (R : Type*) [Field R] : Prop extends IsSemireal R where\n  isSquare_or_isSquare_neg (x : R) : IsSquare x ∨ IsSquare (-x)\n  exists_isRoot_of_odd_natDegree {f : R[X]} (hf : Odd f.natDegree) : ∃ x, f.IsRoot x\n\nattribute [aesop 90% forward] IsRealClosed.isSquare_or_isSquare_neg\n\nnamespace IsRealClosed\n\nuniverse u\n\nvariable {R : Type u} [Field R]\n\ntheorem of_linearOrderedField [LinearOrder R] [IsStrictOrderedRing R]\n    (isSquare_of_nonneg : ∀ {x : R}, 0 ≤ x → IsSquare x)\n    (exists_isRoot_of_odd_natDegree : ∀ {f : R[X]}, Odd f.natDegree → ∃ x, f.IsRoot x) :\n    IsRealClosed R where\n  isSquare_or_isSquare_neg {x} := by\n    rcases le_total x 0 with (neg | pos)\n    · exact .inr <| isSquare_of_nonneg (neg_nonneg_of_nonpos neg)\n    · exact .inl <| isSquare_of_nonneg pos\n  exists_isRoot_of_odd_natDegree := exists_isRoot_of_odd_natDegree\n\nvariable [IsRealClosed R]\n\n@[aesop 50%]\ntheorem _root_.IsSquare.of_not_isSquare_neg {x : R} (hx : ¬ IsSquare (-x)) : IsSquare x := by aesop\n\n@[aesop 80%]\ntheorem isSquare_neg_of_not_isSquare {x : R} (hx : ¬ IsSquare x) : IsSquare (-x) := by aesop\n\ntheorem exists_eq_pow_of_odd (x : R) {n : ℕ} (hn : Odd n) : ∃ r, x = r ^ n := by\n  rcases exists_isRoot_of_odd_natDegree (f := X ^ n - C x) (by simp [hn]) with ⟨r, hr⟩\n  exact ⟨r, by linear_combination - (by simpa using hr : r ^ n - x = 0)⟩\n\ntheorem exists_eq_zpow_of_odd (x : R) {k : ℤ} (hk : Odd k) : ∃ r, x = r ^ k := by\n  rcases k.eq_nat_or_neg with ⟨n, rfl | rfl⟩\n  · simpa using exists_eq_pow_of_odd x (by simpa using hk)\n  · rcases exists_eq_pow_of_odd x (by simpa using hk) with ⟨r, hr⟩\n    exact ⟨r⁻¹, by simpa using hr⟩\n\ntheorem exists_eq_pow_of_isSquare {x : R} (hx : IsSquare x) {n : ℕ} (hn : n ≠ 0) :\n    ∃ r, x = r ^ n := by\n  induction n using Nat.strong_induction_on generalizing x with\n  | h n ih =>\n    rcases Nat.even_or_odd n with (even | odd)\n    · rcases even with ⟨m, hm⟩\n      rcases hx with ⟨s, hs⟩\n      rcases isSquare_or_isSquare_neg s with (h | h) <;>\n        rcases ih m (by lia) h (by lia) with ⟨r, hr⟩ <;>\n        exact ⟨r, by simp [hm, pow_add, ← hr, hs]⟩\n    · exact exists_eq_pow_of_odd x odd\n\ntheorem exists_eq_zpow_of_isSquare {x : R} (hx : IsSquare x) {k : ℤ} (hk : k ≠ 0) :\n    ∃ r, x = r ^ k := by\n  rcases k.eq_nat_or_neg with ⟨n, rfl | rfl⟩\n  · simpa using exists_eq_pow_of_isSquare hx (by simpa using hk)\n  · rcases exists_eq_pow_of_isSquare hx (by simpa using hk) with ⟨r, hr⟩\n    exact ⟨r⁻¹, by simpa using hr⟩\n\nsection LinearOrderedField\n\nvariable [LinearOrder R] [IsStrictOrderedRing R]\n\nTarget:\ntheorem nonneg_iff_isSquare {x : R} : 0 ≤ x ↔ IsSquare x where\n  mp h :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/IsRealClosed","family_id":"nonneg_iff_issquare","file_id":"mathlib/Mathlib/FieldTheory/IsRealClosed/Basic.lean","sample_id":"d3b554a7490cfc938221d5d7fff9423a8ad834bad53b65c4db2638f83535b57e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3973b528cf86a6a3105014eb45539f7af155ea83fbda961dfd591ee49e04f82a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a1d25de80740477526c09d1872d0447c8000fb20b6d0ca785a16aeb63ebdfb95","source_sha256":"6295fc893f54a37fd351974b0ff78fe129cc329f6cbb3ba221c06301535ec6ad","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨p₁, p₂, n, e⟩ := P.cond\n  have : aeval x P.f.derivative ∣ aeval x P.g ^ n :=\n    ⟨_, by simpa [h.1] using congr(aeval x $e.symm)⟩\n  exact isUnit_of_dvd_unit this (.pow _ h.2)","hard_negative":false,"metrics":{"chosen_tokens":67,"rejected_tokens":3,"token_jaccard":0.04878,"token_length_ratio":0.044776},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"4b839d175445379ea5fe88ec52fd89cc193b18e25470f022d7879888e9c297b8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Bivariate\npublic import Mathlib.Algebra.Polynomial.Taylor\npublic import Mathlib.RingTheory.Etale.Basic\npublic import Mathlib.RingTheory.Extension.Presentation.Submersive\npublic import Mathlib.RingTheory.Ideal.IdempotentFG\n\nNamespace:\nStandardEtalePair\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Standard etale maps\n\n## Main definitions\n- `StandardEtalePair`:\n  A pair `f g : R[X]` such that `f` is monic and `f'` is invertible in `R[X][1/g]`.\n- `StandardEtalePair`: The standard etale algebra corresponding to a `StandardEtalePair`.\n- `StandardEtalePair.equivPolynomialQuotient`   : `P.Ring ≃ R[X][Y]/⟨f, Yg-1⟩`\n- `StandardEtalePair.equivAwayAdjoinRoot`       : `P.Ring ≃ (R[X]/f)[1/g]`\n- `StandardEtalePair.equivAwayQuotient`         : `P.Ring ≃ R[X][1/g]/f`\n- `StandardEtalePair.equivMvPolynomialQuotient` : `P.Ring ≃ R[X, Y]/⟨f, Yg-1⟩`\n- `StandardEtalePair.homEquiv`:\n  Maps out of `P.Ring` corresponds to `x` such that `f(x) = 0` and `g(x)` is invertible.\n- We also provide the instance that `P.Ring` is etale over `R`.\n\n- `Algebra.IsStandardEtale`: The class of standard etale algebras.\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen Polynomial\n\nopen scoped Bivariate\n\nnoncomputable section\n\nvariable {R S T : Type*} [CommRing R] [CommRing S] [CommRing T] [Algebra R S] [Algebra R T]\n\nvariable (R) in\n/-- A `StandardEtalePair R` is a pair `f g : R[X]` such that `f` is monic,\nand `f'` is invertible in `R[X][1/g]/f`. -/\nstructure StandardEtalePair : Type _ where\n  /-- The monic polynomial to be quotiented out in a standard etale algebra. -/\n  f : R[X]\n  monic_f : f.Monic\n  /-- The polynomial to be localized away from in a standard etale algebra. -/\n  g : R[X]\n  cond : ∃ p₁ p₂ n, derivative f * p₁ + f * p₂ = g ^ n\n\nvariable (P : StandardEtalePair R)\n\n/-- The standard etale algebra `R[X][Y]/⟨f, Yg-1⟩` associated to a `StandardEtalePair R`.\nAlso see\n`equivPolynomialQuotient   : P.Ring ≃ R[X][Y]/⟨f, Yg-1⟩`\n`equivAwayAdjoinRoot       : P.Ring ≃ (R[X]/f)[1/g]`\n`equivAwayQuotient         : P.Ring ≃ R[X][1/g]/f`\n`equivMvPolynomialQuotient : P.Ring ≃ R[X, Y]/⟨f, Yg-1⟩` -/\nprotected def StandardEtalePair.Ring := R[X][Y] ⧸ Ideal.span {C P.f, Y * C P.g - 1}\n  deriving CommRing, Algebra R\n\nnamespace StandardEtalePair\n\n/-- The `X` in the standard etale algebra `R[X][Y]/⟨f, Yg-1⟩`. -/\nprotected def X : P.Ring := Ideal.Quotient.mk _ (C .X)\n\n/-- There is a map from a standard etale algebra `R[X][Y]/⟨f, Yg-1⟩` to `S` sending `X` to `x` iff\n`f(x) = 0` and `g(x)` is invertible. Also see `StandardEtalePair.homEquiv`. -/\ndef HasMap (x : S) : Prop :=\n  aeval x P.f = 0 ∧ IsUnit (aeval x P.g)\n\n/-- The map `R[X][Y]/⟨f, Yg-1⟩ →ₐ[R] S` sending `X` to `x`, given `P.HasMap x`. -/\ndef lift (x : S) (h : P.HasMap x) : P.Ring →ₐ[R] S :=\n  Ideal.Quotient.liftₐ _ (aevalAeval x ↑(h.2.unit⁻¹))\n    (Ideal.span_le (I := RingHom.ker _).mpr (by simp [Set.pair_subset_iff, h.1]))\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma lift_X (x : S) (h : P.HasMap x) : P.lift x h P.X = x := by\n  simp [lift, StandardEtalePair.Ring, StandardEtalePair.X]\n\nvariable {P} in\nlemma HasMap.map {x : S} (h : P.HasMap x) (f : S →ₐ[R] T) : P.HasMap (f x) :=\n  ⟨by simp [aeval_algHom, h.1], by simpa [aeval_algHom] using h.2.map f⟩\n\nTarget:\nlemma HasMap.isUnit_derivative_f {x : S} (h : P.HasMap x) :\n    IsUnit (P.f.derivative.aeval x) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Etale","family_id":"hasmap","file_id":"mathlib/Mathlib/RingTheory/Etale/StandardEtale.lean","sample_id":"a1d25de80740477526c09d1872d0447c8000fb20b6d0ca785a16aeb63ebdfb95"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2c32995a07b48e16dcaafc728301779b447c776a69f801e281fcde5579068d01","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cf07e5f2d2894cd807b5a1504b06e4111f8b1c912c6a18b539c0705861955d7e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fce701ee6ff6a6606b14ec7394e9ebfdd374d824dea16656e00b6156e86509d7","source_sha256":"7da6267f9f5d264ecd59201d6b4195a7718cac6faa70ee30f8ae303009b58a72","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h ↦ fun _ _ _ ↦ h _ _, fun h ↦ fun i₁ i₂ ↦ ?_⟩\n  by_cases hi : i₁ = i₂\n  · simp [hi]\n  · exact h hi","hard_negative":false,"metrics":{"chosen_tokens":44,"rejected_tokens":49,"token_jaccard":0.846154,"token_length_ratio":1.113636},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"4b85b167521b8fa902566e6a1df148fa862875a224b8587e50be733343da03a3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.MetricSpace.Pseudo.Defs\npublic import Mathlib.Topology.MetricSpace.Isometry\npublic import Mathlib.Topology.MetricSpace.Dilation\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Jovan Gerbscheid. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jovan Gerbscheid, Newell Jensen\n-/\n/-!\n# Congruences\n\nThis file defines `Congruent`, i.e., the equivalence between indexed families of points in a metric\nspace where all corresponding pairwise distances are the same. The motivating example are\ntriangles in the plane.\n\n## Implementation notes\n\nAfter considering two possible approaches to defining congruence — either based on equal pairwise\ndistances or the existence of an isometric equivalence — we have opted for the broader concept of\nequal pairwise distances. This notion is commonly employed in the literature across various metric\nspaces that lack an isometric equivalence.\n\nFor more details see the [Zulip discussion](https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Euclidean.20Geometry).\n\n## Notation\n\n* `v₁ ≅ v₂`: for `Congruent v₁ v₂`.\n-/\n\n@[expose] public section\n\nvariable {ι ι' : Type*} {P₁ P₂ P₃ P₄ : Type*} {v₁ : ι → P₁} {v₂ : ι → P₂} {v₃ : ι → P₃}\n\nsection PseudoEMetricSpace\n\nvariable [PseudoEMetricSpace P₁] [PseudoEMetricSpace P₂]\nvariable [PseudoEMetricSpace P₃] [PseudoEMetricSpace P₄]\n\n/-- A congruence between indexed sets of vertices v₁ and v₂.\nUse `open scoped Congruent` to access the `v₁ ≅ v₂` notation. -/\ndef Congruent (v₁ : ι → P₁) (v₂ : ι → P₂) : Prop :=\n  ∀ i₁ i₂, edist (v₁ i₁) (v₁ i₂) = edist (v₂ i₁) (v₂ i₂)\n\n@[inherit_doc]\nscoped[Congruent] infixl:25 \" ≅ \" => Congruent\n\n/-- Congruence holds if and only if all extended distances are the same. -/\nlemma congruent_iff_edist_eq :\n    Congruent v₁ v₂ ↔ ∀ i₁ i₂, edist (v₁ i₁) (v₁ i₂) = edist (v₂ i₁) (v₂ i₂) :=\n  Iff.rfl\n\n/-- Congruence holds if and only if all extended distances between points with different\nindices are the same. -/\n\nTarget:\nlemma congruent_iff_pairwise_edist_eq :\n    Congruent v₁ v₂ ↔ Pairwise fun i₁ i₂ ↦ edist (v₁ i₁) (v₁ i₂) = edist (v₂ i₁) (v₂ i₂) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  refine ⟨fun h ↦ fun _ _ _ ↦ h _ _, fun h ↦ fun i₁ i₂ ↦ ?_⟩\n  by_cases hi : i₁ = i₂\n  · simp [hi]\n  · exact h hi","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/MetricSpace","family_id":"congruent_iff_pairwise_edist_eq","file_id":"mathlib/Mathlib/Topology/MetricSpace/Congruence.lean","sample_id":"fce701ee6ff6a6606b14ec7394e9ebfdd374d824dea16656e00b6156e86509d7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"124731cfc6276098aa30620798da6c884c01af47d833522f931a5e1fb2b79484","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"45556f8bd9370bd1415c420e560973ba88e4c44babfc2defddcf5c8f39aa00bb","source_sha256":"394c3d8954491754c0a03071ce7aa65ddc93c4783581cf0b4d6869361547187f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_algEquiv_matrix_end_mulOpposite R₀ R\n  have := Module.Finite.equiv e.toLinearEquiv\n  classical exact ⟨n, hn, _, _, _, .of_surjective\n    (Matrix.entryLinearMap R₀ _ (0 : Fin n) (0 : Fin n)) fun f ↦ ⟨fun _ _ ↦ f, rfl⟩, ⟨e⟩⟩","hard_negative":true,"metrics":{"chosen_tokens":82,"rejected_tokens":8,"token_jaccard":0.111111,"token_length_ratio":0.097561},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"4b9406d9f25a023a15e20d3040b09065189dc9712c3c396627e5cd1ee5e0a852","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.FreeModule.Finite.Basic\npublic import Mathlib.RingTheory.FiniteLength\npublic import Mathlib.RingTheory.SimpleModule.Isotypic\npublic import Mathlib.RingTheory.SimpleRing.Congr\npublic import Mathlib.RingTheory.SimpleRing.Matrix\n\nNamespace:\nIsSimpleRing\n\nLocal context:\n/-\nCopyright (c) 2025 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu\n-/\n/-!\n# Wedderburn–Artin Theorem\n\n## Main results\n\n* `IsSimpleRing.tfae`: a simple ring is semisimple iff it is Artinian,\n  iff it has a minimal left ideal.\n\n* `isSimpleRing_isArtinianRing_iff`: a ring is simple Artinian iff it is semisimple, isotypic,\n  and nontrivial.\n\n* `IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite`: a simple Artinian algebra is\n  isomorphic to a (finite-dimensional) matrix algebra over a division algebra. The division\n  algebra is the opposite of the endomorphism algebra of a simple (i.e., minimal) left ideal.\n\n* `IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite`: a semisimple algebra is\n  isomorphic to a finite direct product of matrix algebras over division algebras. The division\n  algebras are the opposites of the endomorphism algebras of the simple (i.e., minimal)\n  left ideals.\n\n* `IsSimpleRing.exists_algEquiv_matrix_divisionRing_finite`,\n  `IsSemisimpleRing.exists_algEquiv_pi_matrix_divisionRing_finite`:\n  if the simple Artinian / semisimple algebra is finite as a module over a base ring, then the\n  division algebra(s) are also finite over the same ring.\n  If the base ring is an algebraically closed field, the only finite-dimensional division algebra\n  over it is itself, and we obtain `IsSimpleRing.exists_algEquiv_matrix_of_isAlgClosed` and\n  `IsSemisimpleRing.exists_algEquiv_pi_matrix_of_isAlgClosed` (in a later file).\n\n-/\n\npublic section\n\nuniverse u\nvariable (R₀ : Type*) {R : Type u} [CommSemiring R₀] [Ring R] [Algebra R₀ R]\n\n/-- A simple ring is semisimple iff it is Artinian, iff it has a minimal left ideal. -/\ntheorem IsSimpleRing.tfae [IsSimpleRing R] : List.TFAE\n    [IsSemisimpleRing R, IsArtinianRing R, ∃ I : Ideal R, IsAtom I] := by\n  tfae_have 1 → 2 := fun _ ↦ inferInstance\n  tfae_have 2 → 3 := fun _ ↦ IsAtomic.exists_atom _\n  tfae_have 3 → 1 := fun ⟨I, hI⟩ ↦ by\n    have ⟨_, h⟩ := isSimpleRing_iff_isTwoSided_imp.mp ‹IsSimpleRing R›\n    simp_rw [← isFullyInvariant_iff_isTwoSided] at h\n    have := isSimpleModule_iff_isAtom.mpr hI\n    obtain eq | eq := h _ (.isotypicComponent R R I)\n    · exact (hI.bot_lt.not_ge <| (le_sSup <| by exact ⟨.refl ..⟩).trans_eq eq).elim\n    exact .congr (.symm <| .trans (.ofEq _ _ eq) Submodule.topEquiv)\n  tfae_finish\n\ntheorem IsSimpleRing.isSemisimpleRing_iff_isArtinianRing [IsSimpleRing R] :\n    IsSemisimpleRing R ↔ IsArtinianRing R := tfae.out 0 1\n\ntheorem isSimpleRing_isArtinianRing_iff :\n    IsSimpleRing R ∧ IsArtinianRing R ↔ IsSemisimpleRing R ∧ IsIsotypic R R ∧ Nontrivial R := by\n  refine ⟨fun ⟨_, _⟩ ↦ ?_, fun ⟨_, _, _⟩ ↦ ?_⟩\n  on_goal 1 => have := IsSimpleRing.isSemisimpleRing_iff_isArtinianRing.mpr ‹_›\n  all_goals simp_rw [isIsotypic_iff_isFullyInvariant_imp_bot_or_top,\n      isFullyInvariant_iff_isTwoSided, isSimpleRing_iff_isTwoSided_imp] at *\n  · exact ⟨this, by rwa [and_comm]⟩\n  · exact ⟨⟨‹_›, ‹_›⟩, inferInstance⟩\n\nnamespace IsSimpleRing\n\nvariable (R) [IsSimpleRing R] [IsArtinianRing R]\n\ninstance (priority := low) : IsSemisimpleRing R :=\n  (isSimpleRing_isArtinianRing_iff.mp ⟨‹_›, ‹_›⟩).1\n\ntheorem isIsotypic (M) [AddCommGroup M] [Module R M] : IsIsotypic R M :=\n  (isSimpleRing_isArtinianRing_iff.mp ⟨‹_›, ‹_›⟩).2.1.of_self M\n\n/-- The **Wedderburn–Artin Theorem**: an Artinian simple ring is isomorphic to a matrix\nring over the opposite of the endomorphism ring of its simple module. -/\ntheorem exists_ringEquiv_matrix_end_mulOpposite :\n    ∃ (n : ℕ) (_ : NeZero n) (I : Ideal R) (_ : IsSimpleModule R I),\n      Nonempty (R ≃+* Matrix (Fin n) (Fin n) (Module.End R I)ᵐᵒᵖ) := by\n  have ⟨n, hn, S, hS, ⟨e⟩⟩ := (isIsotypic R R).linearEquiv_fun\n  refine ⟨n, hn, S, hS, ⟨.trans (.opOp R) <| .trans (.op ?_) (.symm .mopMatrix)⟩⟩\n  exact .trans (.moduleEndSelf R) <| .trans e.conjRingEquiv (endVecRingEquivMatrixEnd ..)\n\n/-- The **Wedderburn–Artin Theorem**: an Artinian simple ring is isomorphic to a matrix\nring over a division ring. -/\ntheorem exists_ringEquiv_matrix_divisionRing :\n    ∃ (n : ℕ) (_ : NeZero n) (D : Type u) (_ : DivisionRing D),\n      Nonempty (R ≃+* Matrix (Fin n) (Fin n) D) := by\n  have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_ringEquiv_matrix_end_mulOpposite R\n  classical exact ⟨n, hn, _, _, ⟨e⟩⟩\n\n/-- The **Wedderburn–Artin Theorem**, algebra form: an Artinian simple algebra is isomorphic\nto a matrix algebra over the opposite of the endomorphism algebra of its simple module. -/\ntheorem exists_algEquiv_matrix_end_mulOpposite :\n    ∃ (n : ℕ) (_ : NeZero n) (I : Ideal R) (_ : IsSimpleModule R I),\n      Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) (Module.End R I)ᵐᵒᵖ) := by\n  have ⟨n, hn, S, hS, ⟨e⟩⟩ := (isIsotypic R R).linearEquiv_fun\n  refine ⟨n, hn, S, hS, ⟨.trans (.opOp R₀ R) <| .trans (.op ?_) (.symm .mopMatrix)⟩⟩\n  exact .trans (.moduleEndSelf R₀) <| .trans (e.conjAlgEquiv R₀) (endVecAlgEquivMatrixEnd ..)\n\n/-- The **Wedderburn–Artin Theorem**, algebra form: an Artinian simple algebra is isomorphic\nto a matrix algebra over a division algebra. -/\ntheorem exists_algEquiv_matrix_divisionRing :\n    ∃ (n : ℕ) (_ : NeZero n) (D : Type u) (_ : DivisionRing D) (_ : Algebra R₀ D),\n      Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) D) := by\n  have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_algEquiv_matrix_end_mulOpposite R₀ R\n  classical exact ⟨n, hn, _, _, _, ⟨e⟩⟩\n\n/-- The **Wedderburn–Artin Theorem**, algebra form, finite case: a finite Artinian simple algebra is\nisomorphic to a matrix algebra over a finite division algebra. -/\n\nTarget:\ntheorem exists_algEquiv_matrix_divisionRing_finite [Module.Finite R₀ R] :\n    ∃ (n : ℕ) (_ : NeZero n) (D : Type u) (_ : DivisionRing D) (_ : Algebra R₀ D)\n      (_ : Module.Finite R₀ D), Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) D) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"82e0fcc22e3556b503f6ac44940706255c58c98e3a2257322c00f2ae0e7388ef","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/SimpleModule","family_id":"exists_algequiv_matrix_divisionring_finite","file_id":"mathlib/Mathlib/RingTheory/SimpleModule/WedderburnArtin.lean","sample_id":"45556f8bd9370bd1415c420e560973ba88e4c44babfc2defddcf5c8f39aa00bb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3f5dfa9a05050a7e8f6e2944c4416e1a85109aff6c3b31b2844a70ac199a91e0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"44d2da225f20cd3b6e6276fb821a5a22fe9ec474c9711a39a7cbc160460c406f","source_sha256":"8d5705aabd3ec1513ff008c85f46f0fb487e906338de45986cd81d4559495758","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨C, D, rfl⟩ := hNM\n  exact hI.of_delete.of_contract","hard_negative":true,"metrics":{"chosen_tokens":17,"rejected_tokens":8,"token_jaccard":0.1,"token_length_ratio":0.470588},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"4bb1015705ff0383fe350ad9a560d2b59676bbcb2d21f846ee82f1d691d438f5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Matroid.Minor.Contract\n\nNamespace:\nMatroid\n\nLocal context:\n/-\nCopyright (c) 2025 Peter Nelson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Peter Nelson\n-/\n/-!\n# Matroid Minors\n\nA matroid `N = M ／ C ＼ D` obtained from a matroid `M` by a contraction then a delete,\n(or equivalently, by any number of contractions/deletions in any order) is a *minor* of `M`.\nThis gives a partial order on `Matroid α` that is ubiquitous in matroid theory,\nand interacts nicely with duality and linear representations.\n\nAlthough we provide a `PartialOrder` instance on `Matroid α` corresponding to the minor order,\nwe do not use the `M ≤ N` / `N < M` notation directly,\ninstead writing `N ≤m M` and `N <m M` for more convenient dot notation.\n\n## Main Declarations\n\n* `Matroid.IsMinor N M`, written `N ≤m M`, means that `N = M ／ C ＼ D` for some\n  subset `C` and `D` of `M.E`.\n* `Matroid.IsStrictMinor N M`, written `N <m M`, means that `N = M ／ C ＼ D`\n  for some subsets `C` and `D` of `M.E` that are not both nonempty.\n* `Matroid.IsMinor.exists_eq_contract_delete_disjoint` : we can choose `C` and `D` disjoint.\n\n-/\n\n@[expose] public section\n\nnamespace Matroid\n\nopen Set\n\nsection Minor\n\nvariable {α : Type*} {M M' N : Matroid α} {e f : α} {I C D : Set α}\n\n/-! ### Minors -/\n\n/-- `N` is a minor of `M` if `N = M ／ C ＼ D` for some `C` and `D`.\nThe definition itself does not require `C` and `D` to be disjoint,\nor even to be subsets of the ground set. See `Matroid.IsMinor.exists_eq_contract_delete_disjoint`\nfor the fact that we can choose `C` and `D` with these properties. -/\ndef IsMinor (N M : Matroid α) : Prop := ∃ C D, N = M ／ C ＼ D\n\n/-- `≤m` denotes the minor relation on matroids. -/\ninfixl:50 \" ≤m \" => Matroid.IsMinor\n\n@[simp]\nlemma contract_delete_isMinor (M : Matroid α) (C D : Set α) : M ／ C ＼ D ≤m M :=\n  ⟨C, D, rfl⟩\n\nlemma IsMinor.exists_eq_contract_delete_disjoint (h : N ≤m M) :\n    ∃ (C D : Set α), C ⊆ M.E ∧ D ⊆ M.E ∧ Disjoint C D ∧ N = M ／ C ＼ D := by\n  obtain ⟨C, D, rfl⟩ := h\n  exact ⟨C ∩ M.E, (D ∩ M.E) \\ C, inter_subset_right, sdiff_subset.trans inter_subset_right,\n    disjoint_sdiff_right.mono_left inter_subset_left,\n    by simp [delete_eq_delete_iff, inter_assoc, inter_sdiff_assoc]⟩\n\n/-- `N` is a strict minor of `M` if `N` is a minor of `M` and `N ≠ M`.\nEquivalently, `N` is obtained from `M` by deleting/contracting subsets of the ground set\nthat are not both empty. -/\ndef IsStrictMinor (N M : Matroid α) : Prop := N ≤m M ∧ ¬ M ≤m N\n\n/-- `<m` denotes the strict minor relation on matroids. -/\ninfixl:50 \" <m \" => Matroid.IsStrictMinor\n\nlemma IsMinor.subset (h : N ≤m M) : N.E ⊆ M.E := by\n  obtain ⟨C, D, rfl⟩ := h\n  exact sdiff_subset.trans sdiff_subset\n\nlemma IsMinor.refl {M : Matroid α} : M ≤m M := ⟨∅, ∅, by simp⟩\n\nlemma IsMinor.trans {M₁ M₂ M₃ : Matroid α} (h : M₁ ≤m M₂) (h' : M₂ ≤m M₃) : M₁ ≤m M₃ := by\n  obtain ⟨C₁, D₁, rfl⟩ := h\n  obtain ⟨C₂, D₂, rfl⟩ := h'\n  exact ⟨C₂ ∪ C₁ \\ D₂, D₂ ∪ D₁, by rw [contract_delete_contract_delete']⟩\n\nlemma IsMinor.eq_of_ground_subset (h : N ≤m M) (hE : M.E ⊆ N.E) : M = N := by\n  obtain ⟨C, D, rfl⟩ := h\n  rw [delete_ground, contract_ground, subset_sdiff, subset_sdiff] at hE\n  rw [← contract_inter_ground_eq, hE.1.2.symm.inter_eq, contract_empty, ← delete_inter_ground_eq,\n    hE.2.symm.inter_eq, delete_empty]\n\nlemma IsMinor.antisymm (h : N ≤m M) (h' : M ≤m N) : N = M :=\n  h'.eq_of_ground_subset h.subset\n\n/-- The minor order is a `PartialOrder` on `Matroid α`.\nWe prefer the spelling `N ≤m M` over `N ≤ M` for the dot notation. -/\ninstance (α : Type*) : PartialOrder (Matroid α) where\n  le N M := N ≤m M\n  lt N M := N <m M\n  le_refl _ := IsMinor.refl\n  le_trans _ _ _ := IsMinor.trans\n  le_antisymm _ _ := IsMinor.antisymm\n\nlemma IsMinor.le (h : N ≤m M) : N ≤ M := h\n\nlemma IsStrictMinor.lt (h : N <m M) : N < M := h\n\n@[simp]\nlemma le_eq_isMinor : (fun M M' : Matroid α ↦ M ≤ M') = Matroid.IsMinor := rfl\n\n@[simp]\nlemma lt_eq_isStrictMinor : (fun M M' : Matroid α ↦ M < M') = Matroid.IsStrictMinor := rfl\n\nlemma isStrictMinor_iff_isMinor_ne : N <m M ↔ N ≤m M ∧ N ≠ M :=\n  lt_iff_le_and_ne (α := Matroid α)\n\nlemma IsStrictMinor.ne (h : N <m M) : N ≠ M :=\n  h.lt.ne\n\nlemma isStrictMinor_irrefl (M : Matroid α) : ¬ (M <m M) :=\n  lt_irrefl M\n\nlemma IsStrictMinor.isMinor (h : N <m M) : N ≤m M :=\n  h.lt.le\n\nlemma IsStrictMinor.not_isMinor (h : N <m M) : ¬ (M ≤m N) :=\n  h.lt.not_ge\n\nlemma IsStrictMinor.ssubset (h : N <m M) : N.E ⊂ M.E :=\n  h.isMinor.subset.ssubset_of_ne (fun hE ↦ h.ne (h.isMinor.eq_of_ground_subset hE.symm.subset).symm)\n\nlemma isStrictMinor_iff_isMinor_ssubset : N <m M ↔ N ≤m M ∧ N.E ⊂ M.E :=\n  ⟨fun h ↦ ⟨h.isMinor, h.ssubset⟩, fun ⟨h, hss⟩ ↦ ⟨h, fun h' ↦ hss.ne <| by rw [h'.antisymm h]⟩⟩\n\nlemma IsStrictMinor.trans_isMinor (h : N <m M) (h' : M ≤m M') : N <m M' :=\n  h.lt.trans_le h'\n\nlemma IsMinor.trans_isStrictMinor (h : N ≤m M) (h' : M <m M') : N <m M' :=\n  h.le.trans_lt h'\n\nlemma IsStrictMinor.trans (h : N <m M) (h' : M <m M') : N <m M' :=\n  h.lt.trans h'\n\nTarget:\nlemma Indep.of_isMinor (hI : N.Indep I) (hNM : N ≤m M) : M.Indep I :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"d8a236761568859e199298ca9787b6884663aa7e010bc5c776ba51f9fbe8a6ae","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Matroid","family_id":"indep","file_id":"mathlib/Mathlib/Combinatorics/Matroid/Minor/Order.lean","sample_id":"44d2da225f20cd3b6e6276fb821a5a22fe9ec474c9711a39a7cbc160460c406f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c601c0ce6ee21cbe086a05718f7a601ae3b0678e21055794ba74900b0b12e072","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"28d10eb9cccf5e65b337e07ea501309eded98562e31eabf192d08d15441bba9d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"15c500daf4b584078896ed64949b949708a161dcfff41d36df74a29d2e07383f","source_sha256":"e2e260981e2cd985149de5ebb4a157aa3216ee7c359bb9bfe83a133b3f869574","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [IsPeriodicPt, hco.comp_iterate] at hfg\n  exact hfg.left_of_comp hg","hard_negative":true,"metrics":{"chosen_tokens":16,"rejected_tokens":2,"token_jaccard":0.066667,"token_length_ratio":0.125},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"4bcf3eace62aa9d57d97c0b2a3da967898101501b484f3dbf749f84ad2e00ad8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Defs\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.Sub.Basic\npublic import Mathlib.Data.List.Cycle\npublic import Mathlib.Data.PNat.Notation\npublic import Mathlib.Dynamics.FixedPoints.Basic\n\nNamespace:\nFunction.IsPeriodicPt\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Periodic points\n\nA point `x : α` is a periodic point of `f : α → α` of period `n` if `f^[n] x = x`.\n\n## Main definitions\n\n* `IsPeriodicPt f n x` : `x` is a periodic point of `f` of period `n`, i.e. `f^[n] x = x`.\n  We do not require `n > 0` in the definition.\n* `ptsOfPeriod f n` : the set `{x | IsPeriodicPt f n x}`. Note that `n` is not required to\n  be the minimal period of `x`.\n* `periodicPts f` : the set of all periodic points of `f`.\n* `minimalPeriod f x` : the minimal period of a point `x` under an endomorphism `f` or zero\n  if `x` is not a periodic point of `f`.\n* `orbit f x`: the cycle `[x, f x, f (f x), ...]` for a periodic point.\n* `MulAction.period g x` : the minimal period of a point `x` under the multiplicative action of `g`;\n  an equivalent `AddAction.period g x` is defined for additive actions.\n\n## Main statements\n\nWe provide “dot syntax”-style operations on terms of the form `h : IsPeriodicPt f n x` including\narithmetic operations on `n` and `h.map (hg : SemiconjBy g f f')`. We also prove that `f`\nis bijective on each set `ptsOfPeriod f n` and on `periodicPts f`. Finally, we prove that `x`\nis a periodic point of `f` of period `n` if and only if `minimalPeriod f x | n`.\n\n## References\n\n* https://en.wikipedia.org/wiki/Periodic_point\n\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero\n\n\nopen Set\n\nnamespace Function\n\nopen Function (Commute)\n\nvariable {α : Type*} {β : Type*} {f fa : α → α} {fb : β → β} {x y : α} {m n : ℕ}\n\n/-- A point `x` is a periodic point of `f : α → α` of period `n` if `f^[n] x = x`.\nNote that we do not require `0 < n` in this definition. Many theorems about periodic points\nneed this assumption. -/\ndef IsPeriodicPt (f : α → α) (n : ℕ) (x : α) :=\n  IsFixedPt f^[n] x\n\n/-- A fixed point of `f` is a periodic point of `f` of any prescribed period. -/\ntheorem IsFixedPt.isPeriodicPt (hf : IsFixedPt f x) (n : ℕ) : IsPeriodicPt f n x :=\n  hf.iterate n\n\n/-- For the identity map, all points are periodic. -/\ntheorem is_periodic_id (n : ℕ) (x : α) : IsPeriodicPt id n x :=\n  (isFixedPt_id x).isPeriodicPt n\n\n/-- Any point is a periodic point of period `0`. -/\ntheorem isPeriodicPt_zero (f : α → α) (x : α) : IsPeriodicPt f 0 x :=\n  isFixedPt_id x\n\nnamespace IsPeriodicPt\n\n@[nontriviality]\ntheorem of_subsingleton [Subsingleton α] (f : α → α) (n : ℕ) (x : α) : IsPeriodicPt f n x :=\n  IsFixedPt.of_subsingleton _ _\n\ninstance [DecidableEq α] {f : α → α} {n : ℕ} {x : α} : Decidable (IsPeriodicPt f n x) :=\n  IsFixedPt.decidable\n\nprotected theorem isFixedPt (hf : IsPeriodicPt f n x) : IsFixedPt f^[n] x :=\n  hf\n\nprotected theorem map (hx : IsPeriodicPt fa n x) {g : α → β} (hg : Semiconj g fa fb) :\n    IsPeriodicPt fb n (g x) :=\n  IsFixedPt.map hx (hg.iterate_right n)\n\ntheorem apply_iterate (hx : IsPeriodicPt f n x) (m : ℕ) : IsPeriodicPt f n (f^[m] x) :=\n  hx.map <| Commute.iterate_self f m\n\nprotected theorem apply (hx : IsPeriodicPt f n x) : IsPeriodicPt f n (f x) :=\n  hx.apply_iterate 1\n\nprotected theorem add (hn : IsPeriodicPt f n x) (hm : IsPeriodicPt f m x) :\n    IsPeriodicPt f (n + m) x := by\n  rw [IsPeriodicPt, iterate_add]\n  exact hn.comp hm\n\ntheorem left_of_add (hn : IsPeriodicPt f (n + m) x) (hm : IsPeriodicPt f m x) :\n    IsPeriodicPt f n x := by\n  rw [IsPeriodicPt, iterate_add] at hn\n  exact hn.left_of_comp hm\n\ntheorem right_of_add (hn : IsPeriodicPt f (n + m) x) (hm : IsPeriodicPt f n x) :\n    IsPeriodicPt f m x := by\n  rw [add_comm] at hn\n  exact hn.left_of_add hm\n\nprotected theorem sub (hm : IsPeriodicPt f m x) (hn : IsPeriodicPt f n x) :\n    IsPeriodicPt f (m - n) x := by\n  rcases le_total n m with h | h\n  · refine left_of_add ?_ hn\n    rwa [tsub_add_cancel_of_le h]\n  · rw [tsub_eq_zero_iff_le.mpr h]\n    apply isPeriodicPt_zero\n\nprotected theorem mul_const (hm : IsPeriodicPt f m x) (n : ℕ) : IsPeriodicPt f (m * n) x := by\n  simp only [IsPeriodicPt, iterate_mul, hm.isFixedPt.iterate n]\n\nprotected theorem const_mul (hm : IsPeriodicPt f m x) (n : ℕ) : IsPeriodicPt f (n * m) x := by\n  simp only [mul_comm n, hm.mul_const n]\n\ntheorem trans_dvd (hm : IsPeriodicPt f m x) {n : ℕ} (hn : m ∣ n) : IsPeriodicPt f n x :=\n  let ⟨k, hk⟩ := hn\n  hk.symm ▸ hm.mul_const k\n\nprotected theorem iterate (hf : IsPeriodicPt f n x) (m : ℕ) : IsPeriodicPt f^[m] n x := by\n  rw [IsPeriodicPt, ← iterate_mul, mul_comm, iterate_mul]\n  exact hf.isFixedPt.iterate m\n\ntheorem comp {g : α → α} (hco : Commute f g) (hf : IsPeriodicPt f n x) (hg : IsPeriodicPt g n x) :\n    IsPeriodicPt (f ∘ g) n x := by\n  rw [IsPeriodicPt, hco.comp_iterate]\n  exact IsFixedPt.comp hf hg\n\ntheorem comp_lcm {g : α → α} (hco : Commute f g) (hf : IsPeriodicPt f m x)\n    (hg : IsPeriodicPt g n x) : IsPeriodicPt (f ∘ g) (Nat.lcm m n) x :=\n  (hf.trans_dvd <| Nat.dvd_lcm_left _ _).comp hco (hg.trans_dvd <| Nat.dvd_lcm_right _ _)\n\nTarget:\ntheorem left_of_comp {g : α → α} (hco : Commute f g) (hfg : IsPeriodicPt (f ∘ g) n x)\n    (hg : IsPeriodicPt g n x) : IsPeriodicPt f n x :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_15c500daf4b5","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"c2e00198ccdbe96a029065a2b718bc4860980a8bbcc8f358f432886ea3680822","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/PeriodicPts","family_id":"left_of_comp","file_id":"mathlib/Mathlib/Dynamics/PeriodicPts/Defs.lean","sample_id":"15c500daf4b584078896ed64949b949708a161dcfff41d36df74a29d2e07383f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0fd4fc28ba9a30a85e46c5d1d3ba799c62c0383a953832f15efb88028c8307ee","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4fe496042d93eaaba67d648ef8798ee607ecf6a9f45506405cbd5d76bcef6689","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a75eaccf6d18b5c2eb5e6918fb406ae54ba35c64367a2abe5f21d18d5b2dac31","source_sha256":"2c2357bdce988b5b8902bab0c6b1d4a599e97cee77b05d9e7215f22e881d81ce","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI := Fintype.ofFinite α\n  induction φ with simp only [BoundedFormula.Realize]\n  | equal t₁ t₂ =>\n    rcases term_realize_eq_add_dotProduct t₁ with ⟨k₁, u₁, ht₁⟩\n    rcases term_realize_eq_add_dotProduct t₂ with ⟨k₂, u₂, ht₂⟩\n    convert! Nat.isSemilinearSet_setOf_mulVec_eq ![k₁] ![k₂] (.of ![u₁]) (.of ![u₂])\n    simp [ht₁, ht₂]\n  | rel f => nomatch f\n  | falsum => exact .empty\n  | imp _ _ ih₁ ih₂ =>\n    convert! (ih₂.compl.inter ih₁).compl using 1\n    simp [setOf_inter_eq_sep, imp_iff_not_or, compl_setOf]\n  | @all n φ ih =>\n    let e := (Equiv.sumAssoc α (Fin n) (Fin 1)).trans (Equiv.sumCongr (.refl α) finSumFinEquiv)\n    rw [← isSemilinearSet_image_iff (LinearEquiv.funCongrLeft ℕ ℕ e)] at ih\n    convert! ih.compl.proj.compl using 1\n    simp_rw [compl_setOf, not_exists, Fin.forall_fin_succ_pi, Fin.forall_fin_zero_pi,\n      mem_compl_iff, mem_image, not_not, ← LinearEquiv.eq_symm_apply, LinearEquiv.funCongrLeft_symm,\n      exists_eq_right, mem_setOf, LinearEquiv.funCongrLeft_apply, LinearMap.funLeft,\n      LinearMap.coe_mk, AddHom.coe_mk]\n    congr! 4\n    ext i\n    cases i using Fin.lastCases <;> simp [e]","hard_negative":false,"metrics":{"chosen_tokens":270,"rejected_tokens":277,"token_jaccard":0.98,"token_length_ratio":1.025926},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"4bef2db32b2a281b4b0efe919fc7887b2b2ba25599d1a53674926d0efc22183e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.ModelTheory.Arithmetic.Presburger.Basic\npublic import Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic\npublic import Mathlib.ModelTheory.Definability\nimport Mathlib.Algebra.Group.Submonoid.Finsupp\nimport Mathlib.LinearAlgebra.Matrix.Notation\n\nNamespace:\nFirstOrder.Language.presburger\n\nLocal context:\n/-\nCopyright (c) 2025 Dexin Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dexin Zhang\n-/\n/-!\n# Presburger definability and semilinear sets\n\nThis file formalizes the classical result that Presburger definable sets are the same as semilinear\nsets. As an application of this result, we show that the graph of multiplication is not Presburger\ndefinable.\n\n## Main Results\n\n- `presburger.definable_iff_isSemilinearSet`: a set is Presburger definable in `ℕ` if and only if it\n  is semilinear.\n- `presburger.definable₁_iff_ultimately_periodic`: in the 1-dimensional case, a set is Presburger\n  arithmetic definable in `ℕ` if and only if it is ultimately periodic, i.e. periodic after some\n  number `k`.\n- `presburger.mul_not_definable`: the graph of multiplication is not Presburger definable in `ℕ`.\n\n## References\n\n* [Seymour Ginsburg and Edwin H. Spanier, *Bounded ALGOL-Like Languages*][ginsburg1964]\n* [Seymour Ginsburg and Edwin H. Spanier, *Semigroups, Presburger Formulas, and\n  Languages*][ginsburg1966]\n* [Samuel Eilenberg and M. P. Schützenberger, *Rational Sets in Commutative Monoids*][eilenberg1969]\n-/\n\npublic section\n\nvariable {α : Type*} {s : Set (α → ℕ)} {A : Set ℕ}\n\nopen Set FirstOrder Language\n\ntheorem IsLinearSet.definable [Finite α] (hs : IsLinearSet s) : A.Definable presburger s := by\n  rw [isLinearSet_iff] at hs\n  rcases hs with ⟨v, t, rfl⟩\n  refine ⟨Formula.iExs t (Formula.iInf fun i : α =>\n    (Term.var (Sum.inl i)).equal\n      (Term.varsToConstants\n        ((v i : presburger.Term _) + presburger.sum Finset.univ fun x : t =>\n          x.1 i • Term.var (Sum.inr (Sum.inr x))))), ?_⟩\n  ext x\n  simp only [mem_vadd_set, SetLike.mem_coe, AddSubmonoid.mem_closure_finset', Finset.univ_eq_attach,\n    nsmul_eq_mul, vadd_eq_add, ↓existsAndEq, true_and, mem_setOf_eq, Formula.realize_iExs,\n    Formula.realize_iInf, Formula.realize_equal, Term.realize_var, Sum.elim_inl,\n    Term.realize_varsToConstants, coe_con, presburger.realize_add, presburger.realize_natCast,\n    Nat.cast_id, presburger.realize_sum, presburger.realize_nsmul, Sum.elim_inr, smul_eq_mul]\n  congr! with a\n  simp_rw [Eq.comm (b := x), fun x : t => mul_comm (a x : α → ℕ) x, funext_iff]\n  congr! 1 with i\n  simp\n\ntheorem IsSemilinearSet.definable [Finite α] (hs : IsSemilinearSet s) :\n    A.Definable presburger s := by\n  rw [isSemilinearSet_iff] at hs\n  rcases hs with ⟨S, hS, rfl⟩\n  choose φ hφ using fun s : S => (hS s.1 s.2).definable\n  refine ⟨Formula.iSup φ, ?_⟩\n  ext x\n  have := fun s hs x => Set.ext_iff.1 (hφ ⟨s, hs⟩).symm x\n  simp only [mem_setOf_eq] at this\n  simp [this]\n\nnamespace FirstOrder.Language.presburger\n\nset_option backward.isDefEq.respectTransparency false in\nlemma term_realize_eq_add_dotProduct [Fintype α] (t : presburger[[A]].Term α) :\n    ∃ (k : ℕ) (u : α → ℕ), ∀ (v : α → ℕ), t.realize v = k + u ⬝ᵥ v := by\n  classical\n  induction t with simp only [Term.realize]\n  | var i =>\n    exact ⟨0, Pi.single i 1, by simp⟩\n  | @func l f ts ih =>\n    cases f with\n    | inl f =>\n      choose k u ih using ih\n      cases f with\n      | zero =>\n        refine ⟨0, 0, fun v => ?_⟩\n        rw [withConstants_funMap_sumInl]\n        simp\n      | one =>\n        refine ⟨1, 0, fun v => ?_⟩\n        rw [withConstants_funMap_sumInl]\n        simp [ih]\n      | add =>\n        refine ⟨k 0 + k 1, u 0 + u 1, fun v => ?_⟩\n        rw [withConstants_funMap_sumInl, add_dotProduct, add_left_comm, add_assoc, add_left_comm,\n          ← add_assoc]\n        simp [ih]\n    | inr f =>\n      cases l with\n      | zero =>\n        refine ⟨f, 0, fun v => ?_⟩\n        rw [withConstants_funMap_sumInr, zero_dotProduct, add_zero]\n        rfl\n      | succ => nomatch f\n\nvariable [Finite α]\n\nTarget:\nlemma isSemilinearSet_boundedFormula_realize {n} (φ : presburger[[A]].BoundedFormula α n) :\n    IsSemilinearSet {v : α ⊕ Fin n → ℕ | φ.Realize (v ∘ Sum.inl) (v ∘ Sum.inr)} :=\n\nProof body:\n","rejected":"```lean\nby\n  haveI := Fintype.ofFinite α\n  induction φ with simp only [BoundedFormula.Realize]\n  | equal t₁ t₂ =>\n    rcases term_realize_eq_add_dotProduct t₁ with ⟨k₁, u₁, ht₁⟩\n    rcases term_realize_eq_add_dotProduct t₂ with ⟨k₂, u₂, ht₂⟩\n    convert! Nat.isSemilinearSet_setOf_mulVec_eq ![k₁] ![k₂] (.of ![u₁]) (.of ![u₂])\n    simp [ht₁, ht₂]\n  | rel f => nomatch f\n  | falsum => exact .empty\n  | imp _ _ ih₁ ih₂ =>\n    convert! (ih₂.compl.inter ih₁).compl using 1\n    simp [setOf_inter_eq_sep, imp_iff_not_or, compl_setOf]\n  | @all n φ ih =>\n    let e := (Equiv.sumAssoc α (Fin n) (Fin 1)).trans (Equiv.sumCongr (.refl α) finSumFinEquiv)\n    rw [← isSemilinearSet_image_iff (LinearEquiv.funCongrLeft ℕ ℕ e)] at ih\n    convert! ih.compl.proj.compl using 1\n    simp_rw [compl_setOf, not_exists, Fin.forall_fin_succ_pi, Fin.forall_fin_zero_pi,\n      mem_compl_iff, mem_image, not_not, ← LinearEquiv.eq_symm_apply, LinearEquiv.funCongrLeft_symm,\n      exists_eq_right, mem_setOf, LinearEquiv.funCongrLeft_apply, LinearMap.funLeft,\n      LinearMap.coe_mk, AddHom.coe_mk]\n    congr! 4\n    ext i\n    cases i using Fin.lastCases <;> simp [e]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"ModelTheory/Arithmetic","family_id":"issemilinearset_boundedformula_realize","file_id":"mathlib/Mathlib/ModelTheory/Arithmetic/Presburger/Definability.lean","sample_id":"a75eaccf6d18b5c2eb5e6918fb406ae54ba35c64367a2abe5f21d18d5b2dac31"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c30b62521a4c15bdccd0e80ec4d2a61314f23daa4911bb4ad6a467ce05c1c053","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5fc66937e637b4b0522e166ab9a57947149372c442431abe243199902f780248","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ff91d6fc0a99e05c49e2fa58e7af54a13ccacbe396be47e3ff0015e67bd86101","source_sha256":"0155ab8151a9daebd303562a6d33da3f5df4ece30ebde16cdbf90ade81f9770b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h I hIX ↦ ?_, fun h ↦ ext_indep rfl fun I (hI : I ⊆ X) ↦ ?_⟩\n  · rw [← and_iff_left (a := (M.Indep I)) hIX, ← and_iff_left (a := (M'.Indep I)) hIX,\n      ← restrict_indep_iff, h, restrict_indep_iff]\n  rw [restrict_indep_iff, and_iff_left hI, restrict_indep_iff, and_iff_left hI, h _ hI]","hard_negative":false,"metrics":{"chosen_tokens":83,"rejected_tokens":88,"token_jaccard":0.891892,"token_length_ratio":1.060241},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"4c34b82311f97c6c46cdaaedd5fd2836247ee1fb5604b6398410b31906b112c3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Matroid.Dual\n\nNamespace:\nMatroid\n\nLocal context:\n/-\nCopyright (c) 2023 Peter Nelson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Peter Nelson\n-/\n/-!\n# Matroid Restriction\n\nGiven `M : Matroid α` and `R : Set α`, the independent sets of `M` that are contained in `R`\nare the independent sets of another matroid `M ↾ R` with ground set `R`,\ncalled the 'restriction' of `M` to `R`.\nFor `I ⊆ R ⊆ M.E`, `I` is a basis of `R` in `M` if and only if `I` is a base\nof the restriction `M ↾ R`, so this construction relates `Matroid.IsBasis` to `Matroid.IsBase`.\n\nIf `N M : Matroid α` satisfy `N = M ↾ R` for some `R ⊆ M.E`,\nthen we call `N` a 'restriction of `M`', and write `N ≤r M`. This is a partial order.\n\nThis file proves that the restriction is a matroid and that the `≤r` order is a partial order,\nand gives related API.\nIt also proves some `Matroid.IsBasis` analogues of `Matroid.IsBase` lemmas that,\nwhile they could be stated in `Data.Matroid.Basic`,\nare hard to prove without `Matroid.restrict` API.\n\n## Main Definitions\n\n* `M.restrict R`, written `M ↾ R`, is the restriction of `M : Matroid α` to `R : Set α`: i.e.\n  the matroid with ground set `R` whose independent sets are the `M`-independent subsets of `R`.\n\n* `Matroid.Restriction N M`, written `N ≤r M`, means that `N = M ↾ R` for some `R ⊆ M.E`.\n\n* `Matroid.IsStrictRestriction N M`, written `N <r M`, means that `N = M ↾ R` for some `R ⊂ M.E`.\n\n* `Matroidᵣ α` is a type synonym for `Matroid α`, equipped with the `PartialOrder` `≤r`.\n\n## Implementation Notes\n\nSince `R` and `M.E` are both terms in `Set α`, to define the restriction `M ↾ R`,\nwe need to either insist that `R ⊆ M.E`, or to say what happens when `R` contains the junk\noutside `M.E`.\n\nIt turns out that `R ⊆ M.E` is just an unnecessary hypothesis; if we say the restriction\n`M ↾ R` has ground set `R` and its independent sets are the `M`-independent subsets of `R`,\nwe always get a matroid, in which the elements of `R \\ M.E` aren't in any independent sets.\nWe could instead define this matroid to always be 'smaller' than `M` by setting\n`(M ↾ R).E := R ∩ M.E`, but this is worse definitionally, and more generally less convenient.\n\nThis makes it possible to actually restrict a matroid 'upwards'; for instance, if `M : Matroid α`\nsatisfies `M.E = ∅`, then `M ↾ Set.univ` is the matroid on `α` whose ground set is all of `α`,\nwhere the empty set is the only independent set.\n(In general, elements of `R \\ M.E` are all 'loops' of the matroid `M ↾ R`;\nsee `Matroid.loops` and `Matroid.restrict_loops_eq'` for a precise version of this statement.)\nThis is mathematically strange, but is useful for API building.\n\nThe cost of allowing a restriction of `M` to be 'bigger' than `M` itself is that\nthe statement `M ↾ R ≤r M` is only true with the hypothesis `R ⊆ M.E`\n(at least, if we want `≤r` to be a partial order).\nBut this isn't too inconvenient in practice. Indeed `(· ⊆ M.E)` proofs\ncan often be automatically provided by `aesop_mat`.\n\nWe define the restriction order `≤r` to give a `PartialOrder` instance on the type synonym\n`Matroidᵣ α` rather than `Matroid α` itself, because the `PartialOrder (Matroid α)` instance is\nreserved for the more mathematically important 'minor' order; see `Matroid.IsMinor`.\n-/\n\n@[expose] public section\n\nassert_not_exists Field\n\nopen Set\n\nnamespace Matroid\n\nvariable {α : Type*} {M : Matroid α} {R I X Y : Set α}\n\nsection restrict\n\n/-- The `IndepMatroid` whose independent sets are the independent subsets of `R`. -/\n@[simps] def restrictIndepMatroid (M : Matroid α) (R : Set α) : IndepMatroid α where\n  E := R\n  Indep I := M.Indep I ∧ I ⊆ R\n  indep_empty := ⟨M.empty_indep, empty_subset _⟩\n  indep_subset := fun _ _ h hIJ ↦ ⟨h.1.subset hIJ, hIJ.trans h.2⟩\n  indep_aug := by\n    rintro I I' ⟨hI, hIY⟩ (hIn : ¬ M.IsBasis' I R) (hI' : M.IsBasis' I' R)\n    rw [isBasis'_iff_isBasis_inter_ground] at hIn hI'\n    obtain ⟨B', hB', rfl⟩ := hI'.exists_isBase\n    obtain ⟨B, hB, hIB, hBIB'⟩ := hI.exists_isBase_subset_union_isBase hB'\n    rw [hB'.inter_isBasis_iff_compl_inter_isBasis_dual, sdiff_inter_sdiff] at hI'\n    have hss : M.E \\ (B' ∪ (R ∩ M.E)) ⊆ M.E \\ (B ∪ (R ∩ M.E)) := by\n      apply sdiff_subset_sdiff_right\n      rw [union_subset_iff, and_iff_left subset_union_right, union_comm]\n      exact hBIB'.trans (union_subset_union_left _ (subset_inter hIY hI.subset_ground))\n    have hi : M✶.Indep (M.E \\ (B ∪ (R ∩ M.E))) := by\n      rw [dual_indep_iff_exists]\n      exact ⟨B, hB, disjoint_of_subset_right subset_union_left disjoint_sdiff_left⟩\n    have h_eq := hI'.eq_of_subset_indep hi hss\n      (sdiff_subset_sdiff_right subset_union_right)\n    rw [h_eq, ← sdiff_inter_sdiff, ← hB.inter_isBasis_iff_compl_inter_isBasis_dual] at hI'\n    obtain ⟨J, hJ, hIJ⟩ := hI.subset_isBasis_of_subset\n      (subset_inter hIB (subset_inter hIY hI.subset_ground))\n    obtain rfl := hI'.indep.eq_of_isBasis hJ\n    have hIJ' : I ⊂ B ∩ (R ∩ M.E) := hIJ.ssubset_of_ne (fun he ↦ hIn (by rwa [he]))\n    obtain ⟨e, he⟩ := exists_of_ssubset hIJ'\n    exact ⟨e, ⟨⟨(hBIB' he.1.1).elim (fun h ↦ (he.2 h).elim) id,he.1.2⟩, he.2⟩,\n      hI'.indep.subset (insert_subset he.1 hIJ), insert_subset he.1.2.1 hIY⟩\n  indep_maximal := by\n    rintro A hAR I ⟨hI, _⟩ hIA\n    obtain ⟨J, hJ, hIJ⟩ := hI.subset_isBasis'_of_subset hIA\n    use J\n    simp only [hIJ, and_assoc, maximal_subset_iff, hJ.indep, hJ.subset, and_imp, true_and,\n      hJ.subset.trans hAR]\n    exact fun K hK _ hKA hJK ↦ hJ.eq_of_subset_indep hK hJK hKA\n  subset_ground _ := And.right\n\n/-- Change the ground set of a matroid to some `R : Set α`. The independent sets of the restriction\nare the independent subsets of the new ground set. Most commonly used when `R ⊆ M.E`,\nbut it is convenient not to require this. The elements of `R \\ M.E` become 'loops'. -/\ndef restrict (M : Matroid α) (R : Set α) : Matroid α := (M.restrictIndepMatroid R).matroid\n\n/-- `M ↾ R` means `M.restrict R`. -/\nscoped infixl:65 \" ↾ \" => Matroid.restrict\n\n@[simp] theorem restrict_indep_iff : (M ↾ R).Indep I ↔ M.Indep I ∧ I ⊆ R := Iff.rfl\n\ntheorem Indep.indep_restrict_of_subset (h : M.Indep I) (hIR : I ⊆ R) : (M ↾ R).Indep I :=\n  restrict_indep_iff.mpr ⟨h,hIR⟩\n\ntheorem Indep.of_restrict (hI : (M ↾ R).Indep I) : M.Indep I :=\n  (restrict_indep_iff.1 hI).1\n\n@[simp] theorem restrict_ground_eq : (M ↾ R).E = R := rfl\n\ntheorem restrict_finite {R : Set α} (hR : R.Finite) : (M ↾ R).Finite :=\n  ⟨hR⟩\n\n@[simp] theorem restrict_dep_iff : (M ↾ R).Dep X ↔ ¬ M.Indep X ∧ X ⊆ R := by\n  rw [Dep, restrict_indep_iff, restrict_ground_eq]; tauto\n\n@[simp] theorem restrict_ground_eq_self (M : Matroid α) : (M ↾ M.E) = M := by\n  refine ext_indep rfl ?_; simp_all\n\ntheorem restrict_restrict_eq {R₁ R₂ : Set α} (M : Matroid α) (hR : R₂ ⊆ R₁) :\n    (M ↾ R₁) ↾ R₂ = M ↾ R₂ := by\n  refine ext_indep rfl ?_\n  simp only [restrict_ground_eq, restrict_indep_iff, and_congr_left_iff, and_iff_left_iff_imp]\n  exact fun _ h _ _ ↦ h.trans hR\n\n@[simp] theorem restrict_idem (M : Matroid α) (R : Set α) : M ↾ R ↾ R = M ↾ R := by\n  rw [M.restrict_restrict_eq Subset.rfl]\n\n@[simp] theorem isBase_restrict_iff (hX : X ⊆ M.E := by aesop_mat) :\n    (M ↾ X).IsBase I ↔ M.IsBasis I X := by\n  simp_rw [isBase_iff_maximal_indep, IsBasis, and_iff_left hX, maximal_iff, restrict_indep_iff]\n\ntheorem isBase_restrict_iff' : (M ↾ X).IsBase I ↔ M.IsBasis' I X := by\n  simp_rw [isBase_iff_maximal_indep, IsBasis', maximal_iff, restrict_indep_iff]\n\ntheorem IsBasis'.isBase_restrict (hI : M.IsBasis' I X) : (M ↾ X).IsBase I :=\n  isBase_restrict_iff'.1 hI\n\ntheorem IsBasis.restrict_isBase (h : M.IsBasis I X) : (M ↾ X).IsBase I :=\n  (isBase_restrict_iff h.subset_ground).2 h\n\ninstance restrict_rankFinite [M.RankFinite] (R : Set α) : (M ↾ R).RankFinite :=\n  let ⟨_, hB⟩ := (M ↾ R).exists_isBase\n  hB.rankFinite_of_finite (hB.indep.of_restrict.finite)\n\ninstance restrict_finitary [Finitary M] (R : Set α) : Finitary (M ↾ R) := by\n  refine ⟨fun I hI ↦ ?_⟩\n  simp only [restrict_indep_iff] at *\n  rw [indep_iff_forall_finite_subset_indep]\n  exact ⟨fun J hJ hJfin ↦ (hI J hJ hJfin).1,\n    fun e heI ↦ singleton_subset_iff.1 (hI _ (by simpa) (toFinite _)).2⟩\n\n@[simp] theorem IsBasis.isBase_restrict (h : M.IsBasis I X) : (M ↾ X).IsBase I :=\n  (isBase_restrict_iff h.subset_ground).mpr h\n\ntheorem IsBasis.isBasis_restrict_of_subset (hI : M.IsBasis I X) (hXY : X ⊆ Y) :\n    (M ↾ Y).IsBasis I X := by\n  rwa [← isBase_restrict_iff, M.restrict_restrict_eq hXY, isBase_restrict_iff]\n\ntheorem isBasis'_restrict_iff : (M ↾ R).IsBasis' I X ↔ M.IsBasis' I (X ∩ R) ∧ I ⊆ R := by\n  simp_rw [IsBasis', maximal_iff, restrict_indep_iff, subset_inter_iff, and_imp]\n  tauto\n\ntheorem isBasis_restrict_iff' : (M ↾ R).IsBasis I X ↔ M.IsBasis I (X ∩ M.E) ∧ X ⊆ R := by\n  rw [isBasis_iff_isBasis'_subset_ground, isBasis'_restrict_iff, restrict_ground_eq,\n    and_congr_left_iff, ← isBasis'_iff_isBasis_inter_ground]\n  intro hXR\n  rw [inter_eq_self_of_subset_left hXR, and_iff_left_iff_imp]\n  exact fun h ↦ h.subset.trans hXR\n\ntheorem isBasis_restrict_iff (hR : R ⊆ M.E := by aesop_mat) :\n    (M ↾ R).IsBasis I X ↔ M.IsBasis I X ∧ X ⊆ R := by\n  rw [isBasis_restrict_iff', and_congr_left_iff]\n  intro hXR\n  rw [← isBasis'_iff_isBasis_inter_ground, isBasis'_iff_isBasis]\n\nlemma isBasis'_iff_isBasis_restrict_univ : M.IsBasis' I X ↔ (M ↾ univ).IsBasis I X := by\n  rw [isBasis_restrict_iff', isBasis'_iff_isBasis_inter_ground, and_iff_left (subset_univ _)]\n\nTarget:\ntheorem restrict_eq_restrict_iff (M M' : Matroid α) (X : Set α) :\n    M ↾ X = M' ↾ X ↔ ∀ I, I ⊆ X → (M.Indep I ↔ M'.Indep I) :=\n\nProof body:\n","rejected":"by\n  refine ⟨fun h I hIX ↦ ?_, fun h ↦ ext_indep rfl fun I (hI : I ⊆ X) ↦ ?_⟩\n  · rw [← and_iff_left (a := (M.Indep I)) hIX, ← and_iff_left (a := (M'.Indep I)) hIX,\n      ← restrict_indep_iff, h, restrict_indep_iff]\n  rw [restrict_indep_iff, and_iff_left hI, restrict_indep_iff, and_iff_left hI, h _ hI]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Matroid","family_id":"restrict_eq_restrict_iff","file_id":"mathlib/Mathlib/Combinatorics/Matroid/Minor/Restrict.lean","sample_id":"ff91d6fc0a99e05c49e2fa58e7af54a13ccacbe396be47e3ff0015e67bd86101"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a3707e1c83ef260ac94852111d4f8128b1e6f031babe4d0a83c426fc4ebac9e6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"176932c1d8e508eb9eba7d81a4615dd6834e3ab24c439e3d1fb20d4e33a48ec9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9944f4492e2edc232793a5f7e57320d19cd2217d990686c0e9b0e255481105bf","source_sha256":"0a9373d9f782e07a7cdfc4c29959b20ea9b224cdd8ac732a23d6364e81e86934","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [Part.fix]\n  rw [assert_neg h']","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.25},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"4c46308a663d4ae9dced66d37cd2e298ec2e8da1563c90d2c2ccd0f348a4d304","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Part\npublic import Mathlib.Data.Nat.Find\npublic import Mathlib.Data.Nat.Upto\npublic import Mathlib.Data.Stream.Defs\n\nNamespace:\nPart\n\nLocal context:\n/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\n/-!\n# Fixed point\n\nThis module defines a generic `fix` operator for defining recursive\ncomputations that are not necessarily well-founded or productive.\nAn instance is defined for `Part`.\n\n## Main definition\n\n* class `Fix`\n* `Part.fix`\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\nvariable {α : Type*} {β : α → Type*}\n\n/-- `Fix α` provides a `fix` operator to define recursive computation\nvia the fixed point of function of type `α → α`. -/\nclass Fix (α : Type*) where\n  /-- `fix f` represents the computation of a fixed point for `f`. -/\n  fix : (α → α) → α\n\nnamespace Part\n\nopen Part Nat Nat.Upto\n\nsection Basic\n\nvariable (f : (∀ a, Part (β a)) → (∀ a, Part (β a)))\n\n/-- A series of successive, finite approximation of the fixed point of `f`, defined by\n`approx f n = f^[n] ⊥`. The limit of this chain is the fixed point of `f`. -/\ndef Fix.approx : Stream' (∀ a, Part (β a))\n  | 0 => ⊥\n  | Nat.succ i => f (Fix.approx i)\n\n/-- loop body for finding the fixed point of `f` -/\ndef fixAux {p : ℕ → Prop} (i : Nat.Upto p) (g : ∀ j : Nat.Upto p, i < j → ∀ a, Part (β a)) :\n    ∀ a, Part (β a) :=\n  f fun x : α => (assert ¬p i.val) fun h : ¬p i.val => g (i.succ h) (Nat.lt_succ_self _) x\n\n/-- The least fixed point of `f`.\n\nIf `f` is a continuous function (according to complete partial orders),\nit satisfies the equations:\n\n  1. `fix f = f (fix f)`          (is a fixed point)\n  2. `∀ X, f X ≤ X → fix f ≤ X`   (least fixed point)\n-/\nprotected def fix (x : α) : Part (β x) :=\n  (Part.assert (∃ i, (Fix.approx f i x).Dom)) fun h =>\n    WellFounded.fix.{1} (Nat.Upto.wf h) (fixAux f) Nat.Upto.zero x\n\nopen Classical in\nprotected theorem fix_def {x : α} (h' : ∃ i, (Fix.approx f i x).Dom) :\n    Part.fix f x = Fix.approx f (Nat.succ (Nat.find h')) x := by\n  let p := fun i : ℕ => (Fix.approx f i x).Dom\n  have : p (Nat.find h') := Nat.find_spec h'\n  generalize hk : Nat.find h' = k\n  replace hk : Nat.find h' = k + (@Upto.zero p).val := hk\n  rw [hk] at this\n  revert hk\n  dsimp [Part.fix]; rw [assert_pos h']; revert this\n  generalize Upto.zero = z; intro _this hk\n  suffices ∀ x' hwf,\n    WellFounded.fix hwf (fixAux f) z x' = Fix.approx f (succ k) x'\n    from this _ _\n  induction k generalizing z with\n  | zero =>\n    intro x' _\n    rw [Fix.approx, WellFounded.fix_eq, fixAux]\n    congr\n    ext x : 1\n    rw [assert_neg]\n    · rfl\n    · rw [Nat.zero_add] at _this\n      simpa only [not_not, Coe]\n  | succ n n_ih =>\n    intro x' _\n    rw [Fix.approx, WellFounded.fix_eq, fixAux]\n    congr\n    ext : 1\n    have hh : ¬(Fix.approx f z.val x).Dom := by\n      apply Nat.find_min h'\n      lia\n    rw [succ_add_eq_add_succ] at _this hk\n    rw [assert_pos hh, n_ih (Upto.succ z hh) _this hk]\n\nTarget:\ntheorem fix_def' {x : α} (h' : ¬∃ i, (Fix.approx f i x).Dom) : Part.fix f x = none :=\n\nProof body:\n","rejected":"by\n  exact fix_def","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"7c044083bcd6223612786074b9ffdb47b01f2ebe117fe03e5ddab86c8c885001","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control","family_id":"fix_def","file_id":"mathlib/Mathlib/Control/Fix.lean","sample_id":"9944f4492e2edc232793a5f7e57320d19cd2217d990686c0e9b0e255481105bf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"61e6f75295dcf0ee0f0edfdb232d6f4ff00a1685b5790d5e3eca26ccc61aea57","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"91a92b19287fd834779cff643bf9db6f489d497152bb848e415ab87fcceac41f","source_sha256":"8a5f3265148069c15c32da8cd4a02af932338089cab2e996aa5a2f61253214b2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Module.finrank_mul_finrank ℝ ℂ E, Complex.finrank_real_complex]","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.0625,"token_length_ratio":0.2},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"4c4c26977612f1a5bbb792f9cd11ddc329542d8435d99a15e967e0d489fa6b19","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Rat\npublic import Mathlib.Analysis.Complex.Cardinality\npublic import Mathlib.LinearAlgebra.Complex.Module\npublic import Mathlib.LinearAlgebra.FiniteDimensional.Defs\npublic import Mathlib.Order.Interval.Set.Infinite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser\n-/\n/-!\n# Complex number as a finite-dimensional vector space over `ℝ`\n\nThis file contains the `FiniteDimensional ℝ ℂ` instance, as well as some results about the rank\n(`finrank` and `Module.rank`).\n-/\n\npublic section\n\nopen Module\n\nnamespace Complex\n\ninstance : FiniteDimensional ℝ ℂ := basisOneI.finiteDimensional_of_finite\n\n/-- `ℂ` is a finite extension of `ℝ` of degree 2, i.e `[ℂ : ℝ] = 2` -/\n@[simp, stacks 09G4]\ntheorem finrank_real_complex : finrank ℝ ℂ = 2 := by\n  rw [finrank_eq_card_basis basisOneI, Fintype.card_fin]\n\n@[simp]\ntheorem rank_real_complex : Module.rank ℝ ℂ = 2 := by simp [← finrank_eq_rank, finrank_real_complex]\n\ntheorem rank_real_complex'.{u} : Cardinal.lift.{u} (Module.rank ℝ ℂ) = 2 := by\n  rw [← finrank_eq_rank, finrank_real_complex, Cardinal.lift_natCast, Nat.cast_ofNat]\n\n/-- `Fact` version of the dimension of `ℂ` over `ℝ`, locally useful in the definition of the\ncircle. -/\ntheorem finrank_real_complex_fact : Fact (finrank ℝ ℂ = 2) :=\n  ⟨finrank_real_complex⟩\n\nend Complex\n\ninstance (priority := 500) FiniteDimensional.complexToReal (E : Type*) [AddCommGroup E]\n    [Module ℂ E] [FiniteDimensional ℂ E] : FiniteDimensional ℝ E :=\n  FiniteDimensional.trans ℝ ℂ E\n\ntheorem rank_real_of_complex (E : Type*) [AddCommGroup E] [Module ℂ E] :\n    Module.rank ℝ E = 2 * Module.rank ℂ E :=\n  Cardinal.lift_inj.{_, 0}.1 <| by\n    rw [← lift_rank_mul_lift_rank ℝ ℂ E, Complex.rank_real_complex']\n    simp only [Cardinal.lift_id']\n\nTarget:\ntheorem finrank_real_of_complex (E : Type*) [AddCommGroup E] [Module ℂ E] :\n    Module.finrank ℝ E = 2 * Module.finrank ℂ E :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Complex","family_id":"finrank_real_of_complex","file_id":"mathlib/Mathlib/LinearAlgebra/Complex/FiniteDimensional.lean","sample_id":"91a92b19287fd834779cff643bf9db6f489d497152bb848e415ab87fcceac41f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d988ca19062348f5a91c0b5ac55ee8cf0c8bfcea69c1997f85ea9823ef01e86a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"21544d2f51f369dc7014292be7e61bf52982b39514987f4c5a1ef51ecf1523b5","source_sha256":"059da84dda9e7cf638670a1c074b01de35e7641bc88d590bf894e2e5fc5223e6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h => Set.infinite_prod.1 ?_, ?_⟩\n  · rw [← image_uncurry_prod] at h\n    exact h.of_image _\n  · rintro (⟨hs, b, hb⟩ | ⟨ht, a, ha⟩)\n    · exact hs.image2_left hb (hfs _ hb)\n    · exact ht.image2_right ha (hft _ ha)","hard_negative":true,"metrics":{"chosen_tokens":71,"rejected_tokens":8,"token_jaccard":0.023256,"token_length_ratio":0.112676},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"4c5597cfca38a7e67b8c5a017fabffb0eee7449caa7da8df58b9c09edb4cbb73","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Basic\npublic import Mathlib.Data.Fintype.Prod\npublic import Mathlib.Data.Fintype.Pi\npublic import Mathlib.Algebra.Order.Group.Multiset\npublic import Mathlib.Data.ULift\npublic import Mathlib.Data.Set.NAry\n\nNamespace:\nSet\n\nLocal context:\n/-\nCopyright (c) 2022 Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kyle Miller\n-/\n/-!\n# Finiteness of products\n-/\n\npublic section\n\nassert_not_exists IsOrderedRing MonoidWithZero\n\nvariable {α β : Type*}\n\nnamespace Finite\n\ninstance [Finite α] [Finite β] : Finite (α × β) := by\n  haveI := Fintype.ofFinite α\n  haveI := Fintype.ofFinite β\n  infer_instance\n\ninstance {α β : Sort*} [Finite α] [Finite β] : Finite (PProd α β) :=\n  of_equiv _ Equiv.pprodEquivProdPLift.symm\n\ntheorem prod_left (β) [Finite (α × β)] [Nonempty β] : Finite α :=\n  of_surjective (Prod.fst : α × β → α) Prod.fst_surjective\n\ntheorem prod_right (α) [Finite (α × β)] [Nonempty α] : Finite β :=\n  of_surjective (Prod.snd : α × β → β) Prod.snd_surjective\n\nend Finite\n\nlemma Prod.finite_iff [Nonempty α] [Nonempty β] : Finite (α × β) ↔ Finite α ∧ Finite β where\n  mp _ := ⟨.prod_left β, .prod_right α⟩\n  mpr | ⟨_, _⟩ => inferInstance\n\ninstance Pi.finite {α : Sort*} {β : α → Sort*} [Finite α] [∀ a, Finite (β a)] :\n    Finite (∀ a, β a) := by\n  classical\n  haveI := Fintype.ofFinite (PLift α)\n  haveI := fun a => Fintype.ofFinite (PLift (β a))\n  exact\n    Finite.of_equiv (∀ a : PLift α, PLift (β (Equiv.plift a)))\n      (Equiv.piCongr Equiv.plift fun _ => Equiv.plift)\n\ninstance Function.Embedding.finite {α β : Sort*} [Finite β] : Finite (α ↪ β) := by\n  rcases isEmpty_or_nonempty (α ↪ β) with _ | h\n  · infer_instance\n  · refine h.elim fun f => ?_\n    haveI : Finite α := Finite.of_injective _ f.injective\n    exact Finite.of_injective _ DFunLike.coe_injective\n\ninstance Equiv.finite_right {α β : Sort*} [Finite β] : Finite (α ≃ β) :=\n  Finite.of_injective Equiv.toEmbedding fun e₁ e₂ h => Equiv.ext <| by\n    convert! DFunLike.congr_fun h using 0\n\ninstance Equiv.finite_left {α β : Sort*} [Finite α] : Finite (α ≃ β) :=\n  Finite.of_equiv _ ⟨Equiv.symm, Equiv.symm, Equiv.symm_symm, Equiv.symm_symm⟩\n\n@[to_additive]\ninstance MulEquiv.finite_left {α β : Type*} [Mul α] [Mul β] [Finite α] : Finite (α ≃* β) :=\n  Finite.of_injective toEquiv toEquiv_injective\n\n@[to_additive]\ninstance MulEquiv.finite_right {α β : Type*} [Mul α] [Mul β] [Finite β] : Finite (α ≃* β) :=\n  Finite.of_injective toEquiv toEquiv_injective\n\nopen Set Function\n\nvariable {γ : Type*}\n\nnamespace Set\n\n/-! ### Fintype instances\n\nEvery instance here should have a corresponding `Set.Finite` constructor in the next section.\n-/\n\nsection FintypeInstances\n\ninstance fintypeProd (s : Set α) (t : Set β) [Fintype s] [Fintype t] :\n    Fintype (s ×ˢ t : Set (α × β)) :=\n  Fintype.ofFinset (s.toFinset ×ˢ t.toFinset) <| by simp\n\ninstance fintypeOffDiag (s : Set α) [Fintype s] : Fintype s.offDiag :=\n  Fintype.ofFinset s.toFinset.offDiag <| by simp\n\n/-- `image2 f s t` is `Fintype` if `s` and `t` are. -/\ninstance fintypeImage2 [DecidableEq γ] (f : α → β → γ) (s : Set α) (t : Set β) [hs : Fintype s]\n    [ht : Fintype t] : Fintype (image2 f s t : Set γ) := by\n  rw [← image_prod]\n  apply Set.fintypeImage\n\nend FintypeInstances\n\nend Set\n\n/-! ### Finite instances\n\nThere is seemingly some overlap between the following instances and the `Fintype` instances\nin `Data.Set.Finite`. While every `Fintype` instance gives a `Finite` instance, those\ninstances that depend on `Fintype` or `Decidable` instances need an additional `Finite` instance\nto be able to generally apply.\n\nSome set instances do not appear here since they are consequences of others, for example\n`Subtype.Finite` for subsets of a finite type.\n-/\n\n\nnamespace Finite.Set\n\ninstance finite_prod (s : Set α) (t : Set β) [Finite s] [Finite t] :\n    Finite (s ×ˢ t : Set (α × β)) :=\n  Finite.of_equiv _ (Equiv.Set.prod s t).symm\n\ninstance finite_image2 (f : α → β → γ) (s : Set α) (t : Set β) [Finite s] [Finite t] :\n    Finite (image2 f s t : Set γ) := by\n  rw [← image_prod]\n  infer_instance\n\nend Finite.Set\n\nnamespace Set\n\n/-! ### Constructors for `Set.Finite`\n\nEvery constructor here should have a corresponding `Fintype` instance in the previous section\n(or in the `Fintype` module).\n\nThe implementation of these constructors ideally should be no more than `Set.toFinite`,\nafter possibly setting up some `Fintype` and classical `Decidable` instances.\n-/\n\n\nsection SetFiniteConstructors\n\nsection Prod\n\nvariable {s : Set α} {t : Set β}\n\nprotected theorem Finite.prod (hs : s.Finite) (ht : t.Finite) : (s ×ˢ t : Set (α × β)).Finite := by\n  have := hs.to_subtype\n  have := ht.to_subtype\n  apply toFinite\n\ntheorem Finite.of_prod_left (h : (s ×ˢ t : Set (α × β)).Finite) : t.Nonempty → s.Finite :=\n  fun ⟨b, hb⟩ => (h.image Prod.fst).subset fun a ha => ⟨(a, b), ⟨ha, hb⟩, rfl⟩\n\ntheorem Finite.of_prod_right (h : (s ×ˢ t : Set (α × β)).Finite) : s.Nonempty → t.Finite :=\n  fun ⟨a, ha⟩ => (h.image Prod.snd).subset fun b hb => ⟨(a, b), ⟨ha, hb⟩, rfl⟩\n\nprotected theorem Infinite.prod_left (hs : s.Infinite) (ht : t.Nonempty) : (s ×ˢ t).Infinite :=\n  fun h => hs <| h.of_prod_left ht\n\nprotected theorem Infinite.prod_right (ht : t.Infinite) (hs : s.Nonempty) : (s ×ˢ t).Infinite :=\n  fun h => ht <| h.of_prod_right hs\n\nprotected theorem infinite_prod :\n    (s ×ˢ t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty := by\n  refine ⟨fun h => ?_, ?_⟩\n  · simp_rw [Set.Infinite, @and_comm ¬_, ← Classical.not_imp]\n    by_contra!\n    exact h ((this.1 h.nonempty.snd).prod <| this.2 h.nonempty.fst)\n  · rintro (h | h)\n    · exact h.1.prod_left h.2\n    · exact h.1.prod_right h.2\n\ntheorem finite_prod : (s ×ˢ t).Finite ↔ (s.Finite ∨ t = ∅) ∧ (t.Finite ∨ s = ∅) := by\n  contrapose! +distrib; exact Set.infinite_prod\n\nprotected theorem Finite.offDiag {s : Set α} (hs : s.Finite) : s.offDiag.Finite :=\n  (hs.prod hs).subset s.offDiag_subset_prod\n\nprotected theorem Finite.image2 (f : α → β → γ) (hs : s.Finite) (ht : t.Finite) :\n    (image2 f s t).Finite := by\n  have := hs.to_subtype\n  have := ht.to_subtype\n  apply toFinite\n\nend Prod\n\nend SetFiniteConstructors\n\n/-! ### Properties -/\n\ntheorem Finite.toFinset_prod {s : Set α} {t : Set β} (hs : s.Finite) (ht : t.Finite) :\n    hs.toFinset ×ˢ ht.toFinset = (hs.prod ht).toFinset :=\n  Finset.ext <| by simp\n\ntheorem Finite.toFinset_offDiag {s : Set α} (hs : s.Finite) :\n    hs.offDiag.toFinset = hs.toFinset.offDiag :=\n  Finset.ext <| by simp\n\ntheorem finite_image_fst_and_snd_iff {s : Set (α × β)} :\n    (Prod.fst '' s).Finite ∧ (Prod.snd '' s).Finite ↔ s.Finite :=\n  ⟨fun h => (h.1.prod h.2).subset fun _ h => ⟨mem_image_of_mem _ h, mem_image_of_mem _ h⟩,\n    fun h => ⟨h.image _, h.image _⟩⟩\n\n/-! ### Infinite sets -/\n\nvariable {s t : Set α}\n\nsection Image2\n\nvariable {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β}\n\nprotected theorem Infinite.image2_left (hs : s.Infinite) (hb : b ∈ t)\n    (hf : InjOn (fun a => f a b) s) : (image2 f s t).Infinite :=\n  (hs.image hf).mono <| image_subset_image2_left hb\n\nprotected theorem Infinite.image2_right (ht : t.Infinite) (ha : a ∈ s) (hf : InjOn (f a) t) :\n    (image2 f s t).Infinite :=\n  (ht.image hf).mono <| image_subset_image2_right ha\n\nTarget:\ntheorem infinite_image2 (hfs : ∀ b ∈ t, InjOn (fun a => f a b) s) (hft : ∀ a ∈ s, InjOn (f a) t) :\n    (image2 f s t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"f83a0232fcbd1ba953a3b544ffbe0b503479aef0db0e59df808b3c79db3e7882","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finite","family_id":"infinite_image","file_id":"mathlib/Mathlib/Data/Finite/Prod.lean","sample_id":"21544d2f51f369dc7014292be7e61bf52982b39514987f4c5a1ef51ecf1523b5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ea63d2911a78e27e3d0efd6769a583eaec219edad30fe20857b3d050d16b7158","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f611f13c4698d3e99009d189350bca949401d584539fd12a5756b6e5b77a146b","source_sha256":"546544902b60193a288cef69b3c4f5d20597fa5d3b2853c4808d2cc29f4c6640","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  letI := (IntermediateField.inclusion H.le).toAlgebra\n  have : IsScalarTower F E L := .of_algebraMap_eq' rfl\n  refine lt_of_le_of_ne ?_ ?_\n  · exact Module.finrank_top_le_finrank_of_isScalarTower _ _ _\n  · exact .symm (mt (eq_of_le_of_finrank_eq' H.le) H.ne)","hard_negative":false,"metrics":{"chosen_tokens":53,"rejected_tokens":2,"token_jaccard":0.03125,"token_length_ratio":0.037736},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"4c5e170dc08e386d7a279385ba3a0a0adcb8e4296d64539a4bf17698bdf66fd4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.IntermediateField.Basic\npublic import Mathlib.FieldTheory.Minpoly.Basic\npublic import Mathlib.FieldTheory.Tower\npublic import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition\npublic import Mathlib.RingTheory.Algebraic.Integral\n\nNamespace:\nIntermediateField\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Results on finite dimensionality and algebraicity of intermediate fields.\n-/\n\n@[expose] public section\n\nopen Module\n\nvariable {K L : Type*} [Field K] [Field L] [Algebra K L]\n  {S : IntermediateField K L}\n\ntheorem IntermediateField.coe_isIntegral_iff {R : Type*} [CommRing R] [Algebra R K] [Algebra R L]\n    [IsScalarTower R K L] {x : S} : IsIntegral R (x : L) ↔ IsIntegral R x :=\n  isIntegral_algHom_iff (S.val.restrictScalars R) Subtype.val_injective\n\n/-- Turn an algebraic subalgebra into an intermediate field, `Subalgebra.IsAlgebraic` version. -/\ndef Subalgebra.IsAlgebraic.toIntermediateField {S : Subalgebra K L} (hS : S.IsAlgebraic) :\n    IntermediateField K L where\n  toSubalgebra := S\n  inv_mem' x hx := Algebra.adjoin_le_iff.mpr\n    (Set.singleton_subset_iff.mpr hx) (hS x hx).isIntegral.inv_mem_adjoin\n\n/-- Turn an algebraic subalgebra into an intermediate field, `Algebra.IsAlgebraic` version. -/\nabbrev Algebra.IsAlgebraic.toIntermediateField (S : Subalgebra K L) [Algebra.IsAlgebraic K S] :\n    IntermediateField K L := (S.isAlgebraic_iff.mpr ‹_›).toIntermediateField\n\nnamespace IntermediateField\n\ninstance isAlgebraic_tower_bot [Algebra.IsAlgebraic K L] : Algebra.IsAlgebraic K S :=\n  Algebra.IsAlgebraic.of_injective S.val S.val.injective\n\ninstance isAlgebraic_tower_top [Algebra.IsAlgebraic K L] : Algebra.IsAlgebraic S L :=\n  Algebra.IsAlgebraic.tower_top (K := K) S\n\nsection FiniteDimensional\n\nvariable (F E : IntermediateField K L)\n\ninstance finiteDimensional_left [FiniteDimensional K L] : FiniteDimensional K F := .left K F L\ninstance finiteDimensional_right [FiniteDimensional K L] : FiniteDimensional F L := .right K F L\n\n@[simp]\ntheorem rank_eq_rank_subalgebra : Module.rank K F.toSubalgebra = Module.rank K F :=\n  rfl\n\n@[simp]\ntheorem finrank_eq_finrank_subalgebra : finrank K F.toSubalgebra = finrank K F :=\n  rfl\n\nvariable {F} {E}\n\n/-- If `F ≤ E` are two intermediate fields of `L / K` such that `[E : K] ≤ [F : K]` are finite,\nthen `F = E`. -/\ntheorem eq_of_le_of_finrank_le [hfin : FiniteDimensional K E] (h_le : F ≤ E)\n    (h_finrank : finrank K E ≤ finrank K F) : F = E :=\n  haveI : Module.Finite K E.toSubalgebra := hfin\n  toSubalgebra_injective <| Subalgebra.eq_of_le_of_finrank_le h_le h_finrank\n\n/-- If `F ≤ E` are two intermediate fields of `L / K` such that `[F : K] = [E : K]` are finite,\nthen `F = E`. -/\ntheorem eq_of_le_of_finrank_eq [FiniteDimensional K E] (h_le : F ≤ E)\n    (h_finrank : finrank K F = finrank K E) : F = E :=\n  eq_of_le_of_finrank_le h_le h_finrank.ge\n\n/-- If `F ≤ E` are two intermediate fields of `L / K` such that `[E : K]` is finite,\nthen `F = E` iff `[F : K] = [E : K]`. -/\ntheorem eq_iff_finrank_eq_of_le [FiniteDimensional K E] (h_le : F ≤ E) :\n    F = E ↔ finrank K F = finrank K E :=\n  ⟨fun h ↦ by rw [h], eq_of_le_of_finrank_eq h_le⟩\n\n-- If `F ≤ E` are two intermediate fields of a finite extension `L / K` such that\n-- `[L : F] ≤ [L : E]`, then `F = E`. Marked as private since it's a direct corollary of\n-- `eq_of_le_of_finrank_le'` (the `FiniteDimensional K L` implies `FiniteDimensional F L`\n-- automatically by typeclass resolution).\nprivate theorem eq_of_le_of_finrank_le'' [FiniteDimensional K L] (h_le : F ≤ E)\n    (h_finrank : finrank F L ≤ finrank E L) : F = E := by\n  apply eq_of_le_of_finrank_le h_le\n  have h1 := finrank_mul_finrank K F L\n  have h2 := finrank_mul_finrank K E L\n  have h3 : 0 < finrank E L := finrank_pos\n  nlinarith\n\n/-- If `F ≤ E` are two intermediate fields of `L / K` such that `[L : F] ≤ [L : E]` are finite,\nthen `F = E`. -/\ntheorem eq_of_le_of_finrank_le' [FiniteDimensional F L] (h_le : F ≤ E)\n    (h_finrank : finrank F L ≤ finrank E L) : F = E := by\n  refine le_antisymm h_le (fun l hl ↦ ?_)\n  rwa [← mem_extendScalars (le_refl F), eq_of_le_of_finrank_le''\n    ((extendScalars_le_extendScalars_iff (le_refl F) h_le).2 h_le) h_finrank, mem_extendScalars]\n\n/-- If `F ≤ E` are two intermediate fields of `L / K` such that `[L : F] = [L : E]` are finite,\nthen `F = E`. -/\ntheorem eq_of_le_of_finrank_eq' [FiniteDimensional F L] (h_le : F ≤ E)\n    (h_finrank : finrank F L = finrank E L) : F = E :=\n  eq_of_le_of_finrank_le' h_le h_finrank.le\n\n/-- If `F ≤ E` are two intermediate fields of `L / K` such that `[L : F]` is finite,\nthen `F = E` iff `[L : F] = [L : E]`. -/\ntheorem eq_iff_finrank_eq_of_le' [FiniteDimensional F L] (h_le : F ≤ E) :\n    F = E ↔ finrank F L = finrank E L :=\n  ⟨fun h ↦ by rw [h], eq_of_le_of_finrank_eq' h_le⟩\n\nTarget:\nlemma finrank_lt_of_gt [FiniteDimensional F L] (H : F < E) :\n    Module.finrank E L < Module.finrank F L :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/IntermediateField","family_id":"finrank_lt_of_gt","file_id":"mathlib/Mathlib/FieldTheory/IntermediateField/Algebraic.lean","sample_id":"f611f13c4698d3e99009d189350bca949401d584539fd12a5756b6e5b77a146b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6ec31665b33411506cda64640e21f6774b047c685e35cff9b518e65219ef78f0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cdb77a9f055c59dec5cff52d796012f86cc44a94acfc76f498ab422b776d820c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"96fed0cd3e0fdbf1ac8825b9ea3ac8717dd91342a9fea198dd325c9dbb8823cc","source_sha256":"7d9672c8f14a833f3923b39b4803102d55499036394ed799a1e56e11ac046c23","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine c.isColimit.hom_ext (fun j ↦ ?_)\n  dsimp\n  induction j using SuccOrder.limitRecOn with\n  | isMin j hj =>\n    obtain rfl := hj.eq_bot\n    simpa [← cancel_epi c.isoBot.inv] using h₀\n  | succ j hj hj' =>\n    apply (c.attachCells j hj).hom_ext\n    · simpa using hj'\n    · intro i\n      simpa only [Category.assoc, Cells.ι] using h ({ hj := hj, k := i, .. })\n  | isSuccLimit j hj hj' =>\n    exact (c.F.isColimitOfIsWellOrderContinuous j hj).hom_ext\n      (fun ⟨k, hk⟩ ↦ by simpa using hj' k hk)","hard_negative":false,"metrics":{"chosen_tokens":129,"rejected_tokens":133,"token_jaccard":0.951613,"token_length_ratio":1.031008},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"4c8218252586fe1977c94c63e6ae6c59ea6d939600c7f2bae42446d1471841a9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.RelativeCellComplex.AttachCells\npublic import Mathlib.CategoryTheory.MorphismProperty.TransfiniteComposition\n\nNamespace:\nHomotopicalAlgebra.RelativeCellComplex\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Relative cell complexes\n\nIn this file, we define a structure `RelativeCellComplex` which expresses\nthat a morphism `f : X ⟶ Y` is a transfinite composition of morphisms,\nall of which consist in attaching cells. Here, we allow a different\nfamily of authorized cells at each step. For example, (relative)\nCW-complexes are defined in the file `Mathlib/Topology/CWComplex/Abstract/Basic.lean`\nby requiring that at the `n`th step, we attach `n`-disks along their\nboundaries.\n\nThis structure `RelativeCellComplex` is also used in the\nformalization of the small object argument,\nsee the file `Mathlib/CategoryTheory/SmallObject/IsCardinalForSmallObjectArgument.lean`.\n\n## References\n* https://ncatlab.org/nlab/show/small+object+argument\n\n-/\n\n@[expose] public section\n\nuniverse w w' t v u\n\nopen CategoryTheory\n\nnamespace HomotopicalAlgebra\n\nvariable {C : Type u} [Category.{v} C]\n  {J : Type w'} [LinearOrder J] [OrderBot J] [SuccOrder J] [WellFoundedLT J]\n  {α : J → Type t} {A B : (j : J) → α j → C}\n  (basicCell : (j : J) → (i : α j) → A j i ⟶ B j i) {X Y : C} (f : X ⟶ Y)\n\n/-- Let `J` be a well-ordered type. Assume that for each `j : J`, we\nhave a family `basicCell j` of morphisms. A relative cell complex\nis a morphism `f : X ⟶ Y` which is a transfinite composition of morphisms\nin such a way that at the step `j : J`, we attach cells in the family `basicCell j`. -/\nstructure RelativeCellComplex\n    extends TransfiniteCompositionOfShape J f where\n  /-- If `j` is not the maximum element, `F.obj (Order.succ j)` is obtained\n  from `F.obj j` by attaching cells in the family of morphisms `basicCell j`. -/\n  attachCells (j : J) (hj : ¬ IsMax j) :\n    AttachCells.{w} (basicCell j) (F.map (homOfLE (Order.le_succ j)))\n\nnamespace RelativeCellComplex\n\nvariable {basicCell f} (c : RelativeCellComplex basicCell f)\n\n/-- The index type of cells in a relative cell complex. -/\nstructure Cells where\n  /-- the step where the cell is added -/\n  j : J\n  hj : ¬ IsMax j\n  /-- the index of the cell -/\n  k : (c.attachCells j hj).ι\n\nvariable {c} in\n/-- Given a cell `γ` in a relative cell complex, this is the corresponding\nindex in the family of morphisms `basicCell γ.j`. -/\ndef Cells.i (γ : Cells c) : α γ.j := (c.attachCells γ.j γ.hj).π γ.k\n\nvariable {c} in\n/-- The inclusion of a cell. -/\ndef Cells.ι (γ : Cells c) : B γ.j γ.i ⟶ Y :=\n  (c.attachCells γ.j γ.hj).cell γ.k ≫ c.incl.app (Order.succ γ.j)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma hom_ext {Z : C} {φ₁ φ₂ : Y ⟶ Z} (h₀ : f ≫ φ₁ = f ≫ φ₂)\n    (h : ∀ (γ : Cells c), γ.ι ≫ φ₁ = γ.ι ≫ φ₂) :\n    φ₁ = φ₂ :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  refine c.isColimit.hom_ext (fun j ↦ ?_)\n  dsimp\n  induction j using SuccOrder.limitRecOn with\n  | isMin j hj =>\n    obtain rfl := hj.eq_bot\n    simpa [← cancel_epi c.isoBot.inv] using h₀\n  | succ j hj hj' =>\n    apply (c.attachCells j hj).hom_ext\n    · simpa using hj'\n    · intro i\n      simpa only [Category.assoc, Cells.ι] using h ({ hj := hj, k := i, .. })\n  | isSuccLimit j hj hj' =>\n    exact (c.F.isColimitOfIsWellOrderContinuous j hj).hom_ext\n      (fun ⟨k, hk⟩ ↦ by simpa using hj' k hk)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/RelativeCellComplex","family_id":"hom_ext","file_id":"mathlib/Mathlib/AlgebraicTopology/RelativeCellComplex/Basic.lean","sample_id":"96fed0cd3e0fdbf1ac8825b9ea3ac8717dd91342a9fea198dd325c9dbb8823cc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bf3037486d0c6b30f0d1754728261534db0d266ca9de3c9d72286f831a1bc900","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"53e21ad5e315c317c8084e011afafd42127d23df856e46c6d95998155a1cbf26","source_sha256":"610b55cc6b034881bfda1b6609f87ef5fc28aecf81f7b61240acfb496bc220e4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI : IsCentral K D :=\n  { out := fun x ↦ show x ∈ Subalgebra.center k D → _ by\n      simp only [center_eq_bot, mem_bot, Set.mem_range, forall_exists_index]\n      rintro x rfl\n      exact ⟨algebraMap k K x, by simp [algebraMap_eq_smul_one, smul_assoc]⟩ }\n  refine ⟨FaithfulSMul.algebraMap_injective k K, fun x => ?_⟩\n  have H : algebraMap K D x ∈ (Subalgebra.center K D : Set D) := Subalgebra.algebraMap_mem _ _\n  rw [show (Subalgebra.center K D : Set D) = Subalgebra.center k D by rfl] at H\n  simp only [center_eq_bot, coe_bot, Set.mem_range] at H\n  obtain ⟨x', H⟩ := H\n  exact ⟨x', (algebraMap K D).injective <| by simp [← H, algebraMap_eq_smul_one]⟩","hard_negative":false,"metrics":{"chosen_tokens":161,"rejected_tokens":2,"token_jaccard":0.016667,"token_length_ratio":0.012422},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"4cc110ac059e95ceba606f8cfb21f827284bcf75f9d1a1b58a60b6e24ff05765","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Central.Defs\nimport Mathlib.Algebra.Module.Torsion.Field\n\nNamespace:\nAlgebra.IsCentral\n\nLocal context:\n/-\nCopyright (c) 2024 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Jujian Zhang, Yunzhou Xie\n-/\n/-!\n# Central Algebras\n\nIn this file, we prove some basic results about central algebras over a commutative ring.\n\n## Main results\n\n- `Algebra.IsCentral.center_eq_bot`: the center of a central algebra over `K` is equal to `K`.\n- `Algebra.IsCentral.self`: a commutative ring is a central algebra over itself.\n- `Algebra.IsCentral.baseField_essentially_unique`: Let `D/K/k` be a tower of scalars where\n  `K` and `k` are fields. If `D` is a nontrivial central algebra over `k`, `K` is isomorphic to `k`.\n-/\n\npublic section\n\nuniverse u v\n\nnamespace Algebra.IsCentral\n\nvariable (K : Type u) [CommSemiring K] (D D' : Type v) [Semiring D] [Algebra K D]\n  [h : IsCentral K D] [Semiring D'] [Algebra K D']\n\n@[simp]\nlemma center_eq_bot : Subalgebra.center K D = ⊥ := eq_bot_iff.2 IsCentral.out\n\nvariable {D} in\nlemma mem_center_iff {x : D} : x ∈ Subalgebra.center K D ↔ ∃ (a : K), x = algebraMap K D a := by\n  rw [center_eq_bot, Algebra.mem_bot]\n  simp [eq_comm]\n\ninstance self : IsCentral K K where\n  out x := by simp [Algebra.mem_bot]\n\nTarget:\nlemma baseField_essentially_unique\n    (k K D : Type*) [Field k] [Field K] [Ring D] [Nontrivial D]\n    [Algebra k K] [Algebra K D] [Algebra k D] [IsScalarTower k K D]\n    [IsCentral k D] :\n    Function.Bijective (algebraMap k K) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Central","family_id":"basefield_essentially_unique","file_id":"mathlib/Mathlib/Algebra/Central/Basic.lean","sample_id":"53e21ad5e315c317c8084e011afafd42127d23df856e46c6d95998155a1cbf26"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ad8808e6ef06989ebffa9f2a94c9e16fc819a971dc924b479659fd09f73634ab","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c0b4f5ee96fca0c0f0a00a8f442ac70878628b5f442f826e65950e0dc0cfc65c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"92326efcf9c32173b0a7545e06842af3e1db8144b2ac5b58b5ccfdd3ce10f509","source_sha256":"016de02d162ccc90d858e39e45ebc4da0ae14e5643087f6444b25344546eb84f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [Ideal.eq_top_iff_one, ← Subtype.range_val (s := s), ← Set.range_comp,\n    Finsupp.mem_ideal_span_range_iff_exists_finsupp] at hs' ⊢\n  choose f hf using hs'\n  have : Fintype s := hs.fintype\n  refine ⟨Finsupp.equivFunOnFinite.symm fun i x ↦ f x i, ?_⟩\n  ext i\n  simpa [Finsupp.sum_fintype] using hf i","hard_negative":true,"metrics":{"chosen_tokens":72,"rejected_tokens":2,"token_jaccard":0.021277,"token_length_ratio":0.027778},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"4d298730cd1f3c306750a768626c598733f8b6b62025ecfb49ff124966ad3d46","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.Immersion\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# `Π Rᵢ`-Points of Schemes\n\nWe show that the canonical map `X(Π Rᵢ) ⟶ Π X(Rᵢ)` (`AlgebraicGeometry.pointsPi`)\nis injective and surjective under various assumptions.\n\n-/\n\n@[expose] public section\n\nopen CategoryTheory Limits PrimeSpectrum\n\nnamespace AlgebraicGeometry\n\nuniverse u v\n\nvariable {ι : Type u} (R : ι → CommRingCat.{u})\n\nTarget:\nlemma Ideal.span_eq_top_of_span_image_evalRingHom\n    {ι} {R : ι → Type*} [∀ i, CommRing (R i)] (s : Set (Π i, R i))\n    (hs : s.Finite) (hs' : ∀ i, Ideal.span (Pi.evalRingHom (R ·) i '' s) = ⊤) :\n    Ideal.span s = ⊤ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_92326efcf9c3","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"4b4970a52a3e0acee569902af02fae78f97589e8cc80ebc362c0c1fa7eabdf87","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry","family_id":"ideal","file_id":"mathlib/Mathlib/AlgebraicGeometry/PointsPi.lean","sample_id":"92326efcf9c32173b0a7545e06842af3e1db8144b2ac5b58b5ccfdd3ce10f509"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c628935d9750505c8adb2f5d44da9ce697c92da40b87a39fbb8e2a552d65d0b4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8fa0d3b17dbb462bc253c216aedf5600d245ea36ea5fef5dc975ff5afcf7c43e","source_sha256":"c7531e909ba94c250fd36a161457eaacef5f4b4e592debc46a5c88309057a427","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext; exact geometricallyReduced_iff _","hard_negative":true,"metrics":{"chosen_tokens":6,"rejected_tokens":8,"token_jaccard":0.083333,"token_length_ratio":1.333333},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"4d3752c44c94b818a92521913f032d1b581843e844e98d7627af9a5c36bad379","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Artinian\npublic import Mathlib.AlgebraicGeometry.Geometrically.Basic\npublic import Mathlib.AlgebraicGeometry.Morphisms.SchemeTheoreticallyDominant\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2026 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Geometrically Reduced Schemes\n\n## Main results\n- `AlgebraicGeometry.GeometricallyReduced`:\n  We say that morphism `f : X ⟶ Y` is geometrically reduced if for all `Spec K ⟶ Y` with `K`\n  a field, `X ×[Y] Spec K` is reduced.\n  We also provide the fact that this is stable under base change (by `infer_instance`)\n- `GeometricallyReduced.iff_geometricallyReduced_fiber`:\n  A scheme is geometrically reduced over `S` iff the fibers of all\n  `s : S` are geometrically reduced.\n- `AlgebraicGeometry.GeometricallyReduced.isReduced_of_flat_of_isLocallyNoetherian`:\n  If `X` is geometrically reduced and flat over a reduced and locally noetherian scheme,\n  then `X` is also reduced.\n  In particular, the base change of a geometrically reduced and flat scheme to an\n  reduced and locally noetherian scheme is reduced (by `infer_instance`).\n\n## TODO\nGet rid of the noetherian assumption.\n-/\n\npublic section\n\nopen CategoryTheory MorphismProperty Limits\n\nnamespace AlgebraicGeometry\n\nvariable {X Y Z S : Scheme} (f : X ⟶ S) (g : Y ⟶ S)\n\n/-- We say that morphism `f : X ⟶ Y` is geometrically reduced if for all `Spec K ⟶ Y` with `K`\na field, `X ×[Y] Spec K` is reduced. -/\n@[mk_iff]\nclass GeometricallyReduced (f : X ⟶ Y) : Prop where\n  geometrically_isReduced : geometrically IsReduced f\n\nTarget:\nlemma GeometricallyReduced.eq_geometrically :\n    @GeometricallyReduced = geometrically IsReduced :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"05650197539227623cd8139f84f7093968cfbcee6139fd125290d396b2b3f0b9","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Geometrically","family_id":"geometricallyreduced","file_id":"mathlib/Mathlib/AlgebraicGeometry/Geometrically/Reduced.lean","sample_id":"8fa0d3b17dbb462bc253c216aedf5600d245ea36ea5fef5dc975ff5afcf7c43e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"355d8928efd1f0f2dd702b5a22182d531523711ce4f995520e27017b7c2e8b46","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8ad925533b8c56254faf2ab0de8790476e52ea1f36cb7dddd76cb2f24ea5a395","source_sha256":"4dc8cbdda13dd66c6fd01c0ae738b0853be3fe2c1370940a1d31f2e403201cc7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩\n  · filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy\n  · intro ε εpos\n    filter_upwards [Icc_mem_nhdsLT <| show x - ε < x by linarith] with y hy\n    rw [closedBall_eq_Icc]\n    exact Icc_subset_Icc hy.1 (by linarith)","hard_negative":false,"metrics":{"chosen_tokens":63,"rejected_tokens":2,"token_jaccard":0.025641,"token_length_ratio":0.031746},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"4d7757c821237c3b682b7375c71b48783fba66866cb20c41a35cafe6acc2426c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Covering.DensityTheorem\npublic import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2022 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Covering theorems for Lebesgue measure in one dimension\n\nWe have a general theory of covering theorems for doubling measures, developed notably\nin `DensityTheorem.lean`. In this file, we expand the API for this theory in one dimension,\nby showing that intervals belong to the relevant Vitali family.\n-/\n\npublic section\n\n\nopen Set MeasureTheory IsUnifLocDoublingMeasure Filter\n\nopen scoped Topology\n\nnamespace Real\n\ntheorem Icc_mem_vitaliFamily_at_right {x y : ℝ} (hxy : x < y) :\n    Icc x y ∈ (vitaliFamily (volume : Measure ℝ) 1).setsAt x := by\n  rw [Icc_eq_closedBall]\n  refine closedBall_mem_vitaliFamily_of_dist_le_mul _ ?_ (by linarith)\n  rw [dist_comm, Real.dist_eq, abs_of_nonneg] <;> linarith\n\ntheorem tendsto_Icc_vitaliFamily_right (x : ℝ) :\n    Tendsto (fun y => Icc x y) (𝓝[>] x) ((vitaliFamily (volume : Measure ℝ) 1).filterAt x) := by\n  refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩\n  · filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy\n  · intro ε εpos\n    filter_upwards [Icc_mem_nhdsGT <| show x < x + ε by linarith] with y hy\n    rw [closedBall_eq_Icc]\n    exact Icc_subset_Icc (by linarith) hy.2\n\ntheorem Icc_mem_vitaliFamily_at_left {x y : ℝ} (hxy : x < y) :\n    Icc x y ∈ (vitaliFamily (volume : Measure ℝ) 1).setsAt y := by\n  rw [Icc_eq_closedBall]\n  refine closedBall_mem_vitaliFamily_of_dist_le_mul _ ?_ (by linarith)\n  rw [Real.dist_eq, abs_of_nonneg] <;> linarith\n\nTarget:\ntheorem tendsto_Icc_vitaliFamily_left (x : ℝ) :\n    Tendsto (fun y => Icc y x) (𝓝[<] x) ((vitaliFamily (volume : Measure ℝ) 1).filterAt x) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Covering","family_id":"tendsto_icc_vitalifamily_left","file_id":"mathlib/Mathlib/MeasureTheory/Covering/OneDim.lean","sample_id":"8ad925533b8c56254faf2ab0de8790476e52ea1f36cb7dddd76cb2f24ea5a395"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8bb9f6292b5b7cdafeb1af9846ff9f4bfcbbf84b87fb669f9431c4e7c51bed33","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8b95081cd7f651acb1a47e7a53ca218a713ae886e12ac7935b03bee7470a950f","source_sha256":"3542347af6e76d4ad663bef58f83e85c4df560564e9c17485c7828e4299be8bb","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [iSup, hf.map_sSup', ← range_comp]\n  rfl","hard_negative":true,"metrics":{"chosen_tokens":14,"rejected_tokens":8,"token_jaccard":0.052632,"token_length_ratio":0.571429},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"4d8bdcb0e4adf3c283cbed0e81e900050e438413613d11a5a28e349e20137752","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\npublic import Mathlib.Order.RelIso.Basic\n\nNamespace:\nLeftOrdContinuous\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Johannes Hölzl\n-/\n/-!\n# Order continuity\n\nWe say that a function is *left order continuous* if it sends all least upper bounds\nto least upper bounds. The order dual notion is called *right order continuity*.\n\nFor monotone functions `ℝ → ℝ` these notions correspond to the usual left and right continuity.\n\nWe prove some basic lemmas (`map_sup`, `map_sSup` etc) and prove that a `RelIso` is both left\nand right order continuous.\n-/\n\n@[expose] public section\n\n\nuniverse u v w x\n\nvariable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}\n\nopen Function OrderDual Set\n\n/-!\n### Definitions\n-/\n\n\n/-- A function `f` between preorders is left order continuous if it preserves all suprema.  We\ndefine it using `IsLUB` instead of `sSup` so that the proof works both for complete lattices and\nconditionally complete lattices. -/\n@[to_dual\n/-- A function `f` between preorders is right order continuous if it preserves all infima.  We\ndefine it using `IsGLB` instead of `sInf` so that the proof works both for complete lattices and\nconditionally complete lattices. -/]\ndef LeftOrdContinuous [Preorder α] [Preorder β] (f : α → β) :=\n  ∀ ⦃s : Set α⦄ ⦃x⦄, IsLUB s x → IsLUB (f '' s) (f x)\n\nnamespace LeftOrdContinuous\n\nsection Preorder\n\nvariable (α) [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β}\n\n@[to_dual]\nprotected theorem id : LeftOrdContinuous (id : α → α) := fun s x h => by\n  simpa only [image_id] using! h\n\nvariable {α}\n\n@[to_dual]\nprotected theorem dual :\n    LeftOrdContinuous f → RightOrdContinuous (toDual ∘ f ∘ ofDual) :=\n  id\n\n@[deprecated (since := \"2026-04-08\")] alias rightOrdContinuous_dual := LeftOrdContinuous.dual\n\n@[deprecated (since := \"2026-04-08\")] alias _root_.RightOrdContinuous.orderDual :=\n  RightOrdContinuous.dual\n\n@[to_dual]\ntheorem map_isGreatest (hf : LeftOrdContinuous f) {s : Set α} {x : α} (h : IsGreatest s x) :\n    IsGreatest (f '' s) (f x) :=\n  ⟨mem_image_of_mem f h.1, (hf h.isLUB).1⟩\n\n@[to_dual]\ntheorem mono (hf : LeftOrdContinuous f) : Monotone f := fun a₁ a₂ h =>\n  have : IsGreatest {a₁, a₂} a₂ := ⟨Or.inr rfl, by simp [*]⟩\n  (hf.map_isGreatest this).2 <| mem_image_of_mem _ (Or.inl rfl)\n\n@[to_dual]\ntheorem comp (hg : LeftOrdContinuous g) (hf : LeftOrdContinuous f) : LeftOrdContinuous (g ∘ f) :=\n  fun s x h => by simpa only [image_image] using! hg (hf h)\n\n@[to_dual]\nprotected theorem iterate {f : α → α} (hf : LeftOrdContinuous f) (n : ℕ) :\n    LeftOrdContinuous f^[n] :=\n  match n with\n  | 0 => LeftOrdContinuous.id α\n  | (n + 1) => (LeftOrdContinuous.iterate hf n).comp hf\n\nend Preorder\n\nsection SemilatticeSup\n\nvariable [SemilatticeSup α] [SemilatticeSup β] {f : α → β}\n\n@[to_dual]\ntheorem map_sup (hf : LeftOrdContinuous f) (x y : α) : f (x ⊔ y) = f x ⊔ f y :=\n  (hf isLUB_pair).unique <| by simp only [image_pair, isLUB_pair]\n\n@[to_dual]\ntheorem le_iff (hf : LeftOrdContinuous f) (h : Injective f) {x y} : f x ≤ f y ↔ x ≤ y := by\n  simp only [← sup_eq_right, ← hf.map_sup, h.eq_iff]\n\n@[to_dual]\ntheorem lt_iff (hf : LeftOrdContinuous f) (h : Injective f) {x y} : f x < f y ↔ x < y := by\n  simp only [lt_iff_le_not_ge, hf.le_iff h]\n\nvariable (f)\n\n/-- Convert an injective left order continuous function to an order embedding. -/\n@[to_dual\n/-- Convert an injective right order continuous function to an order embedding. -/]\ndef toOrderEmbedding (hf : LeftOrdContinuous f) (h : Injective f) : α ↪o β :=\n  ⟨⟨f, h⟩, hf.le_iff h⟩\n\nvariable {f}\n\n@[to_dual (attr := simp)]\ntheorem coe_toOrderEmbedding (hf : LeftOrdContinuous f) (h : Injective f) :\n    ⇑(hf.toOrderEmbedding f h) = f :=\n  rfl\n\nend SemilatticeSup\n\nsection CompleteLattice\n\nvariable [CompleteLattice α] [CompleteLattice β] {f : α → β}\n\n@[to_dual]\ntheorem map_sSup' (hf : LeftOrdContinuous f) (s : Set α) : f (sSup s) = sSup (f '' s) :=\n  (hf <| isLUB_sSup s).sSup_eq.symm\n\n@[to_dual]\ntheorem map_sSup (hf : LeftOrdContinuous f) (s : Set α) : f (sSup s) = ⨆ x ∈ s, f x := by\n  rw [hf.map_sSup', sSup_image]\n\n@[to_dual]\n\nTarget:\ntheorem map_iSup (hf : LeftOrdContinuous f) (g : ι → α) : f (⨆ i, g i) = ⨆ i, f (g i) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"74f024850af0ff304abf422d1458554df0119526ff462bd70f18f633e6107c0c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order","family_id":"map_isup","file_id":"mathlib/Mathlib/Order/OrdContinuous.lean","sample_id":"8b95081cd7f651acb1a47e7a53ca218a713ae886e12ac7935b03bee7470a950f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bc3578532551ba1c8a4369ed77aebd3d2f1775c8fadb93080f656df6af2cd8fe","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f6fe57d0752e0cd229648ee764e154901c2c974592574e9f5ce3b35211409c9e","source_sha256":"2b139890caa5a415f5023de326a4c66559e073e900b5cf8f559558dc034a601c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have ⟨φ⟩ := ne\n  refine h.toIsAlgebraic.of_injective φ φ.injective\n    |>.normalClosure_eq_iSup_adjoin_of_splits fun x ↦ ?_\n  rw [← minpoly.algHom_eq _ φ.injective]\n  apply h.splits","hard_negative":false,"metrics":{"chosen_tokens":41,"rejected_tokens":2,"token_jaccard":0.033333,"token_length_ratio":0.04878},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"4db77acf22f2f5746d1fbcc6a12a9ab4f3d2dfae8eacd6623002eeb4be6def61","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.SimpleRing.Basic\npublic import Mathlib.FieldTheory.Normal.Basic\npublic import Mathlib.Order.Closure\npublic import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Thomas Browning. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Thomas Browning\n-/\n/-!\n# Normal closures\n\n## Main definitions\n\nGiven field extensions `K/F` and `L/F`, the predicate `IsNormalClosure F K L` says that the\nminimal polynomial of every element of `K` over `F` splits in `L`, and that `L` is generated\nby the roots of such minimal polynomials. These conditions uniquely characterize `L/F` up to\n`F`-algebra isomorphisms (`IsNormalClosure.equiv`).\n\nThe explicit construction `IntermediateField.normalClosure F K L` of a field extension `K/F`\ninside another field extension `L/F` is the smallest intermediate field of\n`L/F` that contains the image of every `F`-algebra embedding `K →ₐ[F] L`.\nIt satisfies the `IsNormalClosure` predicate if `L/F` satisfies the\nabovementioned splitting condition, in particular if `L/K/F` form a tower and\n`L/F` is normal.\n-/\n\n@[expose] public section\n\nopen IntermediateField IsScalarTower Polynomial\n\nvariable (F K L : Type*) [Field F] [Field K] [Field L] [Algebra F K] [Algebra F L]\n\n/-- `L/F` is a normal closure of `K/F` if the minimal polynomial of every element of `K` over `F`\n  splits in `L`, and `L` is generated by roots of such minimal polynomials over `F`.\n  (Since the minimal polynomial of a transcendental element is 0,\n  the normal closure of `K/F` is the same as the normal closure over `F`\n  of the algebraic closure of `F` in `K`.) -/\n@[stacks 0BMF \"Predicate version\"]\nclass IsNormalClosure : Prop where\n  splits (x : K) : ((minpoly F x).map (algebraMap F L)).Splits\n  adjoin_rootSet : ⨆ x : K, adjoin F ((minpoly F x).rootSet L) = ⊤\n/- TODO: show `IsNormalClosure F K L ↔ IsNormalClosure F (integralClosure F K) L`; we can't state\n  this yet because `integralClosure F K` needs to have a `Field` instance. -/\n\n/-- The normal closure of `K/F` in `L/F`. -/\n@[stacks 0BMF]\nnoncomputable def IntermediateField.normalClosure : IntermediateField F L :=\n  ⨆ f : K →ₐ[F] L, f.fieldRange\n\nlemma normalClosure_def : normalClosure F K L = ⨆ f : K →ₐ[F] L, f.fieldRange :=\n  rfl\n\nvariable {F K L}\n\n/-- A normal closure is always normal. -/\nlemma IsNormalClosure.normal [h : IsNormalClosure F K L] : Normal F L :=\n  Normal.of_algEquiv topEquiv (h := h.adjoin_rootSet ▸ IntermediateField.normal_iSup (h :=\n    fun _ ↦ Normal.of_isSplittingField (hFEp := adjoin_rootSet_isSplittingField <| h.splits _)))\n\nlemma normalClosure_le_iff {K' : IntermediateField F L} :\n    normalClosure F K L ≤ K' ↔ ∀ f : K →ₐ[F] L, f.fieldRange ≤ K' :=\n  iSup_le_iff\n\nlemma AlgHom.fieldRange_le_normalClosure (f : K →ₐ[F] L) : f.fieldRange ≤ normalClosure F K L :=\n  le_iSup AlgHom.fieldRange f\n\nnamespace Algebra.IsAlgebraic\nvariable [Algebra.IsAlgebraic F K]\n\nlemma normalClosure_le_iSup_adjoin :\n    normalClosure F K L ≤ ⨆ x : K, IntermediateField.adjoin F ((minpoly F x).rootSet L) :=\n  iSup_le fun f _ ⟨x, hx⟩ ↦ le_iSup (α := IntermediateField F L) _ x <|\n    IntermediateField.subset_adjoin F _ <| by\n      rw [mem_rootSet_of_ne (minpoly.ne_zero (Algebra.IsIntegral.isIntegral x)), ← hx,\n        AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom, aeval_algHom_apply, minpoly.aeval, map_zero]\n\nvariable (splits : ∀ x : K, ((minpoly F x).map (algebraMap F L)).Splits)\n\ninclude splits in\nlemma normalClosure_eq_iSup_adjoin_of_splits :\n    normalClosure F K L = ⨆ x : K, IntermediateField.adjoin F ((minpoly F x).rootSet L) :=\n  normalClosure_le_iSup_adjoin.antisymm <|\n    iSup_le fun x ↦ IntermediateField.adjoin_le_iff.mpr fun _ hy ↦\n      let ⟨φ, hφ⟩ := IntermediateField.exists_algHom_of_splits_of_aeval\n        (fun x ↦ ⟨Algebra.IsIntegral.isIntegral x, splits x⟩) (mem_rootSet.mp hy).2\n      le_iSup AlgHom.fieldRange φ ⟨x, hφ⟩\n\n/-- If `K/F` is algebraic, the \"generated by roots\" condition in IsNormalClosure can be replaced\n  by \"generated by images of embeddings\". -/\nlemma isNormalClosure_iff : IsNormalClosure F K L ↔\n    (∀ x : K, ((minpoly F x).map (algebraMap F L)).Splits) ∧ normalClosure F K L = ⊤ := by\n  refine ⟨fun ⟨splits, h⟩ ↦ ⟨splits, ?_⟩, fun ⟨splits, h⟩ ↦ ⟨splits, ?_⟩⟩ <;>\n    simpa only [normalClosure_eq_iSup_adjoin_of_splits splits] using h\n-- TODO: IntermediateField.isNormalClosure_iff similar to IntermediateField.isSplittingField_iff\n\nset_option backward.isDefEq.respectTransparency false in\ninclude splits in\n/-- `normalClosure F K L` is a valid normal closure if `K/F` is algebraic\n  and all minimal polynomials of `K/F` splits in `L/F`. -/\nlemma isNormalClosure_normalClosure : IsNormalClosure F K (normalClosure F K L) := by\n  rw [isNormalClosure_iff]; constructor\n  · rw [normalClosure_eq_iSup_adjoin_of_splits splits]\n    exact fun x ↦ splits_of_splits (splits x) ((IntermediateField.subset_adjoin F _).trans <|\n      SetLike.coe_subset_coe.mpr <| by apply le_iSup _ x)\n  simp_rw [normalClosure, ← top_le_iff]\n  refine fun x _ ↦ ((⨆ f : K →ₐ[F] L, f.fieldRange).val).injective.mem_set_image |>.mp ?_\n  rw [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, coe_val, ← IntermediateField.coe_val,\n    ← IntermediateField.coe_map, IntermediateField.map_iSup]\n  refine (iSup_le fun f ↦ ?_ : normalClosure F K L ≤ _) x.2\n  refine le_iSup_of_le (f.codRestrict _ fun x ↦ f.fieldRange_le_normalClosure ⟨x, rfl⟩) ?_\n  rw [AlgHom.map_fieldRange, val, AlgHom.val_comp_codRestrict]\n\nend Algebra.IsAlgebraic\n\n/-- A normal closure of `K/F` embeds into any `L/F`\n  where the minimal polynomials of `K/F` splits. -/\nnoncomputable def IsNormalClosure.lift [h : IsNormalClosure F K L] {L'} [Field L'] [Algebra F L']\n    (splits : ∀ x : K, ((minpoly F x).map (algebraMap F L')).Splits) : L →ₐ[F] L' := by\n  have := h.adjoin_rootSet; rw [← gc.l_iSup] at this\n  refine Nonempty.some <| nonempty_algHom_of_adjoin_splits\n    (fun x hx ↦ ⟨isAlgebraic_iff_isIntegral.mp ((h.normal).isAlgebraic x), ?_⟩) this\n  obtain ⟨y, hx⟩ := Set.mem_iUnion.mp hx\n  by_cases iy : IsIntegral F y\n  · exact (splits y).of_dvd (map_ne_zero (minpoly.ne_zero iy))\n      ((map_dvd_map' _).mpr (minpoly.dvd F x (mem_rootSet.mp hx).2))\n  · simp [minpoly.eq_zero iy] at hx\n\n/-- Normal closures of `K/F` are unique up to F-algebra isomorphisms. -/\nnoncomputable def IsNormalClosure.equiv {L'} [Field L'] [Algebra F L']\n    [h : IsNormalClosure F K L] [h' : IsNormalClosure F K L'] : L ≃ₐ[F] L' :=\n  have := h.normal\n  AlgEquiv.ofBijective _ <| And.left <|\n    Normal.toIsAlgebraic.algHom_bijective₂\n      (IsNormalClosure.lift fun _ : K ↦ h'.splits _)\n      (IsNormalClosure.lift fun _ : K ↦ h.splits _)\n\nvariable (F K L)\n\ninstance isNormalClosure_normalClosure [ne : Nonempty (K →ₐ[F] L)] [h : Normal F L] :\n    IsNormalClosure F K (normalClosure F K L) := by\n  have ⟨φ⟩ := ne\n  apply (h.toIsAlgebraic.of_injective φ φ.injective).isNormalClosure_normalClosure\n  simp_rw [← minpoly.algHom_eq _ φ.injective]\n  exact fun _ ↦ h.splits _\n\nTarget:\ntheorem normalClosure_eq_iSup_adjoin' [ne : Nonempty (K →ₐ[F] L)] [h : Normal F L] :\n    normalClosure F K L = ⨆ x : K, adjoin F ((minpoly F x).rootSet L) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Normal","family_id":"normalclosure_eq_isup_adjoin","file_id":"mathlib/Mathlib/FieldTheory/Normal/Closure.lean","sample_id":"f6fe57d0752e0cd229648ee764e154901c2c974592574e9f5ce3b35211409c9e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"24379af29dd6a4b6bd06c4a471ab0fb06ebf44b4f629deaf2e40a3cfa29440d2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2f3517d963ac31c8db62c299de697bf8f1869181ac21164e6d958e1f5d7f08dd","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c9a34ccd888b6cf8e9fb2f8c6ffab0caeeb794e571e1779f7f95f329d2895ac6","source_sha256":"e2f1f2e70ec157c61fd85d27c525b29293c57c501ce3057927e134e50f754326","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [← card_univ, ← card_powersetCard]\n  refine (card_le_card fun s hs ↦ mem_biUnion.2 ⟨#s, ?_⟩).trans card_biUnion_le\n  exact ⟨mem_Iic.2 (mem_shatterer.1 hs).card_le_vcDim, mem_powersetCard_univ.2 rfl⟩","hard_negative":false,"metrics":{"chosen_tokens":49,"rejected_tokens":54,"token_jaccard":0.891892,"token_length_ratio":1.102041},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"4dd85ced0cc56d70bfa80e250ff52949c905f4f67b753079d05d7492fee7cfbb","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.SetFamily.Compression.Down\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Order.Interval.Finset.Nat\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Shattering families\n\nThis file defines the shattering property and VC-dimension of set families.\n\n## Main declarations\n\n* `Finset.Shatters`: The shattering property.\n* `Finset.shatterer`: The set family of sets shattered by a set family.\n* `Finset.vcDim`: The Vapnik-Chervonenkis dimension.\n\n## TODO\n\n* Order-shattering\n* Strong shattering\n-/\n\n@[expose] public section\n\nopen scoped FinsetFamily\n\nnamespace Finset\nvariable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s t : Finset α} {a : α}\n\n/-- A set family `𝒜` shatters a set `s` if all subsets of `s` can be obtained as the intersection\nof `s` and some element of the set family, and we denote this `𝒜.Shatters s`. We also say that `s`\nis *traced* by `𝒜`. -/\ndef Shatters (𝒜 : Finset (Finset α)) (s : Finset α) : Prop := ∀ ⦃t⦄, t ⊆ s → ∃ u ∈ 𝒜, s ∩ u = t\n\ninstance : DecidablePred 𝒜.Shatters := fun _s ↦ decidableForallOfDecidableSubsets\n\nlemma Shatters.exists_inter_eq_singleton (hs : Shatters 𝒜 s) (ha : a ∈ s) : ∃ t ∈ 𝒜, s ∩ t = {a} :=\n  hs <| singleton_subset_iff.2 ha\n\nlemma Shatters.mono_left (h : 𝒜 ⊆ ℬ) (h𝒜 : 𝒜.Shatters s) : ℬ.Shatters s :=\n  fun _t ht ↦ let ⟨u, hu, hut⟩ := h𝒜 ht; ⟨u, h hu, hut⟩\n\nlemma Shatters.mono_right (h : t ⊆ s) (hs : 𝒜.Shatters s) : 𝒜.Shatters t := fun u hu ↦ by\n  obtain ⟨v, hv, rfl⟩ := hs (hu.trans h); exact ⟨v, hv, inf_congr_right hu <| inf_le_of_left_le h⟩\n\nlemma Shatters.exists_superset (h : 𝒜.Shatters s) : ∃ t ∈ 𝒜, s ⊆ t :=\n  let ⟨t, ht, hst⟩ := h Subset.rfl; ⟨t, ht, inter_eq_left.1 hst⟩\n\nlemma shatters_of_forall_subset (h : ∀ t, t ⊆ s → t ∈ 𝒜) : 𝒜.Shatters s :=\n  fun t ht ↦ ⟨t, h _ ht, inter_eq_right.2 ht⟩\n\nprotected lemma Shatters.nonempty (h : 𝒜.Shatters s) : 𝒜.Nonempty :=\n  let ⟨t, ht, _⟩ := h Subset.rfl; ⟨t, ht⟩\n\n@[simp] lemma shatters_empty : 𝒜.Shatters ∅ ↔ 𝒜.Nonempty :=\n  ⟨Shatters.nonempty, fun ⟨s, hs⟩ t ht ↦ ⟨s, hs, by rwa [empty_inter, eq_comm, ← subset_empty]⟩⟩\n\nprotected lemma Shatters.subset_iff (h : 𝒜.Shatters s) : t ⊆ s ↔ ∃ u ∈ 𝒜, s ∩ u = t :=\n  ⟨fun ht ↦ h ht, by rintro ⟨u, _, rfl⟩; exact inter_subset_left⟩\n\nlemma shatters_iff : 𝒜.Shatters s ↔ 𝒜.image (fun t ↦ s ∩ t) = s.powerset :=\n  ⟨fun h ↦ by ext t; rw [mem_image, mem_powerset, h.subset_iff],\n    fun h t ht ↦ by rwa [← mem_powerset, ← h, mem_image] at ht⟩\n\nlemma univ_shatters [Fintype α] : univ.Shatters s :=\n  shatters_of_forall_subset fun _ _ ↦ mem_univ _\n\n@[simp] lemma shatters_univ [Fintype α] : 𝒜.Shatters univ ↔ 𝒜 = univ := by\n  rw [shatters_iff, powerset_univ]; simp_rw [univ_inter, image_id']\n\n/-- The set family of sets that are shattered by `𝒜`. -/\ndef shatterer (𝒜 : Finset (Finset α)) : Finset (Finset α) :=\n  {s ∈ 𝒜.biUnion powerset | 𝒜.Shatters s}\n\n@[simp] lemma mem_shatterer : s ∈ 𝒜.shatterer ↔ 𝒜.Shatters s := by\n  refine mem_filter.trans <| and_iff_right_of_imp fun h ↦ ?_\n  simp_rw [mem_biUnion, mem_powerset]\n  exact h.exists_superset\n\n@[gcongr] lemma shatterer_mono (h : 𝒜 ⊆ ℬ) : 𝒜.shatterer ⊆ ℬ.shatterer :=\n  fun _ ↦ by simpa using Shatters.mono_left h\n\nlemma subset_shatterer (h : IsLowerSet (𝒜 : Set (Finset α))) : 𝒜 ⊆ 𝒜.shatterer :=\n  fun _s hs ↦ mem_shatterer.2 fun t ht ↦ ⟨t, h ht hs, inter_eq_right.2 ht⟩\n\n@[simp] lemma isLowerSet_shatterer (𝒜 : Finset (Finset α)) :\n    IsLowerSet (𝒜.shatterer : Set (Finset α)) := fun s t ↦ by simpa using Shatters.mono_right\n\n@[simp] lemma shatterer_eq : 𝒜.shatterer = 𝒜 ↔ IsLowerSet (𝒜 : Set (Finset α)) := by\n  refine ⟨fun h ↦ ?_, fun h ↦ Subset.antisymm (fun s hs ↦ ?_) <| subset_shatterer h⟩\n  · rw [← h]\n    exact isLowerSet_shatterer _\n  · obtain ⟨t, ht, hst⟩ := (mem_shatterer.1 hs).exists_superset\n    exact h hst ht\n\n@[simp] lemma shatterer_idem : 𝒜.shatterer.shatterer = 𝒜.shatterer := by simp\n\n@[simp] lemma shatters_shatterer : 𝒜.shatterer.Shatters s ↔ 𝒜.Shatters s := by\n  simp_rw [← mem_shatterer, shatterer_idem]\n\nprotected alias ⟨_, Shatters.shatterer⟩ := shatters_shatterer\n\nprivate lemma aux (h : ∀ t ∈ 𝒜, a ∉ t) (ht : 𝒜.Shatters t) : a ∉ t := by\n  obtain ⟨u, hu, htu⟩ := ht.exists_superset; exact notMem_mono htu <| h u hu\n\n/-- Pajor's variant of the **Sauer-Shelah lemma**. -/\nlemma card_le_card_shatterer (𝒜 : Finset (Finset α)) : #𝒜 ≤ #𝒜.shatterer := by\n  refine memberFamily_induction_on 𝒜 ?_ ?_ ?_\n  · simp\n  · rfl\n  intro a 𝒜 ih₀ ih₁\n  set ℬ : Finset (Finset α) :=\n    ((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer).image (insert a)\n  have hℬ : #ℬ = #((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer) := by\n    refine card_image_of_injOn <| insert_erase_invOn.2.injOn.mono ?_\n    simp only [coe_inter, Set.subset_def, Set.mem_inter_iff, mem_coe, Set.mem_setOf_eq, and_imp,\n      mem_shatterer]\n    exact fun s _ ↦ aux (fun t ht ↦ (mem_filter.1 ht).2)\n  rw [← card_memberSubfamily_add_card_nonMemberSubfamily a]\n  refine (Nat.add_le_add ih₁ ih₀).trans ?_\n  rw [← card_union_add_card_inter, ← hℬ, ← card_union_of_disjoint]\n  swap\n  · simp only [ℬ, disjoint_left, mem_union, mem_shatterer, mem_image, not_exists, not_and]\n    rintro _ (hs | hs) s - rfl\n    · exact aux (fun t ht ↦ (mem_memberSubfamily.1 ht).2) hs <| mem_insert_self _ _\n    · exact aux (fun t ht ↦ (mem_nonMemberSubfamily.1 ht).2) hs <| mem_insert_self _ _\n  refine card_mono <| union_subset (union_subset ?_ <| shatterer_mono <| filter_subset _ _) ?_\n  · simp only [subset_iff, mem_shatterer]\n    rintro s hs t ht\n    obtain ⟨u, hu, rfl⟩ := hs ht\n    rw [mem_memberSubfamily] at hu\n    refine ⟨insert a u, hu.1, inter_insert_of_notMem fun ha ↦ ?_⟩\n    obtain ⟨v, hv, hsv⟩ := hs.exists_inter_eq_singleton ha\n    rw [mem_memberSubfamily] at hv\n    rw [← singleton_subset_iff (a := a), ← hsv] at hv\n    exact hv.2 inter_subset_right\n  · refine forall_mem_image.2 fun s hs ↦ mem_shatterer.2 fun t ht ↦ ?_\n    simp only [mem_inter, mem_shatterer] at hs\n    rw [subset_insert_iff] at ht\n    by_cases ha : a ∈ t\n    · obtain ⟨u, hu, hsu⟩ := hs.1 ht\n      rw [mem_memberSubfamily] at hu\n      refine ⟨_, hu.1, ?_⟩\n      rw [← insert_inter_distrib, hsu, insert_erase ha]\n    · obtain ⟨u, hu, hsu⟩ := hs.2 ht\n      rw [mem_nonMemberSubfamily] at hu\n      refine ⟨_, hu.1, ?_⟩\n      rwa [insert_inter_of_notMem hu.2, hsu, erase_eq_self]\n\nlemma Shatters.of_compression (hs : (𝓓 a 𝒜).Shatters s) : 𝒜.Shatters s := by\n  intro t ht\n  obtain ⟨u, hu, rfl⟩ := hs ht\n  rw [Down.mem_compression] at hu\n  obtain hu | hu := hu\n  · exact ⟨u, hu.1, rfl⟩\n  by_cases ha : a ∈ s\n  · obtain ⟨v, hv, hsv⟩ := hs <| insert_subset ha ht\n    rw [Down.mem_compression] at hv\n    obtain hv | hv := hv\n    · refine ⟨erase v a, hv.2, ?_⟩\n      rw [inter_erase, hsv, erase_insert]\n      rintro ha\n      rw [insert_eq_self.2 (mem_inter.1 ha).2] at hu\n      exact hu.1 hu.2\n    rw [insert_eq_self.2 <| inter_subset_right (s₁ := s) ?_] at hv\n    cases hv.1 hv.2\n    rw [hsv]\n    exact mem_insert_self _ _\n  · refine ⟨insert a u, hu.2, ?_⟩\n    rw [inter_insert_of_notMem ha]\n\nlemma shatterer_compress_subset_shatterer (a : α) (𝒜 : Finset (Finset α)) :\n    (𝓓 a 𝒜).shatterer ⊆ 𝒜.shatterer := by\n  simp only [subset_iff, mem_shatterer]; exact fun s hs ↦ hs.of_compression\n\n/-! ### Vapnik-Chervonenkis dimension -/\n\n/-- The Vapnik-Chervonenkis dimension of a set family is the maximal size of a set it shatters. -/\ndef vcDim (𝒜 : Finset (Finset α)) : ℕ := 𝒜.shatterer.sup card\n\n@[gcongr] lemma vcDim_mono (h𝒜ℬ : 𝒜 ⊆ ℬ) : 𝒜.vcDim ≤ ℬ.vcDim := by unfold vcDim; gcongr\n\nlemma Shatters.card_le_vcDim (hs : 𝒜.Shatters s) : #s ≤ 𝒜.vcDim := le_sup <| mem_shatterer.2 hs\n\n/-- Down-compressing decreases the VC-dimension. -/\nlemma vcDim_compress_le (a : α) (𝒜 : Finset (Finset α)) : (𝓓 a 𝒜).vcDim ≤ 𝒜.vcDim :=\n  sup_mono <| shatterer_compress_subset_shatterer _ _\n\n/-- The **Sauer-Shelah lemma**. -/\n\nTarget:\nlemma card_shatterer_le_sum_vcDim [Fintype α] :\n    #𝒜.shatterer ≤ ∑ k ∈ Iic 𝒜.vcDim, (Fintype.card α).choose k :=\n\nProof body:\n","rejected":"by\n  simp_rw [← card_univ, ← card_powersetCard]\n  refine (card_le_card fun s hs ↦ mem_biUnion.2 ⟨#s, ?_⟩).trans card_biUnion_le\n  exact ⟨mem_Iic.2 (mem_shatterer.1 hs).card_le_vcDim, mem_powersetCard_univ.2 rfl⟩\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/SetFamily","family_id":"card_shatterer_le_sum_vcdim","file_id":"mathlib/Mathlib/Combinatorics/SetFamily/Shatter.lean","sample_id":"c9a34ccd888b6cf8e9fb2f8c6ffab0caeeb794e571e1779f7f95f329d2895ac6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c91bd3015c099e4be465fbcb7168b44d0092e9266e1434d35ffc1f51e248e636","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4c4e9dcac1de909b93b46f79fa507e9e65d30650d636a7be44d6d6797114f6e3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"befab83d2930909beac8cfad141326e33a75bf337f03f5f78fa5a152c0c13634","source_sha256":"f8317928b1cbc96407d5af706b485b7c6547cb67a94dbd8e0bac5c444710ec85","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  obtain ⟨I, hIC, hI, hst⟩ := hC.sdiff_eq_sUnion' s hs t ht\n  refine ⟨.ofErase I (supIndep_iff_pairwiseDisjoint.mpr hI) ?_, ?_⟩\n  · rw [sup_id_eq_sSup, sSup_eq_sUnion, hst]\n  · grw [Finpartition.ofErase_parts, Finset.erase_subset, hIC]","hard_negative":true,"metrics":{"chosen_tokens":59,"rejected_tokens":3,"token_jaccard":0.025641,"token_length_ratio":0.050847},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"4def5ada9fde120ec6ba300161a75872547282271659bc6b0aa71be041f58a13","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Pairwise.Lattice\npublic import Mathlib.MeasureTheory.PiSystem\npublic import Mathlib.Order.Lattice.Nat\npublic import Mathlib.Order.Partition.Finpartition\npublic import Mathlib.Order.SetAccumulate\npublic import Mathlib.Order.SupClosed\n\nNamespace:\nMeasureTheory.IsSetSemiring\n\nLocal context:\n/-\nCopyright (c) 2023 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne, Peter Pfaffelhuber\n-/\n/-! # Semirings and rings of sets\n\nA semi-ring of sets `C` (in the sense of measure theory) is a family of sets containing `∅`,\nstable by intersection and such that for all `s, t ∈ C`, `t \\ s` is equal to a disjoint union of\nfinitely many sets in `C`. Note that a semi-ring of sets may not contain unions.\n\nAn important example of a semi-ring of sets is intervals in `ℝ`. The intersection of two intervals\nis an interval (possibly empty). The union of two intervals may not be an interval.\nThe set difference of two intervals may not be an interval, but it will be a disjoint union of\ntwo intervals.\n\nA ring of sets is a set of sets containing `∅`, stable by union, set difference and intersection.\n\n## Main definitions\n\n* `MeasureTheory.IsSetSemiring C`: property of being a semi-ring of sets.\n* `MeasureTheory.IsSetSemiring.disjointOfDiff hs ht`: for `s, t` in a semi-ring `C`\n  (with `hC : IsSetSemiring C`) with `hs : s ∈ C`, `ht : t ∈ C`, this is a `Finset` of\n  pairwise disjoint sets such that `s \\ t = ⋃₀ hC.disjointOfDiff hs ht`.\n* `MeasureTheory.IsSetSemiring.disjointOfDiffUnion hs hI`: for `hs : s ∈ C` and a finset\n  `I` of sets in `C` (with `hI : ↑I ⊆ C`), this is a `Finset` of pairwise disjoint sets such that\n  `s \\ ⋃₀ I = ⋃₀ hC.disjointOfDiffUnion hs hI`.\n* `MeasureTheory.IsSetSemiring.disjointOfUnion hJ`: for `hJ ⊆ C`, this is a\n  `Finset` of pairwise disjoint sets such that `⋃₀ J = ⋃₀ hC.disjointOfUnion hJ`.\n\n* `MeasureTheory.IsSetRing`: property of being a ring of sets.\n\n## Main statements\n\n* `MeasureTheory.IsSetSemiring.exists_disjoint_finset_sdiff_eq`: the existence of the `Finset` given\n  by the definition `IsSetSemiring.disjointOfDiffUnion` (see above).\n* `MeasureTheory.IsSetSemiring.disjointOfUnion_props`: In a `hC : IsSetSemiring C`,\n  for a `J : Finset (Set α)` with `J ⊆ C`, there is\n  for every `x in J` some `K x ⊆ C` finite, such that\n  * `⋃ x ∈ J, K x` are pairwise disjoint and do not contain ∅,\n  * `⋃ s ∈ K x, s ⊆ x`,\n  * `⋃ x ∈ J, x = ⋃ x ∈ J, ⋃ s ∈ K x, s`.\n\n-/\n\n@[expose] public section\n\nopen Finset Set\n\nnamespace MeasureTheory\n\nvariable {α : Type*} {C : Set (Set α)} {s t : Set α}\n\n/-- A semi-ring of sets `C` is a family of sets containing `∅`, stable by intersection and such that\nfor all `s, t ∈ C`, `s \\ t` is equal to a disjoint union of finitely many sets in `C`. -/\nstructure IsSetSemiring (C : Set (Set α)) : Prop where\n  empty_mem : ∅ ∈ C\n  inter_mem : ∀ s ∈ C, ∀ t ∈ C, s ∩ t ∈ C\n  sdiff_eq_sUnion' : ∀ s ∈ C, ∀ t ∈ C,\n    ∃ I : Finset (Set α), ↑I ⊆ C ∧ PairwiseDisjoint (I : Set (Set α)) id ∧ s \\ t = ⋃₀ I\n\n/-- A ring of sets `C` is a family of sets containing `∅`, stable by union and set difference.\nIt is then also stable by intersection (see `IsSetRing.inter_mem`). -/\nstructure IsSetRing (C : Set (Set α)) : Prop where\n  empty_mem : ∅ ∈ C\n  union_mem ⦃s t : Set α⦄ : s ∈ C → t ∈ C → s ∪ t ∈ C\n  sdiff_mem ⦃s t : Set α⦄ : s ∈ C → t ∈ C → s \\ t ∈ C\n\nnamespace IsSetSemiring\n\nlemma isPiSystem (hC : IsSetSemiring C) : IsPiSystem C := fun s hs t ht _ ↦ hC.inter_mem s hs t ht\n\nTarget:\ntheorem exists_finpartition_sdiff (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :\n    ∃ P : Finpartition (s \\ t), ↑P.parts ⊆ C :=\n\nProof body:\n","rejected":"by\n  exact exists_finpartition_sdiff","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"16ca00327996b3a9f09ff593366e17352c36ff58f290e435c3d0d360692ab0c6","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory","family_id":"exists_finpartition_sdiff","file_id":"mathlib/Mathlib/MeasureTheory/SetSemiring.lean","sample_id":"befab83d2930909beac8cfad141326e33a75bf337f03f5f78fa5a152c0c13634"},"split":"train"}
{"binding":{"chosen_proof_sha256":"55840963317e3a0d749bfa4ab4253cf6959feaf799aad1f633d62acc28d49b26","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"17501c489548fc579d16232355b4d42b561e12290026de55f9caf2144694a108","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a1e462f08764c6d42623fb769d4457124c492b34cbacfef092d7509dfdd31de7","source_sha256":"02df2dcc4ba0253814e33afb4b0bcd9e1c8a127adf2c967cb9666cee9d9d5264","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by valid) : F.map' i i = 𝟙 _ := F.map_id _","hard_negative":true,"metrics":{"chosen_tokens":17,"rejected_tokens":2,"token_jaccard":0.071429,"token_length_ratio":0.117647},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"4e1057157db780ad8cf263856fa9b13db945c0dc105525f927b69bb1a8f189df","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Nat.Defs\npublic import Mathlib.CategoryTheory.Category.Preorder\npublic import Mathlib.CategoryTheory.Comma.Arrow\npublic import Mathlib.Data.Fintype.Basic\npublic import Mathlib.Tactic.FinCases\npublic import Mathlib.Tactic.SuppressCompilation\n\nNamespace:\nCategoryTheory.ComposableArrows\n\nLocal context:\n/-\nCopyright (c) 2023 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Composable arrows\n\nIf `C` is a category, the type of `n`-simplices in the nerve of `C` identifies\nto the type of functors `Fin (n + 1) ⥤ C`, which can be thought of as families of `n` composable\narrows in `C`. In this file, we introduce and study this category `ComposableArrows C n`\nof `n` composable arrows in `C`.\n\nIf `F : ComposableArrows C n`, we define `F.left` as the leftmost object, `F.right` as the\nrightmost object, and `F.hom : F.left ⟶ F.right` is the canonical map.\n\nThe most significant definition in this file is the constructor\n`F.precomp f : ComposableArrows C (n + 1)` for `F : ComposableArrows C n` and `f : X ⟶ F.left`:\n\"it shifts `F` towards the right and inserts `f` on the left\". This `precomp` has\ngood definitional properties.\n\nIn the namespace `CategoryTheory.ComposableArrows`, we provide constructors\nlike `mk₁ f`, `mk₂ f g`, `mk₃ f g h` for `ComposableArrows C n` for small `n`.\n\nTODO (@joelriou):\n* construct some elements in `ComposableArrows m (Fin (n + 1))` for small `n`\n  the precomposition with which shall induce functors\n  `ComposableArrows C n ⥤ ComposableArrows C m` which correspond to simplicial operations\n  (specifically faces) with good definitional properties (this might be necessary for\n  up to `n = 7` in order to formalize spectral sequences following Verdier)\n\n-/\n\nset_option backward.defeqAttrib.useBackward true\n\n@[expose] public section\n\nset_option backward.privateInPublic true\n\n/-!\nNew `simprocs` that run even in `dsimp` have caused breakages in this file.\n\n(e.g. `dsimp` can now simplify `2 + 3` to `5`)\n\nFor now, we just turn off the offending simprocs in this file.\n\n*However*, hopefully it is possible to refactor the material here so that no disabling of\nsimprocs is needed.\n\nSee issue https://github.com/leanprover-community/mathlib4/issues/27382.\n-/\nattribute [-simp] Fin.reduceFinMk\n\nnamespace CategoryTheory\n\nopen Category\n\nvariable (C : Type*) [Category* C]\n\n/-- `ComposableArrows C n` is the type of functors `Fin (n + 1) ⥤ C`. -/\nabbrev ComposableArrows (n : ℕ) := Fin (n + 1) ⥤ C\n\nnamespace ComposableArrows\n\nvariable {C} {n m : ℕ}\nvariable (F G : ComposableArrows C n)\n\n-- We do not yet replace `omega` with `lia` here, as it is measurably slower.\n/-- A wrapper for `omega` which prefaces it with some quick and useful attempts -/\nmacro \"valid\" : tactic =>\n  `(tactic| first | assumption | apply zero_le | apply le_rfl | transitivity <;> assumption | omega)\n\n/-- The `i`th object (with `i : ℕ` such that `i ≤ n`) of `F : ComposableArrows C n`. -/\n@[simp]\nabbrev obj' (i : ℕ) (hi : i ≤ n := by valid) : C := F.obj ⟨i, by lia⟩\n\n/-- The map `F.obj' i ⟶ F.obj' j` when `F : ComposableArrows C n`, and `i` and `j`\nare natural numbers such that `i ≤ j ≤ n`. -/\n@[simp]\nabbrev map' (i j : ℕ) (hij : i ≤ j := by valid) (hjn : j ≤ n := by valid) :\n    F.obj ⟨i, by lia⟩ ⟶ F.obj ⟨j, by lia⟩ :=\n  F.map (homOfLE (by simp only [Fin.mk_le_mk]; valid))\n\nTarget:\nlemma map'_self (i : ℕ) (hi : i ≤ n :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_a1e462f08764","rejection":{"accepted":false,"failure_type":"unknown_tactic","output_sha256":"4e32ad2f0af4a6e9a38133d76b55b664f63c636db97a765122d4cc21f8be2e4a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/ComposableArrows","family_id":"map'_self","file_id":"mathlib/Mathlib/CategoryTheory/ComposableArrows/Basic.lean","sample_id":"a1e462f08764c6d42623fb769d4457124c492b34cbacfef092d7509dfdd31de7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0c64ae53f4b1abd57d7d7d427b1d8d5ae6a8abade34b2f455335c0dfb08d0184","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f05312ee1b649b954d0ad3d3cedaebdac4a9c0766690dacc1ff7776f9463beba","source_sha256":"0a178ec9526bb9830472977cb3b19da5015b9b3548b786a523adc3dcb0786526","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [subsingleton_iff_support_subset, PrimeSpectrum.basicOpen_eq_zeroLocus_compl,\n    disjoint_compl_left_iff, Set.le_iff_subset]","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":5,"token_jaccard":0.0625,"token_length_ratio":0.333333},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"4e27cddb31488dc9013d8680742a034a762b51f3c1eaecb5c4fb4dec4b1e2b20","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Spectrum.Prime.Topology\npublic import Mathlib.RingTheory.Support\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Subsets of prime spectra related to modules\n\n## Main results\n\n- `LocalizedModule.subsingleton_iff_disjoint` : `M[1/f] = 0 ↔ D(f) ∩ Supp M = 0`.\n- `Module.isClosed_support` : If `M` is a finite `R`-module, then `Supp M` is closed.\n\n## TODO\n- If `M` is finitely presented, the complement of `Supp M` is quasi-compact. (stacks#051B)\n\n-/\n\npublic section\n\nvariable {R A M : Type*} [CommRing R] [AddCommGroup M] [Module R M]\n  [CommRing A] [Algebra R A] [Module A M]\n\nvariable (R M) in\nlemma IsLocalRing.closedPoint_mem_support [IsLocalRing R] [Nontrivial M] :\n    IsLocalRing.closedPoint R ∈ Module.support R M := by\n  obtain ⟨p, hp⟩ := (Module.nonempty_support_iff (R := R)).mpr ‹_›\n  exact Module.mem_support_mono le_top hp\n\n/-- `M[1/f] = 0` if and only if `D(f) ∩ Supp M = 0`. -/\n\nTarget:\nlemma LocalizedModule.subsingleton_iff_disjoint {f : R} :\n    Subsingleton (LocalizedModule.Away f M) ↔\n      Disjoint ↑(PrimeSpectrum.basicOpen f) (Module.support R M) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Spectrum","family_id":"localizedmodule","file_id":"mathlib/Mathlib/RingTheory/Spectrum/Prime/Module.lean","sample_id":"f05312ee1b649b954d0ad3d3cedaebdac4a9c0766690dacc1ff7776f9463beba"},"split":"train"}
{"binding":{"chosen_proof_sha256":"861cfb34c8727a0d4d437e7097497ea9d3f14804aa184568a47ee591868cf5bf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"87030b969f81b483b5bdfa5c63449397f06437d95959ae3498323790baf68ce1","source_sha256":"cd917cebf3f69c14dd2596343577ddfa07783b1b73bd4e0e91a20b1a1be30f59","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Equiv.eq_symm_comp (e := Quiver.symmetrifyStar (φ.obj u))]\n  ext ⟨v, f | g⟩ <;>\n    -- Porting note (https://github.com/leanprover-community/mathlib4/issues/10745): was `simp [Quiver.symmetrifyStar]`\n    simp only [Quiver.symmetrifyStar, Function.comp_apply] <;>\n    erw [Equiv.sigmaSumDistrib_apply, Equiv.sigmaSumDistrib_apply] <;>\n    simp","hard_negative":true,"metrics":{"chosen_tokens":96,"rejected_tokens":8,"token_jaccard":0.053571,"token_length_ratio":0.083333},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"4eb289c4654787f230f26312c46d6db24b48198cd1802389b837ce7e66b898de","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Quiver.Cast\npublic import Mathlib.Combinatorics.Quiver.Symmetric\npublic import Mathlib.Data.Sigma.Basic\npublic import Mathlib.Data.Sum.Basic\npublic import Mathlib.Logic.Equiv.Sum\npublic import Mathlib.Tactic.Common\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Labelle, Rémi Bottinelli\n-/\n/-!\n# Covering\n\nThis file defines coverings of quivers as prefunctors that are bijective on the\nso-called stars and costars at each vertex of the domain.\n\n## Main definitions\n\n* `Quiver.Star u` is the type of all arrows with source `u`;\n* `Quiver.Costar u` is the type of all arrows with target `u`;\n* `Prefunctor.star φ u` is the obvious function `star u → star (φ.obj u)`;\n* `Prefunctor.costar φ u` is the obvious function `costar u → costar (φ.obj u)`;\n* `Prefunctor.IsCovering φ` means that `φ.star u` and `φ.costar u` are bijections for all `u`;\n* `Quiver.PathStar u` is the type of all paths with source `u`;\n* `Prefunctor.pathStar u` is the obvious function `PathStar u → PathStar (φ.obj u)`.\n\n## Main statements\n\n* `Prefunctor.IsCovering.pathStar_bijective` states that if `φ` is a covering,\n  then `φ.pathStar u` is a bijection for all `u`.\n  In other words, every path in the codomain of `φ` lifts uniquely to its domain.\n\n## TODO\n\nClean up the namespaces by renaming `Prefunctor` to `Quiver.Prefunctor`.\n\n## Tags\n\nCover, covering, quiver, path, lift\n-/\n\n@[expose] public section\n\n\nopen Function Quiver\n\nuniverse u v w\n\nvariable {U : Type _} [Quiver.{u} U] {V : Type _} [Quiver.{v} V] (φ : U ⥤q V) {W : Type _}\n  [Quiver.{w} W] (ψ : V ⥤q W)\n\n/-- The `Quiver.Star` at a vertex is the collection of arrows whose source is the vertex.\nThe type `Quiver.Star u` is defined to be `Σ (v : U), (u ⟶ v)`. -/\nabbrev Quiver.Star (u : U) :=\n  Σ v : U, u ⟶ v\n\n/-- Constructor for `Quiver.Star`. Defined to be `Sigma.mk`. -/\nprotected abbrev Quiver.Star.mk {u v : U} (f : u ⟶ v) : Quiver.Star u :=\n  ⟨_, f⟩\n\n/-- The `Quiver.Costar` at a vertex is the collection of arrows whose target is the vertex.\nThe type `Quiver.Costar v` is defined to be `Σ (u : U), (u ⟶ v)`. -/\nabbrev Quiver.Costar (v : U) :=\n  Σ u : U, u ⟶ v\n\n/-- Constructor for `Quiver.Costar`. Defined to be `Sigma.mk`. -/\nprotected abbrev Quiver.Costar.mk {u v : U} (f : u ⟶ v) : Quiver.Costar v :=\n  ⟨_, f⟩\n\n/-- A prefunctor induces a map of `Quiver.Star` at every vertex. -/\n@[simps]\ndef Prefunctor.star (u : U) : Quiver.Star u → Quiver.Star (φ.obj u) := fun F =>\n  Quiver.Star.mk (φ.map F.2)\n\n/-- A prefunctor induces a map of `Quiver.Costar` at every vertex. -/\n@[simps]\ndef Prefunctor.costar (u : U) : Quiver.Costar u → Quiver.Costar (φ.obj u) := fun F =>\n  Quiver.Costar.mk (φ.map F.2)\n\n@[simp]\ntheorem Prefunctor.star_apply {u v : U} (e : u ⟶ v) :\n    φ.star u (Quiver.Star.mk e) = Quiver.Star.mk (φ.map e) :=\n  rfl\n\n@[simp]\ntheorem Prefunctor.costar_apply {u v : U} (e : u ⟶ v) :\n    φ.costar v (Quiver.Costar.mk e) = Quiver.Costar.mk (φ.map e) :=\n  rfl\n\ntheorem Prefunctor.star_comp (u : U) : (φ ⋙q ψ).star u = ψ.star (φ.obj u) ∘ φ.star u :=\n  rfl\n\ntheorem Prefunctor.costar_comp (u : U) : (φ ⋙q ψ).costar u = ψ.costar (φ.obj u) ∘ φ.costar u :=\n  rfl\n\n/-- A prefunctor is a covering of quivers if it defines bijections on all stars and costars. -/\nprotected structure Prefunctor.IsCovering : Prop where\n  star_bijective : ∀ u, Bijective (φ.star u)\n  costar_bijective : ∀ u, Bijective (φ.costar u)\n\n@[simp]\ntheorem Prefunctor.IsCovering.map_injective (hφ : φ.IsCovering) {u v : U} :\n    Injective fun f : u ⟶ v => φ.map f := by\n  rintro f g he\n  have : φ.star u (Quiver.Star.mk f) = φ.star u (Quiver.Star.mk g) := by simpa using he\n  simpa using (hφ.star_bijective u).left this\n\ntheorem Prefunctor.IsCovering.comp (hφ : φ.IsCovering) (hψ : ψ.IsCovering) : (φ ⋙q ψ).IsCovering :=\n  ⟨fun _ => (hψ.star_bijective _).comp (hφ.star_bijective _),\n   fun _ => (hψ.costar_bijective _).comp (hφ.costar_bijective _)⟩\n\ntheorem Prefunctor.IsCovering.of_comp_right (hψ : ψ.IsCovering) (hφψ : (φ ⋙q ψ).IsCovering) :\n    φ.IsCovering :=\n  ⟨fun _ => (Bijective.of_comp_iff' (hψ.star_bijective _) _).mp (hφψ.star_bijective _),\n   fun _ => (Bijective.of_comp_iff' (hψ.costar_bijective _) _).mp (hφψ.costar_bijective _)⟩\n\ntheorem Prefunctor.IsCovering.of_comp_left (hφ : φ.IsCovering) (hφψ : (φ ⋙q ψ).IsCovering)\n    (φsur : Surjective φ.obj) : ψ.IsCovering := by\n  refine ⟨fun v => ?_, fun v => ?_⟩ <;> obtain ⟨u, rfl⟩ := φsur v\n  exacts [(Bijective.of_comp_iff _ (hφ.star_bijective u)).mp (hφψ.star_bijective u),\n    (Bijective.of_comp_iff _ (hφ.costar_bijective u)).mp (hφψ.costar_bijective u)]\n\n/-- The star of the symmetrification of a quiver at a vertex `u` is equivalent to the sum of the\nstar and the costar at `u` in the original quiver. -/\ndef Quiver.symmetrifyStar (u : U) :\n    Quiver.Star (Symmetrify.of.obj u) ≃ Quiver.Star u ⊕ Quiver.Costar u :=\n  Equiv.sigmaSumDistrib _ _\n\n/-- The costar of the symmetrification of a quiver at a vertex `u` is equivalent to the sum of the\ncostar and the star at `u` in the original quiver. -/\ndef Quiver.symmetrifyCostar (u : U) :\n    Quiver.Costar (Symmetrify.of.obj u) ≃ Quiver.Costar u ⊕ Quiver.Star u :=\n  Equiv.sigmaSumDistrib _ _\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem Prefunctor.symmetrifyStar (u : U) :\n    φ.symmetrify.star u =\n      (Quiver.symmetrifyStar _).symm ∘ Sum.map (φ.star u) (φ.costar u) ∘\n        Quiver.symmetrifyStar u :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b736ef4ea45ce21f69cf741c022990c9c33b575d17708de1cfe1e04a2dae968e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Quiver","family_id":"prefunctor","file_id":"mathlib/Mathlib/Combinatorics/Quiver/Covering.lean","sample_id":"87030b969f81b483b5bdfa5c63449397f06437d95959ae3498323790baf68ce1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a0b20474523f493304d2e5c3c4796a60a88d5f5165ad276749d342179c8ed04e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6f6ae3b60a22d3ee8598639d07edd15dc214dc93b32db436d37ba0364ed7ec2a","source_sha256":"3e26e2de40fc89c98c0734f65f6d65c61b9e098984ba72f67bb093182d517ead","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  rw [separatingLeft_iff_ker_eq_bot, ker_eq_bot]\n  let e := LinearEquiv.prodComm R _ _ ≪≫ₗ Module.dualProdDualEquivDual R (Module.Dual R M) M\n  let h_d := e.symm.toLinearMap.comp (dualProd R M)\n  refine (Function.Injective.of_comp_iff e.symm.injective\n    (dualProd R M)).symm.trans ?_\n  rw [← LinearEquiv.coe_toLinearMap, ← coe_comp]\n  change Function.Injective h_d ↔ _\n  have : h_d = prodMap id (Module.Dual.eval R M) := by\n    refine ext fun x => Prod.ext ?_ ?_\n    · ext\n      dsimp [e, h_d, Module.Dual.eval, LinearEquiv.prodComm]\n      simp\n    · ext\n      dsimp [e, h_d, Module.Dual.eval, LinearEquiv.prodComm]\n      simp\n  rw [this, coe_prodMap]\n  refine Prod.map_injective.trans ?_\n  exact and_iff_right Function.injective_id","hard_negative":false,"metrics":{"chosen_tokens":174,"rejected_tokens":2,"token_jaccard":0.032258,"token_length_ratio":0.011494},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"4ec883b8c93f0aeacd0ed82a5be1b2b4ee6b41186b3f77569fdf67528d616959","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Dual.Lemmas\npublic import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv\npublic import Mathlib.LinearAlgebra.QuadraticForm.Prod\n\nNamespace:\nLinearMap\n\nLocal context:\n/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Quadratic form structures related to `Module.Dual`\n\n## Main definitions\n\n* `LinearMap.dualProd R M`, the bilinear form on `(f, x) : Module.Dual R M × M` defined as\n  `f x`.\n* `QuadraticForm.dualProd R M`, the quadratic form on `(f, x) : Module.Dual R M × M` defined as\n  `f x`.\n* `QuadraticForm.toDualProd : (Q.prod <| -Q) →qᵢ QuadraticForm.dualProd R M` a form-preserving map\n  from `(Q.prod <| -Q)` to `QuadraticForm.dualProd R M`.\n\n-/\n\n@[expose] public section\n\nvariable (R M N : Type*)\n\nnamespace LinearMap\n\nsection Semiring\n\nvariable [CommSemiring R] [AddCommMonoid M] [Module R M]\n\n/-- The symmetric bilinear form on `Module.Dual R M × M` defined as\n`B (f, x) (g, y) = f y + g x`. -/\n@[simps!]\ndef dualProd : LinearMap.BilinForm R (Module.Dual R M × M) :=\n    (applyₗ.comp (snd R (Module.Dual R M) M)).compl₂ (fst R (Module.Dual R M) M) +\n      ((applyₗ.comp (snd R (Module.Dual R M) M)).compl₂ (fst R (Module.Dual R M) M)).flip\n\ntheorem isSymm_dualProd : (dualProd R M).IsSymm := ⟨fun _x _y => add_comm _ _⟩\n\nend Semiring\n\nsection Ring\n\nvariable [CommRing R] [AddCommGroup M] [Module R M]\n\nTarget:\ntheorem separatingLeft_dualProd :\n    (dualProd R M).SeparatingLeft ↔ Function.Injective (Module.Dual.eval R M) :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/QuadraticForm","family_id":"separatingleft_dualprod","file_id":"mathlib/Mathlib/LinearAlgebra/QuadraticForm/Dual.lean","sample_id":"6f6ae3b60a22d3ee8598639d07edd15dc214dc93b32db436d37ba0364ed7ec2a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ed61a64bf86bd3e8070ca4701c17ddc605afcea75c9e8aec1615eb96bd1b85a1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8fdf799834e94c10d8af1bf7c477253a01d1a4dd51152b802911ca3f496e8d72","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"591110f7b37cc2daae15e5485ae2d271b10317dae65d4419218d3c8977316858","source_sha256":"830973e88871280f6e5db16d5b576af36be09ea83a1a7629aab6adabb39a26a1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction n with\n  | zero => simp\n  | succ n IH =>\n    rw [iteratedLineDerivOp_succ_left]\n    exact (tsupport_lineDerivOp_subset (m 0) _).trans (IH <| Fin.tail m)","hard_negative":false,"metrics":{"chosen_tokens":37,"rejected_tokens":42,"token_jaccard":0.83871,"token_length_ratio":1.135135},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"4ee16d24cd4b77f035228cc116946fe45150dd58832ba6c6745318ff99d1fdfb","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Distribution.SchwartzSpace.Basic\npublic import Mathlib.Analysis.Calculus.IteratedDeriv.Defs\npublic import Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts\npublic import Mathlib.Analysis.InnerProductSpace.Laplacian\npublic import Mathlib.MeasureTheory.Integral.IntegralEqImproper\n\nNamespace:\nSchwartzMap\n\nLocal context:\n/-\nCopyright (c) 2026 Moritz Doll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Moritz Doll\n-/\n/-!\n# Derivatives of Schwartz functions\n\nIn this file we define the various notions of derivatives of Schwartz functions.\n\n## Main definitions\n\n* `SchwartzMap.fderivCLM`: The differential as a continuous linear map\n  `𝓢(E, F) →L[𝕜] 𝓢(E, E →L[ℝ] F)`\n* `SchwartzMap.derivCLM`: The one-dimensional derivative as a continuous linear map\n  `𝓢(ℝ, F) →L[𝕜] 𝓢(ℝ, F)`\n* `SchwartzMap.instLineDeriv`: The directional derivative with notation `∂_{m} f`\n* `SchwartzMap.instLaplacian`: The Laplacian for `𝓢(E, F)` as an instance of the notation type-class\n  `Laplacian`.\n\n## Main statements\n\n* `SchwartzMap.iteratedLineDerivOp_eq_iteratedFDeriv`: the iterated directional derivative is given\n  by the applied Fréchet derivative of a Schwartz function.\n* `SchwartzMap.laplacian_eq_sum`: the Laplacian is given by the sum of second derivatives in any\n  orthonormal basis.\n* `SchwartzMap.integral_bilinear_lineDerivOp_right_eq_neg_left`: Integration by parts using the\n  directional derivative `∂_{m}`\n* `SchwartzMap.integral_bilinear_laplacian_right_eq_left`: Integration by parts for the Laplacian\n\n-/\n\n@[expose] public noncomputable section\n\nvariable {ι 𝕜 𝕜' D E F V F F₁ F₂ F₃ : Type*}\n\nnamespace SchwartzMap\n\nvariable [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ F]\n\nsection Derivatives\n\n/-! ### Derivatives of Schwartz functions -/\n\nvariable [NormedSpace ℝ E]\n\nvariable (𝕜)\nvariable [RCLike 𝕜] [NormedSpace 𝕜 F]\n\nvariable (F) in\n/-- The 1-dimensional derivative on Schwartz space as a continuous `𝕜`-linear map. -/\ndef derivCLM : 𝓢(ℝ, F) →L[𝕜] 𝓢(ℝ, F) :=\n  mkCLM (deriv ·) (fun f g _ => deriv_add f.differentiableAt g.differentiableAt)\n    (fun a f _ => deriv_const_smul a f.differentiableAt)\n    (fun f => (contDiff_succ_iff_deriv.mp (f.smooth ⊤)).2.2) fun ⟨k, n⟩ =>\n    ⟨{⟨k, n + 1⟩}, 1, zero_le_one, fun f x => by\n      simpa only [Real.norm_eq_abs, Finset.sup_singleton, schwartzSeminormFamily_apply, one_mul,\n        norm_iteratedFDeriv_eq_norm_iteratedDeriv, ← iteratedDeriv_succ'] using\n        f.le_seminorm' 𝕜 k (n + 1) x⟩\n\n@[simp]\ntheorem derivCLM_apply (f : 𝓢(ℝ, F)) (x : ℝ) : derivCLM 𝕜 F f x = deriv f x :=\n  rfl\n\ntheorem hasDerivAt (f : 𝓢(ℝ, F)) (x : ℝ) : HasDerivAt f (deriv f x) x :=\n  f.differentiableAt.hasDerivAt\n\nopen LineDeriv\n\nsection fderiv\n\nvariable [SMulCommClass ℝ 𝕜 F]\n\nvariable (E F) in\n/-- The Fréchet derivative on Schwartz space as a continuous `𝕜`-linear map. -/\ndef fderivCLM : 𝓢(E, F) →L[𝕜] 𝓢(E, E →L[ℝ] F) :=\n  mkCLM (fderiv ℝ ·) (fun f g _ => fderiv_add f.differentiableAt g.differentiableAt)\n    (fun a f _ => fderiv_const_smul f.differentiableAt a)\n    (fun f => (contDiff_succ_iff_fderiv.mp (f.smooth ⊤)).2.2) fun ⟨k, n⟩ =>\n    ⟨{⟨k, n + 1⟩}, 1, zero_le_one, fun f x => by\n      simpa only [schwartzSeminormFamily_apply, Seminorm.comp_apply, Finset.sup_singleton,\n        one_smul, norm_iteratedFDeriv_fderiv, one_mul] using f.le_seminorm 𝕜 k (n + 1) x⟩\n\n@[simp]\ntheorem fderivCLM_apply (f : 𝓢(E, F)) (x : E) : fderivCLM 𝕜 E F f x = fderiv ℝ f x :=\n  rfl\n\ntheorem hasFDerivAt (f : 𝓢(E, F)) (x : E) : HasFDerivAt f (fderiv ℝ f x) x :=\n  f.differentiableAt.hasFDerivAt\n\n/-- The partial derivative (or directional derivative) in the direction `m : E` as a\ncontinuous linear map on Schwartz space. -/\ninstance : LineDeriv E 𝓢(E, F) 𝓢(E, F) where\n  lineDerivOp m f := (SchwartzMap.evalCLM ℝ E F m ∘L fderivCLM ℝ E F) f\n\ntheorem lineDerivOp_apply_eq_fderiv (m : E) (f : 𝓢(E, F)) (x : E) :\n    ∂_{m} f x = fderiv ℝ f x m := rfl\n\ninstance : LineDerivAdd E 𝓢(E, F) 𝓢(E, F) where\n  lineDerivOp_add m := ((SchwartzMap.evalCLM ℝ E F m).comp (fderivCLM ℝ E F)).map_add\n  lineDerivOp_left_add v w f := by\n    ext x\n    simp [lineDerivOp_apply_eq_fderiv]\n\ninstance : LineDerivSMul 𝕜 E 𝓢(E, F) 𝓢(E, F) where\n  lineDerivOp_smul m := (SchwartzMap.evalCLM 𝕜 E F m ∘L fderivCLM 𝕜 E F).map_smul\n\ninstance : LineDerivLeftSMul ℝ E 𝓢(E, F) 𝓢(E, F) where\n  lineDerivOp_left_smul r y f := by\n    ext x\n    simp [lineDerivOp_apply_eq_fderiv]\n\ninstance : ContinuousLineDeriv E 𝓢(E, F) 𝓢(E, F) where\n  continuous_lineDerivOp m := (SchwartzMap.evalCLM ℝ E F m ∘L fderivCLM ℝ E F).continuous\n\nopen LineDeriv\n\ntheorem lineDerivOpCLM_eq (m : E) :\n    lineDerivOpCLM 𝕜 𝓢(E, F) m = SchwartzMap.evalCLM 𝕜 E F m ∘L fderivCLM 𝕜 E F := rfl\n\n@[deprecated (since := \"2025-11-25\")]\nalias pderivCLM := lineDerivOpCLM\n\n@[deprecated (since := \"2025-11-25\")]\nalias pderivCLM_apply := LineDeriv.lineDerivOpCLM_apply\n\ntheorem lineDerivOp_apply (m : E) (f : 𝓢(E, F)) (x : E) : ∂_{m} f x = lineDeriv ℝ f x m :=\n  f.differentiableAt.lineDeriv_eq_fderiv.symm\n\n@[deprecated (since := \"2025-11-25\")]\nalias iteratedPDeriv := LineDeriv.iteratedLineDerivOpCLM\n\n@[deprecated (since := \"2025-11-25\")]\nalias iteratedPDeriv_zero := LineDeriv.iteratedLineDerivOp_zero\n\n@[deprecated (since := \"2025-11-25\")]\nalias iteratedPDeriv_one := LineDeriv.iteratedLineDerivOp_one\n\n@[deprecated (since := \"2025-11-25\")]\nalias iteratedPDeriv_succ_left := LineDeriv.iteratedLineDerivOp_succ_left\n\n@[deprecated (since := \"2025-11-25\")]\nalias iteratedPDeriv_succ_right := LineDeriv.iteratedLineDerivOp_succ_right\n\ntheorem iteratedLineDerivOp_eq_iteratedFDeriv {n : ℕ} {m : Fin n → E} {f : 𝓢(E, F)} {x : E} :\n    ∂^{m} f x = iteratedFDeriv ℝ n f x m := by\n  induction n generalizing x with\n  | zero => simp\n  | succ n ih =>\n    rw [iteratedLineDerivOp_succ_left, iteratedFDeriv_succ_apply_left,\n      ← fderiv_continuousMultilinear_apply_const_apply]\n    · simp only [lineDerivOp_apply_eq_fderiv, ← ih]\n    · exact (f.smooth ⊤).differentiable_iteratedFDeriv (mod_cast ENat.coe_lt_top n) x\n\n@[deprecated (since := \"2025-11-25\")]\nalias iteratedPDeriv_eq_iteratedFDeriv := iteratedLineDerivOp_eq_iteratedFDeriv\n\nend fderiv\n\nvariable [NormedAddCommGroup D] [NormedSpace ℝ D]\n\ntheorem lineDerivOp_compCLMOfContinuousLinearEquiv (m : D) (g : D ≃L[ℝ] E) (f : 𝓢(E, F)) :\n    ∂_{m} (compCLMOfContinuousLinearEquiv 𝕜 g f) =\n    compCLMOfContinuousLinearEquiv 𝕜 g (∂_{g m} f) := by\n  ext x\n  simp [lineDerivOp_apply_eq_fderiv, ContinuousLinearEquiv.comp_right_fderiv]\n\nend Derivatives\n\nsection support\n\nvariable (𝕜)\nvariable [RCLike 𝕜] [NormedSpace 𝕜 F]\n\ntheorem tsupport_derivCLM_subset (f : 𝓢(ℝ, F)) : tsupport (derivCLM 𝕜 F f) ⊆ tsupport f := by\n  change tsupport (deriv f ·) ⊆ _\n  simp_rw [← fderiv_apply_one_eq_deriv]\n  exact tsupport_fderiv_apply_subset ℝ 1\n\nvariable [NormedSpace ℝ E] [SMulCommClass ℝ 𝕜 F]\n\ntheorem tsupport_fderivCLM_subset (f : 𝓢(E, F)) : tsupport (fderivCLM 𝕜 E F f) ⊆ tsupport f :=\n  tsupport_fderiv_subset ℝ\n\nopen LineDeriv\n\ntheorem tsupport_lineDerivOp_subset (m : E) (f : 𝓢(E, F)) :\n    tsupport (∂_{m} f : 𝓢(E, F)) ⊆ tsupport f :=\n  tsupport_fderiv_apply_subset ℝ m\n\nTarget:\ntheorem tsupport_iteratedLineDerivOp_subset {n : ℕ} (m : Fin n → E) (f : 𝓢(E, F)) :\n    tsupport (∂^{m} f : 𝓢(E, F)) ⊆ tsupport f :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  induction n with\n  | zero => simp\n  | succ n IH =>\n    rw [iteratedLineDerivOp_succ_left]\n    exact (tsupport_lineDerivOp_subset (m 0) _).trans (IH <| Fin.tail m)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Distribution","family_id":"tsupport_iteratedlinederivop_subset","file_id":"mathlib/Mathlib/Analysis/Distribution/SchwartzSpace/Deriv.lean","sample_id":"591110f7b37cc2daae15e5485ae2d271b10317dae65d4419218d3c8977316858"},"split":"train"}
{"binding":{"chosen_proof_sha256":"20d35803748fd7d6978af67d3ca14fe96ffdbfdea01cb8bdd6440cb3a20e70bb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e02776207e53734b0bc60bad4f0d74454901295821ad3bc99f5982ff9c34693f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"87698f733f1f8800ecece55747b606be522d5d9ae2965cfc2fd1ae226fa9c49c","source_sha256":"24ffbf256f6f6f7a2617901323c2d532e2d7871c826a8b11f0580b283e994302","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let nop := @neg_one_pow_div_two_of_three_mod_four\n  rw [quadratic_reciprocity', pow_mul, nop hp, nop hq, neg_one_mul] <;>\n  rwa [← Prime.mod_two_eq_one_iff_ne_two Fact.out, odd_of_mod_four_eq_three]","hard_negative":true,"metrics":{"chosen_tokens":35,"rejected_tokens":3,"token_jaccard":0.035714,"token_length_ratio":0.085714},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"4efccae5d697a6e43b27703b33dc396738a7243e7a553dc59835492186548581","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.LegendreSymbol.Basic\npublic import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum\n\nNamespace:\nlegendreSym\n\nLocal context:\n/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Michael Stoll\n-/\n/-!\n# Quadratic reciprocity.\n\n## Main results\n\nWe prove the law of quadratic reciprocity, see `legendreSym.quadratic_reciprocity` and\n`legendreSym.quadratic_reciprocity'`, as well as the\ninterpretations in terms of existence of square roots depending on the congruence mod 4,\n`ZMod.exists_sq_eq_prime_iff_of_mod_four_eq_one` and\n`ZMod.exists_sq_eq_prime_iff_of_mod_four_eq_three`.\n\nWe also prove the supplementary laws that give conditions for when `2` or `-2`\nis a square modulo a prime `p`:\n`legendreSym.at_two` and `ZMod.exists_sq_eq_two_iff` for `2` and\n`legendreSym.at_neg_two` and `ZMod.exists_sq_eq_neg_two_iff` for `-2`.\n\n## Implementation notes\n\nThe proofs use results for quadratic characters on arbitrary finite fields\nfrom `NumberTheory.LegendreSymbol.QuadraticChar.GaussSum`, which in turn are based on\nproperties of quadratic Gauss sums as provided by `NumberTheory.LegendreSymbol.GaussSum`.\n\n## Tags\n\nquadratic residue, quadratic nonresidue, Legendre symbol, quadratic reciprocity\n-/\n\npublic section\n\n\nopen Nat\n\nsection Values\n\nvariable {p : ℕ} [Fact p.Prime]\n\nopen ZMod\n\n/-!\n### The value of the Legendre symbol at `2` and `-2`\n\nSee `jacobiSym.at_two` and `jacobiSym.at_neg_two` for the corresponding statements\nfor the Jacobi symbol.\n-/\n\n\nnamespace legendreSym\n\n/-- `legendreSym p 2` is given by `χ₈ p`. -/\ntheorem at_two (hp : p ≠ 2) : legendreSym p 2 = χ₈ p := by\n  have : (2 : ZMod p) = (2 : ℤ) := by norm_cast\n  rw [legendreSym, ← this, quadraticChar_two ((ringChar_zmod_n p).substr hp), card p]\n\n/-- `legendreSym p (-2)` is given by `χ₈' p`. -/\ntheorem at_neg_two (hp : p ≠ 2) : legendreSym p (-2) = χ₈' p := by\n  have : (-2 : ZMod p) = (-2 : ℤ) := by norm_cast\n  rw [legendreSym, ← this, quadraticChar_neg_two ((ringChar_zmod_n p).substr hp), card p]\n\nend legendreSym\n\nnamespace ZMod\n\n/-- `2` is a square modulo an odd prime `p` iff `p` is congruent to `1` or `7` mod `8`. -/\ntheorem exists_sq_eq_two_iff (hp : p ≠ 2) : IsSquare (2 : ZMod p) ↔ p % 8 = 1 ∨ p % 8 = 7 := by\n  rw [FiniteField.isSquare_two_iff, card p]\n  have h₁ := (Prime.mod_two_eq_one_iff_ne_two Fact.out).mpr hp\n  lia\n\n/-- `-2` is a square modulo an odd prime `p` iff `p` is congruent to `1` or `3` mod `8`. -/\ntheorem exists_sq_eq_neg_two_iff (hp : p ≠ 2) : IsSquare (-2 : ZMod p) ↔ p % 8 = 1 ∨ p % 8 = 3 := by\n  rw [FiniteField.isSquare_neg_two_iff, card p]\n  have h₁ := (Prime.mod_two_eq_one_iff_ne_two Fact.out).mpr hp\n  lia\n\nend ZMod\n\nend Values\n\nsection Reciprocity\n\n/-!\n### The Law of Quadratic Reciprocity\n\nSee `jacobiSym.quadratic_reciprocity` and variants for a version of Quadratic Reciprocity\nfor the Jacobi symbol.\n-/\n\n\nvariable {p q : ℕ} [Fact p.Prime] [Fact q.Prime]\n\nnamespace legendreSym\n\nopen ZMod\n\n/-- **The Law of Quadratic Reciprocity**: if `p` and `q` are distinct odd primes, then\n`(q / p) * (p / q) = (-1)^((p-1)(q-1)/4)`. -/\ntheorem quadratic_reciprocity (hp : p ≠ 2) (hq : q ≠ 2) (hpq : p ≠ q) :\n    legendreSym q p * legendreSym p q = (-1) ^ (p / 2 * (q / 2)) := by\n  have hp₁ := (Prime.eq_two_or_odd <| @Fact.out p.Prime _).resolve_left hp\n  have hq₁ := (Prime.eq_two_or_odd <| @Fact.out q.Prime _).resolve_left hq\n  have hq₂ : ringChar (ZMod q) ≠ 2 := (ringChar_zmod_n q).substr hq\n  have h :=\n    quadraticChar_odd_prime ((ringChar_zmod_n p).substr hp) hq ((ringChar_zmod_n p).substr hpq)\n  rw [card p] at h\n  have nc : ∀ n r : ℕ, ((n : ℤ) : ZMod r) = n := fun n r => by norm_cast\n  have nc' : (((-1) ^ (p / 2) : ℤ) : ZMod q) = (-1) ^ (p / 2) := by norm_cast\n  rw [legendreSym, legendreSym, nc, nc, h, map_mul, mul_rotate', mul_comm (p / 2), ← pow_two,\n    quadraticChar_sq_one (prime_ne_zero q p hpq.symm), mul_one, pow_mul, χ₄_eq_neg_one_pow hp₁, nc',\n    map_pow, quadraticChar_neg_one hq₂, card q, χ₄_eq_neg_one_pow hq₁]\n\n/-- The Law of Quadratic Reciprocity: if `p` and `q` are odd primes, then\n`(q / p) = (-1)^((p-1)(q-1)/4) * (p / q)`. -/\ntheorem quadratic_reciprocity' (hp : p ≠ 2) (hq : q ≠ 2) :\n    legendreSym q p = (-1) ^ (p / 2 * (q / 2)) * legendreSym p q := by\n  rcases eq_or_ne p q with rfl | h\n  · rw [(eq_zero_iff p p).mpr (mod_cast natCast_self p), mul_zero]\n  · have qr := congr_arg (· * legendreSym p q) (quadratic_reciprocity hp hq h)\n    have : ((q : ℤ) : ZMod p) ≠ 0 := mod_cast prime_ne_zero p q h\n    simpa only [mul_assoc, ← pow_two, sq_one p this, mul_one] using qr\n\n/-- The Law of Quadratic Reciprocity: if `p` and `q` are odd primes and `p % 4 = 1`,\nthen `(q / p) = (p / q)`. -/\ntheorem quadratic_reciprocity_one_mod_four (hp : p % 4 = 1) (hq : q ≠ 2) :\n    legendreSym q p = legendreSym p q := by\n  rw [quadratic_reciprocity'\n      ((Prime.mod_two_eq_one_iff_ne_two Fact.out).mp (odd_of_mod_four_eq_one hp)) hq,\n    pow_mul, neg_one_pow_div_two_of_one_mod_four hp, one_pow, one_mul]\n\n/-- The Law of Quadratic Reciprocity: if `p` and `q` are primes that are both congruent\nto `3` mod `4`, then `(q / p) = -(p / q)`. -/\n\nTarget:\ntheorem quadratic_reciprocity_three_mod_four (hp : p % 4 = 3) (hq : q % 4 = 3) :\n    legendreSym q p = -legendreSym p q :=\n\nProof body:\n","rejected":"by\n  exact quadratic_reciprocity_three_mod_four","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"aeb4065d2b8f3bdc8324d3f4cbe041dfd76be3d5d4cbf8b9d4c96dce7b3ba323","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/LegendreSymbol","family_id":"quadratic_reciprocity_three_mod_four","file_id":"mathlib/Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean","sample_id":"87698f733f1f8800ecece55747b606be522d5d9ae2965cfc2fd1ae226fa9c49c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c198f493b5c0a079bb4ea7d468b89dbbd0202be048c196e631696690e8762509","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f83dd1b97c9a129dc210da5c52474019b307010c9a4d0c19de7a0ae4adc2ce13","source_sha256":"5fcf0f4a5da250f9e6a71f1db0e6fc4a861d3fb51b0fa6cda384195f89e3653e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain rfl | hμ := eq_zero_or_isProbabilityMeasure μ\n  · simp\n  simpa [Pi.mul_def, integral_of_ae_eq_zero_or_one, integral_one_sub_of_ae_eq_zero_or_one, mul_comm,\n    *] using condVar_of_ae_eq_zero_or_one bot_le hXmeas hX","hard_negative":false,"metrics":{"chosen_tokens":30,"rejected_tokens":5,"token_jaccard":0.033333,"token_length_ratio":0.166667},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"4f0317c0938615165c65458c4678ddef1b8f25e96c1522eead7bc4274ca554ab","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Integral.Bochner.Basic\npublic import Mathlib.Probability.CondVar\nimport Mathlib.Probability.Notation\n\nNamespace:\nProbabilityTheory\n\nLocal context:\n/-\nCopyright (c) 2024 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Distributions on two values\n\nThis file proves a few lemmas about random variables that take at most two values.\n-/\n\npublic section\n\nopen MeasureTheory\nopen scoped ProbabilityTheory\n\nnamespace MeasureTheory\nvariable {Ω : Type*} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : Measure Ω}\n\n/-- If an `AEMeasurable` function is ae equal to `0` or `1`, then its integral is equal to the\nmeasure of the set where it equals `1`. -/\nlemma integral_of_ae_eq_zero_or_one (hXmeas : AEMeasurable X μ) (hX : ∀ᵐ ω ∂μ, X ω = 0 ∨ X ω = 1) :\n    μ[X] = μ.real {ω | X ω = 1} := by\n  refine (integral_map (f := id) hXmeas <| by fun_prop).symm.trans ?_\n  rw [(Measure.ae_eq_or_eq_iff_map_eq_dirac_add_dirac hXmeas zero_ne_one).1 hX]\n  by_cases h : μ {ω | X ω = 1} = ⊤\n  · simp [h, Measure.real, Set.preimage, integral_undef, Integrable, HasFiniteIntegral]\n  rw [integral_add_measure ⟨by fun_prop, by simp [HasFiniteIntegral]⟩ <|\n    .smul_measure (by simp [integrable_dirac]) h]\n  simp [Measure.real, Set.preimage]\n\n/-- If a random variable is ae equal to `0` or `1`, then one minus its expectation is equal to the\nprobability that it equals `0`. -/\nlemma integral_one_sub_of_ae_eq_zero_or_one (hXmeas : AEMeasurable X μ)\n    (hX : ∀ᵐ ω ∂μ, X ω = 0 ∨ X ω = 1) : ∫ ω, 1 - X ω ∂μ = μ.real {ω | X ω = 0} := by\n  calc\n    _ = μ.real {ω | 1 - X ω = 1} :=\n      integral_of_ae_eq_zero_or_one (aemeasurable_const (b := 1).sub hXmeas)\n        (by simpa [sub_eq_zero, or_comm, eq_comm (a := (1 : ℝ))] using hX)\n    _ = μ.real {ω | X ω = 0} := by simp\n\nend MeasureTheory\n\n\nnamespace ProbabilityTheory\nvariable {Ω : Type*} {m : MeasurableSpace Ω} {X Y : Ω → ℝ} {μ : Measure ℝ} {P : Measure Ω}\n\n/-- If a random variable is ae equal to `0` or `1`, then its conditional variance is the product of\nthe conditional probabilities that it's equal to `0` and that it's equal to `1`. -/\nlemma condVar_of_ae_eq_zero_or_one {m₀ : MeasurableSpace Ω} (hm : m ≤ m₀) {μ : Measure[m₀] Ω}\n    [IsFiniteMeasure μ] (hXmeas : AEMeasurable[m₀] X μ) (hX : ∀ᵐ ω ∂μ, X ω = 0 ∨ X ω = 1) :\n    Var[X; μ | m] =ᵐ[μ] μ[X | m] * μ[1 - X | m] := by\n  wlog hXmeas : Measurable[m₀] X\n  · obtain ⟨Y, hYmeas, hXY⟩ := ‹AEMeasurable[m₀] X μ›\n    calc\n      Var[X; μ | m]\n      _ =ᵐ[μ] Var[Y; μ | m] := condVar_congr_ae hXY\n      _ =ᵐ[μ] μ[Y | m] * μ[1 - Y | m] := by\n        refine this hm hYmeas.aemeasurable ?_ hYmeas\n        filter_upwards [hX, hXY] with ω hXω hXYω\n        simp [hXω, ← hXYω]\n      _ =ᵐ[μ] μ[X | m] * μ[1 - X | m] := by\n        refine .mul ?_ ?_ <;>\n          exact condExp_congr_ae <| by filter_upwards [hXY] with ω hω; simp [hω]\n  calc\n    _ =ᵐ[μ] μ[X ^ 2 | m] - μ[X | m] ^ 2 :=\n      condVar_ae_eq_condExp_sq_sub_sq_condExp hm <| .of_bound hXmeas.aestronglyMeasurable 1 <| by\n        filter_upwards [hX]; rintro ω (hω | hω) <;> simp [hω]\n    _ =ᵐ[μ] μ[X | m] - μ[X | m] ^ 2 := by\n      refine .sub ?_ ae_eq_rfl\n      exact condExp_congr_ae <| by filter_upwards [hX]; rintro ω (hω | hω) <;> simp [hω]\n    _ =ᵐ[μ] μ[X | m] * μ[1 - X | m] := by\n      rw [sq, ← one_sub_mul, mul_comm]\n      refine .mul ae_eq_rfl ?_\n      calc\n        1 - μ[X | m]\n        _ = μ[1 | m] - μ[X | m] := by simp [Pi.one_def, hm]\n        _ =ᵐ[μ] μ[1 - X | m] := by\n          refine (condExp_sub (integrable_const _)\n            (.of_bound (C := 1) hXmeas.aestronglyMeasurable ?_) _).symm\n          filter_upwards [hX]\n          rintro ω (hω | hω) <;> simp [hω]\n\n/-- If a random variable is ae equal to `0` or `1`, then its variance is the product of\nthe probabilities that it's equal to `0` and that it's equal to `1`. -/\n\nTarget:\nlemma variance_of_ae_eq_zero_or_one {μ : Measure Ω} [IsZeroOrProbabilityMeasure μ]\n    (hXmeas : AEMeasurable X μ) (hX : ∀ᵐ ω ∂μ, X ω = 0 ∨ X ω = 1) :\n    Var[X; μ] = μ.real {ω | X ω = 0} * μ.real {ω | X ω = 1} :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Distributions","family_id":"variance_of_ae_eq_zero_or_one","file_id":"mathlib/Mathlib/Probability/Distributions/TwoValued.lean","sample_id":"f83dd1b97c9a129dc210da5c52474019b307010c9a4d0c19de7a0ae4adc2ce13"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fecffccc9af025141c64f561cc932cc9a7f6f3a1d76dd0660a012426d4c05839","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c3604586f2ac7368d223dc67d1199e013bd8135ef158fafca76c177f59a92d99","source_sha256":"00cdc131b09c868f94179716d61273d8fdf897ed023d34f30c7085668a3b31ac","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold xgcdAux\n  exact if_pos rfl","hard_negative":true,"metrics":{"chosen_tokens":6,"rejected_tokens":8,"token_jaccard":0.083333,"token_length_ratio":1.333333},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"4f118d6f3c2732cc08b0a9a589b3088369750c36297a779b9c64f6b791b4da31","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Order.RelClasses\n\nNamespace:\nEuclideanDomain\n\nLocal context:\n/-\nCopyright (c) 2018 Louis Carlin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Louis Carlin, Mario Carneiro\n-/\n/-!\n# Euclidean domains\n\nThis file introduces Euclidean domains and provides the extended Euclidean algorithm. To be precise,\na slightly more general version is provided which is sometimes called a transfinite Euclidean domain\nand differs in the fact that the degree function need not take values in `ℕ` but can take values in\nany well-ordered set. Transfinite Euclidean domains were introduced by Motzkin and examples which\ndon't satisfy the classical notion were provided independently by Hiblot and Nagata.\n\n## Main definitions\n\n* `EuclideanDomain`: Defines Euclidean domain with functions `quotient` and `remainder`. Instances\n  of `Div` and `Mod` are provided, so that one can write `a = b * (a / b) + a % b`.\n* `gcd`: defines the greatest common divisors of two elements of a Euclidean domain.\n* `xgcd`: given two elements `a b : R`, `xgcd a b` defines the pair `(x, y)` such that\n  `x * a + y * b = gcd a b`.\n* `lcm`: defines the lowest common multiple of two elements `a` and `b` of a Euclidean domain as\n  `a * b / (gcd a b)`\n\n## Main statements\n\nSee `Algebra.EuclideanDomain.Basic` for most of the theorems about Euclidean domains,\nincluding Bézout's lemma.\n\nSee `Algebra.EuclideanDomain.Instances` for the fact that `ℤ` is a Euclidean domain,\nas is any field.\n\n## Notation\n\n`≺` denotes the well-founded relation on the Euclidean domain, e.g. in the example of the polynomial\nring over a field, `p ≺ q` for polynomials `p` and `q` if and only if the degree of `p` is less than\nthe degree of `q`.\n\n## Implementation details\n\nInstead of working with a valuation, `EuclideanDomain` is implemented with the existence of a well\nfounded relation `r` on the integral domain `R`, which in the example of `ℤ` would correspond to\nsetting `i ≺ j` for integers `i` and `j` if the absolute value of `i` is smaller than the absolute\nvalue of `j`.\n\n## References\n\n* [Th. Motzkin, *The Euclidean algorithm*][MR32592]\n* [J.-J. Hiblot, *Des anneaux euclidiens dont le plus petit algorithme n'est pas à valeurs finies*]\n  [MR399081]\n* [M. Nagata, *On Euclid algorithm*][MR541021]\n\n\n## Tags\n\nEuclidean domain, transfinite Euclidean domain, Bézout's lemma\n-/\n\n@[expose] public section\n\nuniverse u\n\n/-- A `EuclideanDomain` is a non-trivial commutative ring with a division and a remainder,\n  satisfying `b * (a / b) + a % b = a`.\n  The definition of a Euclidean domain usually includes a valuation function `R → ℕ`.\n  This definition is slightly generalised to include a well-founded relation\n  `r` with the property that `r (a % b) b`, instead of a valuation. -/\n@[wikidata Q867345]\nclass EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R where\n  /-- A division function (denoted `/`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `%` denotes `remainder`. -/\n  protected quotient : R → R → R\n  /-- Division by zero should always give zero by convention. -/\n  protected quotient_zero : ∀ a, quotient a 0 = 0\n  /-- A remainder function (denoted `%`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `/` denotes `quotient`. -/\n  protected remainder : R → R → R\n  /-- The property that links the quotient and remainder functions.\n    This allows us to compute GCDs and LCMs. -/\n  protected quotient_mul_add_remainder_eq : ∀ a b, b * quotient a b + remainder a b = a\n  /-- A well-founded relation on `R`, satisfying `r (a % b) b`.\n    This ensures that the GCD algorithm always terminates. -/\n  protected r : R → R → Prop\n  /-- The relation `r` must be well-founded.\n    This ensures that the GCD algorithm always terminates. -/\n  r_wellFounded : WellFounded r\n  /-- The relation `r` satisfies `r (a % b) b`. -/\n  protected remainder_lt : ∀ (a) {b}, b ≠ 0 → r (remainder a b) b\n  /-- An additional constraint on `r`. -/\n  mul_left_not_lt : ∀ (a) {b}, b ≠ 0 → ¬r (a * b) a\n\n/-\nLean has far more theorems about fields than about Euclidean domains. We thus\nlower the priority of `Euclideandomain.toCommRing`, encouraging typeclass inference\nto try `Field.toCommRing` first. Without this priority-lowering, typeclass inference\nfinds the more inefficient path `Field.toEuclideanDomain.toCommRing` by default. This\npriority change saves over 500G instructions across mathlib. See\nhttps://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/We.20need.20to.20talk.20about.20Euclidean.20Domains/near/594655420\n-/\n-- see Note [lower instance priority]\nattribute [instance 100] EuclideanDomain.toCommRing\n\nnamespace EuclideanDomain\n\nvariable {R : Type u} [EuclideanDomain R]\n\n/-- Abbreviated notation for the well-founded relation `r` in a Euclidean domain. -/\nlocal infixl:50 \" ≺ \" => EuclideanDomain.r\n\nlocal instance wellFoundedRelation : WellFoundedRelation R where\n  rel := EuclideanDomain.r\n  wf := r_wellFounded\n\ninstance isWellFounded : IsWellFounded R (· ≺ ·) where\n  wf := r_wellFounded\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Div R :=\n  ⟨EuclideanDomain.quotient⟩\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Mod R :=\n  ⟨EuclideanDomain.remainder⟩\n\ntheorem div_add_mod (a b : R) : b * (a / b) + a % b = a :=\n  EuclideanDomain.quotient_mul_add_remainder_eq _ _\n\ntheorem mod_add_div (a b : R) : a % b + b * (a / b) = a :=\n  (add_comm _ _).trans (div_add_mod _ _)\n\ntheorem mod_add_div' (m k : R) : m % k + m / k * k = m := by\n  rw [mul_comm]\n  exact mod_add_div _ _\n\ntheorem div_add_mod' (m k : R) : m / k * k + m % k = m := by\n  rw [mul_comm]\n  exact div_add_mod _ _\n\ntheorem mod_lt : ∀ (a) {b : R}, b ≠ 0 → a % b ≺ b :=\n  EuclideanDomain.remainder_lt\n\ntheorem mul_right_not_lt {a : R} (b) (h : a ≠ 0) : ¬a * b ≺ b := by\n  rw [mul_comm]\n  exact mul_left_not_lt b h\n\n@[simp]\ntheorem mod_zero (a : R) : a % 0 = a := by simpa only [zero_mul, zero_add] using div_add_mod a 0\n\ntheorem lt_one (a : R) : a ≺ (1 : R) → a = 0 :=\n  haveI := Classical.dec\n  not_imp_not.1 fun h => by simpa only [one_mul] using mul_left_not_lt 1 h\n\n@[simp]\ntheorem div_zero (a : R) : a / 0 = 0 :=\n  EuclideanDomain.quotient_zero a\n\nsection\n\n@[elab_as_elim]\ntheorem GCD.induction {P : R → R → Prop} (a b : R) (H0 : ∀ x, P 0 x)\n    (H1 : ∀ a b, a ≠ 0 → P (b % a) a → P a b) : P a b := by\n  classical\n  exact if a0 : a = 0 then\n    a0.symm ▸ H0 b\n  else\n    have _ := mod_lt b a0\n    H1 _ _ a0 (GCD.induction (b % a) a H0 H1)\ntermination_by a\n\nend\n\nsection GCD\n\nvariable [DecidableEq R]\n\n/-- `gcd a b` is a (non-unique) element such that `gcd a b ∣ a` `gcd a b ∣ b`, and for\n  any element `c` such that `c ∣ a` and `c ∣ b`, then `c ∣ gcd a b` -/\ndef gcd (a b : R) : R :=\n  if a0 : a = 0 then b\n  else\n    have _ := mod_lt b a0\n    gcd (b % a) a\ntermination_by a\n\n@[simp]\ntheorem gcd_zero_left (a : R) : gcd 0 a = a := by\n  rw [gcd]\n  exact if_pos rfl\n\n/-- An implementation of the extended GCD algorithm.\nAt each step we are computing a triple `(r, s, t)`, where `r` is the next value of the GCD\nalgorithm, to compute the greatest common divisor of the input (say `x` and `y`), and `s` and `t`\nare the coefficients in front of `x` and `y` to obtain `r` (i.e. `r = s * x + t * y`).\nThe function `xgcdAux` takes in two triples, and from these recursively computes the next triple:\n```\nxgcdAux (r, s, t) (r', s', t') = xgcdAux (r' % r, s' - (r' / r) * s, t' - (r' / r) * t) (r, s, t)\n```\n-/\ndef xgcdAux (r s t r' s' t' : R) : R × R × R :=\n  if _hr : r = 0 then (r', s', t')\n  else\n    let q := r' / r\n    have _ := mod_lt r' _hr\n    xgcdAux (r' % r) (s' - q * s) (t' - q * t) r s t\ntermination_by r\n\n@[simp]\n\nTarget:\ntheorem xgcd_zero_left {s t r' s' t' : R} : xgcdAux 0 s t r' s' t' = (r', s', t') :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"50ad3d62bc125ab492492a9f3567c17234fb5eb46e7ef18552c6a12ba608b637","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/EuclideanDomain","family_id":"xgcd_zero_left","file_id":"mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean","sample_id":"c3604586f2ac7368d223dc67d1199e013bd8135ef158fafca76c177f59a92d99"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eff3de0495382d14fb1653bae8a0cc2d9bf54b578da62db37b5ffda72bfdcbdf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"27c82bde5cab40ddeb3cf4fe4ff5388002b1c6d7395440a1ff8ff9c86f617bfd","source_sha256":"e91ba8652f5a9acc4cc194c5b5cca3581b49fb91d7b9b32974af75019938ff7e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [← le_bot_iff, not_le_iff_exists]","hard_negative":false,"metrics":{"chosen_tokens":8,"rejected_tokens":2,"token_jaccard":0.111111,"token_length_ratio":0.25},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"4f86a9d0017a9602ba65901ac9f93b69381344943d74d1ea9d689b315489c6ec","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.ObjectProperty.ClosedUnderIsomorphisms\npublic import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory\npublic import Mathlib.Order.CompleteLattice.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# ObjectProperty is a complete lattice\n\n-/\n\npublic section\n\nuniverse v u\n\nnamespace CategoryTheory.ObjectProperty\n\nvariable {C : Type u} [Category.{v} C]\n\nexample : CompleteLattice (ObjectProperty C) := inferInstance\n\nsection\n\nvariable (P Q : ObjectProperty C) (X : C)\n\n@[simp high] lemma prop_inf_iff : (P ⊓ Q) X ↔ P X ∧ Q X := Iff.rfl\n\n@[simp high] lemma prop_sup_iff : (P ⊔ Q) X ↔ P X ∨ Q X := Iff.rfl\n\ninstance nonempty_sup_left [P.Nonempty] : (P ⊔ Q).Nonempty :=\n  nonempty_of_prop (Or.inl P.prop_arbitrary)\n\ninstance nonempty_sup_right [Q.Nonempty] : (P ⊔ Q).Nonempty :=\n  nonempty_of_prop (Or.inr Q.prop_arbitrary)\n\ninstance nonempty_top [Nonempty C] : (⊤ : ObjectProperty C).Nonempty :=\n  nonempty_of_prop (X := Classical.arbitrary C) (by trivial)\n\nlemma isoClosure_sup : (P ⊔ Q).isoClosure = P.isoClosure ⊔ Q.isoClosure := by\n  ext X\n  simp only [prop_sup_iff]\n  constructor\n  · rintro ⟨Y, hY, ⟨e⟩⟩\n    simp only [prop_sup_iff] at hY\n    obtain hY | hY := hY\n    · exact Or.inl ⟨Y, hY, ⟨e⟩⟩\n    · exact Or.inr ⟨Y, hY, ⟨e⟩⟩\n  · rintro (hY | hY)\n    · exact monotone_isoClosure le_sup_left _ hY\n    · exact monotone_isoClosure le_sup_right _ hY\n\ninstance [P.IsClosedUnderIsomorphisms] [Q.IsClosedUnderIsomorphisms] :\n    (P ⊔ Q).IsClosedUnderIsomorphisms := by\n  simp only [isClosedUnderIsomorphisms_iff_isoClosure_eq_self, isoClosure_sup, isoClosure_eq_self]\n\ninstance [P.IsClosedUnderIsomorphisms] [Q.IsClosedUnderIsomorphisms] :\n    IsClosedUnderIsomorphisms (P ⊓ Q) where\n  of_iso e h := ⟨IsClosedUnderIsomorphisms.of_iso e h.1, IsClosedUnderIsomorphisms.of_iso e h.2⟩\n\ninstance : IsClosedUnderIsomorphisms (⊥ : ObjectProperty C) where\n  of_iso _ h := h\n\ninstance : IsClosedUnderIsomorphisms (⊤ : ObjectProperty C) where\n  of_iso := by simp\n\nend\n\nsection\n\nvariable {α : Sort*} (P : α → ObjectProperty C) (X : C)\n\n@[simp high] lemma prop_iSup_iff :\n    (⨆ (a : α), P a) X ↔ ∃ (a : α), P a X := by simp\n\nlemma nonempty_iSup (a : α) [(P a).Nonempty] : (⨆ a, P a).Nonempty :=\n  nonempty_of_prop ((prop_iSup_iff P _).mpr ⟨a, (P a).prop_arbitrary⟩)\n\nlemma isoClosure_iSup :\n    ((⨆ (a : α), P a)).isoClosure = ⨆ (a : α), (P a).isoClosure := by\n  refine le_antisymm ?_ ?_\n  · rintro X ⟨Y, hY, ⟨e⟩⟩\n    simp only [prop_iSup_iff] at hY ⊢\n    obtain ⟨a, hY⟩ := hY\n    exact ⟨a, _, hY, ⟨e⟩⟩\n  · simp only [iSup_le_iff]\n    intro a\n    rw [isoClosure_le_iff]\n    exact (le_iSup P a).trans (le_isoClosure _)\n\ninstance [∀ a, (P a).IsClosedUnderIsomorphisms] :\n    ((⨆ (a : α), P a)).IsClosedUnderIsomorphisms := by\n  simp only [isClosedUnderIsomorphisms_iff_isoClosure_eq_self,\n    isoClosure_iSup, isoClosure_eq_self]\n\nend\n\n@[push]\n\nTarget:\nlemma ne_bot_iff_exists (P : ObjectProperty C) : ¬ P = ⊥ ↔ ∃ X, P X :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/ObjectProperty","family_id":"ne_bot_iff_exists","file_id":"mathlib/Mathlib/CategoryTheory/ObjectProperty/CompleteLattice.lean","sample_id":"27c82bde5cab40ddeb3cf4fe4ff5388002b1c6d7395440a1ff8ff9c86f617bfd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f18be9dd1c59e0ebc0b61eaff66514bafe594968c74bd37a4cddbaa1b0ea3c89","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9d5cb2b7c8e2dbd23ec1db01a351a3a7f7fd0db2101d28da1c5ca78815614d0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"197f653d4b36fd6849b8b6ebccef410407e7a70fbd810f7a0a75770cb14b42e2","source_sha256":"bf4a6ea9ec041102f48b300b957ea9d15b34197752c294a77f2238db87be2629","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rintro _ (rfl | rfl | rfl) _ (rfl | rfl | rfl) h <;>\n  simp only\n    [cast_neg, cast_one, cast_zero, neg_eq_zero, one_ne_zero, zero_ne_one, zero_eq_neg] at h ⊢\n  · exact ((Ring.neg_one_ne_one_of_char_ne_two hR).symm h).elim\n  · exact ((Ring.neg_one_ne_one_of_char_ne_two hR) h).elim","hard_negative":true,"metrics":{"chosen_tokens":70,"rejected_tokens":3,"token_jaccard":0.058824,"token_length_ratio":0.042857},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"4f997d32f6a9e9c1a1b1a282bf56045dd32333f55af9dd727bd5140b7df4f56c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharP.Defs\npublic import Mathlib.Algebra.Group.Fin.Basic\npublic import Mathlib.Algebra.Ring.ULift\npublic import Mathlib.Algebra.Ring.Opposite\npublic import Mathlib.Data.Int.ModEq\npublic import Mathlib.Data.Nat.Cast.Prod\npublic import Mathlib.Data.ULift\npublic import Mathlib.Order.Interval.Set.Defs\npublic import Mathlib.Algebra.Ring.GrindInstances\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Joey van Langen, Casper Putz\n-/\n/-!\n# Characteristic of semirings\n\nThis file collects some fundamental results on the characteristic of rings that don't need the extra\nimports of `Mathlib/Algebra/CharP/Lemmas.lean`.\n\nAs such, we can probably reorganize and find a better home for most of these lemmas.\n-/\n\npublic section\n\nassert_not_exists Finset TwoSidedIdeal\n\nopen Set\n\nvariable (R : Type*)\n\nnamespace CharP\nsection AddMonoidWithOne\nvariable [AddMonoidWithOne R] (p : ℕ)\n\nvariable [CharP R p] {a b : ℕ}\n\nlemma natCast_eq_natCast' (h : a ≡ b [MOD p]) : (a : R) = b := by\n  wlog hle : a ≤ b\n  · exact (this R p h.symm (le_of_not_ge hle)).symm\n  rw [Nat.modEq_iff_dvd' hle] at h\n  rw [← Nat.sub_add_cancel hle, Nat.cast_add, (cast_eq_zero_iff R p _).mpr h, zero_add]\n\nlemma natCast_eq_natCast_mod (a : ℕ) : (a : R) = a % p :=\n  natCast_eq_natCast' R p (Nat.mod_modEq a p).symm\n\nvariable [IsRightCancelAdd R]\n\nlemma natCast_eq_natCast : (a : R) = b ↔ a ≡ b [MOD p] := by\n  wlog hle : a ≤ b\n  · rw [eq_comm, this R p (le_of_not_ge hle), Nat.ModEq.comm]\n  rw [Nat.modEq_iff_dvd' hle, ← cast_eq_zero_iff R p (b - a),\n    ← add_right_cancel_iff (G := R) (a := a) (b := b - a), zero_add, ← Nat.cast_add,\n    Nat.sub_add_cancel hle, eq_comm]\n\nlemma natCast_injOn_Iio : (Set.Iio p).InjOn ((↑) : ℕ → R) :=\n  fun _a ha _b hb hab ↦ ((natCast_eq_natCast _ _).1 hab).eq_of_lt_of_lt ha hb\n\nend AddMonoidWithOne\n\nsection AddGroupWithOne\nvariable [AddGroupWithOne R] (p : ℕ) [CharP R p] {a b : ℤ}\n\nlemma intCast_eq_intCast : (a : R) = b ↔ a ≡ b [ZMOD p] := by\n  rw [eq_comm, ← sub_eq_zero, ← Int.cast_sub, CharP.intCast_eq_zero_iff R p, Int.modEq_iff_dvd]\n\nlemma intCast_eq_intCast_mod : (a : R) = a % (p : ℤ) :=\n  (CharP.intCast_eq_intCast R p).mpr (Int.mod_modEq a p).symm\n\nlemma intCast_injOn_Ico [IsRightCancelAdd R] : InjOn (Int.cast : ℤ → R) (Ico 0 p) := by\n  rintro a ⟨ha₀, ha⟩ b ⟨hb₀, hb⟩ hab\n  lift a to ℕ using ha₀\n  lift b to ℕ using hb₀\n  norm_cast at *\n  exact natCast_injOn_Iio _ _ ha hb hab\n\nend AddGroupWithOne\nend CharP\n\nnamespace CharP\n\nsection NonAssocSemiring\n\nvariable {R} [NonAssocSemiring R]\n\nvariable (R) in\n/-- If a ring `R` is of characteristic `p`, then for any prime number `q` different from `p`,\nit is not zero in `R`. -/\nlemma cast_ne_zero_of_ne_of_prime [Nontrivial R]\n    {p q : ℕ} [CharP R p] (hq : q.Prime) (hneq : p ≠ q) : (q : R) ≠ 0 := fun h ↦ by\n  rw [cast_eq_zero_iff R p q] at h\n  rcases hq.eq_one_or_self_of_dvd _ h with rfl | h\n  · exact false_of_nontrivial_of_char_one (R := R)\n  · exact hneq h\n\nlemma ringChar_of_prime_eq_zero [Nontrivial R] {p : ℕ} (hprime : Nat.Prime p)\n    (hp0 : (p : R) = 0) : ringChar R = p :=\n  Or.resolve_left ((Nat.dvd_prime hprime).1 (ringChar.dvd hp0)) ringChar_ne_one\n\nlemma charP_iff_prime_eq_zero [Nontrivial R] {p : ℕ} (hp : p.Prime) :\n    CharP R p ↔ (p : R) = 0 :=\n  ⟨fun _ => cast_eq_zero R p,\n   fun hp0 => (ringChar_of_prime_eq_zero hp hp0) ▸ inferInstance⟩\n\nend NonAssocSemiring\nend CharP\n\nsection\n\n/-- We have `2 ≠ 0` in a nontrivial ring whose characteristic is not `2`. -/\nprotected lemma Ring.two_ne_zero {R : Type*} [NonAssocSemiring R] [Nontrivial R]\n    (hR : ringChar R ≠ 2) : (2 : R) ≠ 0 := by\n  rw [Ne, (by norm_cast : (2 : R) = (2 : ℕ)), ringChar.spec, Nat.dvd_prime Nat.prime_two]\n  exact mt (or_iff_left hR).mp CharP.ringChar_ne_one\n\n-- We have `CharP.neg_one_ne_one`, which assumes `[Ring R] (p : ℕ) [CharP R p] [Fact (2 < p)]`.\n-- This is a version using `ringChar` instead.\n/-- Characteristic `≠ 2` and nontrivial implies that `-1 ≠ 1`. -/\nlemma Ring.neg_one_ne_one_of_char_ne_two {R : Type*} [NonAssocRing R] [Nontrivial R]\n    (hR : ringChar R ≠ 2) : (-1 : R) ≠ 1 := fun h =>\n  Ring.two_ne_zero hR (one_add_one_eq_two (R := R) ▸ neg_eq_iff_add_eq_zero.mp h)\n\n/-- Characteristic `≠ 2` in a domain implies that `-a = a` iff `a = 0`. -/\nlemma Ring.eq_self_iff_eq_zero_of_char_ne_two {R : Type*} [NonAssocRing R] [Nontrivial R]\n    [NoZeroDivisors R] (hR : ringChar R ≠ 2) {a : R} : -a = a ↔ a = 0 :=\n  ⟨fun h =>\n    (mul_eq_zero.mp <| (two_mul a).trans <| neg_eq_iff_add_eq_zero.mp h).resolve_left\n      (Ring.two_ne_zero hR),\n    fun h => ((congr_arg (fun x => -x) h).trans neg_zero).trans h.symm⟩\n\nend\n\nsection Prod\nvariable (S : Type*) [AddMonoidWithOne R] [AddMonoidWithOne S] (p q : ℕ) [CharP R p]\n\n/-- The characteristic of the product of rings is the least common multiple of the\ncharacteristics of the two rings. -/\ninstance Nat.lcm.charP [CharP S q] : CharP (R × S) (Nat.lcm p q) where\n  cast_eq_zero_iff := by\n    simp [Prod.ext_iff, CharP.cast_eq_zero_iff R p, CharP.cast_eq_zero_iff S q, Nat.lcm_dvd_iff]\n\n/-- The characteristic of the product of two rings of the same characteristic\n  is the same as the characteristic of the rings -/\ninstance Prod.charP [CharP S p] : CharP (R × S) p := by\n  convert! Nat.lcm.charP R S p p; simp\n\ninstance Prod.charZero_of_left [CharZero R] : CharZero (R × S) where\n  cast_injective _ _ h := CharZero.cast_injective congr(Prod.fst $h)\n\ninstance Prod.charZero_of_right [CharZero S] : CharZero (R × S) where\n  cast_injective _ _ h := CharZero.cast_injective congr(Prod.snd $h)\n\nend Prod\n\ninstance ULift.charP [AddMonoidWithOne R] (p : ℕ) [CharP R p] : CharP (ULift R) p where\n  cast_eq_zero_iff n := Iff.trans ULift.ext_iff <| CharP.cast_eq_zero_iff R p n\n\ninstance MulOpposite.charP [AddMonoidWithOne R] (p : ℕ) [CharP R p] : CharP Rᵐᵒᵖ p where\n  cast_eq_zero_iff n := MulOpposite.unop_inj.symm.trans <| CharP.cast_eq_zero_iff R p n\n\nsection\n\n/-- If two integers from `{0, 1, -1}` result in equal elements in a ring `R`\nthat is nontrivial and of characteristic not `2`, then they are equal. -/\n\nTarget:\nlemma Int.cast_injOn_of_ringChar_ne_two {R : Type*} [NonAssocRing R] [Nontrivial R]\n    (hR : ringChar R ≠ 2) : ({0, 1, -1} : Set ℤ).InjOn ((↑) : ℤ → R) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_197f653d4b36","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"9a7d37ce0fcd0538d172ea0a9f2be0082afe1edc35fbb751a0c4e16175feeb12","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/CharP","family_id":"int","file_id":"mathlib/Mathlib/Algebra/CharP/Basic.lean","sample_id":"197f653d4b36fd6849b8b6ebccef410407e7a70fbd810f7a0a75770cb14b42e2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0a479f8dae478b3bb9603d60f313d74028b2da15cce1a50538dcbe628c6b865b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7710ac4a7094e4ea990b13bcd33210bc3b8c0f405e8e432edd0f2698a46a0814","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2b9e2ca58e134ad4578da49d74b391a4ef05b73b213f399f3d3257947965fc7d","source_sha256":"cbd935405b8857ab21a8766b03dbdc75523269b01677822430d1eb4f04caf0ba","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine (QuadraticMap.associated_rightInverse R).injective ?_\n  ext m₁ m₂ m₁' m₂' m₁'' m₂''\n  simp only [associated_tmul, QuadraticMap.associated_comp]\n  exact mul_assoc _ _ _","hard_negative":true,"metrics":{"chosen_tokens":45,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.066667},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"4fbaf64e58630e632255af040534dde4a60e5559f2af006655541cf23a13a960","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct\npublic import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv\n\nNamespace:\nQuadraticForm\n\nLocal context:\n/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Linear equivalences of tensor products as isometries\n\nThese results are separate from the definition of `QuadraticForm.tmul` as that file is very slow.\n\n## Main definitions\n\n* `QuadraticForm.Isometry.tmul`: `TensorProduct.map` as a `QuadraticForm.Isometry`\n* `QuadraticForm.tensorComm`: `TensorProduct.comm` as a `QuadraticForm.IsometryEquiv`\n* `QuadraticForm.tensorAssoc`: `TensorProduct.assoc` as a `QuadraticForm.IsometryEquiv`\n* `QuadraticForm.tensorRId`: `TensorProduct.rid` as a `QuadraticForm.IsometryEquiv`\n* `QuadraticForm.tensorLId`: `TensorProduct.lid` as a `QuadraticForm.IsometryEquiv`\n-/\n\n@[expose] public section\n\nuniverse uR uM₁ uM₂ uM₃ uM₄\nvariable {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} {M₄ : Type uM₄}\n\nopen scoped TensorProduct\n\nopen QuadraticMap\n\nnamespace QuadraticForm\n\nvariable [CommRing R]\nvariable [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M₄]\nvariable [Module R M₁] [Module R M₂] [Module R M₃] [Module R M₄] [Invertible (2 : R)]\n\n@[simp]\ntheorem tmul_comp_tensorMap\n    {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂}\n    {Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄}\n    (f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) :\n    (Q₂.tmul Q₄).comp (TensorProduct.map f.toLinearMap g.toLinearMap) = Q₁.tmul Q₃ := by\n  have h₁ : Q₁ = Q₂.comp f.toLinearMap := QuadraticMap.ext fun x => (f.map_app x).symm\n  have h₃ : Q₃ = Q₄.comp g.toLinearMap := QuadraticMap.ext fun x => (g.map_app x).symm\n  refine (QuadraticMap.associated_rightInverse R).injective ?_\n  ext m₁ m₃ m₁' m₃'\n  simp [h₁, h₃, associated_tmul]\n\n@[simp]\ntheorem tmul_tensorMap_apply\n    {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂}\n    {Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄}\n    (f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) (x : M₁ ⊗[R] M₃) :\n    Q₂.tmul Q₄ (TensorProduct.map f.toLinearMap g.toLinearMap x) = Q₁.tmul Q₃ x :=\n  DFunLike.congr_fun (tmul_comp_tensorMap f g) x\n\nnamespace Isometry\n\n/-- `TensorProduct.map` for `QuadraticForm.Isometry`s -/\ndef _root_.QuadraticMap.Isometry.tmul\n    {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂}\n    {Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄}\n    (f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) : (Q₁.tmul Q₃) →qᵢ (Q₂.tmul Q₄) where\n  toLinearMap := TensorProduct.map f.toLinearMap g.toLinearMap\n  map_app' := tmul_tensorMap_apply f g\n\n@[simp]\ntheorem _root_.QuadraticMap.Isometry.tmul_apply\n    {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂}\n    {Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄}\n    (f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) (x : M₁ ⊗[R] M₃) :\n    f.tmul g x = TensorProduct.map f.toLinearMap g.toLinearMap x :=\n  rfl\n\nend Isometry\n\nsection tensorComm\n\n@[simp]\ntheorem tmul_comp_tensorComm (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) :\n    (Q₂.tmul Q₁).comp (TensorProduct.comm R M₁ M₂) = Q₁.tmul Q₂ := by\n  refine (QuadraticMap.associated_rightInverse R).injective ?_\n  ext m₁ m₂ m₁' m₂'\n  simp only [associated_tmul, QuadraticMap.associated_comp]\n  exact mul_comm _ _\n\n@[simp]\ntheorem tmul_tensorComm_apply\n    (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (x : M₁ ⊗[R] M₂) :\n    Q₂.tmul Q₁ (TensorProduct.comm R M₁ M₂ x) = Q₁.tmul Q₂ x :=\n  DFunLike.congr_fun (tmul_comp_tensorComm Q₁ Q₂) x\n\n/-- `TensorProduct.comm` preserves tensor products of quadratic forms. -/\n@[simps toLinearEquiv]\ndef tensorComm (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) :\n    (Q₁.tmul Q₂).IsometryEquiv (Q₂.tmul Q₁) where\n  toLinearEquiv := TensorProduct.comm R M₁ M₂\n  map_app' := tmul_tensorComm_apply Q₁ Q₂\n\n@[simp] lemma tensorComm_apply (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂)\n    (x : M₁ ⊗[R] M₂) :\n    tensorComm Q₁ Q₂ x = TensorProduct.comm R M₁ M₂ x :=\n  rfl\n\n@[simp] lemma tensorComm_symm (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) :\n    (tensorComm Q₁ Q₂).symm = tensorComm Q₂ Q₁ :=\n  rfl\n\nend tensorComm\n\nsection tensorAssoc\n\n@[simp]\n\nTarget:\ntheorem tmul_comp_tensorAssoc\n    (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) (Q₃ : QuadraticForm R M₃) :\n    (Q₁.tmul (Q₂.tmul Q₃)).comp (TensorProduct.assoc R M₁ M₂ M₃) = (Q₁.tmul Q₂).tmul Q₃ :=\n\nProof body:\n","rejected":"by\n  exact tmul_comp_tensorAssoc","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"bfa7f0f43e92f80671448652ab65d12f3f9c80d0e762ccaeff7e1e2176fd5e69","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/QuadraticForm","family_id":"tmul_comp_tensorassoc","file_id":"mathlib/Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean","sample_id":"2b9e2ca58e134ad4578da49d74b391a4ef05b73b213f399f3d3257947965fc7d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"576771f2ca0b7fae2b561a23269a331367199c4b2800aa18415fec083cc832ae","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a6cca448626f77dd5c9ef36b63004c75746e4d1ca163562a1109c588ccb9df23","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"be9b8d29bc8008d9f2006f7c0e2e488c12668f4a1e6f2ba4ffcf6937f3ace739","source_sha256":"d0eafab6c9f23d1000e4c23ec8fbc2844b38b0956624818eec3c9dbb37162d5b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [liftOfDerivationToSquareZero_apply, map_add, Ideal.Quotient.eq_zero_iff_mem.mpr (d x).prop,\n    zero_add, Ideal.Quotient.mk_algebraMap]","hard_negative":true,"metrics":{"chosen_tokens":29,"rejected_tokens":3,"token_jaccard":0.047619,"token_length_ratio":0.103448},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"4fdf9fd54987d81d04bd20ee5108d27ba9257b0f1acf56094e413d25cea78119","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Derivation.Basic\npublic import Mathlib.RingTheory.Ideal.Quotient.Operations\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Nicolò Cavalleri. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Nicolò Cavalleri, Andrew Yang\n-/\n/-!\n# Derivations into Square-Zero Ideals\n\n## Main statements\n\n- `derivationToSquareZeroOfLift`: The `R`-derivations from `A` into a square-zero ideal `I`\n  of `B` corresponds to the lifts `A →ₐ[R] B` of the map `A →ₐ[R] B ⧸ I`.\n\n-/\n\n@[expose] public section\n\n\nsection ToSquareZero\n\nuniverse u v w\n\nvariable {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [CommRing B]\nvariable [Algebra R A] [Algebra R B] (I : Ideal B)\n\n/-- If `f₁ f₂ : A →ₐ[R] B` are two lifts of the same `A →ₐ[R] B ⧸ I`,\n  we may define a map `f₁ - f₂ : A →ₗ[R] I`. -/\ndef diffToIdealOfQuotientCompEq (f₁ f₂ : A →ₐ[R] B)\n    (e : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂) : A →ₗ[R] I :=\n  LinearMap.codRestrict (I.restrictScalars _) (f₁.toLinearMap - f₂.toLinearMap)\n    (fun x => by simpa [Ideal.Quotient.eq] using congr($e x))\n\n@[simp]\ntheorem diffToIdealOfQuotientCompEq_apply (f₁ f₂ : A →ₐ[R] B)\n    (e : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂) (x : A) :\n    ((diffToIdealOfQuotientCompEq I f₁ f₂ e) x : B) = f₁ x - f₂ x :=\n  rfl\n\nvariable [Algebra A B]\n\n/-- Given a tower of algebras `R → A → B`, and a square-zero `I : Ideal B`, each lift `A →ₐ[R] B`\nof the canonical map `A →ₐ[R] B ⧸ I` corresponds to an `R`-derivation from `A` to `I`. -/\ndef derivationToSquareZeroOfLift [IsScalarTower R A B] (hI : I ^ 2 = ⊥) (f : A →ₐ[R] B)\n    (e : (Ideal.Quotient.mkₐ R I).comp f = IsScalarTower.toAlgHom R A (B ⧸ I)) :\n    Derivation R A I := by\n  refine\n    { diffToIdealOfQuotientCompEq I f (IsScalarTower.toAlgHom R A B) ?_ with\n      map_one_eq_zero' := ?_\n      leibniz' := ?_ }\n  · ext; simp [e]\n  · ext; simp\n  · intro x y\n    let F := diffToIdealOfQuotientCompEq I f (IsScalarTower.toAlgHom R A B) (by rw [e]; ext; rfl)\n    have : (f x - algebraMap A B x) * (f y - algebraMap A B y) = 0 := by\n      rw [← Ideal.mem_bot, ← hI, pow_two]\n      convert! Ideal.mul_mem_mul (F x).2 (F y).2 using 1\n    ext\n    dsimp only [Submodule.coe_add, Submodule.coe_mk, LinearMap.coe_mk,\n      diffToIdealOfQuotientCompEq_apply, Submodule.coe_smul_of_tower, IsScalarTower.coe_toAlgHom',\n      LinearMap.toFun_eq_coe]\n    simp only [map_mul, sub_mul, mul_sub, Algebra.smul_def] at this ⊢\n    rw [sub_eq_iff_eq_add, sub_eq_iff_eq_add] at this\n    simp only [this]\n    ring\n\nvariable (hI : I ^ 2 = ⊥)\n\ntheorem derivationToSquareZeroOfLift_apply [IsScalarTower R A B] (f : A →ₐ[R] B)\n    (e : (Ideal.Quotient.mkₐ R I).comp f = IsScalarTower.toAlgHom R A (B ⧸ I)) (x : A) :\n    (derivationToSquareZeroOfLift I hI f e x : B) = f x - algebraMap A B x :=\n  rfl\n\n/-- Given a tower of algebras `R → A → B`, and a square-zero `I : Ideal B`, each `R`-derivation\nfrom `A` to `I` corresponds to a lift `A →ₐ[R] B` of the canonical map `A →ₐ[R] B ⧸ I`. -/\n@[simps -isSimp]\ndef liftOfDerivationToSquareZero [IsScalarTower R A B] (hI : I ^ 2 = ⊥) (f : Derivation R A I) :\n    A →ₐ[R] B :=\n  { ((I.restrictScalars R).subtype.comp f.toLinearMap + (IsScalarTower.toAlgHom R A B).toLinearMap :\n      A →ₗ[R] B) with\n    toFun := fun x => f x + algebraMap A B x\n    map_one' := by\n      rw [map_one (algebraMap _ _), f.map_one_eq_zero, Submodule.coe_zero, zero_add]\n    map_mul' := fun x y => by\n      have : (f x : B) * f y = 0 := by\n        rw [← Ideal.mem_bot, ← hI, pow_two]\n        convert! Ideal.mul_mem_mul (f x).2 (f y).2 using 1\n      simp only [map_mul, f.leibniz, add_mul, mul_add, Submodule.coe_add,\n        Submodule.coe_smul_of_tower, Algebra.smul_def, this]\n      ring\n    commutes' := fun r => by\n      simp only [Derivation.map_algebraMap, zero_add, Submodule.coe_zero, ←\n        IsScalarTower.algebraMap_apply R A B r]\n    map_zero' := ((I.restrictScalars R).subtype.comp f.toLinearMap +\n      (IsScalarTower.toAlgHom R A B).toLinearMap).map_zero }\n\n-- simp normal form is `liftOfDerivationToSquareZero_mk_apply'`\n\nTarget:\ntheorem liftOfDerivationToSquareZero_mk_apply [IsScalarTower R A B] (d : Derivation R A I) (x : A) :\n    Ideal.Quotient.mk I (liftOfDerivationToSquareZero I hI d x) = algebraMap A (B ⧸ I) x :=\n\nProof body:\n","rejected":"by\n  exact liftOfDerivationToSquareZero_mk_apply","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"ce2bff483618949a061787dc5297ae0e80e0a2ac0e886162d158743dd1da8aad","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Derivation","family_id":"liftofderivationtosquarezero_mk_apply","file_id":"mathlib/Mathlib/RingTheory/Derivation/ToSquareZero.lean","sample_id":"be9b8d29bc8008d9f2006f7c0e2e488c12668f4a1e6f2ba4ffcf6937f3ace739"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e26bc9c73cc42d866d4dadb94c4f9430e4046c553d052939bf6c8d1e66df3de3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"68566ca8925ab58d94e66a6a28da0a7a652742ea19b39e6ea230c9f35dea4715","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bad13452563a2e4433cf2964ac3661b1745a1d26c8fd4dce023319c5196a9b1e","source_sha256":"1c7994cb953bf15a3e2a75bf5a9f75792705603cc1fc781f9e07bc585e941697","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine hK.induction_on ⟨∅, by simp⟩ (fun s t hst ⟨W, Wc, Wo, hKW, hWU⟩ ↦ ?_) ?_ ?_\n  · use W, Wc, Wo, subset_trans hst hKW, hWU\n  · intro s t ⟨W₁, Wc₁, Wo₁, hKW₁, hWU₁⟩ ⟨W₂, Wc₂, Wo₂, hKW₂, hWU₂⟩\n    exact ⟨W₁ ∪ W₂, Wc₁.union Wc₂, Wo₁.union Wo₂, Set.union_subset_union hKW₁ hKW₂,\n      Set.union_subset hWU₁ hWU₂⟩\n  · intro x hx\n    obtain ⟨V, h, hxV, hVU⟩ :=\n      PrespectralSpace.isTopologicalBasis.exists_subset_of_mem_open (hKU hx) hU\n    exact ⟨V, mem_nhdsWithin.mpr ⟨V, h.1, hxV, Set.inter_subset_left⟩, V, h.2, h.1, subset_rfl, hVU⟩","hard_negative":false,"metrics":{"chosen_tokens":182,"rejected_tokens":189,"token_jaccard":0.964912,"token_length_ratio":1.038462},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"5007e6af0957866990d0d1cb0704b1e97e392129502d020f571c5b416598cfa1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Ideal\npublic import Mathlib.Topology.Sets.Compacts\npublic import Mathlib.Topology.Sets.OpenCover\npublic import Mathlib.Topology.Spectral.Hom\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Prespectral spaces\n\nIn this file, we define prespectral spaces as spaces whose lattice of compact opens forms a basis.\n\n-/\n\n@[expose] public section\n\nopen TopologicalSpace Topology\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n\n/-- A space is prespectral if the lattice of compact opens forms a basis. -/\n@[stacks 08YG \"The last condition for spectral spaces\", mk_iff]\nclass PrespectralSpace (X : Type*) [TopologicalSpace X] : Prop where\n  isTopologicalBasis : IsTopologicalBasis { U : Set X | IsOpen U ∧ IsCompact U }\n\n/-- A space is prespectral if it has a basis consisting of compact opens. -/\nlemma PrespectralSpace.of_isTopologicalBasis {B : Set (Set X)}\n    (basis : IsTopologicalBasis B) (isCompact_basis : ∀ U ∈ B, IsCompact U) :\n    PrespectralSpace X where\n  isTopologicalBasis := basis.of_isOpen_of_subset (fun _ h ↦ h.1)\n    fun s hs ↦ ⟨basis.isOpen hs, isCompact_basis s hs⟩\n\n/-- A space is prespectral if it has a basis consisting of compact opens.\nThis is the variant with an indexed basis instead. -/\nlemma PrespectralSpace.of_isTopologicalBasis' {ι : Type*} {b : ι → Set X}\n    (basis : IsTopologicalBasis (Set.range b)) (isCompact_basis : ∀ i, IsCompact (b i)) :\n    PrespectralSpace X :=\n  .of_isTopologicalBasis basis (by simp_all)\n\ninstance (priority := low) [NoetherianSpace X] : PrespectralSpace X :=\n  .of_isTopologicalBasis isTopologicalBasis_opens fun _ _ ↦ NoetherianSpace.isCompact _\n\ninstance (priority := low) [PrespectralSpace X] : LocallyCompactSpace X where\n  local_compact_nhds _ _ hn :=\n    have ⟨V, ⟨hV₁, hV₂⟩, hxV, hVn⟩ := PrespectralSpace.isTopologicalBasis.mem_nhds_iff.mp hn\n    ⟨V, hV₁.mem_nhds hxV, hVn, hV₂⟩\n\nopen PrespectralSpace in\ninstance (priority := low) [T2Space X] [PrespectralSpace X] : TotallySeparatedSpace X :=\n  totallySeparatedSpace_iff_exists_isClopen.mpr fun _ _ hxy ↦\n    have ⟨U, ⟨hU₁, hU₂⟩, hxU, hyU⟩ :=\n      isTopologicalBasis.exists_subset_of_mem_open hxy isClosed_singleton.isOpen_compl\n    ⟨U, ⟨hU₂.isClosed, hU₁⟩, hxU, fun h ↦ hyU h rfl⟩\n\nlemma PrespectralSpace.of_isOpenCover\n    {ι : Type*} {U : ι → Opens X} (hU : IsOpenCover U) [∀ i, PrespectralSpace (U i)] :\n    PrespectralSpace X := by\n  refine .of_isTopologicalBasis (hU.isTopologicalBasis fun i ↦ isTopologicalBasis) ?_\n  simp only [Set.mem_iUnion, Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp,\n    forall_comm (α := Set _), forall_apply_eq_imp_iff₂]\n  exact fun i V hV hV' ↦ hV'.image continuous_subtype_val\n\nlemma PrespectralSpace.of_isInducing [PrespectralSpace Y]\n    (f : X → Y) (hf : IsInducing f) (hf' : IsSpectralMap f) : PrespectralSpace X :=\n  .of_isTopologicalBasis (PrespectralSpace.isTopologicalBasis.isInducing hf) (by\n    simp only [Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp]\n    rintro _ U h₁ h₂ rfl\n    exact hf'.isCompact_preimage_of_isOpen h₁ h₂)\n\nlemma PrespectralSpace.of_isClosedEmbedding [PrespectralSpace Y]\n    (f : X → Y) (hf : IsClosedEmbedding f) : PrespectralSpace X :=\n  .of_isInducing f hf.isInducing hf.isProperMap.isSpectralMap\n\n/-- Let `f : X → Y` be an open embedding of topological spaces.\nIf `Y` is a prespectral space (i.e., the quasi-compact opens of `Y` form a basis),\nthen `X` is also a prespectral space. -/\nlemma Topology.IsOpenEmbedding.prespectralSpace [PrespectralSpace Y]\n    {f : X → Y} (hf : IsOpenEmbedding f) :\n    PrespectralSpace X where\n  isTopologicalBasis := by\n    apply isTopologicalBasis_of_isOpen_of_nhds (fun U hU ↦ hU.1) <| fun x U hx hU ↦ ?_\n    obtain ⟨V, ⟨hoV, hcV⟩, hfx, hVf⟩ : ∃ V ∈ {V | IsOpen V ∧ IsCompact V}, f x ∈ V ∧ V ⊆ f '' U :=\n      (PrespectralSpace.isTopologicalBasis (X := Y)).isOpen_iff.mp\n        (hf.isOpen_iff_image_isOpen.mp hU) (f x) ⟨x, hx, rfl⟩\n    refine ⟨f ⁻¹' V, ⟨hoV.preimage hf.continuous, ?_⟩, ⟨hfx, fun y hy ↦ ?_⟩⟩\n    · exact hf.toIsInducing.isCompact_preimage' hcV <| Set.SurjOn.subset_range hVf\n    · exact hf.injective.mem_set_image.mp (hVf hy)\n\ninstance PrespectralSpace.sigma {ι : Type*} (X : ι → Type*) [∀ i, TopologicalSpace (X i)]\n    [∀ i, PrespectralSpace (X i)] : PrespectralSpace (Σ i, X i) :=\n  .of_isTopologicalBasis (IsTopologicalBasis.sigma fun i ↦ isTopologicalBasis) fun U hU ↦ by\n    simp_rw [Set.mem_iUnion] at hU\n    obtain ⟨i, V, hV, rfl⟩ := hU\n    exact hV.2.image continuous_sigmaMk\n\nvariable (X) in\nlemma PrespectralSpace.isBasis_opens [PrespectralSpace X] :\n    TopologicalSpace.Opens.IsBasis { U : Opens X | IsCompact (U : Set X) } := by\n  dsimp only [TopologicalSpace.Opens.IsBasis]\n  convert! isTopologicalBasis (X := X)\n  ext s\n  exact ⟨fun ⟨V, hV, heq⟩ ↦ heq ▸ ⟨V.2, hV⟩, fun h ↦ ⟨⟨s, h.1⟩, h.2, rfl⟩⟩\n\n/-- In a prespectral space, the lattice of opens is determined by its lattice of compact opens. -/\ndef PrespectralSpace.opensEquiv [PrespectralSpace X] :\n    Opens X ≃o Order.Ideal (CompactOpens X) where\n  toFun U := ⟨⟨{ V | (V : Set X) ⊆ U }, fun U₁ U₂ h₁ h₂ ↦ subset_trans (α := Set X) h₁ h₂⟩,\n    ⟨⊥, by simp⟩, fun U₁ h₁ U₂ h₂ ↦ ⟨U₁ ⊔ U₂, by aesop, le_sup_left, le_sup_right⟩⟩\n  invFun I := ⨆ U ∈ I, U.toOpens\n  left_inv U := by\n    apply le_antisymm\n    · simp only [iSup_le_iff]\n      exact fun _ ↦ id\n    · intro x hxU\n      obtain ⟨V, ⟨h₁, h₂⟩, hxV, hVU⟩ := isTopologicalBasis.exists_subset_of_mem_open hxU U.2\n      simp only [Opens.mem_iSup]\n      exact ⟨⟨⟨_, h₂⟩, h₁⟩, hVU, hxV⟩\n  right_inv I := by\n    ext U\n    dsimp\n    change U.toOpens ≤ _ ↔ _\n    refine ⟨fun H ↦ ?_, fun h ↦ le_iSup₂ (f := fun U (h : U ∈ I) ↦ U.toOpens) U h⟩\n    simp only [← SetLike.coe_subset_coe, Opens.iSup_mk, Opens.carrier_eq_coe, Opens.coe_mk] at H\n    obtain ⟨s, hsI, hs, hU⟩ := U.isCompact.elim_finite_subcover_image (fun U _ ↦ U.2) H\n    exact I.lower (a := hs.toFinset.sup fun i ↦ i) (by simpa [← SetLike.coe_subset_coe]) (by simpa)\n  map_rel_iff' {U V} := by\n    change (∀ (W : CompactOpens X), (W : Set X) ⊆ U → (W : Set X) ⊆ V) ↔ U ≤ V\n    refine ⟨?_, fun H W ↦ (le_trans · H)⟩\n    intro H x hxU\n    obtain ⟨W, ⟨h₁, h₂⟩, hxW, hWU⟩ := isTopologicalBasis.exists_subset_of_mem_open hxU U.2\n    exact H ⟨⟨W, h₂⟩, h₁⟩ hWU hxW\n\nopen TopologicalSpace Opens in\n/-- If `X` has a basis of compact opens and `f : X → S` is open, every\ncompact open of `S` is the image of a compact open of `X`. -/\nlemma IsOpenMap.exists_opens_image_eq_of_prespectralSpace [PrespectralSpace X] {f : X → Y}\n    (hfc : Continuous f) (h : IsOpenMap f) {U : Set Y} (hs : U ⊆ Set.range f) (hU : IsOpen U)\n    (hc : IsCompact U) : ∃ (V : Opens X), IsCompact V.1 ∧ f '' V = U := by\n  obtain ⟨Us, hUs, heq⟩ := TopologicalSpace.Opens.isBasis_iff_cover.mp\n    (PrespectralSpace.isBasis_opens X) ⟨f ⁻¹' U, hU.preimage hfc⟩\n  obtain ⟨t, ht⟩ := by\n    refine hc.elim_finite_subcover (fun s : Us ↦ f '' s.1) (fun s ↦ h _ s.1.2) (fun x hx ↦ ?_)\n    obtain ⟨x, rfl⟩ := hs hx\n    obtain ⟨i, hi, hx⟩ := mem_sSup.mp <| by rwa [← heq]\n    exact Set.mem_iUnion.mpr ⟨⟨i, hi⟩, x, hx, rfl⟩\n  refine ⟨⨆ s ∈ t, s.1, ?_, ?_⟩\n  · simp only [iSup_mk, carrier_eq_coe, coe_mk]\n    exact t.finite_toSet.isCompact_biUnion fun i _ ↦ hUs i.2\n  · simp only [iSup_mk, carrier_eq_coe, Set.iUnion_coe_set, coe_mk, Set.image_iUnion]\n    convert_to ⋃ i ∈ t, f '' i.1 = U\n    · simp\n    · refine subset_antisymm (fun x ↦ ?_) ht\n      simp_rw [Set.mem_iUnion]\n      rintro ⟨i, hi, x, hx, rfl⟩\n      have := heq ▸ mem_sSup.mpr ⟨i.1, i.2, hx⟩\n      exact this\n\nTarget:\nlemma PrespectralSpace.exists_isCompact_and_isOpen_between [PrespectralSpace X] {K U : Set X}\n    (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :\n    ∃ (W : Set X), IsCompact W ∧ IsOpen W ∧ K ⊆ W ∧ W ⊆ U :=\n\nProof body:\n","rejected":"```lean\nby\n  refine hK.induction_on ⟨∅, by simp⟩ (fun s t hst ⟨W, Wc, Wo, hKW, hWU⟩ ↦ ?_) ?_ ?_\n  · use W, Wc, Wo, subset_trans hst hKW, hWU\n  · intro s t ⟨W₁, Wc₁, Wo₁, hKW₁, hWU₁⟩ ⟨W₂, Wc₂, Wo₂, hKW₂, hWU₂⟩\n    exact ⟨W₁ ∪ W₂, Wc₁.union Wc₂, Wo₁.union Wo₂, Set.union_subset_union hKW₁ hKW₂,\n      Set.union_subset hWU₁ hWU₂⟩\n  · intro x hx\n    obtain ⟨V, h, hxV, hVU⟩ :=\n      PrespectralSpace.isTopologicalBasis.exists_subset_of_mem_open (hKU hx) hU\n    exact ⟨V, mem_nhdsWithin.mpr ⟨V, h.1, hxV, Set.inter_subset_left⟩, V, h.2, h.1, subset_rfl, hVU⟩\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Spectral","family_id":"prespectralspace","file_id":"mathlib/Mathlib/Topology/Spectral/Prespectral.lean","sample_id":"bad13452563a2e4433cf2964ac3661b1745a1d26c8fd4dce023319c5196a9b1e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"74ef8f872938daf0468322013395261e7267b7edbfb838bf7741225d397b7c55","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6f1c9a92ed629452f0151fcb0fc207570a587c02cdb1f6593cc28304e3ab24d9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"de57e8cad85d8b44648cf10f3d51ce2288da48a2805529ae83b12ceae714db81","source_sha256":"250e0bf39b2fceb88cc44db25e786e52b9f1629e4d745089c27db7ba4ae4e60e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← card k V]\n  have : Nat.card V > 0 := Nat.card_pos\n  lia","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":2,"token_jaccard":0.055556,"token_length_ratio":0.095238},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"501de5693f27c2c231cba4f85bd073d76ec4407768c35416bf51a367b6292408","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.GeomSum\npublic import Mathlib.Algebra.GroupWithZero.Units.Fintype\npublic import Mathlib.Data.Finite.Sum\npublic import Mathlib.GroupTheory.GroupAction.Quotient\npublic import Mathlib.LinearAlgebra.Projectivization.Basic\n\nNamespace:\nProjectivization\n\nLocal context:\n/-\nCopyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Judith Ludwig, Christian Merten\n-/\n/-!\n# Cardinality of projective spaces\n\nWe compute the cardinality of `ℙ k V` if `k` is a finite field.\n\n-/\n\n@[expose] public section\n\nnamespace Projectivization\n\nopen scoped LinearAlgebra.Projectivization\n\nsection\n\nvariable (k V : Type*) [DivisionRing k] [AddCommGroup V] [Module k V]\n\n/-- `ℙ k V` is equivalent to the quotient of the non-zero elements of `V` by `kˣ`. -/\ndef equivQuotientOrbitRel : ℙ k V ≃ Quotient (MulAction.orbitRel kˣ { v : V // v ≠ 0 }) :=\n  Quotient.congr (Equiv.refl _) (fun x y ↦ (Units.orbitRel_nonZero_iff k V x y).symm)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The non-zero elements of `V` are equivalent to the product of `ℙ k V` with the units of `k`. -/\nnoncomputable def nonZeroEquivProjectivizationProdUnits : { v : V // v ≠ 0 } ≃ ℙ k V × kˣ :=\n  let e := MulAction.selfEquivOrbitsQuotientProd <| fun b ↦ by\n    rw [(Units.nonZeroSubMul k V).stabilizer_of_subMul,\n      Module.stabilizer_units_eq_bot_of_ne_zero k b.property]\n  e.trans (Equiv.prodCongrLeft (fun _ ↦ (equivQuotientOrbitRel k V).symm))\n\ninstance isEmpty_of_subsingleton [Subsingleton V] : IsEmpty (ℙ k V) := by\n  have : IsEmpty { v : V // v ≠ 0 } := ⟨fun v ↦ v.2 (Subsingleton.elim v.1 0)⟩\n  simpa using (nonZeroEquivProjectivizationProdUnits k V).symm.isEmpty\n\n/-- If `V` is a finite `k`-module and `k` is finite, `ℙ k V` is finite. -/\ninstance finite_of_finite [Finite V] : Finite (ℙ k V) :=\n  have : Finite (ℙ k V × kˣ) := Finite.of_equiv _ (nonZeroEquivProjectivizationProdUnits k V)\n  Finite.prod_left kˣ\n\nlemma finite_iff_of_finite [Finite k] : Finite (ℙ k V) ↔ Finite V := by\n  classical\n  refine ⟨fun h ↦ ?_, fun h ↦ inferInstance⟩\n  let e := nonZeroEquivProjectivizationProdUnits k V\n  have : Finite { v : V // v ≠ 0 } := Finite.of_equiv _ e.symm\n  let eq : { v : V // v ≠ 0 } ⊕ Unit ≃ V :=\n    ⟨(Sum.elim Subtype.val (fun _ ↦ 0)), fun v ↦ if h : v = 0 then Sum.inr () else Sum.inl ⟨v, h⟩,\n      by intro x; aesop, by intro x; aesop⟩\n  exact Finite.of_equiv _ eq\n\n/-- Fraction free cardinality formula for the points of `ℙ k V` if `k` and `V` are finite\n(for silly reasons the formula also holds when `k` and `V` are infinite).\nSee `Projectivization.card'` and `Projectivization.card''` for other spellings of the formula. -/\nlemma card : Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1) := by\n  nontriviality V\n  cases finite_or_infinite k with\n  | inr h =>\n    have : Infinite V := Module.Free.infinite k V\n    simp\n  | inl h =>\n  cases finite_or_infinite V with\n  | inr h =>\n    have := not_iff_not.mpr (finite_iff_of_finite k V)\n    push Not at this\n    have : Infinite (ℙ k V) := by rwa [this]\n    simp\n  | inl h =>\n  classical\n  haveI : Fintype V := Fintype.ofFinite V\n  haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)\n  haveI : Fintype k := Fintype.ofFinite k\n  have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp\n  simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]\n  rw [Fintype.card_congr (nonZeroEquivProjectivizationProdUnits k V), Fintype.card_prod]\n\n/-- Cardinality formula for the points of `ℙ k V` if `k` and `V` are finite with less\nnatural subtraction. -/\n\nTarget:\nlemma card' [Finite V] : Nat.card V = Nat.card (ℙ k V) * (Nat.card k - 1) + 1 :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_de57e8cad85d","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"7b49672f151f29c159ecddc514b6581dfa16438cea634bf1c8b96eff5b787676","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Projectivization","family_id":"card","file_id":"mathlib/Mathlib/LinearAlgebra/Projectivization/Cardinality.lean","sample_id":"de57e8cad85d8b44648cf10f3d51ce2288da48a2805529ae83b12ceae714db81"},"split":"train"}
{"binding":{"chosen_proof_sha256":"80d415d211305782b1e1301d4d0be21abb30589a9327984a8f2cfe8e37267cf8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b851594e49436ef442f014bb3282b6289da2f1ebabe51553e6016a327fa2734c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d55812070999d9cd69897ef7878e60eedf847e5cca72e88cad19720567719cd3","source_sha256":"ed84729d6b60d1d061a43d8f6649c91b9089118e6fc8675bd3d550d272a00e60","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : ∀ᵐ x ∂μ, (f - f') x = 0 := by\n    apply ae_eq_zero_of_integral_contMDiff_smul_eq_zero I (hf.sub hf')\n    intro g g_diff g_supp\n    simp only [Pi.sub_apply, smul_sub]\n    rw [integral_sub, sub_eq_zero]\n    · exact h g g_diff g_supp\n    · exact hf.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp\n    · exact hf'.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp\n  filter_upwards [this] with x hx\n  simpa [sub_eq_zero] using hx","hard_negative":true,"metrics":{"chosen_tokens":84,"rejected_tokens":5,"token_jaccard":0.058824,"token_length_ratio":0.059524},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"5048be38f67859049d4f1f6b9d1c829026ea81ce21f2e8176c34944812f5fb2f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Geometry.Manifold.PartitionOfUnity\npublic import Mathlib.MeasureTheory.Function.AEEqOfIntegral\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Functions which vanish as distributions vanish as functions\n\nIn a finite-dimensional normed real vector space endowed with a Borel measure, consider a locally\nintegrable function whose integral against all compactly supported smooth functions vanishes. Then\nthe function is almost everywhere zero.\nThis is proved in `ae_eq_zero_of_integral_contDiff_smul_eq_zero`.\n\nA version for two functions having the same integral when multiplied by smooth compactly supported\nfunctions is also given in `ae_eq_of_integral_contDiff_smul_eq`.\n\nThese are deduced from the same results on finite-dimensional real manifolds, given respectively\nas `ae_eq_zero_of_integral_contMDiff_smul_eq_zero` and `ae_eq_of_integral_contMDiff_smul_eq`.\n-/\n\npublic section\n\nopen MeasureTheory Filter Metric Function Set TopologicalSpace\n\nopen scoped Topology Manifold ContDiff\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]\n  {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]\n\nsection Manifold\n\nvariable {H : Type*} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)\n  {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]\n  [MeasurableSpace M] [BorelSpace M] [T2Space M]\n  {f f' : M → F} {μ : Measure M}\n\n/-- If a locally integrable function `f` on a finite-dimensional real manifold has zero integral\nwhen multiplied by any smooth compactly supported function, then `f` vanishes almost everywhere. -/\ntheorem ae_eq_zero_of_integral_contMDiff_smul_eq_zero [SigmaCompactSpace M]\n    (hf : LocallyIntegrable f μ)\n    (h : ∀ g : M → ℝ, CMDiff ∞ g → HasCompactSupport g → ∫ x, g x • f x ∂μ = 0) :\n    ∀ᵐ x ∂μ, f x = 0 := by\n  -- record topological properties of `M`\n  have := I.locallyCompactSpace\n  have := ChartedSpace.locallyCompactSpace H M\n  have := I.secondCountableTopology\n  have := ChartedSpace.secondCountable_of_sigmaCompact H M\n  let _ : MetricSpace M := TopologicalSpace.metrizableSpaceMetric M\n  -- it suffices to show that the integral of the function vanishes on any compact set `s`\n  apply ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero' hf (fun s hs ↦ Eq.symm ?_)\n  obtain ⟨δ, δpos, hδ⟩ : ∃ δ, 0 < δ ∧ IsCompact (cthickening δ s) := hs.exists_isCompact_cthickening\n  -- choose a sequence of smooth functions `gₙ` equal to `1` on `s` and vanishing outside of the\n  -- `uₙ`-neighborhood of `s`, where `uₙ` tends to zero. Then each integral `∫ gₙ f` vanishes,\n  -- and by dominated convergence these integrals converge to `∫ x in s, f`.\n  obtain ⟨u, -, u_pos, u_lim⟩ : ∃ u, StrictAnti u ∧ (∀ (n : ℕ), u n ∈ Ioo 0 δ)\n    ∧ Tendsto u atTop (𝓝 0) := exists_seq_strictAnti_tendsto' δpos\n  let v : ℕ → Set M := fun n ↦ thickening (u n) s\n  obtain ⟨K, K_compact, vK⟩ : ∃ K, IsCompact K ∧ ∀ n, v n ⊆ K :=\n    ⟨_, hδ, fun n ↦ thickening_subset_cthickening_of_le (u_pos n).2.le _⟩\n  have : ∀ n, ∃ (g : M → ℝ), support g = v n ∧ CMDiff ∞ g ∧ Set.range g ⊆ Set.Icc 0 1\n          ∧ ∀ x ∈ s, g x = 1 := by\n    intro n\n    rcases exists_contMDiff_support_eq_eq_one_iff I isOpen_thickening hs.isClosed\n      (self_subset_thickening (u_pos n).1 s) with ⟨g, g_smooth, g_range, g_supp, hg⟩\n    exact ⟨g, g_supp, g_smooth, g_range, fun x hx ↦ (hg x).1 hx⟩\n  choose g g_supp g_diff g_range hg using this\n  -- main fact: the integral of `∫ gₙ f` tends to `∫ x in s, f`.\n  have L : Tendsto (fun n ↦ ∫ x, g n x • f x ∂μ) atTop (𝓝 (∫ x in s, f x ∂μ)) := by\n    rw [← integral_indicator hs.measurableSet]\n    let bound : M → ℝ := K.indicator (fun x ↦ ‖f x‖)\n    have A : ∀ n, AEStronglyMeasurable (fun x ↦ g n x • f x) μ :=\n      fun n ↦ (g_diff n).continuous.aestronglyMeasurable.smul hf.aestronglyMeasurable\n    have B : Integrable bound μ := by\n      rw [integrable_indicator_iff K_compact.measurableSet]\n      exact (hf.integrableOn_isCompact K_compact).norm\n    have C : ∀ n, ∀ᵐ x ∂μ, ‖g n x • f x‖ ≤ bound x := by\n      intro n\n      filter_upwards with x\n      rw [norm_smul]\n      refine le_indicator_apply (fun _ ↦ ?_) (fun hxK ↦ ?_)\n      · have : ‖g n x‖ ≤ 1 := by\n          have := g_range n (mem_range_self (f := g n) x)\n          rw [Real.norm_of_nonneg this.1]\n          exact this.2\n        exact mul_le_of_le_one_left (norm_nonneg _) this\n      · have : g n x = 0 := by rw [← notMem_support, g_supp]; contrapose hxK; exact vK n hxK\n        simp [this]\n    have D : ∀ᵐ x ∂μ, Tendsto (fun n => g n x • f x) atTop (𝓝 (s.indicator f x)) := by\n      filter_upwards with x\n      by_cases hxs : x ∈ s\n      · have : ∀ n, g n x = 1 := fun n ↦ hg n x hxs\n        simp [this, indicator_of_mem hxs f]\n      · simp_rw [indicator_of_notMem hxs f]\n        apply tendsto_const_nhds.congr'\n        suffices H : ∀ᶠ n in atTop, g n x = 0 by\n          filter_upwards [H] with n hn using by simp [hn]\n        obtain ⟨ε, εpos, hε⟩ : ∃ ε, 0 < ε ∧ x ∉ thickening ε s := by\n          rw [← hs.isClosed.closure_eq, closure_eq_iInter_thickening s] at hxs\n          simpa using hxs\n        filter_upwards [(tendsto_order.1 u_lim).2 _ εpos] with n hn\n        rw [← notMem_support, g_supp]\n        contrapose hε\n        exact thickening_mono hn.le s hε\n    exact tendsto_integral_of_dominated_convergence bound A B C D\n  -- deduce that `∫ x in s, f = 0` as each integral `∫ gₙ f` vanishes by assumption\n  have : ∀ n, ∫ x, g n x • f x ∂μ = 0 := by\n    refine fun n ↦ h _ (g_diff n) ?_\n    apply HasCompactSupport.of_support_subset_isCompact K_compact\n    simpa [g_supp] using vK n\n  simpa [this] using L\n\n@[deprecated (since := \"2025-12-17\")]\nalias ae_eq_zero_of_integral_smooth_smul_eq_zero := ae_eq_zero_of_integral_contMDiff_smul_eq_zero\n\n-- An instance with keys containing `Opens`\ninstance (U : Opens M) : BorelSpace U := inferInstanceAs (BorelSpace (U : Set M))\n\n/-- If a function `f` locally integrable on an open subset `U` of a finite-dimensional real\n  manifold has zero integral when multiplied by any smooth function compactly supported\n  in `U`, then `f` vanishes almost everywhere in `U`. -/\ntheorem IsOpen.ae_eq_zero_of_integral_contMDiff_smul_eq_zero' {U : Set M} (hU : IsOpen U)\n    (hSig : IsSigmaCompact U) (hf : LocallyIntegrableOn f U μ)\n    (h : ∀ g : M → ℝ,\n      CMDiff ∞ g → HasCompactSupport g → tsupport g ⊆ U → ∫ x, g x • f x ∂μ = 0) :\n    ∀ᵐ x ∂μ, x ∈ U → f x = 0 := by\n  have meas_U := hU.measurableSet\n  rw [← ae_restrict_iff' meas_U, ae_restrict_iff_subtype meas_U]\n  let U : Opens M := ⟨U, hU⟩\n  change ∀ᵐ (x : U) ∂_, _\n  haveI : SigmaCompactSpace U := isSigmaCompact_iff_sigmaCompactSpace.mp hSig\n  refine ae_eq_zero_of_integral_contMDiff_smul_eq_zero I ?_ fun g g_smth g_supp ↦ ?_\n  · exact (locallyIntegrable_comap meas_U).mpr hf\n  specialize h (Subtype.val.extend g 0) (g_smth.extend_zero g_supp)\n    (g_supp.extend_zero continuous_subtype_val) ((g_supp.tsupport_extend_zero_subset\n      continuous_subtype_val).trans <| Subtype.coe_image_subset _ _)\n  rw [← setIntegral_eq_integral_of_forall_compl_eq_zero (s := U) fun x hx ↦ ?_] at h\n  · rw [← integral_subtype_comap] at h\n    · simp_rw [Subtype.val_injective.extend_apply] at h; exact h\n    · exact meas_U\n  rw [Function.extend_apply' _ _ _ (mt _ hx)]\n  · apply zero_smul\n  · rintro ⟨x, rfl⟩; exact x.2\n\n@[deprecated (since := \"2025-12-17\")]\nalias IsOpen.ae_eq_zero_of_integral_smooth_smul_eq_zero' :=\n  IsOpen.ae_eq_zero_of_integral_contMDiff_smul_eq_zero'\n\nvariable [SigmaCompactSpace M]\n\ntheorem IsOpen.ae_eq_zero_of_integral_contMDiff_smul_eq_zero {U : Set M} (hU : IsOpen U)\n    (hf : LocallyIntegrableOn f U μ)\n    (h : ∀ g : M → ℝ,\n      CMDiff ∞ g → HasCompactSupport g → tsupport g ⊆ U → ∫ x, g x • f x ∂μ = 0) :\n    ∀ᵐ x ∂μ, x ∈ U → f x = 0 :=\n  haveI := I.locallyCompactSpace\n  haveI := ChartedSpace.locallyCompactSpace H M\n  haveI := hU.locallyCompactSpace\n  haveI := I.secondCountableTopology\n  haveI := ChartedSpace.secondCountable_of_sigmaCompact H M\n  hU.ae_eq_zero_of_integral_contMDiff_smul_eq_zero' _\n    (isSigmaCompact_iff_sigmaCompactSpace.mpr inferInstance) hf h\n\n@[deprecated (since := \"2025-12-17\")]\nalias IsOpen.ae_eq_zero_of_integral_smooth_smul_eq_zero :=\n  IsOpen.ae_eq_zero_of_integral_contMDiff_smul_eq_zero\n\n/-- If two locally integrable functions on a finite-dimensional real manifold have the same integral\nwhen multiplied by any smooth compactly supported function, then they coincide almost everywhere. -/\n\nTarget:\ntheorem ae_eq_of_integral_contMDiff_smul_eq\n    (hf : LocallyIntegrable f μ) (hf' : LocallyIntegrable f' μ) (h : ∀ (g : M → ℝ),\n      CMDiff ∞ g → HasCompactSupport g → ∫ x, g x • f x ∂μ = ∫ x, g x • f' x ∂μ) :\n    ∀ᵐ x ∂μ, f x = f' x :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_d55812070999","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"edc83f01ae4243a8c26683a1e1085c57beb5351944fecf5be0206597f9224c42","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Distribution","family_id":"ae_eq_of_integral_contmdiff_smul_eq","file_id":"mathlib/Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean","sample_id":"d55812070999d9cd69897ef7878e60eedf847e5cca72e88cad19720567719cd3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"af1766a439a78c83ac802fa03802f5aeae0b2eb140261f6b10e3a3388ff5d5e4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"745a6c18148ff085b3e3109620a641446bbba3c804b4945122922293430144c0","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"22f2feedaac2189bc2f75b488121e482511637bba540133178964abac118fe65","source_sha256":"28c44eebb7ced13bf4be0bd8100be84e866f1a63f874878f0058707234306682","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let f (u : R) : R × R := (star u * u, u * star u)\n  have hf : f ⁻¹' {(1, 1)} = unitary R := by ext u; simp [f, Unitary.mem_iff]\n  rw [← hf]\n  exact isClosed_singleton.preimage (by fun_prop)","hard_negative":true,"metrics":{"chosen_tokens":67,"rejected_tokens":3,"token_jaccard":0.052632,"token_length_ratio":0.044776},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"5066c446b4d8501ee839f04c639d9355b1bd6bace89459a197d9790e5204d9fb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Star.Unitary\npublic import Mathlib.Topology.Algebra.Group.Defs\npublic import Mathlib.Topology.Algebra.Star\npublic import Mathlib.Topology.Algebra.Monoid\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux, Bhavik Mehta\n-/\n/-! # Topological properties of the unitary (sub)group\n\n* In a topological star monoid `R`, `unitary R` is a topological group\n* In a topological star monoid `R` which is T1, `unitary R` is closed as a subset of `R`.\n-/\n\npublic section\n\nvariable {R : Type*} [Monoid R] [StarMul R] [TopologicalSpace R]\n\ninstance [ContinuousStar R] : ContinuousStar (unitary R) where\n  continuous_star := continuous_induced_rng.mpr continuous_subtype_val.star\n\ninstance [ContinuousStar R] : ContinuousInv (unitary R) where\n  continuous_inv := continuous_star\n\ninstance [ContinuousMul R] [ContinuousStar R] : IsTopologicalGroup (unitary R) where\n\nTarget:\nlemma isClosed_unitary [T1Space R] [ContinuousStar R] [ContinuousMul R] :\n    IsClosed (unitary R : Set R) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_22f2feedaac2","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f6a33baac10a779348f19609f946f7357f0483c9037871e3032589c244288a1b","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Algebra","family_id":"isclosed_unitary","file_id":"mathlib/Mathlib/Topology/Algebra/Star/Unitary.lean","sample_id":"22f2feedaac2189bc2f75b488121e482511637bba540133178964abac118fe65"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dab19792c72b6e7680af848aa1b358000086832bd7758b76eb7e5a5172d70aec","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"032679d5999e79cef4404c0d285be7a3e680162cde2c9a860c8b8bb71d184422","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"21c2606d6758e396ea3dedfc301c790b246ae8c817f34e167e84ed4e781597b5","source_sha256":"7745c09f53b57dd9e1369fa3be9c7884e924965f772e3b47328f6b6cb7e756a6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  grw [Sum.dist, ← le_add_of_nonneg_right dist_nonneg, ← le_add_of_nonneg_left dist_nonneg]","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":20,"token_jaccard":0.75,"token_length_ratio":1.333333},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"5080f3d935c50a769094e1e5271d4c0c5342b7fa650518ee9305064fc844a802","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ConditionallyCompleteLattice.Group\npublic import Mathlib.Topology.MetricSpace.Isometry\n\nNamespace:\nMetric\n\nLocal context:\n/-\nCopyright (c) 2019 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Metric space gluing\n\nGluing two metric spaces along a common subset. Formally, we are given\n\n```\n     Φ\n  Z ---> X\n  |\n  |Ψ\n  v\n  Y\n```\nwhere `hΦ : Isometry Φ` and `hΨ : Isometry Ψ`.\nWe want to complete the square by a space `GlueSpace hΦ hΨ` and two isometries\n`toGlueL hΦ hΨ` and `toGlueR hΦ hΨ` that make the square commute.\nWe start by defining a predistance on the disjoint union `X ⊕ Y`, for which\npoints `Φ p` and `Ψ p` are at distance 0. The (quotient) metric space associated\nto this predistance is the desired space.\n\nThis is an instance of a more general construction, where `Φ` and `Ψ` do not have to be isometries,\nbut the distances in the image almost coincide, up to `2ε` say. Then one can almost glue the two\nspaces so that the images of a point under `Φ` and `Ψ` are `ε`-close. If `ε > 0`, this yields a\nmetric space structure on `X ⊕ Y`, without the need to take a quotient. In particular,\nthis gives a natural metric space structure on `X ⊕ Y`, where the basepoints\nare at distance 1, say, and the distances between other points are obtained by going through the two\nbasepoints.\n(We also register the same metric space structure on a general disjoint union `Σ i, E i`).\n\nWe also define the inductive limit of metric spaces. Given\n```\n     f 0        f 1        f 2        f 3\nX 0 -----> X 1 -----> X 2 -----> X 3 -----> ...\n```\nwhere the `X n` are metric spaces and `f n` isometric embeddings, we define the inductive\nlimit of the `X n`, also known as the increasing union of the `X n` in this context, if we\nidentify `X n` and `X (n+1)` through `f n`. This is a metric space in which all `X n` embed\nisometrically and in a way compatible with `f n`.\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nuniverse u v w\n\nopen Function Set Uniformity Topology\n\nnamespace Metric\n\nsection ApproxGluing\n\nvariable {X : Type u} {Y : Type v} {Z : Type w}\nvariable [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} {ε : ℝ}\n\n/-- Define a predistance on `X ⊕ Y`, for which `Φ p` and `Ψ p` are at distance `ε` -/\ndef glueDist (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) : X ⊕ Y → X ⊕ Y → ℝ\n  | .inl x, .inl y => dist x y\n  | .inr x, .inr y => dist x y\n  | .inl x, .inr y => (⨅ p, dist x (Φ p) + dist y (Ψ p)) + ε\n  | .inr x, .inl y => (⨅ p, dist y (Φ p) + dist x (Ψ p)) + ε\n\nset_option backward.privateInPublic true in\nprivate theorem glueDist_self (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) : ∀ x, glueDist Φ Ψ ε x x = 0\n  | .inl _ => dist_self _\n  | .inr _ => dist_self _\n\ntheorem glueDist_glued_points [Nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (p : Z) :\n    glueDist Φ Ψ ε (.inl (Φ p)) (.inr (Ψ p)) = ε := by\n  have : ⨅ q, dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q) = 0 := by\n    have A : ∀ q, 0 ≤ dist (Φ p) (Φ q) + dist (Ψ p) (Ψ q) := fun _ => by positivity\n    refine le_antisymm ?_ (le_ciInf A)\n    have : 0 = dist (Φ p) (Φ p) + dist (Ψ p) (Ψ p) := by simp\n    rw [this]\n    exact ciInf_le ⟨0, forall_mem_range.2 A⟩ p\n  simp only [glueDist, this, zero_add]\n\nset_option backward.privateInPublic true in\nprivate theorem glueDist_comm (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) :\n    ∀ x y, glueDist Φ Ψ ε x y = glueDist Φ Ψ ε y x\n  | .inl _, .inl _ => dist_comm _ _\n  | .inr _, .inr _ => dist_comm _ _\n  | .inl _, .inr _ => rfl\n  | .inr _, .inl _ => rfl\n\ntheorem glueDist_swap (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) :\n    ∀ x y, glueDist Ψ Φ ε x.swap y.swap = glueDist Φ Ψ ε x y\n  | .inl _, .inl _ => rfl\n  | .inr _, .inr _ => rfl\n  | .inl _, .inr _ => by simp only [glueDist, Sum.swap_inl, Sum.swap_inr, add_comm]\n  | .inr _, .inl _ => by simp only [glueDist, Sum.swap_inl, Sum.swap_inr, add_comm]\n\ntheorem le_glueDist_inl_inr (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x y) :\n    ε ≤ glueDist Φ Ψ ε (.inl x) (.inr y) :=\n  le_add_of_nonneg_left <| Real.iInf_nonneg fun _ => by positivity\n\ntheorem le_glueDist_inr_inl (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x y) :\n    ε ≤ glueDist Φ Ψ ε (.inr x) (.inl y) := by\n  rw [glueDist_comm]; apply le_glueDist_inl_inr\n\nsection\nvariable [Nonempty Z]\n\nprivate theorem glueDist_triangle_inl_inr_inr (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x : X) (y z : Y) :\n    glueDist Φ Ψ ε (.inl x) (.inr z) ≤\n      glueDist Φ Ψ ε (.inl x) (.inr y) + glueDist Φ Ψ ε (.inr y) (.inr z) := by\n  simp only [glueDist]\n  rw [add_right_comm, add_le_add_iff_right]\n  refine le_ciInf_add fun p => ciInf_le_of_le ⟨0, ?_⟩ p ?_\n  · exact forall_mem_range.2 fun _ => by positivity\n  · linarith [dist_triangle_left z (Ψ p) y]\n\nprivate theorem glueDist_triangle_inl_inr_inl (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ)\n    (H : ∀ p q, |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) (x : X) (y : Y) (z : X) :\n    glueDist Φ Ψ ε (.inl x) (.inl z) ≤\n      glueDist Φ Ψ ε (.inl x) (.inr y) + glueDist Φ Ψ ε (.inr y) (.inl z) := by\n  simp_rw [glueDist, add_add_add_comm _ ε, add_assoc]\n  refine le_ciInf_add fun p => ?_\n  rw [add_left_comm, add_assoc, ← two_mul]\n  refine le_ciInf_add fun q => ?_\n  rw [dist_comm z]\n  linarith [dist_triangle4 x (Φ p) (Φ q) z, dist_triangle_left (Ψ p) (Ψ q) y, (abs_le.1 (H p q)).2]\n\nset_option backward.privateInPublic true in\nprivate theorem glueDist_triangle (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ)\n    (H : ∀ p q, |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) :\n    ∀ x y z, glueDist Φ Ψ ε x z ≤ glueDist Φ Ψ ε x y + glueDist Φ Ψ ε y z\n  | .inl _, .inl _, .inl _ => dist_triangle _ _ _\n  | .inr _, .inr _, .inr _ => dist_triangle _ _ _\n  | .inr x, .inl y, .inl z => by\n    simp only [← glueDist_swap Φ]\n    apply glueDist_triangle_inl_inr_inr\n  | .inr x, .inr y, .inl z => by\n    simpa only [glueDist_comm, add_comm] using glueDist_triangle_inl_inr_inr _ _ _ z y x\n  | .inl x, .inl y, .inr z => by\n    simpa only [← glueDist_swap Φ, glueDist_comm, add_comm, Sum.swap_inl, Sum.swap_inr]\n      using glueDist_triangle_inl_inr_inr Ψ Φ ε z y x\n  | .inl _, .inr _, .inr _ => glueDist_triangle_inl_inr_inr ..\n  | .inl x, .inr y, .inl z => glueDist_triangle_inl_inr_inl Φ Ψ ε H x y z\n  | .inr x, .inl y, .inr z => by\n    simp only [← glueDist_swap Φ]\n    apply glueDist_triangle_inl_inr_inl\n    simpa only [abs_sub_comm]\n\nend\n\nset_option backward.privateInPublic true in\nprivate theorem eq_of_glueDist_eq_zero (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (ε0 : 0 < ε) :\n    ∀ p q : X ⊕ Y, glueDist Φ Ψ ε p q = 0 → p = q\n  | .inl x, .inl y, h => by rw [eq_of_dist_eq_zero h]\n  | .inl x, .inr y, h => by exfalso; linarith [le_glueDist_inl_inr Φ Ψ ε x y]\n  | .inr x, .inl y, h => by exfalso; linarith [le_glueDist_inr_inl Φ Ψ ε x y]\n  | .inr x, .inr y, h => by rw [eq_of_dist_eq_zero h]\n\ntheorem Sum.mem_uniformity_iff_glueDist (hε : 0 < ε) (s : Set ((X ⊕ Y) × (X ⊕ Y))) :\n    s ∈ 𝓤 (X ⊕ Y) ↔ ∃ δ > 0, ∀ a b, glueDist Φ Ψ ε a b < δ → (a, b) ∈ s := by\n  simp only [Sum.uniformity, Filter.mem_sup, Filter.mem_map, mem_uniformity_dist, mem_preimage]\n  constructor\n  · rintro ⟨⟨δX, δX0, hX⟩, δY, δY0, hY⟩\n    refine ⟨min (min δX δY) ε, lt_min (lt_min δX0 δY0) hε, ?_⟩\n    rintro (a | a) (b | b) h <;> simp only [lt_min_iff] at h\n    · exact hX h.1.1\n    · exact absurd h.2 (le_glueDist_inl_inr _ _ _ _ _).not_gt\n    · exact absurd h.2 (le_glueDist_inr_inl _ _ _ _ _).not_gt\n    · exact hY h.1.2\n  · rintro ⟨ε, ε0, H⟩\n    constructor <;> exact ⟨ε, ε0, fun _ _ h => H _ _ h⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Given two maps `Φ` and `Ψ` intro metric spaces `X` and `Y` such that the distances between\n`Φ p` and `Φ q`, and between `Ψ p` and `Ψ q`, coincide up to `2 ε` where `ε > 0`, one can almost\nglue the two spaces `X` and `Y` along the images of `Φ` and `Ψ`, so that `Φ p` and `Ψ p` are\nat distance `ε`. -/\n@[implicit_reducible]\ndef glueMetricApprox [Nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (ε0 : 0 < ε)\n    (H : ∀ p q, |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) : MetricSpace (X ⊕ Y) where\n  dist := glueDist Φ Ψ ε\n  dist_self := glueDist_self Φ Ψ ε\n  dist_comm := glueDist_comm Φ Ψ ε\n  dist_triangle := glueDist_triangle Φ Ψ ε H\n  eq_of_dist_eq_zero := eq_of_glueDist_eq_zero Φ Ψ ε ε0 _ _\n  toUniformSpace := Sum.instUniformSpace\n  uniformity_dist := uniformity_dist_of_mem_uniformity _ _ <| Sum.mem_uniformity_iff_glueDist ε0\n\nend ApproxGluing\n\nsection Sum\n\n/-!\n### Metric on `X ⊕ Y`\n\nA particular case of the previous construction is when one uses basepoints in `X` and `Y` and one\nglues only along the basepoints, putting them at distance 1. We give a direct definition of\nthe distance, without `iInf`, as it is easier to use in applications, and show that it is equal to\nthe gluing distance defined above to take advantage of the lemmas we have already proved.\n-/\nvariable {X : Type u} {Y : Type v} {Z : Type w}\nvariable [MetricSpace X] [MetricSpace Y]\n\n/-- Distance on a disjoint union. There are many (noncanonical) ways to put a distance compatible\nwith each factor.\nIf the two spaces are bounded, one can say for instance that each point in the first is at distance\n`diam X + diam Y + 1` of each point in the second.\nInstead, we choose a construction that works for unbounded spaces, but requires basepoints,\nchosen arbitrarily.\nWe embed isometrically each factor, set the basepoints at distance 1,\narbitrarily, and say that the distance from `a` to `b` is the sum of the distances of `a` and `b` to\ntheir respective basepoints, plus the distance 1 between the basepoints.\nSince there is an arbitrary choice in this construction, it is not an instance by default. -/\nprotected def Sum.dist : X ⊕ Y → X ⊕ Y → ℝ\n  | .inl a, .inl a' => dist a a'\n  | .inr b, .inr b' => dist b b'\n  | .inl a, .inr b => dist a (Nonempty.some ⟨a⟩) + 1 + dist (Nonempty.some ⟨b⟩) b\n  | .inr b, .inl a => dist b (Nonempty.some ⟨b⟩) + 1 + dist (Nonempty.some ⟨a⟩) a\n\ntheorem Sum.dist_eq_glueDist {p q : X ⊕ Y} (x : X) (y : Y) :\n    Sum.dist p q =\n      glueDist (fun _ : Unit => Nonempty.some ⟨x⟩) (fun _ : Unit => Nonempty.some ⟨y⟩) 1 p q := by\n  cases p <;> cases q <;> first | rfl | simp [Sum.dist, glueDist, dist_comm, add_comm,\n    add_left_comm, add_assoc]\n\nset_option backward.privateInPublic true in\nprivate theorem Sum.dist_comm (x y : X ⊕ Y) : Sum.dist x y = Sum.dist y x := by\n  cases x <;> cases y <;> simp [Sum.dist, _root_.dist_comm, add_comm, add_left_comm]\n\nTarget:\ntheorem Sum.one_le_dist_inl_inr {x : X} {y : Y} : 1 ≤ Sum.dist (.inl x) (.inr y) :=\n\nProof body:\n","rejected":"by\n  grw [Sum.dist, ← le_add_of_nonneg_right dist_nonneg, ← le_add_of_nonneg_left dist_nonneg]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/MetricSpace","family_id":"sum","file_id":"mathlib/Mathlib/Topology/MetricSpace/Gluing.lean","sample_id":"21c2606d6758e396ea3dedfc301c790b246ae8c817f34e167e84ed4e781597b5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"893f9954189be14b6650b8e11183442e164c9e72d10c7e09ce63fc59ebc05f3e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8eea71a463014df8b67da19c424d8a2a9e086496694de1957a25caee92f8b153","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e5021bc99d39f6a7ba2ff9754e2b8736041ba7f4cf828fd50b3785d42ee868a1","source_sha256":"1ff5a6a6007c7fc57cd72503958e574b72763b74e50bd4d47517631475da6ea9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext; simp [equiv]","hard_negative":true,"metrics":{"chosen_tokens":7,"rejected_tokens":5,"token_jaccard":0.090909,"token_length_ratio":0.714286},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"508c362a4cf6edc0b99aff676d8fe77e006fd29170d9a62b60e76c190c07bfef","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Colimit.Module\npublic import Mathlib.RingTheory.Finiteness.Basic\n\nNamespace:\nModule.fgSystem\n\nLocal context:\n/-\nCopyright (c) 2024 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu\n-/\n/-!\n# Modules as direct limits of finitely generated submodules\n\nWe show that every module is the direct limit of its finitely generated submodules.\n\n## Main definitions\n\n* `Module.fgSystem`: the directed system of finitely generated submodules of a module.\n\n* `Module.fgSystem.equiv`: the isomorphism between a module and the direct limit of its\n  finitely generated submodules.\n-/\n\n@[expose] public section\n\nnamespace Module\n\nvariable (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]\n\n/-- The directed system of finitely generated submodules of a module. -/\ndef fgSystem (N₁ N₂ : {N : Submodule R M // N.FG}) (le : N₁ ≤ N₂) : N₁ →ₗ[R] N₂ :=\n  Submodule.inclusion le\n\nopen DirectLimit\n\nnamespace fgSystem\n\ninstance : IsDirectedOrder {N : Submodule R M // N.FG} where\n  directed N₁ N₂ :=\n    ⟨⟨_, N₁.2.sup N₂.2⟩, Subtype.coe_le_coe.mp le_sup_left, Subtype.coe_le_coe.mp le_sup_right⟩\n\ninstance : DirectedSystem _ (fgSystem R M · · · ·) where\n  map_self _ _ := rfl\n  map_map _ _ _ _ _ _ := rfl\n\nvariable [DecidableEq (Submodule R M)]\n\nopen Submodule in\n/-- Every module is the direct limit of its finitely generated submodules. -/\nnoncomputable def equiv : DirectLimit _ (fgSystem R M) ≃ₗ[R] M :=\n  .ofBijective (lift _ _ _ _ (fun _ ↦ Submodule.subtype _) fun _ _ _ _ ↦ rfl)\n    ⟨lift_injective _ _ fun _ ↦ Subtype.val_injective, fun x ↦\n      ⟨of _ _ _ _ ⟨_, fg_span_singleton x⟩ ⟨x, subset_span <| by rfl⟩, lift_of ..⟩⟩\n\nvariable {R M}\n\nTarget:\nlemma equiv_comp_of (N : {N : Submodule R M // N.FG}) :\n    (equiv R M).toLinearMap ∘ₗ of _ _ _ _ N = N.1.subtype :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_e5021bc99d39","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"b207211b5acc4d6caa2214cad1f3e5e32f94e5a9b004b49c3c20cb0bfa6f1ac7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Colimit","family_id":"equiv_comp_of","file_id":"mathlib/Mathlib/Algebra/Colimit/Finiteness.lean","sample_id":"e5021bc99d39f6a7ba2ff9754e2b8736041ba7f4cf828fd50b3785d42ee868a1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"42ee086aa0660c3672751c6ea4362a4dc2000a25c82c17c805dafacc7e49fe69","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"893c0b78dc997ad25e432d8ba1bc777ff19e30748d019f95bceb9d4a488d6e19","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"141687aaa229bc38944346a4f56eb62c28baec7cf1b9f600772bfcea49a063a4","source_sha256":"37e5cf98a4004b5dd772a5a2594f4a59dd8c3a858390b7b2963d50078d06b866","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have eq₁ : _ = (π₁ i, j, k).base := PreservesPullback.iso_hom_fst (forget C) _ _\n  have eq₂ : _ = (π₂ i, j, k).base := PreservesPullback.iso_hom_snd (forget C) _ _\n  rw [← eq₁, ← eq₂, TopCat.coe_comp, Set.image_comp, TopCat.coe_comp, Set.preimage_comp,\n    Set.image_preimage_eq]\n  · simp only [forget_obj, forget_map, TopCat.pullback_snd_image_fst_preimage]\n  rw [← TopCat.epi_iff_surjective]\n  infer_instance","hard_negative":false,"metrics":{"chosen_tokens":105,"rejected_tokens":110,"token_jaccard":0.911111,"token_length_ratio":1.047619},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"508fb31ef39832290414b5d6c1198fa4437874eb06ea8c1e960b0ad0d2a9a960","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Gluing\npublic import Mathlib.Geometry.RingedSpace.OpenImmersion\npublic import Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits\n\nNamespace:\nAlgebraicGeometry.PresheafedSpace.GlueData\n\nLocal context:\n/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Gluing structured spaces\n\nGiven a family of gluing data of structured spaces (presheafed spaces, sheafed spaces, or locally\nringed spaces), we may glue them together.\n\nThe construction should be \"sealed\" and considered as a black box, while only using the API\nprovided.\n\n## Main definitions\n\n* `AlgebraicGeometry.PresheafedSpace.GlueData`: A structure containing the family of gluing data.\n* `CategoryTheory.GlueData.glued`: The glued presheafed space.\n  This is defined as the multicoequalizer of `∐ V i j ⇉ ∐ U i`, so that the general colimit API\n  can be used.\n* `CategoryTheory.GlueData.ι`: The immersion `ι i : U i ⟶ glued` for each `i : J`.\n\n## Main results\n\n* `AlgebraicGeometry.PresheafedSpace.GlueData.ιIsOpenImmersion`: The map `ι i : U i ⟶ glued`\n  is an open immersion for each `i : J`.\n* `AlgebraicGeometry.PresheafedSpace.GlueData.ι_jointly_surjective` : The underlying maps of\n  `ι i : U i ⟶ glued` are jointly surjective.\n* `AlgebraicGeometry.PresheafedSpace.GlueData.vPullbackConeIsLimit` : `V i j` is the pullback\n  (intersection) of `U i` and `U j` over the glued space.\n\nAnalogous results are also provided for `SheafedSpace` and `LocallyRingedSpace`.\n\n## Implementation details\n\nAlmost the whole file is dedicated to showing that `ι i` is an open immersion. The fact that\nthis is an open embedding of topological spaces follows from `Mathlib/Topology/Gluing.lean`, and it\nremains to construct `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_X, ι i '' U)` for each `U ⊆ U i`.\nSince `Γ(𝒪_X, ι i '' U)` is the limit of `diagram_over_open`, the components of the structure\nsheaves of the spaces in the gluing diagram, we need to construct a map\n`ιInvApp_π_app : Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing diagram.\n\nWe will refer to ![this diagram](https://i.imgur.com/P0phrwr.png) in the following docstrings.\nThe `X` is the glued space, and the dotted arrow is a partial inverse guaranteed by the fact\nthat it is an open immersion. The map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_{U_j}, _)` is given by the composition\nof the red arrows, and the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_{V_{jk}}, _)` is given by the composition of the\nblue arrows. To lift this into a map from `Γ(𝒪_X, ι i '' U)`, we also need to show that these\ncommute with the maps in the diagram (the green arrows), which is just a lengthy diagram-chasing.\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen TopologicalSpace CategoryTheory Opposite Topology\n\nopen CategoryTheory.Limits AlgebraicGeometry.PresheafedSpace\n\nopen AlgebraicGeometry.PresheafedSpace.IsOpenImmersion\n\nopen CategoryTheory.GlueData\n\nnamespace AlgebraicGeometry\n\nuniverse v u\n\nvariable (C : Type u) [Category.{v} C]\n\nnamespace PresheafedSpace\n\n/-- A family of gluing data consists of\n1. An index type `J`\n2. A presheafed space `U i` for each `i : J`.\n3. A presheafed space `V i j` for each `i j : J`.\n   (Note that this is `J × J → PresheafedSpace C` rather than `J → J → PresheafedSpace C` to\n   connect to the limits library more easily.)\n4. An open immersion `f i j : V i j ⟶ U i` for each `i j : J`.\n5. A transition map `t i j : V i j ⟶ V j i` for each `i j : J`.\n\nsuch that\n6. `f i i` is an isomorphism.\n7. `t i i` is the identity.\n8. `V i j ×[U i] V i k ⟶ V i j ⟶ V j i` factors through `V j k ×[U j] V j i ⟶ V j i` via some\n   `t' : V i j ×[U i] V i k ⟶ V j k ×[U j] V j i`.\n9. `t' i j k ≫ t' j k i ≫ t' k i j = 𝟙 _`.\n\nWe can then glue the spaces `U i` together by identifying `V i j` with `V j i`, such\nthat the `U i`'s are open subspaces of the glued space.\n-/\nstructure GlueData extends CategoryTheory.GlueData (PresheafedSpace.{v, u, v} C) where\n  f_open : ∀ i j, IsOpenImmersion (f i j)\n\nattribute [instance] GlueData.f_open\n\nnamespace GlueData\n\nvariable {C}\nvariable (D : GlueData.{v, u} C)\n\nlocal notation \"𝖣\" => D.toGlueData\n\nlocal notation \"π₁ \" i \", \" j \", \" k => pullback.fst (D.f i j) (D.f i k)\n\nlocal notation \"π₂ \" i \", \" j \", \" k => pullback.snd (D.f i j) (D.f i k)\n\nset_option quotPrecheck false\nlocal notation \"π₁⁻¹ \" i \", \" j \", \" k =>\n  (PresheafedSpace.IsOpenImmersion.pullbackFstOfRight (D.f i j) (D.f i k)).invApp\n\nset_option quotPrecheck false\nlocal notation \"π₂⁻¹ \" i \", \" j \", \" k =>\n  (PresheafedSpace.IsOpenImmersion.pullbackSndOfLeft (D.f i j) (D.f i k)).invApp\n\n/-- The glue data of topological spaces associated to a family of glue data of PresheafedSpaces. -/\nabbrev toTopGlueData : TopCat.GlueData :=\n  { f_open := fun i j => (D.f_open i j).base_open\n    toGlueData := 𝖣.mapGlueData (forget C) }\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem ι_isOpenEmbedding [HasLimits C] (i : D.J) : IsOpenEmbedding (𝖣.ι i).base := by\n  rw [← show _ = (𝖣.ι i).base from 𝖣.ι_gluedIso_inv (PresheafedSpace.forget _) _, TopCat.coe_comp]\n  exact (TopCat.homeoOfIso (𝖣.gluedIso (PresheafedSpace.forget _)).symm).isOpenEmbedding.comp\n      (D.toTopGlueData.ι_isOpenEmbedding i)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem pullback_base (i j k : D.J) (S : Set (D.V (i, j)).carrier) :\n    (π₂ i, j, k) '' (π₁ i, j, k) ⁻¹' S = D.f i k ⁻¹' D.f i j '' S :=\n\nProof body:\n","rejected":"by\n  have eq₁ : _ = (π₁ i, j, k).base := PreservesPullback.iso_hom_fst (forget C) _ _\n  have eq₂ : _ = (π₂ i, j, k).base := PreservesPullback.iso_hom_snd (forget C) _ _\n  rw [← eq₁, ← eq₂, TopCat.coe_comp, Set.image_comp, TopCat.coe_comp, Set.preimage_comp,\n    Set.image_preimage_eq]\n  · simp only [forget_obj, forget_map, TopCat.pullback_snd_image_fst_preimage]\n  rw [← TopCat.epi_iff_surjective]\n  infer_instance\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/RingedSpace","family_id":"pullback_base","file_id":"mathlib/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean","sample_id":"141687aaa229bc38944346a4f56eb62c28baec7cf1b9f600772bfcea49a063a4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"242a43583e27fd27079936e1ce2e8cce7893c073e1ef4300e30f594b3e61646a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ba42f23de79f77ef3eb81eed5bddcbb4d1d3e1398285c31d837798f7ca1df42e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"91739cefd9b24c38b935a570789d9ce5a71a9f8d6f001da414f1877193c9705f","source_sha256":"3b79245b81821b44a91daf15c8bd7da3bce2a30e3ee9d29e2db6a7d0e0195196","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Ideal.one_eq_top, ← Ideal.add_eq_sup, Ideal.add_eq_one_iff] at h\n  have ⟨y, hy, z, hz, eq⟩ := h\n  have ⟨p, hp⟩ := (mem_ideal_map_adjoin ..).mp hy\n  have ⟨w, hw⟩ := Ideal.mem_span_singleton.mp hz\n  have ⟨q, hq⟩ := adjoin_eq_exists_aeval R x w\n  use (1 - p - X * q).reverse\n  have : (1 - p - X * q).coeff 0 - 1 ∈ I := by simpa using hp.1 0\n  apply_fun (·.1) at eq hw\n  dsimp at eq\n  rw [reverse_leadingCoeff, trailingCoeff_eq_coeff_zero]\n  · exact ⟨this, (eval₂_reverse_eq_zero_iff ..).mpr <| by simp [← aeval_def, hp.2, hq, ← eq, hw]⟩\n  · exact fun h ↦ hI <| by simpa [h, Ideal.eq_top_iff_one]","hard_negative":false,"metrics":{"chosen_tokens":179,"rejected_tokens":183,"token_jaccard":0.958333,"token_length_ratio":1.022346},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"509ece37aaf27539757a138558ca77205771841097aae099161baa2e44a99382","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.RingDivision\npublic import Mathlib.RingTheory.Adjoin.Polynomial.Basic\npublic import Mathlib.RingTheory.Ideal.Maps\n\nNamespace:\nAlgebra\n\nLocal context:\n/-\nCopyright (c) 2019 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n/-!\n# Ideals in polynomial rings\n-/\n\npublic section\n\nnoncomputable section\n\nopen Polynomial\n\nopen Finset\n\nuniverse u v w\n\nnamespace Polynomial\n\nvariable {R : Type*} [CommRing R] {a : R}\n\ntheorem mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero {b : R[X]} {P : R[X][X]} :\n    P ∈ Ideal.span {C (X - C a), X - C b} ↔ (P.eval b).eval a = 0 := by\n  rw [Ideal.mem_span_pair]\n  constructor <;> intro h\n  · rcases h with ⟨_, _, rfl⟩\n    simp\n  · rcases dvd_iff_isRoot.mpr h with ⟨p, hp⟩\n    rcases @X_sub_C_dvd_sub_C_eval _ b _ P with ⟨q, hq⟩\n    exact ⟨C p, q, by rw [mul_comm, mul_comm q, eq_add_of_sub_eq' hq, hp, C_mul]⟩\n\ntheorem ker_evalRingHom (x : R) : RingHom.ker (evalRingHom x) = Ideal.span {X - C x} := by\n  ext y\n  simp [Ideal.mem_span_singleton, dvd_iff_isRoot, RingHom.mem_ker]\n\n@[simp]\ntheorem ker_modByMonicHom {q : R[X]} (hq : q.Monic) :\n    LinearMap.ker (Polynomial.modByMonicHom q) = (Ideal.span {q}).restrictScalars R :=\n  Submodule.ext fun _ => (mem_ker_modByMonic hq).trans Ideal.mem_span_singleton.symm\n\n@[simp]\nlemma ker_constantCoeff : RingHom.ker constantCoeff = .span {(X : R[X])} := by\n  refine le_antisymm (fun p hp ↦ ?_) (by simp [Ideal.span_le])\n  simp only [RingHom.mem_ker, constantCoeff_apply, ← Polynomial.X_dvd_iff] at hp\n  rwa [Ideal.mem_span_singleton]\n\nend Polynomial\n\nnamespace Algebra\n\nvariable {R S : Type*}\n\nlemma mem_ideal_map_adjoin [CommSemiring R] [Semiring S] [Algebra R S] (x : S) (I : Ideal R)\n    {y : R[x]} :\n    y ∈ I.map (algebraMap R (R[x])) ↔\n      ∃ p : R[X], (∀ i, p.coeff i ∈ I) ∧ Polynomial.aeval x p = y := by\n  constructor\n  · intro H\n    induction H using Submodule.span_induction with\n    | mem a ha =>\n      obtain ⟨a, ha, rfl⟩ := ha\n      exact ⟨C a, fun i ↦ by rw [coeff_C]; aesop, aeval_C _ _⟩\n    | zero => exact ⟨0, by simp, aeval_zero _⟩\n    | add a b ha hb ha' hb' =>\n      obtain ⟨a, ha, ha'⟩ := ha'\n      obtain ⟨b, hb, hb'⟩ := hb'\n      exact ⟨a + b, fun i ↦ by simpa using add_mem (ha i) (hb i), by simp [ha', hb']⟩\n    | smul a b hb hb' =>\n      obtain ⟨b', hb, hb'⟩ := hb'\n      have ⟨p, hp⟩ := adjoin_eq_exists_aeval R x a\n      refine ⟨p * b', fun i ↦ ?_, by simp [hp, hb']⟩\n      rw [coeff_mul]\n      exact sum_mem fun i hi ↦ Ideal.mul_mem_left _ _ (hb _)\n  · rintro ⟨p, hp, hp'⟩\n    have : y = ∑ i ∈ p.support, p.coeff i • ⟨_, (X ^ i).aeval_mem_adjoin_singleton _ x⟩ := by\n      trans ∑ i ∈ p.support, ⟨_, (C (p.coeff i) * X ^ i).aeval_mem_adjoin_singleton _ x⟩\n      · ext1\n        simp only [AddSubmonoidClass.coe_finsetSum, ← map_sum, ← hp', ← as_sum_support_C_mul_X_pow]\n      · congr with i\n        simp [Algebra.smul_def]\n    simp_rw [this, Algebra.smul_def]\n    exact sum_mem fun i _ ↦ Ideal.mul_mem_right _ _ (Ideal.mem_map_of_mem _ (hp i))\n\nTarget:\nlemma exists_aeval_invOf_eq_zero_of_idealMap_adjoin_sup_span_eq_top [CommRing R] [CommRing S]\n    [Algebra R S] (x : S) (I : Ideal R) (hI : I ≠ ⊤) [Invertible x]\n    (h : I.map (algebraMap R (R[x])) ⊔ .span {⟨x, subset_adjoin rfl⟩} = ⊤) :\n    ∃ p : R[X], p.leadingCoeff - 1 ∈ I ∧ p.aeval ⅟x = 0 :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [← Ideal.one_eq_top, ← Ideal.add_eq_sup, Ideal.add_eq_one_iff] at h\n  have ⟨y, hy, z, hz, eq⟩ := h\n  have ⟨p, hp⟩ := (mem_ideal_map_adjoin ..).mp hy\n  have ⟨w, hw⟩ := Ideal.mem_span_singleton.mp hz\n  have ⟨q, hq⟩ := adjoin_eq_exists_aeval R x w\n  use (1 - p - X * q).reverse\n  have : (1 - p - X * q).coeff 0 - 1 ∈ I := by simpa using hp.1 0\n  apply_fun (·.1) at eq hw\n  dsimp at eq\n  rw [reverse_leadingCoeff, trailingCoeff_eq_coeff_zero]\n  · exact ⟨this, (eval₂_reverse_eq_zero_iff ..).mpr <| by simp [← aeval_def, hp.2, hq, ← eq, hw]⟩\n  · exact fun h ↦ hI <| by simpa [h, Ideal.eq_top_iff_one]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Polynomial","family_id":"exists_aeval_invof_eq_zero_of_idealmap_adjoin_sup_span_eq_top","file_id":"mathlib/Mathlib/RingTheory/Polynomial/Ideal.lean","sample_id":"91739cefd9b24c38b935a570789d9ce5a71a9f8d6f001da414f1877193c9705f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b03c7a526376dec68a09b72eb773aacc63927789fd0d788813f105124df25955","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"18fb5557e856be8fbe154a34d1876d0a93f5e6b8c2f2b794a98d101e936bdb41","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a86a08c70d72b421064dab529cf5e5d4ac90c62963744456fb1461e7c0e42c96","source_sha256":"4a32472b9e6529573aae59ec6bc77b66591720ace487daac48e6d523127ce834","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : Algebra.IsSeparable (FractionRing A) (FractionRing R₂) := by\n    refine Algebra.IsSeparable.of_equiv_equiv (FractionRing.algEquiv A K).symm.toRingEquiv\n          (FractionRing.algEquiv R₂ F₂).symm.toRingEquiv ?_\n    ext x\n    exact IsFractionRing.algEquiv_commutes (FractionRing.algEquiv A K).symm\n      (FractionRing.algEquiv R₂ ↥F₂).symm _\n  suffices span B (algebraMap F₂ L '' (traceDual A K (1 : Submodule R₂ F₂))) ≤\n      traceDual R₁ F₁ (1 : Submodule B L) by\n    apply le_antisymm\n    · refine SetLike.coe_subset_coe.mp (subset_trans ?_ this)\n      rw [← Submodule.span_span_of_tower R₁ B]\n      exact Submodule.subset_span\n    · exact traceDual_le_span_map_traceDual A B R₁ R₂ h₁ h₂\n  have := dvd_of_eq <|\n    (IsDedekindDomain.differentIdeal_eq_map_differentIdeal A B R₁ R₂ h₁ h₂ h₃).symm\n  rwa [Ideal.dvd_iff_le, ← coeIdeal_le_coeIdeal (K := L), coeIdeal_differentIdeal R₁ F₁,\n    inv_le_comm, ← extendedHom_coeIdeal_eq_map (K := F₂), coeIdeal_differentIdeal A K, map_inv₀,\n    inv_inv, ← coe_le_coe, coe_extendedHom_eq_span, coe_dual_one, ← coeToSet_coeToSubmodule,\n    coe_dual_one] at this\n  · simp\n  · rw [← extendedHom_coeIdeal_eq_map (K := F₂), ne_eq, extendedHom_eq_zero_iff]\n    rw [coeIdeal_eq_zero]\n    exact differentIdeal_ne_bot","hard_negative":false,"metrics":{"chosen_tokens":260,"rejected_tokens":265,"token_jaccard":0.953488,"token_length_ratio":1.019231},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"513a28ed81dfc6bb4498c8dec54b7f6f4271deb8f50776ab29ec9b546796e7d2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.LinearDisjoint\npublic import Mathlib.RingTheory.DedekindDomain.Different\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Xavier Roblot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Xavier Roblot\n-/\n/-!\n# Disjoint extensions with coprime different ideals\n\nLet `A ⊆ B` be a finite extension of Dedekind domains and assume that `A ⊆ R₁, R₂ ⊆ B` are two\nsubrings such that `Frac R₁ ⊔ Frac R₂ = Frac B`, `Frac R₁` and `Frac R₂` are linearly disjoint\nover `Frac A`, and that `𝓓(R₁/A)` and `𝓓(R₂/A)` are coprime where `𝓓` denotes the different ideal\nand `Frac R` denotes the fraction field of a domain `R`.\n\n## Main results and definitions\n\n* `IsDedekindDomain.differentIdeal_eq_map_differentIdeal`: `𝓓(B/R₁) = 𝓓(R₂/A)`\n* `IsDedekindDomain.differentIdeal_eq_differentIdeal_mul_differentIdeal_of_isCoprime`:\n  `𝓓(B/A) = 𝓓(R₁/A) * 𝓓(R₂/A)`.\n* `Module.Basis.ofIsCoprimeDifferentIdeal`: Construct a `R₁`-basis of `B` by lifting an\n  `A`-basis of `R₂`.\n* `IsDedekindDomain.range_sup_range_eq_top_of_isCoprime_differentIdeal`: `B` is generated\n  (as an `A`-algebra) by `R₁` and `R₂`.\n\n-/\n\n@[expose] public section\n\nopen FractionalIdeal nonZeroDivisors IntermediateField Algebra Module Submodule\n\nvariable (A B : Type*) {K L : Type*} [CommRing A] [Field K] [Algebra A K] [IsFractionRing A K]\n  [CommRing B] [Field L] [Algebra B L] [Algebra A L] [Algebra K L] [FiniteDimensional K L]\n  [IsScalarTower A K L]\nvariable (R₁ R₂ : Type*) [CommRing R₁] [CommRing R₂] [IsDomain R₁] [Algebra A R₁] [Algebra A R₂]\n  [Algebra R₁ B] [Algebra R₂ B] [Algebra R₁ L] [Algebra R₂ L]\n  [IsScalarTower A R₁ L] [IsScalarTower R₁ B L] [IsScalarTower R₂ B L] [Module.Finite A R₂]\nvariable {F₁ F₂ : IntermediateField K L} [Algebra R₁ F₁] [Algebra R₂ F₂] [IsTorsionFree R₁ F₁]\n  [IsScalarTower A F₂ L] [IsScalarTower A R₂ F₂] [IsScalarTower R₁ F₁ L] [IsScalarTower R₂ F₂ L]\n  [Algebra.IsSeparable K F₂] [Algebra.IsSeparable F₁ L]\n\ntheorem Submodule.traceDual_le_span_map_traceDual [Module.Free A R₂]\n    [IsLocalization (Algebra.algebraMapSubmonoid R₂ A⁰) F₂] (h₁ : F₁.LinearDisjoint F₂)\n    (h₂ : F₁ ⊔ F₂ = ⊤) :\n    (traceDual R₁ F₁ (1 : Submodule B L)).restrictScalars R₁ ≤\n      span R₁ (algebraMap F₂ L '' (traceDual A K (1 : Submodule R₂ F₂))) := by\n  intro x hx\n  have h₂' : F₁.toSubalgebra ⊔ F₂.toSubalgebra = ⊤ := by\n    simpa [sup_toSubalgebra_of_isAlgebraic_right] using congr_arg IntermediateField.toSubalgebra h₂\n  let b₂ := (Free.chooseBasis A R₂).localizationLocalization K A⁰ F₂\n  let B₁ := h₁.basisOfBasisRight h₂' b₂\n  have h_main : x ∈ span R₁ (Set.range B₁.traceDual) := by\n    rw [B₁.traceDual.mem_span_iff_repr_mem R₁ x]\n    intro i\n    rw [B₁.traceDual_repr_apply]\n    refine mem_traceDual.mp hx _ ?_\n    rw [LinearDisjoint.basisOfBasisRight_apply, Basis.localizationLocalization_apply,\n      ← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply R₂ B L, mem_one]\n    exact ⟨_, rfl⟩\n  have h : Set.range B₁.traceDual =\n      Set.range (IsScalarTower.toAlgHom A F₂ L ∘ b₂.traceDual) := by\n    refine congr_arg Set.range <| B₁.traceDual_eq_iff.mpr fun i j ↦ ?_\n    rw [LinearDisjoint.basisOfBasisRight_apply, traceForm_apply, Function.comp_apply,\n      IsScalarTower.coe_toAlgHom', ← map_mul, h₁.trace_algebraMap h₂, b₂.trace_traceDual_mul,\n      MonoidWithZeroHom.map_ite_one_zero]\n  rwa [← span_span_of_tower A R₁, h, Set.range_comp, ← AlgHom.coe_toLinearMap, ← map_span,\n    ← traceDual_span_of_basis A (1 : Submodule R₂ F₂) b₂\n      (by rw [Basis.localizationLocalization_span K A⁰ F₂]; ext; simp)] at h_main\n\nattribute [local instance] FractionRing.liftAlgebra\n\nvariable [IsDomain A] [IsDedekindDomain B] [IsDedekindDomain R₁] [IsDedekindDomain R₂]\n    [IsFractionRing B L] [IsFractionRing R₁ F₁] [IsFractionRing R₂ F₂] [IsIntegrallyClosed A]\n    [IsIntegralClosure B R₁ L] [IsTorsionFree R₁ B] [IsTorsionFree R₂ B]\n\nset_option linter.overlappingInstances false\n\nnamespace IsDedekindDomain\n\ntheorem differentIdeal_dvd_map_differentIdeal [Algebra.IsIntegral R₂ B]\n    [Module.Free A R₂] [IsLocalization (Algebra.algebraMapSubmonoid R₂ A⁰) F₂]\n    (h₁ : F₁.LinearDisjoint F₂) (h₂ : F₁ ⊔ F₂ = ⊤) :\n    differentIdeal R₁ B ∣ Ideal.map (algebraMap R₂ B) (differentIdeal A R₂) := by\n  have : Algebra.IsSeparable (FractionRing A) (FractionRing R₂) := by\n    refine Algebra.IsSeparable.of_equiv_equiv (FractionRing.algEquiv A K).symm.toRingEquiv\n          (FractionRing.algEquiv R₂ F₂).symm.toRingEquiv ?_\n    ext _\n    exact IsFractionRing.algEquiv_commutes (FractionRing.algEquiv A K).symm\n      (FractionRing.algEquiv R₂ ↥F₂).symm _\n  rw [Ideal.dvd_iff_le, ← coeIdeal_le_coeIdeal L, coeIdeal_differentIdeal R₁ F₁ L B,\n    ← extendedHom_coeIdeal_eq_map L B (K := F₂), le_inv_comm _ (by simp), ← map_inv₀,\n    coeIdeal_differentIdeal A K, inv_inv, ← coe_le_coe, coe_dual_one, coe_extendedHom_eq_span,\n    ← coeToSet_coeToSubmodule, coe_dual_one]\n  · have := Submodule.span_mono (R := B) <| traceDual_le_span_map_traceDual A B R₁ R₂ h₁ h₂\n    rwa [← span_coe_eq_restrictScalars, span_span_of_tower, span_span_of_tower, span_eq] at this\n  · exact (_root_.map_ne_zero _).mpr <| coeIdeal_eq_zero.not.mpr differentIdeal_ne_bot\n\nvariable [Algebra A B] [Module.Finite A B] [IsTorsionFree A B] [IsTorsionFree A R₁]\n  [IsTorsionFree A R₂] [Module.Finite A R₁] [Module.Finite R₂ B] [IsScalarTower A R₂ B]\n  [Module.Finite R₁ B] [Algebra.IsSeparable (FractionRing A) (FractionRing B)]\n  [IsScalarTower A R₁ B]\n\ntheorem map_differentIdeal_dvd_differentIdeal\n    (h : IsCoprime ((differentIdeal A R₁).map (algebraMap R₁ B))\n      ((differentIdeal A R₂).map (algebraMap R₂ B))) :\n    Ideal.map (algebraMap R₂ B) (differentIdeal A R₂) ∣ differentIdeal R₁ B :=\n  have := (differentIdeal_eq_differentIdeal_mul_differentIdeal A R₂ B).symm.trans\n    (differentIdeal_eq_differentIdeal_mul_differentIdeal A R₁ B)\n  h.symm.dvd_of_dvd_mul_right (dvd_of_mul_left_eq _ this)\n\ntheorem differentIdeal_eq_map_differentIdeal [Module.Free A R₂] (h₁ : F₁.LinearDisjoint F₂)\n    (h₂ : F₁ ⊔ F₂ = ⊤)\n    (h₃ : IsCoprime ((differentIdeal A R₁).map (algebraMap R₁ B))\n      ((differentIdeal A R₂).map (algebraMap R₂ B))) :\n    differentIdeal R₁ B = Ideal.map (algebraMap R₂ B) (differentIdeal A R₂) := by\n  apply dvd_antisymm\n  · exact differentIdeal_dvd_map_differentIdeal A B R₁ R₂ h₁ h₂\n  · exact map_differentIdeal_dvd_differentIdeal A B R₁ R₂ h₃\n\n/--\nLet `A ⊆ B` be a finite extension of Dedekind domains and assume that `A ⊆ R₁, R₂ ⊆ B` are two\nsubrings such that `Frac R₁ ⊔ Frac R₂ = Frac B`, `Frac R₁` and `Frac R₂` are linearly disjoint\nover `Frac A`, and that `𝓓(R₁/A)` and `𝓓(R₂/A)` are coprime where `𝓓` denotes the different ideal\nand `Frac R` denotes the fraction field of a domain `R`.\nWe have `𝓓(B/A) = 𝓓(R₁/A) * 𝓓(R₂/A)`.\n-/\ntheorem differentIdeal_eq_differentIdeal_mul_differentIdeal_of_isCoprime\n    [Module.Free A R₂] (h₁ : F₁.LinearDisjoint F₂) (h₂ : F₁ ⊔ F₂ = ⊤)\n    (h₃ : IsCoprime ((differentIdeal A R₁).map (algebraMap R₁ B))\n      ((differentIdeal A R₂).map (algebraMap R₂ B))) :\n    differentIdeal A B = differentIdeal R₁ B * differentIdeal R₂ B := by\n  have := differentIdeal_eq_differentIdeal_mul_differentIdeal A R₂ B\n  rwa [← differentIdeal_eq_map_differentIdeal A B R₁ R₂ h₁ h₂ h₃,\n    mul_comm] at this\n\nend IsDedekindDomain\n\nvariable [Algebra A B] [Module.Finite A B] [IsTorsionFree A B] [IsTorsionFree A R₁]\n  [IsTorsionFree A R₂] [Module.Finite A R₁] [Module.Finite R₂ B] [IsScalarTower A R₂ B]\n  [Module.Finite R₁ B] [Algebra.IsSeparable (FractionRing A) (FractionRing B)]\n  [IsScalarTower A R₁ B]\n\nTarget:\ntheorem Submodule.traceDual_eq_span_map_traceDual_of_linearDisjoint [Module.Free A R₂]\n    [IsLocalization (Algebra.algebraMapSubmonoid R₂ A⁰) F₂] (h₁ : F₁.LinearDisjoint F₂)\n    (h₂ : F₁ ⊔ F₂ = ⊤) (h₃ : IsCoprime ((differentIdeal A R₁).map (algebraMap R₁ B))\n      ((differentIdeal A R₂).map (algebraMap R₂ B))) :\n    span R₁ (algebraMap F₂ L '' (traceDual A K (1 : Submodule R₂ F₂))) =\n      (traceDual R₁ F₁ (1 : Submodule B L)).restrictScalars R₁ :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  have : Algebra.IsSeparable (FractionRing A) (FractionRing R₂) := by\n    refine Algebra.IsSeparable.of_equiv_equiv (FractionRing.algEquiv A K).symm.toRingEquiv\n          (FractionRing.algEquiv R₂ F₂).symm.toRingEquiv ?_\n    ext x\n    exact IsFractionRing.algEquiv_commutes (FractionRing.algEquiv A K).symm\n      (FractionRing.algEquiv R₂ ↥F₂).symm _\n  suffices span B (algebraMap F₂ L '' (traceDual A K (1 : Submodule R₂ F₂))) ≤\n      traceDual R₁ F₁ (1 : Submodule B L) by\n    apply le_antisymm\n    · refine SetLike.coe_subset_coe.mp (subset_trans ?_ this)\n      rw [← Submodule.span_span_of_tower R₁ B]\n      exact Submodule.subset_span\n    · exact traceDual_le_span_map_traceDual A B R₁ R₂ h₁ h₂\n  have := dvd_of_eq <|\n    (IsDedekindDomain.differentIdeal_eq_map_differentIdeal A B R₁ R₂ h₁ h₂ h₃).symm\n  rwa [Ideal.dvd_iff_le, ← coeIdeal_le_coeIdeal (K := L), coeIdeal_differentIdeal R₁ F₁,\n    inv_le_comm, ← extendedHom_coeIdeal_eq_map (K := F₂), coeIdeal_differentIdeal A K, map_inv₀,\n    inv_inv, ← coe_le_coe, coe_extendedHom_eq_span, coe_dual_one, ← coeToSet_coeToSubmodule,\n    coe_dual_one] at this\n  · simp\n  · rw [← extendedHom_coeIdeal_eq_map (K := F₂), ne_eq, extendedHom_eq_zero_iff]\n    rw [coeIdeal_eq_zero]\n    exact differentIdeal_ne_bot","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/DedekindDomain","family_id":"submodule","file_id":"mathlib/Mathlib/RingTheory/DedekindDomain/LinearDisjoint.lean","sample_id":"a86a08c70d72b421064dab529cf5e5d4ac90c62963744456fb1461e7c0e42c96"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b310f14e4f1b5c03181254cbcff43ceec8a366fb3cce3389a0bcba30d3ef880d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9a2df3c5de7a292cbf62d67b3b00525a929d8c851243d6ce0af4fd45360425d2","source_sha256":"69b176e2c392735eaa07198e23b2563fecf64b20f095a33f543e0985594e4963","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases h0 : p = 0\n  · simp [h0]\n  by_cases h_top : p = ∞\n  · simp only [h_top, eLpNorm, eLpNormEssSup, ENNReal.top_ne_zero, if_true,\n      if_false]\n    apply essSup_eq\n    exact h.comp (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)\n  simp only [eLpNorm_eq_eLpNorm' h0 h_top, eLpNorm', one_div]\n  congr 1\n  apply lintegral_eq\n  exact h.comp (Measurable.pow_const (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)\n    p.toReal)","hard_negative":true,"metrics":{"chosen_tokens":81,"rejected_tokens":8,"token_jaccard":0.044444,"token_length_ratio":0.098765},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"51a3806fd13a2e5f6fe0a800f5a88efa7311e04ccac58eebaeb4934dc4df794f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.HasLaw\npublic import Mathlib.Probability.Moments.Variance\npublic import Mathlib.MeasureTheory.Function.UniformIntegrable\n\nNamespace:\nProbabilityTheory.IdentDistrib\n\nLocal context:\n/-\nCopyright (c) 2022 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Identically distributed random variables\n\nTwo random variables defined on two (possibly different) probability spaces but taking value in\nthe same space are *identically distributed* if their distributions (i.e., the image probability\nmeasures on the target space) coincide. We define this concept and establish its basic properties\nin this file.\n\n## Main definitions and results\n\n* `IdentDistrib f g μ ν` registers that the image of `μ` under `f` coincides with the image of `ν`\n  under `g` (and that `f` and `g` are almost everywhere measurable, as otherwise the image measures\n  don't make sense). The measures can be kept implicit as in `IdentDistrib f g` if the spaces\n  are registered as measure spaces.\n* `IdentDistrib.comp`: being identically distributed is stable under composition with measurable\n  maps.\n\nThere are two main kinds of lemmas, under the assumption that `f` and `g` are identically\ndistributed: lemmas saying that two quantities computed for `f` and `g` are the same, and lemmas\nsaying that if `f` has some property then `g` also has it. The first kind is registered as\n`IdentDistrib.foo_fst`, the second one as `IdentDistrib.foo_snd` (in the latter case, to deduce\na property of `f` from one of `g`, use `h.symm.foo_snd` where `h : IdentDistrib f g μ ν`). For\ninstance:\n\n* `IdentDistrib.measure_mem_eq`: if `f` and `g` are identically distributed, then the probabilities\n  that they belong to a given measurable set are the same.\n* `IdentDistrib.integral_eq`: if `f` and `g` are identically distributed, then their integrals\n  are the same.\n* `IdentDistrib.variance_eq`: if `f` and `g` are identically distributed, then their variances\n  are the same.\n\n* `IdentDistrib.aestronglyMeasurable_snd`: if `f` and `g` are identically distributed and `f`\n  is almost everywhere strongly measurable, then so is `g`.\n* `IdentDistrib.memLp_snd`: if `f` and `g` are identically distributed and `f`\n  belongs to `ℒp`, then so does `g`.\n\nWe also register several dot notation shortcuts for convenience.\nFor instance, if `h : IdentDistrib f g μ ν`, then `h.sq` states that `f^2` and `g^2` are\nidentically distributed, and `h.norm` states that `‖f‖` and `‖g‖` are identically distributed, and\nso on.\n-/\n\npublic section\n\n\nopen MeasureTheory Filter Finset\n\nnoncomputable section\n\nopen scoped Topology MeasureTheory ENNReal NNReal\n\nvariable {α β γ δ : Type*} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ]\n  [MeasurableSpace δ]\n\nnamespace ProbabilityTheory\n\n/-- Two functions defined on two (possibly different) measure spaces are identically distributed if\ntheir image measures coincide. This only makes sense when the functions are ae measurable\n(as otherwise the image measures are not defined), so we require this as well in the definition. -/\nstructure IdentDistrib (f : α → γ) (g : β → γ)\n    (μ : Measure α := by volume_tac)\n    (ν : Measure β := by volume_tac) : Prop where\n  aemeasurable_fst : AEMeasurable f μ\n  aemeasurable_snd : AEMeasurable g ν\n  map_eq : Measure.map f μ = Measure.map g ν\n\nnamespace IdentDistrib\n\nopen TopologicalSpace\n\nvariable {μ : Measure α} {ν : Measure β} {f : α → γ} {g : β → γ}\n\nprotected theorem refl (hf : AEMeasurable f μ) : IdentDistrib f f μ μ :=\n  { aemeasurable_fst := hf\n    aemeasurable_snd := hf\n    map_eq := rfl }\n\nprotected theorem symm (h : IdentDistrib f g μ ν) : IdentDistrib g f ν μ :=\n  { aemeasurable_fst := h.aemeasurable_snd\n    aemeasurable_snd := h.aemeasurable_fst\n    map_eq := h.map_eq.symm }\n\nprotected theorem trans {ρ : Measure δ} {h : δ → γ} (h₁ : IdentDistrib f g μ ν)\n    (h₂ : IdentDistrib g h ν ρ) : IdentDistrib f h μ ρ :=\n  { aemeasurable_fst := h₁.aemeasurable_fst\n    aemeasurable_snd := h₂.aemeasurable_snd\n    map_eq := h₁.map_eq.trans h₂.map_eq }\n\nprotected theorem comp_of_aemeasurable {u : γ → δ} (h : IdentDistrib f g μ ν)\n    (hu : AEMeasurable u (Measure.map f μ)) : IdentDistrib (u ∘ f) (u ∘ g) μ ν :=\n  { aemeasurable_fst := hu.comp_aemeasurable h.aemeasurable_fst\n    aemeasurable_snd := by rw [h.map_eq] at hu; exact hu.comp_aemeasurable h.aemeasurable_snd\n    map_eq := by\n      rw [← AEMeasurable.map_map_of_aemeasurable hu h.aemeasurable_fst, ←\n        AEMeasurable.map_map_of_aemeasurable _ h.aemeasurable_snd, h.map_eq]\n      rwa [← h.map_eq] }\n\nprotected theorem comp {u : γ → δ} (h : IdentDistrib f g μ ν) (hu : Measurable u) :\n    IdentDistrib (u ∘ f) (u ∘ g) μ ν :=\n  h.comp_of_aemeasurable hu.aemeasurable\n\nprotected theorem of_ae_eq {g : α → γ} (hf : AEMeasurable f μ) (heq : f =ᵐ[μ] g) :\n    IdentDistrib f g μ μ :=\n  { aemeasurable_fst := hf\n    aemeasurable_snd := hf.congr heq\n    map_eq := Measure.map_congr heq }\n\nlemma _root_.MeasureTheory.AEMeasurable.identDistrib_mk\n    (hf : AEMeasurable f μ) : IdentDistrib f (hf.mk f) μ μ :=\n  IdentDistrib.of_ae_eq hf hf.ae_eq_mk\n\nlemma _root_.MeasureTheory.AEStronglyMeasurable.identDistrib_mk\n    [TopologicalSpace γ] [PseudoMetrizableSpace γ] [BorelSpace γ]\n    (hf : AEStronglyMeasurable f μ) : IdentDistrib f (hf.mk f) μ μ :=\n  IdentDistrib.of_ae_eq hf.aemeasurable hf.ae_eq_mk\n\ntheorem measure_mem_eq (h : IdentDistrib f g μ ν) {s : Set γ} (hs : MeasurableSet s) :\n    μ (f ⁻¹' s) = ν (g ⁻¹' s) := by\n  rw [← Measure.map_apply_of_aemeasurable h.aemeasurable_fst hs, ←\n    Measure.map_apply_of_aemeasurable h.aemeasurable_snd hs, h.map_eq]\n\nalias measure_preimage_eq := measure_mem_eq\n\ntheorem ae_snd (h : IdentDistrib f g μ ν) {p : γ → Prop} (pmeas : MeasurableSet {x | p x})\n    (hp : ∀ᵐ x ∂μ, p (f x)) : ∀ᵐ x ∂ν, p (g x) := by\n  apply (ae_map_iff h.aemeasurable_snd pmeas).1\n  rw [← h.map_eq]\n  exact (ae_map_iff h.aemeasurable_fst pmeas).2 hp\n\ntheorem ae_mem_snd (h : IdentDistrib f g μ ν) {t : Set γ} (tmeas : MeasurableSet t)\n    (ht : ∀ᵐ x ∂μ, f x ∈ t) : ∀ᵐ x ∂ν, g x ∈ t :=\n  h.ae_snd tmeas ht\n\ntheorem _root_.ProbabilityTheory.HasLaw.identDistrib {κ : Measure γ} (h₀ : HasLaw f κ μ)\n    (h₁ : HasLaw g κ ν) : IdentDistrib f g μ ν :=\n  ⟨h₀.aemeasurable, h₁.aemeasurable, by simp [h₀.map_eq, h₁.map_eq]⟩\n\ntheorem hasLaw {κ : Measure γ} (h₀ : IdentDistrib f g μ ν) (h₁ : HasLaw f κ μ) : HasLaw g κ ν :=\n  ⟨h₀.aemeasurable_snd, by simp [h₀.map_eq, ← h₁.map_eq]⟩\n\n/-- In a second countable topology, the first function in an identically distributed pair is a.e.\nstrongly measurable. So is the second function, but use `h.symm.aestronglyMeasurable_fst` as\n`h.aestronglyMeasurable_snd` has a different meaning. -/\ntheorem aestronglyMeasurable_fst [TopologicalSpace γ] [PseudoMetrizableSpace γ]\n    [OpensMeasurableSpace γ] [SecondCountableTopology γ] (h : IdentDistrib f g μ ν) :\n    AEStronglyMeasurable f μ :=\n  h.aemeasurable_fst.aestronglyMeasurable\n\n/-- If `f` and `g` are identically distributed and `f` is a.e. strongly measurable, so is `g`. -/\ntheorem aestronglyMeasurable_snd [TopologicalSpace γ] [PseudoMetrizableSpace γ] [BorelSpace γ]\n    (h : IdentDistrib f g μ ν) (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable g ν := by\n  refine aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨h.aemeasurable_snd, ?_⟩\n  rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 hf).2 with ⟨t, t_sep, ht⟩\n  refine ⟨closure t, t_sep.closure, ?_⟩\n  apply h.ae_mem_snd isClosed_closure.measurableSet\n  filter_upwards [ht] with x hx using subset_closure hx\n\ntheorem aestronglyMeasurable_iff [TopologicalSpace γ] [PseudoMetrizableSpace γ] [BorelSpace γ]\n    (h : IdentDistrib f g μ ν) : AEStronglyMeasurable f μ ↔ AEStronglyMeasurable g ν :=\n  ⟨fun hf => h.aestronglyMeasurable_snd hf, fun hg => h.symm.aestronglyMeasurable_snd hg⟩\n\ntheorem essSup_eq [ConditionallyCompleteLinearOrder γ] [TopologicalSpace γ] [OpensMeasurableSpace γ]\n    [OrderClosedTopology γ] (h : IdentDistrib f g μ ν) : essSup f μ = essSup g ν := by\n  have I : ∀ a, μ {x : α | a < f x} = ν {x : β | a < g x} := fun a =>\n    h.measure_mem_eq measurableSet_Ioi\n  simp_rw [essSup_eq_sInf, I]\n\ntheorem lintegral_eq {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (h : IdentDistrib f g μ ν) :\n    ∫⁻ x, f x ∂μ = ∫⁻ x, g x ∂ν := by\n  change ∫⁻ x, id (f x) ∂μ = ∫⁻ x, id (g x) ∂ν\n  rw [← lintegral_map' aemeasurable_id h.aemeasurable_fst, ←\n    lintegral_map' aemeasurable_id h.aemeasurable_snd, h.map_eq]\n\ntheorem integral_eq [NormedAddCommGroup γ] [NormedSpace ℝ γ] [BorelSpace γ]\n    (h : IdentDistrib f g μ ν) : ∫ x, f x ∂μ = ∫ x, g x ∂ν := by\n  by_cases hf : AEStronglyMeasurable f μ\n  · have A : AEStronglyMeasurable id (Measure.map f μ) := by\n      rw [aestronglyMeasurable_iff_aemeasurable_separable]\n      rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 hf).2 with ⟨t, t_sep, ht⟩\n      refine ⟨aemeasurable_id, ⟨closure t, t_sep.closure, ?_⟩⟩\n      rw [ae_map_iff h.aemeasurable_fst]\n      · filter_upwards [ht] with x hx using subset_closure hx\n      · exact isClosed_closure.measurableSet\n    change ∫ x, id (f x) ∂μ = ∫ x, id (g x) ∂ν\n    rw [← integral_map h.aemeasurable_fst A]\n    rw [h.map_eq] at A\n    rw [← integral_map h.aemeasurable_snd A, h.map_eq]\n  · rw [integral_non_aestronglyMeasurable hf]\n    rw [h.aestronglyMeasurable_iff] at hf\n    rw [integral_non_aestronglyMeasurable hf]\n\nTarget:\ntheorem eLpNorm_eq [NormedAddCommGroup γ] [OpensMeasurableSpace γ] (h : IdentDistrib f g μ ν)\n    (p : ℝ≥0∞) : eLpNorm f p μ = eLpNorm g p ν :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"4bda3c70154c73fd82f4f61e82204ebe6803b1d53f5220b246c409dcdfc6f123","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability","family_id":"elpnorm_eq","file_id":"mathlib/Mathlib/Probability/IdentDistrib.lean","sample_id":"9a2df3c5de7a292cbf62d67b3b00525a929d8c851243d6ce0af4fd45360425d2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2fe6742b90ce6b6b787aeae843d8b7476e8ec34c6e8c54b97436e93c09e1ac78","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1144fa410a8b26bda3677497f6231039c429b8f262693cec88f6d80138aa67f1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9edf83ad0b7479676e136f7279c23e835e46a349748bddabb519ed9ff699acc7","source_sha256":"06fcefd9d2e8fefa05b9d4bdca9149cb45a5aceb7ddd5cd1ef4e639ced848f8f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : IsNoetherianRing A := Algebra.FiniteType.isNoetherianRing R A\n  rw [finite_iff_isArtinianRing, isArtinianRing_iff_isNoetherianRing_krullDimLE_zero,\n    and_iff_right this]","hard_negative":true,"metrics":{"chosen_tokens":22,"rejected_tokens":5,"token_jaccard":0.090909,"token_length_ratio":0.227273},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"51fe28b37bf58dba6860541937fc17444f1e588d8b5508e9c9e40ba714a1badb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.HopkinsLevitzki\npublic import Mathlib.RingTheory.Jacobson.Ring\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu, Andrew Yang\n-/\n/-!\n# Artinian rings over Jacobson rings\n\n## Main results\n- `Module.finite_iff_isArtinianRing`: If `A` is a finite type algebra over an Artinian ring `R`,\n  then `A` is finite over `R` if and only if `A` is an Artinian ring.\n\n-/\n\npublic section\n\nvariable (R A : Type*) [CommRing R] [CommRing A] [Algebra R A] [Algebra.FiniteType R A]\n\nattribute [local instance] IsArtinianRing.fieldOfSubtypeIsMaximal in\nlemma Module.finite_of_isSemisimpleRing [IsJacobsonRing R] [IsSemisimpleRing A] :\n    Module.Finite R A :=\n  have (I : MaximalSpectrum A) := finite_of_finite_type_of_isJacobsonRing R (A ⧸ I.asIdeal)\n  .equiv ((IsArtinianRing.equivPi A).restrictScalars R).toLinearEquiv.symm\n\n/-- If `A` is a finite type algebra over `R`, then `A` is an Artinian ring and `R` is Jacobson\nimplies `A` is finite over `R`. -/\n/- If made an instance, causes timeouts synthesizing `FaithfulSMul R I.ResidueField` at\n`Ideal.algebraMap_residueField_eq_zero` and `Ideal.ker_algebraMap_residueField` during\nsimpNF linting. -/\nlemma Module.finite_of_isArtinianRing [IsJacobsonRing R] [IsArtinianRing A] :\n    Module.Finite R A :=\n  have := finite_of_isSemisimpleRing R (A ⧸ Ring.jacobson A)\n  IsSemiprimaryRing.finite_of_isArtinian R A A\n\n/-- If `A` is a finite type algebra over an Artinian ring `R`,\nthen `A` is finite over `R` if and only if `A` is an Artinian ring. -/\nlemma Module.finite_iff_isArtinianRing [IsArtinianRing R] :\n    Module.Finite R A ↔ IsArtinianRing A :=\n  ⟨isArtinian_of_tower _ ∘ ((IsArtinianRing.tfae R A).out 0 2).mp,\n    fun _ ↦ finite_of_isArtinianRing R A⟩\n\n/-- If `A` is a finite type algebra over an Artinian ring `R`,\nthen `A` is finite over `R` if and only if `dim A = 0`. -/\n\nTarget:\nlemma Module.finite_iff_krullDimLE_zero [IsArtinianRing R] :\n    Module.Finite R A ↔ Ring.KrullDimLE 0 A :=\n\nProof body:\n","rejected":"by\n  exact Module.finite_iff_krullDimLE_zero","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"446cda6c7cb6df039c0674d67b8d800b3e4b8e9491e8d8acb1a599cae761bc37","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Jacobson","family_id":"module","file_id":"mathlib/Mathlib/RingTheory/Jacobson/Artinian.lean","sample_id":"9edf83ad0b7479676e136f7279c23e835e46a349748bddabb519ed9ff699acc7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ed61a64bf86bd3e8070ca4701c17ddc605afcea75c9e8aec1615eb96bd1b85a1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"591110f7b37cc2daae15e5485ae2d271b10317dae65d4419218d3c8977316858","source_sha256":"830973e88871280f6e5db16d5b576af36be09ea83a1a7629aab6adabb39a26a1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction n with\n  | zero => simp\n  | succ n IH =>\n    rw [iteratedLineDerivOp_succ_left]\n    exact (tsupport_lineDerivOp_subset (m 0) _).trans (IH <| Fin.tail m)","hard_negative":true,"metrics":{"chosen_tokens":37,"rejected_tokens":8,"token_jaccard":0.03125,"token_length_ratio":0.216216},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"5224d1e183f7982dd854bbb50f206fa8491ea1bcccb1939de480c55142208494","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Distribution.SchwartzSpace.Basic\npublic import Mathlib.Analysis.Calculus.IteratedDeriv.Defs\npublic import Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts\npublic import Mathlib.Analysis.InnerProductSpace.Laplacian\npublic import Mathlib.MeasureTheory.Integral.IntegralEqImproper\n\nNamespace:\nSchwartzMap\n\nLocal context:\n/-\nCopyright (c) 2026 Moritz Doll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Moritz Doll\n-/\n/-!\n# Derivatives of Schwartz functions\n\nIn this file we define the various notions of derivatives of Schwartz functions.\n\n## Main definitions\n\n* `SchwartzMap.fderivCLM`: The differential as a continuous linear map\n  `𝓢(E, F) →L[𝕜] 𝓢(E, E →L[ℝ] F)`\n* `SchwartzMap.derivCLM`: The one-dimensional derivative as a continuous linear map\n  `𝓢(ℝ, F) →L[𝕜] 𝓢(ℝ, F)`\n* `SchwartzMap.instLineDeriv`: The directional derivative with notation `∂_{m} f`\n* `SchwartzMap.instLaplacian`: The Laplacian for `𝓢(E, F)` as an instance of the notation type-class\n  `Laplacian`.\n\n## Main statements\n\n* `SchwartzMap.iteratedLineDerivOp_eq_iteratedFDeriv`: the iterated directional derivative is given\n  by the applied Fréchet derivative of a Schwartz function.\n* `SchwartzMap.laplacian_eq_sum`: the Laplacian is given by the sum of second derivatives in any\n  orthonormal basis.\n* `SchwartzMap.integral_bilinear_lineDerivOp_right_eq_neg_left`: Integration by parts using the\n  directional derivative `∂_{m}`\n* `SchwartzMap.integral_bilinear_laplacian_right_eq_left`: Integration by parts for the Laplacian\n\n-/\n\n@[expose] public noncomputable section\n\nvariable {ι 𝕜 𝕜' D E F V F F₁ F₂ F₃ : Type*}\n\nnamespace SchwartzMap\n\nvariable [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ F]\n\nsection Derivatives\n\n/-! ### Derivatives of Schwartz functions -/\n\nvariable [NormedSpace ℝ E]\n\nvariable (𝕜)\nvariable [RCLike 𝕜] [NormedSpace 𝕜 F]\n\nvariable (F) in\n/-- The 1-dimensional derivative on Schwartz space as a continuous `𝕜`-linear map. -/\ndef derivCLM : 𝓢(ℝ, F) →L[𝕜] 𝓢(ℝ, F) :=\n  mkCLM (deriv ·) (fun f g _ => deriv_add f.differentiableAt g.differentiableAt)\n    (fun a f _ => deriv_const_smul a f.differentiableAt)\n    (fun f => (contDiff_succ_iff_deriv.mp (f.smooth ⊤)).2.2) fun ⟨k, n⟩ =>\n    ⟨{⟨k, n + 1⟩}, 1, zero_le_one, fun f x => by\n      simpa only [Real.norm_eq_abs, Finset.sup_singleton, schwartzSeminormFamily_apply, one_mul,\n        norm_iteratedFDeriv_eq_norm_iteratedDeriv, ← iteratedDeriv_succ'] using\n        f.le_seminorm' 𝕜 k (n + 1) x⟩\n\n@[simp]\ntheorem derivCLM_apply (f : 𝓢(ℝ, F)) (x : ℝ) : derivCLM 𝕜 F f x = deriv f x :=\n  rfl\n\ntheorem hasDerivAt (f : 𝓢(ℝ, F)) (x : ℝ) : HasDerivAt f (deriv f x) x :=\n  f.differentiableAt.hasDerivAt\n\nopen LineDeriv\n\nsection fderiv\n\nvariable [SMulCommClass ℝ 𝕜 F]\n\nvariable (E F) in\n/-- The Fréchet derivative on Schwartz space as a continuous `𝕜`-linear map. -/\ndef fderivCLM : 𝓢(E, F) →L[𝕜] 𝓢(E, E →L[ℝ] F) :=\n  mkCLM (fderiv ℝ ·) (fun f g _ => fderiv_add f.differentiableAt g.differentiableAt)\n    (fun a f _ => fderiv_const_smul f.differentiableAt a)\n    (fun f => (contDiff_succ_iff_fderiv.mp (f.smooth ⊤)).2.2) fun ⟨k, n⟩ =>\n    ⟨{⟨k, n + 1⟩}, 1, zero_le_one, fun f x => by\n      simpa only [schwartzSeminormFamily_apply, Seminorm.comp_apply, Finset.sup_singleton,\n        one_smul, norm_iteratedFDeriv_fderiv, one_mul] using f.le_seminorm 𝕜 k (n + 1) x⟩\n\n@[simp]\ntheorem fderivCLM_apply (f : 𝓢(E, F)) (x : E) : fderivCLM 𝕜 E F f x = fderiv ℝ f x :=\n  rfl\n\ntheorem hasFDerivAt (f : 𝓢(E, F)) (x : E) : HasFDerivAt f (fderiv ℝ f x) x :=\n  f.differentiableAt.hasFDerivAt\n\n/-- The partial derivative (or directional derivative) in the direction `m : E` as a\ncontinuous linear map on Schwartz space. -/\ninstance : LineDeriv E 𝓢(E, F) 𝓢(E, F) where\n  lineDerivOp m f := (SchwartzMap.evalCLM ℝ E F m ∘L fderivCLM ℝ E F) f\n\ntheorem lineDerivOp_apply_eq_fderiv (m : E) (f : 𝓢(E, F)) (x : E) :\n    ∂_{m} f x = fderiv ℝ f x m := rfl\n\ninstance : LineDerivAdd E 𝓢(E, F) 𝓢(E, F) where\n  lineDerivOp_add m := ((SchwartzMap.evalCLM ℝ E F m).comp (fderivCLM ℝ E F)).map_add\n  lineDerivOp_left_add v w f := by\n    ext x\n    simp [lineDerivOp_apply_eq_fderiv]\n\ninstance : LineDerivSMul 𝕜 E 𝓢(E, F) 𝓢(E, F) where\n  lineDerivOp_smul m := (SchwartzMap.evalCLM 𝕜 E F m ∘L fderivCLM 𝕜 E F).map_smul\n\ninstance : LineDerivLeftSMul ℝ E 𝓢(E, F) 𝓢(E, F) where\n  lineDerivOp_left_smul r y f := by\n    ext x\n    simp [lineDerivOp_apply_eq_fderiv]\n\ninstance : ContinuousLineDeriv E 𝓢(E, F) 𝓢(E, F) where\n  continuous_lineDerivOp m := (SchwartzMap.evalCLM ℝ E F m ∘L fderivCLM ℝ E F).continuous\n\nopen LineDeriv\n\ntheorem lineDerivOpCLM_eq (m : E) :\n    lineDerivOpCLM 𝕜 𝓢(E, F) m = SchwartzMap.evalCLM 𝕜 E F m ∘L fderivCLM 𝕜 E F := rfl\n\n@[deprecated (since := \"2025-11-25\")]\nalias pderivCLM := lineDerivOpCLM\n\n@[deprecated (since := \"2025-11-25\")]\nalias pderivCLM_apply := LineDeriv.lineDerivOpCLM_apply\n\ntheorem lineDerivOp_apply (m : E) (f : 𝓢(E, F)) (x : E) : ∂_{m} f x = lineDeriv ℝ f x m :=\n  f.differentiableAt.lineDeriv_eq_fderiv.symm\n\n@[deprecated (since := \"2025-11-25\")]\nalias iteratedPDeriv := LineDeriv.iteratedLineDerivOpCLM\n\n@[deprecated (since := \"2025-11-25\")]\nalias iteratedPDeriv_zero := LineDeriv.iteratedLineDerivOp_zero\n\n@[deprecated (since := \"2025-11-25\")]\nalias iteratedPDeriv_one := LineDeriv.iteratedLineDerivOp_one\n\n@[deprecated (since := \"2025-11-25\")]\nalias iteratedPDeriv_succ_left := LineDeriv.iteratedLineDerivOp_succ_left\n\n@[deprecated (since := \"2025-11-25\")]\nalias iteratedPDeriv_succ_right := LineDeriv.iteratedLineDerivOp_succ_right\n\ntheorem iteratedLineDerivOp_eq_iteratedFDeriv {n : ℕ} {m : Fin n → E} {f : 𝓢(E, F)} {x : E} :\n    ∂^{m} f x = iteratedFDeriv ℝ n f x m := by\n  induction n generalizing x with\n  | zero => simp\n  | succ n ih =>\n    rw [iteratedLineDerivOp_succ_left, iteratedFDeriv_succ_apply_left,\n      ← fderiv_continuousMultilinear_apply_const_apply]\n    · simp only [lineDerivOp_apply_eq_fderiv, ← ih]\n    · exact (f.smooth ⊤).differentiable_iteratedFDeriv (mod_cast ENat.coe_lt_top n) x\n\n@[deprecated (since := \"2025-11-25\")]\nalias iteratedPDeriv_eq_iteratedFDeriv := iteratedLineDerivOp_eq_iteratedFDeriv\n\nend fderiv\n\nvariable [NormedAddCommGroup D] [NormedSpace ℝ D]\n\ntheorem lineDerivOp_compCLMOfContinuousLinearEquiv (m : D) (g : D ≃L[ℝ] E) (f : 𝓢(E, F)) :\n    ∂_{m} (compCLMOfContinuousLinearEquiv 𝕜 g f) =\n    compCLMOfContinuousLinearEquiv 𝕜 g (∂_{g m} f) := by\n  ext x\n  simp [lineDerivOp_apply_eq_fderiv, ContinuousLinearEquiv.comp_right_fderiv]\n\nend Derivatives\n\nsection support\n\nvariable (𝕜)\nvariable [RCLike 𝕜] [NormedSpace 𝕜 F]\n\ntheorem tsupport_derivCLM_subset (f : 𝓢(ℝ, F)) : tsupport (derivCLM 𝕜 F f) ⊆ tsupport f := by\n  change tsupport (deriv f ·) ⊆ _\n  simp_rw [← fderiv_apply_one_eq_deriv]\n  exact tsupport_fderiv_apply_subset ℝ 1\n\nvariable [NormedSpace ℝ E] [SMulCommClass ℝ 𝕜 F]\n\ntheorem tsupport_fderivCLM_subset (f : 𝓢(E, F)) : tsupport (fderivCLM 𝕜 E F f) ⊆ tsupport f :=\n  tsupport_fderiv_subset ℝ\n\nopen LineDeriv\n\ntheorem tsupport_lineDerivOp_subset (m : E) (f : 𝓢(E, F)) :\n    tsupport (∂_{m} f : 𝓢(E, F)) ⊆ tsupport f :=\n  tsupport_fderiv_apply_subset ℝ m\n\nTarget:\ntheorem tsupport_iteratedLineDerivOp_subset {n : ℕ} (m : Fin n → E) (f : 𝓢(E, F)) :\n    tsupport (∂^{m} f : 𝓢(E, F)) ⊆ tsupport f :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"1c2bcbc35545db0f2f6b4e75cfecda1cfd94a3928875ff4e6ccbfd85b55e07d8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Distribution","family_id":"tsupport_iteratedlinederivop_subset","file_id":"mathlib/Mathlib/Analysis/Distribution/SchwartzSpace/Deriv.lean","sample_id":"591110f7b37cc2daae15e5485ae2d271b10317dae65d4419218d3c8977316858"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0cde5a74deeec0335214ee6e0227e2c1e15f0001cffe80b1335cabf0968874c8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a97e119ca71797897224fbe3e5e9ebae5dd528e846b50145f35d1b114bd3f83c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"80af364b54e4f9d5bc6e01d40bc9d559f947616a68e18a5600b526c2e866a061","source_sha256":"345dde00cb75d3f9bc3f5909b7969856d423edbbc6d0a61a7a141d42b1e88b8d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : (p.map C).LiesOver p := ⟨IsMaximal.eq_of_le inferInstance IsPrime.ne_top' le_comap_map⟩\n  simp [height_eq_height_add_of_liesOver_of_hasGoingDown p]","hard_negative":true,"metrics":{"chosen_tokens":28,"rejected_tokens":3,"token_jaccard":0.04,"token_length_ratio":0.107143},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"523ee8122292988d06872dc4be90f95604e24a4d23992a2c82fc82806d487187","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.FieldDivision\npublic import Mathlib.RingTheory.KrullDimension.PID\npublic import Mathlib.RingTheory.LocalRing.ResidueField.Fiber\npublic import Mathlib.RingTheory.Ideal.KrullsHeightTheorem\npublic import Mathlib.RingTheory.KrullDimension.NonZeroDivisors\n\nNamespace:\nPolynomial\n\nLocal context:\n/-\nCopyright (c) 2025 Jingting Wang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jingting Wang, Sihan Su, Yi Song, Christian Merten\n-/\n/-!\n# Krull dimension of polynomial ring\n\nThis file proves properties of the Krull dimension of the polynomial ring over a commutative ring\n\n## Main results\n\n* `Polynomial.ringKrullDim_le`: the Krull dimension of the polynomial ring over a commutative ring\n  `R` is less than `2 * (ringKrullDim R) + 1`.\n\nFor noetherian rings:\n* `Polynomial.ringKrullDim_of_isNoetherianRing`: the Krull dimension of `R[X]` is `dim R + 1`.\n* `MvPolynomial.ringKrullDim_of_isNoetherianRing`: the Krull dimension of `R[X₁, ..., Xₙ]` is\n  `dim R + n`.\n-/\n\npublic section\n\ntheorem Polynomial.ringKrullDim_le {R : Type*} [CommRing R] :\n    ringKrullDim (Polynomial R) ≤ 2 * (ringKrullDim R) + 1 := by\n  rw [ringKrullDim, ringKrullDim]\n  apply Order.krullDim_le_of_krullDim_preimage_le' (PrimeSpectrum.comap C) ?_ (fun p ↦ ?_)\n  · exact fun {a b} h ↦ Ideal.comap_mono h\n  · rw [show C = (algebraMap R (Polynomial R)) from rfl, Order.krullDim_eq_of_orderIso\n      (PrimeSpectrum.preimageOrderIsoFiber R (Polynomial R) p), ← ringKrullDim,\n      ← ringKrullDim_eq_of_ringEquiv (polyEquivTensor R (p.asIdeal.ResidueField)).toRingEquiv,\n      ← Ring.krullDimLE_iff]\n    infer_instance\n\nvariable {R : Type*} [CommRing R] [IsNoetherianRing R]\n\nnamespace Polynomial\n\nopen Ideal IsLocalization\n\n/--\nLet `p` be a maximal ideal of `A`. If `P` is a maximal ideal of `A[X]` lying above `p`,\nthen `ht(P) = ht(p) + 1`.\nSee `Polynomial.height_eq_height_add_one` for the more general version that does not assume `p` is\nmaximal.\n-/\nprivate lemma height_eq_height_add_one_of_isMaximal (p : Ideal R) [p.IsMaximal] (P : Ideal R[X])\n    [P.IsMaximal] [P.LiesOver p] : P.height = p.height + 1 := by\n  let _ : Field (R ⧸ p) := Quotient.field p\n  suffices h : (P.map (Ideal.Quotient.mk (Ideal.map (algebraMap R R[X]) p))).height = 1 by\n    rw [height_eq_height_add_of_liesOver_of_hasGoingDown p, h]\n  let e : (R[X] ⧸ (p.map (algebraMap R R[X]))) ≃+* (R ⧸ p)[X] :=\n    (polynomialQuotientEquivQuotientPolynomial p).symm\n  let P' : Ideal (R ⧸ p)[X] := Ideal.map e <| P.map (Ideal.Quotient.mk <| p.map (algebraMap R R[X]))\n  have : (P.map (Ideal.Quotient.mk <| p.map (algebraMap R R[X]))).IsMaximal := by\n    refine .map_of_surjective_of_ker_le Quotient.mk_surjective ?_\n    rw [mk_ker, LiesOver.over (P := P) (p := p)]\n    exact map_comap_le\n  have : P'.IsMaximal := map_isMaximal_of_equiv e\n  have : P'.height = 1 := IsPrincipalIdealRing.height_eq_one_of_isMaximal P' polynomial_not_isField\n  rwa [← e.height_map <| P.map (Ideal.Quotient.mk <| p.map (algebraMap R R[X]))]\n\n/-- Let `p` be a maximal ideal of `R`. Then the height of `p[X]` equals the height of `p`. -/\n\nTarget:\nlemma height_map_C (p : Ideal R) [p.IsMaximal] : (p.map C).height = p.height :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_80af364b54e4","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c1b4bda3c312a33e4bb71a0bb74c8b18b61cae9034c6143889889e43465aa846","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/KrullDimension","family_id":"height_map_c","file_id":"mathlib/Mathlib/RingTheory/KrullDimension/Polynomial.lean","sample_id":"80af364b54e4f9d5bc6e01d40bc9d559f947616a68e18a5600b526c2e866a061"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b9ea88f535ce93e2572a2904d87bf732a9a01a29fb1bf1dec81d4ec155726243","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6e268d2dae8bb1ebce0d32eba74344e7b2814801568b87e7044f89199cff8536","source_sha256":"91e010f7f912f8a75b6beeb48fe6ecf2da1ebc9b1ad8cd51916c3d730bd71e3d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [← orderOf_pos_iff, ← Nat.card_zpowers, Nat.card_pos_iff, ← SetLike.coe_sort_coe,\n    nonempty_coe_sort, Nat.card_pos_iff, Set.finite_coe_iff, OneMemClass.coe_nonempty, true_and]","hard_negative":false,"metrics":{"chosen_tokens":37,"rejected_tokens":3,"token_jaccard":0.045455,"token_length_ratio":0.081081},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"5258275ba68f60994e8b5c1eed692ec4cb604e5e395032867b6165db5a829107","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.ZMod.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# `ZMod n` and quotient groups / rings\n\nThis file relates `ZMod n` to the quotient group `ℤ / AddSubgroup.zmultiples (n : ℤ)`.\n\n## Main definitions\n\n- `ZMod.quotientZMultiplesNatEquivZMod` and `ZMod.quotientZMultiplesEquivZMod`:\n  `ZMod n` is the group quotient of `ℤ` by `n ℤ := AddSubgroup.zmultiples (n)`,\n  (where `n : ℕ` and `n : ℤ` respectively)\n- `ZMod.lift n f` is the map from `ZMod n` induced by `f : ℤ →+ A` that maps `n` to `0`.\n\n## Tags\n\nzmod, quotient group\n-/\n\n@[expose] public section\n\nassert_not_exists Ideal TwoSidedIdeal\n\nopen QuotientAddGroup Set ZMod\nopen scoped IsMulCommutative\n\nvariable (n : ℕ) {A R : Type*} [AddGroup A] [Ring R]\n\nnamespace Int\n\n/-- `ℤ` modulo multiples of `n : ℕ` is `ZMod n`. -/\ndef quotientZMultiplesNatEquivZMod : ℤ ⧸ AddSubgroup.zmultiples (n : ℤ) ≃+ ZMod n :=\n  (quotientAddEquivOfEq (ZMod.ker_intCastAddHom _)).symm.trans <|\n    quotientKerEquivOfRightInverse (Int.castAddHom (ZMod n)) cast intCast_zmod_cast\n\n/-- `ℤ` modulo multiples of `a : ℤ` is `ZMod a.natAbs`. -/\ndef quotientZMultiplesEquivZMod (a : ℤ) : ℤ ⧸ AddSubgroup.zmultiples a ≃+ ZMod a.natAbs :=\n  (quotientAddEquivOfEq (zmultiples_natAbs a)).symm.trans (quotientZMultiplesNatEquivZMod a.natAbs)\n\n@[simp]\nlemma index_zmultiples (a : ℤ) : (AddSubgroup.zmultiples a).index = a.natAbs := by\n  rw [AddSubgroup.index, Nat.card_congr (quotientZMultiplesEquivZMod a).toEquiv, Nat.card_zmod]\n\nend Int\n\n\nnamespace AddAction\n\nopen AddSubgroup AddMonoidHom AddEquiv Function\n\nvariable {α β : Type*} [AddGroup α] (a : α) [AddAction α β] (b : β)\n\n/-- The quotient `(ℤ ∙ a) ⧸ (stabilizer b)` is cyclic of order `minimalPeriod (a +ᵥ ·) b`. -/\nnoncomputable def zmultiplesQuotientStabilizerEquiv :\n    zmultiples a ⧸ stabilizer (zmultiples a) b ≃+ ZMod (minimalPeriod (a +ᵥ ·) b) :=\n  (ofBijective\n          (map _ (stabilizer (zmultiples a) b) (zmultiplesHom (zmultiples a) ⟨a, mem_zmultiples a⟩)\n            (by\n              rw [zmultiples_le, mem_comap, mem_stabilizer_iff, zmultiplesHom_apply, natCast_zsmul]\n              simp_rw [← vadd_iterate]\n              exact isPeriodicPt_minimalPeriod (a +ᵥ ·) b))\n          ⟨by\n            rw [← ker_eq_bot_iff, eq_bot_iff]\n            refine fun q => induction_on q fun n hn => ?_\n            rw [mem_bot, eq_zero_iff, Int.mem_zmultiples_iff, ←\n              zsmul_vadd_eq_iff_minimalPeriod_dvd]\n            exact (eq_zero_iff _).mp hn, fun q =>\n            induction_on q fun ⟨_, n, rfl⟩ => ⟨n, rfl⟩⟩).symm.trans\n    (Int.quotientZMultiplesNatEquivZMod (minimalPeriod (a +ᵥ ·) b))\n\ntheorem zmultiplesQuotientStabilizerEquiv_symm_apply (n : ZMod (minimalPeriod (a +ᵥ ·) b)) :\n    (zmultiplesQuotientStabilizerEquiv a b).symm n =\n      (cast n : ℤ) • (⟨a, mem_zmultiples a⟩ : zmultiples a) :=\n  rfl\n\nend AddAction\n\nnamespace MulAction\n\nopen AddAction Subgroup AddSubgroup Function\n\nvariable {α β : Type*} [Group α] (a : α) [MulAction α β] (b : β)\n\n/-- The quotient `(a ^ ℤ) ⧸ (stabilizer b)` is cyclic of order `minimalPeriod ((•) a) b`. -/\nnoncomputable def zpowersQuotientStabilizerEquiv :\n    zpowers a ⧸ stabilizer (zpowers a) b ≃* Multiplicative (ZMod (minimalPeriod (a • ·) b)) :=\n  letI f := zmultiplesQuotientStabilizerEquiv (Additive.ofMul a) b\n  AddEquiv.toMultiplicative f\n\ntheorem zpowersQuotientStabilizerEquiv_symm_apply (n : ZMod (minimalPeriod (a • ·) b)) :\n    (zpowersQuotientStabilizerEquiv a b).symm n = (⟨a, mem_zpowers a⟩ : zpowers a) ^ (cast n : ℤ) :=\n  rfl\n\n/-- The orbit `(a ^ ℤ) • b` is a cycle of order `minimalPeriod ((•) a) b`. -/\nnoncomputable def orbitZPowersEquiv : orbit (zpowers a) b ≃ ZMod (minimalPeriod (a • ·) b) :=\n  (orbitEquivQuotientStabilizer _ b).trans (zpowersQuotientStabilizerEquiv a b).toEquiv\n\n/-- The orbit `(ℤ • a) +ᵥ b` is a cycle of order `minimalPeriod (a +ᵥ ·) b`. -/\nnoncomputable def _root_.AddAction.orbitZMultiplesEquiv {α β : Type*} [AddGroup α] (a : α)\n    [AddAction α β] (b : β) :\n    AddAction.orbit (zmultiples a) b ≃ ZMod (minimalPeriod (a +ᵥ ·) b) :=\n  (AddAction.orbitEquivQuotientStabilizer (zmultiples a) b).trans\n    (zmultiplesQuotientStabilizerEquiv a b).toEquiv\n\nattribute [to_additive existing] orbitZPowersEquiv\n\n@[to_additive]\ntheorem orbitZPowersEquiv_symm_apply (k : ZMod (minimalPeriod (a • ·) b)) :\n    (orbitZPowersEquiv a b).symm k =\n      (⟨a, mem_zpowers a⟩ : zpowers a) ^ (cast k : ℤ) • ⟨b, mem_orbit_self b⟩ :=\n  rfl\n\ntheorem orbitZPowersEquiv_symm_apply' (k : ℤ) :\n    (orbitZPowersEquiv a b).symm k =\n      (⟨a, mem_zpowers a⟩ : zpowers a) ^ k • ⟨b, mem_orbit_self b⟩ := by\n  rw [orbitZPowersEquiv_symm_apply, ZMod.coe_intCast]\n  exact Subtype.ext (zpow_smul_mod_minimalPeriod _ _ k)\n\ntheorem _root_.AddAction.orbitZMultiplesEquiv_symm_apply' {α β : Type*} [AddGroup α] (a : α)\n    [AddAction α β] (b : β) (k : ℤ) :\n    (AddAction.orbitZMultiplesEquiv a b).symm k =\n      k • (⟨a, mem_zmultiples a⟩ : zmultiples a) +ᵥ ⟨b, AddAction.mem_orbit_self b⟩ := by\n  rw [AddAction.orbitZMultiplesEquiv_symm_apply, ZMod.coe_intCast]\n  -- Making `a` explicit turns this from ~190000 heartbeats to ~700.\n  exact Subtype.ext (zsmul_vadd_mod_minimalPeriod a _ k)\n\nattribute [to_additive existing]\n  orbitZPowersEquiv_symm_apply'\n\n@[to_additive]\ntheorem minimalPeriod_eq_card [Fintype (orbit (zpowers a) b)] :\n    minimalPeriod (a • ·) b = Fintype.card (orbit (zpowers a) b) := by\n  rw [← Fintype.ofEquiv_card (orbitZPowersEquiv a b), ZMod.card]\n\n@[to_additive]\ninstance minimalPeriod_pos [Finite <| orbit (zpowers a) b] :\n    NeZero <| minimalPeriod (a • ·) b :=\n  ⟨by\n    cases nonempty_fintype (orbit (zpowers a) b)\n    haveI : Nonempty (orbit (zpowers a) b) := (nonempty_orbit b).to_subtype\n    rw [minimalPeriod_eq_card]\n    exact Fintype.card_ne_zero⟩\n\nend MulAction\n\nsection Group\n\nopen Subgroup\n\nvariable {α : Type*} [Group α] (a : α)\n\n/-- See also `Fintype.card_zpowers`. -/\n@[to_additive (attr := simp) /-- See also `Fintype.card_zmultiples`. -/]\ntheorem Nat.card_zpowers : Nat.card (zpowers a) = orderOf a := by\n  have := Nat.card_congr (MulAction.orbitZPowersEquiv a (1 : α))\n  rwa [Nat.card_zmod, orbit_subgroup_one_eq_self] at this\n\nvariable {a}\n\n@[to_additive (attr := simp)]\n\nTarget:\nlemma finite_zpowers : (zpowers a : Set α).Finite ↔ IsOfFinOrder a :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/ZMod","family_id":"finite_zpowers","file_id":"mathlib/Mathlib/Data/ZMod/QuotientGroup.lean","sample_id":"6e268d2dae8bb1ebce0d32eba74344e7b2814801568b87e7044f89199cff8536"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d0709903e2751d33af1193290b412e76b1fe0c171376fb60a72f4abe9461ccc4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0a2ef5fc379692cad4911bcf497a95e52e7ee8d27df049e9f4ac60929757366d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ea366b04c54fa2d802b07a4ed0e9ad19edd8f2e0e916dddae3f37ee74b249b0d","source_sha256":"b1dde0e82be30e8a9ca87b23c46d04afc92be61818622ba6d4603daa4d4a9f70","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [sub_eq_add_neg, ofCauchy_add, ofCauchy_neg]\n  rfl","hard_negative":true,"metrics":{"chosen_tokens":10,"rejected_tokens":3,"token_jaccard":0.090909,"token_length_ratio":0.3},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"52834c2e64957ca128d71c166d2b4387966bc39f47d04a22fa5642552572c755","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.CauSeq.Completion\npublic import Mathlib.Algebra.Order.Ring.Rat\npublic import Mathlib.Data.Rat.Cast.Defs\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Floris van Doorn\n-/\n/-!\n# Real numbers from Cauchy sequences\n\nThis file defines `ℝ` as the type of equivalence classes of Cauchy sequences of rational numbers.\nThis choice is motivated by how easy it is to prove that `ℝ` is a commutative ring, by simply\nlifting everything to `ℚ`.\n\nThe facts that the real numbers are an Archimedean floor ring,\nand a conditionally complete linear order,\nhave been deferred to the file `Mathlib/Data/Real/Archimedean.lean`,\nin order to keep the imports here simple.\n\nThe fact that the real numbers are a (trivial) \\*-ring has similarly been deferred to\n`Mathlib/Data/Real/Star.lean`.\n-/\n\n@[expose] public section\n\n\nassert_not_exists Finset Module Submonoid FloorRing\n\n/-- The type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational\nnumbers. -/\n@[wikidata Q12916, wikidata Q2584477]\nstructure Real where ofCauchy ::\n  /-- The underlying Cauchy completion -/\n  cauchy : CauSeq.Completion.Cauchy (abs : ℚ → ℚ)\n\n@[inherit_doc]\nnotation \"ℝ\" => Real\n\nnamespace CauSeq.Completion\n\n-- this can't go in `Data.Real.CauSeqCompletion` as the structure on `ℚ` isn't available\n@[simp]\ntheorem ofRat_rat {abv : ℚ → ℚ} [IsAbsoluteValue abv] (q : ℚ) :\n    ofRat (q : ℚ) = (q : Cauchy abv) :=\n  rfl\n\nend CauSeq.Completion\n\nnamespace Real\n\nopen CauSeq CauSeq.Completion\n\nvariable {x : ℝ}\n\ntheorem ext_cauchy_iff : ∀ {x y : Real}, x = y ↔ x.cauchy = y.cauchy\n  | ⟨a⟩, ⟨b⟩ => by rw [ofCauchy.injEq]\n\ntheorem ext_cauchy {x y : Real} : x.cauchy = y.cauchy → x = y :=\n  ext_cauchy_iff.2\n\n/-- The real numbers are isomorphic to the quotient of Cauchy sequences on the rationals. -/\ndef equivCauchy : ℝ ≃ CauSeq.Completion.Cauchy (abs : ℚ → ℚ) :=\n  ⟨Real.cauchy, Real.ofCauchy, fun ⟨_⟩ => rfl, fun _ => rfl⟩\n\nset_option backward.privateInPublic true in\n-- irreducible doesn't work for instances: https://github.com/leanprover-community/lean/issues/511\nprivate irreducible_def zero : ℝ :=\n  ⟨0⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def one : ℝ :=\n  ⟨1⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def add : ℝ → ℝ → ℝ\n  | ⟨a⟩, ⟨b⟩ => ⟨a + b⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def neg : ℝ → ℝ\n  | ⟨a⟩ => ⟨-a⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def mul : ℝ → ℝ → ℝ\n  | ⟨a⟩, ⟨b⟩ => ⟨a * b⟩\n\nset_option backward.privateInPublic true in\nprivate noncomputable irreducible_def inv' : ℝ → ℝ\n  | ⟨a⟩ => ⟨a⁻¹⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Zero ℝ :=\n  ⟨zero⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : One ℝ :=\n  ⟨one⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Add ℝ :=\n  ⟨add⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Neg ℝ :=\n  ⟨neg⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Mul ℝ :=\n  ⟨mul⟩\n\ninstance : Sub ℝ :=\n  ⟨fun a b => a + -b⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nnoncomputable instance : Inv ℝ :=\n  ⟨inv'⟩\n\ntheorem ofCauchy_zero : (⟨0⟩ : ℝ) = 0 :=\n  zero_def.symm\n\ntheorem ofCauchy_one : (⟨1⟩ : ℝ) = 1 :=\n  one_def.symm\n\ntheorem ofCauchy_add (a b) : (⟨a + b⟩ : ℝ) = ⟨a⟩ + ⟨b⟩ :=\n  (add_def _ _).symm\n\ntheorem ofCauchy_neg (a) : (⟨-a⟩ : ℝ) = -⟨a⟩ :=\n  (neg_def _).symm\n\nTarget:\ntheorem ofCauchy_sub (a b) : (⟨a - b⟩ : ℝ) = ⟨a⟩ - ⟨b⟩ :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_ea366b04c54f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"b63fe3f7ff6028433c2f93702001151932c6a3fb64fa4b184c3875f9a106972b","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Real","family_id":"ofcauchy_sub","file_id":"mathlib/Mathlib/Data/Real/Basic.lean","sample_id":"ea366b04c54fa2d802b07a4ed0e9ad19edd8f2e0e916dddae3f37ee74b249b0d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"228e4c456772f8dbd4f2347154e877aedbab995a4c162aadcd1eff84790d42d5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d61440398863b603b3185bd9be1a7f02c573d18ffa317ffa26f3c3cf5dacdb05","source_sha256":"9404d94c318d3af35f74e686f2ec04bdb0f4088fe90fc5c0cd1b8f6b94df8f36","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [mul_app']\n  exact x.naturality (y.app X)","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.133333,"token_length_ratio":0.2},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"52a9226718fc7c6774c376dd924f81969629626d589c802fe6b6bff2cda43cc3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Endomorphism\n\nNamespace:\nCategoryTheory.CatCenter\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The center of a category\n\nGiven a category `C`, we introduce an abbreviation `CatCenter C` for\nthe center of the category `C`, which is `End (𝟭 C)`, the\ntype of endomorphisms of the identity functor of `C`.\n\n## References\n* https://ncatlab.org/nlab/show/center+of+a+category\n\n-/\n\npublic section\nuniverse v u\n\nnamespace CategoryTheory\n\nopen Category\n\nvariable (C : Type u) [Category.{v} C]\n\n/-- The center of a category `C` is the type `End (𝟭 C)` of the endomorphisms\nof the identify functor of `C`. -/\nabbrev CatCenter := End (𝟭 C)\n\nnamespace CatCenter\n\nvariable {C}\n\n/-- The action of the center of a category on an object. (This is necessary as\n`NatTrans.app x X` is syntactically an endomorphism of `(𝟭 C).obj X`\nrather than of `X`.) -/\nabbrev app (x : CatCenter C) (X : C) : X ⟶ X := NatTrans.app x X\n\n@[ext]\nlemma ext (x y : CatCenter C) (h : ∀ (X : C), x.app X = y.app X) : x = y :=\n  NatTrans.ext (funext h)\n\n@[reassoc]\nlemma naturality (z : CatCenter C) {X Y : C} (f : X ⟶ Y) :\n    f ≫ z.app Y = z.app X ≫ f := NatTrans.naturality z f\n\n@[reassoc]\nlemma mul_app' (x y : CatCenter C) (X : C) : (x * y).app X = y.app X ≫ x.app X := rfl\n\n@[reassoc]\n\nTarget:\nlemma mul_app (x y : CatCenter C) (X : C) : (x * y).app X = x.app X ≫ y.app X :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Center","family_id":"mul_app","file_id":"mathlib/Mathlib/CategoryTheory/Center/Basic.lean","sample_id":"d61440398863b603b3185bd9be1a7f02c573d18ffa317ffa26f3c3cf5dacdb05"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f8d69be5788c6d89c35f089a74bce8cf26bd128b40b0aa5d2082d7e6f83ba04a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ae2b1c6974f9fb9803b46bf7d64af5cfa3d2f828b04d3c9226a22b4f5f52c859","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"45175a44a36d53ed6e01fe3580b4a35a13cfe7dd1d0d93468e34de78ad24b899","source_sha256":"c35204fe93e22d960b9e0f1d04e2f2f5f2bf73b540a7e7fec0bfcf5bf1349499","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Int.natCast_dvd]\n  exact Int.Prime.dvd_pow hp h","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":2,"token_jaccard":0.076923,"token_length_ratio":0.133333},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"52bc4acd2e279911accb7a19f4ef162c012319ac160a6bc5dc1e15fb958e4400","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.EuclideanDomain.Basic\npublic import Mathlib.Algebra.EuclideanDomain.Int\npublic import Mathlib.Algebra.GCDMonoid.Nat\npublic import Mathlib.Data.Nat.Prime.Int\npublic import Mathlib.RingTheory.PrincipalIdealDomain\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson\n-/\n/-!\n# Divisibility over ℤ\n\nThis file collects results for the integers that use ring theory in their proofs or\ncases of ℤ being examples of structures in ring theory.\n\n## Main statements\n\n* `Int.Prime.dvd_mul'`: A prime number dividing a product in ℤ divides at least one factor.\n* `Int.exists_prime_and_dvd`: Every non-unit integer has a prime divisor.\n* `Int.prime_iff_natAbs_prime`: Primality in ℤ corresponds to primality of its absolute value in ℕ.\n* `Int.span_natAbs`: The principal ideal generated by `a.natAbs` is equal to that of `a`.\n\n## Tags\n\nprime, irreducible, integers, normalization monoid, gcd monoid, greatest common divisor\n-/\n\npublic section\n\nnamespace Int\n\n\ntheorem isCoprime_iff_nat_coprime {a b : ℤ} : IsCoprime a b ↔ Nat.Coprime a.natAbs b.natAbs := by\n  rw [isCoprime_iff_gcd_eq_one, Nat.coprime_iff_gcd_eq_one, gcd_eq_natAbs]\n\n/-- If `gcd a (m * n) ≠ 1`, then `gcd a m ≠ 1` or `gcd a n ≠ 1`. -/\ntheorem gcd_ne_one_iff_gcd_mul_right_ne_one {a : ℤ} {m n : ℕ} :\n    a.gcd (m * n) ≠ 1 ↔ a.gcd m ≠ 1 ∨ a.gcd n ≠ 1 := by\n  simp only [← isCoprime_iff_gcd_eq_one, ← not_and_or, not_iff_not, IsCoprime.mul_right_iff]\n\ntheorem sq_of_gcd_eq_one {a b c : ℤ} (h : Int.gcd a b = 1) (heq : a * b = c ^ 2) :\n    ∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 := by\n  have h' : IsUnit (GCDMonoid.gcd a b) := by\n    rw [← coe_gcd, h, Int.ofNat_one]\n    exact isUnit_one\n  obtain ⟨d, ⟨u, hu⟩⟩ := exists_associated_pow_of_mul_eq_pow h' heq\n  use d\n  rw [← hu]\n  rcases Int.units_eq_one_or u with hu' | hu' <;>\n    · rw [hu']\n      simp\n\ntheorem sq_of_isCoprime {a b c : ℤ} (h : IsCoprime a b) (heq : a * b = c ^ 2) :\n    ∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 :=\n  sq_of_gcd_eq_one (isCoprime_iff_gcd_eq_one.mp h) heq\n\ntheorem natAbs_euclideanDomain_gcd (a b : ℤ) :\n    Int.natAbs (EuclideanDomain.gcd a b) = Int.gcd a b := by\n  apply Nat.dvd_antisymm <;> rw [← Int.natCast_dvd_natCast]\n  · rw [Int.natAbs_dvd]\n    exact Int.dvd_coe_gcd (EuclideanDomain.gcd_dvd_left _ _) (EuclideanDomain.gcd_dvd_right _ _)\n  · rw [Int.dvd_natAbs]\n    exact EuclideanDomain.dvd_gcd (Int.gcd_dvd_left ..) (Int.gcd_dvd_right ..)\n\nend Int\n\ntheorem Int.Prime.dvd_mul {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ m * n) :\n    p ∣ m.natAbs ∨ p ∣ n.natAbs := by\n  rwa [← hp.dvd_mul, ← Int.natAbs_mul, ← Int.natCast_dvd]\n\ntheorem Int.Prime.dvd_mul' {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ m * n) :\n    (p : ℤ) ∣ m ∨ (p : ℤ) ∣ n := by\n  rw [Int.natCast_dvd, Int.natCast_dvd]\n  exact Int.Prime.dvd_mul hp h\n\ntheorem Int.Prime.dvd_pow {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ n ^ k) :\n    p ∣ n.natAbs := by\n  rw [Int.natCast_dvd, Int.natAbs_pow] at h\n  exact hp.dvd_of_dvd_pow h\n\nTarget:\ntheorem Int.Prime.dvd_pow' {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ n ^ k) :\n    (p : ℤ) ∣ n :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_45175a44a36d","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"2bb370ac6231fd8fe46ed4dc588244fd044b09ec2d7224d9f3475a579711a364","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Int","family_id":"int","file_id":"mathlib/Mathlib/RingTheory/Int/Basic.lean","sample_id":"45175a44a36d53ed6e01fe3580b4a35a13cfe7dd1d0d93468e34de78ad24b899"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d23fb76aeeed9353afe60037f9d9b0091fa23e08b64b45bbb0d2d34201990525","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1311b25412146444bec8792249c758390d61d9ee68ab52ec762c017c5fec5d5b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0e496e19d8169f840fb42721b4ba0ad691cd3e68e48df661f9f6fc16f635b298","source_sha256":"74b0e866ba6855a98543cc59c9d908f25a74c6e6456b4e8b44974d8edb78dfd2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [loopOfHom, ← Category.assoc, IsIso.comp_inv_eq, Category.id_comp]\n  rcases H with H | H\n  · rw [treeHom_eq T (Path.cons default ⟨Sum.inl e, H⟩), homOfPath]\n    rfl\n  · rw [treeHom_eq T (Path.cons default ⟨Sum.inr e, H⟩), homOfPath]\n    simp only [IsIso.inv_hom_id, Category.comp_id, Category.assoc, treeHom]","hard_negative":true,"metrics":{"chosen_tokens":86,"rejected_tokens":2,"token_jaccard":0.025641,"token_length_ratio":0.023256},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"52cb3a03eef21b5c80156e080db996a9619f1f0326fcb3a580794eac2b9565b9","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Action\npublic import Mathlib.Combinatorics.Quiver.Arborescence\npublic import Mathlib.Combinatorics.Quiver.ConnectedComponent\npublic import Mathlib.GroupTheory.FreeGroup.IsFreeGroup\n\nNamespace:\nIsFreeGroupoid.SpanningTree\n\nLocal context:\n/-\nCopyright (c) 2021 David Wärn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David Wärn\n-/\n/-!\n# The Nielsen-Schreier theorem\n\nThis file proves that a subgroup of a free group is itself free.\n\n## Main result\n\n- `subgroupIsFreeOfIsFree H`: an instance saying that a subgroup of a free group is free.\n\n## Proof overview\n\nThe proof is analogous to the proof using covering spaces and fundamental groups of graphs,\nbut we work directly with groupoids instead of topological spaces. Under this analogy,\n\n- `IsFreeGroupoid G` corresponds to saying that a space is a graph.\n- `endMulEquivSubgroup H` plays the role of replacing 'subgroup of fundamental group' with\n  'fundamental group of covering space'.\n- `actionGroupoidIsFree G A` corresponds to the fact that a covering of a (single-vertex)\n  graph is a graph.\n- `endIsFree T` corresponds to the fact that, given a spanning tree `T` of a\n  graph, its fundamental group is free (generated by loops from the complement of the tree).\n\n## Implementation notes\n\nOur definition of `IsFreeGroupoid` is nonstandard. Normally one would require that functors\n`G ⥤ X` to any _groupoid_ `X` are given by graph homomorphisms from the generators, but we only\nconsider _groups_ `X`. This simplifies the argument since functor equality is complicated in\ngeneral, but simple for functors to single object categories.\n\n## References\n\nhttps://ncatlab.org/nlab/show/Nielsen-Schreier+theorem\n\n## Tags\n\nfree group, free groupoid, Nielsen-Schreier\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nuniverse v u\n\nopen CategoryTheory CategoryTheory.ActionCategory CategoryTheory.SingleObj Quiver FreeGroup\n\n/-- `IsFreeGroupoid.Generators G` is a type synonym for `G`. We think of this as\nthe vertices of the generating quiver of `G` when `G` is free. We can't use `G` directly,\nsince `G` already has a quiver instance from being a groupoid. -/\n@[nolint unusedArguments]\ndef IsFreeGroupoid.Generators (G) [Groupoid G] :=\n  G\n\n/--\nA groupoid `G` is free when we have the following data:\n- a quiver on `IsFreeGroupoid.Generators G` (a type synonym for `G`)\n- a function `of` taking a generating arrow to a morphism in `G`\n- such that a functor from `G` to any group `X` is uniquely determined\n  by assigning labels in `X` to the generating arrows.\n\nThis definition is nonstandard. Normally one would require that functors `G ⥤ X`\nto any _groupoid_ `X` are given by graph homomorphisms from `generators`. -/\nclass IsFreeGroupoid (G) [Groupoid.{v} G] where\n  quiverGenerators : Quiver.{v} (IsFreeGroupoid.Generators G)\n  of : ∀ {a b : IsFreeGroupoid.Generators G}, (a ⟶ b) → ((show G from a) ⟶ b)\n  unique_lift :\n    ∀ {X : Type v} [Group X] (f : Labelling (IsFreeGroupoid.Generators G) X),\n      ∃! F : G ⥤ CategoryTheory.SingleObj X, ∀ (a b) (g : a ⟶ b), F.map (of g) = f g\n\nattribute [nolint docBlame] IsFreeGroupoid.of IsFreeGroupoid.unique_lift\n\nnamespace IsFreeGroupoid\n\nattribute [instance_reducible, instance] quiverGenerators\n\n/-- Two functors from a free groupoid to a group are equal when they agree on the generating\nquiver. -/\n@[ext]\ntheorem ext_functor {G} [Groupoid.{v} G] [IsFreeGroupoid G] {X : Type v} [Group X]\n    (f g : G ⥤ CategoryTheory.SingleObj X) (h : ∀ (a b) (e : a ⟶ b), f.map (of e) = g.map (of e)) :\n    f = g :=\n  let ⟨_, _, u⟩ := @unique_lift G _ _ X _ fun (a b : Generators G) (e : a ⟶ b) => g.map (of e)\n  _root_.trans (u _ h) (u _ fun _ _ _ => rfl).symm\n\n/-- An action groupoid over a free group is free. More generally, one could show that the groupoid\nof elements over a free groupoid is free, but this version is easier to prove and suffices for our\npurposes.\n\nAnalogous to the fact that a covering space of a graph is a graph. (A free groupoid is like a graph,\nand a groupoid of elements is like a covering space.) -/\ninstance actionGroupoidIsFree {G A : Type u} [Group G] [IsFreeGroup G] [MulAction G A] :\n    IsFreeGroupoid (ActionCategory G A) where\n  quiverGenerators :=\n    ⟨fun a b => { e : IsFreeGroup.Generators G // IsFreeGroup.of e • a.back = b.back }⟩\n  of := fun (e : Subtype _) => ⟨IsFreeGroup.of e, e.property⟩\n  unique_lift := by\n    intro X _ f\n    let f' : IsFreeGroup.Generators G → (A → X) ⋊[mulAutArrow] G := fun e =>\n      ⟨fun b => @f ⟨(), _⟩ ⟨(), b⟩ ⟨e, smul_inv_smul _ b⟩, IsFreeGroup.of e⟩\n    rcases IsFreeGroup.unique_lift f' with ⟨F', hF', uF'⟩\n    refine ⟨uncurry F' ?_, ?_, ?_⟩\n    · suffices SemidirectProduct.rightHom.comp F' = MonoidHom.id _ by\n        exact DFunLike.ext_iff.mp this\n      apply IsFreeGroup.ext_hom (fun x ↦ ?_)\n      rw [MonoidHom.comp_apply, hF']\n      rfl\n    · rintro ⟨⟨⟩, a : A⟩ ⟨⟨⟩, b⟩ ⟨e, h : IsFreeGroup.of e • a = b⟩\n      change (F' (IsFreeGroup.of _)).left _ = _\n      rw [hF']\n      cases inv_smul_eq_iff.mpr h.symm\n      rfl\n    · intro E hE\n      have : curry E = F' := by\n        apply uF'\n        intro e\n        ext\n        · convert! hE _ _ _\n          rfl\n        · rfl\n      apply Functor.hext\n      · intro\n        apply Unit.ext\n      · refine ActionCategory.cases ?_\n        intros\n        simp only [← this, uncurry_map, curry_apply_left, coe_back, homOfPair.val]\n        rfl\n\nnamespace SpanningTree\n\n/- In this section, we suppose we have a free groupoid with a spanning tree for its generating\nquiver. The goal is to prove that the vertex group at the root is free. A picture to have in mind\nis that we are 'pulling' the endpoints of all the edges of the quiver along the spanning tree to\nthe root. -/\nvariable {G : Type u} [Groupoid.{u} G] [IsFreeGroupoid G]\n  (T : WideSubquiver (Symmetrify <| Generators G)) [Arborescence T]\n\nset_option backward.privateInPublic true in\n/-- The root of `T`, except its type is `G` instead of the type synonym `T`. -/\nprivate def root' : G :=\n  show T from root T\n\n-- this has to be marked noncomputable, see issue https://github.com/leanprover-community/mathlib4/pull/451.\n-- It might be nicer to define this in terms of `composePath`\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- A path in the tree gives a hom, by composition. -/\ndef homOfPath : ∀ {a : G}, Path (root T) a → (root' T ⟶ a)\n  | _, Path.nil => 𝟙 _\n  | _, Path.cons p f => homOfPath p ≫ Sum.recOn f.val (fun e => of e) fun e => inv (of e)\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- For every vertex `a`, there is a canonical hom from the root, given by the path in the tree. -/\ndef treeHom (a : G) : root' T ⟶ a :=\n  homOfPath T default\n\n/-- Any path to `a` gives `treeHom T a`, since paths in the tree are unique. -/\ntheorem treeHom_eq {a : G} (p : Path (root T) a) : treeHom T a = homOfPath T p := by\n  rw [treeHom, Unique.default_eq]\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n@[simp]\ntheorem treeHom_root : treeHom T (root' T) = 𝟙 _ :=\n  -- this should just be `treeHom_eq T Path.nil`, but Lean treats `homOfPath` with suspicion.\n    _root_.trans\n    (treeHom_eq T Path.nil) rfl\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Any hom in `G` can be made into a loop, by conjugating with `treeHom`s. -/\ndef loopOfHom {a b : G} (p : a ⟶ b) : End (root' T) :=\n  treeHom T a ≫ p ≫ inv (treeHom T b)\n\nset_option backward.isDefEq.respectTransparency false in\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Turning an edge in the spanning tree into a loop gives the identity loop. -/\n\nTarget:\ntheorem loopOfHom_eq_id {a b : Generators G} (e) (H : e ∈ wideSubquiverSymmetrify T a b) :\n    loopOfHom T (of e) = 𝟙 (root' T) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_0e496e19d816","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"f68b07bd56d372d9894bc57d82f97ca9532d120d110f8d7c9cc4d1e69fdd6d61","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/FreeGroup","family_id":"loopofhom_eq_id","file_id":"mathlib/Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean","sample_id":"0e496e19d8169f840fb42721b4ba0ad691cd3e68e48df661f9f6fc16f635b298"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cc4265fa4a32e1dda64a7dec7b903dcc9cceebb2f55f6e8ee131b8ddefaf2f73","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"55440f62ede7bbdc77ead5e942a883e7750663cc7506c8d69af849e8b683ebab","source_sha256":"da76ff62b317f067f0d000725a7dfe7652cd6b413da74f1215e83607e9c9c512","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : Ideal.span ({a} : Set α) ≠ ⊤ := by\n    intro H\n    rw [Ideal.span_singleton_eq_top] at H\n    contradiction\n  rcases Ideal.exists_le_maximal _ this with ⟨I, Imax, H⟩\n  use I, Imax\n  apply H\n  apply Ideal.subset_span\n  exact Set.mem_singleton a","hard_negative":true,"metrics":{"chosen_tokens":58,"rejected_tokens":8,"token_jaccard":0.095238,"token_length_ratio":0.137931},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"52eb69386f60e79de25ff7a01786e71f7738c4184e3634ead432a31f5b27a0c3","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Ideal.Maximal\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Chris Hughes, Mario Carneiro\n-/\n/-!\n# The set of non-invertible elements of a monoid\n\n## Main definitions\n\n* `nonunits` is the set of non-invertible elements of a monoid.\n\n## Main results\n\n* `exists_max_ideal_of_mem_nonunits`: every element of `nonunits` is contained in a maximal ideal\n-/\n\n@[expose] public section\n\n\nvariable {F α β : Type*} {a b : α}\n\n/-- The set of non-invertible elements of a monoid. -/\ndef nonunits (α : Type*) [Monoid α] : Set α :=\n  { a | ¬IsUnit a }\n\n@[simp]\ntheorem mem_nonunits_iff [Monoid α] : a ∈ nonunits α ↔ ¬IsUnit a :=\n  Iff.rfl\n\ntheorem mul_mem_nonunits_right [CommMonoid α] : b ∈ nonunits α → a * b ∈ nonunits α :=\n  mt isUnit_of_mul_isUnit_right\n\ntheorem mul_mem_nonunits_left [CommMonoid α] : a ∈ nonunits α → a * b ∈ nonunits α :=\n  mt isUnit_of_mul_isUnit_left\n\ntheorem zero_mem_nonunits [MonoidWithZero α] : 0 ∈ nonunits α ↔ (0 : α) ≠ 1 :=\n  not_congr isUnit_zero_iff\n\n@[simp high] -- High priority shortcut lemma\ntheorem one_notMem_nonunits [Monoid α] : (1 : α) ∉ nonunits α :=\n  not_not_intro isUnit_one\n\n@[simp high] -- High priority shortcut lemma\ntheorem map_mem_nonunits_iff [Monoid α] [Monoid β] [FunLike F α β] [MonoidHomClass F α β] (f : F)\n    [IsLocalHom f] (a) : f a ∈ nonunits β ↔ a ∈ nonunits α :=\n  ⟨fun h ha => h <| ha.map f, fun h ha => h <| ha.of_map⟩\n\ntheorem coe_subset_nonunits [Semiring α] {I : Ideal α} (h : I ≠ ⊤) : (I : Set α) ⊆ nonunits α :=\n  fun _x hx hu => h <| I.eq_top_of_isUnit_mem hx hu\n\nTarget:\ntheorem exists_max_ideal_of_mem_nonunits [CommSemiring α] (h : a ∈ nonunits α) :\n    ∃ I : Ideal α, I.IsMaximal ∧ a ∈ I :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"51ac8d578beefa9bf07d61413ab6a1e0d23403b5550f471af68947946066092c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Ideal","family_id":"exists_max_ideal_of_mem_nonunits","file_id":"mathlib/Mathlib/RingTheory/Ideal/Nonunits.lean","sample_id":"55440f62ede7bbdc77ead5e942a883e7750663cc7506c8d69af849e8b683ebab"},"split":"train"}
{"binding":{"chosen_proof_sha256":"911bfcaad6d4d642cf91de9a824237ea9c93ccf06c46a0091532a586f961081e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"81cc30a11f3d20483daf75e9fda2f6c76ad5ce05ffcfae6c6e3d41ca1b04cd16","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"61d172a6c1c40adedfc16e781f60f95fac4d1f6ec88c11c9a5b57c8a1f496302","source_sha256":"aac9c42310c3c68ba48257d401225da11673aa0693f3a499d296961d7c43ce42","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [SetLike.ext_iff, mem_stabilizer_set]\n  refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_right ha⟩\n  have : 1 * MulOpposite.op a ∈ s := (h 1).2 s.one_mem\n  simpa only [op_smul_eq_mul, SetLike.mem_coe, one_mul] using! this","hard_negative":true,"metrics":{"chosen_tokens":63,"rejected_tokens":5,"token_jaccard":0.044444,"token_length_ratio":0.079365},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"52fe0a921ac477e6baf6a461b12b8f9c6f79ea55ff93ab783b749cdf305cb8fa","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Pointwise.Finset\npublic import Mathlib.GroupTheory.QuotientGroup.Defs\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\n\nNamespace:\nMulAction\n\nLocal context:\n/-\nCopyright (c) 2023 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Stabilizer of a set under a pointwise action\n\nThis file characterises the stabilizer of a set/finset under the pointwise action of a group.\n-/\n\npublic section\n\nopen Function MulOpposite Set\nopen scoped Pointwise\n\nnamespace MulAction\nvariable {G H α : Type*}\n\n/-! ### Stabilizer of a set -/\n\nsection Set\nsection Group\nvariable [Group G] [Group H] [MulAction G α] {a : G} {s t : Set α}\n\n@[to_additive (attr := simp)]\nlemma stabilizer_empty : stabilizer G (∅ : Set α) = ⊤ :=\n  Subgroup.coe_eq_univ.1 <| eq_univ_of_forall fun _a ↦ smul_set_empty\n\n@[to_additive (attr := simp)]\nlemma stabilizer_univ : stabilizer G (Set.univ : Set α) = ⊤ := by\n  ext\n  simp\n\n@[to_additive (attr := simp)]\nlemma stabilizer_singleton (b : α) : stabilizer G ({b} : Set α) = stabilizer G b := by ext; simp\n\n@[to_additive]\nlemma mem_stabilizer_set {s : Set α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s := by\n  refine mem_stabilizer_iff.trans ⟨fun h b ↦ ?_, fun h ↦ ?_⟩\n  · rw [← (smul_mem_smul_set_iff : a • b ∈ _ ↔ _), h]\n  simp_rw [Set.ext_iff, mem_smul_set_iff_inv_smul_mem]\n  exact ((MulAction.toPerm a).forall_congr' <| by simp [Iff.comm]).1 h\n\n@[to_additive]\nlemma map_stabilizer_le (f : G →* H) (s : Set G) :\n    (stabilizer G s).map f ≤ stabilizer H (f '' s) := by\n  rintro a\n  simp only [Subgroup.mem_map, mem_stabilizer_iff, forall_exists_index, and_imp]\n  rintro a ha rfl\n  rw [← image_smul_distrib, ha]\n\n@[to_additive (attr := simp)]\nlemma stabilizer_mul_self (s : Set G) : (stabilizer G s : Set G) * s = s := by\n  ext\n  refine ⟨?_, fun h ↦ ⟨_, (stabilizer G s).one_mem, _, h, one_mul _⟩⟩\n  rintro ⟨a, ha, b, hb, rfl⟩\n  rw [← mem_stabilizer_iff.1 ha]\n  exact smul_mem_smul_set hb\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_apply₂ {f : Set α → Set α → Set α}\n    (hf : ∀ a : G, a • f s t = f (a • s) (a • t)) :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (f s t) := by aesop (add simp [SetLike.le_def])\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_union :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s ∪ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_union\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_inter :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s ∩ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_inter\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_sdiff :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s \\ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_sdiff\n\n@[to_additive]\nlemma stabilizer_union_eq_left (hdisj : Disjoint s t) (hstab : stabilizer G s ≤ stabilizer G t)\n    (hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G t) :\n    stabilizer G (s ∪ t) = stabilizer G s := by\n  refine le_antisymm ?_ ?_\n  · calc\n      stabilizer G (s ∪ t)\n        ≤ stabilizer G (s ∪ t) ⊓ stabilizer G t := by simpa\n      _ ≤ stabilizer G ((s ∪ t) \\ t) := stabilizer_inf_stabilizer_le_stabilizer_sdiff\n      _ = stabilizer G s := by rw [union_sdiff_cancel_right]; simpa [← disjoint_iff_inter_eq_empty]\n  · calc\n      stabilizer G s\n        ≤ stabilizer G s ⊓ stabilizer G t := by simpa\n      _ ≤ stabilizer G (s ∪ t) := stabilizer_inf_stabilizer_le_stabilizer_union\n\n@[to_additive]\nlemma stabilizer_union_eq_right (hdisj : Disjoint s t) (hstab : stabilizer G t ≤ stabilizer G s)\n    (hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G s) :\n    stabilizer G (s ∪ t) = stabilizer G t := by\n  rw [union_comm, stabilizer_union_eq_left hdisj.symm hstab (union_comm .. ▸ hstab_union)]\n\nvariable {s : Set G}\n\nopen scoped RightActions in\n@[to_additive]\nlemma op_smul_set_stabilizer_subset (ha : a ∈ s) : (stabilizer G s : Set G) <• a ⊆ s :=\n  smul_set_subset_iff.2 fun b hb ↦ by rw [← hb]; exact smul_mem_smul_set ha\n\n@[to_additive]\nlemma stabilizer_subset_div_right (ha : a ∈ s) : ↑(stabilizer G s) ⊆ s / {a} := fun b hb ↦\n  ⟨_, by rwa [← smul_eq_mul, mem_stabilizer_set.1 hb], _, mem_singleton _, mul_div_cancel_right _ _⟩\n\n@[to_additive]\nlemma stabilizer_finite (hs₀ : s.Nonempty) (hs : s.Finite) : (stabilizer G s : Set G).Finite := by\n  obtain ⟨a, ha⟩ := hs₀\n  exact (hs.div <| finite_singleton _).subset <| stabilizer_subset_div_right ha\n\nend Group\n\nsection CommGroup\nvariable [CommGroup G] {s t : Set G} {a : G}\n\n@[to_additive]\nlemma smul_set_stabilizer_subset (ha : a ∈ s) : a • (stabilizer G s : Set G) ⊆ s := by\n  simpa using op_smul_set_stabilizer_subset ha\n\nend CommGroup\nend Set\n\nvariable [Group G] [Group H] [MulAction G α] {a : G}\n\n/-! ### Stabilizer of a subgroup -/\n\nsection Subgroup\n\n-- TODO: Is there a lemma that could unify the following three very similar lemmas?\n\n@[to_additive (attr := simp)]\nlemma stabilizer_subgroup (s : Subgroup G) : stabilizer G (s : Set G) = s := by\n  simp_rw [SetLike.ext_iff, mem_stabilizer_set]\n  refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_left ha⟩\n  simpa only [smul_eq_mul, SetLike.mem_coe, mul_one] using (h 1).2 s.one_mem\n\n@[to_additive (attr := simp)]\nlemma stabilizer_op_subgroup (s : Subgroup G) : stabilizer Gᵐᵒᵖ (s : Set G) = s.op := by\n  simp_rw [SetLike.ext_iff, mem_stabilizer_set]\n  simp only [smul_eq_mul_unop, SetLike.mem_coe, Subgroup.mem_op, «forall», unop_op]\n  refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_right ha⟩\n  simpa only [op_smul_eq_mul, SetLike.mem_coe, one_mul] using (h 1).2 s.one_mem\n\n@[to_additive (attr := simp)]\n\nTarget:\nlemma stabilizer_subgroup_op (s : Subgroup Gᵐᵒᵖ) : stabilizer G (s : Set Gᵐᵒᵖ) = s.unop :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_61d172a6c1c4","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"2104f8b3dee36930aa9cd4a99df06874328a55ffe1c940e0b2b2835ece121ddb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Pointwise","family_id":"stabilizer_subgroup_op","file_id":"mathlib/Mathlib/Algebra/Pointwise/Stabilizer.lean","sample_id":"61d172a6c1c40adedfc16e781f60f95fac4d1f6ec88c11c9a5b57c8a1f496302"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a3c59538351f137c4bd7f08ab998d351346f182e72d5bba3bd7e2722c4d90e7a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"10fa2791fbfc644d39bf2d81c08a78f073daf00278ede35b73bb7ef9624d8ac7","source_sha256":"0928fcde12666c1de471a7063a35374f64473f516c8c0caf3bbaebeb9f5bd198","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply injective_of_mono (isoFinYonedaComponents F X).hom\n  simp only [Iso.inv_hom_id_apply]\n  ext x\n  rw [isoFinYonedaComponents_hom_apply]\n  simp only [← Functor.map_comp_apply, ← op_comp, CompHausLike.const_comp,\n    ← isoFinYonedaComponents_hom_apply, Iso.inv_hom_id_apply, Function.comp_apply]","hard_negative":false,"metrics":{"chosen_tokens":49,"rejected_tokens":5,"token_jaccard":0.030303,"token_length_ratio":0.102041},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"530922ee25ebeffbda7f97ba384cd69792eddc059c2e9498949d61ad54d53219","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Condensed.Discrete.LocallyConstant\npublic import Mathlib.Condensed.Equivalence\npublic import Mathlib.Topology.Category.LightProfinite.Extend\n\nNamespace:\nCondensed\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# The condensed set given by left Kan extension from `FintypeCat` to `Profinite`.\n\nThis file provides the necessary API to prove that a condensed set `X` is discrete if and only if\nfor every profinite set `S = limᵢSᵢ`, `X(S) ≅ colimᵢX(Sᵢ)`, and the analogous result for light\ncondensed sets.\n-/\n\n@[expose] public section\n\nuniverse u\n\nnoncomputable section\n\nopen CategoryTheory Functor Limits FintypeCat CompHausLike.LocallyConstant\n\nnamespace Condensed\n\nsection LocallyConstantAsColimit\n\nvariable {I : Type u} [Category.{u} I] [IsCofiltered I] {F : I ⥤ FintypeCat.{u}}\n  (c : Cone <| F ⋙ toProfinite) (X : Type (u + 1))\n\n/-- The presheaf on `Profinite` of locally constant functions to `X`. -/\nabbrev locallyConstantPresheaf : Profinite.{u}ᵒᵖ ⥤ Type (u + 1) :=\n  CompHausLike.LocallyConstant.functorToPresheaves.{u, u + 1}.obj X\n\nset_option backward.defeqAttrib.useBackward true in\n/--\nThe functor `locallyConstantPresheaf` takes cofiltered limits of finite sets with surjective\nprojection maps to colimits.\n-/\nnoncomputable def isColimitLocallyConstantPresheaf (hc : IsLimit c) [∀ i, Epi (c.π.app i)] :\n    IsColimit <| (locallyConstantPresheaf X).mapCocone c.op := by\n  refine Types.FilteredColimit.isColimitOf _ _ ?_ ?_\n  · intro (f : LocallyConstant c.pt X)\n    obtain ⟨j, h⟩ := Profinite.exists_locallyConstant.{_, u} c hc f\n    exact ⟨⟨j⟩, h⟩\n  · intro ⟨i⟩ ⟨j⟩ (fi : LocallyConstant _ _) (fj : LocallyConstant _ _)\n      (h : fi.comap (c.π.app i).hom.hom = fj.comap (c.π.app j).hom.hom)\n    obtain ⟨k, ki, kj, _⟩ := IsCofilteredOrEmpty.cone_objs i j\n    refine ⟨⟨k⟩, ki.op, kj.op, ?_⟩\n    dsimp\n    ext x\n    obtain ⟨x, hx⟩ := ((Profinite.epi_iff_surjective (c.π.app k)).mp inferInstance) x\n    rw [← hx]\n    change fi ((c.π.app k ≫ (F ⋙ toProfinite).map _) x) =\n      fj ((c.π.app k ≫ (F ⋙ toProfinite).map _) x)\n    have h := LocallyConstant.congr_fun h x\n    dsimp\n    rwa [dsimp% c.w, dsimp% c.w]\n\n@[simp]\nlemma isColimitLocallyConstantPresheaf_desc_apply (hc : IsLimit c) [∀ i, Epi (c.π.app i)]\n    (s : Cocone ((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X))\n    (i : I) (f : LocallyConstant (toProfinite.obj (F.obj i)) X) :\n    dsimp% (isColimitLocallyConstantPresheaf c X hc).desc s (f.comap (c.π.app i).hom.hom) =\n      s.ι.app ⟨i⟩ f := by\n  change ((((locallyConstantPresheaf X).mapCocone c.op).ι.app ⟨i⟩) ≫\n    (isColimitLocallyConstantPresheaf c X hc).desc s) _ = _\n  rw [(isColimitLocallyConstantPresheaf c X hc).fac]\n  rfl\n\n/-- `isColimitLocallyConstantPresheaf` in the case of `S.asLimit`. -/\nnoncomputable def isColimitLocallyConstantPresheafDiagram (S : Profinite) :\n    IsColimit <| (locallyConstantPresheaf X).mapCocone S.asLimitCone.op :=\n  isColimitLocallyConstantPresheaf _ _ S.asLimit\n\n@[simp]\nlemma isColimitLocallyConstantPresheafDiagram_desc_apply (S : Profinite)\n    (s : Cocone (S.diagram.op ⋙ locallyConstantPresheaf X))\n    (i : DiscreteQuotient S) (f : LocallyConstant (S.diagram.obj i) X) :\n    dsimp% (isColimitLocallyConstantPresheafDiagram X S).desc s\n      (f.comap (S.asLimitCone.π.app i).hom.hom) = s.ι.app ⟨i⟩ f :=\n  isColimitLocallyConstantPresheaf_desc_apply S.asLimitCone X S.asLimit s i f\n\nend LocallyConstantAsColimit\n\n/--\nGiven a presheaf `F` on `Profinite`, `lanPresheaf F` is the left Kan extension of its\nrestriction to finite sets along the inclusion functor of finite sets into `Profinite`.\n-/\nabbrev lanPresheaf (F : Profinite.{u}ᵒᵖ ⥤ Type (u + 1)) : Profinite.{u}ᵒᵖ ⥤ Type (u + 1) :=\n  pointwiseLeftKanExtension toProfinite.op (toProfinite.op ⋙ F)\n\n/--\nTo presheaves on `Profinite` whose restrictions to finite sets are isomorphic have isomorphic left\nKan extensions.\n-/\ndef lanPresheafExt {F G : Profinite.{u}ᵒᵖ ⥤ Type (u + 1)}\n    (i : toProfinite.op ⋙ F ≅ toProfinite.op ⋙ G) : lanPresheaf F ≅ lanPresheaf G :=\n  leftKanExtensionUniqueOfIso _ (pointwiseLeftKanExtensionUnit _ _) i _\n    (pointwiseLeftKanExtensionUnit _ _)\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma lanPresheafExt_hom {F G : Profinite.{u}ᵒᵖ ⥤ Type (u + 1)} (S : Profinite.{u}ᵒᵖ)\n    (i : toProfinite.op ⋙ F ≅ toProfinite.op ⋙ G) : (lanPresheafExt i).hom.app S =\n      colimMap (whiskerLeft (CostructuredArrow.proj toProfinite.op S) i.hom) := by\n  simp only [lanPresheaf, lanPresheafExt,\n    leftKanExtensionUniqueOfIso_hom, pointwiseLeftKanExtension_desc_app]\n  apply colimit.hom_ext\n  aesop\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma lanPresheafExt_inv {F G : Profinite.{u}ᵒᵖ ⥤ Type (u + 1)} (S : Profinite.{u}ᵒᵖ)\n    (i : toProfinite.op ⋙ F ≅ toProfinite.op ⋙ G) : (lanPresheafExt i).inv.app S =\n      colimMap (whiskerLeft (CostructuredArrow.proj toProfinite.op S) i.inv) := by\n  simp only [lanPresheaf, lanPresheafExt,\n    leftKanExtensionUniqueOfIso_inv, pointwiseLeftKanExtension_desc_app]\n  apply colimit.hom_ext\n  aesop\n\nvariable {S : Profinite.{u}} {F : Profinite.{u}ᵒᵖ ⥤ Type (u + 1)}\n\ninstance : Final <| Profinite.Extend.functorOp S.asLimitCone :=\n  Profinite.Extend.functorOp_final S.asLimitCone S.asLimit\n\n/--\nA presheaf, which takes a profinite set written as a cofiltered limit to the corresponding\ncolimit, agrees with the left Kan extension of its restriction.\n-/\ndef lanPresheafIso (hF : IsColimit <| F.mapCocone S.asLimitCone.op) :\n    (lanPresheaf F).obj ⟨S⟩ ≅ F.obj ⟨S⟩ :=\n  (Functor.Final.colimitIso (Profinite.Extend.functorOp S.asLimitCone) _).symm ≪≫\n    (colimit.isColimit _).coconePointUniqueUpToIso hF\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma lanPresheafIso_hom (hF : IsColimit <| F.mapCocone S.asLimitCone.op) :\n    (lanPresheafIso hF).hom = colimit.desc _ (Profinite.Extend.cocone _ _) := by\n  simp [lanPresheafIso, Final.colimitIso]\n  rfl\n\nset_option backward.isDefEq.respectTransparency false in\n/-- `lanPresheafIso` is natural in `S`. -/\ndef lanPresheafNatIso (hF : ∀ S : Profinite, IsColimit <| F.mapCocone S.asLimitCone.op) :\n    lanPresheaf F ≅ F :=\n  NatIso.ofComponents (fun ⟨S⟩ ↦ (lanPresheafIso (hF S)))\n    fun _ ↦ (by simpa using colimit.hom_ext fun _ ↦ (by simp))\n\n@[simp]\nlemma lanPresheafNatIso_hom_app (hF : ∀ S : Profinite, IsColimit <| F.mapCocone S.asLimitCone.op)\n    (S : Profiniteᵒᵖ) : (lanPresheafNatIso hF).hom.app S =\n      colimit.desc _ (Profinite.Extend.cocone _ _) := by\n  simp [lanPresheafNatIso]\n\n/--\n`lanPresheaf (locallyConstantPresheaf X)` is a sheaf for the coherent topology on `Profinite`.\n-/\ndef lanSheafProfinite (X : Type (u + 1)) :\n    Sheaf (coherentTopology Profinite.{u}) (Type (u + 1)) where\n  obj := lanPresheaf (locallyConstantPresheaf X)\n  property := by\n    rw [Presheaf.isSheaf_of_iso_iff (lanPresheafNatIso\n      fun _ ↦ isColimitLocallyConstantPresheafDiagram _ _)]\n    exact ((CompHausLike.LocallyConstant.functor.{u, u + 1}\n      (hs := fun _ _ _ ↦ ((Profinite.effectiveEpi_tfae _).out 0 2).mp)).obj X).property\n\n/-- `lanPresheaf (locallyConstantPresheaf X)` as a condensed set. -/\ndef lanCondensedSet (X : Type (u + 1)) : CondensedSet.{u} :=\n  (ProfiniteCompHaus.equivalence _).functor.obj (lanSheafProfinite X)\n\nvariable (F : Profinite.{u}ᵒᵖ ⥤ Type (u + 1))\n\n/--\nThe functor which takes a finite set to the set of maps into `F(*)` for a presheaf `F` on\n`Profinite`.\n-/\n@[simps obj map]\ndef finYoneda : FintypeCat.{u}ᵒᵖ ⥤ Type (u + 1) where\n  obj X := X.unop → F.obj (toProfinite.op.obj ⟨of <| PUnit.{u + 1}⟩)\n  map f := ↾fun g ↦ g ∘ f.unop\n\n/-- `locallyConstantPresheaf` restricted to finite sets is isomorphic to `finYoneda F`. -/\n@[simps! hom_app]\ndef locallyConstantIsoFinYoneda :\n    toProfinite.op ⋙ (locallyConstantPresheaf (F.obj (toProfinite.op.obj\n      ⟨of <| PUnit.{u + 1}⟩))) ≅\n    finYoneda F :=\n  NatIso.ofComponents fun Y ↦ {\n    hom := ↾fun f ↦ f.1\n    inv := ↾fun f ↦ ⟨f, @IsLocallyConstant.of_discrete _ _ _ ⟨rfl⟩ _⟩ }\n\n/-- A finite set as a coproduct cocone in `Profinite` over itself. -/\ndef fintypeCatAsCofan (X : Profinite) :\n    Cofan (fun (_ : X) ↦ (Profinite.of (PUnit.{u + 1}))) :=\n  Cofan.mk X (fun x ↦ ConcreteCategory.ofHom (ContinuousMap.const _ x))\n\n/-- A finite set is the coproduct of its points in `Profinite`. -/\ndef fintypeCatAsCofanIsColimit (X : Profinite) [Finite X] :\n    IsColimit (fintypeCatAsCofan X) :=\n  Cofan.IsColimit.mk _ (fun t ↦ ConcreteCategory.ofHom ⟨fun x ↦ t.inj x PUnit.unit,\n    continuous_of_discreteTopology (α := X)⟩) (by aesop)\n    (fun _ _ h ↦ by ext x; exact CategoryTheory.congr_fun (h x) _)\n\nvariable [PreservesFiniteProducts F]\n\nnoncomputable instance (X : Profinite) [Finite X] :\n    PreservesLimitsOfShape (Discrete X) F :=\n  let X' := (Countable.toSmall.{0} X).equiv_small.choose\n  let e : X ≃ X' := (Countable.toSmall X).equiv_small.choose_spec.some\n  have : Finite X' := .of_equiv X e\n  preservesLimitsOfShape_of_equiv (Discrete.equivalence e.symm) F\n\n/-- Auxiliary definition for `isoFinYoneda`. -/\ndef isoFinYonedaComponents (X : Profinite.{u}) [Finite X] :\n    F.obj ⟨X⟩ ≅ (X → F.obj ⟨Profinite.of PUnit.{u + 1}⟩) :=\n  (isLimitFanMkObjOfIsLimit F _ _\n    (Cofan.IsColimit.op (fintypeCatAsCofanIsColimit X))).conePointUniqueUpToIso\n      (Types.productLimitCone.{u, u + 1} fun _ ↦ F.obj ⟨Profinite.of PUnit.{u + 1}⟩).2\n\n@[simp]\nlemma isoFinYonedaComponents_hom (X : Profinite.{u}) [Finite X] :\n    (isoFinYonedaComponents F X).hom =\n    ↾fun y x ↦ F.map ((Profinite.of PUnit.{u + 1}).const x).op y :=\n  rfl\n\nlemma isoFinYonedaComponents_hom_apply (X : Profinite.{u}) [Finite X] (y : F.obj ⟨X⟩) (x : X) :\n    (isoFinYonedaComponents F X).hom y x =\n      F.map ((Profinite.of PUnit.{u + 1}).const x).op y :=\n  rfl\n\nTarget:\nlemma isoFinYonedaComponents_inv_comp {X Y : Profinite.{u}} [Finite X] [Finite Y]\n    (f : Y → F.obj ⟨Profinite.of PUnit⟩) (g : X ⟶ Y) :\n    (isoFinYonedaComponents F X).inv (f ∘ g) = F.map g.op ((isoFinYonedaComponents F Y).inv f) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed/Discrete","family_id":"isofinyonedacomponents_inv_comp","file_id":"mathlib/Mathlib/Condensed/Discrete/Colimit.lean","sample_id":"10fa2791fbfc644d39bf2d81c08a78f073daf00278ede35b73bb7ef9624d8ac7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3b000d981c87ecc371d9ed638c299bf8ec24374d90b8958428a050b3089d276d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c8ff660e3cb075cd21a289a8fe0e908d06e3a660d3865e6edea15898913062b8","source_sha256":"0d1f9dee5bc403406539e90adb439a195f88865e97423c979d76639acfd9231c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases ofSeq_surjective x with ⟨f, rfl⟩\n  rw [tendsto_ofSeq, (nhds_basis_Ioo _).tendsto_right_iff]\n  simp_rw [Set.mem_Ioo, eventually_and, ← ofSeq_lt_ofSeq]\n  refine ⟨fun H ↦ ⟨fun s hs ↦ ?_, fun s hs ↦ ?_⟩, fun H ⟨s, t⟩ ⟨hs, ht⟩ ↦ ⟨?_, ?_⟩⟩\n  · obtain ⟨t, ht⟩ := exists_gt r\n    exact (H ⟨s, t⟩ ⟨hs, ht⟩).1.le\n  · obtain ⟨t, ht⟩ := exists_lt r\n    exact (H ⟨t, s⟩ ⟨ht, hs⟩).2.le\n  · obtain ⟨u, hu, hu'⟩ := exists_between hs\n    exact (coe_lt_coe.2 hu).trans_le (H.1 _ hu')\n  · obtain ⟨u, hu, hu'⟩ := exists_between ht\n    exact (H.2 _ hu).trans_lt (coe_lt_coe.2 hu')","hard_negative":false,"metrics":{"chosen_tokens":186,"rejected_tokens":2,"token_jaccard":0.018868,"token_length_ratio":0.010753},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"531ecd16b44945ecff81e82a8a5a3340ce2baba21949d3356aa7b98b12c13554","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Ring.StandardPart\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.Order.Filter.FilterProduct\n\nNamespace:\nHyperreal\n\nLocal context:\n/-\nCopyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Abhimanyu Pallavi Sudhir, Violeta Hernández Palacios\n-/\n/-!\n# Construction of the hyperreal numbers as an ultraproduct of real sequences\n\nWe define the `Hyperreal` numbers as quotients of sequences `ℕ → ℝ` by an ultrafilter. These form\na field, and we prove some of their basic properties.\n\nNote that most of the machinery that is usually defined for the specific purpose of non-standard\nanalysis (infinitesimal and infinite elements, standard parts) has been generalized to other\nnon-archimedean fields. In particular:\n\n- `ArchimedeanClass` can be used to measure whether an element is infinitesimal (`0 < mk x`) or\n  infinite (`mk x < 0`).\n- `ArchimedeanClass.stdPart` generalizes the standard part function to a general ordered field.\n\n## Todo\n\nUse Łoś's Theorem `FirstOrder.Language.Ultraproduct.sentence_realize` to formalize the transfer\nprinciple on `Hyperreal`.\n-/\n\n@[expose] public section\n\nopen ArchimedeanClass Filter Germ Topology\n\nnoncomputable section\n\n/-- Hyperreal numbers on the ultrafilter extending the cofinite filter. -/\ndef Hyperreal : Type :=\n  Germ (hyperfilter ℕ : Filter ℕ) ℝ\nderiving Inhabited\n\nnamespace Hyperreal\n\n@[inherit_doc] notation \"ℝ*\" => Hyperreal\n\ninstance : Field ℝ* :=\n  inferInstanceAs (Field (Germ _ _))\n\ninstance : LinearOrder ℝ* :=\n  inferInstanceAs (LinearOrder (Germ _ _))\n\ninstance : IsStrictOrderedRing ℝ* :=\n  inferInstanceAs (IsStrictOrderedRing (Germ _ _))\n\n/-- Natural embedding `ℝ → ℝ*`. -/\n@[coe] def ofReal : ℝ → ℝ* := const\n\ninstance : CoeTC ℝ ℝ* := ⟨ofReal⟩\n\n@[simp, norm_cast]\ntheorem coe_eq_coe {x y : ℝ} : (x : ℝ*) = y ↔ x = y :=\n  Germ.const_inj\n\ntheorem coe_ne_coe {x y : ℝ} : (x : ℝ*) ≠ y ↔ x ≠ y :=\n  coe_eq_coe.not\n\n@[simp, norm_cast]\ntheorem coe_eq_zero {x : ℝ} : (x : ℝ*) = 0 ↔ x = 0 :=\n  coe_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_eq_one {x : ℝ} : (x : ℝ*) = 1 ↔ x = 1 :=\n  coe_eq_coe\n\n@[norm_cast]\ntheorem coe_ne_zero {x : ℝ} : (x : ℝ*) ≠ 0 ↔ x ≠ 0 :=\n  coe_ne_coe\n\n@[norm_cast]\ntheorem coe_ne_one {x : ℝ} : (x : ℝ*) ≠ 1 ↔ x ≠ 1 :=\n  coe_ne_coe\n\n@[simp, norm_cast]\ntheorem coe_one : ↑(1 : ℝ) = (1 : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_zero : ↑(0 : ℝ) = (0 : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_inv (x : ℝ) : ↑x⁻¹ = (x⁻¹ : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_neg (x : ℝ) : ↑(-x) = (-x : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_add (x y : ℝ) : ↑(x + y) = (x + y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_ofNat (n : ℕ) [n.AtLeastTwo] :\n    ((ofNat(n) : ℝ) : ℝ*) = OfNat.ofNat n :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_mul (x y : ℝ) : ↑(x * y) = (x * y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_div (x y : ℝ) : ↑(x / y) = (x / y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sub (x y : ℝ) : ↑(x - y) = (x - y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_le_coe {x y : ℝ} : (x : ℝ*) ≤ y ↔ x ≤ y :=\n  Germ.const_le_iff\n\n@[simp, norm_cast]\ntheorem coe_lt_coe {x y : ℝ} : (x : ℝ*) < y ↔ x < y :=\n  Germ.const_lt_iff\n\n@[simp, norm_cast]\ntheorem coe_nonneg {x : ℝ} : 0 ≤ (x : ℝ*) ↔ 0 ≤ x :=\n  coe_le_coe\n\n@[simp, norm_cast]\ntheorem coe_pos {x : ℝ} : 0 < (x : ℝ*) ↔ 0 < x :=\n  coe_lt_coe\n\n@[simp, norm_cast]\ntheorem coe_abs (x : ℝ) : ((|x| : ℝ) : ℝ*) = |↑x| :=\n  const_abs x\n\n@[simp, norm_cast]\ntheorem coe_max (x y : ℝ) : ((max x y : ℝ) : ℝ*) = max ↑x ↑y :=\n  Germ.const_max _ _\n\n@[simp, norm_cast]\ntheorem coe_min (x y : ℝ) : ((min x y : ℝ) : ℝ*) = min ↑x ↑y :=\n  Germ.const_min _ _\n\n/-- The canonical map `ℝ → ℝ*` as an `OrderRingHom`. -/\n@[simps]\ndef coeRingHom : ℝ →+*o ℝ* where\n  toFun x := x\n  map_zero' := rfl\n  map_one' := rfl\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  monotone' _ _ := coe_le_coe.2\n\n@[simp]\ntheorem archimedeanClassMk_coe_nonneg (x : ℝ) : 0 ≤ mk (x : ℝ*) :=\n  mk_map_nonneg_of_archimedean coeRingHom x\n\n@[simp]\ntheorem archimdeanClassMk_coe {x : ℝ} (hx : x ≠ 0) : mk (x : ℝ*) = 0 :=\n  mk_map_of_archimedean' coeRingHom hx\n\n@[simp]\ntheorem stdPart_coe (x : ℝ) : stdPart (x : ℝ*) = x :=\n  stdPart_map_real coeRingHom x\n\n/-! ### Basic constants -/\n\n/-- Construct a hyperreal number from a sequence of real numbers. -/\ndef ofSeq (f : ℕ → ℝ) : ℝ* := (↑f : Germ (hyperfilter ℕ : Filter ℕ) ℝ)\n\ntheorem ofSeq_surjective : Function.Surjective ofSeq := Quot.exists_rep\n\ntheorem ofSeq_lt_ofSeq {f g : ℕ → ℝ} : ofSeq f < ofSeq g ↔ ∀ᶠ n in hyperfilter ℕ, f n < g n :=\n  Germ.coe_lt\n\ntheorem ofSeq_le_ofSeq {f g : ℕ → ℝ} : ofSeq f ≤ ofSeq g ↔ ∀ᶠ n in hyperfilter ℕ, f n ≤ g n :=\n  Germ.coe_le\n\n/-! #### ω -/\n\n/-- A sample infinite hyperreal ω = ⟦(0, 1, 2, 3, ⋯)⟧. -/\ndef omega : ℝ* := ofSeq Nat.cast\n\n@[inherit_doc] scoped notation \"ω\" => Hyperreal.omega\nrecommended_spelling \"omega\" for \"ω\" in [omega, «termω»]\n\ntheorem coe_lt_omega (r : ℝ) : r < ω := by\n  apply ofSeq_lt_ofSeq.2 <| Filter.Eventually.filter_mono Nat.hyperfilter_le_atTop _\n  obtain ⟨n, hn⟩ := exists_nat_gt r\n  rw [eventually_atTop]\n  exact ⟨n, fun m hm ↦ hn.trans_le (mod_cast hm)⟩\n\ntheorem omega_pos : 0 < ω :=\n  coe_lt_omega 0\n\n@[simp]\ntheorem omega_ne_zero : ω ≠ 0 :=\n  omega_pos.ne'\n\n@[simp]\ntheorem abs_omega : |ω| = ω :=\n  abs_of_pos omega_pos\n\n@[simp]\ntheorem archimedeanClassMk_omega_neg : mk ω < 0 :=\n  fun n ↦ by simpa using! coe_lt_omega n\n\n@[simp]\ntheorem stdPart_omega : stdPart ω = 0 := by\n  rw [stdPart_eq_zero]\n  exact archimedeanClassMk_omega_neg.ne\n\n/-! #### ε -/\n\n/-- A sample infinitesimal hyperreal ε = ⟦(0, 1, 1/2, 1/3, ⋯)⟧. -/\ndef epsilon : ℝ* :=\n  ofSeq fun n => n⁻¹\n\n@[inherit_doc] scoped notation \"ε\" => Hyperreal.epsilon\nrecommended_spelling \"epsilon\" for \"ε\" in [epsilon, «termε»]\n\n@[simp]\ntheorem inv_omega : ω⁻¹ = ε :=\n  rfl\n\n@[simp]\ntheorem inv_epsilon : ε⁻¹ = ω :=\n  @inv_inv _ _ ω\n\n@[simp]\ntheorem epsilon_pos : 0 < ε :=\n  inv_pos_of_pos omega_pos\n\n@[simp]\ntheorem epsilon_ne_zero : ε ≠ 0 :=\n  epsilon_pos.ne'\n\n@[simp]\ntheorem epsilon_mul_omega : ε * ω = 1 :=\n  @inv_mul_cancel₀ _ _ ω omega_ne_zero\n\n@[simp]\ntheorem archimedeanClassMk_epsilon_pos : 0 < mk ε := by\n  simp [← inv_omega]\n\n/-!\n### Some facts about `Tendsto`\n-/\n\n@[simp]\ntheorem tendsto_ofSeq {f : ℕ → ℝ} {lb : Filter ℝ} :\n    (ofSeq f).Tendsto lb ↔ Tendsto f (hyperfilter ℕ) lb :=\n  .rfl\n\ntheorem stdPart_map {x : ℝ*} {r : ℝ} {f : ℝ → ℝ} (hf : ContinuousAt f r)\n    (hxr : x.Tendsto (𝓝 r)) : (x.map f).Tendsto (𝓝 (f r)) := by\n  rcases ofSeq_surjective x with ⟨g, rfl⟩\n  exact hf.tendsto.comp hxr\n\ntheorem stdPart_map₂ {x y : ℝ*} {r s : ℝ} {f : ℝ → ℝ → ℝ}\n    (hxr : x.Tendsto (𝓝 r)) (hys : y.Tendsto (𝓝 s))\n    (hf : ContinuousAt (Function.uncurry f) (r, s)) : (x.map₂ f y).Tendsto (𝓝 (f r s)) := by\n  rcases ofSeq_surjective x with ⟨x, rfl⟩\n  rcases ofSeq_surjective y with ⟨y, rfl⟩\n  exact hf.tendsto.comp (hxr.prodMk_nhds hys)\n\nTarget:\ntheorem tendsto_iff_forall {x : ℝ*} {r : ℝ} :\n    x.Tendsto (𝓝 r) ↔ (∀ s < r, s ≤ x) ∧ (∀ s > r, x ≤ s) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Real","family_id":"tendsto_iff_forall","file_id":"mathlib/Mathlib/Analysis/Real/Hyperreal.lean","sample_id":"c8ff660e3cb075cd21a289a8fe0e908d06e3a660d3865e6edea15898913062b8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9c89a7a96cac539d3af0940d447c59a7152af477d925ba54bdc0d7f1377d3647","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"70071016d6f956ee807fdebd39b7e38611eccc815e374ddf4ad147924380a172","source_sha256":"ebde9039546abaab19fae84dcc9dee6e17f2d20f60817bbfa41ae3bdde9f8a83","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext; simp [mem_coe, mem_extended_iff]","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":2,"token_jaccard":0.1,"token_length_ratio":0.222222},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"536b1a01a00eba9f5a204b631097a0450da83bb199cf458f6ca4240c91a063be","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas\n\nNamespace:\nFractionalIdeal\n\nLocal context:\n/-\nCopyright (c) 2024 James Sundstrom. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: James Sundstrom, Xavier Roblot\n-/\n/-!\n# Extension of fractional ideals\n\nThis file defines the extension of a fractional ideal along a ring homomorphism.\n\n## Main definitions\n\n* `FractionalIdeal.extended`: Let `A` and `B` be commutative rings with respective localizations\n  `IsLocalization M K` and `IsLocalization N L`. Let `f : A →+* B` be a ring homomorphism with\n  `hf : M ≤ Submonoid.comap f N`. If `I : FractionalIdeal M K`, then the extension of `I` along\n  `f` is `extended L hf I : FractionalIdeal N L`.\n* `FractionalIdeal.extendedHom'`: The ring homomorphism version of `FractionalIdeal.extended`.\n* `FractionalIdeal.extendedHom`: For `A ⊆ B` an extension of domains, the ring homomorphism that\n  sends a fractional ideal of `A` to a fractional ideal of `B`.\n\n## Main results\n\n* `FractionalIdeal.extendedHom_injective`: the map `FractionalIdeal.extendedHom` is injective.\n* `FractionalIdeal.extended_extended`: extending fractional ideals is compatible with composition\n  of ring homomorphisms.\n* `FractionalIdeal.extendedHom'_comp`: the homomorphisms induced by extension of fractional\n  ideals compose in towers.\n* `Ideal.map_algebraMap_injective`: For `A ⊆ B` an extension of Dedekind domains, the map that\n  sends an ideal `I` of `A` to `I·B` is injective.\n\n## Tags\n\nfractional ideal, fractional ideals, extended, extension\n-/\n\n@[expose] public section\n\nopen IsLocalization FractionalIdeal Module Submodule\n\nnamespace FractionalIdeal\n\nsection RingHom\n\nvariable {A : Type*} [CommRing A] {B : Type*} [CommRing B] {f : A →+* B}\nvariable {K : Type*} {M : Submonoid A} [CommRing K] [Algebra A K] [IsLocalization M K]\nvariable (L : Type*) {N : Submonoid B} [CommRing L] [Algebra B L] [IsLocalization N L]\nvariable (hf : M ≤ Submonoid.comap f N)\nvariable (I : FractionalIdeal M K) (J : FractionalIdeal M K)\n\n/-- Given commutative rings `A` and `B` with respective localizations `IsLocalization M K` and\n`IsLocalization N L`, and a ring homomorphism `f : A →+* B` satisfying `M ≤ Submonoid.comap f N`, a\nfractional ideal `I` of `A` can be extended along `f` to a fractional ideal of `B`. -/\ndef extended (I : FractionalIdeal M K) : FractionalIdeal N L where\n  val := span B <| (IsLocalization.map (S := K) L f hf) '' I\n  property := by\n    have ⟨a, ha, frac⟩ := I.isFractional\n    refine ⟨f a, hf ha, fun b hb ↦ ?_⟩\n    refine span_induction (fun x hx ↦ ?_) ⟨0, by simp⟩\n      (fun x y _ _ hx hy ↦ smul_add (f a) x y ▸ isInteger_add hx hy) (fun b c _ hc ↦ ?_) hb\n    · rcases hx with ⟨k, kI, rfl⟩\n      obtain ⟨c, hc⟩ := frac k kI\n      exact ⟨f c, by simp [← IsLocalization.map_smul, ← hc]⟩\n    · rw [← smul_assoc, smul_eq_mul, mul_comm (f a), ← smul_eq_mul, smul_assoc]\n      exact isInteger_smul hc\n\nlocal notation \"map_f\" => (IsLocalization.map (S := K) L f hf)\n\nlemma mem_extended_iff (x : L) : x ∈ I.extended L hf ↔ x ∈ span B (map_f '' I) := by\n  constructor <;> { intro hx; simpa }\n\n@[simp]\n\nTarget:\nlemma coe_extended_eq_span : I.extended L hf = span B (map_f '' I) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/FractionalIdeal","family_id":"coe_extended_eq_span","file_id":"mathlib/Mathlib/RingTheory/FractionalIdeal/Extended.lean","sample_id":"70071016d6f956ee807fdebd39b7e38611eccc815e374ddf4ad147924380a172"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a27f59db44cea34ebdac75d20e4acc76ed165c06147e346a2eafa82d1ac01d9a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"683dd16c2c5ea5c93eba3da8de5686a9da33699ae926a711d9de06ce67a8dd11","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3236ac1bb6f48e9d26b20fb63c3f2eb332af2bf7a5e2f318caa01e95dce53a89","source_sha256":"3ebef0ce1ff4e8804552a350539c21f4e3856ca570ca14d4062fa9367e8ccafd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨β, _, sgn, g, hg, hβ, hf⟩ := exists_signed_sum s f\n  refine\n    ⟨β ⊕ (Fin (n - ∑ i ∈ s, (f i).natAbs)), inferInstance, Sum.elim sgn 0,\n      Sum.elim g (Classical.arbitrary (Fin (n - Finset.sum s fun i => Int.natAbs (f i)) → α)),\n        ?_, by simp [hβ, h], fun a ha => by simp [hf _ ha]⟩\n  rintro (b | b) hb\n  · cases hb (hg _)\n  · rfl","hard_negative":false,"metrics":{"chosen_tokens":126,"rejected_tokens":130,"token_jaccard":0.947368,"token_length_ratio":1.031746},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"53a001df2e55db9be0ad446cc73bb9431d900d0eaacc9d8b20a523274ffa527b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GroupWithZero.Units.Lemmas\npublic import Mathlib.Algebra.Order.BigOperators.Group.Finset\npublic import Mathlib.Algebra.Order.Ring.Cast\npublic import Mathlib.Data.Fintype.BigOperators\npublic import Mathlib.Data.Sign.Defs\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Eric Rodriguez. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Rodriguez\n-/\n/-!\n# Sign function\n\nThis file defines the sign function for types with zero and a decidable less-than relation, and\nproves some basic theorems about it.\n-/\n\n@[expose] public section\n\nuniverse u\nvariable {α : Type u}\n\nnamespace SignType\n\n/-- Casting `SignType → ℤ → α` is the same as casting directly `SignType → α`. -/\n@[simp, norm_cast]\nlemma intCast_cast {α : Type*} [AddGroupWithOne α] (s : SignType) : ((s : ℤ) : α) = s :=\n  map_cast' _ Int.cast_one Int.cast_zero (@Int.cast_one α _ ▸ Int.cast_neg 1) _\n\ntheorem pow_odd (s : SignType) {n : ℕ} (hn : Odd n) : s ^ n = s := by\n  obtain ⟨k, rfl⟩ := hn\n  rw [pow_add, pow_one, pow_mul, sq]\n  cases s <;> simp\n\ntheorem zpow_odd (s : SignType) {z : ℤ} (hz : Odd z) : s ^ z = s := by\n  obtain rfl | hs := eq_or_ne s 0\n  · rw [zero_zpow]\n    rintro rfl\n    simp at hz\n  obtain ⟨k, rfl⟩ := hz\n  rw [zpow_add₀ hs, zpow_one, zpow_mul, zpow_two]\n  cases s <;> simp\n\nlemma pow_even (s : SignType) {n : ℕ} (hn : Even n) (hs : s ≠ 0) :\n    s ^ n = 1 := by\n  cases s <;> simp_all\n\nlemma zpow_even (s : SignType) {z : ℤ} (hz : Even z) (hs : s ≠ 0) :\n    s ^ z = 1 := by\n  cases s <;> simp_all [Even.neg_one_zpow]\n\n/-- `SignType.cast` as a `MulWithZeroHom`. -/\n@[simps]\ndef castHom {α} [MulZeroOneClass α] [HasDistribNeg α] : SignType →*₀ α where\n  toFun := cast\n  map_zero' := rfl\n  map_one' := rfl\n  map_mul' x y := by cases x <;> cases y <;> simp [zero_eq_zero, pos_eq_one, neg_eq_neg_one]\n\ntheorem univ_eq : (Finset.univ : Finset SignType) = {0, -1, 1} := by\n  decide\n\ntheorem range_eq {α} (f : SignType → α) : Set.range f = {f zero, f neg, f pos} := by\n  classical rw [← Fintype.coe_image_univ, univ_eq]\n  classical simp [Finset.coe_insert]\n\n@[simp, norm_cast] lemma coe_mul {α} [MulZeroOneClass α] [HasDistribNeg α] (a b : SignType) :\n    ↑(a * b) = (a : α) * b :=\n  map_mul SignType.castHom _ _\n\n@[simp, norm_cast] lemma coe_pow {α} [MonoidWithZero α] [HasDistribNeg α] (a : SignType) (k : ℕ) :\n    ↑(a ^ k) = (a : α) ^ k :=\n  map_pow SignType.castHom _ _\n\n@[simp, norm_cast] lemma coe_zpow {α} [GroupWithZero α] [HasDistribNeg α] (a : SignType) (k : ℤ) :\n    ↑(a ^ k) = (a : α) ^ k :=\n  map_zpow₀ SignType.castHom _ _\n\nend SignType\n\nopen SignType\n\nsection OrderedRing\n\n@[simp]\nlemma sign_intCast {α : Type*} [Ring α] [PartialOrder α] [IsOrderedRing α]\n    [Nontrivial α] [DecidableLT α] (n : ℤ) :\n    sign (n : α) = sign n := by\n  simp only [sign_apply, Int.cast_pos, Int.cast_lt_zero]\n\nend OrderedRing\n\nsection LinearOrderedRing\n\nvariable [Ring α] [LinearOrder α] [IsStrictOrderedRing α]\n\ntheorem sign_mul (x y : α) : sign (x * y) = sign x * sign y := by\n  rcases lt_trichotomy x 0 with (hx | hx | hx) <;> rcases lt_trichotomy y 0 with (hy | hy | hy) <;>\n    simp [hx, hy, mul_pos_of_neg_of_neg, mul_neg_of_neg_of_pos, mul_neg_of_pos_of_neg]\n\n@[simp] theorem sign_mul_abs (x : α) : (sign x * |x| : α) = x := by\n  rcases lt_trichotomy x 0 with hx | rfl | hx <;> simp [*, abs_of_pos, abs_of_neg]\n\n@[simp] theorem abs_mul_sign (x : α) : (|x| * sign x : α) = x := by\n  rcases lt_trichotomy x 0 with hx | rfl | hx <;> simp [*, abs_of_pos, abs_of_neg]\n\n@[simp]\ntheorem sign_mul_self (x : α) : sign x * x = |x| := by\n  rcases lt_trichotomy x 0 with hx | rfl | hx <;> simp [*, abs_of_pos, abs_of_neg]\n\n@[simp]\ntheorem self_mul_sign (x : α) : x * sign x = |x| := by\n  rcases lt_trichotomy x 0 with hx | rfl | hx <;> simp [*, abs_of_pos, abs_of_neg]\n\n/-- `SignType.sign` as a `MonoidWithZeroHom` for a nontrivial ordered semiring. Note that linearity\nis required; consider ℂ with the order `z ≤ w` iff they have the same imaginary part and\n`z - w ≤ 0` in the reals; then `1 + I` and `1 - I` are incomparable to zero, and thus we have:\n`0 * 0 = SignType.sign (1 + I) * SignType.sign (1 - I) ≠ SignType.sign 2 = 1`.\n(`Complex.orderedCommRing`) -/\n@[simps -fullyApplied]\ndef signHom : α →*₀ SignType where\n  toFun := sign\n  map_zero' := sign_zero\n  map_one' := sign_one\n  map_mul' := sign_mul\n\ntheorem sign_pow (x : α) (n : ℕ) : sign (x ^ n) = sign x ^ n := map_pow signHom x n\n\nend LinearOrderedRing\n\nsection LinearOrderedAddCommGroup\n\nvariable [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α]\n\ntheorem sign_sum {ι : Type*} {s : Finset ι} {f : ι → α} (hs : s.Nonempty) (t : SignType)\n    (h : ∀ i ∈ s, sign (f i) = t) : sign (∑ i ∈ s, f i) = t := by\n  cases t\n  · simp_rw [zero_eq_zero, sign_eq_zero_iff] at h ⊢\n    exact Finset.sum_eq_zero h\n  · simp_rw [neg_eq_neg_one, sign_eq_neg_one_iff] at h ⊢\n    exact Finset.sum_neg h hs\n  · simp_rw [pos_eq_one, sign_eq_one_iff] at h ⊢\n    exact Finset.sum_pos h hs\n\nend LinearOrderedAddCommGroup\n\nopen Finset Nat\n\nsection exists_signed_sum\n\n/-!\nIn this section we explicitly handle universe variables,\nbecause Lean creates a fresh universe variable for the type whose existence is asserted.\nBut we want the type to live in the same universe as the input type.\n-/\n\nprivate theorem exists_signed_sum_aux [DecidableEq α] (s : Finset α) (f : α → ℤ) :\n    ∃ (β : Type u) (t : Finset β) (sgn : β → SignType) (g : β → α),\n      (∀ b, g b ∈ s) ∧\n        (#t = ∑ a ∈ s, (f a).natAbs) ∧\n          ∀ a ∈ s, (∑ b ∈ t, if g b = a then (sgn b : ℤ) else 0) = f a := by\n  refine\n    ⟨(Σ _ : { x // x ∈ s }, ℕ), Finset.univ.sigma fun a => range (f a).natAbs,\n      fun a => sign (f a.1), fun a => a.1, fun a => a.1.2, ?_, ?_⟩\n  · simp [sum_attach (f := fun a => (f a).natAbs)]\n  · intro x hx\n    simp [sum_sigma, hx, ← Int.sign_eq_sign, Int.sign_mul_abs, mul_comm |f _|,\n      sum_attach (s := s) (f := fun y => if y = x then f y else 0)]\n\n/-- We can decompose a sum of absolute value `n` into a sum of `n` signs. -/\ntheorem exists_signed_sum [DecidableEq α] (s : Finset α) (f : α → ℤ) :\n    ∃ (β : Type u) (_ : Fintype β) (sgn : β → SignType) (g : β → α),\n      (∀ b, g b ∈ s) ∧\n        (Fintype.card β = ∑ a ∈ s, (f a).natAbs) ∧\n          ∀ a ∈ s, (∑ b, if g b = a then (sgn b : ℤ) else 0) = f a :=\n  let ⟨β, t, sgn, g, hg, ht, hf⟩ := exists_signed_sum_aux s f\n  ⟨t, inferInstance, fun b => sgn b, fun b => g b, fun b => hg b, by simp [ht], fun a ha =>\n    (sum_attach t fun b ↦ ite (g b = a) (sgn b : ℤ) 0).trans <| hf _ ha⟩\n\n/-- We can decompose a sum of absolute value less than `n` into a sum of at most `n` signs. -/\n\nTarget:\ntheorem exists_signed_sum' [Nonempty α] [DecidableEq α] (s : Finset α) (f : α → ℤ)\n    (n : ℕ) (h : (∑ i ∈ s, (f i).natAbs) ≤ n) :\n    ∃ (β : Type u) (_ : Fintype β) (sgn : β → SignType) (g : β → α),\n      (∀ b, g b ∉ s → sgn b = 0) ∧\n        Fintype.card β = n ∧ ∀ a ∈ s, (∑ i, if g i = a then (sgn i : ℤ) else 0) = f a :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  obtain ⟨β, _, sgn, g, hg, hβ, hf⟩ := exists_signed_sum s f\n  refine\n    ⟨β ⊕ (Fin (n - ∑ i ∈ s, (f i).natAbs)), inferInstance, Sum.elim sgn 0,\n      Sum.elim g (Classical.arbitrary (Fin (n - Finset.sum s fun i => Int.natAbs (f i)) → α)),\n        ?_, by simp [hβ, h], fun a ha => by simp [hf _ ha]⟩\n  rintro (b | b) hb\n  · cases hb (hg _)\n  · rfl","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sign","family_id":"exists_signed_sum","file_id":"mathlib/Mathlib/Data/Sign/Basic.lean","sample_id":"3236ac1bb6f48e9d26b20fb63c3f2eb332af2bf7a5e2f318caa01e95dce53a89"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2bb1a47c41a1654399bf9970e1dd3f153a3e08560ed360131ac9536f9ce93d74","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d14998e2d236d5892c9ed492b6063a072a0ec193e356245fa60104974e67b50c","source_sha256":"88899366de27f8f7e91c2113d9cd607e3b4b1f4debab3b6b3a2046a9e9f7c31f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction s using Finset.cons_induction with\n  | empty => simp\n  | cons a s ha ihs =>\n    simp_rw [Finset.sum_cons, ihs]\n    refine iSup_add_iSup fun i j ↦ (hf i j).imp fun k hk ↦ ?_\n    gcongr\n    exacts [(hk a).1, (hk _).2]","hard_negative":false,"metrics":{"chosen_tokens":63,"rejected_tokens":5,"token_jaccard":0.047619,"token_length_ratio":0.079365},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"53a41a1a9f07a7a5fb844ca8707cc4a803777bfb25edfee76742d86bf37de416","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Ring.Finset\npublic import Mathlib.Algebra.BigOperators.WithTop\npublic import Mathlib.Algebra.Order.BigOperators.Group.Finset\npublic import Mathlib.Data.ENat.Lattice\n\nNamespace:\nENat\n\nLocal context:\n/-\nCopyright (c) 2024 Joachim Breitner, Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joachim Breitner, Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Sum of suprema in `ENat`\n-/\n\npublic section\n\nassert_not_exists Field\n\nnamespace ENat\n\nvariable {a b c d : ℕ∞} {r p q : ℕ}\n\nsection OperationsAndInfty\n\nvariable {α : Type*}\n\n@[simp]\ntheorem toNat_prod {ι : Type*} {s : Finset ι} {f : ι → ℕ∞} :\n    (∏ i ∈ s, f i).toNat = ∏ i ∈ s, (f i).toNat :=\n  map_prod toNatHom _ _\n\ntheorem iInf_sum {ι α : Type*} {f : ι → α → ℕ∞} {s : Finset α} [Nonempty ι]\n    (h : ∀ (t : Finset α) (i j : ι), ∃ k, ∀ a ∈ t, f k a ≤ f i a ∧ f k a ≤ f j a) :\n    ⨅ i, ∑ a ∈ s, f i a = ∑ a ∈ s, ⨅ i, f i a := by\n  induction s using Finset.cons_induction_on with\n  | empty => simp only [Finset.sum_empty, ciInf_const]\n  | cons a s ha ih =>\n    simp only [Finset.sum_cons, ← ih]\n    refine (iInf_add_iInf fun i j => ?_).symm\n    refine (h (Finset.cons a s ha) i j).imp fun k hk => ?_\n    rw [Finset.forall_mem_cons] at hk\n    exact add_le_add hk.1.1 (Finset.sum_le_sum fun a ha => (hk.2 a ha).2)\n\nend OperationsAndInfty\n\nsection Sum\n\nopen Finset\n\nvariable {α : Type*} {s : Finset α} {f : α → ℕ∞}\n\n/-- A product of finite numbers is still finite. -/\nlemma prod_ne_top (h : ∀ a ∈ s, f a ≠ ⊤) : ∏ a ∈ s, f a ≠ ⊤ := WithTop.prod_ne_top h\n\n/-- A product of finite numbers is still finite. -/\nlemma prod_lt_top (h : ∀ a ∈ s, f a < ⊤) : ∏ a ∈ s, f a < ⊤ := WithTop.prod_lt_top h\n\n/-- A sum is infinite iff one of the summands is infinite. -/\n@[simp] lemma sum_eq_top : ∑ x ∈ s, f x = ⊤ ↔ ∃ a ∈ s, f a = ⊤ := WithTop.sum_eq_top\n\n/-- A sum is finite iff all summands are finite. -/\nlemma sum_ne_top : ∑ a ∈ s, f a ≠ ⊤ ↔ ∀ a ∈ s, f a ≠ ⊤ := WithTop.sum_ne_top\n\n/-- A sum is finite iff all summands are finite. -/\n@[simp] lemma sum_lt_top : ∑ a ∈ s, f a < ⊤ ↔ ∀ a ∈ s, f a < ⊤ := WithTop.sum_lt_top\n\ntheorem lt_top_of_sum_ne_top {s : Finset α} {f : α → ℕ∞} (h : ∑ x ∈ s, f x ≠ ⊤) {a : α}\n    (ha : a ∈ s) : f a < ⊤ :=\n  sum_lt_top.1 h.lt_top a ha\n\n/-- Seeing `ℕ∞` as `ℕ` does not change their sum, unless one of the `ℕ∞` is\ninfinity -/\ntheorem toNat_sum {s : Finset α} {f : α → ℕ∞} (hf : ∀ a ∈ s, f a ≠ ⊤) :\n    ENat.toNat (∑ a ∈ s, f a) = ∑ a ∈ s, ENat.toNat (f a) := by\n  rw [← coe_inj, coe_toNat (sum_ne_top.2 hf), Nat.cast_sum]\n  exact sum_congr rfl fun x hx => (coe_toNat (hf x hx)).symm\n\ntheorem sum_lt_sum_of_nonempty {s : Finset α} (hs : s.Nonempty) {f g : α → ℕ∞}\n    (Hlt : ∀ i ∈ s, f i < g i) : ∑ i ∈ s, f i < ∑ i ∈ s, g i := by\n  induction hs using Nonempty.cons_induction with\n  | singleton => simp [Hlt _ (mem_singleton_self _)]\n  | cons _ _ _ _ ih =>\n    simp only [sum_cons, forall_mem_cons] at Hlt ⊢\n    exact ENat.add_lt_add Hlt.1 (ih Hlt.2)\n\ntheorem exists_le_of_sum_le {s : Finset α} (hs : s.Nonempty) {f g : α → ℕ∞}\n    (Hle : ∑ i ∈ s, f i ≤ ∑ i ∈ s, g i) : ∃ i ∈ s, f i ≤ g i := by\n  contrapose! Hle\n  apply sum_lt_sum_of_nonempty hs Hle\n\nend Sum\n\nTarget:\nlemma sum_iSup {α ι : Type*} {s : Finset α} {f : α → ι → ℕ∞}\n    (hf : ∀ i j, ∃ k, ∀ a, f a i ≤ f a k ∧ f a j ≤ f a k) :\n    ∑ a ∈ s, ⨆ i, f a i = ⨆ i, ∑ a ∈ s, f a i :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/ENat","family_id":"sum_isup","file_id":"mathlib/Mathlib/Data/ENat/BigOperators.lean","sample_id":"d14998e2d236d5892c9ed492b6063a072a0ec193e356245fa60104974e67b50c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b0a03b5526b62c3517381ba5457b6ebaafee0e4da110c931ab1a6ee0269bdd8b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5b0c69fc729fb329df53e747032d70f91d7a865fc26c9ee5566a2b68840871a7","source_sha256":"4e9a1f48a3b50b199e8e6174f030d1362e64d040c2a6826f217f6f03af70994c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases p\n  rw [← ofFractionRing_smul]","hard_negative":true,"metrics":{"chosen_tokens":8,"rejected_tokens":8,"token_jaccard":0.071429,"token_length_ratio":1.0},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"53deb89327ca4871dc01d4ad970146679a1e338876eca5a53f77558e95ae14e2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharP.Algebra\npublic import Mathlib.FieldTheory.RatFunc.Defs\npublic import Mathlib.RingTheory.Polynomial.Content\npublic import Mathlib.RingTheory.Algebraic.Integral\n\nNamespace:\nRatFunc\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# The field structure of rational functions\n\n## Main definitions\nWorking with rational functions as polynomials:\n- `RatFunc.instField` provides a field structure\n\nYou can use `IsFractionRing` API to treat `RatFunc` as the field of fractions of polynomials:\n* `algebraMap K[X] K⟮X⟯` maps polynomials to rational functions\n* `IsFractionRing.algEquiv` maps other fields of fractions of `K[X]` to `K⟮X⟯`.\n\nIn particular:\n* `FractionRing.algEquiv K[X] K⟮X⟯` maps the generic field of\n  fraction construction to `K⟮X⟯`. Combine this with `AlgEquiv.restrictScalars` to change\n  the `FractionRing K[X] ≃ₐ[K[X]] K⟮X⟯` to `FractionRing K[X] ≃ₐ[K] K⟮X⟯`.\n\nWorking with rational functions as fractions:\n- `RatFunc.num` and `RatFunc.denom` give the numerator and denominator.\n  These values are chosen to be coprime and such that `RatFunc.denom` is monic.\n\nLifting homomorphisms of polynomials to other types, by mapping and dividing, as long\nas the homomorphism retains the non-zero-divisor property:\n- `RatFunc.liftMonoidWithZeroHom` lifts a `K[X] →*₀ G₀` to\n  a `K⟮X⟯ →*₀ G₀`, where `[CommRing K] [CommGroupWithZero G₀]`\n- `RatFunc.liftRingHom` lifts a `K[X] →+* L` to a `K⟮X⟯ →+* L`,\n  where `[CommRing K] [Field L]`\n- `RatFunc.liftAlgHom` lifts a `K[X] →ₐ[S] L` to a `K⟮X⟯ →ₐ[S] L`,\n  where `[CommRing K] [Field L] [CommSemiring S] [Algebra S K[X]] [Algebra S L]`\n\nThis is satisfied by injective homs.\n\nWe also have lifting homomorphisms of polynomials to other polynomials,\nwith the same condition on retaining the non-zero-divisor property across the map:\n- `RatFunc.map` lifts `K[X] →* R[X]` when `[CommRing K] [CommRing R]`\n- `RatFunc.mapRingHom` lifts `K[X] →+* R[X]` when `[CommRing K] [CommRing R]`\n- `RatFunc.mapAlgHom` lifts `K[X] →ₐ[S] R[X]` when\n  `[CommRing K] [IsDomain K] [CommRing R] [IsDomain R]`\n-/\n\n@[expose] public section\n\nuniverse u v\n\nnoncomputable section\n\nopen scoped nonZeroDivisors Polynomial\n\nvariable {K : Type u}\n\nnamespace RatFunc\n\nsection Field\n\nvariable [CommRing K]\n\n/-- The zero rational function. -/\nprotected irreducible_def zero : K⟮X⟯ :=\n  ⟨0⟩\n\ninstance : Zero K⟮X⟯ :=\n  ⟨RatFunc.zero⟩\n\ntheorem ofFractionRing_zero : (ofFractionRing 0 : K⟮X⟯) = 0 :=\n  zero_def.symm\n\n/-- Addition of rational functions. -/\nprotected irreducible_def add : K⟮X⟯ → K⟮X⟯ → K⟮X⟯\n  | ⟨p⟩, ⟨q⟩ => ⟨p + q⟩\n\ninstance : Add K⟮X⟯ :=\n  ⟨RatFunc.add⟩\n\ntheorem ofFractionRing_add (p q : FractionRing K[X]) :\n    ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q :=\n  (add_def _ _).symm\n\n/-- Subtraction of rational functions. -/\nprotected irreducible_def sub : K⟮X⟯ → K⟮X⟯ → K⟮X⟯\n  | ⟨p⟩, ⟨q⟩ => ⟨p - q⟩\n\ninstance : Sub K⟮X⟯ :=\n  ⟨RatFunc.sub⟩\n\ntheorem ofFractionRing_sub (p q : FractionRing K[X]) :\n    ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q :=\n  (sub_def _ _).symm\n\n/-- Additive inverse of a rational function. -/\nprotected irreducible_def neg : K⟮X⟯ → K⟮X⟯\n  | ⟨p⟩ => ⟨-p⟩\n\ninstance : Neg K⟮X⟯ :=\n  ⟨RatFunc.neg⟩\n\ntheorem ofFractionRing_neg (p : FractionRing K[X]) :\n    ofFractionRing (-p) = -ofFractionRing p :=\n  (neg_def _).symm\n\n/-- The multiplicative unit of rational functions. -/\nprotected irreducible_def one : K⟮X⟯ :=\n  ⟨1⟩\n\ninstance : One K⟮X⟯ :=\n  ⟨RatFunc.one⟩\n\ntheorem ofFractionRing_one : (ofFractionRing 1 : K⟮X⟯) = 1 :=\n  one_def.symm\n\n/-- Multiplication of rational functions. -/\nprotected irreducible_def mul : K⟮X⟯ → K⟮X⟯ → K⟮X⟯\n  | ⟨p⟩, ⟨q⟩ => ⟨p * q⟩\n\ninstance : Mul K⟮X⟯ :=\n  ⟨RatFunc.mul⟩\n\ntheorem ofFractionRing_mul (p q : FractionRing K[X]) :\n    ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q :=\n  (mul_def _ _).symm\n\nsection IsDomain\n\nvariable [IsDomain K]\n\n/-- Division of rational functions. -/\nprotected irreducible_def div : K⟮X⟯ → K⟮X⟯ → K⟮X⟯\n  | ⟨p⟩, ⟨q⟩ => ⟨p / q⟩\n\ninstance : Div K⟮X⟯ :=\n  ⟨RatFunc.div⟩\n\ntheorem ofFractionRing_div (p q : FractionRing K[X]) :\n    ofFractionRing (p / q) = ofFractionRing p / ofFractionRing q :=\n  (div_def _ _).symm\n\n/-- Multiplicative inverse of a rational function. -/\nprotected irreducible_def inv : K⟮X⟯ → K⟮X⟯\n  | ⟨p⟩ => ⟨p⁻¹⟩\n\ninstance : Inv K⟮X⟯ :=\n  ⟨RatFunc.inv⟩\n\ntheorem ofFractionRing_inv (p : FractionRing K[X]) :\n    ofFractionRing p⁻¹ = (ofFractionRing p)⁻¹ :=\n  (inv_def _).symm\n\n-- Auxiliary lemma for the `Field` instance\ntheorem mul_inv_cancel : ∀ {p : K⟮X⟯}, p ≠ 0 → p * p⁻¹ = 1\n  | ⟨p⟩, h => by\n    have : p ≠ 0 := fun hp => h <| by rw [hp, ofFractionRing_zero]\n    simpa only [← ofFractionRing_inv, ← ofFractionRing_mul, ← ofFractionRing_one,\n        ofFractionRing.injEq] using\n      mul_inv_cancel₀ this\n\nend IsDomain\n\nsection SMul\n\nvariable {R : Type*}\n\n/-- Scalar multiplication of rational functions. -/\nprotected irreducible_def smul [SMul R (FractionRing K[X])] : R → K⟮X⟯ → K⟮X⟯\n  | r, ⟨p⟩ => ⟨r • p⟩\n\ninstance [SMul R (FractionRing K[X])] : SMul R K⟮X⟯ :=\n  ⟨RatFunc.smul⟩\n\ntheorem ofFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : FractionRing K[X]) :\n    ofFractionRing (c • p) = c • ofFractionRing p :=\n  (smul_def _ _).symm\n\nTarget:\ntheorem toFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : K⟮X⟯) :\n    toFractionRing (c • p) = c • toFractionRing p :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"de9878b2bdbca9d755e0906ee6cc2f48bbd71f044a7ac728855c52af26d751a4","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/RatFunc","family_id":"tofractionring_smul","file_id":"mathlib/Mathlib/FieldTheory/RatFunc/Basic.lean","sample_id":"5b0c69fc729fb329df53e747032d70f91d7a865fc26c9ee5566a2b68840871a7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"06f47103349d9ae4b5fa9c701936477e3308e59432a0c09cdb18eb8a9e2de388","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ea19c96b1833a290f3ad4a2c8868771d2cf2aee1459ba1b113ef3ef0cf9c4a4c","source_sha256":"0333a24d4ec160f28f2400f09ad2bc2ec0955c0c7bd92de1261f4a4e19fe5936","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  contrapose!; exact parts_eq_empty_iff","hard_negative":false,"metrics":{"chosen_tokens":6,"rejected_tokens":2,"token_jaccard":0.142857,"token_length_ratio":0.333333},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"53df40ab2673253f5c170c53ccd444d6ff208859f1e5d21d2b9acb337a68a58b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\npublic import Mathlib.Data.Finset.Pairwise\npublic import Mathlib.Data.Finset.Preimage\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Data.Setoid.Basic\npublic import Mathlib.Order.Atoms\npublic import Mathlib.Order.SupIndep\n\nNamespace:\nFinpartition\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Finite partitions\n\nIn this file, we define finite partitions. A finpartition of `a : α` is a finite set of pairwise\ndisjoint parts `parts : Finset α` which does not contain `⊥` and whose supremum is `a`.\n\nFinpartitions of a finset are at the heart of Szemerédi's regularity lemma. They are also studied\npurely order theoretically in Sperner theory.\n\n## Constructions\n\nWe provide many ways to build finpartitions:\n* `Finpartition.ofErase`: Builds a finpartition by erasing `⊥` for you.\n* `Finpartition.ofSubset`: Builds a finpartition from a subset of the parts of a previous\n  finpartition.\n* `Finpartition.empty`: The empty finpartition of `⊥`.\n* `Finpartition.indiscrete`: The indiscrete, aka trivial, aka pure, finpartition made of a single\n  part.\n* `Finpartition.discrete`: The discrete finpartition of `s : Finset α` made of singletons.\n* `Finpartition.toSubtype`: Turns a finpartition of a type to one of a subtype.\n* `Finpartition.bind`: Puts together the finpartitions of the parts of a finpartition into a new\n  finpartition.\n* `Finpartition.extend`: Extends a finpartition of `a` to a finpartition of `a ⊔ b` by adding `b`\n  as a new part.\n* `Finpartition.extendOfLE`: Extends a finpartition of `a` to a finpartition of `b` when `a ≤ b`,\n  by adding `b \\ a` as a new part (if nonempty).\n* `Finpartition.restrict`: Restricts a finpartition of `a` to `b` where `b ≤ a` by intersecting\n  each part with `b`.\n* `Finpartition.ofPairwiseDisjoint`: Builds a finpartition from a finset `parts` of pairwise\n  disjoint elements.\n* `Finpartition.combine`: Combines a family of partitions of pairwise disjoint elements into a\n  partition of their sup.\n* `Finpartition.ofExistsUnique`: Builds a finpartition from a collection of parts such that each\n  element is in exactly one part.\n* `Finpartition.ofSetoid`: With `Fintype α`, constructs the finpartition of `univ : Finset α`\n  induced by the equivalence classes of `s : Setoid α`.\n* `Finpartition.atomise`: Makes a finpartition of `s : Finset α` by breaking `s` along all finsets\n  in `F : Finset (Finset α)`. Two elements of `s` belong to the same part iff they belong to the\n  same elements of `F`.\n\n`Finpartition.indiscrete` and `Finpartition.bind` together form the monadic structure of\n`Finpartition`.\n\n## Implementation notes\n\nForbidding `⊥` as a part follows mathematical tradition and is a pragmatic choice concerning\noperations on `Finpartition`. Not caring about `⊥` being a part or not breaks extensionality (it's\nnot because the parts of `P` and the parts of `Q` have the same elements that `P = Q`). Enforcing\n`⊥` to be a part makes `Finpartition.bind` uglier and doesn't rid us of the need of\n`Finpartition.ofErase`.\n\n## TODO\n\nThe order is the wrong way around to make `Finpartition a` a graded order. Is it bad to depart from\nthe literature and turn the order around?\n\nThe specialisation to `Finset α` could be generalised to atomistic orders.\n-/\n\n@[expose] public section\n\n\nopen Finset Function\n\nvariable {α : Type*}\n\n/-- A finite partition of `a : α` is a pairwise disjoint finite set of elements whose supremum is\n`a`. We forbid `⊥` as a part. -/\n@[ext]\nstructure Finpartition [Lattice α] [OrderBot α] (a : α) where\n  /-- The elements of the finite partition of `a` -/\n  parts : Finset α\n  /-- The partition is supremum-independent -/\n  protected supIndep : parts.SupIndep id\n  /-- The supremum of the partition is `a` -/\n  sup_parts : parts.sup id = a\n  /-- No element of the partition is bottom -/\n  bot_notMem : ⊥ ∉ parts\n  deriving DecidableEq\n\nnamespace Finpartition\n\nsection Lattice\n\nvariable [Lattice α] [OrderBot α]\n\n/-- A `Finpartition` constructor which does not insist on `⊥` not being a part. -/\n@[simps]\ndef ofErase [DecidableEq α] {a : α} (parts : Finset α) (sup_indep : parts.SupIndep id)\n    (sup_parts : parts.sup id = a) : Finpartition a where\n  parts := parts.erase ⊥\n  supIndep := sup_indep.subset (erase_subset _ _)\n  sup_parts := (sup_erase_bot _).trans sup_parts\n  bot_notMem := notMem_erase _ _\n\n/-- A `Finpartition` constructor from a bigger existing finpartition. -/\n@[simps]\ndef ofSubset {a b : α} (P : Finpartition a) {parts : Finset α} (subset : parts ⊆ P.parts)\n    (sup_parts : parts.sup id = b) : Finpartition b :=\n  { parts := parts\n    supIndep := P.supIndep.subset subset\n    sup_parts := sup_parts\n    bot_notMem := fun h ↦ P.bot_notMem (subset h) }\n\nlemma sum_ofSubset_eq_sum {a b : α} (P : Finpartition a) {parts : Finset α}\n    (subset : parts ⊆ P.parts) (sup_parts : parts.sup id = b)\n    {X : Type*} [AddCommMonoid X] (f : α → X) (hf : ∀ p ∈ P.parts, p ∉ parts → f p = 0) :\n    ∑ p ∈ (P.ofSubset subset sup_parts).parts, f p = ∑ p ∈ P.parts, f p :=\n  Finset.sum_subset subset hf\n\n/-- Changes the type of a finpartition to an equal one. -/\n@[simps]\ndef copy {a b : α} (P : Finpartition a) (h : a = b) : Finpartition b where\n  parts := P.parts\n  supIndep := P.supIndep\n  sup_parts := h ▸ P.sup_parts\n  bot_notMem := P.bot_notMem\n\n/-- Transfer a finpartition over an order isomorphism. -/\ndef map {β : Type*} [Lattice β] [OrderBot β] {a : α} (e : α ≃o β) (P : Finpartition a) :\n    Finpartition (e a) where\n  parts := P.parts.map e\n  supIndep u hu _ hb hbu _ hx hxu := by\n    rw [← map_symm_subset] at hu\n    simp only [mem_map_equiv] at hb\n    have := P.supIndep hu hb (by simp [hbu]) (map_rel e.symm hx) ?_\n    · rw [← e.symm.map_bot] at this\n      exact e.symm.map_rel_iff.mp this\n    · convert! e.symm.map_rel_iff.mpr hxu\n      rw [map_finset_sup, sup_map]\n      rfl\n  sup_parts := by simp [← P.sup_parts]\n  bot_notMem := by\n    rw [mem_map_equiv]\n    convert! P.bot_notMem\n    exact e.symm.map_bot\n\n@[simp]\ntheorem parts_map {β : Type*} [Lattice β] [OrderBot β] {a : α} {e : α ≃o β} {P : Finpartition a} :\n    (P.map e).parts = P.parts.map e := rfl\n\nvariable (α)\n\n/-- The empty finpartition. -/\n@[simps]\nprotected def empty : Finpartition (⊥ : α) where\n  parts := ∅\n  supIndep := supIndep_empty _\n  sup_parts := Finset.sup_empty\n  bot_notMem := notMem_empty ⊥\n\ninstance : Inhabited (Finpartition (⊥ : α)) :=\n  ⟨Finpartition.empty α⟩\n\n@[simp]\ntheorem default_eq_empty : (default : Finpartition (⊥ : α)) = Finpartition.empty α :=\n  rfl\n\nvariable {α} {a : α}\n\n/-- The finpartition in one part, aka indiscrete finpartition. -/\n@[simps]\ndef indiscrete (ha : a ≠ ⊥) : Finpartition a where\n  parts := {a}\n  supIndep := supIndep_singleton _ _\n  sup_parts := Finset.sup_singleton\n  bot_notMem h := ha (mem_singleton.1 h).symm\n\nvariable (P : Finpartition a)\n\nprotected theorem le {b : α} (hb : b ∈ P.parts) : b ≤ a :=\n  (le_sup hb).trans P.sup_parts.le\n\ntheorem ne_bot {b : α} (hb : b ∈ P.parts) : b ≠ ⊥ := by\n  intro h\n  refine P.bot_notMem (?_)\n  rw [h] at hb\n  exact hb\n\nprotected theorem disjoint : (P.parts : Set α).PairwiseDisjoint id :=\n  P.supIndep.pairwiseDisjoint\n\nvariable {P}\n\n@[simp]\ntheorem parts_eq_empty_iff : P.parts = ∅ ↔ a = ⊥ := by\n  simp_rw [← P.sup_parts]\n  refine ⟨fun h ↦ ?_, fun h ↦ eq_empty_iff_forall_notMem.2 fun b hb ↦ P.bot_notMem ?_⟩\n  · rw [h]\n    exact Finset.sup_empty\n  · rwa [← le_bot_iff.1 ((le_sup hb).trans h.le)]\n\n@[simp]\n\nTarget:\ntheorem parts_nonempty_iff : P.parts.Nonempty ↔ a ≠ ⊥ :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Partition","family_id":"parts_nonempty_iff","file_id":"mathlib/Mathlib/Order/Partition/Finpartition.lean","sample_id":"ea19c96b1833a290f3ad4a2c8868771d2cf2aee1459ba1b113ef3ef0cf9c4a4c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"82e688aad755eac6f657be89199e8e6fa7e63dacb10bb688605e04b2c9309872","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"63fa03af2984e0af65d6d9483e8e9d588ac0a5984b297b2cbfd8d08c63366ee8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d3b554a7490cfc938221d5d7fff9423a8ad834bad53b65c4db2638f83535b57e","source_sha256":"e2806df73ecfc81f91bdaba3be881cdfeef7117f485c371a23cf9b886b2f1661","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    suffices IsSquare (-x) → x = 0 by aesop\n    exact fun hc ↦ le_antisymm (nonpos_of_neg_nonneg (IsSquare.nonneg hc)) h\n  mpr := IsSquare.nonneg\n\nalias ⟨_root_.IsSquare.of_nonneg, _⟩ := nonneg_iff_isSquare","hard_negative":true,"metrics":{"chosen_tokens":45,"rejected_tokens":2,"token_jaccard":0.032258,"token_length_ratio":0.044444},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"54207e52c7379db8f6b3476978c04053753edc60d1b4596f809603dfbb42caac","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Degree.Domain\npublic import Mathlib.Algebra.Polynomial.Eval.Defs\npublic import Mathlib.Algebra.Ring.Semireal.Defs\npublic import Mathlib.Tactic.LinearCombination\n\nNamespace:\nIsRealClosed\n\nLocal context:\n/-\nCopyright (c) 2025 Artie Khovanov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Artie Khovanov\n-/\n/-!\n# Real Closed Field\n\nA field `R` is real closed if all of the following hold:\n1. `R` is real (that is, `-1` is not a sum of squares in `R`).\n2. for every `x` in `R`, one of `x` or `-x` is a square.\n3. every odd-degree polynomial over `R` has a root in `R`.\n\nA real closed field is an algebraic generalisation of the real numbers.\n\nIn this file we define real closed fields and prove some of their properties.\n\nTODO (Artie Khovanov) : equivalent conditions for a real field to be real closed\nTODO (Artie Khovanov) : real numbers, real algebraic numbers, hyperreals form a real closed field\n\n## Main Definitions\n\n- `IsRealClosed R` is the typeclass saying `R` is a real closed field.\n\n## Tags\n\nreal closed, rcf\n\n-/\n\npublic section\n\nopen Polynomial\n\n/--\nA field `R` is real closed if all of the following hold:\n1. `R` is real (that is, `-1` is not a sum of squares in `R`).\n2. for every `x` in `R`, one of `x` or `-x` is a square.\n3. every odd-degree polynomial over `R` has a root in `R`.\n-/\nclass IsRealClosed (R : Type*) [Field R] : Prop extends IsSemireal R where\n  isSquare_or_isSquare_neg (x : R) : IsSquare x ∨ IsSquare (-x)\n  exists_isRoot_of_odd_natDegree {f : R[X]} (hf : Odd f.natDegree) : ∃ x, f.IsRoot x\n\nattribute [aesop 90% forward] IsRealClosed.isSquare_or_isSquare_neg\n\nnamespace IsRealClosed\n\nuniverse u\n\nvariable {R : Type u} [Field R]\n\ntheorem of_linearOrderedField [LinearOrder R] [IsStrictOrderedRing R]\n    (isSquare_of_nonneg : ∀ {x : R}, 0 ≤ x → IsSquare x)\n    (exists_isRoot_of_odd_natDegree : ∀ {f : R[X]}, Odd f.natDegree → ∃ x, f.IsRoot x) :\n    IsRealClosed R where\n  isSquare_or_isSquare_neg {x} := by\n    rcases le_total x 0 with (neg | pos)\n    · exact .inr <| isSquare_of_nonneg (neg_nonneg_of_nonpos neg)\n    · exact .inl <| isSquare_of_nonneg pos\n  exists_isRoot_of_odd_natDegree := exists_isRoot_of_odd_natDegree\n\nvariable [IsRealClosed R]\n\n@[aesop 50%]\ntheorem _root_.IsSquare.of_not_isSquare_neg {x : R} (hx : ¬ IsSquare (-x)) : IsSquare x := by aesop\n\n@[aesop 80%]\ntheorem isSquare_neg_of_not_isSquare {x : R} (hx : ¬ IsSquare x) : IsSquare (-x) := by aesop\n\ntheorem exists_eq_pow_of_odd (x : R) {n : ℕ} (hn : Odd n) : ∃ r, x = r ^ n := by\n  rcases exists_isRoot_of_odd_natDegree (f := X ^ n - C x) (by simp [hn]) with ⟨r, hr⟩\n  exact ⟨r, by linear_combination - (by simpa using hr : r ^ n - x = 0)⟩\n\ntheorem exists_eq_zpow_of_odd (x : R) {k : ℤ} (hk : Odd k) : ∃ r, x = r ^ k := by\n  rcases k.eq_nat_or_neg with ⟨n, rfl | rfl⟩\n  · simpa using exists_eq_pow_of_odd x (by simpa using hk)\n  · rcases exists_eq_pow_of_odd x (by simpa using hk) with ⟨r, hr⟩\n    exact ⟨r⁻¹, by simpa using hr⟩\n\ntheorem exists_eq_pow_of_isSquare {x : R} (hx : IsSquare x) {n : ℕ} (hn : n ≠ 0) :\n    ∃ r, x = r ^ n := by\n  induction n using Nat.strong_induction_on generalizing x with\n  | h n ih =>\n    rcases Nat.even_or_odd n with (even | odd)\n    · rcases even with ⟨m, hm⟩\n      rcases hx with ⟨s, hs⟩\n      rcases isSquare_or_isSquare_neg s with (h | h) <;>\n        rcases ih m (by lia) h (by lia) with ⟨r, hr⟩ <;>\n        exact ⟨r, by simp [hm, pow_add, ← hr, hs]⟩\n    · exact exists_eq_pow_of_odd x odd\n\ntheorem exists_eq_zpow_of_isSquare {x : R} (hx : IsSquare x) {k : ℤ} (hk : k ≠ 0) :\n    ∃ r, x = r ^ k := by\n  rcases k.eq_nat_or_neg with ⟨n, rfl | rfl⟩\n  · simpa using exists_eq_pow_of_isSquare hx (by simpa using hk)\n  · rcases exists_eq_pow_of_isSquare hx (by simpa using hk) with ⟨r, hr⟩\n    exact ⟨r⁻¹, by simpa using hr⟩\n\nsection LinearOrderedField\n\nvariable [LinearOrder R] [IsStrictOrderedRing R]\n\nTarget:\ntheorem nonneg_iff_isSquare {x : R} : 0 ≤ x ↔ IsSquare x where\n  mp h :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_d3b554a7490c","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"2f24929c52e2499bdb8c260c48f34f98dcf32c92bf85d06536fa8e20bd1afe07","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/IsRealClosed","family_id":"nonneg_iff_issquare","file_id":"mathlib/Mathlib/FieldTheory/IsRealClosed/Basic.lean","sample_id":"d3b554a7490cfc938221d5d7fff9423a8ad834bad53b65c4db2638f83535b57e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f4a24f2bb7a6f0211b6b657b7e7dccc3a2dcf7c7a859099b25ae3a21d107e6f6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"955e34b4be7edd56dadecaa66b57be2059d1e36951adf123ec19009bee070aac","source_sha256":"7d247632b790bfc237d79d3c040fb7f21d9cff00d190a67e120ccb73056460a6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only [← hammingDist_zero_right, hf]\n    using! hammingDist_comp_le_hammingDist f (y := fun _ ↦ 0)","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":3,"token_jaccard":0.043478,"token_length_ratio":0.142857},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"54259e3ecfb005214b7de48f5f4540bea5b1d90ad5807fc619aa08d9dd82d3ae","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Group.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Wrenna Robson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Wrenna Robson\n-/\n/-!\n# Hamming spaces\n\nThe Hamming metric counts the number of places two members of a (finite) Pi type\ndiffer. The Hamming norm is the same as the Hamming metric over additive groups, and\ncounts the number of places a member of a (finite) Pi type differs from zero.\n\nThis is a useful notion in various applications, but in particular it is relevant\nin coding theory, in which it is fundamental for defining the minimum distance of a\ncode.\n\n## Main definitions\n* `hammingDist x y`: the Hamming distance between `x` and `y`, the number of entries which differ.\n* `hammingNorm x`: the Hamming norm of `x`, the number of non-zero entries.\n* `Hamming β`: a type synonym for `Π i, β i` with `dist` and `norm` provided by the above.\n* `Hamming.toHamming`, `Hamming.ofHamming`: functions for casting between `Hamming β` and\n  `Π i, β i`.\n* the Hamming norm forms a normed group on `Hamming β`.\n-/\n\n@[expose] public section\n\n\nsection HammingDistNorm\n\nopen Finset Function\n\nvariable {α ι : Type*} {β : ι → Type*} [Fintype ι] [∀ i, DecidableEq (β i)]\nvariable {γ : ι → Type*} [∀ i, DecidableEq (γ i)]\n\n/-- The Hamming distance function to the naturals. -/\ndef hammingDist (x y : ∀ i, β i) : ℕ := #{i | x i ≠ y i}\n\n/-- Corresponds to `dist_self`. -/\n@[simp]\ntheorem hammingDist_self (x : ∀ i, β i) : hammingDist x x = 0 := by\n  rw [hammingDist, card_eq_zero, filter_eq_empty_iff]\n  exact fun _ _ H => H rfl\n\n-- TODO: this seems unnecessary.\n/-- Corresponds to `dist_nonneg`. -/\ntheorem hammingDist_nonneg {x y : ∀ i, β i} : 0 ≤ hammingDist x y :=\n  zero_le\n\n/-- Corresponds to `dist_comm`. -/\ntheorem hammingDist_comm (x y : ∀ i, β i) : hammingDist x y = hammingDist y x := by\n  simp_rw [hammingDist, ne_comm]\n\n/-- Corresponds to `dist_triangle`. -/\ntheorem hammingDist_triangle (x y z : ∀ i, β i) :\n    hammingDist x z ≤ hammingDist x y + hammingDist y z := by\n  classical\n    unfold hammingDist\n    refine le_trans (card_mono ?_) (card_union_le _ _)\n    rw [← filter_or]\n    exact monotone_filter_right _ fun i _ h ↦ (h.ne_or_ne _).imp_right Ne.symm\n\n/-- Corresponds to `dist_triangle_left`. -/\ntheorem hammingDist_triangle_left (x y z : ∀ i, β i) :\n    hammingDist x y ≤ hammingDist z x + hammingDist z y := by\n  rw [hammingDist_comm z]\n  exact hammingDist_triangle _ _ _\n\n/-- Corresponds to `dist_triangle_right`. -/\ntheorem hammingDist_triangle_right (x y z : ∀ i, β i) :\n    hammingDist x y ≤ hammingDist x z + hammingDist y z := by\n  rw [hammingDist_comm y]\n  exact hammingDist_triangle _ _ _\n\n/-- Corresponds to `swap_dist`. -/\ntheorem swap_hammingDist : swap (@hammingDist _ β _ _) = hammingDist := by\n  funext x y\n  exact hammingDist_comm _ _\n\n/-- Corresponds to `eq_of_dist_eq_zero`. -/\ntheorem eq_of_hammingDist_eq_zero {x y : ∀ i, β i} : hammingDist x y = 0 → x = y := by\n  simp_rw [hammingDist, card_eq_zero, filter_eq_empty_iff, Classical.not_not, funext_iff, mem_univ,\n    forall_true_left, imp_self]\n\n/-- Corresponds to `dist_eq_zero`. -/\n@[simp]\ntheorem hammingDist_eq_zero {x y : ∀ i, β i} : hammingDist x y = 0 ↔ x = y :=\n  ⟨eq_of_hammingDist_eq_zero, fun H => by\n    rw [H]\n    exact hammingDist_self _⟩\n\n/-- Corresponds to `zero_eq_dist`. -/\n@[simp]\ntheorem hamming_zero_eq_dist {x y : ∀ i, β i} : 0 = hammingDist x y ↔ x = y := by\n  rw [eq_comm, hammingDist_eq_zero]\n\n/-- Corresponds to `dist_ne_zero`. -/\ntheorem hammingDist_ne_zero {x y : ∀ i, β i} : hammingDist x y ≠ 0 ↔ x ≠ y :=\n  hammingDist_eq_zero.not\n\n/-- Corresponds to `dist_pos`. -/\n@[simp]\ntheorem hammingDist_pos {x y : ∀ i, β i} : 0 < hammingDist x y ↔ x ≠ y := by\n  rw [← hammingDist_ne_zero, iff_not_comm, not_lt, Nat.le_zero]\n\ntheorem hammingDist_lt_one {x y : ∀ i, β i} : hammingDist x y < 1 ↔ x = y := by\n  rw [Nat.lt_one_iff, hammingDist_eq_zero]\n\ntheorem hammingDist_le_card_fintype {x y : ∀ i, β i} : hammingDist x y ≤ Fintype.card ι :=\n  card_le_univ _\n\ntheorem hammingDist_comp_le_hammingDist (f : ∀ i, γ i → β i) {x y : ∀ i, γ i} :\n    hammingDist (fun i => f i (x i)) (fun i => f i (y i)) ≤ hammingDist x y := by\n  dsimp [hammingDist]; gcongr; simp +contextual\n\ntheorem hammingDist_comp (f : ∀ i, γ i → β i) {x y : ∀ i, γ i} (hf : ∀ i, Injective (f i)) :\n    hammingDist (fun i => f i (x i)) (fun i => f i (y i)) = hammingDist x y :=\n  le_antisymm (hammingDist_comp_le_hammingDist _) <| by dsimp [hammingDist]; gcongr; exact @hf _ _ _\n\ntheorem hammingDist_smul_le_hammingDist [∀ i, SMul α (β i)] {k : α} {x y : ∀ i, β i} :\n    hammingDist (k • x) (k • y) ≤ hammingDist x y :=\n  hammingDist_comp_le_hammingDist fun i => (k • · : β i → β i)\n\n/-- Corresponds to `dist_smul` with the discrete norm on `α`. -/\ntheorem hammingDist_smul [∀ i, SMul α (β i)] {k : α} {x y : ∀ i, β i}\n    (hk : ∀ i, IsSMulRegular (β i) k) : hammingDist (k • x) (k • y) = hammingDist x y :=\n  hammingDist_comp (fun i => (k • · : β i → β i)) hk\n\nsection Zero\n\nvariable [∀ i, Zero (β i)] [∀ i, Zero (γ i)]\n\n/-- The Hamming weight function to the naturals. -/\ndef hammingNorm (x : ∀ i, β i) : ℕ := #{i | x i ≠ 0}\n\n/-- Corresponds to `dist_zero_right`. -/\n@[simp]\ntheorem hammingDist_zero_right (x : ∀ i, β i) : hammingDist x 0 = hammingNorm x :=\n  rfl\n\n/-- Corresponds to `dist_zero_left`. -/\n@[simp]\ntheorem hammingDist_zero_left : hammingDist (0 : ∀ i, β i) = hammingNorm :=\n  funext fun x => by rw [hammingDist_comm, hammingDist_zero_right]\n\n-- TODO: this seems unnecessary.\n/-- Corresponds to `norm_nonneg`. -/\ntheorem hammingNorm_nonneg {x : ∀ i, β i} : 0 ≤ hammingNorm x :=\n  zero_le\n\n/-- Corresponds to `norm_zero`. -/\n@[simp]\ntheorem hammingNorm_zero : hammingNorm (0 : ∀ i, β i) = 0 :=\n  hammingDist_self _\n\n/-- Corresponds to `norm_eq_zero`. -/\n@[simp]\ntheorem hammingNorm_eq_zero {x : ∀ i, β i} : hammingNorm x = 0 ↔ x = 0 :=\n  hammingDist_eq_zero\n\n/-- Corresponds to `norm_ne_zero_iff`. -/\ntheorem hammingNorm_ne_zero_iff {x : ∀ i, β i} : hammingNorm x ≠ 0 ↔ x ≠ 0 :=\n  hammingNorm_eq_zero.not\n\n/-- Corresponds to `norm_pos_iff`. -/\n@[simp]\ntheorem hammingNorm_pos_iff {x : ∀ i, β i} : 0 < hammingNorm x ↔ x ≠ 0 :=\n  hammingDist_pos\n\ntheorem hammingNorm_lt_one {x : ∀ i, β i} : hammingNorm x < 1 ↔ x = 0 :=\n  hammingDist_lt_one\n\ntheorem hammingNorm_le_card_fintype {x : ∀ i, β i} : hammingNorm x ≤ Fintype.card ι :=\n  hammingDist_le_card_fintype\n\nTarget:\ntheorem hammingNorm_comp_le_hammingNorm (f : ∀ i, γ i → β i) {x : ∀ i, γ i} (hf : ∀ i, f i 0 = 0) :\n    (hammingNorm fun i => f i (x i)) ≤ hammingNorm x :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"InformationTheory","family_id":"hammingnorm_comp_le_hammingnorm","file_id":"mathlib/Mathlib/InformationTheory/Hamming.lean","sample_id":"955e34b4be7edd56dadecaa66b57be2059d1e36951adf123ec19009bee070aac"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0c3061002fd6080e1b0c2c6f335b3c8a50d21cd02337f3272c2365c7c29242da","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0ba20bcd7c6a0c24655c4caacb4b81458510a3a5d34649a2bdac4f0ecad7697b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1a320ce7ba94d641553130a5996d892f469c6c053d11bc8e9c32a512ac9b6215","source_sha256":"88c666c6d936856962069aacdb5158224b504be63ef88b462fdb4de48e42b813","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (NeZero.ne n)\n  intro i\n  simpa using (h i).injective.ne (by decide : (0 : Fin 3) ≠ 1)","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":5,"token_jaccard":0.0625,"token_length_ratio":0.119048},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"54811d683d74bd35a80c35c2a04d342258fb9658f8fa224bd40aeedc81e73173","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Convex.Between\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Tactic.Continuity\n\nNamespace:\nPolygon\n\nLocal context:\n/-\nCopyright (c) 2026 A. M. Berns. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: A. M. Berns\n-/\n/-!\n# Polygons\n\nThis file defines polygons in affine spaces.\nFor the special case `n = 3`, an interconversion is provided with `Affine.Triangle`.\n\n## Main definitions\n\n* `Polygon P n`: A polygon with `n` vertices in a type `P`.\n\n-/\n\n@[expose] public section\n\nopen Set\n\n/-- A polygon with `n` vertices in a type `P`. -/\nstructure Polygon (P : Type*) (n : ℕ) where\n  /-- The vertices of the polygon, indexed by `Fin n`. -/\n  vertices : Fin n → P\n\nnamespace Polygon\n\nvariable {R V P : Type*} {n : ℕ}\n\n/-- A coercion to function so that vertices can\nbe written as `poly i` instead of `poly.vertices i` -/\ninstance : CoeFun (Polygon P n) (fun _ => Fin n → P) where\n  coe := Polygon.vertices\n\n/-- A polygon has nondegenerate edges if adjacent vertices are distinct. -/\ndef HasNondegenerateEdges (poly : Polygon P n) : Prop :=\n  ∀ i : Fin n, poly i ≠ poly (finRotate n i)\n\ntheorem HasNondegenerateEdges.two_le [NeZero n] {poly : Polygon P n}\n    (h : poly.HasNondegenerateEdges) : 2 ≤ n := by\n  by_contra! hlt\n  interval_cases n\n  · simp_all only [neZero_zero_iff_false]\n  · exact h 0 (by simp)\n\nvariable [Ring R] [AddCommGroup V] [Module R V] [AddTorsor V P]\n\nvariable (R) in\n/-- The `i`-th edge as an affine map `R →ᵃ[R] P`. -/\ndef edgePath (poly : Polygon P n) (i : Fin n) : R →ᵃ[R] P :=\n  AffineMap.lineMap (poly i) (poly (finRotate n i))\n\nvariable (R) in\n/-- The `i`-th edge as a set of points using an `affineSegment`. -/\ndef edgeSet [PartialOrder R] (poly : Polygon P n) (i : Fin n) : Set P :=\n  affineSegment R (poly i) (poly (finRotate n i))\n\nvariable (R) in\n/-- The `edgeSet` is equivalent to the image of the `edgePath`. -/\ntheorem edgeSet_eq_image_edgePath [PartialOrder R] (poly : Polygon P n) (i : Fin n) :\n    poly.edgeSet R i = poly.edgePath R i '' Icc (0 : R) 1 := rfl\n\nvariable (R) in\n/-- The boundary of the polygon is the union of all its edges. -/\ndef boundary [PartialOrder R] (poly : Polygon P n) : Set P :=\n  ⋃ i, poly.edgeSet R i\n\nvariable (R) in\n/-- A polygon has nondegenerate vertices if any three consecutive vertices\nare affinely independent. -/\ndef HasNondegenerateVertices [NeZero n] (poly : Polygon P n) : Prop :=\n  ∀ i : Fin n, AffineIndependent R ![poly i, poly (i + 1), poly (i + 2)]\n\n/-- Polygons with nondegenerate vertices also have nondegenerate edges. -/\n\nTarget:\ntheorem HasNondegenerateVertices.hasNondegenerateEdges [NeZero n] [Nontrivial R]\n    {poly : Polygon P n}\n    (h : poly.HasNondegenerateVertices R) : poly.HasNondegenerateEdges :=\n\nProof body:\n","rejected":"by\n  exact HasNondegenerateVertices.hasNondegenerateEdges","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"9f2d654676167592250311d4b0a82e279b0df8ca10cddf034881b7271bd0eef0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/Polygon","family_id":"hasnondegeneratevertices","file_id":"mathlib/Mathlib/Geometry/Polygon/Basic.lean","sample_id":"1a320ce7ba94d641553130a5996d892f469c6c053d11bc8e9c32a512ac9b6215"},"split":"train"}
{"binding":{"chosen_proof_sha256":"01a9a5bebe8404a63c3e4dd6c8ed6a63ca7ab9f3512e93006d4e0432218d9a02","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c4b589473919a5f5307e8873d42d412d0380d063128d057e1cc374f50a3fe80d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d99fd6fb9456101e1d28de19e801f40e15b75e0d01ea3c483f56e794b32e60bf","source_sha256":"2ce531595998b973f239f8a83dbac8a4626afb2088d76e997a149cdb9eae73d2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let ⟨⟨κ, b⟩⟩ := Module.Free.exists_basis (R := R) (M := M)\n  rw [(equivFreeAlgebra b).toLinearEquiv.rank_eq, FreeAlgebra.rank_eq, mk_list_eq_sum_pow,\n    Basis.mk_eq_rank'' b]","hard_negative":false,"metrics":{"chosen_tokens":47,"rejected_tokens":51,"token_jaccard":0.896552,"token_length_ratio":1.085106},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"548bcfaee18e6dd3b4b8ee64866abc636af2388cf1a8ad327758a817c3dcc7d1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorAlgebra.Basic\npublic import Mathlib.LinearAlgebra.FreeAlgebra\n\nNamespace:\nTensorAlgebra\n\nLocal context:\n/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# A basis for `TensorAlgebra R M`\n\n## Main definitions\n\n* `TensorAlgebra.equivMonoidAlgebra b : TensorAlgebra R M ≃ₐ[R] FreeAlgebra R κ`:\n  the isomorphism given by a basis `b : Basis κ R M`.\n* `Basis.tensorAlgebra b : Basis (FreeMonoid κ) R (TensorAlgebra R M)`:\n  the basis on the tensor algebra given by a basis `b : Basis κ R M`.\n\n## Main results\n\n* `TensorAlgebra.instFreeModule`: the tensor algebra over `M` is free when `M` is\n* `TensorAlgebra.rank_eq`\n\n-/\n\n@[expose] public section\n\nopen Module\n\nnamespace TensorAlgebra\n\nuniverse uκ uR uM\nvariable {κ : Type uκ} {R : Type uR} {M : Type uM}\n\nsection CommSemiring\nvariable [CommSemiring R] [AddCommMonoid M] [Module R M]\n\n/-- A basis provides an algebra isomorphism with the free algebra, replacing each basis vector\nwith its index. -/\nnoncomputable def equivFreeAlgebra (b : Basis κ R M) :\n    TensorAlgebra R M ≃ₐ[R] FreeAlgebra R κ :=\n  AlgEquiv.ofAlgHom\n    (TensorAlgebra.lift _ (Finsupp.linearCombination _ (FreeAlgebra.ι _) ∘ₗ b.repr.toLinearMap))\n    (FreeAlgebra.lift _ (ι R ∘ b))\n    (by ext; simp)\n    (hom_ext <| b.ext fun i => by simp)\n\n@[simp]\nlemma equivFreeAlgebra_ι_apply (b : Basis κ R M) (i : κ) :\n    equivFreeAlgebra b (ι R (b i)) = FreeAlgebra.ι R i :=\n  (TensorAlgebra.lift_ι_apply _ _).trans <| by simp\n\n@[simp]\nlemma equivFreeAlgebra_symm_ι (b : Basis κ R M) (i : κ) :\n    (equivFreeAlgebra b).symm (FreeAlgebra.ι R i) = ι R (b i) :=\n  (equivFreeAlgebra b).toEquiv.symm_apply_eq.mpr <| equivFreeAlgebra_ι_apply b i |>.symm\n\n/-- A basis on `M` can be lifted to a basis on `TensorAlgebra R M` -/\n@[simps! repr_apply]\nnoncomputable def _root_.Module.Basis.tensorAlgebra (b : Basis κ R M) :\n    Basis (FreeMonoid κ) R (TensorAlgebra R M) :=\n  (FreeAlgebra.basisFreeMonoid R κ).map <| (equivFreeAlgebra b).symm.toLinearEquiv\n\n/-- `TensorAlgebra R M` is free when `M` is. -/\ninstance instModuleFree [Module.Free R M] : Module.Free R (TensorAlgebra R M) :=\n  let ⟨⟨_κ, b⟩⟩ := Module.Free.exists_basis (R := R) (M := M)\n  .of_basis b.tensorAlgebra\n\n/-- The `TensorAlgebra` of a free module over a commutative semiring with no zero-divisors has\nno zero-divisors. -/\ninstance instNoZeroDivisors [NoZeroDivisors R] [Module.Free R M] :\n    NoZeroDivisors (TensorAlgebra R M) :=\n  have ⟨⟨_, b⟩⟩ := ‹Module.Free R M›\n  (equivFreeAlgebra b).toMulEquiv.noZeroDivisors\n\nend CommSemiring\n\nsection CommRing\nvariable [CommRing R] [AddCommGroup M] [Module R M]\n\n/-- The `TensorAlgebra` of a free module over an integral domain is a domain. -/\ninstance instIsDomain [IsDomain R] [Module.Free R M] : IsDomain (TensorAlgebra R M) :=\n  NoZeroDivisors.to_isDomain _\n\nattribute [pp_with_univ] Cardinal.lift\n\nopen Cardinal in\n\nTarget:\nlemma rank_eq [Nontrivial R] [Module.Free R M] :\n    Module.rank R (TensorAlgebra R M) = Cardinal.lift.{uR} (sum fun n ↦ Module.rank R M ^ n) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  let ⟨⟨κ, b⟩⟩ := Module.Free.exists_basis (R := R) (M := M)\n  rw [(equivFreeAlgebra b).toLinearEquiv.rank_eq, FreeAlgebra.rank_eq, mk_list_eq_sum_pow,\n    Basis.mk_eq_rank'' b]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/TensorAlgebra","family_id":"rank_eq","file_id":"mathlib/Mathlib/LinearAlgebra/TensorAlgebra/Basis.lean","sample_id":"d99fd6fb9456101e1d28de19e801f40e15b75e0d01ea3c483f56e794b32e60bf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c8dca49869410f2bacfcd9d441c978765c739555c35836f4fba20bb9b5e2825f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6af85baa030133d662458b29f177a94490847771e8924b2d16e4895df43a5a47","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"34c67d0984704f562f600f9107ee26cdff7c7034435d16dae2284506bc3e8668","source_sha256":"837dba1e44eb02eb7837e8e7b72b59c277e85899454f45d3be66de44ecd98632","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases h : compare a b\n  · simpa using le_of_lt <| compare_lt_iff_lt.1 h\n  · simpa using le_of_eq <| compare_eq_iff_eq.1 h\n  · simpa using compare_gt_iff_gt.1 h","hard_negative":true,"metrics":{"chosen_tokens":34,"rejected_tokens":5,"token_jaccard":0.090909,"token_length_ratio":0.147059},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"54a2ae9b9ca155ce8342a440d78938aab02cdb6bacfbc58687c308e52f85bd11","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Classes.Order\npublic import Batteries.Tactic.Trans\npublic import Mathlib.Data.Ordering.Basic\npublic import Mathlib.Tactic.ExtendDoc\npublic import Mathlib.Tactic.Push.Attr\npublic import Mathlib.Tactic.Simps.Basic\npublic import Mathlib.Tactic.SplitIfs\npublic import Mathlib.Order.Defs.PartialOrder\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Orders\n\nDefines classes for linear orders and proves some basic lemmas about them.\n\nWe intentionally avoid using `grind` in this fundamental file to keep the proofs understandable,\nrather than hiding the reasoning behind automation.\n-/\n\n@[expose] public section\n\nvariable {α : Type*}\n\nsection LinearOrder\n\n/-!\n### Definition of `LinearOrder` and lemmas about types with a linear order\n-/\n\n/-- Default definition of `max`. -/\ndef maxDefault [LE α] [DecidableLE α] (a b : α) :=\n  if a ≤ b then b else a\n\n/-- Default definition of `min`. -/\ndef minDefault [LE α] [DecidableLE α] (a b : α) :=\n  if a ≤ b then a else b\n\n/-- This attempts to prove that a given instance of `compare` is equal to `compareOfLessAndEq` by\nintroducing the arguments and trying the following approaches in order:\n\n1. seeing if `rfl` works\n2. seeing if the `compare` at hand is nonetheless essentially `compareOfLessAndEq`, but, because of\n   implicit arguments, requires us to unfold the defs and split the `if`s in the definition of\n   `compareOfLessAndEq`\n3. seeing if we can split by cases on the arguments, then see if the defs work themselves out\n   (useful when `compare` is defined via a `match` statement, as it is for `Bool`)\n-/\nmacro \"compareOfLessAndEq_rfl\" : tactic =>\n  `(tactic| (intro a b; first | rfl |\n    (simp only [compare, compareOfLessAndEq]; split_ifs <;> rfl) |\n    (induction a <;> induction b <;> simp +decide only)))\n\n/-- A linear order is reflexive, transitive, antisymmetric and total relation `≤`.\nWe assume that every linear ordered type has decidable `(≤)`, `(<)`, and `(=)`. -/\nclass LinearOrder (α : Type*) extends PartialOrder α, Min α, Max α, Ord α where\n  /-- A linear order is total. -/\n  protected le_total (a b : α) : a ≤ b ∨ b ≤ a\n  /-- In a linearly ordered type, we assume the order relations are all decidable. -/\n  toDecidableLE : DecidableLE α\n  /-- In a linearly ordered type, we assume the order relations are all decidable. -/\n  toDecidableEq : DecidableEq α := @decidableEqOfDecidableLE _ _ toDecidableLE\n  /-- In a linearly ordered type, we assume the order relations are all decidable. -/\n  toDecidableLT : DecidableLT α := @decidableLTOfDecidableLE _ _ toDecidableLE\n  min := fun a b => if a ≤ b then a else b\n  max := fun a b => if a ≤ b then b else a\n  /-- The minimum function is equivalent to the one you get from `minOfLe`. -/\n  protected min_def : ∀ a b, min a b = if a ≤ b then a else b := by intros; rfl\n  /-- The minimum function is equivalent to the one you get from `maxOfLe`. -/\n  protected max_def : ∀ a b, max a b = if a ≤ b then b else a := by intros; rfl\n  compare a b := compareOfLessAndEq a b\n  /-- Comparison via `compare` is equal to the canonical comparison given decidable `<` and `=`. -/\n  compare_eq_compareOfLessAndEq : ∀ a b, compare a b = compareOfLessAndEq a b := by\n    compareOfLessAndEq_rfl\n\nattribute [to_dual existing] LinearOrder.toMax\n\nvariable [LinearOrder α] {a b c : α}\n\nattribute [instance_reducible, instance 900] LinearOrder.toDecidableLT\nattribute [instance_reducible, instance 900] LinearOrder.toDecidableLE\nattribute [instance_reducible, instance 900] LinearOrder.toDecidableEq\n\ninstance : Std.IsLinearOrder α where\n  le_total := LinearOrder.le_total\n\n@[to_dual self] lemma le_total : ∀ a b : α, a ≤ b ∨ b ≤ a := LinearOrder.le_total\n\n@[to_dual self] lemma le_of_not_ge : ¬a ≤ b → b ≤ a := (le_total a b).resolve_left\n@[to_dual self] lemma lt_of_not_ge (h : ¬b ≤ a) : a < b := lt_of_le_not_ge (le_of_not_ge h) h\n\n@[to_dual self] lemma lt_or_ge (a b : α) : a < b ∨ b ≤ a :=\n  if hba : b ≤ a then Or.inr hba else Or.inl <| lt_of_not_ge hba\n\n@[to_dual self] lemma le_or_gt (a b : α) : a ≤ b ∨ b < a := (lt_or_ge b a).symm\n\n@[to_dual gt_trichotomy]\nlemma lt_trichotomy (a b : α) : a < b ∨ a = b ∨ b < a :=\n  (lt_or_ge a b).imp_right (fun h ↦ (Decidable.lt_or_eq_of_le' h).symm)\n\n@[to_dual self]\nlemma le_of_not_gt (h : ¬b < a) : a ≤ b := (le_or_gt a b).resolve_right h\n\n@[to_dual gt_or_lt_of_ne]\nlemma lt_or_gt_of_ne (h : a ≠ b) : a < b ∨ b < a :=\n  (lt_trichotomy a b).imp_right (fun h' ↦ h'.resolve_left h)\n\n@[to_dual ne_iff_gt_or_lt]\nlemma ne_iff_lt_or_gt : a ≠ b ↔ a < b ∨ b < a := ⟨lt_or_gt_of_ne, (Or.elim · ne_of_lt ne_of_gt)⟩\n\n@[to_dual self] lemma lt_iff_not_ge : a < b ↔ ¬b ≤ a := ⟨not_le_of_gt, lt_of_not_ge⟩\n\n@[simp, push, to_dual self] lemma not_lt : ¬a < b ↔ b ≤ a := ⟨le_of_not_gt, not_lt_of_ge⟩\n@[simp, push, to_dual self] lemma not_le : ¬a ≤ b ↔ b < a := lt_iff_not_ge.symm\n\n@[to_dual eq_or_lt_of_not_gt]\nlemma eq_or_gt_of_not_lt (h : ¬a < b) : a = b ∨ b < a :=\n  if h₁ : a = b then Or.inl h₁ else Or.inr (lt_of_not_ge fun hge => h (lt_of_le_of_ne hge h₁))\n\n@[to_dual self]\ntheorem le_imp_le_of_lt_imp_lt {α β} [Preorder α] [LinearOrder β] {a b : α} {c d : β}\n    (H : d < c → b < a) (h : a ≤ b) : c ≤ d :=\n  le_of_not_gt fun h' => not_le_of_gt (H h') h\n\n@[grind =]\nlemma min_def (a b : α) : min a b = if a ≤ b then a else b := LinearOrder.min_def a b\n@[grind =]\nlemma max_def (a b : α) : max a b = if a ≤ b then b else a := LinearOrder.max_def a b\n\ntheorem min_ind {motive : α → Prop} (ha : a ≤ b → motive a) (hb : b ≤ a → motive b) :\n    motive (min a b) := by\n  rw [min_def]; split_ifs with h\n  exacts [ha h, hb (le_of_not_ge h)]\n\n@[to_dual existing (attr := elab_as_elim)]\ntheorem max_ind {motive : α → Prop} (ha : b ≤ a → motive a) (hb : a ≤ b → motive b) :\n    motive (max a b) := by\n  rw [max_def]; split_ifs with h\n  exacts [hb h, ha (le_of_not_ge h)]\n\n@[to_dual existing max_def]\ntheorem min_def' (a b : α) : min a b = if b ≤ a then b else a := by\n  obtain h | h | h := lt_trichotomy a b <;> simp [le_of_lt, not_le_of_gt, h, min_def]\n\n@[to_dual existing min_def]\ntheorem max_def' (a b : α) : max a b = if b ≤ a then a else b := by\n  obtain h | h | h := lt_trichotomy a b <;> simp [le_of_lt, not_le_of_gt, h, max_def]\n\n@[to_dual le_max_left]\nlemma min_le_left (a b : α) : min a b ≤ a := by\n  rw [min_def]\n  split_ifs with h <;> simp [h, le_of_not_ge]\n\n@[to_dual le_max_right]\nlemma min_le_right (a b : α) : min a b ≤ b := by\n  rw [min_def]\n  split_ifs with h <;> simp [h]\n\n@[to_dual max_le]\nlemma le_min (h₁ : c ≤ a) (h₂ : c ≤ b) : c ≤ min a b := by\n  rw [min_def]\n  split_ifs <;> assumption\n\n@[to_dual]\nlemma eq_min (h₁ : c ≤ a) (h₂ : c ≤ b) (h₃ : ∀ {d}, d ≤ a → d ≤ b → d ≤ c) : c = min a b :=\n  le_antisymm (le_min h₁ h₂) (h₃ (min_le_left a b) (min_le_right a b))\n\n@[to_dual]\nlemma min_comm (a b : α) : min a b = min b a :=\n  eq_min (min_le_right a b) (min_le_left a b) fun h₁ h₂ => le_min h₂ h₁\n\n@[to_dual]\nlemma min_assoc (a b c : α) : min (min a b) c = min a (min b c) :=\n  eq_min\n    (le_trans (min_le_left ..) (min_le_left ..))\n    (le_min (le_trans (min_le_left ..) (min_le_right ..)) (min_le_right ..))\n    (fun h₁ h₂ ↦\n      le_min (le_min h₁ (le_trans h₂ (min_le_left ..))) (le_trans h₂ (min_le_right ..)))\n\n@[to_dual]\nlemma min_left_comm (a b c : α) : min a (min b c) = min b (min a c) := by\n  rw [← min_assoc, min_comm a, min_assoc]\n\n@[to_dual (attr := simp)] lemma min_self (a : α) : min a a = a := by rw [min_def, ite_id]\n\n@[to_dual]\nlemma min_eq_left (h : a ≤ b) : min a b = a := (eq_min le_rfl h (fun h _ ↦ h)).symm\n\n@[to_dual]\nlemma min_eq_right (h : b ≤ a) : min a b = b := min_comm b a ▸ min_eq_left h\n\n@[to_dual] lemma min_eq_left_of_lt (h : a < b) : min a b = a := min_eq_left (le_of_lt h)\n@[to_dual] lemma min_eq_right_of_lt (h : b < a) : min a b = b := min_eq_right (le_of_lt h)\n\n@[to_dual max_lt]\nlemma lt_min (h₁ : a < b) (h₂ : a < c) : a < min b c := by\n  cases le_total b c <;> simp [min_eq_left, min_eq_right, *]\n\nsection Ord\n\nlemma compare_lt_iff_lt : compare a b = .lt ↔ a < b := by\n  rw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq_eq_lt]\n\nlemma compare_eq_iff_eq : compare a b = .eq ↔ a = b := by\n  rw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq_eq_eq le_refl not_le]\n\nlemma compare_gt_iff_gt : compare a b = .gt ↔ b < a := by\n  rw [LinearOrder.compare_eq_compareOfLessAndEq,\n    compareOfLessAndEq_eq_gt le_antisymm le_total not_le]\n\nTarget:\nlemma compare_le_iff_le : compare a b ≠ .gt ↔ a ≤ b :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_34c67d098470","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"2e1909660ac264933960bb0f0fde3367ca57c38790ec68461a6a41e4dc566e37","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Defs","family_id":"compare_le_iff_le","file_id":"mathlib/Mathlib/Order/Defs/LinearOrder.lean","sample_id":"34c67d0984704f562f600f9107ee26cdff7c7034435d16dae2284506bc3e8668"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8da08d0db4cb5d343e5a5a407377b94360b60c72eeb504f2655147b8545bacfb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"31e49ca21c3f68d25f9457b6c613ab99458689f33e2a68bb7962f4b7d6f0bb4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"24a5ad421e4a7eee0e5feabb06c49c3c686c8f596032a3cb50682135e2ce5773","source_sha256":"f2a66bdccdec789c239da17203e2b07ced0f35d264cc0fb10abbfb178b2db08a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases isEmpty_or_nonempty α <;> simp [FreeCommRing]","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":16,"token_jaccard":0.6875,"token_length_ratio":1.454545},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"54f40f60542f8940ca6cd1683a8e551e9299ecef939a1e02bf77583ef0dcc655","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.FreeAbelianGroup.Finsupp\npublic import Mathlib.Algebra.Ring.TransferInstance\npublic import Mathlib.Data.Finsupp.Fintype\npublic import Mathlib.Data.ZMod.Defs\npublic import Mathlib.GroupTheory.FreeGroup.Reduce\npublic import Mathlib.RingTheory.FreeCommRing\npublic import Mathlib.SetTheory.Cardinal.Arithmetic\npublic import Mathlib.SetTheory.Cardinal.Finsupp\n\nNamespace:\nCardinal\n\nLocal context:\n/-\nCopyright (c) 2024 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser, Daniel Weber\n-/\n/-!\n# Cardinalities of free constructions\n\nThis file shows that all the free constructions over `α` have cardinality `max #α ℵ₀`,\nand are thus infinite, and specifically countable over countable generators.\n\nCombined with the ring `Fin n` for the finite cases, this lets us show that there is a `CommRing` of\nany cardinality.\n-/\n\npublic section\n\nuniverse u\nvariable (α : Type u)\n\nsection Infinite\n\n@[to_additive]\ninstance [Nonempty α] : Infinite (FreeMonoid α) := inferInstanceAs <| Infinite (List α)\n\n@[to_additive]\ninstance [Nonempty α] : Infinite (FreeGroup α) := by\n  classical\n  exact Infinite.of_surjective FreeGroup.norm FreeGroup.norm_surjective\n\ninstance [Nonempty α] : Infinite (FreeAbelianGroup α) :=\n  (FreeAbelianGroup.equivFinsupp α).toEquiv.infinite_iff.2 inferInstance\n\nderiving instance Infinite for FreeRing, FreeCommRing\n\nend Infinite\n\nsection Countable\n\nvariable [Countable α]\n\n@[to_additive]\ninstance : Countable (FreeMonoid α) := inferInstanceAs <| Countable (List α)\n\n@[to_additive]\ninstance : Countable (FreeGroup α) := inferInstanceAs <| Countable (Quot _)\n\ninstance : Countable (FreeAbelianGroup α) := inferInstanceAs <| Countable (Quot _)\n\ninstance : Countable (FreeRing α) := inferInstanceAs <| Countable (Quot _)\n\ninstance : Countable (FreeCommRing α) :=\n  inferInstanceAs <| Countable (FreeAbelianGroup (Multiset α))\n\nend Countable\n\nnamespace Cardinal\n\ntheorem mk_abelianization_le (G : Type u) [Group G] :\n    #(Abelianization G) ≤ #G := Cardinal.mk_le_of_surjective Quotient.mk_surjective\n\n@[to_additive (attr := simp)]\ntheorem mk_freeMonoid [Nonempty α] : #(FreeMonoid α) = max #α ℵ₀ :=\n    Cardinal.mk_list_eq_max_mk_aleph0 _\n\n@[to_additive (attr := simp)]\ntheorem mk_freeGroup [Nonempty α] : #(FreeGroup α) = max #α ℵ₀ := by\n  classical\n  apply le_antisymm\n  · apply (mk_le_of_injective (FreeGroup.toWord_injective (α := α))).trans_eq\n    simp only [mk_list_eq_max_mk_aleph0, mk_prod, lift_uzero, mk_fintype, Fintype.card_bool,\n      Nat.cast_ofNat, lift_ofNat]\n    obtain hα | hα := lt_or_ge #α ℵ₀\n    · simp only [hα.le, max_eq_right, max_eq_right_iff]\n      exact (mul_lt_aleph0 hα natCast_lt_aleph0).le\n    · rw [max_eq_left hα, max_eq_left (hα.trans <| Cardinal.le_mul_right two_ne_zero),\n        Cardinal.mul_eq_left hα _ (by simp)]\n      exact natCast_le_aleph0.trans hα\n  · apply max_le\n    · exact mk_le_of_injective FreeGroup.of_injective\n    · simp\n\n@[simp]\ntheorem mk_freeAbelianGroup [Nonempty α] : #(FreeAbelianGroup α) = max #α ℵ₀ := by\n  rw [Cardinal.mk_congr (FreeAbelianGroup.equivFinsupp α).toEquiv]\n  simp\n\n@[simp]\ntheorem mk_freeRing : #(FreeRing α) = max #α ℵ₀ := by\n  cases isEmpty_or_nonempty α <;> simp [FreeRing]\n\n@[simp]\n\nTarget:\ntheorem mk_freeCommRing : #(FreeCommRing α) = max #α ℵ₀ :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  cases isEmpty_or_nonempty α <;> simp [FreeCommRing]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Cardinal","family_id":"mk_freecommring","file_id":"mathlib/Mathlib/SetTheory/Cardinal/Free.lean","sample_id":"24a5ad421e4a7eee0e5feabb06c49c3c686c8f596032a3cb50682135e2ce5773"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6124388c54e3601f5b392168c48f6a150dee4519aaf6dbd646732ffc8061300b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d59287e3947f846b3ad368db2e503ad4659a8961a5b312354e1ce2722dc1bc50","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"84f48056d33f497341acedeaa75d75fead8d9c6cb4f1d4df07a45458a9009e00","source_sha256":"ab525a505d882c41e1965c3ef070c66d963cb0848742b2b2ba04dd9efd351a5a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x <;> simp","hard_negative":true,"metrics":{"chosen_tokens":7,"rejected_tokens":3,"token_jaccard":0.111111,"token_length_ratio":0.428571},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"5508cb0c2a2461fbfe8be45a8f7d49c26f5c692cf0a886d19f4e484c4fc86e2b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.RootSystem.Basic\npublic import Mathlib.LinearAlgebra.RootSystem.Defs\n\nNamespace:\nRootPairing.Hom\n\nLocal context:\n/-\nCopyright (c) 2024 Scott Carnahan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Carnahan\n-/\n/-!\n# Morphisms of root pairings\nThis file defines morphisms of root pairings, following the definition of morphisms of root data\ngiven in SGA III Exp. 21 Section 6.\n\n## Main definitions:\n* `Hom`: A morphism of root pairings is a linear map of weight spaces, its transverse on coweight\n  spaces, and a bijection on the set that indexes roots and coroots.\n* `Hom.id`: The identity morphism.\n* `Hom.comp`: The composite of two morphisms.\n* `End`: The endomorphism monoid of a root pairing.\n* `Hom.weightHom`: The homomorphism from the endomorphism monoid to linear endomorphisms on the\n  weight space.\n* `Hom.coweightHom`: The homomorphism from the endomorphism monoid to the opposite monoid of linear\n  endomorphisms on the coweight space.\n* `Equiv`: An equivalence of root pairings is a morphism for which the maps on weight spaces and\n  coweight spaces are bijective.\n* `Equiv.toHom`: The morphism underlying an equivalence.\n* `Equiv.weightEquiv`: The linear isomorphism on weight spaces given by an equivalence.\n* `Equiv.coweightEquiv`: The linear isomorphism on coweight spaces given by an equivalence.\n* `Equiv.id`: The identity equivalence.\n* `Equiv.comp`: The composite of two equivalences.\n* `Equiv.symm`: The inverse of an equivalence.\n* `Aut`: The automorphism group of a root pairing.\n* `Equiv.toEndUnit`: The group isomorphism between the automorphism group of a root pairing and the\n  group of invertible endomorphisms.\n* `Equiv.weightHom`: The homomorphism from the automorphism group to linear automorphisms on the\n  weight space.\n* `Equiv.coweightHom`: The homomorphism from the automorphism group to the opposite group of linear\n  automorphisms on the coweight space.\n* `Equiv.reflection`: The automorphism of a root pairing given by reflection in a root and\n  coreflection in the corresponding coroot.\n\n## TODO\n* Special types of morphisms: Isogenies, weight/coweight space embeddings\n* Weyl group reimplementation?\n\n-/\n\n@[expose] public section\n\nopen Set Function\n\nnoncomputable section\n\nvariable {ι R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]\n\nnamespace RootPairing\n\n/-- A morphism of root pairings is a pair of mutually transposed maps of weight and coweight spaces\nthat preserves roots and coroots.  We make the map of indexing sets explicit. -/\n@[ext]\nstructure Hom {ι₂ M₂ N₂ : Type*}\n    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) where\n  /-- A linear map on weight space. -/\n  weightMap : M →ₗ[R] M₂\n  /-- A contravariant linear map on coweight space. -/\n  coweightMap : N₂ →ₗ[R] N\n  /-- A bijection on index sets. -/\n  indexEquiv : ι ≃ ι₂\n  weight_coweight_transpose :\n    weightMap.dualMap ∘ₗ Q.flip.toPerfPair = P.flip.toPerfPair ∘ₗ coweightMap\n  root_weightMap : weightMap ∘ P.root = Q.root ∘ indexEquiv\n  coroot_coweightMap : coweightMap ∘ Q.coroot = P.coroot ∘ indexEquiv.symm\n\nnamespace Hom\n\nlemma weight_coweight_transpose_apply {ι₂ M₂ N₂ : Type*}\n    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (x : N₂) (f : Hom P Q) :\n    f.weightMap.dualMap (Q.flip.toPerfPair x) = P.flip.toPerfPair (f.coweightMap x) :=\n  Eq.mp (propext LinearMap.ext_iff) f.weight_coweight_transpose x\n\nlemma root_weightMap_apply {ι₂ M₂ N₂ : Type*}\n    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (i : ι) (f : Hom P Q) :\n    f.weightMap (P.root i) = Q.root (f.indexEquiv i) :=\n  Eq.mp (propext funext_iff) f.root_weightMap i\n\nlemma coroot_coweightMap_apply {ι₂ M₂ N₂ : Type*}\n    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (i : ι₂) (f : Hom P Q) :\n    f.coweightMap (Q.coroot i) = P.coroot (f.indexEquiv.symm i) :=\n  Eq.mp (propext funext_iff) f.coroot_coweightMap i\n\n/-- The identity morphism of a root pairing. -/\n@[simps!]\ndef id (P : RootPairing ι R M N) : Hom P P where\n  weightMap := LinearMap.id\n  coweightMap := LinearMap.id\n  indexEquiv := Equiv.refl ι\n  weight_coweight_transpose := by simp\n  root_weightMap := by simp\n  coroot_coweightMap := by simp\n\n/-- Composition of morphisms -/\n@[simps!]\ndef comp {ι₁ M₁ N₁ ι₂ M₂ N₂ : Type*} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁]\n    [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂}\n    (g : Hom P₁ P₂) (f : Hom P P₁) : Hom P P₂ where\n  weightMap := g.weightMap ∘ₗ f.weightMap\n  coweightMap := f.coweightMap ∘ₗ g.coweightMap\n  indexEquiv := f.indexEquiv.trans g.indexEquiv\n  weight_coweight_transpose := by\n    ext φ x\n    rw [← LinearMap.dualMap_comp_dualMap, ← LinearMap.comp_assoc _ f.coweightMap,\n      ← f.weight_coweight_transpose, LinearMap.comp_assoc g.coweightMap,\n      ← g.weight_coweight_transpose, ← LinearMap.comp_assoc]\n  root_weightMap := by\n    ext i\n    simp only [LinearMap.coe_comp, Equiv.coe_trans]\n    rw [comp_assoc, f.root_weightMap, ← comp_assoc, g.root_weightMap, comp_assoc]\n  coroot_coweightMap := by\n    ext i\n    simp only [LinearMap.coe_comp]\n    rw [comp_assoc, g.coroot_coweightMap, ← comp_assoc, f.coroot_coweightMap, comp_assoc]\n    simp\n\n@[simp]\nlemma id_comp {ι₂ M₂ N₂ : Type*}\n    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : Hom P Q) :\n    comp f (id P) = f := by\n  ext x <;> simp\n\n@[simp]\n\nTarget:\nlemma comp_id {ι₂ M₂ N₂ : Type*}\n    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]\n    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : Hom P Q) :\n    comp (id Q) f = f :=\n\nProof body:\n","rejected":"by\n  exact comp_id","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"9da21032eee273c05bf700eee8814e0d8c29e06eddb9e1b87b1ca60f25501287","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/RootSystem","family_id":"comp_id","file_id":"mathlib/Mathlib/LinearAlgebra/RootSystem/Hom.lean","sample_id":"84f48056d33f497341acedeaa75d75fead8d9c6cb4f1d4df07a45458a9009e00"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a956e0393455f70ac3f0634d07a69fa1a653d99212eafe809fbf92b858839bc8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a1347b35a1da940b0cf7bb9fa19ec8fb67a45c4aec52bd9f946f2c2f04d77b83","source_sha256":"33b47846d8ad6e41763ee1834aea3dbb18288085daf7f82ed174fd1d15b2a25b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · induction fs generalizing t with\n    | nil =>\n      simp only [sequence, mem_pure, imp_self, forall₂_nil_left_iff, exists_eq_left, Set.pure_def,\n        singleton_subset_iff, traverse_nil]\n    | cons b fs ih =>\n      intro ht\n      rcases mem_seq_iff.1 ht with ⟨u, hu, v, hv, ht⟩\n      rcases mem_map_iff_exists_image.1 hu with ⟨w, hw, hwu⟩\n      rcases ih v hv with ⟨us, hus, hu⟩\n      exact ⟨w::us, Forall₂.cons hw hus, (Set.seq_mono hwu hu).trans ht⟩\n  · rintro ⟨us, hus, hs⟩\n    exact mem_of_superset (mem_traverse _ _ hus) hs","hard_negative":false,"metrics":{"chosen_tokens":127,"rejected_tokens":2,"token_jaccard":0.016393,"token_length_ratio":0.015748},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"554f1933643c5a10f33fcba815ead4f78471ee9fda92bad0578fb8484c07e2de","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Control.Traversable.Instances\npublic import Mathlib.Order.Filter.Map\n\nNamespace:\nFilter\n\nLocal context:\n/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Properties of `Traversable.traverse` on `List`s and `Filter`s\n\nIn this file we prove basic properties (monotonicity, membership)\nfor `Traversable.traverse f l`, where `f : β → Filter α` and `l : List β`.\n-/\n\npublic section\n\nopen Set List\n\nnamespace Filter\n\nuniverse u\n\nvariable {α β γ : Type u} {f : β → Filter α} {s : γ → Set α}\n\ntheorem sequence_mono : ∀ as bs : List (Filter α), Forall₂ (· ≤ ·) as bs → sequence as ≤ sequence bs\n  | [], [], Forall₂.nil => le_rfl\n  | _::as, _::bs, Forall₂.cons h hs => seq_mono (map_mono h) (sequence_mono as bs hs)\n\nTarget:\ntheorem mem_traverse :\n    ∀ (fs : List β) (us : List γ),\n      Forall₂ (fun b c => s c ∈ f b) fs us → traverse s us ∈ traverse f fs\n  | [], [], Forall₂.nil => mem_pure.2 <| mem_singleton _\n  | _::fs, _::us, Forall₂.cons h hs => seq_mem_seq (image_mem_map h) (mem_traverse fs us hs)\n\n-- TODO: add a `Filter.HasBasis` statement\ntheorem mem_traverse_iff (fs : List β) (t : Set (List α)) :\n    t ∈ traverse f fs ↔\n      ∃ us : List (Set α), Forall₂ (fun b (s : Set α) => s ∈ f b) fs us ∧ sequence us ⊆ t :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Filter","family_id":"mem_traverse","file_id":"mathlib/Mathlib/Order/Filter/ListTraverse.lean","sample_id":"a1347b35a1da940b0cf7bb9fa19ec8fb67a45c4aec52bd9f946f2c2f04d77b83"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e13f4e0fbff74fe329154bb27fe6656b950cc8d302959791efca88cd4237a5ab","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"727321c81fb49fb44e787715ddf3c0aafbedfa69ef549affbe7e565a44f0be51","source_sha256":"6c1d4f9fced239f6f8c0c8785a4325f383edabba30747fe08c8262d3408cd815","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h => ?_, fun ⟨h₁, h₂⟩ => Equiv.antisymm h₁ h₂⟩\n  obtain - | ⟨h₁, h₂⟩ := h\n  · simp [Lists'.Subset.refl]\n  · exact ⟨h₁, h₂⟩","hard_negative":false,"metrics":{"chosen_tokens":56,"rejected_tokens":2,"token_jaccard":0.074074,"token_length_ratio":0.035714},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"55ac925c075dda9902643abf20a731e3b1d24eb2aa212a85281012e74a5c4069","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.Monoid.NatCast\npublic import Mathlib.Algebra.Ring.Nat\npublic import Mathlib.Data.Sigma.Basic\npublic import Batteries.Tactic.Lint.TypeClass\n\nNamespace:\nLists\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# A computable model of ZFA without infinity\n\nIn this file we define finite hereditary lists. This is useful for calculations in naive set theory.\n\nWe distinguish two kinds of ZFA lists:\n* Atoms. Directly correspond to an element of the original type.\n* Proper ZFA lists. Can be thought of (but are not implemented) as a list of ZFA lists (not\n  necessarily proper).\n\nFor example, `Lists ℕ` contains stuff like `23`, `[]`, `[37]`, `[1, [[2], 3], 4]`.\n\n## Implementation note\n\nAs we want to be able to append both atoms and proper ZFA lists to proper ZFA lists, it's handy that\natoms and proper ZFA lists belong to the same type, even though atoms of `α` could be modelled as\n`α` directly. But we don't want to be able to append anything to atoms.\n\nThis calls for a two-step definition of ZFA lists:\n* First, define ZFA prelists as atoms and proper ZFA prelists. Those proper ZFA prelists are defined\n  by inductive appending of (not necessarily proper) ZFA lists.\n* Second, define ZFA lists by rubbing out the distinction between atoms and proper lists.\n\n## Main declarations\n\n* `Lists' α false`: Atoms as ZFA prelists. Basically a copy of `α`.\n* `Lists' α true`: Proper ZFA prelists. Defined inductively from the empty ZFA prelist\n  (`Lists'.nil`) and from appending a ZFA prelist to a proper ZFA prelist (`Lists'.cons a l`).\n* `Lists α`: ZFA lists. Sum of the atoms and proper ZFA prelists.\n* `Finsets α`: ZFA sets. Defined as `Lists` quotiented by `Lists.Equiv`, the extensional\n  equivalence.\n-/\n\n@[expose] public section\n\n\nvariable {α : Type*}\n\n/-- Prelists, helper type to define `Lists`. `Lists' α false` are the \"atoms\", a copy of `α`.\n`Lists' α true` are the \"proper\" ZFA prelists, inductively defined from the empty ZFA prelist and\nfrom appending a ZFA prelist to a proper ZFA prelist. It is made so that you can't append anything\nto an atom while having only one appending function for appending both atoms and proper ZFA prelists\nto a proper ZFA prelist. -/\ninductive Lists'.{u} (α : Type u) : Bool → Type u\n  | atom : α → Lists' α false\n  | nil : Lists' α true\n  | cons' {b} : Lists' α b → Lists' α true → Lists' α true\n  deriving DecidableEq\ncompile_inductive% Lists'\n\n/-- Hereditarily finite list, aka ZFA list. A ZFA list is either an \"atom\" (`b = false`),\ncorresponding to an element of `α`, or a \"proper\" ZFA list, inductively defined from the empty ZFA\nlist and from appending a ZFA list to a proper ZFA list. -/\ndef Lists (α : Type*) :=\n  Σ b, Lists' α b\n\nnamespace Lists'\n\ninstance [Inhabited α] : ∀ b, Inhabited (Lists' α b)\n  | true => ⟨nil⟩\n  | false => ⟨atom default⟩\n\n/-- Appending a ZFA list to a proper ZFA prelist. -/\ndef cons : Lists α → Lists' α true → Lists' α true\n  | ⟨_, a⟩, l => cons' a l\n\n/-- Converts a ZFA prelist to a `List` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp]\ndef toList : ∀ {b}, Lists' α b → List (Lists α)\n  | _, atom _ => []\n  | _, nil => []\n  | _, cons' a l => ⟨_, a⟩ :: l.toList\n\n@[simp]\ntheorem toList_cons (a : Lists α) (l) : toList (cons a l) = a :: l.toList := rfl\n\n/-- Converts a `List` of ZFA lists to a proper ZFA prelist. -/\n@[simp]\ndef ofList : List (Lists α) → Lists' α true\n  | [] => nil\n  | a :: l => cons a (ofList l)\n\n@[simp]\ntheorem to_ofList (l : List (Lists α)) : toList (ofList l) = l := by induction l <;> simp [*]\n\n@[simp]\ntheorem of_toList : ∀ l : Lists' α true, ofList (toList l) = l :=\n  suffices ∀ (b) (h : true = b) (l : Lists' α b),\n      let l' : Lists' α true := h ▸ l\n      ofList (toList l') = l'\n    from this _ rfl\n  fun b h l => by\n    induction l with\n    | atom => cases h\n    | nil => simp\n    | cons' b a _ IH => simpa [cons] using IH rfl\n\n/-- Recursion/induction principle for `Lists'.ofList`. -/\n@[elab_as_elim]\ndef recOfList {motive : Lists' α true → Sort*} (ofList : ∀ l, motive (ofList l)) : ∀ l, motive l :=\n  fun l ↦ cast (by simp) <| ofList (l.toList)\n\nend Lists'\n\nmutual\n  /-- Equivalence of ZFA lists. Defined inductively. -/\n  inductive Lists.Equiv : Lists α → Lists α → Prop\n    | refl (l) : Lists.Equiv l l\n    | antisymm {l₁ l₂ : Lists' α true} :\n      Lists'.Subset l₁ l₂ → Lists'.Subset l₂ l₁ → Lists.Equiv ⟨_, l₁⟩ ⟨_, l₂⟩\n\n  /-- Subset relation for ZFA lists. Defined inductively. -/\n  inductive Lists'.Subset : Lists' α true → Lists' α true → Prop\n    | nil {l} : Lists'.Subset Lists'.nil l\n    | cons {a a' l l'} :\n      Lists.Equiv a a' →\n        a' ∈ Lists'.toList l' → Lists'.Subset l l' → Lists'.Subset (Lists'.cons a l) l'\nend\n\nlocal infixl:50 \" ~ \" => Lists.Equiv\n\nnamespace Lists'\n\ninstance : HasSubset (Lists' α true) :=\n  ⟨Lists'.Subset⟩\n\n/-- ZFA prelist membership. A ZFA list is in a ZFA prelist if some element of this ZFA prelist is\nequivalent as a ZFA list to this ZFA list. -/\ninstance {b} : Membership (Lists α) (Lists' α b) :=\n  ⟨fun l a => ∃ a' ∈ l.toList, a ~ a'⟩\n\ntheorem mem_def {b a} {l : Lists' α b} : a ∈ l ↔ ∃ a' ∈ l.toList, a ~ a' :=\n  Iff.rfl\n\n@[simp]\ntheorem mem_cons {a y l} : a ∈ @cons α y l ↔ a ~ y ∨ a ∈ l := by\n  simp [mem_def, or_and_right, exists_or]\n\ntheorem cons_subset {a} {l₁ l₂ : Lists' α true} : Lists'.cons a l₁ ⊆ l₂ ↔ a ∈ l₂ ∧ l₁ ⊆ l₂ := by\n  refine ⟨fun h => ?_, fun ⟨⟨a', m, e⟩, s⟩ => Subset.cons e m s⟩\n  generalize h' : Lists'.cons a l₁ = l₁' at h\n  obtain - | @⟨a', _, _, _, e, m, s⟩ := h\n  · cases a\n    cases h'\n  cases a; cases a'; cases h'; exact ⟨⟨_, m, e⟩, s⟩\n\ntheorem ofList_subset {l₁ l₂ : List (Lists α)} (h : l₁ ⊆ l₂) :\n    Lists'.ofList l₁ ⊆ Lists'.ofList l₂ := by\n  induction l₁ with\n  | nil => exact Subset.nil\n  | cons _ _ l₁_ih =>\n    refine Subset.cons (Lists.Equiv.refl _) ?_ (l₁_ih (List.subset_of_cons_subset h))\n    simp only [List.cons_subset] at h; simp [h]\n\n@[refl]\ntheorem Subset.refl {l : Lists' α true} : l ⊆ l := by\n  rw [← Lists'.of_toList l]; exact ofList_subset (List.Subset.refl _)\n\ntheorem subset_nil {l : Lists' α true} : l ⊆ Lists'.nil → l = Lists'.nil := by\n  rw [← of_toList l]\n  induction toList l <;> intro h\n  · rfl\n  · rcases cons_subset.1 h with ⟨⟨_, ⟨⟩, _⟩, _⟩\n\ntheorem mem_of_subset' {a} : ∀ {l₁ l₂ : Lists' α true} (_ : l₁ ⊆ l₂) (_ : a ∈ l₁.toList), a ∈ l₂\n  | nil, _, Lists'.Subset.nil, h => by cases h\n  | cons' a0 l0, l₂, s, h => by\n    obtain - | ⟨e, m, s⟩ := s\n    simp only [toList, Sigma.eta, List.mem_cons] at h\n    rcases h with (rfl | h)\n    · exact ⟨_, m, e⟩\n    · exact mem_of_subset' s h\n\ntheorem subset_def {l₁ l₂ : Lists' α true} : l₁ ⊆ l₂ ↔ ∀ a ∈ l₁.toList, a ∈ l₂ :=\n  ⟨fun H _ => mem_of_subset' H, fun H => by\n    induction l₁ using recOfList with | _ l₁\n    induction l₁ with\n    | nil => exact Subset.nil\n    | cons h t t_ih =>\n      simp only [to_ofList, ofList, toList_cons, List.mem_cons, forall_eq_or_imp] at *\n      exact cons_subset.2 ⟨H.1, t_ih H.2⟩⟩\n\nend Lists'\n\nnamespace Lists\n\n/-- Sends `a : α` to the corresponding atom in `Lists α`. -/\n@[match_pattern]\ndef atom (a : α) : Lists α :=\n  ⟨_, Lists'.atom a⟩\n\n/-- Converts a proper ZFA prelist to a ZFA list. -/\n@[match_pattern]\ndef of' (l : Lists' α true) : Lists α :=\n  ⟨_, l⟩\n\n/-- Converts a ZFA list to a `List` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp]\ndef toList : Lists α → List (Lists α)\n  | ⟨_, l⟩ => l.toList\n\n/-- Predicate stating that a ZFA list is proper. -/\ndef IsList (l : Lists α) : Prop :=\n  l.1\n\n/-- Converts a `List` of ZFA lists to a ZFA list. -/\ndef ofList (l : List (Lists α)) : Lists α :=\n  of' (Lists'.ofList l)\n\ntheorem isList_toList (l : List (Lists α)) : IsList (ofList l) :=\n  Eq.refl _\n\ntheorem to_ofList (l : List (Lists α)) : toList (ofList l) = l := by simp [ofList, of']\n\ntheorem of_toList : ∀ {l : Lists α}, IsList l → ofList (toList l) = l\n  | ⟨true, l⟩, _ => by simp_all [ofList, of']\n\ninstance : Inhabited (Lists α) :=\n  ⟨of' Lists'.nil⟩\n\ninstance [DecidableEq α] : DecidableEq (Lists α) := inferInstanceAs <| DecidableEq (Sigma _)\n\ninstance [SizeOf α] : SizeOf (Lists α) := inferInstanceAs <| SizeOf (Sigma _)\n\n/-- A recursion principle for pairs of ZFA lists and proper ZFA prelists. -/\ndef inductionMut (C : Lists α → Sort*) (D : Lists' α true → Sort*)\n    (C0 : ∀ a, C (atom a)) (C1 : ∀ l, D l → C (of' l))\n    (D0 : D Lists'.nil) (D1 : ∀ a l, C a → D l → D (Lists'.cons a l)) :\n    PProd (∀ l, C l) (∀ l, D l) := by\n  suffices ∀ {b} (l : Lists' α b),\n      PProd (C ⟨_, l⟩)\n        (match b, l with\n        | true, l => D l\n        | false, _ => PUnit)\n    by exact ⟨fun ⟨b, l⟩ => (this _).1, fun l => (this l).2⟩\n  intro b l\n  induction l with\n  | atom => exact ⟨C0 _, ⟨⟩⟩\n  | nil => exact ⟨C1 _ D0, D0⟩\n  | cons' a l IH₁ IH =>\n    have : D (Lists'.cons' a l) := D1 ⟨_, _⟩ _ IH₁.1 IH.2\n    exact ⟨C1 _ this, this⟩\n\n/-- Membership of ZFA list. A ZFA list belongs to a proper ZFA list if it belongs to the latter as a\nproper ZFA prelist. An atom has no members. -/\ndef mem (a : Lists α) : Lists α → Prop\n  | ⟨false, _⟩ => False\n  | ⟨_, l⟩ => a ∈ l\n\ninstance : Membership (Lists α) (Lists α) where\n  mem ls l := mem l ls\n\nTarget:\ntheorem isList_of_mem {a : Lists α} : ∀ {l : Lists α}, a ∈ l → IsList l\n  | ⟨_, Lists'.nil⟩, _ => rfl\n  | ⟨_, Lists'.cons' _ _⟩, _ => rfl\n\ntheorem Equiv.antisymm_iff {l₁ l₂ : Lists' α true} : of' l₁ ~ of' l₂ ↔ l₁ ⊆ l₂ ∧ l₂ ⊆ l₁ :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory","family_id":"islist_of_mem","file_id":"mathlib/Mathlib/SetTheory/Lists.lean","sample_id":"727321c81fb49fb44e787715ddf3c0aafbedfa69ef549affbe7e565a44f0be51"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f4dbb4c53a6e3a2d5b935a93c9fc29cc9b719e218918f120b3b408ba8d371698","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"da2067aad04e85cd3a30fda8e64f9e051df4cbdc94cfa40eb43e54487398c56f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5a2ece2a22fcf936fd41bf7ec3c1d0ec84ea941a229b765b720040afa6959a7e","source_sha256":"695353e91eba40e45d9ecd0e58f8b00a56d575b25190f682a1689e29c7316a51","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold logDeriv\n  simp [field, Derivation.leibniz_div]","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.272727},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"56752ee8d3d9ab9b8d9bee9c069009f17c220a55ecf516756c69f42ea72c3212","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Derivation.MapCoeffs\npublic import Mathlib.FieldTheory.PrimitiveElement\n\nNamespace:\nDifferential\n\nLocal context:\n/-\nCopyright (c) 2024 Daniel Weber. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Weber\n-/\n/-!\n# Differential Fields\n\nThis file defines the logarithmic derivative `Differential.logDeriv` and proves properties of it.\nThis is defined algebraically, compared to `logDeriv` which is analytical.\n-/\n\n@[expose] public section\n\nnamespace Differential\n\nopen algebraMap Polynomial IntermediateField\n\nvariable {R : Type*} [Field R] [Differential R] (a b : R)\n\n/--\nThe logarithmic derivative of a is a′ / a.\n-/\ndef logDeriv : R := a′ / a\n\n@[simp]\nlemma logDeriv_zero : logDeriv (0 : R) = 0 := by\n  simp [logDeriv]\n\n@[simp]\nlemma logDeriv_one : logDeriv (1 : R) = 0 := by\n  simp [logDeriv]\n\nlemma logDeriv_mul (ha : a ≠ 0) (hb : b ≠ 0) : logDeriv (a * b) = logDeriv a + logDeriv b := by\n  unfold logDeriv\n  simp [field]\n  ring\n\nTarget:\nlemma logDeriv_div (ha : a ≠ 0) (hb : b ≠ 0) : logDeriv (a / b) = logDeriv a - logDeriv b :=\n\nProof body:\n","rejected":"by\n  exact logDeriv_div","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"eb653610184ca293a54e6fc6b94ba94bc9bca30812cf9eab257e3854b221bd61","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Differential","family_id":"logderiv_div","file_id":"mathlib/Mathlib/FieldTheory/Differential/Basic.lean","sample_id":"5a2ece2a22fcf936fd41bf7ec3c1d0ec84ea941a229b765b720040afa6959a7e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e79d2c183534cfffc99b615577f1db04cdc59929d66d45cf7905520be00385d4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1928d6ae6dc61f8cdfa63dfcb56b1f379fb62fee494f4dee240da30bec0a0daa","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5627650f4b1175b07bdab889615a32ef5fc6133ef66c020ba65b75f9c09ed833","source_sha256":"de50c3991d383f79a1e56820d274c7129a795179a5130040268b4911014d87d8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  obtain ⟨F₁, hF₁, hAF₁⟩ := hA.sq_covBySMul\n  obtain ⟨F₂, hF₂, hBF₂⟩ := hB.sq_covBySMul\n  have := hA.one_le\n  choose f hf using exists_smul_inter_smul_subset_smul_inv_mul_inter_inv_mul A B\n  refine ⟨.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1)), ?_, ?_⟩\n  · calc\n      (#(.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1))) : ℝ)\n      _ ≤ #(F₁ ^ (m - 1)) * #(F₂ ^ (n - 1)) := mod_cast Finset.card_image₂_le ..\n      _ ≤ #F₁ ^ (m - 1) * #F₂ ^ (n - 1) := by gcongr <;> exact mod_cast Finset.card_pow_le\n      _ ≤ K ^ (m - 1) * L ^ (n - 1) := by gcongr\n  · calc\n      A ^ m ∩ B ^ n ⊆ (F₁ ^ (m - 1) * A) ∩ (F₂ ^ (n - 1) * B) := by\n        gcongr <;> apply pow_subset_pow_mul_of_sq_subset_mul <;> norm_cast <;> lia\n      _ = ⋃ (a ∈ F₁ ^ (m - 1)) (b ∈ F₂ ^ (n - 1)), a • A ∩ b • B := by\n        simp_rw [← smul_eq_mul, ← iUnion_smul_set, iUnion₂_inter_iUnion₂]; norm_cast\n      _ ⊆ ⋃ (a ∈ F₁ ^ (m - 1)) (b ∈ F₂ ^ (n - 1)), f a b • (A⁻¹ * A ∩ (B⁻¹ * B)) := by\n        gcongr; exact hf ..\n      _ = (Finset.image₂ f (F₁ ^ (m - 1)) (F₂ ^ (n - 1))) * (A ^ 2 ∩ B ^ 2) := by\n        simp_rw [hA.inv_eq_self, hB.inv_eq_self, ← sq]\n        rw [Finset.coe_image₂, ← smul_eq_mul, ← iUnion_smul_set, biUnion_image2]\n        simp_rw [Finset.mem_coe]","hard_negative":true,"metrics":{"chosen_tokens":429,"rejected_tokens":2,"token_jaccard":0.011628,"token_length_ratio":0.004662},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"5692b1e8ac29d5c1a49e338e52e72fc95aef0a9c746d45866b05ddd94030b6da","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Subgroup.Pointwise\npublic import Mathlib.Combinatorics.Additive.CovBySMul\npublic import Mathlib.Combinatorics.Additive.RuzsaCovering\npublic import Mathlib.Combinatorics.Additive.SmallTripling\n\nNamespace:\nIsApproximateSubgroup\n\nLocal context:\n/-\nCopyright (c) 2024 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Approximate subgroups\n\nThis file defines approximate subgroups of a group, namely symmetric sets `A` such that `A * A` can\nbe covered by a small number of translates of `A`.\n\n## Main results\n\nApproximate subgroups are a central concept in additive combinatorics, as a natural weakening and\nflexible substitute of genuine subgroups. As such, they share numerous properties with subgroups:\n* `IsApproximateSubgroup.image`: Group homomorphisms send approximate subgroups to approximate\n  subgroups\n* `IsApproximateSubgroup.pow_inter_pow`: The intersection of (non-trivial powers of) two approximate\n  subgroups is an approximate subgroup. Warning: The intersection of two approximate subgroups isn't\n  an approximate subgroup in general.\n\nApproximate subgroups are close qualitatively and quantitatively to other concepts in additive\ncombinatorics:\n* `IsApproximateSubgroup.card_pow_le`: An approximate subgroup has small powers.\n* `IsApproximateSubgroup.of_small_tripling`: A set of small tripling can be made an approximate\n  subgroup by squaring.\n\nIt can be readily confirmed that approximate subgroups are a weakening of subgroups:\n* `isApproximateSubgroup_one`: A 1-approximate subgroup is the same thing as a subgroup.\n-/\n\npublic section\n\nopen scoped Finset Pointwise\n\nvariable {G : Type*} [Group G] {A B : Set G} {K L : ℝ} {m n : ℕ}\n\n/--\nAn approximate subgroup in a group is a symmetric set `A` containing the identity and such that\n`A + A` can be covered by a small number of translates of `A`.\n\nIn practice, we will take `K` fixed and `A` large but finite.\n-/\nstructure IsApproximateAddSubgroup {G : Type*} [AddGroup G] (K : ℝ) (A : Set G) : Prop where\n  zero_mem : 0 ∈ A\n  neg_eq_self : -A = A\n  two_nsmul_covByVAdd : CovByVAdd G K (2 • A) A\n\n/--\nAn approximate subgroup in a group is a symmetric set `A` containing the identity and such that\n`A * A` can be covered by a small number of translates of `A`.\n\nIn practice, we will take `K` fixed and `A` large but finite.\n-/\n@[to_additive]\nstructure IsApproximateSubgroup (K : ℝ) (A : Set G) : Prop where\n  one_mem : 1 ∈ A\n  inv_eq_self : A⁻¹ = A\n  sq_covBySMul : CovBySMul G K (A ^ 2) A\n\nnamespace IsApproximateSubgroup\n\n@[to_additive] lemma nonempty (hA : IsApproximateSubgroup K A) : A.Nonempty := ⟨1, hA.one_mem⟩\n\n@[to_additive one_le]\nlemma one_le (hA : IsApproximateSubgroup K A) : 1 ≤ K := by\n  obtain ⟨F, hF, hSF⟩ := hA.sq_covBySMul\n  grw [← hF]\n  have : F.Nonempty := by by_contra! rfl; simp [hA.nonempty.ne_empty] at hSF\n  simpa\n\n@[to_additive]\nlemma mono (hKL : K ≤ L) (hA : IsApproximateSubgroup K A) : IsApproximateSubgroup L A where\n  one_mem := hA.one_mem\n  inv_eq_self := hA.inv_eq_self\n  sq_covBySMul := hA.sq_covBySMul.mono hKL\n\n@[to_additive]\nlemma card_pow_le [DecidableEq G] {A : Finset G} (hA : IsApproximateSubgroup K (A : Set G)) :\n    ∀ {n}, #(A ^ n) ≤ K ^ (n - 1) * #A\n  | 0 => by simpa using hA.nonempty\n  | 1 => by simp\n  | n + 2 => by\n    obtain ⟨F, hF, hSF⟩ := hA.sq_covBySMul\n    calc\n      (#(A ^ (n + 2)) : ℝ) ≤ #(F ^ (n + 1) * A) := by\n        gcongr; exact mod_cast Set.pow_subset_pow_mul_of_sq_subset_mul hSF (by lia)\n      _ ≤ #(F ^ (n + 1)) * #A := mod_cast Finset.card_mul_le\n      _ ≤ #F ^ (n + 1) * #A := by gcongr; exact mod_cast Finset.card_pow_le\n      _ ≤ K ^ (n + 1) * #A := by gcongr\n\n@[to_additive]\nlemma card_mul_self_le [DecidableEq G] {A : Finset G} (hA : IsApproximateSubgroup K (A : Set G)) :\n    #(A * A) ≤ K * #A := by simpa [sq] using hA.card_pow_le (n := 2)\n\n@[to_additive]\nlemma image {F H : Type*} [Group H] [FunLike F G H] [MonoidHomClass F G H] (f : F)\n    (hA : IsApproximateSubgroup K A) : IsApproximateSubgroup K (f '' A) where\n  one_mem := ⟨1, hA.one_mem, map_one _⟩\n  inv_eq_self := by simp [← Set.image_inv, hA.inv_eq_self]\n  sq_covBySMul := by\n    classical\n    obtain ⟨F, hF, hAF⟩ := hA.sq_covBySMul\n    refine ⟨F.image f, ?_, ?_⟩\n    · calc\n        (#(F.image f) : ℝ) ≤ #F := mod_cast F.card_image_le\n        _ ≤ K := hF\n    · simp only [← Set.image_pow, Finset.coe_image, ← Set.image_mul, smul_eq_mul] at hAF ⊢\n      gcongr\n\n@[to_additive]\nlemma subgroup {S : Type*} [SetLike S G] [SubgroupClass S G] {H : S} :\n    IsApproximateSubgroup 1 (H : Set G) where\n  one_mem := OneMemClass.one_mem H\n  inv_eq_self := inv_coe_set\n  sq_covBySMul := ⟨{1}, by simp⟩\n\nopen Finset in\n@[to_additive]\nlemma of_small_tripling [DecidableEq G] {A : Finset G} (hA₁ : 1 ∈ A) (hAsymm : A⁻¹ = A)\n    (hA : #(A ^ 3) ≤ K * #A) : IsApproximateSubgroup (K ^ 3) (A ^ 2 : Set G) where\n  one_mem := by rw [sq, ← one_mul 1]; exact Set.mul_mem_mul hA₁ hA₁\n  inv_eq_self := by simp [← inv_pow, hAsymm, ← coe_inv]\n  sq_covBySMul := by\n    replace hA := calc (#(A ^ 4 * A) : ℝ)\n      _ = #(A ^ 5) := by rw [← pow_succ]\n      _ ≤ K ^ 3 * #A := small_pow_of_small_tripling (by lia) hA hAsymm\n    have hA₀ : A.Nonempty := ⟨1, hA₁⟩\n    obtain ⟨F, -, hF, hAF⟩ := ruzsa_covering_mul hA₀ hA\n    exact ⟨F, hF, by norm_cast; simpa [div_eq_mul_inv, pow_succ, mul_assoc, hAsymm] using hAF⟩\n\nopen Set in\n@[to_additive]\n\nTarget:\nlemma pow_inter_pow_covBySMul_sq_inter_sq\n    (hA : IsApproximateSubgroup K A) (hB : IsApproximateSubgroup L B) (hm : 2 ≤ m) (hn : 2 ≤ n) :\n    CovBySMul G (K ^ (m - 1) * L ^ (n - 1)) (A ^ m ∩ B ^ n) (A ^ 2 ∩ B ^ 2) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_5627650f4b11","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"669e6640dcf03570b34e698c51aae8f7b19206fbeb47941dfd0b6b1a8af24901","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Additive","family_id":"pow_inter_pow_covbysmul_sq_inter_sq","file_id":"mathlib/Mathlib/Combinatorics/Additive/ApproximateSubgroup.lean","sample_id":"5627650f4b1175b07bdab889615a32ef5fc6133ef66c020ba65b75f9c09ed833"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f30223a701d84b8cf4910174a2ebd64c60a3db6a916ec5748abb240a8d7ca7c4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"766f3502274c6a836f440a41e6af60197e8da15e11cc99e0e09ed8fb000b1581","source_sha256":"a1ecfd868e0d49b8667b2cdba350ee4f2ceb12fbe0b51094508e312f262dbf5e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using (toInt_le_toInt val j i).not","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":8,"token_jaccard":0.058824,"token_length_ratio":0.727273},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"56ae7c939929edcd14e39dfeb778b8bc5dad11b61816f1feba6cdb641f85ba88","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.List.Pairwise\npublic import Batteries.Tactic.GeneralizeProofs\npublic import Mathlib.Tactic.Order.CollectFacts\n\nNamespace:\nMathlib.Tactic.Order.ToInt\n\nLocal context:\n/-\nCopyright (c) 2025 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\npublic meta import Mathlib.Util.AtomM\npublic meta import Mathlib.Util.Qq\npublic meta import Std.Data.HashMap.AdditionalOperations\n\n/-!\n# Translating linear orders to ℤ\n\nIn this file we implement the translation of a problem in any linearly ordered type to a problem in\n`ℤ`. This allows us to use the `lia` tactic to solve it.\n\nWhile the core algorithm of the `order` tactic is complete for the theory of linear orders in the\nsignature (`<`, `≤`),\nit becomes incomplete in the signature with lattice operations `⊓` and `⊔`. With these operations,\nthe problem becomes NP-hard, and the idea is to reuse a smart and efficient procedure, such as\n`lia`.\n\n## TODO\n\nMigrate to `grind` when it is ready.\n-/\n\npublic meta section\n\nnamespace Mathlib.Tactic.Order.ToInt\n\nvariable {α : Type*} [LinearOrder α] {n : ℕ} (val : Fin n → α)\n\n/-- The main theorem asserting the existence of a translation.\nWe use `Classical.choose` to turn this into a value for use in the `order` tactic,\nsee `toInt`.\n-/\ntheorem exists_translation : ∃ tr : Fin n → ℤ, ∀ i j, val i ≤ val j ↔ tr i ≤ tr j := by\n  let li := List.ofFn val\n  let sli := li.mergeSort\n  have (i : Fin n) : ∃ j : Fin sli.length, sli[j] = val i := by\n    apply List.get_of_mem\n    rw [List.Perm.mem_iff (List.mergeSort_perm _ _)]\n    simp [li]\n  use fun i ↦ (this i).choose\n  intro i j\n  simp only [Fin.getElem_fin, Int.ofNat_le]\n  by_cases h_eq : val i = val j\n  · simp [h_eq]\n  generalize_proofs _ hi hj\n  rw [← hi.choose_spec, ← hj.choose_spec] at h_eq\n  conv_lhs => rw [← hi.choose_spec, ← hj.choose_spec]\n  have := li.pairwise_mergeSort (le := fun a b ↦ decide (a ≤ b))\n      (fun a b c ↦ by simpa using le_trans) (by simpa using le_total)\n  rw [List.pairwise_iff_get] at this\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · contrapose! h\n    exact lt_of_le_of_ne (by simpa using (this hj.choose hi.choose (by simpa)))\n      (fun h ↦ h_eq (h.symm))\n  · simpa using this hi.choose hj.choose (by apply lt_of_le_of_ne h; contrapose h_eq; simp [h_eq])\n\n/-- Auxiliary definition used by the `order` tactic to transfer facts in a linear order to `ℤ`. -/\nnoncomputable def toInt (k : Fin n) : ℤ :=\n  (exists_translation val).choose k\n\nvariable (i j k : Fin n)\n\ntheorem toInt_le_toInt : toInt val i ≤ toInt val j ↔ val i ≤ val j := by\n  simp [toInt, (exists_translation val).choose_spec]\n\nTarget:\ntheorem toInt_lt_toInt : toInt val i < toInt val j ↔ val i < val j :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"a9c18c9a47d73bd29dc8909c0fc1d328f2ce5842f7dcfa21a0df684a9389fc7c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Order","family_id":"toint_lt_toint","file_id":"mathlib/Mathlib/Tactic/Order/ToInt.lean","sample_id":"766f3502274c6a836f440a41e6af60197e8da15e11cc99e0e09ed8fb000b1581"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cff2f50ea6340f1c0d647e96a5ab7bcbcc74041793ed6f6cea3ea0d685f9013d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"abe2f0c7fc30ae281c7ff189951bcaa47281af118fe1bc43e1b8f22db459fdac","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"865430ae7ad7a3af9b967fd8cc81590df7b116ddc79d82d6edeca41aba827e49","source_sha256":"b2e5d2441dae74e6a2f4e8de4b7d4e87dd1e5e1f1bcd7a4ed0737bd7e824e441","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro s hs\n  induction hs with\n  | base t t_mem => exact self_subset_generateSetAlgebra (h t_mem)\n  | empty => exact isSetAlgebra_generateSetAlgebra.empty_mem\n  | compl t _ t_mem => exact isSetAlgebra_generateSetAlgebra.compl_mem t_mem\n  | union t u _ _ t_mem u_mem => exact isSetAlgebra_generateSetAlgebra.union_mem t_mem u_mem","hard_negative":true,"metrics":{"chosen_tokens":51,"rejected_tokens":5,"token_jaccard":0.103448,"token_length_ratio":0.098039},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"56bf80222f5ef214403484d2405998564d1ad9edd7200a06255af78ff1960025","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finite.Prod\npublic import Mathlib.MeasureTheory.SetSemiring\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2024 Etienne Marion. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Etienne Marion\n-/\n/-!\n# Algebra of sets\n\nIn this file we define the notion of algebra of sets and give its basic properties. An algebra\nof sets is a family of sets containing the empty set and closed by complement and binary union.\nIt is therefore similar to a `σ`-algebra, except that it is not necessarily closed\nby countable unions.\n\nWe also define the algebra of sets generated by a family of sets and give its basic properties,\nand we prove that it is countable when it is generated by a countable family. We prove that\nthe `σ`-algebra generated by a family of sets `𝒜` is the same as the one generated by the algebra\nof sets generated by `𝒜`.\n\n## Main definitions\n\n* `MeasureTheory.IsSetAlgebra`: property of being an algebra of sets.\n* `MeasureTheory.generateSetAlgebra`: the algebra of sets generated by a family of sets.\n\n## Main statements\n\n* `MeasureTheory.mem_generateSetAlgebra_elim`: If a set `s` belongs to the algebra of sets\n  generated by `𝒜`, then it can be written as a finite union of finite intersections of sets which\n  are in `𝒜` or have their complement in `𝒜`.\n* `MeasureTheory.countable_generateSetAlgebra`: If a family of sets is countable then so is\n  the algebra of sets generated by it.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Field_of_sets>\n\n## Tags\n\nalgebra of sets, generated algebra of sets\n-/\n\npublic section\n\nopen MeasurableSpace Set\n\nnamespace MeasureTheory\n\nvariable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α}\n\n/-! ### Definition and basic properties of an algebra of sets -/\n\n/-- An algebra of sets is a family of sets containing the empty set and closed by complement and\nunion. Consequently it is also closed by difference (see `IsSetAlgebra.sdiff_mem`) and intersection\n(see `IsSetAlgebra.inter_mem`). -/\nstructure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where\n  empty_mem : ∅ ∈ 𝒜\n  compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜\n  union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜\n\nnamespace IsSetAlgebra\n\n/-- An algebra of sets contains the whole set. -/\ntheorem univ_mem (h𝒜 : IsSetAlgebra 𝒜) : univ ∈ 𝒜 :=\n  compl_empty ▸ h𝒜.compl_mem h𝒜.empty_mem\n\n/-- An algebra of sets is closed by intersection. -/\ntheorem inter_mem (h𝒜 : IsSetAlgebra 𝒜) (s_mem : s ∈ 𝒜) (t_mem : t ∈ 𝒜) :\n    s ∩ t ∈ 𝒜 :=\n  inter_eq_compl_compl_union_compl .. ▸\n    h𝒜.compl_mem (h𝒜.union_mem (h𝒜.compl_mem s_mem) (h𝒜.compl_mem t_mem))\n\n/-- An algebra of sets is closed by difference. -/\ntheorem sdiff_mem (h𝒜 : IsSetAlgebra 𝒜) (s_mem : s ∈ 𝒜) (t_mem : t ∈ 𝒜) :\n    s \\ t ∈ 𝒜 := h𝒜.inter_mem s_mem (h𝒜.compl_mem t_mem)\n\n@[deprecated (since := \"2026-06-03\")] alias diff_mem := sdiff_mem\n\n/-- An algebra of sets is a ring of sets. -/\ntheorem isSetRing (h𝒜 : IsSetAlgebra 𝒜) : IsSetRing 𝒜 where\n  empty_mem := h𝒜.empty_mem\n  union_mem := h𝒜.union_mem\n  sdiff_mem := fun _ _ ↦ h𝒜.sdiff_mem\n\n/-- An algebra of sets is closed by finite unions. -/\ntheorem biUnion_mem {ι : Type*} (h𝒜 : IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι)\n    (hs : ∀ i ∈ S, s i ∈ 𝒜) : ⋃ i ∈ S, s i ∈ 𝒜 := h𝒜.isSetRing.biUnion_mem S hs\n\n/-- An algebra of sets is closed by finite intersections. -/\ntheorem biInter_mem {ι : Type*} (h𝒜 : IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι)\n    (hs : ∀ i ∈ S, s i ∈ 𝒜) : ⋂ i ∈ S, s i ∈ 𝒜 := by\n  by_cases! h : S = ∅\n  · rw [h, ← Finset.set_biInter_coe, Finset.coe_empty, biInter_empty]\n    exact h𝒜.univ_mem\n  · exact h𝒜.isSetRing.biInter_mem S h hs\n\nend IsSetAlgebra\n\nsection generateSetAlgebra\n\n/-! ### Definition and properties of the algebra of sets generated by some family -/\n\n/-- `generateSetAlgebra 𝒜` is the smallest algebra of sets containing `𝒜`. -/\ninductive generateSetAlgebra {α : Type*} (𝒜 : Set (Set α)) : Set (Set α)\n  | base (s : Set α) (s_mem : s ∈ 𝒜) : generateSetAlgebra 𝒜 s\n  | empty : generateSetAlgebra 𝒜 ∅\n  | compl (s : Set α) (hs : generateSetAlgebra 𝒜 s) : generateSetAlgebra 𝒜 sᶜ\n  | union (s t : Set α) (hs : generateSetAlgebra 𝒜 s) (ht : generateSetAlgebra 𝒜 t) :\n      generateSetAlgebra 𝒜 (s ∪ t)\n\n/-- The algebra of sets generated by a family of sets is an algebra of sets. -/\ntheorem isSetAlgebra_generateSetAlgebra :\n    IsSetAlgebra (generateSetAlgebra 𝒜) where\n  empty_mem := generateSetAlgebra.empty\n  compl_mem := fun _ hs ↦ generateSetAlgebra.compl _ hs\n  union_mem := fun _ _ hs ht ↦ generateSetAlgebra.union _ _ hs ht\n\n/-- The algebra of sets generated by `𝒜` contains `𝒜`. -/\ntheorem self_subset_generateSetAlgebra : 𝒜 ⊆ generateSetAlgebra 𝒜 :=\n  fun _ ↦ generateSetAlgebra.base _\n\n/-- The measurable space generated by a family of sets `𝒜` is the same as the one generated\nby the algebra of sets generated by `𝒜`. -/\n@[simp]\ntheorem generateFrom_generateSetAlgebra_eq :\n    generateFrom (generateSetAlgebra 𝒜) = generateFrom 𝒜 := by\n  refine le_antisymm (fun s ms ↦ ?_) (generateFrom_mono self_subset_generateSetAlgebra)\n  induction s, ms using generateFrom_induction with\n  | hC t ht h =>\n    clear h\n    induction ht with\n    | base u u_mem => exact measurableSet_generateFrom u_mem\n    | empty => exact @MeasurableSet.empty _ (generateFrom 𝒜)\n    | compl u _ mu => exact mu.compl\n    | union u v _ _ mu mv => exact MeasurableSet.union mu mv\n  | empty => exact MeasurableSpace.measurableSet_empty _\n  | compl t _ ht => exact ht.compl\n  | iUnion t _ ht => exact .iUnion ht\n\n/-- If a family of sets `𝒜` is contained in `ℬ`, then the algebra of sets generated by `𝒜`\nis contained in the one generated by `ℬ`. -/\n\nTarget:\ntheorem generateSetAlgebra_mono {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ) :\n    generateSetAlgebra 𝒜 ⊆ generateSetAlgebra ℬ :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_865430ae7ad7","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"060547565e9a4dc8913d6bb4c86217cd0dd07ec697d6f20db2a4b4b343a100de","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory","family_id":"generatesetalgebra_mono","file_id":"mathlib/Mathlib/MeasureTheory/SetAlgebra.lean","sample_id":"865430ae7ad7a3af9b967fd8cc81590df7b116ddc79d82d6edeca41aba827e49"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f11a99acaff28e0866b0cf9264b8d87a92a312c57ace97d7c08c3310cdd9276a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5013414c846a303e2d7ae6cf16e87bd0d86e0af5364ad267930855e6631a1174","source_sha256":"2a575e671e878bbb47b82f04d189e9f55eb07313e1f1bf4de528a0907677d433","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Finset.offDiag_filter_lt_eq_filter_le]\n  conv_rhs => rw [← Finset.sum_subtype_eq_sum_filter]\n  refine (Finset.sum_equiv Sym2.sortEquiv.symm ?_ ?_).symm\n  all_goals aesop","hard_negative":false,"metrics":{"chosen_tokens":35,"rejected_tokens":3,"token_jaccard":0.086957,"token_length_ratio":0.085714},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"56c6ef6628ceaf35cf25b4584a3da3371fa8d2b58dd4c39b0364e2a1a833eaa0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Sym\npublic import Mathlib.Data.Sym.Sym2.Order\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2024 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Lemmas on `Finset.sum` and `Finset.prod` involving `Finset.sym2` or `Finset.sym`.\n-/\n\npublic section\n\nnamespace Finset\n\nopen Multiset\n\nTarget:\ntheorem sum_sym2_filter_not_isDiag {ι M} [LinearOrder ι] [AddCommMonoid M]\n    (s : Finset ι) (p : Sym2 ι → M) :\n    ∑ i ∈ s.sym2 with ¬ i.IsDiag, p i = ∑ i ∈ s.offDiag with i.1 < i.2, p s(i.1, i.2) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/BigOperators","family_id":"sum_sym2_filter_not_isdiag","file_id":"mathlib/Mathlib/Algebra/BigOperators/Sym.lean","sample_id":"5013414c846a303e2d7ae6cf16e87bd0d86e0af5364ad267930855e6631a1174"},"split":"train"}
{"binding":{"chosen_proof_sha256":"693a609014e39cacfdbaa86bb0c471973c1d57e2dd7fdea8e3f828ddfc7bc6a5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"01d2f765d25d3c26ead89a2832e8a488c1d450952c409c57941c7491031ad72c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"535e482a49088acaa3ff943705a11086429404a3e1464c08016c546a7309ccb5","source_sha256":"f83ecd09e0c1dc0009269016250874e7bd19aed3be89d47a83b8fee4b9eb7cc4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    refine RelEmbedding.wellFounded_iff_isEmpty.mpr ⟨fun a => ?_⟩\n    suffices sSup (Set.range a) ∈ Set.range a by\n      obtain ⟨n, hn⟩ := Set.mem_range.mp this\n      have h' : sSup (Set.range a) < a (n + 1) := by\n        change _ > _\n        simp [← hn, a.map_rel_iff]\n      apply lt_irrefl (a (n + 1))\n      apply lt_of_le_of_lt _ h'\n      apply le_sSup\n      apply Set.mem_range_self\n    apply h (Set.range a)\n    · use a 37\n      apply Set.mem_range_self\n    · rintro x ⟨m, hm⟩ y ⟨n, hn⟩\n      use m ⊔ n\n      rw [← hm, ← hn]\n      apply RelHomClass.map_sup a","hard_negative":true,"metrics":{"chosen_tokens":141,"rejected_tokens":2,"token_jaccard":0.016393,"token_length_ratio":0.014184},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"56c6ef9aa5a7837ac1fd6948ef0de2a3b32e959c44781abf6df08e9c1bc19ff5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Atoms\npublic import Mathlib.Order.OrderIsoNat\npublic import Mathlib.Order.RelIso.Set\npublic import Mathlib.Order.SupClosed\npublic import Mathlib.Order.SupIndep\npublic import Mathlib.Order.Zorn\npublic import Mathlib.Data.Finset.Order\npublic import Mathlib.Order.Interval.Set.OrderIso\npublic import Mathlib.Data.Finite.Set\npublic import Mathlib.Tactic.TFAE\n\nNamespace:\nCompleteLattice\n\nLocal context:\n/-\nCopyright (c) 2021 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Compactness properties for complete lattices\n\nFor complete lattices, there are numerous equivalent ways to express the fact that the relation `>`\nis well-founded. In this file we define three especially-useful characterisations and provide\nproofs that they are indeed equivalent to well-foundedness.\n\n## Main definitions\n* `CompleteLattice.IsSupClosedCompact`\n* `CompleteLattice.IsSupFiniteCompact`\n* `IsCompactElement`\n* `IsCompactlyGenerated`\n\n## Main results\nThe main result is that the following four conditions are equivalent for a complete lattice:\n* `well_founded (>)`\n* `CompleteLattice.IsSupClosedCompact`\n* `CompleteLattice.IsSupFiniteCompact`\n* `∀ k, IsCompactElement k`\n\nThis is demonstrated by means of the following four lemmas:\n* `CompleteLattice.WellFounded.isSupFiniteCompact`\n* `CompleteLattice.IsSupFiniteCompact.isSupClosedCompact`\n* `CompleteLattice.IsSupClosedCompact.wellFounded`\n* `CompleteLattice.isSupFiniteCompact_iff_all_elements_compact`\n\nWe also show well-founded lattices are compactly generated\n(`CompleteLattice.isCompactlyGenerated_of_wellFounded`).\n\n## References\n- [G. Călugăreanu, *Lattice Concepts of Module Theory*][calugareanu]\n\n## Tags\n\ncomplete lattice, well-founded, compact\n-/\n\n@[expose] public section\n\nopen Set\n/-- An element `k` is compact if any directed set with `LUB` (least upper bound) above\n`k` has already got above `k` at some point in the set.\nSuch an element is also called \"finite\" or \"S-compact\". -/\ndef IsCompactElement {α : Type*} [PartialOrder α] (k : α) :=\n  ∀ (s : Set α) (u : α),\n    s.Nonempty →\n    DirectedOn (· ≤ ·) s →\n    IsLUB s u →\n    k ≤ u →\n    ∃ x ∈ s, k ≤ x\n\nvariable {ι : Sort*} {α : Type*} [CompleteLattice α] {f : ι → α}\n\nnamespace CompleteLattice\n\nvariable (α)\n\n/-- A compactness property for a complete lattice is that any `sup`-closed non-empty subset\ncontains its `sSup`. -/\ndef IsSupClosedCompact : Prop :=\n  ∀ (s : Set α) (_ : s.Nonempty), SupClosed s → sSup s ∈ s\n\n/-- A compactness property for a complete lattice is that any subset has a finite subset with the\nsame `sSup`. -/\ndef IsSupFiniteCompact : Prop :=\n  ∀ s : Set α, ∃ t : Finset α, ↑t ⊆ s ∧ sSup s = t.sup id\n\n/-- An element `k` is compact if and only if any directed set with `sSup` above\n`k` already got above `k` at some point in the set. -/\ntheorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :\n    IsCompactElement k ↔\n      ∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x := by\n  constructor\n  · intro hk s hs hs' h_le\n    exact hk s (sSup s) hs hs' (isLUB_sSup s) h_le\n  · intro h s u hs hs' hu h_le\n    rw [isLUB_iff_sSup_eq] at hu\n    rw [← hu] at h_le\n    exact h s hs hs' h_le\n\n/-- An element `k` of is compact if any set with `sSup`\nabove `k` has a finite subset with `sSup` above `k`. -/\ntheorem isCompactElement_iff_exists_le_sSup_of_le_sSup (k : α) :\n    IsCompactElement k ↔ ∀ s : Set α, k ≤ sSup s → ∃ t : Finset α, ↑t ⊆ s ∧ k ≤ t.sup id := by\n  classical\n    rw [isCompactElement_iff_le_of_directed_sSup_le]\n    constructor\n    · intro hk s hsup\n      -- Consider the set of finite joins of elements of the (plain) set s.\n      let S : Set α := { x | ∃ t : Finset α, ↑t ⊆ s ∧ x = t.sup id }\n      -- S is directed, nonempty, and still has sup above k.\n      have dir_US : DirectedOn (· ≤ ·) S := by\n        rintro x ⟨c, hc⟩ y ⟨d, hd⟩\n        use x ⊔ y\n        constructor\n        · use c ∪ d\n          constructor\n          · simp only [hc.left, hd.left, Set.union_subset_iff, Finset.coe_union, and_self_iff]\n          · simp only [hc.right, hd.right, Finset.sup_union]\n        simp only [and_self_iff, le_sup_left, le_sup_right]\n      have sup_S : sSup s ≤ sSup S := by\n        apply sSup_le_sSup\n        intro x hx\n        use {x}\n        simpa only [and_true, id, Finset.coe_singleton, eq_self_iff_true,\n          Finset.sup_singleton, Set.singleton_subset_iff]\n      have Sne : S.Nonempty := by\n        suffices ⊥ ∈ S from Set.nonempty_of_mem this\n        use ∅\n        simp\n      -- Now apply the defn of compact and finish.\n      obtain ⟨j, ⟨hjS, hjk⟩⟩ := hk S Sne dir_US (le_trans hsup sup_S)\n      obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS\n      use t\n      exact ⟨htS, by rwa [← htsup]⟩\n    · intro hk s hne hdir hsup\n      obtain ⟨t, ht⟩ := hk s hsup\n      -- certainly every element of t is below something in s, since ↑t ⊆ s.\n      have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y := fun x hxt => ⟨x, ht.left hxt, le_rfl⟩\n      obtain ⟨x, ⟨hxs, hsupx⟩⟩ := Finset.sup_le_of_le_directed s hne hdir t t_below_s\n      exact ⟨x, ⟨hxs, le_trans ht.right hsupx⟩⟩\n\ntheorem isCompactElement_iff_exists_le_iSup_of_le_iSup.{u} {α : Type u} [CompleteLattice α]\n    (k : α) : IsCompactElement k ↔\n      ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s := by\n  classical\n    rw [isCompactElement_iff_exists_le_sSup_of_le_sSup]\n    constructor\n    · intro H ι s hs\n      obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs\n      have : ∀ x : t, ∃ i, s i = x := fun x => ht x.prop\n      choose f hf using this\n      refine ⟨Finset.univ.image f, ht'.trans ?_⟩\n      rw [Finset.sup_le_iff]\n      intro b hb\n      rw [← show s (f ⟨b, hb⟩) = id b from hf _]\n      exact Finset.le_sup (Finset.mem_image_of_mem f <| Finset.mem_univ (Subtype.mk b hb))\n    · intro H s hs\n      obtain ⟨t, ht⟩ :=\n        H s Subtype.val\n          (by\n            delta iSup\n            rwa [Subtype.range_coe])\n      refine ⟨t.image Subtype.val, by simp, ht.trans ?_⟩\n      rw [Finset.sup_le_iff]\n      exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem Subtype.val hx)\n\ntheorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type*}\n    (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by\n  classical\n    rw [isCompactElement_iff_le_of_directed_sSup_le] at hk\n    let g : Finset ι → α := fun s => ⨆ i ∈ s, f i\n    have h1 : DirectedOn (· ≤ ·) (Set.range g) := by\n      rintro - ⟨s, rfl⟩ - ⟨t, rfl⟩\n      exact\n        ⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, iSup_le_iSup_of_subset Finset.subset_union_left,\n          iSup_le_iSup_of_subset Finset.subset_union_right⟩\n    have h2 : k ≤ sSup (Set.range g) :=\n      h.trans\n        (iSup_le fun i =>\n          le_sSup_of_le ⟨{i}, rfl⟩\n            (le_iSup_of_le i (le_iSup_of_le (Finset.mem_singleton_self i) le_rfl)))\n    obtain ⟨-, ⟨s, rfl⟩, hs⟩ := hk (Set.range g) (Set.range_nonempty g) h1 h2\n    exact ⟨s, hs⟩\n\n/-- A compact element `k` has the property that any directed set lying strictly below `k` has\nits `sSup` strictly below `k`. -/\ntheorem IsCompactElement.directed_sSup_lt_of_lt {α : Type*} [CompleteLattice α] {k : α}\n    (hk : IsCompactElement k) {s : Set α} (hemp : s.Nonempty) (hdir : DirectedOn (· ≤ ·) s)\n    (hbelow : ∀ x ∈ s, x < k) : sSup s < k := by\n  rw [isCompactElement_iff_le_of_directed_sSup_le] at hk\n  by_contra h\n  have sSup' : sSup s ≤ k := sSup_le fun s hs => (hbelow s hs).le\n  replace sSup : sSup s = k := eq_iff_le_not_lt.mpr ⟨sSup', h⟩\n  obtain ⟨x, hxs, hkx⟩ := hk s hemp hdir sSup.symm.le\n  obtain hxk := hbelow x hxs\n  exact hxk.ne (hxk.le.antisymm hkx)\n\ntheorem isCompactElement_finsetSup {α β : Type*} [CompleteLattice α] {f : β → α} (s : Finset β)\n    (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by\n  classical\n    simp_rw [isCompactElement_iff_le_of_directed_sSup_le] at ⊢ h\n    intro d hemp hdir hsup\n    rw [← Function.id_comp f]\n    rw [← Finset.sup_image]\n    apply Finset.sup_le_of_le_directed d hemp hdir\n    rintro x hx\n    obtain ⟨p, ⟨hps, rfl⟩⟩ := Finset.mem_image.mp hx\n    specialize h p hps\n    specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)\n    simpa only [exists_prop]\n\ntheorem WellFoundedGT.isSupFiniteCompact [WellFoundedGT α] :\n    IsSupFiniteCompact α := fun s => by\n  let S := { x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x }\n  obtain ⟨m, ⟨t, ⟨ht₁, rfl⟩⟩, hm⟩ := wellFounded_gt.has_min S ⟨⊥, ∅, by simp⟩\n  refine ⟨t, ht₁, (sSup_le fun y hy => ?_).antisymm ?_⟩\n  · classical\n    rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))\n        (hm _ ⟨insert y t, by simp [Set.insert_subset_iff, hy, ht₁]⟩)]\n    simp\n  · rw [Finset.sup_id_eq_sSup]\n    exact sSup_le_sSup ht₁\n\ntheorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) :\n    IsSupClosedCompact α := by\n  intro s hne hsc; obtain ⟨t, ht₁, ht₂⟩ := h s; clear h\n  rcases t.eq_empty_or_nonempty with rfl | h\n  · rw [Finset.sup_empty] at ht₂\n    rw [ht₂]\n    simp [eq_singleton_bot_of_sSup_eq_bot_of_nonempty ht₂ hne]\n  · rw [ht₂]\n    exact hsc.finsetSup_mem h ht₁\n\nTarget:\ntheorem IsSupClosedCompact.wellFoundedGT (h : IsSupClosedCompact α) :\n    WellFoundedGT α where\n  wf :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_535e482a4908","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e8d8ba2400f522352e252c3b72c364f3ff83333fd3670dffa56dc5c6ddc78702","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/CompactlyGenerated","family_id":"issupclosedcompact","file_id":"mathlib/Mathlib/Order/CompactlyGenerated/Basic.lean","sample_id":"535e482a49088acaa3ff943705a11086429404a3e1464c08016c546a7309ccb5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7fb3cc1775db12a3f434a99b04b7c0a0f79496406a0eb39713c9f5d39c18a786","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8e9095fa0fd358aa442905acbee67ff7e4183a7fbd628d4a82c82cb398b049c4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b9d6520e23cf7ef508708d905f3af667f91b01c1501bdfb215036dc08f82f360","source_sha256":"3e56b0bca06aab56ed706da176a00f9bb677f8dddf1dc55819d7e75bf8119e45","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [sInter_eq_biInter]\n  exact MeasurableSet.biInter hs h","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.166667,"token_length_ratio":0.272727},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"56ccdf872a2b782017832b60c771462302e196e514f9d1320284f884cb30ccc1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Countable\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\npublic import Mathlib.Tactic.CrossRefAttribute\npublic import Mathlib.Tactic.FunProp.Attr\npublic import Mathlib.Tactic.Measurability\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro\n-/\n/-!\n# Measurable spaces and measurable functions\n\nThis file defines measurable spaces and measurable functions.\n\nA measurable space is a set equipped with a σ-algebra, a collection of\nsubsets closed under complementation and countable union. A function\nbetween measurable spaces is measurable if the preimage of each\nmeasurable subset is measurable.\n\nσ-algebras on a fixed set `α` form a complete lattice. Here we order\nσ-algebras by writing `m₁ ≤ m₂` if every set which is `m₁`-measurable is\nalso `m₂`-measurable (that is, `m₁` is a subset of `m₂`). In particular, any\ncollection of subsets of `α` generates a smallest σ-algebra which\ncontains all of them.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Measurable_space>\n* <https://en.wikipedia.org/wiki/Sigma-algebra>\n* <https://en.wikipedia.org/wiki/Dynkin_system>\n\n## Tags\n\nmeasurable space, σ-algebra, measurable function\n-/\n\n@[expose] public section\n\nassert_not_exists Covariant MonoidWithZero\n\nopen Set Encodable Function Equiv\n\nvariable {α β γ δ δ' : Type*} {ι : Sort*} {s t u : Set α}\n\n/-- A measurable space is a space equipped with a σ-algebra. -/\n@[class] structure MeasurableSpace (α : Type*) where\n  /-- Predicate saying that a given set is measurable. Use `MeasurableSet` in the root namespace\n  instead. -/\n  MeasurableSet' : Set α → Prop\n  /-- The empty set is a measurable set. Use `MeasurableSet.empty` instead. -/\n  measurableSet_empty : MeasurableSet' ∅\n  /-- The complement of a measurable set is a measurable set. Use `MeasurableSet.compl` instead. -/\n  measurableSet_compl : ∀ s, MeasurableSet' s → MeasurableSet' sᶜ\n  /-- The union of a sequence of measurable sets is a measurable set. Use a more general\n  `MeasurableSet.iUnion` instead. -/\n  measurableSet_iUnion : ∀ f : ℕ → Set α, (∀ i, MeasurableSet' (f i)) → MeasurableSet' (⋃ i, f i)\n\ninstance [h : MeasurableSpace α] : MeasurableSpace αᵒᵈ := h\n\n/-- `MeasurableSet s` means that `s` is measurable (in the ambient measure space on `α`) -/\ndef MeasurableSet [MeasurableSpace α] (s : Set α) : Prop :=\n  ‹MeasurableSpace α›.MeasurableSet' s\n\n/-- Notation for `MeasurableSet` with respect to a non-standard σ-algebra. -/\nscoped[MeasureTheory] notation \"MeasurableSet[\" m \"]\" => @MeasurableSet _ m\n\nopen MeasureTheory\n\nsection\n\nopen scoped symmDiff\n\n@[simp, measurability]\ntheorem MeasurableSet.empty [MeasurableSpace α] : MeasurableSet (∅ : Set α) :=\n  MeasurableSpace.measurableSet_empty _\n\nvariable {m : MeasurableSpace α}\n\n@[measurability]\nprotected theorem MeasurableSet.compl : MeasurableSet s → MeasurableSet sᶜ :=\n  MeasurableSpace.measurableSet_compl _ s\n\nprotected theorem MeasurableSet.of_compl (h : MeasurableSet sᶜ) : MeasurableSet s :=\n  compl_compl s ▸ h.compl\n\n@[simp]\ntheorem MeasurableSet.compl_iff : MeasurableSet sᶜ ↔ MeasurableSet s :=\n  ⟨.of_compl, .compl⟩\n\n@[simp, measurability]\nprotected theorem MeasurableSet.univ : MeasurableSet (univ : Set α) :=\n  .of_compl <| by simp\n\n@[nontriviality, measurability]\ntheorem Subsingleton.measurableSet [Subsingleton α] {s : Set α} : MeasurableSet s :=\n  Subsingleton.set_cases MeasurableSet.empty MeasurableSet.univ s\n\ntheorem MeasurableSet.congr {s t : Set α} (hs : MeasurableSet s) (h : s = t) : MeasurableSet t := by\n  rwa [← h]\n\n@[measurability]\nprotected theorem MeasurableSet.iUnion [Countable ι] ⦃f : ι → Set α⦄\n    (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋃ b, f b) := by\n  cases isEmpty_or_nonempty ι\n  · simp\n  · rcases exists_surjective_nat ι with ⟨e, he⟩\n    rw [← iUnion_congr_of_surjective _ he (fun _ => rfl)]\n    exact m.measurableSet_iUnion _ fun _ => h _\n\nprotected theorem MeasurableSet.biUnion {f : β → Set α} {s : Set β} (hs : s.Countable)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) := by\n  rw [biUnion_eq_iUnion]\n  have := hs.to_subtype\n  exact MeasurableSet.iUnion (by simpa using h)\n\ntheorem Set.Finite.measurableSet_biUnion {f : β → Set α} {s : Set β} (hs : s.Finite)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) :=\n  .biUnion hs.countable h\n\ntheorem Finset.measurableSet_biUnion {f : β → Set α} (s : Finset β)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) :=\n  s.finite_toSet.measurableSet_biUnion h\n\nprotected theorem MeasurableSet.sUnion {s : Set (Set α)} (hs : s.Countable)\n    (h : ∀ t ∈ s, MeasurableSet t) : MeasurableSet (⋃₀ s) := by\n  rw [sUnion_eq_biUnion]\n  exact .biUnion hs h\n\ntheorem Set.Finite.measurableSet_sUnion {s : Set (Set α)} (hs : s.Finite)\n    (h : ∀ t ∈ s, MeasurableSet t) : MeasurableSet (⋃₀ s) :=\n  MeasurableSet.sUnion hs.countable h\n\n@[measurability]\ntheorem MeasurableSet.iInter [Countable ι] {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) :\n    MeasurableSet (⋂ b, f b) :=\n  .of_compl <| by rw [compl_iInter]; exact .iUnion fun b => (h b).compl\n\ntheorem MeasurableSet.biInter {f : β → Set α} {s : Set β} (hs : s.Countable)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) :=\n  .of_compl <| by rw [compl_iInter₂]; exact .biUnion hs fun b hb => (h b hb).compl\n\ntheorem Set.Finite.measurableSet_biInter {f : β → Set α} {s : Set β} (hs : s.Finite)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) :=\n  .biInter hs.countable h\n\ntheorem Finset.measurableSet_biInter {f : β → Set α} (s : Finset β)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) :=\n  s.finite_toSet.measurableSet_biInter h\n\nTarget:\ntheorem MeasurableSet.sInter {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s, MeasurableSet t) :\n    MeasurableSet (⋂₀ s) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_b9d6520e23cf","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"a5248173b206fd8a8128faa1c226b843bc44e38d1e25b58dc3db20a5782224c3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/MeasurableSpace","family_id":"measurableset","file_id":"mathlib/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","sample_id":"b9d6520e23cf7ef508708d905f3af667f91b01c1501bdfb215036dc08f82f360"},"split":"train"}
{"binding":{"chosen_proof_sha256":"59cc5684ec22bcbe1a47c40b86884f4ebd0253cacb27e5dc09b1539a4441b454","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dbf6ffda00f48d6763a755f6b1baea6be9639bc7a19fe67ae570fed274326119","source_sha256":"f3eaf728d7cb8f06fa2b67e9feb8de9fb95bffa5aec7e165f2a4546b07059ca7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa [ChainComplex.of.d] using ((inhomogeneousChains A).d_comp_d (n + 2) (n + 1) n)","hard_negative":false,"metrics":{"chosen_tokens":29,"rejected_tokens":3,"token_jaccard":0.05,"token_length_ratio":0.103448},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"575b8df59cf6c8a9c846a363125d9bfbd4a38754ce5172f69ed33cc0a19b8ac4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.ConcreteCategory\npublic import Mathlib.RepresentationTheory.Coinvariants\npublic import Mathlib.RepresentationTheory.Homological.Resolution\npublic import Mathlib.Tactic.CategoryTheory.Slice\npublic import Mathlib.CategoryTheory.Abelian.LeftDerived\n\nNamespace:\ngroupHomology\n\nLocal context:\n/-\nCopyright (c) 2025 Amelia Livingston. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Amelia Livingston\n-/\n/-!\n# The group homology of a `k`-linear `G`-representation\n\nLet `k` be a commutative ring and `G` a group. This file defines the group homology of\n`A : Rep k G` to be the homology of the complex\n$$\\dots \\to \\bigoplus_{G^2} A \\to \\bigoplus_{G^1} A \\to \\bigoplus_{G^0} A$$\nwith differential $d_n$ sending $a\\cdot (g_0, \\dots, g_n)$ to\n$$\\rho(g_0^{-1})(a)\\cdot (g_1, \\dots, g_n)$$\n$$+ \\sum_{i = 0}^{n - 1}(-1)^{i + 1}a\\cdot (g_0, \\dots, g_ig_{i + 1}, \\dots, g_n)$$\n$$+ (-1)^{n + 1}a\\cdot (g_0, \\dots, g_{n - 1})$$ (where `ρ` is the representation attached to `A`).\n\nWe have a `k`-linear isomorphism\n$\\bigoplus_{G^n} A \\cong (A \\otimes_k \\left(\\bigoplus_{G^n} k[G]\\right))_G$ given by\n`Rep.coinvariantsTensorFreeLEquiv`. If we conjugate the $n$th differential in $(A \\otimes_k P)_G$\nby this isomorphism, where `P` is the bar resolution of `k` as a trivial `k`-linear\n`G`-representation, then the resulting map agrees with the differential $d_n$ defined\nabove, a fact we prove.\n\nHence our $d_n$ squares to zero, and we get\n$\\mathrm{H}_n(G, A) \\cong \\mathrm{Tor}_n(A, k),$ where $\\mathrm{Tor}$ is defined by deriving the\nsecond argument of the functor $(A, B) \\mapsto (A \\otimes_k B)_G.$\n\nTo talk about homology in low degree, the file\n`Mathlib/RepresentationTheory/Homological/GroupHomology/LowDegree.lean` provides API specialized to\n`H₀`, `H₁`, `H₂`.\n\n## Main definitions\n\n* `Rep.Tor k G n`: the left-derived functors given by deriving the second argument of\n  $(A, B) \\mapsto (A \\otimes_k B)_G$.\n* `groupHomology.inhomogeneousChains A`: a complex whose objects are\n  $\\bigoplus_{G^n} A$ and whose homology is the group homology $\\mathrm{H}_n(G, A).$\n* `groupHomology.inhomogeneousChainsIso A`: an isomorphism between the above two complexes.\n* `groupHomology A n`: this is $\\mathrm{H}_n(G, A),$ defined as the $n$th homology of the\n  second complex, `inhomogeneousChains A`.\n* `groupHomologyIsoTor A n`: an isomorphism $\\mathrm{H}_n(G, A) \\cong \\mathrm{Tor}_n(A, k)$\n  induced by `inhomogeneousChainsIso A`.\n\n## Implementation notes\n\nGroup homology is typically stated for `G`-modules, or equivalently modules over the group ring\n`ℤ[G].` However, `ℤ` can be generalized to any commutative ring `k`, which is what we use.\nMoreover, we express `k[G]`-module structures on a module `k`-module `A` using the `Rep` definition.\nWe avoid using instances `Module k[G] A` so that we do not run into possible scalar action diamonds.\n\nNote that the existing definition of `Tor` in `Mathlib.CategoryTheory.Monoidal.Tor` is for monoidal\ncategories, and the bifunctor we need to derive here maps to `ModuleCat k`. Hence we define\n`Rep.Tor k G n` by instead left-deriving the second argument of `Rep.coinvariantsTensor k G`:\n$(A, B) \\mapsto (A \\otimes_k B)_G$. The functor `Rep.coinvariantsTensor k G` is naturally\nisomorphic to the functor sending `A, B` to `A ⊗[k[G]] B`, where we give `A` the `k[G]ᵐᵒᵖ`-module\nstructure defined by `g • a := A.ρ g⁻¹ a`, but currently mathlib's `TensorProduct` is only defined\nfor commutative rings.\n\n## TODO\n\n* Upgrading `groupHomologyIsoTor` to an isomorphism of derived functors.\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nuniverse u v w\n\nopen CategoryTheory CategoryTheory.Limits\n\nvariable (k G : Type u) [CommRing k] [Group G]\n\nopen MonoidalCategory Representation Finsupp\n\nsection Tor\n\nvariable {k G} in\n/-- Given `A : Rep k G` and a chain complex `P` in `Rep k G`, this is the chain complex whose\n`n`th object is `(A ⊗ Pₙ)_G`. -/\nabbrev HomologicalComplex.coinvariantsTensorObj {α : Type*} [AddRightCancelSemigroup α] [One α]\n    (A : Rep k G) (P : ChainComplex (Rep k G) α) :\n    ChainComplex (ModuleCat k) α :=\n  (((Rep.coinvariantsTensor k G).obj A).mapHomologicalComplex _).obj P\n\nnamespace Rep\n\n/-- The left-derived functors given by deriving the second argument of `A, B ↦ (A ⊗[k] B)_G`. -/\n@[simps]\ndef Tor (n : ℕ) : Rep k G ⥤ Rep k G ⥤ ModuleCat k where\n  obj X := Functor.leftDerived ((coinvariantsTensor k G).obj X) n\n  map f := NatTrans.leftDerived ((coinvariantsTensor k G).map f) n\n\nvariable {k G} (A : Rep.{w} k G)\n\n/-- `Tor` can be computed using a projective resolution. -/\nabbrev torIso (A : Rep k G) {B : Rep k G} (P : ProjectiveResolution B) (n : ℕ) :\n    ((Rep.Tor k G n).obj A).obj B ≅ (P.complex.coinvariantsTensorObj A).homology n :=\n  P.isoLeftDerivedObj _ n\n\n/-- The higher `Tor` groups for `X` and `Y` are zero if `Y` is projective. -/\nlemma isZero_Tor_succ_of_projective (X Y : Rep k G) [Projective Y] (n : ℕ) :\n    IsZero (((Tor k G (n + 1)).obj X).obj Y) :=\n  Functor.isZero_leftDerived_obj_projective_succ ..\n\nend Rep\nend Tor\n\nnamespace groupHomology\n\nopen Rep Finsupp\n\nvariable {k G : Type u} [CommRing k] [Group G] (A : Rep.{u} k G) (n : ℕ)\n\nnamespace inhomogeneousChains\n\n/-- The differential in the complex of inhomogeneous chains used to calculate group homology. -/\ndef d : ModuleCat.of k ((Fin (n + 1) → G) →₀ A) ⟶ ModuleCat.of k ((Fin n → G) →₀ A) :=\n  ModuleCat.ofHom <| lsum (R := k) k fun g => lsingle (fun i => g i.succ) ∘ₗ A.ρ (g 0)⁻¹ +\n    Finset.univ.sum fun j : Fin (n + 1) =>\n      (-1 : k) ^ ((j : ℕ) + 1) • lsingle (Fin.contractNth j (· * ·) g)\n\nvariable {A n} in\n@[simp]\ntheorem d_single (n : ℕ) (g : Fin (n + 1) → G) (a : A) :\n    d A n (single g a) = single (fun i => g i.succ) (A.ρ (g 0)⁻¹ a) +\n      Finset.univ.sum fun j : Fin (n + 1) =>\n        (-1 : k) ^ ((j : ℕ) + 1) • single (Fin.contractNth j (· * ·) g) a := by\n  simp [d]\n\nopen ModuleCat.MonoidalCategory\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\ntheorem d_eq [DecidableEq G] :\n    d A n = (coinvariantsTensorFreeLEquiv A (Fin (n + 1) → G)).toModuleIso.inv ≫\n      ((barComplex k G).coinvariantsTensorObj A).d (n + 1) n ≫\n      (coinvariantsTensorFreeLEquiv A (Fin n → G)).toModuleIso.hom := by\n  ext : 3\n  simp [d_single (k := k), TensorProduct.tmul_add, TensorProduct.tmul_sum,\n    barComplex.d_single (k := k)]\n\nend inhomogeneousChains\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Given a `k`-linear `G`-representation `A`, this is the complex of inhomogeneous chains\n$$\\dots \\to \\bigoplus_{G^1} A \\to \\bigoplus_{G^0} A \\to 0$$\nwhich calculates the group homology of `A`. -/\nnoncomputable abbrev inhomogeneousChains :\n    ChainComplex (ModuleCat k) ℕ :=\n  ChainComplex.of (fun n => ModuleCat.of k ((Fin n → G) →₀ A))\n    (fun n => inhomogeneousChains.d A n) fun n => by\n    classical\n    rw [inhomogeneousChains.d_eq, inhomogeneousChains.d_eq]\n    slice_lhs 3 4 => rw [Iso.hom_inv_id]\n    slice_lhs 2 4 => rw [Category.id_comp, ((barComplex k G).coinvariantsTensorObj A).d_comp_d]\n    simp\n\nopen inhomogeneousChains\n\nvariable {A n} in\n@[ext]\ntheorem inhomogeneousChains.ext {M : ModuleCat k} {x y : (inhomogeneousChains A).X n ⟶ M}\n    (h : ∀ g, ModuleCat.ofHom (lsingle g) ≫ x = ModuleCat.ofHom (lsingle g) ≫ y) :\n    x = y := ModuleCat.hom_ext <| lhom_ext' fun g => ModuleCat.hom_ext_iff.1 (h g)\n\ntheorem inhomogeneousChains.d_def (n : ℕ) :\n    (inhomogeneousChains A).d (n + 1) n = d A n := by\n  simp [inhomogeneousChains]\n\nset_option backward.defeqAttrib.useBackward true in\n\nTarget:\ntheorem inhomogeneousChains.d_comp_d :\n    d A (n + 1) ≫ d A n = 0 :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RepresentationTheory/Homological","family_id":"inhomogeneouschains","file_id":"mathlib/Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean","sample_id":"dbf6ffda00f48d6763a755f6b1baea6be9639bc7a19fe67ae570fed274326119"},"split":"train"}
{"binding":{"chosen_proof_sha256":"407de8aea2c8c096a4b6c72b70c997e3bb00015ef6148a6a35ed2aaaed552283","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b0eb9dca5eadf93208ff9a93aeaae01b29686cff0a978676d7b76265513743f7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"99a463e8b538cc1129c05262ca34214f41e74d8600f9e3aa84ec10439902b1f8","source_sha256":"fdcf591465bcad30a0f93e23ea718e1d3a47c3c81447ba4e02561aad0c6cb720","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    obtain ⟨T', e, h'⟩ := hF T hT\n    exact (ShortComplex.exact_iff_of_iso\n      (F.mapShortComplex.mapIso ((shortComplexOfDistTriangleIsoOfIso e hT)))).2 h'","hard_negative":true,"metrics":{"chosen_tokens":36,"rejected_tokens":5,"token_jaccard":0.12,"token_length_ratio":0.138889},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"57654030e90807aefc68d361d835d2671f8d382a1dbe9aa740bc461c3db6ab3b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.ShortComplex.Exact\npublic import Mathlib.CategoryTheory.Shift.ShiftSequence\npublic import Mathlib.CategoryTheory.Triangulated.Functor\npublic import Mathlib.CategoryTheory.Triangulated.Subcategory\npublic import Mathlib.Algebra.Homology.ExactSequence\n\nNamespace:\nCategoryTheory.Functor\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-! # Homological functors\n\nIn this file, given a functor `F : C ⥤ A` from a pretriangulated category to\nan abelian category, we define the type class `F.IsHomological`, which is the property\nthat `F` sends distinguished triangles in `C` to exact sequences in `A`.\n\nIf `F` has been endowed with `[F.ShiftSequence ℤ]`, then we may think\nof the functor `F` as a `H^0`, and then the `H^n` functors are the functors `F.shift n : C ⥤ A`:\nwe have isomorphisms `(F.shift n).obj X ≅ F.obj (X⟦n⟧)`, but through the choice of this\n\"shift sequence\", the user may provide functors with better definitional properties.\n\nGiven a triangle `T` in `C`, we define a connecting homomorphism\n`F.homologySequenceδ T n₀ n₁ h : (F.shift n₀).obj T.obj₃ ⟶ (F.shift n₁).obj T.obj₁`\nunder the assumption `h : n₀ + 1 = n₁`. When `T` is distinguished, this connecting\nhomomorphism is part of a long exact sequence\n`... ⟶ (F.shift n₀).obj T.obj₁ ⟶ (F.shift n₀).obj T.obj₂ ⟶ (F.shift n₀).obj T.obj₃ ⟶ ...`\n\nThe exactness of this long exact sequence is given by three lemmas\n`F.homologySequence_exact₁`, `F.homologySequence_exact₂` and `F.homologySequence_exact₃`.\n\nIf `F` is a homological functor, we define the strictly full triangulated subcategory\n`F.homologicalKernel`: it consists of objects `X : C` such that for all `n : ℤ`,\n`(F.shift n).obj X` (or `F.obj (X⟦n⟧)`) is zero. We show that a morphism `f` in `C`\nbelongs to `F.homologicalKernel.trW` (i.e. the cone of `f` is in this kernel) iff\n`(F.shift n).map f` is an isomorphism for all `n : ℤ`.\n\nNote: depending on the sources, homological functors are sometimes\ncalled cohomological functors, while certain authors use \"cohomological functors\"\nfor \"contravariant\" functors (i.e. functors `Cᵒᵖ ⥤ A`).\n\n## References\n* [Jean-Louis Verdier, *Des catégories dérivées des catégories abéliennes*][verdier1996]\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Category Limits Pretriangulated ZeroObject Preadditive\n\nvariable {C D A : Type*} [Category* C] [HasShift C ℤ]\n  [Category* D] [HasZeroObject D] [HasShift D ℤ] [Preadditive D]\n  [∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] [Pretriangulated D]\n  [Category* A]\n\nnamespace Functor\n\nvariable (F : C ⥤ A)\n\n/-- The kernel of a homological functor `F : C ⥤ A` is the strictly full\ntriangulated subcategory consisting of objects `X` such that\nfor all `n : ℤ`, `F.obj (X⟦n⟧)` is zero. -/\ndef homologicalKernel : ObjectProperty C :=\n  fun X ↦ ∀ (n : ℤ), IsZero (F.obj (X⟦n⟧))\n\nlemma mem_homologicalKernel_iff [F.ShiftSequence ℤ] (X : C) :\n    F.homologicalKernel X ↔ ∀ (n : ℤ), IsZero ((F.shift n).obj X) := by\n  simp only [← fun (n : ℤ) => Iso.isZero_iff ((F.isoShift n).app X),\n    homologicalKernel, comp_obj]\n\nsection Pretriangulated\n\nvariable [HasZeroObject C] [Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive]\n  [Pretriangulated C] [Abelian A]\n\n/-- A functor from a pretriangulated category to an abelian category is a homological functor\nif it sends distinguished triangles to exact sequences. -/\nclass IsHomological : Prop extends F.PreservesZeroMorphisms where\n  exact (T : Triangle C) (hT : T ∈ distTriang C) :\n    ((shortComplexOfDistTriangle T hT).map F).Exact\n\nlemma map_distinguished_exact [F.IsHomological] (T : Triangle C) (hT : T ∈ distTriang C) :\n    ((shortComplexOfDistTriangle T hT).map F).Exact :=\n  IsHomological.exact _ hT\n\ninstance (L : C ⥤ D) (F : D ⥤ A) [L.CommShift ℤ] [L.IsTriangulated] [F.IsHomological] :\n    (L ⋙ F).IsHomological where\n  exact T hT := F.map_distinguished_exact _ (L.map_distinguished T hT)\n\nTarget:\nlemma IsHomological.mk' [F.PreservesZeroMorphisms]\n    (hF : ∀ (T : Pretriangulated.Triangle C) (hT : T ∈ distTriang C),\n      ∃ (T' : Pretriangulated.Triangle C) (e : T ≅ T'),\n      ((shortComplexOfDistTriangle T' (isomorphic_distinguished _ hT _ e.symm)).map F).Exact) :\n    F.IsHomological where\n  exact T hT :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_99a463e8b538","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"ce8f3ce39a285957bb0c08bd803f77516ad2c33a66ea13ba64029dc0e6154946","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Triangulated","family_id":"ishomological","file_id":"mathlib/Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean","sample_id":"99a463e8b538cc1129c05262ca34214f41e74d8600f9e3aa84ec10439902b1f8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8b8c47751a3cd16893a65f53a169d429e22504bc42c9299e12e8db816e1165e6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a559373815454d0166c5dab8fbbe7db64a117d1cd999728a36144f94e03e6450","source_sha256":"e05f0de1b38fa7ce1307d0699fed1c367aa4ac0c4de0f78f8d97c1a3bc8f1b0d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  induction s using Finset.induction generalizing j with\n  | empty => simp [coeff_one, apply_ite, isIntegral_zero, isIntegral_one]\n  | insert a s has IH =>\n    rw [Finset.prod_insert has, coeff_mul]\n    exact IsIntegral.sum _ fun i hi ↦ .mul (H _ (by simp) _) (IH (fun _ _ ↦ H _ (by aesop)) _)","hard_negative":false,"metrics":{"chosen_tokens":76,"rejected_tokens":2,"token_jaccard":0.02381,"token_length_ratio":0.026316},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"57a2913553354a4249f36a6e43e5e817282ab87c0ae7bbac929161f04d7a3889","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Multiset.Fintype\npublic import Mathlib.RingTheory.AdjoinRoot\npublic import Mathlib.RingTheory.Polynomial.RationalRoot\npublic import Mathlib.RingTheory.IntegralClosure.IsIntegral.AlmostIntegral\n\nNamespace:\nPolynomial\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Results about coefficients of polynomials being integral\n\n## Main results\n- `Polynomial.isIntegral_coeff_of_dvd`: If a monic polynomial `p` divides another monic polynomial\n  with integral coefficients, then the coefficients of `p` are themselves integral.\n- `Polynomial.isIntegral_iff_isIntegral_coeff`:\n  `p : S[X]` is integral over `R[X]` iff the coefficients of `p` are integral over `R`.\n- `MvPolynomial.isIntegral_iff_isIntegral_coeff`: `p : MvPolynomial σ S` is integral over\n  `MvPolynomial σ R` iff the coefficients of `p` are integral over `R`.\n- We also provide the instance `[IsIntegrallyClosed R] : IsIntegrallyClosed R[X]`.\n\n-/\n\npublic section\n\nvariable {R S ι : Type*} [CommRing R] [CommRing S] [Algebra R S]\n\nnamespace Polynomial\n\nTarget:\nlemma isIntegral_coeff_prod\n    (s : Finset ι) (p : ι → S[X]) (H : ∀ i ∈ s, ∀ j, IsIntegral R ((p i).coeff j)) (j : ℕ) :\n    IsIntegral R ((s.prod p).coeff j) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Polynomial","family_id":"isintegral_coeff_prod","file_id":"mathlib/Mathlib/RingTheory/Polynomial/IsIntegral.lean","sample_id":"a559373815454d0166c5dab8fbbe7db64a117d1cd999728a36144f94e03e6450"},"split":"train"}
{"binding":{"chosen_proof_sha256":"31a2975efba554cce04bfa852066668eb27b98da82a9d33652d228e97ecec653","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e1358bd4ad7cf3e79ebde6f12a407b8ab63bc3e0fb649df13922ddc8c1e08091","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7e8c17610a3baa53fbdd954115a7d7f9ea7c655e0a0e548683cb4072c1194356","source_sha256":"75f644a2aa9ce33f4e3a9b3af6b61387bae9fdea7ce3dbf0ef1ec4c5bea1cda5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [removeFactor, natDegree_divByMonic _ (monic_X_sub_C _), natDegree_map, natDegree_X_sub_C]","hard_negative":true,"metrics":{"chosen_tokens":16,"rejected_tokens":5,"token_jaccard":0.058824,"token_length_ratio":0.3125},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"580b736f64034e0c9dde6c8163e1d78aad7ef24d4073614005343df6c3f7334a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharP.Algebra\npublic import Mathlib.FieldTheory.SplittingField.IsSplittingField\npublic import Mathlib.RingTheory.Algebraic.Basic\n\nNamespace:\nPolynomial\n\nLocal context:\n/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Splitting fields\n\nIn this file we prove the existence and uniqueness of splitting fields.\n\n## Main definitions\n\n* `Polynomial.SplittingField f`: A fixed splitting field of the polynomial `f`.\n\n## Main statements\n\n* `Polynomial.IsSplittingField.algEquiv`: Every splitting field of a polynomial `f` is isomorphic\n  to `SplittingField f` and thus, being a splitting field is unique up to isomorphism.\n\n## Implementation details\nWe construct a `SplittingFieldAux` without worrying about whether the instances satisfy nice\ndefinitional equalities. Then the actual `SplittingField` is defined to be a quotient of a\n`MvPolynomial` ring by the kernel of the obvious map into `SplittingFieldAux`. Because the\nactual `SplittingField` will be a quotient of a `MvPolynomial`, it has nice instances on it.\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nuniverse u v w\n\nvariable {F : Type u} {K : Type v} {L : Type w}\n\nnamespace Polynomial\n\nvariable [Field K] [Field L] [Field F]\n\nopen Polynomial\n\nsection SplittingField\n\nopen Classical in\n/-- Non-computably choose an irreducible factor from a polynomial. -/\ndef factor (f : K[X]) : K[X] :=\n  if H : ∃ g, Irreducible g ∧ g ∣ f then Classical.choose H else X\n\ntheorem irreducible_factor (f : K[X]) : Irreducible (factor f) := by\n  rw [factor]\n  split_ifs with H\n  · exact (Classical.choose_spec H).1\n  · exact irreducible_X\n\n/-- See note [fact non-instances]. -/\ntheorem fact_irreducible_factor (f : K[X]) : Fact (Irreducible (factor f)) :=\n  ⟨irreducible_factor f⟩\n\nattribute [local instance] fact_irreducible_factor\n\ntheorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f := by\n  by_cases hf2 : f = 0; · rw [hf2]; exact dvd_zero _\n  rw [factor, dif_pos (WfDvdMonoid.exists_irreducible_factor hf1 hf2)]\n  exact (Classical.choose_spec <| WfDvdMonoid.exists_irreducible_factor hf1 hf2).2\n\ntheorem factor_dvd_of_degree_ne_zero {f : K[X]} (hf : f.degree ≠ 0) : factor f ∣ f :=\n  factor_dvd_of_not_isUnit (mt degree_eq_zero_of_isUnit hf)\n\ntheorem factor_dvd_of_natDegree_ne_zero {f : K[X]} (hf : f.natDegree ≠ 0) : factor f ∣ f :=\n  factor_dvd_of_degree_ne_zero (mt natDegree_eq_of_degree_eq_some hf)\n\nlemma isCoprime_iff_aeval_ne_zero (f g : K[X]) : IsCoprime f g ↔ ∀ {A : Type v} [CommRing A]\n    [IsDomain A] [Algebra K A] (a : A), aeval a f ≠ 0 ∨ aeval a g ≠ 0 := by\n  refine ⟨fun h => aeval_ne_zero_of_isCoprime h, fun h => isCoprime_of_dvd _ _ ?_ fun x hx _ => ?_⟩\n  · replace h := @h K _ _ _ 0\n    contrapose! h\n    rw [h.left, h.right, map_zero, and_self]\n  · rintro ⟨_, rfl⟩ ⟨_, rfl⟩\n    replace h := not_and_or.mpr <| h <| AdjoinRoot.root x.factor\n    simp only [AdjoinRoot.aeval_eq, AdjoinRoot.mk_eq_zero,\n      dvd_mul_of_dvd_left <| factor_dvd_of_not_isUnit hx, true_and, not_true] at h\n\n/-- Divide a polynomial f by `X - C r` where `r` is a root of `f` in a bigger field extension. -/\ndef removeFactor (f : K[X]) : Polynomial (AdjoinRoot <| factor f) :=\n  map (AdjoinRoot.of f.factor) f /ₘ (X - C (AdjoinRoot.root f.factor))\n\ntheorem X_sub_C_mul_removeFactor (f : K[X]) (hf : f.natDegree ≠ 0) :\n    (X - C (AdjoinRoot.root f.factor)) * f.removeFactor = map (AdjoinRoot.of f.factor) f := by\n  let ⟨g, hg⟩ := factor_dvd_of_natDegree_ne_zero hf\n  apply (mul_divByMonic_eq_iff_isRoot\n    (R := AdjoinRoot f.factor) (a := AdjoinRoot.root f.factor)).mpr\n  rw [IsRoot.def, eval_map, hg, eval₂_mul, ← hg, AdjoinRoot.eval₂_root, zero_mul]\n\nTarget:\ntheorem natDegree_removeFactor (f : K[X]) : f.removeFactor.natDegree = f.natDegree - 1 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_7e8c17610a3b","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c4e097d2e5270f07c3fc95d5c3727ad68846f1a480964a231e43128455b3d092","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/SplittingField","family_id":"natdegree_removefactor","file_id":"mathlib/Mathlib/FieldTheory/SplittingField/Construction.lean","sample_id":"7e8c17610a3baa53fbdd954115a7d7f9ea7c655e0a0e548683cb4072c1194356"},"split":"train"}
{"binding":{"chosen_proof_sha256":"17f240f800994d7074febbde9e3f6dc2b5012103ed8b58582f7bf051f16e73b0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"71b5431732be50bdd6b70dfcf19109b211b0c8cbf137e0a38e4a305dcdc38202","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dd825554f724a499bc7ed68c1db6d9b3cab6e04c69fa70bc2acb5b885250233e","source_sha256":"85d2f87939059f796ffb510eeb3d6af5b97a8cdc2c5223d679b28800dce394e2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro b₁ b₂ h\n  simpa only [and_true, eq_self_iff_true, mk_inj] using h","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":2,"token_jaccard":0.0625,"token_length_ratio":0.111111},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"5867f91fb5358c488a96554e8f024260fb11a11ac630cd3f2b05d8c8c663b0d4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Lean.PrettyPrinter.Delaborator.Builtins\npublic import Mathlib.Logic.Function.Defs\npublic import Mathlib.Logic.Function.Iterate\npublic import Mathlib.Tactic.Inhabit\npublic import Batteries.Tactic.Trans\nimport Mathlib.Tactic.Attr.Register\n\nNamespace:\nProd\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Extra facts about `Prod`\n\nThis file proves various simple lemmas about `Prod`.\nIt also defines better delaborators for product projections.\n-/\n\n@[expose] public section\n\nvariable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}\n\nnamespace Prod\n\nlemma swap_eq_iff_eq_swap {x : α × β} {y : β × α} : x.swap = y ↔ x = y.swap := by grind\n\ndef mk.injArrow {x₁ : α} {y₁ : β} {x₂ : α} {y₂ : β} :\n    (x₁, y₁) = (x₂, y₂) → ∀ ⦃P : Sort*⦄, (x₁ = x₂ → y₁ = y₂ → P) → P := by\n  intros h P w\n  cases h\n  exact w rfl rfl\n\n@[simp]\ntheorem mk.eta : ∀ {p : α × β}, (p.1, p.2) = p\n  | (_, _) => rfl\n\ntheorem forall' {p : α → β → Prop} : (∀ x : α × β, p x.1 x.2) ↔ ∀ a b, p a b :=\n  Prod.forall\n\ntheorem exists' {p : α → β → Prop} : (∃ x : α × β, p x.1 x.2) ↔ ∃ a b, p a b :=\n  Prod.exists\n\n@[simp]\ntheorem snd_comp_mk (x : α) : Prod.snd ∘ (Prod.mk x : β → α × β) = id :=\n  rfl\n\n@[simp]\ntheorem fst_comp_mk (x : α) : Prod.fst ∘ (Prod.mk x : β → α × β) = Function.const β x :=\n  rfl\n\nattribute [mfld_simps] map_apply\n\n-- This was previously a `simp` lemma, but no longer is on the basis that it destructures the pair.\n--  See `map_apply`, `map_fst`, and `map_snd` for slightly weaker lemmas in the `simp` set.\ntheorem map_apply' (f : α → γ) (g : β → δ) (p : α × β) : map f g p = (f p.1, g p.2) :=\n  rfl\n\ntheorem map_fst' (f : α → γ) (g : β → δ) : Prod.fst ∘ map f g = f ∘ Prod.fst :=\n  funext <| map_fst f g\n\ntheorem map_snd' (f : α → γ) (g : β → δ) : Prod.snd ∘ map f g = g ∘ Prod.snd :=\n  funext <| map_snd f g\n\ntheorem mk_inj {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) = (a₂, b₂) ↔ a₁ = a₂ ∧ b₁ = b₂ := by simp\n\ntheorem mk_right_injective {α β : Type*} (a : α) : (mk a : β → α × β).Injective := by\n  intro b₁ b₂ h\n  simpa only [true_and, Prod.mk_inj, eq_self_iff_true] using h\n\nTarget:\ntheorem mk_left_injective {α β : Type*} (b : β) : (fun a ↦ mk a b : α → α × β).Injective :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_dd825554f724","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"3b0b53c6afff56ef38aa0c0f1ea09f08eef30e10813f7c00134bb70bfb00fd41","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Prod","family_id":"mk_left_injective","file_id":"mathlib/Mathlib/Data/Prod/Basic.lean","sample_id":"dd825554f724a499bc7ed68c1db6d9b3cab6e04c69fa70bc2acb5b885250233e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"badd85084558f6da2abee0803d9cfe41d37ed7236c401bd880c419ea46047156","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"37ef8d929e9bcfe001f6766d7178835316111afc2781f679460581bd31bb046f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"73b36450d7beabc8b542d5bd6e54c7d1928d98c7ccc3d417ffbb153021170e98","source_sha256":"3ebf52bd4f56640ff82f3663b47b89c207fb7b51b21d5bfbbe3cfb7edd34865e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [lt_univ']\n  constructor\n  · rintro ⟨β, e⟩\n    exact ⟨#β, lift_mk_eq.{u, _, v + 1}.2 e⟩\n  · rintro ⟨c, hc⟩\n    exact ⟨⟨c.out, lift_mk_eq.{u, _, v + 1}.1 (hc.trans (congr rfl c.mk_out.symm))⟩⟩","hard_negative":false,"metrics":{"chosen_tokens":76,"rejected_tokens":81,"token_jaccard":0.897436,"token_length_ratio":1.065789},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"587b1e609a8abb0fe56a4eb42f42ab21da66f44e1cff5588bbf034106c5df7ab","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.SetTheory.Ordinal.Basic\n\nNamespace:\nCardinal\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Floris van Doorn\n-/\n/-!\n# Universal ordinal and cardinal\n\n`Cardinal.univ` is the cardinality of the cardinals themselves. Likewise, `Ordinal.univ` is the\norder type of the ordinals. These are related via `Cardinal.univ.ord = Ordinal.univ` and\n`Ordinal.univ.card = Cardinal.univ`.\n\nThe cardinal `Cardinal.univ` is strongly inaccessible. This reflects the fact that in ZFC, the\ncardinals form a proper class. See `IsInaccessible.univ` for a proof.\n\n## Implementation notes\n\nWe actually define `Cardinal.univ` as the cardinality of `Ordinal`, rather than that of `Cardinal`.\nThis makes the basic API easier to set up. See `Cardinal.mk_cardinal` for a proof that\n`Cardinal.univ = #Cardinal`.\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nset_option linter.checkUnivs false in\nopen Ordinal in\n-- intended to be used with explicit universe parameters\n/-- The ordinal `univ.{u, v}` is the order type of `Ordinal.{u}` or `Cardinal.{u}`, as an element of\n`Ordinal.{v}` (when `u < v`). -/\n@[pp_with_univ, nolint checkUnivs]\ndef Ordinal.univ : Ordinal.{max (u + 1) v} :=\n  lift.{v, u + 1} (typeLT Ordinal)\n\nset_option linter.checkUnivs false in\nopen Cardinal in\n-- intended to be used with explicit universe parameters\n/-- The cardinal `univ.{u, v}` is the cardinality of `Ordinal.{u}` or `Cardinal.{u}`, as an element\nof `Cardinal.{v}` (when `u < v`). -/\n@[pp_with_univ, nolint checkUnivs]\ndef Cardinal.univ : Cardinal.{max (u + 1) v} :=\n  lift.{v, u + 1} #Ordinal\n\n/-! ### Universal ordinal -/\n\nnamespace Ordinal\n\n@[simp]\ntheorem type_lt_ordinal : typeLT Ordinal = univ.{u, u + 1} :=\n  (lift_id _).symm\n\n@[deprecated type_lt_ordinal (since := \"2026-03-20\")]\ntheorem univ_id : univ.{u, u + 1} = typeLT Ordinal :=\n  lift_id _\n\n@[simp]\ntheorem lift_univ : lift.{w} univ.{u, v} = univ.{u, max v w} :=\n  lift_lift _\n\ntheorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=\n  congr_fun lift_umax _\n\n/-- Principal segment version of the lift operation on ordinals, embedding `Ordinal.{u}` in\n`Ordinal.{v}` as a principal segment when `u < v`. -/\ndef liftPrincipalSeg : Ordinal.{u} <i Ordinal.{max (u + 1) v} :=\n  ⟨liftInitialSeg.{max (u + 1) v, u}, univ.{u, v}, by\n    refine fun b => inductionOn b ?_; intro β s _\n    rw [univ, ← lift_umax]; constructor <;> intro h\n    · obtain ⟨a, e⟩ := h\n      rw [← e]\n      refine inductionOn a ?_\n      intro α r _\n      exact lift_type_lt.{u, u + 1, max (u + 1) v}.2 ⟨typein r⟩\n    · rw [← lift_id (type s)] at h ⊢\n      obtain ⟨f⟩ := lift_type_lt.{_,_,v}.1 h\n      obtain ⟨f, a, hf⟩ := f\n      exists a\n      induction a using inductionOn with | type α r\n      refine lift_type_eq.{u, max (u + 1) v, max (u + 1) v}.2\n        ⟨(RelIso.ofSurjective (RelEmbedding.ofMonotone ?_ ?_) ?_).symm⟩\n      · exact fun b => enum r ⟨f b, (hf _).1 ⟨_, rfl⟩⟩\n      · refine fun a b h => (typein_lt_typein r).1 ?_\n        rw [typein_enum, typein_enum]\n        exact f.map_rel_iff.2 h\n      · intro a'\n        obtain ⟨b, e⟩ := (hf _).2 (typein_lt_type _ a')\n        exists b\n        simp only [RelEmbedding.ofMonotone_coe]\n        simp [e]⟩\n\n@[simp]\ntheorem liftPrincipalSeg_coe :\n    (liftPrincipalSeg.{u, v} : Ordinal → Ordinal) = lift.{max (u + 1) v} :=\n  rfl\n\n@[simp]\ntheorem liftPrincipalSeg_top : (liftPrincipalSeg.{u, v}).top = univ.{u, v} :=\n  rfl\n\n@[deprecated liftPrincipalSeg_top (since := \"2026-03-20\")]\ntheorem liftPrincipalSeg_top' : liftPrincipalSeg.{u, u + 1}.top = typeLT Ordinal := by\n  simp\n\n@[simp]\ntheorem card_univ : card univ.{u, v} = Cardinal.univ.{u, v} :=\n  rfl\n\nend Ordinal\n\n/-! ### Universal cardinal -/\n\nnamespace Cardinal\n\n@[simp]\ntheorem mk_ordinal : #Ordinal = univ.{u, u + 1} :=\n  (lift_id _).symm\n\n@[deprecated mk_ordinal (since := \"2026-04-22\")]\ntheorem univ_id : univ.{u, u + 1} = #Ordinal :=\n  lift_id _\n\n@[simp]\ntheorem lift_univ : lift.{w} univ.{u, v} = univ.{u, max v w} :=\n  lift_lift _\n\ntheorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=\n  congr_fun lift_umax _\n\ntheorem lift_lt_univ (c : Cardinal) : lift.{u + 1, u} c < univ.{u, u + 1} := by\n  simpa only [Ordinal.liftPrincipalSeg_coe, lift_ord, lift_succ, ord_le, Order.succ_le_iff] using!\n    le_of_lt (Ordinal.liftPrincipalSeg.{u, u + 1}.lt_top (Order.succ c).ord)\n\ntheorem lift_lt_univ' (c : Cardinal) : lift.{max (u + 1) v, u} c < univ.{u, v} := by\n  have := lift_lt.{_, max (u + 1) v}.2 (lift_lt_univ c)\n  rw [lift_lift, lift_univ, univ_umax.{u, v}] at this\n  exact this\n\n@[simp]\ntheorem aleph0_lt_univ : ℵ₀ < univ.{u, v} := by\n  simpa using lift_lt_univ' ℵ₀\n\n@[simp]\ntheorem nat_lt_univ (n : ℕ) : n < univ.{u, v} := natCast_lt_aleph0.trans aleph0_lt_univ\n\n@[simp]\ntheorem univ_pos : 0 < univ.{u, v} :=\n  aleph0_lt_univ.pos\n\n@[simp]\ntheorem univ_ne_zero : univ.{u, v} ≠ 0 :=\n  univ_pos.ne'\n\n@[simp]\ntheorem ord_univ : ord univ.{u, v} = Ordinal.univ.{u, v} := by\n  refine le_antisymm (ord_card_le _) <| le_of_forall_lt fun o h => lt_ord.2 ?_\n  have := Ordinal.liftPrincipalSeg.mem_range_of_rel_top (by simpa using h)\n  rcases this with ⟨o, h'⟩\n  rw [← h', Ordinal.liftPrincipalSeg_coe, ← Ordinal.lift_card]\n  apply lift_lt_univ'\n\ntheorem lt_univ {c} : c < univ.{u, u + 1} ↔ ∃ c', c = lift.{u + 1, u} c' :=\n  ⟨fun h => by\n    have := ord_lt_ord.2 h\n    rw [ord_univ] at this\n    obtain ⟨o, e⟩ := Ordinal.liftPrincipalSeg.mem_range_of_rel_top (by simpa)\n    have := card_ord c\n    rw [← e, Ordinal.liftPrincipalSeg_coe, ← Ordinal.lift_card] at this\n    exact ⟨_, this.symm⟩, fun ⟨_, e⟩ => e.symm ▸ lift_lt_univ _⟩\n\ntheorem lt_univ' {c} : c < univ.{u, v} ↔ ∃ c', c = lift.{max (u + 1) v, u} c' :=\n  ⟨fun h => by\n    let ⟨a, h', e⟩ := lt_lift_iff.1 h\n    rw [mk_ordinal] at h'\n    rcases lt_univ.{u}.1 h' with ⟨c', rfl⟩\n    exact ⟨c', by simp only [e.symm, lift_lift]⟩, fun ⟨_, e⟩ => e.symm ▸ lift_lt_univ' _⟩\n\ntheorem IsStrongLimit.univ : IsStrongLimit univ.{u, v} :=\n  ⟨univ_ne_zero, fun c h ↦ let ⟨w, h⟩ := lt_univ'.1 h; lt_univ'.2 ⟨2 ^ w, by simp [h]⟩⟩\n\nTarget:\ntheorem small_iff_lift_mk_lt_univ {α : Type u} :\n    Small.{v} α ↔ Cardinal.lift.{v + 1, _} #α < univ.{v, max u (v + 1)} :=\n\nProof body:\n","rejected":"by\n  rw [lt_univ']\n  constructor\n  · rintro ⟨β, e⟩\n    exact ⟨#β, lift_mk_eq.{u, _, v + 1}.2 e⟩\n  · rintro ⟨c, hc⟩\n    exact ⟨⟨c.out, lift_mk_eq.{u, _, v + 1}.1 (hc.trans (congr rfl c.mk_out.symm))⟩⟩\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Ordinal","family_id":"small_iff_lift_mk_lt_univ","file_id":"mathlib/Mathlib/SetTheory/Ordinal/Univ.lean","sample_id":"73b36450d7beabc8b542d5bd6e54c7d1928d98c7ccc3d417ffbb153021170e98"},"split":"train"}
{"binding":{"chosen_proof_sha256":"38de9a26cb91dd01aaa9b7d35dd028817eb092a585a923cb7e7b911ff5d759bb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"836ef9c3cd55ae2e3f293b4d27ae834545f003ae5a97617e40dffc59a7b9fdae","source_sha256":"98d8a988714b8411039a52ca8cb94b19a051f27bd0e995b1332bddd6c23da9ea","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine fun _ _ => ⟨1, by norm_num, 1, by norm_num, ?_⟩\n  filter_upwards [] with x\n  simp","hard_negative":true,"metrics":{"chosen_tokens":26,"rejected_tokens":8,"token_jaccard":0.043478,"token_length_ratio":0.307692},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"58d5a99c0c6eff27aa92b8e126798666f1d41493e3a14476bb8c35181062a4cb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Pow.Real\npublic import Mathlib.Algebra.Order.ToIntervalMod\npublic import Mathlib.Analysis.SpecialFunctions.Log.Base\nimport Mathlib.Algebra.Order.Interval.Set.Group\n\nNamespace:\nAkraBazziRecurrence\n\nLocal context:\n/-\nCopyright (c) 2023 Frédéric Dupuis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Frédéric Dupuis\n-/\n/-!\n# Akra-Bazzi theorem: the polynomial growth condition\n\nThis file defines and develops an API for the polynomial growth condition that appears in the\nstatement of the Akra-Bazzi theorem: for the theorem to hold, the function `g` must\nsatisfy the condition that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b*n` and `n` for any\nconstant `b ∈ (0,1)`.\n\n## Implementation notes\n\nOur definition requires that the condition hold for any `b ∈ (0,1)`. This is equivalent to requiring\nit only for `b = 1 / 2` (or any other particular value in `(0, 1)`). While this could, in principle,\nmake it harder to prove that a particular function grows polynomially, this issue does not seem to\narise in practice.\n\n-/\n\n@[expose] public section\n\nopen Finset Real Filter Asymptotics\nopen scoped Topology\n\nnamespace AkraBazziRecurrence\n\n/-- The growth condition that the function `g` must satisfy for the Akra-Bazzi theorem to apply.\nIt roughly states that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b * n` and `n`, for any\nconstant `b ∈ (0, 1)`. -/\ndef GrowsPolynomially (f : ℝ → ℝ) : Prop :=\n  ∀ b ∈ Set.Ioo 0 1, ∃ c₁ > 0, ∃ c₂ > 0,\n    ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * (f x)) (c₂ * f x)\n\nnamespace GrowsPolynomially\n\nlemma congr_of_eventuallyEq {f g : ℝ → ℝ} (hfg : f =ᶠ[atTop] g) (hg : GrowsPolynomially g) :\n    GrowsPolynomially f := by\n  intro b hb\n  have hg' := hg b hb\n  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hg'⟩ := hg'\n  refine ⟨c₁, hc₁_mem, c₂, hc₂_mem, ?_⟩\n  filter_upwards [hg', (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg, hfg]\n    with x hx₁ hx₂ hx₃\n  intro u hu\n  rw [hx₂ u hu.1, hx₃]\n  exact hx₁ u hu\n\nlemma iff_eventuallyEq {f g : ℝ → ℝ} (h : f =ᶠ[atTop] g) :\n    GrowsPolynomially f ↔ GrowsPolynomially g :=\n  ⟨fun hf => congr_of_eventuallyEq h.symm hf, fun hg => congr_of_eventuallyEq h hg⟩\n\nvariable {f : ℝ → ℝ}\n\nlemma eventually_atTop_le {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ≤ c * f x := by\n  obtain ⟨c₁, _, c₂, hc₂, h⟩ := hf b hb\n  refine ⟨c₂, hc₂, ?_⟩\n  filter_upwards [h]\n  exact fun _ H u hu => (H u hu).2\n\nlemma eventually_atTop_le_nat {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (b * n) n, f u ≤ c * f n := by\n  obtain ⟨c, hc_mem, hc⟩ := hf.eventually_atTop_le hb\n  exact ⟨c, hc_mem, hc.natCast_atTop⟩\n\nlemma eventually_atTop_ge {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, c * f x ≤ f u := by\n  obtain ⟨c₁, hc₁, c₂, _, h⟩ := hf b hb\n  refine ⟨c₁, hc₁, ?_⟩\n  filter_upwards [h]\n  exact fun _ H u hu => (H u hu).1\n\nlemma eventually_atTop_ge_nat {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (b * n) n, c * f n ≤ f u := by\n  obtain ⟨c, hc_mem, hc⟩ := hf.eventually_atTop_ge hb\n  exact ⟨c, hc_mem, hc.natCast_atTop⟩\n\nlemma eventually_zero_of_frequently_zero (hf : GrowsPolynomially f) (hf' : ∃ᶠ x in atTop, f x = 0) :\n    ∀ᶠ x in atTop, f x = 0 := by\n  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf (1 / 2) (by norm_num)\n  rw [frequently_atTop] at hf'\n  filter_upwards [eventually_forall_ge_atTop.mpr hf, eventually_gt_atTop 0] with x hx hx_pos\n  obtain ⟨x₀, hx₀_ge, hx₀⟩ := hf' (max x 1)\n  have x₀_pos := calc\n    0 < 1 := by norm_num\n    _ ≤ x₀ := le_of_max_le_right hx₀_ge\n  have hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z →\n      z ∈ Set.Icc ((2 : ℝ) ^ (-(m : ℤ) - 1) * x₀) ((2 : ℝ) ^ (-(m : ℤ)) * x₀) → f z = 0 := by\n    intro m\n    induction m with\n    | zero =>\n      simp only [CharP.cast_eq_zero, neg_zero, zero_sub, zpow_zero, one_mul] at *\n      specialize hx x₀ (le_of_max_le_left hx₀_ge)\n      simp only [hx₀, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx\n      refine fun z _ hz => hx _ ?_\n      simp only [zpow_neg, zpow_one] at hz\n      simp only [one_div, hz]\n    | succ k ih =>\n      intro z hxz hz\n      simp only [Nat.cast_add, Nat.cast_one] at *\n      have hx' : x ≤ (2 : ℝ) ^ (-(k : ℤ) - 1) * x₀ := by\n        calc x ≤ z := hxz\n          _ ≤ _ := by simp only [neg_add, ← sub_eq_add_neg] at hz; exact hz.2\n      specialize hx ((2 : ℝ) ^ (-(k : ℤ) - 1) * x₀) hx' z\n      specialize ih ((2 : ℝ) ^ (-(k : ℤ) - 1) * x₀) hx' ?ineq\n      case ineq =>\n        rw [Set.left_mem_Icc]\n        gcongr\n        · norm_num\n        · lia\n      simp only [ih, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx\n      refine hx ⟨?lb₁, ?ub₁⟩\n      case lb₁ =>\n        rw [one_div, ← zpow_neg_one, ← mul_assoc, ← zpow_add₀ (by norm_num)]\n        have h₁ : (-1 : ℤ) + (-k - 1) = -k - 2 := by ring\n        have h₂ : -(k + (1 : ℤ)) - 1 = -k - 2 := by ring\n        rw [h₁]\n        rw [h₂] at hz\n        exact hz.1\n      case ub₁ =>\n        have := hz.2\n        simp only [neg_add, ← sub_eq_add_neg] at this\n        exact this\n  refine hmain ⌊-logb 2 (x / x₀)⌋₊ x le_rfl ⟨?lb, ?ub⟩\n  case lb =>\n    rw [← le_div_iff₀ x₀_pos]\n    refine (logb_le_logb (b := 2) (by norm_num) (zpow_pos (by norm_num) _)\n      (by positivity)).mp ?_\n    rw [← rpow_intCast, logb_rpow (by norm_num) (by norm_num), ← neg_le_neg_iff]\n    simp only [Int.cast_sub, Int.cast_neg, Int.cast_natCast, Int.cast_one, neg_sub, sub_neg_eq_add]\n    calc -logb 2 (x / x₀) ≤ ⌈-logb 2 (x / x₀)⌉₊ := Nat.le_ceil (-logb 2 (x / x₀))\n         _ ≤ _ := by rw [add_comm]; exact_mod_cast Nat.ceil_le_floor_add_one _\n  case ub =>\n    rw [← div_le_iff₀ x₀_pos]\n    refine (logb_le_logb (b := 2) (by norm_num) (by positivity)\n      (zpow_pos (by norm_num) _)).mp ?_\n    rw [← rpow_intCast, logb_rpow (by norm_num) (by norm_num), ← neg_le_neg_iff]\n    simp only [Int.cast_neg, Int.cast_natCast, neg_neg]\n    have : 0 ≤ -logb 2 (x / x₀) := by\n      rw [neg_nonneg]\n      refine logb_nonpos (by norm_num) (by positivity) ?_\n      rw [div_le_one x₀_pos]\n      exact le_of_max_le_left hx₀_ge\n    exact_mod_cast Nat.floor_le this\n\nlemma eventually_atTop_nonneg_or_nonpos (hf : GrowsPolynomially f) :\n    (∀ᶠ x in atTop, 0 ≤ f x) ∨ (∀ᶠ x in atTop, f x ≤ 0) := by\n  obtain ⟨c₁, _, c₂, _, h⟩ := hf (1 / 2) (by norm_num)\n  match lt_trichotomy c₁ c₂ with\n  | .inl hlt => -- c₁ < c₂\n    left\n    filter_upwards [h, eventually_ge_atTop 0] with x hx hx_nonneg\n    have h' : 3 / 4 * x ∈ Set.Icc (1 / 2 * x) x := by\n      rw [Set.mem_Icc]\n      exact ⟨by gcongr ?_ * x; norm_num, by linarith⟩\n    have hu := hx (3 / 4 * x) h'\n    have hu := Set.nonempty_of_mem hu\n    rw [Set.nonempty_Icc] at hu\n    have hu' : 0 ≤ (c₂ - c₁) * f x := by linarith\n    exact nonneg_of_mul_nonneg_right hu' (by linarith)\n  | .inr (.inr hgt) => -- c₂ < c₁\n    right\n    filter_upwards [h, eventually_ge_atTop 0] with x hx hx_nonneg\n    have h' : 3 / 4 * x ∈ Set.Icc (1 / 2 * x) x := by\n      rw [Set.mem_Icc]\n      exact ⟨by gcongr ?_ * x; norm_num, by linarith⟩\n    have hu := hx (3 / 4 * x) h'\n    have hu := Set.nonempty_of_mem hu\n    rw [Set.nonempty_Icc] at hu\n    have hu' : (c₁ - c₂) * f x ≤ 0 := by linarith\n    exact nonpos_of_mul_nonpos_right hu' (by linarith)\n  | .inr (.inl heq) => -- c₁ = c₂\n    have hmain : ∃ c, ∀ᶠ x in atTop, f x = c := by\n      simp only [heq, Set.Icc_self, Set.mem_singleton_iff] at h\n      rw [eventually_atTop] at h\n      obtain ⟨n₀, hn₀⟩ := h\n      refine ⟨f (max n₀ 2), ?_⟩\n      rw [eventually_atTop]\n      refine ⟨max n₀ 2, ?_⟩\n      refine Real.induction_Ico_mul _ 2 (by norm_num) (by positivity) ?base ?step\n      case base => grind\n      case step =>\n        intro n _ _ z _\n        have le_2n : max n₀ 2 ≤ (2 : ℝ) ^ n * max n₀ 2 := by\n          simp [one_le_pow₀ (show (1 : ℝ) ≤ 2 by norm_num1)]\n        have half_z_to_base : f (1 / 2 * z) = f (max n₀ 2) := by\n          grind [mul_assoc]\n        grind\n    obtain ⟨c, hc⟩ := hmain\n    cases le_or_gt 0 c with\n    | inl hpos =>\n      exact Or.inl <| by filter_upwards [hc] with _ hc; simpa only [hc]\n    | inr hneg =>\n      right\n      filter_upwards [hc] with x hc\n      exact le_of_lt <| by simpa only [hc]\n\nlemma eventually_atTop_zero_or_pos_or_neg (hf : GrowsPolynomially f) :\n    (∀ᶠ x in atTop, f x = 0) ∨ (∀ᶠ x in atTop, 0 < f x) ∨ (∀ᶠ x in atTop, f x < 0) := by\n  by_cases! h : ∃ᶠ x in atTop, f x = 0\n  · exact Or.inl <| eventually_zero_of_frequently_zero hf h\n  · cases eventually_atTop_nonneg_or_nonpos hf with\n    | inl h' =>\n      refine Or.inr (Or.inl ?_)\n      simp only [lt_iff_le_and_ne]\n      rw [eventually_and]\n      exact ⟨h', by filter_upwards [h] with x hx; exact hx.symm⟩\n    | inr h' =>\n      refine Or.inr (Or.inr ?_)\n      simp only [lt_iff_le_and_ne]\n      rw [eventually_and]\n      exact ⟨h', h⟩\n\nprotected lemma neg {f : ℝ → ℝ} (hf : GrowsPolynomially f) : GrowsPolynomially (-f) := by\n  intro b hb\n  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf b hb\n  refine ⟨c₂, hc₂_mem, c₁, hc₁_mem, ?_⟩\n  filter_upwards [hf] with x hx\n  intro u hu\n  simp only [Pi.neg_apply, Set.neg_mem_Icc_iff, neg_mul_eq_mul_neg, neg_neg]\n  exact hx u hu\n\nprotected lemma neg_iff {f : ℝ → ℝ} : GrowsPolynomially f ↔ GrowsPolynomially (-f) :=\n  ⟨fun hf => hf.neg, fun hf => by rw [← neg_neg f]; exact hf.neg⟩\n\nprotected lemma abs (hf : GrowsPolynomially f) : GrowsPolynomially (fun x => |f x|) := by\n  cases eventually_atTop_nonneg_or_nonpos hf with\n  | inl hf' =>\n    have hmain : f =ᶠ[atTop] fun x => |f x| := by\n      filter_upwards [hf'] with x hx\n      rw [abs_of_nonneg hx]\n    rw [← iff_eventuallyEq hmain]\n    exact hf\n  | inr hf' =>\n    have hmain : -f =ᶠ[atTop] fun x => |f x| := by\n      filter_upwards [hf'] with x hx\n      simp only [Pi.neg_apply, abs_of_nonpos hx]\n    rw [← iff_eventuallyEq hmain]\n    exact hf.neg\n\nprotected lemma norm (hf : GrowsPolynomially f) : GrowsPolynomially (fun x => ‖f x‖) := by\n  simp only [norm_eq_abs]\n  exact hf.abs\n\nend GrowsPolynomially\n\nvariable {f : ℝ → ℝ}\n\nTarget:\nlemma growsPolynomially_const {c : ℝ} : GrowsPolynomially (fun _ => c) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"2695022722712bd1e5816bd8eef6d9b823f970978310953830938acd5f1e77de","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability/AkraBazzi","family_id":"growspolynomially_const","file_id":"mathlib/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean","sample_id":"836ef9c3cd55ae2e3f293b4d27ae834545f003ae5a97617e40dffc59a7b9fdae"},"split":"train"}
{"binding":{"chosen_proof_sha256":"af1766a439a78c83ac802fa03802f5aeae0b2eb140261f6b10e3a3388ff5d5e4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"22f2feedaac2189bc2f75b488121e482511637bba540133178964abac118fe65","source_sha256":"28c44eebb7ced13bf4be0bd8100be84e866f1a63f874878f0058707234306682","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let f (u : R) : R × R := (star u * u, u * star u)\n  have hf : f ⁻¹' {(1, 1)} = unitary R := by ext u; simp [f, Unitary.mem_iff]\n  rw [← hf]\n  exact isClosed_singleton.preimage (by fun_prop)","hard_negative":false,"metrics":{"chosen_tokens":67,"rejected_tokens":2,"token_jaccard":0.026316,"token_length_ratio":0.029851},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"595b3d9c7858b19deac7542cf57063f8dab1aeea11d5f2f7c27de5567129ef61","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Star.Unitary\npublic import Mathlib.Topology.Algebra.Group.Defs\npublic import Mathlib.Topology.Algebra.Star\npublic import Mathlib.Topology.Algebra.Monoid\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux, Bhavik Mehta\n-/\n/-! # Topological properties of the unitary (sub)group\n\n* In a topological star monoid `R`, `unitary R` is a topological group\n* In a topological star monoid `R` which is T1, `unitary R` is closed as a subset of `R`.\n-/\n\npublic section\n\nvariable {R : Type*} [Monoid R] [StarMul R] [TopologicalSpace R]\n\ninstance [ContinuousStar R] : ContinuousStar (unitary R) where\n  continuous_star := continuous_induced_rng.mpr continuous_subtype_val.star\n\ninstance [ContinuousStar R] : ContinuousInv (unitary R) where\n  continuous_inv := continuous_star\n\ninstance [ContinuousMul R] [ContinuousStar R] : IsTopologicalGroup (unitary R) where\n\nTarget:\nlemma isClosed_unitary [T1Space R] [ContinuousStar R] [ContinuousMul R] :\n    IsClosed (unitary R : Set R) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Algebra","family_id":"isclosed_unitary","file_id":"mathlib/Mathlib/Topology/Algebra/Star/Unitary.lean","sample_id":"22f2feedaac2189bc2f75b488121e482511637bba540133178964abac118fe65"},"split":"train"}
{"binding":{"chosen_proof_sha256":"76fdca8112ad5b228e441ce2f0fc6042386b4c4cdeafde082a85932a16c29411","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"207dc58f8ca84bc6887d256453cd927a0187e4bcc3e853e69b12bdde54f74c00","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8ceba723b8cfca101595c85d0f256daa590d35c274c45371de9577614b0b3c30","source_sha256":"ebea26ad83d186da19f8d5aec5a4532b5e10776580381e96c50d87a2ed0ca5b4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [← toDual_covBy_toDual_iff (α := α)] at hfin ⊢\n  exact exists_covby_infinite_Ici_of_infinite_Ici (α := αᵒᵈ) ha hfin","hard_negative":false,"metrics":{"chosen_tokens":25,"rejected_tokens":32,"token_jaccard":0.9,"token_length_ratio":1.28},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"598816a90977e25addaa7a53face0a23876802e74962e119601ee902534a0f74","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Lattice\npublic import Mathlib.Order.Atoms\npublic import Mathlib.Order.Interval.Finset.Defs\npublic import Mathlib.Order.Preorder.Finite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Atoms, Coatoms, Simple Lattices, and Finiteness\n\nThis module contains some results on atoms and simple lattices in the finite context.\n\n## Main results\n* `Finite.to_isAtomic`, `Finite.to_isCoatomic`: Finite partial orders with bottom resp. top\n  are atomic resp. coatomic.\n\n-/\n\npublic section\n\n\nvariable {α β : Type*}\n\nnamespace IsSimpleOrder\n\nvariable [LE α] [BoundedOrder α] [IsSimpleOrder α]\n\nsection DecidableEq\n\n/-- It is important that `IsSimpleOrder` is the last type-class argument of this instance,\nso that type-class inference fails quickly if it doesn't apply.\n\nNote that as of 2025-08-13, this is false. Could someone investigate? -/\nscoped instance (priority := 200) [DecidableEq α] : Fintype α :=\n  Fintype.ofEquiv Bool equivBool.symm\n\nend DecidableEq\n\nscoped instance (priority := 200) : Finite α := by classical infer_instance\n\nend IsSimpleOrder\n\nnamespace Fintype\n\nnamespace IsSimpleOrder\n\nopen scoped _root_.IsSimpleOrder\n\nvariable [LE α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α]\n\ntheorem univ : (Finset.univ : Finset α) = {⊤, ⊥} := by\n  ext\n  simpa using (eq_bot_or_eq_top _).symm\n\ntheorem card : Fintype.card α = 2 :=\n  (Fintype.ofEquiv_card _).trans Fintype.card_bool\n\nend IsSimpleOrder\n\nend Fintype\n\nnamespace Bool\n\ninstance : IsSimpleOrder Bool :=\n  ⟨fun a => by\n    rw [← Finset.mem_singleton, Or.comm, ← Finset.mem_insert, top_eq_true, bot_eq_false, ←\n      Fintype.univ_bool]\n    apply Finset.mem_univ⟩\n\nend Bool\n\nsection Fintype\n\nopen Finset\n\n-- see Note [lower instance priority]\ninstance (priority := 100) Finite.to_isCoatomic [PartialOrder α] [OrderTop α] [Finite α] :\n    IsCoatomic α :=\n  IsStronglyCoatomic.toIsCoatomic α\n\n-- see Note [lower instance priority]\ninstance (priority := 100) Finite.to_isAtomic [PartialOrder α] [OrderBot α] [Finite α] :\n    IsAtomic α :=\n  isCoatomic_dual_iff_isAtomic.mp Finite.to_isCoatomic\n\nend Fintype\n\nsection LocallyFinite\n\nvariable [Preorder α] [LocallyFiniteOrder α]\n\ninstance : IsStronglyAtomic α where\n  exists_covBy_le_of_lt a b hab := by\n    obtain ⟨x, hx, hxmin⟩ := (LocallyFiniteOrder.finsetIoc a b).exists_minimal\n      ⟨b, by simpa [LocallyFiniteOrder.finset_mem_Ioc]⟩\n    simp only [LocallyFiniteOrder.finset_mem_Ioc] at hx hxmin\n    exact ⟨x, ⟨hx.1, fun c hac hcx ↦ hcx.not_ge <| hxmin ⟨hac, hcx.le.trans hx.2⟩ hcx.le⟩, hx.2⟩\n\ninstance : IsStronglyCoatomic α := by\n  rw [← isStronglyAtomic_dual_iff_is_stronglyCoatomic]; infer_instance\n\nend LocallyFinite\n\nsection IsStronglyAtomic\n\nvariable [PartialOrder α] {a : α}\n\ntheorem exists_covby_infinite_Ici_of_infinite_Ici [IsStronglyAtomic α]\n    (ha : (Set.Ici a).Infinite) (hfin : {x | a ⋖ x}.Finite) :\n    ∃ b, a ⋖ b ∧ (Set.Ici b).Infinite := by\n  by_contra! h\n  refine ((hfin.biUnion (t := Set.Ici) (by simpa using h)).subset (fun b hb ↦ ?_)).not_infinite\n    (ha.sdiff (Set.finite_singleton a))\n  obtain ⟨x, hax, hxb⟩ := ((show a ≤ b from hb.1).lt_of_ne (Ne.symm hb.2)).exists_covby_le\n  exact Set.mem_biUnion hax hxb\n\nTarget:\ntheorem exists_covby_infinite_Iic_of_infinite_Iic [IsStronglyCoatomic α]\n    (ha : (Set.Iic a).Infinite) (hfin : {x | x ⋖ a}.Finite) :\n    ∃ b, b ⋖ a ∧ (Set.Iic b).Infinite :=\n\nProof body:\n","rejected":"```lean\nby\n  simp_rw [← toDual_covBy_toDual_iff (α := α)] at hfin ⊢\n  exact exists_covby_infinite_Ici_of_infinite_Ici (α := αᵒᵈ) ha hfin\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Atoms","family_id":"exists_covby_infinite_iic_of_infinite_iic","file_id":"mathlib/Mathlib/Order/Atoms/Finite.lean","sample_id":"8ceba723b8cfca101595c85d0f256daa590d35c274c45371de9577614b0b3c30"},"split":"train"}
{"binding":{"chosen_proof_sha256":"61f4608135cc80db3b144a8f0b385332ee17b58f1c351db640ceccbb2a9d7cc8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"fe4564cce88cf91d6bce5cf6a3f2448a7315e8735ba8fe17c1cf17a7220f767f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6a4d757b91c5500aa6ea19c37733ffcdf411c7c5b0b8869100541b0d79adff6c","source_sha256":"47c01594d211b3c54399310f0cb3b9f706ed5bf9e27bc1387c290b2a49698c96","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x; grind [eq_one_or_toFreeMonoid x]","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":3,"token_jaccard":0.1,"token_length_ratio":0.333333},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"599ba9e85a7fad7dc5f4665c704b9f0eb621d4f70f2227db14af4acdbe8dd7b7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.FreeMonoid.Basic\npublic import Mathlib.Algebra.Free\npublic import Mathlib.Algebra.Group.WithOne.Basic\npublic import Mathlib.Algebra.Group.Units.Basic\npublic import Mathlib.Data.Set.Operations\nimport Mathlib.Data.Set.Insert\n\nNamespace:\nFreeSemigroup\n\nLocal context:\n/-\nCopyright (c) 2026 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\n/-!\n# Relation between the free semigroup and the free monoid\n\nWe provide some constructions relating the free semigroup and the free monoid on the same type.\n\n## Main definitions\n* `FreeSemigroup.toFreeMonoid`: the natural embedding of the free semigroup into the free monoid.\n* `FreeMonoid.equivWithOneFreeSemigroup`: the free monoid is isomorphic to the free semigroup\n  with a `1` added.\n-/\n\npublic section\n\nvariable {α : Type*}\n\nnamespace FreeSemigroup\n\nopen FreeMonoid\n\n/--\nThe natural embedding of the free semigroup into the free monoid.\nThis is injective (`FreeSemigroup.toFreeMonoid_injective`), and its image\nconsists of all non-`1` elements of the free monoid (`FreeSemigroup.eq_one_or_toFreeMonoid`).\n-/\n@[expose, to_additive /-- The natural embedding of the free additive semigroup into the\nfree additive monoid. This is injective (`FreeAddSemigroup.toFreeAddMonoid_injective`), and its\nimage consists of all non-`0` elements of the free additive monoid\n(`FreeAddSemigroup.eq_zero_or_toFreeAddMonoid`). -/]\ndef toFreeMonoid : FreeSemigroup α →ₙ* FreeMonoid α :=\n  lift FreeMonoid.of\n\n@[to_additive (attr := simp, grind =)]\nlemma toFreeMonoid_of (x : α) : toFreeMonoid (.of x) = .of x := rfl\n\n@[to_additive]\nlemma toFreeMonoid_mk_eq_cons (x : α) (xs : List α) :\n    toFreeMonoid ⟨x, xs⟩ = FreeMonoid.ofList (x :: xs) := by\n  suffices ∀ x : FreeMonoid α, (xs.map FreeMonoid.of).foldl (· * ·) x = x * ofList xs by\n    simpa [← List.foldl_map, lift_mk_eq_foldl, toFreeMonoid, lift] using this (FreeMonoid.of x)\n  induction xs with grind [ofList_nil, ofList_cons]\n\n@[to_additive (attr := grind .)]\nlemma toFreeMonoid_injective : Function.Injective (@toFreeMonoid α) := by\n  rintro ⟨x, xs⟩ ⟨y, ys⟩ h\n  simp only [toFreeMonoid_mk_eq_cons, Equiv.apply_eq_iff_eq] at h\n  simpa using h\n\n@[to_additive (attr := simp, grind .)]\nlemma toFreeMonoid_ne_one (x : FreeSemigroup α) : toFreeMonoid x ≠ 1 := by\n  induction x with simp\n\n@[to_additive]\nlemma eq_one_or_toFreeMonoid (x : FreeMonoid α) : x = 1 ∨ ∃ y, toFreeMonoid y = x :=\n  x.inductionOn' (by simp) <| by\n    rintro b _ (rfl | ⟨y, rfl⟩)\n    · exact Or.inr ⟨of b, by simp⟩\n    · exact Or.inr ⟨of b * y, by simp⟩\n\n@[to_additive (attr := simp)]\n\nTarget:\nlemma range_toFreeMonoid : Set.range (@toFreeMonoid α) = {1}ᶜ :=\n\nProof body:\n","rejected":"by\n  exact range_toFreeMonoid","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"e4f37dc6e55fa6e45fa46154a37c990cc00f400ad3da1e6afa6d451ec054f775","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/FreeMonoid","family_id":"range_tofreemonoid","file_id":"mathlib/Mathlib/Algebra/FreeMonoid/FreeSemigroup.lean","sample_id":"6a4d757b91c5500aa6ea19c37733ffcdf411c7c5b0b8869100541b0d79adff6c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3d02c82bbfbe829ee560bc14711178f8e05f9f6e82051c0e8f4a63b14e8d38c0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"595e91c0da49b46a91c3741f0089b51f2875fc10f72a63c9a56e6dda6a7ca6c2","source_sha256":"f830da4bee5fab2594631f06af881784def836046450586c520e0644c9ae3582","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let φ (i : ι) := separatingFamily c (fun (_ : Unit) ↦ X) ⟨⟨⟩, i⟩\n  refine isSeparator_of_isColimit_cofan\n    (isSeparating_separatingFamily c (X := fun (_ : Unit) ↦ X) (by simpa using! hX))\n      (c := Cofan.mk (∐ φ) (fun ⟨_, i⟩ ↦ Sigma.ι φ i)) ?_\n  exact IsColimit.ofWhiskerEquivalence\n    (Discrete.equivalence (Equiv.punitProd.{0} ι).symm) (coproductIsCoproduct φ)","hard_negative":false,"metrics":{"chosen_tokens":104,"rejected_tokens":2,"token_jaccard":0.022222,"token_length_ratio":0.019231},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"59f24280146f01a5bef105407893c9e7bb3155520b272ec67591552315f0746c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.Double\npublic import Mathlib.Algebra.Homology.HomologicalComplexLimits\npublic import Mathlib.CategoryTheory.Generator.Basic\n\nNamespace:\nHomologicalComplex\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Generators of the category of homological complexes\n\nLet `c : ComplexShape ι` be a complex shape with no loop.\nIf a category `C` has a separator, then `HomologicalComplex C c`\nhas a separating family, and a separator when suitable coproducts exist.\n\n-/\n\n@[expose] public section\n\nuniverse t w v u\n\nopen CategoryTheory Limits\n\nnamespace HomologicalComplex\n\nvariable {C : Type u} [Category.{v} C] {ι : Type w} (c : ComplexShape ι) [c.HasNoLoop]\n\nsection\n\nvariable [HasZeroMorphisms C] [HasZeroObject C]\n\nvariable {α : Type t} {X : α → C} (hX : ObjectProperty.IsSeparating (.ofObj X))\n\nvariable (X) in\n/-- If `X : α → C` is a separating family, and `c : ComplexShape ι` has no loop,\nthen this is a separating family indexed by `α × ι` in `HomologicalComplex C c`,\nwhich consists of homological complexes that are nonzero in at most\ntwo (consecutive) degrees. -/\nnoncomputable def separatingFamily (j : α × ι) : HomologicalComplex C c :=\n  evalCompCoyonedaCorepresentative c (X j.1) j.2\n\nset_option backward.isDefEq.respectTransparency false in\ninclude hX in\nlemma isSeparating_separatingFamily :\n    ObjectProperty.IsSeparating (.ofObj (separatingFamily c X)) := by\n  intro K L f g h\n  ext j\n  apply hX\n  rintro _ ⟨a⟩ p\n  have H := evalCompCoyonedaCorepresentable c (X a) j\n  apply H.homEquiv.symm.injective\n  simpa only [H.homEquiv_symm_comp] using! h _\n    (ObjectProperty.ofObj_apply _ ⟨a, j⟩) (H.homEquiv.symm p)\n\nend\n\nvariable [HasCoproductsOfShape ι C] [Preadditive C] [HasZeroObject C]\n\nTarget:\nlemma isSeparator_coproduct_separatingFamily {X : C} (hX : IsSeparator X) :\n    IsSeparator (∐ (fun i ↦ separatingFamily c (fun (_ : Unit) ↦ X) ⟨⟨⟩, i⟩)) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Generator","family_id":"isseparator_coproduct_separatingfamily","file_id":"mathlib/Mathlib/CategoryTheory/Generator/HomologicalComplex.lean","sample_id":"595e91c0da49b46a91c3741f0089b51f2875fc10f72a63c9a56e6dda6a7ca6c2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3dc6c6b95959f83835d9501fcfa554ee77808794954cb31229e8e1de18edb21b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3f67b4910954a3acaaa3087e7993d00a824bcfea76d08a647dfda9454dd07897","source_sha256":"b4e7e5c120f48c9a1a0c6aa6bf563869c6ea4d08a420d4b58e8dbf288b17256c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : NeZero (Nat.card G) := ⟨Nat.card_pos.ne'⟩\n  have hord := HasEnoughRootsOfUnity.natCard_rootsOfUnity M (Nat.card G)\n  let e := (IsCyclic.monoidHom_mulEquiv_rootsOfUnity G Mˣ).some\n  exact ⟨e.trans (rootsOfUnityUnitsMulEquiv M (Nat.card G)) |>.trans (mulEquivOfCyclicCardEq hord)⟩","hard_negative":false,"metrics":{"chosen_tokens":68,"rejected_tokens":3,"token_jaccard":0.0625,"token_length_ratio":0.044118},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"5a67ea9e6f37cf3f1f1c0badc524bb182372931ea352913157ee9b608a998911","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Commutative monoids with enough roots of unity\n\nWe define a typeclass `HasEnoughRootsOfUnity M n` for a commutative monoid `M` and\na natural number `n` that asserts that `M` contains a primitive `n`th root of unity\nand that the group of `n`th roots of unity in `M` is cyclic. Such monoids are suitable\ntargets for homomorphisms from groups of exponent (dividing) `n`; for example,\nthe homomorphisms can then be used to separate elements of the source group.\n-/\n\npublic section\n\n/-- This is a type class recording that a commutative monoid `M` contains primitive `n`th\nroots of unity and such that the group of `n`th roots of unity is cyclic.\n\nSuch monoids are suitable targets in the context of duality statements for groups\nof exponent `n`. -/\nclass HasEnoughRootsOfUnity (M : Type*) [CommMonoid M] (n : ℕ) where\n  prim : ∃ m : M, IsPrimitiveRoot m n\n  cyc : IsCyclic <| rootsOfUnity n M\n\nnamespace HasEnoughRootsOfUnity\n\nlemma exists_primitiveRoot (M : Type*) [CommMonoid M] (n : ℕ) [HasEnoughRootsOfUnity M n] :\n    ∃ ζ : M, IsPrimitiveRoot ζ n :=\n  HasEnoughRootsOfUnity.prim\n\ninstance rootsOfUnity_isCyclic (M : Type*) [CommMonoid M] (n : ℕ) [HasEnoughRootsOfUnity M n] :\n    IsCyclic (rootsOfUnity n M) :=\n  HasEnoughRootsOfUnity.cyc\n\n/-- If `HasEnoughRootsOfUnity M n` and `m ∣ n`, then also `HasEnoughRootsOfUnity M m`. -/\nlemma of_dvd (M : Type*) [CommMonoid M] {m n : ℕ} [NeZero n] (hmn : m ∣ n)\n    [HasEnoughRootsOfUnity M n] :\n    HasEnoughRootsOfUnity M m where\n  prim :=\n    have ⟨ζ, hζ⟩ := exists_primitiveRoot M n\n    have ⟨k, hk⟩ := hmn\n    ⟨ζ ^ k, IsPrimitiveRoot.pow (NeZero.pos n) hζ (mul_comm m k ▸ hk)⟩\n  cyc := Subgroup.isCyclic_of_le <| rootsOfUnity_le_of_dvd hmn\n\n/-- If `M` satisfies `HasEnoughRootsOfUnity`, then the group of `n`th roots of unity\nin `M` is finite. -/\ninstance finite_rootsOfUnity (M : Type*) [CommMonoid M] (n : ℕ) [NeZero n]\n    [HasEnoughRootsOfUnity M n] :\n    Finite <| rootsOfUnity n M := by\n  have := rootsOfUnity_isCyclic M n\n  obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := rootsOfUnity n M)\n  have hg' : g ^ n = 1 := OneMemClass.coe_eq_one.mp g.prop\n  let f (j : ZMod n) : rootsOfUnity n M := g ^ (j.val : ℤ)\n  refine Finite.of_surjective f fun x ↦ ?_\n  obtain ⟨k, hk⟩ := Subgroup.mem_zpowers_iff.mp <| hg x\n  refine ⟨k, ?_⟩\n  simpa only [ZMod.natCast_val, ← hk, f, ZMod.coe_intCast] using (zpow_eq_zpow_emod' k hg').symm\n\n/-- If `M` satisfies `HasEnoughRootsOfUnity`, then the group of `n`th roots of unity\nin `M` (is cyclic and) has order `n`. -/\nlemma natCard_rootsOfUnity (M : Type*) [CommMonoid M] (n : ℕ) [NeZero n]\n    [HasEnoughRootsOfUnity M n] :\n    Nat.card (rootsOfUnity n M) = n := by\n  obtain ⟨ζ, h⟩ := exists_primitiveRoot M n\n  rw [← IsCyclic.exponent_eq_card]\n  refine dvd_antisymm ?_ ?_\n  · exact Monoid.exponent_dvd_of_forall_pow_eq_one fun g ↦ OneMemClass.coe_eq_one.mp g.prop\n  · nth_rewrite 1 [h.eq_orderOf]\n    rw [← (h.isUnit NeZero.out).unit_spec, orderOf_units]\n    let ζ' : rootsOfUnity n M := ⟨(h.isUnit NeZero.out).unit, ?_⟩\n    · rw [← Subgroup.orderOf_mk]\n      exact Monoid.order_dvd_exponent ζ'\n    simp only [mem_rootsOfUnity]\n    rw [← Units.val_inj, Units.val_pow_eq_pow_val, IsUnit.unit_spec, h.pow_eq_one, Units.val_one]\n\nlemma of_card_le {R : Type*} [CommRing R] [IsDomain R] {n : ℕ} [NeZero n]\n    (h : n ≤ Fintype.card (rootsOfUnity n R)) : HasEnoughRootsOfUnity R n where\n  prim := card_rootsOfUnity_eq_iff_exists_isPrimitiveRoot.mp (le_antisymm (card_rootsOfUnity R n) h)\n  cyc := rootsOfUnity.isCyclic R n\n\nend HasEnoughRootsOfUnity\n\nlemma MulEquiv.hasEnoughRootsOfUnity {n : ℕ} [NeZero n] {M N : Type*} [CommMonoid M]\n    [CommMonoid N] [hm : HasEnoughRootsOfUnity M n] (e : rootsOfUnity n M ≃* rootsOfUnity n N) :\n    HasEnoughRootsOfUnity N n where\n  prim := by\n    obtain ⟨m, hm⟩ := hm.prim\n    use (e hm.toRootsOfUnity).val.val\n    rw [IsPrimitiveRoot.coe_units_iff, IsPrimitiveRoot.coe_submonoidClass_iff]\n    refine .map_of_injective ?_ e.injective\n    rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, ← IsPrimitiveRoot.coe_units_iff]\n  cyc := isCyclic_of_surjective e e.surjective\n\nsection cyclic\n\n/-- The group of group homomorphisms from a finite cyclic group `G` of order `n` into the\ngroup of units of a ring `M` with all roots of unity is isomorphic to `G` -/\n\nTarget:\nlemma IsCyclic.monoidHom_equiv_self (G M : Type*) [CommGroup G] [Finite G]\n    [IsCyclic G] [CommMonoid M] [HasEnoughRootsOfUnity M (Nat.card G)] :\n    Nonempty ((G →* Mˣ) ≃* G) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/RootsOfUnity","family_id":"iscyclic","file_id":"mathlib/Mathlib/RingTheory/RootsOfUnity/EnoughRootsOfUnity.lean","sample_id":"3f67b4910954a3acaaa3087e7993d00a824bcfea76d08a647dfda9454dd07897"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d02b7bd3abbc9b6e64a37811efb4eb652c5a1b34ba47dca636dd232d861f2cb3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0f11cb72d8fdf85708f09496e1d4ec940b7ea868b69526e5d606eb6240aff621","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f90ebae3e238223bb942229ba9072bef620bae81190f54e359d35b11e6164e6b","source_sha256":"ad6816bc3c40144c654f99c6788a9e4aaf794fd4dab9e18db82a79f74bc4d693","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  simpa [Units.ext_iff] using pow_card_sub_one_eq_one (Units.ne_zero _)","hard_negative":false,"metrics":{"chosen_tokens":16,"rejected_tokens":21,"token_jaccard":0.777778,"token_length_ratio":1.3125},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"5a750ad150523f2dee0ece8f31d0d5f35b75f887961d579a9e841a7d8779141c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity\npublic import Mathlib.FieldTheory.Finite.Basic\n\nNamespace:\nZMod\n\nLocal context:\n/-\nCopyright (c) 2025 Yakov Pechersky. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yakov Pechersky\n-/\n/-!\n# Torsion group of `ZMod p` for prime `p`\n\nThis file shows that the `ZMod p` has `p - 1` roots-of-unity.\n\n-/\n\npublic section\n\nnamespace ZMod\n\nTarget:\nlemma rootsOfUnity_eq_top {p : ℕ} [Fact p.Prime] :\n    (rootsOfUnity (p - 1) (ZMod p)) = ⊤ :=\n\nProof body:\n","rejected":"by\n  ext\n  simpa [Units.ext_iff] using pow_card_sub_one_eq_one (Units.ne_zero _)\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/ZMod","family_id":"rootsofunity_eq_top","file_id":"mathlib/Mathlib/RingTheory/ZMod/Torsion.lean","sample_id":"f90ebae3e238223bb942229ba9072bef620bae81190f54e359d35b11e6164e6b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"297365a1d8f1a47429f28c814575b3f64d65dc37c4e0b829e52928a4af8f384b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3aefbf7570c01386e517f76b1bd54bb6dfb0e26362a5aa624680398a23b1819d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7f59f02f1adc338374d83c01438102e2efac30fffd4f3810fcf815de0d1655df","source_sha256":"aac9c42310c3c68ba48257d401225da11673aa0693f3a499d296961d7c43ce42","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine le_antisymm ?_ ?_\n  · calc\n      stabilizer G (s ∪ t)\n        ≤ stabilizer G (s ∪ t) ⊓ stabilizer G t := by simpa\n      _ ≤ stabilizer G ((s ∪ t) \\ t) := stabilizer_inf_stabilizer_le_stabilizer_sdiff\n      _ = stabilizer G s := by rw [union_sdiff_cancel_right]; simpa [← disjoint_iff_inter_eq_empty]\n  · calc\n      stabilizer G s\n        ≤ stabilizer G s ⊓ stabilizer G t := by simpa\n      _ ≤ stabilizer G (s ∪ t) := stabilizer_inf_stabilizer_le_stabilizer_union","hard_negative":true,"metrics":{"chosen_tokens":90,"rejected_tokens":3,"token_jaccard":0.032258,"token_length_ratio":0.033333},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"5abaf2c83647f1f13938464dbaf7ab107edacae50cbc77734f8fbc94e91a50f7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Pointwise.Finset\npublic import Mathlib.GroupTheory.QuotientGroup.Defs\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\n\nNamespace:\nMulAction\n\nLocal context:\n/-\nCopyright (c) 2023 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Stabilizer of a set under a pointwise action\n\nThis file characterises the stabilizer of a set/finset under the pointwise action of a group.\n-/\n\npublic section\n\nopen Function MulOpposite Set\nopen scoped Pointwise\n\nnamespace MulAction\nvariable {G H α : Type*}\n\n/-! ### Stabilizer of a set -/\n\nsection Set\nsection Group\nvariable [Group G] [Group H] [MulAction G α] {a : G} {s t : Set α}\n\n@[to_additive (attr := simp)]\nlemma stabilizer_empty : stabilizer G (∅ : Set α) = ⊤ :=\n  Subgroup.coe_eq_univ.1 <| eq_univ_of_forall fun _a ↦ smul_set_empty\n\n@[to_additive (attr := simp)]\nlemma stabilizer_univ : stabilizer G (Set.univ : Set α) = ⊤ := by\n  ext\n  simp\n\n@[to_additive (attr := simp)]\nlemma stabilizer_singleton (b : α) : stabilizer G ({b} : Set α) = stabilizer G b := by ext; simp\n\n@[to_additive]\nlemma mem_stabilizer_set {s : Set α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s := by\n  refine mem_stabilizer_iff.trans ⟨fun h b ↦ ?_, fun h ↦ ?_⟩\n  · rw [← (smul_mem_smul_set_iff : a • b ∈ _ ↔ _), h]\n  simp_rw [Set.ext_iff, mem_smul_set_iff_inv_smul_mem]\n  exact ((MulAction.toPerm a).forall_congr' <| by simp [Iff.comm]).1 h\n\n@[to_additive]\nlemma map_stabilizer_le (f : G →* H) (s : Set G) :\n    (stabilizer G s).map f ≤ stabilizer H (f '' s) := by\n  rintro a\n  simp only [Subgroup.mem_map, mem_stabilizer_iff, forall_exists_index, and_imp]\n  rintro a ha rfl\n  rw [← image_smul_distrib, ha]\n\n@[to_additive (attr := simp)]\nlemma stabilizer_mul_self (s : Set G) : (stabilizer G s : Set G) * s = s := by\n  ext\n  refine ⟨?_, fun h ↦ ⟨_, (stabilizer G s).one_mem, _, h, one_mul _⟩⟩\n  rintro ⟨a, ha, b, hb, rfl⟩\n  rw [← mem_stabilizer_iff.1 ha]\n  exact smul_mem_smul_set hb\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_apply₂ {f : Set α → Set α → Set α}\n    (hf : ∀ a : G, a • f s t = f (a • s) (a • t)) :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (f s t) := by aesop (add simp [SetLike.le_def])\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_union :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s ∪ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_union\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_inter :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s ∩ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_inter\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_sdiff :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s \\ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_sdiff\n\n@[to_additive]\n\nTarget:\nlemma stabilizer_union_eq_left (hdisj : Disjoint s t) (hstab : stabilizer G s ≤ stabilizer G t)\n    (hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G t) :\n    stabilizer G (s ∪ t) = stabilizer G s :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_7f59f02f1adc","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f8abfe531cdcfbd7bd118f7e9f07557252723c897c758308d7c834717df2a13a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Pointwise","family_id":"stabilizer_union_eq_left","file_id":"mathlib/Mathlib/Algebra/Pointwise/Stabilizer.lean","sample_id":"7f59f02f1adc338374d83c01438102e2efac30fffd4f3810fcf815de0d1655df"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4854fca9d1b2dda857b810abc21527817558defad53eaaee4f2088331645faca","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"dd3f7e3a83fee5c50a2179510fc132b93fa044ac1c9b4a8dbb7b79a0ad04247a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"063710952a3a4d125df5a3e180296e39817ba1f9ff1c0061323053fb46b49fc4","source_sha256":"da03aef6be316c0f1e4d46eb4e8f0617ba6def9cf6bbead1497df151ed7bf988","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by cases x; rfl\n\nnonrec def ReaderT.callCC {ε} [MonadCont m] {α β : Type _}\n    (f : Label α (ReaderT ε m) β → ReaderT ε m α) : ReaderT ε m α :=\n  ReaderT.mk (fun r => callCC fun f' => (f <| ReaderT.mkLabel _ f').run r)","hard_negative":false,"metrics":{"chosen_tokens":72,"rejected_tokens":76,"token_jaccard":0.925,"token_length_ratio":1.055556},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"5ac172c0558f3c2505c7931bbe57e9e7be51241d86f42eac1f3acb82ee65583d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Control.Monad.Basic\npublic import Mathlib.Control.Monad.Writer\npublic import Mathlib.Control.Lawful\npublic import Batteries.Tactic.Congr\npublic import Batteries.Lean.Except\nimport all Init.Control.Option  -- for unfolding `Option.lift`\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\n/-!\n# Continuation Monad\n\nMonad encapsulating continuation passing programming style, similar to\nHaskell's `Cont`, `ContT` and `MonadCont`:\n<https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Cont.html>\n<https://hackage.haskell.org/package/transformers-0.6.2.0/docs/Control-Monad-Trans-Cont.html>\n-/\n\n@[expose] public section\n\nuniverse u v w u₀ u₁ v₀ v₁\n\nstructure MonadCont.Label (α : Type w) (m : Type u → Type v) (β : Type u) where\n  apply : α → m β\n\nabbrev MonadCont.goto {α β} {m : Type u → Type v} (f : MonadCont.Label α m β) (x : α) :=\n  f.apply x\n\nclass MonadCont (m : Type u → Type v) where\n  callCC : ∀ {α β}, (MonadCont.Label α m β → m α) → m α\n\nopen MonadCont\n\nclass LawfulMonadCont (m : Type u → Type v) [Monad m] [MonadCont m] : Prop\n    extends LawfulMonad m where\n  callCC_bind_right {α ω γ} (cmd : m α) (next : Label ω m γ → α → m ω) :\n    (callCC fun f => cmd >>= next f) = cmd >>= fun x => callCC fun f => next f x\n  callCC_bind_left {α} (β) (x : α) (dead : Label α m β → β → m α) :\n    (callCC fun f : Label α m β => goto f x >>= dead f) = pure x\n  callCC_dummy {α β} (dummy : m α) : (callCC fun _ : Label α m β => dummy) = dummy\n\nexport LawfulMonadCont (callCC_bind_right callCC_bind_left callCC_dummy)\n\ndef ContT (r : Type u) (m : Type u → Type v) (α : Type w) :=\n  (α → m r) → m r\n\nabbrev Cont (r : Type u) (α : Type w) :=\n  ContT r Id α\n\nnamespace ContT\n\nexport MonadCont (Label goto)\n\nvariable {r : Type u} {m : Type u → Type v} {α β : Type w}\n\n/-- Build a `ContT` from a function taking a continuation callback. -/\ndef mk (f : (α → m r) → m r) : ContT r m α := f\n\n/-- Run a `ContT` with a provided callback. -/\ndef run (x : ContT r m α) : (α → m r) → m r := x\n\ndef map (f : m r → m r) (x : ContT r m α) : ContT r m α :=\n  f ∘ x\n\ntheorem run_contT_map_contT (f : m r → m r) (x : ContT r m α) : run (map f x) = f ∘ run x :=\n  rfl\n\ndef withContT (f : (β → m r) → α → m r) (x : ContT r m α) : ContT r m β := fun g => x <| f g\n\ntheorem run_withContT (f : (β → m r) → α → m r) (x : ContT r m α) :\n    run (withContT f x) = run x ∘ f :=\n  rfl\n\n@[ext]\nprotected theorem ext {x y : ContT r m α} (h : ∀ f, x.run f = y.run f) : x = y := by\n  unfold ContT; ext; apply h\n\ninstance : Monad (ContT r m) where\n  pure x f := f x\n  bind x f g := x fun i => f i g\n\n@[simp]\ntheorem run_mk (f : (α → m r) → m r) (k : α → m r) : (.mk f : ContT r m α).run k = f k := rfl\n\n@[simp]\ntheorem run_pure (a : α) (k : α → m r) : (pure a : ContT r m α).run k = k a := rfl\n\n@[simp]\ntheorem run_bind (x : ContT r m α) (f : α → ContT r m β) (k : β → m r) :\n    (x >>= f).run k = x.run fun x => (f x).run k := rfl\n\n@[simp]\ntheorem run_map (f : α → β) (x : ContT r m α) (k : β → m r) :\n    (f <$> x).run k = x.run (k ∘ f) := rfl\n\n@[simp]\ntheorem run_seq (f : ContT r m (α → β)) (x : ContT r m α) (k : β → m r) :\n    (f <*> x).run k = f.run fun f => x.run (k ∘ f) := rfl\n\n@[simp]\ntheorem run_seqLeft (x : ContT r m α) (y : ContT r m β) (k : α → m r) :\n    (x <* y).run k = x.run fun x => y.run fun _ => k x := rfl\n\n@[simp]\ntheorem run_seqRight (x : ContT r m α) (y : ContT r m β) (k : β → m r) :\n    (x *> y).run k = x.run fun _ => y.run k := rfl\n\ninstance : LawfulMonad (ContT r m) := LawfulMonad.mk'\n  (id_map := by intros; rfl)\n  (pure_bind := by intros; ext; rfl)\n  (bind_assoc := by intros; ext; rfl)\n\ninstance [Monad m] : MonadLift m (ContT r m) where\n  monadLift x := .mk fun k => x >>= k\n\n@[simp]\ntheorem run_monadLift [Monad m] {α} (x : m α) (k : α → m r) :\n    (monadLift x : ContT r m α).run k = x >>= k := rfl\n\ntheorem monadLift_bind [Monad m] [LawfulMonad m] {α β} (x : m α) (f : α → m β) :\n    (monadLift (x >>= f) : ContT r m β) = monadLift x >>= monadLift ∘ f := by\n  ext\n  simp only [bind_assoc, run_bind, run_monadLift, Function.comp_apply]\n\ninstance : MonadCont (ContT r m) where\n  callCC f := .mk fun k => f ⟨fun x => .mk fun _ => k x⟩ k\n\n@[simp]\ntheorem run_callCC (f : Label α (ContT r m) β → ContT r m α) (k : α → m r) :\n    (callCC f).run k = (f ⟨fun x => .mk fun _ => k x⟩).run k := rfl\n\ninstance : LawfulMonadCont (ContT r m) where\n  callCC_bind_right := by intros; ext; rfl\n  callCC_bind_left := by intros; ext; rfl\n  callCC_dummy := by intros; ext; rfl\n\n/-- Note that `tryCatch` does not have correct behavior in this monad:\n```\ndef foo : ContT Bool (Except String) Bool := do\n  let x ← try\n    pure true\n  catch _ =>\n    return false\n  throw s!\"oh no {x}\"\n#eval foo.run pure\n-- `Except.ok false`, no error\n```\nHere, the `throwError` is being run inside the `try`.\nSee [Zulip](https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/MonadExcept.20in.20the.20ContT.20monad/near/375341221)\nfor further discussion.\n-/\ninstance (ε) [MonadExceptOf ε m] : MonadExceptOf ε (ContT r m) where\n  throw e := .mk fun _ => throw e\n  tryCatch act h := .mk fun k => tryCatch (act.run k) fun e => (h e).run k\n\n@[simp]\ntheorem run_throw {ε} [MonadExceptOf ε m]\n    (e : ε) (f : α → m r) :\n    (throw e : ContT r m α).run f = throw e := rfl\n\n@[simp]\ntheorem run_tryCatch {ε} [MonadExceptOf ε m]\n    (act : ContT r m α) (h : ε → ContT r m α) (f : α → m r) :\n    (tryCatch act h : ContT r m α).run f = tryCatch (act.run f) fun e => (h e).run f := rfl\n\nend ContT\n\nvariable {m : Type u → Type v}\n\nsection\nvariable [Monad m]\n\ndef ExceptT.mkLabel {α β ε} : Label (Except.{u, u} ε α) m β → Label α (ExceptT ε m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (Except.ok a)⟩\n\ntheorem ExceptT.goto_mkLabel {α β ε : Type _} (x : Label (Except.{u, u} ε α) m β) (i : α) :\n    goto (ExceptT.mkLabel x) i = ExceptT.mk (Except.ok <$> goto x (Except.ok i)) := by\n  cases x; rfl\n\nnonrec def ExceptT.callCC {ε} [MonadCont m] {α β : Type _}\n    (f : Label α (ExceptT ε m) β → ExceptT ε m α) : ExceptT ε m α :=\n  ExceptT.mk (callCC fun x : Label _ m β => ExceptT.run <| f (ExceptT.mkLabel x))\n\ninstance {ε} [MonadCont m] : MonadCont (ExceptT ε m) where\n  callCC := ExceptT.callCC\n\ninstance {ε} [MonadCont m] [LawfulMonadCont m] : LawfulMonadCont (ExceptT ε m) where\n  callCC_bind_right := by\n    intros; simp only [callCC, ExceptT.callCC, ExceptT.run_bind, callCC_bind_right]; ext\n    dsimp\n    congr with ⟨⟩ <;> simp [@callCC_dummy m _]\n  callCC_bind_left := by\n    intros\n    simp only [callCC, ExceptT.callCC, ExceptT.goto_mkLabel, map_eq_bind_pure_comp, Function.comp,\n      ExceptT.run_bind, ExceptT.run_mk, bind_assoc, pure_bind, @callCC_bind_left m _]\n    ext; rfl\n  callCC_dummy := by intros; simp only [callCC, ExceptT.callCC, @callCC_dummy m _]; ext; rfl\n\ndef OptionT.mkLabel {α β} : Label (Option.{u} α) m β → Label α (OptionT m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (some a)⟩\n\ntheorem OptionT.goto_mkLabel {α β : Type _} (x : Label (Option.{u} α) m β) (i : α) :\n    goto (OptionT.mkLabel x) i = OptionT.mk (goto x (some i) >>= fun a => pure (some a)) :=\n  (rfl)\n\nnonrec def OptionT.callCC [MonadCont m] {α β : Type _} (f : Label α (OptionT m) β → OptionT m α) :\n    OptionT m α :=\n  OptionT.mk (callCC fun x : Label _ m β => OptionT.run <| f (OptionT.mkLabel x) : m (Option α))\n\n@[simp]\nlemma run_callCC [MonadCont m] {α β : Type _} (f : Label α (OptionT m) β → OptionT m α) :\n    (OptionT.callCC f).run = (callCC fun x => OptionT.run <| f (OptionT.mkLabel x)) := (rfl)\n\ninstance [MonadCont m] : MonadCont (OptionT m) where\n  callCC := OptionT.callCC\n\ninstance [MonadCont m] [LawfulMonadCont m] : LawfulMonadCont (OptionT m) where\n  callCC_bind_right := by\n    refine fun _ _ => OptionT.ext ?_\n    simpa [callCC, Option.elimM, callCC_bind_right] using\n      bind_congr fun | some _ => rfl | none => by simp [@callCC_dummy m _]\n  callCC_bind_left := by\n    intros\n    ext\n    simp [callCC, OptionT.goto_mkLabel, @callCC_bind_left m _]\n  callCC_dummy := by intros; ext; simp [callCC, OptionT.callCC, @callCC_dummy m _]\n\ndef WriterT.mkLabel {α β ω} [EmptyCollection ω] : Label (α × ω) m β → Label α (WriterT ω m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (a, ∅)⟩\n\ndef WriterT.mkLabel' {α β ω} [Monoid ω] : Label (α × ω) m β → Label α (WriterT ω m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (a, 1)⟩\n\ntheorem WriterT.goto_mkLabel {α β ω : Type _} [EmptyCollection ω] (x : Label (α × ω) m β) (i : α) :\n    goto (WriterT.mkLabel x) i = monadLift (goto x (i, ∅)) := by cases x; rfl\n\ntheorem WriterT.goto_mkLabel' {α β ω : Type _} [Monoid ω] (x : Label (α × ω) m β) (i : α) :\n    goto (WriterT.mkLabel' x) i = monadLift (goto x (i, 1)) := by cases x; rfl\n\nnonrec def WriterT.callCC [MonadCont m] {α β ω : Type _} [EmptyCollection ω]\n    (f : Label α (WriterT ω m) β → WriterT ω m α) : WriterT ω m α :=\n  WriterT.mk <| callCC (WriterT.run ∘ f ∘ WriterT.mkLabel : Label (α × ω) m β → m (α × ω))\n\ndef WriterT.callCC' [MonadCont m] {α β ω : Type _} [Monoid ω]\n    (f : Label α (WriterT ω m) β → WriterT ω m α) : WriterT ω m α :=\n  WriterT.mk <|\n    MonadCont.callCC (WriterT.run ∘ f ∘ WriterT.mkLabel' : Label (α × ω) m β → m (α × ω))\n\nend\n\ninstance (ω) [Monad m] [EmptyCollection ω] [MonadCont m] : MonadCont (WriterT ω m) where\n  callCC := WriterT.callCC\n\ninstance (ω) [Monad m] [Monoid ω] [MonadCont m] : MonadCont (WriterT ω m) where\n  callCC := WriterT.callCC'\n\ndef StateT.mkLabel {α β σ : Type u} : Label (α × σ) m (β × σ) → Label α (StateT σ m) β\n  | ⟨f⟩ => ⟨fun a => StateT.mk (fun s => f (a, s))⟩\n\ntheorem StateT.goto_mkLabel {α β σ : Type u} (x : Label (α × σ) m (β × σ)) (i : α) :\n    goto (StateT.mkLabel x) i = StateT.mk (fun s => goto x (i, s)) := by cases x; rfl\n\nnonrec def StateT.callCC {σ} [MonadCont m] {α β : Type _}\n    (f : Label α (StateT σ m) β → StateT σ m α) : StateT σ m α :=\n  StateT.mk (fun r => callCC fun f' => (f <| StateT.mkLabel f').run r)\n\ninstance {σ} [MonadCont m] : MonadCont (StateT σ m) where\n  callCC := StateT.callCC\n\ninstance {σ} [Monad m] [MonadCont m] [LawfulMonadCont m] : LawfulMonadCont (StateT σ m) where\n  callCC_bind_right := by\n    intros\n    simp only [callCC, StateT.callCC, StateT.run_bind, callCC_bind_right]; ext; rfl\n  callCC_bind_left := by\n    intros\n    simp only [callCC, StateT.callCC, StateT.goto_mkLabel, StateT.run_bind, StateT.run_mk,\n      callCC_bind_left]; ext; rfl\n  callCC_dummy := by\n    intros\n    simp only [callCC, StateT.callCC, @callCC_dummy m _]\n    ext; rfl\n\ndef ReaderT.mkLabel {α β} (ρ) : Label α m β → Label α (ReaderT ρ m) β\n  | ⟨f⟩ => ⟨monadLift ∘ f⟩\n\nTarget:\ntheorem ReaderT.goto_mkLabel {α ρ β} (x : Label α m β) (i : α) :\n    goto (ReaderT.mkLabel ρ x) i = monadLift (goto x i) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby cases x; rfl\n\nnonrec def ReaderT.callCC {ε} [MonadCont m] {α β : Type _}\n    (f : Label α (ReaderT ε m) β → ReaderT ε m α) : ReaderT ε m α :=\n  ReaderT.mk (fun r => callCC fun f' => (f <| ReaderT.mkLabel _ f').run r)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control/Monad","family_id":"readert","file_id":"mathlib/Mathlib/Control/Monad/Cont.lean","sample_id":"063710952a3a4d125df5a3e180296e39817ba1f9ff1c0061323053fb46b49fc4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6bde054e8dd9e2e89dc5431f020cd1a2610cdbc6535abeb8845dde9b2fdd1677","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6c18e3a6e36fcd3e1caadefef8b6d28d63c691769c6b3ffbae3c86bf10ef7481","source_sha256":"8134c926e471796d362206cf36f0ce764965a63064ad2f4ac11aa45b1280dd60","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  change\n    Quotient.map (@funMap L M ps.toStructure n f) Prestructure.fun_equiv (Quotient.finChoice _) =\n      _\n  rw [Quotient.finChoice_eq, Quotient.map_mk]","hard_negative":false,"metrics":{"chosen_tokens":37,"rejected_tokens":3,"token_jaccard":0.035714,"token_length_ratio":0.081081},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"5afa3c2bcb3183ecfcee1dc1c3ed85e621d5c8e7bc9146ef287b40d90f4da266","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Quotient\npublic import Mathlib.ModelTheory.Semantics\n\nNamespace:\nFirstOrder.Language\n\nLocal context:\n/-\nCopyright (c) 2022 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Quotients of First-Order Structures\n\nThis file defines prestructures and quotients of first-order structures.\n\n## Main Definitions\n\n- If `s` is a setoid (equivalence relation) on `M`, a `FirstOrder.Language.Prestructure s` is the\n  data for a first-order structure on `M` that will still be a structure when modded out by `s`.\n- The structure `FirstOrder.Language.quotientStructure s` is the resulting structure on\n  `Quotient s`.\n-/\n\npublic section\n\n\nnamespace FirstOrder\n\nnamespace Language\n\nvariable (L : Language) {M : Type*}\n\nopen FirstOrder\n\nopen Structure\n\n/-- A prestructure is a first-order structure with a `Setoid` equivalence relation on it,\n  such that quotienting by that equivalence relation is still a structure. -/\nclass Prestructure (s : Setoid M) where\n  /-- The underlying first-order structure -/\n  toStructure : L.Structure M\n  fun_equiv : ∀ {n} {f : L.Functions n} (x y : Fin n → M), x ≈ y → funMap f x ≈ funMap f y\n  rel_equiv : ∀ {n} {r : L.Relations n} (x y : Fin n → M) (_ : x ≈ y), RelMap r x = RelMap r y\n\nvariable {L} {s : Setoid M}\nvariable [ps : L.Prestructure s]\n\ninstance quotientStructure : L.Structure (Quotient s) where\n  funMap {n} f x :=\n    Quotient.map (@funMap L M ps.toStructure n f) Prestructure.fun_equiv (Quotient.finChoice x)\n  RelMap {n} r x :=\n    Quotient.lift (@RelMap L M ps.toStructure n r) Prestructure.rel_equiv (Quotient.finChoice x)\n\nvariable (s)\n\nTarget:\ntheorem funMap_quotient_mk' {n : ℕ} (f : L.Functions n) (x : Fin n → M) :\n    (funMap f fun i => (⟦x i⟧ : Quotient s)) = ⟦@funMap _ _ ps.toStructure _ f x⟧ :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"ModelTheory","family_id":"funmap_quotient_mk","file_id":"mathlib/Mathlib/ModelTheory/Quotients.lean","sample_id":"6c18e3a6e36fcd3e1caadefef8b6d28d63c691769c6b3ffbae3c86bf10ef7481"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e6c90bca07c73d1bcb9479a5b66cf4f9a197038e47590f39f131d2b92e478d48","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2e55761302eb4d18256d8d74dab0d8cb060597618d19eb424413096193e3d7a8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7b81765a8031ad51037f4259b92e28a5ea92d5e155ac346ebd82828d508f44f4","source_sha256":"ebea26ad83d186da19f8d5aec5a4532b5e10776580381e96c50d87a2ed0ca5b4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_contra! h\n  refine ((hfin.biUnion (t := Set.Ici) (by simpa using h)).subset (fun b hb ↦ ?_)).not_infinite\n    (ha.sdiff (Set.finite_singleton a))\n  obtain ⟨x, hax, hxb⟩ := ((show a ≤ b from hb.1).lt_of_ne (Ne.symm hb.2)).exists_covby_le\n  exact Set.mem_biUnion hax hxb","hard_negative":true,"metrics":{"chosen_tokens":87,"rejected_tokens":3,"token_jaccard":0.042553,"token_length_ratio":0.034483},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"5b1dc0853c16d2801f3f94dfca8599dc0e21f8d01ed7b4b0d9a6ed410dfe3738","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Lattice\npublic import Mathlib.Order.Atoms\npublic import Mathlib.Order.Interval.Finset.Defs\npublic import Mathlib.Order.Preorder.Finite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Atoms, Coatoms, Simple Lattices, and Finiteness\n\nThis module contains some results on atoms and simple lattices in the finite context.\n\n## Main results\n* `Finite.to_isAtomic`, `Finite.to_isCoatomic`: Finite partial orders with bottom resp. top\n  are atomic resp. coatomic.\n\n-/\n\npublic section\n\n\nvariable {α β : Type*}\n\nnamespace IsSimpleOrder\n\nvariable [LE α] [BoundedOrder α] [IsSimpleOrder α]\n\nsection DecidableEq\n\n/-- It is important that `IsSimpleOrder` is the last type-class argument of this instance,\nso that type-class inference fails quickly if it doesn't apply.\n\nNote that as of 2025-08-13, this is false. Could someone investigate? -/\nscoped instance (priority := 200) [DecidableEq α] : Fintype α :=\n  Fintype.ofEquiv Bool equivBool.symm\n\nend DecidableEq\n\nscoped instance (priority := 200) : Finite α := by classical infer_instance\n\nend IsSimpleOrder\n\nnamespace Fintype\n\nnamespace IsSimpleOrder\n\nopen scoped _root_.IsSimpleOrder\n\nvariable [LE α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α]\n\ntheorem univ : (Finset.univ : Finset α) = {⊤, ⊥} := by\n  ext\n  simpa using (eq_bot_or_eq_top _).symm\n\ntheorem card : Fintype.card α = 2 :=\n  (Fintype.ofEquiv_card _).trans Fintype.card_bool\n\nend IsSimpleOrder\n\nend Fintype\n\nnamespace Bool\n\ninstance : IsSimpleOrder Bool :=\n  ⟨fun a => by\n    rw [← Finset.mem_singleton, Or.comm, ← Finset.mem_insert, top_eq_true, bot_eq_false, ←\n      Fintype.univ_bool]\n    apply Finset.mem_univ⟩\n\nend Bool\n\nsection Fintype\n\nopen Finset\n\n-- see Note [lower instance priority]\ninstance (priority := 100) Finite.to_isCoatomic [PartialOrder α] [OrderTop α] [Finite α] :\n    IsCoatomic α :=\n  IsStronglyCoatomic.toIsCoatomic α\n\n-- see Note [lower instance priority]\ninstance (priority := 100) Finite.to_isAtomic [PartialOrder α] [OrderBot α] [Finite α] :\n    IsAtomic α :=\n  isCoatomic_dual_iff_isAtomic.mp Finite.to_isCoatomic\n\nend Fintype\n\nsection LocallyFinite\n\nvariable [Preorder α] [LocallyFiniteOrder α]\n\ninstance : IsStronglyAtomic α where\n  exists_covBy_le_of_lt a b hab := by\n    obtain ⟨x, hx, hxmin⟩ := (LocallyFiniteOrder.finsetIoc a b).exists_minimal\n      ⟨b, by simpa [LocallyFiniteOrder.finset_mem_Ioc]⟩\n    simp only [LocallyFiniteOrder.finset_mem_Ioc] at hx hxmin\n    exact ⟨x, ⟨hx.1, fun c hac hcx ↦ hcx.not_ge <| hxmin ⟨hac, hcx.le.trans hx.2⟩ hcx.le⟩, hx.2⟩\n\ninstance : IsStronglyCoatomic α := by\n  rw [← isStronglyAtomic_dual_iff_is_stronglyCoatomic]; infer_instance\n\nend LocallyFinite\n\nsection IsStronglyAtomic\n\nvariable [PartialOrder α] {a : α}\n\nTarget:\ntheorem exists_covby_infinite_Ici_of_infinite_Ici [IsStronglyAtomic α]\n    (ha : (Set.Ici a).Infinite) (hfin : {x | a ⋖ x}.Finite) :\n    ∃ b, a ⋖ b ∧ (Set.Ici b).Infinite :=\n\nProof body:\n","rejected":"by\n  exact exists_covby_infinite_Ici_of_infinite_Ici","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"11c797f08da7c4ce94e8fa8979a94d78bb4ba6c8e635ffeeda9f6940b78b6258","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Atoms","family_id":"exists_covby_infinite_ici_of_infinite_ici","file_id":"mathlib/Mathlib/Order/Atoms/Finite.lean","sample_id":"7b81765a8031ad51037f4259b92e28a5ea92d5e155ac346ebd82828d508f44f4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1927463d606bf10bea7c2ddb5c31b00ff037d766b129f9620262d1b12ca44ebf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"908ac39382a7f493cb7b06dfb923c7c2bbe26c0171890450cd1f88f1355c7451","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f7a04a0bdfb9132901b240788f15f017596751bca9984db5e9bf9d3c080b9f4c","source_sha256":"b91134b0321c98fdbc750a31ba0314dea5693d2a0ca446a2ffe21e21866c8af5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← tsum_dirichletSummand χ hs]\n  apply eulerProduct_completely_multiplicative <| summable_dirichletSummand χ hs","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":2,"token_jaccard":0.071429,"token_length_ratio":0.133333},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"5b21ff78d3f57636f8c33e569c1fdea243ce383b9e43965f1207010b8d10763e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.EulerProduct.ExpLog\npublic import Mathlib.NumberTheory.LSeries.Dirichlet\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# The Euler Product for the Riemann Zeta Function and Dirichlet L-Series\n\nThe first main result of this file is the Euler Product formula for the Riemann ζ function\n$$\\prod_p \\frac{1}{1 - p^{-s}}\n   = \\lim_{n \\to \\infty} \\prod_{p < n} \\frac{1}{1 - p^{-s}} = \\zeta(s)$$\nfor $s$ with real part $> 1$ ($p$ runs through the primes).\n`riemannZeta_eulerProduct` is the second equality above. There are versions\n`riemannZeta_eulerProduct_hasProd` and `riemannZeta_eulerProduct_tprod` in terms of `HasProd`\nand `tprod`, respectively.\n\nThe second result is `dirichletLSeries_eulerProduct` (with variants\n`dirichletLSeries_eulerProduct_hasProd` and `dirichletLSeries_eulerProduct_tprod`),\nwhich is the analogous statement for Dirichlet L-series.\n-/\n\n@[expose] public section\n\nopen Complex\n\nvariable {s : ℂ}\n\n/-- When `s ≠ 0`, the map `n ↦ n^(-s)` is completely multiplicative and vanishes at zero. -/\nnoncomputable\ndef riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where\n  toFun n := (n : ℂ) ^ (-s)\n  map_zero' := by simp [hs]\n  map_one' := by simp\n  map_mul' m n := by\n    simpa only [Nat.cast_mul, ofReal_natCast]\n      using mul_cpow_ofReal_nonneg m.cast_nonneg n.cast_nonneg _\n\n/-- When `χ` is a Dirichlet character and `s ≠ 0`, the map `n ↦ χ n * n^(-s)` is completely\nmultiplicative and vanishes at zero. -/\nnoncomputable\ndef dirichletSummandHom {n : ℕ} (χ : DirichletCharacter ℂ n) (hs : s ≠ 0) : ℕ →*₀ ℂ where\n  toFun n := χ n * (n : ℂ) ^ (-s)\n  map_zero' := by simp [hs]\n  map_one' := by simp\n  map_mul' m n := by\n    simp_rw [← ofReal_natCast]\n    simpa only [Nat.cast_mul, IsUnit.mul_iff, not_and, map_mul, ofReal_mul,\n      mul_cpow_ofReal_nonneg m.cast_nonneg n.cast_nonneg _]\n      using mul_mul_mul_comm ..\n\n/-- When `s.re > 1`, the map `n ↦ n^(-s)` is norm-summable. -/\nlemma summable_riemannZetaSummand (hs : 1 < s.re) :\n    Summable (fun n ↦ ‖riemannZetaSummandHom (ne_zero_of_one_lt_re hs) n‖) := by\n  simp only [riemannZetaSummandHom, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]\n  convert! Real.summable_nat_rpow_inv.mpr hs with n\n  rw [← ofReal_natCast,\n    norm_cpow_eq_rpow_re_of_nonneg (Nat.cast_nonneg n) <| re_neg_ne_zero_of_one_lt_re hs,\n    neg_re, Real.rpow_neg <| Nat.cast_nonneg n]\n\nlemma tsum_riemannZetaSummand (hs : 1 < s.re) :\n    ∑' (n : ℕ), riemannZetaSummandHom (ne_zero_of_one_lt_re hs) n = riemannZeta s := by\n  have hsum := summable_riemannZetaSummand hs\n  rw [zeta_eq_tsum_one_div_nat_add_one_cpow hs, hsum.of_norm.tsum_eq_zero_add, map_zero, zero_add]\n  simp only [riemannZetaSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk,\n    Nat.cast_add, Nat.cast_one, one_div]\n\n/-- When `s.re > 1`, the map `n ↦ χ(n) * n^(-s)` is norm-summable. -/\nlemma summable_dirichletSummand {N : ℕ} (χ : DirichletCharacter ℂ N) (hs : 1 < s.re) :\n    Summable (fun n ↦ ‖dirichletSummandHom χ (ne_zero_of_one_lt_re hs) n‖) := by\n  simp only [dirichletSummandHom, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, norm_mul]\n  exact (summable_riemannZetaSummand hs).of_nonneg_of_le (fun _ ↦ by positivity)\n    (fun n ↦ mul_le_of_le_one_left (norm_nonneg _) <| χ.norm_le_one n)\n\nopen scoped LSeries.notation in\nlemma tsum_dirichletSummand {N : ℕ} (χ : DirichletCharacter ℂ N) (hs : 1 < s.re) :\n    ∑' (n : ℕ), dirichletSummandHom χ (ne_zero_of_one_lt_re hs) n = L ↗χ s := by\n  simp only [dirichletSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, LSeries,\n    LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs), div_eq_mul_inv]\n\nopen Filter Nat Topology EulerProduct\n\n/-- The Euler product for the Riemann ζ function, valid for `s.re > 1`.\nThis version is stated in terms of `HasProd`. -/\ntheorem riemannZeta_eulerProduct_hasProd (hs : 1 < s.re) :\n    HasProd (fun p : Primes ↦ (1 - (p : ℂ) ^ (-s))⁻¹) (riemannZeta s) := by\n  rw [← tsum_riemannZetaSummand hs]\n  apply eulerProduct_completely_multiplicative_hasProd <| summable_riemannZetaSummand hs\n\n/-- The Euler product for the Riemann ζ function, valid for `s.re > 1`.\nThis version is stated in terms of `tprod`. -/\ntheorem riemannZeta_eulerProduct_tprod (hs : 1 < s.re) :\n    ∏' p : Primes, (1 - (p : ℂ) ^ (-s))⁻¹ = riemannZeta s :=\n  (riemannZeta_eulerProduct_hasProd hs).tprod_eq\n\n/-- The Euler product for the Riemann ζ function, valid for `s.re > 1`.\nThis version is stated in the form of convergence of finite partial products. -/\ntheorem riemannZeta_eulerProduct (hs : 1 < s.re) :\n    Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - (p : ℂ) ^ (-s))⁻¹) atTop\n      (𝓝 (riemannZeta s)) := by\n  rw [← tsum_riemannZetaSummand hs]\n  apply eulerProduct_completely_multiplicative <| summable_riemannZetaSummand hs\n\nopen scoped LSeries.notation\n\n/-- The Euler product for Dirichlet L-series, valid for `s.re > 1`.\nThis version is stated in terms of `HasProd`. -/\ntheorem DirichletCharacter.LSeries_eulerProduct_hasProd {N : ℕ} (χ : DirichletCharacter ℂ N)\n    (hs : 1 < s.re) :\n    HasProd (fun p : Primes ↦ (1 - χ p * (p : ℂ) ^ (-s))⁻¹) (L ↗χ s) := by\n  rw [← tsum_dirichletSummand χ hs]\n  convert! eulerProduct_completely_multiplicative_hasProd <| summable_dirichletSummand χ hs\n\n/-- The Euler product for Dirichlet L-series, valid for `s.re > 1`.\nThis version is stated in terms of `tprod`. -/\ntheorem DirichletCharacter.LSeries_eulerProduct_tprod {N : ℕ} (χ : DirichletCharacter ℂ N)\n    (hs : 1 < s.re) :\n    ∏' p : Primes, (1 - χ p * (p : ℂ) ^ (-s))⁻¹ = L ↗χ s :=\n  (DirichletCharacter.LSeries_eulerProduct_hasProd χ hs).tprod_eq\n\n/-- The Euler product for Dirichlet L-series, valid for `s.re > 1`.\nThis version is stated in the form of convergence of finite partial products. -/\n\nTarget:\ntheorem DirichletCharacter.LSeries_eulerProduct {N : ℕ} (χ : DirichletCharacter ℂ N)\n    (hs : 1 < s.re) :\n    Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - χ p * (p : ℂ) ^ (-s))⁻¹) atTop\n      (𝓝 (L ↗χ s)) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_f7a04a0bdfb9","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"666a68473ce8afb834a0807101d07c13568e348b411fd8f66be7e4f83b04e0c5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/EulerProduct","family_id":"dirichletcharacter","file_id":"mathlib/Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean","sample_id":"f7a04a0bdfb9132901b240788f15f017596751bca9984db5e9bf9d3c080b9f4c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ad208d86006761f39c52958101642c0dd5169f84bd3c5136aba050921aea4aea","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d3597b3bf39cf226418d617a055f2bd47be5b5944c89099dabf82fe678119c83","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0659aa452ee5b4142dc99d160a8e44415976c14c3bb5194948dedcb030d02219","source_sha256":"dd8040e4f8746a8f8bfe236176883e5dd7af73f9152b910f7d596480c7e29802","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let ⟨g, hg⟩ := (ContinuousLinearMap.id 𝕜 S).exist_extension_of_finiteDimensional_range\n  exact ⟨g, DFunLike.congr_fun hg.symm⟩","hard_negative":false,"metrics":{"chosen_tokens":28,"rejected_tokens":35,"token_jaccard":0.909091,"token_length_ratio":1.25},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"5b2435560eb218eecc313172d6c6754c5996fc8be5109235d738d204df01ab5b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Convex.Cone.Extension\npublic import Mathlib.Analysis.LocallyConvex.AbsConvexOpen\npublic import Mathlib.Analysis.LocallyConvex.WeakDual\npublic import Mathlib.Analysis.Normed.Module.RCLike.Extend\npublic import Mathlib.Topology.Algebra.Module.FiniteDimension\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2026 Yongxi Lin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yongxi Lin\n-/\n/-!\n# Hahn-Banach theorem for polynormable spaces\n\nIn this file, we prove the analytic Hahn-Banach theorem for polynormable spaces over a field\nsatisfying `IsRCLikeNormedField`. For any continuous linear functional on a subspace, we can extend\nit to the entire space. Note that we cannot use `LocallyConvexSpace` because an\n`IsRCLikeNormedField` has no order structure.\n\nWe prove\n* `Module.Dual.exists_continuous_extension_of_le_seminorm`: Hahn-Banach theorem for linear\n  functionals dominated by a continuous seminorm on polynormable spaces over a field satisfying\n  `IsRCLikeNormedField`.\n* `StrongDual.exists_extension`: Hahn-Banach theorem for continuous linear functionals on\n  polynormable spaces over fields satisfying `IsRCLikeNormedField`.\n\n-/\n\npublic section\n\nopen Module Topology RCLike\n\nopen scoped ComplexConjugate\n\nvariable {𝕜 E : Type*} [AddCommGroup E]\n\ntheorem Module.Dual.exists_extension_of_le_seminorm_real [Module ℝ E]\n    (S : Subspace ℝ E) (f : Dual ℝ S)\n    {p : Seminorm ℝ E} (hp : ∀ x, f x ≤ p x) :\n    ∃ g : Dual ℝ E, (∀ x : S, g x = f x) ∧ ∀ x, |g x| ≤ p x := by\n  obtain ⟨g, hg, hl⟩ := by\n    refine exists_extension_of_le_sublinear ⟨S, f⟩ p (fun _ hc _ => ?_) ?_ hp\n    · simp [map_smul_eq_mul, abs_of_nonneg hc.le]\n    · exact fun x y => map_add_le_add p x y\n  exact ⟨g, hg, p.abs_le_of_le hl⟩\n\nvariable [NormedField 𝕜] [IsRCLikeNormedField 𝕜]\n\ntheorem Module.Dual.exists_extension_of_le_seminorm [Module 𝕜 E] (S : Submodule 𝕜 E) (f : Dual 𝕜 S)\n    {p : Seminorm 𝕜 E} (hp : ∀ x, ‖f x‖ ≤ p x) :\n    ∃ g : Dual 𝕜 E, (∀ x : S, g x = f x) ∧ ∀ x, ‖g x‖ ≤ p x := by\n  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜\n  letI : Module ℝ E := .restrictScalars ℝ 𝕜 E\n  letI : IsScalarTower ℝ 𝕜 E := .restrictScalars _ _ _\n  let fr : Dual ℝ S := reLm.comp (f.restrictScalars ℝ)\n  obtain ⟨g, (hg : ∀ x : S, g x = fr x), hgp⟩ :=\n    fr.exists_extension_of_le_seminorm_real (S.restrictScalars ℝ) (p := p.restrictScalars ℝ)\n      fun x ↦ (re_le_norm (f x)).trans (hp x)\n  refine ⟨g.extendRCLike, fun x ↦ ?_, fun x ↦ ?_⟩\n  · rw [g.extendRCLike_apply, ← Submodule.coe_smul, hg, hg]\n    simp [fr, mul_comm I]\n  · apply norm_extendRCLike_le_seminorm\n    exact hgp\n\nvariable [TopologicalSpace E]\n\n/-- **Hahn-Banach theorem** for linear functionals dominated by a continuous seminorm on\npolynormable spaces over `ℝ`. -/\ntheorem Module.Dual.exists_continuous_extension_of_le_seminorm_real [IsTopologicalAddGroup E]\n    [Module ℝ E] [ContinuousSMul ℝ E] [PolynormableSpace ℝ E] (S : Subspace ℝ E) (f : Dual ℝ S)\n    {p : Seminorm ℝ E} (hp_cont : Continuous p) (hp : ∀ x, f x ≤ p x) :\n    ∃ g : StrongDual ℝ E, (∀ x : S, g x = f x) ∧ ∀ x, |g x| ≤ p x := by\n  obtain ⟨g, hg, hl⟩ := f.exists_extension_of_le_seminorm_real S hp\n  exact ⟨⟨g, (PolynormableSpace.withSeminorms ℝ E).continuous_real_rng g\n    ⟨{⟨p, hp_cont⟩}, 1, fun x ↦ by simpa using (le_abs_self _).trans (hl x)⟩⟩, hg, hl⟩\n\nvariable [Module 𝕜 E] [PolynormableSpace 𝕜 E]\n\n/-- **Hahn-Banach theorem** for linear functionals dominated by a continuous seminorm on\npolynormable spaces over fields satisfying `IsRCLikeNormedField`. -/\ntheorem Module.Dual.exists_continuous_extension_of_le_seminorm (S : Submodule 𝕜 E) (f : Dual 𝕜 S)\n    {p : Seminorm 𝕜 E} (hp_cont : Continuous p) (hp : ∀ x, ‖f x‖ ≤ p x) :\n    ∃ g : StrongDual 𝕜 E, (∀ x : S, g x = f x) ∧ ∀ x, ‖g x‖ ≤ p x := by\n  obtain ⟨g, hg, hle⟩ := Dual.exists_extension_of_le_seminorm S f hp\n  refine ⟨⟨g, (PolynormableSpace.withSeminorms 𝕜 E).continuous_normedSpace_rng 𝕜 g ?_⟩, hg, hle⟩\n  exact ⟨{⟨p, hp_cont⟩}, 1, by simpa⟩\n\n/-- **Hahn-Banach theorem** for continuous linear functionals on polynormable spaces over a field\nsatisfying `IsRCLikeNormedField`. -/\ntheorem StrongDual.exists_extension {𝕜} [NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜]\n    [Module 𝕜 E] [PolynormableSpace 𝕜 E] (S : Submodule 𝕜 E) (f : StrongDual 𝕜 S) :\n    ∃ g : StrongDual 𝕜 E, ∀ x : S, g x = f x := by\n  obtain ⟨q, hq_cont, hq⟩ := Seminorm.exists_le_comp_of_isInducing (f := S.subtype)\n    (p := f.toSeminorm) f.continuous.norm IsInducing.subtypeVal\n  obtain ⟨g, hg, _⟩ := Dual.exists_continuous_extension_of_le_seminorm S f.toLinearMap hq_cont hq\n  exact ⟨g, hg⟩\n\nvariable {F : Type*} [AddCommGroup F] [TopologicalSpace F] [IsTopologicalAddGroup F] [Module 𝕜 F]\n  [ContinuousSMul 𝕜 F] [T2Space F]\n\n/-- Corollary of the polynormable **Hahn-Banach theorem**: if `f : S → F` is a continuous\nlinear map with finite-dimensional range, then `f` extends to a continuous linear map on the whole\nspace. -/\nlemma ContinuousLinearMap.exist_extension_of_finiteDimensional_range {S : Submodule 𝕜 E}\n    (f : S →L[𝕜] F) [FiniteDimensional 𝕜 f.range] :\n    ∃ g : E →L[𝕜] F, f = g.comp S.subtypeL := by\n  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜\n  let b := Module.finBasis 𝕜 f.range\n  let e := b.equivFunL\n  let fi := fun i ↦ (LinearMap.toContinuousLinearMap (b.coord i)).comp\n    (f.codRestrict _ <| LinearMap.mem_range_self _)\n  choose gi hgf using fun i ↦ StrongDual.exists_extension S (fi i)\n  use f.range.subtypeL.comp <| e.symm.toContinuousLinearMap.comp (.pi gi)\n  ext x\n  simp [fi, e, hgf]\n\n/-- A finite-dimensional submodule of a polynormable space over a field satisfying\n`IsRCLikeNormedField` is `Submodule.ClosedComplemented`. -/\n\nTarget:\nlemma Submodule.ClosedComplemented.of_finiteDimensional [PolynormableSpace 𝕜 F] (S : Submodule 𝕜 F)\n    [FiniteDimensional 𝕜 S] : S.ClosedComplemented :=\n\nProof body:\n","rejected":"```lean\nby\n  let ⟨g, hg⟩ := (ContinuousLinearMap.id 𝕜 S).exist_extension_of_finiteDimensional_range\n  exact ⟨g, DFunLike.congr_fun hg.symm⟩\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/LocallyConvex","family_id":"submodule","file_id":"mathlib/Mathlib/Analysis/LocallyConvex/HahnBanach.lean","sample_id":"0659aa452ee5b4142dc99d160a8e44415976c14c3bb5194948dedcb030d02219"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ccb01e73a5d93776c5ed6830bc49ca880d5f04d2137609fc522a9ded57d8bf1c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6c405ac245686730f2e5f7fda35bbf73c516cea158bccf92ad5f323956571fcb","source_sha256":"e095ac1997df0a4695785cbe0250e8d2d4a54f54912b6bc94870e67de3b0352a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine Multifork.IsLimit.hom_ext hc fun a ↦ ?_\n  have heq := d.condition ⟨⟨(f.s₀ a), (g.s₀ a)⟩, H.H a⟩\n  simp only [multicospanIndex_right, multicospanShape_fst, multicospanIndex_left,\n    multicospanIndex_fst, multicospanShape_snd, multicospanIndex_snd] at heq\n  simp [-Homotopy.wl, -Homotopy.wr, ← H.wl, ← H.wr, reassoc_of% heq]","hard_negative":false,"metrics":{"chosen_tokens":86,"rejected_tokens":3,"token_jaccard":0.021277,"token_length_ratio":0.034884},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"5b9661add22f5b32cbc10c679eee56a9c11d0fa02f33ae7cc26700c037f5c8f0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Quotient\npublic import Mathlib.CategoryTheory.Sites.Hypercover.One\npublic import Mathlib.CategoryTheory.Filtered.Basic\n\nNamespace:\nCategoryTheory.PreOneHypercover\n\nLocal context:\n/-\nCopyright (c) 2025 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten\n-/\n/-!\n# The category of `1`-hypercovers up to homotopy\n\nIn this file we define the category of `1`-hypercovers up to homotopy. This is the category of\n`1`-hypercovers, but where morphisms are considered up to existence of a homotopy.\n\n## Main definitions\n\n- `CategoryTheory.PreOneHypercover.Homotopy`: A homotopy of refinements `E ⟶ F` is a family of\n  morphisms `Xᵢ ⟶ Yₐ` where `Yₐ` is a component of the cover of `X_{f(i)} ×[S] X_{g(i)}`.\n- `CategoryTheory.GrothendieckTopology.HOneHypercover`: The category of `1`-hypercovers\n  with respect to a Grothendieck topology and morphisms up to homotopy.\n\n## Main results\n\n- `CategoryTheory.GrothendieckTopology.HOneHypercover.isCofiltered_of_hasPullbacks`: The\n  category of `1`-hypercovers up to homotopy is cofiltered if `C` has pullbacks.\n-/\n\n@[expose] public section\n\nuniverse w'' w' w v u\n\nnamespace CategoryTheory\n\nopen Limits\n\nvariable {C : Type u} [Category.{v} C]\n\nnamespace PreOneHypercover\n\nvariable {S : C} {E : PreOneHypercover.{w} S} {F : PreOneHypercover.{w'} S}\n\n/-- A homotopy of refinements `E ⟶ F` is a family of morphisms `Xᵢ ⟶ Yₐ` where\n`Yₐ` is a component of the cover of `X_{f(i)} ×[S] X_{g(i)}`. -/\nstructure Homotopy (f g : E.Hom F) where\n  /-- The index map sending `i : E.I₀` to `a` above `(f(i), g(i))`. -/\n  H (i : E.I₀) : F.I₁ (f.s₀ i) (g.s₀ i)\n  /-- The morphism `Xᵢ ⟶ Yₐ`. -/\n  a (i : E.I₀) : E.X i ⟶ F.Y (H i)\n  wl (i : E.I₀) : a i ≫ F.p₁ (H i) = f.h₀ i\n  wr (i : E.I₀) : a i ≫ F.p₂ (H i) = g.h₀ i\n\nattribute [reassoc (attr := simp)] Homotopy.wl Homotopy.wr\n\nsection\n\nvariable {A : Type*} [Category* A]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- Homotopic refinements induce the same map on multiequalizers. -/\n\nTarget:\nlemma Homotopy.mapMultiforkOfIsLimit_eq\n    {E F : PreOneHypercover.{w} S} {f g : E.Hom F} (H : Homotopy f g)\n    (P : Cᵒᵖ ⥤ A) {c : Multifork (E.multicospanIndex P)} (hc : IsLimit c)\n    (d : Multifork (F.multicospanIndex P)) :\n    f.mapMultiforkOfIsLimit P hc d = g.mapMultiforkOfIsLimit P hc d :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Sites","family_id":"homotopy","file_id":"mathlib/Mathlib/CategoryTheory/Sites/Hypercover/Homotopy.lean","sample_id":"6c405ac245686730f2e5f7fda35bbf73c516cea158bccf92ad5f323956571fcb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9117fb79a21990bf1e7a81654768056689d985c1af2446daa090c5cc9faf03cd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3890101cc6d52342ce69e19715aee46bb30db6af8bbb140e8a5fac3718968504","source_sha256":"80f83e637655104ff57f368934350815480bb54de691c8a5aefe35bab1dd7af7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases eq_or_lt_of_le hj with h | h\n  · rw [h]\n  · exact T.row_weak h cell","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":3,"token_jaccard":0.117647,"token_length_ratio":0.15},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"5bcb2aae0b28e1d06d12f60b24640fc7447a50ff412c0a47d7fb58d037c6cc83","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Young.YoungDiagram\n\nNamespace:\nSemistandardYoungTableau\n\nLocal context:\n/-\nCopyright (c) 2022 Jake Levinson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jake Levinson\n-/\n/-!\n# Semistandard Young tableaux\n\nA semistandard Young tableau is a filling of a Young diagram by natural numbers, such that\nthe entries are weakly increasing left-to-right along rows (i.e. for fixed `i`), and\nstrictly-increasing top-to-bottom along columns (i.e. for fixed `j`).\n\nAn example of an SSYT of shape `μ = [4, 2, 1]` is:\n\n```text\n0 0 0 2\n1 1\n2\n```\n\nWe represent a semistandard Young tableau as a function `ℕ → ℕ → ℕ`, which is required to be zero\nfor all pairs `(i, j) ∉ μ` and to satisfy the row-weak and column-strict conditions on `μ`.\n\n\n## Main definitions\n\n- `SemistandardYoungTableau (μ : YoungDiagram)`: semistandard Young tableaux of shape `μ`. There is\n  a `coe` instance such that `T i j` is value of the `(i, j)` entry of the semistandard Young\n  tableau `T`.\n- `SemistandardYoungTableau.highestWeight (μ : YoungDiagram)`: the semistandard Young tableau whose\n  `i`th row consists entirely of `i`s, for each `i`.\n\n## Tags\n\nSemistandard Young tableau\n\n## References\n\n<https://en.wikipedia.org/wiki/Young_tableau>\n\n-/\n\n@[expose] public section\n\n\n/-- A semistandard Young tableau is a filling of the cells of a Young diagram by natural\nnumbers, such that the entries in each row are weakly increasing (left to right), and the entries\nin each column are strictly increasing (top to bottom).\n\nHere, a semistandard Young tableau is represented as an unrestricted function `ℕ → ℕ → ℕ` that, for\nreasons of extensionality, is required to vanish outside `μ`. -/\nstructure SemistandardYoungTableau (μ : YoungDiagram) where\n  /-- `entry i j` is value of the `(i, j)` entry of the SSYT `μ`. -/\n  entry : ℕ → ℕ → ℕ\n  /-- The entries in each row are weakly increasing (left to right). -/\n  row_weak' : ∀ {i j1 j2 : ℕ}, j1 < j2 → (i, j2) ∈ μ → entry i j1 ≤ entry i j2\n  /-- The entries in each column are strictly increasing (top to bottom). -/\n  col_strict' : ∀ {i1 i2 j : ℕ}, i1 < i2 → (i2, j) ∈ μ → entry i1 j < entry i2 j\n  /-- `entry` is required to be zero for all pairs `(i, j) ∉ μ`. -/\n  zeros' : ∀ {i j}, (i, j) ∉ μ → entry i j = 0\n\nnamespace SemistandardYoungTableau\n\ninstance instFunLike {μ : YoungDiagram} : FunLike (SemistandardYoungTableau μ) ℕ (ℕ → ℕ) where\n  coe := SemistandardYoungTableau.entry\n  coe_injective T T' h := by\n    cases T\n    cases T'\n    congr\n\n@[simp]\ntheorem to_fun_eq_coe {μ : YoungDiagram} {T : SemistandardYoungTableau μ} :\n    T.entry = (T : ℕ → ℕ → ℕ) :=\n  rfl\n\n@[ext]\ntheorem ext {μ : YoungDiagram} {T T' : SemistandardYoungTableau μ} (h : ∀ i j, T i j = T' i j) :\n    T = T' :=\n  DFunLike.ext T T' fun _ ↦ by\n    funext\n    apply h\n\n/-- Copy of an `SemistandardYoungTableau μ` with a new `entry` equal to the old one. Useful to fix\ndefinitional equalities. -/\nprotected def copy {μ : YoungDiagram} (T : SemistandardYoungTableau μ) (entry' : ℕ → ℕ → ℕ)\n    (h : entry' = T) : SemistandardYoungTableau μ where\n  entry := entry'\n  row_weak' := h.symm ▸ T.row_weak'\n  col_strict' := h.symm ▸ T.col_strict'\n  zeros' := h.symm ▸ T.zeros'\n\n@[simp]\ntheorem coe_copy {μ : YoungDiagram} (T : SemistandardYoungTableau μ) (entry' : ℕ → ℕ → ℕ)\n    (h : entry' = T) : ⇑(T.copy entry' h) = entry' :=\n  rfl\n\ntheorem copy_eq {μ : YoungDiagram} (T : SemistandardYoungTableau μ) (entry' : ℕ → ℕ → ℕ)\n    (h : entry' = T) : T.copy entry' h = T :=\n  DFunLike.ext' h\n\ntheorem row_weak {μ : YoungDiagram} (T : SemistandardYoungTableau μ) {i j1 j2 : ℕ} (hj : j1 < j2)\n    (hcell : (i, j2) ∈ μ) : T i j1 ≤ T i j2 :=\n  T.row_weak' hj hcell\n\ntheorem col_strict {μ : YoungDiagram} (T : SemistandardYoungTableau μ) {i1 i2 j : ℕ} (hi : i1 < i2)\n    (hcell : (i2, j) ∈ μ) : T i1 j < T i2 j :=\n  T.col_strict' hi hcell\n\ntheorem zeros {μ : YoungDiagram} (T : SemistandardYoungTableau μ) {i j : ℕ}\n    (not_cell : (i, j) ∉ μ) : T i j = 0 :=\n  T.zeros' not_cell\n\nTarget:\ntheorem row_weak_of_le {μ : YoungDiagram} (T : SemistandardYoungTableau μ) {i j1 j2 : ℕ}\n    (hj : j1 ≤ j2) (cell : (i, j2) ∈ μ) : T i j1 ≤ T i j2 :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Young","family_id":"row_weak_of_le","file_id":"mathlib/Mathlib/Combinatorics/Young/SemistandardTableau.lean","sample_id":"3890101cc6d52342ce69e19715aee46bb30db6af8bbb140e8a5fac3718968504"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a57d660deed4ffd6ef566ef7d9ca6c07159f27d62499692c71e882d53f3bfa34","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"de470aca7ab0cd9ed863886604fdb71bf833b261a094a1692705ddb8ff76b1d8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"211d293f722fe1130c34a732f32615615b914ff3488d05c85b24b7fea5b7d189","source_sha256":"6515bdb4a68678b27490ec2b3332f6d7b1e529489ff9d844898ec1eda5637ec5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  grind [= lt_iff_le_not_ge]","hard_negative":true,"metrics":{"chosen_tokens":6,"rejected_tokens":2,"token_jaccard":0.142857,"token_length_ratio":0.333333},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"5c15b65eedaa892cb703a0fc9844f59e52ef7c50a28686f8d6cc9e9e89ae7e63","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Tactic.Alias\npublic import Batteries.Tactic.Trans\npublic import Mathlib.Tactic.ExtendDoc\npublic import Mathlib.Tactic.ToDual\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Orders\n\nDefines classes for preorders and partial orders\nand proves some basic lemmas about them.\n\nWe also define covering relations on a preorder.\nWe say that `b` *covers* `a` if `a < b` and there is no element in between.\nWe say that `b` *weakly covers* `a` if `a ≤ b` and there is no element between `a` and `b`.\nIn a partial order this is equivalent to `a ⋖ b ∨ a = b`,\nin a preorder this is equivalent to `a ⋖ b ∨ (a ≤ b ∧ b ≤ a)`\n\n## Notation\n\n* `a ⋖ b` means that `b` covers `a`.\n* `a ⩿ b` means that `b` weakly covers `a`.\n-/\n\n@[expose] public section\n\nvariable {α : Type*}\n\nsection Preorder\n\n/-!\n### Definition of `Preorder` and lemmas about types with a `Preorder`\n-/\n\n/--\nA preorder is a reflexive, transitive relation `≤`.\nIn a preorder, `a < b` means `a ≤ b ∧ ¬b ≤ a`, and `<` is defined this way by default.\nYou can override this definition to set a better def-eq.\n-/\nclass Preorder (α : Type*) extends LE α, LT α where\n  protected le_refl : ∀ a : α, a ≤ a\n  protected le_trans : ∀ a b c : α, a ≤ b → b ≤ c → a ≤ c\n  lt := fun a b => a ≤ b ∧ ¬b ≤ a\n  protected lt_iff_le_not_ge : ∀ a b : α, a < b ↔ a ≤ b ∧ ¬b ≤ a := by intros; rfl\n\nattribute [to_dual self (reorder := le_trans (a c, 4 5), lt_iff_le_not_ge (a b))] Preorder.mk\n\ninstance [Preorder α] : Std.LawfulOrderLT α where\n  lt_iff := Preorder.lt_iff_le_not_ge\n\ninstance [Preorder α] : Std.IsPreorder α where\n  le_refl := Preorder.le_refl\n  le_trans := Preorder.le_trans\n\nvariable [Preorder α] {a b c : α}\n\n/-- The relation `≤` on a preorder is reflexive. -/\n@[refl] lemma le_refl : ∀ a : α, a ≤ a := Preorder.le_refl\n\n/-- A version of `le_refl` where the argument is implicit -/\nlemma le_rfl : a ≤ a := le_refl a\n\n/-- The relation `≤` on a preorder is transitive. -/\nlemma le_trans : a ≤ b → b ≤ c → a ≤ c := Preorder.le_trans _ _ _\n\n@[to_dual existing le_trans]\nlemma ge_trans : b ≤ a → c ≤ b → c ≤ a := flip le_trans\n\n@[to_dual self]\nlemma lt_iff_le_not_ge : a < b ↔ a ≤ b ∧ ¬b ≤ a := Preorder.lt_iff_le_not_ge _ _\n\n@[to_dual self]\nlemma lt_of_le_not_ge (hab : a ≤ b) (hba : ¬ b ≤ a) : a < b := lt_iff_le_not_ge.2 ⟨hab, hba⟩\n\n@[to_dual ge_of_eq] lemma le_of_eq (hab : a = b) : a ≤ b := by rw [hab]\n@[to_dual self] lemma le_of_lt (hab : a < b) : a ≤ b := (lt_iff_le_not_ge.1 hab).1\n@[to_dual self] lemma not_le_of_gt (hab : a < b) : ¬ b ≤ a := (lt_iff_le_not_ge.1 hab).2\n@[to_dual self] lemma not_lt_of_ge (hab : a ≤ b) : ¬ b < a := imp_not_comm.1 not_le_of_gt hab\n\n@[to_dual self] alias LT.lt.not_ge := not_le_of_gt\n@[to_dual self] alias LE.le.not_gt := not_lt_of_ge\n\nlemma lt_irrefl (a : α) : ¬a < a := fun h ↦ not_le_of_gt h le_rfl\n\n@[to_dual lt_of_lt_of_le']\nlemma lt_of_lt_of_le (hab : a < b) (hbc : b ≤ c) : a < c :=\n  lt_of_le_not_ge (le_trans (le_of_lt hab) hbc) fun hca ↦ not_le_of_gt hab (le_trans hbc hca)\n\n@[to_dual lt_of_le_of_lt']\nlemma lt_of_le_of_lt (hab : a ≤ b) (hbc : b < c) : a < c :=\n  lt_of_le_not_ge (le_trans hab (le_of_lt hbc)) fun hca ↦ not_le_of_gt hbc (le_trans hca hab)\n\n@[to_dual gt_trans]\nlemma lt_trans : a < b → b < c → a < c := fun h₁ h₂ => lt_of_lt_of_le h₁ (le_of_lt h₂)\n\n@[to_dual ne_of_gt]\nlemma ne_of_lt (h : a < b) : a ≠ b := fun he => absurd h (he ▸ lt_irrefl a)\n@[to_dual self]\nlemma lt_asymm (h : a < b) : ¬b < a := fun h1 : b < a => lt_irrefl a (lt_trans h h1)\n\n@[to_dual self] alias not_lt_of_gt := lt_asymm\n\n@[to_dual le_of_lt_or_eq']\nlemma le_of_lt_or_eq (h : a < b ∨ a = b) : a ≤ b := h.elim le_of_lt le_of_eq\n@[to_dual le_of_eq_or_lt']\nlemma le_of_eq_or_lt (h : a = b ∨ a < b) : a ≤ b := h.elim le_of_eq le_of_lt\n\n@[to_dual self]\n\nTarget:\nlemma lt_iff_gt_iff_le_iff_ge : (a < b ↔ b < a) ↔ (a ≤ b ↔ b ≤ a) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_211d293f722f","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"d8f151fa4ec2b750bec1866298c99eefc7bf66ba484608549f30dba80ed0671e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Defs","family_id":"lt_iff_gt_iff_le_iff_ge","file_id":"mathlib/Mathlib/Order/Defs/PartialOrder.lean","sample_id":"211d293f722fe1130c34a732f32615615b914ff3488d05c85b24b7fea5b7d189"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b9ea88f535ce93e2572a2904d87bf732a9a01a29fb1bf1dec81d4ec155726243","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"823540ed3b37d6e59c7f7ee773f32c63d19ea6128bee66cf2ba5a74b1501cce0","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6e268d2dae8bb1ebce0d32eba74344e7b2814801568b87e7044f89199cff8536","source_sha256":"91e010f7f912f8a75b6beeb48fe6ecf2da1ebc9b1ad8cd51916c3d730bd71e3d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [← orderOf_pos_iff, ← Nat.card_zpowers, Nat.card_pos_iff, ← SetLike.coe_sort_coe,\n    nonempty_coe_sort, Nat.card_pos_iff, Set.finite_coe_iff, OneMemClass.coe_nonempty, true_and]","hard_negative":true,"metrics":{"chosen_tokens":37,"rejected_tokens":3,"token_jaccard":0.045455,"token_length_ratio":0.081081},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"5c2e3529ea383f616c73414b201f2ff8a6e4b12f18eb2b249e40028fbfb85847","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.ZMod.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# `ZMod n` and quotient groups / rings\n\nThis file relates `ZMod n` to the quotient group `ℤ / AddSubgroup.zmultiples (n : ℤ)`.\n\n## Main definitions\n\n- `ZMod.quotientZMultiplesNatEquivZMod` and `ZMod.quotientZMultiplesEquivZMod`:\n  `ZMod n` is the group quotient of `ℤ` by `n ℤ := AddSubgroup.zmultiples (n)`,\n  (where `n : ℕ` and `n : ℤ` respectively)\n- `ZMod.lift n f` is the map from `ZMod n` induced by `f : ℤ →+ A` that maps `n` to `0`.\n\n## Tags\n\nzmod, quotient group\n-/\n\n@[expose] public section\n\nassert_not_exists Ideal TwoSidedIdeal\n\nopen QuotientAddGroup Set ZMod\nopen scoped IsMulCommutative\n\nvariable (n : ℕ) {A R : Type*} [AddGroup A] [Ring R]\n\nnamespace Int\n\n/-- `ℤ` modulo multiples of `n : ℕ` is `ZMod n`. -/\ndef quotientZMultiplesNatEquivZMod : ℤ ⧸ AddSubgroup.zmultiples (n : ℤ) ≃+ ZMod n :=\n  (quotientAddEquivOfEq (ZMod.ker_intCastAddHom _)).symm.trans <|\n    quotientKerEquivOfRightInverse (Int.castAddHom (ZMod n)) cast intCast_zmod_cast\n\n/-- `ℤ` modulo multiples of `a : ℤ` is `ZMod a.natAbs`. -/\ndef quotientZMultiplesEquivZMod (a : ℤ) : ℤ ⧸ AddSubgroup.zmultiples a ≃+ ZMod a.natAbs :=\n  (quotientAddEquivOfEq (zmultiples_natAbs a)).symm.trans (quotientZMultiplesNatEquivZMod a.natAbs)\n\n@[simp]\nlemma index_zmultiples (a : ℤ) : (AddSubgroup.zmultiples a).index = a.natAbs := by\n  rw [AddSubgroup.index, Nat.card_congr (quotientZMultiplesEquivZMod a).toEquiv, Nat.card_zmod]\n\nend Int\n\n\nnamespace AddAction\n\nopen AddSubgroup AddMonoidHom AddEquiv Function\n\nvariable {α β : Type*} [AddGroup α] (a : α) [AddAction α β] (b : β)\n\n/-- The quotient `(ℤ ∙ a) ⧸ (stabilizer b)` is cyclic of order `minimalPeriod (a +ᵥ ·) b`. -/\nnoncomputable def zmultiplesQuotientStabilizerEquiv :\n    zmultiples a ⧸ stabilizer (zmultiples a) b ≃+ ZMod (minimalPeriod (a +ᵥ ·) b) :=\n  (ofBijective\n          (map _ (stabilizer (zmultiples a) b) (zmultiplesHom (zmultiples a) ⟨a, mem_zmultiples a⟩)\n            (by\n              rw [zmultiples_le, mem_comap, mem_stabilizer_iff, zmultiplesHom_apply, natCast_zsmul]\n              simp_rw [← vadd_iterate]\n              exact isPeriodicPt_minimalPeriod (a +ᵥ ·) b))\n          ⟨by\n            rw [← ker_eq_bot_iff, eq_bot_iff]\n            refine fun q => induction_on q fun n hn => ?_\n            rw [mem_bot, eq_zero_iff, Int.mem_zmultiples_iff, ←\n              zsmul_vadd_eq_iff_minimalPeriod_dvd]\n            exact (eq_zero_iff _).mp hn, fun q =>\n            induction_on q fun ⟨_, n, rfl⟩ => ⟨n, rfl⟩⟩).symm.trans\n    (Int.quotientZMultiplesNatEquivZMod (minimalPeriod (a +ᵥ ·) b))\n\ntheorem zmultiplesQuotientStabilizerEquiv_symm_apply (n : ZMod (minimalPeriod (a +ᵥ ·) b)) :\n    (zmultiplesQuotientStabilizerEquiv a b).symm n =\n      (cast n : ℤ) • (⟨a, mem_zmultiples a⟩ : zmultiples a) :=\n  rfl\n\nend AddAction\n\nnamespace MulAction\n\nopen AddAction Subgroup AddSubgroup Function\n\nvariable {α β : Type*} [Group α] (a : α) [MulAction α β] (b : β)\n\n/-- The quotient `(a ^ ℤ) ⧸ (stabilizer b)` is cyclic of order `minimalPeriod ((•) a) b`. -/\nnoncomputable def zpowersQuotientStabilizerEquiv :\n    zpowers a ⧸ stabilizer (zpowers a) b ≃* Multiplicative (ZMod (minimalPeriod (a • ·) b)) :=\n  letI f := zmultiplesQuotientStabilizerEquiv (Additive.ofMul a) b\n  AddEquiv.toMultiplicative f\n\ntheorem zpowersQuotientStabilizerEquiv_symm_apply (n : ZMod (minimalPeriod (a • ·) b)) :\n    (zpowersQuotientStabilizerEquiv a b).symm n = (⟨a, mem_zpowers a⟩ : zpowers a) ^ (cast n : ℤ) :=\n  rfl\n\n/-- The orbit `(a ^ ℤ) • b` is a cycle of order `minimalPeriod ((•) a) b`. -/\nnoncomputable def orbitZPowersEquiv : orbit (zpowers a) b ≃ ZMod (minimalPeriod (a • ·) b) :=\n  (orbitEquivQuotientStabilizer _ b).trans (zpowersQuotientStabilizerEquiv a b).toEquiv\n\n/-- The orbit `(ℤ • a) +ᵥ b` is a cycle of order `minimalPeriod (a +ᵥ ·) b`. -/\nnoncomputable def _root_.AddAction.orbitZMultiplesEquiv {α β : Type*} [AddGroup α] (a : α)\n    [AddAction α β] (b : β) :\n    AddAction.orbit (zmultiples a) b ≃ ZMod (minimalPeriod (a +ᵥ ·) b) :=\n  (AddAction.orbitEquivQuotientStabilizer (zmultiples a) b).trans\n    (zmultiplesQuotientStabilizerEquiv a b).toEquiv\n\nattribute [to_additive existing] orbitZPowersEquiv\n\n@[to_additive]\ntheorem orbitZPowersEquiv_symm_apply (k : ZMod (minimalPeriod (a • ·) b)) :\n    (orbitZPowersEquiv a b).symm k =\n      (⟨a, mem_zpowers a⟩ : zpowers a) ^ (cast k : ℤ) • ⟨b, mem_orbit_self b⟩ :=\n  rfl\n\ntheorem orbitZPowersEquiv_symm_apply' (k : ℤ) :\n    (orbitZPowersEquiv a b).symm k =\n      (⟨a, mem_zpowers a⟩ : zpowers a) ^ k • ⟨b, mem_orbit_self b⟩ := by\n  rw [orbitZPowersEquiv_symm_apply, ZMod.coe_intCast]\n  exact Subtype.ext (zpow_smul_mod_minimalPeriod _ _ k)\n\ntheorem _root_.AddAction.orbitZMultiplesEquiv_symm_apply' {α β : Type*} [AddGroup α] (a : α)\n    [AddAction α β] (b : β) (k : ℤ) :\n    (AddAction.orbitZMultiplesEquiv a b).symm k =\n      k • (⟨a, mem_zmultiples a⟩ : zmultiples a) +ᵥ ⟨b, AddAction.mem_orbit_self b⟩ := by\n  rw [AddAction.orbitZMultiplesEquiv_symm_apply, ZMod.coe_intCast]\n  -- Making `a` explicit turns this from ~190000 heartbeats to ~700.\n  exact Subtype.ext (zsmul_vadd_mod_minimalPeriod a _ k)\n\nattribute [to_additive existing]\n  orbitZPowersEquiv_symm_apply'\n\n@[to_additive]\ntheorem minimalPeriod_eq_card [Fintype (orbit (zpowers a) b)] :\n    minimalPeriod (a • ·) b = Fintype.card (orbit (zpowers a) b) := by\n  rw [← Fintype.ofEquiv_card (orbitZPowersEquiv a b), ZMod.card]\n\n@[to_additive]\ninstance minimalPeriod_pos [Finite <| orbit (zpowers a) b] :\n    NeZero <| minimalPeriod (a • ·) b :=\n  ⟨by\n    cases nonempty_fintype (orbit (zpowers a) b)\n    haveI : Nonempty (orbit (zpowers a) b) := (nonempty_orbit b).to_subtype\n    rw [minimalPeriod_eq_card]\n    exact Fintype.card_ne_zero⟩\n\nend MulAction\n\nsection Group\n\nopen Subgroup\n\nvariable {α : Type*} [Group α] (a : α)\n\n/-- See also `Fintype.card_zpowers`. -/\n@[to_additive (attr := simp) /-- See also `Fintype.card_zmultiples`. -/]\ntheorem Nat.card_zpowers : Nat.card (zpowers a) = orderOf a := by\n  have := Nat.card_congr (MulAction.orbitZPowersEquiv a (1 : α))\n  rwa [Nat.card_zmod, orbit_subgroup_one_eq_self] at this\n\nvariable {a}\n\n@[to_additive (attr := simp)]\n\nTarget:\nlemma finite_zpowers : (zpowers a : Set α).Finite ↔ IsOfFinOrder a :=\n\nProof body:\n","rejected":"by\n  exact finite_zpowers","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"70805a4f2bee6a55777f2380d9825fd257154326fdf94ed8b1a65ba5cd01258d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/ZMod","family_id":"finite_zpowers","file_id":"mathlib/Mathlib/Data/ZMod/QuotientGroup.lean","sample_id":"6e268d2dae8bb1ebce0d32eba74344e7b2814801568b87e7044f89199cff8536"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9eee49c5561e216892ae126cb44a6a373a2be5f508e4256a2d5f533053f0d013","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3e916153cf2d028f0fe2211091c80661d43ef75952801649ac6582ce0158842e","source_sha256":"1b33d2ba913b39d2ffb044888cc338bb0c7eb620bd058c3099d88a86d55171a5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [(ofSet s hs).trans_ofSet hs']\n  ext <;> simp [hs'.interior_eq]","hard_negative":false,"metrics":{"chosen_tokens":22,"rejected_tokens":2,"token_jaccard":0.052632,"token_length_ratio":0.090909},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"5c3de58afe1bc0fb8014575e3a99600bbd0769625ceba6abef3278da576c9fa9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.OpenPartialHomeomorph.IsImage\n\nNamespace:\nOpenPartialHomeomorph\n\nLocal context:\n/-\nCopyright (c) 2019 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Partial homeomorphisms: composition\n\n## Main definitions\n\n* `OpenPartialHomeomorph.trans`: the composition of two open partial homeomorphisms\n-/\n\n@[expose] public section\n\nopen Function Set Filter Topology\n\nvariable {X X' : Type*} {Y Y' : Type*} {Z Z' : Type*}\n  [TopologicalSpace X] [TopologicalSpace X'] [TopologicalSpace Y] [TopologicalSpace Y']\n  [TopologicalSpace Z] [TopologicalSpace Z']\n\nnamespace OpenPartialHomeomorph\n\nvariable (e : OpenPartialHomeomorph X Y)\n\n/-!\n## Composition\n\n`trans`: composition of two open partial homeomorphisms\n-/\nsection trans\n\nvariable (e' : OpenPartialHomeomorph Y Z)\n\n/-- Composition of two open partial homeomorphisms when the target of the first and the source of\nthe second coincide. -/\n@[simps! apply symm_apply toPartialHomeomorph, simps! -isSimp source target]\nprotected def trans' (h : e.target = e'.source) : OpenPartialHomeomorph X Z where\n  toPartialEquiv := PartialEquiv.trans' e.toPartialEquiv e'.toPartialEquiv h\n  open_source := e.open_source\n  open_target := e'.open_target\n  continuousOn_toFun := e'.continuousOn.comp e.continuousOn <| h ▸ e.mapsTo\n  continuousOn_invFun := e.continuousOn_symm.comp e'.continuousOn_symm <| h.symm ▸ e'.mapsTo_symm\n\n/-- Composing two open partial homeomorphisms, by restricting to the maximal domain where their\ncomposition is well defined.\nWithin the `Manifold` namespace, there is the notation `e ≫ₕ f` for this. -/\n@[trans]\nprotected def trans : OpenPartialHomeomorph X Z :=\n  OpenPartialHomeomorph.trans' (e.symm.restrOpen e'.source e'.open_source).symm\n    (e'.restrOpen e.target e.open_target) (by simp [inter_comm])\n\n@[simp, mfld_simps]\ntheorem trans_toPartialEquiv :\n    (e.trans e').toPartialEquiv = e.toPartialEquiv.trans e'.toPartialEquiv :=\n  rfl\n\n@[simp, mfld_simps]\ntheorem coe_trans : (e.trans e' : X → Z) = e' ∘ e :=\n  rfl\n\n@[simp, mfld_simps]\ntheorem coe_trans_symm : ((e.trans e').symm : Z → X) = e.symm ∘ e'.symm :=\n  rfl\n\ntheorem trans_apply {x : X} : (e.trans e') x = e' (e x) :=\n  rfl\n\ntheorem trans_symm_eq_symm_trans_symm : (e.trans e').symm = e'.symm.trans e.symm := rfl\n\n/-- This could be considered as a simp lemma, but there are many situations where it makes something\nsimple into something more complicated. -/\ntheorem trans_source : (e.trans e').source = e.source ∩ e ⁻¹' e'.source :=\n  PartialEquiv.trans_source e.toPartialEquiv e'.toPartialEquiv\n\ntheorem trans_source' : (e.trans e').source = e.source ∩ e ⁻¹' (e.target ∩ e'.source) :=\n  PartialEquiv.trans_source' e.toPartialEquiv e'.toPartialEquiv\n\ntheorem trans_source'' : (e.trans e').source = e.symm '' (e.target ∩ e'.source) :=\n  PartialEquiv.trans_source'' e.toPartialEquiv e'.toPartialEquiv\n\ntheorem image_trans_source : e '' (e.trans e').source = e.target ∩ e'.source :=\n  PartialEquiv.image_trans_source e.toPartialEquiv e'.toPartialEquiv\n\ntheorem trans_target : (e.trans e').target = e'.target ∩ e'.symm ⁻¹' e.target :=\n  rfl\n\ntheorem trans_target' : (e.trans e').target = e'.target ∩ e'.symm ⁻¹' (e'.source ∩ e.target) :=\n  trans_source' e'.symm e.symm\n\ntheorem trans_target'' : (e.trans e').target = e' '' (e'.source ∩ e.target) :=\n  trans_source'' e'.symm e.symm\n\ntheorem inv_image_trans_target : e'.symm '' (e.trans e').target = e'.source ∩ e.target :=\n  image_trans_source e'.symm e.symm\n\ntheorem trans_assoc (e'' : OpenPartialHomeomorph Z Z') :\n    (e.trans e').trans e'' = e.trans (e'.trans e'') :=\n  toPartialEquiv_injective <| e.1.trans_assoc _ _\n\n@[simp, mfld_simps]\ntheorem trans_refl : e.trans (OpenPartialHomeomorph.refl Y) = e :=\n  toPartialHomeomorph_injective (PartialHomeomorph.toPartialEquiv_injective e.1.trans_refl)\n\n@[simp, mfld_simps]\ntheorem refl_trans : (OpenPartialHomeomorph.refl X).trans e = e :=\n  toPartialHomeomorph_injective (PartialHomeomorph.toPartialEquiv_injective e.1.refl_trans)\n\ntheorem trans_ofSet {s : Set Y} (hs : IsOpen s) : e.trans (ofSet s hs) = e.restr (e ⁻¹' s) :=\n  OpenPartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) <| by\n    rw [trans_source, restr_source, ofSet_source, ← preimage_interior, hs.interior_eq]\n\ntheorem trans_of_set' {s : Set Y} (hs : IsOpen s) :\n    e.trans (ofSet s hs) = e.restr (e.source ∩ e ⁻¹' s) := by rw [trans_ofSet, restr_source_inter]\n\ntheorem ofSet_trans {s : Set X} (hs : IsOpen s) : (ofSet s hs).trans e = e.restr s :=\n  OpenPartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) <|\n    by simp [hs.interior_eq, inter_comm]\n\ntheorem ofSet_trans' {s : Set X} (hs : IsOpen s) :\n    (ofSet s hs).trans e = e.restr (e.source ∩ s) := by\n  rw [ofSet_trans, restr_source_inter]\n\n@[simp, mfld_simps]\n\nTarget:\ntheorem ofSet_trans_ofSet {s : Set X} (hs : IsOpen s) {s' : Set X} (hs' : IsOpen s') :\n    (ofSet s hs).trans (ofSet s' hs') = ofSet (s ∩ s') (IsOpen.inter hs hs') :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/OpenPartialHomeomorph","family_id":"ofset_trans_ofset","file_id":"mathlib/Mathlib/Topology/OpenPartialHomeomorph/Composition.lean","sample_id":"3e916153cf2d028f0fe2211091c80661d43ef75952801649ac6582ce0158842e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"af8967d58c125507dfdd8a870d8e2d61acb151c1a146b922c5673f800fa162b8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4d42ffb30d467171f90d25061bdb6019ce4a7b2cd47b0c61a8c1baa7e67cd6ef","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2f8dba92a8be5eab58edb05807e69424a9fcd6a33e1126c48358e2bed2791279","source_sha256":"46ad986a41048710426d3296a8284bc983e91e0a000d6a497db675e512528c15","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [mem_disjSum, or_iff_right]\n  · simp only [inr.injEq, exists_eq_right]\n  rintro ⟨a, _, ha⟩\n  exact inl_ne_inr ha","hard_negative":true,"metrics":{"chosen_tokens":28,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.071429},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"5c4929242b552524a8bb8a73b2337ead51dc600282646d90930fe92afd69316f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.Multiset\n\nNamespace:\nMultiset\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Disjoint sum of multisets\n\nThis file defines the disjoint sum of two multisets as `Multiset (α ⊕ β)`. Beware not to confuse\nwith the `Multiset.sum` operation which computes the additive sum.\n\n## Main declarations\n\n* `Multiset.disjSum`: `s.disjSum t` is the disjoint sum of `s` and `t`.\n-/\n\n@[expose] public section\n\n\nopen Sum\n\nnamespace Multiset\n\nvariable {α β γ : Type*} (s : Multiset α) (t : Multiset β)\n\n/-- Disjoint sum of multisets. -/\ndef disjSum : Multiset (α ⊕ β) :=\n  s.map inl + t.map inr\n\n@[simp]\ntheorem zero_disjSum : (0 : Multiset α).disjSum t = t.map inr :=\n  Multiset.zero_add _\n\n@[simp]\ntheorem disjSum_zero : s.disjSum (0 : Multiset β) = s.map inl :=\n  Multiset.add_zero _\n\n@[simp]\ntheorem card_disjSum : Multiset.card (s.disjSum t) = Multiset.card s + Multiset.card t := by\n  rw [disjSum, card_add, card_map, card_map]\n\nvariable {s t} {s₁ s₂ : Multiset α} {t₁ t₂ : Multiset β} {a : α} {b : β} {x : α ⊕ β}\n\ntheorem mem_disjSum : x ∈ s.disjSum t ↔ (∃ a, a ∈ s ∧ inl a = x) ∨ ∃ b, b ∈ t ∧ inr b = x := by\n  simp_rw [disjSum, mem_add, mem_map]\n\n@[simp]\ntheorem inl_mem_disjSum : inl a ∈ s.disjSum t ↔ a ∈ s := by\n  rw [mem_disjSum, or_iff_left]\n  · simp only [inl.injEq, exists_eq_right]\n  rintro ⟨b, _, hb⟩\n  exact inr_ne_inl hb\n\n@[simp]\n\nTarget:\ntheorem inr_mem_disjSum : inr b ∈ s.disjSum t ↔ b ∈ t :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_2f8dba92a8be","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"513d375df8fc0d79246cb01048d64efb2270107cc87fdd011c65e50a5ab53b99","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Multiset","family_id":"inr_mem_disjsum","file_id":"mathlib/Mathlib/Data/Multiset/Sum.lean","sample_id":"2f8dba92a8be5eab58edb05807e69424a9fcd6a33e1126c48358e2bed2791279"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3dc6c6b95959f83835d9501fcfa554ee77808794954cb31229e8e1de18edb21b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"02bca982f4b83936374ddc4eb72ca5b4ec320e9b7aa951058a1f9531889d7707","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3f67b4910954a3acaaa3087e7993d00a824bcfea76d08a647dfda9454dd07897","source_sha256":"b4e7e5c120f48c9a1a0c6aa6bf563869c6ea4d08a420d4b58e8dbf288b17256c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : NeZero (Nat.card G) := ⟨Nat.card_pos.ne'⟩\n  have hord := HasEnoughRootsOfUnity.natCard_rootsOfUnity M (Nat.card G)\n  let e := (IsCyclic.monoidHom_mulEquiv_rootsOfUnity G Mˣ).some\n  exact ⟨e.trans (rootsOfUnityUnitsMulEquiv M (Nat.card G)) |>.trans (mulEquivOfCyclicCardEq hord)⟩","hard_negative":true,"metrics":{"chosen_tokens":68,"rejected_tokens":3,"token_jaccard":0.0625,"token_length_ratio":0.044118},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"5c5b9b3968b04975c62ae33e51c9b7c0ba91798b1d2e0cd380fcb6b7988f59ba","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Commutative monoids with enough roots of unity\n\nWe define a typeclass `HasEnoughRootsOfUnity M n` for a commutative monoid `M` and\na natural number `n` that asserts that `M` contains a primitive `n`th root of unity\nand that the group of `n`th roots of unity in `M` is cyclic. Such monoids are suitable\ntargets for homomorphisms from groups of exponent (dividing) `n`; for example,\nthe homomorphisms can then be used to separate elements of the source group.\n-/\n\npublic section\n\n/-- This is a type class recording that a commutative monoid `M` contains primitive `n`th\nroots of unity and such that the group of `n`th roots of unity is cyclic.\n\nSuch monoids are suitable targets in the context of duality statements for groups\nof exponent `n`. -/\nclass HasEnoughRootsOfUnity (M : Type*) [CommMonoid M] (n : ℕ) where\n  prim : ∃ m : M, IsPrimitiveRoot m n\n  cyc : IsCyclic <| rootsOfUnity n M\n\nnamespace HasEnoughRootsOfUnity\n\nlemma exists_primitiveRoot (M : Type*) [CommMonoid M] (n : ℕ) [HasEnoughRootsOfUnity M n] :\n    ∃ ζ : M, IsPrimitiveRoot ζ n :=\n  HasEnoughRootsOfUnity.prim\n\ninstance rootsOfUnity_isCyclic (M : Type*) [CommMonoid M] (n : ℕ) [HasEnoughRootsOfUnity M n] :\n    IsCyclic (rootsOfUnity n M) :=\n  HasEnoughRootsOfUnity.cyc\n\n/-- If `HasEnoughRootsOfUnity M n` and `m ∣ n`, then also `HasEnoughRootsOfUnity M m`. -/\nlemma of_dvd (M : Type*) [CommMonoid M] {m n : ℕ} [NeZero n] (hmn : m ∣ n)\n    [HasEnoughRootsOfUnity M n] :\n    HasEnoughRootsOfUnity M m where\n  prim :=\n    have ⟨ζ, hζ⟩ := exists_primitiveRoot M n\n    have ⟨k, hk⟩ := hmn\n    ⟨ζ ^ k, IsPrimitiveRoot.pow (NeZero.pos n) hζ (mul_comm m k ▸ hk)⟩\n  cyc := Subgroup.isCyclic_of_le <| rootsOfUnity_le_of_dvd hmn\n\n/-- If `M` satisfies `HasEnoughRootsOfUnity`, then the group of `n`th roots of unity\nin `M` is finite. -/\ninstance finite_rootsOfUnity (M : Type*) [CommMonoid M] (n : ℕ) [NeZero n]\n    [HasEnoughRootsOfUnity M n] :\n    Finite <| rootsOfUnity n M := by\n  have := rootsOfUnity_isCyclic M n\n  obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := rootsOfUnity n M)\n  have hg' : g ^ n = 1 := OneMemClass.coe_eq_one.mp g.prop\n  let f (j : ZMod n) : rootsOfUnity n M := g ^ (j.val : ℤ)\n  refine Finite.of_surjective f fun x ↦ ?_\n  obtain ⟨k, hk⟩ := Subgroup.mem_zpowers_iff.mp <| hg x\n  refine ⟨k, ?_⟩\n  simpa only [ZMod.natCast_val, ← hk, f, ZMod.coe_intCast] using (zpow_eq_zpow_emod' k hg').symm\n\n/-- If `M` satisfies `HasEnoughRootsOfUnity`, then the group of `n`th roots of unity\nin `M` (is cyclic and) has order `n`. -/\nlemma natCard_rootsOfUnity (M : Type*) [CommMonoid M] (n : ℕ) [NeZero n]\n    [HasEnoughRootsOfUnity M n] :\n    Nat.card (rootsOfUnity n M) = n := by\n  obtain ⟨ζ, h⟩ := exists_primitiveRoot M n\n  rw [← IsCyclic.exponent_eq_card]\n  refine dvd_antisymm ?_ ?_\n  · exact Monoid.exponent_dvd_of_forall_pow_eq_one fun g ↦ OneMemClass.coe_eq_one.mp g.prop\n  · nth_rewrite 1 [h.eq_orderOf]\n    rw [← (h.isUnit NeZero.out).unit_spec, orderOf_units]\n    let ζ' : rootsOfUnity n M := ⟨(h.isUnit NeZero.out).unit, ?_⟩\n    · rw [← Subgroup.orderOf_mk]\n      exact Monoid.order_dvd_exponent ζ'\n    simp only [mem_rootsOfUnity]\n    rw [← Units.val_inj, Units.val_pow_eq_pow_val, IsUnit.unit_spec, h.pow_eq_one, Units.val_one]\n\nlemma of_card_le {R : Type*} [CommRing R] [IsDomain R] {n : ℕ} [NeZero n]\n    (h : n ≤ Fintype.card (rootsOfUnity n R)) : HasEnoughRootsOfUnity R n where\n  prim := card_rootsOfUnity_eq_iff_exists_isPrimitiveRoot.mp (le_antisymm (card_rootsOfUnity R n) h)\n  cyc := rootsOfUnity.isCyclic R n\n\nend HasEnoughRootsOfUnity\n\nlemma MulEquiv.hasEnoughRootsOfUnity {n : ℕ} [NeZero n] {M N : Type*} [CommMonoid M]\n    [CommMonoid N] [hm : HasEnoughRootsOfUnity M n] (e : rootsOfUnity n M ≃* rootsOfUnity n N) :\n    HasEnoughRootsOfUnity N n where\n  prim := by\n    obtain ⟨m, hm⟩ := hm.prim\n    use (e hm.toRootsOfUnity).val.val\n    rw [IsPrimitiveRoot.coe_units_iff, IsPrimitiveRoot.coe_submonoidClass_iff]\n    refine .map_of_injective ?_ e.injective\n    rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, ← IsPrimitiveRoot.coe_units_iff]\n  cyc := isCyclic_of_surjective e e.surjective\n\nsection cyclic\n\n/-- The group of group homomorphisms from a finite cyclic group `G` of order `n` into the\ngroup of units of a ring `M` with all roots of unity is isomorphic to `G` -/\n\nTarget:\nlemma IsCyclic.monoidHom_equiv_self (G M : Type*) [CommGroup G] [Finite G]\n    [IsCyclic G] [CommMonoid M] [HasEnoughRootsOfUnity M (Nat.card G)] :\n    Nonempty ((G →* Mˣ) ≃* G) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_3f67b4910954","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"ff66b6e65fee72c5bdf3480afd10653bb3e53e9e7c60648332ced037f0ae6ea3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/RootsOfUnity","family_id":"iscyclic","file_id":"mathlib/Mathlib/RingTheory/RootsOfUnity/EnoughRootsOfUnity.lean","sample_id":"3f67b4910954a3acaaa3087e7993d00a824bcfea76d08a647dfda9454dd07897"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bbbd68780f20b0315d7a36f5a0356a461835a08a2fef3351d5866ce71a89c492","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"550eea517b6f57ead3df265df6cf73d2563e67f3ae71b6c49159805a9ce8003d","source_sha256":"a7f1c7b70d08168190ea72793a69601025e762dd93aea5040d82a7e6ab550627","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_pow] at hxy ⊢\n  rw [hxy]","hard_negative":true,"metrics":{"chosen_tokens":23,"rejected_tokens":8,"token_jaccard":0.043478,"token_length_ratio":0.347826},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"5c7529cda976d85a9a9adf1b9b0f3b53232f647a423d16d2aec41c4810f0c835","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Submodule.Map\npublic import Mathlib.Algebra.Polynomial.Eval.Defs\npublic import Mathlib.RingTheory.Ideal.Quotient.Defs\npublic import Mathlib.Algebra.Module.Submodule.RestrictScalars\npublic import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic\n\nNamespace:\nSModEq\n\nLocal context:\n/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n/-!\n# modular equivalence for submodule\n-/\n\n@[expose] public section\n\n\nopen Submodule\n\nopen Polynomial\n\nvariable {R : Type*} [Ring R]\nvariable {S : Type*} [Ring S]\nvariable {A : Type*} [CommRing A]\nvariable {M : Type*} [AddCommGroup M] [Module R M] [Module S M] (U U₁ U₂ : Submodule R M)\nvariable {x x₁ x₂ y y₁ y₂ z z₁ z₂ : M}\nvariable {N : Type*} [AddCommGroup N] [Module R N] (V V₁ V₂ : Submodule R N)\n\n/-- A predicate saying two elements of a module are equivalent modulo a submodule. -/\ndef SModEq (x y : M) : Prop :=\n  (Submodule.Quotient.mk x : M ⧸ U) = Submodule.Quotient.mk y\n\n@[inherit_doc] notation:50 x \" ≡ \" y \" [SMOD \" N \"]\" => SModEq N x y\n\nvariable {U U₁ U₂}\n\nprotected theorem SModEq.def :\n    x ≡ y [SMOD U] ↔ (Submodule.Quotient.mk x : M ⧸ U) = Submodule.Quotient.mk y :=\n  Iff.rfl\n\nnamespace SModEq\n\ntheorem sub_mem : x ≡ y [SMOD U] ↔ x - y ∈ U := by rw [SModEq.def, Submodule.Quotient.eq]\n\n@[simp]\ntheorem top : x ≡ y [SMOD (⊤ : Submodule R M)] :=\n  (Submodule.Quotient.eq ⊤).2 mem_top\n\n@[simp]\ntheorem bot : x ≡ y [SMOD (⊥ : Submodule R M)] ↔ x = y := by\n  rw [SModEq.def, Submodule.Quotient.eq, mem_bot, sub_eq_zero]\n\n@[gcongr, mono]\ntheorem mono (HU : U₁ ≤ U₂) (hxy : x ≡ y [SMOD U₁]) : x ≡ y [SMOD U₂] :=\n  (Submodule.Quotient.eq U₂).2 <| HU <| (Submodule.Quotient.eq U₁).1 hxy\n\nlemma of_toAddSubgroup_le {U : Submodule R M} {V : Submodule S M}\n    (h : U.toAddSubgroup ≤ V.toAddSubgroup) {x y : M} (hxy : x ≡ y [SMOD U]) : x ≡ y [SMOD V] := by\n  simp only [SModEq, Submodule.Quotient.eq] at hxy ⊢\n  exact h hxy\n\n@[refl, simp]\nprotected theorem refl (x : M) : x ≡ x [SMOD U] :=\n  @rfl _ _\n\nprotected theorem rfl : x ≡ x [SMOD U] :=\n  SModEq.refl _\n\ninstance : Std.Refl (SModEq U) :=\n  ⟨SModEq.refl⟩\n\n@[symm]\nnonrec theorem symm (hxy : x ≡ y [SMOD U]) : y ≡ x [SMOD U] :=\n  hxy.symm\n\ntheorem comm : x ≡ y [SMOD U] ↔ y ≡ x [SMOD U] := ⟨symm, symm⟩\n\n@[trans]\nnonrec theorem trans (hxy : x ≡ y [SMOD U]) (hyz : y ≡ z [SMOD U]) : x ≡ z [SMOD U] :=\n  hxy.trans hyz\n\ninstance instTrans : Trans (SModEq U) (SModEq U) (SModEq U) where\n  trans := trans\n\n@[gcongr]\ntheorem add (hxy₁ : x₁ ≡ y₁ [SMOD U]) (hxy₂ : x₂ ≡ y₂ [SMOD U]) : x₁ + x₂ ≡ y₁ + y₂ [SMOD U] := by\n  rw [SModEq.def] at hxy₁ hxy₂ ⊢\n  simp_rw [Quotient.mk_add, hxy₁, hxy₂]\n\n@[gcongr]\ntheorem sum {ι} {s : Finset ι} {x y : ι → M}\n    (hxy : ∀ i ∈ s, x i ≡ y i [SMOD U]) : ∑ i ∈ s, x i ≡ ∑ i ∈ s, y i [SMOD U] := by\n  classical\n  induction s using Finset.cons_induction with\n  | empty => simp [SModEq.rfl]\n  | cons i s _ ih =>\n    grw [Finset.sum_cons, Finset.sum_cons, hxy i (Finset.mem_cons_self i s),\n      ih (fun j hj ↦ hxy j (Finset.mem_cons_of_mem hj))]\n\n@[gcongr]\ntheorem smul (hxy : x ≡ y [SMOD U]) (c : R) : c • x ≡ c • y [SMOD U] := by\n  rw [SModEq.def] at hxy ⊢\n  simp_rw [Quotient.mk_smul, hxy]\n\n@[gcongr]\nlemma nsmul (hxy : x ≡ y [SMOD U]) (n : ℕ) : n • x ≡ n • y [SMOD U] := by\n  rw [SModEq.def] at hxy ⊢\n  simp_rw [Quotient.mk_smul, hxy]\n\n@[gcongr]\nlemma zsmul (hxy : x ≡ y [SMOD U]) (n : ℤ) : n • x ≡ n • y [SMOD U] := by\n  rw [SModEq.def] at hxy ⊢\n  simp_rw [Quotient.mk_smul, hxy]\n\n@[gcongr]\ntheorem mul {I : Ideal A} {x₁ x₂ y₁ y₂ : A} (hxy₁ : x₁ ≡ y₁ [SMOD I])\n    (hxy₂ : x₂ ≡ y₂ [SMOD I]) : x₁ * x₂ ≡ y₁ * y₂ [SMOD I] := by\n  simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_mul] at hxy₁ hxy₂ ⊢\n  rw [hxy₁, hxy₂]\n\n@[gcongr]\ntheorem prod {I : Ideal A} {ι} {s : Finset ι} {x y : ι → A}\n    (hxy : ∀ i ∈ s, x i ≡ y i [SMOD I]) : ∏ i ∈ s, x i ≡ ∏ i ∈ s, y i [SMOD I] := by\n  classical\n  induction s using Finset.cons_induction with\n  | empty => simp [SModEq.rfl]\n  | cons i s _ ih =>\n    grw [Finset.prod_cons, Finset.prod_cons, hxy i (Finset.mem_cons_self i s),\n      ih (fun j hj ↦ hxy j (Finset.mem_cons_of_mem hj))]\n\n@[gcongr]\n\nTarget:\nlemma pow {I : Ideal A} {x y : A} (n : ℕ) (hxy : x ≡ y [SMOD I]) :\n    x ^ n ≡ y ^ n [SMOD I] :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"504115d569001f0a67a9ce455f0e9cd144d5f834fa1d24a2642020756e258b38","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/SModEq","family_id":"pow","file_id":"mathlib/Mathlib/LinearAlgebra/SModEq/Basic.lean","sample_id":"550eea517b6f57ead3df265df6cf73d2563e67f3ae71b6c49159805a9ce8003d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7333d1a239a893fe44382cfe648dc46c728566269ad1ec5413c976d55e86e914","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b445b02c8a845e7a62745b861f317ebf3f3b0bec322dabeee073597433fb2a4c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dabe8f52856a803fb511e285a4cf7be82ea7a0120495bcf033465a0c9f492e10","source_sha256":"3ebf52bd4f56640ff82f3663b47b89c207fb7b51b21d5bfbbe3cfb7edd34865e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only [Ordinal.liftPrincipalSeg_coe, lift_ord, lift_succ, ord_le, Order.succ_le_iff] using!\n    le_of_lt (Ordinal.liftPrincipalSeg.{u, u + 1}.lt_top (Order.succ c).ord)","hard_negative":false,"metrics":{"chosen_tokens":44,"rejected_tokens":48,"token_jaccard":0.90625,"token_length_ratio":1.090909},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"5cda1d4857e72035b57376b1472fbb443a60d895d5995a95110903e3f63a40b8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.SetTheory.Ordinal.Basic\n\nNamespace:\nCardinal\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Floris van Doorn\n-/\n/-!\n# Universal ordinal and cardinal\n\n`Cardinal.univ` is the cardinality of the cardinals themselves. Likewise, `Ordinal.univ` is the\norder type of the ordinals. These are related via `Cardinal.univ.ord = Ordinal.univ` and\n`Ordinal.univ.card = Cardinal.univ`.\n\nThe cardinal `Cardinal.univ` is strongly inaccessible. This reflects the fact that in ZFC, the\ncardinals form a proper class. See `IsInaccessible.univ` for a proof.\n\n## Implementation notes\n\nWe actually define `Cardinal.univ` as the cardinality of `Ordinal`, rather than that of `Cardinal`.\nThis makes the basic API easier to set up. See `Cardinal.mk_cardinal` for a proof that\n`Cardinal.univ = #Cardinal`.\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nset_option linter.checkUnivs false in\nopen Ordinal in\n-- intended to be used with explicit universe parameters\n/-- The ordinal `univ.{u, v}` is the order type of `Ordinal.{u}` or `Cardinal.{u}`, as an element of\n`Ordinal.{v}` (when `u < v`). -/\n@[pp_with_univ, nolint checkUnivs]\ndef Ordinal.univ : Ordinal.{max (u + 1) v} :=\n  lift.{v, u + 1} (typeLT Ordinal)\n\nset_option linter.checkUnivs false in\nopen Cardinal in\n-- intended to be used with explicit universe parameters\n/-- The cardinal `univ.{u, v}` is the cardinality of `Ordinal.{u}` or `Cardinal.{u}`, as an element\nof `Cardinal.{v}` (when `u < v`). -/\n@[pp_with_univ, nolint checkUnivs]\ndef Cardinal.univ : Cardinal.{max (u + 1) v} :=\n  lift.{v, u + 1} #Ordinal\n\n/-! ### Universal ordinal -/\n\nnamespace Ordinal\n\n@[simp]\ntheorem type_lt_ordinal : typeLT Ordinal = univ.{u, u + 1} :=\n  (lift_id _).symm\n\n@[deprecated type_lt_ordinal (since := \"2026-03-20\")]\ntheorem univ_id : univ.{u, u + 1} = typeLT Ordinal :=\n  lift_id _\n\n@[simp]\ntheorem lift_univ : lift.{w} univ.{u, v} = univ.{u, max v w} :=\n  lift_lift _\n\ntheorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=\n  congr_fun lift_umax _\n\n/-- Principal segment version of the lift operation on ordinals, embedding `Ordinal.{u}` in\n`Ordinal.{v}` as a principal segment when `u < v`. -/\ndef liftPrincipalSeg : Ordinal.{u} <i Ordinal.{max (u + 1) v} :=\n  ⟨liftInitialSeg.{max (u + 1) v, u}, univ.{u, v}, by\n    refine fun b => inductionOn b ?_; intro β s _\n    rw [univ, ← lift_umax]; constructor <;> intro h\n    · obtain ⟨a, e⟩ := h\n      rw [← e]\n      refine inductionOn a ?_\n      intro α r _\n      exact lift_type_lt.{u, u + 1, max (u + 1) v}.2 ⟨typein r⟩\n    · rw [← lift_id (type s)] at h ⊢\n      obtain ⟨f⟩ := lift_type_lt.{_,_,v}.1 h\n      obtain ⟨f, a, hf⟩ := f\n      exists a\n      induction a using inductionOn with | type α r\n      refine lift_type_eq.{u, max (u + 1) v, max (u + 1) v}.2\n        ⟨(RelIso.ofSurjective (RelEmbedding.ofMonotone ?_ ?_) ?_).symm⟩\n      · exact fun b => enum r ⟨f b, (hf _).1 ⟨_, rfl⟩⟩\n      · refine fun a b h => (typein_lt_typein r).1 ?_\n        rw [typein_enum, typein_enum]\n        exact f.map_rel_iff.2 h\n      · intro a'\n        obtain ⟨b, e⟩ := (hf _).2 (typein_lt_type _ a')\n        exists b\n        simp only [RelEmbedding.ofMonotone_coe]\n        simp [e]⟩\n\n@[simp]\ntheorem liftPrincipalSeg_coe :\n    (liftPrincipalSeg.{u, v} : Ordinal → Ordinal) = lift.{max (u + 1) v} :=\n  rfl\n\n@[simp]\ntheorem liftPrincipalSeg_top : (liftPrincipalSeg.{u, v}).top = univ.{u, v} :=\n  rfl\n\n@[deprecated liftPrincipalSeg_top (since := \"2026-03-20\")]\ntheorem liftPrincipalSeg_top' : liftPrincipalSeg.{u, u + 1}.top = typeLT Ordinal := by\n  simp\n\n@[simp]\ntheorem card_univ : card univ.{u, v} = Cardinal.univ.{u, v} :=\n  rfl\n\nend Ordinal\n\n/-! ### Universal cardinal -/\n\nnamespace Cardinal\n\n@[simp]\ntheorem mk_ordinal : #Ordinal = univ.{u, u + 1} :=\n  (lift_id _).symm\n\n@[deprecated mk_ordinal (since := \"2026-04-22\")]\ntheorem univ_id : univ.{u, u + 1} = #Ordinal :=\n  lift_id _\n\n@[simp]\ntheorem lift_univ : lift.{w} univ.{u, v} = univ.{u, max v w} :=\n  lift_lift _\n\ntheorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=\n  congr_fun lift_umax _\n\nTarget:\ntheorem lift_lt_univ (c : Cardinal) : lift.{u + 1, u} c < univ.{u, u + 1} :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  simpa only [Ordinal.liftPrincipalSeg_coe, lift_ord, lift_succ, ord_le, Order.succ_le_iff] using!\n    le_of_lt (Ordinal.liftPrincipalSeg.{u, u + 1}.lt_top (Order.succ c).ord)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Ordinal","family_id":"lift_lt_univ","file_id":"mathlib/Mathlib/SetTheory/Ordinal/Univ.lean","sample_id":"dabe8f52856a803fb511e285a4cf7be82ea7a0120495bcf033465a0c9f492e10"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bbb068c1b254f9a62b1fe49e6c1cd52f49bd7cbde7d3a811059b2617b8439517","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c87ad3ae24029059963e03348c139139f0d135dcb45808f3d2427d4e578980f8","source_sha256":"50e4a4d798ba2dd2872d163b6a5aeb55d8cf128d153e35dcc8c630f4eadcbabe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h0 : v πᵥ ≠ 0 := by simpa using uniformizingPolynomial_ne_zero hle\n  induction f using RatFunc.induction_on with\n  | f p q hq =>\n    use (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {p})).factors -\n      (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {q})).factors\n    simp only [map_div₀, map_pow, zpow_sub₀ h0, zpow_natCast,\n      valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle hq,\n      valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle\n        (p := p) (by aesop)]","hard_negative":false,"metrics":{"chosen_tokens":116,"rejected_tokens":2,"token_jaccard":0.019231,"token_length_ratio":0.017241},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"5d25718484196511abe16bf9004a3b4d745cd58cff3c2facf14b202951e3da9d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.Finite.Valuation\npublic import Mathlib.NumberTheory.FunctionField\npublic import Mathlib.RingTheory.Valuation.Discrete.Basic\n\nNamespace:\nRatFunc\n\nLocal context:\n/-\nCopyright (c) 2025 María Inés de Frutos-Fernández & Xavier Généreux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: María Inés de Frutos-Fernández, Xavier Généreux\n-/\n/-!\n# Ostrowski's theorem for `K(X)`\n\nThis file proves Ostrowski's theorem for the field of rational functions `K(X)`, where `K` is any\nfield: if `v` is a discrete valuation on `K(X)` which is trivial on elements of `K`, then `v` is\nequivalent to either the `I`-adic valuation for some `I : HeightOneSpectrum K[X]`, or to the\nvaluation at infinity `FunctionField.inftyValuation K`.\n\n## Main results\n- `RatFunc.valuation_isEquiv_infty_or_adic`: Ostrowski's theorem for `K(X)`.\n-/\n\n@[expose] public noncomputable section\n\n\nopen Multiplicative WithZero\n\nvariable {K Γ : Type*} [Field K] [LinearOrderedCommGroupWithZero Γ] {v : Valuation (RatFunc K) Γ}\n\nnamespace RatFunc\n\nsection Infinity\n\nopen Polynomial Valuation\n\nlemma valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X {f : RatFunc K}\n    [v.IsTrivialOn K] (hlt : 1 < v X) (hf : f ≠ 0) : v f = v RatFunc.X ^ f.intDegree := by\n  induction f using RatFunc.induction_on with\n  | f p q hq =>\n    rw [intDegree_div (by grind only) (by grind only), v.map_div, zpow_sub₀ (ne_zero_of_lt hlt)]\n    simp_rw [intDegree_polynomial, zpow_natCast, ← coePolynomial_eq_algebraMap]\n    have hp : p ≠ 0 := by contrapose hf; simp [hf]\n    rw [valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X _ hlt hp,\n      valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X _ hlt hq]\n\nvariable [DecidableEq (RatFunc K)]\n\nlemma valuation_isEquiv_inftyValuation_of_one_lt_valuation_X [v.IsTrivialOn K] (hlt : 1 < v X) :\n    v.IsEquiv (inftyValuation K) := by\n  refine isEquiv_iff_val_lt_one.mpr fun {f} ↦ ?_\n  rcases eq_or_ne f 0 with rfl | hf\n  · simp\n  · have hlt' : 1 < inftyValuation K X := by simp [← exp_zero]\n    rw [valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt hf,\n      valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt' hf]\n    grind [one_le_zpow_iff_right₀]\n\nend Infinity\n\nopen IsDedekindDomain HeightOneSpectrum Set Valuation Polynomial\n\nlemma setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty [v.IsNontrivial] [v.IsTrivialOn K]\n    (hle : v RatFunc.X ≤ 1) : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty := by\n  obtain ⟨w, h0, h1⟩ := IsNontrivial.exists_lt_one (v := v)\n  induction w using RatFunc.induction_on with\n  | f p q =>\n    simp only [ne_eq, _root_.div_eq_zero_iff, FaithfulSMul.algebraMap_eq_zero_iff, not_or,\n      map_div₀] at *\n    have hor : ¬v ↑p = 1 ∨ ¬v ↑q = 1 := by rw [← not_and_or]; aesop\n    suffices ∀ r : K[X], v (↑r) ≠ 1 → r ≠ 0 → {p : K[X] | v ↑p < 1 ∧ ¬p = 0}.Nonempty by\n      exact Or.elim hor (fun hp ↦ this p hp h0.1) (fun hq ↦ this q hq h0.2)\n    exact fun r hr hr0 ↦ ⟨r, lt_iff_le_and_ne.mpr\n      ⟨Polynomial.valuation_le_one_of_valuation_X_le_one _ hle r, hr⟩, hr0⟩\n\nprivate lemma one_le_valuation_factor (hne : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty) {a b : K[X]}\n    (hab : v ↑(a * b) < 1 ∧ a ≠ 0 ∧ b ≠ 0) (hπᵥ : degree_lt_wf.min _ hne = a * b)\n    (hb : ¬IsUnit b) : 1 ≤ v ↑a := by\n  set πᵥ := degree_lt_wf.min _ hne\n  have hda : a.degree < πᵥ.degree := by\n    have hbpos := degree_pos_of_ne_zero_of_nonunit hab.2.2 hb\n    simp_rw [hπᵥ, degree_mul, degree_eq_natDegree hab.2.1, degree_eq_natDegree hab.2.2] at hbpos ⊢\n    norm_cast\n    simpa using hbpos\n  have hlea := imp_not_comm.mp (degree_lt_wf.not_lt_min _) hda\n  grind\n\nlemma irreducible_min_polynomial_valuation_lt_one_and_ne_zero [v.IsTrivialOn K]\n    (hne : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty) :\n    Irreducible (degree_lt_wf.min {p : K[X] | v p < 1 ∧ p ≠ 0} hne) := by\n  set πᵥ := degree_lt_wf.min _ hne\n  have hπᵥ : v πᵥ < 1 ∧ πᵥ ≠ 0 := degree_lt_wf.min_mem _ hne\n  refine irreducible_iff.mpr ⟨?_, fun a b hab ↦ ?_⟩\n  · simp only [Polynomial.isUnit_iff, isUnit_iff_ne_zero]\n    intro ⟨a, ha0, ha⟩\n    rw [← ha, coePolynomial, algebraMap_C, ← algebraMap_eq_C] at hπᵥ\n    grind\n  · by_contra! H\n    simp only [hab, ne_eq, mul_eq_zero, not_or] at hπᵥ\n    have hva := one_le_valuation_factor hne hπᵥ hab H.2\n    simp only [mul_comm a b, @and_comm (¬a = 0)] at hπᵥ hab\n    have := Right.one_le_mul (one_le_valuation_factor hne hπᵥ hab H.1) hva\n    simp only [coePolynomial_eq_algebraMap, map_mul] at hπᵥ this\n    grind\n\nsection valuation_X_le_one\n\nvariable [v.IsNontrivial] [v.IsTrivialOn K] (hle : v RatFunc.X ≤ 1)\n\n/-- A uniformizing element for the valuation `v`, as a polynomial in `K[X]`. -/\nabbrev uniformizingPolynomial : K[X] :=\n  WellFounded.min degree_lt_wf _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)\n\n@[inherit_doc]\nlocal notation \"πᵥ\" => uniformizingPolynomial hle\n\nlemma uniformizingPolynomial_ne_zero : πᵥ ≠ 0 := by\n  have := degree_lt_wf.min_mem _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)\n  simp_all [uniformizingPolynomial]\n\nlemma valuation_uniformizingPolynomial_lt_one : v πᵥ < 1 := by\n  simpa using! (degree_lt_wf.min_mem _\n    (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)).1\n\nopen Ideal in\n/-- The maximal ideal of `K[X]` generated by the `uniformizingPolynomial` for `v`. -/\ndef valuationIdeal : HeightOneSpectrum K[X] where\n  asIdeal := Submodule.span K[X] {πᵥ}\n  isPrime := IsMaximal.isPrime (PrincipalIdealRing.isMaximal_of_irreducible\n    (irreducible_min_polynomial_valuation_lt_one_and_ne_zero\n      (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)))\n  ne_bot := by simpa using uniformizingPolynomial_ne_zero hle\n\n@[inherit_doc]\nlocal notation \"Pᵥ\" => RatFunc.valuationIdeal hle\n\nsection Associates\n\nopen EuclideanDomain in\nlemma valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one {p : K[X]}\n    (hp : p ≠ 0) :\n    v (algebraMap K[X] (RatFunc K) p) = v (πᵥ ^ ((Associates.mk (Pᵥ).asIdeal).count\n      (Associates.mk (Ideal.span {p})).factors)) := by\n  set π := πᵥ\n  have hne := setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle\n  have hπirr : Irreducible π := irreducible_min_polynomial_valuation_lt_one_and_ne_zero hne\n  obtain ⟨k, q, hnq, heq⟩ := WfDvdMonoid.max_power_factor hp hπirr\n  have hπ : π ∈ _ := degree_lt_wf.min_mem _ hne\n  simp only [ne_eq, mem_setOf] at hπ\n  nth_rw 1 [heq]\n  simp only [map_mul, map_pow]\n  suffices v (algebraMap K[X] (RatFunc K) q) = 1 by\n    simp only [this, mul_one]\n    congr\n    exact (Ideal.count_associates_eq (irreducible_iff_prime.mp hπirr) hnq heq).symm\n  rw [← mod_add_div q π, map_add]\n  rw [← mod_eq_zero] at hnq\n  suffices v (algebraMap K[X] (RatFunc K) (q % π)) = 1 ∧\n      v (algebraMap K[X] (RatFunc K) (π * (q / π))) < 1 by\n    obtain ⟨h₁, h₂⟩ := this\n    rw [← h₁] at h₂ ⊢\n    exact Valuation.map_add_eq_of_lt_left _ h₂\n  constructor\n  · rw [← coePolynomial_eq_algebraMap]\n    have hnπ : q % π ∉ {p : K[X] | v ↑p < 1 ∧ p ≠ 0} :=\n      imp_not_comm.mp (degree_lt_wf.not_lt_min _) (EuclideanDomain.remainder_lt q hπ.2)\n    have := Polynomial.valuation_le_one_of_valuation_X_le_one _ hle (q % π)\n    grind\n  · simpa only [map_mul, ← coePolynomial_eq_algebraMap]\n      using mul_lt_one_of_lt_of_le hπ.1 <| (q / π).valuation_le_one_of_valuation_X_le_one _ hle\n\nTarget:\nlemma exists_zpow_uniformizingPolynomial {f : RatFunc K} (hf : f ≠ 0) :\n    ∃ (z : ℤ), v f = v πᵥ ^ z :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/RatFunc","family_id":"exists_zpow_uniformizingpolynomial","file_id":"mathlib/Mathlib/NumberTheory/RatFunc/Ostrowski.lean","sample_id":"c87ad3ae24029059963e03348c139139f0d135dcb45808f3d2427d4e578980f8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4525ed279745c1a825a2d966ade6667dde0e9859d5d810c453aa56b8f6012282","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7b0349090d26641392842231a42304f2907b4e65dd3bf3990a7b61ce8b063793","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"06f153f627e2ad13220f5061c7800c30d1bdd6c1da9c7ac7f9963af2c8d4a748","source_sha256":"522c0739e8f7535bf0f954aff339b473dbd0f9e903c0b5a9a4fff87d391d66b3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only using! congr_fun (f.isCompat' φ) x","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":17,"token_jaccard":0.8125,"token_length_ratio":1.307692},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"5d38c44ba3feb80d6feedf76a67ab657983c710f80fe522050ca6aea967f4929","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorProduct.RightExactness\npublic import Mathlib.RingTheory.Congruence.Hom\npublic import Mathlib.RingTheory.FiniteType\npublic import Mathlib.RingTheory.TensorProduct.DirectLimitFG\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir & María-Inés de Frutos-Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir & María-Inés de Frutos-Fernández\n-/\n/-! # Polynomial laws on modules\n\nLet `M` and `N` be a modules over a commutative ring `R`.\nA polynomial law `f : PolynomialLaw R M N`, with notation `f : M →ₚₗₗ[R] N`,\nis a “law” that assigns a natural map `PolynomialLaw.toFun' f S : S ⊗[R] M → S ⊗[R] N`\nfor every `R`-algebra `S`.\n\nFor type-theoretic reasons, if `R : Type u`, then the definition of the polynomial map `f`\nis restricted to `R`-algebras `S` such that `S : Type u`.\nUsing the fact that a module is the direct limit of its finitely generated submodules, that a\nfinitely generated subalgebra is a quotient of a polynomial ring in the universe `u`, plus\nthe commutation of tensor products with direct limits, we extend the functor\nto all `R`-algebras.\n\nThe two fields involving the definition of `PolynomialLaw`,\n`PolynomialLaw.toFun'` and `PolynomialLaw.isCompat'` are primed.\nThey are superseded by their universe-polymorphic counterparts,\nthe definition `PolynomialLaw.toFun` and the lemma `PolynomialLaw.isCompat`\nwhich should be used once the theory is properly stated.\n\nFor constructions of general definitions of `PolynomialLaw`\nat a universe-polymorphic level, one needs to lift\nelements in a tensor product to smaller universes.\nFor this, one can make use of\n`PolynomialLaw.exists_lift` or `PolynomialLaw.exists_lift'`,\nor establish appropriate generalizations.\n\n## Main definitions/lemmas\n\n* Instance : `Module R (M →ₚₗ[R] N)` shows that polynomial laws form an `R`-module.\n\n* `PolynomialLaw.ground f` is the map `M → N` corresponding to `PolynomialLaw.toFun' f R` under\n  the isomorphisms `R ⊗[R] M ≃ₗ[R] M`, and similarly for `N`.\n\nIn further works, we construct the coefficients of a polynomial law and show the relation with\npolynomials (when the module `M` is free and finite).\n\n## Implementation notes\n\nIn the literature, the theory is written for commutative rings, but this implementation\nonly assumes `R` is a commutative semiring.\n\n## References\n\n* [Roby, Norbert. 1963. «Lois polynomes et lois formelles en théorie des modules».\n  Annales scientifiques de l’École Normale Supérieure 80 (3): 213‑348](Roby-1963)\n\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nnoncomputable section PolynomialLaw\n\nopen scoped TensorProduct\n\nopen LinearMap TensorProduct AlgHom RingCon\n\n/-- A polynomial law `M →ₚₗ[R] N` between `R`-modules is a functorial family of maps\n`S ⊗[R] M → S ⊗[R] N`, for all `R`-algebras `S`.\n\nFor universe reasons, `S` has to be restricted to the same universe as `R`. -/\n@[ext]\nstructure PolynomialLaw (R : Type u) [CommSemiring R]\n    (M : Type*) [AddCommMonoid M] [Module R M] (N : Type*) [AddCommMonoid N] [Module R N] where\n  /-- The functions `S ⊗[R] M → S ⊗[R] N` underlying a polynomial law -/\n  toFun' (S : Type u) [CommSemiring S] [Algebra R S] : S ⊗[R] M → S ⊗[R] N\n  /-- The compatibility relations between the functions underlying a polynomial law -/\n  isCompat' {S : Type u} [CommSemiring S] [Algebra R S]\n    {S' : Type u} [CommSemiring S'] [Algebra R S'] (φ : S →ₐ[R] S') :\n    φ.toLinearMap.rTensor N ∘ toFun' S = toFun' S' ∘ φ.toLinearMap.rTensor M := by aesop\n\n/-- `M →ₚₗ[R] N` is the type of `R`-polynomial laws from `M` to `N`. -/\nnotation:25 M \" →ₚₗ[\" R:25 \"] \" N:0 => PolynomialLaw R M N\n\n@[local simp]\n\nTarget:\ntheorem PolynomialLaw.isCompat_apply'\n    {R : Type u} [CommSemiring R] {M : Type*} [AddCommMonoid M] [Module R M]\n    {N : Type*} [AddCommMonoid N] [Module R N] {f : M →ₚₗ[R] N}\n    {S : Type u} [CommSemiring S] [Algebra R S] {S' : Type u} [CommSemiring S'] [Algebra R S']\n    (φ : S →ₐ[R] S') (x : S ⊗[R] M) :\n    (φ.toLinearMap.rTensor N) ((f.toFun' S) x) = (f.toFun' S') (φ.toLinearMap.rTensor M x) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  simpa only using! congr_fun (f.isCompat' φ) x","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/PolynomialLaw","family_id":"polynomiallaw","file_id":"mathlib/Mathlib/RingTheory/PolynomialLaw/Basic.lean","sample_id":"06f153f627e2ad13220f5061c7800c30d1bdd6c1da9c7ac7f9963af2c8d4a748"},"split":"train"}
{"binding":{"chosen_proof_sha256":"00de7867ae9086f2cf9c529fdfa5d87816a8332564eca883373884a7662e8cc7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"85d7ced1570d9b1f378e8a36e8670d4ffef538f394854e06ad6626a33d61df6e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7d66ee4f35ab6035f8dacc7a5cf8575522e26c3c67988759a6c4ebffd4318014","source_sha256":"d494c9de8c64f10c69db3a3e7c2cc9feffbd36585bacb5b41e8c3dcb64729d9b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [rowLen, Nat.lt_find_iff]\n  push Not\n  exact ⟨fun h _ hmj => μ.up_left_mem (by rfl) hmj h, fun h => h _ (by rfl)⟩","hard_negative":false,"metrics":{"chosen_tokens":38,"rejected_tokens":45,"token_jaccard":0.923077,"token_length_ratio":1.184211},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"5d7e4ddf47490630e40bfbde73f1f4c6c8a607b0e9ea032ed0b24da7a730e6d7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Preimage\npublic import Mathlib.Data.Finset.Prod\npublic import Mathlib.Data.SetLike.Basic\npublic import Mathlib.Order.UpperLower.Basic\n\nNamespace:\nYoungDiagram\n\nLocal context:\n/-\nCopyright (c) 2022 Jake Levinson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jake Levinson\n-/\n/-!\n# Young diagrams\n\nA Young diagram is a finite set of up-left justified boxes:\n\n```text\n□□□□□\n□□□\n□□□\n□\n```\nThis Young diagram corresponds to the [5, 3, 3, 1] partition of 12.\n\nWe represent it as a lower set in `ℕ × ℕ` in the product partial order. We write `(i, j) ∈ μ`\nto say that `(i, j)` (in matrix coordinates) is in the Young diagram `μ`.\n\n## Main definitions\n\n- `YoungDiagram` : Young diagrams\n- `YoungDiagram.card` : the number of cells in a Young diagram (its *cardinality*)\n- `YoungDiagram.instDistribLatticeYoungDiagram` : a distributive lattice instance for Young diagrams\n  ordered by containment, with `(⊥ : YoungDiagram)` the empty diagram.\n- `YoungDiagram.row` and `YoungDiagram.rowLen`: rows of a Young diagram and their lengths\n- `YoungDiagram.col` and `YoungDiagram.colLen`: columns of a Young diagram and their lengths\n\n## Notation\n\nIn \"English notation\", a Young diagram is drawn so that (i1, j1) ≤ (i2, j2)\nmeans (i1, j1) is weakly up-and-left of (i2, j2). This terminology is used\nbelow, e.g. in `YoungDiagram.up_left_mem`.\n\n## Tags\n\nYoung diagram\n\n## References\n\n<https://en.wikipedia.org/wiki/Young_tableau>\n\n-/\n\n@[expose] public section\n\n\nopen Function\n\n/-- A Young diagram is a finite collection of cells on the `ℕ × ℕ` grid such that whenever\na cell is present, so are all the ones above and to the left of it. Like matrices, an `(i, j)` cell\nis a cell in row `i` and column `j`, where rows are enumerated downward and columns rightward.\n\nYoung diagrams are modeled as finite sets in `ℕ × ℕ` that are lower sets with respect to the\nstandard order on products. -/\n@[ext]\nstructure YoungDiagram where\n  /-- A finite set which represents a finite collection of cells on the `ℕ × ℕ` grid. -/\n  cells : Finset (ℕ × ℕ)\n  /-- Cells are up-left justified, witnessed by the fact that `cells` is a lower set in `ℕ × ℕ`. -/\n  isLowerSet : IsLowerSet (cells : Set (ℕ × ℕ))\n\nnamespace YoungDiagram\n\ninstance : SetLike YoungDiagram (ℕ × ℕ) where\n  coe y := y.cells\n  coe_injective μ ν h := by rwa [YoungDiagram.ext_iff, ← Finset.coe_inj]\n\ninstance : PartialOrder YoungDiagram := .ofSetLike YoungDiagram (ℕ × ℕ)\n\n@[simp]\ntheorem mem_cells {μ : YoungDiagram} (c : ℕ × ℕ) : c ∈ μ.cells ↔ c ∈ μ :=\n  Iff.rfl\n\n@[simp]\ntheorem mem_mk (c : ℕ × ℕ) (cells) (isLowerSet) :\n    c ∈ YoungDiagram.mk cells isLowerSet ↔ c ∈ cells :=\n  Iff.rfl\n\ninstance decidableMem (μ : YoungDiagram) : DecidablePred (· ∈ μ) :=\n  inferInstanceAs (DecidablePred (· ∈ μ.cells))\n\n/-- In \"English notation\", a Young diagram is drawn so that (i1, j1) ≤ (i2, j2)\nmeans (i1, j1) is weakly up-and-left of (i2, j2). -/\ntheorem up_left_mem (μ : YoungDiagram) {i1 i2 j1 j2 : ℕ} (hi : i1 ≤ i2) (hj : j1 ≤ j2)\n    (hcell : (i2, j2) ∈ μ) : (i1, j1) ∈ μ :=\n  μ.isLowerSet (Prod.mk_le_mk.mpr ⟨hi, hj⟩) hcell\n\nsection DistribLattice\n\n@[simp]\ntheorem cells_subset_iff {μ ν : YoungDiagram} : μ.cells ⊆ ν.cells ↔ μ ≤ ν :=\n  Iff.rfl\n\n@[simp]\ntheorem cells_ssubset_iff {μ ν : YoungDiagram} : μ.cells ⊂ ν.cells ↔ μ < ν :=\n  Iff.rfl\n\ninstance : Max YoungDiagram where\n  max μ ν :=\n    { cells := μ.cells ∪ ν.cells\n      isLowerSet := by\n        rw [Finset.coe_union]\n        exact μ.isLowerSet.union ν.isLowerSet }\n\n@[simp]\ntheorem cells_sup (μ ν : YoungDiagram) : (μ ⊔ ν).cells = μ.cells ∪ ν.cells :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sup (μ ν : YoungDiagram) : ↑(μ ⊔ ν) = (μ ∪ ν : Set (ℕ × ℕ)) :=\n  Finset.coe_union _ _\n\n@[simp]\ntheorem mem_sup {μ ν : YoungDiagram} {x : ℕ × ℕ} : x ∈ μ ⊔ ν ↔ x ∈ μ ∨ x ∈ ν :=\n  Finset.mem_union\n\ninstance : Min YoungDiagram where\n  min μ ν :=\n    { cells := μ.cells ∩ ν.cells\n      isLowerSet := by\n        rw [Finset.coe_inter]\n        exact μ.isLowerSet.inter ν.isLowerSet }\n\n@[simp]\ntheorem cells_inf (μ ν : YoungDiagram) : (μ ⊓ ν).cells = μ.cells ∩ ν.cells :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_inf (μ ν : YoungDiagram) : ↑(μ ⊓ ν) = (μ ∩ ν : Set (ℕ × ℕ)) :=\n  Finset.coe_inter _ _\n\n@[simp]\ntheorem mem_inf {μ ν : YoungDiagram} {x : ℕ × ℕ} : x ∈ μ ⊓ ν ↔ x ∈ μ ∧ x ∈ ν :=\n  Finset.mem_inter\n\n/-- The empty Young diagram is `(⊥ : YoungDiagram)`. -/\ninstance : OrderBot YoungDiagram where\n  bot :=\n    { cells := ∅\n      isLowerSet := by\n        intro a b _ h\n        simp only [Finset.coe_empty, Set.mem_empty_iff_false]\n        simp only [Finset.coe_empty, Set.mem_empty_iff_false] at h }\n  bot_le _ _ := by\n    intro y\n    simp only [mem_mk, Finset.notMem_empty] at y\n\n@[simp]\ntheorem cells_bot : (⊥ : YoungDiagram).cells = ∅ :=\n  rfl\n\n@[simp]\ntheorem notMem_bot (x : ℕ × ℕ) : x ∉ (⊥ : YoungDiagram) :=\n  Finset.notMem_empty x\n\n@[norm_cast]\ntheorem coe_bot : (⊥ : YoungDiagram) = (∅ : Set (ℕ × ℕ)) := by\n  ext; simp\n\ninstance : Inhabited YoungDiagram :=\n  ⟨⊥⟩\n\ninstance : DistribLattice YoungDiagram :=\n  Function.Injective.distribLattice YoungDiagram.cells (fun μ ν h ↦ by rwa [YoungDiagram.ext_iff])\n    .rfl .rfl (fun _ _ ↦ rfl) fun _ _ ↦ rfl\n\nend DistribLattice\n\n/-- Cardinality of a Young diagram -/\nprotected abbrev card (μ : YoungDiagram) : ℕ :=\n  μ.cells.card\n\nsection Transpose\n\n/-- The `transpose` of a Young diagram is obtained by swapping i's with j's. -/\ndef transpose (μ : YoungDiagram) : YoungDiagram where\n  cells := (Equiv.prodComm _ _).finsetCongr μ.cells\n  isLowerSet _ _ h := by\n    simp only [Finset.mem_coe, Equiv.finsetCongr_apply, Finset.mem_map_equiv]\n    intro hcell\n    apply μ.isLowerSet _ hcell\n    simp [h]\n\n@[simp]\ntheorem mem_transpose {μ : YoungDiagram} {c : ℕ × ℕ} : c ∈ μ.transpose ↔ c.swap ∈ μ := by\n  simp [transpose]\n\n@[simp]\ntheorem transpose_transpose (μ : YoungDiagram) : μ.transpose.transpose = μ := by\n  ext x\n  simp\n\ntheorem transpose_eq_iff_eq_transpose {μ ν : YoungDiagram} : μ.transpose = ν ↔ μ = ν.transpose := by\n  constructor <;>\n    · rintro rfl\n      simp\n\n@[simp]\ntheorem transpose_eq_iff {μ ν : YoungDiagram} : μ.transpose = ν.transpose ↔ μ = ν := by\n  rw [transpose_eq_iff_eq_transpose]\n  simp\n\n-- This is effectively both directions of `transpose_le_iff` below.\nprotected theorem le_of_transpose_le {μ ν : YoungDiagram} (h_le : μ.transpose ≤ ν) :\n    μ ≤ ν.transpose := fun c hc => by\n  simp only [mem_transpose]\n  apply h_le\n  simpa\n\n@[simp]\ntheorem transpose_le_iff {μ ν : YoungDiagram} : μ.transpose ≤ ν.transpose ↔ μ ≤ ν :=\n  ⟨fun h => by\n    convert! YoungDiagram.le_of_transpose_le h\n    simp, fun h => by\n    rw [← transpose_transpose μ] at h\n    exact YoungDiagram.le_of_transpose_le h ⟩\n\n@[gcongr, mono]\nprotected theorem transpose_mono {μ ν : YoungDiagram} (h_le : μ ≤ ν) : μ.transpose ≤ ν.transpose :=\n  transpose_le_iff.mpr h_le\n\n/-- Transposing Young diagrams is an `OrderIso`. -/\n@[simps]\ndef transposeOrderIso : YoungDiagram ≃o YoungDiagram :=\n  ⟨⟨transpose, transpose, fun _ => by simp, fun _ => by simp⟩, by simp⟩\n\nend Transpose\n\nsection Rows\n\n/-! ### Rows and row lengths of Young diagrams.\n\nThis section defines `μ.row` and `μ.rowLen`, with the following API:\n      1.  `(i, j) ∈ μ ↔ j < μ.rowLen i`\n      2.  `μ.row i = {i} ×ˢ (Finset.range (μ.rowLen i))`\n      3.  `μ.rowLen i = (μ.row i).card`\n      4.  `∀ {i1 i2}, i1 ≤ i2 → μ.rowLen i2 ≤ μ.rowLen i1`\n\nNote: #3 is not convenient for defining `μ.rowLen`; instead, `μ.rowLen` is defined\nas the smallest `j` such that `(i, j) ∉ μ`. -/\n\n\n/-- The `i`-th row of a Young diagram consists of the cells whose first coordinate is `i`. -/\ndef row (μ : YoungDiagram) (i : ℕ) : Finset (ℕ × ℕ) :=\n  μ.cells.filter fun c => c.fst = i\n\ntheorem mem_row_iff {μ : YoungDiagram} {i : ℕ} {c : ℕ × ℕ} : c ∈ μ.row i ↔ c ∈ μ ∧ c.fst = i := by\n  simp [row]\n\ntheorem mk_mem_row_iff {μ : YoungDiagram} {i j : ℕ} : (i, j) ∈ μ.row i ↔ (i, j) ∈ μ := by simp [row]\n\nprotected theorem exists_notMem_row (μ : YoungDiagram) (i : ℕ) : ∃ j, (i, j) ∉ μ := by\n  obtain ⟨j, hj⟩ :=\n    Infinite.exists_notMem_finset\n      (μ.cells.preimage (Prod.mk i) fun _ _ _ _ h => by\n        cases h\n        rfl)\n  rw [Finset.mem_preimage] at hj\n  exact ⟨j, hj⟩\n\n/-- Length of a row of a Young diagram -/\ndef rowLen (μ : YoungDiagram) (i : ℕ) : ℕ :=\n  Nat.find <| μ.exists_notMem_row i\n\nTarget:\ntheorem mem_iff_lt_rowLen {μ : YoungDiagram} {i j : ℕ} : (i, j) ∈ μ ↔ j < μ.rowLen i :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [rowLen, Nat.lt_find_iff]\n  push Not\n  exact ⟨fun h _ hmj => μ.up_left_mem (by rfl) hmj h, fun h => h _ (by rfl)⟩\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Young","family_id":"mem_iff_lt_rowlen","file_id":"mathlib/Mathlib/Combinatorics/Young/YoungDiagram.lean","sample_id":"7d66ee4f35ab6035f8dacc7a5cf8575522e26c3c67988759a6c4ebffd4318014"},"split":"train"}
{"binding":{"chosen_proof_sha256":"839d48065eb916980f8e3d750584330734f1d7501b43b9c8e7cc3615d69e6548","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"18639f8ebcd25d6e01b9f3933b9d096b5bbfe38692f95895d111811c7ffc0f0a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"04cd638bd1240f2fe7fe9709e6cdc0eae054c396fecbe87a5f13fcb8dac94627","source_sha256":"00cdc131b09c868f94179716d61273d8fdf897ed023d34f30c7085668a3b31ac","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  exact if a0 : a = 0 then\n    a0.symm ▸ H0 b\n  else\n    have _ := mod_lt b a0\n    H1 _ _ a0 (GCD.induction (b % a) a H0 H1)\ntermination_by a","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":5,"token_jaccard":0.185185,"token_length_ratio":0.119048},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"5ddc9cd3ae4188262e96bfd70d39d88718fe5cbeceab145bf34e53bdfa885299","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Order.RelClasses\n\nNamespace:\nEuclideanDomain\n\nLocal context:\n/-\nCopyright (c) 2018 Louis Carlin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Louis Carlin, Mario Carneiro\n-/\n/-!\n# Euclidean domains\n\nThis file introduces Euclidean domains and provides the extended Euclidean algorithm. To be precise,\na slightly more general version is provided which is sometimes called a transfinite Euclidean domain\nand differs in the fact that the degree function need not take values in `ℕ` but can take values in\nany well-ordered set. Transfinite Euclidean domains were introduced by Motzkin and examples which\ndon't satisfy the classical notion were provided independently by Hiblot and Nagata.\n\n## Main definitions\n\n* `EuclideanDomain`: Defines Euclidean domain with functions `quotient` and `remainder`. Instances\n  of `Div` and `Mod` are provided, so that one can write `a = b * (a / b) + a % b`.\n* `gcd`: defines the greatest common divisors of two elements of a Euclidean domain.\n* `xgcd`: given two elements `a b : R`, `xgcd a b` defines the pair `(x, y)` such that\n  `x * a + y * b = gcd a b`.\n* `lcm`: defines the lowest common multiple of two elements `a` and `b` of a Euclidean domain as\n  `a * b / (gcd a b)`\n\n## Main statements\n\nSee `Algebra.EuclideanDomain.Basic` for most of the theorems about Euclidean domains,\nincluding Bézout's lemma.\n\nSee `Algebra.EuclideanDomain.Instances` for the fact that `ℤ` is a Euclidean domain,\nas is any field.\n\n## Notation\n\n`≺` denotes the well-founded relation on the Euclidean domain, e.g. in the example of the polynomial\nring over a field, `p ≺ q` for polynomials `p` and `q` if and only if the degree of `p` is less than\nthe degree of `q`.\n\n## Implementation details\n\nInstead of working with a valuation, `EuclideanDomain` is implemented with the existence of a well\nfounded relation `r` on the integral domain `R`, which in the example of `ℤ` would correspond to\nsetting `i ≺ j` for integers `i` and `j` if the absolute value of `i` is smaller than the absolute\nvalue of `j`.\n\n## References\n\n* [Th. Motzkin, *The Euclidean algorithm*][MR32592]\n* [J.-J. Hiblot, *Des anneaux euclidiens dont le plus petit algorithme n'est pas à valeurs finies*]\n  [MR399081]\n* [M. Nagata, *On Euclid algorithm*][MR541021]\n\n\n## Tags\n\nEuclidean domain, transfinite Euclidean domain, Bézout's lemma\n-/\n\n@[expose] public section\n\nuniverse u\n\n/-- A `EuclideanDomain` is a non-trivial commutative ring with a division and a remainder,\n  satisfying `b * (a / b) + a % b = a`.\n  The definition of a Euclidean domain usually includes a valuation function `R → ℕ`.\n  This definition is slightly generalised to include a well-founded relation\n  `r` with the property that `r (a % b) b`, instead of a valuation. -/\n@[wikidata Q867345]\nclass EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R where\n  /-- A division function (denoted `/`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `%` denotes `remainder`. -/\n  protected quotient : R → R → R\n  /-- Division by zero should always give zero by convention. -/\n  protected quotient_zero : ∀ a, quotient a 0 = 0\n  /-- A remainder function (denoted `%`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `/` denotes `quotient`. -/\n  protected remainder : R → R → R\n  /-- The property that links the quotient and remainder functions.\n    This allows us to compute GCDs and LCMs. -/\n  protected quotient_mul_add_remainder_eq : ∀ a b, b * quotient a b + remainder a b = a\n  /-- A well-founded relation on `R`, satisfying `r (a % b) b`.\n    This ensures that the GCD algorithm always terminates. -/\n  protected r : R → R → Prop\n  /-- The relation `r` must be well-founded.\n    This ensures that the GCD algorithm always terminates. -/\n  r_wellFounded : WellFounded r\n  /-- The relation `r` satisfies `r (a % b) b`. -/\n  protected remainder_lt : ∀ (a) {b}, b ≠ 0 → r (remainder a b) b\n  /-- An additional constraint on `r`. -/\n  mul_left_not_lt : ∀ (a) {b}, b ≠ 0 → ¬r (a * b) a\n\n/-\nLean has far more theorems about fields than about Euclidean domains. We thus\nlower the priority of `Euclideandomain.toCommRing`, encouraging typeclass inference\nto try `Field.toCommRing` first. Without this priority-lowering, typeclass inference\nfinds the more inefficient path `Field.toEuclideanDomain.toCommRing` by default. This\npriority change saves over 500G instructions across mathlib. See\nhttps://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/We.20need.20to.20talk.20about.20Euclidean.20Domains/near/594655420\n-/\n-- see Note [lower instance priority]\nattribute [instance 100] EuclideanDomain.toCommRing\n\nnamespace EuclideanDomain\n\nvariable {R : Type u} [EuclideanDomain R]\n\n/-- Abbreviated notation for the well-founded relation `r` in a Euclidean domain. -/\nlocal infixl:50 \" ≺ \" => EuclideanDomain.r\n\nlocal instance wellFoundedRelation : WellFoundedRelation R where\n  rel := EuclideanDomain.r\n  wf := r_wellFounded\n\ninstance isWellFounded : IsWellFounded R (· ≺ ·) where\n  wf := r_wellFounded\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Div R :=\n  ⟨EuclideanDomain.quotient⟩\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Mod R :=\n  ⟨EuclideanDomain.remainder⟩\n\ntheorem div_add_mod (a b : R) : b * (a / b) + a % b = a :=\n  EuclideanDomain.quotient_mul_add_remainder_eq _ _\n\ntheorem mod_add_div (a b : R) : a % b + b * (a / b) = a :=\n  (add_comm _ _).trans (div_add_mod _ _)\n\ntheorem mod_add_div' (m k : R) : m % k + m / k * k = m := by\n  rw [mul_comm]\n  exact mod_add_div _ _\n\ntheorem div_add_mod' (m k : R) : m / k * k + m % k = m := by\n  rw [mul_comm]\n  exact div_add_mod _ _\n\ntheorem mod_lt : ∀ (a) {b : R}, b ≠ 0 → a % b ≺ b :=\n  EuclideanDomain.remainder_lt\n\ntheorem mul_right_not_lt {a : R} (b) (h : a ≠ 0) : ¬a * b ≺ b := by\n  rw [mul_comm]\n  exact mul_left_not_lt b h\n\n@[simp]\ntheorem mod_zero (a : R) : a % 0 = a := by simpa only [zero_mul, zero_add] using div_add_mod a 0\n\ntheorem lt_one (a : R) : a ≺ (1 : R) → a = 0 :=\n  haveI := Classical.dec\n  not_imp_not.1 fun h => by simpa only [one_mul] using mul_left_not_lt 1 h\n\n@[simp]\ntheorem div_zero (a : R) : a / 0 = 0 :=\n  EuclideanDomain.quotient_zero a\n\nsection\n\n@[elab_as_elim]\n\nTarget:\ntheorem GCD.induction {P : R → R → Prop} (a b : R) (H0 : ∀ x, P 0 x)\n    (H1 : ∀ a b, a ≠ 0 → P (b % a) a → P a b) : P a b :=\n\nProof body:\n","rejected":"by\n  exact GCD.induction","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"044f9cb2078be90a5358d28fe7970f2f36fc21a24ff61936a30a0975b9ddb53d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/EuclideanDomain","family_id":"gcd","file_id":"mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean","sample_id":"04cd638bd1240f2fe7fe9709e6cdc0eae054c396fecbe87a5f13fcb8dac94627"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3e9d48f580176762fa7a5e0ebfd29db0eb21f956c84ea97eca970101095c5878","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"352387881c46e504868b78447c6e9de91b428ed4692476d3bb44b66f8cb11e49","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"57ea28a7b08d8f885b958267bc30048df940f453f6ddd0117c4636da80751a9c","source_sha256":"f6612dc0cd9222234e5394f61b2340672945324a405e24dff4457d1abcc92db2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  letI : MeasurableSpace (ℕ × Ω) := 𝓕.predictable\n  letI : MeasurableSpace Ω := 𝓕 n\n  let X m := (Function.curry (h𝓕.approx m) (n + 1))\n  refine ⟨(fun m ↦ SimpleFunc.mk (X m) ?_ ?_), (fun ω ↦ h𝓕.tendsto_approx ⟨(n + 1), ω⟩)⟩\n  · intro s\n    rw [(by aesop : X m ⁻¹' {s} = {ω | (n + 1, ω) ∈ h𝓕.approx m ⁻¹' {s}})]\n    apply measurableSet_prodMk_add_one_of_predictable\n    apply (h𝓕.approx m).measurableSet_fiber\n  · apply (h𝓕.approx m).finite_range.subset\n    rw [Set.range_subset_iff]\n    aesop","hard_negative":true,"metrics":{"chosen_tokens":154,"rejected_tokens":5,"token_jaccard":0.033333,"token_length_ratio":0.032468},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"5e01c0d2e10bc2323650bf0dd3b56888dba0a4aa7ac264fafd110ff226bee327","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Process.Adapted\n\nNamespace:\nMeasureTheory.IsStronglyPredictable\n\nLocal context:\n/-\nCopyright (c) 2025 Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kexing Ying\n-/\n/-!\n# Predictable σ-algebra\n\nThis file defines the predictable σ-algebra associated to a filtration, as well as the\nnotion of predictable processes. We prove that predictable processes are progressively measurable\nand adapted. We also give an equivalent characterization of predictability for discrete processes.\n\n## Main definitions\n\n* `Filtration.predictable` : The predictable σ-algebra associated to a filtration.\n* `IsStronglyPredictable` : A process is predictable if it is measurable with respect to the\n  predictable σ-algebra.\n\n## Main results\n\n* `IsStronglyPredictable.isStronglyProgressive` : A predictable process is progressively measurable.\n* `IsStronglyPredictable.iff_measurable_add_one` : `u` is a discrete predictable process iff\n  `u (n + 1)` is `𝓕 n`-measurable and `u 0` is `𝓕 0`-measurable.\n\n## Tags\n\npredictable, previsible\n\n-/\n\n@[expose] public section\n\nopen Filter Order TopologicalSpace\n\nopen scoped MeasureTheory NNReal ENNReal Topology\n\nnamespace MeasureTheory\n\nvariable {Ω ι : Type*} {m : MeasurableSpace Ω} {E : Type*}\n\nsection\n\nnamespace Filtration\n\n/-- Given a filtration `𝓕`, the predictable σ-algebra is the σ-algebra on `ι × Ω` generated by\nsets of the form `(t, ∞) × A` for `t ∈ ι` and `A ∈ 𝓕 t` and `{⊥} × A` for `A ∈ 𝓕 ⊥`. -/\n@[implicit_reducible]\ndef predictable [Preorder ι] [OrderBot ι] (𝓕 : Filtration ι m) : MeasurableSpace (ι × Ω) :=\n  MeasurableSpace.generateFrom <|\n    {s | ∃ A, MeasurableSet[𝓕 ⊥] A ∧ s = {⊥} ×ˢ A} ∪\n    {s | ∃ i A, MeasurableSet[𝓕 i] A ∧ s = Set.Ioi i ×ˢ A}\n\nend Filtration\n\nlemma measurableSet_predictable_singleton_bot_prod [LinearOrder ι] [OrderBot ι]\n    {𝓕 : Filtration ι m} {s : Set Ω} (hs : MeasurableSet[𝓕 ⊥] s) :\n    MeasurableSet[𝓕.predictable] <| {⊥} ×ˢ s :=\n  MeasurableSpace.measurableSet_generateFrom <| Or.inl ⟨s, hs, rfl⟩\n\nlemma measurableSet_predictable_Ioi_prod [LinearOrder ι] [OrderBot ι]\n    {𝓕 : Filtration ι m} {i : ι} {s : Set Ω} (hs : MeasurableSet[𝓕 i] s) :\n    MeasurableSet[𝓕.predictable] <| Set.Ioi i ×ˢ s :=\n  MeasurableSpace.measurableSet_generateFrom <| Or.inr ⟨i, s, hs, rfl⟩\n\n/-- Sets of the form `(i, j] × A` for any `A ∈ 𝓕 i` are measurable with respect to the predictable\nσ-algebra. -/\nlemma measurableSet_predictable_Ioc_prod [LinearOrder ι] [OrderBot ι]\n    {𝓕 : Filtration ι m} (i j : ι) {s : Set Ω} (hs : MeasurableSet[𝓕 i] s) :\n    MeasurableSet[𝓕.predictable] <| Set.Ioc i j ×ˢ s := by\n  obtain hij | hij := le_total j i\n  · simp [hij]\n  · rw [← Set.Ioi_sdiff_Ioi, (by simp : (Set.Ioi i \\ Set.Ioi j) ×ˢ s\n      = Set.Ioi i ×ˢ (s \\ s) ∪ (Set.Ioi i \\ Set.Ioi j) ×ˢ s), ← Set.prod_sdiff_prod]\n    exact (measurableSet_predictable_Ioi_prod hs).diff\n      (measurableSet_predictable_Ioi_prod <| 𝓕.mono hij _ hs)\n\nlemma measurableSet_predictable_singleton_prod\n    {𝓕 : Filtration ℕ m} {n : ℕ} {s : Set Ω} (hs : MeasurableSet[𝓕 n] s) :\n    MeasurableSet[𝓕.predictable] <| {n + 1} ×ˢ s := by\n  rw [(_ : {n + 1} = Set.Ioc n (n + 1))]\n  · exact measurableSet_predictable_Ioc_prod _ _ hs\n  · ext m\n    simp only [Set.mem_singleton_iff, Set.mem_Ioc]\n    lia\n\nlemma measurableSet_prodMk_add_one_of_predictable {𝓕 : Filtration ℕ m} {s : Set (ℕ × Ω)}\n    (hs : MeasurableSet[𝓕.predictable] s) (n : ℕ) :\n    MeasurableSet[𝓕 n] {ω | (n + 1, ω) ∈ s} := by\n  rw [(by aesop : {ω | (n + 1, ω) ∈ s} = (Prod.mk (α := Set.singleton (n + 1)) (β := Ω)\n      ⟨n + 1, rfl⟩) ⁻¹' ((fun (p : Set.singleton (n + 1) × Ω) ↦ ((p.1 : ℕ), p.2)) ⁻¹' s))]\n  refine measurableSet_preimage (mβ := Subtype.instMeasurableSpace.prod (𝓕 n))\n    measurable_prodMk_left <| measurableSet_preimage ?_ hs\n  rw [measurable_iff_comap_le, MeasurableSpace.comap_le_iff_le_map]\n  refine MeasurableSpace.generateFrom_le ?_\n  rintro - (⟨A, hA, rfl⟩ | ⟨i, A, hA, rfl⟩)\n  · rw [MeasurableSpace.map_def,\n      (_ : (fun (p : Set.singleton (n + 1) × Ω) ↦ ((p.1 : ℕ), p.2)) ⁻¹' ({⊥} ×ˢ A) = ∅)]\n    · simp\n    · ext p\n      simp only [Nat.bot_eq_zero, Set.mem_preimage, Set.mem_prod, Set.mem_singleton_iff,\n        Set.mem_empty_iff_false, iff_false, not_and]\n      exact fun hp1 ↦ False.elim <| Nat.succ_ne_zero n (hp1 ▸ p.1.2.symm)\n  · rw [MeasurableSpace.map_def]\n    obtain hni | hin := lt_or_ge n i\n    · rw [(_ : (fun (p : Set.singleton (n + 1) × Ω) ↦ ((p.1 : ℕ), p.2)) ⁻¹' (Set.Ioi i ×ˢ A) = ∅)]\n      · simp\n      · ext p\n        simp only [Set.mem_preimage, Set.mem_prod, Set.mem_Ioi, Set.mem_empty_iff_false,\n          iff_false, not_and]\n        rw [p.1.2]\n        grind\n    · rw [(_ : (fun (p : Set.singleton (n + 1) × Ω) ↦ ((p.1 : ℕ), p.2)) ⁻¹' (Set.Ioi i ×ˢ A)\n          = {⟨n + 1, rfl⟩} ×ˢ A)]\n      · exact MeasurableSet.prod (MeasurableSet.of_subtype_image trivial) (𝓕.mono hin _ hA)\n      · ext p\n        simp only [Set.mem_preimage, Set.mem_prod, Set.mem_Ioi, Set.mem_singleton_iff,\n          and_congr_left_iff]\n        intro hp2\n        rw [p.1.2]\n        exact ⟨fun _ ↦ by aesop, fun _ ↦ lt_add_one_iff.2 hin⟩\n\nlemma measurableSpace_le_predictable_of_measurableSet [Preorder ι] [OrderBot ι]\n    {𝓕 : Filtration ι m} {m' : MeasurableSpace (ι × Ω)}\n    (hm'bot : ∀ A, MeasurableSet[𝓕 ⊥] A → MeasurableSet[m'] ({⊥} ×ˢ A))\n    (hm' : ∀ i A, MeasurableSet[𝓕 i] A → MeasurableSet[m'] ((Set.Ioi i) ×ˢ A)) :\n    𝓕.predictable ≤ m' := by\n  refine MeasurableSpace.generateFrom_le ?_\n  rintro - (⟨A, hA, rfl⟩ | ⟨i, A, hA, rfl⟩)\n  · exact hm'bot A hA\n  · exact hm' i A hA\n\n/-- The inclusion map from [0,i] × Ω with the subtype × 𝓕 i σ-algebra) to ι × Ω with the\npredictable σ-algebra is measurable -/\nlemma measurable_inclusion_predictable [LinearOrder ι] [OrderBot ι] [MeasurableSpace ι]\n    [TopologicalSpace ι] [OpensMeasurableSpace ι] [OrderClosedTopology ι] {𝓕 : Filtration ι m} {i} :\n    @Measurable (Set.Iic i × Ω) (ι × Ω) (Subtype.instMeasurableSpace.prod (𝓕 i)) 𝓕.predictable\n    fun x ↦ ⟨x.1.val, x.2⟩ := by\n  rw [measurable_iff_comap_le]\n  refine MeasurableSpace.comap_le_iff_le_map.2 <|\n    measurableSpace_le_predictable_of_measurableSet ?_ ?_\n  · intros A hA\n    simp only [MeasurableSpace.map_def,\n      (by aesop : (fun (p : Set.Iic i × Ω) ↦ ((p.1 : ι), p.2)) ⁻¹' ({⊥} ×ˢ A) = {⊥} ×ˢ A)]\n    exact (measurableSet_singleton _).prod <| 𝓕.mono bot_le _ hA\n  · intros j A hA\n    simp only [MeasurableSpace.map_def]\n    obtain hji | hij := le_total j i\n    · rw [(by grind : (fun (p : Set.Iic i × Ω) ↦ ((p.1 : ι), p.2)) ⁻¹' Set.Ioi j ×ˢ A\n        = (Subtype.val ⁻¹' (Set.Ioc j i)) ×ˢ A)]\n      exact (measurable_subtype_coe measurableSet_Ioc).prod (𝓕.mono hji _ hA)\n    · simp [(by grind : (fun (p : Set.Iic i × Ω) ↦ ((p.1 : ι), p.2)) ⁻¹' Set.Ioi j ×ˢ A = ∅)]\n\nend\n\nvariable [TopologicalSpace E]\n\n/-- A process is said to be predictable if it is measurable with respect to the predictable\nσ-algebra. -/\ndef IsStronglyPredictable [Preorder ι] [OrderBot ι] (𝓕 : Filtration ι m) (u : ι → Ω → E) :=\n  StronglyMeasurable[𝓕.predictable] <| Function.uncurry u\n\nnamespace IsStronglyPredictable\n\nopen Filtration\n\nvariable [LinearOrder ι] [OrderBot ι] [MeasurableSpace ι] [TopologicalSpace ι]\n    [OpensMeasurableSpace ι] [OrderClosedTopology ι]\n\n/-- A predictable process is progressively measurable. -/\nlemma isStronglyProgressive {𝓕 : Filtration ι m} {u : ι → Ω → E} (h𝓕 : IsStronglyPredictable 𝓕 u) :\n    IsStronglyProgressive 𝓕 u := by\n  intro i\n  letI : MeasurableSpace (ι × Ω) := 𝓕.predictable\n  letI : MeasurableSpace (Set.Iic i × Ω) := Subtype.instMeasurableSpace.prod (𝓕 i)\n  let X m (x : Set.Iic i × Ω) := h𝓕.approx m ⟨x.1, x.2⟩\n  refine ⟨fun m ↦ SimpleFunc.mk (X m) ?_ ?_, ?_⟩\n  · exact fun e ↦ measurable_inclusion_predictable <| (h𝓕.approx m).measurableSet_fiber e\n  · exact Set.Finite.subset (h𝓕.approx m).finite_range (by grind)\n  · exact fun n ↦ by apply h𝓕.tendsto_approx\n\n/-- A predictable process is adapted. -/\nlemma stronglyAdapted {𝓕 : Filtration ι m} {u : ι → Ω → E} (h𝓕 : IsStronglyPredictable 𝓕 u) :\n    StronglyAdapted 𝓕 u :=\n  h𝓕.isStronglyProgressive.stronglyAdapted\n\nsection Discrete\n\n/-- If `u` is a discrete predictable process, then `u (n + 1)` is `𝓕 n`-measurable. -/\n\nTarget:\nlemma measurable_add_one {𝓕 : Filtration ℕ m} {u : ℕ → Ω → E}\n    (h𝓕 : IsStronglyPredictable 𝓕 u) (n : ℕ) : StronglyMeasurable[𝓕 n] (u (n + 1)) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_57ea28a7b08d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c306a7148b10c6b61881053d17fe2a9398891b75e0c836e1ab98c3e7021ce396","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Process","family_id":"measurable_add_one","file_id":"mathlib/Mathlib/Probability/Process/Predictable.lean","sample_id":"57ea28a7b08d8f885b958267bc30048df940f453f6ddd0117c4636da80751a9c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"da2d9310157ed9ceff8c58c2de745bed1ed2329391fedb4f8010b3557e4b56ed","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9b4e3e0d6a7653b54818bb9ece9503e507bf5b7abbd8d2d4b99515af39aba54b","source_sha256":"5719c6ab1b79cb92451b632f8631c4fdd8e58d363dc71c3722b36068c3f61537","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h : ∃ n, x ≤ t n := by\n    simp only [tendsto_atTop_atTop_iff_of_monotone ht_mono.monotone] at ht_tendsto\n    exact ht_tendsto x\n  have h' m := Nat.find_min h (m := m)\n  simp only [not_le] at h'\n  exact ⟨Nat.find h - 1, h' _ (by simp [hx]), by simp [Nat.find_spec h, hx]⟩","hard_negative":false,"metrics":{"chosen_tokens":76,"rejected_tokens":2,"token_jaccard":0.026316,"token_length_ratio":0.026316},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"5ecfac8ac7314e2dfdcfa6a3c4aa619ffe9a03c12334bbcf7ce588d77263e0b7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecificLimits.ArithmeticGeometric\npublic import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap\npublic import Mathlib.MeasureTheory.Function.L1Space.Integrable\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n-/\n/-!\n# Fernique's theorem for rotation-invariant measures\n\nLet `μ` be a finite measure on a second-countable normed space `E` such that the product measure\n`μ.prod μ` on `E × E` is invariant by rotation of angle `-π/4`.\nThen there exists a constant `C > 0` such that the function `x ↦ exp (C * ‖x‖ ^ 2)` is integrable\nwith respect to `μ`.\n\n## Sketch of the proof\n\nThe main case of the proof is for `μ` a probability measure such that there exists a positive\n`a : ℝ` such that `2⁻¹ < μ {x | ‖x‖ ≤ a} < 1`. If `μ` is a probability measure and `a` does not\nexist then we can show that there is a ball with finite radius of measure 1, and the result is true\nfor `C = 1` (for example), since `x ↦ exp (‖x‖ ^ 2)` is almost surely bounded.\nWe then choose such an `a`.\n\nIn order to show the existence of `C` such that `x ↦ exp (C * ‖x‖ ^ 2)` is integrable, we prove as\nintermediate result that for `a, c` with `2⁻¹ < c ≤ μ {x | ‖x‖ ≤ a}`,\nthe integral `∫⁻ x, exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2) ∂μ` is bounded by a finite quantity\n(`logRatio c` is a multiple of `log (c / (1 - c))`). We can then take `C = logRatio c * a⁻¹ ^ 2`.\n\nWe now turn to the proof of the intermediate result.\n\nFirst in `measure_le_mul_measure_gt_le_of_map_rotation_eq_self` we prove that if a measure `μ` is\nsuch that `μ.prod μ` is invariant by rotation of angle `-π/4` then\n`μ {x | ‖x‖ ≤ a} * μ {x | b < ‖x‖} ≤ μ {x | (b - a) / √2 < ‖x‖} ^ 2`.\nThe rotation invariance is used only through that inequality.\n\nWe define a sequence of thresholds `t n` inductively by `t 0 = a` and `t (n + 1) = √2 * t n + a`.\nThey are chosen such that the invariance by rotation gives\n`μ {x | ‖x‖ ≤ a} * μ {x | t (n + 1) < ‖x‖} ≤ μ {x | t n < ‖x‖} ^ 2`.\nThanks to that inequality we can show that `μ {x | t n < ‖x‖}` decreases fast with `n`:\nfor `mₐ = μ {x | ‖x‖ ≤ a}`, `μ {x | t n < ‖x‖} ≤ mₐ * exp (- log (mₐ / (1 - mₐ)) * 2 ^ n)`.\n\nWe cut the space into annuli `{x | t n < ‖x‖ ≤ t n + 1}` and bound the integral separately on\neach annulus. On that set the function `exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2)` is bounded by\n`exp (logRatio c * a⁻¹ ^ 2 * t (n + 1) ^ 2)`, which is in turn less than\n`exp (2⁻¹ * log (c / (1 - c)) * 2 ^ n)` (from the definition of the threshold `t` and `logRatio c`).\nThe measure of the annulus is bounded by `μ {x | t n < ‖x‖}`, for which we derived an upper bound\nabove. The function gets exponentially large, but `μ {x | t n < ‖x‖}` decreases even faster, so the\nintegral is bounded by a quantity of the form `exp (- u * 2 ^ n)` for `u>0`.\nSumming over all annuli (over `n`) gives a finite value for the integral.\n\n## Main statements\n\n* `lintegral_exp_mul_sq_norm_le_of_map_rotation_eq_self`: for `μ` a probability measure\n  whose product with itself is invariant by rotation and for `a, c` with\n  `2⁻¹ < c ≤ μ {x | ‖x‖ ≤ a}`, the integral `∫⁻ x, exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2) ∂μ`\n  is bounded by a quantity that does not depend on `a`.\n* `exists_integrable_exp_sq_of_map_rotation_eq_self`: Fernique's theorem for finite measures\n  whose product is invariant by rotation.\n\n## References\n\n* [Xavier Fernique, *Intégrabilité des vecteurs gaussiens*][fernique1970integrabilite]\n* [Martin Hairer, *An introduction to stochastic PDEs*][hairer2009introduction]\n\n## TODO\n\nFrom the intermediate result `lintegral_exp_mul_sq_norm_le_of_map_rotation_eq_self`,\nwe can deduce bounds on all the moments of the measure `μ` as function of powers of\nthe first moment.\n\n-/\n\n@[expose] public section\n\nopen MeasureTheory ProbabilityTheory Complex NormedSpace Filter\nopen scoped ENNReal NNReal Real Topology\n\nsection Aux\n\nTarget:\nlemma StrictMono.exists_between_of_tendsto_atTop {β : Type*} [LinearOrder β] {t : ℕ → β}\n    (ht_mono : StrictMono t) (ht_tendsto : Tendsto t atTop atTop) {x : β} (hx : t 0 < x) :\n    ∃ n, t n < x ∧ x ≤ t (n + 1) :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Distributions","family_id":"strictmono","file_id":"mathlib/Mathlib/Probability/Distributions/Fernique.lean","sample_id":"9b4e3e0d6a7653b54818bb9ece9503e507bf5b7abbd8d2d4b99515af39aba54b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bc3578532551ba1c8a4369ed77aebd3d2f1775c8fadb93080f656df6af2cd8fe","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b97fb9e2cc68aedd758bfae5333c8c15c22cb1035b6305d722b7dd9adf180bf4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f6fe57d0752e0cd229648ee764e154901c2c974592574e9f5ce3b35211409c9e","source_sha256":"2b139890caa5a415f5023de326a4c66559e073e900b5cf8f559558dc034a601c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have ⟨φ⟩ := ne\n  refine h.toIsAlgebraic.of_injective φ φ.injective\n    |>.normalClosure_eq_iSup_adjoin_of_splits fun x ↦ ?_\n  rw [← minpoly.algHom_eq _ φ.injective]\n  apply h.splits","hard_negative":true,"metrics":{"chosen_tokens":41,"rejected_tokens":2,"token_jaccard":0.033333,"token_length_ratio":0.04878},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"5ed4bb05bdf22b4c2a32b645b078308803bebee357a3cb730607d3493fcc0bfe","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.SimpleRing.Basic\npublic import Mathlib.FieldTheory.Normal.Basic\npublic import Mathlib.Order.Closure\npublic import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Thomas Browning. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Thomas Browning\n-/\n/-!\n# Normal closures\n\n## Main definitions\n\nGiven field extensions `K/F` and `L/F`, the predicate `IsNormalClosure F K L` says that the\nminimal polynomial of every element of `K` over `F` splits in `L`, and that `L` is generated\nby the roots of such minimal polynomials. These conditions uniquely characterize `L/F` up to\n`F`-algebra isomorphisms (`IsNormalClosure.equiv`).\n\nThe explicit construction `IntermediateField.normalClosure F K L` of a field extension `K/F`\ninside another field extension `L/F` is the smallest intermediate field of\n`L/F` that contains the image of every `F`-algebra embedding `K →ₐ[F] L`.\nIt satisfies the `IsNormalClosure` predicate if `L/F` satisfies the\nabovementioned splitting condition, in particular if `L/K/F` form a tower and\n`L/F` is normal.\n-/\n\n@[expose] public section\n\nopen IntermediateField IsScalarTower Polynomial\n\nvariable (F K L : Type*) [Field F] [Field K] [Field L] [Algebra F K] [Algebra F L]\n\n/-- `L/F` is a normal closure of `K/F` if the minimal polynomial of every element of `K` over `F`\n  splits in `L`, and `L` is generated by roots of such minimal polynomials over `F`.\n  (Since the minimal polynomial of a transcendental element is 0,\n  the normal closure of `K/F` is the same as the normal closure over `F`\n  of the algebraic closure of `F` in `K`.) -/\n@[stacks 0BMF \"Predicate version\"]\nclass IsNormalClosure : Prop where\n  splits (x : K) : ((minpoly F x).map (algebraMap F L)).Splits\n  adjoin_rootSet : ⨆ x : K, adjoin F ((minpoly F x).rootSet L) = ⊤\n/- TODO: show `IsNormalClosure F K L ↔ IsNormalClosure F (integralClosure F K) L`; we can't state\n  this yet because `integralClosure F K` needs to have a `Field` instance. -/\n\n/-- The normal closure of `K/F` in `L/F`. -/\n@[stacks 0BMF]\nnoncomputable def IntermediateField.normalClosure : IntermediateField F L :=\n  ⨆ f : K →ₐ[F] L, f.fieldRange\n\nlemma normalClosure_def : normalClosure F K L = ⨆ f : K →ₐ[F] L, f.fieldRange :=\n  rfl\n\nvariable {F K L}\n\n/-- A normal closure is always normal. -/\nlemma IsNormalClosure.normal [h : IsNormalClosure F K L] : Normal F L :=\n  Normal.of_algEquiv topEquiv (h := h.adjoin_rootSet ▸ IntermediateField.normal_iSup (h :=\n    fun _ ↦ Normal.of_isSplittingField (hFEp := adjoin_rootSet_isSplittingField <| h.splits _)))\n\nlemma normalClosure_le_iff {K' : IntermediateField F L} :\n    normalClosure F K L ≤ K' ↔ ∀ f : K →ₐ[F] L, f.fieldRange ≤ K' :=\n  iSup_le_iff\n\nlemma AlgHom.fieldRange_le_normalClosure (f : K →ₐ[F] L) : f.fieldRange ≤ normalClosure F K L :=\n  le_iSup AlgHom.fieldRange f\n\nnamespace Algebra.IsAlgebraic\nvariable [Algebra.IsAlgebraic F K]\n\nlemma normalClosure_le_iSup_adjoin :\n    normalClosure F K L ≤ ⨆ x : K, IntermediateField.adjoin F ((minpoly F x).rootSet L) :=\n  iSup_le fun f _ ⟨x, hx⟩ ↦ le_iSup (α := IntermediateField F L) _ x <|\n    IntermediateField.subset_adjoin F _ <| by\n      rw [mem_rootSet_of_ne (minpoly.ne_zero (Algebra.IsIntegral.isIntegral x)), ← hx,\n        AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom, aeval_algHom_apply, minpoly.aeval, map_zero]\n\nvariable (splits : ∀ x : K, ((minpoly F x).map (algebraMap F L)).Splits)\n\ninclude splits in\nlemma normalClosure_eq_iSup_adjoin_of_splits :\n    normalClosure F K L = ⨆ x : K, IntermediateField.adjoin F ((minpoly F x).rootSet L) :=\n  normalClosure_le_iSup_adjoin.antisymm <|\n    iSup_le fun x ↦ IntermediateField.adjoin_le_iff.mpr fun _ hy ↦\n      let ⟨φ, hφ⟩ := IntermediateField.exists_algHom_of_splits_of_aeval\n        (fun x ↦ ⟨Algebra.IsIntegral.isIntegral x, splits x⟩) (mem_rootSet.mp hy).2\n      le_iSup AlgHom.fieldRange φ ⟨x, hφ⟩\n\n/-- If `K/F` is algebraic, the \"generated by roots\" condition in IsNormalClosure can be replaced\n  by \"generated by images of embeddings\". -/\nlemma isNormalClosure_iff : IsNormalClosure F K L ↔\n    (∀ x : K, ((minpoly F x).map (algebraMap F L)).Splits) ∧ normalClosure F K L = ⊤ := by\n  refine ⟨fun ⟨splits, h⟩ ↦ ⟨splits, ?_⟩, fun ⟨splits, h⟩ ↦ ⟨splits, ?_⟩⟩ <;>\n    simpa only [normalClosure_eq_iSup_adjoin_of_splits splits] using h\n-- TODO: IntermediateField.isNormalClosure_iff similar to IntermediateField.isSplittingField_iff\n\nset_option backward.isDefEq.respectTransparency false in\ninclude splits in\n/-- `normalClosure F K L` is a valid normal closure if `K/F` is algebraic\n  and all minimal polynomials of `K/F` splits in `L/F`. -/\nlemma isNormalClosure_normalClosure : IsNormalClosure F K (normalClosure F K L) := by\n  rw [isNormalClosure_iff]; constructor\n  · rw [normalClosure_eq_iSup_adjoin_of_splits splits]\n    exact fun x ↦ splits_of_splits (splits x) ((IntermediateField.subset_adjoin F _).trans <|\n      SetLike.coe_subset_coe.mpr <| by apply le_iSup _ x)\n  simp_rw [normalClosure, ← top_le_iff]\n  refine fun x _ ↦ ((⨆ f : K →ₐ[F] L, f.fieldRange).val).injective.mem_set_image |>.mp ?_\n  rw [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, coe_val, ← IntermediateField.coe_val,\n    ← IntermediateField.coe_map, IntermediateField.map_iSup]\n  refine (iSup_le fun f ↦ ?_ : normalClosure F K L ≤ _) x.2\n  refine le_iSup_of_le (f.codRestrict _ fun x ↦ f.fieldRange_le_normalClosure ⟨x, rfl⟩) ?_\n  rw [AlgHom.map_fieldRange, val, AlgHom.val_comp_codRestrict]\n\nend Algebra.IsAlgebraic\n\n/-- A normal closure of `K/F` embeds into any `L/F`\n  where the minimal polynomials of `K/F` splits. -/\nnoncomputable def IsNormalClosure.lift [h : IsNormalClosure F K L] {L'} [Field L'] [Algebra F L']\n    (splits : ∀ x : K, ((minpoly F x).map (algebraMap F L')).Splits) : L →ₐ[F] L' := by\n  have := h.adjoin_rootSet; rw [← gc.l_iSup] at this\n  refine Nonempty.some <| nonempty_algHom_of_adjoin_splits\n    (fun x hx ↦ ⟨isAlgebraic_iff_isIntegral.mp ((h.normal).isAlgebraic x), ?_⟩) this\n  obtain ⟨y, hx⟩ := Set.mem_iUnion.mp hx\n  by_cases iy : IsIntegral F y\n  · exact (splits y).of_dvd (map_ne_zero (minpoly.ne_zero iy))\n      ((map_dvd_map' _).mpr (minpoly.dvd F x (mem_rootSet.mp hx).2))\n  · simp [minpoly.eq_zero iy] at hx\n\n/-- Normal closures of `K/F` are unique up to F-algebra isomorphisms. -/\nnoncomputable def IsNormalClosure.equiv {L'} [Field L'] [Algebra F L']\n    [h : IsNormalClosure F K L] [h' : IsNormalClosure F K L'] : L ≃ₐ[F] L' :=\n  have := h.normal\n  AlgEquiv.ofBijective _ <| And.left <|\n    Normal.toIsAlgebraic.algHom_bijective₂\n      (IsNormalClosure.lift fun _ : K ↦ h'.splits _)\n      (IsNormalClosure.lift fun _ : K ↦ h.splits _)\n\nvariable (F K L)\n\ninstance isNormalClosure_normalClosure [ne : Nonempty (K →ₐ[F] L)] [h : Normal F L] :\n    IsNormalClosure F K (normalClosure F K L) := by\n  have ⟨φ⟩ := ne\n  apply (h.toIsAlgebraic.of_injective φ φ.injective).isNormalClosure_normalClosure\n  simp_rw [← minpoly.algHom_eq _ φ.injective]\n  exact fun _ ↦ h.splits _\n\nTarget:\ntheorem normalClosure_eq_iSup_adjoin' [ne : Nonempty (K →ₐ[F] L)] [h : Normal F L] :\n    normalClosure F K L = ⨆ x : K, adjoin F ((minpoly F x).rootSet L) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_f6fe57d0752e","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"7b1518385a5ba9d0a83bd1d8e58195e5c9ddaf1c9c9b2760a13f5c857f9ce18a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Normal","family_id":"normalclosure_eq_isup_adjoin","file_id":"mathlib/Mathlib/FieldTheory/Normal/Closure.lean","sample_id":"f6fe57d0752e0cd229648ee764e154901c2c974592574e9f5ce3b35211409c9e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1f1ea0e13f9e5b99d77e0d90716326ac0d10ebd31e81aeaa4cd98f72f2f5c63c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"19249b12421b93ef9aca4c5c6690238801a0a01bd844a6075bfc5c80d4237b2d","source_sha256":"d0e82c6901f418bcc62ae1e69134f5914117f0eba42bc658092940528f8b500f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction n, hn using Nat.le_induction with\n  | base => simp\n  | succ k hk ih =>\n    have hck : 0 ≤ c k := hc k hk\n    have heq : c k * ∑ j ∈ Ico n₀ k, b j * ∏ i ∈ Ico (j + 1) k, c i + b k =\n        ∑ j ∈ Ico n₀ (k + 1), b j * ∏ i ∈ Ico (j + 1) (k + 1), c i := by\n      rw [sum_Ico_succ_top hk, mul_sum, Ico_self, prod_empty, mul_one]\n      refine congr_arg (· + b k) (sum_congr rfl fun j hj ↦ ?_)\n      rw [prod_Ico_succ_top (by have := mem_Ico.mp hj; omega)]; ring\n    calc u (k + 1)\n      _ ≤ c k * u k + b k := hu k hk\n      _ ≤ c k * (u n₀ * ∏ i ∈ Ico n₀ k, c i +\n            ∑ j ∈ Ico n₀ k, b j * ∏ i ∈ Ico (j + 1) k, c i) + b k := by gcongr\n      _ = u n₀ * ∏ i ∈ Ico n₀ (k + 1), c i +\n            ∑ j ∈ Ico n₀ (k + 1), b j * ∏ i ∈ Ico (j + 1) (k + 1), c i := by\n          rw [← heq, ← prod_Ico_mul_eq_prod_Ico_add_one hk]; ring","hard_negative":true,"metrics":{"chosen_tokens":286,"rejected_tokens":8,"token_jaccard":0.054054,"token_length_ratio":0.027972},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"5ed7f629890456a76dc7af2a5eab7208c0fe428066ab25de49288d6fe67797ca","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.ExpDeriv\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2026 Dennj Osele. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dennj Osele\n-/\n/-!\n# Discrete Grönwall inequality\n\nVarious forms of the discrete Grönwall inequality, bounding solutions to recurrence\ninequalities `u (n+1) ≤ c n * u n + b n` and `u (n+1) ≤ (1 + c n) * u n + b n`.\n\n## Main results\n\n* `discrete_gronwall_prod_general`: product form, over any ordered commutative semiring.\n* `discrete_gronwall`: classical exponential bound for the `(1 + c)` form, over `ℝ`.\n* `discrete_gronwall_Ico`: uniform bound over an interval, over `ℝ`.\n\n## References\n\n* [T. H. Grönwall, *Note on the derivatives with respect to a parameter of the solutions of a\n  system of differential equations*][Gronwall_1919]\n\n## See also\n\n* `Mathlib.Analysis.ODE.Gronwall` for the continuous Grönwall inequality for ODEs.\n-/\n\n@[expose] public section\n\nopen Real Finset\n\nsection General\n\n/-! ### Generalized product form -/\n\nvariable {R : Type*} [CommSemiring R] [PartialOrder R] [IsOrderedRing R] {u b c : ℕ → R}\n\n/-- Discrete Grönwall inequality, product form: if `u (n+1) ≤ c n * u n + b n` and `0 ≤ c n`\nthen `u n ≤ u n₀ * ∏ c i + ∑ b k * ∏ c i` over the appropriate ranges. -/\n\nTarget:\ntheorem discrete_gronwall_prod_general {n₀ : ℕ} (hu : ∀ n ≥ n₀, u (n + 1) ≤ c n * u n + b n)\n    (hc : ∀ n ≥ n₀, 0 ≤ c n) ⦃n : ℕ⦄ (hn : n₀ ≤ n) :\n    u n ≤ u n₀ * ∏ i ∈ Ico n₀ n, c i +\n      ∑ k ∈ Ico n₀ n, b k * ∏ i ∈ Ico (k + 1) n, c i :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"af10428edcc082dd3f679ee77f769d82028cd6e459370567dcf0dc593de70cc1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/ODE","family_id":"discrete_gronwall_prod_general","file_id":"mathlib/Mathlib/Analysis/ODE/DiscreteGronwall.lean","sample_id":"19249b12421b93ef9aca4c5c6690238801a0a01bd844a6075bfc5c80d4237b2d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fd406c6900b20879ac4c2cbd6762020da2098ae8a94437f8bd328ff516fb751d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9b1bbd8b0b0a1747cccd2009098a56d3ba588503f823228d885fa9ccd0f30c49","source_sha256":"2330b04bbf9f5616db3c6c30d466b364ee306b290c33249fea3b2f667c7433f7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := Adjunction.rightAdjointUniq_hom_counit (mapPullbackAdj (𝟙 X)) (id X).mapPullbackAdj\n  rw [pullbackId, Adjunction.rightAdjointUniq_hom_counit]","hard_negative":false,"metrics":{"chosen_tokens":27,"rejected_tokens":2,"token_jaccard":0.055556,"token_length_ratio":0.074074},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"5ef28815007bfd414a8fb7a9b009a5a3650e905cdf4e5575bf130ad6334448c3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Comma.Over.Pullback\npublic import Mathlib.CategoryTheory.Monoidal.Cartesian.Basic\npublic import Mathlib.CategoryTheory.Adjunction.Unique\npublic import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Defs\n\nNamespace:\nCategoryTheory.ChosenPullbacksAlong\n\nLocal context:\n/-\nCopyright (c) 2025 Sina Hazratpour. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sina Hazratpour\n-/\n/-!\n# Chosen pullbacks along a morphism\n\n## Main declarations\n\n- `ChosenPullbacksAlong` : For a morphism `f : Y ⟶ X` in `C`, the type class\n  `ChosenPullbacksAlong f` provides the data of a pullback functor `Over X ⥤ Over Y`\n  as a right adjoint to `Over.map f`.\n\n## Main results\n\n- We prove that `ChosenPullbacksAlong` has good closure properties: isos have chosen pullbacks,\n  and composition of morphisms with chosen pullbacks have chosen pullbacks.\n\n- We prove that chosen pullbacks yield usual pullbacks: `ChosenPullbacksAlong.isPullback`\n  proves that for morphisms `f` and `g` with the same codomain, the object\n  `ChosenPullbacksAlong.pullbackObj f g` together with morphisms\n  `ChosenPullbacksAlong.fst f g` and `ChosenPullbacksAlong.snd f g` form a pullback square\n  over `f` and `g`.\n\n- We prove that in cartesian monoidal categories, morphisms to the terminal tensor unit and\n  the product projections have chosen pullbacks.\n\n-/\n\n@[expose] public section\n\nuniverse v₁ v₂ u₁ u₂\n\nnamespace CategoryTheory\n\nopen Category Limits CartesianMonoidalCategory MonoidalCategory Over\n\nvariable {C : Type u₁} [Category.{v₁} C]\n\n/-- A functorial choice of pullbacks along a morphism `f : Y ⟶ X` in `C` given by a functor\n`Over X ⥤ Over Y` which is a right adjoint to the functor `Over.map f`. -/\nclass ChosenPullbacksAlong {Y X : C} (f : Y ⟶ X) where\n  /-- The pullback functor along `f`. -/\n  pullback : Over X ⥤ Over Y\n  /-- The adjunction between `Over.map f` and `pullback f`. -/\n  mapPullbackAdj (f) : Over.map f ⊣ pullback\n\nvariable (C) in\n/-- A category has chosen pullbacks if every morphism has a chosen pullback. -/\nabbrev ChosenPullbacks := Π {X Y : C} (f : Y ⟶ X), ChosenPullbacksAlong f\n\nnamespace ChosenPullbacksAlong\n\n/-- Relating the existing noncomputable `HasPullbacksAlong` typeclass to `ChosenPullbacksAlong`. -/\n@[simps, implicit_reducible]\nnoncomputable def ofHasPullbacksAlong {Y X : C} (f : Y ⟶ X) [HasPullbacksAlong f] :\n    ChosenPullbacksAlong f where\n  pullback := Over.pullback f\n  mapPullbackAdj := Over.mapPullbackAdj f\n\n/-- The identity morphism has a functorial choice of pullbacks. -/\n@[implicit_reducible]\ndef id (X : C) : ChosenPullbacksAlong (𝟙 X) where\n  pullback := 𝟭 _\n  mapPullbackAdj := (Adjunction.id).ofNatIsoLeft (Over.mapId _).symm\n\n/-- Any chosen pullback functor of the identity morphism is naturally isomorphic to the identity\nfunctor. -/\ndef pullbackId (X : C) [ChosenPullbacksAlong (𝟙 X)] :\n    pullback (𝟙 X) ≅ 𝟭 (Over X) :=\n  (mapPullbackAdj (𝟙 X)).rightAdjointUniq (id X).mapPullbackAdj\n\n@[reassoc (attr := simp)]\ntheorem unit_pullbackId_hom_app (X : C) [ChosenPullbacksAlong (𝟙 X)] (Y : Over X) :\n    (mapPullbackAdj (𝟙 X)).unit.app Y ≫ (pullbackId X).hom.app ((Over.map (𝟙 X)).obj Y) =\n      (id X).mapPullbackAdj.unit.app Y := by\n  rw [pullbackId, Adjunction.unit_rightAdjointUniq_hom_app]\n\n@[reassoc (attr := simp)]\ntheorem unit_pullbackId_hom (X : C) [ChosenPullbacksAlong (𝟙 X)] :\n    (mapPullbackAdj (𝟙 X)).unit ≫ (Over.map (𝟙 X)).whiskerLeft (pullbackId X).hom =\n      (id X).mapPullbackAdj.unit := by\n  rw [pullbackId, Adjunction.unit_rightAdjointUniq_hom]\n\n@[reassoc (attr := simp)]\n\nTarget:\ntheorem pullbackId_hom_counit (X : C) [ChosenPullbacksAlong (𝟙 X)] :\n    Functor.whiskerRight (pullbackId X).hom (Over.map (𝟙 X)) ≫ (id X).mapPullbackAdj.counit =\n      (mapPullbackAdj (𝟙 X)).counit :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/LocallyCartesianClosed","family_id":"pullbackid_hom_counit","file_id":"mathlib/Mathlib/CategoryTheory/LocallyCartesianClosed/ChosenPullbacksAlong.lean","sample_id":"9b1bbd8b0b0a1747cccd2009098a56d3ba588503f823228d885fa9ccd0f30c49"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0c64ae53f4b1abd57d7d7d427b1d8d5ae6a8abade34b2f455335c0dfb08d0184","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f05312ee1b649b954d0ad3d3cedaebdac4a9c0766690dacc1ff7776f9463beba","source_sha256":"0a178ec9526bb9830472977cb3b19da5015b9b3548b786a523adc3dcb0786526","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [subsingleton_iff_support_subset, PrimeSpectrum.basicOpen_eq_zeroLocus_compl,\n    disjoint_compl_left_iff, Set.le_iff_subset]","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":8,"token_jaccard":0.055556,"token_length_ratio":0.533333},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"5f349b6c05d09baa879acf42cc9ecb0a4f46f00d4dba0361dcd14f99d23de449","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Spectrum.Prime.Topology\npublic import Mathlib.RingTheory.Support\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Subsets of prime spectra related to modules\n\n## Main results\n\n- `LocalizedModule.subsingleton_iff_disjoint` : `M[1/f] = 0 ↔ D(f) ∩ Supp M = 0`.\n- `Module.isClosed_support` : If `M` is a finite `R`-module, then `Supp M` is closed.\n\n## TODO\n- If `M` is finitely presented, the complement of `Supp M` is quasi-compact. (stacks#051B)\n\n-/\n\npublic section\n\nvariable {R A M : Type*} [CommRing R] [AddCommGroup M] [Module R M]\n  [CommRing A] [Algebra R A] [Module A M]\n\nvariable (R M) in\nlemma IsLocalRing.closedPoint_mem_support [IsLocalRing R] [Nontrivial M] :\n    IsLocalRing.closedPoint R ∈ Module.support R M := by\n  obtain ⟨p, hp⟩ := (Module.nonempty_support_iff (R := R)).mpr ‹_›\n  exact Module.mem_support_mono le_top hp\n\n/-- `M[1/f] = 0` if and only if `D(f) ∩ Supp M = 0`. -/\n\nTarget:\nlemma LocalizedModule.subsingleton_iff_disjoint {f : R} :\n    Subsingleton (LocalizedModule.Away f M) ↔\n      Disjoint ↑(PrimeSpectrum.basicOpen f) (Module.support R M) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"37d4ff0ae14f24830f528636dabca2fbba5b391e7c706d600a69dbb19bd9f263","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Spectrum","family_id":"localizedmodule","file_id":"mathlib/Mathlib/RingTheory/Spectrum/Prime/Module.lean","sample_id":"f05312ee1b649b954d0ad3d3cedaebdac4a9c0766690dacc1ff7776f9463beba"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9427a4321d73c521b7283942cf3246fe882d0cae4ae971903987493e7ed9579b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7c3c6fabe63da232039bfcfad352f1cc6f001b2ce6fe12a8d6099283072984c1","source_sha256":"d54536321e31ab6ecb143f6ff47411119fb9708f05bec728ae60cd91a7b3224f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isUnit_iff_exists]\n  exact ⟨n, inv_mul_of_unit _ h, mul_inv_of_unit _ h⟩","hard_negative":false,"metrics":{"chosen_tokens":17,"rejected_tokens":3,"token_jaccard":0.133333,"token_length_ratio":0.176471},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"5f4f14ece8555daf0ac0a747863fb91a74a3c7bc21a64b2c7748898e4495f8e1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Associated\npublic import Mathlib.Data.ZMod.Basic\npublic import Mathlib.RingTheory.Coprime.Lemmas\n\nNamespace:\nZMod\n\nLocal context:\n/-\nCopyright (c) 2023 Moritz Firsching. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Moritz Firsching, Ashvni Narayanan, Michael Stoll\n-/\n/-!\n# Lemmas about units in `ZMod`.\n-/\n\n@[expose] public section\n\nassert_not_exists TwoSidedIdeal\n\nnamespace ZMod\n\nvariable {n m : ℕ}\n/-- `unitsMap` is a group homomorphism that maps units of `ZMod m` to units of `ZMod n` when `n`\ndivides `m`. -/\ndef unitsMap (hm : n ∣ m) : (ZMod m)ˣ →* (ZMod n)ˣ := Units.map (castHom hm (ZMod n))\n\nlemma unitsMap_def (hm : n ∣ m) : unitsMap hm = Units.map (castHom hm (ZMod n)) := rfl\n\nlemma unitsMap_comp {d : ℕ} (hm : n ∣ m) (hd : m ∣ d) :\n    (unitsMap hm).comp (unitsMap hd) = unitsMap (dvd_trans hm hd) := by\n  simp only [unitsMap_def]\n  rw [← Units.map_comp]\n  exact congr_arg Units.map <| congr_arg RingHom.toMonoidHom <| castHom_comp hm hd\n\n@[simp]\nlemma unitsMap_self (n : ℕ) : unitsMap (dvd_refl n) = MonoidHom.id _ := by\n  simp [unitsMap, castHom_self]\n\n/-- `unitsMap_val` shows that coercing from `(ZMod m)ˣ` to `ZMod n` gives the same result\nwhen going via `(ZMod n)ˣ` and `ZMod m`. -/\nlemma unitsMap_val (h : n ∣ m) (a : (ZMod m)ˣ) :\n    ↑(unitsMap h a) = ((a : ZMod m).cast : ZMod n) := rfl\n\nlemma isUnit_cast_of_dvd (hm : n ∣ m) (a : Units (ZMod m)) : IsUnit (cast (a : ZMod m) : ZMod n) :=\n  Units.isUnit (unitsMap hm a)\ntheorem unitsMap_surjective [hm : NeZero m] (h : n ∣ m) :\n    Function.Surjective (unitsMap h) := by\n  suffices ∀ x : ℕ, x.Coprime n → ∃ k : ℕ, (x + k * n).Coprime m by\n    intro x\n    have ⟨k, hk⟩ := this x.val.val (val_coe_unit_coprime x)\n    refine ⟨unitOfCoprime _ hk, Units.ext ?_⟩\n    have : NeZero n := ⟨fun hn ↦ hm.out (eq_zero_of_zero_dvd (hn ▸ h))⟩\n    simp [unitsMap_def, -castHom_apply]\n  intro x hx\n  let ps : Finset ℕ := {p ∈ m.primeFactors | ¬p ∣ x}\n  use ps.prod id\n  apply Nat.coprime_of_dvd\n  intro p pp hp hpn\n  by_cases hpx : p ∣ x\n  · have h := Nat.dvd_sub hp hpx\n    rw [add_comm, Nat.add_sub_cancel] at h\n    rcases pp.dvd_mul.mp h with h | h\n    · have ⟨q, hq, hq'⟩ := (pp.prime.dvd_finsetProd_iff id).mp h\n      rw [Finset.mem_filter, Nat.mem_primeFactors,\n        ← (Nat.prime_dvd_prime_iff_eq pp hq.1.1).mp hq'] at hq\n      exact hq.2 hpx\n    · exact Nat.Prime.not_coprime_iff_dvd.mpr ⟨p, pp, hpx, h⟩ hx\n  · have pps : p ∈ ps := Finset.mem_filter.mpr ⟨Nat.mem_primeFactors.mpr ⟨pp, hpn, hm.out⟩, hpx⟩\n    have h := Nat.dvd_sub hp ((Finset.dvd_prod_of_mem id pps).mul_right n)\n    rw [Nat.add_sub_cancel] at h\n    contradiction\n\n-- This needs `Nat.primeFactors`, so cannot go into `Mathlib/Data/ZMod/Basic.lean`.\nopen Nat in\nlemma not_isUnit_of_mem_primeFactors {n p : ℕ} (h : p ∈ n.primeFactors) :\n    ¬ IsUnit (p : ZMod n) := by\n  rw [isUnit_iff_coprime]\n  exact (Prime.dvd_iff_not_coprime <| prime_of_mem_primeFactors h).mp <| dvd_of_mem_primeFactors h\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Any element of `ZMod N` has the form `u * d` where `u` is a unit and `d` is a divisor of `N`. -/\nlemma eq_unit_mul_divisor {N : ℕ} (a : ZMod N) :\n    ∃ d : ℕ, d ∣ N ∧ ∃ (u : ZMod N), IsUnit u ∧ a = u * d := by\n  rcases eq_or_ne N 0 with rfl | hN\n  -- Silly special case : N = 0. Of no mathematical interest, but true, so let's prove it.\n  · change ℤ at a\n    rcases eq_or_ne a 0 with rfl | ha\n    · refine ⟨0, dvd_zero _, 1, isUnit_one, by rw [Nat.cast_zero, mul_zero]⟩\n    refine ⟨a.natAbs, dvd_zero _, Int.sign a, ?_, (Int.sign_mul_natAbs a).symm⟩\n    rcases lt_or_gt_of_ne ha with h | h\n    · simp only [Int.sign_eq_neg_one_of_neg h, IsUnit.neg_iff, isUnit_one]\n    · simp only [Int.sign_eq_one_of_pos h, isUnit_one]\n  -- now the interesting case\n  have : NeZero N := ⟨hN⟩\n  -- Define `d` as the GCD of a lift of `a` and `N`.\n  let d := a.val.gcd N\n  have hd : d ≠ 0 := Nat.gcd_ne_zero_right hN\n  obtain ⟨a₀, (ha₀ : _ = d * _)⟩ := a.val.gcd_dvd_left N\n  obtain ⟨N₀, (hN₀ : _ = d * _)⟩ := a.val.gcd_dvd_right N\n  refine ⟨d, ⟨N₀, hN₀⟩, ?_⟩\n  -- Show `a` is a unit mod `N / d`.\n  have hu₀ : IsUnit (a₀ : ZMod N₀) := by\n    refine (isUnit_iff_coprime _ _).mpr (Nat.isCoprime_iff_coprime.mp ?_)\n    obtain ⟨p, q, hpq⟩ : ∃ (p q : ℤ), d = a.val * p + N * q := ⟨_, _, Nat.gcd_eq_gcd_ab _ _⟩\n    rw [ha₀, hN₀, Nat.cast_mul, Nat.cast_mul, mul_assoc, mul_assoc, ← mul_add, eq_comm,\n      mul_comm _ p, mul_comm _ q] at hpq\n    exact ⟨p, q, Int.eq_one_of_mul_eq_self_right (Nat.cast_ne_zero.mpr hd) hpq⟩\n  -- Lift it arbitrarily to a unit mod `N`.\n  obtain ⟨u, hu⟩ := (unitsMap_surjective (⟨d, mul_comm d N₀ ▸ hN₀⟩ : N₀ ∣ N)) hu₀.unit\n  rw [unitsMap_def, ← Units.val_inj, Units.coe_map, IsUnit.unit_spec, MonoidHom.coe_coe] at hu\n  refine ⟨u.val, u.isUnit, ?_⟩\n  rw [← natCast_zmod_val a, ← natCast_zmod_val u.1, ha₀, ← Nat.cast_mul,\n    natCast_eq_natCast_iff, mul_comm _ d, Nat.ModEq]\n  simp only [hN₀, Nat.mul_mod_mul_left, Nat.mul_right_inj hd]\n  rw [← Nat.ModEq, ← natCast_eq_natCast_iff, ← hu, natCast_val, castHom_apply]\n\ntheorem coe_int_mul_inv_eq_one {n : ℕ} {x : ℤ} (h : IsCoprime x n) :\n    (x : ZMod n) * (x : ZMod n)⁻¹ = 1 := by\n  by_cases hn : n = 0\n  · simp only [hn, Nat.cast_zero, isCoprime_zero_right] at h\n    rcases Int.isUnit_eq_one_or h with h | h <;> simp [h]\n  haveI : NeZero n := ⟨hn⟩\n  rw [← natCast_zmod_val x]\n  apply coe_mul_inv_eq_one\n  rwa [Int.isCoprime_iff_gcd_eq_one, ← Int.gcd_emod, ← val_intCast] at h\n\ntheorem coe_int_inv_mul_eq_one {n : ℕ} {x : ℤ} (h : IsCoprime x n) :\n    (x : ZMod n)⁻¹ * (x : ZMod n) = 1 := by\n  rw [mul_comm, coe_int_mul_inv_eq_one h]\n\nlemma coe_int_mul_val_inv {n : ℕ} [NeZero n] {m : ℤ} (h : IsCoprime m n) :\n    (m * (m⁻¹ : ZMod n).val : ZMod n) = 1 := by\n  rw [natCast_zmod_val, coe_int_mul_inv_eq_one h]\n\nlemma coe_int_val_inv_mul {n : ℕ} [NeZero n] {m : ℤ} (h : IsCoprime m n) :\n    ((m⁻¹ : ZMod n).val : ZMod n) * m = 1 := by\n  rw [mul_comm, coe_int_mul_val_inv h]\n\n/-- The unit of `ZMod m` associated with an integer prime to `n`. -/\ndef unitOfIsCoprime {m : ℕ} (n : ℤ)\n    (h : IsCoprime n (m : ℤ)) : (ZMod m)ˣ where\n  val := n\n  inv := n⁻¹\n  val_inv := coe_int_mul_inv_eq_one h\n  inv_val := coe_int_inv_mul_eq_one h\n\n@[simp]\ntheorem coe_unitOfIsCoprime {m : ℕ} (n : ℤ) (h : IsCoprime n ↑m) :\n    (unitOfIsCoprime n h : ZMod m) = n := rfl\n\nTarget:\ntheorem isUnit_inv {m : ℕ} {n : ℤ} (h : IsUnit (n : ZMod m)) :\n    IsUnit (n : ZMod m)⁻¹ :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/ZMod","family_id":"isunit_inv","file_id":"mathlib/Mathlib/Data/ZMod/Units.lean","sample_id":"7c3c6fabe63da232039bfcfad352f1cc6f001b2ce6fe12a8d6099283072984c1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3b83d6405db8d7b67a22120881e91f0e3ce367fb9b3ca08e1bae760a47811d62","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"02dd01e0fea4843490de53f3241bde2b1e0aa756a868ba3c30902810d57955ee","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ae60c1f3772d8f03a4244e5c6a05fff7f8704657d4e3b6abd61ebd01d5c23d05","source_sha256":"ced30280014822262039ce197c9e4ae821d789623b770f4b0e6844d4852109a3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases Nat.eq_zero_or_pos n with\n  | inl hn =>\n    rw [if_pos hn, hn, dp_zero]\n  | inr hn =>\n    rw [if_neg (ne_of_gt hn), ← zero_smul R (0 : M), dp_smul]\n    rw [zero_pow (Nat.pos_iff_ne_zero.mp hn), zero_smul]","hard_negative":false,"metrics":{"chosen_tokens":57,"rejected_tokens":62,"token_jaccard":0.888889,"token_length_ratio":1.087719},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"5f7f5e7698bac394bffbae83e9ad9f779c3209c99213a89ea051e4792f65683f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.Eval\npublic import Mathlib.Algebra.RingQuot\npublic import Mathlib.RingTheory.DividedPowers.Basic\n\nNamespace:\nDividedPowerAlgebra\n\nLocal context:\n/-\nCopyright (c) 2026 Antoine Chambert-Loir & María Inés de Frutos—Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir, María Inés de Frutos—Fernández\n-/\n/-!\n# The universal divided power algebra\n\nLet `R` be a (commutative) semiring and `M` be an `R`-module. In this file we define `Γ_R(M)`,\nthe universal divided power algebra of `M`, as the ring quotient of the polynomial ring\nin the variables `ℕ × M` by the relation `DividedPowerAlgebra.Rel`.\n\n`DividedPowerAlgebra R M` satisfies a weak universal property for morphisms to rings with\ndivided powers (`DividedPowerAlgebra.lift`).\n\n## Main definitions\n\n* `DividedPowerAlgebra.Rel`: the type coding the basic relations that will give rise to the\n  divided power algebra.\n\n* `DividedPowerAlgebra R M`: the universal divided power algebra of the `R`-module `M`,\n  defined as `RingQuot` of `DividedPowerAlgebra.Rel R M`.\n\n* `DividedPowerAlgebra.dp R n m`: for `n : ℕ` and `m : M`, this is the equivalence class of\n  `MvPolynomial.X (⟨n, m⟩)` in `DividedPowerAlgebra R M`.\n\n  When that algebra is endowed with its canonical divided power structure (to be defined),\n  the image of `MvPolynomial.X (n, m)`, for any `n : ℕ` and `m : M`, is equal to\n  the `n`th divided power of the image of `m`.\n\n  The API will be setup so that it is never (never say never…) necessary to lift to `MvPolynomial`.\n\n* `DividedPowerAlgebra.lift`: the weak universal property of `DividedPowerAlgebra R M`.\n\n* `DividedPowerAlgebra.map`: the functoriality map between divided power algebras\n  associated with a linear map of the underlying modules.\n  Given an `R`-algebra `S`, an `S`-module `N` and an `R`-linear map `f : M →ₗ[R] N`,\n  this is the map `DividedPowerAlgebra R M →ₐ[R] DividedPowerAlgebra S N`\n  sending `dp R n m` to `dp S n (f m)`.\n\n## References\n\n* [P. Berthelot (1974), *Cohomologie cristalline des schémas de\n  caractéristique $p$ > 0*][Berthelot-1974]\n\n* [P. Berthelot and A. Ogus (1978), *Notes on crystalline\n  cohomology*][BerthelotOgus-1978]\n\n* [N. Roby (1963), *Lois polynomes et lois formelles en théorie des\n  modules*][Roby-1963]\n\n* [N. Roby (1965), *Les algèbres à puissances dividées*][Roby-1965]\n\n## TODO\n\n* Show in upcoming files that `DividedPowerAlgebra R M` has divided powers.\n\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Finset Ideal MvPolynomial RingQuot\n\nvariable (R M : Type*) [CommSemiring R] [AddCommMonoid M] [Module R M]\n\nnamespace DividedPowerAlgebra\n\n/-- The type coding the basic relations that will give rise to the divided power algebra.\n  The class of `MvPolynomial.X (n, a)` will be equal to `dpow n a`, for `a ∈ M`. -/\ninductive Rel : MvPolynomial (ℕ × M) R → MvPolynomial (ℕ × M) R → Prop\n  | rfl_zero : Rel 0 0 -- Needed for technical reasons.\n  | zero {a : M} : Rel (X (0, a)) 1\n  | smul {r : R} {n : ℕ} {a : M} : Rel (X (n, r • a)) (r ^ n • X (n, a))\n  | mul {m n : ℕ} {a : M} : Rel (X (m, a) * X (n, a)) (Nat.choose (m + n) m • X (m + n, a))\n  | add {n : ℕ} {a b : M} :\n    Rel (X (n, a + b)) ((Finset.antidiagonal n).sum fun k ↦ X (k.1, a) * X (k.2, b))\n\n/-- The ideal of `MvPolynomial (ℕ × M) R` generated by `Rel`. -/\ndef RelI : Ideal (MvPolynomial (ℕ × M) R) := ofRel (DividedPowerAlgebra.Rel R M)\n\nend DividedPowerAlgebra\n\n/-- The divided power algebra of a module M is defined as the ring quotient of the polynomial ring\n  in the variables `ℕ × M` by the ring relation defined by `DividedPowerAlgebra.Rel`.\n  We will later show that that `DividedPowerAlgebra R M` has divided powers.\n  It satisfies a weak universal property for morphisms to rings with divided powers. -/\nabbrev DividedPowerAlgebra := RingQuot (DividedPowerAlgebra.Rel R M)\n\nnamespace DividedPowerAlgebra\n\nopen MvPolynomial\n\nvariable {R M}\n\nlemma mkAlgHom_surjective : Function.Surjective (mkAlgHom R (Rel R M)) :=\n  RingQuot.mkAlgHom_surjective _ _\n\nlemma mkAlgHom_C (a : R) :\n    mkAlgHom R (Rel R M) (C a) = algebraMap R (DividedPowerAlgebra R M) a := by\n  rw [← MvPolynomial.algebraMap_eq, AlgHom.commutes]\n\nlemma mkRingHom_C (a : R) :\n    mkRingHom (Rel R M) (C a) = algebraMap R (DividedPowerAlgebra R M) a := by\n  rw [← mkAlgHom_C, mkAlgHom, AlgHom.coe_mk]\n\nvariable (R) in\n/-- `dp R n m` is the equivalence class of `X (⟨n, m⟩)` in `DividedPowerAlgebra R M`. -/\ndef dp (n : ℕ) (m : M) : DividedPowerAlgebra R M := mkAlgHom R (Rel R M) (X ⟨n, m⟩)\n\ntheorem dp_def (n : ℕ) (m : M) :\n  dp R n m = mkAlgHom R (Rel R M) (X ⟨n, m⟩) := rfl\n\nprotected theorem induction_on' {P : DividedPowerAlgebra R M → Prop} (f : DividedPowerAlgebra R M)\n    (h_C : ∀ a, P (mkAlgHom R (Rel R M) (C a))) (h_add : ∀ f g, P f → P g → P (f + g))\n    (h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)) : P f := by\n  obtain ⟨F, hf⟩ := RingQuot.mkRingHom_surjective (DividedPowerAlgebra.Rel R M) f\n  rw [← hf]\n  induction F using MvPolynomial.induction_on generalizing f with\n  | C a =>\n      convert! h_C a using 1\n      rw [mkAlgHom, AlgHom.coe_mk]\n  | add g1 g2 hg1 hg2 =>\n      rw [map_add]\n      exact h_add _ _ (hg1 ((mkRingHom (Rel R M)) g1) rfl) (hg2 ((mkRingHom (Rel R M)) g2) rfl)\n  | mul_X g nm h =>\n      have h' : (mkRingHom (Rel R M)) (X nm) = dp R nm.1 nm.2 := by\n        simp only [dp_def, Prod.mk.eta, mkAlgHom, AlgHom.coe_mk]\n      rw [_root_.map_mul, h']\n      exact h_dp _ _ _ (h (mkRingHom (Rel R M) g) rfl)\n\n@[elab_as_elim]\nprotected theorem induction_on {P : DividedPowerAlgebra R M → Prop} (f : DividedPowerAlgebra R M)\n    (C : ∀ a, P (algebraMap R _ a)) (add : ∀ f g, P f → P g → P (f + g))\n    (dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)) : P f :=\n  DividedPowerAlgebra.induction_on' f (fun a ↦ by rw [mkAlgHom_C]; exact C a) add dp\n\ntheorem dp_eq_mkRingHom (n : ℕ) (m : M) :\n    dp R n m = mkRingHom (Rel R M) (X (⟨n, m⟩)) := by\n  simp [dp, mkRingHom, mkAlgHom]\n\ntheorem dp_zero {m : M} : dp R 0 m = 1 := by\n  rw [dp_def, ← map_one (mkAlgHom R (Rel R M))]\n  exact RingQuot.mkAlgHom_rel R Rel.zero\n\ntheorem dp_smul {r : R} {n : ℕ} {m : M} : dp R n (r • m) = r ^ n • dp R n m := by\n  rw [dp_def, dp_def, ← map_smul]\n  exact mkAlgHom_rel R Rel.smul\n\nTarget:\ntheorem dp_null {n : ℕ} : dp R n (0 : M) = if n = 0 then 1 else 0 :=\n\nProof body:\n","rejected":"by\n  cases Nat.eq_zero_or_pos n with\n  | inl hn =>\n    rw [if_pos hn, hn, dp_zero]\n  | inr hn =>\n    rw [if_neg (ne_of_gt hn), ← zero_smul R (0 : M), dp_smul]\n    rw [zero_pow (Nat.pos_iff_ne_zero.mp hn), zero_smul]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/DividedPowerAlgebra","family_id":"dp_null","file_id":"mathlib/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean","sample_id":"ae60c1f3772d8f03a4244e5c6a05fff7f8704657d4e3b6abd61ebd01d5c23d05"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b654cbf08d5ade13b5e6184e32b41bf84d2fa139990565902f32727bb1db98ee","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"de40c21181d595bbe1edc7f9ec432ad2cd6e6b77e6caea24a65a445ca6570762","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0eb67f824175dac229b7646063b2648b75a17b30e9821e1a51528362c79353d3","source_sha256":"0722d600b193d93cb9e55becbab169fa607364d0057f53064083850c9da2e6cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun ⟨g, hg⟩ ↦ ?_, fun h ↦ (isCuspForm_iff f).mpr (isZeroAt_of_coeffZero_eq_zero f h)⟩\n  rw [← hg, qExpansion_coeff_zero one_pos\n    (ModularFormClass.analyticAt_cuspFunction_zero _ one_pos one_mem_strictPeriods_SL)\n    (periodic_comp_ofComplex _ one_mem_strictPeriods_SL)]\n  exact (CuspFormClass.zero_at_infty g).valueAtInfty_eq_zero","hard_negative":true,"metrics":{"chosen_tokens":58,"rejected_tokens":3,"token_jaccard":0.058824,"token_length_ratio":0.051724},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"5fe429e9e2acd7222f74c4432aa1608622c4b2dd167763db7f56386493186b9e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.ModularForms.QExpansion\npublic import Mathlib.NumberTheory.ModularForms.LevelOne.Basic\npublic import Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion\n\nNamespace:\nModularForm\n\nLocal context:\n/-\nCopyright (c) 2026 Chris Birkbeck. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Birkbeck\n-/\n/-!\n# Cusp form submodule and IsCuspForm predicate\n\nThis file defines the inclusion of cusp forms into modular forms as a linear map, the cusp form\nsubmodule of modular forms, and the `IsCuspForm` predicate. It also provides a direct constructor\n`ModularForm.toCuspForm` for building cusp forms from modular forms with vanishing constant\nq-expansion coefficient (for `𝒮ℒ`).\n\n## Main definitions\n\n* `CuspForm.toModularFormₗ`: the inclusion `CuspForm Γ k →ₗ[ℂ] ModularForm Γ k`.\n* `ModularForm.cuspFormSubmodule`: the submodule of `ModularForm Γ k` consisting of cusp forms.\n* `ModularForm.IsCuspForm`: predicate that a modular form lies in the cusp form submodule.\n* `ModularForm.toCuspForm`: builds a `CuspForm 𝒮ℒ k` from a `ModularForm` whose q-expansion\n  has vanishing constant term.\n\n## Main results\n\n* `CuspForm.toModularFormₗ_injective`: the inclusion is injective.\n* `CuspForm.equivCuspFormSubmodule`: `CuspForm Γ k ≃ₗ[ℂ] cuspFormSubmodule Γ k`.\n* `ModularForm.isCuspForm_iff_coeffZero_eq_zero`: for `𝒮ℒ`, `IsCuspForm` is equivalent to the\n  q-expansion having vanishing constant term.\n-/\n\n@[expose] public noncomputable section\n\nopen UpperHalfPlane ModularForm Complex SlashInvariantForm SlashInvariantFormClass\n  ModularFormClass MatrixGroups OnePoint Filter Topology\n\nvariable {Γ : Subgroup (GL (Fin 2) ℝ)} {k : ℤ}\n\nnamespace CuspForm\n\n/-- The inclusion of cusp forms into modular forms, as a ℂ-linear map. -/\ndef toModularFormₗ [Γ.HasDetOne] : CuspForm Γ k →ₗ[ℂ] ModularForm Γ k where\n  toFun := ModularFormClass.modularForm\n  map_add' _ _ := rfl\n  map_smul' _ _ := rfl\n\n@[simp]\nlemma toModularFormₗ_apply [Γ.HasDetOne] (f : CuspForm Γ k) (z : ℍ) :\n    (toModularFormₗ f) z = f z := rfl\n\nlemma toModularFormₗ_eq_coe [Γ.HasDetOne] (f : CuspForm Γ k) :\n    toModularFormₗ f = (f : ModularForm Γ k) := rfl\n\nlemma toModularFormₗ_injective [Γ.HasDetOne] :\n    Function.Injective (toModularFormₗ : CuspForm Γ k → ModularForm Γ k) :=\n  fun _ _ h ↦ DFunLike.ext _ _ fun z ↦ DFunLike.congr_fun h z\n\nend CuspForm\n\nnamespace ModularForm\n\n/-- The submodule of `ModularForm Γ k` consisting of cusp forms, defined as the range of\nthe inclusion `CuspForm.toModularFormₗ`. -/\ndef cuspFormSubmodule (Γ : Subgroup (GL (Fin 2) ℝ)) (k : ℤ) [Γ.HasDetOne] :\n    Submodule ℂ (ModularForm Γ k) :=\n  LinearMap.range CuspForm.toModularFormₗ\n\n/-- A modular form is a cusp form if it lies in the cusp form submodule. -/\ndef IsCuspForm [Γ.HasDetOne] (f : ModularForm Γ k) : Prop :=\n  f ∈ cuspFormSubmodule Γ k\n\n@[simp]\nlemma mem_cuspFormSubmodule_iff [Γ.HasDetOne] {f : ModularForm Γ k} :\n    f ∈ cuspFormSubmodule Γ k ↔ IsCuspForm f := Iff.rfl\n\n/-- The cusp form submodule is linearly equivalent to the type of cusp forms. -/\ndef CuspForm.equivCuspFormSubmodule (Γ : Subgroup (GL (Fin 2) ℝ)) (k : ℤ) [Γ.HasDetOne] :\n    CuspForm Γ k ≃ₗ[ℂ] cuspFormSubmodule Γ k :=\n  LinearEquiv.ofInjective CuspForm.toModularFormₗ CuspForm.toModularFormₗ_injective\n\n/-- The underlying modular form (via `toModularFormₗ`) of a `CuspForm` is itself a cusp form. -/\nlemma CuspForm.isCuspForm_toModularFormₗ {Γ : Subgroup (GL (Fin 2) ℝ)} [Γ.HasDetOne]\n    (f : CuspForm Γ k) : ModularForm.IsCuspForm f.toModularFormₗ := by\n  simp [← mem_cuspFormSubmodule_iff, ModularForm.cuspFormSubmodule]\n\n/-- A modular form is a cusp form if and only if it vanishes at every cusp. This is the\ngeneral characterization valid for any subgroup. -/\nlemma isCuspForm_iff [Γ.HasDetOne] (f : ModularForm Γ k) :\n    IsCuspForm f ↔ ∀ {c}, IsCusp c Γ → c.IsZeroAt f k :=\n  ⟨fun ⟨g, hg⟩ _ ↦ hg ▸ g.zero_at_cusps', fun h ↦ ⟨⟨f, f.holo', h⟩, rfl⟩⟩\n\n/-- A modular form with `valueAtInfty f = 0` is zero at infinity. -/\nlemma isZeroAtImInfty_of_valueAtInfty_eq_zero {F : Type*} [FunLike F ℍ ℂ]\n    [DiscreteTopology Γ] [Γ.HasDetPlusMinusOne] [Fact (IsCusp ∞ Γ)] [ModularFormClass F Γ k]\n    (f : F) (h : valueAtInfty f = 0) : IsZeroAtImInfty f := by\n  have hh : 0 < Γ.strictWidthInfty := Γ.strictWidthInfty_pos_iff.mpr Fact.out\n  have hΓ : Γ.strictWidthInfty ∈ Γ.strictPeriods := Γ.strictWidthInfty_mem_strictPeriods\n  have hanal := ModularFormClass.analyticAt_cuspFunction_zero f hh hΓ\n  have hper := periodic_comp_ofComplex f hΓ\n  simp_rw [IsZeroAtImInfty, ZeroAtFilter, ← h, ← cuspFunction_apply_zero hh hanal hper]\n  exact (hanal.continuousAt.tendsto.comp (qParam_tendsto_atImInfty hh)).congr\n    (fun τ ↦ SlashInvariantFormClass.eq_cuspFunction f τ hΓ hh.ne')\n\nsection SL2Z\n\nvariable {k : ℤ}\n\n/-- An `𝒮ℒ` modular form with vanishing q-expansion constant term vanishes at every cusp. -/\nlemma isZeroAt_of_coeffZero_eq_zero (f : ModularForm 𝒮ℒ k)\n    (h : (qExpansion 1 f).coeff 0 = 0) {c : OnePoint ℝ} (hc : IsCusp c 𝒮ℒ) :\n    c.IsZeroAt f k := by\n  rw [Subgroup.IsArithmetic.isCusp_iff_isCusp_SL2Z] at hc\n  rw [isZeroAt_iff_forall_SL2Z hc]\n  intro γ _\n  rw [show (⇑f ∣[k] γ) = ⇑f from f.slash_action_eq' _ ⟨γ, rfl⟩]\n  exact isZeroAtImInfty_of_valueAtInfty_eq_zero f <| by\n    rwa [← qExpansion_coeff_zero one_pos\n      (ModularFormClass.analyticAt_cuspFunction_zero f one_pos one_mem_strictPeriods_SL)\n      (periodic_comp_ofComplex f one_mem_strictPeriods_SL)]\n\n/-- Build a `CuspForm 𝒮ℒ k` from a `ModularForm 𝒮ℒ k` whose q-expansion has vanishing\nconstant term. The resulting cusp form has the same underlying function. -/\ndef toCuspForm (f : ModularForm 𝒮ℒ k) (h : (qExpansion 1 f).coeff 0 = 0) : CuspForm 𝒮ℒ k :=\n  { f with zero_at_cusps' := isZeroAt_of_coeffZero_eq_zero f h }\n\n@[simp]\nlemma toCuspForm_apply (f : ModularForm 𝒮ℒ k) (h : (qExpansion 1 f).coeff 0 = 0)\n    (z : ℍ) : (toCuspForm f h) z = f z := rfl\n\n/-- For `𝒮ℒ` modular forms, `IsCuspForm` is equivalent to the q-expansion having vanishing\nconstant term. -/\n\nTarget:\nlemma isCuspForm_iff_coeffZero_eq_zero (f : ModularForm 𝒮ℒ k) :\n    IsCuspForm f ↔ (qExpansion 1 f).coeff 0 = 0 :=\n\nProof body:\n","rejected":"by\n  exact isCuspForm_iff_coeffZero_eq_zero","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"93960059c6b35838146de0a4e3d0d098404924f5c50df6b54a77f288340c28b7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/ModularForms","family_id":"iscuspform_iff_coeffzero_eq_zero","file_id":"mathlib/Mathlib/NumberTheory/ModularForms/CuspFormSubmodule.lean","sample_id":"0eb67f824175dac229b7646063b2648b75a17b30e9821e1a51528362c79353d3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"17f240f800994d7074febbde9e3f6dc2b5012103ed8b58582f7bf051f16e73b0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c5a217f06acc8ac4d7b7d814d63ee498809bc279335f8650808ef0f452a470af","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dd825554f724a499bc7ed68c1db6d9b3cab6e04c69fa70bc2acb5b885250233e","source_sha256":"85d2f87939059f796ffb510eeb3d6af5b97a8cdc2c5223d679b28800dce394e2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro b₁ b₂ h\n  simpa only [and_true, eq_self_iff_true, mk_inj] using h","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":25,"token_jaccard":0.882353,"token_length_ratio":1.388889},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"604faa073f61840109fecb5a03b4046738a28b6a8f2702eb202cee2a295c533d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Lean.PrettyPrinter.Delaborator.Builtins\npublic import Mathlib.Logic.Function.Defs\npublic import Mathlib.Logic.Function.Iterate\npublic import Mathlib.Tactic.Inhabit\npublic import Batteries.Tactic.Trans\nimport Mathlib.Tactic.Attr.Register\n\nNamespace:\nProd\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Extra facts about `Prod`\n\nThis file proves various simple lemmas about `Prod`.\nIt also defines better delaborators for product projections.\n-/\n\n@[expose] public section\n\nvariable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}\n\nnamespace Prod\n\nlemma swap_eq_iff_eq_swap {x : α × β} {y : β × α} : x.swap = y ↔ x = y.swap := by grind\n\ndef mk.injArrow {x₁ : α} {y₁ : β} {x₂ : α} {y₂ : β} :\n    (x₁, y₁) = (x₂, y₂) → ∀ ⦃P : Sort*⦄, (x₁ = x₂ → y₁ = y₂ → P) → P := by\n  intros h P w\n  cases h\n  exact w rfl rfl\n\n@[simp]\ntheorem mk.eta : ∀ {p : α × β}, (p.1, p.2) = p\n  | (_, _) => rfl\n\ntheorem forall' {p : α → β → Prop} : (∀ x : α × β, p x.1 x.2) ↔ ∀ a b, p a b :=\n  Prod.forall\n\ntheorem exists' {p : α → β → Prop} : (∃ x : α × β, p x.1 x.2) ↔ ∃ a b, p a b :=\n  Prod.exists\n\n@[simp]\ntheorem snd_comp_mk (x : α) : Prod.snd ∘ (Prod.mk x : β → α × β) = id :=\n  rfl\n\n@[simp]\ntheorem fst_comp_mk (x : α) : Prod.fst ∘ (Prod.mk x : β → α × β) = Function.const β x :=\n  rfl\n\nattribute [mfld_simps] map_apply\n\n-- This was previously a `simp` lemma, but no longer is on the basis that it destructures the pair.\n--  See `map_apply`, `map_fst`, and `map_snd` for slightly weaker lemmas in the `simp` set.\ntheorem map_apply' (f : α → γ) (g : β → δ) (p : α × β) : map f g p = (f p.1, g p.2) :=\n  rfl\n\ntheorem map_fst' (f : α → γ) (g : β → δ) : Prod.fst ∘ map f g = f ∘ Prod.fst :=\n  funext <| map_fst f g\n\ntheorem map_snd' (f : α → γ) (g : β → δ) : Prod.snd ∘ map f g = g ∘ Prod.snd :=\n  funext <| map_snd f g\n\ntheorem mk_inj {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) = (a₂, b₂) ↔ a₁ = a₂ ∧ b₁ = b₂ := by simp\n\ntheorem mk_right_injective {α β : Type*} (a : α) : (mk a : β → α × β).Injective := by\n  intro b₁ b₂ h\n  simpa only [true_and, Prod.mk_inj, eq_self_iff_true] using h\n\nTarget:\ntheorem mk_left_injective {α β : Type*} (b : β) : (fun a ↦ mk a b : α → α × β).Injective :=\n\nProof body:\n","rejected":"```lean\nby\n  intro b₁ b₂ h\n  simpa only [and_true, eq_self_iff_true, mk_inj] using h\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Prod","family_id":"mk_left_injective","file_id":"mathlib/Mathlib/Data/Prod/Basic.lean","sample_id":"dd825554f724a499bc7ed68c1db6d9b3cab6e04c69fa70bc2acb5b885250233e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d4554f84e387bbee7a732a6c3f4886d4878a68fc65f7eb3a496b2c61095b621d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"95cc52a04da20a5f5c49fe48e2d7c757cb686f4efaddac32474d2f88abd5ceb5","source_sha256":"55f23f9454fd78a1ab9f5d31dee9f06c983f6e922e17ae4ec8a9c45046a436fc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [-objD, -obj_X]","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":2,"token_jaccard":0.111111,"token_length_ratio":0.222222},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"608812857d9c8de1bda119fba6700eebdc92519d6b923357e61649b0da01a71e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.HomologicalComplex\npublic import Mathlib.AlgebraicTopology.SimplicialObject.Basic\npublic import Mathlib.CategoryTheory.Abelian.Basic\n\nNamespace:\nAlgebraicTopology\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n## Moore complex\n\nWe construct the normalized Moore complex, as a functor\n`SimplicialObject C ⥤ ChainComplex C ℕ`,\nfor any abelian category `C`.\n\nThe `n`-th object is intersection of\nthe kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.\n\nThe differentials are induced from `X.δ 0`,\nwhich maps each of these intersections of kernels to the next.\n\nThis functor is one direction of the Dold-Kan equivalence, which we're still working towards.\n\n### References\n\n* https://stacks.math.columbia.edu/tag/0194\n* https://ncatlab.org/nlab/show/Moore+complex\n-/\n\n@[expose] public section\n\n\nuniverse v u\n\nnoncomputable section\n\nopen CategoryTheory CategoryTheory.Limits\n\nopen Opposite\n\nopen scoped Simplicial\n\nnamespace AlgebraicTopology\n\nvariable {C : Type*} [Category* C] [Abelian C]\n\nattribute [local instance] Abelian.hasPullbacks\n\n/-! The definitions in this namespace are all auxiliary definitions for `NormalizedMooreComplex`\nand should usually only be accessed via that. -/\n\n\nnamespace NormalizedMooreComplex\n\nopen CategoryTheory.Subobject\n\nvariable (X : SimplicialObject C)\n\n/-- The normalized Moore complex in degree `n`, as a subobject of `X n`.\n-/\ndef objX : ∀ n : ℕ, Subobject (X.obj (op ⦋n⦌))\n  | 0 => ⊤\n  | n + 1 => Finset.univ.inf fun k : Fin (n + 1) => kernelSubobject (X.δ k.succ)\n\n@[simp] theorem objX_zero : objX X 0 = ⊤ :=\n  rfl\n\n@[simp] theorem objX_add_one (n) :\n    objX X (n + 1) = Finset.univ.inf fun k : Fin (n + 1) => kernelSubobject (X.δ k.succ) :=\n  rfl\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The differentials in the normalized Moore complex.\n-/\n@[simp]\ndef objD : ∀ n : ℕ, (objX X (n + 1) : C) ⟶ (objX X n : C)\n  | 0 => Subobject.arrow _ ≫ X.δ (0 : Fin 2) ≫ inv (⊤ : Subobject _).arrow\n  | n + 1 => by\n    -- The differential is `Subobject.arrow _ ≫ X.δ (0 : Fin (n+3))`,\n    -- factored through the intersection of the kernels.\n    refine factorThru _ (arrow _ ≫ X.δ (0 : Fin (n + 3))) ?_\n    -- We now need to show that it factors!\n    -- A morphism factors through an intersection of subobjects if it factors through each.\n    refine (finset_inf_factors _).mpr fun i _ => ?_\n    -- A morphism `f` factors through the kernel of `g` exactly if `f ≫ g = 0`.\n    apply kernelSubobject_factors\n    dsimp [objX]\n    -- Use a simplicial identity\n    rw [Category.assoc, ← Fin.castSucc_zero, ← X.δ_comp_δ (Fin.zero_le i.succ)]\n    -- We can rewrite the arrow out of the intersection of all the kernels as a composition\n    -- of a morphism we don't care about with the arrow out of the kernel of `X.δ i.succ.succ`.\n    rw [← factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ i.succ (by simp)),\n      Category.assoc, kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 := by\n  -- It's a pity we need to do a case split here;\n    -- after the first rw the proofs are almost identical\n  rcases n with _ | n <;> dsimp [objD]\n  · rw [Subobject.factorThru_arrow_assoc, Category.assoc, ← Fin.castSucc_zero,\n      ← X.δ_comp_δ_assoc (Fin.zero_le (0 : Fin 2)),\n      ← factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ (0 : Fin 2) (by simp)),\n      Category.assoc, kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]\n  · rw [factorThru_right, factorThru_eq_zero, factorThru_arrow_assoc, Category.assoc,\n      ← Fin.castSucc_zero,\n      ← X.δ_comp_δ (Fin.zero_le (0 : Fin (n + 3))),\n      ← factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ (0 : Fin (n + 3)) (by simp)),\n      Category.assoc, kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]\n\n/-- The normalized Moore complex functor, on objects.\n-/\n@[simps!]\ndef obj (X : SimplicialObject C) : ChainComplex C ℕ :=\n  ChainComplex.of (fun n => (objX X n : C))\n    (-- the coercion here picks a representative of the subobject\n      objD X) (d_squared X)\n\nvariable {X} {Y : SimplicialObject C} (f : X ⟶ Y)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The normalized Moore complex functor, on morphisms.\n-/\n@[simps!]\ndef map (f : X ⟶ Y) : obj X ⟶ obj Y :=\n  ChainComplex.ofHom\n    (fun n => factorThru _ (arrow _ ≫ f.app (op ⦋n⦌)) (by\n      cases n <;> dsimp\n      · apply top_factors\n      · refine (finset_inf_factors _).mpr fun i _ => kernelSubobject_factors _ _ ?_\n        rw [Category.assoc, SimplicialObject.δ, ← f.naturality,\n          ← factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ i (by simp)),\n          Category.assoc]\n        rw [← SimplicialObject.δ_def, kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]))\n    fun n => by cases n <;> dsimp [objD, objX, ChainComplex.of.d] <;> cat_disch\n\nend NormalizedMooreComplex\n\nopen NormalizedMooreComplex\n\nset_option backward.defeqAttrib.useBackward true in\nvariable (C) in\n/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.\n\nThe `n`-th object is intersection of\nthe kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.\n\nThe differentials are induced from `X.δ 0`,\nwhich maps each of these intersections of kernels to the next.\n-/\n@[simps]\ndef normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where\n  obj := obj\n  map f := map f\n\nset_option backward.defeqAttrib.useBackward true in\n-- Not `@[simp]` as `simp` can prove this.\n\nTarget:\ntheorem normalizedMooreComplex_objD (X : SimplicialObject C) (n : ℕ) :\n    ((normalizedMooreComplex C).obj X).d (n + 1) n = NormalizedMooreComplex.objD X n :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology","family_id":"normalizedmoorecomplex_objd","file_id":"mathlib/Mathlib/AlgebraicTopology/MooreComplex.lean","sample_id":"95cc52a04da20a5f5c49fe48e2d7c757cb686f4efaddac32474d2f88abd5ceb5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"280119e14881b51170fc3d0d4d95c5bf43762c9b1e1b31c95c9da1e53082f7ea","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"68756b8c7ed9cf05ae44a5c6d82ef89becd413ab12a17334b54eab418a5bdba8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b84cec6b1d751705167f6f103cf7edb4e8a75633ff7d571654ff5ea97e2ce6d3","source_sha256":"c48d585d22788683dbcc8ee8c25758ae9938a430160d8e35ea71552758ca3f83","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases h with ⟨u, hu⟩\n  rw [← Units.inv_mul_eq_one, mul_add] at hu\n  apply Or.imp _ _ (isUnit_or_isUnit_of_add_one hu) <;> exact isUnit_of_mul_isUnit_right","hard_negative":true,"metrics":{"chosen_tokens":35,"rejected_tokens":3,"token_jaccard":0.064516,"token_length_ratio":0.085714},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"61186375c9c82fd6e361fb61ba2dc721e531b03f1918cda05e874ccdd64363d6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.LocalRing.Defs\npublic import Mathlib.RingTheory.Ideal.Nonunits\n\nNamespace:\nIsLocalRing\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Chris Hughes, Mario Carneiro\n-/\n/-!\n\n# Local rings\n\nWe prove basic properties of local rings.\n\n-/\n\npublic section\n\nvariable {R S : Type*}\n\nnamespace IsLocalRing\n\nsection Semiring\n\nvariable [Semiring R]\n\ntheorem of_isUnit_or_isUnit_of_isUnit_add [Nontrivial R]\n    (h : ∀ a b : R, IsUnit (a + b) → IsUnit a ∨ IsUnit b) : IsLocalRing R :=\n  ⟨fun {a b} hab => h a b <| hab.symm ▸ isUnit_one⟩\n\n/-- A semiring is local if it is nontrivial and the set of nonunits is closed under the addition. -/\ntheorem of_nonunits_add [Nontrivial R]\n    (h : ∀ a b : R, a ∈ nonunits R → b ∈ nonunits R → a + b ∈ nonunits R) : IsLocalRing R where\n  isUnit_or_isUnit_of_add_one {a b} hab :=\n    or_iff_not_and_not.2 fun H => h a b H.1 H.2 <| hab.symm ▸ isUnit_one\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R]\n\n/-- A semiring is local if it has a unique maximal ideal. -/\ntheorem of_unique_max_ideal (h : ∃! I : Ideal R, I.IsMaximal) : IsLocalRing R :=\n  @of_nonunits_add _ _\n    (nontrivial_of_ne (0 : R) 1 <|\n      let ⟨I, Imax, _⟩ := h\n      fun H : 0 = 1 => Imax.1.1 <| I.eq_top_iff_one.2 <| H ▸ I.zero_mem)\n    fun x y hx hy H =>\n    let ⟨I, Imax, Iuniq⟩ := h\n    let ⟨Ix, Ixmax, Hx⟩ := exists_max_ideal_of_mem_nonunits hx\n    let ⟨Iy, Iymax, Hy⟩ := exists_max_ideal_of_mem_nonunits hy\n    have xmemI : x ∈ I := Iuniq Ix Ixmax ▸ Hx\n    have ymemI : y ∈ I := Iuniq Iy Iymax ▸ Hy\n    Imax.1.1 <| I.eq_top_of_isUnit_mem (I.add_mem xmemI ymemI) H\n\ntheorem of_unique_nonzero_prime (h : ∃! P : Ideal R, P ≠ ⊥ ∧ Ideal.IsPrime P) : IsLocalRing R :=\n  of_unique_max_ideal\n    (by\n      rcases h with ⟨P, ⟨hPnonzero, hPnot_top, _⟩, hPunique⟩\n      refine ⟨P, ⟨⟨hPnot_top, ?_⟩⟩, fun M hM => hPunique _ ⟨?_, Ideal.IsMaximal.isPrime hM⟩⟩\n      · refine Ideal.maximal_of_no_maximal fun M hPM hM => ne_of_lt hPM ?_\n        exact (hPunique _ ⟨ne_bot_of_gt hPM, Ideal.IsMaximal.isPrime hM⟩).symm\n      · rintro rfl\n        exact hPnot_top (hM.1.2 P (bot_lt_iff_ne_bot.2 hPnonzero)))\n\nvariable [IsLocalRing R]\n\nTarget:\ntheorem isUnit_or_isUnit_of_isUnit_add {a b : R} (h : IsUnit (a + b)) : IsUnit a ∨ IsUnit b :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_b84cec6b1d75","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"0a26ae39026f0e5b2ca8202bfab8de17cf723df1a9e3af431f76b965b8751387","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/LocalRing","family_id":"isunit_or_isunit_of_isunit_add","file_id":"mathlib/Mathlib/RingTheory/LocalRing/Basic.lean","sample_id":"b84cec6b1d751705167f6f103cf7edb4e8a75633ff7d571654ff5ea97e2ce6d3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5bd7c7167db7f215623bed49eb0f0085e5528c13934e7559ad6540e992b06a8a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2e71596933c22369bea1ce7d19358deda1acb57a1b511e60283fae72b0ed1051","source_sha256":"8d66dfc09aa4bf19e9637e6b2cc5cd204dddcf3c4b5353ddb78dc856eda79a64","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  letI := h𝕜.rclike\n  exact hs.mono_field","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":2,"token_jaccard":0.090909,"token_length_ratio":0.181818},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"6180baa8390b4f66256715cc7521fc44755d22acbdf801a1932536dcc6cf7e7a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.RCLike.Basic\npublic import Mathlib.Topology.Instances.RealVectorSpace\npublic import Mathlib.Analysis.Calculus.TangentCone.Real\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-! # Relationships between unique differentiability over `ℝ` and `ℂ`\n\nA set of unique differentiability for `ℝ` is also a set of unique differentiability for `ℂ`\n(or for a general field satisfying `IsRCLikeNormedField 𝕜`).\n-/\n\npublic section\n\nvariable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [h𝕜 : IsRCLikeNormedField 𝕜]\n  {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace ℝ E]\n  {s : Set E} {x : E}\n\ntheorem tangentConeAt_real_subset_isRCLikeNormedField :\n    tangentConeAt ℝ s x ⊆ tangentConeAt 𝕜 s x := by\n  letI := h𝕜.rclike\n  exact tangentConeAt_mono_field\n\nTarget:\ntheorem UniqueDiffWithinAt.of_real (hs : UniqueDiffWithinAt ℝ s x) :\n    UniqueDiffWithinAt 𝕜 s x :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/RCLike","family_id":"uniquediffwithinat","file_id":"mathlib/Mathlib/Analysis/RCLike/TangentCone.lean","sample_id":"2e71596933c22369bea1ce7d19358deda1acb57a1b511e60283fae72b0ed1051"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ad3461d90bc1a2d6d0744349016f752c25c2128b46b806f78f2e03cb5879efbf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"958dff2c8ca01f64b80069695c6cce9df9835719c5658058e7911b54468bca5a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"45c618e63e4bfb258f2eb5bccb2da6bd795c57ebdbec6a654acc6ef23d1bc7ee","source_sha256":"4b95b9e160e21b716ea1133988a6954cd2d9129042925a6d4bf4809d518f0745","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have uc : (iUnion fun i => Option.elim' sᶜ u i) = univ := by\n    apply Subset.antisymm (subset_univ _)\n    · simp_rw [← compl_union_self s, Option.elim', iUnion_option]\n      apply union_subset_union_right sᶜ us\n  rcases precise_refinement (Option.elim' sᶜ u) (Option.forall.2 ⟨isOpen_compl_iff.2 hs, uo⟩)\n      uc with\n    ⟨v, vo, vc, vf, vu⟩\n  refine ⟨v ∘ some, fun i ↦ vo _, ?_, vf.comp_injective (Option.some_injective _), fun i ↦ vu _⟩\n  · simp only [iUnion_option, ← compl_subset_iff_union] at vc\n    exact Subset.trans (subset_compl_comm.1 <| vu Option.none) vc","hard_negative":true,"metrics":{"chosen_tokens":143,"rejected_tokens":3,"token_jaccard":0.029412,"token_length_ratio":0.020979},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"61836ccf1928cb838a95af86894df1d7fafc1594cdc566807366fbaa41adfec0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Option.Basic\npublic import Mathlib.Topology.Separation.Regular\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton, Yury Kudryashov\n-/\n/-!\n# Paracompact topological spaces\n\nA topological space `X` is said to be paracompact if every open covering of `X` admits a locally\nfinite refinement.\n\nThe definition requires that each set of the new covering is a subset of one of the sets of the\ninitial covering. However, one can ensure that each open covering `s : ι → Set X` admits a *precise*\nlocally finite refinement, i.e., an open covering `t : ι → Set X` with the same index set such that\n`∀ i, t i ⊆ s i`, see lemma `precise_refinement`. We also provide a convenience lemma\n`precise_refinement_set` that deals with open coverings of a closed subset of `X` instead of the\nwhole space.\n\nWe also prove the following facts.\n\n* Every compact space is paracompact, see instance `paracompact_of_compact`.\n\n* A locally compact sigma compact Hausdorff space is paracompact, see instance\n  `paracompact_of_locallyCompact_sigmaCompact`. Moreover, we can choose a locally finite\n  refinement with sets in a given collection of filter bases of `𝓝 x`, `x : X`, see\n  `refinement_of_locallyCompact_sigmaCompact_of_nhds_basis`. For example, in a proper metric space\n  every open covering `⋃ i, s i` admits a refinement `⋃ i, Metric.ball (c i) (r i)`.\n\n* Every paracompact Hausdorff space is normal. This statement is not an instance to avoid loops in\n  the instance graph.\n\n* Every `EMetricSpace` is a paracompact space, see instance `EMetric.instParacompactSpace` in\n  `Topology/EMetricSpace/Paracompact`.\n\n## TODO\n\nProve (some of) [Michael's theorems](https://ncatlab.org/nlab/show/Michael%27s+theorem).\n\n## Tags\n\ncompact space, paracompact space, locally finite covering\n-/\n\npublic section\n\n\nopen Set Filter Function\n\nopen Filter Topology\n\nuniverse u v w\n\n/-- A topological space is called paracompact, if every open covering of this space admits a locally\nfinite refinement. We use the same universe for all types in the definition to avoid creating a\nclass like `ParacompactSpace.{u v}`. Due to lemma `precise_refinement` below, every open covering\n`s : α → Set X` indexed on `α : Type v` has a *precise* locally finite refinement, i.e., a locally\nfinite refinement `t : α → Set X` indexed on the same type such that each `∀ i, t i ⊆ s i`. -/\nclass ParacompactSpace (X : Type v) [TopologicalSpace X] : Prop where\n  /-- Every open cover of a paracompact space assumes a locally finite refinement. -/\n  locallyFinite_refinement :\n    ∀ (α : Type v) (s : α → Set X), (∀ a, IsOpen (s a)) → (⋃ a, s a = univ) →\n      ∃ (β : Type v) (t : β → Set X),\n        (∀ b, IsOpen (t b)) ∧ (⋃ b, t b = univ) ∧ LocallyFinite t ∧ ∀ b, ∃ a, t b ⊆ s a\n\nvariable {ι : Type u} {X : Type v} {Y : Type w} [TopologicalSpace X] [TopologicalSpace Y]\n\n/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,\none indexed on the same type with each open set contained in the corresponding original one. -/\ntheorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))\n    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧\n    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by\n  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`\n  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))\n    (forall_subtype_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])\n  simp only [exists_subtype_range_iff, iUnion_eq_univ_iff] at this\n  choose α t hto hXt htf ind hind using this\n  choose t_inv ht_inv using hXt\n  choose U hxU hU using htf\n  -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`\n  refine ⟨fun i ↦ ⋃ (a : α) (_ : ind a = i), t a, ?_, ?_, ?_, ?_⟩\n  · exact fun a ↦ isOpen_iUnion fun a ↦ isOpen_iUnion fun _ ↦ hto a\n  · simp only [eq_univ_iff_forall, mem_iUnion]\n    exact fun x ↦ ⟨ind (t_inv x), _, rfl, ht_inv _⟩\n  · refine fun x ↦ ⟨U x, hxU x, ((hU x).image ind).subset ?_⟩\n    simp only [subset_def, mem_iUnion, mem_setOf_eq, Set.Nonempty, mem_inter_iff]\n    rintro i ⟨y, ⟨a, rfl, hya⟩, hyU⟩\n    exact mem_image_of_mem _ ⟨y, hya, hyU⟩\n  · simp only [subset_def, mem_iUnion]\n    rintro i x ⟨a, rfl, hxa⟩\n    exact hind _ hxa\n\n/-- In a paracompact space, every open covering of a closed set admits a locally finite refinement\nindexed by the same type. -/\n\nTarget:\ntheorem precise_refinement_set [ParacompactSpace X] {s : Set X} (hs : IsClosed s) (u : ι → Set X)\n    (uo : ∀ i, IsOpen (u i)) (us : s ⊆ ⋃ i, u i) :\n    ∃ v : ι → Set X, (∀ i, IsOpen (v i)) ∧ (s ⊆ ⋃ i, v i) ∧ LocallyFinite v ∧ ∀ i, v i ⊆ u i :=\n\nProof body:\n","rejected":"by\n  exact precise_refinement_set","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"66ebdfc1bef0defc4f2675a58a412d95d58da93a2301f1ea001a9135307ef2d1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Compactness","family_id":"precise_refinement_set","file_id":"mathlib/Mathlib/Topology/Compactness/Paracompact.lean","sample_id":"45c618e63e4bfb258f2eb5bccb2da6bd795c57ebdbec6a654acc6ef23d1bc7ee"},"split":"train"}
{"binding":{"chosen_proof_sha256":"85d4d418bf899b74281aa7eeb44015fb0b192ec7ec0ce946edf50b0cafb9b526","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ed1374ebe28e9b99f5e81874724199c0edfb80ce0f4bc23069e0f6178e331a7a","source_sha256":"4c2fd0b515ef4b06865711215377bd2bbde8a660c25b8d72d68a97c859a1494a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  simp [map, hw]","hard_negative":false,"metrics":{"chosen_tokens":8,"rejected_tokens":3,"token_jaccard":0.1,"token_length_ratio":0.375},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"618debe6a91ef617a7cddbc17196549300383a5942679bf6a64d65d14f661198","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.LocallyFinsupp\npublic import Mathlib.Topology.Spectral.Basic\n\nNamespace:\nFunction.locallyFinsupp\n\nLocal context:\n/-\nCopyright (c) 2026 Raphael Douglas Giles. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Raphael Douglas Giles\n-/\n/-!\n# Pushforward of functions with locally finite support\n\nIn this file we define the notion of the pushforward of a function with locally finite support\nbetween prespectral spaces along a spectral map. This is used for defining the (proper) pushforward\nof algebraic cycles in algebraic geometry.\n\n## Main declarations\n\n- `Function.locallyFinsupp.map`: If `f : X → Y` is a spectral map between spectral spaces and\n  `c : X → R` is locally of finite support, the pushforward of `c` along `f` at `y : Y` is\n  `∑ᶠ x ∈ f ⁻¹' {y}, c x * w x`, where `w : X → R` is a weight function.\n\n## Notes\n\nIn the case of algebraic cycles, the weight function used in `Function.locallyFinsupp.map` will be\nspecialized to the degree of the residue field extension\n(see https://stacks.math.columbia.edu/tag/02R4).\n-/\n\n@[expose] public section\n\nopen Set Order Topology TopologicalSpace\n\nvariable {X Y R : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n  {f : X → Y} (hf : IsSpectralMap f) (w : X → R)\n\nnamespace Function.locallyFinsupp\n\nvariable [Semiring R] {W : Set Y} (hW : IsOpen W) (c : Function.locallyFinsupp X R)\n  [PrespectralSpace Y]\n\nvariable (f) in\n/--\nThe pushforward of a function `c` of locally finite support by a spectral map with respect to a\nweight function `w`.\n-/\nnoncomputable\ndef map (hf : IsSpectralMap f) (c : locallyFinsupp X R) : Function.locallyFinsupp Y R where\n  toFun z := ∑ᶠ x ∈ f ⁻¹' {z}, c x * w x\n  supportWithinDomain' := by simp\n  supportLocallyFiniteWithinDomain' y _ := by\n    obtain ⟨U, hU⟩ := (PrespectralSpace.isTopologicalBasis (X := Y)).exists_subset_of_mem_open\n      (by simp : y ∈ ⊤) (by simp)\n    refine ⟨U, IsOpen.mem_nhds hU.1.1 hU.2.1, ?_⟩\n    suffices h : (U ∩ {z | (f ⁻¹' {z} ∩ support ⇑c).Nonempty}).Finite by\n      refine h.subset (inter_subset_inter_right U fun y hy ↦ ?_)\n      obtain ⟨x, (hx : f x = y), h'⟩ := exists_ne_zero_of_finsum_mem_ne_zero hy\n      use x\n      grind [mem_support]\n    suffices (f ⁻¹' (U ∩ {z | (f ⁻¹' {z} ∩ c.support).Nonempty}) ∩ c.support).Finite from\n      (this.image f).subset (fun a ha ↦ by grind [Set.Nonempty])\n    exact (c.locallyFiniteSupport.finite_inter_support_of_isCompact <| hf.2 hU.1.1 hU.1.2).subset\n      (by simp; grind)\n\n@[simp]\nlemma map_apply (hf : IsSpectralMap f) (c : locallyFinsupp X R) (y : Y) :\n    map f w hf c y = ∑ᶠ x ∈ f ⁻¹' {y}, c x * w x := rfl\n\nlemma support_map_subset_of_forall_mem (s : Set X) (t : Set Y) (hc : c.support ⊆ s)\n    (h : ∀ x : X, x ∈ s → w x ≠ 0 → f x ∈ t) : (map f w hf c).support ⊆ t := by\n  intro y hy\n  obtain ⟨x, (rfl : f x = y), h'⟩ := exists_ne_zero_of_finsum_mem_ne_zero hy\n  grind [mem_support]\n\n@[simp]\n\nTarget:\nlemma map_id [PrespectralSpace X] (hw : ∀ z : X, w z = 1) :\n    map id w isSpectralMap_id c = c :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/LocallyFinsupp","family_id":"map_id","file_id":"mathlib/Mathlib/Topology/LocallyFinsupp/Pushforward.lean","sample_id":"ed1374ebe28e9b99f5e81874724199c0edfb80ce0f4bc23069e0f6178e331a7a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0678ccbc2752fb8380a44655810f825f8a9182914dd6e363b01acde3b055d878","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"501f3f54663e4d1e0765f46e1474477e0318be2a1b8bafd92a7145307568e22e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2d5cb89a9cfda0560e3764a0fa8315a7c93635c11c81d041a427e94157d40461","source_sha256":"1c0dcf6a3ce55bec7a88672314ea6ae6741a6599903854de40abb691f935e16b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction y generalizing x with\n  | nil => simpa using h\n  | cons y ys ih => exact T.prop (ih (by simpa))","hard_negative":true,"metrics":{"chosen_tokens":29,"rejected_tokens":2,"token_jaccard":0.045455,"token_length_ratio":0.068966},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"61b3372f1461b554ffd6e07a6d5ce191b26e2b99d5b44bb6f80de9a043ea9c3b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.CompleteLattice.SetLike\n\nNamespace:\nDescriptive.Tree\n\nLocal context:\n/-\nCopyright (c) 2024 Sven Manthe. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sven Manthe\n-/\n/-!\n# Trees in the sense of descriptive set theory\n\nThis file defines trees of depth `ω` in the sense of descriptive set theory as sets of finite\nsequences that are stable under taking prefixes.\n\n## Main declarations\n\n* `tree A`: a (possibly infinite) tree of depth at most `ω` with nodes in `A`\n-/\n\n@[expose] public section\n\nnamespace Descriptive\n\n/-- A tree is a set of finite sequences, implemented as `List A`, that is stable under\n  taking prefixes. For the definition we use the equivalent property `x ++ [a] ∈ T → x ∈ T`,\n  which is more convenient to check. We define `tree A` as a complete sublattice of\n  `Set (List A)`, which coerces to the type of trees on `A`. -/\ndef tree (A : Type*) : CompleteSublattice (Set (List A)) :=\n  CompleteSublattice.mk' {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T}\n    (by rintro S hS x a ⟨t, ht, hx⟩; use t, ht, hS ht hx)\n    (by rintro S hS x a h T hT; exact hS hT <| h T hT)\n\n@[simps!] instance (A : Type*) : SetLike (tree A) (List A) := SetLike.instSubtypeSet\n\nexample (A : Type*) : PartialOrder (tree A) := inferInstance\n\nnamespace Tree\nvariable {A : Type*} {S T : tree A}\n\nTarget:\nlemma mem_of_append {x y : List A} (h : x ++ y ∈ T) : x ∈ T :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_2d5cb89a9cfd","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"0f354678c0f83971db7925ae6659b7f54eeeac4a8991428d1c82ccdc51e8ad88","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Descriptive","family_id":"mem_of_append","file_id":"mathlib/Mathlib/SetTheory/Descriptive/Tree.lean","sample_id":"2d5cb89a9cfda0560e3764a0fa8315a7c93635c11c81d041a427e94157d40461"},"split":"train"}
{"binding":{"chosen_proof_sha256":"05b7a960b09bb29c3bdb02159f0748a158edd33fe39806e604a0595973e7a4e9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"861b7d8a1fe1c1ac90ef7cb56d31d4772a1a7fac4746b22305f0316ae345cc5c","source_sha256":"2c2357bdce988b5b8902bab0c6b1d4a599e97cee77b05d9e7215f22e881d81ce","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isSemilinearSet_iff] at hs\n  rcases hs with ⟨S, hS, rfl⟩\n  choose φ hφ using fun s : S => (hS s.1 s.2).definable\n  refine ⟨Formula.iSup φ, ?_⟩\n  ext x\n  have := fun s hs x => Set.ext_iff.1 (hφ ⟨s, hs⟩).symm x\n  simp only [mem_setOf_eq] at this\n  simp [this]","hard_negative":false,"metrics":{"chosen_tokens":85,"rejected_tokens":3,"token_jaccard":0.021277,"token_length_ratio":0.035294},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"61be77ea0fb5c21e05ae0939392bda1474e0bdfd53521e34f8bf2a08d19e3453","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.ModelTheory.Arithmetic.Presburger.Basic\npublic import Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic\npublic import Mathlib.ModelTheory.Definability\nimport Mathlib.Algebra.Group.Submonoid.Finsupp\nimport Mathlib.LinearAlgebra.Matrix.Notation\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Dexin Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dexin Zhang\n-/\n/-!\n# Presburger definability and semilinear sets\n\nThis file formalizes the classical result that Presburger definable sets are the same as semilinear\nsets. As an application of this result, we show that the graph of multiplication is not Presburger\ndefinable.\n\n## Main Results\n\n- `presburger.definable_iff_isSemilinearSet`: a set is Presburger definable in `ℕ` if and only if it\n  is semilinear.\n- `presburger.definable₁_iff_ultimately_periodic`: in the 1-dimensional case, a set is Presburger\n  arithmetic definable in `ℕ` if and only if it is ultimately periodic, i.e. periodic after some\n  number `k`.\n- `presburger.mul_not_definable`: the graph of multiplication is not Presburger definable in `ℕ`.\n\n## References\n\n* [Seymour Ginsburg and Edwin H. Spanier, *Bounded ALGOL-Like Languages*][ginsburg1964]\n* [Seymour Ginsburg and Edwin H. Spanier, *Semigroups, Presburger Formulas, and\n  Languages*][ginsburg1966]\n* [Samuel Eilenberg and M. P. Schützenberger, *Rational Sets in Commutative Monoids*][eilenberg1969]\n-/\n\npublic section\n\nvariable {α : Type*} {s : Set (α → ℕ)} {A : Set ℕ}\n\nopen Set FirstOrder Language\n\ntheorem IsLinearSet.definable [Finite α] (hs : IsLinearSet s) : A.Definable presburger s := by\n  rw [isLinearSet_iff] at hs\n  rcases hs with ⟨v, t, rfl⟩\n  refine ⟨Formula.iExs t (Formula.iInf fun i : α =>\n    (Term.var (Sum.inl i)).equal\n      (Term.varsToConstants\n        ((v i : presburger.Term _) + presburger.sum Finset.univ fun x : t =>\n          x.1 i • Term.var (Sum.inr (Sum.inr x))))), ?_⟩\n  ext x\n  simp only [mem_vadd_set, SetLike.mem_coe, AddSubmonoid.mem_closure_finset', Finset.univ_eq_attach,\n    nsmul_eq_mul, vadd_eq_add, ↓existsAndEq, true_and, mem_setOf_eq, Formula.realize_iExs,\n    Formula.realize_iInf, Formula.realize_equal, Term.realize_var, Sum.elim_inl,\n    Term.realize_varsToConstants, coe_con, presburger.realize_add, presburger.realize_natCast,\n    Nat.cast_id, presburger.realize_sum, presburger.realize_nsmul, Sum.elim_inr, smul_eq_mul]\n  congr! with a\n  simp_rw [Eq.comm (b := x), fun x : t => mul_comm (a x : α → ℕ) x, funext_iff]\n  congr! 1 with i\n  simp\n\nTarget:\ntheorem IsSemilinearSet.definable [Finite α] (hs : IsSemilinearSet s) :\n    A.Definable presburger s :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"ModelTheory/Arithmetic","family_id":"issemilinearset","file_id":"mathlib/Mathlib/ModelTheory/Arithmetic/Presburger/Definability.lean","sample_id":"861b7d8a1fe1c1ac90ef7cb56d31d4772a1a7fac4746b22305f0316ae345cc5c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"66e860b3b879ffe015e76af344d6b2852af86ab51c1939b65912c04b3badbf97","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8bccf914e3afe1f74007d8940af4723ed37d027ecbab49b4b237cf2f1ae3ded2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6c05ada6fb867e067b57945bee363d3116da5202db3c69d06f6b508b9b741204","source_sha256":"d43040b92adbb8ebcddab07470e26da05f34a625604831787d2b69fe79c9281e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h => ?_, fun h => h.symm ▸ mulSingle_one i⟩\n  rw [← mulSingle_eq_same i x, h, one_apply]","hard_negative":true,"metrics":{"chosen_tokens":30,"rejected_tokens":3,"token_jaccard":0.041667,"token_length_ratio":0.1},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"620ec99f2740f4221cdd0448a9e955acc76d3351042523d9f4916d9731993ca4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Pi.Defs\npublic import Mathlib.Logic.Function.Basic\n\nNamespace:\nPi\n\nLocal context:\n/-\nCopyright (c) 2020 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot, Eric Wieser\n-/\n/-!\n# Very basic algebraic operations on pi types\n\nThis file provides very basic algebraic operations on functions.\n-/\n\n@[expose] public section\n\nassert_not_exists Monoid Preorder\n\nopen Function\n\nvariable {ι ι' α β : Type*} {G M N O : ι → Type*}\n\nnamespace Pi\nvariable [∀ i, One (M i)] [∀ i, One (N i)] [∀ i, One (O i)] [DecidableEq ι] {i : ι} {x : M i}\n\n/-- The function supported at `i`, with value `x` there, and `1` elsewhere. -/\n@[to_additive /-- The function supported at `i`, with value `x` there, and `0` elsewhere. -/]\ndef mulSingle (i : ι) (x : M i) : ∀ j, M j := Function.update 1 i x\n\n@[to_additive (attr := simp)]\nlemma mulSingle_eq_same (i : ι) (x : M i) : mulSingle i x i = x := Function.update_self i x _\n\n@[to_additive (attr := simp)]\nlemma mulSingle_eq_of_ne {i i' : ι} (h : i' ≠ i) (x : M i) : mulSingle i x i' = 1 :=\n  Function.update_of_ne h x _\n\n/-- Abbreviation for `mulSingle_eq_of_ne h.symm`, for ease of use by `simp`. -/\n@[to_additive (attr := simp)\n  /-- Abbreviation for `single_eq_of_ne h.symm`, for ease of use by `simp`. -/]\nlemma mulSingle_eq_of_ne' {i i' : ι} (h : i ≠ i') (x : M i) : mulSingle i x i' = 1 :=\n  mulSingle_eq_of_ne h.symm x\n\n@[to_additive (attr := simp)]\nlemma mulSingle_one (i : ι) : mulSingle i (1 : M i) = 1 := Function.update_eq_self _ _\n\n@[to_additive (attr := simp)]\n\nTarget:\nlemma mulSingle_eq_one_iff : mulSingle i x = 1 ↔ x = 1 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_6c05ada6fb86","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"28b1d82e3402bc5d5fd255754a4dbc2634054d3077ac89f8a5f7b48a8ebf634d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Notation","family_id":"mulsingle_eq_one_iff","file_id":"mathlib/Mathlib/Algebra/Notation/Pi/Basic.lean","sample_id":"6c05ada6fb867e067b57945bee363d3116da5202db3c69d06f6b508b9b741204"},"split":"train"}
{"binding":{"chosen_proof_sha256":"db5676c2a4baaaa8601683c727e173d0b3b0fe87aa92feb60828f3a50296fa53","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ba002fff735a1eceaabf1a8e0dbcc41b8470b3783bbd4a79e63a536436ef24f3","source_sha256":"cb81d90c8dbabbe33b1b9939fcf99c692948a640de2613d83a1e798cde8653e5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hp : ((f.support : Set G).smulAntidiagonal (Function.support x) p).Finite :=\n    Set.SMulAntidiagonal.finite_of_finite_fst f.support.finite_toSet x.support p\n  set s : Set (G × P) := ↑(Finset.SMulAntidiagonal p hp)\n  have h₁ : s.InjOn Prod.fst := fun _ h₁ _ h₂ h ↦ by\n    rw [Finset.mem_coe, mem_smulAntidiagonal_of_group] at h₁ h₂\n    aesop\n  have h₂ : s.MapsTo Prod.fst ↑f.support := fun g hg ↦ by aesop\n  have h₃ (g : G) (hg : g ∈ f.support) (hgn : g ∉ Prod.fst '' s) : f g • x (g⁻¹ • p) = 0 := by\n    obtain (h | h) : f g = 0 ∨ ∀ q, ¬ x q = 0 → ¬g • q = p := by aesop\n    · simp [h]\n    · have := h (g⁻¹ • p)\n      aesop\n  rw [smul_eq, Finset.sum_of_injOn Prod.fst h₁ h₂ h₃]\n  aesop","hard_negative":false,"metrics":{"chosen_tokens":224,"rejected_tokens":2,"token_jaccard":0.014706,"token_length_ratio":0.008929},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"62368d7d15cb1037595ffb41f0c7fe4914ae356d9bb9dff027d838a28737f86a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MonoidAlgebra.Defs\npublic import Mathlib.Data.Finset.SMulAntidiagonal\n\nNamespace:\nMonoidAlgebra\n\nLocal context:\n/-\nCopyright (c) 2025 Scott Carnahan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Carnahan\n-/\n/-!\n# Scalar multiplication by (additive) monoid rings on formal functions.\nGiven sets `G` and `P`, with a left-cancellative scalar-multiplication (or vector-addition) of `G`\non `P`, together with a module `V` over a semiring `R`, we define a convolution action of the monoid\nalgebra `R[G]` on the set of functions `P → V`.\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nvariable {G P R V : Type*}\n\nnamespace MonoidAlgebra\n\n@[to_additive]\ntheorem mem_smulAntidiagonal_of_group [Group G] [MulAction G P] [Semiring R] [Zero V]\n    (f : R[G]) (x : P → V) (p : P) (gh : G × P) :\n    gh ∈ Finset.SMulAntidiagonal p\n      (Set.SMulAntidiagonal.finite_of_finite_fst f.support.finite_toSet x.support p) ↔\n      f gh.1 ≠ 0 ∧ x gh.2 ≠ 0 ∧ gh.2 = gh.1⁻¹ • p := by\n  rw [Finset.mem_smulAntidiagonal, eq_inv_smul_iff, Function.mem_support, Finset.mem_coe,\n    Finsupp.mem_support_iff]\n\n/-- A convolution-type scalar multiplication of the monoid algebra on the set of formal\nfunctions. -/\n@[to_additive (dont_translate := R) /-- A convolution-type scalar multiplication of the additive\nmonoid algebra on the set of formal functions. -/]\nscoped instance [SMul G P] [IsLeftCancelSMul G P] [Semiring R] [AddCommMonoid V]\n    [SMulWithZero R V] :\n    SMul (R[G]) (P → V) where\n  smul f x p := ∑ gh ∈ Finset.SMulAntidiagonal p\n    (Set.SMulAntidiagonal.finite_of_finite_fst f.support.finite_toSet x.support p), f gh.1 • x gh.2\n\n@[to_additive (dont_translate := R) smul_eq]\ntheorem smul_eq [SMul G P] [IsLeftCancelSMul G P] [Semiring R] [AddCommMonoid V] [SMulWithZero R V]\n    (f : R[G]) (x : P → V) (p : P)\n    (hp : ((f.support : Set G).smulAntidiagonal (Function.support x) p).Finite :=\n      Set.SMulAntidiagonal.finite_of_finite_fst f.support.finite_toSet x.support p) :\n    (f • x) p = ∑ gh ∈ Finset.SMulAntidiagonal p hp, f gh.1 • x gh.2 :=\n  rfl\n\n@[to_additive (dont_translate := R) smul_apply_addAction]\n\nTarget:\ntheorem smul_apply_mulAction [Group G] [MulAction G P] [Semiring R] [AddCommMonoid V]\n    [SMulWithZero R V] (f : MonoidAlgebra R G) (x : P → V) (p : P) :\n    (f • x) p = ∑ i ∈ f.support, (f i) • x (i⁻¹ • p) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/MonoidAlgebra","family_id":"smul_apply_mulaction","file_id":"mathlib/Mathlib/Algebra/MonoidAlgebra/PointwiseSMul.lean","sample_id":"ba002fff735a1eceaabf1a8e0dbcc41b8470b3783bbd4a79e63a536436ef24f3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e99a691cbba97dd50b462f822d8ea54c6c328e4bf1d6437c6b809f75afb37a33","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"02f4eef5948f491574f8f5278e997fcef4df32591c8c8bb4303a7d31f4a5ac8b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"71930e9f307d51dc22d21feef3aebed1568f0f6f3ee70012583f25208fbd5c02","source_sha256":"864301aaa8f40ca7deda0f7c4364088256abebe91b440dcc329dc526e83462dd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have key : ∀ x ∈ hs.toFinset, s.indicator (fun _ ↦ a) x = a := by\n    intro x hx\n    rw [indicator_of_mem (hs.mem_toFinset.1 hx) (fun _ ↦ a)]\n  rw [Nat.card_eq_card_finite_toFinset hs, ← sum_eq_card_nsmul key, eventually_atTop]\n  obtain ⟨m, hm⟩ := hs.bddAbove\n  refine ⟨m + 1, fun n n_m ↦ (sum_subset ?_ ?_).symm⟩ <;> intro x <;> rw [hs.mem_toFinset]\n  · rw [Finset.mem_range]\n    exact fun x_s ↦ ((mem_upperBounds.1 hm) x x_s).trans_lt (Nat.lt_of_succ_le n_m)\n  · exact fun _ x_s ↦ indicator_of_notMem x_s (fun _ ↦ a)","hard_negative":false,"metrics":{"chosen_tokens":146,"rejected_tokens":153,"token_jaccard":0.967213,"token_length_ratio":1.047945},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"62759a6568879ae87178790e10e987d6c4740171c5c1c3c29c7171506bd2f3b4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Indicator\npublic import Mathlib.Algebra.Order.Archimedean.Basic\npublic import Mathlib.Algebra.Order.BigOperators.Group.Finset\npublic import Mathlib.Algebra.Order.Group.Indicator\npublic import Mathlib.Order.LiminfLimsup\npublic import Mathlib.SetTheory.Cardinal.Finite\n\nNamespace:\nSet\n\nLocal context:\n/-\nCopyright (c) 2024 Damien Thomine. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Damien Thomine\n-/\n/-!\n# Cardinality and limit of sum of indicators\nThis file contains results relating the cardinality of subsets of ℕ and limits,\nlimsups of sums of indicators.\n\n## Tags\nfinite, indicator, limsup, tendsto\n-/\n\npublic section\n\nnamespace Set\n\nopen Filter Finset\n\nTarget:\nlemma sum_indicator_eventually_eq_card {α : Type*} [AddCommMonoid α] (a : α) {s : Set ℕ}\n    (hs : s.Finite) :\n    ∀ᶠ n in atTop, ∑ k ∈ Finset.range n, s.indicator (fun _ ↦ a) k = (Nat.card s) • a :=\n\nProof body:\n","rejected":"```lean\nby\n  have key : ∀ x ∈ hs.toFinset, s.indicator (fun _ ↦ a) x = a := by\n    intro x hx\n    rw [indicator_of_mem (hs.mem_toFinset.1 hx) (fun _ ↦ a)]\n  rw [Nat.card_eq_card_finite_toFinset hs, ← sum_eq_card_nsmul key, eventually_atTop]\n  obtain ⟨m, hm⟩ := hs.bddAbove\n  refine ⟨m + 1, fun n n_m ↦ (sum_subset ?_ ?_).symm⟩ <;> intro x <;> rw [hs.mem_toFinset]\n  · rw [Finset.mem_range]\n    exact fun x_s ↦ ((mem_upperBounds.1 hm) x x_s).trans_lt (Nat.lt_of_succ_le n_m)\n  · exact fun _ x_s ↦ indicator_of_notMem x_s (fun _ ↦ a)\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Order","family_id":"sum_indicator_eventually_eq_card","file_id":"mathlib/Mathlib/Algebra/Order/Archimedean/IndicatorCard.lean","sample_id":"71930e9f307d51dc22d21feef3aebed1568f0f6f3ee70012583f25208fbd5c02"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d9b98cd109749b4d841c4e5ec47310732f89bc8df99f9f6e87e3aa11beb61e37","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c35d695e227f0e1f879e46ee2607f0e36709c6a188c63c2e3d7dc3df5e613bba","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"33493801268bbdfd455e186fa50a17ff9c38e950d8c9e9bcab0dd5bfd9827ec7","source_sha256":"7cfbfc972c134864031de87e3b5157be2d7af83e375a2c0e1cc9ff7db598678d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have ⟨hμν_ac, hκη_ac⟩ := Measure.absolutelyContinuous_compProd_iff.mp h_ac\n  rw [← integrable_rnDeriv_mul_log_iff h_ac] at h_int\n  replace h_int := convexOn_mul_log.integrable_apply_rnDeriv_of_integrable_compProd\n    continuous_mul_log.stronglyMeasurable continuous_mul_log.continuousWithinAt h_int hκη_ac\n  exact (integrable_rnDeriv_mul_log_iff hμν_ac).mp h_int","hard_negative":true,"metrics":{"chosen_tokens":56,"rejected_tokens":5,"token_jaccard":0.085714,"token_length_ratio":0.089286},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"62856dbab2d46edb96ea8c66e80ad1efd3877284e4b9f9d7c9dca173d75617a3","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.InformationTheory.KullbackLeibler.Basic\npublic import Mathlib.Probability.Kernel.Composition.MeasureCompProd\npublic import Mathlib.Probability.Notation\nimport Mathlib.Probability.Kernel.Composition.IntegralCompProd\nimport Mathlib.Probability.Kernel.Composition.RadonNikodym\n\nNamespace:\nInformationTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne, Lorenzo Luccioli\n-/\n/-!\n# Chain rule for the Kullback-Leibler divergence\n\nSuppose that we have two finite joint measures on a product `𝓧 × 𝓨`, which can be decomposed as\n`μ ⊗ₘ κ` and `ν ⊗ₘ η`, where `μ` and `ν` are measures on `𝓧` and `κ` and `η` are Markov kernels\nfrom `𝓧` to `𝓨`. Then we can express the Kullback-Leibler divergence between these two joint\nmeasures as a sum of `klDiv μ ν` and the conditional Kullback-Leibler divergence between the kernels\n`κ` and `η`, averaged over `μ`. The resulting equality is most often written as\n`klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + μ[fun x ↦ klDiv (κ x) (η x)]`.\n\nHere we first prove the following version:\n`klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)`.\nThis version avoids the issue of measurability of the function `x ↦ klDiv (κ x) (η x)`, which is not\nalways guaranteed, and thus holds for all measurable spaces `𝓧` and `𝓨`, without any assumptions.\n\n## Main statements\n\n* `klDiv_compProd_eq_add`: `klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)`\n* `klDiv_compProd_left`: `klDiv (μ ⊗ₘ κ) (ν ⊗ₘ κ) = klDiv μ ν`\n\n## Proof\n\nThe main ingredient is the chain rule for Radon-Nikodym derivatives:\n`∂(μ ⊗ₘ κ)/∂(ν ⊗ₘ η) = ∂μ/∂ν * ∂(μ ⊗ₘ κ)/∂(μ ⊗ₘ η)`.\nThen, omitting edge cases, the Kullback-Leibler divergence is an integral of a logarithm of the\nderivative on the left, which decomposes into a sum of two integrals of logarithms.\nWe now give a more detailed outline of the proof.\n\nThe Kullback-Leibler divergence `klDiv μ ν` is defined with an if-then-else statement:\nif the measures are absolutely continuous (`μ ≪ ν`) and the log-likelihood ratio `llr μ ν` is\nintegrable, then it is defined as `∫ x, llr μ ν x ∂μ + ν.real univ - μ.real univ`, otherwise\nit is defined to be `∞`.\n\nWe first deal with the case in which absolute continuity does not hold. The main observation is\nthat `μ ⊗ₘ κ ≪ ν ⊗ₘ η ↔ μ ≪ ν ∧ μ ⊗ₘ κ ≪ μ ⊗ₘ η`, which means that if one of the two sides of the\nKL equality is infinite because of lack of absolute continuity, then the other side is also infinite\nfor the same reason.\n\nThen, we deal with the case in which absolute continuity holds but integrability does not. Again,\nwe can show a similar equivalence for integrability, which allows us to conclude that both sides\nare infinite.\n`Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)` is equivalent to\n`Integrable (llr μ ν) μ ∧ Integrable (llr (μ ⊗ₘ κ) (μ ⊗ₘ η)) (μ ⊗ₘ κ)`.\nThe proof of this equivalence relies on the convexity of the function `x ↦ x * log x`.\n\nFinally, we prove the equality in the case in which both absolute continuity and integrability hold.\nIn that case, `klDiv μ ν = ∫ x, llr μ ν x ∂μ + ν.real univ - μ.real univ` and similarly for\nthe other terms. It is easy to see that it suffices to prove the equality of the integrals parts.\nFinally, the computation for the integral of the log-likelihood ratio is as follows:\n```\n∫ p, llr (μ ⊗ₘ κ) (ν ⊗ₘ η) p ∂(μ ⊗ₘ κ)\n_ = ∫ p, ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal * log ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal ∂(ν ⊗ₘ η)\n_ = ∫ p, ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal *\n    (log ((∂μ/∂ν) p.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal) ∂(ν ⊗ₘ η)\n_ = ∫ p, (log ((∂μ/∂ν) p.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal) ∂(μ ⊗ₘ κ)\n_ = ∫ p, log ((∂μ/∂ν) p.1).toReal ∂(μ ⊗ₘ κ) + ∫ p, log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal ∂(μ ⊗ₘ κ)\n_ = ∫ a, llr μ ν a ∂μ + ∫ p, llr (μ ⊗ₘ κ) (μ ⊗ₘ η) p ∂(μ ⊗ₘ κ)\n```\n\n## TODO\n\nAdd a version of the chain rule for the integral form of the contional KL divergence, i.e.\n`μ[fun x ↦ klDiv (κ x) (η x)]`.\n\n-/\n\npublic section\n\nopen Real MeasureTheory Set ProbabilityTheory\nopen scoped ENNReal\n\nnamespace InformationTheory\n\nvariable {𝓧 𝓨 : Type*} {m𝓧 : MeasurableSpace 𝓧} {m𝓨 : MeasurableSpace 𝓨}\n  {μ ν : Measure 𝓧} {κ η : Kernel 𝓧 𝓨}\n  [IsFiniteMeasure μ] [IsFiniteMeasure ν] [IsMarkovKernel κ] [IsMarkovKernel η]\n\n/-- If the log-likelihood ration between two composition-products is integrable, then so is the\nlog-likelihood ratio between the two measures on the first space. -/\n\nTarget:\nlemma integrable_llr_of_integrable_llr_compProd\n    (h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η) (h_int : Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)) :\n    Integrable (llr μ ν) μ :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_33493801268b","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"a4ba8a8fa0074b81cb1ecf70db7e8657c8312096326e034e9548a6cea40d85f5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"InformationTheory/KullbackLeibler","family_id":"integrable_llr_of_integrable_llr_compprod","file_id":"mathlib/Mathlib/InformationTheory/KullbackLeibler/ChainRule.lean","sample_id":"33493801268bbdfd455e186fa50a17ff9c38e950d8c9e9bcab0dd5bfd9827ec7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"35928b4ec6c4dd8e48530f64dacc44b471422b62b6f0a8ae08f8ed59d2ec96f4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ca8114b0fec5bc25a1e786573058e15d2438da15e74c508243822dc34ade61e1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"325825a927e5735338dd52a28d2dff4f12d1ad248dc0bc8ca3daba7d87f856c6","source_sha256":"65733f0fee285d1da4e1bad2af528d5547ef268aaa4d457452ee23fb9ac9cd1d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  calc\n    _ = 𝟙 _ ⊗≫ (F.map₂ β ▷ η.app b ≫ η.naturality g) ▷ θ.app b ⊗≫\n          η.app a ◁ θ.naturality g ⊗≫ 𝟙 _ := by\n      bicategory\n    _ = 𝟙 _ ⊗≫ η.naturality f ▷ θ.app b ⊗≫\n          η.app a ◁ (G.map₂ β ▷ θ.app b ≫ θ.naturality g) ⊗≫ 𝟙 _ := by\n      rw [η.naturality_naturality]\n      bicategory\n    _ = _ := by\n      rw [θ.naturality_naturality]\n      bicategory","hard_negative":true,"metrics":{"chosen_tokens":112,"rejected_tokens":3,"token_jaccard":0.030303,"token_length_ratio":0.026786},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"632510a50b9f9f135c5f684847fb97e145382bb48e573e530ea5d49d88e71174","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Bicategory.Functor.Lax\npublic import Mathlib.Tactic.CategoryTheory.Bicategory.Basic\n\nNamespace:\nCategoryTheory.Lax.OplaxTrans\n\nLocal context:\n/-\nCopyright (c) 2025 Robin Carlier. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yuma Mizuno, Calle Sönne, Robin Carlier\n-/\n/-!\n# Transformations between lax functors\n\nJust as there are natural transformations between functors, there are transformations\nbetween lax functors. The equality in the naturality condition of a natural transformation gets\nreplaced by a specified 2-morphism. Now, there are three possible types of transformations (between\nlax functors):\n* lax natural transformations;\n* oplax natural transformations;\n* strong natural transformations.\n\nThese differ in the direction (and invertibility) of the 2-morphisms involved in the naturality\ncondition.\n\n## Main definitions\n\n* `Lax.LaxTrans F G`: lax transformations between lax functors `F` and `G`. The naturality\n  condition is given by a 2-morphism `app a ≫ G.map f ⟶ F.map f ≫ app b` for each 1-morphism\n  `f : a ⟶ b`.\n* `Lax.OplaxTrans F G`: oplax transformations between lax functors `F` and `G`. The naturality\n  condition is given by a 2-morphism `F.map f ≫ app b ⟶ app a ≫ G.map f` for each 1-morphism\n  `f : a ⟶ b`.\n* `Lax.StrongTrans F G`: strong transformations between lax functors `F` and `G`. The naturality\n  condition is given by a 2-isomorphism `app a ≫ G.map f ≅ F.map f ≫ app b` for each 1-morphism\n  `f : a ⟶ b`.\n\nUsing these, we define three (scoped) `CategoryStruct` instances on `B ⥤ᴸ C`, in the\n`Lax.LaxTrans`, `Lax.OplaxTrans`, and `Lax.StrongTrans` namespaces. The arrows in these\n`CategoryStruct` instances are given by lax transformations, oplax transformations, and strong\ntransformations respectively.\n\nWe also provide API for going between lax transformations and strong transformations:\n* `LaxTrans.StrongCore η`: a structure on a lax transformation between lax functors that\n  promotes it to a strong transformation.\n* `StrongTrans.mkOfLax η η'`: given a lax transformation `η` such that each component\n  2-morphism is an isomorphism, `mkOfLax` gives the corresponding strong transformation.\n\n## References\n* [Niles Johnson, Donald Yau, *2-Dimensional Categories*](https://arxiv.org/abs/2002.06055),\n  section 4.2.\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory.Lax\n\nopen Category Bicategory\n\nuniverse w₁ w₂ v₁ v₂ u₁ u₂\n\nvariable {B : Type u₁} [Bicategory.{w₁, v₁} B] {C : Type u₂} [Bicategory.{w₂, v₂} C]\n\n/-- If `η` is a lax transformation between `F` and `G`, we have a 1-morphism\n`η.app a : F.obj a ⟶ G.obj a` for each object `a : B`. We also have a 2-morphism\n`η.naturality f : app a ≫ G.map f ⟶ F.map f ≫ app b` for each 1-morphism `f : a ⟶ b`.\nThese 2-morphisms satisfy the naturality condition, and preserve the identities and\nthe compositions modulo some adjustments of domains and codomains of 2-morphisms.\n-/\nstructure LaxTrans (F G : B ⥤ᴸ C) where\n  /-- The component 1-morphisms of a lax transformation. -/\n  app (a : B) : F.obj a ⟶ G.obj a\n  /-- The 2-morphisms underlying the lax naturality constraint. -/\n  naturality {a b : B} (f : a ⟶ b) : app a ≫ G.map f ⟶ F.map f ≫ app b\n  /-- Naturality of the lax naturality constraint. -/\n  naturality_naturality {a b : B} {f g : a ⟶ b} (η : f ⟶ g) :\n      naturality f ≫ F.map₂ η ▷ app b = app a ◁ G.map₂ η ≫ naturality g := by\n    cat_disch\n  /-- Lax unity. -/\n  naturality_id (a : B) :\n      app a ◁ G.mapId a ≫ naturality (𝟙 a) =\n        (ρ_ (app a)).hom ≫ (λ_ (app a)).inv ≫ F.mapId a ▷ app a := by\n    cat_disch\n  /-- Lax functoriality. -/\n  naturality_comp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) :\n      app a ◁ G.mapComp f g ≫ naturality (f ≫ g) =\n      (α_ _ _ _).inv ≫ naturality f ▷ G.map g ≫ (α_ _ _ _).hom ≫\n        F.map f ◁ naturality g ≫ (α_ _ _ _).inv ≫ F.mapComp f g ▷ app c := by\n    cat_disch\n\nattribute [reassoc (attr := simp)] LaxTrans.naturality_naturality LaxTrans.naturality_id\n  LaxTrans.naturality_comp\n\nnamespace LaxTrans\n\nvariable {F G H : B ⥤ᴸ C} (η : LaxTrans F G) (θ : LaxTrans G H)\n\nvariable (F) in\n/-- The identity lax transformation. -/\ndef id : LaxTrans F F where\n  app a := 𝟙 (F.obj a)\n  naturality {_ _} f := (λ_ (F.map f)).hom ≫ (ρ_ (F.map f)).inv\n\ninstance : Inhabited (LaxTrans F F) :=\n  ⟨id F⟩\n\n/-- Auxiliary definition for `vComp`. -/\nabbrev vCompApp (a : B) : F.obj a ⟶ H.obj a :=\n  η.app a ≫ θ.app a\n\n/-- Auxiliary definition for `vComp`. -/\nabbrev vCompNaturality {a b : B} (f : a ⟶ b) :\n    (η.app a ≫ θ.app a) ≫ H.map f ⟶ F.map f ≫ η.app b ≫ θ.app b :=\n  (α_ _ _ _).hom ≫ η.app a ◁ θ.naturality f ≫ (α_ _ _ _).inv ≫\n    η.naturality f ▷ θ.app b ≫ (α_ _ _ _).hom\n\ntheorem vComp_naturality_naturality {a b : B} {f g : a ⟶ b} (β : f ⟶ g) :\n    η.vCompNaturality θ f ≫ F.map₂ β ▷ η.vCompApp θ b =\n      η.vCompApp θ a ◁ H.map₂ β ≫ η.vCompNaturality θ g :=\n  calc\n    _ = 𝟙 _ ⊗≫ η.app a ◁ θ.naturality f ⊗≫\n          (η.naturality f ≫ F.map₂ β ▷ η.app b) ▷ θ.app b ⊗≫ 𝟙 _ := by\n      bicategory\n    _ = 𝟙 _ ⊗≫ η.app a ◁ (θ.naturality f ≫ G.map₂ β ▷ θ.app b) ⊗≫\n          η.naturality g ▷ θ.app b ⊗≫ 𝟙 _ := by\n      rw [naturality_naturality]\n      bicategory\n    _ = _ := by\n      rw [naturality_naturality]\n      bicategory\n\ntheorem vComp_naturality_id (a : B) :\n    η.vCompApp θ a ◁ H.mapId a ≫ η.vCompNaturality θ (𝟙 a) =\n      (ρ_ (η.vCompApp θ a)).hom ≫ (λ_ (η.vCompApp θ a)).inv ≫ F.mapId a ▷ η.vCompApp θ a :=\n  calc\n    _ = 𝟙 _ ⊗≫ η.app a ◁ (θ.app a ◁ H.mapId a ≫ θ.naturality (𝟙 a)) ⊗≫\n          η.naturality (𝟙 a) ▷ θ.app a ⊗≫ 𝟙 _ := by\n      bicategory\n    _ = 𝟙 _ ⊗≫ (η.app a ◁ G.mapId a ≫ η.naturality (𝟙 a)) ▷ θ.app a ⊗≫ 𝟙 _ := by\n      rw [naturality_id]\n      bicategory\n    _ = _ := by\n      rw [naturality_id]\n      bicategory\n\ntheorem vComp_naturality_comp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) :\n    η.vCompApp θ a ◁ H.mapComp f g ≫ η.vCompNaturality θ (f ≫ g) =\n      (α_ (η.vCompApp θ a) (H.map f) (H.map g)).inv ≫\n        η.vCompNaturality θ f ▷ H.map g ≫\n          (α_ (F.map f) (η.vCompApp θ b) (H.map g)).hom ≫\n            F.map f ◁ η.vCompNaturality θ g ≫\n              (α_ (F.map f) (F.map g) (η.vCompApp θ c)).inv ≫ F.mapComp f g ▷ η.vCompApp θ c :=\n  calc\n    _ = 𝟙 _ ⊗≫ η.app a ◁ (θ.app a ◁ H.mapComp f g ≫ θ.naturality (f ≫ g)) ⊗≫\n          η.naturality (f ≫ g) ▷ θ.app c ⊗≫ 𝟙 _ := by\n      bicategory\n    _ = 𝟙 _ ⊗≫ η.app a ◁ (θ.naturality f ▷ (H.map g) ⊗≫ G.map f ◁ θ.naturality g) ⊗≫\n          (η.app a ◁ G.mapComp f g ≫ η.naturality (f ≫ g)) ▷ θ.app c ⊗≫ 𝟙 _ := by\n      rw [naturality_comp θ]\n      bicategory\n    _ = 𝟙 _ ⊗≫ η.app a ◁ θ.naturality f ▷ H.map g ⊗≫\n          ((η.app a ≫ G.map f) ◁ θ.naturality g ≫ η.naturality f ▷ (G.map g ≫ θ.app c)) ⊗≫\n            F.map f ◁ η.naturality g ▷ θ.app c ⊗≫\n              F.mapComp f g ▷ η.app c ▷ θ.app c ⊗≫ 𝟙 _ := by\n      rw [naturality_comp η]\n      bicategory\n    _ = _ := by\n      rw [whisker_exchange]\n      bicategory\n\n/-- Vertical composition of lax transformations. -/\ndef vComp (η : LaxTrans F G) (θ : LaxTrans G H) : LaxTrans F H where\n  app a := vCompApp η θ a\n  naturality := vCompNaturality η θ\n  naturality_naturality := vComp_naturality_naturality η θ\n  naturality_id := vComp_naturality_id η θ\n  naturality_comp := vComp_naturality_comp η θ\n\nattribute [local simp] vCompApp vCompNaturality in\n/-- `CategoryStruct` on `B ⥤ᴸ C` where the (1-)morphisms are given by lax\ntransformations. -/\n@[simps! id_app id_naturality comp_app comp_naturality]\nscoped instance : CategoryStruct (B ⥤ᴸ C) where\n  Hom := LaxTrans\n  id := LaxTrans.id\n  comp := LaxTrans.vComp\n\n@[deprecated (since := \"2026-03-16\")] alias vComp_app := comp_app\n@[deprecated (since := \"2026-03-16\")] alias vComp_naturality := comp_naturality\n\nend LaxTrans\n\n/-- If `η` is an oplax transformation between `F` and `G`, we have a 1-morphism\n`η.app a : F.obj a ⟶ G.obj a` for each object `a : B`. We also have a 2-morphism\n`η.naturality f : F.map f ≫ app b ⟶ app a ≫ G.map f` for each 1-morphism `f : a ⟶ b`.\nThese 2-morphisms satisfy the naturality condition, and preserve the identities and\nthe compositions modulo some adjustments of domains and codomains of 2-morphisms.\n-/\nstructure OplaxTrans (F G : B ⥤ᴸ C) where\n  /-- The component 1-morphisms of an oplax transformation. -/\n  app (a : B) : F.obj a ⟶ G.obj a\n  /-- The 2-morphisms underlying the oplax naturality constraint. -/\n  naturality {a b : B} (f : a ⟶ b) : F.map f ≫ app b ⟶ app a ≫ G.map f\n  /-- Naturality of the oplax naturality constraint. -/\n  naturality_naturality {a b : B} {f g : a ⟶ b} (η : f ⟶ g) :\n      F.map₂ η ▷ app b ≫ naturality g = naturality f ≫ app a ◁ G.map₂ η := by\n    cat_disch\n  naturality_id (a : B) :\n      F.mapId a ▷ app a ≫ naturality (𝟙 a) =\n        (λ_ (app a)).hom ≫ (ρ_ (app a)).inv ≫ app a ◁ G.mapId a := by\n    cat_disch\n  naturality_comp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) :\n      F.mapComp f g ▷ app c ≫ naturality (f ≫ g) =\n        (α_ _ _ _).hom ≫ F.map f ◁ naturality g ≫\n          (α_ _ _ _).inv ≫ naturality f ▷ G.map g ≫ (α_ _ _ _).hom ≫\n            app a ◁ G.mapComp f g := by\n    cat_disch\n\nnamespace OplaxTrans\n\nattribute [reassoc (attr := simp)] naturality_naturality naturality_id naturality_comp\n\nvariable {F G H : B ⥤ᴸ C} (η : OplaxTrans F G) (θ : OplaxTrans G H)\n\nvariable (F) in\n/-- The identity oplax transformation. -/\ndef id : OplaxTrans F F where\n  app a := 𝟙 (F.obj a)\n  naturality {_ _} f := (ρ_ (F.map f)).hom ≫ (λ_ (F.map f)).inv\n\ninstance : Inhabited (OplaxTrans F F) :=\n  ⟨id F⟩\n\n/-- Auxiliary definition for `vComp`. -/\nabbrev vCompApp (a : B) : F.obj a ⟶ H.obj a := η.app a ≫ θ.app a\n\n/-- Auxiliary definition for `vComp`. -/\nabbrev vCompNaturality {a b : B} (f : a ⟶ b) :\n    F.map f ≫ η.app b ≫ θ.app b ⟶ (η.app a ≫ θ.app a) ≫ H.map f :=\n  (α_ _ _ _).inv ≫ η.naturality f ▷ θ.app b ≫ (α_ _ _ _).hom ≫\n    η.app a ◁ θ.naturality f ≫ (α_ _ _ _).inv\n\nTarget:\ntheorem vComp_naturality_naturality {a b : B} {f g : a ⟶ b} (β : f ⟶ g) :\n    F.map₂ β ▷ η.vCompApp θ b ≫ η.vCompNaturality θ g =\n      η.vCompNaturality θ f ≫ η.vCompApp θ a ◁ H.map₂ β :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_325825a927e5","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"a78cef6cfb9e129191d6991a931fff261f59f3f6a2a11e298e4eae50d348cd40","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Bicategory","family_id":"vcomp_naturality_naturality","file_id":"mathlib/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Lax.lean","sample_id":"325825a927e5735338dd52a28d2dff4f12d1ad248dc0bc8ca3daba7d87f856c6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1927463d606bf10bea7c2ddb5c31b00ff037d766b129f9620262d1b12ca44ebf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f7a04a0bdfb9132901b240788f15f017596751bca9984db5e9bf9d3c080b9f4c","source_sha256":"b91134b0321c98fdbc750a31ba0314dea5693d2a0ca446a2ffe21e21866c8af5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← tsum_dirichletSummand χ hs]\n  apply eulerProduct_completely_multiplicative <| summable_dirichletSummand χ hs","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":2,"token_jaccard":0.071429,"token_length_ratio":0.133333},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"634cfe713e0c566ce7a19af7bdb61ece92a7405e61e54ec080facf6a6dfc2c3e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.EulerProduct.ExpLog\npublic import Mathlib.NumberTheory.LSeries.Dirichlet\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# The Euler Product for the Riemann Zeta Function and Dirichlet L-Series\n\nThe first main result of this file is the Euler Product formula for the Riemann ζ function\n$$\\prod_p \\frac{1}{1 - p^{-s}}\n   = \\lim_{n \\to \\infty} \\prod_{p < n} \\frac{1}{1 - p^{-s}} = \\zeta(s)$$\nfor $s$ with real part $> 1$ ($p$ runs through the primes).\n`riemannZeta_eulerProduct` is the second equality above. There are versions\n`riemannZeta_eulerProduct_hasProd` and `riemannZeta_eulerProduct_tprod` in terms of `HasProd`\nand `tprod`, respectively.\n\nThe second result is `dirichletLSeries_eulerProduct` (with variants\n`dirichletLSeries_eulerProduct_hasProd` and `dirichletLSeries_eulerProduct_tprod`),\nwhich is the analogous statement for Dirichlet L-series.\n-/\n\n@[expose] public section\n\nopen Complex\n\nvariable {s : ℂ}\n\n/-- When `s ≠ 0`, the map `n ↦ n^(-s)` is completely multiplicative and vanishes at zero. -/\nnoncomputable\ndef riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where\n  toFun n := (n : ℂ) ^ (-s)\n  map_zero' := by simp [hs]\n  map_one' := by simp\n  map_mul' m n := by\n    simpa only [Nat.cast_mul, ofReal_natCast]\n      using mul_cpow_ofReal_nonneg m.cast_nonneg n.cast_nonneg _\n\n/-- When `χ` is a Dirichlet character and `s ≠ 0`, the map `n ↦ χ n * n^(-s)` is completely\nmultiplicative and vanishes at zero. -/\nnoncomputable\ndef dirichletSummandHom {n : ℕ} (χ : DirichletCharacter ℂ n) (hs : s ≠ 0) : ℕ →*₀ ℂ where\n  toFun n := χ n * (n : ℂ) ^ (-s)\n  map_zero' := by simp [hs]\n  map_one' := by simp\n  map_mul' m n := by\n    simp_rw [← ofReal_natCast]\n    simpa only [Nat.cast_mul, IsUnit.mul_iff, not_and, map_mul, ofReal_mul,\n      mul_cpow_ofReal_nonneg m.cast_nonneg n.cast_nonneg _]\n      using mul_mul_mul_comm ..\n\n/-- When `s.re > 1`, the map `n ↦ n^(-s)` is norm-summable. -/\nlemma summable_riemannZetaSummand (hs : 1 < s.re) :\n    Summable (fun n ↦ ‖riemannZetaSummandHom (ne_zero_of_one_lt_re hs) n‖) := by\n  simp only [riemannZetaSummandHom, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]\n  convert! Real.summable_nat_rpow_inv.mpr hs with n\n  rw [← ofReal_natCast,\n    norm_cpow_eq_rpow_re_of_nonneg (Nat.cast_nonneg n) <| re_neg_ne_zero_of_one_lt_re hs,\n    neg_re, Real.rpow_neg <| Nat.cast_nonneg n]\n\nlemma tsum_riemannZetaSummand (hs : 1 < s.re) :\n    ∑' (n : ℕ), riemannZetaSummandHom (ne_zero_of_one_lt_re hs) n = riemannZeta s := by\n  have hsum := summable_riemannZetaSummand hs\n  rw [zeta_eq_tsum_one_div_nat_add_one_cpow hs, hsum.of_norm.tsum_eq_zero_add, map_zero, zero_add]\n  simp only [riemannZetaSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk,\n    Nat.cast_add, Nat.cast_one, one_div]\n\n/-- When `s.re > 1`, the map `n ↦ χ(n) * n^(-s)` is norm-summable. -/\nlemma summable_dirichletSummand {N : ℕ} (χ : DirichletCharacter ℂ N) (hs : 1 < s.re) :\n    Summable (fun n ↦ ‖dirichletSummandHom χ (ne_zero_of_one_lt_re hs) n‖) := by\n  simp only [dirichletSummandHom, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, norm_mul]\n  exact (summable_riemannZetaSummand hs).of_nonneg_of_le (fun _ ↦ by positivity)\n    (fun n ↦ mul_le_of_le_one_left (norm_nonneg _) <| χ.norm_le_one n)\n\nopen scoped LSeries.notation in\nlemma tsum_dirichletSummand {N : ℕ} (χ : DirichletCharacter ℂ N) (hs : 1 < s.re) :\n    ∑' (n : ℕ), dirichletSummandHom χ (ne_zero_of_one_lt_re hs) n = L ↗χ s := by\n  simp only [dirichletSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, LSeries,\n    LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs), div_eq_mul_inv]\n\nopen Filter Nat Topology EulerProduct\n\n/-- The Euler product for the Riemann ζ function, valid for `s.re > 1`.\nThis version is stated in terms of `HasProd`. -/\ntheorem riemannZeta_eulerProduct_hasProd (hs : 1 < s.re) :\n    HasProd (fun p : Primes ↦ (1 - (p : ℂ) ^ (-s))⁻¹) (riemannZeta s) := by\n  rw [← tsum_riemannZetaSummand hs]\n  apply eulerProduct_completely_multiplicative_hasProd <| summable_riemannZetaSummand hs\n\n/-- The Euler product for the Riemann ζ function, valid for `s.re > 1`.\nThis version is stated in terms of `tprod`. -/\ntheorem riemannZeta_eulerProduct_tprod (hs : 1 < s.re) :\n    ∏' p : Primes, (1 - (p : ℂ) ^ (-s))⁻¹ = riemannZeta s :=\n  (riemannZeta_eulerProduct_hasProd hs).tprod_eq\n\n/-- The Euler product for the Riemann ζ function, valid for `s.re > 1`.\nThis version is stated in the form of convergence of finite partial products. -/\ntheorem riemannZeta_eulerProduct (hs : 1 < s.re) :\n    Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - (p : ℂ) ^ (-s))⁻¹) atTop\n      (𝓝 (riemannZeta s)) := by\n  rw [← tsum_riemannZetaSummand hs]\n  apply eulerProduct_completely_multiplicative <| summable_riemannZetaSummand hs\n\nopen scoped LSeries.notation\n\n/-- The Euler product for Dirichlet L-series, valid for `s.re > 1`.\nThis version is stated in terms of `HasProd`. -/\ntheorem DirichletCharacter.LSeries_eulerProduct_hasProd {N : ℕ} (χ : DirichletCharacter ℂ N)\n    (hs : 1 < s.re) :\n    HasProd (fun p : Primes ↦ (1 - χ p * (p : ℂ) ^ (-s))⁻¹) (L ↗χ s) := by\n  rw [← tsum_dirichletSummand χ hs]\n  convert! eulerProduct_completely_multiplicative_hasProd <| summable_dirichletSummand χ hs\n\n/-- The Euler product for Dirichlet L-series, valid for `s.re > 1`.\nThis version is stated in terms of `tprod`. -/\ntheorem DirichletCharacter.LSeries_eulerProduct_tprod {N : ℕ} (χ : DirichletCharacter ℂ N)\n    (hs : 1 < s.re) :\n    ∏' p : Primes, (1 - χ p * (p : ℂ) ^ (-s))⁻¹ = L ↗χ s :=\n  (DirichletCharacter.LSeries_eulerProduct_hasProd χ hs).tprod_eq\n\n/-- The Euler product for Dirichlet L-series, valid for `s.re > 1`.\nThis version is stated in the form of convergence of finite partial products. -/\n\nTarget:\ntheorem DirichletCharacter.LSeries_eulerProduct {N : ℕ} (χ : DirichletCharacter ℂ N)\n    (hs : 1 < s.re) :\n    Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - χ p * (p : ℂ) ^ (-s))⁻¹) atTop\n      (𝓝 (L ↗χ s)) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/EulerProduct","family_id":"dirichletcharacter","file_id":"mathlib/Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean","sample_id":"f7a04a0bdfb9132901b240788f15f017596751bca9984db5e9bf9d3c080b9f4c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a329996be1763b313924738bf153d1074388b636f52e211888a4d4f3a26dea07","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e335eb5a3b25d52b6c9d5323a6623c91e51f870e7ad986b02c2746a667115775","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1f97d0501e044f00fde760711297ef657cb86ce42d441db370fdf596ff19fb0c","source_sha256":"155010ab3058b3fe6da475eb20a319e6c48a44ac24aa3149ecf584265954195c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using w\n\n-- It would be nice to mark this as `aesop cases` if\n-- https://github.com/leanprover-community/aesop/issues/59\n-- is resolved.","hard_negative":false,"metrics":{"chosen_tokens":44,"rejected_tokens":51,"token_jaccard":0.967742,"token_length_ratio":1.159091},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"634e54a5b06cd9594b6ca2b20da84e8e030c881b55ab4b0e475264e8137f277e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Logic.Equiv.Defs\npublic import Mathlib.Tactic.MkIffOfInductiveProp\npublic import Mathlib.Tactic.PPWithUniv\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Small types\n\nA type is `w`-small if there exists an equivalence to some `S : Type w`.\n\nWe provide a noncomputable model `Shrink α : Type w`, and `equivShrink α : α ≃ Shrink α`.\n\nA subsingleton type is `w`-small for any `w`.\n\nIf `α ≃ β`, then `Small.{w} α ↔ Small.{w} β`.\n\nSee `Mathlib/Logic/Small/Basic.lean` for further instances and theorems.\n-/\n\n@[expose] public section\n\nuniverse u w v v'\n\n/-- A type is `Small.{w}` if there exists an equivalence to some `S : Type w`.\n-/\n-- After https://github.com/leanprover/lean4/pull/12286 and\n-- https://github.com/leanprover/lean4/pull/12423: `v` is a true output (determined by `α`),\n-- but we need the attribute to prevent `w` from also being treated as output.\n-- See Note [universe output parameters and typeclass caching].\n@[univ_out_params v, mk_iff, pp_with_univ]\nclass Small (α : Type v) : Prop where\n  /-- If a type is `Small.{w}`, then there exists an equivalence with some `S : Type w` -/\n  equiv_small : ∃ S : Type w, Nonempty (α ≃ S)\n\n/-- Constructor for `Small α` from an explicit witness type and equivalence.\n-/\ntheorem Small.mk' {α : Type v} {S : Type w} (e : α ≃ S) : Small.{w} α :=\n  ⟨⟨S, ⟨e⟩⟩⟩\n\n/-- An arbitrarily chosen model in `Type w` for a `w`-small type.\n-/\n@[pp_with_univ, no_expose]\ndef Shrink (α : Type v) [Small.{w} α] : Type w :=\n  Classical.choose (@Small.equiv_small α _)\n\n/-- The noncomputable equivalence between a `w`-small type and a model.\n-/\n@[no_expose]\nnoncomputable def equivShrink (α : Type v) [Small.{w} α] : α ≃ Shrink α :=\n  Nonempty.some (Classical.choose_spec (@Small.equiv_small α _))\n\n@[ext]\n\nTarget:\ntheorem Shrink.ext {α : Type v} [Small.{w} α] {x y : Shrink α}\n    (w : (equivShrink _).symm x = (equivShrink _).symm y) : x = y :=\n\nProof body:\n","rejected":"```lean\nby\n  simpa using w\n\n-- It would be nice to mark this as `aesop cases` if\n-- https://github.com/leanprover-community/aesop/issues/59\n-- is resolved.\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Small","family_id":"shrink","file_id":"mathlib/Mathlib/Logic/Small/Defs.lean","sample_id":"1f97d0501e044f00fde760711297ef657cb86ce42d441db370fdf596ff19fb0c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ea338c8c6deda19356f39b2f9dc6bfce64075bf47da045c1a31eb2b19f1fe070","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"67ec5d4508cf381b8aa1ef1018c768d8313e9324cd439b93e03bba971352c00c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7c30455a13ea31c0fdc5ee48caccc8329851df034a9e411c0a2bb1d61bbf2a76","source_sha256":"dc37d7e189f3a4d32e4029e0df16c9eeb5ca8b81845b68289b84bb8e5dd20ee8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h => ?_, fun _ => inferInstance⟩\n  contrapose! h\n  infer_instance","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":2,"token_jaccard":0.066667,"token_length_ratio":0.111111},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"63d2385d914bc74f975fe25fe9cc5282f49069d52970ace258073ad6dc4990b3","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Sets.Closeds\npublic import Mathlib.Topology.QuasiSeparated\n\nNamespace:\nTopologicalSpace.Compacts\n\nLocal context:\n/-\nCopyright (c) 2020 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Yaël Dillies\n-/\n/-!\n# Compact sets\n\nWe define a few types of compact sets in a topological space.\n\n## Main Definitions\n\nFor a topological space `α`,\n* `TopologicalSpace.Compacts α`: The type of compact sets.\n* `TopologicalSpace.NonemptyCompacts α`: The type of non-empty compact sets.\n* `TopologicalSpace.PositiveCompacts α`: The type of compact sets with non-empty interior.\n* `TopologicalSpace.CompactOpens α`: The type of compact open sets. This is a central object in the\n  study of spectral spaces.\n-/\n\n@[expose] public section\n\n\nopen Set\n\nvariable {α β γ : Type*} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]\n\nnamespace TopologicalSpace\n\n/-! ### Compact sets -/\n\n/-- The type of compact sets of a topological space. -/\nstructure Compacts (α : Type*) [TopologicalSpace α] where\n  /-- the carrier set, i.e. the points in this set -/\n  carrier : Set α\n  isCompact' : IsCompact carrier\n\nnamespace Compacts\n\ninstance : SetLike (Compacts α) α where\n  coe := Compacts.carrier\n  coe_injective s t h := by cases s; cases t; congr\n\ninstance : PartialOrder (Compacts α) := .ofSetLike (Compacts α) α\n\n/-- See Note [custom simps projection]. -/\ndef Simps.coe (s : Compacts α) : Set α := s\n\ninitialize_simps_projections Compacts (carrier → coe, as_prefix coe)\n\nprotected theorem isCompact (s : Compacts α) : IsCompact (s : Set α) :=\n  s.isCompact'\n\ninstance (K : Compacts α) : CompactSpace K :=\n  isCompact_iff_compactSpace.1 K.isCompact\n\n/-- Reinterpret a compact as a closed set. -/\n@[simps]\ndef toCloseds [T2Space α] (s : Compacts α) : Closeds α :=\n  ⟨s, s.isCompact.isClosed⟩\n\n@[simp]\ntheorem mem_toCloseds [T2Space α] {x : α} {s : Compacts α} :\n    x ∈ s.toCloseds ↔ x ∈ s :=\n  Iff.rfl\n\ntheorem toCloseds_injective [T2Space α] : Function.Injective (toCloseds (α := α)) :=\n  .of_comp (f := SetLike.coe) SetLike.coe_injective\n\ninstance : CanLift (Set α) (Compacts α) (↑) IsCompact where prf K hK := ⟨⟨K, hK⟩, rfl⟩\n\n@[ext]\nprotected theorem ext {s t : Compacts α} (h : (s : Set α) = t) : s = t :=\n  SetLike.ext' h\n\n@[simp]\ntheorem coe_mk (s : Set α) (h) : (mk s h : Set α) = s :=\n  rfl\n\n@[simp]\ntheorem carrier_eq_coe (s : Compacts α) : s.carrier = s :=\n  rfl\n\ninstance : Max (Compacts α) :=\n  ⟨fun s t => ⟨s ∪ t, s.isCompact.union t.isCompact⟩⟩\n\ninstance [T2Space α] : Min (Compacts α) :=\n  ⟨fun s t => ⟨s ∩ t, s.isCompact.inter t.isCompact⟩⟩\n\ninstance [CompactSpace α] : Top (Compacts α) :=\n  ⟨⟨univ, isCompact_univ⟩⟩\n\ninstance : Bot (Compacts α) :=\n  ⟨⟨∅, isCompact_empty⟩⟩\n\ninstance : SemilatticeSup (Compacts α) :=\n  fast_instance% SetLike.coe_injective.semilatticeSup _ .rfl .rfl fun _ _ ↦ rfl\n\ninstance [T2Space α] : DistribLattice (Compacts α) :=\n  fast_instance% SetLike.coe_injective.distribLattice _ .rfl .rfl (fun _ _ ↦ rfl) fun _ _ ↦ rfl\n\ninstance : OrderBot (Compacts α) :=\n  fast_instance% OrderBot.lift ((↑) : _ → Set α) (fun _ _ => id) rfl\n\ninstance [CompactSpace α] : BoundedOrder (Compacts α) :=\n  fast_instance% BoundedOrder.lift ((↑) : _ → Set α) (fun _ _ => id) rfl rfl\n\n/-- The type of compact sets is inhabited, with default element the empty set. -/\ninstance : Inhabited (Compacts α) := ⟨⊥⟩\n\ninstance [IsEmpty α] : Unique (Compacts α) where\n  uniq _ := Compacts.ext (Subsingleton.elim _ _)\n\n@[simp]\ntheorem coe_sup (s t : Compacts α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t :=\n  rfl\n\n@[simp]\ntheorem coe_inf [T2Space α] (s t : Compacts α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t :=\n  rfl\n\n@[simp]\ntheorem coe_top [CompactSpace α] : (↑(⊤ : Compacts α) : Set α) = univ :=\n  rfl\n\n@[simp]\ntheorem coe_bot : (↑(⊥ : Compacts α) : Set α) = ∅ :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_eq_empty {s : Compacts α} : (s : Set α) = ∅ ↔ s = ⊥ :=\n  SetLike.coe_injective.eq_iff' rfl\n\n@[simp]\ntheorem coe_nonempty {s : Compacts α} : (s : Set α).Nonempty ↔ s ≠ ⊥ :=\n  nonempty_iff_ne_empty.trans coe_eq_empty.not\n\n@[simp]\ntheorem coe_finset_sup {ι : Type*} {s : Finset ι} {f : ι → Compacts α} :\n    (↑(s.sup f) : Set α) = s.sup fun i => ↑(f i) := by\n  refine Finset.cons_induction_on s rfl fun a s _ h => ?_\n  simp_rw [Finset.sup_cons, coe_sup, sup_eq_union]\n  congr\n\n@[simps]\ninstance : Singleton α (Compacts α) where\n  singleton x := ⟨{x}, isCompact_singleton⟩\n\n@[simp]\ntheorem mem_singleton (x y : α) : x ∈ ({y} : Compacts α) ↔ x = y :=\n  Iff.rfl\n\n@[simp]\ntheorem toCloseds_singleton [T2Space α] (x : α) : toCloseds {x} = {x} :=\n  rfl\n\ntheorem singleton_injective : Function.Injective ({·} : α → Compacts α) :=\n  .of_comp (f := SetLike.coe) Set.singleton_injective\n\n@[simp]\ntheorem singleton_inj {x y : α} : ({x} : Compacts α) = {y} ↔ x = y :=\n  singleton_injective.eq_iff\n\ninstance [Nonempty α] : Nontrivial (Compacts α) := by\n  constructor\n  obtain ⟨x⟩ := ‹Nonempty α›\n  exact ⟨⊥, {x}, ne_of_apply_ne SetLike.coe (Set.empty_ne_singleton x)⟩\n\n@[simp]\n\nTarget:\ntheorem subsingleton_iff : Subsingleton (Compacts α) ↔ IsEmpty α :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_7c30455a13ea","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"a6c490ca941c9b42e2ab8981efb38dcc907ea993d97bb32302ae1a52b003eb25","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Sets","family_id":"subsingleton_iff","file_id":"mathlib/Mathlib/Topology/Sets/Compacts.lean","sample_id":"7c30455a13ea31c0fdc5ee48caccc8329851df034a9e411c0a2bb1d61bbf2a76"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0678ccbc2752fb8380a44655810f825f8a9182914dd6e363b01acde3b055d878","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2d5cb89a9cfda0560e3764a0fa8315a7c93635c11c81d041a427e94157d40461","source_sha256":"1c0dcf6a3ce55bec7a88672314ea6ae6741a6599903854de40abb691f935e16b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction y generalizing x with\n  | nil => simpa using h\n  | cons y ys ih => exact T.prop (ih (by simpa))","hard_negative":false,"metrics":{"chosen_tokens":29,"rejected_tokens":2,"token_jaccard":0.045455,"token_length_ratio":0.068966},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"640dcd7df4f7de5a236b33951f7fefab47a70e5015704c0c3904c6f7e803618e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.CompleteLattice.SetLike\n\nNamespace:\nDescriptive.Tree\n\nLocal context:\n/-\nCopyright (c) 2024 Sven Manthe. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sven Manthe\n-/\n/-!\n# Trees in the sense of descriptive set theory\n\nThis file defines trees of depth `ω` in the sense of descriptive set theory as sets of finite\nsequences that are stable under taking prefixes.\n\n## Main declarations\n\n* `tree A`: a (possibly infinite) tree of depth at most `ω` with nodes in `A`\n-/\n\n@[expose] public section\n\nnamespace Descriptive\n\n/-- A tree is a set of finite sequences, implemented as `List A`, that is stable under\n  taking prefixes. For the definition we use the equivalent property `x ++ [a] ∈ T → x ∈ T`,\n  which is more convenient to check. We define `tree A` as a complete sublattice of\n  `Set (List A)`, which coerces to the type of trees on `A`. -/\ndef tree (A : Type*) : CompleteSublattice (Set (List A)) :=\n  CompleteSublattice.mk' {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T}\n    (by rintro S hS x a ⟨t, ht, hx⟩; use t, ht, hS ht hx)\n    (by rintro S hS x a h T hT; exact hS hT <| h T hT)\n\n@[simps!] instance (A : Type*) : SetLike (tree A) (List A) := SetLike.instSubtypeSet\n\nexample (A : Type*) : PartialOrder (tree A) := inferInstance\n\nnamespace Tree\nvariable {A : Type*} {S T : tree A}\n\nTarget:\nlemma mem_of_append {x y : List A} (h : x ++ y ∈ T) : x ∈ T :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Descriptive","family_id":"mem_of_append","file_id":"mathlib/Mathlib/SetTheory/Descriptive/Tree.lean","sample_id":"2d5cb89a9cfda0560e3764a0fa8315a7c93635c11c81d041a427e94157d40461"},"split":"train"}
{"binding":{"chosen_proof_sha256":"de8a32b5f73d36b3c5675db1bfb4e3c6a3d479eee57e39397db99ac101c60702","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"460fd408bbfa2c9d3696e93eaf82711398320704b30a3dd64ec1d4fbb298afc9","source_sha256":"fe20ac9adb6e069125c46c6cc0a23c1dfac91d49367534266b3b31063c05d603","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun H ↦ ?_, isTopologicallyNilpotent_of_constantCoeff⟩\n  replace H : Tendsto (fun n ↦ constantCoeff (f ^ n)) atTop (nhds 0) :=\n    continuous_constantCoeff R |>.tendsto' 0 0 constantCoeff_zero |>.comp H\n  simpa only [map_pow] using! H","hard_negative":false,"metrics":{"chosen_tokens":54,"rejected_tokens":2,"token_jaccard":0.025,"token_length_ratio":0.037037},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"64420a671444cbc235088b592bf837243bbb5a41c7867e300934638a683548c3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finsupp.Interval\npublic import Mathlib.RingTheory.Ideal.Quotient.Defs\npublic import Mathlib.RingTheory.MvPowerSeries.PiTopology\npublic import Mathlib.Topology.Algebra.LinearTopology\npublic import Mathlib.RingTheory.TwoSidedIdeal.Operations\n\nNamespace:\nMvPowerSeries.LinearTopology\n\nLocal context:\n/-\nCopyright (c) 2024 Antoine Chambert-Loir, María Inés de Frutos-Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir, María Inés de Frutos-Fernández\n-/\n/-! # Linear topology on the ring of multivariate power series\n\n- `MvPowerSeries.LinearTopology.basis`: the ideals of the ring of multivariate power series\n  all coefficients the exponent of which is smaller than some bound vanish.\n\n- `MvPowerSeries.LinearTopology.hasBasis_nhds_zero` :\n  the two-sided ideals from `MvPowerSeries.LinearTopology.basis` form a basis\n  of neighborhoods of `0` if the topology of `R` is (left and right) linear.\n\n## Instances :\n\nIf `R` has a linear topology, then the product topology on `MvPowerSeries σ R`\nis a linear topology.\n\nThis applies in particular when `R` has the discrete topology.\n\n## Note\n\nIf we had an analogue of `PolynomialModule` for power series,\nmeaning that we could consider the `R⟦X⟧`-module `M⟦X⟧` when `M` is an `R`-module,\nthen one could prove that `M⟦X⟧` is linearly topologized over `R⟦X⟧`\nwhenever `M` is linearly topologized over `R`.\nTo recover the ring case, it would remain to show that the isomorphism between\n`Rᵐᵒᵖ⟦X⟧` and `R⟦X⟧ᵐᵒᵖ` identifies their respective actions on `R⟦X⟧`.\n(And likewise in the multivariate case.)\n\n-/\n\n@[expose] public section\n\nnamespace MvPowerSeries\n\nnamespace LinearTopology\n\nopen scoped Topology\n\nopen Set SetLike Filter\n\n/-- The underlying family for the basis of ideals in a multivariate power series ring. -/\nnoncomputable def basis (σ : Type*) (R : Type*) [Ring R] (Jd : TwoSidedIdeal R × (σ →₀ ℕ)) :\n    TwoSidedIdeal (MvPowerSeries σ R) :=\n  TwoSidedIdeal.mk' {f | ∀ e ≤ Jd.2, coeff e f ∈ Jd.1}\n    (by simp [coeff_zero])\n    (fun hf hg e he ↦ by rw [map_add]; exact add_mem (hf e he) (hg e he))\n    (fun {f} hf e he ↦ by simp only [map_neg, neg_mem, hf e he])\n    (fun {f g} hg e he ↦ by\n      classical\n      rw [coeff_mul]\n      apply sum_mem\n      rintro uv huv\n      exact TwoSidedIdeal.mul_mem_left _ _ _ (hg _ (le_trans (Finset.antidiagonal.snd_le huv) he)))\n    (fun {f g} hf e he ↦ by\n      classical\n      rw [coeff_mul]\n      apply sum_mem\n      rintro uv huv\n      exact TwoSidedIdeal.mul_mem_right _ _ _ (hf _ (le_trans (Finset.antidiagonal.fst_le huv) he)))\n\nvariable {σ : Type*} {R : Type*} [Ring R]\n\n/-- A power series `f` belongs to the two-sided ideal `basis σ R ⟨J, d⟩`\nif and only if `coeff e f ∈ J` for all `e ≤ d`. -/\ntheorem mem_basis_iff {f : MvPowerSeries σ R} {Jd : TwoSidedIdeal R × (σ →₀ ℕ)} :\n    f ∈ basis σ R Jd ↔ ∀ e ≤ Jd.2, coeff e f ∈ Jd.1 := by\n  simp [basis]\n\n/-- If `J ≤ K` and `e ≤ d`, then we have the inclusion of two-sided ideals\n`basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e,>`. -/\ntheorem basis_le {Jd Ke : TwoSidedIdeal R × (σ →₀ ℕ)} (hJK : Jd.1 ≤ Ke.1) (hed : Ke.2 ≤ Jd.2) :\n    basis σ R Jd ≤ basis σ R Ke :=\n  fun _ ↦ forall_imp (fun _ h hue ↦ hJK (h (le_trans hue hed)))\n\n/-- `basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e⟩` if and only if `J ≤ K` and `e ≤ d`. -/\ntheorem basis_le_iff {J K : TwoSidedIdeal R} {d e : σ →₀ ℕ} (hK : K ≠ ⊤) :\n    basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e⟩ ↔ J ≤ K ∧ e ≤ d := by\n  classical\n  constructor\n  · simp only [basis, TwoSidedIdeal.le_iff, TwoSidedIdeal.coe_mk', setOf_subset_setOf]\n    intro h\n    constructor\n    · intro x hx\n      have (d' : _) : coeff d' (C (σ := σ) x) ∈ J := by\n        rw [coeff_C]; split_ifs <;> [exact hx; exact J.zero_mem]\n      simpa using h (C x) (fun _ _ ↦ this _) _ zero_le\n    · by_contra h'\n      apply hK\n      rw [eq_top_iff]\n      intro x _\n      have (d') (hd'_le : d' ≤ d) : coeff d' (monomial e x) ∈ J := by\n        rw [coeff_monomial]\n        split_ifs with hd' <;> [exact (h' (hd' ▸ hd'_le)).elim; exact J.zero_mem]\n      simpa using h (monomial e x) this _ le_rfl\n  · rintro ⟨hJK, hed⟩\n    exact basis_le hJK hed\n\nvariable [TopologicalSpace R]\n\n-- We endow MvPowerSeries σ R with the product topology.\nopen WithPiTopology\n\nset_option backward.isDefEq.respectTransparency false in\n/-- If the ring `R` is endowed with a linear topology, then the sets `↑basis σ R (J, d)`,\nfor `J : TwoSidedIdeal R` which are neighborhoods of `0 : R` and `d : σ →₀ ℕ`,\nconstitute a basis of neighborhoods of `0 : MvPowerSeries σ R` for the product topology. -/\nlemma hasBasis_nhds_zero [IsLinearTopology R R] [IsLinearTopology Rᵐᵒᵖ R] :\n    (𝓝 0 : Filter (MvPowerSeries σ R)).HasBasis\n      (fun Id : TwoSidedIdeal R × (σ →₀ ℕ) ↦ (Id.1 : Set R) ∈ 𝓝 0)\n      (fun Id ↦ basis _ _ Id) := by\n  classical\n  rw [nhds_pi]\n  refine IsLinearTopology.hasBasis_twoSidedIdeal.pi_self.to_hasBasis ?_ ?_\n  · intro ⟨D, I⟩ ⟨hD, hI⟩\n    refine ⟨⟨I, Finset.sup hD.toFinset id⟩, hI, fun f hf d hd ↦ ?_⟩\n    rw [SetLike.mem_coe, mem_basis_iff] at hf\n    convert! hf _ <| Finset.le_sup (hD.mem_toFinset.mpr hd)\n  · intro ⟨I, d⟩ hI\n    refine ⟨⟨Iic d, I⟩, ⟨finite_Iic d, hI⟩, ?_⟩\n    simpa [basis, coeff_apply, Iic, Set.pi] using! subset_rfl\n\n/-- The topology on `MvPowerSeries` is a left linear topology\n  when the ring of coefficients has a linear topology. -/\ninstance [IsLinearTopology R R] [IsLinearTopology Rᵐᵒᵖ R] :\n    IsLinearTopology (MvPowerSeries σ R) (MvPowerSeries σ R) :=\n  IsLinearTopology.mk_of_hasBasis' _ hasBasis_nhds_zero TwoSidedIdeal.mul_mem_left\n\n/-- The topology on `MvPowerSeries` is a right linear topology\n  when the ring of coefficients has a linear topology. -/\ninstance [IsLinearTopology R R] [IsLinearTopology Rᵐᵒᵖ R] :\n    IsLinearTopology (MvPowerSeries σ R)ᵐᵒᵖ (MvPowerSeries σ R) :=\n  IsLinearTopology.mk_of_hasBasis' _ hasBasis_nhds_zero (fun J _ _ hg ↦ J.mul_mem_right _ _ hg)\n\ntheorem isTopologicallyNilpotent_of_constantCoeff\n    {R : Type*} [CommRing R] [TopologicalSpace R] [IsLinearTopology R R]\n    {f : MvPowerSeries σ R} (hf : IsTopologicallyNilpotent (constantCoeff f)) :\n    IsTopologicallyNilpotent f := by\n  simp_rw [IsTopologicallyNilpotent, tendsto_iff_coeff_tendsto, coeff_zero,\n    IsLinearTopology.hasBasis_ideal.tendsto_right_iff]\n  intro d I hI\n  replace hf := hf.eventually_mem hI\n  simp_rw [eventually_atTop, SetLike.mem_coe, ← Ideal.Quotient.eq_zero_iff_mem,\n    map_pow, ← coeff_map, ← constantCoeff_map] at hf ⊢\n  obtain ⟨N, hN⟩ := hf\n  use N + d.degree\n  intro n hn\n  simpa only [map_pow] using coeff_eq_zero_of_constantCoeff_nilpotent (hN N le_rfl) hn\n\n/-- Assuming the base ring has a linear topology, the powers of a `MvPowerSeries` converge to 0\niff its constant coefficient is topologically nilpotent.\n\nSee also `MvPowerSeries.WithPiTopology.isTopologicallyNilpotent_iff_constantCoeff_isNilpotent`. -/\n\nTarget:\ntheorem isTopologicallyNilpotent_iff_constantCoeff\n    {R : Type*} [CommRing R] [TopologicalSpace R] [IsLinearTopology R R] (f : MvPowerSeries σ R) :\n    Tendsto (fun n : ℕ => f ^ n) atTop (nhds 0) ↔\n      IsTopologicallyNilpotent (constantCoeff f) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/MvPowerSeries","family_id":"istopologicallynilpotent_iff_constantcoeff","file_id":"mathlib/Mathlib/RingTheory/MvPowerSeries/LinearTopology.lean","sample_id":"460fd408bbfa2c9d3696e93eaf82711398320704b30a3dd64ec1d4fbb298afc9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"078c3ba86f93f2f5c0b7638f81f2db59b5ef54f0de3d73360e15d4f2f0ce975a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7825f2858d4abf33c4a06f455ce84c7b780373bc10da71f091ba3ea07fa28616","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5f55f4262f31682a3e12d8625b11abf778bb7c39ed9ebef5aef18cf9e8a8d94d","source_sha256":"5719c6ab1b79cb92451b632f8631c4fdd8e58d363dc71c3722b36068c3f61537","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [normThreshold_zero]\n  calc a / (1 - √2)\n  _ ≤ 0 := div_nonpos_of_nonneg_of_nonpos ha_pos.le (by simp)\n  _ < a := ha_pos","hard_negative":true,"metrics":{"chosen_tokens":32,"rejected_tokens":3,"token_jaccard":0.038462,"token_length_ratio":0.09375},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"6488fb7e4416697b83503b794308f7c1c74f5ec15d365ad6bf10641665f194c6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecificLimits.ArithmeticGeometric\npublic import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap\npublic import Mathlib.MeasureTheory.Function.L1Space.Integrable\n\nNamespace:\nProbabilityTheory.Fernique\n\nLocal context:\n/-\nCopyright (c) 2025 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n-/\n/-!\n# Fernique's theorem for rotation-invariant measures\n\nLet `μ` be a finite measure on a second-countable normed space `E` such that the product measure\n`μ.prod μ` on `E × E` is invariant by rotation of angle `-π/4`.\nThen there exists a constant `C > 0` such that the function `x ↦ exp (C * ‖x‖ ^ 2)` is integrable\nwith respect to `μ`.\n\n## Sketch of the proof\n\nThe main case of the proof is for `μ` a probability measure such that there exists a positive\n`a : ℝ` such that `2⁻¹ < μ {x | ‖x‖ ≤ a} < 1`. If `μ` is a probability measure and `a` does not\nexist then we can show that there is a ball with finite radius of measure 1, and the result is true\nfor `C = 1` (for example), since `x ↦ exp (‖x‖ ^ 2)` is almost surely bounded.\nWe then choose such an `a`.\n\nIn order to show the existence of `C` such that `x ↦ exp (C * ‖x‖ ^ 2)` is integrable, we prove as\nintermediate result that for `a, c` with `2⁻¹ < c ≤ μ {x | ‖x‖ ≤ a}`,\nthe integral `∫⁻ x, exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2) ∂μ` is bounded by a finite quantity\n(`logRatio c` is a multiple of `log (c / (1 - c))`). We can then take `C = logRatio c * a⁻¹ ^ 2`.\n\nWe now turn to the proof of the intermediate result.\n\nFirst in `measure_le_mul_measure_gt_le_of_map_rotation_eq_self` we prove that if a measure `μ` is\nsuch that `μ.prod μ` is invariant by rotation of angle `-π/4` then\n`μ {x | ‖x‖ ≤ a} * μ {x | b < ‖x‖} ≤ μ {x | (b - a) / √2 < ‖x‖} ^ 2`.\nThe rotation invariance is used only through that inequality.\n\nWe define a sequence of thresholds `t n` inductively by `t 0 = a` and `t (n + 1) = √2 * t n + a`.\nThey are chosen such that the invariance by rotation gives\n`μ {x | ‖x‖ ≤ a} * μ {x | t (n + 1) < ‖x‖} ≤ μ {x | t n < ‖x‖} ^ 2`.\nThanks to that inequality we can show that `μ {x | t n < ‖x‖}` decreases fast with `n`:\nfor `mₐ = μ {x | ‖x‖ ≤ a}`, `μ {x | t n < ‖x‖} ≤ mₐ * exp (- log (mₐ / (1 - mₐ)) * 2 ^ n)`.\n\nWe cut the space into annuli `{x | t n < ‖x‖ ≤ t n + 1}` and bound the integral separately on\neach annulus. On that set the function `exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2)` is bounded by\n`exp (logRatio c * a⁻¹ ^ 2 * t (n + 1) ^ 2)`, which is in turn less than\n`exp (2⁻¹ * log (c / (1 - c)) * 2 ^ n)` (from the definition of the threshold `t` and `logRatio c`).\nThe measure of the annulus is bounded by `μ {x | t n < ‖x‖}`, for which we derived an upper bound\nabove. The function gets exponentially large, but `μ {x | t n < ‖x‖}` decreases even faster, so the\nintegral is bounded by a quantity of the form `exp (- u * 2 ^ n)` for `u>0`.\nSumming over all annuli (over `n`) gives a finite value for the integral.\n\n## Main statements\n\n* `lintegral_exp_mul_sq_norm_le_of_map_rotation_eq_self`: for `μ` a probability measure\n  whose product with itself is invariant by rotation and for `a, c` with\n  `2⁻¹ < c ≤ μ {x | ‖x‖ ≤ a}`, the integral `∫⁻ x, exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2) ∂μ`\n  is bounded by a quantity that does not depend on `a`.\n* `exists_integrable_exp_sq_of_map_rotation_eq_self`: Fernique's theorem for finite measures\n  whose product is invariant by rotation.\n\n## References\n\n* [Xavier Fernique, *Intégrabilité des vecteurs gaussiens*][fernique1970integrabilite]\n* [Martin Hairer, *An introduction to stochastic PDEs*][hairer2009introduction]\n\n## TODO\n\nFrom the intermediate result `lintegral_exp_mul_sq_norm_le_of_map_rotation_eq_self`,\nwe can deduce bounds on all the moments of the measure `μ` as function of powers of\nthe first moment.\n\n-/\n\n@[expose] public section\n\nopen MeasureTheory ProbabilityTheory Complex NormedSpace Filter\nopen scoped ENNReal NNReal Real Topology\n\nsection Aux\n\nlemma StrictMono.exists_between_of_tendsto_atTop {β : Type*} [LinearOrder β] {t : ℕ → β}\n    (ht_mono : StrictMono t) (ht_tendsto : Tendsto t atTop atTop) {x : β} (hx : t 0 < x) :\n    ∃ n, t n < x ∧ x ≤ t (n + 1) := by\n  have h : ∃ n, x ≤ t n := by\n    simp only [tendsto_atTop_atTop_iff_of_monotone ht_mono.monotone] at ht_tendsto\n    exact ht_tendsto x\n  have h' m := Nat.find_min h (m := m)\n  simp only [not_le] at h'\n  exact ⟨Nat.find h - 1, h' _ (by simp [hx]), by simp [Nat.find_spec h, hx]⟩\n\nend Aux\n\nnamespace ProbabilityTheory\n\nvariable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ℝ E]\n\n/-- The rotation in `E × E` with angle `θ`, as a continuous linear map. -/\nnoncomputable\ndef _root_.ContinuousLinearMap.rotation (θ : ℝ) : E × E →L[ℝ] E × E where\n  toFun := fun x ↦ (Real.cos θ • x.1 + Real.sin θ • x.2, - Real.sin θ • x.1 + Real.cos θ • x.2)\n  map_add' x y := by\n    simp only [Prod.fst_add, smul_add, Prod.snd_add, neg_smul, Prod.mk_add_mk]\n    abel_nf\n  map_smul' c x := by simp [smul_comm c]\n\nlemma _root_.ContinuousLinearMap.rotation_apply (θ : ℝ) (x : E × E) :\n    ContinuousLinearMap.rotation θ x\n     = (Real.cos θ • x.1 + Real.sin θ • x.2, -Real.sin θ • x.1 + Real.cos θ • x.2) := rfl\n\nvariable [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E] {μ : Measure E} {a : ℝ}\n\n/-- If a measure `μ` is such that `μ.prod μ` is invariant by rotation of angle `-π/4` then\n`μ {x | ‖x‖ ≤ a} * μ {x | b < ‖x‖} ≤ μ {x | (b - a) / √2 < ‖x‖} ^ 2`. -/\nlemma measure_le_mul_measure_gt_le_of_map_rotation_eq_self [SFinite μ]\n    (h : (μ.prod μ).map (ContinuousLinearMap.rotation (-(π / 4))) = μ.prod μ)\n    (a b : ℝ) :\n    μ {x | ‖x‖ ≤ a} * μ {x | b < ‖x‖} ≤ μ {x | (b - a) / √2 < ‖x‖} ^ 2 := by\n  calc μ {x | ‖x‖ ≤ a} * μ {x | b < ‖x‖}\n  _ = (μ.prod μ) ({x | ‖x‖ ≤ a} ×ˢ {y | b < ‖y‖}) := by rw [Measure.prod_prod]\n    -- This is the measure of two bands in the plane (draw a picture!)\n  _ = (μ.prod μ) {p | ‖p.1‖ ≤ a ∧ b < ‖p.2‖} := rfl\n  _ = ((μ.prod μ).map (ContinuousLinearMap.rotation (-(π / 4)))) {p | ‖p.1‖ ≤ a ∧ b < ‖p.2‖} := by\n    -- We can rotate the bands since `μ.prod μ` is invariant under rotation\n    rw [h]\n  _ = (μ.prod μ) {p | ‖p.1 - p.2‖ / √2 ≤ a ∧ b < ‖p.1 + p.2‖ / √2} := by\n    rw [Measure.map_apply (by fun_prop)]\n    swap\n    · refine MeasurableSet.inter ?_ ?_\n      · change MeasurableSet {p : E × E | ‖p.1‖ ≤ a}\n        exact measurableSet_le (by fun_prop) (by fun_prop)\n      · change MeasurableSet {p : E × E | b < ‖p.2‖}\n        exact measurableSet_lt (by fun_prop) (by fun_prop)\n    congr 1\n    simp only [Set.preimage_setOf_eq, ContinuousLinearMap.rotation_apply, Real.cos_neg,\n      Real.cos_pi_div_four, Real.sin_neg, Real.sin_pi_div_four, neg_smul, neg_neg]\n    have h_twos : ‖2⁻¹ * √2‖ = (√2)⁻¹ := by\n      simp only [norm_mul, norm_inv, Real.norm_ofNat, Real.norm_eq_abs]\n      rw [abs_of_nonneg (by positivity)]\n      nth_rw 1 [← Real.sq_sqrt (by simp : (0 : ℝ) ≤ 2)]\n      rw [pow_two, mul_inv, mul_assoc, inv_mul_cancel₀ (by positivity), mul_one]\n    congr! with p\n    · rw [← sub_eq_add_neg, ← smul_sub, norm_smul, div_eq_inv_mul, div_eq_inv_mul, h_twos]\n    · rw [← smul_add, norm_smul, div_eq_inv_mul, div_eq_inv_mul, h_twos]\n  _ ≤ (μ.prod μ) {p | (b - a) / √2 < ‖p.1‖ ∧ (b - a) / √2 < ‖p.2‖} := by\n    -- The rotated bands are contained in quadrants.\n    refine measure_mono fun p ↦ ?_\n    simp only [Set.mem_setOf_eq, and_imp]\n    intro hp1 hp2\n    suffices (b - a) / √2 < min ‖p.1‖ ‖p.2‖ from lt_min_iff.mp this\n    calc (b - a) / √2\n    _ < (‖p.1 + p.2‖ - ‖p.1 - p.2‖) / 2 := by\n      suffices b - a < ‖p.1 + p.2‖ / √2 - ‖p.1 - p.2‖ / √2 by\n        calc (b - a) / √2 < (‖p.1 + p.2‖ / √2 - ‖p.1 - p.2‖ / √2) / √2 := by gcongr\n        _ = (‖p.1 + p.2‖ - ‖p.1 - p.2‖) / 2 := by\n          field_simp; rw [Real.sq_sqrt (by positivity)]; ring\n      calc b - a < ‖p.1 + p.2‖ / √2 - a := by gcongr\n      _ ≤ ‖p.1 + p.2‖ / √2 - ‖p.1 - p.2‖ / √2 := by gcongr\n    _ ≤ min ‖p.1‖ ‖p.2‖ := by\n      have := norm_add_sub_norm_sub_le_two_mul_min p.1 p.2\n      linarith\n  _ = (μ.prod μ) ({x | (b - a) / √2 < ‖x‖} ×ˢ {y | (b - a) / √2 < ‖y‖}) := rfl\n  _ ≤ μ {x | (b - a) / √2 < ‖x‖} ^ 2 := by rw [Measure.prod_prod, pow_two]\n\nnamespace Fernique\n\n/-- A sequence of real thresholds that will be used to cut the space into annuli.\nChosen such that for a rotation invariant measure, an application of lemma\n`measure_le_mul_measure_gt_le_of_map_rotation_eq_self` gives\n`μ {x | ‖x‖ ≤ a} * μ {x | normThreshold a (n + 1) < ‖x‖} ≤ μ {x | normThreshold a n < ‖x‖} ^ 2`. -/\nnoncomputable def normThreshold (a : ℝ) : ℕ → ℝ := arithGeom √2 a a\n\nlemma normThreshold_zero : normThreshold a 0 = a := rfl\n\nlemma normThreshold_add_one (n : ℕ) : normThreshold a (n + 1) = √2 * normThreshold a n + a := rfl\n\nlemma measure_le_mul_measure_gt_normThreshold_le_of_map_rotation_eq_self [SFinite μ]\n    (h_rot : (μ.prod μ).map (ContinuousLinearMap.rotation (-(π / 4))) = μ.prod μ) (a : ℝ) (n : ℕ) :\n    μ {x | ‖x‖ ≤ a} * μ {x | normThreshold a (n + 1) < ‖x‖}\n      ≤ μ {x | normThreshold a n < ‖x‖} ^ 2 := by\n  convert! measure_le_mul_measure_gt_le_of_map_rotation_eq_self h_rot _ _\n  simp [normThreshold_add_one]\n\nTarget:\nlemma lt_normThreshold_zero (ha_pos : 0 < a) : a / (1 - √2) < normThreshold a 0 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_5f55f4262f31","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f01ee2a99ebe44248b4fedba58b2999d353f5813f2b0d44d5b296ed14e75fadb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Distributions","family_id":"lt_normthreshold_zero","file_id":"mathlib/Mathlib/Probability/Distributions/Fernique.lean","sample_id":"5f55f4262f31682a3e12d8625b11abf778bb7c39ed9ebef5aef18cf9e8a8d94d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c165c49ef010a17e120cfdcd1a6cc6afaacc95eed29ab44ee041da167fa437e8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"882301a1333b1d1e59e6d22aa8bb0161b0e3eb5e7d586b2fba0158c278cd4f8d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c0fbb14cf7d5c8d77a74ea9851f8a6d43a29d0444f221fd7a52f0350a1b23692","source_sha256":"e0ea8313840496a509375804bd2710d8b2f93f0c2a8c8922e53c600d1e07dd2a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  -- this follows from a covering argument using the sets satisfying `ρ a ≤ ν a`.\n  apply ENNReal.le_of_forall_pos_le_add fun ε εpos _ => ?_\n  obtain ⟨U, sU, U_open, νU⟩ : ∃ (U : Set α), s ⊆ U ∧ IsOpen U ∧ ν U ≤ ν s + ε :=\n    exists_isOpen_le_add s ν (ENNReal.coe_pos.2 εpos).ne'\n  let f : α → Set (Set α) := fun _ => {a | ρ a ≤ ν a ∧ a ⊆ U}\n  have h : v.FineSubfamilyOn f s := by\n    apply v.fineSubfamilyOn_of_frequently f s fun x hx => ?_\n    have :=\n      (hs x hx).and_eventually\n        ((v.eventually_filterAt_mem_setsAt x).and\n          (v.eventually_filterAt_subset_of_nhds (U_open.mem_nhds (sU hx))))\n    apply Frequently.mono this\n    rintro a ⟨ρa, _, aU⟩\n    exact ⟨ρa, aU⟩\n  haveI : Encodable h.index := h.index_countable.toEncodable\n  calc\n    ρ s ≤ ∑' x : h.index, ρ (h.covering x) := h.measure_le_tsum_of_absolutelyContinuous hρ\n    _ ≤ ∑' x : h.index, ν (h.covering x) := ENNReal.tsum_le_tsum fun x => (h.covering_mem x.2).1\n    _ = ν (⋃ x : h.index, h.covering x) := by\n      rw [measure_iUnion h.covering_disjoint_subtype fun i => h.measurableSet_u i.2]\n    _ ≤ ν U := (measure_mono (iUnion_subset fun i => (h.covering_mem i.2).2))\n    _ ≤ ν s + ε := νU","hard_negative":false,"metrics":{"chosen_tokens":324,"rejected_tokens":328,"token_jaccard":0.969697,"token_length_ratio":1.012346},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"64add9dafbb27d4d82980b7a0a2da1e071227280adb535bec9613782bda38ca7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Covering.VitaliFamily\npublic import Mathlib.MeasureTheory.Function.AEMeasurableOrder\npublic import Mathlib.MeasureTheory.Integral.Average\npublic import Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue\npublic import Mathlib.MeasureTheory.Measure.Regular\n\nNamespace:\nVitaliFamily\n\nLocal context:\n/-\nCopyright (c) 2021 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Differentiation of measures\n\nOn a second countable metric space with a measure `μ`, consider a Vitali family (i.e., for each `x`\none has a family of sets shrinking to `x`, with a good behavior with respect to covering theorems).\nConsider also another measure `ρ`. Then, for almost every `x`, the ratio `ρ a / μ a` converges when\n`a` shrinks to `x` along the Vitali family, towards the Radon-Nikodym derivative of `ρ` with\nrespect to `μ`. This is the main theorem on differentiation of measures.\n\nThis theorem is proved in this file, under the name `VitaliFamily.ae_tendsto_rnDeriv`. Note that,\nalmost surely, `μ a` is eventually positive and finite (see\n`VitaliFamily.ae_eventually_measure_pos` and `VitaliFamily.eventually_measure_lt_top`), so the\nratio really makes sense.\n\nFor concrete applications, one needs concrete instances of Vitali families, as provided for instance\nby `Besicovitch.vitaliFamily` (for balls) or by `Vitali.vitaliFamily` (for doubling measures).\n\nSpecific applications to Lebesgue density points and the Lebesgue differentiation theorem are also\nderived:\n* `VitaliFamily.ae_tendsto_measure_inter_div` states that, for almost every point `x ∈ s`,\n  then `μ (s ∩ a) / μ a` tends to `1` as `a` shrinks to `x` along a Vitali family.\n* `VitaliFamily.ae_tendsto_average_norm_sub` states that, for almost every point `x`, then the\n  average of `y ↦ ‖f y - f x‖` on `a` tends to `0` as `a` shrinks to `x` along a Vitali family.\n\n## Sketch of proof\n\nLet `v` be a Vitali family for `μ`. Assume for simplicity that `ρ` is absolutely continuous with\nrespect to `μ`, as the case of a singular measure is easier.\n\nIt is easy to see that a set `s` on which `liminf ρ a / μ a < q` satisfies `ρ s ≤ q * μ s`, by using\na disjoint subcovering provided by the definition of Vitali families. Similarly for the limsup.\nIt follows that a set on which `ρ a / μ a` oscillates has measure `0`, and therefore that\n`ρ a / μ a` converges almost surely (`VitaliFamily.ae_tendsto_div`). Moreover, on a set where the\nlimit is close to a constant `c`, one gets `ρ s ∼ c μ s`, using again a covering lemma as above.\nIt follows that `ρ` is equal to `μ.withDensity (v.limRatio ρ x)`, where `v.limRatio ρ x` is the\nlimit of `ρ a / μ a` at `x` (which is well defined almost everywhere). By uniqueness of the\nRadon-Nikodym derivative, one gets `v.limRatio ρ x = ρ.rnDeriv μ x` almost everywhere, completing\nthe proof.\n\nThere is a difficulty in this sketch: this argument works well when `v.limRatio ρ` is measurable,\nbut there is no guarantee that this is the case, especially if one doesn't make further assumptions\non the Vitali family. We use an indirect argument to show that `v.limRatio ρ` is always\nalmost everywhere measurable, again based on the disjoint subcovering argument\n(see `VitaliFamily.exists_measurable_supersets_limRatio`), and then proceed as sketched above\nbut replacing `v.limRatio ρ` by a measurable version called `v.limRatioMeas ρ`.\n\n## Counterexample\n\nThe standing assumption in this file is that spaces are second countable. Without this assumption,\nmeasures may be zero locally but nonzero globally, which is not compatible with differentiation\ntheory (which deduces global information from local one). Here is an example displaying this\nbehavior.\n\nDefine a measure `μ` by `μ s = 0` if `s` is covered by countably many balls of radius `1`,\nand `μ s = ∞` otherwise. This is indeed a countably additive measure, which is moreover\nlocally finite and doubling at small scales. It vanishes on every ball of radius `1`, so all the\nquantities in differentiation theory (defined as ratios of measures as the radius tends to zero)\nmake no sense. However, the measure is not globally zero if the space is big enough.\n\n## References\n\n* [Herbert Federer, Geometric Measure Theory, Chapter 2.9][Federer1996]\n-/\n\n@[expose] public section\n\nopen MeasureTheory Metric Set Filter TopologicalSpace MeasureTheory.Measure\n\nopen scoped Filter ENNReal MeasureTheory NNReal Topology\n\nvariable {α : Type*} [PseudoMetricSpace α] {m0 : MeasurableSpace α} {μ : Measure α}\n  (v : VitaliFamily μ)\n  {E : Type*} [NormedAddCommGroup E]\n\nnamespace VitaliFamily\n\n/-- The limit along a Vitali family of `ρ a / μ a` where it makes sense, and garbage otherwise.\nDo *not* use this definition: it is only a temporary device to show that this ratio tends almost\neverywhere to the Radon-Nikodym derivative. -/\nnoncomputable def limRatio (ρ : Measure α) (x : α) : ℝ≥0∞ :=\n  limUnder (v.filterAt x) fun a => ρ a / μ a\n\n/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive\nmeasure. (This is a nontrivial result, following from the covering property of Vitali families). -/\ntheorem ae_eventually_measure_pos [SecondCountableTopology α] :\n    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by\n  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs\n  simp -zeta only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs\n  change μ s = 0\n  let f : α → Set (Set α) := fun _ => {a | μ a = 0}\n  have h : v.FineSubfamilyOn f s := by\n    intro x hx ε εpos\n    rw [hs] at hx\n    simp only [frequently_filterAt_iff, gt_iff_lt, mem_setOf_eq] at hx\n    rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩\n    exact ⟨a, ⟨a_sets, μa⟩, ax⟩\n  refine le_antisymm ?_ bot_le\n  calc\n    μ s ≤ ∑' x : h.index, μ (h.covering x) := h.measure_le_tsum\n    _ = ∑' x : h.index, 0 := by congr; ext1 x; exact h.covering_mem x.2\n    _ = 0 := by simp only [tsum_zero]\n\n/-- For every point `x`, sufficiently small sets in a Vitali family around `x` have finite measure.\n(This is a trivial result, following from the fact that the measure is locally finite). -/\ntheorem eventually_measure_lt_top [IsLocallyFiniteMeasure μ] (x : α) :\n    ∀ᶠ a in v.filterAt x, μ a < ∞ :=\n  (μ.finiteAt_nhds x).eventually.filter_mono inf_le_left\n\n/-- If two measures `ρ` and `ν` have, at every point of a set `s`, arbitrarily small sets in a\nVitali family satisfying `ρ a ≤ ν a`, then `ρ s ≤ ν s` if `ρ ≪ μ`. -/\n\nTarget:\ntheorem measure_le_of_frequently_le [SecondCountableTopology α] [BorelSpace α] {ρ : Measure α}\n    (ν : Measure α) [IsLocallyFiniteMeasure ν] (hρ : ρ ≪ μ) (s : Set α)\n    (hs : ∀ x ∈ s, ∃ᶠ a in v.filterAt x, ρ a ≤ ν a) : ρ s ≤ ν s :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  -- this follows from a covering argument using the sets satisfying `ρ a ≤ ν a`.\n  apply ENNReal.le_of_forall_pos_le_add fun ε εpos _ => ?_\n  obtain ⟨U, sU, U_open, νU⟩ : ∃ (U : Set α), s ⊆ U ∧ IsOpen U ∧ ν U ≤ ν s + ε :=\n    exists_isOpen_le_add s ν (ENNReal.coe_pos.2 εpos).ne'\n  let f : α → Set (Set α) := fun _ => {a | ρ a ≤ ν a ∧ a ⊆ U}\n  have h : v.FineSubfamilyOn f s := by\n    apply v.fineSubfamilyOn_of_frequently f s fun x hx => ?_\n    have :=\n      (hs x hx).and_eventually\n        ((v.eventually_filterAt_mem_setsAt x).and\n          (v.eventually_filterAt_subset_of_nhds (U_open.mem_nhds (sU hx))))\n    apply Frequently.mono this\n    rintro a ⟨ρa, _, aU⟩\n    exact ⟨ρa, aU⟩\n  haveI : Encodable h.index := h.index_countable.toEncodable\n  calc\n    ρ s ≤ ∑' x : h.index, ρ (h.covering x) := h.measure_le_tsum_of_absolutelyContinuous hρ\n    _ ≤ ∑' x : h.index, ν (h.covering x) := ENNReal.tsum_le_tsum fun x => (h.covering_mem x.2).1\n    _ = ν (⋃ x : h.index, h.covering x) := by\n      rw [measure_iUnion h.covering_disjoint_subtype fun i => h.measurableSet_u i.2]\n    _ ≤ ν U := (measure_mono (iUnion_subset fun i => (h.covering_mem i.2).2))\n    _ ≤ ν s + ε := νU","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Covering","family_id":"measure_le_of_frequently_le","file_id":"mathlib/Mathlib/MeasureTheory/Covering/Differentiation.lean","sample_id":"c0fbb14cf7d5c8d77a74ea9851f8a6d43a29d0444f221fd7a52f0350a1b23692"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4acb529aee5f1ece8076fbbdc844587e9d93366665ac8d5f6045ad6127fa4525","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1cd3353ed9a27852a0187d96663634280991314b9277de0a0ac0db0c81973f77","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"beebaf6401da7ba6ae02cab75a18db90132912fa7666bc24027b84d9741587a1","source_sha256":"3ebf52bd4f56640ff82f3663b47b89c207fb7b51b21d5bfbbe3cfb7edd34865e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := lift_lt.{_, max (u + 1) v}.2 (lift_lt_univ c)\n  rw [lift_lift, lift_univ, univ_umax.{u, v}] at this\n  exact this","hard_negative":false,"metrics":{"chosen_tokens":40,"rejected_tokens":45,"token_jaccard":0.848485,"token_length_ratio":1.125},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"64dd6e8952d65c5a85ce27a1fbaa734169e4af4263fffde0142f4ee593a2dac2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.SetTheory.Ordinal.Basic\n\nNamespace:\nCardinal\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Floris van Doorn\n-/\n/-!\n# Universal ordinal and cardinal\n\n`Cardinal.univ` is the cardinality of the cardinals themselves. Likewise, `Ordinal.univ` is the\norder type of the ordinals. These are related via `Cardinal.univ.ord = Ordinal.univ` and\n`Ordinal.univ.card = Cardinal.univ`.\n\nThe cardinal `Cardinal.univ` is strongly inaccessible. This reflects the fact that in ZFC, the\ncardinals form a proper class. See `IsInaccessible.univ` for a proof.\n\n## Implementation notes\n\nWe actually define `Cardinal.univ` as the cardinality of `Ordinal`, rather than that of `Cardinal`.\nThis makes the basic API easier to set up. See `Cardinal.mk_cardinal` for a proof that\n`Cardinal.univ = #Cardinal`.\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nset_option linter.checkUnivs false in\nopen Ordinal in\n-- intended to be used with explicit universe parameters\n/-- The ordinal `univ.{u, v}` is the order type of `Ordinal.{u}` or `Cardinal.{u}`, as an element of\n`Ordinal.{v}` (when `u < v`). -/\n@[pp_with_univ, nolint checkUnivs]\ndef Ordinal.univ : Ordinal.{max (u + 1) v} :=\n  lift.{v, u + 1} (typeLT Ordinal)\n\nset_option linter.checkUnivs false in\nopen Cardinal in\n-- intended to be used with explicit universe parameters\n/-- The cardinal `univ.{u, v}` is the cardinality of `Ordinal.{u}` or `Cardinal.{u}`, as an element\nof `Cardinal.{v}` (when `u < v`). -/\n@[pp_with_univ, nolint checkUnivs]\ndef Cardinal.univ : Cardinal.{max (u + 1) v} :=\n  lift.{v, u + 1} #Ordinal\n\n/-! ### Universal ordinal -/\n\nnamespace Ordinal\n\n@[simp]\ntheorem type_lt_ordinal : typeLT Ordinal = univ.{u, u + 1} :=\n  (lift_id _).symm\n\n@[deprecated type_lt_ordinal (since := \"2026-03-20\")]\ntheorem univ_id : univ.{u, u + 1} = typeLT Ordinal :=\n  lift_id _\n\n@[simp]\ntheorem lift_univ : lift.{w} univ.{u, v} = univ.{u, max v w} :=\n  lift_lift _\n\ntheorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=\n  congr_fun lift_umax _\n\n/-- Principal segment version of the lift operation on ordinals, embedding `Ordinal.{u}` in\n`Ordinal.{v}` as a principal segment when `u < v`. -/\ndef liftPrincipalSeg : Ordinal.{u} <i Ordinal.{max (u + 1) v} :=\n  ⟨liftInitialSeg.{max (u + 1) v, u}, univ.{u, v}, by\n    refine fun b => inductionOn b ?_; intro β s _\n    rw [univ, ← lift_umax]; constructor <;> intro h\n    · obtain ⟨a, e⟩ := h\n      rw [← e]\n      refine inductionOn a ?_\n      intro α r _\n      exact lift_type_lt.{u, u + 1, max (u + 1) v}.2 ⟨typein r⟩\n    · rw [← lift_id (type s)] at h ⊢\n      obtain ⟨f⟩ := lift_type_lt.{_,_,v}.1 h\n      obtain ⟨f, a, hf⟩ := f\n      exists a\n      induction a using inductionOn with | type α r\n      refine lift_type_eq.{u, max (u + 1) v, max (u + 1) v}.2\n        ⟨(RelIso.ofSurjective (RelEmbedding.ofMonotone ?_ ?_) ?_).symm⟩\n      · exact fun b => enum r ⟨f b, (hf _).1 ⟨_, rfl⟩⟩\n      · refine fun a b h => (typein_lt_typein r).1 ?_\n        rw [typein_enum, typein_enum]\n        exact f.map_rel_iff.2 h\n      · intro a'\n        obtain ⟨b, e⟩ := (hf _).2 (typein_lt_type _ a')\n        exists b\n        simp only [RelEmbedding.ofMonotone_coe]\n        simp [e]⟩\n\n@[simp]\ntheorem liftPrincipalSeg_coe :\n    (liftPrincipalSeg.{u, v} : Ordinal → Ordinal) = lift.{max (u + 1) v} :=\n  rfl\n\n@[simp]\ntheorem liftPrincipalSeg_top : (liftPrincipalSeg.{u, v}).top = univ.{u, v} :=\n  rfl\n\n@[deprecated liftPrincipalSeg_top (since := \"2026-03-20\")]\ntheorem liftPrincipalSeg_top' : liftPrincipalSeg.{u, u + 1}.top = typeLT Ordinal := by\n  simp\n\n@[simp]\ntheorem card_univ : card univ.{u, v} = Cardinal.univ.{u, v} :=\n  rfl\n\nend Ordinal\n\n/-! ### Universal cardinal -/\n\nnamespace Cardinal\n\n@[simp]\ntheorem mk_ordinal : #Ordinal = univ.{u, u + 1} :=\n  (lift_id _).symm\n\n@[deprecated mk_ordinal (since := \"2026-04-22\")]\ntheorem univ_id : univ.{u, u + 1} = #Ordinal :=\n  lift_id _\n\n@[simp]\ntheorem lift_univ : lift.{w} univ.{u, v} = univ.{u, max v w} :=\n  lift_lift _\n\ntheorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=\n  congr_fun lift_umax _\n\ntheorem lift_lt_univ (c : Cardinal) : lift.{u + 1, u} c < univ.{u, u + 1} := by\n  simpa only [Ordinal.liftPrincipalSeg_coe, lift_ord, lift_succ, ord_le, Order.succ_le_iff] using!\n    le_of_lt (Ordinal.liftPrincipalSeg.{u, u + 1}.lt_top (Order.succ c).ord)\n\nTarget:\ntheorem lift_lt_univ' (c : Cardinal) : lift.{max (u + 1) v, u} c < univ.{u, v} :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  have := lift_lt.{_, max (u + 1) v}.2 (lift_lt_univ c)\n  rw [lift_lift, lift_univ, univ_umax.{u, v}] at this\n  exact this","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Ordinal","family_id":"lift_lt_univ","file_id":"mathlib/Mathlib/SetTheory/Ordinal/Univ.lean","sample_id":"beebaf6401da7ba6ae02cab75a18db90132912fa7666bc24027b84d9741587a1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"29e959508ac284606b1ea79d1f9c44fa587eb78cdc7fa0d4ba1b2b14a0490f11","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"955ffaab0169b8f61dba8b3842d08064f9ab7cdf6d874b7505c429f20b79b8f8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e5f7ea238c907be9131815ee500db761a2036d4f0c717eb8bb8747df40a1767e","source_sha256":"155010ab3058b3fe6da475eb20a319e6c48a44ac24aa3149ecf584265954195c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [Shrink.rec, eqRec_eq_cast, cast_eq_iff_heq]\n  rw [Equiv.symm_apply_apply]","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":23,"token_jaccard":0.777778,"token_length_ratio":1.277778},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"64f2397ee56431cda2008184d76ea5b1ed7d3227a0e48385730de97374df7702","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Logic.Equiv.Defs\npublic import Mathlib.Tactic.MkIffOfInductiveProp\npublic import Mathlib.Tactic.PPWithUniv\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Small types\n\nA type is `w`-small if there exists an equivalence to some `S : Type w`.\n\nWe provide a noncomputable model `Shrink α : Type w`, and `equivShrink α : α ≃ Shrink α`.\n\nA subsingleton type is `w`-small for any `w`.\n\nIf `α ≃ β`, then `Small.{w} α ↔ Small.{w} β`.\n\nSee `Mathlib/Logic/Small/Basic.lean` for further instances and theorems.\n-/\n\n@[expose] public section\n\nuniverse u w v v'\n\n/-- A type is `Small.{w}` if there exists an equivalence to some `S : Type w`.\n-/\n-- After https://github.com/leanprover/lean4/pull/12286 and\n-- https://github.com/leanprover/lean4/pull/12423: `v` is a true output (determined by `α`),\n-- but we need the attribute to prevent `w` from also being treated as output.\n-- See Note [universe output parameters and typeclass caching].\n@[univ_out_params v, mk_iff, pp_with_univ]\nclass Small (α : Type v) : Prop where\n  /-- If a type is `Small.{w}`, then there exists an equivalence with some `S : Type w` -/\n  equiv_small : ∃ S : Type w, Nonempty (α ≃ S)\n\n/-- Constructor for `Small α` from an explicit witness type and equivalence.\n-/\ntheorem Small.mk' {α : Type v} {S : Type w} (e : α ≃ S) : Small.{w} α :=\n  ⟨⟨S, ⟨e⟩⟩⟩\n\n/-- An arbitrarily chosen model in `Type w` for a `w`-small type.\n-/\n@[pp_with_univ, no_expose]\ndef Shrink (α : Type v) [Small.{w} α] : Type w :=\n  Classical.choose (@Small.equiv_small α _)\n\n/-- The noncomputable equivalence between a `w`-small type and a model.\n-/\n@[no_expose]\nnoncomputable def equivShrink (α : Type v) [Small.{w} α] : α ≃ Shrink α :=\n  Nonempty.some (Classical.choose_spec (@Small.equiv_small α _))\n\n@[ext]\ntheorem Shrink.ext {α : Type v} [Small.{w} α] {x y : Shrink α}\n    (w : (equivShrink _).symm x = (equivShrink _).symm y) : x = y := by\n  simpa using w\n\n-- It would be nice to mark this as `aesop cases` if\n-- https://github.com/leanprover-community/aesop/issues/59\n-- is resolved.\n@[induction_eliminator]\nprotected noncomputable def Shrink.rec {α : Type*} [Small.{w} α] {F : Shrink α → Sort v}\n    (h : ∀ X, F (equivShrink _ X)) : ∀ X, F X :=\n  fun X => ((equivShrink _).apply_symm_apply X) ▸ (h _)\n\n@[simp]\n\nTarget:\nlemma Shrink.rec_equivShrink {α : Type*} [Small.{w} α] {F : Shrink α → Sort v}\n    {f : (a : α) → F (equivShrink α a)} (a : α) : Shrink.rec f (equivShrink _ a) = f a :=\n\nProof body:\n","rejected":"by\n  simp only [Shrink.rec, eqRec_eq_cast, cast_eq_iff_heq]\n  rw [Equiv.symm_apply_apply]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Small","family_id":"shrink","file_id":"mathlib/Mathlib/Logic/Small/Defs.lean","sample_id":"e5f7ea238c907be9131815ee500db761a2036d4f0c717eb8bb8747df40a1767e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1c6acec1b0c2300f662b80f0c922eb6275df7e3daf8325d6423e0848393437e0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1aa534d9cd1750e6e0c840502eacb09027fa2a9c5b17d399098e7ccc8846f802","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"76c80b8cd380741542148048f4f9d53c485cf357afdef4b9fbe25f15cafe33a7","source_sha256":"20cce9f0236f79aab52863f3444c0947034ea25b58342e0def2f63cbf563ddf0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by simpa using! I.ringCon.add hx hy","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":5,"token_jaccard":0.153846,"token_length_ratio":0.454545},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"654c971ae9ea183b29debfc713366287ddebc4ed82e7d009f11bde879cc48575","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Tactic.Abel\npublic import Mathlib.Algebra.Ring.Opposite\npublic import Mathlib.GroupTheory.GroupAction.SubMulAction\npublic import Mathlib.RingTheory.Congruence.Opposite\n\nNamespace:\nTwoSidedIdeal\n\nLocal context:\n/-\nCopyright (c) 2024 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang\n-/\n/-!\n# Two Sided Ideals\n\nIn this file, for any `Ring R`, we reinterpret `I : RingCon R` as a two-sided-ideal of a ring.\n\n## Main definitions and results\n\n* `TwoSidedIdeal`: For any `NonUnitalNonAssocRing R`, `TwoSidedIdeal R` is a wrapper around\n  `RingCon R`.\n* `TwoSidedIdeal.setLike`: Every `I : TwoSidedIdeal R` can be interpreted as a set of `R` where\n  `x ∈ I` if and only if `I.ringCon x 0`.\n* `TwoSidedIdeal.addCommGroup`: Every `I : TwoSidedIdeal R` is an abelian group.\n\n-/\n\n@[expose] public section\n\nopen MulOpposite\n\nsection definitions\n\n/--\nA two-sided ideal of a ring `R` is a subset of `R` that contains `0` and is closed under addition,\nnegation, and absorbs multiplication on both sides.\n-/\nstructure TwoSidedIdeal (R : Type*) [NonUnitalNonAssocRing R] where\n  /-- every two-sided-ideal is induced by a congruence relation on the ring. -/\n  ringCon : RingCon R\n\nend definitions\n\nnamespace TwoSidedIdeal\n\nsection NonUnitalNonAssocRing\n\nvariable {R : Type*} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R)\n\ninstance [Nontrivial R] : Nontrivial (TwoSidedIdeal R) := by\n  obtain ⟨I, J, h⟩ : Nontrivial (RingCon R) := inferInstance\n  exact ⟨⟨I⟩, ⟨J⟩, by contrapose h; aesop⟩\n\ninstance setLike : SetLike (TwoSidedIdeal R) R where\n  coe t := {r | t.ringCon r 0}\n  coe_injective := by\n    rintro ⟨t₁⟩ ⟨t₂⟩ (h : {x | _} = {x | _})\n    congr 1\n    refine RingCon.ext fun a b ↦ ⟨fun H ↦ ?_, fun H ↦ ?_⟩\n    · have H' : a - b ∈ {x | t₁ x 0} := sub_self b ▸ t₁.sub H (t₁.refl b)\n      rw [h] at H'\n      convert! t₂.add H' (t₂.refl b) using 1 <;> abel\n    · have H' : a - b ∈ {x | t₂ x 0} := sub_self b ▸ t₂.sub H (t₂.refl b)\n      rw [← h] at H'\n      convert! t₁.add H' (t₁.refl b) using 1 <;> abel\n\ninstance : PartialOrder (TwoSidedIdeal R) := .ofSetLike (TwoSidedIdeal R) R\n\nlemma mem_iff (x : R) : x ∈ I ↔ I.ringCon x 0 := Iff.rfl\n\n@[simp]\nlemma mem_mk {x : R} {c : RingCon R} : x ∈ mk c ↔ c x 0 := Iff.rfl\n\n@[simp, norm_cast]\nlemma coe_mk {c : RingCon R} : (mk c : Set R) = {x | c x 0} := rfl\n\nlemma rel_iff (x y : R) : I.ringCon x y ↔ x - y ∈ I := by\n  rw [mem_iff]\n  constructor\n  · intro h; convert! I.ringCon.sub h (I.ringCon.refl y); abel\n  · intro h; convert! I.ringCon.add h (I.ringCon.refl y) <;> abel\n\n/--\nthe coercion from two-sided-ideals to sets is an order embedding\n-/\n@[simps]\ndef coeOrderEmbedding : TwoSidedIdeal R ↪o Set R where\n  toFun := SetLike.coe\n  inj' := SetLike.coe_injective\n  map_rel_iff' {I J} := ⟨fun (h : (I : Set R) ⊆ (J : Set R)) _ h' ↦ h h', fun h _ h' ↦ h h'⟩\n\nlemma le_iff {I J : TwoSidedIdeal R} : I ≤ J ↔ (I : Set R) ⊆ (J : Set R) := Iff.rfl\n\n/-- Two-sided-ideals corresponds to congruence relations on a ring. -/\n@[simps apply symm_apply]\ndef orderIsoRingCon : TwoSidedIdeal R ≃o RingCon R where\n  toFun := TwoSidedIdeal.ringCon\n  invFun := .mk\n  map_rel_iff' {I J} := Iff.symm <| le_iff.trans ⟨fun h x y r => by rw [rel_iff] at r ⊢; exact h r,\n    fun h x hx => by rw [SetLike.mem_coe, mem_iff] at hx ⊢; exact h hx⟩\n\nlemma ringCon_injective : Function.Injective (TwoSidedIdeal.ringCon (R := R)) := by\n  rintro ⟨x⟩ ⟨y⟩ rfl; rfl\n\nlemma ringCon_le_iff {I J : TwoSidedIdeal R} : I ≤ J ↔ I.ringCon ≤ J.ringCon :=\n  orderIsoRingCon.map_rel_iff.symm\n\n@[ext]\nlemma ext {I J : TwoSidedIdeal R} (h : ∀ x, x ∈ I ↔ x ∈ J) : I = J :=\n  coeOrderEmbedding.injective (Set.ext h)\n\nlemma lt_iff (I J : TwoSidedIdeal R) : I < J ↔ (I : Set R) ⊂ (J : Set R) := by\n  rw [lt_iff_le_and_ne, Set.ssubset_iff_subset_ne, le_iff]\n  simp\n\nlemma zero_mem : 0 ∈ I := I.ringCon.refl 0\n\nTarget:\nlemma add_mem {x y} (hx : x ∈ I) (hy : y ∈ I) : x + y ∈ I :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_76c80b8cd380","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f3eba94b9e0af848eb1dedaf22b82d19e96f5adcf1b3be938ef972ac295c4bac","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/TwoSidedIdeal","family_id":"add_mem","file_id":"mathlib/Mathlib/RingTheory/TwoSidedIdeal/Basic.lean","sample_id":"76c80b8cd380741542148048f4f9d53c485cf357afdef4b9fbe25f15cafe33a7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cfefd25aa03c8c5b75a26d1fb3bac847d72b8789c719fd94bddb3d949b0e73ab","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"deb583f09156ce6e73c3a31850ee7ecaa774e316228fb62067000f39f1064eb0","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3872ad2e2e46d2587f8e296dc30252bf359d8b645c5d7c7851970aa025ddb54e","source_sha256":"d3c748be9a560274e8be6246cdd5ae4eb345ae9279176c7be340f873d2e38ac6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun hn x ↦ lt_of_le_of_lt (norm_coe_le_norm f x) hn, ?_⟩\n  · obtain (he | he) := isEmpty_or_nonempty α\n    · simpa\n    · obtain ⟨x, hx⟩ := exist_norm_eq h\n      exact fun h ↦ hx ▸ h x","hard_negative":false,"metrics":{"chosen_tokens":48,"rejected_tokens":55,"token_jaccard":0.935484,"token_length_ratio":1.145833},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"65505ed8673c7d32d973ff8a63b30de2f5c4fb9bd575f47eddf9cdc43a009af4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.ContinuousMap.Bounded.Normed\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Yoh Tanimoto. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yoh Tanimoto\n-/\n/-!\n# Compactly supported bounded continuous functions\n\nThe two-sided ideal of compactly supported bounded continuous functions taking values in a metric\nspace, with the uniform distance.\n-/\n\n@[expose] public section\n\nopen Set BoundedContinuousFunction\n\nsection CompactlySupported\n\n/-- The two-sided ideal of compactly supported functions. -/\nnoncomputable def compactlySupported (α γ : Type*) [TopologicalSpace α] [NonUnitalNormedRing γ] :\n    TwoSidedIdeal (α →ᵇ γ) :=\n  .mk' {z | HasCompactSupport z} .zero .add .neg .mul_left .mul_right\n\nvariable {α γ : Type*} [TopologicalSpace α] [NonUnitalNormedRing γ]\n\n@[inherit_doc]\nscoped[BoundedContinuousFunction] notation\n  \"C_cb(\" α \", \" γ \")\" => compactlySupported α γ\n\nlemma mem_compactlySupported {f : α →ᵇ γ} :\n    f ∈ C_cb(α, γ) ↔ HasCompactSupport f :=\n  TwoSidedIdeal.mem_mk' {z : α →ᵇ γ | HasCompactSupport z} .zero .add .neg .mul_left .mul_right f\n\nlemma exist_norm_eq [c : Nonempty α] {f : α →ᵇ γ} (h : f ∈ C_cb(α, γ)) : ∃ (x : α),\n    ‖f x‖ = ‖f‖ := by\n  by_cases hs : (tsupport f).Nonempty\n  · obtain ⟨x, _, hmax⟩ := mem_compactlySupported.mp h |>.exists_isMaxOn hs <|\n      (map_continuous f).norm.continuousOn\n    refine ⟨x, le_antisymm (norm_coe_le_norm f x) (norm_le (norm_nonneg _) |>.mpr fun y ↦ ?_)⟩\n    by_cases hy : y ∈ tsupport f\n    · exact hmax hy\n    · simp [image_eq_zero_of_notMem_tsupport hy]\n  · suffices f = 0 by simp [this]\n    rwa [not_nonempty_iff_eq_empty, tsupport_eq_empty_iff, ← coe_zero, ← DFunLike.ext'_iff] at hs\n\nTarget:\ntheorem norm_lt_iff_of_compactlySupported {f : α →ᵇ γ} (h : f ∈ C_cb(α, γ)) {M : ℝ}\n    (M0 : 0 < M) : ‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M :=\n\nProof body:\n","rejected":"```lean\nby\n  refine ⟨fun hn x ↦ lt_of_le_of_lt (norm_coe_le_norm f x) hn, ?_⟩\n  · obtain (he | he) := isEmpty_or_nonempty α\n    · simpa\n    · obtain ⟨x, hx⟩ := exist_norm_eq h\n      exact fun h ↦ hx ▸ h x\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/ContinuousMap","family_id":"norm_lt_iff_of_compactlysupported","file_id":"mathlib/Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean","sample_id":"3872ad2e2e46d2587f8e296dc30252bf359d8b645c5d7c7851970aa025ddb54e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b656fc39071ddeaac02c29e4861f9ce7d1311a7b76fad806354a82acd6bcd29b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"719c1265d6da97aed3f9b0cf9aa1b7a5fcc8c80c094927f5c0f9b5f00b5bc7b3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"66a9d446584881e6ae181bf60c62b6c4435815bff91f04bcf763c4007fc9aaae","source_sha256":"7bb1b48aebf21c1532e493ca79c87d1bb8ee632b2dfff76d51a712868ddd9efa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases isEmpty_or_nonempty ι\n  · rw [iSup_of_empty', iSup_of_empty']\n  · exact hdf.ciSup_le fun x ↦ hdg.le_ciSup_of_le B x (H x)","hard_negative":false,"metrics":{"chosen_tokens":28,"rejected_tokens":32,"token_jaccard":0.884615,"token_length_ratio":1.142857},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"6556d98de994045a73f4ab436191c1831078216e81b8597188c61414471682cd","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ConditionallyCompletePartialOrder.Basic\npublic import Mathlib.Order.GaloisConnection.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2026 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\n/-!\n# Indexed sup / inf in conditionally complete lattices\n\nThis file proves lemmas about `iSup` and `iInf` for functions valued in a conditionally complete\npartial order, as opposed to a conditionally complete lattice.\n\n## TODO\n\n+ Use `@[to_dual]` in the `GaloisConnection` and `OrderIso` sections.\n\n-/\n\npublic section\n\n-- Guard against import creep\nassert_not_exists Multiset\n\nopen Function OrderDual Set\n\nvariable {α β γ : Type*} {ι : Sort*}\n\nsection ConditionallyCompletePartialOrderSup\n\nvariable [ConditionallyCompletePartialOrderSup α] {a b : α}\n\n@[to_dual]\ntheorem Directed.isLUB_ciSup [Nonempty ι] {f : ι → α} (hd : Directed (· ≤ ·) f)\n    (H : BddAbove (range f)) : IsLUB (range f) (⨆ i, f i) :=\n  hd.directedOn_range.isLUB_csSup (range_nonempty f) H\n\n@[to_dual]\ntheorem DirectedOn.isLUB_ciSup_set {f : β → α} {s : Set β} (hd : DirectedOn (· ≤ ·) (f '' s))\n    (H : BddAbove (f '' s)) (Hne : s.Nonempty) :\n    IsLUB (f '' s) (⨆ i : s, f i) := by\n  rw [← sSup_image']\n  exact hd.isLUB_csSup (Hne.image _) H\n\n@[to_dual Directed.le_ciInf_iff]\ntheorem Directed.ciSup_le_iff [Nonempty ι] {f : ι → α} {a : α}\n    (hd : Directed (· ≤ ·) f) (hf : BddAbove (range f)) :\n    iSup f ≤ a ↔ ∀ i, f i ≤ a :=\n  (isLUB_le_iff <| hd.isLUB_ciSup hf).trans forall_mem_range\n\n@[to_dual DirectedOn.le_ciInf_set_iff]\ntheorem DirectedOn.ciSup_set_le_iff {ι : Type*} {s : Set ι} {f : ι → α} {a : α} (hs : s.Nonempty)\n    (hd : DirectedOn (· ≤ ·) (f '' s)) (hf : BddAbove (f '' s)) :\n    ⨆ i : s, f i ≤ a ↔ ∀ i ∈ s, f i ≤ a :=\n  (isLUB_le_iff <| hd.isLUB_ciSup_set hf hs).trans forall_mem_image\n\n@[to_dual Directed.ciInf_le_of_le]\ntheorem Directed.le_ciSup_of_le {f : ι → α} (hd : Directed (· ≤ ·) f)\n    (H : BddAbove (range f)) (c : ι) (h : a ≤ f c) : a ≤ iSup f :=\n  le_trans h (hd.le_ciSup H c)\n\n/-- The indexed suprema of two functions are comparable if the functions are pointwise comparable -/\n@[to_dual (attr := gcongr low)\n/-- The indexed infimum of two functions are comparable if the functions are pointwise\ncomparable -/]\n\nTarget:\ntheorem Directed.ciSup_mono {f g : ι → α} (hdf : Directed (· ≤ ·) f)\n    (hdg : Directed (· ≤ ·) g) (B : BddAbove (range g)) (H : ∀ x, f x ≤ g x) :\n    iSup f ≤ iSup g :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  cases isEmpty_or_nonempty ι\n  · rw [iSup_of_empty', iSup_of_empty']\n  · exact hdf.ciSup_le fun x ↦ hdg.le_ciSup_of_le B x (H x)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/ConditionallyCompletePartialOrder","family_id":"directed","file_id":"mathlib/Mathlib/Order/ConditionallyCompletePartialOrder/Indexed.lean","sample_id":"66a9d446584881e6ae181bf60c62b6c4435815bff91f04bcf763c4007fc9aaae"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7308aa6374fea597c48250db2901872d2efda7ec2b018c03749527c542943e3a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"581df8840e83eb5edacbd18f72ceb4691509bd9435b9ce1a4d25d13b301c42bd","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"88dc792ee34d82bdbe14dbea6a9bf9efe530e0a2ea1640a11d02fb687a45c35f","source_sha256":"f5c6a6d95455e9edee0a0f7f8fe9448df43f3ee7944da75da2fff4d714325b9b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨n, hn⟩ := exists_rat_eq_convergent h\n  exact ⟨n, hn.symm ▸ convs_eq_convergent ξ n⟩","hard_negative":true,"metrics":{"chosen_tokens":22,"rejected_tokens":3,"token_jaccard":0.117647,"token_length_ratio":0.136364},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"655ec6c808d7df615d43df0d5ba7f03a31b4cf4bd6f5727cb54f8ecb46d1ec1b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries\npublic import Mathlib.Algebra.ContinuedFractions.Computation.Translations\npublic import Mathlib.NumberTheory.DiophantineApproximation.Basic\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2022 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Geißer, Michael Stoll\n-/\n/-!\n# Diophantine Approximation using continued fractions\n\n## Main statements\n\nThere are two versions of Legendre's Theorem.`Real.exists_rat_eq_convergent`,\ndefined in `Mathlib/NumberTheory/DiophantineApproximation/Basic.lean`, uses `Real.convergent`,\na simple recursive definition of the convergents that is also defined in that file.\nThis file provides `Real.exists_convs_eq_rat`, using `GenContFract.convs` of `GenContFract.of ξ`.\n-/\n\npublic section\n\nsection Convergent\n\nnamespace Real\n\nopen Int\n\n/-!\nOur `convergent`s agree with `GenContFract.convs`.\n-/\n\nopen GenContFract\n\n/-- The `n`th convergent of the `GenContFract.of ξ` agrees with `ξ.convergent n`. -/\ntheorem convs_eq_convergent (ξ : ℝ) (n : ℕ) :\n    (GenContFract.of ξ).convs n = ξ.convergent n := by\n  induction n generalizing ξ with\n  | zero => simp only [zeroth_conv_eq_h, of_h_eq_floor, convergent_zero, Rat.cast_intCast]\n  | succ n ih => rw [convs_succ, ih (fract ξ)⁻¹, convergent_succ, one_div]; norm_cast\n\nend Real\n\nend Convergent\n\nnamespace Real\n\nvariable {ξ : ℝ} {u v : ℤ}\n\n/-- The main result, *Legendre's Theorem* on rational approximation:\nif `ξ` is a real number and `q` is a rational number such that `|ξ - q| < 1/(2*q.den^2)`,\nthen `q` is a convergent of the continued fraction expansion of `ξ`.\nThis is the version using `GenContFract.convs`. -/\n\nTarget:\ntheorem exists_convs_eq_rat {q : ℚ}\n    (h : |ξ - q| < 1 / (2 * (q.den : ℝ) ^ 2)) : ∃ n, (GenContFract.of ξ).convs n = q :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_88dc792ee34d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f97653a8c17f20bb89110d4a80754c612af3085d28b5120373ead1492c0ba914","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/DiophantineApproximation","family_id":"exists_convs_eq_rat","file_id":"mathlib/Mathlib/NumberTheory/DiophantineApproximation/ContinuedFractions.lean","sample_id":"88dc792ee34d82bdbe14dbea6a9bf9efe530e0a2ea1640a11d02fb687a45c35f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2b5a7d59d6ab27179c1c1bc56a6c116c83ca6fccc9d560e5875bd86a9aa70480","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8d0fad0c6786d754062706cdf367d3b518319e89dce0b8c02682ad4b8d332126","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"32ce66990d1ffba436ebb7dc106eea97303764f97b8b35040bc36958f838afdb","source_sha256":"2245dbaf366fe900a8a3490e27a54ff47c641f9f75713ae9f1dd06aaebe571c5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [List.mem_iff_get, Fin.exists_iff, eq_comm (b := b)]","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":2,"token_jaccard":0.0625,"token_length_ratio":0.111111},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"657e2a6bb5fc7af0634486c4df803989c7fa4355e29e4e8a00c7c1af0632b500","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Data.List.Defs\n\nNamespace:\nList\n\nLocal context:\n/-\nCopyright (c) 2024 Miyahara Kō. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Miyahara Kō\n-/\n/-!\n# iterate\n\nProves various lemmas about `List.iterate`.\n-/\n\npublic section\n\nvariable {α : Type*}\n\nnamespace List\n\n@[simp]\ntheorem length_iterate (f : α → α) (a : α) (n : ℕ) : length (iterate f a n) = n := by\n  induction n generalizing a <;> simp [*]\n\n@[simp]\ntheorem iterate_eq_nil {f : α → α} {a : α} {n : ℕ} : iterate f a n = [] ↔ n = 0 := by\n  rw [← length_eq_zero_iff, length_iterate]\n\ntheorem getElem?_iterate (f : α → α) (a : α) :\n    ∀ (n i : ℕ), i < n → (iterate f a n)[i]? = f^[i] a\n  | n + 1, 0, _ => by simp\n  | n + 1, i + 1, h => by simp [getElem?_iterate f (f a) n i (by simpa using h)]\n\n@[simp]\ntheorem getElem_iterate (f : α → α) (a : α) (n : ℕ) (i : Nat) (h : i < (iterate f a n).length) :\n    (iterate f a n)[i] = f^[i] a :=\n  (getElem_eq_iff _).2 <| getElem?_iterate _ _ _ _ <| by rwa [length_iterate] at h\n\n@[simp]\n\nTarget:\ntheorem mem_iterate {f : α → α} {a : α} {n : ℕ} {b : α} :\n    b ∈ iterate f a n ↔ ∃ m < n, b = f^[m] a :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_32ce66990d1f","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"df3c87cd4b27ed940158d498e269e644378bdf3d8c1ddad7a4c60c248c2c2b46","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/List","family_id":"mem_iterate","file_id":"mathlib/Mathlib/Data/List/Iterate.lean","sample_id":"32ce66990d1ffba436ebb7dc106eea97303764f97b8b35040bc36958f838afdb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9c6b8baf38f61be33d84aa02acad26a64e1f3aa8d253f05349f099a61ea5244e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f176d26ff830ef796b098e912be2e03d673bb82d8bb1a553aaeef68cdf0760c9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ff5f243c31598fd2aadb8dc36b39e21f486eb66be282ba3b720cbff07478671d","source_sha256":"94021f194f9dbe2b3ff10c0fe9e60d21f920f7ec47e18de0732170ff2d7ff47e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h p hp hp' ↦ hp.not_isUnit (h p hp'), fun h d hd ↦ by_contra fun hdu ↦ ?_⟩\n  have hd₀ : d ≠ 0 := ne_zero_of_dvd_ne_zero (ne_zero_of_dvd_ne_zero hx₀ hd) (dvd_mul_left d d)\n  obtain ⟨p, irr, dvd⟩ := WfDvdMonoid.exists_irreducible_factor hdu hd₀\n  exact h p irr ((mul_dvd_mul dvd dvd).trans hd)","hard_negative":true,"metrics":{"chosen_tokens":79,"rejected_tokens":5,"token_jaccard":0.075,"token_length_ratio":0.063291},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"65d3fff3408719a2ede93cd67e1dabf020bb63d559df78942f238c48f1eaa51e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Coprime.Lemmas\npublic import Mathlib.RingTheory.Nilpotent.Basic\npublic import Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid\npublic import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Squarefree elements of monoids\nAn element of a monoid is squarefree when it is not divisible by any squares\nexcept the squares of units.\n\nResults about squarefree natural numbers are proved in `Data.Nat.Squarefree`.\n\n## Main Definitions\n- `Squarefree r` indicates that `r` is only divisible by `x * x` if `x` is a unit.\n\n## Main Results\n- `multiplicity.squarefree_iff_emultiplicity_le_one`: `x` is `Squarefree` iff for every `y`, either\n  `emultiplicity y x ≤ 1` or `IsUnit y`.\n- `UniqueFactorizationMonoid.squarefree_iff_nodup_factors`: A nonzero element `x` of a unique\n  factorization monoid is squarefree iff `factors x` has no duplicate factors.\n\n## Tags\nsquarefree, multiplicity\n\n-/\n\n@[expose] public section\n\n\nvariable {R : Type*}\n\n/-- An element of a monoid is squarefree if the only squares that\n  divide it are the squares of units. -/\ndef Squarefree [Monoid R] (r : R) : Prop :=\n  ∀ x : R, x * x ∣ r → IsUnit x\n\ntheorem IsRelPrime.of_squarefree_mul [CommMonoid R] {m n : R} (h : Squarefree (m * n)) :\n    IsRelPrime m n := fun c hca hcb ↦ h c (mul_dvd_mul hca hcb)\n\n@[simp]\ntheorem IsUnit.squarefree [CommMonoid R] {x : R} (h : IsUnit x) : Squarefree x := fun _ hdvd =>\n  isUnit_of_mul_isUnit_left (isUnit_of_dvd_unit hdvd h)\n\ntheorem squarefree_one [CommMonoid R] : Squarefree (1 : R) :=\n  isUnit_one.squarefree\n\n@[simp]\ntheorem not_squarefree_zero [MonoidWithZero R] [Nontrivial R] : ¬Squarefree (0 : R) := by\n  rw [Squarefree, not_forall]\n  exact ⟨0, by simp⟩\n\ntheorem Squarefree.ne_zero [MonoidWithZero R] [Nontrivial R] {m : R} (hm : Squarefree (m : R)) :\n    m ≠ 0 := by\n  rintro rfl\n  exact not_squarefree_zero hm\n\n@[simp]\ntheorem Irreducible.squarefree [CommMonoid R] {x : R} (h : Irreducible x) : Squarefree x := by\n  rintro y ⟨z, hz⟩\n  rw [mul_assoc] at hz\n  rcases h.isUnit_or_isUnit hz with (hu | hu)\n  · exact hu\n  · apply isUnit_of_mul_isUnit_left hu\n\n@[simp]\ntheorem Prime.squarefree [CommMonoidWithZero R] [IsCancelMulZero R] {x : R} (h : Prime x) :\n    Squarefree x :=\n  h.irreducible.squarefree\n\ntheorem Squarefree.of_mul_left [Monoid R] {m n : R} (hmn : Squarefree (m * n)) : Squarefree m :=\n  fun p hp => hmn p (dvd_mul_of_dvd_left hp n)\n\ntheorem Squarefree.of_mul_right [CommMonoid R] {m n : R} (hmn : Squarefree (m * n)) :\n    Squarefree n := fun p hp => hmn p (dvd_mul_of_dvd_right hp m)\n\ntheorem Squarefree.squarefree_of_dvd [Monoid R] {x y : R} (hdvd : x ∣ y) (hsq : Squarefree y) :\n    Squarefree x := fun _ h => hsq _ (h.trans hdvd)\n\ntheorem Associated.squarefree_iff [Monoid R] {x y : R} (h : Associated x y) :\n    Squarefree x ↔ Squarefree y :=\n  ⟨fun hx ↦ hx.squarefree_of_dvd h.dvd', fun hy ↦ hy.squarefree_of_dvd h.dvd⟩\n\ntheorem Squarefree.eq_zero_or_one_of_pow_of_not_isUnit [Monoid R] {x : R} {n : ℕ}\n    (h : Squarefree (x ^ n)) (h' : ¬ IsUnit x) :\n    n = 0 ∨ n = 1 := by\n  contrapose! h'\n  replace h' : 2 ≤ n := by lia\n  have : x * x ∣ x ^ n := by rw [← sq]; exact pow_dvd_pow x h'\n  exact h.squarefree_of_dvd this x (refl _)\n\ntheorem Squarefree.pow_dvd_of_pow_dvd [Monoid R] {x y : R} {n : ℕ}\n    (hx : Squarefree y) (h : x ^ n ∣ y) : x ^ n ∣ x := by\n  by_cases hu : IsUnit x\n  · exact (hu.pow n).dvd\n  · rcases (hx.squarefree_of_dvd h).eq_zero_or_one_of_pow_of_not_isUnit hu with rfl | rfl <;> simp\n\nsection SquarefreeGcdOfSquarefree\n\nvariable {α : Type*} [CommMonoidWithZero α] [GCDMonoid α]\n\ntheorem Squarefree.gcd_right (a : α) {b : α} (hb : Squarefree b) : Squarefree (gcd a b) :=\n  hb.squarefree_of_dvd (gcd_dvd_right _ _)\n\ntheorem Squarefree.gcd_left {a : α} (b : α) (ha : Squarefree a) : Squarefree (gcd a b) :=\n  ha.squarefree_of_dvd (gcd_dvd_left _ _)\n\nend SquarefreeGcdOfSquarefree\n\ntheorem squarefree_iff_emultiplicity_le_one [CommMonoid R] (r : R) :\n    Squarefree r ↔ ∀ x : R, emultiplicity x r ≤ 1 ∨ IsUnit x := by\n  refine forall_congr' fun a => ?_\n  rw [← sq, pow_dvd_iff_le_emultiplicity, or_iff_not_imp_left, not_le, imp_congr _ Iff.rfl]\n  norm_cast\n  rw [← one_add_one_eq_two]\n  exact Order.add_one_le_iff_of_not_isMax (by simp)\n\nsection Irreducible\n\nvariable [CommMonoidWithZero R] [WfDvdMonoid R]\n\nTarget:\ntheorem squarefree_iff_no_irreducibles {x : R} (hx₀ : x ≠ 0) :\n    Squarefree x ↔ ∀ p, Irreducible p → ¬ (p * p ∣ x) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_ff5f243c3159","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"3715e83bf682fb1105ead0708dd43592344fab389870a44de509aa50284f0d6e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Squarefree","family_id":"squarefree_iff_no_irreducibles","file_id":"mathlib/Mathlib/Algebra/Squarefree/Basic.lean","sample_id":"ff5f243c31598fd2aadb8dc36b39e21f486eb66be282ba3b720cbff07478671d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1c6acec1b0c2300f662b80f0c922eb6275df7e3daf8325d6423e0848393437e0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2a6176a9db1730286a0ff010222bf3e543128b73fca8a9334a47cf26410cfd21","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"76c80b8cd380741542148048f4f9d53c485cf357afdef4b9fbe25f15cafe33a7","source_sha256":"20cce9f0236f79aab52863f3444c0947034ea25b58342e0def2f63cbf563ddf0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by simpa using! I.ringCon.add hx hy","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":16,"token_jaccard":0.714286,"token_length_ratio":1.454545},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"65db2051249a7f09929bde4a94b45756f68d8ad980f49fc516e07d8f55dc140b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Tactic.Abel\npublic import Mathlib.Algebra.Ring.Opposite\npublic import Mathlib.GroupTheory.GroupAction.SubMulAction\npublic import Mathlib.RingTheory.Congruence.Opposite\n\nNamespace:\nTwoSidedIdeal\n\nLocal context:\n/-\nCopyright (c) 2024 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang\n-/\n/-!\n# Two Sided Ideals\n\nIn this file, for any `Ring R`, we reinterpret `I : RingCon R` as a two-sided-ideal of a ring.\n\n## Main definitions and results\n\n* `TwoSidedIdeal`: For any `NonUnitalNonAssocRing R`, `TwoSidedIdeal R` is a wrapper around\n  `RingCon R`.\n* `TwoSidedIdeal.setLike`: Every `I : TwoSidedIdeal R` can be interpreted as a set of `R` where\n  `x ∈ I` if and only if `I.ringCon x 0`.\n* `TwoSidedIdeal.addCommGroup`: Every `I : TwoSidedIdeal R` is an abelian group.\n\n-/\n\n@[expose] public section\n\nopen MulOpposite\n\nsection definitions\n\n/--\nA two-sided ideal of a ring `R` is a subset of `R` that contains `0` and is closed under addition,\nnegation, and absorbs multiplication on both sides.\n-/\nstructure TwoSidedIdeal (R : Type*) [NonUnitalNonAssocRing R] where\n  /-- every two-sided-ideal is induced by a congruence relation on the ring. -/\n  ringCon : RingCon R\n\nend definitions\n\nnamespace TwoSidedIdeal\n\nsection NonUnitalNonAssocRing\n\nvariable {R : Type*} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R)\n\ninstance [Nontrivial R] : Nontrivial (TwoSidedIdeal R) := by\n  obtain ⟨I, J, h⟩ : Nontrivial (RingCon R) := inferInstance\n  exact ⟨⟨I⟩, ⟨J⟩, by contrapose h; aesop⟩\n\ninstance setLike : SetLike (TwoSidedIdeal R) R where\n  coe t := {r | t.ringCon r 0}\n  coe_injective := by\n    rintro ⟨t₁⟩ ⟨t₂⟩ (h : {x | _} = {x | _})\n    congr 1\n    refine RingCon.ext fun a b ↦ ⟨fun H ↦ ?_, fun H ↦ ?_⟩\n    · have H' : a - b ∈ {x | t₁ x 0} := sub_self b ▸ t₁.sub H (t₁.refl b)\n      rw [h] at H'\n      convert! t₂.add H' (t₂.refl b) using 1 <;> abel\n    · have H' : a - b ∈ {x | t₂ x 0} := sub_self b ▸ t₂.sub H (t₂.refl b)\n      rw [← h] at H'\n      convert! t₁.add H' (t₁.refl b) using 1 <;> abel\n\ninstance : PartialOrder (TwoSidedIdeal R) := .ofSetLike (TwoSidedIdeal R) R\n\nlemma mem_iff (x : R) : x ∈ I ↔ I.ringCon x 0 := Iff.rfl\n\n@[simp]\nlemma mem_mk {x : R} {c : RingCon R} : x ∈ mk c ↔ c x 0 := Iff.rfl\n\n@[simp, norm_cast]\nlemma coe_mk {c : RingCon R} : (mk c : Set R) = {x | c x 0} := rfl\n\nlemma rel_iff (x y : R) : I.ringCon x y ↔ x - y ∈ I := by\n  rw [mem_iff]\n  constructor\n  · intro h; convert! I.ringCon.sub h (I.ringCon.refl y); abel\n  · intro h; convert! I.ringCon.add h (I.ringCon.refl y) <;> abel\n\n/--\nthe coercion from two-sided-ideals to sets is an order embedding\n-/\n@[simps]\ndef coeOrderEmbedding : TwoSidedIdeal R ↪o Set R where\n  toFun := SetLike.coe\n  inj' := SetLike.coe_injective\n  map_rel_iff' {I J} := ⟨fun (h : (I : Set R) ⊆ (J : Set R)) _ h' ↦ h h', fun h _ h' ↦ h h'⟩\n\nlemma le_iff {I J : TwoSidedIdeal R} : I ≤ J ↔ (I : Set R) ⊆ (J : Set R) := Iff.rfl\n\n/-- Two-sided-ideals corresponds to congruence relations on a ring. -/\n@[simps apply symm_apply]\ndef orderIsoRingCon : TwoSidedIdeal R ≃o RingCon R where\n  toFun := TwoSidedIdeal.ringCon\n  invFun := .mk\n  map_rel_iff' {I J} := Iff.symm <| le_iff.trans ⟨fun h x y r => by rw [rel_iff] at r ⊢; exact h r,\n    fun h x hx => by rw [SetLike.mem_coe, mem_iff] at hx ⊢; exact h hx⟩\n\nlemma ringCon_injective : Function.Injective (TwoSidedIdeal.ringCon (R := R)) := by\n  rintro ⟨x⟩ ⟨y⟩ rfl; rfl\n\nlemma ringCon_le_iff {I J : TwoSidedIdeal R} : I ≤ J ↔ I.ringCon ≤ J.ringCon :=\n  orderIsoRingCon.map_rel_iff.symm\n\n@[ext]\nlemma ext {I J : TwoSidedIdeal R} (h : ∀ x, x ∈ I ↔ x ∈ J) : I = J :=\n  coeOrderEmbedding.injective (Set.ext h)\n\nlemma lt_iff (I J : TwoSidedIdeal R) : I < J ↔ (I : Set R) ⊂ (J : Set R) := by\n  rw [lt_iff_le_and_ne, Set.ssubset_iff_subset_ne, le_iff]\n  simp\n\nlemma zero_mem : 0 ∈ I := I.ringCon.refl 0\n\nTarget:\nlemma add_mem {x y} (hx : x ∈ I) (hy : y ∈ I) : x + y ∈ I :=\n\nProof body:\n","rejected":"by simpa using! I.ringCon.add hx hy\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/TwoSidedIdeal","family_id":"add_mem","file_id":"mathlib/Mathlib/RingTheory/TwoSidedIdeal/Basic.lean","sample_id":"76c80b8cd380741542148048f4f9d53c485cf357afdef4b9fbe25f15cafe33a7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a6a0730acc445e18a71efd0b82b6e6d57aa672e42e9dfc4ec8e49f142e23c559","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"afb40734c45bc026b3d01a0cc527b84f0434b661d87c2a26e7cbfccf318f6610","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"510546c198df0e51ff7856781031b1d5536b836338687232036e8eccc1052100","source_sha256":"94021f194f9dbe2b3ff10c0fe9e60d21f920f7ec47e18de0732170ff2d7ff47e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree, ← not_exists]\n  simp only [hr, not_true, false_or, and_false]","hard_negative":false,"metrics":{"chosen_tokens":19,"rejected_tokens":24,"token_jaccard":0.736842,"token_length_ratio":1.263158},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"65dc9b76c0a5b94548647162ba8b5957d753078f0ca43cc7f27727c95a425ece","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Coprime.Lemmas\npublic import Mathlib.RingTheory.Nilpotent.Basic\npublic import Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid\npublic import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Squarefree elements of monoids\nAn element of a monoid is squarefree when it is not divisible by any squares\nexcept the squares of units.\n\nResults about squarefree natural numbers are proved in `Data.Nat.Squarefree`.\n\n## Main Definitions\n- `Squarefree r` indicates that `r` is only divisible by `x * x` if `x` is a unit.\n\n## Main Results\n- `multiplicity.squarefree_iff_emultiplicity_le_one`: `x` is `Squarefree` iff for every `y`, either\n  `emultiplicity y x ≤ 1` or `IsUnit y`.\n- `UniqueFactorizationMonoid.squarefree_iff_nodup_factors`: A nonzero element `x` of a unique\n  factorization monoid is squarefree iff `factors x` has no duplicate factors.\n\n## Tags\nsquarefree, multiplicity\n\n-/\n\n@[expose] public section\n\n\nvariable {R : Type*}\n\n/-- An element of a monoid is squarefree if the only squares that\n  divide it are the squares of units. -/\ndef Squarefree [Monoid R] (r : R) : Prop :=\n  ∀ x : R, x * x ∣ r → IsUnit x\n\ntheorem IsRelPrime.of_squarefree_mul [CommMonoid R] {m n : R} (h : Squarefree (m * n)) :\n    IsRelPrime m n := fun c hca hcb ↦ h c (mul_dvd_mul hca hcb)\n\n@[simp]\ntheorem IsUnit.squarefree [CommMonoid R] {x : R} (h : IsUnit x) : Squarefree x := fun _ hdvd =>\n  isUnit_of_mul_isUnit_left (isUnit_of_dvd_unit hdvd h)\n\ntheorem squarefree_one [CommMonoid R] : Squarefree (1 : R) :=\n  isUnit_one.squarefree\n\n@[simp]\ntheorem not_squarefree_zero [MonoidWithZero R] [Nontrivial R] : ¬Squarefree (0 : R) := by\n  rw [Squarefree, not_forall]\n  exact ⟨0, by simp⟩\n\ntheorem Squarefree.ne_zero [MonoidWithZero R] [Nontrivial R] {m : R} (hm : Squarefree (m : R)) :\n    m ≠ 0 := by\n  rintro rfl\n  exact not_squarefree_zero hm\n\n@[simp]\ntheorem Irreducible.squarefree [CommMonoid R] {x : R} (h : Irreducible x) : Squarefree x := by\n  rintro y ⟨z, hz⟩\n  rw [mul_assoc] at hz\n  rcases h.isUnit_or_isUnit hz with (hu | hu)\n  · exact hu\n  · apply isUnit_of_mul_isUnit_left hu\n\n@[simp]\ntheorem Prime.squarefree [CommMonoidWithZero R] [IsCancelMulZero R] {x : R} (h : Prime x) :\n    Squarefree x :=\n  h.irreducible.squarefree\n\ntheorem Squarefree.of_mul_left [Monoid R] {m n : R} (hmn : Squarefree (m * n)) : Squarefree m :=\n  fun p hp => hmn p (dvd_mul_of_dvd_left hp n)\n\ntheorem Squarefree.of_mul_right [CommMonoid R] {m n : R} (hmn : Squarefree (m * n)) :\n    Squarefree n := fun p hp => hmn p (dvd_mul_of_dvd_right hp m)\n\ntheorem Squarefree.squarefree_of_dvd [Monoid R] {x y : R} (hdvd : x ∣ y) (hsq : Squarefree y) :\n    Squarefree x := fun _ h => hsq _ (h.trans hdvd)\n\ntheorem Associated.squarefree_iff [Monoid R] {x y : R} (h : Associated x y) :\n    Squarefree x ↔ Squarefree y :=\n  ⟨fun hx ↦ hx.squarefree_of_dvd h.dvd', fun hy ↦ hy.squarefree_of_dvd h.dvd⟩\n\ntheorem Squarefree.eq_zero_or_one_of_pow_of_not_isUnit [Monoid R] {x : R} {n : ℕ}\n    (h : Squarefree (x ^ n)) (h' : ¬ IsUnit x) :\n    n = 0 ∨ n = 1 := by\n  contrapose! h'\n  replace h' : 2 ≤ n := by lia\n  have : x * x ∣ x ^ n := by rw [← sq]; exact pow_dvd_pow x h'\n  exact h.squarefree_of_dvd this x (refl _)\n\ntheorem Squarefree.pow_dvd_of_pow_dvd [Monoid R] {x y : R} {n : ℕ}\n    (hx : Squarefree y) (h : x ^ n ∣ y) : x ^ n ∣ x := by\n  by_cases hu : IsUnit x\n  · exact (hu.pow n).dvd\n  · rcases (hx.squarefree_of_dvd h).eq_zero_or_one_of_pow_of_not_isUnit hu with rfl | rfl <;> simp\n\nsection SquarefreeGcdOfSquarefree\n\nvariable {α : Type*} [CommMonoidWithZero α] [GCDMonoid α]\n\ntheorem Squarefree.gcd_right (a : α) {b : α} (hb : Squarefree b) : Squarefree (gcd a b) :=\n  hb.squarefree_of_dvd (gcd_dvd_right _ _)\n\ntheorem Squarefree.gcd_left {a : α} (b : α) (ha : Squarefree a) : Squarefree (gcd a b) :=\n  ha.squarefree_of_dvd (gcd_dvd_left _ _)\n\nend SquarefreeGcdOfSquarefree\n\ntheorem squarefree_iff_emultiplicity_le_one [CommMonoid R] (r : R) :\n    Squarefree r ↔ ∀ x : R, emultiplicity x r ≤ 1 ∨ IsUnit x := by\n  refine forall_congr' fun a => ?_\n  rw [← sq, pow_dvd_iff_le_emultiplicity, or_iff_not_imp_left, not_le, imp_congr _ Iff.rfl]\n  norm_cast\n  rw [← one_add_one_eq_two]\n  exact Order.add_one_le_iff_of_not_isMax (by simp)\n\nsection Irreducible\n\nvariable [CommMonoidWithZero R] [WfDvdMonoid R]\n\ntheorem squarefree_iff_no_irreducibles {x : R} (hx₀ : x ≠ 0) :\n    Squarefree x ↔ ∀ p, Irreducible p → ¬ (p * p ∣ x) := by\n  refine ⟨fun h p hp hp' ↦ hp.not_isUnit (h p hp'), fun h d hd ↦ by_contra fun hdu ↦ ?_⟩\n  have hd₀ : d ≠ 0 := ne_zero_of_dvd_ne_zero (ne_zero_of_dvd_ne_zero hx₀ hd) (dvd_mul_left d d)\n  obtain ⟨p, irr, dvd⟩ := WfDvdMonoid.exists_irreducible_factor hdu hd₀\n  exact h p irr ((mul_dvd_mul dvd dvd).trans hd)\n\ntheorem irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree (r : R) :\n    (∀ x : R, Irreducible x → ¬x * x ∣ r) ↔ (r = 0 ∧ ∀ x : R, ¬Irreducible x) ∨ Squarefree r := by\n  refine ⟨fun h ↦ ?_, ?_⟩\n  · rcases eq_or_ne r 0 with (rfl | hr)\n    · exact .inl (by simpa using h)\n    · exact .inr ((squarefree_iff_no_irreducibles hr).mpr h)\n  · rintro (⟨rfl, h⟩ | h)\n    · simpa using h\n    intro x hx t\n    exact hx.not_isUnit (h x t)\n\ntheorem squarefree_iff_irreducible_sq_not_dvd_of_ne_zero {r : R} (hr : r ≠ 0) :\n    Squarefree r ↔ ∀ x : R, Irreducible x → ¬x * x ∣ r := by\n  simpa [hr] using (irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree r).symm\n\nTarget:\ntheorem squarefree_iff_irreducible_sq_not_dvd_of_exists_irreducible {r : R}\n    (hr : ∃ x : R, Irreducible x) : Squarefree r ↔ ∀ x : R, Irreducible x → ¬x * x ∣ r :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  rw [irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree, ← not_exists]\n  simp only [hr, not_true, false_or, and_false]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Squarefree","family_id":"squarefree_iff_irreducible_sq_not_dvd_of_exists_irreducible","file_id":"mathlib/Mathlib/Algebra/Squarefree/Basic.lean","sample_id":"510546c198df0e51ff7856781031b1d5536b836338687232036e8eccc1052100"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8390e404c259cefd040f4b8484fc4ca8346f13b8985655df805e82d4eca600cf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2b3879b0cad92f3b44a2e7ca566d11d48df147224dd7bfcf52ae47b52191abb9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"923ffa343160dc678018bf65a4585befc8d6632387419cd5ee47dffbd11b76e4","source_sha256":"0aa3275c10d58f2b53efa7a12fc422557c22614d050f0318487bd09968cf8e9f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x\n  induction x <;> rfl","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":14,"token_jaccard":0.615385,"token_length_ratio":1.555556},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"65de2a2715f4002c0b1034c38c8b15ff718a1b421e05bc56f46bd6f3e61dbb02","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Basic\npublic import Mathlib.Algebra.Group.Equiv.Defs\npublic import Mathlib.Algebra.Group.WithOne.Defs\n\nNamespace:\nWithOne\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Johan Commelin\n-/\n/-!\n# More operations on `WithOne` and `WithZero`\n\nThis file defines various bundled morphisms on `WithOne` and `WithZero`\nthat were not available in `Algebra/Group/WithOne/Defs`.\n\n## Main definitions\n\n* `WithOne.lift`, `WithZero.lift`\n* `WithOne.map`, `WithZero.map`\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero DenselyOrdered\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\nnamespace WithOne\n\n@[to_additive]\ninstance instInvolutiveInv [InvolutiveInv α] : InvolutiveInv (WithOne α) where\n  inv_inv a := (Option.map_map _ _ _).trans <| by simp_rw [inv_comp_inv, Option.map_id, id]\n\nsection\n\n/-- `WithOne.coe` as a bundled morphism -/\n@[to_additive (attr := simps apply) /-- `WithZero.coe` as a bundled morphism -/]\ndef coeMulHom [Mul α] : α →ₙ* WithOne α where\n  toFun := coe\n  map_mul' _ _ := rfl\n\nend\n\nsection lift\n\nvariable [Mul α] [MulOneClass β]\n\n/-- Lift a semigroup homomorphism `f` to a bundled monoid homomorphism. -/\n@[to_additive /--\nLift an additive semigroup homomorphism `f` to a bundled additive monoid homomorphism. -/]\ndef lift : (α →ₙ* β) ≃ (WithOne α →* β) where\n  toFun f :=\n    { toFun := WithOne.recOneCoe 1 f, map_one' := rfl,\n      map_mul' := fun x y => x.cases_on (by simp) (fun x => y.cases_on (by simp) (f.map_mul x)) }\n  invFun F := F.toMulHom.comp coeMulHom\n  right_inv F := MonoidHom.ext fun x => WithOne.cases_on x F.map_one.symm (fun _ => rfl)\n\nvariable (f : α →ₙ* β)\n\n@[to_additive (attr := simp)]\ntheorem lift_coe (x : α) : lift f x = f x :=\n  rfl\n\n@[to_additive (attr := simp)]\ntheorem lift_one : lift f 1 = 1 :=\n  rfl\n\n@[to_additive]\ntheorem lift_unique (f : WithOne α →* β) : f = lift (f.toMulHom.comp coeMulHom) :=\n  (lift.apply_symm_apply f).symm\n\n@[to_additive (attr := simp)]\ntheorem lift_symm_apply (f : WithOne α →* β) (x : α) : lift.symm f x = f x := rfl\n\n@[to_additive]\nlemma lift_symm_injective_of_injective {f : WithOne α →* β} (hf : Function.Injective f) :\n    Function.Injective (lift.symm f) :=\n  fun _ _ ↦ by simp [hf.eq_iff]\n\nend lift\n\nsection Map\n\nvariable [Mul α] [Mul β] [Mul γ]\n\n/-- Given a multiplicative map from `α → β` returns a monoid homomorphism\n  from `WithOne α` to `WithOne β` -/\n@[to_additive /-- Given an additive map from `α → β` returns an additive monoid homomorphism from\n`WithZero α` to `WithZero β` -/]\ndef mapMulHom (f : α →ₙ* β) : WithOne α →* WithOne β :=\n  lift (coeMulHom.comp f)\n\n@[to_additive (attr := simp)]\ntheorem mapMulHom_coe (f : α →ₙ* β) (a : α) : mapMulHom f (a : WithOne α) = f a :=\n  rfl\n\n@[to_additive (attr := simp)]\n\nTarget:\ntheorem mapMulHom_id : mapMulHom (MulHom.id α) = MonoidHom.id (WithOne α) :=\n\nProof body:\n","rejected":"by\n  ext x\n  induction x <;> rfl\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Group","family_id":"mapmulhom_id","file_id":"mathlib/Mathlib/Algebra/Group/WithOne/Basic.lean","sample_id":"923ffa343160dc678018bf65a4585befc8d6632387419cd5ee47dffbd11b76e4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d99c571d1e3b54f4a7c947a3bff9621121704efdf01dd61bab5cf6b4a711ab1c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"052befed77426c858368b90b76b9540e8aaef3d27651275eef8d687539481881","source_sha256":"94fce5005b3452fffd0935ed1c8380f811db0b64c02673ffec8176805d9749dd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [Functor.IsDense.leftKanExtensionIso]","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":3,"token_jaccard":0.1,"token_length_ratio":0.333333},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"65df635bab0e9d6233cbf5181ff045a971a88c78a56ba8f7d34bdadfdcf0a4a6","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Functor.KanExtension.DenseAt\npublic import Mathlib.CategoryTheory.Limits.Presheaf\npublic import Mathlib.CategoryTheory.Generator.StrongGenerator\n\nNamespace:\nCategoryTheory.Functor\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Dense functors\n\nA functor `F : C ⥤ D` is dense (`F.IsDense`) if `𝟭 D` is a pointwise\nleft Kan extension of `F` along itself, i.e. any `Y : D` is the\ncolimit of all `F.obj X` for all morphisms `F.obj X ⟶ Y` (which\nis the condition `F.DenseAt Y`).\nWhen `F` is full, we show that this\nis equivalent to saying that the restricted Yoneda functor\n`D ⥤ Cᵒᵖ ⥤ Type _` is fully faithful (see the lemma\n`Functor.isDense_iff_fullyFaithful_restrictedULiftYoneda`).\n\nWe also show that the range of a dense functor is a strong\ngenerator (see `Functor.isStrongGenerator_of_isDense`).\n\n## References\n\n* https://ncatlab.org/nlab/show/dense+subcategory\n\n-/\n\n@[expose] public section\n\nuniverse w v₁ v₂ v₃ u₁ u₂ u₃\n\nnamespace CategoryTheory\n\nopen Limits Opposite Presheaf ConcreteCategory\n\nvariable {C : Type u₁} {D : Type u₂} [Category.{v₁} C] [Category.{v₂} D]\n  {C' : Type u₃} [Category.{v₃} C']\n\nnamespace Functor\n\n/-- A functor `F : C ⥤ D` is dense if any `Y : D` is a canonical colimit\nrelatively to `F`. -/\nclass IsDense (F : C ⥤ D) : Prop where\n  isDenseAt (F) (Y : D) : F.isDenseAt Y\n\n/-- This is a choice of structure `F.DenseAt Y` when `F : C ⥤ D`\nis dense, and `Y : D`. -/\nnoncomputable def denseAt (F : C ⥤ D) [F.IsDense] (Y : D) : F.DenseAt Y :=\n  (IsDense.isDenseAt F Y).some\n\nlemma isDense_iff_nonempty_isPointwiseLeftKanExtension (F : C ⥤ D) :\n    F.IsDense ↔\n      Nonempty ((LeftExtension.mk _ (rightUnitor F).inv).IsPointwiseLeftKanExtension) :=\n  ⟨fun _ ↦ ⟨fun _ ↦ F.denseAt _⟩, fun ⟨h⟩ ↦ ⟨fun _ ↦ ⟨h _⟩⟩⟩\n\ninstance (F : C ⥤ D) [F.IsDense] : Functor.IsLeftKanExtension (𝟭 D) (Functor.rightUnitor F).inv :=\n  ((Functor.isDense_iff_nonempty_isPointwiseLeftKanExtension F).mp ‹_›).some.isLeftKanExtension\n\ninstance (F : C ⥤ D) [F.IsDense] : F.HasPointwiseLeftKanExtension F :=\n  fun X ↦ (Functor.IsDense.isDenseAt F X).some.hasPointwiseLeftKanExtensionAt\n\nlemma IsDense.of_iso {F G : C ⥤ D} (e : F ≅ G) [F.IsDense] :\n    G.IsDense where\n  isDenseAt Y := by\n    rw [← Functor.congr_isDenseAt e]\n    exact ⟨F.denseAt Y⟩\n\nlemma IsDense.iff_of_iso {F G : C ⥤ D} (e : F ≅ G) :\n    F.IsDense ↔ G.IsDense :=\n  ⟨fun _ ↦ of_iso e, fun _ ↦ of_iso e.symm⟩\n\nvariable (F : C ⥤ D)\n\ninstance (G : C' ⥤ C) [F.IsDense] [G.IsEquivalence] :\n    (G ⋙ F).IsDense where\n  isDenseAt Y := ⟨(F.denseAt Y).precompOfFinal G⟩\n\nlemma IsDense.comp_left_iff_of_isEquivalence (G : C' ⥤ C) [G.IsEquivalence] :\n    (G ⋙ F).IsDense ↔ F.IsDense := by\n  refine ⟨fun _ ↦ ?_, fun _ ↦ inferInstance⟩\n  let e : G.inv ⋙ G ⋙ F ≅ F := (associator _ _ _).symm ≪≫\n    isoWhiskerRight (G.asEquivalence.counitIso) _ ≪≫ F.leftUnitor\n  exact of_iso e\n\ninstance (G : D ⥤ C') [F.IsDense] [G.IsEquivalence] :\n    (F ⋙ G).IsDense where\n  isDenseAt Y :=\n    ⟨ letI e : Y ≅ G.obj (G.inv.obj Y) := G.asEquivalence.counitIso.symm.app Y\n      DenseAt.ofIso (F.denseAt (G.inv.obj Y) |>.postcompEquivalence G) e.symm ⟩\n\nlemma IsDense.comp_right_iff_of_isEquivalence (G : D ⥤ C') [G.IsEquivalence] :\n    (F ⋙ G).IsDense ↔ F.IsDense := by\n  refine ⟨fun _ ↦ ?_, fun _ ↦ inferInstance⟩\n  let e : (F ⋙ G) ⋙ G.inv ≅ F := associator .. ≪≫\n    isoWhiskerLeft _ G.asEquivalence.unitIso.symm ≪≫ F.rightUnitor\n  exact of_iso e\n\nset_option backward.defeqAttrib.useBackward true in\ninstance [F.IsDense] : (restrictedULiftYoneda.{w} F).Faithful where\n  map_injective h :=\n    (F.denseAt _).hom_ext' (fun X p ↦ by\n      simpa using! ULift.up_injective (ConcreteCategory.congr_hom (CC := fun X ↦ X)\n        (NatTrans.congr_app h (op X)) (ULift.up p)))\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\ninstance [F.IsDense] : (restrictedULiftYoneda.{w} F).Full where\n  map_surjective {Y Z} f := by\n    let c : Cocone (CostructuredArrow.proj F Y ⋙ F) :=\n      { pt := Z\n        ι :=\n          { app g := ((f.app (op g.left)) (ULift.up g.hom)).down\n            naturality g₁ g₂ φ := by\n              simpa [uliftFunctor, uliftYoneda,\n                restrictedULiftYoneda, ← ULift.down_inj] using\n                ((f.naturality_apply φ.left.op) (ULift.up g₂.hom)).symm } }\n    refine ⟨(F.denseAt Y).desc c, ?_⟩\n    ext ⟨X⟩ ⟨x⟩\n    have := (F.denseAt Y).fac c (.mk x)\n    dsimp [c] at this\n    simpa using ULift.down_injective this\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nvariable {F} in\nlemma IsDense.of_fullyFaithful_restrictedULiftYoneda [F.Full]\n    (h : (restrictedULiftYoneda.{w} F).FullyFaithful) :\n    F.IsDense where\n  isDenseAt Y := by\n    let φ (s : Cocone (CostructuredArrow.proj F Y ⋙ F)) :\n        (restrictedULiftYoneda.{w} F).obj Y ⟶ (restrictedULiftYoneda F).obj s.pt :=\n      { app := fun ⟨X⟩ ↦ ↾fun ⟨x⟩ ↦ ULift.up (s.ι.app (.mk x))\n        naturality := by\n          rintro ⟨X₁⟩ ⟨X₂⟩ ⟨f⟩\n          ext ⟨x⟩\n          let α : CostructuredArrow.mk (F.map f ≫ x) ⟶ CostructuredArrow.mk x :=\n            CostructuredArrow.homMk f\n          exact ULift.down_injective (s.w α).symm }\n    have hφ (s) (j) : (restrictedULiftYoneda F).map j.hom ≫ φ s =\n        (restrictedULiftYoneda F).map (s.ι.app j) := by\n      ext ⟨X⟩ ⟨x⟩\n      let α : .mk (x ≫ j.hom) ⟶ j := CostructuredArrow.homMk (F.preimage x)\n      have := s.w α\n      dsimp [uliftYoneda, φ, α] at this ⊢\n      apply ULift.down_injective\n      simpa using this.symm\n    exact\n      ⟨{desc s := (h.preimage (φ s))\n        fac s j := h.map_injective (by simp [hφ])\n        uniq s m hm := h.map_injective (by\n          ext ⟨_⟩ ⟨_⟩\n          simp [φ, ← hm]) }⟩\n\nlemma isDense_iff_fullyFaithful_restrictedULiftYoneda [F.Full] :\n    F.IsDense ↔ Nonempty (restrictedULiftYoneda.{w} F).FullyFaithful :=\n  ⟨fun _ ↦ ⟨FullyFaithful.ofFullyFaithful _⟩,\n    fun ⟨h⟩ ↦ IsDense.of_fullyFaithful_restrictedULiftYoneda h⟩\n\nopen ObjectProperty in\nlemma isStrongGenerator_of_isDense [F.IsDense] :\n    IsStrongGenerator (.ofObj F.obj) :=\n  (IsStrongGenerator.mk_of_exists_colimitsOfShape.{max u₁ u₂ v₁ v₂,\n      max u₁ v₁ v₂} (fun Y ↦ ⟨_, _, ⟨{\n    ι := _\n    diag := _\n    isColimit := (IsColimit.whiskerEquivalence (F.denseAt Y)\n      ((ShrinkHoms.equivalence _).symm.trans ((Shrink.equivalence _)).symm))\n    prop_diag_obj := by simp }⟩⟩))\n\n/-- If `F` is dense, the left Kan extension of `F` along `F` is isomorphic to the identity. -/\nnoncomputable def IsDense.leftKanExtensionIso (F : C ⥤ D) [F.IsDense] :\n    F.leftKanExtension F ≅ 𝟭 D :=\n  Functor.leftKanExtensionUnique _ (F.leftKanExtensionUnit F) _ F.rightUnitor.inv\n\n@[reassoc (attr := simp)]\n\nTarget:\nlemma IsDense.leftKanExtensionUnit_leftKanExtensionIso_hom (F : C ⥤ D) [F.IsDense] :\n    F.leftKanExtensionUnit F ≫ F.whiskerLeft (Functor.IsDense.leftKanExtensionIso F).hom =\n      F.rightUnitor.inv :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Functor","family_id":"isdense","file_id":"mathlib/Mathlib/CategoryTheory/Functor/KanExtension/Dense.lean","sample_id":"052befed77426c858368b90b76b9540e8aaef3d27651275eef8d687539481881"},"split":"train"}
{"binding":{"chosen_proof_sha256":"16b3be2598502346de2754703a4649aa758c506146fa34fc091635ed0bf9b555","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"41091d79c58a55895294fdd103ee3f3bbfd8236b3858691c14a8a3dfe9a1b2a2","source_sha256":"04666e1a54002d0f7b310bb68c623ecf47eda2066cc741dda3dda2f5fc6f4c64","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases isEmpty_or_nonempty X with h | h\n  · exact ⟨Classical.arbitrary Y, funext <| h.elim⟩\n  · exact ⟨f (Classical.arbitrary X), hf.eq_const _⟩","hard_negative":false,"metrics":{"chosen_tokens":39,"rejected_tokens":5,"token_jaccard":0.111111,"token_length_ratio":0.128205},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"65e3d6b9cf5695a9237acfca994779d50791b309e06b70d11686ba8dd8ed00fc","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Indicator\npublic import Mathlib.Topology.Connected.LocallyConnected\npublic import Mathlib.Topology.Sets.Closeds\n\nNamespace:\nIsLocallyConstant\n\nLocal context:\n/-\nCopyright (c) 2021 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\n/-!\n# Locally constant functions\n\nThis file sets up the theory of locally constant function from a topological space to a type.\n\n## Main definitions and constructions\n\n* `IsLocallyConstant f` : a map `f : X → Y` where `X` is a topological space is locally\n                            constant if every set in `Y` has an open preimage.\n* `LocallyConstant X Y` : the type of locally constant maps from `X` to `Y`\n* `LocallyConstant.map` : push-forward of locally constant maps\n* `LocallyConstant.comap` : pull-back of locally constant maps\n-/\n\n@[expose] public section\n\nvariable {X Y Z α : Type*} [TopologicalSpace X]\n\nopen Set Filter\nopen scoped Topology\n\n/-- A function between topological spaces is locally constant if the preimage of any set is open. -/\ndef IsLocallyConstant (f : X → Y) : Prop :=\n  ∀ s : Set Y, IsOpen (f ⁻¹' s)\n\nnamespace IsLocallyConstant\n\nopen List in\nprotected theorem tfae (f : X → Y) :\n    TFAE [IsLocallyConstant f,\n      ∀ x, ∀ᶠ x' in 𝓝 x, f x' = f x,\n      ∀ x, IsOpen { x' | f x' = f x },\n      ∀ y, IsOpen (f ⁻¹' {y}),\n      ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x] := by\n  tfae_have 1 → 4 := fun h y => h {y}\n  tfae_have 4 → 3 := fun h x => h (f x)\n  tfae_have 3 → 2 := fun h x => IsOpen.mem_nhds (h x) rfl\n  tfae_have 2 → 5\n  | h, x => by\n    rcases mem_nhds_iff.1 (h x) with ⟨U, eq, hU, hx⟩\n    exact ⟨U, hU, hx, eq⟩\n  tfae_have 5 → 1\n  | h, s => by\n    refine isOpen_iff_forall_mem_open.2 fun x hx ↦ ?_\n    rcases h x with ⟨U, hU, hxU, eq⟩\n    exact ⟨U, fun x' hx' => mem_preimage.2 <| (eq x' hx').symm ▸ hx, hU, hxU⟩\n  tfae_finish\n\n@[nontriviality]\ntheorem of_discrete [DiscreteTopology X] (f : X → Y) : IsLocallyConstant f := fun _ =>\n  isOpen_discrete _\n\ntheorem isOpen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsOpen { x | f x = y } :=\n  hf {y}\n\ntheorem isClosed_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClosed { x | f x = y } :=\n  ⟨hf {y}ᶜ⟩\n\ntheorem isClopen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClopen { x | f x = y } :=\n  ⟨isClosed_fiber hf _, isOpen_fiber hf _⟩\n\ntheorem iff_exists_open (f : X → Y) :\n    IsLocallyConstant f ↔ ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x :=\n  (IsLocallyConstant.tfae f).out 0 4\n\ntheorem iff_eventually_eq (f : X → Y) : IsLocallyConstant f ↔ ∀ x, ∀ᶠ y in 𝓝 x, f y = f x :=\n  (IsLocallyConstant.tfae f).out 0 1\n\ntheorem exists_open {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x :=\n  (iff_exists_open f).1 hf x\n\nprotected theorem eventually_eq {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    ∀ᶠ y in 𝓝 x, f y = f x :=\n  (iff_eventually_eq f).1 hf x\n\ntheorem iff_isOpen_fiber_apply {f : X → Y} : IsLocallyConstant f ↔ ∀ x, IsOpen (f ⁻¹' {f x}) :=\n  (IsLocallyConstant.tfae f).out 0 2\n\ntheorem iff_isOpen_fiber {f : X → Y} : IsLocallyConstant f ↔ ∀ y, IsOpen (f ⁻¹' {y}) :=\n  (IsLocallyConstant.tfae f).out 0 3\n\nprotected theorem continuous [TopologicalSpace Y] {f : X → Y} (hf : IsLocallyConstant f) :\n    Continuous f :=\n  ⟨fun _ _ => hf _⟩\n\ntheorem iff_continuous {_ : TopologicalSpace Y} [DiscreteTopology Y] (f : X → Y) :\n    IsLocallyConstant f ↔ Continuous f :=\n  ⟨IsLocallyConstant.continuous, fun h s => h.isOpen_preimage s (isOpen_discrete _)⟩\n\ntheorem of_constant (f : X → Y) (h : ∀ x y, f x = f y) : IsLocallyConstant f :=\n  (iff_eventually_eq f).2 fun _ => Eventually.of_forall fun _ => h _ _\n\nprotected theorem const (y : Y) : IsLocallyConstant (Function.const X y) :=\n  of_constant _ fun _ _ => rfl\n\nprotected theorem comp {f : X → Y} (hf : IsLocallyConstant f) (g : Y → Z) :\n    IsLocallyConstant (g ∘ f) := fun s => by\n  rw [Set.preimage_comp]\n  exact hf _\n\ntheorem prodMk {Y'} {f : X → Y} {f' : X → Y'} (hf : IsLocallyConstant f)\n    (hf' : IsLocallyConstant f') : IsLocallyConstant fun x => (f x, f' x) :=\n  (iff_eventually_eq _).2 fun x =>\n    (hf.eventually_eq x).mp <| (hf'.eventually_eq x).mono fun _ hf' hf => Prod.ext hf hf'\n\ntheorem comp₂ {Y₁ Y₂ Z : Type*} {f : X → Y₁} {g : X → Y₂} (hf : IsLocallyConstant f)\n    (hg : IsLocallyConstant g) (h : Y₁ → Y₂ → Z) : IsLocallyConstant fun x => h (f x) (g x) :=\n  (hf.prodMk hg).comp fun x : Y₁ × Y₂ => h x.1 x.2\n\ntheorem comp_continuous [TopologicalSpace Y] {g : Y → Z} {f : X → Y} (hg : IsLocallyConstant g)\n    (hf : Continuous f) : IsLocallyConstant (g ∘ f) := fun s => by\n  rw [Set.preimage_comp]\n  exact hf.isOpen_preimage _ (hg _)\n\n/-- A locally constant function is constant on any preconnected set. -/\ntheorem apply_eq_of_isPreconnected {f : X → Y} (hf : IsLocallyConstant f) {s : Set X}\n    (hs : IsPreconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by\n  let U := f ⁻¹' {f y}\n  suffices x ∉ Uᶜ from Classical.not_not.1 this\n  intro hxV\n  specialize hs U Uᶜ (hf {f y}) (hf {f y}ᶜ) _ ⟨y, ⟨hy, rfl⟩⟩ ⟨x, ⟨hx, hxV⟩⟩\n  · simp only [union_compl_self, subset_univ]\n  · simp only [inter_empty, Set.not_nonempty_empty, inter_compl_self] at hs\n\ntheorem apply_eq_of_preconnectedSpace [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f)\n    (x y : X) : f x = f y :=\n  hf.apply_eq_of_isPreconnected isPreconnected_univ trivial trivial\n\ntheorem eq_const [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    f = Function.const X (f x) :=\n  funext fun y => hf.apply_eq_of_preconnectedSpace y x\n\nTarget:\ntheorem exists_eq_const [PreconnectedSpace X] [Nonempty Y] {f : X → Y} (hf : IsLocallyConstant f) :\n    ∃ y, f = Function.const X y :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/LocallyConstant","family_id":"exists_eq_const","file_id":"mathlib/Mathlib/Topology/LocallyConstant/Basic.lean","sample_id":"41091d79c58a55895294fdd103ee3f3bbfd8236b3858691c14a8a3dfe9a1b2a2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4c0ce5d128f69a2b99881bee610f584c1a319bb4f885c9faba74ae83efc61828","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c8cd16ca7c7e5a007fb195fa59aed53c5734a477ccb7bfa5ed293e06fc1f996c","source_sha256":"d9cef763bc837b8cf34f04f4dde67c215bb2f264d0a2452e9434fbd1d721514c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    have := hf (equalizer.lift f h) (equalizer.ι p q) (by simp)\n    rw [← cancel_epi (equalizer.ι p q), equalizer.condition]\n  isIso := by tauto","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":5,"token_jaccard":0.035714,"token_length_ratio":0.119048},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"666d03667d0bdc320217bf01ffe15b5005697344c8cf473a49e27d9fd9bddec0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Subobject.Lattice\npublic import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Extremal epimorphisms\n\nAn extremal epimorphism `p : X ⟶ Y` is an epimorphism which does not factor\nthrough any proper subobject of `Y`. In case the category has equalizers,\nwe show that a morphism `p : X ⟶ Y` which does not factor through\nany proper subobject of `Y` is automatically an epimorphism, and also\nan extremal epimorphism. We also show that a strong epimorphism\nis an extremal epimorphism, and that both notions coincide when\nthe category has pullbacks.\n\n## References\n\n* https://ncatlab.org/nlab/show/extremal+epimorphism\n\n-/\n\npublic section\n\nuniverse v u\n\nnamespace CategoryTheory\n\nopen Limits\n\nvariable {C : Type u} [Category.{v} C] {X Y : C}\n\n/-- An extremal epimorphism `f : X ⟶ Y` is an epimorphism which does not\nfactor through any proper subobject of `Y`. -/\nclass ExtremalEpi (f : X ⟶ Y) : Prop extends Epi f where\n  isIso (f) {Z : C} (p : X ⟶ Z) (i : Z ⟶ Y) (fac : p ≫ i = f) [Mono i] : IsIso i\n\nvariable (f : X ⟶ Y)\n\nlemma ExtremalEpi.subobject_eq_top [ExtremalEpi f]\n    {A : Subobject Y} (hA : Subobject.Factors A f) : A = ⊤ := by\n  rw [← Subobject.isIso_arrow_iff_eq_top]\n  exact isIso f (Subobject.factorThru A f hA) _ (by simp)\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma ExtremalEpi.mk_of_hasEqualizers [HasEqualizers C]\n    (hf : ∀ ⦃Z : C⦄ (p : X ⟶ Z) (i : Z ⟶ Y) (_ : p ≫ i = f) [Mono i], IsIso i) :\n    ExtremalEpi f where\n  left_cancellation {Z} p q h :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory","family_id":"extremalepi","file_id":"mathlib/Mathlib/CategoryTheory/ExtremalEpi.lean","sample_id":"c8cd16ca7c7e5a007fb195fa59aed53c5734a477ccb7bfa5ed293e06fc1f996c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d7f9b791b4990ce10c90fe8f0ae29086af43a18750773d3fc0cc6aa2bef28c2c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fd1fd9a53ab94e93ee71cb526c00c66f283a440ab2cd926e2cae6762d9e62179","source_sha256":"c1ecc80ead99a17d01db2984c5f06df1499354702fc192601863dee22fbd7b8f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine hs.induction_on _ (by simp) fun {i s} his hs e ↦ ?_\n  simp only [iInf_insert, e, ideal_inf]","hard_negative":false,"metrics":{"chosen_tokens":30,"rejected_tokens":5,"token_jaccard":0.074074,"token_length_ratio":0.166667},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"6684815c752a340c53cb511cd0fe2d62e976fa1128da9d4d0a30df235753ff55","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact\npublic import Mathlib.AlgebraicGeometry.Properties\npublic import Mathlib.Tactic.DepRewrite\n\nNamespace:\nAlgebraicGeometry.Scheme.IdealSheafData\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Ideal sheaves on schemes\n\nWe define ideal sheaves of schemes and provide various constructors for it.\n\n## Main definition\n* `AlgebraicGeometry.Scheme.IdealSheafData`: A structure that contains the data to uniquely define\n  an ideal sheaf, consisting of\n  1. an ideal `I(U) ≤ Γ(X, U)` for every affine open `U`\n  2. a proof that `I(D(f)) = I(U)_f` for every affine open `U` and every section `f : Γ(X, U)`.\n* `AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals`:\n  The largest ideal sheaf contained in a family of ideals.\n* `AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine`:\n  Over affine schemes, ideal sheaves are in bijection with ideals of the global sections.\n* `AlgebraicGeometry.Scheme.IdealSheafData.support`: The support of an ideal sheaf.\n* `AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal`: The vanishing ideal of a set.\n* `AlgebraicGeometry.Scheme.Hom.ker`: The kernel of a morphism.\n\n## Main results\n* `AlgebraicGeometry.Scheme.IdealSheafData.gc`:\n  `support` and `vanishingIdeal` forms a Galois connection.\n* `AlgebraicGeometry.Scheme.Hom.support_ker`: The support of a kernel of a quasi-compact morphism\n  is the closure of the range.\n\n## Implementation detail\n\nIdeal sheaves are not yet defined in this file as actual subsheaves of `𝒪ₓ`.\nInstead, for the ease of development and application,\nwe define the structure `IdealSheafData` containing all necessary data to uniquely define an\nideal sheaf. This should be refactored as a constructor for ideal sheaves once they are introduced\ninto mathlib.\n\n-/\n\n@[expose] public section\n\nopen CategoryTheory TopologicalSpace\n\nuniverse u\n\nnamespace AlgebraicGeometry.Scheme\n\nvariable {X : Scheme.{u}}\n\n/--\nA structure that contains the data to uniquely define an ideal sheaf, consisting of\n1. an ideal `I(U) ≤ Γ(X, U)` for every affine open `U`\n2. a proof that `I(D(f)) = I(U)_f` for every affine open `U` and every section `f : Γ(X, U)`\n3. a subset of `X` equal to the support.\n\nAlso see `Scheme.IdealSheafData.mkOfMemSupportIff` for a constructor with the condition on the\nsupport being (usually) easier to prove.\n-/\nstructure IdealSheafData (X : Scheme.{u}) : Type u where\n  /-- The component of an ideal sheaf at an affine open. -/\n  ideal : ∀ U : X.affineOpens, Ideal Γ(X, U)\n  /-- Also see `AlgebraicGeometry.Scheme.IdealSheafData.map_ideal` -/\n  map_ideal_basicOpen : ∀ (U : X.affineOpens) (f : Γ(X, U)),\n    (ideal U).map (X.presheaf.map (homOfLE <| X.basicOpen_le f).op).hom =\n      ideal (X.affineBasicOpen f)\n  /-- The support of an ideal sheaf. Use `IdealSheafData.support` instead for most occasions. -/\n  supportSet : Set X := ⋂ U, X.zeroLocus (U := U.1) (ideal U)\n  supportSet_eq_iInter_zeroLocus : supportSet = ⋂ U, X.zeroLocus (U := U.1) (ideal U) := by rfl\n\nnamespace IdealSheafData\n\n@[ext]\nprotected lemma ext {I J : X.IdealSheafData} (h : I.ideal = J.ideal) : I = J := by\n  obtain ⟨i, _, s, hs⟩ := I\n  obtain ⟨j, _, t, ht⟩ := J\n  subst h\n  congr\n  rw [hs, ht]\n\nsection Order\n\ninstance : PartialOrder (IdealSheafData X) := PartialOrder.lift ideal fun _ _ ↦ IdealSheafData.ext\n\nlemma le_def {I J : IdealSheafData X} : I ≤ J ↔ ∀ U, I.ideal U ≤ J.ideal U := .rfl\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : CompleteSemilatticeSup (IdealSheafData X) where\n  sSup s :=\n  { ideal := sSup (ideal '' s),\n    map_ideal_basicOpen := by\n      have : sSup (ideal '' s) = ⨆ i : s, ideal i.1 := by\n        conv_lhs => rw [← Subtype.range_val (s := s), ← Set.range_comp]\n        rfl\n      simp only [this, iSup_apply, Ideal.map_iSup, map_ideal_basicOpen, implies_true] }\n  isLUB_sSup _ := .of_image (f := ideal) le_def (isLUB_sSup _)\n\n/-- The largest ideal sheaf contained in a family of ideals. -/\ndef ofIdeals (I : ∀ U : X.affineOpens, Ideal Γ(X, U)) : IdealSheafData X :=\n  sSup { J : IdealSheafData X | J.ideal ≤ I }\n\nlemma ideal_ofIdeals_le (I : ∀ U : X.affineOpens, Ideal Γ(X, U)) :\n    (ofIdeals I).ideal ≤ I :=\n  sSup_le (Set.forall_mem_image.mpr fun _ ↦ id)\n\n/-- The Galois coinsertion between ideal sheaves and arbitrary families of ideals. -/\nprotected def gci : GaloisCoinsertion ideal (ofIdeals (X := X)) where\n  choice I hI :=\n  { ideal := I\n    map_ideal_basicOpen U f :=\n      (ideal_ofIdeals_le I).antisymm hI ▸ (ofIdeals I).map_ideal_basicOpen U f }\n  gc _ _ := ⟨(le_sSup ·), (le_trans · (ideal_ofIdeals_le _))⟩\n  u_l_le _ := sSup_le fun _ ↦ id\n  choice_eq I hI := IdealSheafData.ext (hI.antisymm (ideal_ofIdeals_le I))\n\nlemma strictMono_ideal : StrictMono (ideal (X := X)) := IdealSheafData.gci.strictMono_l\nlemma ideal_mono : Monotone (ideal (X := X)) := strictMono_ideal.monotone\nlemma ofIdeals_mono : Monotone (ofIdeals (X := X)) := IdealSheafData.gci.gc.monotone_u\nlemma ofIdeals_ideal (I : IdealSheafData X) : ofIdeals I.ideal = I := IdealSheafData.gci.u_l_eq _\nlemma le_ofIdeals_iff {I : IdealSheafData X} {J} : I ≤ ofIdeals J ↔ I.ideal ≤ J :=\n  IdealSheafData.gci.gc.le_iff_le.symm\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : OrderTop (IdealSheafData X) where\n  top.ideal := ⊤\n  top.map_ideal_basicOpen := by simp [Ideal.map_top]\n  top.supportSet := ⊥\n  top.supportSet_eq_iInter_zeroLocus := by\n    ext x\n    simpa using X.isBasis_affineOpens.exists_subset_of_mem_open (Set.mem_univ x) isOpen_univ\n  le_top I U := le_top\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : OrderBot (IdealSheafData X) where\n  bot.ideal := ⊥\n  bot.map_ideal_basicOpen := by simp\n  bot.supportSet := ⊤\n  bot.supportSet_eq_iInter_zeroLocus := by ext; simp\n  bot_le I U := bot_le\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : SemilatticeInf (IdealSheafData X) where\n  inf I J :=\n  { ideal := I.ideal ⊓ J.ideal\n    map_ideal_basicOpen U f := by\n      dsimp\n      have : (X.presheaf.map (homOfLE (X.basicOpen_le f)).op).hom = algebraMap _ _ := rfl\n      have inst := U.2.isLocalization_basicOpen f\n      rw [← I.map_ideal_basicOpen U f, ← J.map_ideal_basicOpen U f, this]\n      ext x\n      obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq (.powers f) x\n      simp only [IsLocalization.mk'_mem_map_algebraMap_iff, Submonoid.mem_powers_iff, Ideal.mem_inf,\n        exists_exists_eq_and]\n      refine ⟨fun ⟨n, h₁, h₂⟩ ↦ ⟨⟨n, h₁⟩, ⟨n, h₂⟩⟩, ?_⟩\n      rintro ⟨⟨n₁, h₁⟩, ⟨n₂, h₂⟩⟩\n      refine ⟨n₁ + n₂, ?_, ?_⟩\n      · rw [add_comm, pow_add, mul_assoc]; exact Ideal.mul_mem_left _ _ h₁\n      · rw [pow_add, mul_assoc]; exact Ideal.mul_mem_left _ _ h₂ }\n  inf_le_left I J U := inf_le_left\n  inf_le_right I J U := inf_le_right\n  le_inf I J K hIJ hIK U := le_inf (hIJ U) (hIK U)\n\ninstance : CompleteLattice (IdealSheafData X) where\n  __ := (inferInstance : OrderTop (IdealSheafData X))\n  __ := (inferInstance : OrderBot (IdealSheafData X))\n  __ := (inferInstance : SemilatticeInf (IdealSheafData X))\n  __ := (inferInstance : CompleteSemilatticeSup (IdealSheafData X))\n  __ := IdealSheafData.gci.liftCompleteLattice\n\n@[simp]\nlemma ideal_top : ideal (X := X) ⊤ = ⊤ := rfl\n\n@[simp]\nlemma ideal_bot : ideal (X := X) ⊥ = ⊥ := rfl\n\n@[simp]\nlemma ideal_sup {I J : IdealSheafData X} : (I ⊔ J).ideal = I.ideal ⊔ J.ideal := rfl\n\n@[simp]\nlemma ideal_sSup {I : Set (IdealSheafData X)} : (sSup I).ideal = sSup (ideal '' I) := rfl\n\n@[simp]\nlemma ideal_iSup {ι : Type*} {I : ι → IdealSheafData X} : (iSup I).ideal = ⨆ i, (I i).ideal := by\n  rw [← sSup_range, ← sSup_range, ideal_sSup, ← Set.range_comp, Function.comp_def]\n\n@[simp]\nlemma ideal_inf {I J : IdealSheafData X} : (I ⊓ J).ideal = I.ideal ⊓ J.ideal := rfl\n\n@[simp]\n\nTarget:\nlemma ideal_biInf {ι : Type*} (I : ι → IdealSheafData X) {s : Set ι} (hs : s.Finite) :\n    (⨅ i ∈ s, I i).ideal = ⨅ i ∈ s, (I i).ideal :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/IdealSheaf","family_id":"ideal_biinf","file_id":"mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/Basic.lean","sample_id":"fd1fd9a53ab94e93ee71cb526c00c66f283a440ab2cd926e2cae6762d9e62179"},"split":"train"}
{"binding":{"chosen_proof_sha256":"76fa0ee83574746f588fc7899220fecd9e66129f439898c7d8ec00fbf1b68c95","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bfdf8a5735fb83291498154fe55688e7c5eabb3111989c0465481333ca9088fd","source_sha256":"231219a77cbb3d5335e2c2c8fee21d9a8c54aa585e63a8a73c88281c1837f6a8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨i', hi'_lub⟩ : ∃ i', IsLUB (Set.Iio i) i' := exists_lub_Iio i\n  rcases lub_Iio_eq_self_or_Iio_eq_Iic i hi'_lub with hi'_eq_i | h_Iio_eq_Iic\n  · rw [← hi'_eq_i] at hi'_lub ⊢\n    exact hτ.measurableSet_lt_of_isLUB i' hi'_lub\n  · have h_lt_eq_preimage : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iio i := rfl\n    have h_Iio_eq_Iic' : Set.Iio (i : WithTop ι) = Set.Iic (i' : WithTop ι) := by\n      rw [← image_coe_Iio, ← image_coe_Iic, h_Iio_eq_Iic]\n    rw [h_lt_eq_preimage, h_Iio_eq_Iic']\n    exact f.mono (le_of_isLUB_Iio i hi'_lub) _ (hτ.measurableSet_le i')","hard_negative":false,"metrics":{"chosen_tokens":128,"rejected_tokens":3,"token_jaccard":0.033898,"token_length_ratio":0.023438},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"66a36a29a184c930f6350812b46806ac0e2e0deecc516ee3520dceb931d92c66","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Process.Adapted\npublic import Mathlib.MeasureTheory.Constructions.BorelSpace.WithTop\npublic import Mathlib.Data.ENat.Lattice\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2021 Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kexing Ying, Rémy Degenne\n-/\n/-!\n# Stopping times, stopped processes and stopped values\n\nDefinition and properties of stopping times.\n\n## Main definitions\n\n* `MeasureTheory.IsStoppingTime`: a stopping time with respect to some filtration `f` on a\n  measurable space `Ω` is a function `τ : Ω → WithTop ι` such that for all `i : ι`,\n  the preimage of `{j | j ≤ i}` along `τ` is `f i`-measurable\n* `MeasureTheory.IsStoppingTime.measurableSpace`: the σ-algebra associated with a stopping time\n\n## Main results\n\n* `IsStronglyProgressive.stoppedProcess`: the stopped process of a progressively measurable process\n  is progressively measurable.\n* `memLp_stoppedProcess`: if a process belongs to `ℒp` at every time in `ℕ`, then its stopped\n  process belongs to `ℒp` as well.\n\n## Implementation notes\n\nFor a filtration on a type `ι`, we define stopping times as functions from the measurable space `Ω`\nto `WithTop ι`, which allows stopping times that can take an infinite value, represented by\n`⊤ : WithTop ι`.\n\nThis means that if we have a process `X : ι → Ω → β` and a stopping time `τ : Ω → WithTop ι`, then\nto consider the value of `X` at the stopping time `τ ω`, we need to write `X (τ ω).untopA ω`,\nin which `(τ ω).untopA` is the value of `τ ω` in `ι` if `τ ω ≠ ⊤` and some arbitrary value if\n`τ ω = ⊤`.\n\nWhile indexing would be more convenient if we defined stopping times as functions from `Ω` to `ι`,\nthis would prevent us from using stopping times as in standard mathematical literature, where a\ntypical example of stopping time is the first time an event occurs, which may never happen.\nConsider for example the first time a coin lands heads when flipping it infinitely many times:\nthis is almost surely finite, but possibly infinite. We could also not use a function `Ω → ι` with\narbitrary value for the infinite case, because this would be incompatible with the stopping time\nproperty.\n\n## Tags\n\nstopping time, stochastic process\n\n-/\n\n@[expose] public section\n\nopen Filter Order TopologicalSpace WithTop\n\nopen scoped MeasureTheory NNReal ENNReal Topology\n\nnamespace MeasureTheory\n\nvariable {Ω β ι : Type*} {m : MeasurableSpace Ω}\n\n/-! ### Stopping times -/\n\n\n/-- A stopping time with respect to some filtration `f` is a function\n`τ` such that for all `i`, the preimage of `{j | j ≤ i}` along `τ` is measurable\nwith respect to `f i`.\n\nIntuitively, the stopping time `τ` describes some stopping rule such that at time\n`i`, we may determine it with the information we have at time `i`. -/\ndef IsStoppingTime [Preorder ι] (f : Filtration ι m) (τ : Ω → WithTop ι) :=\n  ∀ i : ι, MeasurableSet[f i] <| {ω | τ ω ≤ i}\n\ntheorem isStoppingTime_const [Preorder ι] (f : Filtration ι m) (i : ι) :\n    IsStoppingTime f fun _ => i := fun j => by simp only [MeasurableSet.const]\n\nsection MeasurableSet\n\nsection Preorder\n\nvariable [Preorder ι] {f : Filtration ι m} {τ : Ω → WithTop ι}\n\nprotected theorem IsStoppingTime.measurableSet_le (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω ≤ i} :=\n  hτ i\n\ntheorem IsStoppingTime.measurableSet_lt_of_pred [PredOrder ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω < i} := by\n  by_cases hi_min : IsMin i\n  · suffices {ω : Ω | τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i)\n    ext1 ω\n    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]\n    rw [isMin_iff_forall_not_lt] at hi_min\n    cases τ ω with\n    | top => simp\n    | coe t => exact mod_cast hi_min t\n  have : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iic (pred i : ι) := by\n    ext ω\n    push _ ∈ _\n    cases τ ω with\n    | top => simp\n    | coe t =>\n      simp only [coe_lt_coe, coe_le_coe]\n      rw [le_pred_iff_of_not_isMin hi_min]\n  rw [this]\n  exact f.mono (pred_le i) _ (hτ.measurableSet_le <| pred i)\n\nend Preorder\n\nsection CountableStoppingTime\n\nnamespace IsStoppingTime\n\nvariable [PartialOrder ι] {τ : Ω → WithTop ι} {f : Filtration ι m}\n\nprotected theorem measurableSet_eq_of_countable_range (hτ : IsStoppingTime f τ)\n    (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω | τ ω = i} := by\n  have : {ω | τ ω = i} = {ω | τ ω ≤ i} \\ ⋃ (j ∈ Set.range τ) (_ : j < i), {ω | τ ω ≤ j} := by\n    ext1 a\n    simp only [Set.mem_setOf_eq, Set.mem_range, Set.iUnion_exists, Set.iUnion_iUnion_eq',\n      Set.mem_sdiff, Set.mem_iUnion, exists_prop, not_exists, not_and]\n    constructor <;> intro h\n    · simp only [h, lt_iff_le_not_ge, le_refl, and_imp, imp_self, imp_true_iff, and_self_iff]\n    · exact h.1.eq_or_lt.resolve_right fun h_lt => h.2 a h_lt le_rfl\n  rw [this]\n  refine (hτ.measurableSet_le i).diff ?_\n  refine MeasurableSet.biUnion h_countable fun j _ => ?_\n  classical\n  rw [Set.iUnion_eq_if]\n  split_ifs with hji\n  · lift j to ι using (ne_top_of_lt hji)\n    exact f.mono (mod_cast hji.le) _ (hτ.measurableSet_le j)\n  · exact @MeasurableSet.empty _ (f i)\n\nprotected theorem measurableSet_eq_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω = i} :=\n  hτ.measurableSet_eq_of_countable_range (Set.to_countable _) i\n\nprotected theorem measurableSet_lt_of_countable_range (hτ : IsStoppingTime f τ)\n    (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω | τ ω < i} := by\n  have : {ω | τ ω < i} = {ω | τ ω ≤ i} \\ {ω | τ ω = i} := by ext1 ω; simp [lt_iff_le_and_ne]\n  rw [this]\n  exact (hτ.measurableSet_le i).diff (hτ.measurableSet_eq_of_countable_range h_countable i)\n\nprotected theorem measurableSet_lt_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω < i} :=\n  hτ.measurableSet_lt_of_countable_range (Set.to_countable _) i\n\nprotected theorem measurableSet_ge_of_countable_range {ι} [LinearOrder ι] {τ : Ω → WithTop ι}\n    {f : Filtration ι m} (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) :\n    MeasurableSet[f i] {ω | i ≤ τ ω} := by\n  have : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ := by\n    ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt]\n  rw [this]\n  exact (hτ.measurableSet_lt_of_countable_range h_countable i).compl\n\nprotected theorem measurableSet_ge_of_countable {ι} [LinearOrder ι] {τ : Ω → WithTop ι}\n    {f : Filtration ι m} [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | i ≤ τ ω} :=\n  hτ.measurableSet_ge_of_countable_range (Set.to_countable _) i\n\nend IsStoppingTime\n\nend CountableStoppingTime\n\nsection LinearOrder\n\nvariable [LinearOrder ι] {f : Filtration ι m} {τ : Ω → WithTop ι}\n\ntheorem IsStoppingTime.measurableSet_gt (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | i < τ ω} := by\n  have : {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ := by\n    ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_le]\n  rw [this]\n  exact (hτ.measurableSet_le i).compl\n\nsection TopologicalSpace\n\nvariable [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι]\n\n/-- Auxiliary lemma for `MeasureTheory.IsStoppingTime.measurableSet_lt`. -/\ntheorem IsStoppingTime.measurableSet_lt_of_isLUB (hτ : IsStoppingTime f τ) (i : ι)\n    (h_lub : IsLUB (Set.Iio i) i) : MeasurableSet[f i] {ω | τ ω < i} := by\n  by_cases hi_min : IsMin i\n  · suffices {ω | τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i)\n    ext1 ω\n    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]\n    cases τ ω with\n    | top => simp\n    | coe t => norm_cast; exact isMin_iff_forall_not_lt.mp hi_min t\n  obtain ⟨seq, -, -, h_tendsto, h_bound⟩ :\n      ∃ seq : ℕ → ι, Monotone seq ∧ (∀ j, seq j ≤ i) ∧ Tendsto seq atTop (𝓝 i) ∧ ∀ j, seq j < i :=\n    h_lub.exists_seq_monotone_tendsto (not_isMin_iff.mp hi_min)\n  have h_Iio_eq_Union : Set.Iio (i : WithTop ι) = ⋃ j, {k : WithTop ι | k ≤ seq j} := by\n    ext1 k\n    push _ ∈ _\n    refine ⟨fun hk_lt_i => ?_, fun h_exists_k_le_seq => ?_⟩\n    · rw [tendsto_atTop'] at h_tendsto\n      cases k with\n      | top => simp at hk_lt_i\n      | coe k =>\n        norm_cast at hk_lt_i ⊢\n        have h_nhds : Set.Ici k ∈ 𝓝 i :=\n          mem_nhds_iff.mpr ⟨Set.Ioi k, Set.Ioi_subset_Ici le_rfl, isOpen_Ioi, hk_lt_i⟩\n        obtain ⟨a, ha⟩ : ∃ a : ℕ, ∀ b : ℕ, b ≥ a → k ≤ seq b := h_tendsto (Set.Ici k) h_nhds\n        exact ⟨a, ha a le_rfl⟩\n    · obtain ⟨j, hk_seq_j⟩ := h_exists_k_le_seq\n      exact hk_seq_j.trans_lt (mod_cast h_bound j)\n  have h_lt_eq_preimage : {ω | τ ω < i} = τ ⁻¹' Set.Iio i := by\n    ext1 ω; push _ ∈ _; rfl\n  rw [h_lt_eq_preimage, h_Iio_eq_Union]\n  simp only [Set.preimage_iUnion, Set.preimage_setOf_eq]\n  exact MeasurableSet.iUnion fun n => f.mono (h_bound n).le _ (hτ.measurableSet_le (seq n))\n\nTarget:\ntheorem IsStoppingTime.measurableSet_lt (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω < i} :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Process","family_id":"isstoppingtime","file_id":"mathlib/Mathlib/Probability/Process/Stopping.lean","sample_id":"bfdf8a5735fb83291498154fe55688e7c5eabb3111989c0465481333ca9088fd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cff2f50ea6340f1c0d647e96a5ab7bcbcc74041793ed6f6cea3ea0d685f9013d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"865430ae7ad7a3af9b967fd8cc81590df7b116ddc79d82d6edeca41aba827e49","source_sha256":"b2e5d2441dae74e6a2f4e8de4b7d4e87dd1e5e1f1bcd7a4ed0737bd7e824e441","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro s hs\n  induction hs with\n  | base t t_mem => exact self_subset_generateSetAlgebra (h t_mem)\n  | empty => exact isSetAlgebra_generateSetAlgebra.empty_mem\n  | compl t _ t_mem => exact isSetAlgebra_generateSetAlgebra.compl_mem t_mem\n  | union t u _ _ t_mem u_mem => exact isSetAlgebra_generateSetAlgebra.union_mem t_mem u_mem","hard_negative":false,"metrics":{"chosen_tokens":51,"rejected_tokens":2,"token_jaccard":0.035714,"token_length_ratio":0.039216},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"6710b8a328a307d8c286174af117e894bec061d556e9dec71cc8840dbeb34e45","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finite.Prod\npublic import Mathlib.MeasureTheory.SetSemiring\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2024 Etienne Marion. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Etienne Marion\n-/\n/-!\n# Algebra of sets\n\nIn this file we define the notion of algebra of sets and give its basic properties. An algebra\nof sets is a family of sets containing the empty set and closed by complement and binary union.\nIt is therefore similar to a `σ`-algebra, except that it is not necessarily closed\nby countable unions.\n\nWe also define the algebra of sets generated by a family of sets and give its basic properties,\nand we prove that it is countable when it is generated by a countable family. We prove that\nthe `σ`-algebra generated by a family of sets `𝒜` is the same as the one generated by the algebra\nof sets generated by `𝒜`.\n\n## Main definitions\n\n* `MeasureTheory.IsSetAlgebra`: property of being an algebra of sets.\n* `MeasureTheory.generateSetAlgebra`: the algebra of sets generated by a family of sets.\n\n## Main statements\n\n* `MeasureTheory.mem_generateSetAlgebra_elim`: If a set `s` belongs to the algebra of sets\n  generated by `𝒜`, then it can be written as a finite union of finite intersections of sets which\n  are in `𝒜` or have their complement in `𝒜`.\n* `MeasureTheory.countable_generateSetAlgebra`: If a family of sets is countable then so is\n  the algebra of sets generated by it.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Field_of_sets>\n\n## Tags\n\nalgebra of sets, generated algebra of sets\n-/\n\npublic section\n\nopen MeasurableSpace Set\n\nnamespace MeasureTheory\n\nvariable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α}\n\n/-! ### Definition and basic properties of an algebra of sets -/\n\n/-- An algebra of sets is a family of sets containing the empty set and closed by complement and\nunion. Consequently it is also closed by difference (see `IsSetAlgebra.sdiff_mem`) and intersection\n(see `IsSetAlgebra.inter_mem`). -/\nstructure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where\n  empty_mem : ∅ ∈ 𝒜\n  compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜\n  union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜\n\nnamespace IsSetAlgebra\n\n/-- An algebra of sets contains the whole set. -/\ntheorem univ_mem (h𝒜 : IsSetAlgebra 𝒜) : univ ∈ 𝒜 :=\n  compl_empty ▸ h𝒜.compl_mem h𝒜.empty_mem\n\n/-- An algebra of sets is closed by intersection. -/\ntheorem inter_mem (h𝒜 : IsSetAlgebra 𝒜) (s_mem : s ∈ 𝒜) (t_mem : t ∈ 𝒜) :\n    s ∩ t ∈ 𝒜 :=\n  inter_eq_compl_compl_union_compl .. ▸\n    h𝒜.compl_mem (h𝒜.union_mem (h𝒜.compl_mem s_mem) (h𝒜.compl_mem t_mem))\n\n/-- An algebra of sets is closed by difference. -/\ntheorem sdiff_mem (h𝒜 : IsSetAlgebra 𝒜) (s_mem : s ∈ 𝒜) (t_mem : t ∈ 𝒜) :\n    s \\ t ∈ 𝒜 := h𝒜.inter_mem s_mem (h𝒜.compl_mem t_mem)\n\n@[deprecated (since := \"2026-06-03\")] alias diff_mem := sdiff_mem\n\n/-- An algebra of sets is a ring of sets. -/\ntheorem isSetRing (h𝒜 : IsSetAlgebra 𝒜) : IsSetRing 𝒜 where\n  empty_mem := h𝒜.empty_mem\n  union_mem := h𝒜.union_mem\n  sdiff_mem := fun _ _ ↦ h𝒜.sdiff_mem\n\n/-- An algebra of sets is closed by finite unions. -/\ntheorem biUnion_mem {ι : Type*} (h𝒜 : IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι)\n    (hs : ∀ i ∈ S, s i ∈ 𝒜) : ⋃ i ∈ S, s i ∈ 𝒜 := h𝒜.isSetRing.biUnion_mem S hs\n\n/-- An algebra of sets is closed by finite intersections. -/\ntheorem biInter_mem {ι : Type*} (h𝒜 : IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι)\n    (hs : ∀ i ∈ S, s i ∈ 𝒜) : ⋂ i ∈ S, s i ∈ 𝒜 := by\n  by_cases! h : S = ∅\n  · rw [h, ← Finset.set_biInter_coe, Finset.coe_empty, biInter_empty]\n    exact h𝒜.univ_mem\n  · exact h𝒜.isSetRing.biInter_mem S h hs\n\nend IsSetAlgebra\n\nsection generateSetAlgebra\n\n/-! ### Definition and properties of the algebra of sets generated by some family -/\n\n/-- `generateSetAlgebra 𝒜` is the smallest algebra of sets containing `𝒜`. -/\ninductive generateSetAlgebra {α : Type*} (𝒜 : Set (Set α)) : Set (Set α)\n  | base (s : Set α) (s_mem : s ∈ 𝒜) : generateSetAlgebra 𝒜 s\n  | empty : generateSetAlgebra 𝒜 ∅\n  | compl (s : Set α) (hs : generateSetAlgebra 𝒜 s) : generateSetAlgebra 𝒜 sᶜ\n  | union (s t : Set α) (hs : generateSetAlgebra 𝒜 s) (ht : generateSetAlgebra 𝒜 t) :\n      generateSetAlgebra 𝒜 (s ∪ t)\n\n/-- The algebra of sets generated by a family of sets is an algebra of sets. -/\ntheorem isSetAlgebra_generateSetAlgebra :\n    IsSetAlgebra (generateSetAlgebra 𝒜) where\n  empty_mem := generateSetAlgebra.empty\n  compl_mem := fun _ hs ↦ generateSetAlgebra.compl _ hs\n  union_mem := fun _ _ hs ht ↦ generateSetAlgebra.union _ _ hs ht\n\n/-- The algebra of sets generated by `𝒜` contains `𝒜`. -/\ntheorem self_subset_generateSetAlgebra : 𝒜 ⊆ generateSetAlgebra 𝒜 :=\n  fun _ ↦ generateSetAlgebra.base _\n\n/-- The measurable space generated by a family of sets `𝒜` is the same as the one generated\nby the algebra of sets generated by `𝒜`. -/\n@[simp]\ntheorem generateFrom_generateSetAlgebra_eq :\n    generateFrom (generateSetAlgebra 𝒜) = generateFrom 𝒜 := by\n  refine le_antisymm (fun s ms ↦ ?_) (generateFrom_mono self_subset_generateSetAlgebra)\n  induction s, ms using generateFrom_induction with\n  | hC t ht h =>\n    clear h\n    induction ht with\n    | base u u_mem => exact measurableSet_generateFrom u_mem\n    | empty => exact @MeasurableSet.empty _ (generateFrom 𝒜)\n    | compl u _ mu => exact mu.compl\n    | union u v _ _ mu mv => exact MeasurableSet.union mu mv\n  | empty => exact MeasurableSpace.measurableSet_empty _\n  | compl t _ ht => exact ht.compl\n  | iUnion t _ ht => exact .iUnion ht\n\n/-- If a family of sets `𝒜` is contained in `ℬ`, then the algebra of sets generated by `𝒜`\nis contained in the one generated by `ℬ`. -/\n\nTarget:\ntheorem generateSetAlgebra_mono {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ) :\n    generateSetAlgebra 𝒜 ⊆ generateSetAlgebra ℬ :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory","family_id":"generatesetalgebra_mono","file_id":"mathlib/Mathlib/MeasureTheory/SetAlgebra.lean","sample_id":"865430ae7ad7a3af9b967fd8cc81590df7b116ddc79d82d6edeca41aba827e49"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e5473258af32213de8f9e9baf793bc2d791875a7c5122f73158f25e7055ff270","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bff817808a5cc69dcd74325b621cd3b889cc91e9bdb2f5fd6f80c7083b9ec6d9","source_sha256":"f589366b76253f55a5f00af110f0088b3f55c67577d56940b18639c5f1a2a3b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only [sub_eq_add_neg] using!\n    (f.commute_add_nat n).inverses_right (Equiv.addRight _).right_inv (Equiv.addRight _).left_inv","hard_negative":true,"metrics":{"chosen_tokens":32,"rejected_tokens":8,"token_jaccard":0.038462,"token_length_ratio":0.25},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"6713b2fe40bc8084465e68eb63f7b2f9048095eb37d0ab888c597a2d504c31e1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.Order.SemiconjSup\n\nNamespace:\nCircleDeg1Lift\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Translation number of a monotone real map that commutes with `x ↦ x + 1`\n\nLet `f : ℝ → ℝ` be a monotone map such that `f (x + 1) = f x + 1` for all `x`. Then the limit\n$$\n  \\tau(f)=\\lim_{n\\to\\infty}{f^n(x)-x}{n}\n$$\nexists and does not depend on `x`. This number is called the *translation number* of `f`.\nDifferent authors use different notation for this number: `τ`, `ρ`, `rot`, etc\n\nIn this file we define a structure `CircleDeg1Lift` for bundled maps with these properties, define\ntranslation number of `f : CircleDeg1Lift`, prove some estimates relating `f^n(x)-x` to `τ(f)`. In\ncase of a continuous map `f` we also prove that `f` admits a point `x` such that `f^n(x)=x+m` if and\nonly if `τ(f)=m/n`.\n\nMaps of this type naturally appear as lifts of orientation-preserving circle homeomorphisms. More\nprecisely, let `f` be an orientation-preserving homeomorphism of the circle $S^1=ℝ/ℤ$, and\nconsider a real number `a` such that\n`⟦a⟧ = f 0`, where `⟦⟧` means the natural projection `ℝ → ℝ/ℤ`. Then there exists a unique\ncontinuous function `F : ℝ → ℝ` such that `F 0 = a` and `⟦F x⟧ = f ⟦x⟧` for all `x` (this fact is\nnot formalized yet). This function is strictly monotone, continuous, and satisfies\n`F (x + 1) = F x + 1`. The number `⟦τ F⟧ : ℝ / ℤ` is called the *rotation number* of `f`.\nIt does not depend on the choice of `a`.\n\n## Main definitions\n\n* `CircleDeg1Lift`: a monotone map `f : ℝ → ℝ` such that `f (x + 1) = f x + 1` for all `x`;\n  the type `CircleDeg1Lift` is equipped with `Lattice` and `Monoid` structures; the\n  multiplication is given by composition: `(f * g) x = f (g x)`.\n* `CircleDeg1Lift.translationNumber`: translation number of `f : CircleDeg1Lift`.\n\n## Main statements\n\nWe prove the following properties of `CircleDeg1Lift.translationNumber`.\n\n* `CircleDeg1Lift.translationNumber_eq_of_dist_bounded`: if the distance between `(f^n) 0`\n  and `(g^n) 0` is bounded from above uniformly in `n : ℕ`, then `f` and `g` have equal\n  translation numbers.\n\n* `CircleDeg1Lift.translationNumber_eq_of_semiconjBy`: if two `CircleDeg1Lift` maps `f`, `g`\n  are semiconjugate by a `CircleDeg1Lift` map, then `τ f = τ g`.\n\n* `CircleDeg1Lift.translationNumber_units_inv`: if `f` is an invertible `CircleDeg1Lift` map\n  (equivalently, `f` is a lift of an orientation-preserving circle homeomorphism), then\n  the translation number of `f⁻¹` is the negative of the translation number of `f`.\n\n* `CircleDeg1Lift.translationNumber_mul_of_commute`: if `f` and `g` commute, then\n  `τ (f * g) = τ f + τ g`.\n\n* `CircleDeg1Lift.translationNumber_eq_rat_iff`: the translation number of `f` is equal to\n  a rational number `m / n` if and only if `(f^n) x = x + m` for some `x`.\n\n* `CircleDeg1Lift.semiconj_of_bijective_of_translationNumber_eq`: if `f` and `g` are two\n  bijective `CircleDeg1Lift` maps and their translation numbers are equal, then these\n  maps are semiconjugate to each other.\n\n* `CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq`: let `f₁` and `f₂` be\n  two actions of a group `G` on the circle by degree 1 maps (formally, `f₁` and `f₂` are two\n  homomorphisms from `G →* CircleDeg1Lift`). If the translation numbers of `f₁ g` and `f₂ g` are\n  equal to each other for all `g : G`, then these two actions are semiconjugate by some\n  `F : CircleDeg1Lift`. This is a version of Proposition 5.4 from [Étienne Ghys, Groupes\n  d'homéomorphismes du cercle et cohomologie bornée][ghys87:groupes].\n\n## Notation\n\nWe use a local notation `τ` for the translation number of `f : CircleDeg1Lift`.\n\n## Implementation notes\n\nWe define the translation number of `f : CircleDeg1Lift` to be the limit of the sequence\n`(f ^ (2 ^ n)) 0 / (2 ^ n)`, then prove that `((f ^ n) x - x) / n` tends to this number for any `x`.\nThis way it is much easier to prove that the limit exists and basic properties of the limit.\n\nWe define translation number for a wider class of maps `f : ℝ → ℝ` instead of lifts of orientation\npreserving circle homeomorphisms for two reasons:\n\n* non-strictly monotone circle self-maps with discontinuities naturally appear as Poincaré maps\n  for some flows on the two-torus (e.g., one can take a constant flow and glue in a few Cherry\n  cells);\n* definition and some basic properties still work for this class.\n\n## References\n\n* [Étienne Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée][ghys87:groupes]\n\n## TODO\n\nHere are some short-term goals.\n\n* Introduce a structure or a typeclass for lifts of circle homeomorphisms. We use\n  `Units CircleDeg1Lift` for now, but it's better to have a dedicated type (or a typeclass?).\n\n* Prove that the `SemiconjBy` relation on circle homeomorphisms is an equivalence relation.\n\n* Introduce `ConditionallyCompleteLattice` structure, use it in the proof of\n  `CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq`.\n\n* Prove that the orbits of the irrational rotation are dense in the circle. Deduce that a\n  homeomorphism with an irrational rotation is semiconjugate to the corresponding irrational\n  translation by a continuous `CircleDeg1Lift`.\n\n## Tags\n\ncircle homeomorphism, rotation number\n-/\n\n@[expose] public section\n\nopen Filter Set Int Topology\nopen Function hiding Commute\n\n/-!\n### Definition and monoid structure\n-/\n\n/-- A lift of a monotone degree one map `S¹ → S¹`. -/\nstructure CircleDeg1Lift : Type extends ℝ →o ℝ where\n  map_add_one' : ∀ x, toFun (x + 1) = toFun x + 1\n\nnamespace CircleDeg1Lift\n\ninstance : FunLike CircleDeg1Lift ℝ ℝ where\n  coe f := f.toFun\n  coe_injective | ⟨⟨_, _⟩, _⟩, ⟨⟨_, _⟩, _⟩, rfl => rfl\n\ninstance : OrderHomClass CircleDeg1Lift ℝ ℝ where\n  map_rel f _ _ h := f.monotone' h\n\n@[simp] theorem coe_mk (f h) : ⇑(mk f h) = f := rfl\n\nvariable (f g : CircleDeg1Lift)\n\n@[simp] theorem coe_toOrderHom : ⇑f.toOrderHom = f := rfl\n\nprotected theorem monotone : Monotone f := f.monotone'\n\n@[gcongr, mono] theorem mono {x y} (h : x ≤ y) : f x ≤ f y := f.monotone h\n\ntheorem strictMono_iff_injective : StrictMono f ↔ Injective f :=\n  f.monotone.strictMono_iff_injective\n\n@[simp]\ntheorem map_add_one : ∀ x, f (x + 1) = f x + 1 :=\n  f.map_add_one'\n\n@[simp]\ntheorem map_one_add (x : ℝ) : f (1 + x) = 1 + f x := by rw [add_comm, map_add_one, add_comm 1]\n\n@[ext]\ntheorem ext ⦃f g : CircleDeg1Lift⦄ (h : ∀ x, f x = g x) : f = g :=\n  DFunLike.ext f g h\n\ninstance : Monoid CircleDeg1Lift where\n  mul f g :=\n    { toOrderHom := f.1.comp g.1\n      map_add_one' := fun x => by simp [map_add_one] }\n  one := ⟨.id, fun _ => rfl⟩\n  mul_one _ := rfl\n  one_mul _ := rfl\n  mul_assoc _ _ _ := DFunLike.coe_injective rfl\n\ninstance : Inhabited CircleDeg1Lift := ⟨1⟩\n\n@[simp]\ntheorem coe_mul : ⇑(f * g) = f ∘ g :=\n  rfl\n\ntheorem mul_apply (x) : (f * g) x = f (g x) :=\n  rfl\n\n@[simp]\ntheorem coe_one : ⇑(1 : CircleDeg1Lift) = id :=\n  rfl\n\ninstance unitsHasCoeToFun : CoeFun CircleDeg1Liftˣ fun _ => ℝ → ℝ :=\n  ⟨fun f => ⇑(f : CircleDeg1Lift)⟩\n\n@[simp]\ntheorem units_inv_apply_apply (f : CircleDeg1Liftˣ) (x : ℝ) :\n    (f⁻¹ : CircleDeg1Liftˣ) (f x) = x := by simp only [← mul_apply, f.inv_mul, coe_one, id]\n\n@[simp]\ntheorem units_apply_inv_apply (f : CircleDeg1Liftˣ) (x : ℝ) :\n    f ((f⁻¹ : CircleDeg1Liftˣ) x) = x := by simp only [← mul_apply, f.mul_inv, coe_one, id]\n\n/-- If a lift of a circle map is bijective, then it is an order automorphism of the line. -/\ndef toOrderIso : CircleDeg1Liftˣ →* ℝ ≃o ℝ where\n  toFun f :=\n    { toFun := f\n      invFun := ⇑f⁻¹\n      left_inv := units_inv_apply_apply f\n      right_inv := units_apply_inv_apply f\n      map_rel_iff' := ⟨fun h => by simpa using mono (↑f⁻¹) h, mono f⟩ }\n  map_one' := rfl\n  map_mul' _ _ := rfl\n\n@[simp]\ntheorem coe_toOrderIso (f : CircleDeg1Liftˣ) : ⇑(toOrderIso f) = f :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderIso_symm (f : CircleDeg1Liftˣ) :\n    ⇑(toOrderIso f).symm = (f⁻¹ : CircleDeg1Liftˣ) :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderIso_inv (f : CircleDeg1Liftˣ) : ⇑(toOrderIso f)⁻¹ = (f⁻¹ : CircleDeg1Liftˣ) :=\n  rfl\n\ntheorem isUnit_iff_bijective {f : CircleDeg1Lift} : IsUnit f ↔ Bijective f :=\n  ⟨fun ⟨u, h⟩ => h ▸ (toOrderIso u).bijective, fun h =>\n    Units.isUnit\n      { val := f\n        inv :=\n          { toFun := (Equiv.ofBijective f h).symm\n            monotone' := fun x y hxy =>\n              (f.strictMono_iff_injective.2 h.1).le_iff_le.1\n                (by simp only [Equiv.ofBijective_apply_symm_apply f h, hxy])\n            map_add_one' := fun x =>\n              h.1 <| by simp only [Equiv.ofBijective_apply_symm_apply f, f.map_add_one] }\n        val_inv := ext <| Equiv.ofBijective_apply_symm_apply f h\n        inv_val := ext <| Equiv.ofBijective_symm_apply_apply f h }⟩\n\ntheorem coe_pow : ∀ n : ℕ, ⇑(f ^ n) = f^[n]\n  | 0 => rfl\n  | n + 1 => by\n    simp [coe_pow n, pow_succ]\n\ntheorem semiconjBy_iff_semiconj {f g₁ g₂ : CircleDeg1Lift} :\n    SemiconjBy f g₁ g₂ ↔ Semiconj f g₁ g₂ :=\n  CircleDeg1Lift.ext_iff\n\ntheorem commute_iff_commute {f g : CircleDeg1Lift} : Commute f g ↔ Function.Commute f g :=\n  CircleDeg1Lift.ext_iff\n\n/-!\n### Translate by a constant\n-/\n\n\n/-- The map `y ↦ x + y` as a `CircleDeg1Lift`. More precisely, we define a homomorphism from\n`Multiplicative ℝ` to `CircleDeg1Liftˣ`, so the translation by `x` is\n`translation (Multiplicative.ofAdd x)`. -/\ndef translate : Multiplicative ℝ →* CircleDeg1Liftˣ := MonoidHom.toHomUnits <|\n  { toFun x := ⟨⟨fun y => x.toAdd + y, add_right_mono⟩, fun _ => (add_assoc ..).symm⟩\n    map_one' := ext zero_add\n    map_mul' _ _ := ext <| add_assoc _ _ }\n\n@[simp]\ntheorem translate_apply (x y : ℝ) : translate (Multiplicative.ofAdd x) y = x + y :=\n  rfl\n\n@[simp]\ntheorem translate_inv_apply (x y : ℝ) : (translate <| Multiplicative.ofAdd x)⁻¹ y = -x + y :=\n  rfl\n\n@[simp]\ntheorem translate_zpow (x : ℝ) (n : ℤ) :\n    translate (Multiplicative.ofAdd x) ^ n = translate (Multiplicative.ofAdd <| ↑n * x) := by\n  simp only [← zsmul_eq_mul, ofAdd_zsmul, map_zpow]\n\n@[simp]\ntheorem translate_pow (x : ℝ) (n : ℕ) :\n    translate (Multiplicative.ofAdd x) ^ n = translate (Multiplicative.ofAdd <| ↑n * x) :=\n  translate_zpow x n\n\n@[simp]\ntheorem translate_iterate (x : ℝ) (n : ℕ) :\n    (translate (Multiplicative.ofAdd x))^[n] = translate (Multiplicative.ofAdd <| ↑n * x) := by\n  rw [← coe_pow, ← Units.val_pow_eq_pow_val, translate_pow]\n\n/-!\n### Commutativity with integer translations\n\nIn this section we prove that `f` commutes with translations by an integer number.\nFirst we formulate these statements (for a natural or an integer number,\naddition on the left or on the right, addition or subtraction) using `Function.Commute`,\nthen reformulate as `simp` lemmas `map_int_add` etc.\n-/\n\ntheorem commute_nat_add (n : ℕ) : Function.Commute f (n + ·) := by\n  simpa only [nsmul_one, add_left_iterate] using Function.Commute.iterate_right f.map_one_add n\n\ntheorem commute_add_nat (n : ℕ) : Function.Commute f (· + n) := by\n  simp only [add_comm _ (n : ℝ), f.commute_nat_add n]\n\nTarget:\ntheorem commute_sub_nat (n : ℕ) : Function.Commute f (· - n) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"de5527eb48fab754a24ba3ead9d2db7dd1db73cca77b65990218b8090075d3ca","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/Circle","family_id":"commute_sub_nat","file_id":"mathlib/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean","sample_id":"bff817808a5cc69dcd74325b621cd3b889cc91e9bdb2f5fd6f80c7083b9ec6d9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"81f6811c61bda2ac7e16fe95d39238e8cafe9933dcb6310481ee06482505a998","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9ea116b7daa0ff9aa81b54eb159a02ebd9756dce658d8964cedd444ceb6ca675","source_sha256":"957fb86c605e5aefaf558e2e21f21bba2c9e2fd312cbcb361e91615272ce22b4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [fiber, fiberι, Scheme.Pullback.range_fst, Scheme.range_fromSpecResidueField]","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":2,"token_jaccard":0.076923,"token_length_ratio":0.111111},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"67273ce46ef66fa6a713d881ae64136ead2fdd59cfbf30133608db85640e3fc4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.PullbackCarrier\npublic import Mathlib.RingTheory.LocalRing.ResidueField.Fiber\npublic import Mathlib.RingTheory.Spectrum.Prime.Jacobson\npublic import Mathlib.AlgebraicGeometry.Morphisms.Affine\npublic import Mathlib.AlgebraicGeometry.Morphisms.FiniteType\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Scheme-theoretic fiber\n\n## Main result\n- `AlgebraicGeometry.Scheme.Hom.fiber`: `f.fiber y` is the scheme-theoretic fiber of `f` at `y`.\n- `AlgebraicGeometry.Scheme.Hom.fiberHomeo`: `f.fiber y` is homeomorphic to `f ⁻¹' {y}`.\n- `AlgebraicGeometry.Scheme.Hom.finite_preimage`: Finite morphisms have finite fibers.\n- `AlgebraicGeometry.Scheme.Hom.discrete_fiber`: Finite morphisms have discrete fibers.\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nnoncomputable section\n\nopen CategoryTheory Limits\n\nnamespace AlgebraicGeometry\n\nvariable {X Y : Scheme.{u}}\n\n/-- `f.fiber y` is the scheme-theoretic fiber of `f` at `y`. -/\ndef Scheme.Hom.fiber (f : X ⟶ Y) (y : Y) : Scheme := pullback f (Y.fromSpecResidueField y)\n\n/-- `f.fiberι y : f.fiber y ⟶ X` is the embedding of the scheme-theoretic fiber into `X`. -/\ndef Scheme.Hom.fiberι (f : X ⟶ Y) (y : Y) : f.fiber y ⟶ X := pullback.fst _ _\n\ninstance (f : X ⟶ Y) (y : Y) : (f.fiber y).CanonicallyOver X where hom := f.fiberι y\n\n/-- The canonical map from the scheme-theoretic fiber to the residue field. -/\ndef Scheme.Hom.fiberToSpecResidueField (f : X ⟶ Y) (y : Y) :\n    f.fiber y ⟶ Spec (Y.residueField y) :=\n  pullback.snd _ _\n\n@[reassoc]\nlemma Scheme.Hom.fiber_fac (f : X ⟶ Y) (y : Y) :\n    f.fiberι y ≫ f = f.fiberToSpecResidueField y ≫ Y.fromSpecResidueField y :=\n  pullback.condition\n\n/-- The fiber of `f` at `y` is naturally a `κ(y)`-scheme. -/\n@[reducible] def Scheme.Hom.fiberOverSpecResidueField\n    (f : X ⟶ Y) (y : Y) : (f.fiber y).Over (Spec (Y.residueField y)) where\n  hom := f.fiberToSpecResidueField y\n\nlemma Scheme.Hom.fiberToSpecResidueField_apply (f : X ⟶ Y) (y : Y) (x : f.fiber y) :\n    f.fiberToSpecResidueField y x = IsLocalRing.closedPoint (Y.residueField y) :=\n  Subsingleton.elim (α := PrimeSpectrum _) _ _\n\nset_option backward.isDefEq.respectTransparency false in\nlemma isPullback_fiberToSpecResidueField_of_isPullback {P X Y Z : Scheme.{u}} {fst : P ⟶ X}\n    {snd : P ⟶ Y} {f : X ⟶ Z} {g : Y ⟶ Z} (h : IsPullback fst snd f g) (y : Y) :\n    IsPullback (pullback.map _ _ _ _ fst (Spec.map (g.residueFieldMap y)) g h.w.symm (by simp))\n      (snd.fiberToSpecResidueField y)\n      (f.fiberToSpecResidueField (g y))\n      (Spec.map (g.residueFieldMap y)) := by\n  refine .of_right (h₁₂ := pullback.fst _ _) ?_ ?_\n      (IsPullback.of_hasPullback f (Z.fromSpecResidueField (g y)))\n  · simpa using! (IsPullback.of_hasPullback _ _).paste_horiz h\n  · simp [Scheme.Hom.fiberToSpecResidueField]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The morphism from the fiber of `Spec S ⟶ Spec R` at some prime `p` to `Spec κ(p)`\nis isomorphic to the map induced by `κ(p) ⟶ κ(p) ⊗[R] S`. -/\nnoncomputable def Spec.fiberToSpecResidueFieldIso (R S : Type u) [CommRing R] [CommRing S]\n    [Algebra R S] (p : PrimeSpectrum R) :\n    Arrow.mk ((Spec.map (CommRingCat.ofHom <| algebraMap R S)).fiberToSpecResidueField p) ≅\n      Arrow.mk (Spec.map <| CommRingCat.ofHom <|\n        algebraMap p.asIdeal.ResidueField (p.asIdeal.Fiber S)) := by\n  refine Arrow.isoMk' _ _\n    (pullbackSymmetry _ _ ≪≫ ?_ ≪≫ pullbackSpecIso R p.asIdeal.ResidueField S) ?_ ?_\n  · refine pullback.congrHom\n      (Scheme.Spec.map_residueFieldIso_inv_eq_fromSpecResidueField (.of R) p).symm rfl ≪≫ ?_\n    refine asIso <| pullback.map _ _ _ _ (Spec.map <| (Scheme.Spec.residueFieldIso (.of R) _).inv)\n      (𝟙 _) (𝟙 _) (by simp) (by simp)\n  · exact Scheme.Spec.mapIso (Scheme.Spec.residueFieldIso (.of R) _).symm.op\n  · cat_disch\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma Scheme.Hom.range_fiberι (f : X ⟶ Y) (y : Y) :\n    Set.range (f.fiberι y) = f ⁻¹' {y} :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry","family_id":"scheme","file_id":"mathlib/Mathlib/AlgebraicGeometry/Fiber.lean","sample_id":"9ea116b7daa0ff9aa81b54eb159a02ebd9756dce658d8964cedd444ceb6ca675"},"split":"train"}
{"binding":{"chosen_proof_sha256":"baf74685d38b5f76627c288eda6c3033c2f20e6a31a28995ec3f9cc5ab112e2a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"317b87a5beef988d47043cbd23eccd8456a804914b949c3de8ba8b8d221e791d","source_sha256":"ba981a331bca09368a4e9ba1529e46eda80e78747c54ab2e477c7892665030f9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI := Fintype.ofFinite ι\n  classical\n  refine (Basis.piTensorProduct b).ext_elem (fun q ↦ ?_)\n  simp [Finsupp.single_apply, Fintype.prod_ite_zero, ← funext_iff]","hard_negative":false,"metrics":{"chosen_tokens":37,"rejected_tokens":3,"token_jaccard":0.032258,"token_length_ratio":0.081081},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"673f03b25af2f45630eedfcfc586275d00ab166858ba76142234df68f06fc64c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Finsupp.VectorSpace\npublic import Mathlib.LinearAlgebra.PiTensorProduct.Finsupp\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Daniel Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Morrison, Sophie Morel\n-/\n/-!\n# Basis for `PiTensorProduct`\n\nThis file constructs a basis for `PiTensorProduct` given bases on the component spaces.\n-/\n\n@[expose] public section\n\nsection PiTensorProduct\n\nattribute [local ext] PiTensorProduct.ext\n\nopen LinearMap PiTensorProduct Module TensorProduct\n\nvariable {ι R : Type*} {M : ι → Type*} {κ : ι → Type*} [CommSemiring R] [∀ i, AddCommMonoid (M i)]\n  [∀ i, Module R (M i)]\n\nopen Classical in\n/-- Let `ι` be a `Finite` type and `M` be a family of modules indexed by `ι`. If `b i : κ i → M i`\nis a basis for every `i` in `ι`, then `fun (p : Π i, κ i) ↦ ⨂ₜ[R] i, b i (p i)` is a basis\nof `⨂[R] i, M i`.\n-/\nnoncomputable def Basis.piTensorProduct [Finite ι] (b : Π i, Basis (κ i) R (M i)) :\n    Basis (Π i, κ i) R (⨂[R] i, M i) :=\n  haveI := Fintype.ofFinite ι\n  Finsupp.basisSingleOne.map\n    ((PiTensorProduct.congr (fun i ↦ (b i).repr)) ≪≫ₗ\n      ofFinsuppEquiv ≪≫ₗ\n      Finsupp.lcongr (Equiv.refl _) (constantBaseRingEquiv _ R).toLinearEquiv).symm\n\n@[simp]\ntheorem Basis.piTensorProduct_repr_tprod_apply [Fintype ι] (b : Π i, Basis (κ i) R (M i))\n    (x : Π i, M i) (p : Π i, κ i) :\n    (Basis.piTensorProduct b).repr (tprod R x) p = ∏ i : ι, (b i).repr (x i) (p i) := by\n  rw [piTensorProduct, Subsingleton.elim (Fintype.ofFinite ι) ‹_›]\n  simp\n\n@[simp]\n\nTarget:\ntheorem Basis.piTensorProduct_apply [Finite ι] (b : Π i, Basis (κ i) R (M i)) (p : Π i, κ i) :\n    Basis.piTensorProduct b p = ⨂ₜ[R] i, (b i) (p i) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/PiTensorProduct","family_id":"basis","file_id":"mathlib/Mathlib/LinearAlgebra/PiTensorProduct/Basis.lean","sample_id":"317b87a5beef988d47043cbd23eccd8456a804914b949c3de8ba8b8d221e791d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"32e319c4110fd1b6339d57505a0712e65f79dbff575ad63d342ca445e0185468","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5f583e779a0f586168eb801c8f41ed46b5920677578a5ac3171bab04bd27faca","source_sha256":"b76902cf8d4ceba8892c478a05df06c4c7f74a8ad6562beb851b28c9078eec8d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [← HomologicalComplex.comp_f]\n\n/-- The homotopy equivalence from `X.chainComplex R` to `X.normalizedChainComplex R`. -/\nnoncomputable def homotopyEquivNormalizedChainComplex :\n    HomotopyEquiv (X.chainComplex R) (X.normalizedChainComplex R) :=\n  SimplicialObject.Splitting.homotopyEquivNondegComplex _","hard_negative":false,"metrics":{"chosen_tokens":55,"rejected_tokens":2,"token_jaccard":0.030303,"token_length_ratio":0.036364},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"6742fd3b9761a4ba69c34fcd1c66f483db8e770c7eac0dc54e04656af0f16f92","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.QuasiIso\npublic import Mathlib.AlgebraicTopology.SimplicialSet.Homology.Basic\npublic import Mathlib.AlgebraicTopology.SimplicialSet.Splitting\npublic import Mathlib.AlgebraicTopology.SimplicialSet.Dimension\npublic import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject\npublic import Mathlib.CategoryTheory.Limits.Preserves.SigmaConst\n\nNamespace:\nSSet\n\nLocal context:\n/-\nCopyright (c) 2026 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Computing homology using nondegenerate simplices\n\nIn this file, we introduce the normalized chain complex `X.normalizedChainComplex R`\nof a simplicial set `X` with coefficients in `R` (where `R` is an object of a\npreadditive category `C` with coproducts). The `n`-chains of this complex\nidentify to the coproduct of copies of `R` indexed by the nondegenerate\n`n`-simplices of `X`. In particular, we deduce that the homology is zero in degree `≥ d`\nwhen `X` has dimension `< d`.\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nopen CategoryTheory Limits HomologicalComplex Simplicial\n  AlgebraicTopology.DoldKan\n\nnamespace SSet\n\nvariable {C : Type u} [Category.{v} C] [HasCoproducts.{w} C] [Preadditive C]\n  (X Y : SSet.{w}) (f : X ⟶ Y) (R : C)\n\n/-- The normalized chain complex of a simplicial set `X` with coefficients in `R`.\nIn degree `n`, it consists of a coproduct of copies of `R` indexed by the\nnondegenerate `n`-simplices of `X`. -/\nnoncomputable def normalizedChainComplex : ChainComplex C ℕ :=\n  (X.splitting.map (sigmaConst.obj R)).nondegComplex\n\n/-- The split epi `X.chainComplex R ⟶ X.normalizedChainComplex R`. -/\nnoncomputable def toNormalizedChainComplex : X.chainComplex R ⟶ X.normalizedChainComplex R :=\n  (X.splitting.map (sigmaConst.obj R)).toNondegComplex\n\n/-- The split mono `X.normalizedChainComplex R ⟶ X.chainComplex R`. -/\nnoncomputable def fromNormalizedChainComplex : X.normalizedChainComplex R ⟶ X.chainComplex R :=\n  (X.splitting.map (sigmaConst.obj R)).fromNondegComplex\n\n@[reassoc (attr := simp)]\nlemma PInfty_toNormalizedChainComplex :\n    PInfty ≫ X.toNormalizedChainComplex R = X.toNormalizedChainComplex R :=\n  SimplicialObject.Splitting.PInfty_toNondegComplex _\n\ninstance : IsSplitEpi (X.toNormalizedChainComplex R) :=\n  SimplicialObject.Splitting.isSplitEpi_toNondegComplex _\n\ninstance : IsSplitMono (X.fromNormalizedChainComplex R) :=\n  SimplicialObject.Splitting.isSplitMono_fromNondegComplex _\n\n@[reassoc (attr := simp)]\nlemma fromNormalizedChainComplex_toNormalizedChainComplex :\n    X.fromNormalizedChainComplex R ≫ X.toNormalizedChainComplex R = 𝟙 _ :=\n  SimplicialObject.Splitting.fromNondegComplex_toNondegComplex _\n\n@[reassoc (attr := simp)]\nlemma fromNormalizedChainComplex_f_toNormalizedChainComplex_f (n : ℕ) :\n    (X.fromNormalizedChainComplex R).f n ≫ (X.toNormalizedChainComplex R).f n = 𝟙 _ := by\n  simp [← HomologicalComplex.comp_f]\n\n@[reassoc (attr := simp)]\nlemma toNormalizedChainComplex_fromNormalizedChainComplex :\n    X.toNormalizedChainComplex R ≫ X.fromNormalizedChainComplex R = PInfty :=\n  SimplicialObject.Splitting.toNondegComplex_fromNondegComplex _\n\n@[reassoc (attr := simp)]\n\nTarget:\nlemma toNormalizedChainComplex_f_fromNormalizedChainComplex_f (n : ℕ) :\n    (X.toNormalizedChainComplex R).f n ≫ (X.fromNormalizedChainComplex R).f n = PInfty.f n :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplicialSet","family_id":"tonormalizedchaincomplex_f_fromnormalizedchaincomplex_f","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplicialSet/Homology/Nondegenerate.lean","sample_id":"5f583e779a0f586168eb801c8f41ed46b5920677578a5ac3171bab04bd27faca"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eeea18d1418cb641b169bfeb8bce930a2742ccdb2c216d63523e2c130f059c37","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"26ccc8781af1080819c179c8d94b6ddbd480d0e88733e0b400dfc6b8d88a87ee","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"88630b1e90f48b536f58e1419d7c32cb382c083b0e23e75073e0b1e0fdf80749","source_sha256":"44a01634b6653e0d9e33dfb14b6ec5ffa4aa5d523005601762c92770961f9cd1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨χ, hχ⟩ := Finset.rado_selection_subtype (β := β) (fun s ↦ g s s.finite_toSet)\n  refine ⟨χ, fun s hs ↦ ?_⟩\n  obtain ⟨t, ht, hst⟩ := hχ hs.toFinset\n  simp only [Set.Finite.toFinset_subset] at ht\n  exact ⟨t, by simp_all⟩","hard_negative":true,"metrics":{"chosen_tokens":70,"rejected_tokens":5,"token_jaccard":0.076923,"token_length_ratio":0.071429},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"677ae943ce0f5d5b5a0122fe65d9c3bab5252e5ca6d009c13043ec4e3daf4701","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\nimport Mathlib.Topology.Compactness.Compact\npublic import Mathlib.Data.Set.Finite.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\n/-!\n# Combinatorial compactness and the Rado selection lemma\n\nThis file contains compactness arguments for constructing infinite objects from finite\napproximations. The main result is a formalization of Rado's selection principle, as an application\nof compactness to combinatorics.\n\nWe give four versions, depending on whether the \"partial\" functions are defined locally or globally,\nand whether we use `Finset` or `Set.Finite`. The precise formulation of the lemma is therefore\n`Finset.rado_selection_subtype` or `Set.Finite.rado_selection_subtype`, but the versions avoiding\nsubtypes are easier to prove and often easier to apply, so they are provided too.\n\n## Main results\n\n* `Finset.rado_selection`: Given functions `g : Finset α → α → β` where `β` is finite,\n  there exists a single function `χ : α → β` which is constructed out of `g`.\n  More precisely, for each finite set `s`, there exists a larger set `t ⊇ s` such that\n  `χ` and `g t` agree on `s`.\n  In fact, we can more generally allow each `g s` to be a dependent function, as `(a : α) → β a`, so\n  the type of `g` will be `Finset α → (a : α) → β a`.\n\n* `Finset.rado_selection_subtype`: A variant where `g` takes elements in the subtype.\n\n* `Set.Finite.rado_selection`: A variant using `Set.Finite`.\n\n* `Set.Finite.rado_selection`: A variant using `Set.Finite` and where `g` takes elements in the\n  subtype.\n\n## Implementation notes\n\nThe proof uses the fact that the product of finite discrete spaces is compact\n(by Tychonoff's theorem). The closed sets corresponding to \"agreeing with `g s` on `s`\"\nhave the finite intersection property, so their intersection is nonempty.\n\n## References\n\n* de Bruijn, N. G.; Erdős, P. (1951). \"A colour problem for infinite graphs and a problem\n  in the theory of relations\".\n* Rado, R. (1949). \"Axiomatic treatment of rank in infinite sets\".\n\n-/\n\npublic section\n\nvariable {α : Type*} {β : α → Type*} [∀ a, Finite (β a)]\n\n/--\nGiven a (dependent) function `g s : (a : α) → β a` for each finset `s` of `α`, provided that\neach `β a` is finite, we can find another function `χ : (a : α) → β a` such that on every `s`,\nthere is some larger `t` such that `χ` agrees with `g t` on `s`.\nInformally, we are stitching together the local functions `g s` into a global `χ` such that on\neach `s`, `χ` can be expressed in terms of one of the `g`.\n-/\ntheorem Finset.rado_selection (g : Finset α → (a : α) → β a) :\n    ∃ χ : (a : α) → β a, ∀ s : Finset α, ∃ t : Finset α, s ⊆ t ∧ ∀ x ∈ s, χ x = g t x := by\n  classical\n  let instTop (a : α) : TopologicalSpace (β a) := ⊥\n  have instDiscr (a : α) : DiscreteTopology (β a) := discreteTopology_bot _\n  let e (s : Finset α) : Set ((a : α) → β a) := {f | ∃ t, s ⊆ t ∧ ∀ x ∈ s, f x = g t x}\n  have (s : Finset α) : s.restrict ⁻¹' {f | ∃ t, s ⊆ t ∧ ∀ x, f x = g t x} = e s := by simp [e]\n  have he' (s : Finset α) : IsClosed (e s) := by\n    rw [← this]\n    exact (isClosed_discrete _).preimage (by fun_prop)\n  have he'' (B : Finset (Finset α)) : (⋂ i ∈ B, e i).Nonempty := by\n    refine ⟨g (B.biUnion id), ?_⟩\n    simp only [Set.mem_iInter, Set.mem_setOf_eq, e]\n    intro i hi\n    exact ⟨_, subset_biUnion_of_mem id hi, by simp⟩\n  simpa using! CompactSpace.iInter_nonempty he' he''\n\n/--\nGiven a (dependent) function `g s : (a : s) → β a` for each finset `s` of `α`, provided that\neach `β a` is finite, we can find another function `χ : (a : α) → β a` such that on every `s`,\nthere is some larger `t` such that `χ` agrees with `g t` on `s`.\nInformally, we are stitching together the local functions `g s` into a global `χ` such that on\neach `s`, `χ` can be expressed in terms of one of the `g`.\n-/\ntheorem Finset.rado_selection_subtype (g : (s : Finset α) → (a : s) → β a) :\n    ∃ χ : (a : α) → β a, ∀ s : Finset α,\n      ∃ (t : Finset α) (hst : s ⊆ t), ∀ x : s, χ x = g t (Set.inclusion hst x) := by\n  classical\n  have (a : α) : Nonempty (β a) := ⟨g {a} ⟨a, by simp⟩⟩\n  let g' (s) (a : α) : β a := if ha : a ∈ s then g s ⟨a, ha⟩ else Classical.arbitrary (β a)\n  have hg (s : Finset α) (x : s) : g s x = g' s x := by simp [g']\n  simpa [hg] using Finset.rado_selection g'\n\n/--\nGiven a (dependent) function `g s : (a : α) → β a` for each finite set `s` of `α`, provided that\neach `β a` is finite, we can find another function `χ : (a : α) → β a` such that on every `s`,\nthere is some larger `t` such that `χ` agrees with `g t` on `s`.\nInformally, we are stitching together the local functions `g s` into a global `χ` such that on\neach `s`, `χ` can be expressed in terms of one of the `g`.\n-/\ntheorem Set.Finite.rado_selection (g : (s : Set α) → s.Finite → (a : α) → β a) :\n    ∃ χ : (a : α) → β a, ∀ s : Set α, s.Finite →\n      ∃ (t : Set α) (ht : t.Finite), s ⊆ t ∧ ∀ x ∈ s, χ x = g t ht x := by\n  obtain ⟨χ, hχ⟩ := Finset.rado_selection (fun s ↦ g s s.finite_toSet)\n  refine ⟨χ, fun s hs ↦ ?_⟩\n  obtain ⟨t, ht, ht'⟩ := hχ hs.toFinset\n  exact ⟨t, by simp_all⟩\n\n/--\nGiven a (dependent) function `g s : (a : s) → β a` for each finite set `s` of `α`, provided that\neach `β a` is finite, we can find another function `χ : (a : α) → β a` such that on every `s`,\nthere is some larger `t` such that `χ` agrees with `g t` on `s`.\nInformally, we are stitching together the local functions `g s` into a global `χ` such that on\neach `s`, `χ` can be expressed in terms of one of the `g`.\n-/\n\nTarget:\ntheorem Set.Finite.rado_selection_subtype (g : (s : Set α) → s.Finite → (a : s) → β a) :\n    ∃ χ : (a : α) → β a, ∀ s : Set α, s.Finite →\n      ∃ (t : Set α) (ht : t.Finite) (hst : s ⊆ t), ∀ x : s, χ x = g t ht (Set.inclusion hst x) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_88630b1e90f4","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"5e9c1d83ea08a6ddfbb8aa6b76237d5783cdc9f356579e982c7c54e27365a53a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics","family_id":"set","file_id":"mathlib/Mathlib/Combinatorics/Compactness.lean","sample_id":"88630b1e90f48b536f58e1419d7c32cb382c083b0e23e75073e0b1e0fdf80749"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d7f9b791b4990ce10c90fe8f0ae29086af43a18750773d3fc0cc6aa2bef28c2c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fd1fd9a53ab94e93ee71cb526c00c66f283a440ab2cd926e2cae6762d9e62179","source_sha256":"c1ecc80ead99a17d01db2984c5f06df1499354702fc192601863dee22fbd7b8f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine hs.induction_on _ (by simp) fun {i s} his hs e ↦ ?_\n  simp only [iInf_insert, e, ideal_inf]","hard_negative":true,"metrics":{"chosen_tokens":30,"rejected_tokens":8,"token_jaccard":0.033333,"token_length_ratio":0.266667},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"6791e7fedcd4c4f203b942538cc6f8d58f31db99d551ae4e9413ef79853b3fdd","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact\npublic import Mathlib.AlgebraicGeometry.Properties\npublic import Mathlib.Tactic.DepRewrite\n\nNamespace:\nAlgebraicGeometry.Scheme.IdealSheafData\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Ideal sheaves on schemes\n\nWe define ideal sheaves of schemes and provide various constructors for it.\n\n## Main definition\n* `AlgebraicGeometry.Scheme.IdealSheafData`: A structure that contains the data to uniquely define\n  an ideal sheaf, consisting of\n  1. an ideal `I(U) ≤ Γ(X, U)` for every affine open `U`\n  2. a proof that `I(D(f)) = I(U)_f` for every affine open `U` and every section `f : Γ(X, U)`.\n* `AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals`:\n  The largest ideal sheaf contained in a family of ideals.\n* `AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine`:\n  Over affine schemes, ideal sheaves are in bijection with ideals of the global sections.\n* `AlgebraicGeometry.Scheme.IdealSheafData.support`: The support of an ideal sheaf.\n* `AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal`: The vanishing ideal of a set.\n* `AlgebraicGeometry.Scheme.Hom.ker`: The kernel of a morphism.\n\n## Main results\n* `AlgebraicGeometry.Scheme.IdealSheafData.gc`:\n  `support` and `vanishingIdeal` forms a Galois connection.\n* `AlgebraicGeometry.Scheme.Hom.support_ker`: The support of a kernel of a quasi-compact morphism\n  is the closure of the range.\n\n## Implementation detail\n\nIdeal sheaves are not yet defined in this file as actual subsheaves of `𝒪ₓ`.\nInstead, for the ease of development and application,\nwe define the structure `IdealSheafData` containing all necessary data to uniquely define an\nideal sheaf. This should be refactored as a constructor for ideal sheaves once they are introduced\ninto mathlib.\n\n-/\n\n@[expose] public section\n\nopen CategoryTheory TopologicalSpace\n\nuniverse u\n\nnamespace AlgebraicGeometry.Scheme\n\nvariable {X : Scheme.{u}}\n\n/--\nA structure that contains the data to uniquely define an ideal sheaf, consisting of\n1. an ideal `I(U) ≤ Γ(X, U)` for every affine open `U`\n2. a proof that `I(D(f)) = I(U)_f` for every affine open `U` and every section `f : Γ(X, U)`\n3. a subset of `X` equal to the support.\n\nAlso see `Scheme.IdealSheafData.mkOfMemSupportIff` for a constructor with the condition on the\nsupport being (usually) easier to prove.\n-/\nstructure IdealSheafData (X : Scheme.{u}) : Type u where\n  /-- The component of an ideal sheaf at an affine open. -/\n  ideal : ∀ U : X.affineOpens, Ideal Γ(X, U)\n  /-- Also see `AlgebraicGeometry.Scheme.IdealSheafData.map_ideal` -/\n  map_ideal_basicOpen : ∀ (U : X.affineOpens) (f : Γ(X, U)),\n    (ideal U).map (X.presheaf.map (homOfLE <| X.basicOpen_le f).op).hom =\n      ideal (X.affineBasicOpen f)\n  /-- The support of an ideal sheaf. Use `IdealSheafData.support` instead for most occasions. -/\n  supportSet : Set X := ⋂ U, X.zeroLocus (U := U.1) (ideal U)\n  supportSet_eq_iInter_zeroLocus : supportSet = ⋂ U, X.zeroLocus (U := U.1) (ideal U) := by rfl\n\nnamespace IdealSheafData\n\n@[ext]\nprotected lemma ext {I J : X.IdealSheafData} (h : I.ideal = J.ideal) : I = J := by\n  obtain ⟨i, _, s, hs⟩ := I\n  obtain ⟨j, _, t, ht⟩ := J\n  subst h\n  congr\n  rw [hs, ht]\n\nsection Order\n\ninstance : PartialOrder (IdealSheafData X) := PartialOrder.lift ideal fun _ _ ↦ IdealSheafData.ext\n\nlemma le_def {I J : IdealSheafData X} : I ≤ J ↔ ∀ U, I.ideal U ≤ J.ideal U := .rfl\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : CompleteSemilatticeSup (IdealSheafData X) where\n  sSup s :=\n  { ideal := sSup (ideal '' s),\n    map_ideal_basicOpen := by\n      have : sSup (ideal '' s) = ⨆ i : s, ideal i.1 := by\n        conv_lhs => rw [← Subtype.range_val (s := s), ← Set.range_comp]\n        rfl\n      simp only [this, iSup_apply, Ideal.map_iSup, map_ideal_basicOpen, implies_true] }\n  isLUB_sSup _ := .of_image (f := ideal) le_def (isLUB_sSup _)\n\n/-- The largest ideal sheaf contained in a family of ideals. -/\ndef ofIdeals (I : ∀ U : X.affineOpens, Ideal Γ(X, U)) : IdealSheafData X :=\n  sSup { J : IdealSheafData X | J.ideal ≤ I }\n\nlemma ideal_ofIdeals_le (I : ∀ U : X.affineOpens, Ideal Γ(X, U)) :\n    (ofIdeals I).ideal ≤ I :=\n  sSup_le (Set.forall_mem_image.mpr fun _ ↦ id)\n\n/-- The Galois coinsertion between ideal sheaves and arbitrary families of ideals. -/\nprotected def gci : GaloisCoinsertion ideal (ofIdeals (X := X)) where\n  choice I hI :=\n  { ideal := I\n    map_ideal_basicOpen U f :=\n      (ideal_ofIdeals_le I).antisymm hI ▸ (ofIdeals I).map_ideal_basicOpen U f }\n  gc _ _ := ⟨(le_sSup ·), (le_trans · (ideal_ofIdeals_le _))⟩\n  u_l_le _ := sSup_le fun _ ↦ id\n  choice_eq I hI := IdealSheafData.ext (hI.antisymm (ideal_ofIdeals_le I))\n\nlemma strictMono_ideal : StrictMono (ideal (X := X)) := IdealSheafData.gci.strictMono_l\nlemma ideal_mono : Monotone (ideal (X := X)) := strictMono_ideal.monotone\nlemma ofIdeals_mono : Monotone (ofIdeals (X := X)) := IdealSheafData.gci.gc.monotone_u\nlemma ofIdeals_ideal (I : IdealSheafData X) : ofIdeals I.ideal = I := IdealSheafData.gci.u_l_eq _\nlemma le_ofIdeals_iff {I : IdealSheafData X} {J} : I ≤ ofIdeals J ↔ I.ideal ≤ J :=\n  IdealSheafData.gci.gc.le_iff_le.symm\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : OrderTop (IdealSheafData X) where\n  top.ideal := ⊤\n  top.map_ideal_basicOpen := by simp [Ideal.map_top]\n  top.supportSet := ⊥\n  top.supportSet_eq_iInter_zeroLocus := by\n    ext x\n    simpa using X.isBasis_affineOpens.exists_subset_of_mem_open (Set.mem_univ x) isOpen_univ\n  le_top I U := le_top\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : OrderBot (IdealSheafData X) where\n  bot.ideal := ⊥\n  bot.map_ideal_basicOpen := by simp\n  bot.supportSet := ⊤\n  bot.supportSet_eq_iInter_zeroLocus := by ext; simp\n  bot_le I U := bot_le\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : SemilatticeInf (IdealSheafData X) where\n  inf I J :=\n  { ideal := I.ideal ⊓ J.ideal\n    map_ideal_basicOpen U f := by\n      dsimp\n      have : (X.presheaf.map (homOfLE (X.basicOpen_le f)).op).hom = algebraMap _ _ := rfl\n      have inst := U.2.isLocalization_basicOpen f\n      rw [← I.map_ideal_basicOpen U f, ← J.map_ideal_basicOpen U f, this]\n      ext x\n      obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq (.powers f) x\n      simp only [IsLocalization.mk'_mem_map_algebraMap_iff, Submonoid.mem_powers_iff, Ideal.mem_inf,\n        exists_exists_eq_and]\n      refine ⟨fun ⟨n, h₁, h₂⟩ ↦ ⟨⟨n, h₁⟩, ⟨n, h₂⟩⟩, ?_⟩\n      rintro ⟨⟨n₁, h₁⟩, ⟨n₂, h₂⟩⟩\n      refine ⟨n₁ + n₂, ?_, ?_⟩\n      · rw [add_comm, pow_add, mul_assoc]; exact Ideal.mul_mem_left _ _ h₁\n      · rw [pow_add, mul_assoc]; exact Ideal.mul_mem_left _ _ h₂ }\n  inf_le_left I J U := inf_le_left\n  inf_le_right I J U := inf_le_right\n  le_inf I J K hIJ hIK U := le_inf (hIJ U) (hIK U)\n\ninstance : CompleteLattice (IdealSheafData X) where\n  __ := (inferInstance : OrderTop (IdealSheafData X))\n  __ := (inferInstance : OrderBot (IdealSheafData X))\n  __ := (inferInstance : SemilatticeInf (IdealSheafData X))\n  __ := (inferInstance : CompleteSemilatticeSup (IdealSheafData X))\n  __ := IdealSheafData.gci.liftCompleteLattice\n\n@[simp]\nlemma ideal_top : ideal (X := X) ⊤ = ⊤ := rfl\n\n@[simp]\nlemma ideal_bot : ideal (X := X) ⊥ = ⊥ := rfl\n\n@[simp]\nlemma ideal_sup {I J : IdealSheafData X} : (I ⊔ J).ideal = I.ideal ⊔ J.ideal := rfl\n\n@[simp]\nlemma ideal_sSup {I : Set (IdealSheafData X)} : (sSup I).ideal = sSup (ideal '' I) := rfl\n\n@[simp]\nlemma ideal_iSup {ι : Type*} {I : ι → IdealSheafData X} : (iSup I).ideal = ⨆ i, (I i).ideal := by\n  rw [← sSup_range, ← sSup_range, ideal_sSup, ← Set.range_comp, Function.comp_def]\n\n@[simp]\nlemma ideal_inf {I J : IdealSheafData X} : (I ⊓ J).ideal = I.ideal ⊓ J.ideal := rfl\n\n@[simp]\n\nTarget:\nlemma ideal_biInf {ι : Type*} (I : ι → IdealSheafData X) {s : Set ι} (hs : s.Finite) :\n    (⨅ i ∈ s, I i).ideal = ⨅ i ∈ s, (I i).ideal :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"9526e73d9c63205db28236294bbda493881a30958fda9ee27c078ae44ee7adc5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/IdealSheaf","family_id":"ideal_biinf","file_id":"mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/Basic.lean","sample_id":"fd1fd9a53ab94e93ee71cb526c00c66f283a440ab2cd926e2cae6762d9e62179"},"split":"train"}
{"binding":{"chosen_proof_sha256":"be5d4d87008e815276be22f67ac5016b50a8ac860e36c4ba227914947bb79f69","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4c373cfd20145617555f6ba51d8c75290a6c4afc94ff70ac7ad735cc785944a1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9e0d38f023447d9d9affb59a11c3c96b5f0f51e0a6c3e310742ed531f794b9ce","source_sha256":"ebf0fa9c73e678969fdd70bcd10ba88fbd15d277d2467b36c6ff110685cacf37","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext <;> simp; ext <;> simp <;> (rw [← Category.assoc]; congr 1; simp)","hard_negative":false,"metrics":{"chosen_tokens":29,"rejected_tokens":36,"token_jaccard":0.894737,"token_length_ratio":1.241379},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"679a872412b6fc7dda7f6ef14e44c7ba3509764dfbca7d5b142ec1cff274c983","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Subobject.Lattice\npublic import Mathlib.CategoryTheory.Monoidal.Braided.Basic\npublic import Mathlib.CategoryTheory.Dialectica.Basic\n\nNamespace:\nCategoryTheory.Dial\n\nLocal context:\n/-\nCopyright (c) 2024 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# The Dialectica category is symmetric monoidal\n\nWe show that the category `Dial` has a symmetric monoidal category structure.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nnamespace CategoryTheory\n\nopen MonoidalCategory Limits\n\nuniverse v u\nvariable {C : Type u} [Category.{v} C] [HasFiniteProducts C] [HasPullbacks C]\n\nnamespace Dial\n\nlocal notation \"π₁\" => prod.fst\nlocal notation \"π₂\" => prod.snd\nlocal notation \"π(\" a \", \" b \")\" => prod.lift a b\n\n/-- The object `X ⊗ Y` in the `Dial C` category just tuples the left and right components. -/\n@[simps] def tensorObjImpl (X Y : Dial C) : Dial C where\n  src := X.src ⨯ Y.src\n  tgt := X.tgt ⨯ Y.tgt\n  rel :=\n    (Subobject.pullback (prod.map π₁ π₁)).obj X.rel ⊓\n    (Subobject.pullback (prod.map π₂ π₂)).obj Y.rel\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The functorial action of `X ⊗ Y` in `Dial C`. -/\n@[simps] def tensorHomImpl {X₁ X₂ Y₁ Y₂ : Dial C} (f : X₁ ⟶ X₂) (g : Y₁ ⟶ Y₂) :\n    tensorObjImpl X₁ Y₁ ⟶ tensorObjImpl X₂ Y₂ where\n  f := prod.map f.f g.f\n  F := π(prod.map π₁ π₁ ≫ f.F, prod.map π₂ π₂ ≫ g.F)\n  le := by\n    simp only [tensorObjImpl, Subobject.inf_pullback]\n    apply inf_le_inf <;> rw [← Subobject.pullback_comp, ← Subobject.pullback_comp]\n    · have := (Subobject.pullback (prod.map π₁ π₁ :\n        (X₁.src ⨯ Y₁.src) ⨯ X₂.tgt ⨯ Y₂.tgt ⟶ _)).monotone (Hom.le f)\n      rw [← Subobject.pullback_comp, ← Subobject.pullback_comp] at this\n      convert! this using 3 <;> simp\n    · have := (Subobject.pullback (prod.map π₂ π₂ :\n        (X₁.src ⨯ Y₁.src) ⨯ X₂.tgt ⨯ Y₂.tgt ⟶ _)).monotone (Hom.le g)\n      rw [← Subobject.pullback_comp, ← Subobject.pullback_comp] at this\n      convert! this using 3 <;> simp\n\n/-- The unit for the tensor `X ⊗ Y` in `Dial C`. -/\n@[simps] def tensorUnitImpl : Dial C := { src := ⊤_ _, tgt := ⊤_ _, rel := ⊤ }\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- Left unit cancellation `1 ⊗ X ≅ X` in `Dial C`. -/\n@[simps!] def leftUnitorImpl (X : Dial C) : tensorObjImpl tensorUnitImpl X ≅ X :=\n  isoMk (Limits.prod.leftUnitor _) (Limits.prod.leftUnitor _) <| by simp [Subobject.pullback_top]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- Right unit cancellation `X ⊗ 1 ≅ X` in `Dial C`. -/\n@[simps!] def rightUnitorImpl (X : Dial C) : tensorObjImpl X tensorUnitImpl ≅ X :=\n  isoMk (Limits.prod.rightUnitor _) (Limits.prod.rightUnitor _) <| by simp [Subobject.pullback_top]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The associator for tensor, `(X ⊗ Y) ⊗ Z ≅ X ⊗ (Y ⊗ Z)` in `Dial C`. -/\n@[simps!]\ndef associatorImpl (X Y Z : Dial C) :\n    tensorObjImpl (tensorObjImpl X Y) Z ≅ tensorObjImpl X (tensorObjImpl Y Z) :=\n  isoMk (prod.associator ..) (prod.associator ..) <| by\n    simp [Subobject.inf_pullback, ← Subobject.pullback_comp, inf_assoc]\n\n@[simps!]\ninstance : MonoidalCategoryStruct (Dial C) where\n  tensorUnit := tensorUnitImpl\n  tensorObj := tensorObjImpl\n  whiskerLeft X _ _ f := tensorHomImpl (𝟙 X) f\n  whiskerRight f Y := tensorHomImpl f (𝟙 Y)\n  tensorHom := tensorHomImpl\n  leftUnitor := leftUnitorImpl\n  rightUnitor := rightUnitorImpl\n  associator := associatorImpl\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\ntheorem id_tensorHom_id (X₁ X₂ : Dial C) : (𝟙 X₁ ⊗ₘ 𝟙 X₂ : _ ⟶ _) = 𝟙 (X₁ ⊗ X₂ : Dial C) := by\n  cat_disch\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n-- TODO: fix the non-terminal simp\nset_option linter.flexible false in\n\nTarget:\ntheorem tensorHom_comp_tensorHom {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : Dial C}\n    (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁) (g₂ : Y₂ ⟶ Z₂) :\n    (f₁ ⊗ₘ f₂) ≫ (g₁ ⊗ₘ g₂) = (f₁ ≫ g₁) ⊗ₘ (f₂ ≫ g₂) :=\n\nProof body:\n","rejected":"```lean\nby\n  ext <;> simp; ext <;> simp <;> (rw [← Category.assoc]; congr 1; simp)\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Dialectica","family_id":"tensorhom_comp_tensorhom","file_id":"mathlib/Mathlib/CategoryTheory/Dialectica/Monoidal.lean","sample_id":"9e0d38f023447d9d9affb59a11c3c96b5f0f51e0a6c3e310742ed531f794b9ce"},"split":"train"}
{"binding":{"chosen_proof_sha256":"02be6bd02ac451c737251c72bcfefcd2bda23393af373ca35cd3c4b0bb93ffd2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4dda6d8eed8aa9990b9ce99d74443691206268bc0aae2f322f2853e45e3b954a","source_sha256":"a7aa42b03cd6d38aac48f2dc40ba1b38bc0a247c798efe96de58375304c838ee","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [coe_inv, inv_def, normSq_coe, inv_one, ofReal_one, mul_one]","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.2},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"67f27c6a5ce84c4753b9a6f0ca89084d8df644348fd59d40bf89c421ccd9947a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic\npublic import Mathlib.Analysis.Normed.Field.UnitBall\n\nNamespace:\nCircle\n\nLocal context:\n/-\nCopyright (c) 2021 Heather Macbeth. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Heather Macbeth\n-/\n/-!\n# The circle\n\nThis file defines `Circle` to be the metric sphere (`Metric.sphere`) in `ℂ` centred at `0` of\nradius `1`.  We equip it with the following structure:\n\n* a submonoid of `ℂ`\n* a group\n* a topological group\n\nWe furthermore define `Circle.exp` to be the natural map `fun t ↦ exp (t * I)` from `ℝ` to\n`Circle`, and show that this map is a group homomorphism.\n\nWe define two additive characters onto the circle:\n* `Real.fourierChar`: The character `fun x ↦ exp ((2 * π * x) * I)` (for which we introduce the\n  notation `𝐞` in the scope `FourierTransform`). This uses the analyst convention that there is a\n  `2 * π` in the exponent.\n* `Real.probChar`: The character `fun x ↦ exp (x * I)`, which uses the probabilist convention that\n  there is no `2 * π` in the exponent.\n\n## Implementation notes\n\nBecause later (in `Geometry.Manifold.Instances.Sphere`) one wants to equip the circle with a smooth\nmanifold structure borrowed from `Metric.sphere`, the underlying set is\n`{z : ℂ | abs (z - 0) = 1}`.  This prevents certain algebraic facts from working definitionally --\nfor example, the circle is not defeq to `{z : ℂ | abs z = 1}`, which is the kernel of `Complex.abs`\nconsidered as a homomorphism from `ℂ` to `ℝ`, nor is it defeq to `{z : ℂ | normSq z = 1}`, which\nis the kernel of the homomorphism `Complex.normSq` from `ℂ` to `ℝ`.\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen Complex Function Metric ComplexConjugate\n\n/-- The unit circle in `ℂ`. -/\ndef Circle : Type := Submonoid.unitSphere ℂ\nderiving TopologicalSpace\n\nnamespace Circle\nvariable {x y : Circle}\n\ninstance instCoeOut : CoeOut Circle ℂ := subtypeCoe\n\ninstance instCommGroup : CommGroup Circle := inferInstanceAs <| CommGroup (sphere _ _)\ninstance : HasDistribNeg Circle := inferInstanceAs <| HasDistribNeg (sphere _ _)\ninstance : ContinuousNeg Circle := inferInstanceAs <| ContinuousNeg (sphere _ _)\ninstance instMetricSpace : MetricSpace Circle := inferInstanceAs <| MetricSpace (sphere _ _)\n\n@[ext] lemma ext : (x : ℂ) = y → x = y := Subtype.ext\n\nlemma coe_injective : Injective ((↑) : Circle → ℂ) := fun _ _ ↦ ext\n\n-- Not simp because `SetLike.coe_eq_coe` already proves it\nlemma coe_inj : (x : ℂ) = y ↔ x = y := coe_injective.eq_iff\n\nlemma norm_coe (z : Circle) : ‖(z : ℂ)‖ = 1 := mem_sphere_zero_iff_norm.1 z.2\n\n@[simp] lemma normSq_coe (z : Circle) : normSq z = 1 := by simp [normSq_eq_norm_sq]\n@[simp] lemma coe_ne_zero (z : Circle) : (z : ℂ) ≠ 0 := ne_zero_of_mem_unit_sphere z\n@[simp, norm_cast] lemma coe_one : ↑(1 : Circle) = (1 : ℂ) := rfl\n-- Not simp because `OneMemClass.coe_eq_one` already proves it\n@[norm_cast] lemma coe_eq_one : (x : ℂ) = 1 ↔ x = 1 := by rw [← coe_inj, coe_one]\n@[simp, norm_cast] lemma coe_mul (z w : Circle) : ↑(z * w) = (z : ℂ) * w := rfl\n@[simp, norm_cast] lemma coe_inv (z : Circle) : ↑z⁻¹ = (z : ℂ)⁻¹ := rfl\n\nTarget:\nlemma coe_inv_eq_conj (z : Circle) : ↑z⁻¹ = conj (z : ℂ) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Complex","family_id":"coe_inv_eq_conj","file_id":"mathlib/Mathlib/Analysis/Complex/Circle.lean","sample_id":"4dda6d8eed8aa9990b9ce99d74443691206268bc0aae2f322f2853e45e3b954a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f5148887f1a7dfc8799d4c5143dcee9d63a0a776aa7b9be04cad5dfcce2d500a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2200214482fedc100475ea58f565cc5394b158fdc04059d24586139e25ab023b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b53835cfef00f711bf8e92a6529c08ab65e2ebdbf08f272d243eb3a0265fab0b","source_sha256":"ce72ff57b0bb6ff2d5bfc5f2bf6fafec557312ecbed5c6bd326819562f53269d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun {U} hU ↦ ?_⟩\n  exact .of_comp (a := f.s₀) (𝒱.f ·) (f.h₀ ·)\n    (fun _ ↦ Scheme.Hom.continuous _) (fun i ↦ funext <| by simp [← Scheme.Hom.comp_apply])\n    (fun _ ↦ Scheme.Hom.continuous _) U.2 (hU.isCompactOpenCovered 𝒱)","hard_negative":true,"metrics":{"chosen_tokens":85,"rejected_tokens":2,"token_jaccard":0.025,"token_length_ratio":0.023529},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"67ff61369669a0c8a6eeb5d825966cbfee8361b9d31c7841367860f815fa7a7e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.Affine\npublic import Mathlib.AlgebraicGeometry.Properties\npublic import Mathlib.AlgebraicGeometry.PullbackCarrier\npublic import Mathlib.Topology.Sets.CompactOpenCovered\n\nNamespace:\nAlgebraicGeometry.QuasiCompactCover\n\nLocal context:\n/-\nCopyright (c) 2025 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten\n-/\n/-!\n# Quasi-compact covers\n\nA cover of a scheme is quasi-compact if every affine open of the base can be covered\nby a finite union of images of quasi-compact opens of the components.\n\nThis is used to define the fpqc (faithfully flat, quasi-compact) topology, where covers are given by\nflat covers that are quasi-compact.\n-/\n\n@[expose] public section\n\nuniverse w' w u v\n\nopen CategoryTheory Limits MorphismProperty TopologicalSpace.Opens AlgebraicGeometry\n\nnamespace AlgebraicGeometry\n\nvariable {S : Scheme.{u}}\n\n/--\nA cover of a scheme is quasi-compact if every affine open of the base can be covered\nby a finite union of images of quasi-compact opens of the components.\n-/\n@[stacks 022B, mk_iff]\nclass QuasiCompactCover (𝒰 : PreZeroHypercover.{v} S) : Prop where\n  isCompactOpenCovered_of_isAffineOpen {U : S.Opens} (hU : IsAffineOpen U) :\n    IsCompactOpenCovered (𝒰.f ·) (U : Set S)\n\nvariable (𝒰 : PreZeroHypercover.{v} S)\n\nlemma IsAffineOpen.isCompactOpenCovered [QuasiCompactCover 𝒰] {U : S.Opens} (hU : IsAffineOpen U) :\n    IsCompactOpenCovered (𝒰.f ·) (U : Set S) :=\n  QuasiCompactCover.isCompactOpenCovered_of_isAffineOpen hU\n\nnamespace QuasiCompactCover\n\nlemma isCompactOpenCovered_of_isCompact [QuasiCompactCover 𝒰]\n    {U : S.Opens} (hU : IsCompact (U : Set S)) :\n    IsCompactOpenCovered (𝒰.f ·) (U : Set S) := by\n  obtain ⟨Us, hUs, hUf, hUc⟩ := S.isBasis_affineOpens.exists_finite_of_isCompact hU\n  refine .of_biUnion_eq_of_finite (SetLike.coe '' Us) (by simp_all) (hUf.image _) ?_\n  simpa using fun t ht ↦ IsAffineOpen.isCompactOpenCovered 𝒰 (hUs ht)\n\nvariable {𝒰 : PreZeroHypercover.{v} S} {K : Precoverage Scheme.{u}}\n\nvariable (𝒰) in\nlemma exists_isAffineOpen_of_isCompact [QuasiCompactCover 𝒰] {U : S.Opens}\n    (hU : IsCompact (U : Set S)) :\n    ∃ (n : ℕ) (f : Fin n → 𝒰.I₀) (V : ∀ i, (𝒰.X (f i)).Opens),\n      (∀ i, IsAffineOpen (V i)) ∧\n      ⋃ i, 𝒰.f (f i) '' (V i) = U := by\n  obtain ⟨n, a, V, ha, heq⟩ := (isCompactOpenCovered_of_isCompact 𝒰 hU).exists_mem_of_isBasis\n    (fun i ↦ (𝒰.X i).isBasis_affineOpens) (fun _ _ h ↦ h.isCompact)\n  exact ⟨n, a, V, ha, heq⟩\n\n/-- If the component maps of `𝒰` are open, `𝒰` is quasi-compact. This in particular\napplies if `K` is the fppf topology (i.e., flat and of finite presentation) and hence in\nparticular for étale and Zariski covers. -/\n@[stacks 022C]\nlemma of_isOpenMap {𝒰 : S.Cover K} [Scheme.JointlySurjective K] (h : ∀ i, IsOpenMap (𝒰.f i)) :\n    QuasiCompactCover 𝒰.toPreZeroHypercover where\n  isCompactOpenCovered_of_isAffineOpen {U} hU := .of_isOpenMap\n    (fun i ↦ (𝒰.f i).continuous) h (fun x _ ↦ ⟨𝒰.idx x, 𝒰.covers x⟩) U.2 hU.isCompact\n\n/-- Any open cover is quasi-compact. -/\ninstance (𝒰 : S.OpenCover) : QuasiCompactCover 𝒰.toPreZeroHypercover :=\n  of_isOpenMap fun i ↦ (𝒰.f i).isOpenEmbedding.isOpenMap\n\n/-- If `𝒱` is a refinement of `𝒰` such that `𝒱` is quasicompact, also `𝒰` is quasicompact. -/\n@[stacks 03L8]\n\nTarget:\nlemma of_hom {𝒱 : PreZeroHypercover.{w'} S} (f : 𝒱.Hom 𝒰) [QuasiCompactCover 𝒱] :\n    QuasiCompactCover 𝒰 :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_b53835cfef00","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"2822a633e8c431be1dd8a28f37dd57884322e6dc3c13429d6036183992312270","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Cover","family_id":"of_hom","file_id":"mathlib/Mathlib/AlgebraicGeometry/Cover/QuasiCompact.lean","sample_id":"b53835cfef00f711bf8e92a6529c08ab65e2ebdbf08f272d243eb3a0265fab0b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9f76daa2d17794583c7f31dfaabd84b49dfce6e55e1f6c203ac803f9bd29220e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1e270d1bd1eeb14c5bf62361efc62c628e247ff0c8b0d9ddc26f734fc3b353a7","source_sha256":"1c0dcf6a3ce55bec7a88672314ea6ae6741a6599903854de40abb691f935e16b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rw [← subAt_pullSub S x]; gcongr\n  mpr _ := le_trans (by gcongr) (pullSub_subAt T x)","hard_negative":false,"metrics":{"chosen_tokens":23,"rejected_tokens":2,"token_jaccard":0.052632,"token_length_ratio":0.086957},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"6845695eec8ae84259197debf5701f9513a2145885e44fbc1921cf37f216c477","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.CompleteLattice.SetLike\n\nNamespace:\nDescriptive.Tree\n\nLocal context:\n/-\nCopyright (c) 2024 Sven Manthe. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sven Manthe\n-/\n/-!\n# Trees in the sense of descriptive set theory\n\nThis file defines trees of depth `ω` in the sense of descriptive set theory as sets of finite\nsequences that are stable under taking prefixes.\n\n## Main declarations\n\n* `tree A`: a (possibly infinite) tree of depth at most `ω` with nodes in `A`\n-/\n\n@[expose] public section\n\nnamespace Descriptive\n\n/-- A tree is a set of finite sequences, implemented as `List A`, that is stable under\n  taking prefixes. For the definition we use the equivalent property `x ++ [a] ∈ T → x ∈ T`,\n  which is more convenient to check. We define `tree A` as a complete sublattice of\n  `Set (List A)`, which coerces to the type of trees on `A`. -/\ndef tree (A : Type*) : CompleteSublattice (Set (List A)) :=\n  CompleteSublattice.mk' {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T}\n    (by rintro S hS x a ⟨t, ht, hx⟩; use t, ht, hS ht hx)\n    (by rintro S hS x a h T hT; exact hS hT <| h T hT)\n\n@[simps!] instance (A : Type*) : SetLike (tree A) (List A) := SetLike.instSubtypeSet\n\nexample (A : Type*) : PartialOrder (tree A) := inferInstance\n\nnamespace Tree\nvariable {A : Type*} {S T : tree A}\n\nlemma mem_of_append {x y : List A} (h : x ++ y ∈ T) : x ∈ T := by\n  induction y generalizing x with\n  | nil => simpa using h\n  | cons y ys ih => exact T.prop (ih (by simpa))\n\nlemma mem_of_prefix {x y : List A} (h' : x <+: y) (h : y ∈ T) : x ∈ T := by\n  obtain ⟨_, rfl⟩ := h'; exact mem_of_append h\n\ninstance : Trans List.IsPrefix (fun x (T : tree A) ↦ x ∈ T) (fun x T ↦ x ∈ T) where\n  trans := mem_of_prefix\n\nlemma singleton_mem (T : tree A) {a : A} {x : List A} (h : a :: x ∈ T) : [a] ∈ T :=\n  mem_of_prefix ⟨x, rfl⟩ h\n\n@[simp] lemma tree_eq_bot : T = ⊥ ↔ [] ∉ T where\n  mp := by rintro rfl; simp\n  mpr h := by ext x; simpa using fun h' ↦ h <| mem_of_prefix x.nil_prefix h'\n\nlemma take_mem {n : ℕ} (x : T) : x.val.take n ∈ T :=\n  mem_of_prefix (x.val.take_prefix n) x.prop\n\n/-- A variant of `List.take` internally to a tree -/\n@[simps] def take (n : ℕ) (x : T) : T := ⟨x.val.take n, take_mem x⟩\n\n@[simp] lemma take_take (m n : ℕ) (x : T) : take m (take n x) = take (m ⊓ n) x := by\n  simp [Subtype.ext_iff, List.take_take]\n\n@[simp] lemma take_eq_take {x : T} {m n : ℕ} :\n    take m x = take n x ↔ m ⊓ x.val.length = n ⊓ x.val.length := by simp [Subtype.ext_iff]\n\n-- ### `subAt`\n\nvariable (T) (x y : List A)\n\n/-- The residual tree obtained by regarding the node x as new root -/\ndef subAt : tree A :=\n  ⟨(x ++ ·)⁻¹' T, fun _ a _ ↦ mem_of_append (y := [a]) (by rwa [List.append_assoc])⟩\n\n@[simp] lemma mem_subAt : y ∈ subAt T x ↔ x ++ y ∈ T := Iff.rfl\n\n@[simp] lemma subAt_nil : subAt T [] = T := rfl\n\n@[simp] lemma subAt_append : subAt (subAt T x) y = subAt T (x ++ y) := by ext; simp\n\n@[gcongr] lemma subAt_mono (h : S ≤ T) : subAt S x ≤ subAt T x :=\n  Set.preimage_mono h\n\n/-- A variant of `List.drop` that takes values in `subAt` -/\n@[simps] def drop (n : ℕ) (x : T) : subAt T (Tree.take n x).val :=\n  ⟨x.val.drop n, by simp⟩\n\n-- ### `pullSub`\n\n/-- Adjoint of `subAt`, given by pasting x before the root of T. Explicitly,\n  elements are prefixes of x or x with an element of T appended -/\ndef pullSub : tree A where\n  val := { y | y.take x.length <+: x ∧ y.drop x.length ∈ T }\n  property := fun y a ⟨h1, h2⟩ ↦\n    ⟨((y.prefix_append [a]).take x.length).trans h1,\n    mem_of_prefix ((y.prefix_append [a]).drop x.length) h2⟩\n\nvariable {T x y}\n\nlemma mem_pullSub_short (hl : y.length ≤ x.length) : y ∈ pullSub T x ↔ y <+: x ∧ [] ∈ T := by\n  simp [pullSub, List.take_of_length_le hl, List.drop_eq_nil_iff.mpr hl]\n\nlemma mem_pullSub_long (hl : x.length ≤ y.length) : y ∈ pullSub T x ↔ ∃ z ∈ T, y = x ++ z where\n  mp := by\n    intro ⟨h1, h2⟩; use y.drop x.length, h2\n    nth_rw 1 [← List.take_append_drop x.length y]\n    simpa [-List.take_append_drop, List.prefix_iff_eq_take, hl] using h1\n  mpr := by simp +contextual [pullSub]\n\n@[simp] lemma mem_pullSub_append : x ++ y ∈ pullSub T x ↔ y ∈ T := by simp [mem_pullSub_long]\n\n@[simp] lemma mem_pullSub_self : x ∈ pullSub T x ↔ [] ∈ T := by\n  simpa using mem_pullSub_append (y := [])\n\n\nvariable (T x y)\n\nlemma pullSub_subAt : pullSub (subAt T x) x ≤ T := by\n  intro y (h : y ∈ pullSub _ x); rcases le_total y.length x.length with h' | h'\n  · rw [mem_pullSub_short h'] at h; exact mem_of_prefix h.1 (by simpa using h.2)\n  · rw [mem_pullSub_long h'] at h; obtain ⟨_, h, rfl⟩ := h; exact h\n\n@[simp] lemma subAt_pullSub : subAt (pullSub T x) x = T := by\n  ext y; simp\n\n@[gcongr] lemma pullSub_mono (h : S ≤ T) x : pullSub S x ≤ pullSub T x :=\n  fun _ ⟨h1, h2⟩ ↦ ⟨h1, h h2⟩\n\nTarget:\nlemma pullSub_adjunction (S T : tree A) (x : List A) : pullSub S x ≤ T ↔ S ≤ subAt T x where\n  mp _ :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Descriptive","family_id":"pullsub_adjunction","file_id":"mathlib/Mathlib/SetTheory/Descriptive/Tree.lean","sample_id":"1e270d1bd1eeb14c5bf62361efc62c628e247ff0c8b0d9ddc26f734fc3b353a7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"82d206dfe82328407b70971415868cac0e9eb31b2da76433329c285b45739547","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0cdce9457755c2ffaac6240c2fe685201fa89a7182b4a5c18a28a687a06b3355","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f2d217efd8dc67e5b8a5a62c60104ed7db1fe692e856a2299afd32cbd082efda","source_sha256":"20a211529741be444705515bbef40327474b8d60b75b7d4d28a459e31d8d7bbe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply Linear.comp_smul","hard_negative":false,"metrics":{"chosen_tokens":5,"rejected_tokens":10,"token_jaccard":0.5,"token_length_ratio":2.0},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"684c0855b5d0b353cd3f9c82d2ff4a4d2717ca322fe95383dc9c51c0bcdc46f1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Defs\npublic import Mathlib.Algebra.Module.TransferInstance\npublic import Mathlib.Algebra.Group.Invertible.Defs\npublic import Mathlib.Algebra.Module.Equiv.Defs\npublic import Mathlib.CategoryTheory.Preadditive.Basic\n\nNamespace:\nCategoryTheory.Linear\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Linear categories\n\nAn `R`-linear category is a category in which `X ⟶ Y` is an `R`-module in such a way that\ncomposition of morphisms is `R`-linear in both variables.\n\nNote that sometimes in the literature a \"linear category\" is further required to be abelian.\n\n## Implementation\n\nCorresponding to the fact that we need to have an `AddCommGroup X` structure in place\nto talk about a `Module R X` structure,\nwe need `Preadditive C` as a prerequisite typeclass for `Linear R C`.\nThis makes for longer signatures than would be ideal.\n\n## Future work\n\nIt would be nice to have a usable framework of enriched categories in which this would just be\na category enriched in `Module R`.\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nopen CategoryTheory.Limits\n\nopen LinearMap\n\nnamespace CategoryTheory\n\n/-- A category is called `R`-linear if `P ⟶ Q` is an `R`-module such that composition is\n`R`-linear in both variables. -/\nclass Linear (R : Type w) [Semiring R] (C : Type u) [Category.{v} C] [Preadditive C] where\n  homModule : ∀ X Y : C, Module R (X ⟶ Y) := by infer_instance\n  /-- compatibility of the scalar multiplication with the post-composition -/\n  smul_comp : ∀ (X Y Z : C) (r : R) (f : X ⟶ Y) (g : Y ⟶ Z), (r • f) ≫ g = r • f ≫ g := by\n    cat_disch\n  /-- compatibility of the scalar multiplication with the pre-composition -/\n  comp_smul : ∀ (X Y Z : C) (f : X ⟶ Y) (r : R) (g : Y ⟶ Z), f ≫ (r • g) = r • f ≫ g := by\n    cat_disch\n\nattribute [instance_reducible, instance] Linear.homModule\n\nattribute [simp] Linear.smul_comp Linear.comp_smul\n\n-- (the linter doesn't like `simp` on the `_assoc` lemma)\nend CategoryTheory\n\nopen CategoryTheory\n\nnamespace CategoryTheory.Linear\n\nvariable {C : Type u} [Category.{v} C] [Preadditive C]\n\ninstance preadditiveNatLinear : Linear ℕ C where\n  smul_comp X _Y _Z r f g := by exact (Preadditive.rightComp X g).map_nsmul r f\n  comp_smul _X _Y Z f r g := by exact (Preadditive.leftComp Z f).map_nsmul r g\n\ninstance preadditiveIntLinear : Linear ℤ C where\n  smul_comp X _Y _Z r f g := by exact (Preadditive.rightComp X g).map_zsmul r f\n  comp_smul _X _Y Z f r g := by exact (Preadditive.leftComp Z f).map_zsmul r g\n\nsection End\n\nvariable {R : Type w}\n\ninstance [Semiring R] [Linear R C] (X : C) : Module R (End X) :=\n  inferInstanceAs <| Module R (X ⟶ X)\n\ninstance [CommSemiring R] [Linear R C] (X : C) : Algebra R (End X) :=\n  Algebra.ofModule (fun _ _ _ => comp_smul _ _ _ _ _ _) fun _ _ _ => smul_comp _ _ _ _ _ _\n\nend End\n\nsection\n\nvariable {R : Type w} [Semiring R] [Linear R C]\n\nsection InducedCategory\n\nuniverse u'\n\nvariable {D : Type u'} (F : D → C)\n\ninstance inducedCategory : Linear.{w, v} R (InducedCategory C F) where\n  homModule X Y := Equiv.module _ InducedCategory.homEquiv\n  smul_comp _ _ _ _ _ _ := by ext; apply smul_comp\n  comp_smul _ _ _ _ _ _ := by ext; apply comp_smul\n\nvariable {F} in\n/-- The linear equivalence `(X ⟶ Y) ≃+ (F X ⟶ F Y)` when `F : D → C` and\n`C` is a `R`-linear category. -/\n@[simps!]\ndef _root_.CategoryTheory.InducedCategory.homLinearEquiv\n    {X Y : InducedCategory C F} :\n    (X ⟶ Y) ≃ₗ[R] (F X ⟶ F Y) where\n  toAddEquiv := InducedCategory.homAddEquiv\n  map_smul' := by cat_disch\n\nend InducedCategory\n\ninstance fullSubcategory (Z : ObjectProperty C) : Linear.{w, v} R Z.FullSubcategory :=\n  inducedCategory _\n\nvariable (R)\n\n/-- Composition by a fixed left argument as an `R`-linear map. -/\n@[simps]\ndef leftComp {X Y : C} (Z : C) (f : X ⟶ Y) : (Y ⟶ Z) →ₗ[R] X ⟶ Z where\n  toFun g := f ≫ g\n  map_add' := by simp\n  map_smul' := by simp\n\n/-- Composition by a fixed right argument as an `R`-linear map. -/\n@[simps]\ndef rightComp (X : C) {Y Z : C} (g : Y ⟶ Z) : (X ⟶ Y) →ₗ[R] X ⟶ Z where\n  toFun f := f ≫ g\n  map_add' := by simp\n  map_smul' := by simp\n\ninstance {X Y : C} (f : X ⟶ Y) [Epi f] (r : R) [Invertible r] : Epi (r • f) :=\n  ⟨fun g g' H => by\n    rw [smul_comp, smul_comp, ← comp_smul, ← comp_smul, cancel_epi] at H\n    simpa [smul_smul] using congr_arg (fun f => ⅟r • f) H⟩\n\ninstance {X Y : C} (f : X ⟶ Y) [Mono f] (r : R) [Invertible r] : Mono (r • f) :=\n  ⟨fun g g' H => by\n    rw [comp_smul, comp_smul, ← smul_comp, ← smul_comp, cancel_mono] at H\n    simpa [smul_smul] using congr_arg (fun f => ⅟r • f) H⟩\n\n/-- Given isomorphic objects `X ≅ Y, W ≅ Z` in a `k`-linear category, we have a `k`-linear\nisomorphism between `Hom(X, W)` and `Hom(Y, Z).` -/\ndef homCongr (k : Type*) {C : Type*} [Category* C] [Semiring k] [Preadditive C] [Linear k C]\n    {X Y W Z : C} (f₁ : X ≅ Y) (f₂ : W ≅ Z) : (X ⟶ W) ≃ₗ[k] Y ⟶ Z :=\n  {\n    (rightComp k Y f₂.hom).comp\n      (leftComp k W\n        f₁.symm.hom) with\n    invFun := (leftComp k W f₁.hom).comp (rightComp k Y f₂.symm.hom)\n    left_inv := fun x => by\n      simp only [Iso.symm_hom, LinearMap.toFun_eq_coe, LinearMap.coe_comp, Function.comp_apply,\n        leftComp_apply, rightComp_apply, Category.assoc, Iso.hom_inv_id, Category.comp_id,\n        Iso.hom_inv_id_assoc]\n    right_inv := fun x => by\n      simp only [Iso.symm_hom, LinearMap.coe_comp, Function.comp_apply, rightComp_apply,\n        leftComp_apply, LinearMap.toFun_eq_coe, Iso.inv_hom_id_assoc, Category.assoc,\n        Iso.inv_hom_id, Category.comp_id] }\n\ntheorem homCongr_apply (k : Type*) {C : Type*} [Category* C] [Semiring k] [Preadditive C]\n    [Linear k C] {X Y W Z : C} (f₁ : X ≅ Y) (f₂ : W ≅ Z) (f : X ⟶ W) :\n    homCongr k f₁ f₂ f = (f₁.inv ≫ f) ≫ f₂.hom :=\n  rfl\n\ntheorem homCongr_symm_apply (k : Type*) {C : Type*} [Category* C] [Semiring k] [Preadditive C]\n    [Linear k C] {X Y W Z : C} (f₁ : X ≅ Y) (f₂ : W ≅ Z) (f : Y ⟶ Z) :\n    (homCongr k f₁ f₂).symm f = f₁.hom ≫ f ≫ f₂.inv :=\n  rfl\n\nvariable {R}\n\n@[simp]\nlemma units_smul_comp {X Y Z : C} (r : Rˣ) (f : X ⟶ Y) (g : Y ⟶ Z) :\n    (r • f) ≫ g = r • f ≫ g := by\n  apply Linear.smul_comp\n\n@[simp]\n\nTarget:\nlemma comp_units_smul {X Y Z : C} (f : X ⟶ Y) (r : Rˣ) (g : Y ⟶ Z) :\n    f ≫ (r • g) = r • f ≫ g :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  apply Linear.comp_smul","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Linear","family_id":"comp_units_smul","file_id":"mathlib/Mathlib/CategoryTheory/Linear/Basic.lean","sample_id":"f2d217efd8dc67e5b8a5a62c60104ed7db1fe692e856a2299afd32cbd082efda"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ae21492afcd16aacb24a3ebd74e7f36439c097260b4eee6f2b4d2294937fc244","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0aeef97f4cfea55c0640efb59e6d6792baab329447e04559e5b0bd1794a70844","source_sha256":"139083b5f0b4f5b6fe528c0c9bf16219e4a814c177511d56fdc66704924ae51e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases F; rfl","hard_negative":false,"metrics":{"chosen_tokens":5,"rejected_tokens":5,"token_jaccard":0.111111,"token_length_ratio":1.0},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"68791f88bfdeff9ffcdd53a1dcffe6b6ae0b0a311b306542362c396e1289518d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Analysis.Filter\npublic import Mathlib.Topology.Bases\npublic import Mathlib.Topology.LocallyFinite\n\nNamespace:\nCtop\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Computational realization of topological spaces (experimental)\n\nThis file provides infrastructure to compute with topological spaces.\n\n## Main declarations\n\n* `Ctop`: Realization of a topology basis.\n* `Ctop.Realizer`: Realization of a topological space. `Ctop` that generates the given topology.\n* `LocallyFinite.Realizer`: Realization of the local finiteness of an indexed family of sets.\n* `Compact.Realizer`: Realization of the compactness of a set.\n-/\n\n@[expose] public section\n\n\nopen Set\n\nopen Filter hiding Realizer\n\nopen Topology\n\n/-- A `Ctop α σ` is a realization of a topology (basis) on `α`,\n  represented by a type `σ` together with operations for the top element and\n  the intersection operation. -/\nstructure Ctop (α σ : Type*) where\n  f : σ → Set α\n  top : α → σ\n  top_mem : ∀ x : α, x ∈ f (top x)\n  inter : ∀ (a b) (x : α), x ∈ f a ∩ f b → σ\n  inter_mem : ∀ a b x h, x ∈ f (inter a b x h)\n  inter_sub : ∀ a b x h, f (inter a b x h) ⊆ f a ∩ f b\n\nvariable {α : Type*} {β : Type*} {σ : Type*} {τ : Type*}\n\ninstance : Inhabited (Ctop α (Set α)) :=\n  ⟨{  f := id\n      top := singleton\n      top_mem := mem_singleton\n      inter := fun s t _ _ ↦ s ∩ t\n      inter_mem := fun _s _t _a ↦ id\n      inter_sub := fun _s _t _a _ha ↦ Subset.rfl }⟩\n\nnamespace Ctop\n\nsection\n\nvariable (F : Ctop α σ)\n\ninstance : CoeFun (Ctop α σ) fun _ ↦ σ → Set α :=\n  ⟨Ctop.f⟩\n\ntheorem coe_mk (f T h₁ I h₂ h₃ a) : (@Ctop.mk α σ f T h₁ I h₂ h₃) a = f a := rfl\n\n/-- Map a Ctop to an equivalent representation type. -/\ndef ofEquiv (E : σ ≃ τ) : Ctop α σ → Ctop α τ\n  | ⟨f, T, h₁, I, h₂, h₃⟩ =>\n    { f := fun a ↦ f (E.symm a)\n      top := fun x ↦ E (T x)\n      top_mem := fun x ↦ by simpa using h₁ x\n      inter := fun a b x h ↦ E (I (E.symm a) (E.symm b) x h)\n      inter_mem := fun a b x h ↦ by simpa using h₂ (E.symm a) (E.symm b) x h\n      inter_sub := fun a b x h ↦ by simpa using h₃ (E.symm a) (E.symm b) x h }\n\n@[simp]\n\nTarget:\ntheorem ofEquiv_val (E : σ ≃ τ) (F : Ctop α σ) (a : τ) : F.ofEquiv E a = F (E.symm a) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Analysis","family_id":"ofequiv_val","file_id":"mathlib/Mathlib/Data/Analysis/Topology.lean","sample_id":"0aeef97f4cfea55c0640efb59e6d6792baab329447e04559e5b0bd1794a70844"},"split":"train"}
{"binding":{"chosen_proof_sha256":"08d1285c21c0227003f0de3a7a5739a4341408268171b9be42c7528b7aade63d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f7b620c51cfaab544fc583af36da411df3fc24767a9c25ebd60f0ee43a35db07","source_sha256":"d7743ebc410d9ee31a12263ae8d894bf6aecd1d52cd1a367eccb2f5de9ab8de8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Function.isPeriodicPt_iff_minimalPeriod_dvd, Nat.dvd_prime hp.out,\n    ← minimalPeriod_eq_one_iff_isFixedPt.not, or_and_right, and_not_self_iff, false_or,\n    iff_self_and]\n  exact fun h ↦ ne_of_eq_of_ne h hp.out.ne_one","hard_negative":false,"metrics":{"chosen_tokens":38,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.078947},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"68a40bff9b7512c92c56067e25eccc951172a6683864aee191761f917c945c8f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GCDMonoid.Finset\npublic import Mathlib.Data.Nat.Prime.Basic\npublic import Mathlib.Data.PNat.Basic\npublic import Mathlib.Dynamics.PeriodicPts.Defs\npublic import Mathlib.Order.Lattice.Nat\n\nNamespace:\nFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Extra lemmas about periodic points\n-/\n\npublic section\n\nopen Nat Set\n\nnamespace Function\nvariable {α : Type*} {f : α → α} {x y : α}\n\nopen Function (Commute)\n\ntheorem directed_ptsOfPeriod_pnat (f : α → α) : Directed (· ⊆ ·) fun n : ℕ+ => ptsOfPeriod f n :=\n  fun m n => ⟨m * n, fun _ hx => hx.mul_const n, fun _ hx => hx.const_mul m⟩\n\nvariable (f) in\ntheorem bijOn_periodicPts : BijOn f (periodicPts f) (periodicPts f) :=\n  iUnion_pnat_ptsOfPeriod f ▸\n    bijOn_iUnion_of_directed (directed_ptsOfPeriod_pnat f) fun i => bijOn_ptsOfPeriod f i.pos\n\nTarget:\ntheorem minimalPeriod_eq_prime_iff {p : ℕ} [hp : Fact p.Prime] :\n    minimalPeriod f x = p ↔ IsPeriodicPt f p x ∧ ¬IsFixedPt f x :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/PeriodicPts","family_id":"minimalperiod_eq_prime_iff","file_id":"mathlib/Mathlib/Dynamics/PeriodicPts/Lemmas.lean","sample_id":"f7b620c51cfaab544fc583af36da411df3fc24767a9c25ebd60f0ee43a35db07"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d216e83441f49f61f8d591a6008f2029d6a48f6f63ee5482956becb053185e63","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9c77b43c796b9a66a9282957c2b67d46b27f3beda4c44634fa70becc056cd2ca","source_sha256":"ee7b415cab50ba8cea4663cf633679dc26a3069a4531b1546132f03d2045e5e6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [commute_iff_eq, hp.le_iff_mul_eq_right hq |>.mp h, hp.le_iff_mul_eq_left hq |>.mp h]","hard_negative":true,"metrics":{"chosen_tokens":25,"rejected_tokens":8,"token_jaccard":0.047619,"token_length_ratio":0.32},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"68ad3d6f87bb505ac46bb83ed5986e0a31cf1eb79e259cde78fa937d1879e154","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order\npublic import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric\nimport Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Projection\n\nNamespace:\nIsStarProjection\n\nLocal context:\n/-\nCopyright (c) 2025 Monica Omar. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Monica Omar, Jireh Loreaux\n-/\n/-!\n\n# Projections in C⋆-algebras\n\nHere we collect results about projections specific to C⋆-algebras.\n\n## Main results\n\n+ `isStarProjection_iff_isIdempotentElem_and_isStarNormal`: star projections are precisely\n  idempotent normal elements.\n+ `IsStarProjection.le_tfae`: for star projections `p` and `q`, the following are equivalent:\n  - `p ≤ q`\n  - `q * p = p`\n  - `p * q = p`\n  - `q - p` is a star projection\n  - `q - p` is an idempotent element\n\n-/\n\npublic section\n\nopen scoped CStarAlgebra\n\nsection NonUnital\nvariable {A : Type*} [TopologicalSpace A] [NonUnitalRing A] [StarRing A]\n\nlemma isStarProjection_iff_quasispectrum_subset_and_isSelfAdjoint [Module ℝ A] [IsScalarTower ℝ A A]\n    [SMulCommClass ℝ A A] [NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint] {p : A} :\n    IsStarProjection p ↔ quasispectrum ℝ p ⊆ {0, 1} ∧ IsSelfAdjoint p :=\n  (isStarProjection_iff p).eq ▸\n    and_congr_left_iff.mpr fun h ↦ isIdempotentElem_iff_quasispectrum_subset ℝ p h\n\nsection Normal\nvariable [Module ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A]\n  [NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal]\n\n/-- An idempotent element in a non-unital C⋆-algebra is self-adjoint iff it is normal. -/\ntheorem IsIdempotentElem.isSelfAdjoint_iff_isStarNormal {p : A} (hp : IsIdempotentElem p) :\n    IsSelfAdjoint p ↔ IsStarNormal p := by\n  simp only [isSelfAdjoint_iff_isStarNormal_and_quasispectrumRestricts,\n    QuasispectrumRestricts.real_iff, and_iff_left_iff_imp]\n  intro h x hx\n  rcases hp.quasispectrum_subset _ hx with (hx | hx) <;> simp [Set.mem_singleton_iff.mp hx]\n\n/-- An element in a non-unital C⋆-algebra is a star projection\nif and only if it is idempotent and normal. -/\ntheorem isStarProjection_iff_isIdempotentElem_and_isStarNormal {p : A} :\n    IsStarProjection p ↔ IsIdempotentElem p ∧ IsStarNormal p :=\n  (isStarProjection_iff p).eq ▸ and_congr_right_iff.eq ▸ fun h => h.isSelfAdjoint_iff_isStarNormal\n\ntheorem isStarProjection_iff_quasispectrum_subset_and_isStarNormal {p : A} :\n    IsStarProjection p ↔ quasispectrum ℂ p ⊆ {0, 1} ∧ IsStarNormal p :=\n  isStarProjection_iff_isIdempotentElem_and_isStarNormal (p := p).eq ▸\n    and_congr_left_iff.mpr fun h ↦ isIdempotentElem_iff_quasispectrum_subset ℂ p h\n\nend Normal\nend NonUnital\n\nsection Unital\nvariable {A : Type*} [TopologicalSpace A] [Ring A] [StarRing A]\n\nlemma isStarProjection_iff_spectrum_subset_and_isSelfAdjoint [Algebra ℝ A]\n    [NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint] {p : A} :\n    IsStarProjection p ↔ spectrum ℝ p ⊆ {0, 1} ∧ IsSelfAdjoint p :=\n  (isStarProjection_iff p).eq ▸\n    and_congr_left_iff.mpr fun h ↦ isIdempotentElem_iff_spectrum_subset ℝ p h\n\ntheorem isStarProjection_iff_spectrum_subset_and_isStarNormal [Algebra ℂ A]\n    [NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal] {p : A} :\n    IsStarProjection p ↔ spectrum ℂ p ⊆ {0, 1} ∧ IsStarNormal p :=\n  isStarProjection_iff_isIdempotentElem_and_isStarNormal (p := p).eq ▸\n    and_congr_left_iff.mpr fun h ↦ isIdempotentElem_iff_spectrum_subset ℂ p h\n\nend Unital\n\nnamespace IsStarProjection\n\nvariable {A : Type*} [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A] {p q : A}\n\nopen CFC in\nlemma le_tfae (hp : IsStarProjection p) (hq : IsStarProjection q) :\n  List.TFAE\n    [p ≤ q,\n    q * p = p,\n    p * q = p,\n    IsStarProjection (q - p),\n    IsIdempotentElem (q - p)] := by\n  tfae_have 1 → 2 := fun h ↦ (hq.mul_right_and_mul_left_of_nonneg_of_le hp.nonneg h).2\n  tfae_have 2 → 3 := fun h ↦ by\n    simpa [hp.isSelfAdjoint.star_eq, hq.isSelfAdjoint.star_eq] using congr(star $h)\n  tfae_have 3 → 4 := hp.sub_of_mul_eq_left hq\n  tfae_have 4 → 1 := fun h ↦ by simpa using h.nonneg\n  tfae_have 4 ↔ 5 := by simp [isStarProjection_iff, hq.isSelfAdjoint.sub hp.isSelfAdjoint]\n  tfae_finish\n\nlemma le_iff_mul_eq_right (hp : IsStarProjection p) (hq : IsStarProjection q) :\n    p ≤ q ↔ q * p = p :=\n  hp.le_tfae hq |>.out 0 1\n\nlemma le_iff_mul_eq_left (hp : IsStarProjection p) (hq : IsStarProjection q) :\n    p ≤ q ↔ p * q = p :=\n  hp.le_tfae hq |>.out 0 2\n\nlemma le_iff_sub (hp : IsStarProjection p) (hq : IsStarProjection q) :\n    p ≤ q ↔ IsStarProjection (q - p) :=\n  hp.le_tfae hq |>.out 0 3\n\nlemma le_iff_idempotent_sub (hp : IsStarProjection p) (hq : IsStarProjection q) :\n    p ≤ q ↔ IsIdempotentElem (q - p) :=\n  hp.le_tfae hq |>.out 0 4\n\nTarget:\nlemma commute_of_le (hp : IsStarProjection p) (hq : IsStarProjection q) (h : p ≤ q) :\n    Commute p q :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"d86ce97fb9dbeb6d52941debacee15a7914437ecfd602b701b6d053181cf4eca","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/CStarAlgebra","family_id":"commute_of_le","file_id":"mathlib/Mathlib/Analysis/CStarAlgebra/Projection.lean","sample_id":"9c77b43c796b9a66a9282957c2b67d46b27f3beda4c44634fa70becc056cd2ca"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2f2eac7accbcc17b05097bd1c9316c1f6c3d52ca67232fbfd3c0c187a0e1b260","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b373b3ff8a6dcb9dbdb8440546b2bd1badb9717022672d83027f9ba298e428d7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"01224b14a9dc704c2fd71fd678b713d03a65ff9d6ee15bc743185909d5552045","source_sha256":"cee08e35f7a82853091be31ffd641ba79c118d64e6ff9753c07e342b11ecbe8e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← le_div_iff₀ pi_div_two_pos, div_eq_inv_mul (W k) _]\n  rw [W_eq_integral_sin_pow_div_integral_sin_pow, le_div_iff₀ (integral_sin_pow_pos _)]\n  convert! integral_sin_pow_succ_le (2 * k + 1)\n  rw [integral_sin_pow (2 * k)]\n  simp","hard_negative":true,"metrics":{"chosen_tokens":46,"rejected_tokens":3,"token_jaccard":0.035714,"token_length_ratio":0.065217},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"68f4916afcb2098ea0252d2851954499d6aa9a027e4102ae335bfc3e7e7eddae","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Integrals.Basic\n\nNamespace:\nReal.Wallis\n\nLocal context:\n/-\nCopyright (c) 2021 Hanting Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Hanting Zhang\n-/\n/-! # The Wallis formula for Pi\n\nThis file establishes the Wallis product for `π` (`Real.tendsto_prod_pi_div_two`). Our proof is\nlargely about analyzing the behaviour of the sequence `∫ x in 0..π, sin x ^ n` as `n → ∞`.\nSee: https://en.wikipedia.org/wiki/Wallis_product\n\nThe proof can be broken down into two pieces. The first step (carried out in\n`Mathlib/Analysis/SpecialFunctions/Integrals/Basic.lean`) is to use repeated integration by parts to\nobtain an explicit formula for this integral, which is rational if `n` is odd and a rational\nmultiple of `π` if `n` is even.\n\nThe second step, carried out here, is to estimate the ratio\n`∫ (x : ℝ) in 0..π, sin x ^ (2 * k + 1) / ∫ (x : ℝ) in 0..π, sin x ^ (2 * k)` and prove that\nit converges to one using the squeeze theorem. The final product for `π` is obtained after some\nalgebraic manipulation.\n\n## Main statements\n\n* `Real.Wallis.W`: the product of the first `k` terms in Wallis' formula for `π`.\n* `Real.Wallis.W_eq_integral_sin_pow_div_integral_sin_pow`: express `W n` as a ratio of integrals.\n* `Real.Wallis.W_le` and `Real.Wallis.le_W`: upper and lower bounds for `W n`.\n* `Real.tendsto_prod_pi_div_two`: the Wallis product formula.\n-/\n\n@[expose] public section\n\n\nopen scoped Real Topology Nat\n\nopen Filter Finset intervalIntegral\n\nnamespace Real\n\nnamespace Wallis\n\n\n/-- The product of the first `k` terms in Wallis' formula for `π`. -/\nnoncomputable def W (k : ℕ) : ℝ :=\n  ∏ i ∈ range k, (2 * i + 2) / (2 * i + 1) * ((2 * i + 2) / (2 * i + 3))\n\ntheorem W_succ (k : ℕ) :\n    W (k + 1) = W k * ((2 * k + 2) / (2 * k + 1) * ((2 * k + 2) / (2 * k + 3))) :=\n  prod_range_succ _ _\n\ntheorem W_pos (k : ℕ) : 0 < W k := by\n  induction k with\n  | zero => unfold W; simp\n  | succ k hk =>\n    rw [W_succ]\n    refine mul_pos hk (mul_pos (div_pos ?_ ?_) (div_pos ?_ ?_)) <;> positivity\n\ntheorem W_eq_factorial_ratio (n : ℕ) :\n    W n = 2 ^ (4 * n) * n ! ^ 4 / ((2 * n)! ^ 2 * (2 * n + 1)) := by\n  induction n with\n  | zero =>\n    simp only [W, prod_range_zero, Nat.factorial_zero, mul_zero, pow_zero]\n    norm_num\n  | succ n IH =>\n    unfold W at IH ⊢\n    rw [prod_range_succ, IH, _root_.div_mul_div_comm, _root_.div_mul_div_comm]\n    refine (div_eq_div_iff ?_ ?_).mpr ?_\n    any_goals exact ne_of_gt (by positivity)\n    simp_rw [Nat.mul_succ, Nat.factorial_succ, pow_succ]\n    push_cast\n    ring_nf\n\ntheorem W_eq_integral_sin_pow_div_integral_sin_pow (k : ℕ) : (π / 2)⁻¹ * W k =\n    (∫ x : ℝ in 0..π, sin x ^ (2 * k + 1)) / ∫ x : ℝ in 0..π, sin x ^ (2 * k) := by\n  rw [integral_sin_pow_even, integral_sin_pow_odd, mul_div_mul_comm, ← prod_div_distrib, inv_div]\n  simp_rw [div_div_div_comm, div_div_eq_mul_div, mul_div_assoc]\n  rfl\n\ntheorem W_le (k : ℕ) : W k ≤ π / 2 := by\n  rw [← div_le_one pi_div_two_pos, div_eq_inv_mul]\n  rw [W_eq_integral_sin_pow_div_integral_sin_pow, div_le_one (integral_sin_pow_pos _)]\n  apply integral_sin_pow_succ_le\n\nTarget:\ntheorem le_W (k : ℕ) : ((2 : ℝ) * k + 1) / (2 * k + 2) * (π / 2) ≤ W k :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_01224b14a9dc","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d07ddd786dcbe9c2f4a38ffa0a10f91217cb68af13e15bff0af81a4e5bd42ca4","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Real","family_id":"le_w","file_id":"mathlib/Mathlib/Analysis/Real/Pi/Wallis.lean","sample_id":"01224b14a9dc704c2fd71fd678b713d03a65ff9d6ee15bc743185909d5552045"},"split":"train"}
{"binding":{"chosen_proof_sha256":"de4e82cdf62623b5f6eb51d83e7f17e091632bf91d46d2109ff7b01e77e7f85f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8ba782304e1ea50b35011d546ef6e91a73975487ffa818be72dbad863f642987","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"143ab2863e5ef8203ad9cdb078d9a96faf3dccd3f496872d4f958e2b6081b962","source_sha256":"d6e838a5f1b3817af8820eb123fb16847eef4259153bc8aa4506ab114a1382bf","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hinj : Function.Injective (algebraMap F[X] K) := by\n    rw [IsScalarTower.algebraMap_eq F[X] F⟮X⟯ K]\n    exact (algebraMap F⟮X⟯ K).injective.comp (IsFractionRing.injective F[X] F⟮X⟯)\n  rw [injective_iff_map_eq_zero (algebraMap F[X] (↥(ringOfIntegers F K)))]\n  intro p hp\n  rw [← Subtype.coe_inj, Subalgebra.coe_zero] at hp\n  rw [injective_iff_map_eq_zero (algebraMap F[X] K)] at hinj\n  exact hinj p hp","hard_negative":true,"metrics":{"chosen_tokens":111,"rejected_tokens":3,"token_jaccard":0.051282,"token_length_ratio":0.027027},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"68fd92fdd2ad301daa925c724aa59b9c9e4419553a2e801718d641cdc8d877a7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.RatFunc.Degree\npublic import Mathlib.RingTheory.DedekindDomain.IntegralClosure\npublic import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed\npublic import Mathlib.Topology.Algebra.Valued.ValuedField\npublic import Mathlib.Topology.Algebra.InfiniteSum.Defs\npublic import Mathlib.FieldTheory.RatFunc.IntermediateField\npublic import Mathlib.RingTheory.Adjoin.Polynomial.Bivariate\npublic import Mathlib.FieldTheory.RatFunc.Valuation -- for deprecation to `RatFunc.inftyValuation` and `RatFunc.CompletionAtInfty`\n\nNamespace:\nFunctionField.ringOfIntegers\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen, Ashvni Narayanan\n-/\n/-!\n# Function fields\n\nThis file defines a function field and the ring of integers corresponding to it.\n\n## Main definitions\n\n- `FunctionField F K` states that `K` is a function field over the field `F`,\n  i.e. it is a finite extension of the field of rational functions in one variable over `F`.\n- `FunctionField.ringOfIntegers` defines the ring of integers corresponding to a function field\n  as the integral closure of `F[X]` in the function field.\n\n## Implementation notes\nThe definitions that involve a field of fractions choose a canonical field of fractions,\nbut are independent of that choice. We also omit assumptions like\n`IsScalarTower F[X] (FractionRing F[X]) K` in definitions,\nadding them back in lemmas when they are needed.\n\n## References\n* [D. Marcus, *Number Fields*][marcus1977number]\n* [J.W.S. Cassels, A. Fröhlich, *Algebraic Number Theory*][cassels1967algebraic]\n* [P. Samuel, *Algebraic Theory of Numbers*][samuel1967]\n* [M. Rosen, *Number Theory in Function Fields*][rosen2002]\n\n## Tags\nfunction field, ring of integers\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen scoped nonZeroDivisors Polynomial WithZero RatFunc\n\nvariable (F K : Type*) [Field F] [Field K]\n\n/-- `K` is a function field over the field `F` if it is a finite\nextension of the field of rational functions in one variable over `F`.\n\nNote that `K` can be a function field over multiple, non-isomorphic, `F`.\n-/\nabbrev FunctionField [Algebra F⟮X⟯ K] : Prop :=\n  FiniteDimensional F⟮X⟯ K\n\n/-- `K` is a function field over `F` iff it is a finite extension of `F(t)`. -/\ntheorem functionField_iff (Ft : Type*) [Field Ft] [Algebra F[X] Ft]\n    [IsFractionRing F[X] Ft] [Algebra F⟮X⟯ K] [Algebra Ft K] [Algebra F[X] K]\n    [IsScalarTower F[X] Ft K] [IsScalarTower F[X] F⟮X⟯ K] :\n    FunctionField F K ↔ FiniteDimensional Ft K := by\n  let e := IsLocalization.algEquiv F[X]⁰ F⟮X⟯ Ft\n  have : ∀ (c) (x : K), e c • x = c • x := by\n    intro c x\n    rw [Algebra.smul_def, Algebra.smul_def]\n    congr\n    refine congr_fun (f := fun c => algebraMap Ft K (e c)) ?_ c\n    refine IsLocalization.ext (nonZeroDivisors F[X]) _ _ ?_ ?_ ?_ ?_ ?_ <;> intros <;>\n      simp only [map_one, map_mul, AlgEquiv.commutes, ← IsScalarTower.algebraMap_apply]\n  constructor <;> intro h\n  · let b := Module.finBasis F⟮X⟯ K\n    exact (b.mapCoeffs e this).finiteDimensional_of_finite\n  · let b := Module.finBasis Ft K\n    refine (b.mapCoeffs e.symm ?_).finiteDimensional_of_finite\n    intro c x; convert! (this (e.symm c) x).symm; simp only [e.apply_symm_apply]\n\nnamespace FunctionField\n\ntheorem algebraMap_injective [Algebra F[X] K] [Algebra F⟮X⟯ K]\n    [IsScalarTower F[X] F⟮X⟯ K] : Function.Injective (algebraMap F[X] K) := by\n  rw [IsScalarTower.algebraMap_eq F[X] F⟮X⟯ K]\n  exact (algebraMap F⟮X⟯ K).injective.comp (IsFractionRing.injective F[X] F⟮X⟯)\n\n/-- The function field analogue of `NumberField.ringOfIntegers`:\n`FunctionField.ringOfIntegers F K` is the integral closure of `F[X]` in `K`.\n\nWe don't actually assume `K` is a function field over `F` in the definition,\nonly when proving its properties.\n-/\ndef ringOfIntegers [Algebra F[X] K] :=\n  integralClosure F[X] K\n\nnamespace ringOfIntegers\n\nvariable [Algebra F[X] K]\n\ninstance : IsDomain (ringOfIntegers F K) :=\n  (ringOfIntegers F K).isDomain\n\ninstance : IsIntegralClosure (ringOfIntegers F K) F[X] K :=\n  integralClosure.isIntegralClosure _ _\n\nvariable [Algebra F⟮X⟯ K] [IsScalarTower F[X] F⟮X⟯ K]\n\nTarget:\ntheorem algebraMap_injective : Function.Injective (algebraMap F[X] (ringOfIntegers F K)) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_143ab2863e5e","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"292c27b988f5649ff312b116f57c1b3ef448808a632e250b862f34bf016acb19","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory","family_id":"algebramap_injective","file_id":"mathlib/Mathlib/NumberTheory/FunctionField.lean","sample_id":"143ab2863e5ef8203ad9cdb078d9a96faf3dccd3f496872d4f958e2b6081b962"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d93c674782daaf325f075697a98c3704ebb23b2b31510b6d8f8e1890f07f7bba","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b824477d2857f6ad9c20787126a537e7587db4bd2905b36a6db22f978b45743a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dac74140b5028292313ba8ba24d22052ad637c133b6e08b981669e5ff7752981","source_sha256":"5e794e43f06e38bed06c0789971ed6287919f753a812f6be7d0a2ba23c851951","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [noZeroSMulDivisors_iff, or_iff_not_imp_left]\n  exact ⟨fun h r hr m eq ↦ h eq hr, fun h r m eq hr ↦ h r hr m eq⟩","hard_negative":true,"metrics":{"chosen_tokens":33,"rejected_tokens":2,"token_jaccard":0.055556,"token_length_ratio":0.060606},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"690f14e8598a642816c7abeba4d718f382096d1cc402705a725b04455c72108a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Torsion.Free\npublic import Mathlib.Tactic.Contrapose\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2015 Nathaniel Thomas. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen, Yury Kudryashov, Joseph Myers, Heather Macbeth, Kim Morrison, Yaël Dillies\n-/\n/-!\n# `NoZeroSMulDivisors`\n\nThis file defines the `NoZeroSMulDivisors` class, and includes some tests\nfor the vanishing of elements (especially in modules over division rings).\n\n## Usage notes\n\nNote that `NoZeroSMulDivisors` is deprecated in favor of `Module.IsTorsionFree`, which is the\nmathematically correct generalisation to semimodules.\n-/\n\npublic section\n\nassert_not_exists RelIso Multiset Set.indicator Pi.single_smul₀\n\nvariable {R M G : Type*}\n\n/-- `NoZeroSMulDivisors R M` states that a scalar multiple is `0` only if either argument is `0`.\nThis is a version of saying that `M` is torsion free, without assuming `R` is zero-divisor free.\n\nThe main application of `NoZeroSMulDivisors R M`, when `M` is a module,\nis the result `smul_eq_zero`: a scalar multiple is `0` iff either argument is `0`.\n\nIt is a generalization of the `NoZeroDivisors` class to heterogeneous multiplication.\n-/\n@[mk_iff]\nclass NoZeroSMulDivisors (R M : Type*) [Zero R] [Zero M] [SMul R M] : Prop where\n  /-- If scalar multiplication yields zero, either the scalar or the vector was zero. -/\n  eq_zero_or_eq_zero_of_smul_eq_zero : ∀ {c : R} {x : M}, c • x = 0 → c = 0 ∨ x = 0\n\nexport NoZeroSMulDivisors (eq_zero_or_eq_zero_of_smul_eq_zero)\n\n/-- Pullback a `NoZeroSMulDivisors` instance along an injective function. -/\ntheorem Function.Injective.noZeroSMulDivisors {R M N : Type*} [Zero R] [Zero M] [Zero N]\n    [SMul R M] [SMul R N] [NoZeroSMulDivisors R N] (f : M → N) (hf : Function.Injective f)\n    (h0 : f 0 = 0) (hs : ∀ (c : R) (x : M), f (c • x) = c • f x) : NoZeroSMulDivisors R M :=\n  ⟨fun {_ _} h =>\n    Or.imp_right (@hf _ _) <| h0.symm ▸ eq_zero_or_eq_zero_of_smul_eq_zero (by rw [← hs, h, h0])⟩\n\n-- See note [lower instance priority]\ninstance (priority := 100) NoZeroDivisors.toNoZeroSMulDivisors [Zero R] [Mul R]\n    [NoZeroDivisors R] : NoZeroSMulDivisors R R :=\n  ⟨fun {_ _} => eq_zero_or_eq_zero_of_mul_eq_zero⟩\n\ninstance [Semiring R] [IsDomain R] [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] :\n    Module.IsTorsionFree R M where\n  isSMulRegular r hr m₁ m₂ hm := by\n    dsimp at hm\n    rw [← sub_eq_zero, ← smul_sub] at hm\n    simpa [hr.ne_zero, sub_eq_zero] using eq_zero_or_eq_zero_of_smul_eq_zero hm\n\nTarget:\ntheorem noZeroSMulDivisors_iff_right_eq_zero_of_smul [Zero R] [Zero M] [SMul R M] :\n    NoZeroSMulDivisors R M ↔ ∀ r : R, r ≠ 0 → ∀ m : M, r • m = 0 → m = 0 :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_dac74140b502","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e16d571c851e82e00549e48f917c69df3f57df4d1743b52c94e5dcf916423f57","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/NoZeroSMulDivisors","family_id":"nozerosmuldivisors_iff_right_eq_zero_of_smul","file_id":"mathlib/Mathlib/Algebra/NoZeroSMulDivisors/Defs.lean","sample_id":"dac74140b5028292313ba8ba24d22052ad637c133b6e08b981669e5ff7752981"},"split":"train"}
{"binding":{"chosen_proof_sha256":"471c71a8aa6e81210c7b7ea449c4bb32d4af4dfb79ecb5a4425fcd21336ff9fe","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a0a40b53d14535de5a165c9e14107454e6cc6b4afceef9775bdaaf0c31600cd0","source_sha256":"006922c150073d26eccce5bef3353d7015b01e87320ecec635275017a27177d4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [ncoeff, HVertexOperator.coeff, LinearMap.coe_mk, AddHom.coe_mk]\n  exact HahnSeries.coeff_eq_zero_of_lt_order h","hard_negative":false,"metrics":{"chosen_tokens":23,"rejected_tokens":2,"token_jaccard":0.055556,"token_length_ratio":0.086957},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"6922f341ebd1f0184ca530051e47869bb1a13748a9acc47b35d422683bc70f33","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Vertex.HVertexOperator\npublic import Mathlib.Data.Int.Interval\n\nNamespace:\nVertexOperator\n\nLocal context:\n/-\nCopyright (c) 2024 Scott Carnahan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Carnahan\n-/\n/-!\n# Vertex operators\nIn this file we introduce vertex operators as linear maps to Laurent series.\n\n## Definitions\n* `VertexOperator` is an `R`-linear map from an `R`-module `V` to `LaurentSeries V`.\n* `VertexOperator.ncoeff` is the coefficient of a vertex operator under normalized indexing.\n\n## TODO\n* `HasseDerivative` : A divided-power derivative.\n* `Locality` : A weak form of commutativity.\n* `Residue products` : A family of products on `VertexOperator R V` parametrized by integers.\n\n## References\n* [G. Mason, *Vertex rings and Pierce bundles*][mason2017]\n* [A. Matsuo, K. Nagatomo, *On axioms for a vertex algebra and locality of quantum\n  fields*][matsuo1997]\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nvariable {R V : Type*} [CommRing R] [AddCommGroup V] [Module R V]\n\n/-- A vertex operator over a commutative ring `R` is an `R`-linear map from an `R`-module `V` to\nLaurent series with coefficients in `V`.  We write this as a specialization of the heterogeneous\ncase. -/\nabbrev VertexOperator (R : Type*) (V : Type*) [CommRing R] [AddCommGroup V]\n    [Module R V] := HVertexOperator ℤ R V V\n\nnamespace VertexOperator\n\nopen HVertexOperator\n\n@[ext]\ntheorem ext (A B : VertexOperator R V) (h : ∀ v : V, A v = B v) :\n    A = B := LinearMap.ext h\n\n/-- The coefficient of a vertex operator under normalized indexing. -/\ndef ncoeff : VertexOperator R V →ₗ[R] ℤ → Module.End R V where\n  toFun A n := HVertexOperator.coeff A (-n - 1)\n  map_add' _ _ := by ext; simp\n  map_smul' _ _ := by ext; simp\n\ntheorem ncoeff_apply (A : VertexOperator R V) (n : ℤ) : ncoeff A n = coeff A (-n - 1) :=\n  rfl\n\n/-- In the literature, the `n`th normalized coefficient of a vertex operator `A` is written as\neither `Aₙ` or `A(n)`. -/\nscoped[VertexOperator] notation A \"[[\" n \"]]\" => ncoeff A n\n\n@[simp]\ntheorem coeff_eq_ncoeff (A : VertexOperator R V)\n    (n : ℤ) : HVertexOperator.coeff A n = A[[-n - 1]] := by\n  rw [ncoeff_apply, neg_sub, Int.sub_neg, add_sub_cancel_left]\n\nTarget:\ntheorem ncoeff_eq_zero_of_lt_order (A : VertexOperator R V) (n : ℤ) (x : V)\n    (h : -n - 1 < HahnSeries.order ((HahnModule.of R).symm (A x))) : (A[[n]]) x = 0 :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Vertex","family_id":"ncoeff_eq_zero_of_lt_order","file_id":"mathlib/Mathlib/Algebra/Vertex/VertexOperator.lean","sample_id":"a0a40b53d14535de5a165c9e14107454e6cc6b4afceef9775bdaaf0c31600cd0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"46daf5b2e7bf4d90e3013b1a193808b8c73aa5defa9de5956e0736e8e4908309","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9439016a6f4e6a57c85536640298da69f3a589358dd7d886fc77db2d3daea612","source_sha256":"18b2ea13d158ef44c4da5d71e0384f4a95c10d686d0e4d8e534d2cf89fc900ba","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply of_span_eq_top_target _ hs\n  rintro - ⟨i, rfl⟩\n  exact .equiv <| (IsLocalization.algEquiv (.powers <| s i) _ (T i)).symm |>.restrictScalars R","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":8,"token_jaccard":0.029412,"token_length_ratio":0.190476},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"698779a8993fc8c1ef6b8bc616dd712300b2818803d8d409e07e425853219289","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Finiteness.FiniteTypeLocal\npublic import Mathlib.RingTheory.Localization.Away.AdjoinRoot\n\nNamespace:\nAlgebra.FinitePresentation\n\nLocal context:\n/-\nCopyright (c) 2024 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten\n-/\n/-!\n\n# `Algebra.FinitePresentation` is local\n\nIn this file we show that being a finitely presented algebra is local.\n\n## Main results\n\n- `Algebra.FinitePresentation.of_span_eq_top_target`: finite presentation is local on the\n  (algebraic) target\n\n-/\n\npublic section\n\nopen scoped Pointwise TensorProduct\n\nnamespace Algebra.FinitePresentation\n\nvariable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]\n\n/--\nIf `S` is an `R`-algebra with a surjection from a finitely-presented `R`-algebra `A`, such that\nlocalized at a spanning set `{ r }` of elements of `A`, `Sᵣ` is finitely-presented, then\n`S` is finitely presented.\nThis is almost `finitePresentation_ofLocalizationSpanTarget`. The difference is,\nthat here the set `t` generates the unit ideal of `A`, while in the general version,\nit only generates a quotient of `A`.\n-/\nlemma of_span_eq_top_target_aux {A : Type*} [CommRing A] [Algebra R A]\n    [Algebra.FinitePresentation R A] (f : A →ₐ[R] S) (hf : Function.Surjective f)\n    (t : Finset A) (ht : Ideal.span (t : Set A) = ⊤)\n    (H : ∀ g : t, Algebra.FinitePresentation R (Localization.Away (f g))) :\n    Algebra.FinitePresentation R S := by\n  apply Algebra.FinitePresentation.of_surjective hf\n  apply RingHom.ker_fg_of_localizationSpan t ht\n  intro g\n  let f' : Localization.Away g.val →ₐ[R] Localization.Away (f g) :=\n    Localization.awayMapₐ f g.val\n  have (g : t) : Algebra.FinitePresentation R (Localization.Away g.val) :=\n    haveI : Algebra.FinitePresentation A (Localization.Away g.val) :=\n      IsLocalization.Away.finitePresentation g.val\n    Algebra.FinitePresentation.trans R A (Localization.Away g.val)\n  apply Algebra.FinitePresentation.ker_fG_of_surjective f'\n  exact IsLocalization.Away.mapₐ_surjective_of_surjective _ hf\n\nuniverse u\n\n/-- Finite-presentation can be checked on a standard covering of the target. -/\nlemma of_span_eq_top_target (s : Set S) (hs : Ideal.span (s : Set S) = ⊤)\n    (h : ∀ i ∈ s, Algebra.FinitePresentation R (Localization.Away i)) :\n    Algebra.FinitePresentation R S := by\n  obtain ⟨s, h₁, hs⟩ := (Ideal.span_eq_top_iff_finite s).mp hs\n  replace h (i : s) : Algebra.FinitePresentation R (Localization.Away i.val) := h i (h₁ i.property)\n  classical\n  /-\n  We already know that `S` is of finite type over `R`, so we have a surjection\n  `MvPolynomial (Fin n) R →ₐ[R] S`. To reason about the kernel, we want to check it on the stalks\n  of preimages of `s`. But the preimages do not necessarily span `MvPolynomial (Fin n) R`, so\n  we quotient out by an ideal and apply `finitePresentation_ofLocalizationSpanTarget_aux`.\n  -/\n  have hfintype : Algebra.FiniteType R S := by\n    apply Algebra.FiniteType.of_span_eq_top_target s hs\n    intro x hx\n    have := h ⟨x, hx⟩\n    infer_instance\n  obtain ⟨n, f, hf⟩ := Algebra.FiniteType.iff_quotient_mvPolynomial''.mp hfintype\n  obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_linearCombination S (s : Set S) 1).mp\n      (show (1 : S) ∈ Ideal.span (s : Set S) by rw [hs]; trivial)\n  choose g' hg' using (fun g : s ↦ hf g)\n  choose h' hh' using (fun g : s ↦ hf (l g))\n  let I : Ideal (MvPolynomial (Fin n) R) := Ideal.span { ∑ g : s, g' g * h' g - 1 }\n  let A := MvPolynomial (Fin n) R ⧸ I\n  have hfI : ∀ a ∈ I, f a = 0 := by\n    intro p hp\n    simp only [Finset.univ_eq_attach, I, Ideal.mem_span_singleton] at hp\n    obtain ⟨q, rfl⟩ := hp\n    simp only [map_mul, map_sub, map_sum, map_one, hg', hh']\n    rw [Finsupp.linearCombination_apply_of_mem_supported (α := (s : Set S)) S (s := s.attach)] at hl\n    · rw [← hl]\n      simp only [Finset.coe_sort_coe, smul_eq_mul, mul_comm, sub_self, zero_mul]\n    · rintro a -\n      simp\n  let f' : A →ₐ[R] S := Ideal.Quotient.liftₐ I f hfI\n  have hf' : Function.Surjective f' :=\n    Ideal.Quotient.lift_surjective_of_surjective I hfI hf\n  let t : Finset A := Finset.image (fun g ↦ g' g) Finset.univ\n  have ht : Ideal.span (t : Set A) = ⊤ := by\n    rw [Ideal.eq_top_iff_one]\n    have : ∑ g : { x // x ∈ s }, g' g * h' g = (1 : A) := by\n      apply eq_of_sub_eq_zero\n      rw [← map_one (Ideal.Quotient.mk I), ← map_sub, Ideal.Quotient.eq_zero_iff_mem]\n      apply Ideal.subset_span\n      simp\n    simp_rw [← this, Finset.univ_eq_attach, map_sum, map_mul]\n    refine Ideal.sum_mem _ (fun g _ ↦ Ideal.mul_mem_right _ _ <| Ideal.subset_span ?_)\n    simp [t]\n  have : Algebra.FinitePresentation R A := by\n    apply Algebra.FinitePresentation.quotient\n    simp only [Finset.univ_eq_attach, I]\n    exact ⟨{∑ g ∈ s.attach, g' g * h' g - 1}, by simp⟩\n  have Ht (g : t) : Algebra.FinitePresentation R (Localization.Away (f' g)) := by\n    have : ∃ (a : S) (hb : a ∈ s), (Ideal.Quotient.mk I) (g' ⟨a, hb⟩) = g.val := by\n      obtain ⟨g, hg⟩ := g\n      convert! hg\n      simp [A, t]\n    obtain ⟨r, hr, hrr⟩ := this\n    simp only [f']\n    rw [← hrr, Ideal.Quotient.liftₐ_apply, Ideal.Quotient.lift_mk]\n    simp_rw +instances [RingHom.coe_coe]\n    rw [hg']\n    apply h\n  exact of_span_eq_top_target_aux f' hf' t ht Ht\n\n/-- Finite-presentation can be checked on a standard covering of the target. -/\n\nTarget:\nlemma of_span_eq_top_target_of_isLocalizationAway {ι : Type*} (s : ι → S)\n    (hs : Ideal.span (Set.range s) = ⊤) (T : ι → Type*) [∀ i, CommRing (T i)] [∀ i, Algebra R (T i)]\n    [∀ i, Algebra S (T i)] [∀ i, IsScalarTower R S (T i)] [∀ i, IsLocalization.Away (s i) (T i)]\n    [∀ i, Algebra.FinitePresentation R (T i)] :\n    Algebra.FinitePresentation R S :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"bfd92c5e7fda4a0dd3ca7960f5ef17fe540591adb6ec4018c9b213f68f9f8b99","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Finiteness","family_id":"of_span_eq_top_target_of_islocalizationaway","file_id":"mathlib/Mathlib/RingTheory/Finiteness/FinitePresentationLocal.lean","sample_id":"9439016a6f4e6a57c85536640298da69f3a589358dd7d886fc77db2d3daea612"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c66eb3df5899011ef7909b5b28a8dbc7ead7f1cd7984dec7b597a92d174e6203","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bf0246d237d6d9933a85d317c9082d92e62a7779e9ca287e29ec9e8f9ef938b0","source_sha256":"696827f59141a7e073b768692576d4aa28b090a0ebc3937f46b2b10aa2472ecd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [SModEq.sub_mem] at hxy ⊢\n  rw [← smul_sub]\n  exact smul_mem_smul hc hxy","hard_negative":false,"metrics":{"chosen_tokens":19,"rejected_tokens":3,"token_jaccard":0.125,"token_length_ratio":0.157895},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"6995f95510fe39e14f87ea7517134b55811a4d8808f651a17caf3d4a0fc12265","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Operations\npublic import Mathlib.LinearAlgebra.SModEq.Basic\n\nNamespace:\nSModEq\n\nLocal context:\n/-\nCopyright (c) 2025 Jiedong Jiang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jiedong Jiang\n-/\n/-!\n# Pointwise lemmas for modular equivalence\n\nIn this file, we record more lemmas about `SModEq` on elements\nof modules or rings.\n-/\n\npublic section\n\nopen Submodule\n\nopen Polynomial\n\nvariable {R : Type*} [Ring R] {I : Ideal R}\nvariable {M : Type*} [AddCommGroup M] [Module R M] {U : Submodule R M}\nvariable {x y : M}\n\nnamespace SModEq\n\n/--\nA variant of `SModEq.smul`, where the scalar belongs to an ideal.\n-/\n\nTarget:\ntheorem smul' (hxy : x ≡ y [SMOD U])\n    {c : R} (hc : c ∈ I) : c • x ≡ c • y [SMOD (I • U)] :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/SModEq","family_id":"smul","file_id":"mathlib/Mathlib/LinearAlgebra/SModEq/Pointwise.lean","sample_id":"bf0246d237d6d9933a85d317c9082d92e62a7779e9ca287e29ec9e8f9ef938b0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4854fca9d1b2dda857b810abc21527817558defad53eaaee4f2088331645faca","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"506ccf9dfd99be292019dce873814f6f387afab1a0c094bdbb7940416b1f4d0c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"063710952a3a4d125df5a3e180296e39817ba1f9ff1c0061323053fb46b49fc4","source_sha256":"da03aef6be316c0f1e4d46eb4e8f0617ba6def9cf6bbead1497df151ed7bf988","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by cases x; rfl\n\nnonrec def ReaderT.callCC {ε} [MonadCont m] {α β : Type _}\n    (f : Label α (ReaderT ε m) β → ReaderT ε m α) : ReaderT ε m α :=\n  ReaderT.mk (fun r => callCC fun f' => (f <| ReaderT.mkLabel _ f').run r)","hard_negative":true,"metrics":{"chosen_tokens":72,"rejected_tokens":5,"token_jaccard":0.076923,"token_length_ratio":0.069444},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"69b0652e12395bee9c58c776dd46abe21cc386066abad279daba36ae579c072b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Control.Monad.Basic\npublic import Mathlib.Control.Monad.Writer\npublic import Mathlib.Control.Lawful\npublic import Batteries.Tactic.Congr\npublic import Batteries.Lean.Except\nimport all Init.Control.Option  -- for unfolding `Option.lift`\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\n/-!\n# Continuation Monad\n\nMonad encapsulating continuation passing programming style, similar to\nHaskell's `Cont`, `ContT` and `MonadCont`:\n<https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Cont.html>\n<https://hackage.haskell.org/package/transformers-0.6.2.0/docs/Control-Monad-Trans-Cont.html>\n-/\n\n@[expose] public section\n\nuniverse u v w u₀ u₁ v₀ v₁\n\nstructure MonadCont.Label (α : Type w) (m : Type u → Type v) (β : Type u) where\n  apply : α → m β\n\nabbrev MonadCont.goto {α β} {m : Type u → Type v} (f : MonadCont.Label α m β) (x : α) :=\n  f.apply x\n\nclass MonadCont (m : Type u → Type v) where\n  callCC : ∀ {α β}, (MonadCont.Label α m β → m α) → m α\n\nopen MonadCont\n\nclass LawfulMonadCont (m : Type u → Type v) [Monad m] [MonadCont m] : Prop\n    extends LawfulMonad m where\n  callCC_bind_right {α ω γ} (cmd : m α) (next : Label ω m γ → α → m ω) :\n    (callCC fun f => cmd >>= next f) = cmd >>= fun x => callCC fun f => next f x\n  callCC_bind_left {α} (β) (x : α) (dead : Label α m β → β → m α) :\n    (callCC fun f : Label α m β => goto f x >>= dead f) = pure x\n  callCC_dummy {α β} (dummy : m α) : (callCC fun _ : Label α m β => dummy) = dummy\n\nexport LawfulMonadCont (callCC_bind_right callCC_bind_left callCC_dummy)\n\ndef ContT (r : Type u) (m : Type u → Type v) (α : Type w) :=\n  (α → m r) → m r\n\nabbrev Cont (r : Type u) (α : Type w) :=\n  ContT r Id α\n\nnamespace ContT\n\nexport MonadCont (Label goto)\n\nvariable {r : Type u} {m : Type u → Type v} {α β : Type w}\n\n/-- Build a `ContT` from a function taking a continuation callback. -/\ndef mk (f : (α → m r) → m r) : ContT r m α := f\n\n/-- Run a `ContT` with a provided callback. -/\ndef run (x : ContT r m α) : (α → m r) → m r := x\n\ndef map (f : m r → m r) (x : ContT r m α) : ContT r m α :=\n  f ∘ x\n\ntheorem run_contT_map_contT (f : m r → m r) (x : ContT r m α) : run (map f x) = f ∘ run x :=\n  rfl\n\ndef withContT (f : (β → m r) → α → m r) (x : ContT r m α) : ContT r m β := fun g => x <| f g\n\ntheorem run_withContT (f : (β → m r) → α → m r) (x : ContT r m α) :\n    run (withContT f x) = run x ∘ f :=\n  rfl\n\n@[ext]\nprotected theorem ext {x y : ContT r m α} (h : ∀ f, x.run f = y.run f) : x = y := by\n  unfold ContT; ext; apply h\n\ninstance : Monad (ContT r m) where\n  pure x f := f x\n  bind x f g := x fun i => f i g\n\n@[simp]\ntheorem run_mk (f : (α → m r) → m r) (k : α → m r) : (.mk f : ContT r m α).run k = f k := rfl\n\n@[simp]\ntheorem run_pure (a : α) (k : α → m r) : (pure a : ContT r m α).run k = k a := rfl\n\n@[simp]\ntheorem run_bind (x : ContT r m α) (f : α → ContT r m β) (k : β → m r) :\n    (x >>= f).run k = x.run fun x => (f x).run k := rfl\n\n@[simp]\ntheorem run_map (f : α → β) (x : ContT r m α) (k : β → m r) :\n    (f <$> x).run k = x.run (k ∘ f) := rfl\n\n@[simp]\ntheorem run_seq (f : ContT r m (α → β)) (x : ContT r m α) (k : β → m r) :\n    (f <*> x).run k = f.run fun f => x.run (k ∘ f) := rfl\n\n@[simp]\ntheorem run_seqLeft (x : ContT r m α) (y : ContT r m β) (k : α → m r) :\n    (x <* y).run k = x.run fun x => y.run fun _ => k x := rfl\n\n@[simp]\ntheorem run_seqRight (x : ContT r m α) (y : ContT r m β) (k : β → m r) :\n    (x *> y).run k = x.run fun _ => y.run k := rfl\n\ninstance : LawfulMonad (ContT r m) := LawfulMonad.mk'\n  (id_map := by intros; rfl)\n  (pure_bind := by intros; ext; rfl)\n  (bind_assoc := by intros; ext; rfl)\n\ninstance [Monad m] : MonadLift m (ContT r m) where\n  monadLift x := .mk fun k => x >>= k\n\n@[simp]\ntheorem run_monadLift [Monad m] {α} (x : m α) (k : α → m r) :\n    (monadLift x : ContT r m α).run k = x >>= k := rfl\n\ntheorem monadLift_bind [Monad m] [LawfulMonad m] {α β} (x : m α) (f : α → m β) :\n    (monadLift (x >>= f) : ContT r m β) = monadLift x >>= monadLift ∘ f := by\n  ext\n  simp only [bind_assoc, run_bind, run_monadLift, Function.comp_apply]\n\ninstance : MonadCont (ContT r m) where\n  callCC f := .mk fun k => f ⟨fun x => .mk fun _ => k x⟩ k\n\n@[simp]\ntheorem run_callCC (f : Label α (ContT r m) β → ContT r m α) (k : α → m r) :\n    (callCC f).run k = (f ⟨fun x => .mk fun _ => k x⟩).run k := rfl\n\ninstance : LawfulMonadCont (ContT r m) where\n  callCC_bind_right := by intros; ext; rfl\n  callCC_bind_left := by intros; ext; rfl\n  callCC_dummy := by intros; ext; rfl\n\n/-- Note that `tryCatch` does not have correct behavior in this monad:\n```\ndef foo : ContT Bool (Except String) Bool := do\n  let x ← try\n    pure true\n  catch _ =>\n    return false\n  throw s!\"oh no {x}\"\n#eval foo.run pure\n-- `Except.ok false`, no error\n```\nHere, the `throwError` is being run inside the `try`.\nSee [Zulip](https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/MonadExcept.20in.20the.20ContT.20monad/near/375341221)\nfor further discussion.\n-/\ninstance (ε) [MonadExceptOf ε m] : MonadExceptOf ε (ContT r m) where\n  throw e := .mk fun _ => throw e\n  tryCatch act h := .mk fun k => tryCatch (act.run k) fun e => (h e).run k\n\n@[simp]\ntheorem run_throw {ε} [MonadExceptOf ε m]\n    (e : ε) (f : α → m r) :\n    (throw e : ContT r m α).run f = throw e := rfl\n\n@[simp]\ntheorem run_tryCatch {ε} [MonadExceptOf ε m]\n    (act : ContT r m α) (h : ε → ContT r m α) (f : α → m r) :\n    (tryCatch act h : ContT r m α).run f = tryCatch (act.run f) fun e => (h e).run f := rfl\n\nend ContT\n\nvariable {m : Type u → Type v}\n\nsection\nvariable [Monad m]\n\ndef ExceptT.mkLabel {α β ε} : Label (Except.{u, u} ε α) m β → Label α (ExceptT ε m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (Except.ok a)⟩\n\ntheorem ExceptT.goto_mkLabel {α β ε : Type _} (x : Label (Except.{u, u} ε α) m β) (i : α) :\n    goto (ExceptT.mkLabel x) i = ExceptT.mk (Except.ok <$> goto x (Except.ok i)) := by\n  cases x; rfl\n\nnonrec def ExceptT.callCC {ε} [MonadCont m] {α β : Type _}\n    (f : Label α (ExceptT ε m) β → ExceptT ε m α) : ExceptT ε m α :=\n  ExceptT.mk (callCC fun x : Label _ m β => ExceptT.run <| f (ExceptT.mkLabel x))\n\ninstance {ε} [MonadCont m] : MonadCont (ExceptT ε m) where\n  callCC := ExceptT.callCC\n\ninstance {ε} [MonadCont m] [LawfulMonadCont m] : LawfulMonadCont (ExceptT ε m) where\n  callCC_bind_right := by\n    intros; simp only [callCC, ExceptT.callCC, ExceptT.run_bind, callCC_bind_right]; ext\n    dsimp\n    congr with ⟨⟩ <;> simp [@callCC_dummy m _]\n  callCC_bind_left := by\n    intros\n    simp only [callCC, ExceptT.callCC, ExceptT.goto_mkLabel, map_eq_bind_pure_comp, Function.comp,\n      ExceptT.run_bind, ExceptT.run_mk, bind_assoc, pure_bind, @callCC_bind_left m _]\n    ext; rfl\n  callCC_dummy := by intros; simp only [callCC, ExceptT.callCC, @callCC_dummy m _]; ext; rfl\n\ndef OptionT.mkLabel {α β} : Label (Option.{u} α) m β → Label α (OptionT m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (some a)⟩\n\ntheorem OptionT.goto_mkLabel {α β : Type _} (x : Label (Option.{u} α) m β) (i : α) :\n    goto (OptionT.mkLabel x) i = OptionT.mk (goto x (some i) >>= fun a => pure (some a)) :=\n  (rfl)\n\nnonrec def OptionT.callCC [MonadCont m] {α β : Type _} (f : Label α (OptionT m) β → OptionT m α) :\n    OptionT m α :=\n  OptionT.mk (callCC fun x : Label _ m β => OptionT.run <| f (OptionT.mkLabel x) : m (Option α))\n\n@[simp]\nlemma run_callCC [MonadCont m] {α β : Type _} (f : Label α (OptionT m) β → OptionT m α) :\n    (OptionT.callCC f).run = (callCC fun x => OptionT.run <| f (OptionT.mkLabel x)) := (rfl)\n\ninstance [MonadCont m] : MonadCont (OptionT m) where\n  callCC := OptionT.callCC\n\ninstance [MonadCont m] [LawfulMonadCont m] : LawfulMonadCont (OptionT m) where\n  callCC_bind_right := by\n    refine fun _ _ => OptionT.ext ?_\n    simpa [callCC, Option.elimM, callCC_bind_right] using\n      bind_congr fun | some _ => rfl | none => by simp [@callCC_dummy m _]\n  callCC_bind_left := by\n    intros\n    ext\n    simp [callCC, OptionT.goto_mkLabel, @callCC_bind_left m _]\n  callCC_dummy := by intros; ext; simp [callCC, OptionT.callCC, @callCC_dummy m _]\n\ndef WriterT.mkLabel {α β ω} [EmptyCollection ω] : Label (α × ω) m β → Label α (WriterT ω m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (a, ∅)⟩\n\ndef WriterT.mkLabel' {α β ω} [Monoid ω] : Label (α × ω) m β → Label α (WriterT ω m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (a, 1)⟩\n\ntheorem WriterT.goto_mkLabel {α β ω : Type _} [EmptyCollection ω] (x : Label (α × ω) m β) (i : α) :\n    goto (WriterT.mkLabel x) i = monadLift (goto x (i, ∅)) := by cases x; rfl\n\ntheorem WriterT.goto_mkLabel' {α β ω : Type _} [Monoid ω] (x : Label (α × ω) m β) (i : α) :\n    goto (WriterT.mkLabel' x) i = monadLift (goto x (i, 1)) := by cases x; rfl\n\nnonrec def WriterT.callCC [MonadCont m] {α β ω : Type _} [EmptyCollection ω]\n    (f : Label α (WriterT ω m) β → WriterT ω m α) : WriterT ω m α :=\n  WriterT.mk <| callCC (WriterT.run ∘ f ∘ WriterT.mkLabel : Label (α × ω) m β → m (α × ω))\n\ndef WriterT.callCC' [MonadCont m] {α β ω : Type _} [Monoid ω]\n    (f : Label α (WriterT ω m) β → WriterT ω m α) : WriterT ω m α :=\n  WriterT.mk <|\n    MonadCont.callCC (WriterT.run ∘ f ∘ WriterT.mkLabel' : Label (α × ω) m β → m (α × ω))\n\nend\n\ninstance (ω) [Monad m] [EmptyCollection ω] [MonadCont m] : MonadCont (WriterT ω m) where\n  callCC := WriterT.callCC\n\ninstance (ω) [Monad m] [Monoid ω] [MonadCont m] : MonadCont (WriterT ω m) where\n  callCC := WriterT.callCC'\n\ndef StateT.mkLabel {α β σ : Type u} : Label (α × σ) m (β × σ) → Label α (StateT σ m) β\n  | ⟨f⟩ => ⟨fun a => StateT.mk (fun s => f (a, s))⟩\n\ntheorem StateT.goto_mkLabel {α β σ : Type u} (x : Label (α × σ) m (β × σ)) (i : α) :\n    goto (StateT.mkLabel x) i = StateT.mk (fun s => goto x (i, s)) := by cases x; rfl\n\nnonrec def StateT.callCC {σ} [MonadCont m] {α β : Type _}\n    (f : Label α (StateT σ m) β → StateT σ m α) : StateT σ m α :=\n  StateT.mk (fun r => callCC fun f' => (f <| StateT.mkLabel f').run r)\n\ninstance {σ} [MonadCont m] : MonadCont (StateT σ m) where\n  callCC := StateT.callCC\n\ninstance {σ} [Monad m] [MonadCont m] [LawfulMonadCont m] : LawfulMonadCont (StateT σ m) where\n  callCC_bind_right := by\n    intros\n    simp only [callCC, StateT.callCC, StateT.run_bind, callCC_bind_right]; ext; rfl\n  callCC_bind_left := by\n    intros\n    simp only [callCC, StateT.callCC, StateT.goto_mkLabel, StateT.run_bind, StateT.run_mk,\n      callCC_bind_left]; ext; rfl\n  callCC_dummy := by\n    intros\n    simp only [callCC, StateT.callCC, @callCC_dummy m _]\n    ext; rfl\n\ndef ReaderT.mkLabel {α β} (ρ) : Label α m β → Label α (ReaderT ρ m) β\n  | ⟨f⟩ => ⟨monadLift ∘ f⟩\n\nTarget:\ntheorem ReaderT.goto_mkLabel {α ρ β} (x : Label α m β) (i : α) :\n    goto (ReaderT.mkLabel ρ x) i = monadLift (goto x i) :=\n\nProof body:\n","rejected":"by\n  exact ReaderT.goto_mkLabel","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"d489f6e26f60be43fa848a04ea14587c7a614fd6bbff11c171842c866d6688c4","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control/Monad","family_id":"readert","file_id":"mathlib/Mathlib/Control/Monad/Cont.lean","sample_id":"063710952a3a4d125df5a3e180296e39817ba1f9ff1c0061323053fb46b49fc4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2f0e35fce05624abb1b88e07b05ba59ec3da85bd198bf90cf2c93b8503cfca8a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4f3514ad322903372735ab2658d2f373073140704a1b691a7a6fd8affea56371","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"93a0ad9cdbe52633a2a6476e78bca93ac766df3668d5e36441bd4a7ce8340870","source_sha256":"15d1aa45ed96e60a321174c4449c6369f20349d72b788c380b869a678d1f4952","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let g : X ⟶ (⨆ i, U i : Y.Opens) := IsOpenImmersion.lift (Scheme.Opens.ι _) f (by simpa using H)\n  rw [← IsOpenImmersion.lift_fac (⨆ i, U i).ι f (by simpa using H)]\n  apply MorphismProperty.RespectsRight.postcomp (Q := @IsOpenImmersion) _ inferInstance\n  rw [iff_of_iSup_eq_top (P := P) (U := fun i : ι ↦ (⨆ i, U i).ι ⁻¹ᵁ U i)]\n  · intro i\n    have heq : g ⁻¹ᵁ (⨆ i, U i).ι ⁻¹ᵁ U i = f ⁻¹ᵁ U i := by\n      change (g ≫ (⨆ i, U i).ι) ⁻¹ᵁ U i = _\n      simp [g]\n    let e : Arrow.mk (g ∣_ (⨆ i, U i).ι ⁻¹ᵁ U i) ≅ Arrow.mk (f ∣_ U i) :=\n        Arrow.isoMk (X.isoOfEq heq) (Scheme.Opens.isoOfLE (le_iSup U i)) <| by\n      simp [← CategoryTheory.cancel_mono (U i).ι, g]\n    rw [P.arrow_mk_iso_iff e]\n    exact hf i\n  apply (⨆ i, U i).ι.image_injective\n  dsimp\n  rw [Scheme.Hom.image_iSup, Scheme.Hom.image_top_eq_opensRange, Scheme.Opens.opensRange_ι]\n  simp [Scheme.Hom.image_preimage_eq_opensRange_inf, le_iSup U]","hard_negative":true,"metrics":{"chosen_tokens":294,"rejected_tokens":3,"token_jaccard":0.026316,"token_length_ratio":0.010204},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"69cf1ac5948ba337fa46dd69b6ae7453902383f185eae090a740b7c77981300d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Limits\npublic import Mathlib.CategoryTheory.MorphismProperty.Local\npublic import Mathlib.Data.List.TFAE\n\nNamespace:\nAlgebraicGeometry.IsZariskiLocalAtTarget\n\nLocal context:\n/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Properties of morphisms between Schemes\n\nWe provide the basic framework for talking about properties of morphisms between Schemes.\n\nA `MorphismProperty Scheme` is a predicate on morphisms between schemes. For properties local at\nthe target, its behaviour is entirely determined by its definition on morphisms into affine schemes,\nwhich we call an `AffineTargetMorphismProperty`. In this file, we provide API lemmas for properties\nlocal at the target, and special support for those properties whose `AffineTargetMorphismProperty`\ntakes on a simpler form. We also provide API lemmas for properties local at the source.\nThe main interfaces of the API are the typeclasses `IsZariskiLocalAtTarget`,\n`IsZariskiLocalAtSource` and `HasAffineProperty`, which we describe in detail below.\n\n## `IsZariskiLocalAtTarget`\n\n- `AlgebraicGeometry.IsZariskiLocalAtTarget`: We say that `IsZariskiLocalAtTarget P` for\n  `P : MorphismProperty Scheme` if\n  1. `P` respects isomorphisms.\n  2. `P` holds for `f ∣_ U` for an open cover `U` of `Y` if and only if `P` holds for `f`.\n\nFor a morphism property `P` local at the target and `f : X ⟶ Y`, we provide these API lemmas:\n\n- `AlgebraicGeometry.IsZariskiLocalAtTarget.of_isPullback`:\n    `P` is preserved under pullback along open immersions.\n- `AlgebraicGeometry.IsZariskiLocalAtTarget.restrict`:\n    `P f → P (f ∣_ U)` for an open `U` of `Y`.\n- `AlgebraicGeometry.IsZariskiLocalAtTarget.iff_of_iSup_eq_top`:\n    `P f ↔ ∀ i, P (f ∣_ U i)` for a family `U` of open sets covering `Y`.\n- `AlgebraicGeometry.IsZariskiLocalAtTarget.iff_of_openCover`:\n    `P f ↔ ∀ i, P (𝒰.pullbackHom f i)` for `𝒰 : Y.OpenCover`.\n\n## `IsZariskiLocalAtSource`\n\n- `AlgebraicGeometry.IsZariskiLocalAtSource`: We say that `IsZariskiLocalAtSource P` for\n  `P : MorphismProperty Scheme` if\n  1. `P` respects isomorphisms.\n  2. `P` holds for `𝒰.f i ≫ f` for an open cover `𝒰` of `X` iff `P` holds for `f : X ⟶ Y`.\n\nFor a morphism property `P` local at the source and `f : X ⟶ Y`, we provide these API lemmas:\n\n- `AlgebraicGeometry.IsZariskiLocalAtSource.comp`:\n    `P` is preserved under composition with open immersions at the source.\n- `AlgebraicGeometry.IsZariskiLocalAtSource.iff_of_iSup_eq_top`:\n    `P f ↔ ∀ i, P ((U i).ι ≫ f)` for a family `U` of open sets covering `X`.\n- `AlgebraicGeometry.IsZariskiLocalAtSource.iff_of_openCover`:\n    `P f ↔ ∀ i, P (𝒰.f i ≫ f)` for `𝒰 : X.OpenCover`.\n- `AlgebraicGeometry.IsZariskiLocalAtSource.of_isOpenImmersion`: If `P` contains identities then `P`\n    holds for open immersions.\n\n## `AffineTargetMorphismProperty`\n\n- `AlgebraicGeometry.AffineTargetMorphismProperty`:\n    The type of predicates on `f : X ⟶ Y` with `Y` affine.\n- `AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal`: We say that `P.IsLocal` if `P`\n    satisfies the assumptions of the affine communication lemma\n    (`AlgebraicGeometry.of_affine_open_cover`). That is,\n    1. `P` respects isomorphisms.\n    2. If `P` holds for `f : X ⟶ Y`, then `P` holds for `f ∣_ Y.basicOpen r` for any\n      global section `r`.\n    3. If `P` holds for `f ∣_ Y.basicOpen r` for all `r` in a spanning set of the global sections,\n      then `P` holds for `f`.\n\n## `HasAffineProperty`\n\n- `AlgebraicGeometry.HasAffineProperty`:\n  `HasAffineProperty P Q` is a type class asserting that `P` is local at the target,\n  and over affine schemes, it is equivalent to `Q : AffineTargetMorphismProperty`.\n\nFor `HasAffineProperty P Q` and `f : X ⟶ Y`, we provide these API lemmas:\n\n- `AlgebraicGeometry.HasAffineProperty.of_isPullback`:\n    `P` is preserved under pullback along open immersions from affine schemes.\n- `AlgebraicGeometry.HasAffineProperty.restrict`:\n    `P f → Q (f ∣_ U)` for affine `U` of `Y`.\n- `AlgebraicGeometry.HasAffineProperty.iff_of_iSup_eq_top`:\n    `P f ↔ ∀ i, Q (f ∣_ U i)` for a family `U` of affine open sets covering `Y`.\n- `AlgebraicGeometry.HasAffineProperty.iff_of_openCover`:\n    `P f ↔ ∀ i, Q (𝒰.pullbackHom f i)` for affine open covers `𝒰` of `Y`.\n- `AlgebraicGeometry.HasAffineProperty.isStableUnderBaseChange`:\n    If `Q` is stable under affine base change, then `P` is stable under arbitrary base change.\n\n## Implementation details\n\nThe properties `IsZariskiLocalAtTarget` and `IsZariskiLocalAtSource` are defined as abbreviations\nfor the respective local property of morphism properties defined generally for categories equipped\nwith a `Precoverage`.\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\nopen TopologicalSpace CategoryTheory CategoryTheory.Limits Opposite\n\nnoncomputable section\n\nnamespace AlgebraicGeometry\n\n/-- A property is Zariski-local at target if it is local at target in the Zariski topology. -/\nabbrev IsZariskiLocalAtTarget (P : MorphismProperty Scheme.{u}) :=\n  P.IsLocalAtTarget Scheme.zariskiPrecoverage\n\nnamespace IsZariskiLocalAtTarget\n\n/--\n`P` is local at the target if\n1. `P` respects isomorphisms.\n2. If `P` holds for `f : X ⟶ Y`, then `P` holds for `f ∣_ U` for any `U`.\n3. If `P` holds for `f ∣_ U` for an open cover `U` of `Y`, then `P` holds for `f`.\n-/\nprotected lemma mk' {P : MorphismProperty Scheme} [P.RespectsIso]\n    (restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens), P f → P (f ∣_ U))\n    (of_sSup_eq_top :\n      ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), iSup U = ⊤ →\n        (∀ i, P (f ∣_ U i)) → P f) :\n    IsZariskiLocalAtTarget P := by\n  refine .mk_of_iff_of_zeroHypercover fun {X Y} f 𝒰 ↦ ?_\n  refine ⟨fun hf i ↦ (P.arrow_mk_iso_iff (morphismRestrictOpensRange _ _)).mp (restrict _ _ hf),\n    fun h ↦ ?_⟩\n  refine of_sSup_eq_top f _ (Scheme.OpenCover.iSup_opensRange <| .ulift 𝒰) ?_\n  exact fun i ↦ (P.arrow_mk_iso_iff (morphismRestrictOpensRange f _)).mpr (h _)\n\nvariable {P : MorphismProperty Scheme.{u}} [IsZariskiLocalAtTarget P]\n  {X Y : Scheme.{u}} {f : X ⟶ Y} (𝒰 : Y.OpenCover)\n\nlemma of_isPullback {UX UY : Scheme.{u}} {iY : UY ⟶ Y} [IsOpenImmersion iY]\n    {iX : UX ⟶ X} {f' : UX ⟶ UY} (h : IsPullback iX f' f iY) (H : P f) : P f' :=\n  MorphismProperty.IsLocalAtTarget.of_isPullback (Y.affineCover.add iY) .none h H\n\ntheorem restrict (hf : P f) (U : Y.Opens) : P (f ∣_ U) :=\n  of_isPullback (isPullback_morphismRestrict f U).flip hf\n\nlemma of_iSup_eq_top {ι} (U : ι → Y.Opens) (hU : iSup U = ⊤)\n    (H : ∀ i, P (f ∣_ U i)) : P f := by\n  refine (P.iff_of_zeroHypercover_target\n    (Y.openCoverOfIsOpenCover (s := Set.range U) Subtype.val (by ext; simp [← hU]))).mpr fun i ↦ ?_\n  obtain ⟨_, i, rfl⟩ := i\n  refine (P.arrow_mk_iso_iff (morphismRestrictOpensRange f _)).mp ?_\n  change P (f ∣_ (U i).ι.opensRange)\n  rw [Scheme.Opens.opensRange_ι]\n  exact H i\n\ntheorem iff_of_iSup_eq_top {ι} (U : ι → Y.Opens) (hU : iSup U = ⊤) :\n    P f ↔ ∀ i, P (f ∣_ U i) :=\n  ⟨fun H _ ↦ restrict H _, of_iSup_eq_top U hU⟩\n\nlemma of_openCover (H : ∀ i, P (𝒰.pullbackHom f i)) : P f := by\n  apply of_iSup_eq_top (fun i ↦ (𝒰.f i).opensRange) 𝒰.iSup_opensRange\n  exact fun i ↦ (P.arrow_mk_iso_iff (morphismRestrictOpensRange f _)).mpr (H i)\n\ntheorem iff_of_openCover (𝒰 : Y.OpenCover) :\n    P f ↔ ∀ i, P (𝒰.pullbackHom f i) :=\n  ⟨fun H _ ↦ of_isPullback (.of_hasPullback _ _) H, of_openCover _⟩\n\nset_option backward.defeqAttrib.useBackward true in\n\nTarget:\nlemma of_range_subset_iSup [P.RespectsRight @IsOpenImmersion] {ι : Type*} (U : ι → Y.Opens)\n    (H : Set.range f ⊆ (⨆ i, U i : Y.Opens)) (hf : ∀ i, P (f ∣_ U i)) : P f :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_93a0ad9cdbe5","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"10e8a49c43a1fabc5110ea9d976f96c17de8cda685b1c84f147a87943e7d6267","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Morphisms","family_id":"of_range_subset_isup","file_id":"mathlib/Mathlib/AlgebraicGeometry/Morphisms/Basic.lean","sample_id":"93a0ad9cdbe52633a2a6476e78bca93ac766df3668d5e36441bd4a7ce8340870"},"split":"train"}
{"binding":{"chosen_proof_sha256":"de0e9cf3147279392578300fd27bc7777015480616482bce0bced9bd64ff6f36","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7c2cc263cb08cd0ae7b3b6f1e5d4cbec38b9a07797c8b210ba27cbac169d95dc","source_sha256":"67ccbb07cd4eb0a572ef6c3e31cad126033c1f36f3721cbbe9e484682e7571e0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [NatTrans.ext_iff]\n  simp [le_def, Set.subset_def, ConcreteCategory.hom_ext_iff, funext_iff]","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":8,"token_jaccard":0.047619,"token_length_ratio":0.380952},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"69e5edc15184e562bee70be0f06e43f072f345b376d1fe3b9e9009a209aaf5a7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Shapes.Equalizers\npublic import Mathlib.CategoryTheory.Subfunctor.Image\n\nNamespace:\nCategoryTheory.Subfunctor\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The equalizer of two morphisms of functors, as a subfunctor\n\nIf `F₁` and `F₂` are type-valued functors, `A : Subfunctor F₁`, and\n`f` and `g` are two morphisms `A.toFunctor ⟶ F₂`, we introduce\n`Subcomplex.equalizer f g`, which is the subfunctor of `F₁` contained in `A`\nwhere `f` and `g` coincide.\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nnamespace CategoryTheory\n\nvariable {C : Type u} [Category.{v} C] {F₁ F₂ : C ⥤ Type w} {A : Subfunctor F₁}\n  (f g : A.toFunctor ⟶ F₂)\n\nnamespace Subfunctor\n\n/-- The equalizer of two morphisms of type-valued functors of types of the form\n`A.toFunctor ⟶ F₂` with `A : Subfunctor F₁`, as a subcomplex of `F₁`. -/\n@[simps -isSimp]\nprotected def equalizer : Subfunctor F₁ where\n  obj U := setOf (fun x ↦ ∃ (hx : x ∈ A.obj _), f.app _ ⟨x, hx⟩ = g.app _ ⟨x, hx⟩)\n  map φ x := by\n    rintro ⟨hx, h⟩\n    exact ⟨A.map _ hx,\n      (NatTrans.naturality_apply f φ ⟨x, hx⟩).trans (Eq.trans (by rw [h])\n        (NatTrans.naturality_apply g φ ⟨x, hx⟩).symm)⟩\n\nattribute [local simp] equalizer_obj\n\nlemma equalizer_le : Subfunctor.equalizer f g ≤ A :=\n  fun _ _ h ↦ h.1\n\n@[simp]\nlemma equalizer_self : Subfunctor.equalizer f f = A := by aesop\n\nlemma mem_equalizer_iff {i : C} (x : A.toFunctor.obj i) :\n    x.1 ∈ (Subfunctor.equalizer f g).obj i ↔ f.app i x = g.app i x := by\n  simp\n\nTarget:\nlemma range_le_equalizer_iff {G : C ⥤ Type w} (φ : G ⟶ A.toFunctor) :\n    range (φ ≫ A.ι) ≤ Subfunctor.equalizer f g ↔ φ ≫ f = φ ≫ g :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"6558a8ef63f8f3040d5ff8daa4b0aaa1b1e75f13d587e3f6e90d13e3f6a26edb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Subfunctor","family_id":"range_le_equalizer_iff","file_id":"mathlib/Mathlib/CategoryTheory/Subfunctor/Equalizer.lean","sample_id":"7c2cc263cb08cd0ae7b3b6f1e5d4cbec38b9a07797c8b210ba27cbac169d95dc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a40d38439b063352a0a57202db0bbc83f3e167b29ffee4ace96a68c95b1c7096","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"74e41a4b5bb950621cded82f802859d90e618c5ffbeaff846b1d0a22fbcf641d","source_sha256":"158ce289cc94776d7ace3f5f8788d7841da0114f64a958ffadba75342e4312aa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply Subtype.ext; funext n\n  rcases h : s.val n; · rfl\n  refine absurd ?_ H; exact ⟨⟨_, _, h.symm⟩⟩","hard_negative":false,"metrics":{"chosen_tokens":36,"rejected_tokens":2,"token_jaccard":0.038462,"token_length_ratio":0.055556},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"69ebd5249d8f6940aa394412a311a854931b8d15cb413b67a43fef099ed79df0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.Find\npublic import Mathlib.Data.Stream.Init\npublic import Mathlib.Logic.Relator\npublic import Mathlib.Tactic.Common\npublic import Batteries.Tactic.Lint.Simp\n\nNamespace:\nComputation\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Coinductive formalization of unbounded computations.\n\nThis file provides a `Computation` type where `Computation α` is the type of\nunbounded computations returning `α`.\n-/\n\n@[expose] public section\n\nopen Function\n\nuniverse u v w\n\n/-\ncoinductive Computation (α : Type u) : Type u\n| pure : α → Computation α\n| think : Computation α → Computation α\n-/\n/-- `Computation α` is the type of unbounded computations returning `α`.\n  An element of `Computation α` is an infinite sequence of `Option α` such\n  that if `f n = some a` for some `n` then it is constantly `some a` after that. -/\ndef Computation (α : Type u) : Type u :=\n  { f : Stream' (Option α) // ∀ ⦃n a⦄, f n = some a → f (n + 1) = some a }\n\nnamespace Computation\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n-- constructors\n/-- `pure a` is the computation that immediately terminates with result `a`. -/\ndef pure (a : α) : Computation α :=\n  ⟨Stream'.const (some a), fun _ _ => id⟩\n\ninstance : CoeTC α (Computation α) :=\n  ⟨pure⟩\n\n-- note [use has_coe_t]\n/-- `think c` is the computation that delays for one \"tick\" and then performs\n  computation `c`. -/\ndef think (c : Computation α) : Computation α :=\n  ⟨Stream'.cons none c.1, fun n a h => by\n    rcases n with - | n\n    · contradiction\n    · exact c.2 h⟩\n\n/-- `thinkN c n` is the computation that delays for `n` ticks and then performs\n  computation `c`. -/\ndef thinkN (c : Computation α) : ℕ → Computation α\n  | 0 => c\n  | n + 1 => think (thinkN c n)\n\n-- check for immediate result\n/-- `head c` is the first step of computation, either `some a` if `c = pure a`\n  or `none` if `c = think c'`. -/\ndef head (c : Computation α) : Option α :=\n  c.1.head\n\n-- one step of computation\n/-- `tail c` is the remainder of computation, either `c` if `c = pure a`\n  or `c'` if `c = think c'`. -/\ndef tail (c : Computation α) : Computation α :=\n  ⟨c.1.tail, fun _ _ h => c.2 h⟩\n\n/-- `empty α` is the computation that never returns, an infinite sequence of\n  `think`s. -/\ndef empty (α) : Computation α :=\n  ⟨Stream'.const none, fun _ _ => id⟩\n\ninstance : Inhabited (Computation α) :=\n  ⟨empty _⟩\n\n/-- `runFor c n` evaluates `c` for `n` steps and returns the result, or `none`\n  if it did not terminate after `n` steps. -/\ndef runFor : Computation α → ℕ → Option α :=\n  Subtype.val\n\n/-- `destruct c` is the destructor for `Computation α` as a coinductive type.\n  It returns `inl a` if `c = pure a` and `inr c'` if `c = think c'`. -/\ndef destruct (c : Computation α) : α ⊕ (Computation α) :=\n  match c.1 0 with\n  | none => Sum.inr (tail c)\n  | some a => Sum.inl a\n\n/-- `run c` is an unsound meta function that runs `c` to completion, possibly\n  resulting in an infinite loop in the VM. -/\nunsafe def run : Computation α → α\n  | c =>\n    match destruct c with\n    | Sum.inl a => a\n    | Sum.inr ca => run ca\n\ntheorem destruct_eq_pure {s : Computation α} {a : α} : destruct s = Sum.inl a → s = pure a := by\n  dsimp [destruct]\n  cases f0 : s.1 0 <;> intro h\n  · contradiction\n  · apply Subtype.ext\n    funext n\n    induction n with\n    | zero => injection h with h'; rwa [h'] at f0\n    | succ n IH => exact s.2 IH\n\ntheorem destruct_eq_think {s : Computation α} {s'} : destruct s = Sum.inr s' → s = think s' := by\n  dsimp [destruct]\n  rcases f0 : s.1 0 with - | a' <;> intro h\n  · injection h with h'\n    rw [← h']\n    obtain ⟨f, al⟩ := s\n    apply Subtype.ext\n    dsimp [think, tail]\n    rw [← f0]\n    exact (Stream'.eta f).symm\n  · contradiction\n\n@[simp]\ntheorem destruct_pure (a : α) : destruct (pure a) = Sum.inl a :=\n  rfl\n\n@[simp]\ntheorem destruct_think : ∀ s : Computation α, destruct (think s) = Sum.inr s\n  | ⟨_, _⟩ => rfl\n\n@[simp]\ntheorem destruct_empty : destruct (empty α) = Sum.inr (empty α) :=\n  rfl\n\n@[simp]\ntheorem head_pure (a : α) : head (pure a) = some a :=\n  rfl\n\n@[simp]\ntheorem head_think (s : Computation α) : head (think s) = none :=\n  rfl\n\n@[simp]\ntheorem head_empty : head (empty α) = none :=\n  rfl\n\n@[simp]\ntheorem tail_pure (a : α) : tail (pure a) = pure a :=\n  rfl\n\n@[simp]\ntheorem tail_think (s : Computation α) : tail (think s) = s := rfl\n\n@[simp]\ntheorem tail_empty : tail (empty α) = empty α :=\n  rfl\n\ntheorem think_empty : empty α = think (empty α) :=\n  destruct_eq_think destruct_empty\n\n/-- Recursion principle for computations, compare with `List.recOn`. -/\n@[elab_as_elim]\ndef recOn {motive : Computation α → Sort v} (s : Computation α) (pure : ∀ a, motive (pure a))\n    (think : ∀ s, motive (think s)) : motive s :=\n  match H : destruct s with\n  | Sum.inl v => by\n    rw [destruct_eq_pure H]\n    apply pure\n  | Sum.inr v => match v with\n    | ⟨a, s'⟩ => by\n      rw [destruct_eq_think H]\n      apply think\n\n/-- Corecursor constructor for `corec` -/\ndef Corec.f (f : β → α ⊕ β) : α ⊕ β → Option α × (α ⊕ β)\n  | Sum.inl a => (some a, Sum.inl a)\n  | Sum.inr b =>\n    (match f b with\n      | Sum.inl a => some a\n      | Sum.inr _ => none,\n      f b)\n\n/-- `corec f b` is the corecursor for `Computation α` as a coinductive type.\n  If `f b = inl a` then `corec f b = pure a`, and if `f b = inl b'` then\n  `corec f b = think (corec f b')`. -/\ndef corec (f : β → α ⊕ β) (b : β) : Computation α := by\n  refine ⟨Stream'.corec' (Corec.f f) (Sum.inr b), fun n a' h => ?_⟩\n  rw [Stream'.corec'_eq]\n  change Stream'.corec' (Corec.f f) (Corec.f f (Sum.inr b)).2 n = some a'\n  revert h; generalize Sum.inr b = o\n  induction n generalizing o with\n  | zero =>\n    change (Corec.f f o).1 = some a' → (Corec.f f (Corec.f f o).2).1 = some a'\n    rcases o with _ | b <;> intro h\n    · exact h\n    unfold Corec.f at *; split <;> simp_all\n  | succ n IH =>\n    rw [Stream'.corec'_eq (Corec.f f) (Corec.f f o).2, Stream'.corec'_eq (Corec.f f) o]\n    exact IH (Corec.f f o).2\n\n/-- left map of `⊕` -/\ndef lmap (f : α → β) : α ⊕ γ → β ⊕ γ\n  | Sum.inl a => Sum.inl (f a)\n  | Sum.inr b => Sum.inr b\n\n/-- right map of `⊕` -/\ndef rmap (f : β → γ) : α ⊕ β → α ⊕ γ\n  | Sum.inl a => Sum.inl a\n  | Sum.inr b => Sum.inr (f b)\n\nattribute [simp] lmap rmap\n\n@[simp]\ntheorem corec_eq (f : β → α ⊕ β) (b : β) : destruct (corec f b) = rmap (corec f) (f b) := by\n  dsimp [corec, destruct]\n  rw [show Stream'.corec' (Corec.f f) (Sum.inr b) 0 =\n    Sum.rec Option.some (fun _ ↦ none) (f b) by\n    dsimp [Corec.f, Stream'.corec', Stream'.corec, Stream'.map, Stream'.get, Stream'.iterate]\n    match (f b) with\n    | Sum.inl x => rfl\n    | Sum.inr x => rfl]\n  rcases h : f b with a | b'; · rfl\n  dsimp [Corec.f, destruct]\n  apply congr_arg; apply Subtype.ext\n  dsimp [corec, tail]\n  rw [Stream'.corec'_eq, Stream'.tail_cons]\n  dsimp [Corec.f]; rw [h]\n\nsection Bisim\n\nvariable (R : Computation α → Computation α → Prop)\n\n/-- bisimilarity relation -/\nlocal infixl:50 \" ~ \" => R\n\n/-- Bisimilarity over a sum of `Computation`s -/\ndef BisimO : α ⊕ (Computation α) → α ⊕ (Computation α) → Prop\n  | Sum.inl a, Sum.inl a' => a = a'\n  | Sum.inr s, Sum.inr s' => R s s'\n  | _, _ => False\n\nattribute [simp] BisimO\nattribute [nolint simpNF] BisimO.eq_3\n\n/-- Attribute expressing bisimilarity over two `Computation`s -/\ndef IsBisimulation :=\n  ∀ ⦃s₁ s₂⦄, s₁ ~ s₂ → BisimO R (destruct s₁) (destruct s₂)\n\n-- If two computations are bisimilar, then they are equal\ntheorem eq_of_bisim (bisim : IsBisimulation R) {s₁ s₂} (r : s₁ ~ s₂) : s₁ = s₂ := by\n  apply Subtype.ext\n  apply Stream'.eq_of_bisim fun x y => ∃ s s' : Computation α, s.1 = x ∧ s'.1 = y ∧ R s s'\n  · dsimp [Stream'.IsBisimulation]\n    intro t₁ t₂ e\n    match t₁, t₂, e with\n    | _, _, ⟨s, s', rfl, rfl, r⟩ =>\n      suffices head s = head s' ∧ R (tail s) (tail s') from\n        And.imp id (fun r => ⟨tail s, tail s', by cases s; rfl, by cases s'; rfl, r⟩) this\n      have h := bisim r; revert r h\n      refine recOn s ?_ ?_ <;> intro r' <;> refine recOn s' ?_ ?_ <;> intro a' r h\n      · constructor <;> dsimp at h\n        · rw [h]\n        · rw [h] at r\n          rw [tail_pure, tail_pure, h]\n          assumption\n      · rw [destruct_pure, destruct_think] at h\n        exact False.elim h\n      · rw [destruct_pure, destruct_think] at h\n        exact False.elim h\n      · simp_all\n  · exact ⟨s₁, s₂, rfl, rfl, r⟩\n\nend Bisim\n\n-- It's more of a stretch to use ∈ for this relation, but it\n-- asserts that the computation limits to the given value.\n/-- Assertion that a `Computation` limits to a given value -/\nprotected def Mem (s : Computation α) (a : α) :=\n  some a ∈ s.1\n\ninstance : Membership α (Computation α) :=\n  ⟨Computation.Mem⟩\n\ntheorem le_stable (s : Computation α) {a m n} (h : m ≤ n) : s.1 m = some a → s.1 n = some a := by\n  obtain ⟨f, al⟩ := s\n  induction h with\n  | refl => exact id\n  | step _ IH => exact fun h2 ↦ al (IH h2)\n\ntheorem mem_unique {s : Computation α} {a b : α} : a ∈ s → b ∈ s → a = b\n  | ⟨m, ha⟩, ⟨n, hb⟩ => by\n    injection\n      (le_stable s (le_max_left m n) ha.symm).symm.trans (le_stable s (le_max_right m n) hb.symm)\n\ntheorem Mem.left_unique : Relator.LeftUnique ((· ∈ ·) : α → Computation α → Prop) := fun _ _ _ =>\n  mem_unique\n\n/-- `Terminates s` asserts that the computation `s` eventually terminates with some value. -/\nclass Terminates (s : Computation α) : Prop where\n  /-- assertion that there is some term `a` such that the `Computation` terminates -/\n  term : ∃ a, a ∈ s\n\ntheorem terminates_iff (s : Computation α) : Terminates s ↔ ∃ a, a ∈ s :=\n  ⟨fun h => h.1, Terminates.mk⟩\n\ntheorem terminates_of_mem {s : Computation α} {a : α} (h : a ∈ s) : Terminates s :=\n  ⟨⟨a, h⟩⟩\n\ntheorem terminates_def (s : Computation α) : Terminates s ↔ ∃ n, (s.1 n).isSome :=\n  ⟨fun ⟨⟨a, n, h⟩⟩ =>\n    ⟨n, by\n      dsimp [Stream'.get] at h\n      rw [← h]\n      exact rfl⟩,\n    fun ⟨n, h⟩ => ⟨⟨Option.get _ h, n, (Option.eq_some_of_isSome h).symm⟩⟩⟩\n\ntheorem ret_mem (a : α) : a ∈ pure a :=\n  Exists.intro 0 rfl\n\ntheorem eq_of_pure_mem {a a' : α} (h : a' ∈ pure a) : a' = a :=\n  mem_unique h (ret_mem _)\n\n@[simp]\ntheorem mem_pure_iff (a b : α) : a ∈ pure b ↔ a = b :=\n  ⟨eq_of_pure_mem, fun h => h ▸ ret_mem _⟩\n\ninstance ret_terminates (a : α) : Terminates (pure a) :=\n  terminates_of_mem (ret_mem _)\n\ntheorem think_mem {s : Computation α} {a} : a ∈ s → a ∈ think s\n  | ⟨n, h⟩ => ⟨n + 1, h⟩\n\ninstance think_terminates (s : Computation α) : ∀ [Terminates s], Terminates (think s)\n  | ⟨⟨a, n, h⟩⟩ => ⟨⟨a, n + 1, h⟩⟩\n\ntheorem of_think_mem {s : Computation α} {a} : a ∈ think s → a ∈ s\n  | ⟨n, h⟩ => by\n    rcases n with - | n'\n    · contradiction\n    · exact ⟨n', h⟩\n\ntheorem of_think_terminates {s : Computation α} : Terminates (think s) → Terminates s\n  | ⟨⟨a, h⟩⟩ => ⟨⟨a, of_think_mem h⟩⟩\n\ntheorem notMem_empty (a : α) : a ∉ empty α := fun ⟨n, h⟩ => by contradiction\n\ntheorem not_terminates_empty : ¬Terminates (empty α) := fun ⟨⟨a, h⟩⟩ => notMem_empty a h\n\nTarget:\ntheorem eq_empty_of_not_terminates {s} (H : ¬Terminates s) : s = empty α :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Seq","family_id":"eq_empty_of_not_terminates","file_id":"mathlib/Mathlib/Data/Seq/Computation.lean","sample_id":"74e41a4b5bb950621cded82f802859d90e618c5ffbeaff846b1d0a22fbcf641d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f95ac83c92042b8eca33c773ee62f7012472f34ee0a3ebc546b077931f47c21d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"10a43da42697c190a54beae2ffcca52222667a5f6d3aa6ae6e4063ca49960cb0","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"83bbcd0a12ed90d64932bbbf23b8435e8984a966305ad698b755df700c2893b8","source_sha256":"e6501306e0c166f31aa2d54b2053e75c750b45815aae03787ae7a4d5aa54d0d7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction hs using Finset.Nonempty.cons_induction with\n  | singleton i => exact ⟨i, by simp⟩\n  | @cons i s hi hs ih =>\n    obtain ⟨j, hj⟩ := ih\n    by_cases hji : f j ≤ f i\n    · refine ⟨i, mem_cons_self .., ?_⟩\n      simp only [mem_cons, forall_eq_or_imp, imp_self, true_and]\n      exact fun k hk hik ↦ _root_.trans (hj.2 hk <| _root_.trans hji hik) hji\n    · exact ⟨j, mem_cons_of_mem hj.1, by simpa [hji] using hj.2⟩","hard_negative":true,"metrics":{"chosen_tokens":112,"rejected_tokens":3,"token_jaccard":0.033898,"token_length_ratio":0.026786},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"69fe836f2fc4a509a1d269b96ad6f05d2ab30fa4e01aedd3d31c0cce6f45be40","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Basic\npublic import Mathlib.Order.Minimal\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2025 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Finite preorders and finite sets in a preorder\n\nThis file shows that non-empty finite sets in a preorder have minimal/maximal elements, and\ncontrapositively that non-empty sets without minimal or maximal elements are infinite.\n-/\n\npublic section\n\nvariable {ι α β : Type*}\n\nnamespace Finset\nsection IsTrans\nvariable [LE α] [IsTrans α LE.le] {s : Finset α} {a : α}\n\nTarget:\nlemma exists_maximalFor (f : ι → α) (s : Finset ι) (hs : s.Nonempty) :\n    ∃ i, MaximalFor (· ∈ s) f i :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_83bbcd0a12ed","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"ad6db25547f15ab0b280e78eba944e995a578aa92c68be7e3422f6192f6573b0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Preorder","family_id":"exists_maximalfor","file_id":"mathlib/Mathlib/Order/Preorder/Finite.lean","sample_id":"83bbcd0a12ed90d64932bbbf23b8435e8984a966305ad698b755df700c2893b8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"de9ab835bffa0c1a69cfad4d6364d7e5552c21f89d63ae535eda5295f7d4c8b1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7f4f13cee3f4a102def98fffe2db00bfee11b4a8beba869042646774c9863f43","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3a6d8b994748d4e07066dd4b842c5e18a4a267de9cc4efed1e2d52e8ad6a24cb","source_sha256":"354c54d863521e4c2e2cb8eb45baaec320a74608e472a8d10bc34f45771b2911","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨reg.1.of_flat_of_isBaseChange hf, ?_⟩\n  rw [← Ideal.map_ofList]\n  exact ((hf.map_smul_top_ne_top_iff_of_faithfullyFlat R M _).mpr reg.2.symm).symm","hard_negative":true,"metrics":{"chosen_tokens":40,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.075},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"6a1fde328b4eae961e1fdcc614348279a1bf08527d0da39c1eb08c85c84bf412","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Flat.FaithfullyFlat.Basic\npublic import Mathlib.RingTheory.Flat.Localization\npublic import Mathlib.RingTheory.Regular.RegularSequence\n\nNamespace:\nRingTheory.Sequence\n\nLocal context:\n/-\nCopyright (c) 2025 Yongle Hu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yongle Hu, Nailin Guan\n-/\n/-!\n# `RingTheory.Sequence.IsWeaklyRegular` is stable under flat base change\n\n## Main results\n* `RingTheory.Sequence.IsWeaklyRegular.of_flat_of_isBaseChange`: Let `R` be a commutative ring,\n  `M` be an `R`-module, `S` be a flat `R`-algebra, `N` be the base change of `M` to `S`.\n  If `[r₁, …, rₙ]` is a weakly regular `M`-sequence, then its image in `N` is a weakly regular\n  `N`-sequence.\n-/\n\npublic section\n\nnamespace RingTheory.Sequence\n\nopen Module\n\nvariable {R S M N : Type*} [CommRing R] [CommRing S] [Algebra R S]\n  [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower R S N]\n\n/-- Let `R` be a commutative ring, `M` be an `R`-module, `S` be a flat `R`-algebra, `N` be the base\n  change of `M` to `S`. If `[r₁, …, rₙ]` is a weakly regular `M`-sequence, then its image in `N` is\n  a weakly regular `N`-sequence. -/\ntheorem IsWeaklyRegular.of_flat_of_isBaseChange [Flat R S] {f : M →ₗ[R] N} (hf : IsBaseChange S f)\n    {rs : List R} (reg : IsWeaklyRegular M rs) : IsWeaklyRegular N (rs.map (algebraMap R S)) := by\n  induction rs generalizing M N with\n  | nil => simp\n  | cons x _ ih =>\n    simp only [List.map_cons, isWeaklyRegular_cons_iff] at reg ⊢\n    have e := (QuotSMulTop.algebraMapTensorEquivTensorQuotSMulTop x M S).symm ≪≫ₗ\n      QuotSMulTop.congr ((algebraMap R S) x) hf.equiv\n    have hg : IsBaseChange S <|\n        e.toLinearMap.restrictScalars R ∘ₗ TensorProduct.mk R S (QuotSMulTop x M) 1 :=\n      IsBaseChange.of_equiv e (fun _ ↦ by simp)\n    exact ⟨reg.1.of_flat_of_isBaseChange hf, ih hg reg.2⟩\n\ntheorem IsWeaklyRegular.of_flat [Flat R S] {rs : List R} (reg : IsWeaklyRegular R rs) :\n    IsWeaklyRegular S (rs.map (algebraMap R S)) :=\n  reg.of_flat_of_isBaseChange (IsBaseChange.linearMap R S)\n\nvariable (S) (T : Submonoid R) [IsLocalization T S]\n\ntheorem IsWeaklyRegular.of_isLocalizedModule (f : M →ₗ[R] N) [IsLocalizedModule T f]\n    {rs : List R} (reg : IsWeaklyRegular M rs) : IsWeaklyRegular N (rs.map (algebraMap R S)) :=\n  have : Flat R S := IsLocalization.flat S T\n  reg.of_flat_of_isBaseChange (IsLocalizedModule.isBaseChange T S f)\n\ninclude T in\ntheorem IsWeaklyRegular.of_isLocalization {rs : List R} (reg : IsWeaklyRegular R rs) :\n    IsWeaklyRegular S (rs.map (algebraMap R S)) :=\n  reg.of_isLocalizedModule S T (Algebra.linearMap R S)\n\nvariable (p : Ideal R) [p.IsPrime] [IsLocalization.AtPrime S p]\n\ntheorem IsWeaklyRegular.isRegular_of_isLocalizedModule_of_mem\n    [Nontrivial N] [Module.Finite S N] (f : M →ₗ[R] N) [IsLocalizedModule.AtPrime p f]\n    {rs : List R} (reg : IsWeaklyRegular M rs) (mem : ∀ r ∈ rs, r ∈ p) :\n    IsRegular N (rs.map (algebraMap R S)) := by\n  have : IsLocalRing S := IsLocalization.AtPrime.isLocalRing S p\n  refine (IsLocalRing.isRegular_iff_isWeaklyRegular_of_subset_maximalIdeal (fun _ hr ↦ ?_)).mpr <|\n    reg.of_isLocalizedModule S p.primeCompl f\n  rcases List.mem_map.mp hr with ⟨r, hr, eq⟩\n  simpa only [← eq, IsLocalization.AtPrime.to_map_mem_maximal_iff S p] using mem r hr\n\ntheorem IsWeaklyRegular.isRegular_of_isLocalization_of_mem\n    {rs : List R} (reg : IsWeaklyRegular R rs) (mem : ∀ r ∈ rs, r ∈ p) :\n    IsRegular S (rs.map (algebraMap R S)) :=\n  have : Nontrivial S := IsLocalization.AtPrime.nontrivial S p\n  reg.isRegular_of_isLocalizedModule_of_mem S p (Algebra.linearMap R S) mem\n\nvariable {S} [FaithfullyFlat R S]\n\n/-- Let `R` be a commutative ring, `M` be an `R`-module, `S` be a faithfully flat `R`-algebra,\n  `N` be the base change of `M` to `S`. If `[r₁, …, rₙ]` is a regular `M`-sequence, then its image\n  in `N` is a regular `N`-sequence. -/\n\nTarget:\ntheorem IsRegular.of_faithfullyFlat_of_isBaseChange {f : M →ₗ[R] N} (hf : IsBaseChange S f)\n    {rs : List R} (reg : IsRegular M rs) : IsRegular N (rs.map (algebraMap R S)) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_3a6d8b994748","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"88bdca814c3785c377d41f98c9f068f19c050a7f914ac811ab3a3d626c141925","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Regular","family_id":"isregular","file_id":"mathlib/Mathlib/RingTheory/Regular/Flat.lean","sample_id":"3a6d8b994748d4e07066dd4b842c5e18a4a267de9cc4efed1e2d52e8ad6a24cb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0726e99880ae39e0c58cb38c38f992cfa6af39d47cf8ffe5ea36831ca442aef8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9a39f51210f01f82a58423ea9a2a56c213d4c4905dd0974a1b7d4d89547abe73","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"06e54fa94e0402146a1056a8e38c4177d4d7f6c2eb72f119267197e2349465f8","source_sha256":"1c7994cb953bf15a3e2a75bf5a9f75792705603cc1fc781f9e07bc585e941697","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨Us, hUs, heq⟩ := TopologicalSpace.Opens.isBasis_iff_cover.mp\n    (PrespectralSpace.isBasis_opens X) ⟨f ⁻¹' U, hU.preimage hfc⟩\n  obtain ⟨t, ht⟩ := by\n    refine hc.elim_finite_subcover (fun s : Us ↦ f '' s.1) (fun s ↦ h _ s.1.2) (fun x hx ↦ ?_)\n    obtain ⟨x, rfl⟩ := hs hx\n    obtain ⟨i, hi, hx⟩ := mem_sSup.mp <| by rwa [← heq]\n    exact Set.mem_iUnion.mpr ⟨⟨i, hi⟩, x, hx, rfl⟩\n  refine ⟨⨆ s ∈ t, s.1, ?_, ?_⟩\n  · simp only [iSup_mk, carrier_eq_coe, coe_mk]\n    exact t.finite_toSet.isCompact_biUnion fun i _ ↦ hUs i.2\n  · simp only [iSup_mk, carrier_eq_coe, Set.iUnion_coe_set, coe_mk, Set.image_iUnion]\n    convert_to ⋃ i ∈ t, f '' i.1 = U\n    · simp\n    · refine subset_antisymm (fun x ↦ ?_) ht\n      simp_rw [Set.mem_iUnion]\n      rintro ⟨i, hi, x, hx, rfl⟩\n      have := heq ▸ mem_sSup.mpr ⟨i.1, i.2, hx⟩\n      exact this","hard_negative":false,"metrics":{"chosen_tokens":252,"rejected_tokens":257,"token_jaccard":0.951807,"token_length_ratio":1.019841},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"6a3acf6f1286ac55078124496c97862927be9b3a20f7c2b8ebf051369a1f63fe","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Ideal\npublic import Mathlib.Topology.Sets.Compacts\npublic import Mathlib.Topology.Sets.OpenCover\npublic import Mathlib.Topology.Spectral.Hom\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Prespectral spaces\n\nIn this file, we define prespectral spaces as spaces whose lattice of compact opens forms a basis.\n\n-/\n\n@[expose] public section\n\nopen TopologicalSpace Topology\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n\n/-- A space is prespectral if the lattice of compact opens forms a basis. -/\n@[stacks 08YG \"The last condition for spectral spaces\", mk_iff]\nclass PrespectralSpace (X : Type*) [TopologicalSpace X] : Prop where\n  isTopologicalBasis : IsTopologicalBasis { U : Set X | IsOpen U ∧ IsCompact U }\n\n/-- A space is prespectral if it has a basis consisting of compact opens. -/\nlemma PrespectralSpace.of_isTopologicalBasis {B : Set (Set X)}\n    (basis : IsTopologicalBasis B) (isCompact_basis : ∀ U ∈ B, IsCompact U) :\n    PrespectralSpace X where\n  isTopologicalBasis := basis.of_isOpen_of_subset (fun _ h ↦ h.1)\n    fun s hs ↦ ⟨basis.isOpen hs, isCompact_basis s hs⟩\n\n/-- A space is prespectral if it has a basis consisting of compact opens.\nThis is the variant with an indexed basis instead. -/\nlemma PrespectralSpace.of_isTopologicalBasis' {ι : Type*} {b : ι → Set X}\n    (basis : IsTopologicalBasis (Set.range b)) (isCompact_basis : ∀ i, IsCompact (b i)) :\n    PrespectralSpace X :=\n  .of_isTopologicalBasis basis (by simp_all)\n\ninstance (priority := low) [NoetherianSpace X] : PrespectralSpace X :=\n  .of_isTopologicalBasis isTopologicalBasis_opens fun _ _ ↦ NoetherianSpace.isCompact _\n\ninstance (priority := low) [PrespectralSpace X] : LocallyCompactSpace X where\n  local_compact_nhds _ _ hn :=\n    have ⟨V, ⟨hV₁, hV₂⟩, hxV, hVn⟩ := PrespectralSpace.isTopologicalBasis.mem_nhds_iff.mp hn\n    ⟨V, hV₁.mem_nhds hxV, hVn, hV₂⟩\n\nopen PrespectralSpace in\ninstance (priority := low) [T2Space X] [PrespectralSpace X] : TotallySeparatedSpace X :=\n  totallySeparatedSpace_iff_exists_isClopen.mpr fun _ _ hxy ↦\n    have ⟨U, ⟨hU₁, hU₂⟩, hxU, hyU⟩ :=\n      isTopologicalBasis.exists_subset_of_mem_open hxy isClosed_singleton.isOpen_compl\n    ⟨U, ⟨hU₂.isClosed, hU₁⟩, hxU, fun h ↦ hyU h rfl⟩\n\nlemma PrespectralSpace.of_isOpenCover\n    {ι : Type*} {U : ι → Opens X} (hU : IsOpenCover U) [∀ i, PrespectralSpace (U i)] :\n    PrespectralSpace X := by\n  refine .of_isTopologicalBasis (hU.isTopologicalBasis fun i ↦ isTopologicalBasis) ?_\n  simp only [Set.mem_iUnion, Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp,\n    forall_comm (α := Set _), forall_apply_eq_imp_iff₂]\n  exact fun i V hV hV' ↦ hV'.image continuous_subtype_val\n\nlemma PrespectralSpace.of_isInducing [PrespectralSpace Y]\n    (f : X → Y) (hf : IsInducing f) (hf' : IsSpectralMap f) : PrespectralSpace X :=\n  .of_isTopologicalBasis (PrespectralSpace.isTopologicalBasis.isInducing hf) (by\n    simp only [Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp]\n    rintro _ U h₁ h₂ rfl\n    exact hf'.isCompact_preimage_of_isOpen h₁ h₂)\n\nlemma PrespectralSpace.of_isClosedEmbedding [PrespectralSpace Y]\n    (f : X → Y) (hf : IsClosedEmbedding f) : PrespectralSpace X :=\n  .of_isInducing f hf.isInducing hf.isProperMap.isSpectralMap\n\n/-- Let `f : X → Y` be an open embedding of topological spaces.\nIf `Y` is a prespectral space (i.e., the quasi-compact opens of `Y` form a basis),\nthen `X` is also a prespectral space. -/\nlemma Topology.IsOpenEmbedding.prespectralSpace [PrespectralSpace Y]\n    {f : X → Y} (hf : IsOpenEmbedding f) :\n    PrespectralSpace X where\n  isTopologicalBasis := by\n    apply isTopologicalBasis_of_isOpen_of_nhds (fun U hU ↦ hU.1) <| fun x U hx hU ↦ ?_\n    obtain ⟨V, ⟨hoV, hcV⟩, hfx, hVf⟩ : ∃ V ∈ {V | IsOpen V ∧ IsCompact V}, f x ∈ V ∧ V ⊆ f '' U :=\n      (PrespectralSpace.isTopologicalBasis (X := Y)).isOpen_iff.mp\n        (hf.isOpen_iff_image_isOpen.mp hU) (f x) ⟨x, hx, rfl⟩\n    refine ⟨f ⁻¹' V, ⟨hoV.preimage hf.continuous, ?_⟩, ⟨hfx, fun y hy ↦ ?_⟩⟩\n    · exact hf.toIsInducing.isCompact_preimage' hcV <| Set.SurjOn.subset_range hVf\n    · exact hf.injective.mem_set_image.mp (hVf hy)\n\ninstance PrespectralSpace.sigma {ι : Type*} (X : ι → Type*) [∀ i, TopologicalSpace (X i)]\n    [∀ i, PrespectralSpace (X i)] : PrespectralSpace (Σ i, X i) :=\n  .of_isTopologicalBasis (IsTopologicalBasis.sigma fun i ↦ isTopologicalBasis) fun U hU ↦ by\n    simp_rw [Set.mem_iUnion] at hU\n    obtain ⟨i, V, hV, rfl⟩ := hU\n    exact hV.2.image continuous_sigmaMk\n\nvariable (X) in\nlemma PrespectralSpace.isBasis_opens [PrespectralSpace X] :\n    TopologicalSpace.Opens.IsBasis { U : Opens X | IsCompact (U : Set X) } := by\n  dsimp only [TopologicalSpace.Opens.IsBasis]\n  convert! isTopologicalBasis (X := X)\n  ext s\n  exact ⟨fun ⟨V, hV, heq⟩ ↦ heq ▸ ⟨V.2, hV⟩, fun h ↦ ⟨⟨s, h.1⟩, h.2, rfl⟩⟩\n\n/-- In a prespectral space, the lattice of opens is determined by its lattice of compact opens. -/\ndef PrespectralSpace.opensEquiv [PrespectralSpace X] :\n    Opens X ≃o Order.Ideal (CompactOpens X) where\n  toFun U := ⟨⟨{ V | (V : Set X) ⊆ U }, fun U₁ U₂ h₁ h₂ ↦ subset_trans (α := Set X) h₁ h₂⟩,\n    ⟨⊥, by simp⟩, fun U₁ h₁ U₂ h₂ ↦ ⟨U₁ ⊔ U₂, by aesop, le_sup_left, le_sup_right⟩⟩\n  invFun I := ⨆ U ∈ I, U.toOpens\n  left_inv U := by\n    apply le_antisymm\n    · simp only [iSup_le_iff]\n      exact fun _ ↦ id\n    · intro x hxU\n      obtain ⟨V, ⟨h₁, h₂⟩, hxV, hVU⟩ := isTopologicalBasis.exists_subset_of_mem_open hxU U.2\n      simp only [Opens.mem_iSup]\n      exact ⟨⟨⟨_, h₂⟩, h₁⟩, hVU, hxV⟩\n  right_inv I := by\n    ext U\n    dsimp\n    change U.toOpens ≤ _ ↔ _\n    refine ⟨fun H ↦ ?_, fun h ↦ le_iSup₂ (f := fun U (h : U ∈ I) ↦ U.toOpens) U h⟩\n    simp only [← SetLike.coe_subset_coe, Opens.iSup_mk, Opens.carrier_eq_coe, Opens.coe_mk] at H\n    obtain ⟨s, hsI, hs, hU⟩ := U.isCompact.elim_finite_subcover_image (fun U _ ↦ U.2) H\n    exact I.lower (a := hs.toFinset.sup fun i ↦ i) (by simpa [← SetLike.coe_subset_coe]) (by simpa)\n  map_rel_iff' {U V} := by\n    change (∀ (W : CompactOpens X), (W : Set X) ⊆ U → (W : Set X) ⊆ V) ↔ U ≤ V\n    refine ⟨?_, fun H W ↦ (le_trans · H)⟩\n    intro H x hxU\n    obtain ⟨W, ⟨h₁, h₂⟩, hxW, hWU⟩ := isTopologicalBasis.exists_subset_of_mem_open hxU U.2\n    exact H ⟨⟨W, h₂⟩, h₁⟩ hWU hxW\n\nopen TopologicalSpace Opens in\n/-- If `X` has a basis of compact opens and `f : X → S` is open, every\ncompact open of `S` is the image of a compact open of `X`. -/\n\nTarget:\nlemma IsOpenMap.exists_opens_image_eq_of_prespectralSpace [PrespectralSpace X] {f : X → Y}\n    (hfc : Continuous f) (h : IsOpenMap f) {U : Set Y} (hs : U ⊆ Set.range f) (hU : IsOpen U)\n    (hc : IsCompact U) : ∃ (V : Opens X), IsCompact V.1 ∧ f '' V = U :=\n\nProof body:\n","rejected":"by\n  obtain ⟨Us, hUs, heq⟩ := TopologicalSpace.Opens.isBasis_iff_cover.mp\n    (PrespectralSpace.isBasis_opens X) ⟨f ⁻¹' U, hU.preimage hfc⟩\n  obtain ⟨t, ht⟩ := by\n    refine hc.elim_finite_subcover (fun s : Us ↦ f '' s.1) (fun s ↦ h _ s.1.2) (fun x hx ↦ ?_)\n    obtain ⟨x, rfl⟩ := hs hx\n    obtain ⟨i, hi, hx⟩ := mem_sSup.mp <| by rwa [← heq]\n    exact Set.mem_iUnion.mpr ⟨⟨i, hi⟩, x, hx, rfl⟩\n  refine ⟨⨆ s ∈ t, s.1, ?_, ?_⟩\n  · simp only [iSup_mk, carrier_eq_coe, coe_mk]\n    exact t.finite_toSet.isCompact_biUnion fun i _ ↦ hUs i.2\n  · simp only [iSup_mk, carrier_eq_coe, Set.iUnion_coe_set, coe_mk, Set.image_iUnion]\n    convert_to ⋃ i ∈ t, f '' i.1 = U\n    · simp\n    · refine subset_antisymm (fun x ↦ ?_) ht\n      simp_rw [Set.mem_iUnion]\n      rintro ⟨i, hi, x, hx, rfl⟩\n      have := heq ▸ mem_sSup.mpr ⟨i.1, i.2, hx⟩\n      exact this\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Spectral","family_id":"isopenmap","file_id":"mathlib/Mathlib/Topology/Spectral/Prespectral.lean","sample_id":"06e54fa94e0402146a1056a8e38c4177d4d7f6c2eb72f119267197e2349465f8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"74e6e717b4029a5bb695646d264a47c3c431da50efbba97f42340e634e0683b0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7736e8daaa8f61cdcc7d061cb79e4a68846e752fbc7a81c150325816b7000c0c","source_sha256":"88899366de27f8f7e91c2113d9cd607e3b4b1f4debab3b6b3a2046a9e9f7c31f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction hs using Nonempty.cons_induction with\n  | singleton => simp [Hlt _ (mem_singleton_self _)]\n  | cons _ _ _ _ ih =>\n    simp only [sum_cons, forall_mem_cons] at Hlt ⊢\n    exact ENat.add_lt_add Hlt.1 (ih Hlt.2)","hard_negative":false,"metrics":{"chosen_tokens":51,"rejected_tokens":2,"token_jaccard":0.030303,"token_length_ratio":0.039216},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"6a3ad530e9773fd254b1cd97a1d286b8209c699291516714cc2bedb7f6f8190e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Ring.Finset\npublic import Mathlib.Algebra.BigOperators.WithTop\npublic import Mathlib.Algebra.Order.BigOperators.Group.Finset\npublic import Mathlib.Data.ENat.Lattice\n\nNamespace:\nENat\n\nLocal context:\n/-\nCopyright (c) 2024 Joachim Breitner, Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joachim Breitner, Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Sum of suprema in `ENat`\n-/\n\npublic section\n\nassert_not_exists Field\n\nnamespace ENat\n\nvariable {a b c d : ℕ∞} {r p q : ℕ}\n\nsection OperationsAndInfty\n\nvariable {α : Type*}\n\n@[simp]\ntheorem toNat_prod {ι : Type*} {s : Finset ι} {f : ι → ℕ∞} :\n    (∏ i ∈ s, f i).toNat = ∏ i ∈ s, (f i).toNat :=\n  map_prod toNatHom _ _\n\ntheorem iInf_sum {ι α : Type*} {f : ι → α → ℕ∞} {s : Finset α} [Nonempty ι]\n    (h : ∀ (t : Finset α) (i j : ι), ∃ k, ∀ a ∈ t, f k a ≤ f i a ∧ f k a ≤ f j a) :\n    ⨅ i, ∑ a ∈ s, f i a = ∑ a ∈ s, ⨅ i, f i a := by\n  induction s using Finset.cons_induction_on with\n  | empty => simp only [Finset.sum_empty, ciInf_const]\n  | cons a s ha ih =>\n    simp only [Finset.sum_cons, ← ih]\n    refine (iInf_add_iInf fun i j => ?_).symm\n    refine (h (Finset.cons a s ha) i j).imp fun k hk => ?_\n    rw [Finset.forall_mem_cons] at hk\n    exact add_le_add hk.1.1 (Finset.sum_le_sum fun a ha => (hk.2 a ha).2)\n\nend OperationsAndInfty\n\nsection Sum\n\nopen Finset\n\nvariable {α : Type*} {s : Finset α} {f : α → ℕ∞}\n\n/-- A product of finite numbers is still finite. -/\nlemma prod_ne_top (h : ∀ a ∈ s, f a ≠ ⊤) : ∏ a ∈ s, f a ≠ ⊤ := WithTop.prod_ne_top h\n\n/-- A product of finite numbers is still finite. -/\nlemma prod_lt_top (h : ∀ a ∈ s, f a < ⊤) : ∏ a ∈ s, f a < ⊤ := WithTop.prod_lt_top h\n\n/-- A sum is infinite iff one of the summands is infinite. -/\n@[simp] lemma sum_eq_top : ∑ x ∈ s, f x = ⊤ ↔ ∃ a ∈ s, f a = ⊤ := WithTop.sum_eq_top\n\n/-- A sum is finite iff all summands are finite. -/\nlemma sum_ne_top : ∑ a ∈ s, f a ≠ ⊤ ↔ ∀ a ∈ s, f a ≠ ⊤ := WithTop.sum_ne_top\n\n/-- A sum is finite iff all summands are finite. -/\n@[simp] lemma sum_lt_top : ∑ a ∈ s, f a < ⊤ ↔ ∀ a ∈ s, f a < ⊤ := WithTop.sum_lt_top\n\ntheorem lt_top_of_sum_ne_top {s : Finset α} {f : α → ℕ∞} (h : ∑ x ∈ s, f x ≠ ⊤) {a : α}\n    (ha : a ∈ s) : f a < ⊤ :=\n  sum_lt_top.1 h.lt_top a ha\n\n/-- Seeing `ℕ∞` as `ℕ` does not change their sum, unless one of the `ℕ∞` is\ninfinity -/\ntheorem toNat_sum {s : Finset α} {f : α → ℕ∞} (hf : ∀ a ∈ s, f a ≠ ⊤) :\n    ENat.toNat (∑ a ∈ s, f a) = ∑ a ∈ s, ENat.toNat (f a) := by\n  rw [← coe_inj, coe_toNat (sum_ne_top.2 hf), Nat.cast_sum]\n  exact sum_congr rfl fun x hx => (coe_toNat (hf x hx)).symm\n\nTarget:\ntheorem sum_lt_sum_of_nonempty {s : Finset α} (hs : s.Nonempty) {f g : α → ℕ∞}\n    (Hlt : ∀ i ∈ s, f i < g i) : ∑ i ∈ s, f i < ∑ i ∈ s, g i :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/ENat","family_id":"sum_lt_sum_of_nonempty","file_id":"mathlib/Mathlib/Data/ENat/BigOperators.lean","sample_id":"7736e8daaa8f61cdcc7d061cb79e4a68846e752fbc7a81c150325816b7000c0c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"defdc0751f53e30ab813268ab9f2c357764562c8bfc11a4ae2b13dad9c027777","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9a7f488b9072e1c3cd3e743c9cf1c9c590011f4701530db321fda7898e28d6e1","source_sha256":"dc7a83c1eb70da8d3ca703a602b6a96706f55b88f4b16cc5437746f0ac16014e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":5,"token_jaccard":0.066667,"token_length_ratio":0.333333},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"6a7f576cef4e95140a4e5d45e5d10c5272cc752092d430b16d0c8655a04c9687","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.GaloisConnection.Basic\n\nNamespace:\nHeyting\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Heyting regular elements\n\nThis file defines Heyting regular elements, elements of a Heyting algebra that are their own double\ncomplement, and proves that they form a Boolean algebra.\n\nFrom a logic standpoint, this means that we can perform classical logic within intuitionistic logic\nby simply double-negating all propositions. This is practical for synthetic computability theory.\n\n## Main declarations\n\n* `IsRegular`: `a` is Heyting-regular if `aᶜᶜ = a`.\n* `Regular`: The subtype of Heyting-regular elements.\n* `Regular.BooleanAlgebra`: Heyting-regular elements form a Boolean algebra.\n\n## References\n\n* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]\n-/\n\n@[expose] public section\n\n-- We want the theorems in this file to be intuitionistic.\nset_option linter.unusedDecidableInType false\n\nopen Function\n\nvariable {α : Type*}\n\nnamespace Heyting\n\nsection Compl\n\nvariable [Compl α] {a : α}\n\n/-- An element of a Heyting algebra is regular if its double complement is itself. -/\ndef IsRegular (a : α) : Prop :=\n  aᶜᶜ = a\n\nprotected theorem IsRegular.eq : IsRegular a → aᶜᶜ = a :=\n  id\n\ninstance IsRegular.decidablePred [DecidableEq α] : @DecidablePred α IsRegular := fun _ =>\n  ‹DecidableEq α› _ _\n\nend Compl\n\nsection HeytingAlgebra\n\nvariable [HeytingAlgebra α] {a b : α}\n\ntheorem isRegular_bot : IsRegular (⊥ : α) := by rw [IsRegular, compl_bot, compl_top]\n\ntheorem isRegular_top : IsRegular (⊤ : α) := by rw [IsRegular, compl_top, compl_bot]\n\nTarget:\ntheorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a ⊓ b) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Heyting","family_id":"isregular","file_id":"mathlib/Mathlib/Order/Heyting/Regular.lean","sample_id":"9a7f488b9072e1c3cd3e743c9cf1c9c590011f4701530db321fda7898e28d6e1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2f6287791c3459636988da88fa63c514ca770a07a4d1537e256b0a039a8246d2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"04c6531705ec9500b581b2d2a796d16ba02ac2e1427cc0cb5352dd3fc63e7c66","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f6e07b9557819a58269647c6b5189639d71a69d808a07410892e1fb416291719","source_sha256":"29ef0dde93e398484e13c344e681af6a10869ecf4559604c0c8bb7026329d317","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases l.exists_antitone_basis with ⟨u, hu⟩\n  rcases (Filter.hasBasis_self.mpr hl'.exists_measurable_subset).exists_antitone_subbasis with\n    ⟨v, v_meas, hv⟩\n  simp only [hu.tendsto_iff hv.toHasBasis, true_imp_iff, true_and, setOf_forall, setOf_exists]\n  exact .iInter fun n ↦ .iUnion fun _ ↦ .biInter (to_countable _) fun i _ ↦\n    (v_meas n).2.preimage (hf i)","hard_negative":true,"metrics":{"chosen_tokens":83,"rejected_tokens":5,"token_jaccard":0.06383,"token_length_ratio":0.060241},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"6ae91d2e8a07a5f7f30f318fd79ffb0596aba5a13b1aecd7fa41f4135057a965","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.MeasurableSpace.Constructions\npublic import Mathlib.Order.Filter.AtTopBot.CompleteLattice\npublic import Mathlib.Order.Filter.AtTopBot.CountablyGenerated\npublic import Mathlib.Order.Filter.SmallSets\npublic import Mathlib.Order.LiminfLimsup\npublic import Mathlib.Tactic.FinCases\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro\n-/\n/-!\n# Measurably generated filters\n\nWe say that a filter `f` is measurably generated if every set `s ∈ f` includes a measurable\nset `t ∈ f`. This property is useful, e.g., to extract a measurable witness of `Filter.Eventually`.\n-/\n\npublic section\n\nopen Set Filter\n\nuniverse uι\n\nvariable {α β γ δ : Type*} {ι : Sort uι}\n\nnamespace MeasurableSpace\n\n/-- The sigma-algebra generated by a single set `s` is `{∅, s, sᶜ, univ}`. -/\n@[simp] theorem generateFrom_singleton (s : Set α) :\n    generateFrom {s} = MeasurableSpace.comap (· ∈ s) ⊤ := by\n  classical\n  letI : MeasurableSpace α := generateFrom {s}\n  refine le_antisymm (generateFrom_le fun t ht => ⟨{True}, trivial, by simp [ht.symm]⟩) ?_\n  rintro _ ⟨u, -, rfl⟩\n  exact (show MeasurableSet s from GenerateMeasurable.basic _ <| mem_singleton s).mem trivial\n\nlemma generateFrom_singleton_le {m : MeasurableSpace α} {s : Set α} (hs : MeasurableSet s) :\n    MeasurableSpace.generateFrom {s} ≤ m :=\n  generateFrom_le (fun _ ht ↦ mem_singleton_iff.1 ht ▸ hs)\n\nlemma comap_indicator_const_le_generateFrom_singleton {M : Type*} [Zero M] [MeasurableSpace M]\n    (s : Set α) (c : M) :\n    MeasurableSpace.comap (s.indicator (fun _ ↦ c)) inferInstance ≤\n      MeasurableSpace.generateFrom {s} :=\n  (measurable_const.indicator (measurableSet_generateFrom (by simp))).comap_le\n\nend MeasurableSpace\n\nnamespace MeasureTheory\n\ntheorem measurableSet_generateFrom_singleton_iff {s t : Set α} :\n    MeasurableSet[MeasurableSpace.generateFrom {s}] t ↔ t = ∅ ∨ t = s ∨ t = sᶜ ∨ t = univ := by\n  simp_rw +instances [MeasurableSpace.generateFrom_singleton]\n  unfold MeasurableSet MeasurableSpace.MeasurableSet' MeasurableSpace.comap\n  simp_rw [MeasurableSpace.measurableSet_top, true_and]\n  constructor\n  · rintro ⟨x, rfl⟩\n    by_cases hT : True ∈ x\n    · by_cases hF : False ∈ x\n      · suffices x = univ by grind\n        grind [univ_eq_true_false]\n      · grind\n    · by_cases hF : False ∈ x\n      · grind\n      · suffices x ⊆ ∅ by grind\n        intro p hp\n        fin_cases p <;> contradiction\n  · rintro (rfl | rfl | rfl | rfl)\n    on_goal 1 => use ∅\n    on_goal 2 => use {True}\n    on_goal 3 => use {False}\n    on_goal 4 => use Set.univ\n    all_goals\n      simp [compl_def]\n\nend MeasureTheory\n\nnamespace Filter\n\nvariable [MeasurableSpace α]\n\n/-- A filter `f` is measurably generated if each `s ∈ f` includes a measurable `t ∈ f`. -/\nclass IsMeasurablyGenerated (f : Filter α) : Prop where\n  exists_measurable_subset : ∀ ⦃s⦄, s ∈ f → ∃ t ∈ f, MeasurableSet t ∧ t ⊆ s\n\ninstance isMeasurablyGenerated_bot : IsMeasurablyGenerated (⊥ : Filter α) :=\n  ⟨fun _ _ => ⟨∅, mem_bot, MeasurableSet.empty, empty_subset _⟩⟩\n\ninstance isMeasurablyGenerated_top : IsMeasurablyGenerated (⊤ : Filter α) :=\n  ⟨fun _s hs => ⟨univ, univ_mem, MeasurableSet.univ, fun x _ => hs x⟩⟩\n\ntheorem Eventually.exists_measurable_mem {f : Filter α} [IsMeasurablyGenerated f] {p : α → Prop}\n    (h : ∀ᶠ x in f, p x) : ∃ s ∈ f, MeasurableSet s ∧ ∀ x ∈ s, p x :=\n  IsMeasurablyGenerated.exists_measurable_subset h\n\ntheorem Eventually.exists_measurable_mem_of_smallSets {f : Filter α} [IsMeasurablyGenerated f]\n    {p : Set α → Prop} (h : ∀ᶠ s in f.smallSets, p s) : ∃ s ∈ f, MeasurableSet s ∧ p s :=\n  let ⟨_s, hsf, hs⟩ := eventually_smallSets.1 h\n  let ⟨t, htf, htm, hts⟩ := IsMeasurablyGenerated.exists_measurable_subset hsf\n  ⟨t, htf, htm, hs t hts⟩\n\ninstance inf_isMeasurablyGenerated (f g : Filter α) [IsMeasurablyGenerated f]\n    [IsMeasurablyGenerated g] : IsMeasurablyGenerated (f ⊓ g) := by\n  constructor\n  rintro t ⟨sf, hsf, sg, hsg, rfl⟩\n  rcases IsMeasurablyGenerated.exists_measurable_subset hsf with ⟨s'f, hs'f, hmf, hs'sf⟩\n  rcases IsMeasurablyGenerated.exists_measurable_subset hsg with ⟨s'g, hs'g, hmg, hs'sg⟩\n  refine ⟨s'f ∩ s'g, inter_mem_inf hs'f hs'g, hmf.inter hmg, ?_⟩\n  exact inter_subset_inter hs'sf hs'sg\n\ntheorem principal_isMeasurablyGenerated_iff {s : Set α} :\n    IsMeasurablyGenerated (𝓟 s) ↔ MeasurableSet s := by\n  refine ⟨?_, fun hs => ⟨fun t ht => ⟨s, mem_principal_self s, hs, ht⟩⟩⟩\n  rintro ⟨hs⟩\n  rcases hs (mem_principal_self s) with ⟨t, ht, htm, hts⟩\n  have : t = s := hts.antisymm ht\n  rwa [← this]\n\nalias ⟨_, _root_.MeasurableSet.principal_isMeasurablyGenerated⟩ :=\n  principal_isMeasurablyGenerated_iff\n\ninstance iInf_isMeasurablyGenerated {f : ι → Filter α} [∀ i, IsMeasurablyGenerated (f i)] :\n    IsMeasurablyGenerated (⨅ i, f i) := by\n  refine ⟨fun s hs => ?_⟩\n  rw [← Equiv.plift.surjective.iInf_comp, mem_iInf] at hs\n  rcases hs with ⟨t, ht, ⟨V, hVf, rfl⟩⟩\n  choose U hUf hU using fun i => IsMeasurablyGenerated.exists_measurable_subset (hVf i)\n  refine ⟨⋂ i : t, U i, ?_, ?_, ?_⟩\n  · rw [← Equiv.plift.surjective.iInf_comp, mem_iInf]\n    exact ⟨t, ht, U, hUf, rfl⟩\n  · haveI := ht.countable.toEncodable.countable\n    exact MeasurableSet.iInter fun i => (hU i).1\n  · exact iInter_mono fun i => (hU i).2\n\nend Filter\n\n/-- The set of points for which a sequence of measurable functions converges to a given value\nis measurable. -/\n@[measurability]\n\nTarget:\nlemma measurableSet_tendsto {_ : MeasurableSpace β} [MeasurableSpace γ]\n    [Countable δ] {l : Filter δ} [l.IsCountablyGenerated]\n    (l' : Filter γ) [l'.IsCountablyGenerated] [hl' : l'.IsMeasurablyGenerated]\n    {f : δ → β → γ} (hf : ∀ i, Measurable (f i)) :\n    MeasurableSet { x | Tendsto (fun n ↦ f n x) l l' } :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_f6e07b955781","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"49f342f124d6d08c8514ce9671c7728ffe347c2e31f59b41de24d33bf7352c27","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/MeasurableSpace","family_id":"measurableset_tendsto","file_id":"mathlib/Mathlib/MeasureTheory/MeasurableSpace/MeasurablyGenerated.lean","sample_id":"f6e07b9557819a58269647c6b5189639d71a69d808a07410892e1fb416291719"},"split":"train"}
{"binding":{"chosen_proof_sha256":"991657dbd4edd454359c31dd8bdf594c016e0afa8edd3fccac619f800a61e43a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ffd1e1b04297070915458a9e6cd813666c832b8a8356d56ffcadd5fc8def94d3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2adf0561a9917b239f775543158667a626bcb1a48a3b86163579bd9e8eb95116","source_sha256":"aa1352e2143e704eafbedf6a7be68b6abc2920999f9fb6e7acf0e05da957ef08","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [pointOfClosedPoint, ← SpecMap_residueFieldIsoBase_inv, ← Spec.map_comp]","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":17,"token_jaccard":0.785714,"token_length_ratio":1.307692},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"6b0c7332ef0556832036e8c1e4b88e9452d121e887677bd78489e2b34c4c3b0f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.Finite\npublic import Mathlib.FieldTheory.IsAlgClosed.Basic\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2026 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Schemes over algebraically closed fields\n\nWe show that if `X` is locally of finite type over an algebraically closed field `k`,\nthen the closed points of `X` are in bijection with the `k`-points of `X`.\nSee `AlgebraicGeometry.pointEquivClosedPoint`.\n\n-/\n\n@[expose] public noncomputable section\n\nopen CategoryTheory\n\nnamespace AlgebraicGeometry\n\nuniverse u\n\nvariable {X Y : Scheme.{u}} {K : Type u} [Field K] [IsAlgClosed K]\n    (f : X ⟶ Spec (.of K)) [LocallyOfFiniteType f] (x : X) (hx : IsClosed {x})\n\n/-- If `X` is a locally of finite type `k`-scheme and `k` is algebraically closed, then\nthe residue field of any closed point of `x` is isomorphic to `k`. -/\ndef residueFieldIsoBase : X.residueField x ≅ .of K :=\n  letI : IsIso (Spec.preimage (X.fromSpecResidueField x ≫ f)) := by\n    have : IsFinite (X.fromSpecResidueField x ≫ f) := by\n      rw [isClosed_singleton_iff_isClosedImmersion] at hx\n      rw [isFinite_iff_locallyOfFiniteType_of_jacobsonSpace]\n      infer_instance\n    rw [ConcreteCategory.isIso_iff_bijective]\n    refine IsAlgClosed.ringHom_bijective_of_isIntegral _ ?_\n    rw [← IsIntegralHom.SpecMap_iff, Spec.map_preimage]\n    infer_instance\n  (asIso (Spec.preimage (X.fromSpecResidueField x ≫ f))).symm\n\n@[simp, reassoc]\nlemma SpecMap_residueFieldIsoBase_inv :\n    Spec.map (residueFieldIsoBase f x hx).inv = X.fromSpecResidueField x ≫ f :=\n  Spec.map_preimage _\n\n/-- If `k` is algebraically closed, this is the `k`-point of `X` associated to a closed point. -/\nnoncomputable\ndef pointOfClosedPoint : Spec (.of K) ⟶ X :=\n  Spec.map (residueFieldIsoBase f x hx).hom ≫ X.fromSpecResidueField x\n\n@[reassoc (attr := simp)]\n\nTarget:\nlemma pointOfClosedPoint_comp : pointOfClosedPoint f x hx ≫ f = 𝟙 _ :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  simp [pointOfClosedPoint, ← SpecMap_residueFieldIsoBase_inv, ← Spec.map_comp]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/AlgClosed","family_id":"pointofclosedpoint_comp","file_id":"mathlib/Mathlib/AlgebraicGeometry/AlgClosed/Basic.lean","sample_id":"2adf0561a9917b239f775543158667a626bcb1a48a3b86163579bd9e8eb95116"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7749888414d06cd3f5ca86c4419d34b5759e4af27af9cf5a4232c6e45cfd9c6a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"099bd67010bd3bdf0391a625cbe08e92d2569c8b6c447860d4abbc5d4f43fc92","source_sha256":"f589366b76253f55a5f00af110f0088b3f55c67577d56940b18639c5f1a2a3b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only [add_comm _ (n : ℝ)] using f.commute_add_int n","hard_negative":false,"metrics":{"chosen_tokens":17,"rejected_tokens":2,"token_jaccard":0.058824,"token_length_ratio":0.117647},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"6b10aa030836b7e419d3385406285d77f7ef146ba48538967b81391736625f8f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.Order.SemiconjSup\n\nNamespace:\nCircleDeg1Lift\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Translation number of a monotone real map that commutes with `x ↦ x + 1`\n\nLet `f : ℝ → ℝ` be a monotone map such that `f (x + 1) = f x + 1` for all `x`. Then the limit\n$$\n  \\tau(f)=\\lim_{n\\to\\infty}{f^n(x)-x}{n}\n$$\nexists and does not depend on `x`. This number is called the *translation number* of `f`.\nDifferent authors use different notation for this number: `τ`, `ρ`, `rot`, etc\n\nIn this file we define a structure `CircleDeg1Lift` for bundled maps with these properties, define\ntranslation number of `f : CircleDeg1Lift`, prove some estimates relating `f^n(x)-x` to `τ(f)`. In\ncase of a continuous map `f` we also prove that `f` admits a point `x` such that `f^n(x)=x+m` if and\nonly if `τ(f)=m/n`.\n\nMaps of this type naturally appear as lifts of orientation-preserving circle homeomorphisms. More\nprecisely, let `f` be an orientation-preserving homeomorphism of the circle $S^1=ℝ/ℤ$, and\nconsider a real number `a` such that\n`⟦a⟧ = f 0`, where `⟦⟧` means the natural projection `ℝ → ℝ/ℤ`. Then there exists a unique\ncontinuous function `F : ℝ → ℝ` such that `F 0 = a` and `⟦F x⟧ = f ⟦x⟧` for all `x` (this fact is\nnot formalized yet). This function is strictly monotone, continuous, and satisfies\n`F (x + 1) = F x + 1`. The number `⟦τ F⟧ : ℝ / ℤ` is called the *rotation number* of `f`.\nIt does not depend on the choice of `a`.\n\n## Main definitions\n\n* `CircleDeg1Lift`: a monotone map `f : ℝ → ℝ` such that `f (x + 1) = f x + 1` for all `x`;\n  the type `CircleDeg1Lift` is equipped with `Lattice` and `Monoid` structures; the\n  multiplication is given by composition: `(f * g) x = f (g x)`.\n* `CircleDeg1Lift.translationNumber`: translation number of `f : CircleDeg1Lift`.\n\n## Main statements\n\nWe prove the following properties of `CircleDeg1Lift.translationNumber`.\n\n* `CircleDeg1Lift.translationNumber_eq_of_dist_bounded`: if the distance between `(f^n) 0`\n  and `(g^n) 0` is bounded from above uniformly in `n : ℕ`, then `f` and `g` have equal\n  translation numbers.\n\n* `CircleDeg1Lift.translationNumber_eq_of_semiconjBy`: if two `CircleDeg1Lift` maps `f`, `g`\n  are semiconjugate by a `CircleDeg1Lift` map, then `τ f = τ g`.\n\n* `CircleDeg1Lift.translationNumber_units_inv`: if `f` is an invertible `CircleDeg1Lift` map\n  (equivalently, `f` is a lift of an orientation-preserving circle homeomorphism), then\n  the translation number of `f⁻¹` is the negative of the translation number of `f`.\n\n* `CircleDeg1Lift.translationNumber_mul_of_commute`: if `f` and `g` commute, then\n  `τ (f * g) = τ f + τ g`.\n\n* `CircleDeg1Lift.translationNumber_eq_rat_iff`: the translation number of `f` is equal to\n  a rational number `m / n` if and only if `(f^n) x = x + m` for some `x`.\n\n* `CircleDeg1Lift.semiconj_of_bijective_of_translationNumber_eq`: if `f` and `g` are two\n  bijective `CircleDeg1Lift` maps and their translation numbers are equal, then these\n  maps are semiconjugate to each other.\n\n* `CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq`: let `f₁` and `f₂` be\n  two actions of a group `G` on the circle by degree 1 maps (formally, `f₁` and `f₂` are two\n  homomorphisms from `G →* CircleDeg1Lift`). If the translation numbers of `f₁ g` and `f₂ g` are\n  equal to each other for all `g : G`, then these two actions are semiconjugate by some\n  `F : CircleDeg1Lift`. This is a version of Proposition 5.4 from [Étienne Ghys, Groupes\n  d'homéomorphismes du cercle et cohomologie bornée][ghys87:groupes].\n\n## Notation\n\nWe use a local notation `τ` for the translation number of `f : CircleDeg1Lift`.\n\n## Implementation notes\n\nWe define the translation number of `f : CircleDeg1Lift` to be the limit of the sequence\n`(f ^ (2 ^ n)) 0 / (2 ^ n)`, then prove that `((f ^ n) x - x) / n` tends to this number for any `x`.\nThis way it is much easier to prove that the limit exists and basic properties of the limit.\n\nWe define translation number for a wider class of maps `f : ℝ → ℝ` instead of lifts of orientation\npreserving circle homeomorphisms for two reasons:\n\n* non-strictly monotone circle self-maps with discontinuities naturally appear as Poincaré maps\n  for some flows on the two-torus (e.g., one can take a constant flow and glue in a few Cherry\n  cells);\n* definition and some basic properties still work for this class.\n\n## References\n\n* [Étienne Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée][ghys87:groupes]\n\n## TODO\n\nHere are some short-term goals.\n\n* Introduce a structure or a typeclass for lifts of circle homeomorphisms. We use\n  `Units CircleDeg1Lift` for now, but it's better to have a dedicated type (or a typeclass?).\n\n* Prove that the `SemiconjBy` relation on circle homeomorphisms is an equivalence relation.\n\n* Introduce `ConditionallyCompleteLattice` structure, use it in the proof of\n  `CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq`.\n\n* Prove that the orbits of the irrational rotation are dense in the circle. Deduce that a\n  homeomorphism with an irrational rotation is semiconjugate to the corresponding irrational\n  translation by a continuous `CircleDeg1Lift`.\n\n## Tags\n\ncircle homeomorphism, rotation number\n-/\n\n@[expose] public section\n\nopen Filter Set Int Topology\nopen Function hiding Commute\n\n/-!\n### Definition and monoid structure\n-/\n\n/-- A lift of a monotone degree one map `S¹ → S¹`. -/\nstructure CircleDeg1Lift : Type extends ℝ →o ℝ where\n  map_add_one' : ∀ x, toFun (x + 1) = toFun x + 1\n\nnamespace CircleDeg1Lift\n\ninstance : FunLike CircleDeg1Lift ℝ ℝ where\n  coe f := f.toFun\n  coe_injective | ⟨⟨_, _⟩, _⟩, ⟨⟨_, _⟩, _⟩, rfl => rfl\n\ninstance : OrderHomClass CircleDeg1Lift ℝ ℝ where\n  map_rel f _ _ h := f.monotone' h\n\n@[simp] theorem coe_mk (f h) : ⇑(mk f h) = f := rfl\n\nvariable (f g : CircleDeg1Lift)\n\n@[simp] theorem coe_toOrderHom : ⇑f.toOrderHom = f := rfl\n\nprotected theorem monotone : Monotone f := f.monotone'\n\n@[gcongr, mono] theorem mono {x y} (h : x ≤ y) : f x ≤ f y := f.monotone h\n\ntheorem strictMono_iff_injective : StrictMono f ↔ Injective f :=\n  f.monotone.strictMono_iff_injective\n\n@[simp]\ntheorem map_add_one : ∀ x, f (x + 1) = f x + 1 :=\n  f.map_add_one'\n\n@[simp]\ntheorem map_one_add (x : ℝ) : f (1 + x) = 1 + f x := by rw [add_comm, map_add_one, add_comm 1]\n\n@[ext]\ntheorem ext ⦃f g : CircleDeg1Lift⦄ (h : ∀ x, f x = g x) : f = g :=\n  DFunLike.ext f g h\n\ninstance : Monoid CircleDeg1Lift where\n  mul f g :=\n    { toOrderHom := f.1.comp g.1\n      map_add_one' := fun x => by simp [map_add_one] }\n  one := ⟨.id, fun _ => rfl⟩\n  mul_one _ := rfl\n  one_mul _ := rfl\n  mul_assoc _ _ _ := DFunLike.coe_injective rfl\n\ninstance : Inhabited CircleDeg1Lift := ⟨1⟩\n\n@[simp]\ntheorem coe_mul : ⇑(f * g) = f ∘ g :=\n  rfl\n\ntheorem mul_apply (x) : (f * g) x = f (g x) :=\n  rfl\n\n@[simp]\ntheorem coe_one : ⇑(1 : CircleDeg1Lift) = id :=\n  rfl\n\ninstance unitsHasCoeToFun : CoeFun CircleDeg1Liftˣ fun _ => ℝ → ℝ :=\n  ⟨fun f => ⇑(f : CircleDeg1Lift)⟩\n\n@[simp]\ntheorem units_inv_apply_apply (f : CircleDeg1Liftˣ) (x : ℝ) :\n    (f⁻¹ : CircleDeg1Liftˣ) (f x) = x := by simp only [← mul_apply, f.inv_mul, coe_one, id]\n\n@[simp]\ntheorem units_apply_inv_apply (f : CircleDeg1Liftˣ) (x : ℝ) :\n    f ((f⁻¹ : CircleDeg1Liftˣ) x) = x := by simp only [← mul_apply, f.mul_inv, coe_one, id]\n\n/-- If a lift of a circle map is bijective, then it is an order automorphism of the line. -/\ndef toOrderIso : CircleDeg1Liftˣ →* ℝ ≃o ℝ where\n  toFun f :=\n    { toFun := f\n      invFun := ⇑f⁻¹\n      left_inv := units_inv_apply_apply f\n      right_inv := units_apply_inv_apply f\n      map_rel_iff' := ⟨fun h => by simpa using mono (↑f⁻¹) h, mono f⟩ }\n  map_one' := rfl\n  map_mul' _ _ := rfl\n\n@[simp]\ntheorem coe_toOrderIso (f : CircleDeg1Liftˣ) : ⇑(toOrderIso f) = f :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderIso_symm (f : CircleDeg1Liftˣ) :\n    ⇑(toOrderIso f).symm = (f⁻¹ : CircleDeg1Liftˣ) :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderIso_inv (f : CircleDeg1Liftˣ) : ⇑(toOrderIso f)⁻¹ = (f⁻¹ : CircleDeg1Liftˣ) :=\n  rfl\n\ntheorem isUnit_iff_bijective {f : CircleDeg1Lift} : IsUnit f ↔ Bijective f :=\n  ⟨fun ⟨u, h⟩ => h ▸ (toOrderIso u).bijective, fun h =>\n    Units.isUnit\n      { val := f\n        inv :=\n          { toFun := (Equiv.ofBijective f h).symm\n            monotone' := fun x y hxy =>\n              (f.strictMono_iff_injective.2 h.1).le_iff_le.1\n                (by simp only [Equiv.ofBijective_apply_symm_apply f h, hxy])\n            map_add_one' := fun x =>\n              h.1 <| by simp only [Equiv.ofBijective_apply_symm_apply f, f.map_add_one] }\n        val_inv := ext <| Equiv.ofBijective_apply_symm_apply f h\n        inv_val := ext <| Equiv.ofBijective_symm_apply_apply f h }⟩\n\ntheorem coe_pow : ∀ n : ℕ, ⇑(f ^ n) = f^[n]\n  | 0 => rfl\n  | n + 1 => by\n    simp [coe_pow n, pow_succ]\n\ntheorem semiconjBy_iff_semiconj {f g₁ g₂ : CircleDeg1Lift} :\n    SemiconjBy f g₁ g₂ ↔ Semiconj f g₁ g₂ :=\n  CircleDeg1Lift.ext_iff\n\ntheorem commute_iff_commute {f g : CircleDeg1Lift} : Commute f g ↔ Function.Commute f g :=\n  CircleDeg1Lift.ext_iff\n\n/-!\n### Translate by a constant\n-/\n\n\n/-- The map `y ↦ x + y` as a `CircleDeg1Lift`. More precisely, we define a homomorphism from\n`Multiplicative ℝ` to `CircleDeg1Liftˣ`, so the translation by `x` is\n`translation (Multiplicative.ofAdd x)`. -/\ndef translate : Multiplicative ℝ →* CircleDeg1Liftˣ := MonoidHom.toHomUnits <|\n  { toFun x := ⟨⟨fun y => x.toAdd + y, add_right_mono⟩, fun _ => (add_assoc ..).symm⟩\n    map_one' := ext zero_add\n    map_mul' _ _ := ext <| add_assoc _ _ }\n\n@[simp]\ntheorem translate_apply (x y : ℝ) : translate (Multiplicative.ofAdd x) y = x + y :=\n  rfl\n\n@[simp]\ntheorem translate_inv_apply (x y : ℝ) : (translate <| Multiplicative.ofAdd x)⁻¹ y = -x + y :=\n  rfl\n\n@[simp]\ntheorem translate_zpow (x : ℝ) (n : ℤ) :\n    translate (Multiplicative.ofAdd x) ^ n = translate (Multiplicative.ofAdd <| ↑n * x) := by\n  simp only [← zsmul_eq_mul, ofAdd_zsmul, map_zpow]\n\n@[simp]\ntheorem translate_pow (x : ℝ) (n : ℕ) :\n    translate (Multiplicative.ofAdd x) ^ n = translate (Multiplicative.ofAdd <| ↑n * x) :=\n  translate_zpow x n\n\n@[simp]\ntheorem translate_iterate (x : ℝ) (n : ℕ) :\n    (translate (Multiplicative.ofAdd x))^[n] = translate (Multiplicative.ofAdd <| ↑n * x) := by\n  rw [← coe_pow, ← Units.val_pow_eq_pow_val, translate_pow]\n\n/-!\n### Commutativity with integer translations\n\nIn this section we prove that `f` commutes with translations by an integer number.\nFirst we formulate these statements (for a natural or an integer number,\naddition on the left or on the right, addition or subtraction) using `Function.Commute`,\nthen reformulate as `simp` lemmas `map_int_add` etc.\n-/\n\ntheorem commute_nat_add (n : ℕ) : Function.Commute f (n + ·) := by\n  simpa only [nsmul_one, add_left_iterate] using Function.Commute.iterate_right f.map_one_add n\n\ntheorem commute_add_nat (n : ℕ) : Function.Commute f (· + n) := by\n  simp only [add_comm _ (n : ℝ), f.commute_nat_add n]\n\ntheorem commute_sub_nat (n : ℕ) : Function.Commute f (· - n) := by\n  simpa only [sub_eq_add_neg] using!\n    (f.commute_add_nat n).inverses_right (Equiv.addRight _).right_inv (Equiv.addRight _).left_inv\n\nTarget:\ntheorem commute_add_int : ∀ n : ℤ, Function.Commute f (· + n)\n  | (n : ℕ) => f.commute_add_nat n\n  | -[n+1] => by simpa [sub_eq_add_neg] using f.commute_sub_nat (n + 1)\n\ntheorem commute_int_add (n : ℤ) : Function.Commute f (n + ·) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/Circle","family_id":"commute_add_int","file_id":"mathlib/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean","sample_id":"099bd67010bd3bdf0391a625cbe08e92d2569c8b6c447860d4abbc5d4f43fc92"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8e1e8082a0a84f2600d4003971ff82d46d02db0f4b2396697813b11973e2972c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"660e9e4b92da9e1464e29ce063157b6ff472ec59e69123eed2b8146bcadb1cb1","source_sha256":"74b0e866ba6855a98543cc59c9d908f25a74c6e6456b4e8b44974d8edb78dfd2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rintro ⟨p⟩\n  rw [← @WeaklyConnectedComponent.eq (Generators G), eq_comm, ← FreeGroup.of_injective.eq_iff, ←\n    mul_inv_eq_one]\n  let X := FreeGroup (WeaklyConnectedComponent <| Generators G)\n  let f : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)\n  let F : G ⥤ CategoryTheory.SingleObj.{u} (X : Type u) := SingleObj.differenceFunctor f\n  change (F.map p) = ((@CategoryTheory.Functor.const G _ _ (SingleObj.category X)).obj ()).map p\n  congr; ext\n  rw [Functor.const_obj_map, id_as_one, differenceFunctor_map, @mul_inv_eq_one _ _ (f _)]\n  apply congr_arg FreeGroup.of\n  apply (WeaklyConnectedComponent.eq _ _).mpr\n  exact ⟨Hom.toPath (Sum.inr (by assumption))⟩","hard_negative":false,"metrics":{"chosen_tokens":168,"rejected_tokens":2,"token_jaccard":0.014493,"token_length_ratio":0.011905},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"6b2c493e064f4ac62cadbade8561dcd815bd28b9332d632c5a3b79c3d2373a24","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Action\npublic import Mathlib.Combinatorics.Quiver.Arborescence\npublic import Mathlib.Combinatorics.Quiver.ConnectedComponent\npublic import Mathlib.GroupTheory.FreeGroup.IsFreeGroup\n\nNamespace:\nIsFreeGroupoid\n\nLocal context:\n/-\nCopyright (c) 2021 David Wärn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David Wärn\n-/\n/-!\n# The Nielsen-Schreier theorem\n\nThis file proves that a subgroup of a free group is itself free.\n\n## Main result\n\n- `subgroupIsFreeOfIsFree H`: an instance saying that a subgroup of a free group is free.\n\n## Proof overview\n\nThe proof is analogous to the proof using covering spaces and fundamental groups of graphs,\nbut we work directly with groupoids instead of topological spaces. Under this analogy,\n\n- `IsFreeGroupoid G` corresponds to saying that a space is a graph.\n- `endMulEquivSubgroup H` plays the role of replacing 'subgroup of fundamental group' with\n  'fundamental group of covering space'.\n- `actionGroupoidIsFree G A` corresponds to the fact that a covering of a (single-vertex)\n  graph is a graph.\n- `endIsFree T` corresponds to the fact that, given a spanning tree `T` of a\n  graph, its fundamental group is free (generated by loops from the complement of the tree).\n\n## Implementation notes\n\nOur definition of `IsFreeGroupoid` is nonstandard. Normally one would require that functors\n`G ⥤ X` to any _groupoid_ `X` are given by graph homomorphisms from the generators, but we only\nconsider _groups_ `X`. This simplifies the argument since functor equality is complicated in\ngeneral, but simple for functors to single object categories.\n\n## References\n\nhttps://ncatlab.org/nlab/show/Nielsen-Schreier+theorem\n\n## Tags\n\nfree group, free groupoid, Nielsen-Schreier\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nuniverse v u\n\nopen CategoryTheory CategoryTheory.ActionCategory CategoryTheory.SingleObj Quiver FreeGroup\n\n/-- `IsFreeGroupoid.Generators G` is a type synonym for `G`. We think of this as\nthe vertices of the generating quiver of `G` when `G` is free. We can't use `G` directly,\nsince `G` already has a quiver instance from being a groupoid. -/\n@[nolint unusedArguments]\ndef IsFreeGroupoid.Generators (G) [Groupoid G] :=\n  G\n\n/--\nA groupoid `G` is free when we have the following data:\n- a quiver on `IsFreeGroupoid.Generators G` (a type synonym for `G`)\n- a function `of` taking a generating arrow to a morphism in `G`\n- such that a functor from `G` to any group `X` is uniquely determined\n  by assigning labels in `X` to the generating arrows.\n\nThis definition is nonstandard. Normally one would require that functors `G ⥤ X`\nto any _groupoid_ `X` are given by graph homomorphisms from `generators`. -/\nclass IsFreeGroupoid (G) [Groupoid.{v} G] where\n  quiverGenerators : Quiver.{v} (IsFreeGroupoid.Generators G)\n  of : ∀ {a b : IsFreeGroupoid.Generators G}, (a ⟶ b) → ((show G from a) ⟶ b)\n  unique_lift :\n    ∀ {X : Type v} [Group X] (f : Labelling (IsFreeGroupoid.Generators G) X),\n      ∃! F : G ⥤ CategoryTheory.SingleObj X, ∀ (a b) (g : a ⟶ b), F.map (of g) = f g\n\nattribute [nolint docBlame] IsFreeGroupoid.of IsFreeGroupoid.unique_lift\n\nnamespace IsFreeGroupoid\n\nattribute [instance_reducible, instance] quiverGenerators\n\n/-- Two functors from a free groupoid to a group are equal when they agree on the generating\nquiver. -/\n@[ext]\ntheorem ext_functor {G} [Groupoid.{v} G] [IsFreeGroupoid G] {X : Type v} [Group X]\n    (f g : G ⥤ CategoryTheory.SingleObj X) (h : ∀ (a b) (e : a ⟶ b), f.map (of e) = g.map (of e)) :\n    f = g :=\n  let ⟨_, _, u⟩ := @unique_lift G _ _ X _ fun (a b : Generators G) (e : a ⟶ b) => g.map (of e)\n  _root_.trans (u _ h) (u _ fun _ _ _ => rfl).symm\n\n/-- An action groupoid over a free group is free. More generally, one could show that the groupoid\nof elements over a free groupoid is free, but this version is easier to prove and suffices for our\npurposes.\n\nAnalogous to the fact that a covering space of a graph is a graph. (A free groupoid is like a graph,\nand a groupoid of elements is like a covering space.) -/\ninstance actionGroupoidIsFree {G A : Type u} [Group G] [IsFreeGroup G] [MulAction G A] :\n    IsFreeGroupoid (ActionCategory G A) where\n  quiverGenerators :=\n    ⟨fun a b => { e : IsFreeGroup.Generators G // IsFreeGroup.of e • a.back = b.back }⟩\n  of := fun (e : Subtype _) => ⟨IsFreeGroup.of e, e.property⟩\n  unique_lift := by\n    intro X _ f\n    let f' : IsFreeGroup.Generators G → (A → X) ⋊[mulAutArrow] G := fun e =>\n      ⟨fun b => @f ⟨(), _⟩ ⟨(), b⟩ ⟨e, smul_inv_smul _ b⟩, IsFreeGroup.of e⟩\n    rcases IsFreeGroup.unique_lift f' with ⟨F', hF', uF'⟩\n    refine ⟨uncurry F' ?_, ?_, ?_⟩\n    · suffices SemidirectProduct.rightHom.comp F' = MonoidHom.id _ by\n        exact DFunLike.ext_iff.mp this\n      apply IsFreeGroup.ext_hom (fun x ↦ ?_)\n      rw [MonoidHom.comp_apply, hF']\n      rfl\n    · rintro ⟨⟨⟩, a : A⟩ ⟨⟨⟩, b⟩ ⟨e, h : IsFreeGroup.of e • a = b⟩\n      change (F' (IsFreeGroup.of _)).left _ = _\n      rw [hF']\n      cases inv_smul_eq_iff.mpr h.symm\n      rfl\n    · intro E hE\n      have : curry E = F' := by\n        apply uF'\n        intro e\n        ext\n        · convert! hE _ _ _\n          rfl\n        · rfl\n      apply Functor.hext\n      · intro\n        apply Unit.ext\n      · refine ActionCategory.cases ?_\n        intros\n        simp only [← this, uncurry_map, curry_apply_left, coe_back, homOfPair.val]\n        rfl\n\nnamespace SpanningTree\n\n/- In this section, we suppose we have a free groupoid with a spanning tree for its generating\nquiver. The goal is to prove that the vertex group at the root is free. A picture to have in mind\nis that we are 'pulling' the endpoints of all the edges of the quiver along the spanning tree to\nthe root. -/\nvariable {G : Type u} [Groupoid.{u} G] [IsFreeGroupoid G]\n  (T : WideSubquiver (Symmetrify <| Generators G)) [Arborescence T]\n\nset_option backward.privateInPublic true in\n/-- The root of `T`, except its type is `G` instead of the type synonym `T`. -/\nprivate def root' : G :=\n  show T from root T\n\n-- this has to be marked noncomputable, see issue https://github.com/leanprover-community/mathlib4/pull/451.\n-- It might be nicer to define this in terms of `composePath`\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- A path in the tree gives a hom, by composition. -/\ndef homOfPath : ∀ {a : G}, Path (root T) a → (root' T ⟶ a)\n  | _, Path.nil => 𝟙 _\n  | _, Path.cons p f => homOfPath p ≫ Sum.recOn f.val (fun e => of e) fun e => inv (of e)\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- For every vertex `a`, there is a canonical hom from the root, given by the path in the tree. -/\ndef treeHom (a : G) : root' T ⟶ a :=\n  homOfPath T default\n\n/-- Any path to `a` gives `treeHom T a`, since paths in the tree are unique. -/\ntheorem treeHom_eq {a : G} (p : Path (root T) a) : treeHom T a = homOfPath T p := by\n  rw [treeHom, Unique.default_eq]\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n@[simp]\ntheorem treeHom_root : treeHom T (root' T) = 𝟙 _ :=\n  -- this should just be `treeHom_eq T Path.nil`, but Lean treats `homOfPath` with suspicion.\n    _root_.trans\n    (treeHom_eq T Path.nil) rfl\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Any hom in `G` can be made into a loop, by conjugating with `treeHom`s. -/\ndef loopOfHom {a b : G} (p : a ⟶ b) : End (root' T) :=\n  treeHom T a ≫ p ≫ inv (treeHom T b)\n\nset_option backward.isDefEq.respectTransparency false in\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Turning an edge in the spanning tree into a loop gives the identity loop. -/\ntheorem loopOfHom_eq_id {a b : Generators G} (e) (H : e ∈ wideSubquiverSymmetrify T a b) :\n    loopOfHom T (of e) = 𝟙 (root' T) := by\n  rw [loopOfHom, ← Category.assoc, IsIso.comp_inv_eq, Category.id_comp]\n  rcases H with H | H\n  · rw [treeHom_eq T (Path.cons default ⟨Sum.inl e, H⟩), homOfPath]\n    rfl\n  · rw [treeHom_eq T (Path.cons default ⟨Sum.inr e, H⟩), homOfPath]\n    simp only [IsIso.inv_hom_id, Category.comp_id, Category.assoc, treeHom]\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Since a hom gives a loop, any homomorphism from the vertex group at the root\nextends to a functor on the whole groupoid. -/\n@[simps]\ndef functorOfMonoidHom {X} [Monoid X] (f : End (root' T) →* X) :\n    G ⥤ CategoryTheory.SingleObj X where\n  obj _ := ()\n  map p := f (loopOfHom T p)\n  map_id := by\n    intro a\n    dsimp only [loopOfHom]\n    rw [Category.id_comp, IsIso.hom_inv_id, ← End.one_def, f.map_one, id_as_one]\n  map_comp := by\n    intros\n    rw [comp_as_mul, ← f.map_mul]\n    simp only [IsIso.inv_hom_id_assoc, loopOfHom, End.mul_def, Category.assoc]\n\nset_option backward.isDefEq.respectTransparency false in\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nopen scoped Classical in\n/-- Given a free groupoid and an arborescence of its generating quiver, the vertex\ngroup at the root is freely generated by loops coming from generating arrows\nin the complement of the tree. -/\nlemma endIsFree : IsFreeGroup (End (root' T)) :=\n  IsFreeGroup.ofUniqueLift ((wideSubquiverEquivSetTotal <| wideSubquiverSymmetrify T)ᶜ : Set _)\n    (fun e => loopOfHom T (of e.val.hom))\n    (by\n      intro X _ f\n      let f' : Labelling (Generators G) X := fun a b e =>\n        if h : e ∈ wideSubquiverSymmetrify T a b then 1 else f ⟨⟨a, b, e⟩, h⟩\n      rcases unique_lift f' with ⟨F', hF', uF'⟩\n      refine ⟨F'.mapEnd _, ?_, ?_⟩\n      · suffices ∀ {x y} (q : x ⟶ y), F'.map (loopOfHom T q) = (F'.map q : X) by\n          rintro ⟨⟨a, b, e⟩, h⟩\n          simp only [Functor.mapEnd, DFunLike.coe, this, hF']\n          exact dif_neg h\n        intro x y q\n        suffices ∀ {a} (p : Path (root T) a), F'.map (homOfPath T p) = 1 by\n          simp only [this, treeHom, comp_as_mul, inv_as_inv, loopOfHom, inv_one, mul_one,\n            one_mul, Functor.map_inv, Functor.map_comp]\n        intro a p\n        induction p with\n        | nil => rw [homOfPath, F'.map_id, id_as_one]\n        | cons p e ih =>\n          rw [homOfPath, F'.map_comp, comp_as_mul, ih, mul_one]\n          rcases e with ⟨e | e, eT⟩\n          · rw [hF']\n            exact dif_pos (Or.inl eT)\n          · rw [F'.map_inv, inv_as_inv, inv_eq_one, hF']\n            exact dif_pos (Or.inr eT)\n      · intro E hE\n        ext x\n        suffices (functorOfMonoidHom T E).map x = F'.map x by\n          simpa only [loopOfHom, functorOfMonoidHom, IsIso.inv_id, treeHom_root,\n            Category.id_comp, Category.comp_id] using! this\n        congr\n        apply uF'\n        intro a b e\n        change E (loopOfHom T _) = dite _ _ _\n        split_ifs with h\n        · rw [loopOfHom_eq_id T e h, ← End.one_def, E.map_one]\n        · exact hE ⟨⟨a, b, e⟩, h⟩)\n\nend SpanningTree\n\nset_option backward.privateInPublic true in\n/-- Another name for the identity function `G → G`, to help type checking. -/\nprivate def symgen {G : Type u} [Groupoid.{v} G] [IsFreeGroupoid G] :\n    G → Symmetrify (Generators G) :=\n  id\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- If there exists a morphism `a → b` in a free groupoid, then there also exists a zigzag\nfrom `a` to `b` in the generating quiver. -/\n\nTarget:\ntheorem path_nonempty_of_hom {G} [Groupoid.{u, u} G] [IsFreeGroupoid G] {a b : G} :\n    Nonempty (a ⟶ b) → Nonempty (Path (symgen a) (symgen b)) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/FreeGroup","family_id":"path_nonempty_of_hom","file_id":"mathlib/Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean","sample_id":"660e9e4b92da9e1464e29ce063157b6ff472ec59e69123eed2b8146bcadb1cb1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"12eef625fe3216831384e04422da6efdbadfd13d594c33e0c21289712e29f94e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"792bc3b530c92020c59f266814860e6bb84561275f3f929fc99524afe7ca96ed","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3a3c329ac7ef7bd97b6db71b43a7efe1eaaa001c53c4b487b813a42f94dd4a0c","source_sha256":"d7136b89194da5b89655bfe0eeb14ceccd917f2ca80d6a836d008fec03fc59c2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  rw [monomial_def]\n  exact Pi.single_eq_same _ _","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":5,"token_jaccard":0.230769,"token_length_ratio":0.416667},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"6b57fc5b26c9de708910d4bdfe077ff55f862c59480538f36ce537c67415ec35","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Antidiag.Finsupp\npublic import Mathlib.Data.Finsupp.Weight\npublic import Mathlib.LinearAlgebra.Pi\npublic import Mathlib.Algebra.MvPolynomial.Basic\npublic import Mathlib.Tactic.NormNum\n\nNamespace:\nMvPowerSeries\n\nLocal context:\n/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Kenny Lau\n-/\n/-!\n# Formal (multivariate) power series\n\nThis file defines multivariate formal power series\nand develops the basic properties of these objects.\n\nA formal power series is to a polynomial like an infinite sum is to a finite sum.\n\nWe provide the natural inclusion from multivariate polynomials to multivariate formal power series.\n\n## Main definitions\n\n- `MvPowerSeries.C`: constant power series\n\n- `MvPowerSeries.X`: the indeterminates\n\n- `MvPowerSeries.coeff`, `MvPowerSeries.constantCoeff`:\n  the coefficients of a `MvPowerSeries`, its constant coefficient\n\n- `MvPowerSeries.monomial`: the monomials\n\n- `MvPowerSeries.coeff_mul`: computes the coefficients of the product of two `MvPowerSeries`\n\n- `MvPowerSeries.coeff_prod` : computes the coefficients of products of `MvPowerSeries`\n\n- `MvPowerSeries.coeff_pow` : computes the coefficients of powers of a `MvPowerSeries`\n\n- `MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent`: if the constant coefficient\n  of a `MvPowerSeries` is nilpotent, then some coefficients of its powers are automatically zero\n\n- `MvPowerSeries.map`: apply a `RingHom` to the coefficients of a `MvPowerSeries` (as a `RingHom`).\n\n- `MvPowerSeries.X_pow_dvd_iff`, `MvPowerSeries.X_dvd_iff`: equivalent\n  conditions for (a power of) an indeterminate to divide a `MvPowerSeries`\n\n- `MvPolynomial.toMvPowerSeries`: the canonical coercion from `MvPolynomial` to `MvPowerSeries`\n\n\n## Note\n\nThis file sets up the (semi)ring structure on multivariate power series:\nadditional results are in:\n* `Mathlib/RingTheory/MvPowerSeries/Inverse.lean` : invertibility,\n  formal power series over a local ring form a local ring;\n* `Mathlib/RingTheory/MvPowerSeries/Trunc.lean`: truncation of power series.\n\nIn `Mathlib/RingTheory/PowerSeries/Basic.lean`, formal power series in one variable\nwill be obtained as a particular case, defined by\n  `PowerSeries R := MvPowerSeries Unit R`.\nSee that file for a specific description.\n\n## Implementation notes\n\nIn this file we define multivariate formal power series with\nvariables indexed by `σ` and coefficients in `R` as\n`MvPowerSeries σ R := (σ →₀ ℕ) → R`.\nUnfortunately there is not yet enough API to show that they are the completion\nof the ring of multivariate polynomials. However, we provide most of the infrastructure\nthat is needed to do this. Once I-adic completion (topological or algebraic) is available\nit should not be hard to fill in the details.\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen Finset (antidiagonal mem_antidiagonal)\n\n/-- Multivariate formal power series, where `σ` is the index set of the variables\nand `R` is the coefficient ring. -/\ndef MvPowerSeries (σ : Type*) (R : Type*) :=\n  (σ →₀ ℕ) → R\n\nnamespace MvPowerSeries\n\nopen Finsupp\n\nvariable {σ R : Type*}\n\ninstance [Inhabited R] : Inhabited (MvPowerSeries σ R) :=\n  inferInstanceAs <| Inhabited ((σ →₀ ℕ) → R)\n\ninstance [Zero R] : Zero (MvPowerSeries σ R) :=\n  inferInstanceAs <| Zero ((σ →₀ ℕ) → R)\n\ninstance [AddMonoid R] : AddMonoid (MvPowerSeries σ R) :=\n  inferInstanceAs <| AddMonoid ((σ →₀ ℕ) → R)\n\ninstance [AddGroup R] : AddGroup (MvPowerSeries σ R) :=\n  inferInstanceAs <| AddGroup ((σ →₀ ℕ) → R)\n\ninstance [AddCommMonoid R] : AddCommMonoid (MvPowerSeries σ R) :=\n  inferInstanceAs <| AddCommMonoid ((σ →₀ ℕ) → R)\n\ninstance [AddCommGroup R] : AddCommGroup (MvPowerSeries σ R) :=\n  inferInstanceAs <| AddCommGroup ((σ →₀ ℕ) → R)\n\ninstance [Nontrivial R] : Nontrivial (MvPowerSeries σ R) :=\n  inferInstanceAs <| Nontrivial ((σ →₀ ℕ) → R)\n\ninstance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R (MvPowerSeries σ A) :=\n  inferInstanceAs <| Module R ((σ →₀ ℕ) → A)\n\ninstance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S]\n    [IsScalarTower R S A] : IsScalarTower R S (MvPowerSeries σ A) :=\n  inferInstanceAs <| IsScalarTower R S ((σ →₀ ℕ) → A)\n\nsection Semiring\n\nvariable [Semiring R]\n\n/-- The `n`th monomial as multivariate formal power series:\n  it is defined as the `R`-linear map from `R` to the semiring\n  of multivariate formal power series associating to each `a`\n  the map sending `n : σ →₀ ℕ` to the value `a`\n  and sending all other `x : σ →₀ ℕ` different from `n` to `0`. -/\ndef monomial (n : σ →₀ ℕ) : R →ₗ[R] MvPowerSeries σ R :=\n  letI := Classical.decEq σ\n  LinearMap.single R (fun _ ↦ R) n\n\n/-- The `n`th coefficient of a multivariate formal power series. -/\ndef coeff (n : σ →₀ ℕ) : MvPowerSeries σ R →ₗ[R] R :=\n  LinearMap.proj n\n\ntheorem coeff_apply (f : MvPowerSeries σ R) (d : σ →₀ ℕ) : coeff d f = f d :=\n  rfl\n\n/-- Two multivariate formal power series are equal if all their coefficients are equal. -/\n@[ext]\ntheorem ext {φ ψ : MvPowerSeries σ R} (h : ∀ n : σ →₀ ℕ, coeff n φ = coeff n ψ) : φ = ψ :=\n  funext h\n\n/-- Two multivariate formal power series are equal\nif and only if all their coefficients are equal. -/\nadd_decl_doc MvPowerSeries.ext_iff\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem monomial_def [DecidableEq σ] (n : σ →₀ ℕ) :\n    monomial n = LinearMap.single R (fun _ ↦ R) n := by\n  rw [monomial]\n  -- unify the `Decidable` arguments\n  convert! rfl\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem coeff_monomial [DecidableEq σ] (m n : σ →₀ ℕ) (a : R) :\n    coeff m (monomial n a) = if m = n then a else 0 := by\n  dsimp only [coeff, MvPowerSeries]\n  rw [monomial_def, LinearMap.proj_apply (i := m), LinearMap.single_apply, Pi.single_apply]\n\n@[simp]\n\nTarget:\ntheorem coeff_monomial_same (n : σ →₀ ℕ) (a : R) : coeff n (monomial n a) = a :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_3a3c329ac7ef","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"80fc3728752a516898d07abf38ea6c636e6b9d30e9a52332e0117e665d609f1a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/MvPowerSeries","family_id":"coeff_monomial_same","file_id":"mathlib/Mathlib/RingTheory/MvPowerSeries/Basic.lean","sample_id":"3a3c329ac7ef7bd97b6db71b43a7efe1eaaa001c53c4b487b813a42f94dd4a0c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e99a691cbba97dd50b462f822d8ea54c6c328e4bf1d6437c6b809f75afb37a33","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"29d94a2fea1431fff864f71743a56f3cc63550de8f628d8bb502ce52793c91f8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"71930e9f307d51dc22d21feef3aebed1568f0f6f3ee70012583f25208fbd5c02","source_sha256":"864301aaa8f40ca7deda0f7c4364088256abebe91b440dcc329dc526e83462dd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have key : ∀ x ∈ hs.toFinset, s.indicator (fun _ ↦ a) x = a := by\n    intro x hx\n    rw [indicator_of_mem (hs.mem_toFinset.1 hx) (fun _ ↦ a)]\n  rw [Nat.card_eq_card_finite_toFinset hs, ← sum_eq_card_nsmul key, eventually_atTop]\n  obtain ⟨m, hm⟩ := hs.bddAbove\n  refine ⟨m + 1, fun n n_m ↦ (sum_subset ?_ ?_).symm⟩ <;> intro x <;> rw [hs.mem_toFinset]\n  · rw [Finset.mem_range]\n    exact fun x_s ↦ ((mem_upperBounds.1 hm) x x_s).trans_lt (Nat.lt_of_succ_le n_m)\n  · exact fun _ x_s ↦ indicator_of_notMem x_s (fun _ ↦ a)","hard_negative":true,"metrics":{"chosen_tokens":146,"rejected_tokens":2,"token_jaccard":0.016667,"token_length_ratio":0.013699},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"6b71a8d59f9fd0d3e53fa9335ecda90cd757a2ec34da85e4bb882746c0e3fc6c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Indicator\npublic import Mathlib.Algebra.Order.Archimedean.Basic\npublic import Mathlib.Algebra.Order.BigOperators.Group.Finset\npublic import Mathlib.Algebra.Order.Group.Indicator\npublic import Mathlib.Order.LiminfLimsup\npublic import Mathlib.SetTheory.Cardinal.Finite\n\nNamespace:\nSet\n\nLocal context:\n/-\nCopyright (c) 2024 Damien Thomine. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Damien Thomine\n-/\n/-!\n# Cardinality and limit of sum of indicators\nThis file contains results relating the cardinality of subsets of ℕ and limits,\nlimsups of sums of indicators.\n\n## Tags\nfinite, indicator, limsup, tendsto\n-/\n\npublic section\n\nnamespace Set\n\nopen Filter Finset\n\nTarget:\nlemma sum_indicator_eventually_eq_card {α : Type*} [AddCommMonoid α] (a : α) {s : Set ℕ}\n    (hs : s.Finite) :\n    ∀ᶠ n in atTop, ∑ k ∈ Finset.range n, s.indicator (fun _ ↦ a) k = (Nat.card s) • a :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_71930e9f307d","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"3c908ace2349f6a695ba3590395460653d9e87667f67513c1b144d979648c6db","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Order","family_id":"sum_indicator_eventually_eq_card","file_id":"mathlib/Mathlib/Algebra/Order/Archimedean/IndicatorCard.lean","sample_id":"71930e9f307d51dc22d21feef3aebed1568f0f6f3ee70012583f25208fbd5c02"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8af251c6a8eb77af2dfa75888d7c394a97af0ecb8aa758dfff389aa06e6d13fe","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6a8e8150ef6c1755c0f36af56608a5c55a4714f68f327a0666a7f1b00c8049b8","source_sha256":"b6022f061f06753f0797fa4fa7ab43cc6fb4c7d61abaa7b4bb4707d4dc0299c4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← LinearMap.comp_assoc, map_map_comp_assoc_eq]\n  ext\n  rfl","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.25},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"6b7ee87143d14f8efe72be43fea5fb1a15f3de815c197c659eb56bed7306df3f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorProduct.Tower\npublic import Mathlib.RingTheory.Coalgebra.Basic\nimport Mathlib.Tactic.Attr.Register\n\nNamespace:\nCoassocSimps\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang, Yaël Dillies\n-/\n/-!\n# Tactic to reassociate comultiplication in a coalgebra\n\n`coassoc_simps` is a simp set useful to prove tautologies on coalgebras.\n\nThe general algorithm it follows is to push the associators `TensorProduct.assoc` and\ncommutators `TensorProduct.comm` inwards (to the right) until they cancel against\nco-multiplications.\n\nThe simp set makes the following choice of normal form\n* It regards `TensorProduct.map`, `TensorProduct.assoc`, `TensorProduct.comm` as the primitive\n  constructions and rewrites everything else such as `lTensor`, `leftComm` using them.\n* It rewrites both sides into a right associated composition of linear maps.\n  In particular `LinearMap.comp_assoc` and `LinearEquiv.coe_trans` are tagged.\n* It rewrites `(f₂ ⊗ g₂) ∘ (f₁ ⊗ g₁)` into `(f₂ ∘ f₁) ⊗ (g₂ ∘ g₁)`.\n\n## Notes\n\n- It is not confluent with `(ε ⊗ₘ id) ∘ₗ δ = λ⁻¹`.\n  It is often useful to `trans` (or `calc`) with a term containing\n  `(ε ⊗ₘ _) ∘ₗ δ` or `(_ ⊗ₘ ε) ∘ₗ δ`,\n  and use one of `map_counit_comp_comul_left` `map_counit_comp_comul_right`\n  `map_counit_comp_comul_left_assoc` `map_counit_comp_comul_right_assoc` to continue.\n\n- Some lemmas (e.g. `lid_comp_map : λ ∘ₗ (f ⊗ₘ g) = g ∘ₗ λ ∘ₗ (f ⊗ₘ id)`) loops when tagged as simp,\n  so we wrap it inside a rudimentary simproc that only fires when `g ≠ id`.\n-/\n\n@[expose] public section\n\nopen TensorProduct\n\nopen LinearMap (id)\nopen Coalgebra\n\nopen Qq\nnamespace CoassocSimps\n\nvariable {R A M N P M' N' P' Q Q' M₁ M₂ M₃ N₁ N₂ N₃ : Type*}\n    [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A]\n    [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P]\n    [AddCommMonoid M'] [Module R M'] [AddCommMonoid N'] [Module R N']\n    [AddCommMonoid P'] [Module R P'] [AddCommMonoid Q] [Module R Q] [AddCommMonoid Q'] [Module R Q']\n    [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃]\n    [AddCommMonoid N₁] [AddCommMonoid N₂] [AddCommMonoid N₃]\n    [Module R M₁] [Module R M₂] [Module R M₃] [Module R N₁] [Module R N₂] [Module R N₃]\n\nlocal notation3 \"α\" => (TensorProduct.assoc R _ _ _).toLinearMap\nlocal notation3 \"α⁻¹\" => (TensorProduct.assoc R _ _ _).symm.toLinearMap\nlocal notation3 \"λ\" => (TensorProduct.lid R _).toLinearMap\nlocal notation3 \"λ⁻¹\" => (TensorProduct.lid R _).symm.toLinearMap\nlocal notation3 \"ρ\" => (TensorProduct.rid R _).toLinearMap\nlocal notation3 \"ρ⁻¹\" => (TensorProduct.rid R _).symm.toLinearMap\nlocal notation3 \"β\" => (TensorProduct.comm R _ _).toLinearMap\nlocal infix:90 \" ⊗ₘ \" => TensorProduct.map\nlocal notation3 \"δ\" => comul (R := R)\nlocal notation3 \"ε\" => counit (R := R)\n\nattribute [coassoc_simps] LinearMap.comp_id LinearMap.id_comp TensorProduct.map_id\n  LinearMap.lTensor_def LinearMap.rTensor_def LinearMap.comp_assoc\n  LinearEquiv.coe_trans LinearEquiv.trans_symm\n  LinearEquiv.refl_toLinearMap TensorProduct.toLinearMap_congr\n  LinearEquiv.comp_symm LinearEquiv.symm_comp LinearEquiv.symm_symm\n  LinearEquiv.coe_lTensor LinearEquiv.coe_lTensor_symm\n  LinearEquiv.coe_rTensor LinearEquiv.coe_rTensor_symm\n  IsCocomm.comm_comp_comul TensorProduct.AlgebraTensorModule.map_eq\n  TensorProduct.AlgebraTensorModule.assoc_eq TensorProduct.AlgebraTensorModule.rightComm_eq\n  TensorProduct.tensorTensorTensorComm TensorProduct.AlgebraTensorModule.tensorTensorTensorComm\n  TensorProduct.AlgebraTensorModule.congr_eq LinearEquiv.comp_symm_assoc\n  LinearEquiv.symm_comp_assoc TensorProduct.rightComm_def TensorProduct.leftComm_def\n  TensorProduct.comm_symm TensorProduct.comm_comp_comm TensorProduct.comm_comp_comm_assoc\n\nattribute [coassoc_simps← ] TensorProduct.map_comp TensorProduct.map_map_comp_assoc_eq\n  TensorProduct.map_map_comp_assoc_symm_eq\n\n@[coassoc_simps]\nlemma TensorProduct.map_comp_assoc\n    (f : M →ₗ[R] N) (g : N →ₗ[R] P) (f' : M' →ₗ[R] N') (g' : N' →ₗ[R] P') (φ : M₁ →ₗ[R] M ⊗[R] M') :\n    map g g' ∘ₗ map f f' ∘ₗ φ = map (g ∘ₗ f) (g' ∘ₗ f') ∘ₛₗ φ := by\n  rw [← LinearMap.comp_assoc, TensorProduct.map_comp]\n\n@[coassoc_simps← ]\nlemma TensorProduct.map_map_comp_assoc_eq_assoc\n    (f₁ : M₁ →ₗ[R] N₁) (f₂ : M₂ →ₗ[R] N₂) (f₃ : M₃ →ₗ[R] N₃) (f : M →ₗ[R] M₁ ⊗[R] M₂ ⊗[R] M₃) :\n    f₁ ⊗ₘ (f₂ ⊗ₘ f₃) ∘ₗ α ∘ₗ f = α ∘ₗ ((f₁ ⊗ₘ f₂) ⊗ₘ f₃) ∘ₗ f := by\n  rw [← LinearMap.comp_assoc, ← LinearMap.comp_assoc, TensorProduct.map_map_comp_assoc_eq]\n\n@[coassoc_simps← ]\nlemma TensorProduct.map_map_comp_assoc_symm_eq_assoc\n    (f₁ : M₁ →ₗ[R] N₁) (f₂ : M₂ →ₗ[R] N₂) (f₃ : M₃ →ₗ[R] N₃) (f : M →ₗ[R] M₁ ⊗[R] (M₂ ⊗[R] M₃)) :\n    (f₁ ⊗ₘ f₂) ⊗ₘ f₃ ∘ₗ α⁻¹ ∘ₗ f = α⁻¹ ∘ₗ (f₁ ⊗ₘ (f₂ ⊗ₘ f₃)) ∘ₗ f := by\n  rw [← LinearMap.comp_assoc, ← LinearMap.comp_assoc, TensorProduct.map_map_comp_assoc_symm_eq]\n\n@[coassoc_simps]\n\nTarget:\nlemma assoc_comp_map_map_comp\n    (f₁ : M₁ →ₗ[R] N₁) (f₂ : M₂ →ₗ[R] N₂) (f₃ : M₃ →ₗ[R] N₃) (f₁₂ : M →ₗ[R] M₁ ⊗[R] M₂) :\n    α ∘ₗ (((f₁ ⊗ₘ f₂) ∘ₗ f₁₂) ⊗ₘ f₃) = (f₁ ⊗ₘ (f₂ ⊗ₘ f₃)) ∘ₗ α ∘ₗ (f₁₂ ⊗ₘ id) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Coalgebra","family_id":"assoc_comp_map_map_comp","file_id":"mathlib/Mathlib/RingTheory/Coalgebra/CoassocSimps.lean","sample_id":"6a8e8150ef6c1755c0f36af56608a5c55a4714f68f327a0666a7f1b00c8049b8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"68b7446878a3dfdb301fca861ee4b86aba230785df09d24821618a061945374b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a8f96640d4ba327fc7e9e6bd4ae65eab83d12cf3037d5abef2ad29e199cb314c","source_sha256":"f9d3644b75382b6a32e9f16c61996173c9614e3a85719811c47a399dfcb0f697","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n    let ⟨f⟩ := quotient_prod_linearEquiv p\n    exact rank_prod'.symm.trans f.rank_eq","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":2,"token_jaccard":0.0625,"token_length_ratio":0.111111},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"6b8596949063684467dad9ad77e3c53c4ef4e4f3815f36075a288e4e57c3ff10","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Basis.VectorSpace\npublic import Mathlib.LinearAlgebra.Dimension.Finite\npublic import Mathlib.LinearAlgebra.Dimension.RankNullity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Johannes Hölzl, Sander Dahmen,\nKim Morrison, Chris Hughes, Anne Baanen, Junyan Xu\n-/\n/-!\n# Dimension of vector spaces\n\nIn this file we provide results about `Module.rank` and `Module.finrank` of vector spaces\nover division rings.\n\n## Main statements\n\nFor vector spaces (i.e. modules over a field), we have\n\n* `rank_quotient_add_rank_of_divisionRing`: if `V₁` is a submodule of `V`, then\n  `Module.rank (V/V₁) + Module.rank V₁ = Module.rank V`.\n* `rank_range_add_rank_ker`: the rank-nullity theorem.\n\nSee also `Mathlib/LinearAlgebra/Dimension/ErdosKaplansky.lean` for the Erdős-Kaplansky theorem.\n\n-/\n\npublic section\n\n\nnoncomputable section\n\nuniverse u₀ u v v' v'' u₁' w w'\n\nvariable {K : Type u} {V V₁ V₂ V₃ : Type v}\nvariable {ι : Type w}\n\nopen Cardinal Basis Submodule Function Set\n\nsection Module\n\nsection DivisionRing\n\nvariable [DivisionRing K]\nvariable [AddCommGroup V] [Module K V]\nvariable [AddCommGroup V₁] [Module K V₁]\n\n/-- If a vector space has a finite dimension, the index set of `Basis.ofVectorSpace` is finite. -/\ntheorem Module.Basis.finite_ofVectorSpaceIndex_of_rank_lt_aleph0 (h : Module.rank K V < ℵ₀) :\n    (Basis.ofVectorSpaceIndex K V).Finite :=\n  Set.finite_def.2 <| (Basis.ofVectorSpace K V).nonempty_fintype_index_of_rank_lt_aleph0 h\n\n/-- Also see `rank_quotient_add_rank`. -/\n\nTarget:\ntheorem rank_quotient_add_rank_of_divisionRing (p : Submodule K V) :\n    Module.rank K (V ⧸ p) + Module.rank K p = Module.rank K V :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Dimension","family_id":"rank_quotient_add_rank_of_divisionring","file_id":"mathlib/Mathlib/LinearAlgebra/Dimension/DivisionRing.lean","sample_id":"a8f96640d4ba327fc7e9e6bd4ae65eab83d12cf3037d5abef2ad29e199cb314c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a8a91e5033815085839eaad1f6a9d8721d67997ed1799815646887dd0bdc0a37","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e7bd69cfd0dadb9fdaab62111c02ae20ae7bf73cdd0cc212f534d85bcf679dc1","source_sha256":"5446f542892910a2397a84cd0f024393cbca50d14ecc1fa829e9751db035abd3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [AlgEquiv.ext_iff, Subtype.ext_iff, AlgEquiv.restrictNormal_apply]","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":2,"token_jaccard":0.090909,"token_length_ratio":0.133333},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"6b859716db3520013cc49d56dd93f6f55b040b688487c35ee5125302f7cbad45","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Splits\npublic import Mathlib.FieldTheory.Galois.Notation\npublic import Mathlib.FieldTheory.IntermediateField.Basic\npublic import Mathlib.FieldTheory.Minpoly.Field\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Thomas Browning, Patrick Lutz\n-/\n/-!\n# Normal field extensions\n\nIn this file we define normal field extensions.\n\n## Main Definitions\n\n- `Normal F K` where `K` is a field extension of `F`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Polynomial IsScalarTower\n\nvariable (F K : Type*) [Field F] [Field K] [Algebra F K]\n\n/-- Typeclass for normal field extensions: an algebraic extension of fields `K/F` is *normal*\nif the minimal polynomial of every element `x` in `K` splits in `K`, i.e. every `F`-conjugate\nof `x` is in `K`. -/\n@[stacks 09HM]\nclass Normal : Prop extends Algebra.IsAlgebraic F K where\n  splits' (x : K) : Splits ((minpoly F x).map (algebraMap F K))\n\nvariable {F K}\n\ntheorem Normal.isIntegral (_ : Normal F K) (x : K) : IsIntegral F x :=\n  Algebra.IsIntegral.isIntegral x\n\ntheorem Normal.splits (_ : Normal F K) (x : K) : Splits ((minpoly F x).map (algebraMap F K)) :=\n  Normal.splits' x\n\ntheorem normal_iff :\n    Normal F K ↔ ∀ x : K, IsIntegral F x ∧ Splits ((minpoly F x).map (algebraMap F K)) :=\n  ⟨fun h x => ⟨h.isIntegral x, h.splits x⟩, fun h =>\n    { isAlgebraic := fun x => (h x).1.isAlgebraic\n      splits' := fun x => (h x).2 }⟩\n\ntheorem Normal.out :\n    Normal F K → ∀ x : K, IsIntegral F x ∧ Splits ((minpoly F x).map (algebraMap F K)) :=\n  normal_iff.1\n\nvariable (F K)\n\ninstance normal_self : Normal F F where\n  isAlgebraic := fun _ => isIntegral_algebraMap.isAlgebraic\n  splits' := fun x => (minpoly.eq_X_sub_C' x).symm ▸ by simp\n\nsection NormalTower\n\nvariable (E : Type*) [Field E] [Algebra F E] [Algebra K E] [IsScalarTower F K E]\n\n@[stacks 09HN]\ntheorem Normal.tower_top_of_normal [h : Normal F E] : Normal K E :=\n  normal_iff.2 fun x => by\n    obtain ⟨hx, hhx⟩ := h.out x\n    rw [algebraMap_eq F K E, ← map_map] at hhx\n    exact ⟨hx.tower_top, hhx.of_dvd (map_ne_zero (map_ne_zero (minpoly.ne_zero hx)))\n      ((map_dvd_map' _).mpr (minpoly.dvd_map_of_isScalarTower F K x))⟩\n\ninstance IntermediateField.normal (K : IntermediateField F E) [Normal F E] : Normal K E :=\n  Normal.tower_top_of_normal F K E\n\ntheorem AlgHom.normal_bijective [h : Normal F E] (ϕ : E →ₐ[F] K) : Function.Bijective ϕ :=\n  h.toIsAlgebraic.bijective_of_isScalarTower' ϕ\n\nvariable {E F}\nvariable {E' : Type*} [Field E'] [Algebra F E']\n\ntheorem Normal.of_algEquiv [h : Normal F E] (f : E ≃ₐ[F] E') : Normal F E' := by\n  rw [normal_iff] at h ⊢\n  intro x; specialize h (f.symm x)\n  rw [← f.apply_symm_apply x, minpoly.algEquiv_eq, ← f.toAlgHom.comp_algebraMap, ← map_map]\n  exact ⟨h.1.map f, h.2.map _⟩\n\ntheorem AlgEquiv.transfer_normal (f : E ≃ₐ[F] E') : Normal F E ↔ Normal F E' :=\n  ⟨fun _ ↦ Normal.of_algEquiv f, fun _ ↦ Normal.of_algEquiv f.symm⟩\n\ntheorem Normal.of_equiv_equiv {M N : Type*} [Field N] [Field M] [Algebra M N]\n    [h : Normal F E] {f : F ≃+* M} {g : E ≃+* N}\n    (hcomp : (algebraMap M N).comp f = (g : E →+* N).comp (algebraMap F E)) :\n    Normal M N := by\n  have := h\n  rw [normal_iff] at h ⊢\n  intro x\n  rw [← g.apply_symm_apply x]\n  refine ⟨(h (g.symm x)).1.map_of_comp_eq _ _ hcomp, ?_⟩\n  rw [← minpoly.map_eq_of_equiv_equiv hcomp, map_map, hcomp, ← map_map]\n  exact (h (g.symm x)).2.map _\n\nend NormalTower\n\nnamespace IntermediateField\n\nvariable {F K}\nvariable {L : Type*} [Field L] [Algebra F L] [Algebra K L] [IsScalarTower F K L]\n\n@[simp]\ntheorem restrictScalars_normal {E : IntermediateField K L} :\n    Normal F (E.restrictScalars F) ↔ Normal F E :=\n  Iff.rfl\n\nend IntermediateField\n\nvariable {F} {K}\nvariable {K₁ K₂ K₃ : Type*} [Field K₁] [Field K₂] [Field K₃] [Algebra F K₁]\n  [Algebra F K₂] [Algebra F K₃] (ϕ : K₁ →ₐ[F] K₂) (χ : K₁ ≃ₐ[F] K₂) (ψ : K₂ →ₐ[F] K₃)\n  (ω : K₂ ≃ₐ[F] K₃)\n\nsection Restrict\n\nvariable (E : Type*) [Field E] [Algebra F E] [Algebra E K₁] [Algebra E K₂] [Algebra E K₃]\n  [IsScalarTower F E K₁] [IsScalarTower F E K₂] [IsScalarTower F E K₃]\n\n/-- Restrict algebra homomorphism to image of normal subfield -/\ndef AlgHom.restrictNormalAux [h : Normal F E] :\n    (toAlgHom F E K₁).range →ₐ[F] (toAlgHom F E K₂).range where\n  toFun x :=\n    ⟨ϕ x, by\n      suffices (toAlgHom F E K₁).range.map ϕ ≤ _ by exact this ⟨x, Subtype.mem x, rfl⟩\n      rintro x ⟨y, ⟨z, hy⟩, hx⟩\n      rw [← hx, ← hy]\n      apply minpoly.mem_range_of_degree_eq_one E\n      refine ((h.splits z).of_dvd (map_ne_zero (minpoly.ne_zero (h.isIntegral z)))\n        (minpoly.dvd E _ (by simp [aeval_algHom_apply]))).degree_eq_one_of_irreducible\n        (minpoly.irreducible ?_)\n      simp only [AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom]\n      suffices IsIntegral F _ by exact this.tower_top\n      exact ((h.isIntegral z).map <| toAlgHom F E K₁).map ϕ⟩\n  map_zero' := Subtype.ext (map_zero _)\n  map_one' := Subtype.ext (map_one _)\n  map_add' x y := Subtype.ext <| by simp\n  map_mul' x y := Subtype.ext <| by simp\n  commutes' x := Subtype.ext (ϕ.commutes x)\n\n/-- Restrict algebra homomorphism to normal subfield. -/\n@[stacks 0BME \"Part 1\"]\ndef AlgHom.restrictNormal [Normal F E] : E →ₐ[F] E :=\n  ((AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K₂)).symm.toAlgHom.comp\n        (ϕ.restrictNormalAux E)).comp\n    (AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K₁)).toAlgHom\n\n/-- Restrict algebra homomorphism to normal subfield (`AlgEquiv` version) -/\ndef AlgHom.restrictNormal' [Normal F E] : Gal(E/F) :=\n  AlgEquiv.ofBijective (AlgHom.restrictNormal ϕ E) (AlgHom.normal_bijective F E E _)\n\n@[simp]\ntheorem AlgHom.restrictNormal_commutes [Normal F E] (x : E) :\n    algebraMap E K₂ (ϕ.restrictNormal E x) = ϕ (algebraMap E K₁ x) :=\n  Subtype.ext_iff.mp\n    (AlgEquiv.apply_symm_apply (AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K₂))\n      (ϕ.restrictNormalAux E ⟨IsScalarTower.toAlgHom F E K₁ x, x, rfl⟩))\n\ntheorem AlgHom.restrictNormal_comp [Normal F E] :\n    (ψ.restrictNormal E).comp (ϕ.restrictNormal E) = (ψ.comp ϕ).restrictNormal E :=\n  AlgHom.ext fun _ =>\n    (algebraMap E K₃).injective (by simp only [AlgHom.comp_apply, AlgHom.restrictNormal_commutes])\n\n/-- Restrict algebra isomorphism to a normal subfield -/\ndef AlgEquiv.restrictNormal [Normal F E] : Gal(E/F) :=\n  AlgHom.restrictNormal' χ.toAlgHom E\n\n@[simp]\ntheorem AlgEquiv.restrictNormal_commutes [Normal F E] (x : E) :\n    algebraMap E K₂ (χ.restrictNormal E x) = χ (algebraMap E K₁ x) :=\n  χ.toAlgHom.restrictNormal_commutes E x\n\ntheorem AlgEquiv.restrictNormal_apply (L : IntermediateField F K₁) [Normal F L] (σ : Gal(K₁/F))\n    (x : L) : restrictNormal σ L x = σ x :=\n  AlgEquiv.restrictNormal_commutes σ L x\n\nTarget:\ntheorem AlgEquiv.restrictNormal_eq_one_iff (L : IntermediateField F K₁) [Normal F L]\n    (σ : Gal(K₁/F)) : restrictNormal σ L = 1 ↔ ∀ x ∈ L, σ x = x :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Normal","family_id":"algequiv","file_id":"mathlib/Mathlib/FieldTheory/Normal/Defs.lean","sample_id":"e7bd69cfd0dadb9fdaab62111c02ae20ae7bf73cdd0cc212f534d85bcf679dc1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"355d8928efd1f0f2dd702b5a22182d531523711ce4f995520e27017b7c2e8b46","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c7578c9e54d861a424808b1e99539d93867e1a37e2b897c54022f1af2cd7dc2c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8ad925533b8c56254faf2ab0de8790476e52ea1f36cb7dddd76cb2f24ea5a395","source_sha256":"4dc8cbdda13dd66c6fd01c0ae738b0853be3fe2c1370940a1d31f2e403201cc7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩\n  · filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy\n  · intro ε εpos\n    filter_upwards [Icc_mem_nhdsLT <| show x - ε < x by linarith] with y hy\n    rw [closedBall_eq_Icc]\n    exact Icc_subset_Icc hy.1 (by linarith)","hard_negative":true,"metrics":{"chosen_tokens":63,"rejected_tokens":3,"token_jaccard":0.051282,"token_length_ratio":0.047619},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"6b9085aede07a4702af03a394daff29550439a0c92aaaa34e2bc8decf849b527","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Covering.DensityTheorem\npublic import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2022 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Covering theorems for Lebesgue measure in one dimension\n\nWe have a general theory of covering theorems for doubling measures, developed notably\nin `DensityTheorem.lean`. In this file, we expand the API for this theory in one dimension,\nby showing that intervals belong to the relevant Vitali family.\n-/\n\npublic section\n\n\nopen Set MeasureTheory IsUnifLocDoublingMeasure Filter\n\nopen scoped Topology\n\nnamespace Real\n\ntheorem Icc_mem_vitaliFamily_at_right {x y : ℝ} (hxy : x < y) :\n    Icc x y ∈ (vitaliFamily (volume : Measure ℝ) 1).setsAt x := by\n  rw [Icc_eq_closedBall]\n  refine closedBall_mem_vitaliFamily_of_dist_le_mul _ ?_ (by linarith)\n  rw [dist_comm, Real.dist_eq, abs_of_nonneg] <;> linarith\n\ntheorem tendsto_Icc_vitaliFamily_right (x : ℝ) :\n    Tendsto (fun y => Icc x y) (𝓝[>] x) ((vitaliFamily (volume : Measure ℝ) 1).filterAt x) := by\n  refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩\n  · filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy\n  · intro ε εpos\n    filter_upwards [Icc_mem_nhdsGT <| show x < x + ε by linarith] with y hy\n    rw [closedBall_eq_Icc]\n    exact Icc_subset_Icc (by linarith) hy.2\n\ntheorem Icc_mem_vitaliFamily_at_left {x y : ℝ} (hxy : x < y) :\n    Icc x y ∈ (vitaliFamily (volume : Measure ℝ) 1).setsAt y := by\n  rw [Icc_eq_closedBall]\n  refine closedBall_mem_vitaliFamily_of_dist_le_mul _ ?_ (by linarith)\n  rw [Real.dist_eq, abs_of_nonneg] <;> linarith\n\nTarget:\ntheorem tendsto_Icc_vitaliFamily_left (x : ℝ) :\n    Tendsto (fun y => Icc y x) (𝓝[<] x) ((vitaliFamily (volume : Measure ℝ) 1).filterAt x) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_8ad925533b8c","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f3aa7baa2d27b96138f6a76ee851cf06d9c56d057604b3c2c8bc5cf2fee34068","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Covering","family_id":"tendsto_icc_vitalifamily_left","file_id":"mathlib/Mathlib/MeasureTheory/Covering/OneDim.lean","sample_id":"8ad925533b8c56254faf2ab0de8790476e52ea1f36cb7dddd76cb2f24ea5a395"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1e710a9d7e1e6f7c130e5c280474bf460f6d3246c3fb6e99473017f60576e601","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8451c1001183081b0718334cf931fa3560f5cf1ca44e66d4b164528ee6a8ec57","source_sha256":"da03aef6be316c0f1e4d46eb4e8f0617ba6def9cf6bbead1497df151ed7bf988","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by cases x; rfl\n\nnonrec def StateT.callCC {σ} [MonadCont m] {α β : Type _}\n    (f : Label α (StateT σ m) β → StateT σ m α) : StateT σ m α :=\n  StateT.mk (fun r => callCC fun f' => (f <| StateT.mkLabel f').run r)","hard_negative":true,"metrics":{"chosen_tokens":71,"rejected_tokens":8,"token_jaccard":0.073171,"token_length_ratio":0.112676},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"6b93d0267d91c871c2a0115ea1384e4a3c0eccaa373d44bc6b8389a7225158f9","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Control.Monad.Basic\npublic import Mathlib.Control.Monad.Writer\npublic import Mathlib.Control.Lawful\npublic import Batteries.Tactic.Congr\npublic import Batteries.Lean.Except\nimport all Init.Control.Option  -- for unfolding `Option.lift`\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\n/-!\n# Continuation Monad\n\nMonad encapsulating continuation passing programming style, similar to\nHaskell's `Cont`, `ContT` and `MonadCont`:\n<https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Cont.html>\n<https://hackage.haskell.org/package/transformers-0.6.2.0/docs/Control-Monad-Trans-Cont.html>\n-/\n\n@[expose] public section\n\nuniverse u v w u₀ u₁ v₀ v₁\n\nstructure MonadCont.Label (α : Type w) (m : Type u → Type v) (β : Type u) where\n  apply : α → m β\n\nabbrev MonadCont.goto {α β} {m : Type u → Type v} (f : MonadCont.Label α m β) (x : α) :=\n  f.apply x\n\nclass MonadCont (m : Type u → Type v) where\n  callCC : ∀ {α β}, (MonadCont.Label α m β → m α) → m α\n\nopen MonadCont\n\nclass LawfulMonadCont (m : Type u → Type v) [Monad m] [MonadCont m] : Prop\n    extends LawfulMonad m where\n  callCC_bind_right {α ω γ} (cmd : m α) (next : Label ω m γ → α → m ω) :\n    (callCC fun f => cmd >>= next f) = cmd >>= fun x => callCC fun f => next f x\n  callCC_bind_left {α} (β) (x : α) (dead : Label α m β → β → m α) :\n    (callCC fun f : Label α m β => goto f x >>= dead f) = pure x\n  callCC_dummy {α β} (dummy : m α) : (callCC fun _ : Label α m β => dummy) = dummy\n\nexport LawfulMonadCont (callCC_bind_right callCC_bind_left callCC_dummy)\n\ndef ContT (r : Type u) (m : Type u → Type v) (α : Type w) :=\n  (α → m r) → m r\n\nabbrev Cont (r : Type u) (α : Type w) :=\n  ContT r Id α\n\nnamespace ContT\n\nexport MonadCont (Label goto)\n\nvariable {r : Type u} {m : Type u → Type v} {α β : Type w}\n\n/-- Build a `ContT` from a function taking a continuation callback. -/\ndef mk (f : (α → m r) → m r) : ContT r m α := f\n\n/-- Run a `ContT` with a provided callback. -/\ndef run (x : ContT r m α) : (α → m r) → m r := x\n\ndef map (f : m r → m r) (x : ContT r m α) : ContT r m α :=\n  f ∘ x\n\ntheorem run_contT_map_contT (f : m r → m r) (x : ContT r m α) : run (map f x) = f ∘ run x :=\n  rfl\n\ndef withContT (f : (β → m r) → α → m r) (x : ContT r m α) : ContT r m β := fun g => x <| f g\n\ntheorem run_withContT (f : (β → m r) → α → m r) (x : ContT r m α) :\n    run (withContT f x) = run x ∘ f :=\n  rfl\n\n@[ext]\nprotected theorem ext {x y : ContT r m α} (h : ∀ f, x.run f = y.run f) : x = y := by\n  unfold ContT; ext; apply h\n\ninstance : Monad (ContT r m) where\n  pure x f := f x\n  bind x f g := x fun i => f i g\n\n@[simp]\ntheorem run_mk (f : (α → m r) → m r) (k : α → m r) : (.mk f : ContT r m α).run k = f k := rfl\n\n@[simp]\ntheorem run_pure (a : α) (k : α → m r) : (pure a : ContT r m α).run k = k a := rfl\n\n@[simp]\ntheorem run_bind (x : ContT r m α) (f : α → ContT r m β) (k : β → m r) :\n    (x >>= f).run k = x.run fun x => (f x).run k := rfl\n\n@[simp]\ntheorem run_map (f : α → β) (x : ContT r m α) (k : β → m r) :\n    (f <$> x).run k = x.run (k ∘ f) := rfl\n\n@[simp]\ntheorem run_seq (f : ContT r m (α → β)) (x : ContT r m α) (k : β → m r) :\n    (f <*> x).run k = f.run fun f => x.run (k ∘ f) := rfl\n\n@[simp]\ntheorem run_seqLeft (x : ContT r m α) (y : ContT r m β) (k : α → m r) :\n    (x <* y).run k = x.run fun x => y.run fun _ => k x := rfl\n\n@[simp]\ntheorem run_seqRight (x : ContT r m α) (y : ContT r m β) (k : β → m r) :\n    (x *> y).run k = x.run fun _ => y.run k := rfl\n\ninstance : LawfulMonad (ContT r m) := LawfulMonad.mk'\n  (id_map := by intros; rfl)\n  (pure_bind := by intros; ext; rfl)\n  (bind_assoc := by intros; ext; rfl)\n\ninstance [Monad m] : MonadLift m (ContT r m) where\n  monadLift x := .mk fun k => x >>= k\n\n@[simp]\ntheorem run_monadLift [Monad m] {α} (x : m α) (k : α → m r) :\n    (monadLift x : ContT r m α).run k = x >>= k := rfl\n\ntheorem monadLift_bind [Monad m] [LawfulMonad m] {α β} (x : m α) (f : α → m β) :\n    (monadLift (x >>= f) : ContT r m β) = monadLift x >>= monadLift ∘ f := by\n  ext\n  simp only [bind_assoc, run_bind, run_monadLift, Function.comp_apply]\n\ninstance : MonadCont (ContT r m) where\n  callCC f := .mk fun k => f ⟨fun x => .mk fun _ => k x⟩ k\n\n@[simp]\ntheorem run_callCC (f : Label α (ContT r m) β → ContT r m α) (k : α → m r) :\n    (callCC f).run k = (f ⟨fun x => .mk fun _ => k x⟩).run k := rfl\n\ninstance : LawfulMonadCont (ContT r m) where\n  callCC_bind_right := by intros; ext; rfl\n  callCC_bind_left := by intros; ext; rfl\n  callCC_dummy := by intros; ext; rfl\n\n/-- Note that `tryCatch` does not have correct behavior in this monad:\n```\ndef foo : ContT Bool (Except String) Bool := do\n  let x ← try\n    pure true\n  catch _ =>\n    return false\n  throw s!\"oh no {x}\"\n#eval foo.run pure\n-- `Except.ok false`, no error\n```\nHere, the `throwError` is being run inside the `try`.\nSee [Zulip](https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/MonadExcept.20in.20the.20ContT.20monad/near/375341221)\nfor further discussion.\n-/\ninstance (ε) [MonadExceptOf ε m] : MonadExceptOf ε (ContT r m) where\n  throw e := .mk fun _ => throw e\n  tryCatch act h := .mk fun k => tryCatch (act.run k) fun e => (h e).run k\n\n@[simp]\ntheorem run_throw {ε} [MonadExceptOf ε m]\n    (e : ε) (f : α → m r) :\n    (throw e : ContT r m α).run f = throw e := rfl\n\n@[simp]\ntheorem run_tryCatch {ε} [MonadExceptOf ε m]\n    (act : ContT r m α) (h : ε → ContT r m α) (f : α → m r) :\n    (tryCatch act h : ContT r m α).run f = tryCatch (act.run f) fun e => (h e).run f := rfl\n\nend ContT\n\nvariable {m : Type u → Type v}\n\nsection\nvariable [Monad m]\n\ndef ExceptT.mkLabel {α β ε} : Label (Except.{u, u} ε α) m β → Label α (ExceptT ε m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (Except.ok a)⟩\n\ntheorem ExceptT.goto_mkLabel {α β ε : Type _} (x : Label (Except.{u, u} ε α) m β) (i : α) :\n    goto (ExceptT.mkLabel x) i = ExceptT.mk (Except.ok <$> goto x (Except.ok i)) := by\n  cases x; rfl\n\nnonrec def ExceptT.callCC {ε} [MonadCont m] {α β : Type _}\n    (f : Label α (ExceptT ε m) β → ExceptT ε m α) : ExceptT ε m α :=\n  ExceptT.mk (callCC fun x : Label _ m β => ExceptT.run <| f (ExceptT.mkLabel x))\n\ninstance {ε} [MonadCont m] : MonadCont (ExceptT ε m) where\n  callCC := ExceptT.callCC\n\ninstance {ε} [MonadCont m] [LawfulMonadCont m] : LawfulMonadCont (ExceptT ε m) where\n  callCC_bind_right := by\n    intros; simp only [callCC, ExceptT.callCC, ExceptT.run_bind, callCC_bind_right]; ext\n    dsimp\n    congr with ⟨⟩ <;> simp [@callCC_dummy m _]\n  callCC_bind_left := by\n    intros\n    simp only [callCC, ExceptT.callCC, ExceptT.goto_mkLabel, map_eq_bind_pure_comp, Function.comp,\n      ExceptT.run_bind, ExceptT.run_mk, bind_assoc, pure_bind, @callCC_bind_left m _]\n    ext; rfl\n  callCC_dummy := by intros; simp only [callCC, ExceptT.callCC, @callCC_dummy m _]; ext; rfl\n\ndef OptionT.mkLabel {α β} : Label (Option.{u} α) m β → Label α (OptionT m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (some a)⟩\n\ntheorem OptionT.goto_mkLabel {α β : Type _} (x : Label (Option.{u} α) m β) (i : α) :\n    goto (OptionT.mkLabel x) i = OptionT.mk (goto x (some i) >>= fun a => pure (some a)) :=\n  (rfl)\n\nnonrec def OptionT.callCC [MonadCont m] {α β : Type _} (f : Label α (OptionT m) β → OptionT m α) :\n    OptionT m α :=\n  OptionT.mk (callCC fun x : Label _ m β => OptionT.run <| f (OptionT.mkLabel x) : m (Option α))\n\n@[simp]\nlemma run_callCC [MonadCont m] {α β : Type _} (f : Label α (OptionT m) β → OptionT m α) :\n    (OptionT.callCC f).run = (callCC fun x => OptionT.run <| f (OptionT.mkLabel x)) := (rfl)\n\ninstance [MonadCont m] : MonadCont (OptionT m) where\n  callCC := OptionT.callCC\n\ninstance [MonadCont m] [LawfulMonadCont m] : LawfulMonadCont (OptionT m) where\n  callCC_bind_right := by\n    refine fun _ _ => OptionT.ext ?_\n    simpa [callCC, Option.elimM, callCC_bind_right] using\n      bind_congr fun | some _ => rfl | none => by simp [@callCC_dummy m _]\n  callCC_bind_left := by\n    intros\n    ext\n    simp [callCC, OptionT.goto_mkLabel, @callCC_bind_left m _]\n  callCC_dummy := by intros; ext; simp [callCC, OptionT.callCC, @callCC_dummy m _]\n\ndef WriterT.mkLabel {α β ω} [EmptyCollection ω] : Label (α × ω) m β → Label α (WriterT ω m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (a, ∅)⟩\n\ndef WriterT.mkLabel' {α β ω} [Monoid ω] : Label (α × ω) m β → Label α (WriterT ω m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (a, 1)⟩\n\ntheorem WriterT.goto_mkLabel {α β ω : Type _} [EmptyCollection ω] (x : Label (α × ω) m β) (i : α) :\n    goto (WriterT.mkLabel x) i = monadLift (goto x (i, ∅)) := by cases x; rfl\n\ntheorem WriterT.goto_mkLabel' {α β ω : Type _} [Monoid ω] (x : Label (α × ω) m β) (i : α) :\n    goto (WriterT.mkLabel' x) i = monadLift (goto x (i, 1)) := by cases x; rfl\n\nnonrec def WriterT.callCC [MonadCont m] {α β ω : Type _} [EmptyCollection ω]\n    (f : Label α (WriterT ω m) β → WriterT ω m α) : WriterT ω m α :=\n  WriterT.mk <| callCC (WriterT.run ∘ f ∘ WriterT.mkLabel : Label (α × ω) m β → m (α × ω))\n\ndef WriterT.callCC' [MonadCont m] {α β ω : Type _} [Monoid ω]\n    (f : Label α (WriterT ω m) β → WriterT ω m α) : WriterT ω m α :=\n  WriterT.mk <|\n    MonadCont.callCC (WriterT.run ∘ f ∘ WriterT.mkLabel' : Label (α × ω) m β → m (α × ω))\n\nend\n\ninstance (ω) [Monad m] [EmptyCollection ω] [MonadCont m] : MonadCont (WriterT ω m) where\n  callCC := WriterT.callCC\n\ninstance (ω) [Monad m] [Monoid ω] [MonadCont m] : MonadCont (WriterT ω m) where\n  callCC := WriterT.callCC'\n\ndef StateT.mkLabel {α β σ : Type u} : Label (α × σ) m (β × σ) → Label α (StateT σ m) β\n  | ⟨f⟩ => ⟨fun a => StateT.mk (fun s => f (a, s))⟩\n\nTarget:\ntheorem StateT.goto_mkLabel {α β σ : Type u} (x : Label (α × σ) m (β × σ)) (i : α) :\n    goto (StateT.mkLabel x) i = StateT.mk (fun s => goto x (i, s)) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"575ec23ce4a082f312691784f5746a7ae8e8f898dcf4e503b927436679856375","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control/Monad","family_id":"statet","file_id":"mathlib/Mathlib/Control/Monad/Cont.lean","sample_id":"8451c1001183081b0718334cf931fa3560f5cf1ca44e66d4b164528ee6a8ec57"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0c7a4b44bba5c20da7810399c16de1e098a5c39bdee940dd765e6735e6e9d672","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4b02dfc32f7fdee24d8ec38c3b482d9b41a187e47ae0b91f0f84541668392e15","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"af27a105e43a072189a2b49287038416eed37dc52c7226788734b79f2baf138e","source_sha256":"6b6fd17d69ce9cda1c5ca51d87b6241ac80bfb6aa3b060b832bdfe6d51236491","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [IsSelfAdjoint.star_eq, ← sub_eq_add_neg, (CFC.sqrt_nonneg (1 - a ^ 2 : A)).isSelfAdjoint]","hard_negative":false,"metrics":{"chosen_tokens":27,"rejected_tokens":32,"token_jaccard":0.846154,"token_length_ratio":1.185185},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"6bb66eaf4323e4c7deebd13d2072ca1ce8ebb1ce92f587cdd4841fa51a18ce32","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order\npublic import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary\npublic import Mathlib.Analysis.Normed.Module.Normalize\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\n/-! # Unitary elements span C⋆-algebras\n\n## Main results\n\n+ `CStarAlgebra.exists_sum_four_unitary`: every element `x` in a unital C⋆-algebra is a linear\n  combination of four unitary elements, and the norm of each coefficient does not exceed `‖x‖ / 2`.\n+ `CStarAlgebra.span_unitary`: a unital C⋆-algebra is spanned by its unitary elements.\n-/\n\n@[expose] public section\n\nvariable {A : Type*} [CStarAlgebra A]\n\nopen scoped ComplexStarModule\nopen Complex\n\nsection Ordered\n\nvariable [PartialOrder A] [StarOrderedRing A]\n\n/-- If `a : A` is a selfadjoint element in a C⋆-algebra with `‖a‖ ≤ 1`,\nthen `a + I • CFC.sqrt (1 - a ^ 2)` is unitary.\n\nThis is the key tool to show that a C⋆-algebra is spanned by its unitary elements. -/\nlemma IsSelfAdjoint.self_add_I_smul_cfcSqrt_sub_sq_mem_unitary (a : A) (ha : IsSelfAdjoint a)\n    (ha_norm : ‖a‖ ≤ 1) : a + I • CFC.sqrt (1 - a ^ 2) ∈ unitary A := by\n  obtain (_ | _) := subsingleton_or_nontrivial A\n  · simp [Subsingleton.elim (a + I • CFC.sqrt (1 - a ^ 2)) 1, one_mem (unitary A)]\n  have key : a + I • CFC.sqrt (1 - a ^ 2) = cfc (fun x : ℂ ↦ x.re + I * √(1 - x.re ^ 2)) a := by\n    rw [CFC.sqrt_eq_real_sqrt (1 - a ^ 2) ?nonneg]\n    case nonneg =>\n      rwa [sub_nonneg, ← CStarAlgebra.norm_le_one_iff_of_nonneg (a ^ 2), sq, ha.norm_mul_self,\n        sq_le_one_iff₀ (by positivity)]\n    rw [cfc_add .., cfc_const_mul .., ← cfc_real_eq_complex (fun x ↦ x) ha, cfc_id' ℝ a,\n      ← cfc_real_eq_complex (fun x ↦ √(1 - x ^ 2)) ha, cfcₙ_eq_cfc, cfc_comp' (√·) (1 - · ^ 2) a,\n      cfc_sub .., cfc_pow .., cfc_const_one .., cfc_id' ..]\n  rw [key, cfc_unitary_iff ..]\n  intro x hx\n  rw [← starRingEnd_apply, ← Complex.normSq_eq_conj_mul_self,\n    Complex.normSq_ofReal_add_I_mul_sqrt_one_sub, Complex.ofReal_one]\n  exact spectrum.norm_le_norm_of_mem (ha.spectrumRestricts.apply_mem hx) |>.trans ha_norm\n\n/-- For `a` selfadjoint with `‖a‖ ≤ 1`, this is the unitary `a + I • √(1 - a ^ 2)`. -/\n@[simps]\nnoncomputable def selfAdjoint.unitarySelfAddISMul (a : selfAdjoint A) (ha_norm : ‖a‖ ≤ 1) :\n    unitary A :=\n  ⟨(a : A) + I • CFC.sqrt (1 - a ^ 2 : A), a.2.self_add_I_smul_cfcSqrt_sub_sq_mem_unitary _ ha_norm⟩\n\nTarget:\nlemma selfAdjoint.star_coe_unitarySelfAddISMul (a : selfAdjoint A) (ha_norm : ‖a‖ ≤ 1) :\n    (star (unitarySelfAddISMul a ha_norm) : A) = a - I • CFC.sqrt (1 - a ^ 2 : A) :=\n\nProof body:\n","rejected":"by\n  simp [IsSelfAdjoint.star_eq, ← sub_eq_add_neg, (CFC.sqrt_nonneg (1 - a ^ 2 : A)).isSelfAdjoint]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/CStarAlgebra","family_id":"selfadjoint","file_id":"mathlib/Mathlib/Analysis/CStarAlgebra/Unitary/Span.lean","sample_id":"af27a105e43a072189a2b49287038416eed37dc52c7226788734b79f2baf138e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b0a03b5526b62c3517381ba5457b6ebaafee0e4da110c931ab1a6ee0269bdd8b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3a5c41e6c5f8d96bb9ba0e28b19215e24201ceb82fce27001debef23a8ce06bb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5b0c69fc729fb329df53e747032d70f91d7a865fc26c9ee5566a2b68840871a7","source_sha256":"4e9a1f48a3b50b199e8e6174f030d1362e64d040c2a6826f217f6f03af70994c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases p\n  rw [← ofFractionRing_smul]","hard_negative":false,"metrics":{"chosen_tokens":8,"rejected_tokens":13,"token_jaccard":0.615385,"token_length_ratio":1.625},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"6c370c96e94f198256381a159b24f1cac3a35319a9f4b580c77ad8170c7ebf72","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharP.Algebra\npublic import Mathlib.FieldTheory.RatFunc.Defs\npublic import Mathlib.RingTheory.Polynomial.Content\npublic import Mathlib.RingTheory.Algebraic.Integral\n\nNamespace:\nRatFunc\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# The field structure of rational functions\n\n## Main definitions\nWorking with rational functions as polynomials:\n- `RatFunc.instField` provides a field structure\n\nYou can use `IsFractionRing` API to treat `RatFunc` as the field of fractions of polynomials:\n* `algebraMap K[X] K⟮X⟯` maps polynomials to rational functions\n* `IsFractionRing.algEquiv` maps other fields of fractions of `K[X]` to `K⟮X⟯`.\n\nIn particular:\n* `FractionRing.algEquiv K[X] K⟮X⟯` maps the generic field of\n  fraction construction to `K⟮X⟯`. Combine this with `AlgEquiv.restrictScalars` to change\n  the `FractionRing K[X] ≃ₐ[K[X]] K⟮X⟯` to `FractionRing K[X] ≃ₐ[K] K⟮X⟯`.\n\nWorking with rational functions as fractions:\n- `RatFunc.num` and `RatFunc.denom` give the numerator and denominator.\n  These values are chosen to be coprime and such that `RatFunc.denom` is monic.\n\nLifting homomorphisms of polynomials to other types, by mapping and dividing, as long\nas the homomorphism retains the non-zero-divisor property:\n- `RatFunc.liftMonoidWithZeroHom` lifts a `K[X] →*₀ G₀` to\n  a `K⟮X⟯ →*₀ G₀`, where `[CommRing K] [CommGroupWithZero G₀]`\n- `RatFunc.liftRingHom` lifts a `K[X] →+* L` to a `K⟮X⟯ →+* L`,\n  where `[CommRing K] [Field L]`\n- `RatFunc.liftAlgHom` lifts a `K[X] →ₐ[S] L` to a `K⟮X⟯ →ₐ[S] L`,\n  where `[CommRing K] [Field L] [CommSemiring S] [Algebra S K[X]] [Algebra S L]`\n\nThis is satisfied by injective homs.\n\nWe also have lifting homomorphisms of polynomials to other polynomials,\nwith the same condition on retaining the non-zero-divisor property across the map:\n- `RatFunc.map` lifts `K[X] →* R[X]` when `[CommRing K] [CommRing R]`\n- `RatFunc.mapRingHom` lifts `K[X] →+* R[X]` when `[CommRing K] [CommRing R]`\n- `RatFunc.mapAlgHom` lifts `K[X] →ₐ[S] R[X]` when\n  `[CommRing K] [IsDomain K] [CommRing R] [IsDomain R]`\n-/\n\n@[expose] public section\n\nuniverse u v\n\nnoncomputable section\n\nopen scoped nonZeroDivisors Polynomial\n\nvariable {K : Type u}\n\nnamespace RatFunc\n\nsection Field\n\nvariable [CommRing K]\n\n/-- The zero rational function. -/\nprotected irreducible_def zero : K⟮X⟯ :=\n  ⟨0⟩\n\ninstance : Zero K⟮X⟯ :=\n  ⟨RatFunc.zero⟩\n\ntheorem ofFractionRing_zero : (ofFractionRing 0 : K⟮X⟯) = 0 :=\n  zero_def.symm\n\n/-- Addition of rational functions. -/\nprotected irreducible_def add : K⟮X⟯ → K⟮X⟯ → K⟮X⟯\n  | ⟨p⟩, ⟨q⟩ => ⟨p + q⟩\n\ninstance : Add K⟮X⟯ :=\n  ⟨RatFunc.add⟩\n\ntheorem ofFractionRing_add (p q : FractionRing K[X]) :\n    ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q :=\n  (add_def _ _).symm\n\n/-- Subtraction of rational functions. -/\nprotected irreducible_def sub : K⟮X⟯ → K⟮X⟯ → K⟮X⟯\n  | ⟨p⟩, ⟨q⟩ => ⟨p - q⟩\n\ninstance : Sub K⟮X⟯ :=\n  ⟨RatFunc.sub⟩\n\ntheorem ofFractionRing_sub (p q : FractionRing K[X]) :\n    ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q :=\n  (sub_def _ _).symm\n\n/-- Additive inverse of a rational function. -/\nprotected irreducible_def neg : K⟮X⟯ → K⟮X⟯\n  | ⟨p⟩ => ⟨-p⟩\n\ninstance : Neg K⟮X⟯ :=\n  ⟨RatFunc.neg⟩\n\ntheorem ofFractionRing_neg (p : FractionRing K[X]) :\n    ofFractionRing (-p) = -ofFractionRing p :=\n  (neg_def _).symm\n\n/-- The multiplicative unit of rational functions. -/\nprotected irreducible_def one : K⟮X⟯ :=\n  ⟨1⟩\n\ninstance : One K⟮X⟯ :=\n  ⟨RatFunc.one⟩\n\ntheorem ofFractionRing_one : (ofFractionRing 1 : K⟮X⟯) = 1 :=\n  one_def.symm\n\n/-- Multiplication of rational functions. -/\nprotected irreducible_def mul : K⟮X⟯ → K⟮X⟯ → K⟮X⟯\n  | ⟨p⟩, ⟨q⟩ => ⟨p * q⟩\n\ninstance : Mul K⟮X⟯ :=\n  ⟨RatFunc.mul⟩\n\ntheorem ofFractionRing_mul (p q : FractionRing K[X]) :\n    ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q :=\n  (mul_def _ _).symm\n\nsection IsDomain\n\nvariable [IsDomain K]\n\n/-- Division of rational functions. -/\nprotected irreducible_def div : K⟮X⟯ → K⟮X⟯ → K⟮X⟯\n  | ⟨p⟩, ⟨q⟩ => ⟨p / q⟩\n\ninstance : Div K⟮X⟯ :=\n  ⟨RatFunc.div⟩\n\ntheorem ofFractionRing_div (p q : FractionRing K[X]) :\n    ofFractionRing (p / q) = ofFractionRing p / ofFractionRing q :=\n  (div_def _ _).symm\n\n/-- Multiplicative inverse of a rational function. -/\nprotected irreducible_def inv : K⟮X⟯ → K⟮X⟯\n  | ⟨p⟩ => ⟨p⁻¹⟩\n\ninstance : Inv K⟮X⟯ :=\n  ⟨RatFunc.inv⟩\n\ntheorem ofFractionRing_inv (p : FractionRing K[X]) :\n    ofFractionRing p⁻¹ = (ofFractionRing p)⁻¹ :=\n  (inv_def _).symm\n\n-- Auxiliary lemma for the `Field` instance\ntheorem mul_inv_cancel : ∀ {p : K⟮X⟯}, p ≠ 0 → p * p⁻¹ = 1\n  | ⟨p⟩, h => by\n    have : p ≠ 0 := fun hp => h <| by rw [hp, ofFractionRing_zero]\n    simpa only [← ofFractionRing_inv, ← ofFractionRing_mul, ← ofFractionRing_one,\n        ofFractionRing.injEq] using\n      mul_inv_cancel₀ this\n\nend IsDomain\n\nsection SMul\n\nvariable {R : Type*}\n\n/-- Scalar multiplication of rational functions. -/\nprotected irreducible_def smul [SMul R (FractionRing K[X])] : R → K⟮X⟯ → K⟮X⟯\n  | r, ⟨p⟩ => ⟨r • p⟩\n\ninstance [SMul R (FractionRing K[X])] : SMul R K⟮X⟯ :=\n  ⟨RatFunc.smul⟩\n\ntheorem ofFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : FractionRing K[X]) :\n    ofFractionRing (c • p) = c • ofFractionRing p :=\n  (smul_def _ _).symm\n\nTarget:\ntheorem toFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : K⟮X⟯) :\n    toFractionRing (c • p) = c • toFractionRing p :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  cases p\n  rw [← ofFractionRing_smul]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/RatFunc","family_id":"tofractionring_smul","file_id":"mathlib/Mathlib/FieldTheory/RatFunc/Basic.lean","sample_id":"5b0c69fc729fb329df53e747032d70f91d7a865fc26c9ee5566a2b68840871a7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e1c4c37ea1905b854b1db5a6620e0ed4d40197fe584c2e9cb453bf8d89ed1d01","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c2825dbc1dbcd1603529867e278ac4ffcf35262bd65e7baa81ae11e2740b486e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"322ab09370ffc7ba02e5b0d458715ab5e05e1b425cf1b69dc3d33d89b75fd5ba","source_sha256":"c1ecc80ead99a17d01db2984c5f06df1499354702fc192601863dee22fbd7b8f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using ideal_biInf I Set.finite_univ","hard_negative":true,"metrics":{"chosen_tokens":8,"rejected_tokens":3,"token_jaccard":0.1,"token_length_ratio":0.375},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"6c451ec27cae6a09e2ca9ff7543b25c26bb8324dcb0f6e5a6e000449522fef58","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact\npublic import Mathlib.AlgebraicGeometry.Properties\npublic import Mathlib.Tactic.DepRewrite\n\nNamespace:\nAlgebraicGeometry.Scheme.IdealSheafData\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Ideal sheaves on schemes\n\nWe define ideal sheaves of schemes and provide various constructors for it.\n\n## Main definition\n* `AlgebraicGeometry.Scheme.IdealSheafData`: A structure that contains the data to uniquely define\n  an ideal sheaf, consisting of\n  1. an ideal `I(U) ≤ Γ(X, U)` for every affine open `U`\n  2. a proof that `I(D(f)) = I(U)_f` for every affine open `U` and every section `f : Γ(X, U)`.\n* `AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals`:\n  The largest ideal sheaf contained in a family of ideals.\n* `AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine`:\n  Over affine schemes, ideal sheaves are in bijection with ideals of the global sections.\n* `AlgebraicGeometry.Scheme.IdealSheafData.support`: The support of an ideal sheaf.\n* `AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal`: The vanishing ideal of a set.\n* `AlgebraicGeometry.Scheme.Hom.ker`: The kernel of a morphism.\n\n## Main results\n* `AlgebraicGeometry.Scheme.IdealSheafData.gc`:\n  `support` and `vanishingIdeal` forms a Galois connection.\n* `AlgebraicGeometry.Scheme.Hom.support_ker`: The support of a kernel of a quasi-compact morphism\n  is the closure of the range.\n\n## Implementation detail\n\nIdeal sheaves are not yet defined in this file as actual subsheaves of `𝒪ₓ`.\nInstead, for the ease of development and application,\nwe define the structure `IdealSheafData` containing all necessary data to uniquely define an\nideal sheaf. This should be refactored as a constructor for ideal sheaves once they are introduced\ninto mathlib.\n\n-/\n\n@[expose] public section\n\nopen CategoryTheory TopologicalSpace\n\nuniverse u\n\nnamespace AlgebraicGeometry.Scheme\n\nvariable {X : Scheme.{u}}\n\n/--\nA structure that contains the data to uniquely define an ideal sheaf, consisting of\n1. an ideal `I(U) ≤ Γ(X, U)` for every affine open `U`\n2. a proof that `I(D(f)) = I(U)_f` for every affine open `U` and every section `f : Γ(X, U)`\n3. a subset of `X` equal to the support.\n\nAlso see `Scheme.IdealSheafData.mkOfMemSupportIff` for a constructor with the condition on the\nsupport being (usually) easier to prove.\n-/\nstructure IdealSheafData (X : Scheme.{u}) : Type u where\n  /-- The component of an ideal sheaf at an affine open. -/\n  ideal : ∀ U : X.affineOpens, Ideal Γ(X, U)\n  /-- Also see `AlgebraicGeometry.Scheme.IdealSheafData.map_ideal` -/\n  map_ideal_basicOpen : ∀ (U : X.affineOpens) (f : Γ(X, U)),\n    (ideal U).map (X.presheaf.map (homOfLE <| X.basicOpen_le f).op).hom =\n      ideal (X.affineBasicOpen f)\n  /-- The support of an ideal sheaf. Use `IdealSheafData.support` instead for most occasions. -/\n  supportSet : Set X := ⋂ U, X.zeroLocus (U := U.1) (ideal U)\n  supportSet_eq_iInter_zeroLocus : supportSet = ⋂ U, X.zeroLocus (U := U.1) (ideal U) := by rfl\n\nnamespace IdealSheafData\n\n@[ext]\nprotected lemma ext {I J : X.IdealSheafData} (h : I.ideal = J.ideal) : I = J := by\n  obtain ⟨i, _, s, hs⟩ := I\n  obtain ⟨j, _, t, ht⟩ := J\n  subst h\n  congr\n  rw [hs, ht]\n\nsection Order\n\ninstance : PartialOrder (IdealSheafData X) := PartialOrder.lift ideal fun _ _ ↦ IdealSheafData.ext\n\nlemma le_def {I J : IdealSheafData X} : I ≤ J ↔ ∀ U, I.ideal U ≤ J.ideal U := .rfl\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : CompleteSemilatticeSup (IdealSheafData X) where\n  sSup s :=\n  { ideal := sSup (ideal '' s),\n    map_ideal_basicOpen := by\n      have : sSup (ideal '' s) = ⨆ i : s, ideal i.1 := by\n        conv_lhs => rw [← Subtype.range_val (s := s), ← Set.range_comp]\n        rfl\n      simp only [this, iSup_apply, Ideal.map_iSup, map_ideal_basicOpen, implies_true] }\n  isLUB_sSup _ := .of_image (f := ideal) le_def (isLUB_sSup _)\n\n/-- The largest ideal sheaf contained in a family of ideals. -/\ndef ofIdeals (I : ∀ U : X.affineOpens, Ideal Γ(X, U)) : IdealSheafData X :=\n  sSup { J : IdealSheafData X | J.ideal ≤ I }\n\nlemma ideal_ofIdeals_le (I : ∀ U : X.affineOpens, Ideal Γ(X, U)) :\n    (ofIdeals I).ideal ≤ I :=\n  sSup_le (Set.forall_mem_image.mpr fun _ ↦ id)\n\n/-- The Galois coinsertion between ideal sheaves and arbitrary families of ideals. -/\nprotected def gci : GaloisCoinsertion ideal (ofIdeals (X := X)) where\n  choice I hI :=\n  { ideal := I\n    map_ideal_basicOpen U f :=\n      (ideal_ofIdeals_le I).antisymm hI ▸ (ofIdeals I).map_ideal_basicOpen U f }\n  gc _ _ := ⟨(le_sSup ·), (le_trans · (ideal_ofIdeals_le _))⟩\n  u_l_le _ := sSup_le fun _ ↦ id\n  choice_eq I hI := IdealSheafData.ext (hI.antisymm (ideal_ofIdeals_le I))\n\nlemma strictMono_ideal : StrictMono (ideal (X := X)) := IdealSheafData.gci.strictMono_l\nlemma ideal_mono : Monotone (ideal (X := X)) := strictMono_ideal.monotone\nlemma ofIdeals_mono : Monotone (ofIdeals (X := X)) := IdealSheafData.gci.gc.monotone_u\nlemma ofIdeals_ideal (I : IdealSheafData X) : ofIdeals I.ideal = I := IdealSheafData.gci.u_l_eq _\nlemma le_ofIdeals_iff {I : IdealSheafData X} {J} : I ≤ ofIdeals J ↔ I.ideal ≤ J :=\n  IdealSheafData.gci.gc.le_iff_le.symm\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : OrderTop (IdealSheafData X) where\n  top.ideal := ⊤\n  top.map_ideal_basicOpen := by simp [Ideal.map_top]\n  top.supportSet := ⊥\n  top.supportSet_eq_iInter_zeroLocus := by\n    ext x\n    simpa using X.isBasis_affineOpens.exists_subset_of_mem_open (Set.mem_univ x) isOpen_univ\n  le_top I U := le_top\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : OrderBot (IdealSheafData X) where\n  bot.ideal := ⊥\n  bot.map_ideal_basicOpen := by simp\n  bot.supportSet := ⊤\n  bot.supportSet_eq_iInter_zeroLocus := by ext; simp\n  bot_le I U := bot_le\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : SemilatticeInf (IdealSheafData X) where\n  inf I J :=\n  { ideal := I.ideal ⊓ J.ideal\n    map_ideal_basicOpen U f := by\n      dsimp\n      have : (X.presheaf.map (homOfLE (X.basicOpen_le f)).op).hom = algebraMap _ _ := rfl\n      have inst := U.2.isLocalization_basicOpen f\n      rw [← I.map_ideal_basicOpen U f, ← J.map_ideal_basicOpen U f, this]\n      ext x\n      obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq (.powers f) x\n      simp only [IsLocalization.mk'_mem_map_algebraMap_iff, Submonoid.mem_powers_iff, Ideal.mem_inf,\n        exists_exists_eq_and]\n      refine ⟨fun ⟨n, h₁, h₂⟩ ↦ ⟨⟨n, h₁⟩, ⟨n, h₂⟩⟩, ?_⟩\n      rintro ⟨⟨n₁, h₁⟩, ⟨n₂, h₂⟩⟩\n      refine ⟨n₁ + n₂, ?_, ?_⟩\n      · rw [add_comm, pow_add, mul_assoc]; exact Ideal.mul_mem_left _ _ h₁\n      · rw [pow_add, mul_assoc]; exact Ideal.mul_mem_left _ _ h₂ }\n  inf_le_left I J U := inf_le_left\n  inf_le_right I J U := inf_le_right\n  le_inf I J K hIJ hIK U := le_inf (hIJ U) (hIK U)\n\ninstance : CompleteLattice (IdealSheafData X) where\n  __ := (inferInstance : OrderTop (IdealSheafData X))\n  __ := (inferInstance : OrderBot (IdealSheafData X))\n  __ := (inferInstance : SemilatticeInf (IdealSheafData X))\n  __ := (inferInstance : CompleteSemilatticeSup (IdealSheafData X))\n  __ := IdealSheafData.gci.liftCompleteLattice\n\n@[simp]\nlemma ideal_top : ideal (X := X) ⊤ = ⊤ := rfl\n\n@[simp]\nlemma ideal_bot : ideal (X := X) ⊥ = ⊥ := rfl\n\n@[simp]\nlemma ideal_sup {I J : IdealSheafData X} : (I ⊔ J).ideal = I.ideal ⊔ J.ideal := rfl\n\n@[simp]\nlemma ideal_sSup {I : Set (IdealSheafData X)} : (sSup I).ideal = sSup (ideal '' I) := rfl\n\n@[simp]\nlemma ideal_iSup {ι : Type*} {I : ι → IdealSheafData X} : (iSup I).ideal = ⨆ i, (I i).ideal := by\n  rw [← sSup_range, ← sSup_range, ideal_sSup, ← Set.range_comp, Function.comp_def]\n\n@[simp]\nlemma ideal_inf {I J : IdealSheafData X} : (I ⊓ J).ideal = I.ideal ⊓ J.ideal := rfl\n\n@[simp]\nlemma ideal_biInf {ι : Type*} (I : ι → IdealSheafData X) {s : Set ι} (hs : s.Finite) :\n    (⨅ i ∈ s, I i).ideal = ⨅ i ∈ s, (I i).ideal := by\n  refine hs.induction_on _ (by simp) fun {i s} his hs e ↦ ?_\n  simp only [iInf_insert, e, ideal_inf]\n\n@[simp]\n\nTarget:\nlemma ideal_iInf {ι : Type*} (I : ι → IdealSheafData X) [Finite ι] :\n    (⨅ i, I i).ideal = ⨅ i, (I i).ideal :=\n\nProof body:\n","rejected":"by\n  exact ideal_iInf","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"1fd8ad3507d7671b7fe89ee7943e17682e5178be8bbde12ede02413b3aa492a8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/IdealSheaf","family_id":"ideal_iinf","file_id":"mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/Basic.lean","sample_id":"322ab09370ffc7ba02e5b0d458715ab5e05e1b425cf1b69dc3d33d89b75fd5ba"},"split":"train"}
{"binding":{"chosen_proof_sha256":"30f83bf0d483bdb24895df71acc1365551186d762bcd20d14e7f1ac4b162ed7c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"862a0c729b5cd3749b546c796ee66cf2c56a927378f8d912626bfe6707f1fa0a","source_sha256":"2aaaf1d6e3bf8bcc38fe6595b30037cc7d4b213d51f17e0ec2ab4dd92ba3ffe4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨s, -, hscard, hs⟩ := addRothNumber_spec (univ : Finset α)\n  haveI := noAccidental hs\n  rw [← hscard, ← card_triangleIndices, ← card_triangles]\n  exact (locallyLinear hs).le_ruzsaSzemerediNumber","hard_negative":false,"metrics":{"chosen_tokens":41,"rejected_tokens":5,"token_jaccard":0.0625,"token_length_ratio":0.121951},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"6c6bec73a3eab774723079f876f3c0074c3a0fd987e9a27c689bc1097add8305","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Additive.AP.Three.Behrend\npublic import Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite\npublic import Mathlib.Tactic.Rify\npublic import Mathlib.Tactic.Qify\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# The Ruzsa-Szemerédi problem\n\nThis file proves the lower bound of the Ruzsa-Szemerédi problem. The problem is to find the maximum\nnumber of edges that a graph on `n` vertices can have if all edges belong to at most one triangle.\n\nThe lower bound comes from turning the big 3AP-free set from Behrend's construction into a graph\nthat has the property that every triangle gives a (possibly trivial) arithmetic progression on the\noriginal set.\n\n## Main declarations\n\n* `ruzsaSzemerediNumberNat n`: Maximum number of edges a graph on `n` vertices can have such that\n  each edge belongs to exactly one triangle.\n* `ruzsaSzemerediNumberNat_asymptotic_lower_bound`: There exists a graph with `n` vertices and\n  `Ω((n ^ 2 * exp (-4 * √(log n))))` edges such that each edge belongs to exactly one triangle.\n-/\n\n@[expose] public section\n\nopen Finset Nat Real SimpleGraph Sum3 SimpleGraph.TripartiteFromTriangles\nopen Fintype (card)\nopen scoped Pointwise\n\nvariable {α β : Type*}\n\n/-! ### The Ruzsa-Szemerédi number -/\n\nsection ruzsaSzemerediNumber\nvariable [DecidableEq α] [DecidableEq β] [Fintype α] [Fintype β] {G H : SimpleGraph α}\n\nvariable (α) in\n/-- The **Ruzsa-Szemerédi number** of a fintype is the maximum number of edges a locally linear\ngraph on that type can have.\n\nIn other words, `ruzsaSzemerediNumber α` is the maximum number of edges a graph on `α` can have such\nthat each edge belongs to exactly one triangle. -/\nnoncomputable def ruzsaSzemerediNumber : ℕ := by\n  classical\n  exact Nat.findGreatest (fun m ↦ ∃ (G : SimpleGraph α) (_ : DecidableRel G.Adj),\n    #(G.cliqueFinset 3) = m ∧ G.LocallyLinear) ((card α).choose 3)\n\nopen scoped Classical in\nlemma ruzsaSzemerediNumber_le : ruzsaSzemerediNumber α ≤ (card α).choose 3 := Nat.findGreatest_le _\n\nlemma ruzsaSzemerediNumber_spec :\n    ∃ (G : SimpleGraph α) (_ : DecidableRel G.Adj),\n      #(G.cliqueFinset 3) = ruzsaSzemerediNumber α ∧ G.LocallyLinear := by\n  classical\n  exact @Nat.findGreatest_spec _\n    (fun m ↦ ∃ (G : SimpleGraph α) (_ : DecidableRel G.Adj),\n      #(G.cliqueFinset 3) = m ∧ G.LocallyLinear) _ _ (Nat.zero_le _)\n    ⟨⊥, inferInstance, by simp, locallyLinear_bot⟩\n\nvariable {m n : ℕ}\n\nlemma SimpleGraph.LocallyLinear.le_ruzsaSzemerediNumber [DecidableRel G.Adj]\n    (hG : G.LocallyLinear) : #(G.cliqueFinset 3) ≤ ruzsaSzemerediNumber α := by\n  classical\n  exact le_findGreatest card_cliqueFinset_le ⟨G, inferInstance, by congr, hG⟩\n\nlemma ruzsaSzemerediNumber_mono (f : α ↪ β) : ruzsaSzemerediNumber α ≤ ruzsaSzemerediNumber β := by\n  classical\n  refine findGreatest_mono ?_ (choose_mono _ <| Fintype.card_le_of_embedding f)\n  rintro n ⟨G, _, rfl, hG⟩\n  refine ⟨G.map f, inferInstance, ?_, hG.map _⟩\n  rw [← card_map ⟨map f, Finset.map_injective _⟩, ← cliqueFinset_map G f]\n  decide\n\nlemma ruzsaSzemerediNumber_congr (e : α ≃ β) : ruzsaSzemerediNumber α = ruzsaSzemerediNumber β :=\n  (ruzsaSzemerediNumber_mono (e : α ↪ β)).antisymm <| ruzsaSzemerediNumber_mono e.symm\n\n/-- The `n`-th **Ruzsa-Szemerédi number** is the maximum number of edges a locally linear graph on\n`n` vertices can have.\n\nIn other words, `ruzsaSzemerediNumberNat n` is the maximum number of edges a graph on `n` vertices\ncan have such that each edge belongs to exactly one triangle. -/\nnoncomputable def ruzsaSzemerediNumberNat (n : ℕ) : ℕ := ruzsaSzemerediNumber (Fin n)\n\n@[simp]\nlemma ruzsaSzemerediNumberNat_card : ruzsaSzemerediNumberNat (card α) = ruzsaSzemerediNumber α :=\n  ruzsaSzemerediNumber_congr (Fintype.equivFin _).symm\n\n@[gcongr]\nlemma ruzsaSzemerediNumberNat_mono : Monotone ruzsaSzemerediNumberNat := fun _m _n h =>\n  ruzsaSzemerediNumber_mono (Fin.castLEEmb h)\n\nlemma ruzsaSzemerediNumberNat_le : ruzsaSzemerediNumberNat n ≤ n.choose 3 :=\n  ruzsaSzemerediNumber_le.trans_eq <| by rw [Fintype.card_fin]\n\n@[simp] lemma ruzsaSzemerediNumberNat_zero : ruzsaSzemerediNumberNat 0 = 0 :=\n  le_zero_iff.1 ruzsaSzemerediNumberNat_le\n\n@[simp] lemma ruzsaSzemerediNumberNat_one : ruzsaSzemerediNumberNat 1 = 0 :=\n  le_zero_iff.1 ruzsaSzemerediNumberNat_le\n\n@[simp] lemma ruzsaSzemerediNumberNat_two : ruzsaSzemerediNumberNat 2 = 0 :=\n  le_zero_iff.1 ruzsaSzemerediNumberNat_le\n\nend ruzsaSzemerediNumber\n\n/-! ### The Ruzsa-Szemerédi construction -/\n\nsection RuzsaSzemeredi\nvariable [Fintype α] [CommRing α] {s : Finset α} {x : α × α × α}\n\n/-- The triangle indices for the Ruzsa-Szemerédi construction. -/\nprivate def triangleIndices (s : Finset α) : Finset (α × α × α) :=\n  (univ ×ˢ s).map\n    ⟨fun xa ↦ (xa.1, xa.1 + xa.2, xa.1 + 2 * xa.2), by\n      rintro ⟨x, a⟩ ⟨y, b⟩ h\n      simp only [Prod.ext_iff] at h\n      obtain rfl := h.1\n      obtain rfl := add_right_injective _ h.2.1\n      rfl⟩\n\n@[simp]\nprivate lemma mem_triangleIndices :\n    x ∈ triangleIndices s ↔ ∃ y, ∃ a ∈ s, (y, y + a, y + 2 * a) = x := by simp [triangleIndices]\n\n@[simp]\nprivate lemma card_triangleIndices : #(triangleIndices s) = card α * #s := by\n  simp [triangleIndices]\n\nprivate lemma noAccidental (hs : ThreeAPFree (s : Set α)) :\n    NoAccidental (triangleIndices s : Finset (α × α × α)) where\n  eq_or_eq_or_eq := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ _ _ d a ha rfl rfl rfl b' b hb rfl rfl h₁ d' c hc rfl h₂ rfl\n    have : a + c = b + b := by linear_combination h₁.symm - h₂.symm\n    obtain rfl := hs ha hb hc this\n    simp_all\n\nvariable [Fact <| IsUnit (2 : α)]\n\nprivate instance : ExplicitDisjoint (triangleIndices s : Finset (α × α × α)) where\n  inj₀ := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ x a ha rfl rfl rfl y b hb rfl h₁ h₂\n    linear_combination 2 * h₁.symm - h₂.symm\n  inj₁ := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ x a ha rfl rfl rfl y b hb rfl rfl h\n    simpa [(Fact.out (p := IsUnit (2 : α))).mul_right_inj, eq_comm] using h\n  inj₂ := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ x a ha rfl rfl rfl y b hb rfl h rfl\n    simpa [(Fact.out (p := IsUnit (2 : α))).mul_right_inj, eq_comm] using h\n\nprivate lemma locallyLinear (hs : ThreeAPFree (s : Set α)) :\n    (graph <| triangleIndices s).LocallyLinear :=\n  haveI := noAccidental hs; TripartiteFromTriangles.locallyLinear _\n\nprivate lemma card_edgeFinset (hs : ThreeAPFree (s : Set α)) [DecidableEq α] :\n    #(graph <| triangleIndices s).edgeFinset = 3 * card α * #s := by\n  haveI := noAccidental hs\n  rw [(locallyLinear hs).card_edgeFinset, card_triangles, card_triangleIndices, mul_assoc]\n\nend RuzsaSzemeredi\n\nvariable (α) [Fintype α] [DecidableEq α] [CommRing α] [Fact <| IsUnit (2 : α)]\n\nTarget:\nlemma addRothNumber_le_ruzsaSzemerediNumber :\n    card α * addRothNumber (univ : Finset α) ≤ ruzsaSzemerediNumber (Sum α (Sum α α)) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Extremal","family_id":"addrothnumber_le_ruzsaszemeredinumber","file_id":"mathlib/Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean","sample_id":"862a0c729b5cd3749b546c796ee66cf2c56a927378f8d912626bfe6707f1fa0a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6ef68f198f9c3ddb8b2e403677d5bebc09f01883ade66189972cc6ad03aa6c3a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e52881820a620469d9332d0452bd209c49db4dddc5eb8e1aa89aa852e3b4661b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5cac71067938ac4bef77108bfac10db01fab6c38979ad732ebaa5982c97b9231","source_sha256":"148d7eba9bc7c4474f48190ef655e8ff7d2748109cb095b07744bfb109f40298","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let f := (lmul R S).toLinearMap.prodMap (lmul R T).toLinearMap\n  have : (lmul R (S × T)).toLinearMap = (prodMapLinear R S T S T R).comp f :=\n    LinearMap.ext₂ Prod.mul_def\n  simp_rw [trace, this]\n  exact trace_prodMap' _ _","hard_negative":false,"metrics":{"chosen_tokens":64,"rejected_tokens":69,"token_jaccard":0.891892,"token_length_ratio":1.078125},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"6c9003c36ce82bbfe994aa2099238fb105a507356857eef5de44825a2266f4c8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas\npublic import Mathlib.LinearAlgebra.Matrix.BilinearForm\npublic import Mathlib.LinearAlgebra.Trace\n\nNamespace:\nAlgebra\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Trace for (finite) ring extensions.\n\nSuppose we have an `R`-algebra `S` with a finite basis. For each `s : S`,\nthe trace of the linear map given by multiplying by `s` gives information about\nthe roots of the minimal polynomial of `s` over `R`.\n\n## Main definitions\n\n* `Algebra.trace R S x`: the trace of an element `s` of an `R`-algebra `S`\n* `Algebra.traceForm R S`: bilinear form sending `x`, `y` to the trace of `x * y`\n* `Algebra.traceMatrix R b`: the matrix whose `(i j)`-th element is the trace of `b i * b j`.\n\n## Main results\n\n* `trace_algebraMap_of_basis`, `trace_algebraMap`: if `x : K`, then `Tr_{L/K} x = [L : K] x`\n* `trace_trace_of_basis`, `trace_trace`: `Tr_{L/K} (Tr_{F/L} x) = Tr_{F/K} x`\n\n## Implementation notes\n\nTypically, the trace is defined specifically for finite field extensions.\nThe definition is as general as possible and the assumption that the extension is finite\nis added to the lemmas as needed.\n\nWe only define the trace for left multiplication (`Algebra.leftMulMatrix`,\ni.e. `LinearMap.mulLeft`).\nFor now, the definitions assume `S` is commutative, so the choice doesn't matter anyway.\n\n## References\n\n* https://en.wikipedia.org/wiki/Field_trace\n\n-/\n\n@[expose] public section\n\n\nuniverse w\n\nvariable {R S T : Type*} [CommRing R] [CommRing S] [CommRing T]\nvariable [Algebra R S] [Algebra R T]\nvariable {ι : Type w} [Fintype ι]\n\nopen Module\n\nopen LinearMap (BilinForm)\nopen LinearMap\n\nopen Matrix\n\nopen scoped Matrix\n\nnamespace Algebra\n\nvariable (R S)\n\n/-- The trace of an element `s` of an `R`-algebra is the trace of `(s * ·)`,\nas an `R`-linear map. -/\n@[stacks 0BIF \"Trace\"]\nnoncomputable def trace : S →ₗ[R] R :=\n  (LinearMap.trace R S).comp (lmul R S).toLinearMap\n\nvariable {S}\n\n-- Not a `simp` lemma since there are more interesting ways to rewrite `trace R S x`,\n-- for example `trace_trace`\ntheorem trace_apply (x) : trace R S x = LinearMap.trace R S (lmul R S x) :=\n  rfl\n\ntheorem trace_eq_zero_of_not_exists_basis (h : ¬∃ s : Finset S, Nonempty (Basis s R S)) :\n    trace R S = 0 := by ext s; simp [trace_apply, LinearMap.trace, h]\n\nvariable {R}\n\n-- Can't be a `simp` lemma because it depends on a choice of basis\ntheorem trace_eq_matrix_trace [DecidableEq ι] (b : Basis ι R S) (s : S) :\n    trace R S s = Matrix.trace (Algebra.leftMulMatrix b s) := by\n  rw [trace_apply, LinearMap.trace_eq_matrix_trace _ b, ← toMatrix_lmul_eq]; rfl\n\n/-- If `x` is in the base field `K`, then the trace is `[L : K] * x`. -/\ntheorem trace_algebraMap_of_basis (b : Basis ι R S) (x : R) :\n    trace R S (algebraMap R S x) = Fintype.card ι • x := by\n  haveI := Classical.decEq ι\n  rw [trace_apply, LinearMap.trace_eq_matrix_trace R b, Matrix.trace]\n  convert! Finset.sum_const x\n  simp [-coe_lmul_eq_mul]\n\n\n/-- The trace map from `R` to itself is the identity map. -/\n@[simp] theorem trace_self : trace R R = LinearMap.id := by\n  ext; simpa using trace_algebraMap_of_basis (.singleton (Fin 1) R) 1\n\ntheorem trace_self_apply (a) : trace R R a = a := by simp\n\n/-- If `x` is in the base field `K`, then the trace is `[L : K] * x`.\n\n(If `L` is not finite-dimensional over `K`, then `trace` and `finrank` return `0`.)\n-/\n@[simp]\ntheorem trace_algebraMap [StrongRankCondition R] [Module.Free R S] (x : R) :\n    trace R S (algebraMap R S x) = finrank R S • x := by\n  by_cases H : ∃ s : Finset S, Nonempty (Basis s R S)\n  · rw [trace_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some]\n  · simp [trace_eq_zero_of_not_exists_basis R H, finrank_eq_zero_of_not_exists_basis_finset H]\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem trace_trace_of_basis [Algebra S T] [IsScalarTower R S T] {ι κ : Type*} [Finite ι]\n    [Finite κ] (b : Basis ι R S) (c : Basis κ S T) (x : T) :\n    trace R S (trace S T x) = trace R T x := by\n  haveI := Classical.decEq ι\n  haveI := Classical.decEq κ\n  cases nonempty_fintype ι\n  cases nonempty_fintype κ\n  rw [trace_eq_matrix_trace (b.smulTower c), trace_eq_matrix_trace b, trace_eq_matrix_trace c,\n    Matrix.trace, Matrix.trace, Matrix.trace, ← Finset.univ_product_univ, Finset.sum_product]\n  refine Finset.sum_congr rfl fun i _ ↦ ?_\n  simp only [map_sum, smulTower_leftMulMatrix, Finset.sum_apply, Matrix.diag,\n    Finset.sum_apply i (Finset.univ : Finset κ) fun y => leftMulMatrix b (leftMulMatrix c x y y)]\n\ntheorem trace_comp_trace_of_basis [Algebra S T] [IsScalarTower R S T] {ι κ : Type*} [Finite ι]\n    [Finite κ] (b : Basis ι R S) (c : Basis κ S T) :\n    (trace R S).comp ((trace S T).restrictScalars R) = trace R T := by\n  ext\n  rw [LinearMap.comp_apply, LinearMap.restrictScalars_apply, trace_trace_of_basis b c]\n\n@[simp]\ntheorem trace_trace [Algebra S T] [IsScalarTower R S T]\n    [Module.Free R S] [Module.Finite R S] [Module.Free S T] [Module.Finite S T] (x : T) :\n    trace R S (trace S T x) = trace R T x :=\n  trace_trace_of_basis (Module.Free.chooseBasis R S) (Module.Free.chooseBasis S T) x\n\n/-- Let `T / S / R` be a tower of finite extensions of fields. Then\n$\\text{Trace}_{T/R} = \\text{Trace}_{S/R} \\circ \\text{Trace}_{T/S}$. -/\n@[simp, stacks 0BIJ \"Trace\"]\ntheorem trace_comp_trace [Algebra S T] [IsScalarTower R S T]\n    [Module.Free R S] [Module.Finite R S] [Module.Free S T] [Module.Finite S T] :\n    (trace R S).comp ((trace S T).restrictScalars R) = trace R T :=\n  LinearMap.ext trace_trace\n\n@[simp]\n\nTarget:\ntheorem trace_prod_apply [Module.Free R S] [Module.Free R T] [Module.Finite R S] [Module.Finite R T]\n    (x : S × T) : trace R (S × T) x = trace R S x.fst + trace R T x.snd :=\n\nProof body:\n","rejected":"by\n  let f := (lmul R S).toLinearMap.prodMap (lmul R T).toLinearMap\n  have : (lmul R (S × T)).toLinearMap = (prodMapLinear R S T S T R).comp f :=\n    LinearMap.ext₂ Prod.mul_def\n  simp_rw [trace, this]\n  exact trace_prodMap' _ _\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Trace","family_id":"trace_prod_apply","file_id":"mathlib/Mathlib/RingTheory/Trace/Defs.lean","sample_id":"5cac71067938ac4bef77108bfac10db01fab6c38979ad732ebaa5982c97b9231"},"split":"train"}
{"binding":{"chosen_proof_sha256":"667083b2ccb953eab7042f8125e1495f0c41208219db3ae4328356bea107af18","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b1b23128020afe0708f69c540c35c829ecc671bed6b190d153106b1f52a56996","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"76ce3bf080d160d098641f9edded107effa5e468a4e154be6bb90faf62e605f0","source_sha256":"3b1ad71a44a0ba95fe172fff4adb35c1c3d0ab3dacc0cacca37d6024c51fd12b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [div_def, ← @Rat.round_cast ℝ _ _, ← @Rat.round_cast ℝ _ _]\n  simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, add_mul]","hard_negative":true,"metrics":{"chosen_tokens":36,"rejected_tokens":3,"token_jaccard":0.05,"token_length_ratio":0.083333},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"6c9fabc5a718462041620288093484af2a1dae83875e6f5bdce5d0e2dd5f8783","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Archimedean.Real.Basic\npublic import Mathlib.Data.Complex.Basic\npublic import Mathlib.Data.Nat.Prime.Basic\npublic import Mathlib.NumberTheory.Zsqrtd.Basic\n\nNamespace:\nGaussianInt\n\nLocal context:\n/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Gaussian integers\n\nThe Gaussian integers are complex integer, complex numbers whose real and imaginary parts are both\nintegers.\n\n## Main definitions\n\nThe Euclidean domain structure on `ℤ[i]` is defined in this file.\n\nThe homomorphism `GaussianInt.toComplex` into the complex numbers is also defined in this file.\n\n## See also\n\nSee `NumberTheory.Zsqrtd.QuadraticReciprocity` for:\n* `prime_iff_mod_four_eq_three_of_nat_prime`:\n  A prime natural number is prime in `ℤ[i]` if and only if it is `3` mod `4`\n\n## Notation\n\nThis file uses the local notation `ℤ[i]` for `GaussianInt`\n\n## Implementation notes\n\nGaussian integers are implemented using the more general definition `Zsqrtd`, the type of integers\nadjoined a square root of `d`, in this case `-1`. The definition is reducible, so that properties\nand definitions about `Zsqrtd` can easily be used.\n-/\n\n@[expose] public section\n\n\nopen Zsqrtd Complex\n\nopen scoped ComplexConjugate\n\n/-- The Gaussian integers, defined as `ℤ√(-1)`. -/\nabbrev GaussianInt : Type :=\n  Zsqrtd (-1)\n\nlocal notation \"ℤ[i]\" => GaussianInt\n\nnamespace GaussianInt\n\ninstance : Repr ℤ[i] :=\n  ⟨fun x _ => \"⟨\" ++ repr x.re ++ \", \" ++ repr x.im ++ \"⟩\"⟩\n\ninstance instCommRing : CommRing ℤ[i] :=\n  Zsqrtd.commRing\n\nsection\n\nattribute [-instance] Complex.instField -- Avoid making things noncomputable unnecessarily.\n\n/-- The embedding of the Gaussian integers into the complex numbers, as a ring homomorphism. -/\ndef toComplex : ℤ[i] →+* ℂ :=\n  Zsqrtd.lift ⟨I, by simp⟩\n\nend\n\ninstance : Coe ℤ[i] ℂ :=\n  ⟨toComplex⟩\n\ntheorem toComplex_def (x : ℤ[i]) : (x : ℂ) = x.re + x.im * I :=\n  rfl\n\ntheorem toComplex_def' (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ) = x + y * I := by simp [toComplex_def]\n\ntheorem toComplex_def₂ (x : ℤ[i]) : (x : ℂ) = ⟨x.re, x.im⟩ := by\n  apply Complex.ext <;> simp [toComplex_def]\n\n@[simp]\ntheorem intCast_re (x : ℤ[i]) : ((x.re : ℤ) : ℝ) = (x : ℂ).re := by simp [toComplex_def]\n\n@[simp]\ntheorem intCast_im (x : ℤ[i]) : ((x.im : ℤ) : ℝ) = (x : ℂ).im := by simp [toComplex_def]\n\n@[simp]\ntheorem re_toComplex (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).re = x := by simp [toComplex_def]\n\n@[simp]\ntheorem im_toComplex (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).im = y := by simp [toComplex_def]\n\ntheorem toComplex_add (x y : ℤ[i]) : ((x + y : ℤ[i]) : ℂ) = x + y :=\n  toComplex.map_add _ _\n\ntheorem toComplex_mul (x y : ℤ[i]) : ((x * y : ℤ[i]) : ℂ) = x * y :=\n  toComplex.map_mul _ _\n\ntheorem toComplex_one : ((1 : ℤ[i]) : ℂ) = 1 :=\n  toComplex.map_one\n\ntheorem toComplex_zero : ((0 : ℤ[i]) : ℂ) = 0 :=\n  toComplex.map_zero\n\ntheorem toComplex_neg (x : ℤ[i]) : ((-x : ℤ[i]) : ℂ) = -x :=\n  toComplex.map_neg _\n\ntheorem toComplex_sub (x y : ℤ[i]) : ((x - y : ℤ[i]) : ℂ) = x - y :=\n  toComplex.map_sub _ _\n\n@[simp]\ntheorem toComplex_star (x : ℤ[i]) : ((star x : ℤ[i]) : ℂ) = conj (x : ℂ) := by\n  rw [toComplex_def₂, toComplex_def₂]\n  exact congr_arg₂ _ rfl (Int.cast_neg _)\n\n@[simp]\ntheorem toComplex_inj {x y : ℤ[i]} : (x : ℂ) = y ↔ x = y := by\n  cases x; cases y; simp [toComplex_def₂]\n\nlemma toComplex_injective : Function.Injective GaussianInt.toComplex :=\n  fun ⦃_ _⦄ ↦ toComplex_inj.mp\n\n@[simp]\ntheorem toComplex_eq_zero {x : ℤ[i]} : (x : ℂ) = 0 ↔ x = 0 := by\n  rw [← toComplex_zero, toComplex_inj]\n\n@[simp]\ntheorem intCast_real_norm (x : ℤ[i]) : (x.norm : ℝ) = Complex.normSq (x : ℂ) := by\n  rw [Zsqrtd.norm, normSq]; simp\n\n@[simp]\ntheorem intCast_complex_norm (x : ℤ[i]) : (x.norm : ℂ) = Complex.normSq (x : ℂ) := by\n  cases x; rw [Zsqrtd.norm, normSq]; simp\n\ntheorem norm_nonneg (x : ℤ[i]) : 0 ≤ norm x :=\n  Zsqrtd.norm_nonneg (by simp) _\n\n@[simp]\ntheorem norm_eq_zero {x : ℤ[i]} : norm x = 0 ↔ x = 0 := by rw [← @Int.cast_inj ℝ _ _ _]; simp\n\ntheorem norm_pos {x : ℤ[i]} : 0 < norm x ↔ x ≠ 0 := by\n  rw [lt_iff_le_and_ne, Ne, eq_comm, norm_eq_zero]; simp [norm_nonneg]\n\ntheorem abs_natCast_norm (x : ℤ[i]) : (x.norm.natAbs : ℤ) = x.norm :=\n  Int.natAbs_of_nonneg (norm_nonneg _)\n\ntheorem natCast_natAbs_norm {α : Type*} [AddGroupWithOne α] (x : ℤ[i]) :\n    (x.norm.natAbs : α) = x.norm := by\n  simp\n\ntheorem natAbs_norm_eq (x : ℤ[i]) :\n    x.norm.natAbs = x.re.natAbs * x.re.natAbs + x.im.natAbs * x.im.natAbs := by\n  zify\n  rw [abs_norm (by simp)]\n  simp [Zsqrtd.norm]\n\ninstance : Div ℤ[i] :=\n  ⟨fun x y =>\n    let n := (norm y : ℚ)⁻¹\n    let c := star y\n    ⟨round ((x * c).re * n : ℚ), round ((x * c).im * n : ℚ)⟩⟩\n\ntheorem div_def (x y : ℤ[i]) :\n    x / y = ⟨round ((x * star y).re / norm y : ℚ), round ((x * star y).im / norm y : ℚ)⟩ :=\n  show Zsqrtd.mk _ _ = _ by simp [div_eq_mul_inv]\n\ntheorem toComplex_re_div (x y : ℤ[i]) : ((x / y : ℤ[i]) : ℂ).re = round (x / y : ℂ).re := by\n  rw [div_def, ← @Rat.round_cast ℝ _ _]\n  simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, add_mul]\n\nTarget:\ntheorem toComplex_im_div (x y : ℤ[i]) : ((x / y : ℤ[i]) : ℂ).im = round (x / y : ℂ).im :=\n\nProof body:\n","rejected":"by\n  exact toComplex_im_div","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"f245cf6f88286ec979e42f060a828c3dbf251248eda49d47534f93a072f59898","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Zsqrtd","family_id":"tocomplex_im_div","file_id":"mathlib/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean","sample_id":"76ce3bf080d160d098641f9edded107effa5e468a4e154be6bb90faf62e605f0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6bc6459c3657e743f6b65ab2f495aaf259c49629a6dfb3dcfa555bfa29fc68bb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"56582689395b3fd4d53bdd9aa4d7c0b70282e024f7b7737ed578d8946fb979bf","source_sha256":"bc065e5d26147920810f7e88f906504ecfcce0ffd1da2e0b44dc4df7256292f1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hf : Monotone fun n => ⨅ i ≥ n, f i := fun n m h => biInf_mono fun i => h.trans\n  rw [← Monotone.iSup_nat_add hf k]\n  · simp_rw [iInf_ge_eq_iInf_nat_add, ← Nat.add_assoc]\n\n-- Not `@[simp]` since the subterm `?f (i + ?k)` produces an ugly higher-order unification problem.\n-- (Although the `simpNF` linter does not complain.)\n-- See: https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/complete_lattice.20and.20has_sup/near/316497982","hard_negative":false,"metrics":{"chosen_tokens":143,"rejected_tokens":2,"token_jaccard":0.025316,"token_length_ratio":0.013986},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"6caa21cea7054c22cab76a355d75bd6a77c89a6b3d004604d78ba2cb400e424e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Bool.Set\npublic import Mathlib.Data.Nat.Set\npublic import Mathlib.Order.CompleteLattice.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Theory of complete lattices\n\nThis file contains results on complete lattices that need more theory to develop.\n\n## Naming conventions\n\nIn lemma names,\n* `sSup` is called `sSup`\n* `sInf` is called `sInf`\n* `⨆ i, s i` is called `iSup`\n* `⨅ i, s i` is called `iInf`\n* `⨆ i j, s i j` is called `iSup₂`. This is an `iSup` inside an `iSup`.\n* `⨅ i j, s i j` is called `iInf₂`. This is an `iInf` inside an `iInf`.\n* `⨆ i ∈ s, t i` is called `biSup` for \"bounded `iSup`\". This is the special case of `iSup₂`\n  where `j : i ∈ s`.\n* `⨅ i ∈ s, t i` is called `biInf` for \"bounded `iInf`\". This is the special case of `iInf₂`\n  where `j : i ∈ s`.\n\n## Notation\n\n* `⨆ i, f i` : `iSup f`, the supremum of the range of `f`;\n* `⨅ i, f i` : `iInf f`, the infimum of the range of `f`.\n-/\n\npublic section\n\nopen Function OrderDual Set\n\nvariable {α β γ : Type*} {ι ι' : Sort*} {κ : ι → Sort*} {κ' : ι' → Sort*}\n\nopen OrderDual\n\nsection\n\nvariable [CompleteLattice α] {f g s : ι → α} {a b : α}\n\n/-!\n### `iSup` and `iInf` under `Bool`\n-/\n\n@[to_dual]\ntheorem iSup_bool_eq {f : Bool → α} : ⨆ b : Bool, f b = f true ⊔ f false := by\n  rw [iSup, Bool.range_eq, sSup_pair, sup_comm]\n\n@[to_dual]\ntheorem sup_eq_iSup (x y : α) : x ⊔ y = ⨆ b : Bool, cond b x y := by\n  rw [iSup_bool_eq, Bool.cond_true, Bool.cond_false]\n\n/-!\n### `iSup` and `iInf` under `ℕ`\n-/\n\n@[to_dual]\ntheorem iSup_ge_eq_iSup_nat_add (u : ℕ → α) (n : ℕ) : ⨆ i ≥ n, u i = ⨆ i, u (i + n) := by\n  apply le_antisymm <;> simp only [iSup_le_iff]\n  · refine fun i hi => le_sSup ⟨i - n, ?_⟩\n    dsimp only\n    rw [Nat.sub_add_cancel hi]\n  · exact fun i => le_sSup ⟨i + n, iSup_pos (Nat.le_add_left _ _)⟩\n\n-- `to_dual` cannot translate between `Monotone` and `Antitone`.\ntheorem Monotone.iSup_nat_add {f : ℕ → α} (hf : Monotone f) (k : ℕ) : ⨆ n, f (n + k) = ⨆ n, f n :=\n  le_antisymm (iSup_le fun i => le_iSup _ (i + k)) <| iSup_mono fun i => hf <| Nat.le_add_right i k\n\ntheorem Antitone.iInf_nat_add {f : ℕ → α} (hf : Antitone f) (k : ℕ) : ⨅ n, f (n + k) = ⨅ n, f n :=\n  hf.dual_right.iSup_nat_add k\n\n-- Not `@[simp]` since the subterm `?f (i + ?k)` produces an ugly higher-order unification problem.\n-- (Although the `simpNF` linter does not complain.)\n-- See: https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/complete_lattice.20and.20has_sup/near/316497982\n\nTarget:\ntheorem iSup_iInf_ge_nat_add (f : ℕ → α) (k : ℕ) :\n    ⨆ n, ⨅ i ≥ n, f (i + k) = ⨆ n, ⨅ i ≥ n, f i :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/CompleteLattice","family_id":"isup_iinf_ge_nat_add","file_id":"mathlib/Mathlib/Order/CompleteLattice/Lemmas.lean","sample_id":"56582689395b3fd4d53bdd9aa4d7c0b70282e024f7b7737ed578d8946fb979bf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5e0e8b084461b68f53dd226528eec6d4daa60789f672fc8f9edb2d39518f89fb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"73b61e1aea22a359d649ed44c1a7748117734f06b4e1a751b190fe4b4843b1c8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f5cf3adf5b2f007282e6afc982afeb8adfc7d1ca08c6a7becb4328354365d479","source_sha256":"54764080c8f7f97a39b09c795498f3d597d51750fa303452440b9c097a0d0bc6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply c.isPushout.hom_ext h₀\n  apply Cofan.IsColimit.hom_ext c.isColimit₂\n  simpa [cell_def] using h","hard_negative":true,"metrics":{"chosen_tokens":25,"rejected_tokens":5,"token_jaccard":0.1,"token_length_ratio":0.2},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"6cb1943243d158949aa2734503e5b5f5bb33c0df588558990fe08884a6ac429d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.MorphismProperty.Limits\n\nNamespace:\nHomotopicalAlgebra.AttachCells\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Attaching cells\n\nGiven a family of morphisms `g a : A a ⟶ B a` and a morphism `f : X₁ ⟶ X₂`,\nwe introduce a structure `AttachCells g f` which expresses that `X₂`\nis obtained from `X₁` by attaching cells of the form `g a`. It means that\nthere is a pushout diagram of the form\n```\n⨿ i, A (π i) -----> X₁\n  |                 |f\n  v                 v\n⨿ i, B (π i) -----> X₂\n```\nIn other words, the morphism `f` is a pushout of coproducts of morphisms\nof the form `g a : A a ⟶ B a`, see `nonempty_attachCells_iff`.\n\nSee the file `Mathlib/AlgebraicTopology/RelativeCellComplex/Basic.lean` for transfinite compositions\nof morphisms `f` with `AttachCells g f` structures.\n\n-/\n\n@[expose] public section\n\nuniverse w' w t t' v u\n\nopen CategoryTheory Limits\n\nnamespace HomotopicalAlgebra\n\nvariable {C : Type u} [Category.{v} C]\n  {α : Type t} {A B : α → C} (g : ∀ a, A a ⟶ B a)\n  {X₁ X₂ : C} (f : X₁ ⟶ X₂)\n\n/-- Given a family of morphisms `g a : A a ⟶ B a` and a morphism `f : X₁ ⟶ X₂`,\nthis structure contains the data and properties which expresses that `X₂`\nis obtained from `X₁` by attaching cells of the form `g a`. -/\nstructure AttachCells where\n  /-- the index type of the cells -/\n  ι : Type w\n  /-- for each `i : ι`, we shall attach a cell given by the morphism `g (π i)`. -/\n  π : ι → α\n  /-- a colimit cofan which gives the coproduct of the object `A (π i)` -/\n  cofan₁ : Cofan (fun i ↦ A (π i))\n  /-- a colimit cofan which gives the coproduct of the object `B (π i)` -/\n  cofan₂ : Cofan (fun i ↦ B (π i))\n  /-- `cofan₁` is colimit -/\n  isColimit₁ : IsColimit cofan₁\n  /-- `cofan₂` is colimit -/\n  isColimit₂ : IsColimit cofan₂\n  /-- the coproduct of the maps `g (π i) : A (π i) ⟶ B (π i)` for all `i : ι`. -/\n  m : cofan₁.pt ⟶ cofan₂.pt\n  hm (i : ι) : cofan₁.inj i ≫ m = g (π i) ≫ cofan₂.inj i := by cat_disch\n  /-- the top morphism of the pushout square -/\n  g₁ : cofan₁.pt ⟶ X₁\n  /-- the bottom morphism of the pushout square -/\n  g₂ : cofan₂.pt ⟶ X₂\n  isPushout : IsPushout g₁ m f g₂\n\nnamespace AttachCells\n\nopen MorphismProperty\n\nattribute [reassoc (attr := simp)] hm\n\nvariable {g f} (c : AttachCells.{w} g f)\n\ninclude c\n\nlemma pushouts_coproducts : (coproducts.{w} (ofHoms g)).pushouts f := by\n  refine ⟨_, _, _, _, _, ?_, c.isPushout⟩\n  have : c.m = c.isColimit₁.desc\n      (Cocone.mk _ (Discrete.natTrans (fun ⟨i⟩ ↦ by exact g (c.π i)) ≫ c.cofan₂.ι)) :=\n    c.isColimit₁.hom_ext (fun ⟨i⟩ ↦ by rw [IsColimit.fac]; exact c.hm i)\n  rw [this, coproducts_iff]\n  exact ⟨c.ι, ⟨_, _, _, _, c.isColimit₁, c.isColimit₂, _, fun i ↦ ⟨_⟩⟩⟩\n\n/-- The inclusion of a cell. -/\ndef cell (i : c.ι) : B (c.π i) ⟶ X₂ := c.cofan₂.inj i ≫ c.g₂\n\n@[reassoc]\nlemma cell_def (i : c.ι) : c.cell i = c.cofan₂.inj i ≫ c.g₂ := rfl\n\nTarget:\nlemma hom_ext {Z : C} {φ φ' : X₂ ⟶ Z}\n    (h₀ : f ≫ φ = f ≫ φ') (h : ∀ i, c.cell i ≫ φ = c.cell i ≫ φ') :\n    φ = φ' :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_f5cf3adf5b2f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"be2ea9014a88548e9b8f4a61cb0933db339bd78a9aad1e2f24f318c4bdc82922","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/RelativeCellComplex","family_id":"hom_ext","file_id":"mathlib/Mathlib/AlgebraicTopology/RelativeCellComplex/AttachCells.lean","sample_id":"f5cf3adf5b2f007282e6afc982afeb8adfc7d1ca08c6a7becb4328354365d479"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7361ae6d5ff40c196b32b028010a5c56393ce19a055fbd12b76df3365e97ad61","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"108e499b9132e898676c8d66a09fbbcb3ff6a762c243e0f235bc6414a3079513","source_sha256":"d7743ebc410d9ee31a12263ae8d894bf6aecd1d52cd1a367eccb2f5de9ab8de8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h ↦ eq_univ_iff_forall.mpr h.mem_periodicPts, fun h ↦ ?_⟩\n  rw [Finite.injective_iff_surjective, ← range_eq_univ, ← univ_subset_iff, ← h]\n  apply periodicPts_subset_range","hard_negative":false,"metrics":{"chosen_tokens":36,"rejected_tokens":3,"token_jaccard":0.038462,"token_length_ratio":0.083333},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"6cccae1b89db7521eed11bb5941dc56ba8caf5d9ee4a72d1085f431647e49944","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GCDMonoid.Finset\npublic import Mathlib.Data.Nat.Prime.Basic\npublic import Mathlib.Data.PNat.Basic\npublic import Mathlib.Dynamics.PeriodicPts.Defs\npublic import Mathlib.Order.Lattice.Nat\n\nNamespace:\nFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Extra lemmas about periodic points\n-/\n\npublic section\n\nopen Nat Set\n\nnamespace Function\nvariable {α : Type*} {f : α → α} {x y : α}\n\nopen Function (Commute)\n\ntheorem directed_ptsOfPeriod_pnat (f : α → α) : Directed (· ⊆ ·) fun n : ℕ+ => ptsOfPeriod f n :=\n  fun m n => ⟨m * n, fun _ hx => hx.mul_const n, fun _ hx => hx.const_mul m⟩\n\nvariable (f) in\ntheorem bijOn_periodicPts : BijOn f (periodicPts f) (periodicPts f) :=\n  iUnion_pnat_ptsOfPeriod f ▸\n    bijOn_iUnion_of_directed (directed_ptsOfPeriod_pnat f) fun i => bijOn_ptsOfPeriod f i.pos\n\ntheorem minimalPeriod_eq_prime_iff {p : ℕ} [hp : Fact p.Prime] :\n    minimalPeriod f x = p ↔ IsPeriodicPt f p x ∧ ¬IsFixedPt f x := by\n  rw [Function.isPeriodicPt_iff_minimalPeriod_dvd, Nat.dvd_prime hp.out,\n    ← minimalPeriod_eq_one_iff_isFixedPt.not, or_and_right, and_not_self_iff, false_or,\n    iff_self_and]\n  exact fun h ↦ ne_of_eq_of_ne h hp.out.ne_one\n\ntheorem minimalPeriod_eq_sInf_n_pos_IsPeriodicPt :\n    minimalPeriod f x = sInf { n > 0 | IsPeriodicPt f n x } := by\n  dsimp +instances [minimalPeriod, periodicPts, sInf]\n  grind\n\n/-- The backward direction of `minimalPeriod_eq_prime_iff`. -/\ntheorem minimalPeriod_eq_prime {p : ℕ} [hp : Fact p.Prime] (hper : IsPeriodicPt f p x)\n    (hfix : ¬IsFixedPt f x) : minimalPeriod f x = p :=\n  minimalPeriod_eq_prime_iff.mpr ⟨hper, hfix⟩\n\ntheorem minimalPeriod_eq_prime_pow {p k : ℕ} [hp : Fact p.Prime] (hk : ¬IsPeriodicPt f (p ^ k) x)\n    (hk1 : IsPeriodicPt f (p ^ (k + 1)) x) : minimalPeriod f x = p ^ (k + 1) := by\n  apply Nat.eq_prime_pow_of_dvd_least_prime_pow hp.out <;>\n    rwa [← isPeriodicPt_iff_minimalPeriod_dvd]\n\ntheorem Commute.minimalPeriod_of_comp_dvd_mul {g : α → α} (h : Commute f g) :\n    minimalPeriod (f ∘ g) x ∣ minimalPeriod f x * minimalPeriod g x :=\n  dvd_trans h.minimalPeriod_of_comp_dvd_lcm (Nat.lcm_dvd_mul _ _)\n\ntheorem Commute.minimalPeriod_of_comp_eq_mul_of_coprime {g : α → α} (h : Commute f g)\n    (hco : Coprime (minimalPeriod f x) (minimalPeriod g x)) :\n    minimalPeriod (f ∘ g) x = minimalPeriod f x * minimalPeriod g x := by\n  apply h.minimalPeriod_of_comp_dvd_mul.antisymm\n  suffices ∀ {f g : α → α},\n      Commute f g →\n        Coprime (minimalPeriod f x) (minimalPeriod g x) →\n          minimalPeriod f x ∣ minimalPeriod (f ∘ g) x from\n    hco.mul_dvd_of_dvd_of_dvd (this h hco) (h.comp_eq.symm ▸ this h.symm hco.symm)\n  intro f g h hco\n  refine hco.dvd_of_dvd_mul_left (IsPeriodicPt.left_of_comp h ?_ ?_).minimalPeriod_dvd\n  · exact (isPeriodicPt_minimalPeriod _ _).const_mul _\n  · exact (isPeriodicPt_minimalPeriod _ _).mul_const _\n\nsection Fintype\n\nopen Fintype\n\ntheorem minimalPeriod_le_card [Fintype α] : minimalPeriod f x ≤ card α := by\n  rw [← periodicOrbit_length]\n  exact List.Nodup.length_le_card nodup_periodicOrbit\n\ntheorem isPeriodicPt_factorial_card_of_mem_periodicPts [Fintype α] (h : x ∈ periodicPts f) :\n    IsPeriodicPt f (card α)! x :=\n  isPeriodicPt_iff_minimalPeriod_dvd.mpr\n    (Nat.dvd_factorial (minimalPeriod_pos_of_mem_periodicPts h) minimalPeriod_le_card)\n\ntheorem mem_periodicPts_iff_isPeriodicPt_factorial_card [Fintype α] :\n    x ∈ periodicPts f ↔ IsPeriodicPt f (card α)! x where\n  mp := isPeriodicPt_factorial_card_of_mem_periodicPts\n  mpr h := minimalPeriod_pos_iff_mem_periodicPts.mp\n    (IsPeriodicPt.minimalPeriod_pos (Nat.factorial_pos _) h)\n\ntheorem Injective.mem_periodicPts [Finite α] (h : Injective f) (x : α) : x ∈ periodicPts f := by\n  obtain ⟨m, n, heq, hne⟩ : ∃ m n, f^[m] x = f^[n] x ∧ m ≠ n := by\n    simpa [Injective] using not_injective_infinite_finite (f^[·] x)\n  rcases lt_or_gt_of_ne hne with hlt | hlt\n  · exact mk_mem_periodicPts (by lia) (iterate_cancel h heq.symm)\n  · exact mk_mem_periodicPts (by lia) (iterate_cancel h heq)\n\nTarget:\ntheorem injective_iff_periodicPts_eq_univ [Finite α] : Injective f ↔ periodicPts f = univ :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/PeriodicPts","family_id":"injective_iff_periodicpts_eq_univ","file_id":"mathlib/Mathlib/Dynamics/PeriodicPts/Lemmas.lean","sample_id":"108e499b9132e898676c8d66a09fbbcb3ff6a762c243e0f235bc6414a3079513"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bdd71d8bba5ab6bc0aadc26f2ebfd16f9c6d0a1ea732936b96c085c594ae8d5f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4fba185756badbaa3d579461622f7431189517e5fb7071a627d47ad8323f43b0","source_sha256":"1c0dcf6a3ce55bec7a88672314ea6ae6741a6599903854de40abb691f935e16b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [pullSub, List.take_of_length_le hl, List.drop_eq_nil_iff.mpr hl]","hard_negative":false,"metrics":{"chosen_tokens":17,"rejected_tokens":2,"token_jaccard":0.076923,"token_length_ratio":0.117647},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"6cdc61f01dcff11877381f602b1a234b9c8efc39079e4b9d8902c111598b6fa9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.CompleteLattice.SetLike\n\nNamespace:\nDescriptive.Tree\n\nLocal context:\n/-\nCopyright (c) 2024 Sven Manthe. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sven Manthe\n-/\n/-!\n# Trees in the sense of descriptive set theory\n\nThis file defines trees of depth `ω` in the sense of descriptive set theory as sets of finite\nsequences that are stable under taking prefixes.\n\n## Main declarations\n\n* `tree A`: a (possibly infinite) tree of depth at most `ω` with nodes in `A`\n-/\n\n@[expose] public section\n\nnamespace Descriptive\n\n/-- A tree is a set of finite sequences, implemented as `List A`, that is stable under\n  taking prefixes. For the definition we use the equivalent property `x ++ [a] ∈ T → x ∈ T`,\n  which is more convenient to check. We define `tree A` as a complete sublattice of\n  `Set (List A)`, which coerces to the type of trees on `A`. -/\ndef tree (A : Type*) : CompleteSublattice (Set (List A)) :=\n  CompleteSublattice.mk' {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T}\n    (by rintro S hS x a ⟨t, ht, hx⟩; use t, ht, hS ht hx)\n    (by rintro S hS x a h T hT; exact hS hT <| h T hT)\n\n@[simps!] instance (A : Type*) : SetLike (tree A) (List A) := SetLike.instSubtypeSet\n\nexample (A : Type*) : PartialOrder (tree A) := inferInstance\n\nnamespace Tree\nvariable {A : Type*} {S T : tree A}\n\nlemma mem_of_append {x y : List A} (h : x ++ y ∈ T) : x ∈ T := by\n  induction y generalizing x with\n  | nil => simpa using h\n  | cons y ys ih => exact T.prop (ih (by simpa))\n\nlemma mem_of_prefix {x y : List A} (h' : x <+: y) (h : y ∈ T) : x ∈ T := by\n  obtain ⟨_, rfl⟩ := h'; exact mem_of_append h\n\ninstance : Trans List.IsPrefix (fun x (T : tree A) ↦ x ∈ T) (fun x T ↦ x ∈ T) where\n  trans := mem_of_prefix\n\nlemma singleton_mem (T : tree A) {a : A} {x : List A} (h : a :: x ∈ T) : [a] ∈ T :=\n  mem_of_prefix ⟨x, rfl⟩ h\n\n@[simp] lemma tree_eq_bot : T = ⊥ ↔ [] ∉ T where\n  mp := by rintro rfl; simp\n  mpr h := by ext x; simpa using fun h' ↦ h <| mem_of_prefix x.nil_prefix h'\n\nlemma take_mem {n : ℕ} (x : T) : x.val.take n ∈ T :=\n  mem_of_prefix (x.val.take_prefix n) x.prop\n\n/-- A variant of `List.take` internally to a tree -/\n@[simps] def take (n : ℕ) (x : T) : T := ⟨x.val.take n, take_mem x⟩\n\n@[simp] lemma take_take (m n : ℕ) (x : T) : take m (take n x) = take (m ⊓ n) x := by\n  simp [Subtype.ext_iff, List.take_take]\n\n@[simp] lemma take_eq_take {x : T} {m n : ℕ} :\n    take m x = take n x ↔ m ⊓ x.val.length = n ⊓ x.val.length := by simp [Subtype.ext_iff]\n\n-- ### `subAt`\n\nvariable (T) (x y : List A)\n\n/-- The residual tree obtained by regarding the node x as new root -/\ndef subAt : tree A :=\n  ⟨(x ++ ·)⁻¹' T, fun _ a _ ↦ mem_of_append (y := [a]) (by rwa [List.append_assoc])⟩\n\n@[simp] lemma mem_subAt : y ∈ subAt T x ↔ x ++ y ∈ T := Iff.rfl\n\n@[simp] lemma subAt_nil : subAt T [] = T := rfl\n\n@[simp] lemma subAt_append : subAt (subAt T x) y = subAt T (x ++ y) := by ext; simp\n\n@[gcongr] lemma subAt_mono (h : S ≤ T) : subAt S x ≤ subAt T x :=\n  Set.preimage_mono h\n\n/-- A variant of `List.drop` that takes values in `subAt` -/\n@[simps] def drop (n : ℕ) (x : T) : subAt T (Tree.take n x).val :=\n  ⟨x.val.drop n, by simp⟩\n\n-- ### `pullSub`\n\n/-- Adjoint of `subAt`, given by pasting x before the root of T. Explicitly,\n  elements are prefixes of x or x with an element of T appended -/\ndef pullSub : tree A where\n  val := { y | y.take x.length <+: x ∧ y.drop x.length ∈ T }\n  property := fun y a ⟨h1, h2⟩ ↦\n    ⟨((y.prefix_append [a]).take x.length).trans h1,\n    mem_of_prefix ((y.prefix_append [a]).drop x.length) h2⟩\n\nvariable {T x y}\n\nTarget:\nlemma mem_pullSub_short (hl : y.length ≤ x.length) : y ∈ pullSub T x ↔ y <+: x ∧ [] ∈ T :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Descriptive","family_id":"mem_pullsub_short","file_id":"mathlib/Mathlib/SetTheory/Descriptive/Tree.lean","sample_id":"4fba185756badbaa3d579461622f7431189517e5fb7071a627d47ad8323f43b0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e1fc953f2e811d1c999db981871450abfdfcde88bd7349db65e45507d796d374","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c9b478f786784f9ac4583d69de5a7f68ad959e59303de1658982ea618e1c423e","source_sha256":"b2e5d2441dae74e6a2f4e8de4b7d4e87dd1e5e1f1bcd7a4ed0737bd7e824e441","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let ℬ := {s | s ∈ 𝒜} ∪ {s | sᶜ ∈ 𝒜}\n  have count_ℬ : ℬ.Countable := by\n    apply h.union\n    have : compl '' 𝒜 = {s | sᶜ ∈ 𝒜} := by\n      ext s\n      simpa using ⟨fun ⟨x, x_mem, hx⟩ ↦ by simp [← hx, x_mem], fun hs ↦ ⟨sᶜ, hs, by simp⟩⟩\n    exact this ▸ h.image compl\n  let f : Set (Set (Set α)) → Set α := fun A ↦ ⋃ a ∈ A, ⋂ t ∈ a, t\n  let 𝒞 := {a | a.Finite ∧ a ⊆ ℬ}\n  have count_𝒞 : 𝒞.Countable := countable_setOf_finite_subset (countable_coe_iff.1 count_ℬ)\n  let 𝒟 := {A | A.Finite ∧ A ⊆ 𝒞}\n  have count_𝒟 : 𝒟.Countable := countable_setOf_finite_subset (countable_coe_iff.1 count_𝒞)\n  have : generateSetAlgebra 𝒜 ⊆ f '' 𝒟 := by\n    intro s s_mem\n    rcases mem_generateSetAlgebra_elim s_mem with ⟨A, A_fin, mem_A, hA, rfl⟩\n    exact ⟨A, ⟨A_fin, fun a ha ↦ ⟨mem_A a ha, hA a ha⟩⟩, rfl⟩\n  exact (count_𝒟.image f).mono this","hard_negative":false,"metrics":{"chosen_tokens":246,"rejected_tokens":5,"token_jaccard":0.025974,"token_length_ratio":0.020325},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"6cf4f6b3bd66463364c12fdd1fd2931408a93654374ac8b5fad8735853c31a07","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finite.Prod\npublic import Mathlib.MeasureTheory.SetSemiring\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2024 Etienne Marion. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Etienne Marion\n-/\n/-!\n# Algebra of sets\n\nIn this file we define the notion of algebra of sets and give its basic properties. An algebra\nof sets is a family of sets containing the empty set and closed by complement and binary union.\nIt is therefore similar to a `σ`-algebra, except that it is not necessarily closed\nby countable unions.\n\nWe also define the algebra of sets generated by a family of sets and give its basic properties,\nand we prove that it is countable when it is generated by a countable family. We prove that\nthe `σ`-algebra generated by a family of sets `𝒜` is the same as the one generated by the algebra\nof sets generated by `𝒜`.\n\n## Main definitions\n\n* `MeasureTheory.IsSetAlgebra`: property of being an algebra of sets.\n* `MeasureTheory.generateSetAlgebra`: the algebra of sets generated by a family of sets.\n\n## Main statements\n\n* `MeasureTheory.mem_generateSetAlgebra_elim`: If a set `s` belongs to the algebra of sets\n  generated by `𝒜`, then it can be written as a finite union of finite intersections of sets which\n  are in `𝒜` or have their complement in `𝒜`.\n* `MeasureTheory.countable_generateSetAlgebra`: If a family of sets is countable then so is\n  the algebra of sets generated by it.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Field_of_sets>\n\n## Tags\n\nalgebra of sets, generated algebra of sets\n-/\n\npublic section\n\nopen MeasurableSpace Set\n\nnamespace MeasureTheory\n\nvariable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α}\n\n/-! ### Definition and basic properties of an algebra of sets -/\n\n/-- An algebra of sets is a family of sets containing the empty set and closed by complement and\nunion. Consequently it is also closed by difference (see `IsSetAlgebra.sdiff_mem`) and intersection\n(see `IsSetAlgebra.inter_mem`). -/\nstructure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where\n  empty_mem : ∅ ∈ 𝒜\n  compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜\n  union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜\n\nnamespace IsSetAlgebra\n\n/-- An algebra of sets contains the whole set. -/\ntheorem univ_mem (h𝒜 : IsSetAlgebra 𝒜) : univ ∈ 𝒜 :=\n  compl_empty ▸ h𝒜.compl_mem h𝒜.empty_mem\n\n/-- An algebra of sets is closed by intersection. -/\ntheorem inter_mem (h𝒜 : IsSetAlgebra 𝒜) (s_mem : s ∈ 𝒜) (t_mem : t ∈ 𝒜) :\n    s ∩ t ∈ 𝒜 :=\n  inter_eq_compl_compl_union_compl .. ▸\n    h𝒜.compl_mem (h𝒜.union_mem (h𝒜.compl_mem s_mem) (h𝒜.compl_mem t_mem))\n\n/-- An algebra of sets is closed by difference. -/\ntheorem sdiff_mem (h𝒜 : IsSetAlgebra 𝒜) (s_mem : s ∈ 𝒜) (t_mem : t ∈ 𝒜) :\n    s \\ t ∈ 𝒜 := h𝒜.inter_mem s_mem (h𝒜.compl_mem t_mem)\n\n@[deprecated (since := \"2026-06-03\")] alias diff_mem := sdiff_mem\n\n/-- An algebra of sets is a ring of sets. -/\ntheorem isSetRing (h𝒜 : IsSetAlgebra 𝒜) : IsSetRing 𝒜 where\n  empty_mem := h𝒜.empty_mem\n  union_mem := h𝒜.union_mem\n  sdiff_mem := fun _ _ ↦ h𝒜.sdiff_mem\n\n/-- An algebra of sets is closed by finite unions. -/\ntheorem biUnion_mem {ι : Type*} (h𝒜 : IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι)\n    (hs : ∀ i ∈ S, s i ∈ 𝒜) : ⋃ i ∈ S, s i ∈ 𝒜 := h𝒜.isSetRing.biUnion_mem S hs\n\n/-- An algebra of sets is closed by finite intersections. -/\ntheorem biInter_mem {ι : Type*} (h𝒜 : IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι)\n    (hs : ∀ i ∈ S, s i ∈ 𝒜) : ⋂ i ∈ S, s i ∈ 𝒜 := by\n  by_cases! h : S = ∅\n  · rw [h, ← Finset.set_biInter_coe, Finset.coe_empty, biInter_empty]\n    exact h𝒜.univ_mem\n  · exact h𝒜.isSetRing.biInter_mem S h hs\n\nend IsSetAlgebra\n\nsection generateSetAlgebra\n\n/-! ### Definition and properties of the algebra of sets generated by some family -/\n\n/-- `generateSetAlgebra 𝒜` is the smallest algebra of sets containing `𝒜`. -/\ninductive generateSetAlgebra {α : Type*} (𝒜 : Set (Set α)) : Set (Set α)\n  | base (s : Set α) (s_mem : s ∈ 𝒜) : generateSetAlgebra 𝒜 s\n  | empty : generateSetAlgebra 𝒜 ∅\n  | compl (s : Set α) (hs : generateSetAlgebra 𝒜 s) : generateSetAlgebra 𝒜 sᶜ\n  | union (s t : Set α) (hs : generateSetAlgebra 𝒜 s) (ht : generateSetAlgebra 𝒜 t) :\n      generateSetAlgebra 𝒜 (s ∪ t)\n\n/-- The algebra of sets generated by a family of sets is an algebra of sets. -/\ntheorem isSetAlgebra_generateSetAlgebra :\n    IsSetAlgebra (generateSetAlgebra 𝒜) where\n  empty_mem := generateSetAlgebra.empty\n  compl_mem := fun _ hs ↦ generateSetAlgebra.compl _ hs\n  union_mem := fun _ _ hs ht ↦ generateSetAlgebra.union _ _ hs ht\n\n/-- The algebra of sets generated by `𝒜` contains `𝒜`. -/\ntheorem self_subset_generateSetAlgebra : 𝒜 ⊆ generateSetAlgebra 𝒜 :=\n  fun _ ↦ generateSetAlgebra.base _\n\n/-- The measurable space generated by a family of sets `𝒜` is the same as the one generated\nby the algebra of sets generated by `𝒜`. -/\n@[simp]\ntheorem generateFrom_generateSetAlgebra_eq :\n    generateFrom (generateSetAlgebra 𝒜) = generateFrom 𝒜 := by\n  refine le_antisymm (fun s ms ↦ ?_) (generateFrom_mono self_subset_generateSetAlgebra)\n  induction s, ms using generateFrom_induction with\n  | hC t ht h =>\n    clear h\n    induction ht with\n    | base u u_mem => exact measurableSet_generateFrom u_mem\n    | empty => exact @MeasurableSet.empty _ (generateFrom 𝒜)\n    | compl u _ mu => exact mu.compl\n    | union u v _ _ mu mv => exact MeasurableSet.union mu mv\n  | empty => exact MeasurableSpace.measurableSet_empty _\n  | compl t _ ht => exact ht.compl\n  | iUnion t _ ht => exact .iUnion ht\n\n/-- If a family of sets `𝒜` is contained in `ℬ`, then the algebra of sets generated by `𝒜`\nis contained in the one generated by `ℬ`. -/\ntheorem generateSetAlgebra_mono {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ) :\n    generateSetAlgebra 𝒜 ⊆ generateSetAlgebra ℬ := by\n  intro s hs\n  induction hs with\n  | base t t_mem => exact self_subset_generateSetAlgebra (h t_mem)\n  | empty => exact isSetAlgebra_generateSetAlgebra.empty_mem\n  | compl t _ t_mem => exact isSetAlgebra_generateSetAlgebra.compl_mem t_mem\n  | union t u _ _ t_mem u_mem => exact isSetAlgebra_generateSetAlgebra.union_mem t_mem u_mem\n\nnamespace IsSetAlgebra\n\n/-- If a family of sets `𝒜` is contained in an algebra of sets `ℬ`, then so is the algebra of sets\ngenerated by `𝒜`. -/\ntheorem generateSetAlgebra_subset {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ)\n    (hℬ : IsSetAlgebra ℬ) : generateSetAlgebra 𝒜 ⊆ ℬ := by\n  intro s hs\n  induction hs with\n  | base t t_mem => exact h t_mem\n  | empty => exact hℬ.empty_mem\n  | compl t _ t_mem => exact hℬ.compl_mem t_mem\n  | union t u _ _ t_mem u_mem => exact hℬ.union_mem t_mem u_mem\n\n/-- If `𝒜` is an algebra of sets, then it contains the algebra generated by itself. -/\ntheorem generateSetAlgebra_subset_self (h𝒜 : IsSetAlgebra 𝒜) :\n    generateSetAlgebra 𝒜 ⊆ 𝒜 := h𝒜.generateSetAlgebra_subset subset_rfl\n\n/-- If `𝒜` is an algebra of sets, then it is equal to the algebra generated by itself. -/\ntheorem generateSetAlgebra_eq (h𝒜 : IsSetAlgebra 𝒜) : generateSetAlgebra 𝒜 = 𝒜 :=\n  Subset.antisymm h𝒜.generateSetAlgebra_subset_self self_subset_generateSetAlgebra\n\nend IsSetAlgebra\n\n/-- If a set belongs to the algebra of sets generated by `𝒜` then it can be written as a finite\nunion of finite intersections of sets which are in `𝒜` or have their complement in `𝒜`. -/\ntheorem mem_generateSetAlgebra_elim (s_mem : s ∈ generateSetAlgebra 𝒜) :\n    ∃ A : Set (Set (Set α)), A.Finite ∧ (∀ a ∈ A, a.Finite) ∧\n    (∀ᵉ (a ∈ A) (t ∈ a), t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ s = ⋃ a ∈ A, ⋂ t ∈ a, t := by\n  induction s_mem with\n  | base u u_mem =>\n    refine ⟨{{u}}, finite_singleton {u},\n      fun a ha ↦ eq_of_mem_singleton ha ▸ finite_singleton u,\n      fun a ha t ht ↦ ?_, by simp⟩\n    rw [eq_of_mem_singleton ha, ha, eq_of_mem_singleton ht, ht] at *\n    exact Or.inl u_mem\n  | empty => exact ⟨∅, finite_empty, fun _ h ↦ (notMem_empty _ h).elim,\n    fun _ ha _ _ ↦ (notMem_empty _ ha).elim, by simp⟩\n  | compl u _ u_ind =>\n    rcases u_ind with ⟨A, A_fin, mem_A, hA, u_eq⟩\n    have := finite_coe_iff.2 A_fin\n    have := fun a : A ↦ finite_coe_iff.2 <| mem_A a.1 a.2\n    refine ⟨{{(f a).1ᶜ | a : A} | f : (Π a : A, ↑a)}, finite_coe_iff.1 inferInstance,\n      fun a ⟨f, hf⟩ ↦ hf ▸ finite_coe_iff.1 inferInstance, fun a ha t ht ↦ ?_, ?_⟩\n    · rcases ha with ⟨f, rfl⟩\n      rcases ht with ⟨a, rfl⟩\n      rw [compl_compl, or_comm]\n      exact hA a.1 a.2 (f a).1 (f a).2\n    · ext x\n      simp only [u_eq, compl_iUnion, compl_iInter, mem_iInter, mem_iUnion, mem_compl_iff,\n        exists_prop, Subtype.exists, mem_setOf_eq, iUnion_exists, iUnion_iUnion_eq',\n        iInter_exists]\n      constructor <;> intro hx\n      · choose f hf using hx\n        exact ⟨fun ⟨a, ha⟩ ↦ ⟨f a ha, (hf a ha).1⟩, fun _ a ha h ↦ by rw [← h]; exact (hf a ha).2⟩\n      · rcases hx with ⟨f, hf⟩\n        exact fun a ha ↦ ⟨f ⟨a, ha⟩, (f ⟨a, ha⟩).2, hf (f ⟨a, ha⟩)ᶜ a ha rfl⟩\n  | union u v _ _ u_ind v_ind =>\n    rcases u_ind with ⟨Au, Au_fin, mem_Au, hAu, u_eq⟩\n    rcases v_ind with ⟨Av, Av_fin, mem_Av, hAv, v_eq⟩\n    refine ⟨Au ∪ Av, Au_fin.union Av_fin, ?_, ?_, by rw [u_eq, v_eq, ← biUnion_union]⟩\n    · rintro a (ha | ha)\n      · exact mem_Au a ha\n      · exact mem_Av a ha\n    · rintro a (ha | ha) t ht\n      · exact hAu a ha t ht\n      · exact hAv a ha t ht\n\n/-- If a family of sets is countable then so is the algebra of sets generated by it. -/\n\nTarget:\ntheorem countable_generateSetAlgebra (h : 𝒜.Countable) :\n    (generateSetAlgebra 𝒜).Countable :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory","family_id":"countable_generatesetalgebra","file_id":"mathlib/Mathlib/MeasureTheory/SetAlgebra.lean","sample_id":"c9b478f786784f9ac4583d69de5a7f68ad959e59303de1658982ea618e1c423e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"22533728b6f9b3f9588a530b9a97fe6bf76b4cb9f09b93a8510eba1da1bd326d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cc1a3c9a360e868e0958495a5e7eb164e9674884b5e053190acf055700be4b22","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1709493a1e9d915fafb30165d87390b1f52391762f60fb9606bb3e33e9e6a8da","source_sha256":"42683ce9ec3a96f77aab204680a294682617adafa7d756b1f3dbef8d091d30cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  apply Sieve.ext\n  simp [← NatTrans.naturality_apply]","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":18,"token_jaccard":0.733333,"token_length_ratio":1.384615},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"6cfda7002749ef81591ff3a1f47212d48e8594c71478dd59c276b52ab9c924ba","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Sites.Closed\npublic import Mathlib.CategoryTheory.Sites.Equivalence\npublic import Mathlib.CategoryTheory.Subobject.Classifier.Defs\npublic import Mathlib.CategoryTheory.Subfunctor.Image\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Edward van de Meent. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Edward van de Meent\n-/\n/-!\n\n# (Elementary) Sheaf Topos\n\nWe define a subobject classifier for categories of sheaves of (large enough) types.\n\n## Main definitions\n\nLet `C` refer to a category with (when relevant) Grothendieck topology `J`.\n\n* `Presheaf.classifier C` is a construction of a subobject classifier in `Cᵒᵖ ⥤ Type (max u v)`.\n* `Sheaf.classifier J` is a construction of a subobject classifier in `Sheaf J (Type (max u v))`.\n* `inferInstance : HasClassifier (Cᵒᵖ ⥤ Type w)` says that `Cᵒᵖ ⥤ Type w` has a subobject\n  classifier if `C` is `w`-essentially small.\n* `inferInstance : HasClassifier (Sheaf J (Type w))` says that `Sheaf J (Type w)` has a\n  subobject classifier if `C` is `w`-essentially small.\n\n## Main results\n\n* Any category of sheaves of types has a subobject classifier if the site is essentially small.\n* As a consequence, (because categories of sheaves are cartesian monoidal and have finite limits,)\n  such categories are Elementary Topoi.\n\n## TODOS:\n\n* generalize `Presheaf.isClosed_χ_app_apply_of` to only assuming `G` is separated\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nnamespace CategoryTheory\n\nvariable {C : Type u} [Category.{v} C]\nopen Limits\n\nsection presheaf\n\nvariable (C) in\n/-- The truth morphism in the category of presheaves. At each component `X : C`, it is the constant\nmap returning `⊤ : Sieve X`. -/\n@[simps]\ndef Presheaf.truth : (Functor.const _).obj PUnit ⟶ Functor.sieves C where\n  app X := ↾fun _ ↦ (⊤ : Sieve X.unop)\n\nvariable {F G : Cᵒᵖ ⥤ Type (max u v)}\n\nset_option backward.defeqAttrib.useBackward true in\n/--\nThe characteristic map of an inclusion of presheaves.\nGiven a monomorphism of sheaves `m : F ⟶ G`, an object X of the site, map an element `x : G(X)`\nto the (closed) sieve on X where `f : Y → X` is in the sieve iff\n  `∃ a ∈ F(Y), G(f)(x) = m_Y(a)`\n-/\n@[simps app]\ndef Presheaf.χ (m : F ⟶ G) : G ⟶ Functor.sieves C where\n  app X := ↾fun x ↦ ⟨fun Y f => ∃ a, G.map f.op x = m.app (.op Y) a, by\n    intro Y Z f ⟨a, ha⟩ g\n    use F.map g.op a\n    simp [ha, NatTrans.naturality_apply]⟩\n\nTarget:\nlemma Presheaf.comp_χ_eq (m : F ⟶ G) : m ≫ Presheaf.χ m =\n    (Functor.isTerminalConst _ Types.isTerminalPUnit).from F ≫ Presheaf.truth C :=\n\nProof body:\n","rejected":"by\n  ext\n  apply Sieve.ext\n  simp [← NatTrans.naturality_apply]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Topos","family_id":"presheaf","file_id":"mathlib/Mathlib/CategoryTheory/Topos/Sheaf.lean","sample_id":"1709493a1e9d915fafb30165d87390b1f52391762f60fb9606bb3e33e9e6a8da"},"split":"train"}
{"binding":{"chosen_proof_sha256":"82c27cb6e99a7d6e45601e4127587eb4a9e18558df87ff23a7cf9764fabc33f4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e35d2d0cef7ad182c12561cd06aa4dfc586540c8088942f242309bd850f3a822","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"12e4ed73b2164c9034e09e737bb93b021152af848a112fdd9ed5bf5c79a07e22","source_sha256":"6a1fd2aa47744142955f801b29fb06cc46e0d439ea80c9cccba97a19a60fbbe9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [galRestrict', galRestrict', Subalgebra.algebraMap_eq]","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.090909,"token_length_ratio":0.272727},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"6d1bd83e301130d85b93c07a713cf783b622b2e8156c7b756c1ee8e33ec8c0f7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.IntegralClosure\npublic import Mathlib.RingTheory.RingHom.Finite\npublic import Mathlib.RingTheory.Localization.LocalizationLocalization\npublic import Mathlib.RingTheory.Localization.NormTrace\npublic import Mathlib.RingTheory.Norm.Transitivity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Restriction of various maps between fields to integrally closed subrings.\n\nIn this file, we assume `A` is an integrally closed domain; `K` is the fraction ring of `A`;\n`L` is a finite extension of `K`; `B` is the integral closure of `A` in `L`.\nWe call this the AKLB setup.\n\n## Main definitions\n- `galRestrict`: The restriction `Aut(L/K) → Aut(B/A)` as an `MulEquiv` in an AKLB setup.\n- `Algebra.intTrace`: The trace map of a finite extension of integrally closed domains `B/A` is\n  defined to be the restriction of the trace map of `Frac(B)/Frac(A)`.\n- `Algebra.intNorm`: The norm map of a finite extension of integrally closed domains `B/A` is\n  defined to be the restriction of the norm map of `Frac(B)/Frac(A)`.\n\n-/\n\n@[expose] public section\n\nopen Module nonZeroDivisors\n\nvariable (A K L L₂ L₃ B B₂ B₃ : Type*)\nvariable [CommRing A] [CommRing B] [CommRing B₂] [CommRing B₃]\nvariable [Algebra A B] [Algebra A B₂] [Algebra A B₃]\nvariable [Field K] [Field L] [Field L₂] [Field L₃]\nvariable [Algebra A K] [IsFractionRing A K]\nvariable [Algebra K L] [Algebra A L] [IsScalarTower A K L]\nvariable [Algebra K L₂] [Algebra A L₂] [IsScalarTower A K L₂]\nvariable [Algebra K L₃] [Algebra A L₃] [IsScalarTower A K L₃]\nvariable [Algebra B L] [IsScalarTower A B L] [IsIntegralClosure B A L]\nvariable [Algebra B₂ L₂] [IsScalarTower A B₂ L₂] [IsIntegralClosure B₂ A L₂]\nvariable [Algebra B₃ L₃] [IsScalarTower A B₃ L₃] [IsIntegralClosure B₃ A L₃]\n\nsection galois\n\nsection galRestrict'\nvariable {K L L₂ L₃}\nomit [IsFractionRing A K]\n\n/-- A generalization of `galRestrictHom` beyond endomorphisms. -/\nnoncomputable\ndef galRestrict' (f : L →ₐ[K] L₂) : (B →ₐ[A] B₂) :=\n  (IsIntegralClosure.equiv A (integralClosure A L₂) L₂ B₂).toAlgHom.comp\n      (((f.restrictScalars A).comp (IsScalarTower.toAlgHom A B L)).codRestrict\n        (integralClosure A L₂) (fun x ↦ IsIntegral.map _ (IsIntegralClosure.isIntegral A L x)))\n\n@[simp]\n\nTarget:\nlemma algebraMap_galRestrict'_apply (σ : L →ₐ[K] L₂) (x : B) :\n    algebraMap B₂ L₂ (galRestrict' A B B₂ σ x) = σ (algebraMap B L x) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_12e4ed73b216","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"4b2c05a51566015007f9a673b4ce3036f25f1267fab174b95324e402b2aaeb03","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/IntegralClosure","family_id":"algebramap_galrestrict'_apply","file_id":"mathlib/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean","sample_id":"12e4ed73b2164c9034e09e737bb93b021152af848a112fdd9ed5bf5c79a07e22"},"split":"train"}
{"binding":{"chosen_proof_sha256":"792cf305c49d49d53ed8828dada761a3db9769ab9dc9ac7edc92e61179edcded","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"dcee79aa8fce85c29907b9729bf9a202db99fe4db9343277c60032775bea19a1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1d3d2dc551d90c30ba5f909bc69c9cc130fae372a274b0debe56a42386450ace","source_sha256":"287a704d503b237a43f85c1d8d085e3a9c8170677a7fe6bc6d74659961e522de","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · rw [Ne, eq_none_iff', not_not]\n    exact fun h => ⟨o.get h, eq_some_iff.2 (get_mem h)⟩\n  · rintro ⟨x, rfl⟩\n    apply some_ne_none","hard_negative":false,"metrics":{"chosen_tokens":38,"rejected_tokens":43,"token_jaccard":0.852941,"token_length_ratio":1.131579},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"6d243751e6f0f41d31118a5aac00c3dccfc5b19dde46d3cf95624ba14dfefdad","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Defs\npublic import Mathlib.Data.Set.Subsingleton\npublic import Mathlib.Logic.Equiv.Defs\n\nNamespace:\nPart\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Jeremy Avigad, Simon Hudon\n-/\n/-!\n# Partial values of a type\n\nThis file defines `Part α`, the partial values of a type.\n`o : Part α` carries a proposition `o.Dom`, its domain, along with a function `get : o.Dom → α`, its\nvalue. The rule is then that every partial value has a value but, to access it, you need to provide\na proof of the domain.\n`Part α` behaves the same as `Option α` except that `o : Option α` is decidably `none` or `some a`\nfor some `a : α`, while the domain of `o : Part α` doesn't have to be decidable. That means you can\ntranslate back and forth between a partial value with a decidable domain and an option, and\n`Option α` and `Part α` are classically equivalent. In general, `Part α` is bigger than `Option α`.\n\n## Main declarations\n`Option`-like declarations:\n* `Part.none`: The partial value whose domain is `False`.\n* `Part.some a`: The partial value whose domain is `True` and whose value is `a`.\n* `Part.ofOption`: Converts an `Option α` to a `Part α` by sending `none` to `none` and `some a` to\n  `some a`.\n* `Part.toOption`: Converts a `Part α` with a decidable domain to an `Option α`.\n* `Part.equivOption`: Classical equivalence between `Part α` and `Option α`.\n\nMonadic structure:\n* `Part.bind`: `o.bind f` has value `(f (o.get _)).get _` (`f o` morally) and is defined when `o`\n  and `f (o.get _)` are defined.\n* `Part.map`: Maps the value and keeps the same domain.\n\nOther:\n* `Part.restrict`: `Part.restrict p o` replaces the domain of `o : Part α` by `p : Prop` so long as\n  `p → o.Dom`.\n* `Part.assert`: `assert p f` appends `p` to the domains of the values of a partial function.\n* `Part.unwrap`: Gets the value of a partial value regardless of its domain. Unsound.\n\n## Notation\nFor `a : α`, `o : Part α`, `a ∈ o` means that `o` is defined and equal to `a`. Formally, it means\n`o.Dom` and `o.get _ = a`.\n-/\n\n@[expose] public section\n\nassert_not_exists RelIso\n\nopen Function\n\n/-- `Part α` is the type of \"partial values\" of type `α`. It\n  is similar to `Option α` except the domain condition can be an\n  arbitrary proposition, not necessarily decidable. -/\nstructure Part.{u} (α : Type u) : Type u where\n  /-- The domain of a partial value -/\n  Dom : Prop\n  /-- Extract a value from a partial value given a proof of `Dom` -/\n  get : Dom → α\n\nnamespace Part\n\nvariable {α : Type*} {β : Type*} {γ : Type*}\n\n/-- Convert a `Part α` with a decidable domain to an option -/\ndef toOption (o : Part α) [Decidable o.Dom] : Option α :=\n  if h : Dom o then some (o.get h) else none\n\n@[simp] lemma toOption_isSome (o : Part α) [Decidable o.Dom] : o.toOption.isSome ↔ o.Dom := by\n  by_cases h : o.Dom <;> simp [h, toOption]\n\n@[simp] lemma toOption_eq_none (o : Part α) [Decidable o.Dom] : o.toOption = none ↔ ¬o.Dom := by\n  by_cases h : o.Dom <;> simp [h, toOption]\n\n/-- `Part` extensionality -/\ntheorem ext' : ∀ {o p : Part α}, (o.Dom ↔ p.Dom) → (∀ h₁ h₂, o.get h₁ = p.get h₂) → o = p\n  | ⟨od, o⟩, ⟨pd, p⟩, H1, H2 => by\n    have t : od = pd := propext H1\n    cases t; rw [show o = p from funext fun p => H2 p p]\n\n/-- `Part` eta expansion -/\n@[simp]\ntheorem eta : ∀ o : Part α, (⟨o.Dom, fun h => o.get h⟩ : Part α) = o\n  | ⟨_, _⟩ => rfl\n\n/-- `a ∈ o` means that `o` is defined and equal to `a` -/\nprotected def Mem (o : Part α) (a : α) : Prop :=\n  ∃ h, o.get h = a\n\ninstance : Membership α (Part α) :=\n  ⟨Part.Mem⟩\n\ntheorem mem_eq (a : α) (o : Part α) : (a ∈ o) = ∃ h, o.get h = a :=\n  rfl\n\ntheorem dom_iff_mem : ∀ {o : Part α}, o.Dom ↔ ∃ y, y ∈ o\n  | ⟨_, f⟩ => ⟨fun h => ⟨f h, h, rfl⟩, fun ⟨_, h, rfl⟩ => h⟩\n\ntheorem get_mem {o : Part α} (h) : get o h ∈ o :=\n  ⟨_, rfl⟩\n\n@[simp]\ntheorem mem_mk_iff {p : Prop} {o : p → α} {a : α} : a ∈ Part.mk p o ↔ ∃ h, o h = a :=\n  Iff.rfl\n\n/-- `Part` extensionality -/\n@[ext]\ntheorem ext {o p : Part α} (H : ∀ a, a ∈ o ↔ a ∈ p) : o = p :=\n  (ext' ⟨fun h => ((H _).1 ⟨h, rfl⟩).fst, fun h => ((H _).2 ⟨h, rfl⟩).fst⟩) fun _ _ =>\n    ((H _).2 ⟨_, rfl⟩).snd\n\n/-- The `none` value in `Part` has a `False` domain and an empty function. -/\ndef none : Part α :=\n  ⟨False, False.rec⟩\n\ninstance : Inhabited (Part α) :=\n  ⟨none⟩\n\n@[simp]\ntheorem notMem_none (a : α) : a ∉ @none α := fun h => h.fst\n\n/-- The `some a` value in `Part` has a `True` domain and the\n  function returns `a`. -/\ndef some (a : α) : Part α :=\n  ⟨True, fun _ => a⟩\n\n@[simp]\ntheorem some_dom (a : α) : (some a).Dom :=\n  trivial\n\ntheorem mem_unique : ∀ {a b : α} {o : Part α}, a ∈ o → b ∈ o → a = b\n  | _, _, ⟨_, _⟩, ⟨_, rfl⟩, ⟨_, rfl⟩ => rfl\n\ntheorem mem_right_unique : ∀ {a : α} {o p : Part α}, a ∈ o → a ∈ p → o = p\n  | _, _, _, ⟨ho, _⟩, ⟨hp, _⟩ => ext' (iff_of_true ho hp) (by simp [*])\n\ntheorem Mem.left_unique : Relator.LeftUnique ((· ∈ ·) : α → Part α → Prop) := fun _ _ _ =>\n  mem_unique\n\ntheorem Mem.right_unique : Relator.RightUnique ((· ∈ ·) : α → Part α → Prop) := fun _ _ _ =>\n  mem_right_unique\n\ntheorem get_eq_of_mem {o : Part α} {a} (h : a ∈ o) (h') : get o h' = a :=\n  mem_unique ⟨_, rfl⟩ h\n\nprotected theorem subsingleton (o : Part α) : Set.Subsingleton { a | a ∈ o } := fun _ ha _ hb =>\n  mem_unique ha hb\n\n@[simp]\ntheorem get_some {a : α} (ha : (some a).Dom) : get (some a) ha = a :=\n  rfl\n\ntheorem mem_some (a : α) : a ∈ some a :=\n  ⟨trivial, rfl⟩\n\n@[simp]\ntheorem mem_some_iff {a b} : b ∈ (some a : Part α) ↔ b = a :=\n  ⟨fun ⟨_, e⟩ => e.symm, fun e => ⟨trivial, e.symm⟩⟩\n\ntheorem eq_some_iff {a : α} {o : Part α} : o = some a ↔ a ∈ o :=\n  ⟨fun e => e.symm ▸ mem_some _, fun ⟨h, e⟩ => e ▸ ext' (iff_true_intro h) fun _ _ => rfl⟩\n\ntheorem eq_none_iff {o : Part α} : o = none ↔ ∀ a, a ∉ o :=\n  ⟨fun e => e.symm ▸ notMem_none, fun h => ext (by simpa)⟩\n\ntheorem eq_none_iff' {o : Part α} : o = none ↔ ¬o.Dom :=\n  ⟨fun e => e.symm ▸ id, fun h => eq_none_iff.2 fun _ h' => h h'.fst⟩\n\n@[simp]\ntheorem not_none_dom : ¬(none : Part α).Dom :=\n  id\n\n@[simp]\ntheorem some_ne_none (x : α) : some x ≠ none := by\n  intro h\n  exact true_ne_false (congr_arg Dom h)\n\n@[simp]\ntheorem none_ne_some (x : α) : none ≠ some x :=\n  (some_ne_none x).symm\n\nTarget:\ntheorem ne_none_iff {o : Part α} : o ≠ none ↔ ∃ x, o = some x :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  constructor\n  · rw [Ne, eq_none_iff', not_not]\n    exact fun h => ⟨o.get h, eq_some_iff.2 (get_mem h)⟩\n  · rintro ⟨x, rfl⟩\n    apply some_ne_none","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data","family_id":"ne_none_iff","file_id":"mathlib/Mathlib/Data/Part.lean","sample_id":"1d3d2dc551d90c30ba5f909bc69c9cc130fae372a274b0debe56a42386450ace"},"split":"train"}
{"binding":{"chosen_proof_sha256":"242a43583e27fd27079936e1ce2e8cce7893c073e1ef4300e30f594b3e61646a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7af71cb8f41dbf3ff57b02e47f2e80d5b01cc5dcd3b0ad2d88eccb7ce57595a3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"91739cefd9b24c38b935a570789d9ce5a71a9f8d6f001da414f1877193c9705f","source_sha256":"3b79245b81821b44a91daf15c8bd7da3bce2a30e3ee9d29e2db6a7d0e0195196","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Ideal.one_eq_top, ← Ideal.add_eq_sup, Ideal.add_eq_one_iff] at h\n  have ⟨y, hy, z, hz, eq⟩ := h\n  have ⟨p, hp⟩ := (mem_ideal_map_adjoin ..).mp hy\n  have ⟨w, hw⟩ := Ideal.mem_span_singleton.mp hz\n  have ⟨q, hq⟩ := adjoin_eq_exists_aeval R x w\n  use (1 - p - X * q).reverse\n  have : (1 - p - X * q).coeff 0 - 1 ∈ I := by simpa using hp.1 0\n  apply_fun (·.1) at eq hw\n  dsimp at eq\n  rw [reverse_leadingCoeff, trailingCoeff_eq_coeff_zero]\n  · exact ⟨this, (eval₂_reverse_eq_zero_iff ..).mpr <| by simp [← aeval_def, hp.2, hq, ← eq, hw]⟩\n  · exact fun h ↦ hI <| by simpa [h, Ideal.eq_top_iff_one]","hard_negative":true,"metrics":{"chosen_tokens":179,"rejected_tokens":3,"token_jaccard":0.028571,"token_length_ratio":0.01676},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"6d4a7398ba533895214e0e0d0c8ad8640a67dc0a9ce8c0c73669df7fa6879a03","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.RingDivision\npublic import Mathlib.RingTheory.Adjoin.Polynomial.Basic\npublic import Mathlib.RingTheory.Ideal.Maps\n\nNamespace:\nAlgebra\n\nLocal context:\n/-\nCopyright (c) 2019 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n/-!\n# Ideals in polynomial rings\n-/\n\npublic section\n\nnoncomputable section\n\nopen Polynomial\n\nopen Finset\n\nuniverse u v w\n\nnamespace Polynomial\n\nvariable {R : Type*} [CommRing R] {a : R}\n\ntheorem mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero {b : R[X]} {P : R[X][X]} :\n    P ∈ Ideal.span {C (X - C a), X - C b} ↔ (P.eval b).eval a = 0 := by\n  rw [Ideal.mem_span_pair]\n  constructor <;> intro h\n  · rcases h with ⟨_, _, rfl⟩\n    simp\n  · rcases dvd_iff_isRoot.mpr h with ⟨p, hp⟩\n    rcases @X_sub_C_dvd_sub_C_eval _ b _ P with ⟨q, hq⟩\n    exact ⟨C p, q, by rw [mul_comm, mul_comm q, eq_add_of_sub_eq' hq, hp, C_mul]⟩\n\ntheorem ker_evalRingHom (x : R) : RingHom.ker (evalRingHom x) = Ideal.span {X - C x} := by\n  ext y\n  simp [Ideal.mem_span_singleton, dvd_iff_isRoot, RingHom.mem_ker]\n\n@[simp]\ntheorem ker_modByMonicHom {q : R[X]} (hq : q.Monic) :\n    LinearMap.ker (Polynomial.modByMonicHom q) = (Ideal.span {q}).restrictScalars R :=\n  Submodule.ext fun _ => (mem_ker_modByMonic hq).trans Ideal.mem_span_singleton.symm\n\n@[simp]\nlemma ker_constantCoeff : RingHom.ker constantCoeff = .span {(X : R[X])} := by\n  refine le_antisymm (fun p hp ↦ ?_) (by simp [Ideal.span_le])\n  simp only [RingHom.mem_ker, constantCoeff_apply, ← Polynomial.X_dvd_iff] at hp\n  rwa [Ideal.mem_span_singleton]\n\nend Polynomial\n\nnamespace Algebra\n\nvariable {R S : Type*}\n\nlemma mem_ideal_map_adjoin [CommSemiring R] [Semiring S] [Algebra R S] (x : S) (I : Ideal R)\n    {y : R[x]} :\n    y ∈ I.map (algebraMap R (R[x])) ↔\n      ∃ p : R[X], (∀ i, p.coeff i ∈ I) ∧ Polynomial.aeval x p = y := by\n  constructor\n  · intro H\n    induction H using Submodule.span_induction with\n    | mem a ha =>\n      obtain ⟨a, ha, rfl⟩ := ha\n      exact ⟨C a, fun i ↦ by rw [coeff_C]; aesop, aeval_C _ _⟩\n    | zero => exact ⟨0, by simp, aeval_zero _⟩\n    | add a b ha hb ha' hb' =>\n      obtain ⟨a, ha, ha'⟩ := ha'\n      obtain ⟨b, hb, hb'⟩ := hb'\n      exact ⟨a + b, fun i ↦ by simpa using add_mem (ha i) (hb i), by simp [ha', hb']⟩\n    | smul a b hb hb' =>\n      obtain ⟨b', hb, hb'⟩ := hb'\n      have ⟨p, hp⟩ := adjoin_eq_exists_aeval R x a\n      refine ⟨p * b', fun i ↦ ?_, by simp [hp, hb']⟩\n      rw [coeff_mul]\n      exact sum_mem fun i hi ↦ Ideal.mul_mem_left _ _ (hb _)\n  · rintro ⟨p, hp, hp'⟩\n    have : y = ∑ i ∈ p.support, p.coeff i • ⟨_, (X ^ i).aeval_mem_adjoin_singleton _ x⟩ := by\n      trans ∑ i ∈ p.support, ⟨_, (C (p.coeff i) * X ^ i).aeval_mem_adjoin_singleton _ x⟩\n      · ext1\n        simp only [AddSubmonoidClass.coe_finsetSum, ← map_sum, ← hp', ← as_sum_support_C_mul_X_pow]\n      · congr with i\n        simp [Algebra.smul_def]\n    simp_rw [this, Algebra.smul_def]\n    exact sum_mem fun i _ ↦ Ideal.mul_mem_right _ _ (Ideal.mem_map_of_mem _ (hp i))\n\nTarget:\nlemma exists_aeval_invOf_eq_zero_of_idealMap_adjoin_sup_span_eq_top [CommRing R] [CommRing S]\n    [Algebra R S] (x : S) (I : Ideal R) (hI : I ≠ ⊤) [Invertible x]\n    (h : I.map (algebraMap R (R[x])) ⊔ .span {⟨x, subset_adjoin rfl⟩} = ⊤) :\n    ∃ p : R[X], p.leadingCoeff - 1 ∈ I ∧ p.aeval ⅟x = 0 :=\n\nProof body:\n","rejected":"by\n  exact exists_aeval_invOf_eq_zero_of_idealMap_adjoin_sup_span_eq_top","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"b5a3a42f11bd5da62fc5ded7564c9d18ded7fafced5bf19f6c3355b577846a38","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Polynomial","family_id":"exists_aeval_invof_eq_zero_of_idealmap_adjoin_sup_span_eq_top","file_id":"mathlib/Mathlib/RingTheory/Polynomial/Ideal.lean","sample_id":"91739cefd9b24c38b935a570789d9ce5a71a9f8d6f001da414f1877193c9705f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e6c5467020ffc4f18638257b26b5e8e286b407b28f63974096a991066b7eb6a4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6be9545aedd7e6b94e2cba861b00ad8b06f63ac214c443b7f533604392c9fc8e","source_sha256":"34049d18cb594f0a184df593dbd6d4fba0ade793ef81827c21fb1d80f135b5c5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rw [← div_eq_mul_inv, ← h3, mul_div_cancel_right₀ _ h2]","hard_negative":false,"metrics":{"chosen_tokens":14,"rejected_tokens":2,"token_jaccard":0.076923,"token_length_ratio":0.142857},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"6d69c8225815643fa303e0d735cf6393b0272037e38458be8ac8c5582fe33312","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Basic\npublic import Mathlib.Algebra.Order.Ring.Defs\npublic import Mathlib.Data.Tree.Basic\npublic import Mathlib.Tactic.NormNum.Core\npublic import Mathlib.Util.SynthesizeUsing\n\nNamespace:\nMathlib.Tactic.CancelDenoms\n\nLocal context:\n/-\nCopyright (c) 2020 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n-/\npublic meta import Mathlib.Data.Tree.Basic\npublic meta import Mathlib.Algebra.Group.Nat.Defs\n\n/-!\n# A tactic for canceling numeric denominators\n\nThis file defines tactics that cancel numeric denominators from field Expressions.\n\nAs an example, we want to transform a comparison `5*(a/3 + b/4) < c/3` into the equivalent\n`5*(4*a + 3*b) < 4*c`.\n\n## Implementation notes\n\nThe tooling here was originally written for `linarith`, not intended as an interactive tactic.\nThe interactive version has been split off because it is sometimes convenient to use on its own.\nThere are likely some rough edges to it.\n\nImproving this tactic would be a good project for someone interested in learning tactic programming.\n-/\n\npublic meta section\n\nopen Lean Parser Tactic Mathlib Meta NormNum Qq\n\ninitialize registerTraceClass `CancelDenoms\n\nnamespace Mathlib.Tactic.CancelDenoms\n\n/-! ### Lemmas used in the procedure -/\n\ntheorem mul_subst {α} [CommRing α] {n1 n2 k e1 e2 t1 t2 : α}\n    (h1 : n1 * e1 = t1) (h2 : n2 * e2 = t2) (h3 : n1 * n2 = k) : k * (e1 * e2) = t1 * t2 := by\n  rw [← h3, mul_comm n1, mul_assoc n2, ← mul_assoc n1, h1,\n      ← mul_assoc n2, mul_comm n2, mul_assoc, h2]\n\ntheorem div_subst {α} [Field α] {n1 n2 k e1 e2 t1 : α}\n    (h1 : n1 * e1 = t1) (h2 : n2 / e2 = 1) (h3 : n1 * n2 = k) : k * (e1 / e2) = t1 := by\n  rw [← h3, mul_assoc, mul_div_left_comm, h2, ← mul_assoc, h1, mul_comm, one_mul]\n\ntheorem cancel_factors_eq_div {α} [Field α] {n e e' : α}\n    (h : n * e = e') (h2 : n ≠ 0) : e = e' / n :=\n  eq_div_of_mul_eq h2 <| by rwa [mul_comm] at h\n\ntheorem add_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :\n    n * (e1 + e2) = t1 + t2 := by simp [left_distrib, *]\n\ntheorem sub_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :\n    n * (e1 - e2) = t1 - t2 := by simp [left_distrib, *, sub_eq_add_neg]\n\ntheorem neg_subst {α} [Ring α] {n e t : α} (h1 : n * e = t) : n * -e = -t := by simp [*]\n\ntheorem pow_subst {α} [CommRing α] {n e1 t1 k l : α} {e2 : ℕ}\n    (h1 : n * e1 = t1) (h2 : l * n ^ e2 = k) : k * (e1 ^ e2) = l * t1 ^ e2 := by\n  rw [← h2, ← h1, mul_pow, mul_assoc]\n\nTarget:\ntheorem inv_subst {α} [Field α] {n k e : α} (h2 : e ≠ 0) (h3 : n * e = k) :\n    k * (e ⁻¹) = n :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/CancelDenoms","family_id":"inv_subst","file_id":"mathlib/Mathlib/Tactic/CancelDenoms/Core.lean","sample_id":"6be9545aedd7e6b94e2cba861b00ad8b06f63ac214c443b7f533604392c9fc8e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"de3f7865b87871f2b8374f7e15ee4575771052e51846b14aedf16b2bb0b0042a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"84c9e058563163df6246bc78ed6fd10279fecd344d9583f603b14ffcd889ecd9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"986207edb34e921fae14ac5de27f5721ed1936c5cf23f19974ccfb40895a7398","source_sha256":"8e0c46324d60474a3cf61c252575628882e61646e1feb4e93ffa47506153589c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext ⟨a, b⟩\n  simp only [mem_smulAntidiagonal, mem_singleton, Prod.ext_iff]\n  constructor\n  · rintro ⟨has, hat, hst⟩\n    obtain rfl :=\n      (hs.min_le hns has).eq_of_not_lt fun hlt =>\n        (SMul.smul_lt_smul_of_lt_of_le hlt <| ht.min_le hnt hat).ne' hst\n    exact ⟨rfl, IsCancelSMul.left_cancel _ _ _ hst⟩\n  · rintro ⟨rfl, rfl⟩\n    exact ⟨hs.min_mem _, ht.min_mem _, rfl⟩","hard_negative":true,"metrics":{"chosen_tokens":92,"rejected_tokens":3,"token_jaccard":0.042553,"token_length_ratio":0.032609},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"6d87707a649919115a07f8f5acedbd5228641db0b92c86df64af70fff9b032d2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Pointwise.Set.Scalar\npublic import Mathlib.Data.Set.SMulAntidiagonal\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2024 Scott Carnahan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Carnahan\n-/\n/-!\n# Antidiagonal for scalar multiplication as a `Finset`.\n\nGiven sets `G` and `P`, with an action of `G` on `P`, we construct, for any element `a` in `P`,\nthe `Finset` of all pairs of an element in `s` and an element in `t` that scalar-multiply to `a`,\nassuming that set is finite.\n\n## Definitions\n* Finset.SMulAntidiagonal : Finset antidiagonal for PWO inputs.\n* Finset.VAddAntidiagonal : Finset antidiagonal for PWO inputs.\n\n-/\n\n@[expose] public section\n\nvariable {G P : Type*}\n\nopen scoped Pointwise\n\nnamespace Set\n\n@[to_additive]\ntheorem IsPWO.smul [Preorder G] [Preorder P] [SMul G P] [IsOrderedSMul G P]\n    {s : Set G} {t : Set P} (hs : s.IsPWO) (ht : t.IsPWO) : IsPWO (s • t) := by\n  rw [← @image_smul_prod]\n  exact (hs.prod ht).image_of_monotone (monotone_fst.smul monotone_snd)\n\n@[to_additive]\ntheorem IsWF.smul [LinearOrder G] [LinearOrder P] [SMul G P] [IsOrderedSMul G P] {s : Set G}\n    {t : Set P} (hs : s.IsWF) (ht : t.IsWF) : IsWF (s • t) :=\n  (hs.isPWO.smul ht.isPWO).isWF\n\n@[to_additive]\ntheorem IsWF.min_smul [LinearOrder G] [LinearOrder P] [SMul G P] [IsOrderedSMul G P]\n    {s : Set G} {t : Set P} (hs : s.IsWF) (ht : t.IsWF) (hsn : s.Nonempty) (htn : t.Nonempty) :\n    (hs.smul ht).min (hsn.smul htn) = hs.min hsn • ht.min htn := by\n  refine le_antisymm (IsWF.min_le _ _ (mem_smul.2 ⟨_, hs.min_mem _, _, ht.min_mem _, rfl⟩)) ?_\n  rw [IsWF.le_min_iff]\n  rintro _ ⟨x, hx, y, hy, rfl⟩\n  exact IsOrderedSMul.smul_le_smul (hs.min_le _ hx) (ht.min_le _ hy)\n\nend Set\n\nnamespace Finset\n\nsection\n\nopen Set\n\nvariable [SMul G P]\n\n/-- `Finset.SMulAntidiagonal hs ht a` is the set of all pairs of an element in `s` and an\nelement in `t` whose scalar multiplication yields `a`, but its construction requires a proof that\nthe set-theoretic antidiagonal is finite. -/\n@[to_additive /-- `Finset.VAddAntidiagonal hs ht a` is the set of all pairs of an element in `s`\nand an element in `t` whose vector addition yields `a`, but its construction requires proofs that\n`s` and `t` are well-ordered. -/]\nnoncomputable def SMulAntidiagonal {s : Set G}\n    {t : Set P} (a : P) (h : (s.smulAntidiagonal t a).Finite) : Finset (G × P) :=\n  h.toFinset\n\n@[to_additive (attr := simp)]\ntheorem mem_smulAntidiagonal {s : Set G}\n    {t : Set P} (a : P) (h : (s.smulAntidiagonal t a).Finite) {x : G × P} :\n    x ∈ SMulAntidiagonal a h ↔ x.1 ∈ s ∧ x.2 ∈ t ∧ x.1 • x.2 = a := by\n  simp only [SMulAntidiagonal, Set.Finite.mem_toFinset]\n  exact Set.mem_sep_iff\n\n@[to_additive]\ntheorem smulAntidiagonal_mono_left {s u : Set G} {t : Set P} (a : P) (h : u ⊆ s)\n    (hst : (s.smulAntidiagonal t a).Finite) (hut : (u.smulAntidiagonal t a).Finite) :\n    SMulAntidiagonal a hut ⊆ SMulAntidiagonal a hst :=\n  Set.Finite.toFinset_mono <| Set.smulAntidiagonal_mono_left h\n\n@[to_additive]\ntheorem smulAntidiagonal_mono_right {s : Set G}\n    {t v : Set P} (a : P) (hst : (s.smulAntidiagonal t a).Finite)\n    (hsv : (s.smulAntidiagonal v a).Finite) (h : v ⊆ t) :\n    SMulAntidiagonal a hsv ⊆ SMulAntidiagonal a hst :=\n  Set.Finite.toFinset_mono <| Set.smulAntidiagonal_mono_right h\n\n@[to_additive]\ntheorem support_smulAntidiagonal_subset_smul {s : Set G}\n    {t : Set P} (hst : ∀ a, (s.smulAntidiagonal t a).Finite) :\n    { a | (SMulAntidiagonal a (hst a)).Nonempty } ⊆ (s • t) := by\n  grind [mem_smul, mem_smulAntidiagonal]\n\nvariable [PartialOrder G] [PartialOrder P] [IsOrderedCancelSMul G P] {s : Set G}\n    {t : Set P} (hs : s.IsPWO) (ht : t.IsPWO) (a : P) {u : Set G} {hu : u.IsPWO} {v : Set P}\n    {hv : v.IsPWO} {x : G × P}\n\n@[to_additive]\ntheorem isPWO_support_smulAntidiagonal :\n    { a | (SMulAntidiagonal a (Set.SMulAntidiagonal.finite_of_isPWO hs ht a)).Nonempty }.IsPWO :=\n  (hs.smul ht).mono\n    (support_smulAntidiagonal_subset_smul (fun a ↦ (Set.SMulAntidiagonal.finite_of_isPWO hs ht a)))\n\nend\n\n@[to_additive]\n\nTarget:\ntheorem smulAntidiagonal_min_smul_min [LinearOrder G] [LinearOrder P] [SMul G P]\n    [IsOrderedCancelSMul G P] {s : Set G} {t : Set P} (hs : s.IsWF) (ht : t.IsWF) (hns : s.Nonempty)\n    (hnt : t.Nonempty) :\n    SMulAntidiagonal (hs.min hns • ht.min hnt)\n      (Set.SMulAntidiagonal.finite_of_isPWO hs.isPWO ht.isPWO (hs.min hns • ht.min hnt)) =\n      {(hs.min hns, ht.min hnt)} :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_986207edb34e","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"b8f654f6a70a7bd4a9b24154e4351207ca984629053ebbeed0411d9c84244e40","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finset","family_id":"smulantidiagonal_min_smul_min","file_id":"mathlib/Mathlib/Data/Finset/SMulAntidiagonal.lean","sample_id":"986207edb34e921fae14ac5de27f5721ed1936c5cf23f19974ccfb40895a7398"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b4cbd01d6c035dc37c5d6b8fd96656834089520d56a98eb2574c117f82f08ca7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2e83f293bd34554caad077053dfa4b4856eabf4ae072c64cf35623b00558a124","source_sha256":"ea6805b6e04e1a10e4f1800fbf6f645af1b3fbe7b7bc9db5c014508a4b9f528f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [dens, card_biUnion h, sum_div]","hard_negative":false,"metrics":{"chosen_tokens":10,"rejected_tokens":2,"token_jaccard":0.1,"token_length_ratio":0.2},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"6dcdcdafa5a2d032828cdc4f1d2da66942f09f7e14672a629716a2f9019355e8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Ring.Finset\npublic import Mathlib.Algebra.Field.Defs\npublic import Mathlib.Data.Finset.Density\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Daniel Weber\n-/\n/-!\n# Results about big operators with values in a field\n-/\n\npublic section\n\nopen Fintype\n\nvariable {ι K : Type*} [DivisionSemiring K]\n\nlemma Multiset.sum_map_div (s : Multiset ι) (f : ι → K) (a : K) :\n    (s.map (fun x ↦ f x / a)).sum = (s.map f).sum / a := by\n  simp only [div_eq_mul_inv, Multiset.sum_map_mul_right]\n\nlemma Finset.sum_div (s : Finset ι) (f : ι → K) (a : K) :\n    (∑ i ∈ s, f i) / a = ∑ i ∈ s, f i / a := by simp only [div_eq_mul_inv, sum_mul]\n\n-- TODO: Move these to `Algebra.BigOperators.Group.Finset.Basic`, next to the corresponding `card`\n-- lemmas, once `Finset.dens` doesn't depend on `Field` anymore.\nnamespace Finset\nvariable {α β : Type*} [Fintype β]\n\n@[simp]\nlemma dens_disjiUnion (s : Finset α) (t : α → Finset β) (h) :\n    (s.disjiUnion t h).dens = ∑ a ∈ s, (t a).dens := by\n  simp [dens, sum_div]\n\nvariable {s : Finset α} {t : α → Finset β}\n\nTarget:\nlemma dens_biUnion [DecidableEq β] (h : (s : Set α).PairwiseDisjoint t) :\n    (s.biUnion t).dens = ∑ u ∈ s, (t u).dens :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/BigOperators","family_id":"dens_biunion","file_id":"mathlib/Mathlib/Algebra/BigOperators/Field.lean","sample_id":"2e83f293bd34554caad077053dfa4b4856eabf4ae072c64cf35623b00558a124"},"split":"train"}
{"binding":{"chosen_proof_sha256":"faf6bc027aba61a3300a23f2d313e28567ba4f3a90cdcf2560518c60032b7b01","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bb2be5794bc2bef47d57ee2509f8e94d0f35879b91907a617a92bba8e4b1af69","source_sha256":"b6f10e56b5f036623bcf6641814f24797b2faa600c67007e65fd922ddca8f380","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  let ι := Module.Free.ChooseBasisIndex R S\n  let b : Module.Basis ι R S := Module.Free.chooseBasis R S\n  rw [trace_eq_matrix_trace b, trace_eq_matrix_trace (basisQuotient b), AddMonoidHom.map_trace]\n  congr 1\n  ext i j\n  simp only [leftMulMatrix_apply, coe_lmul_eq_mul, LinearMap.toMatrix_apply,\n    basisQuotient_apply, LinearMap.mul_apply', Matrix.map_apply, ← map_mul,\n    basisQuotient_repr]","hard_negative":false,"metrics":{"chosen_tokens":75,"rejected_tokens":3,"token_jaccard":0.022222,"token_length_ratio":0.04},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"6e32e5345b8e94445783f5568fecd3d7e871363a271f7662638cf3c0863140c1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.Dvr\npublic import Mathlib.RingTheory.IntegralClosure.IntegralRestrict\npublic import Mathlib.RingTheory.LocalRing.Quotient\npublic import Mathlib.RingTheory.Localization.AtPrime.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Riccardo Brasca. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang, Riccardo Brasca\n-/\n/-!\n\nWe gather results about the relations between the trace map on `B → A` and the trace map on\nquotients and localizations.\n\n## Main Results\n\n* `Algebra.trace_quotient_eq_of_isDedekindDomain` : The trace map on `B → A` coincides with the\n  trace map on `B⧸pB → A⧸p`.\n\n-/\n\npublic section\n\nvariable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]\n\nopen IsLocalRing FiniteDimensional Module Submodule IsLocalization.AtPrime\n\nsection IsLocalRing\n\nlocal notation \"p\" => maximalIdeal R\nlocal notation \"pS\" => Ideal.map (algebraMap R S) p\n\nvariable [Module.Free R S] [Module.Finite R S]\n\nattribute [local instance] Ideal.Quotient.field\n\nTarget:\nlemma Algebra.trace_quotient_mk [IsLocalRing R] (x : S) :\n    Algebra.trace (R ⧸ p) (S ⧸ pS) (Ideal.Quotient.mk pS x) =\n      Ideal.Quotient.mk p (Algebra.trace R S x) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Trace","family_id":"algebra","file_id":"mathlib/Mathlib/RingTheory/Trace/Quotient.lean","sample_id":"bb2be5794bc2bef47d57ee2509f8e94d0f35879b91907a617a92bba8e4b1af69"},"split":"train"}
{"binding":{"chosen_proof_sha256":"03e21e37518786f0e8f480f42a07a2927cdb9237ca44f9d215146d88cd8ce9cd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7f63bf8b080e808117284c406a2db8ef2f14a5c00152c31fb40b5d94668e2b76","source_sha256":"ccfcbde3f10bc565628c0a834c762e9d5e446934b2c86ddc567af141b0f9967f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n    obtain ⟨s, hs⟩ := hI\n    rw [← adjoin_monomial_eq_reesAlgebra, ← hs]\n    use s.image (monomial 1)\n    rw [Finset.coe_image]\n    change\n      _ =\n        Algebra.adjoin R\n          (Submodule.map (monomial 1 : R →ₗ[R] R[X]) (Submodule.span R ↑s) : Set R[X])\n    rw [Submodule.map_span, Algebra.adjoin_span]","hard_negative":true,"metrics":{"chosen_tokens":83,"rejected_tokens":8,"token_jaccard":0.066667,"token_length_ratio":0.096386},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"6e49e2cd2078731611b7de506027505ec3560f5e845d759c9814e918939b53bb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Ideal.BigOperators\npublic import Mathlib.RingTheory.FiniteType\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Rees algebra\n\nThe Rees algebra of an ideal `I` is the subalgebra `R[It]` of `R[t]` defined as `R[It] = ⨁ₙ Iⁿ tⁿ`.\nThis is used to prove the Artin-Rees lemma, and will potentially enable us to calculate some\nblowup in the future.\n\n## Main definition\n\n- `reesAlgebra` : The Rees algebra of an ideal `I`, defined as a subalgebra of `R[X]`.\n- `adjoin_monomial_eq_reesAlgebra` : The Rees algebra is generated by the degree one elements.\n- `reesAlgebra.fg` : The Rees algebra of a f.g. ideal is of finite type. In particular, this\n  implies that the rees algebra over a Noetherian ring is still Noetherian.\n\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\nvariable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)\n\nopen Polynomial\n\n/-- The Rees algebra of an ideal `I`, defined as the subalgebra of `R[X]` whose `i`-th coefficient\nfalls in `I ^ i`. -/\ndef reesAlgebra : Subalgebra R R[X] where\n  carrier := { f | ∀ i, f.coeff i ∈ I ^ i }\n  mul_mem' hf hg i := by\n    rw [coeff_mul]\n    apply Ideal.sum_mem\n    rintro ⟨j, k⟩ e\n    rw [← Finset.mem_antidiagonal.mp e, pow_add]\n    exact Ideal.mul_mem_mul (hf j) (hg k)\n  one_mem' i := by\n    rw [coeff_one]\n    split_ifs with h\n    · subst h\n      simp\n    · simp\n  add_mem' hf hg i := by\n    rw [coeff_add]\n    exact Ideal.add_mem _ (hf i) (hg i)\n  zero_mem' _ := Ideal.zero_mem _\n  algebraMap_mem' r i := by\n    rw [algebraMap_apply, coeff_C]\n    split_ifs with h\n    · subst h\n      simp\n    · simp\n\ntheorem mem_reesAlgebra_iff (f : R[X]) : f ∈ reesAlgebra I ↔ ∀ i, f.coeff i ∈ I ^ i :=\n  Iff.rfl\n\ntheorem mem_reesAlgebra_iff_support (f : R[X]) :\n    f ∈ reesAlgebra I ↔ ∀ i ∈ f.support, f.coeff i ∈ I ^ i := by\n  apply forall_congr'\n  intro a\n  rw [mem_support_iff, Iff.comm, Classical.imp_iff_right_iff, Ne, ← imp_iff_not_or]\n  exact fun e => e.symm ▸ (I ^ a).zero_mem\n\ntheorem reesAlgebra.monomial_mem {I : Ideal R} {i : ℕ} {r : R} :\n    monomial i r ∈ reesAlgebra I ↔ r ∈ I ^ i := by\n  simp +contextual [mem_reesAlgebra_iff_support, coeff_monomial, ←\n    imp_iff_not_or]\n\ntheorem monomial_mem_adjoin_monomial {I : Ideal R} {n : ℕ} {r : R} (hr : r ∈ I ^ n) :\n    monomial n r ∈ Algebra.adjoin R (Submodule.map (monomial 1 : R →ₗ[R] R[X]) I : Set R[X]) := by\n  induction n generalizing r with\n  | zero => exact Subalgebra.algebraMap_mem _ _\n  | succ n hn =>\n    rw [pow_succ'] at hr\n    refine Submodule.smul_induction_on hr ?_ ?_\n    · intro r hr s hs\n      rw [add_comm n 1, smul_eq_mul, ← monomial_mul_monomial]\n      exact Subalgebra.mul_mem _ (Algebra.subset_adjoin (Set.mem_image_of_mem _ hr)) (hn hs)\n    · intro x y hx hy\n      rw [map_add]\n      exact Subalgebra.add_mem _ hx hy\n\ntheorem adjoin_monomial_eq_reesAlgebra :\n    Algebra.adjoin R (Submodule.map (monomial 1 : R →ₗ[R] R[X]) I : Set R[X]) = reesAlgebra I := by\n  apply le_antisymm\n  · apply Algebra.adjoin_le _\n    rintro _ ⟨r, hr, rfl⟩\n    exact reesAlgebra.monomial_mem.mpr (by rwa [pow_one])\n  · intro p hp\n    rw [p.as_sum_support]\n    apply Subalgebra.sum_mem _ _\n    rintro i -\n    exact monomial_mem_adjoin_monomial (hp i)\n\nvariable {I}\n\nTarget:\ntheorem reesAlgebra.fg (hI : I.FG) : (reesAlgebra I).FG :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"c25a7977c628cdee59a2d045b1399722135e60f2926bfdebed852c938c348ed7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory","family_id":"reesalgebra","file_id":"mathlib/Mathlib/RingTheory/ReesAlgebra.lean","sample_id":"7f63bf8b080e808117284c406a2db8ef2f14a5c00152c31fb40b5d94668e2b76"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5a49187c184286dd02eeb6ca07b11443ba36cdee6d43f2c64edd8e661703bcce","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"394f27a561e1c227c4d6a3e91cad146754404323a259cba6efddc68715f8ebd8","source_sha256":"c0e3fa9405ddf4fdd99cd3aa6c9ff7258178f89496e31a48d0f8f121f656258a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [mem_Icc, insertNth_le_iff, le_insertNth_iff, and_assoc, @and_left_comm (x ≤ q₂ i)]","hard_negative":false,"metrics":{"chosen_tokens":22,"rejected_tokens":2,"token_jaccard":0.05,"token_length_ratio":0.090909},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"6e66c6ac5795dde5aedc215ef6dfefc646ef4b569df750d7e7f42ed20af8e789","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.VecNotation\npublic import Mathlib.Logic.Equiv.Fin.Basic\npublic import Mathlib.Order.Fin.Basic\npublic import Mathlib.Order.PiLex\npublic import Mathlib.Order.Interval.Set.Defs\n\nNamespace:\nFin\n\nLocal context:\n/-\nCopyright (c) 2019 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes\n-/\n/-!\n# Order properties on tuples\n-/\n\n@[expose] public section\n\nassert_not_exists Monoid\n\nopen Function Set\n\nnamespace Fin\nvariable {m n : ℕ} {α : Fin (n + 1) → Type*} (x : α 0) (q : ∀ i, α i) (p : ∀ i : Fin n, α i.succ)\n  (i : Fin n) (y : α i.succ) (z : α 0)\n\nlemma pi_lex_lt_cons_cons {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ}\n    (s : ∀ {i : Fin n.succ}, α i → α i → Prop) :\n    Pi.Lex (· < ·) (@s) (Fin.cons x₀ x) (Fin.cons y₀ y) ↔\n      s x₀ y₀ ∨ x₀ = y₀ ∧ Pi.Lex (· < ·) (@fun i : Fin n ↦ @s i.succ) x y := by\n  simp_rw [Pi.Lex, Fin.exists_fin_succ, Fin.cons_succ, Fin.cons_zero, Fin.forall_iff_succ]\n  simp [and_assoc, exists_and_left]\n\nvariable [∀ i, Preorder (α i)]\n\nTarget:\nlemma insertNth_mem_Icc {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)}\n    {q₁ q₂ : ∀ j, α j} :\n    i.insertNth x p ∈ Icc q₁ q₂ ↔\n      x ∈ Icc (q₁ i) (q₂ i) ∧ p ∈ Icc (fun j ↦ q₁ (i.succAbove j)) fun j ↦ q₂ (i.succAbove j) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Fin","family_id":"insertnth_mem_icc","file_id":"mathlib/Mathlib/Order/Fin/Tuple.lean","sample_id":"394f27a561e1c227c4d6a3e91cad146754404323a259cba6efddc68715f8ebd8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d99c571d1e3b54f4a7c947a3bff9621121704efdf01dd61bab5cf6b4a711ab1c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"116f8e1c9d0741f93e3292389186ed75e7a06ae8523ae29af081b6d96f320580","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"052befed77426c858368b90b76b9540e8aaef3d27651275eef8d687539481881","source_sha256":"94fce5005b3452fffd0935ed1c8380f811db0b64c02673ffec8176805d9749dd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [Functor.IsDense.leftKanExtensionIso]","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":3,"token_jaccard":0.1,"token_length_ratio":0.333333},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"6e786572ddae818e696bb13a8b65b1f22421c160ee57a1f6ef4b8466e3d2066a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Functor.KanExtension.DenseAt\npublic import Mathlib.CategoryTheory.Limits.Presheaf\npublic import Mathlib.CategoryTheory.Generator.StrongGenerator\n\nNamespace:\nCategoryTheory.Functor\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Dense functors\n\nA functor `F : C ⥤ D` is dense (`F.IsDense`) if `𝟭 D` is a pointwise\nleft Kan extension of `F` along itself, i.e. any `Y : D` is the\ncolimit of all `F.obj X` for all morphisms `F.obj X ⟶ Y` (which\nis the condition `F.DenseAt Y`).\nWhen `F` is full, we show that this\nis equivalent to saying that the restricted Yoneda functor\n`D ⥤ Cᵒᵖ ⥤ Type _` is fully faithful (see the lemma\n`Functor.isDense_iff_fullyFaithful_restrictedULiftYoneda`).\n\nWe also show that the range of a dense functor is a strong\ngenerator (see `Functor.isStrongGenerator_of_isDense`).\n\n## References\n\n* https://ncatlab.org/nlab/show/dense+subcategory\n\n-/\n\n@[expose] public section\n\nuniverse w v₁ v₂ v₃ u₁ u₂ u₃\n\nnamespace CategoryTheory\n\nopen Limits Opposite Presheaf ConcreteCategory\n\nvariable {C : Type u₁} {D : Type u₂} [Category.{v₁} C] [Category.{v₂} D]\n  {C' : Type u₃} [Category.{v₃} C']\n\nnamespace Functor\n\n/-- A functor `F : C ⥤ D` is dense if any `Y : D` is a canonical colimit\nrelatively to `F`. -/\nclass IsDense (F : C ⥤ D) : Prop where\n  isDenseAt (F) (Y : D) : F.isDenseAt Y\n\n/-- This is a choice of structure `F.DenseAt Y` when `F : C ⥤ D`\nis dense, and `Y : D`. -/\nnoncomputable def denseAt (F : C ⥤ D) [F.IsDense] (Y : D) : F.DenseAt Y :=\n  (IsDense.isDenseAt F Y).some\n\nlemma isDense_iff_nonempty_isPointwiseLeftKanExtension (F : C ⥤ D) :\n    F.IsDense ↔\n      Nonempty ((LeftExtension.mk _ (rightUnitor F).inv).IsPointwiseLeftKanExtension) :=\n  ⟨fun _ ↦ ⟨fun _ ↦ F.denseAt _⟩, fun ⟨h⟩ ↦ ⟨fun _ ↦ ⟨h _⟩⟩⟩\n\ninstance (F : C ⥤ D) [F.IsDense] : Functor.IsLeftKanExtension (𝟭 D) (Functor.rightUnitor F).inv :=\n  ((Functor.isDense_iff_nonempty_isPointwiseLeftKanExtension F).mp ‹_›).some.isLeftKanExtension\n\ninstance (F : C ⥤ D) [F.IsDense] : F.HasPointwiseLeftKanExtension F :=\n  fun X ↦ (Functor.IsDense.isDenseAt F X).some.hasPointwiseLeftKanExtensionAt\n\nlemma IsDense.of_iso {F G : C ⥤ D} (e : F ≅ G) [F.IsDense] :\n    G.IsDense where\n  isDenseAt Y := by\n    rw [← Functor.congr_isDenseAt e]\n    exact ⟨F.denseAt Y⟩\n\nlemma IsDense.iff_of_iso {F G : C ⥤ D} (e : F ≅ G) :\n    F.IsDense ↔ G.IsDense :=\n  ⟨fun _ ↦ of_iso e, fun _ ↦ of_iso e.symm⟩\n\nvariable (F : C ⥤ D)\n\ninstance (G : C' ⥤ C) [F.IsDense] [G.IsEquivalence] :\n    (G ⋙ F).IsDense where\n  isDenseAt Y := ⟨(F.denseAt Y).precompOfFinal G⟩\n\nlemma IsDense.comp_left_iff_of_isEquivalence (G : C' ⥤ C) [G.IsEquivalence] :\n    (G ⋙ F).IsDense ↔ F.IsDense := by\n  refine ⟨fun _ ↦ ?_, fun _ ↦ inferInstance⟩\n  let e : G.inv ⋙ G ⋙ F ≅ F := (associator _ _ _).symm ≪≫\n    isoWhiskerRight (G.asEquivalence.counitIso) _ ≪≫ F.leftUnitor\n  exact of_iso e\n\ninstance (G : D ⥤ C') [F.IsDense] [G.IsEquivalence] :\n    (F ⋙ G).IsDense where\n  isDenseAt Y :=\n    ⟨ letI e : Y ≅ G.obj (G.inv.obj Y) := G.asEquivalence.counitIso.symm.app Y\n      DenseAt.ofIso (F.denseAt (G.inv.obj Y) |>.postcompEquivalence G) e.symm ⟩\n\nlemma IsDense.comp_right_iff_of_isEquivalence (G : D ⥤ C') [G.IsEquivalence] :\n    (F ⋙ G).IsDense ↔ F.IsDense := by\n  refine ⟨fun _ ↦ ?_, fun _ ↦ inferInstance⟩\n  let e : (F ⋙ G) ⋙ G.inv ≅ F := associator .. ≪≫\n    isoWhiskerLeft _ G.asEquivalence.unitIso.symm ≪≫ F.rightUnitor\n  exact of_iso e\n\nset_option backward.defeqAttrib.useBackward true in\ninstance [F.IsDense] : (restrictedULiftYoneda.{w} F).Faithful where\n  map_injective h :=\n    (F.denseAt _).hom_ext' (fun X p ↦ by\n      simpa using! ULift.up_injective (ConcreteCategory.congr_hom (CC := fun X ↦ X)\n        (NatTrans.congr_app h (op X)) (ULift.up p)))\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\ninstance [F.IsDense] : (restrictedULiftYoneda.{w} F).Full where\n  map_surjective {Y Z} f := by\n    let c : Cocone (CostructuredArrow.proj F Y ⋙ F) :=\n      { pt := Z\n        ι :=\n          { app g := ((f.app (op g.left)) (ULift.up g.hom)).down\n            naturality g₁ g₂ φ := by\n              simpa [uliftFunctor, uliftYoneda,\n                restrictedULiftYoneda, ← ULift.down_inj] using\n                ((f.naturality_apply φ.left.op) (ULift.up g₂.hom)).symm } }\n    refine ⟨(F.denseAt Y).desc c, ?_⟩\n    ext ⟨X⟩ ⟨x⟩\n    have := (F.denseAt Y).fac c (.mk x)\n    dsimp [c] at this\n    simpa using ULift.down_injective this\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nvariable {F} in\nlemma IsDense.of_fullyFaithful_restrictedULiftYoneda [F.Full]\n    (h : (restrictedULiftYoneda.{w} F).FullyFaithful) :\n    F.IsDense where\n  isDenseAt Y := by\n    let φ (s : Cocone (CostructuredArrow.proj F Y ⋙ F)) :\n        (restrictedULiftYoneda.{w} F).obj Y ⟶ (restrictedULiftYoneda F).obj s.pt :=\n      { app := fun ⟨X⟩ ↦ ↾fun ⟨x⟩ ↦ ULift.up (s.ι.app (.mk x))\n        naturality := by\n          rintro ⟨X₁⟩ ⟨X₂⟩ ⟨f⟩\n          ext ⟨x⟩\n          let α : CostructuredArrow.mk (F.map f ≫ x) ⟶ CostructuredArrow.mk x :=\n            CostructuredArrow.homMk f\n          exact ULift.down_injective (s.w α).symm }\n    have hφ (s) (j) : (restrictedULiftYoneda F).map j.hom ≫ φ s =\n        (restrictedULiftYoneda F).map (s.ι.app j) := by\n      ext ⟨X⟩ ⟨x⟩\n      let α : .mk (x ≫ j.hom) ⟶ j := CostructuredArrow.homMk (F.preimage x)\n      have := s.w α\n      dsimp [uliftYoneda, φ, α] at this ⊢\n      apply ULift.down_injective\n      simpa using this.symm\n    exact\n      ⟨{desc s := (h.preimage (φ s))\n        fac s j := h.map_injective (by simp [hφ])\n        uniq s m hm := h.map_injective (by\n          ext ⟨_⟩ ⟨_⟩\n          simp [φ, ← hm]) }⟩\n\nlemma isDense_iff_fullyFaithful_restrictedULiftYoneda [F.Full] :\n    F.IsDense ↔ Nonempty (restrictedULiftYoneda.{w} F).FullyFaithful :=\n  ⟨fun _ ↦ ⟨FullyFaithful.ofFullyFaithful _⟩,\n    fun ⟨h⟩ ↦ IsDense.of_fullyFaithful_restrictedULiftYoneda h⟩\n\nopen ObjectProperty in\nlemma isStrongGenerator_of_isDense [F.IsDense] :\n    IsStrongGenerator (.ofObj F.obj) :=\n  (IsStrongGenerator.mk_of_exists_colimitsOfShape.{max u₁ u₂ v₁ v₂,\n      max u₁ v₁ v₂} (fun Y ↦ ⟨_, _, ⟨{\n    ι := _\n    diag := _\n    isColimit := (IsColimit.whiskerEquivalence (F.denseAt Y)\n      ((ShrinkHoms.equivalence _).symm.trans ((Shrink.equivalence _)).symm))\n    prop_diag_obj := by simp }⟩⟩))\n\n/-- If `F` is dense, the left Kan extension of `F` along `F` is isomorphic to the identity. -/\nnoncomputable def IsDense.leftKanExtensionIso (F : C ⥤ D) [F.IsDense] :\n    F.leftKanExtension F ≅ 𝟭 D :=\n  Functor.leftKanExtensionUnique _ (F.leftKanExtensionUnit F) _ F.rightUnitor.inv\n\n@[reassoc (attr := simp)]\n\nTarget:\nlemma IsDense.leftKanExtensionUnit_leftKanExtensionIso_hom (F : C ⥤ D) [F.IsDense] :\n    F.leftKanExtensionUnit F ≫ F.whiskerLeft (Functor.IsDense.leftKanExtensionIso F).hom =\n      F.rightUnitor.inv :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_052befed7742","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"5b40bdcd560b2e2de5bc2efee713385c2062a2b20c9d229094e132677744d401","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Functor","family_id":"isdense","file_id":"mathlib/Mathlib/CategoryTheory/Functor/KanExtension/Dense.lean","sample_id":"052befed77426c858368b90b76b9540e8aaef3d27651275eef8d687539481881"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3f320a505bb6b269ad39068f91895095fa32692c32719fd89a78bd89e6e3c623","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"308f6cf4c900ea2a1076236867cd7cbcb72edd4a9f392f88b684001d707ae5c6","source_sha256":"220aaaefdf1c03061afd6055e7419dc5fe5b656993ba2d93b412d09922822f9e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  rw [LinearMap.trace_eq_matrix_trace 𝕜 b.toBasis T]\n  apply Fintype.sum_congr\n  intro i\n  rw [Matrix.diag_apply, T.toMatrix_apply, b.coe_toBasis, b.coe_toBasis_repr_apply,\n    b.repr_apply_apply]\n\nvariable [FiniteDimensional 𝕜 E]\nvariable {n : ℕ} (hn : Module.finrank 𝕜 E = n)","hard_negative":false,"metrics":{"chosen_tokens":64,"rejected_tokens":5,"token_jaccard":0.02381,"token_length_ratio":0.078125},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"6eb085285944923e9453294de851a9b54c0f06217cb06f0d3d3c6d771c53e371","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.InnerProductSpace.PiL2\npublic import Mathlib.Analysis.InnerProductSpace.Spectrum\npublic import Mathlib.LinearAlgebra.Eigenspace.Charpoly\n\nNamespace:\nLinearMap\n\nLocal context:\n/-\nCopyright (c) 2025 Iván Renison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Iván Renison\n-/\n/-!\n# Traces in inner product spaces\n\nThis file contains various results about traces of linear operators in inner product spaces.\n-/\n\npublic section\n\nnamespace LinearMap\n\nvariable {𝕜 E ι : Type*} [RCLike 𝕜] [Fintype ι]\nvariable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]\n\nopen scoped InnerProductSpace\n\nTarget:\nlemma trace_eq_sum_inner (T : E →ₗ[𝕜] E) (b : OrthonormalBasis ι 𝕜 E) :\n    T.trace 𝕜 E = ∑ i, ⟪b i, T (b i)⟫_𝕜 :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/InnerProductSpace","family_id":"trace_eq_sum_inner","file_id":"mathlib/Mathlib/Analysis/InnerProductSpace/Trace.lean","sample_id":"308f6cf4c900ea2a1076236867cd7cbcb72edd4a9f392f88b684001d707ae5c6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9346ea21709b5b4adcb76342b23a0fdfec46476e1c52d71821429c4b741b1882","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3d58730a822f37dbe124ac3de8532bdf9e6b3a34db752189dda5130b2510e7eb","source_sha256":"e6ed626f52730f45ca36129d8b77b24071a2cee6973cf9b0b2dfb896ff9281e8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨f, g, h⟩ := exists_pair_ne (α → β)\n  rw [ne_eq, funext_iff, Classical.not_forall] at h\n  obtain ⟨a, h⟩ := h\n  exact nontrivial_of_ne _ _ h","hard_negative":false,"metrics":{"chosen_tokens":41,"rejected_tokens":3,"token_jaccard":0.068966,"token_length_ratio":0.073171},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"6f1a288c07c4d96a8b085094e2664682df61e706471a1c4c26fb17690de98816","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Tactic.Push.Attr\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Nontrivial types\n\nA type is *nontrivial* if it contains at least two elements. This is useful in particular for rings\n(where it is equivalent to the fact that zero is different from one) and for vector spaces\n(where it is equivalent to the fact that the dimension is positive).\n\nWe introduce a typeclass `Nontrivial` formalizing this property.\n\nBasic results about nontrivial types are in `Mathlib/Logic/Nontrivial/Basic.lean`.\n-/\n\npublic section\n\nvariable {α : Type*} {β : Type*}\n\n/-- Predicate typeclass for expressing that a type is not reduced to a single element. In rings,\nthis is equivalent to `0 ≠ 1`. In vector spaces, this is equivalent to positive dimension. -/\nclass Nontrivial (α : Type*) : Prop where\n  /-- In a nontrivial type, there exists a pair of distinct terms. -/\n  exists_pair_ne : ∃ x y : α, x ≠ y\n\ntheorem nontrivial_iff : Nontrivial α ↔ ∃ x y : α, x ≠ y :=\n  ⟨fun h ↦ h.exists_pair_ne, fun h ↦ ⟨h⟩⟩\n\ntheorem exists_pair_ne (α : Type*) [Nontrivial α] : ∃ x y : α, x ≠ y :=\n  Nontrivial.exists_pair_ne\n\n/-- Pushforward a `Nontrivial` instance along an injective function. -/\nprotected theorem Function.Injective.nontrivial [Nontrivial α] {f : α → β}\n    (hf : Function.Injective f) : Nontrivial β :=\n  let ⟨x, y, h⟩ := exists_pair_ne α\n  ⟨⟨f x, f y, hf.ne h⟩⟩\n\n/-- An injective function from a nontrivial type has an argument at\nwhich it does not take a given value. -/\nprotected theorem Function.Injective.exists_ne [Nontrivial α] {f : α → β}\n    (hf : Function.Injective f) (y : β) : ∃ x, f x ≠ y := by\n  rcases exists_pair_ne α with ⟨x₁, x₂, hx⟩\n  by_cases h : f x₂ = y\n  · exact ⟨x₁, (hf.ne_iff' h).2 hx⟩\n  · exact ⟨x₂, h⟩\n\n-- See Note [decidable namespace]\nprotected theorem Decidable.exists_ne [Nontrivial α] [DecidableEq α] (x : α) : ∃ y, y ≠ x := by\n  rcases exists_pair_ne α with ⟨y, y', h⟩\n  by_cases hx : x = y\n  · rw [← hx] at h\n    exact ⟨y', h.symm⟩\n  · exact ⟨y, Ne.symm hx⟩\n\nopen Classical in\ntheorem exists_ne [Nontrivial α] (x : α) : ∃ y, y ≠ x := Decidable.exists_ne x\n\n-- `x` and `y` are explicit here, as they are often needed to guide typechecking of `h`.\ntheorem nontrivial_of_ne (x y : α) (h : x ≠ y) : Nontrivial α :=\n  ⟨⟨x, y, h⟩⟩\n\ntheorem nontrivial_iff_exists_ne (x : α) : Nontrivial α ↔ ∃ y, y ≠ x :=\n  ⟨fun h ↦ @exists_ne α h x, fun ⟨_, hy⟩ ↦ nontrivial_of_ne _ _ hy⟩\n\nTarget:\ntheorem Function.nontrivial_of_nontrivial (α β : Type*) [Nontrivial (α → β)] :\n    Nontrivial β :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Nontrivial","family_id":"function","file_id":"mathlib/Mathlib/Logic/Nontrivial/Defs.lean","sample_id":"3d58730a822f37dbe124ac3de8532bdf9e6b3a34db752189dda5130b2510e7eb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"df3d8d4ac17ee91cfc6c4a0a0a21cbff1216a809c22aaf032985f2bc0080f0a1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"502c601dc704b016dbc827a9cd8efa7a22333a98a499271c49028df90994ce61","source_sha256":"0d4d2f5ac977ed70567b806a9e6298c3776d513de7fc9f5724c6140a65d7d751","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal' (I := ⊤) (f := f) (by simp) hg\n\nvariable [I.IsTwoSided]","hard_negative":false,"metrics":{"chosen_tokens":25,"rejected_tokens":2,"token_jaccard":0.055556,"token_length_ratio":0.08},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"6f35ce548dc8cebee7b14d3ad77827f59343ccdc4881d45ead3d8bd8a7afef75","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Ideal.Operations\npublic import Mathlib.RingTheory.Ideal.BigOperators\npublic import Mathlib.RingTheory.PowerSeries.Basic\n\nNamespace:\nPowerSeries\n\nLocal context:\n/-\nCopyright (c) 2025 Jz Pan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jz Pan\n-/\n/-!\n\n# Some results on the coefficients of multiplication of two power series\n\n## Main results\n\n- `PowerSeries.coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal`,\n  `PowerSeries.coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal'`:\n  if for all `i ≤ n` (resp. for all `i`), the `i`-th coefficients of power series `f` and `g` are\n  in ideals `I` and `J`, respectively, then for all `i ≤ n` (resp. for all `i`), the `i`-th\n  coefficients of `f * g` are in `I * J`.\n\n- `PowerSeries.coeff_mul_mem_ideal_of_coeff_right_mem_ideal`,\n  `PowerSeries.coeff_mul_mem_ideal_of_coeff_right_mem_ideal'`:\n  if for all `i ≤ n` (resp. for all `i`), the `i`-th coefficients of power series `g` are\n  in ideal `I`, then for all `i ≤ n` (resp. for all `i`), the `i`-th coefficients of `f * g` are\n  in `I`.\n\n- `PowerSeries.coeff_mul_mem_ideal_of_coeff_left_mem_ideal`,\n  `PowerSeries.coeff_mul_mem_ideal_of_coeff_left_mem_ideal'`:\n  if for all `i ≤ n` (resp. for all `i`), the `i`-th coefficients of power series `f` are\n  in ideal `I`, then for all `i ≤ n` (resp. for all `i`), the `i`-th coefficients of `f * g` are\n  in `I`.\n\n-/\n\npublic section\n\nnamespace PowerSeries\n\nvariable {A : Type*} [Semiring A] {I J : Ideal A} {f g : A⟦X⟧} (n : ℕ)\n\ntheorem coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal (hf : ∀ i ≤ n, coeff i f ∈ I)\n    (hg : ∀ i ≤ n, coeff i g ∈ J) : ∀ i ≤ n, coeff i (f * g) ∈ I * J := fun i hi ↦ by\n  rw [coeff_mul]\n  exact Ideal.sum_mem _ fun p hp ↦ Ideal.mul_mem_mul\n    (hf _ ((Finset.antidiagonal.fst_le hp).trans hi))\n    (hg _ ((Finset.antidiagonal.snd_le hp).trans hi))\n\ntheorem coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal' (hf : ∀ i, coeff i f ∈ I)\n    (hg : ∀ i, coeff i g ∈ J) : ∀ i, coeff i (f * g) ∈ I * J :=\n  fun i ↦ coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal i\n    (fun i _ ↦ hf i) (fun i _ ↦ hg i) i le_rfl\n\ntheorem coeff_mul_mem_ideal_of_coeff_right_mem_ideal\n    (hg : ∀ i ≤ n, coeff i g ∈ I) : ∀ i ≤ n, coeff i (f * g) ∈ I := by\n  simpa using coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal (I := ⊤) (f := f) n (by simp) hg\n\nTarget:\ntheorem coeff_mul_mem_ideal_of_coeff_right_mem_ideal'\n    (hg : ∀ i, coeff i g ∈ I) : ∀ i, coeff i (f * g) ∈ I :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/PowerSeries","family_id":"coeff_mul_mem_ideal_of_coeff_right_mem_ideal","file_id":"mathlib/Mathlib/RingTheory/PowerSeries/CoeffMulMem.lean","sample_id":"502c601dc704b016dbc827a9cd8efa7a22333a98a499271c49028df90994ce61"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7aefead92a5663164517806ecfcb8b153d186d51071288da4cb411af6d57db41","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c860402497f1b4dc3bdc93021a1e3e3c1ea1267241369ebbfdacfe5570fa2d4c","source_sha256":"7fb2a027e17dd48df4c5d05af7f579e28e12ec18c52dd1aaeae9c6854e93da06","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext v\n  simp [alternatizeUncurryFin_apply, ← map_insertNth]\n\nvariable {S : Type*} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N]","hard_negative":false,"metrics":{"chosen_tokens":32,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.0625},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"6f49b5efa959dbcff18e70db8cebfd582f9f967896300069799c9f6e115ef064","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Alternating.Curry\npublic import Mathlib.GroupTheory.Perm.Fin\npublic import Mathlib.Data.Fin.Parity\n\nNamespace:\nAlternatingMap\n\nLocal context:\n/-\nCopyright (c) 2025 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Uncurrying alternating maps\n\nGiven a function `f` which is linear in the first argument\nand is alternating form in the other `n` arguments,\nthis file defines an alternating form `AlternatingMap.alternatizeUncurryFin f` in `n + 1` arguments.\n\nThis function is given by\n```\nAlternatingMap.alternatizeUncurryFin f v =\n  ∑ i : Fin (n + 1), (-1) ^ (i : ℕ) • f (v i) (removeNth i v)\n```\n\nGiven an alternating map `f` of `n + 1` arguments,\neach term in the sum above written for `f.curryLeft` equals the original map,\nthus `f.curryLeft.alternatizeUncurryFin = (n + 1) • f`.\n\nWe do not multiply the result of `alternatizeUncurryFin` by `(n + 1)⁻¹`\nso that the construction works for `R`-multilinear maps over any commutative ring `R`,\nnot only a field of characteristic zero.\n\n## Main results\n\n- `AlternatingMap.alternatizeUncurryFin_curryLeft`:\n  the round-trip formula for currying/uncurrying, see above.\n\n- `AlternatingMap.alternatizeUncurryFin_alternatizeUncurryFinLM_comp_of_symmetric`:\n  If `f` is a symmetric bilinear map taking values in the space of alternating maps,\n  then the twice uncurried `f` is zero.\n\nA version of the latter theorem for continuous alternating maps\nwill be used to prove that the second exterior derivative of a differential form is zero.\n-/\n\n@[expose] public section\n\nopen Fin Function\n\nnamespace AlternatingMap\n\nvariable {R : Type*} {M M₂ N N₂ : Type*} [CommRing R] [AddCommGroup M]\n  [AddCommGroup M₂] [AddCommGroup N] [AddCommGroup N₂] [Module R M] [Module R M₂]\n  [Module R N] [Module R N₂] {n : ℕ}\n\n/-- If `f` is a `(n + 1)`-multilinear alternating map, `x` is an element of the domain,\nand `v` is an `n`-vector, then the value of `f` at `v` with `x` inserted at the `p`th place\nequals `(-1) ^ p` times the value of `f` at `v` with `x` prepended. -/\ntheorem map_insertNth (f : M [⋀^Fin (n + 1)]→ₗ[R] N) (p : Fin (n + 1)) (x : M) (v : Fin n → M) :\n    f (p.insertNth x v) = (-1) ^ (p : ℕ) • f (Matrix.vecCons x v) := by\n  rw [← cons_comp_cycleRange, map_perm, Matrix.vecCons]\n  simp [Units.smul_def]\n\ntheorem neg_one_pow_smul_map_insertNth (f : M [⋀^Fin (n + 1)]→ₗ[R] N) (p : Fin (n + 1)) (x : M)\n    (v : Fin n → M) :\n    (-1) ^ (p : ℕ) • f (p.insertNth x v) = f (Matrix.vecCons x v) := by\n  rw [map_insertNth, smul_smul, ← pow_add, Even.neg_one_pow, one_smul]\n  use p\n\n/-- Let `v` be an `(n + 1)`-tuple with two equal elements `v i = v j`, `i ≠ j`.\nLet `w i` (resp., `w j`) be the vector `v` with `i`th (resp., `j`th) element removed.\nThen `(-1) ^ i • f (w i) + (-1) ^ j • f (w j) = 0`.\nThis follows from the fact that these two vectors differ by a permutation of sign `(-1) ^ (i + j)`.\n\nThese are the only two nonzero terms in the proof of `map_eq_zero_of_eq`\nin the definition of `alternatizeUncurryFin` below. -/\ntheorem neg_one_pow_smul_map_removeNth_add_eq_zero_of_eq (f : M [⋀^Fin n]→ₗ[R] N)\n    {v : Fin (n + 1) → M} {i j : Fin (n + 1)} (hvij : v i = v j) (hij : i ≠ j) :\n    (-1) ^ (i : ℕ) • f (i.removeNth v) + (-1) ^ (j : ℕ) • f (j.removeNth v) = 0 := by\n  rcases exists_succAbove_eq hij with ⟨i, rfl⟩\n  obtain ⟨m, rfl⟩ : ∃ m, m + 1 = n := by simp [i.pos]\n  rw [← (i.predAbove j).insertNth_self_removeNth (removeNth _ _), ← removeNth_removeNth_eq_swap,\n    removeNth, succAbove_succAbove_predAbove, map_insertNth, ← neg_one_pow_smul_map_insertNth,\n    insertNth_removeNth, update_eq_self_iff.2, smul_smul, ← pow_add,\n    neg_one_pow_succAbove_add_predAbove, neg_smul, pow_add, mul_smul,\n    smul_smul (_ ^ i.val), ← sq, ← pow_mul, pow_mul', neg_one_pow_two, one_pow, one_smul,\n    neg_add_cancel]\n  exact hvij.symm\n\n/-- Given a function which is linear in the first argument\nand is alternating in the other `n` arguments,\nbuild an alternating form in `n + 1` arguments.\n\nThe function is given by\n```\nalternatizeUncurryFin f v = ∑ i : Fin (n + 1), (-1) ^ (i : ℕ) • f (v i) (removeNth i v)\n```\n\nNote that the round-trip with `curryFin` multiplies the form by `n + 1`,\nsince we want to avoid division in this definition. -/\ndef alternatizeUncurryFin (f : M →ₗ[R] M [⋀^Fin n]→ₗ[R] N) :\n    M [⋀^Fin (n + 1)]→ₗ[R] N where\n  toMultilinearMap :=\n    ∑ p : Fin (n + 1), (-1) ^ (p : ℕ) • LinearMap.uncurryMid p (toMultilinearMapLM ∘ₗ f)\n  map_eq_zero_of_eq' := by\n    intro v i j hvij hij\n    suffices ∑ k : Fin (n + 1), (-1) ^ (k : ℕ) • f (v k) (k.removeNth v) = 0 by simpa\n    calc\n      _ = (-1) ^ (i : ℕ) • f (v i) (i.removeNth v) + (-1) ^ (j : ℕ) • f (v j) (j.removeNth v) := by\n        refine Fintype.sum_eq_add _ _ hij fun k ⟨hki, hkj⟩ ↦ ?_\n        rcases exists_succAbove_eq hki.symm with ⟨i, rfl⟩\n        rcases exists_succAbove_eq hkj.symm with ⟨j, rfl⟩\n        rw [(f (v k)).map_eq_zero_of_eq _ hvij (ne_of_apply_ne _ hij), smul_zero]\n      _ = 0 := by\n        rw [hvij, neg_one_pow_smul_map_removeNth_add_eq_zero_of_eq] <;> assumption\n\ntheorem alternatizeUncurryFin_apply (f : M →ₗ[R] M [⋀^Fin n]→ₗ[R] N) (v : Fin (n + 1) → M) :\n    alternatizeUncurryFin f v = ∑ i : Fin (n + 1), (-1) ^ (i : ℕ) • f (v i) (removeNth i v) := by\n  simp [alternatizeUncurryFin]\n\n@[simp]\ntheorem alternatizeUncurryFin_add (f g : M →ₗ[R] M [⋀^Fin n]→ₗ[R] N) :\n    alternatizeUncurryFin (f + g) = alternatizeUncurryFin f + alternatizeUncurryFin g := by\n  ext\n  simp [alternatizeUncurryFin_apply, Finset.sum_add_distrib]\n\n@[simp]\n\nTarget:\nlemma alternatizeUncurryFin_curryLeft (f : M [⋀^Fin (n + 1)]→ₗ[R] N) :\n    alternatizeUncurryFin (curryLeft f) = (n + 1) • f :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Alternating","family_id":"alternatizeuncurryfin_curryleft","file_id":"mathlib/Mathlib/LinearAlgebra/Alternating/Uncurry/Fin.lean","sample_id":"c860402497f1b4dc3bdc93021a1e3e3c1ea1267241369ebbfdacfe5570fa2d4c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"74e6e717b4029a5bb695646d264a47c3c431da50efbba97f42340e634e0683b0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a15d943c4f0d40defc381407b0abb8cc658447fdf0056795f4405219ceae7f15","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7736e8daaa8f61cdcc7d061cb79e4a68846e752fbc7a81c150325816b7000c0c","source_sha256":"88899366de27f8f7e91c2113d9cd607e3b4b1f4debab3b6b3a2046a9e9f7c31f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction hs using Nonempty.cons_induction with\n  | singleton => simp [Hlt _ (mem_singleton_self _)]\n  | cons _ _ _ _ ih =>\n    simp only [sum_cons, forall_mem_cons] at Hlt ⊢\n    exact ENat.add_lt_add Hlt.1 (ih Hlt.2)","hard_negative":true,"metrics":{"chosen_tokens":51,"rejected_tokens":2,"token_jaccard":0.030303,"token_length_ratio":0.039216},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"6f6f8ec55d38042bdf3b4e1f651d62d1db40bd83d50a642fce317a22f72546ce","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Ring.Finset\npublic import Mathlib.Algebra.BigOperators.WithTop\npublic import Mathlib.Algebra.Order.BigOperators.Group.Finset\npublic import Mathlib.Data.ENat.Lattice\n\nNamespace:\nENat\n\nLocal context:\n/-\nCopyright (c) 2024 Joachim Breitner, Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joachim Breitner, Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Sum of suprema in `ENat`\n-/\n\npublic section\n\nassert_not_exists Field\n\nnamespace ENat\n\nvariable {a b c d : ℕ∞} {r p q : ℕ}\n\nsection OperationsAndInfty\n\nvariable {α : Type*}\n\n@[simp]\ntheorem toNat_prod {ι : Type*} {s : Finset ι} {f : ι → ℕ∞} :\n    (∏ i ∈ s, f i).toNat = ∏ i ∈ s, (f i).toNat :=\n  map_prod toNatHom _ _\n\ntheorem iInf_sum {ι α : Type*} {f : ι → α → ℕ∞} {s : Finset α} [Nonempty ι]\n    (h : ∀ (t : Finset α) (i j : ι), ∃ k, ∀ a ∈ t, f k a ≤ f i a ∧ f k a ≤ f j a) :\n    ⨅ i, ∑ a ∈ s, f i a = ∑ a ∈ s, ⨅ i, f i a := by\n  induction s using Finset.cons_induction_on with\n  | empty => simp only [Finset.sum_empty, ciInf_const]\n  | cons a s ha ih =>\n    simp only [Finset.sum_cons, ← ih]\n    refine (iInf_add_iInf fun i j => ?_).symm\n    refine (h (Finset.cons a s ha) i j).imp fun k hk => ?_\n    rw [Finset.forall_mem_cons] at hk\n    exact add_le_add hk.1.1 (Finset.sum_le_sum fun a ha => (hk.2 a ha).2)\n\nend OperationsAndInfty\n\nsection Sum\n\nopen Finset\n\nvariable {α : Type*} {s : Finset α} {f : α → ℕ∞}\n\n/-- A product of finite numbers is still finite. -/\nlemma prod_ne_top (h : ∀ a ∈ s, f a ≠ ⊤) : ∏ a ∈ s, f a ≠ ⊤ := WithTop.prod_ne_top h\n\n/-- A product of finite numbers is still finite. -/\nlemma prod_lt_top (h : ∀ a ∈ s, f a < ⊤) : ∏ a ∈ s, f a < ⊤ := WithTop.prod_lt_top h\n\n/-- A sum is infinite iff one of the summands is infinite. -/\n@[simp] lemma sum_eq_top : ∑ x ∈ s, f x = ⊤ ↔ ∃ a ∈ s, f a = ⊤ := WithTop.sum_eq_top\n\n/-- A sum is finite iff all summands are finite. -/\nlemma sum_ne_top : ∑ a ∈ s, f a ≠ ⊤ ↔ ∀ a ∈ s, f a ≠ ⊤ := WithTop.sum_ne_top\n\n/-- A sum is finite iff all summands are finite. -/\n@[simp] lemma sum_lt_top : ∑ a ∈ s, f a < ⊤ ↔ ∀ a ∈ s, f a < ⊤ := WithTop.sum_lt_top\n\ntheorem lt_top_of_sum_ne_top {s : Finset α} {f : α → ℕ∞} (h : ∑ x ∈ s, f x ≠ ⊤) {a : α}\n    (ha : a ∈ s) : f a < ⊤ :=\n  sum_lt_top.1 h.lt_top a ha\n\n/-- Seeing `ℕ∞` as `ℕ` does not change their sum, unless one of the `ℕ∞` is\ninfinity -/\ntheorem toNat_sum {s : Finset α} {f : α → ℕ∞} (hf : ∀ a ∈ s, f a ≠ ⊤) :\n    ENat.toNat (∑ a ∈ s, f a) = ∑ a ∈ s, ENat.toNat (f a) := by\n  rw [← coe_inj, coe_toNat (sum_ne_top.2 hf), Nat.cast_sum]\n  exact sum_congr rfl fun x hx => (coe_toNat (hf x hx)).symm\n\nTarget:\ntheorem sum_lt_sum_of_nonempty {s : Finset α} (hs : s.Nonempty) {f g : α → ℕ∞}\n    (Hlt : ∀ i ∈ s, f i < g i) : ∑ i ∈ s, f i < ∑ i ∈ s, g i :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_7736e8daaa8f","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"18bbe7497667610435b98cd62fe97151305f047eb40be51b061886d780052244","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/ENat","family_id":"sum_lt_sum_of_nonempty","file_id":"mathlib/Mathlib/Data/ENat/BigOperators.lean","sample_id":"7736e8daaa8f61cdcc7d061cb79e4a68846e752fbc7a81c150325816b7000c0c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"19d0436be488b2c04681a3df70e828f641be531fc0e4601e74671ee295a63fea","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"61f6ec391c59827903fc83f292d6ac851125eff6c87e0601e45a860b03f38ad5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"08fc9f4a5a976873fc6fe96edc2ccf033e6ac6f6554e9759e87483072bcf261d","source_sha256":"d5c87fac893fcf2654818fe7d90a93a609b65e58a2bb593393eb189396ee74f1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rwa [H.ker_flip]","hard_negative":false,"metrics":{"chosen_tokens":7,"rejected_tokens":11,"token_jaccard":0.7,"token_length_ratio":1.571429},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"6f972500216fbf0815cf77e393bf26fd1f42532696f74defa0acb35d57a24fb4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Basis.Basic\npublic import Mathlib.LinearAlgebra.BilinearMap\npublic import Mathlib.LinearAlgebra.LinearIndependent.Lemmas\nimport Mathlib.Algebra.Module.Torsion.Field\n\nNamespace:\nLinearMap.IsRefl\n\nLocal context:\n/-\nCopyright (c) 2018 Andreas Swerdlow. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andreas Swerdlow\n-/\n/-!\n# Sesquilinear maps\n\nThis file provides properties about sesquilinear maps and forms. The maps considered are of the\nform `M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M`, where `I₁ : R₁ →+* R` and `I₂ : R₂ →+* R` are ring homomorphisms and\n`M₁` is a module over `R₁`, `M₂` is a module over `R₂` and `M` is a module over `R`.\nSesquilinear forms are the special case that `M₁ = M₂`, `M = R₁ = R₂ = R`, and `I₁ = RingHom.id R`.\nTaking additionally `I₂ = RingHom.id R`, then one obtains bilinear forms.\n\nSesquilinear maps are a special case of the bilinear maps defined in `BilinearMap.lean`, and many\nbasic lemmas about construction and elementary calculations are found there.\n\n## Main declarations\n\n* `IsSymm`, `IsAlt`: states that a sesquilinear form is symmetric and alternating, respectively\n* `orthogonalBilin` provides the orthogonal complement with respect to a sesquilinear map\n\n## References\n\n* <https://en.wikipedia.org/wiki/Sesquilinear_form#Over_arbitrary_rings>\n\n## Tags\n\nSesquilinear form, Sesquilinear map\n-/\n\n@[expose] public section\n\nopen Module\n\nvariable {R R₁ R₂ R₃ M M₁ M₂ M₃ Mₗ₁ Mₗ₁' Mₗ₂ Mₗ₂' K K₁ K₂ V V₁ V₂ n : Type*}\n\nnamespace LinearMap\n\n/-! ### Orthogonal vectors -/\n\n\nsection CommRing\n\n-- the `ₗ` subscript variables are for special cases about linear (as opposed to semilinear) maps\nvariable [CommSemiring R] [CommSemiring R₁] [AddCommMonoid M₁] [Module R₁ M₁] [CommSemiring R₂]\n  [AddCommMonoid M₂] [Module R₂ M₂] [AddCommMonoid M] [Module R M]\n  {I₁ : R₁ →+* R} {I₂ : R₂ →+* R} {I₁' : R₁ →+* R}\n\n/-- The proposition that two elements of a sesquilinear map space are orthogonal -/\n@[deprecated \"Use `B x y = 0`\" (since := \"2026-03-30\")]\ndef IsOrtho (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x : M₁) (y : M₂) : Prop :=\n  B x y = 0\n\nset_option linter.deprecated false in\n@[deprecated \"`LinearMap.IsOrtho` has been deprecated\" (since := \"2026-03-30\")]\ntheorem isOrtho_def {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M} {x y} : B.IsOrtho x y ↔ B x y = 0 :=\n  Iff.rfl\n\nset_option linter.deprecated false in\n@[deprecated \"`LinearMap.IsOrtho` has been deprecated\" (since := \"2026-03-30\")]\ntheorem isOrtho_zero_left (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x) : IsOrtho B (0 : M₁) x := by\n  dsimp only [IsOrtho]\n  rw [map_zero B, zero_apply]\n\nset_option linter.deprecated false in\n@[deprecated \"`LinearMap.IsOrtho` has been deprecated\" (since := \"2026-03-30\")]\ntheorem isOrtho_zero_right (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x) : IsOrtho B x (0 : M₂) :=\n  map_zero (B x)\n\nset_option linter.deprecated false in\n@[deprecated \"`LinearMap.IsOrtho` has been deprecated\" (since := \"2026-03-30\")]\ntheorem isOrtho_flip {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M} {x y} : B.IsOrtho x y ↔ B.flip.IsOrtho y x := by\n  simp_rw [isOrtho_def, flip_apply]\n\nopen scoped Function in -- required for scoped `on` notation\n/-- A set of vectors `v` is orthogonal with respect to some bilinear map `B` if and only\nif for all `i ≠ j`, `B (v i) (v j) = 0`. -/\ndef IsOrthoᵢ (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M) (v : n → M₁) : Prop :=\n  Pairwise ((fun n m => B n m = 0) on v)\n\ntheorem isOrthoᵢ_def {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M} {v : n → M₁} :\n    B.IsOrthoᵢ v ↔ ∀ i j : n, i ≠ j → B (v i) (v j) = 0 :=\n  Iff.rfl\n\ntheorem isOrthoᵢ_flip (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M) {v : n → M₁} :\n    B.IsOrthoᵢ v ↔ B.flip.IsOrthoᵢ v := by\n  simp_rw [isOrthoᵢ_def]\n  constructor <;> exact fun h i j hij ↦ h j i hij.symm\n\nend CommRing\n\nsection Field\n\nvariable [Field K] [AddCommGroup V] [Module K V] [Field K₁] [AddCommGroup V₁] [Module K₁ V₁]\n  [Field K₂] [AddCommGroup V₂] [Module K₂ V₂]\n  {I₁ : K₁ →+* K} {I₂ : K₂ →+* K} {I₁' : K₁ →+* K} {J₁ : K →+* K} {J₂ : K →+* K}\n\nset_option linter.deprecated false in\n@[deprecated \"`LinearMap.IsOrtho` has been deprecated\" (since := \"2026-03-30\")]\ntheorem ortho_smul_left {B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V} {x y} {a : K₁} (ha : a ≠ 0) :\n    IsOrtho B x y ↔ IsOrtho B (a • x) y := by\n  dsimp only [IsOrtho]\n  constructor <;> intro H\n  · rw [map_smulₛₗ₂, H, smul_zero]\n  · rw [map_smulₛₗ₂, smul_eq_zero] at H\n    rcases H with H | H\n    · rw [map_eq_zero I₁] at H\n      trivial\n    · exact H\n\nset_option linter.deprecated false in\n@[deprecated \"`LinearMap.IsOrtho` has been deprecated\" (since := \"2026-03-30\")]\ntheorem ortho_smul_right {B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V} {x y} {a : K₂} {ha : a ≠ 0} :\n    IsOrtho B x y ↔ IsOrtho B x (a • y) := by\n  simp_all [IsOrtho]\n\n/-- A set of orthogonal vectors `v` with respect to some sesquilinear map `B` is linearly\n  independent if for all `i`, `B (v i) (v i) ≠ 0`. -/\ntheorem linearIndependent_of_isOrthoᵢ {B : V₁ →ₛₗ[I₁] V₁ →ₛₗ[I₁'] V} {v : n → V₁}\n    (hv₁ : B.IsOrthoᵢ v) (hv₂ : ∀ i, B (v i) (v i) ≠ 0) : LinearIndependent K₁ v := by\n  classical\n    rw [linearIndependent_iff']\n    intro s w hs i hi\n    have : B (s.sum fun i : n ↦ w i • v i) (v i) = 0 := by rw [hs, map_zero, zero_apply]\n    have hsum : (s.sum fun j : n ↦ I₁ (w j) • B (v j) (v i)) = I₁ (w i) • B (v i) (v i) := by\n      apply Finset.sum_eq_single_of_mem i hi\n      intro j _hj hij\n      rw [isOrthoᵢ_def.1 hv₁ _ _ hij, smul_zero]\n    simp_rw [B.map_sum₂, map_smulₛₗ₂, hsum] at this\n    apply (map_eq_zero I₁).mp\n    exact (smul_eq_zero.mp this).elim _root_.id (hv₂ i · |>.elim)\n\nend Field\n\n/-! ### Reflexive bilinear maps -/\n\nsection Reflexive\n\nvariable [CommSemiring R] [AddCommMonoid M] [Module R M] [CommSemiring R₁] [AddCommMonoid M₁]\n  [Module R₁ M₁] {I₁ : R₁ →+* R} {I₂ : R₁ →+* R} {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}\n\n/-- The proposition that a sesquilinear map is reflexive -/\ndef IsRefl (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M) : Prop :=\n  ∀ x y, B x y = 0 → B y x = 0\n\nnamespace IsRefl\n\nsection\nvariable (H : B.IsRefl)\ninclude H\n\ntheorem eq_zero : ∀ {x y}, B x y = 0 → B y x = 0 := fun {x y} ↦ H x y\n\ntheorem eq_iff {x y} : B x y = 0 ↔ B y x = 0 := ⟨H x y, H y x⟩\n\n@[deprecated (since := \"2026-03-30\")]\nalias ortho_comm := eq_iff\n\ntheorem domRestrict (p : Submodule R₁ M₁) : (B.domRestrict₁₂ p p).IsRefl :=\n  fun _ _ ↦ by\n  simp_rw [domRestrict₁₂_apply]\n  exact H _ _\nend\n\n@[simp]\ntheorem flip_isRefl_iff : B.flip.IsRefl ↔ B.IsRefl :=\n  forall_comm\n\nlemma ker_flip (H : B.IsRefl) : B.flip.ker = B.ker := by\n  ext x\n  simp [LinearMap.ext_iff, H.eq_iff]\n\nTarget:\ntheorem ker_flip_eq_bot (H : B.IsRefl) (h : LinearMap.ker B = ⊥) : LinearMap.ker B.flip = ⊥ :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rwa [H.ker_flip]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/SesquilinearForm","family_id":"ker_flip_eq_bot","file_id":"mathlib/Mathlib/LinearAlgebra/SesquilinearForm/Basic.lean","sample_id":"08fc9f4a5a976873fc6fe96edc2ccf033e6ac6f6554e9759e87483072bcf261d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fecffccc9af025141c64f561cc932cc9a7f6f3a1d76dd0660a012426d4c05839","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e4758bbc670928f41cb8dee16caeed3cacc5d66c566bf12ad5bf6fe21c451b08","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c3604586f2ac7368d223dc67d1199e013bd8135ef158fafca76c177f59a92d99","source_sha256":"00cdc131b09c868f94179716d61273d8fdf897ed023d34f30c7085668a3b31ac","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold xgcdAux\n  exact if_pos rfl","hard_negative":false,"metrics":{"chosen_tokens":6,"rejected_tokens":11,"token_jaccard":0.545455,"token_length_ratio":1.833333},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"6fa57c5f95bea5a1821df9c84c74977a6a7bf530eba351a423f25381c11f9773","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Order.RelClasses\n\nNamespace:\nEuclideanDomain\n\nLocal context:\n/-\nCopyright (c) 2018 Louis Carlin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Louis Carlin, Mario Carneiro\n-/\n/-!\n# Euclidean domains\n\nThis file introduces Euclidean domains and provides the extended Euclidean algorithm. To be precise,\na slightly more general version is provided which is sometimes called a transfinite Euclidean domain\nand differs in the fact that the degree function need not take values in `ℕ` but can take values in\nany well-ordered set. Transfinite Euclidean domains were introduced by Motzkin and examples which\ndon't satisfy the classical notion were provided independently by Hiblot and Nagata.\n\n## Main definitions\n\n* `EuclideanDomain`: Defines Euclidean domain with functions `quotient` and `remainder`. Instances\n  of `Div` and `Mod` are provided, so that one can write `a = b * (a / b) + a % b`.\n* `gcd`: defines the greatest common divisors of two elements of a Euclidean domain.\n* `xgcd`: given two elements `a b : R`, `xgcd a b` defines the pair `(x, y)` such that\n  `x * a + y * b = gcd a b`.\n* `lcm`: defines the lowest common multiple of two elements `a` and `b` of a Euclidean domain as\n  `a * b / (gcd a b)`\n\n## Main statements\n\nSee `Algebra.EuclideanDomain.Basic` for most of the theorems about Euclidean domains,\nincluding Bézout's lemma.\n\nSee `Algebra.EuclideanDomain.Instances` for the fact that `ℤ` is a Euclidean domain,\nas is any field.\n\n## Notation\n\n`≺` denotes the well-founded relation on the Euclidean domain, e.g. in the example of the polynomial\nring over a field, `p ≺ q` for polynomials `p` and `q` if and only if the degree of `p` is less than\nthe degree of `q`.\n\n## Implementation details\n\nInstead of working with a valuation, `EuclideanDomain` is implemented with the existence of a well\nfounded relation `r` on the integral domain `R`, which in the example of `ℤ` would correspond to\nsetting `i ≺ j` for integers `i` and `j` if the absolute value of `i` is smaller than the absolute\nvalue of `j`.\n\n## References\n\n* [Th. Motzkin, *The Euclidean algorithm*][MR32592]\n* [J.-J. Hiblot, *Des anneaux euclidiens dont le plus petit algorithme n'est pas à valeurs finies*]\n  [MR399081]\n* [M. Nagata, *On Euclid algorithm*][MR541021]\n\n\n## Tags\n\nEuclidean domain, transfinite Euclidean domain, Bézout's lemma\n-/\n\n@[expose] public section\n\nuniverse u\n\n/-- A `EuclideanDomain` is a non-trivial commutative ring with a division and a remainder,\n  satisfying `b * (a / b) + a % b = a`.\n  The definition of a Euclidean domain usually includes a valuation function `R → ℕ`.\n  This definition is slightly generalised to include a well-founded relation\n  `r` with the property that `r (a % b) b`, instead of a valuation. -/\n@[wikidata Q867345]\nclass EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R where\n  /-- A division function (denoted `/`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `%` denotes `remainder`. -/\n  protected quotient : R → R → R\n  /-- Division by zero should always give zero by convention. -/\n  protected quotient_zero : ∀ a, quotient a 0 = 0\n  /-- A remainder function (denoted `%`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `/` denotes `quotient`. -/\n  protected remainder : R → R → R\n  /-- The property that links the quotient and remainder functions.\n    This allows us to compute GCDs and LCMs. -/\n  protected quotient_mul_add_remainder_eq : ∀ a b, b * quotient a b + remainder a b = a\n  /-- A well-founded relation on `R`, satisfying `r (a % b) b`.\n    This ensures that the GCD algorithm always terminates. -/\n  protected r : R → R → Prop\n  /-- The relation `r` must be well-founded.\n    This ensures that the GCD algorithm always terminates. -/\n  r_wellFounded : WellFounded r\n  /-- The relation `r` satisfies `r (a % b) b`. -/\n  protected remainder_lt : ∀ (a) {b}, b ≠ 0 → r (remainder a b) b\n  /-- An additional constraint on `r`. -/\n  mul_left_not_lt : ∀ (a) {b}, b ≠ 0 → ¬r (a * b) a\n\n/-\nLean has far more theorems about fields than about Euclidean domains. We thus\nlower the priority of `Euclideandomain.toCommRing`, encouraging typeclass inference\nto try `Field.toCommRing` first. Without this priority-lowering, typeclass inference\nfinds the more inefficient path `Field.toEuclideanDomain.toCommRing` by default. This\npriority change saves over 500G instructions across mathlib. See\nhttps://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/We.20need.20to.20talk.20about.20Euclidean.20Domains/near/594655420\n-/\n-- see Note [lower instance priority]\nattribute [instance 100] EuclideanDomain.toCommRing\n\nnamespace EuclideanDomain\n\nvariable {R : Type u} [EuclideanDomain R]\n\n/-- Abbreviated notation for the well-founded relation `r` in a Euclidean domain. -/\nlocal infixl:50 \" ≺ \" => EuclideanDomain.r\n\nlocal instance wellFoundedRelation : WellFoundedRelation R where\n  rel := EuclideanDomain.r\n  wf := r_wellFounded\n\ninstance isWellFounded : IsWellFounded R (· ≺ ·) where\n  wf := r_wellFounded\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Div R :=\n  ⟨EuclideanDomain.quotient⟩\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Mod R :=\n  ⟨EuclideanDomain.remainder⟩\n\ntheorem div_add_mod (a b : R) : b * (a / b) + a % b = a :=\n  EuclideanDomain.quotient_mul_add_remainder_eq _ _\n\ntheorem mod_add_div (a b : R) : a % b + b * (a / b) = a :=\n  (add_comm _ _).trans (div_add_mod _ _)\n\ntheorem mod_add_div' (m k : R) : m % k + m / k * k = m := by\n  rw [mul_comm]\n  exact mod_add_div _ _\n\ntheorem div_add_mod' (m k : R) : m / k * k + m % k = m := by\n  rw [mul_comm]\n  exact div_add_mod _ _\n\ntheorem mod_lt : ∀ (a) {b : R}, b ≠ 0 → a % b ≺ b :=\n  EuclideanDomain.remainder_lt\n\ntheorem mul_right_not_lt {a : R} (b) (h : a ≠ 0) : ¬a * b ≺ b := by\n  rw [mul_comm]\n  exact mul_left_not_lt b h\n\n@[simp]\ntheorem mod_zero (a : R) : a % 0 = a := by simpa only [zero_mul, zero_add] using div_add_mod a 0\n\ntheorem lt_one (a : R) : a ≺ (1 : R) → a = 0 :=\n  haveI := Classical.dec\n  not_imp_not.1 fun h => by simpa only [one_mul] using mul_left_not_lt 1 h\n\n@[simp]\ntheorem div_zero (a : R) : a / 0 = 0 :=\n  EuclideanDomain.quotient_zero a\n\nsection\n\n@[elab_as_elim]\ntheorem GCD.induction {P : R → R → Prop} (a b : R) (H0 : ∀ x, P 0 x)\n    (H1 : ∀ a b, a ≠ 0 → P (b % a) a → P a b) : P a b := by\n  classical\n  exact if a0 : a = 0 then\n    a0.symm ▸ H0 b\n  else\n    have _ := mod_lt b a0\n    H1 _ _ a0 (GCD.induction (b % a) a H0 H1)\ntermination_by a\n\nend\n\nsection GCD\n\nvariable [DecidableEq R]\n\n/-- `gcd a b` is a (non-unique) element such that `gcd a b ∣ a` `gcd a b ∣ b`, and for\n  any element `c` such that `c ∣ a` and `c ∣ b`, then `c ∣ gcd a b` -/\ndef gcd (a b : R) : R :=\n  if a0 : a = 0 then b\n  else\n    have _ := mod_lt b a0\n    gcd (b % a) a\ntermination_by a\n\n@[simp]\ntheorem gcd_zero_left (a : R) : gcd 0 a = a := by\n  rw [gcd]\n  exact if_pos rfl\n\n/-- An implementation of the extended GCD algorithm.\nAt each step we are computing a triple `(r, s, t)`, where `r` is the next value of the GCD\nalgorithm, to compute the greatest common divisor of the input (say `x` and `y`), and `s` and `t`\nare the coefficients in front of `x` and `y` to obtain `r` (i.e. `r = s * x + t * y`).\nThe function `xgcdAux` takes in two triples, and from these recursively computes the next triple:\n```\nxgcdAux (r, s, t) (r', s', t') = xgcdAux (r' % r, s' - (r' / r) * s, t' - (r' / r) * t) (r, s, t)\n```\n-/\ndef xgcdAux (r s t r' s' t' : R) : R × R × R :=\n  if _hr : r = 0 then (r', s', t')\n  else\n    let q := r' / r\n    have _ := mod_lt r' _hr\n    xgcdAux (r' % r) (s' - q * s) (t' - q * t) r s t\ntermination_by r\n\n@[simp]\n\nTarget:\ntheorem xgcd_zero_left {s t r' s' t' : R} : xgcdAux 0 s t r' s' t' = (r', s', t') :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  unfold xgcdAux\n  exact if_pos rfl","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/EuclideanDomain","family_id":"xgcd_zero_left","file_id":"mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean","sample_id":"c3604586f2ac7368d223dc67d1199e013bd8135ef158fafca76c177f59a92d99"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a18d962dbf46cb1bc44d7afa74d89d68066f49e5784c2cb2792cd31f2ea27cd6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c4e84a42ec28d598074201ae4fc19260b87334cb9033f8c5a714b5febc8bfa62","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5b6616e952970d213dc2af95acd62437ab9f4ee924f0692e7518320229a74eaa","source_sha256":"f440a69580513b2b91fa2379cabfe0bde1204ddec625489d040b307defdad141","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [H.apply_zero_path, H.apply_one_path, H.evalAt_eq]\n  erw [H.evalAt_eq]\n  dsimp only [prodToProdTopI]\n  constructor\n  · slice_lhs 2 4 => rw [eqToHom_trans, eqToHom_refl] -- Porting note: this ↓ `simp` didn't do this\n    slice_lhs 2 4 => simp [← CategoryTheory.Functor.map_comp]\n    rfl\n  · slice_lhs 2 4 => rw [eqToHom_trans, eqToHom_refl] -- Porting note: this ↓ `simp` didn't do this\n    slice_lhs 2 4 => simp [← CategoryTheory.Functor.map_comp]\n    rfl","hard_negative":false,"metrics":{"chosen_tokens":103,"rejected_tokens":107,"token_jaccard":0.925,"token_length_ratio":1.038835},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"6faf74c3d717469da6af4b5b5a725c7cba7e6e3b58ed15f98e6e3b270dd9bb91","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Homotopy.Equiv\npublic import Mathlib.CategoryTheory.Equivalence\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.Product\n\nNamespace:\nContinuousMap.Homotopy\n\nLocal context:\n/-\nCopyright (c) 2022 Praneeth Kolichala. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Praneeth Kolichala, Yury Kudryashov\n-/\n/-!\n# Homotopic maps induce naturally isomorphic functors\n\n## Main definitions\n\n- `FundamentalGroupoidFunctor.homotopicMapsNatIso H` The natural isomorphism\n  between the induced functors `f : π(X) ⥤ π(Y)` and `g : π(X) ⥤ π(Y)`, given a homotopy\n  `H : f ∼ g`\n\n- `FundamentalGroupoidFunctor.equivOfHomotopyEquiv hequiv` The equivalence of the categories\n  `π(X)` and `π(Y)` given a homotopy equivalence `hequiv : X ≃ₕ Y` between them.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nuniverse u v\n\nopen FundamentalGroupoid CategoryTheory FundamentalGroupoidFunctor\nopen scoped FundamentalGroupoid unitInterval\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Let `F` be a homotopy between two continuous maps `f g : C(X, Y)`.\nGiven a path `p : Path x₁ x₂` in the domain, consider the following two paths in the codomain.\nOne path goes along the image of `p` under `f`, then along the trajectory of `x₂` under `F`.\nThe other path goes along the trajectory of `x₁` under `F`, then along the image of `p` under `g`.\n\nThese two paths are homotopic. -/\ntheorem Path.Homotopic.map_trans_evalAt {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n    {f g : C(X, Y)} (F : f.Homotopy g) {x₁ x₂ : X} (p : Path x₁ x₂) :\n    ((p.map (map_continuous f)).trans (F.evalAt x₂)).Homotopic\n      ((F.evalAt x₁).trans (p.map (map_continuous g))) := by\n  /- Let `G` be the continuous map on the unit square sending `(t, s)` to `F(t, p(s))`.\n  Then our homotopy is the image under `G` of a homotopy\n  between the two paths from `(0, 0)` to `(1, 1)` along the sides of the square. -/\n  set G : C(I × I, Y) := F.toContinuousMap.comp (.prodMap (.id _) p)\n  set p₁ : Path ((0, 0) : I × I) (1, 1) := .prod (.trans (.refl _) .id) (.trans .id (.refl _))\n  set p₂ : Path ((0, 0) : I × I) (1, 1) := .prod (.trans .id (.refl _)) (.trans (.refl _) .id)\n  set Fsq : p₁.Homotopy p₂ :=\n    Path.Homotopic.prodHomotopy (.trans (.reflTrans _) (.symm <| .transRefl _))\n      (.trans (.transRefl _) (.symm <| .reflTrans _))\n  refine ⟨((Fsq.map G).pathCast ?H0 ?H1).cast ?hp ?hq⟩\n  all_goals aesop (add simp Path.trans_apply)\n\nnamespace FundamentalGroupoidFunctor\n\nopen CategoryTheory\nopen scoped FundamentalGroupoid ContinuousMap\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n  {f g : C(X, Y)}\n\nset_option backward.isDefEq.respectTransparency false in\nset_option pp.proofs.withType true in\n/-- Given a homotopy H : f ∼ g, we have an associated natural isomorphism between the induced\nfunctors `map f` and `map g` on fundamental groupoids. -/\ndef homotopicMapsNatIso (H : ContinuousMap.Homotopy f g) : map f ⟶ map g where\n  app x := ⟦H.evalAt x.as⟧\n  naturality := by\n    rintro ⟨x⟩ ⟨y⟩ p\n    rcases Path.Homotopic.Quotient.mk_surjective p with ⟨p, rfl⟩\n    simp only [map_map, Path.Homotopic.Quotient.mk''_eq_mk, comp_eq,\n      ← Path.Homotopic.Quotient.mk_map, ← Path.Homotopic.Quotient.mk_trans]\n    rw [Path.Homotopic.Quotient.eq]\n    exact .map_trans_evalAt _ _\n\ninstance (H : ContinuousMap.Homotopy f g) : IsIso (homotopicMapsNatIso H) :=\n  NatIso.isIso_of_isIso_app _\n\nopen scoped ContinuousMap\n\n/-- Homotopy equivalent topological spaces have equivalent fundamental groupoids. -/\ndef equivOfHomotopyEquiv {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] (hequiv : X ≃ₕ Y) :\n    πₓ (.of X) ≌ πₓ (.of Y) := by\n  apply CategoryTheory.Equivalence.mk (map hequiv.toFun) (map hequiv.invFun)\n  · simpa only [FundamentalGroupoid.map_id, FundamentalGroupoid.map_comp]\n      using (asIso (homotopicMapsNatIso hequiv.left_inv.some)).symm\n  · simpa only [FundamentalGroupoid.map_id, FundamentalGroupoid.map_comp]\n      using asIso (homotopicMapsNatIso hequiv.right_inv.some)\n\nend FundamentalGroupoidFunctor\n\n/-!\n### Old proof\n\nThe rest of the file contains definitions and theorems required to write the same proof\nin a slightly different manner.\n\nThe proof was rewritten in 2025 for two reasons:\n\n- the new proof is much more straightforward;\n- the new proof is fully universe polymorphic.\n\nTODO: review which of these definitions and theorems are useful for other reasons,\nthen deprecate the rest of them.\n-/\n\nnamespace unitInterval\n\n/-- The path 0 ⟶ 1 in `I` -/\ndef path01 : Path (0 : I) 1 where\n  toFun := id\n  source' := rfl\n  target' := rfl\n\n/-- The path 0 ⟶ 1 in `ULift I` -/\ndef upath01 : Path (ULift.up 0 : ULift.{u} I) (ULift.up 1) where\n  toFun := ULift.up\n  source' := rfl\n  target' := rfl\n\n/-- The homotopy path class of 0 → 1 in `ULift I` -/\ndef uhpath01 : @fromTop (TopCat.of <| ULift.{u} I) (ULift.up (0 : I)) ⟶ fromTop (ULift.up 1) :=\n  ⟦upath01⟧\n\nend unitInterval\n\nnamespace ContinuousMap.Homotopy\n\nopen unitInterval (uhpath01)\n\nsection Casts\n\n/-- Abbreviation for `eqToHom` that accepts points in a topological space -/\nabbrev hcast {X : TopCat.{u}} {x₀ x₁ : X} (hx : x₀ = x₁) : fromTop x₀ ⟶ fromTop x₁ :=\n  eqToHom <| FundamentalGroupoid.ext hx\n\n@[simp]\ntheorem hcast_def {X : TopCat.{u}} {x₀ x₁ : X} (hx₀ : x₀ = x₁) :\n    hcast hx₀ = eqToHom (FundamentalGroupoid.ext hx₀) :=\n  rfl\n\nvariable {X₁ X₂ Y : TopCat.{u}} {f : C(X₁, Y)} {g : C(X₂, Y)} {x₀ x₁ : X₁} {x₂ x₃ : X₂}\n  {p : Path x₀ x₁} {q : Path x₂ x₃} (hfg : ∀ t, f (p t) = g (q t))\ninclude hfg\n\n/-- If `f(p(t) = g(q(t))` for two paths `p` and `q`, then the induced path homotopy classes\n`f(p)` and `g(p)` are the same as well, despite having a priori different types -/\ntheorem heq_path_of_eq_image :\n    (πₘ (TopCat.ofHom f)).map ⟦p⟧ ≍ (πₘ (TopCat.ofHom g)).map ⟦q⟧ := by\n  simp only [map_eq]\n  apply Path.Homotopic.hpath_hext\n  exact hfg\n\nset_option backward.privateInPublic true in\nprivate theorem start_path : f x₀ = g x₂ := by convert! hfg 0 <;> simp only [Path.source]\n\nset_option backward.privateInPublic true in\nprivate theorem end_path : f x₁ = g x₃ := by convert! hfg 1 <;> simp only [Path.target]\n\nset_option backward.isDefEq.respectTransparency false in\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ntheorem eq_path_of_eq_image :\n    (πₘ (TopCat.ofHom f)).map ⟦p⟧ =\n        hcast (start_path hfg) ≫ (πₘ (TopCat.ofHom g)).map ⟦q⟧ ≫ hcast (end_path hfg).symm := by\n  rw [conj_eqToHom_iff_heq\n    ((πₘ (TopCat.ofHom f)).map ⟦p⟧) ((πₘ (TopCat.ofHom g)).map ⟦q⟧)\n    (FundamentalGroupoid.ext <| start_path hfg)\n    (FundamentalGroupoid.ext <| end_path hfg)]\n  exact heq_path_of_eq_image hfg\n\nend Casts\n\n-- We let `X` and `Y` be spaces, and `f` and `g` be homotopic maps between them\nvariable {X Y : TopCat.{u}} {f g : C(X, Y)} (H : ContinuousMap.Homotopy f g) {x₀ x₁ : X}\n  (p : fromTop x₀ ⟶ fromTop x₁)\n\n/-!\nThese definitions set up the following diagram, for each path `p`:\n\n```\n            f(p)\n        *--------*\n        | \\      |\n    H₀  |   \\ d  |  H₁\n        |     \\  |\n        *--------*\n            g(p)\n```\n\nHere, `H₀ = H.evalAt x₀` is the path from `f(x₀)` to `g(x₀)`,\nand similarly for `H₁`. Similarly, `f(p)` denotes the\npath in Y that the induced map `f` takes `p`, and similarly for `g(p)`.\n\nFinally, `d`, the diagonal path, is H(0 ⟶ 1, p), the result of the induced `H` on\n`Path.Homotopic.prod (0 ⟶ 1) p`, where `(0 ⟶ 1)` denotes the path from `0` to `1` in `I`.\n\nIt is clear that the diagram commutes (`H₀ ≫ g(p) = d = f(p) ≫ H₁`), but unfortunately,\nmany of the paths do not have defeq starting/ending points, so we end up needing some casting.\n-/\n\n\n/-- Interpret a homotopy `H : C(I × X, Y)` as a map `C(ULift I × X, Y)` -/\ndef uliftMap : C(TopCat.of (ULift.{u} I × X), Y) :=\n  ⟨fun x => H (x.1.down, x.2),\n    H.continuous.comp ((continuous_uliftDown.comp continuous_fst).prodMk continuous_snd)⟩\n\ntheorem ulift_apply (i : ULift.{u} I) (x : X) : H.uliftMap (i, x) = H (i.down, x) :=\n  rfl\n\n/-- An abbreviation for `prodToProdTop`, with some types already in place to help the\ntypechecker. In particular, the first path should be on the ulifted unit interval. -/\nabbrev prodToProdTopI {a₁ a₂ : TopCat.of (ULift I)} {b₁ b₂ : X} (p₁ : fromTop a₁ ⟶ fromTop a₂)\n    (p₂ : fromTop b₁ ⟶ fromTop b₂) :=\n  (prodToProdTop (TopCat.of <| ULift I) X).map (X := (⟨a₁⟩, ⟨b₁⟩)) (Y := (⟨a₂⟩, ⟨b₂⟩)) (p₁, p₂)\n\n/-- The diagonal path `d` of a homotopy `H` on a path `p` -/\ndef diagonalPath : fromTop (H (0, x₀)) ⟶ fromTop (H (1, x₁)) :=\n  (πₘ (TopCat.ofHom H.uliftMap)).map (prodToProdTopI uhpath01 p)\n\n/-- The diagonal path, but starting from `f x₀` and going to `g x₁` -/\ndef diagonalPath' : fromTop (f x₀) ⟶ fromTop (g x₁) :=\n  hcast (H.apply_zero x₀).symm ≫ H.diagonalPath p ≫ hcast (H.apply_one x₁)\n\n/-- Proof that `f(p) = H(0 ⟶ 0, p)`, with the appropriate casts -/\ntheorem apply_zero_path : (πₘ (TopCat.ofHom f)).map p = hcast (H.apply_zero x₀).symm ≫\n    (πₘ (TopCat.ofHom H.uliftMap)).map\n      (prodToProdTopI (𝟙 (@fromTop (TopCat.of _) (ULift.up 0))) p) ≫\n    hcast (H.apply_zero x₁) :=\n  Quotient.inductionOn p fun p' => by\n    apply @eq_path_of_eq_image _ _ _ _ H.uliftMap _ _ _ _ _ ((Path.refl (ULift.up _)).prod p')\n    intros\n    rw [Path.prod_coe, ulift_apply H]\n    simp\n\n/-- Proof that `g(p) = H(1 ⟶ 1, p)`, with the appropriate casts -/\ntheorem apply_one_path : (πₘ (TopCat.ofHom g)).map p = hcast (H.apply_one x₀).symm ≫\n    (πₘ (TopCat.ofHom H.uliftMap)).map\n      (prodToProdTopI (𝟙 (@fromTop (TopCat.of _) (ULift.up 1))) p) ≫\n    hcast (H.apply_one x₁) :=\n  Quotient.inductionOn p fun p' => by\n    apply @eq_path_of_eq_image _ _ _ _ H.uliftMap _ _ _ _ _ ((Path.refl (ULift.up _)).prod p')\n    intros\n    rw [Path.prod_coe, ulift_apply H]\n    simp\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Proof that `H.evalAt x = H(0 ⟶ 1, x ⟶ x)`, with the appropriate casts -/\ntheorem evalAt_eq (x : X) : ⟦H.evalAt x⟧ = hcast (H.apply_zero x).symm ≫\n    (πₘ (TopCat.ofHom H.uliftMap)).map (prodToProdTopI uhpath01 (𝟙 (fromTop x))) ≫\n      hcast (H.apply_one x).symm.symm := by\n  dsimp only [prodToProdTopI, uhpath01, hcast]\n  refine (@conj_eqToHom_iff_heq (πₓ Y) _ _ _ _ _ _ _ _\n    (FundamentalGroupoid.ext <| H.apply_one x).symm).mpr ?_\n  simp only [map_eq]\n  apply Path.Homotopic.hpath_hext; intro; rfl\n\nset_option backward.isDefEq.respectTransparency false in\n-- Finally, we show `d = f(p) ≫ H₁ = H₀ ≫ g(p)`\n\nTarget:\ntheorem eq_diag_path : (πₘ (TopCat.ofHom f)).map p ≫ ⟦H.evalAt x₁⟧ = H.diagonalPath' p ∧\n    (⟦H.evalAt x₀⟧ ≫ (πₘ (TopCat.ofHom g)).map p :\n    fromTop (f x₀) ⟶ fromTop (g x₁)) = H.diagonalPath' p :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [H.apply_zero_path, H.apply_one_path, H.evalAt_eq]\n  erw [H.evalAt_eq]\n  dsimp only [prodToProdTopI]\n  constructor\n  · slice_lhs 2 4 => rw [eqToHom_trans, eqToHom_refl] -- Porting note: this ↓ `simp` didn't do this\n    slice_lhs 2 4 => simp [← CategoryTheory.Functor.map_comp]\n    rfl\n  · slice_lhs 2 4 => rw [eqToHom_trans, eqToHom_refl] -- Porting note: this ↓ `simp` didn't do this\n    slice_lhs 2 4 => simp [← CategoryTheory.Functor.map_comp]\n    rfl","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/FundamentalGroupoid","family_id":"eq_diag_path","file_id":"mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean","sample_id":"5b6616e952970d213dc2af95acd62437ab9f4ee924f0692e7518320229a74eaa"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ebb83eb68b98b09837a3cd4cf60703741f66ac05dee2a408489caf38cd22110c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"dc649d8b167a8fedb6377057872d96f16387445f37b9b72a1b54de0de1dccd5a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9fd02cd1275b92cba5b435d3686fa58fec1fc7b2cad79d852c9e074db6daabbc","source_sha256":"b1ceba138fc84cb225f35cbdb2dcc1af4a948959a8326e3efc95adc26f7cbe28","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold rieszContentAux\n  gcongr\n  apply rieszContentAux_image_nonempty","hard_negative":false,"metrics":{"chosen_tokens":6,"rejected_tokens":11,"token_jaccard":0.545455,"token_length_ratio":1.833333},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"6fccc1678e04898a08c4493ae77f0a3f76d5c5a2fc5225d76d1f7b3a4516a559","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Measure.Content\npublic import Mathlib.Topology.ContinuousMap.CompactlySupported\npublic import Mathlib.Topology.PartitionOfUnity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Jesse Reimann. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jesse Reimann, Kalle Kytölä\n-/\n/-!\n# Riesz–Markov–Kakutani representation theorem\n\nThis file prepares technical definitions and results for the Riesz-Markov-Kakutani representation\ntheorem on a locally compact T2 space `X`. As a special case, the statements about linear\nfunctionals on bounded continuous functions follows. Actual theorems, depending on the\nlinearity (`ℝ`, `ℝ≥0` or `ℂ`), are proven in separate files\n(`Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Real.lean`,\n`Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/NNReal.lean`...)\n\nTo make use of the existing API, the measure is constructed from a content `λ` on the\ncompact subsets of a locally compact space X, rather than the usual construction of open sets in the\nliterature.\n\n## References\n\n* [Walter Rudin, Real and Complex Analysis.][Rud87]\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen scoped BoundedContinuousFunction NNReal ENNReal\nopen Set Function TopologicalSpace CompactlySupported CompactlySupportedContinuousMap\n  MeasureTheory\n\nvariable {X : Type*} [TopologicalSpace X]\nvariable (Λ : C_c(X, ℝ≥0) →ₗ[ℝ≥0] ℝ≥0)\n\n/-! ### Construction of the content: -/\n\nsection Monotone\n\nlemma CompactlySupportedContinuousMap.monotone_of_nnreal : Monotone Λ := by\n  intro f₁ f₂ h\n  obtain ⟨g, hg⟩ := CompactlySupportedContinuousMap.exists_add_of_le h\n  rw [← hg]\n  simp\n\nend Monotone\n\n/-- Given a positive linear functional `Λ` on continuous compactly supported functions on `X`\nwith values in `ℝ≥0`, for `K ⊆ X` compact define `λ(K) = inf {Λf | 1≤f on K}`.\nWhen `X` is a locally compact T2 space, this will be shown to be a\ncontent, and will be shown to agree with the Riesz measure on the compact subsets `K ⊆ X`. -/\ndef rieszContentAux : Compacts X → ℝ≥0 := fun K =>\n  sInf (Λ '' { f : C_c(X, ℝ≥0) | ∀ x ∈ K, (1 : ℝ≥0) ≤ f x })\n\nsection RieszMonotone\n\nvariable [T2Space X] [LocallyCompactSpace X]\n\n/-- For any compact subset `K ⊆ X`, there exist some compactly supported continuous nonnegative\nfunctions `f` on `X` such that `f ≥ 1` on `K`. -/\ntheorem rieszContentAux_image_nonempty (K : Compacts X) :\n    (Λ '' { f : C_c(X, ℝ≥0) | ∀ x ∈ K, (1 : ℝ≥0) ≤ f x }).Nonempty := by\n  rw [image_nonempty]\n  obtain ⟨V, hVcp, hKsubintV⟩ := exists_compact_superset K.2\n  have hIsCompact_closure_interior : IsCompact (closure (interior V)) := by\n    apply IsCompact.of_isClosed_subset hVcp isClosed_closure\n    nth_rw 2 [← closure_eq_iff_isClosed.mpr (IsCompact.isClosed hVcp)]\n    exact closure_mono interior_subset\n  obtain ⟨f, hsuppfsubV, hfeq1onK, hfinicc⟩ :=\n    exists_tsupport_one_of_isOpen_isClosed isOpen_interior hIsCompact_closure_interior\n      (IsCompact.isClosed K.2) hKsubintV\n  have hfHasCompactSupport : HasCompactSupport f :=\n    IsCompact.of_isClosed_subset hVcp (isClosed_tsupport f)\n      (Set.Subset.trans hsuppfsubV interior_subset)\n  use nnrealPart ⟨f, hfHasCompactSupport⟩\n  intro x hx\n  apply le_of_eq\n  simp only [nnrealPart_apply, CompactlySupportedContinuousMap.coe_mk]\n  rw [← Real.toNNReal_one, Real.toNNReal_eq_toNNReal_iff (zero_le_one' ℝ) (hfinicc x).1]\n  exact hfeq1onK.symm hx\n\n/-- Riesz content `λ` (associated with a positive linear functional `Λ`) is\nmonotone: if `K₁ ⊆ K₂` are compact subsets in `X`, then `λ(K₁) ≤ λ(K₂)`. -/\n\nTarget:\ntheorem rieszContentAux_mono {K₁ K₂ : Compacts X} (h : K₁ ≤ K₂) :\n    rieszContentAux Λ K₁ ≤ rieszContentAux Λ K₂ :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  unfold rieszContentAux\n  gcongr\n  apply rieszContentAux_image_nonempty","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Integral","family_id":"rieszcontentaux_mono","file_id":"mathlib/Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Basic.lean","sample_id":"9fd02cd1275b92cba5b435d3686fa58fec1fc7b2cad79d852c9e074db6daabbc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eaff7377a9ad125a2190653d93c4626decbfd214d6bde989ce3ba7f1259b5118","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3c420338d39a0c37f216d1672eac3275cd1da445554d92b1413aa0b8ca6170f7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b748ffc79ac3338e3482458996caeaad3f245a010466e177b0f35271384d2cf9","source_sha256":"a487b8b8276a05b76f39e70e3e24ea2993bb9cba3cee679a6d47a9f00c0fc13a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases nonempty_fintype α\n  simpa using exists_max_image univ f univ_nonempty","hard_negative":false,"metrics":{"chosen_tokens":10,"rejected_tokens":14,"token_jaccard":0.714286,"token_length_ratio":1.4},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"6ff9f9b5eb5c04889ef76297a1c45df506b92a7c44d0594d5ccc0146939ab879","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Max\npublic import Mathlib.Data.Fintype.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Lemmas relating fintypes and order/lattice structure.\n-/\n\npublic section\n\n\nopen Function\n\nopen Nat\n\nuniverse u v\n\nvariable {ι α β : Type*}\n\nnamespace Finset\n\nvariable [Fintype α] {s : Finset α}\n\n/-- A special case of `Finset.sup_eq_iSup` that omits the useless `x ∈ univ` binder. -/\ntheorem sup_univ_eq_iSup [CompleteLattice β] (f : α → β) : Finset.univ.sup f = iSup f :=\n  (sup_eq_iSup _ f).trans <| congr_arg _ <| funext fun _ => iSup_pos (mem_univ _)\n\n/-- A special case of `Finset.inf_eq_iInf` that omits the useless `x ∈ univ` binder. -/\ntheorem inf_univ_eq_iInf [CompleteLattice β] (f : α → β) : Finset.univ.inf f = iInf f :=\n  @sup_univ_eq_iSup _ βᵒᵈ _ _ (f : α → βᵒᵈ)\n\n@[simp]\ntheorem fold_inf_univ [SemilatticeInf α] [OrderBot α] (a : α) :\n    (Finset.univ.fold min a fun x => x) = ⊥ :=\n  eq_bot_iff.2 <|\n    ((Finset.fold_op_rel_iff_and <| @le_inf_iff α _).1 le_rfl).2 ⊥ <| Finset.mem_univ _\n\n@[simp]\ntheorem fold_sup_univ [SemilatticeSup α] [OrderTop α] (a : α) :\n    (Finset.univ.fold max a fun x => x) = ⊤ :=\n  @fold_inf_univ αᵒᵈ _ _ _ _\n\nlemma mem_inf [DecidableEq α] {s : Finset ι} {f : ι → Finset α} {a : α} :\n    a ∈ s.inf f ↔ ∀ i ∈ s, a ∈ f i := by induction s using Finset.cons_induction <;> simp [*]\n\nend Finset\n\nopen Finset\n\nTarget:\ntheorem Finite.exists_max [Finite α] [Nonempty α] [LinearOrder β] (f : α → β) :\n    ∃ x₀ : α, ∀ x, f x ≤ f x₀ :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  cases nonempty_fintype α\n  simpa using exists_max_image univ f univ_nonempty","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Fintype","family_id":"finite","file_id":"mathlib/Mathlib/Data/Fintype/Lattice.lean","sample_id":"b748ffc79ac3338e3482458996caeaad3f245a010466e177b0f35271384d2cf9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"08218aabda46401fe50c2dc9de3ff5011d981b5e7f148f3d58bd8c99abd66739","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0848e9cfd63256eb77b8f8cfb09985072eb7cfe38acec90f8458e1ed27657ac9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"631dd733f33921d58948a0df45ceabb189d48a1055bf4f8fc988734c4839b6cb","source_sha256":"a532f8917e86d9e21f6417bd7287cabd2ab46e6ff2388a4cee138b7afaddc6d0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [simplyConnectedSpace_iff, equiv_punit_iff_unique,\n    FundamentalGroupoid.nonempty_iff X, and_congr_right_iff, Nonempty.forall]\n  intros\n  exact ⟨fun h _ _ => h _ _, fun h _ _ => h _ _⟩","hard_negative":true,"metrics":{"chosen_tokens":40,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.075},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"705afe824e58477343667ebd317fe2449f001c9c760fd1a8bc5c8e69a42ac565","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps\npublic import Mathlib.Topology.Homotopy.Contractible\npublic import Mathlib.CategoryTheory.PUnit\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.PUnit\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Praneeth Kolichala. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Praneeth Kolichala\n-/\n/-!\n# Simply connected spaces\nThis file defines simply connected spaces.\nA topological space is simply connected if its fundamental groupoid is equivalent to `Unit`.\n\nWe also define the corresponding predicate for sets.\n\n## Main theorems\n  - `simply_connected_iff_unique_homotopic` - A space is simply connected if and only if it is\n    nonempty and there is a unique path up to homotopy between any two points\n\n  - `SimplyConnectedSpace.ofContractible` - A contractible space is simply connected\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen CategoryTheory\nopen scoped ContinuousMap\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n\n/-- A simply connected space is one whose fundamental groupoid is equivalent to `Discrete Unit` -/\n@[mk_iff]\nclass SimplyConnectedSpace (X : Type*) [TopologicalSpace X] : Prop where\n  equiv_unit : Nonempty (FundamentalGroupoid X ≌ Discrete Unit)\n\n@[deprecated (since := \"2026-01-08\")]\nalias simply_connected_def := simplyConnectedSpace_iff\n\nTarget:\ntheorem simply_connected_iff_unique_homotopic (X : Type*) [TopologicalSpace X] :\n    SimplyConnectedSpace X ↔\n      Nonempty X ∧ ∀ x y : X, Nonempty (Unique (Path.Homotopic.Quotient x y)) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_631dd733f339","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"60fd3419fb59c04f92b71f03cdaee0d354b07e39135ac3737b9444ffa36e397a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/FundamentalGroupoid","family_id":"simply_connected_iff_unique_homotopic","file_id":"mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean","sample_id":"631dd733f33921d58948a0df45ceabb189d48a1055bf4f8fc988734c4839b6cb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"04ff734dba3e12dd755f41758fae7a61d3a513148d7d550f593f25ac71b54468","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9e767d0b834339320d21f2fe6faa7f7606cdac3e6fc203de41750386839ebf3b","source_sha256":"06b8f3152156917b43eb16aa25f88f706675fd088336840c57324ab27b671934","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  subst coeff_eq deg_eq_deg coeff_eq_deg\n  exact natDegree_eq_of_le_of_coeff_ne_zero ‹_› ‹_›","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":8,"token_jaccard":0.0625,"token_length_ratio":0.615385},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"709ab7efafab3645b7ee4c0887f3218b55c26e2bdcfad4201e1b9808c2c18b1f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Degree.Lemmas\n\nNamespace:\nMathlib.Tactic.ComputeDegree\n\nLocal context:\n/-\nCopyright (c) 2023 Damiano Testa. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Damiano Testa\n-/\n/-!\n\n# `compute_degree` and `monicity`: tactics for explicit polynomials\n\nThis file defines two related tactics: `compute_degree` and `monicity`.\n\nUsing `compute_degree` when the goal is of one of the seven forms\n* `natDegree f ≤ d` (or `<`),\n* `degree f ≤ d` (or `<`),\n* `natDegree f = d`,\n* `degree f = d`,\n* `coeff f d = r`, if `d` is the degree of `f`,\n\ntries to solve the goal.\nIt may leave side-goals, in case it is not entirely successful.\n\nUsing `monicity` when the goal is of the form `Monic f` tries to solve the goal.\nIt may leave side-goals, in case it is not entirely successful.\n\nBoth tactics admit a `!` modifier (`compute_degree!` and `monicity!`) instructing\nLean to try harder to close the goal.\n\nSee the doc-strings for more details.\n\n## Future work\n\n* Currently, `compute_degree` does not deal correctly with some edge cases.  For instance,\n  ```lean\n  example [Semiring R] : natDegree (C 0 : R[X]) = 0 := by\n    compute_degree\n  --  ⊢ 0 ≠ 0\n  ```\n  Still, it may not be worth to provide special support for `natDegree f = 0`.\n* Make sure that numerals in coefficients are treated correctly.\n* Make sure that `compute_degree` works with goals of the form `degree f ≤ ↑d`, with an\n  explicit coercion from `ℕ` on the RHS.\n* Add support for proving goals of the from `natDegree f ≠ 0` and `degree f ≠ 0`.\n* Make sure that `degree`, `natDegree` and `coeff` are equally supported.\n\n## Implementation details\n\nAssume that `f : R[X]` is a polynomial with coefficients in a semiring `R` and\n`d` is either in `ℕ` or in `WithBot ℕ`.\n\nIf the goal has the form `natDegree f < d`, then we convert it to two separate goals:\n* `natDegree f ≤ ?_`, on which we apply the following steps;\n* `?_ < d`;\n\nwhere `?_` is a metavariable that `compute_degree` computes in its process.\nWe proceed similarly for `degree f < d`.\n\nIf the goal has the form `natDegree f = d`, then we convert it to three separate goals:\n* `natDegree f ≤ d`;\n* `coeff f d = r`;\n* `r ≠ 0`.\n\nSimilarly, an initial goal of the form `degree f = d` gives rise to goals of the form\n* `degree f ≤ d`;\n* `coeff f d = r`;\n* `r ≠ 0`.\n\nNext, we apply successively lemmas whose side-goals all have the shape\n* `natDegree f ≤ d`;\n* `degree f ≤ d`;\n* `coeff f d = r`;\n\nplus possibly \"numerical\" identities and choices of elements in `ℕ`, `WithBot ℕ`, and `R`.\n\nRecursing into `f`, we break apart additions, multiplications, powers, subtractions,...\nThe leaves of the process are\n* numerals, `C a`, `X` and `monomial a n`, to which we assign degree `0`, `1` and `a` respectively;\n* `fvar`s `f`, to which we tautologically assign degree `natDegree f`.\n-/\n\npublic meta section\n\nopen Polynomial\n\nnamespace Mathlib.Tactic.ComputeDegree\n\nsection recursion_lemmas\n/-!\n### Simple lemmas about `natDegree`\n\nThe lemmas in this section all have the form `natDegree <some form of cast> ≤ 0`.\nTheir proofs are weakenings of the stronger lemmas `natDegree <same> = 0`.\nThese are the lemmas called by `compute_degree` on (almost) all the leaves of its recursion.\n-/\n\nvariable {R : Type*}\n\nsection semiring\nvariable [Semiring R]\n\ntheorem natDegree_C_le (a : R) : natDegree (C a) ≤ 0 := (natDegree_C a).le\n\ntheorem natDegree_natCast_le (n : ℕ) : natDegree (n : R[X]) ≤ 0 := (natDegree_natCast _).le\ntheorem natDegree_zero_le : natDegree (0 : R[X]) ≤ 0 := natDegree_zero.le\ntheorem natDegree_one_le : natDegree (1 : R[X]) ≤ 0 := natDegree_one.le\n\ntheorem coeff_add_of_eq {n : ℕ} {a b : R} {f g : R[X]}\n    (h_add_left : f.coeff n = a) (h_add_right : g.coeff n = b) :\n    (f + g).coeff n = a + b := by subst ‹_› ‹_›; apply coeff_add\n\ntheorem coeff_mul_add_of_le_natDegree_of_eq_ite {d df dg : ℕ} {a b : R} {f g : R[X]}\n    (h_mul_left : natDegree f ≤ df) (h_mul_right : natDegree g ≤ dg)\n    (h_mul_left : f.coeff df = a) (h_mul_right : g.coeff dg = b) (ddf : df + dg ≤ d) :\n    (f * g).coeff d = if d = df + dg then a * b else 0 := by\n  split_ifs with h\n  · subst h_mul_left h_mul_right h\n    exact coeff_mul_add_eq_of_natDegree_le ‹_› ‹_›\n  · apply coeff_eq_zero_of_natDegree_lt\n    apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ddf ?_)\n    · exact natDegree_mul_le_of_le ‹_› ‹_›\n    · exact ne_comm.mp h\n\ntheorem coeff_pow_of_natDegree_le_of_eq_ite' {m n o : ℕ} {a : R} {p : R[X]}\n    (h_pow : natDegree p ≤ n) (h_exp : m * n ≤ o) (h_pow_bas : coeff p n = a) :\n    coeff (p ^ m) o = if o = m * n then a ^ m else 0 := by\n  split_ifs with h\n  · subst h h_pow_bas\n    exact coeff_pow_of_natDegree_le ‹_›\n  · apply coeff_eq_zero_of_natDegree_lt\n    apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ‹_› ?_)\n    · exact natDegree_pow_le_of_le m ‹_›\n    · exact Iff.mp ne_comm h\n\nsection SMul\n\nvariable {S : Type*} [SMulZeroClass S R] {n : ℕ} {a : S} {f : R[X]}\n\ntheorem natDegree_smul_le_of_le (hf : natDegree f ≤ n) :\n    natDegree (a • f) ≤ n :=\n  (natDegree_smul_le a f).trans hf\n\ntheorem degree_smul_le_of_le (hf : degree f ≤ n) :\n    degree (a • f) ≤ n :=\n  (degree_smul_le a f).trans hf\n\ntheorem coeff_smul : (a • f).coeff n = a • f.coeff n := rfl\n\nend SMul\n\nsection congr_lemmas\n\n/-- The following two lemmas should be viewed as a hand-made \"congr\"-lemmas.\nThey achieve the following goals.\n* They introduce *two* fresh metavariables replacing the given one `deg`,\n  one for the `natDegree ≤` computation and one for the `coeff =` computation.\n  This helps `compute_degree`, since it does not \"pre-estimate\" the degree,\n  but it \"picks it up along the way\".\n* They split checking the inequality `coeff p n ≠ 0` into the task of\n  finding a value `c` for the `coeff` and then\n  proving that this value is non-zero by `coeff_ne_zero`.\n-/\n\nTarget:\ntheorem natDegree_eq_of_le_of_coeff_ne_zero' {deg m o : ℕ} {c : R} {p : R[X]}\n    (h_natDeg_le : natDegree p ≤ m) (coeff_eq : coeff p o = c)\n    (coeff_ne_zero : c ≠ 0) (deg_eq_deg : m = deg) (coeff_eq_deg : o = deg) :\n    natDegree p = deg :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"81ae73f5c87bdbe878e9591419658790651faf62d37a64e02b720b209863c8b7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic","family_id":"natdegree_eq_of_le_of_coeff_ne_zero","file_id":"mathlib/Mathlib/Tactic/ComputeDegree.lean","sample_id":"9e767d0b834339320d21f2fe6faa7f7606cdac3e6fc203de41750386839ebf3b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bb1aa1df4a3618112e80043cd006efbcdf4906086196e285cd31ba1676129c32","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"46f6f8415ade511d6c5719e771f8a7d1cad432a0c7eaf7e5bfc6155da642fd12","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"26a28f745aa7034105aa52a93b588fb63e8e92ad6d8d99ffd3be3558367e64ea","source_sha256":"ffda45eef0509a041f24079788fb5aa3f3f4500975a78975ccc64409c3de362b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← closure_openCell_eq_closedCell, E.closed.closure_subset_iff, ← E.union]\n  apply subset_union_of_subset_right\n  exact subset_iUnion_of_subset n\n    (subset_iUnion (fun (j : ↑(E.I n)) ↦ openCell (C := C) n j) ⟨i, hi⟩)","hard_negative":true,"metrics":{"chosen_tokens":53,"rejected_tokens":2,"token_jaccard":0.029412,"token_length_ratio":0.037736},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"70d262f3eb7cb27a461d247c90f663d95aa2f2b97c4a989d8ee137bbf304e2cb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.CWComplex.Classical.Finite\npublic import Mathlib.Analysis.Normed.Module.RCLike.Real\n\nNamespace:\nTopology\n\nLocal context:\n/-\nCopyright (c) 2025 Floris van Doorn and Hannah Scholz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Hannah Scholz\n-/\n/-!\n# Subcomplexes\n\nIn this file we discuss subcomplexes of CW complexes.\nThe definition of subcomplexes is in the file `Mathlib/Topology/CWComplex/Classical/Basic.lean`.\n\n## Main results\n* `RelCWComplex.Subcomplex.instRelCWComplex`: a subcomplex of a (relative) CW complex is again a\n  (relative) CW complex.\n\n## References\n* [K. Jänich, *Topology*][Janich1984]\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Metric Set\n\nnamespace Topology\n\nvariable {X : Type*} [t : TopologicalSpace X] {C D : Set X}\n\n@[alias_in CWComplex.Subcomplex]\n\nTarget:\nlemma RelCWComplex.Subcomplex.closedCell_subset_of_mem [T2Space X] [RelCWComplex C D]\n    (E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) :\n    closedCell n i ⊆ E :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_26a28f745aa7","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"baf62482116c32afa88114c6c653d725b96cd4611e7a0263d244159888447578","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/CWComplex","family_id":"relcwcomplex","file_id":"mathlib/Mathlib/Topology/CWComplex/Classical/Subcomplex.lean","sample_id":"26a28f745aa7034105aa52a93b588fb63e8e92ad6d8d99ffd3be3558367e64ea"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4fff200b3969b2099ac39b5df9f12ed101d4778e55341e30c66e404ab2007cba","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8c9c68d8ea4801f822185f3282b0bfffb01c51c0ebe70f6b71757e8c9c924c93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"68d51948bcb20f9102720ae7d20bb860c7d2a1e39fd0c3d403bf97713844dd99","source_sha256":"3ebef0ce1ff4e8804552a350539c21f4e3856ca570ca14d4062fa9367e8ccafd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨k, rfl⟩ := hn\n  rw [pow_add, pow_one, pow_mul, sq]\n  cases s <;> simp","hard_negative":true,"metrics":{"chosen_tokens":25,"rejected_tokens":5,"token_jaccard":0.038462,"token_length_ratio":0.2},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"7108b244e01f4fb241f31eea24ddfb5904edd1be2ffc2742e3ece4ab09afa418","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GroupWithZero.Units.Lemmas\npublic import Mathlib.Algebra.Order.BigOperators.Group.Finset\npublic import Mathlib.Algebra.Order.Ring.Cast\npublic import Mathlib.Data.Fintype.BigOperators\npublic import Mathlib.Data.Sign.Defs\n\nNamespace:\nSignType\n\nLocal context:\n/-\nCopyright (c) 2022 Eric Rodriguez. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Rodriguez\n-/\n/-!\n# Sign function\n\nThis file defines the sign function for types with zero and a decidable less-than relation, and\nproves some basic theorems about it.\n-/\n\n@[expose] public section\n\nuniverse u\nvariable {α : Type u}\n\nnamespace SignType\n\n/-- Casting `SignType → ℤ → α` is the same as casting directly `SignType → α`. -/\n@[simp, norm_cast]\nlemma intCast_cast {α : Type*} [AddGroupWithOne α] (s : SignType) : ((s : ℤ) : α) = s :=\n  map_cast' _ Int.cast_one Int.cast_zero (@Int.cast_one α _ ▸ Int.cast_neg 1) _\n\nTarget:\ntheorem pow_odd (s : SignType) {n : ℕ} (hn : Odd n) : s ^ n = s :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_68d51948bcb2","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"68f76d11d3c9ee5a1dd860c2a62b6cc335392300d9d976b7d8b17ab7ba24d5bc","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sign","family_id":"pow_odd","file_id":"mathlib/Mathlib/Data/Sign/Basic.lean","sample_id":"68d51948bcb20f9102720ae7d20bb860c7d2a1e39fd0c3d403bf97713844dd99"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2fa6d6384b3fc37bc32726c7d0dfd505d0e32dc595d3c34173c19db9a5d115ee","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d26cc5be978dead2b623c3a4790cc21af0d92aeaa8715f5d06e969086e47e5d8","source_sha256":"e8c6f00bd5a71ab259d53c29519ad94f338dec18478f71b6e20990ccef2afa1b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := IsClosedUnderBinaryCoproducts.closedUnderIsomorphisms P\n  have := hasFiniteCoproducts_of_has_binary_and_initial (C := P.FullSubcategory)\n  have := PreservesFiniteCoproducts.of_preserves_binary_and_initial P.ι\n  exact ⟨fun J _ ↦ P.isClosedUnderColimitsOfShape_of_preservesColimitsOfShape_ι _⟩","hard_negative":false,"metrics":{"chosen_tokens":37,"rejected_tokens":5,"token_jaccard":0.12,"token_length_ratio":0.135135},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"710c3ab28a5b75a6174a98a9fd32910d4685cf896c0fb0b79049f549c7563792","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts\npublic import Mathlib.CategoryTheory.Limits.FullSubcategory\npublic import Mathlib.CategoryTheory.ObjectProperty.ColimitsClosure\npublic import Mathlib.CategoryTheory.ObjectProperty.ContainsZero\npublic import Mathlib.Data.Fintype.Shrink\n\nNamespace:\nCategoryTheory.ObjectProperty\n\nLocal context:\n/-\nCopyright (c) 2026 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Properties of objects that are stable under finite products\n\nWe introduce typeclasses `IsClosedUnderBinaryProducts` and\n`IsClosedUnderFiniteProducts` expressing that `P : ObjectProperty C`\nis closed under binary products or finite products.\nWe introduce a constructor for `P.IsClosedUnderFiniteProducts`\nassuming `P.IsClosedUnderBinaryProducts`,\n`P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)` and that `C`\nhas finite products.\n\n-/\n\nuniverse w\n\npublic section\n\nnamespace CategoryTheory.ObjectProperty\n\nopen Limits\n\nvariable {C : Type*} [Category* C] (P : ObjectProperty C)\n\n/-- The typeclass saying that `P : ObjectProperty C` is stable under binary products. -/\nabbrev IsClosedUnderBinaryProducts :=\n  P.IsClosedUnderLimitsOfShape (Discrete WalkingPair)\n\nlemma prop_of_isLimit_binaryFan [P.IsClosedUnderBinaryProducts] {X Y : C} {B : BinaryFan X Y}\n    (hB : IsLimit B) (hX : P X) (hY : P Y) :\n    P B.pt :=\n  P.prop_of_isLimit hB (by rintro ⟨_ | _⟩ <;> assumption)\n\nlemma prop_prod [P.IsClosedUnderBinaryProducts] (X Y : C) [HasBinaryProduct X Y]\n    (hX : P X) (hY : P Y) :\n    P (X ⨯ Y) :=\n  P.prop_of_isLimit_binaryFan (limit.isLimit _) hX hY\n\nlemma prop_of_isTerminal [P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)]\n    (X : C) (hX : IsTerminal X) :\n    P X :=\n  P.prop_of_isLimit hX (by rintro ⟨⟨⟩⟩)\n\nlemma prop_terminal [P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)] [HasTerminal C] :\n    P (⊤_ C) :=\n  P.prop_of_isTerminal _ terminalIsTerminal\n\n-- see Note [lower instance priority]\ninstance (priority := 100) [P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)] [HasTerminal C] :\n    P.Nonempty :=\n  nonempty_of_prop P.prop_terminal\n\nlemma IsClosedUnderBinaryProducts.closedUnderIsomorphisms [HasTerminal C]\n    [P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)] [P.IsClosedUnderBinaryProducts] :\n    P.IsClosedUnderIsomorphisms where\n  of_iso {X Y} e hX := by\n    let h : IsLimit (BinaryFan.mk (terminal.from Y) e.inv) :=\n      BinaryFan.IsLimit.mk _ (fun _ f ↦ f ≫ e.hom) (by cat_disch) (by simp) (by cat_disch)\n    exact P.prop_of_isLimit_binaryFan h P.prop_terminal hX\n\n/-- All objects that are binary products of objects in `P`. -/\nabbrev binaryProductsClosure (P : ObjectProperty C) : ObjectProperty C :=\n  P.limitClosure (Discrete WalkingPair)\n\nlemma binaryProductsClosure_le_iff [HasTerminal C] {P Q : ObjectProperty C}\n    [Q.IsClosedUnderBinaryProducts] [Q.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)] :\n    P.binaryProductsClosure ≤ Q ↔ P ≤ Q := by\n  refine ⟨fun h ↦ (P.le_limitsClosure _).trans h, fun h ↦ ?_⟩\n  letI : Q.IsClosedUnderIsomorphisms := IsClosedUnderBinaryProducts.closedUnderIsomorphisms Q\n  exact limitsClosure_le h\n\n/-- The typeclass saying that `P : ObjectProperty C` is stable under finite products. -/\nclass IsClosedUnderFiniteProducts : Prop where\n  isClosedUnderLimitsOfShape (J : Type) [Finite J] :\n    P.IsClosedUnderLimitsOfShape (Discrete J) := by infer_instance\n\nvariable {P} in\n/-- `IsClosedUnderFiniteProducts` may be checked at any universe. -/\nlemma IsClosedUnderFiniteProducts.of_isClosedUnderLimitsOfShape\n    (H : ∀ (J : Type w) [Finite J], P.IsClosedUnderLimitsOfShape (Discrete J)) :\n    P.IsClosedUnderFiniteProducts where\n  isClosedUnderLimitsOfShape J _ := by\n    rw [P.isClosedUnderLimitsOfShape_iff_of_equivalence (Discrete.equivalence (equivShrink.{w} _))]\n    exact H _\n\ninstance [P.IsClosedUnderFiniteProducts] (J : Type*) [Finite J] :\n    P.IsClosedUnderLimitsOfShape (Discrete J) := by\n  obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin J\n  have : P.IsClosedUnderLimitsOfShape (Discrete (Fin n)) :=\n    IsClosedUnderFiniteProducts.isClosedUnderLimitsOfShape _\n  exact IsClosedUnderLimitsOfShape.of_equivalence (Discrete.equivalence e.symm)\n\ninstance [HasFiniteProducts C] [P.IsClosedUnderFiniteProducts] :\n    HasFiniteProducts P.FullSubcategory where\n  out _ := inferInstance\n\nlemma prop_of_isLimit_fan [P.IsClosedUnderFiniteProducts] {J : Type*} [Finite J] {f : J → C}\n    {F : Fan f} (hF : IsLimit F) (h : ∀ j, P (f j)) :\n    P F.pt :=\n  P.prop_of_isLimit hF (by intro ⟨j⟩; exact h j)\n\nlemma prop_product [P.IsClosedUnderFiniteProducts] {J : Type*} [Finite J] {f : J → C}\n    [HasProduct f] (h : ∀ j, P (f j)) :\n    P (∏ᶜ f) :=\n  P.prop_of_isLimit_fan (limit.isLimit (Discrete.functor f)) h\n\ninstance [P.ContainsZero] [P.IsClosedUnderIsomorphisms] :\n    P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty) where\n  limitsOfShape_le := by\n    rintro X ⟨p⟩\n    obtain ⟨Z, hZ, hZ₂⟩ := P.exists_prop_of_containsZero\n    have hX : IsTerminal X :=\n      (IsLimit.equivOfNatIsoOfIso p.diag.uniqueFromEmpty _ _\n        (by exact Cone.ext (Iso.refl _) (by rintro ⟨⟨⟩⟩))).1 p.isLimit\n    exact P.prop_of_isZero (IsZero.of_iso hZ\n      (IsLimit.conePointUniqueUpToIso hX (IsZero.isTerminal hZ)))\n\nvariable {P} in\nlemma IsClosedUnderFiniteProducts.mk' [HasFiniteProducts C]\n    [P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)]\n    [P.IsClosedUnderBinaryProducts] :\n    P.IsClosedUnderFiniteProducts := by\n  have := IsClosedUnderBinaryProducts.closedUnderIsomorphisms P\n  have := hasFiniteProducts_of_has_binary_and_terminal (C := P.FullSubcategory)\n  have := PreservesFiniteProducts.of_preserves_binary_and_terminal P.ι\n  exact ⟨fun J _ ↦ P.isClosedUnderLimitsOfShape_of_preservesLimitsOfShape_ι _⟩\n\n/-- The typeclass saying that `P : ObjectProperty C` is stable under binary coproducts. -/\nabbrev IsClosedUnderBinaryCoproducts :=\n  P.IsClosedUnderColimitsOfShape (Discrete WalkingPair)\n\nlemma prop_of_isColimit_binaryCofan [P.IsClosedUnderBinaryCoproducts] {X Y : C}\n    {B : BinaryCofan X Y} (hB : IsColimit B) (hX : P X) (hY : P Y) :\n    P B.pt :=\n  P.prop_of_isColimit hB (by rintro ⟨_ | _⟩ <;> assumption)\n\nlemma prop_coprod [P.IsClosedUnderBinaryCoproducts] (X Y : C) [HasBinaryCoproduct X Y]\n    (hX : P X) (hY : P Y) :\n    P (X ⨿ Y) :=\n  P.prop_of_isColimit_binaryCofan (colimit.isColimit (Limits.pair X Y)) hX hY\n\nlemma prop_of_isInitial [P.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty)]\n    (X : C) (hX : IsInitial X) :\n    P X :=\n  P.prop_of_isColimit hX (by rintro ⟨⟨⟩⟩)\n\nlemma prop_initial [P.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty)] [HasInitial C] :\n    P (⊥_ C) :=\n  P.prop_of_isInitial _ initialIsInitial\n\n-- see Note [lower instance priority]\ninstance (priority := 100) [P.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty)] [HasInitial C] :\n    P.Nonempty :=\n  nonempty_of_prop P.prop_initial\n\nset_option backward.defeqAttrib.useBackward true in\nlemma IsClosedUnderBinaryCoproducts.closedUnderIsomorphisms [HasInitial C]\n    [P.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty)] [P.IsClosedUnderBinaryCoproducts] :\n    P.IsClosedUnderIsomorphisms where\n  of_iso {X Y} e hX := by\n    let h : IsColimit (BinaryCofan.mk (initial.to Y) e.hom) :=\n      BinaryCofan.IsColimit.mk _ (fun _ f ↦ e.inv ≫ f) (by cat_disch) (by simp) (by cat_disch)\n    exact P.prop_of_isColimit_binaryCofan h P.prop_initial hX\n\n/-- All objects that are binary coproducts of objects in `P`. -/\nabbrev binaryCoproductsClosure (P : ObjectProperty C) : ObjectProperty C :=\n  P.colimitClosure (Discrete WalkingPair)\n\nlemma binaryCoproductsClosure_le_iff [HasInitial C] {P Q : ObjectProperty C}\n    [Q.IsClosedUnderBinaryCoproducts] [Q.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty)] :\n    P.binaryCoproductsClosure ≤ Q ↔ P ≤ Q := by\n  refine ⟨fun h ↦ (P.le_colimitsClosure _).trans h, fun h ↦ ?_⟩\n  letI : Q.IsClosedUnderIsomorphisms := IsClosedUnderBinaryCoproducts.closedUnderIsomorphisms Q\n  exact colimitsClosure_le h\n\n/-- The typeclass saying that `P : ObjectProperty C` is stable under finite coproducts. -/\nclass IsClosedUnderFiniteCoproducts : Prop where\n  isClosedUnderColimitsOfShape (J : Type) [Finite J] :\n    P.IsClosedUnderColimitsOfShape (Discrete J) := by infer_instance\n\nvariable {P} in\n/-- `IsClosedUnderFiniteProducts` may be checked at any universe. -/\nlemma IsClosedUnderFiniteCoproducts.of_isClosedUnderColimitsOfShape\n    (H : ∀ (J : Type w) [Finite J], P.IsClosedUnderColimitsOfShape (Discrete J)) :\n    P.IsClosedUnderFiniteCoproducts where\n  isClosedUnderColimitsOfShape J _ := by\n    rw [P.isClosedUnderColimitsOfShape_iff_of_equivalence\n      (Discrete.equivalence (equivShrink.{w} _))]\n    exact H _\n\ninstance [P.IsClosedUnderFiniteCoproducts] (J : Type*) [Finite J] :\n    P.IsClosedUnderColimitsOfShape (Discrete J) := by\n  obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin J\n  have : P.IsClosedUnderColimitsOfShape (Discrete (Fin n)) :=\n    IsClosedUnderFiniteCoproducts.isClosedUnderColimitsOfShape _\n  exact IsClosedUnderColimitsOfShape.of_equivalence (Discrete.equivalence e.symm)\n\ninstance [HasFiniteCoproducts C] [P.IsClosedUnderFiniteCoproducts] :\n    HasFiniteCoproducts P.FullSubcategory where\n  out _ := inferInstance\n\nlemma prop_of_isColimit_cofan [P.IsClosedUnderFiniteCoproducts] {J : Type*} [Finite J] {f : J → C}\n    {F : Cofan f} (hF : IsColimit F) (h : ∀ j, P (f j)) :\n    P F.pt :=\n  P.prop_of_isColimit hF (by intro ⟨j⟩; exact h j)\n\nlemma prop_coproduct [P.IsClosedUnderFiniteCoproducts] {J : Type*} [Finite J] {f : J → C}\n    [HasCoproduct f] (h : ∀ j, P (f j)) :\n    P (∐ f) :=\n  P.prop_of_isColimit_cofan (colimit.isColimit (Discrete.functor f)) h\n\ninstance [P.ContainsZero] [P.IsClosedUnderIsomorphisms] :\n    P.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty) where\n  colimitsOfShape_le := by\n    rintro X ⟨p⟩\n    obtain ⟨Z, hZ, hZ₂⟩ := P.exists_prop_of_containsZero\n    have hX : IsInitial X :=\n      (IsColimit.equivOfNatIsoOfIso p.diag.uniqueFromEmpty _ _\n        (by exact Cocone.ext (Iso.refl _) (by rintro ⟨⟨⟩⟩))).1 p.isColimit\n    exact P.prop_of_isZero (IsZero.of_iso hZ\n      (IsColimit.coconePointUniqueUpToIso hX (IsZero.isInitial hZ)))\n\nvariable {P} in\n\nTarget:\nlemma IsClosedUnderFiniteCoproducts.mk' [HasFiniteCoproducts C]\n    [P.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty)]\n    [P.IsClosedUnderBinaryCoproducts] :\n    P.IsClosedUnderFiniteCoproducts :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/ObjectProperty","family_id":"isclosedunderfinitecoproducts","file_id":"mathlib/Mathlib/CategoryTheory/ObjectProperty/FiniteProducts.lean","sample_id":"d26cc5be978dead2b623c3a4790cc21af0d92aeaa8715f5d06e969086e47e5d8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8beab1ef316edfb8523c7f6d8a094e22bcc81e68fa42d1a455d344f9254b12c0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"51c303a48a29878142e6b9a5fc566752c563d155cb6a520b4d63f144ec0617c9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"be98270d4e641660001524cb4a6a1e9cfae5fb603cff7ae0acf7c6d8fed02aa2","source_sha256":"39dfdad0e4eff3ed2f02c59604c6ff92849183f69003baf79902622224068963","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [MonomialOrder.lex_lt_iff, Finsupp.Lex.lt_iff_of_unique, ofLex_toLex]","hard_negative":true,"metrics":{"chosen_tokens":16,"rejected_tokens":5,"token_jaccard":0.125,"token_length_ratio":0.3125},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"7111c363f29e854ab2c68ba9341c8701557619ead5eeb6de8c8487bf22dfc307","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finsupp.Lex\npublic import Mathlib.Data.Finsupp.WellFounded\npublic import Mathlib.Data.List.TFAE\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Antoine Chambert-Loir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir\n-/\n/-! # Monomial orders\n\n## Monomial orders\n\nA *monomial order* is well ordering relation on a type of the form `σ →₀ ℕ` which\nis compatible with addition and for which `0` is the smallest element.\nSince several monomial orders may have to be used simultaneously, one cannot\nget them as instances.\n\nIn this formalization, they are presented as a structure `MonomialOrder` which encapsulates\n`MonomialOrder.toSyn`, an additive and monotone isomorphism to a linearly ordered cancellative\nadditive commutative monoid.\nThe entry `MonomialOrder.wf` asserts that `MonomialOrder.syn` is well founded.\n\nThe terminology comes from commutative algebra and algebraic geometry, especially Gröbner bases,\nwhere `c : σ →₀ ℕ` are exponents of monomials.\n\nGiven a monomial order `m : MonomialOrder σ`, we provide the notation\n`c ≼[m] d` and `c ≺[m] d` to compare `c d : σ →₀ ℕ` with respect to `m`.\nIt is activated using `open scoped MonomialOrder`.\n\n## Examples\n\nCommutative algebra defines many monomial orders, with different usefulness ranges.\nIn this file, we provide the basic example of lexicographic ordering.\nFor the graded lexicographic ordering, see `Mathlib/Data/Finsupp/MonomialOrder/DegLex.lean`\n\n* `MonomialOrder.lex` : the lexicographic ordering on `σ →₀ ℕ`.\n  For this, `σ` needs to be embedded with an ordering relation which satisfies `WellFoundedGT σ`.\n  (This last property is automatic when `σ` is finite).\n\nThe type synonym is `Lex (σ →₀ ℕ)` and the two lemmas `MonomialOrder.lex_le_iff`\nand `MonomialOrder.lex_lt_iff` rewrite the ordering as comparisons in the type `Lex (σ →₀ ℕ)`.\n\n## References\n\n* [Cox, Little and O'Shea, *Ideals, varieties, and algorithms*][coxlittleoshea1997]\n* [Becker and Weispfenning, *Gröbner bases*][Becker-Weispfenning1993]\n\n## Note\n\nIn algebraic geometry, when the finitely many variables are indexed by integers,\nit is customary to order them using the opposite order : `MvPolynomial.X 0 > MvPolynomial.X 1 > … `\n\n-/\n\n@[expose] public section\n\n/-- Monomial orders : equivalence of `σ →₀ ℕ` with a well-ordered type -/\nstructure MonomialOrder (σ : Type*) where\n  /-- The synonym type -/\n  syn : Type*\n  /-- `syn` is an additive commutative monoid -/\n  acm : AddCommMonoid syn := by infer_instance\n  /-- `syn` is linearly ordered -/\n  lo : LinearOrder syn := by infer_instance\n  /-- `syn` is a linearly ordered cancellative additive commutative monoid -/\n  iocam : IsOrderedCancelAddMonoid syn := by infer_instance\n  /-- the additive equivalence from `σ →₀ ℕ` to `syn` -/\n  toSyn : (σ →₀ ℕ) ≃+ syn\n  /-- `toSyn` is monotone -/\n  toSyn_monotone : Monotone toSyn\n  /-- `syn` is a well ordering -/\n  wf : WellFoundedLT syn := by infer_instance\n\nattribute [instance] MonomialOrder.acm MonomialOrder.lo MonomialOrder.iocam MonomialOrder.wf\n\nnamespace MonomialOrder\n\nvariable {σ : Type*} (m : MonomialOrder σ)\n\nlemma le_add_right (a b : σ →₀ ℕ) :\n    m.toSyn a ≤ m.toSyn a + m.toSyn b := by\n  rw [← map_add]\n  exact m.toSyn_monotone le_self_add\n\ninstance orderBot : OrderBot (m.syn) where\n  bot := 0\n  bot_le a := by\n    have := m.le_add_right 0 (m.toSyn.symm a)\n    simpa [map_add, zero_add]\n\n@[simp]\ntheorem bot_eq_zero : (⊥ : m.syn) = 0 := rfl\n\n@[simp]\nlemma zero_le (a : m.syn) : 0 ≤ a := bot_le\n\ntheorem eq_zero_iff {a : m.syn} : a = 0 ↔ a ≤ 0 := eq_bot_iff\n\nlemma toSyn_eq_zero_iff (a : σ →₀ ℕ) :\n    m.toSyn a = 0 ↔ a = 0 := AddEquiv.map_eq_zero_iff m.toSyn\n\nlemma toSyn_lt_iff_ne_zero {a : m.syn} :\n    0 < a ↔ a ≠ 0 := bot_lt_iff_ne_bot\n\nlemma toSyn_strictMono : StrictMono (m.toSyn) := by\n  apply m.toSyn_monotone.strictMono_of_injective m.toSyn.injective\n\n/-- Given a monomial order, notation for the corresponding strict order relation on `σ →₀ ℕ` -/\nscoped\nnotation:50 c \" ≺[\" m:25 \"] \" d:50 => (MonomialOrder.toSyn m c < MonomialOrder.toSyn m d)\n\n/-- Given a monomial order, notation for the corresponding order relation on `σ →₀ ℕ` -/\nscoped\nnotation:50 c \" ≼[\" m:25 \"] \" d:50 => (MonomialOrder.toSyn m c ≤ MonomialOrder.toSyn m d)\n\nend MonomialOrder\n\nsection Lex\n\nopen Finsupp\n\nopen scoped MonomialOrder\n\n-- The linear order on `Finsupp`s obtained by the lexicographic ordering. -/\nnoncomputable instance {α N : Type*} [LinearOrder α]\n    [AddCommMonoid N] [PartialOrder N] [IsOrderedCancelAddMonoid N] :\n    IsOrderedCancelAddMonoid (Lex (α →₀ N)) where\n  le_of_add_le_add_left a b c h := by simpa only [add_le_add_iff_left] using h\n  add_le_add_left a b h c := by simpa using h\n\n/-- for the lexicographic ordering, X 0 * X 1 < X 0 ^ 2 -/\nexample : toLex (Finsupp.single 0 2) > toLex (Finsupp.single 0 1 + Finsupp.single 1 1) := by\n  use 0; simp\n\n/-- for the lexicographic ordering, X 1 < X 0 -/\nexample : toLex (Finsupp.single 1 1) < toLex (Finsupp.single 0 1) := by\n  use 0; simp\n\n/-- for the lexicographic ordering, X 1 < X 0 ^ 2 -/\nexample : toLex (Finsupp.single 1 1) < toLex (Finsupp.single 0 2) := by\n  use 0; simp\n\nvariable {σ : Type*} [LinearOrder σ]\n\n/-- The lexicographic order on `σ →₀ ℕ`, as a `MonomialOrder` -/\nnoncomputable def MonomialOrder.lex [WellFoundedGT σ] :\n    MonomialOrder σ where\n  syn := Lex (σ →₀ ℕ)\n  toSyn :=\n  { toEquiv := toLex\n    map_add' := toLex_add }\n  toSyn_monotone := Finsupp.toLex_monotone\n\ntheorem MonomialOrder.lex_le_iff [WellFoundedGT σ] {c d : σ →₀ ℕ} :\n    c ≼[lex] d ↔ toLex c ≤ toLex d := Iff.rfl\n\ntheorem MonomialOrder.lex_lt_iff [WellFoundedGT σ] {c d : σ →₀ ℕ} :\n    c ≺[lex] d ↔ toLex c < toLex d := Iff.rfl\n\nTarget:\ntheorem MonomialOrder.lex_lt_iff_of_unique [Unique σ] {c d : σ →₀ ℕ} :\n    c ≺[lex] d ↔ c default < d default :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_be98270d4e64","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"a102d739eac1f3dd7ed21bb61800a3923f35ee6d358e86e1001f89e0dc92a11a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finsupp","family_id":"monomialorder","file_id":"mathlib/Mathlib/Data/Finsupp/MonomialOrder.lean","sample_id":"be98270d4e641660001524cb4a6a1e9cfae5fb603cff7ae0acf7c6d8fed02aa2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9a5efd93ec86ca2b78ac7e681d48ea70e75ceb2212f655294f2322b5b525cad8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"aad3a3baeb6dacb034a541f0080f4a6bbc685cc200a408a1414f15d0a2cae0b7","source_sha256":"129f2f3aa776146d58955f965fc598f57556bb308d25e687562d21d0c89b2ff2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply isCyclic_of_card_dvd_prime (p := 2)\n  simpa [card_perm] using factorial_dvd_factorial hα","hard_negative":false,"metrics":{"chosen_tokens":16,"rejected_tokens":3,"token_jaccard":0.055556,"token_length_ratio":0.1875},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"711b93fe2d8be1452273f6c56fd69626c7ce8427b002529cf474d9ffa2cbae6d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Perm\npublic import Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic\npublic import Mathlib.SetTheory.Cardinal.Finite\n\nNamespace:\nEquiv.Perm\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir\n-/\n/-! # Properties of `Equiv.Perm` on `Finite` types\n\nLet `α` be a `Finite` type.\n\n* `Nat.card_perm`: cardinality of `Equiv.Perm α`.\n\n* `Equiv.Perm.isCyclic_of_card_le_two`: if `Nat.card α ≤ 2`,\n  then `Equiv.Perm α` is cyclic.\n\n* `Equiv.Perm.isCyclic_iff_card_le_two`: `Equiv.Perm α` is cyclic iff `Nat.card α ≤ 2`.\n\n* `Equiv.Perm.isMulCommutative_iff_card_le_two`: `Equiv.Perm α` is commutative iff `Nat.card α ≤ 2`.\n\n-/\n\npublic section\n\nassert_not_exists Field\n\nopen Equiv Nat\n\nvariable {α : Type*} [Finite α]\n\nnamespace Nat\n\ntheorem card_perm : Nat.card (Perm α) = (Nat.card α)! := by\n  classical\n  have := Fintype.ofFinite α\n  rw [card_eq_fintype_card, card_eq_fintype_card, Fintype.card_perm]\n\nend Nat\n\nnamespace Equiv.Perm\n\nTarget:\ntheorem isCyclic_of_card_le_two (hα : Nat.card α ≤ 2) :\n    IsCyclic (Perm α) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finite","family_id":"iscyclic_of_card_le_two","file_id":"mathlib/Mathlib/Data/Finite/Perm.lean","sample_id":"aad3a3baeb6dacb034a541f0080f4a6bbc685cc200a408a1414f15d0a2cae0b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ebb5a2a7a52df61da4119d7165c562d64bfda64efb43081a7067ed8d5938555e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"40b2cc76633014d1e5dccc9d2cd9aa43dc95f8c9291dfcfb08d96f928e0a655e","source_sha256":"3b1ad71a44a0ba95fe172fff4adb35c1c3d0ab3dacc0cacca37d6024c51fd12b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [toComplex_def₂, toComplex_def₂]\n  exact congr_arg₂ _ rfl (Int.cast_neg _)","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":2,"token_jaccard":0.058824,"token_length_ratio":0.1},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"714c12810245961ba2256119eadc3a82adcdab15bb591b97a452458e355e4255","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Archimedean.Real.Basic\npublic import Mathlib.Data.Complex.Basic\npublic import Mathlib.Data.Nat.Prime.Basic\npublic import Mathlib.NumberTheory.Zsqrtd.Basic\n\nNamespace:\nGaussianInt\n\nLocal context:\n/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Gaussian integers\n\nThe Gaussian integers are complex integer, complex numbers whose real and imaginary parts are both\nintegers.\n\n## Main definitions\n\nThe Euclidean domain structure on `ℤ[i]` is defined in this file.\n\nThe homomorphism `GaussianInt.toComplex` into the complex numbers is also defined in this file.\n\n## See also\n\nSee `NumberTheory.Zsqrtd.QuadraticReciprocity` for:\n* `prime_iff_mod_four_eq_three_of_nat_prime`:\n  A prime natural number is prime in `ℤ[i]` if and only if it is `3` mod `4`\n\n## Notation\n\nThis file uses the local notation `ℤ[i]` for `GaussianInt`\n\n## Implementation notes\n\nGaussian integers are implemented using the more general definition `Zsqrtd`, the type of integers\nadjoined a square root of `d`, in this case `-1`. The definition is reducible, so that properties\nand definitions about `Zsqrtd` can easily be used.\n-/\n\n@[expose] public section\n\n\nopen Zsqrtd Complex\n\nopen scoped ComplexConjugate\n\n/-- The Gaussian integers, defined as `ℤ√(-1)`. -/\nabbrev GaussianInt : Type :=\n  Zsqrtd (-1)\n\nlocal notation \"ℤ[i]\" => GaussianInt\n\nnamespace GaussianInt\n\ninstance : Repr ℤ[i] :=\n  ⟨fun x _ => \"⟨\" ++ repr x.re ++ \", \" ++ repr x.im ++ \"⟩\"⟩\n\ninstance instCommRing : CommRing ℤ[i] :=\n  Zsqrtd.commRing\n\nsection\n\nattribute [-instance] Complex.instField -- Avoid making things noncomputable unnecessarily.\n\n/-- The embedding of the Gaussian integers into the complex numbers, as a ring homomorphism. -/\ndef toComplex : ℤ[i] →+* ℂ :=\n  Zsqrtd.lift ⟨I, by simp⟩\n\nend\n\ninstance : Coe ℤ[i] ℂ :=\n  ⟨toComplex⟩\n\ntheorem toComplex_def (x : ℤ[i]) : (x : ℂ) = x.re + x.im * I :=\n  rfl\n\ntheorem toComplex_def' (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ) = x + y * I := by simp [toComplex_def]\n\ntheorem toComplex_def₂ (x : ℤ[i]) : (x : ℂ) = ⟨x.re, x.im⟩ := by\n  apply Complex.ext <;> simp [toComplex_def]\n\n@[simp]\ntheorem intCast_re (x : ℤ[i]) : ((x.re : ℤ) : ℝ) = (x : ℂ).re := by simp [toComplex_def]\n\n@[simp]\ntheorem intCast_im (x : ℤ[i]) : ((x.im : ℤ) : ℝ) = (x : ℂ).im := by simp [toComplex_def]\n\n@[simp]\ntheorem re_toComplex (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).re = x := by simp [toComplex_def]\n\n@[simp]\ntheorem im_toComplex (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).im = y := by simp [toComplex_def]\n\ntheorem toComplex_add (x y : ℤ[i]) : ((x + y : ℤ[i]) : ℂ) = x + y :=\n  toComplex.map_add _ _\n\ntheorem toComplex_mul (x y : ℤ[i]) : ((x * y : ℤ[i]) : ℂ) = x * y :=\n  toComplex.map_mul _ _\n\ntheorem toComplex_one : ((1 : ℤ[i]) : ℂ) = 1 :=\n  toComplex.map_one\n\ntheorem toComplex_zero : ((0 : ℤ[i]) : ℂ) = 0 :=\n  toComplex.map_zero\n\ntheorem toComplex_neg (x : ℤ[i]) : ((-x : ℤ[i]) : ℂ) = -x :=\n  toComplex.map_neg _\n\ntheorem toComplex_sub (x y : ℤ[i]) : ((x - y : ℤ[i]) : ℂ) = x - y :=\n  toComplex.map_sub _ _\n\n@[simp]\n\nTarget:\ntheorem toComplex_star (x : ℤ[i]) : ((star x : ℤ[i]) : ℂ) = conj (x : ℂ) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Zsqrtd","family_id":"tocomplex_star","file_id":"mathlib/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean","sample_id":"40b2cc76633014d1e5dccc9d2cd9aa43dc95f8c9291dfcfb08d96f928e0a655e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a6a0730acc445e18a71efd0b82b6e6d57aa672e42e9dfc4ec8e49f142e23c559","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"510546c198df0e51ff7856781031b1d5536b836338687232036e8eccc1052100","source_sha256":"94021f194f9dbe2b3ff10c0fe9e60d21f920f7ec47e18de0732170ff2d7ff47e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree, ← not_exists]\n  simp only [hr, not_true, false_or, and_false]","hard_negative":true,"metrics":{"chosen_tokens":19,"rejected_tokens":8,"token_jaccard":0.05,"token_length_ratio":0.421053},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"723b7a339aad6bb628855fecf2ce6e5f1cac375832de65f03ff840d6326a56cc","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Coprime.Lemmas\npublic import Mathlib.RingTheory.Nilpotent.Basic\npublic import Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid\npublic import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Squarefree elements of monoids\nAn element of a monoid is squarefree when it is not divisible by any squares\nexcept the squares of units.\n\nResults about squarefree natural numbers are proved in `Data.Nat.Squarefree`.\n\n## Main Definitions\n- `Squarefree r` indicates that `r` is only divisible by `x * x` if `x` is a unit.\n\n## Main Results\n- `multiplicity.squarefree_iff_emultiplicity_le_one`: `x` is `Squarefree` iff for every `y`, either\n  `emultiplicity y x ≤ 1` or `IsUnit y`.\n- `UniqueFactorizationMonoid.squarefree_iff_nodup_factors`: A nonzero element `x` of a unique\n  factorization monoid is squarefree iff `factors x` has no duplicate factors.\n\n## Tags\nsquarefree, multiplicity\n\n-/\n\n@[expose] public section\n\n\nvariable {R : Type*}\n\n/-- An element of a monoid is squarefree if the only squares that\n  divide it are the squares of units. -/\ndef Squarefree [Monoid R] (r : R) : Prop :=\n  ∀ x : R, x * x ∣ r → IsUnit x\n\ntheorem IsRelPrime.of_squarefree_mul [CommMonoid R] {m n : R} (h : Squarefree (m * n)) :\n    IsRelPrime m n := fun c hca hcb ↦ h c (mul_dvd_mul hca hcb)\n\n@[simp]\ntheorem IsUnit.squarefree [CommMonoid R] {x : R} (h : IsUnit x) : Squarefree x := fun _ hdvd =>\n  isUnit_of_mul_isUnit_left (isUnit_of_dvd_unit hdvd h)\n\ntheorem squarefree_one [CommMonoid R] : Squarefree (1 : R) :=\n  isUnit_one.squarefree\n\n@[simp]\ntheorem not_squarefree_zero [MonoidWithZero R] [Nontrivial R] : ¬Squarefree (0 : R) := by\n  rw [Squarefree, not_forall]\n  exact ⟨0, by simp⟩\n\ntheorem Squarefree.ne_zero [MonoidWithZero R] [Nontrivial R] {m : R} (hm : Squarefree (m : R)) :\n    m ≠ 0 := by\n  rintro rfl\n  exact not_squarefree_zero hm\n\n@[simp]\ntheorem Irreducible.squarefree [CommMonoid R] {x : R} (h : Irreducible x) : Squarefree x := by\n  rintro y ⟨z, hz⟩\n  rw [mul_assoc] at hz\n  rcases h.isUnit_or_isUnit hz with (hu | hu)\n  · exact hu\n  · apply isUnit_of_mul_isUnit_left hu\n\n@[simp]\ntheorem Prime.squarefree [CommMonoidWithZero R] [IsCancelMulZero R] {x : R} (h : Prime x) :\n    Squarefree x :=\n  h.irreducible.squarefree\n\ntheorem Squarefree.of_mul_left [Monoid R] {m n : R} (hmn : Squarefree (m * n)) : Squarefree m :=\n  fun p hp => hmn p (dvd_mul_of_dvd_left hp n)\n\ntheorem Squarefree.of_mul_right [CommMonoid R] {m n : R} (hmn : Squarefree (m * n)) :\n    Squarefree n := fun p hp => hmn p (dvd_mul_of_dvd_right hp m)\n\ntheorem Squarefree.squarefree_of_dvd [Monoid R] {x y : R} (hdvd : x ∣ y) (hsq : Squarefree y) :\n    Squarefree x := fun _ h => hsq _ (h.trans hdvd)\n\ntheorem Associated.squarefree_iff [Monoid R] {x y : R} (h : Associated x y) :\n    Squarefree x ↔ Squarefree y :=\n  ⟨fun hx ↦ hx.squarefree_of_dvd h.dvd', fun hy ↦ hy.squarefree_of_dvd h.dvd⟩\n\ntheorem Squarefree.eq_zero_or_one_of_pow_of_not_isUnit [Monoid R] {x : R} {n : ℕ}\n    (h : Squarefree (x ^ n)) (h' : ¬ IsUnit x) :\n    n = 0 ∨ n = 1 := by\n  contrapose! h'\n  replace h' : 2 ≤ n := by lia\n  have : x * x ∣ x ^ n := by rw [← sq]; exact pow_dvd_pow x h'\n  exact h.squarefree_of_dvd this x (refl _)\n\ntheorem Squarefree.pow_dvd_of_pow_dvd [Monoid R] {x y : R} {n : ℕ}\n    (hx : Squarefree y) (h : x ^ n ∣ y) : x ^ n ∣ x := by\n  by_cases hu : IsUnit x\n  · exact (hu.pow n).dvd\n  · rcases (hx.squarefree_of_dvd h).eq_zero_or_one_of_pow_of_not_isUnit hu with rfl | rfl <;> simp\n\nsection SquarefreeGcdOfSquarefree\n\nvariable {α : Type*} [CommMonoidWithZero α] [GCDMonoid α]\n\ntheorem Squarefree.gcd_right (a : α) {b : α} (hb : Squarefree b) : Squarefree (gcd a b) :=\n  hb.squarefree_of_dvd (gcd_dvd_right _ _)\n\ntheorem Squarefree.gcd_left {a : α} (b : α) (ha : Squarefree a) : Squarefree (gcd a b) :=\n  ha.squarefree_of_dvd (gcd_dvd_left _ _)\n\nend SquarefreeGcdOfSquarefree\n\ntheorem squarefree_iff_emultiplicity_le_one [CommMonoid R] (r : R) :\n    Squarefree r ↔ ∀ x : R, emultiplicity x r ≤ 1 ∨ IsUnit x := by\n  refine forall_congr' fun a => ?_\n  rw [← sq, pow_dvd_iff_le_emultiplicity, or_iff_not_imp_left, not_le, imp_congr _ Iff.rfl]\n  norm_cast\n  rw [← one_add_one_eq_two]\n  exact Order.add_one_le_iff_of_not_isMax (by simp)\n\nsection Irreducible\n\nvariable [CommMonoidWithZero R] [WfDvdMonoid R]\n\ntheorem squarefree_iff_no_irreducibles {x : R} (hx₀ : x ≠ 0) :\n    Squarefree x ↔ ∀ p, Irreducible p → ¬ (p * p ∣ x) := by\n  refine ⟨fun h p hp hp' ↦ hp.not_isUnit (h p hp'), fun h d hd ↦ by_contra fun hdu ↦ ?_⟩\n  have hd₀ : d ≠ 0 := ne_zero_of_dvd_ne_zero (ne_zero_of_dvd_ne_zero hx₀ hd) (dvd_mul_left d d)\n  obtain ⟨p, irr, dvd⟩ := WfDvdMonoid.exists_irreducible_factor hdu hd₀\n  exact h p irr ((mul_dvd_mul dvd dvd).trans hd)\n\ntheorem irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree (r : R) :\n    (∀ x : R, Irreducible x → ¬x * x ∣ r) ↔ (r = 0 ∧ ∀ x : R, ¬Irreducible x) ∨ Squarefree r := by\n  refine ⟨fun h ↦ ?_, ?_⟩\n  · rcases eq_or_ne r 0 with (rfl | hr)\n    · exact .inl (by simpa using h)\n    · exact .inr ((squarefree_iff_no_irreducibles hr).mpr h)\n  · rintro (⟨rfl, h⟩ | h)\n    · simpa using h\n    intro x hx t\n    exact hx.not_isUnit (h x t)\n\ntheorem squarefree_iff_irreducible_sq_not_dvd_of_ne_zero {r : R} (hr : r ≠ 0) :\n    Squarefree r ↔ ∀ x : R, Irreducible x → ¬x * x ∣ r := by\n  simpa [hr] using (irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree r).symm\n\nTarget:\ntheorem squarefree_iff_irreducible_sq_not_dvd_of_exists_irreducible {r : R}\n    (hr : ∃ x : R, Irreducible x) : Squarefree r ↔ ∀ x : R, Irreducible x → ¬x * x ∣ r :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"3eeb917ca4946ca631feaf1139eeb44cb63e6768c58d836ebc2806e74847831d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Squarefree","family_id":"squarefree_iff_irreducible_sq_not_dvd_of_exists_irreducible","file_id":"mathlib/Mathlib/Algebra/Squarefree/Basic.lean","sample_id":"510546c198df0e51ff7856781031b1d5536b836338687232036e8eccc1052100"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5e5876ae05a155ef3db2c120345927956dab48f54ef044b4e48039170ed59c7d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b8a6fc429e8fab84bafaec67f5918625193c8c32c8b4d06caaef778ad29f2b44","source_sha256":"fcd4a8db6b3cd3d3eac0ed5f4bd02b4de008d2f8c57c99f188ee276c869b4813","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases f; cases g\n  dsimp at hfg₁ hfg₂\n  cat_disch","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.153846},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"724348b5bba5c16b637bfc2a008231021f39e73a6adab3a9af6059f65f56fdab","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Bicategory.LocallyDiscrete\npublic import Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo\n\nNamespace:\nCategoryTheory.Pseudofunctor.Grothendieck\n\nLocal context:\n/-\nCopyright (c) 2024 Calle Sönne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Calle Sönne, Joseph Hua\n-/\n/-!\n# The Grothendieck and CoGrothendieck constructions\n\n## The Grothendieck construction\n\nGiven a category `𝒮` and any pseudofunctor `F` from `𝒮` to `Cat`, we associate to it a category\n`∫ F`, defined as follows:\n* Objects: pairs `(S, a)` where `S` is an object of the base category and `a` is an object of the\n  category `F(S)`.\n* Morphisms: morphisms `(R, b) ⟶ (S, a)` are defined as pairs `(f, h)` where `f : R ⟶ S` is a\n  morphism in `𝒮` and `h : F(f)(a) ⟶ b`\n\nThe category `∫ F` is equipped with a projection functor `∫ F ⥤ 𝒮`,\ngiven by projecting to the first factors, i.e.\n* On objects, it sends `(S, a)` to `S`\n* On morphisms, it sends `(f, h)` to `f`\n\n## The CoGrothendieck construction\n\nGiven a category `𝒮` and any pseudofunctor `F` from `𝒮ᵒᵖ` to `Cat`,\nwe associate to it a category `∫ᶜ F`, defined as follows:\n* Objects: pairs `(S, a)` where `S` is an object of the base category and `a` is an object of the\n  category `F(S)`.\n* Morphisms: morphisms `(R, b) ⟶ (S, a)` are defined as pairs `(f, h)` where `f : R ⟶ S` is a\n  morphism in `𝒮` and `h : b ⟶ F(f)(a)`\n\nThe category `∫ᶜ F` is equipped with a functor `∫ᶜ F ⥤ 𝒮`,\ngiven by projecting to the first factors, i.e.\n* On objects, it sends `(S, a)` to `S`\n* On morphisms, it sends `(f, h)` to `f`\n\n## Naming conventions\n\nThe name `Grothendieck` is reserved for the construction on covariant pseudofunctors from `𝒮` to\n`Cat`, whereas the word `CoGrothendieck` is used for the contravariant construction.\nThis is consistent with the convention for the Grothendieck construction on 1-functors\n`CategoryTheory.Grothendieck`.\n\n## Future work / TODO\n\n1. Once the bicategory of pseudofunctors has been defined, show that this construction forms a\n   pseudofunctor from `LocallyDiscrete 𝒮 ⥤ᵖ Catᵒᵖ` to `Cat`.\n2. Deduce the results in `CategoryTheory.Grothendieck` as a specialization of\n   `Pseudofunctor.Grothendieck`.\n\n## References\n[Vistoli2008] \"Notes on Grothendieck Topologies, Fibered Categories and Descent Theory\" by\nAngelo Vistoli\n\n-/\n\nset_option backward.defeqAttrib.useBackward true\n\n@[expose] public section\n\nnamespace CategoryTheory.Pseudofunctor\n\nuniverse w v₁ v₂ v₃ u₁ u₂ u₃\n\nopen Functor Category Opposite Discrete Bicategory StrongTrans\n\nvariable {𝒮 : Type u₁} [Category.{v₁} 𝒮]\n\n/-- The type of objects in the fibered category associated to a pseudofunctor from a\n1-category to Cat. -/\n@[ext]\nstructure Grothendieck (F : LocallyDiscrete 𝒮 ⥤ᵖ Cat.{v₂, u₂}) where\n  /-- The underlying object in the base category. -/\n  base : 𝒮\n  /-- The object in the fiber of the base object. -/\n  fiber : F.obj ⟨base⟩\n\nnamespace Grothendieck\n\nvariable {F : LocallyDiscrete 𝒮 ⥤ᵖ Cat.{v₂, u₂}}\n\n/-- Notation for the Grothendieck category associated to a pseudofunctor `F`. -/\nscoped prefix:75 \"∫ \" => Grothendieck\n\n/-- A morphism in the Grothendieck construction `∫ F` between two points `X Y : ∫ F` consists of\na morphism in the base category `base : X.base ⟶ Y.base` and\na morphism in a fiber `f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber`. -/\nstructure Hom (X Y : ∫ F) where\n  /-- The morphism between base objects. -/\n  base : X.base ⟶ Y.base\n  /-- The morphism in the fiber over the domain. -/\n  fiber : (F.map base.toLoc).toFunctor.obj X.fiber ⟶ Y.fiber\n\n@[simps! id_base id_fiber comp_base comp_fiber]\ninstance categoryStruct : CategoryStruct (∫ F) where\n  Hom X Y := Hom X Y\n  id X := {\n    base := 𝟙 X.base\n    fiber := (F.mapId ⟨X.base⟩).hom.toNatTrans.app X.fiber }\n  comp {X _ _} f g := {\n    base := f.base ≫ g.base\n    fiber := (F.mapComp f.base.toLoc g.base.toLoc).hom.toNatTrans.app X.fiber ≫\n      (F.map g.base.toLoc).toFunctor.map f.fiber ≫ g.fiber }\n\ninstance (X : ∫ F) : Inhabited (Hom X X) :=\n  ⟨𝟙 X⟩\n\nsection\n\nvariable {a b : ∫ F}\n\n@[ext (iff := false)]\n\nTarget:\nlemma Hom.ext (f g : a ⟶ b) (hfg₁ : f.base = g.base)\n    (hfg₂ : eqToHom (hfg₁ ▸ rfl) ≫ f.fiber = g.fiber) : f = g :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Bicategory","family_id":"hom","file_id":"mathlib/Mathlib/CategoryTheory/Bicategory/Grothendieck.lean","sample_id":"b8a6fc429e8fab84bafaec67f5918625193c8c32c8b4d06caaef778ad29f2b44"},"split":"train"}
{"binding":{"chosen_proof_sha256":"61c7560cab56bf0e5d3f545a1a32c8f50733fcb31780bdea8dc6b7263c7d7328","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9d2bf5ac2ac3150f033b3aa1e7ec2bd054d3367fa85eb8a333d0bfe6e07859d8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"35877c6cf79fa66db90e5991ad29f2b3f093444f285a8e92079a5bb760e5281a","source_sha256":"d1220062a6e285767ee3caf437f2e0e411175bb2084db2baf1a505199a4fb98e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI : NumberField K := @NumberField.mk _ _ _ (IsCyclotomicExtension.finiteDimensional {n} ℚ K)\n  have H₁ : (aeval (hζ.powerBasis ℚ).gen) (X - 1 : ℤ[X]) = (hζ.subOnePowerBasis ℚ).gen := by simp\n  have H₂ : (aeval (hζ.subOnePowerBasis ℚ).gen) (X + 1 : ℤ[X]) = (hζ.powerBasis ℚ).gen := by simp\n  refine discr_eq_discr_of_toMatrix_coeff_isIntegral _ (fun i j => toMatrix_isIntegral H₁ ?_ ?_ _ _)\n    fun i j => toMatrix_isIntegral H₂ ?_ ?_ _ _\n  · exact hζ.isIntegral (NeZero.pos _)\n  · refine minpoly.isIntegrallyClosed_eq_field_fractions' (K := ℚ) (hζ.isIntegral (NeZero.pos _))\n  · exact (hζ.isIntegral (NeZero.pos _)).sub isIntegral_one\n  · refine minpoly.isIntegrallyClosed_eq_field_fractions' (K := ℚ) ?_\n    exact (hζ.isIntegral (NeZero.pos _)).sub isIntegral_one","hard_negative":true,"metrics":{"chosen_tokens":211,"rejected_tokens":3,"token_jaccard":0.036364,"token_length_ratio":0.014218},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"729481b6fc06b267d2b7f4a5aa1137432021d685ce9385836a75b793169fa0f3","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots\npublic import Mathlib.RingTheory.DedekindDomain.Dvr\npublic import Mathlib.NumberTheory.NumberField.Discriminant.Defs\n\nNamespace:\nIsPrimitiveRoot\n\nLocal context:\n/-\nCopyright (c) 2022 Riccardo Brasca. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Riccardo Brasca\n-/\n/-!\n# Discriminant of cyclotomic fields\nWe compute the discriminant of a `p ^ n`-th cyclotomic extension.\n\n## Main results\n* `IsCyclotomicExtension.discr_odd_prime` : if `p` is an odd prime such that\n  `IsCyclotomicExtension {p} K L` and `Irreducible (cyclotomic p K)`, then\n  `discr K (hζ.powerBasis K).basis = (-1) ^ ((p - 1) / 2) * p ^ (p - 2)` for any\n  `hζ : IsPrimitiveRoot ζ p`.\n\n-/\n\npublic section\n\n\nuniverse u v\n\nopen Algebra Polynomial Nat IsPrimitiveRoot PowerBasis\n\nopen scoped Polynomial Cyclotomic\n\nnamespace IsPrimitiveRoot\n\nvariable {n : ℕ} [NeZero n] {K : Type u} [Field K] [CharZero K] {ζ : K}\nvariable [ce : IsCyclotomicExtension {n} ℚ K]\n\n/-- The discriminant of the power basis given by a primitive root of unity `ζ` is the same as the\ndiscriminant of the power basis given by `ζ - 1`. -/\n\nTarget:\ntheorem discr_zeta_eq_discr_zeta_sub_one (hζ : IsPrimitiveRoot ζ n) :\n    discr ℚ (hζ.powerBasis ℚ).basis = discr ℚ (hζ.subOnePowerBasis ℚ).basis :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_35877c6cf79f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"3634919b7b582a179393e98c6e2abeeeaffdb548451d95d5ed3d333ebbffe312","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Cyclotomic","family_id":"discr_zeta_eq_discr_zeta_sub_one","file_id":"mathlib/Mathlib/NumberTheory/Cyclotomic/Discriminant.lean","sample_id":"35877c6cf79fa66db90e5991ad29f2b3f093444f285a8e92079a5bb760e5281a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"097944a1a41dd54c4d08892020794d36ebd2632227e02e3aff759ac09d3dd6c6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f7b0ddcb6f37e2b9d3ae884fa2def989d9b43bbe1c28441bdb81f2248248cfed","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b15d779c153730a548af1bcb1242b64da3cdb5f65c04ae8beeb21d386329dcf8","source_sha256":"6ea826f0633d3930f7872110f571f16f76059ec3668e205ef02a1089b9b31873","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI : Nonempty ι := Fintype.card_pos_iff.mp (hn.symm ▸ Nat.succ_pos _)\n  haveI := Classical.decEq ι\n  -- Choose an arbitrary element `x : ι` and `y : t x`.\n  let x := Classical.arbitrary ι\n  have tx_ne : (t x).Nonempty := by\n    rw [← Finset.card_pos]\n    calc\n      0 < 1 := Nat.one_pos\n      _ ≤ #(.biUnion {x} t) := ht {x}\n      _ = (t x).card := by rw [Finset.singleton_biUnion]\n  choose y hy using tx_ne\n  -- Restrict to everything except `x` and `y`.\n  let ι' := { x' : ι | x' ≠ x }\n  let t' : ι' → Finset α := fun x' => (t x').erase y\n  have card_ι' : Fintype.card ι' = n :=\n    calc\n      Fintype.card ι' = Fintype.card ι - 1 := Set.card_ne_eq _\n      _ = n := by rw [hn, Nat.add_succ_sub_one, add_zero]\n  rcases ih t' card_ι'.le (hall_cond_of_erase y ha) with ⟨f', hfinj, hfr⟩\n  -- Extend the resulting function.\n  refine ⟨fun z => if h : z = x then y else f' ⟨z, h⟩, ?_, ?_⟩\n  · rintro z₁ z₂\n    have key : ∀ {x}, y ≠ f' x := by\n      intro x h\n      simpa [t', ← h] using hfr x\n    by_cases h₁ : z₁ = x <;> by_cases h₂ : z₂ = x <;>\n      simp [h₁, h₂, hfinj.eq_iff, key, key.symm]\n  · intro z\n    simp only\n    split_ifs with hz\n    · rwa [hz]\n    · specialize hfr ⟨z, hz⟩\n      rw [mem_erase] at hfr\n      exact hfr.2","hard_negative":true,"metrics":{"chosen_tokens":367,"rejected_tokens":2,"token_jaccard":0.008197,"token_length_ratio":0.00545},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"72b3eb706509af3ab55c19e72abe4d777c44a86a4af0ab6e578833c623d9f6e8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Basic\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Data.Set.Finite.Basic\n\nNamespace:\nHallMarriageTheorem\n\nLocal context:\n/-\nCopyright (c) 2021 Alena Gusakov, Bhavik Mehta, Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alena Gusakov, Bhavik Mehta, Kyle Miller\n-/\n/-!\n# Hall's Marriage Theorem for finite index types\n\nThis module proves the basic form of Hall's theorem.\nIn contrast to the theorem described in `Combinatorics.Hall.Basic`, this\nversion requires that the indexed family `t : ι → Finset α` have `ι` be finite.\nThe `Combinatorics.Hall.Basic` module applies a compactness argument to this version\nto remove the `Finite` constraint on `ι`.\n\nThe modules are split like this since the generalized statement\ndepends on the topology and category theory libraries, but the finite\ncase in this module has few dependencies.\n\nA description of this formalization is in [Gusakov2021].\n\n## Main statements\n\n* `Finset.all_card_le_biUnion_card_iff_existsInjective'` is Hall's theorem with\n  a finite index set.  This is elsewhere generalized to\n  `Finset.all_card_le_biUnion_card_iff_existsInjective`.\n\n## Tags\n\nHall's Marriage Theorem, indexed families\n-/\n\npublic section\n\n\nopen Finset\n\nuniverse u v\n\nnamespace HallMarriageTheorem\n\nvariable {ι : Type u} {α : Type v} [DecidableEq α] {t : ι → Finset α}\n\nsection Fintype\n\nvariable [Fintype ι]\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem hall_cond_of_erase {x : ι} (a : α)\n    (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t))\n    (s' : Finset { x' : ι | x' ≠ x }) : #s' ≤ #(s'.biUnion fun x' => (t x').erase a) := by\n  haveI := Classical.decEq ι\n  specialize ha (s'.image fun z => z.1)\n  rw [image_nonempty, Finset.card_image_of_injective s' Subtype.coe_injective] at ha\n  by_cases! he : s'.Nonempty\n  · have ha' : #s' < #(s'.biUnion fun x => t x) := by\n      convert! ha he fun h => by simpa [← h] using mem_univ x using 2\n      ext x\n      simp only [mem_image, mem_biUnion, SetCoe.exists, exists_and_right,\n        exists_eq_right]\n    rw [← erase_biUnion]\n    by_cases hb : a ∈ s'.biUnion fun x => t x\n    · rw [card_erase_of_mem hb]\n      exact Nat.le_sub_one_of_lt ha'\n    · rw [erase_eq_of_notMem hb]\n      exact Nat.le_of_lt ha'\n  · subst s'\n    simp\n\n/-- First case of the inductive step: assuming that\n`∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)`\nand that the statement of **Hall's Marriage Theorem** is true for all\n`ι'` of cardinality ≤ `n`, then it is true for `ι` of cardinality `n + 1`.\n-/\n\nTarget:\ntheorem hall_hard_inductive_step_A {n : ℕ} (hn : Fintype.card ι = n + 1)\n    (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))\n    (ih :\n      ∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),\n        Fintype.card ι' ≤ n →\n          (∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →\n            ∃ f : ι' → α, Function.Injective f ∧ ∀ x, f x ∈ t' x)\n    (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t)) :\n    ∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_b15d779c1537","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"79b6cb0d082e5f0f9bb4a732a80eef3002e5fe15b6e3012c3b4a72cb5a112dcf","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Hall","family_id":"hall_hard_inductive_step_a","file_id":"mathlib/Mathlib/Combinatorics/Hall/Finite.lean","sample_id":"b15d779c153730a548af1bcb1242b64da3cdb5f65c04ae8beeb21d386329dcf8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d69154e0d9492b7c81193d7944884dcb10496ed41d6438469a8fe12e15c51a76","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"214a60175ac5e85f8cd7bac259c887da208681562d1e346666da9d00bb10432c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6a8521660d4ef846f6eee10f1b6a7111248906a80446120366f799acf8507e31","source_sha256":"6c1d4f9fced239f6f8c0c8785a4325f383edabba30747fe08c8262d3408cd815","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h => ?_, fun ⟨⟨a', m, e⟩, s⟩ => Subset.cons e m s⟩\n  generalize h' : Lists'.cons a l₁ = l₁' at h\n  obtain - | @⟨a', _, _, _, e, m, s⟩ := h\n  · cases a\n    cases h'\n  cases a; cases a'; cases h'; exact ⟨⟨_, m, e⟩, s⟩","hard_negative":true,"metrics":{"chosen_tokens":91,"rejected_tokens":3,"token_jaccard":0.054054,"token_length_ratio":0.032967},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"72e9a62dc1a3921b53bf4b1f349c404504d34a92aad7fdb44286311051c8eb06","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.Monoid.NatCast\npublic import Mathlib.Algebra.Ring.Nat\npublic import Mathlib.Data.Sigma.Basic\npublic import Batteries.Tactic.Lint.TypeClass\n\nNamespace:\nLists'\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# A computable model of ZFA without infinity\n\nIn this file we define finite hereditary lists. This is useful for calculations in naive set theory.\n\nWe distinguish two kinds of ZFA lists:\n* Atoms. Directly correspond to an element of the original type.\n* Proper ZFA lists. Can be thought of (but are not implemented) as a list of ZFA lists (not\n  necessarily proper).\n\nFor example, `Lists ℕ` contains stuff like `23`, `[]`, `[37]`, `[1, [[2], 3], 4]`.\n\n## Implementation note\n\nAs we want to be able to append both atoms and proper ZFA lists to proper ZFA lists, it's handy that\natoms and proper ZFA lists belong to the same type, even though atoms of `α` could be modelled as\n`α` directly. But we don't want to be able to append anything to atoms.\n\nThis calls for a two-step definition of ZFA lists:\n* First, define ZFA prelists as atoms and proper ZFA prelists. Those proper ZFA prelists are defined\n  by inductive appending of (not necessarily proper) ZFA lists.\n* Second, define ZFA lists by rubbing out the distinction between atoms and proper lists.\n\n## Main declarations\n\n* `Lists' α false`: Atoms as ZFA prelists. Basically a copy of `α`.\n* `Lists' α true`: Proper ZFA prelists. Defined inductively from the empty ZFA prelist\n  (`Lists'.nil`) and from appending a ZFA prelist to a proper ZFA prelist (`Lists'.cons a l`).\n* `Lists α`: ZFA lists. Sum of the atoms and proper ZFA prelists.\n* `Finsets α`: ZFA sets. Defined as `Lists` quotiented by `Lists.Equiv`, the extensional\n  equivalence.\n-/\n\n@[expose] public section\n\n\nvariable {α : Type*}\n\n/-- Prelists, helper type to define `Lists`. `Lists' α false` are the \"atoms\", a copy of `α`.\n`Lists' α true` are the \"proper\" ZFA prelists, inductively defined from the empty ZFA prelist and\nfrom appending a ZFA prelist to a proper ZFA prelist. It is made so that you can't append anything\nto an atom while having only one appending function for appending both atoms and proper ZFA prelists\nto a proper ZFA prelist. -/\ninductive Lists'.{u} (α : Type u) : Bool → Type u\n  | atom : α → Lists' α false\n  | nil : Lists' α true\n  | cons' {b} : Lists' α b → Lists' α true → Lists' α true\n  deriving DecidableEq\ncompile_inductive% Lists'\n\n/-- Hereditarily finite list, aka ZFA list. A ZFA list is either an \"atom\" (`b = false`),\ncorresponding to an element of `α`, or a \"proper\" ZFA list, inductively defined from the empty ZFA\nlist and from appending a ZFA list to a proper ZFA list. -/\ndef Lists (α : Type*) :=\n  Σ b, Lists' α b\n\nnamespace Lists'\n\ninstance [Inhabited α] : ∀ b, Inhabited (Lists' α b)\n  | true => ⟨nil⟩\n  | false => ⟨atom default⟩\n\n/-- Appending a ZFA list to a proper ZFA prelist. -/\ndef cons : Lists α → Lists' α true → Lists' α true\n  | ⟨_, a⟩, l => cons' a l\n\n/-- Converts a ZFA prelist to a `List` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp]\ndef toList : ∀ {b}, Lists' α b → List (Lists α)\n  | _, atom _ => []\n  | _, nil => []\n  | _, cons' a l => ⟨_, a⟩ :: l.toList\n\n@[simp]\ntheorem toList_cons (a : Lists α) (l) : toList (cons a l) = a :: l.toList := rfl\n\n/-- Converts a `List` of ZFA lists to a proper ZFA prelist. -/\n@[simp]\ndef ofList : List (Lists α) → Lists' α true\n  | [] => nil\n  | a :: l => cons a (ofList l)\n\n@[simp]\ntheorem to_ofList (l : List (Lists α)) : toList (ofList l) = l := by induction l <;> simp [*]\n\n@[simp]\ntheorem of_toList : ∀ l : Lists' α true, ofList (toList l) = l :=\n  suffices ∀ (b) (h : true = b) (l : Lists' α b),\n      let l' : Lists' α true := h ▸ l\n      ofList (toList l') = l'\n    from this _ rfl\n  fun b h l => by\n    induction l with\n    | atom => cases h\n    | nil => simp\n    | cons' b a _ IH => simpa [cons] using IH rfl\n\n/-- Recursion/induction principle for `Lists'.ofList`. -/\n@[elab_as_elim]\ndef recOfList {motive : Lists' α true → Sort*} (ofList : ∀ l, motive (ofList l)) : ∀ l, motive l :=\n  fun l ↦ cast (by simp) <| ofList (l.toList)\n\nend Lists'\n\nmutual\n  /-- Equivalence of ZFA lists. Defined inductively. -/\n  inductive Lists.Equiv : Lists α → Lists α → Prop\n    | refl (l) : Lists.Equiv l l\n    | antisymm {l₁ l₂ : Lists' α true} :\n      Lists'.Subset l₁ l₂ → Lists'.Subset l₂ l₁ → Lists.Equiv ⟨_, l₁⟩ ⟨_, l₂⟩\n\n  /-- Subset relation for ZFA lists. Defined inductively. -/\n  inductive Lists'.Subset : Lists' α true → Lists' α true → Prop\n    | nil {l} : Lists'.Subset Lists'.nil l\n    | cons {a a' l l'} :\n      Lists.Equiv a a' →\n        a' ∈ Lists'.toList l' → Lists'.Subset l l' → Lists'.Subset (Lists'.cons a l) l'\nend\n\nlocal infixl:50 \" ~ \" => Lists.Equiv\n\nnamespace Lists'\n\ninstance : HasSubset (Lists' α true) :=\n  ⟨Lists'.Subset⟩\n\n/-- ZFA prelist membership. A ZFA list is in a ZFA prelist if some element of this ZFA prelist is\nequivalent as a ZFA list to this ZFA list. -/\ninstance {b} : Membership (Lists α) (Lists' α b) :=\n  ⟨fun l a => ∃ a' ∈ l.toList, a ~ a'⟩\n\ntheorem mem_def {b a} {l : Lists' α b} : a ∈ l ↔ ∃ a' ∈ l.toList, a ~ a' :=\n  Iff.rfl\n\n@[simp]\ntheorem mem_cons {a y l} : a ∈ @cons α y l ↔ a ~ y ∨ a ∈ l := by\n  simp [mem_def, or_and_right, exists_or]\n\nTarget:\ntheorem cons_subset {a} {l₁ l₂ : Lists' α true} : Lists'.cons a l₁ ⊆ l₂ ↔ a ∈ l₂ ∧ l₁ ⊆ l₂ :=\n\nProof body:\n","rejected":"by\n  exact cons_subset","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"fbe7bf6cc9d05a9574443f0e80e225b84d5f7aa72975cbdc09b46367c805e5f4","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory","family_id":"cons_subset","file_id":"mathlib/Mathlib/SetTheory/Lists.lean","sample_id":"6a8521660d4ef846f6eee10f1b6a7111248906a80446120366f799acf8507e31"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bc0a4d32839ad3481e8623af1efc75f8290857aa2e5f1191a4c4a6c87919cec5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7c154fd643bb633d6de220155a96922617dc209b9d9b80785e09dfdb903c1f81","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0e6ca74f069156e403ca37b6a55058b4c1f3d343ff3c45b25117782dc63111e3","source_sha256":"cd2a9f6a6ac4ffdabcb0fdd545e558d6cc8573a01d198e082fa0d6a20deaa548","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply fun s ↦ (residue_ne_zero_iff_isUnit s).mp\n  rw [map_aeval_eq_aeval_map (ψ := residue S) (φ := residue R) rfl,\n    ← derivative_map, minpoly_map_residue hadj]\n  exact (Algebra.IsSeparable.isSeparable _ _).aeval_derivative_ne_zero (minpoly.aeval _ _)","hard_negative":true,"metrics":{"chosen_tokens":53,"rejected_tokens":2,"token_jaccard":0.028571,"token_length_ratio":0.037736},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"73038170996a2c96cef1dd4a9f719857b0936c52e8eeec00a9b44b293f174714","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.IsAdjoinRoot\npublic import Mathlib.RingTheory.LocalRing.Quotient\npublic import Mathlib.RingTheory.Smooth.Flat\npublic import Mathlib.RingTheory.Unramified.LocalRing\n\nNamespace:\nIsLocalRing\n\nLocal context:\n/-\nCopyright (c) 2026 University of Washington Math AI Lab. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bianca Viray, Bryan Boehnke, Grant Yang, George Peykanu, Tianshuo Wang\n-/\n\n/-!\n# Étale extensions of local rings\n\nWe prove that a finite étale extension of local rings is monogenic (generated by a single element),\nand that the derivative of the minimal polynomial evaluated at the generator is a unit.\nThese are parts 1 and 2 of Lemma 3.2 of [arXiv:2503.07846](https://arxiv.org/abs/2503.07846).\n\n## Main results\n\n* `IsLocalRing.exists_adjoin_eq_top`: a finite étale extension of local rings is generated by a\n  single element (Lemma 3.2, part 1).\n* `IsLocalRing.isUnit_aeval_derivative_minpoly_of_adjoin_eq_top`: if `R → S` is étale and\n  `R[β] = S`, then `f'(β)` is a unit, where `f = minpoly R β` (Lemma 3.2, part 2).\n\n## Key intermediate results\n\n* `IsLocalRing.adjoin_residue_eq_top_iff_adjoin_eq_top`: `β` generates `S` over `R` iff\n  `β mod m_S` generates `S/m_S` over `R/m_R`.\n* `IsLocalRing.finrank_eq_finrank_residueField`: for finite étale extensions of local rings,\n  `finrank R S = finrank (ResidueField R) (ResidueField S)`.\n* `IsLocalRing.minpoly_map_residue`: the minimal polynomial of `β` over `R` maps to the\n  minimal polynomial of `β mod m_S` over the residue field.\n\n## Future work\n\nThe following results from [arXiv:2503.07846](https://arxiv.org/abs/2503.07846) (formalized at [uw-math-ai/monogenic-extensions](https://github.com/uw-math-ai/monogenic-extensions)) are planned for\nfuture PRs:\n\n* **Converse**: If `S ≅ R[X]/(f)` with `f` monic and `f'(root)` a unit, then `R → S` is étale.\n* **Lemma 3.1** (partial étale case): If `R` and `S` are local integral domains with `R`\n  integrally closed, `S` a UFD, `R → S` finite and injective, and there exists a height-one\n  prime `q ⊆ S` such that `R/(q ∩ R) → S/q` is étale, then `S ≅ R[X]/(f)` for some monic `f`.\n\n## Tags\n\nétale, monogenic, local ring, minimal polynomial, residue field\n-/\n\n@[expose] public section\n\nnamespace IsLocalRing\n\nvariable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]\n    [IsLocalRing S] [IsLocalRing R] [Module.Finite R S] [FaithfulSMul R S]\n\nopen Polynomial IsLocalRing Algebra\n\n/-- When `β` generates `S` over `R`, the residue `β₀ = β mod m_S`\ngenerates `S/m_S` over `R/m_R`. -/\nlemma adjoin_residue_eq_top_iff_adjoin_eq_top [Algebra.FormallyUnramified R S] (β : S) :\n    Algebra.adjoin (ResidueField R) {residue S β} = ⊤ ↔ Algebra.adjoin R {β} = ⊤ := by\n  constructor\n  · intro hβ\n    refine eq_top_iff.mpr <| Submodule.le_of_le_smul_of_le_jacobson_bot\n      (Module.finite_def.mp inferInstance) (IsLocalRing.maximalIdeal_le_jacobson ⊥)\n      (?_ : ⊤ ≤ (adjoin R {β}).toSubmodule ⊔ maximalIdeal R • ⊤)\n    intro s _\n    rw [adjoin_singleton_eq_range_aeval, AlgHom.range_eq_top] at hβ\n    obtain ⟨p, hp⟩ := hβ (residue S s)\n    obtain ⟨q, rfl⟩ := Polynomial.map_surjective _ residue_surjective p\n    rw [Ideal.smul_top_eq_map]\n    refine Submodule.mem_sup.mpr ⟨aeval β q, ?_, s - aeval β q, ?_, by ring⟩\n    · rw [adjoin_singleton_eq_range_aeval]; exact ⟨q, rfl⟩\n    · rw [Algebra.FormallyUnramified.map_maximalIdeal, Submodule.restrictScalars_mem,\n        ← Ideal.Quotient.eq]\n      -- def eq abuse since IsLocalRing.residue needs to be an abbrev\n      rw [← map_aeval_eq_aeval_map (ψ := residue S) (φ := residue R) rfl] at hp\n      exact hp.symm\n  · intro hβ_gen\n    rw [Algebra.adjoin_singleton_eq_range_aeval, AlgHom.range_eq_top] at *\n    intro x\n    obtain ⟨s, rfl⟩ := residue_surjective (R := S) x\n    obtain ⟨p, rfl⟩ := hβ_gen s\n    exact ⟨p.map (residue R), by\n      rw [← map_aeval_eq_aeval_map (ψ := residue S) (φ := residue R) rfl p β]⟩\n\n/-- A finite étale extension of local rings is generated by a single element.\nThis is Lemma 3.2, part 1 of [arXiv:2503.07846](https://arxiv.org/abs/2503.07846).\nThe proof lifts a primitive element of the residue field extension via Nakayama's lemma. -/\ntheorem exists_adjoin_eq_top [Algebra.FormallyUnramified R S] :\n    ∃ β : S, Algebra.adjoin R {β} = ⊤ := by\n  obtain ⟨β₀, hβ₀⟩ := Field.exists_primitive_element (ResidueField R) (ResidueField S)\n  obtain ⟨β, hβ⟩ := residue_surjective (R := S) β₀\n  refine ⟨β, adjoin_residue_eq_top_iff_adjoin_eq_top β |>.mp ?_⟩\n  rw [hβ,\n    ← IntermediateField.adjoin_simple_toSubalgebra_of_isAlgebraic (IsAlgebraic.of_finite _ _),\n    hβ₀, IntermediateField.top_toSubalgebra]\n\n/-- For finite étale extensions of local rings,\n`finrank R S = finrank (ResidueField R) (ResidueField S)`. -/\nlemma finrank_eq_finrank_residueField [Algebra.Etale R S] :\n    Module.finrank R S =\n      Module.finrank (ResidueField R) (ResidueField S) := by\n  have : Module.Free R S := Module.free_of_flat_of_isLocalRing\n  have e := AddEquiv.toLinearEquiv (R := R ⧸ maximalIdeal R) (Ideal.quotEquivOfEq <|\n      Algebra.FormallyUnramified.map_maximalIdeal (R := R) (S := S)).toAddEquiv\n    ?_\n  · rw [← finrank_quotient_map (R := R) (S := S)]\n    exact e.finrank_eq -- again IsLocalRing.residue should be abbrev\n  · intro r x\n    obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective r\n    obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x\n    simp only [RingEquiv.toAddEquiv_eq_coe]; rfl\n\n/-- For a monogenic étale extension of local rings, the minimal polynomial of `β`\nmaps to the minimal polynomial of `β mod m_S` over the residue field. -/\nlemma minpoly_map_residue [Algebra.Etale R S]\n    {β : S} (hadj : Algebra.adjoin R {β} = ⊤) :\n    (minpoly R β).map (residue R) = minpoly (ResidueField R) (residue S β) := by\n  have h := minpoly.monic <| Algebra.IsIntegral.isIntegral (R := R) β\n  -- Both monic, same degree, divisibility ⟹ equal\n  refine eq_of_monic_of_dvd_of_natDegree_le\n    (minpoly.monic <| Algebra.IsIntegral.isIntegral <| residue S β)\n    (h.map _) (minpoly.dvd (ResidueField R) (residue S β) ?_) ?_\n  · rw [← map_aeval_eq_aeval_map (ψ := residue S) (φ := residue R) rfl]\n    simp\n  · have : Module.Free R S := Module.free_of_flat_of_isLocalRing\n    have hβ₀ := (adjoin_residue_eq_top_iff_adjoin_eq_top β).mpr hadj\n    rw [h.natDegree_map _,\n      ← (IsAdjoinRootMonic.mkOfAdjoinEqTop' hadj).finrank,\n      finrank_eq_finrank_residueField,\n      (IsAdjoinRootMonic.mkOfAdjoinEqTop' hβ₀).finrank]\n\n/-- If `R → S` is étale and `R[β] = S`, then `f'(β)` is a unit in `S`,\nwhere `f = minpoly R β`. The proof reduces to separability of the\nresidue field extension via `minpoly_map_residue`. -/\n\nTarget:\nlemma isUnit_aeval_derivative_minpoly_of_adjoin_eq_top\n    [Algebra.Etale R S] {β : S}\n    (hadj : Algebra.adjoin R {β} = ⊤) :\n    IsUnit (aeval β (minpoly R β).derivative) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_0e6ca74f0691","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"32b73c6080d4ab0a707397b21cf0a8042dda92368c6bc7f07d843ae3b5656ff0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/LocalRing","family_id":"isunit_aeval_derivative_minpoly_of_adjoin_eq_top","file_id":"mathlib/Mathlib/RingTheory/LocalRing/Etale.lean","sample_id":"0e6ca74f069156e403ca37b6a55058b4c1f3d343ff3c45b25117782dc63111e3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d7e6aa18e95f9b73927d19d13b21d8b699befd36d25410b98b9a340426e1233b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b4e99ad4afa8ca8b8a63a7f500ca31ddf489f9cb7a94710418b967810c2e789b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2f91e46e3433b86a1e3699a32e11802a7183cc285fe3179f472be8008081dcf8","source_sha256":"76ff543811de607fc5619ca4ca6c9c5878d83cf85372b02e848f8ebf54663e8c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain rfl | hn := eq_or_ne n 0\n  · simpa [ssubset_iff_subset_not_subset, hX₁, -Finset.subset_singleton_iff]\n      using! hX.not_subset_singleton\n  refine (pow_subset_pow_right hX₁ <| n.le_add_right _).ssubset_of_ne ?_\n  contrapose hXclosure with hXn\n  rw [← closure_pow (mod_cast hX₁) hn]\n  wlog hn₁ : n = 1\n  · simp +contextual only [pow_one] at this\n    replace hXn d : X ^ (n + d) = X ^ n := by\n      induction d with\n      | zero => rw [add_zero]\n      | succ d hd =>\n        rw [pow_add, pow_one] at hXn\n        rw [← add_assoc, pow_add, pow_one, hd, ← hXn]\n    exact mod_cast this (one_mem_pow hX₁) (hX.pow hn) one_ne_zero\n      (by simp [hXn, ← pow_mul, mul_two]) (by simp)\n  subst hn₁\n  simp only [ne_eq, one_ne_zero, not_false_eq_true, Nat.reduceAdd, pow_one] at *\n  let Xgp : Subgroup G :=\n  { carrier := X\n    mul_mem' := fun {x y} hx hy ↦ by\n      norm_cast at *\n      simpa [← hXn, ← sq] using! mul_mem_mul hx hy\n    one_mem' := hX₁\n    inv_mem' := fun {x} hx ↦ by\n      norm_cast at *\n      have : x • X ⊆ X := by\n        simpa [← hXn, add_assoc, ← sq] using! smul_finset_subset_mul (t := X) hx\n      have : x • X = X := eq_of_subset_of_card_le this (card_smul_finset ..).ge\n      rw [← eq_inv_smul_iff] at this\n      rw [this]\n      simpa [mem_inv_smul_finset_iff] }\n  exact subset_closure.antisymm <| (closure_le Xgp).2 subset_rfl","hard_negative":true,"metrics":{"chosen_tokens":307,"rejected_tokens":3,"token_jaccard":0.018018,"token_length_ratio":0.009772},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"732b975916060b779192d7889a54911aecf001716a59de5ad38fef92ada24267","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Pointwise.Finset\npublic import Mathlib.Algebra.Group.Subgroup.Pointwise\npublic import Mathlib.Data.Nat.SuccPred\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2024 Yaël Dillies, Patrick Luo, Eric Rodriguez. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Patrick Luo, Eric Rodriguez\n-/\n/-!\n# Linear lower bound on the growth of a generating set\n\nThis file proves that the growth of a set generating an infinite group is at least linear.\n-/\n\npublic section\n\nopen Subgroup\nopen scoped Pointwise\n\nnamespace Finset\nvariable {G : Type*} [Group G] [DecidableEq G] {X : Finset G} {n : ℕ}\n\n@[to_additive]\n\nTarget:\nlemma pow_ssubset_pow_succ_of_pow_ne_closure (hX₁ : (1 : G) ∈ X) (hX : X.Nontrivial)\n    (hXclosure : (X ^ n : Set G) ≠ closure (X : Set G)) : X ^ n ⊂ X ^ (n + 1) :=\n\nProof body:\n","rejected":"by\n  exact pow_ssubset_pow_succ_of_pow_ne_closure","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"06c8cac30534d0abea709c353e66c0f59f4a94fce155bf36b91bdfc8351f0811","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/Group","family_id":"pow_ssubset_pow_succ_of_pow_ne_closure","file_id":"mathlib/Mathlib/Geometry/Group/Growth/LinearLowerBound.lean","sample_id":"2f91e46e3433b86a1e3699a32e11802a7183cc285fe3179f472be8008081dcf8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d4714964e668b31f0718a64704bfc3c462347f6aac0cd63c4a504fd6654a6944","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bcfe71f4669f39454a991e49045a95b381eb505965239b851eaa93006f10ef03","source_sha256":"3ea4a54ee767d2f1814d8b90accb092fa9ec4a1fd88519664221a22cfd80fa82","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  subst s_card\n  simp [basis_apply]\n\nvariable (s : powersetCard I m) (t : powersetCard I n)","hard_negative":false,"metrics":{"chosen_tokens":22,"rejected_tokens":3,"token_jaccard":0.052632,"token_length_ratio":0.136364},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"733441983324e5f25e03f63cd4ee9898fe2f501939be6abd620a64de9c0903e0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.ExteriorAlgebra.Grading\npublic import Mathlib.LinearAlgebra.ExteriorPower.Basis\n\nNamespace:\nExteriorAlgebra\n\nLocal context:\n/-\nCopyright (c) 2026 Daniel Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Morrison\n-/\n/-!\n# Basis for `ExteriorAlgebra`\n-/\n\n@[expose] public section\n\nnamespace ExteriorAlgebra\n\nopen Module Set Set.powersetCard exteriorPower\n\nvariable {R M : Type*} {m n : ℕ} {I : Type*} [LinearOrder I] [CommRing R]\n  [AddCommGroup M] [Module R M] (b : Module.Basis I R M)\n\n/-- The direct sum decomposition of the exterior algebra from the graded algebra structure. -/\ninstance : DirectSum.Decomposition (fun n ↦ ⋀[R]^n M) :=\n  GradedRing.toDecomposition (self := ExteriorAlgebra.gradedAlgebra R M)\n\n/-- If `b` is a basis of `M` (indexed by a linearly ordered type), the basis of the exterior\nalgebra of `M` formed by the `n`-fold exterior products of elements of `b` for each `n`. -/\nnoncomputable def _root_.Module.Basis.ExteriorAlgebra : Basis (Finset I) R (ExteriorAlgebra R M) :=\n  .reindex\n    ((DirectSum.Decomposition.isInternal (fun n => ⋀[R]^n M)).collectedBasis b.exteriorPower)\n    Set.powersetCard.prodEquiv\n\nlemma basis_apply (s : Finset I) :\n    b.ExteriorAlgebra s = ιMulti_family R s.card b (prodEquiv.symm s).2 := by\n  simp [Basis.ExteriorAlgebra]\n\nTarget:\nlemma basis_apply_ofCard {s : Finset I} (s_card : s.card = n) :\n    b.ExteriorAlgebra s = ιMulti_family R n b (ofCard s_card) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/ExteriorAlgebra","family_id":"basis_apply_ofcard","file_id":"mathlib/Mathlib/LinearAlgebra/ExteriorAlgebra/Basis.lean","sample_id":"bcfe71f4669f39454a991e49045a95b381eb505965239b851eaa93006f10ef03"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a3b30f98b08bb268a88816f205f3d07c5c34faaa5cf59c628e0cdbe5b5a6a78c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"bf06058cb3cfa5ca349bc783b48b8f8528d753a8ce2aebdb55b6dc4d8b537783","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e87fe6db087248fc861a35d6fd880f370abae53489bf00891ff2b411146d68fa","source_sha256":"00cdc131b09c868f94179716d61273d8fdf897ed023d34f30c7085668a3b31ac","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [mul_comm]\n  exact mul_left_not_lt b h","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":16,"token_jaccard":0.818182,"token_length_ratio":1.777778},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"7355e5cb055628c8164914fff035164c22b437301bda2f2a7c3d8ff0b78a2e5f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Order.RelClasses\n\nNamespace:\nEuclideanDomain\n\nLocal context:\n/-\nCopyright (c) 2018 Louis Carlin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Louis Carlin, Mario Carneiro\n-/\n/-!\n# Euclidean domains\n\nThis file introduces Euclidean domains and provides the extended Euclidean algorithm. To be precise,\na slightly more general version is provided which is sometimes called a transfinite Euclidean domain\nand differs in the fact that the degree function need not take values in `ℕ` but can take values in\nany well-ordered set. Transfinite Euclidean domains were introduced by Motzkin and examples which\ndon't satisfy the classical notion were provided independently by Hiblot and Nagata.\n\n## Main definitions\n\n* `EuclideanDomain`: Defines Euclidean domain with functions `quotient` and `remainder`. Instances\n  of `Div` and `Mod` are provided, so that one can write `a = b * (a / b) + a % b`.\n* `gcd`: defines the greatest common divisors of two elements of a Euclidean domain.\n* `xgcd`: given two elements `a b : R`, `xgcd a b` defines the pair `(x, y)` such that\n  `x * a + y * b = gcd a b`.\n* `lcm`: defines the lowest common multiple of two elements `a` and `b` of a Euclidean domain as\n  `a * b / (gcd a b)`\n\n## Main statements\n\nSee `Algebra.EuclideanDomain.Basic` for most of the theorems about Euclidean domains,\nincluding Bézout's lemma.\n\nSee `Algebra.EuclideanDomain.Instances` for the fact that `ℤ` is a Euclidean domain,\nas is any field.\n\n## Notation\n\n`≺` denotes the well-founded relation on the Euclidean domain, e.g. in the example of the polynomial\nring over a field, `p ≺ q` for polynomials `p` and `q` if and only if the degree of `p` is less than\nthe degree of `q`.\n\n## Implementation details\n\nInstead of working with a valuation, `EuclideanDomain` is implemented with the existence of a well\nfounded relation `r` on the integral domain `R`, which in the example of `ℤ` would correspond to\nsetting `i ≺ j` for integers `i` and `j` if the absolute value of `i` is smaller than the absolute\nvalue of `j`.\n\n## References\n\n* [Th. Motzkin, *The Euclidean algorithm*][MR32592]\n* [J.-J. Hiblot, *Des anneaux euclidiens dont le plus petit algorithme n'est pas à valeurs finies*]\n  [MR399081]\n* [M. Nagata, *On Euclid algorithm*][MR541021]\n\n\n## Tags\n\nEuclidean domain, transfinite Euclidean domain, Bézout's lemma\n-/\n\n@[expose] public section\n\nuniverse u\n\n/-- A `EuclideanDomain` is a non-trivial commutative ring with a division and a remainder,\n  satisfying `b * (a / b) + a % b = a`.\n  The definition of a Euclidean domain usually includes a valuation function `R → ℕ`.\n  This definition is slightly generalised to include a well-founded relation\n  `r` with the property that `r (a % b) b`, instead of a valuation. -/\n@[wikidata Q867345]\nclass EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R where\n  /-- A division function (denoted `/`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `%` denotes `remainder`. -/\n  protected quotient : R → R → R\n  /-- Division by zero should always give zero by convention. -/\n  protected quotient_zero : ∀ a, quotient a 0 = 0\n  /-- A remainder function (denoted `%`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `/` denotes `quotient`. -/\n  protected remainder : R → R → R\n  /-- The property that links the quotient and remainder functions.\n    This allows us to compute GCDs and LCMs. -/\n  protected quotient_mul_add_remainder_eq : ∀ a b, b * quotient a b + remainder a b = a\n  /-- A well-founded relation on `R`, satisfying `r (a % b) b`.\n    This ensures that the GCD algorithm always terminates. -/\n  protected r : R → R → Prop\n  /-- The relation `r` must be well-founded.\n    This ensures that the GCD algorithm always terminates. -/\n  r_wellFounded : WellFounded r\n  /-- The relation `r` satisfies `r (a % b) b`. -/\n  protected remainder_lt : ∀ (a) {b}, b ≠ 0 → r (remainder a b) b\n  /-- An additional constraint on `r`. -/\n  mul_left_not_lt : ∀ (a) {b}, b ≠ 0 → ¬r (a * b) a\n\n/-\nLean has far more theorems about fields than about Euclidean domains. We thus\nlower the priority of `Euclideandomain.toCommRing`, encouraging typeclass inference\nto try `Field.toCommRing` first. Without this priority-lowering, typeclass inference\nfinds the more inefficient path `Field.toEuclideanDomain.toCommRing` by default. This\npriority change saves over 500G instructions across mathlib. See\nhttps://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/We.20need.20to.20talk.20about.20Euclidean.20Domains/near/594655420\n-/\n-- see Note [lower instance priority]\nattribute [instance 100] EuclideanDomain.toCommRing\n\nnamespace EuclideanDomain\n\nvariable {R : Type u} [EuclideanDomain R]\n\n/-- Abbreviated notation for the well-founded relation `r` in a Euclidean domain. -/\nlocal infixl:50 \" ≺ \" => EuclideanDomain.r\n\nlocal instance wellFoundedRelation : WellFoundedRelation R where\n  rel := EuclideanDomain.r\n  wf := r_wellFounded\n\ninstance isWellFounded : IsWellFounded R (· ≺ ·) where\n  wf := r_wellFounded\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Div R :=\n  ⟨EuclideanDomain.quotient⟩\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Mod R :=\n  ⟨EuclideanDomain.remainder⟩\n\ntheorem div_add_mod (a b : R) : b * (a / b) + a % b = a :=\n  EuclideanDomain.quotient_mul_add_remainder_eq _ _\n\ntheorem mod_add_div (a b : R) : a % b + b * (a / b) = a :=\n  (add_comm _ _).trans (div_add_mod _ _)\n\ntheorem mod_add_div' (m k : R) : m % k + m / k * k = m := by\n  rw [mul_comm]\n  exact mod_add_div _ _\n\ntheorem div_add_mod' (m k : R) : m / k * k + m % k = m := by\n  rw [mul_comm]\n  exact div_add_mod _ _\n\ntheorem mod_lt : ∀ (a) {b : R}, b ≠ 0 → a % b ≺ b :=\n  EuclideanDomain.remainder_lt\n\nTarget:\ntheorem mul_right_not_lt {a : R} (b) (h : a ≠ 0) : ¬a * b ≺ b :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [mul_comm]\n  exact mul_left_not_lt b h\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/EuclideanDomain","family_id":"mul_right_not_lt","file_id":"mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean","sample_id":"e87fe6db087248fc861a35d6fd880f370abae53489bf00891ff2b411146d68fa"},"split":"train"}
{"binding":{"chosen_proof_sha256":"055208fdff31df9d8fb76bf05bb6a68b4de72a41035c9aa9cc8d72c2b3b25d67","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"029e00c3f6a53264f7ab33ca9285b495e3986d3f7052bd853ac23e6d759b17de","source_sha256":"38b81cd46858f243db931b63ae095bb873edc08d47b08e77c1c254a8ae6a5653","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [Algebra.smul_def, ← nat_rawCast_2]","hard_negative":false,"metrics":{"chosen_tokens":10,"rejected_tokens":2,"token_jaccard":0.090909,"token_length_ratio":0.2},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"73abfdbe560cdbb26633ee5b11db55425d30641d69065182ca351928bf2bbeca","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Defs\npublic import Mathlib.Algebra.Algebra.Basic\npublic import Mathlib.Tactic.Ring.RingNF\n\nNamespace:\nMathlib.Tactic.Algebra\n\nLocal context:\n/-\nCopyright (c) 2025 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Arend Mellendijk\n-/\n/-! # Lemmas for the `algebra` tactic.\n-/\n\n@[expose] public section\n\nopen Mathlib.Meta.NormNum\n\nnamespace Mathlib.Tactic.Algebra\n\nsection ring\n\nvariable {R A : Type*} [CommRing R] [CommRing A] [Algebra R A]\n\n/- evalCast -/\ntheorem isInt_negOfNat_eq {a : A} {lit : ℕ} (h : IsInt a (Int.negOfNat lit)) :\n    a = algebraMap R A (Int.rawCast (Int.negOfNat lit) + 0 : R) + 0 := by\n  simp [h.out]\n\nend ring\n\nsection semifield\n\nvariable {R A : Type*} [Semifield R] [Semifield A] [Algebra R A]\n\n/- evalCast -/\ntheorem isNNRat_eq_rawCast {a : A} {n d : ℕ} (h : IsNNRat a n d) :\n    a = algebraMap R A (NNRat.rawCast n d + 0 : R) + 0 := by\n  simp [Mathlib.Tactic.Ring.cast_nnrat h]\n\nend semifield\n\nsection field\n\nvariable {R A : Type*} [Field R] [Field A] [Algebra R A]\n\n/- evalCast -/\ntheorem isRat_eq_rawCast {a : A} {n d : ℕ} (h : IsRat a (.negOfNat n) d) :\n    a = algebraMap R A (Rat.rawCast (.negOfNat n) d + 0 : R) + 0 := by\n  simp [Mathlib.Tactic.Ring.cast_rat h]\n\nend field\n\nvariable {R A : Type*} [sR : CommSemiring R] [sA : CommSemiring A] [sAlg : Algebra R A]\n\n/- evalCast -/\ntheorem isNat_zero_eq {a : A} (h : IsNat a 0) : a = 0 := by\n  have := h.out\n  simp [this]\n\n/- evalCast -/\ntheorem isNat_eq_rawCast {a : A} {lit : ℕ} (h : IsNat a lit) :\n    a = algebraMap R A (lit + 0 : R) + 0 := by\n  simp [h.out]\n\nsection cleanup\n\nvariable {n d : ℕ}\n\nsection cleanupSMul\n\ntheorem add_assoc_rev (a b c : R) : a + (b + c) = a + b + c := (add_assoc ..).symm\ntheorem mul_assoc_rev (a b c : R) : a * (b * c) = a * b * c := (mul_assoc ..).symm\ntheorem mul_neg {R} [Ring R] (a b : R) : a * -b = -(a * b) := by simp\ntheorem add_neg {R} [Ring R] (a b : R) : a + -b = a - b := (sub_eq_add_neg ..).symm\ntheorem nat_rawCast_0 : (Nat.rawCast 0 : R) = 0 := by simp\ntheorem nat_rawCast_1 : (Nat.rawCast 1 : R) = 1 := by simp\ntheorem nat_rawCast_2 [Nat.AtLeastTwo n] : (Nat.rawCast n : R) = OfNat.ofNat n := rfl\ntheorem int_rawCast_neg {R} [Ring R] : (Int.rawCast (.negOfNat n) : R) = -Nat.rawCast n := by simp\ntheorem nnrat_rawCast {R} [DivisionSemiring R] :\n    (NNRat.rawCast n d : R) = Nat.rawCast n / Nat.rawCast d := by simp\ntheorem rat_rawCast_neg {R} [DivisionRing R] :\n    (Rat.rawCast (.negOfNat n) d : R) = Int.rawCast (.negOfNat n) / Nat.rawCast d := by simp\n\nend cleanupSMul\nsection cleanupConsts\n\ntheorem ofNat_smul {R A} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    [n.AtLeastTwo] {a : A} :\n    (ofNat(n) : R) • a = ofNat(n) * a := by\n  simp_rw [← nat_rawCast_2]\n  simp [Nat.cast_smul_eq_nsmul]\n\ntheorem neg_ofNat_smul {R A} [CommRing R] [CommRing A] [Algebra R A] {a : A} [n.AtLeastTwo] :\n    (- ofNat(n) : R) • a = - (ofNat(n)) * a := by\n  simpa [← nat_rawCast_2] using! ofNat_smul\n\ntheorem neg_1_smul {R A} [CommRing R] [CommRing A] [Algebra R A] {a : A} :\n    (-1 : R) • a = - a := by\n  simp\n\ntheorem nnRat_ofNat_smul_1 {R A} [Semifield R] [Semifield A] [Algebra R A] {a : A}\n    [d.AtLeastTwo] :\n    (1 / ofNat(d) : R) • a = (1 / ofNat(d)) * a := by\n  simp [Algebra.smul_def, ← nat_rawCast_2]\n\ntheorem nnRat_ofNat_smul_2 {R A} [Semifield R] [Semifield A] [Algebra R A] {a : A}\n    [n.AtLeastTwo] [d.AtLeastTwo] :\n    (ofNat(n) / ofNat(d) : R) • a = (ofNat(n) / ofNat(d)) * a := by\n  simp [Algebra.smul_def, ← nat_rawCast_2]\n\nTarget:\ntheorem rat_ofNat_smul_1 {R A} [Field R] [Field A] [Algebra R A] {a : A}\n    [d.AtLeastTwo] :\n    ((- 1) / ofNat(d) : R) • a = ((- 1) / ofNat(d)) * a :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Algebra","family_id":"rat_ofnat_smul","file_id":"mathlib/Mathlib/Tactic/Algebra/Lemmas.lean","sample_id":"029e00c3f6a53264f7ab33ca9285b495e3986d3f7052bd853ac23e6d759b17de"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3d02c82bbfbe829ee560bc14711178f8e05f9f6e82051c0e8f4a63b14e8d38c0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8a646e481e3d4448b1b3d15a701d2ad658e892f1b057e4296ce688e2945247da","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"595e91c0da49b46a91c3741f0089b51f2875fc10f72a63c9a56e6dda6a7ca6c2","source_sha256":"f830da4bee5fab2594631f06af881784def836046450586c520e0644c9ae3582","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let φ (i : ι) := separatingFamily c (fun (_ : Unit) ↦ X) ⟨⟨⟩, i⟩\n  refine isSeparator_of_isColimit_cofan\n    (isSeparating_separatingFamily c (X := fun (_ : Unit) ↦ X) (by simpa using! hX))\n      (c := Cofan.mk (∐ φ) (fun ⟨_, i⟩ ↦ Sigma.ι φ i)) ?_\n  exact IsColimit.ofWhiskerEquivalence\n    (Discrete.equivalence (Equiv.punitProd.{0} ι).symm) (coproductIsCoproduct φ)","hard_negative":true,"metrics":{"chosen_tokens":104,"rejected_tokens":2,"token_jaccard":0.022222,"token_length_ratio":0.019231},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"7440a937d4b1cfcf174438ecb381329039be73cb489f0b83c3846c61a1322b0f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.Double\npublic import Mathlib.Algebra.Homology.HomologicalComplexLimits\npublic import Mathlib.CategoryTheory.Generator.Basic\n\nNamespace:\nHomologicalComplex\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Generators of the category of homological complexes\n\nLet `c : ComplexShape ι` be a complex shape with no loop.\nIf a category `C` has a separator, then `HomologicalComplex C c`\nhas a separating family, and a separator when suitable coproducts exist.\n\n-/\n\n@[expose] public section\n\nuniverse t w v u\n\nopen CategoryTheory Limits\n\nnamespace HomologicalComplex\n\nvariable {C : Type u} [Category.{v} C] {ι : Type w} (c : ComplexShape ι) [c.HasNoLoop]\n\nsection\n\nvariable [HasZeroMorphisms C] [HasZeroObject C]\n\nvariable {α : Type t} {X : α → C} (hX : ObjectProperty.IsSeparating (.ofObj X))\n\nvariable (X) in\n/-- If `X : α → C` is a separating family, and `c : ComplexShape ι` has no loop,\nthen this is a separating family indexed by `α × ι` in `HomologicalComplex C c`,\nwhich consists of homological complexes that are nonzero in at most\ntwo (consecutive) degrees. -/\nnoncomputable def separatingFamily (j : α × ι) : HomologicalComplex C c :=\n  evalCompCoyonedaCorepresentative c (X j.1) j.2\n\nset_option backward.isDefEq.respectTransparency false in\ninclude hX in\nlemma isSeparating_separatingFamily :\n    ObjectProperty.IsSeparating (.ofObj (separatingFamily c X)) := by\n  intro K L f g h\n  ext j\n  apply hX\n  rintro _ ⟨a⟩ p\n  have H := evalCompCoyonedaCorepresentable c (X a) j\n  apply H.homEquiv.symm.injective\n  simpa only [H.homEquiv_symm_comp] using! h _\n    (ObjectProperty.ofObj_apply _ ⟨a, j⟩) (H.homEquiv.symm p)\n\nend\n\nvariable [HasCoproductsOfShape ι C] [Preadditive C] [HasZeroObject C]\n\nTarget:\nlemma isSeparator_coproduct_separatingFamily {X : C} (hX : IsSeparator X) :\n    IsSeparator (∐ (fun i ↦ separatingFamily c (fun (_ : Unit) ↦ X) ⟨⟨⟩, i⟩)) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_595e91c0da49","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e4be2211094dc4b10904b5e40f9d20199f28992057e828a5a9f8fa9f92cc629f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Generator","family_id":"isseparator_coproduct_separatingfamily","file_id":"mathlib/Mathlib/CategoryTheory/Generator/HomologicalComplex.lean","sample_id":"595e91c0da49b46a91c3741f0089b51f2875fc10f72a63c9a56e6dda6a7ca6c2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9347b24919ebf6db54344437810a6552ffb014d358370aa936d28051c4570548","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"81c21537640f160fa2f5e82543efd902153888b9ea5edc7a72bb0057ef52c1a6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"02e40ac0605619b9d35a8cf4395e31d8750e7cb631f2db38ba43f21adcd702b7","source_sha256":"6ccf0444fc2138afeb704541116189c71bea73fd06b5fb4f13fd001240de6123","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [eq_comm, ← Ultrafilter.coe_le_coe]\n  change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, { v : Ultrafilter α | s ∈ v } ∈ u\n  simp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff,\n    mem_setOf_eq]\n  constructor\n  · intro h a ha\n    exact h _ ⟨ha, a, rfl⟩\n  · rintro h a ⟨xi, a, rfl⟩\n    exact h _ xi","hard_negative":true,"metrics":{"chosen_tokens":80,"rejected_tokens":3,"token_jaccard":0.041667,"token_length_ratio":0.0375},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"74a9a8897f4eb65ee88d4ea73e28c5385dad876258bfe3af20d90f707977f21c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Bases\npublic import Mathlib.Topology.DenseEmbedding\npublic import Mathlib.Topology.Connected.TotallyDisconnected\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton\n-/\n/-! # Stone-Čech compactification\n\nConstruction of the Stone-Čech compactification using ultrafilters.\n\nFor any topological space `α`, we build a compact Hausdorff space `StoneCech α` and a continuous\nmap `stoneCechUnit : α → StoneCech α` which is minimal in the sense of the following universal\nproperty: for any compact Hausdorff space `β` and every map `f : α → β` such that\n`hf : Continuous f`, there is a unique map `stoneCechExtend hf : StoneCech α → β` such that\n`stoneCechExtend_extends : stoneCechExtend hf ∘ stoneCechUnit = f`.\nContinuity of this extension is asserted by `continuous_stoneCechExtend` and uniqueness by\n`stoneCech_hom_ext`.\n\nBeware that the terminology “extend” is slightly misleading since `stoneCechUnit` is not always\ninjective, so one cannot always think of `α` as being “inside” its compactification `StoneCech α`.\n\n## Implementation notes\n\nParts of the formalization are based on “Ultrafilters and Topology”\nby Marius Stekelenburg, particularly section 5. However the construction in the general\ncase is different because the equivalence relation on spaces of ultrafilters described\nby Stekelenburg causes issues with universes since it involves a condition\non all compact Hausdorff spaces. We replace it by a two steps construction.\nThe first step called `PreStoneCech` guarantees the expected universal property but\nnot the Hausdorff condition. We then define `StoneCech α` as `T2Quotient (PreStoneCech α)`.\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen Filter Set\n\nopen Topology\n\nuniverse u v\n\nsection Ultrafilter\n\n/- The set of ultrafilters on α carries a natural topology which makes\n  it the Stone-Čech compactification of α (viewed as a discrete space). -/\n/-- Basis for the topology on `Ultrafilter α`. -/\ndef ultrafilterBasis (α : Type u) : Set (Set (Ultrafilter α)) :=\n  range fun s : Set α ↦ { u | s ∈ u }\n\nvariable {α : Type u}\n\ninstance Ultrafilter.topologicalSpace : TopologicalSpace (Ultrafilter α) :=\n  TopologicalSpace.generateFrom (ultrafilterBasis α)\n\ntheorem ultrafilterBasis_is_basis : TopologicalSpace.IsTopologicalBasis (ultrafilterBasis α) :=\n  ⟨by\n    rintro _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ u ⟨ua, ub⟩\n    refine ⟨_, ⟨a ∩ b, rfl⟩, inter_mem ua ub, fun v hv ↦ ⟨?_, ?_⟩⟩ <;> apply mem_of_superset hv <;>\n      simp [inter_subset_right],\n    eq_univ_of_univ_subset <| subset_sUnion_of_mem <| ⟨univ, eq_univ_of_forall fun _ ↦ univ_mem⟩,\n    rfl⟩\n\n/-- The basic open sets for the topology on ultrafilters are open. -/\ntheorem ultrafilter_isOpen_basic (s : Set α) : IsOpen { u : Ultrafilter α | s ∈ u } :=\n  ultrafilterBasis_is_basis.isOpen ⟨s, rfl⟩\n\n/-- The basic open sets for the topology on ultrafilters are also closed. -/\ntheorem ultrafilter_isClosed_basic (s : Set α) : IsClosed { u : Ultrafilter α | s ∈ u } := by\n  rw [← isOpen_compl_iff]\n  convert! ultrafilter_isOpen_basic sᶜ using 1\n  ext u\n  exact Ultrafilter.compl_mem_iff_notMem.symm\n\n/-- Every ultrafilter `u` on `Ultrafilter α` converges to a unique\n  point of `Ultrafilter α`, namely `joinM u`. -/\n\nTarget:\ntheorem ultrafilter_converges_iff {u : Ultrafilter (Ultrafilter α)} {x : Ultrafilter α} :\n    ↑u ≤ 𝓝 x ↔ x = joinM u :=\n\nProof body:\n","rejected":"by\n  exact ultrafilter_converges_iff","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"230a632b0943838c0947592f77cdba6e07961bd01d5f6c829d750204550d4ef3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Compactification","family_id":"ultrafilter_converges_iff","file_id":"mathlib/Mathlib/Topology/Compactification/StoneCech.lean","sample_id":"02e40ac0605619b9d35a8cf4395e31d8750e7cb631f2db38ba43f21adcd702b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f6f5bbf3230230f3a0428b8083f535c36ff227e3eca17b3d7dbf4a73cb1bc39d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ea51963a627d887e53897ff9660cd0137a6ca3fccf9a879fe29c6140902dfe65","source_sha256":"f1d0cf668c2807493d79a9d4c8a96aaed3d5b4da33fe2397162a57706cbf448f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [weaklyEtale_iff, weaklyEtale_iff, Module.Flat.ulift_left_iff, Module.Flat.ulift_right_iff]\n  congr!\n  conv_rhs => rw [← RingHom.Flat.ulift_iff.{u₁, u₂}]\n  rw [TensorProduct.lmul'_ulift, AlgHom.toRingHom_eq_coe, AlgHom.comp_toRingHom]\n  exact RingHom.Flat.comp_iff_of_bijective_right (Equiv.bijective _)","hard_negative":false,"metrics":{"chosen_tokens":66,"rejected_tokens":5,"token_jaccard":0.081081,"token_length_ratio":0.075758},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"74d7bd6d85f89af9b663fd617b42617c14ad918b8743d30c81f1566277f2a0f5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.RingHom.Flat\npublic import Mathlib.RingTheory.Etale.Basic\npublic import Mathlib.RingTheory.Smooth.Flat\n\nNamespace:\nAlgebra.WeaklyEtale\n\nLocal context:\n/-\nCopyright (c) 2026 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten\n-/\n/-!\n# Weakly étale algebras\n\nIn this file we define weakly étale algebras. An `R`-algebra `S` is weakly étale if\n`S` is `R`-flat and the multiplication map `S ⊗[R] S → S` is flat.\n\n## TODOs\n\n- Show that a weakly étale algebra is formally unramified and in particular that\n  a weakly étale algebra of finite presentation is étale (@chrisflav).\n-/\n\npublic section\n\nuniverse u u₁ u₂ u₃\n\nopen TensorProduct\n\nnamespace Algebra\n\nvariable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]\n\n/-- `S` is a weakly-étale `R`-algebra if both `R → S` and `S ⊗[R] S → R` are flat.\nThis is also called absolutely flat. -/\n@[stacks 092B, mk_iff]\nclass WeaklyEtale (R S : Type*) [CommRing R] [CommRing S] [Algebra R S] where\n  flat : Module.Flat R S := by infer_instance\n  flat_lmul' (R S) : (Algebra.TensorProduct.lmul' R (S := S)).Flat\n\nattribute [instance] WeaklyEtale.flat\n\nnamespace WeaklyEtale\n\nattribute [local instance] ULift.algebra' in\n\nTarget:\nlemma ulift_iff : WeaklyEtale (ULift.{u₁} R) (ULift.{u₂} S) ↔ WeaklyEtale R S :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Etale","family_id":"ulift_iff","file_id":"mathlib/Mathlib/RingTheory/Etale/Weakly.lean","sample_id":"ea51963a627d887e53897ff9660cd0137a6ca3fccf9a879fe29c6140902dfe65"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8c5d22170da6aa84840f7344a2c31c2097b39c782de19947a86a24798df1b587","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c360ffd931955f77da36029b2c2c6057afeb7e861c5fc96268f3b26c2470d808","source_sha256":"082955edd445e22305bf58ec1b7c606429333e8cae7ee766f31439d19efb2538","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rintro ⟨f, a, hf⟩ ⟨g, b, hg⟩ rfl\n  congr\n  refine extensional_of_trichotomous_of_irrefl s fun x ↦ ?_\n  rw [← hf, hg]","hard_negative":false,"metrics":{"chosen_tokens":33,"rejected_tokens":3,"token_jaccard":0.037037,"token_length_ratio":0.090909},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"760516a94ff607ee7433aa3e4fcfded531922f7507862fb87dfbc70c7ce7e6d9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Sum.Order\npublic import Mathlib.Order.Hom.Lex\npublic import Mathlib.Order.RelIso.Set\npublic import Mathlib.Order.UpperLower.Basic\npublic import Mathlib.Order.WellFounded\n\nNamespace:\nPrincipalSeg\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios\n-/\n/-!\n# Initial and principal segments\n\nThis file defines initial and principal segment embeddings. Though these definitions make sense for\narbitrary relations, they're intended for use with well orders.\n\nAn initial segment is simply a lower set, i.e. if `x` belongs to the range, then any `y < x` also\nbelongs to the range. A principal segment is a set of the form `Set.Iio x` for some `x`.\n\nAn initial segment embedding `r ≼i s` is an order embedding `r ↪ s` such that its range is an\ninitial segment. Likewise, a principal segment embedding `r ≺i s` has a principal segment for a\nrange.\n\n## Main definitions\n\n* `InitialSeg r s`: Type of initial segment embeddings of `r` into `s`, denoted by `r ≼i s`.\n* `PrincipalSeg r s`: Type of principal segment embeddings of `r` into `s`, denoted by `r ≺i s`.\n\nThe lemmas `Ordinal.type_le_iff` and `Ordinal.type_lt_iff` tell us that `≼i` corresponds to the `≤`\nrelation on ordinals, while `≺i` corresponds to the `<` relation. This prompts us to think of\n`PrincipalSeg` as a \"strict\" version of `InitialSeg`.\n\n## Notation\n\nThese notations belong to the `InitialSeg` locale.\n\n* `r ≼i s`: the type of initial segment embeddings of `r` into `s`.\n* `r ≺i s`: the type of principal segment embeddings of `r` into `s`.\n* `α ≤i β` is an abbreviation for `(· < ·) ≼i (· < ·)`.\n* `α <i β` is an abbreviation for `(· < ·) ≺i (· < ·)`.\n-/\n\n@[expose] public section\n\n/-! ### Initial segment embeddings -/\n\nuniverse u\n\nvariable {α β γ : Type*} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop}\n\nopen Function\n\n/-- If `r` is a relation on `α` and `s` in a relation on `β`, then `f : r ≼i s` is an order\nembedding whose `Set.range` is a lower set. That is, whenever `b < f a` in `β` then `b` is in the\nrange of `f`. -/\nstructure InitialSeg {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends r ↪r s where\n  /-- The order embedding is an initial segment -/\n  mem_range_of_rel' : ∀ a b, s b (toRelEmbedding a) → b ∈ Set.range toRelEmbedding\n\n@[inherit_doc]\nscoped[InitialSeg] infixl:25 \" ≼i \" => InitialSeg\n\n/-- An `InitialSeg` between the `<` relations of two types. -/\nnotation3:25 α:24 \" ≤i \" β:25 => @InitialSeg α β (· < ·) (· < ·)\n\nnamespace InitialSeg\n\ninstance : Coe (r ≼i s) (r ↪r s) :=\n  ⟨InitialSeg.toRelEmbedding⟩\n\ninstance : FunLike (r ≼i s) α β where\n  coe f := f.toFun\n  coe_injective := by\n    rintro ⟨f, hf⟩ ⟨g, hg⟩ h\n    congr with x\n    exact congr_fun h x\n\ninstance : EmbeddingLike (r ≼i s) α β where\n  injective' f := f.inj'\n\ninstance : RelHomClass (r ≼i s) r s where\n  map_rel f := f.map_rel_iff.2\n\n/-- An initial segment embedding between the `<` relations of two partial orders is an order\nembedding. -/\ndef toOrderEmbedding [PartialOrder α] [PartialOrder β] (f : α ≤i β) : α ↪o β :=\n  f.orderEmbeddingOfLTEmbedding\n\n@[simp]\ntheorem toOrderEmbedding_apply [PartialOrder α] [PartialOrder β] (f : α ≤i β) (x : α) :\n    f.toOrderEmbedding x = f x :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderEmbedding [PartialOrder α] [PartialOrder β] (f : α ≤i β) :\n    (f.toOrderEmbedding : α → β) = f :=\n  rfl\n\ninstance [PartialOrder α] [PartialOrder β] : OrderHomClass (α ≤i β) α β where\n  map_rel f := f.toOrderEmbedding.map_rel_iff.2\n\n@[ext] lemma ext {f g : r ≼i s} (h : ∀ x, f x = g x) : f = g :=\n  DFunLike.ext f g h\n\n@[simp]\ntheorem coe_coe_fn (f : r ≼i s) : ((f : r ↪r s) : α → β) = f :=\n  rfl\n\ntheorem mem_range_of_rel (f : r ≼i s) {a : α} {b : β} : s b (f a) → b ∈ Set.range f :=\n  f.mem_range_of_rel' _ _\n\ntheorem map_rel_iff {a b : α} (f : r ≼i s) : s (f a) (f b) ↔ r a b :=\n  f.map_rel_iff'\n\ntheorem inj (f : r ≼i s) {a b : α} : f a = f b ↔ a = b :=\n  f.toRelEmbedding.inj\n\ntheorem exists_eq_iff_rel (f : r ≼i s) {a : α} {b : β} : s b (f a) ↔ ∃ a', f a' = b ∧ r a' a :=\n  ⟨fun h => by\n    rcases f.mem_range_of_rel h with ⟨a', rfl⟩\n    exact ⟨a', rfl, f.map_rel_iff.1 h⟩,\n    fun ⟨_, e, h⟩ => e ▸ f.map_rel_iff.2 h⟩\n\n/-- A relation isomorphism is an initial segment embedding -/\n@[simps!]\ndef _root_.RelIso.toInitialSeg (f : r ≃r s) : r ≼i s :=\n  ⟨f, by simp⟩\n\n/-- The identity function shows that `≼i` is reflexive -/\n@[refl]\nprotected def refl (r : α → α → Prop) : r ≼i r :=\n  (RelIso.refl r).toInitialSeg\n\ninstance (r : α → α → Prop) : Inhabited (r ≼i r) :=\n  ⟨InitialSeg.refl r⟩\n\n/-- Composition of functions shows that `≼i` is transitive -/\n@[trans]\nprotected def trans (f : r ≼i s) (g : s ≼i t) : r ≼i t :=\n  ⟨f.1.trans g.1, fun a c h => by\n    simp only [RelEmbedding.coe_trans, coe_coe_fn, comp_apply] at h ⊢\n    rcases g.2 _ _ h with ⟨b, rfl⟩; have h := g.map_rel_iff.1 h\n    rcases f.2 _ _ h with ⟨a', rfl⟩; exact ⟨a', rfl⟩⟩\n\n@[simp]\ntheorem refl_apply (x : α) : InitialSeg.refl r x = x :=\n  rfl\n\n@[simp]\ntheorem trans_apply (f : r ≼i s) (g : s ≼i t) (a : α) : (f.trans g) a = g (f a) :=\n  rfl\n\ninstance subsingleton_of_trichotomous_of_irrefl [Std.Trichotomous s] [Std.Irrefl s]\n    [IsWellFounded α r] : Subsingleton (r ≼i s) where\n  allEq f g := by\n    ext a\n    refine IsWellFounded.induction r a fun b IH =>\n      extensional_of_trichotomous_of_irrefl s fun x => ?_\n    rw [f.exists_eq_iff_rel, g.exists_eq_iff_rel]\n    exact exists_congr fun x => and_congr_left fun hx => IH _ hx ▸ Iff.rfl\n\n/-- Given a well order `s`, there is at most one initial segment embedding of `r` into `s`. -/\ninstance [IsWellOrder β s] : Subsingleton (r ≼i s) :=\n  ⟨fun a => have := a.isWellFounded; Subsingleton.elim a⟩\n\nprotected theorem eq [IsWellOrder β s] (f g : r ≼i s) (a) : f a = g a := by\n  rw [Subsingleton.elim f g]\n\ntheorem eq_relIso [IsWellOrder β s] (f : r ≼i s) (g : r ≃r s) (a : α) : g a = f a :=\n  InitialSeg.eq g.toInitialSeg f a\n\n/-- If we have order embeddings between `α` and `β` whose ranges are initial segments, and `β` is a\nwell order, then `α` and `β` are order-isomorphic. -/\ndef antisymm [IsWellOrder β s] (f : r ≼i s) (g : s ≼i r) : r ≃r s :=\n  have := f.toRelEmbedding.isWellOrder\n  ⟨⟨f, g, (f.trans g).eq (InitialSeg.refl _), (g.trans f).eq (InitialSeg.refl _)⟩, f.map_rel_iff'⟩\n\n@[simp]\ntheorem antisymm_toFun [IsWellOrder β s] (f : r ≼i s) (g : s ≼i r) : (antisymm f g : α → β) = f :=\n  rfl\n\n@[simp]\ntheorem antisymm_symm [IsWellOrder α r] [IsWellOrder β s] (f : r ≼i s) (g : s ≼i r) :\n    (antisymm f g).symm = antisymm g f :=\n  RelIso.coe_fn_injective rfl\n\n/-- An initial segment embedding is either an isomorphism, or a principal segment embedding.\n\nSee also `InitialSeg.ltOrEq`. -/\ntheorem eq_or_principal [IsWellOrder β s] (f : r ≼i s) :\n    Surjective f ∨ ∃ b, ∀ x, x ∈ Set.range f ↔ s x b := by\n  apply or_iff_not_imp_right.2\n  intro h b\n  push Not at h\n  apply IsWellFounded.induction s b\n  intro x IH\n  obtain ⟨y, ⟨hy, hs⟩ | ⟨hy, hs⟩⟩ := h x\n  · obtain (rfl | h) := (trichotomous y x).resolve_left hs\n    · exact hy\n    · obtain ⟨z, rfl⟩ := hy\n      exact f.mem_range_of_rel h\n  · obtain ⟨z, rfl⟩ := IH y hs\n    cases hy (Set.mem_range_self z)\n\n/-- Restrict the codomain of an initial segment -/\ndef codRestrict (p : Set β) (f : r ≼i s) (H : ∀ a, f a ∈ p) : r ≼i Subrel s (· ∈ p) :=\n  ⟨RelEmbedding.codRestrict p f H, fun a ⟨b, m⟩ h =>\n    let ⟨a', e⟩ := f.mem_range_of_rel h\n    ⟨a', by subst e; rfl⟩⟩\n\n@[simp]\ntheorem codRestrict_apply (p) (f : r ≼i s) (H a) : codRestrict p f H a = ⟨f a, H a⟩ :=\n  rfl\n\n/-- Initial segment embedding from an empty type. -/\ndef ofIsEmpty (r : α → α → Prop) (s : β → β → Prop) [IsEmpty α] : r ≼i s :=\n  ⟨RelEmbedding.ofIsEmpty r s, isEmptyElim⟩\n\n/-- Initial segment embedding of an order `r` into the disjoint union of `r` and `s`. -/\ndef leAdd (r : α → α → Prop) (s : β → β → Prop) : r ≼i Sum.Lex r s :=\n  ⟨⟨⟨Sum.inl, fun _ _ => Sum.inl.inj⟩, Sum.lex_inl_inl⟩, fun a b => by\n    cases b <;> [exact fun _ => ⟨_, rfl⟩; exact False.elim ∘ Sum.lex_inr_inl]⟩\n\n@[simp]\ntheorem leAdd_apply (r : α → α → Prop) (s : β → β → Prop) (a) : leAdd r s a = Sum.inl a :=\n  rfl\n\nprotected theorem acc (f : r ≼i s) (a : α) : Acc r a ↔ Acc s (f a) :=\n  ⟨by\n    refine fun h => Acc.recOn h fun a _ ha => Acc.intro _ fun b hb => ?_\n    obtain ⟨a', rfl⟩ := f.mem_range_of_rel hb\n    exact ha _ (f.map_rel_iff.mp hb), f.toRelEmbedding.acc a⟩\n\nend InitialSeg\n\n/-! ### Principal segments -/\n\n/-- If `r` is a relation on `α` and `s` in a relation on `β`, then `f : r ≺i s` is an initial\nsegment embedding whose range is `Set.Iio x` for some element `x`. If `β` is a well order, this is\nequivalent to the embedding not being surjective. -/\nstructure PrincipalSeg {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends r ↪r s where\n  /-- The supremum of the principal segment -/\n  top : β\n  /-- The range of the order embedding is the set of elements `b` such that `s b top` -/\n  mem_range_iff_rel' : ∀ b, b ∈ Set.range toRelEmbedding ↔ s b top\n\n@[inherit_doc]\nscoped[InitialSeg] infixl:25 \" ≺i \" => PrincipalSeg\n\n/-- A `PrincipalSeg` between the `<` relations of two types. -/\nnotation3:25 α:24 \" <i \" β:25 => @PrincipalSeg α β (· < ·) (· < ·)\n\nopen scoped InitialSeg\n\nnamespace PrincipalSeg\n\ninstance : CoeOut (r ≺i s) (r ↪r s) :=\n  ⟨PrincipalSeg.toRelEmbedding⟩\n\ninstance : CoeFun (r ≺i s) fun _ => α → β :=\n  ⟨fun f => f⟩\n\nTarget:\ntheorem toRelEmbedding_injective [Std.Irrefl s] [Std.Trichotomous s] :\n    Function.Injective (@toRelEmbedding α β r s) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order","family_id":"torelembedding_injective","file_id":"mathlib/Mathlib/Order/InitialSeg.lean","sample_id":"c360ffd931955f77da36029b2c2c6057afeb7e861c5fc96268f3b26c2470d808"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d4e8af4e9759204cb08f477f781bb80c363e8cafacffe36b1eb6bdb2061a0215","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"19e049bd84946cd04ffe0d191841761afe60997cc15dcfb60c1a792ffbbf2701","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4cb237f6549cf75bf0312f10927aa64c8d7dbadae8e86cca221ffb114034bffb","source_sha256":"5a984bf431d6ea343adf1846001af32b3b0dc44766966d2864ca4d0d545aa12b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨?_, Submodule.projectionOnto_comp_surjective_of_exact hfg _ hmap⟩\n  rwa [LinearMap.coe_comp, Set.InjOn.injective_iff ↑(LinearMap.range f) _ subset_rfl]\n  simpa [← LinearMap.disjoint_ker_iff_injOn, ← hfg.linearMap_ker_eq]","hard_negative":false,"metrics":{"chosen_tokens":46,"rejected_tokens":50,"token_jaccard":0.909091,"token_length_ratio":1.086957},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"764b930a80816f3f1fd9ac6872042589b47d9dafced7e4c496b48f970a1a9ce5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Exact.Basic\npublic import Mathlib.LinearAlgebra.Basis.Basic\npublic import Mathlib.LinearAlgebra.Projection\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten\n-/\n/-!\n# Basis from a split exact sequence\n\nLet `0 → K → M → P → 0` be a split exact sequence of `R`-modules, let `s : M → K` be a\nretraction of `f` and `v` be a basis of `M` indexed by `κ ⊕ σ`. Then\nif `s vᵢ = 0` for `i : κ` and `(s vⱼ)ⱼ` is linear independent for `j : σ`, then\nthe images of `vᵢ` for `i : κ` form a basis of `P`.\n\nWe treat linear independence and the span condition separately. For convenience this\nis stated not for `κ ⊕ σ`, but for an arbitrary type `ι` with two maps `κ → ι` and `σ → ι`.\n-/\n\n@[expose] public section\n\nvariable {R M K P : Type*} [Ring R] [AddCommGroup M] [AddCommGroup K] [AddCommGroup P]\nvariable [Module R M] [Module R K] [Module R P]\nvariable {f : K →ₗ[R] M} {g : M →ₗ[R] P} {s : M →ₗ[R] K}\nvariable (hs : s ∘ₗ f = LinearMap.id) (hfg : Function.Exact f g)\nvariable {ι κ σ : Type*} {v : ι → M} {a : κ → ι} {b : σ → ι}\n\nsection\ninclude hs hfg\n\nlemma LinearIndependent.linearIndependent_of_exact_of_retraction\n    (hainj : Function.Injective a) (hsa : ∀ i, s (v (a i)) = 0)\n    (hli : LinearIndependent R v) :\n    LinearIndependent R (g ∘ v ∘ a) := by\n  apply (LinearIndependent.comp hli a hainj).map\n  rw [Submodule.disjoint_def, hfg.linearMap_ker_eq]\n  rintro - hy ⟨y, rfl⟩\n  have hz : s (f y) = 0 := by\n    revert hy\n    generalize f y = x\n    intro hy\n    induction hy using Submodule.span_induction with\n    | mem m hm => obtain ⟨i, rfl⟩ := hm; apply hsa\n    | zero => simp_all\n    | add => simp_all\n    | smul => simp_all\n  replace hs := DFunLike.congr_fun hs y\n  simp only [LinearMap.coe_comp, Function.comp_apply, LinearMap.id_coe, id_eq] at hs\n  rw [← hs, hz, map_zero]\n\nprivate lemma top_le_span_of_aux (v : κ ⊕ σ → M)\n    (hg : Function.Surjective g) (hslzero : ∀ i, s (v (.inl i)) = 0)\n    (hli : LinearIndependent R (s ∘ v ∘ .inr)) (hsp : ⊤ ≤ Submodule.span R (Set.range v)) :\n    ⊤ ≤ Submodule.span R (Set.range <| g ∘ v ∘ .inl) := by\n  rintro p -\n  obtain ⟨m, rfl⟩ := hg p\n  wlog h : m ∈ LinearMap.ker s\n  · let x : M := f (s m)\n    rw [show g m = g (m - f (s m)) by simp [hfg.apply_apply_eq_zero]]\n    apply this hs hfg v hg hslzero hli hsp\n    replace hs := DFunLike.congr_fun hs (s m)\n    simp only [LinearMap.coe_comp, Function.comp_apply, LinearMap.id_coe, id_eq] at hs\n    simp [hs]\n  have : m ∈ Submodule.span R (Set.range v) := hsp Submodule.mem_top\n  obtain ⟨c, rfl⟩ := Finsupp.mem_span_range_iff_exists_finsupp.mp this\n  simp only [LinearMap.mem_ker, Finsupp.sum, map_sum, map_smul,\n    Finset.sum_sum_eq_sum_toLeft_add_sum_toRight, map_add, hslzero, smul_zero,\n    Finset.sum_const_zero, zero_add] at h\n  replace hli := (linearIndependent_iff'.mp hli) c.support.toRight (c ∘ .inr) h\n  simp only [Finset.mem_toRight, Finsupp.mem_support_iff, Function.comp_apply, not_imp_self] at hli\n  simp only [Finsupp.sum, Finset.sum_sum_eq_sum_toLeft_add_sum_toRight, hli, zero_smul,\n    Finset.sum_const_zero, add_zero, map_sum, map_smul]\n  exact Submodule.sum_mem _ (fun i hi ↦ Submodule.smul_mem _ _ <| Submodule.subset_span ⟨i, rfl⟩)\n\nlemma Submodule.top_le_span_of_exact_of_retraction (hg : Function.Surjective g)\n    (hsa : ∀ i, s (v (a i)) = 0) (hlib : LinearIndependent R (s ∘ v ∘ b))\n    (hab : Codisjoint (Set.range a) (Set.range b))\n    (hsp : ⊤ ≤ Submodule.span R (Set.range v)) :\n    ⊤ ≤ Submodule.span R (Set.range <| g ∘ v ∘ a) := by\n  apply top_le_span_of_aux hs hfg (Sum.elim (v ∘ a) (v ∘ b)) hg hsa hlib\n  simp only [codisjoint_iff, Set.sup_eq_union, Set.top_eq_univ] at hab\n  rwa [Set.Sum.elim_range, Set.range_comp, Set.range_comp, ← Set.image_union, hab, Set.image_univ]\n\n/-- Let `0 → K → M → P → 0` be a split exact sequence of `R`-modules, let `s : M → K` be a\nretraction of `f` and `v` be a basis of `M` indexed by `κ ⊕ σ`. Then\nif `s vᵢ = 0` for `i : κ` and `(s vⱼ)ⱼ` is linear independent for `j : σ`, then\nthe images of `vᵢ` for `i : κ` form a basis of `P`.\n\nFor convenience this is stated for an arbitrary type `ι` with two maps `κ → ι` and `σ → ι`. -/\nnoncomputable def Module.Basis.ofSplitExact (hg : Function.Surjective g) (v : Basis ι R M)\n    (hainj : Function.Injective a) (hsa : ∀ i, s (v (a i)) = 0)\n    (hlib : LinearIndependent R (s ∘ v ∘ b))\n    (hab : Codisjoint (Set.range a) (Set.range b)) :\n    Basis κ R P :=\n  .mk (v.linearIndependent.linearIndependent_of_exact_of_retraction hs hfg hainj hsa)\n    (Submodule.top_le_span_of_exact_of_retraction hs hfg hg hsa hlib hab (by rw [v.span_eq]))\n\n@[simp]\nlemma Module.Basis.ofSplitExact_apply (hg : Function.Surjective g) (v : Basis ι R M)\n    (hainj : Function.Injective a) (hsa : ∀ i, s (v (a i)) = 0)\n    (hlib : LinearIndependent R (s ∘ v ∘ b))\n    (hab : Codisjoint (Set.range a) (Set.range b)) (k : κ) :\n    ofSplitExact hs hfg hg v hainj hsa hlib hab k = g (v (a k)) := by\n  simp [ofSplitExact]\n\nend\n\nsection\ninclude hfg\n\nlemma Submodule.projectionOnto_comp_surjective_of_exact\n    {p q : Submodule R M} (hpq : IsCompl p q)\n    (hmap : Submodule.map g q = ⊤) :\n    Function.Surjective (Submodule.projectionOnto p q hpq ∘ₗ f) := by\n  rw [← Set.surjOn_univ, LinearMap.coe_comp, Set.surjOn_comp_iff, Set.image_univ]\n  rw [← LinearMap.coe_range, ← Submodule.top_coe (R := R), surjOn_iff_le_map,\n    ← hfg.linearMap_ker_eq]\n  intro x triv\n  obtain ⟨a, haq, ha⟩ : g x.val ∈ q.map g := by rwa [hmap]\n  exact ⟨x - a, by simp [← ha], by simpa⟩\n\n@[deprecated (since := \"2026-05-05\")] alias\n  Submodule.linearProjOfIsCompl_comp_surjective_of_exact :=\n  Submodule.projectionOnto_comp_surjective_of_exact\n\nTarget:\nlemma Submodule.projectionOnto_comp_bijective_of_exact\n    (hf : Function.Injective f) {p q : Submodule R M} (hpq : IsCompl p q)\n    (hker : Disjoint (LinearMap.ker g) q) (hmap : Submodule.map g q = ⊤) :\n    Function.Bijective (Submodule.projectionOnto p q hpq ∘ₗ f) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  refine ⟨?_, Submodule.projectionOnto_comp_surjective_of_exact hfg _ hmap⟩\n  rwa [LinearMap.coe_comp, Set.InjOn.injective_iff ↑(LinearMap.range f) _ subset_rfl]\n  simpa [← LinearMap.disjoint_ker_iff_injOn, ← hfg.linearMap_ker_eq]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Basis","family_id":"submodule","file_id":"mathlib/Mathlib/LinearAlgebra/Basis/Exact.lean","sample_id":"4cb237f6549cf75bf0312f10927aa64c8d7dbadae8e86cca221ffb114034bffb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0e10a34c52b1aa6caddc8087552eba2fdbebe27ef42d67264d88c7c0b6ced53f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b69b8cf11d81b81760b7e2d01790b8cb08e792859b8026925d68c9b27e00c6b1","source_sha256":"0a590cb1cf4bee7c2ccf453bd99af14d58eb8e9c3d2e696c45f4fcbc49319d23","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction H with\n  | basic h =>\n    cases h\n    · exact Or.inr ⟨Set.univ, cond.univ_mem, by simpa⟩\n    · exact Or.inl (by assumption)\n  | univ => exact Or.inl cond.univ_mem\n  | @inter T1 T2 _ _ h1 h2 =>\n    rcases h1 with (h | ⟨Q, hQ, rfl⟩) <;> rcases h2 with (i | ⟨R, hR, rfl⟩)\n    · exact Or.inl (cond.inter_mem h i)\n    · exact\n        Or.inr ⟨T1 ∩ R, cond.inter_mem h hR, by simp only [← Set.inter_assoc, Set.inter_comm _ A]⟩\n    · exact Or.inr ⟨Q ∩ T2, cond.inter_mem hQ i, by simp only [Set.inter_assoc]⟩\n    · exact\n        Or.inr\n          ⟨Q ∩ R, cond.inter_mem hQ hR, by\n            ext x\n            constructor <;> simp +contextual⟩","hard_negative":true,"metrics":{"chosen_tokens":183,"rejected_tokens":8,"token_jaccard":0.015625,"token_length_ratio":0.043716},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"766cf4c6cf29027f1ffcf23a849ba93aa65998d9f624cd26aee3b2f64cd5c511","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Insert\npublic import Mathlib.Data.Set.Lattice\n\nNamespace:\nFiniteInter\n\nLocal context:\n/-\nCopyright (c) 2020 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\n/-!\n# Constructions involving sets of sets.\n\n## Finite Intersections\n\nWe define a structure `FiniteInter` which asserts that a set `S` of subsets of `α` is\nclosed under finite intersections.\n\nWe define `finiteInterClosure` which, given a set `S` of subsets of `α`, is the smallest\nset of subsets of `α` which is closed under finite intersections.\n\n`finiteInterClosure S` is endowed with a term of type `FiniteInter` using\n`finiteInterClosure_finiteInter`.\n\n-/\n\npublic section\n\n\nvariable {α : Type*} (S : Set (Set α))\n\n/-- A structure encapsulating the fact that a set of sets is closed under finite intersection. -/\nstructure FiniteInter : Prop where\n  /-- `univ_mem` states that `Set.univ` is in `S`. -/\n  univ_mem : Set.univ ∈ S\n  /-- `inter_mem` states that any two intersections of sets in `S` is also in `S`. -/\n  inter_mem : ∀ ⦃s⦄, s ∈ S → ∀ ⦃t⦄, t ∈ S → s ∩ t ∈ S\n\nnamespace FiniteInter\n\n/-- The smallest set of sets containing `S` which is closed under finite intersections. -/\ninductive finiteInterClosure : Set (Set α)\n  | basic {s} : s ∈ S → finiteInterClosure s\n  | univ : finiteInterClosure Set.univ\n  | inter {s t} : finiteInterClosure s → finiteInterClosure t → finiteInterClosure (s ∩ t)\n\ntheorem finiteInterClosure_finiteInter : FiniteInter (finiteInterClosure S) :=\n  { univ_mem := finiteInterClosure.univ\n    inter_mem := fun _ h _ => finiteInterClosure.inter h }\n\nvariable {S}\n\ntheorem finiteInter_mem (cond : FiniteInter S) (F : Finset (Set α)) :\n    ↑F ⊆ S → ⋂₀ (↑F : Set (Set α)) ∈ S := by\n  classical\n    refine Finset.induction_on F (fun _ => ?_) ?_\n    · simp [cond.univ_mem]\n    · intro a s _ h1 h2\n      suffices a ∩ ⋂₀ ↑s ∈ S by simpa\n      exact\n        cond.inter_mem (h2 (Finset.mem_insert_self a s))\n          (h1 fun x hx => h2 <| Finset.mem_insert_of_mem hx)\n\nTarget:\ntheorem finiteInterClosure_insert {A : Set α} (cond : FiniteInter S) (P)\n    (H : P ∈ finiteInterClosure (insert A S)) : P ∈ S ∨ ∃ Q ∈ S, P = A ∩ Q :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"dc1b773b5abda086c7650ef3df3463d8c51c37214f3423c2eaba91e3ec85e619","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Set","family_id":"finiteinterclosure_insert","file_id":"mathlib/Mathlib/Data/Set/Constructions.lean","sample_id":"b69b8cf11d81b81760b7e2d01790b8cb08e792859b8026925d68c9b27e00c6b1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"186eee1641eb21a1d1c5d6e7242727e956490464bf744e56de75eb2d692d16ff","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2fa13ea3ad40a789f283ed35615353cfe1a38d7f8ac516a3850699274be5db06","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"88a199b64f05b0ccb332981cf02b5c961ab80a822fd1dae1175ba359f20a3e82","source_sha256":"f5b68a895eb5e12a4d24474be37696a6e7dfc784571814edb493f00c4110f491","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← singleton_vadd]; exact (h₁ x).add h₂","hard_negative":true,"metrics":{"chosen_tokens":17,"rejected_tokens":5,"token_jaccard":0.166667,"token_length_ratio":0.294118},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"767026fd7fc5c319c882fca799ab7680c88ed05aaf0d594ac5bd4d9da2e47dd4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set\npublic import Mathlib.Algebra.Ring.Action.Pointwise.Set\npublic import Mathlib.Topology.Bornology.Basic\n\nNamespace:\nAbsorbent\n\nLocal context:\n/-\nCopyright (c) 2020 Jean Lo, Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jean Lo, Yury Kudryashov\n-/\n/-!\n# Absorption of sets\n\nLet `M` act on `α`, let `A` and `B` be sets in `α`.\nWe say that `A` *absorbs* `B` if for sufficiently large `a : M`, we have `B ⊆ a • A`.\nFormally, \"for sufficiently large `a : M`\" means \"for all but a bounded set of `a`\".\n\nTraditionally, this definition is formulated\nfor the action of a (semi)normed ring on a module over that ring.\n\nWe formulate it in a more general settings for two reasons:\n\n- this way we don't have to depend on metric spaces, normed rings etc;\n- some proofs look nicer with this definition than with something like\n  `∃ r : ℝ, ∀ a : R, r ≤ ‖a‖ → B ⊆ a • A`.\n\nIf `M` is a `GroupWithZero` (e.g., a division ring),\nthe sets absorbing a given set form a filter, see `Filter.absorbing`.\n\n## Implementation notes\n\nFor now, all theorems assume that we deal with (a generalization of) a module over a division ring.\nSome lemmas have multiplicative versions for `MulDistribMulAction`s.\nThey can be added later when someone needs them.\n\n## Keywords\n\nabsorbs, absorbent\n-/\n\n@[expose] public section\n\nassert_not_exists Real\n\nopen Set Bornology Filter\nopen scoped Pointwise\n\nsection Defs\n\nvariable (M : Type*) {α : Type*} [Bornology M] [SMul M α]\n\n/-- A set `s` absorbs another set `t` if `t` is contained in all scalings of `s`\nby all but a bounded set of elements. -/\ndef Absorbs (s t : Set α) : Prop :=\n  ∀ᶠ a in cobounded M, t ⊆ a • s\n\n/-- A set is *absorbent* if it absorbs every singleton. -/\ndef Absorbent (s : Set α) : Prop :=\n  ∀ x, Absorbs M s {x}\n\nend Defs\n\nnamespace Absorbs\n\nsection SMul\n\nvariable {M α : Type*} [Bornology M] [SMul M α] {s s₁ s₂ t t₁ t₂ : Set α} {S T : Set (Set α)}\n\nprotected lemma empty : Absorbs M s ∅ := by simp [Absorbs]\n\nprotected lemma eventually (h : Absorbs M s t) : ∀ᶠ a in cobounded M, t ⊆ a • s := h\n\n@[simp] lemma of_boundedSpace [BoundedSpace M] : Absorbs M s t := by simp [Absorbs]\n\nlemma mono_left (h : Absorbs M s₁ t) (hs : s₁ ⊆ s₂) : Absorbs M s₂ t :=\n  h.mono fun _a ha ↦ ha.trans <| smul_set_mono hs\n\nlemma mono_right (h : Absorbs M s t₁) (ht : t₂ ⊆ t₁) : Absorbs M s t₂ :=\n  h.mono fun _ ↦ ht.trans\n\nlemma mono (h : Absorbs M s₁ t₁) (hs : s₁ ⊆ s₂) (ht : t₂ ⊆ t₁) : Absorbs M s₂ t₂ :=\n  (h.mono_left hs).mono_right ht\n\n@[simp]\nlemma _root_.absorbs_union : Absorbs M s (t₁ ∪ t₂) ↔ Absorbs M s t₁ ∧ Absorbs M s t₂ := by\n  simp [Absorbs]\n\nprotected lemma union (h₁ : Absorbs M s t₁) (h₂ : Absorbs M s t₂) : Absorbs M s (t₁ ∪ t₂) :=\n  absorbs_union.2 ⟨h₁, h₂⟩\n\nlemma _root_.Set.Finite.absorbs_sUnion {T : Set (Set α)} (hT : T.Finite) :\n    Absorbs M s (⋃₀ T) ↔ ∀ t ∈ T, Absorbs M s t := by\n  simp [Absorbs, hT]\n\nprotected lemma sUnion (hT : T.Finite) (hs : ∀ t ∈ T, Absorbs M s t) :\n    Absorbs M s (⋃₀ T) :=\n  hT.absorbs_sUnion.2 hs\n\n@[simp]\nlemma _root_.absorbs_iUnion {ι : Sort*} [Finite ι] {t : ι → Set α} :\n    Absorbs M s (⋃ i, t i) ↔ ∀ i, Absorbs M s (t i) :=\n  (finite_range t).absorbs_sUnion.trans forall_mem_range\n\nprotected alias ⟨_, iUnion⟩ := absorbs_iUnion\n\nlemma _root_.Set.Finite.absorbs_biUnion {ι : Type*} {t : ι → Set α} {I : Set ι} (hI : I.Finite) :\n    Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i) := by\n  simp [Absorbs, hI]\n\nprotected alias ⟨_, biUnion⟩ := Set.Finite.absorbs_biUnion\n\n@[simp]\nlemma _root_.absorbs_biUnion_finset {ι : Type*} {t : ι → Set α} {I : Finset ι} :\n    Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i) :=\n  I.finite_toSet.absorbs_biUnion\n\nprotected alias ⟨_, biUnion_finset⟩ := absorbs_biUnion_finset\n\nend SMul\n\nsection AddZero\n\nvariable {M E : Type*} [Bornology M] {s₁ s₂ t₁ t₂ : Set E}\n\nprotected lemma add [AddZeroClass E] [DistribSMul M E]\n    (h₁ : Absorbs M s₁ t₁) (h₂ : Absorbs M s₂ t₂) : Absorbs M (s₁ + s₂) (t₁ + t₂) :=\n  h₂.mp <| h₁.eventually.mono fun x hx₁ hx₂ ↦ by rw [smul_add]; exact add_subset_add hx₁ hx₂\n\nprotected lemma zero [Zero E] [SMulZeroClass M E] {s : Set E} (hs : 0 ∈ s) : Absorbs M s 0 :=\n  Eventually.of_forall fun _ ↦ zero_subset.2 <| zero_mem_smul_set hs\n\nend AddZero\n\nend Absorbs\n\nsection GroupWithZero\n\nvariable {G₀ α : Type*} [GroupWithZero G₀] [Bornology G₀] [MulAction G₀ α]\n  {s t u : Set α} {S : Set (Set α)}\n\n@[simp]\nprotected lemma Absorbs.univ : Absorbs G₀ univ s :=\n  (eventually_ne_cobounded 0).mono fun a ha ↦ by rw [smul_set_univ₀ ha]; apply subset_univ\n\nlemma absorbs_iff_eventually_cobounded_mapsTo :\n    Absorbs G₀ s t ↔ ∀ᶠ c in cobounded G₀, MapsTo (c⁻¹ • ·) t s :=\n  eventually_congr <| (eventually_ne_cobounded 0).mono fun c hc ↦ by\n    rw [← preimage_smul_inv₀ hc]; rfl\n\nalias ⟨eventually_cobounded_mapsTo, _⟩ := absorbs_iff_eventually_cobounded_mapsTo\n\n@[simp]\nlemma absorbs_inter : Absorbs G₀ (s ∩ t) u ↔ Absorbs G₀ s u ∧ Absorbs G₀ t u := by\n  simp only [absorbs_iff_eventually_cobounded_mapsTo, mapsTo_inter, eventually_and]\n\nprotected lemma Absorbs.inter (hs : Absorbs G₀ s u) (ht : Absorbs G₀ t u) : Absorbs G₀ (s ∩ t) u :=\n  absorbs_inter.2 ⟨hs, ht⟩\n\nvariable (G₀ u) in\n/-- The filter of sets that absorb `u`. -/\ndef Filter.absorbing : Filter α where\n  sets := {s | Absorbs G₀ s u}\n  univ_sets := .univ\n  sets_of_superset h := h.mono_left\n  inter_sets := .inter\n\n@[simp]\nlemma Filter.mem_absorbing : s ∈ absorbing G₀ u ↔ Absorbs G₀ s u := .rfl\n\nlemma Set.Finite.absorbs_sInter (hS : S.Finite) :\n    Absorbs G₀ (⋂₀ S) t ↔ ∀ s ∈ S, Absorbs G₀ s t :=\n  sInter_mem (f := absorbing G₀ t) hS\n\nprotected alias ⟨_, Absorbs.sInter⟩ := Set.Finite.absorbs_sInter\n\n@[simp]\nlemma absorbs_iInter {ι : Sort*} [Finite ι] {s : ι → Set α} :\n    Absorbs G₀ (⋂ i, s i) t ↔ ∀ i, Absorbs G₀ (s i) t :=\n  iInter_mem (f := absorbing G₀ t)\n\nprotected alias ⟨_, Absorbs.iInter⟩ := absorbs_iInter\n\nlemma Set.Finite.absorbs_biInter {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set α} :\n    Absorbs G₀ (⋂ i ∈ I, s i) t ↔ ∀ i ∈ I, Absorbs G₀ (s i) t :=\n  biInter_mem (f := absorbing G₀ t) hI\n\nprotected alias ⟨_, Absorbs.biInter⟩ := Set.Finite.absorbs_biInter\n\n@[simp]\nlemma absorbs_zero_iff [NeBot (cobounded G₀)]\n    {E : Type*} [AddMonoid E] [DistribMulAction G₀ E] {s : Set E} :\n    Absorbs G₀ s 0 ↔ 0 ∈ s := by\n  simp only [absorbs_iff_eventually_cobounded_mapsTo, ← singleton_zero,\n    mapsTo_singleton, smul_zero, eventually_const]\n\nend GroupWithZero\n\nsection AddGroup\n\nvariable {M E : Type*} [Monoid M] [AddGroup E] [DistribMulAction M E] [Bornology M]\n\n@[simp]\nlemma absorbs_neg_neg {s t : Set E} : Absorbs M (-s) (-t) ↔ Absorbs M s t := by simp [Absorbs]\n\nalias ⟨Absorbs.of_neg_neg, Absorbs.neg_neg⟩ := absorbs_neg_neg\n\nlemma Absorbs.sub {s₁ s₂ t₁ t₂ : Set E} (h₁ : Absorbs M s₁ t₁) (h₂ : Absorbs M s₂ t₂) :\n    Absorbs M (s₁ - s₂) (t₁ - t₂) := by\n  simpa only [sub_eq_add_neg] using h₁.add h₂.neg_neg\n\nend AddGroup\n\nnamespace Absorbent\n\nsection SMul\n\nvariable {M α : Type*} [Bornology M] [SMul M α] {s t : Set α}\n\nprotected theorem mono (ht : Absorbent M s) (hsub : s ⊆ t) : Absorbent M t := fun x ↦\n  (ht x).mono_left hsub\n\ntheorem _root_.absorbent_iff_forall_absorbs_singleton : Absorbent M s ↔ ∀ x, Absorbs M s {x} := .rfl\n\nprotected theorem absorbs (hs : Absorbent M s) {x : α} : Absorbs M s {x} := hs x\n\ntheorem absorbs_finite (hs : Absorbent M s) (ht : t.Finite) : Absorbs M s t := by\n  rw [← Set.biUnion_of_singleton t]\n  exact .biUnion ht fun _ _ => hs.absorbs\n\nend SMul\n\nTarget:\ntheorem vadd_absorbs {M E : Type*} [Bornology M] [AddZeroClass E] [DistribSMul M E]\n    {s₁ s₂ t : Set E} {x : E} (h₁ : Absorbent M s₁) (h₂ : Absorbs M s₂ t) :\n    Absorbs M (s₁ + s₂) (x +ᵥ t) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_88a199b64f05","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"60e06e1d1ea35c6f69ca0321d47428be6844b68ddbf92c23fef098d8d6c86f9f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Bornology","family_id":"vadd_absorbs","file_id":"mathlib/Mathlib/Topology/Bornology/Absorbs.lean","sample_id":"88a199b64f05b0ccb332981cf02b5c961ab80a822fd1dae1175ba359f20a3e82"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dbcf29961f4c4a120ac8530a8dcc28a30bfa6b6db329e1836f9e00211fe2e46d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f8430c85d561b56ecd3591aebb22bff125eef0e6ec20a4eed4b6e849ab9060d6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"405cc27ec849e33c860448f43291f011dd4708a3dd857ace59279e3e185b0743","source_sha256":"8fcecfecdd43a0c830d21a96925630041f2ef381c500d297f5f2d704df5a9911","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa [CochainComplex.of.d] using (inhomogeneousCochains A).d_comp_d n (n + 1) (n + 2)","hard_negative":false,"metrics":{"chosen_tokens":27,"rejected_tokens":34,"token_jaccard":0.9,"token_length_ratio":1.259259},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"769949689ce1580f593341d4bd24d7c7e0a3fae5951eaa317579c0b5034ae0e7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.Opposite\npublic import Mathlib.Algebra.Homology.ConcreteCategory\npublic import Mathlib.RepresentationTheory.Homological.Resolution\npublic import Mathlib.Tactic.CategoryTheory.Slice\n\nNamespace:\ngroupCohomology\n\nLocal context:\n/-\nCopyright (c) 2023 Amelia Livingston. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Amelia Livingston\n-/\n/-!\n# The group cohomology of a `k`-linear `G`-representation\n\nLet `k` be a commutative ring and `G` a group. This file defines the group cohomology of\n`A : Rep k G` to be the cohomology of the complex\n$$0 \\to \\mathrm{Fun}(G^0, A) \\to \\mathrm{Fun}(G^1, A) \\to \\mathrm{Fun}(G^2, A) \\to \\dots$$\nwith differential $d^n$ sending $f: G^n \\to A$ to the function mapping $(g_0, \\dots, g_n)$ to\n$$\\rho(g_0)(f(g_1, \\dots, g_n))$$\n$$+ \\sum_{i = 0}^{n - 1} (-1)^{i + 1}\\cdot f(g_0, \\dots, g_ig_{i + 1}, \\dots, g_n)$$\n$$+ (-1)^{n + 1}\\cdot f(g_0, \\dots, g_{n - 1})$$ (where `ρ` is the representation attached to `A`).\n\nWe have a `k`-linear isomorphism\n$\\mathrm{Fun}(G^n, A) \\cong \\mathrm{Hom}(\\bigoplus_{G^n} k[G], A)$, where\nthe right-hand side is morphisms in `Rep k G`, and $k[G]$ is equipped with the left regular\nrepresentation. If we conjugate the $n$th differential in $\\mathrm{Hom}(P, A)$ by this isomorphism,\nwhere `P` is the bar resolution of `k` as a trivial `k`-linear `G`-representation, then the\nresulting map agrees with the differential $d^n$ defined above, a fact we prove.\n\nThis gives us for free a proof that our $d^n$ squares to zero. It also gives us an isomorphism\n$\\mathrm{H}^n(G, A) \\cong \\mathrm{Ext}^n(k, A),$ where $\\mathrm{Ext}$ is taken in the category\n`Rep k G`.\n\nTo talk about cohomology in low degree, please see the file\n`Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean`, which provides API\nspecialized to `H⁰`, `H¹`, `H²`.\n\n## Main definitions\n\n* `groupCohomology.inhomogeneousCochains A`: a complex whose objects are\n  $\\mathrm{Fun}(G^n, A)$ and whose cohomology is the group cohomology $\\mathrm{H}^n(G, A).$\n* `groupCohomology.inhomogeneousCochainsIso A`: an isomorphism between the above complex and the\n  complex $\\mathrm{Hom}(P, A),$ where `P` is the bar resolution of `k` as a trivial resolution.\n* `groupCohomology A n`: this is $\\mathrm{H}^n(G, A),$ defined as the $n$th cohomology of\n  `inhomogeneousCochains A`.\n* `groupCohomologyIsoExt A n`: an isomorphism $\\mathrm{H}^n(G, A) \\cong \\mathrm{Ext}^n(k, A)$\n  (where $\\mathrm{Ext}$ is taken in the category `Rep k G`) induced by `inhomogeneousCochainsIso A`.\n\n## Implementation notes\n\nGroup cohomology is typically stated for `G`-modules, or equivalently modules over the group ring\n`ℤ[G].` However, `ℤ` can be generalized to any commutative ring `k`, which is what we use.\nMoreover, we express `k[G]`-module structures on a module `k`-module `A` using the `Rep`\ndefinition. We avoid using instances `Module k[G] A` so that we do not run into\npossible scalar action diamonds.\n\n## TODO\n\n* Upgrading `groupCohomologyIsoExt` to an isomorphism of derived functors.\n* Profinite cohomology.\n\nLonger term:\n* The Hochschild-Serre spectral sequence (this is perhaps a good toy example for the theory of\n  spectral sequences in general).\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nuniverse u\n\nvariable {k G : Type u} [CommRing k] {n : ℕ}\n\nopen CategoryTheory\n\nnamespace inhomogeneousCochains\n\nopen Rep\n\n/-- The differential in the complex of inhomogeneous cochains used to\ncalculate group cohomology. -/\n@[simps! -isSimp]\ndef d [Monoid G] (A : Rep k G) (n : ℕ) :\n    ModuleCat.of k ((Fin n → G) → A) ⟶ ModuleCat.of k ((Fin (n + 1) → G) → A) :=\n  ModuleCat.ofHom\n  { toFun f g :=\n      A.ρ (g 0) (f fun i => g i.succ) + Finset.univ.sum fun j : Fin (n + 1) =>\n        (-1 : k) ^ ((j : ℕ) + 1) • f (Fin.contractNth j (· * ·) g)\n    map_add' f g := by\n      ext\n      simp [Finset.sum_add_distrib, add_add_add_comm]\n    map_smul' r f := by\n      ext\n      simp [Finset.smul_sum, ← smul_assoc, mul_comm r] }\n\nvariable [Group G] (A : Rep k G) (n : ℕ)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\ntheorem d_eq :\n    d A n =\n      (freeLiftLEquiv k G (Fin n → G) A).toModuleIso.inv ≫\n        ((barComplex k G).linearYonedaObj k A).d n (n + 1) ≫\n          (freeLiftLEquiv k G (Fin (n + 1) → G) A).toModuleIso.hom := by\n  ext\n  simp [d_hom_apply, map_add, barComplex.d_single (k := k), homEquiv]\n\nend inhomogeneousCochains\n\nnamespace groupCohomology\n\nvariable [Group G] (n) (A : Rep.{u} k G)\n\nopen inhomogeneousCochains Rep\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Given a `k`-linear `G`-representation `A`, this is the complex of inhomogeneous cochains\n$$0 \\to \\mathrm{Fun}(G^0, A) \\to \\mathrm{Fun}(G^1, A) \\to \\mathrm{Fun}(G^2, A) \\to \\dots$$\nwhich calculates the group cohomology of `A`. -/\nnoncomputable abbrev inhomogeneousCochains : CochainComplex (ModuleCat k) ℕ :=\n  CochainComplex.of (fun n => ModuleCat.of k ((Fin n → G) → A))\n    (fun n => inhomogeneousCochains.d A n) fun n => by\n    classical\n    rw [d_eq, d_eq]\n    slice_lhs 3 4 => rw [Iso.hom_inv_id]\n    slice_lhs 2 4 => rw [Category.id_comp, ((barComplex k G).linearYonedaObj k A).d_comp_d]\n    simp\n\nvariable {A n} in\n@[ext]\ntheorem inhomogeneousCochains.ext {x y : (inhomogeneousCochains A).X n} (h : ∀ g, x g = y g) :\n    x = y := funext h\n\ntheorem inhomogeneousCochains.d_def (n : ℕ) :\n    (inhomogeneousCochains A).d n (n + 1) = d A n := by\n  simp\n\nset_option backward.defeqAttrib.useBackward true in\n\nTarget:\ntheorem inhomogeneousCochains.d_comp_d :\n    d A n ≫ d A (n + 1) = 0 :=\n\nProof body:\n","rejected":"```lean\nby\n  simpa [CochainComplex.of.d] using (inhomogeneousCochains A).d_comp_d n (n + 1) (n + 2)\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RepresentationTheory/Homological","family_id":"inhomogeneouscochains","file_id":"mathlib/Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean","sample_id":"405cc27ec849e33c860448f43291f011dd4708a3dd857ace59279e3e185b0743"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a577b07c0d33a825fd2224d959c4010ad6df89972c66448b93e986876b718d2f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"56bb01dbdf1213a5571dfdae532b08e414ed78cc621e447a3cc9e4531d26ca79","source_sha256":"2fde3cadc36a49aaa16d3276079af548a012432072c96911e1dbf6460053e397","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  calc\n    Var[X; μ | m]\n    _ = μ[X ^ 2 - 2 * X * μ[X | m] + μ[X | m] ^ 2 | m] := by rw [condVar, sub_sq]\n    _ =ᵐ[μ] μ[X ^ 2 | m] - 2 * μ[X | m] ^ 2 + μ[X | m] ^ 2 := by\n      have aux₀ : Integrable (X ^ 2) μ := hX.integrable_sq\n      have aux₁ : Integrable (2 * X * μ[X | m]) μ := by\n        rw [mul_assoc]\n        exact (memLp_one_iff_integrable.1 <| hX.condExp.mul hX).const_mul _\n      have aux₂ : Integrable (μ[X | m] ^ 2) μ := hX.condExp.integrable_sq\n      filter_upwards [condExp_add (m := m) (aux₀.sub aux₁) aux₂, condExp_sub (m := m) aux₀ aux₁,\n        condExp_mul_of_stronglyMeasurable_right stronglyMeasurable_condExp aux₁\n          ((hX.integrable one_le_two).const_mul _), condExp_ofNat (m := m) 2 X]\n        with ω hω₀ hω₁ hω₂ hω₃\n      simp [hω₀, hω₁, hω₂, hω₃,\n        condExp_of_stronglyMeasurable hm (stronglyMeasurable_condExp.pow _) aux₂]\n      simp [mul_assoc, sq]\n    _ = μ[X ^ 2 | m] - μ[X | m] ^ 2 := by ring","hard_negative":false,"metrics":{"chosen_tokens":288,"rejected_tokens":3,"token_jaccard":0.031746,"token_length_ratio":0.010417},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"769a9f9a7e0c717f9e434be18c7e99bbc8dd0a838bc01ca595bb87d04e868ce8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut\npublic import Mathlib.MeasureTheory.Integral.Average\npublic import Mathlib.Probability.Moments.Variance\n\nNamespace:\nProbabilityTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Conditional variance\n\nThis file defines the variance of a real-valued random variable conditional to a sigma-algebra.\n\n## TODO\n\nDefine the Lebesgue conditional variance. See\n[GibbsMeasure](https://github.com/james18lpc/GibbsMeasure) for a definition of the Lebesgue\nconditional expectation.\n-/\n\n@[expose] public section\n\nopen MeasureTheory Filter\nopen scoped ENNReal\n\nnamespace ProbabilityTheory\nvariable {Ω : Type*} {m₀ m m' : MeasurableSpace Ω} {hm : m ≤ m₀} {X Y : Ω → ℝ} {μ : Measure[m₀] Ω}\n  {s : Set Ω}\n\nvariable (m X μ) in\n/-- Conditional variance of a real-valued random variable. It is defined as `0` if any one of the\nfollowing conditions is true:\n- `m` is not a sub-σ-algebra of `m₀`,\n- `μ` is not σ-finite with respect to `m`,\n- `X - μ[X | m]` is not square-integrable. -/\nnoncomputable def condVar : Ω → ℝ := μ[(X - μ[X | m]) ^ 2 | m]\n\n@[inherit_doc] scoped notation \"Var[\" X \"; \" μ \" | \" m \"]\" => condVar m X μ\n\n/-- Conditional variance of a real-valued random variable. It is defined as `0` if any one of the\nfollowing conditions is true:\n- `m` is not a sub-σ-algebra of `m₀`,\n- `volume` is not σ-finite with respect to `m`,\n- `X - 𝔼[X | m]` is not square-integrable. -/\nscoped notation \"Var[\" f \"|\" m \"]\" => Var[f; MeasureTheory.volume | m]\n\nlemma condVar_of_not_le (hm : ¬m ≤ m₀) : Var[X; μ | m] = 0 := by rw [condVar, condExp_of_not_le hm]\n\nlemma condVar_of_not_sigmaFinite (hμm : ¬SigmaFinite (μ.trim hm)) :\n    Var[X; μ | m] = 0 := by rw [condVar, condExp_of_not_sigmaFinite hm hμm]\n\nopen scoped Classical in\nlemma condVar_of_sigmaFinite [SigmaFinite (μ.trim hm)] :\n    Var[X; μ | m] =\n      if Integrable (fun ω ↦ (X ω - (μ[X | m]) ω) ^ 2) μ then\n        if StronglyMeasurable[m] (fun ω ↦ (X ω - (μ[X | m]) ω) ^ 2) then\n          fun ω ↦ (X ω - (μ[X | m]) ω) ^ 2\n        else aestronglyMeasurable_condExpL1.mk (condExpL1 hm μ fun ω ↦ (X ω - (μ[X | m]) ω) ^ 2)\n      else 0 := condExp_of_sigmaFinite _\n\nlemma condVar_of_stronglyMeasurable [SigmaFinite (μ.trim hm)]\n    (hX : StronglyMeasurable[m] X) (hXint : Integrable ((X - μ[X | m]) ^ 2) μ) :\n    Var[X; μ | m] = fun ω ↦ (X ω - (μ[X | m]) ω) ^ 2 :=\n  condExp_of_stronglyMeasurable _ ((hX.sub stronglyMeasurable_condExp).pow _) hXint\n\nlemma condVar_of_not_integrable (hXint : ¬ Integrable (fun ω ↦ (X ω - (μ[X | m]) ω) ^ 2) μ) :\n    Var[X; μ | m] = 0 := condExp_of_not_integrable hXint\n\n@[simp] lemma condVar_zero : Var[0; μ | m] = 0 := by simp [condVar]\n\n@[simp]\nlemma condVar_const (hm : m ≤ m₀) (c : ℝ) : Var[fun _ ↦ c; μ | m] = 0 := by\n  obtain rfl | hc := eq_or_ne c 0\n  · simp [← Pi.zero_def]\n  by_cases hμm : IsFiniteMeasure μ\n  · simp [condVar, hm]\n  · simp [condVar, condExp_of_not_integrable, integrable_const_iff_isFiniteMeasure hc,\n      integrable_const_iff_isFiniteMeasure <| pow_ne_zero _ hc, hμm, Pi.pow_def]\n\nlemma stronglyMeasurable_condVar : StronglyMeasurable[m] (Var[X; μ | m]) :=\n  stronglyMeasurable_condExp\n\nlemma condVar_congr_ae (h : X =ᵐ[μ] Y) : Var[X; μ | m] =ᵐ[μ] Var[Y; μ | m] :=\n  condExp_congr_ae <| by filter_upwards [h, condExp_congr_ae h] with ω hω hω'; dsimp; rw [hω, hω']\n\nlemma condVar_of_aestronglyMeasurable [hμm : SigmaFinite (μ.trim hm)]\n    (hX : AEStronglyMeasurable[m] X μ) (hXint : Integrable ((X - μ[X | m]) ^ 2) μ) :\n    Var[X; μ | m] =ᵐ[μ] (X - μ[X | m]) ^ 2 :=\n  condExp_of_aestronglyMeasurable' _ ((continuous_pow _).comp_aestronglyMeasurable\n    (hX.sub stronglyMeasurable_condExp.aestronglyMeasurable)) hXint\n\nlemma integrable_condVar : Integrable Var[X; μ | m] μ := integrable_condExp\n\n/-- The integral of the conditional variance `Var[X | m]` over an `m`-measurable set is equal to\nthe integral of `(X - μ[X | m]) ^ 2` on that set. -/\nlemma setIntegral_condVar [SigmaFinite (μ.trim hm)] (hX : Integrable ((X - μ[X | m]) ^ 2) μ)\n    (hs : MeasurableSet[m] s) :\n    ∫ ω in s, (Var[X; μ | m]) ω ∂μ = ∫ ω in s, (X ω - (μ[X | m]) ω) ^ 2 ∂μ :=\n  setIntegral_condExp _ hX hs\n\n-- `(· ^ 2)` is a postfix operator called `_sq` in lemma names, but\n-- `condVar_ae_eq_condExp_sq_sub_condExp_sq` is a bit ridiculous, so we exceptionally denote it by\n-- `sq_` as it were a prefix.\n\nTarget:\nlemma condVar_ae_eq_condExp_sq_sub_sq_condExp (hm : m ≤ m₀) [IsFiniteMeasure μ] (hX : MemLp X 2 μ) :\n    Var[X; μ | m] =ᵐ[μ] μ[X ^ 2 | m] - μ[X | m] ^ 2 :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability","family_id":"condvar_ae_eq_condexp_sq_sub_sq_condexp","file_id":"mathlib/Mathlib/Probability/CondVar.lean","sample_id":"56bb01dbdf1213a5571dfdae532b08e414ed78cc621e447a3cc9e4531d26ca79"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8c34499c0ebf931c33761750b78ccc8342159b860f73a3673d688956dbdeb67f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"29c7c255fa95ebdb4ea48bea160ffb43d64b92ed35970c94775d8c97a38d4881","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"267c884cb305e890be2d32bd1dd81644507811de08af6ce289cc15754404e6fb","source_sha256":"5efd5852b126afb746be3b592d500b1af5de2649a35029fd57ec7bbb27770fcb","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor <;> rintro ⟨u, h₂⟩\n  · refine ⟨f <$$> u, ?_⟩\n    apply (h₁ u).mp h₂\n  · refine ⟨g <$$> u, ?_⟩\n    rw [h₁]\n    simp only [MvFunctor.map_map, h₀, LawfulMvFunctor.id_map, h₂]","hard_negative":false,"metrics":{"chosen_tokens":70,"rejected_tokens":74,"token_jaccard":0.921053,"token_length_ratio":1.057143},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"769b6febc48a54ca2353f7d71fe0e4110ea25bb5a11850a295bb93d61b3e7250","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.Fin2\npublic import Mathlib.Data.TypeVec\npublic import Mathlib.Logic.Equiv.Defs\n\nNamespace:\nMvFunctor\n\nLocal context:\n/-\nCopyright (c) 2018 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Mario Carneiro, Simon Hudon\n-/\n/-!\n\n# Functors between the category of tuples of types, and the category Type\n\nFeatures:\n\n* `MvFunctor n` : the type class of multivariate functors\n* `f <$$> x`    : notation for map\n\n-/\n\n@[expose] public section\n\n\nuniverse u v w\n\nopen MvFunctor\n\n/-- Multivariate functors, i.e. functor between the category of type vectors\nand the category of Type -/\nclass MvFunctor {n : ℕ} (F : TypeVec n → Type*) where\n  /-- Multivariate map, if `f : α ⟹ β` and `x : F α` then `f <$$> x : F β`. -/\n  map : ∀ {α β : TypeVec n}, α ⟹ β → F α → F β\n\n/-- Multivariate map, if `f : α ⟹ β` and `x : F α` then `f <$$> x : F β` -/\nscoped[MvFunctor] infixr:100 \" <$$> \" => MvFunctor.map\n\nvariable {n : ℕ}\n\nnamespace MvFunctor\n\nvariable {α β : TypeVec.{u} n} {F : TypeVec.{u} n → Type v} [MvFunctor F]\n\n/-- predicate lifting over multivariate functors -/\ndef LiftP {α : TypeVec n} (P : ∀ i, α i → Prop) (x : F α) : Prop :=\n  ∃ u : F (fun i => Subtype (P i)), (fun i => @Subtype.val _ (P i)) <$$> u = x\n\n/-- relational lifting over multivariate functors -/\ndef LiftR {α : TypeVec n} (R : ∀ ⦃i⦄, α i → α i → Prop) (x y : F α) : Prop :=\n  ∃ u : F (fun i => { p : α i × α i // R p.fst p.snd }),\n    (fun i (t : { p : α i × α i // R p.fst p.snd }) => t.val.fst) <$$> u = x ∧\n      (fun i (t : { p : α i × α i // R p.fst p.snd }) => t.val.snd) <$$> u = y\n\n/-- given `x : F α` and a projection `i` of type vector `α`, `supp x i` is the set\nof `α.i` contained in `x` -/\ndef supp {α : TypeVec n} (x : F α) (i : Fin2 n) : Set (α i) :=\n  { y : α i | ∀ ⦃P⦄, LiftP P x → P i y }\n\ntheorem of_mem_supp {α : TypeVec n} {x : F α} {P : ∀ ⦃i⦄, α i → Prop} (h : LiftP P x) (i : Fin2 n) :\n    ∀ y ∈ supp x i, P y := fun _y hy => hy h\n\nend MvFunctor\n\n\n\n/-- laws for `MvFunctor` -/\nclass LawfulMvFunctor {n : ℕ} (F : TypeVec n → Type*) [MvFunctor F] : Prop where\n  /-- `map` preserved identities, i.e., maps identity on `α` to identity on `F α` -/\n  id_map : ∀ {α : TypeVec n} (x : F α), TypeVec.id <$$> x = x\n  /-- `map` preserves compositions -/\n  comp_map :\n    ∀ {α β γ : TypeVec n} (g : α ⟹ β) (h : β ⟹ γ) (x : F α), (h ⊚ g) <$$> x = h <$$> g <$$> x\n\nopen Nat TypeVec\n\nnamespace MvFunctor\n\nexport LawfulMvFunctor (comp_map)\n\nopen LawfulMvFunctor\n\nvariable {α β γ : TypeVec.{u} n}\nvariable {F : TypeVec.{u} n → Type v} [MvFunctor F]\nvariable (P : α ⟹ «repeat» n Prop) (R : α ⊗ α ⟹ «repeat» n Prop)\n\n/-- adapt `MvFunctor.LiftP` to accept predicates as arrows -/\ndef LiftP' : F α → Prop :=\n  MvFunctor.LiftP fun i x => ofRepeat <| P i x\n\n\n/-- adapt `MvFunctor.LiftR` to accept relations as arrows -/\ndef LiftR' : F α → F α → Prop :=\n  MvFunctor.LiftR @fun i x y => ofRepeat <| R i <| TypeVec.prod.mk _ x y\n\nvariable [LawfulMvFunctor F]\n\n@[simp]\ntheorem id_map (x : F α) : TypeVec.id <$$> x = x :=\n  LawfulMvFunctor.id_map x\n\n@[simp]\ntheorem id_map' (x : F α) : (fun _i a => a) <$$> x = x :=\n  id_map x\n\ntheorem map_map (g : α ⟹ β) (h : β ⟹ γ) (x : F α) : h <$$> g <$$> x = (h ⊚ g) <$$> x :=\n  Eq.symm <| comp_map _ _ _\n\nsection LiftP'\n\nvariable (F) in\n\nTarget:\ntheorem exists_iff_exists_of_mono {P : F α → Prop} {q : F β → Prop}\n    (f : α ⟹ β) (g : β ⟹ α)\n    (h₀ : f ⊚ g = TypeVec.id)\n    (h₁ : ∀ u : F α, P u ↔ q (f <$$> u)) :\n    (∃ u : F α, P u) ↔ ∃ u : F β, q u :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  constructor <;> rintro ⟨u, h₂⟩\n  · refine ⟨f <$$> u, ?_⟩\n    apply (h₁ u).mp h₂\n  · refine ⟨g <$$> u, ?_⟩\n    rw [h₁]\n    simp only [MvFunctor.map_map, h₀, LawfulMvFunctor.id_map, h₂]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control/Functor","family_id":"exists_iff_exists_of_mono","file_id":"mathlib/Mathlib/Control/Functor/Multivariate.lean","sample_id":"267c884cb305e890be2d32bd1dd81644507811de08af6ce289cc15754404e6fb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6c44e0bc1aac850175730ade0c94468670de6fadad8303d7beb2df51cfcf9e9e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ca377602111c20dcbd45558310dc275698819fa9f7f029d3c8ffed33548fb0c5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"94c18dd81e6ce7afdea3016639b32636d0e629ff55fd5405a516ea90027133c6","source_sha256":"ba493885a2be935359e1e232e1bbaeb68ad80f382975839ead7d3190a1fce95f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases a <;> cases b\n  exacts [map_subset_map.2 (hf₁ _ _), disjSum_mono (hg₁ _ _) (hg₂ _ _), Subset.rfl,\n    map_subset_map.2 (hf₂ _ _)]","hard_negative":false,"metrics":{"chosen_tokens":48,"rejected_tokens":55,"token_jaccard":0.923077,"token_length_ratio":1.145833},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"76a74d1f75f257a951cce24505fc91343e077154afc9a9b4a226eb34b01c9d27","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Sum\npublic import Mathlib.Data.Sum.Order\npublic import Mathlib.Order.Interval.Finset.Defs\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Finite intervals in a disjoint union\n\nThis file provides the `LocallyFiniteOrder` instance for the disjoint sum and linear sum of two\norders and calculates the cardinality of their finite intervals.\n-/\n\n@[expose] public section\n\n\nopen Function Sum\n\nnamespace Finset\n\nvariable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}\n\nsection SumLift₂\n\nvariable (f f₁ g₁ : α₁ → β₁ → Finset γ₁) (g f₂ g₂ : α₂ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁` and `α₂ → β₂ → Finset γ₂` to a map\n`α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to `Alternative` functors if we can\nmake sure to keep computability and universe polymorphism. -/\n@[simp]\ndef sumLift₂ : ∀ (_ : α₁ ⊕ α₂) (_ : β₁ ⊕ β₂), Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f a b).map Embedding.inl\n  | inl _, inr _ => ∅\n  | inr _, inl _ => ∅\n  | inr a, inr b => (g a b).map Embedding.inr\n\nvariable {f f₁ g₁ g f₂ g₂} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\ntheorem mem_sumLift₂ :\n    c ∈ sumLift₂ f g a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨\n        ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g a₂ b₂ := by\n  constructor\n  · rcases a with a | a <;> rcases b with b | b\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr ⟨a, b, c, rfl, rfl, rfl, hc⟩\n  · rintro (⟨a, b, c, rfl, rfl, rfl, h⟩ | ⟨a, b, c, rfl, rfl, rfl, h⟩) <;> exact mem_map_of_mem _ h\n\ntheorem inl_mem_sumLift₂ {c₁ : γ₁} :\n    inl c₁ ∈ sumLift₂ f g a b ↔ ∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f a₁ b₁ := by\n  rw [mem_sumLift₂, or_iff_left]\n  · simp only [inl.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inl_ne_inr h\n\ntheorem inr_mem_sumLift₂ {c₂ : γ₂} :\n    inr c₂ ∈ sumLift₂ f g a b ↔ ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ g a₂ b₂ := by\n  rw [mem_sumLift₂, or_iff_right]\n  · simp only [inr.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inr_ne_inl h\n\ntheorem sumLift₂_eq_empty :\n    sumLift₂ f g a b = ∅ ↔\n      (∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f a₁ b₁ = ∅) ∧\n        ∀ a₂ b₂, a = inr a₂ → b = inr b₂ → g a₂ b₂ = ∅ := by\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · constructor <;>\n    · rintro a b rfl rfl\n      exact map_eq_empty.1 h\n  cases a <;> cases b\n  · exact map_eq_empty.2 (h.1 _ _ rfl rfl)\n  · rfl\n  · rfl\n  · exact map_eq_empty.2 (h.2 _ _ rfl rfl)\n\ntheorem sumLift₂_nonempty :\n    (sumLift₂ f g a b).Nonempty ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ (f a₁ b₁).Nonempty) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ (g a₂ b₂).Nonempty := by\n  simp only [nonempty_iff_ne_empty, Ne, sumLift₂_eq_empty, not_and_or, not_forall, exists_prop]\n\ntheorem sumLift₂_mono (h₁ : ∀ a b, f₁ a b ⊆ g₁ a b) (h₂ : ∀ a b, f₂ a b ⊆ g₂ a b) :\n    ∀ a b, sumLift₂ f₁ f₂ a b ⊆ sumLift₂ g₁ g₂ a b\n  | inl _, inl _ => map_subset_map.2 (h₁ _ _)\n  | inl _, inr _ => Subset.rfl\n  | inr _, inl _ => Subset.rfl\n  | inr _, inr _ => map_subset_map.2 (h₂ _ _)\n\nend SumLift₂\n\nsection SumLexLift\nvariable (f₁ f₁' : α₁ → β₁ → Finset γ₁) (f₂ f₂' : α₂ → β₂ → Finset γ₂)\n  (g₁ g₁' : α₁ → β₂ → Finset γ₁) (g₂ g₂' : α₁ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁`, `α₂ → β₂ → Finset γ₂`, `α₁ → β₂ → Finset γ₁`,\n`α₂ → β₂ → Finset γ₂`  to a map `α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to\nalternative monads if we can make sure to keep computability and universe polymorphism. -/\ndef sumLexLift : α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f₁ a b).map Embedding.inl\n  | inl a, inr b => (g₁ a b).disjSum (g₂ a b)\n  | inr _, inl _ => ∅\n  | inr a, inr b => (f₂ a b).map ⟨_, inr_injective⟩\n\n@[simp]\nlemma sumLexLift_inl_inl (a : α₁) (b : β₁) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inl b) = (f₁ a b).map Embedding.inl := rfl\n\n@[simp]\nlemma sumLexLift_inl_inr (a : α₁) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b) = (g₁ a b).disjSum (g₂ a b) := rfl\n\n@[simp]\nlemma sumLexLift_inr_inl (a : α₂) (b : β₁) : sumLexLift f₁ f₂ g₁ g₂ (inr a) (inl b) = ∅ := rfl\n\n@[simp]\nlemma sumLexLift_inr_inr (a : α₂) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inr a) (inr b) = (f₂ a b).map ⟨_, inr_injective⟩ := rfl\n\nvariable {f₁ g₁ f₂ g₂ f₁' g₁' f₂' g₂'} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\nlemma mem_sumLexLift :\n    c ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        (∃ a₁ b₂ c₁, a = inl a₁ ∧ b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨\n          (∃ a₁ b₂ c₂, a = inl a₁ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n            ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂ := by\n  constructor\n  · obtain a | a := a <;> obtain b | b := b\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (mem_disjSum.1 h).elim ?_ ?_\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inr <| Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr (Or.inr <| Or.inr <| ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n  · rintro (⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩ |\n      ⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact mem_map_of_mem _ hc\n    · exact inl_mem_disjSum.2 hc\n    · exact inr_mem_disjSum.2 hc\n    · exact mem_map_of_mem _ hc\n\nlemma inl_mem_sumLexLift {c₁ : γ₁} :\n    inl c₁ ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        ∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ c₁ ∈ g₁ a₁ b₂ := by\n  simp [mem_sumLexLift]\n\nlemma inr_mem_sumLexLift {c₂ : γ₂} :\n    inr c₂ ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ f₂ a₂ b₂ := by\n  simp [mem_sumLexLift]\n\nTarget:\nlemma sumLexLift_mono (hf₁ : ∀ a b, f₁ a b ⊆ f₁' a b) (hf₂ : ∀ a b, f₂ a b ⊆ f₂' a b)\n    (hg₁ : ∀ a b, g₁ a b ⊆ g₁' a b) (hg₂ : ∀ a b, g₂ a b ⊆ g₂' a b) (a : α₁ ⊕ α₂)\n    (b : β₁ ⊕ β₂) : sumLexLift f₁ f₂ g₁ g₂ a b ⊆ sumLexLift f₁' f₂' g₁' g₂' a b :=\n\nProof body:\n","rejected":"```lean\nby\n  cases a <;> cases b\n  exacts [map_subset_map.2 (hf₁ _ _), disjSum_mono (hg₁ _ _) (hg₂ _ _), Subset.rfl,\n    map_subset_map.2 (hf₂ _ _)]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sum","family_id":"sumlexlift_mono","file_id":"mathlib/Mathlib/Data/Sum/Interval.lean","sample_id":"94c18dd81e6ce7afdea3016639b32636d0e629ff55fd5405a516ea90027133c6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"97f9e88375230bfbd4354802c179b8f0d71d6327ad14ee730bd5022e5fe5beee","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f38bfad9e7315903887b1c13bd5bbedf019cd02eb7213a0215d41686d086b548","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2a418ca608341e871713a18e1c34bfa17c3846a9b9d6a3f1d347ce4be3076bae","source_sha256":"610b55cc6b034881bfda1b6609f87ef5fc28aecf81f7b61240acfb496bc220e4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [center_eq_bot, Algebra.mem_bot]\n  simp [eq_comm]","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.153846},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"76d7669704ea192a8a46c7f5e863d61e4ccd2d3505bb25290ffae7cfada4a26f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Central.Defs\nimport Mathlib.Algebra.Module.Torsion.Field\n\nNamespace:\nAlgebra.IsCentral\n\nLocal context:\n/-\nCopyright (c) 2024 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Jujian Zhang, Yunzhou Xie\n-/\n/-!\n# Central Algebras\n\nIn this file, we prove some basic results about central algebras over a commutative ring.\n\n## Main results\n\n- `Algebra.IsCentral.center_eq_bot`: the center of a central algebra over `K` is equal to `K`.\n- `Algebra.IsCentral.self`: a commutative ring is a central algebra over itself.\n- `Algebra.IsCentral.baseField_essentially_unique`: Let `D/K/k` be a tower of scalars where\n  `K` and `k` are fields. If `D` is a nontrivial central algebra over `k`, `K` is isomorphic to `k`.\n-/\n\npublic section\n\nuniverse u v\n\nnamespace Algebra.IsCentral\n\nvariable (K : Type u) [CommSemiring K] (D D' : Type v) [Semiring D] [Algebra K D]\n  [h : IsCentral K D] [Semiring D'] [Algebra K D']\n\n@[simp]\nlemma center_eq_bot : Subalgebra.center K D = ⊥ := eq_bot_iff.2 IsCentral.out\n\nvariable {D} in\n\nTarget:\nlemma mem_center_iff {x : D} : x ∈ Subalgebra.center K D ↔ ∃ (a : K), x = algebraMap K D a :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_2a418ca60834","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"ab9a2f582d2df4e1b0208d4ea3460428eef7a79ba049950e32dbe97d8598c9d3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Central","family_id":"mem_center_iff","file_id":"mathlib/Mathlib/Algebra/Central/Basic.lean","sample_id":"2a418ca608341e871713a18e1c34bfa17c3846a9b9d6a3f1d347ce4be3076bae"},"split":"train"}
{"binding":{"chosen_proof_sha256":"edfba660bad11fda81e3675a3dfc917ea4d8a1ad555ebc02b9f4efe9064c3712","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"93f185ad5f3a4564fc85cbc6bdef840bfaf45f6f32a5872e78310d56f271faed","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3b27666f8db369265222b785fd8e88c116b6148890f43b6b80939bde09ac3038","source_sha256":"4c2fd0b515ef4b06865711215377bd2bbde8a660c25b8d72d68a97c859a1494a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro y hy\n  obtain ⟨x, (rfl : f x = y), h'⟩ := exists_ne_zero_of_finsum_mem_ne_zero hy\n  grind [mem_support]","hard_negative":true,"metrics":{"chosen_tokens":26,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.076923},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"772ed1c46d917f0135afb77534fd8f73c673bbc6aa8d4ed53580b9b18e8a0d33","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.LocallyFinsupp\npublic import Mathlib.Topology.Spectral.Basic\n\nNamespace:\nFunction.locallyFinsupp\n\nLocal context:\n/-\nCopyright (c) 2026 Raphael Douglas Giles. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Raphael Douglas Giles\n-/\n/-!\n# Pushforward of functions with locally finite support\n\nIn this file we define the notion of the pushforward of a function with locally finite support\nbetween prespectral spaces along a spectral map. This is used for defining the (proper) pushforward\nof algebraic cycles in algebraic geometry.\n\n## Main declarations\n\n- `Function.locallyFinsupp.map`: If `f : X → Y` is a spectral map between spectral spaces and\n  `c : X → R` is locally of finite support, the pushforward of `c` along `f` at `y : Y` is\n  `∑ᶠ x ∈ f ⁻¹' {y}, c x * w x`, where `w : X → R` is a weight function.\n\n## Notes\n\nIn the case of algebraic cycles, the weight function used in `Function.locallyFinsupp.map` will be\nspecialized to the degree of the residue field extension\n(see https://stacks.math.columbia.edu/tag/02R4).\n-/\n\n@[expose] public section\n\nopen Set Order Topology TopologicalSpace\n\nvariable {X Y R : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n  {f : X → Y} (hf : IsSpectralMap f) (w : X → R)\n\nnamespace Function.locallyFinsupp\n\nvariable [Semiring R] {W : Set Y} (hW : IsOpen W) (c : Function.locallyFinsupp X R)\n  [PrespectralSpace Y]\n\nvariable (f) in\n/--\nThe pushforward of a function `c` of locally finite support by a spectral map with respect to a\nweight function `w`.\n-/\nnoncomputable\ndef map (hf : IsSpectralMap f) (c : locallyFinsupp X R) : Function.locallyFinsupp Y R where\n  toFun z := ∑ᶠ x ∈ f ⁻¹' {z}, c x * w x\n  supportWithinDomain' := by simp\n  supportLocallyFiniteWithinDomain' y _ := by\n    obtain ⟨U, hU⟩ := (PrespectralSpace.isTopologicalBasis (X := Y)).exists_subset_of_mem_open\n      (by simp : y ∈ ⊤) (by simp)\n    refine ⟨U, IsOpen.mem_nhds hU.1.1 hU.2.1, ?_⟩\n    suffices h : (U ∩ {z | (f ⁻¹' {z} ∩ support ⇑c).Nonempty}).Finite by\n      refine h.subset (inter_subset_inter_right U fun y hy ↦ ?_)\n      obtain ⟨x, (hx : f x = y), h'⟩ := exists_ne_zero_of_finsum_mem_ne_zero hy\n      use x\n      grind [mem_support]\n    suffices (f ⁻¹' (U ∩ {z | (f ⁻¹' {z} ∩ c.support).Nonempty}) ∩ c.support).Finite from\n      (this.image f).subset (fun a ha ↦ by grind [Set.Nonempty])\n    exact (c.locallyFiniteSupport.finite_inter_support_of_isCompact <| hf.2 hU.1.1 hU.1.2).subset\n      (by simp; grind)\n\n@[simp]\nlemma map_apply (hf : IsSpectralMap f) (c : locallyFinsupp X R) (y : Y) :\n    map f w hf c y = ∑ᶠ x ∈ f ⁻¹' {y}, c x * w x := rfl\n\nTarget:\nlemma support_map_subset_of_forall_mem (s : Set X) (t : Set Y) (hc : c.support ⊆ s)\n    (h : ∀ x : X, x ∈ s → w x ≠ 0 → f x ∈ t) : (map f w hf c).support ⊆ t :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_3b27666f8db3","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"d30545b10931b8493c0d75457a70cbd499d3def0dc543a6654369515ba9f266e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/LocallyFinsupp","family_id":"support_map_subset_of_forall_mem","file_id":"mathlib/Mathlib/Topology/LocallyFinsupp/Pushforward.lean","sample_id":"3b27666f8db369265222b785fd8e88c116b6148890f43b6b80939bde09ac3038"},"split":"train"}
{"binding":{"chosen_proof_sha256":"89a379c0af7b46c3f249bf357aea100510d92246e204cc5b623f3d9530166237","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6808201512d88e8601fe27d1642ad64d66f2c7aa69022d1b0c666bb304c23120","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"58255fb9a08e3ea4853832b4b2e6754cd8d2484fe722ffff05ba10373b5631a3","source_sha256":"4b6a49b0497716f9a81e32d511b44e7e3917e3a9d925e53fe50fefb2d06728f2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← LieSubalgebra.toSubmodule_inj, range_toSubmodule, LieIdeal.toLieSubalgebra_toSubmodule,\n   ker_toSubmodule]\n  exact LinearMap.range_inl R L₁ L₂","hard_negative":false,"metrics":{"chosen_tokens":25,"rejected_tokens":32,"token_jaccard":0.909091,"token_length_ratio":1.28},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"773329383b851aa154549cf3cdd74922b642b81a49a3edd53a2a6ab68e89dea0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Lie.Ideal\n\nNamespace:\nLieHom\n\nLocal context:\n/-\nCopyright (c) 2026 Leonid Ryvkin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonid Ryvkin\n-/\n/-! ### Products of Lie algebras\n\nThis file defines the Lie algebra structure the Product of two Lie algebras\n\n## Main definitions\n\n- products in the domain:\n  - `LieHom.fst` The first projection of a product is a Lie algebra map.\n  - `LieHom.snd` The second projection of a product is a Lie algebra map.\n  - `LieHom.prod_ext` Split equality of Lie algebra homomorphisms from a product into Lie algebra\n  homomorphism over each component,\n- products in the codomain:\n  - `LieHom.inl` The left injection into a product is a Lie algebra map.\n  - `LieHom.inr` The right injection into a product is a Lie algebra map.\n  - `LieHom.prod` The prod of two Lie algebra homomorphisms is a Lie algebra homomorphism.\n- products in both domain and codomain:\n  - `LieHom.prodMap` the `Prod.map` of two Lie algebra homomorphisms is a Lie algebra homomorphism.\n\n## Todo: Extend to further functionality from LinearMap.prod e.g.\n - Lie Equivalences related to products\n - Lie Submodule statements\n\n-/\n\n@[expose] public section\n\nvariable {R L₁ L₂ L L₃ L₄ L₅ L₆ : Type*}\n  [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂]\n  [LieRing L] [LieAlgebra R L] [LieRing L₃] [LieAlgebra R L₃] [LieRing L₄] [LieAlgebra R L₄]\n  [LieRing L₅] [LieAlgebra R L₅] [LieRing L₆] [LieAlgebra R L₆]\n\nnamespace LieAlgebra.Prod\n\ninstance instLieRing : LieRing (L₁ × L₂) where\n  bracket x y := ⟨⁅x.1, y.1⁆, ⁅x.2, y.2⁆⟩\n  add_lie := by simp\n  lie_add := by simp\n  lie_self := by simp\n  leibniz_lie := by simp\n\n@[simp]\ntheorem bracket_apply (x y : L₁ × L₂) : ⁅x, y⁆ = ⟨⁅x.1, y.1⁆, ⁅x.2, y.2⁆⟩ := rfl\n\ninstance instLieAlgebra : LieAlgebra R (L₁ × L₂) where\n  lie_smul _ _ _ := by simp\n\nend LieAlgebra.Prod\n\nnamespace LieHom\n\nsection\nvariable (R L₁ L₂)\n\n/-- The first projection of a product is a Lie algebra map. -/\ndef fst : L₁ × L₂ →ₗ⁅R⁆ L₁ where\n  toLinearMap := LinearMap.fst R L₁ L₂\n  map_lie' := by simp\n\n/-- The second projection of a product is a Lie algebra map. -/\ndef snd : L₁ × L₂ →ₗ⁅R⁆ L₂ where\n  toLinearMap := LinearMap.snd R L₁ L₂\n  map_lie' := by simp\n\n/-- The left injection into a product is a Lie algebra map. -/\ndef inl : L₁ →ₗ⁅R⁆ L₁ × L₂ where\n  toLinearMap := LinearMap.inl R L₁ L₂\n  map_lie' := by simp\n\n/-- The right injection into a product is a Lie algebra map. -/\ndef inr : L₂ →ₗ⁅R⁆ L₁ × L₂ where\n  toLinearMap := LinearMap.inr R L₁ L₂\n  map_lie' := by simp\n\nend\n\n@[simp] theorem fst_apply (x : L₁ × L₂) : fst R L₁ L₂ x = x.1 := rfl\n\n@[simp] theorem snd_apply (x : L₁ × L₂) : snd R L₁ L₂ x = x.2 := rfl\n\n@[simp, norm_cast] lemma coe_fst : ⇑(fst R L₁ L₂) = Prod.fst := rfl\n\n@[simp, norm_cast] lemma coe_snd : ⇑(snd R L₁ L₂) = Prod.snd := rfl\n\ntheorem fst_surjective : Function.Surjective (fst R L₁ L₂) := fun x => ⟨(x, 0), rfl⟩\n\ntheorem snd_surjective : Function.Surjective (snd R L₁ L₂) := fun x => ⟨(0, x), rfl⟩\n\n/-- The prod of two Lie algebra homomorphisms is a Lie algebra homomorphism. -/\n@[simps!]\ndef prod (f : L →ₗ⁅R⁆ L₁) (g : L →ₗ⁅R⁆ L₂) : L →ₗ⁅R⁆ L₁ × L₂ where\n  toLinearMap := LinearMap.prod f g\n  map_lie' := by simp\n\ntheorem coe_prod (f : L →ₗ⁅R⁆ L₁) (g : L →ₗ⁅R⁆ L₂) : ⇑(f.prod g) = Function.prod f g :=\n  rfl\n\n@[simp]\ntheorem fst_prod (f : L →ₗ⁅R⁆ L₁) (g : L →ₗ⁅R⁆ L₂) : (fst R L₁ L₂).comp (prod f g) = f := rfl\n\n@[simp]\ntheorem snd_prod (f : L →ₗ⁅R⁆ L₁) (g : L →ₗ⁅R⁆ L₂) : (snd R L₁ L₂).comp (prod f g) = g := rfl\n\n@[simp]\ntheorem pair_fst_snd : prod (fst R L₁ L₂) (snd R L₁ L₂) = LieHom.id := rfl\n\ntheorem prod_comp (f : L₁ →ₗ⁅R⁆ L₂) (g : L₁ →ₗ⁅R⁆ L)\n    (h : L →ₗ⁅R⁆ L₁) : (f.prod g).comp h = (f.comp h).prod (g.comp h) :=\n  rfl\n\ntheorem inl_apply (x : L₁) : inl R L₁ L₂ x = (x, 0) := rfl\n\ntheorem inr_apply (x : L₂) : inr R L₁ L₂ x = (0, x) := rfl\n\n@[simp] theorem coe_inl : (inl R L₁ L₂ : L₁ → L₁ × L₂) = fun x => (x, 0) := rfl\n\n@[simp] theorem coe_inr : (inr R L₁ L₂ : L₂ → L₁ × L₂) = Prod.mk 0 := rfl\n\ntheorem inl_injective : Function.Injective (inl R L₁ L₂) := fun _ => by simp\n\ntheorem inr_injective : Function.Injective (inr R L₁ L₂) := fun _ => by simp\n\nsection\nvariable (R L₁ L₂)\n\nTarget:\ntheorem range_inl : range (inl R L₁ L₂) = ker (snd R L₁ L₂) :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [← LieSubalgebra.toSubmodule_inj, range_toSubmodule, LieIdeal.toLieSubalgebra_toSubmodule,\n   ker_toSubmodule]\n  exact LinearMap.range_inl R L₁ L₂\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Lie","family_id":"range_inl","file_id":"mathlib/Mathlib/Algebra/Lie/Prod.lean","sample_id":"58255fb9a08e3ea4853832b4b2e6754cd8d2484fe722ffff05ba10373b5631a3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"36011e8dfa494a26101903f706aa37335eb673289a71da56c05014c7180fb544","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f1fe02c934b4c86a0442cf4e7b420b4af8623b47e1199970324e651f3bf95d48","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e1e83a15a2543ab2c7734d45b0801536ee9f8edb36b69899defb9d3f1b0cfbea","source_sha256":"837c21dde83dde778dcfbbb9710a388801c6131a30965944eb186a17738be704","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction n generalizing g with\n  | zero =>\n    funext (x : Fin 1)\n    simp [diagonalSuccIsoTensorTrivial, diagonalOneIsoLeftRegular, Subsingleton.elim x 0,\n      ofMulAction_V, types_tensorObj_def, types_tensorUnit_def]\n  | succ n hn =>\n    funext x\n    induction x using Fin.cases with\n    | zero => simp; rfl\n    | succ i =>\n      simpa [diagonalSuccIsoTensorTrivial, types_tensorObj_def, mul_assoc, Fin.partialProd_succ',\n        ofMulAction_V] using! congrFun (hn (g * f 0) (Fin.tail f)) i","hard_negative":true,"metrics":{"chosen_tokens":91,"rejected_tokens":5,"token_jaccard":0.042553,"token_length_ratio":0.054945},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"77c6df0f5fa30300c1aee2e5a0fdb7092e617bfc89072e63027081531c520a97","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Fin\npublic import Mathlib.CategoryTheory.Monoidal.Linear\npublic import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory\npublic import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence\npublic import Mathlib.CategoryTheory.Monoidal.Transport\npublic import Mathlib.CategoryTheory.Monoidal.Types.Basic\npublic import Mathlib.CategoryTheory.Action.Concrete\npublic import Mathlib.CategoryTheory.Action.Limits\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Induced monoidal structure on `Action V G`\n\nWe show:\n\n* When `V` is monoidal, braided, or symmetric, so is `Action V G`.\n* When `V` is rigid and `G` is a group, `Action V G` is also rigid.\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen CategoryTheory Limits MonoidalCategory\n\nvariable {V : Type*} [Category* V] {G : Type*} [Monoid G]\n\nnamespace Action\n\nsection Monoidal\n\nopen MonoidalCategory\n\nvariable [MonoidalCategory V]\n\n@[simps! tensorUnit_V tensorObj_V tensorHom_hom whiskerLeft_hom whiskerRight_hom\n  associator_hom_hom associator_inv_hom leftUnitor_hom_hom leftUnitor_inv_hom\n  rightUnitor_hom_hom rightUnitor_inv_hom, reducible]\ninstance instMonoidalCategory : MonoidalCategory (Action V G) where\n  tensorObj X Y := Action.mk (X.V ⊗ Y.V) _\n  tensorUnit := Action.mk (𝟙_ _) _\n  __ := Monoidal.transport (Action.functorCategoryEquivalence _ _).symm\n\n@[simp]\ntheorem tensorUnit_ρ {g : G} :\n    @DFunLike.coe (G →* End (𝟙_ V)) _ _ _ (𝟙_ (Action V G)).ρ g = 𝟙 (𝟙_ V) :=\n  rfl\n\n@[simp]\ntheorem tensor_ρ {X Y : Action V G} {g : G} :\n    @DFunLike.coe (G →* End (X.V ⊗ Y.V)) _ _ _ (X ⊗ Y).ρ g = X.ρ g ⊗ₘ Y.ρ g :=\n  rfl\n\n/-- Given an object `X` isomorphic to the tensor unit of `V`, `X` equipped with the trivial action\nis isomorphic to the tensor unit of `Action V G`. -/\ndef tensorUnitIso {X : V} (f : 𝟙_ V ≅ X) : 𝟙_ (Action V G) ≅ Action.mk X 1 :=\n  Action.mkIso f\n\nvariable (V G)\n\nset_option backward.defeqAttrib.useBackward true in\ninstance : (Action.forget V G).Monoidal :=\n  Functor.CoreMonoidal.toMonoidal\n    { εIso := Iso.refl _\n      μIso := fun _ _ ↦ Iso.refl _ }\n\nopen Functor.LaxMonoidal Functor.OplaxMonoidal\n\n@[simp] lemma forget_ε : ε (Action.forget V G) = 𝟙 _ := rfl\n@[simp] lemma forget_η : η (Action.forget V G) = 𝟙 _ := rfl\n\nvariable {V G}\n\n@[simp] lemma forget_μ (X Y : Action V G) : μ (Action.forget V G) X Y = 𝟙 _ := rfl\n@[simp] lemma forget_δ (X Y : Action V G) : δ (Action.forget V G) X Y = 𝟙 _ := rfl\n\nvariable (V G)\n\nsection\n\nvariable [BraidedCategory V]\n\nset_option backward.defeqAttrib.useBackward true in\ninstance : BraidedCategory (Action V G) :=\n  .ofFaithful (Action.forget V G) fun X Y ↦ mkIso (β_ _ _) fun g ↦ by simp\n\n@[simp]\ntheorem β_hom_hom {X Y : Action V G} : (β_ X Y).hom.hom = (β_ X.V Y.V).hom := rfl\n\n@[simp]\ntheorem β_inv_hom {X Y : Action V G} : (β_ X Y).inv.hom = (β_ X.V Y.V).inv := rfl\n\nset_option backward.defeqAttrib.useBackward true in\n/-- When `V` is braided the forgetful functor `Action V G` to `V` is braided. -/\ninstance : (Action.forget V G).Braided where\n\nend\n\ninstance [SymmetricCategory V] : SymmetricCategory (Action V G) :=\n  .ofFaithful (Action.forget V G)\n\nsection\n\nvariable [Preadditive V] [MonoidalPreadditive V]\n\nattribute [local simp] MonoidalPreadditive.whiskerLeft_add MonoidalPreadditive.add_whiskerRight\n\ninstance : MonoidalPreadditive (Action V G) where\n\nvariable {R : Type*} [Semiring R] [Linear R V] [MonoidalLinear R V]\n\ninstance : MonoidalLinear R (Action V G) where\n\nend\n\nnoncomputable section\n\n/-- Upgrading the functor `Action V G ⥤ (SingleObj G ⥤ V)` to a monoidal functor. -/\ninstance FunctorCategoryEquivalence.functorMonoidal :\n    (FunctorCategoryEquivalence.functor (V := V) (G := G)).Monoidal :=\n  inferInstanceAs (Monoidal.equivalenceTransported\n    (Action.functorCategoryEquivalence V G).symm).inverse.Monoidal\n\ninstance functorCategoryEquivalenceFunctorMonoidal :\n    (functorCategoryEquivalence V G).functor.Monoidal :=\n  inferInstanceAs FunctorCategoryEquivalence.functor.Monoidal\n\n/-- Upgrading the functor `(SingleObj G ⥤ V) ⥤ Action V G` to a monoidal functor. -/\ninstance FunctorCategoryEquivalence.inverseMonoidal :\n    (FunctorCategoryEquivalence.inverse (V := V) (G := G)).Monoidal :=\n  inferInstanceAs (Monoidal.equivalenceTransported\n    (Action.functorCategoryEquivalence V G).symm).functor.Monoidal\n\ninstance functorCategoryEquivalenceInverseMonoidal :\n    (functorCategoryEquivalence V G).inverse.Monoidal :=\n  inferInstanceAs FunctorCategoryEquivalence.inverse.Monoidal\n\n@[simp]\nlemma FunctorCategoryEquivalence.functor_ε :\n    ε (FunctorCategoryEquivalence.functor (V := V) (G := G)) = 𝟙 _ := rfl\n\n@[simp]\nlemma FunctorCategoryEquivalence.functor_η :\n    η (FunctorCategoryEquivalence.functor (V := V) (G := G)) = 𝟙 _ := rfl\n\n@[simp]\nlemma FunctorCategoryEquivalence.functor_μ (A B : Action V G) :\n    μ FunctorCategoryEquivalence.functor A B = 𝟙 _ := rfl\n\n@[simp]\nlemma FunctorCategoryEquivalence.functor_δ (A B : Action V G) :\n    δ FunctorCategoryEquivalence.functor A B = 𝟙 _ := rfl\n\n\nvariable (H : Type*) [Group H]\n\ninstance [RightRigidCategory V] : RightRigidCategory (SingleObj H ⥤ V) := by\n  infer_instance\n\n/-- If `V` is right rigid, so is `Action V G`. -/\ninstance [RightRigidCategory V] : RightRigidCategory (Action V H) :=\n  rightRigidCategoryOfEquivalence\n    (functorCategoryEquivalence V H).toAdjunction\n\ninstance [LeftRigidCategory V] : LeftRigidCategory (SingleObj H ⥤ V) := by\n  infer_instance\n\n/-- If `V` is left rigid, so is `Action V G`. -/\ninstance [LeftRigidCategory V] : LeftRigidCategory (Action V H) :=\n  leftRigidCategoryOfEquivalence (functorCategoryEquivalence V H).toAdjunction\n\ninstance [RigidCategory V] : RigidCategory (SingleObj H ⥤ V) := by\n  infer_instance\n\n/-- If `V` is rigid, so is `Action V G`. -/\ninstance [RigidCategory V] : RigidCategory (Action V H) :=\n  rigidCategoryOfEquivalence (functorCategoryEquivalence V H).toAdjunction\n\nvariable {V H}\nvariable (X : Action V H)\n\n@[simp]\ntheorem rightDual_v [RightRigidCategory V] : Xᘁ.V = X.Vᘁ :=\n  rfl\n\n@[simp]\ntheorem leftDual_v [LeftRigidCategory V] : (ᘁX).V = ᘁX.V :=\n  rfl\n\ntheorem rightDual_ρ [RightRigidCategory V] (h : H) : Xᘁ.ρ h = (X.ρ (h⁻¹ : H))ᘁ := by\n  rw [← SingleObj.inv_as_inv]; rfl\n\ntheorem leftDual_ρ [LeftRigidCategory V] (h : H) : (ᘁX).ρ h = ᘁX.ρ (h⁻¹ : H) := by\n  rw [← SingleObj.inv_as_inv]; rfl\n\nend\n\nend Monoidal\n\nsection\n\nopen MonoidalCategory\n\nvariable (G : Type u)\n\n/-- The natural isomorphism of `G`-sets `Gⁿ⁺¹ ≅ G × Gⁿ`, where `G` acts by left multiplication on\neach factor. -/\n@[simps! hom_hom inv_hom]\nnoncomputable def diagonalSuccIsoTensorDiagonal [Monoid G] (n : ℕ) :\n    diagonal G (n + 1) ≅ leftRegular G ⊗ diagonal G n :=\n  mkIso (Fin.consEquiv _).symm.toIso fun _ => rfl\n\nvariable [Group G]\n\n/-- Given `X : Action (Type u) G` for `G` a group, then `G × X` (with `G` acting as left\nmultiplication on the first factor and by `X.ρ` on the second) is isomorphic as a `G`-set to\n`G × X` (with `G` acting as left multiplication on the first factor and trivially on the second).\nThe isomorphism is given by `(g, x) ↦ (g, g⁻¹ • x)`. -/\n@[simps! hom_hom inv_hom]\nnoncomputable def leftRegularTensorIso (X : Action (Type u) G) :\n    leftRegular G ⊗ X ≅ leftRegular G ⊗ trivial G X.V :=\n  mkIso (Equiv.toIso {\n    toFun g := ⟨g.1, (X.ρ (g.1⁻¹ : G) g.2 : X.V)⟩\n    invFun g := ⟨g.1, X.ρ g.1 g.2⟩\n    left_inv _ := Prod.ext rfl <| by simp\n    right_inv _ := Prod.ext rfl <| by simp }) <| fun _ => by\n      ext _\n      simp only [tensorObj_V, tensor_ρ]\n      simp [types_tensorObj_def]\n      rfl\n\n\n/-- An isomorphism of `G`-sets `Gⁿ⁺¹ ≅ G × Gⁿ`, where `G` acts by left multiplication on `Gⁿ⁺¹` and\n`G` but trivially on `Gⁿ`. The map sends `(g₀, ..., gₙ) ↦ (g₀, (g₀⁻¹g₁, g₁⁻¹g₂, ..., gₙ₋₁⁻¹gₙ))`,\nand the inverse is `(g₀, (g₁, ..., gₙ)) ↦ (g₀, g₀g₁, g₀g₁g₂, ..., g₀g₁...gₙ).` -/\nnoncomputable def diagonalSuccIsoTensorTrivial :\n    ∀ n : ℕ, diagonal G (n + 1) ≅ leftRegular G ⊗ trivial G (Fin n → G)\n  | 0 =>\n    diagonalOneIsoLeftRegular G ≪≫\n      (ρ_ _).symm ≪≫ tensorIso (Iso.refl _) (tensorUnitIso (Equiv.ofUnique PUnit _).toIso)\n  | n + 1 =>\n    diagonalSuccIsoTensorDiagonal _ _ ≪≫\n      tensorIso (Iso.refl _) (diagonalSuccIsoTensorTrivial n) ≪≫\n        leftRegularTensorIso _ _ ≪≫\n          tensorIso (Iso.refl _)\n            (mkIso (Fin.insertNthEquiv (fun _ => G) 0).toIso fun _ => rfl)\n\nvariable {G}\n\n@[simp]\ntheorem diagonalSuccIsoTensorTrivial_hom_hom_apply {n : ℕ} (f : Fin (n + 1) → G) :\n    dsimp% (diagonalSuccIsoTensorTrivial G n).hom.hom f =\n      (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f i.succ) := by\n  induction n with\n  | zero => exact Prod.ext rfl (funext fun x => Fin.elim0 x)\n  | succ n hn =>\n    refine Prod.ext rfl (funext fun x => ?_)\n    induction x using Fin.cases\n    <;> simp_all [diagonalSuccIsoTensorTrivial, types_tensorObj_def]\n    <;> rfl\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\n\nTarget:\ntheorem diagonalSuccIsoTensorTrivial_inv_hom_apply {n : ℕ} (g : G) (f : Fin n → G) :\n    dsimp% (diagonalSuccIsoTensorTrivial G n).inv.hom (g, f) =\n      (g • Fin.partialProd f : Fin (n + 1) → G) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_e1e83a15a254","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"ae5030265a4ed6b8ab61fa2a066d996ed34bb244c22ca6e54f855f2c85999331","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Action","family_id":"diagonalsuccisotensortrivial_inv_hom_apply","file_id":"mathlib/Mathlib/CategoryTheory/Action/Monoidal.lean","sample_id":"e1e83a15a2543ab2c7734d45b0801536ee9f8edb36b69899defb9d3f1b0cfbea"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dfb0cfa54bf916f968be75edf3a87f502ebb8bc73b72d17a24c14c29c6f3e807","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5718bd96cf9f59ca4c83944f21497324dfdb689e13608aa103fc1d5ac63a1635","source_sha256":"3827253dc949c32959719d5a285bec59e92b58138d7cf11fd2dc87768305b52a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  convert! sup_mem_subalgebra_closure A f g\n  apply SetLike.ext'\n  simp","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":8,"token_jaccard":0.055556,"token_length_ratio":0.666667},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"77cebed937e7b6a7708952fa1b2e136c827021a41a80593c9f8486ba9984d1c6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Subalgebra.Tower\npublic import Mathlib.Analysis.RCLike.Basic\npublic import Mathlib.Topology.Algebra.Star.Real\npublic import Mathlib.Topology.Algebra.StarSubalgebra\npublic import Mathlib.Topology.Algebra.NonUnitalStarAlgebra\npublic import Mathlib.Topology.ContinuousMap.ContinuousMapZero\npublic import Mathlib.Topology.ContinuousMap.Lattice\npublic import Mathlib.Topology.ContinuousMap.Weierstrass\npublic import Mathlib.Algebra.Order.Module.Basic\n\nNamespace:\nContinuousMap\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison, Heather Macbeth\n-/\n/-!\n# The Stone-Weierstrass theorem\n\nIf a subalgebra `A` of `C(X, ℝ)`, where `X` is a compact topological space,\nseparates points, then it is dense.\n\nWe argue as follows.\n\n* In any subalgebra `A` of `C(X, ℝ)`, if `f ∈ A`, then `abs f ∈ A.topologicalClosure`.\n  This follows from the Weierstrass approximation theorem on `[-‖f‖, ‖f‖]` by\n  approximating `abs` uniformly thereon by polynomials.\n* This ensures that `A.topologicalClosure` is actually a sublattice:\n  if it contains `f` and `g`, then it contains the pointwise supremum `f ⊔ g`\n  and the pointwise infimum `f ⊓ g`.\n* Any nonempty sublattice `L` of `C(X, ℝ)` which separates points is dense,\n  by a nice argument approximating a given `f` above and below using separating functions.\n  For each `x y : X`, we pick a function `g x y ∈ L` so `g x y x = f x` and `g x y y = f y`.\n  By continuity these functions remain close to `f` on small patches around `x` and `y`.\n  We use compactness to identify a certain finitely indexed infimum of finitely indexed supremums\n  which is then close to `f` everywhere, obtaining the desired approximation.\n* Finally we put these pieces together. `L = A.topologicalClosure` is a nonempty sublattice\n  which separates points since `A` does, and so is dense (in fact equal to `⊤`).\n\nWe then prove the complex version for star subalgebras `A`, by separately approximating\nthe real and imaginary parts using the real subalgebra of real-valued functions in `A`\n(which still separates points, by taking the norm-square of a separating function).\n\n## Future work\n\nExtend to cover the case of subalgebras of the continuous functions vanishing at infinity,\non non-compact spaces.\n\n-/\n\n@[expose] public section\n\nassert_not_exists Unitization\n\nnoncomputable section\n\nnamespace ContinuousMap\n\nvariable {X : Type*} [TopologicalSpace X] [CompactSpace X]\n\nopen scoped Polynomial\n\n/-- Turn a function `f : C(X, ℝ)` into a continuous map into `Set.Icc (-‖f‖) (‖f‖)`,\nthereby explicitly attaching bounds.\n-/\ndef attachBound (f : C(X, ℝ)) : C(X, Set.Icc (-‖f‖) ‖f‖) where\n  toFun x := ⟨f x, ⟨neg_norm_le_apply f x, apply_le_norm f x⟩⟩\n\n@[simp]\ntheorem attachBound_apply_coe (f : C(X, ℝ)) (x : X) : ((attachBound f) x : ℝ) = f x :=\n  rfl\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\ntheorem polynomial_comp_attachBound (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :\n    (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound =\n      Polynomial.aeval f g := by\n  ext\n  simp only [Polynomial.aeval_subalgebra_coe, Polynomial.aeval_continuousMap_apply]\n  simp\n\n/-- Given a continuous function `f` in a subalgebra of `C(X, ℝ)`, postcomposing by a polynomial\ngives another function in `A`.\n\nThis lemma proves something slightly more subtle than this:\nwe take `f`, and think of it as a function into the restricted target `Set.Icc (-‖f‖) ‖f‖)`,\nand then postcompose with a polynomial function on that interval.\nThis is in fact the same situation as above, and so also gives a function in `A`.\n-/\ntheorem polynomial_comp_attachBound_mem (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :\n    (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound ∈ A := by\n  rw [polynomial_comp_attachBound]\n  apply SetLike.coe_mem\n\ntheorem comp_attachBound_mem_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A)\n    (p : C(Set.Icc (-‖f‖) ‖f‖, ℝ)) : p.comp (attachBound (f : C(X, ℝ))) ∈ A.topologicalClosure := by\n  -- `p` itself is in the closure of polynomials, by the Weierstrass theorem,\n  have mem_closure : p ∈ (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)).topologicalClosure :=\n    continuousMap_mem_polynomialFunctions_closure _ _ p\n  -- and so there are polynomials arbitrarily close.\n  have frequently_mem_polynomials := mem_closure_iff_frequently.mp mem_closure\n  -- To prove `p.comp (attachBound f)` is in the closure of `A`,\n  -- we show there are elements of `A` arbitrarily close.\n  apply mem_closure_iff_frequently.mpr\n  -- To show that, we pull back the polynomials close to `p`,\n  refine\n    ((compRightContinuousMap ℝ (attachBound (f : C(X, ℝ)))).continuousAt\n            p).tendsto.frequently_map\n      _ ?_ frequently_mem_polynomials\n  -- but need to show that those pullbacks are actually in `A`.\n  rintro _ ⟨g, ⟨-, rfl⟩⟩\n  simp only [SetLike.mem_coe, AlgHom.coe_toRingHom, compRightContinuousMap_apply,\n    Polynomial.toContinuousMapOnAlgHom_apply]\n  apply polynomial_comp_attachBound_mem\n\ntheorem abs_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A) :\n    |(f : C(X, ℝ))| ∈ A.topologicalClosure := by\n  let f' := attachBound (f : C(X, ℝ))\n  let abs : C(Set.Icc (-‖f‖) ‖f‖, ℝ) := { toFun := fun x : Set.Icc (-‖f‖) ‖f‖ => |(x : ℝ)| }\n  change abs.comp f' ∈ A.topologicalClosure\n  apply comp_attachBound_mem_closure\n\ntheorem inf_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :\n    (f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A.topologicalClosure := by\n  rw [inf_eq_half_smul_add_sub_abs_sub' ℝ]\n  refine\n    A.topologicalClosure.smul_mem\n      (A.topologicalClosure.sub_mem\n        (A.topologicalClosure.add_mem (A.le_topologicalClosure f.property)\n          (A.le_topologicalClosure g.property))\n        ?_)\n      _\n  exact mod_cast abs_mem_subalgebra_closure A _\n\ntheorem inf_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))\n    (f g : A) : (f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A := by\n  convert! inf_mem_subalgebra_closure A f g\n  apply SetLike.ext'\n  symm\n  rw [Subalgebra.topologicalClosure_coe, closure_eq_iff_isClosed]\n  exact h\n\ntheorem sup_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :\n    (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A.topologicalClosure := by\n  rw [sup_eq_half_smul_add_add_abs_sub' ℝ]\n  refine\n    A.topologicalClosure.smul_mem\n      (A.topologicalClosure.add_mem\n        (A.topologicalClosure.add_mem (A.le_topologicalClosure f.property)\n          (A.le_topologicalClosure g.property))\n        ?_)\n      _\n  exact mod_cast abs_mem_subalgebra_closure A _\n\nTarget:\ntheorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))\n    (f g : A) : (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"2a8f205f52a4e03d4c0662e35d4724ca07c42a56f9d74550af5099c739d1d40f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/ContinuousMap","family_id":"sup_mem_closed_subalgebra","file_id":"mathlib/Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean","sample_id":"5718bd96cf9f59ca4c83944f21497324dfdb689e13608aa103fc1d5ac63a1635"},"split":"train"}
{"binding":{"chosen_proof_sha256":"839d48065eb916980f8e3d750584330734f1d7501b43b9c8e7cc3615d69e6548","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"04cd638bd1240f2fe7fe9709e6cdc0eae054c396fecbe87a5f13fcb8dac94627","source_sha256":"00cdc131b09c868f94179716d61273d8fdf897ed023d34f30c7085668a3b31ac","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  exact if a0 : a = 0 then\n    a0.symm ▸ H0 b\n  else\n    have _ := mod_lt b a0\n    H1 _ _ a0 (GCD.induction (b % a) a H0 H1)\ntermination_by a","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.071429},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"77f93ac84efdb30ee989fa4d74a070befb44c5dbbd62c2303eaf2f8b1ae6ca89","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Order.RelClasses\n\nNamespace:\nEuclideanDomain\n\nLocal context:\n/-\nCopyright (c) 2018 Louis Carlin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Louis Carlin, Mario Carneiro\n-/\n/-!\n# Euclidean domains\n\nThis file introduces Euclidean domains and provides the extended Euclidean algorithm. To be precise,\na slightly more general version is provided which is sometimes called a transfinite Euclidean domain\nand differs in the fact that the degree function need not take values in `ℕ` but can take values in\nany well-ordered set. Transfinite Euclidean domains were introduced by Motzkin and examples which\ndon't satisfy the classical notion were provided independently by Hiblot and Nagata.\n\n## Main definitions\n\n* `EuclideanDomain`: Defines Euclidean domain with functions `quotient` and `remainder`. Instances\n  of `Div` and `Mod` are provided, so that one can write `a = b * (a / b) + a % b`.\n* `gcd`: defines the greatest common divisors of two elements of a Euclidean domain.\n* `xgcd`: given two elements `a b : R`, `xgcd a b` defines the pair `(x, y)` such that\n  `x * a + y * b = gcd a b`.\n* `lcm`: defines the lowest common multiple of two elements `a` and `b` of a Euclidean domain as\n  `a * b / (gcd a b)`\n\n## Main statements\n\nSee `Algebra.EuclideanDomain.Basic` for most of the theorems about Euclidean domains,\nincluding Bézout's lemma.\n\nSee `Algebra.EuclideanDomain.Instances` for the fact that `ℤ` is a Euclidean domain,\nas is any field.\n\n## Notation\n\n`≺` denotes the well-founded relation on the Euclidean domain, e.g. in the example of the polynomial\nring over a field, `p ≺ q` for polynomials `p` and `q` if and only if the degree of `p` is less than\nthe degree of `q`.\n\n## Implementation details\n\nInstead of working with a valuation, `EuclideanDomain` is implemented with the existence of a well\nfounded relation `r` on the integral domain `R`, which in the example of `ℤ` would correspond to\nsetting `i ≺ j` for integers `i` and `j` if the absolute value of `i` is smaller than the absolute\nvalue of `j`.\n\n## References\n\n* [Th. Motzkin, *The Euclidean algorithm*][MR32592]\n* [J.-J. Hiblot, *Des anneaux euclidiens dont le plus petit algorithme n'est pas à valeurs finies*]\n  [MR399081]\n* [M. Nagata, *On Euclid algorithm*][MR541021]\n\n\n## Tags\n\nEuclidean domain, transfinite Euclidean domain, Bézout's lemma\n-/\n\n@[expose] public section\n\nuniverse u\n\n/-- A `EuclideanDomain` is a non-trivial commutative ring with a division and a remainder,\n  satisfying `b * (a / b) + a % b = a`.\n  The definition of a Euclidean domain usually includes a valuation function `R → ℕ`.\n  This definition is slightly generalised to include a well-founded relation\n  `r` with the property that `r (a % b) b`, instead of a valuation. -/\n@[wikidata Q867345]\nclass EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R where\n  /-- A division function (denoted `/`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `%` denotes `remainder`. -/\n  protected quotient : R → R → R\n  /-- Division by zero should always give zero by convention. -/\n  protected quotient_zero : ∀ a, quotient a 0 = 0\n  /-- A remainder function (denoted `%`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `/` denotes `quotient`. -/\n  protected remainder : R → R → R\n  /-- The property that links the quotient and remainder functions.\n    This allows us to compute GCDs and LCMs. -/\n  protected quotient_mul_add_remainder_eq : ∀ a b, b * quotient a b + remainder a b = a\n  /-- A well-founded relation on `R`, satisfying `r (a % b) b`.\n    This ensures that the GCD algorithm always terminates. -/\n  protected r : R → R → Prop\n  /-- The relation `r` must be well-founded.\n    This ensures that the GCD algorithm always terminates. -/\n  r_wellFounded : WellFounded r\n  /-- The relation `r` satisfies `r (a % b) b`. -/\n  protected remainder_lt : ∀ (a) {b}, b ≠ 0 → r (remainder a b) b\n  /-- An additional constraint on `r`. -/\n  mul_left_not_lt : ∀ (a) {b}, b ≠ 0 → ¬r (a * b) a\n\n/-\nLean has far more theorems about fields than about Euclidean domains. We thus\nlower the priority of `Euclideandomain.toCommRing`, encouraging typeclass inference\nto try `Field.toCommRing` first. Without this priority-lowering, typeclass inference\nfinds the more inefficient path `Field.toEuclideanDomain.toCommRing` by default. This\npriority change saves over 500G instructions across mathlib. See\nhttps://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/We.20need.20to.20talk.20about.20Euclidean.20Domains/near/594655420\n-/\n-- see Note [lower instance priority]\nattribute [instance 100] EuclideanDomain.toCommRing\n\nnamespace EuclideanDomain\n\nvariable {R : Type u} [EuclideanDomain R]\n\n/-- Abbreviated notation for the well-founded relation `r` in a Euclidean domain. -/\nlocal infixl:50 \" ≺ \" => EuclideanDomain.r\n\nlocal instance wellFoundedRelation : WellFoundedRelation R where\n  rel := EuclideanDomain.r\n  wf := r_wellFounded\n\ninstance isWellFounded : IsWellFounded R (· ≺ ·) where\n  wf := r_wellFounded\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Div R :=\n  ⟨EuclideanDomain.quotient⟩\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Mod R :=\n  ⟨EuclideanDomain.remainder⟩\n\ntheorem div_add_mod (a b : R) : b * (a / b) + a % b = a :=\n  EuclideanDomain.quotient_mul_add_remainder_eq _ _\n\ntheorem mod_add_div (a b : R) : a % b + b * (a / b) = a :=\n  (add_comm _ _).trans (div_add_mod _ _)\n\ntheorem mod_add_div' (m k : R) : m % k + m / k * k = m := by\n  rw [mul_comm]\n  exact mod_add_div _ _\n\ntheorem div_add_mod' (m k : R) : m / k * k + m % k = m := by\n  rw [mul_comm]\n  exact div_add_mod _ _\n\ntheorem mod_lt : ∀ (a) {b : R}, b ≠ 0 → a % b ≺ b :=\n  EuclideanDomain.remainder_lt\n\ntheorem mul_right_not_lt {a : R} (b) (h : a ≠ 0) : ¬a * b ≺ b := by\n  rw [mul_comm]\n  exact mul_left_not_lt b h\n\n@[simp]\ntheorem mod_zero (a : R) : a % 0 = a := by simpa only [zero_mul, zero_add] using div_add_mod a 0\n\ntheorem lt_one (a : R) : a ≺ (1 : R) → a = 0 :=\n  haveI := Classical.dec\n  not_imp_not.1 fun h => by simpa only [one_mul] using mul_left_not_lt 1 h\n\n@[simp]\ntheorem div_zero (a : R) : a / 0 = 0 :=\n  EuclideanDomain.quotient_zero a\n\nsection\n\n@[elab_as_elim]\n\nTarget:\ntheorem GCD.induction {P : R → R → Prop} (a b : R) (H0 : ∀ x, P 0 x)\n    (H1 : ∀ a b, a ≠ 0 → P (b % a) a → P a b) : P a b :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/EuclideanDomain","family_id":"gcd","file_id":"mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean","sample_id":"04cd638bd1240f2fe7fe9709e6cdc0eae054c396fecbe87a5f13fcb8dac94627"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f4514d3004c77d49e14e5cb3065d22f4527708fcd18c9c1d909c571ef396f6f1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"56d1d90b79ded4c653dd4d1569f516fe3da4ae3361b2f34985ba3b4d14a8dd13","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e96a9f2e6f2cd142ce46e8c5c7977a2fb22f0ea1c0daabf15e8cbf9d7c647459","source_sha256":"56c135165b3ce5c6f5239e9a81f21fe0754d5a166fa0d32c24c14c3224af6216","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨n, hn⟩ := hnp\n  obtain ⟨k, hk⟩ := IsNilpotent.nilpotent R I M\n  have hk' : I.lcs M k = ⊥ := by\n    simp only [← toSubmodule_inj, I.coe_lcs_eq, hk, bot_toSubmodule]\n  suffices ∀ l, lowerCentralSeries R L M (l * n) ≤ I.lcs M l by\n    rw [isNilpotent_iff R]\n    use k * n\n    simpa [hk'] using this k\n  intro l\n  induction l with\n  | zero => simp\n  | succ l ih => exact (l.succ_mul n).symm ▸ lcs_le_lcs_of_is_nilpotent_span_sup_eq_top hxI hn ih","hard_negative":true,"metrics":{"chosen_tokens":112,"rejected_tokens":3,"token_jaccard":0.032787,"token_length_ratio":0.026786},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"7809726635ea05f7f73b6575daef09d00dfe5f7012f95baecf033134b35e4c57","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Lie.AdjointAction.Basic\npublic import Mathlib.Algebra.Lie.Nilpotent\npublic import Mathlib.Algebra.Lie.Normalizer\n\nNamespace:\nLieSubmodule\n\nLocal context:\n/-\nCopyright (c) 2022 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Engel's theorem\n\nThis file contains a proof of Engel's theorem providing necessary and sufficient conditions for Lie\nalgebras and Lie modules to be nilpotent.\n\nThe key result `LieModule.isNilpotent_iff_forall` says that if `M` is a Lie module of a\nNoetherian Lie algebra `L`, then `M` is nilpotent iff the image of `L → End(M)` consists of\nnilpotent elements. In the special case that we have the adjoint representation `M = L`, this says\nthat a Lie algebra is nilpotent iff `ad x : End(L)` is nilpotent for all `x : L`.\n\nEngel's theorem is true for any coefficients (i.e., it is really a theorem about Lie rings) and so\nwe work with coefficients in any commutative ring `R` throughout.\n\nOn the other hand, Engel's theorem is not true for infinite-dimensional Lie algebras and so a\nfinite-dimensionality assumption is required. We prove the theorem subject to the assumption\nthat the Lie algebra is Noetherian as an `R`-module, though actually we only need the slightly\nweaker property that the relation `>` is well-founded on the complete lattice of Lie subalgebras.\n\n## Remarks about the proof\n\nEngel's theorem is usually proved in the special case that the coefficients are a field, and uses\nan inductive argument on the dimension of the Lie algebra. One begins by choosing either a maximal\nproper Lie subalgebra (in some proofs) or a maximal nilpotent Lie subalgebra (in other proofs, at\nthe cost of obtaining a weaker end result).\n\nSince we work with general coefficients, we cannot induct on dimension and an alternate approach\nmust be taken. The key ingredient is the concept of nilpotency, not just for Lie algebras, but for\nLie modules. Using this concept, we define an _Engelian Lie algebra_ `LieAlgebra.IsEngelian` to\nbe one for which a Lie module is nilpotent whenever the action consists of nilpotent endomorphisms.\nThe argument then proceeds by selecting a maximal Engelian Lie subalgebra and showing that it cannot\nbe proper.\n\nThe first part of the traditional statement of Engel's theorem consists of the statement that if `M`\nis a non-trivial `R`-module and `L ⊆ End(M)` is a finite-dimensional Lie subalgebra of nilpotent\nelements, then there exists a non-zero element `m : M` that is annihilated by every element of `L`.\nThis follows trivially from the result established here `LieModule.isNilpotent_iff_forall`, that\n`M` is a nilpotent Lie module over `L`, since the last non-zero term in the lower central series\nwill consist of such elements `m` (see: `LieModule.nontrivial_max_triv_of_isNilpotent`). It seems\nthat this result has not previously been established at this level of generality.\n\nThe second part of the traditional statement of Engel's theorem concerns nilpotency of the Lie\nalgebra and a proof of this for general coefficients appeared in the literature as long ago\n[as 1937](zorn1937). This also follows trivially from `LieModule.isNilpotent_iff_forall` simply by\ntaking `M = L`.\n\nIt is pleasing that the two parts of the traditional statements of Engel's theorem are thus unified\ninto a single statement about nilpotency of Lie modules. This is not usually emphasised.\n\n## Main definitions\n\n  * `LieAlgebra.IsEngelian`\n  * `LieAlgebra.isEngelian_of_isNoetherian`\n  * `LieModule.isNilpotent_iff_forall`\n  * `LieAlgebra.isNilpotent_iff_forall`\n\n-/\n\n@[expose] public section\n\n\nuniverse u₁ u₂ u₃ u₄\n\nvariable {R : Type u₁} {L : Type u₂} {L₂ : Type u₃} {M : Type u₄}\nvariable [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L₂] [LieAlgebra R L₂]\nvariable [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]\n\nnamespace LieSubmodule\n\nopen LieModule\n\nvariable {I : LieIdeal R L} {x : L} (hxI : R ∙ x ⊔ I = ⊤)\ninclude hxI\n\ntheorem exists_smul_add_of_span_sup_eq_top (y : L) : ∃ t : R, ∃ z ∈ I, y = t • x + z := by\n  have hy : y ∈ (⊤ : Submodule R L) := Submodule.mem_top\n  simp only [← hxI, Submodule.mem_sup, Submodule.mem_span_singleton] at hy\n  obtain ⟨-, ⟨t, rfl⟩, z, hz, rfl⟩ := hy\n  exact ⟨t, z, hz, rfl⟩\n\ntheorem lie_top_eq_of_span_sup_eq_top (N : LieSubmodule R L M) :\n    (↑⁅(⊤ : LieIdeal R L), N⁆ : Submodule R M) =\n      (N : Submodule R M).map (toEnd R L M x) ⊔ (↑⁅I, N⁆ : Submodule R M) := by\n  simp only [lieIdeal_oper_eq_linear_span', Submodule.sup_span, mem_top, true_and,\n    Submodule.map_coe, toEnd_apply_apply]\n  refine le_antisymm (Submodule.span_le.mpr ?_) (Submodule.span_mono fun z hz => ?_)\n  · rintro z ⟨y, n, hn : n ∈ N, rfl⟩\n    obtain ⟨t, z, hz, rfl⟩ := exists_smul_add_of_span_sup_eq_top hxI y\n    simp only [SetLike.mem_coe, Submodule.span_union, Submodule.mem_sup]\n    exact\n      ⟨t • ⁅x, n⁆, Submodule.subset_span ⟨t • n, N.smul_mem' t hn, lie_smul t x n⟩, ⁅z, n⁆,\n        Submodule.subset_span ⟨z, hz, n, hn, rfl⟩, by simp⟩\n  · rcases hz with (⟨m, hm, rfl⟩ | ⟨y, -, m, hm, rfl⟩)\n    exacts [⟨x, m, hm, rfl⟩, ⟨y, m, hm, rfl⟩]\n\ntheorem lcs_le_lcs_of_is_nilpotent_span_sup_eq_top {n i j : ℕ}\n    (hxn : toEnd R L M x ^ n = 0) (hIM : lowerCentralSeries R L M i ≤ I.lcs M j) :\n    lowerCentralSeries R L M (i + n) ≤ I.lcs M (j + 1) := by\n  suffices\n    ∀ l,\n      ((⊤ : LieIdeal R L).lcs M (i + l) : Submodule R M) ≤\n        (I.lcs M j : Submodule R M).map (toEnd R L M x ^ l) ⊔\n          (I.lcs M (j + 1) : Submodule R M)\n    by simpa only [bot_sup_eq, LieIdeal.incl_coe, Submodule.map_zero, hxn] using! this n\n  intro l\n  induction l with\n  | zero =>\n    simp only [add_zero, LieIdeal.lcs_succ, pow_zero, Module.End.one_eq_id,\n      Submodule.map_id]\n    exact le_sup_of_le_left hIM\n  | succ l ih =>\n    simp only [LieIdeal.lcs_succ, i.add_succ l, lie_top_eq_of_span_sup_eq_top hxI, sup_le_iff]\n    refine ⟨(Submodule.map_mono ih).trans ?_, le_sup_of_le_right ?_⟩\n    · rw [Submodule.map_sup, ← Submodule.map_comp, ← Module.End.mul_eq_comp, ← pow_succ', ←\n        I.lcs_succ]\n      grw [coe_map_toEnd_le]\n    · norm_cast\n      gcongr\n      exact le_trans (antitone_lowerCentralSeries R L M le_self_add) hIM\n\nTarget:\ntheorem isNilpotentOfIsNilpotentSpanSupEqTop (hnp : IsNilpotent <| toEnd R L M x)\n    (hIM : IsNilpotent I M) : IsNilpotent L M :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_e96a9f2e6f2c","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"72262c7d8e2f20094c49fdd0b0ad2dbb2f532a8cf3441b40d13a57ddc9990029","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Lie","family_id":"isnilpotentofisnilpotentspansupeqtop","file_id":"mathlib/Mathlib/Algebra/Lie/Engel.lean","sample_id":"e96a9f2e6f2cd142ce46e8c5c7977a2fb22f0ea1c0daabf15e8cbf9d7c647459"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e9b659a1781e41131bb39de2a0211b972065e3468a0c84eab21865825ba5a5a7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7f3aefe15b0c458cb353fa8b5e262f1eafb8aad181fea17cfdfa3bc8a6bf18f2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9c259b24270f51c0a7679a42e8272d4047f65364bf40859a29277a457689ac84","source_sha256":"b5b8fef319876fc584704ec0902863ab8016fe0ba4dee5dd358f86f4688b13cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Rat.isSquare_intCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg (mod_cast hz),\n    Rat.cast_intCast]","hard_negative":false,"metrics":{"chosen_tokens":19,"rejected_tokens":26,"token_jaccard":0.882353,"token_length_ratio":1.368421},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"784a6cd8456315bf9fbd106fe97999d6027dff661b78faceb2cabce960374c0d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Rat\npublic import Mathlib.Data.Nat.Prime.Int\npublic import Mathlib.Data.Rat.Sqrt\npublic import Mathlib.Analysis.Real.Sqrt\npublic import Mathlib.RingTheory.Algebraic.Basic\npublic import Mathlib.Tactic.CrossRefAttribute\npublic import Mathlib.Tactic.IntervalCases\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov\n-/\n/-!\n# Irrational real numbers\n\nIn this file we define a predicate `Irrational` on `ℝ`, prove that the `n`-th root of an integer\nnumber is irrational if it is not integer, and that `√(q : ℚ)` is irrational if and only if\n`¬IsSquare q ∧ 0 ≤ q`.\n\nWe also provide dot-style constructors like `Irrational.add_ratCast`, `Irrational.ratCast_sub` etc.\n\nWith the `Decidable` instances in this file, is possible to prove `Irrational √n` using `decide`,\nwhen `n` is a numeric literal or cast;\nbut this only works if you `unseal Nat.sqrt.iter in` before the theorem where you use this proof.\n-/\n\n@[expose] public section\n\n\nopen Rat Real\n\n/-- A real number is irrational if it is not equal to any rational number. -/\n@[wikidata Q607728]\ndef Irrational (x : ℝ) :=\n  x ∉ Set.range ((↑) : ℚ → ℝ)\n\ntheorem irrational_iff_ne_rational (x : ℝ) : Irrational x ↔ ∀ a b : ℤ, b ≠ 0 → x ≠ a / b := by\n  simp [Irrational, Rat.forall, eq_comm]\n\ntheorem Irrational.ne_rational {x : ℝ} (hx : Irrational x) (a b : ℤ) : x ≠ a / b := by\n  rintro rfl; exact hx ⟨a / b, by simp⟩\n\ntheorem exists_rat_of_not_irrational {x : ℝ} (hx : ¬ Irrational x) : ∃ (q : ℚ), x = q := by\n  grind [Irrational]\n\n/-- A transcendental real number is irrational. -/\ntheorem Transcendental.irrational {r : ℝ} (tr : Transcendental ℚ r) : Irrational r := by\n  rintro ⟨a, rfl⟩\n  exact tr (isAlgebraic_algebraMap a)\n\n/-!\n### Irrationality of roots of integer and rational numbers\n-/\n\n\n/-- If `x^n`, `n > 0`, is integer and is not the `n`-th power of an integer, then\n`x` is irrational. -/\ntheorem irrational_nrt_of_notint_nrt {x : ℝ} (n : ℕ) (m : ℤ) (hxr : x ^ n = m)\n    (hv : ¬∃ y : ℤ, x = y) (hnpos : 0 < n) : Irrational x := by\n  rintro ⟨⟨N, D, P, C⟩, rfl⟩\n  rw [← cast_pow] at hxr\n  have c1 : ((D : ℤ) : ℝ) ≠ 0 := by\n    rw [Int.cast_ne_zero, Int.natCast_ne_zero]\n    exact P\n  have c2 : ((D : ℤ) : ℝ) ^ n ≠ 0 := pow_ne_zero _ c1\n  rw [mk_eq_divInt, cast_pow, cast_divInt, div_pow, div_eq_iff_mul_eq c2, ← Int.cast_pow,\n    ← Int.cast_pow, ← Int.cast_mul, Int.cast_inj] at hxr\n  have hdivn : (D : ℤ) ^ n ∣ N ^ n := Dvd.intro_left m hxr\n  rw [← Int.dvd_natAbs, ← Int.natCast_pow, Int.natCast_dvd_natCast, Int.natAbs_pow,\n    Nat.pow_dvd_pow_iff hnpos.ne'] at hdivn\n  obtain rfl : D = 1 := by rw [← Nat.gcd_eq_right hdivn, C.gcd_eq_one]\n  refine hv ⟨N, ?_⟩\n  rw [mk_eq_divInt, Int.ofNat_one, divInt_one, cast_intCast]\n\n/-- If `x^n = m` is an integer and `n` does not divide the `multiplicity p m`, then `x`\nis irrational. -/\ntheorem irrational_nrt_of_n_not_dvd_multiplicity {x : ℝ} (n : ℕ) {m : ℤ} (hm : m ≠ 0) (p : ℕ)\n    [hp : Fact p.Prime] (hxr : x ^ n = m)\n    (hv : multiplicity (p : ℤ) m % n ≠ 0) :\n    Irrational x := by\n  rcases Nat.eq_zero_or_pos n with (rfl | hnpos)\n  · rw [eq_comm, pow_zero, ← Int.cast_one, Int.cast_inj] at hxr\n    simp [hxr, multiplicity_of_one_right (mt isUnit_iff_dvd_one.1\n      (mt Int.natCast_dvd_natCast.1 hp.1.not_dvd_one))] at hv\n  refine irrational_nrt_of_notint_nrt _ _ hxr ?_ hnpos\n  rintro ⟨y, rfl⟩\n  rw [← Int.cast_pow, Int.cast_inj] at hxr\n  subst m\n  have : y ≠ 0 := by rintro rfl; rw [zero_pow hnpos.ne'] at hm; exact hm rfl\n  rw [(Int.finiteMultiplicity_iff.2 ⟨by simp [hp.1.ne_one], this⟩).multiplicity_pow\n    (Nat.prime_iff_prime_int.1 hp.1), Nat.mul_mod_right] at hv\n  exact hv rfl\n\ntheorem irrational_sqrt_of_multiplicity_odd (m : ℤ) (hm : 0 < m) (p : ℕ) [hp : Fact p.Prime]\n    (Hpv : multiplicity (p : ℤ) m % 2 = 1) :\n    Irrational (√m) :=\n  @irrational_nrt_of_n_not_dvd_multiplicity _ 2 _ (Ne.symm (ne_of_lt hm)) p hp\n    (sq_sqrt (Int.cast_nonneg hm.le)) (by rw [Hpv]; exact one_ne_zero)\n\n@[simp] theorem not_irrational_zero : ¬Irrational 0 := not_not_intro ⟨0, Rat.cast_zero⟩\n@[simp] theorem not_irrational_one : ¬Irrational 1 := not_not_intro ⟨1, Rat.cast_one⟩\n\ntheorem irrational_sqrt_ratCast_iff_of_nonneg {q : ℚ} (hq : 0 ≤ q) :\n    Irrational (√q) ↔ ¬IsSquare q := by\n  refine Iff.not (?_ : Exists _ ↔ Exists _)\n  constructor\n  · rintro ⟨y, hy⟩\n    refine ⟨y, Rat.cast_injective (α := ℝ) ?_⟩\n    rw [Rat.cast_mul, hy, mul_self_sqrt (Rat.cast_nonneg.2 hq)]\n  · rintro ⟨q', rfl⟩\n    exact ⟨|q'|, mod_cast (sqrt_mul_self_eq_abs q').symm⟩\n\ntheorem irrational_sqrt_ratCast_iff {q : ℚ} :\n    Irrational (√q) ↔ ¬IsSquare q ∧ 0 ≤ q := by\n  obtain hq | hq := le_or_gt 0 q\n  · simp_rw [irrational_sqrt_ratCast_iff_of_nonneg hq, and_iff_left hq]\n  · rw [sqrt_eq_zero_of_nonpos (Rat.cast_nonpos.2 hq.le)]\n    simp_rw [not_irrational_zero, false_iff, not_and, not_le, hq, implies_true]\n\nTarget:\ntheorem irrational_sqrt_intCast_iff_of_nonneg {z : ℤ} (hz : 0 ≤ z) :\n    Irrational (√z) ↔ ¬IsSquare z :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [← Rat.isSquare_intCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg (mod_cast hz),\n    Rat.cast_intCast]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Real","family_id":"irrational_sqrt_intcast_iff_of_nonneg","file_id":"mathlib/Mathlib/NumberTheory/Real/Irrational.lean","sample_id":"9c259b24270f51c0a7679a42e8272d4047f65364bf40859a29277a457689ac84"},"split":"train"}
{"binding":{"chosen_proof_sha256":"93aa954ac038910f3cc0d692da80205ae5a7230c0f479de23b1a6ec2e526a44a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a1c310e6cbed0abac9a29428b8024284b7e24880a10b65826e169b2ec0b0c48","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4158bc4533e6e85b1b86e67b31a3d6eca02e3c8f701cb75f73a81c4775a53885","source_sha256":"5d8f7f707a69d48e10d7817e517c71568b55a688a7dc3473a0fa55b05f45f810","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [FractionalOperation.tt, Multiset.mem_map] at rin\n  rw [show r = ![r 0, r 1] by simp [← List.ofFn_inj]]\n  apply lt_of_le_of_ne (mcf.right (r 0) (r 1)).left\n  intro equ\n  have asymm : r 0 ≠ r 1 := by\n    rcases (mcf.right (r 0) (r 1)).right equ with ⟨ha0, hb1⟩ | ⟨ha1, hb0⟩\n    · rw [ha0, hb1] at hab\n      exact hab\n    · rw [ha1, hb0] at hab\n      exact hab.symm\n  apply asymm\n  obtain ⟨o, in_omega, rfl⟩ := rin\n  change o (fun j => ![![a, b], ![b, a]] j 0) = o (fun j => ![![a, b], ![b, a]] j 1)\n  convert! symmega ![a, b] ![b, a] (by simp [List.Perm.swap]) o in_omega using 2 <;>\n    simp [Matrix.const_fin1_eq]\n\nvariable {C : Type*} [AddCommMonoid C] [PartialOrder C] [IsOrderedCancelAddMonoid C]","hard_negative":true,"metrics":{"chosen_tokens":237,"rejected_tokens":7,"token_jaccard":0.036585,"token_length_ratio":0.029536},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"785bf7be35b27ca08a8cb20294d2641d1097b4a33657c0096757c9e17417f85a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Fin\npublic import Mathlib.Algebra.Order.BigOperators.Group.Multiset\npublic import Mathlib.Data.Fin.VecNotation\npublic import Mathlib.LinearAlgebra.Matrix.Notation\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Martin Dvorak. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Martin Dvorak\n-/\n/-!\n\n# General-Valued Constraint Satisfaction Problems\n\nGeneral-Valued CSP is a very broad class of problems in discrete optimization.\nGeneral-Valued CSP subsumes Min-Cost-Hom (including 3-SAT for example) and Finite-Valued CSP.\n\n## Main definitions\n* `ValuedCSP`: A VCSP template; fixes a domain, a codomain, and allowed cost functions.\n* `ValuedCSP.Term`: One summand in a VCSP instance; calls a concrete function from given template.\n* `ValuedCSP.Term.evalSolution`: An evaluation of the VCSP term for given solution.\n* `ValuedCSP.Instance`: An instance of a VCSP problem over given template.\n* `ValuedCSP.Instance.evalSolution`: An evaluation of the VCSP instance for given solution.\n* `ValuedCSP.Instance.IsOptimumSolution`: Is given solution a minimum of the VCSP instance?\n* `Function.HasMaxCutProperty`: Can given binary function express the Max-Cut problem?\n* `FractionalOperation`: Multiset of operations on given domain of the same arity.\n* `FractionalOperation.IsSymmetricFractionalPolymorphismFor`: Is given fractional operation a\n  symmetric fractional polymorphism for given VCSP template?\n\n## References\n* [D. A. Cohen, M. C. Cooper, P. Creed, P. G. Jeavons, S. Živný,\n  *An Algebraic Theory of Complexity for Discrete Optimisation*][cohen2012]\n\n-/\n\n@[expose] public section\n\n/-- A template for a valued CSP problem over a domain `D` with costs in `C`.\nRegarding `C` we want to support `Bool`, `Nat`, `ENat`, `Int`, `Rat`, `NNRat`,\n`Real`, `NNReal`, `EReal`, `ENNReal`, and tuples made of any of those types. -/\n@[nolint unusedArguments]\nabbrev ValuedCSP (D C : Type*) [AddCommMonoid C] [PartialOrder C] [IsOrderedAddMonoid C] :=\n  Set (Σ (n : ℕ), (Fin n → D) → C) -- Cost functions `D^n → C` for any `n`\n\nvariable {D C : Type*} [AddCommMonoid C] [PartialOrder C] [IsOrderedAddMonoid C]\n\n/-- A term in a valued CSP instance over the template `Γ`. -/\nstructure ValuedCSP.Term (Γ : ValuedCSP D C) (ι : Type*) where\n  /-- Arity of the function -/\n  n : ℕ\n  /-- Which cost function is instantiated -/\n  f : (Fin n → D) → C\n  /-- The cost function comes from the template -/\n  inΓ : ⟨n, f⟩ ∈ Γ\n  /-- Which variables are plugged as arguments to the cost function -/\n  app : Fin n → ι\n\n/-- Evaluation of a `Γ` term `t` for given solution `x`. -/\ndef ValuedCSP.Term.evalSolution {Γ : ValuedCSP D C} {ι : Type*}\n    (t : Γ.Term ι) (x : ι → D) : C :=\n  t.f (x ∘ t.app)\n\n/-- A valued CSP instance over the template `Γ` with variables indexed by `ι`. -/\nabbrev ValuedCSP.Instance (Γ : ValuedCSP D C) (ι : Type*) : Type _ :=\n  Multiset (Γ.Term ι)\n\n/-- Evaluation of a `Γ` instance `I` for given solution `x`. -/\ndef ValuedCSP.Instance.evalSolution {Γ : ValuedCSP D C} {ι : Type*}\n    (I : Γ.Instance ι) (x : ι → D) : C :=\n  (I.map (·.evalSolution x)).sum\n\n/-- Condition for `x` being an optimum solution (min) to given `Γ` instance `I`. -/\ndef ValuedCSP.Instance.IsOptimumSolution {Γ : ValuedCSP D C} {ι : Type*}\n    (I : Γ.Instance ι) (x : ι → D) : Prop :=\n  ∀ y : ι → D, I.evalSolution x ≤ I.evalSolution y\n\n/-- Function `f` has Max-Cut property at labels `a` and `b` when `argmin f` is exactly\n`{ ![a, b], ![b, a] }`. -/\ndef Function.HasMaxCutPropertyAt (f : (Fin 2 → D) → C) (a b : D) : Prop :=\n  f ![a, b] = f ![b, a] ∧\n    ∀ x y : D, f ![a, b] ≤ f ![x, y] ∧ (f ![a, b] = f ![x, y] → a = x ∧ b = y ∨ a = y ∧ b = x)\n\n/-- Function `f` has Max-Cut property at some two non-identical labels. -/\ndef Function.HasMaxCutProperty (f : (Fin 2 → D) → C) : Prop :=\n  ∃ a b : D, a ≠ b ∧ f.HasMaxCutPropertyAt a b\n\n/-- Fractional operation is a finite unordered collection of D^m → D possibly with duplicates. -/\nabbrev FractionalOperation (D : Type*) (m : ℕ) : Type _ :=\n  Multiset ((Fin m → D) → D)\n\nvariable {m : ℕ}\n\n/-- Arity of the \"output\" of the fractional operation. -/\n@[simp]\ndef FractionalOperation.size (ω : FractionalOperation D m) : ℕ := ω.card\n\n/-- Fractional operation is valid iff nonempty. -/\ndef FractionalOperation.IsValid (ω : FractionalOperation D m) : Prop :=\n  ω ≠ ∅\n\n/-- Valid fractional operation contains an operation. -/\nlemma FractionalOperation.IsValid.contains {ω : FractionalOperation D m} (valid : ω.IsValid) :\n    ∃ g : (Fin m → D) → D, g ∈ ω :=\n  Multiset.exists_mem_of_ne_zero valid\n\n/-- Fractional operation applied to a transposed table of values. -/\ndef FractionalOperation.tt {ι : Type*} (ω : FractionalOperation D m) (x : Fin m → ι → D) :\n    Multiset (ι → D) :=\n  ω.map (fun (g : (Fin m → D) → D) (i : ι) => g ((Function.swap x) i))\n\n/-- Cost function admits given fractional operation, i.e., `ω` improves `f` in the `≤` sense. -/\ndef Function.AdmitsFractional {n : ℕ} (f : (Fin n → D) → C) (ω : FractionalOperation D m) : Prop :=\n  ∀ x : (Fin m → (Fin n → D)),\n    m • ((ω.tt x).map f).sum ≤ ω.size • Finset.univ.sum (fun i => f (x i))\n\n/-- Fractional operation is a fractional polymorphism for given VCSP template. -/\ndef FractionalOperation.IsFractionalPolymorphismFor\n    (ω : FractionalOperation D m) (Γ : ValuedCSP D C) : Prop :=\n  ∀ f ∈ Γ, f.snd.AdmitsFractional ω\n\n/-- Fractional operation is symmetric. -/\ndef FractionalOperation.IsSymmetric (ω : FractionalOperation D m) : Prop :=\n  ∀ x y : (Fin m → D), List.Perm (List.ofFn x) (List.ofFn y) → ∀ g ∈ ω, g x = g y\n\n/-- Fractional operation is a symmetric fractional polymorphism for given VCSP template. -/\ndef FractionalOperation.IsSymmetricFractionalPolymorphismFor\n    (ω : FractionalOperation D m) (Γ : ValuedCSP D C) : Prop :=\n  ω.IsFractionalPolymorphismFor Γ ∧ ω.IsSymmetric\n\nTarget:\nlemma Function.HasMaxCutPropertyAt.rows_lt_aux {C : Type*} [PartialOrder C]\n    {f : (Fin 2 → D) → C} {a b : D} (mcf : f.HasMaxCutPropertyAt a b) (hab : a ≠ b)\n    {ω : FractionalOperation D 2} (symmega : ω.IsSymmetric)\n    {r : Fin 2 → D} (rin : r ∈ (ω.tt ![![a, b], ![b, a]])) :\n    f ![a, b] < f r :=\n\nProof body:\n","rejected":"by\n  exact Function.HasMaxCutPropertyAt.rows_lt_aux","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"13bb5694a04c3658ab08a9768a64349de82812aed3f04b5206d6c377a31c8316","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Optimization","family_id":"function","file_id":"mathlib/Mathlib/Combinatorics/Optimization/ValuedCSP.lean","sample_id":"4158bc4533e6e85b1b86e67b31a3d6eca02e3c8f701cb75f73a81c4775a53885"},"split":"train"}
{"binding":{"chosen_proof_sha256":"974123677968aef5cec49d697aa3b1eb6faa233c3680735631e7ed936621e266","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"aab605044ac1a0cc179875c4807508cccde8f4829eb26975d9537a751ad57521","source_sha256":"174951d99036d6995b4ac6d486de807bedb58f340aa902f02502a7216515bf63","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [integral_fintype .of_finite]\n  congr with x\n  rw [measureReal_def]\n  congr 2\n  exact PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _)","hard_negative":false,"metrics":{"chosen_tokens":28,"rejected_tokens":2,"token_jaccard":0.045455,"token_length_ratio":0.071429},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"7867391291474dece28c8cf0604786c759ff179160f6900a956f6e010f2fce7f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.ProbabilityMassFunction.Basic\npublic import Mathlib.Probability.ProbabilityMassFunction.Constructions\npublic import Mathlib.MeasureTheory.Integral.Bochner.SumMeasure\n\nNamespace:\nPMF\n\nLocal context:\n/-\nCopyright (c) 2023 Joachim Breitner. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joachim Breitner\n-/\n/-!\n# Integrals with a measure derived from probability mass functions.\n\nThis file connects `PMF` with `integral`. The main result is that the integral (i.e. the expected\nvalue) with regard to a measure derived from a `PMF` is a sum weighted by the `PMF`.\n\nIt also provides the expected value for specific probability mass functions.\n-/\n\npublic section\n\nnamespace PMF\n\nopen MeasureTheory NNReal ENNReal TopologicalSpace\n\nsection General\n\nvariable {α : Type*} [MeasurableSpace α] [MeasurableSingletonClass α]\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]\n\ntheorem integral_eq_tsum (p : PMF α) (f : α → E) (hf : Integrable f p.toMeasure) :\n    ∫ a, f a ∂(p.toMeasure) = ∑' a, (p a).toReal • f a := calc\n  _ = ∫ a in p.support, f a ∂(p.toMeasure) := by rw [restrict_toMeasure_support p]\n  _ = ∑' (a : support p), (p.toMeasure {a.val}).toReal • f a := by\n    apply setIntegral_countable f p.support_countable\n    rwa [IntegrableOn, restrict_toMeasure_support p]\n  _ = ∑' (a : support p), (p a).toReal • f a := by\n    congr with x; congr 2\n    apply PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _)\n  _ = ∑' a, (p a).toReal • f a :=\n    tsum_subtype_eq_of_support_subset <| calc\n      (fun a ↦ (p a).toReal • f a).support ⊆ (fun a ↦ (p a).toReal).support :=\n        Function.support_smul_subset_left _ _\n      _ ⊆ support p := fun x h1 h2 => h1 (by simp [h2])\n\nTarget:\ntheorem integral_eq_sum [Fintype α] (p : PMF α) (f : α → E) :\n    ∫ a, f a ∂(p.toMeasure) = ∑ a, (p a).toReal • f a :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/ProbabilityMassFunction","family_id":"integral_eq_sum","file_id":"mathlib/Mathlib/Probability/ProbabilityMassFunction/Integrals.lean","sample_id":"aab605044ac1a0cc179875c4807508cccde8f4829eb26975d9537a751ad57521"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5e5876ae05a155ef3db2c120345927956dab48f54ef044b4e48039170ed59c7d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0f51432df11e05e50b8d812ab933e7980dbb5071dde78d4390e4b955274c46f4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b8a6fc429e8fab84bafaec67f5918625193c8c32c8b4d06caaef778ad29f2b44","source_sha256":"fcd4a8db6b3cd3d3eac0ed5f4bd02b4de008d2f8c57c99f188ee276c869b4813","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases f; cases g\n  dsimp at hfg₁ hfg₂\n  cat_disch","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.153846},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"78a2b846914e32b87245c710debec5081424b1241e9ff94111f603e25b322da7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Bicategory.LocallyDiscrete\npublic import Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Pseudo\n\nNamespace:\nCategoryTheory.Pseudofunctor.Grothendieck\n\nLocal context:\n/-\nCopyright (c) 2024 Calle Sönne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Calle Sönne, Joseph Hua\n-/\n/-!\n# The Grothendieck and CoGrothendieck constructions\n\n## The Grothendieck construction\n\nGiven a category `𝒮` and any pseudofunctor `F` from `𝒮` to `Cat`, we associate to it a category\n`∫ F`, defined as follows:\n* Objects: pairs `(S, a)` where `S` is an object of the base category and `a` is an object of the\n  category `F(S)`.\n* Morphisms: morphisms `(R, b) ⟶ (S, a)` are defined as pairs `(f, h)` where `f : R ⟶ S` is a\n  morphism in `𝒮` and `h : F(f)(a) ⟶ b`\n\nThe category `∫ F` is equipped with a projection functor `∫ F ⥤ 𝒮`,\ngiven by projecting to the first factors, i.e.\n* On objects, it sends `(S, a)` to `S`\n* On morphisms, it sends `(f, h)` to `f`\n\n## The CoGrothendieck construction\n\nGiven a category `𝒮` and any pseudofunctor `F` from `𝒮ᵒᵖ` to `Cat`,\nwe associate to it a category `∫ᶜ F`, defined as follows:\n* Objects: pairs `(S, a)` where `S` is an object of the base category and `a` is an object of the\n  category `F(S)`.\n* Morphisms: morphisms `(R, b) ⟶ (S, a)` are defined as pairs `(f, h)` where `f : R ⟶ S` is a\n  morphism in `𝒮` and `h : b ⟶ F(f)(a)`\n\nThe category `∫ᶜ F` is equipped with a functor `∫ᶜ F ⥤ 𝒮`,\ngiven by projecting to the first factors, i.e.\n* On objects, it sends `(S, a)` to `S`\n* On morphisms, it sends `(f, h)` to `f`\n\n## Naming conventions\n\nThe name `Grothendieck` is reserved for the construction on covariant pseudofunctors from `𝒮` to\n`Cat`, whereas the word `CoGrothendieck` is used for the contravariant construction.\nThis is consistent with the convention for the Grothendieck construction on 1-functors\n`CategoryTheory.Grothendieck`.\n\n## Future work / TODO\n\n1. Once the bicategory of pseudofunctors has been defined, show that this construction forms a\n   pseudofunctor from `LocallyDiscrete 𝒮 ⥤ᵖ Catᵒᵖ` to `Cat`.\n2. Deduce the results in `CategoryTheory.Grothendieck` as a specialization of\n   `Pseudofunctor.Grothendieck`.\n\n## References\n[Vistoli2008] \"Notes on Grothendieck Topologies, Fibered Categories and Descent Theory\" by\nAngelo Vistoli\n\n-/\n\nset_option backward.defeqAttrib.useBackward true\n\n@[expose] public section\n\nnamespace CategoryTheory.Pseudofunctor\n\nuniverse w v₁ v₂ v₃ u₁ u₂ u₃\n\nopen Functor Category Opposite Discrete Bicategory StrongTrans\n\nvariable {𝒮 : Type u₁} [Category.{v₁} 𝒮]\n\n/-- The type of objects in the fibered category associated to a pseudofunctor from a\n1-category to Cat. -/\n@[ext]\nstructure Grothendieck (F : LocallyDiscrete 𝒮 ⥤ᵖ Cat.{v₂, u₂}) where\n  /-- The underlying object in the base category. -/\n  base : 𝒮\n  /-- The object in the fiber of the base object. -/\n  fiber : F.obj ⟨base⟩\n\nnamespace Grothendieck\n\nvariable {F : LocallyDiscrete 𝒮 ⥤ᵖ Cat.{v₂, u₂}}\n\n/-- Notation for the Grothendieck category associated to a pseudofunctor `F`. -/\nscoped prefix:75 \"∫ \" => Grothendieck\n\n/-- A morphism in the Grothendieck construction `∫ F` between two points `X Y : ∫ F` consists of\na morphism in the base category `base : X.base ⟶ Y.base` and\na morphism in a fiber `f.fiber : (F.map base).obj X.fiber ⟶ Y.fiber`. -/\nstructure Hom (X Y : ∫ F) where\n  /-- The morphism between base objects. -/\n  base : X.base ⟶ Y.base\n  /-- The morphism in the fiber over the domain. -/\n  fiber : (F.map base.toLoc).toFunctor.obj X.fiber ⟶ Y.fiber\n\n@[simps! id_base id_fiber comp_base comp_fiber]\ninstance categoryStruct : CategoryStruct (∫ F) where\n  Hom X Y := Hom X Y\n  id X := {\n    base := 𝟙 X.base\n    fiber := (F.mapId ⟨X.base⟩).hom.toNatTrans.app X.fiber }\n  comp {X _ _} f g := {\n    base := f.base ≫ g.base\n    fiber := (F.mapComp f.base.toLoc g.base.toLoc).hom.toNatTrans.app X.fiber ≫\n      (F.map g.base.toLoc).toFunctor.map f.fiber ≫ g.fiber }\n\ninstance (X : ∫ F) : Inhabited (Hom X X) :=\n  ⟨𝟙 X⟩\n\nsection\n\nvariable {a b : ∫ F}\n\n@[ext (iff := false)]\n\nTarget:\nlemma Hom.ext (f g : a ⟶ b) (hfg₁ : f.base = g.base)\n    (hfg₂ : eqToHom (hfg₁ ▸ rfl) ≫ f.fiber = g.fiber) : f = g :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_b8a6fc429e8f","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"f11705708e0fe26b1af2fca31fde499b7af56a2b6718f99ad116c1a8060ec8d0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Bicategory","family_id":"hom","file_id":"mathlib/Mathlib/CategoryTheory/Bicategory/Grothendieck.lean","sample_id":"b8a6fc429e8fab84bafaec67f5918625193c8c32c8b4d06caaef778ad29f2b44"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7727f8d1a6315f3170b243f902f0e2ff87bac72c93a1e273349cdbafdd693264","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1205eb9e1bd541c4f97a6b6d682d3d9f05b7c621a6ffc31f4b5fc185468fec86","source_sha256":"ae512c5fcdaa1f75b5ee08453eb6bee821b508070e712fcb1cd05ec73381f2d7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [homEquiv]\n  ext j\n  simp [← uncurry_natural_left]","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":2,"token_jaccard":0.090909,"token_length_ratio":0.166667},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"78a95fb6532aa1d5cefea55265257fbb8fcc883354e5fa588ce4d5bc05683d4e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Monoidal.Closed.Enrichment\npublic import Mathlib.CategoryTheory.Enriched.FunctorCategory\n\nNamespace:\nCategoryTheory.MonoidalClosed.FunctorCategory\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Functor categories are monoidal closed\n\nLet `C` be a monoidal closed category. Let `J` be a category. In this file,\nwe obtain that the category `J ⥤ C` is monoidal closed if `C` has suitable\nlimits.\n\n-/\n\n@[expose] public section\n\nuniverse v₁ v₂ u₁ u₂\n\nnamespace CategoryTheory\n\nopen Category Limits MonoidalCategory\n\nnamespace MonoidalClosed\n\nnamespace FunctorCategory\n\nopen Enriched.FunctorCategory\n\nvariable {C : Type u₁} [Category.{v₁} C] [MonoidalCategory C] [MonoidalClosed C]\n  {J : Type u₂} [Category.{v₂} J]\n  [∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂]\n\nattribute [local simp] enrichedCategorySelf_hom\n\nsection\n\nvariable {F₁ F₂ F₂' F₃ F₃' : J ⥤ C}\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The bijection `(F₁ ⊗ F₂ ⟶ F₃) ≃ (F₂ ⟶ functorEnrichedHom C F₁ F₃)` when `F₁`, `F₂`\nand `F₃` are functors `J ⥤ C`, and `C` is monoidal closed. -/\nnoncomputable def homEquiv : (F₁ ⊗ F₂ ⟶ F₃) ≃ (F₂ ⟶ functorEnrichedHom C F₁ F₃) where\n  toFun f :=\n    { app j := end_.lift (fun k ↦ F₂.map k.hom ≫ curry (f.app k.right))\n        (fun k₁ k₂ φ ↦ by\n          dsimp\n          simp only [enrichedOrdinaryCategorySelf_eHomWhiskerLeft, Category.assoc,\n            enrichedOrdinaryCategorySelf_eHomWhiskerRight]\n          rw [← curry_natural_left_assoc, ← curry_natural_left_assoc,\n            ← curry_natural_right, curry_pre_app, Category.assoc,\n            ← f.naturality φ.right, Monoidal.tensorObj_map, tensorHom_def_assoc,\n            ← Under.w φ, Functor.map_comp, MonoidalCategory.whiskerLeft_comp_assoc,\n            whisker_exchange_assoc]) }\n  invFun g :=\n    { app j := uncurry (g.app j ≫ enrichedHomπ C _ _ (Under.mk (𝟙 j)))\n      naturality j j' φ := by\n        dsimp\n        rw [← uncurry_natural_right, tensorHom_def'_assoc, ← uncurry_pre_app,\n          ← uncurry_natural_left, Category.assoc, Category.assoc,\n          NatTrans.naturality_assoc, functorEnrichedHom_map,\n          end_.lift_π_assoc, enrichedOrdinaryCategorySelf_eHomWhiskerRight]\n        dsimp\n        rw [pre_id, NatTrans.id_app, enrichedOrdinaryCategorySelf_eHomWhiskerLeft,\n          Functor.map_id, Category.comp_id, Category.comp_id]\n        congr 2\n        rw [← enrichedOrdinaryCategorySelf_eHomWhiskerRight,\n          ← enrichedOrdinaryCategorySelf_eHomWhiskerLeft]\n        let α : Under.mk (𝟙 j) ⟶ (Under.map φ).obj (Under.mk (𝟙 j')) := Under.homMk φ\n        exact (enrichedHom_condition C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃) α).symm }\n  left_inv f := by cat_disch\n  right_inv g := by\n    ext j\n    dsimp\n    ext k\n    -- this following list was obtained by\n    -- `simp? [enrichedOrdinaryCategorySelf_eHomWhiskerLeft, Under.map, Comma.mapLeft]`\n    simp only [diagram_obj_obj, Functor.comp_obj, Under.forget_obj, enrichedCategorySelf_hom,\n      curry_uncurry, NatTrans.naturality_assoc, functorEnrichedHom_obj, functorEnrichedHom_map,\n      Under.map, Comma.mapLeft, Functor.const_obj_obj, Functor.id_obj, Discrete.natTrans_app,\n      StructuredArrow.left_eq_id, end_.lift_π, Under.mk_right, Under.mk_hom, Iso.refl_inv,\n      NatTrans.id_app, enrichedOrdinaryCategorySelf_eHomWhiskerRight, pre_id, Iso.refl_hom,\n      enrichedOrdinaryCategorySelf_eHomWhiskerLeft, Functor.map_id, Category.comp_id]\n    congr\n    simp\n\nTarget:\nlemma homEquiv_naturality_two_symm (f₂ : F₂ ⟶ F₂') (g : F₂' ⟶ functorEnrichedHom C F₁ F₃) :\n    homEquiv.symm (f₂ ≫ g) = F₁ ◁ f₂ ≫ homEquiv.symm g :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Monoidal","family_id":"homequiv_naturality_two_symm","file_id":"mathlib/Mathlib/CategoryTheory/Monoidal/Closed/FunctorCategory/Basic.lean","sample_id":"1205eb9e1bd541c4f97a6b6d682d3d9f05b7c621a6ffc31f4b5fc185468fec86"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d74b30d157e4d96b787ba5699f29a97d3d77e2405ded9e8ea9925e08645a5660","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"aad755a0feeacee18eda311e51c7bdd7c4de3325eadaa0977eddb936e76e9ea5","source_sha256":"006922c150073d26eccce5bef3353d7015b01e87320ecec635275017a27177d4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [coeff_eq_ncoeff, ncoeff_eq_zero_of_lt_order A (-n - 1) x]\n  lia\n\n/-- Given an endomorphism-valued function on integers satisfying a pointwise bounded-pole condition,\nwe produce a vertex operator. -/\nnoncomputable def of_coeff (f : ℤ → Module.End R V)\n    (hf : ∀ x, BddBelow (Function.support fun y ↦ f y x)) : VertexOperator R V :=\n  HVertexOperator.of_coeff f fun x ↦ (BddBelow.isWF (hf x)).isPWO","hard_negative":false,"metrics":{"chosen_tokens":99,"rejected_tokens":3,"token_jaccard":0.016667,"token_length_ratio":0.030303},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"78d5208459f8be00b5184cd7b88a491fa71f5430b613a3de97bcd69ad5d3158c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Vertex.HVertexOperator\npublic import Mathlib.Data.Int.Interval\n\nNamespace:\nVertexOperator\n\nLocal context:\n/-\nCopyright (c) 2024 Scott Carnahan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Carnahan\n-/\n/-!\n# Vertex operators\nIn this file we introduce vertex operators as linear maps to Laurent series.\n\n## Definitions\n* `VertexOperator` is an `R`-linear map from an `R`-module `V` to `LaurentSeries V`.\n* `VertexOperator.ncoeff` is the coefficient of a vertex operator under normalized indexing.\n\n## TODO\n* `HasseDerivative` : A divided-power derivative.\n* `Locality` : A weak form of commutativity.\n* `Residue products` : A family of products on `VertexOperator R V` parametrized by integers.\n\n## References\n* [G. Mason, *Vertex rings and Pierce bundles*][mason2017]\n* [A. Matsuo, K. Nagatomo, *On axioms for a vertex algebra and locality of quantum\n  fields*][matsuo1997]\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nvariable {R V : Type*} [CommRing R] [AddCommGroup V] [Module R V]\n\n/-- A vertex operator over a commutative ring `R` is an `R`-linear map from an `R`-module `V` to\nLaurent series with coefficients in `V`.  We write this as a specialization of the heterogeneous\ncase. -/\nabbrev VertexOperator (R : Type*) (V : Type*) [CommRing R] [AddCommGroup V]\n    [Module R V] := HVertexOperator ℤ R V V\n\nnamespace VertexOperator\n\nopen HVertexOperator\n\n@[ext]\ntheorem ext (A B : VertexOperator R V) (h : ∀ v : V, A v = B v) :\n    A = B := LinearMap.ext h\n\n/-- The coefficient of a vertex operator under normalized indexing. -/\ndef ncoeff : VertexOperator R V →ₗ[R] ℤ → Module.End R V where\n  toFun A n := HVertexOperator.coeff A (-n - 1)\n  map_add' _ _ := by ext; simp\n  map_smul' _ _ := by ext; simp\n\ntheorem ncoeff_apply (A : VertexOperator R V) (n : ℤ) : ncoeff A n = coeff A (-n - 1) :=\n  rfl\n\n/-- In the literature, the `n`th normalized coefficient of a vertex operator `A` is written as\neither `Aₙ` or `A(n)`. -/\nscoped[VertexOperator] notation A \"[[\" n \"]]\" => ncoeff A n\n\n@[simp]\ntheorem coeff_eq_ncoeff (A : VertexOperator R V)\n    (n : ℤ) : HVertexOperator.coeff A n = A[[-n - 1]] := by\n  rw [ncoeff_apply, neg_sub, Int.sub_neg, add_sub_cancel_left]\n\ntheorem ncoeff_eq_zero_of_lt_order (A : VertexOperator R V) (n : ℤ) (x : V)\n    (h : -n - 1 < HahnSeries.order ((HahnModule.of R).symm (A x))) : (A[[n]]) x = 0 := by\n  simp only [ncoeff, HVertexOperator.coeff, LinearMap.coe_mk, AddHom.coe_mk]\n  exact HahnSeries.coeff_eq_zero_of_lt_order h\n\nTarget:\ntheorem coeff_eq_zero_of_lt_order (A : VertexOperator R V) (n : ℤ) (x : V)\n    (h : n < HahnSeries.order ((HahnModule.of R).symm (A x))) : coeff A n x = 0 :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Vertex","family_id":"coeff_eq_zero_of_lt_order","file_id":"mathlib/Mathlib/Algebra/Vertex/VertexOperator.lean","sample_id":"aad755a0feeacee18eda311e51c7bdd7c4de3325eadaa0977eddb936e76e9ea5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2fa6d6384b3fc37bc32726c7d0dfd505d0e32dc595d3c34173c19db9a5d115ee","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d26cc5be978dead2b623c3a4790cc21af0d92aeaa8715f5d06e969086e47e5d8","source_sha256":"e8c6f00bd5a71ab259d53c29519ad94f338dec18478f71b6e20990ccef2afa1b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := IsClosedUnderBinaryCoproducts.closedUnderIsomorphisms P\n  have := hasFiniteCoproducts_of_has_binary_and_initial (C := P.FullSubcategory)\n  have := PreservesFiniteCoproducts.of_preserves_binary_and_initial P.ι\n  exact ⟨fun J _ ↦ P.isClosedUnderColimitsOfShape_of_preservesColimitsOfShape_ι _⟩","hard_negative":true,"metrics":{"chosen_tokens":37,"rejected_tokens":8,"token_jaccard":0.111111,"token_length_ratio":0.216216},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"78f514c1615585531791afb98a734528bdd415fe113a78bdfa58360225d9460f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts\npublic import Mathlib.CategoryTheory.Limits.FullSubcategory\npublic import Mathlib.CategoryTheory.ObjectProperty.ColimitsClosure\npublic import Mathlib.CategoryTheory.ObjectProperty.ContainsZero\npublic import Mathlib.Data.Fintype.Shrink\n\nNamespace:\nCategoryTheory.ObjectProperty\n\nLocal context:\n/-\nCopyright (c) 2026 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Properties of objects that are stable under finite products\n\nWe introduce typeclasses `IsClosedUnderBinaryProducts` and\n`IsClosedUnderFiniteProducts` expressing that `P : ObjectProperty C`\nis closed under binary products or finite products.\nWe introduce a constructor for `P.IsClosedUnderFiniteProducts`\nassuming `P.IsClosedUnderBinaryProducts`,\n`P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)` and that `C`\nhas finite products.\n\n-/\n\nuniverse w\n\npublic section\n\nnamespace CategoryTheory.ObjectProperty\n\nopen Limits\n\nvariable {C : Type*} [Category* C] (P : ObjectProperty C)\n\n/-- The typeclass saying that `P : ObjectProperty C` is stable under binary products. -/\nabbrev IsClosedUnderBinaryProducts :=\n  P.IsClosedUnderLimitsOfShape (Discrete WalkingPair)\n\nlemma prop_of_isLimit_binaryFan [P.IsClosedUnderBinaryProducts] {X Y : C} {B : BinaryFan X Y}\n    (hB : IsLimit B) (hX : P X) (hY : P Y) :\n    P B.pt :=\n  P.prop_of_isLimit hB (by rintro ⟨_ | _⟩ <;> assumption)\n\nlemma prop_prod [P.IsClosedUnderBinaryProducts] (X Y : C) [HasBinaryProduct X Y]\n    (hX : P X) (hY : P Y) :\n    P (X ⨯ Y) :=\n  P.prop_of_isLimit_binaryFan (limit.isLimit _) hX hY\n\nlemma prop_of_isTerminal [P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)]\n    (X : C) (hX : IsTerminal X) :\n    P X :=\n  P.prop_of_isLimit hX (by rintro ⟨⟨⟩⟩)\n\nlemma prop_terminal [P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)] [HasTerminal C] :\n    P (⊤_ C) :=\n  P.prop_of_isTerminal _ terminalIsTerminal\n\n-- see Note [lower instance priority]\ninstance (priority := 100) [P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)] [HasTerminal C] :\n    P.Nonempty :=\n  nonempty_of_prop P.prop_terminal\n\nlemma IsClosedUnderBinaryProducts.closedUnderIsomorphisms [HasTerminal C]\n    [P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)] [P.IsClosedUnderBinaryProducts] :\n    P.IsClosedUnderIsomorphisms where\n  of_iso {X Y} e hX := by\n    let h : IsLimit (BinaryFan.mk (terminal.from Y) e.inv) :=\n      BinaryFan.IsLimit.mk _ (fun _ f ↦ f ≫ e.hom) (by cat_disch) (by simp) (by cat_disch)\n    exact P.prop_of_isLimit_binaryFan h P.prop_terminal hX\n\n/-- All objects that are binary products of objects in `P`. -/\nabbrev binaryProductsClosure (P : ObjectProperty C) : ObjectProperty C :=\n  P.limitClosure (Discrete WalkingPair)\n\nlemma binaryProductsClosure_le_iff [HasTerminal C] {P Q : ObjectProperty C}\n    [Q.IsClosedUnderBinaryProducts] [Q.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)] :\n    P.binaryProductsClosure ≤ Q ↔ P ≤ Q := by\n  refine ⟨fun h ↦ (P.le_limitsClosure _).trans h, fun h ↦ ?_⟩\n  letI : Q.IsClosedUnderIsomorphisms := IsClosedUnderBinaryProducts.closedUnderIsomorphisms Q\n  exact limitsClosure_le h\n\n/-- The typeclass saying that `P : ObjectProperty C` is stable under finite products. -/\nclass IsClosedUnderFiniteProducts : Prop where\n  isClosedUnderLimitsOfShape (J : Type) [Finite J] :\n    P.IsClosedUnderLimitsOfShape (Discrete J) := by infer_instance\n\nvariable {P} in\n/-- `IsClosedUnderFiniteProducts` may be checked at any universe. -/\nlemma IsClosedUnderFiniteProducts.of_isClosedUnderLimitsOfShape\n    (H : ∀ (J : Type w) [Finite J], P.IsClosedUnderLimitsOfShape (Discrete J)) :\n    P.IsClosedUnderFiniteProducts where\n  isClosedUnderLimitsOfShape J _ := by\n    rw [P.isClosedUnderLimitsOfShape_iff_of_equivalence (Discrete.equivalence (equivShrink.{w} _))]\n    exact H _\n\ninstance [P.IsClosedUnderFiniteProducts] (J : Type*) [Finite J] :\n    P.IsClosedUnderLimitsOfShape (Discrete J) := by\n  obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin J\n  have : P.IsClosedUnderLimitsOfShape (Discrete (Fin n)) :=\n    IsClosedUnderFiniteProducts.isClosedUnderLimitsOfShape _\n  exact IsClosedUnderLimitsOfShape.of_equivalence (Discrete.equivalence e.symm)\n\ninstance [HasFiniteProducts C] [P.IsClosedUnderFiniteProducts] :\n    HasFiniteProducts P.FullSubcategory where\n  out _ := inferInstance\n\nlemma prop_of_isLimit_fan [P.IsClosedUnderFiniteProducts] {J : Type*} [Finite J] {f : J → C}\n    {F : Fan f} (hF : IsLimit F) (h : ∀ j, P (f j)) :\n    P F.pt :=\n  P.prop_of_isLimit hF (by intro ⟨j⟩; exact h j)\n\nlemma prop_product [P.IsClosedUnderFiniteProducts] {J : Type*} [Finite J] {f : J → C}\n    [HasProduct f] (h : ∀ j, P (f j)) :\n    P (∏ᶜ f) :=\n  P.prop_of_isLimit_fan (limit.isLimit (Discrete.functor f)) h\n\ninstance [P.ContainsZero] [P.IsClosedUnderIsomorphisms] :\n    P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty) where\n  limitsOfShape_le := by\n    rintro X ⟨p⟩\n    obtain ⟨Z, hZ, hZ₂⟩ := P.exists_prop_of_containsZero\n    have hX : IsTerminal X :=\n      (IsLimit.equivOfNatIsoOfIso p.diag.uniqueFromEmpty _ _\n        (by exact Cone.ext (Iso.refl _) (by rintro ⟨⟨⟩⟩))).1 p.isLimit\n    exact P.prop_of_isZero (IsZero.of_iso hZ\n      (IsLimit.conePointUniqueUpToIso hX (IsZero.isTerminal hZ)))\n\nvariable {P} in\nlemma IsClosedUnderFiniteProducts.mk' [HasFiniteProducts C]\n    [P.IsClosedUnderLimitsOfShape (Discrete.{0} PEmpty)]\n    [P.IsClosedUnderBinaryProducts] :\n    P.IsClosedUnderFiniteProducts := by\n  have := IsClosedUnderBinaryProducts.closedUnderIsomorphisms P\n  have := hasFiniteProducts_of_has_binary_and_terminal (C := P.FullSubcategory)\n  have := PreservesFiniteProducts.of_preserves_binary_and_terminal P.ι\n  exact ⟨fun J _ ↦ P.isClosedUnderLimitsOfShape_of_preservesLimitsOfShape_ι _⟩\n\n/-- The typeclass saying that `P : ObjectProperty C` is stable under binary coproducts. -/\nabbrev IsClosedUnderBinaryCoproducts :=\n  P.IsClosedUnderColimitsOfShape (Discrete WalkingPair)\n\nlemma prop_of_isColimit_binaryCofan [P.IsClosedUnderBinaryCoproducts] {X Y : C}\n    {B : BinaryCofan X Y} (hB : IsColimit B) (hX : P X) (hY : P Y) :\n    P B.pt :=\n  P.prop_of_isColimit hB (by rintro ⟨_ | _⟩ <;> assumption)\n\nlemma prop_coprod [P.IsClosedUnderBinaryCoproducts] (X Y : C) [HasBinaryCoproduct X Y]\n    (hX : P X) (hY : P Y) :\n    P (X ⨿ Y) :=\n  P.prop_of_isColimit_binaryCofan (colimit.isColimit (Limits.pair X Y)) hX hY\n\nlemma prop_of_isInitial [P.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty)]\n    (X : C) (hX : IsInitial X) :\n    P X :=\n  P.prop_of_isColimit hX (by rintro ⟨⟨⟩⟩)\n\nlemma prop_initial [P.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty)] [HasInitial C] :\n    P (⊥_ C) :=\n  P.prop_of_isInitial _ initialIsInitial\n\n-- see Note [lower instance priority]\ninstance (priority := 100) [P.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty)] [HasInitial C] :\n    P.Nonempty :=\n  nonempty_of_prop P.prop_initial\n\nset_option backward.defeqAttrib.useBackward true in\nlemma IsClosedUnderBinaryCoproducts.closedUnderIsomorphisms [HasInitial C]\n    [P.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty)] [P.IsClosedUnderBinaryCoproducts] :\n    P.IsClosedUnderIsomorphisms where\n  of_iso {X Y} e hX := by\n    let h : IsColimit (BinaryCofan.mk (initial.to Y) e.hom) :=\n      BinaryCofan.IsColimit.mk _ (fun _ f ↦ e.inv ≫ f) (by cat_disch) (by simp) (by cat_disch)\n    exact P.prop_of_isColimit_binaryCofan h P.prop_initial hX\n\n/-- All objects that are binary coproducts of objects in `P`. -/\nabbrev binaryCoproductsClosure (P : ObjectProperty C) : ObjectProperty C :=\n  P.colimitClosure (Discrete WalkingPair)\n\nlemma binaryCoproductsClosure_le_iff [HasInitial C] {P Q : ObjectProperty C}\n    [Q.IsClosedUnderBinaryCoproducts] [Q.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty)] :\n    P.binaryCoproductsClosure ≤ Q ↔ P ≤ Q := by\n  refine ⟨fun h ↦ (P.le_colimitsClosure _).trans h, fun h ↦ ?_⟩\n  letI : Q.IsClosedUnderIsomorphisms := IsClosedUnderBinaryCoproducts.closedUnderIsomorphisms Q\n  exact colimitsClosure_le h\n\n/-- The typeclass saying that `P : ObjectProperty C` is stable under finite coproducts. -/\nclass IsClosedUnderFiniteCoproducts : Prop where\n  isClosedUnderColimitsOfShape (J : Type) [Finite J] :\n    P.IsClosedUnderColimitsOfShape (Discrete J) := by infer_instance\n\nvariable {P} in\n/-- `IsClosedUnderFiniteProducts` may be checked at any universe. -/\nlemma IsClosedUnderFiniteCoproducts.of_isClosedUnderColimitsOfShape\n    (H : ∀ (J : Type w) [Finite J], P.IsClosedUnderColimitsOfShape (Discrete J)) :\n    P.IsClosedUnderFiniteCoproducts where\n  isClosedUnderColimitsOfShape J _ := by\n    rw [P.isClosedUnderColimitsOfShape_iff_of_equivalence\n      (Discrete.equivalence (equivShrink.{w} _))]\n    exact H _\n\ninstance [P.IsClosedUnderFiniteCoproducts] (J : Type*) [Finite J] :\n    P.IsClosedUnderColimitsOfShape (Discrete J) := by\n  obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin J\n  have : P.IsClosedUnderColimitsOfShape (Discrete (Fin n)) :=\n    IsClosedUnderFiniteCoproducts.isClosedUnderColimitsOfShape _\n  exact IsClosedUnderColimitsOfShape.of_equivalence (Discrete.equivalence e.symm)\n\ninstance [HasFiniteCoproducts C] [P.IsClosedUnderFiniteCoproducts] :\n    HasFiniteCoproducts P.FullSubcategory where\n  out _ := inferInstance\n\nlemma prop_of_isColimit_cofan [P.IsClosedUnderFiniteCoproducts] {J : Type*} [Finite J] {f : J → C}\n    {F : Cofan f} (hF : IsColimit F) (h : ∀ j, P (f j)) :\n    P F.pt :=\n  P.prop_of_isColimit hF (by intro ⟨j⟩; exact h j)\n\nlemma prop_coproduct [P.IsClosedUnderFiniteCoproducts] {J : Type*} [Finite J] {f : J → C}\n    [HasCoproduct f] (h : ∀ j, P (f j)) :\n    P (∐ f) :=\n  P.prop_of_isColimit_cofan (colimit.isColimit (Discrete.functor f)) h\n\ninstance [P.ContainsZero] [P.IsClosedUnderIsomorphisms] :\n    P.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty) where\n  colimitsOfShape_le := by\n    rintro X ⟨p⟩\n    obtain ⟨Z, hZ, hZ₂⟩ := P.exists_prop_of_containsZero\n    have hX : IsInitial X :=\n      (IsColimit.equivOfNatIsoOfIso p.diag.uniqueFromEmpty _ _\n        (by exact Cocone.ext (Iso.refl _) (by rintro ⟨⟨⟩⟩))).1 p.isColimit\n    exact P.prop_of_isZero (IsZero.of_iso hZ\n      (IsColimit.coconePointUniqueUpToIso hX (IsZero.isInitial hZ)))\n\nvariable {P} in\n\nTarget:\nlemma IsClosedUnderFiniteCoproducts.mk' [HasFiniteCoproducts C]\n    [P.IsClosedUnderColimitsOfShape (Discrete.{0} PEmpty)]\n    [P.IsClosedUnderBinaryCoproducts] :\n    P.IsClosedUnderFiniteCoproducts :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"96707b9d8d8fe48a7c0d213dadfb974d362741721dcec2e406783dcae461a9f0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/ObjectProperty","family_id":"isclosedunderfinitecoproducts","file_id":"mathlib/Mathlib/CategoryTheory/ObjectProperty/FiniteProducts.lean","sample_id":"d26cc5be978dead2b623c3a4790cc21af0d92aeaa8715f5d06e969086e47e5d8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"801cfb0a412f7e9f521af786c135901e8a7681f3edb0eee3dd6e24550333a788","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7a185c9e616497e88188303b7a6930b62204a459b2d119ee63aa031c5ee66469","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fcb3cb79819283ab318174f307cb21c4367decc586f83a6b3fb969ef48ad9494","source_sha256":"ba493885a2be935359e1e232e1bbaeb68ad80f382975839ead7d3190a1fce95f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · obtain a | a := a <;> obtain b | b := b\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (mem_disjSum.1 h).elim ?_ ?_\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inr <| Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr (Or.inr <| Or.inr <| ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n  · rintro (⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩ |\n      ⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact mem_map_of_mem _ hc\n    · exact inl_mem_disjSum.2 hc\n    · exact inr_mem_disjSum.2 hc\n    · exact mem_map_of_mem _ hc","hard_negative":true,"metrics":{"chosen_tokens":287,"rejected_tokens":3,"token_jaccard":0.044444,"token_length_ratio":0.010453},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"79f866ff335c1c0f8fbe2f6fca45c300f9404e569e6e9a780c50fe7ac0a5307a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Sum\npublic import Mathlib.Data.Sum.Order\npublic import Mathlib.Order.Interval.Finset.Defs\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Finite intervals in a disjoint union\n\nThis file provides the `LocallyFiniteOrder` instance for the disjoint sum and linear sum of two\norders and calculates the cardinality of their finite intervals.\n-/\n\n@[expose] public section\n\n\nopen Function Sum\n\nnamespace Finset\n\nvariable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}\n\nsection SumLift₂\n\nvariable (f f₁ g₁ : α₁ → β₁ → Finset γ₁) (g f₂ g₂ : α₂ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁` and `α₂ → β₂ → Finset γ₂` to a map\n`α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to `Alternative` functors if we can\nmake sure to keep computability and universe polymorphism. -/\n@[simp]\ndef sumLift₂ : ∀ (_ : α₁ ⊕ α₂) (_ : β₁ ⊕ β₂), Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f a b).map Embedding.inl\n  | inl _, inr _ => ∅\n  | inr _, inl _ => ∅\n  | inr a, inr b => (g a b).map Embedding.inr\n\nvariable {f f₁ g₁ g f₂ g₂} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\ntheorem mem_sumLift₂ :\n    c ∈ sumLift₂ f g a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨\n        ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g a₂ b₂ := by\n  constructor\n  · rcases a with a | a <;> rcases b with b | b\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr ⟨a, b, c, rfl, rfl, rfl, hc⟩\n  · rintro (⟨a, b, c, rfl, rfl, rfl, h⟩ | ⟨a, b, c, rfl, rfl, rfl, h⟩) <;> exact mem_map_of_mem _ h\n\ntheorem inl_mem_sumLift₂ {c₁ : γ₁} :\n    inl c₁ ∈ sumLift₂ f g a b ↔ ∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f a₁ b₁ := by\n  rw [mem_sumLift₂, or_iff_left]\n  · simp only [inl.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inl_ne_inr h\n\ntheorem inr_mem_sumLift₂ {c₂ : γ₂} :\n    inr c₂ ∈ sumLift₂ f g a b ↔ ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ g a₂ b₂ := by\n  rw [mem_sumLift₂, or_iff_right]\n  · simp only [inr.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inr_ne_inl h\n\ntheorem sumLift₂_eq_empty :\n    sumLift₂ f g a b = ∅ ↔\n      (∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f a₁ b₁ = ∅) ∧\n        ∀ a₂ b₂, a = inr a₂ → b = inr b₂ → g a₂ b₂ = ∅ := by\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · constructor <;>\n    · rintro a b rfl rfl\n      exact map_eq_empty.1 h\n  cases a <;> cases b\n  · exact map_eq_empty.2 (h.1 _ _ rfl rfl)\n  · rfl\n  · rfl\n  · exact map_eq_empty.2 (h.2 _ _ rfl rfl)\n\ntheorem sumLift₂_nonempty :\n    (sumLift₂ f g a b).Nonempty ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ (f a₁ b₁).Nonempty) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ (g a₂ b₂).Nonempty := by\n  simp only [nonempty_iff_ne_empty, Ne, sumLift₂_eq_empty, not_and_or, not_forall, exists_prop]\n\ntheorem sumLift₂_mono (h₁ : ∀ a b, f₁ a b ⊆ g₁ a b) (h₂ : ∀ a b, f₂ a b ⊆ g₂ a b) :\n    ∀ a b, sumLift₂ f₁ f₂ a b ⊆ sumLift₂ g₁ g₂ a b\n  | inl _, inl _ => map_subset_map.2 (h₁ _ _)\n  | inl _, inr _ => Subset.rfl\n  | inr _, inl _ => Subset.rfl\n  | inr _, inr _ => map_subset_map.2 (h₂ _ _)\n\nend SumLift₂\n\nsection SumLexLift\nvariable (f₁ f₁' : α₁ → β₁ → Finset γ₁) (f₂ f₂' : α₂ → β₂ → Finset γ₂)\n  (g₁ g₁' : α₁ → β₂ → Finset γ₁) (g₂ g₂' : α₁ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁`, `α₂ → β₂ → Finset γ₂`, `α₁ → β₂ → Finset γ₁`,\n`α₂ → β₂ → Finset γ₂`  to a map `α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to\nalternative monads if we can make sure to keep computability and universe polymorphism. -/\ndef sumLexLift : α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f₁ a b).map Embedding.inl\n  | inl a, inr b => (g₁ a b).disjSum (g₂ a b)\n  | inr _, inl _ => ∅\n  | inr a, inr b => (f₂ a b).map ⟨_, inr_injective⟩\n\n@[simp]\nlemma sumLexLift_inl_inl (a : α₁) (b : β₁) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inl b) = (f₁ a b).map Embedding.inl := rfl\n\n@[simp]\nlemma sumLexLift_inl_inr (a : α₁) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b) = (g₁ a b).disjSum (g₂ a b) := rfl\n\n@[simp]\nlemma sumLexLift_inr_inl (a : α₂) (b : β₁) : sumLexLift f₁ f₂ g₁ g₂ (inr a) (inl b) = ∅ := rfl\n\n@[simp]\nlemma sumLexLift_inr_inr (a : α₂) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inr a) (inr b) = (f₂ a b).map ⟨_, inr_injective⟩ := rfl\n\nvariable {f₁ g₁ f₂ g₂ f₁' g₁' f₂' g₂'} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\nTarget:\nlemma mem_sumLexLift :\n    c ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        (∃ a₁ b₂ c₁, a = inl a₁ ∧ b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨\n          (∃ a₁ b₂ c₂, a = inl a₁ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n            ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂ :=\n\nProof body:\n","rejected":"by\n  exact mem_sumLexLift","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"cd40224adc1b9edad13eda42c1703cff3c0ccae3bb0a5d1a15b040c0dbd79daf","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sum","family_id":"mem_sumlexlift","file_id":"mathlib/Mathlib/Data/Sum/Interval.lean","sample_id":"fcb3cb79819283ab318174f307cb21c4367decc586f83a6b3fb969ef48ad9494"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1ce00e07b29f18d90867caf068379ad134d68969aa30ca0e3ea5aac4b2537419","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b8f5e4251a21078b41e8816acb665bda0d4eb833b828902db199e6b72de33080","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9d858434b5715b6466fd7efd13eba53c42a1ae0530865db41d7c8c0a8af13979","source_sha256":"5a8a601bded85c9c17f5a633b68ceab704a71b9b3614d423fc312dd608794122","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [pullbackShiftFunctorZero_inv_app]\n  simp only [Functor.id_obj, pullbackShiftIso, eqToIso.inv, eqToHom_app, shiftFunctorZero',\n    Iso.trans_inv, NatTrans.comp_app, eqToIso_refl, Iso.refl_inv, NatTrans.id_app, assoc]\n  erw [comp_id]","hard_negative":false,"metrics":{"chosen_tokens":46,"rejected_tokens":50,"token_jaccard":0.896552,"token_length_ratio":1.086957},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"79fdfa8c9ed2667bcea7f52fcf14ca19654801306a9cd76345474d85c9d37ec1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Shift.Adjunction\npublic import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2023 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The pullback of a shift by a monoid morphism\n\nGiven a shift by a monoid `B` on a category `C` and a monoid morphism `φ : A →+ B`,\nwe define a shift by `A` on a category `PullbackShift C φ` which is a type synonym for `C`.\n\nIf `F : C ⥤ D` is a functor between categories equipped with shifts by `B`, we define\na type synonym `PullbackShift.functor F φ` for `F`. When `F` has a `CommShift` structure\nby `B`, we define a pulled back `CommShift` structure by `A` on `PullbackShift.functor F φ`.\n\nSimilarly, if `τ` is a natural transformation between functors `F,G : C ⥤ D`, we define\na type synonym\n`PullbackShift.natTrans τ φ : PullbackShift.functor F φ ⟶ PullbackShift.functor G φ`.\nWhen `τ` has a `CommShift` structure by `B` (i.e. is compatible with `CommShift` structures\non `F` and `G`), we define a pulled back `CommShift` structure by `A` on\n`PullbackShift.natTrans τ φ`.\n\nFinally, if we have an adjunction `F ⊣ G` (with `G : D ⥤ C`), we define a type synonym\n`PullbackShift.adjunction adj φ : PullbackShift.functor F φ ⊣ PullbackShift.functor G φ`\nand we show that, if `adj` is compatible with `CommShift` structures\non `F` and `G`, then `PullbackShift.adjunction adj φ` is also compatible with the pulled back\n`CommShift` structures.\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Limits Category\n\nvariable (C : Type*) [Category* C] {A B : Type*} [AddMonoid A] [AddMonoid B]\n\n/-- The category `PullbackShift C φ` is equipped with a shift such that for all `a`,\nthe shift functor by `a` is `shiftFunctor C (φ a)`. -/\n@[nolint unusedArguments]\ndef PullbackShift [HasShift C B] (_ : A →+ B) := C\nderiving Category\n\nattribute [local instance] endofunctorMonoidalCategory\n\nvariable [HasShift C B] (φ : A →+ B)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The shift on `PullbackShift C φ` is obtained by precomposing the shift on `C` with\nthe monoidal functor `Discrete.addMonoidalFunctor φ : Discrete A ⥤ Discrete B`. -/\ninstance : HasShift (PullbackShift C φ) A where\n  shift := Discrete.addMonoidalFunctor φ ⋙ shiftMonoidalFunctor C B\n\ninstance [HasZeroObject C] : HasZeroObject (PullbackShift C φ) :=\n  inferInstanceAs <| HasZeroObject C\n\ninstance [Preadditive C] : Preadditive (PullbackShift C φ) :=\n  inferInstanceAs <| Preadditive C\n\ninstance [Preadditive C] (a : A) [(shiftFunctor C (φ a)).Additive] :\n    (shiftFunctor (PullbackShift C φ) a).Additive :=\n  inferInstanceAs (shiftFunctor C (φ a)).Additive\n\n/-- When `b = φ a`, this is the canonical\nisomorphism `shiftFunctor (PullbackShift C φ) a ≅ shiftFunctor C b`. -/\ndef pullbackShiftIso (a : A) (b : B) (h : b = φ a) :\n    shiftFunctor (PullbackShift C φ) a ≅ shiftFunctor C b := eqToIso (by subst h; rfl)\n\nvariable {C}\nvariable (X : PullbackShift C φ) (a₁ a₂ a₃ : A) (h : a₁ + a₂ = a₃) (b₁ b₂ b₃ : B)\n  (h₁ : b₁ = φ a₁) (h₂ : b₂ = φ a₂) (h₃ : b₃ = φ a₃)\n\nset_option backward.defeqAttrib.useBackward true in\nlemma pullbackShiftFunctorZero_inv_app :\n    (shiftFunctorZero _ A).inv.app X =\n      (shiftFunctorZero C B).inv.app X ≫ (pullbackShiftIso C φ 0 0 (by simp)).inv.app X := by\n  change (shiftFunctorZero C B).inv.app X ≫ _ = _\n  dsimp [Discrete.eqToHom, Discrete.addMonoidalFunctor_ε]\n  congr 2\n  apply eqToHom_map\n\nset_option backward.isDefEq.respectTransparency false in\nlemma pullbackShiftFunctorZero_hom_app :\n    (shiftFunctorZero _ A).hom.app X =\n      (pullbackShiftIso C φ 0 0 (by simp)).hom.app X ≫ (shiftFunctorZero C B).hom.app X := by\n  rw [← cancel_epi ((shiftFunctorZero _ A).inv.app X), Iso.inv_hom_id_app,\n    pullbackShiftFunctorZero_inv_app, assoc, Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app]\n  rfl\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma pullbackShiftFunctorZero'_inv_app :\n    (shiftFunctorZero _ A).inv.app X = (shiftFunctorZero' C (φ 0) (by rw [map_zero])).inv.app X ≫\n      (pullbackShiftIso C φ 0 (φ 0) rfl).inv.app X :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [pullbackShiftFunctorZero_inv_app]\n  simp only [Functor.id_obj, pullbackShiftIso, eqToIso.inv, eqToHom_app, shiftFunctorZero',\n    Iso.trans_inv, NatTrans.comp_app, eqToIso_refl, Iso.refl_inv, NatTrans.id_app, assoc]\n  erw [comp_id]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Shift","family_id":"pullbackshiftfunctorzero'_inv_app","file_id":"mathlib/Mathlib/CategoryTheory/Shift/Pullback.lean","sample_id":"9d858434b5715b6466fd7efd13eba53c42a1ae0530865db41d7c8c0a8af13979"},"split":"train"}
{"binding":{"chosen_proof_sha256":"de4e82cdf62623b5f6eb51d83e7f17e091632bf91d46d2109ff7b01e77e7f85f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"143ab2863e5ef8203ad9cdb078d9a96faf3dccd3f496872d4f958e2b6081b962","source_sha256":"d6e838a5f1b3817af8820eb123fb16847eef4259153bc8aa4506ab114a1382bf","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hinj : Function.Injective (algebraMap F[X] K) := by\n    rw [IsScalarTower.algebraMap_eq F[X] F⟮X⟯ K]\n    exact (algebraMap F⟮X⟯ K).injective.comp (IsFractionRing.injective F[X] F⟮X⟯)\n  rw [injective_iff_map_eq_zero (algebraMap F[X] (↥(ringOfIntegers F K)))]\n  intro p hp\n  rw [← Subtype.coe_inj, Subalgebra.coe_zero] at hp\n  rw [injective_iff_map_eq_zero (algebraMap F[X] K)] at hinj\n  exact hinj p hp","hard_negative":false,"metrics":{"chosen_tokens":111,"rejected_tokens":2,"token_jaccard":0.025641,"token_length_ratio":0.018018},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"7a765987aa3fd156f39bd0a76dcaf480aa8b517f1c327d4428c89bdce8fa77c7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.RatFunc.Degree\npublic import Mathlib.RingTheory.DedekindDomain.IntegralClosure\npublic import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed\npublic import Mathlib.Topology.Algebra.Valued.ValuedField\npublic import Mathlib.Topology.Algebra.InfiniteSum.Defs\npublic import Mathlib.FieldTheory.RatFunc.IntermediateField\npublic import Mathlib.RingTheory.Adjoin.Polynomial.Bivariate\npublic import Mathlib.FieldTheory.RatFunc.Valuation -- for deprecation to `RatFunc.inftyValuation` and `RatFunc.CompletionAtInfty`\n\nNamespace:\nFunctionField.ringOfIntegers\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen, Ashvni Narayanan\n-/\n/-!\n# Function fields\n\nThis file defines a function field and the ring of integers corresponding to it.\n\n## Main definitions\n\n- `FunctionField F K` states that `K` is a function field over the field `F`,\n  i.e. it is a finite extension of the field of rational functions in one variable over `F`.\n- `FunctionField.ringOfIntegers` defines the ring of integers corresponding to a function field\n  as the integral closure of `F[X]` in the function field.\n\n## Implementation notes\nThe definitions that involve a field of fractions choose a canonical field of fractions,\nbut are independent of that choice. We also omit assumptions like\n`IsScalarTower F[X] (FractionRing F[X]) K` in definitions,\nadding them back in lemmas when they are needed.\n\n## References\n* [D. Marcus, *Number Fields*][marcus1977number]\n* [J.W.S. Cassels, A. Fröhlich, *Algebraic Number Theory*][cassels1967algebraic]\n* [P. Samuel, *Algebraic Theory of Numbers*][samuel1967]\n* [M. Rosen, *Number Theory in Function Fields*][rosen2002]\n\n## Tags\nfunction field, ring of integers\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen scoped nonZeroDivisors Polynomial WithZero RatFunc\n\nvariable (F K : Type*) [Field F] [Field K]\n\n/-- `K` is a function field over the field `F` if it is a finite\nextension of the field of rational functions in one variable over `F`.\n\nNote that `K` can be a function field over multiple, non-isomorphic, `F`.\n-/\nabbrev FunctionField [Algebra F⟮X⟯ K] : Prop :=\n  FiniteDimensional F⟮X⟯ K\n\n/-- `K` is a function field over `F` iff it is a finite extension of `F(t)`. -/\ntheorem functionField_iff (Ft : Type*) [Field Ft] [Algebra F[X] Ft]\n    [IsFractionRing F[X] Ft] [Algebra F⟮X⟯ K] [Algebra Ft K] [Algebra F[X] K]\n    [IsScalarTower F[X] Ft K] [IsScalarTower F[X] F⟮X⟯ K] :\n    FunctionField F K ↔ FiniteDimensional Ft K := by\n  let e := IsLocalization.algEquiv F[X]⁰ F⟮X⟯ Ft\n  have : ∀ (c) (x : K), e c • x = c • x := by\n    intro c x\n    rw [Algebra.smul_def, Algebra.smul_def]\n    congr\n    refine congr_fun (f := fun c => algebraMap Ft K (e c)) ?_ c\n    refine IsLocalization.ext (nonZeroDivisors F[X]) _ _ ?_ ?_ ?_ ?_ ?_ <;> intros <;>\n      simp only [map_one, map_mul, AlgEquiv.commutes, ← IsScalarTower.algebraMap_apply]\n  constructor <;> intro h\n  · let b := Module.finBasis F⟮X⟯ K\n    exact (b.mapCoeffs e this).finiteDimensional_of_finite\n  · let b := Module.finBasis Ft K\n    refine (b.mapCoeffs e.symm ?_).finiteDimensional_of_finite\n    intro c x; convert! (this (e.symm c) x).symm; simp only [e.apply_symm_apply]\n\nnamespace FunctionField\n\ntheorem algebraMap_injective [Algebra F[X] K] [Algebra F⟮X⟯ K]\n    [IsScalarTower F[X] F⟮X⟯ K] : Function.Injective (algebraMap F[X] K) := by\n  rw [IsScalarTower.algebraMap_eq F[X] F⟮X⟯ K]\n  exact (algebraMap F⟮X⟯ K).injective.comp (IsFractionRing.injective F[X] F⟮X⟯)\n\n/-- The function field analogue of `NumberField.ringOfIntegers`:\n`FunctionField.ringOfIntegers F K` is the integral closure of `F[X]` in `K`.\n\nWe don't actually assume `K` is a function field over `F` in the definition,\nonly when proving its properties.\n-/\ndef ringOfIntegers [Algebra F[X] K] :=\n  integralClosure F[X] K\n\nnamespace ringOfIntegers\n\nvariable [Algebra F[X] K]\n\ninstance : IsDomain (ringOfIntegers F K) :=\n  (ringOfIntegers F K).isDomain\n\ninstance : IsIntegralClosure (ringOfIntegers F K) F[X] K :=\n  integralClosure.isIntegralClosure _ _\n\nvariable [Algebra F⟮X⟯ K] [IsScalarTower F[X] F⟮X⟯ K]\n\nTarget:\ntheorem algebraMap_injective : Function.Injective (algebraMap F[X] (ringOfIntegers F K)) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory","family_id":"algebramap_injective","file_id":"mathlib/Mathlib/NumberTheory/FunctionField.lean","sample_id":"143ab2863e5ef8203ad9cdb078d9a96faf3dccd3f496872d4f958e2b6081b962"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5d0edc0cc6cc1668b437d5481146309b64fe078c9ece4c0df2b3120bce3ca534","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"30c038fad28d31c8e5506864f2ea72186e5c203c4362484a9ce9e1246c21ddcf","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"80671f5e23cc162f2c4d734bcae6d4b38501c2182a9dd22e1a1e199c28f65d78","source_sha256":"139083b5f0b4f5b6fe528c0c9bf16219e4a814c177511d56fdc66704924ae51e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine TopologicalSpace.ext_nhds fun x ↦ ?_\n  ext s\n  rw [mem_nhds_toTopsp, H]","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":3,"token_jaccard":0.05,"token_length_ratio":0.166667},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"7a7fe09111004f23eed4d079c9847a0a776051036109959571d3c4a0c1900823","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Analysis.Filter\npublic import Mathlib.Topology.Bases\npublic import Mathlib.Topology.LocallyFinite\n\nNamespace:\nCtop.Realizer\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Computational realization of topological spaces (experimental)\n\nThis file provides infrastructure to compute with topological spaces.\n\n## Main declarations\n\n* `Ctop`: Realization of a topology basis.\n* `Ctop.Realizer`: Realization of a topological space. `Ctop` that generates the given topology.\n* `LocallyFinite.Realizer`: Realization of the local finiteness of an indexed family of sets.\n* `Compact.Realizer`: Realization of the compactness of a set.\n-/\n\n@[expose] public section\n\n\nopen Set\n\nopen Filter hiding Realizer\n\nopen Topology\n\n/-- A `Ctop α σ` is a realization of a topology (basis) on `α`,\n  represented by a type `σ` together with operations for the top element and\n  the intersection operation. -/\nstructure Ctop (α σ : Type*) where\n  f : σ → Set α\n  top : α → σ\n  top_mem : ∀ x : α, x ∈ f (top x)\n  inter : ∀ (a b) (x : α), x ∈ f a ∩ f b → σ\n  inter_mem : ∀ a b x h, x ∈ f (inter a b x h)\n  inter_sub : ∀ a b x h, f (inter a b x h) ⊆ f a ∩ f b\n\nvariable {α : Type*} {β : Type*} {σ : Type*} {τ : Type*}\n\ninstance : Inhabited (Ctop α (Set α)) :=\n  ⟨{  f := id\n      top := singleton\n      top_mem := mem_singleton\n      inter := fun s t _ _ ↦ s ∩ t\n      inter_mem := fun _s _t _a ↦ id\n      inter_sub := fun _s _t _a _ha ↦ Subset.rfl }⟩\n\nnamespace Ctop\n\nsection\n\nvariable (F : Ctop α σ)\n\ninstance : CoeFun (Ctop α σ) fun _ ↦ σ → Set α :=\n  ⟨Ctop.f⟩\n\ntheorem coe_mk (f T h₁ I h₂ h₃ a) : (@Ctop.mk α σ f T h₁ I h₂ h₃) a = f a := rfl\n\n/-- Map a Ctop to an equivalent representation type. -/\ndef ofEquiv (E : σ ≃ τ) : Ctop α σ → Ctop α τ\n  | ⟨f, T, h₁, I, h₂, h₃⟩ =>\n    { f := fun a ↦ f (E.symm a)\n      top := fun x ↦ E (T x)\n      top_mem := fun x ↦ by simpa using h₁ x\n      inter := fun a b x h ↦ E (I (E.symm a) (E.symm b) x h)\n      inter_mem := fun a b x h ↦ by simpa using h₂ (E.symm a) (E.symm b) x h\n      inter_sub := fun a b x h ↦ by simpa using h₃ (E.symm a) (E.symm b) x h }\n\n@[simp]\ntheorem ofEquiv_val (E : σ ≃ τ) (F : Ctop α σ) (a : τ) : F.ofEquiv E a = F (E.symm a) := by\n  cases F; rfl\n\nend\n\n/-- Every `Ctop` is a topological space. -/\n@[implicit_reducible]\ndef toTopsp (F : Ctop α σ) : TopologicalSpace α := TopologicalSpace.generateFrom (Set.range F.f)\n\ntheorem toTopsp_isTopologicalBasis (F : Ctop α σ) :\n    @TopologicalSpace.IsTopologicalBasis _ F.toTopsp (Set.range F.f) :=\n  letI := F.toTopsp\n  ⟨fun _u ⟨a, e₁⟩ _v ⟨b, e₂⟩ ↦\n    e₁ ▸ e₂ ▸ fun x h ↦ ⟨_, ⟨_, rfl⟩, F.inter_mem a b x h, F.inter_sub a b x h⟩,\n    eq_univ_iff_forall.2 fun x ↦ ⟨_, ⟨_, rfl⟩, F.top_mem x⟩, rfl⟩\n\n@[simp]\ntheorem mem_nhds_toTopsp (F : Ctop α σ) {s : Set α} {a : α} :\n    s ∈ @nhds _ F.toTopsp a ↔ ∃ b, a ∈ F b ∧ F b ⊆ s :=\n  (@TopologicalSpace.IsTopologicalBasis.mem_nhds_iff _ F.toTopsp _ _ _\n        F.toTopsp_isTopologicalBasis).trans <|\n    ⟨fun ⟨_, ⟨x, rfl⟩, h⟩ ↦ ⟨x, h⟩, fun ⟨x, h⟩ ↦ ⟨_, ⟨x, rfl⟩, h⟩⟩\n\nend Ctop\n\n/-- A `Ctop` realizer for the topological space `T` is a `Ctop`\n  which generates `T`. -/\nstructure Ctop.Realizer (α) [T : TopologicalSpace α] where\n  σ : Type*\n  F : Ctop α σ\n  eq : F.toTopsp = T\n\nopen Ctop\n\n/-- A `Ctop` realizes the topological space it generates. -/\nprotected def Ctop.toRealizer (F : Ctop α σ) : @Ctop.Realizer _ F.toTopsp :=\n  @Ctop.Realizer.mk _ F.toTopsp σ F rfl\n\ninstance (F : Ctop α σ) : Inhabited (@Ctop.Realizer _ F.toTopsp) :=\n  ⟨F.toRealizer⟩\n\nnamespace Ctop.Realizer\n\nprotected theorem is_basis [T : TopologicalSpace α] (F : Realizer α) :\n    TopologicalSpace.IsTopologicalBasis (Set.range F.F.f) := by\n  have := toTopsp_isTopologicalBasis F.F; rwa [F.eq] at this\n\nprotected theorem mem_nhds [T : TopologicalSpace α] (F : Realizer α) {s : Set α} {a : α} :\n    s ∈ 𝓝 a ↔ ∃ b, a ∈ F.F b ∧ F.F b ⊆ s := by\n  have := @mem_nhds_toTopsp _ _ F.F s a; rwa [F.eq] at this\n\ntheorem isOpen_iff [TopologicalSpace α] (F : Realizer α) {s : Set α} :\n    IsOpen s ↔ ∀ a ∈ s, ∃ b, a ∈ F.F b ∧ F.F b ⊆ s :=\n  isOpen_iff_mem_nhds.trans <| forall₂_congr fun _a _h ↦ F.mem_nhds\n\ntheorem isClosed_iff [TopologicalSpace α] (F : Realizer α) {s : Set α} :\n    IsClosed s ↔ ∀ a, (∀ b, a ∈ F.F b → ∃ z, z ∈ F.F b ∩ s) → a ∈ s :=\n  isOpen_compl_iff.symm.trans <|\n    F.isOpen_iff.trans <|\n      forall_congr' fun a ↦\n        show (a ∉ s → ∃ b : F.σ, a ∈ F.F b ∧ ∀ z ∈ F.F b, z ∉ s) ↔ _ by\n          haveI := Classical.propDecidable; rw [not_imp_comm]\n          simp [not_exists, not_and, not_forall, and_comm]\n\ntheorem mem_interior_iff [TopologicalSpace α] (F : Realizer α) {s : Set α} {a : α} :\n    a ∈ interior s ↔ ∃ b, a ∈ F.F b ∧ F.F b ⊆ s :=\n  mem_interior_iff_mem_nhds.trans F.mem_nhds\n\nprotected theorem isOpen [TopologicalSpace α] (F : Realizer α) (s : F.σ) : IsOpen (F.F s) :=\n  isOpen_iff_nhds.2 fun a m ↦ by simpa using F.mem_nhds.2 ⟨s, m, Subset.refl _⟩\n\nTarget:\ntheorem ext' [T : TopologicalSpace α] {σ : Type*} {F : Ctop α σ}\n    (H : ∀ a s, s ∈ 𝓝 a ↔ ∃ b, a ∈ F b ∧ F b ⊆ s) : F.toTopsp = T :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_80671f5e23cc","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"8cb8a10397e47036930b32d832cda0ffd4352bb5adfb7fd8ce5eddc29d366b5e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Analysis","family_id":"ext","file_id":"mathlib/Mathlib/Data/Analysis/Topology.lean","sample_id":"80671f5e23cc162f2c4d734bcae6d4b38501c2182a9dd22e1a1e199c28f65d78"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f13ac1fd7bb3c4deec76c60552efe11696b97aca72a742edff11d90154604aae","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f8994d13abe40d9fcb9cf9840e64c01799efafd833a4f14b14aa97d04d698d68","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1de4a2ae584ee9a9376192b0c305c1c4427057eda902e8636079376cebce05ed","source_sha256":"a9ec2e65ed93a7fc2e8ad578160c3cad69b1b5b9ffe2a184226c12a8f804dde0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply elim_of_ne\n  rintro rfl\n  exact hl.notMem hj","hard_negative":false,"metrics":{"chosen_tokens":10,"rejected_tokens":15,"token_jaccard":0.666667,"token_length_ratio":1.5},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"7ab76bfbab8411713a9a44b7afd6c60279ef39ba91a61a2003990272cba5b9b0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.List.Nodup\npublic import Mathlib.Data.Set.Prod\n\nNamespace:\nList.TProd\n\nLocal context:\n/-\nCopyright (c) 2020 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn\n-/\n/-!\n# Finite products of types\n\nThis file defines the product of types over a list. For `l : List ι` and `α : ι → Type v` we define\n`List.TProd α l = l.foldr (fun i β ↦ α i × β) PUnit`.\nThis type should not be used if `∀ i, α i` or `∀ i ∈ l, α i` can be used instead\n(in the last expression, we could also replace the list `l` by a set or a finset).\nThis type is used as an intermediary between binary products and finitary products.\nThe application of this type is finitary product measures, but it could be used in any\nconstruction/theorem that is easier to define/prove on binary products than on finitary products.\n\n* Once we have the construction on binary products (like binary product measures in\n  `MeasureTheory.prod`), we can easily define a finitary version on the type `TProd l α`\n  by iterating. Properties can also be easily extended from the binary case to the finitary case\n  by iterating.\n* Then we can use the equivalence `List.TProd.piEquivTProd` below (or enhanced versions of it,\n  like a `MeasurableEquiv` for product measures) to get the construction on `∀ i : ι, α i`, at\n  least when assuming `[Fintype ι] [Encodable ι]` (using `Encodable.sortedUniv`).\n  Using `attribute [local instance] Fintype.toEncodable` we can get rid of the argument\n  `[Encodable ι]`.\n\n## Main definitions\n\n* We have the equivalence `TProd.piEquivTProd : (∀ i, α i) ≃ TProd α l`\n  if `l` contains every element of `ι` exactly once.\n* The product of sets is `Set.tprod : (∀ i, Set (α i)) → Set (TProd α l)`.\n-/\n\n@[expose] public section\n\n\nopen List Function\nuniverse u v\nvariable {ι : Type u} {α : ι → Type v} {i j : ι} {l : List ι}\n\nnamespace List\n\nvariable (α) in\n/-- The product of a family of types over a list. -/\nabbrev TProd (l : List ι) : Type v :=\n  l.foldr (fun i β => α i × β) PUnit\n\nnamespace TProd\n\n/-- Turning a function `f : ∀ i, α i` into an element of the iterated product `TProd α l`. -/\nprotected def mk : ∀ (l : List ι) (_f : ∀ i, α i), TProd α l\n  | [] => fun _ => PUnit.unit\n  | i :: is => fun f => (f i, TProd.mk is f)\n\ninstance [∀ i, Inhabited (α i)] : Inhabited (TProd α l) :=\n  ⟨TProd.mk l default⟩\n\n@[simp]\ntheorem fst_mk (i : ι) (l : List ι) (f : ∀ i, α i) : (TProd.mk (i :: l) f).1 = f i :=\n  rfl\n\n@[simp]\ntheorem snd_mk (i : ι) (l : List ι) (f : ∀ i, α i) :\n    (TProd.mk.{u, v} (i :: l) f).2 = TProd.mk.{u, v} l f :=\n  rfl\n\nvariable [DecidableEq ι]\n\n/-- Given an element of the iterated product `l.Prod α`, take a projection into direction `i`.\n  If `i` appears multiple times in `l`, this chooses the first component in direction `i`. -/\nprotected def elim : ∀ {l : List ι} (_ : TProd α l) {i : ι} (_ : i ∈ l), α i\n  | i :: is, v, j, hj =>\n    if hji : j = i then by\n      subst hji\n      exact v.1\n    else TProd.elim v.2 ((List.mem_cons.mp hj).resolve_left hji)\n\n@[simp]\ntheorem elim_self (v : TProd α (i :: l)) : v.elim mem_cons_self = v.1 := by simp [TProd.elim]\n\n@[simp]\ntheorem elim_of_ne (hj : j ∈ i :: l) (hji : j ≠ i) (v : TProd α (i :: l)) :\n    v.elim hj = TProd.elim v.2 ((List.mem_cons.mp hj).resolve_left hji) := by simp [TProd.elim, hji]\n\n@[simp]\n\nTarget:\ntheorem elim_of_mem (hl : (i :: l).Nodup) (hj : j ∈ l) (v : TProd α (i :: l)) :\n    v.elim (mem_cons_of_mem _ hj) = TProd.elim v.2 hj :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  apply elim_of_ne\n  rintro rfl\n  exact hl.notMem hj","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Prod","family_id":"elim_of_mem","file_id":"mathlib/Mathlib/Data/Prod/TProd.lean","sample_id":"1de4a2ae584ee9a9376192b0c305c1c4427057eda902e8636079376cebce05ed"},"split":"train"}
{"binding":{"chosen_proof_sha256":"73a3ff17197caa13c9881ebe96a98ed8cb7dce96f5c0218437447afab1f809f0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2d8dea88dd355420151a905a1257de8aac829b897e12eb600a797e8418da7629","source_sha256":"20bfccc248cbbadfa2532d19d3f23df6035aaa8042662f551c3dac73c46bbeb2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine Finset.measurable_sup' _ ?_\n  simpa [Finset.mem_range]","hard_negative":false,"metrics":{"chosen_tokens":14,"rejected_tokens":3,"token_jaccard":0.166667,"token_length_ratio":0.214286},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"7ab8392dc2388d9be4a2edb5ec4deea45d8a894a4b1c0620c1e67841595e9353","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Measure.AEMeasurable\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n-/\n/-!\n# Typeclasses for measurability of lattice operations\n\nIn this file we define classes `MeasurableSup` and `MeasurableInf` and prove dot-style\nlemmas (`Measurable.sup`, `AEMeasurable.sup` etc). For binary operations we define two typeclasses:\n\n- `MeasurableSup` says that both left and right sup are measurable;\n- `MeasurableSup₂` says that `fun p : α × α => p.1 ⊔ p.2` is measurable,\n\nand similarly for other binary operations. The reason for introducing these classes is that in case\nof topological space `α` equipped with the Borel `σ`-algebra, instances for `MeasurableSup₂`\netc. require `α` to have a second countable topology.\n\nFor instances relating, e.g., `ContinuousSup` to `MeasurableSup` see file\n`MeasureTheory.BorelSpace`.\n\n## Tags\n\nmeasurable function, lattice operation\n\n-/\n\npublic section\n\n\nopen MeasureTheory\n\n/-- We say that a type has `MeasurableSup` if `(c ⊔ ·)` and `(· ⊔ c)` are measurable functions.\nFor a typeclass assuming measurability of `uncurry (· ⊔ ·)` see `MeasurableSup₂`. -/\nclass MeasurableSup (M : Type*) [MeasurableSpace M] [Max M] : Prop where\n  measurable_const_sup : ∀ c : M, Measurable (c ⊔ ·) := by intro c; fun_prop\n  measurable_sup_const : ∀ c : M, Measurable (· ⊔ c) := by intro c; fun_prop\n\n/-- We say that a type has `MeasurableSup₂` if `uncurry (· ⊔ ·)` is a measurable functions.\nFor a typeclass assuming measurability of `(c ⊔ ·)` and `(· ⊔ c)` see `MeasurableSup`. -/\nclass MeasurableSup₂ (M : Type*) [MeasurableSpace M] [Max M] : Prop where\n  measurable_sup : Measurable fun p : M × M => p.1 ⊔ p.2 := by intro p; fun_prop\n\nexport MeasurableSup₂ (measurable_sup)\n\nexport MeasurableSup (measurable_const_sup measurable_sup_const)\n\n/-- We say that a type has `MeasurableInf` if `(c ⊓ ·)` and `(· ⊓ c)` are measurable functions.\nFor a typeclass assuming measurability of `uncurry (· ⊓ ·)` see `MeasurableInf₂`. -/\nclass MeasurableInf (M : Type*) [MeasurableSpace M] [Min M] : Prop where\n  measurable_const_inf : ∀ c : M, Measurable (c ⊓ ·) := by intro c; fun_prop\n  measurable_inf_const : ∀ c : M, Measurable (· ⊓ c) := by intro c; fun_prop\n\n/-- We say that a type has `MeasurableInf₂` if `uncurry (· ⊓ ·)` is a measurable functions.\nFor a typeclass assuming measurability of `(c ⊓ ·)` and `(· ⊓ c)` see `MeasurableInf`. -/\nclass MeasurableInf₂ (M : Type*) [MeasurableSpace M] [Min M] : Prop where\n  measurable_inf : Measurable fun p : M × M => p.1 ⊓ p.2 := by intro p; fun_prop\n\nexport MeasurableInf₂ (measurable_inf)\n\nexport MeasurableInf (measurable_const_inf measurable_inf_const)\n\nvariable {M : Type*} [MeasurableSpace M]\n\nsection OrderDual\n\ninstance (priority := 100) OrderDual.instMeasurableSup [Min M] [MeasurableInf M] :\n    MeasurableSup Mᵒᵈ :=\n  ⟨@measurable_const_inf M _ _ _, @measurable_inf_const M _ _ _⟩\n\ninstance (priority := 100) OrderDual.instMeasurableInf [Max M] [MeasurableSup M] :\n    MeasurableInf Mᵒᵈ :=\n  ⟨@measurable_const_sup M _ _ _, @measurable_sup_const M _ _ _⟩\n\ninstance (priority := 100) OrderDual.instMeasurableSup₂ [Min M] [MeasurableInf₂ M] :\n    MeasurableSup₂ Mᵒᵈ :=\n  ⟨@measurable_inf M _ _ _⟩\n\ninstance (priority := 100) OrderDual.instMeasurableInf₂ [Max M] [MeasurableSup₂ M] :\n    MeasurableInf₂ Mᵒᵈ :=\n  ⟨@measurable_sup M _ _ _⟩\n\nend OrderDual\n\nvariable {α : Type*} {m : MeasurableSpace α} {μ : Measure α} {f g : α → M}\n\nsection Sup\n\nvariable [Max M]\n\nsection MeasurableSup\n\nvariable [MeasurableSup M]\n\n@[fun_prop]\ntheorem Measurable.const_sup (hf : Measurable f) (c : M) : Measurable fun x => c ⊔ f x :=\n  (measurable_const_sup c).comp hf\n\n@[fun_prop]\ntheorem AEMeasurable.const_sup (hf : AEMeasurable f μ) (c : M) :\n    AEMeasurable (fun x => c ⊔ f x) μ :=\n  (MeasurableSup.measurable_const_sup c).comp_aemeasurable hf\n\n@[fun_prop]\ntheorem Measurable.sup_const (hf : Measurable f) (c : M) : Measurable fun x => f x ⊔ c :=\n  (measurable_sup_const c).comp hf\n\n@[fun_prop]\ntheorem AEMeasurable.sup_const (hf : AEMeasurable f μ) (c : M) :\n    AEMeasurable (fun x => f x ⊔ c) μ :=\n  (measurable_sup_const c).comp_aemeasurable hf\n\nend MeasurableSup\n\nsection MeasurableSup₂\n\nvariable [MeasurableSup₂ M]\n\n@[fun_prop]\ntheorem Measurable.sup' (hf : Measurable f) (hg : Measurable g) : Measurable (f ⊔ g) :=\n  measurable_sup.comp (hf.prodMk hg)\n\n@[fun_prop]\ntheorem Measurable.sup (hf : Measurable f) (hg : Measurable g) : Measurable fun a => f a ⊔ g a :=\n  measurable_sup.comp (hf.prodMk hg)\n\n@[fun_prop]\ntheorem AEMeasurable.sup' (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :\n    AEMeasurable (f ⊔ g) μ :=\n  measurable_sup.comp_aemeasurable (hf.prodMk hg)\n\n@[fun_prop]\ntheorem AEMeasurable.sup (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :\n    AEMeasurable (fun a => f a ⊔ g a) μ :=\n  measurable_sup.comp_aemeasurable (hf.prodMk hg)\n\ninstance (priority := 100) MeasurableSup₂.toMeasurableSup : MeasurableSup M where\n\nend MeasurableSup₂\n\nend Sup\n\nsection Inf\n\nvariable [Min M]\n\nsection MeasurableInf\n\nvariable [MeasurableInf M]\n\n@[fun_prop]\ntheorem Measurable.const_inf (hf : Measurable f) (c : M) : Measurable fun x => c ⊓ f x :=\n  (measurable_const_inf c).comp hf\n\n@[fun_prop]\ntheorem AEMeasurable.const_inf (hf : AEMeasurable f μ) (c : M) :\n    AEMeasurable (fun x => c ⊓ f x) μ :=\n  (MeasurableInf.measurable_const_inf c).comp_aemeasurable hf\n\n@[fun_prop]\ntheorem Measurable.inf_const (hf : Measurable f) (c : M) : Measurable fun x => f x ⊓ c :=\n  (measurable_inf_const c).comp hf\n\n@[fun_prop]\ntheorem AEMeasurable.inf_const (hf : AEMeasurable f μ) (c : M) :\n    AEMeasurable (fun x => f x ⊓ c) μ :=\n  (measurable_inf_const c).comp_aemeasurable hf\n\nend MeasurableInf\n\nsection MeasurableInf₂\n\nvariable [MeasurableInf₂ M]\n\n@[fun_prop]\ntheorem Measurable.inf' (hf : Measurable f) (hg : Measurable g) : Measurable (f ⊓ g) :=\n  measurable_inf.comp (hf.prodMk hg)\n\n@[fun_prop]\ntheorem Measurable.inf (hf : Measurable f) (hg : Measurable g) : Measurable fun a => f a ⊓ g a :=\n  measurable_inf.comp (hf.prodMk hg)\n\n@[fun_prop]\ntheorem AEMeasurable.inf' (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :\n    AEMeasurable (f ⊓ g) μ :=\n  measurable_inf.comp_aemeasurable (hf.prodMk hg)\n\n@[fun_prop]\ntheorem AEMeasurable.inf (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :\n    AEMeasurable (fun a => f a ⊓ g a) μ :=\n  measurable_inf.comp_aemeasurable (hf.prodMk hg)\n\ninstance (priority := 100) MeasurableInf₂.to_hasMeasurableInf : MeasurableInf M where\n\nend MeasurableInf₂\n\nend Inf\n\nsection SemilatticeSup\n\nopen Finset\n\nvariable {δ : Type*} [MeasurableSpace δ] [SemilatticeSup α] [MeasurableSup₂ α]\n\n@[fun_prop]\ntheorem Finset.measurable_sup' {ι : Type*} {s : Finset ι} (hs : s.Nonempty) {f : ι → δ → α}\n    (hf : ∀ n ∈ s, Measurable (f n)) : Measurable (s.sup' hs f) :=\n  Finset.sup'_induction hs _ (fun _f hf _g hg => hf.sup hg) fun n hn => hf n hn\n\n@[fun_prop]\n\nTarget:\ntheorem Finset.measurable_range_sup' {f : ℕ → δ → α} {n : ℕ} (hf : ∀ k ≤ n, Measurable (f k)) :\n    Measurable ((range (n + 1)).sup' nonempty_range_add_one f) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Order","family_id":"finset","file_id":"mathlib/Mathlib/MeasureTheory/Order/Lattice.lean","sample_id":"2d8dea88dd355420151a905a1257de8aac829b897e12eb600a797e8418da7629"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9fa7af5d48ddb6db2529f448df0e30aa81c953ad207f9980f8c0f3df88f4e48e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"66542818e479c89882f725bfc7d9605b37b594439204873e38030237ce086893","source_sha256":"c1ecc80ead99a17d01db2984c5f06df1499354702fc192601863dee22fbd7b8f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro V\n  have : ∀ x : V.1, ∃ (i : ι) (r : Γ(X, V.1)) (rU : Γ(X, U i)),\n      X.basicOpen r = X.basicOpen rU ∧ x.1 ∈ X.basicOpen r := by\n    intro ⟨x, hxV⟩\n    obtain ⟨i, hi⟩ := TopologicalSpace.Opens.mem_iSup.mp (hU.ge (Set.mem_univ x))\n    exact ⟨i, exists_basicOpen_le_affine_inter V.2 (U i).2 _ ⟨hxV, hi⟩⟩\n  choose i r rU e hxr using this\n  have : Ideal.span (Set.range r) = ⊤ := by\n    rw [← V.2.self_le_iSup_basicOpen_iff]\n    exact fun x hxV ↦ TopologicalSpace.Opens.mem_iSup.mpr ⟨⟨_, _, rfl⟩, hxr ⟨x, hxV⟩⟩\n  have inst := V.2.isLocalization_basicOpen\n  refine Submodule.le_of_isLocalized_span _ this (fun i ↦ Γ(X, X.basicOpen i.1))\n    (fun i ↦ Algebra.linearMap Γ(X, V.1) Γ(X, X.basicOpen i.1)) ?_\n  rintro ⟨_, j, rfl⟩\n  dsimp\n  simp +instances only [← Submodule.restrictScalars_localized' Γ(X, X.basicOpen (r j)),\n    Ideal.localized'_eq_map, RingHom.algebraMap_toAlgebra]\n  erw [I.map_ideal (U := ⟨_, V.2.basicOpen _⟩) (X.basicOpen_le (r j)),\n    J.map_ideal (U := ⟨_, V.2.basicOpen _⟩) (X.basicOpen_le (r j))]\n  delta algebra_section_section_basicOpen\n  rw! [e]\n  rw [← I.map_ideal (V := (U _)) (X.basicOpen_le _), ← J.map_ideal (V := (U _)) (X.basicOpen_le _)]\n  exact Ideal.map_mono (f := (X.presheaf.map (homOfLE (X.basicOpen_le (rU j))).op).hom) (H (i j))","hard_negative":false,"metrics":{"chosen_tokens":413,"rejected_tokens":3,"token_jaccard":0.020833,"token_length_ratio":0.007264},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"7ace3179a83da61a6a37593706de68e9c2a71800ef040c8a431c5e8af521e82d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact\npublic import Mathlib.AlgebraicGeometry.Properties\npublic import Mathlib.Tactic.DepRewrite\n\nNamespace:\nAlgebraicGeometry.Scheme.IdealSheafData\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Ideal sheaves on schemes\n\nWe define ideal sheaves of schemes and provide various constructors for it.\n\n## Main definition\n* `AlgebraicGeometry.Scheme.IdealSheafData`: A structure that contains the data to uniquely define\n  an ideal sheaf, consisting of\n  1. an ideal `I(U) ≤ Γ(X, U)` for every affine open `U`\n  2. a proof that `I(D(f)) = I(U)_f` for every affine open `U` and every section `f : Γ(X, U)`.\n* `AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals`:\n  The largest ideal sheaf contained in a family of ideals.\n* `AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine`:\n  Over affine schemes, ideal sheaves are in bijection with ideals of the global sections.\n* `AlgebraicGeometry.Scheme.IdealSheafData.support`: The support of an ideal sheaf.\n* `AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal`: The vanishing ideal of a set.\n* `AlgebraicGeometry.Scheme.Hom.ker`: The kernel of a morphism.\n\n## Main results\n* `AlgebraicGeometry.Scheme.IdealSheafData.gc`:\n  `support` and `vanishingIdeal` forms a Galois connection.\n* `AlgebraicGeometry.Scheme.Hom.support_ker`: The support of a kernel of a quasi-compact morphism\n  is the closure of the range.\n\n## Implementation detail\n\nIdeal sheaves are not yet defined in this file as actual subsheaves of `𝒪ₓ`.\nInstead, for the ease of development and application,\nwe define the structure `IdealSheafData` containing all necessary data to uniquely define an\nideal sheaf. This should be refactored as a constructor for ideal sheaves once they are introduced\ninto mathlib.\n\n-/\n\n@[expose] public section\n\nopen CategoryTheory TopologicalSpace\n\nuniverse u\n\nnamespace AlgebraicGeometry.Scheme\n\nvariable {X : Scheme.{u}}\n\n/--\nA structure that contains the data to uniquely define an ideal sheaf, consisting of\n1. an ideal `I(U) ≤ Γ(X, U)` for every affine open `U`\n2. a proof that `I(D(f)) = I(U)_f` for every affine open `U` and every section `f : Γ(X, U)`\n3. a subset of `X` equal to the support.\n\nAlso see `Scheme.IdealSheafData.mkOfMemSupportIff` for a constructor with the condition on the\nsupport being (usually) easier to prove.\n-/\nstructure IdealSheafData (X : Scheme.{u}) : Type u where\n  /-- The component of an ideal sheaf at an affine open. -/\n  ideal : ∀ U : X.affineOpens, Ideal Γ(X, U)\n  /-- Also see `AlgebraicGeometry.Scheme.IdealSheafData.map_ideal` -/\n  map_ideal_basicOpen : ∀ (U : X.affineOpens) (f : Γ(X, U)),\n    (ideal U).map (X.presheaf.map (homOfLE <| X.basicOpen_le f).op).hom =\n      ideal (X.affineBasicOpen f)\n  /-- The support of an ideal sheaf. Use `IdealSheafData.support` instead for most occasions. -/\n  supportSet : Set X := ⋂ U, X.zeroLocus (U := U.1) (ideal U)\n  supportSet_eq_iInter_zeroLocus : supportSet = ⋂ U, X.zeroLocus (U := U.1) (ideal U) := by rfl\n\nnamespace IdealSheafData\n\n@[ext]\nprotected lemma ext {I J : X.IdealSheafData} (h : I.ideal = J.ideal) : I = J := by\n  obtain ⟨i, _, s, hs⟩ := I\n  obtain ⟨j, _, t, ht⟩ := J\n  subst h\n  congr\n  rw [hs, ht]\n\nsection Order\n\ninstance : PartialOrder (IdealSheafData X) := PartialOrder.lift ideal fun _ _ ↦ IdealSheafData.ext\n\nlemma le_def {I J : IdealSheafData X} : I ≤ J ↔ ∀ U, I.ideal U ≤ J.ideal U := .rfl\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : CompleteSemilatticeSup (IdealSheafData X) where\n  sSup s :=\n  { ideal := sSup (ideal '' s),\n    map_ideal_basicOpen := by\n      have : sSup (ideal '' s) = ⨆ i : s, ideal i.1 := by\n        conv_lhs => rw [← Subtype.range_val (s := s), ← Set.range_comp]\n        rfl\n      simp only [this, iSup_apply, Ideal.map_iSup, map_ideal_basicOpen, implies_true] }\n  isLUB_sSup _ := .of_image (f := ideal) le_def (isLUB_sSup _)\n\n/-- The largest ideal sheaf contained in a family of ideals. -/\ndef ofIdeals (I : ∀ U : X.affineOpens, Ideal Γ(X, U)) : IdealSheafData X :=\n  sSup { J : IdealSheafData X | J.ideal ≤ I }\n\nlemma ideal_ofIdeals_le (I : ∀ U : X.affineOpens, Ideal Γ(X, U)) :\n    (ofIdeals I).ideal ≤ I :=\n  sSup_le (Set.forall_mem_image.mpr fun _ ↦ id)\n\n/-- The Galois coinsertion between ideal sheaves and arbitrary families of ideals. -/\nprotected def gci : GaloisCoinsertion ideal (ofIdeals (X := X)) where\n  choice I hI :=\n  { ideal := I\n    map_ideal_basicOpen U f :=\n      (ideal_ofIdeals_le I).antisymm hI ▸ (ofIdeals I).map_ideal_basicOpen U f }\n  gc _ _ := ⟨(le_sSup ·), (le_trans · (ideal_ofIdeals_le _))⟩\n  u_l_le _ := sSup_le fun _ ↦ id\n  choice_eq I hI := IdealSheafData.ext (hI.antisymm (ideal_ofIdeals_le I))\n\nlemma strictMono_ideal : StrictMono (ideal (X := X)) := IdealSheafData.gci.strictMono_l\nlemma ideal_mono : Monotone (ideal (X := X)) := strictMono_ideal.monotone\nlemma ofIdeals_mono : Monotone (ofIdeals (X := X)) := IdealSheafData.gci.gc.monotone_u\nlemma ofIdeals_ideal (I : IdealSheafData X) : ofIdeals I.ideal = I := IdealSheafData.gci.u_l_eq _\nlemma le_ofIdeals_iff {I : IdealSheafData X} {J} : I ≤ ofIdeals J ↔ I.ideal ≤ J :=\n  IdealSheafData.gci.gc.le_iff_le.symm\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : OrderTop (IdealSheafData X) where\n  top.ideal := ⊤\n  top.map_ideal_basicOpen := by simp [Ideal.map_top]\n  top.supportSet := ⊥\n  top.supportSet_eq_iInter_zeroLocus := by\n    ext x\n    simpa using X.isBasis_affineOpens.exists_subset_of_mem_open (Set.mem_univ x) isOpen_univ\n  le_top I U := le_top\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : OrderBot (IdealSheafData X) where\n  bot.ideal := ⊥\n  bot.map_ideal_basicOpen := by simp\n  bot.supportSet := ⊤\n  bot.supportSet_eq_iInter_zeroLocus := by ext; simp\n  bot_le I U := bot_le\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : SemilatticeInf (IdealSheafData X) where\n  inf I J :=\n  { ideal := I.ideal ⊓ J.ideal\n    map_ideal_basicOpen U f := by\n      dsimp\n      have : (X.presheaf.map (homOfLE (X.basicOpen_le f)).op).hom = algebraMap _ _ := rfl\n      have inst := U.2.isLocalization_basicOpen f\n      rw [← I.map_ideal_basicOpen U f, ← J.map_ideal_basicOpen U f, this]\n      ext x\n      obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq (.powers f) x\n      simp only [IsLocalization.mk'_mem_map_algebraMap_iff, Submonoid.mem_powers_iff, Ideal.mem_inf,\n        exists_exists_eq_and]\n      refine ⟨fun ⟨n, h₁, h₂⟩ ↦ ⟨⟨n, h₁⟩, ⟨n, h₂⟩⟩, ?_⟩\n      rintro ⟨⟨n₁, h₁⟩, ⟨n₂, h₂⟩⟩\n      refine ⟨n₁ + n₂, ?_, ?_⟩\n      · rw [add_comm, pow_add, mul_assoc]; exact Ideal.mul_mem_left _ _ h₁\n      · rw [pow_add, mul_assoc]; exact Ideal.mul_mem_left _ _ h₂ }\n  inf_le_left I J U := inf_le_left\n  inf_le_right I J U := inf_le_right\n  le_inf I J K hIJ hIK U := le_inf (hIJ U) (hIK U)\n\ninstance : CompleteLattice (IdealSheafData X) where\n  __ := (inferInstance : OrderTop (IdealSheafData X))\n  __ := (inferInstance : OrderBot (IdealSheafData X))\n  __ := (inferInstance : SemilatticeInf (IdealSheafData X))\n  __ := (inferInstance : CompleteSemilatticeSup (IdealSheafData X))\n  __ := IdealSheafData.gci.liftCompleteLattice\n\n@[simp]\nlemma ideal_top : ideal (X := X) ⊤ = ⊤ := rfl\n\n@[simp]\nlemma ideal_bot : ideal (X := X) ⊥ = ⊥ := rfl\n\n@[simp]\nlemma ideal_sup {I J : IdealSheafData X} : (I ⊔ J).ideal = I.ideal ⊔ J.ideal := rfl\n\n@[simp]\nlemma ideal_sSup {I : Set (IdealSheafData X)} : (sSup I).ideal = sSup (ideal '' I) := rfl\n\n@[simp]\nlemma ideal_iSup {ι : Type*} {I : ι → IdealSheafData X} : (iSup I).ideal = ⨆ i, (I i).ideal := by\n  rw [← sSup_range, ← sSup_range, ideal_sSup, ← Set.range_comp, Function.comp_def]\n\n@[simp]\nlemma ideal_inf {I J : IdealSheafData X} : (I ⊓ J).ideal = I.ideal ⊓ J.ideal := rfl\n\n@[simp]\nlemma ideal_biInf {ι : Type*} (I : ι → IdealSheafData X) {s : Set ι} (hs : s.Finite) :\n    (⨅ i ∈ s, I i).ideal = ⨅ i ∈ s, (I i).ideal := by\n  refine hs.induction_on _ (by simp) fun {i s} his hs e ↦ ?_\n  simp only [iInf_insert, e, ideal_inf]\n\n@[simp]\nlemma ideal_iInf {ι : Type*} (I : ι → IdealSheafData X) [Finite ι] :\n    (⨅ i, I i).ideal = ⨅ i, (I i).ideal := by\n  simpa using ideal_biInf I Set.finite_univ\n\nend Order\n\nvariable (I : IdealSheafData X)\n\nsection map_ideal\n\n/-- subsumed by `IdealSheafData.map_ideal` below. -/\nprivate lemma map_ideal_basicOpen_of_eq\n    {U V : X.affineOpens} (f : Γ(X, U)) (hV : V = X.affineBasicOpen f) :\n    (I.ideal U).map (X.presheaf.map\n        (homOfLE (X := X.Opens) (hV.trans_le (X.affineBasicOpen_le f))).op).hom =\n      I.ideal V := by\n  subst hV; exact I.map_ideal_basicOpen _ _\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma map_ideal {U V : X.affineOpens} (h : U ≤ V) :\n    (I.ideal V).map (X.presheaf.map (homOfLE h).op).hom = I.ideal U := by\n  rw [U.2.ideal_ext_iff]\n  intro x hxU\n  obtain ⟨f, g, hfg, hxf⟩ := exists_basicOpen_le_affine_inter U.2 V.2 x ⟨hxU, h hxU⟩\n  have := I.map_ideal_basicOpen_of_eq (V := X.affineBasicOpen g) f (Subtype.ext hfg.symm)\n  rw [← I.map_ideal_basicOpen] at this\n  apply_fun Ideal.map (X.presheaf.germ (X.basicOpen g) x (hfg ▸ hxf)).hom at this\n  simp only [Ideal.map_map, ← CommRingCat.hom_comp, affineBasicOpen_coe, X.presheaf.germ_res]\n    at this ⊢\n  simp only [homOfLE_leOfHom, TopCat.Presheaf.germ_res', this]\n\n/-- A form of `map_ideal` that is easier to rewrite with. -/\nlemma map_ideal' {U V : X.affineOpens} (h : Opposite.op V.1 ⟶ .op U.1) :\n    (I.ideal V).map (X.presheaf.map h).hom = I.ideal U :=\n  map_ideal _ _\n\nlemma ideal_le_comap_ideal {U V : X.affineOpens} (h : U ≤ V) :\n    I.ideal V ≤ (I.ideal U).comap (X.presheaf.map (homOfLE h).op).hom := by\n  rw [← Ideal.map_le_iff_le_comap, ← I.map_ideal h]\n\nTarget:\nlemma le_of_iSup_eq_top {I J : X.IdealSheafData} {ι : Type*}\n    (U : ι → X.affineOpens) (hU : ⨆ i, (U i).1 = ⊤) (H : ∀ i, I.ideal (U i) ≤ J.ideal (U i)) :\n    I ≤ J :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/IdealSheaf","family_id":"le_of_isup_eq_top","file_id":"mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/Basic.lean","sample_id":"66542818e479c89882f725bfc7d9605b37b594439204873e38030237ce086893"},"split":"train"}
{"binding":{"chosen_proof_sha256":"426d456c7ecb6c1bdb729eb16ba1456459d7dc0d59c4a4d027a77d78ce78493f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a7eb67679c9a628af80fcc71372a23c70dcb24c10c7fdef621c0478749eaa61d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"56f2b093801a2a41875bf30e118429a5cc73ba8976a1adb97741fabeeedbed7b","source_sha256":"c86276a84562d270f98c4177786987e3596479c9ba6b4dd02bd972b6ae1c24e5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [← Nat.not_odd_iff_even]\n  constructor <;> intro h\n  · intro _ w ho\n    have := (w.three_le_chromaticNumber_of_odd_loop ho).trans h.chromaticNumber_le\n    norm_cast\n  · apply colorable_iff_forall_connectedComponents.2\n    intro c\n    obtain ⟨_, hv⟩ := c.nonempty_supp\n    use fun a ↦ Fin.ofNat 2 (c.connected_toSimpleGraph ⟨_, hv⟩ a).some.length\n    intro a b hab he\n    apply h _ <| (((c.connected_toSimpleGraph ⟨_, hv⟩ a).some.concat hab).append\n                 (c.connected_toSimpleGraph ⟨_, hv⟩ b).some.reverse).map c.toSimpleGraph_hom\n    rw [length_map, length_append, length_concat, length_reverse, add_right_comm]\n    have : ((Nonempty.some (c.connected_toSimpleGraph ⟨_, hv⟩ a)).length) % 2 =\n        (Nonempty.some (c.connected_toSimpleGraph ⟨_, hv⟩ b)).length % 2 := by\n      simp_rw [← Fin.val_natCast, ← Fin.ofNat_eq_cast, he]\n    exact (Nat.even_iff.mpr (by lia)).add_one","hard_negative":true,"metrics":{"chosen_tokens":212,"rejected_tokens":3,"token_jaccard":0.027778,"token_length_ratio":0.014151},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"7ad6ebfa679381e0a4e9f7eab3119d7436611438ae49e5e073cac174b02d6a91","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.SimpleGraph.Bipartite\npublic import Mathlib.Combinatorics.SimpleGraph.Circulant\npublic import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex\npublic import Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite\npublic import Mathlib.Combinatorics.SimpleGraph.Hasse\npublic import Mathlib.Data.Fin.Parity\n\nNamespace:\nSimpleGraph\n\nLocal context:\n/-\nCopyright (c) 2023 Iván Renison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Iván Renison\n-/\n/-!\n# Concrete colorings of common graphs\n\nThis file defines colorings for some common graphs.\n\n## Main declarations\n\n* `SimpleGraph.pathGraph.bicoloring`: Bicoloring of a path graph.\n\n-/\n\n@[expose] public section\n\nassert_not_exists Field\n\nnamespace SimpleGraph\n\n/-- Bicoloring of a path graph -/\ndef pathGraph.bicoloring (n : ℕ) :\n    Coloring (pathGraph n) Bool :=\n  Coloring.mk (fun u ↦ u.val % 2 = 0) <| by\n    intro u v\n    rw [pathGraph_adj]\n    rintro (h | h) <;> simp [← h, not_iff, Nat.succ_mod_two_eq_zero_iff]\n\n/-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/\ndef pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where\n  toFun v := ⟨v, trans v.2 h⟩\n  inj' := by\n    rintro v w\n    rw [Fin.mk.injEq]\n    exact Fin.ext\n  map_rel_iff' := by simp [pathGraph]\n\ntheorem chromaticNumber_pathGraph (n : ℕ) (h : 2 ≤ n) :\n    (pathGraph n).chromaticNumber = 2 := by\n  have hc := (pathGraph.bicoloring n).colorable\n  apply le_antisymm\n  · exact hc.chromaticNumber_le\n  · have hadj : (pathGraph n).Adj ⟨0, Nat.zero_lt_of_lt h⟩ ⟨1, h⟩ := by simp [pathGraph_adj]\n    exact two_le_chromaticNumber_of_adj hadj\n\ntheorem Coloring.even_length_iff_congr {α} {G : SimpleGraph α}\n    (c : G.Coloring Bool) {u v : α} (p : G.Walk u v) :\n    Even p.length ↔ (c u ↔ c v) := by\n  induction p with\n  | nil => simp\n  | @cons u v w h p ih =>\n    simp only [Walk.length_cons, Nat.even_add_one]\n    have : ¬ c u = true ↔ c v = true := by\n      rw [← not_iff, ← Bool.eq_iff_iff]\n      exact c.valid h\n    tauto\n\ntheorem Coloring.odd_length_iff_not_congr {α} {G : SimpleGraph α}\n    (c : G.Coloring Bool) {u v : α} (p : G.Walk u v) :\n    Odd p.length ↔ (¬c u ↔ c v) := by\n  rw [← Nat.not_even_iff_odd, c.even_length_iff_congr p]\n  tauto\n\ntheorem Walk.three_le_chromaticNumber_of_odd_loop {α} {G : SimpleGraph α} {u : α} (p : G.Walk u u)\n    (hOdd : Odd p.length) : 3 ≤ G.chromaticNumber := Classical.by_contradiction <| by\n  intro h\n  have h' : G.chromaticNumber ≤ 2 := Order.le_of_lt_add_one <| not_le.mp h\n  let c : G.Coloring (Fin 2) := (chromaticNumber_le_iff_colorable.mp h').some\n  let c' : G.Coloring Bool := recolorOfEquiv G finTwoEquiv c\n  have : ¬c' u ↔ c' u := (c'.odd_length_iff_not_congr p).mp hOdd\n  simp_all\n\n/-- Bicoloring of a cycle graph of even size -/\ndef cycleGraph.bicoloring_of_even (n : ℕ) (h : Even n) : Coloring (cycleGraph n) Bool :=\n  Coloring.mk (fun u ↦ u.val % 2 = 0) <| by\n    intro u v hadj\n    match n with\n    | 0 => exact u.elim0\n    | 1 => simp at h\n    | n + 2 =>\n      simp only [ne_eq, decide_eq_decide]\n      simp only [cycleGraph_adj] at hadj\n      cases hadj with\n      | inl huv | inr huv =>\n        rw [← add_eq_of_eq_sub' huv.symm, ← Fin.even_iff_mod_of_even h,\n          ← Fin.even_iff_mod_of_even h, Fin.even_add_one_iff_odd]\n        apply Classical.not_iff.mpr\n        simp [Fin.not_odd_iff_even_of_even h, Fin.not_even_iff_odd_of_even h]\n\ntheorem chromaticNumber_cycleGraph_of_even (n : ℕ) (h : 2 ≤ n) (hEven : Even n) :\n    (cycleGraph n).chromaticNumber = 2 := by\n  have hc := (cycleGraph.bicoloring_of_even n hEven).colorable\n  apply le_antisymm\n  · apply hc.chromaticNumber_le\n  · have hadj : (cycleGraph n).Adj ⟨0, Nat.zero_lt_of_lt h⟩ ⟨1, h⟩ := by\n      simp [cycleGraph_adj', Fin.sub_val_of_le]\n    exact two_le_chromaticNumber_of_adj hadj\n\n/-- Tricoloring of a cycle graph -/\ndef cycleGraph.tricoloring (n : ℕ) (h : 2 ≤ n) : Coloring (cycleGraph n)\n    (Fin 3) := Coloring.mk (fun u ↦ if u.val = n - 1 then 2 else ⟨u.val % 2, by lia⟩) <| by\n    intro u v hadj\n    match n with\n    | 0 => exact u.elim0\n    | 1 => simp at h\n    | n + 2 =>\n      simp only [cycleGraph_adj] at hadj\n      split_ifs with hu hv\n      · simp [Fin.eq_mk_iff_val_eq.mpr hu, Fin.eq_mk_iff_val_eq.mpr hv] at hadj\n      · refine (Fin.ne_of_lt (Fin.mk_lt_of_lt_val (?_))).symm\n        exact v.val.mod_lt Nat.zero_lt_two\n      · refine (Fin.ne_of_lt (Fin.mk_lt_of_lt_val ?_))\n        exact u.val.mod_lt Nat.zero_lt_two\n      · simp only [ne_eq, Fin.ext_iff]\n        have hu' : u.val + (1 : Fin (n + 2)) < n + 2 := by fin_omega\n        have hv' : v.val + (1 : Fin (n + 2)) < n + 2 := by fin_omega\n        cases hadj with\n        | inl huv | inr huv =>\n          rw [← add_eq_of_eq_sub' huv.symm]\n          simp only [Fin.val_add_eq_of_add_lt hv', Fin.val_add_eq_of_add_lt hu', Fin.val_one]\n          rw [show ∀ x y : ℕ, x % 2 = y % 2 ↔ (Even x ↔ Even y) by simp [Nat.even_iff]; lia,\n            Nat.even_add]\n          simp only [Nat.not_even_one, iff_false, not_iff_self, iff_not_self]\n          exact id\n\ntheorem chromaticNumber_cycleGraph_of_odd (n : ℕ) (h : 2 ≤ n) (hOdd : Odd n) :\n    (cycleGraph n).chromaticNumber = 3 := by\n  have hc := (cycleGraph.tricoloring n h).colorable\n  apply le_antisymm\n  · apply hc.chromaticNumber_le\n  · have hn3 : n - 3 + 3 = n := by\n      refine Nat.sub_add_cancel (Nat.succ_le_of_lt (Nat.lt_of_le_of_ne h ?_))\n      intro h2\n      rw [← h2] at hOdd\n      exact (Nat.not_odd_iff.mpr rfl) hOdd\n    let w : (cycleGraph (n - 3 + 3)).Walk 0 0 := cycleGraph.cycle (n - 3)\n    have hOdd' : Odd w.length := by\n      rw [cycleGraph.length_cycle, hn3]\n      exact hOdd\n    rw [← hn3]\n    exact Walk.three_le_chromaticNumber_of_odd_loop w hOdd'\n\nsection CompleteEquipartiteGraph\n\nvariable {r t : ℕ}\n\n/-- The injection `(x₁, x₂) ↦ x₁` is always an `r`-coloring of a `completeEquipartiteGraph r ·`. -/\ndef Coloring.completeEquipartiteGraph :\n  (completeEquipartiteGraph r t).Coloring (Fin r) := ⟨Prod.fst, id⟩\n\n/-- The `completeEquipartiteGraph r t` is always `r`-colorable. -/\ntheorem completeEquipartiteGraph_colorable :\n  (completeEquipartiteGraph r t).Colorable r := ⟨Coloring.completeEquipartiteGraph⟩\n\nend CompleteEquipartiteGraph\n\nopen Walk\n\nTarget:\nlemma two_colorable_iff_forall_loop_even {α : Type*} {G : SimpleGraph α} :\n    G.Colorable 2 ↔ ∀ u, ∀ (w : G.Walk u u), Even w.length :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_56f2b093801a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"54887d10e51425f1367cd7fc2841bf3c9ec4676eca97686e6e382e08cdd6c4f3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/SimpleGraph","family_id":"two_colorable_iff_forall_loop_even","file_id":"mathlib/Mathlib/Combinatorics/SimpleGraph/Coloring/Constructions.lean","sample_id":"56f2b093801a2a41875bf30e118429a5cc73ba8976a1adb97741fabeeedbed7b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"95f0a046edcc88c2f0897d1f675b7af88bcf9e64df19c8ace7d838acef7a93e4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a2dfa0842d99cd20b40e19c92c20cbe360ba4180b0dbf370f6a305f8e5d3ac8b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"929feb6854c6bd754df0ddca63934028a11faf17c7ad84065207bb38e2efb282","source_sha256":"394c3d8954491754c0a03071ce7aa65ddc93c4783581cf0b4d6869361547187f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have ⟨n, S, d, _, hd, ⟨e⟩⟩ := exists_end_algEquiv_pi_matrix_end R₀ R M\n  classical exact ⟨n, _, d, inferInstance, inferInstance, hd, ⟨e⟩⟩","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":2,"token_jaccard":0.05,"token_length_ratio":0.047619},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"7aee4eea1a9784b2e66436e0d5fa531950b9d61c22e340009cf2b884bb79cae4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.FreeModule.Finite.Basic\npublic import Mathlib.RingTheory.FiniteLength\npublic import Mathlib.RingTheory.SimpleModule.Isotypic\npublic import Mathlib.RingTheory.SimpleRing.Congr\npublic import Mathlib.RingTheory.SimpleRing.Matrix\n\nNamespace:\nIsSemisimpleModule\n\nLocal context:\n/-\nCopyright (c) 2025 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu\n-/\n/-!\n# Wedderburn–Artin Theorem\n\n## Main results\n\n* `IsSimpleRing.tfae`: a simple ring is semisimple iff it is Artinian,\n  iff it has a minimal left ideal.\n\n* `isSimpleRing_isArtinianRing_iff`: a ring is simple Artinian iff it is semisimple, isotypic,\n  and nontrivial.\n\n* `IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite`: a simple Artinian algebra is\n  isomorphic to a (finite-dimensional) matrix algebra over a division algebra. The division\n  algebra is the opposite of the endomorphism algebra of a simple (i.e., minimal) left ideal.\n\n* `IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite`: a semisimple algebra is\n  isomorphic to a finite direct product of matrix algebras over division algebras. The division\n  algebras are the opposites of the endomorphism algebras of the simple (i.e., minimal)\n  left ideals.\n\n* `IsSimpleRing.exists_algEquiv_matrix_divisionRing_finite`,\n  `IsSemisimpleRing.exists_algEquiv_pi_matrix_divisionRing_finite`:\n  if the simple Artinian / semisimple algebra is finite as a module over a base ring, then the\n  division algebra(s) are also finite over the same ring.\n  If the base ring is an algebraically closed field, the only finite-dimensional division algebra\n  over it is itself, and we obtain `IsSimpleRing.exists_algEquiv_matrix_of_isAlgClosed` and\n  `IsSemisimpleRing.exists_algEquiv_pi_matrix_of_isAlgClosed` (in a later file).\n\n-/\n\npublic section\n\nuniverse u\nvariable (R₀ : Type*) {R : Type u} [CommSemiring R₀] [Ring R] [Algebra R₀ R]\n\n/-- A simple ring is semisimple iff it is Artinian, iff it has a minimal left ideal. -/\ntheorem IsSimpleRing.tfae [IsSimpleRing R] : List.TFAE\n    [IsSemisimpleRing R, IsArtinianRing R, ∃ I : Ideal R, IsAtom I] := by\n  tfae_have 1 → 2 := fun _ ↦ inferInstance\n  tfae_have 2 → 3 := fun _ ↦ IsAtomic.exists_atom _\n  tfae_have 3 → 1 := fun ⟨I, hI⟩ ↦ by\n    have ⟨_, h⟩ := isSimpleRing_iff_isTwoSided_imp.mp ‹IsSimpleRing R›\n    simp_rw [← isFullyInvariant_iff_isTwoSided] at h\n    have := isSimpleModule_iff_isAtom.mpr hI\n    obtain eq | eq := h _ (.isotypicComponent R R I)\n    · exact (hI.bot_lt.not_ge <| (le_sSup <| by exact ⟨.refl ..⟩).trans_eq eq).elim\n    exact .congr (.symm <| .trans (.ofEq _ _ eq) Submodule.topEquiv)\n  tfae_finish\n\ntheorem IsSimpleRing.isSemisimpleRing_iff_isArtinianRing [IsSimpleRing R] :\n    IsSemisimpleRing R ↔ IsArtinianRing R := tfae.out 0 1\n\ntheorem isSimpleRing_isArtinianRing_iff :\n    IsSimpleRing R ∧ IsArtinianRing R ↔ IsSemisimpleRing R ∧ IsIsotypic R R ∧ Nontrivial R := by\n  refine ⟨fun ⟨_, _⟩ ↦ ?_, fun ⟨_, _, _⟩ ↦ ?_⟩\n  on_goal 1 => have := IsSimpleRing.isSemisimpleRing_iff_isArtinianRing.mpr ‹_›\n  all_goals simp_rw [isIsotypic_iff_isFullyInvariant_imp_bot_or_top,\n      isFullyInvariant_iff_isTwoSided, isSimpleRing_iff_isTwoSided_imp] at *\n  · exact ⟨this, by rwa [and_comm]⟩\n  · exact ⟨⟨‹_›, ‹_›⟩, inferInstance⟩\n\nnamespace IsSimpleRing\n\nvariable (R) [IsSimpleRing R] [IsArtinianRing R]\n\ninstance (priority := low) : IsSemisimpleRing R :=\n  (isSimpleRing_isArtinianRing_iff.mp ⟨‹_›, ‹_›⟩).1\n\ntheorem isIsotypic (M) [AddCommGroup M] [Module R M] : IsIsotypic R M :=\n  (isSimpleRing_isArtinianRing_iff.mp ⟨‹_›, ‹_›⟩).2.1.of_self M\n\n/-- The **Wedderburn–Artin Theorem**: an Artinian simple ring is isomorphic to a matrix\nring over the opposite of the endomorphism ring of its simple module. -/\ntheorem exists_ringEquiv_matrix_end_mulOpposite :\n    ∃ (n : ℕ) (_ : NeZero n) (I : Ideal R) (_ : IsSimpleModule R I),\n      Nonempty (R ≃+* Matrix (Fin n) (Fin n) (Module.End R I)ᵐᵒᵖ) := by\n  have ⟨n, hn, S, hS, ⟨e⟩⟩ := (isIsotypic R R).linearEquiv_fun\n  refine ⟨n, hn, S, hS, ⟨.trans (.opOp R) <| .trans (.op ?_) (.symm .mopMatrix)⟩⟩\n  exact .trans (.moduleEndSelf R) <| .trans e.conjRingEquiv (endVecRingEquivMatrixEnd ..)\n\n/-- The **Wedderburn–Artin Theorem**: an Artinian simple ring is isomorphic to a matrix\nring over a division ring. -/\ntheorem exists_ringEquiv_matrix_divisionRing :\n    ∃ (n : ℕ) (_ : NeZero n) (D : Type u) (_ : DivisionRing D),\n      Nonempty (R ≃+* Matrix (Fin n) (Fin n) D) := by\n  have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_ringEquiv_matrix_end_mulOpposite R\n  classical exact ⟨n, hn, _, _, ⟨e⟩⟩\n\n/-- The **Wedderburn–Artin Theorem**, algebra form: an Artinian simple algebra is isomorphic\nto a matrix algebra over the opposite of the endomorphism algebra of its simple module. -/\ntheorem exists_algEquiv_matrix_end_mulOpposite :\n    ∃ (n : ℕ) (_ : NeZero n) (I : Ideal R) (_ : IsSimpleModule R I),\n      Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) (Module.End R I)ᵐᵒᵖ) := by\n  have ⟨n, hn, S, hS, ⟨e⟩⟩ := (isIsotypic R R).linearEquiv_fun\n  refine ⟨n, hn, S, hS, ⟨.trans (.opOp R₀ R) <| .trans (.op ?_) (.symm .mopMatrix)⟩⟩\n  exact .trans (.moduleEndSelf R₀) <| .trans (e.conjAlgEquiv R₀) (endVecAlgEquivMatrixEnd ..)\n\n/-- The **Wedderburn–Artin Theorem**, algebra form: an Artinian simple algebra is isomorphic\nto a matrix algebra over a division algebra. -/\ntheorem exists_algEquiv_matrix_divisionRing :\n    ∃ (n : ℕ) (_ : NeZero n) (D : Type u) (_ : DivisionRing D) (_ : Algebra R₀ D),\n      Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) D) := by\n  have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_algEquiv_matrix_end_mulOpposite R₀ R\n  classical exact ⟨n, hn, _, _, _, ⟨e⟩⟩\n\n/-- The **Wedderburn–Artin Theorem**, algebra form, finite case: a finite Artinian simple algebra is\nisomorphic to a matrix algebra over a finite division algebra. -/\ntheorem exists_algEquiv_matrix_divisionRing_finite [Module.Finite R₀ R] :\n    ∃ (n : ℕ) (_ : NeZero n) (D : Type u) (_ : DivisionRing D) (_ : Algebra R₀ D)\n      (_ : Module.Finite R₀ D), Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) D) := by\n  have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_algEquiv_matrix_end_mulOpposite R₀ R\n  have := Module.Finite.equiv e.toLinearEquiv\n  classical exact ⟨n, hn, _, _, _, .of_surjective\n    (Matrix.entryLinearMap R₀ _ (0 : Fin n) (0 : Fin n)) fun f ↦ ⟨fun _ _ ↦ f, rfl⟩, ⟨e⟩⟩\n\nend IsSimpleRing\n\nnamespace IsSemisimpleModule\n\nopen Module (End)\n\nuniverse v\nvariable (R) (M : Type v) [AddCommGroup M] [Module R₀ M] [Module R M] [IsScalarTower R₀ R M]\n  [IsSemisimpleModule R M] [Module.Finite R M]\n\ntheorem exists_end_algEquiv_pi_matrix_end :\n    ∃ (n : ℕ) (S : Fin n → Submodule R M) (d : Fin n → ℕ),\n      (∀ i, IsSimpleModule R (S i)) ∧ (∀ i, NeZero (d i)) ∧\n      Nonempty (End R M ≃ₐ[R₀] Π i, Matrix (Fin (d i)) (Fin (d i)) (End R (S i))) := by\n  choose d pos S _ simple e using fun c : isotypicComponents R M ↦\n    (IsIsotypic.isotypicComponents c.2).submodule_linearEquiv_fun\n  classical exact ⟨_, _, _, fun _ ↦ simple _, fun _ ↦ pos _, ⟨.trans (endAlgEquiv R₀ R M) <| .trans\n    (.piCongrRight fun c ↦ ((e c).some.conjAlgEquiv R₀).trans (endVecAlgEquivMatrixEnd ..)) <|\n    (.piCongrLeft' R₀ _ (Finite.equivFin _))⟩⟩\n\ntheorem exists_end_ringEquiv_pi_matrix_end :\n    ∃ (n : ℕ) (S : Fin n → Submodule R M) (d : Fin n → ℕ),\n      (∀ i, IsSimpleModule R (S i)) ∧ (∀ i, NeZero (d i)) ∧\n      Nonempty (End R M ≃+* Π i, Matrix (Fin (d i)) (Fin (d i)) (End R (S i))) :=\n  have ⟨n, S, d, hS, hd, ⟨e⟩⟩ := exists_end_algEquiv_pi_matrix_end ℕ R M; ⟨n, S, d, hS, hd, ⟨e⟩⟩\n\n@[deprecated (since := \"2025-11-16\")] alias exists_end_algEquiv := exists_end_algEquiv_pi_matrix_end\n@[deprecated (since := \"2025-11-16\")]\nalias exists_end_ringEquiv := exists_end_ringEquiv_pi_matrix_end\n\n-- TODO: can also require D be in `Type u`, since every simple module is the quotient by an ideal.\n\nTarget:\ntheorem exists_end_algEquiv_pi_matrix_divisionRing :\n    ∃ (n : ℕ) (D : Fin n → Type v) (d : Fin n → ℕ) (_ : ∀ i, DivisionRing (D i))\n      (_ : ∀ i, Algebra R₀ (D i)), (∀ i, NeZero (d i)) ∧\n      Nonempty (End R M ≃ₐ[R₀] Π i, Matrix (Fin (d i)) (Fin (d i)) (D i)) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_929feb6854c6","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"19ebaad7edbd149f730fbe585a0e531111b5046f001ff44f2e087c7b0d6dbb07","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/SimpleModule","family_id":"exists_end_algequiv_pi_matrix_divisionring","file_id":"mathlib/Mathlib/RingTheory/SimpleModule/WedderburnArtin.lean","sample_id":"929feb6854c6bd754df0ddca63934028a11faf17c7ad84065207bb38e2efb282"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2b82d4fddfb467900b75edb5d85dcce54ace3fac2a5c050d4837c5985aa49b27","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"71a417e8c511aea0386249ac90264f5677e17bc513575d22f27edf03f7f31296","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"08ef20f28dae9ac8a4b11319080aa916285ab90a007cfc4fe216ca408e450b14","source_sha256":"e6501306e0c166f31aa2d54b2053e75c750b45815aae03787ae7a4d5aa54d0d7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  lift s to Finset α using hs; exact s.exists_le_minimal ha\n\nvariable [Nonempty α]","hard_negative":true,"metrics":{"chosen_tokens":19,"rejected_tokens":3,"token_jaccard":0.111111,"token_length_ratio":0.157895},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"7afe8763db9b202c6b8bb11207f330b9572fefe4daf67b126a08003b453ca72b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Basic\npublic import Mathlib.Order.Minimal\n\nNamespace:\nSet\n\nLocal context:\n/-\nCopyright (c) 2025 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Finite preorders and finite sets in a preorder\n\nThis file shows that non-empty finite sets in a preorder have minimal/maximal elements, and\ncontrapositively that non-empty sets without minimal or maximal elements are infinite.\n-/\n\npublic section\n\nvariable {ι α β : Type*}\n\nnamespace Finset\nsection IsTrans\nvariable [LE α] [IsTrans α LE.le] {s : Finset α} {a : α}\n\nlemma exists_maximalFor (f : ι → α) (s : Finset ι) (hs : s.Nonempty) :\n    ∃ i, MaximalFor (· ∈ s) f i := by\n  induction hs using Finset.Nonempty.cons_induction with\n  | singleton i => exact ⟨i, by simp⟩\n  | @cons i s hi hs ih =>\n    obtain ⟨j, hj⟩ := ih\n    by_cases hji : f j ≤ f i\n    · refine ⟨i, mem_cons_self .., ?_⟩\n      simp only [mem_cons, forall_eq_or_imp, imp_self, true_and]\n      exact fun k hk hik ↦ _root_.trans (hj.2 hk <| _root_.trans hji hik) hji\n    · exact ⟨j, mem_cons_of_mem hj.1, by simpa [hji] using hj.2⟩\n\nlemma exists_minimalFor (f : ι → α) (s : Finset ι) (hs : s.Nonempty) :\n    ∃ i, MinimalFor (· ∈ s) f i := exists_maximalFor (α := αᵒᵈ) f s hs\n\nlemma exists_maximal (hs : s.Nonempty) : ∃ i, Maximal (· ∈ s) i := s.exists_maximalFor id hs\nlemma exists_minimal (hs : s.Nonempty) : ∃ i, Minimal (· ∈ s) i := s.exists_minimalFor id hs\n\nend IsTrans\n\nsection Preorder\nvariable [Preorder α] {s : Finset α} {a : α}\n\nlemma exists_le_maximal (s : Finset α) (ha : a ∈ s) : ∃ b, a ≤ b ∧ Maximal (· ∈ s) b := by\n  classical\n  obtain ⟨b, hb, hab, hbmin⟩ : ∃ b ∈ s, a ≤ b ∧ _ := by\n    simpa [Maximal, and_assoc] using {x ∈ s | a ≤ x}.exists_maximal ⟨a, mem_filter.2 ⟨ha, le_rfl⟩⟩\n  exact ⟨b, hab, hb, fun c hc hbc ↦ hbmin hc (hab.trans hbc) hbc⟩\n\nlemma exists_le_minimal (s : Finset α) (ha : a ∈ s) : ∃ b ≤ a, Minimal (· ∈ s) b :=\n  exists_le_maximal (α := αᵒᵈ) s ha\n\nend Preorder\nend Finset\n\nnamespace Set\nsection IsTrans\nvariable [LE α] [IsTrans α LE.le] {s : Set α} {a : α}\n\nlemma Finite.exists_maximalFor (f : ι → α) (s : Set ι) (h : s.Finite) (hs : s.Nonempty) :\n    ∃ i, MaximalFor (· ∈ s) f i := by\n  lift s to Finset ι using h; exact s.exists_maximalFor f hs\n\nlemma Finite.exists_minimalFor (f : ι → α) (s : Set ι) (h : s.Finite) (hs : s.Nonempty) :\n    ∃ i, MinimalFor (· ∈ s) f i := Finite.exists_maximalFor (α := αᵒᵈ) f s h hs\n\nlemma Finite.exists_maximal (h : s.Finite) (hs : s.Nonempty) : ∃ i, Maximal (· ∈ s) i :=\n  h.exists_maximalFor id _ hs\n\nlemma Finite.exists_minimal (h : s.Finite) (hs : s.Nonempty) : ∃ i, Minimal (· ∈ s) i :=\n  h.exists_minimalFor id _ hs\n\n/-- A version of `Finite.exists_maximalFor` with the (weaker) hypothesis that the image of `s`\nis finite rather than `s` itself. -/\nlemma Finite.exists_maximalFor' (f : ι → α) (s : Set ι) (h : (f '' s).Finite) (hs : s.Nonempty) :\n    ∃ i, MaximalFor (· ∈ s) f i := by\n  obtain ⟨_, ⟨a, ha, rfl⟩, hmax⟩ := Finite.exists_maximalFor id (f '' s) h (hs.image f)\n  exact ⟨a, ha, fun a' ha' hf ↦ hmax (mem_image_of_mem f ha') hf⟩\n\n/-- A version of `Finite.exists_minimalFor` with the (weaker) hypothesis that the image of `s`\nis finite rather than `s` itself. -/\nlemma Finite.exists_minimalFor' (f : ι → α) (s : Set ι) (h : (f '' s).Finite) (hs : s.Nonempty) :\n    ∃ i, MinimalFor (· ∈ s) f i := h.exists_maximalFor' (α := αᵒᵈ) f s hs\n\nend IsTrans\n\nsection Preorder\nvariable [Preorder α] {s : Set α} {a : α}\n\nlemma Finite.exists_le_maximal (hs : s.Finite) (ha : a ∈ s) : ∃ b, a ≤ b ∧ Maximal (· ∈ s) b := by\n  lift s to Finset α using hs; exact s.exists_le_maximal ha\n\nTarget:\nlemma Finite.exists_le_minimal (hs : s.Finite) (ha : a ∈ s) : ∃ b, b ≤ a ∧ Minimal (· ∈ s) b :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_08ef20f28dae","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"0329e9573368f94c00729ce33841202d50dea9e6b5cec2a19be51132c1c02bf9","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Preorder","family_id":"finite","file_id":"mathlib/Mathlib/Order/Preorder/Finite.lean","sample_id":"08ef20f28dae9ac8a4b11319080aa916285ab90a007cfc4fe216ca408e450b14"},"split":"train"}
{"binding":{"chosen_proof_sha256":"88c077efecbeeb27504c4875f1630a5efd1eaa0f4e0d6e798f302270b541c3d4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c296264677213e7c36d5f393049036aa80f4325b33a08beccf06b71101a4451","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5594b1e027625be77c21574d7888f7501782d701404f92b08cf130a4ddbabfb7","source_sha256":"58fa9eef28170e41707520794ca2c1648cebed0e86fef9c10333ef5d54e6c390","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rw [norm_apply, LinearMap.det]; split_ifs <;> trivial\n\nvariable {R}","hard_negative":true,"metrics":{"chosen_tokens":19,"rejected_tokens":2,"token_jaccard":0.052632,"token_length_ratio":0.105263},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"7b3e9f2d62747c31c648253fb17f69c71997de6b9743edd9610bebd74176f915","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Determinant\n\nNamespace:\nAlgebra\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Norm for (finite) ring extensions\n\nSuppose we have an `R`-algebra `S` with a finite basis. For each `s : S`,\nthe determinant of the linear map given by multiplying by `s` gives information\nabout the roots of the minimal polynomial of `s` over `R`.\n\n## Implementation notes\n\nTypically, the norm is defined specifically for finite field extensions.\nThe current definition is as general as possible and the assumption that we have\nfields or that the extension is finite is added to the lemmas as needed.\n\nWe only define the norm for left multiplication (`Algebra.leftMulMatrix`,\ni.e. `LinearMap.mulLeft`).\nFor now, the definitions assume `S` is commutative, so the choice doesn't\nmatter anyway.\n\nSee also `Algebra.trace`, which is defined similarly as the trace of\n`Algebra.leftMulMatrix`.\n\n## References\n\n* https://en.wikipedia.org/wiki/Field_norm\n\n-/\n\n@[expose] public section\n\n\nuniverse u v w\n\nvariable {R S : Type*} [CommRing R] [Ring S]\nvariable [Algebra R S]\nvariable {K : Type*} [Field K]\nvariable {ι : Type w}\n\nopen Module\n\nopen LinearMap\n\nopen Matrix Polynomial\n\nopen scoped Matrix\n\nnamespace Algebra\n\nvariable (R)\n\n/-- The norm of an element `s` of an `R`-algebra is the determinant of `(*) s`. -/\n@[stacks 0BIF \"Norm\"]\nnoncomputable def norm : S →* R :=\n  LinearMap.det.comp (lmul R S).toRingHom.toMonoidHom\n\ntheorem norm_apply (x : S) : norm R x = LinearMap.det (lmul R S x) := rfl\n\n@[simp]\ntheorem norm_self : Algebra.norm R = MonoidHom.id R := by\n  ext\n  simp [norm_apply]\n\nTarget:\ntheorem norm_eq_one_of_not_exists_basis (h : ¬∃ s : Finset S, Nonempty (Basis s R S)) (x : S) :\n    norm R x = 1 :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_5594b1e02762","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"f6dd3613bd478679bd2aae85ad7abecb284ef7ef70cd44660e7a1dee0d304b85","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Norm","family_id":"norm_eq_one_of_not_exists_basis","file_id":"mathlib/Mathlib/RingTheory/Norm/Defs.lean","sample_id":"5594b1e027625be77c21574d7888f7501782d701404f92b08cf130a4ddbabfb7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9346ea21709b5b4adcb76342b23a0fdfec46476e1c52d71821429c4b741b1882","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f15b94388f7d99ac88f3a9765d8d2dff99b870d49de549bcfbfe49f4f267b151","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3d58730a822f37dbe124ac3de8532bdf9e6b3a34db752189dda5130b2510e7eb","source_sha256":"e6ed626f52730f45ca36129d8b77b24071a2cee6973cf9b0b2dfb896ff9281e8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨f, g, h⟩ := exists_pair_ne (α → β)\n  rw [ne_eq, funext_iff, Classical.not_forall] at h\n  obtain ⟨a, h⟩ := h\n  exact nontrivial_of_ne _ _ h","hard_negative":true,"metrics":{"chosen_tokens":41,"rejected_tokens":3,"token_jaccard":0.068966,"token_length_ratio":0.073171},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"7b438a2b21f1827e8d7f8be5239ac160e3e01d5069e2cbb2fec82001e10f505c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Tactic.Push.Attr\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Nontrivial types\n\nA type is *nontrivial* if it contains at least two elements. This is useful in particular for rings\n(where it is equivalent to the fact that zero is different from one) and for vector spaces\n(where it is equivalent to the fact that the dimension is positive).\n\nWe introduce a typeclass `Nontrivial` formalizing this property.\n\nBasic results about nontrivial types are in `Mathlib/Logic/Nontrivial/Basic.lean`.\n-/\n\npublic section\n\nvariable {α : Type*} {β : Type*}\n\n/-- Predicate typeclass for expressing that a type is not reduced to a single element. In rings,\nthis is equivalent to `0 ≠ 1`. In vector spaces, this is equivalent to positive dimension. -/\nclass Nontrivial (α : Type*) : Prop where\n  /-- In a nontrivial type, there exists a pair of distinct terms. -/\n  exists_pair_ne : ∃ x y : α, x ≠ y\n\ntheorem nontrivial_iff : Nontrivial α ↔ ∃ x y : α, x ≠ y :=\n  ⟨fun h ↦ h.exists_pair_ne, fun h ↦ ⟨h⟩⟩\n\ntheorem exists_pair_ne (α : Type*) [Nontrivial α] : ∃ x y : α, x ≠ y :=\n  Nontrivial.exists_pair_ne\n\n/-- Pushforward a `Nontrivial` instance along an injective function. -/\nprotected theorem Function.Injective.nontrivial [Nontrivial α] {f : α → β}\n    (hf : Function.Injective f) : Nontrivial β :=\n  let ⟨x, y, h⟩ := exists_pair_ne α\n  ⟨⟨f x, f y, hf.ne h⟩⟩\n\n/-- An injective function from a nontrivial type has an argument at\nwhich it does not take a given value. -/\nprotected theorem Function.Injective.exists_ne [Nontrivial α] {f : α → β}\n    (hf : Function.Injective f) (y : β) : ∃ x, f x ≠ y := by\n  rcases exists_pair_ne α with ⟨x₁, x₂, hx⟩\n  by_cases h : f x₂ = y\n  · exact ⟨x₁, (hf.ne_iff' h).2 hx⟩\n  · exact ⟨x₂, h⟩\n\n-- See Note [decidable namespace]\nprotected theorem Decidable.exists_ne [Nontrivial α] [DecidableEq α] (x : α) : ∃ y, y ≠ x := by\n  rcases exists_pair_ne α with ⟨y, y', h⟩\n  by_cases hx : x = y\n  · rw [← hx] at h\n    exact ⟨y', h.symm⟩\n  · exact ⟨y, Ne.symm hx⟩\n\nopen Classical in\ntheorem exists_ne [Nontrivial α] (x : α) : ∃ y, y ≠ x := Decidable.exists_ne x\n\n-- `x` and `y` are explicit here, as they are often needed to guide typechecking of `h`.\ntheorem nontrivial_of_ne (x y : α) (h : x ≠ y) : Nontrivial α :=\n  ⟨⟨x, y, h⟩⟩\n\ntheorem nontrivial_iff_exists_ne (x : α) : Nontrivial α ↔ ∃ y, y ≠ x :=\n  ⟨fun h ↦ @exists_ne α h x, fun ⟨_, hy⟩ ↦ nontrivial_of_ne _ _ hy⟩\n\nTarget:\ntheorem Function.nontrivial_of_nontrivial (α β : Type*) [Nontrivial (α → β)] :\n    Nontrivial β :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_3d58730a822f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"7e9fecea4822192c8e292de1f579931f44ecce6da6fda2db520a491a5c077287","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Nontrivial","family_id":"function","file_id":"mathlib/Mathlib/Logic/Nontrivial/Defs.lean","sample_id":"3d58730a822f37dbe124ac3de8532bdf9e6b3a34db752189dda5130b2510e7eb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d56cb08de95550c2d5c8c73e4133b8fd22dbbf83f22597912cb3044a1cf97d24","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"229e61ac4950e38ed3ed9109978b62baa053738db377c99ce4e1a8e4582f0232","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f15d582cb509e0a13008bd362768f09d19cdeaf2278912231d988aa0bbeaa66e","source_sha256":"e0af099a1dfedc907a6ae420e8f9c06d8630aad18426a643f2d72445c5f1b960","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [IsTranscendenceBasis, algebraicIndependent_equiv, EquivLike.range_comp]\n\nalias ⟨_, IsTranscendenceBasis.comp_equiv⟩ := isTranscendenceBasis_equiv","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":25,"token_jaccard":0.85,"token_length_ratio":1.190476},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"7b59689ac4bf95b99be558066d58271b6c8fd3c08e0e5f8803dbc442a09b0628","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.CommRing\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Algebraic Independence\n\nThis file defines algebraic independence of a family of elements of an `R` algebra.\n\n## Main definitions\n\n* `AlgebraicIndependent` - `AlgebraicIndependent R x` states the family of elements `x`\n  is algebraically independent over `R`, meaning that the canonical map out of the multivariable\n  polynomial ring is injective.\n\n* `AlgebraicIndependent.aevalEquiv` - The canonical isomorphism from the polynomial ring to the\n  subalgebra generated by an algebraic independent family.\n\n* `AlgebraicIndependent.repr` - The canonical map from the subalgebra generated by an\n  algebraic independent family into the polynomial ring. It is the inverse of\n  `AlgebraicIndependent.aevalEquiv`.\n\n* `IsTranscendenceBasis R x` - a family `x` is a transcendence basis over `R` if it is a maximal\n  algebraically independent subset.\n\n## Main results\n\nWe show that algebraic independence is preserved under injective maps of the indices.\n\n## References\n\n* [Stacks: Transcendence](https://stacks.math.columbia.edu/tag/030D)\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen Function Set Subalgebra MvPolynomial Algebra\n\nvariable {ι ι' : Type*} (R : Type*) {K A A' : Type*} (x : ι → A)\nvariable [CommRing R] [CommRing A] [CommRing A'] [Algebra R A] [Algebra R A']\n\n/-- `AlgebraicIndependent R x` states the family of elements `x`\n  is algebraically independent over `R`, meaning that the canonical\n  map out of the multivariable polynomial ring is injective. -/\n@[stacks 030E \"(1)\"] def AlgebraicIndependent : Prop :=\n  Injective (MvPolynomial.aeval x : MvPolynomial ι R →ₐ[R] A)\n\n/-- `AlgebraicIndepOn R v s` states that the elements in the family `v` that are indexed by the\nelements of `s` are algebraically independent over `R`. -/\nabbrev AlgebraicIndepOn (s : Set ι) : Prop := AlgebraicIndependent R fun i : s ↦ x i\n\nvariable {R} {x}\n\ntheorem algebraicIndependent_iff :\n    AlgebraicIndependent R x ↔\n      ∀ p : MvPolynomial ι R, MvPolynomial.aeval (x : ι → A) p = 0 → p = 0 :=\n  injective_iff_map_eq_zero _\n\ntheorem AlgebraicIndependent.eq_zero_of_aeval_eq_zero (h : AlgebraicIndependent R x) :\n    ∀ p : MvPolynomial ι R, MvPolynomial.aeval (x : ι → A) p = 0 → p = 0 :=\n  algebraicIndependent_iff.1 h\n\ntheorem algebraicIndependent_iff_injective_aeval :\n    AlgebraicIndependent R x ↔ Injective (MvPolynomial.aeval x : MvPolynomial ι R →ₐ[R] A) :=\n  Iff.rfl\n\nnamespace AlgebraicIndependent\n\ntheorem of_comp (f : A →ₐ[R] A') (hfv : AlgebraicIndependent R (f ∘ x)) :\n    AlgebraicIndependent R x := by\n  have : aeval (f ∘ x) = f.comp (aeval x) := by ext; simp\n  rw [AlgebraicIndependent, this, AlgHom.coe_comp] at hfv\n  exact hfv.of_comp\n\nvariable (hx : AlgebraicIndependent R x)\ninclude hx\n\ntheorem comp (f : ι' → ι) (hf : Function.Injective f) : AlgebraicIndependent R (x ∘ f) := by\n  intro p q\n  simpa [aeval_rename, (rename_injective f hf).eq_iff] using @hx (rename f p) (rename f q)\n\ntheorem coe_range : AlgebraicIndependent R ((↑) : range x → A) := by\n  simpa using hx.comp _ (rangeSplitting_injective x)\n\nend AlgebraicIndependent\n\nopen AlgebraicIndependent\n\ntheorem algebraicIndependent_equiv (e : ι ≃ ι') {f : ι' → A} :\n    AlgebraicIndependent R (f ∘ e) ↔ AlgebraicIndependent R f :=\n  ⟨fun h => Function.comp_id f ▸ e.self_comp_symm ▸ h.comp _ e.symm.injective,\n    fun h => h.comp _ e.injective⟩\n\ntheorem algebraicIndependent_equiv' (e : ι ≃ ι') {f : ι' → A} {g : ι → A} (h : f ∘ e = g) :\n    AlgebraicIndependent R g ↔ AlgebraicIndependent R f :=\n  h ▸ algebraicIndependent_equiv e\n\ntheorem algebraicIndependent_subtype_range {ι} {f : ι → A} (hf : Injective f) :\n    AlgebraicIndependent R ((↑) : range f → A) ↔ AlgebraicIndependent R f :=\n  Iff.symm <| algebraicIndependent_equiv' (Equiv.ofInjective f hf) rfl\n\nalias ⟨AlgebraicIndependent.of_subtype_range, _⟩ := algebraicIndependent_subtype_range\n\ntheorem algebraicIndependent_image {ι} {s : Set ι} {f : ι → A} (hf : Set.InjOn f s) :\n    (AlgebraicIndependent R fun x : s => f x) ↔ AlgebraicIndependent R fun x : f '' s => (x : A) :=\n  algebraicIndependent_equiv' (Equiv.Set.imageOfInjOn _ _ hf) rfl\n\nlemma AlgebraicIndepOn.mono {s t : Set ι} (H : AlgebraicIndepOn R x t) (hst : s ⊆ t) :\n    AlgebraicIndepOn R x s := by\n  simpa [Function.comp] using! H.comp (Set.inclusion hst) (Set.inclusion_injective hst)\n\n@[simp]\nlemma AlgebraicIndepOn.univ : AlgebraicIndepOn R x .univ ↔ AlgebraicIndependent R x :=\n  algebraicIndependent_equiv (Equiv.Set.univ ι)\n\nnamespace AlgebraicIndependent\n\ntheorem mono {t s : Set A} (h : t ⊆ s)\n    (hx : AlgebraicIndependent R ((↑) : s → A)) : AlgebraicIndependent R ((↑) : t → A) :=\n  AlgebraicIndepOn.mono (x := id) hx h\n\nsection repr\n\nvariable (hx : AlgebraicIndependent R x)\ninclude hx\n\n/-- Canonical isomorphism between polynomials and the subalgebra generated by\n  algebraically independent elements. -/\n@[simps! apply_coe]\ndef aevalEquiv : MvPolynomial ι R ≃ₐ[R] Algebra.adjoin R (range x) :=\n  (AlgEquiv.ofInjective (aeval x) (algebraicIndependent_iff_injective_aeval.1 hx)).trans\n    (Subalgebra.equivOfEq _ _ (Algebra.adjoin_range_eq_range_aeval R x).symm)\n\ntheorem algebraMap_aevalEquiv (p : MvPolynomial ι R) :\n    algebraMap (Algebra.adjoin R (range x)) A (hx.aevalEquiv p) = aeval x p :=\n  rfl\n\n/-- The canonical map from the subalgebra generated by an algebraic independent family\n  into the polynomial ring. -/\ndef repr : Algebra.adjoin R (range x) →ₐ[R] MvPolynomial ι R :=\n  hx.aevalEquiv.symm\n\n@[simp]\ntheorem aeval_repr (p) : aeval x (hx.repr p) = p :=\n  Subtype.ext_iff.1 (AlgEquiv.apply_symm_apply hx.aevalEquiv p)\n\ntheorem aeval_comp_repr : (aeval x).comp hx.repr = Subalgebra.val _ :=\n  AlgHom.ext hx.aeval_repr\n\n/-- Canonical isomorphism between polynomials and the subalgebra generated by\n  algebraically independent elements.\n  `AlgebraicIndepOn` version of `AlgebraicIndependent.aevalEquiv`. -/\ndef _root_.AlgebraicIndepOn.aevalEquiv {s : Set ι} (hx : AlgebraicIndepOn R x s) :\n    MvPolynomial s R ≃ₐ[R] Algebra.adjoin R (x '' s) :=\n  (AlgebraicIndependent.aevalEquiv hx).trans\n    (Subalgebra.equivOfEq _ _ congr(Algebra.adjoin _ $(by aesop)))\n\nend repr\n\nend AlgebraicIndependent\n\nvariable (R) in\n/-- A family is a transcendence basis if it is a maximal algebraically independent subset. -/\n@[stacks 030E \"(4)\"] def IsTranscendenceBasis (x : ι → A) : Prop :=\n  AlgebraicIndependent R x ∧\n    ∀ (s : Set A) (_ : AlgebraicIndepOn R id s) (_ : range x ⊆ s), range x = s\n\ntheorem isTranscendenceBasis_iff_maximal {s : Set A} :\n    IsTranscendenceBasis R ((↑) : s → A) ↔ Maximal (AlgebraicIndepOn R id) s := by\n  rw [IsTranscendenceBasis, maximal_iff, Subtype.range_val]; rfl\n\nTarget:\ntheorem isTranscendenceBasis_equiv (e : ι ≃ ι') {f : ι' → A} :\n    IsTranscendenceBasis R (f ∘ e) ↔ IsTranscendenceBasis R f :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  simp_rw [IsTranscendenceBasis, algebraicIndependent_equiv, EquivLike.range_comp]\n\nalias ⟨_, IsTranscendenceBasis.comp_equiv⟩ := isTranscendenceBasis_equiv","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/AlgebraicIndependent","family_id":"istranscendencebasis_equiv","file_id":"mathlib/Mathlib/RingTheory/AlgebraicIndependent/Defs.lean","sample_id":"f15d582cb509e0a13008bd362768f09d19cdeaf2278912231d988aa0bbeaa66e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"164db8212cd6dbe524941c096782dd0230af417452ad9fde38bafe58fdc70918","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e4c90edf46e133292ddad63e3da005b52fed9b0d4f1ff8887a9d93701bc44602","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"470ba23482b167ed1d27d1364fdd8b69790cc7acc5d9ae4345731e59f6a235df","source_sha256":"6a1fd2aa47744142955f801b29fb06cc46e0d439ea80c9cccba97a19a60fbbe9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  apply IsIntegralClosure.algebraMap_injective B A L\n  simp","hard_negative":false,"metrics":{"chosen_tokens":10,"rejected_tokens":14,"token_jaccard":0.769231,"token_length_ratio":1.4},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"7bc69a1d15fae03f06c3dc0f18b3a0768bd7b09beff0cecc251bc76ac8eee245","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.IntegralClosure\npublic import Mathlib.RingTheory.RingHom.Finite\npublic import Mathlib.RingTheory.Localization.LocalizationLocalization\npublic import Mathlib.RingTheory.Localization.NormTrace\npublic import Mathlib.RingTheory.Norm.Transitivity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Restriction of various maps between fields to integrally closed subrings.\n\nIn this file, we assume `A` is an integrally closed domain; `K` is the fraction ring of `A`;\n`L` is a finite extension of `K`; `B` is the integral closure of `A` in `L`.\nWe call this the AKLB setup.\n\n## Main definitions\n- `galRestrict`: The restriction `Aut(L/K) → Aut(B/A)` as an `MulEquiv` in an AKLB setup.\n- `Algebra.intTrace`: The trace map of a finite extension of integrally closed domains `B/A` is\n  defined to be the restriction of the trace map of `Frac(B)/Frac(A)`.\n- `Algebra.intNorm`: The norm map of a finite extension of integrally closed domains `B/A` is\n  defined to be the restriction of the norm map of `Frac(B)/Frac(A)`.\n\n-/\n\n@[expose] public section\n\nopen Module nonZeroDivisors\n\nvariable (A K L L₂ L₃ B B₂ B₃ : Type*)\nvariable [CommRing A] [CommRing B] [CommRing B₂] [CommRing B₃]\nvariable [Algebra A B] [Algebra A B₂] [Algebra A B₃]\nvariable [Field K] [Field L] [Field L₂] [Field L₃]\nvariable [Algebra A K] [IsFractionRing A K]\nvariable [Algebra K L] [Algebra A L] [IsScalarTower A K L]\nvariable [Algebra K L₂] [Algebra A L₂] [IsScalarTower A K L₂]\nvariable [Algebra K L₃] [Algebra A L₃] [IsScalarTower A K L₃]\nvariable [Algebra B L] [IsScalarTower A B L] [IsIntegralClosure B A L]\nvariable [Algebra B₂ L₂] [IsScalarTower A B₂ L₂] [IsIntegralClosure B₂ A L₂]\nvariable [Algebra B₃ L₃] [IsScalarTower A B₃ L₃] [IsIntegralClosure B₃ A L₃]\n\nsection galois\n\nsection galRestrict'\nvariable {K L L₂ L₃}\nomit [IsFractionRing A K]\n\n/-- A generalization of `galRestrictHom` beyond endomorphisms. -/\nnoncomputable\ndef galRestrict' (f : L →ₐ[K] L₂) : (B →ₐ[A] B₂) :=\n  (IsIntegralClosure.equiv A (integralClosure A L₂) L₂ B₂).toAlgHom.comp\n      (((f.restrictScalars A).comp (IsScalarTower.toAlgHom A B L)).codRestrict\n        (integralClosure A L₂) (fun x ↦ IsIntegral.map _ (IsIntegralClosure.isIntegral A L x)))\n\n@[simp]\nlemma algebraMap_galRestrict'_apply (σ : L →ₐ[K] L₂) (x : B) :\n    algebraMap B₂ L₂ (galRestrict' A B B₂ σ x) = σ (algebraMap B L x) := by\n  simp [galRestrict', galRestrict', Subalgebra.algebraMap_eq]\n\n@[simp]\n\nTarget:\ntheorem galRestrict'_id : galRestrict' A B B (.id K L) = .id A B :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  ext\n  apply IsIntegralClosure.algebraMap_injective B A L\n  simp","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/IntegralClosure","family_id":"galrestrict'_id","file_id":"mathlib/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean","sample_id":"470ba23482b167ed1d27d1364fdd8b69790cc7acc5d9ae4345731e59f6a235df"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e65faed5f7af3fe66504654d3728889128f1a2b5c77ec046ad6fc37f70906b6d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"104ce9cbaebdaa87f4904559e1ae0c57adfb3fa19b70fd7be28fa5278b9ac29a","source_sha256":"34049d18cb594f0a184df593dbd6d4fba0ade793ef81827c21fb1d80f135b5c5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by simp [left_distrib, *, sub_eq_add_neg]","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.555556},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"7bda13087b592b3ce73fc3eeea9aed3afcdfb4a03ddb42c3424c9aa1d16c20d1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Basic\npublic import Mathlib.Algebra.Order.Ring.Defs\npublic import Mathlib.Data.Tree.Basic\npublic import Mathlib.Tactic.NormNum.Core\npublic import Mathlib.Util.SynthesizeUsing\n\nNamespace:\nMathlib.Tactic.CancelDenoms\n\nLocal context:\n/-\nCopyright (c) 2020 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n-/\npublic meta import Mathlib.Data.Tree.Basic\npublic meta import Mathlib.Algebra.Group.Nat.Defs\n\n/-!\n# A tactic for canceling numeric denominators\n\nThis file defines tactics that cancel numeric denominators from field Expressions.\n\nAs an example, we want to transform a comparison `5*(a/3 + b/4) < c/3` into the equivalent\n`5*(4*a + 3*b) < 4*c`.\n\n## Implementation notes\n\nThe tooling here was originally written for `linarith`, not intended as an interactive tactic.\nThe interactive version has been split off because it is sometimes convenient to use on its own.\nThere are likely some rough edges to it.\n\nImproving this tactic would be a good project for someone interested in learning tactic programming.\n-/\n\npublic meta section\n\nopen Lean Parser Tactic Mathlib Meta NormNum Qq\n\ninitialize registerTraceClass `CancelDenoms\n\nnamespace Mathlib.Tactic.CancelDenoms\n\n/-! ### Lemmas used in the procedure -/\n\ntheorem mul_subst {α} [CommRing α] {n1 n2 k e1 e2 t1 t2 : α}\n    (h1 : n1 * e1 = t1) (h2 : n2 * e2 = t2) (h3 : n1 * n2 = k) : k * (e1 * e2) = t1 * t2 := by\n  rw [← h3, mul_comm n1, mul_assoc n2, ← mul_assoc n1, h1,\n      ← mul_assoc n2, mul_comm n2, mul_assoc, h2]\n\ntheorem div_subst {α} [Field α] {n1 n2 k e1 e2 t1 : α}\n    (h1 : n1 * e1 = t1) (h2 : n2 / e2 = 1) (h3 : n1 * n2 = k) : k * (e1 / e2) = t1 := by\n  rw [← h3, mul_assoc, mul_div_left_comm, h2, ← mul_assoc, h1, mul_comm, one_mul]\n\ntheorem cancel_factors_eq_div {α} [Field α] {n e e' : α}\n    (h : n * e = e') (h2 : n ≠ 0) : e = e' / n :=\n  eq_div_of_mul_eq h2 <| by rwa [mul_comm] at h\n\ntheorem add_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :\n    n * (e1 + e2) = t1 + t2 := by simp [left_distrib, *]\n\nTarget:\ntheorem sub_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :\n    n * (e1 - e2) = t1 - t2 :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/CancelDenoms","family_id":"sub_subst","file_id":"mathlib/Mathlib/Tactic/CancelDenoms/Core.lean","sample_id":"104ce9cbaebdaa87f4904559e1ae0c57adfb3fa19b70fd7be28fa5278b9ac29a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2b82d4fddfb467900b75edb5d85dcce54ace3fac2a5c050d4837c5985aa49b27","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"08ef20f28dae9ac8a4b11319080aa916285ab90a007cfc4fe216ca408e450b14","source_sha256":"e6501306e0c166f31aa2d54b2053e75c750b45815aae03787ae7a4d5aa54d0d7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  lift s to Finset α using hs; exact s.exists_le_minimal ha\n\nvariable [Nonempty α]","hard_negative":false,"metrics":{"chosen_tokens":19,"rejected_tokens":3,"token_jaccard":0.111111,"token_length_ratio":0.157895},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"7bf34e0d186234ac9305e9a9c462944de2d7683321cd783b890cb2b1ef51df83","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Basic\npublic import Mathlib.Order.Minimal\n\nNamespace:\nSet\n\nLocal context:\n/-\nCopyright (c) 2025 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Finite preorders and finite sets in a preorder\n\nThis file shows that non-empty finite sets in a preorder have minimal/maximal elements, and\ncontrapositively that non-empty sets without minimal or maximal elements are infinite.\n-/\n\npublic section\n\nvariable {ι α β : Type*}\n\nnamespace Finset\nsection IsTrans\nvariable [LE α] [IsTrans α LE.le] {s : Finset α} {a : α}\n\nlemma exists_maximalFor (f : ι → α) (s : Finset ι) (hs : s.Nonempty) :\n    ∃ i, MaximalFor (· ∈ s) f i := by\n  induction hs using Finset.Nonempty.cons_induction with\n  | singleton i => exact ⟨i, by simp⟩\n  | @cons i s hi hs ih =>\n    obtain ⟨j, hj⟩ := ih\n    by_cases hji : f j ≤ f i\n    · refine ⟨i, mem_cons_self .., ?_⟩\n      simp only [mem_cons, forall_eq_or_imp, imp_self, true_and]\n      exact fun k hk hik ↦ _root_.trans (hj.2 hk <| _root_.trans hji hik) hji\n    · exact ⟨j, mem_cons_of_mem hj.1, by simpa [hji] using hj.2⟩\n\nlemma exists_minimalFor (f : ι → α) (s : Finset ι) (hs : s.Nonempty) :\n    ∃ i, MinimalFor (· ∈ s) f i := exists_maximalFor (α := αᵒᵈ) f s hs\n\nlemma exists_maximal (hs : s.Nonempty) : ∃ i, Maximal (· ∈ s) i := s.exists_maximalFor id hs\nlemma exists_minimal (hs : s.Nonempty) : ∃ i, Minimal (· ∈ s) i := s.exists_minimalFor id hs\n\nend IsTrans\n\nsection Preorder\nvariable [Preorder α] {s : Finset α} {a : α}\n\nlemma exists_le_maximal (s : Finset α) (ha : a ∈ s) : ∃ b, a ≤ b ∧ Maximal (· ∈ s) b := by\n  classical\n  obtain ⟨b, hb, hab, hbmin⟩ : ∃ b ∈ s, a ≤ b ∧ _ := by\n    simpa [Maximal, and_assoc] using {x ∈ s | a ≤ x}.exists_maximal ⟨a, mem_filter.2 ⟨ha, le_rfl⟩⟩\n  exact ⟨b, hab, hb, fun c hc hbc ↦ hbmin hc (hab.trans hbc) hbc⟩\n\nlemma exists_le_minimal (s : Finset α) (ha : a ∈ s) : ∃ b ≤ a, Minimal (· ∈ s) b :=\n  exists_le_maximal (α := αᵒᵈ) s ha\n\nend Preorder\nend Finset\n\nnamespace Set\nsection IsTrans\nvariable [LE α] [IsTrans α LE.le] {s : Set α} {a : α}\n\nlemma Finite.exists_maximalFor (f : ι → α) (s : Set ι) (h : s.Finite) (hs : s.Nonempty) :\n    ∃ i, MaximalFor (· ∈ s) f i := by\n  lift s to Finset ι using h; exact s.exists_maximalFor f hs\n\nlemma Finite.exists_minimalFor (f : ι → α) (s : Set ι) (h : s.Finite) (hs : s.Nonempty) :\n    ∃ i, MinimalFor (· ∈ s) f i := Finite.exists_maximalFor (α := αᵒᵈ) f s h hs\n\nlemma Finite.exists_maximal (h : s.Finite) (hs : s.Nonempty) : ∃ i, Maximal (· ∈ s) i :=\n  h.exists_maximalFor id _ hs\n\nlemma Finite.exists_minimal (h : s.Finite) (hs : s.Nonempty) : ∃ i, Minimal (· ∈ s) i :=\n  h.exists_minimalFor id _ hs\n\n/-- A version of `Finite.exists_maximalFor` with the (weaker) hypothesis that the image of `s`\nis finite rather than `s` itself. -/\nlemma Finite.exists_maximalFor' (f : ι → α) (s : Set ι) (h : (f '' s).Finite) (hs : s.Nonempty) :\n    ∃ i, MaximalFor (· ∈ s) f i := by\n  obtain ⟨_, ⟨a, ha, rfl⟩, hmax⟩ := Finite.exists_maximalFor id (f '' s) h (hs.image f)\n  exact ⟨a, ha, fun a' ha' hf ↦ hmax (mem_image_of_mem f ha') hf⟩\n\n/-- A version of `Finite.exists_minimalFor` with the (weaker) hypothesis that the image of `s`\nis finite rather than `s` itself. -/\nlemma Finite.exists_minimalFor' (f : ι → α) (s : Set ι) (h : (f '' s).Finite) (hs : s.Nonempty) :\n    ∃ i, MinimalFor (· ∈ s) f i := h.exists_maximalFor' (α := αᵒᵈ) f s hs\n\nend IsTrans\n\nsection Preorder\nvariable [Preorder α] {s : Set α} {a : α}\n\nlemma Finite.exists_le_maximal (hs : s.Finite) (ha : a ∈ s) : ∃ b, a ≤ b ∧ Maximal (· ∈ s) b := by\n  lift s to Finset α using hs; exact s.exists_le_maximal ha\n\nTarget:\nlemma Finite.exists_le_minimal (hs : s.Finite) (ha : a ∈ s) : ∃ b, b ≤ a ∧ Minimal (· ∈ s) b :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Preorder","family_id":"finite","file_id":"mathlib/Mathlib/Order/Preorder/Finite.lean","sample_id":"08ef20f28dae9ac8a4b11319080aa916285ab90a007cfc4fe216ca408e450b14"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1246a108b01c281615de5c0ac2e3b8d9e0aa2fea03820a7bb20910f1f044ae33","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4180d4b77b71dc3fb2a1ccac790bfafb68b3d5a563a499f05b950658539e98e7","source_sha256":"5f13df63ef1a33fb7d6444e769a7e9b9cf3ab09860a26a91c473157d6239794f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [he.isIdempotentElem.mem_iff, he.isIdempotentElem.range_mem_invtSubmodule_iff,\n    he.isIdempotentElem.ker_mem_invtSubmodule_iff, forall_and, and_iff_left_iff_imp, ← mul_def]\n  intro h x hx\n  simpa [he.isSelfAdjoint.star_eq] using! congr(star $(h _ (star_mem hx)))","hard_negative":true,"metrics":{"chosen_tokens":55,"rejected_tokens":8,"token_jaccard":0.027027,"token_length_ratio":0.145455},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"7c0b81179dfdac553c55db4c6ca378ce1fb10fe02723f57388753a9c3d832dbb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.CStarAlgebra.Classes\npublic import Mathlib.Analysis.InnerProductSpace.Adjoint\n\nNamespace:\nVonNeumannAlgebra\n\nLocal context:\n/-\nCopyright (c) 2022 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Von Neumann algebras\n\nWe give the \"abstract\" and \"concrete\" definitions of a von Neumann algebra.\nWe still have a major project ahead of us to show the equivalence between these definitions!\n\nAn abstract von Neumann algebra `WStarAlgebra M` is a C⋆ algebra with a Banach space predual,\nper Sakai (1971).\n\nA concrete von Neumann algebra `VonNeumannAlgebra H` (where `H` is a Hilbert space)\nis a \\*-closed subalgebra of bounded operators on `H` which is equal to its double commutant.\n\nWe'll also need to prove the von Neumann double commutant theorem,\nthat the concrete definition is equivalent to a \\*-closed subalgebra which is weakly closed.\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\n/-- Sakai's definition of a von Neumann algebra as a C⋆ algebra with a Banach space predual.\n\nSo that we can unambiguously talk about these \"abstract\" von Neumann algebras\nin parallel with the \"concrete\" ones (weakly closed \\*-subalgebras of B(H)),\nwe name this definition `WStarAlgebra`.\n\nNote that for now we only assert the mere existence of predual, rather than picking one.\nThis may later prove problematic, and need to be revisited.\nPicking one may cause problems with definitional unification of different instances.\nOne the other hand, not picking one means that the weak-\\* topology\n(which depends on a choice of predual) must be defined using the choice,\nand we may be unhappy with the resulting opaqueness of the definition.\n-/\nclass WStarAlgebra (M : Type u) [CStarAlgebra M] : Prop where\n  /-- There is a Banach space `X` whose dual is isometrically (conjugate-linearly) isomorphic\n  to the `WStarAlgebra`. -/\n  exists_predual :\n    ∃ (X : Type u) (_ : NormedAddCommGroup X) (_ : NormedSpace ℂ X) (_ : CompleteSpace X),\n      Nonempty (StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M)\n\n-- TODO: Without this, `VonNeumannAlgebra` times out. Why?\n/-- The double commutant definition of a von Neumann algebra,\nas a \\*-closed subalgebra of bounded operators on a Hilbert space,\nwhich is equal to its double commutant.\n\nNote that this definition is parameterised by the Hilbert space\non which the algebra faithfully acts, as is standard in the literature.\nSee `WStarAlgebra` for the abstract notion (a C⋆-algebra with Banach space predual).\n\nNote this is a bundled structure, parameterised by the Hilbert space `H`,\nrather than a typeclass on the type of elements.\nThus we can't say that the bounded operators `H →L[ℂ] H` form a `VonNeumannAlgebra`\n(although we will later construct the instance `WStarAlgebra (H →L[ℂ] H)`),\nand instead will use `⊤ : VonNeumannAlgebra H`.\n-/\nstructure VonNeumannAlgebra (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H]\n    [CompleteSpace H] extends StarSubalgebra ℂ (H →L[ℂ] H) where\n  /-- The double commutant (a.k.a. centralizer) of a `VonNeumannAlgebra` is itself. -/\n  centralizer_centralizer' : Set.centralizer (Set.centralizer carrier) = carrier\n\n/-- Consider a von Neumann algebra acting on a Hilbert space `H` as a \\*-subalgebra of `H →L[ℂ] H`.\n(That is, we forget that it is equal to its double commutant\nor equivalently that it is closed in the weak and strong operator topologies.)\n-/\nadd_decl_doc VonNeumannAlgebra.toStarSubalgebra\n\nnamespace VonNeumannAlgebra\n\nvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n\ninstance instSetLike : SetLike (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  coe S := S.carrier\n  coe_injective S T h := by obtain ⟨⟨⟨⟨⟨⟨_, _⟩, _⟩, _⟩, _⟩, _⟩, _⟩ := S; cases T; congr\n\ninstance : PartialOrder (VonNeumannAlgebra H) := .ofSetLike (VonNeumannAlgebra H) (H →L[ℂ] H)\n\nnoncomputable instance instStarMemClass : StarMemClass (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  star_mem {s} := s.star_mem'\n\ninstance instSubringClass : SubringClass (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  one_mem {s} := s.one_mem'\n  zero_mem {s} := s.zero_mem'\n  neg_mem {s} a ha := show -a ∈ s.toStarSubalgebra from neg_mem ha\n\n@[simp]\ntheorem mem_carrier {S : VonNeumannAlgebra H} {x : H →L[ℂ] H} :\n    x ∈ S.toStarSubalgebra ↔ x ∈ (S : Set (H →L[ℂ] H)) :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toStarSubalgebra (S : VonNeumannAlgebra H) :\n    (S.toStarSubalgebra : Set (H →L[ℂ] H)) = S :=\n  rfl\n\n@[simp]\ntheorem coe_mk (S : StarSubalgebra ℂ (H →L[ℂ] H)) (h) :\n    ((⟨S, h⟩ : VonNeumannAlgebra H) : Set (H →L[ℂ] H)) = S :=\n  rfl\n\n@[ext]\ntheorem ext {S T : VonNeumannAlgebra H} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem centralizer_centralizer (S : VonNeumannAlgebra H) :\n    Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S :=\n  S.centralizer_centralizer'\n\n/-- The centralizer of a `VonNeumannAlgebra`, as a `VonNeumannAlgebra`. -/\nnoncomputable def commutant (S : VonNeumannAlgebra H) : VonNeumannAlgebra H where\n  toStarSubalgebra := StarSubalgebra.centralizer ℂ (S : Set (H →L[ℂ] H))\n  centralizer_centralizer' := by simp\n\n@[simp]\ntheorem coe_commutant (S : VonNeumannAlgebra H) :\n    ↑S.commutant = Set.centralizer (S : Set (H →L[ℂ] H)) := by\n  simp [commutant]\n\n@[simp]\ntheorem mem_commutant_iff {S : VonNeumannAlgebra H} {z : H →L[ℂ] H} :\n    z ∈ S.commutant ↔ ∀ g ∈ S, g * z = z * g := by\n  rw [← SetLike.mem_coe, coe_commutant]\n  rfl\n\n@[simp]\ntheorem commutant_commutant (S : VonNeumannAlgebra H) : S.commutant.commutant = S :=\n  SetLike.coe_injective <| by simp\n\nopen ContinuousLinearMap in\n/-- An idempotent is an element in a von Neumann algebra if and only if\nits range and kernel are invariant under the commutant. -/\ntheorem IsIdempotentElem.mem_iff {e : H →L[ℂ] H} (h : IsIdempotentElem e)\n    (S : VonNeumannAlgebra H) :\n    e ∈ S ↔ ∀ y ∈ S.commutant,\n      e.range ∈ Module.End.invtSubmodule y ∧ e.ker ∈ Module.End.invtSubmodule y := by\n  conv_rhs => simp [← h.commute_iff, Commute.symm_iff (a := e), commute_iff_eq, ← mem_commutant_iff]\n\nopen VonNeumannAlgebra ContinuousLinearMap in\n/-- A star projection is an element in a von Neumann algebra if and only if\nits range is invariant under the commutant. -/\n\nTarget:\ntheorem IsStarProjection.mem_iff {e : H →L[ℂ] H} (he : IsStarProjection e)\n    (S : VonNeumannAlgebra H) :\n    e ∈ S ↔ ∀ y ∈ S.commutant, e.range ∈ Module.End.invtSubmodule y :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"ef05cc6f370400e5f1321b2b0491b694916374fa91d27049a433d689a3d749e4","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/VonNeumannAlgebra","family_id":"isstarprojection","file_id":"mathlib/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean","sample_id":"4180d4b77b71dc3fb2a1ccac790bfafb68b3d5a563a499f05b950658539e98e7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5403bbdd6b8db64d0404aea07e7fcb0b3cc956807fed1c2cfdeeee9bac43ff23","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"17bf13272abd6adb07ae2877731b7c8fc4a266a142d05c92c67a6b19e4cb620a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a3a4139f0052b0a32136e52937242d5f55eef48c36e8634d6c5292950c12886e","source_sha256":"e6e1b45f74734f944289eb9bfce978e134476af3b67ff11012d6e865d086c8c1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  simp [h.neg_to_eq, div_nonneg]","hard_negative":true,"metrics":{"chosen_tokens":10,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.3},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"7c412effa71635470fe3834a98760c6e7e300495d485bdff8d92d21704f92aa2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Floor.Semiring\npublic import Mathlib.Data.NNRat.Order\npublic import Mathlib.Data.Rat.Floor\n\nNamespace:\nMathlib.Meta.NormNum\n\nLocal context:\n/-\nCopyright (c) 2024 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\npublic meta import Mathlib.Data.Rat.Floor\n\n/-!\n# Floor Function for Non-negative Rational Numbers\n\n## Summary\n\nWe define the `FloorSemiring` instance on `ℚ≥0`, and relate its operators to `NNRat.cast`.\n\nNote that we cannot talk about `Int.fract`, which currently only works for rings.\n\n## Tags\n\nnnrat, rationals, ℚ≥0, floor\n-/\n\n@[expose] public section\n\nassert_not_exists Finset\n\nnamespace NNRat\n\ninstance : FloorSemiring ℚ≥0 where\n  floor q := ⌊q.val⌋₊\n  ceil q := ⌈q.val⌉₊\n  floor_of_neg h := by simpa using h.trans zero_lt_one\n  gc_floor {a n} h := by rw [← NNRat.coe_le_coe, Nat.le_floor_iff] <;> norm_cast\n  gc_ceil {a b} := by rw [← NNRat.coe_le_coe, Nat.ceil_le]; norm_cast\n\n@[simp, norm_cast]\ntheorem floor_coe (q : ℚ≥0) : ⌊(q : ℚ)⌋₊ = ⌊q⌋₊ := rfl\n\n@[simp, norm_cast]\ntheorem ceil_coe (q : ℚ≥0) : ⌈(q : ℚ)⌉₊ = ⌈q⌉₊ := rfl\n\n@[simp, norm_cast]\ntheorem coe_floor (q : ℚ≥0) : ↑⌊q⌋₊ = ⌊(q : ℚ)⌋ := Int.natCast_floor_eq_floor q.coe_nonneg\n\n@[simp, norm_cast]\ntheorem coe_ceil (q : ℚ≥0) : ↑⌈q⌉₊ = ⌈(q : ℚ)⌉ := Int.natCast_ceil_eq_ceil q.coe_nonneg\n\nprotected theorem floor_def (q : ℚ≥0) : ⌊q⌋₊ = q.num / q.den := by\n  rw [← Int.natCast_inj, NNRat.coe_floor, Rat.floor_def', Int.natCast_ediv, den_coe, num_coe]\n\nsection Semifield\n\nvariable {K} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [FloorSemiring K]\n\n@[simp, norm_cast]\ntheorem floor_cast (x : ℚ≥0) : ⌊(x : K)⌋₊ = ⌊x⌋₊ :=\n  (Nat.floor_eq_iff x.cast_nonneg).2 (mod_cast (Nat.floor_eq_iff x.cast_nonneg).1 (Eq.refl ⌊x⌋₊))\n\n@[simp, norm_cast]\ntheorem ceil_cast (x : ℚ≥0) : ⌈(x : K)⌉₊ = ⌈x⌉₊ := by\n  obtain rfl | hx := eq_or_ne x 0\n  · simp\n  · refine (Nat.ceil_eq_iff ?_).2 (mod_cast (Nat.ceil_eq_iff ?_).1 (Eq.refl ⌈x⌉₊)) <;> simpa\n\nend Semifield\n\nsection Field\n\nvariable {K} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [FloorRing K]\n\n@[simp, norm_cast]\ntheorem intFloor_cast (x : ℚ≥0) : ⌊(x : K)⌋ = ⌊(x : ℚ)⌋ := by\n  rw [Int.floor_eq_iff, ← coe_floor]\n  norm_cast\n  norm_cast\n  rw [Nat.cast_add_one, ← Nat.floor_eq_iff zero_le]\n\n@[simp, norm_cast]\ntheorem intCeil_cast (x : ℚ≥0) : ⌈(x : K)⌉ = ⌈(x : ℚ)⌉ := by\n  rw [Int.ceil_eq_iff, ← coe_ceil, sub_lt_iff_lt_add]\n  constructor\n  · exact_mod_cast NNRat.cast_strictMono <| Nat.ceil_lt_add_one zero_le\n  · rw [Int.cast_natCast, NNRat.cast_le_natCast]\n    exact Nat.le_ceil _\n\nend Field\n\n@[norm_cast]\ntheorem floor_natCast_div_natCast (n d : ℕ) : ⌊(↑n / ↑d : ℚ≥0)⌋₊ = n / d :=\n  Rat.natFloor_natCast_div_natCast n d\n\nend NNRat\n\nnamespace Mathlib.Meta.NormNum\n\nopen Qq\n\n/-!\n### `norm_num` extension for `Nat.ceil`\n-/\n\ntheorem IsNat.natCeil {R : Type*} [Semiring R] [LinearOrder R] [IsStrictOrderedRing R]\n    [FloorSemiring R] (r : R) (m : ℕ) : IsNat r m → IsNat (⌈r⌉₊) m := by\n  rintro ⟨⟨⟩⟩\n  exact ⟨by simp⟩\n\ntheorem IsInt.natCeil {R : Type*} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] [FloorSemiring R]\n    (r : R) (m : ℕ) : IsInt r (.negOfNat m) → IsNat (⌈r⌉₊) 0 := by\n  rintro ⟨⟨⟩⟩\n  exact ⟨by simp⟩\n\ntheorem IsNNRat.natCeil {R : Type*} [Semifield R] [LinearOrder R] [IsStrictOrderedRing R]\n    [FloorSemiring R] (r : R) (n d : ℕ) (h : IsNNRat r n d) (res : ℕ)\n    (hres : ⌈(n / d : ℚ≥0)⌉₊ = res) : IsNat ⌈r⌉₊ res := by\n  constructor\n  rw [← hres, h.to_eq rfl rfl, ← @NNRat.ceil_cast R]\n  simp\n\nTarget:\ntheorem IsRat.natCeil {R : Type*} [Field R] [LinearOrder R] [IsStrictOrderedRing R]\n    [FloorSemiring R] (r : R) (n d : ℕ) (h : IsRat r (.negOfNat n) d) : IsNat ⌈r⌉₊ 0 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_a3a4139f0052","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"11387462df0846358ae2630fb5a0f5c197b3182eb5bf98c0dfa89dd14270be55","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/NNRat","family_id":"israt","file_id":"mathlib/Mathlib/Data/NNRat/Floor.lean","sample_id":"a3a4139f0052b0a32136e52937242d5f55eef48c36e8634d6c5292950c12886e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e9b602b11908c749a9de1394766214188bf5ab4658bf596bbcd37cd55e75b8a2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d0879bf6448c6e4a9ff954465980f5fa1f80f58fa3aef9de69945f93f5b51233","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e2bfab7a5a1b1912f82cb1a9e10696dbb413a79fc583a7da1b4cad05b058c943","source_sha256":"c0e006eb7064864ca13410a03d1b3208f2b5aa17e594e64bbe3404463f2e5f7f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← S.toNatSubmodule_toAddSubmonoid, ← Submodule.fg_iff_addSubmonoid_fg,\n    Submodule.fg_iff_exists_fin_linearMap]\n  exact exists_congr fun n => ⟨fun ⟨f, hf⟩ => ⟨f, hf ▸ LinearMap.range_toAddSubmonoid _⟩,\n    fun ⟨f, hf⟩ => ⟨f.toNatLinearMap, Submodule.toAddSubmonoid_inj.mp <|\n      hf ▸ LinearMap.range_toAddSubmonoid _⟩⟩","hard_negative":true,"metrics":{"chosen_tokens":68,"rejected_tokens":2,"token_jaccard":0.032258,"token_length_ratio":0.029412},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"7c43df5c6766e2fb7bd5a0048520ab1e8d6a50bf147420c236206a1c902b960b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Congruence.Defs\npublic import Mathlib.LinearAlgebra.Basis.Cardinality\npublic import Mathlib.LinearAlgebra.DFinsupp\npublic import Mathlib.LinearAlgebra.Isomorphisms\npublic import Mathlib.LinearAlgebra.StdBasis\npublic import Mathlib.RingTheory.Finiteness.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\n/-!\n# Finite modules and types with finitely many elements\n\nThis file relates `Module.Finite` and `_root_.Finite`.\n\n-/\n\n@[expose] public section\n\nopen Function (Surjective)\nopen Finsupp\n\nsection ModuleAndAlgebra\n\nvariable (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]\n\nopen Module in\ntheorem Submodule.fg_iff_exists_fin_linearMap {N : Submodule R M} :\n    N.FG ↔ ∃ (n : ℕ) (f : (Fin n → R) →ₗ[R] M), LinearMap.range f = N := by\n  simp_rw [fg_iff_exists_fin_generating_family, ← ((Pi.basisFun R _).constr ℕ).exists_congr_right]\n  simp [Basis.constr_range]\n\nTarget:\ntheorem AddSubmonoid.fg_iff_exists_fin_addMonoidHom {M : Type*} [AddCommMonoid M]\n    {S : AddSubmonoid M} : S.FG ↔ ∃ (n : ℕ) (f : (Fin n → ℕ) →+ M), AddMonoidHom.mrange f = S :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_e2bfab7a5a1b","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"c1a88b654de93427400df4afd2d45901f3bf64d9e66c3db262c1946cf9d5dc4a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Finiteness","family_id":"addsubmonoid","file_id":"mathlib/Mathlib/RingTheory/Finiteness/Cardinality.lean","sample_id":"e2bfab7a5a1b1912f82cb1a9e10696dbb413a79fc583a7da1b4cad05b058c943"},"split":"train"}
{"binding":{"chosen_proof_sha256":"75232f0c801e33fc6a2f2b42c9fbde78aef8e1d36674a56a7714ae3434228c02","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cbf345f5a8e8911a533324abd4e16d374179e7aaeb7e3f7ffe78a7d81a48841b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"419897952885489eef42100d7198d985d0ed456c5f3c123051bf15c5b6fac476","source_sha256":"5fd82395c755ee110eacac536435f2600396b3d8488a7cfc9d34b3e8c0d5cde4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [(v.filterAt_basis_closedBall x).mem_iff, ← and_imp, subset_def, mem_setOf]","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":5,"token_jaccard":0.1,"token_length_ratio":0.25},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"7c7b3b1b246aa21901d77c05ee1d8ae8dc7ba25edd50d3358e3b79a84d94a310","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Measure.AbsolutelyContinuous\n\nNamespace:\nVitaliFamily\n\nLocal context:\n/-\nCopyright (c) 2021 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Vitali families\n\nOn a metric space `X` with a measure `μ`, consider for each `x : X` a family of measurable sets with\nnonempty interiors, called `setsAt x`. This family is a Vitali family if it satisfies the following\nproperty: consider a (possibly non-measurable) set `s`, and for any `x` in `s` a\nsubfamily `f x` of `setsAt x` containing sets of arbitrarily small diameter. Then one can extract\na disjoint subfamily covering almost all `s`.\n\nVitali families are provided by covering theorems such as the Besicovitch covering theorem or the\nVitali covering theorem. They make it possible to formulate general versions of theorems on\ndifferentiations of measure that apply in both contexts.\n\nThis file gives the basic definition of Vitali families. More interesting developments of this\nnotion are deferred to other files:\n* constructions of specific Vitali families are provided by the Besicovitch covering theorem, in\n  `Besicovitch.vitaliFamily`, and by the Vitali covering theorem, in `Vitali.vitaliFamily`.\n* The main theorem on differentiation of measures along a Vitali family is proved in\n  `VitaliFamily.ae_tendsto_rnDeriv`.\n\n## Main definitions\n\n* `VitaliFamily μ` is a structure made, for each `x : X`, of a family of sets around `x`, such that\n  one can extract an almost everywhere disjoint covering from any subfamily containing sets of\n  arbitrarily small diameters.\n\nLet `v` be such a Vitali family.\n* `v.FineSubfamilyOn` describes the subfamilies of `v` from which one can extract almost\n  everywhere disjoint coverings. This property, called\n  `v.FineSubfamilyOn.exists_disjoint_covering_ae`, is essentially a restatement of the definition\n  of a Vitali family. We also provide an API to use efficiently such a disjoint covering.\n* `v.filterAt x` is a filter on sets of `X`, such that convergence with respect to this filter\n  means convergence when sets in the Vitali family shrink towards `x`.\n\n## References\n\n* [Herbert Federer, Geometric Measure Theory, Chapter 2.8][Federer1996]\n  (Vitali families are called Vitali relations there)\n-/\n\n@[expose] public section\n\n\nopen MeasureTheory Metric Set Filter TopologicalSpace MeasureTheory.Measure\nopen scoped Topology\n\nvariable {X : Type*} [PseudoMetricSpace X]\n\n/-- On a metric space `X` with a measure `μ`, consider for each `x : X` a family of measurable sets\nwith nonempty interiors, called `setsAt x`. This family is a Vitali family if it satisfies the\nfollowing property: consider a (possibly non-measurable) set `s`, and for any `x` in `s` a\nsubfamily `f x` of `setsAt x` containing sets of arbitrarily small diameter. Then one can extract\na disjoint subfamily covering almost all `s`.\n\nVitali families are provided by covering theorems such as the Besicovitch covering theorem or the\nVitali covering theorem. They make it possible to formulate general versions of theorems on\ndifferentiations of measure that apply in both contexts.\n-/\nstructure VitaliFamily {m : MeasurableSpace X} (μ : Measure X) where\n  /-- Sets of the family \"centered\" at a given point. -/\n  setsAt : X → Set (Set X)\n  /-- All sets of the family are measurable. -/\n  measurableSet : ∀ x : X, ∀ s ∈ setsAt x, MeasurableSet s\n  /-- All sets of the family have nonempty interior. -/\n  nonempty_interior : ∀ x : X, ∀ s ∈ setsAt x, (interior s).Nonempty\n  /-- For any closed ball around `x`, there exists a set of the family contained in this ball. -/\n  nontrivial : ∀ (x : X), ∀ ε > (0 : ℝ), ∃ s ∈ setsAt x, s ⊆ closedBall x ε\n  /-- Consider a (possibly non-measurable) set `s`,\n  and for any `x` in `s` a subfamily `f x` of `setsAt x`\n  containing sets of arbitrarily small diameter.\n  Then one can extract a disjoint subfamily covering almost all `s`. -/\n  covering : ∀ (s : Set X) (f : X → Set (Set X)),\n    (∀ x ∈ s, f x ⊆ setsAt x) → (∀ x ∈ s, ∀ ε > (0 : ℝ), ∃ t ∈ f x, t ⊆ closedBall x ε) →\n    ∃ t : Set (X × Set X), (∀ p ∈ t, p.1 ∈ s) ∧ (t.PairwiseDisjoint fun p ↦ p.2) ∧\n      (∀ p ∈ t, p.2 ∈ f p.1) ∧ μ (s \\ ⋃ p ∈ t, p.2) = 0\n\nnamespace VitaliFamily\n\nvariable {m0 : MeasurableSpace X} {μ : Measure X}\n\n/-- A Vitali family for a measure `μ` is also a Vitali family for any measure absolutely continuous\nwith respect to `μ`. -/\ndef mono (v : VitaliFamily μ) (ν : Measure X) (hν : ν ≪ μ) : VitaliFamily ν where\n  __ := v\n  covering s f h h' :=\n    let ⟨t, ts, disj, mem_f, hμ⟩ := v.covering s f h h'\n    ⟨t, ts, disj, mem_f, hν hμ⟩\n\n/-- Given a Vitali family `v` for a measure `μ`, a family `f` is a fine subfamily on a set `s` if\nevery point `x` in `s` belongs to arbitrarily small sets in `v.setsAt x ∩ f x`. This is precisely\nthe subfamilies for which the Vitali family definition ensures that one can extract a disjoint\ncovering of almost all `s`. -/\ndef FineSubfamilyOn (v : VitaliFamily μ) (f : X → Set (Set X)) (s : Set X) : Prop :=\n  ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε\n\nnamespace FineSubfamilyOn\n\nvariable {v : VitaliFamily μ} {f : X → Set (Set X)} {s : Set X} (h : v.FineSubfamilyOn f s)\ninclude h\n\ntheorem exists_disjoint_covering_ae :\n    ∃ t : Set (X × Set X),\n      (∀ p : X × Set X, p ∈ t → p.1 ∈ s) ∧\n      (t.PairwiseDisjoint fun p => p.2) ∧\n      (∀ p : X × Set X, p ∈ t → p.2 ∈ v.setsAt p.1 ∩ f p.1) ∧\n      μ (s \\ ⋃ (p : X × Set X) (_ : p ∈ t), p.2) = 0 :=\n  v.covering s (fun x => v.setsAt x ∩ f x) (fun _ _ => inter_subset_left) h\n\n/-- Given `h : v.FineSubfamilyOn f s`, then `h.index` is a set parametrizing a disjoint\ncovering of almost every `s`. -/\nprotected def index : Set (X × Set X) :=\n  h.exists_disjoint_covering_ae.choose\n\n/-- Given `h : v.FineSubfamilyOn f s`, then `h.covering p` is a set in the family,\nfor `p ∈ h.index`, such that these sets form a disjoint covering of almost every `s`. -/\n@[nolint unusedArguments]\nprotected def covering (_h : FineSubfamilyOn v f s) : X × Set X → Set X :=\n  fun p => p.2\n\ntheorem index_subset : ∀ p : X × Set X, p ∈ h.index → p.1 ∈ s :=\n  h.exists_disjoint_covering_ae.choose_spec.1\n\ntheorem covering_disjoint : h.index.PairwiseDisjoint h.covering :=\n  h.exists_disjoint_covering_ae.choose_spec.2.1\n\nopen scoped Function in -- required for scoped `on` notation\ntheorem covering_disjoint_subtype : Pairwise (Disjoint on fun x : h.index => h.covering x) :=\n  (pairwise_subtype_iff_pairwise_set _ _).2 h.covering_disjoint\n\ntheorem covering_mem {p : X × Set X} (hp : p ∈ h.index) : h.covering p ∈ f p.1 :=\n  (h.exists_disjoint_covering_ae.choose_spec.2.2.1 p hp).2\n\ntheorem covering_mem_family {p : X × Set X} (hp : p ∈ h.index) : h.covering p ∈ v.setsAt p.1 :=\n  (h.exists_disjoint_covering_ae.choose_spec.2.2.1 p hp).1\n\ntheorem measure_sdiff_biUnion : μ (s \\ ⋃ p ∈ h.index, h.covering p) = 0 :=\n  h.exists_disjoint_covering_ae.choose_spec.2.2.2\n\n@[deprecated (since := \"2026-06-03\")] alias measure_diff_biUnion := measure_sdiff_biUnion\n\ntheorem index_countable [SecondCountableTopology X] : h.index.Countable :=\n  h.covering_disjoint.countable_of_nonempty_interior fun _ hx =>\n    v.nonempty_interior _ _ (h.covering_mem_family hx)\n\nprotected theorem measurableSet_u {p : X × Set X} (hp : p ∈ h.index) :\n    MeasurableSet (h.covering p) :=\n  v.measurableSet p.1 _ (h.covering_mem_family hp)\n\ntheorem measure_le_tsum_of_absolutelyContinuous [SecondCountableTopology X] {ρ : Measure X}\n    (hρ : ρ ≪ μ) : ρ s ≤ ∑' p : h.index, ρ (h.covering p) :=\n  calc\n    ρ s ≤ ρ ((s \\ ⋃ p ∈ h.index, h.covering p) ∪ ⋃ p ∈ h.index, h.covering p) :=\n      measure_mono (by simp only [subset_union_left, sdiff_union_self])\n    _ ≤ ρ (s \\ ⋃ p ∈ h.index, h.covering p) + ρ (⋃ p ∈ h.index, h.covering p) :=\n      (measure_union_le _ _)\n    _ = ∑' p : h.index, ρ (h.covering p) := by\n      rw [hρ h.measure_sdiff_biUnion, zero_add,\n        measure_biUnion h.index_countable h.covering_disjoint fun x hx => h.measurableSet_u hx]\n\ntheorem measure_le_tsum [SecondCountableTopology X] : μ s ≤ ∑' x : h.index, μ (h.covering x) :=\n  h.measure_le_tsum_of_absolutelyContinuous Measure.AbsolutelyContinuous.rfl\n\nend FineSubfamilyOn\n\n/-- One can enlarge a Vitali family by adding to the sets `f x` at `x` all sets which are not\ncontained in a `δ`-neighborhood on `x`. This does not change the local filter at a point, but it\ncan be convenient to get a nicer global behavior. -/\ndef enlarge (v : VitaliFamily μ) (δ : ℝ) (δpos : 0 < δ) : VitaliFamily μ where\n  setsAt x := v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}\n  measurableSet := by\n    rintro x s (hs | hs)\n    exacts [v.measurableSet _ _ hs, hs.1]\n  nonempty_interior := by\n    rintro x s (hs | hs)\n    exacts [v.nonempty_interior _ _ hs, hs.2.1]\n  nontrivial := by\n    intro x ε εpos\n    rcases v.nontrivial x ε εpos with ⟨s, hs, h's⟩\n    exact ⟨s, mem_union_left _ hs, h's⟩\n  covering := by\n    intro s f fset ffine\n    let g : X → Set (Set X) := fun x => f x ∩ v.setsAt x\n    have : ∀ x ∈ s, ∀ ε : ℝ, ε > 0 → ∃ t ∈ g x, t ⊆ closedBall x ε := by\n      intro x hx ε εpos\n      obtain ⟨t, tf, ht⟩ : ∃ t ∈ f x, t ⊆ closedBall x (min ε δ) :=\n        ffine x hx (min ε δ) (lt_min εpos δpos)\n      rcases fset x hx tf with (h't | h't)\n      · exact ⟨t, ⟨tf, h't⟩, ht.trans (closedBall_subset_closedBall (min_le_left _ _))⟩\n      · refine False.elim (h't.2.2 ?_)\n        exact ht.trans (closedBall_subset_closedBall (min_le_right _ _))\n    rcases v.covering s g (fun x _ => inter_subset_right) this with ⟨t, ts, tdisj, tg, μt⟩\n    exact ⟨t, ts, tdisj, fun p hp => (tg p hp).1, μt⟩\n\nvariable (v : VitaliFamily μ)\n\n/-- Given a vitali family `v`, then `v.filterAt x` is the filter on `Set X` made of those families\nthat contain all sets of `v.setsAt x` of a sufficiently small diameter. This filter makes it\npossible to express limiting behavior when sets in `v.setsAt x` shrink to `x`. -/\ndef filterAt (x : X) : Filter (Set X) := (𝓝 x).smallSets ⊓ 𝓟 (v.setsAt x)\n\ntheorem _root_.Filter.HasBasis.vitaliFamily {ι : Sort*} {p : ι → Prop} {s : ι → Set X} {x : X}\n    (h : (𝓝 x).HasBasis p s) : (v.filterAt x).HasBasis p (fun i ↦ {t ∈ v.setsAt x | t ⊆ s i}) := by\n  simpa only [← Set.setOf_inter_eq_sep] using! h.smallSets.inf_principal _\n\ntheorem filterAt_basis_closedBall (x : X) :\n    (v.filterAt x).HasBasis (0 < ·) ({t ∈ v.setsAt x | t ⊆ closedBall x ·}) :=\n  nhds_basis_closedBall.vitaliFamily v\n\nTarget:\ntheorem mem_filterAt_iff {x : X} {s : Set (Set X)} :\n    s ∈ v.filterAt x ↔ ∃ ε > (0 : ℝ), ∀ t ∈ v.setsAt x, t ⊆ closedBall x ε → t ∈ s :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_419897952885","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c94b0565bfb1a82c34506202c469b89ae8173cffa5a6f678f3dcc56b3386fc18","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Covering","family_id":"mem_filterat_iff","file_id":"mathlib/Mathlib/MeasureTheory/Covering/VitaliFamily.lean","sample_id":"419897952885489eef42100d7198d985d0ed456c5f3c123051bf15c5b6fac476"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4a41798036ff76446dc750a37ef284895b5d38963c6bb7b381d67845e6108343","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"564b84906f8963f8e9e5daabed9de260d2399deb247e9144540e30ecf6f1c0bd","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"da422d22caf26f3e5568440ff4f4ce39cf647f59520cb7416b45b8017ec51f47","source_sha256":"aa1352e2143e704eafbedf6a7be68b6abc2920999f9fb6e7acf0e05da957ef08","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : JacobsonSpace ↥X := LocallyOfFiniteType.jacobsonSpace (f ≫ i)\n  refine ext_of_fromSpecResidueField_eq f g i (S ∩ closedPoints X) ?_ ?_ H'\n  · rwa [dense_iff_closure_eq, JacobsonSpace.closure_inter_closedPoints_eq_closure hS,\n      ← dense_iff_closure_eq]\n  · intro x ⟨hxS, hx⟩\n    rw [← cancel_epi (Spec.map (residueFieldIsoBase (f ≫ i) x hx).hom)]\n    refine ext_of_apply_closedPoint_eq i ?_ ?_ (by simpa using H x hxS hx) <;>\n      simp only [Category.assoc, ← SpecMap_residueFieldIsoBase_inv (f ≫ i) x hx, ← Spec.map_comp,\n        Iso.inv_hom_id, Spec.map_id, ← H']","hard_negative":true,"metrics":{"chosen_tokens":126,"rejected_tokens":3,"token_jaccard":0.015873,"token_length_ratio":0.02381},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"7cdd831064333d1d904deb43e69d1278fab84be0963d04b5a20ce8472d0ad07f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.Finite\npublic import Mathlib.FieldTheory.IsAlgClosed.Basic\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2026 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Schemes over algebraically closed fields\n\nWe show that if `X` is locally of finite type over an algebraically closed field `k`,\nthen the closed points of `X` are in bijection with the `k`-points of `X`.\nSee `AlgebraicGeometry.pointEquivClosedPoint`.\n\n-/\n\n@[expose] public noncomputable section\n\nopen CategoryTheory\n\nnamespace AlgebraicGeometry\n\nuniverse u\n\nvariable {X Y : Scheme.{u}} {K : Type u} [Field K] [IsAlgClosed K]\n    (f : X ⟶ Spec (.of K)) [LocallyOfFiniteType f] (x : X) (hx : IsClosed {x})\n\n/-- If `X` is a locally of finite type `k`-scheme and `k` is algebraically closed, then\nthe residue field of any closed point of `x` is isomorphic to `k`. -/\ndef residueFieldIsoBase : X.residueField x ≅ .of K :=\n  letI : IsIso (Spec.preimage (X.fromSpecResidueField x ≫ f)) := by\n    have : IsFinite (X.fromSpecResidueField x ≫ f) := by\n      rw [isClosed_singleton_iff_isClosedImmersion] at hx\n      rw [isFinite_iff_locallyOfFiniteType_of_jacobsonSpace]\n      infer_instance\n    rw [ConcreteCategory.isIso_iff_bijective]\n    refine IsAlgClosed.ringHom_bijective_of_isIntegral _ ?_\n    rw [← IsIntegralHom.SpecMap_iff, Spec.map_preimage]\n    infer_instance\n  (asIso (Spec.preimage (X.fromSpecResidueField x ≫ f))).symm\n\n@[simp, reassoc]\nlemma SpecMap_residueFieldIsoBase_inv :\n    Spec.map (residueFieldIsoBase f x hx).inv = X.fromSpecResidueField x ≫ f :=\n  Spec.map_preimage _\n\n/-- If `k` is algebraically closed, this is the `k`-point of `X` associated to a closed point. -/\nnoncomputable\ndef pointOfClosedPoint : Spec (.of K) ⟶ X :=\n  Spec.map (residueFieldIsoBase f x hx).hom ≫ X.fromSpecResidueField x\n\n@[reassoc (attr := simp)]\nlemma pointOfClosedPoint_comp : pointOfClosedPoint f x hx ≫ f = 𝟙 _ := by\n  simp [pointOfClosedPoint, ← SpecMap_residueFieldIsoBase_inv, ← Spec.map_comp]\n\n@[simp]\nlemma pointOfClosedPoint_apply (a : _) : pointOfClosedPoint f x hx a = x := by\n  simp [pointOfClosedPoint]\n\nset_option backward.defeqAttrib.useBackward true in\n/-- If `k` is algebraically closed,\nthen the closed points of `X` are in bijection with the `k`-points of `X`. -/\n@[simps]\ndef pointEquivClosedPoint :\n    {p : Spec (.of K) ⟶ X // p ≫ f = 𝟙 _} ≃ closedPoints X where\n  toFun p := ⟨p.1 (IsLocalRing.closedPoint K), by\n    have := isClosedImmersion_of_comp_eq_id _ _ p.2\n    have := p.1.isClosedEmbedding.isClosed_range\n    rwa [Set.range_eq_singleton] at this\n    exact fun x ↦ congr(p.1 $(Subsingleton.elim _ _))⟩\n  invFun x := ⟨pointOfClosedPoint f x.1 x.2, pointOfClosedPoint_comp f x.1 x.2⟩\n  left_inv p := by\n    ext\n    refine ((Scheme.SpecToEquivOfField _ _).symm_apply_eq (x := ⟨_, _⟩)).mpr ?_\n    rw [Scheme.SpecToEquivOfField_eq_iff]\n    dsimp [Scheme.SpecToEquivOfField]\n    simp only [Category.id_comp, exists_const]\n    generalize_proofs _ h\n    refine (Category.comp_id _).symm.trans (((residueFieldIsoBase f _ h).eq_inv_comp).mp ?_)\n    rw [← Spec.map_injective.eq_iff]\n    simp only [Spec.map_id, Spec.map_comp, SpecMap_residueFieldIsoBase_inv]\n    rw [reassoc_of% Scheme.descResidueField_stalkClosedPointTo_fromSpecResidueField, p.2]\n  right_inv x := by simp\n\nlemma ext_of_apply_closedPoint_eq\n    {f g : Spec (.of K) ⟶ X} (h : X ⟶ Spec (.of K))\n    [LocallyOfFiniteType h]\n    (hf : f ≫ h = 𝟙 _) (hg : g ≫ h = 𝟙 _)\n    (H : f (IsLocalRing.closedPoint K) = g (IsLocalRing.closedPoint K)) : f = g :=\n  congr($((pointEquivClosedPoint h).injective (a₁ := ⟨f, hf⟩) (a₂ := ⟨g, hg⟩) (Subtype.ext H)).1)\n\n/-- Let `X` and `Y` be locally of finite type `K`-schemes with `K` algebraically closed and `Y`\nseparated over `K`. Suppose `X` is reduced, then two `K`-morphisms `f g : X ⟶ Y` are equal if\nthey are equal on the closed points of a dense locally closed subset of `X`. -/\n\nTarget:\nlemma ext_of_apply_eq {f g : X ⟶ Y} (i : Y ⟶ Spec (.of K)) [IsSeparated i] [LocallyOfFiniteType i]\n    [IsReduced X] [LocallyOfFiniteType (f ≫ i)]\n    (S : Set X) (hS : IsLocallyClosed S) (hS' : Dense S)\n    (H : ∀ x ∈ S, IsClosed {x} → f x = g x)\n    (H' : f ≫ i = g ≫ i) : f = g :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_da422d22caf2","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"0587da8e3db98e7a7c6a5712f673ed109d6b2e894c25a4028d373e091cba4795","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/AlgClosed","family_id":"ext_of_apply_eq","file_id":"mathlib/Mathlib/AlgebraicGeometry/AlgClosed/Basic.lean","sample_id":"da422d22caf26f3e5568440ff4f4ce39cf647f59520cb7416b45b8017ec51f47"},"split":"train"}
{"binding":{"chosen_proof_sha256":"048280cfb8f07df3e6d2a0a1b3323465e6476c927989956effb4e1caeb51283e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c32bfd18b1eacc9948f2069f737c203f16059abd126861ca53c4f4fff6855765","source_sha256":"d1ba22e22a26245161ec2703bf29aa4e6e9c0ade6d5458aee3e70435922bd387","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨x, w₁, w₂⟩ := x\n  obtain ⟨⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩, -⟩ := h\n  simp [Prod.toFun', Prod.invFun', h₁, h₂]","hard_negative":false,"metrics":{"chosen_tokens":57,"rejected_tokens":2,"token_jaccard":0.041667,"token_length_ratio":0.035088},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"7ce0fc1c53a078a52ab51fc2a4c4cd0fed623635f03401657ad90c211345df81","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.FiberBundle.Basic\n\nNamespace:\nBundle.Trivialization\n\nLocal context:\n/-\nCopyright (c) 2022 Nicolò Cavalleri. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn\n-/\n/-!\n# Standard constructions on fiber bundles\n\nThis file contains several standard constructions on fiber bundles:\n\n* `Bundle.Trivial.fiberBundle 𝕜 B F`: the trivial fiber bundle with model fiber `F` over the base\n  `B`\n\n* `FiberBundle.prod`: for fiber bundles `E₁` and `E₂` over a common base, a fiber bundle structure\n  on their fiberwise product `E₁ ×ᵇ E₂` (the notation stands for `fun x ↦ E₁ x × E₂ x`).\n\n* `FiberBundle.pullback`: for a fiber bundle `E` over `B`, a fiber bundle structure on its\n  pullback `f *ᵖ E` by a map `f : B' → B` (the notation is a type synonym for `E ∘ f`).\n\n## Tags\n\nfiber bundle, fibre bundle, fiberwise product, pullback\n\n-/\n\n@[expose] public section\n\nopen Bundle Filter Set TopologicalSpace Topology\n\n/-! ### The trivial bundle -/\n\nnamespace Bundle\n\nnamespace Trivial\n\nvariable (B : Type*) (F : Type*)\n\n-- TODO: use `TotalSpace.toProd`\ninstance topologicalSpace [t₁ : TopologicalSpace B]\n    [t₂ : TopologicalSpace F] : TopologicalSpace (TotalSpace F (Trivial B F)) :=\n  induced TotalSpace.proj t₁ ⊓ induced (TotalSpace.trivialSnd B F) t₂\n\nvariable [TopologicalSpace B] [TopologicalSpace F]\n\ntheorem isInducing_toProd : IsInducing (TotalSpace.toProd B F) :=\n  ⟨by simp only [instTopologicalSpaceProd, induced_inf, induced_compose]; rfl⟩\n\n/-- Homeomorphism between the total space of the trivial bundle and the Cartesian product. -/\n@[simps!]\ndef homeomorphProd : TotalSpace F (Trivial B F) ≃ₜ B × F :=\n  (TotalSpace.toProd _ _).toHomeomorphOfIsInducing (isInducing_toProd B F)\n\n/-- Local trivialization for trivial bundle. -/\n@[simps!]\ndef trivialization : Trivialization F (π F (Bundle.Trivial B F)) where\n  toOpenPartialHomeomorph := (homeomorphProd B F).toOpenPartialHomeomorph\n  baseSet := univ\n  open_baseSet := isOpen_univ\n  source_eq := rfl\n  target_eq := univ_prod_univ.symm\n  proj_toFun _ _ := rfl\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp] lemma trivialization_symm_apply [Zero F] (b : B) (f : F) :\n    (trivialization B F).symm b f = f := by\n  simp [trivialization, homeomorphProd, TotalSpace.toProd, Trivialization.symm,\n    Pretrivialization.symm, Trivialization.toPretrivialization]\n\n@[simp] lemma toOpenPartialHomeomorph_trivialization_symm_apply (v : B × F) :\n    (trivialization B F).toOpenPartialHomeomorph.symm v = ⟨v.1, v.2⟩ := rfl\n\n/-- Fiber bundle instance on the trivial bundle. -/\n@[simps] instance fiberBundle : FiberBundle F (Bundle.Trivial B F) where\n  trivializationAtlas' := {trivialization B F}\n  trivializationAt' _ := trivialization B F\n  mem_baseSet_trivializationAt' := mem_univ\n  trivialization_mem_atlas' _ := mem_singleton _\n  totalSpaceMk_isInducing' _ := (homeomorphProd B F).symm.isInducing.comp\n    (isInducing_const_prod.2 .id)\n\ntheorem eq_trivialization (e : Trivialization F (π F (Bundle.Trivial B F)))\n    [i : MemTrivializationAtlas e] : e = trivialization B F := i.out\n\nend Trivial\n\nend Bundle\n\n/-! ### Fibrewise product of two bundles -/\n\n\nsection Prod\n\nvariable {B : Type*}\n\nsection Defs\n\nvariable (F₁ : Type*) (E₁ : B → Type*) (F₂ : Type*) (E₂ : B → Type*)\nvariable [TopologicalSpace (TotalSpace F₁ E₁)] [TopologicalSpace (TotalSpace F₂ E₂)]\n\n/-- Equip the total space of the fiberwise product of two fiber bundles `E₁`, `E₂` with\nthe induced topology from the diagonal embedding into `TotalSpace F₁ E₁ × TotalSpace F₂ E₂`. -/\ninstance FiberBundle.Prod.topologicalSpace : TopologicalSpace (TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂)) :=\n  TopologicalSpace.induced\n    (fun p ↦ ((⟨p.1, p.2.1⟩ : TotalSpace F₁ E₁), (⟨p.1, p.2.2⟩ : TotalSpace F₂ E₂)))\n    inferInstance\n\n/-- The diagonal map from the total space of the fiberwise product of two fiber bundles\n`E₁`, `E₂` into `TotalSpace F₁ E₁ × TotalSpace F₂ E₂` is an inducing map. -/\ntheorem FiberBundle.Prod.isInducing_diag :\n    IsInducing (fun p ↦ (⟨p.1, p.2.1⟩, ⟨p.1, p.2.2⟩) :\n      TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂) :=\n  ⟨rfl⟩\n\nend Defs\n\nopen FiberBundle\n\nvariable [TopologicalSpace B] (F₁ : Type*) [TopologicalSpace F₁] (E₁ : B → Type*)\n  [TopologicalSpace (TotalSpace F₁ E₁)] (F₂ : Type*) [TopologicalSpace F₂] (E₂ : B → Type*)\n  [TopologicalSpace (TotalSpace F₂ E₂)]\n\nnamespace Bundle.Trivialization\n\nvariable {F₁ E₁ F₂ E₂}\nvariable (e₁ : Trivialization F₁ (π F₁ E₁)) (e₂ : Trivialization F₂ (π F₂ E₂))\n\n/-- Given trivializations `e₁`, `e₂` for fiber bundles `E₁`, `E₂` over a base `B`, the forward\nfunction for the construction `Trivialization.prod`, the induced\ntrivialization for the fiberwise product of `E₁` and `E₂`. -/\ndef Prod.toFun' : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) → B × F₁ × F₂ :=\n  fun p ↦ ⟨p.1, (e₁ ⟨p.1, p.2.1⟩).2, (e₂ ⟨p.1, p.2.2⟩).2⟩\n\nvariable {e₁ e₂}\n\ntheorem Prod.continuous_to_fun : ContinuousOn (Prod.toFun' e₁ e₂)\n    (π (F₁ × F₂) (E₁ ×ᵇ E₂) ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) := by\n  let f₁ : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ :=\n    fun p ↦ ((⟨p.1, p.2.1⟩ : TotalSpace F₁ E₁), (⟨p.1, p.2.2⟩ : TotalSpace F₂ E₂))\n  let f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p ↦ ⟨e₁ p.1, e₂ p.2⟩\n  let f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p ↦ ⟨p.1.1, p.1.2, p.2.2⟩\n  have hf₁ : Continuous f₁ := (Prod.isInducing_diag F₁ E₁ F₂ E₂).continuous\n  have hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) :=\n    e₁.toOpenPartialHomeomorph.continuousOn.prodMap e₂.toOpenPartialHomeomorph.continuousOn\n  have hf₃ : Continuous f₃ := by fun_prop\n  refine ((hf₃.comp_continuousOn hf₂).comp hf₁.continuousOn ?_).congr ?_\n  · rw [e₁.source_eq, e₂.source_eq]\n    exact mapsTo_preimage _ _\n  rintro ⟨b, v₁, v₂⟩ ⟨hb₁, _⟩\n  simp only [f₁, f₂, f₃, Prod.toFun', Prod.mk_inj, Function.comp_apply, and_true]\n  rw [e₁.coe_fst]\n  rw [e₁.source_eq, mem_preimage]\n  exact hb₁\n\nvariable (e₁ e₂) [∀ x, Zero (E₁ x)] [∀ x, Zero (E₂ x)]\n\n/-- Given trivializations `e₁`, `e₂` for fiber bundles `E₁`, `E₂` over a base `B`, the inverse\nfunction for the construction `Trivialization.prod`, the induced\ntrivialization for the fiberwise product of `E₁` and `E₂`. -/\nnoncomputable def Prod.invFun' (p : B × F₁ × F₂) : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) :=\n  ⟨p.1, e₁.symm p.1 p.2.1, e₂.symm p.1 p.2.2⟩\n\nvariable {e₁ e₂}\n\ntheorem Prod.left_inv {x : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂)}\n    (h : x ∈ π (F₁ × F₂) (E₁ ×ᵇ E₂) ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) :\n    Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x := by\n  obtain ⟨x, v₁, v₂⟩ := x\n  obtain ⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩ := h\n  simp [Prod.toFun', Prod.invFun', h₁, h₂]\n\nTarget:\ntheorem Prod.right_inv {x : B × F₁ × F₂}\n    (h : x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ (univ : Set (F₁ × F₂))) :\n    Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/FiberBundle","family_id":"prod","file_id":"mathlib/Mathlib/Topology/FiberBundle/Constructions.lean","sample_id":"c32bfd18b1eacc9948f2069f737c203f16059abd126861ca53c4f4fff6855765"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b82467ef621f5a21aaae8eaa62dc55df65f51c9c9619a918a2bac8a08b7ed281","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d0d0f38b9e3e428b2ae38db46c5384471141fd719dfd1fd904da355d5cb234fc","source_sha256":"b5b8fef319876fc584704ec0902863ab8016fe0ba4dee5dd358f86f4688b13cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  delta Irrational\n  contrapose!\n  rintro ⟨⟨rx, rfl⟩, ⟨ry, rfl⟩⟩\n  exact ⟨rx + ry, cast_add rx ry⟩","hard_negative":true,"metrics":{"chosen_tokens":29,"rejected_tokens":8,"token_jaccard":0.047619,"token_length_ratio":0.275862},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"7d1bf5685297cc486c32cac4762f20a9fa7f8a034a7081d38e7fa26a9e789427","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Rat\npublic import Mathlib.Data.Nat.Prime.Int\npublic import Mathlib.Data.Rat.Sqrt\npublic import Mathlib.Analysis.Real.Sqrt\npublic import Mathlib.RingTheory.Algebraic.Basic\npublic import Mathlib.Tactic.CrossRefAttribute\npublic import Mathlib.Tactic.IntervalCases\n\nNamespace:\nIrrational\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov\n-/\n/-!\n# Irrational real numbers\n\nIn this file we define a predicate `Irrational` on `ℝ`, prove that the `n`-th root of an integer\nnumber is irrational if it is not integer, and that `√(q : ℚ)` is irrational if and only if\n`¬IsSquare q ∧ 0 ≤ q`.\n\nWe also provide dot-style constructors like `Irrational.add_ratCast`, `Irrational.ratCast_sub` etc.\n\nWith the `Decidable` instances in this file, is possible to prove `Irrational √n` using `decide`,\nwhen `n` is a numeric literal or cast;\nbut this only works if you `unseal Nat.sqrt.iter in` before the theorem where you use this proof.\n-/\n\n@[expose] public section\n\n\nopen Rat Real\n\n/-- A real number is irrational if it is not equal to any rational number. -/\n@[wikidata Q607728]\ndef Irrational (x : ℝ) :=\n  x ∉ Set.range ((↑) : ℚ → ℝ)\n\ntheorem irrational_iff_ne_rational (x : ℝ) : Irrational x ↔ ∀ a b : ℤ, b ≠ 0 → x ≠ a / b := by\n  simp [Irrational, Rat.forall, eq_comm]\n\ntheorem Irrational.ne_rational {x : ℝ} (hx : Irrational x) (a b : ℤ) : x ≠ a / b := by\n  rintro rfl; exact hx ⟨a / b, by simp⟩\n\ntheorem exists_rat_of_not_irrational {x : ℝ} (hx : ¬ Irrational x) : ∃ (q : ℚ), x = q := by\n  grind [Irrational]\n\n/-- A transcendental real number is irrational. -/\ntheorem Transcendental.irrational {r : ℝ} (tr : Transcendental ℚ r) : Irrational r := by\n  rintro ⟨a, rfl⟩\n  exact tr (isAlgebraic_algebraMap a)\n\n/-!\n### Irrationality of roots of integer and rational numbers\n-/\n\n\n/-- If `x^n`, `n > 0`, is integer and is not the `n`-th power of an integer, then\n`x` is irrational. -/\ntheorem irrational_nrt_of_notint_nrt {x : ℝ} (n : ℕ) (m : ℤ) (hxr : x ^ n = m)\n    (hv : ¬∃ y : ℤ, x = y) (hnpos : 0 < n) : Irrational x := by\n  rintro ⟨⟨N, D, P, C⟩, rfl⟩\n  rw [← cast_pow] at hxr\n  have c1 : ((D : ℤ) : ℝ) ≠ 0 := by\n    rw [Int.cast_ne_zero, Int.natCast_ne_zero]\n    exact P\n  have c2 : ((D : ℤ) : ℝ) ^ n ≠ 0 := pow_ne_zero _ c1\n  rw [mk_eq_divInt, cast_pow, cast_divInt, div_pow, div_eq_iff_mul_eq c2, ← Int.cast_pow,\n    ← Int.cast_pow, ← Int.cast_mul, Int.cast_inj] at hxr\n  have hdivn : (D : ℤ) ^ n ∣ N ^ n := Dvd.intro_left m hxr\n  rw [← Int.dvd_natAbs, ← Int.natCast_pow, Int.natCast_dvd_natCast, Int.natAbs_pow,\n    Nat.pow_dvd_pow_iff hnpos.ne'] at hdivn\n  obtain rfl : D = 1 := by rw [← Nat.gcd_eq_right hdivn, C.gcd_eq_one]\n  refine hv ⟨N, ?_⟩\n  rw [mk_eq_divInt, Int.ofNat_one, divInt_one, cast_intCast]\n\n/-- If `x^n = m` is an integer and `n` does not divide the `multiplicity p m`, then `x`\nis irrational. -/\ntheorem irrational_nrt_of_n_not_dvd_multiplicity {x : ℝ} (n : ℕ) {m : ℤ} (hm : m ≠ 0) (p : ℕ)\n    [hp : Fact p.Prime] (hxr : x ^ n = m)\n    (hv : multiplicity (p : ℤ) m % n ≠ 0) :\n    Irrational x := by\n  rcases Nat.eq_zero_or_pos n with (rfl | hnpos)\n  · rw [eq_comm, pow_zero, ← Int.cast_one, Int.cast_inj] at hxr\n    simp [hxr, multiplicity_of_one_right (mt isUnit_iff_dvd_one.1\n      (mt Int.natCast_dvd_natCast.1 hp.1.not_dvd_one))] at hv\n  refine irrational_nrt_of_notint_nrt _ _ hxr ?_ hnpos\n  rintro ⟨y, rfl⟩\n  rw [← Int.cast_pow, Int.cast_inj] at hxr\n  subst m\n  have : y ≠ 0 := by rintro rfl; rw [zero_pow hnpos.ne'] at hm; exact hm rfl\n  rw [(Int.finiteMultiplicity_iff.2 ⟨by simp [hp.1.ne_one], this⟩).multiplicity_pow\n    (Nat.prime_iff_prime_int.1 hp.1), Nat.mul_mod_right] at hv\n  exact hv rfl\n\ntheorem irrational_sqrt_of_multiplicity_odd (m : ℤ) (hm : 0 < m) (p : ℕ) [hp : Fact p.Prime]\n    (Hpv : multiplicity (p : ℤ) m % 2 = 1) :\n    Irrational (√m) :=\n  @irrational_nrt_of_n_not_dvd_multiplicity _ 2 _ (Ne.symm (ne_of_lt hm)) p hp\n    (sq_sqrt (Int.cast_nonneg hm.le)) (by rw [Hpv]; exact one_ne_zero)\n\n@[simp] theorem not_irrational_zero : ¬Irrational 0 := not_not_intro ⟨0, Rat.cast_zero⟩\n@[simp] theorem not_irrational_one : ¬Irrational 1 := not_not_intro ⟨1, Rat.cast_one⟩\n\ntheorem irrational_sqrt_ratCast_iff_of_nonneg {q : ℚ} (hq : 0 ≤ q) :\n    Irrational (√q) ↔ ¬IsSquare q := by\n  refine Iff.not (?_ : Exists _ ↔ Exists _)\n  constructor\n  · rintro ⟨y, hy⟩\n    refine ⟨y, Rat.cast_injective (α := ℝ) ?_⟩\n    rw [Rat.cast_mul, hy, mul_self_sqrt (Rat.cast_nonneg.2 hq)]\n  · rintro ⟨q', rfl⟩\n    exact ⟨|q'|, mod_cast (sqrt_mul_self_eq_abs q').symm⟩\n\ntheorem irrational_sqrt_ratCast_iff {q : ℚ} :\n    Irrational (√q) ↔ ¬IsSquare q ∧ 0 ≤ q := by\n  obtain hq | hq := le_or_gt 0 q\n  · simp_rw [irrational_sqrt_ratCast_iff_of_nonneg hq, and_iff_left hq]\n  · rw [sqrt_eq_zero_of_nonpos (Rat.cast_nonpos.2 hq.le)]\n    simp_rw [not_irrational_zero, false_iff, not_and, not_le, hq, implies_true]\n\ntheorem irrational_sqrt_intCast_iff_of_nonneg {z : ℤ} (hz : 0 ≤ z) :\n    Irrational (√z) ↔ ¬IsSquare z := by\n  rw [← Rat.isSquare_intCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg (mod_cast hz),\n    Rat.cast_intCast]\n\ntheorem irrational_sqrt_intCast_iff {z : ℤ} :\n    Irrational (√z) ↔ ¬IsSquare z ∧ 0 ≤ z := by\n  rw [← Rat.cast_intCast, irrational_sqrt_ratCast_iff, Rat.isSquare_intCast_iff,\n    Int.cast_nonneg_iff]\n\ntheorem irrational_sqrt_natCast_iff {n : ℕ} : Irrational (√n) ↔ ¬IsSquare n := by\n  rw [← Rat.isSquare_natCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg n.cast_nonneg,\n    Rat.cast_natCast]\n\ntheorem irrational_sqrt_ofNat_iff {n : ℕ} [n.AtLeastTwo] :\n    Irrational √(ofNat(n)) ↔ ¬IsSquare ofNat(n) :=\n  irrational_sqrt_natCast_iff\n\ntheorem Nat.Prime.irrational_sqrt {p : ℕ} (hp : Nat.Prime p) : Irrational (√p) :=\n  irrational_sqrt_natCast_iff.mpr hp.not_isSquare\n\n/-- **Irrationality of the Square Root of 2** -/\ntheorem irrational_sqrt_two : Irrational (√2) := by\n  simpa using Nat.prime_two.irrational_sqrt\n\n/--\nThis can be used as\n```lean\nunseal Nat.sqrt.iter in\nexample : Irrational √24 := by decide\n```\n-/\ninstance {n : ℕ} [n.AtLeastTwo] : Decidable (Irrational √(ofNat(n))) :=\n  decidable_of_iff' _ irrational_sqrt_ofNat_iff\n\ninstance (n : ℕ) : Decidable (Irrational (√n)) :=\n  decidable_of_iff' _ irrational_sqrt_natCast_iff\n\ninstance (z : ℤ) : Decidable (Irrational (√z)) :=\n  decidable_of_iff' _ irrational_sqrt_intCast_iff\n\ninstance (q : ℚ) : Decidable (Irrational (√q)) :=\n  decidable_of_iff' _ irrational_sqrt_ratCast_iff\n\n/-!\n### Dot-style operations on `Irrational`\n\n#### Coercion of a rational/integer/natural number is not irrational\n-/\n\n\nnamespace Irrational\n\nvariable {x : ℝ}\n\n/-!\n#### Irrational number is not equal to a rational/integer/natural number\n-/\n\n\ntheorem ne_rat (h : Irrational x) (q : ℚ) : x ≠ q := fun hq => h ⟨q, hq.symm⟩\n\ntheorem ne_int (h : Irrational x) (m : ℤ) : x ≠ m := by\n  rw [← Rat.cast_intCast]\n  exact h.ne_rat _\n\ntheorem ne_nat (h : Irrational x) (m : ℕ) : x ≠ m :=\n  h.ne_int m\n\ntheorem ne_zero (h : Irrational x) : x ≠ 0 := mod_cast h.ne_nat 0\n\ntheorem ne_one (h : Irrational x) : x ≠ 1 := by simpa only [Nat.cast_one] using h.ne_nat 1\n\n@[simp] theorem ne_ofNat (h : Irrational x) (n : ℕ) [n.AtLeastTwo] : x ≠ ofNat(n) :=\n  h.ne_nat n\n\nend Irrational\n\n@[simp]\ntheorem Rat.not_irrational (q : ℚ) : ¬Irrational q := fun h => h ⟨q, rfl⟩\n\n@[simp]\ntheorem Int.not_irrational (m : ℤ) : ¬Irrational m := fun h => h.ne_int m rfl\n\n@[simp]\ntheorem Nat.not_irrational (m : ℕ) : ¬Irrational m := fun h => h.ne_nat m rfl\n\n@[simp] theorem not_irrational_ofNat (n : ℕ) [n.AtLeastTwo] : ¬Irrational ofNat(n) :=\n  n.not_irrational\nnamespace Irrational\n\nvariable (q : ℚ) {x y : ℝ}\n\n/-!\n#### Addition of rational/integer/natural numbers\n-/\n\n\n/-- If `x + y` is irrational, then at least one of `x` and `y` is irrational. -/\n\nTarget:\ntheorem add_cases : Irrational (x + y) → Irrational x ∨ Irrational y :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"fe8a752cc83edb72233070545843b2739325238a18eb5dede21517407ec6ca46","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Real","family_id":"add_cases","file_id":"mathlib/Mathlib/NumberTheory/Real/Irrational.lean","sample_id":"d0d0f38b9e3e428b2ae38db46c5384471141fd719dfd1fd904da355d5cb234fc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b9e374e8c54a58f587ed34013c46e429e34757a52725bc1a7e7b25ce0f218592","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5b180b286925047b6265b1d6b6387babba0a99b025076644a75ae77bbbe0b4d6","source_sha256":"9b749e1f06681e2aad37931b0d304960c8762eef95a6f961195bf175c696c109","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply injective_of_mono (X.mapIso (sigmaIso f).op).hom\n  apply injective_of_mono (sigmaComparison X (fun a ↦ (fiber f a).1))\n  ext a\n  specialize h a\n  rw [← sigmaComparison_comp_sigmaIso] at h\n  exact h","hard_negative":false,"metrics":{"chosen_tokens":48,"rejected_tokens":2,"token_jaccard":0.034483,"token_length_ratio":0.041667},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"7d650e4065a0065272be273a5187b4539fb5779b8660d31e01bd30e3137f47b6","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Condensed.Discrete.Basic\npublic import Mathlib.Condensed.TopComparison\npublic import Mathlib.Topology.Category.CompHausLike.SigmaComparison\npublic import Mathlib.Topology.FiberPartition\n\nNamespace:\nCompHausLike.LocallyConstant\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# The sheaf of locally constant maps on `CompHausLike P`\n\nThis file proves that under suitable conditions, the functor from the category of sets to the\ncategory of sheaves for the coherent topology on `CompHausLike P`, given by mapping a set to the\nsheaf of locally constant maps to it, is left adjoint to the \"underlying set\" functor (evaluation\nat the point).\n\nWe apply this to prove that the constant sheaf functor into (light) condensed sets is isomorphic to\nthe functor of sheaves of locally constant maps described above.\n\n## Proof sketch\n\nThe hard part of this adjunction is to define the counit. Its components are defined as follows:\n\nLet `S : CompHausLike P` and let `Y` be a finite-product-preserving presheaf on `CompHausLike P`\n(e.g. a sheaf for the coherent topology). We need to define a map `LocallyConstant S Y(*) ⟶ Y(S)`.\nGiven a locally constant map `f : S → Y(*)`, let `S = S₁ ⊔ ⋯ ⊔ Sₙ` be the corresponding\ndecomposition of `S` into the fibers. Let `yᵢ ∈ Y(*)` denote the value of `f` on `Sᵢ` and denote\nby `gᵢ` the canonical map `Y(*) → Y(Sᵢ)`. Our map then takes `f` to the image of\n`(g₁(y₁), ⋯, gₙ(yₙ))` under the isomorphism `Y(S₁) × ⋯ × Y(Sₙ) ≅ Y(S₁ ⊔ ⋯ ⊔ Sₙ) = Y(S)`.\n\nNow we need to prove that the counit is natural in `S : CompHausLike P` and\n`Y : Sheaf  (coherentTopology (CompHausLike P)) (Type _)`. There are two key lemmas in all\nnaturality proofs in this file (both lemmas are in the `CompHausLike.LocallyConstant` namespace):\n\n* `presheaf_ext`: given `S`, `Y` and `f : LocallyConstant S Y(*)` like above, another presheaf\n  `X`, and two elements `x y : X(S)`, to prove that `x = y` it suffices to prove that for every\n  inclusion map `ιᵢ : Sᵢ ⟶ S`, `X(ιᵢ)(x) = X(ιᵢ)(y)`.\n  Here it is important that we set everything up in such a way that the `Sᵢ` are literally subtypes\n  of `S`.\n\n* `incl_of_counitAppApp`: given  `S`, `Y` and `f : LocallyConstant S Y(*)` like above, we have\n  `Y(ιᵢ)(ε_{S, Y}(f)) = gᵢ(yᵢ)` where `ε` denotes the counit and the other notation is like above.\n\n## Main definitions\n\n* `CompHausLike.LocallyConstant.functor`: the functor from the category of sets to the category of\n  sheaves for the coherent topology on `CompHausLike P`, which takes a set `X` to\n  `LocallyConstant - X`\n  - `CondensedSet.LocallyConstant.functor` is the above functor in the case of condensed sets.\n  - `LightCondSet.LocallyConstant.functor` is the above functor in the case of light condensed sets.\n\n* `CompHausLike.LocallyConstant.adjunction`: the functor described above is left adjoint to the\n  \"underlying set\" functor `(sheafSections _ _).obj ⟨CompHausLike.of P PUnit.{u + 1}⟩`, which takes\n  a sheaf `X` to the set `X(*)`.\n\n* `CondensedSet.LocallyConstant.iso`: the functor `CondensedSet.LocallyConstant.functor` is\n  isomorphic to the functor `Condensed.discrete (Type _)` (the constant sheaf functor from sets to\n  condensed sets).\n\n* `LightCondSet.LocallyConstant.iso`: the functor `LightCondSet.LocallyConstant.functor` is\n  isomorphic to the functor `LightCondensed.discrete (Type _)` (the constant sheaf functor from sets\n  to light condensed sets).\n\n-/\n\n@[expose] public section\n\nuniverse u w\n\nopen CategoryTheory Limits LocallyConstant TopologicalSpace.Fiber Opposite Function Fiber\n\nvariable {P : TopCat.{u} → Prop}\n\nnamespace CompHausLike.LocallyConstant\n\n/--\nThe functor from the category of sets to presheaves on `CompHausLike P` given by locally constant\nmaps.\n-/\n@[simps obj_obj obj_map map_app]\ndef functorToPresheaves : Type (max u w) ⥤ ((CompHausLike.{u} P)ᵒᵖ ⥤ Type (max u w)) where\n  obj X := {\n    obj := fun ⟨S⟩ ↦ (LocallyConstant S X)\n    map f := ↾fun g ↦ g.comap f.unop.hom.hom }\n  map f := { app _ := ↾fun t ↦ t.map f }\n\n/--\nLocally constant maps are the same as continuous maps when the target is equipped with the discrete\ntopology\n-/\n@[simps]\ndef locallyConstantIsoContinuousMap (Y X : Type*) [TopologicalSpace Y] :\n    LocallyConstant Y X ≅ C(Y, TopCat.discrete.obj X) :=\n  letI : TopologicalSpace X := ⊥\n  haveI : DiscreteTopology X := ⟨rfl⟩\n  { hom := ↾fun f ↦ (f : C(Y, X))\n    inv := ↾fun f ↦ ⟨f, (IsLocallyConstant.iff_continuous f).mpr f.2⟩ }\n\nsection Adjunction\n\nvariable [∀ (S : CompHausLike.{u} P) (p : S → Prop), HasProp P (Subtype p)]\n\nsection\n\nvariable {Q : CompHausLike.{u} P} {Z : Type max u w} (r : LocallyConstant Q Z) (a : Fiber r)\n\n/-- A fiber of a locally constant map as a `CompHausLike P`. -/\nabbrev fiber : CompHausLike.{u} P := CompHausLike.of P a.val\n\n/-- The inclusion map from a component of the coproduct induced by `f` into `S`. -/\ndef sigmaIncl : fiber r a ⟶ Q := ofHom _ (TopologicalSpace.Fiber.sigmaIncl _ a)\n\n/-- The canonical map from the coproduct induced by `f` to `S` as an isomorphism in\n`CompHausLike P`. -/\nnoncomputable def sigmaIso [HasExplicitFiniteCoproducts.{u} P] : (finiteCoproduct (fiber r)) ≅ Q :=\n  isoOfBijective (ofHom _ (sigmaIsoHom r)) ⟨sigmaIsoHom_inj r, sigmaIsoHom_surj r⟩\n\nlemma sigmaComparison_comp_sigmaIso [HasExplicitFiniteCoproducts.{u} P]\n    (X : (CompHausLike.{u} P)ᵒᵖ ⥤ Type (max u w)) :\n    (X.mapIso (sigmaIso r).op).hom ≫ sigmaComparison X (fun a ↦ (fiber r a).1) ≫\n      (↾fun g ↦ g a) = X.map (sigmaIncl r a).op := by\n  ext\n  simp only [Functor.mapIso_hom, Iso.op_hom, sigmaComparison, TypeCat.Fun.toFun_apply,\n    CategoryTheory.comp_apply, ConcreteCategory.hom_ofHom, TypeCat.Fun.coe_mk,\n    ← X.map_comp_apply]\n  rfl\n\nend\n\nvariable {S : CompHausLike.{u} P} {Y : (CompHausLike.{u} P)ᵒᵖ ⥤ Type (max u w)}\n  [HasProp P PUnit.{u + 1}] (f : LocallyConstant S (Y.obj (op (CompHausLike.of P PUnit.{u + 1}))))\n\n/-- The projection of the counit. -/\nnoncomputable def counitAppAppImage : (a : Fiber f) → Y.obj ⟨fiber f a⟩ :=\n  fun a ↦ Y.map (CompHausLike.isTerminalPUnit.from _).op a.image\n\n/--\nThe counit is defined as follows: given a locally constant map `f : S → Y(*)`, let\n`S = S₁ ⊔ ⋯ ⊔ Sₙ` be the corresponding decomposition of `S` into the fibers. We need to provide an\nelement of `Y(S)`. It suffices to provide an element of `Y(Sᵢ)` for all `i`. Let `yᵢ ∈ Y(*)` denote\nthe value of `f` on `Sᵢ`. Our desired element is the image of `yᵢ` under the canonical map\n`Y(*) → Y(Sᵢ)`.\n-/\nnoncomputable def counitAppApp (S : CompHausLike.{u} P)\n    (Y : (CompHausLike.{u} P)ᵒᵖ ⥤ Type (max u w))\n    [PreservesFiniteProducts Y] [HasExplicitFiniteCoproducts.{u} P] :\n    LocallyConstant S (Y.obj (op (CompHausLike.of P PUnit.{u + 1}))) ⟶ Y.obj ⟨S⟩ :=\n  ↾fun r ↦ (inv (sigmaComparison Y (fun a ↦ (fiber r a).1)) ≫\n    (Y.mapIso (sigmaIso r).op).inv) (counitAppAppImage r)\n\n-- This is the key lemma to prove naturality of the counit:\n/--\nTo check equality of two elements of `X(S)`, it suffices to check equality after composing with\neach `X(S) → X(Sᵢ)`.\n-/\n\nTarget:\nlemma presheaf_ext (X : (CompHausLike.{u} P)ᵒᵖ ⥤ Type (max u w))\n    [PreservesFiniteProducts X] (x y : X.obj ⟨S⟩)\n    [HasExplicitFiniteCoproducts.{u} P]\n    (h : ∀ (a : Fiber f), X.map (sigmaIncl f a).op x = X.map (sigmaIncl f a).op y) : x = y :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed/Discrete","family_id":"presheaf_ext","file_id":"mathlib/Mathlib/Condensed/Discrete/LocallyConstant.lean","sample_id":"5b180b286925047b6265b1d6b6387babba0a99b025076644a75ae77bbbe0b4d6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f994609f95dbe1be796aaafe12c37132a7b266348fd2421e6b7d6339c7e0ee00","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"184e16fbe9e5eec732502dd0a1ed9c46036bcaecb5ded625d98d24b0ac54477f","source_sha256":"ff10564e9930cd55392330030cf618a47034f00b752e4ff3cc347c8e7c823f58","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using! (H.toChainHomotopy).homologyMap_eq n","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.25},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"7d65d3c3cf51d0cdebd30857be9621ec84caa31c0bee3d12af219887f2fbc423","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.SimplicialObject.Homotopy\npublic import Mathlib.AlgebraicTopology.AlternatingFaceMapComplex\npublic import Mathlib.Algebra.Homology.Homotopy\n\nNamespace:\nCategoryTheory.SimplicialObject.Homotopy\n\nLocal context:\n/-\nCopyright (c) 2025 Fabian Odermatt. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Fabian Odermatt\n-/\n/-!\n# Simplicial homotopies induce chain homotopies\n\nGiven a simplicial homotopy between morphisms of simplicial objects in a preadditive category,\nwe construct a chain homotopy between the induced morphisms on the alternating face map complexes.\n\nConcretely, if `H : Homotopy f g` gives maps\n`H.h i : X _⦋n⦌ ⟶ Y _⦋n+1⦌` indexed by `i : Fin (n + 1)`, we define the degree-`n` component\nof the chain homotopy as the opposite of alternating sum `∑ i, (-1)^i • H.h i`.\n-/\n\n@[expose] public section\n\nuniverse v u\n\nopen CategoryTheory CategoryTheory.SimplicialObject\nopen SimplexCategory Simplicial Opposite AlgebraicTopology\n\nnamespace CategoryTheory.SimplicialObject.Homotopy\n\nvariable {C : Type u} [Category.{v} C] [Preadditive C]\nvariable {X Y : SimplicialObject C} {f g : X ⟶ Y}\nvariable (H : Homotopy f g)\n\nnamespace ToChainHomotopy\n\n/-- The family of components of the induced chain homotopy -/\nnoncomputable def hom (p q : ℕ) : X _⦋p⦌ ⟶ Y _⦋q⦌ :=\n  if h : p + 1 = q then\n    -∑ k : Fin (p + 1), ((-1 : ℤ) ^ (k : ℕ)) • H.h k ≫ eqToHom (by simp [h])\n  else 0\n\n@[simp]\nlemma hom_eq (p : ℕ) :\n    hom H p (p + 1) = -∑ k : Fin (p + 1), ((-1 : ℤ) ^ (k : ℕ)) • H.h k := by\n  simp [hom]\n\n@[simp]\nlemma hom_eq_zero (p q : ℕ) (hpq : p + 1 ≠ q) :\n    hom H p q = 0 :=\n  dif_neg hpq\n\nprivate lemma comm_zero :\n    letI d : Y _⦋1⦌ ⟶ Y _⦋0⦌ := ((alternatingFaceMapComplex C).obj Y).d 1 0\n    f.app (op ⦋0⦌) = hom H 0 1 ≫ d + g.app (op ⦋0⦌) := by\n  simp [← H.h_last_comp_δ_last 0]\n\nprivate lemma comm_succ (n : ℕ) :\n    letI α : X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌ :=\n      ((alternatingFaceMapComplex C).obj X).d (n + 1) n ≫ ToChainHomotopy.hom H n (n + 1)\n    letI β : X _⦋n + 1⦌ ⟶ Y _⦋n + 1⦌ := hom H (n + 1) (n + 2) ≫\n      ((alternatingFaceMapComplex C).obj Y).d (n + 2) (n + 1)\n    f.app (op ⦋n + 1⦌) = α + β + g.app (op ⦋n + 1⦌) := by\n  rw [← H.h_zero_comp_δ_zero, ← H.h_last_comp_δ_last]\n  dsimp\n  simp only [alternatingFaceMapComplex_obj_d, AlternatingFaceMapComplex.objD, hom_eq,\n    Preadditive.comp_neg, Preadditive.neg_comp, Preadditive.comp_sum,\n    Preadditive.sum_comp, Preadditive.comp_zsmul, Preadditive.zsmul_comp,\n    smul_neg, Finset.sum_neg_distrib, ← Finset.sum_zsmul, smul_smul, ← pow_add]\n  let α (x : Fin (n + 1) × Fin (n + 2)) := (-1) ^ ((x.1 + x.2 : ℕ)) • X.δ x.2 ≫ H.h x.1\n  let β (x : Fin (n + 3) × Fin (n + 2)) := (-1) ^ ((x.1 + x.2 : ℕ)) • H.h x.2 ≫ Y.δ x.1\n  have h₂ (x : Fin (n + 1) × Fin (n + 2)) (hx : x.1.castSucc < x.2) :\n      α x = -β ⟨x.2.succ, x.1.castSucc⟩ := by\n    dsimp [α, β]\n    simp only [← H.h_castSucc_comp_δ_succ_of_lt x.2 x.1 hx,\n      pow_add, pow_one, mul_neg, mul_one, neg_mul, neg_smul, neg_neg]\n    rw [mul_comm]\n  rw [← Finset.sum_product .univ .univ α, ← Finset.sum_product .univ .univ β,\n    Finset.univ_product_univ, Finset.univ_product_univ]\n  let S : Finset (Fin (n + 1) × Fin (n + 2)) := { x | x.1.castSucc < x.2 }\n  let γ₁ (x : Fin (n + 1) × Fin (n + 2)) := (x.2.castSucc, x.1.succ)\n  let γ₂ (x : Fin (n + 1) × Fin (n + 2)) := (x.2.succ, x.1.castSucc)\n  let γ₃ (i : Fin (n + 1)) := (i.castSucc.succ, i.succ)\n  let γ₄ (i : Fin (n + 1)) := (i.castSucc.succ, i.castSucc)\n  have hγ₁ : Function.Injective γ₁ := fun _ _ ↦ by aesop\n  have hγ₂ : Function.Injective γ₂ := fun _ _ ↦ by aesop\n  have hγ₃ : Function.Injective γ₃ := fun _ _ ↦ by aesop\n  have hγ₄ : Function.Injective γ₄ := fun _ _ ↦ by aesop\n  have eq₁ : H.h 0 ≫ Y.δ 0 = β ⟨0, 0⟩ := by simp [β]\n  have eq₂ : H.h (Fin.last _) ≫ Y.δ (Fin.last _) = - β ⟨Fin.last _, Fin.last _⟩ := by\n    dsimp [β]\n    simp only [pow_add, even_two, Even.neg_pow, one_pow, mul_one,\n      pow_one, mul_neg, neg_smul, neg_neg]\n    rw [← pow_add, (Even.add_self n).neg_one_pow, one_smul]\n  have eq₃ : ∑ x ∈ Sᶜ, α x = - ∑ y ∈ Finset.image γ₁ Sᶜ, β y := by\n    rw [← Finset.sum_neg_distrib, Finset.sum_image hγ₁.injOn]\n    refine Finset.sum_congr rfl (fun x hx ↦ ?_)\n    dsimp [α, β, γ₁]\n    simp only [← H.h_succ_comp_δ_castSucc_of_lt x.2 x.1 (by simpa [S] using hx),\n      pow_add, pow_one, mul_neg, mul_one, neg_smul, neg_neg]\n    rw [mul_comm]\n  have eq₄ : ∑ x ∈ S, α x = - ∑ y ∈ Finset.image γ₂ S, β y := by\n    rw [← Finset.sum_neg_distrib, Finset.sum_image hγ₂.injOn]\n    refine Finset.sum_congr rfl (fun x hx ↦ ?_)\n    dsimp [α, β, γ₂]\n    simp only [← H.h_castSucc_comp_δ_succ_of_lt x.2 x.1 (by simpa [S] using hx),\n      pow_add, pow_one, mul_neg, mul_one, neg_mul, neg_smul, neg_neg]\n    rw [mul_comm]\n  have eq₅ : ∑ x, β (γ₄ x) = - ∑ x, β (γ₃ x) := by\n    rw [← Finset.sum_neg_distrib]\n    exact Finset.sum_congr rfl (fun x hx ↦ by simp [h_succ_comp_δ_castSucc_succ, β, γ₃, γ₄])\n  have h₁ : Disjoint (Finset.image γ₁ Sᶜ) (Finset.image γ₂ S) := by\n    rw [Finset.disjoint_iff_ne]\n    grind [Finset.mem_compl]\n  have h₂ : Disjoint (Finset.image γ₃ .univ) (Finset.image γ₄ .univ) := by\n    rw [Finset.disjoint_iff_ne]\n    grind\n  have h₃ : Disjoint (Finset.disjUnion _ _ h₂) {(0, 0), (Fin.last _, Fin.last _)} := by\n    rw [Finset.disjoint_iff_ne]\n    simp only [Finset.mem_insert, forall_eq_or_imp, Prod.forall]\n    rintro ⟨a, _⟩ ⟨b, _⟩\n    simp\n    grind\n  have h₄ : Disjoint (Finset.disjUnion _ _ h₁) (Finset.disjUnion _ _ h₃) := by\n    rw [Finset.disjoint_iff_ne]\n    simp only [Finset.compl_filter, not_lt, Finset.disjUnion_eq_union, Finset.mem_union,\n      Finset.mem_image, Finset.mem_filter, Finset.mem_univ, true_and, Prod.exists, ne_eq,\n      Finset.mem_insert, Finset.mem_singleton, Prod.forall, Prod.mk.injEq, not_and,\n      S, γ₁, γ₂, γ₃, γ₄]\n    rintro ⟨a, _⟩ ⟨b, _⟩ (⟨⟨j, _⟩, ⟨k, _⟩, h₁, h₂, h₃⟩ | ⟨⟨j, _⟩, ⟨k, _⟩, h₁, h₂, h₃⟩) _ _\n      ((⟨⟨i, _⟩, h₄, h₅⟩ | ⟨⟨i, _⟩, h₄, h₅⟩) | (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)) <;>\n        simp [Fin.ext_iff] at h₁ h₂ h₃ ⊢ <;> grind\n  have H : (Finset.disjUnion _ _ h₁)ᶜ = Finset.disjUnion _ _ h₃ :=\n    Finset.compl_eq_of_disjoint_of_card_add_eq h₄ (by\n      rw [Finset.card_disjUnion, Finset.card_disjUnion, Finset.card_disjUnion,\n        Finset.card_image_of_injective _ hγ₁, Finset.card_image_of_injective _ hγ₂,\n        Finset.card_image_of_injective _ hγ₃, Finset.card_image_of_injective _ hγ₄]\n      simp\n      lia)\n  rw [eq₁, eq₂, ← S.sum_add_sum_compl, eq₃, eq₄,\n    neg_add_rev, neg_neg, neg_neg, ← Finset.sum_disjUnion h₁,\n    ← (Finset.disjUnion _ _ h₁).sum_add_sum_compl, neg_add,\n    ← add_assoc, add_neg_cancel, zero_add, H,\n    Finset.sum_disjUnion, Finset.sum_disjUnion,\n    Finset.sum_image hγ₃.injOn, Finset.sum_image hγ₄.injOn,\n    Finset.sum_insert (by simp), Finset.sum_singleton,\n    neg_add_rev, neg_add_rev, neg_add_rev, eq₅]\n  simp\n\nend ToChainHomotopy\n\nset_option backward.isDefEq.respectTransparency false in\n/-- A simplicial homotopy between `f` and `g` induces a chain homotopy\nbetween the induced morphisms on the alternating face map complexes. -/\nnoncomputable def toChainHomotopy (H : Homotopy f g) :\n    _root_.Homotopy\n      ((alternatingFaceMapComplex C).map f)\n      ((alternatingFaceMapComplex C).map g) where\n  hom := ToChainHomotopy.hom H\n  zero i j hij := ToChainHomotopy.hom_eq_zero _ _ _ hij\n  comm n := by\n    cases n with\n    | zero =>\n      rw [prevD_eq (j' := 1) (w := by simp), dNext_eq_zero _ _ (by simp), zero_add]\n      simp [ToChainHomotopy.comm_zero H]\n    | succ n =>\n      rw [dNext_eq (i' := n) (w := by simp), prevD_eq (j' := n + 2) (w := by simp)]\n      simp [ToChainHomotopy.comm_succ H]\n\nTarget:\ntheorem map_homology_eq [CategoryWithHomology C] (H : Homotopy f g) (n : ℕ) :\n    (HomologicalComplex.homologyFunctor C _ n).map ((alternatingFaceMapComplex C).map f) =\n    (HomologicalComplex.homologyFunctor C _ n).map ((alternatingFaceMapComplex C).map g) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplicialObject","family_id":"map_homology_eq","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplicialObject/ChainHomotopy.lean","sample_id":"184e16fbe9e5eec732502dd0a1ed9c46036bcaecb5ded625d98d24b0ac54477f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d7c3832f7227fb403f537f073b6c62458e059e39149445706fd5cbe0c34cde70","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4a94ca7d62a9fac81c4789f5e6cb34fc62a67f792fb4fbe276c40abbf6855644","source_sha256":"1ddcf1b99bc8b3242f5c0a7e676d50f1f12f248d40327684682203ae3ae4dc4a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only [tendsto_iff_comap] using le_iff_ultrafilter","hard_negative":true,"metrics":{"chosen_tokens":8,"rejected_tokens":8,"token_jaccard":0.071429,"token_length_ratio":1.0},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"7da3e9a37f1065c110270f3c6d015594699a0f0028b79b729328e2d0c07dd17d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Filter.Ultrafilter.Defs\npublic import Mathlib.Order.Filter.Cofinite\npublic import Mathlib.Order.ZornAtoms\n\nNamespace:\nFilter\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov\n-/\n/-!\n# Ultrafilters\n\nAn ultrafilter is a minimal (maximal in the set order) proper filter.\nIn this file we define\n\n* `hyperfilter`: the ultrafilter extending the cofinite filter.\n-/\n\n@[expose] public section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type v}\n\nopen Set Filter\n\nnamespace Ultrafilter\n\nvariable {f : Ultrafilter α} {s : Set α}\n\ntheorem finite_sUnion_mem_iff {s : Set (Set α)} (hs : s.Finite) : ⋃₀ s ∈ f ↔ ∃ t ∈ s, t ∈ f := by\n  induction s, hs using Set.Finite.induction_on with\n  | empty => simp\n  | insert _ _ his => simp [union_mem_iff, his, or_and_right, exists_or]\n\ntheorem finite_biUnion_mem_iff {is : Set β} {s : β → Set α} (his : is.Finite) :\n    (⋃ i ∈ is, s i) ∈ f ↔ ∃ i ∈ is, s i ∈ f := by\n  simp only [← sUnion_image, finite_sUnion_mem_iff (his.image s), exists_mem_image]\n\nlemma eventually_exists_mem_iff {is : Set β} {P : β → α → Prop} (his : is.Finite) :\n    (∀ᶠ i in f, ∃ a ∈ is, P a i) ↔ ∃ a ∈ is, ∀ᶠ i in f, P a i := by\n  simp only [Filter.Eventually, Ultrafilter.mem_coe]\n  convert! f.finite_biUnion_mem_iff his (s := P) with i\n  aesop\n\nlemma eventually_exists_iff [Finite β] {P : β → α → Prop} :\n    (∀ᶠ i in f, ∃ a, P a i) ↔ ∃ a, ∀ᶠ i in f, P a i := by\n  simpa using eventually_exists_mem_iff (f := f) (P := P) Set.finite_univ\n\ntheorem eq_pure_of_finite_mem (h : s.Finite) (h' : s ∈ f) : ∃ x ∈ s, f = pure x := by\n  rw [← biUnion_of_singleton s] at h'\n  rcases (Ultrafilter.finite_biUnion_mem_iff h).mp h' with ⟨a, has, haf⟩\n  exact ⟨a, has, eq_of_le (Filter.le_pure_iff.2 haf)⟩\n\ntheorem eq_pure_of_finite [Finite α] (f : Ultrafilter α) : ∃ a, f = pure a :=\n  (eq_pure_of_finite_mem finite_univ univ_mem).imp fun _ ⟨_, ha⟩ => ha\n\ntheorem le_cofinite_or_eq_pure (f : Ultrafilter α) : (f : Filter α) ≤ cofinite ∨ ∃ a, f = pure a :=\n  or_iff_not_imp_left.2 fun h =>\n    let ⟨_, hs, hfin⟩ := Filter.disjoint_cofinite_right.1 (disjoint_iff_not_le.2 h)\n    let ⟨a, _, hf⟩ := eq_pure_of_finite_mem hfin hs\n    ⟨a, hf⟩\n\ntheorem exists_ultrafilter_of_finite_inter_nonempty (S : Set (Set α))\n    (cond : ∀ T : Finset (Set α), (↑T : Set (Set α)) ⊆ S → (⋂₀ (↑T : Set (Set α))).Nonempty) :\n    ∃ F : Ultrafilter α, S ⊆ F.sets :=\n  haveI : NeBot (generate S) :=\n    generate_neBot_iff.2 fun _ hts ht =>\n      ht.coe_toFinset ▸ cond ht.toFinset (ht.coe_toFinset.symm ▸ hts)\n  ⟨of (generate S), fun _ ht => (of_le <| generate S) <| GenerateSets.basic ht⟩\n\nend Ultrafilter\n\nnamespace Filter\n\nopen Ultrafilter\n\n@[to_dual]\nlemma atTop_eq_pure_of_isTop [PartialOrder α] {x : α} (hx : IsTop x) :\n    (atTop : Filter α) = pure x :=\n  { top := x, le_top := hx : OrderTop α }.atTop_eq\n\n/-- The `tendsto` relation can be checked on ultrafilters. -/\n\nTarget:\ntheorem tendsto_iff_ultrafilter (f : α → β) (l₁ : Filter α) (l₂ : Filter β) :\n    Tendsto f l₁ l₂ ↔ ∀ g : Ultrafilter α, ↑g ≤ l₁ → Tendsto f g l₂ :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e422fccc2758c173c5740b3634c733b26082230bfcb6f40aabeeb831ee6bb135","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Filter","family_id":"tendsto_iff_ultrafilter","file_id":"mathlib/Mathlib/Order/Filter/Ultrafilter/Basic.lean","sample_id":"4a94ca7d62a9fac81c4789f5e6cb34fc62a67f792fb4fbe276c40abbf6855644"},"split":"train"}
{"binding":{"chosen_proof_sha256":"37b0177ca46b549523222694ca3ff53bf01def907bba860633ebbd21d6d3186d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6e5dde6af19e66e06a4ae4317b8eb4e8eb4862f32bca8c9663051c5849e4b141","source_sha256":"bda612cd6d1ad1cbdd6e02bafd2e7e77c11e8cfa412747033a52bbe8062964f2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x\n  obtain ⟨I, rfl⟩ := ClassGroup.mk0_surjective x\n  rw [extendedHom_mk0, MonoidHom.one_apply]\n  exact (ClassGroup.mk0_eq_one_iff (extendedIdeal A B I).2).mpr (by simpa using h I)","hard_negative":false,"metrics":{"chosen_tokens":45,"rejected_tokens":2,"token_jaccard":0.03125,"token_length_ratio":0.044444},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"7dd1945e6d25e702d82772c0d74968a8fdd7b09534a732810731a72101ca0d3b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.FractionalIdeal.Extended\npublic import Mathlib.RingTheory.ClassGroup.Basic\n\nNamespace:\nClassGroup\n\nLocal context:\n/-\nCopyright (c) 2026 Riccardo Brasca. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Birkbeck, Riccardo Brasca\n-/\n/-!\n# Class group map induced by an extension of domains\n\nFor an injective extension `A → B` of commutative domains (equivalently `Module.IsTorsionFree A B`),\nwe construct the group homomorphism `ClassGroup.extendedHom : ClassGroup A →* ClassGroup B` given by\npushing fractional ideals forward along the algebra map.\n\n## Main definitions\n\n- `ClassGroup.extendedHom A B`: the induced map between the class groups.\n- `ClassGroup.extendedIdeal A B`: the extension of a nonzero integral ideal.\n\n## Main results\n\n- `ClassGroup.extendedHom_mk`: compatibility with representatives as fractional ideals.\n- `ClassGroup.extendedHom_mk0`: compatibility with representatives as nonzero integral ideals.\n- `ClassGroup.extendedHom_comp`: compatibility of extension in a tower `A → B → C`.\n- `ClassGroup.extendedHom_eq_one_of_forall_isPrincipal`: if the extension of every ideal is\n  principal, then `ClassGroup.extendedHom A B` is trivial.\n-/\n\npublic section\n\nopen scoped nonZeroDivisors\n\nvariable (A B : Type*) [CommRing A] [CommRing B] [Algebra A B]\n  [Module.IsTorsionFree A B]\n\nnamespace ClassGroup\n\nsection CommRing\n\nvariable [IsDomain A] [IsDomain B]\n\n/-- The monoid homomorphism `ClassGroup A → ClassGroup B` induced by an\ninjective extension of domains `A → B`. -/\nnoncomputable def extendedHom : ClassGroup A →* ClassGroup B :=\n  QuotientGroup.map _ _\n    (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom)\n    (by\n      rintro _ ⟨α, rfl⟩\n      refine ⟨Units.mk0 (IsFractionRing.map (j := algebraMap A B)\n        (FaithfulSMul.algebraMap_injective _ _) (α : FractionRing A))\n        (by simp [α.ne_zero]), ?_⟩\n      simpa [coe_toPrincipalIdeal, Units.coe_map, Units.val_mk0] using!\n        (FractionalIdeal.extendedHom_spanSingleton (FractionRing B) B _).symm)\n\n@[simp]\nlemma extendedHom_quotientMk (α : (FractionalIdeal A⁰ (FractionRing A))ˣ) :\n    extendedHom A B (QuotientGroup.mk α) = QuotientGroup.mk\n      (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom α) := by\n  rfl\n\n@[simp]\ntheorem extendedHom_mk (I : (FractionalIdeal A⁰ (FractionRing A))ˣ) :\n    extendedHom A B (ClassGroup.mk _ I) = ClassGroup.mk _\n        (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom I) := by\n  rw [← ClassGroup.Quot_mk_eq_mk, ← ClassGroup.Quot_mk_eq_mk]\n  exact extendedHom_quotientMk A B I\n\n/-- The extension of a nonzero integral ideal along an injective extension of domains. -/\nabbrev extendedIdeal (I : (Ideal A)⁰) : (Ideal B)⁰ :=\n  ⟨I.1.map (algebraMap A B), mem_nonZeroDivisors_iff_ne_zero.mpr <|\n    (Ideal.map_eq_bot_iff_of_injective (FaithfulSMul.algebraMap_injective A B)).not.mpr\n      (mem_nonZeroDivisors_iff_ne_zero.mp I.2)⟩\n\n@[simp]\ntheorem extendedIdeal_extendedIdeal (C : Type*) [CommRing C] [IsDomain C] [Algebra B C]\n    [Algebra A C] [IsScalarTower A B C] [Module.IsTorsionFree B C]\n    [Module.IsTorsionFree A C] (I : (Ideal A)⁰) :\n    extendedIdeal B C (extendedIdeal A B I) = extendedIdeal A C I := by\n  simp [Ideal.map_map, IsScalarTower.algebraMap_eq A B C]\n\nend CommRing\n\nsection DedekindDomain\n\nvariable [IsDedekindDomain A] (C : Type*) [CommRing C] [Algebra B C] [Algebra A C]\n  [IsScalarTower A B C] [Module.IsTorsionFree B C] [Module.IsTorsionFree A C]\n  [IsDedekindDomain C]\n\ntheorem extendedHom_mk0' [IsDomain B] (I : (Ideal A)⁰) :\n    extendedHom A B (ClassGroup.mk0 I) =\n      ClassGroup.mk _ (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom\n      (FractionalIdeal.mk0 (FractionRing A) I)) := by\n  rw [← ClassGroup.mk_mk0 (FractionRing A), extendedHom_mk]\n\nvariable [IsDedekindDomain B]\n\nlemma extendedHom_mk0 (I : (Ideal A)⁰) :\n    extendedHom A B (ClassGroup.mk0 I) = ClassGroup.mk0 (extendedIdeal A B I) := by\n  rw [mk0_eq_quotientMk, mk0_eq_quotientMk, extendedHom_quotientMk]\n  congr; ext : 1\n  exact FractionalIdeal.extendedHom_coeIdeal_eq_map (L := FractionRing B) (B := B) _\n\n\n@[simp]\ntheorem extendedHom_comp_apply (x : ClassGroup A) :\n      extendedHom B C (extendedHom A B x) = extendedHom A C x := by\n  obtain ⟨I, rfl⟩ := ClassGroup.mk0_surjective x\n  rw [extendedHom_mk0 A B I, extendedHom_mk0 B C (extendedIdeal A B I),\n    extendedHom_mk0 A C I, extendedIdeal_extendedIdeal]\n\ntheorem extendedHom_comp : (extendedHom B C).comp (extendedHom A B) = extendedHom A C := by\n  ext x\n  exact extendedHom_comp_apply A B C x\n\nTarget:\ntheorem extendedHom_eq_one_of_forall_isPrincipal\n    (h : ∀ I : (Ideal A), (I.map (algebraMap A B)).IsPrincipal) : extendedHom A B = 1 :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/ClassGroup","family_id":"extendedhom_eq_one_of_forall_isprincipal","file_id":"mathlib/Mathlib/RingTheory/ClassGroup/ExtendedHom.lean","sample_id":"6e5dde6af19e66e06a4ae4317b8eb4e8eb4862f32bca8c9663051c5849e4b141"},"split":"train"}
{"binding":{"chosen_proof_sha256":"30f83bf0d483bdb24895df71acc1365551186d762bcd20d14e7f1ac4b162ed7c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"862a0c729b5cd3749b546c796ee66cf2c56a927378f8d912626bfe6707f1fa0a","source_sha256":"2aaaf1d6e3bf8bcc38fe6595b30037cc7d4b213d51f17e0ec2ab4dd92ba3ffe4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨s, -, hscard, hs⟩ := addRothNumber_spec (univ : Finset α)\n  haveI := noAccidental hs\n  rw [← hscard, ← card_triangleIndices, ← card_triangles]\n  exact (locallyLinear hs).le_ruzsaSzemerediNumber","hard_negative":true,"metrics":{"chosen_tokens":41,"rejected_tokens":8,"token_jaccard":0.090909,"token_length_ratio":0.195122},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"7df2068fdd647fbbc9300618148aa09b4c3f1d0391e48be4e3f75ccd4059e47c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Additive.AP.Three.Behrend\npublic import Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite\npublic import Mathlib.Tactic.Rify\npublic import Mathlib.Tactic.Qify\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# The Ruzsa-Szemerédi problem\n\nThis file proves the lower bound of the Ruzsa-Szemerédi problem. The problem is to find the maximum\nnumber of edges that a graph on `n` vertices can have if all edges belong to at most one triangle.\n\nThe lower bound comes from turning the big 3AP-free set from Behrend's construction into a graph\nthat has the property that every triangle gives a (possibly trivial) arithmetic progression on the\noriginal set.\n\n## Main declarations\n\n* `ruzsaSzemerediNumberNat n`: Maximum number of edges a graph on `n` vertices can have such that\n  each edge belongs to exactly one triangle.\n* `ruzsaSzemerediNumberNat_asymptotic_lower_bound`: There exists a graph with `n` vertices and\n  `Ω((n ^ 2 * exp (-4 * √(log n))))` edges such that each edge belongs to exactly one triangle.\n-/\n\n@[expose] public section\n\nopen Finset Nat Real SimpleGraph Sum3 SimpleGraph.TripartiteFromTriangles\nopen Fintype (card)\nopen scoped Pointwise\n\nvariable {α β : Type*}\n\n/-! ### The Ruzsa-Szemerédi number -/\n\nsection ruzsaSzemerediNumber\nvariable [DecidableEq α] [DecidableEq β] [Fintype α] [Fintype β] {G H : SimpleGraph α}\n\nvariable (α) in\n/-- The **Ruzsa-Szemerédi number** of a fintype is the maximum number of edges a locally linear\ngraph on that type can have.\n\nIn other words, `ruzsaSzemerediNumber α` is the maximum number of edges a graph on `α` can have such\nthat each edge belongs to exactly one triangle. -/\nnoncomputable def ruzsaSzemerediNumber : ℕ := by\n  classical\n  exact Nat.findGreatest (fun m ↦ ∃ (G : SimpleGraph α) (_ : DecidableRel G.Adj),\n    #(G.cliqueFinset 3) = m ∧ G.LocallyLinear) ((card α).choose 3)\n\nopen scoped Classical in\nlemma ruzsaSzemerediNumber_le : ruzsaSzemerediNumber α ≤ (card α).choose 3 := Nat.findGreatest_le _\n\nlemma ruzsaSzemerediNumber_spec :\n    ∃ (G : SimpleGraph α) (_ : DecidableRel G.Adj),\n      #(G.cliqueFinset 3) = ruzsaSzemerediNumber α ∧ G.LocallyLinear := by\n  classical\n  exact @Nat.findGreatest_spec _\n    (fun m ↦ ∃ (G : SimpleGraph α) (_ : DecidableRel G.Adj),\n      #(G.cliqueFinset 3) = m ∧ G.LocallyLinear) _ _ (Nat.zero_le _)\n    ⟨⊥, inferInstance, by simp, locallyLinear_bot⟩\n\nvariable {m n : ℕ}\n\nlemma SimpleGraph.LocallyLinear.le_ruzsaSzemerediNumber [DecidableRel G.Adj]\n    (hG : G.LocallyLinear) : #(G.cliqueFinset 3) ≤ ruzsaSzemerediNumber α := by\n  classical\n  exact le_findGreatest card_cliqueFinset_le ⟨G, inferInstance, by congr, hG⟩\n\nlemma ruzsaSzemerediNumber_mono (f : α ↪ β) : ruzsaSzemerediNumber α ≤ ruzsaSzemerediNumber β := by\n  classical\n  refine findGreatest_mono ?_ (choose_mono _ <| Fintype.card_le_of_embedding f)\n  rintro n ⟨G, _, rfl, hG⟩\n  refine ⟨G.map f, inferInstance, ?_, hG.map _⟩\n  rw [← card_map ⟨map f, Finset.map_injective _⟩, ← cliqueFinset_map G f]\n  decide\n\nlemma ruzsaSzemerediNumber_congr (e : α ≃ β) : ruzsaSzemerediNumber α = ruzsaSzemerediNumber β :=\n  (ruzsaSzemerediNumber_mono (e : α ↪ β)).antisymm <| ruzsaSzemerediNumber_mono e.symm\n\n/-- The `n`-th **Ruzsa-Szemerédi number** is the maximum number of edges a locally linear graph on\n`n` vertices can have.\n\nIn other words, `ruzsaSzemerediNumberNat n` is the maximum number of edges a graph on `n` vertices\ncan have such that each edge belongs to exactly one triangle. -/\nnoncomputable def ruzsaSzemerediNumberNat (n : ℕ) : ℕ := ruzsaSzemerediNumber (Fin n)\n\n@[simp]\nlemma ruzsaSzemerediNumberNat_card : ruzsaSzemerediNumberNat (card α) = ruzsaSzemerediNumber α :=\n  ruzsaSzemerediNumber_congr (Fintype.equivFin _).symm\n\n@[gcongr]\nlemma ruzsaSzemerediNumberNat_mono : Monotone ruzsaSzemerediNumberNat := fun _m _n h =>\n  ruzsaSzemerediNumber_mono (Fin.castLEEmb h)\n\nlemma ruzsaSzemerediNumberNat_le : ruzsaSzemerediNumberNat n ≤ n.choose 3 :=\n  ruzsaSzemerediNumber_le.trans_eq <| by rw [Fintype.card_fin]\n\n@[simp] lemma ruzsaSzemerediNumberNat_zero : ruzsaSzemerediNumberNat 0 = 0 :=\n  le_zero_iff.1 ruzsaSzemerediNumberNat_le\n\n@[simp] lemma ruzsaSzemerediNumberNat_one : ruzsaSzemerediNumberNat 1 = 0 :=\n  le_zero_iff.1 ruzsaSzemerediNumberNat_le\n\n@[simp] lemma ruzsaSzemerediNumberNat_two : ruzsaSzemerediNumberNat 2 = 0 :=\n  le_zero_iff.1 ruzsaSzemerediNumberNat_le\n\nend ruzsaSzemerediNumber\n\n/-! ### The Ruzsa-Szemerédi construction -/\n\nsection RuzsaSzemeredi\nvariable [Fintype α] [CommRing α] {s : Finset α} {x : α × α × α}\n\n/-- The triangle indices for the Ruzsa-Szemerédi construction. -/\nprivate def triangleIndices (s : Finset α) : Finset (α × α × α) :=\n  (univ ×ˢ s).map\n    ⟨fun xa ↦ (xa.1, xa.1 + xa.2, xa.1 + 2 * xa.2), by\n      rintro ⟨x, a⟩ ⟨y, b⟩ h\n      simp only [Prod.ext_iff] at h\n      obtain rfl := h.1\n      obtain rfl := add_right_injective _ h.2.1\n      rfl⟩\n\n@[simp]\nprivate lemma mem_triangleIndices :\n    x ∈ triangleIndices s ↔ ∃ y, ∃ a ∈ s, (y, y + a, y + 2 * a) = x := by simp [triangleIndices]\n\n@[simp]\nprivate lemma card_triangleIndices : #(triangleIndices s) = card α * #s := by\n  simp [triangleIndices]\n\nprivate lemma noAccidental (hs : ThreeAPFree (s : Set α)) :\n    NoAccidental (triangleIndices s : Finset (α × α × α)) where\n  eq_or_eq_or_eq := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ _ _ d a ha rfl rfl rfl b' b hb rfl rfl h₁ d' c hc rfl h₂ rfl\n    have : a + c = b + b := by linear_combination h₁.symm - h₂.symm\n    obtain rfl := hs ha hb hc this\n    simp_all\n\nvariable [Fact <| IsUnit (2 : α)]\n\nprivate instance : ExplicitDisjoint (triangleIndices s : Finset (α × α × α)) where\n  inj₀ := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ x a ha rfl rfl rfl y b hb rfl h₁ h₂\n    linear_combination 2 * h₁.symm - h₂.symm\n  inj₁ := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ x a ha rfl rfl rfl y b hb rfl rfl h\n    simpa [(Fact.out (p := IsUnit (2 : α))).mul_right_inj, eq_comm] using h\n  inj₂ := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ x a ha rfl rfl rfl y b hb rfl h rfl\n    simpa [(Fact.out (p := IsUnit (2 : α))).mul_right_inj, eq_comm] using h\n\nprivate lemma locallyLinear (hs : ThreeAPFree (s : Set α)) :\n    (graph <| triangleIndices s).LocallyLinear :=\n  haveI := noAccidental hs; TripartiteFromTriangles.locallyLinear _\n\nprivate lemma card_edgeFinset (hs : ThreeAPFree (s : Set α)) [DecidableEq α] :\n    #(graph <| triangleIndices s).edgeFinset = 3 * card α * #s := by\n  haveI := noAccidental hs\n  rw [(locallyLinear hs).card_edgeFinset, card_triangles, card_triangleIndices, mul_assoc]\n\nend RuzsaSzemeredi\n\nvariable (α) [Fintype α] [DecidableEq α] [CommRing α] [Fact <| IsUnit (2 : α)]\n\nTarget:\nlemma addRothNumber_le_ruzsaSzemerediNumber :\n    card α * addRothNumber (univ : Finset α) ≤ ruzsaSzemerediNumber (Sum α (Sum α α)) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e220252dbe14e1cd3c9e09821ebb0f2d54f4d3217f44dfff02eee9ac1553012f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Extremal","family_id":"addrothnumber_le_ruzsaszemeredinumber","file_id":"mathlib/Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean","sample_id":"862a0c729b5cd3749b546c796ee66cf2c56a927378f8d912626bfe6707f1fa0a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"42771a2147cb91bca0f9e576050654f9c7f45762679c01f645ed998108481986","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e2222f0bad75d2e7deb1fd25fdb3ceb2a09b7f0046dfa22182f45b77e039767c","source_sha256":"ec4c2033e9d1a1249c37ea9aa929a55ae2775b2733369f392ca904414b15e64c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    simpa using hκ.irreducible hs <| Std.lt_of_lt_of_le hsp (hφ s)","hard_negative":false,"metrics":{"chosen_tokens":19,"rejected_tokens":2,"token_jaccard":0.055556,"token_length_ratio":0.105263},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"7e805872ff2776d4eba6c7d08fa5e7d04e415d972a0d7e7d770e708fd4d2f147","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Kernel.Composition.Comp\n\nNamespace:\nProbabilityTheory.Kernel\n\nLocal context:\n/-\nCopyright (c) 2025 Bryan Wang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bryan Wang\n-/\n/-!\n# Irreducibility of kernels\n\nA kernel `κ : Kernel α α` is `φ`-irreducible, for a given measure `φ` on `α`,\nif for every measurable set `A` with positive measure under `φ`, and for every `a : α`,\nthere exists a positive integer `n` such that we have `(κ ^ n) a A > 0`.\n\nWhen the kernel `κ` is the transition kernel of a Markov chain,\nthis precisely means that the Markov chain is `φ`-irreducible,\nthat is, there is a positive probability of reaching any\n(`φ`-positive measure) set of states from any other state within a finite number of steps.\n\n## Main definitions\n\n* `ProbabilityTheory.Kernel.IsIrreducible`:\n  irreducibility of a given kernel with respect to a measure `φ`.\n\n## Main statements\n\n* `isIrreducible_of_le_measure`: If a kernel `κ` is irreducible with respect to a measure `φ₂`,\n  then it is also irreducible with respect to any measure `φ₁` with `φ₁ ≤ φ₂`.\n\n## References\n\n* [Meyn, S.P. and Tweedie, R.L., *Markov Chains and Stochastic Stability*][meyntweedie1993]\n* [C Robert, G Casella, *Monte Carlo Statistical Methods*][robertcasella2004]\n\n-/\n\npublic section\n\nopen MeasureTheory\n\nopen scoped MeasureTheory ENNReal ProbabilityTheory\n\nnamespace ProbabilityTheory\n\nvariable {α β : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}\n\nnamespace Kernel\n\n/-- A kernel `κ : Kernel α α` is `φ`-irreducible (w.r.t. a given measure `φ` on `α`),\nif for every measurable set `A` with positive measure under `φ`,\nand for every `a : α`, there exists an integer `n` such that `(κ ^ n) a A > 0`.\nRef. *Meyn-Tweedie* Proposition 4.2.1(ii), page 89 -/\n@[mk_iff]\nclass IsIrreducible (φ : Measure α) (κ : Kernel α α) : Prop where\n  irreducible ⦃A⦄ (hA : MeasurableSet A) (hφA : φ A > 0) a :\n    ∃ (n : ℕ), (κ ^ n) a A > 0\n\ninstance {φ : Measure α} [Subsingleton α] :\n    IsIrreducible φ Kernel.id where\n  irreducible s hs hsp a := by\n    use 1;\n    have ha : a ∈ s :=\n      Subsingleton.mem_iff_nonempty.mpr\n        <| MeasureTheory.nonempty_of_measure_ne_zero (μ := φ) (ne_of_lt hsp).symm\n    simp [id_apply, ha]\n\ninstance {c : ℝ≥0∞} {φ : Measure α} {κ : Kernel α α} [hκ : IsIrreducible φ κ] :\n    IsIrreducible (c • φ) κ where\n  irreducible s hs hsp := by\n    simpa using hκ.irreducible hs <| by simp_all\n\nTarget:\nlemma isIrreducible_of_le_measure {φ₁ φ₂ : Measure α} (hφ : φ₁ ≤ φ₂)\n    {κ : Kernel α α} [hκ : IsIrreducible φ₂ κ] :\n    IsIrreducible φ₁ κ where\n  irreducible s hs hsp :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Kernel","family_id":"isirreducible_of_le_measure","file_id":"mathlib/Mathlib/Probability/Kernel/Irreducible.lean","sample_id":"e2222f0bad75d2e7deb1fd25fdb3ceb2a09b7f0046dfa22182f45b77e039767c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8b5027e289eea500a3bcfb155a81791b1a51bc31d201ba8ef228cee84d9c2f28","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1328c6946c14123810833c46f6825053fab738cfcded4ee858a97fc4229fb889","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d2e80bfd09f7c8ffd630b3233e0baf00abf1e879c638b96895f9ef4ded614180","source_sha256":"d4d995f1a37fa9568f07e686f3f287ad8080507cb2490d13806922b7a824453c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [IsIdempotentElem, ← star_mul]\n\nalias ⟨_, IsIdempotentElem.star⟩ := IsIdempotentElem.star_iff","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":5,"token_jaccard":0.105263,"token_length_ratio":0.25},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"7e97c6bc4b773534d536b29be342f92ad773170c983893bd34f22ba1642aeb73","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Star.SelfAdjoint\npublic import Mathlib.Algebra.Module.Basic\npublic import Mathlib.Algebra.Module.Equiv.Defs\npublic import Mathlib.Algebra.Module.LinearMap.Star\npublic import Mathlib.Algebra.Module.Rat\npublic import Mathlib.LinearAlgebra.Prod\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser, Frédéric Dupuis\n-/\n/-!\n# The star operation, bundled as a star-linear equiv\n\nWe define `starLinearEquiv`, which is the star operation bundled as a star-linear map.\nIt is defined on a star algebra `A` over the base ring `R`.\n\nThis file also provides some lemmas that need `Algebra.Module.Basic` imported to prove.\n\n## TODO\n\n- Define `starLinearEquiv` for noncommutative `R`. We only the commutative case for now since,\n  in the noncommutative case, the ring hom needs to reverse the order of multiplication. This\n  requires a ring hom of type `R →+* Rᵐᵒᵖ`, which is very undesirable in the commutative case.\n  One way out would be to define a new typeclass `IsOp R S` and have an instance `IsOp R R`\n  for commutative `R`.\n- Also note that such a definition involving `Rᵐᵒᵖ` or `is_op R S` would require adding\n  the appropriate `RingHomInvPair` instances to be able to define the semilinear\n  equivalence.\n-/\n\n@[expose] public section\n\n\nsection SMulLemmas\n\nvariable {R M : Type*}\n\n@[simp]\ntheorem star_natCast_smul [Semiring R] [AddCommMonoid M] [Module R M] [StarAddMonoid M] (n : ℕ)\n    (x : M) : star ((n : R) • x) = (n : R) • star x :=\n  map_natCast_smul (starAddEquiv : M ≃+ M) R R n x\n\n@[simp]\ntheorem star_intCast_smul [Ring R] [AddCommGroup M] [Module R M] [StarAddMonoid M] (n : ℤ)\n    (x : M) : star ((n : R) • x) = (n : R) • star x :=\n  map_intCast_smul (starAddEquiv : M ≃+ M) R R n x\n\n@[simp]\ntheorem star_inv_natCast_smul [DivisionSemiring R] [AddCommMonoid M] [Module R M] [StarAddMonoid M]\n    (n : ℕ) (x : M) : star ((n⁻¹ : R) • x) = (n⁻¹ : R) • star x :=\n  map_inv_natCast_smul (starAddEquiv : M ≃+ M) R R n x\n\n@[simp]\ntheorem star_inv_intCast_smul [DivisionRing R] [AddCommGroup M] [Module R M] [StarAddMonoid M]\n    (n : ℤ) (x : M) : star ((n⁻¹ : R) • x) = (n⁻¹ : R) • star x :=\n  map_inv_intCast_smul (starAddEquiv : M ≃+ M) R R n x\n\n@[simp]\ntheorem star_ratCast_smul [DivisionRing R] [AddCommGroup M] [Module R M] [StarAddMonoid M] (n : ℚ)\n    (x : M) : star ((n : R) • x) = (n : R) • star x :=\n  map_ratCast_smul (starAddEquiv : M ≃+ M) _ _ _ x\n\n/-!\nPer the naming convention, these two lemmas call `(q • ·)` `nnrat_smul` and `rat_smul` respectively,\nrather than `nnqsmul` and `qsmul` because the latter are reserved to the actions coming from\n`DivisionSemiring` and `DivisionRing`. We provide aliases with `nnqsmul` and `qsmul` for\ndiscoverability.\n-/\n\n/-- Note that this lemma holds for an arbitrary `ℚ≥0`-action, rather than merely one coming from a\n`DivisionSemiring`. We keep both the `nnqsmul` and `nnrat_smul` naming conventions for\ndiscoverability. See `star_nnqsmul`. -/\n@[simp high]\nlemma star_nnrat_smul [AddCommMonoid R] [StarAddMonoid R] [Module ℚ≥0 R] (q : ℚ≥0) (x : R) :\n    star (q • x) = q • star x := map_nnrat_smul (starAddEquiv : R ≃+ R) _ _\n\n/-- Note that this lemma holds for an arbitrary `ℚ`-action, rather than merely one coming from a\n`DivisionRing`. We keep both the `qsmul` and `rat_smul` naming conventions for discoverability.\nSee `star_qsmul`. -/\n@[simp high] lemma star_rat_smul [AddCommGroup R] [StarAddMonoid R] [Module ℚ R] (q : ℚ) (x : R) :\n    star (q • x) = q • star x :=\n  map_rat_smul (starAddEquiv : R ≃+ R) _ _\n\n/-- Note that this lemma holds for an arbitrary `ℚ≥0`-action, rather than merely one coming from a\n`DivisionSemiring`. We keep both the `nnqsmul` and `nnrat_smul` naming conventions for\ndiscoverability. See `star_nnrat_smul`. -/\nalias star_nnqsmul := star_nnrat_smul\n\n/-- Note that this lemma holds for an arbitrary `ℚ`-action, rather than merely one coming from a\n`DivisionRing`. We keep both the `qsmul` and `rat_smul` naming conventions for\ndiscoverability. See `star_rat_smul`. -/\nalias star_qsmul := star_rat_smul\n\ninstance StarAddMonoid.toStarModuleNNRat [AddCommMonoid R] [Module ℚ≥0 R] [StarAddMonoid R] :\n    StarModule ℚ≥0 R where star_smul := star_nnrat_smul\n\ninstance StarAddMonoid.toStarModuleRat [AddCommGroup R] [Module ℚ R] [StarAddMonoid R] :\n    StarModule ℚ R where star_smul := star_rat_smul\n\nend SMulLemmas\n\nsection starLinearEquiv\n\nvariable (R : Type*) {A : Type*}\n  [CommSemiring R] [StarRing R] [AddCommMonoid A] [StarAddMonoid A] [Module R A] [StarModule R A]\n\n/-- If `A` is a module over a commutative `R` with compatible actions,\nthen `star` is a semilinear equivalence. -/\n@[simps! apply]\ndef starLinearEquiv : A ≃ₗ⋆[R] A where\n  __ := starAddEquiv\n  map_smul' := star_smul\n\n@[simp]\ntheorem toAddEquiv_starLinearEquiv :\n    (starLinearEquiv R : A ≃ₗ⋆[R] A).toAddEquiv = starAddEquiv :=\n  rfl\n\n@[simp]\ntheorem symm_starLinearEquiv : (starLinearEquiv R : A ≃ₗ⋆[R] A).symm = starLinearEquiv R :=\n  rfl\n\n@[deprecated \"Use `symm_starLinearEquiv` and `starLinearEquiv_apply` instead\"\n  (since := \"2026-06-03\")]\ntheorem starLinearEquiv_symm_apply (x : A) :\n    (starLinearEquiv R).symm x = starAddEquiv.invFun x := by\n  simp\n\nend starLinearEquiv\n\nsection SelfSkewAdjoint\n\nvariable (R : Type*) (A : Type*) [Semiring R] [StarMul R] [TrivialStar R] [AddCommGroup A]\n  [Module R A] [StarAddMonoid A] [StarModule R A]\n\n/-- The self-adjoint elements of a star module, as a submodule. -/\ndef selfAdjoint.submodule : Submodule R A :=\n  { selfAdjoint A with smul_mem' := fun _ _ => (IsSelfAdjoint.all _).smul }\n\n/-- The skew-adjoint elements of a star module, as a submodule. -/\ndef skewAdjoint.submodule : Submodule R A :=\n  { skewAdjoint A with smul_mem' := skewAdjoint.smul_mem }\n\nvariable {A} [Invertible (2 : R)]\n\n/-- The self-adjoint part of an element of a star module, as a linear map. -/\n@[simps]\ndef selfAdjointPart : A →ₗ[R] selfAdjoint A where\n  toFun x :=\n    ⟨(⅟2 : R) • (x + star x), by\n      rw [selfAdjoint.mem_iff, star_smul, star_trivial, star_add, star_star, add_comm]⟩\n  map_add' x y := by\n    ext\n    simp [add_add_add_comm]\n  map_smul' r x := by\n    ext\n    simp [← mul_smul, show ⅟2 * r = r * ⅟2 from Commute.invOf_left <| (2 : ℕ).cast_commute r]\n\n/-- The skew-adjoint part of an element of a star module, as a linear map. -/\n@[simps]\ndef skewAdjointPart : A →ₗ[R] skewAdjoint A where\n  toFun x :=\n    ⟨(⅟2 : R) • (x - star x), by\n      simp only [skewAdjoint.mem_iff, star_smul, star_sub, star_star, star_trivial, ← smul_neg,\n        neg_sub]⟩\n  map_add' x y := by\n    ext\n    simp only [sub_add, ← smul_add, sub_sub_eq_add_sub, star_add, AddSubgroup.coe_add]\n  map_smul' r x := by\n    ext\n    simp [← mul_smul, ← smul_sub,\n      show r * ⅟2 = ⅟2 * r from Commute.invOf_right <| (2 : ℕ).commute_cast r]\n\ntheorem StarModule.selfAdjointPart_add_skewAdjointPart (x : A) :\n    (selfAdjointPart R x : A) + skewAdjointPart R x = x := by\n  simp only [smul_sub, selfAdjointPart_apply_coe, smul_add, skewAdjointPart_apply_coe,\n    add_add_sub_cancel, invOf_two_smul_add_invOf_two_smul]\n\ntheorem IsSelfAdjoint.coe_selfAdjointPart_apply {x : A} (hx : IsSelfAdjoint x) :\n    (selfAdjointPart R x : A) = x := by\n  rw [selfAdjointPart_apply_coe, hx.star_eq, smul_add, invOf_two_smul_add_invOf_two_smul]\n\ntheorem IsSelfAdjoint.selfAdjointPart_apply {x : A} (hx : IsSelfAdjoint x) :\n    selfAdjointPart R x = ⟨x, hx⟩ :=\n  Subtype.ext (hx.coe_selfAdjointPart_apply R)\n\n@[simp]\ntheorem selfAdjointPart_comp_subtype_selfAdjoint :\n    (selfAdjointPart R).comp (selfAdjoint.submodule R A).subtype = .id :=\n  LinearMap.ext fun x ↦ x.2.selfAdjointPart_apply R\n\ntheorem IsSelfAdjoint.skewAdjointPart_apply {x : A} (hx : IsSelfAdjoint x) :\n    skewAdjointPart R x = 0 := Subtype.ext <| by\n  rw [skewAdjointPart_apply_coe, hx.star_eq, sub_self, smul_zero, ZeroMemClass.coe_zero]\n\n@[simp]\ntheorem skewAdjointPart_comp_subtype_selfAdjoint :\n    (skewAdjointPart R).comp (selfAdjoint.submodule R A).subtype = 0 :=\n  LinearMap.ext fun x ↦ x.2.skewAdjointPart_apply R\n\n@[simp]\ntheorem selfAdjointPart_comp_subtype_skewAdjoint :\n    (selfAdjointPart R).comp (skewAdjoint.submodule R A).subtype = 0 :=\n  LinearMap.ext fun ⟨x, (hx : _ = _)⟩ ↦ Subtype.ext <| by simp [hx]\n\n@[simp]\ntheorem skewAdjointPart_comp_subtype_skewAdjoint :\n    (skewAdjointPart R).comp (skewAdjoint.submodule R A).subtype = .id :=\n  LinearMap.ext fun ⟨x, (hx : _ = _)⟩ ↦ Subtype.ext <| by\n    simp only [LinearMap.comp_apply, Submodule.subtype_apply, skewAdjointPart_apply_coe, hx,\n      sub_neg_eq_add, smul_add, invOf_two_smul_add_invOf_two_smul]; rfl\n\nvariable (A)\n\n/-- The decomposition of elements of a star module into their self- and skew-adjoint parts,\nas a linear equivalence. -/\n@[simps!]\ndef StarModule.decomposeProdAdjoint : A ≃ₗ[R] selfAdjoint A × skewAdjoint A := by\n  refine LinearEquiv.ofLinear ((selfAdjointPart R).prod (skewAdjointPart R))\n    (LinearMap.coprod ((selfAdjoint.submodule R A).subtype) (skewAdjoint.submodule R A).subtype)\n    ?_ (LinearMap.ext <| StarModule.selfAdjointPart_add_skewAdjointPart R)\n  -- Note: with https://github.com/leanprover-community/mathlib4/pull/6965 `Submodule.coe_subtype` doesn't fire in `dsimp` or `simp`\n  ext x <;> dsimp <;> erw [Submodule.coe_subtype, Submodule.coe_subtype] <;> simp\n\nend SelfSkewAdjoint\n\nsection algebraMap\n\nvariable {R A : Type*} [CommSemiring R] [StarRing R] [Semiring A]\nvariable [StarMul A] [Algebra R A] [StarModule R A]\n\n@[simp]\ntheorem algebraMap_star_comm (r : R) : algebraMap R A (star r) = star (algebraMap R A r) := by\n  simp only [Algebra.algebraMap_eq_smul_one, star_smul, star_one]\n\nvariable (A) in\nprotected lemma IsSelfAdjoint.algebraMap {r : R} (hr : IsSelfAdjoint r) :\n    IsSelfAdjoint (algebraMap R A r) := by\n  simpa using! congr(algebraMap R A $(hr.star_eq))\n\nlemma isSelfAdjoint_algebraMap_iff {r : R} (h : Function.Injective (algebraMap R A)) :\n    IsSelfAdjoint (algebraMap R A r) ↔ IsSelfAdjoint r :=\n  ⟨fun hr ↦ h <| algebraMap_star_comm r (A := A) ▸ hr.star_eq, IsSelfAdjoint.algebraMap A⟩\n\nend algebraMap\n\nTarget:\ntheorem IsIdempotentElem.star_iff {R : Type*} [Mul R] [StarMul R] {a : R} :\n    IsIdempotentElem (star a) ↔ IsIdempotentElem a :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_d2e80bfd09f7","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"ab23365fd8632d286b08ca615458823568e27c5f2c70594b84fff7963333f482","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Star","family_id":"isidempotentelem","file_id":"mathlib/Mathlib/Algebra/Star/Module.lean","sample_id":"d2e80bfd09f7c8ffd630b3233e0baf00abf1e879c638b96895f9ef4ded614180"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b925b94bb1cfb3ff34ceacbe250abd23143fbe7540078964bfa526bb642e3c9f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f7bbb727f1ea351360c7052b952d4fc3113423ef399aac4de4fe2b1d680daf36","source_sha256":"31cc90ac880af63d7e71cce55cea648ad2ff660c87dd9690b4a90bd3c1b360e1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  simpa [dotProduct_comm] using! (isSymmetric_toEuclideanLin_iff.mpr hA).im_inner_self_apply _","hard_negative":false,"metrics":{"chosen_tokens":17,"rejected_tokens":5,"token_jaccard":0.105263,"token_length_ratio":0.294118},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"7f0a291701f1a7784e69e2424170e04b16d28b4a90eb53af80e67cdab8aa14d6","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.InnerProductSpace.PiL2\npublic import Mathlib.LinearAlgebra.Matrix.Hermitian\nimport Mathlib.Analysis.InnerProductSpace.Adjoint\n\nNamespace:\nMatrix\n\nLocal context:\n/-\nCopyright (c) 2022 Alexander Bentkamp. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alexander Bentkamp\n-/\n/-!\n# Hermitian matrices over ℝ and ℂ\n\nThis file proves that Hermitian matrices over ℝ and ℂ are exactly the ones whose corresponding\nlinear map is self-adjoint.\n\n## Tags\n\nself-adjoint matrix, hermitian matrix\n-/\n\npublic section\n\n-- TODO:\n-- assert_not_exists MonoidAlgebra\n\nopen RCLike\n\nnamespace Matrix\n\nvariable {𝕜 m n : Type*} {A : Matrix n n 𝕜} [RCLike 𝕜]\n\n/-- The diagonal elements of a complex Hermitian matrix are real. -/\nlemma IsHermitian.coe_re_apply_self (h : A.IsHermitian) (i : n) : (re (A i i) : 𝕜) = A i i := by\n  rw [← conj_eq_iff_re, ← star_def, ← conjTranspose_apply, h.eq]\n\n/-- The diagonal elements of a complex Hermitian matrix are real. -/\nlemma IsHermitian.coe_re_diag (h : A.IsHermitian) : (fun i => (re (A.diag i) : 𝕜)) = A.diag :=\n  funext h.coe_re_apply_self\n\n/-- A matrix is Hermitian iff the corresponding linear map with an orthonormal basis is\nsymmetric. -/\n@[simp]\nlemma isSymmetric_toLin_iff [Fintype n] [DecidableEq n] {E : Type*}\n    [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] (b : OrthonormalBasis n 𝕜 E) :\n    (A.toLin b.toBasis b.toBasis).IsSymmetric ↔ A.IsHermitian := by\n  have : FiniteDimensional 𝕜 E := b.toBasis.finiteDimensional_of_finite\n  simp_rw [LinearMap.IsSymmetric, ← LinearMap.adjoint_inner_left, ← toLin_conjTranspose]\n  refine ⟨fun h ↦ ?_, fun h _ _ ↦ by rw [h.eq]⟩\n  simpa using! (LinearMap.ext fun x ↦ ext_inner_right _ (h x)).symm\n\n/-- A matrix is Hermitian iff the corresponding linear map on the Euclidean space is\nsymmetric. -/\n@[simp]\nlemma isSymmetric_toEuclideanLin_iff [Fintype n] [DecidableEq n] :\n    A.toEuclideanLin.IsSymmetric ↔ A.IsHermitian :=\n  isSymmetric_toLin_iff (EuclideanSpace.basisFun n 𝕜)\n\n@[deprecated isSymmetric_toEuclideanLin_iff \"use isSymmetric_toEuclideanLin_iff.symm\"\n  (since := \"2026-03-30\")]\nlemma isHermitian_iff_isSymmetric [Fintype n] [DecidableEq n] :\n    IsHermitian A ↔ A.toEuclideanLin.IsSymmetric := isSymmetric_toEuclideanLin_iff.symm\n\nTarget:\nlemma IsHermitian.im_star_dotProduct_mulVec_self [Fintype n] (hA : A.IsHermitian) (x : n → 𝕜) :\n     RCLike.im (star x ⬝ᵥ A *ᵥ x) = 0 :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Matrix","family_id":"ishermitian","file_id":"mathlib/Mathlib/Analysis/Matrix/Hermitian.lean","sample_id":"f7bbb727f1ea351360c7052b952d4fc3113423ef399aac4de4fe2b1d680daf36"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a90544dc513a05353e26becaa7c94b77ef02a2abe517eb6e4b51662aba7c3cd3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"74398be95deeaf66769f719ac90f40d24903709435864c5bbbcaaeedc04f5507","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5c25dc9430d8645a6917c9b8ba0f572387c246c6713ecceebd925b758bc2e637","source_sha256":"b8e096f7469de9f8271b2d043d71be202bf790e32cee2afb83b7783c6d94bbb7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa [← Basis.coord_apply] using LinearMap.congr_fun (basis_coord R n b s) x","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":3,"token_jaccard":0.047619,"token_length_ratio":0.15},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"7f182e9f1ef28adea082b090eccd216911e16e2af053a3301ac9f68e6308d1c8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.ExteriorPower.Basic\npublic import Mathlib.LinearAlgebra.ExteriorPower.Pairing\npublic import Mathlib.RingTheory.Finiteness.Subalgebra\npublic import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition\n\nNamespace:\nexteriorPower\n\nLocal context:\n/-\nCopyright (c) 2025 Daniel Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sophie Morel, Daniel Morrison\n-/\n/-!\n# Constructs a basis for exterior powers\n-/\n\n@[expose] public section\n\nvariable {R K M E : Type*} {n : ℕ}\n  [CommRing R] [Field K] [AddCommGroup M] [Module R M] [AddCommGroup E] [Module K E]\n\nnamespace exteriorPower\n\n/-! Finiteness of the exterior power. -/\n\n/-- The `n`th exterior power of a finite module is a finite module. -/\ninstance instFinite [Module.Finite R M] : Module.Finite R (⋀[R]^n M) := by\n  rw [Module.Finite.iff_fg, ExteriorAlgebra.exteriorPower, LinearMap.range_eq_map]\n  exact Submodule.FG.pow (Submodule.FG.map _ Module.Finite.fg_top) n\n\n/-! We construct a basis of `⋀[R]^n M` from a basis of `M`. -/\n\nopen Module Set Set.powersetCard\n\nvariable (R n)\n\n/-- If `b` is a basis of `M` indexed by a linearly ordered type `I` and `s` is a finset of\n`I` of cardinality `n`, then we get a linear form on the `n`th exterior power of `M` by\napplying the `exteriorPower.linearForm` construction to the family of linear forms\ngiven by the coordinates of `b` indexed by elements of `s` (ordered using the given order on\n`I`). -/\nnoncomputable def ιMultiDual {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s : powersetCard I n) : Module.Dual R (⋀[R]^n M) :=\n  pairingDual R M n (ιMulti_family R n b.coord s)\n\n@[simp]\nlemma ιMultiDual_apply_ιMulti {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s : powersetCard I n) (v : Fin n → M) :\n    ιMultiDual R n b s (ιMulti R n v) =\n    (Matrix.of fun i j => b.coord (powersetCard.ofFinEmbEquiv.symm s j) (v i)).det := by\n  simp [ιMultiDual, ιMulti_family, pairingDual_ιMulti_ιMulti]\n\n/-- Let `b` be a basis of `M` indexed by a linearly ordered type `I` and `s` be a finset of `I`\nof cardinality `n`. If we apply the linear form on `⋀[R]^n M` defined by `b` and `s`\nto the exterior product of the `b i` for `i ∈ s`, then we get `1`. -/\nlemma ιMultiDual_apply_diag {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s : powersetCard I n) :\n    ιMultiDual R n b s (ιMulti_family R n b s) = 1 := by\n  rw [ιMulti_family, ιMultiDual_apply_ιMulti]\n  suffices Matrix.of (fun i j => b.coord (powersetCard.ofFinEmbEquiv.symm s j)\n    (b (powersetCard.ofFinEmbEquiv.symm s i))) = 1 by\n    simp_rw [Function.comp_apply, this, Matrix.det_one]\n  ext\n  simp [Matrix.one_apply, Finsupp.single_apply]\n\n/-- Let `b` be a basis of `M` indexed by a linearly ordered type `I` and `s` be a finset of `I`\nof cardinality `n`. Let `t` be a finset of `I` of cardinality `n` such that `s ≠ t`. If we apply\nthe linear form on `⋀[R]^n M` defined by `b` and `s` to the exterior product of the\n`b i` for `i ∈ t`, then we get `0`. -/\nlemma ιMultiDual_apply_nondiag {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s t : powersetCard I n) (hst : s ≠ t) :\n    ιMultiDual R n b s (ιMulti_family R n b t) = 0 := by\n  rw [ιMulti_family, ιMultiDual_apply_ιMulti]\n  obtain ⟨i, his, hit⟩ := (exists_mem_notMem_iff_ne s t).mp hst\n  obtain ⟨k, rfl⟩ := (mem_range_ofFinEmbEquiv_symm_iff_mem s i).mpr his\n  apply Matrix.det_eq_zero_of_column_eq_zero k\n  simp_rw [Matrix.of_apply, Basis.coord_apply, Function.comp_apply, Basis.repr_self]\n  intro j\n  apply Finsupp.single_eq_of_ne\n  by_contra! h\n  apply hit\n  rw [h, powersetCard.ofFinEmbEquiv_symm_apply, ← powersetCard.mem_coe_iff]\n  exact Finset.orderEmbOfFin_mem t.val t.prop j\n\n/-- If `b` is a basis of `M` (indexed by a linearly ordered type), then the family\n`exteriorPower.ιMulti R n b` of the `n`-fold exterior products of its elements is linearly\nindependent in the `n`th exterior power of `M`. -/\nlemma ιMulti_family_linearIndependent_ofBasis {I : Type*} [LinearOrder I] (b : Basis I R M) :\n    LinearIndependent R (ιMulti_family R n b) :=\n  LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun s ↦ ιMultiDual R n b s)\n    (fun _ _ h => ιMultiDual_apply_nondiag R n b _ _ h)\n    (fun _ => ιMultiDual_apply_diag _ _ _ _)\n\nvariable {R} in\n/-- If `b` is a basis of `M` (indexed by a linearly ordered type), the basis of the `n`th\nexterior power of `M` formed by the `n`-fold exterior products of elements of `b`. -/\nnoncomputable def _root_.Module.Basis.exteriorPower {I : Type*} [LinearOrder I] (b : Basis I R M) :\n    Basis (powersetCard I n) R (⋀[R]^n M) :=\n  Basis.mk (ιMulti_family_linearIndependent_ofBasis _ _ _)\n    (eq_top_iff.mp <| ιMulti_family_span_of_span R b.span_eq)\n\n@[simp]\nlemma coe_basis {I : Type*} [LinearOrder I] (b : Basis I R M) :\n    DFunLike.coe (b.exteriorPower n) = ιMulti_family R n b :=\n  Basis.coe_mk _ _\n\nlemma basis_apply {I : Type*} [LinearOrder I] (b : Basis I R M) (s : powersetCard I n) :\n    b.exteriorPower n s = ιMulti_family R n b s := by\n  rw [coe_basis]\n\n/-- If `b` is a basis of `M` indexed by a linearly ordered type `I` and `B` is the corresponding\nbasis of the `n`th exterior power of `M`, indexed by the set of finsets `s` of `I` of cardinality\n`n`, then the coordinate function of `B` at `s` is the linear form on the `n`th exterior power\ndefined by `b` and `s` in `exteriorPower.ιMultiDual`. -/\nlemma basis_coord {I : Type*} [LinearOrder I] (b : Basis I R M) (s : powersetCard I n) :\n    Basis.coord (b.exteriorPower n) s = ιMultiDual R n b s := by\n  apply LinearMap.ext_on (ιMulti_family_span_of_span R (Basis.span_eq b))\n  rintro x ⟨t, rfl⟩\n  rw [Basis.coord_apply]\n  by_cases! hst : s = t\n  · rw [hst, ιMultiDual_apply_diag, ← basis_apply, Basis.repr_self, Finsupp.single_eq_same]\n  · rw [ιMultiDual_apply_nondiag R n b s t hst, ← basis_apply, Basis.repr_self,\n      Finsupp.single_eq_of_ne hst]\n\nTarget:\nlemma basis_repr_apply {I : Type*} [LinearOrder I] (b : Basis I R M) (x : ⋀[R]^n M)\n    (s : powersetCard I n) :\n    Basis.repr (b.exteriorPower n) x s = ιMultiDual R n b s x :=\n\nProof body:\n","rejected":"by\n  exact basis_repr_apply","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"1fba34c09993a37d176b36897273bd81ec922867e74ae405f6efd33a8f8fd1d5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/ExteriorPower","family_id":"basis_repr_apply","file_id":"mathlib/Mathlib/LinearAlgebra/ExteriorPower/Basis.lean","sample_id":"5c25dc9430d8645a6917c9b8ba0f572387c246c6713ecceebd925b758bc2e637"},"split":"train"}
{"binding":{"chosen_proof_sha256":"de3f7865b87871f2b8374f7e15ee4575771052e51846b14aedf16b2bb0b0042a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"986207edb34e921fae14ac5de27f5721ed1936c5cf23f19974ccfb40895a7398","source_sha256":"8e0c46324d60474a3cf61c252575628882e61646e1feb4e93ffa47506153589c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext ⟨a, b⟩\n  simp only [mem_smulAntidiagonal, mem_singleton, Prod.ext_iff]\n  constructor\n  · rintro ⟨has, hat, hst⟩\n    obtain rfl :=\n      (hs.min_le hns has).eq_of_not_lt fun hlt =>\n        (SMul.smul_lt_smul_of_lt_of_le hlt <| ht.min_le hnt hat).ne' hst\n    exact ⟨rfl, IsCancelSMul.left_cancel _ _ _ hst⟩\n  · rintro ⟨rfl, rfl⟩\n    exact ⟨hs.min_mem _, ht.min_mem _, rfl⟩","hard_negative":false,"metrics":{"chosen_tokens":92,"rejected_tokens":3,"token_jaccard":0.042553,"token_length_ratio":0.032609},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"7f6b2ef97c2fd4f7c9c55e9f8886229ba6f7eb65e68003e24550f0aa58309bed","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Pointwise.Set.Scalar\npublic import Mathlib.Data.Set.SMulAntidiagonal\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2024 Scott Carnahan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Carnahan\n-/\n/-!\n# Antidiagonal for scalar multiplication as a `Finset`.\n\nGiven sets `G` and `P`, with an action of `G` on `P`, we construct, for any element `a` in `P`,\nthe `Finset` of all pairs of an element in `s` and an element in `t` that scalar-multiply to `a`,\nassuming that set is finite.\n\n## Definitions\n* Finset.SMulAntidiagonal : Finset antidiagonal for PWO inputs.\n* Finset.VAddAntidiagonal : Finset antidiagonal for PWO inputs.\n\n-/\n\n@[expose] public section\n\nvariable {G P : Type*}\n\nopen scoped Pointwise\n\nnamespace Set\n\n@[to_additive]\ntheorem IsPWO.smul [Preorder G] [Preorder P] [SMul G P] [IsOrderedSMul G P]\n    {s : Set G} {t : Set P} (hs : s.IsPWO) (ht : t.IsPWO) : IsPWO (s • t) := by\n  rw [← @image_smul_prod]\n  exact (hs.prod ht).image_of_monotone (monotone_fst.smul monotone_snd)\n\n@[to_additive]\ntheorem IsWF.smul [LinearOrder G] [LinearOrder P] [SMul G P] [IsOrderedSMul G P] {s : Set G}\n    {t : Set P} (hs : s.IsWF) (ht : t.IsWF) : IsWF (s • t) :=\n  (hs.isPWO.smul ht.isPWO).isWF\n\n@[to_additive]\ntheorem IsWF.min_smul [LinearOrder G] [LinearOrder P] [SMul G P] [IsOrderedSMul G P]\n    {s : Set G} {t : Set P} (hs : s.IsWF) (ht : t.IsWF) (hsn : s.Nonempty) (htn : t.Nonempty) :\n    (hs.smul ht).min (hsn.smul htn) = hs.min hsn • ht.min htn := by\n  refine le_antisymm (IsWF.min_le _ _ (mem_smul.2 ⟨_, hs.min_mem _, _, ht.min_mem _, rfl⟩)) ?_\n  rw [IsWF.le_min_iff]\n  rintro _ ⟨x, hx, y, hy, rfl⟩\n  exact IsOrderedSMul.smul_le_smul (hs.min_le _ hx) (ht.min_le _ hy)\n\nend Set\n\nnamespace Finset\n\nsection\n\nopen Set\n\nvariable [SMul G P]\n\n/-- `Finset.SMulAntidiagonal hs ht a` is the set of all pairs of an element in `s` and an\nelement in `t` whose scalar multiplication yields `a`, but its construction requires a proof that\nthe set-theoretic antidiagonal is finite. -/\n@[to_additive /-- `Finset.VAddAntidiagonal hs ht a` is the set of all pairs of an element in `s`\nand an element in `t` whose vector addition yields `a`, but its construction requires proofs that\n`s` and `t` are well-ordered. -/]\nnoncomputable def SMulAntidiagonal {s : Set G}\n    {t : Set P} (a : P) (h : (s.smulAntidiagonal t a).Finite) : Finset (G × P) :=\n  h.toFinset\n\n@[to_additive (attr := simp)]\ntheorem mem_smulAntidiagonal {s : Set G}\n    {t : Set P} (a : P) (h : (s.smulAntidiagonal t a).Finite) {x : G × P} :\n    x ∈ SMulAntidiagonal a h ↔ x.1 ∈ s ∧ x.2 ∈ t ∧ x.1 • x.2 = a := by\n  simp only [SMulAntidiagonal, Set.Finite.mem_toFinset]\n  exact Set.mem_sep_iff\n\n@[to_additive]\ntheorem smulAntidiagonal_mono_left {s u : Set G} {t : Set P} (a : P) (h : u ⊆ s)\n    (hst : (s.smulAntidiagonal t a).Finite) (hut : (u.smulAntidiagonal t a).Finite) :\n    SMulAntidiagonal a hut ⊆ SMulAntidiagonal a hst :=\n  Set.Finite.toFinset_mono <| Set.smulAntidiagonal_mono_left h\n\n@[to_additive]\ntheorem smulAntidiagonal_mono_right {s : Set G}\n    {t v : Set P} (a : P) (hst : (s.smulAntidiagonal t a).Finite)\n    (hsv : (s.smulAntidiagonal v a).Finite) (h : v ⊆ t) :\n    SMulAntidiagonal a hsv ⊆ SMulAntidiagonal a hst :=\n  Set.Finite.toFinset_mono <| Set.smulAntidiagonal_mono_right h\n\n@[to_additive]\ntheorem support_smulAntidiagonal_subset_smul {s : Set G}\n    {t : Set P} (hst : ∀ a, (s.smulAntidiagonal t a).Finite) :\n    { a | (SMulAntidiagonal a (hst a)).Nonempty } ⊆ (s • t) := by\n  grind [mem_smul, mem_smulAntidiagonal]\n\nvariable [PartialOrder G] [PartialOrder P] [IsOrderedCancelSMul G P] {s : Set G}\n    {t : Set P} (hs : s.IsPWO) (ht : t.IsPWO) (a : P) {u : Set G} {hu : u.IsPWO} {v : Set P}\n    {hv : v.IsPWO} {x : G × P}\n\n@[to_additive]\ntheorem isPWO_support_smulAntidiagonal :\n    { a | (SMulAntidiagonal a (Set.SMulAntidiagonal.finite_of_isPWO hs ht a)).Nonempty }.IsPWO :=\n  (hs.smul ht).mono\n    (support_smulAntidiagonal_subset_smul (fun a ↦ (Set.SMulAntidiagonal.finite_of_isPWO hs ht a)))\n\nend\n\n@[to_additive]\n\nTarget:\ntheorem smulAntidiagonal_min_smul_min [LinearOrder G] [LinearOrder P] [SMul G P]\n    [IsOrderedCancelSMul G P] {s : Set G} {t : Set P} (hs : s.IsWF) (ht : t.IsWF) (hns : s.Nonempty)\n    (hnt : t.Nonempty) :\n    SMulAntidiagonal (hs.min hns • ht.min hnt)\n      (Set.SMulAntidiagonal.finite_of_isPWO hs.isPWO ht.isPWO (hs.min hns • ht.min hnt)) =\n      {(hs.min hns, ht.min hnt)} :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finset","family_id":"smulantidiagonal_min_smul_min","file_id":"mathlib/Mathlib/Data/Finset/SMulAntidiagonal.lean","sample_id":"986207edb34e921fae14ac5de27f5721ed1936c5cf23f19974ccfb40895a7398"},"split":"train"}
{"binding":{"chosen_proof_sha256":"30015a6154bd2aef8d0848639c98385965744ef4b2fb839692a5ae2632c8bacb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3d18d12968db25682253382ebf7b0db0c54c8dc606538f62d7d239d83c68d6b5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"61faa742462a962beb76f3f1545ae04bd1db702ab9662b4c763eedf05c97a9ff","source_sha256":"ebbd09c0a09c3969b7a62f3c013274751c70d4ab23cc8901bc28ba4a69c7b881","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← List.filterMap_eq_filter]\n  apply listFilterMap .id\n  simp only [Primrec₂, Option.guard, decide_eq_true_eq]\n  exact ite (hf.comp snd) (option_some_iff.mpr snd) (const none)","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":47,"token_jaccard":0.857143,"token_length_ratio":1.119048},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"7f88fd78b03b057bcc0d570e84d8488a24669c8097a86df580bfcef16cf6d99d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Computability.Primrec.Basic\npublic import Mathlib.Logic.Encodable.Pi\n\nNamespace:\nPrimrec\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Primitive recursive functions on Lists\n\nThe primitive recursive functions are defined in `Mathlib.Computability.Primrec.Basic`.\nThis file contains definitions and theorems about primitive recursive functions that\nrelate to operation on lists.\n\n## References\n\n* [Mario Carneiro, *Formalizing computability theory via partial recursive functions*][carneiro2019]\n-/\n\n@[expose] public section\n\nopen List (Vector)\nopen Denumerable Encodable Function\n\n\nsection\n\nvariable {α : Type*} {β : Type*} {σ : Type*}\nvariable [Primcodable α] [Primcodable β] [Primcodable σ]\nvariable (H : Nat.Primrec fun n => Encodable.encode (@decode (List β) _ n))\n\nopen Primrec\n\nset_option backward.privateInPublic true in\n@[implicit_reducible]\nprivate def prim : Primcodable (List β) := ⟨H⟩\n\nprivate theorem list_casesOn' {f : α → List β} {g : α → σ} {h : α → β × List β → σ}\n    (hf : haveI := prim H; Primrec f) (hg : Primrec g) (hh : haveI := prim H; Primrec₂ h) :\n    @Primrec _ σ _ _ fun a => List.casesOn (f a) (g a) fun b l => h a (b, l) :=\n  letI := prim H\n  have :\n    @Primrec _ (Option σ) _ _ fun a =>\n      (@decode (Option (β × List β)) _ (encode (f a))).map fun o => Option.casesOn o (g a) (h a) :=\n    ((@map_decode_iff _ (Option (β × List β)) _ _ _ _ _).2 <|\n      to₂ <|\n        option_casesOn snd (hg.comp fst) (hh.comp₂ (fst.comp₂ Primrec₂.left) Primrec₂.right)).comp\n      .id (encode_iff.2 hf)\n  option_some_iff.1 <| this.of_eq fun a => by rcases f a with - | ⟨b, l⟩ <;> simp [encodek]\n\nset_option backward.privateInPublic true in\nprivate theorem list_foldl' {f : α → List β} {g : α → σ} {h : α → σ × β → σ}\n    (hf : haveI := prim H; Primrec f) (hg : Primrec g) (hh : haveI := prim H; Primrec₂ h) :\n    Primrec fun a => (f a).foldl (fun s b => h a (s, b)) (g a) := by\n  letI := prim H\n  let G (a : α) (IH : σ × List β) : σ × List β := List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)\n  have hG : Primrec₂ G := list_casesOn' H (snd.comp snd) snd <|\n    to₂ <|\n    pair (hh.comp (fst.comp fst) <| pair ((fst.comp snd).comp fst) (fst.comp snd))\n      (snd.comp snd)\n  let F := fun (a : α) (n : ℕ) => (G a)^[n] (g a, f a)\n  have hF : Primrec fun a => (F a (encode (f a))).1 :=\n    (fst.comp <|\n      nat_iterate (encode_iff.2 hf) (pair hg hf) <|\n      hG)\n  suffices ∀ a n, F a n = (((f a).take n).foldl (fun s b => h a (s, b)) (g a), (f a).drop n) by\n    refine hF.of_eq fun a => ?_\n    rw [this, List.take_of_length_le (length_le_encode _)]\n  introv\n  dsimp only [F]\n  generalize f a = l\n  generalize g a = x\n  induction n generalizing l x with\n  | zero => rfl\n  | succ n IH =>\n    simp only [iterate_succ, comp_apply]\n    rcases l with - | ⟨b, l⟩ <;> simp [G, IH]\n\nset_option backward.privateInPublic true in\nprivate theorem list_cons' : (haveI := prim H; Primrec₂ (@List.cons β)) :=\n  letI := prim H\n  encode_iff.1 (succ.comp <| Primrec₂.natPair.comp (encode_iff.2 fst) (encode_iff.2 snd))\n\nset_option backward.privateInPublic true in\nprivate theorem list_reverse' :\n    haveI := prim H\n    Primrec (@List.reverse β) :=\n  letI := prim H\n  (list_foldl' H .id (const []) <| to₂ <| ((list_cons' H).comp snd fst).comp snd).of_eq\n    (suffices ∀ l r, List.foldl (fun (s : List β) (b : β) => b :: s) r l = List.reverseAux l r from\n      fun l => this l []\n    fun l => by induction l <;> simp [*, List.reverseAux])\n\nend\n\nnamespace Primcodable\n\nvariable {α : Type*} {β : Type*}\nvariable [Primcodable α] [Primcodable β]\n\nopen Primrec\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nset_option linter.flexible false in -- TODO: revisit this after #13791 is merged\ninstance list : Primcodable (List α) :=\n  ⟨letI H := Primcodable.prim (List ℕ)\n    have : Primrec₂ fun (a : α) (o : Option (List ℕ)) => o.map (List.cons (encode a)) :=\n      option_map snd <| (list_cons' H).comp ((@Primrec.encode α _).comp (fst.comp fst)) snd\n    have :\n      Primrec fun n =>\n        (ofNat (List ℕ) n).reverse.foldl\n          (fun o m => (@decode α _ m).bind fun a => o.map (List.cons (encode a))) (some []) :=\n      list_foldl' H ((list_reverse' H).comp (.ofNat (List ℕ))) (const (some []))\n        (Primrec.comp₂ (bind_decode_iff.2 <| .swap this) Primrec₂.right)\n    nat_iff.1 <|\n      (encode_iff.2 this).of_eq fun n => by\n        rw [List.foldl_reverse]\n        apply Nat.case_strong_induction_on n; · simp\n        intro n IH; simp\n        rcases @decode α _ n.unpair.1 with - | a; · rfl\n        simp only [Option.bind_some, Option.map_some]\n        suffices ∀ (o : Option (List ℕ)) (p), encode o = encode p →\n            encode (Option.map (List.cons (encode a)) o) = encode (Option.map (List.cons a) p) from\n          this _ _ (IH _ (Nat.unpair_right_le n))\n        intro o p IH\n        cases o <;> cases p\n        · rfl\n        · injection IH\n        · injection IH\n        · exact congr_arg (fun k => (Nat.pair (encode a) k).succ.succ) (Nat.succ.inj IH)⟩\nend Primcodable\n\nnamespace Primrec\n\nvariable {α : Type*} {β : Type*} {γ : Type*} {σ : Type*}\nvariable [Primcodable α] [Primcodable β] [Primcodable γ] [Primcodable σ]\n\ntheorem list_cons : Primrec₂ (@List.cons α) :=\n  list_cons' (Primcodable.prim _)\n\ntheorem list_casesOn {f : α → List β} {g : α → σ} {h : α → β × List β → σ} :\n    Primrec f →\n      Primrec g →\n        Primrec₂ h → @Primrec _ σ _ _ fun a => List.casesOn (f a) (g a) fun b l => h a (b, l) :=\n  list_casesOn' (Primcodable.prim _)\n\ntheorem list_foldl {f : α → List β} {g : α → σ} {h : α → σ × β → σ} :\n    Primrec f →\n      Primrec g → Primrec₂ h → Primrec fun a => (f a).foldl (fun s b => h a (s, b)) (g a) :=\n  list_foldl' (Primcodable.prim _)\n\ntheorem list_reverse : Primrec (@List.reverse α) :=\n  list_reverse' (Primcodable.prim _)\n\ntheorem list_foldr {f : α → List β} {g : α → σ} {h : α → β × σ → σ} (hf : Primrec f)\n    (hg : Primrec g) (hh : Primrec₂ h) :\n    Primrec fun a => (f a).foldr (fun b s => h a (b, s)) (g a) :=\n  (list_foldl (list_reverse.comp hf) hg <| to₂ <| hh.comp fst <| (pair snd fst).comp snd).of_eq\n    fun a => by simp [List.foldl_reverse]\n\ntheorem list_head? : Primrec (@List.head? α) :=\n  (list_casesOn .id (const none) (option_some_iff.2 <| fst.comp snd).to₂).of_eq fun l => by\n    cases l <;> rfl\n\ntheorem list_headI [Inhabited α] : Primrec (@List.headI α _) :=\n  (option_getD_default.comp list_head?).of_eq fun l => l.head!_eq_head?_getD.symm\n\ntheorem list_tail : Primrec (@List.tail α) :=\n  (list_casesOn .id (const []) (snd.comp snd).to₂).of_eq fun l => by cases l <;> rfl\n\ntheorem list_rec {f : α → List β} {g : α → σ} {h : α → β × List β × σ → σ} (hf : Primrec f)\n    (hg : Primrec g) (hh : Primrec₂ h) :\n    @Primrec _ σ _ _ fun a => List.recOn (f a) (g a) fun b l IH => h a (b, l, IH) :=\n  let F (a : α) := (f a).foldr (fun (b : β) (s : List β × σ) => (b :: s.1, h a (b, s))) ([], g a)\n  have : Primrec F :=\n    list_foldr hf (pair (const []) hg) <|\n      to₂ <| pair ((list_cons.comp fst (fst.comp snd)).comp snd) hh\n  (snd.comp this).of_eq fun a => by\n    suffices F a = (f a, List.recOn (f a) (g a) fun b l IH => h a (b, l, IH)) by rw [this]\n    dsimp [F]\n    induction f a <;> simp [*]\n\ntheorem list_getElem? : Primrec₂ ((·[·]? : List α → ℕ → Option α)) :=\n  let F (l : List α) (n : ℕ) :=\n    l.foldl\n      (fun (s : ℕ ⊕ α) (a : α) =>\n        Sum.casesOn s (@Nat.casesOn (fun _ => ℕ ⊕ α) · (Sum.inr a) Sum.inl) Sum.inr)\n      (Sum.inl n)\n  have hF : Primrec₂ F :=\n    (list_foldl fst (sumInl.comp snd)\n      ((sumCasesOn fst (nat_casesOn snd (sumInr.comp <| snd.comp fst) (sumInl.comp snd).to₂).to₂\n              (sumInr.comp snd).to₂).comp\n          snd).to₂).to₂\n  have :\n    @Primrec _ (Option α) _ _ fun p : List α × ℕ => Sum.casesOn (F p.1 p.2) (fun _ => none) some :=\n    sumCasesOn hF (const none).to₂ (option_some.comp snd).to₂\n  this.to₂.of_eq fun l n => by\n    dsimp; symm\n    induction l generalizing n with\n    | nil => rfl\n    | cons a l IH =>\n      rcases n with - | n\n      · dsimp [F]\n        clear IH\n        induction l <;> simp_all\n      · simpa using! IH ..\n\ntheorem list_getD (d : α) : Primrec₂ fun l n => List.getD l n d := by\n  simp only [List.getD_eq_getElem?_getD]\n  exact option_getD.comp₂ list_getElem? (const _)\n\ntheorem list_getI [Inhabited α] : Primrec₂ (@List.getI α _) :=\n  list_getD _\n\ntheorem list_append : Primrec₂ ((· ++ ·) : List α → List α → List α) :=\n  (list_foldr fst snd <| to₂ <| comp (@list_cons α _) snd).to₂.of_eq fun l₁ l₂ => by\n    induction l₁ <;> simp [*]\n\ntheorem list_concat : Primrec₂ fun l (a : α) => l ++ [a] :=\n  list_append.comp fst (list_cons.comp snd (const []))\n\ntheorem list_map {f : α → List β} {g : α → β → σ} (hf : Primrec f) (hg : Primrec₂ g) :\n    Primrec fun a => (f a).map (g a) :=\n  (list_foldr hf (const []) <|\n        to₂ <| list_cons.comp (hg.comp fst (fst.comp snd)) (snd.comp snd)).of_eq\n    fun a => by induction f a <;> simp [*]\n\ntheorem list_range : Primrec List.range :=\n  (nat_rec' .id (const []) ((list_concat.comp snd fst).comp snd).to₂).of_eq fun n => by\n    simp; induction n <;> simp [*, List.range_succ]\n\ntheorem list_flatten : Primrec (@List.flatten α) :=\n  (list_foldr .id (const []) <| to₂ <| comp (@list_append α _) snd).of_eq fun l => by\n    dsimp; induction l <;> simp [*]\n\ntheorem list_flatMap {f : α → List β} {g : α → β → List σ} (hf : Primrec f) (hg : Primrec₂ g) :\n    Primrec (fun a => (f a).flatMap (g a)) := list_flatten.comp (list_map hf hg)\n\ntheorem optionToList : Primrec (Option.toList : Option α → List α) :=\n  (option_casesOn Primrec.id (const [])\n    ((list_cons.comp Primrec.id (const [])).comp₂ Primrec₂.right)).of_eq\n  (fun o => by rcases o <;> simp)\n\ntheorem listFilterMap {f : α → List β} {g : α → β → Option σ}\n    (hf : Primrec f) (hg : Primrec₂ g) : Primrec fun a => (f a).filterMap (g a) :=\n  (list_flatMap hf (comp₂ optionToList hg)).of_eq\n    fun _ ↦ Eq.symm <| List.filterMap_eq_flatMap_toList _ _\n\nvariable {p : α → Prop} [DecidablePred p]\n\ntheorem list_length : Primrec (@List.length α) :=\n  (list_foldr (@Primrec.id (List α) _) (const 0) <| to₂ <| (succ.comp <| snd.comp snd).to₂).of_eq\n    fun l => by dsimp; induction l <;> simp [*]\n\n/-- Filtering a list for elements that satisfy a decidable predicate is primitive recursive. -/\n\nTarget:\ntheorem listFilter (hf : PrimrecPred p) : Primrec fun L ↦ List.filter (p ·) L :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  rw [← List.filterMap_eq_filter]\n  apply listFilterMap .id\n  simp only [Primrec₂, Option.guard, decide_eq_true_eq]\n  exact ite (hf.comp snd) (option_some_iff.mpr snd) (const none)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability/Primrec","family_id":"listfilter","file_id":"mathlib/Mathlib/Computability/Primrec/List.lean","sample_id":"61faa742462a962beb76f3f1545ae04bd1db702ab9662b4c763eedf05c97a9ff"},"split":"train"}
{"binding":{"chosen_proof_sha256":"aa83c7abdbb25c29bbdc8a05c929751eafeeab66442c5c16368b86eecdc282c9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"630ba8af62ca18a3656e76cecc9507d63e2e85a534fe0e97435b418273f39f2c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"423b546f69a419e2fd6b478d8c8a356a73c2e1fde067e823b879f61db25761a8","source_sha256":"087c4fdfa1fe64dbc38821c1fa8aea61c46d7fa4bef6ee724aa05ce31b21dbc2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← image_singleton, preimage_image_mk_eq_mul]\n  simp","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":14,"token_jaccard":0.642857,"token_length_ratio":1.555556},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"7fd556fb9d465df0ddb5d92b7d4beb52f4923831b49d8b961bbb1a1462e11f4e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Quotient\npublic import Mathlib.Algebra.Group.Action.Opposite\npublic import Mathlib.Algebra.Group.Subgroup.MulOpposite\npublic import Mathlib.GroupTheory.GroupAction.Defs\npublic import Mathlib.Algebra.Group.Pointwise.Set.Basic\n\nNamespace:\nQuotientGroup\n\nLocal context:\n/-\nCopyright (c) 2018 Mitchell Rowett. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mitchell Rowett, Kim Morrison\n-/\n/-!\n# Cosets\n\nThis file develops the basic theory of left and right cosets.\n\nWhen `G` is a group and `a : G`, `s : Set G`, with  `open scoped Pointwise` we can write:\n* the left coset of `s` by `a` as `a • s`\n* the right coset of `s` by `a` as `MulOpposite.op a • s` (or `op a • s` with `open MulOpposite`,\n  or `s <• a` with `open scoped Pointwise RightActions`)\n\nIf instead `G` is an additive group, we can write (with  `open scoped Pointwise` still)\n* the left coset of `s` by `a` as `a +ᵥ s`\n* the right coset of `s` by `a` as `AddOpposite.op a +ᵥ s` (or `op a +ᵥ s` with `open AddOpposite`,\n  or `s <+ᵥ a` with `open scoped Pointwise RightActions`)\n\n## Main definitions\n\n* `QuotientGroup.quotient s`: the quotient type representing the left cosets with respect to a\n  subgroup `s`, for an `AddGroup` this is `QuotientAddGroup.quotient s`.\n* `QuotientGroup.mk`: the canonical map from `α` to `α/s` for a subgroup `s` of `α`, for an\n  `AddGroup` this is `QuotientAddGroup.mk`.\n\n## Notation\n\n* `G ⧸ H` is the quotient of the (additive) group `G` by the (additive) subgroup `H`\n\n## TODO\n\nProperly merge with pointwise actions on sets, by renaming and deduplicating lemmas as appropriate.\n-/\n\n@[expose] public section\n\nassert_not_exists Cardinal\n\nopen Function Set\nopen scoped Pointwise\n\nvariable {α : Type*}\n\n/- Ensure that `@[to_additive]` uses the right namespace. -/\ninsert_to_additive_translation QuotientGroup QuotientAddGroup\n\nnamespace QuotientGroup\n\nvariable [Group α] (s : Subgroup α)\n\n/-- The equivalence relation corresponding to the partition of a group by left cosets\nof a subgroup. -/\n@[to_additive (attr := implicit_reducible)\n  /-- The equivalence relation corresponding to the partition of a group by left cosets\nof a subgroup. -/]\ndef leftRel : Setoid α :=\n  MulAction.orbitRel s.op α\n\nvariable {s} in\n@[to_additive]\ntheorem leftRel_apply {x y : α} : leftRel s x y ↔ x⁻¹ * y ∈ s :=\n  calc\n    (∃ a : s.op, y * MulOpposite.unop a = x) ↔ ∃ a : s, y * a = x :=\n      s.equivOp.symm.exists_congr_left\n    _ ↔ ∃ a : s, x⁻¹ * y = a⁻¹ := by\n      simp only [inv_mul_eq_iff_eq_mul, Subgroup.coe_inv, eq_mul_inv_iff_mul_eq]\n    _ ↔ x⁻¹ * y ∈ s := by simp [exists_inv_mem_iff_exists_mem]\n\n@[to_additive]\ntheorem leftRel_eq : ⇑(leftRel s) = fun x y => x⁻¹ * y ∈ s :=\n  funext₂ <| by\n    simp only [eq_iff_iff]\n    apply leftRel_apply\n\n@[to_additive]\ninstance leftRelDecidable [DecidablePred (· ∈ s)] : DecidableRel (leftRel s).r := fun x y => by\n  rw [leftRel_eq]\n  exact ‹DecidablePred (· ∈ s)› _\n\n/-- `α ⧸ s` is the quotient type representing the left cosets of `s`. If `s` is a normal subgroup,\n`α ⧸ s` is a group -/\n@[to_additive /-- `α ⧸ s` is the quotient type representing the left cosets of `s`. If `s` is a\nnormal subgroup, `α ⧸ s` is a group -/]\ninstance instHasQuotientSubgroup : HasQuotient α (Subgroup α) :=\n  ⟨fun s => Quotient (leftRel s)⟩\n\n@[to_additive]\ninstance [DecidablePred (· ∈ s)] : DecidableEq (α ⧸ s) :=\n  @Quotient.decidableEq _ _ (leftRelDecidable _)\n\n/-- The equivalence relation corresponding to the partition of a group by right cosets of a\nsubgroup. -/\n@[to_additive (attr := implicit_reducible)\n  /-- The equivalence relation corresponding to the partition of a group by right cosets\n  of a subgroup. -/]\ndef rightRel : Setoid α :=\n  MulAction.orbitRel s α\n\nvariable {s} in\n@[to_additive]\ntheorem rightRel_apply {x y : α} : rightRel s x y ↔ y * x⁻¹ ∈ s :=\n  calc\n    (∃ a : s, (a : α) * y = x) ↔ ∃ a : s, y * x⁻¹ = a⁻¹ := by\n      simp only [mul_inv_eq_iff_eq_mul, Subgroup.coe_inv, eq_inv_mul_iff_mul_eq]\n    _ ↔ y * x⁻¹ ∈ s := by simp [exists_inv_mem_iff_exists_mem]\n\n@[to_additive]\ntheorem rightRel_eq : ⇑(rightRel s) = fun x y => y * x⁻¹ ∈ s :=\n  funext₂ <| by\n    simp only [eq_iff_iff]\n    apply rightRel_apply\n\n@[to_additive]\ninstance rightRelDecidable [DecidablePred (· ∈ s)] : DecidableRel (rightRel s).r := fun x y => by\n  rw [rightRel_eq]\n  exact ‹DecidablePred (· ∈ s)› _\n\n/-- Right cosets are in bijection with left cosets. -/\n@[to_additive /-- Right cosets are in bijection with left cosets. -/]\ndef quotientRightRelEquivQuotientLeftRel : Quotient (QuotientGroup.rightRel s) ≃ α ⧸ s where\n  toFun :=\n    Quotient.map' (fun g => g⁻¹) fun a b => by\n      rw [leftRel_apply, rightRel_apply]\n      exact fun h => (congr_arg (· ∈ s) (by simp)).mp (s.inv_mem h)\n  invFun :=\n    Quotient.map' (fun g => g⁻¹) fun a b => by\n      rw [leftRel_apply, rightRel_apply]\n      exact fun h => (congr_arg (· ∈ s) (by simp)).mp (s.inv_mem h)\n  left_inv g :=\n    Quotient.inductionOn' g fun g =>\n      Quotient.sound'\n        (by\n          simp only [inv_inv]\n          exact Quotient.exact' rfl)\n  right_inv g :=\n    Quotient.inductionOn' g fun g =>\n      Quotient.sound'\n        (by\n          simp only [inv_inv]\n          exact Quotient.exact' rfl)\n\nend QuotientGroup\n\nnamespace QuotientGroup\n\nvariable [Group α] {s : Subgroup α}\n\n/-- The canonical map from a group `α` to the quotient `α ⧸ s`. -/\n@[to_additive (attr := coe)\n/-- The canonical map from an `AddGroup` `α` to the quotient `α ⧸ s`. -/]\nabbrev mk (a : α) : α ⧸ s :=\n  Quotient.mk'' a\n\n@[to_additive]\ntheorem mk_surjective : Function.Surjective <| @mk _ _ s :=\n  Quotient.mk''_surjective\n\n@[to_additive (attr := simp)]\nlemma range_mk : range (QuotientGroup.mk (s := s)) = univ := range_eq_univ.mpr mk_surjective\n\n@[to_additive (attr := elab_as_elim)]\ntheorem induction_on {C : α ⧸ s → Prop} (x : α ⧸ s) (H : ∀ z, C (QuotientGroup.mk z)) : C x :=\n  Quotient.inductionOn' x H\n\n@[to_additive]\ninstance : Coe α (α ⧸ s) :=\n  ⟨mk⟩\n\n@[to_additive] alias induction_on' := induction_on\n\n@[to_additive (attr := simp)]\ntheorem quotient_liftOn_mk {β} (f : α → β) (h) (x : α) : Quotient.liftOn' (x : α ⧸ s) f h = f x :=\n  rfl\n\n@[to_additive]\ntheorem forall_mk {C : α ⧸ s → Prop} : (∀ x : α ⧸ s, C x) ↔ ∀ x : α, C x :=\n  mk_surjective.forall\n\n@[to_additive]\ntheorem exists_mk {C : α ⧸ s → Prop} : (∃ x : α ⧸ s, C x) ↔ ∃ x : α, C x :=\n  mk_surjective.exists\n\n@[to_additive]\ninstance (s : Subgroup α) : Inhabited (α ⧸ s) :=\n  ⟨((1 : α) : α ⧸ s)⟩\n\n@[to_additive]\nprotected theorem eq {a b : α} : (a : α ⧸ s) = b ↔ a⁻¹ * b ∈ s :=\n  calc\n    _ ↔ leftRel s a b := Quotient.eq''\n    _ ↔ _ := by rw [leftRel_apply]\n\n@[to_additive]\ntheorem out_eq' (a : α ⧸ s) : mk a.out = a :=\n  Quotient.out_eq' a\n\nvariable (s)\n\n/-- It can be useful to write `obtain ⟨h, H⟩ := mk_out_eq_mul ...`, and then `rw [H]` or\n`simp_rw [H]` or `simp only [H]`. In order for `simp_rw` and `simp only` to work, this lemma is\nstated in terms of an arbitrary `h : s`, rather than the specific `h = g⁻¹ * (mk g).out`. -/\n@[to_additive QuotientAddGroup.mk_out_eq_mul]\ntheorem mk_out_eq_mul (g : α) : ∃ h : s, (mk g : α ⧸ s).out = g * h :=\n  ⟨⟨g⁻¹ * (mk g).out, QuotientGroup.eq.mp (mk g).out_eq'.symm⟩, by rw [mul_inv_cancel_left]⟩\n\nvariable {s} {a b : α}\n\n@[to_additive (attr := simp)]\ntheorem mk_mul_of_mem (a : α) (hb : b ∈ s) : (mk (a * b) : α ⧸ s) = mk a := by\n  rwa [QuotientGroup.eq, mul_inv_rev, inv_mul_cancel_right, s.inv_mem_iff]\n\n@[to_additive]\ntheorem preimage_image_mk (N : Subgroup α) (s : Set α) :\n    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = ⋃ x : N, (· * (x : α)) ⁻¹' s := by\n  ext x\n  simp only [QuotientGroup.eq, SetLike.exists, exists_prop, Set.mem_preimage, Set.mem_iUnion,\n    Set.mem_image]\n  exact\n    ⟨fun ⟨y, hs, hN⟩ => ⟨_, N.inv_mem hN, by simpa using hs⟩, fun ⟨z, hz, hxz⟩ =>\n      ⟨x * z, hxz, by simpa using hz⟩⟩\n\n@[to_additive]\ntheorem preimage_image_mk_eq_iUnion_image (N : Subgroup α) (s : Set α) :\n    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = ⋃ x : N, (· * (x : α)) '' s := by\n  rw [preimage_image_mk, iUnion_congr_of_surjective (·⁻¹) inv_surjective]\n  exact fun x ↦ image_mul_right'\n\n@[to_additive]\ntheorem preimage_image_mk_eq_mul (N : Subgroup α) (s : Set α) :\n    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = s * N := by\n  rw [preimage_image_mk_eq_iUnion_image, iUnion_subtype, ← image2_mul, ← iUnion_image_right]\n  simp only [SetLike.mem_coe]\n\n@[to_additive]\n\nTarget:\ntheorem preimage_mk_one (N : Subgroup α) :\n    mk ⁻¹' {(mk : α → α ⧸ N) 1} = N :=\n\nProof body:\n","rejected":"by\n  rw [← image_singleton, preimage_image_mk_eq_mul]\n  simp\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Coset","family_id":"preimage_mk_one","file_id":"mathlib/Mathlib/GroupTheory/Coset/Defs.lean","sample_id":"423b546f69a419e2fd6b478d8c8a356a73c2e1fde067e823b879f61db25761a8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ba5868d82bf97e5ff1d70a97d543dc36082af3a6a7d0e0038bdd9326c14ca5d6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c5320067570b9bfc10d7ea602085cbc08d55ba3d758c561b8444485698919964","source_sha256":"05b71bb08d5e0d5da2d7d8beb37172741e5772665eaf54d30c9d72516996fb96","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [CompleteLattice.isCompactElement_iff_exists_le_iSup_of_le_iSup,\n    Finset.sup_eq_iSup] at h ⊢\n  intro ι s hb\n  replace hb : (b : α) ≤ iSup ((↑) ∘ s) := le_trans hb <| (coe_iSup s) ▸ le_refl _\n  obtain ⟨t, ht⟩ := h ι ((↑) ∘ s) hb\n  exact ⟨t, (by simpa using ht : (b : α) ≤ _)⟩","hard_negative":true,"metrics":{"chosen_tokens":84,"rejected_tokens":8,"token_jaccard":0.0625,"token_length_ratio":0.095238},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"7ffcdc5e4f9f2e135eca5a9e2407f35e95c75bee1c8aeffb0030ca1ea00aa864","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.CompleteLatticeIntervals\npublic import Mathlib.Order.CompactlyGenerated.Basic\n\nNamespace:\nSet.Iic\n\nLocal context:\n/-\nCopyright (c) 2024 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Results about compactness properties for intervals in complete lattices\n-/\n\npublic section\n\nvariable {ι α : Type*} [CompleteLattice α]\n\nnamespace Set.Iic\n\nTarget:\ntheorem isCompactElement {a : α} {b : Iic a} (h : IsCompactElement (b : α)) :\n    IsCompactElement b :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"c76f8519aac34e8686ed3330142233ec4c1d3e28a09eede6f93203b2525e1c56","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/CompactlyGenerated","family_id":"iscompactelement","file_id":"mathlib/Mathlib/Order/CompactlyGenerated/Intervals.lean","sample_id":"c5320067570b9bfc10d7ea602085cbc08d55ba3d758c561b8444485698919964"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a3c59538351f137c4bd7f08ab998d351346f182e72d5bba3bd7e2722c4d90e7a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"10fa2791fbfc644d39bf2d81c08a78f073daf00278ede35b73bb7ef9624d8ac7","source_sha256":"0928fcde12666c1de471a7063a35374f64473f516c8c0caf3bbaebeb9f5bd198","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply injective_of_mono (isoFinYonedaComponents F X).hom\n  simp only [Iso.inv_hom_id_apply]\n  ext x\n  rw [isoFinYonedaComponents_hom_apply]\n  simp only [← Functor.map_comp_apply, ← op_comp, CompHausLike.const_comp,\n    ← isoFinYonedaComponents_hom_apply, Iso.inv_hom_id_apply, Function.comp_apply]","hard_negative":true,"metrics":{"chosen_tokens":49,"rejected_tokens":8,"token_jaccard":0.028571,"token_length_ratio":0.163265},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"80364c3aaa050a503407d4cd07901f98f402326d6a4e1516dcba1cb85bd9b392","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Condensed.Discrete.LocallyConstant\npublic import Mathlib.Condensed.Equivalence\npublic import Mathlib.Topology.Category.LightProfinite.Extend\n\nNamespace:\nCondensed\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# The condensed set given by left Kan extension from `FintypeCat` to `Profinite`.\n\nThis file provides the necessary API to prove that a condensed set `X` is discrete if and only if\nfor every profinite set `S = limᵢSᵢ`, `X(S) ≅ colimᵢX(Sᵢ)`, and the analogous result for light\ncondensed sets.\n-/\n\n@[expose] public section\n\nuniverse u\n\nnoncomputable section\n\nopen CategoryTheory Functor Limits FintypeCat CompHausLike.LocallyConstant\n\nnamespace Condensed\n\nsection LocallyConstantAsColimit\n\nvariable {I : Type u} [Category.{u} I] [IsCofiltered I] {F : I ⥤ FintypeCat.{u}}\n  (c : Cone <| F ⋙ toProfinite) (X : Type (u + 1))\n\n/-- The presheaf on `Profinite` of locally constant functions to `X`. -/\nabbrev locallyConstantPresheaf : Profinite.{u}ᵒᵖ ⥤ Type (u + 1) :=\n  CompHausLike.LocallyConstant.functorToPresheaves.{u, u + 1}.obj X\n\nset_option backward.defeqAttrib.useBackward true in\n/--\nThe functor `locallyConstantPresheaf` takes cofiltered limits of finite sets with surjective\nprojection maps to colimits.\n-/\nnoncomputable def isColimitLocallyConstantPresheaf (hc : IsLimit c) [∀ i, Epi (c.π.app i)] :\n    IsColimit <| (locallyConstantPresheaf X).mapCocone c.op := by\n  refine Types.FilteredColimit.isColimitOf _ _ ?_ ?_\n  · intro (f : LocallyConstant c.pt X)\n    obtain ⟨j, h⟩ := Profinite.exists_locallyConstant.{_, u} c hc f\n    exact ⟨⟨j⟩, h⟩\n  · intro ⟨i⟩ ⟨j⟩ (fi : LocallyConstant _ _) (fj : LocallyConstant _ _)\n      (h : fi.comap (c.π.app i).hom.hom = fj.comap (c.π.app j).hom.hom)\n    obtain ⟨k, ki, kj, _⟩ := IsCofilteredOrEmpty.cone_objs i j\n    refine ⟨⟨k⟩, ki.op, kj.op, ?_⟩\n    dsimp\n    ext x\n    obtain ⟨x, hx⟩ := ((Profinite.epi_iff_surjective (c.π.app k)).mp inferInstance) x\n    rw [← hx]\n    change fi ((c.π.app k ≫ (F ⋙ toProfinite).map _) x) =\n      fj ((c.π.app k ≫ (F ⋙ toProfinite).map _) x)\n    have h := LocallyConstant.congr_fun h x\n    dsimp\n    rwa [dsimp% c.w, dsimp% c.w]\n\n@[simp]\nlemma isColimitLocallyConstantPresheaf_desc_apply (hc : IsLimit c) [∀ i, Epi (c.π.app i)]\n    (s : Cocone ((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X))\n    (i : I) (f : LocallyConstant (toProfinite.obj (F.obj i)) X) :\n    dsimp% (isColimitLocallyConstantPresheaf c X hc).desc s (f.comap (c.π.app i).hom.hom) =\n      s.ι.app ⟨i⟩ f := by\n  change ((((locallyConstantPresheaf X).mapCocone c.op).ι.app ⟨i⟩) ≫\n    (isColimitLocallyConstantPresheaf c X hc).desc s) _ = _\n  rw [(isColimitLocallyConstantPresheaf c X hc).fac]\n  rfl\n\n/-- `isColimitLocallyConstantPresheaf` in the case of `S.asLimit`. -/\nnoncomputable def isColimitLocallyConstantPresheafDiagram (S : Profinite) :\n    IsColimit <| (locallyConstantPresheaf X).mapCocone S.asLimitCone.op :=\n  isColimitLocallyConstantPresheaf _ _ S.asLimit\n\n@[simp]\nlemma isColimitLocallyConstantPresheafDiagram_desc_apply (S : Profinite)\n    (s : Cocone (S.diagram.op ⋙ locallyConstantPresheaf X))\n    (i : DiscreteQuotient S) (f : LocallyConstant (S.diagram.obj i) X) :\n    dsimp% (isColimitLocallyConstantPresheafDiagram X S).desc s\n      (f.comap (S.asLimitCone.π.app i).hom.hom) = s.ι.app ⟨i⟩ f :=\n  isColimitLocallyConstantPresheaf_desc_apply S.asLimitCone X S.asLimit s i f\n\nend LocallyConstantAsColimit\n\n/--\nGiven a presheaf `F` on `Profinite`, `lanPresheaf F` is the left Kan extension of its\nrestriction to finite sets along the inclusion functor of finite sets into `Profinite`.\n-/\nabbrev lanPresheaf (F : Profinite.{u}ᵒᵖ ⥤ Type (u + 1)) : Profinite.{u}ᵒᵖ ⥤ Type (u + 1) :=\n  pointwiseLeftKanExtension toProfinite.op (toProfinite.op ⋙ F)\n\n/--\nTo presheaves on `Profinite` whose restrictions to finite sets are isomorphic have isomorphic left\nKan extensions.\n-/\ndef lanPresheafExt {F G : Profinite.{u}ᵒᵖ ⥤ Type (u + 1)}\n    (i : toProfinite.op ⋙ F ≅ toProfinite.op ⋙ G) : lanPresheaf F ≅ lanPresheaf G :=\n  leftKanExtensionUniqueOfIso _ (pointwiseLeftKanExtensionUnit _ _) i _\n    (pointwiseLeftKanExtensionUnit _ _)\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma lanPresheafExt_hom {F G : Profinite.{u}ᵒᵖ ⥤ Type (u + 1)} (S : Profinite.{u}ᵒᵖ)\n    (i : toProfinite.op ⋙ F ≅ toProfinite.op ⋙ G) : (lanPresheafExt i).hom.app S =\n      colimMap (whiskerLeft (CostructuredArrow.proj toProfinite.op S) i.hom) := by\n  simp only [lanPresheaf, lanPresheafExt,\n    leftKanExtensionUniqueOfIso_hom, pointwiseLeftKanExtension_desc_app]\n  apply colimit.hom_ext\n  aesop\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma lanPresheafExt_inv {F G : Profinite.{u}ᵒᵖ ⥤ Type (u + 1)} (S : Profinite.{u}ᵒᵖ)\n    (i : toProfinite.op ⋙ F ≅ toProfinite.op ⋙ G) : (lanPresheafExt i).inv.app S =\n      colimMap (whiskerLeft (CostructuredArrow.proj toProfinite.op S) i.inv) := by\n  simp only [lanPresheaf, lanPresheafExt,\n    leftKanExtensionUniqueOfIso_inv, pointwiseLeftKanExtension_desc_app]\n  apply colimit.hom_ext\n  aesop\n\nvariable {S : Profinite.{u}} {F : Profinite.{u}ᵒᵖ ⥤ Type (u + 1)}\n\ninstance : Final <| Profinite.Extend.functorOp S.asLimitCone :=\n  Profinite.Extend.functorOp_final S.asLimitCone S.asLimit\n\n/--\nA presheaf, which takes a profinite set written as a cofiltered limit to the corresponding\ncolimit, agrees with the left Kan extension of its restriction.\n-/\ndef lanPresheafIso (hF : IsColimit <| F.mapCocone S.asLimitCone.op) :\n    (lanPresheaf F).obj ⟨S⟩ ≅ F.obj ⟨S⟩ :=\n  (Functor.Final.colimitIso (Profinite.Extend.functorOp S.asLimitCone) _).symm ≪≫\n    (colimit.isColimit _).coconePointUniqueUpToIso hF\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma lanPresheafIso_hom (hF : IsColimit <| F.mapCocone S.asLimitCone.op) :\n    (lanPresheafIso hF).hom = colimit.desc _ (Profinite.Extend.cocone _ _) := by\n  simp [lanPresheafIso, Final.colimitIso]\n  rfl\n\nset_option backward.isDefEq.respectTransparency false in\n/-- `lanPresheafIso` is natural in `S`. -/\ndef lanPresheafNatIso (hF : ∀ S : Profinite, IsColimit <| F.mapCocone S.asLimitCone.op) :\n    lanPresheaf F ≅ F :=\n  NatIso.ofComponents (fun ⟨S⟩ ↦ (lanPresheafIso (hF S)))\n    fun _ ↦ (by simpa using colimit.hom_ext fun _ ↦ (by simp))\n\n@[simp]\nlemma lanPresheafNatIso_hom_app (hF : ∀ S : Profinite, IsColimit <| F.mapCocone S.asLimitCone.op)\n    (S : Profiniteᵒᵖ) : (lanPresheafNatIso hF).hom.app S =\n      colimit.desc _ (Profinite.Extend.cocone _ _) := by\n  simp [lanPresheafNatIso]\n\n/--\n`lanPresheaf (locallyConstantPresheaf X)` is a sheaf for the coherent topology on `Profinite`.\n-/\ndef lanSheafProfinite (X : Type (u + 1)) :\n    Sheaf (coherentTopology Profinite.{u}) (Type (u + 1)) where\n  obj := lanPresheaf (locallyConstantPresheaf X)\n  property := by\n    rw [Presheaf.isSheaf_of_iso_iff (lanPresheafNatIso\n      fun _ ↦ isColimitLocallyConstantPresheafDiagram _ _)]\n    exact ((CompHausLike.LocallyConstant.functor.{u, u + 1}\n      (hs := fun _ _ _ ↦ ((Profinite.effectiveEpi_tfae _).out 0 2).mp)).obj X).property\n\n/-- `lanPresheaf (locallyConstantPresheaf X)` as a condensed set. -/\ndef lanCondensedSet (X : Type (u + 1)) : CondensedSet.{u} :=\n  (ProfiniteCompHaus.equivalence _).functor.obj (lanSheafProfinite X)\n\nvariable (F : Profinite.{u}ᵒᵖ ⥤ Type (u + 1))\n\n/--\nThe functor which takes a finite set to the set of maps into `F(*)` for a presheaf `F` on\n`Profinite`.\n-/\n@[simps obj map]\ndef finYoneda : FintypeCat.{u}ᵒᵖ ⥤ Type (u + 1) where\n  obj X := X.unop → F.obj (toProfinite.op.obj ⟨of <| PUnit.{u + 1}⟩)\n  map f := ↾fun g ↦ g ∘ f.unop\n\n/-- `locallyConstantPresheaf` restricted to finite sets is isomorphic to `finYoneda F`. -/\n@[simps! hom_app]\ndef locallyConstantIsoFinYoneda :\n    toProfinite.op ⋙ (locallyConstantPresheaf (F.obj (toProfinite.op.obj\n      ⟨of <| PUnit.{u + 1}⟩))) ≅\n    finYoneda F :=\n  NatIso.ofComponents fun Y ↦ {\n    hom := ↾fun f ↦ f.1\n    inv := ↾fun f ↦ ⟨f, @IsLocallyConstant.of_discrete _ _ _ ⟨rfl⟩ _⟩ }\n\n/-- A finite set as a coproduct cocone in `Profinite` over itself. -/\ndef fintypeCatAsCofan (X : Profinite) :\n    Cofan (fun (_ : X) ↦ (Profinite.of (PUnit.{u + 1}))) :=\n  Cofan.mk X (fun x ↦ ConcreteCategory.ofHom (ContinuousMap.const _ x))\n\n/-- A finite set is the coproduct of its points in `Profinite`. -/\ndef fintypeCatAsCofanIsColimit (X : Profinite) [Finite X] :\n    IsColimit (fintypeCatAsCofan X) :=\n  Cofan.IsColimit.mk _ (fun t ↦ ConcreteCategory.ofHom ⟨fun x ↦ t.inj x PUnit.unit,\n    continuous_of_discreteTopology (α := X)⟩) (by aesop)\n    (fun _ _ h ↦ by ext x; exact CategoryTheory.congr_fun (h x) _)\n\nvariable [PreservesFiniteProducts F]\n\nnoncomputable instance (X : Profinite) [Finite X] :\n    PreservesLimitsOfShape (Discrete X) F :=\n  let X' := (Countable.toSmall.{0} X).equiv_small.choose\n  let e : X ≃ X' := (Countable.toSmall X).equiv_small.choose_spec.some\n  have : Finite X' := .of_equiv X e\n  preservesLimitsOfShape_of_equiv (Discrete.equivalence e.symm) F\n\n/-- Auxiliary definition for `isoFinYoneda`. -/\ndef isoFinYonedaComponents (X : Profinite.{u}) [Finite X] :\n    F.obj ⟨X⟩ ≅ (X → F.obj ⟨Profinite.of PUnit.{u + 1}⟩) :=\n  (isLimitFanMkObjOfIsLimit F _ _\n    (Cofan.IsColimit.op (fintypeCatAsCofanIsColimit X))).conePointUniqueUpToIso\n      (Types.productLimitCone.{u, u + 1} fun _ ↦ F.obj ⟨Profinite.of PUnit.{u + 1}⟩).2\n\n@[simp]\nlemma isoFinYonedaComponents_hom (X : Profinite.{u}) [Finite X] :\n    (isoFinYonedaComponents F X).hom =\n    ↾fun y x ↦ F.map ((Profinite.of PUnit.{u + 1}).const x).op y :=\n  rfl\n\nlemma isoFinYonedaComponents_hom_apply (X : Profinite.{u}) [Finite X] (y : F.obj ⟨X⟩) (x : X) :\n    (isoFinYonedaComponents F X).hom y x =\n      F.map ((Profinite.of PUnit.{u + 1}).const x).op y :=\n  rfl\n\nTarget:\nlemma isoFinYonedaComponents_inv_comp {X Y : Profinite.{u}} [Finite X] [Finite Y]\n    (f : Y → F.obj ⟨Profinite.of PUnit⟩) (g : X ⟶ Y) :\n    (isoFinYonedaComponents F X).inv (f ∘ g) = F.map g.op ((isoFinYonedaComponents F Y).inv f) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"1f22f8636252d432d5335989443b28b5235de1f454d5ae6b2b4d00bd0037deba","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed/Discrete","family_id":"isofinyonedacomponents_inv_comp","file_id":"mathlib/Mathlib/Condensed/Discrete/Colimit.lean","sample_id":"10fa2791fbfc644d39bf2d81c08a78f073daf00278ede35b73bb7ef9624d8ac7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"76e3cedf98327795879092160501512043429427fee2f5b0e827ab58125ee1b5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"fd12a4593741e5435daa603c9da043cf9d1aa8fa84fc53681487a9589609596a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"856d27b4c3ec825fbdd58bf9dd767cc218a09754dd1106fc2031d94545f79fef","source_sha256":"a532f8917e86d9e21f6417bd7287cabd2ab46e6ff2388a4cee138b7afaddc6d0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [IsSimplyConnected, simply_connected_iff_loops_nullhomotopic,\n    ← isPathConnected_iff_pathConnectedSpace]\n  refine .and .rfl ⟨fun h x p hp ↦ ?_, fun h x p ↦ ?_⟩\n  · lift x to s using by simpa using hp 0\n    rcases h x {\n      toFun := fun t ↦ ⟨p t, hp t⟩\n      source' := by simp\n      target' := by simp\n    } with ⟨F⟩\n    exact ⟨F.map (.restrict s (.id _)), fun t ↦ (F t).2⟩\n  · rcases h x (p.map continuous_subtype_val) (fun t ↦ (p t).2) with ⟨F, hF⟩\n    exact ⟨{\n      toFun t := ⟨F t, hF t⟩\n      map_zero_left := by simp\n      map_one_left := by simp\n      prop' := by simp\n    }⟩","hard_negative":true,"metrics":{"chosen_tokens":153,"rejected_tokens":5,"token_jaccard":0.054545,"token_length_ratio":0.03268},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"805c69aa145db9c04a7bf2bb1c191df87c25c463faadf4a442133214af7ed10a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps\npublic import Mathlib.Topology.Homotopy.Contractible\npublic import Mathlib.CategoryTheory.PUnit\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.PUnit\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Praneeth Kolichala. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Praneeth Kolichala\n-/\n/-!\n# Simply connected spaces\nThis file defines simply connected spaces.\nA topological space is simply connected if its fundamental groupoid is equivalent to `Unit`.\n\nWe also define the corresponding predicate for sets.\n\n## Main theorems\n  - `simply_connected_iff_unique_homotopic` - A space is simply connected if and only if it is\n    nonempty and there is a unique path up to homotopy between any two points\n\n  - `SimplyConnectedSpace.ofContractible` - A contractible space is simply connected\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen CategoryTheory\nopen scoped ContinuousMap\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n\n/-- A simply connected space is one whose fundamental groupoid is equivalent to `Discrete Unit` -/\n@[mk_iff]\nclass SimplyConnectedSpace (X : Type*) [TopologicalSpace X] : Prop where\n  equiv_unit : Nonempty (FundamentalGroupoid X ≌ Discrete Unit)\n\n@[deprecated (since := \"2026-01-08\")]\nalias simply_connected_def := simplyConnectedSpace_iff\n\ntheorem simply_connected_iff_unique_homotopic (X : Type*) [TopologicalSpace X] :\n    SimplyConnectedSpace X ↔\n      Nonempty X ∧ ∀ x y : X, Nonempty (Unique (Path.Homotopic.Quotient x y)) := by\n  simp only [simplyConnectedSpace_iff, equiv_punit_iff_unique,\n    FundamentalGroupoid.nonempty_iff X, and_congr_right_iff, Nonempty.forall]\n  intros\n  exact ⟨fun h _ _ => h _ _, fun h _ _ => h _ _⟩\n\ntheorem ContinuousMap.HomotopyEquiv.simplyConnectedSpace [hY : SimplyConnectedSpace Y]\n    (e : X ≃ₕ Y) : SimplyConnectedSpace X :=\n  ⟨hY.1.map (FundamentalGroupoidFunctor.equivOfHomotopyEquiv e).trans⟩\n\ntheorem ContinuousMap.HomotopyEquiv.simplyConnectedSpace_iff (e : X ≃ₕ Y) :\n    SimplyConnectedSpace X ↔ SimplyConnectedSpace Y :=\n  ⟨fun _ ↦ e.symm.simplyConnectedSpace, fun _ ↦ e.simplyConnectedSpace⟩\n\nnamespace SimplyConnectedSpace\n\nvariable {X : Type*} [TopologicalSpace X] [SimplyConnectedSpace X]\n\ninstance (x y : X) : Subsingleton (Path.Homotopic.Quotient x y) :=\n  @Unique.instSubsingleton _ (Nonempty.some (by\n    rw [simply_connected_iff_unique_homotopic] at *; tauto))\n\ninstance (x : X) : Subsingleton (FundamentalGroup X x) :=\n  show Subsingleton (Path.Homotopic.Quotient x x) from inferInstance\n\ninstance (priority := 100) : PathConnectedSpace X :=\n  let unique_homotopic := (simply_connected_iff_unique_homotopic X).mp inferInstance\n  { nonempty := unique_homotopic.1\n    joined := fun x y => ⟨(unique_homotopic.2 x y).some.default.out⟩ }\n\n/-- In a simply connected space, any two paths are homotopic -/\ntheorem paths_homotopic {x y : X} (p₁ p₂ : Path x y) : Path.Homotopic p₁ p₂ :=\n  Quotient.eq.mp (@Subsingleton.elim (Path.Homotopic.Quotient x y) _ _ _)\n\ninstance (priority := 100) ofContractible (Y : Type*) [TopologicalSpace Y] [ContractibleSpace Y] :\n    SimplyConnectedSpace Y :=\n  haveI : SimplyConnectedSpace Unit := ⟨⟨FundamentalGroupoid.punitEquivDiscretePUnit⟩⟩\n  (ContractibleSpace.hequiv Y Unit).some.simplyConnectedSpace\n\nend SimplyConnectedSpace\n\n/-- A space is simply connected iff it is path connected, and there is at most one path\n  up to homotopy between any two points. -/\ntheorem simply_connected_iff_paths_homotopic :\n    SimplyConnectedSpace Y ↔\n      PathConnectedSpace Y ∧ ∀ x y : Y, Subsingleton (Path.Homotopic.Quotient x y) :=\n  ⟨by intro; constructor <;> infer_instance, fun h => by\n    cases h; rw [simply_connected_iff_unique_homotopic]\n    exact ⟨inferInstance, fun x y => ⟨uniqueOfSubsingleton ⟦PathConnectedSpace.somePath x y⟧⟩⟩⟩\n\n/-- Another version of `simply_connected_iff_paths_homotopic` -/\ntheorem simply_connected_iff_paths_homotopic' :\n    SimplyConnectedSpace Y ↔\n      PathConnectedSpace Y ∧ ∀ {x y : Y} (p₁ p₂ : Path x y), Path.Homotopic p₁ p₂ := by\n  convert! simply_connected_iff_paths_homotopic (Y := Y)\n  simp [Path.Homotopic.Quotient, Setoid.eq_top_iff]; rfl\n\nset_option backward.isDefEq.respectTransparency false in\nopen Path.Homotopic.Quotient in\n/-- A space is simply connected if and only if it is path-connected and every loop\n    at any basepoint is null-homotopic (i.e., homotopic to the constant loop). -/\ntheorem simply_connected_iff_loops_nullhomotopic :\n    SimplyConnectedSpace Y ↔\n      PathConnectedSpace Y ∧ ∀ (x : Y) (γ : Path x x), Path.Homotopic γ (Path.refl x) := by\n  rw [simply_connected_iff_paths_homotopic']\n  constructor\n  · -- Forward: all paths homotopic implies all loops null-homotopic\n    intro ⟨hpc, hall⟩\n    exact ⟨hpc, fun x γ => hall γ (Path.refl x)⟩\n  · -- Backward: all loops null-homotopic implies all paths homotopic\n    intro ⟨hpc, hloops⟩\n    refine ⟨hpc, fun {x y} p₁ p₂ => ?_⟩\n    -- Work in the quotient where structural steps can be done by simp\n    rw [← eq]\n    replace hloops : ∀ (x : Y) (γ : Path x x),\n        (⟦γ⟧ : Path.Homotopic.Quotient x x) = ⟦Path.refl x⟧ :=\n      fun x γ => Quotient.sound (hloops x γ)\n    have h : trans ⟦p₁⟧ (symm ⟦p₂⟧) = refl x := by\n      simpa using hloops x (p₁.trans p₂.symm)\n    calc ⟦p₁⟧\n      _ = trans (trans ⟦p₁⟧ (symm ⟦p₂⟧)) ⟦p₂⟧ := by simp\n      _ = ⟦p₂⟧ := by grind\n\n/-!\n### Simply connected sets\n-/\n\n/-- We say that a set is simply connected if it's a simply connected topological space\nin the induced topology. -/\ndef IsSimplyConnected (s : Set X) : Prop := SimplyConnectedSpace s\n\ntheorem IsSimplyConnected.simplyConnectedSpace {s : Set X} (hs : IsSimplyConnected s) :\n    SimplyConnectedSpace s := hs\n\ntheorem IsSimplyConnected.isPathConnected {s : Set X} (hs : IsSimplyConnected s) :\n    IsPathConnected s :=\n  have := hs.simplyConnectedSpace\n  isPathConnected_iff_pathConnectedSpace.mpr inferInstance\n\ntheorem IsSimplyConnected.nonempty {s : Set X} (hs : IsSimplyConnected s) : s.Nonempty :=\n  hs.isPathConnected.nonempty\n\ntheorem Topology.IsEmbedding.isSimplyConnected_image {f : X → Y} (hf : Topology.IsEmbedding f)\n    {s : Set X} :\n    IsSimplyConnected (f '' s) ↔ IsSimplyConnected s :=\n  hf.homeomorphImage s |>.toHomotopyEquiv |>.simplyConnectedSpace_iff |>.symm\n\n@[simp]\ntheorem Homeomorph.isSimplyConnected_image (f : X ≃ₜ Y) {s : Set X} :\n    IsSimplyConnected (f '' s) ↔ IsSimplyConnected s :=\n  f.isEmbedding.isSimplyConnected_image\n\n@[simp]\ntheorem Homeomorph.isSimplyConnected_preimage (f : X ≃ₜ Y) {s : Set Y} :\n    IsSimplyConnected (f ⁻¹' s) ↔ IsSimplyConnected s := by\n  rw [← image_symm, isSimplyConnected_image]\n\n/-- A set is simply connected iff it's path connected\nand any loop is homotopic to the constant path within `s`. -/\n\nTarget:\ntheorem isSimplyConnected_iff_exists_homotopy_refl_forall_mem {s : Set X} :\n    IsSimplyConnected s ↔ IsPathConnected s ∧ ∀ x, ∀ p : Path x x, (∀ t, p t ∈ s) →\n      ∃ F : p.Homotopy (.refl x), ∀ t, F t ∈ s :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_856d27b4c3ec","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"715b96cd58ba26356397f11bc1b092b98a1a6db681d4a3160f7f414cbffd6ff8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/FundamentalGroupoid","family_id":"issimplyconnected_iff_exists_homotopy_refl_forall_mem","file_id":"mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean","sample_id":"856d27b4c3ec825fbdd58bf9dd767cc218a09754dd1106fc2031d94545f79fef"},"split":"train"}
{"binding":{"chosen_proof_sha256":"55204f99d585e2efb5ba03ffeb23f7cf47ecd4bbf63bb94951cbe11d0f4291b8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8a6273c23225b18d4154b0a3b8a34f15622fc02fe611e77262b93d044012a494","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a1caf932105bba53888c1df521c9cec9b3321d4d5015df07eac43fb607350537","source_sha256":"d9780449d2613fd42914cf0df8a7f4380debc01adc843be01bb83642da361938","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [NormNum.IsInt.neg_to_eq h rfl]\n  simp only [ne_eq, neg_eq_zero]\n  apply ne_of_gt\n  simpa using! w","hard_negative":false,"metrics":{"chosen_tokens":24,"rejected_tokens":29,"token_jaccard":0.84,"token_length_ratio":1.208333},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"80b45cf81caf0fa5b6f3a975fbef18e26c5862c7c281965b2db2efd3f90da015","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Invertible\npublic import Mathlib.Algebra.Order.Ring.Cast\npublic import Mathlib.Tactic.HaveI\npublic import Mathlib.Tactic.NormNum.Core\n\nNamespace:\nMathlib.Meta.Positivity\n\nLocal context:\n/-\nCopyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Heather Macbeth, Yaël Dillies\n-/\npublic meta import Mathlib.Control.Basic\n\n/-!\n## `positivity` core functionality\n\nThis file sets up the `positivity` tactic and the `@[positivity]` attribute,\nwhich allow for plugging in new positivity functionality around a positivity-based driver.\nThe actual behavior is in `@[positivity]`-tagged definitions in `Tactic.Positivity.Basic`\nand elsewhere.\n-/\n\npublic meta section\n\nopen Lean\nopen Lean.Meta Qq Lean.Elab Term\n\n/-- A definition of type `PositivityExt` tagged `@[positivity t]` extends the `positivity` tactic.\nThe term (with underscores) `t` indicates which expressions this extension accepts.\nAn extension will be given an expression `e : α`, together with hypotheses\n`[Zero α] [PartialOrder α]` and attempts to prove `e > 0`, `e ≥ 0`, or `e ≠ 0`.\n\nWhen `Positivity.core` calls this extension on an expression `e`, it does not guarantee that `e`\nmatches `t` perfectly: validate the form of the expression (using e.g.\n`match_expr (← withReducible (whnf e))`) before building a proof. See also the\n`let .app ... ← withReducible (whnf e) | throwError ...` lines in the example below.\n\nAn extension can call `Mathlib.Meta.Positivity.core` to recursively solve subgoals.\n\nExample:\n```lean\n@[positivity ite _ _ _] def evalIte : PositivityExt where eval {u α} zα pα e := do\n  let .app (.app (.app (.app f (p : Q(Prop))) (_ : Q(Decidable $p))) (a : Q($α))) (b : Q($α))\n    ← withReducible (whnf e) | throwError \"not ite\"\n  haveI' : $e =Q ite $p $a $b := ⟨⟩\n  guard <| ← withDefault <| withNewMCtxDepth <| isDefEq f q(ite (α := $α))\n  let ra ← core zα pα a; let rb ← core zα pα b\n  ...\n```\n-/\nsyntax (name := positivity) \"positivity \" term,+ : attr\n\nlemma ne_of_ne_of_eq' {α : Sort*} {a c b : α} (hab : (a : α) ≠ c) (hbc : a = b) : b ≠ c := hbc ▸ hab\n\nnamespace Mathlib.Meta.Positivity\n\nvariable {u : Level} {α : Q(Type u)} (zα : Q(Zero $α))\n\n/-- The result of `positivity` running on an expression `e` of type `α`. -/\ninductive Strictness (e : Q($α)) : Option Q(PartialOrder $α) → Type where\n  | positive {pα : Q(PartialOrder $α)} (pf : Q(0 < $e)) : Strictness e pα\n  | nonnegative {pα : Q(PartialOrder $α)} (pf : Q(0 ≤ $e)) : Strictness e pα\n  | nonzero {pα?} (pf : Q($e ≠ 0)) : Strictness e pα?\n  | none {pα?} : Strictness e pα?\n\n/-- Gives a generic description of the `positivity` result. -/\ndef Strictness.toString {e pα?} : Strictness zα e pα? → String\n  | positive _ => \"positive\"\n  | nonnegative _ => \"nonnegative\"\n  | nonzero _ => \"nonzero\"\n  | none => \"none\"\n\n/-- Extract a proof that `e` is positive, if possible, from `Strictness` information about `e`. -/\ndef Strictness.toPositive {e pα} : Strictness zα e (some pα) → Option Q(0 < $e)\n  | .positive pf => some pf\n  | _ => .none\n\n/-- Extract a proof that `e` is nonnegative, if possible, from `Strictness` information about `e`.\n-/\ndef Strictness.toNonneg {e pα} : Strictness zα e (some pα) → Option Q(0 ≤ $e)\n  | .positive pf => some q(le_of_lt $pf)\n  | .nonnegative pf => some pf\n  | _ => .none\n\n/-- Extract a proof that `e` is nonzero, if possible, from `Strictness` information about `e`. -/\ndef Strictness.toNonzero {e pα?} : Strictness zα e pα? → Option Q($e ≠ 0)\n  | .positive pf => some q(ne_of_gt $pf)\n  | .nonzero pf => some pf\n  | _ => .none\n\n/-- An extension for `positivity`. -/\nstructure PositivityExt where\n  /-- Attempts to prove an expression `e : α` is `>0`, `≥0`, or `≠0`. -/\n  eval {u : Level} {α : Q(Type u)} (zα : Q(Zero $α)) (pα? : Option Q(PartialOrder $α)) (e : Q($α)) :\n    MetaM (Strictness zα e pα?)\n\n/-- Read a `positivity` extension from a declaration of the right type. -/\ndef mkPositivityExt (n : Name) : ImportM PositivityExt := do\n  let { env, opts, .. } ← read\n  IO.ofExcept <| unsafe env.evalConstCheck PositivityExt opts ``PositivityExt n\n\n/-- Each `positivity` extension is labelled with a collection of patterns\nwhich determine the expressions to which it should be applied. -/\nabbrev Entry := Array (Array DiscrTree.Key) × Name\n\n/-- Environment extensions for `positivity` declarations -/\ninitialize positivityExt : PersistentEnvExtension Entry (Entry × PositivityExt)\n    (List Entry × DiscrTree PositivityExt) ←\n  -- we only need this to deduplicate entries in the DiscrTree\n  have : BEq PositivityExt := ⟨fun _ _ => false⟩\n  let insert kss v dt := kss.foldl (fun dt ks => dt.insertKeyValue ks v) dt\n  registerPersistentEnvExtension {\n    mkInitial := pure ([], {})\n    addImportedFn := fun s => do\n      let dt ← s.foldlM (init := {}) fun dt s => s.foldlM (init := dt) fun dt (kss, n) => do\n        pure (insert kss (← mkPositivityExt n) dt)\n      pure ([], dt)\n    addEntryFn := fun (entries, s) ((kss, n), ext) => ((kss, n) :: entries, insert kss ext s)\n    exportEntriesFn := fun s => s.1.reverse.toArray\n  }\n\ninitialize registerBuiltinAttribute {\n  name := `positivity\n  descr := \"adds a positivity extension\"\n  applicationTime := .afterCompilation\n  add := fun declName stx kind => match stx with\n    | `(attr| positivity $es,*) => do\n      ensureAttrDeclIsMeta `positivity declName kind\n      unless kind == AttributeKind.global do\n        throwError \"invalid attribute 'positivity', must be global\"\n      let env ← getEnv\n      unless (env.getModuleIdxFor? declName).isNone do\n        throwError \"invalid attribute 'positivity', declaration is in an imported module\"\n      if (IR.getSorryDep env declName).isSome then return -- ignore in progress definitions\n      let ext ← mkPositivityExt declName\n      let keys ← MetaM.run' <| es.getElems.mapM fun stx => do\n        let e ← TermElabM.run' <| withSaveInfoContext <| withAutoBoundImplicit <|\n          withReader ({ · with ignoreTCFailures := true }) do\n            let e ← elabTerm stx none\n            let (_, _, e) ← lambdaMetaTelescope (← mkLambdaFVars (← getLCtx).getFVars e)\n            return e\n        DiscrTree.mkPath e\n      setEnv <| positivityExt.addEntry env ((keys, declName), ext)\n      -- TODO: track what `[positivity]` decls are actually used at use sites\n      recordExtraRevUseOfCurrentModule\n    | _ => throwUnsupportedSyntax\n}\n\nvariable {A : Type*} {e : A}\n\nlemma pos_of_isNat {n : ℕ} [Semiring A] [PartialOrder A] [IsOrderedRing A] [Nontrivial A]\n    (h : NormNum.IsNat e n) (w : Nat.ble 1 n = true) : 0 < (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  apply Nat.cast_pos.2\n  simpa using! w\n\nlemma pos_of_isNat' {n : ℕ}\n    [AddMonoidWithOne A] [PartialOrder A] [AddLeftMono A] [ZeroLEOneClass A] [h'' : NeZero (1 : A)]\n    (h : NormNum.IsNat e n) (w : Nat.ble 1 n = true) : 0 < (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  apply Nat.cast_pos'.2\n  simpa using! w\n\nlemma nonneg_of_isNat {n : ℕ} [Semiring A] [PartialOrder A] [IsOrderedRing A]\n    (h : NormNum.IsNat e n) : 0 ≤ (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  exact Nat.cast_nonneg n\n\nlemma nonneg_of_isNat' {n : ℕ}\n    [AddMonoidWithOne A] [PartialOrder A] [AddLeftMono A] [ZeroLEOneClass A]\n    (h : NormNum.IsNat e n) : 0 ≤ (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  exact Nat.cast_nonneg' n\n\nTarget:\nlemma nz_of_isNegNat {n : ℕ} [Ring A] [PartialOrder A] [IsStrictOrderedRing A]\n    (h : NormNum.IsInt e (.negOfNat n)) (w : Nat.ble 1 n = true) : (e : A) ≠ 0 :=\n\nProof body:\n","rejected":"by\n  rw [NormNum.IsInt.neg_to_eq h rfl]\n  simp only [ne_eq, neg_eq_zero]\n  apply ne_of_gt\n  simpa using! w\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Positivity","family_id":"nz_of_isnegnat","file_id":"mathlib/Mathlib/Tactic/Positivity/Core.lean","sample_id":"a1caf932105bba53888c1df521c9cec9b3321d4d5015df07eac43fb607350537"},"split":"train"}
{"binding":{"chosen_proof_sha256":"aea622174bd7323409020f456ba921f86430f2be7a1ef46c09da2cf27c53cb7b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e8c807a815f30c0f1ea7687200d8d50d2f1132128320257a62e0aeb2c0ad3a77","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"79ff1c5746539b95fb7b7bd18831f461eb7a59f72b537902f6b1acccf75b210c","source_sha256":"0722d600b193d93cb9e55becbab169fa607364d0057f53064083850c9da2e6cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hh : 0 < Γ.strictWidthInfty := Γ.strictWidthInfty_pos_iff.mpr Fact.out\n  have hΓ : Γ.strictWidthInfty ∈ Γ.strictPeriods := Γ.strictWidthInfty_mem_strictPeriods\n  have hanal := ModularFormClass.analyticAt_cuspFunction_zero f hh hΓ\n  have hper := periodic_comp_ofComplex f hΓ\n  simp_rw [IsZeroAtImInfty, ZeroAtFilter, ← h, ← cuspFunction_apply_zero hh hanal hper]\n  exact (hanal.continuousAt.tendsto.comp (qParam_tendsto_atImInfty hh)).congr\n    (fun τ ↦ SlashInvariantFormClass.eq_cuspFunction f τ hΓ hh.ne')","hard_negative":true,"metrics":{"chosen_tokens":96,"rejected_tokens":5,"token_jaccard":0.0625,"token_length_ratio":0.052083},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"80c043ee395f85e0fcbaac03c01b4eefe0b5a936e4fb690089a1fb37a8431cb1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.ModularForms.QExpansion\npublic import Mathlib.NumberTheory.ModularForms.LevelOne.Basic\npublic import Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion\n\nNamespace:\nModularForm\n\nLocal context:\n/-\nCopyright (c) 2026 Chris Birkbeck. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Birkbeck\n-/\n/-!\n# Cusp form submodule and IsCuspForm predicate\n\nThis file defines the inclusion of cusp forms into modular forms as a linear map, the cusp form\nsubmodule of modular forms, and the `IsCuspForm` predicate. It also provides a direct constructor\n`ModularForm.toCuspForm` for building cusp forms from modular forms with vanishing constant\nq-expansion coefficient (for `𝒮ℒ`).\n\n## Main definitions\n\n* `CuspForm.toModularFormₗ`: the inclusion `CuspForm Γ k →ₗ[ℂ] ModularForm Γ k`.\n* `ModularForm.cuspFormSubmodule`: the submodule of `ModularForm Γ k` consisting of cusp forms.\n* `ModularForm.IsCuspForm`: predicate that a modular form lies in the cusp form submodule.\n* `ModularForm.toCuspForm`: builds a `CuspForm 𝒮ℒ k` from a `ModularForm` whose q-expansion\n  has vanishing constant term.\n\n## Main results\n\n* `CuspForm.toModularFormₗ_injective`: the inclusion is injective.\n* `CuspForm.equivCuspFormSubmodule`: `CuspForm Γ k ≃ₗ[ℂ] cuspFormSubmodule Γ k`.\n* `ModularForm.isCuspForm_iff_coeffZero_eq_zero`: for `𝒮ℒ`, `IsCuspForm` is equivalent to the\n  q-expansion having vanishing constant term.\n-/\n\n@[expose] public noncomputable section\n\nopen UpperHalfPlane ModularForm Complex SlashInvariantForm SlashInvariantFormClass\n  ModularFormClass MatrixGroups OnePoint Filter Topology\n\nvariable {Γ : Subgroup (GL (Fin 2) ℝ)} {k : ℤ}\n\nnamespace CuspForm\n\n/-- The inclusion of cusp forms into modular forms, as a ℂ-linear map. -/\ndef toModularFormₗ [Γ.HasDetOne] : CuspForm Γ k →ₗ[ℂ] ModularForm Γ k where\n  toFun := ModularFormClass.modularForm\n  map_add' _ _ := rfl\n  map_smul' _ _ := rfl\n\n@[simp]\nlemma toModularFormₗ_apply [Γ.HasDetOne] (f : CuspForm Γ k) (z : ℍ) :\n    (toModularFormₗ f) z = f z := rfl\n\nlemma toModularFormₗ_eq_coe [Γ.HasDetOne] (f : CuspForm Γ k) :\n    toModularFormₗ f = (f : ModularForm Γ k) := rfl\n\nlemma toModularFormₗ_injective [Γ.HasDetOne] :\n    Function.Injective (toModularFormₗ : CuspForm Γ k → ModularForm Γ k) :=\n  fun _ _ h ↦ DFunLike.ext _ _ fun z ↦ DFunLike.congr_fun h z\n\nend CuspForm\n\nnamespace ModularForm\n\n/-- The submodule of `ModularForm Γ k` consisting of cusp forms, defined as the range of\nthe inclusion `CuspForm.toModularFormₗ`. -/\ndef cuspFormSubmodule (Γ : Subgroup (GL (Fin 2) ℝ)) (k : ℤ) [Γ.HasDetOne] :\n    Submodule ℂ (ModularForm Γ k) :=\n  LinearMap.range CuspForm.toModularFormₗ\n\n/-- A modular form is a cusp form if it lies in the cusp form submodule. -/\ndef IsCuspForm [Γ.HasDetOne] (f : ModularForm Γ k) : Prop :=\n  f ∈ cuspFormSubmodule Γ k\n\n@[simp]\nlemma mem_cuspFormSubmodule_iff [Γ.HasDetOne] {f : ModularForm Γ k} :\n    f ∈ cuspFormSubmodule Γ k ↔ IsCuspForm f := Iff.rfl\n\n/-- The cusp form submodule is linearly equivalent to the type of cusp forms. -/\ndef CuspForm.equivCuspFormSubmodule (Γ : Subgroup (GL (Fin 2) ℝ)) (k : ℤ) [Γ.HasDetOne] :\n    CuspForm Γ k ≃ₗ[ℂ] cuspFormSubmodule Γ k :=\n  LinearEquiv.ofInjective CuspForm.toModularFormₗ CuspForm.toModularFormₗ_injective\n\n/-- The underlying modular form (via `toModularFormₗ`) of a `CuspForm` is itself a cusp form. -/\nlemma CuspForm.isCuspForm_toModularFormₗ {Γ : Subgroup (GL (Fin 2) ℝ)} [Γ.HasDetOne]\n    (f : CuspForm Γ k) : ModularForm.IsCuspForm f.toModularFormₗ := by\n  simp [← mem_cuspFormSubmodule_iff, ModularForm.cuspFormSubmodule]\n\n/-- A modular form is a cusp form if and only if it vanishes at every cusp. This is the\ngeneral characterization valid for any subgroup. -/\nlemma isCuspForm_iff [Γ.HasDetOne] (f : ModularForm Γ k) :\n    IsCuspForm f ↔ ∀ {c}, IsCusp c Γ → c.IsZeroAt f k :=\n  ⟨fun ⟨g, hg⟩ _ ↦ hg ▸ g.zero_at_cusps', fun h ↦ ⟨⟨f, f.holo', h⟩, rfl⟩⟩\n\n/-- A modular form with `valueAtInfty f = 0` is zero at infinity. -/\n\nTarget:\nlemma isZeroAtImInfty_of_valueAtInfty_eq_zero {F : Type*} [FunLike F ℍ ℂ]\n    [DiscreteTopology Γ] [Γ.HasDetPlusMinusOne] [Fact (IsCusp ∞ Γ)] [ModularFormClass F Γ k]\n    (f : F) (h : valueAtInfty f = 0) : IsZeroAtImInfty f :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_79ff1c574653","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"8a404881c04003945ba301479e23ff10d15390c814fde9caf061ea27cac82de3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/ModularForms","family_id":"iszeroatiminfty_of_valueatinfty_eq_zero","file_id":"mathlib/Mathlib/NumberTheory/ModularForms/CuspFormSubmodule.lean","sample_id":"79ff1c5746539b95fb7b7bd18831f461eb7a59f72b537902f6b1acccf75b210c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8debc390e3c9d99e3175632da6d0d7fd484f1961b70777e58cda8dfa8a7eb610","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8a042b931a223bd86aa1c34f5d07f04bc050d5bc14fa26db508bebe4d1bc04b0","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"950388e963953fd7e6eb950b29af28e1f498743a5197639d25cdf72094ea0aae","source_sha256":"127dc67ffbb65bf1977104fb0e1029785baeff1fe0e2b525c55e8f479bde944b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let a := (lfp.comp h).lfp\n  refine (lfp_le _ ?_).antisymm (lfp_le _ (Eq.le ?_))\n  · exact lfp_le _ h.onDiag.map_lfp.le\n  have ha : (lfp ∘ h) a = a := (lfp.comp h).map_lfp\n  calc\n    h a a = h a (h a).lfp := congr_arg (h a) ha.symm\n    _ = (h a).lfp := (h a).map_lfp\n    _ = a := ha","hard_negative":false,"metrics":{"chosen_tokens":105,"rejected_tokens":112,"token_jaccard":0.935484,"token_length_ratio":1.066667},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"80d0d9dd7db71d924e88033babeb58d9edcaa41785b93dcfef72da271250c033","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Dynamics.FixedPoints.Basic\npublic import Mathlib.Order.Hom.Order\npublic import Mathlib.Order.BourbakiWitt\n\nNamespace:\nOrderHom\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Kenny Lau, Yury Kudryashov\n-/\n/-!\n# Fixed point construction on complete lattices\n\nThis file sets up the basic theory of fixed points of a monotone function in a complete lattice.\n\n## Main definitions\n\n* `OrderHom.lfp`: The least fixed point of a bundled monotone function.\n* `OrderHom.gfp`: The greatest fixed point of a bundled monotone function.\n* `OrderHom.prevFixed`: The greatest fixed point of a bundled monotone function smaller than or\n  equal to a given element.\n* `OrderHom.nextFixed`: The least fixed point of a bundled monotone function greater than or\n  equal to a given element.\n* `fixedPoints.completeLattice`: The Knaster-Tarski theorem: fixed points of a monotone\n  self-map of a complete lattice form themselves a complete lattice.\n\n## Tags\n\nfixed point, complete lattice, monotone function\n-/\n\n@[expose] public section\n\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\nopen Function (fixedPoints IsFixedPt)\n\nnamespace OrderHom\n\nsection Basic\n\nvariable [CompleteLattice α] (f : α →o α)\n\n/-- Least fixed point of a monotone function -/\ndef lfp : (α →o α) →o α where\n  toFun f := sInf { a | f a ≤ a }\n  monotone' _ _ hle := sInf_le_sInf fun a ha => (hle a).trans ha\n\n/-- Greatest fixed point of a monotone function -/\ndef gfp : (α →o α) →o α where\n  toFun f := sSup { a | a ≤ f a }\n  monotone' _ _ hle := sSup_le_sSup fun a ha => le_trans ha (hle a)\n\ntheorem lfp_le {a : α} (h : f a ≤ a) : f.lfp ≤ a :=\n  sInf_le h\n\ntheorem lfp_le_fixed {a : α} (h : f a = a) : f.lfp ≤ a :=\n  f.lfp_le h.le\n\ntheorem le_lfp {a : α} (h : ∀ b, f b ≤ b → a ≤ b) : a ≤ f.lfp :=\n  le_sInf h\n\ntheorem map_le_lfp {a : α} (ha : a ≤ f.lfp) : f a ≤ f.lfp :=\n  f.le_lfp fun _ hb => (f.mono <| le_sInf_iff.1 ha _ hb).trans hb\n\n@[simp]\ntheorem map_lfp : f f.lfp = f.lfp :=\n  have h : f f.lfp ≤ f.lfp := f.map_le_lfp le_rfl\n  h.antisymm <| f.lfp_le <| f.mono h\n\ntheorem isFixedPt_lfp : IsFixedPt f f.lfp :=\n  f.map_lfp\n\ntheorem lfp_le_map {a : α} (ha : f.lfp ≤ a) : f.lfp ≤ f a :=\n  calc\n    f.lfp = f f.lfp := f.map_lfp.symm\n    _ ≤ f a := f.mono ha\n\ntheorem isLeast_lfp_le : IsLeast { a | f a ≤ a } f.lfp :=\n  ⟨f.map_lfp.le, fun _ => f.lfp_le⟩\n\ntheorem isLeast_lfp : IsLeast (fixedPoints f) f.lfp :=\n  ⟨f.isFixedPt_lfp, fun _ => f.lfp_le_fixed⟩\n\ntheorem lfp_induction {p : α → Prop} (step : ∀ a, p a → a ≤ f.lfp → p (f a))\n    (hSup : ∀ s, (∀ a ∈ s, p a) → p (sSup s)) : p f.lfp := by\n  set s := { a | a ≤ f.lfp ∧ p a }\n  specialize hSup s fun a => And.right\n  suffices sSup s = f.lfp from this ▸ hSup\n  have h : sSup s ≤ f.lfp := sSup_le fun b => And.left\n  have hmem : f (sSup s) ∈ s := ⟨f.map_le_lfp h, step _ hSup h⟩\n  exact h.antisymm (f.lfp_le <| le_sSup hmem)\n\ntheorem le_gfp {a : α} (h : a ≤ f a) : a ≤ f.gfp :=\n  le_sSup h\n\ntheorem gfp_le {a : α} (h : ∀ b, b ≤ f b → b ≤ a) : f.gfp ≤ a :=\n  sSup_le h\n\ntheorem isFixedPt_gfp : IsFixedPt f f.gfp :=\n  f.dual.isFixedPt_lfp\n\n@[simp]\ntheorem map_gfp : f f.gfp = f.gfp :=\n  f.dual.map_lfp\n\ntheorem map_le_gfp {a : α} (ha : a ≤ f.gfp) : f a ≤ f.gfp :=\n  f.dual.lfp_le_map ha\n\ntheorem gfp_le_map {a : α} (ha : f.gfp ≤ a) : f.gfp ≤ f a :=\n  f.dual.map_le_lfp ha\n\ntheorem isGreatest_gfp_le : IsGreatest { a | a ≤ f a } f.gfp :=\n  f.dual.isLeast_lfp_le\n\ntheorem isGreatest_gfp : IsGreatest (fixedPoints f) f.gfp :=\n  f.dual.isLeast_lfp\n\ntheorem gfp_induction {p : α → Prop} (step : ∀ a, p a → f.gfp ≤ a → p (f a))\n    (hInf : ∀ s, (∀ a ∈ s, p a) → p (sInf s)) : p f.gfp :=\n  f.dual.lfp_induction step hInf\n\ntheorem lfp_le_gfp : f.lfp ≤ f.gfp :=\n  f.lfp_le_fixed f.isFixedPt_gfp\n\nend Basic\n\nsection Eqn\n\nvariable [CompleteLattice α] [CompleteLattice β] (f : β →o α) (g : α →o β)\n\n-- Rolling rule\ntheorem map_lfp_comp : f (g.comp f).lfp = (f.comp g).lfp :=\n  le_antisymm ((f.comp g).map_lfp ▸ f.mono (lfp_le_fixed _ <| congr_arg g (f.comp g).map_lfp)) <|\n    lfp_le _ (congr_arg f (g.comp f).map_lfp).le\n\ntheorem map_gfp_comp : f (g.comp f).gfp = (f.comp g).gfp :=\n  f.dual.map_lfp_comp g.dual\n\n-- Diagonal rule\n\nTarget:\ntheorem lfp_lfp (h : α →o α →o α) : (lfp.comp h).lfp = h.onDiag.lfp :=\n\nProof body:\n","rejected":"```lean\nby\n  let a := (lfp.comp h).lfp\n  refine (lfp_le _ ?_).antisymm (lfp_le _ (Eq.le ?_))\n  · exact lfp_le _ h.onDiag.map_lfp.le\n  have ha : (lfp ∘ h) a = a := (lfp.comp h).map_lfp\n  calc\n    h a a = h a (h a).lfp := congr_arg (h a) ha.symm\n    _ = (h a).lfp := (h a).map_lfp\n    _ = a := ha\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order","family_id":"lfp_lfp","file_id":"mathlib/Mathlib/Order/FixedPoints.lean","sample_id":"950388e963953fd7e6eb950b29af28e1f498743a5197639d25cdf72094ea0aae"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bb1d914f2c85ca879a4ab7ab44641a41b11803eff45090adf0fe5af8fb8479b6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f45a84b72ed006d2229b6b12ac3c6213f31b977445439387e620eb45989aaed9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"26f5274475c00cabc639fcde1f5b699c28f967e875673ef858d4e214040ea235","source_sha256":"8deb8a50d24f74f7392dcccfc5a64bc688ef77075e655438ef9523a05e0e1481","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction n <;> grind [AllZero, sign]","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.25},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"811ac0e59280edbc88efe5b4908bafc2eca30cca87b7b419ca960f73b0c76b0f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Real.Basic\n\nNamespace:\nTactic.ComputeAsymptotics.UnitMonomial.AllZero\n\nLocal context:\n/-\nCopyright (c) 2026 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\n/-!\n\n# Predicates on monomials\n\nIn this file we define `UnitMonomial`: type to represent monomials without coefficient as a list of\nits exponents.  `[e₁, e₂, ..., eₙ]` corresponds to `basis[0] ^ e₁ * ... * basis[n] ^ eₙ` where\n`basis` is the basis of functions.\n\nThen we define some predicates for these lists:\n1. `FirstNonzeroIsPos li` means that the first non-zero element of the list `li` is positive.\n2. `FirstNonzeroIsNeg li` means that the first non-zero element of the list `li` is negative.\n3. `AllZero li` means that all elements in `li` are zero.\n\nThis trichotomy determines the asymptotic behaviour of a monomial:\n`FirstNonzeroIsPos` means it tends to infinity, `FirstNonzeroIsNeg` means it tends to zero and\n`AllZero` means it tends to a constant.\n-/\n\n@[expose] public section\n\nnamespace Tactic.ComputeAsymptotics\n\n/-- Unit monomial, represented as a list of its exponents. `[e₁, e₂, ..., eₙ]` corresponds to\n`basis[0] ^ e₁ * ... * basis[n] ^ eₙ` where `basis` is the basis of functions. -/\nabbrev UnitMonomial := List ℝ\n\nnamespace UnitMonomial\n\n/-- Type representing a sign of the first non-zero exponent, returned by `sign`. -/\ninductive Sign\n| pos | neg | zero\n\n/-- Sign of the first non-zero exponent of a unit monomial. -/\nnoncomputable def sign : UnitMonomial → Sign\n  | [] => .zero\n  | hd :: tl =>\n    if 0 < hd then\n      .pos\n    else if hd < 0 then\n      .neg\n    else\n      sign tl\n\n/-- Predicate stating that the first non-zero exponent is positive. -/\ndef FirstNonzeroIsPos (m : UnitMonomial) : Prop := m.sign = .pos\n\n/-- Predicate stating that the first non-zero exponent is negative. -/\ndef FirstNonzeroIsNeg (m : UnitMonomial) : Prop := m.sign = .neg\n\n/-- Predicate stating that all exponents are zero. -/\ndef AllZero (m : UnitMonomial) : Prop := m.sign = .zero\n\nnamespace AllZero\n\ntheorem nil : AllZero [] :=\n  rfl\n\n@[simp]\ntheorem cons_iff {hd : ℝ} {tl : UnitMonomial} :\n    AllZero (hd :: tl) ↔ hd = 0 ∧ AllZero tl := by\n  grind [AllZero, sign]\n\ntheorem of_tail {hd : ℝ} {tl : UnitMonomial} (h_hd : hd = 0) (h_tl : AllZero tl) :\n    AllZero (hd :: tl) :=\n  cons_iff.mpr ⟨h_hd, h_tl⟩\n\nTarget:\ntheorem replicate {n : ℕ} : AllZero (List.replicate n 0) :=\n\nProof body:\n","rejected":"by\n  exact replicate","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"10a81969658925fb3d5e868b101ed603a8a778e798c7181dc8348d53927a6491","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/ComputeAsymptotics","family_id":"replicate","file_id":"mathlib/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Predicates.lean","sample_id":"26f5274475c00cabc639fcde1f5b699c28f967e875673ef858d4e214040ea235"},"split":"train"}
{"binding":{"chosen_proof_sha256":"12279e85ffc4d0d8cf16eae5dea8e0e2916e98350ea6156d422f2d5856bb9f1e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"004b239a5e44a3546f0f75cadaa65c18122de1beabd1034b95549cdd672fb0e3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"15e4f134fc4396fd42f4d434d7605aca7a909498c0d0844c2e35d252b2adb530","source_sha256":"7bb1b48aebf21c1532e493ca79c87d1bb8ee632b2dfff76d51a712868ddd9efa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← sSup_image']\n  exact hd.isLUB_csSup (Hne.image _) H","hard_negative":true,"metrics":{"chosen_tokens":17,"rejected_tokens":3,"token_jaccard":0.117647,"token_length_ratio":0.176471},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"81b7fd513fcb9efa4f0e9aa8bbde845ce8380abe6dce7748846d2dc911967017","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ConditionallyCompletePartialOrder.Basic\npublic import Mathlib.Order.GaloisConnection.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2026 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\n/-!\n# Indexed sup / inf in conditionally complete lattices\n\nThis file proves lemmas about `iSup` and `iInf` for functions valued in a conditionally complete\npartial order, as opposed to a conditionally complete lattice.\n\n## TODO\n\n+ Use `@[to_dual]` in the `GaloisConnection` and `OrderIso` sections.\n\n-/\n\npublic section\n\n-- Guard against import creep\nassert_not_exists Multiset\n\nopen Function OrderDual Set\n\nvariable {α β γ : Type*} {ι : Sort*}\n\nsection ConditionallyCompletePartialOrderSup\n\nvariable [ConditionallyCompletePartialOrderSup α] {a b : α}\n\n@[to_dual]\ntheorem Directed.isLUB_ciSup [Nonempty ι] {f : ι → α} (hd : Directed (· ≤ ·) f)\n    (H : BddAbove (range f)) : IsLUB (range f) (⨆ i, f i) :=\n  hd.directedOn_range.isLUB_csSup (range_nonempty f) H\n\n@[to_dual]\n\nTarget:\ntheorem DirectedOn.isLUB_ciSup_set {f : β → α} {s : Set β} (hd : DirectedOn (· ≤ ·) (f '' s))\n    (H : BddAbove (f '' s)) (Hne : s.Nonempty) :\n    IsLUB (f '' s) (⨆ i : s, f i) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_15e4f134fc43","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c26714ee1632f16052aba31ec318325473c1f4a1a2b53c56fd9a75d48ff78df1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/ConditionallyCompletePartialOrder","family_id":"directedon","file_id":"mathlib/Mathlib/Order/ConditionallyCompletePartialOrder/Indexed.lean","sample_id":"15e4f134fc4396fd42f4d434d7605aca7a909498c0d0844c2e35d252b2adb530"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fcb2f3f3485ce6a70b3afb69269f67e2beb686789a095f74d88d7d21aa424579","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9b1bc91f5f2c46f4c33f074e71ad609e2c8759ced8861ac7f87ef1f11e0dca63","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"502371bcc584859610c49145091eb701ac3e0cd90f5beb5f633781b964813dd7","source_sha256":"522c0739e8f7535bf0f954aff339b473dbd0f9e903c0b5a9a4fff87d391d66b3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext S; simp only [smul_def, _root_.zero_smul, zero_def, Pi.zero_apply]","hard_negative":true,"metrics":{"chosen_tokens":19,"rejected_tokens":5,"token_jaccard":0.105263,"token_length_ratio":0.263158},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"81d3407b3d28547a738e18544fd20b29e55ea7d849313642f457bd6e5acd14cc","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorProduct.RightExactness\npublic import Mathlib.RingTheory.Congruence.Hom\npublic import Mathlib.RingTheory.FiniteType\npublic import Mathlib.RingTheory.TensorProduct.DirectLimitFG\n\nNamespace:\nPolynomialLaw\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir & María-Inés de Frutos-Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir & María-Inés de Frutos-Fernández\n-/\n/-! # Polynomial laws on modules\n\nLet `M` and `N` be a modules over a commutative ring `R`.\nA polynomial law `f : PolynomialLaw R M N`, with notation `f : M →ₚₗₗ[R] N`,\nis a “law” that assigns a natural map `PolynomialLaw.toFun' f S : S ⊗[R] M → S ⊗[R] N`\nfor every `R`-algebra `S`.\n\nFor type-theoretic reasons, if `R : Type u`, then the definition of the polynomial map `f`\nis restricted to `R`-algebras `S` such that `S : Type u`.\nUsing the fact that a module is the direct limit of its finitely generated submodules, that a\nfinitely generated subalgebra is a quotient of a polynomial ring in the universe `u`, plus\nthe commutation of tensor products with direct limits, we extend the functor\nto all `R`-algebras.\n\nThe two fields involving the definition of `PolynomialLaw`,\n`PolynomialLaw.toFun'` and `PolynomialLaw.isCompat'` are primed.\nThey are superseded by their universe-polymorphic counterparts,\nthe definition `PolynomialLaw.toFun` and the lemma `PolynomialLaw.isCompat`\nwhich should be used once the theory is properly stated.\n\nFor constructions of general definitions of `PolynomialLaw`\nat a universe-polymorphic level, one needs to lift\nelements in a tensor product to smaller universes.\nFor this, one can make use of\n`PolynomialLaw.exists_lift` or `PolynomialLaw.exists_lift'`,\nor establish appropriate generalizations.\n\n## Main definitions/lemmas\n\n* Instance : `Module R (M →ₚₗ[R] N)` shows that polynomial laws form an `R`-module.\n\n* `PolynomialLaw.ground f` is the map `M → N` corresponding to `PolynomialLaw.toFun' f R` under\n  the isomorphisms `R ⊗[R] M ≃ₗ[R] M`, and similarly for `N`.\n\nIn further works, we construct the coefficients of a polynomial law and show the relation with\npolynomials (when the module `M` is free and finite).\n\n## Implementation notes\n\nIn the literature, the theory is written for commutative rings, but this implementation\nonly assumes `R` is a commutative semiring.\n\n## References\n\n* [Roby, Norbert. 1963. «Lois polynomes et lois formelles en théorie des modules».\n  Annales scientifiques de l’École Normale Supérieure 80 (3): 213‑348](Roby-1963)\n\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nnoncomputable section PolynomialLaw\n\nopen scoped TensorProduct\n\nopen LinearMap TensorProduct AlgHom RingCon\n\n/-- A polynomial law `M →ₚₗ[R] N` between `R`-modules is a functorial family of maps\n`S ⊗[R] M → S ⊗[R] N`, for all `R`-algebras `S`.\n\nFor universe reasons, `S` has to be restricted to the same universe as `R`. -/\n@[ext]\nstructure PolynomialLaw (R : Type u) [CommSemiring R]\n    (M : Type*) [AddCommMonoid M] [Module R M] (N : Type*) [AddCommMonoid N] [Module R N] where\n  /-- The functions `S ⊗[R] M → S ⊗[R] N` underlying a polynomial law -/\n  toFun' (S : Type u) [CommSemiring S] [Algebra R S] : S ⊗[R] M → S ⊗[R] N\n  /-- The compatibility relations between the functions underlying a polynomial law -/\n  isCompat' {S : Type u} [CommSemiring S] [Algebra R S]\n    {S' : Type u} [CommSemiring S'] [Algebra R S'] (φ : S →ₐ[R] S') :\n    φ.toLinearMap.rTensor N ∘ toFun' S = toFun' S' ∘ φ.toLinearMap.rTensor M := by aesop\n\n/-- `M →ₚₗ[R] N` is the type of `R`-polynomial laws from `M` to `N`. -/\nnotation:25 M \" →ₚₗ[\" R:25 \"] \" N:0 => PolynomialLaw R M N\n\n@[local simp]\ntheorem PolynomialLaw.isCompat_apply'\n    {R : Type u} [CommSemiring R] {M : Type*} [AddCommMonoid M] [Module R M]\n    {N : Type*} [AddCommMonoid N] [Module R N] {f : M →ₚₗ[R] N}\n    {S : Type u} [CommSemiring S] [Algebra R S] {S' : Type u} [CommSemiring S'] [Algebra R S']\n    (φ : S →ₐ[R] S') (x : S ⊗[R] M) :\n    (φ.toLinearMap.rTensor N) ((f.toFun' S) x) = (f.toFun' S') (φ.toLinearMap.rTensor M x) := by\n  simpa only using! congr_fun (f.isCompat' φ) x\n\nattribute [local simp] PolynomialLaw.isCompat_apply'\n\nnamespace PolynomialLaw\n\nsection Module\n\nsection CommSemiring\n\nvariable {R : Type u} [CommSemiring R] {M : Type*} [AddCommMonoid M] [Module R M]\n  {N : Type*} [AddCommMonoid N] [Module R N] (r a b : R) (f g : M →ₚₗ[R] N)\n\ninstance : Zero (M →ₚₗ[R] N) := ⟨{ toFun' _ := 0 }⟩\n\n@[simp]\ntheorem zero_def (S : Type u) [CommSemiring S] [Algebra R S] :\n    (0 : PolynomialLaw R M N).toFun' S = 0 := rfl\n\ninstance : Inhabited (PolynomialLaw R M N) := ⟨Zero.zero⟩\n\n/-- The identity as a polynomial law -/\ndef id : M →ₚₗ[R] M where\n  toFun' S _ _ := _root_.id\n\ntheorem id_apply' {S : Type u} [CommSemiring S] [Algebra R S] :\n    (id : M →ₚₗ[R] M).toFun' S = _root_.id := rfl\n\n/-- The sum of two polynomial laws -/\nnoncomputable def add : M →ₚₗ[R] N where\n  toFun' S _ _ := f.toFun' S + g.toFun' S\n\ninstance : Add (PolynomialLaw R M N) := ⟨add⟩\n\n@[simp]\ntheorem add_def (S : Type u) [CommSemiring S] [Algebra R S] :\n    (f + g).toFun' S = f.toFun' S + g.toFun' S := rfl\n\ntheorem add_def_apply (S : Type u) [CommSemiring S] [Algebra R S] (m : S ⊗[R] M) :\n    (f + g).toFun' S m = f.toFun' S m + g.toFun' S m := rfl\n\n/-- External multiplication of a `f : M →ₚₗ[R] N` by `r : R` -/\ndef smul : M →ₚₗ[R] N where\n  toFun' S _ _ := r • f.toFun' S\n\ninstance : SMul R (M →ₚₗ[R] N) := ⟨smul⟩\n\n@[simp]\ntheorem smul_def (S : Type u) [CommSemiring S] [Algebra R S] :\n    (r • f).toFun' S = r • f.toFun' S := rfl\n\ntheorem smul_def_apply (S : Type u) [CommSemiring S] [Algebra R S] (m : S ⊗[R] M) :\n    (r • f).toFun' S m = r • f.toFun' S m := rfl\n\ntheorem add_smul : (a + b) • f = a • f + b • f := by\n  ext; simp only [add_def, smul_def, _root_.add_smul]\n\nTarget:\ntheorem zero_smul : (0 : R) • f = 0 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_502371bcc584","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"a09456493b6fb28b4218e93f6ee55e2f087e188c43ded7aab93ec3ad13e98b82","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/PolynomialLaw","family_id":"zero_smul","file_id":"mathlib/Mathlib/RingTheory/PolynomialLaw/Basic.lean","sample_id":"502371bcc584859610c49145091eb701ac3e0cd90f5beb5f633781b964813dd7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"61e6f75295dcf0ee0f0edfdb232d6f4ff00a1685b5790d5e3eca26ccc61aea57","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"62a92841f7b0315f37f375dabbb5c7f5c1a02e9440658c8976f773452e890d7a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"91a92b19287fd834779cff643bf9db6f489d497152bb848e415ab87fcceac41f","source_sha256":"8a5f3265148069c15c32da8cd4a02af932338089cab2e996aa5a2f61253214b2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Module.finrank_mul_finrank ℝ ℂ E, Complex.finrank_real_complex]","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.0625,"token_length_ratio":0.2},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"81f711ad1b8ac02184f4441743bc797007071394584bf6ec2a41d3c53b4cfea1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Rat\npublic import Mathlib.Analysis.Complex.Cardinality\npublic import Mathlib.LinearAlgebra.Complex.Module\npublic import Mathlib.LinearAlgebra.FiniteDimensional.Defs\npublic import Mathlib.Order.Interval.Set.Infinite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser\n-/\n/-!\n# Complex number as a finite-dimensional vector space over `ℝ`\n\nThis file contains the `FiniteDimensional ℝ ℂ` instance, as well as some results about the rank\n(`finrank` and `Module.rank`).\n-/\n\npublic section\n\nopen Module\n\nnamespace Complex\n\ninstance : FiniteDimensional ℝ ℂ := basisOneI.finiteDimensional_of_finite\n\n/-- `ℂ` is a finite extension of `ℝ` of degree 2, i.e `[ℂ : ℝ] = 2` -/\n@[simp, stacks 09G4]\ntheorem finrank_real_complex : finrank ℝ ℂ = 2 := by\n  rw [finrank_eq_card_basis basisOneI, Fintype.card_fin]\n\n@[simp]\ntheorem rank_real_complex : Module.rank ℝ ℂ = 2 := by simp [← finrank_eq_rank, finrank_real_complex]\n\ntheorem rank_real_complex'.{u} : Cardinal.lift.{u} (Module.rank ℝ ℂ) = 2 := by\n  rw [← finrank_eq_rank, finrank_real_complex, Cardinal.lift_natCast, Nat.cast_ofNat]\n\n/-- `Fact` version of the dimension of `ℂ` over `ℝ`, locally useful in the definition of the\ncircle. -/\ntheorem finrank_real_complex_fact : Fact (finrank ℝ ℂ = 2) :=\n  ⟨finrank_real_complex⟩\n\nend Complex\n\ninstance (priority := 500) FiniteDimensional.complexToReal (E : Type*) [AddCommGroup E]\n    [Module ℂ E] [FiniteDimensional ℂ E] : FiniteDimensional ℝ E :=\n  FiniteDimensional.trans ℝ ℂ E\n\ntheorem rank_real_of_complex (E : Type*) [AddCommGroup E] [Module ℂ E] :\n    Module.rank ℝ E = 2 * Module.rank ℂ E :=\n  Cardinal.lift_inj.{_, 0}.1 <| by\n    rw [← lift_rank_mul_lift_rank ℝ ℂ E, Complex.rank_real_complex']\n    simp only [Cardinal.lift_id']\n\nTarget:\ntheorem finrank_real_of_complex (E : Type*) [AddCommGroup E] [Module ℂ E] :\n    Module.finrank ℝ E = 2 * Module.finrank ℂ E :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_91a92b19287f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"4fa09e8b777917f0fc855de1a67270a8b209d731327ca48aeb7bab0287d9fa86","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Complex","family_id":"finrank_real_of_complex","file_id":"mathlib/Mathlib/LinearAlgebra/Complex/FiniteDimensional.lean","sample_id":"91a92b19287fd834779cff643bf9db6f489d497152bb848e415ab87fcceac41f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9ee4d424ebdd38c7fae4d49084f0c22e4992fdbfa58721d1092ad5e07b192bfb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"bb91f322cc7379a2e09992630fffcf6b4834d16b255e5056a81a962a96c37e57","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ea7f7ba297977c69513c6abcc1bfadf80c88bd3e61590c3a43e8307b939f76d2","source_sha256":"e2ead2d6b414091f83a91e8561b014394407eaedab6a9801bc467d9fa54fc95c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  exact div_nonneg (Finset.sum_nonneg fun _ _ => sq_nonneg _) <| sq_nonneg _","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":22,"token_jaccard":0.823529,"token_length_ratio":1.222222},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"824f583e5d71aaedc7228d34a02a05b2fbf8cff1e2e51a75089ed1fbc9e987a0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.NatInt\npublic import Mathlib.Algebra.Order.BigOperators.Group.Finset\npublic import Mathlib.Combinatorics.SimpleGraph.Density\npublic import Mathlib.Data.Rat.BigOperators\n\nNamespace:\nFinpartition\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Energy of a partition\n\nThis file defines the energy of a partition.\n\nThe energy is the auxiliary quantity that drives the induction process in the proof of Szemerédi's\nRegularity Lemma. As long as we do not have a suitable equipartition, we will find a new one that\nhas an energy greater than the previous one plus some fixed constant.\n\n## References\n\n[Yaël Dillies, Bhavik Mehta, *Formalising Szemerédi’s Regularity Lemma in Lean*][srl_itp]\n-/\n\n@[expose] public section\n\n\nopen Finset\n\nvariable {α : Type*} [DecidableEq α] {s : Finset α} (P : Finpartition s) (G : SimpleGraph α)\n  [DecidableRel G.Adj]\n\nnamespace Finpartition\n\n/-- The energy of a partition, also known as index. Auxiliary quantity for Szemerédi's regularity\nlemma. -/\ndef energy : ℚ :=\n  ((∑ uv ∈ P.parts.offDiag, G.edgeDensity uv.1 uv.2 ^ 2) : ℚ) / (#P.parts : ℚ) ^ 2\n\nTarget:\ntheorem energy_nonneg : 0 ≤ P.energy G :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  exact div_nonneg (Finset.sum_nonneg fun _ _ => sq_nonneg _) <| sq_nonneg _","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/SimpleGraph","family_id":"energy_nonneg","file_id":"mathlib/Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean","sample_id":"ea7f7ba297977c69513c6abcc1bfadf80c88bd3e61590c3a43e8307b939f76d2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4644b39b47ec31a8d61ff06e763a66da7cadd0e1fb2b3309d9d1db8d535d144c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"184989b8ace5b0c3b0fc6a3e72a06309693583fa52690d69dba247b7c7c635f8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6d45385fed9b0ad68c7032cd158ad8e6ed294ff9a6d585f7ef6679ce0eea1fd4","source_sha256":"45bff0b3bef16dadf0bda36c7fa30586e758b281d4f3d18c6409891ebf942e27","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← sub_eq_zero]; exact hS.eq_zero_of_inner_left 𝕜 (by simpa [inner_sub_left, sub_eq_zero])","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":2,"token_jaccard":0.055556,"token_length_ratio":0.095238},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"8258f8821e7aa337285f5fc86a9124c987d838608cb3e9715e0683c906b2705e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.InnerProductSpace.Basic\npublic import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Zhouhang Zhou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis\n-/\n/-!\n# Continuity of inner product\n\nWe show that the inner product is continuous, `continuous_inner`.\n\n## Tags\n\ninner product space, Hilbert space, norm\n\n-/\n\npublic section\n\nnoncomputable section\n\nopen RCLike Real Filter Topology ComplexConjugate Finsupp\nopen LinearMap renaming BilinForm → BilinForm\n\nvariable {𝕜 E F : Type*} [RCLike 𝕜]\n\n\nsection Continuous\n\nvariable [SeminormedAddCommGroup E] [InnerProductSpace 𝕜 E]\n\nlocal notation \"⟪\" x \", \" y \"⟫\" => inner 𝕜 x y\n\n/-!\n### Continuity of the inner product\n-/\n\n/-- When an inner product space `E` over `𝕜` is considered as a real normed space, its inner\nproduct satisfies `IsBoundedBilinearMap`.\n\nIn order to state these results, we need a `NormedSpace ℝ E` instance. We will later establish\nsuch an instance by restriction-of-scalars, `InnerProductSpace.rclikeToReal 𝕜 E`, but this\ninstance may be not definitionally equal to some other “natural” instance. So, we assume\n`[NormedSpace ℝ E]`.\n-/\ntheorem _root_.isBoundedBilinearMap_inner [NormedSpace ℝ E] [IsScalarTower ℝ 𝕜 E] :\n    IsBoundedBilinearMap ℝ fun p : E × E => ⟪p.1, p.2⟫ :=\n  { add_left := inner_add_left\n    smul_left := fun r x y => by\n      simp only [← algebraMap_smul 𝕜 r x, algebraMap_eq_ofReal, inner_smul_real_left]\n    add_right := inner_add_right\n    smul_right := fun r x y => by\n      simp only [← algebraMap_smul 𝕜 r y, algebraMap_eq_ofReal, inner_smul_real_right]\n    bound :=\n      ⟨1, zero_lt_one, fun x y => by\n        rw [one_mul]\n        exact norm_inner_le_norm x y⟩ }\n\ntheorem continuous_inner : Continuous fun p : E × E => ⟪p.1, p.2⟫ :=\n  letI : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E\n  haveI := IsScalarTower.restrictScalars ℝ 𝕜 E\n  isBoundedBilinearMap_inner.continuous\n\nvariable {α : Type*}\n\ntheorem Filter.Tendsto.inner {f g : α → E} {l : Filter α} {x y : E} (hf : Tendsto f l (𝓝 x))\n    (hg : Tendsto g l (𝓝 y)) : Tendsto (fun t => ⟪f t, g t⟫) l (𝓝 ⟪x, y⟫) :=\n  (continuous_inner.tendsto _).comp (hf.prodMk_nhds hg)\n\nvariable [TopologicalSpace α] {f g : α → E} {x : α} {s : Set α}\n\ntheorem ContinuousWithinAt.inner (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :\n    ContinuousWithinAt (fun t => ⟪f t, g t⟫) s x :=\n  Filter.Tendsto.inner hf hg\n\n@[fun_prop]\ntheorem ContinuousAt.inner (hf : ContinuousAt f x) (hg : ContinuousAt g x) :\n    ContinuousAt (fun t => ⟪f t, g t⟫) x :=\n  Filter.Tendsto.inner hf hg\n\n@[fun_prop]\ntheorem ContinuousOn.inner (hf : ContinuousOn f s) (hg : ContinuousOn g s) :\n    ContinuousOn (fun t => ⟪f t, g t⟫) s := fun x hx => (hf x hx).inner (hg x hx)\n\n@[continuity, fun_prop]\ntheorem Continuous.inner (hf : Continuous f) (hg : Continuous g) : Continuous fun t => ⟪f t, g t⟫ :=\n  continuous_iff_continuousAt.2 fun _x => by fun_prop\n\nend Continuous\n\nopen Submodule\n\nvariable {E F ι : Type*}\nvariable (𝕜 : Type*) [RCLike 𝕜]\nvariable [NormedAddCommGroup E] [NormedAddCommGroup F]\nvariable [InnerProductSpace 𝕜 E] [InnerProductSpace ℝ F]\nvariable {x y : E} {S : Set E} {f : ι → E}\n\nlocal notation \"⟪\" x \", \" y \"⟫\" => inner 𝕜 x y\n\ntheorem Dense.eq_zero_of_inner_left (hS : Dense S) (h : ∀ v ∈ S, ⟪x, v⟫ = 0) : x = 0 := by\n  let K := span 𝕜 S\n  have hK : Dense (K : Set E) := hS.mono subset_span\n  have : (⟪x, ·⟫) = 0 := (continuous_const.inner continuous_id).ext_on\n    hK continuous_const fun v ↦ Submodule.span_induction h (by simp)\n      (by simp +contextual [inner_add_right]) (by simp +contextual [inner_smul_right])\n  simpa using congr_fun this x\n\ntheorem Dense.eq_zero_of_inner_right (hS : Dense S) (h : ∀ v ∈ S, ⟪v, x⟫ = 0) : x = 0 :=\n  hS.eq_zero_of_inner_left 𝕜 fun v hv ↦ by rw! [← inner_conj_symm]; simp [-inner_conj_symm, h, hv]\n\nTarget:\ntheorem Dense.eq_of_inner_left (hS : Dense S) (h : ∀ v ∈ S, ⟪x, v⟫ = ⟪y, v⟫) : x = y :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_6d45385fed9b","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"d4242e7a02db1ff956958cd35827bf6e4715831809d32899ca9b582a24b7d8f0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/InnerProductSpace","family_id":"dense","file_id":"mathlib/Mathlib/Analysis/InnerProductSpace/Continuous.lean","sample_id":"6d45385fed9b0ad68c7032cd158ad8e6ed294ff9a6d585f7ef6679ce0eea1fd4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d20eb1ad8941b8e34a808b1966083034fd1512741d71702f610e71a800689ea8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ac967f3faf553306a5c24145a1c9caf628fb00eae9e83bd29d882cbc48ac201d","source_sha256":"5596724266063c95bd7c3b79e894544045556a2fdd0d7b0136189582e7829a9b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← single_zero_one, orderTop_single one_ne_zero, WithTop.coe_eq_zero]","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":5,"token_jaccard":0.0625,"token_length_ratio":0.384615},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"825f0451a744941818f042894dece27e6912208a08a507a917d8635af3a28303","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Subalgebra.Lattice\npublic import Mathlib.Algebra.GroupWithZero.Regular\npublic import Mathlib.Algebra.Module.BigOperators\npublic import Mathlib.Data.Finset.MulAntidiagonal\npublic import Mathlib.Data.Finset.SMulAntidiagonal\npublic import Mathlib.GroupTheory.GroupAction.Ring\npublic import Mathlib.RingTheory.HahnSeries.Addition\n\nNamespace:\nHahnSeries\n\nLocal context:\n/-\nCopyright (c) 2021 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson, Scott Carnahan\n-/\n/-!\n# Multiplicative properties of Hahn series\nIf `Γ` is ordered and `R` has zero, then `R⟦Γ⟧` consists of formal series over `Γ` with\ncoefficients in `R`, whose supports are partially well-ordered. This module introduces\nmultiplication and scalar multiplication on Hahn series. If `Γ` is an ordered cancellative\ncommutative additive monoid and `R` is a semiring, then we get a semiring structure on\n`R⟦Γ⟧`. If `Γ` has an ordered vector-addition on `Γ'` and `R` has a scalar multiplication\non `V`, we define `HahnModule Γ' R V` as a type alias for `V⟦Γ'⟧` that admits a scalar\nmultiplication from `R⟦Γ⟧`. The scalar action of `R` on `R⟦Γ⟧` is compatible\nwith the action of `R⟦Γ⟧` on `HahnModule Γ' R V`.\n\n## Main Definitions\n* `HahnModule` is a type alias for `HahnSeries`, which we use for defining scalar multiplication\n  of `R⟦Γ⟧` on `HahnModule Γ' R V` for an `R`-module `V`, where `Γ'` admits an ordered\n  cancellative vector addition operation from `Γ`. The type alias allows us to avoid a potential\n  instance diamond.\n* `HahnModule.of` is the isomorphism from `V⟦Γ⟧` to `HahnModule Γ R V`.\n* `HahnSeries.C` is the `constant term` ring homomorphism `R →+* R⟦Γ⟧`.\n* `HahnSeries.embDomainRingHom` is the ring homomorphism `R⟦Γ⟧ →+* R⟦Γ'⟧`\n  induced by an order embedding `Γ ↪o Γ'`.\n* `HahnSeries.orderTopSubOnePos` is the group of invertible Hahn series close to 1, i.e., those\n  series such that subtracting one yields a series with strictly positive `orderTop`.\n\n## Main results\n* If `R` is a (commutative) (semi-)ring, then so is `R⟦Γ⟧`.\n* If `V` is an `R`-module, then `HahnModule Γ' R V` is a `R⟦Γ⟧`-module.\n\n## TODO\nThe following may be useful for composing vertex operators, but they seem to take time.\n* rightTensorMap: `HahnModule Γ' R U ⊗[R] V →ₗ[R] HahnModule Γ' R (U ⊗[R] V)`\n* leftTensorMap: `U ⊗[R] HahnModule Γ' R V →ₗ[R] HahnModule Γ' R (U ⊗[R] V)`\n\n## References\n- [J. van der Hoeven, *Operators on Generalized Power Series*][van_der_hoeven]\n-/\n\n@[expose] public section\n\nopen Finset Function HahnSeries Pointwise\n\nnoncomputable section\n\nvariable {Γ Γ' R S V : Type*}\n\nnamespace HahnSeries\n\nvariable [Zero Γ] [PartialOrder Γ]\n\ninstance [Zero R] [One R] : One R⟦Γ⟧ where one := single 0 1\ninstance [Zero R] [NatCast R] : NatCast R⟦Γ⟧ where natCast n := single 0 n\ninstance [Zero R] [IntCast R] : IntCast R⟦Γ⟧ where intCast z := single 0 z\ninstance [Zero R] [NNRatCast R] : NNRatCast R⟦Γ⟧ where nnratCast q := single 0 q\ninstance [Zero R] [RatCast R] : RatCast R⟦Γ⟧ where ratCast q := single 0 q\n\nopen Classical in\n@[simp]\ntheorem coeff_one [Zero R] [One R] {a : Γ} : (1 : R⟦Γ⟧).coeff a = if a = 0 then 1 else 0 :=\n  coeff_single\n\n@[simp] theorem single_zero_one [Zero R] [One R] : single (0 : Γ) (1 : R) = 1 := rfl\ntheorem single_zero_natCast [Zero R] [NatCast R] (n : ℕ) : single (0 : Γ) (n : R) = n := rfl\ntheorem single_zero_intCast [Zero R] [IntCast R] (z : ℤ) : single (0 : Γ) (z : R) = z := rfl\ntheorem single_zero_nnratCast [Zero R] [NNRatCast R] (q : ℚ≥0) : single (0 : Γ) (q : R) = q := rfl\ntheorem single_zero_ratCast [Zero R] [RatCast R] (q : ℚ) : single (0 : Γ) (q : R) = q := rfl\n\ntheorem single_zero_ofNat [Zero R] [NatCast R] (n : ℕ) [n.AtLeastTwo] :\n    single (0 : Γ) (ofNat(n) : R) = ofNat(n) := rfl\n\n@[simp]\ntheorem support_one [MulZeroOneClass R] [Nontrivial R] : support (1 : R⟦Γ⟧) = {0} :=\n  support_single_of_ne one_ne_zero\n\n@[simp]\n\nTarget:\ntheorem orderTop_one [Zero R] [One R] [NeZero (1 : R)] : orderTop (1 : R⟦Γ⟧) = 0 :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/HahnSeries","family_id":"ordertop_one","file_id":"mathlib/Mathlib/RingTheory/HahnSeries/Multiplication.lean","sample_id":"ac967f3faf553306a5c24145a1c9caf628fb00eae9e83bd29d882cbc48ac201d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"defdc0751f53e30ab813268ab9f2c357764562c8bfc11a4ae2b13dad9c027777","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9a7f488b9072e1c3cd3e743c9cf1c9c590011f4701530db321fda7898e28d6e1","source_sha256":"dc7a83c1eb70da8d3ca703a602b6a96706f55b88f4b16cc5437746f0ac16014e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [IsRegular, compl_compl_inf_distrib, ha.eq, hb.eq]","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":8,"token_jaccard":0.058824,"token_length_ratio":0.533333},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"82702d71d4c3a4688a47cbd3d41448c5e847cbcc737bdf179b4c4b8133b90af3","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.GaloisConnection.Basic\n\nNamespace:\nHeyting\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Heyting regular elements\n\nThis file defines Heyting regular elements, elements of a Heyting algebra that are their own double\ncomplement, and proves that they form a Boolean algebra.\n\nFrom a logic standpoint, this means that we can perform classical logic within intuitionistic logic\nby simply double-negating all propositions. This is practical for synthetic computability theory.\n\n## Main declarations\n\n* `IsRegular`: `a` is Heyting-regular if `aᶜᶜ = a`.\n* `Regular`: The subtype of Heyting-regular elements.\n* `Regular.BooleanAlgebra`: Heyting-regular elements form a Boolean algebra.\n\n## References\n\n* [Francis Borceux, *Handbook of Categorical Algebra III*][borceux-vol3]\n-/\n\n@[expose] public section\n\n-- We want the theorems in this file to be intuitionistic.\nset_option linter.unusedDecidableInType false\n\nopen Function\n\nvariable {α : Type*}\n\nnamespace Heyting\n\nsection Compl\n\nvariable [Compl α] {a : α}\n\n/-- An element of a Heyting algebra is regular if its double complement is itself. -/\ndef IsRegular (a : α) : Prop :=\n  aᶜᶜ = a\n\nprotected theorem IsRegular.eq : IsRegular a → aᶜᶜ = a :=\n  id\n\ninstance IsRegular.decidablePred [DecidableEq α] : @DecidablePred α IsRegular := fun _ =>\n  ‹DecidableEq α› _ _\n\nend Compl\n\nsection HeytingAlgebra\n\nvariable [HeytingAlgebra α] {a b : α}\n\ntheorem isRegular_bot : IsRegular (⊥ : α) := by rw [IsRegular, compl_bot, compl_top]\n\ntheorem isRegular_top : IsRegular (⊤ : α) := by rw [IsRegular, compl_top, compl_bot]\n\nTarget:\ntheorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a ⊓ b) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"9ba45b74c5d26f2825bc366283e624541d904b72ae7707c39a38e5070be2b0fa","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Heyting","family_id":"isregular","file_id":"mathlib/Mathlib/Order/Heyting/Regular.lean","sample_id":"9a7f488b9072e1c3cd3e743c9cf1c9c590011f4701530db321fda7898e28d6e1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2cd47198bc2f9b64bb28556c6ab7308c7324a109d2a3891b6dff94f0ce2c3121","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1c86ceea114f8b21399d14e3c8eb46ce1772e6d6c23bbea1baca50fead5f88c8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2b6b462ef4cebc5cf1be9ae0f138ed0a50ba4a689e61f2c8626b10e8cc24c3b7","source_sha256":"a29151d12db11777bb9d805b9535e76a5c2a8f2d2a791e6ae7bfa4c562e67f6b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [Nat.lt_add_one_iff, biSup_le_eq_sup]","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":13,"token_jaccard":0.75,"token_length_ratio":1.444444},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"82736011456515ad527910285993062f0da327ba64fbd8a2de39792be9e207c2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ConditionallyCompleteLattice.Finset\npublic import Mathlib.Order.Interval.Finset.Nat\n\nNamespace:\nNat\n\nLocal context:\n/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Floris van Doorn, Gabriel Ebner, Yury Kudryashov\n-/\n/-!\n# Conditionally complete linear order structure on `ℕ`\n\nIn this file we\n\n* define a `ConditionallyCompleteLinearOrderBot` structure on `ℕ`;\n* prove a few lemmas about `iSup`/`iInf`/`Set.iUnion`/`Set.iInter` and natural numbers.\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero\n\nopen Set\n\nnamespace Nat\n\nopen scoped Classical in\nnoncomputable instance : InfSet ℕ :=\n  ⟨fun s ↦ if h : ∃ n, n ∈ s then @Nat.find (fun n ↦ n ∈ s) _ h else 0⟩\n\nopen scoped Classical in\nnoncomputable instance : SupSet ℕ :=\n  ⟨fun s ↦ if h : ∃ n, ∀ a ∈ s, a ≤ n then @Nat.find (fun n ↦ ∀ a ∈ s, a ≤ n) _ h else 0⟩\n\nopen scoped Classical in\ntheorem sInf_def {s : Set ℕ} (h : s.Nonempty) : sInf s = @Nat.find (fun n ↦ n ∈ s) _ h :=\n  dif_pos _\n\nopen scoped Classical in\ntheorem sSup_def {s : Set ℕ} (h : ∃ n, ∀ a ∈ s, a ≤ n) :\n    sSup s = @Nat.find (fun n ↦ ∀ a ∈ s, a ≤ n) _ h :=\n  dif_pos _\n\ntheorem _root_.Set.Infinite.Nat.sSup_eq_zero {s : Set ℕ} (h : s.Infinite) : sSup s = 0 :=\n  dif_neg fun ⟨n, hn⟩ ↦\n    let ⟨k, hks, hk⟩ := h.exists_gt n\n    (hn k hks).not_gt hk\n\ntheorem sSup_of_not_bddAbove {s : Set ℕ} (h : ¬BddAbove s) : sSup s = 0 :=\n  Set.Infinite.Nat.sSup_eq_zero <| Set.infinite_of_not_bddAbove h\n\nlemma iSup_of_not_bddAbove {ι : Sort*} {f : ι → ℕ} (h : ¬ BddAbove (Set.range f)) :\n    (⨆ i, f i : ℕ) = 0 := Nat.sSup_of_not_bddAbove h\n\n@[simp]\ntheorem sInf_eq_zero {s : Set ℕ} : sInf s = 0 ↔ 0 ∈ s ∨ s = ∅ := by\n  cases eq_empty_or_nonempty s with\n  | inl h => subst h\n             simp only [or_true, InfSet.sInf,\n                        mem_empty_iff_false, exists_false, dif_neg, not_false_iff]\n  | inr h => simp only [h.ne_empty, or_false, Nat.sInf_def, h, Nat.find_eq_zero]\n\n@[simp]\ntheorem sInf_empty : sInf ∅ = 0 := by\n  rw [sInf_eq_zero]\n  right\n  rfl\n\n@[simp]\ntheorem iInf_of_empty {ι : Sort*} [IsEmpty ι] (f : ι → ℕ) : iInf f = 0 := by\n  rw [iInf_of_isEmpty, sInf_empty]\n\n/-- This combines `Nat.iInf_of_empty` with `ciInf_const`. -/\n@[simp]\nlemma iInf_const_zero {ι : Sort*} : ⨅ _ : ι, 0 = 0 :=\n  (isEmpty_or_nonempty ι).elim (fun h ↦ by simp) fun h ↦ sInf_eq_zero.2 <| by simp\n\ntheorem sInf_mem {s : Set ℕ} (h : s.Nonempty) : sInf s ∈ s := by\n  classical\n  rw [Nat.sInf_def h]\n  exact Nat.find_spec h\n\ntheorem notMem_of_lt_sInf {s : Set ℕ} {m : ℕ} (hm : m < sInf s) : m ∉ s := by\n  classical\n  cases eq_empty_or_nonempty s with\n  | inl h => subst h; apply notMem_empty\n  | inr h => rw [Nat.sInf_def h] at hm; exact Nat.find_min h hm\n\nprotected theorem sInf_le {s : Set ℕ} {m : ℕ} (hm : m ∈ s) : sInf s ≤ m := by\n  classical\n  rw [Nat.sInf_def ⟨m, hm⟩]\n  exact Nat.find_min' ⟨m, hm⟩ hm\n\ntheorem nonempty_of_pos_sInf {s : Set ℕ} (h : 0 < sInf s) : s.Nonempty := by\n  by_contra contra\n  rw [Set.not_nonempty_iff_eq_empty] at contra\n  have h' : sInf s ≠ 0 := ne_of_gt h\n  apply h'\n  rw [Nat.sInf_eq_zero]\n  right\n  assumption\n\ntheorem nonempty_of_sInf_eq_succ {s : Set ℕ} {k : ℕ} (h : sInf s = k + 1) : s.Nonempty :=\n  nonempty_of_pos_sInf (h.symm ▸ succ_pos k : sInf s > 0)\n\ntheorem eq_Ici_of_nonempty_of_upward_closed {s : Set ℕ} (hs : s.Nonempty)\n    (hs' : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s) : s = Ici (sInf s) :=\n  ext fun n ↦ ⟨fun H ↦ Nat.sInf_le H, fun H ↦ hs' (sInf s) n H (sInf_mem hs)⟩\n\ntheorem sInf_upward_closed_eq_succ_iff {s : Set ℕ} (hs : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s)\n    (k : ℕ) : sInf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s := by\n  classical\n  constructor\n  · intro H\n    rw [eq_Ici_of_nonempty_of_upward_closed (nonempty_of_sInf_eq_succ _) hs, H, mem_Ici, mem_Ici]\n    · exact ⟨le_rfl, k.not_succ_le_self⟩\n    · exact k\n    · assumption\n  · rintro ⟨H, H'⟩\n    rw [sInf_def (⟨_, H⟩ : s.Nonempty), find_eq_iff]\n    exact ⟨H, fun n hnk hns ↦ H' <| hs n k (Nat.lt_succ_iff.mp hnk) hns⟩\n\n/-- This instance is necessary, otherwise the lattice operations would be derived via\n`ConditionallyCompleteLinearOrderBot` and marked as noncomputable. -/\ninstance : Lattice ℕ :=\n  LinearOrder.toLattice\n\nopen scoped Classical in\nnoncomputable instance : ConditionallyCompleteLinearOrderBot ℕ :=\n  { (inferInstance : OrderBot ℕ), (LinearOrder.toLattice : Lattice ℕ),\n    (inferInstance : LinearOrder ℕ) with\n    isLUB_csSup _ hn hb := sSup_def hb ▸ Nat.isLeast_find hb\n    isGLB_csInf _ hn hb := sInf_def hn ▸ (Nat.isLeast_find hn).isGLB\n    csSup_empty := by\n      simp only [sSup_def, Set.mem_empty_iff_false, forall_const, forall_prop_of_false,\n        not_false_iff, exists_const]\n      apply bot_unique (Nat.find_min' _ _)\n      trivial\n    csSup_of_not_bddAbove := by\n      intro s hs\n      simp only [sSup,\n        mem_empty_iff_false, IsEmpty.forall_iff, forall_const, exists_const, dite_true]\n      rw [dif_neg]\n      · exact le_antisymm (zero_le _) (find_le trivial)\n      · exact hs\n    csInf_of_not_bddBelow := fun s hs ↦ by simp at hs }\n\ntheorem sSup_mem {s : Set ℕ} (h₁ : s.Nonempty) (h₂ : BddAbove s) : sSup s ∈ s :=\n  let ⟨k, hk⟩ := h₂\n  h₁.csSup_mem ((finite_le_nat k).subset hk)\n\ntheorem sInf_add {n : ℕ} {p : ℕ → Prop} (hn : n ≤ sInf { m | p m }) :\n    sInf { m | p (m + n) } + n = sInf { m | p m } := by\n  classical\n  obtain h | ⟨m, hm⟩ := { m | p (m + n) }.eq_empty_or_nonempty\n  · rw [h, Nat.sInf_empty, zero_add]\n    obtain hnp | hnp := hn.eq_or_lt\n    · exact hnp\n    suffices hp : p (sInf { m | p m } - n + n) from (h.subset hp).elim\n    rw [Nat.sub_add_cancel hn]\n    exact csInf_mem (nonempty_of_pos_sInf <| n.zero_le.trans_lt hnp)\n  · have hp : ∃ n, n ∈ { m | p m } := ⟨_, hm⟩\n    rw [Nat.sInf_def ⟨m, hm⟩, Nat.sInf_def hp]\n    rw [Nat.sInf_def hp] at hn\n    exact find_add hn\n\ntheorem sInf_add' {n : ℕ} {p : ℕ → Prop} (h : 0 < sInf { m | p m }) :\n    sInf { m | p m } + n = sInf { m | p (m - n) } := by\n  suffices h₁ : n ≤ sInf {m | p (m - n)} by\n    convert! sInf_add h₁\n    simp_rw [Nat.add_sub_cancel_right]\n  obtain ⟨m, hm⟩ := nonempty_of_pos_sInf h\n  refine\n    le_csInf ⟨m + n, ?_⟩ fun b hb ↦\n      le_of_not_gt fun hbn ↦\n        ne_of_mem_of_not_mem ?_ (notMem_of_lt_sInf h) (Nat.sub_eq_zero_of_le hbn.le)\n  · dsimp\n    rwa [Nat.add_sub_cancel_right]\n  · exact hb\n\nsection\n\nvariable {α : Type*} [CompleteLattice α]\n\nTarget:\ntheorem iSup_lt_succ (u : ℕ → α) (n : ℕ) : ⨆ k < n + 1, u k = (⨆ k < n, u k) ⊔ u n :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  simp_rw [Nat.lt_add_one_iff, biSup_le_eq_sup]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Lattice","family_id":"isup_lt_succ","file_id":"mathlib/Mathlib/Order/Lattice/Nat.lean","sample_id":"2b6b462ef4cebc5cf1be9ae0f138ed0a50ba4a689e61f2c8626b10e8cc24c3b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"11fbcca75f5b8a4dbad7c2490be392157767b93ac85f1ccad1aff23708a841e9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8c84f0871128efc9bcf9d29301aeb6213d25f58a7d79db843c43cab6078afc5a","source_sha256":"4a480f7815d84fa6cbaa2c651b3b8967b4642ebe2873904b75dbadaa0411117b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro x hx\n  simp only [Subalgebra.mem_map, Subalgebra.mem_center_iff] at hx ⊢\n  obtain ⟨b, hb0, rfl⟩ := hx\n  intro bc\n  induction bc using TensorProduct.induction_on with\n  | zero => simp\n  | tmul b' c => simp [hb0]\n  | add _ _ _ _ => simp_all [add_mul, mul_add]","hard_negative":false,"metrics":{"chosen_tokens":63,"rejected_tokens":3,"token_jaccard":0.02439,"token_length_ratio":0.047619},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"82b32dfb15871df66c50b11ba4ad9c9ab48298c8adab941b4bb128f52eaae57b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Central.Basic\npublic import Mathlib.RingTheory.Flat.Basic\npublic import Mathlib.LinearAlgebra.Basis.VectorSpace\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang, Yunzhou Xie\n-/\n/-!\n\n# Lemmas about tensor products of central algebras\n\nIn this file we prove for algebras `B` and `C` over a field `K` that if `B ⊗[K] C` is a central\nalgebra and `B, C` nontrivial, then both `B` and `C` are central algebras.\n\n## Main Results\n\n- `Algebra.IsCentral.left_of_tensor_of_field`: If `B` `C` are `K`-algebras where `K` is a field,\n  `C` is nontrivial and `B ⊗[K] C` is a central algebra over `K`, then `B` is a\n  central algebra over `K`.\n- `Algebra.IsCentral.right_of_tensor_of_field`: If `B` `C` are `K`-algebras where `K` is a field,\n  `B` is nontrivial and `B ⊗[K] C` is a central algebra over `K`, then `C` is a\n  central algebra over `K`.\n\n## Tags\nCentral Algebras, Central Simple Algebras, Noncommutative Algebra\n-/\n\npublic section\n\nuniverse u v\n\nopen TensorProduct\n\nvariable (K B C : Type*) [CommSemiring K] [Semiring B] [Semiring C] [Algebra K B] [Algebra K C]\n\nTarget:\nlemma Algebra.TensorProduct.includeLeft_map_center_le :\n    (Subalgebra.center K B).map includeLeft ≤ Subalgebra.center K (B ⊗[K] C) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Central","family_id":"algebra","file_id":"mathlib/Mathlib/Algebra/Central/TensorProduct.lean","sample_id":"8c84f0871128efc9bcf9d29301aeb6213d25f58a7d79db843c43cab6078afc5a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b663b020e08f48e798d48b7ce39be8da0dce1601bdd6b1231b5d9860f49a1922","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"65b86c085b15012e6279a23598357f93290dc69ef9e7a9ff9f5e6d9dd4aee4c4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9464b08287d95c785301c54058651202fff8047c0ca4e3764eb7ffff3e6775f3","source_sha256":"c6fb2ee19236e152d132e9c3d3ef383faafb8161f5d12e957a193a47621136bd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by lia) :\n    t.IsGE ((t.truncGT a).obj X) b :=\n  t.isGE_truncGE_obj ..","hard_negative":false,"metrics":{"chosen_tokens":25,"rejected_tokens":30,"token_jaccard":0.789474,"token_length_ratio":1.2},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"833bbbe0bd9cbe1bb75a1000bd3c2faf5e13acf507f0e06421bca5213a2fd26a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE\n\nNamespace:\nCategoryTheory.Triangulated.TStructure\n\nLocal context:\n/-\nCopyright (c) 2026 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Truncations for a t-structure\n\nLet `t` be a t-structure on a (pre)triangulated category `C`.\nIn this file, for any `n : ℤ`, we introduce the truncation functors\n`t.truncLE n : C ⥤ C` and `t.truncGT n : C ⥤ C`, as variants of the functors\n`t.truncLT n : C ⥤ C` and `t.truncGE n : C ⥤ C` introduced in the file\n`Mathlib/CategoryTheory/Triangulated/TStructure/TruncLTGE.lean`.\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Limits Pretriangulated\n\nvariable {C : Type*} [Category* C] [Preadditive C] [HasZeroObject C] [HasShift C ℤ]\n  [∀ (n : ℤ), (shiftFunctor C n).Additive] [Pretriangulated C]\n\nnamespace Triangulated\n\nnamespace TStructure\n\nvariable (t : TStructure C)\n\n/-- Given a t-structure `t` on a pretriangulated category `C` and `n : ℤ`, this\nis the `≤ n`-truncation functor. See also the natural transformation `truncLEι`. -/\nnoncomputable def truncLE (n : ℤ) : C ⥤ C := t.truncLT (n + 1)\n\ninstance (n : ℤ) : (t.truncLE n).Additive := by\n  dsimp only [truncLE]\n  infer_instance\n\nlemma isLE_truncLE_obj (X : C) (a b : ℤ) (hn : a ≤ b := by lia) :\n    t.IsLE ((t.truncLE a).obj X) b :=\n  t.isLE_truncLT_obj ..\n\ninstance (n : ℤ) (X : C) : t.IsLE ((t.truncLE n).obj X) n :=\n  t.isLE_truncLE_obj ..\n\n/-- Given a t-structure `t` on a pretriangulated category `C` and `n : ℤ`, this\nis the `> n`-truncation functor. See also the natural transformation `truncGTπ`. -/\nnoncomputable def truncGT (n : ℤ) : C ⥤ C := t.truncGE (n + 1)\n\ninstance (n : ℤ) : (t.truncGT n).Additive := by\n  dsimp only [truncGT]\n  infer_instance\n\nTarget:\nlemma isGE_truncGT_obj (X : C) (a b : ℤ) (hn : b ≤ a + 1 :=\n\nProof body:\n","rejected":"Here is the proof:\nby lia) :\n    t.IsGE ((t.truncGT a).obj X) b :=\n  t.isGE_truncGE_obj ..","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Triangulated","family_id":"isge_truncgt_obj","file_id":"mathlib/Mathlib/CategoryTheory/Triangulated/TStructure/TruncLEGT.lean","sample_id":"9464b08287d95c785301c54058651202fff8047c0ca4e3764eb7ffff3e6775f3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"05c1e20ccd52d20617d06c19bb3d65322091abd072a38ef854172815fc9654f9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"15e83a19ac7a4ffb734f70f90cc5cd72e116c7c5b68a279718e9f6a477143d88","source_sha256":"129f2f3aa776146d58955f965fc598f57556bb308d25e687562d21d0c89b2ff2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨?_, fun h ↦ (isCyclic_of_card_le_two h).isMulCommutative⟩\n  classical\n  rintro ⟨⟨h⟩⟩\n  rw [← not_lt, ← Set.ncard_univ, Set.two_lt_ncard_iff]\n  rintro ⟨a, b, c, _, _, _, hab, hac, hbc⟩\n  apply hbc\n  simp_rw [Perm.ext_iff] at h\n  simpa [swap_apply_of_ne_of_ne hab hac] using h (swap a b) (swap b c) a","hard_negative":false,"metrics":{"chosen_tokens":86,"rejected_tokens":3,"token_jaccard":0.02381,"token_length_ratio":0.034884},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"834852a1421aaa28d92fe15e578e114ec5461e05c680616ae398e9e99ed2db70","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Perm\npublic import Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic\npublic import Mathlib.SetTheory.Cardinal.Finite\n\nNamespace:\nEquiv.Perm\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir\n-/\n/-! # Properties of `Equiv.Perm` on `Finite` types\n\nLet `α` be a `Finite` type.\n\n* `Nat.card_perm`: cardinality of `Equiv.Perm α`.\n\n* `Equiv.Perm.isCyclic_of_card_le_two`: if `Nat.card α ≤ 2`,\n  then `Equiv.Perm α` is cyclic.\n\n* `Equiv.Perm.isCyclic_iff_card_le_two`: `Equiv.Perm α` is cyclic iff `Nat.card α ≤ 2`.\n\n* `Equiv.Perm.isMulCommutative_iff_card_le_two`: `Equiv.Perm α` is commutative iff `Nat.card α ≤ 2`.\n\n-/\n\npublic section\n\nassert_not_exists Field\n\nopen Equiv Nat\n\nvariable {α : Type*} [Finite α]\n\nnamespace Nat\n\ntheorem card_perm : Nat.card (Perm α) = (Nat.card α)! := by\n  classical\n  have := Fintype.ofFinite α\n  rw [card_eq_fintype_card, card_eq_fintype_card, Fintype.card_perm]\n\nend Nat\n\nnamespace Equiv.Perm\n\ntheorem isCyclic_of_card_le_two (hα : Nat.card α ≤ 2) :\n    IsCyclic (Perm α) := by\n  apply isCyclic_of_card_dvd_prime (p := 2)\n  simpa [card_perm] using factorial_dvd_factorial hα\n\nTarget:\ntheorem isMulCommutative_iff_card_le_two :\n    IsMulCommutative (Perm α) ↔ Nat.card α ≤ 2 :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finite","family_id":"ismulcommutative_iff_card_le_two","file_id":"mathlib/Mathlib/Data/Finite/Perm.lean","sample_id":"15e83a19ac7a4ffb734f70f90cc5cd72e116c7c5b68a279718e9f6a477143d88"},"split":"train"}
{"binding":{"chosen_proof_sha256":"19a0b796b5a123aa0fa4da0bb1ec6d1209352b44dfd80eed24e61f22f0837fb5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3a8030466b32fb2ee6b5297792fab6b2663e75df46ac0410e9a7f60e6198b783","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0a46fac18823464095b42916006166b41806c313503a9332eebd632a234896e3","source_sha256":"9204017e76fd98a1b3e65ba16535a42898573646b8f44e5554ab4317200de227","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine measure_mono_null ?_ hs.ordConnected.null_frontier\n  rw [← closure_sdiff_interior, hs.interior_eq_empty, sdiff_empty]\n  exact subset_closure","hard_negative":false,"metrics":{"chosen_tokens":23,"rejected_tokens":28,"token_jaccard":0.791667,"token_length_ratio":1.217391},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"83570c25cdc8369a047445ec25ace77abf7faaa3e71a5bf9078737e40c8a9815","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Order.UpperLower\npublic import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace\npublic import Mathlib.Topology.Order.DenselyOrdered\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies, Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Kexing Ying\n-/\n/-!\n# Order-connected sets are null-measurable\n\nThis file proves that order-connected sets in `ℝⁿ` under the pointwise order are null-measurable.\nRecall that `x ≤ y` iff `∀ i, x i ≤ y i`, and `s` is order-connected iff\n`∀ x y ∈ s, ∀ z, x ≤ z → z ≤ y → z ∈ s`.\n\n## Main declarations\n\n* `Set.OrdConnected.null_frontier`: The frontier of an order-connected set in `ℝⁿ` has measure `0`.\n\n## Notes\n\nWe prove null-measurability in `ℝⁿ` with the `∞`-metric, but this transfers directly to `ℝⁿ` with\nthe Euclidean metric because they have the same measurable sets.\n\nNull-measurability can't be strengthened to measurability because any antichain (and in particular\nany subset of the antidiagonal `{(x, y) | x + y = 0}`) is order-connected.\n\n## Sketch proof\n\n1. To show an order-connected set is null-measurable, it is enough to show it has null frontier.\n2. Since an order-connected set is the intersection of its upper and lower closure, it's enough to\n  show that upper and lower sets have null frontier.\n3. WLOG let's prove it for an upper set `s`.\n4. By the Lebesgue density theorem, it is enough to show that any frontier point `x` of `s` is not a\n  Lebesgue point, namely we want the density of `s` over small balls centered at `x` to not tend to\n  either `0` or `1`.\n5. This is true, since by the upper setness of `s` we can intercalate a ball of radius `δ / 4` in\n  `s` intersected with the upper quadrant of the ball of radius `δ` centered at `x` (recall that the\n  balls are taken in the ∞-norm, so they are cubes), and another ball of radius `δ / 4` in `sᶜ` and\n  the lower quadrant of the ball of radius `δ` centered at `x`.\n\n## TODO\n\nGeneralize so that it also applies to `ℝ × ℝ`, for example.\n-/\n\npublic section\n\nopen Filter MeasureTheory Metric Set\nopen scoped Topology\n\nvariable {ι : Type*} [Fintype ι] {s : Set (ι → ℝ)} {x : ι → ℝ}\n\n/-- If we can fit a small ball inside a set `s` intersected with any neighborhood of `x`, then the\ndensity of `s` near `x` is not `0`.\n\nAlong with `aux₁`, this proves that `x` is not a Lebesgue point of `s`. This will be used to prove\nthat the frontier of an order-connected set is null. -/\nprivate lemma aux₀\n    (h : ∀ δ, 0 < δ →\n      ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior s) :\n    ¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)\n        (𝓝 0) := by\n  choose f hf₀ hf₁ using h\n  intro H\n  obtain ⟨ε, -, hε', hε₀⟩ := exists_seq_strictAnti_tendsto_nhdsWithin (0 : ℝ)\n  refine not_eventually.2\n    (Frequently.of_forall fun _ ↦ lt_irrefl <| ENNReal.ofReal <| 4⁻¹ ^ Fintype.card ι)\n    ((Filter.Tendsto.eventually_lt (H.comp hε₀) tendsto_const_nhds ?_).mono fun n ↦\n      lt_of_le_of_lt ?_)\n  on_goal 2 =>\n    calc\n      ENNReal.ofReal (4⁻¹ ^ Fintype.card ι)\n        = volume (closedBall (f (ε n) (hε' n)) (ε n / 4)) / volume (closedBall x (ε n)) := ?_\n      _ ≤ volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n)) := by\n        gcongr\n        exact subset_inter ((hf₁ _ <| hε' n).trans interior_subset_closure) <| hf₀ _ <| hε' n\n    have := hε' n\n    rw [Real.volume_pi_closedBall, Real.volume_pi_closedBall, ← ENNReal.ofReal_div_of_pos,\n      ← div_pow, mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]\n  all_goals positivity\n\n/-- If we can fit a small ball inside a set `sᶜ` intersected with any neighborhood of `x`, then the\ndensity of `s` near `x` is not `1`.\n\nAlong with `aux₀`, this proves that `x` is not a Lebesgue point of `s`. This will be used to prove\nthat the frontier of an order-connected set is null. -/\nprivate lemma aux₁\n    (h : ∀ δ, 0 < δ →\n      ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior sᶜ) :\n    ¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)\n        (𝓝 1) := by\n  choose f hf₀ hf₁ using h\n  intro H\n  obtain ⟨ε, -, hε', hε₀⟩ := exists_seq_strictAnti_tendsto_nhdsWithin (0 : ℝ)\n  refine not_eventually.2\n      (Frequently.of_forall fun _ ↦ lt_irrefl <| 1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι))\n      ((Filter.Tendsto.eventually_lt tendsto_const_nhds (H.comp hε₀) <|\n            ENNReal.sub_lt_self ENNReal.one_ne_top one_ne_zero ?_).mono\n        fun n ↦ lt_of_le_of_lt' ?_)\n  on_goal 2 =>\n    calc\n      volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n))\n        ≤ volume (closedBall x (ε n) \\ closedBall (f (ε n) <| hε' n) (ε n / 4)) /\n          volume (closedBall x (ε n)) := by\n        gcongr\n        rw [sdiff_eq_compl_inter]\n        refine inter_subset_inter_left _ ?_\n        rw [subset_compl_comm, ← interior_compl]\n        exact hf₁ _ _\n      _ = 1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι) := ?_\n    have := hε' n\n    rw [measure_sdiff (hf₀ _ _) _ ((Real.volume_pi_closedBall _ _).trans_ne ENNReal.ofReal_ne_top),\n      Real.volume_pi_closedBall, Real.volume_pi_closedBall, ENNReal.sub_div fun _ _ ↦ _,\n      ENNReal.div_self _ ENNReal.ofReal_ne_top, ← ENNReal.ofReal_div_of_pos, ← div_pow,\n      mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]\n  all_goals try positivity\n  · simp_all\n  · exact measurableSet_closedBall.nullMeasurableSet\n\ntheorem IsUpperSet.null_frontier (hs : IsUpperSet s) : volume (frontier s) = 0 := by\n  refine measure_mono_null (fun x hx ↦ ?_)\n    (Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet _\n      (isClosed_closure (s := s)).measurableSet)\n  by_cases h : x ∈ closure s <;>\n    simp only [mem_compl_iff, mem_setOf, h, not_false_eq_true, indicator_of_notMem,\n      indicator_of_mem, Pi.one_apply]\n  · refine aux₁ fun _ ↦ hs.compl.exists_subset_ball <| frontier_subset_closure ?_\n    rwa [frontier_compl]\n  · exact aux₀ fun _ ↦ hs.exists_subset_ball <| frontier_subset_closure hx\n\ntheorem IsLowerSet.null_frontier (hs : IsLowerSet s) : volume (frontier s) = 0 := by\n  refine measure_mono_null (fun x hx ↦ ?_)\n    (Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet _\n      (isClosed_closure (s := s)).measurableSet)\n  by_cases h : x ∈ closure s <;>\n    simp only [mem_compl_iff, mem_setOf, h, not_false_eq_true, indicator_of_notMem,\n      indicator_of_mem, Pi.one_apply]\n  · refine aux₁ fun _ ↦ hs.compl.exists_subset_ball <| frontier_subset_closure ?_\n    rwa [frontier_compl]\n  · exact aux₀ fun _ ↦ hs.exists_subset_ball <| frontier_subset_closure hx\n\ntheorem Set.OrdConnected.null_frontier (hs : s.OrdConnected) : volume (frontier s) = 0 := by\n  rw [← hs.upperClosure_inter_lowerClosure]\n  exact measure_mono_null (frontier_inter_subset _ _) <| measure_union_null\n    (measure_inter_null_of_null_left _ (UpperSet.upper _).null_frontier)\n    (measure_inter_null_of_null_right _ (LowerSet.lower _).null_frontier)\n\nprotected theorem Set.OrdConnected.nullMeasurableSet (hs : s.OrdConnected) : NullMeasurableSet s :=\n  nullMeasurableSet_of_null_frontier hs.null_frontier\n\nTarget:\ntheorem IsAntichain.volume_eq_zero [Nonempty ι] (hs : IsAntichain (· ≤ ·) s) : volume s = 0 :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  refine measure_mono_null ?_ hs.ordConnected.null_frontier\n  rw [← closure_sdiff_interior, hs.interior_eq_empty, sdiff_empty]\n  exact subset_closure","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Order","family_id":"isantichain","file_id":"mathlib/Mathlib/MeasureTheory/Order/UpperLower.lean","sample_id":"0a46fac18823464095b42916006166b41806c313503a9332eebd632a234896e3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f55e39083a696313fcfdd6bb61c3e62ddbca985973b59bf86a8381597fddf755","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9044b1a9ec5361ec717fcb988a173f7c747b987f4a363348924f9ee3f41d531b","source_sha256":"aa9a2627fd36aceaf056da61c61373399c62e613d54771bd790c452c0c13aad7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : Nontrivial (Subobject X) := nontrivial_of_not_isZero h\n  obtain ⟨Y, s⟩ := (IsAtomic.eq_bot_or_exists_atom_le (⊤ : Subobject X)).resolve_left top_ne_bot\n  exact ⟨Y, (subobject_simple_iff_isAtom _).mpr s.1⟩\n\n/-- Choose an arbitrary simple subobject of a non-zero Artinian object. -/\nnoncomputable def simpleSubobject {X : C} [IsArtinianObject X] (h : ¬IsZero X) : C :=\n  (exists_simple_subobject h).choose\n\n/-- The monomorphism from the arbitrary simple subobject of a non-zero Artinian object. -/\nnoncomputable def simpleSubobjectArrow {X : C} [IsArtinianObject X] (h : ¬IsZero X) :\n    simpleSubobject h ⟶ X :=\n  (exists_simple_subobject h).choose.arrow","hard_negative":false,"metrics":{"chosen_tokens":145,"rejected_tokens":3,"token_jaccard":0.032258,"token_length_ratio":0.02069},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"837d65b12a123529352d38eceeff9dd87743af17b3c21b525ab59c96392451e8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Subobject.Lattice\npublic import Mathlib.CategoryTheory.ObjectProperty.ContainsZero\npublic import Mathlib.CategoryTheory.ObjectProperty.EpiMono\npublic import Mathlib.CategoryTheory.Limits.Constructions.EventuallyConstant\npublic import Mathlib.Order.OrderIsoNat\npublic import Mathlib.CategoryTheory.Simple\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou, Kim Morrison\n-/\n/-!\n# Artinian objects\n\nWe shall say that an object `X` in a category `C` is Artinian\n(type class `IsArtinianObject X`) if the ordered type `Subobject X`\nsatisfies the descending chain condition. The corresponding property of\nobjects `isArtinianObject : ObjectProperty C` is always\nclosed under subobjects.\n\n## Future work\n\n* when `C` is an abelian category, relate `IsArtinianObject` in `C`\n  with `IsNoetherianObject` in `Cᵒᵖ`.\n\n-/\n\n@[expose] public section\n\nuniverse v u\n\nnamespace CategoryTheory\n\nopen Limits ZeroObject\n\nvariable {C : Type u} [Category.{v} C]\n\n/-- An object `X` in a category `C` is Artinian if `Subobject X`\nsatisfies the descending chain condition. This definition is a\nterm in `ObjectProperty C` which allows to study the stability\nproperties of Artinian objects. For statements regarding\nspecific objects, it is advisable to use the type class\n`IsArtinianObject` instead. -/\n@[stacks 0FCF]\ndef isArtinianObject : ObjectProperty C :=\n  fun X ↦ WellFoundedLT (Subobject X)\n\nvariable (X Y : C)\n\n/-- An object `X` in a category `C` is Artinian if `Subobject X`\nsatisfies the descending chain condition. -/\n@[stacks 0FCF]\nabbrev IsArtinianObject : Prop := isArtinianObject.Is X\n\ninstance [IsArtinianObject X] : WellFoundedLT (Subobject X) :=\n  isArtinianObject.prop_of_is X\n\nlemma isArtinianObject_iff_antitone_chain_condition :\n    IsArtinianObject X ↔ ∀ (f : ℕ →o (Subobject X)ᵒᵈ),\n      ∃ (n : ℕ), ∀ (m : ℕ), n ≤ m → f n = f m := by\n  dsimp only [IsArtinianObject]\n  rw [ObjectProperty.is_iff, isArtinianObject,\n    ← wellFoundedGT_dual_iff,\n    wellFoundedGT_iff_monotone_chain_condition]\n\nvariable {X} in\nlemma antitone_chain_condition_of_isArtinianObject\n    [IsArtinianObject X] (f : ℕ →o (Subobject X)ᵒᵈ) :\n    ∃ (n : ℕ), ∀ (m : ℕ), n ≤ m → f n = f m :=\n  (isArtinianObject_iff_antitone_chain_condition X).1 inferInstance f\n\nlemma isArtinianObject_iff_not_strictAnti :\n    IsArtinianObject X ↔ ∀ (f : ℕ → Subobject X), ¬ StrictAnti f := by\n  refine ⟨fun _ ↦ not_strictAnti_of_wellFoundedLT, fun h ↦ ?_⟩\n  dsimp only [IsArtinianObject]\n  rw [ObjectProperty.is_iff, isArtinianObject, WellFoundedLT,\n    isWellFounded_iff, RelEmbedding.wellFounded_iff_isEmpty]\n  exact ⟨fun f ↦ h f.toFun (fun a b h ↦ f.map_rel_iff.2 h)⟩\n\nvariable {X} in\nlemma not_strictAnti_of_isArtinianObject\n    [IsArtinianObject X] (f : ℕ → Subobject X) :\n    ¬ StrictAnti f :=\n  (isArtinianObject_iff_not_strictAnti X).1 inferInstance f\n\nset_option backward.defeqAttrib.useBackward true in\nlemma isArtinianObject_iff_isEventuallyConstant :\n    IsArtinianObject X ↔ ∀ (F : ℕ ⥤ (MonoOver X)ᵒᵖ),\n      IsFiltered.IsEventuallyConstant F := by\n  rw [isArtinianObject_iff_antitone_chain_condition]\n  refine ⟨fun h G ↦ ?_, fun h F ↦ ?_⟩\n  · obtain ⟨n, hn⟩ := h ⟨_, (G ⋙ (Subobject.equivMonoOver X).inverse.op ⋙\n      (orderDualEquivalence _).inverse).monotone⟩\n    refine ⟨n, fun m hm ↦ ?_⟩\n    rw [← isIso_unop_iff, MonoOver.isIso_iff_subobjectMk_eq]\n    exact (hn m (leOfHom hm)).symm\n  · obtain ⟨n, hn⟩ := h (F.monotone.functor ⋙ (orderDualEquivalence _).functor ⋙\n      Subobject.representative.op)\n    refine ⟨n, fun m hm ↦ Eq.symm ?_⟩\n    simpa [isIso_op_iff, isIso_iff_of_reflects_iso, PartialOrder.isIso_iff_eq]\n      using hn (homOfLE hm)\n\nvariable {X} in\nlemma isEventuallyConstant_of_isArtinianObject [IsArtinianObject X]\n    (F : ℕ ⥤ (MonoOver X)ᵒᵖ) : IsFiltered.IsEventuallyConstant F :=\n  (isArtinianObject_iff_isEventuallyConstant X).1 inferInstance F\n\nvariable {X Y}\n\nlemma isArtinianObject_of_isZero (hX : IsZero X) : IsArtinianObject X := by\n  rw [isArtinianObject_iff_antitone_chain_condition]\n  have := Subobject.subsingleton_of_isZero hX\n  intro f\n  exact ⟨0, fun m hm ↦ Subsingleton.elim _ _⟩\n\ninstance [HasZeroObject C] : (isArtinianObject (C := C)).ContainsZero where\n  exists_zero := ⟨0, isZero_zero _, by\n    rw [← isArtinianObject.is_iff]\n    exact isArtinianObject_of_isZero (isZero_zero C)⟩\n\nlemma isArtinianObject_of_mono (i : X ⟶ Y) [Mono i] [IsArtinianObject Y] :\n    IsArtinianObject X := by\n  rw [isArtinianObject_iff_antitone_chain_condition]\n  intro f\n  obtain ⟨n, hn⟩ := antitone_chain_condition_of_isArtinianObject\n    ⟨fun n ↦ (Subobject.map i).obj (f n),\n      fun _ _ h ↦ (Subobject.map i).monotone (f.2 h)⟩\n  exact ⟨n, fun m hm ↦ Subobject.map_obj_injective i (hn m hm)⟩\n\ninstance : (isArtinianObject (C := C)).IsClosedUnderSubobjects where\n  prop_of_mono f _ hY := by\n    rw [← isArtinianObject.is_iff] at hY ⊢\n    exact isArtinianObject_of_mono f\n\nopen Subobject\n\nvariable [HasZeroMorphisms C] [HasZeroObject C]\n\nTarget:\ntheorem exists_simple_subobject {X : C} [IsArtinianObject X] (h : ¬IsZero X) :\n    ∃ Y : Subobject X, Simple (Y : C) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Subobject","family_id":"exists_simple_subobject","file_id":"mathlib/Mathlib/CategoryTheory/Subobject/ArtinianObject.lean","sample_id":"9044b1a9ec5361ec717fcb988a173f7c747b987f4a363348924f9ee3f41d531b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0220babf450eb445fb686e51233d9dc7bd43814cdb2cf6da4315eb591a53bef9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a86ece7891869444431a13676c31d6de05fdf4898f0e667ba12ab2a36fc2b894","source_sha256":"f5d92c67f281e03d72da06b390e360bd28b86e3bb7c33267f68c3c873cd9a4b6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← natCast_zsmul r, zsmul_mem_zmultiples_iff_exists_sub_div (Int.natCast_ne_zero.mpr hn),\n    Int.cast_natCast]\n  rfl","hard_negative":false,"metrics":{"chosen_tokens":22,"rejected_tokens":3,"token_jaccard":0.05,"token_length_ratio":0.136364},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"8388cb4255eac8c2f5327b5f0d77ac20a67ffa98e7b3b6a393a06cccaf8fe873","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Basic\npublic import Mathlib.Algebra.Order.Group.Unbundled.Int\npublic import Mathlib.Algebra.Module.NatInt\npublic import Mathlib.GroupTheory.QuotientGroup.Defs\npublic import Mathlib.Algebra.Group.Subgroup.ZPowers.Basic\n\nNamespace:\nAddSubgroup\n\nLocal context:\n/-\nCopyright (c) 2022 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Lemmas about quotients in characteristic zero\n-/\n\npublic section\n\n\nvariable {R : Type*} [DivisionRing R] [CharZero R] {p : R}\n\nnamespace AddSubgroup\n\n/-- `z • r` is a multiple of `p` iff `r` is `k * (p / z)` above a multiple of `p`, where\n`0 ≤ k < |z|`. -/\ntheorem zsmul_mem_zmultiples_iff_exists_sub_div {r : R} {z : ℤ} (hz : z ≠ 0) :\n    z • r ∈ AddSubgroup.zmultiples p ↔\n      ∃ k : Fin z.natAbs, r - (k : ℕ) • (p / z : R) ∈ AddSubgroup.zmultiples p := by\n  rw [AddSubgroup.mem_zmultiples_iff]\n  simp_rw [AddSubgroup.mem_zmultiples_iff, div_eq_mul_inv, ← smul_mul_assoc, eq_sub_iff_add_eq]\n  have hz' : (z : R) ≠ 0 := Int.cast_ne_zero.mpr hz\n  conv_rhs => simp +singlePass only [← (mul_right_injective₀ hz').eq_iff]\n  simp_rw [← zsmul_eq_mul, smul_add, ← mul_smul_comm, zsmul_eq_mul (z : R)⁻¹, mul_inv_cancel₀ hz',\n    mul_one, ← natCast_zsmul, smul_smul, ← add_smul]\n  constructor\n  · rintro ⟨k, h⟩\n    simp_rw [← h]\n    refine ⟨⟨(k % z).toNat, ?_⟩, k / z, ?_⟩\n    · rw [← Int.ofNat_lt, Int.toNat_of_nonneg (Int.emod_nonneg _ hz)]\n      exact (Int.emod_lt_abs _ hz).trans_eq (Int.abs_eq_natAbs _)\n    rw [Fin.val_mk, Int.toNat_of_nonneg (Int.emod_nonneg _ hz)]\n    nth_rewrite 3 [← Int.mul_ediv_add_emod k z]\n    rfl\n  · rintro ⟨k, n, h⟩\n    exact ⟨_, h⟩\n\nTarget:\ntheorem nsmul_mem_zmultiples_iff_exists_sub_div {r : R} {n : ℕ} (hn : n ≠ 0) :\n    n • r ∈ AddSubgroup.zmultiples p ↔\n      ∃ k : Fin n, r - (k : ℕ) • (p / n : R) ∈ AddSubgroup.zmultiples p :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/CharZero","family_id":"nsmul_mem_zmultiples_iff_exists_sub_div","file_id":"mathlib/Mathlib/Algebra/CharZero/Quotient.lean","sample_id":"a86ece7891869444431a13676c31d6de05fdf4898f0e667ba12ab2a36fc2b894"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a8a91e5033815085839eaad1f6a9d8721d67997ed1799815646887dd0bdc0a37","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2287237bfe25cfcf9e2c3471e496f268610b768110199fad21903ee8054d200c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e7bd69cfd0dadb9fdaab62111c02ae20ae7bf73cdd0cc212f534d85bcf679dc1","source_sha256":"5446f542892910a2397a84cd0f024393cbca50d14ecc1fa829e9751db035abd3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [AlgEquiv.ext_iff, Subtype.ext_iff, AlgEquiv.restrictNormal_apply]","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.2},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"83ac361c840ae2058b083e274477c29f521dc80a428b1d7c78a7613284d2b6f6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Splits\npublic import Mathlib.FieldTheory.Galois.Notation\npublic import Mathlib.FieldTheory.IntermediateField.Basic\npublic import Mathlib.FieldTheory.Minpoly.Field\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Thomas Browning, Patrick Lutz\n-/\n/-!\n# Normal field extensions\n\nIn this file we define normal field extensions.\n\n## Main Definitions\n\n- `Normal F K` where `K` is a field extension of `F`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Polynomial IsScalarTower\n\nvariable (F K : Type*) [Field F] [Field K] [Algebra F K]\n\n/-- Typeclass for normal field extensions: an algebraic extension of fields `K/F` is *normal*\nif the minimal polynomial of every element `x` in `K` splits in `K`, i.e. every `F`-conjugate\nof `x` is in `K`. -/\n@[stacks 09HM]\nclass Normal : Prop extends Algebra.IsAlgebraic F K where\n  splits' (x : K) : Splits ((minpoly F x).map (algebraMap F K))\n\nvariable {F K}\n\ntheorem Normal.isIntegral (_ : Normal F K) (x : K) : IsIntegral F x :=\n  Algebra.IsIntegral.isIntegral x\n\ntheorem Normal.splits (_ : Normal F K) (x : K) : Splits ((minpoly F x).map (algebraMap F K)) :=\n  Normal.splits' x\n\ntheorem normal_iff :\n    Normal F K ↔ ∀ x : K, IsIntegral F x ∧ Splits ((minpoly F x).map (algebraMap F K)) :=\n  ⟨fun h x => ⟨h.isIntegral x, h.splits x⟩, fun h =>\n    { isAlgebraic := fun x => (h x).1.isAlgebraic\n      splits' := fun x => (h x).2 }⟩\n\ntheorem Normal.out :\n    Normal F K → ∀ x : K, IsIntegral F x ∧ Splits ((minpoly F x).map (algebraMap F K)) :=\n  normal_iff.1\n\nvariable (F K)\n\ninstance normal_self : Normal F F where\n  isAlgebraic := fun _ => isIntegral_algebraMap.isAlgebraic\n  splits' := fun x => (minpoly.eq_X_sub_C' x).symm ▸ by simp\n\nsection NormalTower\n\nvariable (E : Type*) [Field E] [Algebra F E] [Algebra K E] [IsScalarTower F K E]\n\n@[stacks 09HN]\ntheorem Normal.tower_top_of_normal [h : Normal F E] : Normal K E :=\n  normal_iff.2 fun x => by\n    obtain ⟨hx, hhx⟩ := h.out x\n    rw [algebraMap_eq F K E, ← map_map] at hhx\n    exact ⟨hx.tower_top, hhx.of_dvd (map_ne_zero (map_ne_zero (minpoly.ne_zero hx)))\n      ((map_dvd_map' _).mpr (minpoly.dvd_map_of_isScalarTower F K x))⟩\n\ninstance IntermediateField.normal (K : IntermediateField F E) [Normal F E] : Normal K E :=\n  Normal.tower_top_of_normal F K E\n\ntheorem AlgHom.normal_bijective [h : Normal F E] (ϕ : E →ₐ[F] K) : Function.Bijective ϕ :=\n  h.toIsAlgebraic.bijective_of_isScalarTower' ϕ\n\nvariable {E F}\nvariable {E' : Type*} [Field E'] [Algebra F E']\n\ntheorem Normal.of_algEquiv [h : Normal F E] (f : E ≃ₐ[F] E') : Normal F E' := by\n  rw [normal_iff] at h ⊢\n  intro x; specialize h (f.symm x)\n  rw [← f.apply_symm_apply x, minpoly.algEquiv_eq, ← f.toAlgHom.comp_algebraMap, ← map_map]\n  exact ⟨h.1.map f, h.2.map _⟩\n\ntheorem AlgEquiv.transfer_normal (f : E ≃ₐ[F] E') : Normal F E ↔ Normal F E' :=\n  ⟨fun _ ↦ Normal.of_algEquiv f, fun _ ↦ Normal.of_algEquiv f.symm⟩\n\ntheorem Normal.of_equiv_equiv {M N : Type*} [Field N] [Field M] [Algebra M N]\n    [h : Normal F E] {f : F ≃+* M} {g : E ≃+* N}\n    (hcomp : (algebraMap M N).comp f = (g : E →+* N).comp (algebraMap F E)) :\n    Normal M N := by\n  have := h\n  rw [normal_iff] at h ⊢\n  intro x\n  rw [← g.apply_symm_apply x]\n  refine ⟨(h (g.symm x)).1.map_of_comp_eq _ _ hcomp, ?_⟩\n  rw [← minpoly.map_eq_of_equiv_equiv hcomp, map_map, hcomp, ← map_map]\n  exact (h (g.symm x)).2.map _\n\nend NormalTower\n\nnamespace IntermediateField\n\nvariable {F K}\nvariable {L : Type*} [Field L] [Algebra F L] [Algebra K L] [IsScalarTower F K L]\n\n@[simp]\ntheorem restrictScalars_normal {E : IntermediateField K L} :\n    Normal F (E.restrictScalars F) ↔ Normal F E :=\n  Iff.rfl\n\nend IntermediateField\n\nvariable {F} {K}\nvariable {K₁ K₂ K₃ : Type*} [Field K₁] [Field K₂] [Field K₃] [Algebra F K₁]\n  [Algebra F K₂] [Algebra F K₃] (ϕ : K₁ →ₐ[F] K₂) (χ : K₁ ≃ₐ[F] K₂) (ψ : K₂ →ₐ[F] K₃)\n  (ω : K₂ ≃ₐ[F] K₃)\n\nsection Restrict\n\nvariable (E : Type*) [Field E] [Algebra F E] [Algebra E K₁] [Algebra E K₂] [Algebra E K₃]\n  [IsScalarTower F E K₁] [IsScalarTower F E K₂] [IsScalarTower F E K₃]\n\n/-- Restrict algebra homomorphism to image of normal subfield -/\ndef AlgHom.restrictNormalAux [h : Normal F E] :\n    (toAlgHom F E K₁).range →ₐ[F] (toAlgHom F E K₂).range where\n  toFun x :=\n    ⟨ϕ x, by\n      suffices (toAlgHom F E K₁).range.map ϕ ≤ _ by exact this ⟨x, Subtype.mem x, rfl⟩\n      rintro x ⟨y, ⟨z, hy⟩, hx⟩\n      rw [← hx, ← hy]\n      apply minpoly.mem_range_of_degree_eq_one E\n      refine ((h.splits z).of_dvd (map_ne_zero (minpoly.ne_zero (h.isIntegral z)))\n        (minpoly.dvd E _ (by simp [aeval_algHom_apply]))).degree_eq_one_of_irreducible\n        (minpoly.irreducible ?_)\n      simp only [AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom]\n      suffices IsIntegral F _ by exact this.tower_top\n      exact ((h.isIntegral z).map <| toAlgHom F E K₁).map ϕ⟩\n  map_zero' := Subtype.ext (map_zero _)\n  map_one' := Subtype.ext (map_one _)\n  map_add' x y := Subtype.ext <| by simp\n  map_mul' x y := Subtype.ext <| by simp\n  commutes' x := Subtype.ext (ϕ.commutes x)\n\n/-- Restrict algebra homomorphism to normal subfield. -/\n@[stacks 0BME \"Part 1\"]\ndef AlgHom.restrictNormal [Normal F E] : E →ₐ[F] E :=\n  ((AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K₂)).symm.toAlgHom.comp\n        (ϕ.restrictNormalAux E)).comp\n    (AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K₁)).toAlgHom\n\n/-- Restrict algebra homomorphism to normal subfield (`AlgEquiv` version) -/\ndef AlgHom.restrictNormal' [Normal F E] : Gal(E/F) :=\n  AlgEquiv.ofBijective (AlgHom.restrictNormal ϕ E) (AlgHom.normal_bijective F E E _)\n\n@[simp]\ntheorem AlgHom.restrictNormal_commutes [Normal F E] (x : E) :\n    algebraMap E K₂ (ϕ.restrictNormal E x) = ϕ (algebraMap E K₁ x) :=\n  Subtype.ext_iff.mp\n    (AlgEquiv.apply_symm_apply (AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F E K₂))\n      (ϕ.restrictNormalAux E ⟨IsScalarTower.toAlgHom F E K₁ x, x, rfl⟩))\n\ntheorem AlgHom.restrictNormal_comp [Normal F E] :\n    (ψ.restrictNormal E).comp (ϕ.restrictNormal E) = (ψ.comp ϕ).restrictNormal E :=\n  AlgHom.ext fun _ =>\n    (algebraMap E K₃).injective (by simp only [AlgHom.comp_apply, AlgHom.restrictNormal_commutes])\n\n/-- Restrict algebra isomorphism to a normal subfield -/\ndef AlgEquiv.restrictNormal [Normal F E] : Gal(E/F) :=\n  AlgHom.restrictNormal' χ.toAlgHom E\n\n@[simp]\ntheorem AlgEquiv.restrictNormal_commutes [Normal F E] (x : E) :\n    algebraMap E K₂ (χ.restrictNormal E x) = χ (algebraMap E K₁ x) :=\n  χ.toAlgHom.restrictNormal_commutes E x\n\ntheorem AlgEquiv.restrictNormal_apply (L : IntermediateField F K₁) [Normal F L] (σ : Gal(K₁/F))\n    (x : L) : restrictNormal σ L x = σ x :=\n  AlgEquiv.restrictNormal_commutes σ L x\n\nTarget:\ntheorem AlgEquiv.restrictNormal_eq_one_iff (L : IntermediateField F K₁) [Normal F L]\n    (σ : Gal(K₁/F)) : restrictNormal σ L = 1 ↔ ∀ x ∈ L, σ x = x :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_e7bd69cfd0da","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"b721a9559137952d5e2b9a7140f0550004513700b5dc57a45125b203e13779aa","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Normal","family_id":"algequiv","file_id":"mathlib/Mathlib/FieldTheory/Normal/Defs.lean","sample_id":"e7bd69cfd0dadb9fdaab62111c02ae20ae7bf73cdd0cc212f534d85bcf679dc1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9a21e4324d10fe830df7d8256fcb74edf6972d935383079b80c071c0dccb073e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b92eb11d145737a83591d6e333398970dc71dbfbf846930df01755ff9e54e9ba","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c7eeb279213fe096be67e549c7a7f69ed83061d62764c53bfce17ab42221096a","source_sha256":"a1ecfd868e0d49b8667b2cdba350ee4f2ceb12fbe0b51094508e312f262dbf5e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using toInt_le_toInt val j i","hard_negative":true,"metrics":{"chosen_tokens":7,"rejected_tokens":5,"token_jaccard":0.090909,"token_length_ratio":0.714286},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"848fb2f04ce053b871ba8ac16046a3e30d71cb24386781bc08ab076e160bbaea","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.List.Pairwise\npublic import Batteries.Tactic.GeneralizeProofs\npublic import Mathlib.Tactic.Order.CollectFacts\n\nNamespace:\nMathlib.Tactic.Order.ToInt\n\nLocal context:\n/-\nCopyright (c) 2025 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\npublic meta import Mathlib.Util.AtomM\npublic meta import Mathlib.Util.Qq\npublic meta import Std.Data.HashMap.AdditionalOperations\n\n/-!\n# Translating linear orders to ℤ\n\nIn this file we implement the translation of a problem in any linearly ordered type to a problem in\n`ℤ`. This allows us to use the `lia` tactic to solve it.\n\nWhile the core algorithm of the `order` tactic is complete for the theory of linear orders in the\nsignature (`<`, `≤`),\nit becomes incomplete in the signature with lattice operations `⊓` and `⊔`. With these operations,\nthe problem becomes NP-hard, and the idea is to reuse a smart and efficient procedure, such as\n`lia`.\n\n## TODO\n\nMigrate to `grind` when it is ready.\n-/\n\npublic meta section\n\nnamespace Mathlib.Tactic.Order.ToInt\n\nvariable {α : Type*} [LinearOrder α] {n : ℕ} (val : Fin n → α)\n\n/-- The main theorem asserting the existence of a translation.\nWe use `Classical.choose` to turn this into a value for use in the `order` tactic,\nsee `toInt`.\n-/\ntheorem exists_translation : ∃ tr : Fin n → ℤ, ∀ i j, val i ≤ val j ↔ tr i ≤ tr j := by\n  let li := List.ofFn val\n  let sli := li.mergeSort\n  have (i : Fin n) : ∃ j : Fin sli.length, sli[j] = val i := by\n    apply List.get_of_mem\n    rw [List.Perm.mem_iff (List.mergeSort_perm _ _)]\n    simp [li]\n  use fun i ↦ (this i).choose\n  intro i j\n  simp only [Fin.getElem_fin, Int.ofNat_le]\n  by_cases h_eq : val i = val j\n  · simp [h_eq]\n  generalize_proofs _ hi hj\n  rw [← hi.choose_spec, ← hj.choose_spec] at h_eq\n  conv_lhs => rw [← hi.choose_spec, ← hj.choose_spec]\n  have := li.pairwise_mergeSort (le := fun a b ↦ decide (a ≤ b))\n      (fun a b c ↦ by simpa using le_trans) (by simpa using le_total)\n  rw [List.pairwise_iff_get] at this\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · contrapose! h\n    exact lt_of_le_of_ne (by simpa using (this hj.choose hi.choose (by simpa)))\n      (fun h ↦ h_eq (h.symm))\n  · simpa using this hi.choose hj.choose (by apply lt_of_le_of_ne h; contrapose h_eq; simp [h_eq])\n\n/-- Auxiliary definition used by the `order` tactic to transfer facts in a linear order to `ℤ`. -/\nnoncomputable def toInt (k : Fin n) : ℤ :=\n  (exists_translation val).choose k\n\nvariable (i j k : Fin n)\n\ntheorem toInt_le_toInt : toInt val i ≤ toInt val j ↔ val i ≤ val j := by\n  simp [toInt, (exists_translation val).choose_spec]\n\ntheorem toInt_lt_toInt : toInt val i < toInt val j ↔ val i < val j := by\n  simpa using (toInt_le_toInt val j i).not\n\ntheorem toInt_eq_toInt : toInt val i = toInt val j ↔ val i = val j := by\n  simp [toInt_le_toInt, le_antisymm_iff]\n\ntheorem toInt_ne_toInt : toInt val i ≠ toInt val j ↔ val i ≠ val j := by\n  simpa using (toInt_eq_toInt val i j).not\n\ntheorem toInt_nle_toInt : ¬toInt val i ≤ toInt val j ↔ ¬val i ≤ val j := by\n  simpa using toInt_lt_toInt val j i\n\nTarget:\ntheorem toInt_nlt_toInt : ¬toInt val i < toInt val j ↔ ¬val i < val j :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_c7eeb279213f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"31b2b05362dc0f6eb2a89a2c2f404f4c574e1b7101c98049288b9856ed547a8c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Order","family_id":"toint_nlt_toint","file_id":"mathlib/Mathlib/Tactic/Order/ToInt.lean","sample_id":"c7eeb279213fe096be67e549c7a7f69ed83061d62764c53bfce17ab42221096a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8f3b01149f94ae9d113cd2afe1cda1c376344890d5f097114694fdbd6fd42eb6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1079d2160a585c07511be441b8c3dc809238a8e95538c5a860932de4e1b57b56","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c7abf66d2d061cad451c9a404bc80bcd09d739b8aa88cc477da70de900f24a00","source_sha256":"bdcd2022da453373ffbdfe5bdbc273db0a9656383ede2bdc6575e68ee03c9aa0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases s.eq_empty_or_nonempty with (rfl | hs); · simp\n  rcases t.eq_empty_or_nonempty with (rfl | ht); · simp\n  simp only [hs.ne_empty, ht.ne_empty, isBounded_prod_of_nonempty (hs.prod ht), false_or]","hard_negative":true,"metrics":{"chosen_tokens":48,"rejected_tokens":3,"token_jaccard":0.038462,"token_length_ratio":0.0625},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"849cf0126fc02ac9cc3b7dda1ffe4168e847bad38adcf930bd2916580332e14f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.TypeTags.Basic\npublic import Mathlib.Topology.Bornology.Basic\n\nNamespace:\nBornology\n\nLocal context:\n/-\nCopyright (c) 2022 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Bornology structure on products and subtypes\n\nIn this file we define `Bornology` and `BoundedSpace` instances on `α × β`, `Π i, X i`, and\n`{x // p x}`. We also prove basic lemmas about `Bornology.cobounded` and `Bornology.IsBounded`\non these types.\n-/\n\npublic section\n\n\nopen Set Filter Bornology Function\n\nopen Filter\n\nvariable {α β ι : Type*} {X : ι → Type*} [Bornology α] [Bornology β]\n  [∀ i, Bornology (X i)]\n\ninstance Prod.instBornology : Bornology (α × β) where\n  cobounded := (cobounded α).coprod (cobounded β)\n  le_cofinite :=\n    @coprod_cofinite α β ▸ coprod_mono ‹Bornology α›.le_cofinite ‹Bornology β›.le_cofinite\n\ninstance Pi.instBornology : Bornology (∀ i, X i) where\n  cobounded := Filter.coprodᵢ fun i => cobounded (X i)\n  le_cofinite := iSup_le fun _ ↦ (comap_mono (Bornology.le_cofinite _)).trans (comap_cofinite_le _)\n\n/-- Inverse image of a bornology. -/\nabbrev Bornology.induced {α β : Type*} [Bornology β] (f : α → β) : Bornology α where\n  cobounded := comap f (cobounded β)\n  le_cofinite := (comap_mono (Bornology.le_cofinite β)).trans (comap_cofinite_le _)\n\ninstance {p : α → Prop} : Bornology (Subtype p) :=\n  Bornology.induced (Subtype.val : Subtype p → α)\n\nnamespace Bornology\n\n/-!\n### Bounded sets in `α × β`\n-/\n\n\ntheorem cobounded_prod : cobounded (α × β) = (cobounded α).coprod (cobounded β) :=\n  rfl\n\ntheorem isBounded_image_fst_and_snd {s : Set (α × β)} :\n    IsBounded (Prod.fst '' s) ∧ IsBounded (Prod.snd '' s) ↔ IsBounded s :=\n  compl_mem_coprod.symm\n\nlemma IsBounded.image_fst {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.fst '' s) :=\n  (isBounded_image_fst_and_snd.2 hs).1\n\nlemma IsBounded.image_snd {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.snd '' s) :=\n  (isBounded_image_fst_and_snd.2 hs).2\n\nvariable {s : Set α} {t : Set β} {S : ∀ i, Set (X i)}\n\ntheorem IsBounded.fst_of_prod (h : IsBounded (s ×ˢ t)) (ht : t.Nonempty) : IsBounded s :=\n  fst_image_prod s ht ▸ h.image_fst\n\ntheorem IsBounded.snd_of_prod (h : IsBounded (s ×ˢ t)) (hs : s.Nonempty) : IsBounded t :=\n  snd_image_prod hs t ▸ h.image_snd\n\ntheorem IsBounded.prod (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s ×ˢ t) :=\n  isBounded_image_fst_and_snd.1\n    ⟨hs.subset <| fst_image_prod_subset _ _, ht.subset <| snd_image_prod_subset _ _⟩\n\ntheorem isBounded_prod_of_nonempty (hne : Set.Nonempty (s ×ˢ t)) :\n    IsBounded (s ×ˢ t) ↔ IsBounded s ∧ IsBounded t :=\n  ⟨fun h => ⟨h.fst_of_prod hne.snd, h.snd_of_prod hne.fst⟩, fun h => h.1.prod h.2⟩\n\nTarget:\ntheorem isBounded_prod : IsBounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ IsBounded s ∧ IsBounded t :=\n\nProof body:\n","rejected":"by\n  exact isBounded_prod","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"83b0d7c437e53ea57ece7ad820efd8174deaf3baae8e0c589578749f4643ba86","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Bornology","family_id":"isbounded_prod","file_id":"mathlib/Mathlib/Topology/Bornology/Constructions.lean","sample_id":"c7abf66d2d061cad451c9a404bc80bcd09d739b8aa88cc477da70de900f24a00"},"split":"train"}
{"binding":{"chosen_proof_sha256":"35928b4ec6c4dd8e48530f64dacc44b471422b62b6f0a8ae08f8ed59d2ec96f4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"325825a927e5735338dd52a28d2dff4f12d1ad248dc0bc8ca3daba7d87f856c6","source_sha256":"65733f0fee285d1da4e1bad2af528d5547ef268aaa4d457452ee23fb9ac9cd1d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  calc\n    _ = 𝟙 _ ⊗≫ (F.map₂ β ▷ η.app b ≫ η.naturality g) ▷ θ.app b ⊗≫\n          η.app a ◁ θ.naturality g ⊗≫ 𝟙 _ := by\n      bicategory\n    _ = 𝟙 _ ⊗≫ η.naturality f ▷ θ.app b ⊗≫\n          η.app a ◁ (G.map₂ β ▷ θ.app b ≫ θ.naturality g) ⊗≫ 𝟙 _ := by\n      rw [η.naturality_naturality]\n      bicategory\n    _ = _ := by\n      rw [θ.naturality_naturality]\n      bicategory","hard_negative":false,"metrics":{"chosen_tokens":112,"rejected_tokens":2,"token_jaccard":0.03125,"token_length_ratio":0.017857},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"8519be5fe12fe566f5218ffa1a27e65eda9d07c8eae710fcb62b90533538e12c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Bicategory.Functor.Lax\npublic import Mathlib.Tactic.CategoryTheory.Bicategory.Basic\n\nNamespace:\nCategoryTheory.Lax.OplaxTrans\n\nLocal context:\n/-\nCopyright (c) 2025 Robin Carlier. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yuma Mizuno, Calle Sönne, Robin Carlier\n-/\n/-!\n# Transformations between lax functors\n\nJust as there are natural transformations between functors, there are transformations\nbetween lax functors. The equality in the naturality condition of a natural transformation gets\nreplaced by a specified 2-morphism. Now, there are three possible types of transformations (between\nlax functors):\n* lax natural transformations;\n* oplax natural transformations;\n* strong natural transformations.\n\nThese differ in the direction (and invertibility) of the 2-morphisms involved in the naturality\ncondition.\n\n## Main definitions\n\n* `Lax.LaxTrans F G`: lax transformations between lax functors `F` and `G`. The naturality\n  condition is given by a 2-morphism `app a ≫ G.map f ⟶ F.map f ≫ app b` for each 1-morphism\n  `f : a ⟶ b`.\n* `Lax.OplaxTrans F G`: oplax transformations between lax functors `F` and `G`. The naturality\n  condition is given by a 2-morphism `F.map f ≫ app b ⟶ app a ≫ G.map f` for each 1-morphism\n  `f : a ⟶ b`.\n* `Lax.StrongTrans F G`: strong transformations between lax functors `F` and `G`. The naturality\n  condition is given by a 2-isomorphism `app a ≫ G.map f ≅ F.map f ≫ app b` for each 1-morphism\n  `f : a ⟶ b`.\n\nUsing these, we define three (scoped) `CategoryStruct` instances on `B ⥤ᴸ C`, in the\n`Lax.LaxTrans`, `Lax.OplaxTrans`, and `Lax.StrongTrans` namespaces. The arrows in these\n`CategoryStruct` instances are given by lax transformations, oplax transformations, and strong\ntransformations respectively.\n\nWe also provide API for going between lax transformations and strong transformations:\n* `LaxTrans.StrongCore η`: a structure on a lax transformation between lax functors that\n  promotes it to a strong transformation.\n* `StrongTrans.mkOfLax η η'`: given a lax transformation `η` such that each component\n  2-morphism is an isomorphism, `mkOfLax` gives the corresponding strong transformation.\n\n## References\n* [Niles Johnson, Donald Yau, *2-Dimensional Categories*](https://arxiv.org/abs/2002.06055),\n  section 4.2.\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory.Lax\n\nopen Category Bicategory\n\nuniverse w₁ w₂ v₁ v₂ u₁ u₂\n\nvariable {B : Type u₁} [Bicategory.{w₁, v₁} B] {C : Type u₂} [Bicategory.{w₂, v₂} C]\n\n/-- If `η` is a lax transformation between `F` and `G`, we have a 1-morphism\n`η.app a : F.obj a ⟶ G.obj a` for each object `a : B`. We also have a 2-morphism\n`η.naturality f : app a ≫ G.map f ⟶ F.map f ≫ app b` for each 1-morphism `f : a ⟶ b`.\nThese 2-morphisms satisfy the naturality condition, and preserve the identities and\nthe compositions modulo some adjustments of domains and codomains of 2-morphisms.\n-/\nstructure LaxTrans (F G : B ⥤ᴸ C) where\n  /-- The component 1-morphisms of a lax transformation. -/\n  app (a : B) : F.obj a ⟶ G.obj a\n  /-- The 2-morphisms underlying the lax naturality constraint. -/\n  naturality {a b : B} (f : a ⟶ b) : app a ≫ G.map f ⟶ F.map f ≫ app b\n  /-- Naturality of the lax naturality constraint. -/\n  naturality_naturality {a b : B} {f g : a ⟶ b} (η : f ⟶ g) :\n      naturality f ≫ F.map₂ η ▷ app b = app a ◁ G.map₂ η ≫ naturality g := by\n    cat_disch\n  /-- Lax unity. -/\n  naturality_id (a : B) :\n      app a ◁ G.mapId a ≫ naturality (𝟙 a) =\n        (ρ_ (app a)).hom ≫ (λ_ (app a)).inv ≫ F.mapId a ▷ app a := by\n    cat_disch\n  /-- Lax functoriality. -/\n  naturality_comp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) :\n      app a ◁ G.mapComp f g ≫ naturality (f ≫ g) =\n      (α_ _ _ _).inv ≫ naturality f ▷ G.map g ≫ (α_ _ _ _).hom ≫\n        F.map f ◁ naturality g ≫ (α_ _ _ _).inv ≫ F.mapComp f g ▷ app c := by\n    cat_disch\n\nattribute [reassoc (attr := simp)] LaxTrans.naturality_naturality LaxTrans.naturality_id\n  LaxTrans.naturality_comp\n\nnamespace LaxTrans\n\nvariable {F G H : B ⥤ᴸ C} (η : LaxTrans F G) (θ : LaxTrans G H)\n\nvariable (F) in\n/-- The identity lax transformation. -/\ndef id : LaxTrans F F where\n  app a := 𝟙 (F.obj a)\n  naturality {_ _} f := (λ_ (F.map f)).hom ≫ (ρ_ (F.map f)).inv\n\ninstance : Inhabited (LaxTrans F F) :=\n  ⟨id F⟩\n\n/-- Auxiliary definition for `vComp`. -/\nabbrev vCompApp (a : B) : F.obj a ⟶ H.obj a :=\n  η.app a ≫ θ.app a\n\n/-- Auxiliary definition for `vComp`. -/\nabbrev vCompNaturality {a b : B} (f : a ⟶ b) :\n    (η.app a ≫ θ.app a) ≫ H.map f ⟶ F.map f ≫ η.app b ≫ θ.app b :=\n  (α_ _ _ _).hom ≫ η.app a ◁ θ.naturality f ≫ (α_ _ _ _).inv ≫\n    η.naturality f ▷ θ.app b ≫ (α_ _ _ _).hom\n\ntheorem vComp_naturality_naturality {a b : B} {f g : a ⟶ b} (β : f ⟶ g) :\n    η.vCompNaturality θ f ≫ F.map₂ β ▷ η.vCompApp θ b =\n      η.vCompApp θ a ◁ H.map₂ β ≫ η.vCompNaturality θ g :=\n  calc\n    _ = 𝟙 _ ⊗≫ η.app a ◁ θ.naturality f ⊗≫\n          (η.naturality f ≫ F.map₂ β ▷ η.app b) ▷ θ.app b ⊗≫ 𝟙 _ := by\n      bicategory\n    _ = 𝟙 _ ⊗≫ η.app a ◁ (θ.naturality f ≫ G.map₂ β ▷ θ.app b) ⊗≫\n          η.naturality g ▷ θ.app b ⊗≫ 𝟙 _ := by\n      rw [naturality_naturality]\n      bicategory\n    _ = _ := by\n      rw [naturality_naturality]\n      bicategory\n\ntheorem vComp_naturality_id (a : B) :\n    η.vCompApp θ a ◁ H.mapId a ≫ η.vCompNaturality θ (𝟙 a) =\n      (ρ_ (η.vCompApp θ a)).hom ≫ (λ_ (η.vCompApp θ a)).inv ≫ F.mapId a ▷ η.vCompApp θ a :=\n  calc\n    _ = 𝟙 _ ⊗≫ η.app a ◁ (θ.app a ◁ H.mapId a ≫ θ.naturality (𝟙 a)) ⊗≫\n          η.naturality (𝟙 a) ▷ θ.app a ⊗≫ 𝟙 _ := by\n      bicategory\n    _ = 𝟙 _ ⊗≫ (η.app a ◁ G.mapId a ≫ η.naturality (𝟙 a)) ▷ θ.app a ⊗≫ 𝟙 _ := by\n      rw [naturality_id]\n      bicategory\n    _ = _ := by\n      rw [naturality_id]\n      bicategory\n\ntheorem vComp_naturality_comp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) :\n    η.vCompApp θ a ◁ H.mapComp f g ≫ η.vCompNaturality θ (f ≫ g) =\n      (α_ (η.vCompApp θ a) (H.map f) (H.map g)).inv ≫\n        η.vCompNaturality θ f ▷ H.map g ≫\n          (α_ (F.map f) (η.vCompApp θ b) (H.map g)).hom ≫\n            F.map f ◁ η.vCompNaturality θ g ≫\n              (α_ (F.map f) (F.map g) (η.vCompApp θ c)).inv ≫ F.mapComp f g ▷ η.vCompApp θ c :=\n  calc\n    _ = 𝟙 _ ⊗≫ η.app a ◁ (θ.app a ◁ H.mapComp f g ≫ θ.naturality (f ≫ g)) ⊗≫\n          η.naturality (f ≫ g) ▷ θ.app c ⊗≫ 𝟙 _ := by\n      bicategory\n    _ = 𝟙 _ ⊗≫ η.app a ◁ (θ.naturality f ▷ (H.map g) ⊗≫ G.map f ◁ θ.naturality g) ⊗≫\n          (η.app a ◁ G.mapComp f g ≫ η.naturality (f ≫ g)) ▷ θ.app c ⊗≫ 𝟙 _ := by\n      rw [naturality_comp θ]\n      bicategory\n    _ = 𝟙 _ ⊗≫ η.app a ◁ θ.naturality f ▷ H.map g ⊗≫\n          ((η.app a ≫ G.map f) ◁ θ.naturality g ≫ η.naturality f ▷ (G.map g ≫ θ.app c)) ⊗≫\n            F.map f ◁ η.naturality g ▷ θ.app c ⊗≫\n              F.mapComp f g ▷ η.app c ▷ θ.app c ⊗≫ 𝟙 _ := by\n      rw [naturality_comp η]\n      bicategory\n    _ = _ := by\n      rw [whisker_exchange]\n      bicategory\n\n/-- Vertical composition of lax transformations. -/\ndef vComp (η : LaxTrans F G) (θ : LaxTrans G H) : LaxTrans F H where\n  app a := vCompApp η θ a\n  naturality := vCompNaturality η θ\n  naturality_naturality := vComp_naturality_naturality η θ\n  naturality_id := vComp_naturality_id η θ\n  naturality_comp := vComp_naturality_comp η θ\n\nattribute [local simp] vCompApp vCompNaturality in\n/-- `CategoryStruct` on `B ⥤ᴸ C` where the (1-)morphisms are given by lax\ntransformations. -/\n@[simps! id_app id_naturality comp_app comp_naturality]\nscoped instance : CategoryStruct (B ⥤ᴸ C) where\n  Hom := LaxTrans\n  id := LaxTrans.id\n  comp := LaxTrans.vComp\n\n@[deprecated (since := \"2026-03-16\")] alias vComp_app := comp_app\n@[deprecated (since := \"2026-03-16\")] alias vComp_naturality := comp_naturality\n\nend LaxTrans\n\n/-- If `η` is an oplax transformation between `F` and `G`, we have a 1-morphism\n`η.app a : F.obj a ⟶ G.obj a` for each object `a : B`. We also have a 2-morphism\n`η.naturality f : F.map f ≫ app b ⟶ app a ≫ G.map f` for each 1-morphism `f : a ⟶ b`.\nThese 2-morphisms satisfy the naturality condition, and preserve the identities and\nthe compositions modulo some adjustments of domains and codomains of 2-morphisms.\n-/\nstructure OplaxTrans (F G : B ⥤ᴸ C) where\n  /-- The component 1-morphisms of an oplax transformation. -/\n  app (a : B) : F.obj a ⟶ G.obj a\n  /-- The 2-morphisms underlying the oplax naturality constraint. -/\n  naturality {a b : B} (f : a ⟶ b) : F.map f ≫ app b ⟶ app a ≫ G.map f\n  /-- Naturality of the oplax naturality constraint. -/\n  naturality_naturality {a b : B} {f g : a ⟶ b} (η : f ⟶ g) :\n      F.map₂ η ▷ app b ≫ naturality g = naturality f ≫ app a ◁ G.map₂ η := by\n    cat_disch\n  naturality_id (a : B) :\n      F.mapId a ▷ app a ≫ naturality (𝟙 a) =\n        (λ_ (app a)).hom ≫ (ρ_ (app a)).inv ≫ app a ◁ G.mapId a := by\n    cat_disch\n  naturality_comp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) :\n      F.mapComp f g ▷ app c ≫ naturality (f ≫ g) =\n        (α_ _ _ _).hom ≫ F.map f ◁ naturality g ≫\n          (α_ _ _ _).inv ≫ naturality f ▷ G.map g ≫ (α_ _ _ _).hom ≫\n            app a ◁ G.mapComp f g := by\n    cat_disch\n\nnamespace OplaxTrans\n\nattribute [reassoc (attr := simp)] naturality_naturality naturality_id naturality_comp\n\nvariable {F G H : B ⥤ᴸ C} (η : OplaxTrans F G) (θ : OplaxTrans G H)\n\nvariable (F) in\n/-- The identity oplax transformation. -/\ndef id : OplaxTrans F F where\n  app a := 𝟙 (F.obj a)\n  naturality {_ _} f := (ρ_ (F.map f)).hom ≫ (λ_ (F.map f)).inv\n\ninstance : Inhabited (OplaxTrans F F) :=\n  ⟨id F⟩\n\n/-- Auxiliary definition for `vComp`. -/\nabbrev vCompApp (a : B) : F.obj a ⟶ H.obj a := η.app a ≫ θ.app a\n\n/-- Auxiliary definition for `vComp`. -/\nabbrev vCompNaturality {a b : B} (f : a ⟶ b) :\n    F.map f ≫ η.app b ≫ θ.app b ⟶ (η.app a ≫ θ.app a) ≫ H.map f :=\n  (α_ _ _ _).inv ≫ η.naturality f ▷ θ.app b ≫ (α_ _ _ _).hom ≫\n    η.app a ◁ θ.naturality f ≫ (α_ _ _ _).inv\n\nTarget:\ntheorem vComp_naturality_naturality {a b : B} {f g : a ⟶ b} (β : f ⟶ g) :\n    F.map₂ β ▷ η.vCompApp θ b ≫ η.vCompNaturality θ g =\n      η.vCompNaturality θ f ≫ η.vCompApp θ a ◁ H.map₂ β :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Bicategory","family_id":"vcomp_naturality_naturality","file_id":"mathlib/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Lax.lean","sample_id":"325825a927e5735338dd52a28d2dff4f12d1ad248dc0bc8ca3daba7d87f856c6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ecfe891855dc120f1040ab8b4113f995f2ce6be82fd87d86c20a75c2f1e66b44","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6381270961bbd859431dc91f4de11a424555613e987c739bd7c51cfea5347942","source_sha256":"d6e838a5f1b3817af8820eb123fb16847eef4259153bc8aa4506ab114a1382bf","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  exact .tower_top (K := F⟮(X : F⟮X⟯)⟯) _\n\nvariable {y : K}","hard_negative":true,"metrics":{"chosen_tokens":26,"rejected_tokens":8,"token_jaccard":0.136364,"token_length_ratio":0.307692},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"851fedd9fafba0caffca0cec300529e36dde29069b3fdf0d1cbd659c86146bab","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.RatFunc.Degree\npublic import Mathlib.RingTheory.DedekindDomain.IntegralClosure\npublic import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed\npublic import Mathlib.Topology.Algebra.Valued.ValuedField\npublic import Mathlib.Topology.Algebra.InfiniteSum.Defs\npublic import Mathlib.FieldTheory.RatFunc.IntermediateField\npublic import Mathlib.RingTheory.Adjoin.Polynomial.Bivariate\npublic import Mathlib.FieldTheory.RatFunc.Valuation -- for deprecation to `RatFunc.inftyValuation` and `RatFunc.CompletionAtInfty`\n\nNamespace:\nFunctionField\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen, Ashvni Narayanan\n-/\n/-!\n# Function fields\n\nThis file defines a function field and the ring of integers corresponding to it.\n\n## Main definitions\n\n- `FunctionField F K` states that `K` is a function field over the field `F`,\n  i.e. it is a finite extension of the field of rational functions in one variable over `F`.\n- `FunctionField.ringOfIntegers` defines the ring of integers corresponding to a function field\n  as the integral closure of `F[X]` in the function field.\n\n## Implementation notes\nThe definitions that involve a field of fractions choose a canonical field of fractions,\nbut are independent of that choice. We also omit assumptions like\n`IsScalarTower F[X] (FractionRing F[X]) K` in definitions,\nadding them back in lemmas when they are needed.\n\n## References\n* [D. Marcus, *Number Fields*][marcus1977number]\n* [J.W.S. Cassels, A. Fröhlich, *Algebraic Number Theory*][cassels1967algebraic]\n* [P. Samuel, *Algebraic Theory of Numbers*][samuel1967]\n* [M. Rosen, *Number Theory in Function Fields*][rosen2002]\n\n## Tags\nfunction field, ring of integers\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen scoped nonZeroDivisors Polynomial WithZero RatFunc\n\nvariable (F K : Type*) [Field F] [Field K]\n\n/-- `K` is a function field over the field `F` if it is a finite\nextension of the field of rational functions in one variable over `F`.\n\nNote that `K` can be a function field over multiple, non-isomorphic, `F`.\n-/\nabbrev FunctionField [Algebra F⟮X⟯ K] : Prop :=\n  FiniteDimensional F⟮X⟯ K\n\n/-- `K` is a function field over `F` iff it is a finite extension of `F(t)`. -/\ntheorem functionField_iff (Ft : Type*) [Field Ft] [Algebra F[X] Ft]\n    [IsFractionRing F[X] Ft] [Algebra F⟮X⟯ K] [Algebra Ft K] [Algebra F[X] K]\n    [IsScalarTower F[X] Ft K] [IsScalarTower F[X] F⟮X⟯ K] :\n    FunctionField F K ↔ FiniteDimensional Ft K := by\n  let e := IsLocalization.algEquiv F[X]⁰ F⟮X⟯ Ft\n  have : ∀ (c) (x : K), e c • x = c • x := by\n    intro c x\n    rw [Algebra.smul_def, Algebra.smul_def]\n    congr\n    refine congr_fun (f := fun c => algebraMap Ft K (e c)) ?_ c\n    refine IsLocalization.ext (nonZeroDivisors F[X]) _ _ ?_ ?_ ?_ ?_ ?_ <;> intros <;>\n      simp only [map_one, map_mul, AlgEquiv.commutes, ← IsScalarTower.algebraMap_apply]\n  constructor <;> intro h\n  · let b := Module.finBasis F⟮X⟯ K\n    exact (b.mapCoeffs e this).finiteDimensional_of_finite\n  · let b := Module.finBasis Ft K\n    refine (b.mapCoeffs e.symm ?_).finiteDimensional_of_finite\n    intro c x; convert! (this (e.symm c) x).symm; simp only [e.apply_symm_apply]\n\nnamespace FunctionField\n\ntheorem algebraMap_injective [Algebra F[X] K] [Algebra F⟮X⟯ K]\n    [IsScalarTower F[X] F⟮X⟯ K] : Function.Injective (algebraMap F[X] K) := by\n  rw [IsScalarTower.algebraMap_eq F[X] F⟮X⟯ K]\n  exact (algebraMap F⟮X⟯ K).injective.comp (IsFractionRing.injective F[X] F⟮X⟯)\n\n/-- The function field analogue of `NumberField.ringOfIntegers`:\n`FunctionField.ringOfIntegers F K` is the integral closure of `F[X]` in `K`.\n\nWe don't actually assume `K` is a function field over `F` in the definition,\nonly when proving its properties.\n-/\ndef ringOfIntegers [Algebra F[X] K] :=\n  integralClosure F[X] K\n\nnamespace ringOfIntegers\n\nvariable [Algebra F[X] K]\n\ninstance : IsDomain (ringOfIntegers F K) :=\n  (ringOfIntegers F K).isDomain\n\ninstance : IsIntegralClosure (ringOfIntegers F K) F[X] K :=\n  integralClosure.isIntegralClosure _ _\n\nvariable [Algebra F⟮X⟯ K] [IsScalarTower F[X] F⟮X⟯ K]\n\ntheorem algebraMap_injective : Function.Injective (algebraMap F[X] (ringOfIntegers F K)) := by\n  have hinj : Function.Injective (algebraMap F[X] K) := by\n    rw [IsScalarTower.algebraMap_eq F[X] F⟮X⟯ K]\n    exact (algebraMap F⟮X⟯ K).injective.comp (IsFractionRing.injective F[X] F⟮X⟯)\n  rw [injective_iff_map_eq_zero (algebraMap F[X] (↥(ringOfIntegers F K)))]\n  intro p hp\n  rw [← Subtype.coe_inj, Subalgebra.coe_zero] at hp\n  rw [injective_iff_map_eq_zero (algebraMap F[X] K)] at hinj\n  exact hinj p hp\n\ntheorem not_isField : ¬IsField (ringOfIntegers F K) := by\n  simpa [← (IsIntegralClosure.isIntegral_algebra F[X] K).isField_iff_isField\n      (algebraMap_injective F K)] using\n    Polynomial.not_isField F\n\nvariable [FunctionField F K]\n\ninstance : IsFractionRing (ringOfIntegers F K) K :=\n  integralClosure.isFractionRing_of_finite_extension F⟮X⟯ K\n\ninstance : IsIntegrallyClosed (ringOfIntegers F K) :=\n  integralClosure.isIntegrallyClosedOfFiniteExtension F⟮X⟯\n\ninstance [Algebra.IsSeparable F⟮X⟯ K] : IsNoetherian F[X] (ringOfIntegers F K) :=\n  IsIntegralClosure.isNoetherian _ F⟮X⟯ K _\n\ninstance [Algebra.IsSeparable F⟮X⟯ K] : IsDedekindDomain (ringOfIntegers F K) :=\n  IsIntegralClosure.isDedekindDomain F[X] F⟮X⟯ K _\n\nend ringOfIntegers\n\nsection deprecated\n\n@[deprecated RatFunc.inftyValuationDef (since := \"2026-04-14\")]\nalias inftyValuationDef := RatFunc.inftyValuationDef\n\n@[deprecated RatFunc.InftyValuation.map_zero' (since := \"2026-04-14\")]\nalias InftyValuation.map_zero' := RatFunc.InftyValuation.map_zero'\n\n@[deprecated RatFunc.InftyValuation.map_one' (since := \"2026-04-14\")]\nalias InftyValuation.map_one' := RatFunc.InftyValuation.map_one'\n\n@[deprecated RatFunc.InftyValuation.map_mul' (since := \"2026-04-14\")]\nalias InftyValuation.map_mul' := RatFunc.InftyValuation.map_mul'\n\n@[deprecated RatFunc.InftyValuation.map_add_le_max' (since := \"2026-04-14\")]\nalias InftyValuation.map_add_le_max' := RatFunc.InftyValuation.map_add_le_max'\n\n@[deprecated RatFunc.inftyValuation_of_nonzero (since := \"2026-04-14\")]\nalias inftyValuation_of_nonzero := RatFunc.inftyValuation_of_nonzero\n\n@[deprecated RatFunc.inftyValuation (since := \"2026-04-14\")]\nalias inftyValuation := RatFunc.inftyValuation\n\n@[deprecated RatFunc.inftyValuation_apply (since := \"2026-04-14\")]\nalias inftyValuation_apply := RatFunc.inftyValuation_apply\n\n@[deprecated RatFunc.inftyValuation.C (since := \"2026-04-14\")]\nalias inftyValuation.C := RatFunc.inftyValuation.C\n\n@[deprecated RatFunc.inftyValuation.X (since := \"2026-04-14\")]\nalias inftyValuation.X := RatFunc.inftyValuation.X\n\n@[deprecated RatFunc.inftyValuation.X_zpow (since := \"2026-04-14\")]\nalias inftyValuation.X_zpow := RatFunc.inftyValuation.X_zpow\n\n@[deprecated RatFunc.inftyValuation.X_inv (since := \"2026-04-14\")]\nalias inftyValuation.X_inv := RatFunc.inftyValuation.X_inv\n\n@[deprecated RatFunc.inftyValuation.polynomial (since := \"2026-04-14\")]\nalias inftyValuation.polynomial := RatFunc.inftyValuation.polynomial\n\n@[deprecated RatFunc.inftyValued (since := \"2026-04-14\")]\nalias inftyValuedFqt := RatFunc.inftyValued\n\n@[deprecated RatFunc.inftyValued.def (since := \"2026-04-14\")]\nalias inftyValuedFqt.def := RatFunc.inftyValued.def\n\n@[deprecated RatFunc.CompletionAtInfty (since := \"2026-04-14\")]\nalias FqtInfty := RatFunc.CompletionAtInfty\n\n@[deprecated \"Use the anonymous `Valued` instance on `RatFunc.CompletionAtInfty`\"\n(since := \"2026-04-14\")]\ninstance valuedFqtInfty [DecidableEq F⟮X⟯] :\n    Valued (RatFunc.CompletionAtInfty F) ℤᵐ⁰ :=\n  inferInstance\n\n@[deprecated RatFunc.valuedCompletionAtInfty.def (since := \"2026-04-14\")]\nalias valuedFqtInfty.def := RatFunc.valuedCompletionAtInfty.def\n\nend deprecated\n\nsection AdjoinTranscendental\n\nopen IntermediateField RatFunc\n\nvariable {F K : Type*} [Field F] [Field K] [Algebra F⟮X⟯ K] [FunctionField F K]\n\ninstance FiniteDimensional.adjoin_X : FiniteDimensional F⟮(X : F⟮X⟯)⟯ K :=\n  have : Module.Finite (⊤ : IntermediateField F F⟮X⟯) F⟮X⟯ :=\n    .top_left F⟮X⟯ F⟮X⟯\n  RatFunc.adjoin_X (K := F) ▸ Module.Finite.trans F⟮X⟯ _\n\nvariable [Algebra F K] [IsScalarTower F F⟮X⟯ K]\n\ntheorem FiniteDimensional.adjoin_algebraMap_X :\n    FiniteDimensional F⟮algebraMap _ K (X : F⟮X⟯)⟯ K :=\n  .of_restrictScalars_finite F⟮(X : F⟮X⟯)⟯ _ _\n\nTarget:\ntheorem Algebra.IsAlgebraic.adjoin_algebraMap_X :\n    Algebra.IsAlgebraic F⟮algebraMap _ K (X : F⟮X⟯)⟯ K :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"6fb4b0b0f18f0e288eb0e9ec48b6b6953124fdb7096f3426fcdb563cf7205245","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory","family_id":"algebra","file_id":"mathlib/Mathlib/NumberTheory/FunctionField.lean","sample_id":"6381270961bbd859431dc91f4de11a424555613e987c739bd7c51cfea5347942"},"split":"train"}
{"binding":{"chosen_proof_sha256":"de8a32b5f73d36b3c5675db1bfb4e3c6a3d479eee57e39397db99ac101c60702","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"310883994a3c9ca44f3b346c085fc78418585c16b9347c0db65c86639692393f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"460fd408bbfa2c9d3696e93eaf82711398320704b30a3dd64ec1d4fbb298afc9","source_sha256":"fe20ac9adb6e069125c46c6cc0a23c1dfac91d49367534266b3b31063c05d603","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun H ↦ ?_, isTopologicallyNilpotent_of_constantCoeff⟩\n  replace H : Tendsto (fun n ↦ constantCoeff (f ^ n)) atTop (nhds 0) :=\n    continuous_constantCoeff R |>.tendsto' 0 0 constantCoeff_zero |>.comp H\n  simpa only [map_pow] using! H","hard_negative":true,"metrics":{"chosen_tokens":54,"rejected_tokens":2,"token_jaccard":0.025,"token_length_ratio":0.037037},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"8541ae21b91d12147dfb7eae302420078a1af539eef3e8cf9ff8b35acedf1643","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finsupp.Interval\npublic import Mathlib.RingTheory.Ideal.Quotient.Defs\npublic import Mathlib.RingTheory.MvPowerSeries.PiTopology\npublic import Mathlib.Topology.Algebra.LinearTopology\npublic import Mathlib.RingTheory.TwoSidedIdeal.Operations\n\nNamespace:\nMvPowerSeries.LinearTopology\n\nLocal context:\n/-\nCopyright (c) 2024 Antoine Chambert-Loir, María Inés de Frutos-Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir, María Inés de Frutos-Fernández\n-/\n/-! # Linear topology on the ring of multivariate power series\n\n- `MvPowerSeries.LinearTopology.basis`: the ideals of the ring of multivariate power series\n  all coefficients the exponent of which is smaller than some bound vanish.\n\n- `MvPowerSeries.LinearTopology.hasBasis_nhds_zero` :\n  the two-sided ideals from `MvPowerSeries.LinearTopology.basis` form a basis\n  of neighborhoods of `0` if the topology of `R` is (left and right) linear.\n\n## Instances :\n\nIf `R` has a linear topology, then the product topology on `MvPowerSeries σ R`\nis a linear topology.\n\nThis applies in particular when `R` has the discrete topology.\n\n## Note\n\nIf we had an analogue of `PolynomialModule` for power series,\nmeaning that we could consider the `R⟦X⟧`-module `M⟦X⟧` when `M` is an `R`-module,\nthen one could prove that `M⟦X⟧` is linearly topologized over `R⟦X⟧`\nwhenever `M` is linearly topologized over `R`.\nTo recover the ring case, it would remain to show that the isomorphism between\n`Rᵐᵒᵖ⟦X⟧` and `R⟦X⟧ᵐᵒᵖ` identifies their respective actions on `R⟦X⟧`.\n(And likewise in the multivariate case.)\n\n-/\n\n@[expose] public section\n\nnamespace MvPowerSeries\n\nnamespace LinearTopology\n\nopen scoped Topology\n\nopen Set SetLike Filter\n\n/-- The underlying family for the basis of ideals in a multivariate power series ring. -/\nnoncomputable def basis (σ : Type*) (R : Type*) [Ring R] (Jd : TwoSidedIdeal R × (σ →₀ ℕ)) :\n    TwoSidedIdeal (MvPowerSeries σ R) :=\n  TwoSidedIdeal.mk' {f | ∀ e ≤ Jd.2, coeff e f ∈ Jd.1}\n    (by simp [coeff_zero])\n    (fun hf hg e he ↦ by rw [map_add]; exact add_mem (hf e he) (hg e he))\n    (fun {f} hf e he ↦ by simp only [map_neg, neg_mem, hf e he])\n    (fun {f g} hg e he ↦ by\n      classical\n      rw [coeff_mul]\n      apply sum_mem\n      rintro uv huv\n      exact TwoSidedIdeal.mul_mem_left _ _ _ (hg _ (le_trans (Finset.antidiagonal.snd_le huv) he)))\n    (fun {f g} hf e he ↦ by\n      classical\n      rw [coeff_mul]\n      apply sum_mem\n      rintro uv huv\n      exact TwoSidedIdeal.mul_mem_right _ _ _ (hf _ (le_trans (Finset.antidiagonal.fst_le huv) he)))\n\nvariable {σ : Type*} {R : Type*} [Ring R]\n\n/-- A power series `f` belongs to the two-sided ideal `basis σ R ⟨J, d⟩`\nif and only if `coeff e f ∈ J` for all `e ≤ d`. -/\ntheorem mem_basis_iff {f : MvPowerSeries σ R} {Jd : TwoSidedIdeal R × (σ →₀ ℕ)} :\n    f ∈ basis σ R Jd ↔ ∀ e ≤ Jd.2, coeff e f ∈ Jd.1 := by\n  simp [basis]\n\n/-- If `J ≤ K` and `e ≤ d`, then we have the inclusion of two-sided ideals\n`basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e,>`. -/\ntheorem basis_le {Jd Ke : TwoSidedIdeal R × (σ →₀ ℕ)} (hJK : Jd.1 ≤ Ke.1) (hed : Ke.2 ≤ Jd.2) :\n    basis σ R Jd ≤ basis σ R Ke :=\n  fun _ ↦ forall_imp (fun _ h hue ↦ hJK (h (le_trans hue hed)))\n\n/-- `basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e⟩` if and only if `J ≤ K` and `e ≤ d`. -/\ntheorem basis_le_iff {J K : TwoSidedIdeal R} {d e : σ →₀ ℕ} (hK : K ≠ ⊤) :\n    basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e⟩ ↔ J ≤ K ∧ e ≤ d := by\n  classical\n  constructor\n  · simp only [basis, TwoSidedIdeal.le_iff, TwoSidedIdeal.coe_mk', setOf_subset_setOf]\n    intro h\n    constructor\n    · intro x hx\n      have (d' : _) : coeff d' (C (σ := σ) x) ∈ J := by\n        rw [coeff_C]; split_ifs <;> [exact hx; exact J.zero_mem]\n      simpa using h (C x) (fun _ _ ↦ this _) _ zero_le\n    · by_contra h'\n      apply hK\n      rw [eq_top_iff]\n      intro x _\n      have (d') (hd'_le : d' ≤ d) : coeff d' (monomial e x) ∈ J := by\n        rw [coeff_monomial]\n        split_ifs with hd' <;> [exact (h' (hd' ▸ hd'_le)).elim; exact J.zero_mem]\n      simpa using h (monomial e x) this _ le_rfl\n  · rintro ⟨hJK, hed⟩\n    exact basis_le hJK hed\n\nvariable [TopologicalSpace R]\n\n-- We endow MvPowerSeries σ R with the product topology.\nopen WithPiTopology\n\nset_option backward.isDefEq.respectTransparency false in\n/-- If the ring `R` is endowed with a linear topology, then the sets `↑basis σ R (J, d)`,\nfor `J : TwoSidedIdeal R` which are neighborhoods of `0 : R` and `d : σ →₀ ℕ`,\nconstitute a basis of neighborhoods of `0 : MvPowerSeries σ R` for the product topology. -/\nlemma hasBasis_nhds_zero [IsLinearTopology R R] [IsLinearTopology Rᵐᵒᵖ R] :\n    (𝓝 0 : Filter (MvPowerSeries σ R)).HasBasis\n      (fun Id : TwoSidedIdeal R × (σ →₀ ℕ) ↦ (Id.1 : Set R) ∈ 𝓝 0)\n      (fun Id ↦ basis _ _ Id) := by\n  classical\n  rw [nhds_pi]\n  refine IsLinearTopology.hasBasis_twoSidedIdeal.pi_self.to_hasBasis ?_ ?_\n  · intro ⟨D, I⟩ ⟨hD, hI⟩\n    refine ⟨⟨I, Finset.sup hD.toFinset id⟩, hI, fun f hf d hd ↦ ?_⟩\n    rw [SetLike.mem_coe, mem_basis_iff] at hf\n    convert! hf _ <| Finset.le_sup (hD.mem_toFinset.mpr hd)\n  · intro ⟨I, d⟩ hI\n    refine ⟨⟨Iic d, I⟩, ⟨finite_Iic d, hI⟩, ?_⟩\n    simpa [basis, coeff_apply, Iic, Set.pi] using! subset_rfl\n\n/-- The topology on `MvPowerSeries` is a left linear topology\n  when the ring of coefficients has a linear topology. -/\ninstance [IsLinearTopology R R] [IsLinearTopology Rᵐᵒᵖ R] :\n    IsLinearTopology (MvPowerSeries σ R) (MvPowerSeries σ R) :=\n  IsLinearTopology.mk_of_hasBasis' _ hasBasis_nhds_zero TwoSidedIdeal.mul_mem_left\n\n/-- The topology on `MvPowerSeries` is a right linear topology\n  when the ring of coefficients has a linear topology. -/\ninstance [IsLinearTopology R R] [IsLinearTopology Rᵐᵒᵖ R] :\n    IsLinearTopology (MvPowerSeries σ R)ᵐᵒᵖ (MvPowerSeries σ R) :=\n  IsLinearTopology.mk_of_hasBasis' _ hasBasis_nhds_zero (fun J _ _ hg ↦ J.mul_mem_right _ _ hg)\n\ntheorem isTopologicallyNilpotent_of_constantCoeff\n    {R : Type*} [CommRing R] [TopologicalSpace R] [IsLinearTopology R R]\n    {f : MvPowerSeries σ R} (hf : IsTopologicallyNilpotent (constantCoeff f)) :\n    IsTopologicallyNilpotent f := by\n  simp_rw [IsTopologicallyNilpotent, tendsto_iff_coeff_tendsto, coeff_zero,\n    IsLinearTopology.hasBasis_ideal.tendsto_right_iff]\n  intro d I hI\n  replace hf := hf.eventually_mem hI\n  simp_rw [eventually_atTop, SetLike.mem_coe, ← Ideal.Quotient.eq_zero_iff_mem,\n    map_pow, ← coeff_map, ← constantCoeff_map] at hf ⊢\n  obtain ⟨N, hN⟩ := hf\n  use N + d.degree\n  intro n hn\n  simpa only [map_pow] using coeff_eq_zero_of_constantCoeff_nilpotent (hN N le_rfl) hn\n\n/-- Assuming the base ring has a linear topology, the powers of a `MvPowerSeries` converge to 0\niff its constant coefficient is topologically nilpotent.\n\nSee also `MvPowerSeries.WithPiTopology.isTopologicallyNilpotent_iff_constantCoeff_isNilpotent`. -/\n\nTarget:\ntheorem isTopologicallyNilpotent_iff_constantCoeff\n    {R : Type*} [CommRing R] [TopologicalSpace R] [IsLinearTopology R R] (f : MvPowerSeries σ R) :\n    Tendsto (fun n : ℕ => f ^ n) atTop (nhds 0) ↔\n      IsTopologicallyNilpotent (constantCoeff f) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_460fd408bbfa","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"9a37532cc48611faa8ea55e49db29cfffe19fc3244dda83b6c6f1a9949db522d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/MvPowerSeries","family_id":"istopologicallynilpotent_iff_constantcoeff","file_id":"mathlib/Mathlib/RingTheory/MvPowerSeries/LinearTopology.lean","sample_id":"460fd408bbfa2c9d3696e93eaf82711398320704b30a3dd64ec1d4fbb298afc9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6f31eac0331d8119b977de849699fd5e0d1fb2f77898bc029e261ff94821e033","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4793854c2e3af40b9d027f7b421519fea77582f2694be3b8462681fa81f0c2f6","source_sha256":"00cdc131b09c868f94179716d61273d8fdf897ed023d34f30c7085668a3b31ac","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by simpa only [zero_mul, zero_add] using div_add_mod a 0","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.25},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"8542cf2283119839b815e3b2167bc8c6346e8c4e46d5416a0798437c93dbcfea","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Order.RelClasses\n\nNamespace:\nEuclideanDomain\n\nLocal context:\n/-\nCopyright (c) 2018 Louis Carlin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Louis Carlin, Mario Carneiro\n-/\n/-!\n# Euclidean domains\n\nThis file introduces Euclidean domains and provides the extended Euclidean algorithm. To be precise,\na slightly more general version is provided which is sometimes called a transfinite Euclidean domain\nand differs in the fact that the degree function need not take values in `ℕ` but can take values in\nany well-ordered set. Transfinite Euclidean domains were introduced by Motzkin and examples which\ndon't satisfy the classical notion were provided independently by Hiblot and Nagata.\n\n## Main definitions\n\n* `EuclideanDomain`: Defines Euclidean domain with functions `quotient` and `remainder`. Instances\n  of `Div` and `Mod` are provided, so that one can write `a = b * (a / b) + a % b`.\n* `gcd`: defines the greatest common divisors of two elements of a Euclidean domain.\n* `xgcd`: given two elements `a b : R`, `xgcd a b` defines the pair `(x, y)` such that\n  `x * a + y * b = gcd a b`.\n* `lcm`: defines the lowest common multiple of two elements `a` and `b` of a Euclidean domain as\n  `a * b / (gcd a b)`\n\n## Main statements\n\nSee `Algebra.EuclideanDomain.Basic` for most of the theorems about Euclidean domains,\nincluding Bézout's lemma.\n\nSee `Algebra.EuclideanDomain.Instances` for the fact that `ℤ` is a Euclidean domain,\nas is any field.\n\n## Notation\n\n`≺` denotes the well-founded relation on the Euclidean domain, e.g. in the example of the polynomial\nring over a field, `p ≺ q` for polynomials `p` and `q` if and only if the degree of `p` is less than\nthe degree of `q`.\n\n## Implementation details\n\nInstead of working with a valuation, `EuclideanDomain` is implemented with the existence of a well\nfounded relation `r` on the integral domain `R`, which in the example of `ℤ` would correspond to\nsetting `i ≺ j` for integers `i` and `j` if the absolute value of `i` is smaller than the absolute\nvalue of `j`.\n\n## References\n\n* [Th. Motzkin, *The Euclidean algorithm*][MR32592]\n* [J.-J. Hiblot, *Des anneaux euclidiens dont le plus petit algorithme n'est pas à valeurs finies*]\n  [MR399081]\n* [M. Nagata, *On Euclid algorithm*][MR541021]\n\n\n## Tags\n\nEuclidean domain, transfinite Euclidean domain, Bézout's lemma\n-/\n\n@[expose] public section\n\nuniverse u\n\n/-- A `EuclideanDomain` is a non-trivial commutative ring with a division and a remainder,\n  satisfying `b * (a / b) + a % b = a`.\n  The definition of a Euclidean domain usually includes a valuation function `R → ℕ`.\n  This definition is slightly generalised to include a well-founded relation\n  `r` with the property that `r (a % b) b`, instead of a valuation. -/\n@[wikidata Q867345]\nclass EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R where\n  /-- A division function (denoted `/`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `%` denotes `remainder`. -/\n  protected quotient : R → R → R\n  /-- Division by zero should always give zero by convention. -/\n  protected quotient_zero : ∀ a, quotient a 0 = 0\n  /-- A remainder function (denoted `%`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `/` denotes `quotient`. -/\n  protected remainder : R → R → R\n  /-- The property that links the quotient and remainder functions.\n    This allows us to compute GCDs and LCMs. -/\n  protected quotient_mul_add_remainder_eq : ∀ a b, b * quotient a b + remainder a b = a\n  /-- A well-founded relation on `R`, satisfying `r (a % b) b`.\n    This ensures that the GCD algorithm always terminates. -/\n  protected r : R → R → Prop\n  /-- The relation `r` must be well-founded.\n    This ensures that the GCD algorithm always terminates. -/\n  r_wellFounded : WellFounded r\n  /-- The relation `r` satisfies `r (a % b) b`. -/\n  protected remainder_lt : ∀ (a) {b}, b ≠ 0 → r (remainder a b) b\n  /-- An additional constraint on `r`. -/\n  mul_left_not_lt : ∀ (a) {b}, b ≠ 0 → ¬r (a * b) a\n\n/-\nLean has far more theorems about fields than about Euclidean domains. We thus\nlower the priority of `Euclideandomain.toCommRing`, encouraging typeclass inference\nto try `Field.toCommRing` first. Without this priority-lowering, typeclass inference\nfinds the more inefficient path `Field.toEuclideanDomain.toCommRing` by default. This\npriority change saves over 500G instructions across mathlib. See\nhttps://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/We.20need.20to.20talk.20about.20Euclidean.20Domains/near/594655420\n-/\n-- see Note [lower instance priority]\nattribute [instance 100] EuclideanDomain.toCommRing\n\nnamespace EuclideanDomain\n\nvariable {R : Type u} [EuclideanDomain R]\n\n/-- Abbreviated notation for the well-founded relation `r` in a Euclidean domain. -/\nlocal infixl:50 \" ≺ \" => EuclideanDomain.r\n\nlocal instance wellFoundedRelation : WellFoundedRelation R where\n  rel := EuclideanDomain.r\n  wf := r_wellFounded\n\ninstance isWellFounded : IsWellFounded R (· ≺ ·) where\n  wf := r_wellFounded\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Div R :=\n  ⟨EuclideanDomain.quotient⟩\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Mod R :=\n  ⟨EuclideanDomain.remainder⟩\n\ntheorem div_add_mod (a b : R) : b * (a / b) + a % b = a :=\n  EuclideanDomain.quotient_mul_add_remainder_eq _ _\n\ntheorem mod_add_div (a b : R) : a % b + b * (a / b) = a :=\n  (add_comm _ _).trans (div_add_mod _ _)\n\ntheorem mod_add_div' (m k : R) : m % k + m / k * k = m := by\n  rw [mul_comm]\n  exact mod_add_div _ _\n\ntheorem div_add_mod' (m k : R) : m / k * k + m % k = m := by\n  rw [mul_comm]\n  exact div_add_mod _ _\n\ntheorem mod_lt : ∀ (a) {b : R}, b ≠ 0 → a % b ≺ b :=\n  EuclideanDomain.remainder_lt\n\ntheorem mul_right_not_lt {a : R} (b) (h : a ≠ 0) : ¬a * b ≺ b := by\n  rw [mul_comm]\n  exact mul_left_not_lt b h\n\n@[simp]\n\nTarget:\ntheorem mod_zero (a : R) : a % 0 = a :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/EuclideanDomain","family_id":"mod_zero","file_id":"mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean","sample_id":"4793854c2e3af40b9d027f7b421519fea77582f2694be3b8462681fa81f0c2f6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f534ed3effb9755a614b851ef33384815c516806a3924c7986894d2e4e795cd3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"467d6115e45bfeaa364bc7d6dc84e27f5374de312ed692a0238a4a35b99ebeeb","source_sha256":"66c20bb7bca6fa55e98ef3e2d1bcc17c8251e92551f3085706c9cdfd3343f5b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← isRegular_pairing_iff, Pairing.isRegular_iff_nonempty_weakRankFunction]\n  exact (P.weakRankFunctionEquiv ℕ).symm.nonempty_congr","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":2,"token_jaccard":0.052632,"token_length_ratio":0.095238},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"85691a281fc98ab670b6ffcea885b3d7e47578d51d10b62ff20358577f1bc7f3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Rank\npublic import Mathlib.Data.Finite.Sigma\n\nNamespace:\nSSet.Subcomplex.PairingCore\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Existence of a rank function to natural numbers\n\nIn this file, we show that if `P : A.Pairing` is\na regular pairing of subcomplex `A` of a simplicial set `X`,\nthen there exists a rank function for `P` with values in `ℕ`.\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen Simplicial\n\nnamespace SSet.Subcomplex\n\nvariable {X : SSet.{u}} {A : X.Subcomplex}\n\nnamespace Pairing\n\nvariable (P : A.Pairing)\n\ninstance (y : P.II) : Finite { x // P.AncestralRel x y } := by\n  let T := { x : P.II // P.AncestralRel x y }\n  let U := Σ (d : Fin (P.p y).1.dim), ⦋d⦌ ⟶ ⦋(P.p y).1.1.1.1⦌\n  let ψ : U → X.S := fun ⟨d, f⟩ ↦ S.mk (X.map f.op (P.p y).1.simplex)\n  have h (t : T) : ∃ u, ψ u = t.1.1.toS := by\n    obtain ⟨f, _, hf⟩ := N.le_iff_exists_mono.1 t.2.2.le\n    refine ⟨⟨⟨t.1.1.dim, ?_⟩, f⟩, ?_⟩\n    · simpa using SSet.N.dim_lt_of_lt t.2.2\n    · rwa [SSet.S.ext_iff]\n  choose φ hφ using h\n  apply Finite.of_injective φ\n  intro t₁ t₂ h\n  rw [Subtype.ext_iff, Subtype.ext_iff, N.ext_iff, SSet.N.ext_iff, ← hφ, ← hφ, h]\n\nsection\n\nvariable {y : P.II} (hy : Acc P.AncestralRel y)\n\n/-- Auxiliary definition for `SSet.Subcomplex.Pairing.Rank`. -/\nnoncomputable def rank' : ℕ :=\n  Acc.recOn hy (fun y _ r ↦ ⨆ (x : { x // P.AncestralRel x y }), r x x.2 + 1)\n\nlemma rank'_eq :\n    P.rank' hy = ⨆ (x : { x // P.AncestralRel x y }), P.rank' (hy.inv x.2) + 1 := by\n  change P.rank' (Acc.intro y fun _ => hy.inv) = _\n  rfl\n\nlemma rank'_lt {x : P.II} (r : P.AncestralRel x y) :\n    P.rank' (hy.inv r) < P.rank' hy := by\n  rw [P.rank'_eq hy, ← Nat.add_one_le_iff]\n  exact le_csSup (Finite.bddAbove_range _) ⟨⟨x, r⟩, rfl⟩\n\nend\n\nsection IsRegular\n\nvariable [P.IsRegular]\n\n/-- The rank function with values in `ℕ` relative to the well founded\nancestrality relation of a regular pairing. -/\nnoncomputable def rank (x : P.II) : ℕ :=\n  P.rank' (P.wf.apply x)\n\nvariable {P} in\nlemma rank_lt {x y : P.II} (h : P.AncestralRel x y) :\n    P.rank x < P.rank y :=\n  P.rank'_lt _ h\n\n/-- The canonical rank function with values in `ℕ` of a regular pairing. -/\nnoncomputable def rankFunction : P.RankFunction ℕ where\n  rank := P.rank\n  lt := P.rank_lt\n\ninstance : Nonempty (P.RankFunction ℕ) := ⟨P.rankFunction⟩\n\ninstance : Nonempty (P.WeakRankFunction ℕ) := ⟨P.rankFunction.toWeakRankFunction⟩\n\nend IsRegular\n\nlemma isRegular_iff_nonempty_rankFunction [P.IsProper] :\n    P.IsRegular ↔ Nonempty (P.RankFunction ℕ) :=\n  ⟨fun _ ↦ inferInstance, fun ⟨h⟩ ↦ h.isRegular⟩\n\nlemma isRegular_iff_nonempty_weakRankFunction [P.IsProper] :\n    P.IsRegular ↔ Nonempty (P.WeakRankFunction ℕ) :=\n  ⟨fun _ ↦ inferInstance, fun ⟨h⟩ ↦ h.isRegular⟩\n\nend Pairing\n\nnamespace PairingCore\n\nvariable (P : A.PairingCore)\n\nlemma isRegular_iff_nonempty_rankFunction [P.IsProper] :\n    P.IsRegular ↔ Nonempty (P.RankFunction ℕ) := by\n  rw [← isRegular_pairing_iff, Pairing.isRegular_iff_nonempty_rankFunction]\n  exact (P.rankFunctionEquiv ℕ).symm.nonempty_congr\n\nTarget:\nlemma isRegular_iff_nonempty_weakRankFunction [P.IsProper] :\n    P.IsRegular ↔ Nonempty (P.WeakRankFunction ℕ) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplicialSet","family_id":"isregular_iff_nonempty_weakrankfunction","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/RankNat.lean","sample_id":"467d6115e45bfeaa364bc7d6dc84e27f5374de312ed692a0238a4a35b99ebeeb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0b0b9e8a744c1d7e4020f039217d12486b9369dcf62ab6b9ae3cf10e259b4ee9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"aeb71f8e68a66dc07c62978a48769bca8b7ac587e71fccaeb6caa13c22e6a3db","source_sha256":"68dce35114cf22d288a51cfbefe877f5725b4487e27909680b1aae1c2b647510","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [← factorPowSucc_comp_fontaineThetaModPPow n]","hard_negative":false,"metrics":{"chosen_tokens":7,"rejected_tokens":3,"token_jaccard":0.111111,"token_length_ratio":0.428571},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"85a31c5481f8ee992a826ad2569167e8f1a28dba5748da45f4f8064c74939f46","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.AdicCompletion.Functoriality\npublic import Mathlib.RingTheory.AdicCompletion.RingHom\npublic import Mathlib.RingTheory.Perfectoid.Untilt\npublic import Mathlib.RingTheory.WittVector.TeichmullerSeries\n\nNamespace:\nWittVector\n\nLocal context:\n/-\nCopyright (c) 2025 Jiedong Jiang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jiedong Jiang\n-/\n/-!\n# Fontaine's θ map\nIn this file, we define Fontaine's `θ` map, which is a ring\nhomomorphism from the Witt vector `𝕎 R♭` of the tilt of a perfectoid ring `R`\nto `R` itself. Our definition of `θ` does not require that `R` is perfectoid in the first place.\nWe only need `R` to be `p`-adically complete.\n\n## Main Definitions\n* `fontaineTheta` : Fontaine's θ map, which is a ring homomorphism from `𝕎 R♭` to `R`.\n\n## Main Theorems\n* `fontaineTheta_teichmuller` : `θ([x])` is the untilt of `x`.\n* `fontaineTheta_surjective` : Fontaine's θ map is surjective.\n\n## TODO\nEstablish that our definition (explicit construction of `θ mod p ^ n`) agrees with the\ndeformation-theoretic approach via the cotangent complex, as in\n[Bhatt, *Lecture notes for a class on perfectoid spaces*.\nRemark 6.1.7](https://www.math.ias.edu/~bhatt/teaching/mat679w17/lectures.pdf).\n\n## Tags\nFontaine's theta map, perfectoid theory, p-adic Hodge theory\n\n## Reference\n\n* [Fontaine, *Sur Certains Types de Représentations p-Adiques du Groupe de Galois d'un Corps Local;\n  Construction d'un Anneau de Barsotti-Tate*][fontaine1982certains]\n* [Fontaine, *Le corps des périodes p-adiques*][fontaine1994corps]\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen Ideal Quotient PreTilt WittVector\n\nnoncomputable section\n\nvariable {R : Type u} [CommRing R] {p : ℕ} [Fact p.Prime]\n\nlocal notation \"𝕎 \" A:100 => WittVector p A\nlocal notation A \"♭\" => PreTilt A p\nlocal notation3 \"𝔭\" => span {(p : R)}\n\nnamespace WittVector\n\n/-!\n## θ as a ring homomorphism\nLet `𝔭` denote the ideal of `R` generated by the prime number `p`. In this section, we first\ndefine the ring homomorphism `fontaineThetaModPPow : 𝕎 R♭ →+* R ⧸ 𝔭 ^ (n + 1)`.\nThen we show they are compatible with each other and lift to a\nring homomorphism `fontaineTheta : 𝕎 R♭ →+* R`.\n\nTo prove this, we define `fontaineThetaModPPow` as a composition of the following ring\nhomomorphisms.\n\n`𝕎 R♭ --𝕎(Frob^-n)-> 𝕎 R♭ --𝕎(coeff 0)-> 𝕎(R/𝔭) --gh_n-> R/𝔭^(n+1)`\n\nHere, the ring map `gh_n` fits in the following diagram.\n\n```\n𝕎(R)  --ghost_n->   R\n|                   |\nv                   v\n𝕎(R/𝔭) --gh_n-> R/𝔭^(n+1)\n```\n-/\n\ntheorem ker_map_le_ker_mk_comp_ghostComponent (n : ℕ) :\n    RingHom.ker (WittVector.map (Ideal.Quotient.mk 𝔭)) ≤\n    RingHom.ker (((Ideal.Quotient.mk (𝔭 ^ (n + 1)))).comp\n    (WittVector.ghostComponent (p := p) n)) := by\n  intro x\n  simp only [RingHom.mem_ker, map_eq_zero_iff, RingHom.comp_apply]\n  intro h\n  simp only [ghostComponent]\n  apply_fun Ideal.quotEquivOfEq (Ideal.span_singleton_pow _ (n + 1))\n  simp only [RingHom.coe_comp, Function.comp_apply, Pi.evalRingHom_apply, ghostMap_apply,\n    quotEquivOfEq_mk, map_zero]\n  simp only [eq_zero_iff_dvd] at h ⊢\n  exact pow_dvd_ghostComponent_of_dvd_coeff (fun _ _ ↦ h _)\n\n/--\nThe lift ring map `gh_n : 𝕎(R/𝔭) →+* R/𝔭^(n+1)` of the `n`-th ghost component\n`𝕎(R) →+* R` along the surjective ring map `𝕎(R) →+* 𝕎(R/𝔭)`.\n-/\ndef ghostComponentModPPow (n : ℕ) : 𝕎 (R ⧸ 𝔭) →+* R ⧸ 𝔭 ^ (n + 1) :=\n  RingHom.liftOfSurjective (WittVector.map (Ideal.Quotient.mk 𝔭))\n    (map_surjective _ Ideal.Quotient.mk_surjective) ⟨((Ideal.Quotient.mk (𝔭 ^ (n + 1)))).comp\n      (WittVector.ghostComponent n), ker_map_le_ker_mk_comp_ghostComponent n⟩\n\n@[simp]\ntheorem ghostComponentModPPow_map_mk (n : ℕ) (x : 𝕎 R) :\n    ghostComponentModPPow n (WittVector.map (Ideal.Quotient.mk 𝔭) x) =\n    WittVector.ghostComponent n x :=\n  RingHom.liftOfSurjective_comp_apply ..\n\n@[simp]\ntheorem quotEquivOfEq_ghostComponentModPPow (x : 𝕎 (R ⧸ 𝔭)) (h : 𝔭 ^ (0 + 1) = 𝔭) :\n    quotEquivOfEq h (ghostComponentModPPow 0 x) = ghostComponent 0 x := by\n  obtain ⟨y, hy⟩ := map_surjective _ Ideal.Quotient.mk_surjective x\n  simp [← hy, ghostComponent_apply]\n\nvariable [Fact ¬IsUnit (p : R)] [IsAdicComplete (span {(p : R)}) R]\n-- local notation 𝔭 does not work in [IsAdicComplete (span {(p : R)}) R]\n\n@[simp]\ntheorem ghostComponentModPPow_teichmuller_coeff (n : ℕ) (x : R♭) :\n    ghostComponentModPPow n (teichmuller p (PreTilt.coeff n x)) =\n    Ideal.Quotient.mk (𝔭 ^ (n + 1)) x.untilt := by\n  simpa using ghostComponentModPPow_map_mk n\n    (teichmuller p ((((_root_.frobeniusEquiv _ p).symm ^ n) x).untilt))\n\nvariable (R p) in\n/--\nThe Fontaine's theta map modulo `p^(n+1)`.\nIt is the composition of the following ring homomorphisms.\n`𝕎 R♭ --𝕎(Frob^-n)-> 𝕎 R♭ --𝕎(coeff 0)-> 𝕎(R/p) --gh_n-> R/p^(n+1)`\n-/\ndef fontaineThetaModPPow (n : ℕ) : 𝕎 R♭ →+* R ⧸ 𝔭 ^ (n + 1) :=\n  (ghostComponentModPPow n).comp (((WittVector.map (PreTilt.coeff 0))).comp\n    (WittVector.map ((_root_.frobeniusEquiv (R♭) p).symm ^ n : R♭ →+* R♭)))\n\n@[simp]\ntheorem fontaineThetaModPPow_teichmuller (n : ℕ) (x : R♭) :\n    fontaineThetaModPPow R p n (teichmuller p x) = Ideal.Quotient.mk _ x.untilt := by\n  simp [fontaineThetaModPPow]\n\ntheorem factorPowSucc_comp_fontaineThetaModPPow (n : ℕ) :\n    (factorPowSucc _ _).comp (fontaineThetaModPPow R p (n + 1)) = fontaineThetaModPPow R p n := by\n  apply eq_of_apply_teichmuller_eq ((factorPowSucc _ _).comp (fontaineThetaModPPow R p (n + 1)))\n    (fontaineThetaModPPow R p n)\n  · use n + 1\n    have : p = Ideal.Quotient.mk (𝔭 ^ (n + 1)) p := by\n      simp [map_natCast]\n    rw [this, ← map_pow, Ideal.Quotient.eq_zero_iff_mem]\n    exact Ideal.pow_mem_pow (mem_span_singleton_self _) _\n  simp [fontaineThetaModPPow]\n\nTarget:\ntheorem factorPowSucc_fontaineThetaModPPow_eq (n : ℕ) (x : 𝕎 R♭) :\n    factorPowSucc _ _ ((fontaineThetaModPPow R p (n + 1)) x) = fontaineThetaModPPow R p n x :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Perfectoid","family_id":"factorpowsucc_fontainethetamodppow_eq","file_id":"mathlib/Mathlib/RingTheory/Perfectoid/FontaineTheta.lean","sample_id":"aeb71f8e68a66dc07c62978a48769bca8b7ac587e71fccaeb6caa13c22e6a3db"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b33ee369bdfae6b8f7816eb5661ea638ec7dc5ec9125117e507e6aaa3923f7b0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d437f1656af8a39ef7edd102454e4e05fab4a6655ab8447864fc6876b244d350","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cfce958e665e03e21c1db8b838eb32bd3a0604379292b7cebef2b66be097f2db","source_sha256":"dcc51bf41b9226e46d854f6feb1d45a4ec3794545cf9d910bab23bf813dc4f3f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro a b h\n  refine le_of_lt_or_eq (or_iff_not_imp_right.2 fun hne ↦ ?_)\n  classical\n  exact ⟨Finset.min' _ (nonempty_neLocus_iff.2 hne),\n    fun j hj ↦ notMem_neLocus.1 fun h ↦ (Finset.min'_le _ _ h).not_gt hj,\n    (h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)⟩","hard_negative":true,"metrics":{"chosen_tokens":72,"rejected_tokens":5,"token_jaccard":0.078947,"token_length_ratio":0.069444},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"85b546c09616adf940e085829b82398d88ec6e4e2fbd45570c1ba38831a2e236","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.PiLex\npublic import Mathlib.Data.DFinsupp.Order\npublic import Mathlib.Data.DFinsupp.NeLocus\npublic import Mathlib.Order.WellFoundedSet\n\nNamespace:\nDFinsupp\n\nLocal context:\n/-\nCopyright (c) 2022 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Damiano Testa, Junyan Xu\n-/\n/-!\n# Lexicographic order on finitely supported dependent functions\n\nThis file defines the lexicographic order on `DFinsupp`.\n-/\n\n@[expose] public section\n\n\nvariable {ι : Type*} {α : ι → Type*}\n\nnamespace DFinsupp\n\nsection Zero\n\nvariable [∀ i, Zero (α i)]\n\n/-- `DFinsupp.Lex r s` is the lexicographic relation on `Π₀ i, α i`, where `ι` is ordered by `r`,\nand `α i` is ordered by `s i`.\n\nThe type synonym `Lex (Π₀ i, α i)` has an order given by `DFinsupp.Lex (· < ·) (· < ·)`, whereas\n`Colex (Π₀ i, α i)` has an order given by `DFinsupp.Lex (· > ·) (· < ·)`.\n-/\nprotected def Lex (r : ι → ι → Prop) (s : ∀ i, α i → α i → Prop) (x y : Π₀ i, α i) : Prop :=\n  Pi.Lex r (s _) x y\n\ntheorem _root_.Pi.lex_eq_dfinsupp_lex {r : ι → ι → Prop} {s : ∀ i, α i → α i → Prop}\n    (a b : Π₀ i, α i) : Pi.Lex r (s _) (a : ∀ i, α i) b = DFinsupp.Lex r s a b :=\n  rfl\n\ntheorem lex_def {r : ι → ι → Prop} {s : ∀ i, α i → α i → Prop} {a b : Π₀ i, α i} :\n    DFinsupp.Lex r s a b ↔ ∃ j, (∀ d, r d j → a d = b d) ∧ s j (a j) (b j) :=\n  .rfl\n\ninstance [LT ι] [∀ i, LT (α i)] : LT (Lex (Π₀ i, α i)) :=\n  ⟨fun f g ↦ DFinsupp.Lex (· < ·) (fun _ ↦ (· < ·)) (ofLex f) (ofLex g)⟩\n\ninstance [LT ι] [∀ i, LT (α i)] : LT (Colex (Π₀ i, α i)) :=\n  ⟨fun f g ↦ DFinsupp.Lex (· > ·) (fun _ ↦ (· < ·)) (ofColex f) (ofColex g)⟩\n\ntheorem Lex.lt_iff [LT ι] [∀ i, LT (α i)] {a b : Lex (Π₀ i, α i)} :\n    a < b ↔ ∃ i, (∀ j, j < i → a j = b j) ∧ a i < b i :=\n  .rfl\n\n@[deprecated (since := \"2025-11-29\")]\nalias lex_lt_iff := Lex.lt_iff\n\ntheorem Colex.lt_iff [LT ι] [∀ i, LT (α i)] {a b : Colex (Π₀ i, α i)} :\n    a < b ↔ ∃ i, (∀ j, i < j → a j = b j) ∧ a i < b i :=\n  .rfl\n\ntheorem lex_lt_of_lt_of_preorder [∀ i, Preorder (α i)] (r) [IsStrictOrder ι r] {x y : Π₀ i, α i}\n    (hlt : x < y) : ∃ i, (∀ j, r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i := by\n  obtain ⟨hle, j, hlt⟩ := Pi.lt_def.1 hlt\n  classical\n  have : (x.neLocus y : Set ι).WellFoundedOn r := (x.neLocus y).finite_toSet.wellFoundedOn\n  obtain ⟨i, hi, hl⟩ := this.has_min { i | x i < y i } ⟨⟨j, mem_neLocus.2 hlt.ne⟩, hlt⟩\n  refine ⟨i, fun k hk ↦ ⟨hle k, ?_⟩, hi⟩\n  exact of_not_not fun h ↦ hl ⟨k, mem_neLocus.2 (ne_of_not_le h).symm⟩ ((hle k).lt_of_not_ge h) hk\n\ntheorem lex_lt_of_lt [∀ i, PartialOrder (α i)] (r) [IsStrictOrder ι r] {x y : Π₀ i, α i}\n    (hlt : x < y) : Pi.Lex r (· < ·) x y := by\n  simp_rw [Pi.Lex, le_antisymm_iff]\n  exact lex_lt_of_lt_of_preorder r hlt\n\ntheorem lex_iff_of_unique [Unique ι] [∀ i, LT (α i)] {r} [Std.Irrefl r] {x y : Π₀ i, α i} :\n    DFinsupp.Lex r (fun _ ↦ (· < ·)) x y ↔ x default < y default :=\n  Pi.lex_iff_of_unique\n\ntheorem Lex.lt_iff_of_unique [Unique ι] [∀ i, LT (α i)] [Preorder ι] {x y : Lex (Π₀ i, α i)} :\n    x < y ↔ x default < y default :=\n  lex_iff_of_unique\n\n@[deprecated (since := \"2025-11-29\")]\nalias lex_lt_iff_of_unique := Lex.lt_iff_of_unique\n\ntheorem colex_lt_iff_of_unique [Unique ι] [∀ i, LT (α i)] [Preorder ι] {x y : Colex (Π₀ i, α i)} :\n    x < y ↔ x default < y default :=\n  lex_iff_of_unique\n\nvariable [LinearOrder ι]\n\ninstance Lex.isStrictOrder [∀ i, PartialOrder (α i)] :\n    IsStrictOrder (Lex (Π₀ i, α i)) (· < ·) where\n  irrefl _ := lt_irrefl (α := Lex (∀ i, α i)) _\n  trans _ _ _ := lt_trans (α := Lex (∀ i, α i))\n\ninstance Colex.isStrictOrder [∀ i, PartialOrder (α i)] :\n    IsStrictOrder (Colex (Π₀ i, α i)) (· < ·) :=\n  Lex.isStrictOrder (ι := ιᵒᵈ)\n\n/-- The partial order on `DFinsupp`s obtained by the lexicographic ordering.\nSee `DFinsupp.Lex.linearOrder` for a proof that this partial order is in fact linear. -/\ninstance Lex.partialOrder [∀ i, PartialOrder (α i)] : PartialOrder (Lex (Π₀ i, α i)) where\n  le x y := ⇑(ofLex x) = ⇑(ofLex y) ∨ x < y\n  __ := PartialOrder.lift (fun x : Lex (Π₀ i, α i) ↦ toLex (⇑(ofLex x)))\n    (DFunLike.coe_injective (F := DFinsupp α))\n\n/-- The partial order on `DFinsupp`s obtained by the colexicographic ordering.\nSee `DFinsupp.Colex.linearOrder` for a proof that this partial order is in fact linear. -/\ninstance Colex.partialOrder [∀ i, PartialOrder (α i)] : PartialOrder (Colex (Π₀ i, α i)) where\n  le x y := ⇑(ofColex x) = ⇑(ofColex y) ∨ x < y\n  __ := PartialOrder.lift (fun x : Colex (Π₀ i, α i) ↦ toColex (⇑(ofColex x)))\n    (DFunLike.coe_injective (F := DFinsupp α))\n\ntheorem Lex.le_iff_of_unique [Unique ι] [∀ i, PartialOrder (α i)] {x y : Lex (Π₀ i, α i)} :\n    x ≤ y ↔ x default ≤ y default :=\n  Pi.lex_le_iff_of_unique\n\n@[deprecated (since := \"2025-11-29\")]\nalias lex_le_iff_of_unique := Lex.le_iff_of_unique\n\ntheorem Colex.le_iff_of_unique [Unique ι] [∀ i, PartialOrder (α i)] {x y : Colex (Π₀ i, α i)} :\n    x ≤ y ↔ x default ≤ y default :=\n  Lex.le_iff_of_unique (ι := ιᵒᵈ)\n\nsection LinearOrder\n\nvariable [∀ i, LinearOrder (α i)]\n\nset_option backward.privateInPublic true in\n/-- Auxiliary helper to case split computably. There is no need for this to be public, as it\ncan be written with `Or.by_cases` on `lt_trichotomy` once the instances below are constructed. -/\nprivate def lt_trichotomy_rec {P : Lex (Π₀ i, α i) → Lex (Π₀ i, α i) → Sort*}\n    (h_lt : ∀ {f g}, toLex f < toLex g → P (toLex f) (toLex g))\n    (h_eq : ∀ {f g}, toLex f = toLex g → P (toLex f) (toLex g))\n    (h_gt : ∀ {f g}, toLex g < toLex f → P (toLex f) (toLex g)) : ∀ f g, P f g :=\n  Lex.rec fun f ↦ Lex.rec fun g ↦ match (motive := ∀ y, (f.neLocus g).min = y → _) _, rfl with\n  | ⊤, h => h_eq (neLocus_eq_empty.mp <| Finset.min_eq_top.mp h)\n  | (wit : ι), h => by\n    apply (mem_neLocus.mp <| Finset.mem_of_min h).lt_or_gt.by_cases <;> intro hwit\n    · exact h_lt ⟨wit, fun j hj ↦ notMem_neLocus.mp (Finset.notMem_of_lt_min hj h), hwit⟩\n    · exact h_gt ⟨wit, fun j hj ↦\n        notMem_neLocus.mp (Finset.notMem_of_lt_min hj <| by rwa [neLocus_comm]), hwit⟩\n\ninstance Lex.total_le : @Std.Total (Lex (Π₀ i, α i)) (· ≤ ·) where\n  total := lt_trichotomy_rec (fun h ↦ Or.inl h.le) (fun h ↦ Or.inl h.le) fun h ↦ Or.inr h.le\n\ninstance Colex.total_le : @Std.Total (Colex (Π₀ i, α i)) (· ≤ ·) :=\n  Lex.total_le (ι := ιᵒᵈ)\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- The less-or-equal relation for the lexicographic ordering is decidable. -/\ninstance Lex.decidableLE : DecidableLE (Lex (Π₀ i, α i)) :=\n  lt_trichotomy_rec (fun h ↦ isTrue <| Or.inr h)\n    (fun h ↦ isTrue <| Or.inl <| congr_arg _ h)\n    fun h ↦ isFalse fun h' ↦ lt_irrefl _ (h.trans_le h')\n\n/-- The less-or-equal relation for the colexicographic ordering is decidable. -/\ninstance Colex.decidableLE : DecidableLE (Colex (Π₀ i, α i)) :=\n  Lex.decidableLE (ι := ιᵒᵈ)\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- The less-than relation for the lexicographic ordering is decidable. -/\ninstance Lex.decidableLT : DecidableLT (Lex (Π₀ i, α i)) :=\n  lt_trichotomy_rec (fun h ↦ isTrue h) (fun h ↦ isFalse h.not_lt) fun h ↦ isFalse h.asymm\n\n/-- The less-than relation for the colexicographic ordering is decidable. -/\ninstance Colex.decidableLT : DecidableLT (Colex (Π₀ i, α i)) :=\n  Lex.decidableLT (ι := ιᵒᵈ)\n\n/-- The linear order on `DFinsupp`s obtained by the lexicographic ordering. -/\ninstance Lex.linearOrder : LinearOrder (Lex (Π₀ i, α i)) where\n  __ := Lex.partialOrder\n  le_total := total_of _\n  toDecidableLT := decidableLT\n  toDecidableLE := decidableLE\n\n/-- The linear order on `DFinsupp`s obtained by the colexicographic ordering. -/\ninstance Colex.linearOrder : LinearOrder (Colex (Π₀ i, α i)) where\n  __ := Colex.partialOrder\n  le_total := total_of _\n  toDecidableLT := decidableLT\n  toDecidableLE := decidableLE\n\nend LinearOrder\n\nvariable [∀ i, PartialOrder (α i)]\n\nTarget:\ntheorem toLex_monotone : Monotone (@toLex (Π₀ i, α i)) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_cfce958e665e","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"58d2bf85a439cdef1547b2e84e32b33ea7fcd7104bfda5ecb205683046c17cc5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/DFinsupp","family_id":"tolex_monotone","file_id":"mathlib/Mathlib/Data/DFinsupp/Lex.lean","sample_id":"cfce958e665e03e21c1db8b838eb32bd3a0604379292b7cebef2b66be097f2db"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ab4cb4aa5cd70f70198092337097cb05b18354e67218420118c17e78a73513e0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9fe3719b8a587cc320eadf69f2619355af22157136ae9400bafc484518741ba5","source_sha256":"eceb96a26dccbd8f8abcd83874539b49b8b7e797f195a864cff85c1bbe8476b2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h ↦ ?_, fun ⟨h₁, h₂⟩ ↦ ?_⟩\n  · refine evalInduction h fun a h IH ↦ ?_\n    rcases e : f a with - | a'\n    · rw [Part.mem_unique h\n          (PFun.mem_fix_iff.2 <| Or.inl <| Part.mem_some_iff.2 <| by rw [e]; rfl)]\n      exact ⟨ReflTransGen.refl, e⟩\n    · rcases PFun.mem_fix_iff.1 h with (h | ⟨_, h, _⟩) <;> rw [e] at h <;>\n        cases Part.mem_some_iff.1 h\n      obtain ⟨h₁, h₂⟩ := IH a' e\n      exact ⟨ReflTransGen.head e h₁, h₂⟩\n  · refine ReflTransGen.head_induction_on h₁ ?_ fun h _ IH ↦ ?_\n    · refine PFun.mem_fix_iff.2 (Or.inl ?_)\n      rw [h₂]\n      apply Part.mem_some\n    · refine PFun.mem_fix_iff.2 (Or.inr ⟨_, ?_, IH⟩)\n      rw [h]\n      apply Part.mem_some","hard_negative":false,"metrics":{"chosen_tokens":215,"rejected_tokens":3,"token_jaccard":0.035714,"token_length_ratio":0.013953},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"85d018d97b9595f3f0fba702e19c5821b8c8fb7fff7fb8d44c70afd17b05a972","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Data.PFun\nimport Mathlib.Data.Finset.Option\nimport Mathlib.Data.Finset.Prod\nimport Mathlib.Data.Fintype.Defs\n\nNamespace:\nStateTransition\n\nLocal context:\n/-\nCopyright (c) 2025 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Pim Spelier, Daan van Gent\n-/\n/-!\n# State Transition Systems\n\nThis file contains simple definitions and lemmas for reasoning about state transition systems\ndefined by a function `σ → Option σ`, where `σ` is the type of states.\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero\n\nopen Relation\n\nnamespace StateTransition\n\n/-- Run a state transition function `σ → Option σ` \"to completion\". The return value is the last\nstate returned before a `none` result. If the state transition function always returns `some`,\nthen the computation diverges, returning `Part.none`. -/\ndef eval {σ} (f : σ → Option σ) : σ → Part σ :=\n  PFun.fix fun s ↦ Part.some <| (f s).elim (Sum.inl s) Sum.inr\n\n/-- The reflexive transitive closure of a state transition function. `Reaches f a b` means\nthere is a finite sequence of steps `f a = some a₁`, `f a₁ = some a₂`, ... such that `aₙ = b`.\nThis relation permits zero steps of the state transition function. -/\ndef Reaches {σ} (f : σ → Option σ) : σ → σ → Prop :=\n  ReflTransGen fun a b ↦ b ∈ f a\n\n/-- The transitive closure of a state transition function. `Reaches₁ f a b` means there is a\nnonempty finite sequence of steps `f a = some a₁`, `f a₁ = some a₂`, ... such that `aₙ = b`.\nThis relation does not permit zero steps of the state transition function. -/\ndef Reaches₁ {σ} (f : σ → Option σ) : σ → σ → Prop :=\n  TransGen fun a b ↦ b ∈ f a\n\ntheorem reaches₁_eq {σ} {f : σ → Option σ} {a b c} (h : f a = f b) :\n    Reaches₁ f a c ↔ Reaches₁ f b c :=\n  TransGen.head'_iff.trans (TransGen.head'_iff.trans <| by rw [h]).symm\n\ntheorem reaches_total {σ} {f : σ → Option σ} {a b c} (hab : Reaches f a b) (hac : Reaches f a c) :\n    Reaches f b c ∨ Reaches f c b :=\n  ReflTransGen.total_of_right_unique (fun _ _ _ ↦ Option.mem_unique) hab hac\n\ntheorem reaches₁_fwd {σ} {f : σ → Option σ} {a b c} (h₁ : Reaches₁ f a c) (h₂ : b ∈ f a) :\n    Reaches f b c := by\n  rcases TransGen.head'_iff.1 h₁ with ⟨b', hab, hbc⟩\n  cases Option.mem_unique hab h₂; exact hbc\n\n/-- A variation on `Reaches`. `Reaches₀ f a b` holds if whenever `Reaches₁ f b c` then\n`Reaches₁ f a c`. This is a weaker property than `Reaches` and is useful for replacing states with\nequivalent states without taking a step. -/\ndef Reaches₀ {σ} (f : σ → Option σ) (a b : σ) : Prop :=\n  ∀ c, Reaches₁ f b c → Reaches₁ f a c\n\ntheorem Reaches₀.trans {σ} {f : σ → Option σ} {a b c : σ} (h₁ : Reaches₀ f a b)\n    (h₂ : Reaches₀ f b c) : Reaches₀ f a c\n  | _, h₃ => h₁ _ (h₂ _ h₃)\n\n@[refl]\ntheorem Reaches₀.refl {σ} {f : σ → Option σ} (a : σ) : Reaches₀ f a a\n  | _, h => h\n\ntheorem Reaches₀.single {σ} {f : σ → Option σ} {a b : σ} (h : b ∈ f a) : Reaches₀ f a b\n  | _, h₂ => h₂.head h\n\ntheorem Reaches₀.head {σ} {f : σ → Option σ} {a b c : σ} (h : b ∈ f a) (h₂ : Reaches₀ f b c) :\n    Reaches₀ f a c :=\n  (Reaches₀.single h).trans h₂\n\ntheorem Reaches₀.tail {σ} {f : σ → Option σ} {a b c : σ} (h₁ : Reaches₀ f a b) (h : c ∈ f b) :\n    Reaches₀ f a c :=\n  h₁.trans (Reaches₀.single h)\n\ntheorem reaches₀_eq {σ} {f : σ → Option σ} {a b} (e : f a = f b) : Reaches₀ f a b\n  | _, h => (reaches₁_eq e).2 h\n\ntheorem Reaches₁.to₀ {σ} {f : σ → Option σ} {a b : σ} (h : Reaches₁ f a b) : Reaches₀ f a b\n  | _, h₂ => h.trans h₂\n\ntheorem Reaches.to₀ {σ} {f : σ → Option σ} {a b : σ} (h : Reaches f a b) : Reaches₀ f a b\n  | _, h₂ => h₂.trans_right h\n\ntheorem Reaches₀.tail' {σ} {f : σ → Option σ} {a b c : σ} (h : Reaches₀ f a b) (h₂ : c ∈ f b) :\n    Reaches₁ f a c :=\n  h _ (TransGen.single h₂)\n\n/-- (co-)Induction principle for `eval`. If a property `C` holds of any point `a` evaluating to `b`\nwhich is either terminal (meaning `a = b`) or where the next point also satisfies `C`, then it\nholds of any point where `eval f a` evaluates to `b`. This formalizes the notion that if\n`eval f a` evaluates to `b` then it reaches terminal state `b` in finitely many steps. -/\n@[elab_as_elim]\ndef evalInduction {σ} {f : σ → Option σ} {b : σ} {C : σ → Sort*} {a : σ}\n    (h : b ∈ eval f a) (H : ∀ a, b ∈ eval f a → (∀ a', f a = some a' → C a') → C a) : C a :=\n  PFun.fixInduction h fun a' ha' h' ↦\n    H _ ha' fun b' e ↦ h' _ <| Part.mem_some_iff.2 <| by rw [e]; rfl\n\nTarget:\ntheorem mem_eval {σ} {f : σ → Option σ} {a b} : b ∈ eval f a ↔ Reaches f a b ∧ f b = none :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability","family_id":"mem_eval","file_id":"mathlib/Mathlib/Computability/StateTransition.lean","sample_id":"9fe3719b8a587cc320eadf69f2619355af22157136ae9400bafc484518741ba5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"306e987910ff023d43690d85687bf5ae9a7b00372cbbe2300a8443541f4fbbe4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8597d33dd3f22278ae75c80beaa1edceebfdbe6779385f7d1a492914d0b34404","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"20b48f91a7f4ad693944c040d924608c454aa3ef8bafb869719f264b357fd194","source_sha256":"42683ce9ec3a96f77aab204680a294682617adafa7d756b1f3dbef8d091d30cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro Y f hf\n  simp only [Presheaf.χ_app, Opposite.op_unop] at hf ⊢\n  choose a ha using fun Z (g : Z ⟶ Y) (hg : (Sieve.pullback f ((χ m).app X x)).arrows g) => hg\n  refine ⟨(hF _ hf).amalgamate a ?_, ?_⟩\n  · introv Y₁ h\n    apply (mono_iff_injective (m.app (.op Z))).mp inferInstance\n    simp_rw [NatTrans.naturality_apply, ← ha, ← Functor.map_comp_apply, ← op_comp,\n      reassoc_of% h]\n  · refine (hG _ hf).ext fun Z f' hf' => ?_\n    rw [← NatTrans.naturality_apply, (hF _ hf).valid_glue _ _ hf', ← (ha _ _ _),\n      op_comp, Functor.map_comp_apply]","hard_negative":true,"metrics":{"chosen_tokens":165,"rejected_tokens":4,"token_jaccard":0.028169,"token_length_ratio":0.024242},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"85d69378d0fcf3cd112215a2b5a2e27365b71dad0802d33f028338e6b64901c4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Sites.Closed\npublic import Mathlib.CategoryTheory.Sites.Equivalence\npublic import Mathlib.CategoryTheory.Subobject.Classifier.Defs\npublic import Mathlib.CategoryTheory.Subfunctor.Image\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Edward van de Meent. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Edward van de Meent\n-/\n/-!\n\n# (Elementary) Sheaf Topos\n\nWe define a subobject classifier for categories of sheaves of (large enough) types.\n\n## Main definitions\n\nLet `C` refer to a category with (when relevant) Grothendieck topology `J`.\n\n* `Presheaf.classifier C` is a construction of a subobject classifier in `Cᵒᵖ ⥤ Type (max u v)`.\n* `Sheaf.classifier J` is a construction of a subobject classifier in `Sheaf J (Type (max u v))`.\n* `inferInstance : HasClassifier (Cᵒᵖ ⥤ Type w)` says that `Cᵒᵖ ⥤ Type w` has a subobject\n  classifier if `C` is `w`-essentially small.\n* `inferInstance : HasClassifier (Sheaf J (Type w))` says that `Sheaf J (Type w)` has a\n  subobject classifier if `C` is `w`-essentially small.\n\n## Main results\n\n* Any category of sheaves of types has a subobject classifier if the site is essentially small.\n* As a consequence, (because categories of sheaves are cartesian monoidal and have finite limits,)\n  such categories are Elementary Topoi.\n\n## TODOS:\n\n* generalize `Presheaf.isClosed_χ_app_apply_of` to only assuming `G` is separated\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nnamespace CategoryTheory\n\nvariable {C : Type u} [Category.{v} C]\nopen Limits\n\nsection presheaf\n\nvariable (C) in\n/-- The truth morphism in the category of presheaves. At each component `X : C`, it is the constant\nmap returning `⊤ : Sieve X`. -/\n@[simps]\ndef Presheaf.truth : (Functor.const _).obj PUnit ⟶ Functor.sieves C where\n  app X := ↾fun _ ↦ (⊤ : Sieve X.unop)\n\nvariable {F G : Cᵒᵖ ⥤ Type (max u v)}\n\nset_option backward.defeqAttrib.useBackward true in\n/--\nThe characteristic map of an inclusion of presheaves.\nGiven a monomorphism of sheaves `m : F ⟶ G`, an object X of the site, map an element `x : G(X)`\nto the (closed) sieve on X where `f : Y → X` is in the sieve iff\n  `∃ a ∈ F(Y), G(f)(x) = m_Y(a)`\n-/\n@[simps app]\ndef Presheaf.χ (m : F ⟶ G) : G ⟶ Functor.sieves C where\n  app X := ↾fun x ↦ ⟨fun Y f => ∃ a, G.map f.op x = m.app (.op Y) a, by\n    intro Y Z f ⟨a, ha⟩ g\n    use F.map g.op a\n    simp [ha, NatTrans.naturality_apply]⟩\n\nlemma Presheaf.comp_χ_eq (m : F ⟶ G) : m ≫ Presheaf.χ m =\n    (Functor.isTerminalConst _ Types.isTerminalPUnit).from F ≫ Presheaf.truth C := by\n  ext\n  apply Sieve.ext\n  simp [← NatTrans.naturality_apply]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma Presheaf.isPullback_χ_truth (m : F ⟶ G) [Mono m] :\n    IsPullback m ((Functor.isTerminalConst _ Types.isTerminalPUnit).from F) (χ m) (truth C) := by\n  refine IsPullback.of_forall_isPullback_app fun X => ?_\n  rw [Types.isPullback_iff]\n  refine ⟨congr(($(comp_χ_eq m)).app X), ?_, ?_⟩\n  · simpa using! (mono_iff_injective (m.app X)).mp (inferInstance)\n  · simp only [Functor.const_obj_obj, Functor.sieves_obj, χ_app, Opposite.op_unop,\n      TypeCat.hom_ofHom, TypeCat.Fun.coe_mk, truth_app, Functor.isTerminalConst_from_app,\n      Types.isTerminalPUnit_from_apply, and_true, forall_const]\n    intro p hp\n    simpa [eq_comm] using! congr($(hp).arrows (𝟙 _))\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma Presheaf.χ_unique (m : F ⟶ G) (χ' : G ⟶ Functor.sieves C)\n    (hχ' : IsPullback m ((Functor.isTerminalConst _ Types.isTerminalPUnit).from _) χ' (truth C)) :\n    χ' = χ m := by\n  ext X x\n  simp only [IsPullback.iff_app, Functor.const_obj_obj, Functor.sieves_obj,\n    Functor.isTerminalConst_from_app, truth_app, Types.isPullback_iff,\n    Types.isTerminalPUnit_from_apply, and_true, TypeCat.hom_ofHom, TypeCat.Fun.coe_mk,\n    forall_const, forall_and] at hχ'\n  obtain ⟨h₁, h₂, h₃⟩ := hχ'\n  refine Sieve.ext fun Y f => ?_\n  simp only [χ_app, Opposite.op_unop]\n  rw [Sieve.mem_iff_pullback_eq_top, ← Quiver.Hom.unop_op f]\n  dsimp\n  have := ConcreteCategory.congr_hom (Functor.sieves_map C (f.op)) (χ'.app X x)\n  rw [← dsimp% this, ← dsimp% NatTrans.naturality_apply χ' f.op x]\n  constructor\n  · intro h\n    obtain ⟨z, hz⟩ := h₃ _ _ h\n    use z, hz.symm\n  · rintro ⟨a, h⟩\n    rw [h]\n    simpa using congr($(h₁ (.op Y)) a)\n\nvariable (C) in\n/-- A construction of a subject classifier in a category of presheaves. -/\n@[simps! Ω truth Ω₀ χ χ₀]\ndef Presheaf.classifier : Subobject.Classifier (Cᵒᵖ ⥤ Type (max u v)) :=\n  .mkOfTerminalΩ₀ ((Functor.const Cᵒᵖ).obj PUnit)\n    (Functor.isTerminalConst _ (Types.isTerminalPUnit)) (Functor.sieves C) (Presheaf.truth C)\n    (Presheaf.χ ·) Presheaf.isPullback_χ_truth (Presheaf.χ_unique ·)\n\n/-- Presheaf categories on an essentially small domain have a subobject classifier. -/\ninstance [EssentiallySmall.{w} C] : HasSubobjectClassifier (Cᵒᵖ ⥤ Type w) where\n  exists_classifier := ⟨(Presheaf.classifier (SmallModel C)).ofEquivalence\n    (Equivalence.congrLeft (E := Type w) (equivSmallModel C).op).symm⟩\n\nend presheaf\n\nvariable {J : GrothendieckTopology C}\n\nopen Presheaf in\n\nTarget:\nlemma GrothendieckTopology.isClosed_χ_app_apply_of_isSheaf_of_isSeparated\n    {F G : Cᵒᵖ ⥤ Type (max u v)} (m : F ⟶ G) [Mono m] (hF : Presieve.IsSheaf J F)\n    (hG : Presieve.IsSeparated J G) (X : Cᵒᵖ) (x : G.obj X) :\n    J.IsClosed ((Presheaf.χ m).app X x) :=\n\nProof body:\n","rejected":"by\n  exact GrothendieckTopology.","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"403071b67998f3f9ee5a39109f0e1362913ce18d8cb8c1abeb22226054d18083","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Topos","family_id":"grothendiecktopology","file_id":"mathlib/Mathlib/CategoryTheory/Topos/Sheaf.lean","sample_id":"20b48f91a7f4ad693944c040d924608c454aa3ef8bafb869719f264b357fd194"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f8689006584708e8f5f2ea5a3447138e87630e033d57129a1fcb9fca25aae75e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1b257d253a1d99e1ebaccfa098b639ce8bc9949faf2e55ad6ca74bb6e26b343e","source_sha256":"d05b0d2ddc5c920ffdbae14bb142f5937424e23790dd69cbc6cb4ae7e530bb6a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let h := MorphismProperty.factorizationData (trivialCofibrations C) (fibrations C)\n      (terminal.from X.obj)\n  have := isCofibrant_of_cofibration h.i\n  have : IsFibrant h.Z := by\n    rw [isFibrant_iff_of_isTerminal h.p terminalIsTerminal]\n    infer_instance\n  exact ⟨BifibrantObject.mk h.Z, homMk h.i, inferInstanceAs (Cofibration h.i),\n    inferInstanceAs (WeakEquivalence h.i)⟩\n\n/-- Given `X : CofibrantObject C`, this is a choice of bifibrant resolution of `X`. -/\nnoncomputable def bifibrantResolutionObj (X : CofibrantObject C) :\n    BifibrantObject C :=\n  (exists_bifibrant X).choose\n\n/-- Given `X : CofibrantObject C`, this is a trivial cofibration\nfrom `X` to a choice of bifibrant resolution. -/\nnoncomputable def iBifibrantResolutionObj (X : CofibrantObject C) :\n    X ⟶ BifibrantObject.ιCofibrantObject.obj (bifibrantResolutionObj X) :=\n  (exists_bifibrant X).choose_spec.choose","hard_negative":true,"metrics":{"chosen_tokens":180,"rejected_tokens":8,"token_jaccard":0.076923,"token_length_ratio":0.044444},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"85e43e0345e51df9ddc6d8c30318b5d4950f3ca3fcf989370622d2ef1d0285f7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy\npublic import Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy\npublic import Mathlib.CategoryTheory.Localization.CalculusOfFractions.OfAdjunction\npublic import Mathlib.CategoryTheory.Quotient.LocallySmall\n\nNamespace:\nHomotopicalAlgebra.CofibrantObject\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The homotopy category of bifibrant objects\n\nWe construct the homotopy category `BifibrantObject.HoCat C` of bifibrant\nobjects in a model category `C` and show that the functor\n`BifibrantObject.toHoCat : BifibrantObject C ⥤ BifibrantObject.HoCat C`\nis a localization functor with respect to weak equivalences.\nWe also show that certain localizer morphisms are localized weak equivalences,\nwhich can be understood by saying that we obtain the same localized\ncategory (up to equivalence) by inverting weak equivalences in `C`,\n`CofibrantObject C`, `FibrantObject C` or `BifibrantObject C`.\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nopen CategoryTheory Limits\n\nnamespace HomotopicalAlgebra\n\nvariable {C : Type u} [Category.{v} C] [ModelCategory C]\n\nnamespace BifibrantObject\n\nvariable (C) in\n/-- The homotopy relation on the category of bifibrant objects. -/\ndef homRel : HomRel (BifibrantObject C) :=\n  fun _ _ f g ↦ RightHomotopyRel f.hom g.hom\n\nlemma homRel_iff_rightHomotopyRel {X Y : BifibrantObject C} {f g : X ⟶ Y} :\n    homRel C f g ↔ RightHomotopyRel f.hom g.hom := Iff.rfl\n\nlemma homRel_iff_leftHomotopyRel {X Y : BifibrantObject C} {f g : X ⟶ Y} :\n    homRel C f g ↔ LeftHomotopyRel f.hom g.hom := by\n  rw [homRel_iff_rightHomotopyRel, leftHomotopyRel_iff_rightHomotopyRel]\n\ninstance : HomRel.IsStableUnderPostcomp (homRel C) where\n  comp_right _ h := h.postcomp _\n\ninstance : HomRel.IsStableUnderPrecomp (homRel C) where\n  comp_left _ _ _ h := h.precomp _\n\ninstance : Congruence (homRel C) where\n  equivalence :=\n    { refl _ := .refl _\n      symm h := .symm h\n      trans h₁ h₂ := .trans h₁ h₂ }\n\nvariable (C) in\n/-- The homotopy category of bifibrant objects. -/\nabbrev HoCat := Quotient (BifibrantObject.homRel C)\n\n/-- The quotient functor from the category of bifibrant objects to its\nhomotopy category. -/\ndef toHoCat : BifibrantObject C ⥤ HoCat C := Quotient.functor _\n\nlemma toHoCat_obj_surjective : Function.Surjective (toHoCat (C := C)).obj :=\n  fun ⟨_⟩ ↦ ⟨_, rfl⟩\n\ninstance : Functor.Full (toHoCat (C := C)) := by dsimp [toHoCat]; infer_instance\n\nlemma toHoCat_map_eq {X Y : BifibrantObject C} {f g : X ⟶ Y}\n    (h : homRel C f g) :\n    toHoCat.map f = toHoCat.map g :=\n  CategoryTheory.Quotient.sound _ h\n\nlemma toHoCat_map_eq_iff {X Y : BifibrantObject C} (f g : X ⟶ Y) :\n    toHoCat.map f = toHoCat.map g ↔ homRel C f g :=\n  Quotient.functor_map_eq_iff _ _ _\n\ninstance [LocallySmall.{w} C] : LocallySmall.{w} (HoCat C) := by\n  dsimp [HoCat]\n  infer_instance\n\nsection\n\nvariable {D : Type*} [Category* D]\n\nlemma inverts_iff_factors (F : BifibrantObject C ⥤ D) :\n    (weakEquivalences _).IsInvertedBy F ↔\n    ∀ ⦃K L : BifibrantObject C⦄ (f g : K ⟶ L),\n      homRel C f g → F.map f = F.map g := by\n  refine ⟨fun H K L f g h ↦ ?_, fun h X Y f hf ↦ ?_⟩\n  · obtain ⟨P, _, ⟨h⟩⟩ := h.exists_very_good_pathObject\n    have := isCofibrant_of_cofibration P.ι\n    have : IsIso (F.map (homMk P.ι)) := H _ (by\n      rw [← weakEquivalence_iff, weakEquivalence_iff_of_objectProperty]\n      exact inferInstanceAs (WeakEquivalence P.ι))\n    simp only [show f = homMk h.h ≫ homMk P.p₀ by cat_disch,\n      show g = homMk h.h ≫ homMk P.p₁ by cat_disch, Functor.map_comp]\n    congr 1\n    simp [← cancel_epi (F.map (homMk P.ι)), ← Functor.map_comp]\n  · rw [← weakEquivalence_iff, weakEquivalence_iff_of_objectProperty] at hf\n    obtain ⟨g', h₁, h₂⟩ := RightHomotopyClass.whitehead f.hom\n    refine ⟨F.map (homMk g'), ?_, ?_⟩\n    all_goals\n      rw [← F.map_comp, ← F.map_id]\n      apply h\n      assumption\n\n/-- The strict universal property of the localization with respect\nto weak equivalences for the quotient functor\n`toHoCat : BifibrantObject C ⥤ BifibrantObject.HoCat C`. -/\ndef strictUniversalPropertyFixedTargetToHoCat :\n    Localization.StrictUniversalPropertyFixedTarget\n      toHoCat (weakEquivalences (BifibrantObject C)) D where\n  inverts := by\n    rw [inverts_iff_factors]\n    intro K L f g h\n    exact CategoryTheory.Quotient.sound _ h\n  lift F hF := CategoryTheory.Quotient.lift _ F\n    (by rwa [inverts_iff_factors] at hF)\n  fac F hF := rfl\n  uniq _ _ h := Quotient.lift_unique' _ _ _ h\n\nend\n\ninstance : toHoCat.IsLocalization (weakEquivalences (BifibrantObject C)) :=\n  .mk' _ _ strictUniversalPropertyFixedTargetToHoCat\n    strictUniversalPropertyFixedTargetToHoCat\n\ninstance {X Y : BifibrantObject C} (f : X ⟶ Y) [hf : WeakEquivalence f] :\n    IsIso (toHoCat.map f) :=\n  Localization.inverts toHoCat (weakEquivalences _) f (by rwa [weakEquivalence_iff] at hf)\n\nsection\n\nvariable {X Y : C} [IsCofibrant X] [IsCofibrant Y] [IsFibrant X] [IsFibrant Y]\n\n/-- Right homotopy classes of maps between bifibrant objects identify\nto morphisms in the homotopy category `BifibrantObject.HoCat`. -/\ndef HoCat.homEquivRight :\n    RightHomotopyClass X Y ≃ (toHoCat.obj (mk X) ⟶ toHoCat.obj (mk Y)) where\n  toFun := Quot.lift (fun f ↦ toHoCat.map (homMk f)) (fun _ _ h ↦ by rwa [toHoCat_map_eq_iff])\n  invFun := Quot.lift (fun f ↦ .mk f.hom) (fun _ _ h ↦ by\n    simpa [RightHomotopyClass.mk_eq_mk_iff] using! h)\n  left_inv := by rintro ⟨f⟩; rfl\n  right_inv := by rintro ⟨f⟩; rfl\n\n@[simp]\nlemma HoCat.homEquivRight_apply (f : X ⟶ Y) :\n    HoCat.homEquivRight (.mk f) = toHoCat.map (homMk f) := rfl\n\n@[simp]\nlemma HoCat.homEquivRight_symm_apply (f : X ⟶ Y) :\n    HoCat.homEquivRight.symm (toHoCat.map (homMk f)) = .mk f := rfl\n\n/-- Left homotopy classes of maps between bifibrant objects identify\nto morphisms in the homotopy category `BifibrantObject.HoCat`. -/\ndef HoCat.homEquivLeft :\n    LeftHomotopyClass X Y ≃ (toHoCat.obj (mk X) ⟶ toHoCat.obj (mk Y)) :=\n  leftHomotopyClassEquivRightHomotopyClass.trans HoCat.homEquivRight\n\n@[simp]\nlemma HoCat.homEquivLeft_apply (f : X ⟶ Y) :\n    HoCat.homEquivLeft (.mk f) = toHoCat.map (homMk f) := by\n  simp [homEquivLeft]\n\n@[simp]\nlemma HoCat.homEquivLeft_symm_apply (f : X ⟶ Y) :\n    HoCat.homEquivRight.symm (toHoCat.map (homMk f)) = .mk f := rfl\n\nend\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The inclusion functor `BifibrantObject.HoCat C ⥤ FibrantObject.HoCat C`. -/\ndef HoCat.ιFibrantObject : HoCat C ⥤ FibrantObject.HoCat C :=\n  CategoryTheory.Quotient.lift _\n    (BifibrantObject.ιFibrantObject ⋙ FibrantObject.toHoCat) (fun _ _ _ _ h ↦ by\n      simpa [FibrantObject.toHoCat_map_eq_iff, FibrantObject.homRel_iff_leftHomotopyRel,\n        homRel_iff_leftHomotopyRel] using h)\n\n@[simp]\nlemma HoCat.ιFibrantObject_obj (X : BifibrantObject C) :\n    HoCat.ιFibrantObject.obj (toHoCat.obj X) =\n      FibrantObject.toHoCat.obj (BifibrantObject.ιFibrantObject.obj X) :=\n  rfl\n\n@[simp]\nlemma HoCat.ιFibrantObject_map_toHoCat_map {X Y : BifibrantObject C} (f : X ⟶ Y) :\n    HoCat.ιFibrantObject.map (toHoCat.map f) =\n      FibrantObject.toHoCat.map (FibrantObject.homMk f.hom) :=\n  rfl\n\n/-- The isomorphism `toHoCat ⋙ HoCat.ιFibrantObject ≅ ιFibrantObject ⋙ FibrantObject.toHoCat`\nbetween functors `BifibrantObject C ⥤ FibrantObject.HoCat C`. -/\ndef toHoCatCompιFibrantObject :\n    toHoCat (C := C) ⋙ HoCat.ιFibrantObject ≅\n      ιFibrantObject ⋙ FibrantObject.toHoCat := Iso.refl _\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The inclusion functor `BifibrantObject.HoCat C ⥤ CofibrantObject.HoCat C`. -/\ndef HoCat.ιCofibrantObject : HoCat C ⥤ CofibrantObject.HoCat C :=\n  CategoryTheory.Quotient.lift _\n    (BifibrantObject.ιCofibrantObject ⋙ CofibrantObject.toHoCat) (fun _ _ _ _ h ↦ by\n      simpa [CofibrantObject.toHoCat_map_eq_iff])\n\n@[simp]\nlemma HoCat.ιCofibrantObject_obj (X : BifibrantObject C) :\n    HoCat.ιCofibrantObject.obj (toHoCat.obj X) =\n      CofibrantObject.toHoCat.obj (BifibrantObject.ιCofibrantObject.obj X) :=\n  rfl\n\n@[simp]\nlemma HoCat.ιCofibrantObject_map_toHoCat_map {X Y : BifibrantObject C} (f : X ⟶ Y) :\n    HoCat.ιCofibrantObject.map (toHoCat.map f) =\n      CofibrantObject.toHoCat.map (CofibrantObject.homMk f.hom) :=\n  rfl\n\n/-- The isomorphism\n`toHoCat ⋙ HoCat.ιCofibrantObject ≅ ιCofibrantObject ⋙ CofibrantObject.toHoCat`\nbetween functors `BifibrantObject C ⥤ CofibrantObject.HoCat C`. -/\ndef toHoCatCompιCofibrantObject :\n    toHoCat (C := C) ⋙ HoCat.ιCofibrantObject ≅\n      ιCofibrantObject ⋙ CofibrantObject.toHoCat := Iso.refl _\n\nend BifibrantObject\n\nnamespace CofibrantObject\n\nTarget:\nlemma exists_bifibrant (X : CofibrantObject C) :\n    ∃ (Y : BifibrantObject C) (i : X ⟶ BifibrantObject.ιCofibrantObject.obj Y),\n      Cofibration (ι.map i) ∧ WeakEquivalence (ι.map i) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"3c17cf65f1141443d5fda3c6491ee0aa1e1052b0307eca31b1a84ac0eade55e9","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/ModelCategory","family_id":"exists_bifibrant","file_id":"mathlib/Mathlib/AlgebraicTopology/ModelCategory/BifibrantObjectHomotopy.lean","sample_id":"1b257d253a1d99e1ebaccfa098b639ce8bc9949faf2e55ad6ca74bb6e26b343e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4bbd86e73cbf3bb6938d81696c0fac5d64455a44efb1861863f6e712d6a46845","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8a9a5d45eecf81f80cc6ac9acad10ad3865265d34cf144e568617dadd24602fc","source_sha256":"e073d0499bdb4f94f221cf1b385cb626e3018e60f9a1a6981e11696230ed0c39","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Finset.disjoint_left]\n  rintro x hx hxc\n  obtain ⟨x, hx', rfl⟩ := mem_map.mp hxc\n  exact hs.compl_notMem hx' hx","hard_negative":false,"metrics":{"chosen_tokens":30,"rejected_tokens":2,"token_jaccard":0.041667,"token_length_ratio":0.066667},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"85e4e72cfdeb240029ac27503970f1f447ef6cdfb91b0dcd9e0e1b6eab4a42a0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Card\npublic import Mathlib.Order.UpperLower.Basic\n\nNamespace:\nSet\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Intersecting families\n\nThis file defines intersecting families and proves their basic properties.\n\n## Main declarations\n\n* `Set.Intersecting`: Predicate for a set of elements in a generalized Boolean algebra to be an\n  intersecting family.\n* `Set.Intersecting.card_le`: An intersecting family can only take up to half the elements, because\n  `a` and `aᶜ` cannot simultaneously be in it.\n* `Set.Intersecting.is_max_iff_card_eq`: Any maximal intersecting family takes up half the elements.\n* `Set.IsIntersectingOf`: Predicate stating that a family `𝒜` of finsets is `L`-intersecting, i.e.,\n  meaning the intersection size of every pair of distinct members of `𝒜` belongs to `L ⊆ ℕ`.\n\n## References\n\n* [D. J. Kleitman, *Families of non-disjoint subsets*][kleitman1966]\n-/\n\n@[expose] public section\n\nassert_not_exists Monoid\n\nopen Finset\n\nnamespace Set\n\nsection SemilatticeInf\n\nvariable {α : Type*}\n\nvariable [SemilatticeInf α] [OrderBot α] {s t : Set α} {a b c : α}\n\n/-- A set family is intersecting if every pair of elements is non-disjoint. -/\ndef Intersecting (s : Set α) : Prop :=\n  ∀ ⦃a⦄, a ∈ s → ∀ ⦃b⦄, b ∈ s → ¬Disjoint a b\n\n@[gcongr, mono]\ntheorem Intersecting.mono (h : t ⊆ s) (hs : s.Intersecting) : t.Intersecting := fun _a ha _b hb =>\n  hs (h ha) (h hb)\n\ntheorem Intersecting.bot_notMem (hs : s.Intersecting) : ⊥ ∉ s := fun h => hs h h disjoint_bot_left\n\ntheorem Intersecting.ne_bot (hs : s.Intersecting) (ha : a ∈ s) : a ≠ ⊥ :=\n  ne_of_mem_of_not_mem ha hs.bot_notMem\n\ntheorem intersecting_empty : (∅ : Set α).Intersecting := fun _ => False.elim\n\n@[simp]\ntheorem intersecting_singleton : ({a} : Set α).Intersecting ↔ a ≠ ⊥ := by simp [Intersecting]\n\nprotected theorem Intersecting.insert (hs : s.Intersecting) (ha : a ≠ ⊥)\n    (h : ∀ b ∈ s, ¬Disjoint a b) : (insert a s).Intersecting := by\n  rintro b (rfl | hb) c (rfl | hc)\n  · rwa [disjoint_self]\n  · exact h _ hc\n  · exact fun H => h _ hb H.symm\n  · exact hs hb hc\n\ntheorem intersecting_insert :\n    (insert a s).Intersecting ↔ s.Intersecting ∧ a ≠ ⊥ ∧ ∀ b ∈ s, ¬Disjoint a b :=\n  ⟨fun h =>\n    ⟨h.mono <| subset_insert _ _, h.ne_bot <| mem_insert _ _, fun _b hb =>\n      h (mem_insert _ _) <| mem_insert_of_mem _ hb⟩,\n    fun h => h.1.insert h.2.1 h.2.2⟩\n\ntheorem intersecting_iff_pairwise_not_disjoint :\n    s.Intersecting ↔ (s.Pairwise fun a b => ¬Disjoint a b) ∧ s ≠ {⊥} := by\n  refine ⟨fun h => ⟨fun a ha b hb _ => h ha hb, ?_⟩, fun h a ha b hb hab => ?_⟩\n  · rintro rfl\n    exact intersecting_singleton.1 h rfl\n  have := h.1.eq ha hb (Classical.not_not.2 hab)\n  rw [this, disjoint_self] at hab\n  rw [hab] at hb\n  exact\n    h.2\n      (eq_singleton_iff_unique_mem.2\n        ⟨hb, fun c hc => not_ne_iff.1 fun H => h.1 hb hc H.symm disjoint_bot_left⟩)\n\nprotected theorem Subsingleton.intersecting (hs : s.Subsingleton) : s.Intersecting ↔ s ≠ {⊥} :=\n  intersecting_iff_pairwise_not_disjoint.trans <| and_iff_right <| hs.pairwise _\n\ntheorem intersecting_iff_eq_empty_of_subsingleton [Subsingleton α] (s : Set α) :\n    s.Intersecting ↔ s = ∅ := by\n  refine\n    subsingleton_of_subsingleton.intersecting.trans\n      ⟨not_imp_comm.2 fun h => subsingleton_of_subsingleton.eq_singleton_of_mem ?_, ?_⟩\n  · obtain ⟨a, ha⟩ := nonempty_iff_ne_empty.2 h\n    rwa [Subsingleton.elim ⊥ a]\n  · rintro rfl\n    exact (Set.singleton_nonempty _).ne_empty.symm\n\n/-- Maximal intersecting families are upper sets. -/\nprotected theorem Intersecting.isUpperSet (hs : s.Intersecting)\n    (h : ∀ t : Set α, t.Intersecting → s ⊆ t → s = t) : IsUpperSet s := by\n  classical\n    rintro a b hab ha\n    rw [h (Insert.insert b s) _ (subset_insert _ _)]\n    · exact mem_insert _ _\n    exact\n      hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab\n\n/-- Maximal intersecting families are upper sets. Finset version. -/\ntheorem Intersecting.isUpperSet' {s : Finset α} (hs : (s : Set α).Intersecting)\n    (h : ∀ t : Finset α, (t : Set α).Intersecting → s ⊆ t → s = t) : IsUpperSet (s : Set α) := by\n  classical\n    rintro a b hab ha\n    rw [h (Insert.insert b s) _ (Finset.subset_insert _ _)]\n    · exact mem_insert_self _ _\n    rw [coe_insert]\n    exact\n      hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab\n\nend SemilatticeInf\n\nsection\n\nvariable {α : Type*}\n\ntheorem Intersecting.exists_mem_set {𝒜 : Set (Set α)} (h𝒜 : 𝒜.Intersecting) {s t : Set α}\n    (hs : s ∈ 𝒜) (ht : t ∈ 𝒜) : ∃ a, a ∈ s ∧ a ∈ t :=\n  not_disjoint_iff.1 <| h𝒜 hs ht\n\ntheorem Intersecting.exists_mem_finset [DecidableEq α] {𝒜 : Set (Finset α)} (h𝒜 : 𝒜.Intersecting)\n    {s t : Finset α} (hs : s ∈ 𝒜) (ht : t ∈ 𝒜) : ∃ a, a ∈ s ∧ a ∈ t :=\n  not_disjoint_iff.1 <| disjoint_coe.not.2 <| h𝒜 hs ht\n\nvariable [BooleanAlgebra α]\n\ntheorem Intersecting.compl_notMem {s : Set α} (hs : s.Intersecting) {a : α} (ha : a ∈ s) :\n    aᶜ ∉ s := fun h => hs ha h disjoint_compl_right\n\ntheorem Intersecting.notMem {s : Set α} (hs : s.Intersecting) {a : α} (ha : aᶜ ∈ s) : a ∉ s :=\n  fun h => hs ha h disjoint_compl_left\n\nTarget:\ntheorem Intersecting.disjoint_map_compl {s : Finset α} (hs : (s : Set α).Intersecting) :\n    Disjoint s (s.map ⟨compl, compl_injective⟩) :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/SetFamily","family_id":"intersecting","file_id":"mathlib/Mathlib/Combinatorics/SetFamily/Intersecting.lean","sample_id":"8a9a5d45eecf81f80cc6ac9acad10ad3865265d34cf144e568617dadd24602fc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bb1aa1df4a3618112e80043cd006efbcdf4906086196e285cd31ba1676129c32","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"26a28f745aa7034105aa52a93b588fb63e8e92ad6d8d99ffd3be3558367e64ea","source_sha256":"ffda45eef0509a041f24079788fb5aa3f3f4500975a78975ccc64409c3de362b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← closure_openCell_eq_closedCell, E.closed.closure_subset_iff, ← E.union]\n  apply subset_union_of_subset_right\n  exact subset_iUnion_of_subset n\n    (subset_iUnion (fun (j : ↑(E.I n)) ↦ openCell (C := C) n j) ⟨i, hi⟩)","hard_negative":false,"metrics":{"chosen_tokens":53,"rejected_tokens":2,"token_jaccard":0.029412,"token_length_ratio":0.037736},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"860ef14fc007eef49c917f1fe1f9b337e9f77d9b1a85bbd9af0558a67d046115","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.CWComplex.Classical.Finite\npublic import Mathlib.Analysis.Normed.Module.RCLike.Real\n\nNamespace:\nTopology\n\nLocal context:\n/-\nCopyright (c) 2025 Floris van Doorn and Hannah Scholz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Hannah Scholz\n-/\n/-!\n# Subcomplexes\n\nIn this file we discuss subcomplexes of CW complexes.\nThe definition of subcomplexes is in the file `Mathlib/Topology/CWComplex/Classical/Basic.lean`.\n\n## Main results\n* `RelCWComplex.Subcomplex.instRelCWComplex`: a subcomplex of a (relative) CW complex is again a\n  (relative) CW complex.\n\n## References\n* [K. Jänich, *Topology*][Janich1984]\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Metric Set\n\nnamespace Topology\n\nvariable {X : Type*} [t : TopologicalSpace X] {C D : Set X}\n\n@[alias_in CWComplex.Subcomplex]\n\nTarget:\nlemma RelCWComplex.Subcomplex.closedCell_subset_of_mem [T2Space X] [RelCWComplex C D]\n    (E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) :\n    closedCell n i ⊆ E :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/CWComplex","family_id":"relcwcomplex","file_id":"mathlib/Mathlib/Topology/CWComplex/Classical/Subcomplex.lean","sample_id":"26a28f745aa7034105aa52a93b588fb63e8e92ad6d8d99ffd3be3558367e64ea"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eb3d60d437f15cfc69d1d37a56f9a9b73ef1279b8f1ffb7d0327dabbc53ce4fd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7e5e76fd4da0d1114a6376ee390941064e398ed5015cd292495ed362e6c5a84e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b5eed30bb36819b24fa9a164628ee2a2c0338152109146fc3f9ca3ebec531d4e","source_sha256":"9cb41eaef9fea958cbbd1f98ac6fcd22922a81e7fe77437a7ce0c4aae8ecfaf2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  simp only [alternatingMapLinearEquiv_comp_ιMulti, LinearMap.compAlternatingMap_apply,\n    LinearMap.coe_comp, comp_apply, alternatingMapLinearEquiv_apply_ιMulti]\n\n/-! Functoriality of the exterior powers. -/\n\nvariable (n) in\n/-- The linear map between `n`th exterior powers induced by a linear map between the modules. -/\nnoncomputable def map (f : M →ₗ[R] N) : ⋀[R]^n M →ₗ[R] ⋀[R]^n N :=\n  alternatingMapLinearEquiv ((ιMulti R n).compLinearMap f)","hard_negative":true,"metrics":{"chosen_tokens":110,"rejected_tokens":2,"token_jaccard":0.018868,"token_length_ratio":0.018182},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"861f04404ba855c66c6ebf4baabaedc5cd86ba903a61932406b89f608dcaadec","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Presentation.Basic\npublic import Mathlib.LinearAlgebra.ExteriorAlgebra.OfAlternating\n\nNamespace:\nexteriorPower\n\nLocal context:\n/-\nCopyright (c) 2024 Sophie Morel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sophie Morel, Joël Riou\n-/\n/-!\n# Exterior powers\n\nWe study the exterior powers of a module `M` over a commutative ring `R`.\n\n## Definitions\n\n* `exteriorPower.ιMulti` is the canonical alternating map on `M` with values in `⋀[R]^n M`.\n\n* `exteriorPower.presentation R n M` is the standard presentation of the `R`-module `⋀[R]^n M`.\n\n* `exteriorPower.map n f : ⋀[R]^n M →ₗ[R] ⋀[R]^n N` is the linear map on `nth` exterior powers\n  induced by a linear map `f : M →ₗ[R] N`. (See the file\n  `Mathlib/Algebra/Category/ModuleCat/ExteriorPower.lean` for the corresponding functor\n  `ModuleCat R ⥤ ModuleCat R`.)\n\n## Theorems\n* `exteriorPower.ιMulti_span`: The image of `exteriorPower.ιMulti` spans `⋀[R]^n M`.\n\n* We construct `exteriorPower.alternatingMapLinearEquiv` which\n  expresses the universal property of the exterior power as a\n  linear equivalence `(M [⋀^Fin n]→ₗ[R] N) ≃ₗ[R] ⋀[R]^n M →ₗ[R] N` between\n  alternating maps and linear maps from the exterior power.\n\n-/\n\n@[expose] public section\n\nopen scoped TensorProduct\n\nuniverse u\n\nvariable (R : Type u) [CommRing R] (n : ℕ) {M N N' : Type*}\n  [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]\n  [AddCommGroup N'] [Module R N']\n\nnamespace exteriorPower\n\nopen Function Set Set.powersetCard\n\n/-! The canonical alternating map from `Fin n → M` to `⋀[R]^n M`. -/\n\n/-- `exteriorAlgebra.ιMulti` is the alternating map from `Fin n → M` to `⋀[r]^n M`\ninduced by `exteriorAlgebra.ιMulti`, i.e. sending a family of vectors `m : Fin n → M` to the\nproduct of its entries. -/\ndef ιMulti : M [⋀^Fin n]→ₗ[R] (⋀[R]^n M) :=\n  (ExteriorAlgebra.ιMulti R n).codRestrict (⋀[R]^n M) fun _ =>\n    ExteriorAlgebra.ιMulti_range R n <| Set.mem_range_self _\n\n@[simp] lemma ιMulti_apply_coe (a : Fin n → M) : ιMulti R n a = ExteriorAlgebra.ιMulti R n a := rfl\n\n/-- Given a linearly ordered family `v` of vectors of `M` and a natural number `n`, produce the\nfamily of `n`fold exterior products of elements of `v`, seen as members of the\n`n`th exterior power. -/\nnoncomputable def ιMulti_family {I : Type*} [LinearOrder I] (v : I → M)\n    (s : powersetCard I n) : ⋀[R]^n M :=\n  ιMulti R n (v ∘ (ofFinEmbEquiv.symm s))\n\nlemma ιMulti_family_eq_coe_comp {I : Type*} [LinearOrder I] (v : I → M) :\n    ExteriorAlgebra.ιMulti_family R n v = (↑) ∘ ιMulti_family R n v :=\n  rfl\n\n@[simp] lemma ιMulti_family_apply_coe {I : Type*} [LinearOrder I] (v : I → M)\n    (s : powersetCard I n) :\n    ιMulti_family R n v s = ExteriorAlgebra.ιMulti_family R n v s := rfl\n\nvariable (M)\n/-- The image of `ExteriorAlgebra.ιMulti R n` spans the `n`th exterior power. Variant of\n`ExteriorAlgebra.ιMulti_span_fixedDegree`, useful in rewrites. -/\nlemma ιMulti_span_fixedDegree :\n    Submodule.span R (Set.range (ExteriorAlgebra.ιMulti R n)) = ⋀[R]^n M :=\n  ExteriorAlgebra.ιMulti_span_fixedDegree R n\n\nopen Set Submodule in\n/-- If a set `s` spans the module `M`, then the set of all elements of the form `x₁ ∧ ⋯ ∧ xₙ`\nwhere `xᵢ ∈ s` spans `⋀ⁿ M`. -/\nlemma ιMulti_span_fixedDegree_of_span_eq_top {s : Set M} (hs : span R s = ⊤) :\n    span R (ExteriorAlgebra.ιMulti R n '' {a | range a ⊆ s}) = ⋀[R]^n M := by\n  apply le_antisymm\n  · rw [span_le]\n    rintro - ⟨y, ⟨y_mem, rfl⟩⟩\n    apply ExteriorAlgebra.ιMulti_range R n\n    simp\n  · rw [ExteriorAlgebra.exteriorPower, LinearMap.range_eq_map, ← hs, map_span, span_pow, span_le]\n    rintro x hx\n    obtain ⟨f, rfl⟩ := Set.mem_pow.mp hx\n    refine mem_span_of_mem ⟨ExteriorAlgebra.ιInv ∘ Subtype.val ∘ f, ?_, ?_⟩\n    · rw [Set.mem_setOf_eq, Set.range_comp, Set.image_subset_iff]\n      apply Subset.trans ?_ (s.image_subset_preimage_of_inverse ExteriorAlgebra.ι_leftInverse)\n      grind\n    · rw [ExteriorAlgebra.ιMulti_apply]\n      apply congrArg (List.prod ∘ List.ofFn)\n      ext i\n      obtain ⟨m, -, hm⟩ := (Set.mem_image _ _ _).mp (f i).2\n      rw [Function.comp_apply, Function.comp_apply, ← hm, ExteriorAlgebra.ι_leftInverse]\n\n/-- The image of `exteriorPower.ιMulti` spans `⋀[R]^n M`. -/\nlemma ιMulti_span :\n    Submodule.span R (Set.range (ιMulti R n)) = (⊤ : Submodule R (⋀[R]^n M)) := by\n  apply LinearMap.map_injective (Submodule.ker_subtype (⋀[R]^n M))\n  rw [LinearMap.map_span, ← Set.image_univ, Set.image_image]\n  simp only [Submodule.coe_subtype, ιMulti_apply_coe, Set.image_univ, Submodule.map_top,\n    Submodule.range_subtype]\n  exact ExteriorAlgebra.ιMulti_span_fixedDegree R n\n\nopen Set Submodule in\n/-- A version of `ιMulti_span_fixedDegree_of_span_eq_top` that works in the exterior power. -/\nlemma ιMulti_span_of_span {s : Set M} (hs : span R s = ⊤) :\n    span R (ιMulti R n '' {a | range a ⊆ s}) = ⊤ := by\n  apply LinearMap.map_injective (ker_subtype (⋀[R]^n M))\n  simpa [LinearMap.map_span, Set.image_image] using ιMulti_span_fixedDegree_of_span_eq_top R n M hs\n\nnamespace presentation\n\n/-- The index type for the relations in the standard presentation of `⋀[R]^n M`,\nin the particular case `ι` is `Fin n`. -/\ninductive Rels (ι : Type*) (M : Type*)\n  | add (m : ι → M) (i : ι) (x y : M)\n  | smul (m : ι → M) (i : ι) (r : R) (x : M)\n  | alt (m : ι → M) (i j : ι) (hm : m i = m j) (hij : i ≠ j)\n\n/-- The relations in the standard presentation of `⋀[R]^n M` with generators and relations. -/\n@[simps]\nnoncomputable def relations (ι : Type*) [DecidableEq ι] (M : Type*)\n    [AddCommGroup M] [Module R M] :\n    Module.Relations R where\n  G := ι → M\n  R := Rels R ι M\n  relation\n    | .add m i x y => Finsupp.single (update m i x) 1 +\n        Finsupp.single (update m i y) 1 -\n        Finsupp.single (update m i (x + y)) 1\n    | .smul m i r x => Finsupp.single (update m i (r • x)) 1 -\n        r • Finsupp.single (update m i x) 1\n    | .alt m _ _ _ _ => Finsupp.single m 1\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nvariable {R} in\n/-- The solutions in a module `N` to the linear equations\ngiven by `exteriorPower.relations R ι M` identify to alternating maps to `N`. -/\n@[simps!]\nnoncomputable def relationsSolutionEquiv {ι : Type*} [DecidableEq ι] {M : Type*}\n    [AddCommGroup M] [Module R M] :\n    (relations R ι M).Solution N ≃ AlternatingMap R M N ι where\n  toFun s :=\n    { toFun := fun m ↦ s.var m\n      map_update_add' := fun m i x y ↦ by\n        have := s.linearCombination_var_relation (.add m i x y)\n        dsimp at this ⊢\n        rw [map_sub, map_add, Finsupp.linearCombination_single, one_smul,\n          Finsupp.linearCombination_single, one_smul,\n          Finsupp.linearCombination_single, one_smul, sub_eq_zero] at this\n        convert! this.symm -- `convert` is necessary due to the implementation of `MultilinearMap`\n      map_update_smul' := fun m i r x ↦ by\n        have := s.linearCombination_var_relation (.smul m i r x)\n        dsimp at this ⊢\n        rw [Finsupp.smul_single, smul_eq_mul, mul_one, map_sub,\n          Finsupp.linearCombination_single, one_smul,\n          Finsupp.linearCombination_single, sub_eq_zero] at this\n        convert! this\n      map_eq_zero_of_eq' := fun v i j hm hij ↦\n        by simpa using s.linearCombination_var_relation (.alt v i j hm hij) }\n  invFun f :=\n    { var := fun m ↦ f m\n      linearCombination_var_relation := by\n        rintro (⟨m, i, x, y⟩ | ⟨m, i, r, x⟩ | ⟨v, i, j, hm, hij⟩)\n        · simp\n        · simp\n        · simpa using f.map_eq_zero_of_eq v hm hij }\n\n/-- The universal property of the exterior power. -/\nnoncomputable def isPresentationCore :\n    (relationsSolutionEquiv.symm (ιMulti R n (M := M))).IsPresentationCore where\n  desc s := LinearMap.comp (ExteriorAlgebra.liftAlternating\n      (Function.update 0 n (relationsSolutionEquiv s))) (Submodule.subtype _)\n  postcomp_desc s := by aesop\n  postcomp_injective {N _ _ f f' h} := by\n    rw [Submodule.linearMap_eq_iff_of_span_eq_top _ _ (ιMulti_span R n M)]\n    rintro ⟨_, ⟨f, rfl⟩⟩\n    exact Module.Relations.Solution.congr_var h f\n\nend presentation\n\n/-- The standard presentation of the `R`-module `⋀[R]^n M`. -/\n@[simps! G R relation var]\nnoncomputable def presentation : Module.Presentation R (⋀[R]^n M) :=\n  .ofIsPresentation (presentation.isPresentationCore R n M).isPresentation\n\nvariable {R M n}\n\n/-- Two linear maps on `⋀[R]^n M` that agree on the image of `exteriorPower.ιMulti`\nare equal. -/\n@[ext]\nlemma linearMap_ext {f : ⋀[R]^n M →ₗ[R] N} {g : ⋀[R]^n M →ₗ[R] N}\n    (heq : f.compAlternatingMap (ιMulti R n) = g.compAlternatingMap (ιMulti R n)) : f = g :=\n  (presentation R n M).postcomp_injective (by ext f; apply DFunLike.congr_fun heq)\n\n/-- The linear equivalence between `n`-fold alternating maps from `M` to `N` and linear maps from\n`⋀[R]^n M` to `N`: this is the universal property of the `n`th exterior power of `M`. -/\nnoncomputable def alternatingMapLinearEquiv : (M [⋀^Fin n]→ₗ[R] N) ≃ₗ[R] ⋀[R]^n M →ₗ[R] N :=\n  LinearEquiv.symm\n    (Equiv.toLinearEquiv\n      ((presentation R n M).linearMapEquiv.trans presentation.relationsSolutionEquiv)\n      { map_add := fun _ _ => rfl\n        map_smul := fun _ _ => rfl })\n\n@[simp]\nlemma alternatingMapLinearEquiv_comp_ιMulti (f : M [⋀^Fin n]→ₗ[R] N) :\n    (alternatingMapLinearEquiv f).compAlternatingMap (ιMulti R n) = f := by\n  obtain ⟨φ, rfl⟩ := alternatingMapLinearEquiv.symm.surjective f\n  dsimp [alternatingMapLinearEquiv]\n  simp only [LinearEquiv.symm_apply_apply]\n  rfl\n\n@[simp]\nlemma alternatingMapLinearEquiv_apply_ιMulti (f : M [⋀^Fin n]→ₗ[R] N) (a : Fin n → M) :\n    alternatingMapLinearEquiv f (ιMulti R n a) = f a :=\n  DFunLike.congr_fun (alternatingMapLinearEquiv_comp_ιMulti f) a\n\n@[simp]\nlemma alternatingMapLinearEquiv_symm_apply (F : ⋀[R]^n M →ₗ[R] N) (m : Fin n → M) :\n    alternatingMapLinearEquiv.symm F m = F.compAlternatingMap (ιMulti R n) m := by\n  obtain ⟨f, rfl⟩ := alternatingMapLinearEquiv.surjective F\n  simp only [LinearEquiv.symm_apply_apply, alternatingMapLinearEquiv_comp_ιMulti]\n\n@[simp]\nlemma alternatingMapLinearEquiv_ιMulti :\n    alternatingMapLinearEquiv (ιMulti R n (M := M)) = LinearMap.id := by\n  ext\n  simp only [alternatingMapLinearEquiv_comp_ιMulti, ιMulti_apply_coe,\n    LinearMap.compAlternatingMap_apply, LinearMap.id_coe, id_eq]\n\n/-- If `f` is an alternating map from `M` to `N`,\n`alternatingMapLinearEquiv f` is the corresponding linear map from `⋀[R]^n M` to `N`,\nand if `g` is a linear map from `N` to `N'`, then\nthe alternating map `g.compAlternatingMap f` from `M` to `N'` corresponds to the linear\nmap `g.comp (alternatingMapLinearEquiv f)` on `⋀[R]^n M`. -/\n\nTarget:\nlemma alternatingMapLinearEquiv_comp (g : N →ₗ[R] N') (f : M [⋀^Fin n]→ₗ[R] N) :\n    alternatingMapLinearEquiv (g.compAlternatingMap f) = g.comp (alternatingMapLinearEquiv f) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_b5eed30bb368","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"fe0c2b73be9cc65a08142380ca28f736ea8ee6eac8890f031adc91dd21ffbabc","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/ExteriorPower","family_id":"alternatingmaplinearequiv_comp","file_id":"mathlib/Mathlib/LinearAlgebra/ExteriorPower/Basic.lean","sample_id":"b5eed30bb36819b24fa9a164628ee2a2c0338152109146fc3f9ca3ebec531d4e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4967cf04180cb0748d25ecdf64a662a7f6a1d6bdd998e61ce6f3d7d2c6f082a5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"214074a4258bfc13547146011a31a2753af760415fa3e6aa45c6a208c1192fbd","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"756685a57c1c40f2d410134b5fc2580aacd3ec04b425aa2b56ef0fc77bf58bf8","source_sha256":"39dfdad0e4eff3ed2f02c59604c6ff92849183f69003baf79902622224068963","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← map_add]\n  exact m.toSyn_monotone le_self_add","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.181818},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"8639c8abe673a0bcacd37e4e68a5bd2786c424da66d8dd4de038b3dae33a22b5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finsupp.Lex\npublic import Mathlib.Data.Finsupp.WellFounded\npublic import Mathlib.Data.List.TFAE\n\nNamespace:\nMonomialOrder\n\nLocal context:\n/-\nCopyright (c) 2024 Antoine Chambert-Loir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir\n-/\n/-! # Monomial orders\n\n## Monomial orders\n\nA *monomial order* is well ordering relation on a type of the form `σ →₀ ℕ` which\nis compatible with addition and for which `0` is the smallest element.\nSince several monomial orders may have to be used simultaneously, one cannot\nget them as instances.\n\nIn this formalization, they are presented as a structure `MonomialOrder` which encapsulates\n`MonomialOrder.toSyn`, an additive and monotone isomorphism to a linearly ordered cancellative\nadditive commutative monoid.\nThe entry `MonomialOrder.wf` asserts that `MonomialOrder.syn` is well founded.\n\nThe terminology comes from commutative algebra and algebraic geometry, especially Gröbner bases,\nwhere `c : σ →₀ ℕ` are exponents of monomials.\n\nGiven a monomial order `m : MonomialOrder σ`, we provide the notation\n`c ≼[m] d` and `c ≺[m] d` to compare `c d : σ →₀ ℕ` with respect to `m`.\nIt is activated using `open scoped MonomialOrder`.\n\n## Examples\n\nCommutative algebra defines many monomial orders, with different usefulness ranges.\nIn this file, we provide the basic example of lexicographic ordering.\nFor the graded lexicographic ordering, see `Mathlib/Data/Finsupp/MonomialOrder/DegLex.lean`\n\n* `MonomialOrder.lex` : the lexicographic ordering on `σ →₀ ℕ`.\n  For this, `σ` needs to be embedded with an ordering relation which satisfies `WellFoundedGT σ`.\n  (This last property is automatic when `σ` is finite).\n\nThe type synonym is `Lex (σ →₀ ℕ)` and the two lemmas `MonomialOrder.lex_le_iff`\nand `MonomialOrder.lex_lt_iff` rewrite the ordering as comparisons in the type `Lex (σ →₀ ℕ)`.\n\n## References\n\n* [Cox, Little and O'Shea, *Ideals, varieties, and algorithms*][coxlittleoshea1997]\n* [Becker and Weispfenning, *Gröbner bases*][Becker-Weispfenning1993]\n\n## Note\n\nIn algebraic geometry, when the finitely many variables are indexed by integers,\nit is customary to order them using the opposite order : `MvPolynomial.X 0 > MvPolynomial.X 1 > … `\n\n-/\n\n@[expose] public section\n\n/-- Monomial orders : equivalence of `σ →₀ ℕ` with a well-ordered type -/\nstructure MonomialOrder (σ : Type*) where\n  /-- The synonym type -/\n  syn : Type*\n  /-- `syn` is an additive commutative monoid -/\n  acm : AddCommMonoid syn := by infer_instance\n  /-- `syn` is linearly ordered -/\n  lo : LinearOrder syn := by infer_instance\n  /-- `syn` is a linearly ordered cancellative additive commutative monoid -/\n  iocam : IsOrderedCancelAddMonoid syn := by infer_instance\n  /-- the additive equivalence from `σ →₀ ℕ` to `syn` -/\n  toSyn : (σ →₀ ℕ) ≃+ syn\n  /-- `toSyn` is monotone -/\n  toSyn_monotone : Monotone toSyn\n  /-- `syn` is a well ordering -/\n  wf : WellFoundedLT syn := by infer_instance\n\nattribute [instance] MonomialOrder.acm MonomialOrder.lo MonomialOrder.iocam MonomialOrder.wf\n\nnamespace MonomialOrder\n\nvariable {σ : Type*} (m : MonomialOrder σ)\n\nTarget:\nlemma le_add_right (a b : σ →₀ ℕ) :\n    m.toSyn a ≤ m.toSyn a + m.toSyn b :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_756685a57c1c","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"ec7f85bb7233429929ffade30851ddc7543cd543ceb3fcbe6221c80ef48e51dd","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finsupp","family_id":"le_add_right","file_id":"mathlib/Mathlib/Data/Finsupp/MonomialOrder.lean","sample_id":"756685a57c1c40f2d410134b5fc2580aacd3ec04b425aa2b56ef0fc77bf58bf8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"238c36aa3d5b83b0f47d94bdb4d2f15b92abc10f86ad4c7efd31a50fddc87c83","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d594da1416489b60a7466e1d8f39b122bd55afcaa513107653ab63deea58dfe7","source_sha256":"e0af099a1dfedc907a6ae420e8f9c06d8630aad18426a643f2d72445c5f1b960","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [IsTranscendenceBasis, maximal_iff, Subtype.range_val]; rfl","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":2,"token_jaccard":0.076923,"token_length_ratio":0.153846},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"864c38833336b66d32a5b98a1a228dd20750ab13e8a5f5c257ade35911cacaca","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.CommRing\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Algebraic Independence\n\nThis file defines algebraic independence of a family of elements of an `R` algebra.\n\n## Main definitions\n\n* `AlgebraicIndependent` - `AlgebraicIndependent R x` states the family of elements `x`\n  is algebraically independent over `R`, meaning that the canonical map out of the multivariable\n  polynomial ring is injective.\n\n* `AlgebraicIndependent.aevalEquiv` - The canonical isomorphism from the polynomial ring to the\n  subalgebra generated by an algebraic independent family.\n\n* `AlgebraicIndependent.repr` - The canonical map from the subalgebra generated by an\n  algebraic independent family into the polynomial ring. It is the inverse of\n  `AlgebraicIndependent.aevalEquiv`.\n\n* `IsTranscendenceBasis R x` - a family `x` is a transcendence basis over `R` if it is a maximal\n  algebraically independent subset.\n\n## Main results\n\nWe show that algebraic independence is preserved under injective maps of the indices.\n\n## References\n\n* [Stacks: Transcendence](https://stacks.math.columbia.edu/tag/030D)\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen Function Set Subalgebra MvPolynomial Algebra\n\nvariable {ι ι' : Type*} (R : Type*) {K A A' : Type*} (x : ι → A)\nvariable [CommRing R] [CommRing A] [CommRing A'] [Algebra R A] [Algebra R A']\n\n/-- `AlgebraicIndependent R x` states the family of elements `x`\n  is algebraically independent over `R`, meaning that the canonical\n  map out of the multivariable polynomial ring is injective. -/\n@[stacks 030E \"(1)\"] def AlgebraicIndependent : Prop :=\n  Injective (MvPolynomial.aeval x : MvPolynomial ι R →ₐ[R] A)\n\n/-- `AlgebraicIndepOn R v s` states that the elements in the family `v` that are indexed by the\nelements of `s` are algebraically independent over `R`. -/\nabbrev AlgebraicIndepOn (s : Set ι) : Prop := AlgebraicIndependent R fun i : s ↦ x i\n\nvariable {R} {x}\n\ntheorem algebraicIndependent_iff :\n    AlgebraicIndependent R x ↔\n      ∀ p : MvPolynomial ι R, MvPolynomial.aeval (x : ι → A) p = 0 → p = 0 :=\n  injective_iff_map_eq_zero _\n\ntheorem AlgebraicIndependent.eq_zero_of_aeval_eq_zero (h : AlgebraicIndependent R x) :\n    ∀ p : MvPolynomial ι R, MvPolynomial.aeval (x : ι → A) p = 0 → p = 0 :=\n  algebraicIndependent_iff.1 h\n\ntheorem algebraicIndependent_iff_injective_aeval :\n    AlgebraicIndependent R x ↔ Injective (MvPolynomial.aeval x : MvPolynomial ι R →ₐ[R] A) :=\n  Iff.rfl\n\nnamespace AlgebraicIndependent\n\ntheorem of_comp (f : A →ₐ[R] A') (hfv : AlgebraicIndependent R (f ∘ x)) :\n    AlgebraicIndependent R x := by\n  have : aeval (f ∘ x) = f.comp (aeval x) := by ext; simp\n  rw [AlgebraicIndependent, this, AlgHom.coe_comp] at hfv\n  exact hfv.of_comp\n\nvariable (hx : AlgebraicIndependent R x)\ninclude hx\n\ntheorem comp (f : ι' → ι) (hf : Function.Injective f) : AlgebraicIndependent R (x ∘ f) := by\n  intro p q\n  simpa [aeval_rename, (rename_injective f hf).eq_iff] using @hx (rename f p) (rename f q)\n\ntheorem coe_range : AlgebraicIndependent R ((↑) : range x → A) := by\n  simpa using hx.comp _ (rangeSplitting_injective x)\n\nend AlgebraicIndependent\n\nopen AlgebraicIndependent\n\ntheorem algebraicIndependent_equiv (e : ι ≃ ι') {f : ι' → A} :\n    AlgebraicIndependent R (f ∘ e) ↔ AlgebraicIndependent R f :=\n  ⟨fun h => Function.comp_id f ▸ e.self_comp_symm ▸ h.comp _ e.symm.injective,\n    fun h => h.comp _ e.injective⟩\n\ntheorem algebraicIndependent_equiv' (e : ι ≃ ι') {f : ι' → A} {g : ι → A} (h : f ∘ e = g) :\n    AlgebraicIndependent R g ↔ AlgebraicIndependent R f :=\n  h ▸ algebraicIndependent_equiv e\n\ntheorem algebraicIndependent_subtype_range {ι} {f : ι → A} (hf : Injective f) :\n    AlgebraicIndependent R ((↑) : range f → A) ↔ AlgebraicIndependent R f :=\n  Iff.symm <| algebraicIndependent_equiv' (Equiv.ofInjective f hf) rfl\n\nalias ⟨AlgebraicIndependent.of_subtype_range, _⟩ := algebraicIndependent_subtype_range\n\ntheorem algebraicIndependent_image {ι} {s : Set ι} {f : ι → A} (hf : Set.InjOn f s) :\n    (AlgebraicIndependent R fun x : s => f x) ↔ AlgebraicIndependent R fun x : f '' s => (x : A) :=\n  algebraicIndependent_equiv' (Equiv.Set.imageOfInjOn _ _ hf) rfl\n\nlemma AlgebraicIndepOn.mono {s t : Set ι} (H : AlgebraicIndepOn R x t) (hst : s ⊆ t) :\n    AlgebraicIndepOn R x s := by\n  simpa [Function.comp] using! H.comp (Set.inclusion hst) (Set.inclusion_injective hst)\n\n@[simp]\nlemma AlgebraicIndepOn.univ : AlgebraicIndepOn R x .univ ↔ AlgebraicIndependent R x :=\n  algebraicIndependent_equiv (Equiv.Set.univ ι)\n\nnamespace AlgebraicIndependent\n\ntheorem mono {t s : Set A} (h : t ⊆ s)\n    (hx : AlgebraicIndependent R ((↑) : s → A)) : AlgebraicIndependent R ((↑) : t → A) :=\n  AlgebraicIndepOn.mono (x := id) hx h\n\nsection repr\n\nvariable (hx : AlgebraicIndependent R x)\ninclude hx\n\n/-- Canonical isomorphism between polynomials and the subalgebra generated by\n  algebraically independent elements. -/\n@[simps! apply_coe]\ndef aevalEquiv : MvPolynomial ι R ≃ₐ[R] Algebra.adjoin R (range x) :=\n  (AlgEquiv.ofInjective (aeval x) (algebraicIndependent_iff_injective_aeval.1 hx)).trans\n    (Subalgebra.equivOfEq _ _ (Algebra.adjoin_range_eq_range_aeval R x).symm)\n\ntheorem algebraMap_aevalEquiv (p : MvPolynomial ι R) :\n    algebraMap (Algebra.adjoin R (range x)) A (hx.aevalEquiv p) = aeval x p :=\n  rfl\n\n/-- The canonical map from the subalgebra generated by an algebraic independent family\n  into the polynomial ring. -/\ndef repr : Algebra.adjoin R (range x) →ₐ[R] MvPolynomial ι R :=\n  hx.aevalEquiv.symm\n\n@[simp]\ntheorem aeval_repr (p) : aeval x (hx.repr p) = p :=\n  Subtype.ext_iff.1 (AlgEquiv.apply_symm_apply hx.aevalEquiv p)\n\ntheorem aeval_comp_repr : (aeval x).comp hx.repr = Subalgebra.val _ :=\n  AlgHom.ext hx.aeval_repr\n\n/-- Canonical isomorphism between polynomials and the subalgebra generated by\n  algebraically independent elements.\n  `AlgebraicIndepOn` version of `AlgebraicIndependent.aevalEquiv`. -/\ndef _root_.AlgebraicIndepOn.aevalEquiv {s : Set ι} (hx : AlgebraicIndepOn R x s) :\n    MvPolynomial s R ≃ₐ[R] Algebra.adjoin R (x '' s) :=\n  (AlgebraicIndependent.aevalEquiv hx).trans\n    (Subalgebra.equivOfEq _ _ congr(Algebra.adjoin _ $(by aesop)))\n\nend repr\n\nend AlgebraicIndependent\n\nvariable (R) in\n/-- A family is a transcendence basis if it is a maximal algebraically independent subset. -/\n@[stacks 030E \"(4)\"] def IsTranscendenceBasis (x : ι → A) : Prop :=\n  AlgebraicIndependent R x ∧\n    ∀ (s : Set A) (_ : AlgebraicIndepOn R id s) (_ : range x ⊆ s), range x = s\n\nTarget:\ntheorem isTranscendenceBasis_iff_maximal {s : Set A} :\n    IsTranscendenceBasis R ((↑) : s → A) ↔ Maximal (AlgebraicIndepOn R id) s :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/AlgebraicIndependent","family_id":"istranscendencebasis_iff_maximal","file_id":"mathlib/Mathlib/RingTheory/AlgebraicIndependent/Defs.lean","sample_id":"d594da1416489b60a7466e1d8f39b122bd55afcaa513107653ab63deea58dfe7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f8d69be5788c6d89c35f089a74bce8cf26bd128b40b0aa5d2082d7e6f83ba04a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cad11ba3cb57fb0c810af6512d2fa884f4e6bc86d49a85deee94f342f0bbcf00","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"45175a44a36d53ed6e01fe3580b4a35a13cfe7dd1d0d93468e34de78ad24b899","source_sha256":"c35204fe93e22d960b9e0f1d04e2f2f5f2bf73b540a7e7fec0bfcf5bf1349499","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Int.natCast_dvd]\n  exact Int.Prime.dvd_pow hp h","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":22,"token_jaccard":0.857143,"token_length_ratio":1.466667},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"865ef7df37e4bca74f813a495d9a8243842d9e8b229c8e0cd9de36f56ae776d5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.EuclideanDomain.Basic\npublic import Mathlib.Algebra.EuclideanDomain.Int\npublic import Mathlib.Algebra.GCDMonoid.Nat\npublic import Mathlib.Data.Nat.Prime.Int\npublic import Mathlib.RingTheory.PrincipalIdealDomain\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson\n-/\n/-!\n# Divisibility over ℤ\n\nThis file collects results for the integers that use ring theory in their proofs or\ncases of ℤ being examples of structures in ring theory.\n\n## Main statements\n\n* `Int.Prime.dvd_mul'`: A prime number dividing a product in ℤ divides at least one factor.\n* `Int.exists_prime_and_dvd`: Every non-unit integer has a prime divisor.\n* `Int.prime_iff_natAbs_prime`: Primality in ℤ corresponds to primality of its absolute value in ℕ.\n* `Int.span_natAbs`: The principal ideal generated by `a.natAbs` is equal to that of `a`.\n\n## Tags\n\nprime, irreducible, integers, normalization monoid, gcd monoid, greatest common divisor\n-/\n\npublic section\n\nnamespace Int\n\n\ntheorem isCoprime_iff_nat_coprime {a b : ℤ} : IsCoprime a b ↔ Nat.Coprime a.natAbs b.natAbs := by\n  rw [isCoprime_iff_gcd_eq_one, Nat.coprime_iff_gcd_eq_one, gcd_eq_natAbs]\n\n/-- If `gcd a (m * n) ≠ 1`, then `gcd a m ≠ 1` or `gcd a n ≠ 1`. -/\ntheorem gcd_ne_one_iff_gcd_mul_right_ne_one {a : ℤ} {m n : ℕ} :\n    a.gcd (m * n) ≠ 1 ↔ a.gcd m ≠ 1 ∨ a.gcd n ≠ 1 := by\n  simp only [← isCoprime_iff_gcd_eq_one, ← not_and_or, not_iff_not, IsCoprime.mul_right_iff]\n\ntheorem sq_of_gcd_eq_one {a b c : ℤ} (h : Int.gcd a b = 1) (heq : a * b = c ^ 2) :\n    ∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 := by\n  have h' : IsUnit (GCDMonoid.gcd a b) := by\n    rw [← coe_gcd, h, Int.ofNat_one]\n    exact isUnit_one\n  obtain ⟨d, ⟨u, hu⟩⟩ := exists_associated_pow_of_mul_eq_pow h' heq\n  use d\n  rw [← hu]\n  rcases Int.units_eq_one_or u with hu' | hu' <;>\n    · rw [hu']\n      simp\n\ntheorem sq_of_isCoprime {a b c : ℤ} (h : IsCoprime a b) (heq : a * b = c ^ 2) :\n    ∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 :=\n  sq_of_gcd_eq_one (isCoprime_iff_gcd_eq_one.mp h) heq\n\ntheorem natAbs_euclideanDomain_gcd (a b : ℤ) :\n    Int.natAbs (EuclideanDomain.gcd a b) = Int.gcd a b := by\n  apply Nat.dvd_antisymm <;> rw [← Int.natCast_dvd_natCast]\n  · rw [Int.natAbs_dvd]\n    exact Int.dvd_coe_gcd (EuclideanDomain.gcd_dvd_left _ _) (EuclideanDomain.gcd_dvd_right _ _)\n  · rw [Int.dvd_natAbs]\n    exact EuclideanDomain.dvd_gcd (Int.gcd_dvd_left ..) (Int.gcd_dvd_right ..)\n\nend Int\n\ntheorem Int.Prime.dvd_mul {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ m * n) :\n    p ∣ m.natAbs ∨ p ∣ n.natAbs := by\n  rwa [← hp.dvd_mul, ← Int.natAbs_mul, ← Int.natCast_dvd]\n\ntheorem Int.Prime.dvd_mul' {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ m * n) :\n    (p : ℤ) ∣ m ∨ (p : ℤ) ∣ n := by\n  rw [Int.natCast_dvd, Int.natCast_dvd]\n  exact Int.Prime.dvd_mul hp h\n\ntheorem Int.Prime.dvd_pow {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ n ^ k) :\n    p ∣ n.natAbs := by\n  rw [Int.natCast_dvd, Int.natAbs_pow] at h\n  exact hp.dvd_of_dvd_pow h\n\nTarget:\ntheorem Int.Prime.dvd_pow' {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ n ^ k) :\n    (p : ℤ) ∣ n :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [Int.natCast_dvd]\n  exact Int.Prime.dvd_pow hp h\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Int","family_id":"int","file_id":"mathlib/Mathlib/RingTheory/Int/Basic.lean","sample_id":"45175a44a36d53ed6e01fe3580b4a35a13cfe7dd1d0d93468e34de78ad24b899"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ad3461d90bc1a2d6d0744349016f752c25c2128b46b806f78f2e03cb5879efbf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"45c618e63e4bfb258f2eb5bccb2da6bd795c57ebdbec6a654acc6ef23d1bc7ee","source_sha256":"4b95b9e160e21b716ea1133988a6954cd2d9129042925a6d4bf4809d518f0745","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have uc : (iUnion fun i => Option.elim' sᶜ u i) = univ := by\n    apply Subset.antisymm (subset_univ _)\n    · simp_rw [← compl_union_self s, Option.elim', iUnion_option]\n      apply union_subset_union_right sᶜ us\n  rcases precise_refinement (Option.elim' sᶜ u) (Option.forall.2 ⟨isOpen_compl_iff.2 hs, uo⟩)\n      uc with\n    ⟨v, vo, vc, vf, vu⟩\n  refine ⟨v ∘ some, fun i ↦ vo _, ?_, vf.comp_injective (Option.some_injective _), fun i ↦ vu _⟩\n  · simp only [iUnion_option, ← compl_subset_iff_union] at vc\n    exact Subset.trans (subset_compl_comm.1 <| vu Option.none) vc","hard_negative":false,"metrics":{"chosen_tokens":143,"rejected_tokens":3,"token_jaccard":0.044776,"token_length_ratio":0.020979},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"867a110679ce44369a2bc513f21d2996347eff46bb1abb0b61558cb88d74dbb5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Option.Basic\npublic import Mathlib.Topology.Separation.Regular\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton, Yury Kudryashov\n-/\n/-!\n# Paracompact topological spaces\n\nA topological space `X` is said to be paracompact if every open covering of `X` admits a locally\nfinite refinement.\n\nThe definition requires that each set of the new covering is a subset of one of the sets of the\ninitial covering. However, one can ensure that each open covering `s : ι → Set X` admits a *precise*\nlocally finite refinement, i.e., an open covering `t : ι → Set X` with the same index set such that\n`∀ i, t i ⊆ s i`, see lemma `precise_refinement`. We also provide a convenience lemma\n`precise_refinement_set` that deals with open coverings of a closed subset of `X` instead of the\nwhole space.\n\nWe also prove the following facts.\n\n* Every compact space is paracompact, see instance `paracompact_of_compact`.\n\n* A locally compact sigma compact Hausdorff space is paracompact, see instance\n  `paracompact_of_locallyCompact_sigmaCompact`. Moreover, we can choose a locally finite\n  refinement with sets in a given collection of filter bases of `𝓝 x`, `x : X`, see\n  `refinement_of_locallyCompact_sigmaCompact_of_nhds_basis`. For example, in a proper metric space\n  every open covering `⋃ i, s i` admits a refinement `⋃ i, Metric.ball (c i) (r i)`.\n\n* Every paracompact Hausdorff space is normal. This statement is not an instance to avoid loops in\n  the instance graph.\n\n* Every `EMetricSpace` is a paracompact space, see instance `EMetric.instParacompactSpace` in\n  `Topology/EMetricSpace/Paracompact`.\n\n## TODO\n\nProve (some of) [Michael's theorems](https://ncatlab.org/nlab/show/Michael%27s+theorem).\n\n## Tags\n\ncompact space, paracompact space, locally finite covering\n-/\n\npublic section\n\n\nopen Set Filter Function\n\nopen Filter Topology\n\nuniverse u v w\n\n/-- A topological space is called paracompact, if every open covering of this space admits a locally\nfinite refinement. We use the same universe for all types in the definition to avoid creating a\nclass like `ParacompactSpace.{u v}`. Due to lemma `precise_refinement` below, every open covering\n`s : α → Set X` indexed on `α : Type v` has a *precise* locally finite refinement, i.e., a locally\nfinite refinement `t : α → Set X` indexed on the same type such that each `∀ i, t i ⊆ s i`. -/\nclass ParacompactSpace (X : Type v) [TopologicalSpace X] : Prop where\n  /-- Every open cover of a paracompact space assumes a locally finite refinement. -/\n  locallyFinite_refinement :\n    ∀ (α : Type v) (s : α → Set X), (∀ a, IsOpen (s a)) → (⋃ a, s a = univ) →\n      ∃ (β : Type v) (t : β → Set X),\n        (∀ b, IsOpen (t b)) ∧ (⋃ b, t b = univ) ∧ LocallyFinite t ∧ ∀ b, ∃ a, t b ⊆ s a\n\nvariable {ι : Type u} {X : Type v} {Y : Type w} [TopologicalSpace X] [TopologicalSpace Y]\n\n/-- Any open cover of a paracompact space has a locally finite *precise* refinement, that is,\none indexed on the same type with each open set contained in the corresponding original one. -/\ntheorem precise_refinement [ParacompactSpace X] (u : ι → Set X) (uo : ∀ a, IsOpen (u a))\n    (uc : ⋃ i, u i = univ) : ∃ v : ι → Set X, (∀ a, IsOpen (v a)) ∧ ⋃ i, v i = univ ∧\n    LocallyFinite v ∧ ∀ a, v a ⊆ u a := by\n  -- Apply definition to `range u`, then turn existence quantifiers into functions using `choose`\n  have := ParacompactSpace.locallyFinite_refinement (range u) (fun r ↦ (r : Set X))\n    (forall_subtype_range_iff.2 uo) (by rwa [← sUnion_range, Subtype.range_coe])\n  simp only [exists_subtype_range_iff, iUnion_eq_univ_iff] at this\n  choose α t hto hXt htf ind hind using this\n  choose t_inv ht_inv using hXt\n  choose U hxU hU using htf\n  -- Send each `i` to the union of `t a` over `a ∈ ind ⁻¹' {i}`\n  refine ⟨fun i ↦ ⋃ (a : α) (_ : ind a = i), t a, ?_, ?_, ?_, ?_⟩\n  · exact fun a ↦ isOpen_iUnion fun a ↦ isOpen_iUnion fun _ ↦ hto a\n  · simp only [eq_univ_iff_forall, mem_iUnion]\n    exact fun x ↦ ⟨ind (t_inv x), _, rfl, ht_inv _⟩\n  · refine fun x ↦ ⟨U x, hxU x, ((hU x).image ind).subset ?_⟩\n    simp only [subset_def, mem_iUnion, mem_setOf_eq, Set.Nonempty, mem_inter_iff]\n    rintro i ⟨y, ⟨a, rfl, hya⟩, hyU⟩\n    exact mem_image_of_mem _ ⟨y, hya, hyU⟩\n  · simp only [subset_def, mem_iUnion]\n    rintro i x ⟨a, rfl, hxa⟩\n    exact hind _ hxa\n\n/-- In a paracompact space, every open covering of a closed set admits a locally finite refinement\nindexed by the same type. -/\n\nTarget:\ntheorem precise_refinement_set [ParacompactSpace X] {s : Set X} (hs : IsClosed s) (u : ι → Set X)\n    (uo : ∀ i, IsOpen (u i)) (us : s ⊆ ⋃ i, u i) :\n    ∃ v : ι → Set X, (∀ i, IsOpen (v i)) ∧ (s ⊆ ⋃ i, v i) ∧ LocallyFinite v ∧ ∀ i, v i ⊆ u i :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Compactness","family_id":"precise_refinement_set","file_id":"mathlib/Mathlib/Topology/Compactness/Paracompact.lean","sample_id":"45c618e63e4bfb258f2eb5bccb2da6bd795c57ebdbec6a654acc6ef23d1bc7ee"},"split":"train"}
{"binding":{"chosen_proof_sha256":"91dab1cd2e2e53fc5f8d17a5c93b18cccb419f8155621f1a414c77fee50249a1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9f0aba5e8c958a1721ba7a16d006099909f296bcaa86d09dd7b80102ebf2e9d4","source_sha256":"0d1f9dee5bc403406539e90adb439a195f88865e97423c979d76639acfd9231c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : x < 0 := lt_of_tendsto_atBot 0 hx\n  intro n\n  simpa [abs_of_neg this, lt_neg] using! lt_of_tendsto_atBot (-n) hx","hard_negative":false,"metrics":{"chosen_tokens":27,"rejected_tokens":5,"token_jaccard":0.037037,"token_length_ratio":0.185185},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"8683fd2297a82fa5a159122bd27c609e352a3b6f48f3e560b8d0eb4d3cd5434b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Ring.StandardPart\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.Order.Filter.FilterProduct\n\nNamespace:\nHyperreal\n\nLocal context:\n/-\nCopyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Abhimanyu Pallavi Sudhir, Violeta Hernández Palacios\n-/\n/-!\n# Construction of the hyperreal numbers as an ultraproduct of real sequences\n\nWe define the `Hyperreal` numbers as quotients of sequences `ℕ → ℝ` by an ultrafilter. These form\na field, and we prove some of their basic properties.\n\nNote that most of the machinery that is usually defined for the specific purpose of non-standard\nanalysis (infinitesimal and infinite elements, standard parts) has been generalized to other\nnon-archimedean fields. In particular:\n\n- `ArchimedeanClass` can be used to measure whether an element is infinitesimal (`0 < mk x`) or\n  infinite (`mk x < 0`).\n- `ArchimedeanClass.stdPart` generalizes the standard part function to a general ordered field.\n\n## Todo\n\nUse Łoś's Theorem `FirstOrder.Language.Ultraproduct.sentence_realize` to formalize the transfer\nprinciple on `Hyperreal`.\n-/\n\n@[expose] public section\n\nopen ArchimedeanClass Filter Germ Topology\n\nnoncomputable section\n\n/-- Hyperreal numbers on the ultrafilter extending the cofinite filter. -/\ndef Hyperreal : Type :=\n  Germ (hyperfilter ℕ : Filter ℕ) ℝ\nderiving Inhabited\n\nnamespace Hyperreal\n\n@[inherit_doc] notation \"ℝ*\" => Hyperreal\n\ninstance : Field ℝ* :=\n  inferInstanceAs (Field (Germ _ _))\n\ninstance : LinearOrder ℝ* :=\n  inferInstanceAs (LinearOrder (Germ _ _))\n\ninstance : IsStrictOrderedRing ℝ* :=\n  inferInstanceAs (IsStrictOrderedRing (Germ _ _))\n\n/-- Natural embedding `ℝ → ℝ*`. -/\n@[coe] def ofReal : ℝ → ℝ* := const\n\ninstance : CoeTC ℝ ℝ* := ⟨ofReal⟩\n\n@[simp, norm_cast]\ntheorem coe_eq_coe {x y : ℝ} : (x : ℝ*) = y ↔ x = y :=\n  Germ.const_inj\n\ntheorem coe_ne_coe {x y : ℝ} : (x : ℝ*) ≠ y ↔ x ≠ y :=\n  coe_eq_coe.not\n\n@[simp, norm_cast]\ntheorem coe_eq_zero {x : ℝ} : (x : ℝ*) = 0 ↔ x = 0 :=\n  coe_eq_coe\n\n@[simp, norm_cast]\ntheorem coe_eq_one {x : ℝ} : (x : ℝ*) = 1 ↔ x = 1 :=\n  coe_eq_coe\n\n@[norm_cast]\ntheorem coe_ne_zero {x : ℝ} : (x : ℝ*) ≠ 0 ↔ x ≠ 0 :=\n  coe_ne_coe\n\n@[norm_cast]\ntheorem coe_ne_one {x : ℝ} : (x : ℝ*) ≠ 1 ↔ x ≠ 1 :=\n  coe_ne_coe\n\n@[simp, norm_cast]\ntheorem coe_one : ↑(1 : ℝ) = (1 : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_zero : ↑(0 : ℝ) = (0 : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_inv (x : ℝ) : ↑x⁻¹ = (x⁻¹ : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_neg (x : ℝ) : ↑(-x) = (-x : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_add (x y : ℝ) : ↑(x + y) = (x + y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_ofNat (n : ℕ) [n.AtLeastTwo] :\n    ((ofNat(n) : ℝ) : ℝ*) = OfNat.ofNat n :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_mul (x y : ℝ) : ↑(x * y) = (x * y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_div (x y : ℝ) : ↑(x / y) = (x / y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_sub (x y : ℝ) : ↑(x - y) = (x - y : ℝ*) :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_le_coe {x y : ℝ} : (x : ℝ*) ≤ y ↔ x ≤ y :=\n  Germ.const_le_iff\n\n@[simp, norm_cast]\ntheorem coe_lt_coe {x y : ℝ} : (x : ℝ*) < y ↔ x < y :=\n  Germ.const_lt_iff\n\n@[simp, norm_cast]\ntheorem coe_nonneg {x : ℝ} : 0 ≤ (x : ℝ*) ↔ 0 ≤ x :=\n  coe_le_coe\n\n@[simp, norm_cast]\ntheorem coe_pos {x : ℝ} : 0 < (x : ℝ*) ↔ 0 < x :=\n  coe_lt_coe\n\n@[simp, norm_cast]\ntheorem coe_abs (x : ℝ) : ((|x| : ℝ) : ℝ*) = |↑x| :=\n  const_abs x\n\n@[simp, norm_cast]\ntheorem coe_max (x y : ℝ) : ((max x y : ℝ) : ℝ*) = max ↑x ↑y :=\n  Germ.const_max _ _\n\n@[simp, norm_cast]\ntheorem coe_min (x y : ℝ) : ((min x y : ℝ) : ℝ*) = min ↑x ↑y :=\n  Germ.const_min _ _\n\n/-- The canonical map `ℝ → ℝ*` as an `OrderRingHom`. -/\n@[simps]\ndef coeRingHom : ℝ →+*o ℝ* where\n  toFun x := x\n  map_zero' := rfl\n  map_one' := rfl\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n  monotone' _ _ := coe_le_coe.2\n\n@[simp]\ntheorem archimedeanClassMk_coe_nonneg (x : ℝ) : 0 ≤ mk (x : ℝ*) :=\n  mk_map_nonneg_of_archimedean coeRingHom x\n\n@[simp]\ntheorem archimdeanClassMk_coe {x : ℝ} (hx : x ≠ 0) : mk (x : ℝ*) = 0 :=\n  mk_map_of_archimedean' coeRingHom hx\n\n@[simp]\ntheorem stdPart_coe (x : ℝ) : stdPart (x : ℝ*) = x :=\n  stdPart_map_real coeRingHom x\n\n/-! ### Basic constants -/\n\n/-- Construct a hyperreal number from a sequence of real numbers. -/\ndef ofSeq (f : ℕ → ℝ) : ℝ* := (↑f : Germ (hyperfilter ℕ : Filter ℕ) ℝ)\n\ntheorem ofSeq_surjective : Function.Surjective ofSeq := Quot.exists_rep\n\ntheorem ofSeq_lt_ofSeq {f g : ℕ → ℝ} : ofSeq f < ofSeq g ↔ ∀ᶠ n in hyperfilter ℕ, f n < g n :=\n  Germ.coe_lt\n\ntheorem ofSeq_le_ofSeq {f g : ℕ → ℝ} : ofSeq f ≤ ofSeq g ↔ ∀ᶠ n in hyperfilter ℕ, f n ≤ g n :=\n  Germ.coe_le\n\n/-! #### ω -/\n\n/-- A sample infinite hyperreal ω = ⟦(0, 1, 2, 3, ⋯)⟧. -/\ndef omega : ℝ* := ofSeq Nat.cast\n\n@[inherit_doc] scoped notation \"ω\" => Hyperreal.omega\nrecommended_spelling \"omega\" for \"ω\" in [omega, «termω»]\n\ntheorem coe_lt_omega (r : ℝ) : r < ω := by\n  apply ofSeq_lt_ofSeq.2 <| Filter.Eventually.filter_mono Nat.hyperfilter_le_atTop _\n  obtain ⟨n, hn⟩ := exists_nat_gt r\n  rw [eventually_atTop]\n  exact ⟨n, fun m hm ↦ hn.trans_le (mod_cast hm)⟩\n\ntheorem omega_pos : 0 < ω :=\n  coe_lt_omega 0\n\n@[simp]\ntheorem omega_ne_zero : ω ≠ 0 :=\n  omega_pos.ne'\n\n@[simp]\ntheorem abs_omega : |ω| = ω :=\n  abs_of_pos omega_pos\n\n@[simp]\ntheorem archimedeanClassMk_omega_neg : mk ω < 0 :=\n  fun n ↦ by simpa using! coe_lt_omega n\n\n@[simp]\ntheorem stdPart_omega : stdPart ω = 0 := by\n  rw [stdPart_eq_zero]\n  exact archimedeanClassMk_omega_neg.ne\n\n/-! #### ε -/\n\n/-- A sample infinitesimal hyperreal ε = ⟦(0, 1, 1/2, 1/3, ⋯)⟧. -/\ndef epsilon : ℝ* :=\n  ofSeq fun n => n⁻¹\n\n@[inherit_doc] scoped notation \"ε\" => Hyperreal.epsilon\nrecommended_spelling \"epsilon\" for \"ε\" in [epsilon, «termε»]\n\n@[simp]\ntheorem inv_omega : ω⁻¹ = ε :=\n  rfl\n\n@[simp]\ntheorem inv_epsilon : ε⁻¹ = ω :=\n  @inv_inv _ _ ω\n\n@[simp]\ntheorem epsilon_pos : 0 < ε :=\n  inv_pos_of_pos omega_pos\n\n@[simp]\ntheorem epsilon_ne_zero : ε ≠ 0 :=\n  epsilon_pos.ne'\n\n@[simp]\ntheorem epsilon_mul_omega : ε * ω = 1 :=\n  @inv_mul_cancel₀ _ _ ω omega_ne_zero\n\n@[simp]\ntheorem archimedeanClassMk_epsilon_pos : 0 < mk ε := by\n  simp [← inv_omega]\n\n/-!\n### Some facts about `Tendsto`\n-/\n\n@[simp]\ntheorem tendsto_ofSeq {f : ℕ → ℝ} {lb : Filter ℝ} :\n    (ofSeq f).Tendsto lb ↔ Tendsto f (hyperfilter ℕ) lb :=\n  .rfl\n\ntheorem stdPart_map {x : ℝ*} {r : ℝ} {f : ℝ → ℝ} (hf : ContinuousAt f r)\n    (hxr : x.Tendsto (𝓝 r)) : (x.map f).Tendsto (𝓝 (f r)) := by\n  rcases ofSeq_surjective x with ⟨g, rfl⟩\n  exact hf.tendsto.comp hxr\n\ntheorem stdPart_map₂ {x y : ℝ*} {r s : ℝ} {f : ℝ → ℝ → ℝ}\n    (hxr : x.Tendsto (𝓝 r)) (hys : y.Tendsto (𝓝 s))\n    (hf : ContinuousAt (Function.uncurry f) (r, s)) : (x.map₂ f y).Tendsto (𝓝 (f r s)) := by\n  rcases ofSeq_surjective x with ⟨x, rfl⟩\n  rcases ofSeq_surjective y with ⟨y, rfl⟩\n  exact hf.tendsto.comp (hxr.prodMk_nhds hys)\n\ntheorem tendsto_iff_forall {x : ℝ*} {r : ℝ} :\n    x.Tendsto (𝓝 r) ↔ (∀ s < r, s ≤ x) ∧ (∀ s > r, x ≤ s) := by\n  rcases ofSeq_surjective x with ⟨f, rfl⟩\n  rw [tendsto_ofSeq, (nhds_basis_Ioo _).tendsto_right_iff]\n  simp_rw [Set.mem_Ioo, eventually_and, ← ofSeq_lt_ofSeq]\n  refine ⟨fun H ↦ ⟨fun s hs ↦ ?_, fun s hs ↦ ?_⟩, fun H ⟨s, t⟩ ⟨hs, ht⟩ ↦ ⟨?_, ?_⟩⟩\n  · obtain ⟨t, ht⟩ := exists_gt r\n    exact (H ⟨s, t⟩ ⟨hs, ht⟩).1.le\n  · obtain ⟨t, ht⟩ := exists_lt r\n    exact (H ⟨t, s⟩ ⟨ht, hs⟩).2.le\n  · obtain ⟨u, hu, hu'⟩ := exists_between hs\n    exact (coe_lt_coe.2 hu).trans_le (H.1 _ hu')\n  · obtain ⟨u, hu, hu'⟩ := exists_between ht\n    exact (H.2 _ hu).trans_lt (coe_lt_coe.2 hu')\n\ntheorem archimedeanClassMk_nonneg_of_tendsto {x : ℝ*} {r : ℝ} (hx : x.Tendsto (𝓝 r)) :\n    0 ≤ mk x := by\n  rw [tendsto_iff_forall] at hx\n  obtain ⟨s, hs⟩ := exists_lt r\n  obtain ⟨t, ht⟩ := exists_gt r\n  exact mk_nonneg_of_le_of_le_of_archimedean coeRingHom (hx.1 s hs) (hx.2 t ht)\n\ntheorem stdPart_of_tendsto {x : ℝ*} {r : ℝ} (hx : x.Tendsto (𝓝 r)) : stdPart x = r := by\n  rw [tendsto_iff_forall] at hx\n  exact stdPart_eq coeRingHom hx.1 hx.2\n\ntheorem archimedeanClassMk_pos_of_tendsto {x : ℝ*} (hx : x.Tendsto (𝓝 0)) : 0 < mk x := by\n  apply (archimedeanClassMk_nonneg_of_tendsto hx).lt_of_ne'\n  rw [← stdPart_eq_zero, stdPart_of_tendsto hx]\n\n@[simp]\ntheorem stdPart_epsilon : stdPart ε = 0 :=\n  stdPart_eq_zero.2 <| archimedeanClassMk_epsilon_pos.ne'\n\ntheorem epsilon_lt_of_pos {r : ℝ} : 0 < r → ε < r :=\n  lt_of_pos_of_archimedean coeRingHom archimedeanClassMk_epsilon_pos\n\ntheorem epsilon_lt_of_neg {r : ℝ} : r < 0 → r < ε :=\n  lt_of_neg_of_archimedean coeRingHom archimedeanClassMk_epsilon_pos\n\n@[deprecated (since := \"2026-01-05\")]\nalias epsilon_lt_pos := epsilon_lt_of_pos\n\n@[deprecated archimedeanClassMk_pos_of_tendsto (since := \"2026-01-05\")]\ntheorem lt_of_tendsto_zero_of_pos {f : ℕ → ℝ} (hf : Tendsto f atTop (𝓝 0)) :\n    ∀ {r : ℝ}, 0 < r → ofSeq f < (r : ℝ*) := fun hr ↦\n  ofSeq_lt_ofSeq.2 <| (hf.eventually <| gt_mem_nhds hr).filter_mono Nat.hyperfilter_le_atTop\n\n@[deprecated archimedeanClassMk_pos_of_tendsto (since := \"2026-01-05\")]\ntheorem neg_lt_of_tendsto_zero_of_pos {f : ℕ → ℝ} (hf : Tendsto f atTop (𝓝 0)) :\n    ∀ {r : ℝ}, 0 < r → (-r : ℝ*) < ofSeq f := fun hr =>\n  have hg := hf.neg\n  neg_lt_of_neg_lt (by rw [neg_zero] at hg; exact lt_of_tendsto_zero_of_pos hg hr)\n\n@[deprecated archimedeanClassMk_pos_of_tendsto (since := \"2026-01-05\")]\ntheorem gt_of_tendsto_zero_of_neg {f : ℕ → ℝ} (hf : Tendsto f atTop (𝓝 0)) :\n    ∀ {r : ℝ}, r < 0 → (r : ℝ*) < ofSeq f := fun {r} hr => by\n  rw [← neg_neg r, coe_neg]; exact neg_lt_of_tendsto_zero_of_pos hf (neg_pos.mpr hr)\n\ntheorem lt_of_tendsto_atTop {x : ℝ*} (r : ℝ) (hx : x.Tendsto atTop) : r < x := by\n  rcases ofSeq_surjective x with ⟨f, rfl⟩\n  rw [tendsto_ofSeq] at hx\n  exact ofSeq_lt_ofSeq.2 <| hx.eventually_mem (Ioi_mem_atTop r)\n\ntheorem lt_of_tendsto_atBot {x : ℝ*} (r : ℝ) (hx : x.Tendsto atBot) : x < r := by\n  rcases ofSeq_surjective x with ⟨f, rfl⟩\n  rw [tendsto_ofSeq] at hx\n  exact ofSeq_lt_ofSeq.2 <| hx.eventually_mem (Iio_mem_atBot r)\n\ntheorem archimedeanClassMk_neg_of_tendsto_atTop {x : ℝ*} (hx : x.Tendsto atTop) : mk x < 0 := by\n  have : 0 < x := lt_of_tendsto_atTop 0 hx\n  intro n\n  simpa [abs_of_pos this] using! lt_of_tendsto_atTop n hx\n\nTarget:\ntheorem archimedeanClassMk_neg_of_tendsto_atBot {x : ℝ*} (hx : x.Tendsto atBot) : mk x < 0 :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Real","family_id":"archimedeanclassmk_neg_of_tendsto_atbot","file_id":"mathlib/Mathlib/Analysis/Real/Hyperreal.lean","sample_id":"9f0aba5e8c958a1721ba7a16d006099909f296bcaa86d09dd7b80102ebf2e9d4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1791f6bfaef40a4ae8b1dcdf06c489870e6b8a0fe97f09d60381a63c54a541cd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9b3f45e527412ab937dbda0aad67341cd01387028790afa9d9ec5b4d5e084b9c","source_sha256":"171194abb27453412c000e21c620bfe335cc5467bbff820dbe99451d3255de0a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [irreducibleComponents_eq_singleton, Set.mem_singleton_iff] at hZ\n  simp [irreducibleComponentOpen, irreducibleComponents_eq_singleton, hZ]","hard_negative":false,"metrics":{"chosen_tokens":19,"rejected_tokens":2,"token_jaccard":0.071429,"token_length_ratio":0.105263},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"86b7c1647e0109de7c408a53b5a8250876a1ef731a5a8c2f6e78be93135cef8c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion\npublic import Mathlib.AlgebraicGeometry.Noetherian\n\nNamespace:\nAlgebraicGeometry.Scheme\n\nLocal context:\n/-\nCopyright (c) 2026 Thomas Browning, Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Thomas Browning, Andrew Yang\n-/\n/-!\n# Subscheme structure on an irreducible component\n\nWe define the subscheme structure on an irreducible component of a Noetherian scheme. Typically,\none takes the reduced induced subscheme structure, but this will throw away information if the\nirreducible component is not already reduced. Instead, we take the closed subscheme defined by\nthe kernel of the restriction to the complement of the union of the other irreducible components.\nFor example, if `X` is irreducible then this will give back the original scheme `X`.\n\n## Main definition\n* `AlgebraicGeometry.Scheme.irreducibleComponentIdeal`: The ideal sheaf data associated to an\n  irreducible component of a Noetherian scheme.\n* `AlgebraicGeometry.Scheme.irreducibleComponent`: The subscheme structure on an irreducible\n  component of a Noetherian scheme.\n\n## TODO\n\nProve that for affine schemes this subscheme structure is defined by the kernel of the\nlocalization away from the union of the other minimal prime ideals.\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nnamespace AlgebraicGeometry.Scheme\n\nvariable (X : Scheme.{u}) (Z : Set X) (hZ : Z ∈ irreducibleComponents X) [IsNoetherian X]\n\n/-- The complement of the irreducible components unequal to `Z` of a Noetherian scheme. -/\ndef irreducibleComponentOpen : Opens X :=\n  ⟨(⋃₀ (irreducibleComponents X \\ {Z}))ᶜ, by\n    rw [Set.sUnion_eq_biUnion, isOpen_compl_iff]\n    exact TopologicalSpace.NoetherianSpace.finite_irreducibleComponents.sdiff.isClosed_biUnion\n      fun W hW ↦ isClosed_of_mem_irreducibleComponents W hW.1⟩\n\n/-- The ideal sheaf data associated to an irreducible component of a Noetherian scheme. -/\ndef irreducibleComponentIdeal : X.IdealSheafData where\n  __ := (irreducibleComponentOpen X Z).ι.ker\n  supportSet := Z\n  supportSet_eq_iInter_zeroLocus := by\n    rw [← IdealSheafData.coe_support_eq_eq_iInter_zeroLocus, Hom.support_ker, Opens.range_ι]\n    exact (closure_sUnion_irreducibleComponents_sdiff_singleton\n      TopologicalSpace.NoetherianSpace.finite_irreducibleComponents Z hZ).symm\n\ntheorem irreducibleComponentIdeal_def :\n    irreducibleComponentIdeal X Z hZ = (irreducibleComponentOpen X Z).ι.ker := by\n  ext\n  rfl\n\n/-- The subscheme structure on an irreducible component of a Noetherian scheme. -/\nnoncomputable def irreducibleComponent : Scheme :=\n  (X.irreducibleComponentIdeal Z hZ).subscheme\n\n/-- The inclusion from an irreducible component of a Noetherian scheme. -/\nnoncomputable def irreducibleComponentι : X.irreducibleComponent Z hZ ⟶ X :=\n  (X.irreducibleComponentIdeal Z hZ).subschemeι\n\nlemma irreducibleComponentι_apply (x : X.irreducibleComponent Z hZ) :\n    X.irreducibleComponentι Z hZ x = x.1 :=\n  rfl\n\ninstance : IsClosedImmersion (X.irreducibleComponentι Z hZ) :=\n  inferInstanceAs (IsClosedImmersion (X.irreducibleComponentIdeal Z hZ).subschemeι)\n\ninstance : IrreducibleSpace (X.irreducibleComponent Z hZ) :=\n  Subtype.irreducibleSpace hZ.1\n\ninclude hZ in\n\nTarget:\ntheorem irreducibleComponentOpen_eq_top [IrreducibleSpace X] :\n    irreducibleComponentOpen X Z = ⊤ :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/IdealSheaf","family_id":"irreduciblecomponentopen_eq_top","file_id":"mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/IrreducibleComponent.lean","sample_id":"9b3f45e527412ab937dbda0aad67341cd01387028790afa9d9ec5b4d5e084b9c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3fc07c2c4acb95dfe246db6ffb6277493987fabf1c2b630f887c08fb23242d47","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"879654c433cff299cfe89ef259733d4899e296266c7758cbde162533660dcf70","source_sha256":"d6accbe9b37eb545d619e89f4675da5baa0eb79327269932bbf60850434ef205","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.toShHom\n    g.toShHom)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]\n  erw [← TopCat.coe_comp]\n  rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]\n  dsimp only [SheafedSpace.forget]\n  rw [imageBasicOpen_image_preimage]\n  exact (imageBasicOpen f g U s).2","hard_negative":false,"metrics":{"chosen_tokens":75,"rejected_tokens":2,"token_jaccard":0.027027,"token_length_ratio":0.026667},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"86d035f7ed3d1bfdbfd379070140b91720be9d2ff1ba1d017ca01a4b5cad7ced","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Geometry.RingedSpace.LocallyRingedSpace\npublic import Mathlib.Algebra.Category.Ring.Constructions\npublic import Mathlib.Geometry.RingedSpace.OpenImmersion\npublic import Mathlib.CategoryTheory.Limits.Types.Coequalizers\npublic import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers\n\nNamespace:\nAlgebraicGeometry.LocallyRingedSpace.HasCoequalizer\n\nLocal context:\n/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Colimits of LocallyRingedSpace\n\nWe construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.\nIt then follows that `LocallyRingedSpace` has all colimits, and\n`forgetToSheafedSpace` preserves them.\n\n-/\n\n@[expose] public section\n\n\nnamespace AlgebraicGeometry\n\nuniverse w' w v u\n\nopen CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace\n\nattribute [local instance] Opposite.small\n\nnamespace SheafedSpace\n\nvariable {C : Type u} [Category.{v} C]\nvariable {J : Type w} [Category.{w'} J] [Small.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)\n\ntheorem isColimit_exists_rep [HasLimitsOfShape Jᵒᵖ C] {c : Cocone F} (hc : IsColimit c) (x : c.pt) :\n    ∃ (i : J) (y : F.obj i), (c.ι.app i).hom.base y = x :=\n  Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x\n\n-- Porting note: argument `C` of colimit need to be made explicit, otherwise we get universe issues\ntheorem colimit_exists_rep [HasLimitsOfShape Jᵒᵖ C] (x : colimit (C := SheafedSpace C) F) :\n    ∃ (i : J) (y : F.obj i), (colimit.ι F i).hom.base y = x :=\n  Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)\n    (isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x\n\nset_option backward.isDefEq.respectTransparency false in\ninstance [HasLimits C] {X Y : SheafedSpace C} (f g : X ⟶ Y) :\n    Epi (coequalizer.π f g).hom.base := by\n  rw [← show _ = (coequalizer.π f g).hom.base from\n      ι_comp_coequalizerComparison f g (SheafedSpace.forget C),\n      ← PreservesCoequalizer.iso_hom]\n  apply epi_comp\n\nend SheafedSpace\n\nnamespace LocallyRingedSpace\n\nsection HasCoproducts\n\nvariable {ι : Type v} [Small.{u} ι] (F : Discrete ι ⥤ LocallyRingedSpace.{u})\n\n/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/\nnoncomputable def coproduct : LocallyRingedSpace where\n  toSheafedSpace := colimit (C := SheafedSpace.{u + 1, u, u} CommRingCat.{u})\n    (F ⋙ forgetToSheafedSpace)\n  isLocalRing x := by\n    obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x\n    haveI : IsLocalRing (((F ⋙ forgetToSheafedSpace).obj i).presheaf.stalk y) :=\n      (F.obj i).isLocalRing _\n    exact\n      (asIso ((colimit.ι (C := SheafedSpace.{u + 1, u, u} CommRingCat.{u})\n        (F ⋙ forgetToSheafedSpace) i :).hom.stalkMap y)).symm.commRingCatIsoToRingEquiv.isLocalRing\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/\nnoncomputable def coproductCofan : Cocone F where\n  pt := coproduct F\n  ι :=\n    { app j := LocallyRingedSpace.homMk (colimit.ι (F ⋙ forgetToSheafedSpace) j)\n      naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; simp }\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The explicit coproduct cofan constructed in `coproductCofan` is indeed a colimit. -/\nnoncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where\n  desc s :=\n    LocallyRingedSpace.homMk (colimit.desc\n      (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) (by\n        intro x\n        obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x\n        have := PresheafedSpace.stalkMap.comp\n          (colimit.ι (F ⋙ forgetToSheafedSpace) i).hom\n          (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)).hom y\n        simp only [← IsIso.comp_inv_eq,\n          ← InducedCategory.comp_hom,\n          PresheafedSpace.stalkMap.congr_hom _ _\n            (congr_arg (InducedCategory.Hom.hom) (colimit.ι_desc\n              (forgetToSheafedSpace.mapCocone s) i))] at this\n        rw [← this]\n        dsimp\n        infer_instance)\n  fac _ _ :=\n    LocallyRingedSpace.forgetToSheafedSpace.map_injective\n     (colimit.ι_desc (C := SheafedSpace _) _ _)\n  uniq s f h :=\n    LocallyRingedSpace.forgetToSheafedSpace.map_injective\n      (IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.toShHom fun j =>\n        congr_arg LocallyRingedSpace.Hom.toShHom (h j))\n\ninstance : HasColimitsOfShape (Discrete ι) LocallyRingedSpace.{u} :=\n  ⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩\n\nnoncomputable instance : PreservesColimitsOfShape (Discrete.{v} ι) forgetToSheafedSpace.{u} :=\n  ⟨fun {G} =>\n    preservesColimit_of_preserves_colimit_cocone (coproductCofanIsColimit G)\n      ((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCat.{u}) _).ofIsoColimit\n        (Cocone.ext (Iso.refl _) fun _ => Category.comp_id _))⟩\n\nend HasCoproducts\n\nsection HasCoequalizer\n\nvariable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)\n\nnamespace HasCoequalizer\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[instance]\ntheorem coequalizer_π_app_isLocalHom\n    (U : TopologicalSpace.Opens (coequalizer f.toShHom g.toShHom).carrier) :\n    IsLocalHom ((coequalizer.π f.toShHom g.toShHom :).hom.c.app (op U)).hom := by\n  have := ι_comp_coequalizerComparison f.toShHom g.toShHom SheafedSpace.forgetToPresheafedSpace\n  dsimp at this\n  rw [← PreservesCoequalizer.iso_hom] at this\n  rw [← this, PresheafedSpace.comp_c_app,\n    ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]\n  -- Porting note (https://github.com/leanprover-community/mathlib4/issues/10754): this instance has to be manually added\n  haveI : IsIso (PreservesCoequalizer.iso\n      SheafedSpace.forgetToPresheafedSpace f.toShHom g.toShHom).hom.c :=\n    inferInstance\n  apply +allowSynthFailures RingHom.isLocalHom_comp\n  · apply +allowSynthFailures RingHom.isLocalHom_comp\n    · apply CommRingCat.equalizer_ι_isLocalHom'\n    · apply isLocalHom_of_isIso\n  · apply isLocalHom_of_isIso\n\n/-!\nWe roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms\nof locally ringed spaces, we want to show that the stalk map of\n`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that\n`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of\nlocally ringed space.\n\nGiven a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show\nthat `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the\nrestriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the\nbasic open set of `π⋆x`.\n\nSince `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have\n`π ⁻¹' π '' V = V` (as the underlying set map is merely the set-theoretic coequalizer).\nThis shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`\nare local ring homs.\n-/\n\n\nvariable (U : Opens (coequalizer f.toShHom g.toShHom).carrier)\nvariable (s : (coequalizer f.toShHom g.toShHom).presheaf.obj (op U))\n\n/-- (Implementation). The basic open set of the section `π꙳ s`. -/\nnoncomputable def imageBasicOpen : Opens Y :=\n  Y.toRingedSpace.basicOpen\n    (show Y.presheaf.obj (op (unop _)) from\n      ((coequalizer.π f.toShHom g.toShHom).hom.c.app (op U)) s)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\ntheorem imageBasicOpen_image_preimage :\n    (coequalizer.π f.toShHom g.toShHom).hom.base ⁻¹'\n      ((coequalizer.π f.toShHom g.toShHom).hom.base ''\n        (imageBasicOpen f g U s).1) = (imageBasicOpen f g U s).1 := by\n  fapply Types.coequalizer_preimage_image_eq_of_preimage_eq (↾f.base)\n    (↾g.base) (↾(coequalizer.π f.toShHom g.toShHom).hom.base)\n  · ext\n    simp only [TypeCat.Fun.toFun_apply, comp_apply, ConcreteCategory.hom_ofHom,\n      TypeCat.Fun.coe_mk, ← TopCat.comp_app, ← PresheafedSpace.comp_base]\n    congr 3\n    exact SheafedSpace.forgetToPresheafedSpace.congr_map\n      (coequalizer.condition f.toShHom g.toShHom)\n  · exact isColimitCoforkMapOfIsColimit (forget TopCat) _\n      (isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)\n      _ (coequalizerIsCoequalizer f.toShHom g.toShHom))\n  · suffices\n      (TopologicalSpace.Opens.map f.base).obj (imageBasicOpen f g U s) =\n        (TopologicalSpace.Opens.map g.base).obj (imageBasicOpen f g U s)\n      by injection this\n    delta imageBasicOpen\n    rw [preimage_basicOpen f, preimage_basicOpen g]\n    dsimp\n    rw [← ConcreteCategory.comp_apply, ← PresheafedSpace.comp_c_app,\n      ← CommRingCat.comp_apply, ← PresheafedSpace.comp_c_app]\n    -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11224): change `rw` to `erw`\n    erw [SheafedSpace.congr_hom_app (coequalizer.condition f.toShHom g.toShHom),\n      CommRingCat.comp_apply, X.toRingedSpace.basicOpen_res]\n    apply inf_eq_right.mpr\n    refine (RingedSpace.basicOpen_le _ _).trans ?_\n    rw [coequalizer.condition f.toShHom g.toShHom]\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem imageBasicOpen_image_open :\n    IsOpen ((coequalizer.π f.toShHom g.toShHom).hom.base '' (imageBasicOpen f g U s).1) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/RingedSpace","family_id":"imagebasicopen_image_open","file_id":"mathlib/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean","sample_id":"879654c433cff299cfe89ef259733d4899e296266c7758cbde162533660dcf70"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5b8c55401ccfdd0326db1a35194aa3affc02ada3be7e2075a756a2a2bba77cb4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fad9f1ea8d6654621705b02879012779787dbbf5d7900dc540f712fce41505c9","source_sha256":"51daf98e481b660f2839034f65b84ec1148fc7fc2de859367b6b6e85ce12e57c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isCoskeletal_iff]\n  exact isRightKanExtension_iff_isIso ((coskAdj n).unit.app X)\n    ((coskAdj n).counit.app _) (𝟙 _) ((coskAdj n).left_triangle_components X)","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":2,"token_jaccard":0.05,"token_length_ratio":0.047619},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"86f84706a1673fda3d7710499e69ff590ca87fb180d35386e48ccc93cb68dcf9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.SimplicialObject.Basic\npublic import Mathlib.CategoryTheory.Functor.KanExtension.Adjunction\npublic import Mathlib.CategoryTheory.Functor.KanExtension.Basic\n\nNamespace:\nCategoryTheory.SimplicialObject\n\nLocal context:\n/-\nCopyright (c) 2024 Emily Riehl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Emily Riehl, Joël Riou\n-/\n/-!\n# Coskeletal simplicial objects\n\nThe identity natural transformation exhibits a simplicial object `X` as a right extension of its\nrestriction along `(Truncated.inclusion n).op` recorded by `rightExtensionInclusion X n`.\n\nThe simplicial object `X` is *n-coskeletal* if `rightExtensionInclusion X n` is a right Kan\nextension.\n\nWhen the ambient category admits right Kan extensions along `(Truncated.inclusion n).op`,\nthen when `X` is `n`-coskeletal, the unit of `coskAdj n` defines an isomorphism:\n`isoCoskOfIsCoskeletal : X ≅ (cosk n).obj X`.\n\nTODO: Prove that `X` is `n`-coskeletal whenever a certain canonical cone is a limit cone.\n-/\n\n@[expose] public section\n\nopen Opposite\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits CategoryTheory.Functor SimplexCategory\n\nuniverse v u v' u'\n\nnamespace CategoryTheory\n\nnamespace SimplicialObject\nvariable {C : Type u} [Category.{v} C]\nvariable (X : SimplicialObject C) (n : ℕ)\n\nnamespace Truncated\n\n/-- The identity natural transformation exhibits a simplicial set as a right extension of its\nrestriction along `(Truncated.inclusion n).op`. -/\n@[simps!]\ndef rightExtensionInclusion :\n    RightExtension (Truncated.inclusion n).op\n      ((Truncated.inclusion n).op ⋙ X) := RightExtension.mk _ (𝟙 _)\n\nend Truncated\n\nopen Truncated\n\n/-- A simplicial object `X` is `n`-coskeletal when it is the right Kan extension of its restriction\nalong `(Truncated.inclusion n).op` via the identity natural transformation. -/\n@[mk_iff]\nclass IsCoskeletal : Prop where\n  isRightKanExtension : IsRightKanExtension X (𝟙 ((Truncated.inclusion n).op ⋙ X))\n\nattribute [instance] IsCoskeletal.isRightKanExtension\n\nsection\n\nvariable [∀ (F : (SimplexCategory.Truncated n)ᵒᵖ ⥤ C),\n    (SimplexCategory.Truncated.inclusion n).op.HasRightKanExtension F]\n\n/-- If `X` is `n`-coskeletal, then `Truncated.rightExtensionInclusion X n` is a terminal object in\nthe category `RightExtension (Truncated.inclusion n).op (Truncated.inclusion.op ⋙ X)`. -/\nnoncomputable def IsCoskeletal.isUniversalOfIsRightKanExtension [X.IsCoskeletal n] :\n    (rightExtensionInclusion X n).IsUniversal := by\n  apply Functor.isUniversalOfIsRightKanExtension\n\nTarget:\ntheorem isCoskeletal_iff_isIso : X.IsCoskeletal n ↔ IsIso ((coskAdj n).unit.app X) :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplicialObject","family_id":"iscoskeletal_iff_isiso","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplicialObject/Coskeletal.lean","sample_id":"fad9f1ea8d6654621705b02879012779787dbbf5d7900dc540f712fce41505c9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5b4476fdc1c0c0f51888b9b0f5b1dfa03fd37018bf6998a42fadc18efa75285e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"eee74c135583607225b77544c2b5eec1a05c13b644ed38b7c5521ccbd22a5bdf","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3681767d9aba4e16656d0211cddaff6efe2e454378f2ebc6b91e0ff3858dab3f","source_sha256":"7e0891f362b30145c7c3c4e0ec7d7143a1d3e97fd040572f98d49bd047596de1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [quasicategory_iff_hasLiftingProperty.{u} _ terminalIsTerminal, innerFibration_iff]\n  exact ⟨fun h _ _ _ ⟨i, h0, hn⟩ ↦ h h0 hn,\n    fun h _ _ h0 hn ↦ h _ (horn_ι_mem_innerHornInclusions h0 hn)⟩","hard_negative":true,"metrics":{"chosen_tokens":49,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.061224},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"8734960e11c18e84e901f8ead8eaac0301bd01aef6200970255c5419ec97ef97","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.Quasicategory.Basic\n\nNamespace:\nSSet\n\nLocal context:\n/-\nCopyright (c) 2026 Jack McKoen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jack McKoen\n-/\n/-!\n# Inner fibrations\n\nInner fibrations of simplicial sets are the morphisms in `SSet` which have the right lifting\nproperty with respect to all inner horn inclusions.\n\nBasic consequences of inner fibrations with respect to the definition of quasi-categories are\nformalized.\n\n-/\n\npublic section\n\nopen CategoryTheory MorphismProperty Simplicial Limits\n\nuniverse u\n\nnamespace SSet\n\n/-- The family of morphisms in `SSet` which consists of inner horn inclusions\n`Λ[n, i].ι : Λ[n, i] ⟶ Δ[n]` (for `0 < i < n`). -/\ninductive innerHornInclusions : MorphismProperty SSet.{u} where\n  | intro {n : ℕ} (i : Fin (n + 3)) (h0 : 0 < i) (hn : i < Fin.last (n + 2)) :\n    innerHornInclusions Λ[n + 2, i].ι\n\nlemma horn_ι_mem_innerHornInclusions {n : ℕ} {i : Fin (n + 1)}\n    (h0 : 0 < i) (hn : i < Fin.last n) : innerHornInclusions (horn.{u} n i).ι := by\n  obtain _ | _ | k := n\n  · grind\n  · grind\n  · exact ⟨i, h0, hn⟩\n\nlemma innerHornInclusions_eq_iSup :\n    innerHornInclusions.{u} =\n    ⨆ n, .ofHoms (fun p : {p : Fin (n + 3) // 0 < p ∧ p < Fin.last (n + 2)} ↦ Λ[n + 2, p].ι) := by\n  ext\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · obtain @⟨n, i, h0, hn⟩ := h\n    simp only [iSup_iff, ofHoms_iff, Subtype.exists, exists_prop]\n    use n, i\n  · simp only [iSup_iff, ofHoms_iff] at h\n    obtain ⟨n, ⟨i, h0, hn⟩, _, _⟩ := h\n    exact horn_ι_mem_innerHornInclusions h0 hn\n\nlemma innerHornInclusions_le_J : innerHornInclusions.{u} ≤ modelCategoryQuillen.J :=\n  fun _ _ _ ⟨_, _, _⟩ ↦ modelCategoryQuillen.horn_ι_mem_J ..\n\nlemma innerHornInclusions_le_monomorphisms :\n    innerHornInclusions.{u} ≤ monomorphisms SSet :=\n  innerHornInclusions_le_J.trans modelCategoryQuillen.J_le_monomorphisms\n\n/-- The inner fibrations are the morphisms which have the right lifting property\nwith respect to inner horn inclusions. -/\n@[expose, kerodon 01BA]\ndef innerFibrations : MorphismProperty SSet.{u} := innerHornInclusions.rlp\nderiving IsMultiplicative, RespectsIso, IsStableUnderBaseChange,\n  IsStableUnderRetracts\n\n/-- A morphism `q` satisfies `[InnerFibration q]` if it belongs to `innerFibrations`. -/\n@[mk_iff]\nclass InnerFibration {X Y : SSet} (q : X ⟶ Y) : Prop where\n  mem : innerFibrations q\n\nlemma mem_innerFibrations {X Y : SSet} (q : X ⟶ Y) [InnerFibration q] : innerFibrations q :=\n  InnerFibration.mem\n\nTarget:\nlemma quasicategory_iff_innerFibration (X : SSet.{u}) :\n    Quasicategory X ↔ InnerFibration (terminal.from X) :=\n\nProof body:\n","rejected":"by\n  exact quasicategory_iff_innerFibration","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"660abf679bef5d3627f03de3886ddb25eb20fb82b18df0e3e679a54bfa950ee8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/Quasicategory","family_id":"quasicategory_iff_innerfibration","file_id":"mathlib/Mathlib/AlgebraicTopology/Quasicategory/InnerFibration.lean","sample_id":"3681767d9aba4e16656d0211cddaff6efe2e454378f2ebc6b91e0ff3858dab3f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fa7ef57b668670c0961ba6cf31ace864279e791239ac9d2e2fff89a3656780bf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f4231e8dda39dee1bbddb1e4ffb21d16d698adcaa0cf76876d45ddbd2a042b9f","source_sha256":"9edb2b00301b8c3a46b08790b6da8897248a760fa9e13f7ddfca795725728403","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":2,"token_jaccard":0.071429,"token_length_ratio":0.111111},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"873e10cdbc89dae4fa9f90d08aa80a42d79fac437736f6bdb0b7aeb24be1f4f3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Products.Basic\n\nNamespace:\nCategoryTheory.Bifunctor\n\nLocal context:\n/-\nCopyright (c) 2017 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Kim Morrison\n-/\n/-!\n# Lemmas about functors out of product categories.\n-/\n\npublic section\n\n\nopen CategoryTheory\n\nnamespace CategoryTheory.Bifunctor\n\nuniverse v₁ v₂ v₃ u₁ u₂ u₃\n\nvariable {C : Type u₁} {D : Type u₂} {E : Type u₃}\nvariable [Category.{v₁} C] [Category.{v₂} D] [Category.{v₃} E]\n\nopen scoped Prod\n\n@[simp]\ntheorem map_id (F : C × D ⥤ E) (X : C) (Y : D) :\n    F.map ((𝟙 X) ×ₘ (𝟙 Y)) = 𝟙 (F.obj (X, Y)) :=\n  F.map_id (X, Y)\n\n@[simp]\ntheorem map_id_comp (F : C × D ⥤ E) (W : C) {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z) :\n    F.map (𝟙 W ×ₘ (f ≫ g)) = F.map (𝟙 W ×ₘ f) ≫ F.map (𝟙 W ×ₘ g) := by\n  rw [← Functor.map_comp, prod_comp, Category.comp_id]\n\n@[simp]\ntheorem map_comp_id (F : C × D ⥤ E) (X Y Z : C) (W : D) (f : X ⟶ Y) (g : Y ⟶ Z) :\n    F.map ((f ≫ g) ×ₘ 𝟙 W) = F.map (f ×ₘ 𝟙 W) ≫ F.map (g ×ₘ 𝟙 W) := by\n  rw [← Functor.map_comp, prod_comp, Category.comp_id]\n\n@[simp]\ntheorem diagonal (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :\n    F.map (𝟙 X ×ₘ g) ≫ F.map (f ×ₘ 𝟙 Y') = F.map (f ×ₘ g) := by\n  rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]\n\n@[simp]\n\nTarget:\ntheorem diagonal' (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :\n    F.map (f ×ₘ 𝟙 Y) ≫ F.map (𝟙 X' ×ₘ g) = F.map (f ×ₘ g) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Products","family_id":"diagonal","file_id":"mathlib/Mathlib/CategoryTheory/Products/Bifunctor.lean","sample_id":"f4231e8dda39dee1bbddb1e4ffb21d16d698adcaa0cf76876d45ddbd2a042b9f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"73a3ff17197caa13c9881ebe96a98ed8cb7dce96f5c0218437447afab1f809f0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5f156dd91316fb432ab42faa62bdffc2ee44cc5bb43426e7854c11a200a754f4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2d8dea88dd355420151a905a1257de8aac829b897e12eb600a797e8418da7629","source_sha256":"20bfccc248cbbadfa2532d19d3f23df6035aaa8042662f551c3dac73c46bbeb2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine Finset.measurable_sup' _ ?_\n  simpa [Finset.mem_range]","hard_negative":true,"metrics":{"chosen_tokens":14,"rejected_tokens":5,"token_jaccard":0.230769,"token_length_ratio":0.357143},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"873f14f3f7947175f8af1db0fe704907bb19351becb93343e19007a26929a973","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Measure.AEMeasurable\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n-/\n/-!\n# Typeclasses for measurability of lattice operations\n\nIn this file we define classes `MeasurableSup` and `MeasurableInf` and prove dot-style\nlemmas (`Measurable.sup`, `AEMeasurable.sup` etc). For binary operations we define two typeclasses:\n\n- `MeasurableSup` says that both left and right sup are measurable;\n- `MeasurableSup₂` says that `fun p : α × α => p.1 ⊔ p.2` is measurable,\n\nand similarly for other binary operations. The reason for introducing these classes is that in case\nof topological space `α` equipped with the Borel `σ`-algebra, instances for `MeasurableSup₂`\netc. require `α` to have a second countable topology.\n\nFor instances relating, e.g., `ContinuousSup` to `MeasurableSup` see file\n`MeasureTheory.BorelSpace`.\n\n## Tags\n\nmeasurable function, lattice operation\n\n-/\n\npublic section\n\n\nopen MeasureTheory\n\n/-- We say that a type has `MeasurableSup` if `(c ⊔ ·)` and `(· ⊔ c)` are measurable functions.\nFor a typeclass assuming measurability of `uncurry (· ⊔ ·)` see `MeasurableSup₂`. -/\nclass MeasurableSup (M : Type*) [MeasurableSpace M] [Max M] : Prop where\n  measurable_const_sup : ∀ c : M, Measurable (c ⊔ ·) := by intro c; fun_prop\n  measurable_sup_const : ∀ c : M, Measurable (· ⊔ c) := by intro c; fun_prop\n\n/-- We say that a type has `MeasurableSup₂` if `uncurry (· ⊔ ·)` is a measurable functions.\nFor a typeclass assuming measurability of `(c ⊔ ·)` and `(· ⊔ c)` see `MeasurableSup`. -/\nclass MeasurableSup₂ (M : Type*) [MeasurableSpace M] [Max M] : Prop where\n  measurable_sup : Measurable fun p : M × M => p.1 ⊔ p.2 := by intro p; fun_prop\n\nexport MeasurableSup₂ (measurable_sup)\n\nexport MeasurableSup (measurable_const_sup measurable_sup_const)\n\n/-- We say that a type has `MeasurableInf` if `(c ⊓ ·)` and `(· ⊓ c)` are measurable functions.\nFor a typeclass assuming measurability of `uncurry (· ⊓ ·)` see `MeasurableInf₂`. -/\nclass MeasurableInf (M : Type*) [MeasurableSpace M] [Min M] : Prop where\n  measurable_const_inf : ∀ c : M, Measurable (c ⊓ ·) := by intro c; fun_prop\n  measurable_inf_const : ∀ c : M, Measurable (· ⊓ c) := by intro c; fun_prop\n\n/-- We say that a type has `MeasurableInf₂` if `uncurry (· ⊓ ·)` is a measurable functions.\nFor a typeclass assuming measurability of `(c ⊓ ·)` and `(· ⊓ c)` see `MeasurableInf`. -/\nclass MeasurableInf₂ (M : Type*) [MeasurableSpace M] [Min M] : Prop where\n  measurable_inf : Measurable fun p : M × M => p.1 ⊓ p.2 := by intro p; fun_prop\n\nexport MeasurableInf₂ (measurable_inf)\n\nexport MeasurableInf (measurable_const_inf measurable_inf_const)\n\nvariable {M : Type*} [MeasurableSpace M]\n\nsection OrderDual\n\ninstance (priority := 100) OrderDual.instMeasurableSup [Min M] [MeasurableInf M] :\n    MeasurableSup Mᵒᵈ :=\n  ⟨@measurable_const_inf M _ _ _, @measurable_inf_const M _ _ _⟩\n\ninstance (priority := 100) OrderDual.instMeasurableInf [Max M] [MeasurableSup M] :\n    MeasurableInf Mᵒᵈ :=\n  ⟨@measurable_const_sup M _ _ _, @measurable_sup_const M _ _ _⟩\n\ninstance (priority := 100) OrderDual.instMeasurableSup₂ [Min M] [MeasurableInf₂ M] :\n    MeasurableSup₂ Mᵒᵈ :=\n  ⟨@measurable_inf M _ _ _⟩\n\ninstance (priority := 100) OrderDual.instMeasurableInf₂ [Max M] [MeasurableSup₂ M] :\n    MeasurableInf₂ Mᵒᵈ :=\n  ⟨@measurable_sup M _ _ _⟩\n\nend OrderDual\n\nvariable {α : Type*} {m : MeasurableSpace α} {μ : Measure α} {f g : α → M}\n\nsection Sup\n\nvariable [Max M]\n\nsection MeasurableSup\n\nvariable [MeasurableSup M]\n\n@[fun_prop]\ntheorem Measurable.const_sup (hf : Measurable f) (c : M) : Measurable fun x => c ⊔ f x :=\n  (measurable_const_sup c).comp hf\n\n@[fun_prop]\ntheorem AEMeasurable.const_sup (hf : AEMeasurable f μ) (c : M) :\n    AEMeasurable (fun x => c ⊔ f x) μ :=\n  (MeasurableSup.measurable_const_sup c).comp_aemeasurable hf\n\n@[fun_prop]\ntheorem Measurable.sup_const (hf : Measurable f) (c : M) : Measurable fun x => f x ⊔ c :=\n  (measurable_sup_const c).comp hf\n\n@[fun_prop]\ntheorem AEMeasurable.sup_const (hf : AEMeasurable f μ) (c : M) :\n    AEMeasurable (fun x => f x ⊔ c) μ :=\n  (measurable_sup_const c).comp_aemeasurable hf\n\nend MeasurableSup\n\nsection MeasurableSup₂\n\nvariable [MeasurableSup₂ M]\n\n@[fun_prop]\ntheorem Measurable.sup' (hf : Measurable f) (hg : Measurable g) : Measurable (f ⊔ g) :=\n  measurable_sup.comp (hf.prodMk hg)\n\n@[fun_prop]\ntheorem Measurable.sup (hf : Measurable f) (hg : Measurable g) : Measurable fun a => f a ⊔ g a :=\n  measurable_sup.comp (hf.prodMk hg)\n\n@[fun_prop]\ntheorem AEMeasurable.sup' (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :\n    AEMeasurable (f ⊔ g) μ :=\n  measurable_sup.comp_aemeasurable (hf.prodMk hg)\n\n@[fun_prop]\ntheorem AEMeasurable.sup (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :\n    AEMeasurable (fun a => f a ⊔ g a) μ :=\n  measurable_sup.comp_aemeasurable (hf.prodMk hg)\n\ninstance (priority := 100) MeasurableSup₂.toMeasurableSup : MeasurableSup M where\n\nend MeasurableSup₂\n\nend Sup\n\nsection Inf\n\nvariable [Min M]\n\nsection MeasurableInf\n\nvariable [MeasurableInf M]\n\n@[fun_prop]\ntheorem Measurable.const_inf (hf : Measurable f) (c : M) : Measurable fun x => c ⊓ f x :=\n  (measurable_const_inf c).comp hf\n\n@[fun_prop]\ntheorem AEMeasurable.const_inf (hf : AEMeasurable f μ) (c : M) :\n    AEMeasurable (fun x => c ⊓ f x) μ :=\n  (MeasurableInf.measurable_const_inf c).comp_aemeasurable hf\n\n@[fun_prop]\ntheorem Measurable.inf_const (hf : Measurable f) (c : M) : Measurable fun x => f x ⊓ c :=\n  (measurable_inf_const c).comp hf\n\n@[fun_prop]\ntheorem AEMeasurable.inf_const (hf : AEMeasurable f μ) (c : M) :\n    AEMeasurable (fun x => f x ⊓ c) μ :=\n  (measurable_inf_const c).comp_aemeasurable hf\n\nend MeasurableInf\n\nsection MeasurableInf₂\n\nvariable [MeasurableInf₂ M]\n\n@[fun_prop]\ntheorem Measurable.inf' (hf : Measurable f) (hg : Measurable g) : Measurable (f ⊓ g) :=\n  measurable_inf.comp (hf.prodMk hg)\n\n@[fun_prop]\ntheorem Measurable.inf (hf : Measurable f) (hg : Measurable g) : Measurable fun a => f a ⊓ g a :=\n  measurable_inf.comp (hf.prodMk hg)\n\n@[fun_prop]\ntheorem AEMeasurable.inf' (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :\n    AEMeasurable (f ⊓ g) μ :=\n  measurable_inf.comp_aemeasurable (hf.prodMk hg)\n\n@[fun_prop]\ntheorem AEMeasurable.inf (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :\n    AEMeasurable (fun a => f a ⊓ g a) μ :=\n  measurable_inf.comp_aemeasurable (hf.prodMk hg)\n\ninstance (priority := 100) MeasurableInf₂.to_hasMeasurableInf : MeasurableInf M where\n\nend MeasurableInf₂\n\nend Inf\n\nsection SemilatticeSup\n\nopen Finset\n\nvariable {δ : Type*} [MeasurableSpace δ] [SemilatticeSup α] [MeasurableSup₂ α]\n\n@[fun_prop]\ntheorem Finset.measurable_sup' {ι : Type*} {s : Finset ι} (hs : s.Nonempty) {f : ι → δ → α}\n    (hf : ∀ n ∈ s, Measurable (f n)) : Measurable (s.sup' hs f) :=\n  Finset.sup'_induction hs _ (fun _f hf _g hg => hf.sup hg) fun n hn => hf n hn\n\n@[fun_prop]\n\nTarget:\ntheorem Finset.measurable_range_sup' {f : ℕ → δ → α} {n : ℕ} (hf : ∀ k ≤ n, Measurable (f k)) :\n    Measurable ((range (n + 1)).sup' nonempty_range_add_one f) :=\n\nProof body:\n","rejected":"by\n  exact Finset.measurable_range_sup","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"8dd5f00dd49941eface0a084621b7a30d227ef1957b3e3a428fa2a857c379b66","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Order","family_id":"finset","file_id":"mathlib/Mathlib/MeasureTheory/Order/Lattice.lean","sample_id":"2d8dea88dd355420151a905a1257de8aac829b897e12eb600a797e8418da7629"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6ef68f198f9c3ddb8b2e403677d5bebc09f01883ade66189972cc6ad03aa6c3a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3b80855cb929e86418fc523ac9ee1b0eb5805373b26bbeb6d26681f147c0be66","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5cac71067938ac4bef77108bfac10db01fab6c38979ad732ebaa5982c97b9231","source_sha256":"148d7eba9bc7c4474f48190ef655e8ff7d2748109cb095b07744bfb109f40298","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let f := (lmul R S).toLinearMap.prodMap (lmul R T).toLinearMap\n  have : (lmul R (S × T)).toLinearMap = (prodMapLinear R S T S T R).comp f :=\n    LinearMap.ext₂ Prod.mul_def\n  simp_rw [trace, this]\n  exact trace_prodMap' _ _","hard_negative":true,"metrics":{"chosen_tokens":64,"rejected_tokens":5,"token_jaccard":0.085714,"token_length_ratio":0.078125},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"875627d90e3c53e50284bae13a69a6d892700a2436ccefeb02a6a4739805a54c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas\npublic import Mathlib.LinearAlgebra.Matrix.BilinearForm\npublic import Mathlib.LinearAlgebra.Trace\n\nNamespace:\nAlgebra\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Trace for (finite) ring extensions.\n\nSuppose we have an `R`-algebra `S` with a finite basis. For each `s : S`,\nthe trace of the linear map given by multiplying by `s` gives information about\nthe roots of the minimal polynomial of `s` over `R`.\n\n## Main definitions\n\n* `Algebra.trace R S x`: the trace of an element `s` of an `R`-algebra `S`\n* `Algebra.traceForm R S`: bilinear form sending `x`, `y` to the trace of `x * y`\n* `Algebra.traceMatrix R b`: the matrix whose `(i j)`-th element is the trace of `b i * b j`.\n\n## Main results\n\n* `trace_algebraMap_of_basis`, `trace_algebraMap`: if `x : K`, then `Tr_{L/K} x = [L : K] x`\n* `trace_trace_of_basis`, `trace_trace`: `Tr_{L/K} (Tr_{F/L} x) = Tr_{F/K} x`\n\n## Implementation notes\n\nTypically, the trace is defined specifically for finite field extensions.\nThe definition is as general as possible and the assumption that the extension is finite\nis added to the lemmas as needed.\n\nWe only define the trace for left multiplication (`Algebra.leftMulMatrix`,\ni.e. `LinearMap.mulLeft`).\nFor now, the definitions assume `S` is commutative, so the choice doesn't matter anyway.\n\n## References\n\n* https://en.wikipedia.org/wiki/Field_trace\n\n-/\n\n@[expose] public section\n\n\nuniverse w\n\nvariable {R S T : Type*} [CommRing R] [CommRing S] [CommRing T]\nvariable [Algebra R S] [Algebra R T]\nvariable {ι : Type w} [Fintype ι]\n\nopen Module\n\nopen LinearMap (BilinForm)\nopen LinearMap\n\nopen Matrix\n\nopen scoped Matrix\n\nnamespace Algebra\n\nvariable (R S)\n\n/-- The trace of an element `s` of an `R`-algebra is the trace of `(s * ·)`,\nas an `R`-linear map. -/\n@[stacks 0BIF \"Trace\"]\nnoncomputable def trace : S →ₗ[R] R :=\n  (LinearMap.trace R S).comp (lmul R S).toLinearMap\n\nvariable {S}\n\n-- Not a `simp` lemma since there are more interesting ways to rewrite `trace R S x`,\n-- for example `trace_trace`\ntheorem trace_apply (x) : trace R S x = LinearMap.trace R S (lmul R S x) :=\n  rfl\n\ntheorem trace_eq_zero_of_not_exists_basis (h : ¬∃ s : Finset S, Nonempty (Basis s R S)) :\n    trace R S = 0 := by ext s; simp [trace_apply, LinearMap.trace, h]\n\nvariable {R}\n\n-- Can't be a `simp` lemma because it depends on a choice of basis\ntheorem trace_eq_matrix_trace [DecidableEq ι] (b : Basis ι R S) (s : S) :\n    trace R S s = Matrix.trace (Algebra.leftMulMatrix b s) := by\n  rw [trace_apply, LinearMap.trace_eq_matrix_trace _ b, ← toMatrix_lmul_eq]; rfl\n\n/-- If `x` is in the base field `K`, then the trace is `[L : K] * x`. -/\ntheorem trace_algebraMap_of_basis (b : Basis ι R S) (x : R) :\n    trace R S (algebraMap R S x) = Fintype.card ι • x := by\n  haveI := Classical.decEq ι\n  rw [trace_apply, LinearMap.trace_eq_matrix_trace R b, Matrix.trace]\n  convert! Finset.sum_const x\n  simp [-coe_lmul_eq_mul]\n\n\n/-- The trace map from `R` to itself is the identity map. -/\n@[simp] theorem trace_self : trace R R = LinearMap.id := by\n  ext; simpa using trace_algebraMap_of_basis (.singleton (Fin 1) R) 1\n\ntheorem trace_self_apply (a) : trace R R a = a := by simp\n\n/-- If `x` is in the base field `K`, then the trace is `[L : K] * x`.\n\n(If `L` is not finite-dimensional over `K`, then `trace` and `finrank` return `0`.)\n-/\n@[simp]\ntheorem trace_algebraMap [StrongRankCondition R] [Module.Free R S] (x : R) :\n    trace R S (algebraMap R S x) = finrank R S • x := by\n  by_cases H : ∃ s : Finset S, Nonempty (Basis s R S)\n  · rw [trace_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some]\n  · simp [trace_eq_zero_of_not_exists_basis R H, finrank_eq_zero_of_not_exists_basis_finset H]\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem trace_trace_of_basis [Algebra S T] [IsScalarTower R S T] {ι κ : Type*} [Finite ι]\n    [Finite κ] (b : Basis ι R S) (c : Basis κ S T) (x : T) :\n    trace R S (trace S T x) = trace R T x := by\n  haveI := Classical.decEq ι\n  haveI := Classical.decEq κ\n  cases nonempty_fintype ι\n  cases nonempty_fintype κ\n  rw [trace_eq_matrix_trace (b.smulTower c), trace_eq_matrix_trace b, trace_eq_matrix_trace c,\n    Matrix.trace, Matrix.trace, Matrix.trace, ← Finset.univ_product_univ, Finset.sum_product]\n  refine Finset.sum_congr rfl fun i _ ↦ ?_\n  simp only [map_sum, smulTower_leftMulMatrix, Finset.sum_apply, Matrix.diag,\n    Finset.sum_apply i (Finset.univ : Finset κ) fun y => leftMulMatrix b (leftMulMatrix c x y y)]\n\ntheorem trace_comp_trace_of_basis [Algebra S T] [IsScalarTower R S T] {ι κ : Type*} [Finite ι]\n    [Finite κ] (b : Basis ι R S) (c : Basis κ S T) :\n    (trace R S).comp ((trace S T).restrictScalars R) = trace R T := by\n  ext\n  rw [LinearMap.comp_apply, LinearMap.restrictScalars_apply, trace_trace_of_basis b c]\n\n@[simp]\ntheorem trace_trace [Algebra S T] [IsScalarTower R S T]\n    [Module.Free R S] [Module.Finite R S] [Module.Free S T] [Module.Finite S T] (x : T) :\n    trace R S (trace S T x) = trace R T x :=\n  trace_trace_of_basis (Module.Free.chooseBasis R S) (Module.Free.chooseBasis S T) x\n\n/-- Let `T / S / R` be a tower of finite extensions of fields. Then\n$\\text{Trace}_{T/R} = \\text{Trace}_{S/R} \\circ \\text{Trace}_{T/S}$. -/\n@[simp, stacks 0BIJ \"Trace\"]\ntheorem trace_comp_trace [Algebra S T] [IsScalarTower R S T]\n    [Module.Free R S] [Module.Finite R S] [Module.Free S T] [Module.Finite S T] :\n    (trace R S).comp ((trace S T).restrictScalars R) = trace R T :=\n  LinearMap.ext trace_trace\n\n@[simp]\n\nTarget:\ntheorem trace_prod_apply [Module.Free R S] [Module.Free R T] [Module.Finite R S] [Module.Finite R T]\n    (x : S × T) : trace R (S × T) x = trace R S x.fst + trace R T x.snd :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_5cac71067938","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"8e8679bdeed7ea13ff2a32ba7670c633a64d5bad144fbb9581b064a7ff435d6e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Trace","family_id":"trace_prod_apply","file_id":"mathlib/Mathlib/RingTheory/Trace/Defs.lean","sample_id":"5cac71067938ac4bef77108bfac10db01fab6c38979ad732ebaa5982c97b9231"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bcf496a337f71ad6a5e3548ca780931dad5f52b6127e92076ca6f453d0ce3736","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a4baaccefd765a94402a3364d09d4e0620026c78c0048a0db42c437c896fe73","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"444126766a6d3867417667c40dd37f00d68870fb70c1031516cd452dd27d2227","source_sha256":"3d16eac88c0b4f07fc5014be8b3e025e599c272f2809936f3c6e3dad4757fa05","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine le_trans (measure_iUnion_le _) (le_trans\n    (ENNReal.tsum_le_tsum <| notConvergentSeqLTIndex_spec (half_pos hε) hf hsm hs hfg) ?_)\n  simp_rw [ENNReal.ofReal_mul (half_pos hε).le]\n  rw [ENNReal.tsum_mul_left, ← ENNReal.ofReal_tsum_of_nonneg, inv_eq_one_div, tsum_geometric_two,\n    ← ENNReal.ofReal_mul (half_pos hε).le, div_mul_cancel₀ ε two_ne_zero]\n  · intro n; positivity\n  · rw [inv_eq_one_div]\n    exact summable_geometric_two","hard_negative":true,"metrics":{"chosen_tokens":86,"rejected_tokens":3,"token_jaccard":0.045455,"token_length_ratio":0.034884},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"875e4c8fc4e9bde14b4fa45ddc1b7519936c4af9f6fdb067008dcd04e3d384cb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic\n\nNamespace:\nMeasureTheory.Egorov\n\nLocal context:\n/-\nCopyright (c) 2022 Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kexing Ying\n-/\n/-!\n# Egorov theorem\n\nThis file contains the Egorov theorem which states that an almost everywhere convergent\nsequence on a finite measure space converges uniformly except on an arbitrarily small set.\nThis theorem is useful for the Vitali convergence theorem as well as theorems regarding\nconvergence in measure.\n\n## Main results\n\n* `MeasureTheory.tendstoUniformlyOn_of_ae_tendsto`: Egorov's theorem which shows that a sequence of\n  almost everywhere convergent functions converges uniformly except on an arbitrarily small set.\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen MeasureTheory NNReal ENNReal Topology\n\nnamespace MeasureTheory\n\nopen Set Filter TopologicalSpace\n\nvariable {α β ι : Type*} {m : MeasurableSpace α} [PseudoEMetricSpace β] {μ : Measure α}\n\nnamespace Egorov\n\n/-- Given a sequence of functions `f` and a function `g`, `notConvergentSeq f g n j` is the\nset of elements such that `f k x` and `g x` are separated by at least `1 / (n + 1)` for some\n`k ≥ j`.\n\nThis definition is useful for Egorov's theorem. -/\ndef notConvergentSeq [Preorder ι] (f : ι → α → β) (g : α → β) (n : ℕ) (j : ι) : Set α :=\n  ⋃ (k) (_ : j ≤ k), { x | (n : ℝ≥0∞)⁻¹ < edist (f k x) (g x) }\n\nvariable {n : ℕ} {j : ι} {s : Set α} {ε : ℝ} {f : ι → α → β} {g : α → β}\n\ntheorem mem_notConvergentSeq_iff [Preorder ι] {x : α} :\n    x ∈ notConvergentSeq f g n j ↔ ∃ k ≥ j, (n : ℝ≥0∞)⁻¹ < edist (f k x) (g x) := by\n  simp_rw [notConvergentSeq, Set.mem_iUnion, exists_prop, mem_setOf]\n\ntheorem notConvergentSeq_antitone [Preorder ι] : Antitone (notConvergentSeq f g n) :=\n  fun _ _ hjk => Set.iUnion₂_mono' fun l hl => ⟨l, le_trans hjk hl, Set.Subset.rfl⟩\n\ntheorem measure_inter_notConvergentSeq_eq_zero [SemilatticeSup ι] [Nonempty ι]\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :\n    μ (s ∩ ⋂ j, notConvergentSeq f g n j) = 0 := by\n  simp_rw [EMetric.tendsto_atTop, ae_iff] at hfg\n  rw [← nonpos_iff_eq_zero, ← hfg]\n  refine measure_mono fun x => ?_\n  simp only [Set.mem_inter_iff, Set.mem_iInter, mem_notConvergentSeq_iff]\n  push Not\n  rintro ⟨hmem, hx⟩\n  refine ⟨hmem, (n : ℝ≥0∞)⁻¹, by simp, fun N => ?_⟩\n  obtain ⟨n, hn₁, hn₂⟩ := hx N\n  exact ⟨n, hn₁, hn₂.le⟩\n\ntheorem notConvergentSeq_measurableSet [Preorder ι] [Countable ι]\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a))) :\n    MeasurableSet (notConvergentSeq f g n j) :=\n  MeasurableSet.iUnion fun k ↦ MeasurableSet.iUnion fun _ ↦\n      measurableSet_lt measurable_const <| hf k\n\ntheorem measure_notConvergentSeq_tendsto_zero [SemilatticeSup ι] [Countable ι]\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a))) (hsm : MeasurableSet s)\n    (hs : μ s ≠ ∞) (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :\n    Tendsto (fun j => μ (s ∩ notConvergentSeq f g n j)) atTop (𝓝 0) := by\n  rcases isEmpty_or_nonempty ι with h | h\n  · have : (fun j => μ (s ∩ notConvergentSeq f g n j)) = fun j => 0 := by\n      simp only [eq_iff_true_of_subsingleton]\n    rw [this]\n    exact tendsto_const_nhds\n  rw [← measure_inter_notConvergentSeq_eq_zero hfg n, Set.inter_iInter]\n  refine tendsto_measure_iInter_atTop\n    (fun n ↦ (hsm.inter <| notConvergentSeq_measurableSet hf).nullMeasurableSet)\n    (fun k l hkl => Set.inter_subset_inter_right _ <| notConvergentSeq_antitone hkl)\n    ⟨h.some, ne_top_of_le_ne_top hs (measure_mono Set.inter_subset_left)⟩\n\nvariable [SemilatticeSup ι] [Nonempty ι] [Countable ι]\n\ntheorem exists_notConvergentSeq_lt (hε : 0 < ε)\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a)))\n    (hsm : MeasurableSet s) (hs : μ s ≠ ∞)\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :\n    ∃ j : ι, μ (s ∩ notConvergentSeq f g n j) ≤ ENNReal.ofReal (ε * 2⁻¹ ^ n) := by\n  have ⟨N, hN⟩ := (ENNReal.tendsto_atTop ENNReal.zero_ne_top).1\n    (measure_notConvergentSeq_tendsto_zero hf hsm hs hfg n) (.ofReal (ε * 2⁻¹ ^ n))\n      (by positivity)\n  rw [zero_add] at hN\n  exact ⟨N, (hN N le_rfl).2⟩\n\n/-- Given some `ε > 0`, `notConvergentSeqLTIndex` provides the index such that\n`notConvergentSeq` (intersected with a set of finite measure) has measure less than\n`ε * 2⁻¹ ^ n`.\n\nThis definition is useful for Egorov's theorem. -/\ndef notConvergentSeqLTIndex (hε : 0 < ε)\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a)))\n    (hsm : MeasurableSet s) (hs : μ s ≠ ∞)\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) : ι :=\n  Classical.choose <| exists_notConvergentSeq_lt hε hf hsm hs hfg n\n\ntheorem notConvergentSeqLTIndex_spec (hε : 0 < ε)\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a)))\n    (hsm : MeasurableSet s) (hs : μ s ≠ ∞)\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :\n    μ (s ∩ notConvergentSeq f g n (notConvergentSeqLTIndex hε hf hsm hs hfg n)) ≤\n      ENNReal.ofReal (ε * 2⁻¹ ^ n) :=\n  Classical.choose_spec <| exists_notConvergentSeq_lt hε hf hsm hs hfg n\n\n/-- Given some `ε > 0`, `iUnionNotConvergentSeq` is the union of `notConvergentSeq` with\nspecific indices such that `iUnionNotConvergentSeq` has measure less equal than `ε`.\n\nThis definition is useful for Egorov's theorem. -/\ndef iUnionNotConvergentSeq (hε : 0 < ε)\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a)))\n    (hsm : MeasurableSet s) (hs : μ s ≠ ∞)\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) : Set α :=\n  ⋃ n, s ∩ notConvergentSeq f g n (notConvergentSeqLTIndex (half_pos hε) hf hsm hs hfg n)\n\ntheorem iUnionNotConvergentSeq_measurableSet (hε : 0 < ε)\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a)))\n    (hsm : MeasurableSet s) (hs : μ s ≠ ∞)\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) :\n    MeasurableSet <| iUnionNotConvergentSeq hε hf hsm hs hfg :=\n  MeasurableSet.iUnion fun _ ↦ hsm.inter <| notConvergentSeq_measurableSet hf\n\nTarget:\ntheorem measure_iUnionNotConvergentSeq (hε : 0 < ε)\n    (hf : ∀ n, Measurable (fun a ↦ edist (f n a) (g a)))\n    (hsm : MeasurableSet s) (hs : μ s ≠ ∞)\n    (hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) :\n    μ (iUnionNotConvergentSeq hε hf hsm hs hfg) ≤ ENNReal.ofReal ε :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_444126766a6d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"042283deed19845832ef4975582f82762a28e3551c82a6b4088f02f15fb26016","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Function","family_id":"measure_iunionnotconvergentseq","file_id":"mathlib/Mathlib/MeasureTheory/Function/Egorov.lean","sample_id":"444126766a6d3867417667c40dd37f00d68870fb70c1031516cd452dd27d2227"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a2aaf585a4a9909c497f3f34f264608d1efd5c7508490b5e6cf98acb7351c7d2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e0c5330d2514224d882fef475861bb09dfae3c5924cf8a420d81d8d903a10d54","source_sha256":"6558239b343add6f929fcdae9db74c785e40b87bcf52cfd3889e1691cf3f4d62","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  ext x'\n  simp_rw [LieIdeal.mem_killingCompl, LieModule.traceForm_comm]\n  refine ⟨fun hx' ↦ ?_, fun hx y _ ↦ hx ▸ LinearMap.map_zero₂ (killingForm R L') y⟩\n  suffices e.symm x' ∈ LinearMap.ker (killingForm R L) by\n    rw [IsKilling.ker_killingForm_eq_bot] at this\n    simpa [map_zero] using (e : L ≃ₗ[R] L').congr_arg this\n  ext y\n  replace hx' : ∀ y', killingForm R L' x' y' = 0 := by simpa using hx'\n  specialize hx' (e y)\n  rwa [← e.apply_symm_apply x', killingForm_of_equiv_apply] at hx'\n\nalias _root_.LieEquiv.isKilling := LieAlgebra.isKilling_of_equiv","hard_negative":false,"metrics":{"chosen_tokens":130,"rejected_tokens":3,"token_jaccard":0.029851,"token_length_ratio":0.023077},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"8798cd3cd389a2a9689d5042e6452c921a933fa3f97e7e4f093d31d91b7ea500","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Lie.InvariantForm\npublic import Mathlib.Algebra.Lie.Semisimple.Basic\npublic import Mathlib.Algebra.Lie.TraceForm\n\nNamespace:\nLieAlgebra\n\nLocal context:\n/-\nCopyright (c) 2023 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Lie algebras with non-degenerate Killing forms.\n\nIn characteristic zero, the following three conditions are equivalent:\n 1. The solvable radical of a Lie algebra is trivial\n 2. A Lie algebra is a direct sum of its simple ideals\n 3. A Lie algebra has non-degenerate Killing form\n\nIn positive characteristic, it is still true that 3 implies 2, and that 2 implies 1, but there are\ncounterexamples to the remaining implications. Thus condition 3 is the strongest assumption.\nFurthermore, much of the Cartan-Killing classification of semisimple Lie algebras in characteristic\nzero, continues to hold in positive characteristic (over a perfect field) if the Lie algebra has a\nnon-degenerate Killing form.\n\nThis file contains basic definitions and results for such Lie algebras.\n\n## Main declarations\n\n* `LieAlgebra.IsKilling`: a typeclass encoding the fact that a Lie algebra has a non-singular\n  Killing form.\n* `LieAlgebra.IsKilling.instSemisimple`: if a finite-dimensional Lie algebra over a field\n  has non-singular Killing form then it is semisimple.\n* `LieAlgebra.IsKilling.instHasTrivialRadical`: if a Lie algebra over a PID\n  has non-singular Killing form then it has trivial radical.\n* `LieIdeal.isCompl_killingCompl`: if a Lie algebra has non-singular Killing form then for all\n  ideals, an ideal and its Killing orthogonal complement are complements.\n\n-/\n\npublic section\n\nvariable (R K L : Type*) [CommRing R] [Field K] [LieRing L] [LieAlgebra R L] [LieAlgebra K L]\n\nnamespace LieAlgebra\n\n/-- We say a Lie algebra is Killing if its Killing form is non-singular.\n\nNB: This is not standard terminology (the literature does not seem to name Lie algebras with this\nproperty). -/\nclass IsKilling : Prop where\n  /-- We say a Lie algebra is Killing if its Killing form is non-singular. -/\n  killingCompl_top_eq_bot : LieIdeal.killingCompl R L ⊤ = ⊥\n\nattribute [simp] IsKilling.killingCompl_top_eq_bot\n\nnamespace IsKilling\n\nvariable [IsKilling R L]\n\n@[simp] lemma ker_killingForm_eq_bot :\n    LinearMap.ker (killingForm R L) = ⊥ := by\n  simp [← LieIdeal.coe_killingCompl_top, killingCompl_top_eq_bot]\n\nlemma killingForm_nondegenerate :\n    (killingForm R L).Nondegenerate := by\n  refine (LieModule.traceForm_isSymm R L L).isRefl.nondegenerate_iff_separatingLeft.mpr ?_\n  simp [LinearMap.separatingLeft_iff_ker_eq_bot]\n\nvariable {R L} in\nlemma ideal_eq_bot_of_isLieAbelian\n    [Module.Free R L] [Module.Finite R L] [IsDomain R] [IsPrincipalIdealRing R]\n    (I : LieIdeal R L) [IsLieAbelian I] : I = ⊥ := by\n  rw [eq_bot_iff, ← killingCompl_top_eq_bot]\n  exact I.le_killingCompl_top_of_isLieAbelian\n\ninstance instSemisimple [IsKilling K L] [Module.Finite K L] : IsSemisimple K L := by\n  apply InvariantForm.isSemisimple_of_nondegenerate (Φ := killingForm K L)\n  · exact IsKilling.killingForm_nondegenerate _ _\n  · exact LieModule.traceForm_lieInvariant _ _ _\n  · exact (LieModule.traceForm_isSymm K L L).isRefl\n  · intro I h₁ h₂\n    exact h₁.1 <| IsKilling.ideal_eq_bot_of_isLieAbelian I\n\n/-- The converse of this is true in characteristic zero; it is\n`LieAlgebra.HasTrivialRadical.instIsKilling`. There are counterexamples\nover fields with positive characteristic.\n\nNote that when the coefficients are a field this instance is redundant since we have\n`LieAlgebra.IsKilling.instSemisimple` and `LieAlgebra.IsSemisimple.instHasTrivialRadical`. -/\ninstance instHasTrivialRadical\n    [Module.Free R L] [Module.Finite R L] [IsDomain R] [IsPrincipalIdealRing R] :\n    HasTrivialRadical R L :=\n  (hasTrivialRadical_iff_no_abelian_ideals R L).mpr IsKilling.ideal_eq_bot_of_isLieAbelian\n\ntheorem isLieAbelian_iff_subsingleton\n    [Module.Free R L] [Module.Finite R L] [IsDomain R] [IsPrincipalIdealRing R] :\n    IsLieAbelian L ↔ Subsingleton L := by\n  constructor\n  · intro h\n    rw [isLieAbelian_iff_center_eq_top R] at h\n    have hc : (⊤ : LieIdeal R L) = ⊥ := by rw [← center_eq_bot R L, h]\n    exact (LieSubmodule.subsingleton_iff R L L).mp (subsingleton_of_top_eq_bot hc)\n  · exact fun _ => inferInstance\n\nend IsKilling\n\nsection LieEquiv\n\nvariable {R L}\nvariable {L' : Type*} [LieRing L'] [LieAlgebra R L']\n\n/-- Given an equivalence `e` of Lie algebras from `L` to `L'`, and elements `x y : L`, the\nrespective Killing forms of `L` and `L'` satisfy `κ'(e x, e y) = κ(x, y)`. -/\n@[simp] lemma killingForm_of_equiv_apply (e : L ≃ₗ⁅R⁆ L') (x y : L) :\n    killingForm R L' (e x) (e y) = killingForm R L x y := by\n  simp_rw [killingForm_apply_apply, ← LieAlgebra.conj_ad_apply, ← LinearEquiv.conj_comp,\n    LinearMap.trace_conj']\n\n/-- Given a Killing Lie algebra `L`, if `L'` is isomorphic to `L`, then `L'` is Killing too. -/\n\nTarget:\nlemma isKilling_of_equiv [IsKilling R L] (e : L ≃ₗ⁅R⁆ L') : IsKilling R L' :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Lie","family_id":"iskilling_of_equiv","file_id":"mathlib/Mathlib/Algebra/Lie/Killing.lean","sample_id":"e0c5330d2514224d882fef475861bb09dfae3c5924cf8a420d81d8d903a10d54"},"split":"train"}
{"binding":{"chosen_proof_sha256":"31a79f6998a821c6e1751ba23882d5c421febc08424475fbe1cf68e517fa33e6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"51957d7dd68893e414fd4283c1aee53565c116ed7a559f131582a626a2d99c20","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6c68bdabc3643e303bfd886e5b6ece2de0d41819d4c4fb2cea020fa9d7f7d7c8","source_sha256":"c611981e5313e82f67b996518241b02b4fd5776e6033546107f5407921b773b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases L <;> grind","hard_negative":false,"metrics":{"chosen_tokens":7,"rejected_tokens":11,"token_jaccard":0.636364,"token_length_ratio":1.571429},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"87a1d22ecb84c0459be5625941b2b527076649799dd5536d564fc1290f5586a4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono\n\nNamespace:\nSimplexCategoryGenRel\n\nLocal context:\n/-\nCopyright (c) 2025 Robin Carlier. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robin Carlier\n-/\n/-! # Normal forms for morphisms in `SimplexCategoryGenRel`.\n\nIn this file, we establish that `P_δ` and `P_σ` morphisms in `SimplexCategoryGenRel`\neach admits a normal form.\n\nIn both cases, the normal forms are encoded as an integer `m`, and a strictly increasing\nlist of integers `[i₀,…,iₙ]` such that `iₖ ≤ m + k` for all `k`. We define a predicate\n`isAdmissible m : List ℕ → Prop` encoding this property. And provide some lemmas to help\nwork with such lists.\n\nNormal forms for `P_σ` morphisms are encoded by `m`-admissible lists, in which case the list\n`[i₀,…,iₙ]` represents the morphism `σ iₙ ≫ ⋯ ≫ σ i₀ : .mk (m + n) ⟶ .mk n`.\n\nNormal forms for `P_δ` morphisms are encoded by `(m + 1)`-admissible lists, in which case the list\n`[i₀,…,iₙ]` represents the morphism `δ i₀ ≫ ⋯ ≫ δ iₙ : .mk n ⟶ .mk (m + n)`.\n\nThe results in this file are to be treated as implementation-only, and they only serve as stepping\nstones towards proving that the canonical functor\n`toSimplexCategory : SimplexCategoryGenRel ⥤ SimplexCategory` is an equivalence.\n\n## References:\n* [Kerodon Tag 04FQ](https://kerodon.net/tag/04FQ)\n* [Kerodon Tag 04FT](https://kerodon.net/tag/04FT)\n\n## TODOs:\n- Show that every `P_δ` admits a unique normal form.\n-/\n\n@[expose] public section\n\nnamespace SimplexCategoryGenRel\n\nopen CategoryTheory\n\nsection AdmissibleLists\n-- Impl. note: We are not bundling admissible lists as a subtype of `List ℕ` so that it remains\n-- easier to perform inductive constructions and proofs on such lists, and we instead bundle\n-- propositions asserting that various List constructions produce admissible lists.\n\nvariable (m : ℕ)\n/-- A list of natural numbers `[i₀, ⋯, iₙ]` is said to be `m`-admissible (for `m : ℕ`) if\n`i₀ < ⋯ < iₙ` and `iₖ ≤ m + k` for all `k`. This would suggest the definition\n`L.IsChain (· < ·) ∧ ∀ k, (h : k < L.length) → L[k] ≤ m + k`.\nHowever, we instead define `IsAdmissible` inductively and show, in\n`isAdmissible_iff_isChain_and_le`, that this is equivalent to the non-inductive definition.\n-/\n@[mk_iff]\ninductive IsAdmissible : (m : ℕ) → (L : List ℕ) → Prop\n  | nil (m : ℕ) : IsAdmissible m []\n  | singleton {m a} (ha : a ≤ m) : IsAdmissible m [a]\n  | cons_cons {m a b L'} (hab : a < b) (hbL : IsAdmissible (m + 1) (b :: L'))\n      (ha : a ≤ m) : IsAdmissible m (a :: b :: L')\n\nattribute [simp, grind ←] IsAdmissible.nil\nattribute [grind →] IsAdmissible.cons_cons\n\nsection IsAdmissible\n\nvariable {m a b : ℕ} {L : List ℕ}\n\n@[simp, grind =]\ntheorem isAdmissible_singleton_iff : IsAdmissible m [a] ↔ a ≤ m :=\n  ⟨fun | .singleton h => h, .singleton⟩\n\n@[simp, grind =]\ntheorem isAdmissible_cons_cons_iff : IsAdmissible m (a :: b :: L) ↔\n    a < b ∧ IsAdmissible (m + 1) (b :: L) ∧ a ≤ m :=\n  ⟨fun | .cons_cons hab hbL ha => ⟨hab, hbL, ha⟩, by grind⟩\n\nTarget:\ntheorem isAdmissible_cons_iff : IsAdmissible m (a :: L) ↔\n    a ≤ m ∧ ((_ : 0 < L.length) → a < L[0]) ∧ IsAdmissible (m + 1) L :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  cases L <;> grind","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplexCategory","family_id":"isadmissible_cons_iff","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/NormalForms.lean","sample_id":"6c68bdabc3643e303bfd886e5b6ece2de0d41819d4c4fb2cea020fa9d7f7d7c8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fcfc00391c408d2d6a402d4b95c161c6a253181ebc839f73404ed0063d916643","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"662af12bfb4b038d7217e684e813afa0288e30d1adb398ad43bc110a38cae151","source_sha256":"3fe8edf97192007f578bcd737a321443da36bb151be21df440f94427a855ebb7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun m hm ↦ ?_⟩\n  obtain ⟨m', hm'⟩ := h ⟨m, hm.isPrime⟩\n  have : m = Ideal.comap (algebraMap A B) m'.asIdeal := by\n    rw [← PrimeSpectrum.comap_asIdeal (algebraMap A B) m', hm']\n  rw [Ideal.smul_top_eq_map, this]\n  exact (Submodule.restrictScalars_eq_top_iff _ _ _).ne.mpr\n    fun top ↦ m'.isPrime.ne_top <| top_le_iff.mp <| top ▸ Ideal.map_comap_le","hard_negative":false,"metrics":{"chosen_tokens":98,"rejected_tokens":3,"token_jaccard":0.039216,"token_length_ratio":0.030612},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"87d5dc0d0e6ffd3a8508426afc710d2fb51b8f53a4e84f33b63cf1fd5755e54d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Flat.FaithfullyFlat.Basic\npublic import Mathlib.RingTheory.Ideal.GoingUp\npublic import Mathlib.RingTheory.Spectrum.Prime.RingHom\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Christian Merten, Yi Song, Sihan Su. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten, Yi Song, Sihan Su\n-/\n/-!\n# Properties of faithfully flat algebras\n\nAn `A`-algebra `B` is faithfully flat if `B` is faithfully flat as an `A`-module. In this\nfile we give equivalent characterizations of faithful flatness in the algebra case.\n\n## Main results\n\nLet `B` be a faithfully flat `A`-algebra:\n\n- `Ideal.comap_map_eq_self_of_faithfullyFlat`: the contraction of the extension of any ideal of\n  `A` to `B` is the ideal itself.\n- `Module.FaithfullyFlat.tensorProduct_mk_injective`: The natural map `M →ₗ[A] B ⊗[A] M` is\n  injective for any `A`-module `M`.\n- `PrimeSpectrum.comap_surjective_of_faithfullyFlat`: The map on prime spectra induced by\n  a faithfully flat ring map is surjective. See also\n  `Ideal.exists_isPrime_liesOver_of_faithfullyFlat` for a version stated in terms of\n  `Ideal.LiesOver`.\n\nConversely, let `B` be a flat `A`-algebra:\n\n- `Module.FaithfullyFlat.of_comap_surjective`: `B` is faithfully flat over `A`,\n  if the induced map on prime spectra is surjective.\n- `Module.FaithfullyFlat.of_flat_of_isLocalHom`: flat + local implies faithfully flat\n\n-/\n\npublic section\n\nuniverse u v\n\nvariable {A B : Type*} [CommRing A] [CommRing B] [Algebra A B]\n\nopen TensorProduct LinearMap\n\n/-- If `A →+* B` is flat and surjective on prime spectra, `B` is a faithfully flat `A`-algebra. -/\n\nTarget:\nlemma Module.FaithfullyFlat.of_comap_surjective [Flat A B]\n    (h : Function.Surjective (PrimeSpectrum.comap (algebraMap A B))) :\n    Module.FaithfullyFlat A B :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Flat","family_id":"module","file_id":"mathlib/Mathlib/RingTheory/Flat/FaithfullyFlat/Algebra.lean","sample_id":"662af12bfb4b038d7217e684e813afa0288e30d1adb398ad43bc110a38cae151"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2c01520ab51bc25fbe7980ca7811c6f4fd2614f49e05e36aaa18fc8557eeaddd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"19062733be5acb02f6c35df456458662dd844817288a82e968c0a1b485bc64b5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f57b40da55b42c3f0a2ab8a4512907a91408e6b8713cf272d9f3041b94ceb710","source_sha256":"21c0805d79514be5c2fec0b605f66875bb5bf5631a7a75dc4887fad5491c11a0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [modEq_comm]\n  simp_rw [modEq_iff_eq_add_zsmul, QuotientAddGroup.eq_iff_sub_mem, AddSubgroup.mem_zmultiples_iff,\n    eq_sub_iff_add_eq', eq_comm]","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":28,"token_jaccard":0.882353,"token_length_ratio":1.333333},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"88242f0ca172f0c6e3b63b8465baeb6b1ce4b73627b9cda529177aaf51922138","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.ModEq\npublic import Mathlib.GroupTheory.QuotientGroup.Defs\npublic import Mathlib.Algebra.Group.Subgroup.ZPowers.Basic\n\nNamespace:\nAddCommGroup\n\nLocal context:\n/-\nCopyright (c) 2023 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n\n/-!\n# Congruence modulo multiples and congruence modulo `AddSubgroup.zmultiples _`\n\nIn this file we show that in an additive commutative group, the congruence relation `a ≡ b [PMOD p]`\nis equivalent to the coercions of `a` and `b` to `G ⧸ AddSubgroup.zmultiples p` being equal.\n-/\n\npublic section\n\nnamespace AddCommGroup\n\nvariable {G : Type*} [AddCommGroup G] {a b p : G}\n\nTarget:\ntheorem modEq_iff_eq_mod_zmultiples : a ≡ b [PMOD p] ↔ (a : G ⧸ AddSubgroup.zmultiples p) = b :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [modEq_comm]\n  simp_rw [modEq_iff_eq_add_zsmul, QuotientAddGroup.eq_iff_sub_mem, AddSubgroup.mem_zmultiples_iff,\n    eq_sub_iff_add_eq', eq_comm]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/QuotientGroup","family_id":"modeq_iff_eq_mod_zmultiples","file_id":"mathlib/Mathlib/GroupTheory/QuotientGroup/ModEq.lean","sample_id":"f57b40da55b42c3f0a2ab8a4512907a91408e6b8713cf272d9f3041b94ceb710"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5bc8e347ea35f5e54b983043b84f723d4bb439695fcffd6e054bbe0bc3c13d4f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0ccc7aefbf796c9bc413bd34f90036d969463ddb4ebea0b7afe1f034f8ff048d","source_sha256":"fe20ac9adb6e069125c46c6cc0a23c1dfac91d49367534266b3b31063c05d603","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  constructor\n  · simp only [basis, TwoSidedIdeal.le_iff, TwoSidedIdeal.coe_mk', setOf_subset_setOf]\n    intro h\n    constructor\n    · intro x hx\n      have (d' : _) : coeff d' (C (σ := σ) x) ∈ J := by\n        rw [coeff_C]; split_ifs <;> [exact hx; exact J.zero_mem]\n      simpa using h (C x) (fun _ _ ↦ this _) _ zero_le\n    · by_contra h'\n      apply hK\n      rw [eq_top_iff]\n      intro x _\n      have (d') (hd'_le : d' ≤ d) : coeff d' (monomial e x) ∈ J := by\n        rw [coeff_monomial]\n        split_ifs with hd' <;> [exact (h' (hd' ▸ hd'_le)).elim; exact J.zero_mem]\n      simpa using h (monomial e x) this _ le_rfl\n  · rintro ⟨hJK, hed⟩\n    exact basis_le hJK hed\n\nvariable [TopologicalSpace R]\n\n-- We endow MvPowerSeries σ R with the product topology.","hard_negative":false,"metrics":{"chosen_tokens":185,"rejected_tokens":5,"token_jaccard":0.037975,"token_length_ratio":0.027027},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"8895a94106f673784bd151c69008c029eae1eb9492a55ca89c5fc9d639718115","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finsupp.Interval\npublic import Mathlib.RingTheory.Ideal.Quotient.Defs\npublic import Mathlib.RingTheory.MvPowerSeries.PiTopology\npublic import Mathlib.Topology.Algebra.LinearTopology\npublic import Mathlib.RingTheory.TwoSidedIdeal.Operations\n\nNamespace:\nMvPowerSeries.LinearTopology\n\nLocal context:\n/-\nCopyright (c) 2024 Antoine Chambert-Loir, María Inés de Frutos-Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir, María Inés de Frutos-Fernández\n-/\n/-! # Linear topology on the ring of multivariate power series\n\n- `MvPowerSeries.LinearTopology.basis`: the ideals of the ring of multivariate power series\n  all coefficients the exponent of which is smaller than some bound vanish.\n\n- `MvPowerSeries.LinearTopology.hasBasis_nhds_zero` :\n  the two-sided ideals from `MvPowerSeries.LinearTopology.basis` form a basis\n  of neighborhoods of `0` if the topology of `R` is (left and right) linear.\n\n## Instances :\n\nIf `R` has a linear topology, then the product topology on `MvPowerSeries σ R`\nis a linear topology.\n\nThis applies in particular when `R` has the discrete topology.\n\n## Note\n\nIf we had an analogue of `PolynomialModule` for power series,\nmeaning that we could consider the `R⟦X⟧`-module `M⟦X⟧` when `M` is an `R`-module,\nthen one could prove that `M⟦X⟧` is linearly topologized over `R⟦X⟧`\nwhenever `M` is linearly topologized over `R`.\nTo recover the ring case, it would remain to show that the isomorphism between\n`Rᵐᵒᵖ⟦X⟧` and `R⟦X⟧ᵐᵒᵖ` identifies their respective actions on `R⟦X⟧`.\n(And likewise in the multivariate case.)\n\n-/\n\n@[expose] public section\n\nnamespace MvPowerSeries\n\nnamespace LinearTopology\n\nopen scoped Topology\n\nopen Set SetLike Filter\n\n/-- The underlying family for the basis of ideals in a multivariate power series ring. -/\nnoncomputable def basis (σ : Type*) (R : Type*) [Ring R] (Jd : TwoSidedIdeal R × (σ →₀ ℕ)) :\n    TwoSidedIdeal (MvPowerSeries σ R) :=\n  TwoSidedIdeal.mk' {f | ∀ e ≤ Jd.2, coeff e f ∈ Jd.1}\n    (by simp [coeff_zero])\n    (fun hf hg e he ↦ by rw [map_add]; exact add_mem (hf e he) (hg e he))\n    (fun {f} hf e he ↦ by simp only [map_neg, neg_mem, hf e he])\n    (fun {f g} hg e he ↦ by\n      classical\n      rw [coeff_mul]\n      apply sum_mem\n      rintro uv huv\n      exact TwoSidedIdeal.mul_mem_left _ _ _ (hg _ (le_trans (Finset.antidiagonal.snd_le huv) he)))\n    (fun {f g} hf e he ↦ by\n      classical\n      rw [coeff_mul]\n      apply sum_mem\n      rintro uv huv\n      exact TwoSidedIdeal.mul_mem_right _ _ _ (hf _ (le_trans (Finset.antidiagonal.fst_le huv) he)))\n\nvariable {σ : Type*} {R : Type*} [Ring R]\n\n/-- A power series `f` belongs to the two-sided ideal `basis σ R ⟨J, d⟩`\nif and only if `coeff e f ∈ J` for all `e ≤ d`. -/\ntheorem mem_basis_iff {f : MvPowerSeries σ R} {Jd : TwoSidedIdeal R × (σ →₀ ℕ)} :\n    f ∈ basis σ R Jd ↔ ∀ e ≤ Jd.2, coeff e f ∈ Jd.1 := by\n  simp [basis]\n\n/-- If `J ≤ K` and `e ≤ d`, then we have the inclusion of two-sided ideals\n`basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e,>`. -/\ntheorem basis_le {Jd Ke : TwoSidedIdeal R × (σ →₀ ℕ)} (hJK : Jd.1 ≤ Ke.1) (hed : Ke.2 ≤ Jd.2) :\n    basis σ R Jd ≤ basis σ R Ke :=\n  fun _ ↦ forall_imp (fun _ h hue ↦ hJK (h (le_trans hue hed)))\n\n/-- `basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e⟩` if and only if `J ≤ K` and `e ≤ d`. -/\n\nTarget:\ntheorem basis_le_iff {J K : TwoSidedIdeal R} {d e : σ →₀ ℕ} (hK : K ≠ ⊤) :\n    basis σ R ⟨J, d⟩ ≤ basis σ R ⟨K, e⟩ ↔ J ≤ K ∧ e ≤ d :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/MvPowerSeries","family_id":"basis_le_iff","file_id":"mathlib/Mathlib/RingTheory/MvPowerSeries/LinearTopology.lean","sample_id":"0ccc7aefbf796c9bc413bd34f90036d969463ddb4ebea0b7afe1f034f8ff048d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cc4265fa4a32e1dda64a7dec7b903dcc9cceebb2f55f6e8ee131b8ddefaf2f73","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"55440f62ede7bbdc77ead5e942a883e7750663cc7506c8d69af849e8b683ebab","source_sha256":"da76ff62b317f067f0d000725a7dfe7652cd6b413da74f1215e83607e9c9c512","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : Ideal.span ({a} : Set α) ≠ ⊤ := by\n    intro H\n    rw [Ideal.span_singleton_eq_top] at H\n    contradiction\n  rcases Ideal.exists_le_maximal _ this with ⟨I, Imax, H⟩\n  use I, Imax\n  apply H\n  apply Ideal.subset_span\n  exact Set.mem_singleton a","hard_negative":false,"metrics":{"chosen_tokens":58,"rejected_tokens":5,"token_jaccard":0.073171,"token_length_ratio":0.086207},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"88b4242280a4e3ecff2ef868c4f87eecc65c3991810eb6fef38c6f3944b8251f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Ideal.Maximal\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Chris Hughes, Mario Carneiro\n-/\n/-!\n# The set of non-invertible elements of a monoid\n\n## Main definitions\n\n* `nonunits` is the set of non-invertible elements of a monoid.\n\n## Main results\n\n* `exists_max_ideal_of_mem_nonunits`: every element of `nonunits` is contained in a maximal ideal\n-/\n\n@[expose] public section\n\n\nvariable {F α β : Type*} {a b : α}\n\n/-- The set of non-invertible elements of a monoid. -/\ndef nonunits (α : Type*) [Monoid α] : Set α :=\n  { a | ¬IsUnit a }\n\n@[simp]\ntheorem mem_nonunits_iff [Monoid α] : a ∈ nonunits α ↔ ¬IsUnit a :=\n  Iff.rfl\n\ntheorem mul_mem_nonunits_right [CommMonoid α] : b ∈ nonunits α → a * b ∈ nonunits α :=\n  mt isUnit_of_mul_isUnit_right\n\ntheorem mul_mem_nonunits_left [CommMonoid α] : a ∈ nonunits α → a * b ∈ nonunits α :=\n  mt isUnit_of_mul_isUnit_left\n\ntheorem zero_mem_nonunits [MonoidWithZero α] : 0 ∈ nonunits α ↔ (0 : α) ≠ 1 :=\n  not_congr isUnit_zero_iff\n\n@[simp high] -- High priority shortcut lemma\ntheorem one_notMem_nonunits [Monoid α] : (1 : α) ∉ nonunits α :=\n  not_not_intro isUnit_one\n\n@[simp high] -- High priority shortcut lemma\ntheorem map_mem_nonunits_iff [Monoid α] [Monoid β] [FunLike F α β] [MonoidHomClass F α β] (f : F)\n    [IsLocalHom f] (a) : f a ∈ nonunits β ↔ a ∈ nonunits α :=\n  ⟨fun h ha => h <| ha.map f, fun h ha => h <| ha.of_map⟩\n\ntheorem coe_subset_nonunits [Semiring α] {I : Ideal α} (h : I ≠ ⊤) : (I : Set α) ⊆ nonunits α :=\n  fun _x hx hu => h <| I.eq_top_of_isUnit_mem hx hu\n\nTarget:\ntheorem exists_max_ideal_of_mem_nonunits [CommSemiring α] (h : a ∈ nonunits α) :\n    ∃ I : Ideal α, I.IsMaximal ∧ a ∈ I :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Ideal","family_id":"exists_max_ideal_of_mem_nonunits","file_id":"mathlib/Mathlib/RingTheory/Ideal/Nonunits.lean","sample_id":"55440f62ede7bbdc77ead5e942a883e7750663cc7506c8d69af849e8b683ebab"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fcfc00391c408d2d6a402d4b95c161c6a253181ebc839f73404ed0063d916643","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0c9c2d914957c16bba37e08631ce2d43c82e8a1dd039dc64c865fd7184a25907","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"662af12bfb4b038d7217e684e813afa0288e30d1adb398ad43bc110a38cae151","source_sha256":"3fe8edf97192007f578bcd737a321443da36bb151be21df440f94427a855ebb7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun m hm ↦ ?_⟩\n  obtain ⟨m', hm'⟩ := h ⟨m, hm.isPrime⟩\n  have : m = Ideal.comap (algebraMap A B) m'.asIdeal := by\n    rw [← PrimeSpectrum.comap_asIdeal (algebraMap A B) m', hm']\n  rw [Ideal.smul_top_eq_map, this]\n  exact (Submodule.restrictScalars_eq_top_iff _ _ _).ne.mpr\n    fun top ↦ m'.isPrime.ne_top <| top_le_iff.mp <| top ▸ Ideal.map_comap_le","hard_negative":true,"metrics":{"chosen_tokens":98,"rejected_tokens":3,"token_jaccard":0.039216,"token_length_ratio":0.030612},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"88bdbbce4eac1ec873fc556a2bda2fbdb8015b7460d5c9c7a17def4af444bf11","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Flat.FaithfullyFlat.Basic\npublic import Mathlib.RingTheory.Ideal.GoingUp\npublic import Mathlib.RingTheory.Spectrum.Prime.RingHom\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Christian Merten, Yi Song, Sihan Su. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten, Yi Song, Sihan Su\n-/\n/-!\n# Properties of faithfully flat algebras\n\nAn `A`-algebra `B` is faithfully flat if `B` is faithfully flat as an `A`-module. In this\nfile we give equivalent characterizations of faithful flatness in the algebra case.\n\n## Main results\n\nLet `B` be a faithfully flat `A`-algebra:\n\n- `Ideal.comap_map_eq_self_of_faithfullyFlat`: the contraction of the extension of any ideal of\n  `A` to `B` is the ideal itself.\n- `Module.FaithfullyFlat.tensorProduct_mk_injective`: The natural map `M →ₗ[A] B ⊗[A] M` is\n  injective for any `A`-module `M`.\n- `PrimeSpectrum.comap_surjective_of_faithfullyFlat`: The map on prime spectra induced by\n  a faithfully flat ring map is surjective. See also\n  `Ideal.exists_isPrime_liesOver_of_faithfullyFlat` for a version stated in terms of\n  `Ideal.LiesOver`.\n\nConversely, let `B` be a flat `A`-algebra:\n\n- `Module.FaithfullyFlat.of_comap_surjective`: `B` is faithfully flat over `A`,\n  if the induced map on prime spectra is surjective.\n- `Module.FaithfullyFlat.of_flat_of_isLocalHom`: flat + local implies faithfully flat\n\n-/\n\npublic section\n\nuniverse u v\n\nvariable {A B : Type*} [CommRing A] [CommRing B] [Algebra A B]\n\nopen TensorProduct LinearMap\n\n/-- If `A →+* B` is flat and surjective on prime spectra, `B` is a faithfully flat `A`-algebra. -/\n\nTarget:\nlemma Module.FaithfullyFlat.of_comap_surjective [Flat A B]\n    (h : Function.Surjective (PrimeSpectrum.comap (algebraMap A B))) :\n    Module.FaithfullyFlat A B :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_662af12bfb4b","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"2b9090b64041f4789e1239499a299e6c1e18735db440e142237a7840a9af5716","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Flat","family_id":"module","file_id":"mathlib/Mathlib/RingTheory/Flat/FaithfullyFlat/Algebra.lean","sample_id":"662af12bfb4b038d7217e684e813afa0288e30d1adb398ad43bc110a38cae151"},"split":"train"}
{"binding":{"chosen_proof_sha256":"107f7ce7855f94c8cc4f2d6c868596a255522f4e82b0b2ecfe00fd289a468b21","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b8e10a38198c67a1563aa7f4d5d8b714b3aa6db3fd6c5c7edac0dbbaeca0cebe","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a1e841521763274b65199f7d26a8e6592c777a58ce061b317bd17dcc7a139bd0","source_sha256":"ea093bfbcaf5876f15935057214ef966d86293bd44542573e87b5ce1a814f0db","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical exact continuous_piecewise hs hf continuousOn_const","hard_negative":false,"metrics":{"chosen_tokens":7,"rejected_tokens":11,"token_jaccard":0.636364,"token_length_ratio":1.571429},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"891664579de121a6b1e6568a783e33f758946bef5ed5177f721e2e7ae78c4625","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Indicator\npublic import Mathlib.Topology.Piecewise\npublic import Mathlib.Topology.Clopen\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 PFR contributors. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: PFR contributors\n-/\n/-!\n# Continuity of indicator functions\n-/\n\npublic section\n\nopen Set\nopen scoped Topology\n\nvariable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} [One β]\n\n@[to_additive]\n\nTarget:\nlemma continuous_mulIndicator (hs : ∀ a ∈ frontier s, f a = 1) (hf : ContinuousOn f (closure s)) :\n    Continuous (mulIndicator s f) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  classical exact continuous_piecewise hs hf continuousOn_const","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Algebra","family_id":"continuous_mulindicator","file_id":"mathlib/Mathlib/Topology/Algebra/Indicator.lean","sample_id":"a1e841521763274b65199f7d26a8e6592c777a58ce061b317bd17dcc7a139bd0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8dd1af80a36345fc761ec289040691b52374d545c9e7b00957a6cfdf28435160","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"550986a1cd495105b0fc378dff54d8c33efe006adb7c332ecd7abda3b23ecd1e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"48750c221d7b3b3a511e093001b0678d1d19b0dff2bd9c9bd5031532b6945ec8","source_sha256":"f92ad0e7d471b925eba21d3dc40708d34e9a126b63fb834c30a1fe2061fbbcfe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [intDegree, add_sub, sub_add, sub_sub_eq_add_sub, sub_sub, sub_eq_sub_iff_add_eq_add]\n  norm_cast\n  rw [← Polynomial.natDegree_mul x.denom_ne_zero y.denom_ne_zero, ←\n    Polynomial.natDegree_mul (RatFunc.num_ne_zero (mul_ne_zero hx hy))\n      (mul_ne_zero x.denom_ne_zero y.denom_ne_zero),\n    ← Polynomial.natDegree_mul (RatFunc.num_ne_zero hx) (RatFunc.num_ne_zero hy), ←\n    Polynomial.natDegree_mul (mul_ne_zero (RatFunc.num_ne_zero hx) (RatFunc.num_ne_zero hy))\n      (x * y).denom_ne_zero,\n    RatFunc.num_denom_mul]","hard_negative":false,"metrics":{"chosen_tokens":102,"rejected_tokens":109,"token_jaccard":0.9375,"token_length_ratio":1.068627},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"891beb6b22aaad6840193c24a4a6097472ed3d684c2712ba41fec48db9f09a38","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.RatFunc.AsPolynomial\npublic import Mathlib.RingTheory.EuclideanDomain\npublic import Mathlib.RingTheory.Localization.FractionRing\npublic import Mathlib.RingTheory.Polynomial.Content\n\nNamespace:\nRatFunc\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# The degree of rational functions\n\n## Main definitions\nWe define the degree of a rational function, with values in `ℤ`:\n- `intDegree` is the degree of a rational function, defined as the difference between the\n  `natDegree` of its numerator and the `natDegree` of its denominator. In particular,\n  `intDegree 0 = 0`.\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nuniverse u\n\nvariable {K : Type u}\n\nnamespace RatFunc\n\nsection IntDegree\n\nopen Polynomial\n\nvariable [Field K]\n\n/-- `intDegree x` is the degree of the rational function `x`, defined as the difference between\nthe `natDegree` of its numerator and the `natDegree` of its denominator. In particular,\n`intDegree 0 = 0`. -/\ndef intDegree (x : K⟮X⟯) : ℤ :=\n  natDegree x.num - natDegree x.denom\n\n@[simp]\ntheorem intDegree_zero : intDegree (0 : K⟮X⟯) = 0 := by\n  rw [intDegree, num_zero, natDegree_zero, denom_zero, natDegree_one, sub_self]\n\n@[simp]\ntheorem intDegree_one : intDegree (1 : K⟮X⟯) = 0 := by\n  rw [intDegree, num_one, denom_one, sub_self]\n\n@[simp]\ntheorem intDegree_C (k : K) : intDegree (C k) = 0 := by\n  rw [intDegree, num_C, natDegree_C, denom_C, natDegree_one, sub_self]\n\n@[simp]\ntheorem intDegree_X : intDegree (X : K⟮X⟯) = 1 := by\n  rw [intDegree, num_X, Polynomial.natDegree_X, denom_X, Polynomial.natDegree_one,\n    Int.ofNat_one, Int.ofNat_zero, sub_zero]\n\n@[simp]\ntheorem intDegree_polynomial {p : K[X]} :\n    intDegree (algebraMap K[X] K⟮X⟯ p) = natDegree p := by\n  rw [intDegree, RatFunc.num_algebraMap, RatFunc.denom_algebraMap, Polynomial.natDegree_one,\n    Int.ofNat_zero, sub_zero]\n\nTarget:\ntheorem intDegree_mul {x y : K⟮X⟯} (hx : x ≠ 0) (hy : y ≠ 0) :\n    intDegree (x * y) = intDegree x + intDegree y :=\n\nProof body:\n","rejected":"```lean\nby\n  simp only [intDegree, add_sub, sub_add, sub_sub_eq_add_sub, sub_sub, sub_eq_sub_iff_add_eq_add]\n  norm_cast\n  rw [← Polynomial.natDegree_mul x.denom_ne_zero y.denom_ne_zero, ←\n    Polynomial.natDegree_mul (RatFunc.num_ne_zero (mul_ne_zero hx hy))\n      (mul_ne_zero x.denom_ne_zero y.denom_ne_zero),\n    ← Polynomial.natDegree_mul (RatFunc.num_ne_zero hx) (RatFunc.num_ne_zero hy), ←\n    Polynomial.natDegree_mul (mul_ne_zero (RatFunc.num_ne_zero hx) (RatFunc.num_ne_zero hy))\n      (x * y).denom_ne_zero,\n    RatFunc.num_denom_mul]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/RatFunc","family_id":"intdegree_mul","file_id":"mathlib/Mathlib/FieldTheory/RatFunc/Degree.lean","sample_id":"48750c221d7b3b3a511e093001b0678d1d19b0dff2bd9c9bd5031532b6945ec8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3431d3593784b843a6d2cba92338ea3a61fb0adeecb9190810dad7f00a815cd9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"80e66e4358b5a047cdbe0254803df37a3bda8284661c9db0a5c755ad758dbabf","source_sha256":"0609bc9d79421b5690dca5429a67571ab70d1ca9db34b7e4c3dbe920598d0eb2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction N with\n  | zero => simp\n  | succ N ih => calc\n    _ ⊆ Ico (a 0) (a N) ∪ Ico (a N) (a (N + 1)) := Ico_subset_Ico_union_Ico\n    _ ⊆ _ := by simpa [Finset.range_add_one] using\n                  union_subset_union_right (Ico (a N) (a (N + 1))) ih","hard_negative":true,"metrics":{"chosen_tokens":70,"rejected_tokens":8,"token_jaccard":0.055556,"token_length_ratio":0.114286},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"89c012e812267083ef624b42bdd20ee6b7dd5ff90df0e63a0ddb196564580a9c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Range\npublic import Mathlib.Data.Set.Lattice\npublic import Mathlib.Order.Interval.Set.LinearOrder\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Oliver Butterley. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Butterley\n-/\n/-!\n# Extra lemmas about unions of intervals\n\nThis file contains lemmas about finite unions of intervals which can't be included with the lemmas\nconcerning infinite unions in `Mathlib/Order/Interval/Set/Disjoint.lean` because we use\n`Finset.range`.\n-/\n\npublic section\n\nopen Set\n\n/-- Union of consecutive intervals contains the interval defined by the initial and final points. -/\ntheorem Ioc_subset_biUnion_Ioc {X : Type*} [LinearOrder X] (N : ℕ) (a : ℕ → X) :\n    Ioc (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ioc (a i) (a (i + 1)) := by\n  induction N with\n  | zero => simp\n  | succ N ih => calc\n    _ ⊆ Ioc (a 0) (a N) ∪ Ioc (a N) (a (N + 1)) := Ioc_subset_Ioc_union_Ioc\n    _ ⊆ _ := by simpa [Finset.range_add_one] using\n                  union_subset_union_right (Ioc (a N) (a (N + 1))) ih\n\n/-- Union of consecutive intervals contains the interval defined by the initial and final points. -/\n\nTarget:\ntheorem Ico_subset_biUnion_Ico {X : Type*} [LinearOrder X] (N : ℕ) (a : ℕ → X) :\n    Ico (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ico (a i) (a (i + 1)) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"a4c3ce6ba60bf3a9d3f93399aedb672d5db3ed2f4faa6dd553b16f09485a3e39","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Interval","family_id":"ico_subset_biunion_ico","file_id":"mathlib/Mathlib/Order/Interval/Set/Union.lean","sample_id":"80e66e4358b5a047cdbe0254803df37a3bda8284661c9db0a5c755ad758dbabf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"86252bc5f575ff25cd53bd8cf62a6e1c8e93248c81bafebff2879ba682468c22","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"24bec75d4c990cfaa562f6c11f1e1ed4950885ceeb40f6408a0260da05969bf6","source_sha256":"155f9e8471ee4b30d28b3dbc51f2958ab69f6c864e7fa3081250ca89e1bfbb0f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply toFinsupp_injective\n  ext x\n  simp_rw [toFinsupp_equivMapDomain, Finsupp.equivMapDomain_apply, toFinsupp_mapDomain,\n    Finsupp.mapDomain_equiv_apply]","hard_negative":false,"metrics":{"chosen_tokens":19,"rejected_tokens":3,"token_jaccard":0.058824,"token_length_ratio":0.157895},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"89d006c93a0e419660c54d6276de49753c9d08567ec42c62f76eea206af3c562","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.SkewMonoidAlgebra.Basic\npublic import Mathlib.Algebra.Module.BigOperators\npublic import Mathlib.Algebra.Algebra.Equiv\n\nNamespace:\nSkewMonoidAlgebra\n\nLocal context:\n/-\nCopyright (c) 2025 Xavier Généreux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: María Inés de Frutos Fernández, Xavier Généreux\n-/\n/-!\n# Lemmas about different kinds of \"lifts\" to `SkewMonoidAlgebra`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nnamespace SkewMonoidAlgebra\n\nvariable {k G H : Type*}\n\nsection lift\n\nvariable [CommSemiring k] [Monoid G] [Monoid H]\nvariable {A B : Type*} [Semiring A] [Algebra k A] [Semiring B] [Algebra k B]\n\n/-- `liftNCRingHom` as an `AlgHom`, for when `f` is an `AlgHom` -/\ndef liftNCAlgHom [MulSemiringAction G A] [SMulCommClass G k A] (f : A →ₐ[k] B)\n    (g : G →* B) (h_comm : ∀ {x y}, (f (y • x)) * g y = (g y) * (f x)) :\n    SkewMonoidAlgebra A G →ₐ[k] B where\n  __ := liftNCRingHom (f : A →+* B) g h_comm\n  commutes' := by simp [liftNCRingHom]\n\n/- Hypotheses needed for `k`-algebra homomorphism from `SkewMonoidAlgebra k G`-/\nvariable [MulSemiringAction G k] [SMulCommClass G k k]\n\nvariable (k G A)\n\n/-- Any monoid homomorphism `G →* A` can be lifted to an algebra homomorphism\n  `SkewMonoidAlgebra k G →ₐ[k] A`. -/\ndef lift : (G →* A) ≃ (AlgHom k (SkewMonoidAlgebra k G) A) where\n  invFun f := (f : SkewMonoidAlgebra k G →* A).comp (of k G)\n  toFun F := by\n    apply liftNCAlgHom (Algebra.ofId k A) F\n    simp_rw [show ∀ (g : G) (r : k), g • r = r by\n        exact fun _ _ ↦ smul_algebraMap _ (algebraMap k k _)]\n    exact Algebra.commutes _ _\n  left_inv f := by\n    ext\n    simp [liftNCAlgHom, liftNCRingHom]\n  right_inv F := by\n    ext\n    simp [liftNCAlgHom, liftNCRingHom]\n\nvariable {k G A}\n\ntheorem lift_apply' (F : G →* A) (f : SkewMonoidAlgebra k G) :\n    lift k G A F f = f.sum fun a b ↦ algebraMap k A b * F a := rfl\n\ntheorem lift_apply (F : G →* A) (f : SkewMonoidAlgebra k G) :\n    lift k G A F f = f.sum fun a b ↦ b • F a := by simp [lift_apply', Algebra.smul_def]\n\ntheorem lift_def (F : G →* A) : (lift k G A F : SkewMonoidAlgebra k G → A) =\n    liftNC ((algebraMap k A : k →+* A) : k →+ A) F := rfl\n\n@[simp]\ntheorem lift_symm_apply (F : AlgHom k (SkewMonoidAlgebra k G) A) (x : G) :\n    (lift k G A).symm F x = F (single x 1) := rfl\n\ntheorem lift_of (F : G →* A) (x) : lift k G A F (of k G x) = F x := by\n  rw [of_apply, ← lift_symm_apply, Equiv.symm_apply_apply]\n\n@[simp]\ntheorem lift_single (F : G →* A) (a b) : lift k G A F (single a b) = b • F a := by\n  rw [lift_def, liftNC_single, Algebra.smul_def, AddMonoidHom.coe_coe]\n\ntheorem lift_unique' (F : AlgHom k (SkewMonoidAlgebra k G) A) :\n    F = lift k G A ((F : SkewMonoidAlgebra k G →* A).comp (of k G)) :=\n  ((lift k G A).apply_symm_apply F).symm\n\n/-- Decomposition of a `k`-algebra homomorphism from `SkewMonoidAlgebra k G` by\n  its values on `F (single a 1)`. -/\ntheorem lift_unique (F : AlgHom k (SkewMonoidAlgebra k G) A)\n    (f : SkewMonoidAlgebra k G) : F f = f.sum fun a b ↦ b • F (single a 1) := by\n  conv_lhs =>\n    rw [lift_unique' F]\n    simp [lift_apply]\n\n/-- If `f : G → H` is a multiplicative homomorphism between two monoids, then\n`mapDomain f` is an algebra homomorphism between their monoid algebras. -/\n@[simps!]\ndef mapDomainAlgHom (k A : Type*) [CommSemiring k] [Semiring A] [Algebra k A] {H F : Type*}\n    [Monoid H] [FunLike F G H] [MonoidHomClass F G H] [MulSemiringAction G A]\n    [MulSemiringAction H A] [SMulCommClass G k A] [SMulCommClass H k A] {f : F}\n    (hf : ∀ (a : G) (x : A), a • x = (f a) • x) :\n    SkewMonoidAlgebra A G →ₐ[k] SkewMonoidAlgebra A H where\n  __ := mapDomainRingHom hf\n  commutes' := by simp [mapDomainRingHom]\n\nend lift\n\nsection equivMapDomain\n\nvariable [AddCommMonoid k]\n\n/-- Given `f : G ≃ H`, we can map `l : SkewMonoidAlgebra k G` to\n`equivMapDomain f l : SkewMonoidAlgebra k H` (computably) by mapping the support forwards\nand the function backwards. -/\ndef equivMapDomain (f : G ≃ H) (l : SkewMonoidAlgebra k G) : SkewMonoidAlgebra k H where\n  toFinsupp := ⟨l.support.map f.toEmbedding, fun a ↦ l.coeff (f.symm a), by simp⟩\n\n@[simp]\ntheorem coeff_equivMapDomain (f : G ≃ H) (l : SkewMonoidAlgebra k G) (b : H) :\n    (equivMapDomain f l).coeff b = l.coeff (f.symm b) :=\n  rfl\n\nlemma toFinsupp_equivMapDomain (f : G ≃ H) (l : SkewMonoidAlgebra k G) :\n    (equivMapDomain f l).toFinsupp = Finsupp.equivMapDomain f l.toFinsupp := rfl\n\nTarget:\ntheorem equivMapDomain_eq_mapDomain (f : G ≃ H) (l : SkewMonoidAlgebra k G) :\n    equivMapDomain f l = mapDomain f l :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/SkewMonoidAlgebra","family_id":"equivmapdomain_eq_mapdomain","file_id":"mathlib/Mathlib/Algebra/SkewMonoidAlgebra/Lift.lean","sample_id":"24bec75d4c990cfaa562f6c11f1e1ed4950885ceeb40f6408a0260da05969bf6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"732ed3f932374a610e819e8e805afcfdf95b627321f8ae3cc0670b9127888807","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"fdbdc32f2293a3472c9b736355b763db029fc1933de04a5317dc35b5d8f39224","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5b7bc29de3804ac7bf45079e461f320c398aef01effbb7aab20e4f7fde575a9c","source_sha256":"5262ad7ee8fe0f47e01721cbd60f82b52bb0748fdb1cf1263478016dd35847f5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨n, rfl⟩ | ⟨n, rfl⟩ := Int.even_or_odd n\n  · simp [zpow_add, ← mul_zpow, Int.negOnePow_even _ (Even.add_self n)]\n  · rw [Int.negOnePow_odd _ (by exact odd_two_mul_add_one n)]\n    simp [Units.smul_def, zpow_add, Int.two_mul, ← mul_zpow]","hard_negative":true,"metrics":{"chosen_tokens":66,"rejected_tokens":3,"token_jaccard":0.060606,"token_length_ratio":0.045455},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"89daae060675c49c88f4af38955d2d9f9516f6dd8a3d81cbb91ce8051de098c4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Center.Preadditive\npublic import Mathlib.Algebra.Ring.NegOnePow\n\nNamespace:\nCategoryTheory.CatCenter\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Powers of `-1` in the center of a preadditive category\n\n-/\n\npublic section\n\nuniverse v u\n\nnamespace CategoryTheory.CatCenter\n\nvariable {C : Type u} [Category.{v} C] [Preadditive C]\n\nopen scoped IsMulCommutative in\n@[simp]\n\nTarget:\nlemma app_neg_one_zpow (n : ℤ) (X : C) :\n    ((-1) ^ n : (CatCenter C)ˣ).val.app X = n.negOnePow • 𝟙 X :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_5b7bc29de380","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"ceb0564fd9b4e3e71593d18a55f0c9d098707d0a92004dcfee2a187859e3feb2","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Center","family_id":"app_neg_one_zpow","file_id":"mathlib/Mathlib/CategoryTheory/Center/NegOnePow.lean","sample_id":"5b7bc29de3804ac7bf45079e461f320c398aef01effbb7aab20e4f7fde575a9c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"389a8937a3f2af4740351819b93e71736e399c56154cd81ab2e2e646a0bfda28","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c471d34833437ae28a7e54bc2a541395cc21cdc7c7863be44db609030e5d9676","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b0b58afdb4cef1d362a08061526d43713fac33d2fd2c151db38d446b2e4f58cd","source_sha256":"36cdfa8e1740ff8584851a3963e63d4eb571008501c17fdeb244e2a7e9d81b8d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [eq_comm, prod_eq_prod_of_nonempty h, eq_comm, @eq_comm _ eY']","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.166667,"token_length_ratio":0.2},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"89eaf3d120f1e250a0eddbad69fb7548a43725362bc820609cbf420a6cbb0313","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.OpenPartialHomeomorph.Composition\n\nNamespace:\nOpenPartialHomeomorph\n\nLocal context:\n/-\nCopyright (c) 2019 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Constructions of new partial homeomorphisms from old\n\n## Main definitions\n\n* `OpenPartialHomeomorph.const`: an open partial homeomorphism which is a constant map,\n  whose source and target are necessarily singleton sets\n* `OpenPartialHomeomorph.subtypeRestr`: restriction to a subtype\n* `OpenPartialHomeomorph.prod`: the product of two open partial homeomorphisms,\n  as an open partial homeomorphism on the product space\n* `OpenPartialHomeomorph.pi`: the product of a finite family of open partial homeomorphisms\n* `OpenPartialHomeomorph.disjointUnion`: combine two open partial homeomorphisms with disjoint\n  sources and disjoint targets\n* `OpenPartialHomeomorph.lift_openEmbedding`: extend an open partial homeomorphism `X → Y`\n  under an open embedding `X → X'`, to an open partial homeomorphism `X' → Z`.\n  (This is used to define the disjoint union of charted spaces.)\n-/\n\n@[expose] public section\n\nopen Function Set Filter Topology\n\nvariable {X X' : Type*} {Y Y' : Type*} {Z Z' : Type*}\n  [TopologicalSpace X] [TopologicalSpace X'] [TopologicalSpace Y] [TopologicalSpace Y']\n  [TopologicalSpace Z] [TopologicalSpace Z']\n\nnamespace OpenPartialHomeomorph\n\nvariable (e : OpenPartialHomeomorph X Y)\n\n/-!\n## Constants\n\n`PartialEquiv.const` as an open partial homeomorphism\n-/\nsection const\n\nvariable {a : X} {b : Y}\n\n/--\nThis is `PartialEquiv.single` as an open partial homeomorphism: a constant map,\nwhose source and target are necessarily singleton sets.\n-/\ndef const (ha : IsOpen {a}) (hb : IsOpen {b}) : OpenPartialHomeomorph X Y where\n  toPartialEquiv := PartialEquiv.single a b\n  open_source := ha\n  open_target := hb\n  continuousOn_toFun := by simp\n  continuousOn_invFun := by simp\n\n@[simp, mfld_simps]\nlemma const_apply (ha : IsOpen {a}) (hb : IsOpen {b}) (x : X) : (const ha hb) x = b := rfl\n\n@[simp, mfld_simps]\nlemma const_source (ha : IsOpen {a}) (hb : IsOpen {b}) : (const ha hb).source = {a} := rfl\n\n@[simp, mfld_simps]\nlemma const_target (ha : IsOpen {a}) (hb : IsOpen {b}) : (const ha hb).target = {b} := rfl\n\nend const\n\n/-!\n## Products\n\nProduct of two open partial homeomorphisms\n-/\nsection Prod\n\n/-- The product of two open partial homeomorphisms, as an open partial homeomorphism on the product\nspace. -/\n@[simps! (attr := mfld_simps) -fullyApplied toPartialHomeomorph apply,\n  simps! -isSimp source target symm_apply]\ndef prod (eX : OpenPartialHomeomorph X X') (eY : OpenPartialHomeomorph Y Y') :\n    OpenPartialHomeomorph (X × Y) (X' × Y') where\n  open_source := eX.open_source.prod eY.open_source\n  open_target := eX.open_target.prod eY.open_target\n  continuousOn_toFun := eX.continuousOn.prodMap eY.continuousOn\n  continuousOn_invFun := eX.continuousOn_symm.prodMap eY.continuousOn_symm\n  toPartialEquiv := eX.toPartialEquiv.prod eY.toPartialEquiv\n\n@[simp, mfld_simps]\ntheorem prod_symm (eX : OpenPartialHomeomorph X X') (eY : OpenPartialHomeomorph Y Y') :\n    (eX.prod eY).symm = eX.symm.prod eY.symm :=\n  rfl\n\n@[simp]\ntheorem refl_prod_refl : (OpenPartialHomeomorph.refl X).prod (OpenPartialHomeomorph.refl Y) =\n    OpenPartialHomeomorph.refl (X × Y) :=\n  OpenPartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) univ_prod_univ\n\n@[simp, mfld_simps]\ntheorem prod_trans (e : OpenPartialHomeomorph X Y) (f : OpenPartialHomeomorph Y Z)\n    (e' : OpenPartialHomeomorph X' Y') (f' : OpenPartialHomeomorph Y' Z') :\n    (e.prod e').trans (f.prod f') = (e.trans f).prod (e'.trans f') :=\n  toPartialEquiv_injective <| e.1.prod_trans ..\n\ntheorem prod_eq_prod_of_nonempty {eX eX' : OpenPartialHomeomorph X X'}\n    {eY eY' : OpenPartialHomeomorph Y Y'} (h : (eX.prod eY).source.Nonempty) :\n    eX.prod eY = eX'.prod eY' ↔ eX = eX' ∧ eY = eY' := by\n  obtain ⟨⟨x, y⟩, -⟩ := id h\n  haveI : Nonempty X := ⟨x⟩\n  haveI : Nonempty X' := ⟨eX x⟩\n  haveI : Nonempty Y := ⟨y⟩\n  haveI : Nonempty Y' := ⟨eY y⟩\n  simp_rw [OpenPartialHomeomorph.ext_iff, prod_apply, prod_symm_apply, prod_source, Prod.ext_iff,\n    Set.prod_eq_prod_iff_of_nonempty h, forall_and, Prod.forall, forall_const,\n    and_assoc, and_left_comm]\n\nTarget:\ntheorem prod_eq_prod_of_nonempty'\n    {eX eX' : OpenPartialHomeomorph X X'} {eY eY' : OpenPartialHomeomorph Y Y'}\n    (h : (eX'.prod eY').source.Nonempty) : eX.prod eY = eX'.prod eY' ↔ eX = eX' ∧ eY = eY' :=\n\nProof body:\n","rejected":"by\n  exact prod_eq_prod_of_nonempty","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"bd5c8c099274d8353551f9dcafe15d7e0a8126aa044c206f751796988aaa77cd","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/OpenPartialHomeomorph","family_id":"prod_eq_prod_of_nonempty","file_id":"mathlib/Mathlib/Topology/OpenPartialHomeomorph/Constructions.lean","sample_id":"b0b58afdb4cef1d362a08061526d43713fac33d2fd2c151db38d446b2e4f58cd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d4554f84e387bbee7a732a6c3f4886d4878a68fc65f7eb3a496b2c61095b621d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"190a0c5110351f1f4b9f147746258119b2602d5c8767ce676607a80e801b0a64","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"95cc52a04da20a5f5c49fe48e2d7c757cb686f4efaddac32474d2f88abd5ceb5","source_sha256":"55f23f9454fd78a1ab9f5d31dee9f06c983f6e922e17ae4ec8a9c45046a436fc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [-objD, -obj_X]","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":2,"token_jaccard":0.111111,"token_length_ratio":0.222222},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"8a09dcdccf5304c3479c0286a781a7159ac02df764e9ef7957985fbf2bcb1d97","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.HomologicalComplex\npublic import Mathlib.AlgebraicTopology.SimplicialObject.Basic\npublic import Mathlib.CategoryTheory.Abelian.Basic\n\nNamespace:\nAlgebraicTopology\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n## Moore complex\n\nWe construct the normalized Moore complex, as a functor\n`SimplicialObject C ⥤ ChainComplex C ℕ`,\nfor any abelian category `C`.\n\nThe `n`-th object is intersection of\nthe kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.\n\nThe differentials are induced from `X.δ 0`,\nwhich maps each of these intersections of kernels to the next.\n\nThis functor is one direction of the Dold-Kan equivalence, which we're still working towards.\n\n### References\n\n* https://stacks.math.columbia.edu/tag/0194\n* https://ncatlab.org/nlab/show/Moore+complex\n-/\n\n@[expose] public section\n\n\nuniverse v u\n\nnoncomputable section\n\nopen CategoryTheory CategoryTheory.Limits\n\nopen Opposite\n\nopen scoped Simplicial\n\nnamespace AlgebraicTopology\n\nvariable {C : Type*} [Category* C] [Abelian C]\n\nattribute [local instance] Abelian.hasPullbacks\n\n/-! The definitions in this namespace are all auxiliary definitions for `NormalizedMooreComplex`\nand should usually only be accessed via that. -/\n\n\nnamespace NormalizedMooreComplex\n\nopen CategoryTheory.Subobject\n\nvariable (X : SimplicialObject C)\n\n/-- The normalized Moore complex in degree `n`, as a subobject of `X n`.\n-/\ndef objX : ∀ n : ℕ, Subobject (X.obj (op ⦋n⦌))\n  | 0 => ⊤\n  | n + 1 => Finset.univ.inf fun k : Fin (n + 1) => kernelSubobject (X.δ k.succ)\n\n@[simp] theorem objX_zero : objX X 0 = ⊤ :=\n  rfl\n\n@[simp] theorem objX_add_one (n) :\n    objX X (n + 1) = Finset.univ.inf fun k : Fin (n + 1) => kernelSubobject (X.δ k.succ) :=\n  rfl\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The differentials in the normalized Moore complex.\n-/\n@[simp]\ndef objD : ∀ n : ℕ, (objX X (n + 1) : C) ⟶ (objX X n : C)\n  | 0 => Subobject.arrow _ ≫ X.δ (0 : Fin 2) ≫ inv (⊤ : Subobject _).arrow\n  | n + 1 => by\n    -- The differential is `Subobject.arrow _ ≫ X.δ (0 : Fin (n+3))`,\n    -- factored through the intersection of the kernels.\n    refine factorThru _ (arrow _ ≫ X.δ (0 : Fin (n + 3))) ?_\n    -- We now need to show that it factors!\n    -- A morphism factors through an intersection of subobjects if it factors through each.\n    refine (finset_inf_factors _).mpr fun i _ => ?_\n    -- A morphism `f` factors through the kernel of `g` exactly if `f ≫ g = 0`.\n    apply kernelSubobject_factors\n    dsimp [objX]\n    -- Use a simplicial identity\n    rw [Category.assoc, ← Fin.castSucc_zero, ← X.δ_comp_δ (Fin.zero_le i.succ)]\n    -- We can rewrite the arrow out of the intersection of all the kernels as a composition\n    -- of a morphism we don't care about with the arrow out of the kernel of `X.δ i.succ.succ`.\n    rw [← factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ i.succ (by simp)),\n      Category.assoc, kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 := by\n  -- It's a pity we need to do a case split here;\n    -- after the first rw the proofs are almost identical\n  rcases n with _ | n <;> dsimp [objD]\n  · rw [Subobject.factorThru_arrow_assoc, Category.assoc, ← Fin.castSucc_zero,\n      ← X.δ_comp_δ_assoc (Fin.zero_le (0 : Fin 2)),\n      ← factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ (0 : Fin 2) (by simp)),\n      Category.assoc, kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]\n  · rw [factorThru_right, factorThru_eq_zero, factorThru_arrow_assoc, Category.assoc,\n      ← Fin.castSucc_zero,\n      ← X.δ_comp_δ (Fin.zero_le (0 : Fin (n + 3))),\n      ← factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ (0 : Fin (n + 3)) (by simp)),\n      Category.assoc, kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]\n\n/-- The normalized Moore complex functor, on objects.\n-/\n@[simps!]\ndef obj (X : SimplicialObject C) : ChainComplex C ℕ :=\n  ChainComplex.of (fun n => (objX X n : C))\n    (-- the coercion here picks a representative of the subobject\n      objD X) (d_squared X)\n\nvariable {X} {Y : SimplicialObject C} (f : X ⟶ Y)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The normalized Moore complex functor, on morphisms.\n-/\n@[simps!]\ndef map (f : X ⟶ Y) : obj X ⟶ obj Y :=\n  ChainComplex.ofHom\n    (fun n => factorThru _ (arrow _ ≫ f.app (op ⦋n⦌)) (by\n      cases n <;> dsimp\n      · apply top_factors\n      · refine (finset_inf_factors _).mpr fun i _ => kernelSubobject_factors _ _ ?_\n        rw [Category.assoc, SimplicialObject.δ, ← f.naturality,\n          ← factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ i (by simp)),\n          Category.assoc]\n        rw [← SimplicialObject.δ_def, kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]))\n    fun n => by cases n <;> dsimp [objD, objX, ChainComplex.of.d] <;> cat_disch\n\nend NormalizedMooreComplex\n\nopen NormalizedMooreComplex\n\nset_option backward.defeqAttrib.useBackward true in\nvariable (C) in\n/-- The (normalized) Moore complex of a simplicial object `X` in an abelian category `C`.\n\nThe `n`-th object is intersection of\nthe kernels of `X.δ i : X.obj n ⟶ X.obj (n-1)`, for `i = 1, ..., n`.\n\nThe differentials are induced from `X.δ 0`,\nwhich maps each of these intersections of kernels to the next.\n-/\n@[simps]\ndef normalizedMooreComplex : SimplicialObject C ⥤ ChainComplex C ℕ where\n  obj := obj\n  map f := map f\n\nset_option backward.defeqAttrib.useBackward true in\n-- Not `@[simp]` as `simp` can prove this.\n\nTarget:\ntheorem normalizedMooreComplex_objD (X : SimplicialObject C) (n : ℕ) :\n    ((normalizedMooreComplex C).obj X).d (n + 1) n = NormalizedMooreComplex.objD X n :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_95cc52a04da2","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"f04dfaf7362ad040bed028810a6c1634fb4f4fcd15f0c11645811b7670ec516a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology","family_id":"normalizedmoorecomplex_objd","file_id":"mathlib/Mathlib/AlgebraicTopology/MooreComplex.lean","sample_id":"95cc52a04da20a5f5c49fe48e2d7c757cb686f4efaddac32474d2f88abd5ceb5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"24379af29dd6a4b6bd06c4a471ab0fb06ebf44b4f629deaf2e40a3cfa29440d2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c3ce2be1cb1418609db10af95ac35411dba0b67ebeaa8bca6a88aca7f317bb24","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c9a34ccd888b6cf8e9fb2f8c6ffab0caeeb794e571e1779f7f95f329d2895ac6","source_sha256":"e2f1f2e70ec157c61fd85d27c525b29293c57c501ce3057927e134e50f754326","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [← card_univ, ← card_powersetCard]\n  refine (card_le_card fun s hs ↦ mem_biUnion.2 ⟨#s, ?_⟩).trans card_biUnion_le\n  exact ⟨mem_Iic.2 (mem_shatterer.1 hs).card_le_vcDim, mem_powersetCard_univ.2 rfl⟩","hard_negative":true,"metrics":{"chosen_tokens":49,"rejected_tokens":5,"token_jaccard":0.085714,"token_length_ratio":0.102041},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"8a22826dde8043386b093afb4528c86d63d6d40dd8e350f40d4428e07f7685d5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.SetFamily.Compression.Down\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Order.Interval.Finset.Nat\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Shattering families\n\nThis file defines the shattering property and VC-dimension of set families.\n\n## Main declarations\n\n* `Finset.Shatters`: The shattering property.\n* `Finset.shatterer`: The set family of sets shattered by a set family.\n* `Finset.vcDim`: The Vapnik-Chervonenkis dimension.\n\n## TODO\n\n* Order-shattering\n* Strong shattering\n-/\n\n@[expose] public section\n\nopen scoped FinsetFamily\n\nnamespace Finset\nvariable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s t : Finset α} {a : α}\n\n/-- A set family `𝒜` shatters a set `s` if all subsets of `s` can be obtained as the intersection\nof `s` and some element of the set family, and we denote this `𝒜.Shatters s`. We also say that `s`\nis *traced* by `𝒜`. -/\ndef Shatters (𝒜 : Finset (Finset α)) (s : Finset α) : Prop := ∀ ⦃t⦄, t ⊆ s → ∃ u ∈ 𝒜, s ∩ u = t\n\ninstance : DecidablePred 𝒜.Shatters := fun _s ↦ decidableForallOfDecidableSubsets\n\nlemma Shatters.exists_inter_eq_singleton (hs : Shatters 𝒜 s) (ha : a ∈ s) : ∃ t ∈ 𝒜, s ∩ t = {a} :=\n  hs <| singleton_subset_iff.2 ha\n\nlemma Shatters.mono_left (h : 𝒜 ⊆ ℬ) (h𝒜 : 𝒜.Shatters s) : ℬ.Shatters s :=\n  fun _t ht ↦ let ⟨u, hu, hut⟩ := h𝒜 ht; ⟨u, h hu, hut⟩\n\nlemma Shatters.mono_right (h : t ⊆ s) (hs : 𝒜.Shatters s) : 𝒜.Shatters t := fun u hu ↦ by\n  obtain ⟨v, hv, rfl⟩ := hs (hu.trans h); exact ⟨v, hv, inf_congr_right hu <| inf_le_of_left_le h⟩\n\nlemma Shatters.exists_superset (h : 𝒜.Shatters s) : ∃ t ∈ 𝒜, s ⊆ t :=\n  let ⟨t, ht, hst⟩ := h Subset.rfl; ⟨t, ht, inter_eq_left.1 hst⟩\n\nlemma shatters_of_forall_subset (h : ∀ t, t ⊆ s → t ∈ 𝒜) : 𝒜.Shatters s :=\n  fun t ht ↦ ⟨t, h _ ht, inter_eq_right.2 ht⟩\n\nprotected lemma Shatters.nonempty (h : 𝒜.Shatters s) : 𝒜.Nonempty :=\n  let ⟨t, ht, _⟩ := h Subset.rfl; ⟨t, ht⟩\n\n@[simp] lemma shatters_empty : 𝒜.Shatters ∅ ↔ 𝒜.Nonempty :=\n  ⟨Shatters.nonempty, fun ⟨s, hs⟩ t ht ↦ ⟨s, hs, by rwa [empty_inter, eq_comm, ← subset_empty]⟩⟩\n\nprotected lemma Shatters.subset_iff (h : 𝒜.Shatters s) : t ⊆ s ↔ ∃ u ∈ 𝒜, s ∩ u = t :=\n  ⟨fun ht ↦ h ht, by rintro ⟨u, _, rfl⟩; exact inter_subset_left⟩\n\nlemma shatters_iff : 𝒜.Shatters s ↔ 𝒜.image (fun t ↦ s ∩ t) = s.powerset :=\n  ⟨fun h ↦ by ext t; rw [mem_image, mem_powerset, h.subset_iff],\n    fun h t ht ↦ by rwa [← mem_powerset, ← h, mem_image] at ht⟩\n\nlemma univ_shatters [Fintype α] : univ.Shatters s :=\n  shatters_of_forall_subset fun _ _ ↦ mem_univ _\n\n@[simp] lemma shatters_univ [Fintype α] : 𝒜.Shatters univ ↔ 𝒜 = univ := by\n  rw [shatters_iff, powerset_univ]; simp_rw [univ_inter, image_id']\n\n/-- The set family of sets that are shattered by `𝒜`. -/\ndef shatterer (𝒜 : Finset (Finset α)) : Finset (Finset α) :=\n  {s ∈ 𝒜.biUnion powerset | 𝒜.Shatters s}\n\n@[simp] lemma mem_shatterer : s ∈ 𝒜.shatterer ↔ 𝒜.Shatters s := by\n  refine mem_filter.trans <| and_iff_right_of_imp fun h ↦ ?_\n  simp_rw [mem_biUnion, mem_powerset]\n  exact h.exists_superset\n\n@[gcongr] lemma shatterer_mono (h : 𝒜 ⊆ ℬ) : 𝒜.shatterer ⊆ ℬ.shatterer :=\n  fun _ ↦ by simpa using Shatters.mono_left h\n\nlemma subset_shatterer (h : IsLowerSet (𝒜 : Set (Finset α))) : 𝒜 ⊆ 𝒜.shatterer :=\n  fun _s hs ↦ mem_shatterer.2 fun t ht ↦ ⟨t, h ht hs, inter_eq_right.2 ht⟩\n\n@[simp] lemma isLowerSet_shatterer (𝒜 : Finset (Finset α)) :\n    IsLowerSet (𝒜.shatterer : Set (Finset α)) := fun s t ↦ by simpa using Shatters.mono_right\n\n@[simp] lemma shatterer_eq : 𝒜.shatterer = 𝒜 ↔ IsLowerSet (𝒜 : Set (Finset α)) := by\n  refine ⟨fun h ↦ ?_, fun h ↦ Subset.antisymm (fun s hs ↦ ?_) <| subset_shatterer h⟩\n  · rw [← h]\n    exact isLowerSet_shatterer _\n  · obtain ⟨t, ht, hst⟩ := (mem_shatterer.1 hs).exists_superset\n    exact h hst ht\n\n@[simp] lemma shatterer_idem : 𝒜.shatterer.shatterer = 𝒜.shatterer := by simp\n\n@[simp] lemma shatters_shatterer : 𝒜.shatterer.Shatters s ↔ 𝒜.Shatters s := by\n  simp_rw [← mem_shatterer, shatterer_idem]\n\nprotected alias ⟨_, Shatters.shatterer⟩ := shatters_shatterer\n\nprivate lemma aux (h : ∀ t ∈ 𝒜, a ∉ t) (ht : 𝒜.Shatters t) : a ∉ t := by\n  obtain ⟨u, hu, htu⟩ := ht.exists_superset; exact notMem_mono htu <| h u hu\n\n/-- Pajor's variant of the **Sauer-Shelah lemma**. -/\nlemma card_le_card_shatterer (𝒜 : Finset (Finset α)) : #𝒜 ≤ #𝒜.shatterer := by\n  refine memberFamily_induction_on 𝒜 ?_ ?_ ?_\n  · simp\n  · rfl\n  intro a 𝒜 ih₀ ih₁\n  set ℬ : Finset (Finset α) :=\n    ((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer).image (insert a)\n  have hℬ : #ℬ = #((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer) := by\n    refine card_image_of_injOn <| insert_erase_invOn.2.injOn.mono ?_\n    simp only [coe_inter, Set.subset_def, Set.mem_inter_iff, mem_coe, Set.mem_setOf_eq, and_imp,\n      mem_shatterer]\n    exact fun s _ ↦ aux (fun t ht ↦ (mem_filter.1 ht).2)\n  rw [← card_memberSubfamily_add_card_nonMemberSubfamily a]\n  refine (Nat.add_le_add ih₁ ih₀).trans ?_\n  rw [← card_union_add_card_inter, ← hℬ, ← card_union_of_disjoint]\n  swap\n  · simp only [ℬ, disjoint_left, mem_union, mem_shatterer, mem_image, not_exists, not_and]\n    rintro _ (hs | hs) s - rfl\n    · exact aux (fun t ht ↦ (mem_memberSubfamily.1 ht).2) hs <| mem_insert_self _ _\n    · exact aux (fun t ht ↦ (mem_nonMemberSubfamily.1 ht).2) hs <| mem_insert_self _ _\n  refine card_mono <| union_subset (union_subset ?_ <| shatterer_mono <| filter_subset _ _) ?_\n  · simp only [subset_iff, mem_shatterer]\n    rintro s hs t ht\n    obtain ⟨u, hu, rfl⟩ := hs ht\n    rw [mem_memberSubfamily] at hu\n    refine ⟨insert a u, hu.1, inter_insert_of_notMem fun ha ↦ ?_⟩\n    obtain ⟨v, hv, hsv⟩ := hs.exists_inter_eq_singleton ha\n    rw [mem_memberSubfamily] at hv\n    rw [← singleton_subset_iff (a := a), ← hsv] at hv\n    exact hv.2 inter_subset_right\n  · refine forall_mem_image.2 fun s hs ↦ mem_shatterer.2 fun t ht ↦ ?_\n    simp only [mem_inter, mem_shatterer] at hs\n    rw [subset_insert_iff] at ht\n    by_cases ha : a ∈ t\n    · obtain ⟨u, hu, hsu⟩ := hs.1 ht\n      rw [mem_memberSubfamily] at hu\n      refine ⟨_, hu.1, ?_⟩\n      rw [← insert_inter_distrib, hsu, insert_erase ha]\n    · obtain ⟨u, hu, hsu⟩ := hs.2 ht\n      rw [mem_nonMemberSubfamily] at hu\n      refine ⟨_, hu.1, ?_⟩\n      rwa [insert_inter_of_notMem hu.2, hsu, erase_eq_self]\n\nlemma Shatters.of_compression (hs : (𝓓 a 𝒜).Shatters s) : 𝒜.Shatters s := by\n  intro t ht\n  obtain ⟨u, hu, rfl⟩ := hs ht\n  rw [Down.mem_compression] at hu\n  obtain hu | hu := hu\n  · exact ⟨u, hu.1, rfl⟩\n  by_cases ha : a ∈ s\n  · obtain ⟨v, hv, hsv⟩ := hs <| insert_subset ha ht\n    rw [Down.mem_compression] at hv\n    obtain hv | hv := hv\n    · refine ⟨erase v a, hv.2, ?_⟩\n      rw [inter_erase, hsv, erase_insert]\n      rintro ha\n      rw [insert_eq_self.2 (mem_inter.1 ha).2] at hu\n      exact hu.1 hu.2\n    rw [insert_eq_self.2 <| inter_subset_right (s₁ := s) ?_] at hv\n    cases hv.1 hv.2\n    rw [hsv]\n    exact mem_insert_self _ _\n  · refine ⟨insert a u, hu.2, ?_⟩\n    rw [inter_insert_of_notMem ha]\n\nlemma shatterer_compress_subset_shatterer (a : α) (𝒜 : Finset (Finset α)) :\n    (𝓓 a 𝒜).shatterer ⊆ 𝒜.shatterer := by\n  simp only [subset_iff, mem_shatterer]; exact fun s hs ↦ hs.of_compression\n\n/-! ### Vapnik-Chervonenkis dimension -/\n\n/-- The Vapnik-Chervonenkis dimension of a set family is the maximal size of a set it shatters. -/\ndef vcDim (𝒜 : Finset (Finset α)) : ℕ := 𝒜.shatterer.sup card\n\n@[gcongr] lemma vcDim_mono (h𝒜ℬ : 𝒜 ⊆ ℬ) : 𝒜.vcDim ≤ ℬ.vcDim := by unfold vcDim; gcongr\n\nlemma Shatters.card_le_vcDim (hs : 𝒜.Shatters s) : #s ≤ 𝒜.vcDim := le_sup <| mem_shatterer.2 hs\n\n/-- Down-compressing decreases the VC-dimension. -/\nlemma vcDim_compress_le (a : α) (𝒜 : Finset (Finset α)) : (𝓓 a 𝒜).vcDim ≤ 𝒜.vcDim :=\n  sup_mono <| shatterer_compress_subset_shatterer _ _\n\n/-- The **Sauer-Shelah lemma**. -/\n\nTarget:\nlemma card_shatterer_le_sum_vcDim [Fintype α] :\n    #𝒜.shatterer ≤ ∑ k ∈ Iic 𝒜.vcDim, (Fintype.card α).choose k :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_c9a34ccd888b","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c61ee19905c3e382b9084cd22061d10972bf819cdea3262d9da34cd7b04b7458","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/SetFamily","family_id":"card_shatterer_le_sum_vcdim","file_id":"mathlib/Mathlib/Combinatorics/SetFamily/Shatter.lean","sample_id":"c9a34ccd888b6cf8e9fb2f8c6ffab0caeeb794e571e1779f7f95f329d2895ac6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"badc607969163b460b5422c403e504297c9199325e39a0ffb23b23e6ce679da2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e3c482c18dafac02bf8a0fd6128bd4ffa9ee2fc18b435651f9c2883c55628c8a","source_sha256":"006922c150073d26eccce5bef3353d7015b01e87320ecec635275017a27177d4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [ncoeff_apply, neg_sub, Int.sub_neg, add_sub_cancel_left]","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":5,"token_jaccard":0.066667,"token_length_ratio":0.384615},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"8a3859f44eb7fd9803b827128c5877ff2c70585c213b125790fc7e67415fd881","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Vertex.HVertexOperator\npublic import Mathlib.Data.Int.Interval\n\nNamespace:\nVertexOperator\n\nLocal context:\n/-\nCopyright (c) 2024 Scott Carnahan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Carnahan\n-/\n/-!\n# Vertex operators\nIn this file we introduce vertex operators as linear maps to Laurent series.\n\n## Definitions\n* `VertexOperator` is an `R`-linear map from an `R`-module `V` to `LaurentSeries V`.\n* `VertexOperator.ncoeff` is the coefficient of a vertex operator under normalized indexing.\n\n## TODO\n* `HasseDerivative` : A divided-power derivative.\n* `Locality` : A weak form of commutativity.\n* `Residue products` : A family of products on `VertexOperator R V` parametrized by integers.\n\n## References\n* [G. Mason, *Vertex rings and Pierce bundles*][mason2017]\n* [A. Matsuo, K. Nagatomo, *On axioms for a vertex algebra and locality of quantum\n  fields*][matsuo1997]\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nvariable {R V : Type*} [CommRing R] [AddCommGroup V] [Module R V]\n\n/-- A vertex operator over a commutative ring `R` is an `R`-linear map from an `R`-module `V` to\nLaurent series with coefficients in `V`.  We write this as a specialization of the heterogeneous\ncase. -/\nabbrev VertexOperator (R : Type*) (V : Type*) [CommRing R] [AddCommGroup V]\n    [Module R V] := HVertexOperator ℤ R V V\n\nnamespace VertexOperator\n\nopen HVertexOperator\n\n@[ext]\ntheorem ext (A B : VertexOperator R V) (h : ∀ v : V, A v = B v) :\n    A = B := LinearMap.ext h\n\n/-- The coefficient of a vertex operator under normalized indexing. -/\ndef ncoeff : VertexOperator R V →ₗ[R] ℤ → Module.End R V where\n  toFun A n := HVertexOperator.coeff A (-n - 1)\n  map_add' _ _ := by ext; simp\n  map_smul' _ _ := by ext; simp\n\ntheorem ncoeff_apply (A : VertexOperator R V) (n : ℤ) : ncoeff A n = coeff A (-n - 1) :=\n  rfl\n\n/-- In the literature, the `n`th normalized coefficient of a vertex operator `A` is written as\neither `Aₙ` or `A(n)`. -/\nscoped[VertexOperator] notation A \"[[\" n \"]]\" => ncoeff A n\n\n@[simp]\n\nTarget:\ntheorem coeff_eq_ncoeff (A : VertexOperator R V)\n    (n : ℤ) : HVertexOperator.coeff A n = A[[-n - 1]] :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Vertex","family_id":"coeff_eq_ncoeff","file_id":"mathlib/Mathlib/Algebra/Vertex/VertexOperator.lean","sample_id":"e3c482c18dafac02bf8a0fd6128bd4ffa9ee2fc18b435651f9c2883c55628c8a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"65f2254f5d37c07b087f477ca102b136468a343c041fb416c4b1c505f1281550","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"561e76fe0eca757e1f6084da3b5fc3cffca14b522807da8603a4dd41f7c3682d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fb692754d646fb760ef043b5d79501f41f1d28e50ad2da85d3e2605daffd725b","source_sha256":"a6234c28d344c069679c6c490f57e0cad723a087d9f05f1fbee4f95afdc863a6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← linearIndepOn_univ_iff, ← Finset.coe_univ, show @Finset.univ (Fin 2) _ = {0,1} from rfl,\n    Finset.coe_insert, Finset.coe_singleton, LinearIndepOn.pair_iff _ (by trivial)]\n  simp","hard_negative":true,"metrics":{"chosen_tokens":48,"rejected_tokens":5,"token_jaccard":0.058824,"token_length_ratio":0.104167},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"8aa8dc8f9ef2593b0e8bfb2ce23b4b8dc18bb951c1627dca517c5f8d351cbb24","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.Tuple.Reflection\npublic import Mathlib.LinearAlgebra.Dual.Defs\npublic import Mathlib.LinearAlgebra.Finsupp.SumProd\npublic import Mathlib.LinearAlgebra.LinearIndependent.Basic\npublic import Mathlib.LinearAlgebra.Pi\npublic import Mathlib.Logic.Equiv.Fin.Rotate\npublic import Mathlib.Tactic.FinCases\npublic import Mathlib.Tactic.Module\npublic import Mathlib.Tactic.Abel\npublic import Mathlib.Tactic.NormNum.Ineq\nimport Mathlib.Algebra.Module.Torsion.Field\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen\n-/\n/-!\n# Linear independence\n\nThis file collects consequences of linear (in)dependence and includes specialized tests for\nspecific families of vectors, requiring more theory to state.\n\n## Main statements\n\nWe prove several specialized tests for linear independence of families of vectors and of sets of\nvectors.\n\n* `linearIndependent_option`, `linearIndependent_finCons`,\n  `linearIndependent_finSucc`, `linearIndependent_finSnoc`: type-specific tests for linear\n  independence of families of vector fields;\n* `linearIndependent_insert`, `linearIndependent_pair`: linear independence tests for set operations\n\nIn many cases we additionally provide dot-style operations (e.g., `LinearIndependent.union`) to\nmake the linear independence tests usable as `hv.insert ha` etc.\n\nWe also prove that, when working over a division ring,\nany family of vectors includes a linear independent subfamily spanning the same subspace.\n\n## TODO\n\nRework proofs to hold in semirings, by avoiding the path through\n`ker (Finsupp.linearCombination R v) = ⊥`.\n\n## Tags\n\nlinearly dependent, linear dependence, linearly independent, linear independence\n\n-/\n\n@[expose] public section\n\n\nassert_not_exists Cardinal\n\nnoncomputable section\n\nopen Function Module Set Submodule\n\nuniverse u' u\n\nvariable {ι : Type u'} {ι' : Type*} {R : Type*} {K : Type*} {s : Set ι}\nvariable {M : Type*} {M' : Type*} {V : Type u}\n\nsection Semiring\n\n\nvariable {v : ι → M}\nvariable [Semiring R] [AddCommMonoid M] [AddCommMonoid M']\nvariable [Module R M] [Module R M']\nvariable (R) (v)\n\nvariable {R v}\n\n/-- A finite family of vectors `v i` is linear independent iff the linear map that sends\n`c : ι → R` to `∑ i, c i • v i` is injective. -/\ntheorem Fintype.linearIndependent_iff'ₛ [Fintype ι] [DecidableEq ι] :\n    LinearIndependent R v ↔\n      Injective (LinearMap.lsum R (fun _ ↦ R) ℕ fun i ↦ LinearMap.id.smulRight (v i)) := by\n  simp [Fintype.linearIndependent_iffₛ, Injective, funext_iff]\n\nlemma LinearIndependent.pair_iffₛ {x y : M} :\n    LinearIndependent R ![x, y] ↔\n      ∀ (s t s' t' : R), s • x + t • y = s' • x + t' • y → s = s' ∧ t = t' := by\n  simp [Fintype.linearIndependent_iffₛ, Fin.forall_fin_two, ← FinVec.forall_iff]; rfl\n\nlemma LinearIndependent.eq_of_pair {x y : M} (h : LinearIndependent R ![x, y])\n    {s t s' t' : R} (h' : s • x + t • y = s' • x + t' • y) : s = s' ∧ t = t' :=\n  pair_iffₛ.mp h _ _ _ _ h'\n\nlemma LinearIndependent.eq_zero_of_pair' {x y : M} (h : LinearIndependent R ![x, y])\n    {s t : R} (h' : s • x = t • y) : s = 0 ∧ t = 0 := by\n  suffices H : s = 0 ∧ 0 = t from ⟨H.1, H.2.symm⟩\n  exact h.eq_of_pair (by simpa using h')\n\nlemma LinearIndependent.eq_zero_of_pair {x y : M} (h : LinearIndependent R ![x, y])\n    {s t : R} (h' : s • x + t • y = 0) : s = 0 ∧ t = 0 := by\n  replace h := @h (.single 0 s + .single 1 t) 0 ?_\n  · exact ⟨by simpa using congr($h 0), by simpa using congr($h 1)⟩\n  simpa\n\nsection Indexed\n\ntheorem linearIndepOn_iUnion_of_directed {η : Type*} {s : η → Set ι} (hs : Directed (· ⊆ ·) s)\n    (h : ∀ i, LinearIndepOn R v (s i)) : LinearIndepOn R v (⋃ i, s i) := by\n  by_cases hη : Nonempty η\n  · refine linearIndepOn_of_finite (⋃ i, s i) fun t ht ft => ?_\n    rcases finite_subset_iUnion ft ht with ⟨I, fi, hI⟩\n    rcases hs.finset_le fi.toFinset with ⟨i, hi⟩\n    exact (h i).mono (Subset.trans hI <| iUnion₂_subset fun j hj => hi j (fi.mem_toFinset.2 hj))\n  · refine (linearIndepOn_empty R v).mono (t := iUnion (s ·)) ?_\n    rintro _ ⟨_, ⟨i, _⟩, _⟩\n    exact hη ⟨i⟩\n\ntheorem linearIndepOn_sUnion_of_directed {s : Set (Set ι)} (hs : DirectedOn (· ⊆ ·) s)\n    (h : ∀ a ∈ s, LinearIndepOn R v a) : LinearIndepOn R v (⋃₀ s) := by\n  rw [sUnion_eq_iUnion]\n  exact linearIndepOn_iUnion_of_directed hs.directed_val (by simpa using h)\n\ntheorem linearIndepOn_biUnion_of_directed {η} {s : Set η} {t : η → Set ι}\n    (hs : DirectedOn (t ⁻¹'o (· ⊆ ·)) s) (h : ∀ a ∈ s, LinearIndepOn R v (t a)) :\n    LinearIndepOn R v (⋃ a ∈ s, t a) := by\n  rw [biUnion_eq_iUnion]\n  exact linearIndepOn_iUnion_of_directed (directed_comp.2 <| hs.directed_val) (by simpa using h)\n\nend Indexed\n\nsection repr\n\nvariable (ι R M) in\ntheorem iSupIndep_range_lsingle :\n    iSupIndep fun i : ι ↦ LinearMap.range (Finsupp.lsingle (R := R) (M := M) i) := by\n  refine fun i ↦ disjoint_iff_inf_le.mpr ?_\n  rintro x ⟨⟨m, rfl⟩, hm⟩\n  suffices ⨆ j ≠ i, LinearMap.range (Finsupp.lsingle j) ≤ Finsupp.supported M R {i}ᶜ by\n    have := (Finsupp.mem_supported ..).mp (this hm); simp_all\n  refine iSup₂_le fun j ne ↦ ?_\n  rintro _ ⟨m, rfl⟩\n  simp [Finsupp.mem_supported, ne]\n\ntheorem LinearMap.iSupIndep_map (f : M →ₗ[R] M') (inj : Injective f) {m : ι → Submodule R M}\n    (ind : iSupIndep m) : iSupIndep fun i ↦ (m i).map f := by\n  simp_rw [iSupIndep, disjoint_iff_inf_le] at ind ⊢\n  rintro i _ ⟨⟨x, hxi, rfl⟩, hx⟩\n  rw [ind i ⟨hxi, _⟩]; · simp\n  simp_rw [← Submodule.map_iSup] at hx\n  have ⟨y, hy, eq⟩ := hx\n  simpa [← inj eq]\n\nvariable (hv : LinearIndependent R v)\n\n/-- See also `iSupIndep_iff_linearIndependent_of_ne_zero`. -/\ntheorem LinearIndependent.iSupIndep_span_singleton (hv : LinearIndependent R v) :\n    iSupIndep fun i => R ∙ v i := by\n  convert! LinearMap.iSupIndep_map _ hv (iSupIndep_range_lsingle ι R R)\n  ext; simp [mem_span_singleton]\n\nend repr\n\nsection union\n\nopen LinearMap Finsupp\n\ntheorem linearIndependent_inl_union_inr' {v : ι → M} {v' : ι' → M'}\n    (hv : LinearIndependent R v) (hv' : LinearIndependent R v') :\n    LinearIndependent R (Sum.elim (inl R M M' ∘ v) (inr R M M' ∘ v')) := by\n  have : linearCombination R (Sum.elim (inl R M M' ∘ v) (inr R M M' ∘ v')) =\n      .prodMap (linearCombination R v) (linearCombination R v') ∘ₗ\n      (sumFinsuppLEquivProdFinsupp R).toLinearMap := by ext (_ | _) <;> simp\n  rw [LinearIndependent, this]\n  simpa [LinearMap.coe_prodMap] using ⟨hv, hv'⟩\n\ntheorem LinearIndependent.inl_union_inr {s : Set M} {t : Set M'}\n    (hs : LinearIndependent R (fun x => x : s → M))\n    (ht : LinearIndependent R (fun x => x : t → M')) :\n    LinearIndependent R (fun x => x : ↥(inl R M M' '' s ∪ inr R M M' '' t) → M × M') := by\n  nontriviality R\n  let e : s ⊕ t ≃ ↥(inl R M M' '' s ∪ inr R M M' '' t) :=\n    .ofBijective (Sum.elim (fun i ↦ ⟨_, .inl ⟨_, i.2, rfl⟩⟩) fun i ↦ ⟨_, .inr ⟨_, i.2, rfl⟩⟩)\n      ⟨by rintro (_ | _) (_ | _) eq <;> simp [hs.ne_zero, ht.ne_zero] at eq <;> aesop,\n        by rintro ⟨_, ⟨_, _, rfl⟩ | ⟨_, _, rfl⟩⟩ <;> aesop⟩\n  refine (linearIndependent_equiv' e ?_).mp (linearIndependent_inl_union_inr' hs ht)\n  ext (_ | _) <;> rfl\n\nend union\n\nsection Maximal\n\nuniverse v w\n\nvariable (R)\n\n/-- TODO : refactor to use `Maximal`. -/\ntheorem exists_maximal_linearIndepOn' (v : ι → M) :\n    ∃ s : Set ι, (LinearIndepOn R v s) ∧ ∀ t : Set ι, s ⊆ t → (LinearIndepOn R v t) → s = t := by\n  let indep : Set ι → Prop := fun s => LinearIndepOn R v s\n  let X := { I : Set ι // indep I }\n  let r : X → X → Prop := fun I J => I.1 ⊆ J.1\n  have key : ∀ c : Set X, IsChain r c → indep (⋃ (I : X) (_ : I ∈ c), I) := by\n    intro c hc\n    dsimp [indep]\n    rw [linearIndepOn_iffₛ]\n    intro f hfsupp g hgsupp hsum\n    rcases eq_empty_or_nonempty c with (rfl | hn)\n    · rw [show f = 0 by simpa using! hfsupp, show g = 0 by simpa using! hgsupp]\n    haveI : Std.Refl r := ⟨fun _ => Set.Subset.refl _⟩\n    classical\n    obtain ⟨I, _I_mem, hI⟩ : ∃ I ∈ c, (f.support ∪ g.support : Set ι) ⊆ I :=\n      f.support.coe_union _ ▸ hc.directedOn.exists_mem_subset_of_finset_subset_biUnion hn <| by\n        simpa using! And.intro hfsupp hgsupp\n    exact linearIndepOn_iffₛ.mp I.2 f (subset_union_left.trans hI)\n      g (subset_union_right.trans hI) hsum\n  obtain ⟨⟨I, hli : indep I⟩, hmax : ∀ a, r ⟨I, hli⟩ a → r a ⟨I, hli⟩⟩ :=\n    exists_maximal_of_chains_bounded (r := r)\n      (fun c hc => ⟨⟨⋃ I ∈ c, (I : Set ι), key c hc⟩, fun I => Set.subset_biUnion_of_mem⟩)\n      Set.Subset.trans\n  exact ⟨I, hli, fun J hsub hli => Set.Subset.antisymm hsub (hmax ⟨J, hli⟩ hsub)⟩\n\nend Maximal\n\nlemma Submodule.codisjoint_span_image_of_codisjoint (hv : Submodule.span R (Set.range v) = ⊤)\n    {s t : Set ι} (hst : Codisjoint s t) :\n    Codisjoint (Submodule.span R (v '' s)) (Submodule.span R (v '' t)) := by\n  rw [Finsupp.span_image_eq_map_linearCombination, Finsupp.span_image_eq_map_linearCombination]\n  refine Submodule.codisjoint_map ?_ (Finsupp.codisjoint_supported_supported hst)\n  rwa [← LinearMap.range_eq_top, Finsupp.range_linearCombination]\n\nlemma LinearIndependent.isCompl_span_image (h₁ : LinearIndependent R v)\n    (h₂ : Submodule.span R (Set.range v) = ⊤) {s t : Set ι} (hst : IsCompl s t) :\n    IsCompl (Submodule.span R (v '' s)) (Submodule.span R (v '' t)) :=\n  ⟨h₁.disjoint_span_image hst.1, Submodule.codisjoint_span_image_of_codisjoint h₂ hst.2⟩\n\nend Semiring\n\nsection Module\n\nvariable {v : ι → M}\nvariable [Ring R] [AddCommGroup M] [AddCommGroup M']\nvariable [Module R M] [Module R M']\n\n/-- A finite family of vectors `v i` is linear independent iff the linear map that sends\n`c : ι → R` to `∑ i, c i • v i` has the trivial kernel. -/\ntheorem Fintype.linearIndependent_iff' [Fintype ι] [DecidableEq ι] :\n    LinearIndependent R v ↔\n      LinearMap.ker (LinearMap.lsum R (fun _ ↦ R) ℕ fun i ↦ LinearMap.id.smulRight (v i)) = ⊥ := by\n  simp [Fintype.linearIndependent_iff, LinearMap.ker_eq_bot', funext_iff]\n\n/-- `linearIndepOn_pair_iff` is a simpler version over fields. -/\nlemma LinearIndepOn.pair_iff {i j : ι} (f : ι → M) (hij : i ≠ j) :\n    LinearIndepOn R f {i,j} ↔ ∀ c d : R, c • f i + d • f j = 0 → c = 0 ∧ d = 0 := by\n  classical\n  rw [linearIndepOn_iff'']\n  refine ⟨fun h c d hcd ↦ ?_, fun h t g ht hg0 h0 ↦ ?_⟩\n  · specialize h {i, j} (Pi.single i c + Pi.single j d)\n    simpa +contextual [Finset.sum_pair, Pi.single_apply, hij, hij.symm, hcd] using h\n  have ht' : t ⊆ {i, j} := by simpa [← Finset.coe_subset]\n  rw [Finset.sum_subset ht', Finset.sum_pair hij] at h0\n  · obtain ⟨hi0, hj0⟩ := h _ _ h0\n    exact fun k hkt ↦ Or.elim (ht hkt) (fun h ↦ h ▸ hi0) (fun h ↦ h ▸ hj0)\n  simp +contextual [hg0]\n\nsection Pair\n\nvariable {x y : M}\n\n/-- Also see `LinearIndependent.pair_iff'` for a simpler version over fields. -/\n\nTarget:\nlemma LinearIndependent.pair_iff :\n    LinearIndependent R ![x, y] ↔ ∀ (s t : R), s • x + t • y = 0 → s = 0 ∧ t = 0 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_fb692754d646","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"6f2c1da2ad6f74fe22e4613ebb4e0abb7459c858831fe188a01ec8b7102900cb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/LinearIndependent","family_id":"linearindependent","file_id":"mathlib/Mathlib/LinearAlgebra/LinearIndependent/Lemmas.lean","sample_id":"fb692754d646fb760ef043b5d79501f41f1d28e50ad2da85d3e2605daffd725b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e1906f3243bcdc4427e534120589c5393cc550f48188ef4c67cb950df5b01321","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"dfc66e404d868f74aa4cfd0272a599ee1a7c3adbc5db8637ab2a7e7d2fd46b33","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2640774717b5cc5cf0106fb4ef5d3f69a7c6539d5084db036f9ef862ac01288e","source_sha256":"823d2b692e062acc3ec291c52e92559d9b37f8adc1d4d307badd142b4bd43de9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h : whiskerLeft F (add F s i a b).hom = _ :=\n    ((whiskeringLeft C D D).obj F).map_preimage _\n  exact (NatTrans.congr_app h X).trans (by simp)","hard_negative":true,"metrics":{"chosen_tokens":47,"rejected_tokens":2,"token_jaccard":0.033333,"token_length_ratio":0.042553},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"8adcc2261a2b57dadc2a1fc442926a1e528501db6aa9f1f3bcc49dfb46f1bbf6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Shift.CommShift\n\nNamespace:\nCategoryTheory.HasShift.Induced\n\nLocal context:\n/-\nCopyright (c) 2023 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Shift induced from a category to another\n\nIn this file, we introduce a sufficient condition on a functor\n`F : C ⥤ D` so that a shift on `C` by a monoid `A` induces a shift on `D`.\nMore precisely, when the functor `(D ⥤ D) ⥤ C ⥤ D` given\nby the precomposition with `F` is fully faithful, and that\nall the shift functors on `C` can be lifted to functors `D ⥤ D`\n(i.e. we have functors `s a : D ⥤ D` for all `a : A`, and isomorphisms\n`F ⋙ s a ≅ shiftFunctor C a ⋙ F`), then these functors `s a` are\nthe shift functors of a term of type `HasShift D A`.\n\nAs this condition on the functor `F` is satisfied for quotient and localization\nfunctors, the main construction `HasShift.induced` in this file shall be\nused for both quotient and localized shifts.\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Functor\n\nvariable {C D : Type _} [Category* C] [Category* D]\n  (F : C ⥤ D) {A : Type _} [AddMonoid A] [HasShift C A]\n  (s : A → D ⥤ D) (i : ∀ a, F ⋙ s a ≅ shiftFunctor C a ⋙ F)\n  [((whiskeringLeft C D D).obj F).Full] [((whiskeringLeft C D D).obj F).Faithful]\n\nnamespace HasShift\n\nnamespace Induced\n\n/-- The `zero` field of the `ShiftMkCore` structure for the induced shift. -/\nnoncomputable def zero : s 0 ≅ 𝟭 D :=\n  ((whiskeringLeft C D D).obj F).preimageIso ((i 0) ≪≫\n    isoWhiskerRight (shiftFunctorZero C A) F ≪≫ F.leftUnitor ≪≫ F.rightUnitor.symm)\n\n/-- The `add` field of the `ShiftMkCore` structure for the induced shift. -/\nnoncomputable def add (a b : A) : s (a + b) ≅ s a ⋙ s b :=\n  ((whiskeringLeft C D D).obj F).preimageIso\n    (i (a + b) ≪≫ isoWhiskerRight (shiftFunctorAdd C a b) F ≪≫\n      Functor.associator _ _ _ ≪≫\n        isoWhiskerLeft _ (i b).symm ≪≫ (Functor.associator _ _ _).symm ≪≫\n        isoWhiskerRight (i a).symm _ ≪≫ Functor.associator _ _ _)\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp]\nlemma zero_hom_app_obj (X : C) :\n    (zero F s i).hom.app (F.obj X) =\n      (i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X) := by\n  have h : whiskerLeft F (zero F s i).hom = _ :=\n    ((whiskeringLeft C D D).obj F).map_preimage _\n  exact (NatTrans.congr_app h X).trans (by simp)\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp]\nlemma zero_inv_app_obj (X : C) :\n    (zero F s i).inv.app (F.obj X) =\n      F.map ((shiftFunctorZero C A).inv.app X) ≫ (i 0).inv.app X := by\n  have h : whiskerLeft F (zero F s i).inv = _ :=\n    ((whiskeringLeft C D D).obj F).map_preimage _\n  exact (NatTrans.congr_app h X).trans (by simp)\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp]\n\nTarget:\nlemma add_hom_app_obj (a b : A) (X : C) :\n    (add F s i a b).hom.app (F.obj X) =\n      (i (a + b)).hom.app X ≫ F.map ((shiftFunctorAdd C a b).hom.app X) ≫\n        (i b).inv.app ((shiftFunctor C a).obj X) ≫ (s b).map ((i a).inv.app X) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_2640774717b5","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b2cff663fa0a790bae7ee563db3fad18f8e8ab0c202fd9a6dea0d8738ab1f407","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Shift","family_id":"add_hom_app_obj","file_id":"mathlib/Mathlib/CategoryTheory/Shift/Induced.lean","sample_id":"2640774717b5cc5cf0106fb4ef5d3f69a7c6539d5084db036f9ef862ac01288e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8855b3aa1fcebcd1bfc8f7c993730be08249feb4e40c04eec6356b1306bf4c78","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ced9a8be21b100f92f5cf1865633692067c84047c80974cbc15b46518f499a9c","source_sha256":"7fb54be1edb74454dd74d15ac42b7739b2a4aa95bc5649191f339f2298801887","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases o₁ <;> rfl","hard_negative":false,"metrics":{"chosen_tokens":8,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.625},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"8b8406d66747866baaa5e51558aa08f4676774e0d17e90214cbc8acec45ef18d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Ordering.Basic\npublic import Mathlib.Order.Defs.Unbundled\n\nNamespace:\nOrdering\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Some `Ordering` lemmas\n-/\n\npublic section\n\nuniverse u\n\nnamespace Ordering\n\n@[simp]\ntheorem ite_eq_lt_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.lt) = if c then a = Ordering.lt else b = Ordering.lt := by\n  by_cases c <;> simp [*]\n\n@[simp]\ntheorem ite_eq_eq_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.eq) = if c then a = Ordering.eq else b = Ordering.eq := by\n  by_cases c <;> simp [*]\n\n@[simp]\ntheorem ite_eq_gt_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.gt) = if c then a = Ordering.gt else b = Ordering.gt := by\n  by_cases c <;> simp [*]\n\n@[simp]\n\nTarget:\nlemma dthen_eq_then (o₁ o₂ : Ordering) : o₁.dthen (fun _ => o₂) = o₁.then o₂ :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Ordering","family_id":"dthen_eq_then","file_id":"mathlib/Mathlib/Data/Ordering/Lemmas.lean","sample_id":"ced9a8be21b100f92f5cf1865633692067c84047c80974cbc15b46518f499a9c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c3e85a55413d2bdbfb981811de0e7022d287d440d63c91a6c4ae1530e90621b9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e09c9698121c193ab550d6a3eaeef59bc9845b72f4707c7cea7edbbd63692aad","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8dd69de99772e4b60d7d7c33b72ef5655efe8caa0668569bdbff7dc2b9a6a47c","source_sha256":"da1f22be161c5bc7cbd676252bea6b38bb4aa0497872bcdf52a178b9d4951829","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa [beta, unbeta, coprimes] using mod_eq_of_modEq\n    ((chineseRemainderOfFinset (l[·]) (coprimes (l[·])) Finset.univ\n      (by simp [coprimes])\n      (by simpa using Set.pairwise_univ.mpr (pairwise_coprime_coprimes _))).prop i (by simp))\n    (coprimes_lt _ _)","hard_negative":true,"metrics":{"chosen_tokens":67,"rejected_tokens":3,"token_jaccard":0.034483,"token_length_ratio":0.044776},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"8b944cd418996cd9b3ccadd6de74d1e5f6fe9bc56f5050979dd752ef8ee4e425","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.ModEq\npublic import Mathlib.Data.Nat.ChineseRemainder\npublic import Mathlib.Data.Nat.Prime.Defs\npublic import Mathlib.Data.Nat.Pairing\npublic import Mathlib.Order.Fin.Basic\npublic import Mathlib.Data.Finset.Lattice.Fold\npublic import Mathlib.Data.Fintype.Basic\npublic import Mathlib.Data.Nat.Factorial.Basic\n\nNamespace:\nNat\n\nLocal context:\n/-\nCopyright (c) 2023 Shogo Saito. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Shogo Saito. Adapted for mathlib by Hunter Monroe\n-/\n/-!\n# Gödel's Beta Function Lemma\n\nThis file proves Gödel's Beta Function Lemma, used to prove the First Incompleteness Theorem. It\npermits quantification over finite sequences of natural numbers in formal theories of arithmetic.\nThis Beta Function has no connection with the unrelated Beta Function defined in analysis. Note\nthat `Nat.beta` and `Nat.unbeta` provide similar functionality to `Encodable.encodeList` and\n`Encodable.decodeList`. We define these separately, because it is easier to prove that `Nat.beta`\nand `Nat.unbeta` are arithmetically definable, and this is hard to prove that for\n`Encodable.encodeList` and `Encodable.decodeList` directly. The arithmetic\ndefinability is needed for the proof of the First Incompleteness Theorem.\n\n## Main result\n\n- `beta_unbeta_coe`: Gödel's Beta Function Lemma.\n\n## Implementation note\n\nThis code is a step towards eventually including a proof of Gödel's First Incompleteness Theorem\nand other key results from the repository https://github.com/iehality/lean4-logic.\n\n## References\n\n* [R. Kaye, *Models of Peano arithmetic*][kaye1991]\n* <https://en.wikipedia.org/wiki/G%C3%B6del%27s_%CE%B2_function>\n\n## Tags\n\nGödel, beta function\n-/\n\n@[expose] public section\n\nnamespace Nat\n\nlemma coprime_mul_succ {n m a} (ha : m - n ∣ a) : Coprime (n * a + 1) (m * a + 1) :=\n  Nat.coprime_of_dvd fun p pp hn hm => by\n    have : p ∣ (m - n) * a := by\n      simpa [Nat.succ_sub_succ, ← Nat.mul_sub_right_distrib] using\n        Nat.dvd_sub hm hn\n    have : p ∣ a := by\n      rcases (Nat.Prime.dvd_mul pp).mp this with (hp | hp)\n      · exact Nat.dvd_trans hp ha\n      · exact hp\n    apply pp.ne_one\n    simpa [Nat.add_sub_cancel_left] using Nat.dvd_sub hn (this.mul_left n)\n\nvariable {m : ℕ}\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nprivate def supOfSeq (a : Fin m → ℕ) : ℕ := max m (Finset.sup .univ a) + 1\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nprivate def coprimes (a : Fin m → ℕ) : Fin m → ℕ := fun i => (i + 1) * (supOfSeq a)! + 1\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nlemma coprimes_lt (a : Fin m → ℕ) (i) : a i < coprimes a i := by\n  have h₁ : a i < supOfSeq a :=\n    Nat.lt_add_one_iff.mpr (le_max_of_le_right <| Finset.le_sup (by simp))\n  have h₂ : supOfSeq a ≤ (i + 1) * (supOfSeq a)! + 1 :=\n    le_trans (self_le_factorial _) (le_trans (Nat.le_mul_of_pos_left (supOfSeq a)! (succ_pos i))\n      (le_add_right _ _))\n  simpa only [coprimes] using lt_of_lt_of_le h₁ h₂\n\nopen scoped Function in -- required for scoped `on` notation\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nprivate lemma pairwise_coprime_coprimes (a : Fin m → ℕ) : Pairwise (Coprime on coprimes a) := by\n  intro i j hij\n  wlog! ltij : i < j\n  · exact (this a hij.symm (lt_of_le_of_ne ltij hij.symm)).symm\n  unfold Function.onFun coprimes\n  have hja : j < supOfSeq a := lt_of_lt_of_le j.prop (le_succ_of_le (le_max_left _ _))\n  exact coprime_mul_succ\n    (Nat.dvd_factorial (by lia)\n      (by simpa only [Nat.succ_sub_succ] using le_of_lt (lt_of_le_of_lt (sub_le j i) hja)))\n\n/-- Gödel's Beta Function. This is similar to `(Encodable.decodeList)[i]`, but it is easier to\nprove that it is arithmetically definable. -/\ndef beta (n i : ℕ) : ℕ := n.unpair.1 % ((i + 1) * n.unpair.2 + 1)\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Inverse of Gödel's Beta Function. This is similar to `Encodable.encodeList`, but it is easier\nto prove that it is arithmetically definable. -/\ndef unbeta (l : List ℕ) : ℕ :=\n  (chineseRemainderOfFinset (ι := Fin l.length) (l[·]) (coprimes (l[·])) Finset.univ\n    (by simp [coprimes])\n    (by simpa using Set.pairwise_univ.mpr (pairwise_coprime_coprimes _)) : ℕ).pair\n  (supOfSeq (m := l.length) (l[·]))!\n\n/-- **Gödel's Beta Function Lemma** -/\n\nTarget:\nlemma beta_unbeta_coe (l : List ℕ) (i : Fin l.length) : beta (unbeta l) i = l[i] :=\n\nProof body:\n","rejected":"by\n  exact beta_unbeta_coe","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"6c209f2942e1f9bc05708f42d396566b4571827d54ae1ff7daeee1fbeb68846a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Godel","family_id":"beta_unbeta_coe","file_id":"mathlib/Mathlib/Logic/Godel/GodelBetaFunction.lean","sample_id":"8dd69de99772e4b60d7d7c33b72ef5655efe8caa0668569bdbff7dc2b9a6a47c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"11fbcca75f5b8a4dbad7c2490be392157767b93ac85f1ccad1aff23708a841e9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9d02c26fa4752bc8511f5fa4db9b7d94bfbc70719e13a7e3abd5064d08f2bbe1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8c84f0871128efc9bcf9d29301aeb6213d25f58a7d79db843c43cab6078afc5a","source_sha256":"4a480f7815d84fa6cbaa2c651b3b8967b4642ebe2873904b75dbadaa0411117b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro x hx\n  simp only [Subalgebra.mem_map, Subalgebra.mem_center_iff] at hx ⊢\n  obtain ⟨b, hb0, rfl⟩ := hx\n  intro bc\n  induction bc using TensorProduct.induction_on with\n  | zero => simp\n  | tmul b' c => simp [hb0]\n  | add _ _ _ _ => simp_all [add_mul, mul_add]","hard_negative":true,"metrics":{"chosen_tokens":63,"rejected_tokens":7,"token_jaccard":0.071429,"token_length_ratio":0.111111},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"8c01ab0fce227ae148c346fdedff0f02cfd1761370d03dd7a48f2b4b25984245","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Central.Basic\npublic import Mathlib.RingTheory.Flat.Basic\npublic import Mathlib.LinearAlgebra.Basis.VectorSpace\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang, Yunzhou Xie\n-/\n/-!\n\n# Lemmas about tensor products of central algebras\n\nIn this file we prove for algebras `B` and `C` over a field `K` that if `B ⊗[K] C` is a central\nalgebra and `B, C` nontrivial, then both `B` and `C` are central algebras.\n\n## Main Results\n\n- `Algebra.IsCentral.left_of_tensor_of_field`: If `B` `C` are `K`-algebras where `K` is a field,\n  `C` is nontrivial and `B ⊗[K] C` is a central algebra over `K`, then `B` is a\n  central algebra over `K`.\n- `Algebra.IsCentral.right_of_tensor_of_field`: If `B` `C` are `K`-algebras where `K` is a field,\n  `B` is nontrivial and `B ⊗[K] C` is a central algebra over `K`, then `C` is a\n  central algebra over `K`.\n\n## Tags\nCentral Algebras, Central Simple Algebras, Noncommutative Algebra\n-/\n\npublic section\n\nuniverse u v\n\nopen TensorProduct\n\nvariable (K B C : Type*) [CommSemiring K] [Semiring B] [Semiring C] [Algebra K B] [Algebra K C]\n\nTarget:\nlemma Algebra.TensorProduct.includeLeft_map_center_le :\n    (Subalgebra.center K B).map includeLeft ≤ Subalgebra.center K (B ⊗[K] C) :=\n\nProof body:\n","rejected":"by\n  exact Algebra.TensorProduct.includeLeft_map_center_le","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"f0b6b6cb9aa1c41f55eac8bd950cdef6b42e1091b3b4e18ade0942d044c2b6eb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Central","family_id":"algebra","file_id":"mathlib/Mathlib/Algebra/Central/TensorProduct.lean","sample_id":"8c84f0871128efc9bcf9d29301aeb6213d25f58a7d79db843c43cab6078afc5a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a14f9a077bdd5b6f8dbacb2f648d28154c7ace639c1cada2afcd1dc3728bb20d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"934a7df138e5d7d9c932ccbe25af68483645b672c40effd2f9069d03cb8c42ea","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"36f6292b2bea01ba2949cab99df0942afa3d01db01b2c1e30655e18e62dad252","source_sha256":"e94c5e4561a46cbb726026eb6950cfdf4d135e8e5c0b4f8f268b42813d064fee","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  intro Z ⟨g, gc⟩ ⟨h, hc⟩ e\n  apply_fun InducedCategory.Hom.hom at e\n  obtain rfl : g = h := ConcreteCategory.hom_ext _ _ fun y ↦ by\n    rw [← (h₁ y).choose_spec]\n    simpa using congr(($e).base.hom (h₁ y).choose)\n  refine SheafedSpace.hom_stalk_ext ⟨g, gc⟩ ⟨g, hc⟩ rfl fun y ↦ ?_\n  rw [← (h₁ y).choose_spec, ← cancel_mono (f.hom.stalkMap (h₁ y).choose), stalkCongr_hom,\n    stalkSpecializes_refl, Category.id_comp, ← PresheafedSpace.stalkMap.comp f.hom ⟨g, gc⟩,\n    ← PresheafedSpace.stalkMap.comp f.hom ⟨g, hc⟩]\n  congr 1","hard_negative":true,"metrics":{"chosen_tokens":158,"rejected_tokens":2,"token_jaccard":0.017241,"token_length_ratio":0.012658},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"8c2e8344bf1a0380244746b8c6fa37aa3f588797df8bed056b9bd6e225a54259","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits\npublic import Mathlib.Geometry.RingedSpace.Stalks\npublic import Mathlib.Topology.Sheaves.Functors\n\nNamespace:\nAlgebraicGeometry.SheafedSpace\n\nLocal context:\n/-\nCopyright (c) 2019 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Sheafed spaces\n\nIntroduces the category of topological spaces equipped with a sheaf (taking values in an\narbitrary target category `C`).\n\nWe further describe how to apply functors and natural transformations to the values of the\npresheaves.\n-/\n\n@[expose] public section\n\nopen CategoryTheory TopCat TopologicalSpace Opposite CategoryTheory.Limits CategoryTheory.Category\n  CategoryTheory.Functor Topology\n\nuniverse u v w' w\n\nvariable (C : Type u) [Category.{v} C]\n\n\n-- We could enable the following line:\n-- attribute [local aesop safe cases (rule_sets := [CategoryTheory])] Opposite\n-- but may need\n-- https://github.com/leanprover-community/aesop/issues/59\n\nnamespace AlgebraicGeometry\n\n/-- A `SheafedSpace C` is a topological space equipped with a sheaf of `C`s. -/\nstructure SheafedSpace extends PresheafedSpace C where\n  /-- A sheafed space is a presheafed space which happens to be a sheaf. -/\n  IsSheaf : presheaf.IsSheaf\n\nvariable {C}\n\nnamespace SheafedSpace\n\ninstance coeCarrier : CoeOut (SheafedSpace C) TopCat where coe X := X.carrier\n\ninstance coeSort : CoeSort (SheafedSpace C) Type* where\n  coe X := X.1\n\n/-- Extract the `sheaf C (X : Top)` from a `SheafedSpace C`. -/\ndef sheaf (X : SheafedSpace C) : Sheaf C (X : TopCat) :=\n  ⟨X.presheaf, X.IsSheaf⟩\n\n/-- Not `@[simp]` since it already reduces to `carrier = carrier`. -/\ntheorem mk_coe (carrier) (presheaf) (h) :\n    (({ carrier\n        presheaf\n        IsSheaf := h } : SheafedSpace C) : TopCat) = carrier :=\n  rfl\n\ninstance (X : SheafedSpace C) : TopologicalSpace X :=\n  X.carrier.str\n\n/-- The trivial `unit`-valued sheaf on any topological space. -/\ndef unit (X : TopCat) : SheafedSpace (Discrete Unit) :=\n  { @PresheafedSpace.const (Discrete Unit) _ X ⟨⟨⟩⟩ with IsSheaf := Presheaf.isSheaf_unit _ }\n\ninstance : Inhabited (SheafedSpace (Discrete Unit)) :=\n  ⟨unit (TopCat.of PEmpty)⟩\n\ninstance : Category (SheafedSpace C) :=\n  inferInstanceAs <| Category (InducedCategory (PresheafedSpace C) SheafedSpace.toPresheafedSpace)\n\n@[ext (iff := false)]\ntheorem ext {X Y : SheafedSpace C} (α β : X ⟶ Y) (w : α.hom.base = β.hom.base)\n    (h : α.hom.c ≫ whiskerRight (eqToHom (by rw [w])) _ = β.hom.c) : α = β :=\n  InducedCategory.hom_ext (PresheafedSpace.ext _ _ w h)\n\n/-- Constructor for isomorphisms in the category `SheafedSpace C`. -/\n@[simps]\ndef isoMk {X Y : SheafedSpace C} (e : X.toPresheafedSpace ≅ Y.toPresheafedSpace) : X ≅ Y where\n  hom := InducedCategory.homMk e.hom\n  inv := InducedCategory.homMk e.inv\n  hom_inv_id := InducedCategory.hom_ext e.hom_inv_id\n  inv_hom_id := InducedCategory.hom_ext e.inv_hom_id\n\n/-- Forgetting the sheaf condition is a functor from `SheafedSpace C` to `PresheafedSpace C`. -/\n@[simps! obj map]\ndef forgetToPresheafedSpace : SheafedSpace C ⥤ PresheafedSpace C :=\n  inducedFunctor _\n-- The `Full, Faithful` instances should be constructed by a deriving handler.\n-- https://github.com/leanprover-community/mathlib4/issues/380\n\n/-- The functor `forgetToPresheafedSpace : SheafedSpace C ⥤ PresheafedSpace C`\nis fully faithful. -/\ndef fullyFaithfulForgetToPresheafedSpace :\n    (forgetToPresheafedSpace (C := C)).FullyFaithful where\n  preimage f := InducedCategory.homMk f\n\n@[simp]\nlemma fullyFaithfulForgetToPresheafedSpace_preimage_hom {X Y : SheafedSpace C}\n    (f : forgetToPresheafedSpace.obj X ⟶ forgetToPresheafedSpace.obj Y) :\n    (fullyFaithfulForgetToPresheafedSpace.preimage f).hom = f := rfl\n\ninstance forgetToPresheafedSpace_full : (forgetToPresheafedSpace (C := C)).Full :=\n  fullyFaithfulForgetToPresheafedSpace.full\n\ninstance forgetToPresheafedSpace_faithful : (forgetToPresheafedSpace (C := C)).Faithful :=\n  fullyFaithfulForgetToPresheafedSpace.faithful\n\ninstance is_presheafedSpace_iso {X Y : SheafedSpace C} (f : X ⟶ Y) [IsIso f] :\n    IsIso f.hom :=\n  SheafedSpace.forgetToPresheafedSpace.map_isIso f\n\nsection\n\nattribute [local simp] id comp\n\n@[simp]\ntheorem id_hom (X : SheafedSpace C) : (𝟙 X : X ⟶ X).hom = 𝟙 X.toPresheafedSpace :=\n  rfl\n\n@[simp]\ntheorem id_hom_base (X : SheafedSpace C) : (𝟙 X : X ⟶ X).hom.base = 𝟙 (X : TopCat) :=\n  rfl\n\ntheorem id_hom_c (X : SheafedSpace C) :\n    (𝟙 X : X ⟶ X).hom.c = eqToHom (Presheaf.Pushforward.id_eq X.presheaf).symm :=\n  rfl\n\ntheorem id_hom_c_app (X : SheafedSpace C) (U) :\n    (𝟙 X : X ⟶ X).hom.c.app U = 𝟙 _ := rfl\n\n@[simp]\ntheorem comp_hom_base {X Y Z : SheafedSpace C} (f : X ⟶ Y) (g : Y ⟶ Z) :\n    (f ≫ g).hom.base = f.hom.base ≫ g.hom.base :=\n  rfl\n\n@[simp]\ntheorem comp_hom_c_app {X Y Z : SheafedSpace C} (α : X ⟶ Y) (β : Y ⟶ Z) (U) :\n    (α ≫ β).hom.c.app U =\n      β.hom.c.app U ≫ α.hom.c.app (op ((Opens.map β.hom.base).obj (unop U))) :=\n  rfl\n\ntheorem comp_hom_c_app' {X Y Z : SheafedSpace C} (α : X ⟶ Y) (β : Y ⟶ Z) (U) :\n    (α ≫ β).hom.c.app (op U) =\n      β.hom.c.app (op U) ≫ α.hom.c.app (op ((Opens.map β.hom.base).obj U)) :=\n  rfl\n\ntheorem congr_hom_app {X Y : SheafedSpace C} {α β : X ⟶ Y} (h : α = β) (U) :\n    α.hom.c.app U = β.hom.c.app U ≫ X.presheaf.map (eqToHom (by subst h; rfl)) :=\n  (PresheafedSpace.congr_app (by rw [h]) U)\n\n@[deprecated (since := \"2025-12-18\")] alias id_base := id_hom_base\n@[deprecated (since := \"2025-12-18\")] alias id_c := id_hom_c\n@[deprecated (since := \"2025-12-18\")] alias id_c_app := id_hom_c_app\n@[deprecated (since := \"2025-12-18\")] alias comp_base := comp_hom_base\n@[deprecated (since := \"2025-12-18\")] alias comp_c_app := comp_hom_c_app\n@[deprecated (since := \"2025-12-18\")] alias comp_c_app' := comp_hom_c_app'\n@[deprecated (since := \"2025-12-18\")] alias congr_app := congr_hom_app\n\nvariable (C)\n\n/-- The forgetful functor from `SheafedSpace` to `Top`. -/\ndef forget : SheafedSpace C ⥤ TopCat where\n  obj X := (X : TopCat)\n  map {_ _} f := f.hom.base\n\nend\n\nopen TopCat.Presheaf\n\n/-- The restriction of a sheafed space along an open embedding into the space.\n-/\ndef restrict {U : TopCat} (X : SheafedSpace C) {f : U ⟶ (X : TopCat)} (h : IsOpenEmbedding f) :\n    SheafedSpace C :=\n  { X.toPresheafedSpace.restrict h with IsSheaf := isSheaf_of_isOpenEmbedding h X.IsSheaf }\n\n/-- The map from the restriction of a presheafed space.\n-/\n@[simps!]\ndef ofRestrict {U : TopCat} (X : SheafedSpace C) {f : U ⟶ (X : TopCat)}\n    (h : IsOpenEmbedding f) : X.restrict h ⟶ X :=\n  InducedCategory.homMk (X.toPresheafedSpace.ofRestrict h)\n\n/-- The restriction of a sheafed space `X` to the top subspace is isomorphic to `X` itself.\n-/\n@[simps! hom inv]\ndef restrictTopIso (X : SheafedSpace C) : X.restrict (Opens.isOpenEmbedding ⊤) ≅ X :=\n  isoMk (X.toPresheafedSpace.restrictTopIso)\n\n/-- The global sections, notated Gamma.\n-/\ndef Γ : (SheafedSpace C)ᵒᵖ ⥤ C :=\n  forgetToPresheafedSpace.op ⋙ PresheafedSpace.Γ\n\ntheorem Γ_def : (Γ : _ ⥤ C) = forgetToPresheafedSpace.op ⋙ PresheafedSpace.Γ :=\n  rfl\n\n@[simp]\ntheorem Γ_obj (X : (SheafedSpace C)ᵒᵖ) : Γ.obj X = (unop X).presheaf.obj (op ⊤) :=\n  rfl\n\ntheorem Γ_obj_op (X : SheafedSpace C) : Γ.obj (op X) = X.presheaf.obj (op ⊤) :=\n  rfl\n\n@[simp]\ntheorem Γ_map {X Y : (SheafedSpace C)ᵒᵖ} (f : X ⟶ Y) : Γ.map f = f.unop.hom.c.app (op ⊤) :=\n  rfl\n\ntheorem Γ_map_op {X Y : SheafedSpace C} (f : X ⟶ Y) : Γ.map f.op = f.hom.c.app (op ⊤) :=\n  rfl\n\nnoncomputable instance (J : Type w) [Category.{w'} J] [Small.{v} J] [HasLimitsOfShape Jᵒᵖ C] :\n    CreatesColimitsOfShape J (forgetToPresheafedSpace : SheafedSpace.{_, _, v} C ⥤ _) :=\n  ⟨fun {K} =>\n    createsColimitOfFullyFaithfulOfIso\n      ⟨(PresheafedSpace.colimitCocone (K ⋙ forgetToPresheafedSpace)).pt,\n        limit_isSheaf _ fun j ↦ Sheaf.pushforward_sheaf_of_sheaf _ (K.obj (unop j)).2⟩\n      (colimit.isoColimitCocone ⟨_, PresheafedSpace.colimitCoconeIsColimit _⟩).symm⟩\n\nnoncomputable instance [HasLimits C] :\n    CreatesColimits (forgetToPresheafedSpace : SheafedSpace C ⥤ _) where\n\ninstance (J : Type w) [Category.{w'} J] [Small.{v} J] [HasLimitsOfShape Jᵒᵖ C] :\n    HasColimitsOfShape J (SheafedSpace.{_, _, v} C) :=\n  hasColimitsOfShape_of_hasColimitsOfShape_createsColimitsOfShape forgetToPresheafedSpace\n\ninstance [HasLimits C] : HasColimits.{v} (SheafedSpace C) where\n\ninstance (J : Type w) [Category.{w'} J] [Small.{v} J] [HasLimitsOfShape Jᵒᵖ C] :\n    PreservesColimitsOfShape J (forget.{_, _, v} C) :=\n  Limits.comp_preservesColimitsOfShape forgetToPresheafedSpace (PresheafedSpace.forget C)\n\nnoncomputable instance [HasLimits C] : PreservesColimits (forget.{_, _, v} C) where\n\nsection ConcreteCategory\n\nvariable {FC : C → C → Type*} {CC : C → Type v} [∀ X Y, FunLike (FC X Y) (CC X) (CC Y)]\nvariable [instCC : ConcreteCategory.{v} C FC] [HasColimits C] [HasLimits C]\nvariable [PreservesLimits (CategoryTheory.forget C)]\nvariable [PreservesFilteredColimits (CategoryTheory.forget C)]\nvariable [(CategoryTheory.forget C).ReflectsIsomorphisms]\n\nattribute [local ext] DFunLike.ext in\ninclude instCC in\nlemma hom_stalk_ext {X Y : SheafedSpace C} (f g : X ⟶ Y) (h : f.hom.base = g.hom.base)\n    (h' : ∀ x, f.hom.stalkMap x = (Y.presheaf.stalkCongr (h ▸ rfl)).hom ≫ g.hom.stalkMap x) :\n    f = g := by\n  obtain ⟨f, fc⟩ := f\n  obtain ⟨g, gc⟩ := g\n  obtain rfl : f = g := h\n  congr\n  ext U s\n  refine section_ext X.sheaf _ _ _ fun x hx ↦\n    show X.presheaf.germ _ x _ _ = X.presheaf.germ _ x _ _ from ?_\n  erw [← PresheafedSpace.stalkMap_germ_apply ⟨f, fc⟩, ← PresheafedSpace.stalkMap_germ_apply ⟨f, gc⟩]\n  simp [h']\n\nattribute [local ext] DFunLike.ext in\ninclude instCC in\nlemma mono_of_base_injective_of_stalk_epi {X Y : SheafedSpace C} (f : X ⟶ Y)\n    (h₁ : Function.Injective f.hom.base)\n    (h₂ : ∀ x, Epi (f.hom.stalkMap x)) : Mono f := by\n  constructor\n  intro Z ⟨g, gc⟩ ⟨h, hc⟩ e\n  obtain rfl : g = h := ConcreteCategory.hom_ext _ _ fun x ↦ h₁ congr(($e).hom.base x)\n  refine SheafedSpace.hom_stalk_ext ⟨g, gc⟩ ⟨g, hc⟩ rfl fun x ↦ ?_\n  rw [← cancel_epi (f.hom.stalkMap (g x)), stalkCongr_hom, stalkSpecializes_refl, Category.id_comp,\n    ← PresheafedSpace.stalkMap.comp ⟨g, gc⟩ f.hom, ← PresheafedSpace.stalkMap.comp ⟨g, hc⟩ f.hom]\n  replace e := congr_arg InducedCategory.Hom.hom e\n  congr 1\n\nattribute [local ext] DFunLike.ext in\ninclude instCC in\n\nTarget:\nlemma epi_of_base_surjective_of_stalk_mono {X Y : SheafedSpace C} (f : X ⟶ Y)\n    (h₁ : Function.Surjective f.hom.base)\n    (h₂ : ∀ x, Mono (f.hom.stalkMap x)) : Epi f :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_36f6292b2bea","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b689190bd4e840c05368912875a34f6d5ff7ae6f3b456f4fe6cde9653bc2d625","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/RingedSpace","family_id":"epi_of_base_surjective_of_stalk_mono","file_id":"mathlib/Mathlib/Geometry/RingedSpace/SheafedSpace.lean","sample_id":"36f6292b2bea01ba2949cab99df0942afa3d01db01b2c1e30655e18e62dad252"},"split":"train"}
{"binding":{"chosen_proof_sha256":"31b27ce5f99a3bf78bf18a3e524b1460389b6e7ac4458283ab4e0eee4a7b4805","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"def358f035864fb5fca9334490b9dd72335d01609cd1d97f4732cac12499a95d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"767863c4f273ffd20ab08a8048315c1504cb395ceb5b970fd18d5025077b58cc","source_sha256":"f31e0ea5c5dcaaef4fe7a595be40e4dcc670dcec12bcae3dfce625de497aea8b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold unfolds; rw [corec_eq]","hard_negative":false,"metrics":{"chosen_tokens":8,"rejected_tokens":13,"token_jaccard":0.615385,"token_length_ratio":1.625},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"8c671c3dfeac12d7f8141e13a69eb10674a2bc90ac5145510ca7145f9fa044b7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Stream.Defs\npublic import Mathlib.Logic.Function.Basic\npublic import Mathlib.Data.Nat.Basic\npublic import Mathlib.Tactic.Common\n\nNamespace:\nStream'\n\nLocal context:\n/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Streams a.k.a. infinite lists a.k.a. infinite sequences\n-/\n\n@[expose] public section\n\nopen Nat Function Option\n\nnamespace Stream'\n\nuniverse u v w\nvariable {α : Type u} {β : Type v} {δ : Type w}\nvariable (m n : ℕ) (x y : List α) (a b : Stream' α)\n\ninstance [Inhabited α] : Inhabited (Stream' α) :=\n  ⟨Stream'.const default⟩\n\n@[simp] protected theorem eta (s : Stream' α) : head s :: tail s = s :=\n  funext fun i => by cases i <;> rfl\n\n/-- Alias for `Stream'.eta` to match `List` API. -/\nalias cons_head_tail := Stream'.eta\n\n@[ext]\nprotected theorem ext {s₁ s₂ : Stream' α} : (∀ n, get s₁ n = get s₂ n) → s₁ = s₂ :=\n  fun h => funext h\n\n@[simp]\ntheorem get_zero_cons (a : α) (s : Stream' α) : get (a::s) 0 = a :=\n  rfl\n\n@[simp]\ntheorem head_cons (a : α) (s : Stream' α) : head (a::s) = a :=\n  rfl\n\n@[simp]\ntheorem tail_cons (a : α) (s : Stream' α) : tail (a::s) = s :=\n  rfl\n\n@[simp]\ntheorem get_drop (n m : ℕ) (s : Stream' α) : get (drop m s) n = get s (m + n) := by\n  rw [Nat.add_comm]\n  rfl\n\ntheorem tail_eq_drop (s : Stream' α) : tail s = drop 1 s :=\n  rfl\n\n@[simp]\ntheorem drop_drop (n m : ℕ) (s : Stream' α) : drop n (drop m s) = drop (m + n) s := by\n  ext; simp [Nat.add_assoc]\n\n@[simp] theorem get_tail {n : ℕ} {s : Stream' α} : s.tail.get n = s.get (n + 1) := rfl\n\n@[simp] theorem tail_drop' {i : ℕ} {s : Stream' α} : tail (drop i s) = s.drop (i + 1) := by\n  ext; simp [Nat.add_comm, Nat.add_left_comm]\n\n@[simp] theorem drop_tail' {i : ℕ} {s : Stream' α} : drop i (tail s) = s.drop (i + 1) := rfl\n\ntheorem tail_drop (n : ℕ) (s : Stream' α) : tail (drop n s) = drop n (tail s) := by simp\n\ntheorem get_succ (n : ℕ) (s : Stream' α) : get s (succ n) = get (tail s) n :=\n  rfl\n\n@[simp]\ntheorem get_succ_cons (n : ℕ) (s : Stream' α) (x : α) : get (x :: s) n.succ = get s n :=\n  rfl\n\n@[simp] lemma get_cons_append_zero {a : α} {x : List α} {s : Stream' α} :\n    (a :: x ++ₛ s).get 0 = a := rfl\n\n@[simp] lemma append_eq_cons {a : α} {as : Stream' α} : [a] ++ₛ as = a :: as := rfl\n\n@[simp] theorem drop_zero {s : Stream' α} : s.drop 0 = s := rfl\n\ntheorem drop_succ (n : ℕ) (s : Stream' α) : drop (succ n) s = drop n (tail s) :=\n  rfl\n\ntheorem head_drop (a : Stream' α) (n : ℕ) : (a.drop n).head = a.get n := by simp\n\ntheorem cons_injective2 : Function.Injective2 (cons : α → Stream' α → Stream' α) := fun x y s t h =>\n  ⟨by rw [← get_zero_cons x s, h, get_zero_cons],\n    Stream'.ext fun n => by rw [← get_succ_cons n _ x, h, get_succ_cons]⟩\n\ntheorem cons_injective_left (s : Stream' α) : Function.Injective fun x => cons x s :=\n  cons_injective2.left _\n\ntheorem cons_injective_right (x : α) : Function.Injective (cons x) :=\n  cons_injective2.right _\n\ntheorem all_def (p : α → Prop) (s : Stream' α) : All p s = ∀ n, p (get s n) :=\n  rfl\n\ntheorem any_def (p : α → Prop) (s : Stream' α) : Any p s = ∃ n, p (get s n) :=\n  rfl\n\n@[simp]\ntheorem mem_cons (a : α) (s : Stream' α) : a ∈ a::s :=\n  Exists.intro 0 rfl\n\ntheorem mem_cons_of_mem {a : α} {s : Stream' α} (b : α) : a ∈ s → a ∈ b::s := fun ⟨n, h⟩ =>\n  Exists.intro (succ n) (by rw [get_succ, tail_cons, h])\n\ntheorem eq_or_mem_of_mem_cons {a b : α} {s : Stream' α} : (a ∈ b::s) → a = b ∨ a ∈ s :=\n    fun ⟨n, h⟩ => by\n  rcases n with - | n'\n  · left\n    exact h\n  · right\n    rw [get_succ, tail_cons] at h\n    exact ⟨n', h⟩\n\ntheorem mem_of_get_eq {n : ℕ} {s : Stream' α} {a : α} : a = get s n → a ∈ s := fun h =>\n  Exists.intro n h\n\ntheorem mem_iff_exists_get_eq {s : Stream' α} {a : α} : a ∈ s ↔ ∃ n, a = s.get n where\n  mp := by simp [Membership.mem, any_def]\n  mpr h := mem_of_get_eq h.choose_spec\n\nsection Map\n\nvariable (f : α → β)\n\ntheorem drop_map (n : ℕ) (s : Stream' α) : drop n (map f s) = map f (drop n s) :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem get_map (n : ℕ) (s : Stream' α) : get (map f s) n = f (get s n) :=\n  rfl\n\ntheorem tail_map (s : Stream' α) : tail (map f s) = map f (tail s) := rfl\n\n@[simp]\ntheorem head_map (s : Stream' α) : head (map f s) = f (head s) :=\n  rfl\n\ntheorem map_eq (s : Stream' α) : map f s = f (head s)::map f (tail s) := by\n  rw [← Stream'.eta (map f s), tail_map, head_map]\n\ntheorem map_cons (a : α) (s : Stream' α) : map f (a::s) = f a::map f s := by\n  rw [← Stream'.eta (map f (a::s)), map_eq]; rfl\n\n@[simp]\ntheorem map_id (s : Stream' α) : map id s = s :=\n  rfl\n\n@[simp]\ntheorem map_map (g : β → δ) (f : α → β) (s : Stream' α) : map g (map f s) = map (g ∘ f) s :=\n  rfl\n\n@[simp]\ntheorem map_tail (s : Stream' α) : map f (tail s) = tail (map f s) :=\n  rfl\n\ntheorem mem_map {a : α} {s : Stream' α} : a ∈ s → f a ∈ map f s := fun ⟨n, h⟩ =>\n  Exists.intro n (by rw [get_map, h])\n\ntheorem exists_of_mem_map {f} {b : β} {s : Stream' α} : b ∈ map f s → ∃ a, a ∈ s ∧ f a = b :=\n  fun ⟨n, h⟩ => ⟨get s n, ⟨n, rfl⟩, h.symm⟩\n\nend Map\n\nsection Zip\n\nvariable (f : α → β → δ)\n\ntheorem drop_zip (n : ℕ) (s₁ : Stream' α) (s₂ : Stream' β) :\n    drop n (zip f s₁ s₂) = zip f (drop n s₁) (drop n s₂) :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem get_zip (n : ℕ) (s₁ : Stream' α) (s₂ : Stream' β) :\n    get (zip f s₁ s₂) n = f (get s₁ n) (get s₂ n) :=\n  rfl\n\ntheorem head_zip (s₁ : Stream' α) (s₂ : Stream' β) : head (zip f s₁ s₂) = f (head s₁) (head s₂) :=\n  rfl\n\ntheorem tail_zip (s₁ : Stream' α) (s₂ : Stream' β) :\n    tail (zip f s₁ s₂) = zip f (tail s₁) (tail s₂) :=\n  rfl\n\ntheorem zip_eq (s₁ : Stream' α) (s₂ : Stream' β) :\n    zip f s₁ s₂ = f (head s₁) (head s₂)::zip f (tail s₁) (tail s₂) := by\n  rw [← Stream'.eta (zip f s₁ s₂)]; rfl\n\n@[simp]\ntheorem get_enum (s : Stream' α) (n : ℕ) : get (enum s) n = (n, s.get n) :=\n  rfl\n\ntheorem enum_eq_zip (s : Stream' α) : enum s = zip Prod.mk nats s :=\n  rfl\n\nend Zip\n\n@[simp]\ntheorem mem_const (a : α) : a ∈ const a :=\n  Exists.intro 0 rfl\n\ntheorem const_eq (a : α) : const a = a::const a := by\n  apply Stream'.ext; intro n\n  cases n <;> rfl\n\n@[simp]\ntheorem tail_const (a : α) : tail (const a) = const a :=\n  suffices tail (a::const a) = const a by rwa [← const_eq] at this\n  rfl\n\n@[simp]\ntheorem map_const (f : α → β) (a : α) : map f (const a) = const (f a) :=\n  rfl\n\n@[simp]\ntheorem get_const (n : ℕ) (a : α) : get (const a) n = a :=\n  rfl\n\n@[simp]\ntheorem drop_const (n : ℕ) (a : α) : drop n (const a) = const a :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem head_iterate (f : α → α) (a : α) : head (iterate f a) = a :=\n  rfl\n\ntheorem get_succ_iterate' (n : ℕ) (f : α → α) (a : α) :\n    get (iterate f a) (succ n) = f (get (iterate f a) n) := rfl\n\ntheorem tail_iterate (f : α → α) (a : α) : tail (iterate f a) = iterate f (f a) := by\n  ext n\n  rw [get_tail]\n  induction n with\n  | zero => rfl\n  | succ n ih => rw [get_succ_iterate', ih, get_succ_iterate']\n\ntheorem iterate_eq (f : α → α) (a : α) : iterate f a = a::iterate f (f a) := by\n  rw [← Stream'.eta (iterate f a)]\n  rw [tail_iterate]; rfl\n\n@[simp]\ntheorem get_zero_iterate (f : α → α) (a : α) : get (iterate f a) 0 = a :=\n  rfl\n\ntheorem get_succ_iterate (n : ℕ) (f : α → α) (a : α) :\n    get (iterate f a) (succ n) = get (iterate f (f a)) n := by rw [get_succ, tail_iterate]\n\nsection Bisim\n\nvariable (R : Stream' α → Stream' α → Prop)\n\n/-- equivalence relation -/\nlocal infixl:50 \" ~ \" => R\n\n/-- Streams `s₁` and `s₂` are defined to be bisimulations if\ntheir heads are equal and tails are bisimulations. -/\ndef IsBisimulation :=\n  ∀ ⦃s₁ s₂⦄, s₁ ~ s₂ →\n      head s₁ = head s₂ ∧ tail s₁ ~ tail s₂\n\ntheorem get_of_bisim (bisim : IsBisimulation R) {s₁ s₂} :\n    ∀ n, s₁ ~ s₂ → get s₁ n = get s₂ n ∧ drop (n + 1) s₁ ~ drop (n + 1) s₂\n  | 0, h => bisim h\n  | n + 1, h =>\n    match bisim h with\n    | ⟨_, trel⟩ => get_of_bisim bisim n trel\n\n-- If two streams are bisimilar, then they are equal\ntheorem eq_of_bisim (bisim : IsBisimulation R) {s₁ s₂} : s₁ ~ s₂ → s₁ = s₂ := fun r =>\n  Stream'.ext fun n => And.left (get_of_bisim R bisim n r)\n\nend Bisim\n\ntheorem bisim_simple (s₁ s₂ : Stream' α) :\n    head s₁ = head s₂ → s₁ = tail s₁ → s₂ = tail s₂ → s₁ = s₂ := fun hh ht₁ ht₂ =>\n  eq_of_bisim (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂)\n    (fun s₁ s₂ ⟨h₁, h₂, h₃⟩ => by grind)\n    (And.intro hh (And.intro ht₁ ht₂))\n\ntheorem coinduction {s₁ s₂ : Stream' α} :\n    head s₁ = head s₂ →\n      (∀ (β : Type u) (fr : Stream' α → β),\n      fr s₁ = fr s₂ → fr (tail s₁) = fr (tail s₂)) → s₁ = s₂ :=\n  fun hh ht =>\n  eq_of_bisim\n    (fun s₁ s₂ =>\n      head s₁ = head s₂ ∧\n        ∀ (β : Type u) (fr : Stream' α → β), fr s₁ = fr s₂ → fr (tail s₁) = fr (tail s₂))\n    (fun s₁ s₂ h =>\n      have h₁ : head s₁ = head s₂ := And.left h\n      have h₂ : head (tail s₁) = head (tail s₂) := And.right h α (@head α) h₁\n      have h₃ :\n        ∀ (β : Type u) (fr : Stream' α → β),\n          fr (tail s₁) = fr (tail s₂) → fr (tail (tail s₁)) = fr (tail (tail s₂)) :=\n        fun β fr => And.right h β fun s => fr (tail s)\n      And.intro h₁ (And.intro h₂ h₃))\n    (And.intro hh ht)\n\n@[simp]\ntheorem iterate_id (a : α) : iterate id a = const a :=\n  coinduction rfl fun β fr ch => by rw [tail_iterate, tail_const]; exact ch\n\ntheorem map_iterate (f : α → α) (a : α) : iterate f (f a) = map f (iterate f a) := by\n  funext n\n  induction n with\n  | zero => rfl\n  | succ n ih =>\n    unfold map iterate get\n    rw [map, get] at ih\n    rw [iterate]\n    exact congrArg f ih\n\nsection Corec\n\ntheorem corec_def (f : α → β) (g : α → α) (a : α) : corec f g a = map f (iterate g a) :=\n  rfl\n\ntheorem corec_eq (f : α → β) (g : α → α) (a : α) : corec f g a = f a :: corec f g (g a) := by\n  rw [corec_def, map_eq, head_iterate, tail_iterate]; rfl\n\ntheorem corec_id_id_eq_const (a : α) : corec id id a = const a := by\n  rw [corec_def, map_id, iterate_id]\n\ntheorem corec_id_f_eq_iterate (f : α → α) (a : α) : corec id f a = iterate f a :=\n  rfl\n\nend Corec\n\nsection Corec'\n\ntheorem corec'_eq (f : α → β × α) (a : α) : corec' f a = (f a).1 :: corec' f (f a).2 :=\n  corec_eq _ _ _\n\nend Corec'\n\nTarget:\ntheorem unfolds_eq (g : α → β) (f : α → α) (a : α) : unfolds g f a = g a :: unfolds g f (f a) :=\n\nProof body:\n","rejected":"by\n  unfold unfolds; rw [corec_eq]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Stream","family_id":"unfolds_eq","file_id":"mathlib/Mathlib/Data/Stream/Init.lean","sample_id":"767863c4f273ffd20ab08a8048315c1504cb395ceb5b970fd18d5025077b58cc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"53c62f34c0746917df7ca2e7b0151f3d1cc2f5a2e7dd8a4133dc4a6cf7d81680","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"21df6f68f8686f6d7b4ab378fa3f7d2564d582425a9b504a6d726062d4362c74","source_sha256":"96d763a04d380b57c61dd8bcbe51e54cac3fe376ddcd3d5a1be83055f4846a12","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro f g hfg\n  have : f.toAlgHom = g.toAlgHom := by\n    apply (hμ.powerBasis K).algHom_ext\n    rw [AlgEquiv.coe_algHom, AlgEquiv.coe_algHom, powerBasis_gen,\n      ← autToPow_spec K hμ g, ← autToPow_spec K hμ f, hfg]\n  exact AlgEquiv.coe_algHom_injective this","hard_negative":false,"metrics":{"chosen_tokens":59,"rejected_tokens":2,"token_jaccard":0.03125,"token_length_ratio":0.033898},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"8c87b79e6f7a98f4cd2e16b9be5cf4a47374f051214cdebb0452b49782d7433e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots\npublic import Mathlib.FieldTheory.PolynomialGaloisGroup\n\nNamespace:\nIsPrimitiveRoot\n\nLocal context:\n/-\nCopyright (c) 2022 Eric Rodriguez. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Rodriguez\n-/\n/-!\n# Galois group of cyclotomic extensions\n\nIn this file, we show the relationship between the Galois group of `K(ζₙ)` and `(ZMod n)ˣ`;\nit is always a subgroup, and if the `n`th cyclotomic polynomial is irreducible, they are isomorphic.\n\n## Main results\n\n* `IsPrimitiveRoot.autToPow_injective`: `IsPrimitiveRoot.autToPow` is injective\n  in the case that it's considered over a cyclotomic field extension.\n* `IsCyclotomicExtension.autEquivPow`: If the `n`th cyclotomic polynomial is irreducible in `K`,\n  then `IsPrimitiveRoot.autToPow` is a `MulEquiv` (for example, in `ℚ` and certain `𝔽ₚ`).\n* `galXPowEquivUnitsZMod`, `galCyclotomicEquivUnitsZMod`: Repackage\n  `IsCyclotomicExtension.autEquivPow` in terms of `Polynomial.Gal`.\n* `IsCyclotomicExtension.Aut.commGroup`: Cyclotomic extensions are abelian.\n\n## References\n\n* https://kconrad.math.uconn.edu/blurbs/galoistheory/cyclotomic.pdf\n\n## TODO\n\n* We currently can get away with the fact that the power of a primitive root is a primitive root,\n  but the correct long-term solution for computing other explicit Galois groups is creating\n  `PowerBasis.map_conjugate`; but figuring out the exact correct assumptions + proof for this is\n  mathematically nontrivial. (Current thoughts: the correct condition is that the annihilating\n  ideal of both elements is equal. This may not hold in an ID, and definitely holds in an ICD.)\n\n-/\n\n@[expose] public section\n\n\nvariable {n : ℕ} [NeZero n] (K : Type*) [Field K] {L : Type*} {μ : L}\n\nopen Polynomial IsCyclotomicExtension\n\nopen scoped Cyclotomic\n\nnamespace IsPrimitiveRoot\n\nvariable [CommRing L] [IsDomain L] (hμ : IsPrimitiveRoot μ n) [Algebra K L]\n  [IsCyclotomicExtension {n} K L]\n\n/-- `IsPrimitiveRoot.autToPow` is injective in the case that it's considered over a cyclotomic\nfield extension. -/\n\nTarget:\ntheorem autToPow_injective : Function.Injective <| hμ.autToPow K :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Cyclotomic","family_id":"auttopow_injective","file_id":"mathlib/Mathlib/NumberTheory/Cyclotomic/Gal.lean","sample_id":"21df6f68f8686f6d7b4ab378fa3f7d2564d582425a9b504a6d726062d4362c74"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e6c5467020ffc4f18638257b26b5e8e286b407b28f63974096a991066b7eb6a4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6c168c8eafbed5003e492ac3a2d6e9416d222021ddccc0a76f9c341e5755dc56","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6be9545aedd7e6b94e2cba861b00ad8b06f63ac214c443b7f533604392c9fc8e","source_sha256":"34049d18cb594f0a184df593dbd6d4fba0ade793ef81827c21fb1d80f135b5c5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rw [← div_eq_mul_inv, ← h3, mul_div_cancel_right₀ _ h2]","hard_negative":true,"metrics":{"chosen_tokens":14,"rejected_tokens":5,"token_jaccard":0.0625,"token_length_ratio":0.357143},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"8c986f7e48b70d93115000d38d4dc4d8bc21850cfd23fccd4c99d72400d450a8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Basic\npublic import Mathlib.Algebra.Order.Ring.Defs\npublic import Mathlib.Data.Tree.Basic\npublic import Mathlib.Tactic.NormNum.Core\npublic import Mathlib.Util.SynthesizeUsing\n\nNamespace:\nMathlib.Tactic.CancelDenoms\n\nLocal context:\n/-\nCopyright (c) 2020 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n-/\npublic meta import Mathlib.Data.Tree.Basic\npublic meta import Mathlib.Algebra.Group.Nat.Defs\n\n/-!\n# A tactic for canceling numeric denominators\n\nThis file defines tactics that cancel numeric denominators from field Expressions.\n\nAs an example, we want to transform a comparison `5*(a/3 + b/4) < c/3` into the equivalent\n`5*(4*a + 3*b) < 4*c`.\n\n## Implementation notes\n\nThe tooling here was originally written for `linarith`, not intended as an interactive tactic.\nThe interactive version has been split off because it is sometimes convenient to use on its own.\nThere are likely some rough edges to it.\n\nImproving this tactic would be a good project for someone interested in learning tactic programming.\n-/\n\npublic meta section\n\nopen Lean Parser Tactic Mathlib Meta NormNum Qq\n\ninitialize registerTraceClass `CancelDenoms\n\nnamespace Mathlib.Tactic.CancelDenoms\n\n/-! ### Lemmas used in the procedure -/\n\ntheorem mul_subst {α} [CommRing α] {n1 n2 k e1 e2 t1 t2 : α}\n    (h1 : n1 * e1 = t1) (h2 : n2 * e2 = t2) (h3 : n1 * n2 = k) : k * (e1 * e2) = t1 * t2 := by\n  rw [← h3, mul_comm n1, mul_assoc n2, ← mul_assoc n1, h1,\n      ← mul_assoc n2, mul_comm n2, mul_assoc, h2]\n\ntheorem div_subst {α} [Field α] {n1 n2 k e1 e2 t1 : α}\n    (h1 : n1 * e1 = t1) (h2 : n2 / e2 = 1) (h3 : n1 * n2 = k) : k * (e1 / e2) = t1 := by\n  rw [← h3, mul_assoc, mul_div_left_comm, h2, ← mul_assoc, h1, mul_comm, one_mul]\n\ntheorem cancel_factors_eq_div {α} [Field α] {n e e' : α}\n    (h : n * e = e') (h2 : n ≠ 0) : e = e' / n :=\n  eq_div_of_mul_eq h2 <| by rwa [mul_comm] at h\n\ntheorem add_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :\n    n * (e1 + e2) = t1 + t2 := by simp [left_distrib, *]\n\ntheorem sub_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :\n    n * (e1 - e2) = t1 - t2 := by simp [left_distrib, *, sub_eq_add_neg]\n\ntheorem neg_subst {α} [Ring α] {n e t : α} (h1 : n * e = t) : n * -e = -t := by simp [*]\n\ntheorem pow_subst {α} [CommRing α] {n e1 t1 k l : α} {e2 : ℕ}\n    (h1 : n * e1 = t1) (h2 : l * n ^ e2 = k) : k * (e1 ^ e2) = l * t1 ^ e2 := by\n  rw [← h2, ← h1, mul_pow, mul_assoc]\n\nTarget:\ntheorem inv_subst {α} [Field α] {n k e : α} (h2 : e ≠ 0) (h3 : n * e = k) :\n    k * (e ⁻¹) = n :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_6be9545aedd7","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"16c8ed511401576fde6d5955bda2452974a0ec0c08a14a799e3d7590d19f5fab","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/CancelDenoms","family_id":"inv_subst","file_id":"mathlib/Mathlib/Tactic/CancelDenoms/Core.lean","sample_id":"6be9545aedd7e6b94e2cba861b00ad8b06f63ac214c443b7f533604392c9fc8e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5914fc3ed55a874cb691c1f7b7c08b38f9e056106005fb4efe3ad9109dba7f6a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"911fa94098dd26004ff50169dbf98303f1037215846ac0fa1ed69397bdb0b739","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"636539d6c0febb4c83b00bdc22993462b52f9a53f319750f1f167f122ce30681","source_sha256":"56dc78571f595294b2d5c06f0925083625898756f6d4c19bdd95afa1a756b177","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply (Ideal.homogeneousCore 𝒜 I).isHomogeneous.isPrime_of_homogeneous_mem_or_mem\n  · exact ne_top_of_le_ne_top h.ne_top (Ideal.toIdeal_homogeneousCore_le 𝒜 I)\n  rintro x y hx hy hxy\n  have H := h.mem_or_mem (Ideal.toIdeal_homogeneousCore_le 𝒜 I hxy)\n  refine H.imp ?_ ?_\n  · exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hx\n  · exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hy","hard_negative":true,"metrics":{"chosen_tokens":66,"rejected_tokens":5,"token_jaccard":0.088235,"token_length_ratio":0.075758},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"8cd3e6b23c491d3580dc0cacb0eee01ad14116d848301c335c7d6e9ce1ad6f50","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang, Eric Wieser\n-/\n/-!\n\nThis file contains a proof that the radical of any homogeneous ideal is a homogeneous ideal\n\n## Main statements\n\n* `Ideal.IsHomogeneous.isPrime_iff`: for any `I : Ideal A`, if `I` is homogeneous, then\n  `I` is prime if and only if `I` is homogeneously prime, i.e. `I ≠ ⊤` and if `x, y` are\n  homogeneous elements such that `x * y ∈ I`, then at least one of `x,y` is in `I`.\n* `Ideal.IsPrime.homogeneousCore`: for any `I : Ideal A`, if `I` is prime, then\n  `I.homogeneous_core 𝒜` (i.e. the largest homogeneous ideal contained in `I`) is also prime.\n* `Ideal.IsHomogeneous.radical`: for any `I : Ideal A`, if `I` is homogeneous, then the\n  radical of `I` is homogeneous as well.\n* `HomogeneousIdeal.radical`: for any `I : HomogeneousIdeal 𝒜`, `I.radical` is the\n  radical of `I` as a `HomogeneousIdeal 𝒜`.\n\n## Implementation details\n\nThroughout this file, the indexing type `ι` of grading is assumed to be a\nlinearly ordered cancellative monoid. This might be stronger than necessary but cancelling\nproperty is strictly necessary; for a counterexample of how `Ideal.IsHomogeneous.isPrime_iff`\nfails for a non-cancellative set see `Counterexamples/HomogeneousPrimeNotPrime.lean`.\n\n## Tags\n\nhomogeneous, radical\n-/\n\n@[expose] public section\n\n\nopen GradedRing DirectSum SetLike Finset\n\nvariable {ι σ A : Type*}\nvariable [CommRing A]\nvariable [AddCommMonoid ι] [LinearOrder ι] [IsOrderedCancelAddMonoid ι]\nvariable [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜]\n\ntheorem Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem {I : Ideal A} (hI : I.IsHomogeneous 𝒜)\n    (I_ne_top : I ≠ ⊤)\n    (homogeneous_mem_or_mem :\n      ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I) :\n    Ideal.IsPrime I :=\n  ⟨I_ne_top, by\n    intro x y hxy\n    by_contra! ⟨rid₁, rid₂⟩\n    classical\n      /-\n        The idea of the proof is the following :\n        since `x * y ∈ I` and `I` homogeneous, then `proj i (x * y) ∈ I` for any `i : ι`.\n        Then consider two sets `{i ∈ x.support | xᵢ ∉ I}` and `{j ∈ y.support | yⱼ ∉ J}`;\n        let `max₁, max₂` be the maximum of the two sets, then `proj (max₁ + max₂) (x * y) ∈ I`.\n        Then, `proj max₁ x ∉ I` and `proj max₂ j ∉ I`\n        but `proj i x ∈ I` for all `max₁ < i` and `proj j y ∈ I` for all `max₂ < j`.\n        `  proj (max₁ + max₂) (x * y)`\n        `= ∑ {(i, j) ∈ supports | i + j = max₁ + max₂}, xᵢ * yⱼ`\n        `= proj max₁ x * proj max₂ y`\n        `  + ∑ {(i, j) ∈ supports \\ {(max₁, max₂)} | i + j = max₁ + max₂}, xᵢ * yⱼ`.\n        This is a contradiction, because both `proj (max₁ + max₂) (x * y) ∈ I` and the sum on the\n        right-hand side is in `I` however `proj max₁ x * proj max₂ y` is not in `I`.\n        -/\n      set set₁ := {i ∈ (decompose 𝒜 x).support | proj 𝒜 i x ∉ I} with set₁_eq\n      set set₂ := {i ∈ (decompose 𝒜 y).support | proj 𝒜 i y ∉ I} with set₂_eq\n      have nonempty :\n        ∀ x : A, x ∉ I → {i ∈ (decompose 𝒜 x).support | proj 𝒜 i x ∉ I}.Nonempty := by\n        intro x hx\n        rw [filter_nonempty_iff]\n        contrapose! hx\n        simp_rw [proj_apply] at hx\n        rw [← sum_support_decompose 𝒜 x]\n        exact Ideal.sum_mem _ hx\n      set max₁ := set₁.max' (nonempty x rid₁)\n      set max₂ := set₂.max' (nonempty y rid₂)\n      have mem_max₁ : max₁ ∈ set₁ := max'_mem set₁ (nonempty x rid₁)\n      have mem_max₂ : max₂ ∈ set₂ := max'_mem set₂ (nonempty y rid₂)\n      replace hxy : proj 𝒜 (max₁ + max₂) (x * y) ∈ I := hI _ hxy\n      have mem_I : proj 𝒜 max₁ x * proj 𝒜 max₂ y ∈ I := by\n        set antidiag :=\n          {z ∈ (decompose 𝒜 x).support ×ˢ (decompose 𝒜 y).support | z.1 + z.2 = max₁ + max₂}\n           with ha\n        have mem_antidiag : (max₁, max₂) ∈ antidiag := by\n          simp only [antidiag, mem_filter, mem_product]\n          exact ⟨⟨mem_of_mem_filter _ mem_max₁, mem_of_mem_filter _ mem_max₂⟩, trivial⟩\n        have eq_add_sum :=\n          calc\n            proj 𝒜 (max₁ + max₂) (x * y) = ∑ ij ∈ antidiag, proj 𝒜 ij.1 x * proj 𝒜 ij.2 y := by\n              simp_rw [ha, proj_apply, DirectSum.decompose_mul, DirectSum.coe_mul_apply 𝒜]\n            _ =\n                proj 𝒜 max₁ x * proj 𝒜 max₂ y +\n                  ∑ ij ∈ antidiag.erase (max₁, max₂), proj 𝒜 ij.1 x * proj 𝒜 ij.2 y :=\n              (add_sum_erase _ _ mem_antidiag).symm\n        rw [eq_sub_of_add_eq eq_add_sum.symm]\n        refine Ideal.sub_mem _ hxy (Ideal.sum_mem _ fun z H => ?_)\n        rcases z with ⟨i, j⟩\n        simp only [antidiag, mem_erase, Prod.mk_inj, Ne, mem_filter, mem_product] at H\n        rcases H with ⟨H₁, ⟨H₂, H₃⟩, H₄⟩\n        have max_lt : max₁ < i ∨ max₂ < j := by\n          convert! le_or_lt_of_add_le_add H₄.ge using 1\n          rw [Ne.le_iff_lt]\n          rintro rfl\n          cases H₁ ⟨rfl, add_left_cancel H₄⟩\n        rcases max_lt with max_lt | max_lt\n        · -- in this case `max₁ < i`, then `xᵢ ∈ I`; for otherwise `i ∈ set₁` then `i ≤ max₁`.\n          have notMem : i ∉ set₁ := fun h =>\n            lt_irrefl _ ((max'_lt_iff set₁ (nonempty x rid₁)).mp max_lt i h)\n          rw [set₁_eq] at notMem\n          simp only [not_and, Classical.not_not, mem_filter] at notMem\n          exact Ideal.mul_mem_right _ I (notMem H₂)\n        · -- in this case `max₂ < j`, then `yⱼ ∈ I`; for otherwise `j ∈ set₂`, then `j ≤ max₂`.\n          have notMem : j ∉ set₂ := fun h =>\n            lt_irrefl _ ((max'_lt_iff set₂ (nonempty y rid₂)).mp max_lt j h)\n          rw [set₂_eq] at notMem\n          simp only [not_and, Classical.not_not, mem_filter] at notMem\n          exact Ideal.mul_mem_left I _ (notMem H₃)\n      have notMem_I : proj 𝒜 max₁ x * proj 𝒜 max₂ y ∉ I := by\n        have neither_mem : proj 𝒜 max₁ x ∉ I ∧ proj 𝒜 max₂ y ∉ I := by\n          rw [mem_filter] at mem_max₁ mem_max₂\n          exact ⟨mem_max₁.2, mem_max₂.2⟩\n        intro _rid\n        rcases homogeneous_mem_or_mem ⟨max₁, SetLike.coe_mem _⟩ ⟨max₂, SetLike.coe_mem _⟩ mem_I\n          with h | h\n        · apply neither_mem.1 h\n        · apply neither_mem.2 h\n      exact notMem_I mem_I⟩\n\ntheorem Ideal.IsHomogeneous.isPrime_iff {I : Ideal A} (h : I.IsHomogeneous 𝒜) :\n    I.IsPrime ↔\n      I ≠ ⊤ ∧\n        ∀ {x y : A},\n          IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I :=\n  ⟨fun HI => ⟨HI.ne_top, fun _ _ hxy => Ideal.IsPrime.mem_or_mem HI hxy⟩,\n    fun ⟨I_ne_top, homogeneous_mem_or_mem⟩ =>\n    h.isPrime_of_homogeneous_mem_or_mem I_ne_top @homogeneous_mem_or_mem⟩\n\nTarget:\ntheorem Ideal.IsPrime.homogeneousCore {I : Ideal A} (h : I.IsPrime) :\n    (I.homogeneousCore 𝒜).toIdeal.IsPrime :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_636539d6c0fe","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1064428c677591b347891e8362c2b706c0bf93a9a75cdac46c739b114c5cfa38","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/GradedAlgebra","family_id":"ideal","file_id":"mathlib/Mathlib/RingTheory/GradedAlgebra/Radical.lean","sample_id":"636539d6c0febb4c83b00bdc22993462b52f9a53f319750f1f167f122ce30681"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0520ccfd6af7e55e37364ebbda9a263098edd7a0ec6ad9af960d91cce2f0e2d9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b16015c2c8cb752b26e08d47468536e58a1c1a2c7cc95ef8b2185f3dd54359d8","source_sha256":"bda612cd6d1ad1cbdd6e02bafd2e7e77c11e8cfa412747033a52bbe8062964f2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← ClassGroup.mk_mk0 (FractionRing A), extendedHom_mk]\n\nvariable [IsDedekindDomain B]","hard_negative":false,"metrics":{"chosen_tokens":19,"rejected_tokens":3,"token_jaccard":0.052632,"token_length_ratio":0.157895},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"8d36f7bf3cc6649e2f83ad2ce23db280d0e98e6a968e00d5505b0f89f38f47ba","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.FractionalIdeal.Extended\npublic import Mathlib.RingTheory.ClassGroup.Basic\n\nNamespace:\nClassGroup\n\nLocal context:\n/-\nCopyright (c) 2026 Riccardo Brasca. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Birkbeck, Riccardo Brasca\n-/\n/-!\n# Class group map induced by an extension of domains\n\nFor an injective extension `A → B` of commutative domains (equivalently `Module.IsTorsionFree A B`),\nwe construct the group homomorphism `ClassGroup.extendedHom : ClassGroup A →* ClassGroup B` given by\npushing fractional ideals forward along the algebra map.\n\n## Main definitions\n\n- `ClassGroup.extendedHom A B`: the induced map between the class groups.\n- `ClassGroup.extendedIdeal A B`: the extension of a nonzero integral ideal.\n\n## Main results\n\n- `ClassGroup.extendedHom_mk`: compatibility with representatives as fractional ideals.\n- `ClassGroup.extendedHom_mk0`: compatibility with representatives as nonzero integral ideals.\n- `ClassGroup.extendedHom_comp`: compatibility of extension in a tower `A → B → C`.\n- `ClassGroup.extendedHom_eq_one_of_forall_isPrincipal`: if the extension of every ideal is\n  principal, then `ClassGroup.extendedHom A B` is trivial.\n-/\n\npublic section\n\nopen scoped nonZeroDivisors\n\nvariable (A B : Type*) [CommRing A] [CommRing B] [Algebra A B]\n  [Module.IsTorsionFree A B]\n\nnamespace ClassGroup\n\nsection CommRing\n\nvariable [IsDomain A] [IsDomain B]\n\n/-- The monoid homomorphism `ClassGroup A → ClassGroup B` induced by an\ninjective extension of domains `A → B`. -/\nnoncomputable def extendedHom : ClassGroup A →* ClassGroup B :=\n  QuotientGroup.map _ _\n    (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom)\n    (by\n      rintro _ ⟨α, rfl⟩\n      refine ⟨Units.mk0 (IsFractionRing.map (j := algebraMap A B)\n        (FaithfulSMul.algebraMap_injective _ _) (α : FractionRing A))\n        (by simp [α.ne_zero]), ?_⟩\n      simpa [coe_toPrincipalIdeal, Units.coe_map, Units.val_mk0] using!\n        (FractionalIdeal.extendedHom_spanSingleton (FractionRing B) B _).symm)\n\n@[simp]\nlemma extendedHom_quotientMk (α : (FractionalIdeal A⁰ (FractionRing A))ˣ) :\n    extendedHom A B (QuotientGroup.mk α) = QuotientGroup.mk\n      (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom α) := by\n  rfl\n\n@[simp]\ntheorem extendedHom_mk (I : (FractionalIdeal A⁰ (FractionRing A))ˣ) :\n    extendedHom A B (ClassGroup.mk _ I) = ClassGroup.mk _\n        (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom I) := by\n  rw [← ClassGroup.Quot_mk_eq_mk, ← ClassGroup.Quot_mk_eq_mk]\n  exact extendedHom_quotientMk A B I\n\n/-- The extension of a nonzero integral ideal along an injective extension of domains. -/\nabbrev extendedIdeal (I : (Ideal A)⁰) : (Ideal B)⁰ :=\n  ⟨I.1.map (algebraMap A B), mem_nonZeroDivisors_iff_ne_zero.mpr <|\n    (Ideal.map_eq_bot_iff_of_injective (FaithfulSMul.algebraMap_injective A B)).not.mpr\n      (mem_nonZeroDivisors_iff_ne_zero.mp I.2)⟩\n\n@[simp]\ntheorem extendedIdeal_extendedIdeal (C : Type*) [CommRing C] [IsDomain C] [Algebra B C]\n    [Algebra A C] [IsScalarTower A B C] [Module.IsTorsionFree B C]\n    [Module.IsTorsionFree A C] (I : (Ideal A)⁰) :\n    extendedIdeal B C (extendedIdeal A B I) = extendedIdeal A C I := by\n  simp [Ideal.map_map, IsScalarTower.algebraMap_eq A B C]\n\nend CommRing\n\nsection DedekindDomain\n\nvariable [IsDedekindDomain A] (C : Type*) [CommRing C] [Algebra B C] [Algebra A C]\n  [IsScalarTower A B C] [Module.IsTorsionFree B C] [Module.IsTorsionFree A C]\n  [IsDedekindDomain C]\n\nTarget:\ntheorem extendedHom_mk0' [IsDomain B] (I : (Ideal A)⁰) :\n    extendedHom A B (ClassGroup.mk0 I) =\n      ClassGroup.mk _ (Units.map (FractionalIdeal.extendedHom (FractionRing B) B).toMonoidHom\n      (FractionalIdeal.mk0 (FractionRing A) I)) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/ClassGroup","family_id":"extendedhom_mk","file_id":"mathlib/Mathlib/RingTheory/ClassGroup/ExtendedHom.lean","sample_id":"b16015c2c8cb752b26e08d47468536e58a1c1a2c7cc95ef8b2185f3dd54359d8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c975c088fa8efb7c0cc15ea499939f263cd071ef49bfcc3a4f706c833af7cab0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e7de025d50f83819819386b2f880ed49a8eebee597e932160cded70716ac05cb","source_sha256":"394c3d8954491754c0a03071ce7aa65ddc93c4783581cf0b4d6869361547187f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  tfae_have 1 → 2 := fun _ ↦ inferInstance\n  tfae_have 2 → 3 := fun _ ↦ IsAtomic.exists_atom _\n  tfae_have 3 → 1 := fun ⟨I, hI⟩ ↦ by\n    have ⟨_, h⟩ := isSimpleRing_iff_isTwoSided_imp.mp ‹IsSimpleRing R›\n    simp_rw [← isFullyInvariant_iff_isTwoSided] at h\n    have := isSimpleModule_iff_isAtom.mpr hI\n    obtain eq | eq := h _ (.isotypicComponent R R I)\n    · exact (hI.bot_lt.not_ge <| (le_sSup <| by exact ⟨.refl ..⟩).trans_eq eq).elim\n    exact .congr (.symm <| .trans (.ofEq _ _ eq) Submodule.topEquiv)\n  tfae_finish","hard_negative":false,"metrics":{"chosen_tokens":127,"rejected_tokens":2,"token_jaccard":0.017241,"token_length_ratio":0.015748},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"8d3c4dae441a9770b09caa9f889a65bafc5d6c877c079cada1fe2d4a603cb18e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.FreeModule.Finite.Basic\npublic import Mathlib.RingTheory.FiniteLength\npublic import Mathlib.RingTheory.SimpleModule.Isotypic\npublic import Mathlib.RingTheory.SimpleRing.Congr\npublic import Mathlib.RingTheory.SimpleRing.Matrix\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu\n-/\n/-!\n# Wedderburn–Artin Theorem\n\n## Main results\n\n* `IsSimpleRing.tfae`: a simple ring is semisimple iff it is Artinian,\n  iff it has a minimal left ideal.\n\n* `isSimpleRing_isArtinianRing_iff`: a ring is simple Artinian iff it is semisimple, isotypic,\n  and nontrivial.\n\n* `IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite`: a simple Artinian algebra is\n  isomorphic to a (finite-dimensional) matrix algebra over a division algebra. The division\n  algebra is the opposite of the endomorphism algebra of a simple (i.e., minimal) left ideal.\n\n* `IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite`: a semisimple algebra is\n  isomorphic to a finite direct product of matrix algebras over division algebras. The division\n  algebras are the opposites of the endomorphism algebras of the simple (i.e., minimal)\n  left ideals.\n\n* `IsSimpleRing.exists_algEquiv_matrix_divisionRing_finite`,\n  `IsSemisimpleRing.exists_algEquiv_pi_matrix_divisionRing_finite`:\n  if the simple Artinian / semisimple algebra is finite as a module over a base ring, then the\n  division algebra(s) are also finite over the same ring.\n  If the base ring is an algebraically closed field, the only finite-dimensional division algebra\n  over it is itself, and we obtain `IsSimpleRing.exists_algEquiv_matrix_of_isAlgClosed` and\n  `IsSemisimpleRing.exists_algEquiv_pi_matrix_of_isAlgClosed` (in a later file).\n\n-/\n\npublic section\n\nuniverse u\nvariable (R₀ : Type*) {R : Type u} [CommSemiring R₀] [Ring R] [Algebra R₀ R]\n\n/-- A simple ring is semisimple iff it is Artinian, iff it has a minimal left ideal. -/\n\nTarget:\ntheorem IsSimpleRing.tfae [IsSimpleRing R] : List.TFAE\n    [IsSemisimpleRing R, IsArtinianRing R, ∃ I : Ideal R, IsAtom I] :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/SimpleModule","family_id":"issimplering","file_id":"mathlib/Mathlib/RingTheory/SimpleModule/WedderburnArtin.lean","sample_id":"e7de025d50f83819819386b2f880ed49a8eebee597e932160cded70716ac05cb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eba6e30ed08f1c8872a40c988023ca98c14ec7e63590064e2ee0d7b74b74430f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c18ef6777010a60698f317842829f54e2ccdf8303265b8b8fa77f321e63a2045","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"382d9c391f850ba4635e3d83c4c92384a1fa17b247df5a67f6099368f1582fb9","source_sha256":"bc35074d1a6058ec18619c6ed537b3d817e9e8ac77cd437323327cafd172da85","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine\n    Computation.eq_of_bisim\n      (fun c1 c2 =>\n        ∃ (l : List α) (s : WSeq α),\n          c1 = Computation.corec (fun ⟨n, s⟩ =>\n            match Seq.destruct s with\n            | none => Sum.inl n\n            | some (none, s') => Sum.inr (n, s')\n            | some (some _, s') => Sum.inr (n + 1, s')) (l.length, s) ∧\n            c2 = Computation.map List.length (Computation.corec (fun ⟨l, s⟩ =>\n              match Seq.destruct s with\n              | none => Sum.inl l.reverse\n              | some (none, s') => Sum.inr (l, s')\n              | some (some a, s') => Sum.inr (a::l, s')) (l, s)))\n      ?_ ⟨[], s, rfl, rfl⟩\n  intro s1 s2 h; rcases h with ⟨l, s, h⟩; rw [h.left, h.right]\n  induction s using WSeq.recOn with\n  | nil => simp [nil]\n  | cons a s => simpa using ⟨a::l, s, by simp, by simp⟩\n  | think s => simpa using ⟨l, s, by simp, by simp⟩","hard_negative":false,"metrics":{"chosen_tokens":263,"rejected_tokens":268,"token_jaccard":0.941176,"token_length_ratio":1.019011},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"8d46d352b0acc02d5fd7931b9809cc9af7987e9fb76bbc152105915aa41d29b1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.DList.Basic\npublic import Mathlib.Data.WSeq.Basic\n\nNamespace:\nStream'.WSeq\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Miscellaneous definitions concerning weak sequences\n\nThese definitions, as well as those in `Mathlib/Data/WSeq/Productive.lean`, are not needed for the\ndevelopment of `Mathlib/Data/Seq/Parallel.lean`.\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nnamespace Stream'.WSeq\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\nopen Function\n\n/-- Get the length of `s` (if it is finite and completes in finite time). -/\ndef length (s : WSeq α) : Computation ℕ :=\n  @Computation.corec ℕ (ℕ × WSeq α)\n    (fun ⟨n, s⟩ =>\n      match Seq.destruct s with\n      | none => Sum.inl n\n      | some (none, s') => Sum.inr (n, s')\n      | some (some _, s') => Sum.inr (n + 1, s'))\n    (0, s)\n\n/-- A weak sequence is finite if `toList s` terminates. Equivalently,\n  it is a finite number of `think` and `cons` applied to `nil`. -/\nclass IsFinite (s : WSeq α) : Prop where\n  out : (toList s).Terminates\n\ninstance toList_terminates (s : WSeq α) [h : IsFinite s] : (toList s).Terminates :=\n  h.out\n\n/-- Get the list corresponding to a finite weak sequence. -/\ndef get (s : WSeq α) [IsFinite s] : List α :=\n  (toList s).get\n\n/-- Replace the `n`th element of `s` with `a`. -/\ndef updateNth (s : WSeq α) (n : ℕ) (a : α) : WSeq α :=\n  @Seq.corec (Option α) (ℕ × WSeq α)\n    (fun ⟨n, s⟩ =>\n      match Seq.destruct s, n with\n      | none, _ => none\n      | some (none, s'), n => some (none, n, s')\n      | some (some a', s'), 0 => some (some a', 0, s')\n      | some (some _, s'), 1 => some (some a, 0, s')\n      | some (some a', s'), n + 2 => some (some a', n + 1, s'))\n    (n + 1, s)\n\n/-- Remove the `n`th element of `s`. -/\ndef removeNth (s : WSeq α) (n : ℕ) : WSeq α :=\n  @Seq.corec (Option α) (ℕ × WSeq α)\n    (fun ⟨n, s⟩ =>\n      match Seq.destruct s, n with\n      | none, _ => none\n      | some (none, s'), n => some (none, n, s')\n      | some (some a', s'), 0 => some (some a', 0, s')\n      | some (some _, s'), 1 => some (none, 0, s')\n      | some (some a', s'), n + 2 => some (some a', n + 1, s'))\n    (n + 1, s)\n\n/-- Map the elements of `s` over `f`, removing any values that yield `none`. -/\ndef filterMap (f : α → Option β) : WSeq α → WSeq β :=\n  Seq.corec fun s =>\n    match Seq.destruct s with\n    | none => none\n    | some (none, s') => some (none, s')\n    | some (some a, s') => some (f a, s')\n\n/-- Select the elements of `s` that satisfy `p`. -/\ndef filter (p : α → Prop) [DecidablePred p] : WSeq α → WSeq α :=\n  filterMap fun a => if p a then some a else none\n\n-- example of infinite list manipulations\n/-- Get the first element of `s` satisfying `p`. -/\ndef find (p : α → Prop) [DecidablePred p] (s : WSeq α) : Computation (Option α) :=\n  head <| filter p s\n\n/-- Zip a function over two weak sequences -/\ndef zipWith (f : α → β → γ) (s1 : WSeq α) (s2 : WSeq β) : WSeq γ :=\n  @Seq.corec (Option γ) (WSeq α × WSeq β)\n    (fun ⟨s1, s2⟩ =>\n      match Seq.destruct s1, Seq.destruct s2 with\n      | some (none, s1'), some (none, s2') => some (none, s1', s2')\n      | some (some _, _), some (none, s2') => some (none, s1, s2')\n      | some (none, s1'), some (some _, _) => some (none, s1', s2)\n      | some (some a1, s1'), some (some a2, s2') => some (some (f a1 a2), s1', s2')\n      | _, _ => none)\n    (s1, s2)\n\n/-- Zip two weak sequences into a single sequence of pairs -/\ndef zip : WSeq α → WSeq β → WSeq (α × β) :=\n  zipWith Prod.mk\n\n/-- Get the list of indexes of elements of `s` satisfying `p` -/\ndef findIndexes (p : α → Prop) [DecidablePred p] (s : WSeq α) : WSeq ℕ :=\n  (zip s (Stream'.nats : WSeq ℕ)).filterMap fun ⟨a, n⟩ => if p a then some n else none\n\n/-- Get the index of the first element of `s` satisfying `p` -/\ndef findIndex (p : α → Prop) [DecidablePred p] (s : WSeq α) : Computation ℕ :=\n  (fun o => Option.getD o 0) <$> head (findIndexes p s)\n\n/-- Get the index of the first occurrence of `a` in `s` -/\ndef indexOf [DecidableEq α] (a : α) : WSeq α → Computation ℕ :=\n  findIndex (Eq a)\n\n/-- Get the indexes of occurrences of `a` in `s` -/\ndef indexesOf [DecidableEq α] (a : α) : WSeq α → WSeq ℕ :=\n  findIndexes (Eq a)\n\n/-- `union s1 s2` is a weak sequence which interleaves `s1` and `s2` in\n  some order (nondeterministically). -/\ndef union (s1 s2 : WSeq α) : WSeq α :=\n  @Seq.corec (Option α) (WSeq α × WSeq α)\n    (fun ⟨s1, s2⟩ =>\n      match Seq.destruct s1, Seq.destruct s2 with\n      | none, none => none\n      | some (a1, s1'), none => some (a1, s1', nil)\n      | none, some (a2, s2') => some (a2, nil, s2')\n      | some (none, s1'), some (none, s2') => some (none, s1', s2')\n      | some (some a1, s1'), some (none, s2') => some (some a1, s1', s2')\n      | some (none, s1'), some (some a2, s2') => some (some a2, s1', s2')\n      | some (some a1, s1'), some (some a2, s2') => some (some a1, cons a2 s1', s2'))\n    (s1, s2)\n\n/-- Returns `true` if `s` is `nil` and `false` if `s` has an element -/\ndef isEmpty (s : WSeq α) : Computation Bool :=\n  Computation.map Option.isNone <| head s\n\n/-- Calculate one step of computation -/\ndef compute (s : WSeq α) : WSeq α :=\n  match Seq.destruct s with\n  | some (none, s') => s'\n  | _ => s\n\n/-- Get the first `n` elements of a weak sequence -/\ndef take (s : WSeq α) (n : ℕ) : WSeq α :=\n  @Seq.corec (Option α) (ℕ × WSeq α)\n    (fun ⟨n, s⟩ =>\n      match n, Seq.destruct s with\n      | 0, _ => none\n      | _ + 1, none => none\n      | m + 1, some (none, s') => some (none, m + 1, s')\n      | m + 1, some (some a, s') => some (some a, m, s'))\n    (n, s)\n\n/-- Split the sequence at position `n` into a finite initial segment\n  and the weak sequence tail -/\ndef splitAt (s : WSeq α) (n : ℕ) : Computation (List α × WSeq α) :=\n  @Computation.corec (List α × WSeq α) (ℕ × List α × WSeq α)\n    (fun ⟨n, l, s⟩ =>\n      match n, Seq.destruct s with\n      | 0, _ => Sum.inl (l.reverse, s)\n      | _ + 1, none => Sum.inl (l.reverse, s)\n      | _ + 1, some (none, s') => Sum.inr (n, l, s')\n      | m + 1, some (some a, s') => Sum.inr (m, a::l, s'))\n    (n, [], s)\n\n/-- Returns `true` if any element of `s` satisfies `p` -/\ndef any (s : WSeq α) (p : α → Bool) : Computation Bool :=\n  Computation.corec\n    (fun s : WSeq α =>\n      match Seq.destruct s with\n      | none => Sum.inl false\n      | some (none, s') => Sum.inr s'\n      | some (some a, s') => if p a then Sum.inl true else Sum.inr s')\n    s\n\n/-- Returns `true` if every element of `s` satisfies `p` -/\ndef all (s : WSeq α) (p : α → Bool) : Computation Bool :=\n  Computation.corec\n    (fun s : WSeq α =>\n      match Seq.destruct s with\n      | none => Sum.inl true\n      | some (none, s') => Sum.inr s'\n      | some (some a, s') => if p a then Sum.inr s' else Sum.inl false)\n    s\n\n/-- Apply a function to the elements of the sequence to produce a sequence\n  of partial results. (There is no `scanr` because this would require\n  working from the end of the sequence, which may not exist.) -/\ndef scanl (f : α → β → α) (a : α) (s : WSeq β) : WSeq α :=\n  cons a <|\n    @Seq.corec (Option α) (α × WSeq β)\n      (fun ⟨a, s⟩ =>\n        match Seq.destruct s with\n        | none => none\n        | some (none, s') => some (none, a, s')\n        | some (some b, s') =>\n          let a' := f a b\n          some (some a', a', s'))\n      (a, s)\n\n/-- Get the weak sequence of initial segments of the input sequence -/\ndef inits (s : WSeq α) : WSeq (List α) :=\n  cons [] <|\n    @Seq.corec (Option (List α)) (Batteries.DList α × WSeq α)\n      (fun ⟨l, s⟩ =>\n        match Seq.destruct s with\n        | none => none\n        | some (none, s') => some (none, l, s')\n        | some (some a, s') =>\n          let l' := l.push a\n          some (some l'.toList, l', s'))\n      (Batteries.DList.empty, s)\n\n/-- Like take, but does not wait for a result. Calculates `n` steps of\n  computation and returns the sequence computed so far -/\ndef collect (s : WSeq α) (n : ℕ) : List α :=\n  (Seq.take n s).filterMap id\n\nTarget:\ntheorem length_eq_map (s : WSeq α) : length s = Computation.map List.length (toList s) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  refine\n    Computation.eq_of_bisim\n      (fun c1 c2 =>\n        ∃ (l : List α) (s : WSeq α),\n          c1 = Computation.corec (fun ⟨n, s⟩ =>\n            match Seq.destruct s with\n            | none => Sum.inl n\n            | some (none, s') => Sum.inr (n, s')\n            | some (some _, s') => Sum.inr (n + 1, s')) (l.length, s) ∧\n            c2 = Computation.map List.length (Computation.corec (fun ⟨l, s⟩ =>\n              match Seq.destruct s with\n              | none => Sum.inl l.reverse\n              | some (none, s') => Sum.inr (l, s')\n              | some (some a, s') => Sum.inr (a::l, s')) (l, s)))\n      ?_ ⟨[], s, rfl, rfl⟩\n  intro s1 s2 h; rcases h with ⟨l, s, h⟩; rw [h.left, h.right]\n  induction s using WSeq.recOn with\n  | nil => simp [nil]\n  | cons a s => simpa using ⟨a::l, s, by simp, by simp⟩\n  | think s => simpa using ⟨l, s, by simp, by simp⟩","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/WSeq","family_id":"length_eq_map","file_id":"mathlib/Mathlib/Data/WSeq/Defs.lean","sample_id":"382d9c391f850ba4635e3d83c4c92384a1fa17b247df5a67f6099368f1582fb9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"097944a1a41dd54c4d08892020794d36ebd2632227e02e3aff759ac09d3dd6c6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b15d779c153730a548af1bcb1242b64da3cdb5f65c04ae8beeb21d386329dcf8","source_sha256":"6ea826f0633d3930f7872110f571f16f76059ec3668e205ef02a1089b9b31873","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI : Nonempty ι := Fintype.card_pos_iff.mp (hn.symm ▸ Nat.succ_pos _)\n  haveI := Classical.decEq ι\n  -- Choose an arbitrary element `x : ι` and `y : t x`.\n  let x := Classical.arbitrary ι\n  have tx_ne : (t x).Nonempty := by\n    rw [← Finset.card_pos]\n    calc\n      0 < 1 := Nat.one_pos\n      _ ≤ #(.biUnion {x} t) := ht {x}\n      _ = (t x).card := by rw [Finset.singleton_biUnion]\n  choose y hy using tx_ne\n  -- Restrict to everything except `x` and `y`.\n  let ι' := { x' : ι | x' ≠ x }\n  let t' : ι' → Finset α := fun x' => (t x').erase y\n  have card_ι' : Fintype.card ι' = n :=\n    calc\n      Fintype.card ι' = Fintype.card ι - 1 := Set.card_ne_eq _\n      _ = n := by rw [hn, Nat.add_succ_sub_one, add_zero]\n  rcases ih t' card_ι'.le (hall_cond_of_erase y ha) with ⟨f', hfinj, hfr⟩\n  -- Extend the resulting function.\n  refine ⟨fun z => if h : z = x then y else f' ⟨z, h⟩, ?_, ?_⟩\n  · rintro z₁ z₂\n    have key : ∀ {x}, y ≠ f' x := by\n      intro x h\n      simpa [t', ← h] using hfr x\n    by_cases h₁ : z₁ = x <;> by_cases h₂ : z₂ = x <;>\n      simp [h₁, h₂, hfinj.eq_iff, key, key.symm]\n  · intro z\n    simp only\n    split_ifs with hz\n    · rwa [hz]\n    · specialize hfr ⟨z, hz⟩\n      rw [mem_erase] at hfr\n      exact hfr.2","hard_negative":false,"metrics":{"chosen_tokens":367,"rejected_tokens":2,"token_jaccard":0.008197,"token_length_ratio":0.00545},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"8d515ec863331c8a96979a98637ec693b7094aa681b6362d5a27056376a0be92","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Basic\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Data.Set.Finite.Basic\n\nNamespace:\nHallMarriageTheorem\n\nLocal context:\n/-\nCopyright (c) 2021 Alena Gusakov, Bhavik Mehta, Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alena Gusakov, Bhavik Mehta, Kyle Miller\n-/\n/-!\n# Hall's Marriage Theorem for finite index types\n\nThis module proves the basic form of Hall's theorem.\nIn contrast to the theorem described in `Combinatorics.Hall.Basic`, this\nversion requires that the indexed family `t : ι → Finset α` have `ι` be finite.\nThe `Combinatorics.Hall.Basic` module applies a compactness argument to this version\nto remove the `Finite` constraint on `ι`.\n\nThe modules are split like this since the generalized statement\ndepends on the topology and category theory libraries, but the finite\ncase in this module has few dependencies.\n\nA description of this formalization is in [Gusakov2021].\n\n## Main statements\n\n* `Finset.all_card_le_biUnion_card_iff_existsInjective'` is Hall's theorem with\n  a finite index set.  This is elsewhere generalized to\n  `Finset.all_card_le_biUnion_card_iff_existsInjective`.\n\n## Tags\n\nHall's Marriage Theorem, indexed families\n-/\n\npublic section\n\n\nopen Finset\n\nuniverse u v\n\nnamespace HallMarriageTheorem\n\nvariable {ι : Type u} {α : Type v} [DecidableEq α] {t : ι → Finset α}\n\nsection Fintype\n\nvariable [Fintype ι]\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem hall_cond_of_erase {x : ι} (a : α)\n    (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t))\n    (s' : Finset { x' : ι | x' ≠ x }) : #s' ≤ #(s'.biUnion fun x' => (t x').erase a) := by\n  haveI := Classical.decEq ι\n  specialize ha (s'.image fun z => z.1)\n  rw [image_nonempty, Finset.card_image_of_injective s' Subtype.coe_injective] at ha\n  by_cases! he : s'.Nonempty\n  · have ha' : #s' < #(s'.biUnion fun x => t x) := by\n      convert! ha he fun h => by simpa [← h] using mem_univ x using 2\n      ext x\n      simp only [mem_image, mem_biUnion, SetCoe.exists, exists_and_right,\n        exists_eq_right]\n    rw [← erase_biUnion]\n    by_cases hb : a ∈ s'.biUnion fun x => t x\n    · rw [card_erase_of_mem hb]\n      exact Nat.le_sub_one_of_lt ha'\n    · rw [erase_eq_of_notMem hb]\n      exact Nat.le_of_lt ha'\n  · subst s'\n    simp\n\n/-- First case of the inductive step: assuming that\n`∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)`\nand that the statement of **Hall's Marriage Theorem** is true for all\n`ι'` of cardinality ≤ `n`, then it is true for `ι` of cardinality `n + 1`.\n-/\n\nTarget:\ntheorem hall_hard_inductive_step_A {n : ℕ} (hn : Fintype.card ι = n + 1)\n    (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))\n    (ih :\n      ∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),\n        Fintype.card ι' ≤ n →\n          (∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →\n            ∃ f : ι' → α, Function.Injective f ∧ ∀ x, f x ∈ t' x)\n    (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t)) :\n    ∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Hall","family_id":"hall_hard_inductive_step_a","file_id":"mathlib/Mathlib/Combinatorics/Hall/Finite.lean","sample_id":"b15d779c153730a548af1bcb1242b64da3cdb5f65c04ae8beeb21d386329dcf8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"886ef528de3066c5206bdbcb8b42b2d7a0a7e9340445f6b246cca5e2a1bdcafc","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cfdd53edfe1acc3d2b51c43a1c120daa3d309e493c2f938aa9d41f5faeec220c","source_sha256":"edb52479240700af7f6c572a1b6c456a3b7b18b85aa47a4adb83e507587b6553","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [smul_Spec_def, smul_Spec_def]\n  simp_rw [smul_restrictAppIso_hom_apply, ← ConcreteCategory.comp_apply, Category.assoc]\n  have :\n      f ≫ (ΓSpecIso S).inv ≫ (Spec S).presheaf.map U.leTop.op ≫ (Hom.appIso (Spec.map f) U).inv =\n        (ΓSpecIso R).inv ≫ (Spec R).presheaf.map (Spec.map f ''ᵁ U).leTop.op := by\n    simp [Iso.cancel_iso_inv_left, Hom.app_eq_appLE]\n    rfl\n  rw [this]","hard_negative":false,"metrics":{"chosen_tokens":108,"rejected_tokens":5,"token_jaccard":0.021277,"token_length_ratio":0.046296},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"8d689755e613a7e5178eb46b6d8643d68c07df78ecf69a3bbd5905335e069127","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Category.ModuleCat.Localization\npublic import Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent\npublic import Mathlib.AlgebraicGeometry.AffineScheme\npublic import Mathlib.AlgebraicGeometry.Modules.Sheaf\n\nNamespace:\nAlgebraicGeometry.Scheme.Modules\n\nLocal context:\n/-\nCopyright (c) 2024 Weihong Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Johan Commelin, Amelia Livingston, Sophie Morel, Jujian Zhang, Weihong Xu,\n  Andrew Yang, Brian Nugent\n-/\n/-!\n\n# Construction of M^~\n\nGiven any commutative ring `R` and `R`-module `M`, we construct the sheaf `M^~` of `𝒪_SpecR`-modules\nsuch that `M^~(U)` is the set of dependent functions that are locally fractions.\n\n## Main definitions\n* `AlgebraicGeometry.tilde` : `M^~` as a sheaf of `𝒪_{Spec R}`-modules.\n* `AlgebraicGeometry.tilde.adjunction` : `~` is left adjoint to taking global sections.\n\n-/\n\n@[expose] public noncomputable section\n\nuniverse u\n\nopen TopCat AlgebraicGeometry TopologicalSpace CategoryTheory Opposite\n\nvariable {R : CommRingCat.{u}} (M : ModuleCat.{u} R)\n\nnamespace AlgebraicGeometry\n\nopen _root_.PrimeSpectrum\n\n/-- The forgetful functor from `𝒪_{Spec R}` modules to sheaves of `R`-modules. -/\ndef modulesSpecToSheaf :\n    (Spec R).Modules ⥤ TopCat.Sheaf (ModuleCat R) (Spec R) :=\n  SheafOfModules.forgetToSheafModuleCat (Spec R).ringCatSheaf (.op ⊤)\n    (Limits.initialOpOfTerminal Limits.isTerminalTop) ⋙\n  sheafCompose _ (ModuleCat.restrictScalars (Scheme.ΓSpecIso R).inv.hom)\n\n/-- The global section functor for `𝒪_{Spec R}` modules -/\nnoncomputable\ndef moduleSpecΓFunctor : (Spec (.of R)).Modules ⥤ ModuleCat R :=\n  modulesSpecToSheaf ⋙ TopCat.Sheaf.forget _ _ ⋙ (evaluation _ _).obj (.op ⊤)\n\nset_option backward.isDefEq.respectTransparency false in\nopen PrimeSpectrum in\n/-- The forgetful functor from `𝒪_{Spec R}` modules to sheaves of `R`-modules is fully faithful. -/\ndef SpecModulesToSheafFullyFaithful : (modulesSpecToSheaf (R := R)).FullyFaithful where\n  preimage {M N} f := ⟨fun U ↦ ModuleCat.ofHom ⟨(f.1.app U).hom.toAddHom, by\n    intro t m\n    apply TopCat.Presheaf.IsSheaf.section_ext (modulesSpecToSheaf.obj N).2\n    intro x hxU\n    obtain ⟨a, ⟨_, ⟨r, rfl⟩, rfl⟩, hxr, hrU : basicOpen _ ≤ _⟩ :=\n      PrimeSpectrum.isBasis_basic_opens.exists_subset_of_mem_open hxU U.unop.2\n    refine ⟨_, hrU, hxr, ?_⟩\n    refine Eq.trans ?_ (N.val.map_smul (homOfLE hrU).op t _).symm\n    change N.1.map (homOfLE hrU).op (f.1.app _ _) = _ • N.1.map (homOfLE hrU).op (f.1.app _ _)\n    have (x : _) :\n        f.1.app _ (M.1.map (homOfLE hrU).op _) = N.1.map (homOfLE hrU).op (f.1.app _ x) :=\n      congr($(f.1.naturality (homOfLE hrU).op).hom x)\n    rw [← this, ← this, M.val.map_smul]\n    generalize (Spec R).ringCatSheaf.obj.map (homOfLE hrU).op t = t\n    letI := Module.compHom (R := Γ(Spec R, basicOpen r)) Γ(M, basicOpen r)\n      (algebraMap R Γ(Spec R, basicOpen r))\n    haveI : IsScalarTower R Γ(Spec R, basicOpen r) Γ(M, basicOpen r) :=\n      .of_algebraMap_smul fun _ _ ↦ rfl\n    letI := Module.compHom Γ(N, basicOpen r) (algebraMap R Γ(Spec R, basicOpen r))\n    haveI : IsScalarTower R Γ(Spec R, basicOpen r) Γ(N, basicOpen r) :=\n      .of_algebraMap_smul fun _ _ ↦ rfl\n    exact (IsLocalization.linearMap_compatibleSMul (.powers (M := R) r)\n      Γ(Spec R, basicOpen r) Γ(M, basicOpen r) Γ(N, basicOpen r)).map_smul\n      (f.hom.app _).hom _ _⟩, fun i ↦ by ext x; exact congr($(f.1.naturality i).hom x)⟩\n  map_preimage f := rfl\n  preimage_map f := rfl\n\ninstance : (modulesSpecToSheaf (R := R)).Faithful := SpecModulesToSheafFullyFaithful.faithful\n\ninstance : (modulesSpecToSheaf (R := R)).Full := SpecModulesToSheafFullyFaithful.full\n\nnamespace Scheme.Modules\n\nvariable {M : (Spec R).Modules} {U V : (Spec R).Opens}\n\ninstance : Module R Γ(M, U) :=\n  inferInstanceAs <| Module R ((modulesSpecToSheaf.obj M).obj.obj (.op U))\n\ninstance : IsScalarTower R Γ(Spec R, U) Γ(M, U) :=\n  IsScalarTower.of_compHom R Γ(Spec R, U) Γ(M, U)\n\nlemma smul_Spec_def (r : R) (x : Γ(M, U)) :\n    r • x = ((Spec R).presheaf.map U.leTop.op) ((Scheme.ΓSpecIso R).inv r) • x :=\n  rfl\n\n@[simp]\nlemma map_smul_Spec (hUV : .op V ⟶ .op U) (f : R) (x : Γ(M, V)) :\n    dsimp% M.presheaf.map hUV (f • x) = f • M.presheaf.map hUV x :=\n  ((modulesSpecToSheaf.obj M).obj.map hUV).hom.map_smul f x\n\nlemma isUnit_algebraMap_end_of_le_basicOpen (f : R) (hf : U ≤ PrimeSpectrum.basicOpen f) :\n    IsUnit (algebraMap R (Module.End R Γ(M, U)) f) := by\n  rw [Module.End.isUnit_iff]\n  have : ⇑((algebraMap R (Module.End ↑R ↑Γ(M, U))) f) =\n      algebraMap (Γ(Spec R, U)) (Module.End Γ(Spec R, U) Γ(M, U))\n        (((Spec R).presheaf.map (homOfLE hf).op) <| algebraMap R _ f) :=\n    rfl\n  rw [this, ← Module.End.isUnit_iff]\n  exact ((IsLocalization.Away.algebraMap_isUnit _).map _).map _\n\nlemma isSMulRegular_of_le_basicOpen {f : R} (hle : U ≤ PrimeSpectrum.basicOpen f) :\n    IsSMulRegular Γ(M, U) f := by\n  intro x y hxy\n  have := M.isUnit_algebraMap_end_of_le_basicOpen _ hle\n  rw [Module.End.isUnit_iff] at this\n  exact this.injective hxy\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\n\nTarget:\nlemma restrictAppIso_smul_Spec {S : CommRingCat.{u}} (f : R ⟶ S)\n    [IsOpenImmersion (Spec.map f)] {U : (Spec S).Opens} (r : R)\n    (x : Γ(M.restrict (Spec.map f), U)) :\n    dsimp% (M.restrictAppIso (Spec.map f) U).hom (f r • x) =\n      r • (M.restrictAppIso (Spec.map f) U).hom x :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Modules","family_id":"restrictappiso_smul_spec","file_id":"mathlib/Mathlib/AlgebraicGeometry/Modules/Tilde.lean","sample_id":"cfdd53edfe1acc3d2b51c43a1c120daa3d309e493c2f938aa9d41f5faeec220c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e1fc953f2e811d1c999db981871450abfdfcde88bd7349db65e45507d796d374","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c9b478f786784f9ac4583d69de5a7f68ad959e59303de1658982ea618e1c423e","source_sha256":"b2e5d2441dae74e6a2f4e8de4b7d4e87dd1e5e1f1bcd7a4ed0737bd7e824e441","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let ℬ := {s | s ∈ 𝒜} ∪ {s | sᶜ ∈ 𝒜}\n  have count_ℬ : ℬ.Countable := by\n    apply h.union\n    have : compl '' 𝒜 = {s | sᶜ ∈ 𝒜} := by\n      ext s\n      simpa using ⟨fun ⟨x, x_mem, hx⟩ ↦ by simp [← hx, x_mem], fun hs ↦ ⟨sᶜ, hs, by simp⟩⟩\n    exact this ▸ h.image compl\n  let f : Set (Set (Set α)) → Set α := fun A ↦ ⋃ a ∈ A, ⋂ t ∈ a, t\n  let 𝒞 := {a | a.Finite ∧ a ⊆ ℬ}\n  have count_𝒞 : 𝒞.Countable := countable_setOf_finite_subset (countable_coe_iff.1 count_ℬ)\n  let 𝒟 := {A | A.Finite ∧ A ⊆ 𝒞}\n  have count_𝒟 : 𝒟.Countable := countable_setOf_finite_subset (countable_coe_iff.1 count_𝒞)\n  have : generateSetAlgebra 𝒜 ⊆ f '' 𝒟 := by\n    intro s s_mem\n    rcases mem_generateSetAlgebra_elim s_mem with ⟨A, A_fin, mem_A, hA, rfl⟩\n    exact ⟨A, ⟨A_fin, fun a ha ↦ ⟨mem_A a ha, hA a ha⟩⟩, rfl⟩\n  exact (count_𝒟.image f).mono this","hard_negative":true,"metrics":{"chosen_tokens":246,"rejected_tokens":8,"token_jaccard":0.051948,"token_length_ratio":0.03252},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"8d73cfb0505474da0386e5518697b1a4a0db5091c9bdd21d27ee3ef02bef68ae","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finite.Prod\npublic import Mathlib.MeasureTheory.SetSemiring\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2024 Etienne Marion. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Etienne Marion\n-/\n/-!\n# Algebra of sets\n\nIn this file we define the notion of algebra of sets and give its basic properties. An algebra\nof sets is a family of sets containing the empty set and closed by complement and binary union.\nIt is therefore similar to a `σ`-algebra, except that it is not necessarily closed\nby countable unions.\n\nWe also define the algebra of sets generated by a family of sets and give its basic properties,\nand we prove that it is countable when it is generated by a countable family. We prove that\nthe `σ`-algebra generated by a family of sets `𝒜` is the same as the one generated by the algebra\nof sets generated by `𝒜`.\n\n## Main definitions\n\n* `MeasureTheory.IsSetAlgebra`: property of being an algebra of sets.\n* `MeasureTheory.generateSetAlgebra`: the algebra of sets generated by a family of sets.\n\n## Main statements\n\n* `MeasureTheory.mem_generateSetAlgebra_elim`: If a set `s` belongs to the algebra of sets\n  generated by `𝒜`, then it can be written as a finite union of finite intersections of sets which\n  are in `𝒜` or have their complement in `𝒜`.\n* `MeasureTheory.countable_generateSetAlgebra`: If a family of sets is countable then so is\n  the algebra of sets generated by it.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Field_of_sets>\n\n## Tags\n\nalgebra of sets, generated algebra of sets\n-/\n\npublic section\n\nopen MeasurableSpace Set\n\nnamespace MeasureTheory\n\nvariable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α}\n\n/-! ### Definition and basic properties of an algebra of sets -/\n\n/-- An algebra of sets is a family of sets containing the empty set and closed by complement and\nunion. Consequently it is also closed by difference (see `IsSetAlgebra.sdiff_mem`) and intersection\n(see `IsSetAlgebra.inter_mem`). -/\nstructure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where\n  empty_mem : ∅ ∈ 𝒜\n  compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜\n  union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜\n\nnamespace IsSetAlgebra\n\n/-- An algebra of sets contains the whole set. -/\ntheorem univ_mem (h𝒜 : IsSetAlgebra 𝒜) : univ ∈ 𝒜 :=\n  compl_empty ▸ h𝒜.compl_mem h𝒜.empty_mem\n\n/-- An algebra of sets is closed by intersection. -/\ntheorem inter_mem (h𝒜 : IsSetAlgebra 𝒜) (s_mem : s ∈ 𝒜) (t_mem : t ∈ 𝒜) :\n    s ∩ t ∈ 𝒜 :=\n  inter_eq_compl_compl_union_compl .. ▸\n    h𝒜.compl_mem (h𝒜.union_mem (h𝒜.compl_mem s_mem) (h𝒜.compl_mem t_mem))\n\n/-- An algebra of sets is closed by difference. -/\ntheorem sdiff_mem (h𝒜 : IsSetAlgebra 𝒜) (s_mem : s ∈ 𝒜) (t_mem : t ∈ 𝒜) :\n    s \\ t ∈ 𝒜 := h𝒜.inter_mem s_mem (h𝒜.compl_mem t_mem)\n\n@[deprecated (since := \"2026-06-03\")] alias diff_mem := sdiff_mem\n\n/-- An algebra of sets is a ring of sets. -/\ntheorem isSetRing (h𝒜 : IsSetAlgebra 𝒜) : IsSetRing 𝒜 where\n  empty_mem := h𝒜.empty_mem\n  union_mem := h𝒜.union_mem\n  sdiff_mem := fun _ _ ↦ h𝒜.sdiff_mem\n\n/-- An algebra of sets is closed by finite unions. -/\ntheorem biUnion_mem {ι : Type*} (h𝒜 : IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι)\n    (hs : ∀ i ∈ S, s i ∈ 𝒜) : ⋃ i ∈ S, s i ∈ 𝒜 := h𝒜.isSetRing.biUnion_mem S hs\n\n/-- An algebra of sets is closed by finite intersections. -/\ntheorem biInter_mem {ι : Type*} (h𝒜 : IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι)\n    (hs : ∀ i ∈ S, s i ∈ 𝒜) : ⋂ i ∈ S, s i ∈ 𝒜 := by\n  by_cases! h : S = ∅\n  · rw [h, ← Finset.set_biInter_coe, Finset.coe_empty, biInter_empty]\n    exact h𝒜.univ_mem\n  · exact h𝒜.isSetRing.biInter_mem S h hs\n\nend IsSetAlgebra\n\nsection generateSetAlgebra\n\n/-! ### Definition and properties of the algebra of sets generated by some family -/\n\n/-- `generateSetAlgebra 𝒜` is the smallest algebra of sets containing `𝒜`. -/\ninductive generateSetAlgebra {α : Type*} (𝒜 : Set (Set α)) : Set (Set α)\n  | base (s : Set α) (s_mem : s ∈ 𝒜) : generateSetAlgebra 𝒜 s\n  | empty : generateSetAlgebra 𝒜 ∅\n  | compl (s : Set α) (hs : generateSetAlgebra 𝒜 s) : generateSetAlgebra 𝒜 sᶜ\n  | union (s t : Set α) (hs : generateSetAlgebra 𝒜 s) (ht : generateSetAlgebra 𝒜 t) :\n      generateSetAlgebra 𝒜 (s ∪ t)\n\n/-- The algebra of sets generated by a family of sets is an algebra of sets. -/\ntheorem isSetAlgebra_generateSetAlgebra :\n    IsSetAlgebra (generateSetAlgebra 𝒜) where\n  empty_mem := generateSetAlgebra.empty\n  compl_mem := fun _ hs ↦ generateSetAlgebra.compl _ hs\n  union_mem := fun _ _ hs ht ↦ generateSetAlgebra.union _ _ hs ht\n\n/-- The algebra of sets generated by `𝒜` contains `𝒜`. -/\ntheorem self_subset_generateSetAlgebra : 𝒜 ⊆ generateSetAlgebra 𝒜 :=\n  fun _ ↦ generateSetAlgebra.base _\n\n/-- The measurable space generated by a family of sets `𝒜` is the same as the one generated\nby the algebra of sets generated by `𝒜`. -/\n@[simp]\ntheorem generateFrom_generateSetAlgebra_eq :\n    generateFrom (generateSetAlgebra 𝒜) = generateFrom 𝒜 := by\n  refine le_antisymm (fun s ms ↦ ?_) (generateFrom_mono self_subset_generateSetAlgebra)\n  induction s, ms using generateFrom_induction with\n  | hC t ht h =>\n    clear h\n    induction ht with\n    | base u u_mem => exact measurableSet_generateFrom u_mem\n    | empty => exact @MeasurableSet.empty _ (generateFrom 𝒜)\n    | compl u _ mu => exact mu.compl\n    | union u v _ _ mu mv => exact MeasurableSet.union mu mv\n  | empty => exact MeasurableSpace.measurableSet_empty _\n  | compl t _ ht => exact ht.compl\n  | iUnion t _ ht => exact .iUnion ht\n\n/-- If a family of sets `𝒜` is contained in `ℬ`, then the algebra of sets generated by `𝒜`\nis contained in the one generated by `ℬ`. -/\ntheorem generateSetAlgebra_mono {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ) :\n    generateSetAlgebra 𝒜 ⊆ generateSetAlgebra ℬ := by\n  intro s hs\n  induction hs with\n  | base t t_mem => exact self_subset_generateSetAlgebra (h t_mem)\n  | empty => exact isSetAlgebra_generateSetAlgebra.empty_mem\n  | compl t _ t_mem => exact isSetAlgebra_generateSetAlgebra.compl_mem t_mem\n  | union t u _ _ t_mem u_mem => exact isSetAlgebra_generateSetAlgebra.union_mem t_mem u_mem\n\nnamespace IsSetAlgebra\n\n/-- If a family of sets `𝒜` is contained in an algebra of sets `ℬ`, then so is the algebra of sets\ngenerated by `𝒜`. -/\ntheorem generateSetAlgebra_subset {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ)\n    (hℬ : IsSetAlgebra ℬ) : generateSetAlgebra 𝒜 ⊆ ℬ := by\n  intro s hs\n  induction hs with\n  | base t t_mem => exact h t_mem\n  | empty => exact hℬ.empty_mem\n  | compl t _ t_mem => exact hℬ.compl_mem t_mem\n  | union t u _ _ t_mem u_mem => exact hℬ.union_mem t_mem u_mem\n\n/-- If `𝒜` is an algebra of sets, then it contains the algebra generated by itself. -/\ntheorem generateSetAlgebra_subset_self (h𝒜 : IsSetAlgebra 𝒜) :\n    generateSetAlgebra 𝒜 ⊆ 𝒜 := h𝒜.generateSetAlgebra_subset subset_rfl\n\n/-- If `𝒜` is an algebra of sets, then it is equal to the algebra generated by itself. -/\ntheorem generateSetAlgebra_eq (h𝒜 : IsSetAlgebra 𝒜) : generateSetAlgebra 𝒜 = 𝒜 :=\n  Subset.antisymm h𝒜.generateSetAlgebra_subset_self self_subset_generateSetAlgebra\n\nend IsSetAlgebra\n\n/-- If a set belongs to the algebra of sets generated by `𝒜` then it can be written as a finite\nunion of finite intersections of sets which are in `𝒜` or have their complement in `𝒜`. -/\ntheorem mem_generateSetAlgebra_elim (s_mem : s ∈ generateSetAlgebra 𝒜) :\n    ∃ A : Set (Set (Set α)), A.Finite ∧ (∀ a ∈ A, a.Finite) ∧\n    (∀ᵉ (a ∈ A) (t ∈ a), t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ s = ⋃ a ∈ A, ⋂ t ∈ a, t := by\n  induction s_mem with\n  | base u u_mem =>\n    refine ⟨{{u}}, finite_singleton {u},\n      fun a ha ↦ eq_of_mem_singleton ha ▸ finite_singleton u,\n      fun a ha t ht ↦ ?_, by simp⟩\n    rw [eq_of_mem_singleton ha, ha, eq_of_mem_singleton ht, ht] at *\n    exact Or.inl u_mem\n  | empty => exact ⟨∅, finite_empty, fun _ h ↦ (notMem_empty _ h).elim,\n    fun _ ha _ _ ↦ (notMem_empty _ ha).elim, by simp⟩\n  | compl u _ u_ind =>\n    rcases u_ind with ⟨A, A_fin, mem_A, hA, u_eq⟩\n    have := finite_coe_iff.2 A_fin\n    have := fun a : A ↦ finite_coe_iff.2 <| mem_A a.1 a.2\n    refine ⟨{{(f a).1ᶜ | a : A} | f : (Π a : A, ↑a)}, finite_coe_iff.1 inferInstance,\n      fun a ⟨f, hf⟩ ↦ hf ▸ finite_coe_iff.1 inferInstance, fun a ha t ht ↦ ?_, ?_⟩\n    · rcases ha with ⟨f, rfl⟩\n      rcases ht with ⟨a, rfl⟩\n      rw [compl_compl, or_comm]\n      exact hA a.1 a.2 (f a).1 (f a).2\n    · ext x\n      simp only [u_eq, compl_iUnion, compl_iInter, mem_iInter, mem_iUnion, mem_compl_iff,\n        exists_prop, Subtype.exists, mem_setOf_eq, iUnion_exists, iUnion_iUnion_eq',\n        iInter_exists]\n      constructor <;> intro hx\n      · choose f hf using hx\n        exact ⟨fun ⟨a, ha⟩ ↦ ⟨f a ha, (hf a ha).1⟩, fun _ a ha h ↦ by rw [← h]; exact (hf a ha).2⟩\n      · rcases hx with ⟨f, hf⟩\n        exact fun a ha ↦ ⟨f ⟨a, ha⟩, (f ⟨a, ha⟩).2, hf (f ⟨a, ha⟩)ᶜ a ha rfl⟩\n  | union u v _ _ u_ind v_ind =>\n    rcases u_ind with ⟨Au, Au_fin, mem_Au, hAu, u_eq⟩\n    rcases v_ind with ⟨Av, Av_fin, mem_Av, hAv, v_eq⟩\n    refine ⟨Au ∪ Av, Au_fin.union Av_fin, ?_, ?_, by rw [u_eq, v_eq, ← biUnion_union]⟩\n    · rintro a (ha | ha)\n      · exact mem_Au a ha\n      · exact mem_Av a ha\n    · rintro a (ha | ha) t ht\n      · exact hAu a ha t ht\n      · exact hAv a ha t ht\n\n/-- If a family of sets is countable then so is the algebra of sets generated by it. -/\n\nTarget:\ntheorem countable_generateSetAlgebra (h : 𝒜.Countable) :\n    (generateSetAlgebra 𝒜).Countable :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"9e97f920a60e2743317f63040a29640c46c1dce68696b1498e54be6c66ee1018","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory","family_id":"countable_generatesetalgebra","file_id":"mathlib/Mathlib/MeasureTheory/SetAlgebra.lean","sample_id":"c9b478f786784f9ac4583d69de5a7f68ad959e59303de1658982ea618e1c423e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"38369068946cc9341941259052323d13d916cefa46cf4a0009f06b6990667aa7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"73a3cbacd26aabe1c9fcecf97006df24efbc5f39af21452f7d6e23076614bde6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dfb013882af3dab10b8a5b0e2b5f55bfe7bbceeec4f48cee972a6e94a5340486","source_sha256":"3ea4a54ee767d2f1814d8b90accb092fa9ec4a1fd88519664221a22cfd80fa82","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [basis_apply_powersetCard, exteriorPower.basis_apply, ιMulti_family_apply_coe]","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":19,"token_jaccard":0.846154,"token_length_ratio":1.583333},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"8deff741aa8c675df5047b3166efd52dedc66aea6e82d8e13d7059f74a8f3ed4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.ExteriorAlgebra.Grading\npublic import Mathlib.LinearAlgebra.ExteriorPower.Basis\n\nNamespace:\nExteriorAlgebra\n\nLocal context:\n/-\nCopyright (c) 2026 Daniel Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Morrison\n-/\n/-!\n# Basis for `ExteriorAlgebra`\n-/\n\n@[expose] public section\n\nnamespace ExteriorAlgebra\n\nopen Module Set Set.powersetCard exteriorPower\n\nvariable {R M : Type*} {m n : ℕ} {I : Type*} [LinearOrder I] [CommRing R]\n  [AddCommGroup M] [Module R M] (b : Module.Basis I R M)\n\n/-- The direct sum decomposition of the exterior algebra from the graded algebra structure. -/\ninstance : DirectSum.Decomposition (fun n ↦ ⋀[R]^n M) :=\n  GradedRing.toDecomposition (self := ExteriorAlgebra.gradedAlgebra R M)\n\n/-- If `b` is a basis of `M` (indexed by a linearly ordered type), the basis of the exterior\nalgebra of `M` formed by the `n`-fold exterior products of elements of `b` for each `n`. -/\nnoncomputable def _root_.Module.Basis.ExteriorAlgebra : Basis (Finset I) R (ExteriorAlgebra R M) :=\n  .reindex\n    ((DirectSum.Decomposition.isInternal (fun n => ⋀[R]^n M)).collectedBasis b.exteriorPower)\n    Set.powersetCard.prodEquiv\n\nlemma basis_apply (s : Finset I) :\n    b.ExteriorAlgebra s = ιMulti_family R s.card b (prodEquiv.symm s).2 := by\n  simp [Basis.ExteriorAlgebra]\n\nlemma basis_apply_ofCard {s : Finset I} (s_card : s.card = n) :\n    b.ExteriorAlgebra s = ιMulti_family R n b (ofCard s_card) := by\n  subst s_card\n  simp [basis_apply]\n\nvariable (s : powersetCard I m) (t : powersetCard I n)\n\nlemma basis_apply_powersetCard :\n    b.ExteriorAlgebra s = ιMulti_family R m b s := by\n  simp [basis_apply_ofCard]\n\nTarget:\nlemma basis_eq_coe_basis :\n    b.ExteriorAlgebra s = (b.exteriorPower m s : ExteriorAlgebra R M) :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [basis_apply_powersetCard, exteriorPower.basis_apply, ιMulti_family_apply_coe]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/ExteriorAlgebra","family_id":"basis_eq_coe_basis","file_id":"mathlib/Mathlib/LinearAlgebra/ExteriorAlgebra/Basis.lean","sample_id":"dfb013882af3dab10b8a5b0e2b5f55bfe7bbceeec4f48cee972a6e94a5340486"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8269b551fa1e3baafeb590db545d184ccf387498c0689c5533fadd958fae33b7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"56dfb71d32b180653fe5db01d6101ad74566b666462b27c7889ed2d552e18dc1","source_sha256":"2330b04bbf9f5616db3c6c30d466b364ee306b290c33249fea3b2f667c7433f7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let adj := mapPullbackAdj g\n  let U : Over Z := Over.mk (φ₁ ≫ snd f g)\n  let φ₁' : U ⟶ (pullback g).obj (Over.mk f) := Over.homMk φ₁\n  let φ₂' : U ⟶ (pullback g).obj (Over.mk f) := Over.homMk φ₂ (by simpa using! h₂.symm)\n  have : φ₁' = φ₂' := by\n    apply (adj.homEquiv U _).symm.injective\n    apply (Over.forget X).map_injective\n    simpa using! h₁\n  exact congr_arg CommaMorphism.left this","hard_negative":false,"metrics":{"chosen_tokens":126,"rejected_tokens":3,"token_jaccard":0.044444,"token_length_ratio":0.02381},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"8df32fd3bb374f85976cd0159e1cf56fe77b4f01aa3709430427890e65319927","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Comma.Over.Pullback\npublic import Mathlib.CategoryTheory.Monoidal.Cartesian.Basic\npublic import Mathlib.CategoryTheory.Adjunction.Unique\npublic import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Defs\n\nNamespace:\nCategoryTheory.ChosenPullbacksAlong\n\nLocal context:\n/-\nCopyright (c) 2025 Sina Hazratpour. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sina Hazratpour\n-/\n/-!\n# Chosen pullbacks along a morphism\n\n## Main declarations\n\n- `ChosenPullbacksAlong` : For a morphism `f : Y ⟶ X` in `C`, the type class\n  `ChosenPullbacksAlong f` provides the data of a pullback functor `Over X ⥤ Over Y`\n  as a right adjoint to `Over.map f`.\n\n## Main results\n\n- We prove that `ChosenPullbacksAlong` has good closure properties: isos have chosen pullbacks,\n  and composition of morphisms with chosen pullbacks have chosen pullbacks.\n\n- We prove that chosen pullbacks yield usual pullbacks: `ChosenPullbacksAlong.isPullback`\n  proves that for morphisms `f` and `g` with the same codomain, the object\n  `ChosenPullbacksAlong.pullbackObj f g` together with morphisms\n  `ChosenPullbacksAlong.fst f g` and `ChosenPullbacksAlong.snd f g` form a pullback square\n  over `f` and `g`.\n\n- We prove that in cartesian monoidal categories, morphisms to the terminal tensor unit and\n  the product projections have chosen pullbacks.\n\n-/\n\n@[expose] public section\n\nuniverse v₁ v₂ u₁ u₂\n\nnamespace CategoryTheory\n\nopen Category Limits CartesianMonoidalCategory MonoidalCategory Over\n\nvariable {C : Type u₁} [Category.{v₁} C]\n\n/-- A functorial choice of pullbacks along a morphism `f : Y ⟶ X` in `C` given by a functor\n`Over X ⥤ Over Y` which is a right adjoint to the functor `Over.map f`. -/\nclass ChosenPullbacksAlong {Y X : C} (f : Y ⟶ X) where\n  /-- The pullback functor along `f`. -/\n  pullback : Over X ⥤ Over Y\n  /-- The adjunction between `Over.map f` and `pullback f`. -/\n  mapPullbackAdj (f) : Over.map f ⊣ pullback\n\nvariable (C) in\n/-- A category has chosen pullbacks if every morphism has a chosen pullback. -/\nabbrev ChosenPullbacks := Π {X Y : C} (f : Y ⟶ X), ChosenPullbacksAlong f\n\nnamespace ChosenPullbacksAlong\n\n/-- Relating the existing noncomputable `HasPullbacksAlong` typeclass to `ChosenPullbacksAlong`. -/\n@[simps, implicit_reducible]\nnoncomputable def ofHasPullbacksAlong {Y X : C} (f : Y ⟶ X) [HasPullbacksAlong f] :\n    ChosenPullbacksAlong f where\n  pullback := Over.pullback f\n  mapPullbackAdj := Over.mapPullbackAdj f\n\n/-- The identity morphism has a functorial choice of pullbacks. -/\n@[implicit_reducible]\ndef id (X : C) : ChosenPullbacksAlong (𝟙 X) where\n  pullback := 𝟭 _\n  mapPullbackAdj := (Adjunction.id).ofNatIsoLeft (Over.mapId _).symm\n\n/-- Any chosen pullback functor of the identity morphism is naturally isomorphic to the identity\nfunctor. -/\ndef pullbackId (X : C) [ChosenPullbacksAlong (𝟙 X)] :\n    pullback (𝟙 X) ≅ 𝟭 (Over X) :=\n  (mapPullbackAdj (𝟙 X)).rightAdjointUniq (id X).mapPullbackAdj\n\n@[reassoc (attr := simp)]\ntheorem unit_pullbackId_hom_app (X : C) [ChosenPullbacksAlong (𝟙 X)] (Y : Over X) :\n    (mapPullbackAdj (𝟙 X)).unit.app Y ≫ (pullbackId X).hom.app ((Over.map (𝟙 X)).obj Y) =\n      (id X).mapPullbackAdj.unit.app Y := by\n  rw [pullbackId, Adjunction.unit_rightAdjointUniq_hom_app]\n\n@[reassoc (attr := simp)]\ntheorem unit_pullbackId_hom (X : C) [ChosenPullbacksAlong (𝟙 X)] :\n    (mapPullbackAdj (𝟙 X)).unit ≫ (Over.map (𝟙 X)).whiskerLeft (pullbackId X).hom =\n      (id X).mapPullbackAdj.unit := by\n  rw [pullbackId, Adjunction.unit_rightAdjointUniq_hom]\n\n@[reassoc (attr := simp)]\ntheorem pullbackId_hom_counit (X : C) [ChosenPullbacksAlong (𝟙 X)] :\n    Functor.whiskerRight (pullbackId X).hom (Over.map (𝟙 X)) ≫ (id X).mapPullbackAdj.counit =\n      (mapPullbackAdj (𝟙 X)).counit := by\n  have := Adjunction.rightAdjointUniq_hom_counit (mapPullbackAdj (𝟙 X)) (id X).mapPullbackAdj\n  rw [pullbackId, Adjunction.rightAdjointUniq_hom_counit]\n\nset_option backward.defeqAttrib.useBackward true in\n/-- Every isomorphism has a functorial choice of pullbacks. -/\n@[simps, implicit_reducible]\ndef iso {Y X : C} (f : Y ≅ X) : ChosenPullbacksAlong f.hom where\n  pullback.obj Z := Over.mk (Z.hom ≫ f.inv)\n  pullback.map {Y Z} g := Over.homMk (g.left)\n  mapPullbackAdj.unit.app T := Over.homMk (𝟙 T.left)\n  mapPullbackAdj.counit.app U := Over.homMk (𝟙 _)\n\n/-- The inverse of an isomorphism has a functorial choice of pullbacks. -/\n@[simps!, implicit_reducible]\ndef isoInv {Y X : C} (f : Y ≅ X) : ChosenPullbacksAlong f.inv := iso f.symm\n\n/-- The composition of morphisms with chosen pullbacks has a chosen pullback. -/\n@[implicit_reducible]\ndef comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z)\n    [ChosenPullbacksAlong f] [ChosenPullbacksAlong g] : ChosenPullbacksAlong (f ≫ g) where\n  pullback := pullback g ⋙ pullback f\n  mapPullbackAdj := ((mapPullbackAdj f).comp (mapPullbackAdj g)).ofNatIsoLeft\n    (Over.mapComp f g).symm\n\n/-- Any chosen pullback of a composite of morphisms is naturally isomorphic to the composition of\nchosen pullback functors. -/\ndef pullbackComp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z)\n    [ChosenPullbacksAlong f] [ChosenPullbacksAlong g] [ChosenPullbacksAlong (f ≫ g)] :\n    pullback (f ≫ g) ≅ pullback g ⋙ pullback f :=\n  Adjunction.rightAdjointUniq (mapPullbackAdj (f ≫ g)) ((comp f g).mapPullbackAdj)\n\n@[reassoc (attr := simp)]\ntheorem unit_pullbackComp_hom {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z)\n    [ChosenPullbacksAlong f] [ChosenPullbacksAlong g] [ChosenPullbacksAlong (f ≫ g)] :\n    (mapPullbackAdj (f ≫ g)).unit ≫ (Over.map (f ≫ g)).whiskerLeft (pullbackComp f g).hom =\n      (comp f g).mapPullbackAdj.unit := by\n  rw [pullbackComp, Adjunction.unit_rightAdjointUniq_hom]\n\n@[reassoc (attr := simp)]\ntheorem pullbackComp_hom_counit {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z)\n    [ChosenPullbacksAlong f] [ChosenPullbacksAlong g] [ChosenPullbacksAlong (f ≫ g)] :\n    Functor.whiskerRight (pullbackComp f g).hom (Over.map (f ≫ g)) ≫\n      (comp f g).mapPullbackAdj.counit =\n      (mapPullbackAdj (f ≫ g)).counit := by\n  rw [pullbackComp, Adjunction.rightAdjointUniq_hom_counit]\n\nset_option backward.defeqAttrib.useBackward true in\n/-- In cartesian monoidal categories, any morphism to the terminal tensor unit has a functorial\nchoice of pullbacks. -/\n@[instance_reducible, simps]\ndef cartesianMonoidalCategoryToUnit [CartesianMonoidalCategory C] {X : C} (f : X ⟶ 𝟙_ C) :\n    ChosenPullbacksAlong f where\n  pullback.obj Y := Over.mk (snd Y.left X)\n  pullback.map {Y Z} g := Over.homMk (g.left ▷ X)\n  mapPullbackAdj.unit.app T := Over.homMk (lift (𝟙 _) (T.hom))\n  mapPullbackAdj.counit.app U := Over.homMk (fst _ _)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- In cartesian monoidal categories, the first product projections `fst` have a functorial choice\nof pullbacks. -/\n@[simps, implicit_reducible]\ndef cartesianMonoidalCategoryFst [CartesianMonoidalCategory C] (X Y : C) :\n    ChosenPullbacksAlong (fst X Y : X ⊗ Y ⟶ X) where\n  pullback.obj Z := Over.mk (Z.hom ▷ Y)\n  pullback.map g := Over.homMk (g.left ▷ Y)\n  mapPullbackAdj.unit.app T := Over.homMk (lift (𝟙 _) (T.hom ≫ snd _ _))\n  mapPullbackAdj.counit.app U := Over.homMk (fst _ _)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- In cartesian monoidal categories, the second product projections `snd` have a functorial choice\nof pullbacks. -/\n@[simps, implicit_reducible]\ndef cartesianMonoidalCategorySnd [CartesianMonoidalCategory C] (X Y : C) :\n    ChosenPullbacksAlong (snd X Y : X ⊗ Y ⟶ Y) where\n  pullback.obj Z := Over.mk (X ◁ Z.hom)\n  pullback.map g := Over.homMk (X ◁ g.left)\n  mapPullbackAdj.unit.app T := Over.homMk (lift (T.hom ≫ fst _ _) (𝟙 _))\n  mapPullbackAdj.counit.app U := Over.homMk (snd _ _)\n\nsection PullbackFromChosenPullbacksAlongs\n\nvariable {Y Z X : C} (f : Y ⟶ X) (g : Z ⟶ X) [ChosenPullbacksAlong g]\n\n/-- The underlying object of the chosen pullback along `g` of `f`. -/\nabbrev pullbackObj : C := ((pullback g).obj (Over.mk f)).left\n\n/-- A morphism in `Over X` from the chosen pullback along `g` of `f` to `Over.mk f`. -/\nabbrev fst' : (Over.map g).obj ((pullback g).obj (Over.mk f)) ⟶ Over.mk f :=\n  (mapPullbackAdj g).counit.app <| Over.mk f\n\n/-- The first projection from the chosen pullback along `g` of `f` to the domain of `f`. -/\nabbrev fst : pullbackObj f g ⟶ Y := fst' f g |>.left\n\ntheorem fst'_left : (fst' f g).left = fst f g := rfl\n\n/-- The second projection from the chosen pullback along `g` of `f` to the domain of `g`. -/\nabbrev snd : pullbackObj f g ⟶ Z := (pullback g).obj (Over.mk f) |>.hom\n\n/-- A morphism in `Over X` from the chosen pullback along `g` of `f` to `Over.mk g`. -/\nabbrev snd' : (Over.map g).obj ((pullback g).obj (Over.mk f)) ⟶ (Over.mk g) :=\n  Over.homMk (snd f g)\n\ntheorem snd'_left : (snd' f g).left = snd f g := rfl\n\nvariable {f g}\n\n@[reassoc]\ntheorem condition : fst f g ≫ f = snd f g ≫ g :=\n  Over.w (fst' f g)\n\nvariable (f g) in\n@[ext]\n\nTarget:\ntheorem hom_ext {W : C} {φ₁ φ₂ : W ⟶ pullbackObj f g} (h₁ : φ₁ ≫ fst _ _ = φ₂ ≫ fst _ _)\n    (h₂ : φ₁ ≫ snd _ _ = φ₂ ≫ snd _ _) :\n    φ₁ = φ₂ :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/LocallyCartesianClosed","family_id":"hom_ext","file_id":"mathlib/Mathlib/CategoryTheory/LocallyCartesianClosed/ChosenPullbacksAlong.lean","sample_id":"56dfb71d32b180653fe5db01d6101ad74566b666462b27c7889ed2d552e18dc1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d56cb08de95550c2d5c8c73e4133b8fd22dbbf83f22597912cb3044a1cf97d24","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1afe6ea91860f1c986d85a12f7c2ff36c9860475193eed60e6c3a988e8a563f6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f15d582cb509e0a13008bd362768f09d19cdeaf2278912231d988aa0bbeaa66e","source_sha256":"e0af099a1dfedc907a6ae420e8f9c06d8630aad18426a643f2d72445c5f1b960","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [IsTranscendenceBasis, algebraicIndependent_equiv, EquivLike.range_comp]\n\nalias ⟨_, IsTranscendenceBasis.comp_equiv⟩ := isTranscendenceBasis_equiv","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":3,"token_jaccard":0.111111,"token_length_ratio":0.142857},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"8ea0c3702e806018da88c6a9f44d0d9352283c826d2b73924e83c4f8e50cd862","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.CommRing\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Algebraic Independence\n\nThis file defines algebraic independence of a family of elements of an `R` algebra.\n\n## Main definitions\n\n* `AlgebraicIndependent` - `AlgebraicIndependent R x` states the family of elements `x`\n  is algebraically independent over `R`, meaning that the canonical map out of the multivariable\n  polynomial ring is injective.\n\n* `AlgebraicIndependent.aevalEquiv` - The canonical isomorphism from the polynomial ring to the\n  subalgebra generated by an algebraic independent family.\n\n* `AlgebraicIndependent.repr` - The canonical map from the subalgebra generated by an\n  algebraic independent family into the polynomial ring. It is the inverse of\n  `AlgebraicIndependent.aevalEquiv`.\n\n* `IsTranscendenceBasis R x` - a family `x` is a transcendence basis over `R` if it is a maximal\n  algebraically independent subset.\n\n## Main results\n\nWe show that algebraic independence is preserved under injective maps of the indices.\n\n## References\n\n* [Stacks: Transcendence](https://stacks.math.columbia.edu/tag/030D)\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen Function Set Subalgebra MvPolynomial Algebra\n\nvariable {ι ι' : Type*} (R : Type*) {K A A' : Type*} (x : ι → A)\nvariable [CommRing R] [CommRing A] [CommRing A'] [Algebra R A] [Algebra R A']\n\n/-- `AlgebraicIndependent R x` states the family of elements `x`\n  is algebraically independent over `R`, meaning that the canonical\n  map out of the multivariable polynomial ring is injective. -/\n@[stacks 030E \"(1)\"] def AlgebraicIndependent : Prop :=\n  Injective (MvPolynomial.aeval x : MvPolynomial ι R →ₐ[R] A)\n\n/-- `AlgebraicIndepOn R v s` states that the elements in the family `v` that are indexed by the\nelements of `s` are algebraically independent over `R`. -/\nabbrev AlgebraicIndepOn (s : Set ι) : Prop := AlgebraicIndependent R fun i : s ↦ x i\n\nvariable {R} {x}\n\ntheorem algebraicIndependent_iff :\n    AlgebraicIndependent R x ↔\n      ∀ p : MvPolynomial ι R, MvPolynomial.aeval (x : ι → A) p = 0 → p = 0 :=\n  injective_iff_map_eq_zero _\n\ntheorem AlgebraicIndependent.eq_zero_of_aeval_eq_zero (h : AlgebraicIndependent R x) :\n    ∀ p : MvPolynomial ι R, MvPolynomial.aeval (x : ι → A) p = 0 → p = 0 :=\n  algebraicIndependent_iff.1 h\n\ntheorem algebraicIndependent_iff_injective_aeval :\n    AlgebraicIndependent R x ↔ Injective (MvPolynomial.aeval x : MvPolynomial ι R →ₐ[R] A) :=\n  Iff.rfl\n\nnamespace AlgebraicIndependent\n\ntheorem of_comp (f : A →ₐ[R] A') (hfv : AlgebraicIndependent R (f ∘ x)) :\n    AlgebraicIndependent R x := by\n  have : aeval (f ∘ x) = f.comp (aeval x) := by ext; simp\n  rw [AlgebraicIndependent, this, AlgHom.coe_comp] at hfv\n  exact hfv.of_comp\n\nvariable (hx : AlgebraicIndependent R x)\ninclude hx\n\ntheorem comp (f : ι' → ι) (hf : Function.Injective f) : AlgebraicIndependent R (x ∘ f) := by\n  intro p q\n  simpa [aeval_rename, (rename_injective f hf).eq_iff] using @hx (rename f p) (rename f q)\n\ntheorem coe_range : AlgebraicIndependent R ((↑) : range x → A) := by\n  simpa using hx.comp _ (rangeSplitting_injective x)\n\nend AlgebraicIndependent\n\nopen AlgebraicIndependent\n\ntheorem algebraicIndependent_equiv (e : ι ≃ ι') {f : ι' → A} :\n    AlgebraicIndependent R (f ∘ e) ↔ AlgebraicIndependent R f :=\n  ⟨fun h => Function.comp_id f ▸ e.self_comp_symm ▸ h.comp _ e.symm.injective,\n    fun h => h.comp _ e.injective⟩\n\ntheorem algebraicIndependent_equiv' (e : ι ≃ ι') {f : ι' → A} {g : ι → A} (h : f ∘ e = g) :\n    AlgebraicIndependent R g ↔ AlgebraicIndependent R f :=\n  h ▸ algebraicIndependent_equiv e\n\ntheorem algebraicIndependent_subtype_range {ι} {f : ι → A} (hf : Injective f) :\n    AlgebraicIndependent R ((↑) : range f → A) ↔ AlgebraicIndependent R f :=\n  Iff.symm <| algebraicIndependent_equiv' (Equiv.ofInjective f hf) rfl\n\nalias ⟨AlgebraicIndependent.of_subtype_range, _⟩ := algebraicIndependent_subtype_range\n\ntheorem algebraicIndependent_image {ι} {s : Set ι} {f : ι → A} (hf : Set.InjOn f s) :\n    (AlgebraicIndependent R fun x : s => f x) ↔ AlgebraicIndependent R fun x : f '' s => (x : A) :=\n  algebraicIndependent_equiv' (Equiv.Set.imageOfInjOn _ _ hf) rfl\n\nlemma AlgebraicIndepOn.mono {s t : Set ι} (H : AlgebraicIndepOn R x t) (hst : s ⊆ t) :\n    AlgebraicIndepOn R x s := by\n  simpa [Function.comp] using! H.comp (Set.inclusion hst) (Set.inclusion_injective hst)\n\n@[simp]\nlemma AlgebraicIndepOn.univ : AlgebraicIndepOn R x .univ ↔ AlgebraicIndependent R x :=\n  algebraicIndependent_equiv (Equiv.Set.univ ι)\n\nnamespace AlgebraicIndependent\n\ntheorem mono {t s : Set A} (h : t ⊆ s)\n    (hx : AlgebraicIndependent R ((↑) : s → A)) : AlgebraicIndependent R ((↑) : t → A) :=\n  AlgebraicIndepOn.mono (x := id) hx h\n\nsection repr\n\nvariable (hx : AlgebraicIndependent R x)\ninclude hx\n\n/-- Canonical isomorphism between polynomials and the subalgebra generated by\n  algebraically independent elements. -/\n@[simps! apply_coe]\ndef aevalEquiv : MvPolynomial ι R ≃ₐ[R] Algebra.adjoin R (range x) :=\n  (AlgEquiv.ofInjective (aeval x) (algebraicIndependent_iff_injective_aeval.1 hx)).trans\n    (Subalgebra.equivOfEq _ _ (Algebra.adjoin_range_eq_range_aeval R x).symm)\n\ntheorem algebraMap_aevalEquiv (p : MvPolynomial ι R) :\n    algebraMap (Algebra.adjoin R (range x)) A (hx.aevalEquiv p) = aeval x p :=\n  rfl\n\n/-- The canonical map from the subalgebra generated by an algebraic independent family\n  into the polynomial ring. -/\ndef repr : Algebra.adjoin R (range x) →ₐ[R] MvPolynomial ι R :=\n  hx.aevalEquiv.symm\n\n@[simp]\ntheorem aeval_repr (p) : aeval x (hx.repr p) = p :=\n  Subtype.ext_iff.1 (AlgEquiv.apply_symm_apply hx.aevalEquiv p)\n\ntheorem aeval_comp_repr : (aeval x).comp hx.repr = Subalgebra.val _ :=\n  AlgHom.ext hx.aeval_repr\n\n/-- Canonical isomorphism between polynomials and the subalgebra generated by\n  algebraically independent elements.\n  `AlgebraicIndepOn` version of `AlgebraicIndependent.aevalEquiv`. -/\ndef _root_.AlgebraicIndepOn.aevalEquiv {s : Set ι} (hx : AlgebraicIndepOn R x s) :\n    MvPolynomial s R ≃ₐ[R] Algebra.adjoin R (x '' s) :=\n  (AlgebraicIndependent.aevalEquiv hx).trans\n    (Subalgebra.equivOfEq _ _ congr(Algebra.adjoin _ $(by aesop)))\n\nend repr\n\nend AlgebraicIndependent\n\nvariable (R) in\n/-- A family is a transcendence basis if it is a maximal algebraically independent subset. -/\n@[stacks 030E \"(4)\"] def IsTranscendenceBasis (x : ι → A) : Prop :=\n  AlgebraicIndependent R x ∧\n    ∀ (s : Set A) (_ : AlgebraicIndepOn R id s) (_ : range x ⊆ s), range x = s\n\ntheorem isTranscendenceBasis_iff_maximal {s : Set A} :\n    IsTranscendenceBasis R ((↑) : s → A) ↔ Maximal (AlgebraicIndepOn R id) s := by\n  rw [IsTranscendenceBasis, maximal_iff, Subtype.range_val]; rfl\n\nTarget:\ntheorem isTranscendenceBasis_equiv (e : ι ≃ ι') {f : ι' → A} :\n    IsTranscendenceBasis R (f ∘ e) ↔ IsTranscendenceBasis R f :=\n\nProof body:\n","rejected":"by\n  exact isTranscendenceBasis_equiv","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"349c187611a38322513dd8291f4544125dc67d61a1c0a92c78b256e97edc90b3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/AlgebraicIndependent","family_id":"istranscendencebasis_equiv","file_id":"mathlib/Mathlib/RingTheory/AlgebraicIndependent/Defs.lean","sample_id":"f15d582cb509e0a13008bd362768f09d19cdeaf2278912231d988aa0bbeaa66e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4d69ce6ddd118e205dc838354a84f5c72260ded77b2d93b7242fcdda08dda204","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e904f2b2bc2150eaeb5f082549806fb4c52d3efb6a424a313f952e0e67dc031d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"039e1da8660af549f0b84de8a8d3a1a21828f8928f6164e5d53b92bab06d1d90","source_sha256":"bfb63160eae740c943f595c7368534b4a070f971456c777a0ade88e8c2da9349","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rwa [← hasPointwiseRightDerivedFunctorAt_iff F L W]","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":2,"token_jaccard":0.1,"token_length_ratio":0.222222},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"8ecb61d91a32633b256de9adaf691ff844d6ae55a7079a39dc61fb75b60575a9","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Functor.Derived.RightDerived\npublic import Mathlib.CategoryTheory.Functor.KanExtension.Pointwise\npublic import Mathlib.CategoryTheory.Localization.StructuredArrow\n\nNamespace:\nCategoryTheory.Functor\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Pointwise right derived functors\n\nWe define pointwise right derived functors using the notion\nof pointwise left Kan extensions.\n\nWe show that if `F : C ⥤ H` inverts `W : MorphismProperty C`,\nthen it has a pointwise right derived functor.\n\nNote: the file `Mathlib/CategoryTheory/Functor/Derived/PointwiseLeftDerived.lean` was obtained\nby dualizing this file. These two files should be kept in sync.\n\n-/\n\n@[expose] public section\n\nuniverse v₁ v₂ v₃ u₁ u₂ u₃\n\nnamespace CategoryTheory\n\nopen Category Limits\n\nnamespace Functor\n\nvariable {C : Type u₁} {D : Type u₂} {H : Type u₃}\n  [Category.{v₁} C] [Category.{v₂} D] [Category.{v₃} H]\n  (F' : D ⥤ H) (F : C ⥤ H) (L : C ⥤ D) (α : F ⟶ L ⋙ F') (W : MorphismProperty C)\n\n/-- Given `F : C ⥤ H`, `W : MorphismProperty C` and `X : C`, we say that `F` has a\npointwise right derived functor at `X` if `F` has a left Kan extension\nat `L.obj X` for any localization functor `L : C ⥤ D` for `W`. In the\ndefinition, this is stated for `L := W.Q`, see `hasPointwiseRightDerivedFunctorAt_iff`\nfor the more general equivalence. -/\nclass HasPointwiseRightDerivedFunctorAt (X : C) : Prop where\n  /-- Use the more general `hasColimit` lemma instead, see also\n  `hasPointwiseRightDerivedFunctorAt_iff` -/\n  hasColimit' : HasPointwiseLeftKanExtensionAt W.Q F (W.Q.obj X)\n\n/-- A functor `F : C ⥤ H` has a pointwise right derived functor with respect to\n`W : MorphismProperty C` if it has a pointwise right derived functor at `X`\nfor any `X : C`. -/\nabbrev HasPointwiseRightDerivedFunctor := ∀ (X : C), F.HasPointwiseRightDerivedFunctorAt W X\n\nlemma hasPointwiseRightDerivedFunctorAt_iff [L.IsLocalization W] (X : C) :\n    F.HasPointwiseRightDerivedFunctorAt W X ↔\n      HasPointwiseLeftKanExtensionAt L F (L.obj X) := by\n  rw [← hasPointwiseLeftKanExtensionAt_iff_of_equivalence W.Q L F\n    (Localization.uniq W.Q L W) (Localization.compUniqFunctor W.Q L W) (W.Q.obj X) (L.obj X)\n    ((Localization.compUniqFunctor W.Q L W).app X)]\n  exact ⟨fun h ↦ h.hasColimit', fun h ↦ ⟨h⟩⟩\n\nTarget:\nlemma HasPointwiseRightDerivedFunctorAt.hasColimit\n    [L.IsLocalization W] (X : C) [F.HasPointwiseRightDerivedFunctorAt W X] :\n    HasPointwiseLeftKanExtensionAt L F (L.obj X) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_039e1da8660a","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"580e8f348462902d745bc6e1c9348bd01aa0463216dab73ed3236bdfde086779","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Functor","family_id":"haspointwiserightderivedfunctorat","file_id":"mathlib/Mathlib/CategoryTheory/Functor/Derived/PointwiseRightDerived.lean","sample_id":"039e1da8660af549f0b84de8a8d3a1a21828f8928f6164e5d53b92bab06d1d90"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9c3779b7663f368283b7a9eee3adc6092b2e62b6c24d85d914985cdd33c9ca16","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"68abd3e690faef28c6d313fc42ae55322e5fb837407140a7bbfd2e3103e9c9b1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b7454ed33b7067c1c405e0b58237396fb00e395f6ff19576d9916ffdee3bc4d9","source_sha256":"4d4d0117550f0a508e4a45c091460a207bd2706a42ac9b83d9699e528b57c020","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun H => H.van_kampen' c₀ hc₀, fun H => ?_⟩\n  constructor\n  simp_rw [BinaryCofan.isVanKampen_iff] at H ⊢\n  intro X Y c hc X' Y' c' αX αY f hX hY\n  obtain ⟨d, hd, hd'⟩ :=\n    Limits.BinaryCofan.IsColimit.desc' hc (HT.from _ ≫ c₀.inl) (HT.from _ ≫ c₀.inr)\n  rw [H c' (αX ≫ HT.from _) (αY ≫ HT.from _) (f ≫ d) (by rw [← reassoc_of% hX, hd, Category.assoc])\n      (by rw [← reassoc_of% hY, hd', Category.assoc])]\n  obtain ⟨hl, hr⟩ := (H c (HT.from _) (HT.from _) d hd.symm hd'.symm).mp ⟨hc⟩\n  rw [hl.paste_vert_iff hX.symm, hr.paste_vert_iff hY.symm]","hard_negative":true,"metrics":{"chosen_tokens":195,"rejected_tokens":2,"token_jaccard":0.016667,"token_length_ratio":0.010256},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"8ee48d16076ce0efbb35cc9651a3611904e557ea2382155c61a34f3275f07f91","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Types.Coproducts\npublic import Mathlib.CategoryTheory.Limits.Types.Products\npublic import Mathlib.CategoryTheory.Limits.Types.Pullbacks\npublic import Mathlib.Topology.Category.TopCat.Limits.Pullbacks\npublic import Mathlib.CategoryTheory.Limits.VanKampen\npublic import Mathlib.CategoryTheory.Limits.MonoCoprod\npublic import Mathlib.CategoryTheory.Limits.Shapes.DisjointCoproduct\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Extensive categories\n\n## Main definitions\n- `CategoryTheory.FinitaryExtensive`: A category is (finitary) extensive if it has finite\n  coproducts, and binary coproducts are van Kampen.\n\n## Main Results\n- `CategoryTheory.hasStrictInitialObjects_of_finitaryExtensive`: The initial object\n  in extensive categories is strict.\n- `CategoryTheory.FinitaryExtensive.mono_inr_of_isColimit`: Coproduct injections are monic in\n  extensive categories.\n- `CategoryTheory.BinaryCofan.isPullback_initial_to_of_isVanKampen`: In extensive categories,\n  sums are disjoint, i.e. the pullback of `X ⟶ X ⨿ Y` and `Y ⟶ X ⨿ Y` is the initial object.\n- `CategoryTheory.types.finitaryExtensive`: The category of types is extensive.\n- `CategoryTheory.FinitaryExtensive_TopCat`:\n  The category `Top` is extensive.\n- `CategoryTheory.FinitaryExtensive_functor`: The category `C ⥤ D` is extensive if `D`\n  has all pullbacks and is extensive.\n- `CategoryTheory.FinitaryExtensive.isVanKampen_finiteCoproducts`: Finite coproducts in a\n  finitary extensive category are van Kampen.\n\n## References\n- https://ncatlab.org/nlab/show/extensive+category\n- [Carboni et al, Introduction to extensive and distributive categories][CARBONI1993145]\n\n-/\n\n@[expose] public section\n\nopen CategoryTheory.Limits Topology\n\nnamespace CategoryTheory\n\nuniverse v' u' v u v'' u''\n\nvariable {J : Type v'} [Category.{u'} J] {C : Type u} [Category.{v} C]\nvariable {D : Type u''} [Category.{v''} D]\n\nsection Extensive\n\nvariable {X Y : C}\n\n/-- A category has pullback of inclusions if it has all pullbacks along coproduct injections. -/\nclass HasPullbacksOfInclusions (C : Type u) [Category.{v} C] [HasBinaryCoproducts C] : Prop where\n  [hasPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), HasPullback coprod.inl f]\n\nattribute [instance] HasPullbacksOfInclusions.hasPullbackInl\n\n/--\nA functor preserves pullback of inclusions if it preserves all pullbacks along coproduct injections.\n-/\nclass PreservesPullbacksOfInclusions {C : Type*} [Category* C] {D : Type*} [Category* D]\n    (F : C ⥤ D) [HasBinaryCoproducts C] where\n  [preservesPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), PreservesLimit (cospan coprod.inl f) F]\n\nattribute [instance] PreservesPullbacksOfInclusions.preservesPullbackInl\n\n/-- A category is (finitary) pre-extensive if it has finite coproducts,\nand binary coproducts are universal. -/\nclass FinitaryPreExtensive (C : Type u) [Category.{v} C] : Prop where\n  [hasFiniteCoproducts : HasFiniteCoproducts C]\n  [hasPullbacksOfInclusions : HasPullbacksOfInclusions C]\n  /-- In a finitary extensive category, all coproducts are van Kampen -/\n  universal' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsUniversalColimit c\n\nattribute [instance] FinitaryPreExtensive.hasFiniteCoproducts\nattribute [instance] FinitaryPreExtensive.hasPullbacksOfInclusions\n\n/-- A category is (finitary) extensive if it has finite coproducts,\nand binary coproducts are van Kampen. -/\nclass FinitaryExtensive (C : Type u) [Category.{v} C] : Prop where\n  [hasFiniteCoproducts : HasFiniteCoproducts C]\n  [hasPullbacksOfInclusions : HasPullbacksOfInclusions C]\n  /-- In a finitary extensive category, all coproducts are van Kampen -/\n  van_kampen' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsVanKampenColimit c\n\nattribute [instance] FinitaryExtensive.hasFiniteCoproducts\nattribute [instance] FinitaryExtensive.hasPullbacksOfInclusions\n\ntheorem FinitaryExtensive.vanKampen [FinitaryExtensive C] {F : Discrete WalkingPair ⥤ C}\n    (c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c := by\n  let X := F.obj ⟨WalkingPair.left⟩\n  let Y := F.obj ⟨WalkingPair.right⟩\n  have : F = pair X Y := by\n    apply Functor.hext\n    · rintro ⟨⟨⟩⟩ <;> rfl\n    · rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simp [X, Y]\n  clear_value X Y\n  subst this\n  exact FinitaryExtensive.van_kampen' c hc\n\nnamespace HasPullbacksOfInclusions\n\ninstance (priority := 100) [HasBinaryCoproducts C] [HasPullbacks C] :\n    HasPullbacksOfInclusions C := ⟨⟩\n\nvariable [HasBinaryCoproducts C] [HasPullbacksOfInclusions C] {X Y Z : C} (f : Z ⟶ X ⨿ Y)\n\ninstance preservesPullbackInl' :\n    HasPullback f coprod.inl :=\n  hasPullback_symmetry _ _\n\nset_option backward.isDefEq.respectTransparency false in\ninstance hasPullbackInr' :\n    HasPullback f coprod.inr := by\n  have : IsPullback (𝟙 _) (f ≫ (coprod.braiding X Y).hom) f (coprod.braiding Y X).hom :=\n    IsPullback.of_horiz_isIso ⟨by simp⟩\n  have := (IsPullback.of_hasPullback (f ≫ (coprod.braiding X Y).hom) coprod.inl).paste_horiz this\n  simp only [coprod.braiding_hom, Category.comp_id, colimit.ι_desc,\n    BinaryCofan.ι_app_left, BinaryCofan.mk_inl] at this\n  exact ⟨⟨⟨_, this.isLimit⟩⟩⟩\n\ninstance hasPullbackInr :\n    HasPullback coprod.inr f :=\n  hasPullback_symmetry _ _\n\nend HasPullbacksOfInclusions\n\nnamespace PreservesPullbacksOfInclusions\n\nvariable {D : Type*} [Category* D] [HasBinaryCoproducts C] (F : C ⥤ D)\n\nnoncomputable\ninstance (priority := 100) [PreservesLimitsOfShape WalkingCospan F] :\n    PreservesPullbacksOfInclusions F := ⟨⟩\n\nvariable [PreservesPullbacksOfInclusions F] {X Y Z : C} (f : Z ⟶ X ⨿ Y)\n\nnoncomputable\ninstance preservesPullbackInl' :\n    PreservesLimit (cospan f coprod.inl) F :=\n  preservesPullback_symmetry _ _ _\n\nset_option backward.isDefEq.respectTransparency false in\nnoncomputable\ninstance preservesPullbackInr' :\n    PreservesLimit (cospan f coprod.inr) F := by\n  apply preservesLimit_of_iso_diagram (K₁ := cospan (f ≫ (coprod.braiding X Y).hom) coprod.inl)\n  apply cospanExt (Iso.refl _) (Iso.refl _) (coprod.braiding X Y).symm <;> simp\n\nnoncomputable\ninstance preservesPullbackInr :\n    PreservesLimit (cospan coprod.inr f) F :=\n  preservesPullback_symmetry _ _ _\n\nend PreservesPullbacksOfInclusions\n\ninstance (priority := 100) FinitaryExtensive.toFinitaryPreExtensive [FinitaryExtensive C] :\n    FinitaryPreExtensive C :=\n  ⟨fun c hc ↦ (FinitaryExtensive.van_kampen' c hc).isUniversal⟩\n\ntheorem FinitaryExtensive.mono_inr_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}\n    (hc : IsColimit c) : Mono c.inr :=\n  BinaryCofan.mono_inr_of_isVanKampen (FinitaryExtensive.vanKampen c hc)\n\ntheorem FinitaryExtensive.mono_inl_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}\n    (hc : IsColimit c) : Mono c.inl :=\n  FinitaryExtensive.mono_inr_of_isColimit (BinaryCofan.isColimitFlip hc)\n\ninstance (priority := low) [FinitaryExtensive C] : MonoCoprod C where\n  binaryCofan_inl _ _ _ hc := BinaryCofan.mono_inr_of_isVanKampen\n    (FinitaryExtensive.vanKampen _ (BinaryCofan.isColimitFlip hc))\n\ntheorem FinitaryExtensive.isPullback_initial_to_binaryCofan [FinitaryExtensive C]\n    {c : BinaryCofan X Y} (hc : IsColimit c) :\n    IsPullback (initial.to _) (initial.to _) c.inl c.inr :=\n  BinaryCofan.isPullback_initial_to_of_isVanKampen (FinitaryExtensive.vanKampen c hc)\n\nset_option backward.defeqAttrib.useBackward true in\ninstance (priority := 100) hasStrictInitialObjects_of_finitaryPreExtensive\n    [FinitaryPreExtensive C] : HasStrictInitialObjects C :=\n  hasStrictInitial_of_isUniversal (FinitaryPreExtensive.universal' _\n    ((BinaryCofan.isColimit_iff_isIso_inr initialIsInitial _).mpr (by\n      dsimp\n      infer_instance)).some)\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem finitaryExtensive_iff_of_isTerminal (C : Type u) [Category.{v} C] [HasFiniteCoproducts C]\n    [HasPullbacksOfInclusions C]\n    (T : C) (HT : IsTerminal T) (c₀ : BinaryCofan T T) (hc₀ : IsColimit c₀) :\n    FinitaryExtensive C ↔ IsVanKampenColimit c₀ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_b7454ed33b70","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"08f75b99c3c4c91ed068afee8bd658ec64932b7b66848b1fb33a3c1e788bb012","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory","family_id":"finitaryextensive_iff_of_isterminal","file_id":"mathlib/Mathlib/CategoryTheory/Extensive.lean","sample_id":"b7454ed33b7067c1c405e0b58237396fb00e395f6ff19576d9916ffdee3bc4d9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d0f64a8fe631e03ccedfddafbaffe13122e5547990701f69b60f4ed018241b86","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fbfa2ca2104fb09021e16e7f3542762a6ce83fb6ca653dc82a4df2bcb75e0375","source_sha256":"f3daaba8f2638bcf797b824f67c87ade5dbc78d42b30d85f5c7e88fd1a4627c0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext f : 1\n  apply (Matrix.stdBasis _ _ _).ext\n  intro ⟨i, j⟩\n  simp only [LinearMap.coe_comp, LinearMap.coe_mk, AddHom.coe_mk, Function.comp_apply, map_sum,\n    map_smul, stdBasis_eq_single, LinearMap.coe_sum, Finset.sum_apply,\n    LinearMap.smul_apply, LinearMap.id_coe, id_eq]\n  ext k l\n  simp [sum_apply, Matrix.mul_apply, single, Fintype.sum_prod_type, ite_and]","hard_negative":false,"metrics":{"chosen_tokens":84,"rejected_tokens":2,"token_jaccard":0.022222,"token_length_ratio":0.02381},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"8f5d730d1adf0f3a3e57d7755282e4455eb1c30aafabc0acc72e5afc29499a07","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Azumaya.Defs\npublic import Mathlib.LinearAlgebra.FreeModule.Finite.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Yunzhou Xie. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yunzhou Xie, Jujian Zhang\n-/\n/-!\n# Matrix algebra is an Azumaya algebra over R\n\nIn this file we prove that finite-dimensional matrix algebra `Matrix n n R` over `R`\nis an Azumaya algebra where `R` is a commutative ring.\n\n## Main Results\n\n- `IsAzumaya.Matrix`: Finite-dimensional matrix algebra over `R` is Azumaya.\n\n-/\n\npublic section\nopen scoped TensorProduct\n\nvariable (R n : Type*) [CommSemiring R] [Fintype n] [DecidableEq n]\n\nnoncomputable section\n\nopen Matrix MulOpposite\n\n/-- `AlgHom.mulLeftRight` for matrix algebra sends basis Eᵢⱼ⊗Eₖₗ to\n  the map `f : Eₛₜ ↦ Eᵢⱼ * Eₛₜ * Eₖₗ = δⱼₛδₜₖEᵢₗ`, therefore we construct the inverse\n  by sending `f` to `∑ᵢₗₛₜ f(Eₛₜ)ᵢₗ • Eᵢₛ⊗Eₜₗ`. -/\nabbrev AlgHom.mulLeftRightMatrix_inv :\n    Module.End R (Matrix n n R) →ₗ[R] Matrix n n R ⊗[R] (Matrix n n R)ᵐᵒᵖ where\n  toFun f := ∑ ⟨⟨i, j⟩, k, l⟩ : (n × n) × n × n,\n    f (single j k 1) i l • (single i j 1) ⊗ₜ[R] op (single k l 1)\n  map_add' f1 f2 := by simp [add_smul, Finset.sum_add_distrib]\n  map_smul' r f := by simp [mul_smul, Finset.smul_sum]\n\nlemma AlgHom.mulLeftRightMatrix.inv_comp :\n    (AlgHom.mulLeftRightMatrix_inv R n).comp\n    (AlgHom.mulLeftRight R (Matrix n n R)).toLinearMap = .id :=\n  ((Matrix.stdBasis _ _ _).tensorProduct ((Matrix.stdBasis _ _ _).map (opLinearEquiv ..))).ext\n  fun ⟨⟨i0, j0⟩, k0, l0⟩ ↦ by\n    simp [stdBasis_eq_single, ite_and, Fintype.sum_prod_type,\n      mulLeftRight_apply, single, Matrix.mul_apply]\n\nTarget:\nlemma AlgHom.mulLeftRightMatrix.comp_inv :\n    (AlgHom.mulLeftRight R (Matrix n n R)).toLinearMap.comp\n    (AlgHom.mulLeftRightMatrix_inv R n) = .id :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Azumaya","family_id":"alghom","file_id":"mathlib/Mathlib/Algebra/Azumaya/Matrix.lean","sample_id":"fbfa2ca2104fb09021e16e7f3542762a6ce83fb6ca653dc82a4df2bcb75e0375"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bb5993596ab6f0322c25fd89be3f755c85b89f6e4d2f8ad57eb17900f2915564","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"28000438c5c8b56a4e4bd3586216319d20a4d6b0a20e68d72c07cc61c75371d1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4aa461ab228ed2d6d40d0634d68fa0428d1672b0b417d2b773b3a8831f202fa2","source_sha256":"522c0739e8f7535bf0f954aff339b473dbd0f9e903c0b5a9a4fff87d391d66b3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext; simp only [add_def, smul_def, _root_.add_smul]","hard_negative":true,"metrics":{"chosen_tokens":14,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.214286},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"8f5e88ad3a6c2cdd8f8cb9e68a3aafaa3f9c7e9367a418a9866dde032e626ba5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorProduct.RightExactness\npublic import Mathlib.RingTheory.Congruence.Hom\npublic import Mathlib.RingTheory.FiniteType\npublic import Mathlib.RingTheory.TensorProduct.DirectLimitFG\n\nNamespace:\nPolynomialLaw\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir & María-Inés de Frutos-Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir & María-Inés de Frutos-Fernández\n-/\n/-! # Polynomial laws on modules\n\nLet `M` and `N` be a modules over a commutative ring `R`.\nA polynomial law `f : PolynomialLaw R M N`, with notation `f : M →ₚₗₗ[R] N`,\nis a “law” that assigns a natural map `PolynomialLaw.toFun' f S : S ⊗[R] M → S ⊗[R] N`\nfor every `R`-algebra `S`.\n\nFor type-theoretic reasons, if `R : Type u`, then the definition of the polynomial map `f`\nis restricted to `R`-algebras `S` such that `S : Type u`.\nUsing the fact that a module is the direct limit of its finitely generated submodules, that a\nfinitely generated subalgebra is a quotient of a polynomial ring in the universe `u`, plus\nthe commutation of tensor products with direct limits, we extend the functor\nto all `R`-algebras.\n\nThe two fields involving the definition of `PolynomialLaw`,\n`PolynomialLaw.toFun'` and `PolynomialLaw.isCompat'` are primed.\nThey are superseded by their universe-polymorphic counterparts,\nthe definition `PolynomialLaw.toFun` and the lemma `PolynomialLaw.isCompat`\nwhich should be used once the theory is properly stated.\n\nFor constructions of general definitions of `PolynomialLaw`\nat a universe-polymorphic level, one needs to lift\nelements in a tensor product to smaller universes.\nFor this, one can make use of\n`PolynomialLaw.exists_lift` or `PolynomialLaw.exists_lift'`,\nor establish appropriate generalizations.\n\n## Main definitions/lemmas\n\n* Instance : `Module R (M →ₚₗ[R] N)` shows that polynomial laws form an `R`-module.\n\n* `PolynomialLaw.ground f` is the map `M → N` corresponding to `PolynomialLaw.toFun' f R` under\n  the isomorphisms `R ⊗[R] M ≃ₗ[R] M`, and similarly for `N`.\n\nIn further works, we construct the coefficients of a polynomial law and show the relation with\npolynomials (when the module `M` is free and finite).\n\n## Implementation notes\n\nIn the literature, the theory is written for commutative rings, but this implementation\nonly assumes `R` is a commutative semiring.\n\n## References\n\n* [Roby, Norbert. 1963. «Lois polynomes et lois formelles en théorie des modules».\n  Annales scientifiques de l’École Normale Supérieure 80 (3): 213‑348](Roby-1963)\n\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nnoncomputable section PolynomialLaw\n\nopen scoped TensorProduct\n\nopen LinearMap TensorProduct AlgHom RingCon\n\n/-- A polynomial law `M →ₚₗ[R] N` between `R`-modules is a functorial family of maps\n`S ⊗[R] M → S ⊗[R] N`, for all `R`-algebras `S`.\n\nFor universe reasons, `S` has to be restricted to the same universe as `R`. -/\n@[ext]\nstructure PolynomialLaw (R : Type u) [CommSemiring R]\n    (M : Type*) [AddCommMonoid M] [Module R M] (N : Type*) [AddCommMonoid N] [Module R N] where\n  /-- The functions `S ⊗[R] M → S ⊗[R] N` underlying a polynomial law -/\n  toFun' (S : Type u) [CommSemiring S] [Algebra R S] : S ⊗[R] M → S ⊗[R] N\n  /-- The compatibility relations between the functions underlying a polynomial law -/\n  isCompat' {S : Type u} [CommSemiring S] [Algebra R S]\n    {S' : Type u} [CommSemiring S'] [Algebra R S'] (φ : S →ₐ[R] S') :\n    φ.toLinearMap.rTensor N ∘ toFun' S = toFun' S' ∘ φ.toLinearMap.rTensor M := by aesop\n\n/-- `M →ₚₗ[R] N` is the type of `R`-polynomial laws from `M` to `N`. -/\nnotation:25 M \" →ₚₗ[\" R:25 \"] \" N:0 => PolynomialLaw R M N\n\n@[local simp]\ntheorem PolynomialLaw.isCompat_apply'\n    {R : Type u} [CommSemiring R] {M : Type*} [AddCommMonoid M] [Module R M]\n    {N : Type*} [AddCommMonoid N] [Module R N] {f : M →ₚₗ[R] N}\n    {S : Type u} [CommSemiring S] [Algebra R S] {S' : Type u} [CommSemiring S'] [Algebra R S']\n    (φ : S →ₐ[R] S') (x : S ⊗[R] M) :\n    (φ.toLinearMap.rTensor N) ((f.toFun' S) x) = (f.toFun' S') (φ.toLinearMap.rTensor M x) := by\n  simpa only using! congr_fun (f.isCompat' φ) x\n\nattribute [local simp] PolynomialLaw.isCompat_apply'\n\nnamespace PolynomialLaw\n\nsection Module\n\nsection CommSemiring\n\nvariable {R : Type u} [CommSemiring R] {M : Type*} [AddCommMonoid M] [Module R M]\n  {N : Type*} [AddCommMonoid N] [Module R N] (r a b : R) (f g : M →ₚₗ[R] N)\n\ninstance : Zero (M →ₚₗ[R] N) := ⟨{ toFun' _ := 0 }⟩\n\n@[simp]\ntheorem zero_def (S : Type u) [CommSemiring S] [Algebra R S] :\n    (0 : PolynomialLaw R M N).toFun' S = 0 := rfl\n\ninstance : Inhabited (PolynomialLaw R M N) := ⟨Zero.zero⟩\n\n/-- The identity as a polynomial law -/\ndef id : M →ₚₗ[R] M where\n  toFun' S _ _ := _root_.id\n\ntheorem id_apply' {S : Type u} [CommSemiring S] [Algebra R S] :\n    (id : M →ₚₗ[R] M).toFun' S = _root_.id := rfl\n\n/-- The sum of two polynomial laws -/\nnoncomputable def add : M →ₚₗ[R] N where\n  toFun' S _ _ := f.toFun' S + g.toFun' S\n\ninstance : Add (PolynomialLaw R M N) := ⟨add⟩\n\n@[simp]\ntheorem add_def (S : Type u) [CommSemiring S] [Algebra R S] :\n    (f + g).toFun' S = f.toFun' S + g.toFun' S := rfl\n\ntheorem add_def_apply (S : Type u) [CommSemiring S] [Algebra R S] (m : S ⊗[R] M) :\n    (f + g).toFun' S m = f.toFun' S m + g.toFun' S m := rfl\n\n/-- External multiplication of a `f : M →ₚₗ[R] N` by `r : R` -/\ndef smul : M →ₚₗ[R] N where\n  toFun' S _ _ := r • f.toFun' S\n\ninstance : SMul R (M →ₚₗ[R] N) := ⟨smul⟩\n\n@[simp]\ntheorem smul_def (S : Type u) [CommSemiring S] [Algebra R S] :\n    (r • f).toFun' S = r • f.toFun' S := rfl\n\ntheorem smul_def_apply (S : Type u) [CommSemiring S] [Algebra R S] (m : S ⊗[R] M) :\n    (r • f).toFun' S m = r • f.toFun' S m := rfl\n\nTarget:\ntheorem add_smul : (a + b) • f = a • f + b • f :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_4aa461ab228e","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"75f60bd80f07d9be411f0e03023ac57fd7a647f3128712d968d0a447d00c9ca9","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/PolynomialLaw","family_id":"add_smul","file_id":"mathlib/Mathlib/RingTheory/PolynomialLaw/Basic.lean","sample_id":"4aa461ab228ed2d6d40d0634d68fa0428d1672b0b417d2b773b3a8831f202fa2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"19a0b796b5a123aa0fa4da0bb1ec6d1209352b44dfd80eed24e61f22f0837fb5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0a46fac18823464095b42916006166b41806c313503a9332eebd632a234896e3","source_sha256":"9204017e76fd98a1b3e65ba16535a42898573646b8f44e5554ab4317200de227","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine measure_mono_null ?_ hs.ordConnected.null_frontier\n  rw [← closure_sdiff_interior, hs.interior_eq_empty, sdiff_empty]\n  exact subset_closure","hard_negative":true,"metrics":{"chosen_tokens":23,"rejected_tokens":8,"token_jaccard":0.04,"token_length_ratio":0.347826},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"8fc52e7a2c60288ccc21487b9ec9b76463104f940224ac42d05a03ab78a23f83","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Order.UpperLower\npublic import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace\npublic import Mathlib.Topology.Order.DenselyOrdered\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies, Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Kexing Ying\n-/\n/-!\n# Order-connected sets are null-measurable\n\nThis file proves that order-connected sets in `ℝⁿ` under the pointwise order are null-measurable.\nRecall that `x ≤ y` iff `∀ i, x i ≤ y i`, and `s` is order-connected iff\n`∀ x y ∈ s, ∀ z, x ≤ z → z ≤ y → z ∈ s`.\n\n## Main declarations\n\n* `Set.OrdConnected.null_frontier`: The frontier of an order-connected set in `ℝⁿ` has measure `0`.\n\n## Notes\n\nWe prove null-measurability in `ℝⁿ` with the `∞`-metric, but this transfers directly to `ℝⁿ` with\nthe Euclidean metric because they have the same measurable sets.\n\nNull-measurability can't be strengthened to measurability because any antichain (and in particular\nany subset of the antidiagonal `{(x, y) | x + y = 0}`) is order-connected.\n\n## Sketch proof\n\n1. To show an order-connected set is null-measurable, it is enough to show it has null frontier.\n2. Since an order-connected set is the intersection of its upper and lower closure, it's enough to\n  show that upper and lower sets have null frontier.\n3. WLOG let's prove it for an upper set `s`.\n4. By the Lebesgue density theorem, it is enough to show that any frontier point `x` of `s` is not a\n  Lebesgue point, namely we want the density of `s` over small balls centered at `x` to not tend to\n  either `0` or `1`.\n5. This is true, since by the upper setness of `s` we can intercalate a ball of radius `δ / 4` in\n  `s` intersected with the upper quadrant of the ball of radius `δ` centered at `x` (recall that the\n  balls are taken in the ∞-norm, so they are cubes), and another ball of radius `δ / 4` in `sᶜ` and\n  the lower quadrant of the ball of radius `δ` centered at `x`.\n\n## TODO\n\nGeneralize so that it also applies to `ℝ × ℝ`, for example.\n-/\n\npublic section\n\nopen Filter MeasureTheory Metric Set\nopen scoped Topology\n\nvariable {ι : Type*} [Fintype ι] {s : Set (ι → ℝ)} {x : ι → ℝ}\n\n/-- If we can fit a small ball inside a set `s` intersected with any neighborhood of `x`, then the\ndensity of `s` near `x` is not `0`.\n\nAlong with `aux₁`, this proves that `x` is not a Lebesgue point of `s`. This will be used to prove\nthat the frontier of an order-connected set is null. -/\nprivate lemma aux₀\n    (h : ∀ δ, 0 < δ →\n      ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior s) :\n    ¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)\n        (𝓝 0) := by\n  choose f hf₀ hf₁ using h\n  intro H\n  obtain ⟨ε, -, hε', hε₀⟩ := exists_seq_strictAnti_tendsto_nhdsWithin (0 : ℝ)\n  refine not_eventually.2\n    (Frequently.of_forall fun _ ↦ lt_irrefl <| ENNReal.ofReal <| 4⁻¹ ^ Fintype.card ι)\n    ((Filter.Tendsto.eventually_lt (H.comp hε₀) tendsto_const_nhds ?_).mono fun n ↦\n      lt_of_le_of_lt ?_)\n  on_goal 2 =>\n    calc\n      ENNReal.ofReal (4⁻¹ ^ Fintype.card ι)\n        = volume (closedBall (f (ε n) (hε' n)) (ε n / 4)) / volume (closedBall x (ε n)) := ?_\n      _ ≤ volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n)) := by\n        gcongr\n        exact subset_inter ((hf₁ _ <| hε' n).trans interior_subset_closure) <| hf₀ _ <| hε' n\n    have := hε' n\n    rw [Real.volume_pi_closedBall, Real.volume_pi_closedBall, ← ENNReal.ofReal_div_of_pos,\n      ← div_pow, mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]\n  all_goals positivity\n\n/-- If we can fit a small ball inside a set `sᶜ` intersected with any neighborhood of `x`, then the\ndensity of `s` near `x` is not `1`.\n\nAlong with `aux₀`, this proves that `x` is not a Lebesgue point of `s`. This will be used to prove\nthat the frontier of an order-connected set is null. -/\nprivate lemma aux₁\n    (h : ∀ δ, 0 < δ →\n      ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior sᶜ) :\n    ¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)\n        (𝓝 1) := by\n  choose f hf₀ hf₁ using h\n  intro H\n  obtain ⟨ε, -, hε', hε₀⟩ := exists_seq_strictAnti_tendsto_nhdsWithin (0 : ℝ)\n  refine not_eventually.2\n      (Frequently.of_forall fun _ ↦ lt_irrefl <| 1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι))\n      ((Filter.Tendsto.eventually_lt tendsto_const_nhds (H.comp hε₀) <|\n            ENNReal.sub_lt_self ENNReal.one_ne_top one_ne_zero ?_).mono\n        fun n ↦ lt_of_le_of_lt' ?_)\n  on_goal 2 =>\n    calc\n      volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n))\n        ≤ volume (closedBall x (ε n) \\ closedBall (f (ε n) <| hε' n) (ε n / 4)) /\n          volume (closedBall x (ε n)) := by\n        gcongr\n        rw [sdiff_eq_compl_inter]\n        refine inter_subset_inter_left _ ?_\n        rw [subset_compl_comm, ← interior_compl]\n        exact hf₁ _ _\n      _ = 1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι) := ?_\n    have := hε' n\n    rw [measure_sdiff (hf₀ _ _) _ ((Real.volume_pi_closedBall _ _).trans_ne ENNReal.ofReal_ne_top),\n      Real.volume_pi_closedBall, Real.volume_pi_closedBall, ENNReal.sub_div fun _ _ ↦ _,\n      ENNReal.div_self _ ENNReal.ofReal_ne_top, ← ENNReal.ofReal_div_of_pos, ← div_pow,\n      mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]\n  all_goals try positivity\n  · simp_all\n  · exact measurableSet_closedBall.nullMeasurableSet\n\ntheorem IsUpperSet.null_frontier (hs : IsUpperSet s) : volume (frontier s) = 0 := by\n  refine measure_mono_null (fun x hx ↦ ?_)\n    (Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet _\n      (isClosed_closure (s := s)).measurableSet)\n  by_cases h : x ∈ closure s <;>\n    simp only [mem_compl_iff, mem_setOf, h, not_false_eq_true, indicator_of_notMem,\n      indicator_of_mem, Pi.one_apply]\n  · refine aux₁ fun _ ↦ hs.compl.exists_subset_ball <| frontier_subset_closure ?_\n    rwa [frontier_compl]\n  · exact aux₀ fun _ ↦ hs.exists_subset_ball <| frontier_subset_closure hx\n\ntheorem IsLowerSet.null_frontier (hs : IsLowerSet s) : volume (frontier s) = 0 := by\n  refine measure_mono_null (fun x hx ↦ ?_)\n    (Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet _\n      (isClosed_closure (s := s)).measurableSet)\n  by_cases h : x ∈ closure s <;>\n    simp only [mem_compl_iff, mem_setOf, h, not_false_eq_true, indicator_of_notMem,\n      indicator_of_mem, Pi.one_apply]\n  · refine aux₁ fun _ ↦ hs.compl.exists_subset_ball <| frontier_subset_closure ?_\n    rwa [frontier_compl]\n  · exact aux₀ fun _ ↦ hs.exists_subset_ball <| frontier_subset_closure hx\n\ntheorem Set.OrdConnected.null_frontier (hs : s.OrdConnected) : volume (frontier s) = 0 := by\n  rw [← hs.upperClosure_inter_lowerClosure]\n  exact measure_mono_null (frontier_inter_subset _ _) <| measure_union_null\n    (measure_inter_null_of_null_left _ (UpperSet.upper _).null_frontier)\n    (measure_inter_null_of_null_right _ (LowerSet.lower _).null_frontier)\n\nprotected theorem Set.OrdConnected.nullMeasurableSet (hs : s.OrdConnected) : NullMeasurableSet s :=\n  nullMeasurableSet_of_null_frontier hs.null_frontier\n\nTarget:\ntheorem IsAntichain.volume_eq_zero [Nonempty ι] (hs : IsAntichain (· ≤ ·) s) : volume s = 0 :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"6584f9190fc28c49db534da1daea2b8245c9b0b6d90bde339b735a88af7d1b63","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Order","family_id":"isantichain","file_id":"mathlib/Mathlib/MeasureTheory/Order/UpperLower.lean","sample_id":"0a46fac18823464095b42916006166b41806c313503a9332eebd632a234896e3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ea6d6b98ce6bb5df330d0c3825d495329a7a811e1188283f31f2efe51dfb0168","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f06c65161176fa9df19e9c48097892fb382db84876f67c1ee262a2a6d26ba0ee","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4ea2d4d2d7f00624026e8f321443e14dc6b69d2bf90adb74245fe472882d93fb","source_sha256":"b6022f061f06753f0797fa4fa7ab43cc6fb4c7d61abaa7b4bb4707d4dc0299c4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← LinearMap.comp_assoc, ← LinearMap.comp_assoc, TensorProduct.map_map_comp_assoc_eq]","hard_negative":false,"metrics":{"chosen_tokens":17,"rejected_tokens":24,"token_jaccard":0.846154,"token_length_ratio":1.411765},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"9028b0f52f1a3867e2d5f02aa6188dab55c13c079d81060736539aec15aca578","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorProduct.Tower\npublic import Mathlib.RingTheory.Coalgebra.Basic\nimport Mathlib.Tactic.Attr.Register\n\nNamespace:\nCoassocSimps\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang, Yaël Dillies\n-/\n/-!\n# Tactic to reassociate comultiplication in a coalgebra\n\n`coassoc_simps` is a simp set useful to prove tautologies on coalgebras.\n\nThe general algorithm it follows is to push the associators `TensorProduct.assoc` and\ncommutators `TensorProduct.comm` inwards (to the right) until they cancel against\nco-multiplications.\n\nThe simp set makes the following choice of normal form\n* It regards `TensorProduct.map`, `TensorProduct.assoc`, `TensorProduct.comm` as the primitive\n  constructions and rewrites everything else such as `lTensor`, `leftComm` using them.\n* It rewrites both sides into a right associated composition of linear maps.\n  In particular `LinearMap.comp_assoc` and `LinearEquiv.coe_trans` are tagged.\n* It rewrites `(f₂ ⊗ g₂) ∘ (f₁ ⊗ g₁)` into `(f₂ ∘ f₁) ⊗ (g₂ ∘ g₁)`.\n\n## Notes\n\n- It is not confluent with `(ε ⊗ₘ id) ∘ₗ δ = λ⁻¹`.\n  It is often useful to `trans` (or `calc`) with a term containing\n  `(ε ⊗ₘ _) ∘ₗ δ` or `(_ ⊗ₘ ε) ∘ₗ δ`,\n  and use one of `map_counit_comp_comul_left` `map_counit_comp_comul_right`\n  `map_counit_comp_comul_left_assoc` `map_counit_comp_comul_right_assoc` to continue.\n\n- Some lemmas (e.g. `lid_comp_map : λ ∘ₗ (f ⊗ₘ g) = g ∘ₗ λ ∘ₗ (f ⊗ₘ id)`) loops when tagged as simp,\n  so we wrap it inside a rudimentary simproc that only fires when `g ≠ id`.\n-/\n\n@[expose] public section\n\nopen TensorProduct\n\nopen LinearMap (id)\nopen Coalgebra\n\nopen Qq\nnamespace CoassocSimps\n\nvariable {R A M N P M' N' P' Q Q' M₁ M₂ M₃ N₁ N₂ N₃ : Type*}\n    [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A]\n    [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P]\n    [AddCommMonoid M'] [Module R M'] [AddCommMonoid N'] [Module R N']\n    [AddCommMonoid P'] [Module R P'] [AddCommMonoid Q] [Module R Q] [AddCommMonoid Q'] [Module R Q']\n    [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃]\n    [AddCommMonoid N₁] [AddCommMonoid N₂] [AddCommMonoid N₃]\n    [Module R M₁] [Module R M₂] [Module R M₃] [Module R N₁] [Module R N₂] [Module R N₃]\n\nlocal notation3 \"α\" => (TensorProduct.assoc R _ _ _).toLinearMap\nlocal notation3 \"α⁻¹\" => (TensorProduct.assoc R _ _ _).symm.toLinearMap\nlocal notation3 \"λ\" => (TensorProduct.lid R _).toLinearMap\nlocal notation3 \"λ⁻¹\" => (TensorProduct.lid R _).symm.toLinearMap\nlocal notation3 \"ρ\" => (TensorProduct.rid R _).toLinearMap\nlocal notation3 \"ρ⁻¹\" => (TensorProduct.rid R _).symm.toLinearMap\nlocal notation3 \"β\" => (TensorProduct.comm R _ _).toLinearMap\nlocal infix:90 \" ⊗ₘ \" => TensorProduct.map\nlocal notation3 \"δ\" => comul (R := R)\nlocal notation3 \"ε\" => counit (R := R)\n\nattribute [coassoc_simps] LinearMap.comp_id LinearMap.id_comp TensorProduct.map_id\n  LinearMap.lTensor_def LinearMap.rTensor_def LinearMap.comp_assoc\n  LinearEquiv.coe_trans LinearEquiv.trans_symm\n  LinearEquiv.refl_toLinearMap TensorProduct.toLinearMap_congr\n  LinearEquiv.comp_symm LinearEquiv.symm_comp LinearEquiv.symm_symm\n  LinearEquiv.coe_lTensor LinearEquiv.coe_lTensor_symm\n  LinearEquiv.coe_rTensor LinearEquiv.coe_rTensor_symm\n  IsCocomm.comm_comp_comul TensorProduct.AlgebraTensorModule.map_eq\n  TensorProduct.AlgebraTensorModule.assoc_eq TensorProduct.AlgebraTensorModule.rightComm_eq\n  TensorProduct.tensorTensorTensorComm TensorProduct.AlgebraTensorModule.tensorTensorTensorComm\n  TensorProduct.AlgebraTensorModule.congr_eq LinearEquiv.comp_symm_assoc\n  LinearEquiv.symm_comp_assoc TensorProduct.rightComm_def TensorProduct.leftComm_def\n  TensorProduct.comm_symm TensorProduct.comm_comp_comm TensorProduct.comm_comp_comm_assoc\n\nattribute [coassoc_simps← ] TensorProduct.map_comp TensorProduct.map_map_comp_assoc_eq\n  TensorProduct.map_map_comp_assoc_symm_eq\n\n@[coassoc_simps]\nlemma TensorProduct.map_comp_assoc\n    (f : M →ₗ[R] N) (g : N →ₗ[R] P) (f' : M' →ₗ[R] N') (g' : N' →ₗ[R] P') (φ : M₁ →ₗ[R] M ⊗[R] M') :\n    map g g' ∘ₗ map f f' ∘ₗ φ = map (g ∘ₗ f) (g' ∘ₗ f') ∘ₛₗ φ := by\n  rw [← LinearMap.comp_assoc, TensorProduct.map_comp]\n\n@[coassoc_simps← ]\n\nTarget:\nlemma TensorProduct.map_map_comp_assoc_eq_assoc\n    (f₁ : M₁ →ₗ[R] N₁) (f₂ : M₂ →ₗ[R] N₂) (f₃ : M₃ →ₗ[R] N₃) (f : M →ₗ[R] M₁ ⊗[R] M₂ ⊗[R] M₃) :\n    f₁ ⊗ₘ (f₂ ⊗ₘ f₃) ∘ₗ α ∘ₗ f = α ∘ₗ ((f₁ ⊗ₘ f₂) ⊗ₘ f₃) ∘ₗ f :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [← LinearMap.comp_assoc, ← LinearMap.comp_assoc, TensorProduct.map_map_comp_assoc_eq]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Coalgebra","family_id":"tensorproduct","file_id":"mathlib/Mathlib/RingTheory/Coalgebra/CoassocSimps.lean","sample_id":"4ea2d4d2d7f00624026e8f321443e14dc6b69d2bf90adb74245fe472882d93fb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2f2eac7accbcc17b05097bd1c9316c1f6c3d52ca67232fbfd3c0c187a0e1b260","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"01224b14a9dc704c2fd71fd678b713d03a65ff9d6ee15bc743185909d5552045","source_sha256":"cee08e35f7a82853091be31ffd641ba79c118d64e6ff9753c07e342b11ecbe8e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← le_div_iff₀ pi_div_two_pos, div_eq_inv_mul (W k) _]\n  rw [W_eq_integral_sin_pow_div_integral_sin_pow, le_div_iff₀ (integral_sin_pow_pos _)]\n  convert! integral_sin_pow_succ_le (2 * k + 1)\n  rw [integral_sin_pow (2 * k)]\n  simp","hard_negative":false,"metrics":{"chosen_tokens":46,"rejected_tokens":2,"token_jaccard":0.037037,"token_length_ratio":0.043478},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"90cadcbf3987c56f87e9ee007189a10be8229725aad39bf7b70bf8bdb495d4d6","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Integrals.Basic\n\nNamespace:\nReal.Wallis\n\nLocal context:\n/-\nCopyright (c) 2021 Hanting Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Hanting Zhang\n-/\n/-! # The Wallis formula for Pi\n\nThis file establishes the Wallis product for `π` (`Real.tendsto_prod_pi_div_two`). Our proof is\nlargely about analyzing the behaviour of the sequence `∫ x in 0..π, sin x ^ n` as `n → ∞`.\nSee: https://en.wikipedia.org/wiki/Wallis_product\n\nThe proof can be broken down into two pieces. The first step (carried out in\n`Mathlib/Analysis/SpecialFunctions/Integrals/Basic.lean`) is to use repeated integration by parts to\nobtain an explicit formula for this integral, which is rational if `n` is odd and a rational\nmultiple of `π` if `n` is even.\n\nThe second step, carried out here, is to estimate the ratio\n`∫ (x : ℝ) in 0..π, sin x ^ (2 * k + 1) / ∫ (x : ℝ) in 0..π, sin x ^ (2 * k)` and prove that\nit converges to one using the squeeze theorem. The final product for `π` is obtained after some\nalgebraic manipulation.\n\n## Main statements\n\n* `Real.Wallis.W`: the product of the first `k` terms in Wallis' formula for `π`.\n* `Real.Wallis.W_eq_integral_sin_pow_div_integral_sin_pow`: express `W n` as a ratio of integrals.\n* `Real.Wallis.W_le` and `Real.Wallis.le_W`: upper and lower bounds for `W n`.\n* `Real.tendsto_prod_pi_div_two`: the Wallis product formula.\n-/\n\n@[expose] public section\n\n\nopen scoped Real Topology Nat\n\nopen Filter Finset intervalIntegral\n\nnamespace Real\n\nnamespace Wallis\n\n\n/-- The product of the first `k` terms in Wallis' formula for `π`. -/\nnoncomputable def W (k : ℕ) : ℝ :=\n  ∏ i ∈ range k, (2 * i + 2) / (2 * i + 1) * ((2 * i + 2) / (2 * i + 3))\n\ntheorem W_succ (k : ℕ) :\n    W (k + 1) = W k * ((2 * k + 2) / (2 * k + 1) * ((2 * k + 2) / (2 * k + 3))) :=\n  prod_range_succ _ _\n\ntheorem W_pos (k : ℕ) : 0 < W k := by\n  induction k with\n  | zero => unfold W; simp\n  | succ k hk =>\n    rw [W_succ]\n    refine mul_pos hk (mul_pos (div_pos ?_ ?_) (div_pos ?_ ?_)) <;> positivity\n\ntheorem W_eq_factorial_ratio (n : ℕ) :\n    W n = 2 ^ (4 * n) * n ! ^ 4 / ((2 * n)! ^ 2 * (2 * n + 1)) := by\n  induction n with\n  | zero =>\n    simp only [W, prod_range_zero, Nat.factorial_zero, mul_zero, pow_zero]\n    norm_num\n  | succ n IH =>\n    unfold W at IH ⊢\n    rw [prod_range_succ, IH, _root_.div_mul_div_comm, _root_.div_mul_div_comm]\n    refine (div_eq_div_iff ?_ ?_).mpr ?_\n    any_goals exact ne_of_gt (by positivity)\n    simp_rw [Nat.mul_succ, Nat.factorial_succ, pow_succ]\n    push_cast\n    ring_nf\n\ntheorem W_eq_integral_sin_pow_div_integral_sin_pow (k : ℕ) : (π / 2)⁻¹ * W k =\n    (∫ x : ℝ in 0..π, sin x ^ (2 * k + 1)) / ∫ x : ℝ in 0..π, sin x ^ (2 * k) := by\n  rw [integral_sin_pow_even, integral_sin_pow_odd, mul_div_mul_comm, ← prod_div_distrib, inv_div]\n  simp_rw [div_div_div_comm, div_div_eq_mul_div, mul_div_assoc]\n  rfl\n\ntheorem W_le (k : ℕ) : W k ≤ π / 2 := by\n  rw [← div_le_one pi_div_two_pos, div_eq_inv_mul]\n  rw [W_eq_integral_sin_pow_div_integral_sin_pow, div_le_one (integral_sin_pow_pos _)]\n  apply integral_sin_pow_succ_le\n\nTarget:\ntheorem le_W (k : ℕ) : ((2 : ℝ) * k + 1) / (2 * k + 2) * (π / 2) ≤ W k :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Real","family_id":"le_w","file_id":"mathlib/Mathlib/Analysis/Real/Pi/Wallis.lean","sample_id":"01224b14a9dc704c2fd71fd678b713d03a65ff9d6ee15bc743185909d5552045"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bb5b4eccf3c9035f221ecef92ea8d7966625741540da04e726a173d5b6907beb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3033f816b6132c5bc0a106b5e710753aa1e911f087b5ce36db9d94f7c0352b85","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f1e96e086d107cbb628f9333e3a0d90e5f01c368a40a55b24a768acd38a2bd9b","source_sha256":"69a9767d2da3b234c9bb0ca0a2aec7e4918b4afa1e5dd9c10eea2ad03e813fc9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply Functor.ext; rotate_left\n  · intro X\n    ext i\n    specialize h i\n    have := congr_obj h X\n    simpa\n  · intro X Y g\n    funext i\n    specialize h i\n    have := congr_hom h g\n    simpa","hard_negative":true,"metrics":{"chosen_tokens":37,"rejected_tokens":3,"token_jaccard":0.043478,"token_length_ratio":0.081081},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"90ddf0958de3b822d1ab6500bc962212a8dde106118f9d93fbe41afe145f71e9","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.EqToHom\npublic import Mathlib.CategoryTheory.NatIso\npublic import Mathlib.CategoryTheory.Products.Basic\n\nNamespace:\nFunctor\n\nLocal context:\n/-\nCopyright (c) 2020 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Kim Morrison\n-/\n/-!\n# Categories of indexed families of objects.\n\nWe define the pointwise category structure on indexed families of objects in a category\n(and also the dependent generalization).\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Functor\n\nuniverse w₀ w₁ w₂ v₁ v₂ v₃ u₁ u₂ u₃\n\nvariable {I : Type w₀} {J : Type w₁} (C : I → Type u₁) [∀ i, Category.{v₁} (C i)]\n\n\n/-- `pi C` gives the Cartesian product of an indexed family of categories.\n-/\ninstance pi : Category.{max w₀ v₁} (∀ i, C i) where\n  Hom X Y := ∀ i, X i ⟶ Y i\n  id X i := 𝟙 (X i)\n  comp f g i := f i ≫ g i\n\nnamespace Pi\n\n@[simp]\ntheorem id_apply (X : ∀ i, C i) (i) : (𝟙 X : ∀ i, X i ⟶ X i) i = 𝟙 (X i) :=\n  rfl\n\n@[simp]\ntheorem comp_apply {X Y Z : ∀ i, C i} (f : X ⟶ Y) (g : Y ⟶ Z) (i) :\n    (f ≫ g : ∀ i, X i ⟶ Z i) i = f i ≫ g i :=\n  rfl\n\n@[ext]\nlemma ext {X Y : ∀ i, C i} {f g : X ⟶ Y} (w : ∀ i, f i = g i) : f = g :=\n  funext (w ·)\n\n/--\nThe evaluation functor at `i : I`, sending an `I`-indexed family of objects to the object over `i`.\n-/\n@[simps]\ndef eval (i : I) : (∀ i, C i) ⥤ C i where\n  obj f := f i\n  map α := α i\n\nsection\n\nvariable {J : Type w₁}\n\ninstance (f : J → I) : (j : J) → Category ((C ∘ f) j) :=\n  inferInstanceAs <| (j : J) → Category (C (f j))\n\n/-- Pull back an `I`-indexed family of objects to a `J`-indexed family, along a function `J → I`.\n-/\n@[simps]\ndef comap (h : J → I) : (∀ i, C i) ⥤ (∀ j, C (h j)) where\n  obj f i := f (h i)\n  map α i := α (h i)\n\nvariable (I)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The natural isomorphism between\npulling back a grading along the identity function,\nand the identity functor. -/\n@[simps]\ndef comapId : comap C (id : I → I) ≅ 𝟭 (∀ i, C i) where\n  hom := { app := fun X => 𝟙 X }\n  inv := { app := fun X => 𝟙 X }\n\nexample (g : J → I) : (j : J) → Category (C (g j)) := by infer_instance\n\nvariable {I}\nvariable {K : Type w₂}\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The natural isomorphism comparing between\npulling back along two successive functions, and\npulling back along their composition\n-/\n@[simps!]\ndef comapComp (f : K → J) (g : J → I) : comap C g ⋙ comap (C ∘ g) f ≅ comap C (g ∘ f) where\n  hom :=\n  { app := fun X b => 𝟙 (X (g (f b)))\n    naturality := fun X Y f' => by simp only [comap, Function.comp]; funext; simp }\n  inv :=\n  { app := fun X b => 𝟙 (X (g (f b)))\n    naturality := fun X Y f' => by simp only [comap, Function.comp]; funext; simp }\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The natural isomorphism between pulling back then evaluating, and just evaluating. -/\n@[simps!]\ndef comapEvalIsoEval (h : J → I) (j : J) : comap C h ⋙ eval (C ∘ h) j ≅ eval C (h j) :=\n  NatIso.ofComponents (fun _ => Iso.refl _) (by simp)\n\nend\n\nsection\n\nvariable {J : Type w₀} {D : J → Type u₁} [∀ j, Category.{v₁} (D j)]\n\ninstance sumElimCategory : ∀ s : I ⊕ J, Category.{v₁} (Sum.elim C D s)\n  | Sum.inl i => inferInstanceAs <| Category (C i)\n  | Sum.inr j => inferInstanceAs <| Category (D j)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The bifunctor combining an `I`-indexed family of objects with a `J`-indexed family of objects\nto obtain an `I ⊕ J`-indexed family of objects.\n-/\n@[simps]\ndef sum : (∀ i, C i) ⥤ (∀ j, D j) ⥤ ∀ s : I ⊕ J, Sum.elim C D s where\n  obj X :=\n    { obj := fun Y s =>\n        match s with\n        | .inl i => X i\n        | .inr j => Y j\n      map := fun {_} {_} f s =>\n        match s with\n        | .inl i => 𝟙 (X i)\n        | .inr j => f j }\n  map {X} {X'} f :=\n    { app := fun Y s =>\n        match s with\n        | .inl i => f i\n        | .inr j => 𝟙 (Y j) }\n\nend\n\nvariable {C}\n\n/-- A family of isomorphisms gives rise to an isomorphism of families. -/\n@[simps]\ndef isoMk {X Y : ∀ i, C i} (iso : ∀ i, X i ≅ Y i) :\n    X ≅ Y where\n  hom := fun i => (iso i).hom\n  inv := fun i => (iso i).inv\n\n/-- An isomorphism between `I`-indexed objects gives an isomorphism between each\npair of corresponding components. -/\n@[simps]\ndef isoApp {X Y : ∀ i, C i} (f : X ≅ Y) (i : I) : X i ≅ Y i :=\n  ⟨f.hom i, f.inv i,\n    by rw [← comp_apply, Iso.hom_inv_id, id_apply], by rw [← comp_apply, Iso.inv_hom_id, id_apply]⟩\n\n@[simp]\ntheorem isoApp_refl (X : ∀ i, C i) (i : I) : isoApp (Iso.refl X) i = Iso.refl (X i) :=\n  rfl\n\n@[simp]\ntheorem isoApp_symm {X Y : ∀ i, C i} (f : X ≅ Y) (i : I) : isoApp f.symm i = (isoApp f i).symm :=\n  rfl\n\n@[simp]\ntheorem isoApp_trans {X Y Z : ∀ i, C i} (f : X ≅ Y) (g : Y ≅ Z) (i : I) :\n    isoApp (f ≪≫ g) i = isoApp f i ≪≫ isoApp g i :=\n  rfl\n\nend Pi\n\nnamespace Functor\n\nvariable {C}\nvariable {D : I → Type u₂} [∀ i, Category.{v₂} (D i)] {A : Type u₃} [Category.{v₃} A]\n\n/-- Assemble an `I`-indexed family of functors into a functor between the pi types.\n-/\n@[simps]\ndef pi (F : ∀ i, C i ⥤ D i) : (∀ i, C i) ⥤ ∀ i, D i where\n  obj f i := (F i).obj (f i)\n  map α i := (F i).map (α i)\n\n/-- Similar to `pi`, but all functors come from the same category `A`\n-/\n@[simps]\ndef pi' (f : ∀ i, A ⥤ C i) : A ⥤ ∀ i, C i where\n  obj a i := (f i).obj a\n  map h i := (f i).map h\n\n/-- The projections of `Functor.pi' F` are isomorphic to the functors of the family `F` -/\n@[simps!]\ndef pi'CompEval {A : Type*} [Category* A] (F : ∀ i, A ⥤ C i) (i : I) :\n    pi' F ⋙ Pi.eval C i ≅ F i :=\n  Iso.refl _\n\nsection EqToHom\n\n@[simp]\ntheorem eqToHom_proj {x x' : ∀ i, C i} (h : x = x') (i : I) :\n    (eqToHom h : x ⟶ x') i = eqToHom (funext_iff.mp h i) := by\n  subst h\n  rfl\n\nend EqToHom\n\n-- One could add some natural isomorphisms showing\n-- how `Functor.pi` commutes with `Pi.eval` and `Pi.comap`.\n@[simp]\ntheorem pi'_eval (f : ∀ i, A ⥤ C i) (i : I) : pi' f ⋙ Pi.eval C i = f i :=\n  rfl\n\n/-- Two functors to a product category are equal iff they agree on every coordinate. -/\n\nTarget:\ntheorem pi_ext (f f' : A ⥤ ∀ i, C i) (h : ∀ i, f ⋙ (Pi.eval C i) = f' ⋙ (Pi.eval C i)) :\n    f = f' :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_f1e96e086d10","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"fc9d3852ad7e151c810a9e75948272c407eba1e7140266dd2fa683583bdb5011","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Pi","family_id":"pi_ext","file_id":"mathlib/Mathlib/CategoryTheory/Pi/Basic.lean","sample_id":"f1e96e086d107cbb628f9333e3a0d90e5f01c368a40a55b24a768acd38a2bd9b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"343a166fea2e90fed4c3043781c3d8f443aa011a6cdb85e131c437406e24ed78","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6e4084086462a436ac3eef56faeb185418f62005673e33be1a9b1b6af321fbca","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f98433b4daad566e797079ce130b71338bc97d14ae011a049aadb8ae222adbed","source_sha256":"92db59c0cca19f9c28529fd17474c1f344e67bd83f08b4344e7abe51502b9b08","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : C a • D f = D (C a * f) := by simp\n  rw [this, C_mul', D.map_smul]","hard_negative":true,"metrics":{"chosen_tokens":29,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.068966},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"919c954efc4d20c6c697ccfb4d207fe70728f71ae645df32f1ab67982423d4cd","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.AlgebraMap\npublic import Mathlib.Algebra.Polynomial.Derivative\npublic import Mathlib.Algebra.Polynomial.Module.AEval\npublic import Mathlib.RingTheory.Adjoin.Polynomial.Basic\npublic import Mathlib.RingTheory.Derivation.Basic\n\nNamespace:\nPolynomial\n\nLocal context:\n/-\nCopyright (c) 2023 Kevin Buzzard. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Richard M. Hill\n-/\n/-!\n# Derivations of univariate polynomials\n\nIn this file we prove that an `R`-derivation of `Polynomial R` is determined by its value on `X`.\nWe also provide a constructor `Polynomial.mkDerivation` that\nbuilds a derivation from its value on `X`, and a linear equivalence\n`Polynomial.mkDerivationEquiv` between `A` and `Derivation (Polynomial R) A`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nnamespace Polynomial\n\nsection CommSemiring\n\nvariable {R A : Type*} [CommSemiring R]\n\n/-- `Polynomial.derivative` as a derivation. -/\n@[simps]\ndef derivative' : Derivation R R[X] R[X] where\n  toFun := derivative\n  map_add' _ _ := derivative_add\n  map_smul' := derivative_smul\n  map_one_eq_zero' := derivative_one\n  leibniz' f g := by simp [mul_comm, add_comm, derivative_mul]\n\nvariable [AddCommMonoid A] [Module R A] [Module (Polynomial R) A]\n\n@[simp]\ntheorem derivation_C (D : Derivation R R[X] A) (a : R) : D (C a) = 0 :=\n  D.map_algebraMap a\n\n@[simp]\n\nTarget:\ntheorem C_smul_derivation_apply (D : Derivation R R[X] A) (a : R) (f : R[X]) :\n    C a • D f = a • D f :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_f98433b4daad","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"c8ef849e945657333aeef2466182d2d216545329efc3a0a42a5425090fbe6a45","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Polynomial","family_id":"c_smul_derivation_apply","file_id":"mathlib/Mathlib/Algebra/Polynomial/Derivation.lean","sample_id":"f98433b4daad566e797079ce130b71338bc97d14ae011a049aadb8ae222adbed"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9ed2efcc5bc8272d2a70d89cfb0ba3dc939ae6d8c6a39b0504302e5c2afc9b8d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b765d1892775d5e6e6a62ecd85051d3ac26c69fedc8cb19fe0955539acfca9f3","source_sha256":"d1ba22e22a26245161ec2703bf29aa4e6e9c0ade6d5458aee3e70435922bd387","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨x, v₁, v₂⟩ := x\n  obtain ⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩ := h\n  simp [Prod.toFun', Prod.invFun', h₁, h₂]","hard_negative":true,"metrics":{"chosen_tokens":53,"rejected_tokens":8,"token_jaccard":0.115385,"token_length_ratio":0.150943},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"91d3837e7475eb285102d03d528dcfbdd9784e43a76e6ccb4f83c38f698f1659","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.FiberBundle.Basic\n\nNamespace:\nBundle.Trivialization\n\nLocal context:\n/-\nCopyright (c) 2022 Nicolò Cavalleri. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn\n-/\n/-!\n# Standard constructions on fiber bundles\n\nThis file contains several standard constructions on fiber bundles:\n\n* `Bundle.Trivial.fiberBundle 𝕜 B F`: the trivial fiber bundle with model fiber `F` over the base\n  `B`\n\n* `FiberBundle.prod`: for fiber bundles `E₁` and `E₂` over a common base, a fiber bundle structure\n  on their fiberwise product `E₁ ×ᵇ E₂` (the notation stands for `fun x ↦ E₁ x × E₂ x`).\n\n* `FiberBundle.pullback`: for a fiber bundle `E` over `B`, a fiber bundle structure on its\n  pullback `f *ᵖ E` by a map `f : B' → B` (the notation is a type synonym for `E ∘ f`).\n\n## Tags\n\nfiber bundle, fibre bundle, fiberwise product, pullback\n\n-/\n\n@[expose] public section\n\nopen Bundle Filter Set TopologicalSpace Topology\n\n/-! ### The trivial bundle -/\n\nnamespace Bundle\n\nnamespace Trivial\n\nvariable (B : Type*) (F : Type*)\n\n-- TODO: use `TotalSpace.toProd`\ninstance topologicalSpace [t₁ : TopologicalSpace B]\n    [t₂ : TopologicalSpace F] : TopologicalSpace (TotalSpace F (Trivial B F)) :=\n  induced TotalSpace.proj t₁ ⊓ induced (TotalSpace.trivialSnd B F) t₂\n\nvariable [TopologicalSpace B] [TopologicalSpace F]\n\ntheorem isInducing_toProd : IsInducing (TotalSpace.toProd B F) :=\n  ⟨by simp only [instTopologicalSpaceProd, induced_inf, induced_compose]; rfl⟩\n\n/-- Homeomorphism between the total space of the trivial bundle and the Cartesian product. -/\n@[simps!]\ndef homeomorphProd : TotalSpace F (Trivial B F) ≃ₜ B × F :=\n  (TotalSpace.toProd _ _).toHomeomorphOfIsInducing (isInducing_toProd B F)\n\n/-- Local trivialization for trivial bundle. -/\n@[simps!]\ndef trivialization : Trivialization F (π F (Bundle.Trivial B F)) where\n  toOpenPartialHomeomorph := (homeomorphProd B F).toOpenPartialHomeomorph\n  baseSet := univ\n  open_baseSet := isOpen_univ\n  source_eq := rfl\n  target_eq := univ_prod_univ.symm\n  proj_toFun _ _ := rfl\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp] lemma trivialization_symm_apply [Zero F] (b : B) (f : F) :\n    (trivialization B F).symm b f = f := by\n  simp [trivialization, homeomorphProd, TotalSpace.toProd, Trivialization.symm,\n    Pretrivialization.symm, Trivialization.toPretrivialization]\n\n@[simp] lemma toOpenPartialHomeomorph_trivialization_symm_apply (v : B × F) :\n    (trivialization B F).toOpenPartialHomeomorph.symm v = ⟨v.1, v.2⟩ := rfl\n\n/-- Fiber bundle instance on the trivial bundle. -/\n@[simps] instance fiberBundle : FiberBundle F (Bundle.Trivial B F) where\n  trivializationAtlas' := {trivialization B F}\n  trivializationAt' _ := trivialization B F\n  mem_baseSet_trivializationAt' := mem_univ\n  trivialization_mem_atlas' _ := mem_singleton _\n  totalSpaceMk_isInducing' _ := (homeomorphProd B F).symm.isInducing.comp\n    (isInducing_const_prod.2 .id)\n\ntheorem eq_trivialization (e : Trivialization F (π F (Bundle.Trivial B F)))\n    [i : MemTrivializationAtlas e] : e = trivialization B F := i.out\n\nend Trivial\n\nend Bundle\n\n/-! ### Fibrewise product of two bundles -/\n\n\nsection Prod\n\nvariable {B : Type*}\n\nsection Defs\n\nvariable (F₁ : Type*) (E₁ : B → Type*) (F₂ : Type*) (E₂ : B → Type*)\nvariable [TopologicalSpace (TotalSpace F₁ E₁)] [TopologicalSpace (TotalSpace F₂ E₂)]\n\n/-- Equip the total space of the fiberwise product of two fiber bundles `E₁`, `E₂` with\nthe induced topology from the diagonal embedding into `TotalSpace F₁ E₁ × TotalSpace F₂ E₂`. -/\ninstance FiberBundle.Prod.topologicalSpace : TopologicalSpace (TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂)) :=\n  TopologicalSpace.induced\n    (fun p ↦ ((⟨p.1, p.2.1⟩ : TotalSpace F₁ E₁), (⟨p.1, p.2.2⟩ : TotalSpace F₂ E₂)))\n    inferInstance\n\n/-- The diagonal map from the total space of the fiberwise product of two fiber bundles\n`E₁`, `E₂` into `TotalSpace F₁ E₁ × TotalSpace F₂ E₂` is an inducing map. -/\ntheorem FiberBundle.Prod.isInducing_diag :\n    IsInducing (fun p ↦ (⟨p.1, p.2.1⟩, ⟨p.1, p.2.2⟩) :\n      TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂) :=\n  ⟨rfl⟩\n\nend Defs\n\nopen FiberBundle\n\nvariable [TopologicalSpace B] (F₁ : Type*) [TopologicalSpace F₁] (E₁ : B → Type*)\n  [TopologicalSpace (TotalSpace F₁ E₁)] (F₂ : Type*) [TopologicalSpace F₂] (E₂ : B → Type*)\n  [TopologicalSpace (TotalSpace F₂ E₂)]\n\nnamespace Bundle.Trivialization\n\nvariable {F₁ E₁ F₂ E₂}\nvariable (e₁ : Trivialization F₁ (π F₁ E₁)) (e₂ : Trivialization F₂ (π F₂ E₂))\n\n/-- Given trivializations `e₁`, `e₂` for fiber bundles `E₁`, `E₂` over a base `B`, the forward\nfunction for the construction `Trivialization.prod`, the induced\ntrivialization for the fiberwise product of `E₁` and `E₂`. -/\ndef Prod.toFun' : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) → B × F₁ × F₂ :=\n  fun p ↦ ⟨p.1, (e₁ ⟨p.1, p.2.1⟩).2, (e₂ ⟨p.1, p.2.2⟩).2⟩\n\nvariable {e₁ e₂}\n\ntheorem Prod.continuous_to_fun : ContinuousOn (Prod.toFun' e₁ e₂)\n    (π (F₁ × F₂) (E₁ ×ᵇ E₂) ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) := by\n  let f₁ : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ :=\n    fun p ↦ ((⟨p.1, p.2.1⟩ : TotalSpace F₁ E₁), (⟨p.1, p.2.2⟩ : TotalSpace F₂ E₂))\n  let f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p ↦ ⟨e₁ p.1, e₂ p.2⟩\n  let f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p ↦ ⟨p.1.1, p.1.2, p.2.2⟩\n  have hf₁ : Continuous f₁ := (Prod.isInducing_diag F₁ E₁ F₂ E₂).continuous\n  have hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) :=\n    e₁.toOpenPartialHomeomorph.continuousOn.prodMap e₂.toOpenPartialHomeomorph.continuousOn\n  have hf₃ : Continuous f₃ := by fun_prop\n  refine ((hf₃.comp_continuousOn hf₂).comp hf₁.continuousOn ?_).congr ?_\n  · rw [e₁.source_eq, e₂.source_eq]\n    exact mapsTo_preimage _ _\n  rintro ⟨b, v₁, v₂⟩ ⟨hb₁, _⟩\n  simp only [f₁, f₂, f₃, Prod.toFun', Prod.mk_inj, Function.comp_apply, and_true]\n  rw [e₁.coe_fst]\n  rw [e₁.source_eq, mem_preimage]\n  exact hb₁\n\nvariable (e₁ e₂) [∀ x, Zero (E₁ x)] [∀ x, Zero (E₂ x)]\n\n/-- Given trivializations `e₁`, `e₂` for fiber bundles `E₁`, `E₂` over a base `B`, the inverse\nfunction for the construction `Trivialization.prod`, the induced\ntrivialization for the fiberwise product of `E₁` and `E₂`. -/\nnoncomputable def Prod.invFun' (p : B × F₁ × F₂) : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) :=\n  ⟨p.1, e₁.symm p.1 p.2.1, e₂.symm p.1 p.2.2⟩\n\nvariable {e₁ e₂}\n\nTarget:\ntheorem Prod.left_inv {x : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂)}\n    (h : x ∈ π (F₁ × F₂) (E₁ ×ᵇ E₂) ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) :\n    Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b337a49e0b685a15b078a4f54b9e32486cb59f04899bfda1ec89e61d8da7837d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/FiberBundle","family_id":"prod","file_id":"mathlib/Mathlib/Topology/FiberBundle/Constructions.lean","sample_id":"b765d1892775d5e6e6a62ecd85051d3ac26c69fedc8cb19fe0955539acfca9f3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cc860a95c409479a8d192fe8d26a650d014f625436d7b17247ecda1e72c0ab68","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c6688be02d1db8888f8f73bfb3dc49d8f7954b075b4191fe5618031c2b9977e0","source_sha256":"f5d92c67f281e03d72da06b390e360bd28b86e3bb7c33267f68c3c873cd9a4b6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction ψ using Quotient.inductionOn\n  induction θ using Quotient.inductionOn\n  simp_rw [← QuotientAddGroup.mk_zsmul, ← QuotientAddGroup.mk_add,\n    QuotientAddGroup.eq_iff_sub_mem, ← smul_sub, ← sub_sub]\n  exact AddSubgroup.zsmul_mem_zmultiples_iff_exists_sub_div hz","hard_negative":false,"metrics":{"chosen_tokens":40,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.075},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"91e894c7c61d4403480af9a1cbbb184b6badf43ec7a37c996284e51aa0e6924a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Basic\npublic import Mathlib.Algebra.Order.Group.Unbundled.Int\npublic import Mathlib.Algebra.Module.NatInt\npublic import Mathlib.GroupTheory.QuotientGroup.Defs\npublic import Mathlib.Algebra.Group.Subgroup.ZPowers.Basic\n\nNamespace:\nQuotientAddGroup\n\nLocal context:\n/-\nCopyright (c) 2022 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Lemmas about quotients in characteristic zero\n-/\n\npublic section\n\n\nvariable {R : Type*} [DivisionRing R] [CharZero R] {p : R}\n\nnamespace AddSubgroup\n\n/-- `z • r` is a multiple of `p` iff `r` is `k * (p / z)` above a multiple of `p`, where\n`0 ≤ k < |z|`. -/\ntheorem zsmul_mem_zmultiples_iff_exists_sub_div {r : R} {z : ℤ} (hz : z ≠ 0) :\n    z • r ∈ AddSubgroup.zmultiples p ↔\n      ∃ k : Fin z.natAbs, r - (k : ℕ) • (p / z : R) ∈ AddSubgroup.zmultiples p := by\n  rw [AddSubgroup.mem_zmultiples_iff]\n  simp_rw [AddSubgroup.mem_zmultiples_iff, div_eq_mul_inv, ← smul_mul_assoc, eq_sub_iff_add_eq]\n  have hz' : (z : R) ≠ 0 := Int.cast_ne_zero.mpr hz\n  conv_rhs => simp +singlePass only [← (mul_right_injective₀ hz').eq_iff]\n  simp_rw [← zsmul_eq_mul, smul_add, ← mul_smul_comm, zsmul_eq_mul (z : R)⁻¹, mul_inv_cancel₀ hz',\n    mul_one, ← natCast_zsmul, smul_smul, ← add_smul]\n  constructor\n  · rintro ⟨k, h⟩\n    simp_rw [← h]\n    refine ⟨⟨(k % z).toNat, ?_⟩, k / z, ?_⟩\n    · rw [← Int.ofNat_lt, Int.toNat_of_nonneg (Int.emod_nonneg _ hz)]\n      exact (Int.emod_lt_abs _ hz).trans_eq (Int.abs_eq_natAbs _)\n    rw [Fin.val_mk, Int.toNat_of_nonneg (Int.emod_nonneg _ hz)]\n    nth_rewrite 3 [← Int.mul_ediv_add_emod k z]\n    rfl\n  · rintro ⟨k, n, h⟩\n    exact ⟨_, h⟩\n\ntheorem nsmul_mem_zmultiples_iff_exists_sub_div {r : R} {n : ℕ} (hn : n ≠ 0) :\n    n • r ∈ AddSubgroup.zmultiples p ↔\n      ∃ k : Fin n, r - (k : ℕ) • (p / n : R) ∈ AddSubgroup.zmultiples p := by\n  rw [← natCast_zsmul r, zsmul_mem_zmultiples_iff_exists_sub_div (Int.natCast_ne_zero.mpr hn),\n    Int.cast_natCast]\n  rfl\n\nend AddSubgroup\n\nnamespace QuotientAddGroup\n\nTarget:\ntheorem zmultiples_zsmul_eq_zsmul_iff {ψ θ : R ⧸ AddSubgroup.zmultiples p} {z : ℤ} (hz : z ≠ 0) :\n    z • ψ = z • θ ↔ ∃ k : Fin z.natAbs, ψ = θ + ((k : ℕ) • (p / z) : R) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/CharZero","family_id":"zmultiples_zsmul_eq_zsmul_iff","file_id":"mathlib/Mathlib/Algebra/CharZero/Quotient.lean","sample_id":"c6688be02d1db8888f8f73bfb3dc49d8f7954b075b4191fe5618031c2b9977e0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a76aedca09d06687f23aabb71a976948a3f5261d4be2092b4354b20a250ef4d4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d52a3705d55c120cb4dee3283877b5e3a6ed1469bc9f96ae5e2c891149346d8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c124e4499748820a17df65d5a170a2889e6ded7f6f676ef140d015398e461d38","source_sha256":"38b81cd46858f243db931b63ae095bb873edc08d47b08e77c1c254a8ae6a5653","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [h]\n\n/- ExProd.equateZero -/","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.25},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"92015ca04c744068c1ebbc8e79cb35a3ef189d8c38fedfefedbfc0c6032ab1e2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Defs\npublic import Mathlib.Algebra.Algebra.Basic\npublic import Mathlib.Tactic.Ring.RingNF\n\nNamespace:\nMathlib.Tactic.Algebra\n\nLocal context:\n/-\nCopyright (c) 2025 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Arend Mellendijk\n-/\n/-! # Lemmas for the `algebra` tactic.\n-/\n\n@[expose] public section\n\nopen Mathlib.Meta.NormNum\n\nnamespace Mathlib.Tactic.Algebra\n\nsection ring\n\nvariable {R A : Type*} [CommRing R] [CommRing A] [Algebra R A]\n\n/- evalCast -/\ntheorem isInt_negOfNat_eq {a : A} {lit : ℕ} (h : IsInt a (Int.negOfNat lit)) :\n    a = algebraMap R A (Int.rawCast (Int.negOfNat lit) + 0 : R) + 0 := by\n  simp [h.out]\n\nend ring\n\nsection semifield\n\nvariable {R A : Type*} [Semifield R] [Semifield A] [Algebra R A]\n\n/- evalCast -/\ntheorem isNNRat_eq_rawCast {a : A} {n d : ℕ} (h : IsNNRat a n d) :\n    a = algebraMap R A (NNRat.rawCast n d + 0 : R) + 0 := by\n  simp [Mathlib.Tactic.Ring.cast_nnrat h]\n\nend semifield\n\nsection field\n\nvariable {R A : Type*} [Field R] [Field A] [Algebra R A]\n\n/- evalCast -/\ntheorem isRat_eq_rawCast {a : A} {n d : ℕ} (h : IsRat a (.negOfNat n) d) :\n    a = algebraMap R A (Rat.rawCast (.negOfNat n) d + 0 : R) + 0 := by\n  simp [Mathlib.Tactic.Ring.cast_rat h]\n\nend field\n\nvariable {R A : Type*} [sR : CommSemiring R] [sA : CommSemiring A] [sAlg : Algebra R A]\n\n/- evalCast -/\ntheorem isNat_zero_eq {a : A} (h : IsNat a 0) : a = 0 := by\n  have := h.out\n  simp [this]\n\n/- evalCast -/\ntheorem isNat_eq_rawCast {a : A} {lit : ℕ} (h : IsNat a lit) :\n    a = algebraMap R A (lit + 0 : R) + 0 := by\n  simp [h.out]\n\nsection cleanup\n\nvariable {n d : ℕ}\n\nsection cleanupSMul\n\ntheorem add_assoc_rev (a b c : R) : a + (b + c) = a + b + c := (add_assoc ..).symm\ntheorem mul_assoc_rev (a b c : R) : a * (b * c) = a * b * c := (mul_assoc ..).symm\ntheorem mul_neg {R} [Ring R] (a b : R) : a * -b = -(a * b) := by simp\ntheorem add_neg {R} [Ring R] (a b : R) : a + -b = a - b := (sub_eq_add_neg ..).symm\ntheorem nat_rawCast_0 : (Nat.rawCast 0 : R) = 0 := by simp\ntheorem nat_rawCast_1 : (Nat.rawCast 1 : R) = 1 := by simp\ntheorem nat_rawCast_2 [Nat.AtLeastTwo n] : (Nat.rawCast n : R) = OfNat.ofNat n := rfl\ntheorem int_rawCast_neg {R} [Ring R] : (Int.rawCast (.negOfNat n) : R) = -Nat.rawCast n := by simp\ntheorem nnrat_rawCast {R} [DivisionSemiring R] :\n    (NNRat.rawCast n d : R) = Nat.rawCast n / Nat.rawCast d := by simp\ntheorem rat_rawCast_neg {R} [DivisionRing R] :\n    (Rat.rawCast (.negOfNat n) d : R) = Int.rawCast (.negOfNat n) / Nat.rawCast d := by simp\n\nend cleanupSMul\nsection cleanupConsts\n\ntheorem ofNat_smul {R A} [CommSemiring R] [CommSemiring A] [Algebra R A]\n    [n.AtLeastTwo] {a : A} :\n    (ofNat(n) : R) • a = ofNat(n) * a := by\n  simp_rw [← nat_rawCast_2]\n  simp [Nat.cast_smul_eq_nsmul]\n\ntheorem neg_ofNat_smul {R A} [CommRing R] [CommRing A] [Algebra R A] {a : A} [n.AtLeastTwo] :\n    (- ofNat(n) : R) • a = - (ofNat(n)) * a := by\n  simpa [← nat_rawCast_2] using! ofNat_smul\n\ntheorem neg_1_smul {R A} [CommRing R] [CommRing A] [Algebra R A] {a : A} :\n    (-1 : R) • a = - a := by\n  simp\n\ntheorem nnRat_ofNat_smul_1 {R A} [Semifield R] [Semifield A] [Algebra R A] {a : A}\n    [d.AtLeastTwo] :\n    (1 / ofNat(d) : R) • a = (1 / ofNat(d)) * a := by\n  simp [Algebra.smul_def, ← nat_rawCast_2]\n\ntheorem nnRat_ofNat_smul_2 {R A} [Semifield R] [Semifield A] [Algebra R A] {a : A}\n    [n.AtLeastTwo] [d.AtLeastTwo] :\n    (ofNat(n) / ofNat(d) : R) • a = (ofNat(n) / ofNat(d)) * a := by\n  simp [Algebra.smul_def, ← nat_rawCast_2]\n\ntheorem rat_ofNat_smul_1 {R A} [Field R] [Field A] [Algebra R A] {a : A}\n    [d.AtLeastTwo] :\n    ((- 1) / ofNat(d) : R) • a = ((- 1) / ofNat(d)) * a := by\n  simp [Algebra.smul_def, ← nat_rawCast_2]\n\ntheorem rat_ofNat_smul_2 {R A} [Field R] [Field A] [Algebra R A] {a : A}\n    [n.AtLeastTwo] [d.AtLeastTwo] :\n    ((- ofNat(n)) / ofNat(d) : R) • a = ((- ofNat(n)) / ofNat(d)) * a := by\n  simp [Algebra.smul_def, ← nat_rawCast_2]\n\nend cleanupConsts\n\nend cleanup\n\nsection equateScalars\n\n/- ExProd.equateZero -/\n\nTarget:\ntheorem smul_one_eq_zero {r : R} (h : r = 0) :\n    r • (1 : A) = 0 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_c124e4499748","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"ae447a9876d057381754ebdd1fb19ee91c512b403b660778146a79cec1765eba","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Algebra","family_id":"smul_one_eq_zero","file_id":"mathlib/Mathlib/Tactic/Algebra/Lemmas.lean","sample_id":"c124e4499748820a17df65d5a170a2889e6ded7f6f676ef140d015398e461d38"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b77e7e870da0ed2ed6f8b7ac1969e927be1488555d4e8afb15051655e1aaced2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a5285c68d16fa2a2346f4fd34255ffe7bd4944930627d370001a13fcff0527b9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"105c96e643efa1f749e66cefffdea4f338def0d05aec0167af11a6685aed3ac3","source_sha256":"85fcf92cc8640c9f278d7a2006b517c6ae93694387f3542b94cbe753427dd693","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [← Function.comp_def, Tendsto, ← map_map, map_inv_atTop]","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.2},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"92515cc23625d70d50c68f9519769e85bca419ac3ec1235b245cdf8e5619d855","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.MinMax\npublic import Mathlib.Order.Filter.AtTopBot.Basic\npublic import Mathlib.Order.Filter.AtTopBot.Map\npublic import Mathlib.Order.Filter.AtTopBot.Monoid\n\nNamespace:\nFilter\n\nLocal context:\n/-\nCopyright (c) 2019 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Convergence to ±infinity in ordered commutative groups\n-/\n\npublic section\n\nvariable {α G : Type*}\nopen Set\n\nnamespace Filter\n\nsection OrderedCommGroup\n\nvariable [CommGroup G] [PartialOrder G] [IsOrderedMonoid G] (l : Filter α) {f g : α → G}\n\n@[to_additive]\ntheorem tendsto_atTop_mul_left_of_le' (C : G) (hf : ∀ᶠ x in l, C ≤ f x) (hg : Tendsto g l atTop) :\n    Tendsto (fun x => f x * g x) l atTop :=\n  .atTop_of_isBoundedUnder_le_mul (f := f⁻¹) ⟨C⁻¹, by simpa⟩ (by simpa)\n\n@[to_additive]\ntheorem tendsto_atBot_mul_left_of_ge' (C : G) (hf : ∀ᶠ x in l, f x ≤ C) (hg : Tendsto g l atBot) :\n    Tendsto (fun x => f x * g x) l atBot :=\n  tendsto_atTop_mul_left_of_le' (G := Gᵒᵈ) _ C hf hg\n\n@[to_additive]\ntheorem tendsto_atTop_mul_left_of_le (C : G) (hf : ∀ x, C ≤ f x) (hg : Tendsto g l atTop) :\n    Tendsto (fun x => f x * g x) l atTop :=\n  tendsto_atTop_mul_left_of_le' l C (univ_mem' hf) hg\n\n@[to_additive]\ntheorem tendsto_atBot_mul_left_of_ge (C : G) (hf : ∀ x, f x ≤ C) (hg : Tendsto g l atBot) :\n    Tendsto (fun x => f x * g x) l atBot :=\n  tendsto_atTop_mul_left_of_le (G := Gᵒᵈ) _ C hf hg\n\n@[to_additive]\ntheorem tendsto_atTop_mul_right_of_le' (C : G) (hf : Tendsto f l atTop) (hg : ∀ᶠ x in l, C ≤ g x) :\n    Tendsto (fun x => f x * g x) l atTop :=\n  .atTop_of_mul_isBoundedUnder_le (g := g⁻¹) ⟨C⁻¹, by simpa⟩ (by simpa)\n\n@[to_additive]\ntheorem tendsto_atBot_mul_right_of_ge' (C : G) (hf : Tendsto f l atBot) (hg : ∀ᶠ x in l, g x ≤ C) :\n    Tendsto (fun x => f x * g x) l atBot :=\n  tendsto_atTop_mul_right_of_le' (G := Gᵒᵈ) _ C hf hg\n\n@[to_additive]\ntheorem tendsto_atTop_mul_right_of_le (C : G) (hf : Tendsto f l atTop) (hg : ∀ x, C ≤ g x) :\n    Tendsto (fun x => f x * g x) l atTop :=\n  tendsto_atTop_mul_right_of_le' l C hf (univ_mem' hg)\n\n@[to_additive]\ntheorem tendsto_atBot_mul_right_of_ge (C : G) (hf : Tendsto f l atBot) (hg : ∀ x, g x ≤ C) :\n    Tendsto (fun x => f x * g x) l atBot :=\n  tendsto_atTop_mul_right_of_le (G := Gᵒᵈ) _ C hf hg\n\n@[to_additive]\ntheorem tendsto_atTop_mul_const_left (C : G) (hf : Tendsto f l atTop) :\n    Tendsto (fun x => C * f x) l atTop :=\n  tendsto_atTop_mul_left_of_le' l C (univ_mem' fun _ => le_refl C) hf\n\n@[to_additive]\ntheorem tendsto_atBot_mul_const_left (C : G) (hf : Tendsto f l atBot) :\n    Tendsto (fun x => C * f x) l atBot :=\n  tendsto_atTop_mul_const_left (G := Gᵒᵈ) _ C hf\n\n@[to_additive]\ntheorem tendsto_atTop_mul_const_right (C : G) (hf : Tendsto f l atTop) :\n    Tendsto (fun x => f x * C) l atTop :=\n  tendsto_atTop_mul_right_of_le' l C hf (univ_mem' fun _ => le_refl C)\n\n@[to_additive]\ntheorem tendsto_atBot_mul_const_right (C : G) (hf : Tendsto f l atBot) :\n    Tendsto (fun x => f x * C) l atBot :=\n  tendsto_atTop_mul_const_right (G := Gᵒᵈ) _ C hf\n\n@[to_additive]\ntheorem map_inv_atBot : map (Inv.inv : G → G) atBot = atTop :=\n  (OrderIso.inv G).map_atBot\n\n@[to_additive]\ntheorem map_inv_atTop : map (Inv.inv : G → G) atTop = atBot :=\n  (OrderIso.inv G).map_atTop\n\n@[to_additive]\ntheorem comap_inv_atBot : comap (Inv.inv : G → G) atBot = atTop :=\n  (OrderIso.inv G).comap_atTop\n\n@[to_additive]\ntheorem comap_inv_atTop : comap (Inv.inv : G → G) atTop = atBot :=\n  (OrderIso.inv G).comap_atBot\n\n@[to_additive]\ntheorem tendsto_inv_atTop_atBot : Tendsto (Inv.inv : G → G) atTop atBot :=\n  (OrderIso.inv G).tendsto_atTop\n\n@[to_additive]\ntheorem tendsto_inv_atBot_atTop : Tendsto (Inv.inv : G → G) atBot atTop :=\n  tendsto_inv_atTop_atBot (G := Gᵒᵈ)\n\nvariable {l}\n\n@[to_additive (attr := simp)]\ntheorem tendsto_inv_atTop_iff : Tendsto (fun x => (f x)⁻¹) l atTop ↔ Tendsto f l atBot :=\n  (OrderIso.inv G).tendsto_atBot_iff\n\n@[to_additive (attr := simp)]\ntheorem tendsto_inv_atBot_iff : Tendsto (fun x => (f x)⁻¹) l atBot ↔ Tendsto f l atTop :=\n  (OrderIso.inv G).tendsto_atTop_iff\n\n@[to_additive (attr := simp)]\n\nTarget:\ntheorem tendsto_comp_inv_atTop_iff {f : G → α} :\n    Tendsto (fun x ↦ f (x⁻¹)) atTop l ↔ Tendsto f atBot l :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_105c96e643ef","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"6640fdd6c5b666a8f20d623043460993fb36c90bd44a5169b972f1d8d873c286","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Filter","family_id":"tendsto_comp_inv_attop_iff","file_id":"mathlib/Mathlib/Order/Filter/AtTopBot/Group.lean","sample_id":"105c96e643efa1f749e66cefffdea4f338def0d05aec0167af11a6685aed3ac3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"603b10c8e8d8264cdfe3ea1be9c171071aba17b8e3411dbc942a8512b4e99c56","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"57919ee9fe831200d4f5697405ab93efe8a6295ef8776bc2c7eff0fee53e76ad","source_sha256":"b81d767611ae69fe62afcbf3affb9ca909a694d39d19ca39528661161a4334bc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI := Fintype.ofFinite ι\n  haveI := fun i => Fintype.ofFinite (κ i)\n  classical\n  refine (Basis.piTensorProduct b).dualBasis.ext (fun p ↦ ?_)\n  refine (Basis.piTensorProduct b).ext (fun q ↦ ?_)\n  simp [Finsupp.single_apply, Fintype.prod_ite_zero, ← funext_iff]","hard_negative":true,"metrics":{"chosen_tokens":67,"rejected_tokens":8,"token_jaccard":0.051282,"token_length_ratio":0.119403},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"926dcba658fcf33b5e50cacd2845a67e2362fd2216d9bc0917af1c5bd2dca98d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Dual.Basis\npublic import Mathlib.LinearAlgebra.FreeModule.Finite.Basic\npublic import Mathlib.LinearAlgebra.PiTensorProduct.Basis\n\nNamespace:\nPiTensorProduct\n\nLocal context:\n/-\nCopyright (c) 2025 Daniel Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Morrison, Sophie Morel\n-/\n/-!\n# Tensor products of dual spaces\n\n## Main definitions\n\n* `PiTensorProduct.dualDistrib`: The canonical linear map from `⨂[R] i, Dual R (M i)` to\n  `Dual R (⨂[R] i, M i)`, sending `⨂ₜ[R] i, f i` to the composition of\n  `PiTensorProduct.map f` with the linear equivalence `⨂[R] i, R →ₗ R` given by multiplication.\n\n* `PiTensorProduct.dualDistribEquiv`: A linear equivalence between `⨂[R] i, Dual R (M i)`\n  and `Dual R (⨂[R] i, M i)` when all `M i` are finite free modules. If\n  `f : (i : ι) → Dual R (M i)`, then this equivalence sends `⨂ₜ[R] i, f i` to the composition of\n  `PiTensorProduct.map f` with the natural isomorphism `⨂[R] i, R ≃ R` given by multiplication.\n-/\n\n@[expose] public section\n\nnamespace PiTensorProduct\n\nopen PiTensorProduct LinearMap Module TensorProduct\n\nvariable {ι : Type*}\n\nsection SemiRing\n\nvariable {R : Type*} {M : ι → Type*} [CommSemiring R] [Π i, AddCommMonoid (M i)]\n  [Π i, Module R (M i)]\n\n/-- The canonical linear map from `⨂[R] i, Dual R (M i)` to `Dual R (⨂[R] i, M i)`,\nsending `⨂ₜ[R] i, f i` to the composition of `PiTensorProduct.map f` with\nthe linear equivalence `⨂[R] i, R →ₗ R` given by multiplication. -/\nnoncomputable def dualDistrib [Finite ι] : (⨂[R] i, Dual R (M i)) →ₗ[R] Dual R (⨂[R] i, M i) :=\n  haveI := Fintype.ofFinite ι\n  (LinearMap.compRight _ (constantBaseRingEquiv ι R).toLinearMap) ∘ₗ piTensorHomMap\n\n@[simp]\ntheorem dualDistrib_apply [Fintype ι] (f : Π i, Dual R (M i)) (m : Π i, M i) :\n    dualDistrib (⨂ₜ[R] i, f i) (⨂ₜ[R] i, m i) = ∏ i, (f i) (m i) := by\n  rw [dualDistrib, Subsingleton.elim (Fintype.ofFinite ι) ‹_›]\n  simp\n\nend SemiRing\n\nsection Ring\n\nvariable {R : Type*} {κ : ι → Type*} {M : ι → Type*} [CommRing R] [Π i, AddCommGroup (M i)]\n  [Π i, Module R (M i)]\n\nopen Classical in\n/-- An inverse to `PiTensorProduct.dualDistrib` given bases. -/\nnoncomputable def dualDistribInvOfBasis [Finite ι] [∀ i, Finite (κ i)]\n    (b : Π i, Basis (κ i) R (M i)) :\n    Dual R (⨂[R] i, M i) →ₗ[R] ⨂[R] i, Dual R (M i) :=\n  haveI := Fintype.ofFinite ι\n  haveI := fun i => Fintype.ofFinite (κ i)\n  ∑ p : (Π i, κ i), (ringLmapEquivSelf R ℕ _).symm (⨂ₜ[R] i, (b i).dualBasis (p i)) ∘ₗ\n    (applyₗ (⨂ₜ[R] i, b i (p i)))\n\nopen Classical in\n@[simp]\ntheorem dualDistribInvOfBasis_apply [Fintype ι] [∀ i, Fintype (κ i)] (b : Π i, Basis (κ i) R (M i))\n    (f : Dual R (⨂[R] i, M i)) : dualDistribInvOfBasis b f =\n    ∑ p : (Π i, κ i), f (⨂ₜ[R] i, b i (p i)) • (⨂ₜ[R] i, (b i).dualBasis (p i)) := by\n  simp only [dualDistribInvOfBasis, Basis.coe_dualBasis, ringLmapEquivSelf_symm_apply, coe_sum,\n    coe_comp, coe_smulRight, End.one_apply, Finset.sum_apply, Function.comp_apply,\n    applyₗ_apply_apply]\n  convert! rfl\n\nTarget:\ntheorem dualDistrib_dualDistribInvOfBasis_left_inverse [Finite ι] [∀ i, Finite (κ i)]\n    (b : Π i, Basis (κ i) R (M i)) :\n    (dualDistrib) ∘ₗ (dualDistribInvOfBasis b) = LinearMap.id :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"cb05b30a78cf980bf36ff77c6e80f66db4663fa5980f198787430dfdc4567007","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/PiTensorProduct","family_id":"dualdistrib_dualdistribinvofbasis_left_inverse","file_id":"mathlib/Mathlib/LinearAlgebra/PiTensorProduct/Dual.lean","sample_id":"57919ee9fe831200d4f5697405ab93efe8a6295ef8776bc2c7eff0fee53e76ad"},"split":"train"}
{"binding":{"chosen_proof_sha256":"91037b5a5d4107cfc6e47498527b6da8e785a774a3e33005feebe3831fc4a2a4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2031f82d6bd5b2086402ba4895d772d5452bfbf03175b7504ea5ff92425b4486","source_sha256":"113aaf52a8970f3d5731b3bef38b5eb29be69585d68fdd5fc22f4868fa9f35e6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← tendsto_norm_atTop_iff_cobounded]\n  simpa [norm_intCast_eq_abs_mul_norm_one] using tendsto_intCast_atTop_atTop\n    |>.comp (tendsto_abs_atBot_atTop.sup tendsto_abs_atTop_atTop)\n    |>.atTop_mul_const (norm_pos_iff.mpr one_ne_zero)","hard_negative":false,"metrics":{"chosen_tokens":32,"rejected_tokens":2,"token_jaccard":0.041667,"token_length_ratio":0.0625},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"92a944633a45622cf5f4fd7400117cfbba626757da487dd20f23729f6cc67f22","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Module\npublic import Mathlib.Algebra.Order.Field.Power\npublic import Mathlib.Algebra.Polynomial.Monic\npublic import Mathlib.Analysis.Asymptotics.Lemmas\npublic import Mathlib.Analysis.Normed.Ring.InfiniteSum\npublic import Mathlib.Analysis.Normed.Module.Basic\npublic import Mathlib.Analysis.Normed.Order.Lattice\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.Data.List.TFAE\npublic import Mathlib.Data.Nat.Choose.Bounds\npublic import Mathlib.Order.Filter.AtTopBot.ModEq\npublic import Mathlib.RingTheory.Polynomial.Pochhammer\npublic import Mathlib.Tactic.NoncommRing\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie, Patrick Massot\n-/\n/-!\n# A collection of specific limit computations\n\nThis file contains important specific limit computations in (semi-)normed groups/rings/spaces, as\nwell as such computations in `ℝ` when the natural proof passes through a fact about normed spaces.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Set Function Filter Finset Metric Module Asymptotics Topology Nat NNReal ENNReal\nopen scoped Ring\n\nvariable {α : Type*}\n\ntheorem tendsto_natCast_atTop_cobounded\n    [NormedRing α] [NormSMulClass ℤ α] [Nontrivial α] :\n    Tendsto Nat.cast atTop (Bornology.cobounded α) := by\n  rw [← tendsto_norm_atTop_iff_cobounded]\n  simpa [norm_natCast_eq_mul_norm_one] using tendsto_natCast_atTop_atTop\n    |>.atTop_mul_const (norm_pos_iff.mpr one_ne_zero)\n\nTarget:\ntheorem tendsto_intCast_atBot_sup_atTop_cobounded\n    [NormedRing α] [NormSMulClass ℤ α] [Nontrivial α] :\n    Tendsto Int.cast (atBot ⊔ atTop) (Bornology.cobounded α) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/SpecificLimits","family_id":"tendsto_intcast_atbot_sup_attop_cobounded","file_id":"mathlib/Mathlib/Analysis/SpecificLimits/Normed.lean","sample_id":"2031f82d6bd5b2086402ba4895d772d5452bfbf03175b7504ea5ff92425b4486"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d39e8a584cb9cadcbcf3f438cba249cdb73b9b66b3d2c3c40bd6185edf5ea6c9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"75d34468da2034a80fca64c4e5869a12c798eeacfe03a8618a187bb63ae913dc","source_sha256":"2f00d04aba5c305cbf490ef92448e4a2acde5744e8e9969803e8bbd6f318fa29","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h : f = 𝟙 j := by cat_disch\n  rw [h]\n  simp","hard_negative":false,"metrics":{"chosen_tokens":16,"rejected_tokens":3,"token_jaccard":0.0625,"token_length_ratio":0.1875},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"92dc07b66192b973ad262a1c76c1392e51822ffbe94f4efe4c817f85ad7f35b5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Pi.Basic\n\nNamespace:\nCategoryTheory.Discrete\n\nLocal context:\n/-\nCopyright (c) 2017 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Kim Morrison, Floris van Doorn\n-/\n/-!\n# Discrete categories\n\nWe define `Discrete α` as a structure containing a term `a : α` for any type `α`,\nand use this type alias to provide a `SmallCategory` instance\nwhose only morphisms are the identities.\n\nThere is an annoying technical difficulty that it has turned out to be inconvenient\nto allow categories with morphisms living in `Prop`,\nso instead of defining `X ⟶ Y` in `Discrete α` as `X = Y`,\none might define it as `PLift (X = Y)`.\nIn fact, to allow `Discrete α` to be a `SmallCategory`\n(i.e. with morphisms in the same universe as the objects),\nwe actually define the hom type `X ⟶ Y` as `ULift (PLift (X = Y))`.\n\n`Discrete.functor` promotes a function `f : I → C` (for any category `C`) to a functor\n`Discrete.functor f : Discrete I ⥤ C`.\n\nSimilarly, `Discrete.natTrans` and `Discrete.natIso` promote `I`-indexed families of morphisms,\nor `I`-indexed families of isomorphisms to natural transformations or natural isomorphism.\n\nWe show equivalences of types are the same as (categorical) equivalences of the corresponding\ndiscrete categories.\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\n-- morphism levels before object levels. See note [category theory universes].\nuniverse v₁ v₂ v₃ u₁ u₁' u₂ u₃\n\n-- This is intentionally a structure rather than a type synonym\n-- to enforce using `DiscreteEquiv` (or `Discrete.mk` and `Discrete.as`) to move between\n-- `Discrete α` and `α`. Otherwise there is too much API leakage.\n/-- A wrapper for promoting any type to a category,\nwith the only morphisms being equalities.\n-/\n@[ext, aesop safe cases (rule_sets := [CategoryTheory])]\nstructure Discrete (α : Type u₁) where\n  /-- A wrapper for promoting any type to a category,\n  with the only morphisms being equalities. -/\n  as : α\n\n@[simp]\ntheorem Discrete.mk_as {α : Type u₁} (X : Discrete α) : Discrete.mk X.as = X :=\n  rfl\n\n/-- `Discrete α` is equivalent to the original type `α`. -/\n@[simps]\ndef discreteEquiv {α : Type u₁} : Discrete α ≃ α where\n  toFun := Discrete.as\n  invFun := Discrete.mk\n  left_inv := by cat_disch\n  right_inv := by cat_disch\n\ninstance {α : Type u₁} [DecidableEq α] : DecidableEq (Discrete α) :=\n  discreteEquiv.decidableEq\n\n/-- The \"Discrete\" category on a type, whose morphisms are equalities.\n\nBecause we do not allow morphisms in `Prop` (only in `Type`),\nsomewhat annoyingly we have to define `X ⟶ Y` as `ULift (PLift (X = Y))`. -/\n@[stacks 001A]\ninstance discreteCategory (α : Type u₁) : SmallCategory (Discrete α) where\n  Hom X Y := ULift (PLift (X.as = Y.as))\n  id _ := ULift.up (PLift.up rfl)\n  comp {X Y Z} g f := by\n    cases X\n    cases Y\n    cases Z\n    rcases f with ⟨⟨⟨⟩⟩⟩\n    exact g\n\nnamespace Discrete\n\nvariable {α : Type u₁}\n\ninstance [Inhabited α] : Inhabited (Discrete α) :=\n  ⟨⟨default⟩⟩\n\ninstance [Subsingleton α] : Subsingleton (Discrete α) :=\n  ⟨by cat_disch⟩\n\ninstance instSubsingletonDiscreteHom (X Y : Discrete α) : Subsingleton (X ⟶ Y) :=\n  show Subsingleton (ULift (PLift _)) from inferInstance\n\n/-- A simple tactic to run `cases` on any `Discrete α` hypotheses. -/\nmacro \"discrete_cases\" : tactic =>\n  `(tactic| fail_if_no_progress casesm* Discrete _, (_ : Discrete _) ⟶ (_ : Discrete _), PLift _)\n\nopen Lean Elab Tactic in\n/--\nUse:\n```\nattribute [local aesop safe tactic (rule_sets := [CategoryTheory])]\n  CategoryTheory.Discrete.discreteCases\n```\nto locally give `cat_disch` the ability to call `cases` on\n`Discrete` and `(_ : Discrete _) ⟶ (_ : Discrete _)` hypotheses.\n-/\nmeta def discreteCases : TacticM Unit := do\n  evalTactic (← `(tactic| discrete_cases))\n\n-- TODO: investigate turning on either\n-- `attribute [aesop safe cases (rule_sets := [CategoryTheory])] Discrete`\n-- or\n-- `attribute [aesop safe tactic (rule_sets := [CategoryTheory])] discreteCases`\n-- globally.\n\ninstance [Unique α] : Unique (Discrete α) :=\n  Unique.mk' (Discrete α)\n\n/-- Extract the equation from a morphism in a discrete category. -/\ntheorem eq_of_hom {X Y : Discrete α} (i : X ⟶ Y) : X.as = Y.as :=\n  i.down.down\n\n/-- Promote an equation between the wrapped terms in `X Y : Discrete α` to a morphism `X ⟶ Y`\nin the discrete category. -/\nprotected abbrev eqToHom {X Y : Discrete α} (h : X.as = Y.as) : X ⟶ Y :=\n  eqToHom (by cat_disch)\n\n/-- Promote an equation between the wrapped terms in `X Y : Discrete α` to an isomorphism `X ≅ Y`\nin the discrete category. -/\nprotected abbrev eqToIso {X Y : Discrete α} (h : X.as = Y.as) : X ≅ Y :=\n  eqToIso (by cat_disch)\n\n/-- A variant of `eqToHom` that lifts terms to the discrete category. -/\nabbrev eqToHom' {a b : α} (h : a = b) : Discrete.mk a ⟶ Discrete.mk b :=\n  Discrete.eqToHom h\n\n/-- A variant of `eqToIso` that lifts terms to the discrete category. -/\nabbrev eqToIso' {a b : α} (h : a = b) : Discrete.mk a ≅ Discrete.mk b :=\n  Discrete.eqToIso h\n\n@[simp]\ntheorem id_def (X : Discrete α) : ULift.up (PLift.up (Eq.refl X.as)) = 𝟙 X :=\n  rfl\n\n@[simp]\ntheorem id_def' (X : α) : ULift.up (PLift.up (Eq.refl X)) = 𝟙 (⟨X⟩ : Discrete α) :=\n  rfl\n\nvariable {C : Type u₂} [Category.{v₂} C]\n\ninstance {I : Type u₁} {i j : Discrete I} (f : i ⟶ j) : IsIso f :=\n  ⟨⟨Discrete.eqToHom (eq_of_hom f).symm, by cat_disch⟩⟩\n\nattribute [local aesop safe tactic (rule_sets := [CategoryTheory])]\n  CategoryTheory.Discrete.discreteCases\n\n/-- Any function `I → C` gives a functor `Discrete I ⥤ C`. -/\ndef functor {I : Type u₁} (F : I → C) : Discrete I ⥤ C where\n  obj := F ∘ Discrete.as\n  map {X Y} f := by\n    dsimp\n    rcases f with ⟨⟨h⟩⟩\n    exact eqToHom (congrArg _ h)\n\n@[simp]\ntheorem functor_obj {I : Type u₁} (F : I → C) (i : I) :\n    (Discrete.functor F).obj (Discrete.mk i) = F i :=\n  rfl\n\ntheorem functor_map {I : Type u₁} (F : I → C) {i : Discrete I} (f : i ⟶ i) :\n    (Discrete.functor F).map f = 𝟙 (F i.as) := by cat_disch\n\n@[simp]\ntheorem functor_obj_eq_as {I : Type u₁} (F : I → C) (X : Discrete I) :\n    (Discrete.functor F).obj X = F X.as :=\n  rfl\n\n@[ext]\nlemma functor_ext {I : Type u₁} {G F : Discrete I ⥤ C} (h : (i : I) → G.obj ⟨i⟩ = F.obj ⟨i⟩) :\n    G = F := by\n  fapply Functor.ext\n  · intro I; rw [h]\n  · intro ⟨X⟩ ⟨Y⟩ ⟨⟨p⟩⟩; simp only at p; induction p; simp\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The discrete functor induced by a composition of maps can be written as a\ncomposition of two discrete functors.\n-/\n@[simps!]\ndef functorComp {I : Type u₁} {J : Type u₁'} (f : J → C) (g : I → J) :\n    Discrete.functor (f ∘ g) ≅ Discrete.functor (Discrete.mk ∘ g) ⋙ Discrete.functor f :=\n  NatIso.ofComponents fun _ => Iso.refl _\n\n/-- For functors out of a discrete category,\na natural transformation is just a collection of maps,\nas the naturality squares are trivial.\n-/\n@[simps]\ndef natTrans {I : Type u₁} {F G : Discrete I ⥤ C} (f : ∀ i : Discrete I, F.obj i ⟶ G.obj i) :\n    F ⟶ G where\n  app := f\n  naturality := fun {X Y} ⟨⟨g⟩⟩ => by\n    discrete_cases\n    rcases g\n    change F.map (𝟙 _) ≫ _ = _ ≫ G.map (𝟙 _)\n    simp\n\n/-- For functors out of a discrete category,\na natural isomorphism is just a collection of isomorphisms,\nas the naturality squares are trivial.\n-/\n@[simps!]\ndef natIso {I : Type u₁} {F G : Discrete I ⥤ C} (f : ∀ i : Discrete I, F.obj i ≅ G.obj i) :\n    F ≅ G :=\n  NatIso.ofComponents f fun ⟨⟨g⟩⟩ => by\n    discrete_cases\n    rcases g\n    change F.map (𝟙 _) ≫ _ = _ ≫ G.map (𝟙 _)\n    simp\n\ninstance {I : Type*} {F G : Discrete I ⥤ C} (f : ∀ i, F.obj i ⟶ G.obj i) [∀ i, IsIso (f i)] :\n    IsIso (Discrete.natTrans f) := by\n  change IsIso (Discrete.natIso (fun i => asIso (f i))).hom\n  infer_instance\n\n@[simp]\ntheorem natIso_app {I : Type u₁} {F G : Discrete I ⥤ C} (f : ∀ i : Discrete I, F.obj i ≅ G.obj i)\n    (i : Discrete I) : (Discrete.natIso f).app i = f i := by cat_disch\n\n/-- Every functor `F` from a discrete category is naturally isomorphic (actually, equal) to\n  `Discrete.functor (F.obj)`. -/\n@[simps!]\ndef natIsoFunctor {I : Type u₁} {F : Discrete I ⥤ C} : F ≅ Discrete.functor (F.obj ∘ Discrete.mk) :=\n  natIso fun _ => Iso.refl _\n\n/-- Composing `Discrete.functor F` with another functor `G` amounts to composing `F` with `G.obj` -/\n@[simps!]\ndef compNatIsoDiscrete {I : Type u₁} {D : Type u₃} [Category.{v₃} D] (F : I → C) (G : C ⥤ D) :\n    Discrete.functor F ⋙ G ≅ Discrete.functor (G.obj ∘ F) :=\n  natIso fun _ => Iso.refl _\n\n/-- We can promote a type-level `Equiv` to\nan equivalence between the corresponding `discrete` categories.\n-/\n@[simps]\ndef equivalence {I : Type u₁} {J : Type u₂} (e : I ≃ J) : Discrete I ≌ Discrete J where\n  functor := Discrete.functor (Discrete.mk ∘ (e : I → J))\n  inverse := Discrete.functor (Discrete.mk ∘ (e.symm : J → I))\n  unitIso :=\n    Discrete.natIso fun i => eqToIso (by simp)\n  counitIso :=\n    Discrete.natIso fun j => eqToIso (by simp)\n\n/-- We can convert an equivalence of `discrete` categories to a type-level `Equiv`. -/\n@[simps]\ndef equivOfEquivalence {α : Type u₁} {β : Type u₂} (h : Discrete α ≌ Discrete β) : α ≃ β where\n  toFun := Discrete.as ∘ h.functor.obj ∘ Discrete.mk\n  invFun := Discrete.as ∘ h.inverse.obj ∘ Discrete.mk\n  left_inv a := by simpa using eq_of_hom (h.unitIso.app (Discrete.mk a)).2\n  right_inv a := by simpa using eq_of_hom (h.counitIso.app (Discrete.mk a)).1\n\nend Discrete\n\nnamespace Discrete\n\nvariable {J : Type v₁}\n\nopen Opposite\n\n/-- A discrete category is equivalent to its opposite category. -/\n@[simps! functor_obj_as inverse_obj]\nprotected def opposite (α : Type u₁) : (Discrete α)ᵒᵖ ≌ Discrete α :=\n  let F : Discrete α ⥤ (Discrete α)ᵒᵖ := Discrete.functor fun x => op (Discrete.mk x)\n  { functor := F.leftOp\n    inverse := F\n    unitIso := NatIso.ofComponents fun ⟨_⟩ => Iso.refl _\n    counitIso := Discrete.natIso fun ⟨_⟩ => Iso.refl _ }\n\nvariable {C : Type u₂} [Category.{v₂} C]\n\n@[simp]\n\nTarget:\ntheorem functor_map_id (F : Discrete J ⥤ C) {j : Discrete J} (f : j ⟶ j) :\n    F.map f = 𝟙 (F.obj j) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Discrete","family_id":"functor_map_id","file_id":"mathlib/Mathlib/CategoryTheory/Discrete/Basic.lean","sample_id":"75d34468da2034a80fca64c4e5869a12c798eeacfe03a8618a187bb63ae913dc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3939293b987d00efdc8989f961d8fe481cabdbec2fac5708f8344374642f9236","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dd448d7bddd39539db63b6005ceef026bb709e5272ecd6e5386e656fa08db32f","source_sha256":"6ccf0444fc2138afeb704541116189c71bea73fd06b5fb4f13fd001240de6123","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [TopologicalSpace.nhds_generateFrom]\n  simp only [comap_iInf, comap_principal]\n  intro s hs\n  rw [← le_principal_iff]\n  refine iInf_le_of_le { u | s ∈ u } ?_\n  refine iInf_le_of_le ⟨hs, ⟨s, rfl⟩⟩ ?_\n  exact principal_mono.2 fun _ ↦ id","hard_negative":false,"metrics":{"chosen_tokens":54,"rejected_tokens":2,"token_jaccard":0.027778,"token_length_ratio":0.037037},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"933a2df7f7791c3680d346b31c5f9372773340ca48a4ddd8f3ec567f2a0d3ce8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Bases\npublic import Mathlib.Topology.DenseEmbedding\npublic import Mathlib.Topology.Connected.TotallyDisconnected\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton\n-/\n/-! # Stone-Čech compactification\n\nConstruction of the Stone-Čech compactification using ultrafilters.\n\nFor any topological space `α`, we build a compact Hausdorff space `StoneCech α` and a continuous\nmap `stoneCechUnit : α → StoneCech α` which is minimal in the sense of the following universal\nproperty: for any compact Hausdorff space `β` and every map `f : α → β` such that\n`hf : Continuous f`, there is a unique map `stoneCechExtend hf : StoneCech α → β` such that\n`stoneCechExtend_extends : stoneCechExtend hf ∘ stoneCechUnit = f`.\nContinuity of this extension is asserted by `continuous_stoneCechExtend` and uniqueness by\n`stoneCech_hom_ext`.\n\nBeware that the terminology “extend” is slightly misleading since `stoneCechUnit` is not always\ninjective, so one cannot always think of `α` as being “inside” its compactification `StoneCech α`.\n\n## Implementation notes\n\nParts of the formalization are based on “Ultrafilters and Topology”\nby Marius Stekelenburg, particularly section 5. However the construction in the general\ncase is different because the equivalence relation on spaces of ultrafilters described\nby Stekelenburg causes issues with universes since it involves a condition\non all compact Hausdorff spaces. We replace it by a two steps construction.\nThe first step called `PreStoneCech` guarantees the expected universal property but\nnot the Hausdorff condition. We then define `StoneCech α` as `T2Quotient (PreStoneCech α)`.\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen Filter Set\n\nopen Topology\n\nuniverse u v\n\nsection Ultrafilter\n\n/- The set of ultrafilters on α carries a natural topology which makes\n  it the Stone-Čech compactification of α (viewed as a discrete space). -/\n/-- Basis for the topology on `Ultrafilter α`. -/\ndef ultrafilterBasis (α : Type u) : Set (Set (Ultrafilter α)) :=\n  range fun s : Set α ↦ { u | s ∈ u }\n\nvariable {α : Type u}\n\ninstance Ultrafilter.topologicalSpace : TopologicalSpace (Ultrafilter α) :=\n  TopologicalSpace.generateFrom (ultrafilterBasis α)\n\ntheorem ultrafilterBasis_is_basis : TopologicalSpace.IsTopologicalBasis (ultrafilterBasis α) :=\n  ⟨by\n    rintro _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ u ⟨ua, ub⟩\n    refine ⟨_, ⟨a ∩ b, rfl⟩, inter_mem ua ub, fun v hv ↦ ⟨?_, ?_⟩⟩ <;> apply mem_of_superset hv <;>\n      simp [inter_subset_right],\n    eq_univ_of_univ_subset <| subset_sUnion_of_mem <| ⟨univ, eq_univ_of_forall fun _ ↦ univ_mem⟩,\n    rfl⟩\n\n/-- The basic open sets for the topology on ultrafilters are open. -/\ntheorem ultrafilter_isOpen_basic (s : Set α) : IsOpen { u : Ultrafilter α | s ∈ u } :=\n  ultrafilterBasis_is_basis.isOpen ⟨s, rfl⟩\n\n/-- The basic open sets for the topology on ultrafilters are also closed. -/\ntheorem ultrafilter_isClosed_basic (s : Set α) : IsClosed { u : Ultrafilter α | s ∈ u } := by\n  rw [← isOpen_compl_iff]\n  convert! ultrafilter_isOpen_basic sᶜ using 1\n  ext u\n  exact Ultrafilter.compl_mem_iff_notMem.symm\n\n/-- Every ultrafilter `u` on `Ultrafilter α` converges to a unique\n  point of `Ultrafilter α`, namely `joinM u`. -/\ntheorem ultrafilter_converges_iff {u : Ultrafilter (Ultrafilter α)} {x : Ultrafilter α} :\n    ↑u ≤ 𝓝 x ↔ x = joinM u := by\n  rw [eq_comm, ← Ultrafilter.coe_le_coe]\n  change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, { v : Ultrafilter α | s ∈ v } ∈ u\n  simp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff,\n    mem_setOf_eq]\n  constructor\n  · intro h a ha\n    exact h _ ⟨ha, a, rfl⟩\n  · rintro h a ⟨xi, a, rfl⟩\n    exact h _ xi\n\ninstance ultrafilter_compact : CompactSpace (Ultrafilter α) :=\n  ⟨isCompact_iff_ultrafilter_le_nhds.mpr fun f _ ↦\n      ⟨joinM f, trivial, ultrafilter_converges_iff.mpr rfl⟩⟩\n\ninstance Ultrafilter.t2Space : T2Space (Ultrafilter α) :=\n  t2_iff_ultrafilter.mpr fun {x y} f fx fy ↦\n    have hx : x = joinM f := ultrafilter_converges_iff.mp fx\n    have hy : y = joinM f := ultrafilter_converges_iff.mp fy\n    hx.trans hy.symm\n\ninstance : TotallyDisconnectedSpace (Ultrafilter α) := by\n  rw [totallyDisconnectedSpace_iff_connectedComponent_singleton]\n  intro A\n  simp only [Set.eq_singleton_iff_unique_mem, mem_connectedComponent, true_and]\n  intro B hB\n  rw [← Ultrafilter.coe_le_coe]\n  intro s hs\n  rw [connectedComponent_eq_iInter_isClopen, Set.mem_iInter] at hB\n  let Z := { F : Ultrafilter α | s ∈ F }\n  have hZ : IsClopen Z := ⟨ultrafilter_isClosed_basic s, ultrafilter_isOpen_basic s⟩\n  exact hB ⟨Z, hZ, hs⟩\n\n@[simp] theorem Ultrafilter.tendsto_pure_self (b : Ultrafilter α) : Tendsto pure b (𝓝 b) := by\n  rw [Tendsto, ← coe_map, ultrafilter_converges_iff]\n  ext s\n  change s ∈ b ↔ {t | s ∈ t} ∈ map pure b\n  simp_rw [mem_map, preimage_setOf_eq, mem_pure, setOf_mem_eq]\n\nTarget:\ntheorem ultrafilter_comap_pure_nhds (b : Ultrafilter α) : comap pure (𝓝 b) ≤ b :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Compactification","family_id":"ultrafilter_comap_pure_nhds","file_id":"mathlib/Mathlib/Topology/Compactification/StoneCech.lean","sample_id":"dd448d7bddd39539db63b6005ceef026bb709e5272ecd6e5386e656fa08db32f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ab4cb4aa5cd70f70198092337097cb05b18354e67218420118c17e78a73513e0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"88d7c6c796cca4e10e9b92b371cecd50104434faf2272571f53777bb0490dda7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9fe3719b8a587cc320eadf69f2619355af22157136ae9400bafc484518741ba5","source_sha256":"eceb96a26dccbd8f8abcd83874539b49b8b7e797f195a864cff85c1bbe8476b2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h ↦ ?_, fun ⟨h₁, h₂⟩ ↦ ?_⟩\n  · refine evalInduction h fun a h IH ↦ ?_\n    rcases e : f a with - | a'\n    · rw [Part.mem_unique h\n          (PFun.mem_fix_iff.2 <| Or.inl <| Part.mem_some_iff.2 <| by rw [e]; rfl)]\n      exact ⟨ReflTransGen.refl, e⟩\n    · rcases PFun.mem_fix_iff.1 h with (h | ⟨_, h, _⟩) <;> rw [e] at h <;>\n        cases Part.mem_some_iff.1 h\n      obtain ⟨h₁, h₂⟩ := IH a' e\n      exact ⟨ReflTransGen.head e h₁, h₂⟩\n  · refine ReflTransGen.head_induction_on h₁ ?_ fun h _ IH ↦ ?_\n    · refine PFun.mem_fix_iff.2 (Or.inl ?_)\n      rw [h₂]\n      apply Part.mem_some\n    · refine PFun.mem_fix_iff.2 (Or.inr ⟨_, ?_, IH⟩)\n      rw [h]\n      apply Part.mem_some","hard_negative":true,"metrics":{"chosen_tokens":215,"rejected_tokens":3,"token_jaccard":0.035714,"token_length_ratio":0.013953},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"935af83f304e69d2db817e7d37adbf8b99b6ef96754aba393851550da6f32d0f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Data.PFun\nimport Mathlib.Data.Finset.Option\nimport Mathlib.Data.Finset.Prod\nimport Mathlib.Data.Fintype.Defs\n\nNamespace:\nStateTransition\n\nLocal context:\n/-\nCopyright (c) 2025 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Pim Spelier, Daan van Gent\n-/\n/-!\n# State Transition Systems\n\nThis file contains simple definitions and lemmas for reasoning about state transition systems\ndefined by a function `σ → Option σ`, where `σ` is the type of states.\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero\n\nopen Relation\n\nnamespace StateTransition\n\n/-- Run a state transition function `σ → Option σ` \"to completion\". The return value is the last\nstate returned before a `none` result. If the state transition function always returns `some`,\nthen the computation diverges, returning `Part.none`. -/\ndef eval {σ} (f : σ → Option σ) : σ → Part σ :=\n  PFun.fix fun s ↦ Part.some <| (f s).elim (Sum.inl s) Sum.inr\n\n/-- The reflexive transitive closure of a state transition function. `Reaches f a b` means\nthere is a finite sequence of steps `f a = some a₁`, `f a₁ = some a₂`, ... such that `aₙ = b`.\nThis relation permits zero steps of the state transition function. -/\ndef Reaches {σ} (f : σ → Option σ) : σ → σ → Prop :=\n  ReflTransGen fun a b ↦ b ∈ f a\n\n/-- The transitive closure of a state transition function. `Reaches₁ f a b` means there is a\nnonempty finite sequence of steps `f a = some a₁`, `f a₁ = some a₂`, ... such that `aₙ = b`.\nThis relation does not permit zero steps of the state transition function. -/\ndef Reaches₁ {σ} (f : σ → Option σ) : σ → σ → Prop :=\n  TransGen fun a b ↦ b ∈ f a\n\ntheorem reaches₁_eq {σ} {f : σ → Option σ} {a b c} (h : f a = f b) :\n    Reaches₁ f a c ↔ Reaches₁ f b c :=\n  TransGen.head'_iff.trans (TransGen.head'_iff.trans <| by rw [h]).symm\n\ntheorem reaches_total {σ} {f : σ → Option σ} {a b c} (hab : Reaches f a b) (hac : Reaches f a c) :\n    Reaches f b c ∨ Reaches f c b :=\n  ReflTransGen.total_of_right_unique (fun _ _ _ ↦ Option.mem_unique) hab hac\n\ntheorem reaches₁_fwd {σ} {f : σ → Option σ} {a b c} (h₁ : Reaches₁ f a c) (h₂ : b ∈ f a) :\n    Reaches f b c := by\n  rcases TransGen.head'_iff.1 h₁ with ⟨b', hab, hbc⟩\n  cases Option.mem_unique hab h₂; exact hbc\n\n/-- A variation on `Reaches`. `Reaches₀ f a b` holds if whenever `Reaches₁ f b c` then\n`Reaches₁ f a c`. This is a weaker property than `Reaches` and is useful for replacing states with\nequivalent states without taking a step. -/\ndef Reaches₀ {σ} (f : σ → Option σ) (a b : σ) : Prop :=\n  ∀ c, Reaches₁ f b c → Reaches₁ f a c\n\ntheorem Reaches₀.trans {σ} {f : σ → Option σ} {a b c : σ} (h₁ : Reaches₀ f a b)\n    (h₂ : Reaches₀ f b c) : Reaches₀ f a c\n  | _, h₃ => h₁ _ (h₂ _ h₃)\n\n@[refl]\ntheorem Reaches₀.refl {σ} {f : σ → Option σ} (a : σ) : Reaches₀ f a a\n  | _, h => h\n\ntheorem Reaches₀.single {σ} {f : σ → Option σ} {a b : σ} (h : b ∈ f a) : Reaches₀ f a b\n  | _, h₂ => h₂.head h\n\ntheorem Reaches₀.head {σ} {f : σ → Option σ} {a b c : σ} (h : b ∈ f a) (h₂ : Reaches₀ f b c) :\n    Reaches₀ f a c :=\n  (Reaches₀.single h).trans h₂\n\ntheorem Reaches₀.tail {σ} {f : σ → Option σ} {a b c : σ} (h₁ : Reaches₀ f a b) (h : c ∈ f b) :\n    Reaches₀ f a c :=\n  h₁.trans (Reaches₀.single h)\n\ntheorem reaches₀_eq {σ} {f : σ → Option σ} {a b} (e : f a = f b) : Reaches₀ f a b\n  | _, h => (reaches₁_eq e).2 h\n\ntheorem Reaches₁.to₀ {σ} {f : σ → Option σ} {a b : σ} (h : Reaches₁ f a b) : Reaches₀ f a b\n  | _, h₂ => h.trans h₂\n\ntheorem Reaches.to₀ {σ} {f : σ → Option σ} {a b : σ} (h : Reaches f a b) : Reaches₀ f a b\n  | _, h₂ => h₂.trans_right h\n\ntheorem Reaches₀.tail' {σ} {f : σ → Option σ} {a b c : σ} (h : Reaches₀ f a b) (h₂ : c ∈ f b) :\n    Reaches₁ f a c :=\n  h _ (TransGen.single h₂)\n\n/-- (co-)Induction principle for `eval`. If a property `C` holds of any point `a` evaluating to `b`\nwhich is either terminal (meaning `a = b`) or where the next point also satisfies `C`, then it\nholds of any point where `eval f a` evaluates to `b`. This formalizes the notion that if\n`eval f a` evaluates to `b` then it reaches terminal state `b` in finitely many steps. -/\n@[elab_as_elim]\ndef evalInduction {σ} {f : σ → Option σ} {b : σ} {C : σ → Sort*} {a : σ}\n    (h : b ∈ eval f a) (H : ∀ a, b ∈ eval f a → (∀ a', f a = some a' → C a') → C a) : C a :=\n  PFun.fixInduction h fun a' ha' h' ↦\n    H _ ha' fun b' e ↦ h' _ <| Part.mem_some_iff.2 <| by rw [e]; rfl\n\nTarget:\ntheorem mem_eval {σ} {f : σ → Option σ} {a b} : b ∈ eval f a ↔ Reaches f a b ∧ f b = none :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_9fe3719b8a58","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"7680089adf39523b071822545c9422f78acfbc1c1267b530ac9d11d5bb1ed0bb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability","family_id":"mem_eval","file_id":"mathlib/Mathlib/Computability/StateTransition.lean","sample_id":"9fe3719b8a587cc320eadf69f2619355af22157136ae9400bafc484518741ba5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a923c894044509b83369abcd9431b69d970849ba97f75437eb89f230df9823e5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0884596d77461ad97de009c6f8b8d539833c319659cef7be0f8610e17da4299a","source_sha256":"68dce35114cf22d288a51cfbefe877f5725b4487e27909680b1aae1c2b647510","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [IsHausdorff.eq_iff_smodEq (I := 𝔭)]\n  simp only [smul_eq_mul, mul_top]\n  intro n\n  cases n\n  · simp\n  · simp [SModEq, mk_pow_fontaineTheta]","hard_negative":false,"metrics":{"chosen_tokens":32,"rejected_tokens":2,"token_jaccard":0.041667,"token_length_ratio":0.0625},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"93941d41ab48a4f62798eb27e56dce2fd3fdad510c71bc294c6f872a87fe2998","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.AdicCompletion.Functoriality\npublic import Mathlib.RingTheory.AdicCompletion.RingHom\npublic import Mathlib.RingTheory.Perfectoid.Untilt\npublic import Mathlib.RingTheory.WittVector.TeichmullerSeries\n\nNamespace:\nWittVector\n\nLocal context:\n/-\nCopyright (c) 2025 Jiedong Jiang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jiedong Jiang\n-/\n/-!\n# Fontaine's θ map\nIn this file, we define Fontaine's `θ` map, which is a ring\nhomomorphism from the Witt vector `𝕎 R♭` of the tilt of a perfectoid ring `R`\nto `R` itself. Our definition of `θ` does not require that `R` is perfectoid in the first place.\nWe only need `R` to be `p`-adically complete.\n\n## Main Definitions\n* `fontaineTheta` : Fontaine's θ map, which is a ring homomorphism from `𝕎 R♭` to `R`.\n\n## Main Theorems\n* `fontaineTheta_teichmuller` : `θ([x])` is the untilt of `x`.\n* `fontaineTheta_surjective` : Fontaine's θ map is surjective.\n\n## TODO\nEstablish that our definition (explicit construction of `θ mod p ^ n`) agrees with the\ndeformation-theoretic approach via the cotangent complex, as in\n[Bhatt, *Lecture notes for a class on perfectoid spaces*.\nRemark 6.1.7](https://www.math.ias.edu/~bhatt/teaching/mat679w17/lectures.pdf).\n\n## Tags\nFontaine's theta map, perfectoid theory, p-adic Hodge theory\n\n## Reference\n\n* [Fontaine, *Sur Certains Types de Représentations p-Adiques du Groupe de Galois d'un Corps Local;\n  Construction d'un Anneau de Barsotti-Tate*][fontaine1982certains]\n* [Fontaine, *Le corps des périodes p-adiques*][fontaine1994corps]\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen Ideal Quotient PreTilt WittVector\n\nnoncomputable section\n\nvariable {R : Type u} [CommRing R] {p : ℕ} [Fact p.Prime]\n\nlocal notation \"𝕎 \" A:100 => WittVector p A\nlocal notation A \"♭\" => PreTilt A p\nlocal notation3 \"𝔭\" => span {(p : R)}\n\nnamespace WittVector\n\n/-!\n## θ as a ring homomorphism\nLet `𝔭` denote the ideal of `R` generated by the prime number `p`. In this section, we first\ndefine the ring homomorphism `fontaineThetaModPPow : 𝕎 R♭ →+* R ⧸ 𝔭 ^ (n + 1)`.\nThen we show they are compatible with each other and lift to a\nring homomorphism `fontaineTheta : 𝕎 R♭ →+* R`.\n\nTo prove this, we define `fontaineThetaModPPow` as a composition of the following ring\nhomomorphisms.\n\n`𝕎 R♭ --𝕎(Frob^-n)-> 𝕎 R♭ --𝕎(coeff 0)-> 𝕎(R/𝔭) --gh_n-> R/𝔭^(n+1)`\n\nHere, the ring map `gh_n` fits in the following diagram.\n\n```\n𝕎(R)  --ghost_n->   R\n|                   |\nv                   v\n𝕎(R/𝔭) --gh_n-> R/𝔭^(n+1)\n```\n-/\n\ntheorem ker_map_le_ker_mk_comp_ghostComponent (n : ℕ) :\n    RingHom.ker (WittVector.map (Ideal.Quotient.mk 𝔭)) ≤\n    RingHom.ker (((Ideal.Quotient.mk (𝔭 ^ (n + 1)))).comp\n    (WittVector.ghostComponent (p := p) n)) := by\n  intro x\n  simp only [RingHom.mem_ker, map_eq_zero_iff, RingHom.comp_apply]\n  intro h\n  simp only [ghostComponent]\n  apply_fun Ideal.quotEquivOfEq (Ideal.span_singleton_pow _ (n + 1))\n  simp only [RingHom.coe_comp, Function.comp_apply, Pi.evalRingHom_apply, ghostMap_apply,\n    quotEquivOfEq_mk, map_zero]\n  simp only [eq_zero_iff_dvd] at h ⊢\n  exact pow_dvd_ghostComponent_of_dvd_coeff (fun _ _ ↦ h _)\n\n/--\nThe lift ring map `gh_n : 𝕎(R/𝔭) →+* R/𝔭^(n+1)` of the `n`-th ghost component\n`𝕎(R) →+* R` along the surjective ring map `𝕎(R) →+* 𝕎(R/𝔭)`.\n-/\ndef ghostComponentModPPow (n : ℕ) : 𝕎 (R ⧸ 𝔭) →+* R ⧸ 𝔭 ^ (n + 1) :=\n  RingHom.liftOfSurjective (WittVector.map (Ideal.Quotient.mk 𝔭))\n    (map_surjective _ Ideal.Quotient.mk_surjective) ⟨((Ideal.Quotient.mk (𝔭 ^ (n + 1)))).comp\n      (WittVector.ghostComponent n), ker_map_le_ker_mk_comp_ghostComponent n⟩\n\n@[simp]\ntheorem ghostComponentModPPow_map_mk (n : ℕ) (x : 𝕎 R) :\n    ghostComponentModPPow n (WittVector.map (Ideal.Quotient.mk 𝔭) x) =\n    WittVector.ghostComponent n x :=\n  RingHom.liftOfSurjective_comp_apply ..\n\n@[simp]\ntheorem quotEquivOfEq_ghostComponentModPPow (x : 𝕎 (R ⧸ 𝔭)) (h : 𝔭 ^ (0 + 1) = 𝔭) :\n    quotEquivOfEq h (ghostComponentModPPow 0 x) = ghostComponent 0 x := by\n  obtain ⟨y, hy⟩ := map_surjective _ Ideal.Quotient.mk_surjective x\n  simp [← hy, ghostComponent_apply]\n\nvariable [Fact ¬IsUnit (p : R)] [IsAdicComplete (span {(p : R)}) R]\n-- local notation 𝔭 does not work in [IsAdicComplete (span {(p : R)}) R]\n\n@[simp]\ntheorem ghostComponentModPPow_teichmuller_coeff (n : ℕ) (x : R♭) :\n    ghostComponentModPPow n (teichmuller p (PreTilt.coeff n x)) =\n    Ideal.Quotient.mk (𝔭 ^ (n + 1)) x.untilt := by\n  simpa using ghostComponentModPPow_map_mk n\n    (teichmuller p ((((_root_.frobeniusEquiv _ p).symm ^ n) x).untilt))\n\nvariable (R p) in\n/--\nThe Fontaine's theta map modulo `p^(n+1)`.\nIt is the composition of the following ring homomorphisms.\n`𝕎 R♭ --𝕎(Frob^-n)-> 𝕎 R♭ --𝕎(coeff 0)-> 𝕎(R/p) --gh_n-> R/p^(n+1)`\n-/\ndef fontaineThetaModPPow (n : ℕ) : 𝕎 R♭ →+* R ⧸ 𝔭 ^ (n + 1) :=\n  (ghostComponentModPPow n).comp (((WittVector.map (PreTilt.coeff 0))).comp\n    (WittVector.map ((_root_.frobeniusEquiv (R♭) p).symm ^ n : R♭ →+* R♭)))\n\n@[simp]\ntheorem fontaineThetaModPPow_teichmuller (n : ℕ) (x : R♭) :\n    fontaineThetaModPPow R p n (teichmuller p x) = Ideal.Quotient.mk _ x.untilt := by\n  simp [fontaineThetaModPPow]\n\ntheorem factorPowSucc_comp_fontaineThetaModPPow (n : ℕ) :\n    (factorPowSucc _ _).comp (fontaineThetaModPPow R p (n + 1)) = fontaineThetaModPPow R p n := by\n  apply eq_of_apply_teichmuller_eq ((factorPowSucc _ _).comp (fontaineThetaModPPow R p (n + 1)))\n    (fontaineThetaModPPow R p n)\n  · use n + 1\n    have : p = Ideal.Quotient.mk (𝔭 ^ (n + 1)) p := by\n      simp [map_natCast]\n    rw [this, ← map_pow, Ideal.Quotient.eq_zero_iff_mem]\n    exact Ideal.pow_mem_pow (mem_span_singleton_self _) _\n  simp [fontaineThetaModPPow]\n\ntheorem factorPowSucc_fontaineThetaModPPow_eq (n : ℕ) (x : 𝕎 R♭) :\n    factorPowSucc _ _ ((fontaineThetaModPPow R p (n + 1)) x) = fontaineThetaModPPow R p n x := by\n  simp [← factorPowSucc_comp_fontaineThetaModPPow n]\n\nopen IsAdicComplete\n\nvariable (R p) in\n/--\nThe Fontaine's θ map from `𝕎 R♭` to `R`.\nIt is the limit of the ring maps `fontaineThetaModPPow n` from `𝕎 R♭` to `R/p^(n+1)`.\n-/\ndef fontaineTheta : 𝕎 R♭ →+* R :=\n  Order.succ_strictMono.liftRingHom 𝔭 _ (factorPowSucc_comp_fontaineThetaModPPow _)\n\ntheorem mk_pow_fontaineTheta (n : ℕ) (x : 𝕎 R♭) :\n    Ideal.Quotient.mk (𝔭 ^ (n + 1)) (fontaineTheta R p x) = fontaineThetaModPPow R p n x :=\n  Order.succ_strictMono.mk_liftRingHom 𝔭 _ (factorPowSucc_comp_fontaineThetaModPPow _) x\n\ntheorem mk_fontaineTheta (x : 𝕎 R♭) :\n    Ideal.Quotient.mk 𝔭 (fontaineTheta R p x) = PreTilt.coeff 0 (x.coeff 0) := by\n  have := mk_pow_fontaineTheta 0 x\n  simp only [Nat.reduceAdd] at this\n  apply_fun Ideal.quotEquivOfEq (pow_one (p : R) ▸ Ideal.span_singleton_pow (p : R) 1) at this\n  simp only [quotEquivOfEq_mk] at this\n  rw [this]\n  simp [fontaineThetaModPPow, ghostComponent_apply, RingHom.one_def]\n\n@[simp]\n\nTarget:\ntheorem fontaineTheta_teichmuller (x : R♭) : fontaineTheta R p (teichmuller p x) = x.untilt :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Perfectoid","family_id":"fontainetheta_teichmuller","file_id":"mathlib/Mathlib/RingTheory/Perfectoid/FontaineTheta.lean","sample_id":"0884596d77461ad97de009c6f8b8d539833c319659cef7be0f8610e17da4299a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8c34499c0ebf931c33761750b78ccc8342159b860f73a3673d688956dbdeb67f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8c140a2ef00922a349e0f673ab6bb93e99770e04dd1c1e704236676b4e2767c1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"267c884cb305e890be2d32bd1dd81644507811de08af6ce289cc15754404e6fb","source_sha256":"5efd5852b126afb746be3b592d500b1af5de2649a35029fd57ec7bbb27770fcb","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor <;> rintro ⟨u, h₂⟩\n  · refine ⟨f <$$> u, ?_⟩\n    apply (h₁ u).mp h₂\n  · refine ⟨g <$$> u, ?_⟩\n    rw [h₁]\n    simp only [MvFunctor.map_map, h₀, LawfulMvFunctor.id_map, h₂]","hard_negative":true,"metrics":{"chosen_tokens":70,"rejected_tokens":3,"token_jaccard":0.027027,"token_length_ratio":0.042857},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"93e4531309605192a328a98a9a127a64aa25eb2271a781e647aae687eb70835b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.Fin2\npublic import Mathlib.Data.TypeVec\npublic import Mathlib.Logic.Equiv.Defs\n\nNamespace:\nMvFunctor\n\nLocal context:\n/-\nCopyright (c) 2018 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Mario Carneiro, Simon Hudon\n-/\n/-!\n\n# Functors between the category of tuples of types, and the category Type\n\nFeatures:\n\n* `MvFunctor n` : the type class of multivariate functors\n* `f <$$> x`    : notation for map\n\n-/\n\n@[expose] public section\n\n\nuniverse u v w\n\nopen MvFunctor\n\n/-- Multivariate functors, i.e. functor between the category of type vectors\nand the category of Type -/\nclass MvFunctor {n : ℕ} (F : TypeVec n → Type*) where\n  /-- Multivariate map, if `f : α ⟹ β` and `x : F α` then `f <$$> x : F β`. -/\n  map : ∀ {α β : TypeVec n}, α ⟹ β → F α → F β\n\n/-- Multivariate map, if `f : α ⟹ β` and `x : F α` then `f <$$> x : F β` -/\nscoped[MvFunctor] infixr:100 \" <$$> \" => MvFunctor.map\n\nvariable {n : ℕ}\n\nnamespace MvFunctor\n\nvariable {α β : TypeVec.{u} n} {F : TypeVec.{u} n → Type v} [MvFunctor F]\n\n/-- predicate lifting over multivariate functors -/\ndef LiftP {α : TypeVec n} (P : ∀ i, α i → Prop) (x : F α) : Prop :=\n  ∃ u : F (fun i => Subtype (P i)), (fun i => @Subtype.val _ (P i)) <$$> u = x\n\n/-- relational lifting over multivariate functors -/\ndef LiftR {α : TypeVec n} (R : ∀ ⦃i⦄, α i → α i → Prop) (x y : F α) : Prop :=\n  ∃ u : F (fun i => { p : α i × α i // R p.fst p.snd }),\n    (fun i (t : { p : α i × α i // R p.fst p.snd }) => t.val.fst) <$$> u = x ∧\n      (fun i (t : { p : α i × α i // R p.fst p.snd }) => t.val.snd) <$$> u = y\n\n/-- given `x : F α` and a projection `i` of type vector `α`, `supp x i` is the set\nof `α.i` contained in `x` -/\ndef supp {α : TypeVec n} (x : F α) (i : Fin2 n) : Set (α i) :=\n  { y : α i | ∀ ⦃P⦄, LiftP P x → P i y }\n\ntheorem of_mem_supp {α : TypeVec n} {x : F α} {P : ∀ ⦃i⦄, α i → Prop} (h : LiftP P x) (i : Fin2 n) :\n    ∀ y ∈ supp x i, P y := fun _y hy => hy h\n\nend MvFunctor\n\n\n\n/-- laws for `MvFunctor` -/\nclass LawfulMvFunctor {n : ℕ} (F : TypeVec n → Type*) [MvFunctor F] : Prop where\n  /-- `map` preserved identities, i.e., maps identity on `α` to identity on `F α` -/\n  id_map : ∀ {α : TypeVec n} (x : F α), TypeVec.id <$$> x = x\n  /-- `map` preserves compositions -/\n  comp_map :\n    ∀ {α β γ : TypeVec n} (g : α ⟹ β) (h : β ⟹ γ) (x : F α), (h ⊚ g) <$$> x = h <$$> g <$$> x\n\nopen Nat TypeVec\n\nnamespace MvFunctor\n\nexport LawfulMvFunctor (comp_map)\n\nopen LawfulMvFunctor\n\nvariable {α β γ : TypeVec.{u} n}\nvariable {F : TypeVec.{u} n → Type v} [MvFunctor F]\nvariable (P : α ⟹ «repeat» n Prop) (R : α ⊗ α ⟹ «repeat» n Prop)\n\n/-- adapt `MvFunctor.LiftP` to accept predicates as arrows -/\ndef LiftP' : F α → Prop :=\n  MvFunctor.LiftP fun i x => ofRepeat <| P i x\n\n\n/-- adapt `MvFunctor.LiftR` to accept relations as arrows -/\ndef LiftR' : F α → F α → Prop :=\n  MvFunctor.LiftR @fun i x y => ofRepeat <| R i <| TypeVec.prod.mk _ x y\n\nvariable [LawfulMvFunctor F]\n\n@[simp]\ntheorem id_map (x : F α) : TypeVec.id <$$> x = x :=\n  LawfulMvFunctor.id_map x\n\n@[simp]\ntheorem id_map' (x : F α) : (fun _i a => a) <$$> x = x :=\n  id_map x\n\ntheorem map_map (g : α ⟹ β) (h : β ⟹ γ) (x : F α) : h <$$> g <$$> x = (h ⊚ g) <$$> x :=\n  Eq.symm <| comp_map _ _ _\n\nsection LiftP'\n\nvariable (F) in\n\nTarget:\ntheorem exists_iff_exists_of_mono {P : F α → Prop} {q : F β → Prop}\n    (f : α ⟹ β) (g : β ⟹ α)\n    (h₀ : f ⊚ g = TypeVec.id)\n    (h₁ : ∀ u : F α, P u ↔ q (f <$$> u)) :\n    (∃ u : F α, P u) ↔ ∃ u : F β, q u :=\n\nProof body:\n","rejected":"by\n  exact exists_iff_exists_of_mono","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"cb185bb97261df45a50d1ec28e331d7692cc2a91b9d22399172cd8a6b49d2cc3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control/Functor","family_id":"exists_iff_exists_of_mono","file_id":"mathlib/Mathlib/Control/Functor/Multivariate.lean","sample_id":"267c884cb305e890be2d32bd1dd81644507811de08af6ce289cc15754404e6fb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"36e2743b3c488bb6e8cf2bf83c88a0f4aa1e0278b4cda351f399f8fa4b568278","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a3683d3fecf5a78a2e088915121e34b1d7028592a27520975260840e3084875a","source_sha256":"30e96c1aa03db850a53c7f6d12a34be8e99ae1b3e4a1595e2543d99d5c075ab6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine OpenPartialHomeomorph.ext _ _ (fun x => rfl) (fun x => rfl) ?_\n  simp [e.open_source.interior_eq, ← inter_assoc]","hard_negative":false,"metrics":{"chosen_tokens":32,"rejected_tokens":3,"token_jaccard":0.041667,"token_length_ratio":0.09375},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"93f777d45c6e104ae37303811b60c8847d91e243e0c791cb8b986b21c96b9170","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.OpenPartialHomeomorph.Continuity\n\nNamespace:\nOpenPartialHomeomorph\n\nLocal context:\n/-\nCopyright (c) 2019 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Partial homeomorphisms: Images of sets\n\n## Main definitions\n\n* `OpenPartialHomeomorph.IsImage`: predicate for when one set is an image of another\n* `OpenPartialHomeomorph.ofSet`: the identity on a set `s`\n* `OpenPartialHomeomorph.EqOnSource`: equivalence relation describing the \"right\" notion of equality\n  for open partial homeomorphisms\n\n## Implementation notes\n\nMost statements are copied from their `PartialEquiv` versions, although some care is required\nespecially when restricting to subsets, as these should be open subsets.\n\nFor design notes, see `PartialEquiv.lean`.\n\n### Local coding conventions\n\nIf a lemma deals with the intersection of a set with either source or target of a `PartialEquiv`,\nthen it should use `e.source ∩ s` or `e.target ∩ t`, not `s ∩ e.source` or `t ∩ e.target`.\n-/\n\n@[expose] public section\n\nopen Function Set Filter Topology\n\nvariable {X X' : Type*} {Y Y' : Type*} {Z Z' : Type*}\n  [TopologicalSpace X] [TopologicalSpace X'] [TopologicalSpace Y] [TopologicalSpace Y']\n  [TopologicalSpace Z] [TopologicalSpace Z']\n\nnamespace OpenPartialHomeomorph\n\nvariable (e : OpenPartialHomeomorph X Y)\n\nsection IsImage\n\n/-!\n## `OpenPartialHomeomorph.IsImage` relation\n\nWe say that `t : Set Y` is an image of `s : Set X` under an open partial homeomorphism `e` if any of\nthe following equivalent conditions hold:\n\n* `e '' (e.source ∩ s) = e.target ∩ t`;\n* `e.source ∩ e ⁻¹ t = e.source ∩ s`;\n* `∀ x ∈ e.source, e x ∈ t ↔ x ∈ s` (this one is used in the definition).\n\nThis definition is a restatement of `PartialEquiv.IsImage` for open partial homeomorphisms.\nIn this section we transfer API about `PartialEquiv.IsImage` to open partial homeomorphisms and\nadd a few `OpenPartialHomeomorph`-specific lemmas like `OpenPartialHomeomorph.IsImage.closure`.\n-/\n\n/-- We say that `t : Set Y` is an image of `s : Set X` under an open partial homeomorphism `e`\nif any of the following equivalent conditions hold:\n\n* `e '' (e.source ∩ s) = e.target ∩ t`;\n* `e.source ∩ e ⁻¹ t = e.source ∩ s`;\n* `∀ x ∈ e.source, e x ∈ t ↔ x ∈ s` (this one is used in the definition).\n-/\ndef IsImage (s : Set X) (t : Set Y) : Prop :=\n  ∀ ⦃x⦄, x ∈ e.source → (e x ∈ t ↔ x ∈ s)\n\nnamespace IsImage\n\nvariable {e} {s : Set X} {t : Set Y} {x : X} {y : Y}\n\ntheorem toPartialEquiv (h : e.IsImage s t) : e.toPartialEquiv.IsImage s t :=\n  h\n\ntheorem apply_mem_iff (h : e.IsImage s t) (hx : x ∈ e.source) : e x ∈ t ↔ x ∈ s :=\n  h hx\n\nprotected theorem symm (h : e.IsImage s t) : e.symm.IsImage t s :=\n  h.toPartialEquiv.symm\n\ntheorem symm_apply_mem_iff (h : e.IsImage s t) (hy : y ∈ e.target) : e.symm y ∈ s ↔ y ∈ t :=\n  h.symm hy\n\n@[simp]\ntheorem symm_iff : e.symm.IsImage t s ↔ e.IsImage s t :=\n  ⟨fun h => h.symm, fun h => h.symm⟩\n\nprotected theorem mapsTo (h : e.IsImage s t) : MapsTo e (e.source ∩ s) (e.target ∩ t) :=\n  h.toPartialEquiv.mapsTo\n\ntheorem symm_mapsTo (h : e.IsImage s t) : MapsTo e.symm (e.target ∩ t) (e.source ∩ s) :=\n  h.symm.mapsTo\n\ntheorem image_eq (h : e.IsImage s t) : e '' (e.source ∩ s) = e.target ∩ t :=\n  h.toPartialEquiv.image_eq\n\ntheorem symm_image_eq (h : e.IsImage s t) : e.symm '' (e.target ∩ t) = e.source ∩ s :=\n  h.symm.image_eq\n\ntheorem iff_preimage_eq : e.IsImage s t ↔ e.source ∩ e ⁻¹' t = e.source ∩ s :=\n  PartialEquiv.IsImage.iff_preimage_eq\n\nalias ⟨preimage_eq, of_preimage_eq⟩ := iff_preimage_eq\n\ntheorem iff_symm_preimage_eq : e.IsImage s t ↔ e.target ∩ e.symm ⁻¹' s = e.target ∩ t :=\n  symm_iff.symm.trans iff_preimage_eq\n\nalias ⟨symm_preimage_eq, of_symm_preimage_eq⟩ := iff_symm_preimage_eq\n\ntheorem iff_symm_preimage_eq' :\n    e.IsImage s t ↔ e.target ∩ e.symm ⁻¹' (e.source ∩ s) = e.target ∩ t := by\n  rw [iff_symm_preimage_eq, ← image_source_inter_eq, ← image_source_inter_eq']\n\nalias ⟨symm_preimage_eq', of_symm_preimage_eq'⟩ := iff_symm_preimage_eq'\n\ntheorem iff_preimage_eq' : e.IsImage s t ↔ e.source ∩ e ⁻¹' (e.target ∩ t) = e.source ∩ s :=\n  symm_iff.symm.trans iff_symm_preimage_eq'\n\nalias ⟨preimage_eq', of_preimage_eq'⟩ := iff_preimage_eq'\n\ntheorem of_image_eq (h : e '' (e.source ∩ s) = e.target ∩ t) : e.IsImage s t :=\n  PartialEquiv.IsImage.of_image_eq h\n\ntheorem of_symm_image_eq (h : e.symm '' (e.target ∩ t) = e.source ∩ s) : e.IsImage s t :=\n  PartialEquiv.IsImage.of_symm_image_eq h\n\nprotected theorem compl (h : e.IsImage s t) : e.IsImage sᶜ tᶜ := fun _ hx => (h hx).not\n\nprotected theorem inter {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :\n    e.IsImage (s ∩ s') (t ∩ t') := fun _ hx => (h hx).and (h' hx)\n\nprotected theorem union {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :\n    e.IsImage (s ∪ s') (t ∪ t') := fun _ hx => (h hx).or (h' hx)\n\nprotected theorem diff {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :\n    e.IsImage (s \\ s') (t \\ t') :=\n  h.inter h'.compl\n\ntheorem leftInvOn_piecewise {e' : OpenPartialHomeomorph X Y} [∀ i, Decidable (i ∈ s)]\n    [∀ i, Decidable (i ∈ t)] (h : e.IsImage s t) (h' : e'.IsImage s t) :\n    LeftInvOn (t.piecewise e.symm e'.symm) (s.piecewise e e') (s.ite e.source e'.source) :=\n  h.toPartialEquiv.leftInvOn_piecewise h'\n\ntheorem inter_eq_of_inter_eq_of_eqOn {e' : OpenPartialHomeomorph X Y} (h : e.IsImage s t)\n    (h' : e'.IsImage s t) (hs : e.source ∩ s = e'.source ∩ s) (Heq : EqOn e e' (e.source ∩ s)) :\n    e.target ∩ t = e'.target ∩ t :=\n  h.toPartialEquiv.inter_eq_of_inter_eq_of_eqOn h' hs Heq\n\ntheorem symm_eqOn_of_inter_eq_of_eqOn {e' : OpenPartialHomeomorph X Y} (h : e.IsImage s t)\n    (hs : e.source ∩ s = e'.source ∩ s) (Heq : EqOn e e' (e.source ∩ s)) :\n    EqOn e.symm e'.symm (e.target ∩ t) :=\n  h.toPartialEquiv.symm_eq_on_of_inter_eq_of_eqOn hs Heq\n\ntheorem map_nhdsWithin_eq (h : e.IsImage s t) (hx : x ∈ e.source) : map e (𝓝[s] x) = 𝓝[t] e x := by\n  rw [e.map_nhdsWithin_eq hx, h.image_eq, e.nhdsWithin_target_inter (e.map_source hx)]\n\nprotected theorem closure (h : e.IsImage s t) : e.IsImage (closure s) (closure t) := fun x hx => by\n  simp only [mem_closure_iff_nhdsWithin_neBot, ← h.map_nhdsWithin_eq hx, map_neBot_iff]\n\nprotected theorem interior (h : e.IsImage s t) : e.IsImage (interior s) (interior t) := by\n  simpa only [closure_compl, compl_compl] using h.compl.closure.compl\n\nprotected theorem frontier (h : e.IsImage s t) : e.IsImage (frontier s) (frontier t) :=\n  h.closure.diff h.interior\n\ntheorem isOpen_iff (h : e.IsImage s t) : IsOpen (e.source ∩ s) ↔ IsOpen (e.target ∩ t) :=\n  ⟨fun hs => h.symm_preimage_eq' ▸ e.symm.isOpen_inter_preimage hs, fun hs =>\n    h.preimage_eq' ▸ e.isOpen_inter_preimage hs⟩\n\n/-- Restrict an `OpenPartialHomeomorph` to a pair of corresponding open sets. -/\n@[simps! -fullyApplied apply symm_apply toPartialHomeomorph]\ndef restr (h : e.IsImage s t) (hs : IsOpen (e.source ∩ s)) : OpenPartialHomeomorph X Y where\n  toPartialEquiv := h.toPartialEquiv.restr\n  open_source := hs\n  open_target := h.isOpen_iff.1 hs\n  continuousOn_toFun := e.continuousOn.mono inter_subset_left\n  continuousOn_invFun := e.symm.continuousOn.mono inter_subset_left\n\nend IsImage\n\ntheorem isImage_source_target : e.IsImage e.source e.target :=\n  e.toPartialEquiv.isImage_source_target\n\ntheorem isImage_source_target_of_disjoint (e' : OpenPartialHomeomorph X Y)\n    (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) :\n    e.IsImage e'.source e'.target :=\n  e.toPartialEquiv.isImage_source_target_of_disjoint e'.toPartialEquiv hs ht\n\n/-- Preimage of interior or interior of preimage coincide for open partial homeomorphisms,\nwhen restricted to the source. -/\ntheorem preimage_interior (s : Set Y) :\n    e.source ∩ e ⁻¹' interior s = e.source ∩ interior (e ⁻¹' s) :=\n  (IsImage.of_preimage_eq rfl).interior.preimage_eq\n\ntheorem preimage_closure (s : Set Y) : e.source ∩ e ⁻¹' closure s = e.source ∩ closure (e ⁻¹' s) :=\n  (IsImage.of_preimage_eq rfl).closure.preimage_eq\n\ntheorem preimage_frontier (s : Set Y) :\n    e.source ∩ e ⁻¹' frontier s = e.source ∩ frontier (e ⁻¹' s) :=\n  (IsImage.of_preimage_eq rfl).frontier.preimage_eq\n\nend IsImage\n\n\nsection restrOpen\n/-!\n## Restriction\n-/\n\n/-- Restricting an open partial homeomorphism `e` to `e.source ∩ s` when `s` is open.\nThis is sometimes hard to use because of the openness assumption, but it has the advantage that\nwhen it can be used then its `PartialEquiv` is defeq to `PartialEquiv.restr`. -/\nprotected def restrOpen (s : Set X) (hs : IsOpen s) : OpenPartialHomeomorph X Y :=\n  (@IsImage.of_symm_preimage_eq X Y _ _ e s (e.symm ⁻¹' s) rfl).restr\n    (IsOpen.inter e.open_source hs)\n\n@[simp, mfld_simps]\ntheorem restrOpen_toPartialEquiv (s : Set X) (hs : IsOpen s) :\n    (e.restrOpen s hs).toPartialEquiv = e.toPartialEquiv.restr s :=\n  rfl\n\n-- Already simp via `PartialEquiv`\ntheorem restrOpen_source (s : Set X) (hs : IsOpen s) : (e.restrOpen s hs).source = e.source ∩ s :=\n  rfl\n\n@[simp] theorem coe_restrOpen {s : Set X} (hs : IsOpen s) : ⇑(e.restrOpen s hs) = e := rfl\n\n@[simp]\ntheorem coe_restrOpen_symm {s : Set X} (hs : IsOpen s) : ⇑(e.restrOpen s hs).symm = e.symm := rfl\n\n/-- Restricting an open partial homeomorphism `e` to `e.source ∩ interior s`. We use the interior to\nmake sure that the restriction is well defined whatever the set s, since open partial homeomorphisms\nare by definition defined on open sets. In applications where `s` is open, this coincides with the\nrestriction of partial equivalences. -/\n@[simps! (attr := mfld_simps) -fullyApplied apply symm_apply,\n  simps! (attr := grind =) -isSimp source target]\nprotected def restr (s : Set X) : OpenPartialHomeomorph X Y :=\n  e.restrOpen (interior s) isOpen_interior\n\n@[simp, mfld_simps]\ntheorem restr_toPartialEquiv (s : Set X) :\n    (e.restr s).toPartialEquiv = e.toPartialEquiv.restr (interior s) :=\n  rfl\n\ntheorem restr_source' (s : Set X) (hs : IsOpen s) : (e.restr s).source = e.source ∩ s := by\n  grind\n\ntheorem restr_toPartialEquiv' (s : Set X) (hs : IsOpen s) :\n    (e.restr s).toPartialEquiv = e.toPartialEquiv.restr s := by\n  rw [e.restr_toPartialEquiv, hs.interior_eq]\n\ntheorem restr_eq_of_source_subset {e : OpenPartialHomeomorph X Y} {s : Set X} (h : e.source ⊆ s) :\n    e.restr s = e :=\n  toPartialEquiv_injective <| PartialEquiv.restr_eq_of_source_subset <|\n    interior_maximal h e.open_source\n\n@[simp, mfld_simps]\ntheorem restr_univ {e : OpenPartialHomeomorph X Y} : e.restr univ = e :=\n  restr_eq_of_source_subset (subset_univ _)\n\n@[simp, grind =]\n\nTarget:\ntheorem restr_source_inter (s : Set X) : e.restr (e.source ∩ s) = e.restr s :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/OpenPartialHomeomorph","family_id":"restr_source_inter","file_id":"mathlib/Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean","sample_id":"a3683d3fecf5a78a2e088915121e34b1d7028592a27520975260840e3084875a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"230ad1a22636c034f298a25415dfe821b738da09682e6993796fa2ce84eb0a0a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"977e118700d760b10bf27f87641e8ca5e4f2c391faefcb40db1140feefdc4453","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"16bdb0569806200171e13b3e0d9b0d935a38c6f276941a0946af8ff38e8cee10","source_sha256":"25ee7ccac39d94bf37419f1553f1c5ec483939108b5bb4af754da5f173eef853","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← nsmul_one, D.map_smul_of_tower n, map_one_eq_zero, smul_zero]","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":5,"token_jaccard":0.125,"token_length_ratio":0.333333},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"942452b9c7e5dc65573672e93beca690224414f6371b92fa26fcf0a06c49e9c9","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.AlgebraMap\npublic import Mathlib.Algebra.Polynomial.Derivative\n\nNamespace:\nDerivation\n\nLocal context:\n/-\nCopyright (c) 2020 Nicolò Cavalleri. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Nicolò Cavalleri, Andrew Yang\n-/\n/-!\n# Derivations\n\nThis file defines derivation. A derivation `D` from the `R`-algebra `A` to the `A`-module `M` is an\n`R`-linear map that satisfy the Leibniz rule `D (a * b) = a * D b + D a * b`.\n\n## Main results\n\n- `Derivation`: The type of `R`-derivations from `A` to `M`. This has an `A`-module structure.\n- `Derivation.llcomp`: We may compose linear maps and derivations to obtain a derivation,\n  and the composition is bilinear.\n\nSee `Mathlib/RingTheory/Derivation/Lie.lean` for\n- `Derivation.instLieAlgebra`: The `R`-derivations from `A` to `A` form a Lie algebra over `R`.\n\nand `Mathlib/RingTheory/Derivation/ToSquareZero.lean` for\n- `derivationToSquareZeroEquivLift`: The `R`-derivations from `A` into a square-zero ideal `I`\n  of `B` corresponds to the lifts `A →ₐ[R] B` of the map `A →ₐ[R] B ⧸ I`.\n\n## Future project\n\n- Generalize derivations into bimodules.\n\n-/\n\n@[expose] public section\n\nopen Algebra\n\n/-- `D : Derivation R A M` is an `R`-linear map from `A` to `M` that satisfies the `leibniz`\nequality. We also require that `D 1 = 0`. See `Derivation.mk'` for a constructor that deduces this\nassumption from the Leibniz rule when `M` is cancellative.\n\nTODO: update this when bimodules are defined. -/\nstructure Derivation (R : Type*) (A : Type*) (M : Type*)\n    [CommSemiring R] [CommSemiring A] [AddCommMonoid M] [Algebra R A] [Module A M] [Module R M]\n    extends A →ₗ[R] M where\n  protected map_one_eq_zero' : toLinearMap 1 = 0\n  protected leibniz' (a b : A) : toLinearMap (a * b) = a • toLinearMap b + b • toLinearMap a\n\n/-- The `LinearMap` underlying a `Derivation`. -/\nadd_decl_doc Derivation.toLinearMap\n\nnamespace Derivation\n\nsection\n\nvariable {R : Type*} {A : Type*} {B : Type*} {M : Type*}\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B] [AddCommMonoid M]\nvariable [Algebra R A] [Algebra R B]\nvariable [Module A M] [Module B M] [Module R M]\n\n\nvariable (D : Derivation R A M) {D1 D2 : Derivation R A M} (r : R) (a b : A)\n\ninstance : FunLike (Derivation R A M) A M where\n  coe D := D.toFun\n  coe_injective D1 D2 h := by cases D1; cases D2; congr; exact DFunLike.coe_injective h\n\ninstance : AddMonoidHomClass (Derivation R A M) A M where\n  map_add D := D.toLinearMap.map_add'\n  map_zero D := D.toLinearMap.map_zero\n\n-- Not a simp lemma because it can be proved via `coeFn_coe` + `toLinearMap_eq_coe`\ntheorem toFun_eq_coe : D.toFun = ⇑D :=\n  rfl\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply (D : Derivation R A M) : A → M := D\n\ninitialize_simps_projections Derivation (toFun → apply)\n\nattribute [coe] toLinearMap\n\ninstance hasCoeToLinearMap : Coe (Derivation R A M) (A →ₗ[R] M) :=\n  ⟨fun D => D.toLinearMap⟩\n\n@[simp]\ntheorem mk_coe (f : A →ₗ[R] M) (h₁ h₂) : ((⟨f, h₁, h₂⟩ : Derivation R A M) : A → M) = f :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coeFn_coe (f : Derivation R A M) : ⇑(f : A →ₗ[R] M) = f :=\n  rfl\n\ntheorem coe_injective : @Function.Injective (Derivation R A M) (A → M) DFunLike.coe :=\n  DFunLike.coe_injective\n\n@[ext]\ntheorem ext (H : ∀ a, D1 a = D2 a) : D1 = D2 :=\n  DFunLike.ext _ _ H\n\ntheorem congr_fun (h : D1 = D2) (a : A) : D1 a = D2 a :=\n  DFunLike.congr_fun h a\n\nprotected theorem map_add : D (a + b) = D a + D b :=\n  map_add D a b\n\nprotected theorem map_zero : D 0 = 0 :=\n  map_zero D\n\n@[simp]\ntheorem map_smul : D (r • a) = r • D a :=\n  D.toLinearMap.map_smul r a\n\n@[simp]\ntheorem leibniz : D (a * b) = a • D b + b • D a :=\n  D.leibniz' _ _\n\n@[simp]\ntheorem map_smul_of_tower {S : Type*} [SMul S A] [SMul S M] [LinearMap.CompatibleSMul A M S R]\n    (D : Derivation R A M) (r : S) (a : A) : D (r • a) = r • D a :=\n  D.toLinearMap.map_smul_of_tower r a\n\n@[simp]\ntheorem map_one_eq_zero : D 1 = 0 :=\n  D.map_one_eq_zero'\n\n@[simp]\ntheorem map_algebraMap : D (algebraMap R A r) = 0 := by\n  rw [← mul_one r, map_mul, map_one, ← smul_def, map_smul, map_one_eq_zero, smul_zero]\n\n@[simp]\n\nTarget:\ntheorem map_natCast (n : ℕ) : D (n : A) = 0 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_16bdb0569806","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1406c1d6d59b2bd47e9ab3099314aab3ff0677f666c8ff57863ae067cb244c1a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Derivation","family_id":"map_natcast","file_id":"mathlib/Mathlib/RingTheory/Derivation/Basic.lean","sample_id":"16bdb0569806200171e13b3e0d9b0d935a38c6f276941a0946af8ff38e8cee10"},"split":"train"}
{"binding":{"chosen_proof_sha256":"96180be036a5aa5c1b27a5d6a0c89b53acf0cf1ee820da9c2ba106ab55ce193e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"39bb38d8237a1e91907a351e01371e5a019c7e54e6d2d170cec7b49921fb560e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1daa9f465ccaa4038e45b7b45c48af664510fd77f6ef72028805211ad957a3be","source_sha256":"089de13c5349089f72463c4ad215460456301c821effdb48ac9998fc0a8a4b16","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold traverse\n  simp only [Functor.map_map, fmap_def, map_comp_coe_apply, Multiset.fmap_def, ←\n    Multiset.map_traverse]","hard_negative":true,"metrics":{"chosen_tokens":23,"rejected_tokens":3,"token_jaccard":0.117647,"token_length_ratio":0.130435},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"942d06e75eed46f68f0e61a9e78f8d790bd50ef4f174e4a05e2f26dd3cdc5c89","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Control.AlternativeMonad\npublic import Mathlib.Data.Finset.Lattice.Union\npublic import Mathlib.Data.Finset.NAry\npublic import Mathlib.Data.Multiset.Functor\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2021 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Kim Morrison\n-/\n/-!\n# Functoriality of `Finset`\n\nThis file defines the functor structure of `Finset`.\n\n## TODO\n\nCurrently, all instances are classical because the functor classes want to run over all types. If\ninstead we could state that a functor is lawful/applicative/traversable... between two given types,\nthen we could provide the instances for types with decidable equality.\n-/\n\n@[expose] public section\n\n\nuniverse u\n\nopen Function\n\nnamespace Finset\n\n/-! ### Functor -/\n\nsection Functor\n\nvariable {α β : Type u} [∀ P, Decidable P]\n\n/-- Because `Finset.image` requires a `DecidableEq` instance for the target type, we can only\nconstruct `Functor Finset` when working classically. -/\nprotected instance functor : Functor Finset where map f s := s.image f\n\ninstance lawfulFunctor : LawfulFunctor Finset where\n  id_map _ := image_id\n  comp_map _ _ _ := image_image.symm\n  map_const {α} {β} := by simp only [Functor.mapConst, Functor.map]\n\n@[simp]\ntheorem fmap_def {s : Finset α} (f : α → β) : f <$> s = s.image f := rfl\n\nend Functor\n\n/-! ### Pure -/\n\n\nprotected instance pure : Pure Finset :=\n  ⟨fun x => {x}⟩\n\n@[simp]\ntheorem pure_def {α} : (pure : α → Finset α) = singleton := rfl\n\n/-! ### Applicative functor -/\n\n\nsection Applicative\n\nvariable {α β : Type u} [∀ P, Decidable P]\n\nprotected instance applicative : Applicative Finset :=\n  { Finset.functor, Finset.pure with\n    seq := fun t s => t.sup fun f => (s ()).image f\n    seqLeft := fun s t => if t () = ∅ then ∅ else s\n    seqRight := fun s t => if s = ∅ then ∅ else t () }\n\n@[simp]\ntheorem seq_def (s : Finset α) (t : Finset (α → β)) : t <*> s = t.sup fun f => s.image f :=\n  rfl\n\n@[simp]\ntheorem seqLeft_def (s : Finset α) (t : Finset β) : s <* t = if t = ∅ then ∅ else s :=\n  rfl\n\n@[simp]\ntheorem seqRight_def (s : Finset α) (t : Finset β) : s *> t = if s = ∅ then ∅ else t :=\n  rfl\n\n/-- `Finset.image₂` in terms of monadic operations. Note that this can't be taken as the definition\nbecause of the lack of universe polymorphism. -/\ntheorem image₂_def {α β γ : Type u} (f : α → β → γ) (s : Finset α) (t : Finset β) :\n    image₂ f s t = f <$> s <*> t := by\n  ext\n  simp [mem_sup]\n\ninstance lawfulApplicative : LawfulApplicative Finset :=\n  { Finset.lawfulFunctor with\n    seqLeft_eq := fun s t => by\n      rw [seq_def, fmap_def, seqLeft_def]\n      obtain rfl | ht := t.eq_empty_or_nonempty\n      · simp_rw [image_empty, if_true]\n        exact (sup_bot _).symm\n      · ext a\n        rw [if_neg ht.ne_empty, mem_sup]\n        refine ⟨fun ha => ⟨const _ a, mem_image_of_mem _ ha, mem_image_const_self.2 ht⟩, ?_⟩\n        rintro ⟨f, hf, ha⟩\n        rw [mem_image] at hf ha\n        obtain ⟨b, hb, rfl⟩ := hf\n        obtain ⟨_, _, rfl⟩ := ha\n        exact hb\n    seqRight_eq := fun s t => by\n      rw [seq_def, fmap_def, seqRight_def]\n      obtain rfl | hs := s.eq_empty_or_nonempty\n      · rw [if_pos rfl, image_empty, sup_empty, bot_eq_empty]\n      · ext a\n        rw [if_neg hs.ne_empty, mem_sup]\n        refine ⟨fun ha => ⟨id, mem_image_const_self.2 hs, by rwa [image_id]⟩, ?_⟩\n        rintro ⟨f, hf, ha⟩\n        rw [mem_image] at hf ha\n        obtain ⟨b, hb, rfl⟩ := ha\n        obtain ⟨_, _, rfl⟩ := hf\n        exact hb\n    pure_seq := fun f s => by simp only [pure_def, seq_def, sup_singleton, fmap_def]\n    map_pure := fun _ _ => image_singleton _ _\n    seq_pure := fun _ _ => sup_singleton_apply _ _\n    seq_assoc := fun s t u => by\n      ext a\n      simp_rw [seq_def, fmap_def]\n      simp only [mem_sup, mem_image]\n      constructor\n      · rintro ⟨g, hg, b, ⟨f, hf, a, ha, rfl⟩, rfl⟩\n        exact ⟨g ∘ f, ⟨comp g, ⟨g, hg, rfl⟩, f, hf, rfl⟩, a, ha, rfl⟩\n      · rintro ⟨c, ⟨_, ⟨g, hg, rfl⟩, f, hf, rfl⟩, a, ha, rfl⟩\n        exact ⟨g, hg, f a, ⟨f, hf, a, ha, rfl⟩, rfl⟩ }\n\ninstance commApplicative : CommApplicative Finset :=\n  { Finset.lawfulApplicative with\n    commutative_prod := fun s t => by\n      simp_rw [seq_def, fmap_def, sup_image, sup_eq_biUnion]\n      change (s.biUnion fun a => t.image fun b => (a, b))\n        = t.biUnion fun b => s.image fun a => (a, b)\n      trans s ×ˢ t <;> [rw [product_eq_biUnion]; rw [product_eq_biUnion_right]] }\n\nend Applicative\n\n/-! ### Monad -/\n\n\nsection Monad\n\nvariable [∀ P, Decidable P]\n\ninstance : Monad Finset :=\n  { Finset.applicative with bind := sup }\n\n@[simp]\ntheorem bind_def {α β} : (· >>= ·) = sup (α := Finset α) (β := β) :=\n  rfl\n\ninstance : LawfulMonad Finset :=\n  { Finset.lawfulApplicative with\n    bind_pure_comp := fun _ _ => sup_singleton_apply _ _\n    bind_map := fun _ _ => rfl\n    pure_bind := fun _ _ => sup_singleton\n    bind_assoc := fun s f g => by simp only [bind, sup_eq_biUnion, biUnion_biUnion] }\n\nend Monad\n\n/-! ### Alternative functor -/\n\n\nsection Alternative\n\nvariable [∀ P, Decidable P]\n\ninstance : AlternativeMonad Finset where\n  orElse s t := s ∪ t ()\n  failure := ∅\n\ninstance : LawfulAlternative Finset where\n  map_failure _ := Finset.image_empty _\n  failure_seq _ := Finset.sup_empty\n  orElse_failure _ := Finset.union_empty _\n  failure_orElse _ := Finset.empty_union _\n  orElse_assoc _ _ _ := Finset.union_assoc _ _ _ |>.symm\n  map_orElse _ _ _ := Finset.image_union _ _\n\nend Alternative\n\n/-! ### Traversable functor -/\n\n\nsection Traversable\n\nvariable {α β γ : Type u} {F G : Type u → Type u} [Applicative F] [Applicative G]\n  [CommApplicative F] [CommApplicative G]\n\n/-- Traverse function for `Finset`. -/\ndef traverse [DecidableEq β] (f : α → F β) (s : Finset α) : F (Finset β) :=\n  Multiset.toFinset <$> Multiset.traverse f s.1\n\n@[simp]\ntheorem id_traverse [DecidableEq α] (s : Finset α) : traverse (pure : α → Id α) s = pure s := by\n  rw [traverse, Multiset.id_traverse]\n  exact s.val_toFinset\n\nopen scoped Classical in\n@[simp]\ntheorem map_comp_coe (h : α → β) :\n    Functor.map h ∘ Multiset.toFinset = Multiset.toFinset ∘ Functor.map h :=\n  funext fun _ => image_toFinset\n\nopen scoped Classical in\n@[simp]\ntheorem map_comp_coe_apply (h : α → β) (s : Multiset α) :\n    s.toFinset.image h = (h <$> s).toFinset :=\n  congrFun (map_comp_coe h) s\n\nopen scoped Classical in\n\nTarget:\ntheorem map_traverse (g : α → G β) (h : β → γ) (s : Finset α) :\n    Functor.map h <$> traverse g s = traverse (Functor.map h ∘ g) s :=\n\nProof body:\n","rejected":"by\n  exact map_traverse","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"c44e8da96e746b4df7a03133b15fb3addc95b1b7f2d5d5908c42221cd369ccfe","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finset","family_id":"map_traverse","file_id":"mathlib/Mathlib/Data/Finset/Functor.lean","sample_id":"1daa9f465ccaa4038e45b7b45c48af664510fd77f6ef72028805211ad957a3be"},"split":"train"}
{"binding":{"chosen_proof_sha256":"69cb17d2f43d6924d494a8be87b9a52df302f4a4d1c3a416993a289830e67b47","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c2d4f1022a58d69b30e8702a765bce86c348d7763c7eb21b990dbfec6f34049e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2c9653db60a3ddae63cedc1cc9e2ba6e8d6ce8800e7284483f64df67132c5de6","source_sha256":"368f0cb56789c53c6820ea0e367b83289d651593fc19d224d3364287d2adc7a4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases hR : R = 0\n  · simp_all\n  have H : ContinuousOn (circleAverage f c) (Set.Ioc 0 |R|) := by\n    refine (h₁f.2.mono ?_).circleAverage (fun z hz ↦ hz.1.le)\n    intro x hx\n    rw [closure_ball _ (by aesop), mem_closedBall_iff_norm]\n    exact hx.2\n  rw [← circleAverage_abs_radius]\n  apply H.eq_of_eqOn_Ioo (by aesop)\n  · intro r hr\n    apply HarmonicOnNhd.circleAverage_eq\n    · apply h₁f.1.mono\n      rw [abs_of_pos hr.1]\n      exact closedBall_subset_ball hr.2","hard_negative":true,"metrics":{"chosen_tokens":114,"rejected_tokens":5,"token_jaccard":0.051724,"token_length_ratio":0.04386},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"944cdb2952f47c03bb39cb7e8557f652cd05e2c3393ec3cac075c2a707bc59f8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Complex.Harmonic.Analytic\npublic import Mathlib.Analysis.Complex.MeanValue\npublic import Mathlib.Analysis.InnerProductSpace.Harmonic.HarmonicContOnCl\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Stefan Kebekus. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stefan Kebekus\n-/\n/-!\n# The Mean Value Property of Harmonic Functions on the Complex Plane\n-/\n\npublic section\n\nopen InnerProductSpace Metric Real\n\nvariable {f : ℂ → ℝ} {c : ℂ} {R : ℝ}\n\n/--\nThe **Mean Value Property** of harmonic functions: If `f : ℂ → ℝ` is harmonic in a neighborhood of a\nclosed disc of radius `R` and center `c`, then the circle average `circleAverage f c R` equals\n`f c`.\n-/\ntheorem HarmonicOnNhd.circleAverage_eq (hf : HarmonicOnNhd f (closedBall c |R|)) :\n    circleAverage f c R = f c := by\n  obtain ⟨e, h₁e, h₂e⟩ := (isCompact_closedBall c |R|).exists_thickening_subset_open\n    (isOpen_setOf_harmonicAt f) hf\n  rw [thickening_closedBall h₁e (abs_nonneg R)] at h₂e\n  obtain ⟨F, h₁F, h₂F⟩ := InnerProductSpace.HarmonicOnNhd.exists_analyticOnNhd_ball_re_eq h₂e\n  have h₃F : DifferentiableOn ℂ F (closure (ball c |R|)) := by\n    intro x hx\n    apply (h₁F x _).differentiableWithinAt\n    grind [mem_ball, mem_closedBall.1 (closure_ball_subset_closedBall hx)]\n  have h₄F : Set.EqOn (Complex.reCLM ∘ F) f (sphere c |R|) :=\n    fun x hx ↦ h₂F (sphere_subset_ball (lt_add_of_pos_left |R| h₁e) hx)\n  rw [← circleAverage_congr_sphere h₄F, Complex.reCLM.circleAverage_comp_comm,\n    h₃F.diffContOnCl.circleAverage]\n  · apply h₂F\n    simp [mem_ball, dist_self, add_pos_of_pos_of_nonneg h₁e (abs_nonneg R)]\n  · apply (h₁F.continuousOn.mono (fun _ _ ↦ by simp_all [dist_eq_norm])).circleIntegrable'\n\n/--\nThe **Mean Value Property** of harmonic functions: If `f : ℂ → ℝ` is harmonic on a disc of radius\n`|R|` and center `c` and continuous on its closure, then the circle average `circleAverage f c R`\nequals `f c`.\n-/\n\nTarget:\ntheorem HarmonicContOnCl.circleAverage_eq {f : ℂ → ℝ} {c : ℂ} {R : ℝ}\n    (h₁f : HarmonicContOnCl f (ball c |R|)) :\n    circleAverage f c R = f c :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_2c9653db60a3","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"3d104c18db9b972e50396b9a392cda3ff9b2c9dded66cd0da7d76145b7678814","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Complex","family_id":"harmoniccontoncl","file_id":"mathlib/Mathlib/Analysis/Complex/Harmonic/MeanValue.lean","sample_id":"2c9653db60a3ddae63cedc1cc9e2ba6e8d6ce8800e7284483f64df67132c5de6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"536871cc5d7d8fd19b1680026dd5fcc3bae6cf4953adabbd17e9ab3ecd407260","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7fa1587a449af9bd025ff7fcf8e9e6160d9d026e5e59b7be7c90ab324dc63b3b","source_sha256":"ba493885a2be935359e1e232e1bbaeb68ad80f382975839ead7d3190a1fce95f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h ↦ ⟨?_, ?_, ?_⟩, fun h ↦ ?_⟩\n  any_goals rintro a b rfl rfl; exact map_eq_empty.1 h\n  · rintro a b rfl rfl; exact disjSum_eq_empty.1 h\n  cases a <;> cases b\n  · exact map_eq_empty.2 (h.1 _ _ rfl rfl)\n  · simp [h.2.1 _ _ rfl rfl]\n  · rfl\n  · exact map_eq_empty.2 (h.2.2 _ _ rfl rfl)","hard_negative":true,"metrics":{"chosen_tokens":99,"rejected_tokens":8,"token_jaccard":0.027027,"token_length_ratio":0.080808},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"946357e3723c2286b6e2e31701f41d9cf43873c3591fae492ea919389abe64bc","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Sum\npublic import Mathlib.Data.Sum.Order\npublic import Mathlib.Order.Interval.Finset.Defs\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Finite intervals in a disjoint union\n\nThis file provides the `LocallyFiniteOrder` instance for the disjoint sum and linear sum of two\norders and calculates the cardinality of their finite intervals.\n-/\n\n@[expose] public section\n\n\nopen Function Sum\n\nnamespace Finset\n\nvariable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}\n\nsection SumLift₂\n\nvariable (f f₁ g₁ : α₁ → β₁ → Finset γ₁) (g f₂ g₂ : α₂ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁` and `α₂ → β₂ → Finset γ₂` to a map\n`α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to `Alternative` functors if we can\nmake sure to keep computability and universe polymorphism. -/\n@[simp]\ndef sumLift₂ : ∀ (_ : α₁ ⊕ α₂) (_ : β₁ ⊕ β₂), Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f a b).map Embedding.inl\n  | inl _, inr _ => ∅\n  | inr _, inl _ => ∅\n  | inr a, inr b => (g a b).map Embedding.inr\n\nvariable {f f₁ g₁ g f₂ g₂} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\ntheorem mem_sumLift₂ :\n    c ∈ sumLift₂ f g a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨\n        ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g a₂ b₂ := by\n  constructor\n  · rcases a with a | a <;> rcases b with b | b\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr ⟨a, b, c, rfl, rfl, rfl, hc⟩\n  · rintro (⟨a, b, c, rfl, rfl, rfl, h⟩ | ⟨a, b, c, rfl, rfl, rfl, h⟩) <;> exact mem_map_of_mem _ h\n\ntheorem inl_mem_sumLift₂ {c₁ : γ₁} :\n    inl c₁ ∈ sumLift₂ f g a b ↔ ∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f a₁ b₁ := by\n  rw [mem_sumLift₂, or_iff_left]\n  · simp only [inl.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inl_ne_inr h\n\ntheorem inr_mem_sumLift₂ {c₂ : γ₂} :\n    inr c₂ ∈ sumLift₂ f g a b ↔ ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ g a₂ b₂ := by\n  rw [mem_sumLift₂, or_iff_right]\n  · simp only [inr.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inr_ne_inl h\n\ntheorem sumLift₂_eq_empty :\n    sumLift₂ f g a b = ∅ ↔\n      (∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f a₁ b₁ = ∅) ∧\n        ∀ a₂ b₂, a = inr a₂ → b = inr b₂ → g a₂ b₂ = ∅ := by\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · constructor <;>\n    · rintro a b rfl rfl\n      exact map_eq_empty.1 h\n  cases a <;> cases b\n  · exact map_eq_empty.2 (h.1 _ _ rfl rfl)\n  · rfl\n  · rfl\n  · exact map_eq_empty.2 (h.2 _ _ rfl rfl)\n\ntheorem sumLift₂_nonempty :\n    (sumLift₂ f g a b).Nonempty ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ (f a₁ b₁).Nonempty) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ (g a₂ b₂).Nonempty := by\n  simp only [nonempty_iff_ne_empty, Ne, sumLift₂_eq_empty, not_and_or, not_forall, exists_prop]\n\ntheorem sumLift₂_mono (h₁ : ∀ a b, f₁ a b ⊆ g₁ a b) (h₂ : ∀ a b, f₂ a b ⊆ g₂ a b) :\n    ∀ a b, sumLift₂ f₁ f₂ a b ⊆ sumLift₂ g₁ g₂ a b\n  | inl _, inl _ => map_subset_map.2 (h₁ _ _)\n  | inl _, inr _ => Subset.rfl\n  | inr _, inl _ => Subset.rfl\n  | inr _, inr _ => map_subset_map.2 (h₂ _ _)\n\nend SumLift₂\n\nsection SumLexLift\nvariable (f₁ f₁' : α₁ → β₁ → Finset γ₁) (f₂ f₂' : α₂ → β₂ → Finset γ₂)\n  (g₁ g₁' : α₁ → β₂ → Finset γ₁) (g₂ g₂' : α₁ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁`, `α₂ → β₂ → Finset γ₂`, `α₁ → β₂ → Finset γ₁`,\n`α₂ → β₂ → Finset γ₂`  to a map `α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to\nalternative monads if we can make sure to keep computability and universe polymorphism. -/\ndef sumLexLift : α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f₁ a b).map Embedding.inl\n  | inl a, inr b => (g₁ a b).disjSum (g₂ a b)\n  | inr _, inl _ => ∅\n  | inr a, inr b => (f₂ a b).map ⟨_, inr_injective⟩\n\n@[simp]\nlemma sumLexLift_inl_inl (a : α₁) (b : β₁) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inl b) = (f₁ a b).map Embedding.inl := rfl\n\n@[simp]\nlemma sumLexLift_inl_inr (a : α₁) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b) = (g₁ a b).disjSum (g₂ a b) := rfl\n\n@[simp]\nlemma sumLexLift_inr_inl (a : α₂) (b : β₁) : sumLexLift f₁ f₂ g₁ g₂ (inr a) (inl b) = ∅ := rfl\n\n@[simp]\nlemma sumLexLift_inr_inr (a : α₂) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inr a) (inr b) = (f₂ a b).map ⟨_, inr_injective⟩ := rfl\n\nvariable {f₁ g₁ f₂ g₂ f₁' g₁' f₂' g₂'} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\nlemma mem_sumLexLift :\n    c ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        (∃ a₁ b₂ c₁, a = inl a₁ ∧ b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨\n          (∃ a₁ b₂ c₂, a = inl a₁ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n            ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂ := by\n  constructor\n  · obtain a | a := a <;> obtain b | b := b\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (mem_disjSum.1 h).elim ?_ ?_\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inr <| Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr (Or.inr <| Or.inr <| ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n  · rintro (⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩ |\n      ⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact mem_map_of_mem _ hc\n    · exact inl_mem_disjSum.2 hc\n    · exact inr_mem_disjSum.2 hc\n    · exact mem_map_of_mem _ hc\n\nlemma inl_mem_sumLexLift {c₁ : γ₁} :\n    inl c₁ ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        ∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ c₁ ∈ g₁ a₁ b₂ := by\n  simp [mem_sumLexLift]\n\nlemma inr_mem_sumLexLift {c₂ : γ₂} :\n    inr c₂ ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ f₂ a₂ b₂ := by\n  simp [mem_sumLexLift]\n\nlemma sumLexLift_mono (hf₁ : ∀ a b, f₁ a b ⊆ f₁' a b) (hf₂ : ∀ a b, f₂ a b ⊆ f₂' a b)\n    (hg₁ : ∀ a b, g₁ a b ⊆ g₁' a b) (hg₂ : ∀ a b, g₂ a b ⊆ g₂' a b) (a : α₁ ⊕ α₂)\n    (b : β₁ ⊕ β₂) : sumLexLift f₁ f₂ g₁ g₂ a b ⊆ sumLexLift f₁' f₂' g₁' g₂' a b := by\n  cases a <;> cases b\n  exacts [map_subset_map.2 (hf₁ _ _), disjSum_mono (hg₁ _ _) (hg₂ _ _), Subset.rfl,\n    map_subset_map.2 (hf₂ _ _)]\n\nTarget:\nlemma sumLexLift_eq_empty :\n    sumLexLift f₁ f₂ g₁ g₂ a b = ∅ ↔\n      (∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f₁ a₁ b₁ = ∅) ∧\n        (∀ a₁ b₂, a = inl a₁ → b = inr b₂ → g₁ a₁ b₂ = ∅ ∧ g₂ a₁ b₂ = ∅) ∧\n          ∀ a₂ b₂, a = inr a₂ → b = inr b₂ → f₂ a₂ b₂ = ∅ :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"53d569c0052ce94b94afc3253716bce29a821b82b471b97a92dcf3254037f6c2","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sum","family_id":"sumlexlift_eq_empty","file_id":"mathlib/Mathlib/Data/Sum/Interval.lean","sample_id":"7fa1587a449af9bd025ff7fcf8e9e6160d9d026e5e59b7be7c90ab324dc63b3b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c7cf3138eb7d775d3fb111ddb9cf2f6f27656c166d40a0c93cd557614d2fd3af","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c2d1819f1557f27ef928e9cc08a0a1fce51d5e749491f60664a68f2b5ce28d95","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2b323eed78bdd6ac7a8d6ff6b4eecb4cf02033f76acf0f63ce6e3ba7ef4dfad0","source_sha256":"f94f55c4ba2322bc4c471438f5b24643adb8c39a2c5e74b34a6830fdd6741d84","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases xs\n  simp only [reverse, snoc, cons, toList_mk]\n  congr\n  simp [toList, append_def]","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":28,"token_jaccard":0.882353,"token_length_ratio":1.333333},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"94c14bb6dbd5e734777f02dad5535744faa179a9efc6544bc526098d0b9ba230","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Vector.Basic\n\nNamespace:\nList.Vector\n\nLocal context:\n/-\nCopyright (c) 2023 Alex Keizer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alex Keizer\n-/\n/-!\n  This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single\n  element to the back of a vector.\n\n  It provides a collection of lemmas that show how different `Vector` operations reduce when their\n  argument is `snoc xs x`.\n\n  Also, an alternative, reverse, induction principle is added, that breaks down a vector into\n  `snoc xs x` for its inductive case. Effectively doing induction from right-to-left\n-/\n\n@[expose] public section\n\nnamespace List\n\nnamespace Vector\n\nvariable {α β σ φ : Type*} {n : ℕ} {x : α} {s : σ} (xs : Vector α n)\n\n/-- Append a single element to the end of a vector -/\ndef snoc : Vector α n → α → Vector α (n + 1) :=\n  fun xs x => xs ++ x ::ᵥ Vector.nil\n\n/-! ## Simplification lemmas -/\n\nsection Simp\n\nvariable {y : α}\n\n@[simp]\ntheorem snoc_cons : (x ::ᵥ xs).snoc y = x ::ᵥ (xs.snoc y) :=\n  rfl\n\n@[simp]\ntheorem snoc_nil : (nil.snoc x) = x ::ᵥ nil :=\n  rfl\n\n@[simp]\ntheorem reverse_cons : reverse (x ::ᵥ xs) = (reverse xs).snoc x := by\n  cases xs\n  simp only [reverse, cons, toList_mk, List.reverse_cons, snoc]\n  congr\n\n@[simp]\n\nTarget:\ntheorem reverse_snoc : reverse (xs.snoc x) = x ::ᵥ (reverse xs) :=\n\nProof body:\n","rejected":"```lean\nby\n  cases xs\n  simp only [reverse, snoc, cons, toList_mk]\n  congr\n  simp [toList, append_def]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Vector","family_id":"reverse_snoc","file_id":"mathlib/Mathlib/Data/Vector/Snoc.lean","sample_id":"2b323eed78bdd6ac7a8d6ff6b4eecb4cf02033f76acf0f63ce6e3ba7ef4dfad0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3bdeb5d69c9d68024e63a03b42978b4ed5db36efc7c46e5dd1c026762b0bff70","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"97fbd19fb2bbaefce301241f4a558e69285e0f5cbfe17d565bbbc90ee9f0c29c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0499f9ddadeadf4353cd9529cac7c5ef8cb81587f40f09d0d81cd202b497e41b","source_sha256":"7e0891f362b30145c7c3c4e0ec7d7143a1d3e97fd040572f98d49bd047596de1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · obtain @⟨n, i, h0, hn⟩ := h\n    simp only [iSup_iff, ofHoms_iff, Subtype.exists, exists_prop]\n    use n, i\n  · simp only [iSup_iff, ofHoms_iff] at h\n    obtain ⟨n, ⟨i, h0, hn⟩, _, _⟩ := h\n    exact horn_ι_mem_innerHornInclusions h0 hn","hard_negative":true,"metrics":{"chosen_tokens":81,"rejected_tokens":3,"token_jaccard":0.055556,"token_length_ratio":0.037037},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"94c6e1e545462c81213d784e2df1d795227dc44992a5722b28bc9ccfcd084744","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.Quasicategory.Basic\n\nNamespace:\nSSet\n\nLocal context:\n/-\nCopyright (c) 2026 Jack McKoen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jack McKoen\n-/\n/-!\n# Inner fibrations\n\nInner fibrations of simplicial sets are the morphisms in `SSet` which have the right lifting\nproperty with respect to all inner horn inclusions.\n\nBasic consequences of inner fibrations with respect to the definition of quasi-categories are\nformalized.\n\n-/\n\npublic section\n\nopen CategoryTheory MorphismProperty Simplicial Limits\n\nuniverse u\n\nnamespace SSet\n\n/-- The family of morphisms in `SSet` which consists of inner horn inclusions\n`Λ[n, i].ι : Λ[n, i] ⟶ Δ[n]` (for `0 < i < n`). -/\ninductive innerHornInclusions : MorphismProperty SSet.{u} where\n  | intro {n : ℕ} (i : Fin (n + 3)) (h0 : 0 < i) (hn : i < Fin.last (n + 2)) :\n    innerHornInclusions Λ[n + 2, i].ι\n\nlemma horn_ι_mem_innerHornInclusions {n : ℕ} {i : Fin (n + 1)}\n    (h0 : 0 < i) (hn : i < Fin.last n) : innerHornInclusions (horn.{u} n i).ι := by\n  obtain _ | _ | k := n\n  · grind\n  · grind\n  · exact ⟨i, h0, hn⟩\n\nTarget:\nlemma innerHornInclusions_eq_iSup :\n    innerHornInclusions.{u} =\n    ⨆ n, .ofHoms (fun p : {p : Fin (n + 3) // 0 < p ∧ p < Fin.last (n + 2)} ↦ Λ[n + 2, p].ι) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_0499f9ddadea","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"eb85c3ff5246f5f01dca8603d7f065d6a0e0f60b154ec42e4fdd05a4649cdb27","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/Quasicategory","family_id":"innerhorninclusions_eq_isup","file_id":"mathlib/Mathlib/AlgebraicTopology/Quasicategory/InnerFibration.lean","sample_id":"0499f9ddadeadf4353cd9529cac7c5ef8cb81587f40f09d0d81cd202b497e41b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"feace96534351d67a98372e9c910b5669a41f603a705ba51814a94d00a6c6ea8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5bb17713869ded8211fecb07cd342899842c304348ff9eddfbbe96f8cb169aa9","source_sha256":"91f7a9e43cd159c4e5d0d1764bbe2e984a1eb627973fe04a06238fa29e76fd73","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro _ ha _ hb hne\n  have := hrc ha hb (fun h ↦ hne (hf h))\n  grind","hard_negative":false,"metrics":{"chosen_tokens":23,"rejected_tokens":2,"token_jaccard":0.058824,"token_length_ratio":0.086957},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"95bc59b4c6500a353a17c85a8ea26f8f60ee1d9dfa1ab3fe427f5eeba5722740","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Notation\npublic import Mathlib.Data.Set.Pairwise.Basic\npublic import Mathlib.Data.SetLike.Basic\npublic import Mathlib.Order.Directed\npublic import Mathlib.Order.Hom.Set\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Chains and flags\n\nThis file defines chains for an arbitrary relation and flags for an order.\n\n## Main declarations\n\n* `IsChain s`: A chain `s` is a set of comparable elements.\n* `Flag`: The type of flags, aka maximal chains, of an order.\n\n## Notes\n\nOriginally ported from Isabelle/HOL. The\n[original file](https://isabelle.in.tum.de/dist/library/HOL/HOL/Zorn.html) was written by Jacques D.\nFleuriot, Tobias Nipkow, Christian Sternagel.\n-/\n\n@[expose] public section\n\nassert_not_exists CompleteLattice\n\nopen Set Set.Notation\n\nvariable {α β F : Type*}\n\n/-! ### Chains -/\n\n\nsection Chain\n\nvariable (r : α → α → Prop)\n\n/-- In this file, we use `≺` as a local notation for any relation `r`. -/\nlocal infixl:50 \" ≺ \" => r\n\n/-- A chain is a set `s` satisfying `x ≺ y ∨ x = y ∨ y ≺ x` for all `x y ∈ s`. -/\ndef IsChain (s : Set α) : Prop :=\n  s.Pairwise fun x y => x ≺ y ∨ y ≺ x\n\n/-- `SuperChain s t` means that `t` is a chain that strictly includes `s`. -/\ndef SuperChain (s t : Set α) : Prop :=\n  IsChain r t ∧ s ⊂ t\n\n/-- A chain `s` is a maximal chain if there does not exists a chain strictly including `s`. -/\ndef IsMaxChain (s : Set α) : Prop :=\n  IsChain r s ∧ ∀ ⦃t⦄, IsChain r t → s ⊆ t → s = t\n\nvariable {r} {c c₁ c₂ s t : Set α} {a b x y : α}\n\n@[simp] lemma IsChain.empty : IsChain r ∅ := pairwise_empty _\n@[simp] lemma IsChain.singleton : IsChain r {a} := pairwise_singleton ..\n\ntheorem Set.Subsingleton.isChain (hs : s.Subsingleton) : IsChain r s :=\n  hs.pairwise _\n\ntheorem IsChain.mono : s ⊆ t → IsChain r t → IsChain r s :=\n  Set.Pairwise.mono\n\ntheorem IsChain.mono_rel {r' : α → α → Prop} (h : IsChain r s) (h_imp : ∀ x y, r x y → r' x y) :\n    IsChain r' s :=\n  h.mono' fun x y => Or.imp (h_imp x y) (h_imp y x)\n\n/-- This can be used to turn `IsChain (≥)` into `IsChain (≤)` and vice-versa. -/\ntheorem IsChain.symm (h : IsChain r s) : IsChain (flip r) s :=\n  h.mono' fun _ _ => Or.symm\n\ntheorem isChain_of_trichotomous [Std.Trichotomous r] (s : Set α) : IsChain r s :=\n  fun a _ b _ hab => (trichotomous_of r a b).imp_right fun h => h.resolve_left hab\n\nprotected theorem IsChain.insert (hs : IsChain r s) (ha : ∀ b ∈ s, a ≠ b → a ≺ b ∨ b ≺ a) :\n    IsChain r (insert a s) :=\n  have : Std.Symm fun a b ↦ a ≺ b ∨ b ≺ a := { symm _ _ := Or.symm }\n  hs.insert_of_symm ha\n\nlemma IsChain.pair (h : r a b) : IsChain r {a, b} :=\n  IsChain.singleton.insert fun _ hb _ ↦ .inl <| (eq_of_mem_singleton hb).symm.recOn ‹_›\n\ntheorem isChain_univ_iff : IsChain r (univ : Set α) ↔ Std.Trichotomous r := by\n  refine ⟨fun h => ⟨fun a b => ?_⟩, fun h => @isChain_of_trichotomous _ _ h univ⟩\n  have : a ≠ b → (r a b ∨ r b a) := h trivial trivial\n  grind\n\ntheorem IsChain.image_of_map_rel (r : α → α → Prop) (s : β → β → Prop) (f : α → β)\n    (h : ∀ x y, r x y → s (f x) (f y)) {c : Set α} (hrc : IsChain r c) : IsChain s (f '' c) :=\n  fun _ ⟨_, ha₁, ha₂⟩ _ ⟨_, hb₁, hb₂⟩ =>\n  ha₂ ▸ hb₂ ▸ fun hxy => (hrc ha₁ hb₁ <| ne_of_apply_ne f hxy).imp (h _ _) (h _ _)\n\nTarget:\ntheorem IsChain.preimage (r : α → α → Prop) (s : β → β → Prop) (f : α → β)\n    (hf : Function.Injective f) (h : ∀ x y, s (f x) (f y) → r x y) {c : Set β} (hrc : IsChain s c) :\n    IsChain r (f ⁻¹' c) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Preorder","family_id":"ischain","file_id":"mathlib/Mathlib/Order/Preorder/Chain.lean","sample_id":"5bb17713869ded8211fecb07cd342899842c304348ff9eddfbbe96f8cb169aa9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4bc1dfb496047d2c2f53f0ca55577975a296dcb639bf7ace11b6e910a7a97f44","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"452212369db5a32584e53a4244413eb62217af8a259514599237694d17941ea9","source_sha256":"5f13df63ef1a33fb7d6444e769a7e9b9cf3ab09860a26a91c473157d6239794f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← SetLike.mem_coe, coe_commutant]\n  rfl","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.272727},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"95c0e2a1d3caa42aa4f85980b24053ba69d137c4abd8c07579f401368524423c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.CStarAlgebra.Classes\npublic import Mathlib.Analysis.InnerProductSpace.Adjoint\n\nNamespace:\nVonNeumannAlgebra\n\nLocal context:\n/-\nCopyright (c) 2022 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Von Neumann algebras\n\nWe give the \"abstract\" and \"concrete\" definitions of a von Neumann algebra.\nWe still have a major project ahead of us to show the equivalence between these definitions!\n\nAn abstract von Neumann algebra `WStarAlgebra M` is a C⋆ algebra with a Banach space predual,\nper Sakai (1971).\n\nA concrete von Neumann algebra `VonNeumannAlgebra H` (where `H` is a Hilbert space)\nis a \\*-closed subalgebra of bounded operators on `H` which is equal to its double commutant.\n\nWe'll also need to prove the von Neumann double commutant theorem,\nthat the concrete definition is equivalent to a \\*-closed subalgebra which is weakly closed.\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\n/-- Sakai's definition of a von Neumann algebra as a C⋆ algebra with a Banach space predual.\n\nSo that we can unambiguously talk about these \"abstract\" von Neumann algebras\nin parallel with the \"concrete\" ones (weakly closed \\*-subalgebras of B(H)),\nwe name this definition `WStarAlgebra`.\n\nNote that for now we only assert the mere existence of predual, rather than picking one.\nThis may later prove problematic, and need to be revisited.\nPicking one may cause problems with definitional unification of different instances.\nOne the other hand, not picking one means that the weak-\\* topology\n(which depends on a choice of predual) must be defined using the choice,\nand we may be unhappy with the resulting opaqueness of the definition.\n-/\nclass WStarAlgebra (M : Type u) [CStarAlgebra M] : Prop where\n  /-- There is a Banach space `X` whose dual is isometrically (conjugate-linearly) isomorphic\n  to the `WStarAlgebra`. -/\n  exists_predual :\n    ∃ (X : Type u) (_ : NormedAddCommGroup X) (_ : NormedSpace ℂ X) (_ : CompleteSpace X),\n      Nonempty (StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M)\n\n-- TODO: Without this, `VonNeumannAlgebra` times out. Why?\n/-- The double commutant definition of a von Neumann algebra,\nas a \\*-closed subalgebra of bounded operators on a Hilbert space,\nwhich is equal to its double commutant.\n\nNote that this definition is parameterised by the Hilbert space\non which the algebra faithfully acts, as is standard in the literature.\nSee `WStarAlgebra` for the abstract notion (a C⋆-algebra with Banach space predual).\n\nNote this is a bundled structure, parameterised by the Hilbert space `H`,\nrather than a typeclass on the type of elements.\nThus we can't say that the bounded operators `H →L[ℂ] H` form a `VonNeumannAlgebra`\n(although we will later construct the instance `WStarAlgebra (H →L[ℂ] H)`),\nand instead will use `⊤ : VonNeumannAlgebra H`.\n-/\nstructure VonNeumannAlgebra (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H]\n    [CompleteSpace H] extends StarSubalgebra ℂ (H →L[ℂ] H) where\n  /-- The double commutant (a.k.a. centralizer) of a `VonNeumannAlgebra` is itself. -/\n  centralizer_centralizer' : Set.centralizer (Set.centralizer carrier) = carrier\n\n/-- Consider a von Neumann algebra acting on a Hilbert space `H` as a \\*-subalgebra of `H →L[ℂ] H`.\n(That is, we forget that it is equal to its double commutant\nor equivalently that it is closed in the weak and strong operator topologies.)\n-/\nadd_decl_doc VonNeumannAlgebra.toStarSubalgebra\n\nnamespace VonNeumannAlgebra\n\nvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n\ninstance instSetLike : SetLike (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  coe S := S.carrier\n  coe_injective S T h := by obtain ⟨⟨⟨⟨⟨⟨_, _⟩, _⟩, _⟩, _⟩, _⟩, _⟩ := S; cases T; congr\n\ninstance : PartialOrder (VonNeumannAlgebra H) := .ofSetLike (VonNeumannAlgebra H) (H →L[ℂ] H)\n\nnoncomputable instance instStarMemClass : StarMemClass (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  star_mem {s} := s.star_mem'\n\ninstance instSubringClass : SubringClass (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  one_mem {s} := s.one_mem'\n  zero_mem {s} := s.zero_mem'\n  neg_mem {s} a ha := show -a ∈ s.toStarSubalgebra from neg_mem ha\n\n@[simp]\ntheorem mem_carrier {S : VonNeumannAlgebra H} {x : H →L[ℂ] H} :\n    x ∈ S.toStarSubalgebra ↔ x ∈ (S : Set (H →L[ℂ] H)) :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toStarSubalgebra (S : VonNeumannAlgebra H) :\n    (S.toStarSubalgebra : Set (H →L[ℂ] H)) = S :=\n  rfl\n\n@[simp]\ntheorem coe_mk (S : StarSubalgebra ℂ (H →L[ℂ] H)) (h) :\n    ((⟨S, h⟩ : VonNeumannAlgebra H) : Set (H →L[ℂ] H)) = S :=\n  rfl\n\n@[ext]\ntheorem ext {S T : VonNeumannAlgebra H} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem centralizer_centralizer (S : VonNeumannAlgebra H) :\n    Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S :=\n  S.centralizer_centralizer'\n\n/-- The centralizer of a `VonNeumannAlgebra`, as a `VonNeumannAlgebra`. -/\nnoncomputable def commutant (S : VonNeumannAlgebra H) : VonNeumannAlgebra H where\n  toStarSubalgebra := StarSubalgebra.centralizer ℂ (S : Set (H →L[ℂ] H))\n  centralizer_centralizer' := by simp\n\n@[simp]\ntheorem coe_commutant (S : VonNeumannAlgebra H) :\n    ↑S.commutant = Set.centralizer (S : Set (H →L[ℂ] H)) := by\n  simp [commutant]\n\n@[simp]\n\nTarget:\ntheorem mem_commutant_iff {S : VonNeumannAlgebra H} {z : H →L[ℂ] H} :\n    z ∈ S.commutant ↔ ∀ g ∈ S, g * z = z * g :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/VonNeumannAlgebra","family_id":"mem_commutant_iff","file_id":"mathlib/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean","sample_id":"452212369db5a32584e53a4244413eb62217af8a259514599237694d17941ea9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0220babf450eb445fb686e51233d9dc7bd43814cdb2cf6da4315eb591a53bef9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b57d7d0c5ab9e4702f316c1b1e6199c14d35cafa6b002fbbe3edd9c4549f0e01","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a86ece7891869444431a13676c31d6de05fdf4898f0e667ba12ab2a36fc2b894","source_sha256":"f5d92c67f281e03d72da06b390e360bd28b86e3bb7c33267f68c3c873cd9a4b6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← natCast_zsmul r, zsmul_mem_zmultiples_iff_exists_sub_div (Int.natCast_ne_zero.mpr hn),\n    Int.cast_natCast]\n  rfl","hard_negative":true,"metrics":{"chosen_tokens":22,"rejected_tokens":3,"token_jaccard":0.05,"token_length_ratio":0.136364},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"95ccc628dd08a7e7fde8db89401fa53c9cd03294a350cd6a2688966422b3a367","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Basic\npublic import Mathlib.Algebra.Order.Group.Unbundled.Int\npublic import Mathlib.Algebra.Module.NatInt\npublic import Mathlib.GroupTheory.QuotientGroup.Defs\npublic import Mathlib.Algebra.Group.Subgroup.ZPowers.Basic\n\nNamespace:\nAddSubgroup\n\nLocal context:\n/-\nCopyright (c) 2022 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Lemmas about quotients in characteristic zero\n-/\n\npublic section\n\n\nvariable {R : Type*} [DivisionRing R] [CharZero R] {p : R}\n\nnamespace AddSubgroup\n\n/-- `z • r` is a multiple of `p` iff `r` is `k * (p / z)` above a multiple of `p`, where\n`0 ≤ k < |z|`. -/\ntheorem zsmul_mem_zmultiples_iff_exists_sub_div {r : R} {z : ℤ} (hz : z ≠ 0) :\n    z • r ∈ AddSubgroup.zmultiples p ↔\n      ∃ k : Fin z.natAbs, r - (k : ℕ) • (p / z : R) ∈ AddSubgroup.zmultiples p := by\n  rw [AddSubgroup.mem_zmultiples_iff]\n  simp_rw [AddSubgroup.mem_zmultiples_iff, div_eq_mul_inv, ← smul_mul_assoc, eq_sub_iff_add_eq]\n  have hz' : (z : R) ≠ 0 := Int.cast_ne_zero.mpr hz\n  conv_rhs => simp +singlePass only [← (mul_right_injective₀ hz').eq_iff]\n  simp_rw [← zsmul_eq_mul, smul_add, ← mul_smul_comm, zsmul_eq_mul (z : R)⁻¹, mul_inv_cancel₀ hz',\n    mul_one, ← natCast_zsmul, smul_smul, ← add_smul]\n  constructor\n  · rintro ⟨k, h⟩\n    simp_rw [← h]\n    refine ⟨⟨(k % z).toNat, ?_⟩, k / z, ?_⟩\n    · rw [← Int.ofNat_lt, Int.toNat_of_nonneg (Int.emod_nonneg _ hz)]\n      exact (Int.emod_lt_abs _ hz).trans_eq (Int.abs_eq_natAbs _)\n    rw [Fin.val_mk, Int.toNat_of_nonneg (Int.emod_nonneg _ hz)]\n    nth_rewrite 3 [← Int.mul_ediv_add_emod k z]\n    rfl\n  · rintro ⟨k, n, h⟩\n    exact ⟨_, h⟩\n\nTarget:\ntheorem nsmul_mem_zmultiples_iff_exists_sub_div {r : R} {n : ℕ} (hn : n ≠ 0) :\n    n • r ∈ AddSubgroup.zmultiples p ↔\n      ∃ k : Fin n, r - (k : ℕ) • (p / n : R) ∈ AddSubgroup.zmultiples p :=\n\nProof body:\n","rejected":"by\n  exact nsmul_mem_zmultiples_iff_exists_sub_div","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"6ef9a5a4ed12c70089f813f6df4ce39ed255fbb003ad6d1e279c9986230c6369","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/CharZero","family_id":"nsmul_mem_zmultiples_iff_exists_sub_div","file_id":"mathlib/Mathlib/Algebra/CharZero/Quotient.lean","sample_id":"a86ece7891869444431a13676c31d6de05fdf4898f0e667ba12ab2a36fc2b894"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f55e39083a696313fcfdd6bb61c3e62ddbca985973b59bf86a8381597fddf755","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d380a35b9ecdb938e51a37459a7daa51513ecd0013a70d98a3d700254b3c760f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9044b1a9ec5361ec717fcb988a173f7c747b987f4a363348924f9ee3f41d531b","source_sha256":"aa9a2627fd36aceaf056da61c61373399c62e613d54771bd790c452c0c13aad7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : Nontrivial (Subobject X) := nontrivial_of_not_isZero h\n  obtain ⟨Y, s⟩ := (IsAtomic.eq_bot_or_exists_atom_le (⊤ : Subobject X)).resolve_left top_ne_bot\n  exact ⟨Y, (subobject_simple_iff_isAtom _).mpr s.1⟩\n\n/-- Choose an arbitrary simple subobject of a non-zero Artinian object. -/\nnoncomputable def simpleSubobject {X : C} [IsArtinianObject X] (h : ¬IsZero X) : C :=\n  (exists_simple_subobject h).choose\n\n/-- The monomorphism from the arbitrary simple subobject of a non-zero Artinian object. -/\nnoncomputable def simpleSubobjectArrow {X : C} [IsArtinianObject X] (h : ¬IsZero X) :\n    simpleSubobject h ⟶ X :=\n  (exists_simple_subobject h).choose.arrow","hard_negative":true,"metrics":{"chosen_tokens":145,"rejected_tokens":3,"token_jaccard":0.04918,"token_length_ratio":0.02069},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"95d1bab8f3e93b2e8c13e1dbf50c71a773e9322fb2cfc579c802c5a4d7823e21","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Subobject.Lattice\npublic import Mathlib.CategoryTheory.ObjectProperty.ContainsZero\npublic import Mathlib.CategoryTheory.ObjectProperty.EpiMono\npublic import Mathlib.CategoryTheory.Limits.Constructions.EventuallyConstant\npublic import Mathlib.Order.OrderIsoNat\npublic import Mathlib.CategoryTheory.Simple\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou, Kim Morrison\n-/\n/-!\n# Artinian objects\n\nWe shall say that an object `X` in a category `C` is Artinian\n(type class `IsArtinianObject X`) if the ordered type `Subobject X`\nsatisfies the descending chain condition. The corresponding property of\nobjects `isArtinianObject : ObjectProperty C` is always\nclosed under subobjects.\n\n## Future work\n\n* when `C` is an abelian category, relate `IsArtinianObject` in `C`\n  with `IsNoetherianObject` in `Cᵒᵖ`.\n\n-/\n\n@[expose] public section\n\nuniverse v u\n\nnamespace CategoryTheory\n\nopen Limits ZeroObject\n\nvariable {C : Type u} [Category.{v} C]\n\n/-- An object `X` in a category `C` is Artinian if `Subobject X`\nsatisfies the descending chain condition. This definition is a\nterm in `ObjectProperty C` which allows to study the stability\nproperties of Artinian objects. For statements regarding\nspecific objects, it is advisable to use the type class\n`IsArtinianObject` instead. -/\n@[stacks 0FCF]\ndef isArtinianObject : ObjectProperty C :=\n  fun X ↦ WellFoundedLT (Subobject X)\n\nvariable (X Y : C)\n\n/-- An object `X` in a category `C` is Artinian if `Subobject X`\nsatisfies the descending chain condition. -/\n@[stacks 0FCF]\nabbrev IsArtinianObject : Prop := isArtinianObject.Is X\n\ninstance [IsArtinianObject X] : WellFoundedLT (Subobject X) :=\n  isArtinianObject.prop_of_is X\n\nlemma isArtinianObject_iff_antitone_chain_condition :\n    IsArtinianObject X ↔ ∀ (f : ℕ →o (Subobject X)ᵒᵈ),\n      ∃ (n : ℕ), ∀ (m : ℕ), n ≤ m → f n = f m := by\n  dsimp only [IsArtinianObject]\n  rw [ObjectProperty.is_iff, isArtinianObject,\n    ← wellFoundedGT_dual_iff,\n    wellFoundedGT_iff_monotone_chain_condition]\n\nvariable {X} in\nlemma antitone_chain_condition_of_isArtinianObject\n    [IsArtinianObject X] (f : ℕ →o (Subobject X)ᵒᵈ) :\n    ∃ (n : ℕ), ∀ (m : ℕ), n ≤ m → f n = f m :=\n  (isArtinianObject_iff_antitone_chain_condition X).1 inferInstance f\n\nlemma isArtinianObject_iff_not_strictAnti :\n    IsArtinianObject X ↔ ∀ (f : ℕ → Subobject X), ¬ StrictAnti f := by\n  refine ⟨fun _ ↦ not_strictAnti_of_wellFoundedLT, fun h ↦ ?_⟩\n  dsimp only [IsArtinianObject]\n  rw [ObjectProperty.is_iff, isArtinianObject, WellFoundedLT,\n    isWellFounded_iff, RelEmbedding.wellFounded_iff_isEmpty]\n  exact ⟨fun f ↦ h f.toFun (fun a b h ↦ f.map_rel_iff.2 h)⟩\n\nvariable {X} in\nlemma not_strictAnti_of_isArtinianObject\n    [IsArtinianObject X] (f : ℕ → Subobject X) :\n    ¬ StrictAnti f :=\n  (isArtinianObject_iff_not_strictAnti X).1 inferInstance f\n\nset_option backward.defeqAttrib.useBackward true in\nlemma isArtinianObject_iff_isEventuallyConstant :\n    IsArtinianObject X ↔ ∀ (F : ℕ ⥤ (MonoOver X)ᵒᵖ),\n      IsFiltered.IsEventuallyConstant F := by\n  rw [isArtinianObject_iff_antitone_chain_condition]\n  refine ⟨fun h G ↦ ?_, fun h F ↦ ?_⟩\n  · obtain ⟨n, hn⟩ := h ⟨_, (G ⋙ (Subobject.equivMonoOver X).inverse.op ⋙\n      (orderDualEquivalence _).inverse).monotone⟩\n    refine ⟨n, fun m hm ↦ ?_⟩\n    rw [← isIso_unop_iff, MonoOver.isIso_iff_subobjectMk_eq]\n    exact (hn m (leOfHom hm)).symm\n  · obtain ⟨n, hn⟩ := h (F.monotone.functor ⋙ (orderDualEquivalence _).functor ⋙\n      Subobject.representative.op)\n    refine ⟨n, fun m hm ↦ Eq.symm ?_⟩\n    simpa [isIso_op_iff, isIso_iff_of_reflects_iso, PartialOrder.isIso_iff_eq]\n      using hn (homOfLE hm)\n\nvariable {X} in\nlemma isEventuallyConstant_of_isArtinianObject [IsArtinianObject X]\n    (F : ℕ ⥤ (MonoOver X)ᵒᵖ) : IsFiltered.IsEventuallyConstant F :=\n  (isArtinianObject_iff_isEventuallyConstant X).1 inferInstance F\n\nvariable {X Y}\n\nlemma isArtinianObject_of_isZero (hX : IsZero X) : IsArtinianObject X := by\n  rw [isArtinianObject_iff_antitone_chain_condition]\n  have := Subobject.subsingleton_of_isZero hX\n  intro f\n  exact ⟨0, fun m hm ↦ Subsingleton.elim _ _⟩\n\ninstance [HasZeroObject C] : (isArtinianObject (C := C)).ContainsZero where\n  exists_zero := ⟨0, isZero_zero _, by\n    rw [← isArtinianObject.is_iff]\n    exact isArtinianObject_of_isZero (isZero_zero C)⟩\n\nlemma isArtinianObject_of_mono (i : X ⟶ Y) [Mono i] [IsArtinianObject Y] :\n    IsArtinianObject X := by\n  rw [isArtinianObject_iff_antitone_chain_condition]\n  intro f\n  obtain ⟨n, hn⟩ := antitone_chain_condition_of_isArtinianObject\n    ⟨fun n ↦ (Subobject.map i).obj (f n),\n      fun _ _ h ↦ (Subobject.map i).monotone (f.2 h)⟩\n  exact ⟨n, fun m hm ↦ Subobject.map_obj_injective i (hn m hm)⟩\n\ninstance : (isArtinianObject (C := C)).IsClosedUnderSubobjects where\n  prop_of_mono f _ hY := by\n    rw [← isArtinianObject.is_iff] at hY ⊢\n    exact isArtinianObject_of_mono f\n\nopen Subobject\n\nvariable [HasZeroMorphisms C] [HasZeroObject C]\n\nTarget:\ntheorem exists_simple_subobject {X : C} [IsArtinianObject X] (h : ¬IsZero X) :\n    ∃ Y : Subobject X, Simple (Y : C) :=\n\nProof body:\n","rejected":"by\n  exact exists_simple_subobject","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"bdc71e8c5e534085dbdb9990398b70f7d2e10e12b69d4c15900767c955adb26e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Subobject","family_id":"exists_simple_subobject","file_id":"mathlib/Mathlib/CategoryTheory/Subobject/ArtinianObject.lean","sample_id":"9044b1a9ec5361ec717fcb988a173f7c747b987f4a363348924f9ee3f41d531b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"212d1f84cff2b596f7c1f93804fd9c282408c326066650fb7feba4fd785cf41f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a53b7c1c6dc7380167b7eef2678e6391bfc432a0b3abcb53f2f3e41f881be729","source_sha256":"52cc50e04f911b3cb7014b8c73cd594d90bc999d3e9e3dfd7bf800868c589bb6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [max_def, val_le_val, ← apply_ite]\n  rfl","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":8,"token_jaccard":0.0625,"token_length_ratio":0.727273},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"963b2498c5963f3336d9f0e776e6de80aa4f4585fc818971222980e418db7e85","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Hom.Basic\npublic import Mathlib.Algebra.Group.Units.Defs\n\nNamespace:\nUnits\n\nLocal context:\n/-\nCopyright (c) 2016 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl\n-/\n/-!\n# Units in ordered monoids\n-/\n\n@[expose] public section\n\nnamespace Units\n\nvariable {α : Type*}\n\n@[to_additive]\ninstance [Monoid α] [Preorder α] : Preorder αˣ :=\n  Preorder.lift val\n\n@[to_additive (attr := simp, norm_cast)]\ntheorem val_le_val [Monoid α] [Preorder α] {a b : αˣ} : (a : α) ≤ b ↔ a ≤ b :=\n  Iff.rfl\n\n@[to_additive (attr := simp, norm_cast)]\ntheorem val_lt_val [Monoid α] [Preorder α] {a b : αˣ} : (a : α) < b ↔ a < b :=\n  Iff.rfl\n\n@[to_additive]\ninstance instPartialOrderUnits [Monoid α] [PartialOrder α] : PartialOrder αˣ :=\n  PartialOrder.lift val val_injective\n\n@[to_additive]\ninstance [Monoid α] [LinearOrder α] : Max αˣ where\n  max a b := if a ≤ b then b else a\n\n@[to_additive]\ninstance [Monoid α] [LinearOrder α] : Min αˣ where\n  min a b := if a ≤ b then a else b\n\n\n@[to_additive (attr := simp, norm_cast)]\n\nTarget:\ntheorem max_val [Monoid α] [LinearOrder α] (a b : αˣ) : (max a b).val = max a.val b.val :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"ded86b16da47f9d82c38ee6c592a843cf0becb292209ed58b815e439fe400d26","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Order","family_id":"max_val","file_id":"mathlib/Mathlib/Algebra/Order/Monoid/Units.lean","sample_id":"a53b7c1c6dc7380167b7eef2678e6391bfc432a0b3abcb53f2f3e41f881be729"},"split":"train"}
{"binding":{"chosen_proof_sha256":"252205b7b358cac3e7fbb348dc9a6e22e848772b9ebec8a5f89dd1c88ca1cccb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9293c8a0161f13a7f11277c44e7e1055a37901d09786b61e3ffd70354bbe72a8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"932a2516a9f3261c5acb0efee2adbd6bd7291142ae5d6a59e51bba4d80664223","source_sha256":"99fe4affc634c570de70587e929805d16958622f49de95b17da87beb4716efa3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction t with\n  | nil => rw [traverse, map]\n  | node v l r hl hr =>\n    rw [traverse, map, hl, hr, Function.comp_apply, map_pure, pure_seq, map_pure, pure_seq,\n      map_pure]","hard_negative":false,"metrics":{"chosen_tokens":45,"rejected_tokens":50,"token_jaccard":0.857143,"token_length_ratio":1.111111},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"967ecca4df948b08929d730358f8d6375d9a3974b635dff02b90e02298f1504f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Tree.Basic\npublic import Mathlib.Control.Applicative\npublic import Mathlib.Control.Traversable.Basic\n\nNamespace:\nBinaryTree\n\nLocal context:\n/-\nCopyright (c) 2025 Edward van de Meent. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Edward van de Meent\n-/\n/-!\n# Traversable Binary Tree\n\nProvides a `Traversable` instance for the `Tree` type.\n-/\n\npublic section\n\nuniverse u v w\n\nnamespace BinaryTree\nsection Traverse\nvariable {α β : Type*}\n\ninstance : Traversable BinaryTree where\n  map := map\n  traverse := traverse\n\nlemma comp_traverse\n    {F : Type u → Type v} {G : Type v → Type w} [Applicative F] [Applicative G]\n    [LawfulApplicative G] {β : Type v} {γ : Type u} (f : β → F γ) (g : α → G β)\n    (t : BinaryTree α) : t.traverse (Functor.Comp.mk ∘ (f <$> ·) ∘ g) =\n      Functor.Comp.mk ((·.traverse f) <$> (t.traverse g)) := by\n  induction t with\n  | nil => rw [traverse, traverse, map_pure, traverse]; rfl\n  | node v l r hl hr =>\n    rw [traverse, hl, hr, traverse]\n    simp only [Function.comp_def, Function.comp_apply, Functor.Comp.map_mk, Functor.map_map,\n      Comp.seq_mk, seq_map_assoc, map_seq]\n    rfl\n\nTarget:\nlemma traverse_eq_map_id (f : α → β) (t : BinaryTree α) :\n    t.traverse ((pure : β → Id β) ∘ f) = pure (t.map f) :=\n\nProof body:\n","rejected":"by\n  induction t with\n  | nil => rw [traverse, map]\n  | node v l r hl hr =>\n    rw [traverse, map, hl, hr, Function.comp_apply, map_pure, pure_seq, map_pure, pure_seq,\n      map_pure]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Tree","family_id":"traverse_eq_map_id","file_id":"mathlib/Mathlib/Data/Tree/Traversable.lean","sample_id":"932a2516a9f3261c5acb0efee2adbd6bd7291142ae5d6a59e51bba4d80664223"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8690ef82dc2fef54f22a4e762da7bbf0fe4e6499a977029b3fad0c51781aa987","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"61af39f4fdc1dd36cdc67664e32ec30006784c866e4e6f32bab0e3ab1b4f9212","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"29db24e141a0e3e320d003b92675922c58df588b3780d126668224314302b43a","source_sha256":"b77307aadbea1488a93479741afc5f9d1f9e6198ff638b8b6035869012641ebc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  rw [ofDirectSumEquiv]\n  convert! ofDFinsuppEquiv_symm_single_tprod p x","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.181818},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"968d0e3de119c581ca76b720d9e63432bacf43339c4bfa462602dc861b4d7f7e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.PiTensorProduct.Basic\npublic import Mathlib.LinearAlgebra.PiTensorProduct.DFinsupp\npublic import Mathlib.Algebra.DirectSum.Module\n\nNamespace:\nPiTensorProduct\n\nLocal context:\n/-\nCopyright (c) 2024 Sophie Morel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sophie Morel, Eric Wieser\n-/\n/-!\n# Tensor products of direct sums\n\nThis file shows that taking `PiTensorProduct`s commutes with taking `DirectSum`s in all arguments.\n\n## Main results\n\n* `ofDirectSumEquiv`: the linear equivalence between a `PiTensorProduct` of `DirectSum`s\n  and the `DirectSum` of the `PiTensorProduct`s.\n-/\n\n@[expose] public section\n\nnamespace PiTensorProduct\n\nopen PiTensorProduct DirectSum TensorProduct\n\nvariable {R ι : Type*} {κ : ι → Type*} {M : (i : ι) → κ i → Type*}\n  [CommSemiring R] [Π i (j : κ i), AddCommMonoid (M i j)] [Π i (j : κ i), Module R (M i j)]\n\nopen scoped Classical in\n/-- The n-ary tensor product distributes over m-ary direct sums. -/\nnoncomputable def ofDirectSumEquiv [Finite ι] :\n    (⨂[R] i, (⨁ j : κ i, M i j)) ≃ₗ[R] ⨁ p : Π i, κ i, ⨂[R] i, M i (p i) :=\n  have : Fintype ι := Fintype.ofFinite ι\n  ofDFinsuppEquiv\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\ntheorem ofDirectSumEquiv_tprod_lof [Fintype ι] [(i : ι) → DecidableEq (κ i)]\n    (p : Π i, κ i) (x : Π i, M i (p i)) :\n    ofDirectSumEquiv (⨂ₜ[R] i, DirectSum.lof R _ _ (p i) (x i)) =\n      DirectSum.lof R _ _ p (⨂ₜ[R] i, x i) := by\n  classical\n  rw [ofDirectSumEquiv]\n  convert! ofDFinsuppEquiv_tprod_single p x\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\n\nTarget:\ntheorem ofDirectSumEquiv_symm_lof_tprod [Fintype ι] [(i : ι) → DecidableEq (κ i)]\n    (p : Π i, κ i) (x : Π i, M i (p i)) :\n    ofDirectSumEquiv.symm (DirectSum.lof R _ _ p (tprod R x)) =\n      (⨂ₜ[R] i, DirectSum.lof R _ _ (p i) (x i)) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_29db24e141a0","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"60e98fc682da6c40c20f1888a7dac37a54624c1127574650db49dc7a58e7dc6b","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/PiTensorProduct","family_id":"ofdirectsumequiv_symm_lof_tprod","file_id":"mathlib/Mathlib/LinearAlgebra/PiTensorProduct/DirectSum.lean","sample_id":"29db24e141a0e3e320d003b92675922c58df588b3780d126668224314302b43a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9a21e4324d10fe830df7d8256fcb74edf6972d935383079b80c071c0dccb073e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"bbdda3acdd91e6499b48bf858ce581113adcc0e6a46020585b33ac523c641699","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c7eeb279213fe096be67e549c7a7f69ed83061d62764c53bfce17ab42221096a","source_sha256":"a1ecfd868e0d49b8667b2cdba350ee4f2ceb12fbe0b51094508e312f262dbf5e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using toInt_le_toInt val j i","hard_negative":false,"metrics":{"chosen_tokens":7,"rejected_tokens":12,"token_jaccard":0.583333,"token_length_ratio":1.714286},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"96c0a6bb3fccd66d890e9aa1139345b35d4bb3b840cbec98a20bfa55e0e25d81","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.List.Pairwise\npublic import Batteries.Tactic.GeneralizeProofs\npublic import Mathlib.Tactic.Order.CollectFacts\n\nNamespace:\nMathlib.Tactic.Order.ToInt\n\nLocal context:\n/-\nCopyright (c) 2025 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\npublic meta import Mathlib.Util.AtomM\npublic meta import Mathlib.Util.Qq\npublic meta import Std.Data.HashMap.AdditionalOperations\n\n/-!\n# Translating linear orders to ℤ\n\nIn this file we implement the translation of a problem in any linearly ordered type to a problem in\n`ℤ`. This allows us to use the `lia` tactic to solve it.\n\nWhile the core algorithm of the `order` tactic is complete for the theory of linear orders in the\nsignature (`<`, `≤`),\nit becomes incomplete in the signature with lattice operations `⊓` and `⊔`. With these operations,\nthe problem becomes NP-hard, and the idea is to reuse a smart and efficient procedure, such as\n`lia`.\n\n## TODO\n\nMigrate to `grind` when it is ready.\n-/\n\npublic meta section\n\nnamespace Mathlib.Tactic.Order.ToInt\n\nvariable {α : Type*} [LinearOrder α] {n : ℕ} (val : Fin n → α)\n\n/-- The main theorem asserting the existence of a translation.\nWe use `Classical.choose` to turn this into a value for use in the `order` tactic,\nsee `toInt`.\n-/\ntheorem exists_translation : ∃ tr : Fin n → ℤ, ∀ i j, val i ≤ val j ↔ tr i ≤ tr j := by\n  let li := List.ofFn val\n  let sli := li.mergeSort\n  have (i : Fin n) : ∃ j : Fin sli.length, sli[j] = val i := by\n    apply List.get_of_mem\n    rw [List.Perm.mem_iff (List.mergeSort_perm _ _)]\n    simp [li]\n  use fun i ↦ (this i).choose\n  intro i j\n  simp only [Fin.getElem_fin, Int.ofNat_le]\n  by_cases h_eq : val i = val j\n  · simp [h_eq]\n  generalize_proofs _ hi hj\n  rw [← hi.choose_spec, ← hj.choose_spec] at h_eq\n  conv_lhs => rw [← hi.choose_spec, ← hj.choose_spec]\n  have := li.pairwise_mergeSort (le := fun a b ↦ decide (a ≤ b))\n      (fun a b c ↦ by simpa using le_trans) (by simpa using le_total)\n  rw [List.pairwise_iff_get] at this\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · contrapose! h\n    exact lt_of_le_of_ne (by simpa using (this hj.choose hi.choose (by simpa)))\n      (fun h ↦ h_eq (h.symm))\n  · simpa using this hi.choose hj.choose (by apply lt_of_le_of_ne h; contrapose h_eq; simp [h_eq])\n\n/-- Auxiliary definition used by the `order` tactic to transfer facts in a linear order to `ℤ`. -/\nnoncomputable def toInt (k : Fin n) : ℤ :=\n  (exists_translation val).choose k\n\nvariable (i j k : Fin n)\n\ntheorem toInt_le_toInt : toInt val i ≤ toInt val j ↔ val i ≤ val j := by\n  simp [toInt, (exists_translation val).choose_spec]\n\ntheorem toInt_lt_toInt : toInt val i < toInt val j ↔ val i < val j := by\n  simpa using (toInt_le_toInt val j i).not\n\ntheorem toInt_eq_toInt : toInt val i = toInt val j ↔ val i = val j := by\n  simp [toInt_le_toInt, le_antisymm_iff]\n\ntheorem toInt_ne_toInt : toInt val i ≠ toInt val j ↔ val i ≠ val j := by\n  simpa using (toInt_eq_toInt val i j).not\n\ntheorem toInt_nle_toInt : ¬toInt val i ≤ toInt val j ↔ ¬val i ≤ val j := by\n  simpa using toInt_lt_toInt val j i\n\nTarget:\ntheorem toInt_nlt_toInt : ¬toInt val i < toInt val j ↔ ¬val i < val j :=\n\nProof body:\n","rejected":"by\n  simpa using toInt_le_toInt val j i\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Order","family_id":"toint_nlt_toint","file_id":"mathlib/Mathlib/Tactic/Order/ToInt.lean","sample_id":"c7eeb279213fe096be67e549c7a7f69ed83061d62764c53bfce17ab42221096a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"084e758470934f76ebf17be847788f3818be1118ff79efcd27a52969227aebb9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2357e78341a07f48e92532f87e0bfabcf4fba0a7567069cbcd02ca8502dd6972","source_sha256":"4b34e7f92543da53c331f629ae697e47be9c96439c6558dd62e8cb4d9bb21d9b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h₁ : x⁻¹ ≠ 1 := by rwa [inv_eq_one_div, Ne, div_eq_iff_mul_eq hx0, one_mul]\n  have h₂ : x⁻¹ - 1 ≠ 0 := mt sub_eq_zero.1 h₁\n  have h₃ : x - 1 ≠ 0 := mt sub_eq_zero.1 hx1\n  have h₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x :=\n    Nat.recOn n (by simp) fun n h => by\n      rw [pow_succ', mul_inv_rev, ← mul_assoc, h, mul_assoc, mul_inv_cancel₀ hx0, mul_assoc,\n        inv_mul_cancel₀ hx0]\n  rw [geom_sum_eq h₁, div_eq_iff_mul_eq h₂, ← mul_right_inj' h₃, ← mul_assoc, ← mul_assoc,\n    mul_inv_cancel₀ h₃]\n  simp only [inv_pow, sub_eq_add_neg, mul_add, one_mul, mul_neg, add_mul, mul_inv_cancel₀ hx0,\n    neg_mul, mul_assoc, mul_one, add_comm, neg_add_rev, neg_neg, h₄, add_left_comm]\n  rw [add_comm _ (-x), add_assoc, add_assoc _ _ 1]","hard_negative":true,"metrics":{"chosen_tokens":194,"rejected_tokens":8,"token_jaccard":0.059701,"token_length_ratio":0.041237},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"970e9b469f8efd896abe579511e6d0d37fec5a2ec1805815c11c52293eaf140a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Basic\npublic import Mathlib.Algebra.Ring.GeomSum\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Neil Strickland. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Neil Strickland\n-/\n/-!\n# Partial sums of geometric series in a field\n\nThis file determines the values of the geometric series $\\sum_{i=0}^{n-1} x^i$ and\n$\\sum_{i=0}^{n-1} x^i y^{n-1-i}$ and variants thereof.\n\n## Main statements\n\n* `geom_sum_Ico` proves that $\\sum_{i=m}^{n-1} x^i=\\frac{x^n-x^m}{x-1}$ in a division ring.\n* `geom_sum₂_Ico` proves that $\\sum_{i=m}^{n-1} x^iy^{n - 1 - i}=\\frac{x^n-y^{n-m}x^m}{x-y}$\n  in a field.\n\nSeveral variants are recorded, generalising in particular to the case of a division ring in\nwhich `x` and `y` commute.\n-/\n\npublic section\n\nassert_not_exists IsOrderedRing\n\nvariable {R K : Type*}\n\nopen Finset MulOpposite\n\nsection DivisionRing\nvariable [DivisionRing K] {x y : K}\n\nprotected theorem Commute.geom_sum₂ (h' : Commute x y) (h : x ≠ y)\n    (n : ℕ) : ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ n) / (x - y) := by\n  have : x - y ≠ 0 := by simp_all [sub_eq_iff_eq_add]\n  rw [← h'.geom_sum₂_mul, mul_div_cancel_right₀ _ this]\n\ntheorem geom_sum_eq (h : x ≠ 1) (n : ℕ) : ∑ i ∈ range n, x ^ i = (x ^ n - 1) / (x - 1) := by\n  have : x - 1 ≠ 0 := by simp_all [sub_eq_iff_eq_add]\n  rw [← geom_sum_mul, mul_div_cancel_right₀ _ this]\n\nprotected theorem Commute.geom_sum₂_Ico (h : Commute x y) (hxy : x ≠ y) {m n : ℕ} (hmn : m ≤ n) :\n    ∑ i ∈ Finset.Ico m n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ (n - m) * x ^ m) / (x - y) := by\n  have : x - y ≠ 0 := by simp_all [sub_eq_iff_eq_add]\n  rw [← h.geom_sum₂_Ico_mul hmn, mul_div_cancel_right₀ _ this]\n\nlemma geom_sum_Ico (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) :\n    ∑ i ∈ Finset.Ico m n, x ^ i = (x ^ n - x ^ m) / (x - 1) := by\n  simp only [sum_Ico_eq_sub _ hmn, geom_sum_eq hx, div_sub_div_same, sub_sub_sub_cancel_right]\n\nlemma geom_sum_Ico' (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) :\n    ∑ i ∈ Finset.Ico m n, x ^ i = (x ^ m - x ^ n) / (1 - x) := by\n  simpa [geom_sum_Ico hx hmn] using neg_div_neg_eq (x ^ m - x ^ n) (1 - x)\n\nTarget:\nlemma geom_sum_inv (hx1 : x ≠ 1) (hx0 : x ≠ 0) (n : ℕ) :\n    ∑ i ∈ range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"f96373c1c75b4007a6925536ea2b0e6fdb6d46243df83d9435f508ad9378de14","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Field","family_id":"geom_sum_inv","file_id":"mathlib/Mathlib/Algebra/Field/GeomSum.lean","sample_id":"2357e78341a07f48e92532f87e0bfabcf4fba0a7567069cbcd02ca8502dd6972"},"split":"train"}
{"binding":{"chosen_proof_sha256":"24551563a4e7d2141b3625f37d2a2f5a066ec7e2be8f2231f8bc3aa32f433342","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8d47a007615a250bf24cd20685fe57ad901c62e3e8a8706c9506725d94de99c7","source_sha256":"da76ff62b317f067f0d000725a7dfe7652cd6b413da74f1215e83607e9c9c512","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isUnit_iff_and, and_iff_right ha]","hard_negative":false,"metrics":{"chosen_tokens":8,"rejected_tokens":3,"token_jaccard":0.1,"token_length_ratio":0.375},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"97b39bb023980107fa94eec4a97fcacbae69f242147a5828b14a1b8d4e680fe7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Ideal.Maximal\n\nNamespace:\nSubmonoid\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Chris Hughes, Mario Carneiro\n-/\n/-!\n# The set of non-invertible elements of a monoid\n\n## Main definitions\n\n* `nonunits` is the set of non-invertible elements of a monoid.\n\n## Main results\n\n* `exists_max_ideal_of_mem_nonunits`: every element of `nonunits` is contained in a maximal ideal\n-/\n\n@[expose] public section\n\n\nvariable {F α β : Type*} {a b : α}\n\n/-- The set of non-invertible elements of a monoid. -/\ndef nonunits (α : Type*) [Monoid α] : Set α :=\n  { a | ¬IsUnit a }\n\n@[simp]\ntheorem mem_nonunits_iff [Monoid α] : a ∈ nonunits α ↔ ¬IsUnit a :=\n  Iff.rfl\n\ntheorem mul_mem_nonunits_right [CommMonoid α] : b ∈ nonunits α → a * b ∈ nonunits α :=\n  mt isUnit_of_mul_isUnit_right\n\ntheorem mul_mem_nonunits_left [CommMonoid α] : a ∈ nonunits α → a * b ∈ nonunits α :=\n  mt isUnit_of_mul_isUnit_left\n\ntheorem zero_mem_nonunits [MonoidWithZero α] : 0 ∈ nonunits α ↔ (0 : α) ≠ 1 :=\n  not_congr isUnit_zero_iff\n\n@[simp high] -- High priority shortcut lemma\ntheorem one_notMem_nonunits [Monoid α] : (1 : α) ∉ nonunits α :=\n  not_not_intro isUnit_one\n\n@[simp high] -- High priority shortcut lemma\ntheorem map_mem_nonunits_iff [Monoid α] [Monoid β] [FunLike F α β] [MonoidHomClass F α β] (f : F)\n    [IsLocalHom f] (a) : f a ∈ nonunits β ↔ a ∈ nonunits α :=\n  ⟨fun h ha => h <| ha.map f, fun h ha => h <| ha.of_map⟩\n\ntheorem coe_subset_nonunits [Semiring α] {I : Ideal α} (h : I ≠ ⊤) : (I : Set α) ⊆ nonunits α :=\n  fun _x hx hu => h <| I.eq_top_of_isUnit_mem hx hu\n\ntheorem exists_max_ideal_of_mem_nonunits [CommSemiring α] (h : a ∈ nonunits α) :\n    ∃ I : Ideal α, I.IsMaximal ∧ a ∈ I := by\n  have : Ideal.span ({a} : Set α) ≠ ⊤ := by\n    intro H\n    rw [Ideal.span_singleton_eq_top] at H\n    contradiction\n  rcases Ideal.exists_le_maximal _ this with ⟨I, Imax, H⟩\n  use I, Imax\n  apply H\n  apply Ideal.subset_span\n  exact Set.mem_singleton a\n\nnamespace Submonoid\n\nvariable {C : Type*} [SetLike C α]\n\ntheorem inv_mem_of_isUnit [DivisionMonoid α] [SubmonoidClass C α] {S : C} {a : S} (ha : IsUnit a) :\n    (a : α)⁻¹ ∈ S := by\n  obtain ⟨u, rfl⟩ := ha\n  convert! u⁻¹.1.2\n  exact (map_inv ((subtype <| ofClass S).comp <| Units.coeHom S) u).symm\n\nsection Group\n\nvariable [Group α] [SubmonoidClass C α] {S : C} {a : S}\n\ntheorem isUnit_iff : IsUnit a ↔ (a : α)⁻¹ ∈ S where\n  mp := inv_mem_of_isUnit\n  mpr h := ⟨⟨a, ⟨_, h⟩, Subtype.ext (mul_inv_cancel _), Subtype.ext (inv_mul_cancel _)⟩, rfl⟩\n\nprotected theorem mem_nonunits_iff : a ∈ nonunits S ↔ (a : α)⁻¹ ∉ S := by\n  rw [mem_nonunits_iff, isUnit_iff]\n\nend Group\n\nsection GroupWithZero\n\nvariable [GroupWithZero α] [SubmonoidClass C α] {S : C} {a : S}\n\ntheorem isUnit_iff_and : IsUnit a ↔ (a : α) ≠ 0 ∧ (a : α)⁻¹ ∈ S where\n  mp h := ⟨(h.map <| subtype <| ofClass S).ne_zero, inv_mem_of_isUnit h⟩\n  mpr h :=\n    ⟨⟨a, ⟨_, h.2⟩, Subtype.ext (mul_inv_cancel₀ h.1), Subtype.ext (inv_mul_cancel₀ h.1)⟩, rfl⟩\n\nTarget:\ntheorem isUnit_iff_of_ne_zero (ha : (a : α) ≠ 0) : IsUnit a ↔ (a : α)⁻¹ ∈ S :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Ideal","family_id":"isunit_iff_of_ne_zero","file_id":"mathlib/Mathlib/RingTheory/Ideal/Nonunits.lean","sample_id":"8d47a007615a250bf24cd20685fe57ad901c62e3e8a8706c9506725d94de99c7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4ba70e5cd3f178f8ef12d7930e49952b41e1f76063cf50cc900cfa723b52ba14","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e0f31a2fa4b167750af0775ad92818537ced61f211e8620b229e5bbd135fd43d","source_sha256":"a9f42fb2a650677b7f31f134a9fdb63cf294a003ee4e030eb51993a29af692d7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Iff.comm]\n  obtain ⟨s₁, rfl⟩ := s₁.exists_eq_ofList\n  obtain ⟨s₂, rfl⟩ := s₂.exists_eq_ofList\n  simp only [lt_iff_toList_lt, String.Legacy.iter, String.Legacy.mkIterator, String.toList_ofList]\n  induction s₁ generalizing s₂ <;> cases s₂\n  · unfold ltb; decide\n  · rename_i c₂ cs₂; apply iff_of_true\n    · unfold ltb\n      simp [Legacy.Iterator.hasNext, Char.utf8Size_pos]\n    · apply List.nil_lt_cons\n  · rename_i c₁ cs₁ ih; apply iff_of_false\n    · unfold ltb\n      simp [Legacy.Iterator.hasNext]\n    · apply not_lt_of_gt; apply List.nil_lt_cons\n  · rename_i c₁ cs₁ ih c₂ cs₂; unfold ltb\n    simp only [Legacy.Iterator.hasNext, Pos.Raw.byteIdx_zero, rawEndPos_ofList, utf8Len_cons,\n      add_pos_iff, Char.utf8Size_pos, or_true, decide_true, ↓reduceIte, Legacy.Iterator.curr,\n      Pos.Raw.get, String.toList_ofList, Pos.Raw.utf8GetAux, Legacy.Iterator.next, Pos.Raw.next,\n      Bool.ite_eq_true_distrib, decide_eq_true_eq]\n    split_ifs with h\n    · subst c₂\n      suffices ltb ⟨ofList (c₁ :: cs₁), (0 : Pos.Raw) + c₁⟩\n          ⟨ofList (c₁ :: cs₂), (0 : Pos.Raw) + c₁⟩ =\n            ltb ⟨ofList cs₁, 0⟩ ⟨ofList cs₂, 0⟩ by\n        rw [this]; exact (ih cs₂).trans List.lex_cons_iff.symm\n      apply ltb_cons_addChar\n    · refine ⟨List.Lex.rel, fun e ↦ ?_⟩\n      cases e <;> rename_i h'\n      · assumption\n      · contradiction","hard_negative":true,"metrics":{"chosen_tokens":329,"rejected_tokens":8,"token_jaccard":0.030303,"token_length_ratio":0.024316},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"97b860f5cab14977a1271c9c93d88b68bae13b8792ccc8e0b9508ca3957652c8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.String.Lemmas\npublic import Mathlib.Data.List.Lex\npublic import Mathlib.Data.Char\npublic import Mathlib.Algebra.Order.Group.Nat\nimport all Init.Data.String.Iterator  -- for unfolding `Iterator.curr`\nimport all Init.Data.Ord.String  -- for unfolding `String.compare`\n\nNamespace:\nString\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Strings\n\nSupplementary theorems about the `String` type.\n-/\n\n@[expose] public section\n\nnamespace String\n\n/-- `<` on string iterators. This coincides with `<` on strings as lists. -/\ndef ltb (s₁ s₂ : Legacy.Iterator) : Bool :=\n  if s₂.hasNext then\n    if s₁.hasNext then\n      if s₁.curr = s₂.curr then\n        ltb s₁.next s₂.next\n      else s₁.curr < s₂.curr\n    else true\n  else false\n\n/-- Induction on `String.ltb`. -/\n@[no_expose] def ltb.inductionOn.{u} {motive : Legacy.Iterator → Legacy.Iterator → Sort u}\n    (it₁ it₂ : Legacy.Iterator)\n    (ind : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      i₁.get s₁ = i₂.get s₂ →\n        motive (Legacy.Iterator.next ⟨s₁, i₁⟩) (Legacy.Iterator.next ⟨s₂, i₂⟩) →\n          motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (eq : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      ¬ i₁.get s₁ = i₂.get s₂ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (base₁ : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → ¬ Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (base₂ : ∀ s₁ s₂ i₁ i₂, ¬ Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩) :\n    motive it₁ it₂ :=\n  if h₂ : it₂.hasNext then\n    if h₁ : it₁.hasNext then\n      if heq : it₁.curr = it₂.curr then\n        ind it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq (inductionOn it₁.next it₂.next ind eq base₁ base₂)\n      else eq it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq\n    else base₁ it₁.s it₂.s it₁.i it₂.i h₂ h₁\n  else base₂ it₁.s it₂.s it₁.i it₂.i h₂\n\ntheorem ltb_cons_addChar' (c : Char) (s₁ s₂ : Legacy.Iterator) :\n    ltb ⟨ofList (c :: s₁.s.toList), s₁.i + c⟩ ⟨ofList (c :: s₂.s.toList), s₂.i + c⟩ =\n      ltb s₁ s₂ := by\n  fun_induction ltb s₁ s₂ with\n  | case1 s₁ s₂ h₁ h₂ h ih =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_pos (by simpa using h₂), if_pos, ← ih]\n    · simp only [Legacy.Iterator.next, Pos.Raw.next, get_cons_addChar, ofList_toList]\n      congr 2 <;> apply Pos.Raw.add_char_right_comm\n    · simpa only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList] using h\n  | case2 s₁ s₂ h₁ h₂ h =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_pos (by simpa using h₂), if_neg]\n    · simp only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList, decide_eq_decide]\n    · simpa only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList] using h\n  | case3 s₁ s₂ h₁ h₂ =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_neg (by simpa using h₂)]\n  | case4 s₁ s₂ h₁ =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, if_neg (by simpa using h₁)]\n\ntheorem ltb_cons_addChar (c : Char) (cs₁ cs₂ : List Char) (i₁ i₂ : Pos.Raw) :\n    ltb ⟨ofList (c :: cs₁), i₁ + c⟩ ⟨ofList (c :: cs₂), i₂ + c⟩ =\n      ltb ⟨ofList cs₁, i₁⟩ ⟨ofList cs₂, i₂⟩ := by\n  rw [eq_comm, ← ltb_cons_addChar' c]\n  simp\n\ntheorem lt_iff_toList_lt {s₁ s₂ : String} : s₁ < s₂ ↔ s₁.toList < s₂.toList :=\n  Iff.rfl\n\n@[simp]\n\nTarget:\ntheorem lt_iff_ltb {s₁ s₂ : String} :\n    s₁ < s₂ ↔ ltb (String.Legacy.iter s₁) (String.Legacy.iter s₂) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"048a0061ab271ef253655400b9b43ee00fad1618e82571b7a796b34b47ff1cd6","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/String","family_id":"lt_iff_ltb","file_id":"mathlib/Mathlib/Data/String/Basic.lean","sample_id":"e0f31a2fa4b167750af0775ad92818537ced61f211e8620b229e5bbd135fd43d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5416e2b8428ca9ff8cc9626edea3896864ca77c250a86dc31c56a1ba22b8daf5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0a9c671be9f78b6b39ea625e04972d8197ddb17307e1821a039f2e97cb91c77d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"91fa9e01efb72d4bb46c66a3c3321fad75748aa4aba9170a103de81feb5b067d","source_sha256":"c9b2b17bec68389df88ab608b3808c19ca8eaaffb1ef5c857eb1b5ade046a67e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold Measure.mconv\n  rw [integral_map (by fun_prop) hf.1, integral_prod]\n  exact (integrable_map_measure hf.1 (by fun_prop)).mp hf","hard_negative":false,"metrics":{"chosen_tokens":32,"rejected_tokens":39,"token_jaccard":0.904762,"token_length_ratio":1.21875},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"97d4c78baa86f065ced236a308ed68b6dc4da72b4f37d9840a8b30e6cc788a08","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Group.Convolution\npublic import Mathlib.MeasureTheory.Integral.Prod\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n-/\n/-!\n# Bochner integrals of convolutions\n\nThis file contains results about the Bochner integrals of convolutions of measures.\n\nThese results are not placed in the main convolution file because we don't want to import Bochner\nintegrals over there.\n\n## Main statements\n\n* `integrable_conv_iff`: A function is integrable with respect to the convolution `μ ∗ ν` iff\n  the function `y ↦ f (x + y)` is integrable with respect to `ν` for `μ`-almost every `x` and\n  the function `x ↦ ∫ y, ‖f (x + y)‖ ∂ν` is integrable with respect to `μ`.\n* `integral_conv`: if `f` is integrable with respect to the convolution `μ ∗ ν`, then\n  `∫ x, f x ∂(μ ∗ₘ ν) = ∫ x, ∫ y, f (x + y) ∂ν ∂μ`.\n-/\n\npublic section\n\nnamespace MeasureTheory\n\nvariable {M F : Type*} [Monoid M] {mM : MeasurableSpace M} [MeasurableMul₂ M]\n  [NormedAddCommGroup F] {μ ν : Measure M} {f : M → F}\n\n@[to_additive]\nlemma integrable_mconv_iff [SFinite ν] (hf : AEStronglyMeasurable f (μ ∗ₘ ν)) :\n    Integrable f (μ ∗ₘ ν)\n      ↔ (∀ᵐ x ∂μ, Integrable (fun y ↦ f (x * y)) ν)\n        ∧ Integrable (fun x ↦ ∫ y, ‖f (x * y)‖ ∂ν) μ := by\n  simp [Measure.mconv, integrable_map_measure hf (by fun_prop),\n    integrable_prod_iff (hf.comp_measurable (by fun_prop))]\n\n@[to_additive]\n\nTarget:\nlemma integral_mconv [NormedSpace ℝ F] [SFinite μ] [SFinite ν] (hf : Integrable f (μ ∗ₘ ν)) :\n    ∫ x, f x ∂(μ ∗ₘ ν) = ∫ x, ∫ y, f (x * y) ∂ν ∂μ :=\n\nProof body:\n","rejected":"```lean\nby\n  unfold Measure.mconv\n  rw [integral_map (by fun_prop) hf.1, integral_prod]\n  exact (integrable_map_measure hf.1 (by fun_prop)).mp hf\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Group","family_id":"integral_mconv","file_id":"mathlib/Mathlib/MeasureTheory/Group/IntegralConvolution.lean","sample_id":"91fa9e01efb72d4bb46c66a3c3321fad75748aa4aba9170a103de81feb5b067d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c337966156f7fa0b44c661383f2baec35f6094ce1c36181a113cc67c17fd2ac2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"27381825cd323e32e19bcfb43a0af2db0042d824fc11ccf2a2aaa45cfb2bd84a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4771507a2edaa490165ad6dbbb8bbee190b6099c2dfd923f1e5c8b47e63ba2b7","source_sha256":"7fb54be1edb74454dd74d15ac42b7739b2a4aa95bc5649191f339f2298801887","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases c <;> simp [*]","hard_negative":true,"metrics":{"chosen_tokens":10,"rejected_tokens":5,"token_jaccard":0.071429,"token_length_ratio":0.5},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"9820b6c1733f1ef51279c5a3fea94bec792e240dfc7e1191548ee1d3eec3bb2c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Ordering.Basic\npublic import Mathlib.Order.Defs.Unbundled\n\nNamespace:\nOrdering\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Some `Ordering` lemmas\n-/\n\npublic section\n\nuniverse u\n\nnamespace Ordering\n\n@[simp]\ntheorem ite_eq_lt_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.lt) = if c then a = Ordering.lt else b = Ordering.lt := by\n  by_cases c <;> simp [*]\n\n@[simp]\ntheorem ite_eq_eq_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.eq) = if c then a = Ordering.eq else b = Ordering.eq := by\n  by_cases c <;> simp [*]\n\n@[simp]\n\nTarget:\ntheorem ite_eq_gt_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.gt) = if c then a = Ordering.gt else b = Ordering.gt :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_4771507a2eda","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"7978311a381272ea846814facc81a9663890bface912b0c8eb134e18a5bfb5cc","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Ordering","family_id":"ite_eq_gt_distrib","file_id":"mathlib/Mathlib/Data/Ordering/Lemmas.lean","sample_id":"4771507a2edaa490165ad6dbbb8bbee190b6099c2dfd923f1e5c8b47e63ba2b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d6d98f4113b14782373b73a8bb3f9c78ad64b883beec8e3deb78731e5a49b554","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9864f227c35a74c0fbf4907c25450f6f1a0580309851c1704d9ed1eaa95ed2e7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7e250c2a0a366c6dc511bcc2bdae4c79d5eb964d8cab919edc4393813525710c","source_sha256":"a9f42fb2a650677b7f31f134a9fdb63cf294a003ee4e030eb51993a29af692d7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [← toList_inj]","hard_negative":true,"metrics":{"chosen_tokens":6,"rejected_tokens":2,"token_jaccard":0.142857,"token_length_ratio":0.333333},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"9877080e2db5e3aec6f116c9b7bde8bbb00c6997655155cbf7a88118471a81ef","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.String.Lemmas\npublic import Mathlib.Data.List.Lex\npublic import Mathlib.Data.Char\npublic import Mathlib.Algebra.Order.Group.Nat\nimport all Init.Data.String.Iterator  -- for unfolding `Iterator.curr`\nimport all Init.Data.Ord.String  -- for unfolding `String.compare`\n\nNamespace:\nString\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Strings\n\nSupplementary theorems about the `String` type.\n-/\n\n@[expose] public section\n\nnamespace String\n\n/-- `<` on string iterators. This coincides with `<` on strings as lists. -/\ndef ltb (s₁ s₂ : Legacy.Iterator) : Bool :=\n  if s₂.hasNext then\n    if s₁.hasNext then\n      if s₁.curr = s₂.curr then\n        ltb s₁.next s₂.next\n      else s₁.curr < s₂.curr\n    else true\n  else false\n\n/-- Induction on `String.ltb`. -/\n@[no_expose] def ltb.inductionOn.{u} {motive : Legacy.Iterator → Legacy.Iterator → Sort u}\n    (it₁ it₂ : Legacy.Iterator)\n    (ind : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      i₁.get s₁ = i₂.get s₂ →\n        motive (Legacy.Iterator.next ⟨s₁, i₁⟩) (Legacy.Iterator.next ⟨s₂, i₂⟩) →\n          motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (eq : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      ¬ i₁.get s₁ = i₂.get s₂ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (base₁ : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → ¬ Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (base₂ : ∀ s₁ s₂ i₁ i₂, ¬ Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩) :\n    motive it₁ it₂ :=\n  if h₂ : it₂.hasNext then\n    if h₁ : it₁.hasNext then\n      if heq : it₁.curr = it₂.curr then\n        ind it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq (inductionOn it₁.next it₂.next ind eq base₁ base₂)\n      else eq it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq\n    else base₁ it₁.s it₂.s it₁.i it₂.i h₂ h₁\n  else base₂ it₁.s it₂.s it₁.i it₂.i h₂\n\ntheorem ltb_cons_addChar' (c : Char) (s₁ s₂ : Legacy.Iterator) :\n    ltb ⟨ofList (c :: s₁.s.toList), s₁.i + c⟩ ⟨ofList (c :: s₂.s.toList), s₂.i + c⟩ =\n      ltb s₁ s₂ := by\n  fun_induction ltb s₁ s₂ with\n  | case1 s₁ s₂ h₁ h₂ h ih =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_pos (by simpa using h₂), if_pos, ← ih]\n    · simp only [Legacy.Iterator.next, Pos.Raw.next, get_cons_addChar, ofList_toList]\n      congr 2 <;> apply Pos.Raw.add_char_right_comm\n    · simpa only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList] using h\n  | case2 s₁ s₂ h₁ h₂ h =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_pos (by simpa using h₂), if_neg]\n    · simp only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList, decide_eq_decide]\n    · simpa only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList] using h\n  | case3 s₁ s₂ h₁ h₂ =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_neg (by simpa using h₂)]\n  | case4 s₁ s₂ h₁ =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, if_neg (by simpa using h₁)]\n\ntheorem ltb_cons_addChar (c : Char) (cs₁ cs₂ : List Char) (i₁ i₂ : Pos.Raw) :\n    ltb ⟨ofList (c :: cs₁), i₁ + c⟩ ⟨ofList (c :: cs₂), i₂ + c⟩ =\n      ltb ⟨ofList cs₁, i₁⟩ ⟨ofList cs₂, i₂⟩ := by\n  rw [eq_comm, ← ltb_cons_addChar' c]\n  simp\n\ntheorem lt_iff_toList_lt {s₁ s₂ : String} : s₁ < s₂ ↔ s₁.toList < s₂.toList :=\n  Iff.rfl\n\n@[simp]\ntheorem lt_iff_ltb {s₁ s₂ : String} :\n    s₁ < s₂ ↔ ltb (String.Legacy.iter s₁) (String.Legacy.iter s₂) := by\n  rw [Iff.comm]\n  obtain ⟨s₁, rfl⟩ := s₁.exists_eq_ofList\n  obtain ⟨s₂, rfl⟩ := s₂.exists_eq_ofList\n  simp only [lt_iff_toList_lt, String.Legacy.iter, String.Legacy.mkIterator, String.toList_ofList]\n  induction s₁ generalizing s₂ <;> cases s₂\n  · unfold ltb; decide\n  · rename_i c₂ cs₂; apply iff_of_true\n    · unfold ltb\n      simp [Legacy.Iterator.hasNext, Char.utf8Size_pos]\n    · apply List.nil_lt_cons\n  · rename_i c₁ cs₁ ih; apply iff_of_false\n    · unfold ltb\n      simp [Legacy.Iterator.hasNext]\n    · apply not_lt_of_gt; apply List.nil_lt_cons\n  · rename_i c₁ cs₁ ih c₂ cs₂; unfold ltb\n    simp only [Legacy.Iterator.hasNext, Pos.Raw.byteIdx_zero, rawEndPos_ofList, utf8Len_cons,\n      add_pos_iff, Char.utf8Size_pos, or_true, decide_true, ↓reduceIte, Legacy.Iterator.curr,\n      Pos.Raw.get, String.toList_ofList, Pos.Raw.utf8GetAux, Legacy.Iterator.next, Pos.Raw.next,\n      Bool.ite_eq_true_distrib, decide_eq_true_eq]\n    split_ifs with h\n    · subst c₂\n      suffices ltb ⟨ofList (c₁ :: cs₁), (0 : Pos.Raw) + c₁⟩\n          ⟨ofList (c₁ :: cs₂), (0 : Pos.Raw) + c₁⟩ =\n            ltb ⟨ofList cs₁, 0⟩ ⟨ofList cs₂, 0⟩ by\n        rw [this]; exact (ih cs₂).trans List.lex_cons_iff.symm\n      apply ltb_cons_addChar\n    · refine ⟨List.Lex.rel, fun e ↦ ?_⟩\n      cases e <;> rename_i h'\n      · assumption\n      · contradiction\n\n@[deprecated \"Use the new String API\" (since := \"2026-04-01\")]\ntheorem toList_nonempty :\n    ∀ {s : String}, s ≠ \"\" → s.toList = String.Legacy.front s :: (String.Legacy.drop s 1).toList\n  | s, h => by\n    obtain ⟨l, rfl⟩ := s.exists_eq_ofList\n    match l with\n    | [] => simp at h\n    | c::cs => simp [Legacy.front, Pos.Raw.get, Pos.Raw.utf8GetAux]\n\n@[simp]\ntheorem head_empty : \"\".toList.head! = default :=\n  rfl\n\nprivate theorem le_iff_not_lt {s₁ s₂ : String} : s₁ ≤ s₂ ↔ ¬ s₂ < s₁ :=\n  Iff.rfl\n\ntheorem le_iff_toList_le {s₁ s₂ : String} : s₁ ≤ s₂ ↔ s₁.toList ≤ s₂.toList := by\n  rw [String.le_iff_not_lt, lt_iff_toList_lt, not_lt]\n\ninstance : LinearOrder String where\n  le_refl _ := le_iff_toList_le.mpr le_rfl\n  le_trans a b c := by\n    simp only [le_iff_toList_le]\n    apply le_trans\n  lt_iff_le_not_ge a b := by\n    simp only [lt_iff_toList_lt, le_iff_toList_le, lt_iff_le_not_ge]\n  le_antisymm a b := by\n    simp only [le_iff_toList_le, ← toList_inj]\n    apply le_antisymm\n  le_total a b := by\n    simp only [le_iff_toList_le]\n    apply le_total\n  toDecidableLE := inferInstance\n  toDecidableEq := inferInstance\n  toDecidableLT := String.decidableLT\n  compare_eq_compareOfLessAndEq a b := by simp [Ord.compare, String.compare]\n\nTarget:\ntheorem ofList_eq {l : List Char} {s : String} : ofList l = s ↔ l = s.toList :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_7e250c2a0a36","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"93521dac595be4f3b0345f5f9145f1b7c69e5813d522c0e2b542bab2a0a04911","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/String","family_id":"oflist_eq","file_id":"mathlib/Mathlib/Data/String/Basic.lean","sample_id":"7e250c2a0a366c6dc511bcc2bdae4c79d5eb964d8cab919edc4393813525710c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e1c4c37ea1905b854b1db5a6620e0ed4d40197fe584c2e9cb453bf8d89ed1d01","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"322ab09370ffc7ba02e5b0d458715ab5e05e1b425cf1b69dc3d33d89b75fd5ba","source_sha256":"c1ecc80ead99a17d01db2984c5f06df1499354702fc192601863dee22fbd7b8f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using ideal_biInf I Set.finite_univ","hard_negative":false,"metrics":{"chosen_tokens":8,"rejected_tokens":3,"token_jaccard":0.1,"token_length_ratio":0.375},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"987729c212f0231cb03a5f250fbf2feef9c28d2b82c41baad07d44f7da76c554","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact\npublic import Mathlib.AlgebraicGeometry.Properties\npublic import Mathlib.Tactic.DepRewrite\n\nNamespace:\nAlgebraicGeometry.Scheme.IdealSheafData\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Ideal sheaves on schemes\n\nWe define ideal sheaves of schemes and provide various constructors for it.\n\n## Main definition\n* `AlgebraicGeometry.Scheme.IdealSheafData`: A structure that contains the data to uniquely define\n  an ideal sheaf, consisting of\n  1. an ideal `I(U) ≤ Γ(X, U)` for every affine open `U`\n  2. a proof that `I(D(f)) = I(U)_f` for every affine open `U` and every section `f : Γ(X, U)`.\n* `AlgebraicGeometry.Scheme.IdealSheafData.ofIdeals`:\n  The largest ideal sheaf contained in a family of ideals.\n* `AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine`:\n  Over affine schemes, ideal sheaves are in bijection with ideals of the global sections.\n* `AlgebraicGeometry.Scheme.IdealSheafData.support`: The support of an ideal sheaf.\n* `AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal`: The vanishing ideal of a set.\n* `AlgebraicGeometry.Scheme.Hom.ker`: The kernel of a morphism.\n\n## Main results\n* `AlgebraicGeometry.Scheme.IdealSheafData.gc`:\n  `support` and `vanishingIdeal` forms a Galois connection.\n* `AlgebraicGeometry.Scheme.Hom.support_ker`: The support of a kernel of a quasi-compact morphism\n  is the closure of the range.\n\n## Implementation detail\n\nIdeal sheaves are not yet defined in this file as actual subsheaves of `𝒪ₓ`.\nInstead, for the ease of development and application,\nwe define the structure `IdealSheafData` containing all necessary data to uniquely define an\nideal sheaf. This should be refactored as a constructor for ideal sheaves once they are introduced\ninto mathlib.\n\n-/\n\n@[expose] public section\n\nopen CategoryTheory TopologicalSpace\n\nuniverse u\n\nnamespace AlgebraicGeometry.Scheme\n\nvariable {X : Scheme.{u}}\n\n/--\nA structure that contains the data to uniquely define an ideal sheaf, consisting of\n1. an ideal `I(U) ≤ Γ(X, U)` for every affine open `U`\n2. a proof that `I(D(f)) = I(U)_f` for every affine open `U` and every section `f : Γ(X, U)`\n3. a subset of `X` equal to the support.\n\nAlso see `Scheme.IdealSheafData.mkOfMemSupportIff` for a constructor with the condition on the\nsupport being (usually) easier to prove.\n-/\nstructure IdealSheafData (X : Scheme.{u}) : Type u where\n  /-- The component of an ideal sheaf at an affine open. -/\n  ideal : ∀ U : X.affineOpens, Ideal Γ(X, U)\n  /-- Also see `AlgebraicGeometry.Scheme.IdealSheafData.map_ideal` -/\n  map_ideal_basicOpen : ∀ (U : X.affineOpens) (f : Γ(X, U)),\n    (ideal U).map (X.presheaf.map (homOfLE <| X.basicOpen_le f).op).hom =\n      ideal (X.affineBasicOpen f)\n  /-- The support of an ideal sheaf. Use `IdealSheafData.support` instead for most occasions. -/\n  supportSet : Set X := ⋂ U, X.zeroLocus (U := U.1) (ideal U)\n  supportSet_eq_iInter_zeroLocus : supportSet = ⋂ U, X.zeroLocus (U := U.1) (ideal U) := by rfl\n\nnamespace IdealSheafData\n\n@[ext]\nprotected lemma ext {I J : X.IdealSheafData} (h : I.ideal = J.ideal) : I = J := by\n  obtain ⟨i, _, s, hs⟩ := I\n  obtain ⟨j, _, t, ht⟩ := J\n  subst h\n  congr\n  rw [hs, ht]\n\nsection Order\n\ninstance : PartialOrder (IdealSheafData X) := PartialOrder.lift ideal fun _ _ ↦ IdealSheafData.ext\n\nlemma le_def {I J : IdealSheafData X} : I ≤ J ↔ ∀ U, I.ideal U ≤ J.ideal U := .rfl\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : CompleteSemilatticeSup (IdealSheafData X) where\n  sSup s :=\n  { ideal := sSup (ideal '' s),\n    map_ideal_basicOpen := by\n      have : sSup (ideal '' s) = ⨆ i : s, ideal i.1 := by\n        conv_lhs => rw [← Subtype.range_val (s := s), ← Set.range_comp]\n        rfl\n      simp only [this, iSup_apply, Ideal.map_iSup, map_ideal_basicOpen, implies_true] }\n  isLUB_sSup _ := .of_image (f := ideal) le_def (isLUB_sSup _)\n\n/-- The largest ideal sheaf contained in a family of ideals. -/\ndef ofIdeals (I : ∀ U : X.affineOpens, Ideal Γ(X, U)) : IdealSheafData X :=\n  sSup { J : IdealSheafData X | J.ideal ≤ I }\n\nlemma ideal_ofIdeals_le (I : ∀ U : X.affineOpens, Ideal Γ(X, U)) :\n    (ofIdeals I).ideal ≤ I :=\n  sSup_le (Set.forall_mem_image.mpr fun _ ↦ id)\n\n/-- The Galois coinsertion between ideal sheaves and arbitrary families of ideals. -/\nprotected def gci : GaloisCoinsertion ideal (ofIdeals (X := X)) where\n  choice I hI :=\n  { ideal := I\n    map_ideal_basicOpen U f :=\n      (ideal_ofIdeals_le I).antisymm hI ▸ (ofIdeals I).map_ideal_basicOpen U f }\n  gc _ _ := ⟨(le_sSup ·), (le_trans · (ideal_ofIdeals_le _))⟩\n  u_l_le _ := sSup_le fun _ ↦ id\n  choice_eq I hI := IdealSheafData.ext (hI.antisymm (ideal_ofIdeals_le I))\n\nlemma strictMono_ideal : StrictMono (ideal (X := X)) := IdealSheafData.gci.strictMono_l\nlemma ideal_mono : Monotone (ideal (X := X)) := strictMono_ideal.monotone\nlemma ofIdeals_mono : Monotone (ofIdeals (X := X)) := IdealSheafData.gci.gc.monotone_u\nlemma ofIdeals_ideal (I : IdealSheafData X) : ofIdeals I.ideal = I := IdealSheafData.gci.u_l_eq _\nlemma le_ofIdeals_iff {I : IdealSheafData X} {J} : I ≤ ofIdeals J ↔ I.ideal ≤ J :=\n  IdealSheafData.gci.gc.le_iff_le.symm\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : OrderTop (IdealSheafData X) where\n  top.ideal := ⊤\n  top.map_ideal_basicOpen := by simp [Ideal.map_top]\n  top.supportSet := ⊥\n  top.supportSet_eq_iInter_zeroLocus := by\n    ext x\n    simpa using X.isBasis_affineOpens.exists_subset_of_mem_open (Set.mem_univ x) isOpen_univ\n  le_top I U := le_top\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : OrderBot (IdealSheafData X) where\n  bot.ideal := ⊥\n  bot.map_ideal_basicOpen := by simp\n  bot.supportSet := ⊤\n  bot.supportSet_eq_iInter_zeroLocus := by ext; simp\n  bot_le I U := bot_le\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : SemilatticeInf (IdealSheafData X) where\n  inf I J :=\n  { ideal := I.ideal ⊓ J.ideal\n    map_ideal_basicOpen U f := by\n      dsimp\n      have : (X.presheaf.map (homOfLE (X.basicOpen_le f)).op).hom = algebraMap _ _ := rfl\n      have inst := U.2.isLocalization_basicOpen f\n      rw [← I.map_ideal_basicOpen U f, ← J.map_ideal_basicOpen U f, this]\n      ext x\n      obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq (.powers f) x\n      simp only [IsLocalization.mk'_mem_map_algebraMap_iff, Submonoid.mem_powers_iff, Ideal.mem_inf,\n        exists_exists_eq_and]\n      refine ⟨fun ⟨n, h₁, h₂⟩ ↦ ⟨⟨n, h₁⟩, ⟨n, h₂⟩⟩, ?_⟩\n      rintro ⟨⟨n₁, h₁⟩, ⟨n₂, h₂⟩⟩\n      refine ⟨n₁ + n₂, ?_, ?_⟩\n      · rw [add_comm, pow_add, mul_assoc]; exact Ideal.mul_mem_left _ _ h₁\n      · rw [pow_add, mul_assoc]; exact Ideal.mul_mem_left _ _ h₂ }\n  inf_le_left I J U := inf_le_left\n  inf_le_right I J U := inf_le_right\n  le_inf I J K hIJ hIK U := le_inf (hIJ U) (hIK U)\n\ninstance : CompleteLattice (IdealSheafData X) where\n  __ := (inferInstance : OrderTop (IdealSheafData X))\n  __ := (inferInstance : OrderBot (IdealSheafData X))\n  __ := (inferInstance : SemilatticeInf (IdealSheafData X))\n  __ := (inferInstance : CompleteSemilatticeSup (IdealSheafData X))\n  __ := IdealSheafData.gci.liftCompleteLattice\n\n@[simp]\nlemma ideal_top : ideal (X := X) ⊤ = ⊤ := rfl\n\n@[simp]\nlemma ideal_bot : ideal (X := X) ⊥ = ⊥ := rfl\n\n@[simp]\nlemma ideal_sup {I J : IdealSheafData X} : (I ⊔ J).ideal = I.ideal ⊔ J.ideal := rfl\n\n@[simp]\nlemma ideal_sSup {I : Set (IdealSheafData X)} : (sSup I).ideal = sSup (ideal '' I) := rfl\n\n@[simp]\nlemma ideal_iSup {ι : Type*} {I : ι → IdealSheafData X} : (iSup I).ideal = ⨆ i, (I i).ideal := by\n  rw [← sSup_range, ← sSup_range, ideal_sSup, ← Set.range_comp, Function.comp_def]\n\n@[simp]\nlemma ideal_inf {I J : IdealSheafData X} : (I ⊓ J).ideal = I.ideal ⊓ J.ideal := rfl\n\n@[simp]\nlemma ideal_biInf {ι : Type*} (I : ι → IdealSheafData X) {s : Set ι} (hs : s.Finite) :\n    (⨅ i ∈ s, I i).ideal = ⨅ i ∈ s, (I i).ideal := by\n  refine hs.induction_on _ (by simp) fun {i s} his hs e ↦ ?_\n  simp only [iInf_insert, e, ideal_inf]\n\n@[simp]\n\nTarget:\nlemma ideal_iInf {ι : Type*} (I : ι → IdealSheafData X) [Finite ι] :\n    (⨅ i, I i).ideal = ⨅ i, (I i).ideal :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/IdealSheaf","family_id":"ideal_iinf","file_id":"mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/Basic.lean","sample_id":"322ab09370ffc7ba02e5b0d458715ab5e05e1b425cf1b69dc3d33d89b75fd5ba"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5a23d360fd69064fcfa72d36f2b8acb7fa78696675236423d19ef91035657ba8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"45bdf565a874c8590b8b9e375ecaaa19d5c42795fb168fac5a8abcf519632618","source_sha256":"7e17e88add6ba664803471632353df2ce7dad3fce436e36f21eedb5e3c301b76","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have _ := t.adj₂.isLeftAdjoint\n  refine ⟨fun h ↦ by rw [← map_rightToLeft_app]; exact G.map_epi _, fun h ↦ ?_⟩\n  rw [epi_rightToLeft_app_iff_epi_map_adj₁_unit_app]\n  simpa using epi_comp (t.adj₂.unit.app (H.obj X)) (H.map (t.adj₂.counit.app X ≫ t.adj₁.unit.app X))","hard_negative":false,"metrics":{"chosen_tokens":85,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.035294},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"9880c5cb4932518794dcfdd1f10f8bb74767dbc571e68a39f0c79916c117e868","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Adjunction.Opposites\npublic import Mathlib.CategoryTheory.Adjunction.Unique\npublic import Mathlib.CategoryTheory.Monad.Adjunction\n\nNamespace:\nCategoryTheory.Adjunction.Triple\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson, Ben Eltschig\n-/\n/-!\n\n# Adjoint triples\n\nThis file concerns adjoint triples `F ⊣ G ⊣ H` of functors `F H : C ⥤ D`, `G : D ⥤ C`. We first\nprove that `F` is fully faithful iff `H` is, and then prove results about the two special cases\nwhere `G` is fully faithful or `F` and `H` are.\n\n## Main results\n\nAll results are about an adjoint triple `F ⊣ G ⊣ H` where `adj₁ : F ⊣ G` and `adj₂ : G ⊣ H`. We\nbundle the adjunctions in a structure `Triple F G H`.\n* `fullyFaithfulEquiv`: `F` is fully faithful iff `H` is.\n* `rightToLeft`: the canonical natural transformation `H ⟶ F` that exists whenever `G` is fully\n  faithful. This is defined as the preimage of `adj₂.counit ≫ adj₁.unit` under whiskering with `G`,\n  but formulas in terms of the units resp. counits of the adjunctions are also given.\n* `whiskerRight_rightToLeft`: whiskering `rightToLeft : H ⟶ F` with `G` yields\n  `adj₂.counit ≫ adj₁.unit : H ⋙ G ⟶ F ⋙ G`.\n* `epi_rightToLeft_app_iff_epi_map_adj₁_unit_app`: `rightToLeft : H ⟶ F` is epic at `X` iff the\n  image of `adj₁.unit.app X` under `H` is.\n* `epi_rightToLeft_app_iff_epi_map_adj₂_counit_app`: `rightToLeft : H ⟶ F` is epic at `X` iff the\n  image of `adj₂.counit.app X` under `F` is.\n* `epi_rightToLeft_app_iff`: when `H` preserves epimorphisms, `rightToLeft : H ⟶ F` is epic at `X`\n  iff `adj₂.counit ≫ adj₁.unit : H ⋙ G ⟶ F ⋙ G` is.\n* `leftToRight`: the canonical natural transformation `F ⟶ H` that exists whenever `F` and `H` are\n  fully faithful. This is defined in terms of the units of the adjunctions, but a formula in terms\n  of the counits is also given.\n* `whiskerLeft_leftToRight`: whiskering `G` with `leftToRight : F ⟶ H` yields\n  `adj₁.counit ≫ adj₂.unit : G ⋙ F ⟶ G ⋙ H`.\n* `mono_leftToRight_app_iff_mono_adj₂_unit_app`: `leftToRight : F ⟶ H` is monic at `X` iff\n  `adj₂.unit` is monic at `F.obj X`.\n* `mono_leftToRight_app_iff_mono_adj₁_counit_app`: `leftToRight : F ⟶ H` is monic at `X` iff\n  `adj₁.counit` is monic at `H.obj X`.\n* `mono_leftToRight_app_iff`: `leftToRight : F ⟶ H` is componentwise monic iff\n  `adj₁.counit ≫ adj₂.unit : G ⋙ F ⟶ G ⋙ H` is.\n-/\n\n@[expose] public section\n\nopen CategoryTheory Functor\n\nvariable {C D : Type*} [Category* C] [Category* D]\nvariable (F : C ⥤ D) (G : D ⥤ C) (H : C ⥤ D)\n\n/-- Structure containing the two adjunctions of an adjoint triple `F ⊣ G ⊣ H`. -/\nstructure CategoryTheory.Adjunction.Triple where\n  /-- Adjunction `F ⊣ G` of the adjoint triple `F ⊣ G ⊣ H`. -/\n  adj₁ : F ⊣ G\n  /-- Adjunction `G ⊣ H` of the adjoint triple `F ⊣ G ⊣ H`. -/\n  adj₂ : G ⊣ H\n\nnamespace CategoryTheory.Adjunction.Triple\n\nvariable {F G H} (t : Triple F G H)\n\nlemma isIso_unit_iff_isIso_counit : IsIso t.adj₁.unit ↔ IsIso t.adj₂.counit := by\n  let adj : F ⋙ G ⊣ H ⋙ G := t.adj₁.comp t.adj₂\n  constructor\n  · intro h\n    let idAdj : 𝟭 C ⊣ H ⋙ G := adj.ofNatIsoLeft (asIso t.adj₁.unit).symm\n    exact t.adj₂.isIso_counit_of_iso (idAdj.rightAdjointUniq id)\n  · intro h\n    let adjId : F ⋙ G ⊣ 𝟭 C := adj.ofNatIsoRight (asIso t.adj₂.counit)\n    exact t.adj₁.isIso_unit_of_iso (adjId.leftAdjointUniq id)\n\n/--\nGiven an adjoint triple `F ⊣ G ⊣ H`, the left adjoint `F` is fully faithful if and only if the\nright adjoint `H` is fully faithful.\n-/\nnoncomputable def fullyFaithfulEquiv : F.FullyFaithful ≃ H.FullyFaithful where\n  toFun h :=\n    haveI := h.full\n    haveI := h.faithful\n    haveI : IsIso t.adj₂.counit := by\n      rw [← t.isIso_unit_iff_isIso_counit]\n      infer_instance\n    t.adj₂.fullyFaithfulROfIsIsoCounit\n  invFun h :=\n    haveI := h.full\n    haveI := h.faithful\n    haveI : IsIso t.adj₁.unit := by\n      rw [t.isIso_unit_iff_isIso_counit]\n      infer_instance\n    t.adj₁.fullyFaithfulLOfIsIsoUnit\n  left_inv _ := Subsingleton.elim _ _\n  right_inv _ := Subsingleton.elim _ _\n\n/-- The adjoint triple `H.op ⊣ G.op ⊣ F.op` dual to an adjoint triple `F ⊣ G ⊣ H`. -/\n@[simps]\nprotected def op : Triple H.op G.op F.op where\n  adj₁ := t.adj₂.op\n  adj₂ := t.adj₁.op\n\nsection InnerFullyFaithful\n\nvariable [G.Full] [G.Faithful]\n\n/-- The natural transformation `H ⟶ F` that exists for every adjoint triple `F ⊣ G ⊣ H` where `G`\nis fully faithful, given here as the preimage of `adj₂.counit ≫ adj₁.unit : H ⋙ G ⟶ F ⋙ G`\nunder whiskering with `G`. -/\nnoncomputable def rightToLeft : H ⟶ F :=\n  ((FullyFaithful.ofFullyFaithful G).whiskeringRight _).preimage (t.adj₂.counit ≫ t.adj₁.unit)\n\n/-- For an adjoint triple `F ⊣ G ⊣ H` where `G` is fully faithful, whiskering the natural\ntransformation `H ⟶ F` with `G` yields the composition of the counit of the second adjunction with\nthe unit of the first adjunction. -/\n@[simp, reassoc]\nlemma whiskerRight_rightToLeft : whiskerRight t.rightToLeft G = t.adj₂.counit ≫ t.adj₁.unit :=\n  ((FullyFaithful.ofFullyFaithful G).whiskeringRight _).map_preimage _\n\n/-- For an adjoint triple `F ⊣ G ⊣ H` where `G` is fully faithful, the images of the components of\nthe natural transformation `H ⟶ F` under `G` are the components of the composition of counit of the\nsecond adjunction with the unit of the first adjunction. -/\n@[simp, reassoc]\nlemma map_rightToLeft_app (X : C) :\n    G.map (t.rightToLeft.app X) = t.adj₂.counit.app X ≫ t.adj₁.unit.app X :=\n  congr_app t.whiskerRight_rightToLeft X\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The natural transformation `H ⟶ F` for an adjoint triple `F ⊣ G ⊣ H` with `G` fully faithful\nis also equal to the whiskered unit `H ⟶ F ⋙ G ⋙ H` of the first adjunction followed by the\ninverse of the whiskered unit `F ⟶ F ⋙ G ⋙ H` of the second. -/\nlemma rightToLeft_eq_units :\n    t.rightToLeft = H.leftUnitor.inv ≫ whiskerRight t.adj₁.unit H ≫ (Functor.associator _ _ _).hom ≫\n    inv (whiskerLeft F t.adj₂.unit) ≫ F.rightUnitor.hom := by\n  ext X; apply G.map_injective; simp [rightToLeft]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The natural transformation `H ⟶ F` for an adjoint triple `F ⊣ G ⊣ H` with `G` fully faithful\nis also equal to the inverse of the whiskered counit `H ⋙ G ⋙ F ⟶ H` of the first adjunction\nfollowed by the whiskered counit `H ⋙ G ⋙ F ⟶ F` of the second. -/\nlemma rightToLeft_eq_counits :\n    t.rightToLeft = H.rightUnitor.inv ≫ inv (whiskerLeft H t.adj₁.counit) ≫\n    (Functor.associator _ _ _).inv ≫ whiskerRight t.adj₂.counit F ≫ F.leftUnitor.hom := by\n  ext X; apply G.map_injective; simp [rightToLeft]\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma adj₁_counit_app_rightToLeft_app (X : C) :\n    t.adj₁.counit.app (H.obj X) ≫ t.rightToLeft.app X = F.map (t.adj₂.counit.app X) :=\n  G.map_injective (by simp [← cancel_epi (t.adj₁.unit.app _)])\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma rightToLeft_app_adj₂_unit_app (X : C) :\n    t.rightToLeft.app X ≫ t.adj₂.unit.app (F.obj X) = H.map (t.adj₁.unit.app X) :=\n  G.map_injective (by simp [← cancel_mono (t.adj₂.counit.app _)])\n\nset_option backward.defeqAttrib.useBackward true in\n/-- For an adjoint triple `F ⊣ G ⊣ H` where `G` is fully faithful, the natural transformation\n`F.op ⟶ H.op` obtained from the dual adjoint triple `H.op ⊣ G.op ⊣ F.op` is dual to the natural\ntransformation `H ⟶ F`. -/\n@[simp]\nlemma op_rightToLeft : t.op.rightToLeft = NatTrans.op t.rightToLeft := by\n  ext\n  rw [rightToLeft_eq_units, rightToLeft_eq_counits]\n  simp\n\nset_option backward.isDefEq.respectTransparency false in\n/-- For an adjoint triple `F ⊣ G ⊣ H` where `G` is fully faithful, the natural transformation\n`H ⟶ F` is epic at `X` iff the image of the unit of the adjunction `F ⊣ G` under `H` is. -/\nlemma epi_rightToLeft_app_iff_epi_map_adj₁_unit_app {X : C} :\n    Epi (t.rightToLeft.app X) ↔ Epi (H.map (t.adj₁.unit.app X)) := by\n  rw [← epi_comp_iff_of_isIso _ (t.adj₂.unit.app (F.obj X)), rightToLeft_app_adj₂_unit_app]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- For an adjoint triple `F ⊣ G ⊣ H` where `G` is fully faithful, the natural transformation\n`H ⟶ F` is epic at `X` iff the image of the counit of the adjunction `G ⊣ H` under `F` is. -/\nlemma epi_rightToLeft_app_iff_epi_map_adj₂_counit_app {X : C} :\n    Epi (t.rightToLeft.app X) ↔ Epi (F.map (t.adj₂.counit.app X)) := by\n  rw [← epi_comp_iff_of_epi (t.adj₁.counit.app (H.obj X)), adj₁_counit_app_rightToLeft_app]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- For an adjoint triple `F ⊣ G ⊣ H` where `G` is fully faithful and `H` preserves epimorphisms\n(which is for example the case if `H` has a further right adjoint), the components of the natural\ntransformation `H ⟶ F` are epic iff the respective components of the natural transformation\n`H ⋙ G ⟶ F ⋙ G` obtained from the units and counits of the adjunctions are. -/\n\nTarget:\nlemma epi_rightToLeft_app_iff [H.PreservesEpimorphisms] {X : C} :\n    Epi (t.rightToLeft.app X) ↔ Epi (t.adj₂.counit.app X ≫ t.adj₁.unit.app X) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Adjunction","family_id":"epi_righttoleft_app_iff","file_id":"mathlib/Mathlib/CategoryTheory/Adjunction/Triple.lean","sample_id":"45bdf565a874c8590b8b9e375ecaaa19d5c42795fb168fac5a8abcf519632618"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f258ff7f229ca2a8c793329e07d0fcda1df2d4bda138c89c5b29fe00a0404c2f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"76dc56f0319fe7f1f9c6eb862aa810c366c15b329cfaa84cfe66011828446e86","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1caba13c5bfb2e48e2aa824af4b9e227bed2165274d63dd634f59f34ffe47e2e","source_sha256":"e357f4d4f71c23dea84a87d11c8330d1561110f75700ec18541bc2103a8420d2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine summable_geometric_iff_norm_lt_one.mp ?_\n  simp_rw [← map_pow]\n  exact hsum.comp_injective <| Nat.pow_right_injective hp","hard_negative":true,"metrics":{"chosen_tokens":22,"rejected_tokens":3,"token_jaccard":0.095238,"token_length_ratio":0.136364},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"98a1a82616377f6b815d79483c3738fb673bd9d4081509e8ad626a1dd8987e4e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Ring.InfiniteSum\npublic import Mathlib.Analysis.SpecificLimits.Normed\npublic import Mathlib.NumberTheory.ArithmeticFunction.Defs\npublic import Mathlib.NumberTheory.SmoothNumbers\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Euler Products\n\nThe main result in this file is `EulerProduct.eulerProduct_hasProd`, which says that\nif `f : ℕ → R` is norm-summable, where `R` is a complete normed commutative ring and `f` is\nmultiplicative on coprime arguments with `f 0 = 0`, then\n`∏' p : Primes, ∑' e : ℕ, f (p^e)` converges to `∑' n, f n`.\n\n`ArithmeticFunction.IsMultiplicative.eulerProduct_hasProd` is a version\nfor multiplicative arithmetic functions in the sense of\n`ArithmeticFunction.IsMultiplicative`.\n\nThere is also a version `EulerProduct.eulerProduct_completely_multiplicative_hasProd`,\nwhich states that `∏' p : Primes, (1 - f p)⁻¹` converges to `∑' n, f n`\nwhen `f` is completely multiplicative with values in a complete normed field `F`\n(implemented as `f : ℕ →*₀ F`).\n\nThere are variants stating the equality of the infinite product and the infinite sum\n(`EulerProduct.eulerProduct_tprod`, `ArithmeticFunction.IsMultiplicative.eulerProduct_tprod`,\n`EulerProduct.eulerProduct_completely_multiplicative_tprod`) and also variants stating\nthe convergence of the sequence of partial products over primes `< n`\n(`EulerProduct.eulerProduct`, `ArithmeticFunction.IsMultiplicative.eulerProduct`,\n`EulerProduct.eulerProduct_completely_multiplicative`.)\n\nAn intermediate step is `EulerProduct.summable_and_hasSum_factoredNumbers_prod_filter_prime_tsum`\n(and its variant `EulerProduct.summable_and_hasSum_factoredNumbers_prod_filter_prime_geometric`),\nwhich relates the finite product over primes `p ∈ s` to the sum of `f n` over `s`-factored `n`,\nfor `s : Finset ℕ`.\n\n## Tags\n\nEuler product, multiplicative function\n-/\n\npublic section\n\n/-- If `f` is multiplicative and summable, then its values at natural numbers `> 1`\nhave norm strictly less than `1`. -/\n\nTarget:\nlemma Summable.norm_lt_one {F : Type*} [NormedDivisionRing F] [CompleteSpace F] {f : ℕ →* F}\n    (hsum : Summable f) {p : ℕ} (hp : 1 < p) :\n    ‖f p‖ < 1 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_1caba13c5bfb","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1fc684596d6aa267652ee86ed26f91783864f05a42f9eefb5edf36459b4cf348","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/EulerProduct","family_id":"summable","file_id":"mathlib/Mathlib/NumberTheory/EulerProduct/Basic.lean","sample_id":"1caba13c5bfb2e48e2aa824af4b9e227bed2165274d63dd634f59f34ffe47e2e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"62c33a874b3140b3e0d71815632c5a3a67b39c9eeeb9eb02b51c7aa26e877fd4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4291a8efb1c861482a7cda69054b40fffe2867a0a2bf93ebb3e84c54147a1c52","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b7d8bbdbc675ed1b504fd154e9b2c440cdd8d44f35c4f66773b08147245061fc","source_sha256":"e6c6110c32737b6ae741871eb4fd44ef0a8fec8abe5efc331215d91b55ffecbc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor <;> grind [IsZero]","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":5,"token_jaccard":0.076923,"token_length_ratio":0.555556},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"998ecf59e80ec1dbfc22b47e8643ad91a2cf909c15aedfa44adaef524b0f8d5b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs\n\nNamespace:\nTactic.ComputeAsymptotics.MultiseriesExpansion.IsZero\n\nLocal context:\n/-\nCopyright (c) 2026 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\n/-!\n# Trimming of multiseries\n\nA multiseries is *trimmed* when its leading coefficient (the head of its expansion) is itself\ntrimmed and non-zero. For a trimmed multiseries, the leading monomial captures the main\nasymptotic behavior of the approximated function.\n\n## Main definitions\n\n* `IsZero`: a multiseries represents the zero function — it is either the real number `0`\n  (for the empty basis) or has an empty underlying sequence (`.nil`).\n* `Trimmed` and `Multiseries.Trimmed`: a multiseries is trimmed in the sense above. The former\n  is defined inductively for `MultiseriesExpansion`, and the latter for `Multiseries` is\n  derived from it.\n\nWe also prove structural lemmas relating these predicates to `seq` and to the `cons`/`nil`\nconstructors.\n\n-/\n\n@[expose] public section\n\nnamespace Tactic.ComputeAsymptotics\n\nnamespace MultiseriesExpansion\n\nopen Filter Topology Stream'\n\n/-- A multiseries is zero if it is the real constant `0` or has an empty sequence. -/\ninductive IsZero : {basis : Basis} → MultiseriesExpansion basis → Prop\n| const {c : MultiseriesExpansion []} (hc : c.toReal = 0) : IsZero c\n| nil {basis_hd} {basis_tl} (f) : @IsZero (basis_hd :: basis_tl) (mk .nil f)\n\nnamespace IsZero\n\n@[simp]\n\nTarget:\ntheorem const_iff {c : MultiseriesExpansion []} : IsZero c ↔ c.toReal = 0 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_b7d8bbdbc675","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"05ad2133f9a9623f30c52faa0929a33e147e84c8e5ec6777c2d4a145bb09ce7d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/ComputeAsymptotics","family_id":"const_iff","file_id":"mathlib/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Trimming.lean","sample_id":"b7d8bbdbc675ed1b504fd154e9b2c440cdd8d44f35c4f66773b08147245061fc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d02b7bd3abbc9b6e64a37811efb4eb652c5a1b34ba47dca636dd232d861f2cb3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3a981c0403dd9005b3944835a67adab49096e346a9a2154bc095379a66b415bf","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f90ebae3e238223bb942229ba9072bef620bae81190f54e359d35b11e6164e6b","source_sha256":"ad6816bc3c40144c654f99c6788a9e4aaf794fd4dab9e18db82a79f74bc4d693","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  simpa [Units.ext_iff] using pow_card_sub_one_eq_one (Units.ne_zero _)","hard_negative":true,"metrics":{"chosen_tokens":16,"rejected_tokens":5,"token_jaccard":0.117647,"token_length_ratio":0.3125},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"99909038c68bf16bd11cd2b9484f8b25af96fc969863a9b506d026c5356590c8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity\npublic import Mathlib.FieldTheory.Finite.Basic\n\nNamespace:\nZMod\n\nLocal context:\n/-\nCopyright (c) 2025 Yakov Pechersky. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yakov Pechersky\n-/\n/-!\n# Torsion group of `ZMod p` for prime `p`\n\nThis file shows that the `ZMod p` has `p - 1` roots-of-unity.\n\n-/\n\npublic section\n\nnamespace ZMod\n\nTarget:\nlemma rootsOfUnity_eq_top {p : ℕ} [Fact p.Prime] :\n    (rootsOfUnity (p - 1) (ZMod p)) = ⊤ :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_f90ebae3e238","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1304ea7bce7ecaab3e83cf83d81bb69172bcf1cd1f54b6dc0b909bd563eeebff","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/ZMod","family_id":"rootsofunity_eq_top","file_id":"mathlib/Mathlib/RingTheory/ZMod/Torsion.lean","sample_id":"f90ebae3e238223bb942229ba9072bef620bae81190f54e359d35b11e6164e6b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"988485a8c756571de67c1e7554cc8678471929e0b79e0159e23042969b87deb3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f392082ed9ba8b8ad90648b283f41e7a07a54e3606f0737a18859bf2fe8ce79b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5dc82f08b0f93626082032ca9aa64a8e948617dc0aa5acf4d820aa7f216565eb","source_sha256":"8bbd9fce2d15ad8ea50e5683616b40f12621dcf6a628ad12261a1d432f9bccf6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction p with\n  | nil => rfl\n  | cons _ _ ih => dsimp [pathToList] at *; rw [ih]","hard_negative":false,"metrics":{"chosen_tokens":25,"rejected_tokens":32,"token_jaccard":0.904762,"token_length_ratio":1.28},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"9a3408f22216413ea087729ec80ea551a048c0779415db79701e88bbb9f99dc5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Quiver.Cast\npublic import Mathlib.Combinatorics.Quiver.Symmetric\n\nNamespace:\nQuiver.SingleObj\n\nLocal context:\n/-\nCopyright (c) 2023 Antoine Labelle. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Labelle\n-/\n/-!\n# Single-object quiver\n\nSingle object quiver with a given arrows type.\n\n## Main definitions\n\nGiven a type `α`, `SingleObj α` is the `Unit` type, whose single object is called `star α`, with\n`Quiver` structure such that `star α ⟶ star α` is the type `α`.\nAn element `x : α` can be reinterpreted as an element of `star α ⟶ star α` using\n`toHom`.\nMore generally, a list of elements of `a` can be reinterpreted as a path from `star α` to\nitself using `pathEquivList`.\n-/\n\n@[expose] public section\n\nnamespace Quiver\n\n/-- Type tag on `Unit` used to define single-object quivers. -/\n@[nolint unusedArguments]\ndef SingleObj (_ : Type*) : Type :=\n  Unit\nderiving Unique\n\nnamespace SingleObj\n\nvariable (α β γ : Type*)\n\ninstance : Quiver (SingleObj α) :=\n  ⟨fun _ _ => α⟩\n\n/-- The single object in `SingleObj α`. -/\ndef star : SingleObj α := default\n\nvariable {α β γ}\n\nlemma ext {x y : SingleObj α} : x = y := Unit.ext x y\n\n-- See note [reducible non-instances]\n/-- Equip `SingleObj α` with a reverse operation. -/\nabbrev hasReverse (rev : α → α) : HasReverse (SingleObj α) := ⟨rev⟩\n\n-- See note [reducible non-instances]\n/-- Equip `SingleObj α` with an involutive reverse operation. -/\nabbrev hasInvolutiveReverse (rev : α → α) (h : Function.Involutive rev) :\n    HasInvolutiveReverse (SingleObj α) where\n  toHasReverse := hasReverse rev\n  inv' := h\n\n/-- The type of arrows from `star α` to itself is equivalent to the original type `α`. -/\n@[simps!]\ndef toHom : α ≃ (star α ⟶ star α) :=\n  Equiv.refl _\n\n/-- Prefunctors between two `SingleObj` quivers correspond to functions between the corresponding\narrows types.\n-/\n@[simps]\ndef toPrefunctor : (α → β) ≃ SingleObj α ⥤q SingleObj β where\n  toFun f := ⟨id, f⟩\n  invFun f a := f.map (toHom a)\n\ntheorem toPrefunctor_id : toPrefunctor id = 𝟭q (SingleObj α) :=\n  rfl\n\n@[simp]\ntheorem toPrefunctor_symm_id : toPrefunctor.symm (𝟭q (SingleObj α)) = id :=\n  rfl\n\ntheorem toPrefunctor_comp (f : α → β) (g : β → γ) :\n    toPrefunctor (g ∘ f) = toPrefunctor f ⋙q toPrefunctor g :=\n  rfl\n\n@[simp]\ntheorem toPrefunctor_symm_comp (f : SingleObj α ⥤q SingleObj β) (g : SingleObj β ⥤q SingleObj γ) :\n    toPrefunctor.symm (f ⋙q g) = toPrefunctor.symm g ∘ toPrefunctor.symm f := by\n  simp only [Equiv.symm_apply_eq, toPrefunctor_comp, Equiv.apply_symm_apply]\n\n/-- Auxiliary definition for `quiver.SingleObj.pathEquivList`.\nConverts a path in the quiver `single_obj α` into a list of elements of type `a`.\n-/\ndef pathToList : ∀ {x : SingleObj α}, Path (star α) x → List α\n  | _, Path.nil => []\n  | _, Path.cons p a => a :: pathToList p\n\n/-- Auxiliary definition for `quiver.SingleObj.pathEquivList`.\nConverts a list of elements of type `α` into a path in the quiver `SingleObj α`.\n-/\n@[simp]\ndef listToPath : List α → Path (star α) (star α)\n  | [] => Path.nil\n  | a :: l => (listToPath l).cons a\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem listToPath_pathToList {x : SingleObj α} (p : Path (star α) x) :\n    listToPath (pathToList p) = p.cast rfl ext :=\n\nProof body:\n","rejected":"```lean\nby\n  induction p with\n  | nil => rfl\n  | cons _ _ ih => dsimp [pathToList] at *; rw [ih]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Quiver","family_id":"listtopath_pathtolist","file_id":"mathlib/Mathlib/Combinatorics/Quiver/SingleObj.lean","sample_id":"5dc82f08b0f93626082032ca9aa64a8e948617dc0aa5acf4d820aa7f216565eb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"107f7ce7855f94c8cc4f2d6c868596a255522f4e82b0b2ecfe00fd289a468b21","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"00be9ebd31a702d21957e4d4af4dc0b35459f241ac0a49ac16267b79365cd8b8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a1e841521763274b65199f7d26a8e6592c777a58ce061b317bd17dcc7a139bd0","source_sha256":"ea093bfbcaf5876f15935057214ef966d86293bd44542573e87b5ce1a814f0db","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical exact continuous_piecewise hs hf continuousOn_const","hard_negative":true,"metrics":{"chosen_tokens":7,"rejected_tokens":3,"token_jaccard":0.25,"token_length_ratio":0.428571},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"9a344deac6ec155c9d19bc7b6e53f219b5374acbcdf18793b511c44adeb62500","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Indicator\npublic import Mathlib.Topology.Piecewise\npublic import Mathlib.Topology.Clopen\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 PFR contributors. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: PFR contributors\n-/\n/-!\n# Continuity of indicator functions\n-/\n\npublic section\n\nopen Set\nopen scoped Topology\n\nvariable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} [One β]\n\n@[to_additive]\n\nTarget:\nlemma continuous_mulIndicator (hs : ∀ a ∈ frontier s, f a = 1) (hf : ContinuousOn f (closure s)) :\n    Continuous (mulIndicator s f) :=\n\nProof body:\n","rejected":"by\n  exact continuous_mulIndicator","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"48f7006cee49963e7e0e0aed59a97bd4dc5ac7b2640ac2c1dfb5bb709993f14c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Algebra","family_id":"continuous_mulindicator","file_id":"mathlib/Mathlib/Topology/Algebra/Indicator.lean","sample_id":"a1e841521763274b65199f7d26a8e6592c777a58ce061b317bd17dcc7a139bd0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4db2e60816a954672941e1fdfd4788e81ed7fe826b34170d024d9dedb9483621","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"00a4fa01aa3988f187453211eea6fee689951bcc17ccb3f8913ec66689b3d85f","source_sha256":"f43f20b9ce00ce089e6a539edbfd4564f6ceb6ee57c06e50a1aa6a1ddd738246","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    have := W.isClosedUnderColimitsOfShape_isLocal J κ hW\n    infer_instance","hard_negative":false,"metrics":{"chosen_tokens":10,"rejected_tokens":2,"token_jaccard":0.090909,"token_length_ratio":0.2},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"9a4347faff1eeaa1541ca5f18cd5b8947856b7717766b1d9c045d2429329ceb2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Adjunction.PartialAdjoint\npublic import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer\npublic import Mathlib.CategoryTheory.Localization.BousfieldTransfiniteComposition\npublic import Mathlib.CategoryTheory.MorphismProperty.IsSmall\npublic import Mathlib.CategoryTheory.Presentable.Adjunction\npublic import Mathlib.CategoryTheory.SmallObject.TransfiniteIteration\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The Orthogonal-reflection construction\n\nGiven `W : MorphismProperty C` (which should be small) and assuming the existence\nof certain colimits in `C`, we construct a morphism `toSucc W Z : Z ⟶ succ W Z` for\nany `Z : C`. This morphism belongs to `W.isLocal.isLocal` and\nis an isomorphism iff `Z` belongs to `W.isLocal` (see the lemma `isIso_toSucc_iff`).\nThe morphism `toSucc W Z : Z ⟶ succ W Z` is defined as a composition\nof two morphisms that are roughly described as follows:\n* `toStep W Z : Z ⟶ step W Z`: for any morphism `f : X ⟶ Y` satisfying `W`\n  and any morphism `X ⟶ Z`, we \"attach\" a morphism `Y ⟶ step W Z` (using\n  coproducts and a pushout in essentially the same way as it is done in\n  the file `Mathlib/CategoryTheory/SmallObject/Construction.lean` for the small object\n  argument);\n* `fromStep W Z : step W Z ⟶ succ W Z`: this morphism coequalizes all pairs\n  of morphisms `g₁ g₂ : Y ⟶ step W Z` such that there is a `f : X ⟶ Y`\n  satisfying `W` such that `f ≫ g₁ = f ≫ g₂`.\n\nThe morphism `toSucc W Z : Z ⟶ succ W Z` is a variant of the (wrong) definition\np. 32 in the book by Adámek and Rosický. In this book, a slightly different object\nthan `succ W Z` is defined directly as a colimit of an intricate diagram, but\ncontrary to what is stated on p. 33, it does not satisfy `isIso_toSucc_iff`.\nThe author of this file was unable to understand the attempt of the authors\nto fix this mistake in the errata to this book. This led to the definition\nin two steps outlined above.\n\n## Main results\n\nThe morphisms described above `toSucc W Z : Z ⟶ succ W Z` for all `Z : C` allow to\ndefine `succStruct W Z₀ : SuccStruct C` for any `Z₀ : C`. By applying\na transfinite iteration to this `SuccStruct`, we obtain the following results\nunder the assumption that `W : MorphismProperty C` is a `w`-small property\nof morphisms in a locally `κ`-presentable category `C` (with `κ : Cardinal.{w}`\na regular cardinal) such that the domains and codomains of the morphisms\nsatisfying `W` are `κ`-presentable:\n* `MorphismProperty.isRightAdjoint_ι_isLocal`: existence of the left adjoint\n  of the inclusion `W.isLocal ⥤ C`;\n* `MorphismProperty.isLocallyPresentable_isLocal`: the full subcategory\n  `W.isLocal` is locally presentable.\n\nThis is essentially the implication (i) → (ii) in Theorem 1.39 (and the corollary 1.40)\nin the book by Adámek and Rosický (note that according to the\nerrata to this book, the implication (ii) → (i) is wrong when `κ = ℵ₀`).\n\n## References\n* [Adámek, J. and Rosický, J., *Locally presentable and accessible categories*][Adamek_Rosicky_1994]\n\n-/\n\n@[expose] public section\n\nuniverse w v' u' v u\n\nnamespace CategoryTheory\n\nopen Limits Localization Opposite\n\nvariable {C : Type u} [Category.{v} C] (W : MorphismProperty C)\n\nset_option backward.isDefEq.respectTransparency false in\nlemma MorphismProperty.isClosedUnderColimitsOfShape_isLocal\n    (J : Type u') [Category.{v'} J] [EssentiallySmall.{w} J]\n    (κ : Cardinal.{w}) [Fact κ.IsRegular] [IsCardinalFiltered J κ]\n    (hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ) :\n    W.isLocal.IsClosedUnderColimitsOfShape J where\n  colimitsOfShape_le := fun Z ⟨p⟩ X Y f hf ↦ by\n    obtain ⟨_, _⟩ := hW f hf\n    refine ⟨fun g₁ g₂ h ↦ ?_, fun g ↦ ?_⟩\n    · obtain ⟨j₁, g₁, rfl⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ p.isColimit g₁\n      obtain ⟨j₂, g₂, rfl⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ p.isColimit g₂\n      dsimp at h ⊢\n      obtain ⟨j₃, u, v, huv⟩ :=\n        IsCardinalPresentable.exists_eq_of_isColimit κ p.isColimit (f ≫ g₁) (f ≫ g₂)\n          (by simpa)\n      simp only [Category.assoc] at huv\n      rw [← p.w u, ← p.w v, reassoc_of% ((p.prop_diag_obj j₃ _ hf).1 huv)]\n    · obtain ⟨j, g, rfl⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ p.isColimit g\n      obtain ⟨g, rfl⟩ := (p.prop_diag_obj j _ hf).2 g\n      exact ⟨g ≫ p.ι.app j, by simp⟩\n\nTarget:\nlemma MorphismProperty.isCardinalAccessible_ι_isLocal\n    (κ : Cardinal.{w}) [Fact κ.IsRegular]\n    [HasCardinalFilteredColimits C κ]\n    (hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ) :\n    W.isLocal.ι.IsCardinalAccessible κ where\n  preservesColimitOfShape J _ _ :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Presentable","family_id":"morphismproperty","file_id":"mathlib/Mathlib/CategoryTheory/Presentable/OrthogonalReflection.lean","sample_id":"00a4fa01aa3988f187453211eea6fee689951bcc17ccb3f8913ec66689b3d85f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3431d3593784b843a6d2cba92338ea3a61fb0adeecb9190810dad7f00a815cd9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"88c32e9180e9f0dad07f72ecceffc95651170f32912ce272463f0613bb38d701","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"80e66e4358b5a047cdbe0254803df37a3bda8284661c9db0a5c755ad758dbabf","source_sha256":"0609bc9d79421b5690dca5429a67571ab70d1ca9db34b7e4c3dbe920598d0eb2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction N with\n  | zero => simp\n  | succ N ih => calc\n    _ ⊆ Ico (a 0) (a N) ∪ Ico (a N) (a (N + 1)) := Ico_subset_Ico_union_Ico\n    _ ⊆ _ := by simpa [Finset.range_add_one] using\n                  union_subset_union_right (Ico (a N) (a (N + 1))) ih","hard_negative":false,"metrics":{"chosen_tokens":70,"rejected_tokens":75,"token_jaccard":0.861111,"token_length_ratio":1.071429},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"9a44dc810c68526fc6f62790d7b53e23564780145a5afd1607383043dfa93d04","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Range\npublic import Mathlib.Data.Set.Lattice\npublic import Mathlib.Order.Interval.Set.LinearOrder\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Oliver Butterley. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Butterley\n-/\n/-!\n# Extra lemmas about unions of intervals\n\nThis file contains lemmas about finite unions of intervals which can't be included with the lemmas\nconcerning infinite unions in `Mathlib/Order/Interval/Set/Disjoint.lean` because we use\n`Finset.range`.\n-/\n\npublic section\n\nopen Set\n\n/-- Union of consecutive intervals contains the interval defined by the initial and final points. -/\ntheorem Ioc_subset_biUnion_Ioc {X : Type*} [LinearOrder X] (N : ℕ) (a : ℕ → X) :\n    Ioc (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ioc (a i) (a (i + 1)) := by\n  induction N with\n  | zero => simp\n  | succ N ih => calc\n    _ ⊆ Ioc (a 0) (a N) ∪ Ioc (a N) (a (N + 1)) := Ioc_subset_Ioc_union_Ioc\n    _ ⊆ _ := by simpa [Finset.range_add_one] using\n                  union_subset_union_right (Ioc (a N) (a (N + 1))) ih\n\n/-- Union of consecutive intervals contains the interval defined by the initial and final points. -/\n\nTarget:\ntheorem Ico_subset_biUnion_Ico {X : Type*} [LinearOrder X] (N : ℕ) (a : ℕ → X) :\n    Ico (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ico (a i) (a (i + 1)) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  induction N with\n  | zero => simp\n  | succ N ih => calc\n    _ ⊆ Ico (a 0) (a N) ∪ Ico (a N) (a (N + 1)) := Ico_subset_Ico_union_Ico\n    _ ⊆ _ := by simpa [Finset.range_add_one] using\n                  union_subset_union_right (Ico (a N) (a (N + 1))) ih","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Interval","family_id":"ico_subset_biunion_ico","file_id":"mathlib/Mathlib/Order/Interval/Set/Union.lean","sample_id":"80e66e4358b5a047cdbe0254803df37a3bda8284661c9db0a5c755ad758dbabf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e65faed5f7af3fe66504654d3728889128f1a2b5c77ec046ad6fc37f70906b6d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"104ce9cbaebdaa87f4904559e1ae0c57adfb3fa19b70fd7be28fa5278b9ac29a","source_sha256":"34049d18cb594f0a184df593dbd6d4fba0ade793ef81827c21fb1d80f135b5c5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by simp [left_distrib, *, sub_eq_add_neg]","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":8,"token_jaccard":0.071429,"token_length_ratio":0.888889},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"9a5da8a7ec02dedf77a68b414659a7ee5fb69a03b55acb1a6894df268ee9d1d5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Basic\npublic import Mathlib.Algebra.Order.Ring.Defs\npublic import Mathlib.Data.Tree.Basic\npublic import Mathlib.Tactic.NormNum.Core\npublic import Mathlib.Util.SynthesizeUsing\n\nNamespace:\nMathlib.Tactic.CancelDenoms\n\nLocal context:\n/-\nCopyright (c) 2020 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n-/\npublic meta import Mathlib.Data.Tree.Basic\npublic meta import Mathlib.Algebra.Group.Nat.Defs\n\n/-!\n# A tactic for canceling numeric denominators\n\nThis file defines tactics that cancel numeric denominators from field Expressions.\n\nAs an example, we want to transform a comparison `5*(a/3 + b/4) < c/3` into the equivalent\n`5*(4*a + 3*b) < 4*c`.\n\n## Implementation notes\n\nThe tooling here was originally written for `linarith`, not intended as an interactive tactic.\nThe interactive version has been split off because it is sometimes convenient to use on its own.\nThere are likely some rough edges to it.\n\nImproving this tactic would be a good project for someone interested in learning tactic programming.\n-/\n\npublic meta section\n\nopen Lean Parser Tactic Mathlib Meta NormNum Qq\n\ninitialize registerTraceClass `CancelDenoms\n\nnamespace Mathlib.Tactic.CancelDenoms\n\n/-! ### Lemmas used in the procedure -/\n\ntheorem mul_subst {α} [CommRing α] {n1 n2 k e1 e2 t1 t2 : α}\n    (h1 : n1 * e1 = t1) (h2 : n2 * e2 = t2) (h3 : n1 * n2 = k) : k * (e1 * e2) = t1 * t2 := by\n  rw [← h3, mul_comm n1, mul_assoc n2, ← mul_assoc n1, h1,\n      ← mul_assoc n2, mul_comm n2, mul_assoc, h2]\n\ntheorem div_subst {α} [Field α] {n1 n2 k e1 e2 t1 : α}\n    (h1 : n1 * e1 = t1) (h2 : n2 / e2 = 1) (h3 : n1 * n2 = k) : k * (e1 / e2) = t1 := by\n  rw [← h3, mul_assoc, mul_div_left_comm, h2, ← mul_assoc, h1, mul_comm, one_mul]\n\ntheorem cancel_factors_eq_div {α} [Field α] {n e e' : α}\n    (h : n * e = e') (h2 : n ≠ 0) : e = e' / n :=\n  eq_div_of_mul_eq h2 <| by rwa [mul_comm] at h\n\ntheorem add_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :\n    n * (e1 + e2) = t1 + t2 := by simp [left_distrib, *]\n\nTarget:\ntheorem sub_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :\n    n * (e1 - e2) = t1 - t2 :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"3d6e41fedb96281cd87efe84955437aef603bb2faead83f7993cb089c0299ee9","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/CancelDenoms","family_id":"sub_subst","file_id":"mathlib/Mathlib/Tactic/CancelDenoms/Core.lean","sample_id":"104ce9cbaebdaa87f4904559e1ae0c57adfb3fa19b70fd7be28fa5278b9ac29a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fb278dc04d580276be7c164bc304823032754bdaf50110401d324e4d50eff42b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"96dbde4f4151db83b5ae77d34dc896b57800591a2988fbcdab30f0f92312396b","source_sha256":"6c1d4f9fced239f6f8c0c8785a4325f383edabba30747fe08c8262d3408cd815","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [mem_def, or_and_right, exists_or]","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":8,"token_jaccard":0.071429,"token_length_ratio":0.888889},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"9a62d60bcdf5a9db49bb7e3c6fc6d9845aae008fcbb95fa76f607bc15afe57d0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.Monoid.NatCast\npublic import Mathlib.Algebra.Ring.Nat\npublic import Mathlib.Data.Sigma.Basic\npublic import Batteries.Tactic.Lint.TypeClass\n\nNamespace:\nLists'\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# A computable model of ZFA without infinity\n\nIn this file we define finite hereditary lists. This is useful for calculations in naive set theory.\n\nWe distinguish two kinds of ZFA lists:\n* Atoms. Directly correspond to an element of the original type.\n* Proper ZFA lists. Can be thought of (but are not implemented) as a list of ZFA lists (not\n  necessarily proper).\n\nFor example, `Lists ℕ` contains stuff like `23`, `[]`, `[37]`, `[1, [[2], 3], 4]`.\n\n## Implementation note\n\nAs we want to be able to append both atoms and proper ZFA lists to proper ZFA lists, it's handy that\natoms and proper ZFA lists belong to the same type, even though atoms of `α` could be modelled as\n`α` directly. But we don't want to be able to append anything to atoms.\n\nThis calls for a two-step definition of ZFA lists:\n* First, define ZFA prelists as atoms and proper ZFA prelists. Those proper ZFA prelists are defined\n  by inductive appending of (not necessarily proper) ZFA lists.\n* Second, define ZFA lists by rubbing out the distinction between atoms and proper lists.\n\n## Main declarations\n\n* `Lists' α false`: Atoms as ZFA prelists. Basically a copy of `α`.\n* `Lists' α true`: Proper ZFA prelists. Defined inductively from the empty ZFA prelist\n  (`Lists'.nil`) and from appending a ZFA prelist to a proper ZFA prelist (`Lists'.cons a l`).\n* `Lists α`: ZFA lists. Sum of the atoms and proper ZFA prelists.\n* `Finsets α`: ZFA sets. Defined as `Lists` quotiented by `Lists.Equiv`, the extensional\n  equivalence.\n-/\n\n@[expose] public section\n\n\nvariable {α : Type*}\n\n/-- Prelists, helper type to define `Lists`. `Lists' α false` are the \"atoms\", a copy of `α`.\n`Lists' α true` are the \"proper\" ZFA prelists, inductively defined from the empty ZFA prelist and\nfrom appending a ZFA prelist to a proper ZFA prelist. It is made so that you can't append anything\nto an atom while having only one appending function for appending both atoms and proper ZFA prelists\nto a proper ZFA prelist. -/\ninductive Lists'.{u} (α : Type u) : Bool → Type u\n  | atom : α → Lists' α false\n  | nil : Lists' α true\n  | cons' {b} : Lists' α b → Lists' α true → Lists' α true\n  deriving DecidableEq\ncompile_inductive% Lists'\n\n/-- Hereditarily finite list, aka ZFA list. A ZFA list is either an \"atom\" (`b = false`),\ncorresponding to an element of `α`, or a \"proper\" ZFA list, inductively defined from the empty ZFA\nlist and from appending a ZFA list to a proper ZFA list. -/\ndef Lists (α : Type*) :=\n  Σ b, Lists' α b\n\nnamespace Lists'\n\ninstance [Inhabited α] : ∀ b, Inhabited (Lists' α b)\n  | true => ⟨nil⟩\n  | false => ⟨atom default⟩\n\n/-- Appending a ZFA list to a proper ZFA prelist. -/\ndef cons : Lists α → Lists' α true → Lists' α true\n  | ⟨_, a⟩, l => cons' a l\n\n/-- Converts a ZFA prelist to a `List` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp]\ndef toList : ∀ {b}, Lists' α b → List (Lists α)\n  | _, atom _ => []\n  | _, nil => []\n  | _, cons' a l => ⟨_, a⟩ :: l.toList\n\n@[simp]\ntheorem toList_cons (a : Lists α) (l) : toList (cons a l) = a :: l.toList := rfl\n\n/-- Converts a `List` of ZFA lists to a proper ZFA prelist. -/\n@[simp]\ndef ofList : List (Lists α) → Lists' α true\n  | [] => nil\n  | a :: l => cons a (ofList l)\n\n@[simp]\ntheorem to_ofList (l : List (Lists α)) : toList (ofList l) = l := by induction l <;> simp [*]\n\n@[simp]\ntheorem of_toList : ∀ l : Lists' α true, ofList (toList l) = l :=\n  suffices ∀ (b) (h : true = b) (l : Lists' α b),\n      let l' : Lists' α true := h ▸ l\n      ofList (toList l') = l'\n    from this _ rfl\n  fun b h l => by\n    induction l with\n    | atom => cases h\n    | nil => simp\n    | cons' b a _ IH => simpa [cons] using IH rfl\n\n/-- Recursion/induction principle for `Lists'.ofList`. -/\n@[elab_as_elim]\ndef recOfList {motive : Lists' α true → Sort*} (ofList : ∀ l, motive (ofList l)) : ∀ l, motive l :=\n  fun l ↦ cast (by simp) <| ofList (l.toList)\n\nend Lists'\n\nmutual\n  /-- Equivalence of ZFA lists. Defined inductively. -/\n  inductive Lists.Equiv : Lists α → Lists α → Prop\n    | refl (l) : Lists.Equiv l l\n    | antisymm {l₁ l₂ : Lists' α true} :\n      Lists'.Subset l₁ l₂ → Lists'.Subset l₂ l₁ → Lists.Equiv ⟨_, l₁⟩ ⟨_, l₂⟩\n\n  /-- Subset relation for ZFA lists. Defined inductively. -/\n  inductive Lists'.Subset : Lists' α true → Lists' α true → Prop\n    | nil {l} : Lists'.Subset Lists'.nil l\n    | cons {a a' l l'} :\n      Lists.Equiv a a' →\n        a' ∈ Lists'.toList l' → Lists'.Subset l l' → Lists'.Subset (Lists'.cons a l) l'\nend\n\nlocal infixl:50 \" ~ \" => Lists.Equiv\n\nnamespace Lists'\n\ninstance : HasSubset (Lists' α true) :=\n  ⟨Lists'.Subset⟩\n\n/-- ZFA prelist membership. A ZFA list is in a ZFA prelist if some element of this ZFA prelist is\nequivalent as a ZFA list to this ZFA list. -/\ninstance {b} : Membership (Lists α) (Lists' α b) :=\n  ⟨fun l a => ∃ a' ∈ l.toList, a ~ a'⟩\n\ntheorem mem_def {b a} {l : Lists' α b} : a ∈ l ↔ ∃ a' ∈ l.toList, a ~ a' :=\n  Iff.rfl\n\n@[simp]\n\nTarget:\ntheorem mem_cons {a y l} : a ∈ @cons α y l ↔ a ~ y ∨ a ∈ l :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"7e75af5eef05ad6aa7c56ce700d8affc4ba550a5dbebbb3e54c1afd93be140f0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory","family_id":"mem_cons","file_id":"mathlib/Mathlib/SetTheory/Lists.lean","sample_id":"96dbde4f4151db83b5ae77d34dc896b57800591a2988fbcdab30f0f92312396b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9c5374a4a76445b21ceb5288576943f3dfabe49605e7c1eedae05176a4a561c7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"dd7e7c7724f251b0bbfef38f1495fa3c1b68a51711c3d0d5237f2996e98353df","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cb787666449dd4a52c1a3d0c14107bfed4df4b3d94d031fd1a98bc977a411bf3","source_sha256":"be2af13256fec9c1b55ab43e0b0388a9f91c486ebeddde5998dfc4d97729095b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← isLocallySurjective_iff_epi']\n  exact LightCondensed.isLocallySurjective_iff_locallySurjective_on_lightProfinite _ f","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.153846,"token_length_ratio":0.25},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"9a995ba4b83be33a1b021bf1f80920c591fd17b22f89ee0a6785e57b243a9eff","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Shapes.SequentialProduct\npublic import Mathlib.CategoryTheory.Sites.Coherent.SequentialLimit\npublic import Mathlib.Condensed.Light.Functors\npublic import Mathlib.Condensed.Light.Limits\n\nNamespace:\nLightCondSet\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# Epimorphisms of light condensed objects\n\nThis file characterises epimorphisms in light condensed sets and modules as the locally surjective\nmorphisms. Here, the condition of locally surjective is phrased in terms of continuous surjections\nof light profinite sets.\n\nFurther, we prove that the functor `lim : Discrete ℕ ⥤ LightCondMod R` preserves epimorphisms.\n-/\n\npublic section\n\nuniverse v u w u' v'\n\nopen CategoryTheory Sheaf Limits GrothendieckTopology\n\nnamespace LightCondensed\n\nvariable (A : Type u') [Category.{v'} A] {FA : A → A → Type*} {CA : A → Type w}\nvariable [∀ X Y, FunLike (FA X Y) (CA X) (CA Y)] [ConcreteCategory.{w} A FA]\n  [PreservesFiniteProducts (CategoryTheory.forget A)]\n\nvariable {X Y : LightCondensed.{u} A} (f : X ⟶ Y)\n\nlemma isLocallySurjective_iff_locallySurjective_on_lightProfinite : IsLocallySurjective f ↔\n    ∀ (S : LightProfinite) (y : ToType (Y.obj.obj ⟨S⟩)),\n      (∃ (S' : LightProfinite) (φ : S' ⟶ S) (_ : Function.Surjective φ)\n        (x : ToType (X.obj.obj ⟨S'⟩)),\n        f.hom.app ⟨S'⟩ x = Y.obj.map ⟨φ⟩ y) := by\n  rw [coherentTopology.isLocallySurjective_iff,\n    regularTopology.isLocallySurjective_iff]\n  simp_rw [LightProfinite.effectiveEpi_iff_surjective]\n\nend LightCondensed\n\nnamespace LightCondSet\n\nvariable {X Y : LightCondSet.{u}} (f : X ⟶ Y)\n\nTarget:\nlemma epi_iff_locallySurjective_on_lightProfinite : Epi f ↔\n    ∀ (S : LightProfinite) (y : Y.obj.obj ⟨S⟩),\n      (∃ (S' : LightProfinite) (φ : S' ⟶ S) (_ : Function.Surjective φ) (x : X.obj.obj ⟨S'⟩),\n        f.hom.app ⟨S'⟩ x = Y.obj.map ⟨φ⟩ y) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_cb787666449d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d29eb7e8dd70c805f3e054806d51e06406fa46814cc85869619e939ff0b78244","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed/Light","family_id":"epi_iff_locallysurjective_on_lightprofinite","file_id":"mathlib/Mathlib/Condensed/Light/Epi.lean","sample_id":"cb787666449dd4a52c1a3d0c14107bfed4df4b3d94d031fd1a98bc977a411bf3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f3a8fc7a4995882f5a643c316d6ec43e868e48edcba54745dfaa0649321d1a67","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cf1a2f0ff6036835a4e6c170684ceaf7b1ce5866e070ddd0ef073566ab378368","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"28eb45ede6284ff2c969d0ba34870b558934c40303d056e5740787b98d7dcd28","source_sha256":"287a704d503b237a43f85c1d8d085e3a9c8170677a7fe6bc6d74659961e522de","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hb' : b.Dom := Part.dom_iff_mem.mpr ⟨a.get ha, hb⟩\n  rwa [← eq_get_iff_mem hb', eq_iff_of_dom ha hb'] at hb","hard_negative":true,"metrics":{"chosen_tokens":33,"rejected_tokens":5,"token_jaccard":0.071429,"token_length_ratio":0.151515},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"9ad0e24a5382afc758cd382915951af34eabab401d09f09c74dda861a4700552","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Defs\npublic import Mathlib.Data.Set.Subsingleton\npublic import Mathlib.Logic.Equiv.Defs\n\nNamespace:\nPart\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Jeremy Avigad, Simon Hudon\n-/\n/-!\n# Partial values of a type\n\nThis file defines `Part α`, the partial values of a type.\n`o : Part α` carries a proposition `o.Dom`, its domain, along with a function `get : o.Dom → α`, its\nvalue. The rule is then that every partial value has a value but, to access it, you need to provide\na proof of the domain.\n`Part α` behaves the same as `Option α` except that `o : Option α` is decidably `none` or `some a`\nfor some `a : α`, while the domain of `o : Part α` doesn't have to be decidable. That means you can\ntranslate back and forth between a partial value with a decidable domain and an option, and\n`Option α` and `Part α` are classically equivalent. In general, `Part α` is bigger than `Option α`.\n\n## Main declarations\n`Option`-like declarations:\n* `Part.none`: The partial value whose domain is `False`.\n* `Part.some a`: The partial value whose domain is `True` and whose value is `a`.\n* `Part.ofOption`: Converts an `Option α` to a `Part α` by sending `none` to `none` and `some a` to\n  `some a`.\n* `Part.toOption`: Converts a `Part α` with a decidable domain to an `Option α`.\n* `Part.equivOption`: Classical equivalence between `Part α` and `Option α`.\n\nMonadic structure:\n* `Part.bind`: `o.bind f` has value `(f (o.get _)).get _` (`f o` morally) and is defined when `o`\n  and `f (o.get _)` are defined.\n* `Part.map`: Maps the value and keeps the same domain.\n\nOther:\n* `Part.restrict`: `Part.restrict p o` replaces the domain of `o : Part α` by `p : Prop` so long as\n  `p → o.Dom`.\n* `Part.assert`: `assert p f` appends `p` to the domains of the values of a partial function.\n* `Part.unwrap`: Gets the value of a partial value regardless of its domain. Unsound.\n\n## Notation\nFor `a : α`, `o : Part α`, `a ∈ o` means that `o` is defined and equal to `a`. Formally, it means\n`o.Dom` and `o.get _ = a`.\n-/\n\n@[expose] public section\n\nassert_not_exists RelIso\n\nopen Function\n\n/-- `Part α` is the type of \"partial values\" of type `α`. It\n  is similar to `Option α` except the domain condition can be an\n  arbitrary proposition, not necessarily decidable. -/\nstructure Part.{u} (α : Type u) : Type u where\n  /-- The domain of a partial value -/\n  Dom : Prop\n  /-- Extract a value from a partial value given a proof of `Dom` -/\n  get : Dom → α\n\nnamespace Part\n\nvariable {α : Type*} {β : Type*} {γ : Type*}\n\n/-- Convert a `Part α` with a decidable domain to an option -/\ndef toOption (o : Part α) [Decidable o.Dom] : Option α :=\n  if h : Dom o then some (o.get h) else none\n\n@[simp] lemma toOption_isSome (o : Part α) [Decidable o.Dom] : o.toOption.isSome ↔ o.Dom := by\n  by_cases h : o.Dom <;> simp [h, toOption]\n\n@[simp] lemma toOption_eq_none (o : Part α) [Decidable o.Dom] : o.toOption = none ↔ ¬o.Dom := by\n  by_cases h : o.Dom <;> simp [h, toOption]\n\n/-- `Part` extensionality -/\ntheorem ext' : ∀ {o p : Part α}, (o.Dom ↔ p.Dom) → (∀ h₁ h₂, o.get h₁ = p.get h₂) → o = p\n  | ⟨od, o⟩, ⟨pd, p⟩, H1, H2 => by\n    have t : od = pd := propext H1\n    cases t; rw [show o = p from funext fun p => H2 p p]\n\n/-- `Part` eta expansion -/\n@[simp]\ntheorem eta : ∀ o : Part α, (⟨o.Dom, fun h => o.get h⟩ : Part α) = o\n  | ⟨_, _⟩ => rfl\n\n/-- `a ∈ o` means that `o` is defined and equal to `a` -/\nprotected def Mem (o : Part α) (a : α) : Prop :=\n  ∃ h, o.get h = a\n\ninstance : Membership α (Part α) :=\n  ⟨Part.Mem⟩\n\ntheorem mem_eq (a : α) (o : Part α) : (a ∈ o) = ∃ h, o.get h = a :=\n  rfl\n\ntheorem dom_iff_mem : ∀ {o : Part α}, o.Dom ↔ ∃ y, y ∈ o\n  | ⟨_, f⟩ => ⟨fun h => ⟨f h, h, rfl⟩, fun ⟨_, h, rfl⟩ => h⟩\n\ntheorem get_mem {o : Part α} (h) : get o h ∈ o :=\n  ⟨_, rfl⟩\n\n@[simp]\ntheorem mem_mk_iff {p : Prop} {o : p → α} {a : α} : a ∈ Part.mk p o ↔ ∃ h, o h = a :=\n  Iff.rfl\n\n/-- `Part` extensionality -/\n@[ext]\ntheorem ext {o p : Part α} (H : ∀ a, a ∈ o ↔ a ∈ p) : o = p :=\n  (ext' ⟨fun h => ((H _).1 ⟨h, rfl⟩).fst, fun h => ((H _).2 ⟨h, rfl⟩).fst⟩) fun _ _ =>\n    ((H _).2 ⟨_, rfl⟩).snd\n\n/-- The `none` value in `Part` has a `False` domain and an empty function. -/\ndef none : Part α :=\n  ⟨False, False.rec⟩\n\ninstance : Inhabited (Part α) :=\n  ⟨none⟩\n\n@[simp]\ntheorem notMem_none (a : α) : a ∉ @none α := fun h => h.fst\n\n/-- The `some a` value in `Part` has a `True` domain and the\n  function returns `a`. -/\ndef some (a : α) : Part α :=\n  ⟨True, fun _ => a⟩\n\n@[simp]\ntheorem some_dom (a : α) : (some a).Dom :=\n  trivial\n\ntheorem mem_unique : ∀ {a b : α} {o : Part α}, a ∈ o → b ∈ o → a = b\n  | _, _, ⟨_, _⟩, ⟨_, rfl⟩, ⟨_, rfl⟩ => rfl\n\ntheorem mem_right_unique : ∀ {a : α} {o p : Part α}, a ∈ o → a ∈ p → o = p\n  | _, _, _, ⟨ho, _⟩, ⟨hp, _⟩ => ext' (iff_of_true ho hp) (by simp [*])\n\ntheorem Mem.left_unique : Relator.LeftUnique ((· ∈ ·) : α → Part α → Prop) := fun _ _ _ =>\n  mem_unique\n\ntheorem Mem.right_unique : Relator.RightUnique ((· ∈ ·) : α → Part α → Prop) := fun _ _ _ =>\n  mem_right_unique\n\ntheorem get_eq_of_mem {o : Part α} {a} (h : a ∈ o) (h') : get o h' = a :=\n  mem_unique ⟨_, rfl⟩ h\n\nprotected theorem subsingleton (o : Part α) : Set.Subsingleton { a | a ∈ o } := fun _ ha _ hb =>\n  mem_unique ha hb\n\n@[simp]\ntheorem get_some {a : α} (ha : (some a).Dom) : get (some a) ha = a :=\n  rfl\n\ntheorem mem_some (a : α) : a ∈ some a :=\n  ⟨trivial, rfl⟩\n\n@[simp]\ntheorem mem_some_iff {a b} : b ∈ (some a : Part α) ↔ b = a :=\n  ⟨fun ⟨_, e⟩ => e.symm, fun e => ⟨trivial, e.symm⟩⟩\n\ntheorem eq_some_iff {a : α} {o : Part α} : o = some a ↔ a ∈ o :=\n  ⟨fun e => e.symm ▸ mem_some _, fun ⟨h, e⟩ => e ▸ ext' (iff_true_intro h) fun _ _ => rfl⟩\n\ntheorem eq_none_iff {o : Part α} : o = none ↔ ∀ a, a ∉ o :=\n  ⟨fun e => e.symm ▸ notMem_none, fun h => ext (by simpa)⟩\n\ntheorem eq_none_iff' {o : Part α} : o = none ↔ ¬o.Dom :=\n  ⟨fun e => e.symm ▸ id, fun h => eq_none_iff.2 fun _ h' => h h'.fst⟩\n\n@[simp]\ntheorem not_none_dom : ¬(none : Part α).Dom :=\n  id\n\n@[simp]\ntheorem some_ne_none (x : α) : some x ≠ none := by\n  intro h\n  exact true_ne_false (congr_arg Dom h)\n\n@[simp]\ntheorem none_ne_some (x : α) : none ≠ some x :=\n  (some_ne_none x).symm\n\ntheorem ne_none_iff {o : Part α} : o ≠ none ↔ ∃ x, o = some x := by\n  constructor\n  · rw [Ne, eq_none_iff', not_not]\n    exact fun h => ⟨o.get h, eq_some_iff.2 (get_mem h)⟩\n  · rintro ⟨x, rfl⟩\n    apply some_ne_none\n\ntheorem eq_none_or_eq_some (o : Part α) : o = none ∨ ∃ x, o = some x :=\n  or_iff_not_imp_left.2 ne_none_iff.1\n\ntheorem some_injective : Injective (@Part.some α) := fun _ _ h =>\n  congr_fun (eq_of_heq (Part.mk.inj h).2) trivial\n\n@[simp]\ntheorem some_inj {a b : α} : Part.some a = some b ↔ a = b :=\n  some_injective.eq_iff\n\n@[simp]\ntheorem some_get {a : Part α} (ha : a.Dom) : Part.some (Part.get a ha) = a :=\n  Eq.symm (eq_some_iff.2 ⟨ha, rfl⟩)\n\ntheorem get_eq_iff_eq_some {a : Part α} {ha : a.Dom} {b : α} : a.get ha = b ↔ a = some b :=\n  ⟨fun h => by simp [h.symm], fun h => by simp [h]⟩\n\ntheorem get_eq_get_of_eq (a : Part α) (ha : a.Dom) {b : Part α} (h : a = b) :\n    a.get ha = b.get (h ▸ ha) := by\n  congr\n\ntheorem get_eq_iff_mem {o : Part α} {a : α} (h : o.Dom) : o.get h = a ↔ a ∈ o :=\n  ⟨fun H => ⟨h, H⟩, fun ⟨_, H⟩ => H⟩\n\ntheorem eq_get_iff_mem {o : Part α} {a : α} (h : o.Dom) : a = o.get h ↔ a ∈ o :=\n  eq_comm.trans (get_eq_iff_mem h)\n\ntheorem eq_of_get_eq_get {a b : Part α} (ha : a.Dom) (hb : b.Dom) (hab : a.get ha = b.get hb) :\n    a = b :=\n  ext' (iff_of_true ha hb) fun _ _ => hab\n\ntheorem eq_iff_of_dom {a b : Part α} (ha : a.Dom) (hb : b.Dom) : a.get ha = b.get hb ↔ a = b :=\n  ⟨eq_of_get_eq_get ha hb, get_eq_get_of_eq a ha⟩\n\nTarget:\ntheorem eq_of_mem {a b : Part α} (ha : a.Dom) (hb : a.get ha ∈ b) : a = b :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_28eb45ede628","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"420c3f3ce3306fb489c0309a0d061c8cb39ccfa13f433fd6b2b617199a97f64a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data","family_id":"eq_of_mem","file_id":"mathlib/Mathlib/Data/Part.lean","sample_id":"28eb45ede6284ff2c969d0ba34870b558934c40303d056e5740787b98d7dcd28"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9d69af709cb1a99e4edeafcd19d4f6a19e8a85efb124f30d847acee7dc29b2e3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e01fdda376cb29cd87e36b0bb679b1859daf960d6f711670a8a4d20ee1851336","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b0369a5a54c56e81b58aafefa0901fd2dbc75fdc809ee1103b64d4a18103c12d","source_sha256":"47c01594d211b3c54399310f0cb3b9f706ed5bf9e27bc1387c290b2a49698c96","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  suffices ∀ x : FreeMonoid α, (xs.map FreeMonoid.of).foldl (· * ·) x = x * ofList xs by\n    simpa [← List.foldl_map, lift_mk_eq_foldl, toFreeMonoid, lift] using this (FreeMonoid.of x)\n  induction xs with grind [ofList_nil, ofList_cons]","hard_negative":true,"metrics":{"chosen_tokens":60,"rejected_tokens":3,"token_jaccard":0.027027,"token_length_ratio":0.05},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"9b1dcfe8fff3a797fc57e600936a39c4bee7b6a9202f893c0c9945adae545eeb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.FreeMonoid.Basic\npublic import Mathlib.Algebra.Free\npublic import Mathlib.Algebra.Group.WithOne.Basic\npublic import Mathlib.Algebra.Group.Units.Basic\npublic import Mathlib.Data.Set.Operations\nimport Mathlib.Data.Set.Insert\n\nNamespace:\nFreeSemigroup\n\nLocal context:\n/-\nCopyright (c) 2026 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\n/-!\n# Relation between the free semigroup and the free monoid\n\nWe provide some constructions relating the free semigroup and the free monoid on the same type.\n\n## Main definitions\n* `FreeSemigroup.toFreeMonoid`: the natural embedding of the free semigroup into the free monoid.\n* `FreeMonoid.equivWithOneFreeSemigroup`: the free monoid is isomorphic to the free semigroup\n  with a `1` added.\n-/\n\npublic section\n\nvariable {α : Type*}\n\nnamespace FreeSemigroup\n\nopen FreeMonoid\n\n/--\nThe natural embedding of the free semigroup into the free monoid.\nThis is injective (`FreeSemigroup.toFreeMonoid_injective`), and its image\nconsists of all non-`1` elements of the free monoid (`FreeSemigroup.eq_one_or_toFreeMonoid`).\n-/\n@[expose, to_additive /-- The natural embedding of the free additive semigroup into the\nfree additive monoid. This is injective (`FreeAddSemigroup.toFreeAddMonoid_injective`), and its\nimage consists of all non-`0` elements of the free additive monoid\n(`FreeAddSemigroup.eq_zero_or_toFreeAddMonoid`). -/]\ndef toFreeMonoid : FreeSemigroup α →ₙ* FreeMonoid α :=\n  lift FreeMonoid.of\n\n@[to_additive (attr := simp, grind =)]\nlemma toFreeMonoid_of (x : α) : toFreeMonoid (.of x) = .of x := rfl\n\n@[to_additive]\n\nTarget:\nlemma toFreeMonoid_mk_eq_cons (x : α) (xs : List α) :\n    toFreeMonoid ⟨x, xs⟩ = FreeMonoid.ofList (x :: xs) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_b0369a5a54c5","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"9f44ebac9cf93fac99ed2d7a2b320900c5324a3d8680b6c03c27d0a55dac15ed","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/FreeMonoid","family_id":"tofreemonoid_mk_eq_cons","file_id":"mathlib/Mathlib/Algebra/FreeMonoid/FreeSemigroup.lean","sample_id":"b0369a5a54c56e81b58aafefa0901fd2dbc75fdc809ee1103b64d4a18103c12d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"280119e14881b51170fc3d0d4d95c5bf43762c9b1e1b31c95c9da1e53082f7ea","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b84cec6b1d751705167f6f103cf7edb4e8a75633ff7d571654ff5ea97e2ce6d3","source_sha256":"c48d585d22788683dbcc8ee8c25758ae9938a430160d8e35ea71552758ca3f83","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases h with ⟨u, hu⟩\n  rw [← Units.inv_mul_eq_one, mul_add] at hu\n  apply Or.imp _ _ (isUnit_or_isUnit_of_add_one hu) <;> exact isUnit_of_mul_isUnit_right","hard_negative":false,"metrics":{"chosen_tokens":35,"rejected_tokens":3,"token_jaccard":0.064516,"token_length_ratio":0.085714},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"9b25e5872f7c33656ac93d706c40fdfee0148e4fcfd27967933aaa24ab5e8a9d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.LocalRing.Defs\npublic import Mathlib.RingTheory.Ideal.Nonunits\n\nNamespace:\nIsLocalRing\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Chris Hughes, Mario Carneiro\n-/\n/-!\n\n# Local rings\n\nWe prove basic properties of local rings.\n\n-/\n\npublic section\n\nvariable {R S : Type*}\n\nnamespace IsLocalRing\n\nsection Semiring\n\nvariable [Semiring R]\n\ntheorem of_isUnit_or_isUnit_of_isUnit_add [Nontrivial R]\n    (h : ∀ a b : R, IsUnit (a + b) → IsUnit a ∨ IsUnit b) : IsLocalRing R :=\n  ⟨fun {a b} hab => h a b <| hab.symm ▸ isUnit_one⟩\n\n/-- A semiring is local if it is nontrivial and the set of nonunits is closed under the addition. -/\ntheorem of_nonunits_add [Nontrivial R]\n    (h : ∀ a b : R, a ∈ nonunits R → b ∈ nonunits R → a + b ∈ nonunits R) : IsLocalRing R where\n  isUnit_or_isUnit_of_add_one {a b} hab :=\n    or_iff_not_and_not.2 fun H => h a b H.1 H.2 <| hab.symm ▸ isUnit_one\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R]\n\n/-- A semiring is local if it has a unique maximal ideal. -/\ntheorem of_unique_max_ideal (h : ∃! I : Ideal R, I.IsMaximal) : IsLocalRing R :=\n  @of_nonunits_add _ _\n    (nontrivial_of_ne (0 : R) 1 <|\n      let ⟨I, Imax, _⟩ := h\n      fun H : 0 = 1 => Imax.1.1 <| I.eq_top_iff_one.2 <| H ▸ I.zero_mem)\n    fun x y hx hy H =>\n    let ⟨I, Imax, Iuniq⟩ := h\n    let ⟨Ix, Ixmax, Hx⟩ := exists_max_ideal_of_mem_nonunits hx\n    let ⟨Iy, Iymax, Hy⟩ := exists_max_ideal_of_mem_nonunits hy\n    have xmemI : x ∈ I := Iuniq Ix Ixmax ▸ Hx\n    have ymemI : y ∈ I := Iuniq Iy Iymax ▸ Hy\n    Imax.1.1 <| I.eq_top_of_isUnit_mem (I.add_mem xmemI ymemI) H\n\ntheorem of_unique_nonzero_prime (h : ∃! P : Ideal R, P ≠ ⊥ ∧ Ideal.IsPrime P) : IsLocalRing R :=\n  of_unique_max_ideal\n    (by\n      rcases h with ⟨P, ⟨hPnonzero, hPnot_top, _⟩, hPunique⟩\n      refine ⟨P, ⟨⟨hPnot_top, ?_⟩⟩, fun M hM => hPunique _ ⟨?_, Ideal.IsMaximal.isPrime hM⟩⟩\n      · refine Ideal.maximal_of_no_maximal fun M hPM hM => ne_of_lt hPM ?_\n        exact (hPunique _ ⟨ne_bot_of_gt hPM, Ideal.IsMaximal.isPrime hM⟩).symm\n      · rintro rfl\n        exact hPnot_top (hM.1.2 P (bot_lt_iff_ne_bot.2 hPnonzero)))\n\nvariable [IsLocalRing R]\n\nTarget:\ntheorem isUnit_or_isUnit_of_isUnit_add {a b : R} (h : IsUnit (a + b)) : IsUnit a ∨ IsUnit b :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/LocalRing","family_id":"isunit_or_isunit_of_isunit_add","file_id":"mathlib/Mathlib/RingTheory/LocalRing/Basic.lean","sample_id":"b84cec6b1d751705167f6f103cf7edb4e8a75633ff7d571654ff5ea97e2ce6d3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4ba70e5cd3f178f8ef12d7930e49952b41e1f76063cf50cc900cfa723b52ba14","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e0f31a2fa4b167750af0775ad92818537ced61f211e8620b229e5bbd135fd43d","source_sha256":"a9f42fb2a650677b7f31f134a9fdb63cf294a003ee4e030eb51993a29af692d7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Iff.comm]\n  obtain ⟨s₁, rfl⟩ := s₁.exists_eq_ofList\n  obtain ⟨s₂, rfl⟩ := s₂.exists_eq_ofList\n  simp only [lt_iff_toList_lt, String.Legacy.iter, String.Legacy.mkIterator, String.toList_ofList]\n  induction s₁ generalizing s₂ <;> cases s₂\n  · unfold ltb; decide\n  · rename_i c₂ cs₂; apply iff_of_true\n    · unfold ltb\n      simp [Legacy.Iterator.hasNext, Char.utf8Size_pos]\n    · apply List.nil_lt_cons\n  · rename_i c₁ cs₁ ih; apply iff_of_false\n    · unfold ltb\n      simp [Legacy.Iterator.hasNext]\n    · apply not_lt_of_gt; apply List.nil_lt_cons\n  · rename_i c₁ cs₁ ih c₂ cs₂; unfold ltb\n    simp only [Legacy.Iterator.hasNext, Pos.Raw.byteIdx_zero, rawEndPos_ofList, utf8Len_cons,\n      add_pos_iff, Char.utf8Size_pos, or_true, decide_true, ↓reduceIte, Legacy.Iterator.curr,\n      Pos.Raw.get, String.toList_ofList, Pos.Raw.utf8GetAux, Legacy.Iterator.next, Pos.Raw.next,\n      Bool.ite_eq_true_distrib, decide_eq_true_eq]\n    split_ifs with h\n    · subst c₂\n      suffices ltb ⟨ofList (c₁ :: cs₁), (0 : Pos.Raw) + c₁⟩\n          ⟨ofList (c₁ :: cs₂), (0 : Pos.Raw) + c₁⟩ =\n            ltb ⟨ofList cs₁, 0⟩ ⟨ofList cs₂, 0⟩ by\n        rw [this]; exact (ih cs₂).trans List.lex_cons_iff.symm\n      apply ltb_cons_addChar\n    · refine ⟨List.Lex.rel, fun e ↦ ?_⟩\n      cases e <;> rename_i h'\n      · assumption\n      · contradiction","hard_negative":false,"metrics":{"chosen_tokens":329,"rejected_tokens":5,"token_jaccard":0.030928,"token_length_ratio":0.015198},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"9b3dd045183297fbcc2ed9b1d6e7a18363637b6fa592ed2885577c023f9e4c39","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.String.Lemmas\npublic import Mathlib.Data.List.Lex\npublic import Mathlib.Data.Char\npublic import Mathlib.Algebra.Order.Group.Nat\nimport all Init.Data.String.Iterator  -- for unfolding `Iterator.curr`\nimport all Init.Data.Ord.String  -- for unfolding `String.compare`\n\nNamespace:\nString\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Strings\n\nSupplementary theorems about the `String` type.\n-/\n\n@[expose] public section\n\nnamespace String\n\n/-- `<` on string iterators. This coincides with `<` on strings as lists. -/\ndef ltb (s₁ s₂ : Legacy.Iterator) : Bool :=\n  if s₂.hasNext then\n    if s₁.hasNext then\n      if s₁.curr = s₂.curr then\n        ltb s₁.next s₂.next\n      else s₁.curr < s₂.curr\n    else true\n  else false\n\n/-- Induction on `String.ltb`. -/\n@[no_expose] def ltb.inductionOn.{u} {motive : Legacy.Iterator → Legacy.Iterator → Sort u}\n    (it₁ it₂ : Legacy.Iterator)\n    (ind : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      i₁.get s₁ = i₂.get s₂ →\n        motive (Legacy.Iterator.next ⟨s₁, i₁⟩) (Legacy.Iterator.next ⟨s₂, i₂⟩) →\n          motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (eq : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      ¬ i₁.get s₁ = i₂.get s₂ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (base₁ : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → ¬ Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (base₂ : ∀ s₁ s₂ i₁ i₂, ¬ Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩) :\n    motive it₁ it₂ :=\n  if h₂ : it₂.hasNext then\n    if h₁ : it₁.hasNext then\n      if heq : it₁.curr = it₂.curr then\n        ind it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq (inductionOn it₁.next it₂.next ind eq base₁ base₂)\n      else eq it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq\n    else base₁ it₁.s it₂.s it₁.i it₂.i h₂ h₁\n  else base₂ it₁.s it₂.s it₁.i it₂.i h₂\n\ntheorem ltb_cons_addChar' (c : Char) (s₁ s₂ : Legacy.Iterator) :\n    ltb ⟨ofList (c :: s₁.s.toList), s₁.i + c⟩ ⟨ofList (c :: s₂.s.toList), s₂.i + c⟩ =\n      ltb s₁ s₂ := by\n  fun_induction ltb s₁ s₂ with\n  | case1 s₁ s₂ h₁ h₂ h ih =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_pos (by simpa using h₂), if_pos, ← ih]\n    · simp only [Legacy.Iterator.next, Pos.Raw.next, get_cons_addChar, ofList_toList]\n      congr 2 <;> apply Pos.Raw.add_char_right_comm\n    · simpa only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList] using h\n  | case2 s₁ s₂ h₁ h₂ h =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_pos (by simpa using h₂), if_neg]\n    · simp only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList, decide_eq_decide]\n    · simpa only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList] using h\n  | case3 s₁ s₂ h₁ h₂ =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_neg (by simpa using h₂)]\n  | case4 s₁ s₂ h₁ =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, if_neg (by simpa using h₁)]\n\ntheorem ltb_cons_addChar (c : Char) (cs₁ cs₂ : List Char) (i₁ i₂ : Pos.Raw) :\n    ltb ⟨ofList (c :: cs₁), i₁ + c⟩ ⟨ofList (c :: cs₂), i₂ + c⟩ =\n      ltb ⟨ofList cs₁, i₁⟩ ⟨ofList cs₂, i₂⟩ := by\n  rw [eq_comm, ← ltb_cons_addChar' c]\n  simp\n\ntheorem lt_iff_toList_lt {s₁ s₂ : String} : s₁ < s₂ ↔ s₁.toList < s₂.toList :=\n  Iff.rfl\n\n@[simp]\n\nTarget:\ntheorem lt_iff_ltb {s₁ s₂ : String} :\n    s₁ < s₂ ↔ ltb (String.Legacy.iter s₁) (String.Legacy.iter s₂) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/String","family_id":"lt_iff_ltb","file_id":"mathlib/Mathlib/Data/String/Basic.lean","sample_id":"e0f31a2fa4b167750af0775ad92818537ced61f211e8620b229e5bbd135fd43d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5e0e8b084461b68f53dd226528eec6d4daa60789f672fc8f9edb2d39518f89fb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f5cf3adf5b2f007282e6afc982afeb8adfc7d1ca08c6a7becb4328354365d479","source_sha256":"54764080c8f7f97a39b09c795498f3d597d51750fa303452440b9c097a0d0bc6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply c.isPushout.hom_ext h₀\n  apply Cofan.IsColimit.hom_ext c.isColimit₂\n  simpa [cell_def] using h","hard_negative":false,"metrics":{"chosen_tokens":25,"rejected_tokens":2,"token_jaccard":0.055556,"token_length_ratio":0.08},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"9b42f9cc2e63d691c00b76901b42ccb95064875fe4a90268959961fadd1cf126","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.MorphismProperty.Limits\n\nNamespace:\nHomotopicalAlgebra.AttachCells\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Attaching cells\n\nGiven a family of morphisms `g a : A a ⟶ B a` and a morphism `f : X₁ ⟶ X₂`,\nwe introduce a structure `AttachCells g f` which expresses that `X₂`\nis obtained from `X₁` by attaching cells of the form `g a`. It means that\nthere is a pushout diagram of the form\n```\n⨿ i, A (π i) -----> X₁\n  |                 |f\n  v                 v\n⨿ i, B (π i) -----> X₂\n```\nIn other words, the morphism `f` is a pushout of coproducts of morphisms\nof the form `g a : A a ⟶ B a`, see `nonempty_attachCells_iff`.\n\nSee the file `Mathlib/AlgebraicTopology/RelativeCellComplex/Basic.lean` for transfinite compositions\nof morphisms `f` with `AttachCells g f` structures.\n\n-/\n\n@[expose] public section\n\nuniverse w' w t t' v u\n\nopen CategoryTheory Limits\n\nnamespace HomotopicalAlgebra\n\nvariable {C : Type u} [Category.{v} C]\n  {α : Type t} {A B : α → C} (g : ∀ a, A a ⟶ B a)\n  {X₁ X₂ : C} (f : X₁ ⟶ X₂)\n\n/-- Given a family of morphisms `g a : A a ⟶ B a` and a morphism `f : X₁ ⟶ X₂`,\nthis structure contains the data and properties which expresses that `X₂`\nis obtained from `X₁` by attaching cells of the form `g a`. -/\nstructure AttachCells where\n  /-- the index type of the cells -/\n  ι : Type w\n  /-- for each `i : ι`, we shall attach a cell given by the morphism `g (π i)`. -/\n  π : ι → α\n  /-- a colimit cofan which gives the coproduct of the object `A (π i)` -/\n  cofan₁ : Cofan (fun i ↦ A (π i))\n  /-- a colimit cofan which gives the coproduct of the object `B (π i)` -/\n  cofan₂ : Cofan (fun i ↦ B (π i))\n  /-- `cofan₁` is colimit -/\n  isColimit₁ : IsColimit cofan₁\n  /-- `cofan₂` is colimit -/\n  isColimit₂ : IsColimit cofan₂\n  /-- the coproduct of the maps `g (π i) : A (π i) ⟶ B (π i)` for all `i : ι`. -/\n  m : cofan₁.pt ⟶ cofan₂.pt\n  hm (i : ι) : cofan₁.inj i ≫ m = g (π i) ≫ cofan₂.inj i := by cat_disch\n  /-- the top morphism of the pushout square -/\n  g₁ : cofan₁.pt ⟶ X₁\n  /-- the bottom morphism of the pushout square -/\n  g₂ : cofan₂.pt ⟶ X₂\n  isPushout : IsPushout g₁ m f g₂\n\nnamespace AttachCells\n\nopen MorphismProperty\n\nattribute [reassoc (attr := simp)] hm\n\nvariable {g f} (c : AttachCells.{w} g f)\n\ninclude c\n\nlemma pushouts_coproducts : (coproducts.{w} (ofHoms g)).pushouts f := by\n  refine ⟨_, _, _, _, _, ?_, c.isPushout⟩\n  have : c.m = c.isColimit₁.desc\n      (Cocone.mk _ (Discrete.natTrans (fun ⟨i⟩ ↦ by exact g (c.π i)) ≫ c.cofan₂.ι)) :=\n    c.isColimit₁.hom_ext (fun ⟨i⟩ ↦ by rw [IsColimit.fac]; exact c.hm i)\n  rw [this, coproducts_iff]\n  exact ⟨c.ι, ⟨_, _, _, _, c.isColimit₁, c.isColimit₂, _, fun i ↦ ⟨_⟩⟩⟩\n\n/-- The inclusion of a cell. -/\ndef cell (i : c.ι) : B (c.π i) ⟶ X₂ := c.cofan₂.inj i ≫ c.g₂\n\n@[reassoc]\nlemma cell_def (i : c.ι) : c.cell i = c.cofan₂.inj i ≫ c.g₂ := rfl\n\nTarget:\nlemma hom_ext {Z : C} {φ φ' : X₂ ⟶ Z}\n    (h₀ : f ≫ φ = f ≫ φ') (h : ∀ i, c.cell i ≫ φ = c.cell i ≫ φ') :\n    φ = φ' :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/RelativeCellComplex","family_id":"hom_ext","file_id":"mathlib/Mathlib/AlgebraicTopology/RelativeCellComplex/AttachCells.lean","sample_id":"f5cf3adf5b2f007282e6afc982afeb8adfc7d1ca08c6a7becb4328354365d479"},"split":"train"}
{"binding":{"chosen_proof_sha256":"95f0a046edcc88c2f0897d1f675b7af88bcf9e64df19c8ace7d838acef7a93e4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"929feb6854c6bd754df0ddca63934028a11faf17c7ad84065207bb38e2efb282","source_sha256":"394c3d8954491754c0a03071ce7aa65ddc93c4783581cf0b4d6869361547187f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have ⟨n, S, d, _, hd, ⟨e⟩⟩ := exists_end_algEquiv_pi_matrix_end R₀ R M\n  classical exact ⟨n, _, d, inferInstance, inferInstance, hd, ⟨e⟩⟩","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":2,"token_jaccard":0.05,"token_length_ratio":0.047619},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"9bb33355c2a4fd8075be3c51c0cb1c6be0fa271188e64ea622f8051e40cd631e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.FreeModule.Finite.Basic\npublic import Mathlib.RingTheory.FiniteLength\npublic import Mathlib.RingTheory.SimpleModule.Isotypic\npublic import Mathlib.RingTheory.SimpleRing.Congr\npublic import Mathlib.RingTheory.SimpleRing.Matrix\n\nNamespace:\nIsSemisimpleModule\n\nLocal context:\n/-\nCopyright (c) 2025 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu\n-/\n/-!\n# Wedderburn–Artin Theorem\n\n## Main results\n\n* `IsSimpleRing.tfae`: a simple ring is semisimple iff it is Artinian,\n  iff it has a minimal left ideal.\n\n* `isSimpleRing_isArtinianRing_iff`: a ring is simple Artinian iff it is semisimple, isotypic,\n  and nontrivial.\n\n* `IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite`: a simple Artinian algebra is\n  isomorphic to a (finite-dimensional) matrix algebra over a division algebra. The division\n  algebra is the opposite of the endomorphism algebra of a simple (i.e., minimal) left ideal.\n\n* `IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite`: a semisimple algebra is\n  isomorphic to a finite direct product of matrix algebras over division algebras. The division\n  algebras are the opposites of the endomorphism algebras of the simple (i.e., minimal)\n  left ideals.\n\n* `IsSimpleRing.exists_algEquiv_matrix_divisionRing_finite`,\n  `IsSemisimpleRing.exists_algEquiv_pi_matrix_divisionRing_finite`:\n  if the simple Artinian / semisimple algebra is finite as a module over a base ring, then the\n  division algebra(s) are also finite over the same ring.\n  If the base ring is an algebraically closed field, the only finite-dimensional division algebra\n  over it is itself, and we obtain `IsSimpleRing.exists_algEquiv_matrix_of_isAlgClosed` and\n  `IsSemisimpleRing.exists_algEquiv_pi_matrix_of_isAlgClosed` (in a later file).\n\n-/\n\npublic section\n\nuniverse u\nvariable (R₀ : Type*) {R : Type u} [CommSemiring R₀] [Ring R] [Algebra R₀ R]\n\n/-- A simple ring is semisimple iff it is Artinian, iff it has a minimal left ideal. -/\ntheorem IsSimpleRing.tfae [IsSimpleRing R] : List.TFAE\n    [IsSemisimpleRing R, IsArtinianRing R, ∃ I : Ideal R, IsAtom I] := by\n  tfae_have 1 → 2 := fun _ ↦ inferInstance\n  tfae_have 2 → 3 := fun _ ↦ IsAtomic.exists_atom _\n  tfae_have 3 → 1 := fun ⟨I, hI⟩ ↦ by\n    have ⟨_, h⟩ := isSimpleRing_iff_isTwoSided_imp.mp ‹IsSimpleRing R›\n    simp_rw [← isFullyInvariant_iff_isTwoSided] at h\n    have := isSimpleModule_iff_isAtom.mpr hI\n    obtain eq | eq := h _ (.isotypicComponent R R I)\n    · exact (hI.bot_lt.not_ge <| (le_sSup <| by exact ⟨.refl ..⟩).trans_eq eq).elim\n    exact .congr (.symm <| .trans (.ofEq _ _ eq) Submodule.topEquiv)\n  tfae_finish\n\ntheorem IsSimpleRing.isSemisimpleRing_iff_isArtinianRing [IsSimpleRing R] :\n    IsSemisimpleRing R ↔ IsArtinianRing R := tfae.out 0 1\n\ntheorem isSimpleRing_isArtinianRing_iff :\n    IsSimpleRing R ∧ IsArtinianRing R ↔ IsSemisimpleRing R ∧ IsIsotypic R R ∧ Nontrivial R := by\n  refine ⟨fun ⟨_, _⟩ ↦ ?_, fun ⟨_, _, _⟩ ↦ ?_⟩\n  on_goal 1 => have := IsSimpleRing.isSemisimpleRing_iff_isArtinianRing.mpr ‹_›\n  all_goals simp_rw [isIsotypic_iff_isFullyInvariant_imp_bot_or_top,\n      isFullyInvariant_iff_isTwoSided, isSimpleRing_iff_isTwoSided_imp] at *\n  · exact ⟨this, by rwa [and_comm]⟩\n  · exact ⟨⟨‹_›, ‹_›⟩, inferInstance⟩\n\nnamespace IsSimpleRing\n\nvariable (R) [IsSimpleRing R] [IsArtinianRing R]\n\ninstance (priority := low) : IsSemisimpleRing R :=\n  (isSimpleRing_isArtinianRing_iff.mp ⟨‹_›, ‹_›⟩).1\n\ntheorem isIsotypic (M) [AddCommGroup M] [Module R M] : IsIsotypic R M :=\n  (isSimpleRing_isArtinianRing_iff.mp ⟨‹_›, ‹_›⟩).2.1.of_self M\n\n/-- The **Wedderburn–Artin Theorem**: an Artinian simple ring is isomorphic to a matrix\nring over the opposite of the endomorphism ring of its simple module. -/\ntheorem exists_ringEquiv_matrix_end_mulOpposite :\n    ∃ (n : ℕ) (_ : NeZero n) (I : Ideal R) (_ : IsSimpleModule R I),\n      Nonempty (R ≃+* Matrix (Fin n) (Fin n) (Module.End R I)ᵐᵒᵖ) := by\n  have ⟨n, hn, S, hS, ⟨e⟩⟩ := (isIsotypic R R).linearEquiv_fun\n  refine ⟨n, hn, S, hS, ⟨.trans (.opOp R) <| .trans (.op ?_) (.symm .mopMatrix)⟩⟩\n  exact .trans (.moduleEndSelf R) <| .trans e.conjRingEquiv (endVecRingEquivMatrixEnd ..)\n\n/-- The **Wedderburn–Artin Theorem**: an Artinian simple ring is isomorphic to a matrix\nring over a division ring. -/\ntheorem exists_ringEquiv_matrix_divisionRing :\n    ∃ (n : ℕ) (_ : NeZero n) (D : Type u) (_ : DivisionRing D),\n      Nonempty (R ≃+* Matrix (Fin n) (Fin n) D) := by\n  have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_ringEquiv_matrix_end_mulOpposite R\n  classical exact ⟨n, hn, _, _, ⟨e⟩⟩\n\n/-- The **Wedderburn–Artin Theorem**, algebra form: an Artinian simple algebra is isomorphic\nto a matrix algebra over the opposite of the endomorphism algebra of its simple module. -/\ntheorem exists_algEquiv_matrix_end_mulOpposite :\n    ∃ (n : ℕ) (_ : NeZero n) (I : Ideal R) (_ : IsSimpleModule R I),\n      Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) (Module.End R I)ᵐᵒᵖ) := by\n  have ⟨n, hn, S, hS, ⟨e⟩⟩ := (isIsotypic R R).linearEquiv_fun\n  refine ⟨n, hn, S, hS, ⟨.trans (.opOp R₀ R) <| .trans (.op ?_) (.symm .mopMatrix)⟩⟩\n  exact .trans (.moduleEndSelf R₀) <| .trans (e.conjAlgEquiv R₀) (endVecAlgEquivMatrixEnd ..)\n\n/-- The **Wedderburn–Artin Theorem**, algebra form: an Artinian simple algebra is isomorphic\nto a matrix algebra over a division algebra. -/\ntheorem exists_algEquiv_matrix_divisionRing :\n    ∃ (n : ℕ) (_ : NeZero n) (D : Type u) (_ : DivisionRing D) (_ : Algebra R₀ D),\n      Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) D) := by\n  have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_algEquiv_matrix_end_mulOpposite R₀ R\n  classical exact ⟨n, hn, _, _, _, ⟨e⟩⟩\n\n/-- The **Wedderburn–Artin Theorem**, algebra form, finite case: a finite Artinian simple algebra is\nisomorphic to a matrix algebra over a finite division algebra. -/\ntheorem exists_algEquiv_matrix_divisionRing_finite [Module.Finite R₀ R] :\n    ∃ (n : ℕ) (_ : NeZero n) (D : Type u) (_ : DivisionRing D) (_ : Algebra R₀ D)\n      (_ : Module.Finite R₀ D), Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) D) := by\n  have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_algEquiv_matrix_end_mulOpposite R₀ R\n  have := Module.Finite.equiv e.toLinearEquiv\n  classical exact ⟨n, hn, _, _, _, .of_surjective\n    (Matrix.entryLinearMap R₀ _ (0 : Fin n) (0 : Fin n)) fun f ↦ ⟨fun _ _ ↦ f, rfl⟩, ⟨e⟩⟩\n\nend IsSimpleRing\n\nnamespace IsSemisimpleModule\n\nopen Module (End)\n\nuniverse v\nvariable (R) (M : Type v) [AddCommGroup M] [Module R₀ M] [Module R M] [IsScalarTower R₀ R M]\n  [IsSemisimpleModule R M] [Module.Finite R M]\n\ntheorem exists_end_algEquiv_pi_matrix_end :\n    ∃ (n : ℕ) (S : Fin n → Submodule R M) (d : Fin n → ℕ),\n      (∀ i, IsSimpleModule R (S i)) ∧ (∀ i, NeZero (d i)) ∧\n      Nonempty (End R M ≃ₐ[R₀] Π i, Matrix (Fin (d i)) (Fin (d i)) (End R (S i))) := by\n  choose d pos S _ simple e using fun c : isotypicComponents R M ↦\n    (IsIsotypic.isotypicComponents c.2).submodule_linearEquiv_fun\n  classical exact ⟨_, _, _, fun _ ↦ simple _, fun _ ↦ pos _, ⟨.trans (endAlgEquiv R₀ R M) <| .trans\n    (.piCongrRight fun c ↦ ((e c).some.conjAlgEquiv R₀).trans (endVecAlgEquivMatrixEnd ..)) <|\n    (.piCongrLeft' R₀ _ (Finite.equivFin _))⟩⟩\n\ntheorem exists_end_ringEquiv_pi_matrix_end :\n    ∃ (n : ℕ) (S : Fin n → Submodule R M) (d : Fin n → ℕ),\n      (∀ i, IsSimpleModule R (S i)) ∧ (∀ i, NeZero (d i)) ∧\n      Nonempty (End R M ≃+* Π i, Matrix (Fin (d i)) (Fin (d i)) (End R (S i))) :=\n  have ⟨n, S, d, hS, hd, ⟨e⟩⟩ := exists_end_algEquiv_pi_matrix_end ℕ R M; ⟨n, S, d, hS, hd, ⟨e⟩⟩\n\n@[deprecated (since := \"2025-11-16\")] alias exists_end_algEquiv := exists_end_algEquiv_pi_matrix_end\n@[deprecated (since := \"2025-11-16\")]\nalias exists_end_ringEquiv := exists_end_ringEquiv_pi_matrix_end\n\n-- TODO: can also require D be in `Type u`, since every simple module is the quotient by an ideal.\n\nTarget:\ntheorem exists_end_algEquiv_pi_matrix_divisionRing :\n    ∃ (n : ℕ) (D : Fin n → Type v) (d : Fin n → ℕ) (_ : ∀ i, DivisionRing (D i))\n      (_ : ∀ i, Algebra R₀ (D i)), (∀ i, NeZero (d i)) ∧\n      Nonempty (End R M ≃ₐ[R₀] Π i, Matrix (Fin (d i)) (Fin (d i)) (D i)) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/SimpleModule","family_id":"exists_end_algequiv_pi_matrix_divisionring","file_id":"mathlib/Mathlib/RingTheory/SimpleModule/WedderburnArtin.lean","sample_id":"929feb6854c6bd754df0ddca63934028a11faf17c7ad84065207bb38e2efb282"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b9e374e8c54a58f587ed34013c46e429e34757a52725bc1a7e7b25ce0f218592","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"adc475a92322c09535483773caf67bedfb86217c3d55743dea0acb6da8df2e6f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5b180b286925047b6265b1d6b6387babba0a99b025076644a75ae77bbbe0b4d6","source_sha256":"9b749e1f06681e2aad37931b0d304960c8762eef95a6f961195bf175c696c109","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply injective_of_mono (X.mapIso (sigmaIso f).op).hom\n  apply injective_of_mono (sigmaComparison X (fun a ↦ (fiber f a).1))\n  ext a\n  specialize h a\n  rw [← sigmaComparison_comp_sigmaIso] at h\n  exact h","hard_negative":true,"metrics":{"chosen_tokens":48,"rejected_tokens":5,"token_jaccard":0.1,"token_length_ratio":0.104167},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"9bdb705348dc55b3a87d665eeb42b1227cd3508e37f3fbe270ddcd680196e5a6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Condensed.Discrete.Basic\npublic import Mathlib.Condensed.TopComparison\npublic import Mathlib.Topology.Category.CompHausLike.SigmaComparison\npublic import Mathlib.Topology.FiberPartition\n\nNamespace:\nCompHausLike.LocallyConstant\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# The sheaf of locally constant maps on `CompHausLike P`\n\nThis file proves that under suitable conditions, the functor from the category of sets to the\ncategory of sheaves for the coherent topology on `CompHausLike P`, given by mapping a set to the\nsheaf of locally constant maps to it, is left adjoint to the \"underlying set\" functor (evaluation\nat the point).\n\nWe apply this to prove that the constant sheaf functor into (light) condensed sets is isomorphic to\nthe functor of sheaves of locally constant maps described above.\n\n## Proof sketch\n\nThe hard part of this adjunction is to define the counit. Its components are defined as follows:\n\nLet `S : CompHausLike P` and let `Y` be a finite-product-preserving presheaf on `CompHausLike P`\n(e.g. a sheaf for the coherent topology). We need to define a map `LocallyConstant S Y(*) ⟶ Y(S)`.\nGiven a locally constant map `f : S → Y(*)`, let `S = S₁ ⊔ ⋯ ⊔ Sₙ` be the corresponding\ndecomposition of `S` into the fibers. Let `yᵢ ∈ Y(*)` denote the value of `f` on `Sᵢ` and denote\nby `gᵢ` the canonical map `Y(*) → Y(Sᵢ)`. Our map then takes `f` to the image of\n`(g₁(y₁), ⋯, gₙ(yₙ))` under the isomorphism `Y(S₁) × ⋯ × Y(Sₙ) ≅ Y(S₁ ⊔ ⋯ ⊔ Sₙ) = Y(S)`.\n\nNow we need to prove that the counit is natural in `S : CompHausLike P` and\n`Y : Sheaf  (coherentTopology (CompHausLike P)) (Type _)`. There are two key lemmas in all\nnaturality proofs in this file (both lemmas are in the `CompHausLike.LocallyConstant` namespace):\n\n* `presheaf_ext`: given `S`, `Y` and `f : LocallyConstant S Y(*)` like above, another presheaf\n  `X`, and two elements `x y : X(S)`, to prove that `x = y` it suffices to prove that for every\n  inclusion map `ιᵢ : Sᵢ ⟶ S`, `X(ιᵢ)(x) = X(ιᵢ)(y)`.\n  Here it is important that we set everything up in such a way that the `Sᵢ` are literally subtypes\n  of `S`.\n\n* `incl_of_counitAppApp`: given  `S`, `Y` and `f : LocallyConstant S Y(*)` like above, we have\n  `Y(ιᵢ)(ε_{S, Y}(f)) = gᵢ(yᵢ)` where `ε` denotes the counit and the other notation is like above.\n\n## Main definitions\n\n* `CompHausLike.LocallyConstant.functor`: the functor from the category of sets to the category of\n  sheaves for the coherent topology on `CompHausLike P`, which takes a set `X` to\n  `LocallyConstant - X`\n  - `CondensedSet.LocallyConstant.functor` is the above functor in the case of condensed sets.\n  - `LightCondSet.LocallyConstant.functor` is the above functor in the case of light condensed sets.\n\n* `CompHausLike.LocallyConstant.adjunction`: the functor described above is left adjoint to the\n  \"underlying set\" functor `(sheafSections _ _).obj ⟨CompHausLike.of P PUnit.{u + 1}⟩`, which takes\n  a sheaf `X` to the set `X(*)`.\n\n* `CondensedSet.LocallyConstant.iso`: the functor `CondensedSet.LocallyConstant.functor` is\n  isomorphic to the functor `Condensed.discrete (Type _)` (the constant sheaf functor from sets to\n  condensed sets).\n\n* `LightCondSet.LocallyConstant.iso`: the functor `LightCondSet.LocallyConstant.functor` is\n  isomorphic to the functor `LightCondensed.discrete (Type _)` (the constant sheaf functor from sets\n  to light condensed sets).\n\n-/\n\n@[expose] public section\n\nuniverse u w\n\nopen CategoryTheory Limits LocallyConstant TopologicalSpace.Fiber Opposite Function Fiber\n\nvariable {P : TopCat.{u} → Prop}\n\nnamespace CompHausLike.LocallyConstant\n\n/--\nThe functor from the category of sets to presheaves on `CompHausLike P` given by locally constant\nmaps.\n-/\n@[simps obj_obj obj_map map_app]\ndef functorToPresheaves : Type (max u w) ⥤ ((CompHausLike.{u} P)ᵒᵖ ⥤ Type (max u w)) where\n  obj X := {\n    obj := fun ⟨S⟩ ↦ (LocallyConstant S X)\n    map f := ↾fun g ↦ g.comap f.unop.hom.hom }\n  map f := { app _ := ↾fun t ↦ t.map f }\n\n/--\nLocally constant maps are the same as continuous maps when the target is equipped with the discrete\ntopology\n-/\n@[simps]\ndef locallyConstantIsoContinuousMap (Y X : Type*) [TopologicalSpace Y] :\n    LocallyConstant Y X ≅ C(Y, TopCat.discrete.obj X) :=\n  letI : TopologicalSpace X := ⊥\n  haveI : DiscreteTopology X := ⟨rfl⟩\n  { hom := ↾fun f ↦ (f : C(Y, X))\n    inv := ↾fun f ↦ ⟨f, (IsLocallyConstant.iff_continuous f).mpr f.2⟩ }\n\nsection Adjunction\n\nvariable [∀ (S : CompHausLike.{u} P) (p : S → Prop), HasProp P (Subtype p)]\n\nsection\n\nvariable {Q : CompHausLike.{u} P} {Z : Type max u w} (r : LocallyConstant Q Z) (a : Fiber r)\n\n/-- A fiber of a locally constant map as a `CompHausLike P`. -/\nabbrev fiber : CompHausLike.{u} P := CompHausLike.of P a.val\n\n/-- The inclusion map from a component of the coproduct induced by `f` into `S`. -/\ndef sigmaIncl : fiber r a ⟶ Q := ofHom _ (TopologicalSpace.Fiber.sigmaIncl _ a)\n\n/-- The canonical map from the coproduct induced by `f` to `S` as an isomorphism in\n`CompHausLike P`. -/\nnoncomputable def sigmaIso [HasExplicitFiniteCoproducts.{u} P] : (finiteCoproduct (fiber r)) ≅ Q :=\n  isoOfBijective (ofHom _ (sigmaIsoHom r)) ⟨sigmaIsoHom_inj r, sigmaIsoHom_surj r⟩\n\nlemma sigmaComparison_comp_sigmaIso [HasExplicitFiniteCoproducts.{u} P]\n    (X : (CompHausLike.{u} P)ᵒᵖ ⥤ Type (max u w)) :\n    (X.mapIso (sigmaIso r).op).hom ≫ sigmaComparison X (fun a ↦ (fiber r a).1) ≫\n      (↾fun g ↦ g a) = X.map (sigmaIncl r a).op := by\n  ext\n  simp only [Functor.mapIso_hom, Iso.op_hom, sigmaComparison, TypeCat.Fun.toFun_apply,\n    CategoryTheory.comp_apply, ConcreteCategory.hom_ofHom, TypeCat.Fun.coe_mk,\n    ← X.map_comp_apply]\n  rfl\n\nend\n\nvariable {S : CompHausLike.{u} P} {Y : (CompHausLike.{u} P)ᵒᵖ ⥤ Type (max u w)}\n  [HasProp P PUnit.{u + 1}] (f : LocallyConstant S (Y.obj (op (CompHausLike.of P PUnit.{u + 1}))))\n\n/-- The projection of the counit. -/\nnoncomputable def counitAppAppImage : (a : Fiber f) → Y.obj ⟨fiber f a⟩ :=\n  fun a ↦ Y.map (CompHausLike.isTerminalPUnit.from _).op a.image\n\n/--\nThe counit is defined as follows: given a locally constant map `f : S → Y(*)`, let\n`S = S₁ ⊔ ⋯ ⊔ Sₙ` be the corresponding decomposition of `S` into the fibers. We need to provide an\nelement of `Y(S)`. It suffices to provide an element of `Y(Sᵢ)` for all `i`. Let `yᵢ ∈ Y(*)` denote\nthe value of `f` on `Sᵢ`. Our desired element is the image of `yᵢ` under the canonical map\n`Y(*) → Y(Sᵢ)`.\n-/\nnoncomputable def counitAppApp (S : CompHausLike.{u} P)\n    (Y : (CompHausLike.{u} P)ᵒᵖ ⥤ Type (max u w))\n    [PreservesFiniteProducts Y] [HasExplicitFiniteCoproducts.{u} P] :\n    LocallyConstant S (Y.obj (op (CompHausLike.of P PUnit.{u + 1}))) ⟶ Y.obj ⟨S⟩ :=\n  ↾fun r ↦ (inv (sigmaComparison Y (fun a ↦ (fiber r a).1)) ≫\n    (Y.mapIso (sigmaIso r).op).inv) (counitAppAppImage r)\n\n-- This is the key lemma to prove naturality of the counit:\n/--\nTo check equality of two elements of `X(S)`, it suffices to check equality after composing with\neach `X(S) → X(Sᵢ)`.\n-/\n\nTarget:\nlemma presheaf_ext (X : (CompHausLike.{u} P)ᵒᵖ ⥤ Type (max u w))\n    [PreservesFiniteProducts X] (x y : X.obj ⟨S⟩)\n    [HasExplicitFiniteCoproducts.{u} P]\n    (h : ∀ (a : Fiber f), X.map (sigmaIncl f a).op x = X.map (sigmaIncl f a).op y) : x = y :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_5b180b286925","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"fbb4490cc5cc1702b1ab3cd7930c7b7c59d7bdf7f526f7b9bc324b8689c3989f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed/Discrete","family_id":"presheaf_ext","file_id":"mathlib/Mathlib/Condensed/Discrete/LocallyConstant.lean","sample_id":"5b180b286925047b6265b1d6b6387babba0a99b025076644a75ae77bbbe0b4d6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"886ef528de3066c5206bdbcb8b42b2d7a0a7e9340445f6b246cca5e2a1bdcafc","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cfdd53edfe1acc3d2b51c43a1c120daa3d309e493c2f938aa9d41f5faeec220c","source_sha256":"edb52479240700af7f6c572a1b6c456a3b7b18b85aa47a4adb83e507587b6553","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [smul_Spec_def, smul_Spec_def]\n  simp_rw [smul_restrictAppIso_hom_apply, ← ConcreteCategory.comp_apply, Category.assoc]\n  have :\n      f ≫ (ΓSpecIso S).inv ≫ (Spec S).presheaf.map U.leTop.op ≫ (Hom.appIso (Spec.map f) U).inv =\n        (ΓSpecIso R).inv ≫ (Spec R).presheaf.map (Spec.map f ''ᵁ U).leTop.op := by\n    simp [Iso.cancel_iso_inv_left, Hom.app_eq_appLE]\n    rfl\n  rw [this]","hard_negative":true,"metrics":{"chosen_tokens":108,"rejected_tokens":8,"token_jaccard":0.086957,"token_length_ratio":0.074074},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"9c24c402ff87d2ee49592979f5cfce060dc04de150731a97f57382f97e3ebbfe","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Category.ModuleCat.Localization\npublic import Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent\npublic import Mathlib.AlgebraicGeometry.AffineScheme\npublic import Mathlib.AlgebraicGeometry.Modules.Sheaf\n\nNamespace:\nAlgebraicGeometry.Scheme.Modules\n\nLocal context:\n/-\nCopyright (c) 2024 Weihong Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Johan Commelin, Amelia Livingston, Sophie Morel, Jujian Zhang, Weihong Xu,\n  Andrew Yang, Brian Nugent\n-/\n/-!\n\n# Construction of M^~\n\nGiven any commutative ring `R` and `R`-module `M`, we construct the sheaf `M^~` of `𝒪_SpecR`-modules\nsuch that `M^~(U)` is the set of dependent functions that are locally fractions.\n\n## Main definitions\n* `AlgebraicGeometry.tilde` : `M^~` as a sheaf of `𝒪_{Spec R}`-modules.\n* `AlgebraicGeometry.tilde.adjunction` : `~` is left adjoint to taking global sections.\n\n-/\n\n@[expose] public noncomputable section\n\nuniverse u\n\nopen TopCat AlgebraicGeometry TopologicalSpace CategoryTheory Opposite\n\nvariable {R : CommRingCat.{u}} (M : ModuleCat.{u} R)\n\nnamespace AlgebraicGeometry\n\nopen _root_.PrimeSpectrum\n\n/-- The forgetful functor from `𝒪_{Spec R}` modules to sheaves of `R`-modules. -/\ndef modulesSpecToSheaf :\n    (Spec R).Modules ⥤ TopCat.Sheaf (ModuleCat R) (Spec R) :=\n  SheafOfModules.forgetToSheafModuleCat (Spec R).ringCatSheaf (.op ⊤)\n    (Limits.initialOpOfTerminal Limits.isTerminalTop) ⋙\n  sheafCompose _ (ModuleCat.restrictScalars (Scheme.ΓSpecIso R).inv.hom)\n\n/-- The global section functor for `𝒪_{Spec R}` modules -/\nnoncomputable\ndef moduleSpecΓFunctor : (Spec (.of R)).Modules ⥤ ModuleCat R :=\n  modulesSpecToSheaf ⋙ TopCat.Sheaf.forget _ _ ⋙ (evaluation _ _).obj (.op ⊤)\n\nset_option backward.isDefEq.respectTransparency false in\nopen PrimeSpectrum in\n/-- The forgetful functor from `𝒪_{Spec R}` modules to sheaves of `R`-modules is fully faithful. -/\ndef SpecModulesToSheafFullyFaithful : (modulesSpecToSheaf (R := R)).FullyFaithful where\n  preimage {M N} f := ⟨fun U ↦ ModuleCat.ofHom ⟨(f.1.app U).hom.toAddHom, by\n    intro t m\n    apply TopCat.Presheaf.IsSheaf.section_ext (modulesSpecToSheaf.obj N).2\n    intro x hxU\n    obtain ⟨a, ⟨_, ⟨r, rfl⟩, rfl⟩, hxr, hrU : basicOpen _ ≤ _⟩ :=\n      PrimeSpectrum.isBasis_basic_opens.exists_subset_of_mem_open hxU U.unop.2\n    refine ⟨_, hrU, hxr, ?_⟩\n    refine Eq.trans ?_ (N.val.map_smul (homOfLE hrU).op t _).symm\n    change N.1.map (homOfLE hrU).op (f.1.app _ _) = _ • N.1.map (homOfLE hrU).op (f.1.app _ _)\n    have (x : _) :\n        f.1.app _ (M.1.map (homOfLE hrU).op _) = N.1.map (homOfLE hrU).op (f.1.app _ x) :=\n      congr($(f.1.naturality (homOfLE hrU).op).hom x)\n    rw [← this, ← this, M.val.map_smul]\n    generalize (Spec R).ringCatSheaf.obj.map (homOfLE hrU).op t = t\n    letI := Module.compHom (R := Γ(Spec R, basicOpen r)) Γ(M, basicOpen r)\n      (algebraMap R Γ(Spec R, basicOpen r))\n    haveI : IsScalarTower R Γ(Spec R, basicOpen r) Γ(M, basicOpen r) :=\n      .of_algebraMap_smul fun _ _ ↦ rfl\n    letI := Module.compHom Γ(N, basicOpen r) (algebraMap R Γ(Spec R, basicOpen r))\n    haveI : IsScalarTower R Γ(Spec R, basicOpen r) Γ(N, basicOpen r) :=\n      .of_algebraMap_smul fun _ _ ↦ rfl\n    exact (IsLocalization.linearMap_compatibleSMul (.powers (M := R) r)\n      Γ(Spec R, basicOpen r) Γ(M, basicOpen r) Γ(N, basicOpen r)).map_smul\n      (f.hom.app _).hom _ _⟩, fun i ↦ by ext x; exact congr($(f.1.naturality i).hom x)⟩\n  map_preimage f := rfl\n  preimage_map f := rfl\n\ninstance : (modulesSpecToSheaf (R := R)).Faithful := SpecModulesToSheafFullyFaithful.faithful\n\ninstance : (modulesSpecToSheaf (R := R)).Full := SpecModulesToSheafFullyFaithful.full\n\nnamespace Scheme.Modules\n\nvariable {M : (Spec R).Modules} {U V : (Spec R).Opens}\n\ninstance : Module R Γ(M, U) :=\n  inferInstanceAs <| Module R ((modulesSpecToSheaf.obj M).obj.obj (.op U))\n\ninstance : IsScalarTower R Γ(Spec R, U) Γ(M, U) :=\n  IsScalarTower.of_compHom R Γ(Spec R, U) Γ(M, U)\n\nlemma smul_Spec_def (r : R) (x : Γ(M, U)) :\n    r • x = ((Spec R).presheaf.map U.leTop.op) ((Scheme.ΓSpecIso R).inv r) • x :=\n  rfl\n\n@[simp]\nlemma map_smul_Spec (hUV : .op V ⟶ .op U) (f : R) (x : Γ(M, V)) :\n    dsimp% M.presheaf.map hUV (f • x) = f • M.presheaf.map hUV x :=\n  ((modulesSpecToSheaf.obj M).obj.map hUV).hom.map_smul f x\n\nlemma isUnit_algebraMap_end_of_le_basicOpen (f : R) (hf : U ≤ PrimeSpectrum.basicOpen f) :\n    IsUnit (algebraMap R (Module.End R Γ(M, U)) f) := by\n  rw [Module.End.isUnit_iff]\n  have : ⇑((algebraMap R (Module.End ↑R ↑Γ(M, U))) f) =\n      algebraMap (Γ(Spec R, U)) (Module.End Γ(Spec R, U) Γ(M, U))\n        (((Spec R).presheaf.map (homOfLE hf).op) <| algebraMap R _ f) :=\n    rfl\n  rw [this, ← Module.End.isUnit_iff]\n  exact ((IsLocalization.Away.algebraMap_isUnit _).map _).map _\n\nlemma isSMulRegular_of_le_basicOpen {f : R} (hle : U ≤ PrimeSpectrum.basicOpen f) :\n    IsSMulRegular Γ(M, U) f := by\n  intro x y hxy\n  have := M.isUnit_algebraMap_end_of_le_basicOpen _ hle\n  rw [Module.End.isUnit_iff] at this\n  exact this.injective hxy\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\n\nTarget:\nlemma restrictAppIso_smul_Spec {S : CommRingCat.{u}} (f : R ⟶ S)\n    [IsOpenImmersion (Spec.map f)] {U : (Spec S).Opens} (r : R)\n    (x : Γ(M.restrict (Spec.map f), U)) :\n    dsimp% (M.restrictAppIso (Spec.map f) U).hom (f r • x) =\n      r • (M.restrictAppIso (Spec.map f) U).hom x :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"3cc5681fb2aadb1c7d4fe839814f0ad6ebf6c93803fcff4920efd664a92338a8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Modules","family_id":"restrictappiso_smul_spec","file_id":"mathlib/Mathlib/AlgebraicGeometry/Modules/Tilde.lean","sample_id":"cfdd53edfe1acc3d2b51c43a1c120daa3d309e493c2f938aa9d41f5faeec220c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ca01e87830c1462b512c1bad10dd02834fa50907bb2a15cdffba13757dee4a07","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"85cb31cb85bde4e36fe5f40e3de45ee20802c0640781b7ef0983baaaae36e2ea","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"32a4aa28ca53ec5f375880fb329a6c92a05a17c1c9d5e8c664a77a4bde982211","source_sha256":"dbbf637f35822ee42ab01ded669b703900488e4b298d437d2dd114913838068f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine mt ((injective_iff_map_eq_zero _).mp bS.algebraMap_injective _) ?_\n  simp only [Finset.prod_eq_zero_iff, not_exists]\n  rintro x ⟨hx, rfl⟩\n  exact finsetApprox.zero_notMem bS adm hx","hard_negative":true,"metrics":{"chosen_tokens":40,"rejected_tokens":5,"token_jaccard":0.09375,"token_length_ratio":0.125},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"9c59fe52dcae363d069c8b6dde13987209b6e7c96dc6c7418b033677f2cdace0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Pow.Real\npublic import Mathlib.LinearAlgebra.FreeModule.PID\npublic import Mathlib.LinearAlgebra.Matrix.AbsoluteValue\npublic import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue\npublic import Mathlib.RingTheory.ClassGroup.Basic\npublic import Mathlib.RingTheory.DedekindDomain.IntegralClosure\npublic import Mathlib.RingTheory.Norm.Basic\n\nNamespace:\nClassGroup\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Class numbers of global fields\nIn this file, we use the notion of \"admissible absolute value\" to prove\nfiniteness of the class group for number fields and function fields.\n\n## Main definitions\n- `ClassGroup.fintypeOfAdmissibleOfAlgebraic`: if `R` has an admissible absolute value,\n  its integral closure has a finite class group\n-/\n\n@[expose] public section\n\nopen Module Ring\nopen scoped nonZeroDivisors\n\nnamespace ClassGroup\nsection EuclideanDomain\n\nvariable {R S : Type*} (K L : Type*) [EuclideanDomain R] [CommRing S] [IsDomain S]\nvariable [Field K] [Field L]\nvariable [Algebra R K] [IsFractionRing R K]\nvariable [Algebra K L] [FiniteDimensional K L] [Algebra.IsSeparable K L]\nvariable [algRL : Algebra R L] [IsScalarTower R K L]\nvariable [Algebra R S] [Algebra S L]\nvariable [ist : IsScalarTower R S L]\nvariable (abv : AbsoluteValue R ℤ)\nvariable {ι : Type*} [DecidableEq ι] [Fintype ι] (bS : Basis ι R S)\n\n/-- If `b` is an `R`-basis of `S` of cardinality `n`, then `normBound abv b` is an integer\nsuch that for every `R`-integral element `a : S` with coordinates `≤ y`,\nwe have `Algebra.norm a ≤ normBound abv b * y ^ n`. (See also `norm_le` and `norm_lt`). -/\nnoncomputable def normBound : ℤ :=\n  let n := Fintype.card ι\n  let i : ι := Nonempty.some bS.index_nonempty\n  let m : ℤ :=\n    Finset.max'\n      (Finset.univ.image fun ijk : ι × ι × ι =>\n        abv (Algebra.leftMulMatrix bS (bS ijk.1) ijk.2.1 ijk.2.2))\n      ⟨_, Finset.mem_image.mpr ⟨⟨i, i, i⟩, Finset.mem_univ _, rfl⟩⟩\n  Nat.factorial n • (n • m) ^ n\n\ntheorem normBound_pos : 0 < normBound abv bS := by\n  obtain ⟨i, j, k, hijk⟩ : ∃ i j k, Algebra.leftMulMatrix bS (bS i) j k ≠ 0 := by\n    by_contra! h\n    obtain ⟨i⟩ := bS.index_nonempty\n    apply bS.ne_zero i\n    apply\n      (injective_iff_map_eq_zero (Algebra.leftMulMatrix bS)).mp (Algebra.leftMulMatrix_injective bS)\n    ext j k\n    simp [h]\n  simp only [normBound, Algebra.smul_def, eq_natCast]\n  apply mul_pos (Int.natCast_pos.mpr (Nat.factorial_pos _))\n  refine pow_pos (mul_pos (Int.natCast_pos.mpr (Fintype.card_pos_iff.mpr ⟨i⟩)) ?_) _\n  refine lt_of_lt_of_le (abv.pos hijk) (Finset.le_max' _ _ ?_)\n  exact Finset.mem_image.mpr ⟨⟨i, j, k⟩, Finset.mem_univ _, rfl⟩\n\n/-- If the `R`-integral element `a : S` has coordinates `≤ y` with respect to some basis `b`,\nits norm is less than `normBound abv b * y ^ dim S`. -/\ntheorem norm_le (a : S) {y : ℤ} (hy : ∀ k, abv (bS.repr a k) ≤ y) :\n    abv (Algebra.norm R a) ≤ normBound abv bS * y ^ Fintype.card ι := by\n  conv_lhs => rw [← bS.sum_repr a]\n  rw [Algebra.norm_apply, ← LinearMap.det_toMatrix bS]\n  simp only [map_sum, map_smul, map_sum, map_smul,\n    normBound, smul_mul_assoc, ← mul_pow]\n  convert! Matrix.det_sum_smul_le Finset.univ _ hy using 3\n  · rw [Finset.card_univ, smul_mul_assoc, mul_comm]\n  · intro i j k\n    apply Finset.le_max'\n    exact Finset.mem_image.mpr ⟨⟨i, j, k⟩, Finset.mem_univ _, rfl⟩\n\n/-- If the `R`-integral element `a : S` has coordinates `< y` with respect to some basis `b`,\nits norm is strictly less than `normBound abv b * y ^ dim S`. -/\ntheorem norm_lt {T : Type*} [Ring T] [LinearOrder T] [IsStrictOrderedRing T] (a : S) {y : T}\n    (hy : ∀ k, (abv (bS.repr a k) : T) < y) :\n    (abv (Algebra.norm R a) : T) < normBound abv bS * y ^ Fintype.card ι := by\n  obtain ⟨i⟩ := bS.index_nonempty\n  have him : (Finset.univ.image fun k => abv (bS.repr a k)).Nonempty :=\n    ⟨_, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩\n  set y' : ℤ := Finset.max' _ him with y'_def\n  have hy' : ∀ k, abv (bS.repr a k) ≤ y' := by\n    intro k\n    exact @Finset.le_max' ℤ _ _ _ (Finset.mem_image.mpr ⟨k, Finset.mem_univ _, rfl⟩)\n  have : (y' : T) < y := by\n    rw [y'_def,\n      ← Finset.max'_image (show Monotone (_ : ℤ → T) from fun x y h => Int.cast_le.mpr h)\n          _ (him.image _)]\n    apply (Finset.max'_lt_iff _ (him.image _)).mpr\n    simp only [Finset.mem_image]\n    rintro _ ⟨x, ⟨k, -, rfl⟩, rfl⟩\n    exact hy k\n  have y'_nonneg : 0 ≤ y' := le_trans (abv.nonneg _) (hy' i)\n  apply (Int.cast_le.mpr (norm_le abv bS a hy')).trans_lt\n  simp only [Int.cast_mul, Int.cast_pow]\n  apply mul_lt_mul' le_rfl\n  · exact pow_lt_pow_left₀ this (by positivity) (@Fintype.card_ne_zero _ _ ⟨i⟩)\n  · positivity\n  · exact Int.cast_pos.mpr (normBound_pos abv bS)\n\n\n/-- A nonzero ideal has an element of minimal norm. -/\ntheorem exists_min (I : (Ideal S)⁰) :\n    ∃ b ∈ (I : Ideal S),\n      b ≠ 0 ∧ ∀ c ∈ (I : Ideal S), abv (Algebra.norm R c) < abv (Algebra.norm R b) → c =\n      (0 : S) := by\n  obtain ⟨_, ⟨b, b_mem, b_ne_zero, rfl⟩, min⟩ := @Int.exists_least_of_bdd\n      (fun a => ∃ b ∈ (I : Ideal S), b ≠ (0 : S) ∧ abv (Algebra.norm R b) = a)\n    (by\n      use 0\n      rintro _ ⟨b, _, _, rfl⟩\n      apply abv.nonneg)\n    (by\n      obtain ⟨b, b_mem, b_ne_zero⟩ := (I : Ideal S).ne_bot_iff.mp (nonZeroDivisors.coe_ne_zero I)\n      exact ⟨_, ⟨b, b_mem, b_ne_zero, rfl⟩⟩)\n  refine ⟨b, b_mem, b_ne_zero, ?_⟩\n  intro c hc lt\n  contrapose! lt with c_ne_zero\n  exact min _ ⟨c, hc, c_ne_zero, rfl⟩\n\nsection IsAdmissible\n\nvariable {abv}\nvariable (adm : abv.IsAdmissible)\n\n/-- If we have a large enough set of elements in `R^ι`, then there will be a pair\nwhose remainders are close together. We'll show that all sets of cardinality\nat least `cardM bS adm` elements satisfy this condition.\n\nThe value of `cardM` is not at all optimal: for specific choices of `R`,\nthe minimum cardinality can be exponentially smaller.\n-/\nnoncomputable def cardM : ℕ :=\n  adm.card (normBound abv bS ^ (-1 / Fintype.card ι : ℝ)) ^ Fintype.card ι\n\nvariable [Infinite R]\n\n/-- In the following results, we need a large set of distinct elements of `R`. -/\nnoncomputable def distinctElems : Fin (cardM bS adm).succ ↪ R :=\n  Fin.valEmbedding.trans (Infinite.natEmbedding R)\n\nvariable [DecidableEq R]\n\n/-- `finsetApprox` is a finite set such that each fractional ideal in the integral closure\ncontains an element close to `finsetApprox`. -/\nnoncomputable def finsetApprox : Finset R :=\n  (Finset.univ.image fun xy : _ × _ => distinctElems bS adm xy.1 - distinctElems bS adm xy.2).erase\n    0\n\ntheorem finsetApprox.zero_notMem : (0 : R) ∉ finsetApprox bS adm :=\n  Finset.notMem_erase _ _\n\n@[simp]\ntheorem mem_finsetApprox {x : R} :\n    x ∈ finsetApprox bS adm ↔ ∃ i j, i ≠ j ∧ distinctElems bS adm i - distinctElems bS adm j =\n    x := by\n  simp only [finsetApprox, Finset.mem_erase, Finset.mem_image]\n  constructor\n  · rintro ⟨hx, ⟨i, j⟩, _, rfl⟩\n    refine ⟨i, j, ?_, rfl⟩\n    rintro rfl\n    simp at hx\n  · rintro ⟨i, j, hij, rfl⟩\n    refine ⟨?_, ⟨i, j⟩, Finset.mem_univ _, rfl⟩\n    rw [Ne, sub_eq_zero]\n    exact fun h => hij ((distinctElems bS adm).injective h)\n\nsection Real\n\nopen Real\n\nattribute [-instance] Real.decidableEq\n\n/-- We can approximate `a / b : L` with `q / r`, where `r` has finitely many options for `L`. -/\ntheorem exists_mem_finsetApprox (a : S) {b} (hb : b ≠ (0 : R)) :\n    ∃ q : S,\n      ∃ r ∈ finsetApprox bS adm, abv (Algebra.norm R (r • a - b • q)) <\n      abv (Algebra.norm R (algebraMap R S b)) := by\n  have dim_pos := Fintype.card_pos_iff.mpr bS.index_nonempty\n  set ε : ℝ := normBound abv bS ^ (-1 / Fintype.card ι : ℝ) with ε_eq\n  have hε : 0 < ε := Real.rpow_pos_of_pos (Int.cast_pos.mpr (normBound_pos abv bS)) _\n  have ε_le : (normBound abv bS : ℝ) * (abv b • ε) ^ (Fintype.card ι : ℝ)\n                ≤ abv b ^ (Fintype.card ι : ℝ) := by\n    have := normBound_pos abv bS\n    have := abv.nonneg b\n    rw [ε_eq, Algebra.smul_def, eq_intCast, mul_rpow, ← rpow_mul, div_mul_cancel₀, rpow_neg_one,\n      mul_left_comm, mul_inv_cancel₀, mul_one, rpow_natCast] <;>\n      try norm_cast; lia\n    · exact Int.cast_nonneg this\n    · linarith\n  set μ : Fin (cardM bS adm).succ ↪ R := distinctElems bS adm\n  let s : ι →₀ R := bS.repr a\n  have s_eq : ∀ i, s i = bS.repr a i := fun i => rfl\n  let qs : Fin (cardM bS adm).succ → ι → R := fun j i => μ j * s i / b\n  let rs : Fin (cardM bS adm).succ → ι → R := fun j i => μ j * s i % b\n  have r_eq : ∀ j i, rs j i = μ j * s i % b := fun i j => rfl\n  have μ_eq : ∀ i j, μ j * s i = b * qs j i + rs j i := by\n    intro i j\n    rw [r_eq, EuclideanDomain.div_add_mod]\n  have μ_mul_a_eq : ∀ j, μ j • a = b • ∑ i, qs j i • bS i + ∑ i, rs j i • bS i := by\n    intro j\n    rw [← bS.sum_repr a]\n    simp only [μ, qs, rs, Finset.smul_sum, ← Finset.sum_add_distrib]\n    refine Finset.sum_congr rfl fun i _ => ?_\n    rw [← s_eq, ← mul_smul, μ_eq, add_smul, mul_smul, ← μ_eq]\n  obtain ⟨j, k, j_ne_k, hjk⟩ := adm.exists_approx hε hb fun j i => μ j * s i\n  have hjk' : ∀ i, (abv (rs k i - rs j i) : ℝ) < abv b • ε := by simpa only [r_eq] using hjk\n  let q := ∑ i, (qs k i - qs j i) • bS i\n  set r := μ k - μ j with r_eq\n  refine ⟨q, r, (mem_finsetApprox bS adm).mpr ?_, ?_⟩\n  · exact ⟨k, j, j_ne_k.symm, rfl⟩\n  have : r • a - b • q = ∑ x : ι, (rs k x • bS x - rs j x • bS x) := by\n    simp only [q, r_eq, sub_smul, μ_mul_a_eq, Finset.smul_sum, ← Finset.sum_add_distrib,\n      ← Finset.sum_sub_distrib, smul_sub]\n    refine Finset.sum_congr rfl fun x _ => ?_\n    ring\n  rw [this, Algebra.norm_algebraMap_of_basis bS, abv.map_pow]\n  refine Int.cast_lt.mp ((norm_lt abv bS _ fun i => lt_of_le_of_lt ?_ (hjk' i)).trans_le ?_)\n  · apply le_of_eq\n    congr\n    simp_rw [map_sum, map_sub, map_smul, Finset.sum_apply',\n      Finsupp.sub_apply, Finsupp.smul_apply, Finset.sum_sub_distrib, Basis.repr_self_apply,\n      smul_eq_mul, mul_boole, Finset.sum_ite_eq', Finset.mem_univ, if_true]\n  · exact mod_cast ε_le\n\n/-- We can approximate `a / b : L` with `q / r`, where `r` has finitely many options for `L`. -/\ntheorem exists_mem_finset_approx' [Algebra.IsAlgebraic R S] (a : S) {b : S} (hb : b ≠ 0) :\n    ∃ q : S,\n      ∃ r ∈ finsetApprox bS adm, abv (Algebra.norm R (r • a - q * b)) < abv (Algebra.norm R b) := by\n  obtain ⟨a', b', hb', h⟩ := Algebra.IsAlgebraic.exists_smul_eq_mul R a hb\n  obtain ⟨q, r, hr, hqr⟩ := exists_mem_finsetApprox bS adm a' hb'\n  refine ⟨q, r, hr, ?_⟩\n  refine\n    lt_of_mul_lt_mul_left ?_ (show 0 ≤ abv (Algebra.norm R (algebraMap R S b')) from abv.nonneg _)\n  refine\n    lt_of_le_of_lt (le_of_eq ?_)\n      (mul_lt_mul hqr le_rfl (abv.pos ((Algebra.norm_ne_zero_iff_of_basis bS).mpr hb))\n        (abv.nonneg _))\n  rw [← abv.map_mul, ← map_mul, ← abv.map_mul, ← map_mul, ← Algebra.smul_def,\n    smul_sub b', sub_mul, smul_comm, h, mul_comm b a', Algebra.smul_mul_assoc r a' b,\n    Algebra.smul_mul_assoc b' q b]\n\nend Real\n\nTarget:\ntheorem prod_finsetApprox_ne_zero : algebraMap R S (∏ m ∈ finsetApprox bS adm, m) ≠ 0 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_32a4aa28ca53","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"78f27bb7efc29cdeff6f19b6f4a5035916aa4fb1a7484ad25cdf828673e36092","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/ClassNumber","family_id":"prod_finsetapprox_ne_zero","file_id":"mathlib/Mathlib/NumberTheory/ClassNumber/Finite.lean","sample_id":"32a4aa28ca53ec5f375880fb329a6c92a05a17c1c9d5e8c664a77a4bde982211"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c849c5df0799eac828ab19895ffb8d4649eaf0e6c519838aa60e6b56e2b819cc","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a6cc210366796b93c3087a5747f28bf3396e3f7d848298d6e55848e06cc71e8d","source_sha256":"279951b6d00a5d584aef7e70a8e7410771dfbbc000b64122d58d64bf59f2a387","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have a := hP.card_parts_eq_average ht\n  lia","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":5,"token_jaccard":0.076923,"token_length_ratio":0.555556},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"9c63ab419bdd2c0d33b5d4adb5f70ce08314517b4fdfe1928c31d828b40958b0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Ring.Nat\npublic import Mathlib.Data.Set.Equitable\npublic import Mathlib.Logic.Equiv.Fin.Basic\npublic import Mathlib.Order.Partition.Finpartition\npublic import Mathlib.Tactic.ApplyFun\n\nNamespace:\nFinpartition\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Finite equipartitions\n\nThis file defines finite equipartitions, the partitions whose parts all are the same size up to a\ndifference of `1`.\n\n## Main declarations\n\n* `Finpartition.IsEquipartition`: Predicate for a `Finpartition` to be an equipartition.\n* `Finpartition.IsEquipartition.exists_partPreservingEquiv`: part-preserving enumeration of a finset\n  equipped with an equipartition. Indices of elements in the same part are congruent modulo\n  the number of parts.\n-/\n\n@[expose] public section\n\n\nopen Finset Fintype\n\nnamespace Finpartition\n\nvariable {α : Type*} [DecidableEq α] {s t : Finset α} (P : Finpartition s)\n\n/-- An equipartition is a partition whose parts are all the same size, up to a difference of `1`. -/\ndef IsEquipartition : Prop :=\n  (P.parts : Set (Finset α)).EquitableOn card\n\ntheorem isEquipartition_iff_card_parts_eq_average :\n    P.IsEquipartition ↔\n      ∀ a : Finset α, a ∈ P.parts → #a = #s / #P.parts ∨ #a = #s / #P.parts + 1 := by\n  simp_rw [IsEquipartition, Finset.equitableOn_iff, P.sum_card_parts]\n\nvariable {P}\n\nlemma not_isEquipartition :\n    ¬P.IsEquipartition ↔ ∃ a ∈ P.parts, ∃ b ∈ P.parts, #b + 1 < #a := Set.not_equitableOn\n\ntheorem _root_.Set.Subsingleton.isEquipartition (h : (P.parts : Set (Finset α)).Subsingleton) :\n    P.IsEquipartition :=\n  Set.Subsingleton.equitableOn h _\n\ntheorem IsEquipartition.card_parts_eq_average (hP : P.IsEquipartition) (ht : t ∈ P.parts) :\n    #t = #s / #P.parts ∨ #t = #s / #P.parts + 1 :=\n  P.isEquipartition_iff_card_parts_eq_average.1 hP _ ht\n\nTarget:\ntheorem IsEquipartition.card_part_eq_average_iff (hP : P.IsEquipartition) (ht : t ∈ P.parts) :\n    #t = #s / #P.parts ↔ #t ≠ #s / #P.parts + 1 :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Partition","family_id":"isequipartition","file_id":"mathlib/Mathlib/Order/Partition/Equipartition.lean","sample_id":"a6cc210366796b93c3087a5747f28bf3396e3f7d848298d6e55848e06cc71e8d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"08f7647cfd9ab05023d94bc2fcca4631870d5f71d7f24de348f6c58aa5102a1d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9b67a823dcea17b7fc9288545c47193c34fe1b90d1bd9cf3ef9be1964ef22f5d","source_sha256":"bf073edff8092ae47149eb64a7d94002d4cd84fb4941057816c6cf0f37fc36aa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isPosSemidef_def, Matrix.posSemidef_iff_dotProduct_mulVec]\n  apply and_congr (B.isSymm_iff_isHermitian_toMatrix b)\n  rw [isNonneg_def]\n  refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩\n  · rw [star_dotProduct_toMatrix₂_mulVec]\n    exact h _\n  · rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]\n    exact h _","hard_negative":true,"metrics":{"chosen_tokens":58,"rejected_tokens":8,"token_jaccard":0.026316,"token_length_ratio":0.137931},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"9c63c5ecad1eedb815533e5209838e77eb95557b83334c565ffb3c1f7431d13a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Matrix.PosDef\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Etienne Marion. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Etienne Marion\n-/\n/-!\n# Sesquilinear forms over a star ring\n\nThis file provides some properties about sesquilinear forms `M →ₗ⋆[R] M →ₗ[R] R` when `R` is a\n`StarRing`.\n-/\n\npublic section\n\nopen Module LinearMap\n\nvariable {R M n : Type*} [CommSemiring R] [StarRing R] [AddCommMonoid M] [Module R M]\n  [Fintype n] [DecidableEq n]\n  {B : M →ₗ⋆[R] M →ₗ[R] R} (b : Basis n R M)\n\nlemma LinearMap.isSymm_iff_basis {ι : Type*} (b : Basis ι R M) :\n    IsSymm B ↔ ∀ i j, star (B (b i) (b j)) = B (b j) (b i) where\n  mp h i j := h.eq _ _\n  mpr := by\n    refine fun h ↦ ⟨fun x y ↦ ?_⟩\n    obtain ⟨fx, tx, ix, -, hx⟩ := Submodule.mem_span_iff_exists_finset_subset.1\n      (by simp : x ∈ Submodule.span R (Set.range b))\n    obtain ⟨fy, ty, iy, -, hy⟩ := Submodule.mem_span_iff_exists_finset_subset.1\n      (by simp : y ∈ Submodule.span R (Set.range b))\n    rw [← hx, ← hy]\n    simp only [map_sum, LinearMap.map_smulₛₗ, starRingEnd_apply, map_smul, coe_sum,\n      Finset.sum_apply, smul_apply, smul_eq_mul, Finset.mul_sum, map_mul, star_star]\n    rw [Finset.sum_comm]\n    refine Finset.sum_congr rfl (fun b₁ h₁ ↦ Finset.sum_congr rfl fun b₂ h₂ ↦ ?_)\n    rw [mul_left_comm]\n    obtain ⟨i, rfl⟩ := ix h₁\n    obtain ⟨j, rfl⟩ := iy h₂\n    rw [h]\n\nlemma LinearMap.isSymm_iff_isHermitian_toMatrix : B.IsSymm ↔ (toMatrix₂ b b B).IsHermitian := by\n  rw [isSymm_iff_basis b, Matrix.IsHermitian.ext_iff, forall_comm]\n  simp [Eq.comm]\n\nlemma star_dotProduct_toMatrix₂_mulVec (x y : n → R) :\n    star x ⬝ᵥ (toMatrix₂ b b B).mulVec y = B (b.equivFun.symm x) (b.equivFun.symm y) :=\n  dotProduct_toMatrix₂_mulVec b b B x y\n\nlemma apply_eq_star_dotProduct_toMatrix₂_mulVec (x y : M) :\n    B x y = star (b.repr x) ⬝ᵥ (toMatrix₂ b b B).mulVec (b.repr y) :=\n  apply_eq_dotProduct_toMatrix₂_mulVec b b B x y\n\nvariable {R : Type*} [CommRing R] [StarRing R] [PartialOrder R] [Module R M]\n  {B : M →ₗ⋆[R] M →ₗ[R] R} (b : Basis n R M)\n\nTarget:\nlemma LinearMap.isPosSemidef_iff_posSemidef_toMatrix :\n    B.IsPosSemidef ↔ (toMatrix₂ b b B).PosSemidef :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"a9386e192ec9ab948f6d25238318c978127a3e7797faf0be0ba97a5e688dcb74","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/SesquilinearForm","family_id":"linearmap","file_id":"mathlib/Mathlib/LinearAlgebra/SesquilinearForm/Star.lean","sample_id":"9b67a823dcea17b7fc9288545c47193c34fe1b90d1bd9cf3ef9be1964ef22f5d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3486bfbcb341bbc6602e3e4c5334d8af2187c1a15d03540967ecccbb4983efe2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9547247b6d603996c4b3fb961825dcb69ba77ace42e69a17a558d84380f3ec34","source_sha256":"a4ded0c37af8d7d4bd1dca4a7e378a31082efe29498bab67ebba55f071776588","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← cancel_mono (mapBifunctorLeftUnitor F X e p hp Y').inv, assoc, assoc, Iso.hom_inv_id,\n    comp_id, mapBifunctorLeftUnitor_inv_naturality, Iso.hom_inv_id_assoc]","hard_negative":false,"metrics":{"chosen_tokens":33,"rejected_tokens":3,"token_jaccard":0.038462,"token_length_ratio":0.090909},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"9c7dd68cebbe856a6934a266678842835bccedaf321335716d34c5692ddbd0f8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.GradedObject.Associator\npublic import Mathlib.CategoryTheory.GradedObject.Single\n\nNamespace:\nCategoryTheory.GradedObject\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The left and right unitors\n\nGiven a bifunctor `F : C ⥤ D ⥤ D`, an object `X : C` such that `F.obj X ≅ 𝟭 D` and a\nmap `p : I × J → J` such that `hp : ∀ (j : J), p ⟨0, j⟩ = j`,\nwe define an isomorphism of `J`-graded objects for any `Y : GradedObject J D`.\n`mapBifunctorLeftUnitor F X e p hp Y : mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`.\nUnder similar assumptions, we also obtain a right unitor isomorphism\n`mapBifunctorMapObj F p X ((single₀ I).obj Y) ≅ X`. Finally,\nthe lemma `mapBifunctor_triangle` promotes a triangle identity involving functors\nto a triangle identity for the induced functors on graded objects.\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Category Limits\n\nnamespace GradedObject\n\nsection LeftUnitor\n\nvariable {C D I J : Type*} [Category* C] [Category* D]\n  [Zero I] [DecidableEq I] [HasInitial C]\n  (F : C ⥤ D ⥤ D) (X : C) (e : F.obj X ≅ 𝟭 D)\n  [∀ (Y : D), PreservesColimit (Functor.empty.{0} C) (F.flip.obj Y)]\n  (p : I × J → J) (hp : ∀ (j : J), p ⟨0, j⟩ = j)\n  (Y Y' : GradedObject J D) (φ : Y ⟶ Y')\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D` and `Y : GradedObject J D`,\nthis is the isomorphism `((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a ≅ Y a.2`\nwhen `a : I × J` is such that `a.1 = 0`. -/\n@[simps!]\nnoncomputable def mapBifunctorObjSingle₀ObjIso (a : I × J) (ha : a.1 = 0) :\n    ((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a ≅ Y a.2 :=\n  (F.mapIso (singleObjApplyIsoOfEq _ X _ ha)).app _ ≪≫ e.app (Y a.2)\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C` and `Y : GradedObject J D`,\n`((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a` is an initial object\nwhen `a : I × J` is such that `a.1 ≠ 0`. -/\nnoncomputable def mapBifunctorObjSingle₀ObjIsInitial (a : I × J) (ha : a.1 ≠ 0) :\n    IsInitial (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a) :=\n  IsInitial.isInitialObj (F.flip.obj (Y a.2)) _ (isInitialSingleObjApply _ _ _ ha)\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D`, `Y : GradedObject J D` and\n`p : I × J → J` such that `p ⟨0, j⟩ = j` for all `j`,\nthis is the (colimit) cofan which shall be used to construct the isomorphism\n`mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`, see `mapBifunctorLeftUnitor`. -/\nnoncomputable def mapBifunctorLeftUnitorCofan (hp : ∀ (j : J), p ⟨0, j⟩ = j) (Y) (j : J) :\n    (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y).CofanMapObjFun p j :=\n  CofanMapObjFun.mk _ _ _ (Y j) (fun a ha =>\n    if ha : a.1 = 0 then\n      (mapBifunctorObjSingle₀ObjIso F X e Y a ha).hom ≫ eqToHom (by aesop)\n    else\n      (mapBifunctorObjSingle₀ObjIsInitial F X Y a ha).to _)\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp, reassoc]\nlemma mapBifunctorLeftUnitorCofan_inj (j : J) :\n    (mapBifunctorLeftUnitorCofan F X e p hp Y j).inj ⟨⟨0, j⟩, hp j⟩ =\n      (F.map (singleObjApplyIso (0 : I) X).hom).app (Y j) ≫ e.hom.app (Y j) := by\n  simp [mapBifunctorLeftUnitorCofan]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The cofan `mapBifunctorLeftUnitorCofan F X e p hp Y j` is a colimit. -/\nnoncomputable def mapBifunctorLeftUnitorCofanIsColimit (j : J) :\n    IsColimit (mapBifunctorLeftUnitorCofan F X e p hp Y j) :=\n  Cofan.IsColimit.mk _\n    (fun s => e.inv.app (Y j) ≫\n      (F.map (singleObjApplyIso (0 : I) X).inv).app (Y j) ≫ s.inj ⟨⟨0, j⟩, hp j⟩)\n    (fun s => by\n      rintro ⟨⟨i, j'⟩, h⟩\n      by_cases hi : i = 0\n      · subst hi\n        simp only [Set.mem_preimage, hp, Set.mem_singleton_iff] at h\n        subst h\n        simp\n      · apply IsInitial.hom_ext\n        exact mapBifunctorObjSingle₀ObjIsInitial _ _ _ _ hi)\n    (fun s m hm => by simp [← hm ⟨⟨0, j⟩, hp j⟩])\n\ninclude e hp in\nlemma mapBifunctorLeftUnitor_hasMap :\n    HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y) p :=\n  CofanMapObjFun.hasMap _ _ _ (mapBifunctorLeftUnitorCofanIsColimit F X e p hp Y)\n\nvariable [HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y) p]\n  [HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y') p]\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D`, `Y : GradedObject J D` and\n`p : I × J → J` such that `p ⟨0, j⟩ = j` for all `j`,\nthis is the left unitor isomorphism `mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`. -/\nnoncomputable def mapBifunctorLeftUnitor : mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y :=\n  isoMk _ _ (fun j => (CofanMapObjFun.iso\n    (mapBifunctorLeftUnitorCofanIsColimit F X e p hp Y j)).symm)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma ι_mapBifunctorLeftUnitor_hom_apply (j : J) :\n    ιMapBifunctorMapObj F p ((single₀ I).obj X) Y 0 j j (hp j) ≫\n      (mapBifunctorLeftUnitor F X e p hp Y).hom j =\n      (F.map (singleObjApplyIso (0 : I) X).hom).app _ ≫ e.hom.app (Y j) := by\n  dsimp [mapBifunctorLeftUnitor]\n  erw [CofanMapObjFun.ιMapObj_iso_inv]\n  rw [mapBifunctorLeftUnitorCofan_inj]\n\nlemma mapBifunctorLeftUnitor_inv_apply (j : J) :\n    (mapBifunctorLeftUnitor F X e p hp Y).inv j =\n      e.inv.app (Y j) ≫ (F.map (singleObjApplyIso (0 : I) X).inv).app (Y j) ≫\n      ιMapBifunctorMapObj F p ((single₀ I).obj X) Y 0 j j (hp j) := rfl\n\nvariable {Y Y'}\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc]\nlemma mapBifunctorLeftUnitor_inv_naturality :\n    φ ≫ (mapBifunctorLeftUnitor F X e p hp Y').inv =\n      (mapBifunctorLeftUnitor F X e p hp Y).inv ≫ mapBifunctorMapMap F p (𝟙 _) φ := by\n  ext j\n  dsimp\n  rw [mapBifunctorLeftUnitor_inv_apply, mapBifunctorLeftUnitor_inv_apply, assoc, assoc,\n    ι_mapBifunctorMapMap]\n  dsimp\n  rw [Functor.map_id, NatTrans.id_app, id_comp, ← NatTrans.naturality_assoc,\n    ← NatTrans.naturality_assoc]\n  rfl\n\n@[reassoc]\n\nTarget:\nlemma mapBifunctorLeftUnitor_naturality :\n    mapBifunctorMapMap F p (𝟙 _) φ ≫ (mapBifunctorLeftUnitor F X e p hp Y').hom =\n      (mapBifunctorLeftUnitor F X e p hp Y).hom ≫ φ :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/GradedObject","family_id":"mapbifunctorleftunitor_naturality","file_id":"mathlib/Mathlib/CategoryTheory/GradedObject/Unitor.lean","sample_id":"9547247b6d603996c4b3fb961825dcb69ba77ace42e69a17a558d84380f3ec34"},"split":"train"}
{"binding":{"chosen_proof_sha256":"65460242af6486d44c1e55ce08a5b548dbca87ef42697a0dbcd7738054ac5b9c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f2228c66df4392ccdb499fe036ba44562c2e5624b785a18fa483e469975763b6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8415a8306797307a1399dfa46e7bc4bffb6c158dd15399817b536ab5989fba40","source_sha256":"79526dcfa55da9c5f7685aed6cead7ebb3b44c830f8cd589df3db791b7c151e7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  convert_to Tendsto (f ∘ (fun x ↦ c + x) ∘ Neg.neg ∘ Inv.inv) atTop l at h\n  · ext\n    simp [AddGroupWithOne.sub_eq_add_neg]\n  simpa [Tendsto, ← Filter.map_map] using h","hard_negative":false,"metrics":{"chosen_tokens":46,"rejected_tokens":53,"token_jaccard":0.944444,"token_length_ratio":1.152174},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"9c8ea8ce79ce4a6881c98d6e3a7624e0b7ae160840d324b2bf5f82001a404ac1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Algebra.Order.Field\npublic import Mathlib.Topology.Maps.Basic\npublic import Mathlib.Analysis.Asymptotics.Lemmas\n\nNamespace:\nTactic.ComputeAsymptotics\n\nLocal context:\n/-\nCopyright (c) 2026 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\n/-!\n# Conversion lemmas\n\nThe main procedure of the `compute_asymptotics` tactic is able to compute limits of functions at\n`atTop` filter. This file contains lemmas we use to reduce other asymptotic goals to\nthe case `Tendsto f atTop l`.\n\n## Main theorems\n\nThis file contains the following lemmas:\n* `tendsto_nhdsGT_of_tendsto_atTop` for `Tendsto f (𝓝[>] c) l`\n* `tendsto_nhdsLT_of_tendsto_atTop` for `Tendsto f (𝓝[<] c) l`\n* `tendsto_nhdsNE_of_tendsto_atTop` for `Tendsto f (𝓝[≠] c) l`\n* `isBigO_of_div_tendsto_atTop` and `isBigO_of_div_tendsto_atBot` for `f =O[l] g`\n\nWe also use lemmas from other files:\n* `tendsto_comp_neg_atTop_iff` for `Tendsto f atBot l`\n* `IsLittleO.of_tendsto_div_atBot` and `IsLittleO.of_tendsto_div_atTop` for `f =o[l] g`\n* `isEquivalent_of_tendsto_one` for `f ∼ g`\n-/\n\npublic section\n\nopen Filter Topology Asymptotics\n\nnamespace Tactic.ComputeAsymptotics\n\nvariable {α 𝕜 : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [TopologicalSpace 𝕜]\n  [OrderTopology 𝕜] {l : Filter α} (f : 𝕜 → α) (c : 𝕜)\n\ntheorem tendsto_nhdsGT_of_tendsto_atTop (h : Tendsto (fun x ↦ f (c + x⁻¹)) atTop l) :\n    Tendsto f (𝓝[>] c) l := by\n  simpa [← Function.comp_def, Tendsto, ← Filter.map_map] using h\n\nTarget:\ntheorem tendsto_nhdsLT_of_tendsto_atTop (h : Tendsto (fun x ↦ f (c - x⁻¹)) atTop l) :\n    Tendsto f (𝓝[<] c) l :=\n\nProof body:\n","rejected":"```lean\nby\n  convert_to Tendsto (f ∘ (fun x ↦ c + x) ∘ Neg.neg ∘ Inv.inv) atTop l at h\n  · ext\n    simp [AddGroupWithOne.sub_eq_add_neg]\n  simpa [Tendsto, ← Filter.map_map] using h\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/ComputeAsymptotics","family_id":"tendsto_nhdslt_of_tendsto_attop","file_id":"mathlib/Mathlib/Tactic/ComputeAsymptotics/Lemmas.lean","sample_id":"8415a8306797307a1399dfa46e7bc4bffb6c158dd15399817b536ab5989fba40"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4db2e60816a954672941e1fdfd4788e81ed7fe826b34170d024d9dedb9483621","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"102237e83b0e666517b452145f83069b8f06962b8bb7633b044f825abf5d81c3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"00a4fa01aa3988f187453211eea6fee689951bcc17ccb3f8913ec66689b3d85f","source_sha256":"f43f20b9ce00ce089e6a539edbfd4564f6ceb6ee57c06e50a1aa6a1ddd738246","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    have := W.isClosedUnderColimitsOfShape_isLocal J κ hW\n    infer_instance","hard_negative":true,"metrics":{"chosen_tokens":10,"rejected_tokens":2,"token_jaccard":0.090909,"token_length_ratio":0.2},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"9cae314c6bf42d5a2de6529c1d251aa190ee4f8c2b1772263f47f694889796b7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Adjunction.PartialAdjoint\npublic import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer\npublic import Mathlib.CategoryTheory.Localization.BousfieldTransfiniteComposition\npublic import Mathlib.CategoryTheory.MorphismProperty.IsSmall\npublic import Mathlib.CategoryTheory.Presentable.Adjunction\npublic import Mathlib.CategoryTheory.SmallObject.TransfiniteIteration\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The Orthogonal-reflection construction\n\nGiven `W : MorphismProperty C` (which should be small) and assuming the existence\nof certain colimits in `C`, we construct a morphism `toSucc W Z : Z ⟶ succ W Z` for\nany `Z : C`. This morphism belongs to `W.isLocal.isLocal` and\nis an isomorphism iff `Z` belongs to `W.isLocal` (see the lemma `isIso_toSucc_iff`).\nThe morphism `toSucc W Z : Z ⟶ succ W Z` is defined as a composition\nof two morphisms that are roughly described as follows:\n* `toStep W Z : Z ⟶ step W Z`: for any morphism `f : X ⟶ Y` satisfying `W`\n  and any morphism `X ⟶ Z`, we \"attach\" a morphism `Y ⟶ step W Z` (using\n  coproducts and a pushout in essentially the same way as it is done in\n  the file `Mathlib/CategoryTheory/SmallObject/Construction.lean` for the small object\n  argument);\n* `fromStep W Z : step W Z ⟶ succ W Z`: this morphism coequalizes all pairs\n  of morphisms `g₁ g₂ : Y ⟶ step W Z` such that there is a `f : X ⟶ Y`\n  satisfying `W` such that `f ≫ g₁ = f ≫ g₂`.\n\nThe morphism `toSucc W Z : Z ⟶ succ W Z` is a variant of the (wrong) definition\np. 32 in the book by Adámek and Rosický. In this book, a slightly different object\nthan `succ W Z` is defined directly as a colimit of an intricate diagram, but\ncontrary to what is stated on p. 33, it does not satisfy `isIso_toSucc_iff`.\nThe author of this file was unable to understand the attempt of the authors\nto fix this mistake in the errata to this book. This led to the definition\nin two steps outlined above.\n\n## Main results\n\nThe morphisms described above `toSucc W Z : Z ⟶ succ W Z` for all `Z : C` allow to\ndefine `succStruct W Z₀ : SuccStruct C` for any `Z₀ : C`. By applying\na transfinite iteration to this `SuccStruct`, we obtain the following results\nunder the assumption that `W : MorphismProperty C` is a `w`-small property\nof morphisms in a locally `κ`-presentable category `C` (with `κ : Cardinal.{w}`\na regular cardinal) such that the domains and codomains of the morphisms\nsatisfying `W` are `κ`-presentable:\n* `MorphismProperty.isRightAdjoint_ι_isLocal`: existence of the left adjoint\n  of the inclusion `W.isLocal ⥤ C`;\n* `MorphismProperty.isLocallyPresentable_isLocal`: the full subcategory\n  `W.isLocal` is locally presentable.\n\nThis is essentially the implication (i) → (ii) in Theorem 1.39 (and the corollary 1.40)\nin the book by Adámek and Rosický (note that according to the\nerrata to this book, the implication (ii) → (i) is wrong when `κ = ℵ₀`).\n\n## References\n* [Adámek, J. and Rosický, J., *Locally presentable and accessible categories*][Adamek_Rosicky_1994]\n\n-/\n\n@[expose] public section\n\nuniverse w v' u' v u\n\nnamespace CategoryTheory\n\nopen Limits Localization Opposite\n\nvariable {C : Type u} [Category.{v} C] (W : MorphismProperty C)\n\nset_option backward.isDefEq.respectTransparency false in\nlemma MorphismProperty.isClosedUnderColimitsOfShape_isLocal\n    (J : Type u') [Category.{v'} J] [EssentiallySmall.{w} J]\n    (κ : Cardinal.{w}) [Fact κ.IsRegular] [IsCardinalFiltered J κ]\n    (hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ) :\n    W.isLocal.IsClosedUnderColimitsOfShape J where\n  colimitsOfShape_le := fun Z ⟨p⟩ X Y f hf ↦ by\n    obtain ⟨_, _⟩ := hW f hf\n    refine ⟨fun g₁ g₂ h ↦ ?_, fun g ↦ ?_⟩\n    · obtain ⟨j₁, g₁, rfl⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ p.isColimit g₁\n      obtain ⟨j₂, g₂, rfl⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ p.isColimit g₂\n      dsimp at h ⊢\n      obtain ⟨j₃, u, v, huv⟩ :=\n        IsCardinalPresentable.exists_eq_of_isColimit κ p.isColimit (f ≫ g₁) (f ≫ g₂)\n          (by simpa)\n      simp only [Category.assoc] at huv\n      rw [← p.w u, ← p.w v, reassoc_of% ((p.prop_diag_obj j₃ _ hf).1 huv)]\n    · obtain ⟨j, g, rfl⟩ := IsCardinalPresentable.exists_hom_of_isColimit κ p.isColimit g\n      obtain ⟨g, rfl⟩ := (p.prop_diag_obj j _ hf).2 g\n      exact ⟨g ≫ p.ι.app j, by simp⟩\n\nTarget:\nlemma MorphismProperty.isCardinalAccessible_ι_isLocal\n    (κ : Cardinal.{w}) [Fact κ.IsRegular]\n    [HasCardinalFilteredColimits C κ]\n    (hW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ) :\n    W.isLocal.ι.IsCardinalAccessible κ where\n  preservesColimitOfShape J _ _ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_00a4fa01aa39","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"fe7ecb9d9e91afc254d08c07d93f9e830ea33bbc5a327e56e3ff714888f4f813","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Presentable","family_id":"morphismproperty","file_id":"mathlib/Mathlib/CategoryTheory/Presentable/OrthogonalReflection.lean","sample_id":"00a4fa01aa3988f187453211eea6fee689951bcc17ccb3f8913ec66689b3d85f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2b60f1251ce380c3c88892d3140d40e19399072a863e960ad4ea10cd3795106f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"031a4f415c5dae30902ba08057eae47c3ac931dc807e7c5e56759fa5665485f9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9a258eabbca29a5c3771ee588ea1b0dec5082cf5ef32ba82797313921b1c69f8","source_sha256":"3f2d058c8c184ba4c5b0f27fc93d3aa940869e09cbde2b006e4860db27ae9682","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases exists_pair_ne α with ⟨x, y, hxy⟩\n  cases lt_or_gt_of_ne hxy <;> exact ⟨_, _, ‹_›⟩","hard_negative":false,"metrics":{"chosen_tokens":28,"rejected_tokens":33,"token_jaccard":0.8,"token_length_ratio":1.178571},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"9cc69174c315e968d49d9d961dd9e0821a1e4c73f6999e8ca4a73f061acd75a7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Prod.Basic\npublic import Mathlib.Logic.Function.Basic\npublic import Mathlib.Logic.Nontrivial.Defs\npublic import Mathlib.Logic.Unique\npublic import Mathlib.Order.Defs.LinearOrder\nimport Mathlib.Tactic.Attr.Register\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Nontrivial types\n\nResults about `Nontrivial`.\n-/\n\n@[expose] public section\n\nvariable {α : Type*} {β : Type*}\n\n-- `x` and `y` are explicit here, as they are often needed to guide typechecking of `h`.\ntheorem nontrivial_of_lt [Preorder α] (x y : α) (h : x < y) : Nontrivial α :=\n  ⟨⟨x, y, ne_of_lt h⟩⟩\n\nTarget:\ntheorem exists_pair_lt (α : Type*) [Nontrivial α] [LinearOrder α] : ∃ x y : α, x < y :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  rcases exists_pair_ne α with ⟨x, y, hxy⟩\n  cases lt_or_gt_of_ne hxy <;> exact ⟨_, _, ‹_›⟩","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Nontrivial","family_id":"exists_pair_lt","file_id":"mathlib/Mathlib/Logic/Nontrivial/Basic.lean","sample_id":"9a258eabbca29a5c3771ee588ea1b0dec5082cf5ef32ba82797313921b1c69f8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7f6de06b7c519dc56f72cfe0e561560f9ccb6c57f8383a711dc7fbaa5be562ae","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"04175e8f69832579c88e53e7defc7abb906d1749bebae6932b35a3cc3ac66b47","source_sha256":"5dcbaf49b7e0422efc9e44781278719441afae74990c650101eb7518f775293b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [absNorm_eq' 1 I₀ (by rw [one_smul]; rfl), OneMemClass.coe_one, map_one, abs_one,\n    Int.cast_one, _root_.div_one]","hard_negative":false,"metrics":{"chosen_tokens":33,"rejected_tokens":2,"token_jaccard":0.041667,"token_length_ratio":0.060606},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"9cf7c55109a65d05cbfd1d40e54ae8152ae76b531282ce05b2e7897fcc6a9fda","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.FractionalIdeal.Basic\npublic import Mathlib.RingTheory.Ideal.Norm.AbsNorm\npublic import Mathlib.RingTheory.Localization.NormTrace\n\nNamespace:\nFractionalIdeal\n\nLocal context:\n/-\nCopyright (c) 2024 Xavier Roblot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Xavier Roblot\n-/\n/-!\n\n# Fractional ideal norms\n\nThis file defines the absolute ideal norm of a fractional ideal `I : FractionalIdeal R⁰ K` where\n`K` is a fraction field of `R`. The norm is defined by\n`FractionalIdeal.absNorm I = Ideal.absNorm I.num / |Algebra.norm ℤ I.den|` where `I.num` is an\nideal of `R` and `I.den` an element of `R⁰` such that `I.den • I = I.num`.\n\n## Main definitions and results\n\n* `FractionalIdeal.absNorm`: the norm as a zero-preserving morphism with values in `ℚ`.\n* `FractionalIdeal.absNorm_eq'`: the value of the norm does not depend on the choice of\n  `I.num` and `I.den`.\n* `FractionalIdeal.abs_det_basis_change`: the norm is given by the determinant\n  of the basis change matrix.\n* `FractionalIdeal.absNorm_span_singleton`: the norm of a principal fractional ideal is the\n  norm of its generator\n-/\n\n@[expose] public section\n\nopen Module\nopen scoped Pointwise nonZeroDivisors\n\nnamespace FractionalIdeal\nvariable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free ℤ R] [Module.Finite ℤ R]\nvariable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]\n\ntheorem absNorm_div_norm_eq_absNorm_div_norm {I : FractionalIdeal R⁰ K} (a : R⁰) (I₀ : Ideal R)\n    (h : a • (I : Submodule R K) = Submodule.map (Algebra.linearMap R K) I₀) :\n    (Ideal.absNorm I.num : ℚ) / |Algebra.norm ℤ (I.den : R)| =\n      (Ideal.absNorm I₀ : ℚ) / |Algebra.norm ℤ (a : R)| := by\n  rw [div_eq_div_iff]\n  · replace h := congr_arg (I.den • ·) h\n    have h' := congr_arg (a • ·) (den_mul_self_eq_num I)\n    rw [smul_comm] at h\n    rw [h, Submonoid.smul_def, Submonoid.smul_def, ← Submodule.ideal_span_singleton_smul,\n      ← Submodule.ideal_span_singleton_smul, ← Submodule.map_smul'', ← Submodule.map_smul'',\n      (LinearMap.map_injective ?_).eq_iff, smul_eq_mul, smul_eq_mul] at h'\n    · simp_rw [← Nat.cast_natAbs, ← Nat.cast_mul, ← Ideal.absNorm_span_singleton]\n      rw [← map_mul, ← map_mul, mul_comm, ← h', mul_comm]\n    · exact LinearMap.ker_eq_bot.mpr (IsFractionRing.injective R K)\n  all_goals simp [Algebra.norm_eq_zero_iff]\n\n/-- The absolute norm of the fractional ideal `I` extending by multiplicativity the absolute norm\non (integral) ideals. -/\nnoncomputable def absNorm : FractionalIdeal R⁰ K →*₀ ℚ where\n  toFun I := (Ideal.absNorm I.num : ℚ) / |Algebra.norm ℤ (I.den : R)|\n  map_zero' := by\n    rw [num_zero_eq, Submodule.zero_eq_bot, Ideal.absNorm_bot, Nat.cast_zero, zero_div]\n    exact IsFractionRing.injective R K\n  map_one' := by\n    rw [absNorm_div_norm_eq_absNorm_div_norm 1 ⊤ (by simp [Submodule.one_eq_range]),\n      Ideal.absNorm_top, Nat.cast_one, OneMemClass.coe_one, map_one, abs_one,\n      Int.cast_one,\n      one_div_one]\n  map_mul' I J := by\n    rw [absNorm_div_norm_eq_absNorm_div_norm (I.den * J.den) (I.num * J.num) (by\n        have : Algebra.linearMap R K = (IsScalarTower.toAlgHom R R K).toLinearMap := rfl\n        rw [coe_mul, this, Submodule.map_mul, ← this, ← den_mul_self_eq_num, ← den_mul_self_eq_num]\n        exact Submodule.mul_smul_mul_eq_smul_mul_smul _ _ _ _),\n      Submonoid.coe_mul, map_mul, map_mul, Nat.cast_mul, div_mul_div_comm,\n      Int.cast_abs, Int.cast_abs, Int.cast_abs, ← abs_mul, Int.cast_mul]\n\ntheorem absNorm_eq (I : FractionalIdeal R⁰ K) :\n    absNorm I = (Ideal.absNorm I.num : ℚ) / |Algebra.norm ℤ (I.den : R)| := rfl\n\ntheorem absNorm_eq' {I : FractionalIdeal R⁰ K} (a : R⁰) (I₀ : Ideal R)\n    (h : a • (I : Submodule R K) = Submodule.map (Algebra.linearMap R K) I₀) :\n    absNorm I = (Ideal.absNorm I₀ : ℚ) / |Algebra.norm ℤ (a : R)| := by\n  rw [absNorm, ← absNorm_div_norm_eq_absNorm_div_norm a I₀ h, MonoidWithZeroHom.coe_mk,\n    ZeroHom.coe_mk]\n\ntheorem absNorm_nonneg (I : FractionalIdeal R⁰ K) : 0 ≤ absNorm I := by dsimp [absNorm]; positivity\n\ntheorem absNorm_bot : absNorm (⊥ : FractionalIdeal R⁰ K) = 0 := absNorm.map_zero'\n\ntheorem absNorm_one : absNorm (1 : FractionalIdeal R⁰ K) = 1 := by convert! absNorm.map_one'\n\ntheorem absNorm_eq_zero_iff [IsDomain K] {I : FractionalIdeal R⁰ K} :\n    absNorm I = 0 ↔ I = 0 := by\n  refine ⟨fun h ↦ zero_of_num_eq_bot zero_notMem_nonZeroDivisors ?_, fun h ↦ h ▸ absNorm_bot⟩\n  rw [absNorm_eq, div_eq_zero_iff] at h\n  refine Ideal.absNorm_eq_zero_iff.mp <| Nat.cast_eq_zero.mp <| h.resolve_right ?_\n  simp [Algebra.norm_eq_zero_iff]\n\nTarget:\ntheorem coeIdeal_absNorm (I₀ : Ideal R) :\n    absNorm (I₀ : FractionalIdeal R⁰ K) = Ideal.absNorm I₀ :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/FractionalIdeal","family_id":"coeideal_absnorm","file_id":"mathlib/Mathlib/RingTheory/FractionalIdeal/Norm.lean","sample_id":"04175e8f69832579c88e53e7defc7abb906d1749bebae6932b35a3cc3ac66b47"},"split":"train"}
{"binding":{"chosen_proof_sha256":"08d1285c21c0227003f0de3a7a5739a4341408268171b9be42c7528b7aade63d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3a1459c8dec68bfc213585aa1c5fccaa2c9aa6555d13cba2a8d0e701bfec6aff","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f7b620c51cfaab544fc583af36da411df3fc24767a9c25ebd60f0ee43a35db07","source_sha256":"d7743ebc410d9ee31a12263ae8d894bf6aecd1d52cd1a367eccb2f5de9ab8de8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Function.isPeriodicPt_iff_minimalPeriod_dvd, Nat.dvd_prime hp.out,\n    ← minimalPeriod_eq_one_iff_isFixedPt.not, or_and_right, and_not_self_iff, false_or,\n    iff_self_and]\n  exact fun h ↦ ne_of_eq_of_ne h hp.out.ne_one","hard_negative":true,"metrics":{"chosen_tokens":38,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.078947},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"9cf89843f5e5126abe2fefb4179117a4e62fe8d9c418dca1e3425a2243117877","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GCDMonoid.Finset\npublic import Mathlib.Data.Nat.Prime.Basic\npublic import Mathlib.Data.PNat.Basic\npublic import Mathlib.Dynamics.PeriodicPts.Defs\npublic import Mathlib.Order.Lattice.Nat\n\nNamespace:\nFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Extra lemmas about periodic points\n-/\n\npublic section\n\nopen Nat Set\n\nnamespace Function\nvariable {α : Type*} {f : α → α} {x y : α}\n\nopen Function (Commute)\n\ntheorem directed_ptsOfPeriod_pnat (f : α → α) : Directed (· ⊆ ·) fun n : ℕ+ => ptsOfPeriod f n :=\n  fun m n => ⟨m * n, fun _ hx => hx.mul_const n, fun _ hx => hx.const_mul m⟩\n\nvariable (f) in\ntheorem bijOn_periodicPts : BijOn f (periodicPts f) (periodicPts f) :=\n  iUnion_pnat_ptsOfPeriod f ▸\n    bijOn_iUnion_of_directed (directed_ptsOfPeriod_pnat f) fun i => bijOn_ptsOfPeriod f i.pos\n\nTarget:\ntheorem minimalPeriod_eq_prime_iff {p : ℕ} [hp : Fact p.Prime] :\n    minimalPeriod f x = p ↔ IsPeriodicPt f p x ∧ ¬IsFixedPt f x :=\n\nProof body:\n","rejected":"by\n  exact minimalPeriod_eq_prime_iff","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"63614e997eb4c88fc6d4055bd7b38fb697773987f80bfa9be0ccf703b218e0b2","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/PeriodicPts","family_id":"minimalperiod_eq_prime_iff","file_id":"mathlib/Mathlib/Dynamics/PeriodicPts/Lemmas.lean","sample_id":"f7b620c51cfaab544fc583af36da411df3fc24767a9c25ebd60f0ee43a35db07"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8af251c6a8eb77af2dfa75888d7c394a97af0ecb8aa758dfff389aa06e6d13fe","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7ef421e6d8431cec7f0feeba135adb69e42126737ccecc01930d4952229f4cbf","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6a8e8150ef6c1755c0f36af56608a5c55a4714f68f327a0666a7f1b00c8049b8","source_sha256":"b6022f061f06753f0797fa4fa7ab43cc6fb4c7d61abaa7b4bb4707d4dc0299c4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← LinearMap.comp_assoc, map_map_comp_assoc_eq]\n  ext\n  rfl","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.25},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"9d2b2d5072a78df278d479cb392b8e2cc67921b37ff2797076f094cedfb4bf3d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorProduct.Tower\npublic import Mathlib.RingTheory.Coalgebra.Basic\nimport Mathlib.Tactic.Attr.Register\n\nNamespace:\nCoassocSimps\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang, Yaël Dillies\n-/\n/-!\n# Tactic to reassociate comultiplication in a coalgebra\n\n`coassoc_simps` is a simp set useful to prove tautologies on coalgebras.\n\nThe general algorithm it follows is to push the associators `TensorProduct.assoc` and\ncommutators `TensorProduct.comm` inwards (to the right) until they cancel against\nco-multiplications.\n\nThe simp set makes the following choice of normal form\n* It regards `TensorProduct.map`, `TensorProduct.assoc`, `TensorProduct.comm` as the primitive\n  constructions and rewrites everything else such as `lTensor`, `leftComm` using them.\n* It rewrites both sides into a right associated composition of linear maps.\n  In particular `LinearMap.comp_assoc` and `LinearEquiv.coe_trans` are tagged.\n* It rewrites `(f₂ ⊗ g₂) ∘ (f₁ ⊗ g₁)` into `(f₂ ∘ f₁) ⊗ (g₂ ∘ g₁)`.\n\n## Notes\n\n- It is not confluent with `(ε ⊗ₘ id) ∘ₗ δ = λ⁻¹`.\n  It is often useful to `trans` (or `calc`) with a term containing\n  `(ε ⊗ₘ _) ∘ₗ δ` or `(_ ⊗ₘ ε) ∘ₗ δ`,\n  and use one of `map_counit_comp_comul_left` `map_counit_comp_comul_right`\n  `map_counit_comp_comul_left_assoc` `map_counit_comp_comul_right_assoc` to continue.\n\n- Some lemmas (e.g. `lid_comp_map : λ ∘ₗ (f ⊗ₘ g) = g ∘ₗ λ ∘ₗ (f ⊗ₘ id)`) loops when tagged as simp,\n  so we wrap it inside a rudimentary simproc that only fires when `g ≠ id`.\n-/\n\n@[expose] public section\n\nopen TensorProduct\n\nopen LinearMap (id)\nopen Coalgebra\n\nopen Qq\nnamespace CoassocSimps\n\nvariable {R A M N P M' N' P' Q Q' M₁ M₂ M₃ N₁ N₂ N₃ : Type*}\n    [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A]\n    [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P]\n    [AddCommMonoid M'] [Module R M'] [AddCommMonoid N'] [Module R N']\n    [AddCommMonoid P'] [Module R P'] [AddCommMonoid Q] [Module R Q] [AddCommMonoid Q'] [Module R Q']\n    [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃]\n    [AddCommMonoid N₁] [AddCommMonoid N₂] [AddCommMonoid N₃]\n    [Module R M₁] [Module R M₂] [Module R M₃] [Module R N₁] [Module R N₂] [Module R N₃]\n\nlocal notation3 \"α\" => (TensorProduct.assoc R _ _ _).toLinearMap\nlocal notation3 \"α⁻¹\" => (TensorProduct.assoc R _ _ _).symm.toLinearMap\nlocal notation3 \"λ\" => (TensorProduct.lid R _).toLinearMap\nlocal notation3 \"λ⁻¹\" => (TensorProduct.lid R _).symm.toLinearMap\nlocal notation3 \"ρ\" => (TensorProduct.rid R _).toLinearMap\nlocal notation3 \"ρ⁻¹\" => (TensorProduct.rid R _).symm.toLinearMap\nlocal notation3 \"β\" => (TensorProduct.comm R _ _).toLinearMap\nlocal infix:90 \" ⊗ₘ \" => TensorProduct.map\nlocal notation3 \"δ\" => comul (R := R)\nlocal notation3 \"ε\" => counit (R := R)\n\nattribute [coassoc_simps] LinearMap.comp_id LinearMap.id_comp TensorProduct.map_id\n  LinearMap.lTensor_def LinearMap.rTensor_def LinearMap.comp_assoc\n  LinearEquiv.coe_trans LinearEquiv.trans_symm\n  LinearEquiv.refl_toLinearMap TensorProduct.toLinearMap_congr\n  LinearEquiv.comp_symm LinearEquiv.symm_comp LinearEquiv.symm_symm\n  LinearEquiv.coe_lTensor LinearEquiv.coe_lTensor_symm\n  LinearEquiv.coe_rTensor LinearEquiv.coe_rTensor_symm\n  IsCocomm.comm_comp_comul TensorProduct.AlgebraTensorModule.map_eq\n  TensorProduct.AlgebraTensorModule.assoc_eq TensorProduct.AlgebraTensorModule.rightComm_eq\n  TensorProduct.tensorTensorTensorComm TensorProduct.AlgebraTensorModule.tensorTensorTensorComm\n  TensorProduct.AlgebraTensorModule.congr_eq LinearEquiv.comp_symm_assoc\n  LinearEquiv.symm_comp_assoc TensorProduct.rightComm_def TensorProduct.leftComm_def\n  TensorProduct.comm_symm TensorProduct.comm_comp_comm TensorProduct.comm_comp_comm_assoc\n\nattribute [coassoc_simps← ] TensorProduct.map_comp TensorProduct.map_map_comp_assoc_eq\n  TensorProduct.map_map_comp_assoc_symm_eq\n\n@[coassoc_simps]\nlemma TensorProduct.map_comp_assoc\n    (f : M →ₗ[R] N) (g : N →ₗ[R] P) (f' : M' →ₗ[R] N') (g' : N' →ₗ[R] P') (φ : M₁ →ₗ[R] M ⊗[R] M') :\n    map g g' ∘ₗ map f f' ∘ₗ φ = map (g ∘ₗ f) (g' ∘ₗ f') ∘ₛₗ φ := by\n  rw [← LinearMap.comp_assoc, TensorProduct.map_comp]\n\n@[coassoc_simps← ]\nlemma TensorProduct.map_map_comp_assoc_eq_assoc\n    (f₁ : M₁ →ₗ[R] N₁) (f₂ : M₂ →ₗ[R] N₂) (f₃ : M₃ →ₗ[R] N₃) (f : M →ₗ[R] M₁ ⊗[R] M₂ ⊗[R] M₃) :\n    f₁ ⊗ₘ (f₂ ⊗ₘ f₃) ∘ₗ α ∘ₗ f = α ∘ₗ ((f₁ ⊗ₘ f₂) ⊗ₘ f₃) ∘ₗ f := by\n  rw [← LinearMap.comp_assoc, ← LinearMap.comp_assoc, TensorProduct.map_map_comp_assoc_eq]\n\n@[coassoc_simps← ]\nlemma TensorProduct.map_map_comp_assoc_symm_eq_assoc\n    (f₁ : M₁ →ₗ[R] N₁) (f₂ : M₂ →ₗ[R] N₂) (f₃ : M₃ →ₗ[R] N₃) (f : M →ₗ[R] M₁ ⊗[R] (M₂ ⊗[R] M₃)) :\n    (f₁ ⊗ₘ f₂) ⊗ₘ f₃ ∘ₗ α⁻¹ ∘ₗ f = α⁻¹ ∘ₗ (f₁ ⊗ₘ (f₂ ⊗ₘ f₃)) ∘ₗ f := by\n  rw [← LinearMap.comp_assoc, ← LinearMap.comp_assoc, TensorProduct.map_map_comp_assoc_symm_eq]\n\n@[coassoc_simps]\n\nTarget:\nlemma assoc_comp_map_map_comp\n    (f₁ : M₁ →ₗ[R] N₁) (f₂ : M₂ →ₗ[R] N₂) (f₃ : M₃ →ₗ[R] N₃) (f₁₂ : M →ₗ[R] M₁ ⊗[R] M₂) :\n    α ∘ₗ (((f₁ ⊗ₘ f₂) ∘ₗ f₁₂) ⊗ₘ f₃) = (f₁ ⊗ₘ (f₂ ⊗ₘ f₃)) ∘ₗ α ∘ₗ (f₁₂ ⊗ₘ id) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_6a8e8150ef6c","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c503fdb4c044847a6cd79274f5ada48e49be7c7ef35657c384aa33c4ebf099e7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Coalgebra","family_id":"assoc_comp_map_map_comp","file_id":"mathlib/Mathlib/RingTheory/Coalgebra/CoassocSimps.lean","sample_id":"6a8e8150ef6c1755c0f36af56608a5c55a4714f68f327a0666a7f1b00c8049b8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ec2ccc87cb1b0580e84707f89c57da330dcbae57aa0cfba902f545e3a992cb11","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c2db1f63c6a5a140290dd7cda06b1d0b049c0a469f71f463a4280885109ef69b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d885a32647abb9166c82b5317470ebda2b4ddd48a31b71c34892848d64e4f116","source_sha256":"50e4a4d798ba2dd2872d163b6a5aeb55d8cf128d153e35dcc8c630f4eadcbabe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_contra h\n  exact absurd Valuation.IsEquiv.refl (h ▸ adicValuation_not_isEquiv_infty_valuation p)","hard_negative":false,"metrics":{"chosen_tokens":16,"rejected_tokens":20,"token_jaccard":0.823529,"token_length_ratio":1.25},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"9d58290ca2321094e9261d6f984f309bc96734b8326058e91ea9a6767e0db4e3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.Finite.Valuation\npublic import Mathlib.NumberTheory.FunctionField\npublic import Mathlib.RingTheory.Valuation.Discrete.Basic\n\nNamespace:\nRatFunc\n\nLocal context:\n/-\nCopyright (c) 2025 María Inés de Frutos-Fernández & Xavier Généreux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: María Inés de Frutos-Fernández, Xavier Généreux\n-/\n/-!\n# Ostrowski's theorem for `K(X)`\n\nThis file proves Ostrowski's theorem for the field of rational functions `K(X)`, where `K` is any\nfield: if `v` is a discrete valuation on `K(X)` which is trivial on elements of `K`, then `v` is\nequivalent to either the `I`-adic valuation for some `I : HeightOneSpectrum K[X]`, or to the\nvaluation at infinity `FunctionField.inftyValuation K`.\n\n## Main results\n- `RatFunc.valuation_isEquiv_infty_or_adic`: Ostrowski's theorem for `K(X)`.\n-/\n\n@[expose] public noncomputable section\n\n\nopen Multiplicative WithZero\n\nvariable {K Γ : Type*} [Field K] [LinearOrderedCommGroupWithZero Γ] {v : Valuation (RatFunc K) Γ}\n\nnamespace RatFunc\n\nsection Infinity\n\nopen Polynomial Valuation\n\nlemma valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X {f : RatFunc K}\n    [v.IsTrivialOn K] (hlt : 1 < v X) (hf : f ≠ 0) : v f = v RatFunc.X ^ f.intDegree := by\n  induction f using RatFunc.induction_on with\n  | f p q hq =>\n    rw [intDegree_div (by grind only) (by grind only), v.map_div, zpow_sub₀ (ne_zero_of_lt hlt)]\n    simp_rw [intDegree_polynomial, zpow_natCast, ← coePolynomial_eq_algebraMap]\n    have hp : p ≠ 0 := by contrapose hf; simp [hf]\n    rw [valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X _ hlt hp,\n      valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X _ hlt hq]\n\nvariable [DecidableEq (RatFunc K)]\n\nlemma valuation_isEquiv_inftyValuation_of_one_lt_valuation_X [v.IsTrivialOn K] (hlt : 1 < v X) :\n    v.IsEquiv (inftyValuation K) := by\n  refine isEquiv_iff_val_lt_one.mpr fun {f} ↦ ?_\n  rcases eq_or_ne f 0 with rfl | hf\n  · simp\n  · have hlt' : 1 < inftyValuation K X := by simp [← exp_zero]\n    rw [valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt hf,\n      valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt' hf]\n    grind [one_le_zpow_iff_right₀]\n\nend Infinity\n\nopen IsDedekindDomain HeightOneSpectrum Set Valuation Polynomial\n\nlemma setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty [v.IsNontrivial] [v.IsTrivialOn K]\n    (hle : v RatFunc.X ≤ 1) : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty := by\n  obtain ⟨w, h0, h1⟩ := IsNontrivial.exists_lt_one (v := v)\n  induction w using RatFunc.induction_on with\n  | f p q =>\n    simp only [ne_eq, _root_.div_eq_zero_iff, FaithfulSMul.algebraMap_eq_zero_iff, not_or,\n      map_div₀] at *\n    have hor : ¬v ↑p = 1 ∨ ¬v ↑q = 1 := by rw [← not_and_or]; aesop\n    suffices ∀ r : K[X], v (↑r) ≠ 1 → r ≠ 0 → {p : K[X] | v ↑p < 1 ∧ ¬p = 0}.Nonempty by\n      exact Or.elim hor (fun hp ↦ this p hp h0.1) (fun hq ↦ this q hq h0.2)\n    exact fun r hr hr0 ↦ ⟨r, lt_iff_le_and_ne.mpr\n      ⟨Polynomial.valuation_le_one_of_valuation_X_le_one _ hle r, hr⟩, hr0⟩\n\nprivate lemma one_le_valuation_factor (hne : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty) {a b : K[X]}\n    (hab : v ↑(a * b) < 1 ∧ a ≠ 0 ∧ b ≠ 0) (hπᵥ : degree_lt_wf.min _ hne = a * b)\n    (hb : ¬IsUnit b) : 1 ≤ v ↑a := by\n  set πᵥ := degree_lt_wf.min _ hne\n  have hda : a.degree < πᵥ.degree := by\n    have hbpos := degree_pos_of_ne_zero_of_nonunit hab.2.2 hb\n    simp_rw [hπᵥ, degree_mul, degree_eq_natDegree hab.2.1, degree_eq_natDegree hab.2.2] at hbpos ⊢\n    norm_cast\n    simpa using hbpos\n  have hlea := imp_not_comm.mp (degree_lt_wf.not_lt_min _) hda\n  grind\n\nlemma irreducible_min_polynomial_valuation_lt_one_and_ne_zero [v.IsTrivialOn K]\n    (hne : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty) :\n    Irreducible (degree_lt_wf.min {p : K[X] | v p < 1 ∧ p ≠ 0} hne) := by\n  set πᵥ := degree_lt_wf.min _ hne\n  have hπᵥ : v πᵥ < 1 ∧ πᵥ ≠ 0 := degree_lt_wf.min_mem _ hne\n  refine irreducible_iff.mpr ⟨?_, fun a b hab ↦ ?_⟩\n  · simp only [Polynomial.isUnit_iff, isUnit_iff_ne_zero]\n    intro ⟨a, ha0, ha⟩\n    rw [← ha, coePolynomial, algebraMap_C, ← algebraMap_eq_C] at hπᵥ\n    grind\n  · by_contra! H\n    simp only [hab, ne_eq, mul_eq_zero, not_or] at hπᵥ\n    have hva := one_le_valuation_factor hne hπᵥ hab H.2\n    simp only [mul_comm a b, @and_comm (¬a = 0)] at hπᵥ hab\n    have := Right.one_le_mul (one_le_valuation_factor hne hπᵥ hab H.1) hva\n    simp only [coePolynomial_eq_algebraMap, map_mul] at hπᵥ this\n    grind\n\nsection valuation_X_le_one\n\nvariable [v.IsNontrivial] [v.IsTrivialOn K] (hle : v RatFunc.X ≤ 1)\n\n/-- A uniformizing element for the valuation `v`, as a polynomial in `K[X]`. -/\nabbrev uniformizingPolynomial : K[X] :=\n  WellFounded.min degree_lt_wf _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)\n\n@[inherit_doc]\nlocal notation \"πᵥ\" => uniformizingPolynomial hle\n\nlemma uniformizingPolynomial_ne_zero : πᵥ ≠ 0 := by\n  have := degree_lt_wf.min_mem _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)\n  simp_all [uniformizingPolynomial]\n\nlemma valuation_uniformizingPolynomial_lt_one : v πᵥ < 1 := by\n  simpa using! (degree_lt_wf.min_mem _\n    (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)).1\n\nopen Ideal in\n/-- The maximal ideal of `K[X]` generated by the `uniformizingPolynomial` for `v`. -/\ndef valuationIdeal : HeightOneSpectrum K[X] where\n  asIdeal := Submodule.span K[X] {πᵥ}\n  isPrime := IsMaximal.isPrime (PrincipalIdealRing.isMaximal_of_irreducible\n    (irreducible_min_polynomial_valuation_lt_one_and_ne_zero\n      (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)))\n  ne_bot := by simpa using uniformizingPolynomial_ne_zero hle\n\n@[inherit_doc]\nlocal notation \"Pᵥ\" => RatFunc.valuationIdeal hle\n\nsection Associates\n\nopen EuclideanDomain in\nlemma valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one {p : K[X]}\n    (hp : p ≠ 0) :\n    v (algebraMap K[X] (RatFunc K) p) = v (πᵥ ^ ((Associates.mk (Pᵥ).asIdeal).count\n      (Associates.mk (Ideal.span {p})).factors)) := by\n  set π := πᵥ\n  have hne := setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle\n  have hπirr : Irreducible π := irreducible_min_polynomial_valuation_lt_one_and_ne_zero hne\n  obtain ⟨k, q, hnq, heq⟩ := WfDvdMonoid.max_power_factor hp hπirr\n  have hπ : π ∈ _ := degree_lt_wf.min_mem _ hne\n  simp only [ne_eq, mem_setOf] at hπ\n  nth_rw 1 [heq]\n  simp only [map_mul, map_pow]\n  suffices v (algebraMap K[X] (RatFunc K) q) = 1 by\n    simp only [this, mul_one]\n    congr\n    exact (Ideal.count_associates_eq (irreducible_iff_prime.mp hπirr) hnq heq).symm\n  rw [← mod_add_div q π, map_add]\n  rw [← mod_eq_zero] at hnq\n  suffices v (algebraMap K[X] (RatFunc K) (q % π)) = 1 ∧\n      v (algebraMap K[X] (RatFunc K) (π * (q / π))) < 1 by\n    obtain ⟨h₁, h₂⟩ := this\n    rw [← h₁] at h₂ ⊢\n    exact Valuation.map_add_eq_of_lt_left _ h₂\n  constructor\n  · rw [← coePolynomial_eq_algebraMap]\n    have hnπ : q % π ∉ {p : K[X] | v ↑p < 1 ∧ p ≠ 0} :=\n      imp_not_comm.mp (degree_lt_wf.not_lt_min _) (EuclideanDomain.remainder_lt q hπ.2)\n    have := Polynomial.valuation_le_one_of_valuation_X_le_one _ hle (q % π)\n    grind\n  · simpa only [map_mul, ← coePolynomial_eq_algebraMap]\n      using mul_lt_one_of_lt_of_le hπ.1 <| (q / π).valuation_le_one_of_valuation_X_le_one _ hle\n\nlemma exists_zpow_uniformizingPolynomial {f : RatFunc K} (hf : f ≠ 0) :\n    ∃ (z : ℤ), v f = v πᵥ ^ z := by\n  have h0 : v πᵥ ≠ 0 := by simpa using uniformizingPolynomial_ne_zero hle\n  induction f using RatFunc.induction_on with\n  | f p q hq =>\n    use (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {p})).factors -\n      (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {q})).factors\n    simp only [map_div₀, map_pow, zpow_sub₀ h0, zpow_natCast,\n      valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle hq,\n      valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle\n        (p := p) (by aesop)]\n\nlemma uniformizingPolynomial_isUniformizer [hv : IsRankOneDiscrete v] :\n    v.IsUniformizer πᵥ := by\n  have h0 : v πᵥ ≠ 0 := by simpa using uniformizingPolynomial_ne_zero hle\n  rw [IsUniformizer, ← hv.valueGroup_genLTOne_eq_generator, ← h0.isUnit.unit_spec, Units.val_inj]\n  apply LinearOrderedCommGroup.Subgroup.genLTOne_unique\n  · rw [← Units.val_lt_val, h0.isUnit.unit_spec, Units.val_one]\n    exact valuation_uniformizingPolynomial_lt_one hle\n  · ext γ\n    simp only [coePolynomial_eq_algebraMap, MonoidWithZeroHom.mem_valueGroup_iff_of_comm, ne_eq,\n      map_eq_zero, Subgroup.mem_zpowers_iff]\n    refine ⟨fun ⟨k, hk⟩ ↦ ?_, fun ⟨a, ha, b, hab⟩ ↦ ?_⟩\n    · use 1, one_ne_zero, πᵥ ^ k\n      simp only [← Units.val_inj, Units.val_zpow_eq_zpow_val] at hk\n      simp [← hk]\n    · obtain ⟨ka, hka⟩ := exists_zpow_uniformizingPolynomial hle ha\n      obtain ⟨kb, hkb⟩ := exists_zpow_uniformizingPolynomial hle (f := b) (by aesop)\n      rw [MonoidWithZeroHom.coe_ofClass, hka, hkb] at hab\n      use kb - ka\n      have : v ↑πᵥ ^ ka ≠ 0 := zpow_ne_zero _ h0\n      simp [zpow_sub, ← Units.val_inj, ← coePolynomial_eq_algebraMap, field, ← hab]\n\nlemma valuation_isEquiv_valuationIdeal_adic_of_valuation_X_le_one [IsRankOneDiscrete v] :\n    v.IsEquiv ((Pᵥ).valuation (RatFunc K)) := by\n  rw [isEquiv_iff_val_le_one]\n  intro f\n  rcases eq_or_ne f 0 with rfl | hf0\n  · simp\n  · induction f using RatFunc.induction_on with\n    | f p q hq0 =>\n      have hp0 : p ≠ 0 := by simp_all\n      set pi := πᵥ with hpi_def\n      have hpi : v.IsUniformizer (pi : RatFunc K) := uniformizingPolynomial_isUniformizer hle\n      simp only [map_div₀, valuation_of_algebraMap, intValuation_def, exp_neg, if_neg hp0,\n        if_neg hq0, div_inv_eq_mul]\n      rw [valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle hp0,\n        valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle hq0]\n      simp_all [div_le_one₀, inv_mul_le_one₀,\n        (pow_le_pow_iff_right_of_lt_one₀ (by simp_all) (IsRankOneDiscrete.generator_lt_one v))]\n\nend Associates\n\nend valuation_X_le_one\n\nlemma adicValuation_not_isEquiv_infty_valuation [DecidableEq (RatFunc K)]\n    (p : IsDedekindDomain.HeightOneSpectrum K[X]) :\n    ¬ (p.valuation (RatFunc K)).IsEquiv (inftyValuation K) := by\n  simp only [isEquiv_iff_val_le_one]\n  push Not\n  refine ⟨X, .inl ⟨p.valuation_le_one _, ?_⟩⟩\n  rw [inftyValuation.X, ← log_lt_iff_lt_exp one_ne_zero, log_one]\n  exact zero_lt_one\n\nTarget:\nlemma adicValuation_ne_inftyValuation [DecidableEq (RatFunc K)]\n   (p : IsDedekindDomain.HeightOneSpectrum K[X]) :\n    p.valuation (RatFunc K) ≠ inftyValuation K :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  by_contra h\n  exact absurd Valuation.IsEquiv.refl (h ▸ adicValuation_not_isEquiv_infty_valuation p)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/RatFunc","family_id":"adicvaluation_ne_inftyvaluation","file_id":"mathlib/Mathlib/NumberTheory/RatFunc/Ostrowski.lean","sample_id":"d885a32647abb9166c82b5317470ebda2b4ddd48a31b71c34892848d64e4f116"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cc860a95c409479a8d192fe8d26a650d014f625436d7b17247ecda1e72c0ab68","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"548db6b888ddc843c2de1b43678e5e79b3c43c2207a07c350d00bd579f6a5ccc","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c6688be02d1db8888f8f73bfb3dc49d8f7954b075b4191fe5618031c2b9977e0","source_sha256":"f5d92c67f281e03d72da06b390e360bd28b86e3bb7c33267f68c3c873cd9a4b6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction ψ using Quotient.inductionOn\n  induction θ using Quotient.inductionOn\n  simp_rw [← QuotientAddGroup.mk_zsmul, ← QuotientAddGroup.mk_add,\n    QuotientAddGroup.eq_iff_sub_mem, ← smul_sub, ← sub_sub]\n  exact AddSubgroup.zsmul_mem_zmultiples_iff_exists_sub_div hz","hard_negative":true,"metrics":{"chosen_tokens":40,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.075},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"9d850d5b8966c198b519969a9841abbbba8cebfc8d449c1ef4e88859573c8cd5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Basic\npublic import Mathlib.Algebra.Order.Group.Unbundled.Int\npublic import Mathlib.Algebra.Module.NatInt\npublic import Mathlib.GroupTheory.QuotientGroup.Defs\npublic import Mathlib.Algebra.Group.Subgroup.ZPowers.Basic\n\nNamespace:\nQuotientAddGroup\n\nLocal context:\n/-\nCopyright (c) 2022 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Lemmas about quotients in characteristic zero\n-/\n\npublic section\n\n\nvariable {R : Type*} [DivisionRing R] [CharZero R] {p : R}\n\nnamespace AddSubgroup\n\n/-- `z • r` is a multiple of `p` iff `r` is `k * (p / z)` above a multiple of `p`, where\n`0 ≤ k < |z|`. -/\ntheorem zsmul_mem_zmultiples_iff_exists_sub_div {r : R} {z : ℤ} (hz : z ≠ 0) :\n    z • r ∈ AddSubgroup.zmultiples p ↔\n      ∃ k : Fin z.natAbs, r - (k : ℕ) • (p / z : R) ∈ AddSubgroup.zmultiples p := by\n  rw [AddSubgroup.mem_zmultiples_iff]\n  simp_rw [AddSubgroup.mem_zmultiples_iff, div_eq_mul_inv, ← smul_mul_assoc, eq_sub_iff_add_eq]\n  have hz' : (z : R) ≠ 0 := Int.cast_ne_zero.mpr hz\n  conv_rhs => simp +singlePass only [← (mul_right_injective₀ hz').eq_iff]\n  simp_rw [← zsmul_eq_mul, smul_add, ← mul_smul_comm, zsmul_eq_mul (z : R)⁻¹, mul_inv_cancel₀ hz',\n    mul_one, ← natCast_zsmul, smul_smul, ← add_smul]\n  constructor\n  · rintro ⟨k, h⟩\n    simp_rw [← h]\n    refine ⟨⟨(k % z).toNat, ?_⟩, k / z, ?_⟩\n    · rw [← Int.ofNat_lt, Int.toNat_of_nonneg (Int.emod_nonneg _ hz)]\n      exact (Int.emod_lt_abs _ hz).trans_eq (Int.abs_eq_natAbs _)\n    rw [Fin.val_mk, Int.toNat_of_nonneg (Int.emod_nonneg _ hz)]\n    nth_rewrite 3 [← Int.mul_ediv_add_emod k z]\n    rfl\n  · rintro ⟨k, n, h⟩\n    exact ⟨_, h⟩\n\ntheorem nsmul_mem_zmultiples_iff_exists_sub_div {r : R} {n : ℕ} (hn : n ≠ 0) :\n    n • r ∈ AddSubgroup.zmultiples p ↔\n      ∃ k : Fin n, r - (k : ℕ) • (p / n : R) ∈ AddSubgroup.zmultiples p := by\n  rw [← natCast_zsmul r, zsmul_mem_zmultiples_iff_exists_sub_div (Int.natCast_ne_zero.mpr hn),\n    Int.cast_natCast]\n  rfl\n\nend AddSubgroup\n\nnamespace QuotientAddGroup\n\nTarget:\ntheorem zmultiples_zsmul_eq_zsmul_iff {ψ θ : R ⧸ AddSubgroup.zmultiples p} {z : ℤ} (hz : z ≠ 0) :\n    z • ψ = z • θ ↔ ∃ k : Fin z.natAbs, ψ = θ + ((k : ℕ) • (p / z) : R) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_c6688be02d1d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"3723345e40798c4ab175004908b05fa9883a3ae63476a1a7621409b451e089fc","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/CharZero","family_id":"zmultiples_zsmul_eq_zsmul_iff","file_id":"mathlib/Mathlib/Algebra/CharZero/Quotient.lean","sample_id":"c6688be02d1db8888f8f73bfb3dc49d8f7954b075b4191fe5618031c2b9977e0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b03c7a526376dec68a09b72eb773aacc63927789fd0d788813f105124df25955","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a86a08c70d72b421064dab529cf5e5d4ac90c62963744456fb1461e7c0e42c96","source_sha256":"4a32472b9e6529573aae59ec6bc77b66591720ace487daac48e6d523127ce834","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : Algebra.IsSeparable (FractionRing A) (FractionRing R₂) := by\n    refine Algebra.IsSeparable.of_equiv_equiv (FractionRing.algEquiv A K).symm.toRingEquiv\n          (FractionRing.algEquiv R₂ F₂).symm.toRingEquiv ?_\n    ext x\n    exact IsFractionRing.algEquiv_commutes (FractionRing.algEquiv A K).symm\n      (FractionRing.algEquiv R₂ ↥F₂).symm _\n  suffices span B (algebraMap F₂ L '' (traceDual A K (1 : Submodule R₂ F₂))) ≤\n      traceDual R₁ F₁ (1 : Submodule B L) by\n    apply le_antisymm\n    · refine SetLike.coe_subset_coe.mp (subset_trans ?_ this)\n      rw [← Submodule.span_span_of_tower R₁ B]\n      exact Submodule.subset_span\n    · exact traceDual_le_span_map_traceDual A B R₁ R₂ h₁ h₂\n  have := dvd_of_eq <|\n    (IsDedekindDomain.differentIdeal_eq_map_differentIdeal A B R₁ R₂ h₁ h₂ h₃).symm\n  rwa [Ideal.dvd_iff_le, ← coeIdeal_le_coeIdeal (K := L), coeIdeal_differentIdeal R₁ F₁,\n    inv_le_comm, ← extendedHom_coeIdeal_eq_map (K := F₂), coeIdeal_differentIdeal A K, map_inv₀,\n    inv_inv, ← coe_le_coe, coe_extendedHom_eq_span, coe_dual_one, ← coeToSet_coeToSubmodule,\n    coe_dual_one] at this\n  · simp\n  · rw [← extendedHom_coeIdeal_eq_map (K := F₂), ne_eq, extendedHom_eq_zero_iff]\n    rw [coeIdeal_eq_zero]\n    exact differentIdeal_ne_bot","hard_negative":true,"metrics":{"chosen_tokens":260,"rejected_tokens":8,"token_jaccard":0.047059,"token_length_ratio":0.030769},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"9e20db1b0e31edaf703b947788146350fa89589f3a5654f489ea19e2111c48ee","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.LinearDisjoint\npublic import Mathlib.RingTheory.DedekindDomain.Different\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Xavier Roblot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Xavier Roblot\n-/\n/-!\n# Disjoint extensions with coprime different ideals\n\nLet `A ⊆ B` be a finite extension of Dedekind domains and assume that `A ⊆ R₁, R₂ ⊆ B` are two\nsubrings such that `Frac R₁ ⊔ Frac R₂ = Frac B`, `Frac R₁` and `Frac R₂` are linearly disjoint\nover `Frac A`, and that `𝓓(R₁/A)` and `𝓓(R₂/A)` are coprime where `𝓓` denotes the different ideal\nand `Frac R` denotes the fraction field of a domain `R`.\n\n## Main results and definitions\n\n* `IsDedekindDomain.differentIdeal_eq_map_differentIdeal`: `𝓓(B/R₁) = 𝓓(R₂/A)`\n* `IsDedekindDomain.differentIdeal_eq_differentIdeal_mul_differentIdeal_of_isCoprime`:\n  `𝓓(B/A) = 𝓓(R₁/A) * 𝓓(R₂/A)`.\n* `Module.Basis.ofIsCoprimeDifferentIdeal`: Construct a `R₁`-basis of `B` by lifting an\n  `A`-basis of `R₂`.\n* `IsDedekindDomain.range_sup_range_eq_top_of_isCoprime_differentIdeal`: `B` is generated\n  (as an `A`-algebra) by `R₁` and `R₂`.\n\n-/\n\n@[expose] public section\n\nopen FractionalIdeal nonZeroDivisors IntermediateField Algebra Module Submodule\n\nvariable (A B : Type*) {K L : Type*} [CommRing A] [Field K] [Algebra A K] [IsFractionRing A K]\n  [CommRing B] [Field L] [Algebra B L] [Algebra A L] [Algebra K L] [FiniteDimensional K L]\n  [IsScalarTower A K L]\nvariable (R₁ R₂ : Type*) [CommRing R₁] [CommRing R₂] [IsDomain R₁] [Algebra A R₁] [Algebra A R₂]\n  [Algebra R₁ B] [Algebra R₂ B] [Algebra R₁ L] [Algebra R₂ L]\n  [IsScalarTower A R₁ L] [IsScalarTower R₁ B L] [IsScalarTower R₂ B L] [Module.Finite A R₂]\nvariable {F₁ F₂ : IntermediateField K L} [Algebra R₁ F₁] [Algebra R₂ F₂] [IsTorsionFree R₁ F₁]\n  [IsScalarTower A F₂ L] [IsScalarTower A R₂ F₂] [IsScalarTower R₁ F₁ L] [IsScalarTower R₂ F₂ L]\n  [Algebra.IsSeparable K F₂] [Algebra.IsSeparable F₁ L]\n\ntheorem Submodule.traceDual_le_span_map_traceDual [Module.Free A R₂]\n    [IsLocalization (Algebra.algebraMapSubmonoid R₂ A⁰) F₂] (h₁ : F₁.LinearDisjoint F₂)\n    (h₂ : F₁ ⊔ F₂ = ⊤) :\n    (traceDual R₁ F₁ (1 : Submodule B L)).restrictScalars R₁ ≤\n      span R₁ (algebraMap F₂ L '' (traceDual A K (1 : Submodule R₂ F₂))) := by\n  intro x hx\n  have h₂' : F₁.toSubalgebra ⊔ F₂.toSubalgebra = ⊤ := by\n    simpa [sup_toSubalgebra_of_isAlgebraic_right] using congr_arg IntermediateField.toSubalgebra h₂\n  let b₂ := (Free.chooseBasis A R₂).localizationLocalization K A⁰ F₂\n  let B₁ := h₁.basisOfBasisRight h₂' b₂\n  have h_main : x ∈ span R₁ (Set.range B₁.traceDual) := by\n    rw [B₁.traceDual.mem_span_iff_repr_mem R₁ x]\n    intro i\n    rw [B₁.traceDual_repr_apply]\n    refine mem_traceDual.mp hx _ ?_\n    rw [LinearDisjoint.basisOfBasisRight_apply, Basis.localizationLocalization_apply,\n      ← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply R₂ B L, mem_one]\n    exact ⟨_, rfl⟩\n  have h : Set.range B₁.traceDual =\n      Set.range (IsScalarTower.toAlgHom A F₂ L ∘ b₂.traceDual) := by\n    refine congr_arg Set.range <| B₁.traceDual_eq_iff.mpr fun i j ↦ ?_\n    rw [LinearDisjoint.basisOfBasisRight_apply, traceForm_apply, Function.comp_apply,\n      IsScalarTower.coe_toAlgHom', ← map_mul, h₁.trace_algebraMap h₂, b₂.trace_traceDual_mul,\n      MonoidWithZeroHom.map_ite_one_zero]\n  rwa [← span_span_of_tower A R₁, h, Set.range_comp, ← AlgHom.coe_toLinearMap, ← map_span,\n    ← traceDual_span_of_basis A (1 : Submodule R₂ F₂) b₂\n      (by rw [Basis.localizationLocalization_span K A⁰ F₂]; ext; simp)] at h_main\n\nattribute [local instance] FractionRing.liftAlgebra\n\nvariable [IsDomain A] [IsDedekindDomain B] [IsDedekindDomain R₁] [IsDedekindDomain R₂]\n    [IsFractionRing B L] [IsFractionRing R₁ F₁] [IsFractionRing R₂ F₂] [IsIntegrallyClosed A]\n    [IsIntegralClosure B R₁ L] [IsTorsionFree R₁ B] [IsTorsionFree R₂ B]\n\nset_option linter.overlappingInstances false\n\nnamespace IsDedekindDomain\n\ntheorem differentIdeal_dvd_map_differentIdeal [Algebra.IsIntegral R₂ B]\n    [Module.Free A R₂] [IsLocalization (Algebra.algebraMapSubmonoid R₂ A⁰) F₂]\n    (h₁ : F₁.LinearDisjoint F₂) (h₂ : F₁ ⊔ F₂ = ⊤) :\n    differentIdeal R₁ B ∣ Ideal.map (algebraMap R₂ B) (differentIdeal A R₂) := by\n  have : Algebra.IsSeparable (FractionRing A) (FractionRing R₂) := by\n    refine Algebra.IsSeparable.of_equiv_equiv (FractionRing.algEquiv A K).symm.toRingEquiv\n          (FractionRing.algEquiv R₂ F₂).symm.toRingEquiv ?_\n    ext _\n    exact IsFractionRing.algEquiv_commutes (FractionRing.algEquiv A K).symm\n      (FractionRing.algEquiv R₂ ↥F₂).symm _\n  rw [Ideal.dvd_iff_le, ← coeIdeal_le_coeIdeal L, coeIdeal_differentIdeal R₁ F₁ L B,\n    ← extendedHom_coeIdeal_eq_map L B (K := F₂), le_inv_comm _ (by simp), ← map_inv₀,\n    coeIdeal_differentIdeal A K, inv_inv, ← coe_le_coe, coe_dual_one, coe_extendedHom_eq_span,\n    ← coeToSet_coeToSubmodule, coe_dual_one]\n  · have := Submodule.span_mono (R := B) <| traceDual_le_span_map_traceDual A B R₁ R₂ h₁ h₂\n    rwa [← span_coe_eq_restrictScalars, span_span_of_tower, span_span_of_tower, span_eq] at this\n  · exact (_root_.map_ne_zero _).mpr <| coeIdeal_eq_zero.not.mpr differentIdeal_ne_bot\n\nvariable [Algebra A B] [Module.Finite A B] [IsTorsionFree A B] [IsTorsionFree A R₁]\n  [IsTorsionFree A R₂] [Module.Finite A R₁] [Module.Finite R₂ B] [IsScalarTower A R₂ B]\n  [Module.Finite R₁ B] [Algebra.IsSeparable (FractionRing A) (FractionRing B)]\n  [IsScalarTower A R₁ B]\n\ntheorem map_differentIdeal_dvd_differentIdeal\n    (h : IsCoprime ((differentIdeal A R₁).map (algebraMap R₁ B))\n      ((differentIdeal A R₂).map (algebraMap R₂ B))) :\n    Ideal.map (algebraMap R₂ B) (differentIdeal A R₂) ∣ differentIdeal R₁ B :=\n  have := (differentIdeal_eq_differentIdeal_mul_differentIdeal A R₂ B).symm.trans\n    (differentIdeal_eq_differentIdeal_mul_differentIdeal A R₁ B)\n  h.symm.dvd_of_dvd_mul_right (dvd_of_mul_left_eq _ this)\n\ntheorem differentIdeal_eq_map_differentIdeal [Module.Free A R₂] (h₁ : F₁.LinearDisjoint F₂)\n    (h₂ : F₁ ⊔ F₂ = ⊤)\n    (h₃ : IsCoprime ((differentIdeal A R₁).map (algebraMap R₁ B))\n      ((differentIdeal A R₂).map (algebraMap R₂ B))) :\n    differentIdeal R₁ B = Ideal.map (algebraMap R₂ B) (differentIdeal A R₂) := by\n  apply dvd_antisymm\n  · exact differentIdeal_dvd_map_differentIdeal A B R₁ R₂ h₁ h₂\n  · exact map_differentIdeal_dvd_differentIdeal A B R₁ R₂ h₃\n\n/--\nLet `A ⊆ B` be a finite extension of Dedekind domains and assume that `A ⊆ R₁, R₂ ⊆ B` are two\nsubrings such that `Frac R₁ ⊔ Frac R₂ = Frac B`, `Frac R₁` and `Frac R₂` are linearly disjoint\nover `Frac A`, and that `𝓓(R₁/A)` and `𝓓(R₂/A)` are coprime where `𝓓` denotes the different ideal\nand `Frac R` denotes the fraction field of a domain `R`.\nWe have `𝓓(B/A) = 𝓓(R₁/A) * 𝓓(R₂/A)`.\n-/\ntheorem differentIdeal_eq_differentIdeal_mul_differentIdeal_of_isCoprime\n    [Module.Free A R₂] (h₁ : F₁.LinearDisjoint F₂) (h₂ : F₁ ⊔ F₂ = ⊤)\n    (h₃ : IsCoprime ((differentIdeal A R₁).map (algebraMap R₁ B))\n      ((differentIdeal A R₂).map (algebraMap R₂ B))) :\n    differentIdeal A B = differentIdeal R₁ B * differentIdeal R₂ B := by\n  have := differentIdeal_eq_differentIdeal_mul_differentIdeal A R₂ B\n  rwa [← differentIdeal_eq_map_differentIdeal A B R₁ R₂ h₁ h₂ h₃,\n    mul_comm] at this\n\nend IsDedekindDomain\n\nvariable [Algebra A B] [Module.Finite A B] [IsTorsionFree A B] [IsTorsionFree A R₁]\n  [IsTorsionFree A R₂] [Module.Finite A R₁] [Module.Finite R₂ B] [IsScalarTower A R₂ B]\n  [Module.Finite R₁ B] [Algebra.IsSeparable (FractionRing A) (FractionRing B)]\n  [IsScalarTower A R₁ B]\n\nTarget:\ntheorem Submodule.traceDual_eq_span_map_traceDual_of_linearDisjoint [Module.Free A R₂]\n    [IsLocalization (Algebra.algebraMapSubmonoid R₂ A⁰) F₂] (h₁ : F₁.LinearDisjoint F₂)\n    (h₂ : F₁ ⊔ F₂ = ⊤) (h₃ : IsCoprime ((differentIdeal A R₁).map (algebraMap R₁ B))\n      ((differentIdeal A R₂).map (algebraMap R₂ B))) :\n    span R₁ (algebraMap F₂ L '' (traceDual A K (1 : Submodule R₂ F₂))) =\n      (traceDual R₁ F₁ (1 : Submodule B L)).restrictScalars R₁ :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"a4b71ad099ac2394ce9495b0a3866c1ec8daf5ad15e3394048e1ec89dd27e8b1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/DedekindDomain","family_id":"submodule","file_id":"mathlib/Mathlib/RingTheory/DedekindDomain/LinearDisjoint.lean","sample_id":"a86a08c70d72b421064dab529cf5e5d4ac90c62963744456fb1461e7c0e42c96"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6dbb4280aff812e88179a089ca7e30391af8146bdc02e8489a28db8f88a2918d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5879b2913a081f78e72683e0e490903029f8e16dc55de2933a3d338f56fd0681","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f56b4dc966c61de3f4b4118e398dfcc1e44b0ddfc9b9e3ab8b9799975c8612d7","source_sha256":"482679c638fc474fa50b011d709adb8dcdc790257773e34a00e39190b4d36d68","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext a\n  apply Subtype.ext\n  ext ⟨i, hi⟩\n  obtain ⟨⟨n, a, ⟨b, hb'⟩, i, rfl : _ = b⟩, rfl⟩ := mk_surjective a\n  simp only [homOfLE_leOfHom, CommRingCat.hom_comp, RingHom.coe_comp, Function.comp_apply]\n  erw [ProjectiveSpectrum.Proj.awayToSection_apply]\n  rw [CommRingCat.hom_ofHom, val_awayMap_mk, Localization.mk_eq_mk', IsLocalization.map_mk',\n    ← Localization.mk_eq_mk']\n  refine Localization.mk_eq_mk_iff.mpr ?_\n  rw [Localization.r_iff_exists]\n  use 1\n  simp [hx]\n  ring","hard_negative":false,"metrics":{"chosen_tokens":107,"rejected_tokens":112,"token_jaccard":0.929825,"token_length_ratio":1.046729},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"9e3b71242247ee3589ac6c094170efd21dc949539acd28cc1c8b1266a300a89a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme\npublic import Mathlib.AlgebraicGeometry.AffineScheme\npublic import Mathlib.AlgebraicGeometry.Gluing\n\nNamespace:\nAlgebraicGeometry.Proj\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Basic properties of the scheme `Proj A`\n\nThe scheme `Proj 𝒜` for a graded ring `𝒜` is constructed in\n`Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean`.\nIn this file we provide basic properties of the scheme.\n\n## Main results\n- `AlgebraicGeometry.Proj.toSpecZero`: The structure map `Proj A ⟶ Spec (A 0)`.\n- `AlgebraicGeometry.Proj.basicOpenIsoSpec`:\n  The canonical isomorphism `Proj A |_ D₊(f) ≅ Spec (A_f)₀`\n  when `f` is homogeneous of positive degree.\n- `AlgebraicGeometry.Proj.awayι`: The open immersion `Spec (A_f)₀ ⟶ Proj A`.\n- `AlgebraicGeometry.Proj.affineOpenCover`: The open cover of `Proj A` by `Spec (A_f)₀` for all\n  homogeneous `f` of positive degree.\n- `AlgebraicGeometry.Proj.stalkIso`:\n  The stalk of `Proj A` at `x` is the degree `0` part of the localization of `A` at `x`.\n- `AlgebraicGeometry.Proj.fromOfGlobalSections`:\n  Given a map `f : A →+* Γ(X, ⊤)` such that the image of the irrelevant ideal under `f`\n  generates the whole ring, we can construct a map `X ⟶ Proj 𝒜`.\n\n-/\n\n@[expose] public section\n\nnamespace AlgebraicGeometry.Proj\n\nopen HomogeneousLocalization CategoryTheory\n\nuniverse u\n\nvariable {σ : Type*} {A : Type u}\nvariable [CommRing A] [SetLike σ A] [AddSubgroupClass σ A]\nvariable (𝒜 : ℕ → σ)\nvariable [GradedRing 𝒜]\n\nsection basicOpen\n\nvariable (f g : A)\n\n/-- The basic open set `D₊(f)` associated to `f : A`. -/\ndef basicOpen : (Proj 𝒜).Opens :=\n  ProjectiveSpectrum.basicOpen 𝒜 f\n\n@[simp]\ntheorem mem_basicOpen (x : Proj 𝒜) :\n    x ∈ basicOpen 𝒜 f ↔ f ∉ x.asHomogeneousIdeal :=\n  Iff.rfl\n\n@[simp] theorem basicOpen_one : basicOpen 𝒜 1 = ⊤ := ProjectiveSpectrum.basicOpen_one ..\n\n@[simp] theorem basicOpen_zero : basicOpen 𝒜 0 = ⊥ := ProjectiveSpectrum.basicOpen_zero ..\n\n@[simp] theorem basicOpen_pow (n) (hn : 0 < n) : basicOpen 𝒜 (f ^ n) = basicOpen 𝒜 f :=\n  ProjectiveSpectrum.basicOpen_pow 𝒜 f n hn\n\ntheorem basicOpen_mul : basicOpen 𝒜 (f * g) = basicOpen 𝒜 f ⊓ basicOpen 𝒜 g :=\n  ProjectiveSpectrum.basicOpen_mul ..\n\ntheorem basicOpen_mono (hfg : f ∣ g) : basicOpen 𝒜 g ≤ basicOpen 𝒜 f :=\n  (hfg.choose_spec ▸ basicOpen_mul 𝒜 f _).trans_le inf_le_left\n\ntheorem basicOpen_eq_iSup_proj (f : A) :\n    basicOpen 𝒜 f = ⨆ i : ℕ, basicOpen 𝒜 (GradedRing.proj 𝒜 i f) :=\n  ProjectiveSpectrum.basicOpen_eq_union_of_projection ..\n\ntheorem isBasis_basicOpen :\n    TopologicalSpace.Opens.IsBasis (Set.range (basicOpen 𝒜)) := by\n  delta TopologicalSpace.Opens.IsBasis\n  convert! ProjectiveSpectrum.isTopologicalBasis_basic_opens 𝒜\n  exact (Set.range_comp _ _).symm\n\n/-- If `{ xᵢ }` spans the irrelevant ideal of `A`, then `D₊(xᵢ)` covers `Proj A`. -/\nlemma iSup_basicOpen_eq_top {ι : Type*} (f : ι → A)\n    (hf : (HomogeneousIdeal.irrelevant 𝒜).toIdeal ≤ Ideal.span (Set.range f)) :\n    ⨆ i, Proj.basicOpen 𝒜 (f i) = ⊤ := by\n  classical\n  refine top_le_iff.mp fun x hx ↦ TopologicalSpace.Opens.mem_iSup.mpr ?_\n  by_contra! H\n  simp only [mem_basicOpen, Decidable.not_not] at H\n  refine x.not_irrelevant_le (hf.trans ?_)\n  rwa [Ideal.span_le, Set.range_subset_iff]\n\n/-- If `{ xᵢ }` are homogeneous and span `A` as an `A₀` algebra, then `D₊(xᵢ)` covers `Proj A`. -/\nlemma iSup_basicOpen_eq_top' {ι : Type*} (f : ι → A)\n    (hfn : ∀ i, ∃ n, f i ∈ 𝒜 n)\n    (hf : Algebra.adjoin (𝒜 0) (Set.range f) = ⊤) :\n    ⨆ i, Proj.basicOpen 𝒜 (f i) = ⊤ := by\n  classical\n  apply Proj.iSup_basicOpen_eq_top\n  intro x hx\n  convert_to x - GradedRing.projZeroRingHom 𝒜 x ∈ _\n  · rw [GradedRing.projZeroRingHom_apply, ← GradedRing.proj_apply,\n      (HomogeneousIdeal.mem_irrelevant_iff _ _).mp hx, sub_zero]\n  clear hx\n  have := (eq_iff_iff.mp congr(x ∈ $hf)).mpr trivial\n  induction this using Algebra.adjoin_induction with\n  | mem x hx =>\n    obtain ⟨i, rfl⟩ := hx\n    obtain ⟨n, hn⟩ := hfn i\n    rw [GradedRing.projZeroRingHom_apply]\n    by_cases hn' : n = 0\n    · rw [DirectSum.decompose_of_mem_same 𝒜 (hn' ▸ hn), sub_self]\n      exact zero_mem _\n    · rw [DirectSum.decompose_of_mem_ne 𝒜 hn hn', sub_zero]\n      exact Ideal.subset_span ⟨_, rfl⟩\n  | algebraMap r =>\n    convert! zero_mem (Ideal.span _)\n    rw [sub_eq_zero]\n    exact (DirectSum.decompose_of_mem_same 𝒜 r.2).symm\n  | add x y hx hy _ _ =>\n    rw [map_add, add_sub_add_comm]\n    exact add_mem ‹_› ‹_›\n  | mul x y hx hy hx' hy' =>\n    convert!\n      add_mem (Ideal.mul_mem_left _ x hy')\n        (Ideal.mul_mem_right (GradedRing.projZeroRingHom 𝒜 y) _ hx') using 1\n    rw [map_mul]\n    ring\n\n/-- The canonical map `(A_f)₀ ⟶ Γ(Proj A, D₊(f))`.\nThis is an isomorphism when `f` is homogeneous of positive degree. See `basicOpenIsoAway` below. -/\ndef awayToSection : CommRingCat.of (Away 𝒜 f) ⟶ Γ(Proj 𝒜, basicOpen 𝒜 f) :=\n  ProjectiveSpectrum.Proj.awayToSection ..\n\n/-- The canonical map `Proj A |_ D₊(f) ⟶ Spec (A_f)₀`.\nThis is an isomorphism when `f` is homogeneous of positive degree. See `basicOpenIsoSpec` below. -/\nnoncomputable\ndef basicOpenToSpec : (basicOpen 𝒜 f).toScheme ⟶ Spec (.of <| Away 𝒜 f) :=\n  (basicOpen 𝒜 f).toSpecΓ ≫ Spec.map (awayToSection 𝒜 f)\n\nlemma basicOpenToSpec_app_top :\n    (basicOpenToSpec 𝒜 f).app ⊤ = (Scheme.ΓSpecIso _).hom ≫ awayToSection 𝒜 f ≫\n      (basicOpen 𝒜 f).topIso.inv := by\n  simp [basicOpenToSpec, Scheme.Opens.toSpecΓ_appTop]\n\n/-- The structure map `Proj A ⟶ Spec A₀`. -/\nnoncomputable\ndef toSpecZero : Proj 𝒜 ⟶ Spec (.of <| 𝒜 0) :=\n  (Scheme.topIso _).inv ≫ (Scheme.isoOfEq _ (basicOpen_one _)).inv ≫\n    basicOpenToSpec 𝒜 1 ≫ Spec.map (CommRingCat.ofHom (fromZeroRingHom 𝒜 _))\n\nvariable {m} (f_deg : f ∈ 𝒜 m) (hm : 0 < m)\n\n/-- The canonical isomorphism `Proj A |_ D₊(f) ≅ Spec (A_f)₀`\nwhen `f` is homogeneous of positive degree. -/\n@[simps! -isSimp hom]\nnoncomputable\ndef basicOpenIsoSpec : (basicOpen 𝒜 f).toScheme ≅ Spec (.of <| Away 𝒜 f) :=\n  have : IsIso (basicOpenToSpec 𝒜 f) := by\n    apply (isIso_iff_of_reflects_iso _ Scheme.forgetToLocallyRingedSpace).mp ?_\n    convert! ProjectiveSpectrum.Proj.isIso_toSpec 𝒜 f f_deg hm using 1\n    refine Eq.trans ?_ (ΓSpec.locallyRingedSpaceAdjunction.homEquiv_apply _ _ _).symm\n    dsimp [basicOpenToSpec, Scheme.Opens.toSpecΓ]\n    simp only [Category.assoc, ← Spec.map_comp]\n    rfl\n  asIso (basicOpenToSpec 𝒜 f)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The canonical isomorphism `(A_f)₀ ≅ Γ(Proj A, D₊(f))`\nwhen `f` is homogeneous of positive degree. -/\n@[simps! -isSimp hom]\nnoncomputable\ndef basicOpenIsoAway : CommRingCat.of (Away 𝒜 f) ≅ Γ(Proj 𝒜, basicOpen 𝒜 f) :=\n  have : IsIso (awayToSection 𝒜 f) := by\n    have := basicOpenToSpec_app_top 𝒜 f\n    rw [← Iso.inv_comp_eq, Iso.eq_comp_inv] at this\n    rw [← this, ← basicOpenIsoSpec_hom 𝒜 f f_deg hm]\n    infer_instance\n  asIso (awayToSection 𝒜 f)\n\n/-- The open immersion `Spec (A_f)₀ ⟶ Proj A`. -/\nnoncomputable\ndef awayι : Spec (.of <| Away 𝒜 f) ⟶ Proj 𝒜 :=\n  (basicOpenIsoSpec 𝒜 f f_deg hm).inv ≫ (Proj.basicOpen 𝒜 f).ι\n\n@[reassoc]\nlemma basicOpenIsoSpec_inv_ι :\n    (basicOpenIsoSpec 𝒜 f f_deg hm).inv ≫ (Proj.basicOpen 𝒜 f).ι = awayι 𝒜 f f_deg hm := rfl\n\ninstance : IsOpenImmersion (Proj.awayι 𝒜 f f_deg hm) :=\n  IsOpenImmersion.comp _ _\n\nlemma opensRange_awayι :\n    (Proj.awayι 𝒜 f f_deg hm).opensRange = Proj.basicOpen 𝒜 f :=\n  (Scheme.Hom.opensRange_comp_of_isIso _ _).trans (basicOpen 𝒜 f).opensRange_ι\n\ninclude f_deg hm in\nlemma isAffineOpen_basicOpen : IsAffineOpen (basicOpen 𝒜 f) := by\n  rw [← opensRange_awayι 𝒜 f f_deg hm]\n  exact isAffineOpen_opensRange (awayι _ _ _ _)\n\n@[reassoc]\nlemma awayι_toSpecZero : awayι 𝒜 f f_deg hm ≫ toSpecZero 𝒜 =\n    Spec.map (CommRingCat.ofHom (fromZeroRingHom 𝒜 _)) := by\n  rw [toSpecZero, basicOpenToSpec, awayι]\n  simp only [Category.assoc, Iso.inv_comp_eq, basicOpenIsoSpec_hom]\n  have (U) (e : U = ⊤) : (basicOpen 𝒜 f).ι ≫ (Scheme.topIso _).inv ≫ (Scheme.isoOfEq _ e).inv =\n      Scheme.homOfLE _ (le_top.trans_eq e.symm) := by\n    simp only [← Category.assoc, Iso.comp_inv_eq]\n    simp only [Scheme.topIso_hom, Category.assoc, Scheme.isoOfEq_hom_ι, Scheme.homOfLE_ι]\n  rw [reassoc_of% this, ← Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_assoc, basicOpenToSpec,\n    Category.assoc, ← Spec.map_comp, ← Spec.map_comp, ← Spec.map_comp]\n  rfl\n\nvariable {f}\nvariable {m' : ℕ} {g : A} (g_deg : g ∈ 𝒜 m') (hm' : 0 < m') {x : A} (hx : x = f * g)\n\n@[reassoc]\n\nTarget:\nlemma awayMap_awayToSection :\n    CommRingCat.ofHom (awayMap 𝒜 g_deg hx) ≫ awayToSection 𝒜 x =\n      awayToSection 𝒜 f ≫ (Proj 𝒜).presheaf.map (homOfLE (basicOpen_mono _ _ _ ⟨_, hx⟩)).op :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  ext a\n  apply Subtype.ext\n  ext ⟨i, hi⟩\n  obtain ⟨⟨n, a, ⟨b, hb'⟩, i, rfl : _ = b⟩, rfl⟩ := mk_surjective a\n  simp only [homOfLE_leOfHom, CommRingCat.hom_comp, RingHom.coe_comp, Function.comp_apply]\n  erw [ProjectiveSpectrum.Proj.awayToSection_apply]\n  rw [CommRingCat.hom_ofHom, val_awayMap_mk, Localization.mk_eq_mk', IsLocalization.map_mk',\n    ← Localization.mk_eq_mk']\n  refine Localization.mk_eq_mk_iff.mpr ?_\n  rw [Localization.r_iff_exists]\n  use 1\n  simp [hx]\n  ring","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/ProjectiveSpectrum","family_id":"awaymap_awaytosection","file_id":"mathlib/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.lean","sample_id":"f56b4dc966c61de3f4b4118e398dfcc1e44b0ddfc9b9e3ab8b9799975c8612d7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"76978dc753664885955d4019ab183b19aab997594734243a4e6b99373e308b2b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4ea77ade5bcaf82f263f63d2e35e31706e2c9a6c7ae719c3b1558eff0efaaa66","source_sha256":"f83ecd09e0c1dc0009269016250874e7bd19aed3be89d47a83b8fee4b9eb7cc4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [isCompactElement_iff_exists_le_sSup_of_le_sSup]\n  refine ⟨fun h k s hs => ?_, fun h s => ?_⟩\n  · obtain ⟨t, ⟨hts, htsup⟩⟩ := h s\n    use t, hts\n    rwa [← htsup]\n  · obtain ⟨t, ⟨hts, htsup⟩⟩ := h (sSup s) s (by rfl)\n    have : sSup s = t.sup id := by\n      suffices t.sup id ≤ sSup s by apply le_antisymm <;> assumption\n      simp only [id, Finset.sup_le_iff]\n      intro x hx\n      exact le_sSup (hts hx)\n    exact ⟨t, hts, this⟩","hard_negative":false,"metrics":{"chosen_tokens":120,"rejected_tokens":2,"token_jaccard":0.037037,"token_length_ratio":0.016667},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"9e449cadd4c0806f41fbc179f26ca137d3d5ce5dafdae5489cac128ea8e573dd","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Atoms\npublic import Mathlib.Order.OrderIsoNat\npublic import Mathlib.Order.RelIso.Set\npublic import Mathlib.Order.SupClosed\npublic import Mathlib.Order.SupIndep\npublic import Mathlib.Order.Zorn\npublic import Mathlib.Data.Finset.Order\npublic import Mathlib.Order.Interval.Set.OrderIso\npublic import Mathlib.Data.Finite.Set\npublic import Mathlib.Tactic.TFAE\n\nNamespace:\nCompleteLattice\n\nLocal context:\n/-\nCopyright (c) 2021 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Compactness properties for complete lattices\n\nFor complete lattices, there are numerous equivalent ways to express the fact that the relation `>`\nis well-founded. In this file we define three especially-useful characterisations and provide\nproofs that they are indeed equivalent to well-foundedness.\n\n## Main definitions\n* `CompleteLattice.IsSupClosedCompact`\n* `CompleteLattice.IsSupFiniteCompact`\n* `IsCompactElement`\n* `IsCompactlyGenerated`\n\n## Main results\nThe main result is that the following four conditions are equivalent for a complete lattice:\n* `well_founded (>)`\n* `CompleteLattice.IsSupClosedCompact`\n* `CompleteLattice.IsSupFiniteCompact`\n* `∀ k, IsCompactElement k`\n\nThis is demonstrated by means of the following four lemmas:\n* `CompleteLattice.WellFounded.isSupFiniteCompact`\n* `CompleteLattice.IsSupFiniteCompact.isSupClosedCompact`\n* `CompleteLattice.IsSupClosedCompact.wellFounded`\n* `CompleteLattice.isSupFiniteCompact_iff_all_elements_compact`\n\nWe also show well-founded lattices are compactly generated\n(`CompleteLattice.isCompactlyGenerated_of_wellFounded`).\n\n## References\n- [G. Călugăreanu, *Lattice Concepts of Module Theory*][calugareanu]\n\n## Tags\n\ncomplete lattice, well-founded, compact\n-/\n\n@[expose] public section\n\nopen Set\n/-- An element `k` is compact if any directed set with `LUB` (least upper bound) above\n`k` has already got above `k` at some point in the set.\nSuch an element is also called \"finite\" or \"S-compact\". -/\ndef IsCompactElement {α : Type*} [PartialOrder α] (k : α) :=\n  ∀ (s : Set α) (u : α),\n    s.Nonempty →\n    DirectedOn (· ≤ ·) s →\n    IsLUB s u →\n    k ≤ u →\n    ∃ x ∈ s, k ≤ x\n\nvariable {ι : Sort*} {α : Type*} [CompleteLattice α] {f : ι → α}\n\nnamespace CompleteLattice\n\nvariable (α)\n\n/-- A compactness property for a complete lattice is that any `sup`-closed non-empty subset\ncontains its `sSup`. -/\ndef IsSupClosedCompact : Prop :=\n  ∀ (s : Set α) (_ : s.Nonempty), SupClosed s → sSup s ∈ s\n\n/-- A compactness property for a complete lattice is that any subset has a finite subset with the\nsame `sSup`. -/\ndef IsSupFiniteCompact : Prop :=\n  ∀ s : Set α, ∃ t : Finset α, ↑t ⊆ s ∧ sSup s = t.sup id\n\n/-- An element `k` is compact if and only if any directed set with `sSup` above\n`k` already got above `k` at some point in the set. -/\ntheorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :\n    IsCompactElement k ↔\n      ∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x := by\n  constructor\n  · intro hk s hs hs' h_le\n    exact hk s (sSup s) hs hs' (isLUB_sSup s) h_le\n  · intro h s u hs hs' hu h_le\n    rw [isLUB_iff_sSup_eq] at hu\n    rw [← hu] at h_le\n    exact h s hs hs' h_le\n\n/-- An element `k` of is compact if any set with `sSup`\nabove `k` has a finite subset with `sSup` above `k`. -/\ntheorem isCompactElement_iff_exists_le_sSup_of_le_sSup (k : α) :\n    IsCompactElement k ↔ ∀ s : Set α, k ≤ sSup s → ∃ t : Finset α, ↑t ⊆ s ∧ k ≤ t.sup id := by\n  classical\n    rw [isCompactElement_iff_le_of_directed_sSup_le]\n    constructor\n    · intro hk s hsup\n      -- Consider the set of finite joins of elements of the (plain) set s.\n      let S : Set α := { x | ∃ t : Finset α, ↑t ⊆ s ∧ x = t.sup id }\n      -- S is directed, nonempty, and still has sup above k.\n      have dir_US : DirectedOn (· ≤ ·) S := by\n        rintro x ⟨c, hc⟩ y ⟨d, hd⟩\n        use x ⊔ y\n        constructor\n        · use c ∪ d\n          constructor\n          · simp only [hc.left, hd.left, Set.union_subset_iff, Finset.coe_union, and_self_iff]\n          · simp only [hc.right, hd.right, Finset.sup_union]\n        simp only [and_self_iff, le_sup_left, le_sup_right]\n      have sup_S : sSup s ≤ sSup S := by\n        apply sSup_le_sSup\n        intro x hx\n        use {x}\n        simpa only [and_true, id, Finset.coe_singleton, eq_self_iff_true,\n          Finset.sup_singleton, Set.singleton_subset_iff]\n      have Sne : S.Nonempty := by\n        suffices ⊥ ∈ S from Set.nonempty_of_mem this\n        use ∅\n        simp\n      -- Now apply the defn of compact and finish.\n      obtain ⟨j, ⟨hjS, hjk⟩⟩ := hk S Sne dir_US (le_trans hsup sup_S)\n      obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS\n      use t\n      exact ⟨htS, by rwa [← htsup]⟩\n    · intro hk s hne hdir hsup\n      obtain ⟨t, ht⟩ := hk s hsup\n      -- certainly every element of t is below something in s, since ↑t ⊆ s.\n      have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y := fun x hxt => ⟨x, ht.left hxt, le_rfl⟩\n      obtain ⟨x, ⟨hxs, hsupx⟩⟩ := Finset.sup_le_of_le_directed s hne hdir t t_below_s\n      exact ⟨x, ⟨hxs, le_trans ht.right hsupx⟩⟩\n\ntheorem isCompactElement_iff_exists_le_iSup_of_le_iSup.{u} {α : Type u} [CompleteLattice α]\n    (k : α) : IsCompactElement k ↔\n      ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s := by\n  classical\n    rw [isCompactElement_iff_exists_le_sSup_of_le_sSup]\n    constructor\n    · intro H ι s hs\n      obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs\n      have : ∀ x : t, ∃ i, s i = x := fun x => ht x.prop\n      choose f hf using this\n      refine ⟨Finset.univ.image f, ht'.trans ?_⟩\n      rw [Finset.sup_le_iff]\n      intro b hb\n      rw [← show s (f ⟨b, hb⟩) = id b from hf _]\n      exact Finset.le_sup (Finset.mem_image_of_mem f <| Finset.mem_univ (Subtype.mk b hb))\n    · intro H s hs\n      obtain ⟨t, ht⟩ :=\n        H s Subtype.val\n          (by\n            delta iSup\n            rwa [Subtype.range_coe])\n      refine ⟨t.image Subtype.val, by simp, ht.trans ?_⟩\n      rw [Finset.sup_le_iff]\n      exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem Subtype.val hx)\n\ntheorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type*}\n    (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by\n  classical\n    rw [isCompactElement_iff_le_of_directed_sSup_le] at hk\n    let g : Finset ι → α := fun s => ⨆ i ∈ s, f i\n    have h1 : DirectedOn (· ≤ ·) (Set.range g) := by\n      rintro - ⟨s, rfl⟩ - ⟨t, rfl⟩\n      exact\n        ⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, iSup_le_iSup_of_subset Finset.subset_union_left,\n          iSup_le_iSup_of_subset Finset.subset_union_right⟩\n    have h2 : k ≤ sSup (Set.range g) :=\n      h.trans\n        (iSup_le fun i =>\n          le_sSup_of_le ⟨{i}, rfl⟩\n            (le_iSup_of_le i (le_iSup_of_le (Finset.mem_singleton_self i) le_rfl)))\n    obtain ⟨-, ⟨s, rfl⟩, hs⟩ := hk (Set.range g) (Set.range_nonempty g) h1 h2\n    exact ⟨s, hs⟩\n\n/-- A compact element `k` has the property that any directed set lying strictly below `k` has\nits `sSup` strictly below `k`. -/\ntheorem IsCompactElement.directed_sSup_lt_of_lt {α : Type*} [CompleteLattice α] {k : α}\n    (hk : IsCompactElement k) {s : Set α} (hemp : s.Nonempty) (hdir : DirectedOn (· ≤ ·) s)\n    (hbelow : ∀ x ∈ s, x < k) : sSup s < k := by\n  rw [isCompactElement_iff_le_of_directed_sSup_le] at hk\n  by_contra h\n  have sSup' : sSup s ≤ k := sSup_le fun s hs => (hbelow s hs).le\n  replace sSup : sSup s = k := eq_iff_le_not_lt.mpr ⟨sSup', h⟩\n  obtain ⟨x, hxs, hkx⟩ := hk s hemp hdir sSup.symm.le\n  obtain hxk := hbelow x hxs\n  exact hxk.ne (hxk.le.antisymm hkx)\n\ntheorem isCompactElement_finsetSup {α β : Type*} [CompleteLattice α] {f : β → α} (s : Finset β)\n    (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by\n  classical\n    simp_rw [isCompactElement_iff_le_of_directed_sSup_le] at ⊢ h\n    intro d hemp hdir hsup\n    rw [← Function.id_comp f]\n    rw [← Finset.sup_image]\n    apply Finset.sup_le_of_le_directed d hemp hdir\n    rintro x hx\n    obtain ⟨p, ⟨hps, rfl⟩⟩ := Finset.mem_image.mp hx\n    specialize h p hps\n    specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)\n    simpa only [exists_prop]\n\ntheorem WellFoundedGT.isSupFiniteCompact [WellFoundedGT α] :\n    IsSupFiniteCompact α := fun s => by\n  let S := { x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x }\n  obtain ⟨m, ⟨t, ⟨ht₁, rfl⟩⟩, hm⟩ := wellFounded_gt.has_min S ⟨⊥, ∅, by simp⟩\n  refine ⟨t, ht₁, (sSup_le fun y hy => ?_).antisymm ?_⟩\n  · classical\n    rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))\n        (hm _ ⟨insert y t, by simp [Set.insert_subset_iff, hy, ht₁]⟩)]\n    simp\n  · rw [Finset.sup_id_eq_sSup]\n    exact sSup_le_sSup ht₁\n\ntheorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) :\n    IsSupClosedCompact α := by\n  intro s hne hsc; obtain ⟨t, ht₁, ht₂⟩ := h s; clear h\n  rcases t.eq_empty_or_nonempty with rfl | h\n  · rw [Finset.sup_empty] at ht₂\n    rw [ht₂]\n    simp [eq_singleton_bot_of_sSup_eq_bot_of_nonempty ht₂ hne]\n  · rw [ht₂]\n    exact hsc.finsetSup_mem h ht₁\n\ntheorem IsSupClosedCompact.wellFoundedGT (h : IsSupClosedCompact α) :\n    WellFoundedGT α where\n  wf := by\n    refine RelEmbedding.wellFounded_iff_isEmpty.mpr ⟨fun a => ?_⟩\n    suffices sSup (Set.range a) ∈ Set.range a by\n      obtain ⟨n, hn⟩ := Set.mem_range.mp this\n      have h' : sSup (Set.range a) < a (n + 1) := by\n        change _ > _\n        simp [← hn, a.map_rel_iff]\n      apply lt_irrefl (a (n + 1))\n      apply lt_of_le_of_lt _ h'\n      apply le_sSup\n      apply Set.mem_range_self\n    apply h (Set.range a)\n    · use a 37\n      apply Set.mem_range_self\n    · rintro x ⟨m, hm⟩ y ⟨n, hn⟩\n      use m ⊔ n\n      rw [← hm, ← hn]\n      apply RelHomClass.map_sup a\n\nTarget:\ntheorem isSupFiniteCompact_iff_all_elements_compact :\n    IsSupFiniteCompact α ↔ ∀ k : α, IsCompactElement k :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/CompactlyGenerated","family_id":"issupfinitecompact_iff_all_elements_compact","file_id":"mathlib/Mathlib/Order/CompactlyGenerated/Basic.lean","sample_id":"4ea77ade5bcaf82f263f63d2e35e31706e2c9a6c7ae719c3b1558eff0efaaa66"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4d69ce6ddd118e205dc838354a84f5c72260ded77b2d93b7242fcdda08dda204","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3e1c2bd9695ae8682b25bbfbf9b301964e52b391a2327abba05974eeeed4a9d5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"039e1da8660af549f0b84de8a8d3a1a21828f8928f6164e5d53b92bab06d1d90","source_sha256":"bfb63160eae740c943f595c7368534b4a070f971456c777a0ade88e8c2da9349","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rwa [← hasPointwiseRightDerivedFunctorAt_iff F L W]","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":16,"token_jaccard":0.818182,"token_length_ratio":1.777778},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"9e8cfabca8d02169b5e0ea7b5a3ea70d33985d84b8fa47d0b0a74c5a09906a59","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Functor.Derived.RightDerived\npublic import Mathlib.CategoryTheory.Functor.KanExtension.Pointwise\npublic import Mathlib.CategoryTheory.Localization.StructuredArrow\n\nNamespace:\nCategoryTheory.Functor\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Pointwise right derived functors\n\nWe define pointwise right derived functors using the notion\nof pointwise left Kan extensions.\n\nWe show that if `F : C ⥤ H` inverts `W : MorphismProperty C`,\nthen it has a pointwise right derived functor.\n\nNote: the file `Mathlib/CategoryTheory/Functor/Derived/PointwiseLeftDerived.lean` was obtained\nby dualizing this file. These two files should be kept in sync.\n\n-/\n\n@[expose] public section\n\nuniverse v₁ v₂ v₃ u₁ u₂ u₃\n\nnamespace CategoryTheory\n\nopen Category Limits\n\nnamespace Functor\n\nvariable {C : Type u₁} {D : Type u₂} {H : Type u₃}\n  [Category.{v₁} C] [Category.{v₂} D] [Category.{v₃} H]\n  (F' : D ⥤ H) (F : C ⥤ H) (L : C ⥤ D) (α : F ⟶ L ⋙ F') (W : MorphismProperty C)\n\n/-- Given `F : C ⥤ H`, `W : MorphismProperty C` and `X : C`, we say that `F` has a\npointwise right derived functor at `X` if `F` has a left Kan extension\nat `L.obj X` for any localization functor `L : C ⥤ D` for `W`. In the\ndefinition, this is stated for `L := W.Q`, see `hasPointwiseRightDerivedFunctorAt_iff`\nfor the more general equivalence. -/\nclass HasPointwiseRightDerivedFunctorAt (X : C) : Prop where\n  /-- Use the more general `hasColimit` lemma instead, see also\n  `hasPointwiseRightDerivedFunctorAt_iff` -/\n  hasColimit' : HasPointwiseLeftKanExtensionAt W.Q F (W.Q.obj X)\n\n/-- A functor `F : C ⥤ H` has a pointwise right derived functor with respect to\n`W : MorphismProperty C` if it has a pointwise right derived functor at `X`\nfor any `X : C`. -/\nabbrev HasPointwiseRightDerivedFunctor := ∀ (X : C), F.HasPointwiseRightDerivedFunctorAt W X\n\nlemma hasPointwiseRightDerivedFunctorAt_iff [L.IsLocalization W] (X : C) :\n    F.HasPointwiseRightDerivedFunctorAt W X ↔\n      HasPointwiseLeftKanExtensionAt L F (L.obj X) := by\n  rw [← hasPointwiseLeftKanExtensionAt_iff_of_equivalence W.Q L F\n    (Localization.uniq W.Q L W) (Localization.compUniqFunctor W.Q L W) (W.Q.obj X) (L.obj X)\n    ((Localization.compUniqFunctor W.Q L W).app X)]\n  exact ⟨fun h ↦ h.hasColimit', fun h ↦ ⟨h⟩⟩\n\nTarget:\nlemma HasPointwiseRightDerivedFunctorAt.hasColimit\n    [L.IsLocalization W] (X : C) [F.HasPointwiseRightDerivedFunctorAt W X] :\n    HasPointwiseLeftKanExtensionAt L F (L.obj X) :=\n\nProof body:\n","rejected":"```lean\nby\n  rwa [← hasPointwiseRightDerivedFunctorAt_iff F L W]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Functor","family_id":"haspointwiserightderivedfunctorat","file_id":"mathlib/Mathlib/CategoryTheory/Functor/Derived/PointwiseRightDerived.lean","sample_id":"039e1da8660af549f0b84de8a8d3a1a21828f8928f6164e5d53b92bab06d1d90"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0dd5faeb64813fa427f25f46617ddcf71a676309c87e033eb80368aa47d425e1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"734c8700a702ad20f6bf7de6d00c2776a234790674c9e8194e96bb935ac2b232","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fc41a1bfba603ad2786ab361571962d38f904641b8aa32668047f0fd3d10b946","source_sha256":"06b8f3152156917b43eb16aa25f88f706675fd088336840c57324ab27b671934","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [← C_eq_intCast, coeff_C, Int.cast_ite, Int.cast_zero]","hard_negative":true,"metrics":{"chosen_tokens":17,"rejected_tokens":5,"token_jaccard":0.125,"token_length_ratio":0.294118},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"9eb7357adebead7df1c91c91320390c086c901b96a074d91332458347d46991c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Degree.Lemmas\n\nNamespace:\nMathlib.Tactic.ComputeDegree\n\nLocal context:\n/-\nCopyright (c) 2023 Damiano Testa. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Damiano Testa\n-/\n/-!\n\n# `compute_degree` and `monicity`: tactics for explicit polynomials\n\nThis file defines two related tactics: `compute_degree` and `monicity`.\n\nUsing `compute_degree` when the goal is of one of the seven forms\n* `natDegree f ≤ d` (or `<`),\n* `degree f ≤ d` (or `<`),\n* `natDegree f = d`,\n* `degree f = d`,\n* `coeff f d = r`, if `d` is the degree of `f`,\n\ntries to solve the goal.\nIt may leave side-goals, in case it is not entirely successful.\n\nUsing `monicity` when the goal is of the form `Monic f` tries to solve the goal.\nIt may leave side-goals, in case it is not entirely successful.\n\nBoth tactics admit a `!` modifier (`compute_degree!` and `monicity!`) instructing\nLean to try harder to close the goal.\n\nSee the doc-strings for more details.\n\n## Future work\n\n* Currently, `compute_degree` does not deal correctly with some edge cases.  For instance,\n  ```lean\n  example [Semiring R] : natDegree (C 0 : R[X]) = 0 := by\n    compute_degree\n  --  ⊢ 0 ≠ 0\n  ```\n  Still, it may not be worth to provide special support for `natDegree f = 0`.\n* Make sure that numerals in coefficients are treated correctly.\n* Make sure that `compute_degree` works with goals of the form `degree f ≤ ↑d`, with an\n  explicit coercion from `ℕ` on the RHS.\n* Add support for proving goals of the from `natDegree f ≠ 0` and `degree f ≠ 0`.\n* Make sure that `degree`, `natDegree` and `coeff` are equally supported.\n\n## Implementation details\n\nAssume that `f : R[X]` is a polynomial with coefficients in a semiring `R` and\n`d` is either in `ℕ` or in `WithBot ℕ`.\n\nIf the goal has the form `natDegree f < d`, then we convert it to two separate goals:\n* `natDegree f ≤ ?_`, on which we apply the following steps;\n* `?_ < d`;\n\nwhere `?_` is a metavariable that `compute_degree` computes in its process.\nWe proceed similarly for `degree f < d`.\n\nIf the goal has the form `natDegree f = d`, then we convert it to three separate goals:\n* `natDegree f ≤ d`;\n* `coeff f d = r`;\n* `r ≠ 0`.\n\nSimilarly, an initial goal of the form `degree f = d` gives rise to goals of the form\n* `degree f ≤ d`;\n* `coeff f d = r`;\n* `r ≠ 0`.\n\nNext, we apply successively lemmas whose side-goals all have the shape\n* `natDegree f ≤ d`;\n* `degree f ≤ d`;\n* `coeff f d = r`;\n\nplus possibly \"numerical\" identities and choices of elements in `ℕ`, `WithBot ℕ`, and `R`.\n\nRecursing into `f`, we break apart additions, multiplications, powers, subtractions,...\nThe leaves of the process are\n* numerals, `C a`, `X` and `monomial a n`, to which we assign degree `0`, `1` and `a` respectively;\n* `fvar`s `f`, to which we tautologically assign degree `natDegree f`.\n-/\n\npublic meta section\n\nopen Polynomial\n\nnamespace Mathlib.Tactic.ComputeDegree\n\nsection recursion_lemmas\n/-!\n### Simple lemmas about `natDegree`\n\nThe lemmas in this section all have the form `natDegree <some form of cast> ≤ 0`.\nTheir proofs are weakenings of the stronger lemmas `natDegree <same> = 0`.\nThese are the lemmas called by `compute_degree` on (almost) all the leaves of its recursion.\n-/\n\nvariable {R : Type*}\n\nsection semiring\nvariable [Semiring R]\n\ntheorem natDegree_C_le (a : R) : natDegree (C a) ≤ 0 := (natDegree_C a).le\n\ntheorem natDegree_natCast_le (n : ℕ) : natDegree (n : R[X]) ≤ 0 := (natDegree_natCast _).le\ntheorem natDegree_zero_le : natDegree (0 : R[X]) ≤ 0 := natDegree_zero.le\ntheorem natDegree_one_le : natDegree (1 : R[X]) ≤ 0 := natDegree_one.le\n\ntheorem coeff_add_of_eq {n : ℕ} {a b : R} {f g : R[X]}\n    (h_add_left : f.coeff n = a) (h_add_right : g.coeff n = b) :\n    (f + g).coeff n = a + b := by subst ‹_› ‹_›; apply coeff_add\n\ntheorem coeff_mul_add_of_le_natDegree_of_eq_ite {d df dg : ℕ} {a b : R} {f g : R[X]}\n    (h_mul_left : natDegree f ≤ df) (h_mul_right : natDegree g ≤ dg)\n    (h_mul_left : f.coeff df = a) (h_mul_right : g.coeff dg = b) (ddf : df + dg ≤ d) :\n    (f * g).coeff d = if d = df + dg then a * b else 0 := by\n  split_ifs with h\n  · subst h_mul_left h_mul_right h\n    exact coeff_mul_add_eq_of_natDegree_le ‹_› ‹_›\n  · apply coeff_eq_zero_of_natDegree_lt\n    apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ddf ?_)\n    · exact natDegree_mul_le_of_le ‹_› ‹_›\n    · exact ne_comm.mp h\n\ntheorem coeff_pow_of_natDegree_le_of_eq_ite' {m n o : ℕ} {a : R} {p : R[X]}\n    (h_pow : natDegree p ≤ n) (h_exp : m * n ≤ o) (h_pow_bas : coeff p n = a) :\n    coeff (p ^ m) o = if o = m * n then a ^ m else 0 := by\n  split_ifs with h\n  · subst h h_pow_bas\n    exact coeff_pow_of_natDegree_le ‹_›\n  · apply coeff_eq_zero_of_natDegree_lt\n    apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ‹_› ?_)\n    · exact natDegree_pow_le_of_le m ‹_›\n    · exact Iff.mp ne_comm h\n\nsection SMul\n\nvariable {S : Type*} [SMulZeroClass S R] {n : ℕ} {a : S} {f : R[X]}\n\ntheorem natDegree_smul_le_of_le (hf : natDegree f ≤ n) :\n    natDegree (a • f) ≤ n :=\n  (natDegree_smul_le a f).trans hf\n\ntheorem degree_smul_le_of_le (hf : degree f ≤ n) :\n    degree (a • f) ≤ n :=\n  (degree_smul_le a f).trans hf\n\ntheorem coeff_smul : (a • f).coeff n = a • f.coeff n := rfl\n\nend SMul\n\nsection congr_lemmas\n\n/-- The following two lemmas should be viewed as a hand-made \"congr\"-lemmas.\nThey achieve the following goals.\n* They introduce *two* fresh metavariables replacing the given one `deg`,\n  one for the `natDegree ≤` computation and one for the `coeff =` computation.\n  This helps `compute_degree`, since it does not \"pre-estimate\" the degree,\n  but it \"picks it up along the way\".\n* They split checking the inequality `coeff p n ≠ 0` into the task of\n  finding a value `c` for the `coeff` and then\n  proving that this value is non-zero by `coeff_ne_zero`.\n-/\ntheorem natDegree_eq_of_le_of_coeff_ne_zero' {deg m o : ℕ} {c : R} {p : R[X]}\n    (h_natDeg_le : natDegree p ≤ m) (coeff_eq : coeff p o = c)\n    (coeff_ne_zero : c ≠ 0) (deg_eq_deg : m = deg) (coeff_eq_deg : o = deg) :\n    natDegree p = deg := by\n  subst coeff_eq deg_eq_deg coeff_eq_deg\n  exact natDegree_eq_of_le_of_coeff_ne_zero ‹_› ‹_›\n\ntheorem degree_eq_of_le_of_coeff_ne_zero' {deg m o : WithBot ℕ} {c : R} {p : R[X]}\n    (h_deg_le : degree p ≤ m) (coeff_eq : coeff p (WithBot.unbotD 0 deg) = c)\n    (coeff_ne_zero : c ≠ 0) (deg_eq_deg : m = deg) (coeff_eq_deg : o = deg) :\n    degree p = deg := by\n  subst coeff_eq coeff_eq_deg deg_eq_deg\n  rcases eq_or_ne m ⊥ with rfl | hh\n  · exact bot_unique h_deg_le\n  · obtain ⟨m, rfl⟩ := WithBot.ne_bot_iff_exists.mp hh\n    exact degree_eq_of_le_of_coeff_ne_zero ‹_› ‹_›\n\nvariable {m n : ℕ} {f : R[X]} {r : R}\n\ntheorem coeff_congr_lhs (h : coeff f m = r) (natDeg_eq_coeff : m = n) : coeff f n = r :=\n  natDeg_eq_coeff ▸ h\ntheorem coeff_congr (h : coeff f m = r) (natDeg_eq_coeff : m = n) {s : R} (rs : r = s) :\n    coeff f n = s :=\n  natDeg_eq_coeff ▸ rs ▸ h\n\nend congr_lemmas\n\nend semiring\n\nsection ring\nvariable [Ring R]\n\ntheorem natDegree_intCast_le (n : ℤ) : natDegree (n : R[X]) ≤ 0 := (natDegree_intCast _).le\n\ntheorem coeff_sub_of_eq {n : ℕ} {a b : R} {f g : R[X]} (hf : f.coeff n = a) (hg : g.coeff n = b) :\n    (f - g).coeff n = a - b := by subst hf hg; apply coeff_sub\n\nTarget:\ntheorem coeff_intCast_ite {n : ℕ} {a : ℤ} : (Int.cast a : R[X]).coeff n = ite (n = 0) a 0 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_fc41a1bfba60","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"86215d490ba04ebddcafbf9574b975189850bfea0add5aea2dd7ee4e9a9da4fe","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic","family_id":"coeff_intcast_ite","file_id":"mathlib/Mathlib/Tactic/ComputeDegree.lean","sample_id":"fc41a1bfba603ad2786ab361571962d38f904641b8aa32668047f0fd3d10b946"},"split":"train"}
{"binding":{"chosen_proof_sha256":"94092e3c9ae7aafc8bfd4b060b03f1f7bf7ca38bd35810157e3b4c1cdb58864b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bd9359a574dcaca31acf15cd4ed4c49c4962b2ea1c2de798364f401e03500718","source_sha256":"5b5f327a6479776df18ad9f770e834a46924f37fc00aca2b6d876cc1986c4651","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext1\n  apply map_C\n\n-- mixing the two monad structures","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":5,"token_jaccard":0.071429,"token_length_ratio":0.454545},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"9eea0249dbc312bea1db7020140451fc160cb511f9cb6fa65fecb5201089dfa4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.Rename\npublic import Mathlib.Algebra.MvPolynomial.Variables\n\nNamespace:\nMvPolynomial\n\nLocal context:\n/-\nCopyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Robert Y. Lewis\n-/\n/-!\n\n# Monad operations on `MvPolynomial`\n\nThis file defines two monadic operations on `MvPolynomial`. Given `p : MvPolynomial σ R`,\n\n* `MvPolynomial.bind₁` and `MvPolynomial.join₁` operate on the variable type `σ`.\n* `MvPolynomial.bind₂` and `MvPolynomial.join₂` operate on the coefficient type `R`.\n\n- `MvPolynomial.bind₁ f φ` with `f : σ → MvPolynomial τ R` and `φ : MvPolynomial σ R`,\n  is the polynomial `φ(f 1, ..., f i, ...) : MvPolynomial τ R`.\n- `MvPolynomial.join₁ φ` with `φ : MvPolynomial (MvPolynomial σ R) R` collapses `φ` to\n  a `MvPolynomial σ R`, by evaluating `φ` under the map `X f ↦ f` for `f : MvPolynomial σ R`.\n  In other words, if you have a polynomial `φ` in a set of variables indexed by a polynomial ring,\n  you evaluate the polynomial in these indexing polynomials.\n- `MvPolynomial.bind₂ f φ` with `f : R →+* MvPolynomial σ S` and `φ : MvPolynomial σ R`\n  is the `MvPolynomial σ S` obtained from `φ` by mapping the coefficients of `φ` through `f`\n  and considering the resulting polynomial as polynomial expression in `MvPolynomial σ R`.\n- `MvPolynomial.join₂ φ` with `φ : MvPolynomial σ (MvPolynomial σ R)` collapses `φ` to\n  a `MvPolynomial σ R`, by considering `φ` as polynomial expression in `MvPolynomial σ R`.\n\nThese operations themselves have algebraic structure: `MvPolynomial.bind₁`\nand `MvPolynomial.join₁` are algebra homs and\n`MvPolynomial.bind₂` and `MvPolynomial.join₂` are ring homs.\n\nThey interact in convenient ways with `MvPolynomial.rename`, `MvPolynomial.map`,\n`MvPolynomial.vars`, and other polynomial operations.\nIndeed, `MvPolynomial.rename` is the \"map\" operation for the (`bind₁`, `join₁`) pair,\nwhereas `MvPolynomial.map` is the \"map\" operation for the other pair.\n\n## Implementation notes\n\nWe add a `LawfulMonad` instance for the (`bind₁`, `join₁`) pair.\nThe second pair cannot be instantiated as a `Monad`,\nsince it is not a monad in `Type` but in `CommRingCat` (or rather `CommSemiRingCat`).\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nnamespace MvPolynomial\n\nopen Finsupp\n\nvariable {σ : Type*} {τ : Type*}\nvariable {R S T : Type*} [CommSemiring R] [CommSemiring S] [CommSemiring T]\n\n/--\n`bind₁` is the \"left-hand side\" bind operation on `MvPolynomial`, operating on the variable type.\nGiven a polynomial `p : MvPolynomial σ R` and a map `f : σ → MvPolynomial τ R` taking variables\nin `p` to polynomials in the variable type `τ`, `bind₁ f p` replaces each variable in `p` with\nits value under `f`, producing a new polynomial in `τ`. The coefficient type remains the same.\nThis operation is an algebra hom.\n-/\ndef bind₁ (f : σ → MvPolynomial τ R) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=\n  aeval f\n\n/-- `bind₂` is the \"right-hand side\" bind operation on `MvPolynomial`,\noperating on the coefficient type.\nGiven a polynomial `p : MvPolynomial σ R` and\na map `f : R → MvPolynomial σ S` taking coefficients in `p` to polynomials over a new ring `S`,\n`bind₂ f p` replaces each coefficient in `p` with its value under `f`,\nproducing a new polynomial over `S`.\nThe variable type remains the same. This operation is a ring hom.\n-/\ndef bind₂ (f : R →+* MvPolynomial σ S) : MvPolynomial σ R →+* MvPolynomial σ S :=\n  eval₂Hom f X\n\n/--\n`join₁` is the monadic join operation corresponding to `MvPolynomial.bind₁`. Given a polynomial `p`\nwith coefficients in `R` whose variables are polynomials in `σ` with coefficients in `R`,\n`join₁ p` collapses `p` to a polynomial with variables in `σ` and coefficients in `R`.\nThis operation is an algebra hom.\n-/\ndef join₁ : MvPolynomial (MvPolynomial σ R) R →ₐ[R] MvPolynomial σ R :=\n  aeval id\n\n/--\n`join₂` is the monadic join operation corresponding to `MvPolynomial.bind₂`. Given a polynomial `p`\nwith variables in `σ` whose coefficients are polynomials in `σ` with coefficients in `R`,\n`join₂ p` collapses `p` to a polynomial with variables in `σ` and coefficients in `R`.\nThis operation is a ring hom.\n-/\ndef join₂ : MvPolynomial σ (MvPolynomial σ R) →+* MvPolynomial σ R :=\n  eval₂Hom (RingHom.id _) X\n\n@[simp]\ntheorem aeval_eq_bind₁ (f : σ → MvPolynomial τ R) : aeval f = bind₁ f :=\n  rfl\n\n@[simp]\ntheorem eval₂Hom_C_eq_bind₁ (f : σ → MvPolynomial τ R) : eval₂Hom C f = bind₁ f :=\n  rfl\n\n@[simp]\ntheorem eval₂Hom_eq_bind₂ (f : R →+* MvPolynomial σ S) : eval₂Hom f X = bind₂ f :=\n  rfl\n\nsection\n\nvariable (σ R)\n\n@[simp]\ntheorem aeval_id_eq_join₁ : aeval id = @join₁ σ R _ :=\n  rfl\n\ntheorem eval₂Hom_C_id_eq_join₁ (φ : MvPolynomial (MvPolynomial σ R) R) :\n    eval₂Hom C id φ = join₁ φ :=\n  rfl\n\n@[simp]\ntheorem eval₂Hom_id_X_eq_join₂ : eval₂Hom (RingHom.id _) X = @join₂ σ R _ :=\n  rfl\n\nend\n\n-- In this file, we don't want to use these simp lemmas,\n-- because we first need to show how these new definitions interact\n-- and the proofs fall back on unfolding the definitions and call simp afterwards\nattribute [-simp]\n  aeval_eq_bind₁ eval₂Hom_C_eq_bind₁ eval₂Hom_eq_bind₂ aeval_id_eq_join₁ eval₂Hom_id_X_eq_join₂\n\n@[simp]\ntheorem bind₁_X_right (f : σ → MvPolynomial τ R) (i : σ) : bind₁ f (X i) = f i :=\n  aeval_X f i\n\n@[simp]\ntheorem bind₂_X_right (f : R →+* MvPolynomial σ S) (i : σ) : bind₂ f (X i) = X i :=\n  eval₂Hom_X' f X i\n\n@[simp]\ntheorem bind₁_X_left : bind₁ (X : σ → MvPolynomial σ R) = AlgHom.id R _ := by\n  ext1 i\n  simp\n\nvariable (f : σ → MvPolynomial τ R)\n\ntheorem bind₁_C_right (f : σ → MvPolynomial τ R) (x) : bind₁ f (C x) = C x := algHom_C _ _\n\n@[simp]\ntheorem bind₂_C_right (f : R →+* MvPolynomial σ S) (r : R) : bind₂ f (C r) = f r :=\n  eval₂Hom_C f X r\n\n@[simp]\ntheorem bind₂_C_left : bind₂ (C : R →+* MvPolynomial σ R) = RingHom.id _ := by ext : 2 <;> simp\n\n@[simp]\ntheorem bind₂_comp_C (f : R →+* MvPolynomial σ S) : (bind₂ f).comp C = f :=\n  RingHom.ext <| bind₂_C_right _\n\n@[simp]\ntheorem join₂_map (f : R →+* MvPolynomial σ S) (φ : MvPolynomial σ R) :\n    join₂ (map f φ) = bind₂ f φ := by simp only [join₂, bind₂, eval₂Hom_map_hom, RingHom.id_comp]\n\n@[simp]\ntheorem join₂_comp_map (f : R →+* MvPolynomial σ S) : join₂.comp (map f) = bind₂ f :=\n  RingHom.ext <| join₂_map _\n\ntheorem aeval_id_rename (f : σ → MvPolynomial τ R) (p : MvPolynomial σ R) :\n    aeval id (rename f p) = aeval f p := by rw [aeval_rename, Function.id_comp]\n\n@[simp]\ntheorem join₁_rename (f : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) :\n    join₁ (rename f φ) = bind₁ f φ :=\n  aeval_id_rename _ _\n\n@[simp]\ntheorem bind₁_id : bind₁ (@id (MvPolynomial σ R)) = join₁ :=\n  rfl\n\n@[simp]\ntheorem bind₂_id : bind₂ (RingHom.id (MvPolynomial σ R)) = join₂ :=\n  rfl\n\ntheorem bind₁_bind₁ {υ : Type*} (f : σ → MvPolynomial τ R) (g : τ → MvPolynomial υ R)\n    (φ : MvPolynomial σ R) : (bind₁ g) (bind₁ f φ) = bind₁ (fun i => bind₁ g (f i)) φ := by\n  simp [bind₁, ← comp_aeval]\n\ntheorem bind₁_comp_bind₁ {υ : Type*} (f : σ → MvPolynomial τ R) (g : τ → MvPolynomial υ R) :\n    (bind₁ g).comp (bind₁ f) = bind₁ fun i => bind₁ g (f i) := by\n  ext1\n  apply bind₁_bind₁\n\ntheorem bind₂_comp_bind₂ (f : R →+* MvPolynomial σ S) (g : S →+* MvPolynomial σ T) :\n    (bind₂ g).comp (bind₂ f) = bind₂ ((bind₂ g).comp f) := by ext : 2 <;> simp\n\ntheorem bind₂_bind₂ (f : R →+* MvPolynomial σ S) (g : S →+* MvPolynomial σ T)\n    (φ : MvPolynomial σ R) : (bind₂ g) (bind₂ f φ) = bind₂ ((bind₂ g).comp f) φ :=\n  RingHom.congr_fun (bind₂_comp_bind₂ f g) φ\n\ntheorem rename_comp_bind₁ {υ : Type*} (f : σ → MvPolynomial τ R) (g : τ → υ) :\n    (rename g).comp (bind₁ f) = bind₁ fun i => rename g <| f i := by\n  ext1 i\n  simp\n\ntheorem rename_bind₁ {υ : Type*} (f : σ → MvPolynomial τ R) (g : τ → υ) (φ : MvPolynomial σ R) :\n    rename g (bind₁ f φ) = bind₁ (fun i => rename g <| f i) φ :=\n  AlgHom.congr_fun (rename_comp_bind₁ f g) φ\n\ntheorem map_bind₂ (f : R →+* MvPolynomial σ S) (g : S →+* T) (φ : MvPolynomial σ R) :\n    map g (bind₂ f φ) = bind₂ ((map g).comp f) φ := by\n  simp only [bind₂, eval₂_comp_right, coe_eval₂Hom, eval₂_map]\n  congr 1 with : 1\n  simp only [Function.comp_apply, map_X]\n\ntheorem bind₁_comp_rename {υ : Type*} (f : τ → MvPolynomial υ R) (g : σ → τ) :\n    (bind₁ f).comp (rename g) = bind₁ (f ∘ g) := by\n  ext1 i\n  simp\n\ntheorem bind₁_rename {υ : Type*} (f : τ → MvPolynomial υ R) (g : σ → τ) (φ : MvPolynomial σ R) :\n    bind₁ f (rename g φ) = bind₁ (f ∘ g) φ :=\n  AlgHom.congr_fun (bind₁_comp_rename f g) φ\n\ntheorem bind₂_map (f : S →+* MvPolynomial σ T) (g : R →+* S) (φ : MvPolynomial σ R) :\n    bind₂ f (map g φ) = bind₂ (f.comp g) φ := by simp [bind₂]\n\n@[simp]\n\nTarget:\ntheorem map_comp_C (f : R →+* S) : (map f).comp (C : R →+* MvPolynomial σ R) = C.comp f :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/MvPolynomial","family_id":"map_comp_c","file_id":"mathlib/Mathlib/Algebra/MvPolynomial/Monad.lean","sample_id":"bd9359a574dcaca31acf15cd4ed4c49c4962b2ea1c2de798364f401e03500718"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dae1c5b0ed2fe337b0ef71d64dc3ef20222db1f81ac15117533aa09e94ca0f93","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1f8f5e1811f5aaefcb9cbe3919d7531d35046b70386cd414dba023e2c12f918a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f8bf5611d48800247cbe0df58105517cb729c0618719c6fcdd68a62dbe1a0152","source_sha256":"913b3712895896b9c1defc42d8c9355d3cd43a62b5cc81738e459895eb085302","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨u, u_cov, u_props⟩ := sc_st\n  exact\n    ⟨u,\n     s_sub.trans u_cov,\n     fun n ↦\n       ⟨(u_props n).1,\n        disjoint_of_subset (fun ⦃_⦄ a ↦ a) t_sub (u_props n).2⟩⟩","hard_negative":true,"metrics":{"chosen_tokens":50,"rejected_tokens":5,"token_jaccard":0.103448,"token_length_ratio":0.1},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"9f0bf0cf60f05c9f74d2a6e900a1cd4125d598ef8d66ef3c79cdb9d2d3e3f032","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Continuous\npublic import Mathlib.Topology.NhdsSet\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro\n-/\n/-!\n# Separated neighbourhoods\n\nThis file defines the predicates `SeparatedNhds` and `HasSeparatingCover`, which are used in\nformulating separation axioms for topological spaces.\n\n## Main definitions\n\n* `SeparatedNhds`: Two `Set`s are separated by neighbourhoods if they are contained in disjoint\n  open sets.\n* `HasSeparatingCover`: A set has a countable cover that can be used with\n  `hasSeparatingCovers_iff_separatedNhds` to witness when two `Set`s have `SeparatedNhds`.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Separation_axiom>\n* [Willard's *General Topology*][zbMATH02107988]\n-/\n\n@[expose] public section\n\nopen Function Set Filter Topology TopologicalSpace\n\nuniverse u v\n\nvariable {X : Type*} {Y : Type*} [TopologicalSpace X]\n\nsection Separation\n\n/--\n`SeparatedNhds` is a predicate on pairs of sub`Set`s of a topological space.  It holds if the two\nsub`Set`s are contained in disjoint open sets.\n-/\ndef SeparatedNhds : Set X → Set X → Prop := fun s t : Set X =>\n  ∃ U V : Set X, IsOpen U ∧ IsOpen V ∧ s ⊆ U ∧ t ⊆ V ∧ Disjoint U V\n\ntheorem separatedNhds_iff_disjoint {s t : Set X} : SeparatedNhds s t ↔ Disjoint (𝓝ˢ s) (𝓝ˢ t) := by\n  simp only [(hasBasis_nhdsSet s).disjoint_iff (hasBasis_nhdsSet t), SeparatedNhds, ←\n    exists_and_left, and_assoc, and_comm, and_left_comm]\n\nalias ⟨SeparatedNhds.disjoint_nhdsSet, _⟩ := separatedNhds_iff_disjoint\n\n/-- `HasSeparatingCover`s can be useful witnesses for `SeparatedNhds`. -/\ndef HasSeparatingCover : Set X → Set X → Prop := fun s t ↦\n  ∃ u : ℕ → Set X, s ⊆ ⋃ n, u n ∧ ∀ n, IsOpen (u n) ∧ Disjoint (closure (u n)) t\n\n/-- Used to prove that a regular topological space with Lindelöf topology is a normal space,\nand a perfectly normal space is a completely normal space. -/\ntheorem hasSeparatingCovers_iff_separatedNhds {s t : Set X} :\n    HasSeparatingCover s t ∧ HasSeparatingCover t s ↔ SeparatedNhds s t := by\n  constructor\n  · rintro ⟨⟨u, u_cov, u_props⟩, ⟨v, v_cov, v_props⟩⟩\n    have open_lemma : ∀ (u₀ a : ℕ → Set X), (∀ n, IsOpen (u₀ n)) →\n      IsOpen (⋃ n, u₀ n \\ closure (a n)) := fun _ _ u₀i_open ↦\n        isOpen_iUnion fun i ↦ (u₀i_open i).sdiff isClosed_closure\n    have cover_lemma : ∀ (h₀ : Set X) (u₀ v₀ : ℕ → Set X),\n        (h₀ ⊆ ⋃ n, u₀ n) → (∀ n, Disjoint (closure (v₀ n)) h₀) →\n        (h₀ ⊆ ⋃ n, u₀ n \\ closure (⋃ m ≤ n, v₀ m)) :=\n        fun h₀ u₀ v₀ h₀_cov dis x xinh ↦ by\n      rcases h₀_cov xinh with ⟨un, ⟨n, rfl⟩, xinun⟩\n      simp only [mem_iUnion]\n      refine ⟨n, xinun, ?_⟩\n      simp_all only [closure_iUnion₂_le_nat, disjoint_right, mem_iUnion,\n        exists_false, not_false_eq_true]\n    refine\n      ⟨⋃ n : ℕ, u n \\ (closure (⋃ m ≤ n, v m)),\n       ⋃ n : ℕ, v n \\ (closure (⋃ m ≤ n, u m)),\n       open_lemma u (fun n ↦ ⋃ m ≤ n, v m) (fun n ↦ (u_props n).1),\n       open_lemma v (fun n ↦ ⋃ m ≤ n, u m) (fun n ↦ (v_props n).1),\n       cover_lemma s u v u_cov (fun n ↦ (v_props n).2),\n       cover_lemma t v u v_cov (fun n ↦ (u_props n).2),\n       ?_⟩\n    rw [Set.disjoint_left]\n    rintro x ⟨un, ⟨n, rfl⟩, xinun⟩\n    suffices ∀ (m : ℕ), x ∈ v m → x ∈ closure (⋃ m' ∈ {m' | m' ≤ m}, u m') by simpa\n    intro m xinvm\n    have n_le_m : n ≤ m := by\n      by_contra m_gt_n\n      exact xinun.2 (subset_closure (mem_biUnion (le_of_lt (not_le.mp m_gt_n)) xinvm))\n    exact subset_closure (mem_biUnion n_le_m xinun.1)\n  · rintro ⟨U, V, U_open, V_open, h_sub_U, k_sub_V, UV_dis⟩\n    exact\n      ⟨⟨fun _ ↦ U,\n        h_sub_U.trans (iUnion_const U).symm.subset,\n        fun _ ↦\n          ⟨U_open, disjoint_of_subset (fun ⦃a⦄ a ↦ a) k_sub_V (UV_dis.closure_left V_open)⟩⟩,\n       ⟨fun _ ↦ V,\n        k_sub_V.trans (iUnion_const V).symm.subset,\n        fun _ ↦\n          ⟨V_open, disjoint_of_subset (fun ⦃a⦄ a ↦ a) h_sub_U (UV_dis.closure_right U_open).symm⟩⟩⟩\n\ntheorem Set.hasSeparatingCover_empty_left (s : Set X) : HasSeparatingCover ∅ s :=\n  ⟨fun _ ↦ ∅, empty_subset (⋃ _, ∅),\n   fun _ ↦ ⟨isOpen_empty, by simp only [closure_empty, empty_disjoint]⟩⟩\n\ntheorem Set.hasSeparatingCover_empty_right (s : Set X) : HasSeparatingCover s ∅ :=\n  ⟨fun _ ↦ univ, (subset_univ s).trans univ.iUnion_const.symm.subset,\n   fun _ ↦ ⟨isOpen_univ, by apply disjoint_empty⟩⟩\n\nTarget:\ntheorem HasSeparatingCover.mono {s₁ s₂ t₁ t₂ : Set X} (sc_st : HasSeparatingCover s₂ t₂)\n    (s_sub : s₁ ⊆ s₂) (t_sub : t₁ ⊆ t₂) : HasSeparatingCover s₁ t₁ :=\n\nProof body:\n","rejected":"by\n  exact HasSeparatingCover.mono","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"ac9c6e1008ff2b0129ae0cfabce8d936833e1cbb736dda7e778c65da8daa1a91","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Separation","family_id":"hasseparatingcover","file_id":"mathlib/Mathlib/Topology/Separation/SeparatedNhds.lean","sample_id":"f8bf5611d48800247cbe0df58105517cb729c0618719c6fcdd68a62dbe1a0152"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3628ed809b6ca5e99b1c81051989cfa58d947fe9238f4a04564fff2b92ccbda9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9fd7dac785c481bc273cd04cdc4c36718863148334346a95619a47d1bb75e484","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0b655641f0fbcc76ef7401981695bc0221f56804bae473d2205adf3e718b621a","source_sha256":"b36923632397960a715546258f21a9493171dd1ee07ef68d71942c93bbc53c1e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨W, hW, hWl, hWr, h⟩ := h\n  have e₁ := isDominant_hom_iff_isDominant_restrict_hom f W hW hWl\n  have e₂ := isDominant_hom_iff_isDominant_restrict_hom g W hW hWr\n  dsimp only [restrict_domain, restrict_hom] at ⊢ e₁ e₂ h\n  rw [e₁, h, ← e₂]","hard_negative":false,"metrics":{"chosen_tokens":58,"rejected_tokens":63,"token_jaccard":0.848485,"token_length_ratio":1.086207},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"9f215292416da8302890ca8acb37a62ec213ead2ba931f96fe1f5c01d4c962b7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Birational.RationalMap\n\nNamespace:\nAlgebraicGeometry.Scheme.PartialMap\n\nLocal context:\n/-\nCopyright (c) 2026 Justus Springer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Justus Springer\n-/\n/-!\n\n# Dominant rational maps\n\nThis file defines `RationalMap.IsDominant` and establishes its connection to\n`IsDominant` on the underlying partial maps.\n\n## Main definition\n\n- `Scheme.RationalMap.IsDominant`: a rational map is dominant if some (equivalently, any)\n  representative partial map has dominant underlying morphism.\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen CategoryTheory\n\nnamespace AlgebraicGeometry\n\nvariable {X Y : Scheme.{u}}\n\nnamespace Scheme\n\nnamespace PartialMap\n\nset_option backward.defeqAttrib.useBackward true in\n/-- Restricting a dominant partial map to a dense open yields a dominant partial map. -/\nlemma isDominant_restrict_hom (f : X.PartialMap Y) [IsDominant f.hom] (U : X.Opens)\n    (hU : Dense (U : Set X)) (hU' : U ≤ f.domain) : IsDominant (f.restrict U hU hU').hom := by\n  dsimp only [restrict_domain, restrict_hom]\n  have : IsDominant (X.homOfLE hU') := Opens.isDominant_homOfLE hU hU'\n  rwa [IsDominant.comp_iff]\n\n/-- If a restriction of `f` is dominant, then `f` is dominant. -/\nlemma isDominant_hom_of_isDominant_restrict_hom (f : X.PartialMap Y) (U : X.Opens)\n    (hU : Dense (U : Set X)) (hU' : U ≤ f.domain) [H : IsDominant (f.restrict U hU hU').hom] :\n    IsDominant f.hom :=\n  IsDominant.of_comp (X.homOfLE hU') f.hom (H := H)\n\n/-- `f.hom` is dominant iff any restriction of `f` is. -/\nlemma isDominant_hom_iff_isDominant_restrict_hom (f : X.PartialMap Y) (U : X.Opens)\n    (hU : Dense (U : Set X)) (hU' : U ≤ f.domain) :\n    IsDominant f.hom ↔ IsDominant (f.restrict U hU hU').hom :=\n  ⟨fun _ ↦ f.isDominant_restrict_hom U hU hU',\n    fun _ ↦ f.isDominant_hom_of_isDominant_restrict_hom U hU hU'⟩\n\nset_option backward.defeqAttrib.useBackward true in\n/-- Dominance of the underlying morphism is invariant under equivalence of partial maps. -/\n\nTarget:\nlemma isDominant_hom_iff_of_equiv (f g : X.PartialMap Y) (h : f.equiv g) :\n    IsDominant f.hom ↔ IsDominant g.hom :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  obtain ⟨W, hW, hWl, hWr, h⟩ := h\n  have e₁ := isDominant_hom_iff_isDominant_restrict_hom f W hW hWl\n  have e₂ := isDominant_hom_iff_isDominant_restrict_hom g W hW hWr\n  dsimp only [restrict_domain, restrict_hom] at ⊢ e₁ e₂ h\n  rw [e₁, h, ← e₂]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Birational","family_id":"isdominant_hom_iff_of_equiv","file_id":"mathlib/Mathlib/AlgebraicGeometry/Birational/Dominant.lean","sample_id":"0b655641f0fbcc76ef7401981695bc0221f56804bae473d2205adf3e718b621a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4c0ce5d128f69a2b99881bee610f584c1a319bb4f885c9faba74ae83efc61828","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c8cd16ca7c7e5a007fb195fa59aed53c5734a477ccb7bfa5ed293e06fc1f996c","source_sha256":"d9cef763bc837b8cf34f04f4dde67c215bb2f264d0a2452e9434fbd1d721514c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    have := hf (equalizer.lift f h) (equalizer.ι p q) (by simp)\n    rw [← cancel_epi (equalizer.ι p q), equalizer.condition]\n  isIso := by tauto","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":8,"token_jaccard":0.107143,"token_length_ratio":0.190476},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"9f574bf755d6e38061b37b87010e639fa37a64a0ad4f390f7daa4ffd4534305f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Subobject.Lattice\npublic import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Extremal epimorphisms\n\nAn extremal epimorphism `p : X ⟶ Y` is an epimorphism which does not factor\nthrough any proper subobject of `Y`. In case the category has equalizers,\nwe show that a morphism `p : X ⟶ Y` which does not factor through\nany proper subobject of `Y` is automatically an epimorphism, and also\nan extremal epimorphism. We also show that a strong epimorphism\nis an extremal epimorphism, and that both notions coincide when\nthe category has pullbacks.\n\n## References\n\n* https://ncatlab.org/nlab/show/extremal+epimorphism\n\n-/\n\npublic section\n\nuniverse v u\n\nnamespace CategoryTheory\n\nopen Limits\n\nvariable {C : Type u} [Category.{v} C] {X Y : C}\n\n/-- An extremal epimorphism `f : X ⟶ Y` is an epimorphism which does not\nfactor through any proper subobject of `Y`. -/\nclass ExtremalEpi (f : X ⟶ Y) : Prop extends Epi f where\n  isIso (f) {Z : C} (p : X ⟶ Z) (i : Z ⟶ Y) (fac : p ≫ i = f) [Mono i] : IsIso i\n\nvariable (f : X ⟶ Y)\n\nlemma ExtremalEpi.subobject_eq_top [ExtremalEpi f]\n    {A : Subobject Y} (hA : Subobject.Factors A f) : A = ⊤ := by\n  rw [← Subobject.isIso_arrow_iff_eq_top]\n  exact isIso f (Subobject.factorThru A f hA) _ (by simp)\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma ExtremalEpi.mk_of_hasEqualizers [HasEqualizers C]\n    (hf : ∀ ⦃Z : C⦄ (p : X ⟶ Z) (i : Z ⟶ Y) (_ : p ≫ i = f) [Mono i], IsIso i) :\n    ExtremalEpi f where\n  left_cancellation {Z} p q h :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"c48e85f3a0dd1885e32e041cdf5815ff823cca7a66bae88376559c0383284a5b","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory","family_id":"extremalepi","file_id":"mathlib/Mathlib/CategoryTheory/ExtremalEpi.lean","sample_id":"c8cd16ca7c7e5a007fb195fa59aed53c5734a477ccb7bfa5ed293e06fc1f996c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4a140838d46443a223247967ef703140b4be6ea2808c8d4f93f75968349bd809","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8490e25c33757674cf12d7337a393540598194fbe44274dc1fe644fbfc191efe","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b579425a25a44dbf03d7aec5fcfc9f633b6db243ba8c3f42fbd8c2796e7b54f2","source_sha256":"38b81cd46858f243db931b63ae095bb873edc08d47b08e77c1c254a8ae6a5653","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := h.out\n  simp [this]\n\n/- evalCast -/","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.2},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"9f8ef36fc6f1a9b3e76c8106fd82701a6fbc31c08206d24aa3e8542d43bbafb5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Defs\npublic import Mathlib.Algebra.Algebra.Basic\npublic import Mathlib.Tactic.Ring.RingNF\n\nNamespace:\nMathlib.Tactic.Algebra\n\nLocal context:\n/-\nCopyright (c) 2025 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Arend Mellendijk\n-/\n/-! # Lemmas for the `algebra` tactic.\n-/\n\n@[expose] public section\n\nopen Mathlib.Meta.NormNum\n\nnamespace Mathlib.Tactic.Algebra\n\nsection ring\n\nvariable {R A : Type*} [CommRing R] [CommRing A] [Algebra R A]\n\n/- evalCast -/\ntheorem isInt_negOfNat_eq {a : A} {lit : ℕ} (h : IsInt a (Int.negOfNat lit)) :\n    a = algebraMap R A (Int.rawCast (Int.negOfNat lit) + 0 : R) + 0 := by\n  simp [h.out]\n\nend ring\n\nsection semifield\n\nvariable {R A : Type*} [Semifield R] [Semifield A] [Algebra R A]\n\n/- evalCast -/\ntheorem isNNRat_eq_rawCast {a : A} {n d : ℕ} (h : IsNNRat a n d) :\n    a = algebraMap R A (NNRat.rawCast n d + 0 : R) + 0 := by\n  simp [Mathlib.Tactic.Ring.cast_nnrat h]\n\nend semifield\n\nsection field\n\nvariable {R A : Type*} [Field R] [Field A] [Algebra R A]\n\n/- evalCast -/\ntheorem isRat_eq_rawCast {a : A} {n d : ℕ} (h : IsRat a (.negOfNat n) d) :\n    a = algebraMap R A (Rat.rawCast (.negOfNat n) d + 0 : R) + 0 := by\n  simp [Mathlib.Tactic.Ring.cast_rat h]\n\nend field\n\nvariable {R A : Type*} [sR : CommSemiring R] [sA : CommSemiring A] [sAlg : Algebra R A]\n\n/- evalCast -/\n\nTarget:\ntheorem isNat_zero_eq {a : A} (h : IsNat a 0) : a = 0 :=\n\nProof body:\n","rejected":"by\n  exact isNat_zero_eq","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"707b7c8a83869e6b8cd1e78df8a7605e31b2246bee96abc87d99aa16187fd170","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Algebra","family_id":"isnat_zero_eq","file_id":"mathlib/Mathlib/Tactic/Algebra/Lemmas.lean","sample_id":"b579425a25a44dbf03d7aec5fcfc9f633b6db243ba8c3f42fbd8c2796e7b54f2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"252205b7b358cac3e7fbb348dc9a6e22e848772b9ebec8a5f89dd1c88ca1cccb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6654eaee1efe2f7a4c8a60caecda062301f55e6f4592e02b3c536de51707ac17","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"932a2516a9f3261c5acb0efee2adbd6bd7291142ae5d6a59e51bba4d80664223","source_sha256":"99fe4affc634c570de70587e929805d16958622f49de95b17da87beb4716efa3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction t with\n  | nil => rw [traverse, map]\n  | node v l r hl hr =>\n    rw [traverse, map, hl, hr, Function.comp_apply, map_pure, pure_seq, map_pure, pure_seq,\n      map_pure]","hard_negative":true,"metrics":{"chosen_tokens":45,"rejected_tokens":5,"token_jaccard":0.074074,"token_length_ratio":0.111111},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"a019863c0476ef5f5db56248c4c9c6dc75a2a33fed0bc89af19fde77466f6454","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Tree.Basic\npublic import Mathlib.Control.Applicative\npublic import Mathlib.Control.Traversable.Basic\n\nNamespace:\nBinaryTree\n\nLocal context:\n/-\nCopyright (c) 2025 Edward van de Meent. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Edward van de Meent\n-/\n/-!\n# Traversable Binary Tree\n\nProvides a `Traversable` instance for the `Tree` type.\n-/\n\npublic section\n\nuniverse u v w\n\nnamespace BinaryTree\nsection Traverse\nvariable {α β : Type*}\n\ninstance : Traversable BinaryTree where\n  map := map\n  traverse := traverse\n\nlemma comp_traverse\n    {F : Type u → Type v} {G : Type v → Type w} [Applicative F] [Applicative G]\n    [LawfulApplicative G] {β : Type v} {γ : Type u} (f : β → F γ) (g : α → G β)\n    (t : BinaryTree α) : t.traverse (Functor.Comp.mk ∘ (f <$> ·) ∘ g) =\n      Functor.Comp.mk ((·.traverse f) <$> (t.traverse g)) := by\n  induction t with\n  | nil => rw [traverse, traverse, map_pure, traverse]; rfl\n  | node v l r hl hr =>\n    rw [traverse, hl, hr, traverse]\n    simp only [Function.comp_def, Function.comp_apply, Functor.Comp.map_mk, Functor.map_map,\n      Comp.seq_mk, seq_map_assoc, map_seq]\n    rfl\n\nTarget:\nlemma traverse_eq_map_id (f : α → β) (t : BinaryTree α) :\n    t.traverse ((pure : β → Id β) ∘ f) = pure (t.map f) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_932a2516a9f3","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"6f6643d507164b618da6939ed9b473049553adc6abcf0577ddcbfafbfd3bf48c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Tree","family_id":"traverse_eq_map_id","file_id":"mathlib/Mathlib/Data/Tree/Traversable.lean","sample_id":"932a2516a9f3261c5acb0efee2adbd6bd7291142ae5d6a59e51bba4d80664223"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f1c76c9c21cec19c82ec1c29aff36596a09a140cc79532bb60d64518174b61b0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2f46fd8127f7a4707cce4dc2e25df1a6780c736d06d1699ff4e93a369a16eb74","source_sha256":"a6234c28d344c069679c6c490f57e0cad723a087d9f05f1fbee4f95afdc863a6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  convert! LinearMap.iSupIndep_map _ hv (iSupIndep_range_lsingle ι R R)\n  ext; simp [mem_span_singleton]","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":3,"token_jaccard":0.047619,"token_length_ratio":0.15},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"a14e2bf7cbbdc701323bb0ea40b270c73feea423df4f617c9e24a091b21b60b2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.Tuple.Reflection\npublic import Mathlib.LinearAlgebra.Dual.Defs\npublic import Mathlib.LinearAlgebra.Finsupp.SumProd\npublic import Mathlib.LinearAlgebra.LinearIndependent.Basic\npublic import Mathlib.LinearAlgebra.Pi\npublic import Mathlib.Logic.Equiv.Fin.Rotate\npublic import Mathlib.Tactic.FinCases\npublic import Mathlib.Tactic.Module\npublic import Mathlib.Tactic.Abel\npublic import Mathlib.Tactic.NormNum.Ineq\nimport Mathlib.Algebra.Module.Torsion.Field\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen\n-/\n/-!\n# Linear independence\n\nThis file collects consequences of linear (in)dependence and includes specialized tests for\nspecific families of vectors, requiring more theory to state.\n\n## Main statements\n\nWe prove several specialized tests for linear independence of families of vectors and of sets of\nvectors.\n\n* `linearIndependent_option`, `linearIndependent_finCons`,\n  `linearIndependent_finSucc`, `linearIndependent_finSnoc`: type-specific tests for linear\n  independence of families of vector fields;\n* `linearIndependent_insert`, `linearIndependent_pair`: linear independence tests for set operations\n\nIn many cases we additionally provide dot-style operations (e.g., `LinearIndependent.union`) to\nmake the linear independence tests usable as `hv.insert ha` etc.\n\nWe also prove that, when working over a division ring,\nany family of vectors includes a linear independent subfamily spanning the same subspace.\n\n## TODO\n\nRework proofs to hold in semirings, by avoiding the path through\n`ker (Finsupp.linearCombination R v) = ⊥`.\n\n## Tags\n\nlinearly dependent, linear dependence, linearly independent, linear independence\n\n-/\n\n@[expose] public section\n\n\nassert_not_exists Cardinal\n\nnoncomputable section\n\nopen Function Module Set Submodule\n\nuniverse u' u\n\nvariable {ι : Type u'} {ι' : Type*} {R : Type*} {K : Type*} {s : Set ι}\nvariable {M : Type*} {M' : Type*} {V : Type u}\n\nsection Semiring\n\n\nvariable {v : ι → M}\nvariable [Semiring R] [AddCommMonoid M] [AddCommMonoid M']\nvariable [Module R M] [Module R M']\nvariable (R) (v)\n\nvariable {R v}\n\n/-- A finite family of vectors `v i` is linear independent iff the linear map that sends\n`c : ι → R` to `∑ i, c i • v i` is injective. -/\ntheorem Fintype.linearIndependent_iff'ₛ [Fintype ι] [DecidableEq ι] :\n    LinearIndependent R v ↔\n      Injective (LinearMap.lsum R (fun _ ↦ R) ℕ fun i ↦ LinearMap.id.smulRight (v i)) := by\n  simp [Fintype.linearIndependent_iffₛ, Injective, funext_iff]\n\nlemma LinearIndependent.pair_iffₛ {x y : M} :\n    LinearIndependent R ![x, y] ↔\n      ∀ (s t s' t' : R), s • x + t • y = s' • x + t' • y → s = s' ∧ t = t' := by\n  simp [Fintype.linearIndependent_iffₛ, Fin.forall_fin_two, ← FinVec.forall_iff]; rfl\n\nlemma LinearIndependent.eq_of_pair {x y : M} (h : LinearIndependent R ![x, y])\n    {s t s' t' : R} (h' : s • x + t • y = s' • x + t' • y) : s = s' ∧ t = t' :=\n  pair_iffₛ.mp h _ _ _ _ h'\n\nlemma LinearIndependent.eq_zero_of_pair' {x y : M} (h : LinearIndependent R ![x, y])\n    {s t : R} (h' : s • x = t • y) : s = 0 ∧ t = 0 := by\n  suffices H : s = 0 ∧ 0 = t from ⟨H.1, H.2.symm⟩\n  exact h.eq_of_pair (by simpa using h')\n\nlemma LinearIndependent.eq_zero_of_pair {x y : M} (h : LinearIndependent R ![x, y])\n    {s t : R} (h' : s • x + t • y = 0) : s = 0 ∧ t = 0 := by\n  replace h := @h (.single 0 s + .single 1 t) 0 ?_\n  · exact ⟨by simpa using congr($h 0), by simpa using congr($h 1)⟩\n  simpa\n\nsection Indexed\n\ntheorem linearIndepOn_iUnion_of_directed {η : Type*} {s : η → Set ι} (hs : Directed (· ⊆ ·) s)\n    (h : ∀ i, LinearIndepOn R v (s i)) : LinearIndepOn R v (⋃ i, s i) := by\n  by_cases hη : Nonempty η\n  · refine linearIndepOn_of_finite (⋃ i, s i) fun t ht ft => ?_\n    rcases finite_subset_iUnion ft ht with ⟨I, fi, hI⟩\n    rcases hs.finset_le fi.toFinset with ⟨i, hi⟩\n    exact (h i).mono (Subset.trans hI <| iUnion₂_subset fun j hj => hi j (fi.mem_toFinset.2 hj))\n  · refine (linearIndepOn_empty R v).mono (t := iUnion (s ·)) ?_\n    rintro _ ⟨_, ⟨i, _⟩, _⟩\n    exact hη ⟨i⟩\n\ntheorem linearIndepOn_sUnion_of_directed {s : Set (Set ι)} (hs : DirectedOn (· ⊆ ·) s)\n    (h : ∀ a ∈ s, LinearIndepOn R v a) : LinearIndepOn R v (⋃₀ s) := by\n  rw [sUnion_eq_iUnion]\n  exact linearIndepOn_iUnion_of_directed hs.directed_val (by simpa using h)\n\ntheorem linearIndepOn_biUnion_of_directed {η} {s : Set η} {t : η → Set ι}\n    (hs : DirectedOn (t ⁻¹'o (· ⊆ ·)) s) (h : ∀ a ∈ s, LinearIndepOn R v (t a)) :\n    LinearIndepOn R v (⋃ a ∈ s, t a) := by\n  rw [biUnion_eq_iUnion]\n  exact linearIndepOn_iUnion_of_directed (directed_comp.2 <| hs.directed_val) (by simpa using h)\n\nend Indexed\n\nsection repr\n\nvariable (ι R M) in\ntheorem iSupIndep_range_lsingle :\n    iSupIndep fun i : ι ↦ LinearMap.range (Finsupp.lsingle (R := R) (M := M) i) := by\n  refine fun i ↦ disjoint_iff_inf_le.mpr ?_\n  rintro x ⟨⟨m, rfl⟩, hm⟩\n  suffices ⨆ j ≠ i, LinearMap.range (Finsupp.lsingle j) ≤ Finsupp.supported M R {i}ᶜ by\n    have := (Finsupp.mem_supported ..).mp (this hm); simp_all\n  refine iSup₂_le fun j ne ↦ ?_\n  rintro _ ⟨m, rfl⟩\n  simp [Finsupp.mem_supported, ne]\n\ntheorem LinearMap.iSupIndep_map (f : M →ₗ[R] M') (inj : Injective f) {m : ι → Submodule R M}\n    (ind : iSupIndep m) : iSupIndep fun i ↦ (m i).map f := by\n  simp_rw [iSupIndep, disjoint_iff_inf_le] at ind ⊢\n  rintro i _ ⟨⟨x, hxi, rfl⟩, hx⟩\n  rw [ind i ⟨hxi, _⟩]; · simp\n  simp_rw [← Submodule.map_iSup] at hx\n  have ⟨y, hy, eq⟩ := hx\n  simpa [← inj eq]\n\nvariable (hv : LinearIndependent R v)\n\n/-- See also `iSupIndep_iff_linearIndependent_of_ne_zero`. -/\n\nTarget:\ntheorem LinearIndependent.iSupIndep_span_singleton (hv : LinearIndependent R v) :\n    iSupIndep fun i => R ∙ v i :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/LinearIndependent","family_id":"linearindependent","file_id":"mathlib/Mathlib/LinearAlgebra/LinearIndependent/Lemmas.lean","sample_id":"2f46fd8127f7a4707cce4dc2e25df1a6780c736d06d1699ff4e93a369a16eb74"},"split":"train"}
{"binding":{"chosen_proof_sha256":"871db309f2a7e188f5170224e4eada7ffe3aa87ef44b3d70e3a18c5c524571b5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a04ab4cdf2e6dd7b879e0c3709f89419da5004b0ffae6a33a5dbda530d452046","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d3baf770853c9de1742f6952ed6591411750a3d98ef5c34871eada480093b869","source_sha256":"138c7bba1e136375811c6bc0d5e62f29dc75b67b8e14507dcd9ecb3045cecbfd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine le_antisymm (map_continuous <| toWeakSpaceCLM 𝕜 E).continuousOn.image_closure\n    (Set.compl_subset_compl.mp fun x hx ↦ ?_)\n  obtain ⟨x, -, rfl⟩ := (toWeakSpace 𝕜 E).toEquiv.image_compl (closure s) |>.symm.subset hx\n  have : ContinuousSMul ℝ E := IsScalarTower.continuousSMul 𝕜\n  obtain ⟨f, u, hus, hux⟩ := RCLike.geometric_hahn_banach_closed_point (𝕜 := 𝕜)\n    hs.closure isClosed_closure (by simpa using hx)\n  let f' : StrongDual 𝕜 (WeakSpace 𝕜 E) :=\n    { toLinearMap := (f : E →ₗ[𝕜] 𝕜).comp ((toWeakSpace 𝕜 E).symm : WeakSpace 𝕜 E →ₗ[𝕜] E)\n      cont := WeakBilin.eval_continuous (topDualPairing 𝕜 E).flip _ }\n  have hux' : u < RCLike.reCLM.comp (f'.restrictScalars ℝ) (toWeakSpace 𝕜 E x) := by simpa [f']\n  have hus' : closure (toWeakSpace 𝕜 E '' s) ⊆\n      {y | RCLike.reCLM.comp (f'.restrictScalars ℝ) y ≤ u} := by\n    refine closure_minimal ?_ <| isClosed_le (by fun_prop) (by fun_prop)\n    rintro - ⟨y, hy, rfl⟩\n    simpa [f'] using (hus y <| subset_closure hy).le\n  exact (hux'.not_ge <| hus' ·)","hard_negative":true,"metrics":{"chosen_tokens":266,"rejected_tokens":3,"token_jaccard":0.022222,"token_length_ratio":0.011278},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"a154114063fd163044a9c149f00709d301e20f7afab29154863c14412d91aaf1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.LocallyConvex.Separation\npublic import Mathlib.Analysis.LocallyConvex.SeparatingDual\npublic import Mathlib.LinearAlgebra.Dual.Defs\npublic import Mathlib.Topology.Algebra.Module.Spaces.WeakDual\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\n/-! # Closures of convex sets in locally convex spaces\n\nThis file contains the standard result that if `E` is a vector space with two locally convex\ntopologies, then the closure of a convex set is the same in either topology, provided they have the\nsame collection of continuous linear functionals. In particular, the weak closure of a convex set\nin a locally convex space coincides with the closure in the original topology.\nOf course, we phrase this in terms of linear maps between locally convex spaces, rather than\ncreating two separate topologies on the same space.\n-/\n\npublic section\n\nvariable {𝕜 E F : Type*}\nvariable [RCLike 𝕜] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜 F]\nvariable [Module ℝ E] [IsScalarTower ℝ 𝕜 E] [Module ℝ F] [IsScalarTower ℝ 𝕜 F]\nvariable [TopologicalSpace E] [IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E]\n  [LocallyConvexSpace ℝ E]\nvariable [TopologicalSpace F] [IsTopologicalAddGroup F] [ContinuousSMul 𝕜 F]\n  [LocallyConvexSpace ℝ F]\n\nvariable (𝕜) in\n/-- If `E` is a locally convex space over `𝕜` (with `RCLike 𝕜`), and `s : Set E` is `ℝ`-convex, then\nthe closure of `s` and the weak closure of `s` coincide. More precisely, the topological closure\ncommutes with `toWeakSpace 𝕜 E`.\n\nThis holds more generally for any linear equivalence `e : E ≃ₗ[𝕜] F` between locally convex spaces\nsuch that precomposition with `e` and `e.symm` preserves continuity of linear functionals. See\n`LinearEquiv.image_closure_of_convex`. -/\n\nTarget:\ntheorem Convex.toWeakSpace_closure {s : Set E} (hs : Convex ℝ s) :\n    (toWeakSpace 𝕜 E) '' (closure s) = closure (toWeakSpace 𝕜 E '' s) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_d3baf770853c","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"89ba6f2bcfc1b178d9d73811da45d89a3dd56b21b3beea3228eb1bd32e984d45","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/LocallyConvex","family_id":"convex","file_id":"mathlib/Mathlib/Analysis/LocallyConvex/WeakSpace.lean","sample_id":"d3baf770853c9de1742f6952ed6591411750a3d98ef5c34871eada480093b869"},"split":"train"}
{"binding":{"chosen_proof_sha256":"843286c15dd0e5c3687da59e28ddbf32542077397522d384d533e4f852e034a9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2c90f2e32be3da6fb260df3bb94bf0c15d7de35d2f1513e19789e1136caf26c6","source_sha256":"fe6f78e60ae83f5b6cc4fe056a4280594015e555feafeac8a4ec205bff00b8f8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [map_comp]\n  exact hg.trans <| map_mono _ hf","hard_negative":false,"metrics":{"chosen_tokens":14,"rejected_tokens":3,"token_jaccard":0.133333,"token_length_ratio":0.214286},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"a1b4e9eb194cfcda8927f2db8dbd9e46ef20345db6b77b3d60529558c528fdd1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal\npublic import Mathlib.RingTheory.GradedAlgebra.RingHom\n\nNamespace:\nHomogeneousIdeal\n\nLocal context:\n/-\nCopyright (c) 2025 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n/-!\n# Maps on homogeneous ideals\n\nIn this file we define `HomogeneousIdeal.map` and `HomogeneousIdeal.comap`.\n-/\n\n@[expose] public section\n\nnamespace HomogeneousIdeal\n\nsection arbitrary_grading\n\nvariable {A B C σ τ ω ι F G : Type*}\n  [Semiring A] [Semiring B] [Semiring C]\n  [SetLike σ A] [SetLike τ B] [SetLike ω C]\n  [AddSubmonoidClass σ A] [AddSubmonoidClass τ B] [AddSubmonoidClass ω C]\n  [DecidableEq ι] [AddMonoid ι]\n  {𝒜 : ι → σ} {ℬ : ι → τ} {𝒞 : ι → ω}\n  [GradedRing 𝒜] [GradedRing ℬ] [GradedRing 𝒞]\n  (f : 𝒜 →+*ᵍ ℬ) (g : ℬ →+*ᵍ 𝒞)\n\n/-- Map a homogeneous ideal along a graded ring homomorphism. The underlying ideal is\n(definitionally) equal to `Ideal.map`. -/\ndef map (I : HomogeneousIdeal 𝒜) : HomogeneousIdeal ℬ where\n  __ := I.toIdeal.map f\n  is_homogeneous' i b hb := by\n    rw [Ideal.map] at hb\n    induction hb using Submodule.span_induction generalizing i with\n    | zero => simp\n    | add => simp [*, Ideal.add_mem]\n    | mem a ha =>\n      obtain ⟨a, ha, rfl⟩ := ha\n      rw [← f.map_directSumDecompose]\n      exact Ideal.mem_map_of_mem _ (I.2 _ ha)\n    | smul a₁ a₂ ha₂ ih =>\n      classical rw [smul_eq_mul, DirectSum.decompose_mul, DirectSum.coe_mul_apply]\n      exact sum_mem fun ij hij ↦ Ideal.mul_mem_left _ _ <| ih _\n\n/-- Pull back a homogeneous ideal along a graded ring homomorphism.\nThe underlying ideal is (definitionally) equal to `Ideal.comap`, whose underlying set is\ndefinitionally equal to the preimage. -/\ndef comap (I : HomogeneousIdeal ℬ) : HomogeneousIdeal 𝒜 where\n  __ := I.toIdeal.comap f\n  is_homogeneous' n a ha := by\n    rw [Ideal.mem_comap, HomogeneousIdeal.mem_iff, f.map_directSumDecompose]\n    exact I.2 _ ha\n\nvariable {I I₁ I₂ I₃ : HomogeneousIdeal 𝒜} {J J₁ J₂ J₃ : HomogeneousIdeal ℬ}\n  {K : HomogeneousIdeal 𝒞}\n\nlemma map_le_iff_le_comap : I.map f ≤ J ↔ I ≤ J.comap f := Ideal.map_le_iff_le_comap\n\nalias ⟨le_comap_of_map_le, map_le_of_le_comap⟩ := map_le_iff_le_comap\n\ntheorem gc_map_comap : GaloisConnection (map f) (comap f) := fun _ _ ↦\n  map_le_iff_le_comap f\n\n@[mono, aesop safe apply] lemma map_mono : Monotone (map f) := (gc_map_comap f).monotone_l\n\n@[gcongr, mono] lemma comap_mono : Monotone (comap f) := (gc_map_comap f).monotone_u\n\n@[simp] lemma toIdeal_comap : (J.comap f).toIdeal = J.toIdeal.comap f := rfl\n\n@[simp] lemma coe_comap : J.comap f = f ⁻¹' J := rfl\n\n@[simp] lemma toIdeal_map : (I.map f).toIdeal = I.toIdeal.map f := rfl\n\ninstance isPrime_comap [J.toIdeal.IsPrime] : (J.comap f).toIdeal.IsPrime :=\n  inferInstanceAs (J.toIdeal.comap f).IsPrime -- this shows that the simpNF already has the instance\n\n@[simp] lemma map_id : I.map (GradedRingHom.id 𝒜) = I := ext <| Ideal.map_id _\n\nlemma map_map : (I.map f).map g = I.map (g.comp f) := ext <| Ideal.map_map _ _\n\nlemma map_comp : I.map (g.comp f) = (I.map f).map g := (map_map f g).symm\n\n@[simp] lemma comap_id : I.comap (GradedRingHom.id 𝒜) = I := rfl\n\nlemma comap_comap : (K.comap g).comap f = K.comap (g.comp f) := rfl\n\nend arbitrary_grading\n\nsection canonical_grading\n\nvariable {A B C σ τ ω ι F G : Type*}\n  [Semiring A] [Semiring B] [Semiring C]\n  [SetLike σ A] [SetLike τ B] [SetLike ω C]\n  [AddSubmonoidClass σ A] [AddSubmonoidClass τ B] [AddSubmonoidClass ω C]\n  [DecidableEq ι] [AddCommMonoid ι] [PartialOrder ι] [CanonicallyOrderedAdd ι]\n  {𝒜 : ι → σ} {ℬ : ι → τ} {𝒞 : ι → ω}\n  [GradedRing 𝒜] [GradedRing ℬ] [GradedRing 𝒞]\n  {f : 𝒜 →+*ᵍ ℬ} {g : ℬ →+*ᵍ 𝒞}\n\nTarget:\ntheorem irrelevant_le_map_comp\n    (hf : ℬ₊ ≤ 𝒜₊.map f) (hg : 𝒞₊ ≤ ℬ₊.map g) : 𝒞₊ ≤ 𝒜₊.map (g.comp f) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/GradedAlgebra","family_id":"irrelevant_le_map_comp","file_id":"mathlib/Mathlib/RingTheory/GradedAlgebra/Homogeneous/Maps.lean","sample_id":"2c90f2e32be3da6fb260df3bb94bf0c15d7de35d2f1513e19789e1136caf26c6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"281c14df73dbd3d58dd1e8053eb70fe2ed484fdf0d2d7eb3b862de0b5a2b7009","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d3834ad3829318d82a78287b79886e2fbb44e9c1c83e11604c22f0673aed1dfc","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e9c86bc71f86c9474c5b0e70638d93a26490f015f3c3db0caf5be6d93809c4ae","source_sha256":"76bf26908fe5bb3f22e0b1de59c408f5480271bc0d25bef1d614882cce4d1085","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : Nonempty J := IsFiltered.nonempty\n  let hc' := Under.isColimitLiftCocone D s c (p ≫ f) h hc\n  obtain ⟨j, q, hq⟩ := exists_hom_of_isColimit (X := Under.mk p) hc' (Under.homMk f rfl)\n  use j, q.right, Under.w q, congr($(hq).right)","hard_negative":true,"metrics":{"chosen_tokens":72,"rejected_tokens":5,"token_jaccard":0.046512,"token_length_ratio":0.069444},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"a1bb0b6fdb4bc45df4dad7ea100332fb12284ef9e5bf54a30e74158a222c342e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Filtered\npublic import Mathlib.CategoryTheory.Limits.Preserves.Filtered\npublic import Mathlib.CategoryTheory.Limits.Types.Filtered\npublic import Mathlib.CategoryTheory.MorphismProperty.Basic\npublic import Mathlib.CategoryTheory.Presentable.Basic\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Finitely Presentable Objects\n\nWe define finitely presentable objects as a synonym for `ℵ₀`-presentable objects,\nand link this definition with the preservation of filtered colimits.\n\n-/\n\n@[expose] public section\n\n\nuniverse w v' v u' u\n\nnamespace CategoryTheory\n\nopen Limits Opposite Cardinal\n\nvariable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]\n\nattribute [local instance] fact_isRegular_aleph0\n\n/-- A functor `F : C ⥤ D` is finitely accessible if it is `ℵ₀`-accessible.\nEquivalently, it preserves all filtered colimits.\nSee `CategoryTheory.Functor.IsFinitelyAccessible_iff_preservesFilteredColimits`. -/\nabbrev Functor.IsFinitelyAccessible (F : C ⥤ D) : Prop := IsCardinalAccessible.{w} F ℵ₀\n\nlemma Functor.IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize {F : C ⥤ D} :\n    IsFinitelyAccessible.{w} F ↔ PreservesFilteredColimitsOfSize.{w, w} F := by\n  refine ⟨fun ⟨H⟩ ↦ ⟨?_⟩, fun ⟨H⟩ ↦ ⟨?_⟩⟩ <;>\n    simp only [isCardinalFiltered_aleph0_iff] at * <;>\n    exact H\n\nlemma Functor.isFinitelyAccessible_iff_preservesFilteredColimits {F : C ⥤ D} :\n    IsFinitelyAccessible.{v'} F ↔ PreservesFilteredColimits F :=\n  IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize\n\n/-- An object `X` is finitely presentable if `Hom(X, -)` preserves all filtered colimits. -/\nabbrev IsFinitelyPresentable (X : C) : Prop :=\n  IsCardinalPresentable.{w} X ℵ₀\n\nvariable (C) in\n/-- `IsFinitelyPresentable` as an `ObjectProperty` on `C`. This is sometimes called \"compact\". -/\ndef ObjectProperty.isFinitelyPresentable : ObjectProperty C := fun X ↦ IsFinitelyPresentable.{w} X\n\nlemma ObjectProperty.isFinitelyPresentable_eq_isCardinalPresentable :\n    isFinitelyPresentable.{w} C = isCardinalPresentable.{w} C ℵ₀ :=\n  rfl\n\nvariable (C) in\n/-- A morphism `f : X ⟶ Y` is finitely presentable if it is so as an object of `Under X`. -/\ndef MorphismProperty.isFinitelyPresentable : MorphismProperty C :=\n  fun _ _ f ↦ ObjectProperty.isFinitelyPresentable.{w} _ (CategoryTheory.Under.mk f)\n\nlemma isFinitelyPresentable_iff_preservesFilteredColimitsOfSize {X : C} :\n    IsFinitelyPresentable.{w} X ↔ PreservesFilteredColimitsOfSize.{w, w} (coyoneda.obj (op X)) :=\n  Functor.IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize\n\nlemma isFinitelyPresentable_iff_preservesFilteredColimits {X : C} :\n    IsFinitelyPresentable.{v} X ↔ PreservesFilteredColimits (coyoneda.obj (op X)) :=\n  Functor.IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize\n\ninstance (X : C) [IsFinitelyPresentable.{w} X] :\n    PreservesFilteredColimitsOfSize.{w, w} (coyoneda.obj (op X)) := by\n  rw [← isFinitelyPresentable_iff_preservesFilteredColimitsOfSize]\n  infer_instance\n\ninstance (X : (ObjectProperty.isFinitelyPresentable.{w} C).FullSubcategory) :\n    IsFinitelyPresentable.{w} ((ObjectProperty.isFinitelyPresentable.{w} C).ι.obj X) :=\n  X.property\n\nlemma IsFinitelyPresentable.exists_hom_of_isColimit {J : Type w} [SmallCategory J] [IsFiltered J]\n    {D : J ⥤ C} {c : Cocone D} (hc : IsColimit c) {X : C} [IsFinitelyPresentable.{w} X]\n    (f : X ⟶ c.pt) :\n    ∃ (j : J) (p : X ⟶ D.obj j), p ≫ c.ι.app j = f :=\n  Types.jointly_surjective_of_isColimit (isColimitOfPreserves (coyoneda.obj (op X)) hc) f\n\nlemma IsFinitelyPresentable.exists_eq_of_isColimit {J : Type w} [SmallCategory J] [IsFiltered J]\n    {D : J ⥤ C} {c : Cocone D} (hc : IsColimit c) {X : C} [IsFinitelyPresentable.{w} X]\n    {i j : J} (f : X ⟶ D.obj i) (g : X ⟶ D.obj j) (h : f ≫ c.ι.app i = g ≫ c.ι.app j) :\n    ∃ (k : J) (u : i ⟶ k) (v : j ⟶ k), f ≫ D.map u = g ≫ D.map v :=\n  (Types.FilteredColimit.isColimit_eq_iff _ (isColimitOfPreserves (coyoneda.obj (op X)) hc)).mp h\n\nTarget:\nlemma IsFinitelyPresentable.exists_hom_of_isColimit_under\n    {J : Type w} [SmallCategory J] [IsFiltered J] {D : J ⥤ C} {c : Cocone D} (hc : IsColimit c)\n    {X A : C} (p : X ⟶ A) (s : (Functor.const J).obj X ⟶ D)\n    [IsFinitelyPresentable.{w} (Under.mk p)]\n    (f : A ⟶ c.pt) (h : ∀ (j : J), s.app j ≫ c.ι.app j = p ≫ f) :\n    ∃ (j : J) (q : A ⟶ D.obj j), p ≫ q = s.app j ∧ q ≫ c.ι.app j = f :=\n\nProof body:\n","rejected":"by\n  exact IsFinitelyPresentable.exists_hom_of_isColimit_under","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"af1ddfa5d3d5dd1978afb2e241b6e0883a3df9c61d341d229e9dcf838133591f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Presentable","family_id":"isfinitelypresentable","file_id":"mathlib/Mathlib/CategoryTheory/Presentable/Finite.lean","sample_id":"e9c86bc71f86c9474c5b0e70638d93a26490f015f3c3db0caf5be6d93809c4ae"},"split":"train"}
{"binding":{"chosen_proof_sha256":"90ae162d9072def41fe3c1f6b71c6fb09d8794f02fb213178dc3f4878196efbb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3129c4684252f138a34ea19d6ab7a0e7965bec43e42276b25a947e2819c0b66a","source_sha256":"bed89a7fbd00d1b7ec6db8c1417583c0ed6fb3ca4d333358b3c5339034072c9b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← cancel_epi (prodComparisonIso (Over.pullback f) _ _).hom.left,\n    Over.hom_left_inv_left_assoc]\n  simp [CartesianMonoidalCategory.prodComparison, snd]","hard_negative":true,"metrics":{"chosen_tokens":33,"rejected_tokens":8,"token_jaccard":0.035714,"token_length_ratio":0.242424},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"a1fe908d8c94804196228f945e342fa8fc94d2cdd114882879712b5808532b04","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Adjunction.Limits\npublic import Mathlib.CategoryTheory.Comma.Over.Pullback\npublic import Mathlib.CategoryTheory.Limits.Constructions.Over.Products\npublic import Mathlib.CategoryTheory.Monoidal.CommMon_\npublic import Mathlib.CategoryTheory.Monoidal.Grp\npublic import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic\n\nNamespace:\nCategoryTheory.Over\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# `CartesianMonoidalCategory` for `Over X`\n\nWe provide a `CartesianMonoidalCategory (Over X)` instance via pullbacks, and provide simp lemmas\nfor the induced `MonoidalCategory (Over X)` instance.\n\n-/\n\npublic noncomputable section\n\nnamespace CategoryTheory.Over\n\nopen Functor Limits CartesianMonoidalCategory\n\nvariable {C : Type*} [Category* C] [HasPullbacks C]\n\nset_option backward.defeqAttrib.useBackward true in\n/-- A choice of finite products of `Over X` given by `Limits.pullback`. -/\nabbrev cartesianMonoidalCategory (X : C) : CartesianMonoidalCategory (Over X) :=\n  .ofChosenFiniteProducts\n    ⟨asEmptyCone (Over.mk (𝟙 X)), IsTerminal.ofUniqueHom (fun Y ↦ Over.homMk Y.hom)\n      fun Y m ↦ Over.OverMorphism.ext (by simpa using m.w)⟩\n    fun Y Z ↦ ⟨pullbackConeEquivBinaryFan.functor.obj (pullback.cone Y.hom Z.hom),\n    (pullback.isLimit _ _).pullbackConeEquivBinaryFanFunctor⟩\n\nattribute [local instance] cartesianMonoidalCategory\n\n/-- `Over X` is braided w.r.t. the Cartesian monoidal structure given by `Limits.pullback`. -/\nabbrev braidedCategory (X : C) : BraidedCategory (Over X) :=\n  .ofCartesianMonoidalCategory\n\nattribute [local instance] braidedCategory\n\nopen MonoidalCategory\n\nvariable {X : C}\n\n@[ext]\nlemma tensorObj_ext {R : C} {S T : Over X} (f₁ f₂ : R ⟶ (S ⊗ T).left)\n    (e₁ : f₁ ≫ pullback.fst _ _ = f₂ ≫ pullback.fst _ _)\n    (e₂ : f₁ ≫ pullback.snd _ _ = f₂ ≫ pullback.snd _ _) : f₁ = f₂ :=\n  pullback.hom_ext e₁ e₂\n\n@[simp]\nlemma tensorObj_left (R S : Over X) : (R ⊗ S).left = Limits.pullback R.hom S.hom := rfl\n\n@[simp]\nlemma tensorObj_hom (R S : Over X) : (R ⊗ S).hom = pullback.fst R.hom S.hom ≫ R.hom := rfl\n\n@[simp]\nlemma tensorUnit_left : (𝟙_ (Over X)).left = X := rfl\n\n@[simp]\nlemma tensorUnit_hom : (𝟙_ (Over X)).hom = 𝟙 X := rfl\n\n@[simp]\nlemma lift_left {R S T : Over X} (f : R ⟶ S) (g : R ⟶ T) :\n    (lift f g).left = pullback.lift f.left g.left (f.w.trans g.w.symm) := rfl\n\n@[simp]\nlemma fst_left {R S : Over X} : (fst R S).left = pullback.fst _ _ := rfl\n\n@[simp]\nlemma snd_left {R S : Over X} : (snd R S).left = pullback.snd _ _ := rfl\n\n@[simp]\nlemma toUnit_left {R : Over X} : (toUnit R).left = R.hom := rfl\n\n@[reassoc (attr := simp)]\nlemma associator_hom_left_fst (R S T : Over X) :\n    (α_ R S T).hom.left ≫ pullback.fst _ (pullback.fst _ _ ≫ _) =\n      pullback.fst _ _ ≫ pullback.fst _ _ :=\n  limit.lift_π _ _\n\n@[reassoc (attr := simp)]\nlemma associator_hom_left_snd_fst (R S T : Over X) :\n    (α_ R S T).hom.left ≫ pullback.snd _ (pullback.fst _ _ ≫ _) ≫ pullback.fst _ _ =\n      pullback.fst _ _ ≫ pullback.snd _ _ :=\n  (limit.lift_π_assoc _ _ _).trans (limit.lift_π _ _)\n\n@[reassoc (attr := simp)]\nlemma associator_hom_left_snd_snd (R S T : Over X) :\n    (α_ R S T).hom.left ≫ pullback.snd _ (pullback.fst _ _ ≫ _) ≫ pullback.snd _ _ =\n      pullback.snd _ _ :=\n  (limit.lift_π_assoc _ _ _).trans (limit.lift_π _ _)\n\n@[reassoc (attr := simp)]\nlemma associator_inv_left_fst_fst (R S T : Over X) :\n    (α_ R S T).inv.left ≫ pullback.fst (pullback.fst _ _ ≫ _) _ ≫ pullback.fst _ _ =\n      pullback.fst _ _ :=\n  (limit.lift_π_assoc _ _ _).trans (limit.lift_π _ _)\n\n@[reassoc (attr := simp)]\nlemma associator_inv_left_fst_snd (R S T : Over X) :\n    (α_ R S T).inv.left ≫ pullback.fst (pullback.fst _ _ ≫ _) _ ≫ pullback.snd _ _ =\n      pullback.snd _ _ ≫ pullback.fst _ _ :=\n  (limit.lift_π_assoc _ _ _).trans (limit.lift_π _ _)\n\n@[reassoc (attr := simp)]\nlemma associator_inv_left_snd (R S T : Over X) :\n    (α_ R S T).inv.left ≫ pullback.snd (pullback.fst _ _ ≫ _) _ =\n      pullback.snd _ _ ≫ pullback.snd _ _ :=\n  limit.lift_π _ _\n\n@[simp]\nlemma leftUnitor_hom_left (Y : Over X) :\n    (λ_ Y).hom.left = pullback.snd _ _ := rfl\n\n@[reassoc (attr := simp)]\nlemma leftUnitor_inv_left_fst (Y : Over X) :\n    (λ_ Y).inv.left ≫ pullback.fst (𝟙 X) _ = Y.hom :=\n  limit.lift_π _ _\n\n@[reassoc (attr := simp)]\nlemma leftUnitor_inv_left_snd (Y : Over X) :\n    (λ_ Y).inv.left ≫ pullback.snd (𝟙 X) _ = 𝟙 Y.left :=\n  limit.lift_π _ _\n\n@[simp]\nlemma rightUnitor_hom_left (Y : Over X) :\n    (ρ_ Y).hom.left = pullback.fst _ (𝟙 X) := rfl\n\n@[reassoc (attr := simp)]\nlemma rightUnitor_inv_left_fst (Y : Over X) :\n    (ρ_ Y).inv.left ≫ pullback.fst _ (𝟙 X) = 𝟙 _ :=\n  limit.lift_π _ _\n\n@[reassoc (attr := simp)]\nlemma rightUnitor_inv_left_snd (Y : Over X) :\n    (ρ_ Y).inv.left ≫ pullback.snd _ (𝟙 X) = Y.hom :=\n  limit.lift_π _ _\n\nlemma whiskerLeft_left {R S T : Over X} (f : S ⟶ T) :\n    (R ◁ f).left = pullback.map _ _ _ _ (𝟙 _) f.left (𝟙 _) (by simp) (by simp) := rfl\n\n@[reassoc (attr := simp)]\nlemma whiskerLeft_left_fst {R S T : Over X} (f : S ⟶ T) :\n    (R ◁ f).left ≫ pullback.fst _ _ = pullback.fst _ _ :=\n  (limit.lift_π _ _).trans (Category.comp_id _)\n\n@[reassoc (attr := simp)]\nlemma whiskerLeft_left_snd {R S T : Over X} (f : S ⟶ T) :\n    (R ◁ f).left ≫ pullback.snd _ _ = pullback.snd _ _ ≫ f.left :=\n  limit.lift_π _ _\n\nlemma whiskerRight_left {R S T : Over X} (f : S ⟶ T) :\n    (f ▷ R).left = pullback.map _ _ _ _ f.left (𝟙 _) (𝟙 _) (by simp) (by simp) := rfl\n\n@[reassoc (attr := simp)]\nlemma whiskerRight_left_fst {R S T : Over X} (f : S ⟶ T) :\n    (f ▷ R).left ≫ pullback.fst _ _ = pullback.fst _ _ ≫ f.left :=\n  limit.lift_π _ _\n\n@[reassoc (attr := simp)]\nlemma whiskerRight_left_snd {R S T : Over X} (f : S ⟶ T) :\n    (f ▷ R).left ≫ pullback.snd _ _ = pullback.snd _ _ :=\n  (limit.lift_π _ _).trans (Category.comp_id _)\n\nlemma tensorHom_left {R S T U : Over X} (f : R ⟶ S) (g : T ⟶ U) :\n    (f ⊗ₘ g).left = pullback.map _ _ _ _ f.left g.left (𝟙 _) (by simp) (by simp) := rfl\n\n@[reassoc (attr := simp)]\nlemma tensorHom_left_fst {S U : C} {R T : Over X} (fS : S ⟶ X) (fU : U ⟶ X)\n    (f : R ⟶ mk fS) (g : T ⟶ mk fU) :\n    (f ⊗ₘ g).left ≫ pullback.fst fS fU = pullback.fst R.hom T.hom ≫ f.left :=\n  limit.lift_π _ _\n\n@[reassoc (attr := simp)]\nlemma tensorHom_left_snd {S U : C} {R T : Over X} (fS : S ⟶ X) (fU : U ⟶ X)\n    (f : R ⟶ mk fS) (g : T ⟶ mk fU) :\n    (f ⊗ₘ g).left ≫ pullback.snd fS fU = pullback.snd R.hom T.hom ≫ g.left :=\n  limit.lift_π _ _\n\n@[simp]\nlemma braiding_hom_left {R S : Over X} :\n    (β_ R S).hom.left = (pullbackSymmetry _ _).hom := rfl\n\n@[simp]\nlemma braiding_inv_left {R S : Over X} :\n    (β_ R S).inv.left = (pullbackSymmetry _ _).hom := rfl\n\nvariable {A B R S Y Z : C} {f : R ⟶ X} {g : S ⟶ X}\n\ninstance : (Over.pullback f).Braided := .ofChosenFiniteProducts _\n\n@[simp]\nlemma η_pullback_left : (OplaxMonoidal.η (Over.pullback f)).left = (pullback.snd (𝟙 _) f) := rfl\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma ε_pullback_left : (LaxMonoidal.ε (Over.pullback f)).left = inv (pullback.snd (𝟙 _) f) := by\n  apply IsIso.eq_inv_of_hom_inv_id\n  rw [← η_pullback_left, ← Over.comp_left, Monoidal.η_ε, Over.id_left]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma μ_pullback_left_fst_fst (R S : Over X) :\n    (LaxMonoidal.μ (Over.pullback f) R S).left ≫\n      pullback.fst _ _ ≫ pullback.fst _ _ = pullback.fst _ _ ≫ pullback.fst _ _ := by\n  rw [Monoidal.μ_of_cartesianMonoidalCategory,\n    ← cancel_epi (prodComparisonIso (Over.pullback f) R S).hom.left, ← Over.comp_left_assoc,\n    Iso.hom_inv_id]\n  simp [CartesianMonoidalCategory.prodComparison, fst]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma μ_pullback_left_fst_snd (R S : Over X) :\n    (LaxMonoidal.μ (Over.pullback f) R S).left ≫\n      pullback.fst _ _ ≫ pullback.snd _ _ = pullback.snd _ _ ≫ pullback.fst _ _ := by\n  rw [Monoidal.μ_of_cartesianMonoidalCategory,\n    ← cancel_epi (prodComparisonIso (Over.pullback f) R S).hom.left,\n    ← Over.comp_left_assoc, Iso.hom_inv_id]\n  simp [CartesianMonoidalCategory.prodComparison, snd]\n\nset_option backward.isDefEq.respectTransparency false in\nlemma μ_pullback_left_snd (R S : Over X) :\n    (LaxMonoidal.μ (Over.pullback f) R S).left ≫ pullback.snd _ _ =\n      pullback.snd _ _ ≫ pullback.snd _ _ := by\n  rw [Monoidal.μ_of_cartesianMonoidalCategory,\n    ← cancel_epi (prodComparisonIso (Over.pullback f) R S).hom.left,\n    ← Over.comp_left_assoc, Iso.hom_inv_id]\n  simp [CartesianMonoidalCategory.prodComparison]\n\n@[simp]\nlemma μ_pullback_left_fst_fst' (g₁ : Y ⟶ X) (g₂ : Z ⟶ X) :\n    (LaxMonoidal.μ (Over.pullback f) (.mk g₁) (.mk g₂)).left ≫\n      pullback.fst (pullback.fst g₁ g₂ ≫ g₁) f ≫ pullback.fst g₁ g₂ =\n        pullback.fst _ _ ≫ pullback.fst _ _ :=\n  μ_pullback_left_fst_fst ..\n\n@[simp]\nlemma μ_pullback_left_fst_snd' (g₁ : Y ⟶ X) (g₂ : Z ⟶ X) :\n    (LaxMonoidal.μ (Over.pullback f) (.mk g₁) (.mk g₂)).left ≫\n      pullback.fst (pullback.fst g₁ g₂ ≫ g₁) f ≫ pullback.snd g₁ g₂ =\n        pullback.snd _ _ ≫ pullback.fst _ _ :=\n  μ_pullback_left_fst_snd ..\n\n@[simp]\nlemma μ_pullback_left_snd' (g₁ : Y ⟶ X) (g₂ : Z ⟶ X) :\n    (LaxMonoidal.μ (Over.pullback f) (.mk g₁) (.mk g₂)).left ≫\n      pullback.snd (pullback.fst g₁ g₂ ≫ g₁) f =\n        pullback.snd _ _ ≫ pullback.snd _ _ := μ_pullback_left_snd ..\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma preservesTerminalIso_pullback (f : R ⟶ S) :\n    preservesTerminalIso (Over.pullback f) =\n      Over.isoMk (asIso (pullback.snd (𝟙 _) f)) (by simp) := by\n  ext1; exact toUnit_unique _ _\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma prodComparisonIso_pullback_inv_left_fst_fst (f : X ⟶ Y) (A B : Over Y) :\n    (prodComparisonIso (Over.pullback f) A B).inv.left ≫\n      pullback.fst (pullback.fst A.hom B.hom ≫ A.hom) f ≫ pullback.fst _ _ =\n        pullback.fst (pullback.snd A.hom f) (pullback.snd B.hom f) ≫ pullback.fst _ _ := by\n  rw [← cancel_epi (prodComparisonIso (Over.pullback f) A B).hom.left,\n    Over.hom_left_inv_left_assoc]\n  simp [CartesianMonoidalCategory.prodComparison, fst]\n\n@[simp]\nlemma prodComparisonIso_pullback_Spec_inv_left_fst_fst' (f : X ⟶ Y) (gA : A ⟶ Y) (gB : B ⟶ Y) :\n    (prodComparisonIso (Over.pullback f) (.mk gA) (.mk gB)).inv.left ≫\n      pullback.fst (pullback.fst gA gB ≫ gA) f ≫ pullback.fst _ _ =\n        pullback.fst (pullback.snd gA f) (pullback.snd gB f) ≫ pullback.fst _ _ :=\n  prodComparisonIso_pullback_inv_left_fst_fst ..\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\n\nTarget:\nlemma prodComparisonIso_pullback_inv_left_fst_snd' (f : X ⟶ Y) (gA : A ⟶ Y) (gB : B ⟶ Y) :\n    (prodComparisonIso (Over.pullback f) (.mk gA) (.mk gB)).inv.left ≫\n      pullback.fst (pullback.fst gA gB ≫ gA) f ≫ pullback.snd _ _ =\n        pullback.snd _ _ ≫ pullback.fst _ _ :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"fa711ec91a0c6b5ff1ef81a1aec0a6032c4b166288b7aff2c36b94b7e6f99218","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Monoidal","family_id":"prodcomparisoniso_pullback_inv_left_fst_snd","file_id":"mathlib/Mathlib/CategoryTheory/Monoidal/Cartesian/Over.lean","sample_id":"3129c4684252f138a34ea19d6ab7a0e7965bec43e42276b25a947e2819c0b66a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dbcf29961f4c4a120ac8530a8dcc28a30bfa6b6db329e1836f9e00211fe2e46d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f7bf80e04d0a2e1788d695cec2bec9e4d9ad07000508ae788bbe0e829f642af9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"405cc27ec849e33c860448f43291f011dd4708a3dd857ace59279e3e185b0743","source_sha256":"8fcecfecdd43a0c830d21a96925630041f2ef381c500d297f5f2d704df5a9911","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa [CochainComplex.of.d] using (inhomogeneousCochains A).d_comp_d n (n + 1) (n + 2)","hard_negative":true,"metrics":{"chosen_tokens":27,"rejected_tokens":2,"token_jaccard":0.052632,"token_length_ratio":0.074074},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"a2067ac654d697a1f792b395ccc80fc1673291eda74fdcb42b0f59422849c8c3","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.Opposite\npublic import Mathlib.Algebra.Homology.ConcreteCategory\npublic import Mathlib.RepresentationTheory.Homological.Resolution\npublic import Mathlib.Tactic.CategoryTheory.Slice\n\nNamespace:\ngroupCohomology\n\nLocal context:\n/-\nCopyright (c) 2023 Amelia Livingston. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Amelia Livingston\n-/\n/-!\n# The group cohomology of a `k`-linear `G`-representation\n\nLet `k` be a commutative ring and `G` a group. This file defines the group cohomology of\n`A : Rep k G` to be the cohomology of the complex\n$$0 \\to \\mathrm{Fun}(G^0, A) \\to \\mathrm{Fun}(G^1, A) \\to \\mathrm{Fun}(G^2, A) \\to \\dots$$\nwith differential $d^n$ sending $f: G^n \\to A$ to the function mapping $(g_0, \\dots, g_n)$ to\n$$\\rho(g_0)(f(g_1, \\dots, g_n))$$\n$$+ \\sum_{i = 0}^{n - 1} (-1)^{i + 1}\\cdot f(g_0, \\dots, g_ig_{i + 1}, \\dots, g_n)$$\n$$+ (-1)^{n + 1}\\cdot f(g_0, \\dots, g_{n - 1})$$ (where `ρ` is the representation attached to `A`).\n\nWe have a `k`-linear isomorphism\n$\\mathrm{Fun}(G^n, A) \\cong \\mathrm{Hom}(\\bigoplus_{G^n} k[G], A)$, where\nthe right-hand side is morphisms in `Rep k G`, and $k[G]$ is equipped with the left regular\nrepresentation. If we conjugate the $n$th differential in $\\mathrm{Hom}(P, A)$ by this isomorphism,\nwhere `P` is the bar resolution of `k` as a trivial `k`-linear `G`-representation, then the\nresulting map agrees with the differential $d^n$ defined above, a fact we prove.\n\nThis gives us for free a proof that our $d^n$ squares to zero. It also gives us an isomorphism\n$\\mathrm{H}^n(G, A) \\cong \\mathrm{Ext}^n(k, A),$ where $\\mathrm{Ext}$ is taken in the category\n`Rep k G`.\n\nTo talk about cohomology in low degree, please see the file\n`Mathlib/RepresentationTheory/Homological/GroupCohomology/LowDegree.lean`, which provides API\nspecialized to `H⁰`, `H¹`, `H²`.\n\n## Main definitions\n\n* `groupCohomology.inhomogeneousCochains A`: a complex whose objects are\n  $\\mathrm{Fun}(G^n, A)$ and whose cohomology is the group cohomology $\\mathrm{H}^n(G, A).$\n* `groupCohomology.inhomogeneousCochainsIso A`: an isomorphism between the above complex and the\n  complex $\\mathrm{Hom}(P, A),$ where `P` is the bar resolution of `k` as a trivial resolution.\n* `groupCohomology A n`: this is $\\mathrm{H}^n(G, A),$ defined as the $n$th cohomology of\n  `inhomogeneousCochains A`.\n* `groupCohomologyIsoExt A n`: an isomorphism $\\mathrm{H}^n(G, A) \\cong \\mathrm{Ext}^n(k, A)$\n  (where $\\mathrm{Ext}$ is taken in the category `Rep k G`) induced by `inhomogeneousCochainsIso A`.\n\n## Implementation notes\n\nGroup cohomology is typically stated for `G`-modules, or equivalently modules over the group ring\n`ℤ[G].` However, `ℤ` can be generalized to any commutative ring `k`, which is what we use.\nMoreover, we express `k[G]`-module structures on a module `k`-module `A` using the `Rep`\ndefinition. We avoid using instances `Module k[G] A` so that we do not run into\npossible scalar action diamonds.\n\n## TODO\n\n* Upgrading `groupCohomologyIsoExt` to an isomorphism of derived functors.\n* Profinite cohomology.\n\nLonger term:\n* The Hochschild-Serre spectral sequence (this is perhaps a good toy example for the theory of\n  spectral sequences in general).\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nuniverse u\n\nvariable {k G : Type u} [CommRing k] {n : ℕ}\n\nopen CategoryTheory\n\nnamespace inhomogeneousCochains\n\nopen Rep\n\n/-- The differential in the complex of inhomogeneous cochains used to\ncalculate group cohomology. -/\n@[simps! -isSimp]\ndef d [Monoid G] (A : Rep k G) (n : ℕ) :\n    ModuleCat.of k ((Fin n → G) → A) ⟶ ModuleCat.of k ((Fin (n + 1) → G) → A) :=\n  ModuleCat.ofHom\n  { toFun f g :=\n      A.ρ (g 0) (f fun i => g i.succ) + Finset.univ.sum fun j : Fin (n + 1) =>\n        (-1 : k) ^ ((j : ℕ) + 1) • f (Fin.contractNth j (· * ·) g)\n    map_add' f g := by\n      ext\n      simp [Finset.sum_add_distrib, add_add_add_comm]\n    map_smul' r f := by\n      ext\n      simp [Finset.smul_sum, ← smul_assoc, mul_comm r] }\n\nvariable [Group G] (A : Rep k G) (n : ℕ)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\ntheorem d_eq :\n    d A n =\n      (freeLiftLEquiv k G (Fin n → G) A).toModuleIso.inv ≫\n        ((barComplex k G).linearYonedaObj k A).d n (n + 1) ≫\n          (freeLiftLEquiv k G (Fin (n + 1) → G) A).toModuleIso.hom := by\n  ext\n  simp [d_hom_apply, map_add, barComplex.d_single (k := k), homEquiv]\n\nend inhomogeneousCochains\n\nnamespace groupCohomology\n\nvariable [Group G] (n) (A : Rep.{u} k G)\n\nopen inhomogeneousCochains Rep\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Given a `k`-linear `G`-representation `A`, this is the complex of inhomogeneous cochains\n$$0 \\to \\mathrm{Fun}(G^0, A) \\to \\mathrm{Fun}(G^1, A) \\to \\mathrm{Fun}(G^2, A) \\to \\dots$$\nwhich calculates the group cohomology of `A`. -/\nnoncomputable abbrev inhomogeneousCochains : CochainComplex (ModuleCat k) ℕ :=\n  CochainComplex.of (fun n => ModuleCat.of k ((Fin n → G) → A))\n    (fun n => inhomogeneousCochains.d A n) fun n => by\n    classical\n    rw [d_eq, d_eq]\n    slice_lhs 3 4 => rw [Iso.hom_inv_id]\n    slice_lhs 2 4 => rw [Category.id_comp, ((barComplex k G).linearYonedaObj k A).d_comp_d]\n    simp\n\nvariable {A n} in\n@[ext]\ntheorem inhomogeneousCochains.ext {x y : (inhomogeneousCochains A).X n} (h : ∀ g, x g = y g) :\n    x = y := funext h\n\ntheorem inhomogeneousCochains.d_def (n : ℕ) :\n    (inhomogeneousCochains A).d n (n + 1) = d A n := by\n  simp\n\nset_option backward.defeqAttrib.useBackward true in\n\nTarget:\ntheorem inhomogeneousCochains.d_comp_d :\n    d A n ≫ d A (n + 1) = 0 :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_405cc27ec849","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"13168a73d4b76a5139b9aa1f337c85935681cfb54b706059f94f1f39e7156fe5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RepresentationTheory/Homological","family_id":"inhomogeneouscochains","file_id":"mathlib/Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean","sample_id":"405cc27ec849e33c860448f43291f011dd4708a3dd857ace59279e3e185b0743"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9740f164ce7086b555af4b7554cbcc58f11573780fea0b6812b9fc64a3703b2b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"35a2e7b8e31ff1274ea9232d6e5039032bbe8e5095bd975831050868a39630fa","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"27c7b061eb2df69ed4438c60ac705043bf7b1c99567810c2e2cbfe95dec0ba49","source_sha256":"c27083b8f8cde0437299c837bd1f7a4c8446e30ca2254817bcb2958bf94cb26d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply MonoidHom.eq_of_eqOn_denseM (closure_range_of _)\n  grind [Set.eqOn_range]","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":2,"token_jaccard":0.066667,"token_length_ratio":0.133333},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"a220e6a934593c52d646236b01d1ff4279126d1687666f9757cbf19059dc9eee","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.FreeMonoid.Basic\npublic import Mathlib.Algebra.Group.Submonoid.Operations\npublic import Mathlib.GroupTheory.Congruence.Hom\n\nNamespace:\nPresentedMonoid\n\nLocal context:\n/-\nCopyright (c) 2024 Hannah Fechtner. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Hannah Fechtner\n-/\n/-!\n# Defining a monoid given by generators and relations\n\nGiven relations `rels` on the free monoid on a type `α`, this file constructs the monoid\ngiven by generators `x : α` and relations `rels`.\n\n## Main definitions\n\n* `PresentedMonoid rels`: the quotient of the free monoid on a type `α` by the closure of one-step\n  reductions (arising from a binary relation on free monoid elements `rels`).\n* `PresentedMonoid.of`: The canonical map from `α` to a presented monoid with generators `α`.\n* `PresentedMonoid.lift f`: the canonical monoid homomorphism `PresentedMonoid rels → M`, given\n  a function `f : α → G` from a type `α` to a monoid `M` which satisfies the relations `rels`.\n\n## Tags\n\ngenerators, relations, monoid presentations\n-/\n\n@[expose] public section\n\nvariable {α : Type*}\n\n/-- Given a set of relations, `rels`, over a type `α`, `PresentedMonoid` constructs the monoid with\ngenerators `x : α` and relations `rels` as a quotient of a congruence structure over rels. -/\n@[to_additive /-- Given a set of relations, `rels`, over a type `α`, `PresentedAddMonoid` constructs\nthe monoid with generators `x : α` and relations `rels` as a quotient of an AddCon structure over\nrels -/]\ndef PresentedMonoid (rels : FreeMonoid α → FreeMonoid α → Prop) := (conGen rels).Quotient\n\nnamespace PresentedMonoid\n\nopen Set Submonoid\n\n@[to_additive]\ninstance {rels : FreeMonoid α → FreeMonoid α → Prop} : Monoid (PresentedMonoid rels) :=\n  inferInstanceAs <| Monoid (conGen rels).Quotient\n\n/-- The quotient map from the free monoid on `α` to the presented monoid with the same generators\nand the given relations `rels`. -/\n@[to_additive /-- The quotient map from the free additive monoid on `α` to the presented additive\nmonoid with the same generators and the given relations `rels` -/]\ndef mk (rels : FreeMonoid α → FreeMonoid α → Prop) : FreeMonoid α →* PresentedMonoid rels where\n  toFun := Quotient.mk (conGen rels).toSetoid\n  map_one' := rfl\n  map_mul' := fun _ _ => rfl\n\n/-- `of` is the canonical map from `α` to a presented monoid with generators `x : α`. The term `x`\nis mapped to the equivalence class of the image of `x` in `FreeMonoid α`. -/\n@[to_additive\n/-- `of` is the canonical map from `α` to a presented additive monoid with generators `x : α`. The\nterm `x` is mapped to the equivalence class of the image of `x` in `FreeAddMonoid α`. -/]\ndef of (rels : FreeMonoid α → FreeMonoid α → Prop) (x : α) : PresentedMonoid rels :=\n  mk rels (.of x)\n\nsection inductionOn\n\nvariable {α₁ α₂ α₃ : Type*} {rels₁ : FreeMonoid α₁ → FreeMonoid α₁ → Prop}\n  {rels₂ : FreeMonoid α₂ → FreeMonoid α₂ → Prop} {rels₃ : FreeMonoid α₃ → FreeMonoid α₃ → Prop}\n\nlocal notation \"P₁\" => PresentedMonoid rels₁\nlocal notation \"P₂\" => PresentedMonoid rels₂\nlocal notation \"P₃\" => PresentedMonoid rels₃\n\n@[to_additive (attr := elab_as_elim), induction_eliminator]\nprotected theorem inductionOn {δ : P₁ → Prop} (q : P₁) (h : ∀ a, δ (mk rels₁ a)) : δ q :=\n  Quotient.ind h q\n\n@[to_additive (attr := elab_as_elim)]\nprotected theorem inductionOn₂ {δ : P₁ → P₂ → Prop} (q₁ : P₁) (q₂ : P₂)\n    (h : ∀ a b, δ (mk rels₁ a) (mk rels₂ b)) : δ q₁ q₂ :=\n  Quotient.inductionOn₂ q₁ q₂ h\n\n@[to_additive (attr := elab_as_elim)]\nprotected theorem inductionOn₃ {δ : P₁ → P₂ → P₃ → Prop} (q₁ : P₁)\n    (q₂ : P₂) (q₃ : P₃) (h : ∀ a b c, δ (mk rels₁ a) (mk rels₂ b) (mk rels₃ c)) :\n    δ q₁ q₂ q₃ :=\n  Quotient.inductionOn₃ q₁ q₂ q₃ h\n\nend inductionOn\n\nvariable {α : Type*} {rels : FreeMonoid α → FreeMonoid α → Prop} {x y : FreeMonoid α}\n\nlemma mk_eq_mk_iff : mk rels x = mk rels y ↔ conGen rels x y := Quotient.eq\n\nlemma mk_eq_mk_of_rel (h : rels x y) : mk rels x = mk rels y := mk_eq_mk_iff.2 (.of _ _ h)\n\n/-- The generators of a presented monoid generate the presented monoid. That is, the submonoid\nclosure of the set of generators equals `⊤`. -/\n@[to_additive (attr := simp) /-- The generators of a presented additive monoid generate the\npresented additive monoid. That is, the additive submonoid closure of the set of generators equals\n`⊤`. -/]\ntheorem closure_range_of (rels : FreeMonoid α → FreeMonoid α → Prop) :\n    Submonoid.closure (Set.range (of rels)) = ⊤ := by\n  rw [Submonoid.eq_top_iff']\n  intro x\n  induction x with | _ a\n  induction a with\n  | one => exact Submonoid.one_mem _\n  | of x => exact subset_closure <| by simp [range, of]\n  | mul x y hx hy => exact Submonoid.mul_mem _ hx hy\n\n@[to_additive]\ntheorem surjective_mk {rels : FreeMonoid α → FreeMonoid α → Prop} :\n    Function.Surjective (mk rels) := fun x ↦ PresentedMonoid.inductionOn x fun a ↦ .intro a rfl\n\nsection ToMonoid\nvariable {α M : Type*} [Monoid M] (f : α → M)\nvariable {rels : FreeMonoid α → FreeMonoid α → Prop}\nvariable (h : ∀ a b : FreeMonoid α, rels a b → FreeMonoid.lift f a = FreeMonoid.lift f b)\n\n/-- The extension of a map `f : α → M` that satisfies the given relations to a monoid homomorphism\nfrom `PresentedMonoid rels → M`. -/\n@[to_additive /-- The extension of a map `f : α → M` that satisfies the given relations to an\nadditive-monoid homomorphism from `PresentedAddMonoid rels → M` -/]\ndef lift : PresentedMonoid rels →* M :=\n  Con.lift _ (FreeMonoid.lift f) (Con.conGen_le.2 h)\n\n@[to_additive]\ntheorem toMonoid.unique (g : MonoidHom (conGen rels).Quotient M)\n    (hg : ∀ a : α, g (of rels a) = f a) : g = lift f h :=\n  Con.lift_unique (Con.conGen_le.2 h) g (FreeMonoid.hom_eq hg)\n\n@[to_additive (attr := simp)]\ntheorem lift_of {x : α} : lift f h (of rels x) = f x := rfl\n\nend ToMonoid\n\n@[to_additive (attr := ext)]\n\nTarget:\ntheorem ext {M : Type*} [Monoid M] (rels : FreeMonoid α → FreeMonoid α → Prop)\n    {φ ψ : PresentedMonoid rels →* M} (hx : ∀ (x : α), φ (.of rels x) = ψ (.of rels x)) :\n    φ = ψ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_27c7b061eb2d","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"f81701716254788fe52c04e43bcebb225d8becb879d53705942d7de3afa7dc28","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/PresentedMonoid","family_id":"ext","file_id":"mathlib/Mathlib/Algebra/PresentedMonoid/Basic.lean","sample_id":"27c7b061eb2df69ed4438c60ac705043bf7b1c99567810c2e2cbfe95dec0ba49"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d7c3832f7227fb403f537f073b6c62458e059e39149445706fd5cbe0c34cde70","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4a94ca7d62a9fac81c4789f5e6cb34fc62a67f792fb4fbe276c40abbf6855644","source_sha256":"1ddcf1b99bc8b3242f5c0a7e676d50f1f12f248d40327684682203ae3ae4dc4a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only [tendsto_iff_comap] using le_iff_ultrafilter","hard_negative":false,"metrics":{"chosen_tokens":8,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.625},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"a27a528238bbf853f57a70a677fecc3574a3c31b16b7ceb272a0a8bb9c189594","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Filter.Ultrafilter.Defs\npublic import Mathlib.Order.Filter.Cofinite\npublic import Mathlib.Order.ZornAtoms\n\nNamespace:\nFilter\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov\n-/\n/-!\n# Ultrafilters\n\nAn ultrafilter is a minimal (maximal in the set order) proper filter.\nIn this file we define\n\n* `hyperfilter`: the ultrafilter extending the cofinite filter.\n-/\n\n@[expose] public section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type v}\n\nopen Set Filter\n\nnamespace Ultrafilter\n\nvariable {f : Ultrafilter α} {s : Set α}\n\ntheorem finite_sUnion_mem_iff {s : Set (Set α)} (hs : s.Finite) : ⋃₀ s ∈ f ↔ ∃ t ∈ s, t ∈ f := by\n  induction s, hs using Set.Finite.induction_on with\n  | empty => simp\n  | insert _ _ his => simp [union_mem_iff, his, or_and_right, exists_or]\n\ntheorem finite_biUnion_mem_iff {is : Set β} {s : β → Set α} (his : is.Finite) :\n    (⋃ i ∈ is, s i) ∈ f ↔ ∃ i ∈ is, s i ∈ f := by\n  simp only [← sUnion_image, finite_sUnion_mem_iff (his.image s), exists_mem_image]\n\nlemma eventually_exists_mem_iff {is : Set β} {P : β → α → Prop} (his : is.Finite) :\n    (∀ᶠ i in f, ∃ a ∈ is, P a i) ↔ ∃ a ∈ is, ∀ᶠ i in f, P a i := by\n  simp only [Filter.Eventually, Ultrafilter.mem_coe]\n  convert! f.finite_biUnion_mem_iff his (s := P) with i\n  aesop\n\nlemma eventually_exists_iff [Finite β] {P : β → α → Prop} :\n    (∀ᶠ i in f, ∃ a, P a i) ↔ ∃ a, ∀ᶠ i in f, P a i := by\n  simpa using eventually_exists_mem_iff (f := f) (P := P) Set.finite_univ\n\ntheorem eq_pure_of_finite_mem (h : s.Finite) (h' : s ∈ f) : ∃ x ∈ s, f = pure x := by\n  rw [← biUnion_of_singleton s] at h'\n  rcases (Ultrafilter.finite_biUnion_mem_iff h).mp h' with ⟨a, has, haf⟩\n  exact ⟨a, has, eq_of_le (Filter.le_pure_iff.2 haf)⟩\n\ntheorem eq_pure_of_finite [Finite α] (f : Ultrafilter α) : ∃ a, f = pure a :=\n  (eq_pure_of_finite_mem finite_univ univ_mem).imp fun _ ⟨_, ha⟩ => ha\n\ntheorem le_cofinite_or_eq_pure (f : Ultrafilter α) : (f : Filter α) ≤ cofinite ∨ ∃ a, f = pure a :=\n  or_iff_not_imp_left.2 fun h =>\n    let ⟨_, hs, hfin⟩ := Filter.disjoint_cofinite_right.1 (disjoint_iff_not_le.2 h)\n    let ⟨a, _, hf⟩ := eq_pure_of_finite_mem hfin hs\n    ⟨a, hf⟩\n\ntheorem exists_ultrafilter_of_finite_inter_nonempty (S : Set (Set α))\n    (cond : ∀ T : Finset (Set α), (↑T : Set (Set α)) ⊆ S → (⋂₀ (↑T : Set (Set α))).Nonempty) :\n    ∃ F : Ultrafilter α, S ⊆ F.sets :=\n  haveI : NeBot (generate S) :=\n    generate_neBot_iff.2 fun _ hts ht =>\n      ht.coe_toFinset ▸ cond ht.toFinset (ht.coe_toFinset.symm ▸ hts)\n  ⟨of (generate S), fun _ ht => (of_le <| generate S) <| GenerateSets.basic ht⟩\n\nend Ultrafilter\n\nnamespace Filter\n\nopen Ultrafilter\n\n@[to_dual]\nlemma atTop_eq_pure_of_isTop [PartialOrder α] {x : α} (hx : IsTop x) :\n    (atTop : Filter α) = pure x :=\n  { top := x, le_top := hx : OrderTop α }.atTop_eq\n\n/-- The `tendsto` relation can be checked on ultrafilters. -/\n\nTarget:\ntheorem tendsto_iff_ultrafilter (f : α → β) (l₁ : Filter α) (l₂ : Filter β) :\n    Tendsto f l₁ l₂ ↔ ∀ g : Ultrafilter α, ↑g ≤ l₁ → Tendsto f g l₂ :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Filter","family_id":"tendsto_iff_ultrafilter","file_id":"mathlib/Mathlib/Order/Filter/Ultrafilter/Basic.lean","sample_id":"4a94ca7d62a9fac81c4789f5e6cb34fc62a67f792fb4fbe276c40abbf6855644"},"split":"train"}
{"binding":{"chosen_proof_sha256":"543d94c4b8beec8d8219800c1a531bf78be218b6d235d8e934ed1722e2e75902","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4561b215385a78d721dfe36030198c0459e29f4db8031b7c27e613ae90c4a078","source_sha256":"9204017e76fd98a1b3e65ba16535a42898573646b8f44e5554ab4317200de227","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine measure_mono_null (fun x hx ↦ ?_)\n    (Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet _\n      (isClosed_closure (s := s)).measurableSet)\n  by_cases h : x ∈ closure s <;>\n    simp only [mem_compl_iff, mem_setOf, h, not_false_eq_true, indicator_of_notMem,\n      indicator_of_mem, Pi.one_apply]\n  · refine aux₁ fun _ ↦ hs.compl.exists_subset_ball <| frontier_subset_closure ?_\n    rwa [frontier_compl]\n  · exact aux₀ fun _ ↦ hs.exists_subset_ball <| frontier_subset_closure hx","hard_negative":false,"metrics":{"chosen_tokens":91,"rejected_tokens":5,"token_jaccard":0.057692,"token_length_ratio":0.054945},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"a2df7fbcc1d1983151a230f218fab8275b725132466812adedbabdbf1f5fee25","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Order.UpperLower\npublic import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace\npublic import Mathlib.Topology.Order.DenselyOrdered\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies, Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Kexing Ying\n-/\n/-!\n# Order-connected sets are null-measurable\n\nThis file proves that order-connected sets in `ℝⁿ` under the pointwise order are null-measurable.\nRecall that `x ≤ y` iff `∀ i, x i ≤ y i`, and `s` is order-connected iff\n`∀ x y ∈ s, ∀ z, x ≤ z → z ≤ y → z ∈ s`.\n\n## Main declarations\n\n* `Set.OrdConnected.null_frontier`: The frontier of an order-connected set in `ℝⁿ` has measure `0`.\n\n## Notes\n\nWe prove null-measurability in `ℝⁿ` with the `∞`-metric, but this transfers directly to `ℝⁿ` with\nthe Euclidean metric because they have the same measurable sets.\n\nNull-measurability can't be strengthened to measurability because any antichain (and in particular\nany subset of the antidiagonal `{(x, y) | x + y = 0}`) is order-connected.\n\n## Sketch proof\n\n1. To show an order-connected set is null-measurable, it is enough to show it has null frontier.\n2. Since an order-connected set is the intersection of its upper and lower closure, it's enough to\n  show that upper and lower sets have null frontier.\n3. WLOG let's prove it for an upper set `s`.\n4. By the Lebesgue density theorem, it is enough to show that any frontier point `x` of `s` is not a\n  Lebesgue point, namely we want the density of `s` over small balls centered at `x` to not tend to\n  either `0` or `1`.\n5. This is true, since by the upper setness of `s` we can intercalate a ball of radius `δ / 4` in\n  `s` intersected with the upper quadrant of the ball of radius `δ` centered at `x` (recall that the\n  balls are taken in the ∞-norm, so they are cubes), and another ball of radius `δ / 4` in `sᶜ` and\n  the lower quadrant of the ball of radius `δ` centered at `x`.\n\n## TODO\n\nGeneralize so that it also applies to `ℝ × ℝ`, for example.\n-/\n\npublic section\n\nopen Filter MeasureTheory Metric Set\nopen scoped Topology\n\nvariable {ι : Type*} [Fintype ι] {s : Set (ι → ℝ)} {x : ι → ℝ}\n\n/-- If we can fit a small ball inside a set `s` intersected with any neighborhood of `x`, then the\ndensity of `s` near `x` is not `0`.\n\nAlong with `aux₁`, this proves that `x` is not a Lebesgue point of `s`. This will be used to prove\nthat the frontier of an order-connected set is null. -/\nprivate lemma aux₀\n    (h : ∀ δ, 0 < δ →\n      ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior s) :\n    ¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)\n        (𝓝 0) := by\n  choose f hf₀ hf₁ using h\n  intro H\n  obtain ⟨ε, -, hε', hε₀⟩ := exists_seq_strictAnti_tendsto_nhdsWithin (0 : ℝ)\n  refine not_eventually.2\n    (Frequently.of_forall fun _ ↦ lt_irrefl <| ENNReal.ofReal <| 4⁻¹ ^ Fintype.card ι)\n    ((Filter.Tendsto.eventually_lt (H.comp hε₀) tendsto_const_nhds ?_).mono fun n ↦\n      lt_of_le_of_lt ?_)\n  on_goal 2 =>\n    calc\n      ENNReal.ofReal (4⁻¹ ^ Fintype.card ι)\n        = volume (closedBall (f (ε n) (hε' n)) (ε n / 4)) / volume (closedBall x (ε n)) := ?_\n      _ ≤ volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n)) := by\n        gcongr\n        exact subset_inter ((hf₁ _ <| hε' n).trans interior_subset_closure) <| hf₀ _ <| hε' n\n    have := hε' n\n    rw [Real.volume_pi_closedBall, Real.volume_pi_closedBall, ← ENNReal.ofReal_div_of_pos,\n      ← div_pow, mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]\n  all_goals positivity\n\n/-- If we can fit a small ball inside a set `sᶜ` intersected with any neighborhood of `x`, then the\ndensity of `s` near `x` is not `1`.\n\nAlong with `aux₀`, this proves that `x` is not a Lebesgue point of `s`. This will be used to prove\nthat the frontier of an order-connected set is null. -/\nprivate lemma aux₁\n    (h : ∀ δ, 0 < δ →\n      ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior sᶜ) :\n    ¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)\n        (𝓝 1) := by\n  choose f hf₀ hf₁ using h\n  intro H\n  obtain ⟨ε, -, hε', hε₀⟩ := exists_seq_strictAnti_tendsto_nhdsWithin (0 : ℝ)\n  refine not_eventually.2\n      (Frequently.of_forall fun _ ↦ lt_irrefl <| 1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι))\n      ((Filter.Tendsto.eventually_lt tendsto_const_nhds (H.comp hε₀) <|\n            ENNReal.sub_lt_self ENNReal.one_ne_top one_ne_zero ?_).mono\n        fun n ↦ lt_of_le_of_lt' ?_)\n  on_goal 2 =>\n    calc\n      volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n))\n        ≤ volume (closedBall x (ε n) \\ closedBall (f (ε n) <| hε' n) (ε n / 4)) /\n          volume (closedBall x (ε n)) := by\n        gcongr\n        rw [sdiff_eq_compl_inter]\n        refine inter_subset_inter_left _ ?_\n        rw [subset_compl_comm, ← interior_compl]\n        exact hf₁ _ _\n      _ = 1 - ENNReal.ofReal (4⁻¹ ^ Fintype.card ι) := ?_\n    have := hε' n\n    rw [measure_sdiff (hf₀ _ _) _ ((Real.volume_pi_closedBall _ _).trans_ne ENNReal.ofReal_ne_top),\n      Real.volume_pi_closedBall, Real.volume_pi_closedBall, ENNReal.sub_div fun _ _ ↦ _,\n      ENNReal.div_self _ ENNReal.ofReal_ne_top, ← ENNReal.ofReal_div_of_pos, ← div_pow,\n      mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]\n  all_goals try positivity\n  · simp_all\n  · exact measurableSet_closedBall.nullMeasurableSet\n\nTarget:\ntheorem IsUpperSet.null_frontier (hs : IsUpperSet s) : volume (frontier s) = 0 :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Order","family_id":"isupperset","file_id":"mathlib/Mathlib/MeasureTheory/Order/UpperLower.lean","sample_id":"4561b215385a78d721dfe36030198c0459e29f4db8031b7c27e613ae90c4a078"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f258ff7f229ca2a8c793329e07d0fcda1df2d4bda138c89c5b29fe00a0404c2f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1caba13c5bfb2e48e2aa824af4b9e227bed2165274d63dd634f59f34ffe47e2e","source_sha256":"e357f4d4f71c23dea84a87d11c8330d1561110f75700ec18541bc2103a8420d2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine summable_geometric_iff_norm_lt_one.mp ?_\n  simp_rw [← map_pow]\n  exact hsum.comp_injective <| Nat.pow_right_injective hp","hard_negative":false,"metrics":{"chosen_tokens":22,"rejected_tokens":2,"token_jaccard":0.047619,"token_length_ratio":0.090909},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"a2ec5e38c441b6f51bdf897b02b35374df0d2b6d2b9d901769846c28559ab449","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Ring.InfiniteSum\npublic import Mathlib.Analysis.SpecificLimits.Normed\npublic import Mathlib.NumberTheory.ArithmeticFunction.Defs\npublic import Mathlib.NumberTheory.SmoothNumbers\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Euler Products\n\nThe main result in this file is `EulerProduct.eulerProduct_hasProd`, which says that\nif `f : ℕ → R` is norm-summable, where `R` is a complete normed commutative ring and `f` is\nmultiplicative on coprime arguments with `f 0 = 0`, then\n`∏' p : Primes, ∑' e : ℕ, f (p^e)` converges to `∑' n, f n`.\n\n`ArithmeticFunction.IsMultiplicative.eulerProduct_hasProd` is a version\nfor multiplicative arithmetic functions in the sense of\n`ArithmeticFunction.IsMultiplicative`.\n\nThere is also a version `EulerProduct.eulerProduct_completely_multiplicative_hasProd`,\nwhich states that `∏' p : Primes, (1 - f p)⁻¹` converges to `∑' n, f n`\nwhen `f` is completely multiplicative with values in a complete normed field `F`\n(implemented as `f : ℕ →*₀ F`).\n\nThere are variants stating the equality of the infinite product and the infinite sum\n(`EulerProduct.eulerProduct_tprod`, `ArithmeticFunction.IsMultiplicative.eulerProduct_tprod`,\n`EulerProduct.eulerProduct_completely_multiplicative_tprod`) and also variants stating\nthe convergence of the sequence of partial products over primes `< n`\n(`EulerProduct.eulerProduct`, `ArithmeticFunction.IsMultiplicative.eulerProduct`,\n`EulerProduct.eulerProduct_completely_multiplicative`.)\n\nAn intermediate step is `EulerProduct.summable_and_hasSum_factoredNumbers_prod_filter_prime_tsum`\n(and its variant `EulerProduct.summable_and_hasSum_factoredNumbers_prod_filter_prime_geometric`),\nwhich relates the finite product over primes `p ∈ s` to the sum of `f n` over `s`-factored `n`,\nfor `s : Finset ℕ`.\n\n## Tags\n\nEuler product, multiplicative function\n-/\n\npublic section\n\n/-- If `f` is multiplicative and summable, then its values at natural numbers `> 1`\nhave norm strictly less than `1`. -/\n\nTarget:\nlemma Summable.norm_lt_one {F : Type*} [NormedDivisionRing F] [CompleteSpace F] {f : ℕ →* F}\n    (hsum : Summable f) {p : ℕ} (hp : 1 < p) :\n    ‖f p‖ < 1 :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/EulerProduct","family_id":"summable","file_id":"mathlib/Mathlib/NumberTheory/EulerProduct/Basic.lean","sample_id":"1caba13c5bfb2e48e2aa824af4b9e227bed2165274d63dd634f59f34ffe47e2e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0dd5faeb64813fa427f25f46617ddcf71a676309c87e033eb80368aa47d425e1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1f592e66efeee00906ccbae7089ea605c8923ff6c257bc54ee50b7626367f3eb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fc41a1bfba603ad2786ab361571962d38f904641b8aa32668047f0fd3d10b946","source_sha256":"06b8f3152156917b43eb16aa25f88f706675fd088336840c57324ab27b671934","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [← C_eq_intCast, coeff_C, Int.cast_ite, Int.cast_zero]","hard_negative":false,"metrics":{"chosen_tokens":17,"rejected_tokens":22,"token_jaccard":0.764706,"token_length_ratio":1.294118},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"a331ea818521809f6ab59e5a011aa6065e295974751e1f77b5649b65f721a37f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Degree.Lemmas\n\nNamespace:\nMathlib.Tactic.ComputeDegree\n\nLocal context:\n/-\nCopyright (c) 2023 Damiano Testa. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Damiano Testa\n-/\n/-!\n\n# `compute_degree` and `monicity`: tactics for explicit polynomials\n\nThis file defines two related tactics: `compute_degree` and `monicity`.\n\nUsing `compute_degree` when the goal is of one of the seven forms\n* `natDegree f ≤ d` (or `<`),\n* `degree f ≤ d` (or `<`),\n* `natDegree f = d`,\n* `degree f = d`,\n* `coeff f d = r`, if `d` is the degree of `f`,\n\ntries to solve the goal.\nIt may leave side-goals, in case it is not entirely successful.\n\nUsing `monicity` when the goal is of the form `Monic f` tries to solve the goal.\nIt may leave side-goals, in case it is not entirely successful.\n\nBoth tactics admit a `!` modifier (`compute_degree!` and `monicity!`) instructing\nLean to try harder to close the goal.\n\nSee the doc-strings for more details.\n\n## Future work\n\n* Currently, `compute_degree` does not deal correctly with some edge cases.  For instance,\n  ```lean\n  example [Semiring R] : natDegree (C 0 : R[X]) = 0 := by\n    compute_degree\n  --  ⊢ 0 ≠ 0\n  ```\n  Still, it may not be worth to provide special support for `natDegree f = 0`.\n* Make sure that numerals in coefficients are treated correctly.\n* Make sure that `compute_degree` works with goals of the form `degree f ≤ ↑d`, with an\n  explicit coercion from `ℕ` on the RHS.\n* Add support for proving goals of the from `natDegree f ≠ 0` and `degree f ≠ 0`.\n* Make sure that `degree`, `natDegree` and `coeff` are equally supported.\n\n## Implementation details\n\nAssume that `f : R[X]` is a polynomial with coefficients in a semiring `R` and\n`d` is either in `ℕ` or in `WithBot ℕ`.\n\nIf the goal has the form `natDegree f < d`, then we convert it to two separate goals:\n* `natDegree f ≤ ?_`, on which we apply the following steps;\n* `?_ < d`;\n\nwhere `?_` is a metavariable that `compute_degree` computes in its process.\nWe proceed similarly for `degree f < d`.\n\nIf the goal has the form `natDegree f = d`, then we convert it to three separate goals:\n* `natDegree f ≤ d`;\n* `coeff f d = r`;\n* `r ≠ 0`.\n\nSimilarly, an initial goal of the form `degree f = d` gives rise to goals of the form\n* `degree f ≤ d`;\n* `coeff f d = r`;\n* `r ≠ 0`.\n\nNext, we apply successively lemmas whose side-goals all have the shape\n* `natDegree f ≤ d`;\n* `degree f ≤ d`;\n* `coeff f d = r`;\n\nplus possibly \"numerical\" identities and choices of elements in `ℕ`, `WithBot ℕ`, and `R`.\n\nRecursing into `f`, we break apart additions, multiplications, powers, subtractions,...\nThe leaves of the process are\n* numerals, `C a`, `X` and `monomial a n`, to which we assign degree `0`, `1` and `a` respectively;\n* `fvar`s `f`, to which we tautologically assign degree `natDegree f`.\n-/\n\npublic meta section\n\nopen Polynomial\n\nnamespace Mathlib.Tactic.ComputeDegree\n\nsection recursion_lemmas\n/-!\n### Simple lemmas about `natDegree`\n\nThe lemmas in this section all have the form `natDegree <some form of cast> ≤ 0`.\nTheir proofs are weakenings of the stronger lemmas `natDegree <same> = 0`.\nThese are the lemmas called by `compute_degree` on (almost) all the leaves of its recursion.\n-/\n\nvariable {R : Type*}\n\nsection semiring\nvariable [Semiring R]\n\ntheorem natDegree_C_le (a : R) : natDegree (C a) ≤ 0 := (natDegree_C a).le\n\ntheorem natDegree_natCast_le (n : ℕ) : natDegree (n : R[X]) ≤ 0 := (natDegree_natCast _).le\ntheorem natDegree_zero_le : natDegree (0 : R[X]) ≤ 0 := natDegree_zero.le\ntheorem natDegree_one_le : natDegree (1 : R[X]) ≤ 0 := natDegree_one.le\n\ntheorem coeff_add_of_eq {n : ℕ} {a b : R} {f g : R[X]}\n    (h_add_left : f.coeff n = a) (h_add_right : g.coeff n = b) :\n    (f + g).coeff n = a + b := by subst ‹_› ‹_›; apply coeff_add\n\ntheorem coeff_mul_add_of_le_natDegree_of_eq_ite {d df dg : ℕ} {a b : R} {f g : R[X]}\n    (h_mul_left : natDegree f ≤ df) (h_mul_right : natDegree g ≤ dg)\n    (h_mul_left : f.coeff df = a) (h_mul_right : g.coeff dg = b) (ddf : df + dg ≤ d) :\n    (f * g).coeff d = if d = df + dg then a * b else 0 := by\n  split_ifs with h\n  · subst h_mul_left h_mul_right h\n    exact coeff_mul_add_eq_of_natDegree_le ‹_› ‹_›\n  · apply coeff_eq_zero_of_natDegree_lt\n    apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ddf ?_)\n    · exact natDegree_mul_le_of_le ‹_› ‹_›\n    · exact ne_comm.mp h\n\ntheorem coeff_pow_of_natDegree_le_of_eq_ite' {m n o : ℕ} {a : R} {p : R[X]}\n    (h_pow : natDegree p ≤ n) (h_exp : m * n ≤ o) (h_pow_bas : coeff p n = a) :\n    coeff (p ^ m) o = if o = m * n then a ^ m else 0 := by\n  split_ifs with h\n  · subst h h_pow_bas\n    exact coeff_pow_of_natDegree_le ‹_›\n  · apply coeff_eq_zero_of_natDegree_lt\n    apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ‹_› ?_)\n    · exact natDegree_pow_le_of_le m ‹_›\n    · exact Iff.mp ne_comm h\n\nsection SMul\n\nvariable {S : Type*} [SMulZeroClass S R] {n : ℕ} {a : S} {f : R[X]}\n\ntheorem natDegree_smul_le_of_le (hf : natDegree f ≤ n) :\n    natDegree (a • f) ≤ n :=\n  (natDegree_smul_le a f).trans hf\n\ntheorem degree_smul_le_of_le (hf : degree f ≤ n) :\n    degree (a • f) ≤ n :=\n  (degree_smul_le a f).trans hf\n\ntheorem coeff_smul : (a • f).coeff n = a • f.coeff n := rfl\n\nend SMul\n\nsection congr_lemmas\n\n/-- The following two lemmas should be viewed as a hand-made \"congr\"-lemmas.\nThey achieve the following goals.\n* They introduce *two* fresh metavariables replacing the given one `deg`,\n  one for the `natDegree ≤` computation and one for the `coeff =` computation.\n  This helps `compute_degree`, since it does not \"pre-estimate\" the degree,\n  but it \"picks it up along the way\".\n* They split checking the inequality `coeff p n ≠ 0` into the task of\n  finding a value `c` for the `coeff` and then\n  proving that this value is non-zero by `coeff_ne_zero`.\n-/\ntheorem natDegree_eq_of_le_of_coeff_ne_zero' {deg m o : ℕ} {c : R} {p : R[X]}\n    (h_natDeg_le : natDegree p ≤ m) (coeff_eq : coeff p o = c)\n    (coeff_ne_zero : c ≠ 0) (deg_eq_deg : m = deg) (coeff_eq_deg : o = deg) :\n    natDegree p = deg := by\n  subst coeff_eq deg_eq_deg coeff_eq_deg\n  exact natDegree_eq_of_le_of_coeff_ne_zero ‹_› ‹_›\n\ntheorem degree_eq_of_le_of_coeff_ne_zero' {deg m o : WithBot ℕ} {c : R} {p : R[X]}\n    (h_deg_le : degree p ≤ m) (coeff_eq : coeff p (WithBot.unbotD 0 deg) = c)\n    (coeff_ne_zero : c ≠ 0) (deg_eq_deg : m = deg) (coeff_eq_deg : o = deg) :\n    degree p = deg := by\n  subst coeff_eq coeff_eq_deg deg_eq_deg\n  rcases eq_or_ne m ⊥ with rfl | hh\n  · exact bot_unique h_deg_le\n  · obtain ⟨m, rfl⟩ := WithBot.ne_bot_iff_exists.mp hh\n    exact degree_eq_of_le_of_coeff_ne_zero ‹_› ‹_›\n\nvariable {m n : ℕ} {f : R[X]} {r : R}\n\ntheorem coeff_congr_lhs (h : coeff f m = r) (natDeg_eq_coeff : m = n) : coeff f n = r :=\n  natDeg_eq_coeff ▸ h\ntheorem coeff_congr (h : coeff f m = r) (natDeg_eq_coeff : m = n) {s : R} (rs : r = s) :\n    coeff f n = s :=\n  natDeg_eq_coeff ▸ rs ▸ h\n\nend congr_lemmas\n\nend semiring\n\nsection ring\nvariable [Ring R]\n\ntheorem natDegree_intCast_le (n : ℤ) : natDegree (n : R[X]) ≤ 0 := (natDegree_intCast _).le\n\ntheorem coeff_sub_of_eq {n : ℕ} {a b : R} {f g : R[X]} (hf : f.coeff n = a) (hg : g.coeff n = b) :\n    (f - g).coeff n = a - b := by subst hf hg; apply coeff_sub\n\nTarget:\ntheorem coeff_intCast_ite {n : ℕ} {a : ℤ} : (Int.cast a : R[X]).coeff n = ite (n = 0) a 0 :=\n\nProof body:\n","rejected":"by\n  simp only [← C_eq_intCast, coeff_C, Int.cast_ite, Int.cast_zero]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic","family_id":"coeff_intcast_ite","file_id":"mathlib/Mathlib/Tactic/ComputeDegree.lean","sample_id":"fc41a1bfba603ad2786ab361571962d38f904641b8aa32668047f0fd3d10b946"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5710d934b0159d30d30972bfc5b7ddfd3eb8cd27b8c73a16ac7a346cdcff98c4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"df1deb7781ab53caab0682db75c3859714f67ed27453bd1a013d385737e04efd","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"519e1718b1b4803bcd063f65500a31317c48e550e0d65b418a9de8968712bef5","source_sha256":"c2ae48c6bd398849186d6b191b0d4c349dab84b9865f7c6ccfbd7ab5b2b3ca24","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rfl\n\nvariable (φ : A →ₐc[R] B)","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":5,"token_jaccard":0.052632,"token_length_ratio":0.333333},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"a340e681742a6ee9487c30fbfe743b01a3b74e26509fa694a1669b0fbf5bf999","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Coalgebra.Hom\npublic import Mathlib.RingTheory.Bialgebra.Basic\n\nNamespace:\nBialgHom\n\nLocal context:\n/-\nCopyright (c) 2024 Amelia Livingston. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov, Amelia Livingston\n-/\n/-!\n# Homomorphisms of `R`-bialgebras\n\nThis file defines bundled homomorphisms of `R`-bialgebras. We simply mimic\n`Mathlib/Algebra/Algebra/Hom.lean`.\n\n## Main definitions\n\n* `BialgHom R A B`: the type of `R`-bialgebra morphisms from `A` to `B`.\n* `Bialgebra.counitBialgHom R A : A →ₐc[R] R`: the counit of a bialgebra as a bialgebra\n  homomorphism.\n\n## Notation\n\n* `A →ₐc[R] B` : `R`-bialgebra homomorphism from `A` to `B`.\n\n-/\n\n@[expose] public section\n\nopen TensorProduct Bialgebra Coalgebra Function\n\nuniverse u v w\n\n/-- Given `R`-algebras `A, B` with comultiplication maps `Δ_A, Δ_B` and counit maps\n`ε_A, ε_B`, an `R`-bialgebra homomorphism `A →ₐc[R] B` is an `R`-algebra map `f` such that\n`ε_B ∘ f = ε_A` and `(f ⊗ f) ∘ Δ_A = Δ_B ∘ f`. -/\nstructure BialgHom (R A B : Type*) [CommSemiring R]\n    [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]\n    [CoalgebraStruct R A] [CoalgebraStruct R B] extends A →ₗc[R] B, A →* B\n\n/-- Reinterpret a `BialgHom` as a `MonoidHom` -/\nadd_decl_doc BialgHom.toMonoidHom\n\n@[inherit_doc BialgHom]\ninfixr:25 \" →ₐc \" => BialgHom _\n\n@[inherit_doc]\nnotation:25 A \" →ₐc[\" R \"] \" B => BialgHom R A B\n\n/-- `BialgHomClass F R A B` asserts `F` is a type of bundled bialgebra homomorphisms\nfrom `A` to `B`. -/\nclass BialgHomClass (F : Type*) (R A B : outParam Type*)\n    [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]\n    [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B] : Prop\n    extends CoalgHomClass F R A B, MonoidHomClass F A B\n\nnamespace BialgHomClass\n\nvariable {R A B F : Type*}\n\nsection\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]\n  [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B]\n  [BialgHomClass F R A B]\n\ninstance (priority := 100) toAlgHomClass : AlgHomClass F R A B where\n  map_mul := map_mul\n  map_one := map_one\n  map_add := map_add\n  map_zero := map_zero\n  commutes := fun c r => by\n    simp only [Algebra.algebraMap_eq_smul_one, map_smul, map_one]\n\n/-- Turn an element of a type `F` satisfying `BialgHomClass F R A B` into an actual\n`BialgHom`. This is declared as the default coercion from `F` to `A →ₐc[R] B`. -/\n@[coe]\ndef toBialgHom (f : F) : A →ₐc[R] B :=\n  { CoalgHomClass.toCoalgHom f, AlgHomClass.toAlgHom f with\n    toFun := f }\n\ninstance instCoeToBialgHom :\n    CoeHead F (A →ₐc[R] B) :=\n  ⟨BialgHomClass.toBialgHom⟩\n\nend\nsection\nvariable [CommSemiring R] [Semiring A] [Bialgebra R A] [Semiring B] [Bialgebra R B]\n  [FunLike F A B] [BialgHomClass F R A B]\n\n@[simp]\ntheorem counitAlgHom_comp (f : F) :\n    (counitAlgHom R B).comp (AlgHomClass.toAlgHom f) = counitAlgHom R A :=\n  AlgHom.toLinearMap_injective (CoalgHomClass.counit_comp f)\n\n@[simp]\ntheorem map_comp_comulAlgHom (f : F) :\n    (Algebra.TensorProduct.map (AlgHomClass.toAlgHom f) (AlgHomClass.toAlgHom f)).comp\n      (comulAlgHom R A) = (comulAlgHom R B).comp (AlgHomClass.toAlgHom f) :=\n  AlgHom.toLinearMap_injective (CoalgHomClass.map_comp_comul f)\n\nend\nend BialgHomClass\n\nnamespace BialgHom\n\nvariable {R A B C D : Type*} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D]\n\nsection AlgebraCoalgebra\n\nvariable [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D]\n  [CoalgebraStruct R A] [CoalgebraStruct R B] [CoalgebraStruct R C] [CoalgebraStruct R D]\n\ninstance funLike : FunLike (A →ₐc[R] B) A B where\n  coe f := f.toFun\n  coe_injective f g h := by\n    rcases f with ⟨_, _⟩\n    rcases g with ⟨_, _⟩\n    simp_all\n\ninstance bialgHomClass : BialgHomClass (A →ₐc[R] B) R A B where\n  map_add := fun f => f.map_add'\n  map_smulₛₗ := fun f => f.map_smul'\n  counit_comp := fun f => f.counit_comp\n  map_comp_comul := fun f => f.map_comp_comul\n  map_mul := fun f => f.map_mul'\n  map_one := fun f => f.map_one'\n\n/-- See Note [custom simps projection] -/\ndef Simps.apply {R α β : Type*} [CommSemiring R]\n    [Semiring α] [Algebra R α] [Semiring β]\n    [Algebra R β] [CoalgebraStruct R α] [CoalgebraStruct R β]\n    (f : α →ₐc[R] β) : α → β := f\n\ninitialize_simps_projections BialgHom (toFun → apply, as_prefix toCoalgHom)\n\n@[simp]\nprotected theorem coe_coe {F : Type*} [FunLike F A B] [BialgHomClass F R A B] (f : F) :\n    ⇑(f : A →ₐc[R] B) = f :=\n  rfl\n\n@[simp]\ntheorem coe_mk {f : A →ₗc[R] B} (h h₁) : ((⟨f, h, h₁⟩ : A →ₐc[R] B) : A → B) = f :=\n  rfl\n\n@[norm_cast]\ntheorem coe_mks {f : A → B} (h₀ h₁ h₂ h₃ h₄ h₅) :\n    ⇑(⟨⟨⟨⟨f, h₀⟩, h₁⟩, h₂, h₃⟩, h₄, h₅⟩ : A →ₐc[R] B) = f :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_coalgHom_mk {f : A →ₗc[R] B} (h h₁) :\n    ((⟨f, h, h₁⟩ : A →ₐc[R] B) : A →ₗc[R] B) = f := by\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_toCoalgHom (f : A →ₐc[R] B) : ⇑(f : A →ₗc[R] B) = f :=\n  rfl\n\nlemma toCoalgHom_apply (f : A →ₐc[R] B) (a : A) : f.toCoalgHom a = f a := rfl\n\n@[simp, norm_cast]\ntheorem coe_toLinearMap (f : A →ₐc[R] B) : ⇑(f : A →ₗ[R] B) = f :=\n  rfl\n\n/-- Turn a bialgebra homomorphism into an algebra homomorphism. -/\n@[coe]\ndef toAlgHom (f : A →ₐc[R] B) : A →ₐ[R] B where\n  __ := f\n  map_zero' := f.map_zero\n  commutes' := by\n    simp [Algebra.algebraMap_eq_smul_one, toCoalgHom_apply]\n\ninstance : Coe (A →ₐc[R] B) (A →ₐ[R] B) := ⟨toAlgHom⟩\n\n@[simp, norm_cast]\ntheorem coe_toAlgHom (f : A →ₐc[R] B) : ⇑(f : A →ₐ[R] B) = f :=\n  rfl\n\nTarget:\ntheorem toAlgHom_toLinearMap (f : A →ₐc[R] B) :\n    ((f : A →ₐ[R] B) : A →ₗ[R] B) = f :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_519e1718b1b4","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"bff67f76e82abb9c386c8872c74302d76034ec0f62fa7fe37fc18c8690890f71","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Bialgebra","family_id":"toalghom_tolinearmap","file_id":"mathlib/Mathlib/RingTheory/Bialgebra/Hom.lean","sample_id":"519e1718b1b4803bcd063f65500a31317c48e550e0d65b418a9de8968712bef5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"679574657f773b5361b65d8bc217854b5dee221b7dde4fd6f07b322f738d2333","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f352361c545b1f8b216789e3adf4cb0dda1a44b18ee9c8950bd563e6230dd79d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bf23546e5597548a8a703fba0f21b06b38585a3dc1e1cc164336dbb8b96373a4","source_sha256":"a9cefe337b8a57141453affd06f1d59efc30b22cebabbd270f973d0add5ab80b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    rwa [← map_natCast' _ he, ← map_natCast' _ he, he'.eq_iff, Nat.cast_inj] at h","hard_negative":true,"metrics":{"chosen_tokens":23,"rejected_tokens":2,"token_jaccard":0.058824,"token_length_ratio":0.086957},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"a3a33a4e720737d60648e64d87942495f4ceeeee8903db1b00e6a784e4be22ac","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharZero.Defs\npublic import Mathlib.Algebra.Group.Equiv.Defs\npublic import Mathlib.Data.Nat.Cast.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Paul Lezeau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Paul Lezeau\n-/\n/-!\n# Transporting `CharZero` across injective `AddMonoidHom`s\n\nThis file exists in order to avoid adding extra imports to other files in this subdirectory.\n-/\n\npublic section\n\nTarget:\ntheorem CharZero.of_addMonoidHom {M N : Type*} [AddCommMonoidWithOne M] [AddCommMonoidWithOne N]\n    [CharZero M] (e : M →+ N) (he : e 1 = 1) (he' : Function.Injective e) : CharZero N where\n  cast_injective n m h :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_bf23546e5597","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b48a9cf12347d77462a3bba311a51487fe50c86a6c82fefce933044f59edb2bb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/CharZero","family_id":"charzero","file_id":"mathlib/Mathlib/Algebra/CharZero/AddMonoidHom.lean","sample_id":"bf23546e5597548a8a703fba0f21b06b38585a3dc1e1cc164336dbb8b96373a4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"63a4845b3fb713f383d6aefe7ca814d2f2062db02f00a1f2f03bbbb49cf53c38","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2b78fa8807ec50bfc8ab5c93ca7eeaa86e8f4d73fc76cde03273374bc7683948","source_sha256":"e433c304165198b6af36e2c9a124b401b67821948088461657a4c34ad256bb4a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  replace h (a m) : Q₁ (a ⊗ₜ m) = Q₂ (a ⊗ₜ m) := by\n    rw [← mul_one a, ← smul_eq_mul, ← smul_tmul', QuadraticMap.map_smul, QuadraticMap.map_smul, h]\n  ext x\n  induction x with\n  | tmul => simp [h]\n  | zero => simp\n  | add x y hx hy =>\n    have : Q₁.polarBilin = Q₂.polarBilin := by\n      ext\n      dsimp [polar]\n      rw [← TensorProduct.tmul_add, h, h, h]\n    replace := congr($this x y)\n    dsimp [polar] at this\n    linear_combination (norm := module) this + hx + hy","hard_negative":false,"metrics":{"chosen_tokens":129,"rejected_tokens":5,"token_jaccard":0.017544,"token_length_ratio":0.03876},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"a3d9c38dfc97419de8d1b30a70a23a68125200039ca1ea1a7676abafdb9aabb0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.BilinearForm.TensorProduct\npublic import Mathlib.LinearAlgebra.QuadraticForm.Basic\npublic import Mathlib.Tactic.LinearCombination\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# The quadratic form on a tensor product\n\n## Main definitions\n\n* `QuadraticForm.tensorDistrib (Q₁ ⊗ₜ Q₂)`: the quadratic form on `M₁ ⊗ M₂` constructed by applying\n  `Q₁` on `M₁` and `Q₂` on `M₂`. This construction is not available in characteristic two.\n\n-/\n\n@[expose] public section\n\nuniverse uR uA uM₁ uM₂ uN₁ uN₂\n\nvariable {R : Type uR} {A : Type uA} {M₁ : Type uM₁} {M₂ : Type uM₂} {N₁ : Type uN₁} {N₂ : Type uN₂}\n\nopen LinearMap (BilinMap BilinForm)\nopen TensorProduct QuadraticMap\n\nsection CommRing\nvariable [CommRing R] [CommRing A]\nvariable [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup N₁] [AddCommGroup N₂]\nvariable [Algebra R A] [Module R M₁] [Module A M₁] [Module R N₁] [Module A N₁]\nvariable [SMulCommClass R A M₁] [IsScalarTower R A M₁] [IsScalarTower R A N₁]\nvariable [Module R M₂] [Module R N₂]\n\nsection InvertibleTwo\nvariable [Invertible (2 : R)]\n\nnamespace QuadraticMap\n\nvariable (R A) in\n/-- The tensor product of two quadratic maps injects into quadratic maps on tensor products.\n\nNote this is heterobasic; the quadratic map on the left can take values in a module over a larger\nring than the one on the right. -/\ndef tensorDistrib :\n    QuadraticMap A M₁ N₁ ⊗[R] QuadraticMap R M₂ N₂ →ₗ[A] QuadraticMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=\n  letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm\n  -- while `letI`s would produce a better term than `let`, they would make this already-slow\n  -- definition even slower.\n  let toQ := BilinMap.toQuadraticMapLinearMap A A (M₁ ⊗[R] M₂)\n  let tmulB := BilinMap.tensorDistrib R A (M₁ := M₁) (M₂ := M₂)\n  let toB := AlgebraTensorModule.map\n      (QuadraticMap.associated : QuadraticMap A M₁ N₁ →ₗ[A] BilinMap A M₁ N₁)\n      (QuadraticMap.associated : QuadraticMap R M₂ N₂ →ₗ[R] BilinMap R M₂ N₂)\n  toQ ∘ₗ tmulB ∘ₗ toB\n\n@[simp]\ntheorem tensorDistrib_tmul (Q₁ : QuadraticMap A M₁ N₁) (Q₂ : QuadraticMap R M₂ N₂) (m₁ : M₁)\n    (m₂ : M₂) : tensorDistrib R A (Q₁ ⊗ₜ Q₂) (m₁ ⊗ₜ m₂) = Q₁ m₁ ⊗ₜ Q₂ m₂ :=\n  letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm\n  (BilinMap.tensorDistrib_tmul _ _ _ _ _ _).trans <| congr_arg₂ _\n    (associated_eq_self_apply _ _ _) (associated_eq_self_apply _ _ _)\n\n/-- The tensor product of two quadratic maps, a shorthand for dot notation. -/\nprotected abbrev tmul (Q₁ : QuadraticMap A M₁ N₁)\n    (Q₂ : QuadraticMap R M₂ N₂) : QuadraticMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=\n  tensorDistrib R A (Q₁ ⊗ₜ[R] Q₂)\n\ntheorem associated_tmul [Invertible (2 : A)]\n    (Q₁ : QuadraticMap A M₁ N₁) (Q₂ : QuadraticMap R M₂ N₂) :\n    (Q₁.tmul Q₂).associated = Q₁.associated.tmul Q₂.associated := by\n  letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm\n  rw [QuadraticMap.tmul, BilinMap.tmul]\n  have : Subsingleton (Invertible (2 : A)) := inferInstance\n  convert!\n    associated_left_inverse A\n      (LinearMap.BilinMap.tmul_isSymm (QuadraticMap.associated_isSymm A Q₁)\n        (QuadraticMap.associated_isSymm R Q₂))\n\nend QuadraticMap\n\nnamespace QuadraticForm\n\nvariable (R A) in\n/-- The tensor product of two quadratic forms injects into quadratic forms on tensor products.\n\nNote this is heterobasic; the quadratic form on the left can take values in a larger ring than\nthe one on the right. -/\ndef tensorDistrib :\n    QuadraticForm A M₁ ⊗[R] QuadraticForm R M₂ →ₗ[A] QuadraticForm A (M₁ ⊗[R] M₂) :=\n  (AlgebraTensorModule.rid R A A).congrQuadraticMap.toLinearMap ∘ₗ QuadraticMap.tensorDistrib R A\n\n-- TODO: make the RHS `MulOpposite.op (Q₂ m₂) • Q₁ m₁` so that this has a nicer defeq for\n-- `R = A` of `Q₁ m₁ * Q₂ m₂`.\n@[simp]\ntheorem tensorDistrib_tmul (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) (m₁ : M₁) (m₂ : M₂) :\n    tensorDistrib R A (Q₁ ⊗ₜ Q₂) (m₁ ⊗ₜ m₂) = Q₂ m₂ • Q₁ m₁ :=\n  letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm\n  (LinearMap.BilinForm.tensorDistrib_tmul _ _ _ _ _ _ _ _).trans <| congr_arg₂ _\n    (associated_eq_self_apply _ _ _) (associated_eq_self_apply _ _ _)\n\n/-- The tensor product of two quadratic forms, a shorthand for dot notation. -/\nprotected abbrev tmul (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) :\n    QuadraticForm A (M₁ ⊗[R] M₂) :=\n  tensorDistrib R A (Q₁ ⊗ₜ[R] Q₂)\n\ntheorem associated_tmul [Invertible (2 : A)] (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) :\n    (Q₁.tmul Q₂).associated = BilinForm.tmul Q₁.associated Q₂.associated := by\n  rw [BilinForm.tmul, BilinForm.tensorDistrib, LinearMap.comp_apply, ← BilinMap.tmul,\n    ← QuadraticMap.associated_tmul Q₁ Q₂, LinearEquiv.coe_coe, LinearEquiv.congrRight₂_apply]\n  ext : 6\n  simp [associated_apply]\n  rfl\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem polarBilin_tmul [Invertible (2 : A)] (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) :\n    polarBilin (Q₁.tmul Q₂) = ⅟(2 : A) • BilinForm.tmul (polarBilin Q₁) (polarBilin Q₂) := by\n  simp_rw [← two_nsmul_associated A, ← two_nsmul_associated R, BilinForm.tmul, tmul_smul,\n    ← smul_tmul', map_nsmul, associated_tmul]\n  rw [smul_comm (_ : A) (_ : ℕ), ← smul_assoc, two_smul _ (_ : A), invOf_two_add_invOf_two,\n    one_smul]\n\nvariable (A) in\n/-- The base change of a quadratic form. -/\nprotected def baseChange (Q : QuadraticForm R M₂) : QuadraticForm A (A ⊗[R] M₂) :=\n  QuadraticForm.tmul (R := R) (A := A) (M₁ := A) (M₂ := M₂) (QuadraticMap.sq (R := A)) Q\n\n@[simp]\ntheorem baseChange_tmul (Q : QuadraticForm R M₂) (a : A) (m₂ : M₂) :\n    Q.baseChange A (a ⊗ₜ m₂) = Q m₂ • (a * a) :=\n  tensorDistrib_tmul _ _ _ _\n\ntheorem associated_baseChange [Invertible (2 : A)] (Q : QuadraticForm R M₂) :\n    associated (R := A) (Q.baseChange A) = BilinForm.baseChange A (associated (R := R) Q) := by\n  dsimp only [QuadraticForm.baseChange, LinearMap.baseChange]\n  rw [associated_tmul (QuadraticMap.sq (R := A)) Q, associated_sq]\n  exact rfl\n\ntheorem polarBilin_baseChange [Invertible (2 : A)] (Q : QuadraticForm R M₂) :\n    polarBilin (Q.baseChange A) = BilinForm.baseChange A (polarBilin Q) := by\n  rw [QuadraticForm.baseChange, BilinForm.baseChange, polarBilin_tmul, BilinForm.tmul,\n    ← map_smul, smul_tmul', ← two_nsmul_associated R, coe_associatedHom, associated_sq,\n    smul_comm, ← smul_assoc, two_smul, invOf_two_add_invOf_two, one_smul]\n\nend QuadraticForm\n\nend InvertibleTwo\n\nset_option backward.defeqAttrib.useBackward true in\n/-- If two quadratic maps from `A ⊗[R] M₂` agree on elements of the form `1 ⊗ m`, they are equal.\n\nIn other words, if a base change exists for a quadratic map, it is unique.\n\nNote that unlike `QuadraticForm.baseChange`, this does not need `Invertible (2 : R)`. -/\n@[ext]\n\nTarget:\ntheorem baseChange_ext ⦃Q₁ Q₂ : QuadraticMap A (A ⊗[R] M₂) N₁⦄\n    (h : ∀ m, Q₁ (1 ⊗ₜ m) = Q₂ (1 ⊗ₜ m)) :\n    Q₁ = Q₂ :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/QuadraticForm","family_id":"basechange_ext","file_id":"mathlib/Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean","sample_id":"2b78fa8807ec50bfc8ab5c93ca7eeaa86e8f4d73fc76cde03273374bc7683948"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1a3ab110d23e9c5a2fcf5c8f847d9eb4eca9aceac610399fa60f5ab4f28d4c1a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"59745f19014a0741162cbebbbb6846c12b9cbeb6957d67bc1fc52b981bcc4ac1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a2b4875ff5d193c274945a397a372732cc39a1c971c7cc22bbea2e4368fe786d","source_sha256":"98d8a988714b8411039a52ca8cb94b19a051f27bd0e995b1332bddd6c23da9ea","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨c₁, hc₁, c₂, _, h⟩ := hf b hb\n  refine ⟨c₁, hc₁, ?_⟩\n  filter_upwards [h]\n  exact fun _ H u hu => (H u hu).1","hard_negative":false,"metrics":{"chosen_tokens":49,"rejected_tokens":54,"token_jaccard":0.882353,"token_length_ratio":1.102041},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"a3ff053968b3576f47dda96b8c08f173f4de721127156565c9f02189d990f8be","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Pow.Real\npublic import Mathlib.Algebra.Order.ToIntervalMod\npublic import Mathlib.Analysis.SpecialFunctions.Log.Base\nimport Mathlib.Algebra.Order.Interval.Set.Group\n\nNamespace:\nAkraBazziRecurrence.GrowsPolynomially\n\nLocal context:\n/-\nCopyright (c) 2023 Frédéric Dupuis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Frédéric Dupuis\n-/\n/-!\n# Akra-Bazzi theorem: the polynomial growth condition\n\nThis file defines and develops an API for the polynomial growth condition that appears in the\nstatement of the Akra-Bazzi theorem: for the theorem to hold, the function `g` must\nsatisfy the condition that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b*n` and `n` for any\nconstant `b ∈ (0,1)`.\n\n## Implementation notes\n\nOur definition requires that the condition hold for any `b ∈ (0,1)`. This is equivalent to requiring\nit only for `b = 1 / 2` (or any other particular value in `(0, 1)`). While this could, in principle,\nmake it harder to prove that a particular function grows polynomially, this issue does not seem to\narise in practice.\n\n-/\n\n@[expose] public section\n\nopen Finset Real Filter Asymptotics\nopen scoped Topology\n\nnamespace AkraBazziRecurrence\n\n/-- The growth condition that the function `g` must satisfy for the Akra-Bazzi theorem to apply.\nIt roughly states that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b * n` and `n`, for any\nconstant `b ∈ (0, 1)`. -/\ndef GrowsPolynomially (f : ℝ → ℝ) : Prop :=\n  ∀ b ∈ Set.Ioo 0 1, ∃ c₁ > 0, ∃ c₂ > 0,\n    ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * (f x)) (c₂ * f x)\n\nnamespace GrowsPolynomially\n\nlemma congr_of_eventuallyEq {f g : ℝ → ℝ} (hfg : f =ᶠ[atTop] g) (hg : GrowsPolynomially g) :\n    GrowsPolynomially f := by\n  intro b hb\n  have hg' := hg b hb\n  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hg'⟩ := hg'\n  refine ⟨c₁, hc₁_mem, c₂, hc₂_mem, ?_⟩\n  filter_upwards [hg', (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg, hfg]\n    with x hx₁ hx₂ hx₃\n  intro u hu\n  rw [hx₂ u hu.1, hx₃]\n  exact hx₁ u hu\n\nlemma iff_eventuallyEq {f g : ℝ → ℝ} (h : f =ᶠ[atTop] g) :\n    GrowsPolynomially f ↔ GrowsPolynomially g :=\n  ⟨fun hf => congr_of_eventuallyEq h.symm hf, fun hg => congr_of_eventuallyEq h hg⟩\n\nvariable {f : ℝ → ℝ}\n\nlemma eventually_atTop_le {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ≤ c * f x := by\n  obtain ⟨c₁, _, c₂, hc₂, h⟩ := hf b hb\n  refine ⟨c₂, hc₂, ?_⟩\n  filter_upwards [h]\n  exact fun _ H u hu => (H u hu).2\n\nlemma eventually_atTop_le_nat {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (b * n) n, f u ≤ c * f n := by\n  obtain ⟨c, hc_mem, hc⟩ := hf.eventually_atTop_le hb\n  exact ⟨c, hc_mem, hc.natCast_atTop⟩\n\nTarget:\nlemma eventually_atTop_ge {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, c * f x ≤ f u :=\n\nProof body:\n","rejected":"by\n  obtain ⟨c₁, hc₁, c₂, _, h⟩ := hf b hb\n  refine ⟨c₁, hc₁, ?_⟩\n  filter_upwards [h]\n  exact fun _ H u hu => (H u hu).1\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability/AkraBazzi","family_id":"eventually_attop_ge","file_id":"mathlib/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean","sample_id":"a2b4875ff5d193c274945a397a372732cc39a1c971c7cc22bbea2e4368fe786d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0f3c5c923c48eece1af3a7c60f1fa94f43ad2e6d58c10a4a8ef6046b414d8933","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"738f9ce2b5be5545426b3823582006b3d80a983396927414435ba8e36978d090","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d0df3140da7fa364792a7ce4f3fa0fb068f28c3068cdfb76bb78b78e0d7a5c60","source_sha256":"d5350ee23560ecfe7472ddc39d464fb66256d2af6cef5b34446a54a647e9b358","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain rfl | hs := s.eq_empty_or_nonempty\n  · rw [smul_set_empty, Real.sSup_empty, Real.sInf_empty, smul_zero]\n  obtain rfl | ha' := ha.eq_or_lt\n  · rw [zero_smul_set hs, zero_smul]\n    exact csSup_singleton 0\n  by_cases h : BddBelow s\n  · exact ((OrderIso.smulRightDual ℝ ha').map_csInf' hs h).symm\n  · rw [Real.sSup_of_not_bddAbove (mt (bddAbove_smul_iff_of_neg ha').1 h),\n        Real.sInf_of_not_bddBelow h, smul_zero]","hard_negative":true,"metrics":{"chosen_tokens":89,"rejected_tokens":5,"token_jaccard":0.090909,"token_length_ratio":0.05618},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"a42a69cd28452e347ff46141da72ccf84bc27094ef054c76ac0253cd455de258","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set\npublic import Mathlib.Algebra.Order.Archimedean.Real.Basic\npublic import Mathlib.Algebra.Order.Module.Pointwise\npublic import Mathlib.Order.ConditionallyCompleteLattice.Indexed\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Eric Wieser\n-/\n/-!\n# Pointwise operations on sets of reals\n\nThis file relates `sInf (a • s)`/`sSup (a • s)` with `a • sInf s`/`a • sSup s` for `s : Set ℝ`.\n\nFrom these, it relates `⨅ i, a • f i` / `⨆ i, a • f i` with `a • (⨅ i, f i)` / `a • (⨆ i, f i)`,\nand provides lemmas about distributing `*` over `⨅` and `⨆`.\n\n## TODO\n\nThis is true more generally for conditionally complete linear order whose default value is `0`. We\ndon't have those yet.\n-/\n\npublic section\n\nassert_not_exists Finset\n\nopen Set\n\nopen scoped Pointwise\n\nvariable {ι : Sort*} {α : Type*} [Field α] [LinearOrder α] [IsStrictOrderedRing α]\n\nsection MulActionWithZero\n\nvariable [MulActionWithZero α ℝ] [IsOrderedModule α ℝ] {a : α}\n\ntheorem Real.sInf_smul_of_nonneg (ha : 0 ≤ a) (s : Set ℝ) : sInf (a • s) = a • sInf s := by\n  obtain rfl | hs := s.eq_empty_or_nonempty\n  · rw [smul_set_empty, Real.sInf_empty, smul_zero]\n  obtain rfl | ha' := ha.eq_or_lt\n  · rw [zero_smul_set hs, zero_smul]\n    exact csInf_singleton 0\n  by_cases h : BddBelow s\n  · exact ((OrderIso.smulRight ha').map_csInf' hs h).symm\n  · rw [Real.sInf_of_not_bddBelow (mt (bddBelow_smul_iff_of_pos ha').1 h),\n        Real.sInf_of_not_bddBelow h, smul_zero]\n\ntheorem Real.smul_iInf_of_nonneg (ha : 0 ≤ a) (f : ι → ℝ) : (a • ⨅ i, f i) = ⨅ i, a • f i :=\n  (Real.sInf_smul_of_nonneg ha _).symm.trans <| congr_arg sInf <| (range_comp _ _).symm\n\ntheorem Real.sSup_smul_of_nonneg (ha : 0 ≤ a) (s : Set ℝ) : sSup (a • s) = a • sSup s := by\n  obtain rfl | hs := s.eq_empty_or_nonempty\n  · rw [smul_set_empty, Real.sSup_empty, smul_zero]\n  obtain rfl | ha' := ha.eq_or_lt\n  · rw [zero_smul_set hs, zero_smul]\n    exact csSup_singleton 0\n  by_cases h : BddAbove s\n  · exact ((OrderIso.smulRight ha').map_csSup' hs h).symm\n  · rw [Real.sSup_of_not_bddAbove (mt (bddAbove_smul_iff_of_pos ha').1 h),\n        Real.sSup_of_not_bddAbove h, smul_zero]\n\ntheorem Real.smul_iSup_of_nonneg (ha : 0 ≤ a) (f : ι → ℝ) : (a • ⨆ i, f i) = ⨆ i, a • f i :=\n  (Real.sSup_smul_of_nonneg ha _).symm.trans <| congr_arg sSup <| (range_comp _ _).symm\n\nend MulActionWithZero\n\nsection Module\n\nvariable [Module α ℝ] [IsOrderedModule α ℝ] {a : α}\n\ntheorem Real.sInf_smul_of_nonpos (ha : a ≤ 0) (s : Set ℝ) : sInf (a • s) = a • sSup s := by\n  obtain rfl | hs := s.eq_empty_or_nonempty\n  · rw [smul_set_empty, Real.sInf_empty, Real.sSup_empty, smul_zero]\n  obtain rfl | ha' := ha.eq_or_lt\n  · rw [zero_smul_set hs, zero_smul]\n    exact csInf_singleton 0\n  by_cases h : BddAbove s\n  · exact ((OrderIso.smulRightDual ℝ ha').map_csSup' hs h).symm\n  · rw [Real.sInf_of_not_bddBelow (mt (bddBelow_smul_iff_of_neg ha').1 h),\n        Real.sSup_of_not_bddAbove h, smul_zero]\n\ntheorem Real.smul_iSup_of_nonpos (ha : a ≤ 0) (f : ι → ℝ) : (a • ⨆ i, f i) = ⨅ i, a • f i :=\n  (Real.sInf_smul_of_nonpos ha _).symm.trans <| congr_arg sInf <| (range_comp _ _).symm\n\nTarget:\ntheorem Real.sSup_smul_of_nonpos (ha : a ≤ 0) (s : Set ℝ) : sSup (a • s) = a • sInf s :=\n\nProof body:\n","rejected":"by\n  exact Real.sSup_smul_of_nonpos","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"06cfa16a2f30bef1f3fa2b63ba647fcb6efa532049c95193258886fc0080f3ad","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Real","family_id":"real","file_id":"mathlib/Mathlib/Data/Real/Pointwise.lean","sample_id":"d0df3140da7fa364792a7ce4f3fa0fb068f28c3068cdfb76bb78b78e0d7a5c60"},"split":"train"}
{"binding":{"chosen_proof_sha256":"871db309f2a7e188f5170224e4eada7ffe3aa87ef44b3d70e3a18c5c524571b5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d3baf770853c9de1742f6952ed6591411750a3d98ef5c34871eada480093b869","source_sha256":"138c7bba1e136375811c6bc0d5e62f29dc75b67b8e14507dcd9ecb3045cecbfd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine le_antisymm (map_continuous <| toWeakSpaceCLM 𝕜 E).continuousOn.image_closure\n    (Set.compl_subset_compl.mp fun x hx ↦ ?_)\n  obtain ⟨x, -, rfl⟩ := (toWeakSpace 𝕜 E).toEquiv.image_compl (closure s) |>.symm.subset hx\n  have : ContinuousSMul ℝ E := IsScalarTower.continuousSMul 𝕜\n  obtain ⟨f, u, hus, hux⟩ := RCLike.geometric_hahn_banach_closed_point (𝕜 := 𝕜)\n    hs.closure isClosed_closure (by simpa using hx)\n  let f' : StrongDual 𝕜 (WeakSpace 𝕜 E) :=\n    { toLinearMap := (f : E →ₗ[𝕜] 𝕜).comp ((toWeakSpace 𝕜 E).symm : WeakSpace 𝕜 E →ₗ[𝕜] E)\n      cont := WeakBilin.eval_continuous (topDualPairing 𝕜 E).flip _ }\n  have hux' : u < RCLike.reCLM.comp (f'.restrictScalars ℝ) (toWeakSpace 𝕜 E x) := by simpa [f']\n  have hus' : closure (toWeakSpace 𝕜 E '' s) ⊆\n      {y | RCLike.reCLM.comp (f'.restrictScalars ℝ) y ≤ u} := by\n    refine closure_minimal ?_ <| isClosed_le (by fun_prop) (by fun_prop)\n    rintro - ⟨y, hy, rfl⟩\n    simpa [f'] using (hus y <| subset_closure hy).le\n  exact (hux'.not_ge <| hus' ·)","hard_negative":false,"metrics":{"chosen_tokens":266,"rejected_tokens":2,"token_jaccard":0.011111,"token_length_ratio":0.007519},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"a44b982519654340f86b214831d62a92748b3435479141cb5dcdcf19fb02f84e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.LocallyConvex.Separation\npublic import Mathlib.Analysis.LocallyConvex.SeparatingDual\npublic import Mathlib.LinearAlgebra.Dual.Defs\npublic import Mathlib.Topology.Algebra.Module.Spaces.WeakDual\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\n/-! # Closures of convex sets in locally convex spaces\n\nThis file contains the standard result that if `E` is a vector space with two locally convex\ntopologies, then the closure of a convex set is the same in either topology, provided they have the\nsame collection of continuous linear functionals. In particular, the weak closure of a convex set\nin a locally convex space coincides with the closure in the original topology.\nOf course, we phrase this in terms of linear maps between locally convex spaces, rather than\ncreating two separate topologies on the same space.\n-/\n\npublic section\n\nvariable {𝕜 E F : Type*}\nvariable [RCLike 𝕜] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜 F]\nvariable [Module ℝ E] [IsScalarTower ℝ 𝕜 E] [Module ℝ F] [IsScalarTower ℝ 𝕜 F]\nvariable [TopologicalSpace E] [IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E]\n  [LocallyConvexSpace ℝ E]\nvariable [TopologicalSpace F] [IsTopologicalAddGroup F] [ContinuousSMul 𝕜 F]\n  [LocallyConvexSpace ℝ F]\n\nvariable (𝕜) in\n/-- If `E` is a locally convex space over `𝕜` (with `RCLike 𝕜`), and `s : Set E` is `ℝ`-convex, then\nthe closure of `s` and the weak closure of `s` coincide. More precisely, the topological closure\ncommutes with `toWeakSpace 𝕜 E`.\n\nThis holds more generally for any linear equivalence `e : E ≃ₗ[𝕜] F` between locally convex spaces\nsuch that precomposition with `e` and `e.symm` preserves continuity of linear functionals. See\n`LinearEquiv.image_closure_of_convex`. -/\n\nTarget:\ntheorem Convex.toWeakSpace_closure {s : Set E} (hs : Convex ℝ s) :\n    (toWeakSpace 𝕜 E) '' (closure s) = closure (toWeakSpace 𝕜 E '' s) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/LocallyConvex","family_id":"convex","file_id":"mathlib/Mathlib/Analysis/LocallyConvex/WeakSpace.lean","sample_id":"d3baf770853c9de1742f6952ed6591411750a3d98ef5c34871eada480093b869"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5110e356531b5aca6a5cc8c637eddfe770b9e7e43c44858448f93b443e747ce9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"35aaf71a5609ad5967e4c59b74a8bb922bd20f66df9f7505cfd11f40e5557975","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"948388cf0ecec93fe348cbe9fadcd98478b5894a8ffcc1cd67d7ba5e880077f1","source_sha256":"76bf26908fe5bb3f22e0b1de59c408f5480271bc0d25bef1d614882cce4d1085","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun ⟨H⟩ ↦ ⟨?_⟩, fun ⟨H⟩ ↦ ⟨?_⟩⟩ <;>\n    simp only [isCardinalFiltered_aleph0_iff] at * <;>\n    exact H","hard_negative":true,"metrics":{"chosen_tokens":38,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.131579},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"a46ac2d5f7bced63d57a0fa800eb58638242929bffa0cff5ac5b2529b1564b30","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Filtered\npublic import Mathlib.CategoryTheory.Limits.Preserves.Filtered\npublic import Mathlib.CategoryTheory.Limits.Types.Filtered\npublic import Mathlib.CategoryTheory.MorphismProperty.Basic\npublic import Mathlib.CategoryTheory.Presentable.Basic\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Finitely Presentable Objects\n\nWe define finitely presentable objects as a synonym for `ℵ₀`-presentable objects,\nand link this definition with the preservation of filtered colimits.\n\n-/\n\n@[expose] public section\n\n\nuniverse w v' v u' u\n\nnamespace CategoryTheory\n\nopen Limits Opposite Cardinal\n\nvariable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]\n\nattribute [local instance] fact_isRegular_aleph0\n\n/-- A functor `F : C ⥤ D` is finitely accessible if it is `ℵ₀`-accessible.\nEquivalently, it preserves all filtered colimits.\nSee `CategoryTheory.Functor.IsFinitelyAccessible_iff_preservesFilteredColimits`. -/\nabbrev Functor.IsFinitelyAccessible (F : C ⥤ D) : Prop := IsCardinalAccessible.{w} F ℵ₀\n\nlemma Functor.IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize {F : C ⥤ D} :\n    IsFinitelyAccessible.{w} F ↔ PreservesFilteredColimitsOfSize.{w, w} F := by\n  refine ⟨fun ⟨H⟩ ↦ ⟨?_⟩, fun ⟨H⟩ ↦ ⟨?_⟩⟩ <;>\n    simp only [isCardinalFiltered_aleph0_iff] at * <;>\n    exact H\n\nlemma Functor.isFinitelyAccessible_iff_preservesFilteredColimits {F : C ⥤ D} :\n    IsFinitelyAccessible.{v'} F ↔ PreservesFilteredColimits F :=\n  IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize\n\n/-- An object `X` is finitely presentable if `Hom(X, -)` preserves all filtered colimits. -/\nabbrev IsFinitelyPresentable (X : C) : Prop :=\n  IsCardinalPresentable.{w} X ℵ₀\n\nvariable (C) in\n/-- `IsFinitelyPresentable` as an `ObjectProperty` on `C`. This is sometimes called \"compact\". -/\ndef ObjectProperty.isFinitelyPresentable : ObjectProperty C := fun X ↦ IsFinitelyPresentable.{w} X\n\nlemma ObjectProperty.isFinitelyPresentable_eq_isCardinalPresentable :\n    isFinitelyPresentable.{w} C = isCardinalPresentable.{w} C ℵ₀ :=\n  rfl\n\nvariable (C) in\n/-- A morphism `f : X ⟶ Y` is finitely presentable if it is so as an object of `Under X`. -/\ndef MorphismProperty.isFinitelyPresentable : MorphismProperty C :=\n  fun _ _ f ↦ ObjectProperty.isFinitelyPresentable.{w} _ (CategoryTheory.Under.mk f)\n\nlemma isFinitelyPresentable_iff_preservesFilteredColimitsOfSize {X : C} :\n    IsFinitelyPresentable.{w} X ↔ PreservesFilteredColimitsOfSize.{w, w} (coyoneda.obj (op X)) :=\n  Functor.IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize\n\nlemma isFinitelyPresentable_iff_preservesFilteredColimits {X : C} :\n    IsFinitelyPresentable.{v} X ↔ PreservesFilteredColimits (coyoneda.obj (op X)) :=\n  Functor.IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize\n\ninstance (X : C) [IsFinitelyPresentable.{w} X] :\n    PreservesFilteredColimitsOfSize.{w, w} (coyoneda.obj (op X)) := by\n  rw [← isFinitelyPresentable_iff_preservesFilteredColimitsOfSize]\n  infer_instance\n\ninstance (X : (ObjectProperty.isFinitelyPresentable.{w} C).FullSubcategory) :\n    IsFinitelyPresentable.{w} ((ObjectProperty.isFinitelyPresentable.{w} C).ι.obj X) :=\n  X.property\n\nlemma IsFinitelyPresentable.exists_hom_of_isColimit {J : Type w} [SmallCategory J] [IsFiltered J]\n    {D : J ⥤ C} {c : Cocone D} (hc : IsColimit c) {X : C} [IsFinitelyPresentable.{w} X]\n    (f : X ⟶ c.pt) :\n    ∃ (j : J) (p : X ⟶ D.obj j), p ≫ c.ι.app j = f :=\n  Types.jointly_surjective_of_isColimit (isColimitOfPreserves (coyoneda.obj (op X)) hc) f\n\nlemma IsFinitelyPresentable.exists_eq_of_isColimit {J : Type w} [SmallCategory J] [IsFiltered J]\n    {D : J ⥤ C} {c : Cocone D} (hc : IsColimit c) {X : C} [IsFinitelyPresentable.{w} X]\n    {i j : J} (f : X ⟶ D.obj i) (g : X ⟶ D.obj j) (h : f ≫ c.ι.app i = g ≫ c.ι.app j) :\n    ∃ (k : J) (u : i ⟶ k) (v : j ⟶ k), f ≫ D.map u = g ≫ D.map v :=\n  (Types.FilteredColimit.isColimit_eq_iff _ (isColimitOfPreserves (coyoneda.obj (op X)) hc)).mp h\n\nlemma IsFinitelyPresentable.exists_hom_of_isColimit_under\n    {J : Type w} [SmallCategory J] [IsFiltered J] {D : J ⥤ C} {c : Cocone D} (hc : IsColimit c)\n    {X A : C} (p : X ⟶ A) (s : (Functor.const J).obj X ⟶ D)\n    [IsFinitelyPresentable.{w} (Under.mk p)]\n    (f : A ⟶ c.pt) (h : ∀ (j : J), s.app j ≫ c.ι.app j = p ≫ f) :\n    ∃ (j : J) (q : A ⟶ D.obj j), p ≫ q = s.app j ∧ q ≫ c.ι.app j = f := by\n  have : Nonempty J := IsFiltered.nonempty\n  let hc' := Under.isColimitLiftCocone D s c (p ≫ f) h hc\n  obtain ⟨j, q, hq⟩ := exists_hom_of_isColimit (X := Under.mk p) hc' (Under.homMk f rfl)\n  use j, q.right, Under.w q, congr($(hq).right)\n\nTarget:\nlemma HasCardinalFilteredColimits_iff_hasFilteredColimitsOfSize :\n    HasCardinalFilteredColimits.{w} C ℵ₀ ↔ HasFilteredColimitsOfSize.{w, w} C :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_948388cf0ece","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"0157dc8d36be3b8cc1ad5eeb90d9ff4eeae1dce4cd0baf91ac2ecc4674ffadf3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Presentable","family_id":"hascardinalfilteredcolimits_iff_hasfilteredcolimitsofsize","file_id":"mathlib/Mathlib/CategoryTheory/Presentable/Finite.lean","sample_id":"948388cf0ecec93fe348cbe9fadcd98478b5894a8ffcc1cd67d7ba5e880077f1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"90ae162d9072def41fe3c1f6b71c6fb09d8794f02fb213178dc3f4878196efbb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3129c4684252f138a34ea19d6ab7a0e7965bec43e42276b25a947e2819c0b66a","source_sha256":"bed89a7fbd00d1b7ec6db8c1417583c0ed6fb3ca4d333358b3c5339034072c9b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← cancel_epi (prodComparisonIso (Over.pullback f) _ _).hom.left,\n    Over.hom_left_inv_left_assoc]\n  simp [CartesianMonoidalCategory.prodComparison, snd]","hard_negative":false,"metrics":{"chosen_tokens":33,"rejected_tokens":5,"token_jaccard":0.08,"token_length_ratio":0.151515},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"a4aa8834606200485227db4654d309cea0b3c7d938e82f7e9186ec5227c6f509","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Adjunction.Limits\npublic import Mathlib.CategoryTheory.Comma.Over.Pullback\npublic import Mathlib.CategoryTheory.Limits.Constructions.Over.Products\npublic import Mathlib.CategoryTheory.Monoidal.CommMon_\npublic import Mathlib.CategoryTheory.Monoidal.Grp\npublic import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic\n\nNamespace:\nCategoryTheory.Over\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# `CartesianMonoidalCategory` for `Over X`\n\nWe provide a `CartesianMonoidalCategory (Over X)` instance via pullbacks, and provide simp lemmas\nfor the induced `MonoidalCategory (Over X)` instance.\n\n-/\n\npublic noncomputable section\n\nnamespace CategoryTheory.Over\n\nopen Functor Limits CartesianMonoidalCategory\n\nvariable {C : Type*} [Category* C] [HasPullbacks C]\n\nset_option backward.defeqAttrib.useBackward true in\n/-- A choice of finite products of `Over X` given by `Limits.pullback`. -/\nabbrev cartesianMonoidalCategory (X : C) : CartesianMonoidalCategory (Over X) :=\n  .ofChosenFiniteProducts\n    ⟨asEmptyCone (Over.mk (𝟙 X)), IsTerminal.ofUniqueHom (fun Y ↦ Over.homMk Y.hom)\n      fun Y m ↦ Over.OverMorphism.ext (by simpa using m.w)⟩\n    fun Y Z ↦ ⟨pullbackConeEquivBinaryFan.functor.obj (pullback.cone Y.hom Z.hom),\n    (pullback.isLimit _ _).pullbackConeEquivBinaryFanFunctor⟩\n\nattribute [local instance] cartesianMonoidalCategory\n\n/-- `Over X` is braided w.r.t. the Cartesian monoidal structure given by `Limits.pullback`. -/\nabbrev braidedCategory (X : C) : BraidedCategory (Over X) :=\n  .ofCartesianMonoidalCategory\n\nattribute [local instance] braidedCategory\n\nopen MonoidalCategory\n\nvariable {X : C}\n\n@[ext]\nlemma tensorObj_ext {R : C} {S T : Over X} (f₁ f₂ : R ⟶ (S ⊗ T).left)\n    (e₁ : f₁ ≫ pullback.fst _ _ = f₂ ≫ pullback.fst _ _)\n    (e₂ : f₁ ≫ pullback.snd _ _ = f₂ ≫ pullback.snd _ _) : f₁ = f₂ :=\n  pullback.hom_ext e₁ e₂\n\n@[simp]\nlemma tensorObj_left (R S : Over X) : (R ⊗ S).left = Limits.pullback R.hom S.hom := rfl\n\n@[simp]\nlemma tensorObj_hom (R S : Over X) : (R ⊗ S).hom = pullback.fst R.hom S.hom ≫ R.hom := rfl\n\n@[simp]\nlemma tensorUnit_left : (𝟙_ (Over X)).left = X := rfl\n\n@[simp]\nlemma tensorUnit_hom : (𝟙_ (Over X)).hom = 𝟙 X := rfl\n\n@[simp]\nlemma lift_left {R S T : Over X} (f : R ⟶ S) (g : R ⟶ T) :\n    (lift f g).left = pullback.lift f.left g.left (f.w.trans g.w.symm) := rfl\n\n@[simp]\nlemma fst_left {R S : Over X} : (fst R S).left = pullback.fst _ _ := rfl\n\n@[simp]\nlemma snd_left {R S : Over X} : (snd R S).left = pullback.snd _ _ := rfl\n\n@[simp]\nlemma toUnit_left {R : Over X} : (toUnit R).left = R.hom := rfl\n\n@[reassoc (attr := simp)]\nlemma associator_hom_left_fst (R S T : Over X) :\n    (α_ R S T).hom.left ≫ pullback.fst _ (pullback.fst _ _ ≫ _) =\n      pullback.fst _ _ ≫ pullback.fst _ _ :=\n  limit.lift_π _ _\n\n@[reassoc (attr := simp)]\nlemma associator_hom_left_snd_fst (R S T : Over X) :\n    (α_ R S T).hom.left ≫ pullback.snd _ (pullback.fst _ _ ≫ _) ≫ pullback.fst _ _ =\n      pullback.fst _ _ ≫ pullback.snd _ _ :=\n  (limit.lift_π_assoc _ _ _).trans (limit.lift_π _ _)\n\n@[reassoc (attr := simp)]\nlemma associator_hom_left_snd_snd (R S T : Over X) :\n    (α_ R S T).hom.left ≫ pullback.snd _ (pullback.fst _ _ ≫ _) ≫ pullback.snd _ _ =\n      pullback.snd _ _ :=\n  (limit.lift_π_assoc _ _ _).trans (limit.lift_π _ _)\n\n@[reassoc (attr := simp)]\nlemma associator_inv_left_fst_fst (R S T : Over X) :\n    (α_ R S T).inv.left ≫ pullback.fst (pullback.fst _ _ ≫ _) _ ≫ pullback.fst _ _ =\n      pullback.fst _ _ :=\n  (limit.lift_π_assoc _ _ _).trans (limit.lift_π _ _)\n\n@[reassoc (attr := simp)]\nlemma associator_inv_left_fst_snd (R S T : Over X) :\n    (α_ R S T).inv.left ≫ pullback.fst (pullback.fst _ _ ≫ _) _ ≫ pullback.snd _ _ =\n      pullback.snd _ _ ≫ pullback.fst _ _ :=\n  (limit.lift_π_assoc _ _ _).trans (limit.lift_π _ _)\n\n@[reassoc (attr := simp)]\nlemma associator_inv_left_snd (R S T : Over X) :\n    (α_ R S T).inv.left ≫ pullback.snd (pullback.fst _ _ ≫ _) _ =\n      pullback.snd _ _ ≫ pullback.snd _ _ :=\n  limit.lift_π _ _\n\n@[simp]\nlemma leftUnitor_hom_left (Y : Over X) :\n    (λ_ Y).hom.left = pullback.snd _ _ := rfl\n\n@[reassoc (attr := simp)]\nlemma leftUnitor_inv_left_fst (Y : Over X) :\n    (λ_ Y).inv.left ≫ pullback.fst (𝟙 X) _ = Y.hom :=\n  limit.lift_π _ _\n\n@[reassoc (attr := simp)]\nlemma leftUnitor_inv_left_snd (Y : Over X) :\n    (λ_ Y).inv.left ≫ pullback.snd (𝟙 X) _ = 𝟙 Y.left :=\n  limit.lift_π _ _\n\n@[simp]\nlemma rightUnitor_hom_left (Y : Over X) :\n    (ρ_ Y).hom.left = pullback.fst _ (𝟙 X) := rfl\n\n@[reassoc (attr := simp)]\nlemma rightUnitor_inv_left_fst (Y : Over X) :\n    (ρ_ Y).inv.left ≫ pullback.fst _ (𝟙 X) = 𝟙 _ :=\n  limit.lift_π _ _\n\n@[reassoc (attr := simp)]\nlemma rightUnitor_inv_left_snd (Y : Over X) :\n    (ρ_ Y).inv.left ≫ pullback.snd _ (𝟙 X) = Y.hom :=\n  limit.lift_π _ _\n\nlemma whiskerLeft_left {R S T : Over X} (f : S ⟶ T) :\n    (R ◁ f).left = pullback.map _ _ _ _ (𝟙 _) f.left (𝟙 _) (by simp) (by simp) := rfl\n\n@[reassoc (attr := simp)]\nlemma whiskerLeft_left_fst {R S T : Over X} (f : S ⟶ T) :\n    (R ◁ f).left ≫ pullback.fst _ _ = pullback.fst _ _ :=\n  (limit.lift_π _ _).trans (Category.comp_id _)\n\n@[reassoc (attr := simp)]\nlemma whiskerLeft_left_snd {R S T : Over X} (f : S ⟶ T) :\n    (R ◁ f).left ≫ pullback.snd _ _ = pullback.snd _ _ ≫ f.left :=\n  limit.lift_π _ _\n\nlemma whiskerRight_left {R S T : Over X} (f : S ⟶ T) :\n    (f ▷ R).left = pullback.map _ _ _ _ f.left (𝟙 _) (𝟙 _) (by simp) (by simp) := rfl\n\n@[reassoc (attr := simp)]\nlemma whiskerRight_left_fst {R S T : Over X} (f : S ⟶ T) :\n    (f ▷ R).left ≫ pullback.fst _ _ = pullback.fst _ _ ≫ f.left :=\n  limit.lift_π _ _\n\n@[reassoc (attr := simp)]\nlemma whiskerRight_left_snd {R S T : Over X} (f : S ⟶ T) :\n    (f ▷ R).left ≫ pullback.snd _ _ = pullback.snd _ _ :=\n  (limit.lift_π _ _).trans (Category.comp_id _)\n\nlemma tensorHom_left {R S T U : Over X} (f : R ⟶ S) (g : T ⟶ U) :\n    (f ⊗ₘ g).left = pullback.map _ _ _ _ f.left g.left (𝟙 _) (by simp) (by simp) := rfl\n\n@[reassoc (attr := simp)]\nlemma tensorHom_left_fst {S U : C} {R T : Over X} (fS : S ⟶ X) (fU : U ⟶ X)\n    (f : R ⟶ mk fS) (g : T ⟶ mk fU) :\n    (f ⊗ₘ g).left ≫ pullback.fst fS fU = pullback.fst R.hom T.hom ≫ f.left :=\n  limit.lift_π _ _\n\n@[reassoc (attr := simp)]\nlemma tensorHom_left_snd {S U : C} {R T : Over X} (fS : S ⟶ X) (fU : U ⟶ X)\n    (f : R ⟶ mk fS) (g : T ⟶ mk fU) :\n    (f ⊗ₘ g).left ≫ pullback.snd fS fU = pullback.snd R.hom T.hom ≫ g.left :=\n  limit.lift_π _ _\n\n@[simp]\nlemma braiding_hom_left {R S : Over X} :\n    (β_ R S).hom.left = (pullbackSymmetry _ _).hom := rfl\n\n@[simp]\nlemma braiding_inv_left {R S : Over X} :\n    (β_ R S).inv.left = (pullbackSymmetry _ _).hom := rfl\n\nvariable {A B R S Y Z : C} {f : R ⟶ X} {g : S ⟶ X}\n\ninstance : (Over.pullback f).Braided := .ofChosenFiniteProducts _\n\n@[simp]\nlemma η_pullback_left : (OplaxMonoidal.η (Over.pullback f)).left = (pullback.snd (𝟙 _) f) := rfl\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma ε_pullback_left : (LaxMonoidal.ε (Over.pullback f)).left = inv (pullback.snd (𝟙 _) f) := by\n  apply IsIso.eq_inv_of_hom_inv_id\n  rw [← η_pullback_left, ← Over.comp_left, Monoidal.η_ε, Over.id_left]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma μ_pullback_left_fst_fst (R S : Over X) :\n    (LaxMonoidal.μ (Over.pullback f) R S).left ≫\n      pullback.fst _ _ ≫ pullback.fst _ _ = pullback.fst _ _ ≫ pullback.fst _ _ := by\n  rw [Monoidal.μ_of_cartesianMonoidalCategory,\n    ← cancel_epi (prodComparisonIso (Over.pullback f) R S).hom.left, ← Over.comp_left_assoc,\n    Iso.hom_inv_id]\n  simp [CartesianMonoidalCategory.prodComparison, fst]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma μ_pullback_left_fst_snd (R S : Over X) :\n    (LaxMonoidal.μ (Over.pullback f) R S).left ≫\n      pullback.fst _ _ ≫ pullback.snd _ _ = pullback.snd _ _ ≫ pullback.fst _ _ := by\n  rw [Monoidal.μ_of_cartesianMonoidalCategory,\n    ← cancel_epi (prodComparisonIso (Over.pullback f) R S).hom.left,\n    ← Over.comp_left_assoc, Iso.hom_inv_id]\n  simp [CartesianMonoidalCategory.prodComparison, snd]\n\nset_option backward.isDefEq.respectTransparency false in\nlemma μ_pullback_left_snd (R S : Over X) :\n    (LaxMonoidal.μ (Over.pullback f) R S).left ≫ pullback.snd _ _ =\n      pullback.snd _ _ ≫ pullback.snd _ _ := by\n  rw [Monoidal.μ_of_cartesianMonoidalCategory,\n    ← cancel_epi (prodComparisonIso (Over.pullback f) R S).hom.left,\n    ← Over.comp_left_assoc, Iso.hom_inv_id]\n  simp [CartesianMonoidalCategory.prodComparison]\n\n@[simp]\nlemma μ_pullback_left_fst_fst' (g₁ : Y ⟶ X) (g₂ : Z ⟶ X) :\n    (LaxMonoidal.μ (Over.pullback f) (.mk g₁) (.mk g₂)).left ≫\n      pullback.fst (pullback.fst g₁ g₂ ≫ g₁) f ≫ pullback.fst g₁ g₂ =\n        pullback.fst _ _ ≫ pullback.fst _ _ :=\n  μ_pullback_left_fst_fst ..\n\n@[simp]\nlemma μ_pullback_left_fst_snd' (g₁ : Y ⟶ X) (g₂ : Z ⟶ X) :\n    (LaxMonoidal.μ (Over.pullback f) (.mk g₁) (.mk g₂)).left ≫\n      pullback.fst (pullback.fst g₁ g₂ ≫ g₁) f ≫ pullback.snd g₁ g₂ =\n        pullback.snd _ _ ≫ pullback.fst _ _ :=\n  μ_pullback_left_fst_snd ..\n\n@[simp]\nlemma μ_pullback_left_snd' (g₁ : Y ⟶ X) (g₂ : Z ⟶ X) :\n    (LaxMonoidal.μ (Over.pullback f) (.mk g₁) (.mk g₂)).left ≫\n      pullback.snd (pullback.fst g₁ g₂ ≫ g₁) f =\n        pullback.snd _ _ ≫ pullback.snd _ _ := μ_pullback_left_snd ..\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma preservesTerminalIso_pullback (f : R ⟶ S) :\n    preservesTerminalIso (Over.pullback f) =\n      Over.isoMk (asIso (pullback.snd (𝟙 _) f)) (by simp) := by\n  ext1; exact toUnit_unique _ _\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma prodComparisonIso_pullback_inv_left_fst_fst (f : X ⟶ Y) (A B : Over Y) :\n    (prodComparisonIso (Over.pullback f) A B).inv.left ≫\n      pullback.fst (pullback.fst A.hom B.hom ≫ A.hom) f ≫ pullback.fst _ _ =\n        pullback.fst (pullback.snd A.hom f) (pullback.snd B.hom f) ≫ pullback.fst _ _ := by\n  rw [← cancel_epi (prodComparisonIso (Over.pullback f) A B).hom.left,\n    Over.hom_left_inv_left_assoc]\n  simp [CartesianMonoidalCategory.prodComparison, fst]\n\n@[simp]\nlemma prodComparisonIso_pullback_Spec_inv_left_fst_fst' (f : X ⟶ Y) (gA : A ⟶ Y) (gB : B ⟶ Y) :\n    (prodComparisonIso (Over.pullback f) (.mk gA) (.mk gB)).inv.left ≫\n      pullback.fst (pullback.fst gA gB ≫ gA) f ≫ pullback.fst _ _ =\n        pullback.fst (pullback.snd gA f) (pullback.snd gB f) ≫ pullback.fst _ _ :=\n  prodComparisonIso_pullback_inv_left_fst_fst ..\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\n\nTarget:\nlemma prodComparisonIso_pullback_inv_left_fst_snd' (f : X ⟶ Y) (gA : A ⟶ Y) (gB : B ⟶ Y) :\n    (prodComparisonIso (Over.pullback f) (.mk gA) (.mk gB)).inv.left ≫\n      pullback.fst (pullback.fst gA gB ≫ gA) f ≫ pullback.snd _ _ =\n        pullback.snd _ _ ≫ pullback.fst _ _ :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Monoidal","family_id":"prodcomparisoniso_pullback_inv_left_fst_snd","file_id":"mathlib/Mathlib/CategoryTheory/Monoidal/Cartesian/Over.lean","sample_id":"3129c4684252f138a34ea19d6ab7a0e7965bec43e42276b25a947e2819c0b66a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"baf74685d38b5f76627c288eda6c3033c2f20e6a31a28995ec3f9cc5ab112e2a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c41500350895bfb16aa2300c4a21bbaf6628c028644d00ba78df5591f31dfd9d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"317b87a5beef988d47043cbd23eccd8456a804914b949c3de8ba8b8d221e791d","source_sha256":"ba981a331bca09368a4e9ba1529e46eda80e78747c54ab2e477c7892665030f9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI := Fintype.ofFinite ι\n  classical\n  refine (Basis.piTensorProduct b).ext_elem (fun q ↦ ?_)\n  simp [Finsupp.single_apply, Fintype.prod_ite_zero, ← funext_iff]","hard_negative":true,"metrics":{"chosen_tokens":37,"rejected_tokens":3,"token_jaccard":0.032258,"token_length_ratio":0.081081},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"a50eb09c5dc13b9e7baffe35b1cbd2d4f9fa916c62c67d91bc4b8c94100a413c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Finsupp.VectorSpace\npublic import Mathlib.LinearAlgebra.PiTensorProduct.Finsupp\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Daniel Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Morrison, Sophie Morel\n-/\n/-!\n# Basis for `PiTensorProduct`\n\nThis file constructs a basis for `PiTensorProduct` given bases on the component spaces.\n-/\n\n@[expose] public section\n\nsection PiTensorProduct\n\nattribute [local ext] PiTensorProduct.ext\n\nopen LinearMap PiTensorProduct Module TensorProduct\n\nvariable {ι R : Type*} {M : ι → Type*} {κ : ι → Type*} [CommSemiring R] [∀ i, AddCommMonoid (M i)]\n  [∀ i, Module R (M i)]\n\nopen Classical in\n/-- Let `ι` be a `Finite` type and `M` be a family of modules indexed by `ι`. If `b i : κ i → M i`\nis a basis for every `i` in `ι`, then `fun (p : Π i, κ i) ↦ ⨂ₜ[R] i, b i (p i)` is a basis\nof `⨂[R] i, M i`.\n-/\nnoncomputable def Basis.piTensorProduct [Finite ι] (b : Π i, Basis (κ i) R (M i)) :\n    Basis (Π i, κ i) R (⨂[R] i, M i) :=\n  haveI := Fintype.ofFinite ι\n  Finsupp.basisSingleOne.map\n    ((PiTensorProduct.congr (fun i ↦ (b i).repr)) ≪≫ₗ\n      ofFinsuppEquiv ≪≫ₗ\n      Finsupp.lcongr (Equiv.refl _) (constantBaseRingEquiv _ R).toLinearEquiv).symm\n\n@[simp]\ntheorem Basis.piTensorProduct_repr_tprod_apply [Fintype ι] (b : Π i, Basis (κ i) R (M i))\n    (x : Π i, M i) (p : Π i, κ i) :\n    (Basis.piTensorProduct b).repr (tprod R x) p = ∏ i : ι, (b i).repr (x i) (p i) := by\n  rw [piTensorProduct, Subsingleton.elim (Fintype.ofFinite ι) ‹_›]\n  simp\n\n@[simp]\n\nTarget:\ntheorem Basis.piTensorProduct_apply [Finite ι] (b : Π i, Basis (κ i) R (M i)) (p : Π i, κ i) :\n    Basis.piTensorProduct b p = ⨂ₜ[R] i, (b i) (p i) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_317b87a5beef","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"51b630ec23df456c39a3f66e21e6af079c4c2cd5981346cc82443591834c6712","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/PiTensorProduct","family_id":"basis","file_id":"mathlib/Mathlib/LinearAlgebra/PiTensorProduct/Basis.lean","sample_id":"317b87a5beef988d47043cbd23eccd8456a804914b949c3de8ba8b8d221e791d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"54c464d6ce26bdef8908aa10edaaa4627d275c464a5704aa3ef40ec7a4c12203","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"39b50e23b82ce5537ed61a7d157a011c0e7700b054d0497f971696526a5f7387","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d0ac2882ec427829126b8598a0819730b07105bce20b1228031035f70c0b76c9","source_sha256":"0c7310750ac9c48298092f0639863d1202fe8a8e7eda47b7ee8abc6e42090355","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rintro ⟨m, hm⟩\n  have hm' : ∀ a, f a ≤ m := fun a ↦ hm <| Set.mem_range_self _\n  obtain ⟨a₀⟩ := ‹Nonempty α›\n  suffices ∀ b, f a₀ ≤ b → ∃ a, b < f a by\n    obtain ⟨a, ha⟩ : ∃ a, m < f a := this m (hm' a₀)\n    exact ha.not_ge (hm' a)\n  have h : ∀ a, ∃ a', f a < f a' := fun a ↦ (exists_gt a).imp (fun a' h ↦ hf h)\n  apply Succ.rec\n  · exact h a₀\n  rintro b _ ⟨a, hba⟩\n  exact (h a).imp (fun a' ↦ (succ_le_of_lt hba).trans_lt)","hard_negative":true,"metrics":{"chosen_tokens":153,"rejected_tokens":5,"token_jaccard":0.056604,"token_length_ratio":0.03268},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"a51bdaecde2fd85d06fd381bca6bbf98e63583d37902c33a2a7cbe7040ee0fd5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.SuccPred.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Archimedean successor and predecessor\n\n* `IsSuccArchimedean`: `SuccOrder` where `succ` iterated to an element gives all the greater\n  ones.\n* `IsPredArchimedean`: `PredOrder` where `pred` iterated to an element gives all the smaller\n  ones.\n-/\n\npublic section\n\nvariable {α β : Type*}\n\nopen Order Function\n\n/-- A `SuccOrder` is succ-archimedean if one can go from any two comparable elements by iterating\n`succ` -/\nclass IsSuccArchimedean (α : Type*) [Preorder α] [SuccOrder α] : Prop where\n  /-- If `a ≤ b` then one can get to `a` from `b` by iterating `succ` -/\n  exists_succ_iterate_of_le {a b : α} (h : a ≤ b) : ∃ n, succ^[n] a = b\n\n/-- A `PredOrder` is pred-archimedean if one can go from any two comparable elements by iterating\n`pred` -/\n@[to_dual existing]\nclass IsPredArchimedean (α : Type*) [Preorder α] [PredOrder α] : Prop where\n  /-- If `a ≤ b` then one can get to `b` from `a` by iterating `pred` -/\n  exists_pred_iterate_of_le {a b : α} (h : a ≤ b) : ∃ n, pred^[n] b = a\n\nexport IsSuccArchimedean (exists_succ_iterate_of_le)\nexport IsPredArchimedean (exists_pred_iterate_of_le)\n\nattribute [to_dual existing] exists_succ_iterate_of_le\n\nsection Preorder\n\nvariable [Preorder α]\n\n-- `to_dual` cannot yet reorder arguments of arguments\ninstance [SuccOrder α] [IsSuccArchimedean α] : IsPredArchimedean αᵒᵈ :=\n  ⟨fun {a b} h => by convert! exists_succ_iterate_of_le h.ofDual⟩\n\n@[to_dual existing]\ninstance [PredOrder α] [IsPredArchimedean α] : IsSuccArchimedean αᵒᵈ :=\n  ⟨fun {a b} h => by convert! exists_pred_iterate_of_le h.ofDual⟩\n\nsection SuccOrder\n\nvariable [SuccOrder α] [IsSuccArchimedean α] {a b : α}\n\n@[to_dual]\ntheorem LE.le.exists_succ_iterate (h : a ≤ b) : ∃ n, succ^[n] a = b :=\n  exists_succ_iterate_of_le h\n\n@[to_dual]\ntheorem exists_succ_iterate_iff_le : (∃ n, succ^[n] a = b) ↔ a ≤ b := by\n  refine ⟨?_, exists_succ_iterate_of_le⟩\n  rintro ⟨n, rfl⟩\n  exact id_le_iterate_of_id_le le_succ n a\n\n-- TODO: rename to `Order.succ_rec`?\n/-- Induction principle on a type with a `SuccOrder` for all elements above a given element `m`. -/\n@[to_dual (attr := elab_as_elim) Pred.rec\n/-- Induction principle on a type with a `PredOrder` for all elements below a given element `m`. -/]\ntheorem Succ.rec {m : α} {P : ∀ n, m ≤ n → Prop} (rfl : P m le_rfl)\n    (succ : ∀ n (hmn : m ≤ n), P n hmn → P (succ n) (hmn.trans <| le_succ _)) ⦃n : α⦄\n    (hmn : m ≤ n) : P n hmn := by\n  obtain ⟨n, rfl⟩ := hmn.exists_succ_iterate\n  induction n with\n  | zero => exact rfl\n  | succ n ih =>\n    simp_rw [Function.iterate_succ_apply']\n    exact succ _ (id_le_iterate_of_id_le le_succ n m) (ih _)\n\n@[to_dual Pred.rec_iff]\ntheorem Succ.rec_iff {p : α → Prop} (hsucc : ∀ a, p a ↔ p (succ a)) {a b : α} (h : a ≤ b) :\n    p a ↔ p b := by\n  obtain ⟨n, rfl⟩ := h.exists_succ_iterate\n  exact Iterate.rec (fun b => p a ↔ p b) Iff.rfl (fun c hc => hc.trans (hsucc _)) n\n\n@[to_dual le_total_of_directed]\nlemma le_total_of_codirected {r v₁ v₂ : α} (h₁ : r ≤ v₁) (h₂ : r ≤ v₂) : v₁ ≤ v₂ ∨ v₂ ≤ v₁ := by\n  obtain ⟨n, rfl⟩ := h₁.exists_succ_iterate\n  obtain ⟨m, rfl⟩ := h₂.exists_succ_iterate\n  clear h₁ h₂\n  wlog h : n ≤ m\n  · rw [Or.comm]\n    apply this\n    exact Nat.le_of_not_ge h\n  left\n  obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le h\n  rw [Nat.add_comm, Function.iterate_add, Function.comp_apply]\n  apply Order.le_succ_iterate\n\nend SuccOrder\n\nend Preorder\n\nsection PartialOrder\n\nvariable [PartialOrder α]\n\n@[to_dual (reorder := h₁ h₂) lt_or_le_of_directed]\nlemma lt_or_le_of_codirected [SuccOrder α] [IsSuccArchimedean α] {r v₁ v₂ : α} (h₁ : r ≤ v₁)\n    (h₂ : r ≤ v₂) : v₁ < v₂ ∨ v₂ ≤ v₁ := by\n  rw [Classical.or_iff_not_imp_right]\n  intro nh\n  rcases le_total_of_codirected h₁ h₂ with h | h\n  · apply lt_of_le_of_ne h (ne_of_not_le nh).symm\n  · contradiction\n\n-- `to_dual` cannot yet reorder arguments of arguments\n/--\nThis isn't an instance due to a loop with `LinearOrder`.\n-/\n-- See note [reducible non-instances]\nabbrev IsSuccArchimedean.linearOrder [SuccOrder α] [IsSuccArchimedean α]\n     [DecidableEq α] [DecidableLE α] [DecidableLT α]\n     [IsCodirectedOrder α] : LinearOrder α where\n  le_total a b :=\n    have ⟨c, ha, hb⟩ := directed_of (· ≥ ·) a b\n    le_total_of_codirected ha hb\n  toDecidableEq := inferInstance\n  toDecidableLE := inferInstance\n  toDecidableLT := inferInstance\n\n/--\nThis isn't an instance due to a loop with `LinearOrder`.\n-/\n-- See note [reducible non-instances]\n@[to_dual existing]\nabbrev IsPredArchimedean.linearOrder [PredOrder α] [IsPredArchimedean α]\n     [DecidableEq α] [DecidableLE α] [DecidableLT α]\n     [IsDirectedOrder α] : LinearOrder α :=\n  letI : LinearOrder αᵒᵈ := IsSuccArchimedean.linearOrder\n  inferInstanceAs (LinearOrder αᵒᵈᵒᵈ)\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [LinearOrder α]\n\nsection SuccOrder\nvariable [SuccOrder α]\n\n@[deprecated (since := \"2026-02-05\")] alias succ_max := Order.succ_max\n@[deprecated (since := \"2026-02-05\")] alias succ_min := Order.succ_min\n\n@[deprecated (since := \"2026-02-05\")] alias pred_max := Order.pred_max\n@[deprecated (since := \"2026-02-05\")] alias pred_min := Order.pred_min\n\nvariable [IsSuccArchimedean α] {a b : α}\n\n@[to_dual]\ntheorem exists_succ_iterate_or : (∃ n, succ^[n] a = b) ∨ ∃ n, succ^[n] b = a :=\n  (le_total a b).imp exists_succ_iterate_of_le exists_succ_iterate_of_le\n\n@[to_dual Pred.rec_linear]\ntheorem Succ.rec_linear {p : α → Prop} (hsucc : ∀ a, p a ↔ p (succ a)) (a b : α) : p a ↔ p b :=\n  (le_total a b).elim (Succ.rec_iff hsucc) fun h => (Succ.rec_iff hsucc h).symm\n\nend SuccOrder\n\nend LinearOrder\n\nsection bdd_range\nvariable [Preorder α] [Nonempty α] [Preorder β] {f : α → β}\n\n@[to_dual]\n\nTarget:\nlemma StrictMono.not_bddAbove_range_of_isSuccArchimedean [NoMaxOrder α] [SuccOrder β]\n    [IsSuccArchimedean β] (hf : StrictMono f) : ¬ BddAbove (Set.range f) :=\n\nProof body:\n","rejected":"by\n  exact StrictMono.not_bddAbove_range_of_isSuccArchimedean","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"d695199e13f0b965550c9f5560525db137aecb9838811b90f0a2ba7be0022612","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/SuccPred","family_id":"strictmono","file_id":"mathlib/Mathlib/Order/SuccPred/Archimedean.lean","sample_id":"d0ac2882ec427829126b8598a0819730b07105bce20b1228031035f70c0b76c9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fa7ef57b668670c0961ba6cf31ace864279e791239ac9d2e2fff89a3656780bf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b98968742f45cb65d12f6f4c121c37ed0ece6ebd6dff7331351e7a584617940a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f4231e8dda39dee1bbddb1e4ffb21d16d698adcaa0cf76876d45ddbd2a042b9f","source_sha256":"9edb2b00301b8c3a46b08790b6da8897248a760fa9e13f7ddfca795725728403","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.166667},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"a52ea9dda4cf5510a1acdc902343ddd6470392bcaef241db5416810ca1f94da9","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Products.Basic\n\nNamespace:\nCategoryTheory.Bifunctor\n\nLocal context:\n/-\nCopyright (c) 2017 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Kim Morrison\n-/\n/-!\n# Lemmas about functors out of product categories.\n-/\n\npublic section\n\n\nopen CategoryTheory\n\nnamespace CategoryTheory.Bifunctor\n\nuniverse v₁ v₂ v₃ u₁ u₂ u₃\n\nvariable {C : Type u₁} {D : Type u₂} {E : Type u₃}\nvariable [Category.{v₁} C] [Category.{v₂} D] [Category.{v₃} E]\n\nopen scoped Prod\n\n@[simp]\ntheorem map_id (F : C × D ⥤ E) (X : C) (Y : D) :\n    F.map ((𝟙 X) ×ₘ (𝟙 Y)) = 𝟙 (F.obj (X, Y)) :=\n  F.map_id (X, Y)\n\n@[simp]\ntheorem map_id_comp (F : C × D ⥤ E) (W : C) {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z) :\n    F.map (𝟙 W ×ₘ (f ≫ g)) = F.map (𝟙 W ×ₘ f) ≫ F.map (𝟙 W ×ₘ g) := by\n  rw [← Functor.map_comp, prod_comp, Category.comp_id]\n\n@[simp]\ntheorem map_comp_id (F : C × D ⥤ E) (X Y Z : C) (W : D) (f : X ⟶ Y) (g : Y ⟶ Z) :\n    F.map ((f ≫ g) ×ₘ 𝟙 W) = F.map (f ×ₘ 𝟙 W) ≫ F.map (g ×ₘ 𝟙 W) := by\n  rw [← Functor.map_comp, prod_comp, Category.comp_id]\n\n@[simp]\ntheorem diagonal (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :\n    F.map (𝟙 X ×ₘ g) ≫ F.map (f ×ₘ 𝟙 Y') = F.map (f ×ₘ g) := by\n  rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]\n\n@[simp]\n\nTarget:\ntheorem diagonal' (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :\n    F.map (f ×ₘ 𝟙 Y) ≫ F.map (𝟙 X' ×ₘ g) = F.map (f ×ₘ g) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_f4231e8dda39","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"aa8c8af28b716053b8fa5ffe35ff9f072b8b4fbdeeceed70db27f2990957b21f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Products","family_id":"diagonal","file_id":"mathlib/Mathlib/CategoryTheory/Products/Bifunctor.lean","sample_id":"f4231e8dda39dee1bbddb1e4ffb21d16d698adcaa0cf76876d45ddbd2a042b9f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9347b24919ebf6db54344437810a6552ffb014d358370aa936d28051c4570548","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d8c92fb88ac14308e19c63829929f1118eaa49d7f95506754d04cde5201048ea","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"02e40ac0605619b9d35a8cf4395e31d8750e7cb631f2db38ba43f21adcd702b7","source_sha256":"6ccf0444fc2138afeb704541116189c71bea73fd06b5fb4f13fd001240de6123","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [eq_comm, ← Ultrafilter.coe_le_coe]\n  change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, { v : Ultrafilter α | s ∈ v } ∈ u\n  simp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff,\n    mem_setOf_eq]\n  constructor\n  · intro h a ha\n    exact h _ ⟨ha, a, rfl⟩\n  · rintro h a ⟨xi, a, rfl⟩\n    exact h _ xi","hard_negative":false,"metrics":{"chosen_tokens":80,"rejected_tokens":84,"token_jaccard":0.94,"token_length_ratio":1.05},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"a553b6a93311ea378b1a9d2cb2c686abfa56cdf93685b941e0d54af26521c693","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Bases\npublic import Mathlib.Topology.DenseEmbedding\npublic import Mathlib.Topology.Connected.TotallyDisconnected\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton\n-/\n/-! # Stone-Čech compactification\n\nConstruction of the Stone-Čech compactification using ultrafilters.\n\nFor any topological space `α`, we build a compact Hausdorff space `StoneCech α` and a continuous\nmap `stoneCechUnit : α → StoneCech α` which is minimal in the sense of the following universal\nproperty: for any compact Hausdorff space `β` and every map `f : α → β` such that\n`hf : Continuous f`, there is a unique map `stoneCechExtend hf : StoneCech α → β` such that\n`stoneCechExtend_extends : stoneCechExtend hf ∘ stoneCechUnit = f`.\nContinuity of this extension is asserted by `continuous_stoneCechExtend` and uniqueness by\n`stoneCech_hom_ext`.\n\nBeware that the terminology “extend” is slightly misleading since `stoneCechUnit` is not always\ninjective, so one cannot always think of `α` as being “inside” its compactification `StoneCech α`.\n\n## Implementation notes\n\nParts of the formalization are based on “Ultrafilters and Topology”\nby Marius Stekelenburg, particularly section 5. However the construction in the general\ncase is different because the equivalence relation on spaces of ultrafilters described\nby Stekelenburg causes issues with universes since it involves a condition\non all compact Hausdorff spaces. We replace it by a two steps construction.\nThe first step called `PreStoneCech` guarantees the expected universal property but\nnot the Hausdorff condition. We then define `StoneCech α` as `T2Quotient (PreStoneCech α)`.\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen Filter Set\n\nopen Topology\n\nuniverse u v\n\nsection Ultrafilter\n\n/- The set of ultrafilters on α carries a natural topology which makes\n  it the Stone-Čech compactification of α (viewed as a discrete space). -/\n/-- Basis for the topology on `Ultrafilter α`. -/\ndef ultrafilterBasis (α : Type u) : Set (Set (Ultrafilter α)) :=\n  range fun s : Set α ↦ { u | s ∈ u }\n\nvariable {α : Type u}\n\ninstance Ultrafilter.topologicalSpace : TopologicalSpace (Ultrafilter α) :=\n  TopologicalSpace.generateFrom (ultrafilterBasis α)\n\ntheorem ultrafilterBasis_is_basis : TopologicalSpace.IsTopologicalBasis (ultrafilterBasis α) :=\n  ⟨by\n    rintro _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ u ⟨ua, ub⟩\n    refine ⟨_, ⟨a ∩ b, rfl⟩, inter_mem ua ub, fun v hv ↦ ⟨?_, ?_⟩⟩ <;> apply mem_of_superset hv <;>\n      simp [inter_subset_right],\n    eq_univ_of_univ_subset <| subset_sUnion_of_mem <| ⟨univ, eq_univ_of_forall fun _ ↦ univ_mem⟩,\n    rfl⟩\n\n/-- The basic open sets for the topology on ultrafilters are open. -/\ntheorem ultrafilter_isOpen_basic (s : Set α) : IsOpen { u : Ultrafilter α | s ∈ u } :=\n  ultrafilterBasis_is_basis.isOpen ⟨s, rfl⟩\n\n/-- The basic open sets for the topology on ultrafilters are also closed. -/\ntheorem ultrafilter_isClosed_basic (s : Set α) : IsClosed { u : Ultrafilter α | s ∈ u } := by\n  rw [← isOpen_compl_iff]\n  convert! ultrafilter_isOpen_basic sᶜ using 1\n  ext u\n  exact Ultrafilter.compl_mem_iff_notMem.symm\n\n/-- Every ultrafilter `u` on `Ultrafilter α` converges to a unique\n  point of `Ultrafilter α`, namely `joinM u`. -/\n\nTarget:\ntheorem ultrafilter_converges_iff {u : Ultrafilter (Ultrafilter α)} {x : Ultrafilter α} :\n    ↑u ≤ 𝓝 x ↔ x = joinM u :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [eq_comm, ← Ultrafilter.coe_le_coe]\n  change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, { v : Ultrafilter α | s ∈ v } ∈ u\n  simp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff,\n    mem_setOf_eq]\n  constructor\n  · intro h a ha\n    exact h _ ⟨ha, a, rfl⟩\n  · rintro h a ⟨xi, a, rfl⟩\n    exact h _ xi","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Compactification","family_id":"ultrafilter_converges_iff","file_id":"mathlib/Mathlib/Topology/Compactification/StoneCech.lean","sample_id":"02e40ac0605619b9d35a8cf4395e31d8750e7cb631f2db38ba43f21adcd702b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d23fb76aeeed9353afe60037f9d9b0091fa23e08b64b45bbb0d2d34201990525","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0e496e19d8169f840fb42721b4ba0ad691cd3e68e48df661f9f6fc16f635b298","source_sha256":"74b0e866ba6855a98543cc59c9d908f25a74c6e6456b4e8b44974d8edb78dfd2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [loopOfHom, ← Category.assoc, IsIso.comp_inv_eq, Category.id_comp]\n  rcases H with H | H\n  · rw [treeHom_eq T (Path.cons default ⟨Sum.inl e, H⟩), homOfPath]\n    rfl\n  · rw [treeHom_eq T (Path.cons default ⟨Sum.inr e, H⟩), homOfPath]\n    simp only [IsIso.inv_hom_id, Category.comp_id, Category.assoc, treeHom]","hard_negative":false,"metrics":{"chosen_tokens":86,"rejected_tokens":2,"token_jaccard":0.025641,"token_length_ratio":0.023256},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"a571e3c35aa7440511981858dcd61caa6cc8c057635cfa26894f9ad9199d865f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Action\npublic import Mathlib.Combinatorics.Quiver.Arborescence\npublic import Mathlib.Combinatorics.Quiver.ConnectedComponent\npublic import Mathlib.GroupTheory.FreeGroup.IsFreeGroup\n\nNamespace:\nIsFreeGroupoid.SpanningTree\n\nLocal context:\n/-\nCopyright (c) 2021 David Wärn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David Wärn\n-/\n/-!\n# The Nielsen-Schreier theorem\n\nThis file proves that a subgroup of a free group is itself free.\n\n## Main result\n\n- `subgroupIsFreeOfIsFree H`: an instance saying that a subgroup of a free group is free.\n\n## Proof overview\n\nThe proof is analogous to the proof using covering spaces and fundamental groups of graphs,\nbut we work directly with groupoids instead of topological spaces. Under this analogy,\n\n- `IsFreeGroupoid G` corresponds to saying that a space is a graph.\n- `endMulEquivSubgroup H` plays the role of replacing 'subgroup of fundamental group' with\n  'fundamental group of covering space'.\n- `actionGroupoidIsFree G A` corresponds to the fact that a covering of a (single-vertex)\n  graph is a graph.\n- `endIsFree T` corresponds to the fact that, given a spanning tree `T` of a\n  graph, its fundamental group is free (generated by loops from the complement of the tree).\n\n## Implementation notes\n\nOur definition of `IsFreeGroupoid` is nonstandard. Normally one would require that functors\n`G ⥤ X` to any _groupoid_ `X` are given by graph homomorphisms from the generators, but we only\nconsider _groups_ `X`. This simplifies the argument since functor equality is complicated in\ngeneral, but simple for functors to single object categories.\n\n## References\n\nhttps://ncatlab.org/nlab/show/Nielsen-Schreier+theorem\n\n## Tags\n\nfree group, free groupoid, Nielsen-Schreier\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nuniverse v u\n\nopen CategoryTheory CategoryTheory.ActionCategory CategoryTheory.SingleObj Quiver FreeGroup\n\n/-- `IsFreeGroupoid.Generators G` is a type synonym for `G`. We think of this as\nthe vertices of the generating quiver of `G` when `G` is free. We can't use `G` directly,\nsince `G` already has a quiver instance from being a groupoid. -/\n@[nolint unusedArguments]\ndef IsFreeGroupoid.Generators (G) [Groupoid G] :=\n  G\n\n/--\nA groupoid `G` is free when we have the following data:\n- a quiver on `IsFreeGroupoid.Generators G` (a type synonym for `G`)\n- a function `of` taking a generating arrow to a morphism in `G`\n- such that a functor from `G` to any group `X` is uniquely determined\n  by assigning labels in `X` to the generating arrows.\n\nThis definition is nonstandard. Normally one would require that functors `G ⥤ X`\nto any _groupoid_ `X` are given by graph homomorphisms from `generators`. -/\nclass IsFreeGroupoid (G) [Groupoid.{v} G] where\n  quiverGenerators : Quiver.{v} (IsFreeGroupoid.Generators G)\n  of : ∀ {a b : IsFreeGroupoid.Generators G}, (a ⟶ b) → ((show G from a) ⟶ b)\n  unique_lift :\n    ∀ {X : Type v} [Group X] (f : Labelling (IsFreeGroupoid.Generators G) X),\n      ∃! F : G ⥤ CategoryTheory.SingleObj X, ∀ (a b) (g : a ⟶ b), F.map (of g) = f g\n\nattribute [nolint docBlame] IsFreeGroupoid.of IsFreeGroupoid.unique_lift\n\nnamespace IsFreeGroupoid\n\nattribute [instance_reducible, instance] quiverGenerators\n\n/-- Two functors from a free groupoid to a group are equal when they agree on the generating\nquiver. -/\n@[ext]\ntheorem ext_functor {G} [Groupoid.{v} G] [IsFreeGroupoid G] {X : Type v} [Group X]\n    (f g : G ⥤ CategoryTheory.SingleObj X) (h : ∀ (a b) (e : a ⟶ b), f.map (of e) = g.map (of e)) :\n    f = g :=\n  let ⟨_, _, u⟩ := @unique_lift G _ _ X _ fun (a b : Generators G) (e : a ⟶ b) => g.map (of e)\n  _root_.trans (u _ h) (u _ fun _ _ _ => rfl).symm\n\n/-- An action groupoid over a free group is free. More generally, one could show that the groupoid\nof elements over a free groupoid is free, but this version is easier to prove and suffices for our\npurposes.\n\nAnalogous to the fact that a covering space of a graph is a graph. (A free groupoid is like a graph,\nand a groupoid of elements is like a covering space.) -/\ninstance actionGroupoidIsFree {G A : Type u} [Group G] [IsFreeGroup G] [MulAction G A] :\n    IsFreeGroupoid (ActionCategory G A) where\n  quiverGenerators :=\n    ⟨fun a b => { e : IsFreeGroup.Generators G // IsFreeGroup.of e • a.back = b.back }⟩\n  of := fun (e : Subtype _) => ⟨IsFreeGroup.of e, e.property⟩\n  unique_lift := by\n    intro X _ f\n    let f' : IsFreeGroup.Generators G → (A → X) ⋊[mulAutArrow] G := fun e =>\n      ⟨fun b => @f ⟨(), _⟩ ⟨(), b⟩ ⟨e, smul_inv_smul _ b⟩, IsFreeGroup.of e⟩\n    rcases IsFreeGroup.unique_lift f' with ⟨F', hF', uF'⟩\n    refine ⟨uncurry F' ?_, ?_, ?_⟩\n    · suffices SemidirectProduct.rightHom.comp F' = MonoidHom.id _ by\n        exact DFunLike.ext_iff.mp this\n      apply IsFreeGroup.ext_hom (fun x ↦ ?_)\n      rw [MonoidHom.comp_apply, hF']\n      rfl\n    · rintro ⟨⟨⟩, a : A⟩ ⟨⟨⟩, b⟩ ⟨e, h : IsFreeGroup.of e • a = b⟩\n      change (F' (IsFreeGroup.of _)).left _ = _\n      rw [hF']\n      cases inv_smul_eq_iff.mpr h.symm\n      rfl\n    · intro E hE\n      have : curry E = F' := by\n        apply uF'\n        intro e\n        ext\n        · convert! hE _ _ _\n          rfl\n        · rfl\n      apply Functor.hext\n      · intro\n        apply Unit.ext\n      · refine ActionCategory.cases ?_\n        intros\n        simp only [← this, uncurry_map, curry_apply_left, coe_back, homOfPair.val]\n        rfl\n\nnamespace SpanningTree\n\n/- In this section, we suppose we have a free groupoid with a spanning tree for its generating\nquiver. The goal is to prove that the vertex group at the root is free. A picture to have in mind\nis that we are 'pulling' the endpoints of all the edges of the quiver along the spanning tree to\nthe root. -/\nvariable {G : Type u} [Groupoid.{u} G] [IsFreeGroupoid G]\n  (T : WideSubquiver (Symmetrify <| Generators G)) [Arborescence T]\n\nset_option backward.privateInPublic true in\n/-- The root of `T`, except its type is `G` instead of the type synonym `T`. -/\nprivate def root' : G :=\n  show T from root T\n\n-- this has to be marked noncomputable, see issue https://github.com/leanprover-community/mathlib4/pull/451.\n-- It might be nicer to define this in terms of `composePath`\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- A path in the tree gives a hom, by composition. -/\ndef homOfPath : ∀ {a : G}, Path (root T) a → (root' T ⟶ a)\n  | _, Path.nil => 𝟙 _\n  | _, Path.cons p f => homOfPath p ≫ Sum.recOn f.val (fun e => of e) fun e => inv (of e)\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- For every vertex `a`, there is a canonical hom from the root, given by the path in the tree. -/\ndef treeHom (a : G) : root' T ⟶ a :=\n  homOfPath T default\n\n/-- Any path to `a` gives `treeHom T a`, since paths in the tree are unique. -/\ntheorem treeHom_eq {a : G} (p : Path (root T) a) : treeHom T a = homOfPath T p := by\n  rw [treeHom, Unique.default_eq]\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n@[simp]\ntheorem treeHom_root : treeHom T (root' T) = 𝟙 _ :=\n  -- this should just be `treeHom_eq T Path.nil`, but Lean treats `homOfPath` with suspicion.\n    _root_.trans\n    (treeHom_eq T Path.nil) rfl\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Any hom in `G` can be made into a loop, by conjugating with `treeHom`s. -/\ndef loopOfHom {a b : G} (p : a ⟶ b) : End (root' T) :=\n  treeHom T a ≫ p ≫ inv (treeHom T b)\n\nset_option backward.isDefEq.respectTransparency false in\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Turning an edge in the spanning tree into a loop gives the identity loop. -/\n\nTarget:\ntheorem loopOfHom_eq_id {a b : Generators G} (e) (H : e ∈ wideSubquiverSymmetrify T a b) :\n    loopOfHom T (of e) = 𝟙 (root' T) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/FreeGroup","family_id":"loopofhom_eq_id","file_id":"mathlib/Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean","sample_id":"0e496e19d8169f840fb42721b4ba0ad691cd3e68e48df661f9f6fc16f635b298"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b2fc7f29216c9249b22ab675eb45a0b56030097a0051e019b9740cd584e92cb3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"90a524481a706873c8f8be57d17806351124c549a6813c5356b1c63017ebd820","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"34e8994f5ec220f3f31f1cf7f6b1b16bc75863bde20089fcb254304fbec368fa","source_sha256":"c0e3fa9405ddf4fdd99cd3aa6c9ff7258178f89496e31a48d0f8f121f656258a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [liftFun_iff_succ r, forall_iff_succ, cons_val_succ, cons_val_zero, ← succ_castSucc,\n    castSucc_zero]\n\nvariable [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α}","hard_negative":true,"metrics":{"chosen_tokens":45,"rejected_tokens":3,"token_jaccard":0.03125,"token_length_ratio":0.066667},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"a5a98b2b82056923ba1ae5cc9b220874a47e971c32b4e864ca7ab0b417211dcc","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.VecNotation\npublic import Mathlib.Logic.Equiv.Fin.Basic\npublic import Mathlib.Order.Fin.Basic\npublic import Mathlib.Order.PiLex\npublic import Mathlib.Order.Interval.Set.Defs\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes\n-/\n/-!\n# Order properties on tuples\n-/\n\n@[expose] public section\n\nassert_not_exists Monoid\n\nopen Function Set\n\nnamespace Fin\nvariable {m n : ℕ} {α : Fin (n + 1) → Type*} (x : α 0) (q : ∀ i, α i) (p : ∀ i : Fin n, α i.succ)\n  (i : Fin n) (y : α i.succ) (z : α 0)\n\nlemma pi_lex_lt_cons_cons {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ}\n    (s : ∀ {i : Fin n.succ}, α i → α i → Prop) :\n    Pi.Lex (· < ·) (@s) (Fin.cons x₀ x) (Fin.cons y₀ y) ↔\n      s x₀ y₀ ∨ x₀ = y₀ ∧ Pi.Lex (· < ·) (@fun i : Fin n ↦ @s i.succ) x y := by\n  simp_rw [Pi.Lex, Fin.exists_fin_succ, Fin.cons_succ, Fin.cons_zero, Fin.forall_iff_succ]\n  simp [and_assoc, exists_and_left]\n\nvariable [∀ i, Preorder (α i)]\n\nlemma insertNth_mem_Icc {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)}\n    {q₁ q₂ : ∀ j, α j} :\n    i.insertNth x p ∈ Icc q₁ q₂ ↔\n      x ∈ Icc (q₁ i) (q₂ i) ∧ p ∈ Icc (fun j ↦ q₁ (i.succAbove j)) fun j ↦ q₂ (i.succAbove j) := by\n  simp only [mem_Icc, insertNth_le_iff, le_insertNth_iff, and_assoc, @and_left_comm (x ≤ q₂ i)]\n\nlemma preimage_insertNth_Icc_of_mem {i : Fin (n + 1)} {x : α i} {q₁ q₂ : ∀ j, α j}\n    (hx : x ∈ Icc (q₁ i) (q₂ i)) :\n    i.insertNth x ⁻¹' Icc q₁ q₂ = Icc (fun j ↦ q₁ (i.succAbove j)) fun j ↦ q₂ (i.succAbove j) :=\n  Set.ext fun p ↦ by simp only [mem_preimage, insertNth_mem_Icc, hx, true_and]\n\nlemma preimage_insertNth_Icc_of_notMem {i : Fin (n + 1)} {x : α i} {q₁ q₂ : ∀ j, α j}\n    (hx : x ∉ Icc (q₁ i) (q₂ i)) : i.insertNth x ⁻¹' Icc q₁ q₂ = ∅ :=\n  Set.ext fun p ↦ by\n    simp only [mem_preimage, insertNth_mem_Icc, hx, false_and, mem_empty_iff_false]\n\nend Fin\n\nopen Fin Matrix\n\nvariable {α : Type*}\n\nopen scoped Relator in\n\nTarget:\nlemma liftFun_vecCons {n : ℕ} (r : α → α → Prop) [IsTrans α r] {f : Fin (n + 1) → α} {a : α} :\n    ((· < ·) ⇒ r) (vecCons a f) (vecCons a f) ↔ r a (f 0) ∧ ((· < ·) ⇒ r) f f :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_34e8994f5ec2","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"4545c51f78b10c4746e0a81afaa7ab82f17a56df42cf05e6f1c5d0c4f8059431","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Fin","family_id":"liftfun_veccons","file_id":"mathlib/Mathlib/Order/Fin/Tuple.lean","sample_id":"34e8994f5ec220f3f31f1cf7f6b1b16bc75863bde20089fcb254304fbec368fa"},"split":"train"}
{"binding":{"chosen_proof_sha256":"991657dbd4edd454359c31dd8bdf594c016e0afa8edd3fccac619f800a61e43a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8b50f5b0dc46f4363a12bc6de0e2753f76c6021e160003cbe6dee0fdd74d0e52","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2adf0561a9917b239f775543158667a626bcb1a48a3b86163579bd9e8eb95116","source_sha256":"aa1352e2143e704eafbedf6a7be68b6abc2920999f9fb6e7acf0e05da957ef08","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [pointOfClosedPoint, ← SpecMap_residueFieldIsoBase_inv, ← Spec.map_comp]","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.230769},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"a5c81dc0255fab260b6e440824d90dcb075b869f129ad952c0695073e0c0a6ec","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.Finite\npublic import Mathlib.FieldTheory.IsAlgClosed.Basic\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2026 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Schemes over algebraically closed fields\n\nWe show that if `X` is locally of finite type over an algebraically closed field `k`,\nthen the closed points of `X` are in bijection with the `k`-points of `X`.\nSee `AlgebraicGeometry.pointEquivClosedPoint`.\n\n-/\n\n@[expose] public noncomputable section\n\nopen CategoryTheory\n\nnamespace AlgebraicGeometry\n\nuniverse u\n\nvariable {X Y : Scheme.{u}} {K : Type u} [Field K] [IsAlgClosed K]\n    (f : X ⟶ Spec (.of K)) [LocallyOfFiniteType f] (x : X) (hx : IsClosed {x})\n\n/-- If `X` is a locally of finite type `k`-scheme and `k` is algebraically closed, then\nthe residue field of any closed point of `x` is isomorphic to `k`. -/\ndef residueFieldIsoBase : X.residueField x ≅ .of K :=\n  letI : IsIso (Spec.preimage (X.fromSpecResidueField x ≫ f)) := by\n    have : IsFinite (X.fromSpecResidueField x ≫ f) := by\n      rw [isClosed_singleton_iff_isClosedImmersion] at hx\n      rw [isFinite_iff_locallyOfFiniteType_of_jacobsonSpace]\n      infer_instance\n    rw [ConcreteCategory.isIso_iff_bijective]\n    refine IsAlgClosed.ringHom_bijective_of_isIntegral _ ?_\n    rw [← IsIntegralHom.SpecMap_iff, Spec.map_preimage]\n    infer_instance\n  (asIso (Spec.preimage (X.fromSpecResidueField x ≫ f))).symm\n\n@[simp, reassoc]\nlemma SpecMap_residueFieldIsoBase_inv :\n    Spec.map (residueFieldIsoBase f x hx).inv = X.fromSpecResidueField x ≫ f :=\n  Spec.map_preimage _\n\n/-- If `k` is algebraically closed, this is the `k`-point of `X` associated to a closed point. -/\nnoncomputable\ndef pointOfClosedPoint : Spec (.of K) ⟶ X :=\n  Spec.map (residueFieldIsoBase f x hx).hom ≫ X.fromSpecResidueField x\n\n@[reassoc (attr := simp)]\n\nTarget:\nlemma pointOfClosedPoint_comp : pointOfClosedPoint f x hx ≫ f = 𝟙 _ :=\n\nProof body:\n","rejected":"by\n  exact pointOfClosedPoint_comp","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"48fe44c62faf4a71a9dae781c9c0b690b8e9bf64729deabd5648596f0f4ecb2b","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/AlgClosed","family_id":"pointofclosedpoint_comp","file_id":"mathlib/Mathlib/AlgebraicGeometry/AlgClosed/Basic.lean","sample_id":"2adf0561a9917b239f775543158667a626bcb1a48a3b86163579bd9e8eb95116"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0bd39f4463d7de80fb25819949e6b30d4389ae1eaeb6979f0ff98507bcbb9bbf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b95a85964efe4bfd41af2d45319f1c087d8c6031912de58a65333126f7c0afac","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f97af12d0eda400dcad66bb72ae73c904310f1b01ffca3c206a7820ced091c83","source_sha256":"a2d058104de73af2ef7925ee923e8edeb8d6ca8dc9be066178103a6d13175051","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cat_disch\n\nvariable {D : Type u'} [𝒟 : Category.{u'} D] (I J : D ⥤ C) (ρ : I ⟶ J) {W : D}","hard_negative":false,"metrics":{"chosen_tokens":39,"rejected_tokens":43,"token_jaccard":0.884615,"token_length_ratio":1.102564},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"a630f0cff0b0ca9510bc167e5a818c3f816f7577dfbc55013e371950544f4953","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Elementwise\npublic import Mathlib.CategoryTheory.EpiMono\npublic import Mathlib.Tactic.PPWithUniv\npublic import Mathlib.Tactic.ToAdditive\n\nNamespace:\nCategoryTheory.FunctorToTypes\n\nLocal context:\n/-\nCopyright (c) 2017 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Kim Morrison, Johannes Hölzl, Dagur Asgeirsson\n-/\n/-!\n# The category `Type`.\n\nIn this section we define a `LargeCategory` structure on `Type u`, in such a way that it becomes a\n`ConcreteCategory`.\n\n## Implementation\n\nWe define the one-field structure `TypeCat.Fun` to wrap a function between types, and a `FunLike`\ninstance on it. Then we define a one-field structure `TypeCat.Hom` which wraps a `Fun`. The\nmorphisms in the category `Type u` are defined to be `TypeCat.Hom`, and the `FC` parameter of\nthe `ConcreteCategory` instance is `TypeCat.Fun`. `TypeCat.Fun` serves as a layer of separation\nbetween the `FC` parameter of the `ConcreteCategory` instance and bare functions, to avoid defining\na `FunLike` instance on the latter (which would give two non-reducibly defeq coercions from\nmorphisms in `Type` to functions), and the outer nesting `TypeCat.Hom` gives a layer of separation\nbetween morphisms and `FC`, as is done for all concrete categories in mathlib.\n\nTo promote a function to a morphism in this category, we provide the abbreviation `↾f`,\nas well as a corresponding notation `↾f`. (Entered as `\\upr `.)\n\n## Main definitions\n\nWe define `uliftFunctor`, from `Type u` to `Type (max u v)`, and show that it is fully faithful\n(but not, of course, essentially surjective).\n\nWe prove some basic facts about the category `Type`:\n*  epimorphisms are surjections and monomorphisms are injections,\n* `Iso` is both `Iso` and `Equiv` to `Equiv` (at least within a fixed universe),\n* every type level `IsLawfulFunctor` gives a categorical functor `Type ⥤ Type`\n  (the corresponding fact about monads is in `Mathlib/CategoryTheory/Monad/Types.lean`).\n-/\n\n@[expose] public section\n\n-- morphism levels before object levels. See note [category theory universes].\nuniverse v w u u'\n\nnamespace TypeCat\n\n/-- A one-field structure wrapping a function between types. -/\n@[ext]\nstructure Fun (X Y : Type*) where\n  /-- The underlying function. -/\n  toFun : X → Y\n\ninstance instFunLikeFun {X Y : Type*} : FunLike (Fun X Y) X Y where\n  coe f x := f.toFun x\n  coe_injective _ := by aesop\n\ninitialize_simps_projections Fun (toFun → apply)\n\nlemma Fun.mk_apply {X Y : Type*} (f : X → Y) (x : X) : (Fun.mk f) x = f x :=\n  rfl\n\n@[simp]\nlemma Fun.coe_mk {X Y : Type*} (f : X → Y) : (Fun.mk f : X → Y) = f :=\n  rfl\n\n/-- The identity function as a `Fun`. -/\n@[simps! +dsimpLhs]\ndef Fun.id (X : Type*) : Fun X X := Fun.mk _root_.id\n\n/-- Composition of `Fun`s. -/\n@[simps! +dsimpLhs]\ndef Fun.comp {X Y Z : Type*} (f : Fun Y Z) (g : Fun X Y) : Fun X Z := mk (f.toFun ∘ g.toFun)\n\n/-- The equivalence between `Fun`s and functions between types. -/\ndef Fun.homEquiv (X Y : Type u) : (Fun X Y) ≃ (X → Y) where\n  toFun f := f\n  invFun f := ⟨f⟩\n  left_inv := by intro; rfl\n  right_inv := by intro; rfl\n\n/-- The type of morphisms in `Type`. -/\n@[ext]\nstructure Hom (X Y : Type u) where\n  private mk ::\n  /-- The underlying function -/\n  hom' : Fun X Y\n\nend TypeCat\n\nopen TypeCat CategoryTheory\n\nset_option backward.privateInPublic true in\n@[to_additive_do_translate] -- Expressions involving this instance can still be additivized.\ninstance CategoryTheory.types : Category.{u} (Type u) where\n  Hom := Hom\n  id X := .mk <| .id X\n  comp f g := .mk <| g.hom'.comp f.hom'\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/--\nThe concrete category instance on `Type u`.\n\nNote: sometimes one needs to specify explicitly `(CC := fun X ↦ X)` to help typeclass inference.\n-/\ninstance : ConcreteCategory.{u} (Type u) Fun where\n  hom := Hom.hom'\n  ofHom := Hom.mk\n\nexample (X Y : Type u) (f : X ⟶ Y) : (f : X → Y) = (ConcreteCategory.hom f : X → Y) := by\n  with_reducible rfl\n\nexample (X Y : Type u) (f : X ⟶ Y) (x : X) : f x = (f : X → Y) x := by\n  with_reducible rfl\n\nexample (X Y : Type*) (f : Fun X Y) : (f : X → Y) = f := by\n  with_reducible rfl\n\nexample (X Y : Type*) (f : Fun X Y) (x : X) : f x = (f : X → Y) x := by\n  with_reducible rfl\n\nnamespace TypeCat\n\n/-- Turn a morphism in `Type` back into a function. -/\nabbrev Hom.hom {X Y : Type u} (f : Hom X Y) : Fun X Y :=\n  ConcreteCategory.hom (C := Type u) f\n\n/-- Typecheck a function as a morphism in `Type`. -/\nabbrev ofHom {X Y : Type u} (f : X → Y) : X ⟶ Y :=\n  ConcreteCategory.ofHom (Fun.mk f)\n\nend TypeCat\n\nnamespace CategoryTheory\n\n@[inherit_doc]\nscoped notation \"↾\" f:200 => TypeCat.ofHom f\n\nend CategoryTheory\n\nnamespace TypeCat\n\n/-- Use the `ConcreteCategory.hom` projection for `@[simps]` lemmas. -/\ndef Hom.Simps.hom (X Y : Type u) (f : X ⟶ Y) :=\n  ConcreteCategory.hom f\n\ninitialize_simps_projections Hom (hom' → hom)\n\n@[simp]\nlemma Fun.toFun_apply {X Y : Type u} (f : Fun X Y) (x : X) : f.toFun x = f x :=\n  rfl\n\nexample (X : Type u) : CategoryTheory.ToType X = X := by with_reducible rfl\n\n@[simp]\nlemma ofHom_eq {X Y : Type u} (f : X ⟶ Y) : ofHom f = f :=\n  rfl\n\n@[simp high]\nlemma hom_ofHom {X Y : Type u} (f : X → Y) : Hom.hom (ofHom f) = Fun.mk f := rfl\n\n@[simp]\nlemma ofHom_hom {X Y : Type u} (f : X ⟶ Y) : ofHom (Hom.hom f) = f := rfl\n\n@[simp]\nlemma ofHom_apply {X Y : Type u} (f : X → Y) (x : X) :\n    (↾f) x = f x :=\n  rfl\n\n/-- `TypeCat.Hom.hom` bundled as an `Equiv`. -/\ndef homEquiv {X Y : Type u} : (X ⟶ Y) ≃ (X → Y) :=\n  (ConcreteCategory.homEquiv (C := Type u)).trans (Fun.homEquiv _ _)\n\n@[simp]\nlemma homEquiv_apply {X Y : Type u} (f : X ⟶ Y) :\n    homEquiv f = f :=\n  rfl\n\n@[simp]\nlemma homEquiv_symm_apply {X Y : Type u} (f : X → Y) :\n    homEquiv.symm f = ofHom f :=\n  rfl\n\nlemma congr_arg {X Y : Type u} (f : X ⟶ Y) {x x' : X} (h : x = x') : f x = f x' := by\n  rw [h]\n\nend TypeCat\n\nnamespace CategoryTheory\n\ntheorem types_id (X : Type u) : (𝟙 X : _ → _) = id :=\n  rfl\n\ntheorem types_comp {X Y Z : Type u} (f : X ⟶ Y) (g : Y ⟶ Z) :\n    ConcreteCategory.hom (f ≫ g) = g ∘ f :=\n  rfl\n\n@[simp]\nlemma types_id_apply (X : Type u) (x : X) : 𝟙 X x = x :=\n  rfl\n\n@[simp]\nlemma types_comp_apply {X Y Z : Type u} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :\n    (f ≫ g) x = g (f x) :=\n  rfl\n\n@[congr]\nlemma types_congr_hom {X Y : Type u} {f g : X ⟶ Y} (h : f = g) (x : X) : f x = g x :=\n  ConcreteCategory.congr_hom h x\n\n@[deprecated (since := \"2026-02-09\")] alias hom_inv_id_apply := Iso.hom_inv_id_apply\n@[deprecated (since := \"2026-02-09\")] alias inv_hom_id_apply := Iso.inv_hom_id_apply\n@[deprecated (since := \"2026-02-09\")] alias asHom := ofHom\n\nnamespace Functor\n\nvariable {J : Type u} [Category.{v} J]\n\n/-- The sections of a functor `F : J ⥤ Type` are\nthe choices of a point `u j : F.obj j` for each `j`,\nsuch that `F.map f (u j) = u j'` for every morphism `f : j ⟶ j'`.\n\nWe later use these to define limits in `Type` and in many concrete categories.\n-/\ndef sections (F : J ⥤ Type w) : Set (∀ j, F.obj j) :=\n  { u | ∀ {j j'} (f : j ⟶ j'), F.map f (u j) = u j' }\n\n@[simp]\nlemma sections_property {F : J ⥤ Type w} (s : F.sections)\n    {j j' : J} (f : j ⟶ j') : F.map f (s.val j) = s.val j' :=\n  s.property f\n\nlemma sections_ext_iff {F : J ⥤ Type w} {x y : F.sections} : x = y ↔ ∀ j, x.val j = y.val j :=\n  Subtype.ext_iff.trans funext_iff\n\nvariable (J)\n\n/-- The functor which sends a functor to types to its sections. -/\n@[simps]\ndef sectionsFunctor : (J ⥤ Type w) ⥤ Type max u w where\n  obj F := F.sections\n  map {F G} φ := ↾fun x ↦ ⟨fun j => φ.app j (x.1 j), fun {j j'} f =>\n    by simp [← NatTrans.naturality_apply, x.2 f]⟩\n\nend Functor\n\nnamespace FunctorToTypes\n\nvariable {C : Type u} [Category.{v} C] (F G H : C ⥤ Type w) {X Y Z : C}\nvariable (σ : F ⟶ G) (τ : G ⟶ H)\n\nattribute [elementwise nosimp] Functor.map_comp Functor.map_id NatTrans.comp_app\n\n@[deprecated Functor.map_comp_apply (since := \"2026-03-09\")]\ntheorem map_comp_apply (f : X ⟶ Y) (g : Y ⟶ Z) (a : F.obj X) :\n    (F.map (f ≫ g)) a = (F.map g) ((F.map f) a) :=\n  F.map_comp_apply f g a\n\n@[deprecated Functor.map_id_apply (since := \"2026-03-09\")]\ntheorem map_id_apply (a : F.obj X) : (F.map (𝟙 X)) a = a :=\n  F.map_id_apply X a\n\n@[deprecated (since := \"2026-02-09\")] alias naturality := NatTrans.naturality_apply\n\n@[deprecated NatTrans.comp_app_apply (since := \"2026-03-09\")]\ntheorem comp (x : F.obj X) : (σ ≫ τ).app X x = τ.app X (σ.app X x) :=\n  σ.comp_app_apply τ X x\n\nattribute [elementwise (attr := simp)] eqToHom_map_comp\n\n@[deprecated \"Use `elementwise_of% eqToHom_map_comp` instead\" (since := \"2026-02-09\")]\n\nTarget:\ntheorem eqToHom_map_comp_apply (p : X = Y) (q : Y = Z) (x : F.obj X) :\n    F.map (eqToHom q) (F.map (eqToHom p) x) = F.map (eqToHom <| p.trans q) x :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  cat_disch\n\nvariable {D : Type u'} [𝒟 : Category.{u'} D] (I J : D ⥤ C) (ρ : I ⟶ J) {W : D}","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Types","family_id":"eqtohom_map_comp_apply","file_id":"mathlib/Mathlib/CategoryTheory/Types/Basic.lean","sample_id":"f97af12d0eda400dcad66bb72ae73c904310f1b01ffca3c206a7820ced091c83"},"split":"train"}
{"binding":{"chosen_proof_sha256":"61dd4a4b69745b7e0fa68c4103cf026764280161c8ce4459f05b5e5caa88e47f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"113a4b35611419334ebc80fdab8a9b41f3954008d94e0bf336640232d6661a89","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7de96c1ef7802d716a78d8f34f82e51e820ae7e56b3d2af666ad2f6428d02fa0","source_sha256":"15e2b4cca4919764d55bc958e94a932ae425f5e9b6e7dc6036a891b4490f43b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [univLE_iff_cardinal_le]; apply le_total","hard_negative":true,"metrics":{"chosen_tokens":8,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.625},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"a64d1f7253bedebff92cced156c06005fa6b375b79f2836e32b8d55d6759bf9e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Logic.UnivLE\npublic import Mathlib.SetTheory.Ordinal.Univ\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu\n-/\n/-!\n# UnivLE and cardinals\n-/\n\npublic section\n\nnoncomputable section\n\nuniverse u v\n\nopen Cardinal\n\ntheorem univLE_iff_cardinal_le : UnivLE.{u, v} ↔ univ.{u, v + 1} ≤ univ.{v, u + 1} := by\n  simp_rw [univLE_iff, small_iff_lift_mk_lt_univ]\n  contrapose!\n  -- strange: simp_rw [univ_umax.{v,u}] doesn't work\n  refine ⟨fun ⟨α, le⟩ ↦ ?_, fun h ↦ ?_⟩\n  · rw [univ_umax.{v, u}, ← lift_le.{u + 1}, lift_univ, lift_lift] at le\n    exact le.trans_lt (lift_lt_univ'.{u, v + 1} #α)\n  · obtain ⟨⟨α⟩, h⟩ := lt_univ'.mp h; use α\n    rw [univ_umax.{v, u}, ← lift_le.{u + 1}, lift_univ, lift_lift]\n    exact h.le\n\ntheorem univLE_iff_exists_embedding : UnivLE.{u, v} ↔ Nonempty (Ordinal.{u} ↪ Ordinal.{v}) := by\n  rw [univLE_iff_cardinal_le]\n  exact lift_mk_le'\n\ntheorem Ordinal.univLE_of_injective {f : Ordinal.{u} → Ordinal.{v}} (h : f.Injective) :\n    UnivLE.{u, v} :=\n  univLE_iff_exists_embedding.2 ⟨f, h⟩\n\n/-- Together with transitivity, this shows `UnivLE` is a total preorder. -/\n\nTarget:\ntheorem univLE_total : UnivLE.{u, v} ∨ UnivLE.{v, u} :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_7de96c1ef780","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"a7cef40ca6bea23817455a518594ece5bb094d201f96f748c81aa510036c97df","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Cardinal","family_id":"univle_total","file_id":"mathlib/Mathlib/SetTheory/Cardinal/UnivLE.lean","sample_id":"7de96c1ef7802d716a78d8f34f82e51e820ae7e56b3d2af666ad2f6428d02fa0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"41ba997d60350b8410456fab7fd86644a15e80166bdc8dbee3ebfacac2920661","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"869b7b26a3015184cd23c518f62b2057658f45868fdff44e1d035d1f7987c1ca","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ab793cebc21f139d2eab524640ed552866c6f6bd28b0f8c110edadda18cfb9c9","source_sha256":"2fda79aa400532989a3a948709c014243f7a3167b1476a14416ee8c8dd7c852b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction l with\n  | nil => simp only [List.not_mem_nil, false_and, exists_false] at hl\n  | cons x l ih =>\n    simp only [List.mem_cons, exists_eq_or_imp] at hl\n    rcases hl with h | hal\n    · simpa only [List.map_cons, List.prod_cons] using I.mul_mem_right _ _ h\n    · simpa only [List.map_cons, List.prod_cons] using I.mul_mem_left _ _ <| ih hal","hard_negative":false,"metrics":{"chosen_tokens":84,"rejected_tokens":89,"token_jaccard":0.880952,"token_length_ratio":1.059524},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"a68115044813e79127693707677f47018f1c1270241d0e99b4d86216d7770ffb","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Congruence.BigOperators\npublic import Mathlib.RingTheory.TwoSidedIdeal.Basic\n\nNamespace:\nTwoSidedIdeal\n\nLocal context:\n/-\nCopyright (c) 2024 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang\n-/\n/-!\n# Interactions between `∑, ∏` and two-sided ideals\n\n-/\n\npublic section\n\nnamespace TwoSidedIdeal\n\nsection sum\n\nvariable {R : Type*} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R)\n\nlemma listSum_mem {ι : Type*} (l : List ι) (f : ι → R) (hl : ∀ x ∈ l, f x ∈ I) :\n    (l.map f).sum ∈ I := by\n  rw [mem_iff, ← List.sum_map_zero]\n  exact I.ringCon.listSum l hl\n\nlemma multisetSum_mem {ι : Type*} (s : Multiset ι) (f : ι → R) (hs : ∀ x ∈ s, f x ∈ I) :\n    (s.map f).sum ∈ I := by\n  rw [mem_iff, ← Multiset.sum_map_zero]\n  exact I.ringCon.multisetSum s hs\n\nlemma finsetSum_mem {ι : Type*} (s : Finset ι) (f : ι → R) (hs : ∀ x ∈ s, f x ∈ I) :\n    s.sum f ∈ I := by\n  rw [mem_iff, ← Finset.sum_const_zero]\n  exact I.ringCon.finsetSum s hs\n\nlemma finsuppSum_mem {ι : Type*} {β : Type*} [Zero β]\n    {f : ι →₀ β} (g : ι → β → R) (h : ∀ i ∈ f.support, g i (f i) ∈ I) :\n    f.sum g ∈ I :=\n  finsetSum_mem _ _ _ h\n\nlemma dfinsuppSum_mem {ι : Type*} {β : ι → Type*}\n    [DecidableEq ι] [∀ i, Zero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)]\n    {f : Π₀ i, β i} (g : (i : ι) → β i → R) (h : ∀ i ∈ f.support, g i (f i) ∈ I) :\n    f.sum g ∈ I :=\n  finsetSum_mem _ _ _ h\n\nend sum\n\nsection prod\n\nsection ring\n\nvariable {R : Type*} [Ring R] (I : TwoSidedIdeal R)\n\nTarget:\nlemma listProd_mem {ι : Type*} (l : List ι) (f : ι → R) (hl : ∃ x ∈ l, f x ∈ I) :\n    (l.map f).prod ∈ I :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  induction l with\n  | nil => simp only [List.not_mem_nil, false_and, exists_false] at hl\n  | cons x l ih =>\n    simp only [List.mem_cons, exists_eq_or_imp] at hl\n    rcases hl with h | hal\n    · simpa only [List.map_cons, List.prod_cons] using I.mul_mem_right _ _ h\n    · simpa only [List.map_cons, List.prod_cons] using I.mul_mem_left _ _ <| ih hal","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/TwoSidedIdeal","family_id":"listprod_mem","file_id":"mathlib/Mathlib/RingTheory/TwoSidedIdeal/BigOperators.lean","sample_id":"ab793cebc21f139d2eab524640ed552866c6f6bd28b0f8c110edadda18cfb9c9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"19d0436be488b2c04681a3df70e828f641be531fc0e4601e74671ee295a63fea","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"327cb8fff8f9cfd0e40105ba93b9c9cf5b2158a5ae99497570a25e89ca40759c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"08fc9f4a5a976873fc6fe96edc2ccf033e6ac6f6554e9759e87483072bcf261d","source_sha256":"d5c87fac893fcf2654818fe7d90a93a609b65e58a2bb593393eb189396ee74f1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rwa [H.ker_flip]","hard_negative":true,"metrics":{"chosen_tokens":7,"rejected_tokens":3,"token_jaccard":0.111111,"token_length_ratio":0.428571},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"a6be3fd05195e6f878a64aeb8a696eeb1047161431e49a7709e89178422881e3","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Basis.Basic\npublic import Mathlib.LinearAlgebra.BilinearMap\npublic import Mathlib.LinearAlgebra.LinearIndependent.Lemmas\nimport Mathlib.Algebra.Module.Torsion.Field\n\nNamespace:\nLinearMap.IsRefl\n\nLocal context:\n/-\nCopyright (c) 2018 Andreas Swerdlow. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andreas Swerdlow\n-/\n/-!\n# Sesquilinear maps\n\nThis file provides properties about sesquilinear maps and forms. The maps considered are of the\nform `M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M`, where `I₁ : R₁ →+* R` and `I₂ : R₂ →+* R` are ring homomorphisms and\n`M₁` is a module over `R₁`, `M₂` is a module over `R₂` and `M` is a module over `R`.\nSesquilinear forms are the special case that `M₁ = M₂`, `M = R₁ = R₂ = R`, and `I₁ = RingHom.id R`.\nTaking additionally `I₂ = RingHom.id R`, then one obtains bilinear forms.\n\nSesquilinear maps are a special case of the bilinear maps defined in `BilinearMap.lean`, and many\nbasic lemmas about construction and elementary calculations are found there.\n\n## Main declarations\n\n* `IsSymm`, `IsAlt`: states that a sesquilinear form is symmetric and alternating, respectively\n* `orthogonalBilin` provides the orthogonal complement with respect to a sesquilinear map\n\n## References\n\n* <https://en.wikipedia.org/wiki/Sesquilinear_form#Over_arbitrary_rings>\n\n## Tags\n\nSesquilinear form, Sesquilinear map\n-/\n\n@[expose] public section\n\nopen Module\n\nvariable {R R₁ R₂ R₃ M M₁ M₂ M₃ Mₗ₁ Mₗ₁' Mₗ₂ Mₗ₂' K K₁ K₂ V V₁ V₂ n : Type*}\n\nnamespace LinearMap\n\n/-! ### Orthogonal vectors -/\n\n\nsection CommRing\n\n-- the `ₗ` subscript variables are for special cases about linear (as opposed to semilinear) maps\nvariable [CommSemiring R] [CommSemiring R₁] [AddCommMonoid M₁] [Module R₁ M₁] [CommSemiring R₂]\n  [AddCommMonoid M₂] [Module R₂ M₂] [AddCommMonoid M] [Module R M]\n  {I₁ : R₁ →+* R} {I₂ : R₂ →+* R} {I₁' : R₁ →+* R}\n\n/-- The proposition that two elements of a sesquilinear map space are orthogonal -/\n@[deprecated \"Use `B x y = 0`\" (since := \"2026-03-30\")]\ndef IsOrtho (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x : M₁) (y : M₂) : Prop :=\n  B x y = 0\n\nset_option linter.deprecated false in\n@[deprecated \"`LinearMap.IsOrtho` has been deprecated\" (since := \"2026-03-30\")]\ntheorem isOrtho_def {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M} {x y} : B.IsOrtho x y ↔ B x y = 0 :=\n  Iff.rfl\n\nset_option linter.deprecated false in\n@[deprecated \"`LinearMap.IsOrtho` has been deprecated\" (since := \"2026-03-30\")]\ntheorem isOrtho_zero_left (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x) : IsOrtho B (0 : M₁) x := by\n  dsimp only [IsOrtho]\n  rw [map_zero B, zero_apply]\n\nset_option linter.deprecated false in\n@[deprecated \"`LinearMap.IsOrtho` has been deprecated\" (since := \"2026-03-30\")]\ntheorem isOrtho_zero_right (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x) : IsOrtho B x (0 : M₂) :=\n  map_zero (B x)\n\nset_option linter.deprecated false in\n@[deprecated \"`LinearMap.IsOrtho` has been deprecated\" (since := \"2026-03-30\")]\ntheorem isOrtho_flip {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M} {x y} : B.IsOrtho x y ↔ B.flip.IsOrtho y x := by\n  simp_rw [isOrtho_def, flip_apply]\n\nopen scoped Function in -- required for scoped `on` notation\n/-- A set of vectors `v` is orthogonal with respect to some bilinear map `B` if and only\nif for all `i ≠ j`, `B (v i) (v j) = 0`. -/\ndef IsOrthoᵢ (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M) (v : n → M₁) : Prop :=\n  Pairwise ((fun n m => B n m = 0) on v)\n\ntheorem isOrthoᵢ_def {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M} {v : n → M₁} :\n    B.IsOrthoᵢ v ↔ ∀ i j : n, i ≠ j → B (v i) (v j) = 0 :=\n  Iff.rfl\n\ntheorem isOrthoᵢ_flip (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M) {v : n → M₁} :\n    B.IsOrthoᵢ v ↔ B.flip.IsOrthoᵢ v := by\n  simp_rw [isOrthoᵢ_def]\n  constructor <;> exact fun h i j hij ↦ h j i hij.symm\n\nend CommRing\n\nsection Field\n\nvariable [Field K] [AddCommGroup V] [Module K V] [Field K₁] [AddCommGroup V₁] [Module K₁ V₁]\n  [Field K₂] [AddCommGroup V₂] [Module K₂ V₂]\n  {I₁ : K₁ →+* K} {I₂ : K₂ →+* K} {I₁' : K₁ →+* K} {J₁ : K →+* K} {J₂ : K →+* K}\n\nset_option linter.deprecated false in\n@[deprecated \"`LinearMap.IsOrtho` has been deprecated\" (since := \"2026-03-30\")]\ntheorem ortho_smul_left {B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V} {x y} {a : K₁} (ha : a ≠ 0) :\n    IsOrtho B x y ↔ IsOrtho B (a • x) y := by\n  dsimp only [IsOrtho]\n  constructor <;> intro H\n  · rw [map_smulₛₗ₂, H, smul_zero]\n  · rw [map_smulₛₗ₂, smul_eq_zero] at H\n    rcases H with H | H\n    · rw [map_eq_zero I₁] at H\n      trivial\n    · exact H\n\nset_option linter.deprecated false in\n@[deprecated \"`LinearMap.IsOrtho` has been deprecated\" (since := \"2026-03-30\")]\ntheorem ortho_smul_right {B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V} {x y} {a : K₂} {ha : a ≠ 0} :\n    IsOrtho B x y ↔ IsOrtho B x (a • y) := by\n  simp_all [IsOrtho]\n\n/-- A set of orthogonal vectors `v` with respect to some sesquilinear map `B` is linearly\n  independent if for all `i`, `B (v i) (v i) ≠ 0`. -/\ntheorem linearIndependent_of_isOrthoᵢ {B : V₁ →ₛₗ[I₁] V₁ →ₛₗ[I₁'] V} {v : n → V₁}\n    (hv₁ : B.IsOrthoᵢ v) (hv₂ : ∀ i, B (v i) (v i) ≠ 0) : LinearIndependent K₁ v := by\n  classical\n    rw [linearIndependent_iff']\n    intro s w hs i hi\n    have : B (s.sum fun i : n ↦ w i • v i) (v i) = 0 := by rw [hs, map_zero, zero_apply]\n    have hsum : (s.sum fun j : n ↦ I₁ (w j) • B (v j) (v i)) = I₁ (w i) • B (v i) (v i) := by\n      apply Finset.sum_eq_single_of_mem i hi\n      intro j _hj hij\n      rw [isOrthoᵢ_def.1 hv₁ _ _ hij, smul_zero]\n    simp_rw [B.map_sum₂, map_smulₛₗ₂, hsum] at this\n    apply (map_eq_zero I₁).mp\n    exact (smul_eq_zero.mp this).elim _root_.id (hv₂ i · |>.elim)\n\nend Field\n\n/-! ### Reflexive bilinear maps -/\n\nsection Reflexive\n\nvariable [CommSemiring R] [AddCommMonoid M] [Module R M] [CommSemiring R₁] [AddCommMonoid M₁]\n  [Module R₁ M₁] {I₁ : R₁ →+* R} {I₂ : R₁ →+* R} {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M}\n\n/-- The proposition that a sesquilinear map is reflexive -/\ndef IsRefl (B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M) : Prop :=\n  ∀ x y, B x y = 0 → B y x = 0\n\nnamespace IsRefl\n\nsection\nvariable (H : B.IsRefl)\ninclude H\n\ntheorem eq_zero : ∀ {x y}, B x y = 0 → B y x = 0 := fun {x y} ↦ H x y\n\ntheorem eq_iff {x y} : B x y = 0 ↔ B y x = 0 := ⟨H x y, H y x⟩\n\n@[deprecated (since := \"2026-03-30\")]\nalias ortho_comm := eq_iff\n\ntheorem domRestrict (p : Submodule R₁ M₁) : (B.domRestrict₁₂ p p).IsRefl :=\n  fun _ _ ↦ by\n  simp_rw [domRestrict₁₂_apply]\n  exact H _ _\nend\n\n@[simp]\ntheorem flip_isRefl_iff : B.flip.IsRefl ↔ B.IsRefl :=\n  forall_comm\n\nlemma ker_flip (H : B.IsRefl) : B.flip.ker = B.ker := by\n  ext x\n  simp [LinearMap.ext_iff, H.eq_iff]\n\nTarget:\ntheorem ker_flip_eq_bot (H : B.IsRefl) (h : LinearMap.ker B = ⊥) : LinearMap.ker B.flip = ⊥ :=\n\nProof body:\n","rejected":"by\n  exact ker_flip_eq_bot","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"61f421ed6f5d7dab3aafd8a69443d41e38c44d8201ae06d203c8fa6f3bd08f69","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/SesquilinearForm","family_id":"ker_flip_eq_bot","file_id":"mathlib/Mathlib/LinearAlgebra/SesquilinearForm/Basic.lean","sample_id":"08fc9f4a5a976873fc6fe96edc2ccf033e6ac6f6554e9759e87483072bcf261d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"82023a5a0373446dea510bd79a74d7b917c1be62462ff0113fe4a4cf3e1520cd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"87399eed90a7a9d71500444beb6af36405b8a0740724037a49d189fcab7e9413","source_sha256":"49729ab7657bf5b96e3883707dabc942ba77c2ab9a085e0f2d0959061ea37dde","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by simp) :\n    ∃ b : B, b ∉ s := by\n  simpa [-SetLike.coe_set_eq, SetLike.ext'_iff, h_top, Set.ne_univ_iff_exists_notMem] using hs","hard_negative":true,"metrics":{"chosen_tokens":33,"rejected_tokens":8,"token_jaccard":0.107143,"token_length_ratio":0.242424},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"a6bf8381dc5784204fa7d80f0373bdd2e3b60ca28ad08e557734fd62d0f90608","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Tactic.Monotonicity.Attr\npublic import Mathlib.Tactic.SetLike\npublic import Mathlib.Data.Set.Basic\n\nNamespace:\nSetLike\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Typeclass for types with a set-like extensionality property\n\nThe `Membership` typeclass is used to let terms of a type have elements.\nMany instances of `Membership` have a set-like extensionality property:\nthings are equal iff they have the same elements.  The `SetLike`\ntypeclass provides a unified interface to define a `Membership` that is\nextensional in this way.\n\nThe main use of `SetLike` is for algebraic subobjects (such as\n`Submonoid` and `Submodule`), whose non-proof data consists only of a\ncarrier set.  In such a situation, the projection to the carrier set\nis injective.\n\nIn general, a type `A` is `SetLike` with elements of type `B` if it\nhas an injective map to `Set B`.  This module provides standard\nboilerplate for every `SetLike`: a `coe_sort`, a `coe` to set,\nand various extensionality and simp lemmas. The order induced by set inclusion is\ncalled `PartialOrder.ofSetlike`: this is not an instance for flexibility in choosing orders.\nThe class `IsConcreteLE` abstractly states the order is equal to that induced by set inclusion;\nan instance is automatically available when defining a `PartialOrder` as\n`.ofSetLike (MySubobject X) X`.\n\nA typical subobject should be declared as:\n```\nstructure MySubobject (X : Type*) [ObjectTypeclass X] where\n  (carrier : Set X)\n  (op_mem' : ∀ {x : X}, x ∈ carrier → sorry ∈ carrier)\n\nnamespace MySubobject\n\nvariable {X : Type*} [ObjectTypeclass X] {x : X}\n\ninstance : SetLike (MySubobject X) X :=\n  ⟨MySubobject.carrier, fun p q h => by cases p; cases q; congr!⟩\n\ninstance : PartialOrder (MySubobject X) := .ofSetLike (MySubobject X) X\n\n@[simp] lemma mem_carrier {p : MySubobject X} : x ∈ p.carrier ↔ x ∈ (p : Set X) := Iff.rfl\n\n@[ext] theorem ext {p q : MySubobject X} (h : ∀ x, x ∈ p ↔ x ∈ q) : p = q := SetLike.ext h\n\n/-- Copy of a `MySubobject` with a new `carrier` equal to the old one. Useful to fix definitional\nequalities. See Note [range copy pattern]. -/\nprotected def copy (p : MySubobject X) (s : Set X) (hs : s = ↑p) : MySubobject X :=\n  { carrier := s\n    op_mem' := hs.symm ▸ p.op_mem' }\n\n@[simp] lemma coe_copy (p : MySubobject X) (s : Set X) (hs : s = ↑p) :\n  (p.copy s hs : Set X) = s := rfl\n\nlemma copy_eq (p : MySubobject X) (s : Set X) (hs : s = ↑p) : p.copy s hs = p :=\n  SetLike.coe_injective hs\n\nend MySubobject\n```\n\nAn alternative to `SetLike` could have been an extensional `Membership` typeclass:\n```\nclass ExtMembership (α : out_param <| Type u) (β : Type v) extends Membership α β where\n  (ext_iff : ∀ {s t : β}, s = t ↔ ∀ (x : α), x ∈ s ↔ x ∈ t)\n```\nWhile this is equivalent, `SetLike` conveniently uses a carrier set projection directly.\n\n## Tags\n\nsubobjects\n-/\n\n@[expose] public section\n\nassert_not_exists RelIso\n\n/-- A class to indicate that there is a canonical injection between `A` and `Set B`.\n\nThis has the effect of giving terms of `A` elements of type `B` (through a `Membership`\ninstance) and a compatible coercion to `Type*` as a subtype.\n\nNote: if `SetLike.coe` is a projection, implementers should create a simp lemma such as\n```\n@[simp] lemma mem_carrier {p : MySubobject X} : x ∈ p.carrier ↔ x ∈ (p : Set X) := Iff.rfl\n```\nto normalize terms.\n\nIf you declare an unbundled subclass of `SetLike`, for example:\n```\nclass MulMemClass (S : Type*) (M : Type*) [Mul M] [SetLike S M] where\n  ...\n```\nThen you should *not* repeat the `outParam` declaration so `SetLike` will supply the value instead.\nThis ensures your subclass will not have issues with synthesis of the `[Mul M]` parameter starting\nbefore the value of `M` is known.\n-/\n@[notation_class* carrier Simps.findCoercionArgs]\nclass SetLike (A : Type*) (B : outParam Type*) where\n  /-- The coercion from a term of a `SetLike` to its corresponding `Set`. -/\n  protected coe : A → Set B\n  /-- The coercion from a term of a `SetLike` to its corresponding `Set` is injective. -/\n  coe_injective : Function.Injective coe\n\nattribute [coe] SetLike.coe\n\nnamespace SetLike\n\nvariable {A : Type*} {B : Type*} [i : SetLike A B]\n\n@[deprecated (since := \"2026-06-04\")] alias coe_injective' := coe_injective\n\ninstance : CoeTC A (Set B) where coe := SetLike.coe\n\ninstance (priority := 100) instMembership : Membership B A :=\n  ⟨fun p x => x ∈ (p : Set B)⟩\n\ninstance (priority := 100) : CoeSort A (Type _) :=\n  ⟨fun p => { x : B // x ∈ p }⟩\n\nsection Delab\nopen Lean PrettyPrinter.Delaborator SubExpr\n\n/-- For terms that match the `CoeSort` instance's body, pretty print as `↥S`\nrather than as `{ x // x ∈ S }`. The discriminating feature is that membership\nuses the `SetLike.instMembership` instance. -/\n@[app_delab Subtype]\nmeta def delabSubtypeSetLike : Delab := whenPPOption getPPNotation do\n  let #[_, .lam n _ body _] := (← getExpr).getAppArgs | failure\n  guard <| body.isAppOf ``Membership.mem\n  let #[_, _, inst, _, .bvar 0] := body.getAppArgs | failure\n  guard <| inst.isAppOfArity ``instMembership 3\n  let S ← withAppArg <| withBindingBody n <| withNaryArg 3 delab\n  `(↥$S)\n\nend Delab\n\nvariable (p q : A)\n\n@[simp, norm_cast]\ntheorem coe_sort_coe : ((p : Set B) : Type _) = p :=\n  rfl\n\nvariable {p q}\n\nprotected theorem «exists» {q : p → Prop} : (∃ x, q x) ↔ ∃ (x : B) (h : x ∈ p), q ⟨x, ‹_›⟩ :=\n  SetCoe.exists\n\nprotected theorem «forall» {q : p → Prop} : (∀ x, q x) ↔ ∀ (x : B) (h : x ∈ p), q ⟨x, ‹_›⟩ :=\n  SetCoe.forall\n\n@[simp, norm_cast]\ntheorem coe_set_eq : (p : Set B) = q ↔ p = q :=\n  coe_injective.eq_iff\n\n@[norm_cast] lemma coe_ne_coe : (p : Set B) ≠ q ↔ p ≠ q := coe_injective.ne_iff\n\ntheorem ext' (h : (p : Set B) = q) : p = q :=\n  coe_injective h\n\ntheorem ext'_iff : p = q ↔ (p : Set B) = q :=\n  coe_set_eq.symm\n\n/-- Note: implementers of `SetLike` must copy this lemma in order to tag it with `@[ext]`. -/\ntheorem ext (h : ∀ x, x ∈ p ↔ x ∈ q) : p = q :=\n  coe_injective <| Set.ext h\n\ntheorem ext_iff : p = q ↔ ∀ x, x ∈ p ↔ x ∈ q :=\n  coe_injective.eq_iff.symm.trans Set.ext_iff\n\n@[simp, push]\ntheorem mem_coe {x : B} : x ∈ (p : Set B) ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_eq_coe {x y : p} : (x : B) = y ↔ x = y :=\n  Subtype.ext_iff.symm\n\n@[simp]\ntheorem coe_mem (x : p) : (x : B) ∈ p :=\n  x.2\n\n@[aesop 5% (rule_sets := [SetLike!])]\nlemma mem_of_subset {s : Set B} (hp : s ⊆ p) {x : B} (hx : x ∈ s) : x ∈ p := hp hx\n\n@[simp]\nprotected theorem eta (x : p) (hx : (x : B) ∈ p) : (⟨x, hx⟩ : p) = x := rfl\n\n@[simp] lemma setOf_mem_eq (a : A) : {b | b ∈ a} = a := rfl\n\n@[nontriviality]\nlemma mem_of_subsingleton [Subsingleton B] (S : A) [h : Nonempty S] {b : B} : b ∈ S := by\n  obtain ⟨s, hs⟩ := nonempty_subtype.mp h\n  simpa [Subsingleton.elim b s]\n\n/-- If `s` is a proper element of a `SetLike` structure (i.e., `s ≠ ⊤`) and the top element\ncoerces to the universal set, then there exists an element not in `s`. -/\n\nTarget:\nlemma exists_not_mem_of_ne_top [LE A] [OrderTop A] (s : A) (hs : s ≠ ⊤)\n    (h_top : ((⊤ : A) : Set B) = Set.univ :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"a67f4427ae01713593cfaa5327c7aa720a8f6fce6519800c29c98aef13ec9b95","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/SetLike","family_id":"exists_not_mem_of_ne_top","file_id":"mathlib/Mathlib/Data/SetLike/Basic.lean","sample_id":"87399eed90a7a9d71500444beb6af36405b8a0740724037a49d189fcab7e9413"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5137effd66997cbfcff7aea301f3e52b8dc7efd3fad26dd29be5d337e1b39c36","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"71eeb2193c99aa08af51083f81cf199e0ad54765f7ba1c48d888d09f69a20da3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0830875cbcc8e3b2fa3c485a8dfb76651d4e973c898e771f3c9e1024b82d5806","source_sha256":"1727a112351b769f97d17a6ca3e660167f05fda3d2180a8ca1a53c6a4fe0c013","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp","hard_negative":false,"metrics":{"chosen_tokens":16,"rejected_tokens":21,"token_jaccard":0.789474,"token_length_ratio":1.3125},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"a6c314f2114a80bc895a2931a261a91d262111c6a654964a00f8d52ff1dfb4a3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Basic\npublic import Mathlib.Algebra.Order.AbsoluteValue.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Absolute values and the integers\n\nThis file contains some results on absolute values applied to integers.\n\n## Main results\n\n* `AbsoluteValue.map_units_int`: an absolute value sends all units of `ℤ` to `1`\n-/\n\npublic section\n\nvariable {R S : Type*} [Ring R] [CommRing S] [LinearOrder S] [IsStrictOrderedRing S]\n\n@[simp]\n\nTarget:\ntheorem AbsoluteValue.map_units_int (abv : AbsoluteValue ℤ S) (x : ℤˣ) : abv x = 1 :=\n\nProof body:\n","rejected":"by\n  rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Int","family_id":"absolutevalue","file_id":"mathlib/Mathlib/Data/Int/AbsoluteValue.lean","sample_id":"0830875cbcc8e3b2fa3c485a8dfb76651d4e973c898e771f3c9e1024b82d5806"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ecfe891855dc120f1040ab8b4113f995f2ce6be82fd87d86c20a75c2f1e66b44","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ff05fe702b3e5167892ee225a104797e7e400770ea91cb68e62132356c190a2b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6381270961bbd859431dc91f4de11a424555613e987c739bd7c51cfea5347942","source_sha256":"d6e838a5f1b3817af8820eb123fb16847eef4259153bc8aa4506ab114a1382bf","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  exact .tower_top (K := F⟮(X : F⟮X⟯)⟯) _\n\nvariable {y : K}","hard_negative":false,"metrics":{"chosen_tokens":26,"rejected_tokens":31,"token_jaccard":0.818182,"token_length_ratio":1.192308},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"a6e8d9ac3f939c2590c037a083c71103f272572c5283744a978617e1480d2fb1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.RatFunc.Degree\npublic import Mathlib.RingTheory.DedekindDomain.IntegralClosure\npublic import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed\npublic import Mathlib.Topology.Algebra.Valued.ValuedField\npublic import Mathlib.Topology.Algebra.InfiniteSum.Defs\npublic import Mathlib.FieldTheory.RatFunc.IntermediateField\npublic import Mathlib.RingTheory.Adjoin.Polynomial.Bivariate\npublic import Mathlib.FieldTheory.RatFunc.Valuation -- for deprecation to `RatFunc.inftyValuation` and `RatFunc.CompletionAtInfty`\n\nNamespace:\nFunctionField\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen, Ashvni Narayanan\n-/\n/-!\n# Function fields\n\nThis file defines a function field and the ring of integers corresponding to it.\n\n## Main definitions\n\n- `FunctionField F K` states that `K` is a function field over the field `F`,\n  i.e. it is a finite extension of the field of rational functions in one variable over `F`.\n- `FunctionField.ringOfIntegers` defines the ring of integers corresponding to a function field\n  as the integral closure of `F[X]` in the function field.\n\n## Implementation notes\nThe definitions that involve a field of fractions choose a canonical field of fractions,\nbut are independent of that choice. We also omit assumptions like\n`IsScalarTower F[X] (FractionRing F[X]) K` in definitions,\nadding them back in lemmas when they are needed.\n\n## References\n* [D. Marcus, *Number Fields*][marcus1977number]\n* [J.W.S. Cassels, A. Fröhlich, *Algebraic Number Theory*][cassels1967algebraic]\n* [P. Samuel, *Algebraic Theory of Numbers*][samuel1967]\n* [M. Rosen, *Number Theory in Function Fields*][rosen2002]\n\n## Tags\nfunction field, ring of integers\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen scoped nonZeroDivisors Polynomial WithZero RatFunc\n\nvariable (F K : Type*) [Field F] [Field K]\n\n/-- `K` is a function field over the field `F` if it is a finite\nextension of the field of rational functions in one variable over `F`.\n\nNote that `K` can be a function field over multiple, non-isomorphic, `F`.\n-/\nabbrev FunctionField [Algebra F⟮X⟯ K] : Prop :=\n  FiniteDimensional F⟮X⟯ K\n\n/-- `K` is a function field over `F` iff it is a finite extension of `F(t)`. -/\ntheorem functionField_iff (Ft : Type*) [Field Ft] [Algebra F[X] Ft]\n    [IsFractionRing F[X] Ft] [Algebra F⟮X⟯ K] [Algebra Ft K] [Algebra F[X] K]\n    [IsScalarTower F[X] Ft K] [IsScalarTower F[X] F⟮X⟯ K] :\n    FunctionField F K ↔ FiniteDimensional Ft K := by\n  let e := IsLocalization.algEquiv F[X]⁰ F⟮X⟯ Ft\n  have : ∀ (c) (x : K), e c • x = c • x := by\n    intro c x\n    rw [Algebra.smul_def, Algebra.smul_def]\n    congr\n    refine congr_fun (f := fun c => algebraMap Ft K (e c)) ?_ c\n    refine IsLocalization.ext (nonZeroDivisors F[X]) _ _ ?_ ?_ ?_ ?_ ?_ <;> intros <;>\n      simp only [map_one, map_mul, AlgEquiv.commutes, ← IsScalarTower.algebraMap_apply]\n  constructor <;> intro h\n  · let b := Module.finBasis F⟮X⟯ K\n    exact (b.mapCoeffs e this).finiteDimensional_of_finite\n  · let b := Module.finBasis Ft K\n    refine (b.mapCoeffs e.symm ?_).finiteDimensional_of_finite\n    intro c x; convert! (this (e.symm c) x).symm; simp only [e.apply_symm_apply]\n\nnamespace FunctionField\n\ntheorem algebraMap_injective [Algebra F[X] K] [Algebra F⟮X⟯ K]\n    [IsScalarTower F[X] F⟮X⟯ K] : Function.Injective (algebraMap F[X] K) := by\n  rw [IsScalarTower.algebraMap_eq F[X] F⟮X⟯ K]\n  exact (algebraMap F⟮X⟯ K).injective.comp (IsFractionRing.injective F[X] F⟮X⟯)\n\n/-- The function field analogue of `NumberField.ringOfIntegers`:\n`FunctionField.ringOfIntegers F K` is the integral closure of `F[X]` in `K`.\n\nWe don't actually assume `K` is a function field over `F` in the definition,\nonly when proving its properties.\n-/\ndef ringOfIntegers [Algebra F[X] K] :=\n  integralClosure F[X] K\n\nnamespace ringOfIntegers\n\nvariable [Algebra F[X] K]\n\ninstance : IsDomain (ringOfIntegers F K) :=\n  (ringOfIntegers F K).isDomain\n\ninstance : IsIntegralClosure (ringOfIntegers F K) F[X] K :=\n  integralClosure.isIntegralClosure _ _\n\nvariable [Algebra F⟮X⟯ K] [IsScalarTower F[X] F⟮X⟯ K]\n\ntheorem algebraMap_injective : Function.Injective (algebraMap F[X] (ringOfIntegers F K)) := by\n  have hinj : Function.Injective (algebraMap F[X] K) := by\n    rw [IsScalarTower.algebraMap_eq F[X] F⟮X⟯ K]\n    exact (algebraMap F⟮X⟯ K).injective.comp (IsFractionRing.injective F[X] F⟮X⟯)\n  rw [injective_iff_map_eq_zero (algebraMap F[X] (↥(ringOfIntegers F K)))]\n  intro p hp\n  rw [← Subtype.coe_inj, Subalgebra.coe_zero] at hp\n  rw [injective_iff_map_eq_zero (algebraMap F[X] K)] at hinj\n  exact hinj p hp\n\ntheorem not_isField : ¬IsField (ringOfIntegers F K) := by\n  simpa [← (IsIntegralClosure.isIntegral_algebra F[X] K).isField_iff_isField\n      (algebraMap_injective F K)] using\n    Polynomial.not_isField F\n\nvariable [FunctionField F K]\n\ninstance : IsFractionRing (ringOfIntegers F K) K :=\n  integralClosure.isFractionRing_of_finite_extension F⟮X⟯ K\n\ninstance : IsIntegrallyClosed (ringOfIntegers F K) :=\n  integralClosure.isIntegrallyClosedOfFiniteExtension F⟮X⟯\n\ninstance [Algebra.IsSeparable F⟮X⟯ K] : IsNoetherian F[X] (ringOfIntegers F K) :=\n  IsIntegralClosure.isNoetherian _ F⟮X⟯ K _\n\ninstance [Algebra.IsSeparable F⟮X⟯ K] : IsDedekindDomain (ringOfIntegers F K) :=\n  IsIntegralClosure.isDedekindDomain F[X] F⟮X⟯ K _\n\nend ringOfIntegers\n\nsection deprecated\n\n@[deprecated RatFunc.inftyValuationDef (since := \"2026-04-14\")]\nalias inftyValuationDef := RatFunc.inftyValuationDef\n\n@[deprecated RatFunc.InftyValuation.map_zero' (since := \"2026-04-14\")]\nalias InftyValuation.map_zero' := RatFunc.InftyValuation.map_zero'\n\n@[deprecated RatFunc.InftyValuation.map_one' (since := \"2026-04-14\")]\nalias InftyValuation.map_one' := RatFunc.InftyValuation.map_one'\n\n@[deprecated RatFunc.InftyValuation.map_mul' (since := \"2026-04-14\")]\nalias InftyValuation.map_mul' := RatFunc.InftyValuation.map_mul'\n\n@[deprecated RatFunc.InftyValuation.map_add_le_max' (since := \"2026-04-14\")]\nalias InftyValuation.map_add_le_max' := RatFunc.InftyValuation.map_add_le_max'\n\n@[deprecated RatFunc.inftyValuation_of_nonzero (since := \"2026-04-14\")]\nalias inftyValuation_of_nonzero := RatFunc.inftyValuation_of_nonzero\n\n@[deprecated RatFunc.inftyValuation (since := \"2026-04-14\")]\nalias inftyValuation := RatFunc.inftyValuation\n\n@[deprecated RatFunc.inftyValuation_apply (since := \"2026-04-14\")]\nalias inftyValuation_apply := RatFunc.inftyValuation_apply\n\n@[deprecated RatFunc.inftyValuation.C (since := \"2026-04-14\")]\nalias inftyValuation.C := RatFunc.inftyValuation.C\n\n@[deprecated RatFunc.inftyValuation.X (since := \"2026-04-14\")]\nalias inftyValuation.X := RatFunc.inftyValuation.X\n\n@[deprecated RatFunc.inftyValuation.X_zpow (since := \"2026-04-14\")]\nalias inftyValuation.X_zpow := RatFunc.inftyValuation.X_zpow\n\n@[deprecated RatFunc.inftyValuation.X_inv (since := \"2026-04-14\")]\nalias inftyValuation.X_inv := RatFunc.inftyValuation.X_inv\n\n@[deprecated RatFunc.inftyValuation.polynomial (since := \"2026-04-14\")]\nalias inftyValuation.polynomial := RatFunc.inftyValuation.polynomial\n\n@[deprecated RatFunc.inftyValued (since := \"2026-04-14\")]\nalias inftyValuedFqt := RatFunc.inftyValued\n\n@[deprecated RatFunc.inftyValued.def (since := \"2026-04-14\")]\nalias inftyValuedFqt.def := RatFunc.inftyValued.def\n\n@[deprecated RatFunc.CompletionAtInfty (since := \"2026-04-14\")]\nalias FqtInfty := RatFunc.CompletionAtInfty\n\n@[deprecated \"Use the anonymous `Valued` instance on `RatFunc.CompletionAtInfty`\"\n(since := \"2026-04-14\")]\ninstance valuedFqtInfty [DecidableEq F⟮X⟯] :\n    Valued (RatFunc.CompletionAtInfty F) ℤᵐ⁰ :=\n  inferInstance\n\n@[deprecated RatFunc.valuedCompletionAtInfty.def (since := \"2026-04-14\")]\nalias valuedFqtInfty.def := RatFunc.valuedCompletionAtInfty.def\n\nend deprecated\n\nsection AdjoinTranscendental\n\nopen IntermediateField RatFunc\n\nvariable {F K : Type*} [Field F] [Field K] [Algebra F⟮X⟯ K] [FunctionField F K]\n\ninstance FiniteDimensional.adjoin_X : FiniteDimensional F⟮(X : F⟮X⟯)⟯ K :=\n  have : Module.Finite (⊤ : IntermediateField F F⟮X⟯) F⟮X⟯ :=\n    .top_left F⟮X⟯ F⟮X⟯\n  RatFunc.adjoin_X (K := F) ▸ Module.Finite.trans F⟮X⟯ _\n\nvariable [Algebra F K] [IsScalarTower F F⟮X⟯ K]\n\ntheorem FiniteDimensional.adjoin_algebraMap_X :\n    FiniteDimensional F⟮algebraMap _ K (X : F⟮X⟯)⟯ K :=\n  .of_restrictScalars_finite F⟮(X : F⟮X⟯)⟯ _ _\n\nTarget:\ntheorem Algebra.IsAlgebraic.adjoin_algebraMap_X :\n    Algebra.IsAlgebraic F⟮algebraMap _ K (X : F⟮X⟯)⟯ K :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  exact .tower_top (K := F⟮(X : F⟮X⟯)⟯) _\n\nvariable {y : K}","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory","family_id":"algebra","file_id":"mathlib/Mathlib/NumberTheory/FunctionField.lean","sample_id":"6381270961bbd859431dc91f4de11a424555613e987c739bd7c51cfea5347942"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9a1a0b80645445024d8bb550e3e8810e677e51b85aec8657ff2f9f66fe77107a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0e6aeac8925875b0126fdb46099ecbd051eb9edc7113b1f3443398d464d4bce4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"33d6750ddf63884a3db77c1ac0c0a14e4b5da0e25cd340d2ef99ffb5417edbe1","source_sha256":"70cf8c7f92181f101a609663b51cd6a85c94239ea2fd1fac014099a79ca682e9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isProperMap_iff_isClosedMap_and_compact_fibers]\n  refine ⟨f.continuous, f.isClosedMap, fun y ↦ ?_⟩\n  have := compactSpace_of_universallyClosed (pullback.snd f (Y.fromSpecResidueField y))\n  rw [← Scheme.range_fromSpecResidueField, ← Scheme.Pullback.range_fst]\n  exact isCompact_range (Scheme.Hom.continuous _)","hard_negative":false,"metrics":{"chosen_tokens":60,"rejected_tokens":65,"token_jaccard":0.875,"token_length_ratio":1.083333},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"a6ea4037d7c0c5752a1d704af2ca4710d5af87718f428399d57368cfbd34695d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion\npublic import Mathlib.AlgebraicGeometry.PullbackCarrier\npublic import Mathlib.Topology.LocalAtTarget\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Universally closed morphism\n\nA morphism of schemes `f : X ⟶ Y` is universally closed if `X ×[Y] Y' ⟶ Y'` is a closed map\nfor all base change `Y' ⟶ Y`.\nThis implies that `f` is topologically proper (`AlgebraicGeometry.Scheme.Hom.isProperMap`).\n\nWe show that being universally closed is local at the target, and is stable under compositions and\nbase changes.\n\n-/\n\npublic section\n\n\nnoncomputable section\n\nopen CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace\n\nuniverse v u\n\nnamespace AlgebraicGeometry\n\nvariable {X Y : Scheme.{u}} (f : X ⟶ Y)\n\nopen CategoryTheory.MorphismProperty\n\n/-- A morphism of schemes `f : X ⟶ Y` is universally closed if the base change `X ×[Y] Y' ⟶ Y'`\nalong any morphism `Y' ⟶ Y` is (topologically) a closed map.\n-/\n@[mk_iff]\nclass UniversallyClosed (f : X ⟶ Y) : Prop where\n  universally_isClosedMap : universally (topologically @IsClosedMap) f\n\n@[deprecated (since := \"2026-01-20\")]\nalias UniversallyClosed.out := UniversallyClosed.universally_isClosedMap\n\nlemma Scheme.Hom.isClosedMap {X Y : Scheme} (f : X ⟶ Y) [UniversallyClosed f] :\n    IsClosedMap f := UniversallyClosed.universally_isClosedMap _ _ _ IsPullback.of_id_snd\n\ntheorem universallyClosed_eq : @UniversallyClosed = universally (topologically @IsClosedMap) := by\n  ext X Y f; rw [universallyClosed_iff]\n\ninstance (priority := 900) [IsClosedImmersion f] : UniversallyClosed f := by\n  rw [universallyClosed_eq]\n  intro X' Y' i₁ i₂ f' hf\n  have hf' : IsClosedImmersion f' :=\n    MorphismProperty.of_isPullback hf.flip inferInstance\n  exact f'.isClosedEmbedding.isClosedMap\n\ntheorem universallyClosed_respectsIso : RespectsIso @UniversallyClosed :=\n  universallyClosed_eq.symm ▸ universally_respectsIso (topologically @IsClosedMap)\n\ninstance universallyClosed_isStableUnderBaseChange : IsStableUnderBaseChange @UniversallyClosed :=\n  universallyClosed_eq.symm ▸ universally_isStableUnderBaseChange (topologically @IsClosedMap)\n\ninstance isClosedMap_isStableUnderComposition :\n    IsStableUnderComposition (topologically @IsClosedMap) where\n  comp_mem f g hf hg := IsClosedMap.comp (f := f) (g := g) hg hf\n\ninstance universallyClosed_isStableUnderComposition :\n    IsStableUnderComposition @UniversallyClosed := by\n  rw [universallyClosed_eq]\n  infer_instance\n\nlemma UniversallyClosed.of_comp_surjective {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)\n    [UniversallyClosed (f ≫ g)] [Surjective f] : UniversallyClosed g := by\n  constructor\n  intro X' Y' i₁ i₂ f' H\n  have := UniversallyClosed.universally_isClosedMap _ _ _\n    ((IsPullback.of_hasPullback i₁ f).paste_horiz H)\n  exact IsClosedMap.of_comp_surjective (MorphismProperty.pullback_fst (P := @Surjective) _ _ ‹_›).1\n    (Scheme.Hom.continuous _) this\n\ninstance universallyClosedTypeComp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)\n    [hf : UniversallyClosed f] [hg : UniversallyClosed g] : UniversallyClosed (f ≫ g) :=\n  comp_mem _ _ _ hf hg\n\ninstance : MorphismProperty.IsMultiplicative @UniversallyClosed where\n  id_mem _ := inferInstance\n\ninstance universallyClosed_fst {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hg : UniversallyClosed g] :\n    UniversallyClosed (pullback.fst f g) :=\n  MorphismProperty.pullback_fst f g hg\n\ninstance universallyClosed_snd {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hf : UniversallyClosed f] :\n    UniversallyClosed (pullback.snd f g) :=\n  MorphismProperty.pullback_snd f g hf\n\ninstance universallyClosed_isZariskiLocalAtTarget : IsZariskiLocalAtTarget @UniversallyClosed := by\n  rw [universallyClosed_eq]\n  apply universally_isZariskiLocalAtTarget\n  intro X Y f ι U hU H\n  simp_rw [topologically, morphismRestrict_base] at H\n  exact hU.isClosedMap_iff_restrictPreimage.mpr H\n\ninstance (f : X ⟶ Y) (V : Y.Opens) [UniversallyClosed f] : UniversallyClosed (f ∣_ V) :=\n  IsZariskiLocalAtTarget.restrict ‹_› V\n\nopen Scheme.Pullback _root_.PrimeSpectrum MvPolynomial in\n/-- If `X` is universally closed over a field, then `X` is quasi-compact. -/\nlemma compactSpace_of_universallyClosed\n    {K} [Field K] (f : X ⟶ Spec (.of K)) [UniversallyClosed f] : CompactSpace X := by\n  classical\n  let 𝒰 : X.OpenCover := X.affineCover\n  let U (i : 𝒰.I₀) : X.Opens := (𝒰.f i).opensRange\n  let T : Scheme := Spec (.of <| MvPolynomial 𝒰.I₀ K)\n  let q : T ⟶ Spec (.of K) := Spec.map (CommRingCat.ofHom MvPolynomial.C)\n  let Ti (i : 𝒰.I₀) : T.Opens := basicOpen (MvPolynomial.X i)\n  let fT : pullback f q ⟶ T := pullback.snd f q\n  let p : pullback f q ⟶ X := pullback.fst f q\n  let Z : Set (pullback f q :) := (⨆ i, fT ⁻¹ᵁ (Ti i) ⊓ p ⁻¹ᵁ (U i) : (pullback f q).Opens)ᶜ\n  have hZ : IsClosed Z := by\n    simp only [Z, isClosed_compl_iff, Opens.coe_iSup, Opens.coe_inf, Opens.map_coe]\n    exact isOpen_iUnion fun i ↦ (fT.continuous.1 _ (Ti i).2).inter (p.continuous.1 _ (U i).2)\n  let Zc : T.Opens := ⟨(fT '' Z)ᶜ, (fT.isClosedMap _ hZ).isOpen_compl⟩\n  let ψ : MvPolynomial 𝒰.I₀ K →ₐ[K] K := MvPolynomial.aeval (fun _ ↦ 1)\n  let t : T := Spec.map (CommRingCat.ofHom ψ.toRingHom) default\n  have ht (i : 𝒰.I₀) : t ∈ Ti i := show ψ (.X i) ≠ 0 by simp [ψ]\n  have htZc : t ∈ Zc := by\n    intro ⟨z, hz, hzt⟩\n    suffices ∃ i, fT z ∈ Ti i ∧ p z ∈ U i from hz (by simpa)\n    exact ⟨𝒰.idx (p z), hzt ▸ ht _, by simpa [U] using 𝒰.covers (p z)⟩\n  obtain ⟨U', ⟨g, rfl⟩, htU', hU'le⟩ := Opens.isBasis_iff_nbhd.mp isBasis_basic_opens htZc\n  let σ : Finset 𝒰.I₀ := MvPolynomial.vars g\n  let φ : MvPolynomial 𝒰.I₀ K →+* MvPolynomial 𝒰.I₀ K :=\n    (MvPolynomial.aeval fun i : 𝒰.I₀ ↦ if i ∈ σ then MvPolynomial.X i else 0).toRingHom\n  let t' : T := Spec.map (CommRingCat.ofHom φ) t\n  have ht'g : t' ∈ PrimeSpectrum.basicOpen g :=\n    show φ g ∉ t.asIdeal from (show φ g = g from aeval_ite_mem_eq_self g subset_rfl).symm ▸ htU'\n  have h : t' ∉ fT '' Z := hU'le ht'g\n  suffices ⋃ i ∈ σ, (U i).1 = Set.univ from\n    ⟨this ▸ Finset.isCompact_biUnion _ fun i _ ↦ isCompact_range (𝒰.f i).continuous⟩\n  rw [Set.iUnion₂_eq_univ_iff]\n  contrapose! h\n  obtain ⟨x, hx⟩ := h\n  obtain ⟨z, rfl, hzr⟩ := exists_preimage_pullback x t' (Subsingleton.elim (f x) (q t'))\n  suffices ∀ i, t ∈ (Ti i).comap ⟨_, continuous_comap φ⟩ → p z ∉ U i from\n    ⟨z, by simpa [Z, p, fT, hzr], hzr⟩\n  intro i hi₁ hi₂\n  rw [comap_basicOpen, show φ (.X i) = 0 by simpa [φ] using (hx i · hi₂), basicOpen_zero] at hi₁\n  cases hi₁\n\nset_option backward.isDefEq.respectTransparency false in\n@[stacks 04XU]\n\nTarget:\nlemma Scheme.Hom.isProperMap (f : X ⟶ Y) [UniversallyClosed f] : IsProperMap f :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  rw [isProperMap_iff_isClosedMap_and_compact_fibers]\n  refine ⟨f.continuous, f.isClosedMap, fun y ↦ ?_⟩\n  have := compactSpace_of_universallyClosed (pullback.snd f (Y.fromSpecResidueField y))\n  rw [← Scheme.range_fromSpecResidueField, ← Scheme.Pullback.range_fst]\n  exact isCompact_range (Scheme.Hom.continuous _)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Morphisms","family_id":"scheme","file_id":"mathlib/Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean","sample_id":"33d6750ddf63884a3db77c1ac0c0a14e4b5da0e25cd340d2ef99ffb5417edbe1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a2aaf585a4a9909c497f3f34f264608d1efd5c7508490b5e6cf98acb7351c7d2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1bd1111a82684ac3770181e9ed442f50e677c4547815bf8225a858fdf741f489","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e0c5330d2514224d882fef475861bb09dfae3c5924cf8a420d81d8d903a10d54","source_sha256":"6558239b343add6f929fcdae9db74c785e40b87bcf52cfd3889e1691cf3f4d62","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  ext x'\n  simp_rw [LieIdeal.mem_killingCompl, LieModule.traceForm_comm]\n  refine ⟨fun hx' ↦ ?_, fun hx y _ ↦ hx ▸ LinearMap.map_zero₂ (killingForm R L') y⟩\n  suffices e.symm x' ∈ LinearMap.ker (killingForm R L) by\n    rw [IsKilling.ker_killingForm_eq_bot] at this\n    simpa [map_zero] using (e : L ≃ₗ[R] L').congr_arg this\n  ext y\n  replace hx' : ∀ y', killingForm R L' x' y' = 0 := by simpa using hx'\n  specialize hx' (e y)\n  rwa [← e.apply_symm_apply x', killingForm_of_equiv_apply] at hx'\n\nalias _root_.LieEquiv.isKilling := LieAlgebra.isKilling_of_equiv","hard_negative":true,"metrics":{"chosen_tokens":130,"rejected_tokens":3,"token_jaccard":0.029851,"token_length_ratio":0.023077},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"a6ee258fa4a9dc8939b2ce74cef0692fffbfef8f09b471f5c628b481e84ea576","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Lie.InvariantForm\npublic import Mathlib.Algebra.Lie.Semisimple.Basic\npublic import Mathlib.Algebra.Lie.TraceForm\n\nNamespace:\nLieAlgebra\n\nLocal context:\n/-\nCopyright (c) 2023 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Lie algebras with non-degenerate Killing forms.\n\nIn characteristic zero, the following three conditions are equivalent:\n 1. The solvable radical of a Lie algebra is trivial\n 2. A Lie algebra is a direct sum of its simple ideals\n 3. A Lie algebra has non-degenerate Killing form\n\nIn positive characteristic, it is still true that 3 implies 2, and that 2 implies 1, but there are\ncounterexamples to the remaining implications. Thus condition 3 is the strongest assumption.\nFurthermore, much of the Cartan-Killing classification of semisimple Lie algebras in characteristic\nzero, continues to hold in positive characteristic (over a perfect field) if the Lie algebra has a\nnon-degenerate Killing form.\n\nThis file contains basic definitions and results for such Lie algebras.\n\n## Main declarations\n\n* `LieAlgebra.IsKilling`: a typeclass encoding the fact that a Lie algebra has a non-singular\n  Killing form.\n* `LieAlgebra.IsKilling.instSemisimple`: if a finite-dimensional Lie algebra over a field\n  has non-singular Killing form then it is semisimple.\n* `LieAlgebra.IsKilling.instHasTrivialRadical`: if a Lie algebra over a PID\n  has non-singular Killing form then it has trivial radical.\n* `LieIdeal.isCompl_killingCompl`: if a Lie algebra has non-singular Killing form then for all\n  ideals, an ideal and its Killing orthogonal complement are complements.\n\n-/\n\npublic section\n\nvariable (R K L : Type*) [CommRing R] [Field K] [LieRing L] [LieAlgebra R L] [LieAlgebra K L]\n\nnamespace LieAlgebra\n\n/-- We say a Lie algebra is Killing if its Killing form is non-singular.\n\nNB: This is not standard terminology (the literature does not seem to name Lie algebras with this\nproperty). -/\nclass IsKilling : Prop where\n  /-- We say a Lie algebra is Killing if its Killing form is non-singular. -/\n  killingCompl_top_eq_bot : LieIdeal.killingCompl R L ⊤ = ⊥\n\nattribute [simp] IsKilling.killingCompl_top_eq_bot\n\nnamespace IsKilling\n\nvariable [IsKilling R L]\n\n@[simp] lemma ker_killingForm_eq_bot :\n    LinearMap.ker (killingForm R L) = ⊥ := by\n  simp [← LieIdeal.coe_killingCompl_top, killingCompl_top_eq_bot]\n\nlemma killingForm_nondegenerate :\n    (killingForm R L).Nondegenerate := by\n  refine (LieModule.traceForm_isSymm R L L).isRefl.nondegenerate_iff_separatingLeft.mpr ?_\n  simp [LinearMap.separatingLeft_iff_ker_eq_bot]\n\nvariable {R L} in\nlemma ideal_eq_bot_of_isLieAbelian\n    [Module.Free R L] [Module.Finite R L] [IsDomain R] [IsPrincipalIdealRing R]\n    (I : LieIdeal R L) [IsLieAbelian I] : I = ⊥ := by\n  rw [eq_bot_iff, ← killingCompl_top_eq_bot]\n  exact I.le_killingCompl_top_of_isLieAbelian\n\ninstance instSemisimple [IsKilling K L] [Module.Finite K L] : IsSemisimple K L := by\n  apply InvariantForm.isSemisimple_of_nondegenerate (Φ := killingForm K L)\n  · exact IsKilling.killingForm_nondegenerate _ _\n  · exact LieModule.traceForm_lieInvariant _ _ _\n  · exact (LieModule.traceForm_isSymm K L L).isRefl\n  · intro I h₁ h₂\n    exact h₁.1 <| IsKilling.ideal_eq_bot_of_isLieAbelian I\n\n/-- The converse of this is true in characteristic zero; it is\n`LieAlgebra.HasTrivialRadical.instIsKilling`. There are counterexamples\nover fields with positive characteristic.\n\nNote that when the coefficients are a field this instance is redundant since we have\n`LieAlgebra.IsKilling.instSemisimple` and `LieAlgebra.IsSemisimple.instHasTrivialRadical`. -/\ninstance instHasTrivialRadical\n    [Module.Free R L] [Module.Finite R L] [IsDomain R] [IsPrincipalIdealRing R] :\n    HasTrivialRadical R L :=\n  (hasTrivialRadical_iff_no_abelian_ideals R L).mpr IsKilling.ideal_eq_bot_of_isLieAbelian\n\ntheorem isLieAbelian_iff_subsingleton\n    [Module.Free R L] [Module.Finite R L] [IsDomain R] [IsPrincipalIdealRing R] :\n    IsLieAbelian L ↔ Subsingleton L := by\n  constructor\n  · intro h\n    rw [isLieAbelian_iff_center_eq_top R] at h\n    have hc : (⊤ : LieIdeal R L) = ⊥ := by rw [← center_eq_bot R L, h]\n    exact (LieSubmodule.subsingleton_iff R L L).mp (subsingleton_of_top_eq_bot hc)\n  · exact fun _ => inferInstance\n\nend IsKilling\n\nsection LieEquiv\n\nvariable {R L}\nvariable {L' : Type*} [LieRing L'] [LieAlgebra R L']\n\n/-- Given an equivalence `e` of Lie algebras from `L` to `L'`, and elements `x y : L`, the\nrespective Killing forms of `L` and `L'` satisfy `κ'(e x, e y) = κ(x, y)`. -/\n@[simp] lemma killingForm_of_equiv_apply (e : L ≃ₗ⁅R⁆ L') (x y : L) :\n    killingForm R L' (e x) (e y) = killingForm R L x y := by\n  simp_rw [killingForm_apply_apply, ← LieAlgebra.conj_ad_apply, ← LinearEquiv.conj_comp,\n    LinearMap.trace_conj']\n\n/-- Given a Killing Lie algebra `L`, if `L'` is isomorphic to `L`, then `L'` is Killing too. -/\n\nTarget:\nlemma isKilling_of_equiv [IsKilling R L] (e : L ≃ₗ⁅R⁆ L') : IsKilling R L' :=\n\nProof body:\n","rejected":"by\n  exact isKilling_of_equiv","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"3d95404cc25321a319496f7f09e072a42f0a91641b11f41c4713bd1a11288f79","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Lie","family_id":"iskilling_of_equiv","file_id":"mathlib/Mathlib/Algebra/Lie/Killing.lean","sample_id":"e0c5330d2514224d882fef475861bb09dfae3c5924cf8a420d81d8d903a10d54"},"split":"train"}
{"binding":{"chosen_proof_sha256":"68a591d0520d49abb38f3df228ef3dd797bb020e537249311751475dbc1d52d8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"44ece7bf253c5f603c55e81415554957d2378036c79d5b76487dc473ce633518","source_sha256":"837dba1e44eb02eb7837e8e7b72b59c277e85899454f45d3be66de44ecd98632","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [max_def]; split_ifs with h\n  exacts [hb h, ha (le_of_not_ge h)]","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":8,"token_jaccard":0.045455,"token_length_ratio":0.4},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"a7128f0600c5e39124039f1f51aebd89c9722f9e1da9460864b06b4d50f2fd90","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Classes.Order\npublic import Batteries.Tactic.Trans\npublic import Mathlib.Data.Ordering.Basic\npublic import Mathlib.Tactic.ExtendDoc\npublic import Mathlib.Tactic.Push.Attr\npublic import Mathlib.Tactic.Simps.Basic\npublic import Mathlib.Tactic.SplitIfs\npublic import Mathlib.Order.Defs.PartialOrder\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Orders\n\nDefines classes for linear orders and proves some basic lemmas about them.\n\nWe intentionally avoid using `grind` in this fundamental file to keep the proofs understandable,\nrather than hiding the reasoning behind automation.\n-/\n\n@[expose] public section\n\nvariable {α : Type*}\n\nsection LinearOrder\n\n/-!\n### Definition of `LinearOrder` and lemmas about types with a linear order\n-/\n\n/-- Default definition of `max`. -/\ndef maxDefault [LE α] [DecidableLE α] (a b : α) :=\n  if a ≤ b then b else a\n\n/-- Default definition of `min`. -/\ndef minDefault [LE α] [DecidableLE α] (a b : α) :=\n  if a ≤ b then a else b\n\n/-- This attempts to prove that a given instance of `compare` is equal to `compareOfLessAndEq` by\nintroducing the arguments and trying the following approaches in order:\n\n1. seeing if `rfl` works\n2. seeing if the `compare` at hand is nonetheless essentially `compareOfLessAndEq`, but, because of\n   implicit arguments, requires us to unfold the defs and split the `if`s in the definition of\n   `compareOfLessAndEq`\n3. seeing if we can split by cases on the arguments, then see if the defs work themselves out\n   (useful when `compare` is defined via a `match` statement, as it is for `Bool`)\n-/\nmacro \"compareOfLessAndEq_rfl\" : tactic =>\n  `(tactic| (intro a b; first | rfl |\n    (simp only [compare, compareOfLessAndEq]; split_ifs <;> rfl) |\n    (induction a <;> induction b <;> simp +decide only)))\n\n/-- A linear order is reflexive, transitive, antisymmetric and total relation `≤`.\nWe assume that every linear ordered type has decidable `(≤)`, `(<)`, and `(=)`. -/\nclass LinearOrder (α : Type*) extends PartialOrder α, Min α, Max α, Ord α where\n  /-- A linear order is total. -/\n  protected le_total (a b : α) : a ≤ b ∨ b ≤ a\n  /-- In a linearly ordered type, we assume the order relations are all decidable. -/\n  toDecidableLE : DecidableLE α\n  /-- In a linearly ordered type, we assume the order relations are all decidable. -/\n  toDecidableEq : DecidableEq α := @decidableEqOfDecidableLE _ _ toDecidableLE\n  /-- In a linearly ordered type, we assume the order relations are all decidable. -/\n  toDecidableLT : DecidableLT α := @decidableLTOfDecidableLE _ _ toDecidableLE\n  min := fun a b => if a ≤ b then a else b\n  max := fun a b => if a ≤ b then b else a\n  /-- The minimum function is equivalent to the one you get from `minOfLe`. -/\n  protected min_def : ∀ a b, min a b = if a ≤ b then a else b := by intros; rfl\n  /-- The minimum function is equivalent to the one you get from `maxOfLe`. -/\n  protected max_def : ∀ a b, max a b = if a ≤ b then b else a := by intros; rfl\n  compare a b := compareOfLessAndEq a b\n  /-- Comparison via `compare` is equal to the canonical comparison given decidable `<` and `=`. -/\n  compare_eq_compareOfLessAndEq : ∀ a b, compare a b = compareOfLessAndEq a b := by\n    compareOfLessAndEq_rfl\n\nattribute [to_dual existing] LinearOrder.toMax\n\nvariable [LinearOrder α] {a b c : α}\n\nattribute [instance_reducible, instance 900] LinearOrder.toDecidableLT\nattribute [instance_reducible, instance 900] LinearOrder.toDecidableLE\nattribute [instance_reducible, instance 900] LinearOrder.toDecidableEq\n\ninstance : Std.IsLinearOrder α where\n  le_total := LinearOrder.le_total\n\n@[to_dual self] lemma le_total : ∀ a b : α, a ≤ b ∨ b ≤ a := LinearOrder.le_total\n\n@[to_dual self] lemma le_of_not_ge : ¬a ≤ b → b ≤ a := (le_total a b).resolve_left\n@[to_dual self] lemma lt_of_not_ge (h : ¬b ≤ a) : a < b := lt_of_le_not_ge (le_of_not_ge h) h\n\n@[to_dual self] lemma lt_or_ge (a b : α) : a < b ∨ b ≤ a :=\n  if hba : b ≤ a then Or.inr hba else Or.inl <| lt_of_not_ge hba\n\n@[to_dual self] lemma le_or_gt (a b : α) : a ≤ b ∨ b < a := (lt_or_ge b a).symm\n\n@[to_dual gt_trichotomy]\nlemma lt_trichotomy (a b : α) : a < b ∨ a = b ∨ b < a :=\n  (lt_or_ge a b).imp_right (fun h ↦ (Decidable.lt_or_eq_of_le' h).symm)\n\n@[to_dual self]\nlemma le_of_not_gt (h : ¬b < a) : a ≤ b := (le_or_gt a b).resolve_right h\n\n@[to_dual gt_or_lt_of_ne]\nlemma lt_or_gt_of_ne (h : a ≠ b) : a < b ∨ b < a :=\n  (lt_trichotomy a b).imp_right (fun h' ↦ h'.resolve_left h)\n\n@[to_dual ne_iff_gt_or_lt]\nlemma ne_iff_lt_or_gt : a ≠ b ↔ a < b ∨ b < a := ⟨lt_or_gt_of_ne, (Or.elim · ne_of_lt ne_of_gt)⟩\n\n@[to_dual self] lemma lt_iff_not_ge : a < b ↔ ¬b ≤ a := ⟨not_le_of_gt, lt_of_not_ge⟩\n\n@[simp, push, to_dual self] lemma not_lt : ¬a < b ↔ b ≤ a := ⟨le_of_not_gt, not_lt_of_ge⟩\n@[simp, push, to_dual self] lemma not_le : ¬a ≤ b ↔ b < a := lt_iff_not_ge.symm\n\n@[to_dual eq_or_lt_of_not_gt]\nlemma eq_or_gt_of_not_lt (h : ¬a < b) : a = b ∨ b < a :=\n  if h₁ : a = b then Or.inl h₁ else Or.inr (lt_of_not_ge fun hge => h (lt_of_le_of_ne hge h₁))\n\n@[to_dual self]\ntheorem le_imp_le_of_lt_imp_lt {α β} [Preorder α] [LinearOrder β] {a b : α} {c d : β}\n    (H : d < c → b < a) (h : a ≤ b) : c ≤ d :=\n  le_of_not_gt fun h' => not_le_of_gt (H h') h\n\n@[grind =]\nlemma min_def (a b : α) : min a b = if a ≤ b then a else b := LinearOrder.min_def a b\n@[grind =]\nlemma max_def (a b : α) : max a b = if a ≤ b then b else a := LinearOrder.max_def a b\n\ntheorem min_ind {motive : α → Prop} (ha : a ≤ b → motive a) (hb : b ≤ a → motive b) :\n    motive (min a b) := by\n  rw [min_def]; split_ifs with h\n  exacts [ha h, hb (le_of_not_ge h)]\n\n@[to_dual existing (attr := elab_as_elim)]\n\nTarget:\ntheorem max_ind {motive : α → Prop} (ha : b ≤ a → motive a) (hb : a ≤ b → motive b) :\n    motive (max a b) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e23c888cba8807f65e06dfd4008129842890a6ecabebded29e3371b45091dd80","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Defs","family_id":"max_ind","file_id":"mathlib/Mathlib/Order/Defs/LinearOrder.lean","sample_id":"44ece7bf253c5f603c55e81415554957d2378036c79d5b76487dc473ce633518"},"split":"train"}
{"binding":{"chosen_proof_sha256":"021873745f79b607afe2480d1a3ec9b92a92bc4d1a53d9616a37575a2e4fbf1f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"88184057873a0a559cc6d5510520e38470290a3d57980287af7d4cf570e8c138","source_sha256":"554524b5a9af2e4b62f0d83e3222b642d629dcdc418afd3afc6a8cda3e670128","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rw [← top_sdiff', ← top_sdiff', map_sdiff, map_top]","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":3,"token_jaccard":0.090909,"token_length_ratio":0.230769},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"a74d81b107708f9ec4fc0d1b61568bdc29a4cc3adab3a13ee0d3175967f8f0c9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Hom.BoundedLattice\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Heyting algebra morphisms\n\nA Heyting homomorphism between two Heyting algebras is a bounded lattice homomorphism that preserves\nHeyting implication.\n\nWe use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to\nbe satisfied by itself and all stricter types.\n\n## Types of morphisms\n\n* `HeytingHom`: Heyting homomorphisms.\n* `CoheytingHom`: Co-Heyting homomorphisms.\n* `BiheytingHom`: Bi-Heyting homomorphisms.\n\n## Typeclasses\n\n* `HeytingHomClass`\n* `CoheytingHomClass`\n* `BiheytingHomClass`\n-/\n\n@[expose] public section\n\n\nopen Function\n\nvariable {F α β γ δ : Type*}\n\n/-- The type of Heyting homomorphisms from `α` to `β`. Bounded lattice homomorphisms that preserve\nHeyting implication. -/\nstructure HeytingHom (α β : Type*) [HeytingAlgebra α] [HeytingAlgebra β] extends\n  LatticeHom α β where\n  /-- The proposition that a Heyting homomorphism preserves the bottom element. -/\n  protected map_bot' : toFun ⊥ = ⊥\n  /-- The proposition that a Heyting homomorphism preserves the Heyting implication. -/\n  protected map_himp' : ∀ a b, toFun (a ⇨ b) = toFun a ⇨ toFun b\n\n/-- The type of co-Heyting homomorphisms from `α` to `β`. Bounded lattice homomorphisms that\npreserve difference. -/\nstructure CoheytingHom (α β : Type*) [CoheytingAlgebra α] [CoheytingAlgebra β] extends\n  LatticeHom α β where\n  /-- The proposition that a co-Heyting homomorphism preserves the top element. -/\n  protected map_top' : toFun ⊤ = ⊤\n  /-- The proposition that a co-Heyting homomorphism preserves the difference operation. -/\n  protected map_sdiff' : ∀ a b, toFun (a \\ b) = toFun a \\ toFun b\n\n/-- The type of bi-Heyting homomorphisms from `α` to `β`. Bounded lattice homomorphisms that\npreserve Heyting implication and difference. -/\nstructure BiheytingHom (α β : Type*) [BiheytingAlgebra α] [BiheytingAlgebra β] extends\n  LatticeHom α β where\n  /-- The proposition that a bi-Heyting homomorphism preserves the Heyting implication. -/\n  protected map_himp' : ∀ a b, toFun (a ⇨ b) = toFun a ⇨ toFun b\n  /-- The proposition that a bi-Heyting homomorphism preserves the difference operation. -/\n  protected map_sdiff' : ∀ a b, toFun (a \\ b) = toFun a \\ toFun b\n\n/-- `HeytingHomClass F α β` states that `F` is a type of Heyting homomorphisms.\n\nYou should extend this class when you extend `HeytingHom`. -/\nclass HeytingHomClass (F α β : Type*) [HeytingAlgebra α] [HeytingAlgebra β] [FunLike F α β] : Prop\n    extends LatticeHomClass F α β where\n  /-- The proposition that a Heyting homomorphism preserves the bottom element. -/\n  map_bot (f : F) : f ⊥ = ⊥\n  /-- The proposition that a Heyting homomorphism preserves the Heyting implication. -/\n  map_himp (f : F) : ∀ a b, f (a ⇨ b) = f a ⇨ f b\n\n/-- `CoheytingHomClass F α β` states that `F` is a type of co-Heyting homomorphisms.\n\nYou should extend this class when you extend `CoheytingHom`. -/\nclass CoheytingHomClass (F α β : Type*) [CoheytingAlgebra α] [CoheytingAlgebra β] [FunLike F α β] :\n    Prop\n  extends LatticeHomClass F α β where\n  /-- The proposition that a co-Heyting homomorphism preserves the top element. -/\n  map_top (f : F) : f ⊤ = ⊤\n  /-- The proposition that a co-Heyting homomorphism preserves the difference operation. -/\n  map_sdiff (f : F) : ∀ a b, f (a \\ b) = f a \\ f b\n\n/-- `BiheytingHomClass F α β` states that `F` is a type of bi-Heyting homomorphisms.\n\nYou should extend this class when you extend `BiheytingHom`. -/\nclass BiheytingHomClass (F α β : Type*) [BiheytingAlgebra α] [BiheytingAlgebra β] [FunLike F α β] :\n    Prop\n  extends LatticeHomClass F α β where\n  /-- The proposition that a bi-Heyting homomorphism preserves the Heyting implication. -/\n  map_himp (f : F) : ∀ a b, f (a ⇨ b) = f a ⇨ f b\n  /-- The proposition that a bi-Heyting homomorphism preserves the difference operation. -/\n  map_sdiff (f : F) : ∀ a b, f (a \\ b) = f a \\ f b\n\nexport HeytingHomClass (map_himp)\n\nexport CoheytingHomClass (map_sdiff)\n\nattribute [simp] map_himp map_sdiff\n\nsection Hom\n\nvariable [FunLike F α β]\n\n/-! This section passes in some instances implicitly. See note [implicit instance arguments] -/\n\n-- See note [lower instance priority]\ninstance (priority := 100) HeytingHomClass.toBoundedLatticeHomClass [HeytingAlgebra α]\n    {_ : HeytingAlgebra β} [HeytingHomClass F α β] : BoundedLatticeHomClass F α β :=\n  { ‹HeytingHomClass F α β› with\n    map_top := fun f => by rw [← @himp_self α _ ⊥, ← himp_self, map_himp] }\n\n-- See note [lower instance priority]\ninstance (priority := 100) CoheytingHomClass.toBoundedLatticeHomClass [CoheytingAlgebra α]\n    {_ : CoheytingAlgebra β} [CoheytingHomClass F α β] : BoundedLatticeHomClass F α β :=\n  { ‹CoheytingHomClass F α β› with\n    map_bot := fun f => by rw [← @sdiff_self α _ ⊤, ← sdiff_self, map_sdiff] }\n\n-- See note [lower instance priority]\ninstance (priority := 100) BiheytingHomClass.toHeytingHomClass [BiheytingAlgebra α]\n    {_ : BiheytingAlgebra β} [BiheytingHomClass F α β] : HeytingHomClass F α β :=\n  { ‹BiheytingHomClass F α β› with\n    map_bot := fun f => by rw [← @sdiff_self α _ ⊤, ← sdiff_self, BiheytingHomClass.map_sdiff] }\n\n-- See note [lower instance priority]\ninstance (priority := 100) BiheytingHomClass.toCoheytingHomClass [BiheytingAlgebra α]\n    {_ : BiheytingAlgebra β} [BiheytingHomClass F α β] : CoheytingHomClass F α β :=\n  { ‹BiheytingHomClass F α β› with\n    map_top := fun f => by rw [← @himp_self α _ ⊥, ← himp_self, map_himp] }\n\nend Hom\n\nsection Equiv\n\nvariable [EquivLike F α β]\n\n-- See note [lower instance priority]\ninstance (priority := 100) OrderIsoClass.toHeytingHomClass [HeytingAlgebra α]\n    {_ : HeytingAlgebra β} [OrderIsoClass F α β] : HeytingHomClass F α β :=\n  { OrderIsoClass.toBoundedLatticeHomClass with\n    map_himp := fun f a b =>\n      eq_of_forall_le_iff fun c => by\n        simp only [← map_inv_le_iff, le_himp_iff]\n        rw [← OrderIsoClass.map_le_map_iff f]\n        simp }\n\n-- See note [lower instance priority]\ninstance (priority := 100) OrderIsoClass.toCoheytingHomClass [CoheytingAlgebra α]\n    {_ : CoheytingAlgebra β} [OrderIsoClass F α β] : CoheytingHomClass F α β :=\n  { OrderIsoClass.toBoundedLatticeHomClass with\n    map_sdiff := fun f a b =>\n      eq_of_forall_ge_iff fun c => by\n        simp only [← le_map_inv_iff, sdiff_le_iff]\n        rw [← OrderIsoClass.map_le_map_iff f]\n        simp }\n\n-- See note [lower instance priority]\ninstance (priority := 100) OrderIsoClass.toBiheytingHomClass [BiheytingAlgebra α]\n    {_ : BiheytingAlgebra β} [OrderIsoClass F α β] : BiheytingHomClass F α β :=\n  { OrderIsoClass.toLatticeHomClass with\n    map_himp := fun f a b =>\n      eq_of_forall_le_iff fun c => by\n        simp only [← map_inv_le_iff, le_himp_iff]\n        rw [← OrderIsoClass.map_le_map_iff f]\n        simp\n    map_sdiff := fun f a b =>\n      eq_of_forall_ge_iff fun c => by\n        simp only [← le_map_inv_iff, sdiff_le_iff]\n        rw [← OrderIsoClass.map_le_map_iff f]\n        simp }\n\nend Equiv\n\nvariable [FunLike F α β]\n\ninstance BoundedLatticeHomClass.toBiheytingHomClass [BooleanAlgebra α] [BooleanAlgebra β]\n    [BoundedLatticeHomClass F α β] : BiheytingHomClass F α β :=\n  { ‹BoundedLatticeHomClass F α β› with\n    map_himp := fun f a b => by rw [himp_eq, himp_eq, map_sup, (isCompl_compl.map _).compl_eq]\n    map_sdiff := fun f a b => by rw [sdiff_eq, sdiff_eq, map_inf, (isCompl_compl.map _).compl_eq] }\n\nsection HeytingAlgebra\n\nopen scoped symmDiff\n\nvariable [HeytingAlgebra α] [HeytingAlgebra β] [HeytingHomClass F α β] (f : F)\n\n@[simp]\ntheorem map_compl (a : α) : f aᶜ = (f a)ᶜ := by rw [← himp_bot, ← himp_bot, map_himp, map_bot]\n\n@[simp]\ntheorem map_bihimp (a b : α) : f (a ⇔ b) = f a ⇔ f b := by simp_rw [bihimp, map_inf, map_himp]\n\nend HeytingAlgebra\n\nsection CoheytingAlgebra\n\nopen scoped symmDiff\n\nvariable [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingHomClass F α β] (f : F)\n\n@[simp]\n\nTarget:\ntheorem map_hnot (a : α) : f (￢a) = ￢f a :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Heyting","family_id":"map_hnot","file_id":"mathlib/Mathlib/Order/Heyting/Hom.lean","sample_id":"88184057873a0a559cc6d5510520e38470290a3d57980287af7d4cf570e8c138"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e08724fba84126ab1ada747c46865f50f539e3028cb4c27da440863f309c14d1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"92784161800bc7703887ac5638e35236b09a0de489f670f0ce2dac8a7f91006c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5d6d44820ec47fa0236df113ccc6391b9ab3172c1aa71f5359c1564641b4264c","source_sha256":"b397fa221cf5b605f3212baf045d24b21f44907b35b75875e7daf454ffb5e047","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  borelize G\n  rw [modularCharacterFun_eq_haarScalarFactor MeasureTheory.Measure.haar g]\n  exact haarScalarFactor_pos_of_isHaarMeasure _ _","hard_negative":false,"metrics":{"chosen_tokens":17,"rejected_tokens":22,"token_jaccard":0.789474,"token_length_ratio":1.294118},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"a76c53a866dfa50e9f8228aa2e2306f5da4746d6db35490b77167f10fc3c8a34","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Function.LocallyIntegrable\npublic import Mathlib.MeasureTheory.Group.Integral\npublic import Mathlib.MeasureTheory.Group.Measure\npublic import Mathlib.Topology.Metrizable.Urysohn\npublic import Mathlib.MeasureTheory.Measure.Haar.Unique\npublic import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic\n\nNamespace:\nMeasureTheory.Measure\n\nLocal context:\n/-\nCopyright (c) 2025 Noam Atar. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Noam Atar\n-/\n/-!\n# Modular character of a locally compact group\n\nOn a locally compact group, there is a natural homomorphism `G → ℝ≥0*`, which for `g : G` gives the\nvalue `μ (· * g⁻¹) / μ`, where `μ` is an (inner regular) Haar measure. This file defines this\nhomomorphism, called the modular character, and shows that it is independent of the chosen Haar\nmeasure.\n\nTODO: Show that the character is continuous.\n\n## Main Declarations\n\n* `modularCharacterFun`: Define the modular character function. If `μ` is a left Haar measure on `G`\n  and `g : G`, the measure `A ↦ μ (A g⁻¹)` is also a left Haar measure, so by uniqueness is of the\n  form `Δ(g) μ`, for `Δ(g) ∈ ℝ≥0`. This `Δ` is the modular character. The result that this does not\n  depend on the measure chosen is `modularCharacterFun_eq_haarScalarFactor`.\n* `modularCharacter`: The homomorphism G →* ℝ≥0 whose toFun is `modularCharacterFun`.\n-/\n\n@[expose] public section\n\nopen MeasureTheory\nopen scoped NNReal\n\nnamespace MeasureTheory\n\nnamespace Measure\n\nvariable {G : Type*} [TopologicalSpace G] [Group G] [IsTopologicalGroup G] [LocallyCompactSpace G]\n\n/-- The modular character as a map is `g ↦ μ (· * g⁻¹) / μ`, where `μ` is a left Haar measure.\n\n  See also `modularCharacter` that defines the map as a homomorphism. -/\n@[to_additive /-- The additive modular character as a map is `g ↦ μ (· - g) / μ`, where `μ` is an\n  left additive Haar measure. -/]\nnoncomputable def modularCharacterFun (g : G) : ℝ≥0 :=\n  letI : MeasurableSpace G := borel G\n  haveI : BorelSpace G := ⟨rfl⟩\n  haarScalarFactor (map (· * g) MeasureTheory.Measure.haar) MeasureTheory.Measure.haar\n\n/-- Independence of modularCharacterFun from the chosen Haar measure. -/\n@[to_additive /-- Independence of addModularCharacterFun from the chosen Haar measure -/]\nlemma modularCharacterFun_eq_haarScalarFactor [MeasurableSpace G] [BorelSpace G] (μ : Measure G)\n    [IsHaarMeasure μ] (g : G) : modularCharacterFun g = haarScalarFactor (map (· * g) μ) μ := by\n  let ν := MeasureTheory.Measure.haar (G := G)\n  obtain ⟨⟨f, f_cont⟩, f_comp, f_nonneg, f_one⟩ :\n    ∃ f : C(G, ℝ), HasCompactSupport f ∧ 0 ≤ f ∧ f 1 ≠ 0 := exists_continuous_nonneg_pos 1\n  have int_f_ne_zero (μ₀ : Measure G) [IsHaarMeasure μ₀] : ∫ x, f x ∂μ₀ ≠ 0 :=\n    ne_of_gt (f_cont.integral_pos_of_hasCompactSupport_nonneg_nonzero f_comp f_nonneg f_one)\n  apply NNReal.coe_injective\n  have t : (∫ x, f (x * g) ∂ν) = (∫ x, f (x * g) ∂(haarScalarFactor ν μ • μ)) := by\n    refine integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport ν μ ?_ ?_\n    · exact Continuous.comp' f_cont (continuous_mul_const g)\n    · have j : (fun x ↦ f (x * g)) = (f ∘ (Homeomorph.mulRight g)) := rfl\n      rw [j]\n      exact HasCompactSupport.comp_homeomorph f_comp _\n  have r : (haarScalarFactor ν μ : ℝ) / (haarScalarFactor ν μ) = 1 := by\n    refine div_self ?_\n    rw [NNReal.coe_ne_zero]\n    apply (ne_of_lt (haarScalarFactor_pos_of_isHaarMeasure _ _)).symm\n  calc\n  ↑(modularCharacterFun g) = ↑(haarScalarFactor (map (· * g) ν) ν) := by borelize G; rfl\n  _ = (∫ x, f x ∂(map (· * g) ν)) / ∫ x, f x ∂ν :=\n    haarScalarFactor_eq_integral_div _ _ f_cont f_comp (int_f_ne_zero ν)\n  _ = (∫ x, f (x * g) ∂ν) / ∫ x, f x ∂ν := by\n    rw [integral_map (AEMeasurable.mul_const aemeasurable_id' _)\n    (Continuous.aestronglyMeasurable f_cont)]\n  _ = (∫ x, f (x * g) ∂(haarScalarFactor ν μ • μ)) / ∫ x, f x ∂ν := by rw [t]\n  _ = (∫ x, f (x * g) ∂(haarScalarFactor ν μ • μ)) / ∫ x, f x ∂(haarScalarFactor ν μ • μ) := by\n    rw [integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport ν μ f_cont f_comp]\n  _ = (haarScalarFactor ν μ • ∫ x, f (x * g) ∂μ) / (haarScalarFactor ν μ • ∫ x, f x ∂μ) := by\n    rw [integral_smul_nnreal_measure, integral_smul_nnreal_measure]\n  _ = (haarScalarFactor ν μ / haarScalarFactor ν μ) * ((∫ x, f (x * g) ∂μ) / ∫ x, f x ∂μ) :=\n    mul_div_mul_comm _ _ _ _\n  _ = 1 * ((∫ x, f (x * g) ∂μ) / ∫ x, f x ∂μ) := by rw [r]\n  _ = (∫ x, f (x * g) ∂μ) / ∫ x, f x ∂μ := by rw [one_mul]\n  _ = (∫ x, f x ∂(map (· * g) μ)) / ∫ x, f x ∂μ := by\n    rw [integral_map (AEMeasurable.mul_const aemeasurable_id' _)\n    (Continuous.aestronglyMeasurable f_cont)]\n  _ = haarScalarFactor (map (· * g) μ) μ :=\n    (haarScalarFactor_eq_integral_div _ _ f_cont f_comp (int_f_ne_zero μ)).symm\n\n@[to_additive]\nlemma map_right_mul_eq_modularCharacterFun_smul [MeasurableSpace G] [BorelSpace G] (μ : Measure G)\n    [IsHaarMeasure μ] [InnerRegular μ] (g : G) : map (· * g) μ = modularCharacterFun g • μ := by\n  rw [modularCharacterFun_eq_haarScalarFactor μ _]\n  exact isMulLeftInvariant_eq_smul_of_innerRegular _ μ\n\n@[to_additive]\n\nTarget:\nlemma modularCharacterFun_pos (g : G) : 0 < modularCharacterFun g :=\n\nProof body:\n","rejected":"by\n  borelize G\n  rw [modularCharacterFun_eq_haarScalarFactor MeasureTheory.Measure.haar g]\n  exact haarScalarFactor_pos_of_isHaarMeasure _ _\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Group","family_id":"modularcharacterfun_pos","file_id":"mathlib/Mathlib/MeasureTheory/Group/ModularCharacter.lean","sample_id":"5d6d44820ec47fa0236df113ccc6391b9ab3172c1aa71f5359c1564641b4264c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"22008102f7d34b93c9ef351ad767f156bb7824bb62e00233f134ea27d1c44cf6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f1be1ab398e8e6f5f51fea76b3b1c755d097ca53fd2fb60f4b3eb6a00e32ab92","source_sha256":"2ecbfb606a976f2e47686b1a1c471e54098f1481b48ac0c349b703cde7d54204","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [eq_geometrically, GeometricallyReduced.eq_geometrically,\n    GeometricallyIrreducible.eq_geometrically, ← geometrically_inf]\n  eta_expand\n  simp [isIntegral_iff_irreducibleSpace_and_isReduced, and_comm]","hard_negative":false,"metrics":{"chosen_tokens":23,"rejected_tokens":2,"token_jaccard":0.0625,"token_length_ratio":0.086957},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"a78b35603ca4c7cb4ae39a5eb3c0e491d9d41b6027b8e2fd7ad441a719f86f11","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Geometrically.Reduced\npublic import Mathlib.AlgebraicGeometry.Geometrically.Irreducible\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2026 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Geometrically Integral Schemes\n\n## Main results\n- `AlgebraicGeometry.GeometricallyIntegral`:\n  We say that morphism `f : X ⟶ Y` is geometrically integral if for all `Spec K ⟶ Y` with `K`\n  a field, `X ×[Y] Spec K` is integral.\n  We also provide the fact that this is stable under base change (`by infer_instance`)\n- `GeometricallyIntegral.iff_geometricallyIntegral_fiber`:\n  A scheme is geometrically integral over `S` iff the fibers of all\n  `s : S` are geometrically integral.\n- `AlgebraicGeometry.GeometricallyIntegral.isIntegral_of_isLocallyNoetherian`:\n  If `X` is geometrically integral, flat, and universally open (e.g. when over a field),\n  over an integral locally noetherian scheme, then `X` is also integral.\n- `AlgebraicGeometry.GeometricallyIntegral.isIntegral_of_subsingleton`:\n  If `X` is geometrically integral over a field, then it is integral.\n-/\n\npublic section\n\nopen CategoryTheory MorphismProperty Limits\n\nnamespace AlgebraicGeometry\n\nvariable {X Y Z S : Scheme} (f : X ⟶ S) (g : Y ⟶ S)\n\n/-- We say that morphism `f : X ⟶ Y` is geometrically integral if for all `Spec K ⟶ Y` with `K`\na field, `X ×[Y] Spec K` is integral. -/\n@[mk_iff]\nclass GeometricallyIntegral (f : X ⟶ Y) : Prop where\n  geometrically_isIntegral : geometrically IsIntegral f\n\nlemma GeometricallyIntegral.eq_geometrically :\n    @GeometricallyIntegral = geometrically IsIntegral := by\n  ext; exact geometricallyIntegral_iff _\n\nTarget:\nlemma GeometricallyIntegral.eq_geometricallyReduced_inf_geometricallyIrreducible :\n    @GeometricallyIntegral =\n      (@GeometricallyReduced ⊓ @GeometricallyIrreducible : MorphismProperty Scheme) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Geometrically","family_id":"geometricallyintegral","file_id":"mathlib/Mathlib/AlgebraicGeometry/Geometrically/Integral.lean","sample_id":"f1be1ab398e8e6f5f51fea76b3b1c755d097ca53fd2fb60f4b3eb6a00e32ab92"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f11a99acaff28e0866b0cf9264b8d87a92a312c57ace97d7c08c3310cdd9276a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2a683c5a45e5862414cab6197a75e171596dc41fe8b3fc3ce9824a5cf737be85","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5013414c846a303e2d7ae6cf16e87bd0d86e0af5364ad267930855e6631a1174","source_sha256":"2a575e671e878bbb47b82f04d189e9f55eb07313e1f1bf4de528a0907677d433","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Finset.offDiag_filter_lt_eq_filter_le]\n  conv_rhs => rw [← Finset.sum_subtype_eq_sum_filter]\n  refine (Finset.sum_equiv Sym2.sortEquiv.symm ?_ ?_).symm\n  all_goals aesop","hard_negative":true,"metrics":{"chosen_tokens":35,"rejected_tokens":3,"token_jaccard":0.041667,"token_length_ratio":0.085714},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"a7b86764b0bf626186de26d89f7c50b29f965d248e1f1954cae7c6e0376d56fb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Sym\npublic import Mathlib.Data.Sym.Sym2.Order\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2024 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Lemmas on `Finset.sum` and `Finset.prod` involving `Finset.sym2` or `Finset.sym`.\n-/\n\npublic section\n\nnamespace Finset\n\nopen Multiset\n\nTarget:\ntheorem sum_sym2_filter_not_isDiag {ι M} [LinearOrder ι] [AddCommMonoid M]\n    (s : Finset ι) (p : Sym2 ι → M) :\n    ∑ i ∈ s.sym2 with ¬ i.IsDiag, p i = ∑ i ∈ s.offDiag with i.1 < i.2, p s(i.1, i.2) :=\n\nProof body:\n","rejected":"by\n  exact sum_sym2_filter_not_isDiag","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"4eedc8db06b12feaa92ce9e448b3ef1f7c512da87e83434cf9ad27506ca57378","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/BigOperators","family_id":"sum_sym2_filter_not_isdiag","file_id":"mathlib/Mathlib/Algebra/BigOperators/Sym.lean","sample_id":"5013414c846a303e2d7ae6cf16e87bd0d86e0af5364ad267930855e6631a1174"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0d95bb02e9e9a9b98bdd4fbdf7abc16ec6edaa269b3beda79e93adb84be4b936","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"930f23c5bc98778cfbbd2c8d89384e1922ceebe378e5552b76e975a024be70ca","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"32b115c85276c93388c62a93ae6422efd911c7d0d06efd966d995ffe632b53e9","source_sha256":"53d8534063fc3b6cb81173ac579623e7fbacb797af1fdf83974b842dd64042c1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases h : k = 0\n  · simp [h]\n  exact (isPrimitiveRoot_exp k h).card_primitiveRoots","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":2,"token_jaccard":0.055556,"token_length_ratio":0.1},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"a7b8db17588a9b6faeb1539523a8d6b5d3f308bb1f00c64ef82aa480a29c172c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Complex.Log\npublic import Mathlib.RingTheory.Int.Basic\npublic import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots\npublic import Mathlib.Tactic.Rify\npublic import Mathlib.Tactic.Qify\n\nNamespace:\nComplex\n\nLocal context:\n/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Snir Broshi\n-/\n/-!\n# Complex roots of unity\n\nIn this file we show that the `n`-th complex roots of unity\nare exactly the complex numbers `exp (2 * π * I * (i / n))` for `i ∈ Finset.range n`.\n\n## Main declarations\n\n* `Complex.mem_rootsOfUnity`: the complex `n`-th roots of unity are exactly the\n  complex numbers of the form `exp (2 * π * I * (i / n))` for some `i < n`.\n* `Complex.card_rootsOfUnity`: the number of `n`-th roots of unity is exactly `n`.\n* `Complex.norm_rootOfUnity_eq_one`: A complex root of unity has norm `1`.\n\n-/\n\npublic section\n\n\nnamespace Complex\n\nopen Polynomial Real\n\nopen scoped Nat Real\n\ntheorem isPrimitiveRoot_exp_of_isCoprime (i : ℤ) (n : ℕ) (h0 : n ≠ 0) (hi : IsCoprime i n) :\n    IsPrimitiveRoot (exp (2 * π * I * (i / n))) n := by\n  rw [IsPrimitiveRoot.iff_def]\n  simp only [← exp_nat_mul, exp_eq_one_iff]\n  constructor\n  · use i\n    simp (discharger := norm_cast) [field]\n  · simp only [forall_exists_index]\n    have hn0 : (n : ℂ) ≠ 0 := mod_cast h0\n    rintro l k hk\n    field_simp at hk\n    norm_cast at hk\n    exact Int.natCast_dvd_natCast.mp <| hi.symm.dvd_of_dvd_mul_right <| hk ▸ dvd_mul_right ..\n\ntheorem isPrimitiveRoot_exp_of_coprime (i n : ℕ) (h0 : n ≠ 0) (hi : i.Coprime n) :\n    IsPrimitiveRoot (exp (2 * π * I * (i / n))) n :=\n  isPrimitiveRoot_exp_of_isCoprime _ _ h0 hi.isCoprime\n\ntheorem isPrimitiveRoot_exp_rat (q : ℚ) : IsPrimitiveRoot (exp (2 * π * I * q)) q.den := by\n  convert!\n    isPrimitiveRoot_exp_of_isCoprime _ _ q.den_nz <| Int.isCoprime_iff_nat_coprime.mpr q.reduced\n  nth_rw 1 [← Rat.num_div_den q]\n  simp\n\ntheorem isPrimitiveRoot_exp_rat_of_even_num (q : ℚ) (h : Even q.num) :\n    IsPrimitiveRoot (exp (π * I * q)) q.den := by\n  have ⟨n, hn⟩ := even_iff_exists_two_nsmul _ |>.mp h\n  convert! isPrimitiveRoot_exp_rat (n / q.den) using 1\n  · nth_rw 1 [← q.num_div_den, hn, Int.nsmul_eq_mul]\n    push_cast\n    ring_nf\n  · rw [← Int.cast_natCast, ← Rat.divInt_eq_div, ← Rat.mk_eq_divInt (nz := by simp)]\n    apply Nat.Coprime.coprime_mul_left (k := 2)\n    convert! q.reduced\n    grind\n\ntheorem isPrimitiveRoot_exp_rat_of_odd_num (q : ℚ) (h : Odd q.num) :\n    IsPrimitiveRoot (exp (π * I * q)) (2 * q.den) := by\n  convert! isPrimitiveRoot_exp_rat (q / 2) using 1\n  · push_cast\n    ring_nf\n  · nth_rw 2 [← q.num_div_den]\n    rw [mul_comm, div_div, ← Int.cast_ofNat, ← Int.cast_natCast, ← Int.cast_mul,\n      ← Rat.divInt_eq_div, ← Nat.cast_ofNat (R := ℤ), ← Nat.cast_mul,\n      ← Rat.mk_eq_divInt (nz := by simp)\n        (c := Nat.Coprime.mul_right q.reduced h.natAbs.coprime_two_right)]\n\ntheorem isPrimitiveRoot_exp (n : ℕ) (h0 : n ≠ 0) : IsPrimitiveRoot (exp (2 * π * I / n)) n := by\n  simpa only [Nat.cast_one, one_div] using!\n    isPrimitiveRoot_exp_of_coprime 1 n h0 n.coprime_one_left\n\ntheorem isPrimitiveRoot_iff (ζ : ℂ) (n : ℕ) (hn : n ≠ 0) :\n    IsPrimitiveRoot ζ n ↔ ∃ i < n, ∃ _ : i.Coprime n, exp (2 * π * I * (i / n)) = ζ := by\n  have hn0 : (n : ℂ) ≠ 0 := mod_cast hn\n  constructor; swap\n  · rintro ⟨i, -, hi, rfl⟩; exact isPrimitiveRoot_exp_of_coprime i n hn hi\n  intro h\n  have : NeZero n := ⟨hn⟩\n  obtain ⟨i, hi, rfl⟩ :=\n    (isPrimitiveRoot_exp n hn).eq_pow_of_pow_eq_one h.pow_eq_one\n  refine ⟨i, hi, ((isPrimitiveRoot_exp n hn).pow_iff_coprime (Nat.pos_of_ne_zero hn) i).mp h, ?_⟩\n  rw [← exp_nat_mul]\n  congr 1\n  ring\n\n/-- The complex `n`-th roots of unity are exactly the\ncomplex numbers of the form `exp (2 * Real.pi * Complex.I * (i / n))` for some `i < n`. -/\nnonrec theorem mem_rootsOfUnity (n : ℕ) [NeZero n] (x : Units ℂ) :\n    x ∈ rootsOfUnity n ℂ ↔ ∃ i < n, exp (2 * π * I * (i / n)) = x := by\n  rw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one]\n  have hn0 : (n : ℂ) ≠ 0 := mod_cast NeZero.out\n  constructor\n  · intro h\n    obtain ⟨i, hi, H⟩ : ∃ i < (n : ℕ), exp (2 * π * I / n) ^ i = x := by\n      simpa only using (isPrimitiveRoot_exp n NeZero.out).eq_pow_of_pow_eq_one h\n    refine ⟨i, hi, ?_⟩\n    rw [← H, ← exp_nat_mul]\n    congr 1\n    ring\n  · rintro ⟨i, _, H⟩\n    rw [← H, ← exp_nat_mul, exp_eq_one_iff]\n    use i\n    simp [field]\n\ntheorem card_rootsOfUnity (n : ℕ) [NeZero n] : Fintype.card (rootsOfUnity n ℂ) = n :=\n  (isPrimitiveRoot_exp n NeZero.out).card_rootsOfUnity\n\nTarget:\ntheorem card_primitiveRoots (k : ℕ) : (primitiveRoots k ℂ).card = φ k :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_32b115c85276","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"3684b97611187606a3954cb932b3a6b2019b1c74af5e0005a6aad3cd2516644a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/RootsOfUnity","family_id":"card_primitiveroots","file_id":"mathlib/Mathlib/RingTheory/RootsOfUnity/Complex.lean","sample_id":"32b115c85276c93388c62a93ae6422efd911c7d0d06efd966d995ffe632b53e9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f534ed3effb9755a614b851ef33384815c516806a3924c7986894d2e4e795cd3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3007d31a80186b26e9b8c1ae8d1dca50183ed5d5581191513a6261d467cbfeb8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"467d6115e45bfeaa364bc7d6dc84e27f5374de312ed692a0238a4a35b99ebeeb","source_sha256":"66c20bb7bca6fa55e98ef3e2d1bcc17c8251e92551f3085706c9cdfd3343f5b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← isRegular_pairing_iff, Pairing.isRegular_iff_nonempty_weakRankFunction]\n  exact (P.weakRankFunctionEquiv ℕ).symm.nonempty_congr","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":3,"token_jaccard":0.105263,"token_length_ratio":0.142857},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"a82f4f859a5e768f63cf9d1ab88ad7a78e3e6ee0939745cefd9466b50e2b8f83","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Rank\npublic import Mathlib.Data.Finite.Sigma\n\nNamespace:\nSSet.Subcomplex.PairingCore\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Existence of a rank function to natural numbers\n\nIn this file, we show that if `P : A.Pairing` is\na regular pairing of subcomplex `A` of a simplicial set `X`,\nthen there exists a rank function for `P` with values in `ℕ`.\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen Simplicial\n\nnamespace SSet.Subcomplex\n\nvariable {X : SSet.{u}} {A : X.Subcomplex}\n\nnamespace Pairing\n\nvariable (P : A.Pairing)\n\ninstance (y : P.II) : Finite { x // P.AncestralRel x y } := by\n  let T := { x : P.II // P.AncestralRel x y }\n  let U := Σ (d : Fin (P.p y).1.dim), ⦋d⦌ ⟶ ⦋(P.p y).1.1.1.1⦌\n  let ψ : U → X.S := fun ⟨d, f⟩ ↦ S.mk (X.map f.op (P.p y).1.simplex)\n  have h (t : T) : ∃ u, ψ u = t.1.1.toS := by\n    obtain ⟨f, _, hf⟩ := N.le_iff_exists_mono.1 t.2.2.le\n    refine ⟨⟨⟨t.1.1.dim, ?_⟩, f⟩, ?_⟩\n    · simpa using SSet.N.dim_lt_of_lt t.2.2\n    · rwa [SSet.S.ext_iff]\n  choose φ hφ using h\n  apply Finite.of_injective φ\n  intro t₁ t₂ h\n  rw [Subtype.ext_iff, Subtype.ext_iff, N.ext_iff, SSet.N.ext_iff, ← hφ, ← hφ, h]\n\nsection\n\nvariable {y : P.II} (hy : Acc P.AncestralRel y)\n\n/-- Auxiliary definition for `SSet.Subcomplex.Pairing.Rank`. -/\nnoncomputable def rank' : ℕ :=\n  Acc.recOn hy (fun y _ r ↦ ⨆ (x : { x // P.AncestralRel x y }), r x x.2 + 1)\n\nlemma rank'_eq :\n    P.rank' hy = ⨆ (x : { x // P.AncestralRel x y }), P.rank' (hy.inv x.2) + 1 := by\n  change P.rank' (Acc.intro y fun _ => hy.inv) = _\n  rfl\n\nlemma rank'_lt {x : P.II} (r : P.AncestralRel x y) :\n    P.rank' (hy.inv r) < P.rank' hy := by\n  rw [P.rank'_eq hy, ← Nat.add_one_le_iff]\n  exact le_csSup (Finite.bddAbove_range _) ⟨⟨x, r⟩, rfl⟩\n\nend\n\nsection IsRegular\n\nvariable [P.IsRegular]\n\n/-- The rank function with values in `ℕ` relative to the well founded\nancestrality relation of a regular pairing. -/\nnoncomputable def rank (x : P.II) : ℕ :=\n  P.rank' (P.wf.apply x)\n\nvariable {P} in\nlemma rank_lt {x y : P.II} (h : P.AncestralRel x y) :\n    P.rank x < P.rank y :=\n  P.rank'_lt _ h\n\n/-- The canonical rank function with values in `ℕ` of a regular pairing. -/\nnoncomputable def rankFunction : P.RankFunction ℕ where\n  rank := P.rank\n  lt := P.rank_lt\n\ninstance : Nonempty (P.RankFunction ℕ) := ⟨P.rankFunction⟩\n\ninstance : Nonempty (P.WeakRankFunction ℕ) := ⟨P.rankFunction.toWeakRankFunction⟩\n\nend IsRegular\n\nlemma isRegular_iff_nonempty_rankFunction [P.IsProper] :\n    P.IsRegular ↔ Nonempty (P.RankFunction ℕ) :=\n  ⟨fun _ ↦ inferInstance, fun ⟨h⟩ ↦ h.isRegular⟩\n\nlemma isRegular_iff_nonempty_weakRankFunction [P.IsProper] :\n    P.IsRegular ↔ Nonempty (P.WeakRankFunction ℕ) :=\n  ⟨fun _ ↦ inferInstance, fun ⟨h⟩ ↦ h.isRegular⟩\n\nend Pairing\n\nnamespace PairingCore\n\nvariable (P : A.PairingCore)\n\nlemma isRegular_iff_nonempty_rankFunction [P.IsProper] :\n    P.IsRegular ↔ Nonempty (P.RankFunction ℕ) := by\n  rw [← isRegular_pairing_iff, Pairing.isRegular_iff_nonempty_rankFunction]\n  exact (P.rankFunctionEquiv ℕ).symm.nonempty_congr\n\nTarget:\nlemma isRegular_iff_nonempty_weakRankFunction [P.IsProper] :\n    P.IsRegular ↔ Nonempty (P.WeakRankFunction ℕ) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_467d6115e45b","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d2b24cb184e9e1cf34f1d055c3cdaad9b7c56790093b3b8d88ff248695a993c5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplicialSet","family_id":"isregular_iff_nonempty_weakrankfunction","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/RankNat.lean","sample_id":"467d6115e45bfeaa364bc7d6dc84e27f5374de312ed692a0238a4a35b99ebeeb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"caf7c1208514616ff661b06fa1617859cfde2ba543e0b78c240ce9e176c8198e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2afb39b3691bc4ab78a4e23759e1c825b68cb3a96649f5d69c1247d3fbea884e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"50e5e16cfcfd38d6568b5e49b24fb4dda95d8927c47ec6721aa44339e4393525","source_sha256":"1d77f6aaa753df530e96edf4819026a453d32ae6fc473ac307ea722c26a6c807","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using h [[]] [[], []]","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":20,"token_jaccard":0.583333,"token_length_ratio":1.333333},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"a85abf74b32c8c10a31e79c7d5fcb0accd6bc00266cb63c14c57ae0d70d3edb1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Basic\n\nNamespace:\nInformationTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Elazar Gershuni. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Elazar Gershuni\n-/\n/-!\n# Uniquely Decodable Codes\n\nThis file defines uniquely decodable codes and proves basic properties.\n\n## Main definitions\n\n* `UniquelyDecodable`: A set of codewords is uniquely decodable if distinct concatenations\n  of codewords yield distinct strings.\n\n## Main results\n\n* `UniquelyDecodable.epsilon_not_mem`: Uniquely decodable codes cannot contain the empty\n  string.\n* `UniquelyDecodable.flatten_injective`: The flatten function is injective on lists of\n  codewords from a uniquely decodable code.\n-/\n\n@[expose] public section\n\nnamespace InformationTheory\n\nvariable {α : Type*}\n\n/-- A set of lists is uniquely decodable if distinct concatenations yield distinct strings. -/\ndef UniquelyDecodable (S : Set (List α)) : Prop :=\n  ∀ (L₁ L₂ : List (List α)),\n    (∀ w ∈ L₁, w ∈ S) → (∀ w ∈ L₂, w ∈ S) →\n    L₁.flatten = L₂.flatten → L₁ = L₂\n\nvariable {S : Set (List α)}\n\n/-- If a code is uniquely decodable, it does not contain the empty string.\n\nThe empty string can be \"decoded\" as either zero or two copies of itself,\nviolating unique decodability. -/\n\nTarget:\nlemma UniquelyDecodable.epsilon_not_mem\n    (h : UniquelyDecodable S) :\n    [] ∉ S :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  simpa using h [[]] [[], []]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"InformationTheory/Coding","family_id":"uniquelydecodable","file_id":"mathlib/Mathlib/InformationTheory/Coding/UniquelyDecodable.lean","sample_id":"50e5e16cfcfd38d6568b5e49b24fb4dda95d8927c47ec6721aa44339e4393525"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5ca120de9531c196478bb85a2839de944f04348088dc99c356e217a11b64504f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"34db14d5b84ad45bdcf1fb3d5c33e842b363f371673a1cfe9291b8e92cf2e21f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e2cd3c1a8076b3af47b98c32f101f8a48aa6a72e4acab38020fc2356c4738a6b","source_sha256":"5fbd6e0c9aed4ae0af15e49b0a292eb4e252a42e876e277383b0144f2f430dad","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction xs generalizing ys i with\n  | nil => cases h₂\n  | cons x' xs xs_ih =>\n    cases i\n    · simp only [length_cons, lt_add_iff_pos_left, add_pos_iff, Nat.lt_add_one, or_true,\n        getElem?_eq_getElem, getElem_cons_zero, Option.some.injEq] at h₂\n      subst h₂\n      cases ys\n      · cases h₁\n      · simp\n    · cases ys\n      · cases h₁\n      · obtain - | ⟨h₀, h₁⟩ := h₀\n        simp only [getElem?_cons_succ, zip_cons_cons, applyId_cons] at h₂ ⊢\n        rw [if_neg]\n        · apply xs_ih <;> solve_by_elim [Nat.succ.inj]\n        · apply h₀; apply List.mem_of_getElem? h₂","hard_negative":false,"metrics":{"chosen_tokens":129,"rejected_tokens":133,"token_jaccard":0.952381,"token_length_ratio":1.031008},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"a861aaa4b13802099045e3372e2ac5f0118ab3cc7961bbe40c28b46cbc2929df","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.MLList.Basic\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.ZeroLEOne\npublic import Mathlib.Data.DFinsupp.Defs\npublic import Mathlib.Data.Finsupp.Defs\npublic import Mathlib.Data.Int.Range\npublic import Mathlib.Data.List.Sigma\npublic import Mathlib.Data.PNat.Notation\npublic import Mathlib.Tactic.Bound.Init\n\nNamespace:\nPlausible.InjectiveFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\npublic meta import Mathlib.Data.Finsupp.ToDFinsupp\npublic meta import Mathlib.Data.Int.Range\npublic meta import Mathlib.Data.List.Sigma\npublic meta import Plausible.Functions\n\n/-!\n## `Plausible`: generators for functions\n\nThis file defines `Sampleable` instances for `ℤ → ℤ` injective functions.\n\nInjective functions are generated by creating a list of numbers and\na permutation of that list. The permutation insures that every input\nis mapped to a unique output. When an input is not found in the list\nthe input itself is used as an output.\n\nInjective functions `f : α → α` could be generated easily instead of\n`ℤ → ℤ` by generating a `List α`, removing duplicates and creating a\npermutation. One has to be careful when generating the domain to make\nit vast enough that, when generating arguments to apply `f` to,\nthey argument should be likely to lie in the domain of `f`. This is\nthe reason that injective functions `f : ℤ → ℤ` are generated by\nfixing the domain to the range `[-2*size .. 2*size]`, with `size`\nthe size parameter of the `gen` monad.\n\nMuch of the machinery provided in this file is applicable to generate\ninjective functions of type `α → α` and new instances should be easy\nto define.\n\nOther classes of functions such as monotone functions can generated using\nsimilar techniques. For monotone functions, generating two lists, sorting them\nand matching them should suffice, with appropriate default values.\nSome care must be taken for shrinking such functions to make sure\ntheir defining property is invariant through shrinking. Injective\nfunctions are an example of how complicated it can get.\n-/\n\n@[expose] public meta section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type v}\n\nnamespace Plausible\n\nnamespace TotalFunction\n\nsection Finsupp\n\nvariable [DecidableEq α]\n\n/--\nThis theorem exists because plausible does not have access to dlookup but\nmathlib has all the theory for it and wants to use it. We probably want to\nbring these two together at some point.\n-/\nprivate theorem apply_eq_dlookup (m : List (Σ _ : α, β)) (y : β) (x : α) :\n     (withDefault m y).apply x = (m.dlookup x).getD y := by\n  dsimp only [apply]\n  congr 1\n  induction m with\n  | nil => simp\n  | cons p m ih =>\n    rcases p with ⟨fst, snd⟩\n    by_cases heq : fst = x\n    · simp [heq]\n    · rw [List.dlookup_cons_ne]\n      · simp [heq, ih]\n      · symm\n        simp [heq]\n\nvariable [Zero β] [DecidableEq β]\n\n/-- Map a `TotalFunction` to one whose default value is zero so that it represents a `Finsupp`. -/\n@[simp]\ndef zeroDefault : TotalFunction α β → TotalFunction α β\n  | .withDefault A _ => .withDefault A 0\n\n/-- The support of a zero default `TotalFunction`. -/\ndef zeroDefaultSupp : TotalFunction α β → Finset α\n  | .withDefault A _ =>\n    List.toFinset <| (A.dedupKeys.filter fun ab => Sigma.snd ab ≠ 0).map Sigma.fst\n\n/-- Create a finitely supported function from a total function by taking the default value to\nzero. -/\ndef applyFinsupp (tf : TotalFunction α β) : α →₀ β where\n  support := zeroDefaultSupp tf\n  toFun := tf.zeroDefault.apply\n  mem_support_toFun := by\n    intro a\n    rcases tf with ⟨A, y⟩\n    simp only [zeroDefaultSupp, List.mem_map, List.mem_filter, exists_and_right,\n      List.mem_toFinset, exists_eq_right, Sigma.exists, Ne, zeroDefault]\n    rw [apply_eq_dlookup]\n    constructor\n    · rintro ⟨od, hval, hod⟩\n      have := List.mem_dlookup (List.nodupKeys_dedupKeys A) hval\n      rw [(_ : List.dlookup a A = od)]\n      · simpa using hod\n      · simpa [List.dlookup_dedupKeys]\n    · intro h\n      use (A.dlookup a).getD (0 : β)\n      rw [← List.dlookup_dedupKeys] at h ⊢\n      simp only [h, ← List.mem_dlookup_iff A.nodupKeys_dedupKeys, not_false_iff, Option.mem_def]\n      cases haA : List.dlookup a A.dedupKeys\n      · simp [haA] at h\n      · simp\n\nvariable [SampleableExt α] [SampleableExt β] [Repr α]\n\ninstance Finsupp.sampleableExt : SampleableExt (α →₀ β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\n-- TODO: support a non-constant codomain type\ninstance DFinsupp.sampleableExt : SampleableExt (Π₀ _ : α, β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp.toDFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\nend Finsupp\nend TotalFunction\n\nopen _root_.List\n\n/-- Data structure specifying a total function using a list of pairs\nand a default value returned when the input is not in the domain of\nthe partial function.\n\n`mapToSelf f` encodes `x ↦ f x` when `x ∈ f` and `x ↦ x`,\ni.e. `x` to itself, otherwise.\n\nWe use `Σ` to encode mappings instead of `×` because we\nrely on the association list API defined in `Mathlib/Data/List/Sigma.lean`.\n-/\ninductive InjectiveFunction (α : Type u) : Type u\n  | mapToSelf (xs : List (Σ _ : α, α)) :\n      xs.map Sigma.fst ~ xs.map Sigma.snd → List.Nodup (xs.map Sigma.snd) → InjectiveFunction α\n\ninstance : Inhabited (InjectiveFunction α) :=\n  ⟨⟨[], List.Perm.nil, List.nodup_nil⟩⟩\n\nnamespace InjectiveFunction\n\n/-- Apply a total function to an argument. -/\ndef apply [DecidableEq α] : InjectiveFunction α → α → α\n  | InjectiveFunction.mapToSelf m _ _, x => (m.dlookup x).getD x\n\n/-- Produce a string for a given `InjectiveFunction`.\nThe output is of the form `[x₀ ↦ f x₀, .. xₙ ↦ f xₙ, x ↦ x]`.\nUnlike for `TotalFunction`, the default value is not a constant\nbut the identity function.\n-/\nprotected def repr [Repr α] : InjectiveFunction α → String\n  | InjectiveFunction.mapToSelf m _ _ => s! \"[{TotalFunction.reprAux m}x ↦ x]\"\n\ninstance (α : Type u) [Repr α] : Repr (InjectiveFunction α) where\n  reprPrec f _p := InjectiveFunction.repr f\n\n/-- Interpret a list of pairs as a total function, defaulting to\nthe identity function when no entries are found for a given function -/\ndef List.applyId [DecidableEq α] (xs : List (α × α)) (x : α) : α :=\n  ((xs.map Prod.toSigma).dlookup x).getD x\n\n@[simp]\ntheorem List.applyId_cons [DecidableEq α] (xs : List (α × α)) (x y z : α) :\n    List.applyId ((y, z)::xs) x = if y = x then z else List.applyId xs x := by\n  simp only [List.applyId, List.dlookup, eq_rec_constant, Prod.toSigma, List.map]\n  split_ifs <;> rfl\n\nopen Function\nopen List\n\nopen Nat\n\nTarget:\ntheorem List.applyId_zip_eq [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs)\n    (h₁ : xs.length = ys.length) (x y : α) (i : ℕ) (h₂ : xs[i]? = some x) :\n    List.applyId.{u} (xs.zip ys) x = y ↔ ys[i]? = some y :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  induction xs generalizing ys i with\n  | nil => cases h₂\n  | cons x' xs xs_ih =>\n    cases i\n    · simp only [length_cons, lt_add_iff_pos_left, add_pos_iff, Nat.lt_add_one, or_true,\n        getElem?_eq_getElem, getElem_cons_zero, Option.some.injEq] at h₂\n      subst h₂\n      cases ys\n      · cases h₁\n      · simp\n    · cases ys\n      · cases h₁\n      · obtain - | ⟨h₀, h₁⟩ := h₀\n        simp only [getElem?_cons_succ, zip_cons_cons, applyId_cons] at h₂ ⊢\n        rw [if_neg]\n        · apply xs_ih <;> solve_by_elim [Nat.succ.inj]\n        · apply h₀; apply List.mem_of_getElem? h₂","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Testing/Plausible","family_id":"list","file_id":"mathlib/Mathlib/Testing/Plausible/Functions.lean","sample_id":"e2cd3c1a8076b3af47b98c32f101f8a48aa6a72e4acab38020fc2356c4738a6b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2139989fd770c5936c4b547ad3ae043d8606b002f3ccc3b366913c36130fefc8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1dd4201e2b46ac0426635902442743acfd866e49ec23d6fbab9739e8f5773d09","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"89245e196e8f1c3866de6d9410eff906de2a4f7ec9bd4db4b755d68b297fae0f","source_sha256":"7cfbfc972c134864031de87e3b5157be2d7af83e375a2c0e1cc9ff7db598678d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  filter_upwards [rnDeriv_compProd h_ac ν] with p hp\n  simp_rw [hp, ENNReal.toReal_mul]\n  by_cases h_zero1 : ((∂μ/∂ν) p.1).toReal = 0\n  · simp [h_zero1]\n  by_cases h_zero2 : ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal = 0\n  · simp [h_zero2]\n  simp [log_mul h_zero1 h_zero2]","hard_negative":true,"metrics":{"chosen_tokens":76,"rejected_tokens":3,"token_jaccard":0.027027,"token_length_ratio":0.039474},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"a87df9a29cbf70ee58c5b300def1e0a51624bb89ab8e003b48da3dc1dc3397d6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.InformationTheory.KullbackLeibler.Basic\npublic import Mathlib.Probability.Kernel.Composition.MeasureCompProd\npublic import Mathlib.Probability.Notation\nimport Mathlib.Probability.Kernel.Composition.IntegralCompProd\nimport Mathlib.Probability.Kernel.Composition.RadonNikodym\n\nNamespace:\nInformationTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne, Lorenzo Luccioli\n-/\n/-!\n# Chain rule for the Kullback-Leibler divergence\n\nSuppose that we have two finite joint measures on a product `𝓧 × 𝓨`, which can be decomposed as\n`μ ⊗ₘ κ` and `ν ⊗ₘ η`, where `μ` and `ν` are measures on `𝓧` and `κ` and `η` are Markov kernels\nfrom `𝓧` to `𝓨`. Then we can express the Kullback-Leibler divergence between these two joint\nmeasures as a sum of `klDiv μ ν` and the conditional Kullback-Leibler divergence between the kernels\n`κ` and `η`, averaged over `μ`. The resulting equality is most often written as\n`klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + μ[fun x ↦ klDiv (κ x) (η x)]`.\n\nHere we first prove the following version:\n`klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)`.\nThis version avoids the issue of measurability of the function `x ↦ klDiv (κ x) (η x)`, which is not\nalways guaranteed, and thus holds for all measurable spaces `𝓧` and `𝓨`, without any assumptions.\n\n## Main statements\n\n* `klDiv_compProd_eq_add`: `klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)`\n* `klDiv_compProd_left`: `klDiv (μ ⊗ₘ κ) (ν ⊗ₘ κ) = klDiv μ ν`\n\n## Proof\n\nThe main ingredient is the chain rule for Radon-Nikodym derivatives:\n`∂(μ ⊗ₘ κ)/∂(ν ⊗ₘ η) = ∂μ/∂ν * ∂(μ ⊗ₘ κ)/∂(μ ⊗ₘ η)`.\nThen, omitting edge cases, the Kullback-Leibler divergence is an integral of a logarithm of the\nderivative on the left, which decomposes into a sum of two integrals of logarithms.\nWe now give a more detailed outline of the proof.\n\nThe Kullback-Leibler divergence `klDiv μ ν` is defined with an if-then-else statement:\nif the measures are absolutely continuous (`μ ≪ ν`) and the log-likelihood ratio `llr μ ν` is\nintegrable, then it is defined as `∫ x, llr μ ν x ∂μ + ν.real univ - μ.real univ`, otherwise\nit is defined to be `∞`.\n\nWe first deal with the case in which absolute continuity does not hold. The main observation is\nthat `μ ⊗ₘ κ ≪ ν ⊗ₘ η ↔ μ ≪ ν ∧ μ ⊗ₘ κ ≪ μ ⊗ₘ η`, which means that if one of the two sides of the\nKL equality is infinite because of lack of absolute continuity, then the other side is also infinite\nfor the same reason.\n\nThen, we deal with the case in which absolute continuity holds but integrability does not. Again,\nwe can show a similar equivalence for integrability, which allows us to conclude that both sides\nare infinite.\n`Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)` is equivalent to\n`Integrable (llr μ ν) μ ∧ Integrable (llr (μ ⊗ₘ κ) (μ ⊗ₘ η)) (μ ⊗ₘ κ)`.\nThe proof of this equivalence relies on the convexity of the function `x ↦ x * log x`.\n\nFinally, we prove the equality in the case in which both absolute continuity and integrability hold.\nIn that case, `klDiv μ ν = ∫ x, llr μ ν x ∂μ + ν.real univ - μ.real univ` and similarly for\nthe other terms. It is easy to see that it suffices to prove the equality of the integrals parts.\nFinally, the computation for the integral of the log-likelihood ratio is as follows:\n```\n∫ p, llr (μ ⊗ₘ κ) (ν ⊗ₘ η) p ∂(μ ⊗ₘ κ)\n_ = ∫ p, ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal * log ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal ∂(ν ⊗ₘ η)\n_ = ∫ p, ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal *\n    (log ((∂μ/∂ν) p.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal) ∂(ν ⊗ₘ η)\n_ = ∫ p, (log ((∂μ/∂ν) p.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal) ∂(μ ⊗ₘ κ)\n_ = ∫ p, log ((∂μ/∂ν) p.1).toReal ∂(μ ⊗ₘ κ) + ∫ p, log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal ∂(μ ⊗ₘ κ)\n_ = ∫ a, llr μ ν a ∂μ + ∫ p, llr (μ ⊗ₘ κ) (μ ⊗ₘ η) p ∂(μ ⊗ₘ κ)\n```\n\n## TODO\n\nAdd a version of the chain rule for the integral form of the contional KL divergence, i.e.\n`μ[fun x ↦ klDiv (κ x) (η x)]`.\n\n-/\n\npublic section\n\nopen Real MeasureTheory Set ProbabilityTheory\nopen scoped ENNReal\n\nnamespace InformationTheory\n\nvariable {𝓧 𝓨 : Type*} {m𝓧 : MeasurableSpace 𝓧} {m𝓨 : MeasurableSpace 𝓨}\n  {μ ν : Measure 𝓧} {κ η : Kernel 𝓧 𝓨}\n  [IsFiniteMeasure μ] [IsFiniteMeasure ν] [IsMarkovKernel κ] [IsMarkovKernel η]\n\n/-- If the log-likelihood ration between two composition-products is integrable, then so is the\nlog-likelihood ratio between the two measures on the first space. -/\nlemma integrable_llr_of_integrable_llr_compProd\n    (h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η) (h_int : Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)) :\n    Integrable (llr μ ν) μ := by\n  have ⟨hμν_ac, hκη_ac⟩ := Measure.absolutelyContinuous_compProd_iff.mp h_ac\n  rw [← integrable_rnDeriv_mul_log_iff h_ac] at h_int\n  replace h_int := convexOn_mul_log.integrable_apply_rnDeriv_of_integrable_compProd\n    continuous_mul_log.stronglyMeasurable continuous_mul_log.continuousWithinAt h_int hκη_ac\n  exact (integrable_rnDeriv_mul_log_iff hμν_ac).mp h_int\n\nTarget:\nlemma rnDeriv_compProd_mul_log_eq_mul_add (h_ac : μ ⊗ₘ κ ≪ μ ⊗ₘ η) :\n    ∀ᵐ p ∂(ν ⊗ₘ η), ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal * log ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal =\n      (((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal * (log ((∂μ/∂ν) p.1).toReal +\n        log ((∂(μ ⊗ₘ κ)/∂(μ ⊗ₘ η)) p).toReal)) :=\n\nProof body:\n","rejected":"by\n  exact rnDeriv_compProd_mul_log_eq_mul_add","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"da79b9c499ec0efbd5b73ce0827793aa31e143febc99a4f77847dd213dc393dd","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"InformationTheory/KullbackLeibler","family_id":"rnderiv_compprod_mul_log_eq_mul_add","file_id":"mathlib/Mathlib/InformationTheory/KullbackLeibler/ChainRule.lean","sample_id":"89245e196e8f1c3866de6d9410eff906de2a4f7ec9bd4db4b755d68b297fae0f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c55c96a734207c6a61a412e0cab744c2c07b34209605ac6a4b05c1a6bd545ce9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a0803f253ced08eb971dedf8185abdf61b52cd78e4ed89a12a08a78af65ea62a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4d49f41524ca78215a8298981fc8795e5f250086d26cc6a75dc11a9339f59a8c","source_sha256":"5a251c5b9da944aa52b3b3c01ae0c840fa23c2c1f9c9e5ace9139527d3f0e3e9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [tprod_apply, map_list_prod, List.map_ofFn]\n  simp_rw [Function.comp_def, toDirectSum_ι]\n  rw [DirectSum.list_prod_ofFn_of_eq_dProd]\n  apply DirectSum.of_eq_of_gradedMonoid_eq\n  rw [GradedMonoid.mk_list_dProd]\n  rw [TensorPower.list_prod_gradedMonoid_mk_single]","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":5,"token_jaccard":0.076923,"token_length_ratio":0.119048},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"a88ef5d0251a728382583723908a8f2e6d68c1c8c8fa8220647b931ca2b5057a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorAlgebra.Basic\npublic import Mathlib.LinearAlgebra.TensorPower.Basic\n\nNamespace:\nTensorAlgebra\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Tensor algebras as direct sums of tensor powers\n\nIn this file we show that `TensorAlgebra R M` is isomorphic to a direct sum of tensor powers, as\n`TensorAlgebra.equivDirectSum`.\n-/\n\n@[expose] public section\n\nopen scoped DirectSum TensorProduct\n\nvariable {R M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M]\n\nnamespace TensorPower\n\n/-- The canonical embedding from a tensor power to the tensor algebra -/\ndef toTensorAlgebra {n} : ⨂[R]^n M →ₗ[R] TensorAlgebra R M :=\n  PiTensorProduct.lift (TensorAlgebra.tprod R M n)\n\n@[simp]\ntheorem toTensorAlgebra_tprod {n} (x : Fin n → M) :\n    TensorPower.toTensorAlgebra (PiTensorProduct.tprod R x) = TensorAlgebra.tprod R M n x :=\n  PiTensorProduct.lift.tprod _\n\n@[simp]\ntheorem toTensorAlgebra_gOne :\n    TensorPower.toTensorAlgebra (@GradedMonoid.GOne.one _ (fun n => ⨂[R]^n M) _ _) = 1 := by\n  simp [GradedMonoid.GOne.one, TensorPower.toTensorAlgebra_tprod]\n\n@[simp]\ntheorem toTensorAlgebra_gMul {i j} (a : (⨂[R]^i) M) (b : (⨂[R]^j) M) :\n    TensorPower.toTensorAlgebra (@GradedMonoid.GMul.mul _ (fun n => ⨂[R]^n M) _ _ _ _ a b) =\n      TensorPower.toTensorAlgebra a * TensorPower.toTensorAlgebra b := by\n  -- change `a` and `b` to `tprod R a` and `tprod R b`\n  rw [TensorPower.gMul_eq_coe_linearMap, ← LinearMap.compr₂_apply, ← @LinearMap.mul_apply' R, ←\n    LinearMap.compl₂_apply, ← LinearMap.comp_apply]\n  refine LinearMap.congr_fun (LinearMap.congr_fun ?_ a) b\n  clear! a b\n  ext (a b)\n  simp only [LinearMap.compMultilinearMap_apply, LinearMap.compr₂_apply, ← gMul_def,\n    TensorProduct.mk_apply, LinearEquiv.coe_coe, tprod_mul_tprod, toTensorAlgebra_tprod,\n    TensorAlgebra.tprod_apply, LinearMap.comp_apply, LinearMap.compl₂_apply]\n  refine Eq.trans ?_ List.prod_append\n  congr\n  rw [List.ofFn_comp' _ (TensorAlgebra.ι R), List.ofFn_comp' _ (TensorAlgebra.ι R),\n    List.ofFn_comp' _ (TensorAlgebra.ι R), ← List.map_append, List.ofFn_fin_append]\n\n@[simp]\ntheorem toTensorAlgebra_galgebra_toFun (r : R) :\n    TensorPower.toTensorAlgebra (DirectSum.GAlgebra.toFun (R := R) (A := fun n => ⨂[R]^n M) r) =\n      algebraMap _ _ r := by\n  rw [TensorPower.galgebra_toFun_def, TensorPower.algebraMap₀_eq_smul_one, map_smul,\n    TensorPower.toTensorAlgebra_gOne, Algebra.algebraMap_eq_smul_one]\n\nend TensorPower\n\nnamespace TensorAlgebra\n\n/-- The canonical map from a direct sum of tensor powers to the tensor algebra. -/\ndef ofDirectSum : (⨁ n, ⨂[R]^n M) →ₐ[R] TensorAlgebra R M :=\n  DirectSum.toAlgebra _ _ (fun _ => TensorPower.toTensorAlgebra) TensorPower.toTensorAlgebra_gOne\n    (fun {_ _} => TensorPower.toTensorAlgebra_gMul)\n\n@[simp]\ntheorem ofDirectSum_of_tprod {n} (x : Fin n → M) :\n    ofDirectSum (DirectSum.of _ n (PiTensorProduct.tprod R x)) = tprod R M n x :=\n  (DirectSum.toAddMonoid_of\n    (fun _ ↦ LinearMap.toAddMonoidHom TensorPower.toTensorAlgebra) _ _).trans\n  (TensorPower.toTensorAlgebra_tprod _)\n\n/-- The canonical map from the tensor algebra to a direct sum of tensor powers. -/\ndef toDirectSum : TensorAlgebra R M →ₐ[R] ⨁ n, ⨂[R]^n M :=\n  TensorAlgebra.lift R <|\n    DirectSum.lof R ℕ (fun n => ⨂[R]^n M) _ ∘ₗ\n      (LinearEquiv.symm <| PiTensorProduct.subsingletonEquiv (0 : Fin 1) : M ≃ₗ[R] _).toLinearMap\n\n@[simp]\ntheorem toDirectSum_ι (x : M) :\n    toDirectSum (ι R x) =\n      DirectSum.of (fun n => ⨂[R]^n M) _ (PiTensorProduct.tprod R fun _ : Fin 1 => x) := by\n  simp [toDirectSum, TensorAlgebra.lift_ι_apply, DirectSum.lof_eq_of]\n\ntheorem ofDirectSum_comp_toDirectSum :\n    ofDirectSum.comp toDirectSum = AlgHom.id R (TensorAlgebra R M) := by\n  ext\n  simp [tprod_apply]\n\n@[simp]\ntheorem ofDirectSum_toDirectSum (x : TensorAlgebra R M) :\n    ofDirectSum (TensorAlgebra.toDirectSum x) = x :=\n  AlgHom.congr_fun ofDirectSum_comp_toDirectSum x\n\n@[simp]\ntheorem mk_reindex_cast {n m : ℕ} (h : n = m) (x : ⨂[R]^n M) :\n    GradedMonoid.mk (A := fun i => (⨂[R]^i) M) m\n    (PiTensorProduct.reindex R (fun _ ↦ M) (Equiv.cast <| congr_arg Fin h) x) =\n    GradedMonoid.mk n x :=\n  Eq.symm (PiTensorProduct.gradedMonoid_eq_of_reindex_cast h rfl)\n\n@[simp]\ntheorem mk_reindex_fin_cast {n m : ℕ} (h : n = m) (x : ⨂[R]^n M) :\n    GradedMonoid.mk (A := fun i => (⨂[R]^i) M) m\n    (PiTensorProduct.reindex R (fun _ ↦ M) (finCongr h) x) = GradedMonoid.mk n x := by\n  rw [finCongr_eq_equivCast, mk_reindex_cast h]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The product of tensor products made of a single vector is the same as a single product of\nall the vectors. -/\ntheorem _root_.TensorPower.list_prod_gradedMonoid_mk_single (n : ℕ) (x : Fin n → M) :\n    ((List.finRange n).map fun a =>\n          (GradedMonoid.mk _ (PiTensorProduct.tprod R fun _ : Fin 1 => x a) :\n            GradedMonoid fun n => ⨂[R]^n M)).prod =\n      GradedMonoid.mk n (PiTensorProduct.tprod R x) := by\n  refine Fin.consInduction ?_ ?_ x <;> clear x\n  · rw [List.finRange_zero, List.map_nil, List.prod_nil]\n    rfl\n  · intro n x₀ x ih\n    rw [List.finRange_succ, List.map_cons, List.prod_cons, List.map_map]\n    simp_rw [Function.comp_def, Fin.cons_zero, Fin.cons_succ]\n    rw [ih, GradedMonoid.mk_mul_mk, TensorPower.tprod_mul_tprod]\n    refine TensorPower.gradedMonoid_eq_of_cast (add_comm _ _) ?_\n    dsimp only [GradedMonoid.mk]\n    rw [TensorPower.cast_tprod]\n    simp_rw [Fin.append_left_eq_cons, Function.comp_def]\n    congr 1 with i\n\nTarget:\ntheorem toDirectSum_tensorPower_tprod {n} (x : Fin n → M) :\n    toDirectSum (tprod R M n x) = DirectSum.of _ n (PiTensorProduct.tprod R x) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_4d49f41524ca","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"2410e7311e2da84b2802f73a11dbd6ac88448e508bfc1ae122e9ae76303dd099","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/TensorAlgebra","family_id":"todirectsum_tensorpower_tprod","file_id":"mathlib/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean","sample_id":"4d49f41524ca78215a8298981fc8795e5f250086d26cc6a75dc11a9339f59a8c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2d27cea8f3ca8caf10e9d8775b5b6885d300093e2302453f512d2957b166201a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4e9ba5114733e3b96be27c5b4d01de8367929d2bf280e4296d00c4e5e9fb55bb","source_sha256":"73ebc2e72f7678b20eb20bcb8222b112e9c86c209dc6aa3492fcf8a265aa9fa7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let : Field (A ⧸ p) := Ideal.Quotient.field p\n  rw [Ideal.isRadical_iff_quotient_reduced]\n  exact Algebra.FormallyUnramified.isReduced_of_field (A ⧸ p) (B ⧸ p.map (algebraMap A B))","hard_negative":true,"metrics":{"chosen_tokens":45,"rejected_tokens":8,"token_jaccard":0.103448,"token_length_ratio":0.177778},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"a8fe70e31fccd97cc78d9b9f8d9336c008c804bb4891fb281c5a333ef46c36cf","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.PurelyInseparable.Basic\npublic import Mathlib.RingTheory.Artinian.Ring\npublic import Mathlib.RingTheory.Unramified.Finite\npublic import Mathlib.RingTheory.Unramified.Locus\n\nNamespace:\nAlgebra.FormallyUnramified\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Unramified algebras over fields\n\n## Main results\n\nLet `K` be a field, `A` be a `K`-algebra and `L` be a field extension of `K`.\n\n- `Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_isLocalRing`:\n    If `A` is `K`-unramified and `K` is alg-closed, then `K = A`.\n- `Algebra.FormallyUnramified.isReduced_of_field`:\n    If `A` is `K`-unramified then `A` is reduced.\n- `Algebra.FormallyUnramified.iff_isSeparable`:\n    `L` is unramified over `K` iff `L` is separable over `K`.\n\n## References\n\n- [B. Iversen, *Generic Local Structure of the Morphisms in Commutative Algebra*][iversen]\n\n-/\n\npublic section\n\nopen Algebra Module Polynomial\nopen scoped TensorProduct\n\nuniverse u\n\nvariable (K A L : Type*) [Field K] [Field L] [CommRing A] [Algebra K A] [Algebra K L]\n\nnamespace Algebra.FormallyUnramified\n\ntheorem of_isSeparable [Algebra.IsSeparable K L] : FormallyUnramified K L := by\n  rw [iff_comp_injective]\n  intro B _ _ I hI f₁ f₂ e\n  ext x\n  have : f₁ x - f₂ x ∈ I := by\n    simpa [Ideal.Quotient.mk_eq_mk_iff_sub_mem] using AlgHom.congr_fun e x\n  have := Polynomial.eval_add_of_sq_eq_zero ((minpoly K x).map (algebraMap K B)) (f₂ x)\n    (f₁ x - f₂ x) (show (f₁ x - f₂ x) ^ 2 ∈ ⊥ from hI ▸ Ideal.pow_mem_pow this 2)\n  simp only [add_sub_cancel, eval_map_algebraMap, aeval_algHom_apply, minpoly.aeval, map_zero,\n    derivative_map, zero_add] at this\n  rwa [eq_comm, ((isUnit_iff_ne_zero.mpr\n    ((Algebra.IsSeparable.isSeparable K x).aeval_derivative_ne_zero\n      (minpoly.aeval K x))).map f₂).mul_right_eq_zero, sub_eq_zero] at this\n\nvariable [FormallyUnramified K A] [EssFiniteType K A]\nvariable [FormallyUnramified K L] [EssFiniteType K L]\n\ntheorem bijective_of_isAlgClosed_of_isLocalRing\n    [IsAlgClosed K] [IsLocalRing A] :\n    Function.Bijective (algebraMap K A) := by\n  have := finite_of_free (R := K) (S := A)\n  have : IsArtinianRing A := isArtinian_of_tower K inferInstance\n  have hA : IsNilpotent (IsLocalRing.maximalIdeal A) := by\n    rw [← IsLocalRing.jacobson_eq_maximalIdeal ⊥]\n    · exact IsArtinianRing.isNilpotent_jacobson_bot\n    · exact bot_ne_top\n  let e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A := {\n    __ := Algebra.ofId K (A ⧸ IsLocalRing.maximalIdeal A)\n    __ := Equiv.ofBijective _ IsAlgClosed.algebraMap_bijective_of_isIntegral }\n  let e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=\n    (Algebra.TensorProduct.congr AlgEquiv.refl e.symm).trans (Algebra.TensorProduct.rid K A A)\n  let f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap.comp (sec K A _)\n  have hf : (Algebra.ofId _ _).toLinearMap ∘ₗ f = LinearMap.id := by\n    dsimp [f]\n    rw [← LinearMap.comp_assoc, ← comp_sec K A]\n    congr 1\n    apply LinearMap.restrictScalars_injective K\n    apply _root_.TensorProduct.ext'\n    intro r s\n    obtain ⟨s, rfl⟩ := e.surjective s\n    suffices s • (Ideal.Quotient.mk (IsLocalRing.maximalIdeal A)) r = r • e s by\n      simpa [ofId, e']\n    simp [Algebra.smul_def, e, ofId, mul_comm]\n  have hf₁ : f 1 • (1 : A ⧸ IsLocalRing.maximalIdeal A) = 1 := by\n    rw [← algebraMap_eq_smul_one]\n    exact LinearMap.congr_fun hf 1\n  have hf₂ : 1 - f 1 ∈ IsLocalRing.maximalIdeal A := by\n    rw [← Ideal.Quotient.eq_zero_iff_mem, map_sub, map_one, ← Ideal.Quotient.algebraMap_eq,\n     algebraMap_eq_smul_one, hf₁, sub_self]\n  have hf₃ : IsIdempotentElem (1 - f 1) := by\n    apply IsIdempotentElem.one_sub\n    rw [IsIdempotentElem, ← smul_eq_mul, ← map_smul, hf₁]\n  have hf₄ : f 1 = 1 := by\n    obtain ⟨n, hn⟩ := hA\n    have : (1 - f 1) ^ n = 0 := by\n      rw [← Ideal.mem_bot, ← Ideal.zero_eq_bot, ← hn]\n      exact Ideal.pow_mem_pow hf₂ n\n    rw [eq_comm, ← sub_eq_zero, ← hf₃.pow_succ_eq n, pow_succ, this, zero_mul]\n  refine Equiv.bijective ⟨algebraMap K A, ⇑e.symm ∘ ⇑(algebraMap A _), fun x ↦ by simp, fun x ↦ ?_⟩\n  have : ⇑(algebraMap K A) = ⇑f ∘ ⇑e := by\n    ext k\n    conv_rhs => rw [← mul_one k, ← smul_eq_mul, Function.comp_apply, map_smul,\n      LinearMap.map_smul_of_tower, map_one, hf₄, ← algebraMap_eq_smul_one]\n  rw [this]\n  simp only [Function.comp_apply, AlgEquiv.apply_symm_apply, algebraMap_eq_smul_one,\n    map_smul, hf₄, smul_eq_mul, mul_one]\n\ntheorem isField_of_isAlgClosed_of_isLocalRing\n    [IsAlgClosed K] [IsLocalRing A] : IsField A := by\n  rw [IsLocalRing.isField_iff_maximalIdeal_eq, eq_bot_iff]\n  intro x hx\n  obtain ⟨x, rfl⟩ := (bijective_of_isAlgClosed_of_isLocalRing K A).surjective x\n  change _ = 0\n  rw [← (algebraMap K A).map_zero]\n  by_contra hx'\n  exact hx ((isUnit_iff_ne_zero.mpr\n    (fun e ↦ hx' ((algebraMap K A).congr_arg e))).map (algebraMap K A))\n\ninclude K in\ntheorem isReduced_of_field :\n    IsReduced A := by\n  constructor\n  intro x hx\n  let f := (Algebra.TensorProduct.includeRight (R := K) (A := AlgebraicClosure K) (B := A))\n  have : Function.Injective f := by\n    have : ⇑f = (LinearMap.rTensor A (Algebra.ofId K (AlgebraicClosure K)).toLinearMap).comp\n        (Algebra.TensorProduct.lid K A).symm.toLinearMap := by\n      ext x; simp [f]\n    rw [this]\n    suffices Function.Injective\n        (LinearMap.rTensor A (Algebra.ofId K (AlgebraicClosure K)).toLinearMap) by\n      exact this.comp (Algebra.TensorProduct.lid K A).symm.injective\n    apply Module.Flat.rTensor_preserves_injective_linearMap\n    exact (algebraMap K _).injective\n  apply this\n  rw [map_zero]\n  apply eq_zero_of_localization\n  intro M hM\n  have hy := (hx.map f).map (algebraMap _ (Localization.AtPrime M))\n  generalize algebraMap _ (Localization.AtPrime M) (f x) = y at *\n  have := EssFiniteType.of_isLocalization (Localization.AtPrime M) M.primeCompl\n  have := of_isLocalization (Rₘ := Localization.AtPrime M) M.primeCompl\n  have := EssFiniteType.comp (AlgebraicClosure K) (AlgebraicClosure K ⊗[K] A)\n    (Localization.AtPrime M)\n  have := comp (AlgebraicClosure K) (AlgebraicClosure K ⊗[K] A)\n    (Localization.AtPrime M)\n  letI := (isField_of_isAlgClosed_of_isLocalRing (AlgebraicClosure K)\n    (A := Localization.AtPrime M)).toField\n  exact hy.eq_zero\n\nTarget:\ntheorem isRadical_map_isMaximal (B : Type*) [CommRing B] [Algebra A B]\n    [Algebra.EssFiniteType A B] [Algebra.FormallyUnramified A B] (p : Ideal A) [p.IsMaximal] :\n    (p.map (algebraMap A B)).IsRadical :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"bc74fb8be802f8eab74cd49d8f6d29d8f8313b18b5ea70ee4311c716d6e9129e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Unramified","family_id":"isradical_map_ismaximal","file_id":"mathlib/Mathlib/RingTheory/Unramified/Field.lean","sample_id":"4e9ba5114733e3b96be27c5b4d01de8367929d2bf280e4296d00c4e5e9fb55bb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"36e2743b3c488bb6e8cf2bf83c88a0f4aa1e0278b4cda351f399f8fa4b568278","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b387da58bcb61e7e635a1a81f3607072af846e0dc1fa8c2c2a1a64370dcf8c7d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a3683d3fecf5a78a2e088915121e34b1d7028592a27520975260840e3084875a","source_sha256":"30e96c1aa03db850a53c7f6d12a34be8e99ae1b3e4a1595e2543d99d5c075ab6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine OpenPartialHomeomorph.ext _ _ (fun x => rfl) (fun x => rfl) ?_\n  simp [e.open_source.interior_eq, ← inter_assoc]","hard_negative":true,"metrics":{"chosen_tokens":32,"rejected_tokens":3,"token_jaccard":0.041667,"token_length_ratio":0.09375},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"a90eefd368bf7dde767f679839e2e224eb61ee544782833f3268eb64a10f33ee","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.OpenPartialHomeomorph.Continuity\n\nNamespace:\nOpenPartialHomeomorph\n\nLocal context:\n/-\nCopyright (c) 2019 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Partial homeomorphisms: Images of sets\n\n## Main definitions\n\n* `OpenPartialHomeomorph.IsImage`: predicate for when one set is an image of another\n* `OpenPartialHomeomorph.ofSet`: the identity on a set `s`\n* `OpenPartialHomeomorph.EqOnSource`: equivalence relation describing the \"right\" notion of equality\n  for open partial homeomorphisms\n\n## Implementation notes\n\nMost statements are copied from their `PartialEquiv` versions, although some care is required\nespecially when restricting to subsets, as these should be open subsets.\n\nFor design notes, see `PartialEquiv.lean`.\n\n### Local coding conventions\n\nIf a lemma deals with the intersection of a set with either source or target of a `PartialEquiv`,\nthen it should use `e.source ∩ s` or `e.target ∩ t`, not `s ∩ e.source` or `t ∩ e.target`.\n-/\n\n@[expose] public section\n\nopen Function Set Filter Topology\n\nvariable {X X' : Type*} {Y Y' : Type*} {Z Z' : Type*}\n  [TopologicalSpace X] [TopologicalSpace X'] [TopologicalSpace Y] [TopologicalSpace Y']\n  [TopologicalSpace Z] [TopologicalSpace Z']\n\nnamespace OpenPartialHomeomorph\n\nvariable (e : OpenPartialHomeomorph X Y)\n\nsection IsImage\n\n/-!\n## `OpenPartialHomeomorph.IsImage` relation\n\nWe say that `t : Set Y` is an image of `s : Set X` under an open partial homeomorphism `e` if any of\nthe following equivalent conditions hold:\n\n* `e '' (e.source ∩ s) = e.target ∩ t`;\n* `e.source ∩ e ⁻¹ t = e.source ∩ s`;\n* `∀ x ∈ e.source, e x ∈ t ↔ x ∈ s` (this one is used in the definition).\n\nThis definition is a restatement of `PartialEquiv.IsImage` for open partial homeomorphisms.\nIn this section we transfer API about `PartialEquiv.IsImage` to open partial homeomorphisms and\nadd a few `OpenPartialHomeomorph`-specific lemmas like `OpenPartialHomeomorph.IsImage.closure`.\n-/\n\n/-- We say that `t : Set Y` is an image of `s : Set X` under an open partial homeomorphism `e`\nif any of the following equivalent conditions hold:\n\n* `e '' (e.source ∩ s) = e.target ∩ t`;\n* `e.source ∩ e ⁻¹ t = e.source ∩ s`;\n* `∀ x ∈ e.source, e x ∈ t ↔ x ∈ s` (this one is used in the definition).\n-/\ndef IsImage (s : Set X) (t : Set Y) : Prop :=\n  ∀ ⦃x⦄, x ∈ e.source → (e x ∈ t ↔ x ∈ s)\n\nnamespace IsImage\n\nvariable {e} {s : Set X} {t : Set Y} {x : X} {y : Y}\n\ntheorem toPartialEquiv (h : e.IsImage s t) : e.toPartialEquiv.IsImage s t :=\n  h\n\ntheorem apply_mem_iff (h : e.IsImage s t) (hx : x ∈ e.source) : e x ∈ t ↔ x ∈ s :=\n  h hx\n\nprotected theorem symm (h : e.IsImage s t) : e.symm.IsImage t s :=\n  h.toPartialEquiv.symm\n\ntheorem symm_apply_mem_iff (h : e.IsImage s t) (hy : y ∈ e.target) : e.symm y ∈ s ↔ y ∈ t :=\n  h.symm hy\n\n@[simp]\ntheorem symm_iff : e.symm.IsImage t s ↔ e.IsImage s t :=\n  ⟨fun h => h.symm, fun h => h.symm⟩\n\nprotected theorem mapsTo (h : e.IsImage s t) : MapsTo e (e.source ∩ s) (e.target ∩ t) :=\n  h.toPartialEquiv.mapsTo\n\ntheorem symm_mapsTo (h : e.IsImage s t) : MapsTo e.symm (e.target ∩ t) (e.source ∩ s) :=\n  h.symm.mapsTo\n\ntheorem image_eq (h : e.IsImage s t) : e '' (e.source ∩ s) = e.target ∩ t :=\n  h.toPartialEquiv.image_eq\n\ntheorem symm_image_eq (h : e.IsImage s t) : e.symm '' (e.target ∩ t) = e.source ∩ s :=\n  h.symm.image_eq\n\ntheorem iff_preimage_eq : e.IsImage s t ↔ e.source ∩ e ⁻¹' t = e.source ∩ s :=\n  PartialEquiv.IsImage.iff_preimage_eq\n\nalias ⟨preimage_eq, of_preimage_eq⟩ := iff_preimage_eq\n\ntheorem iff_symm_preimage_eq : e.IsImage s t ↔ e.target ∩ e.symm ⁻¹' s = e.target ∩ t :=\n  symm_iff.symm.trans iff_preimage_eq\n\nalias ⟨symm_preimage_eq, of_symm_preimage_eq⟩ := iff_symm_preimage_eq\n\ntheorem iff_symm_preimage_eq' :\n    e.IsImage s t ↔ e.target ∩ e.symm ⁻¹' (e.source ∩ s) = e.target ∩ t := by\n  rw [iff_symm_preimage_eq, ← image_source_inter_eq, ← image_source_inter_eq']\n\nalias ⟨symm_preimage_eq', of_symm_preimage_eq'⟩ := iff_symm_preimage_eq'\n\ntheorem iff_preimage_eq' : e.IsImage s t ↔ e.source ∩ e ⁻¹' (e.target ∩ t) = e.source ∩ s :=\n  symm_iff.symm.trans iff_symm_preimage_eq'\n\nalias ⟨preimage_eq', of_preimage_eq'⟩ := iff_preimage_eq'\n\ntheorem of_image_eq (h : e '' (e.source ∩ s) = e.target ∩ t) : e.IsImage s t :=\n  PartialEquiv.IsImage.of_image_eq h\n\ntheorem of_symm_image_eq (h : e.symm '' (e.target ∩ t) = e.source ∩ s) : e.IsImage s t :=\n  PartialEquiv.IsImage.of_symm_image_eq h\n\nprotected theorem compl (h : e.IsImage s t) : e.IsImage sᶜ tᶜ := fun _ hx => (h hx).not\n\nprotected theorem inter {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :\n    e.IsImage (s ∩ s') (t ∩ t') := fun _ hx => (h hx).and (h' hx)\n\nprotected theorem union {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :\n    e.IsImage (s ∪ s') (t ∪ t') := fun _ hx => (h hx).or (h' hx)\n\nprotected theorem diff {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :\n    e.IsImage (s \\ s') (t \\ t') :=\n  h.inter h'.compl\n\ntheorem leftInvOn_piecewise {e' : OpenPartialHomeomorph X Y} [∀ i, Decidable (i ∈ s)]\n    [∀ i, Decidable (i ∈ t)] (h : e.IsImage s t) (h' : e'.IsImage s t) :\n    LeftInvOn (t.piecewise e.symm e'.symm) (s.piecewise e e') (s.ite e.source e'.source) :=\n  h.toPartialEquiv.leftInvOn_piecewise h'\n\ntheorem inter_eq_of_inter_eq_of_eqOn {e' : OpenPartialHomeomorph X Y} (h : e.IsImage s t)\n    (h' : e'.IsImage s t) (hs : e.source ∩ s = e'.source ∩ s) (Heq : EqOn e e' (e.source ∩ s)) :\n    e.target ∩ t = e'.target ∩ t :=\n  h.toPartialEquiv.inter_eq_of_inter_eq_of_eqOn h' hs Heq\n\ntheorem symm_eqOn_of_inter_eq_of_eqOn {e' : OpenPartialHomeomorph X Y} (h : e.IsImage s t)\n    (hs : e.source ∩ s = e'.source ∩ s) (Heq : EqOn e e' (e.source ∩ s)) :\n    EqOn e.symm e'.symm (e.target ∩ t) :=\n  h.toPartialEquiv.symm_eq_on_of_inter_eq_of_eqOn hs Heq\n\ntheorem map_nhdsWithin_eq (h : e.IsImage s t) (hx : x ∈ e.source) : map e (𝓝[s] x) = 𝓝[t] e x := by\n  rw [e.map_nhdsWithin_eq hx, h.image_eq, e.nhdsWithin_target_inter (e.map_source hx)]\n\nprotected theorem closure (h : e.IsImage s t) : e.IsImage (closure s) (closure t) := fun x hx => by\n  simp only [mem_closure_iff_nhdsWithin_neBot, ← h.map_nhdsWithin_eq hx, map_neBot_iff]\n\nprotected theorem interior (h : e.IsImage s t) : e.IsImage (interior s) (interior t) := by\n  simpa only [closure_compl, compl_compl] using h.compl.closure.compl\n\nprotected theorem frontier (h : e.IsImage s t) : e.IsImage (frontier s) (frontier t) :=\n  h.closure.diff h.interior\n\ntheorem isOpen_iff (h : e.IsImage s t) : IsOpen (e.source ∩ s) ↔ IsOpen (e.target ∩ t) :=\n  ⟨fun hs => h.symm_preimage_eq' ▸ e.symm.isOpen_inter_preimage hs, fun hs =>\n    h.preimage_eq' ▸ e.isOpen_inter_preimage hs⟩\n\n/-- Restrict an `OpenPartialHomeomorph` to a pair of corresponding open sets. -/\n@[simps! -fullyApplied apply symm_apply toPartialHomeomorph]\ndef restr (h : e.IsImage s t) (hs : IsOpen (e.source ∩ s)) : OpenPartialHomeomorph X Y where\n  toPartialEquiv := h.toPartialEquiv.restr\n  open_source := hs\n  open_target := h.isOpen_iff.1 hs\n  continuousOn_toFun := e.continuousOn.mono inter_subset_left\n  continuousOn_invFun := e.symm.continuousOn.mono inter_subset_left\n\nend IsImage\n\ntheorem isImage_source_target : e.IsImage e.source e.target :=\n  e.toPartialEquiv.isImage_source_target\n\ntheorem isImage_source_target_of_disjoint (e' : OpenPartialHomeomorph X Y)\n    (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) :\n    e.IsImage e'.source e'.target :=\n  e.toPartialEquiv.isImage_source_target_of_disjoint e'.toPartialEquiv hs ht\n\n/-- Preimage of interior or interior of preimage coincide for open partial homeomorphisms,\nwhen restricted to the source. -/\ntheorem preimage_interior (s : Set Y) :\n    e.source ∩ e ⁻¹' interior s = e.source ∩ interior (e ⁻¹' s) :=\n  (IsImage.of_preimage_eq rfl).interior.preimage_eq\n\ntheorem preimage_closure (s : Set Y) : e.source ∩ e ⁻¹' closure s = e.source ∩ closure (e ⁻¹' s) :=\n  (IsImage.of_preimage_eq rfl).closure.preimage_eq\n\ntheorem preimage_frontier (s : Set Y) :\n    e.source ∩ e ⁻¹' frontier s = e.source ∩ frontier (e ⁻¹' s) :=\n  (IsImage.of_preimage_eq rfl).frontier.preimage_eq\n\nend IsImage\n\n\nsection restrOpen\n/-!\n## Restriction\n-/\n\n/-- Restricting an open partial homeomorphism `e` to `e.source ∩ s` when `s` is open.\nThis is sometimes hard to use because of the openness assumption, but it has the advantage that\nwhen it can be used then its `PartialEquiv` is defeq to `PartialEquiv.restr`. -/\nprotected def restrOpen (s : Set X) (hs : IsOpen s) : OpenPartialHomeomorph X Y :=\n  (@IsImage.of_symm_preimage_eq X Y _ _ e s (e.symm ⁻¹' s) rfl).restr\n    (IsOpen.inter e.open_source hs)\n\n@[simp, mfld_simps]\ntheorem restrOpen_toPartialEquiv (s : Set X) (hs : IsOpen s) :\n    (e.restrOpen s hs).toPartialEquiv = e.toPartialEquiv.restr s :=\n  rfl\n\n-- Already simp via `PartialEquiv`\ntheorem restrOpen_source (s : Set X) (hs : IsOpen s) : (e.restrOpen s hs).source = e.source ∩ s :=\n  rfl\n\n@[simp] theorem coe_restrOpen {s : Set X} (hs : IsOpen s) : ⇑(e.restrOpen s hs) = e := rfl\n\n@[simp]\ntheorem coe_restrOpen_symm {s : Set X} (hs : IsOpen s) : ⇑(e.restrOpen s hs).symm = e.symm := rfl\n\n/-- Restricting an open partial homeomorphism `e` to `e.source ∩ interior s`. We use the interior to\nmake sure that the restriction is well defined whatever the set s, since open partial homeomorphisms\nare by definition defined on open sets. In applications where `s` is open, this coincides with the\nrestriction of partial equivalences. -/\n@[simps! (attr := mfld_simps) -fullyApplied apply symm_apply,\n  simps! (attr := grind =) -isSimp source target]\nprotected def restr (s : Set X) : OpenPartialHomeomorph X Y :=\n  e.restrOpen (interior s) isOpen_interior\n\n@[simp, mfld_simps]\ntheorem restr_toPartialEquiv (s : Set X) :\n    (e.restr s).toPartialEquiv = e.toPartialEquiv.restr (interior s) :=\n  rfl\n\ntheorem restr_source' (s : Set X) (hs : IsOpen s) : (e.restr s).source = e.source ∩ s := by\n  grind\n\ntheorem restr_toPartialEquiv' (s : Set X) (hs : IsOpen s) :\n    (e.restr s).toPartialEquiv = e.toPartialEquiv.restr s := by\n  rw [e.restr_toPartialEquiv, hs.interior_eq]\n\ntheorem restr_eq_of_source_subset {e : OpenPartialHomeomorph X Y} {s : Set X} (h : e.source ⊆ s) :\n    e.restr s = e :=\n  toPartialEquiv_injective <| PartialEquiv.restr_eq_of_source_subset <|\n    interior_maximal h e.open_source\n\n@[simp, mfld_simps]\ntheorem restr_univ {e : OpenPartialHomeomorph X Y} : e.restr univ = e :=\n  restr_eq_of_source_subset (subset_univ _)\n\n@[simp, grind =]\n\nTarget:\ntheorem restr_source_inter (s : Set X) : e.restr (e.source ∩ s) = e.restr s :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_a3683d3fecf5","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"9c92b0b7255664f814ece45769a71af6937ed11bd12e371f8574781d60d32251","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/OpenPartialHomeomorph","family_id":"restr_source_inter","file_id":"mathlib/Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean","sample_id":"a3683d3fecf5a78a2e088915121e34b1d7028592a27520975260840e3084875a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"65d184d09b9f9abbbd2d10bfaea51c9a0bf698460f9d72b6fbec0c7ce1a8ad5d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2b26d3e25a280904d8577d75d632bf5a13763ff21493b59168968dcb1e0a5669","source_sha256":"f94f55c4ba2322bc4c471438f5b24643adb8c39a2c5e74b34a6830fdd6741d84","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction n with\n  | zero => rfl\n  | succ n ih =>\n    rw [replicate_succ]\n    conv => rhs; rw [replicate_succ]\n    rw [snoc_cons, ih]","hard_negative":false,"metrics":{"chosen_tokens":31,"rejected_tokens":3,"token_jaccard":0.047619,"token_length_ratio":0.096774},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"a95134a1fcf39e2d807349b5f66ca5156122aedc5ed47172b8840365e622ddab","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Vector.Basic\n\nNamespace:\nList.Vector\n\nLocal context:\n/-\nCopyright (c) 2023 Alex Keizer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alex Keizer\n-/\n/-!\n  This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single\n  element to the back of a vector.\n\n  It provides a collection of lemmas that show how different `Vector` operations reduce when their\n  argument is `snoc xs x`.\n\n  Also, an alternative, reverse, induction principle is added, that breaks down a vector into\n  `snoc xs x` for its inductive case. Effectively doing induction from right-to-left\n-/\n\n@[expose] public section\n\nnamespace List\n\nnamespace Vector\n\nvariable {α β σ φ : Type*} {n : ℕ} {x : α} {s : σ} (xs : Vector α n)\n\n/-- Append a single element to the end of a vector -/\ndef snoc : Vector α n → α → Vector α (n + 1) :=\n  fun xs x => xs ++ x ::ᵥ Vector.nil\n\n/-! ## Simplification lemmas -/\n\nsection Simp\n\nvariable {y : α}\n\n@[simp]\ntheorem snoc_cons : (x ::ᵥ xs).snoc y = x ::ᵥ (xs.snoc y) :=\n  rfl\n\n@[simp]\ntheorem snoc_nil : (nil.snoc x) = x ::ᵥ nil :=\n  rfl\n\n@[simp]\ntheorem reverse_cons : reverse (x ::ᵥ xs) = (reverse xs).snoc x := by\n  cases xs\n  simp only [reverse, cons, toList_mk, List.reverse_cons, snoc]\n  congr\n\n@[simp]\ntheorem reverse_snoc : reverse (xs.snoc x) = x ::ᵥ (reverse xs) := by\n  cases xs\n  simp only [reverse, snoc, cons, toList_mk]\n  congr\n  simp [toList, append_def]\n\nTarget:\ntheorem replicate_succ_to_snoc (val : α) :\n    replicate (n + 1) val = (replicate n val).snoc val :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Vector","family_id":"replicate_succ_to_snoc","file_id":"mathlib/Mathlib/Data/Vector/Snoc.lean","sample_id":"2b26d3e25a280904d8577d75d632bf5a13763ff21493b59168968dcb1e0a5669"},"split":"train"}
{"binding":{"chosen_proof_sha256":"88c077efecbeeb27504c4875f1630a5efd1eaa0f4e0d6e798f302270b541c3d4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"72da71e251950deea88fc047accda33cc2ed86ab5ee18076c9caf8bc847bc42e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5594b1e027625be77c21574d7888f7501782d701404f92b08cf130a4ddbabfb7","source_sha256":"58fa9eef28170e41707520794ca2c1648cebed0e86fef9c10333ef5d54e6c390","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rw [norm_apply, LinearMap.det]; split_ifs <;> trivial\n\nvariable {R}","hard_negative":false,"metrics":{"chosen_tokens":19,"rejected_tokens":26,"token_jaccard":0.9,"token_length_ratio":1.368421},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"a9a50f63d9d475fe67e33ac1dba2dbd60b957dc2eecbc0bcf26bd1f79574dfe9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Determinant\n\nNamespace:\nAlgebra\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Norm for (finite) ring extensions\n\nSuppose we have an `R`-algebra `S` with a finite basis. For each `s : S`,\nthe determinant of the linear map given by multiplying by `s` gives information\nabout the roots of the minimal polynomial of `s` over `R`.\n\n## Implementation notes\n\nTypically, the norm is defined specifically for finite field extensions.\nThe current definition is as general as possible and the assumption that we have\nfields or that the extension is finite is added to the lemmas as needed.\n\nWe only define the norm for left multiplication (`Algebra.leftMulMatrix`,\ni.e. `LinearMap.mulLeft`).\nFor now, the definitions assume `S` is commutative, so the choice doesn't\nmatter anyway.\n\nSee also `Algebra.trace`, which is defined similarly as the trace of\n`Algebra.leftMulMatrix`.\n\n## References\n\n* https://en.wikipedia.org/wiki/Field_norm\n\n-/\n\n@[expose] public section\n\n\nuniverse u v w\n\nvariable {R S : Type*} [CommRing R] [Ring S]\nvariable [Algebra R S]\nvariable {K : Type*} [Field K]\nvariable {ι : Type w}\n\nopen Module\n\nopen LinearMap\n\nopen Matrix Polynomial\n\nopen scoped Matrix\n\nnamespace Algebra\n\nvariable (R)\n\n/-- The norm of an element `s` of an `R`-algebra is the determinant of `(*) s`. -/\n@[stacks 0BIF \"Norm\"]\nnoncomputable def norm : S →* R :=\n  LinearMap.det.comp (lmul R S).toRingHom.toMonoidHom\n\ntheorem norm_apply (x : S) : norm R x = LinearMap.det (lmul R S x) := rfl\n\n@[simp]\ntheorem norm_self : Algebra.norm R = MonoidHom.id R := by\n  ext\n  simp [norm_apply]\n\nTarget:\ntheorem norm_eq_one_of_not_exists_basis (h : ¬∃ s : Finset S, Nonempty (Basis s R S)) (x : S) :\n    norm R x = 1 :=\n\nProof body:\n","rejected":"```lean\nby rw [norm_apply, LinearMap.det]; split_ifs <;> trivial\n\nvariable {R}\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Norm","family_id":"norm_eq_one_of_not_exists_basis","file_id":"mathlib/Mathlib/RingTheory/Norm/Defs.lean","sample_id":"5594b1e027625be77c21574d7888f7501782d701404f92b08cf130a4ddbabfb7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6296f9ab27c166fefa518a334c1d06c276278c6742ae2b92e22ac06c24c52565","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c3d5b7f22e02615e5f628e9cf5747323821ccde022154296e79b921c69ff62e8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c88d0c7a08db8e0ee629366fac952f34c781e69b50a4902cc54b66d0c91187ae","source_sha256":"20106a17e82d1783ea6f9971b872fe8ba3aa3b18f09bbcdf6c093b13edf6c381","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  conv_rhs => rw [← sum_single a, ← sum_single b]\n  simp_rw [mul_def, map_finsuppSum, liftNC_single, Finsupp.sum_mul, Finsupp.mul_sum]\n  refine Finset.sum_congr rfl fun y hy => Finset.sum_congr rfl fun x _hx => ?_\n  simp [mul_assoc, (h_comm hy).left_comm]","hard_negative":false,"metrics":{"chosen_tokens":59,"rejected_tokens":64,"token_jaccard":0.9,"token_length_ratio":1.084746},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"a9bb9890fc5dab9ccfa25e726a57d03ab2a4a606082c1ffceff6378c0fffa565","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MonoidAlgebra.Defs\n\nNamespace:\nMonoidAlgebra\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Yury Kudryashov, Kim Morrison\n-/\n/-!\n# Lifting monoid algebras\n\nThis file defines `liftNC`. For the definition of `MonoidAlgebra.lift`, see\n`Mathlib/Algebra/MonoidAlgebra/Basic.lean`.\n\n## Main results\n* `MonoidAlgebra.liftNC`, `AddMonoidAlgebra.liftNC`: lift a homomorphism `f : k →+ R` and a\n  function `g : G → R` to a homomorphism `k[G] →+ R`.\n-/\n\n@[expose] public section\n\nassert_not_exists NonUnitalAlgHom AlgEquiv\n\nnoncomputable section\n\nopen Finsupp hiding single\n\nuniverse u₁ u₂ u₃ u₄\n\nvariable (k : Type u₁) (G : Type u₂) (H : Type*) {R S T M : Type*}\n\n/-! ### Multiplicative monoids -/\n\nnamespace MonoidAlgebra\n\nvariable {k G}\n\nsection\n\nvariable [Semiring k] [NonUnitalNonAssocSemiring R]\n\n/-- A non-commutative version of `MonoidAlgebra.lift`: given an additive homomorphism `f : k →+ R`\nand a homomorphism `g : G → R`, returns the additive homomorphism from\n`k[G]` such that `liftNC f g (single a b) = f b * g a`. If `f` is a ring homomorphism\nand the range of either `f` or `g` is in center of `R`, then the result is a ring homomorphism.  If\n`R` is a `k`-algebra and `f = algebraMap k R`, then the result is an algebra homomorphism called\n`MonoidAlgebra.lift`. -/\ndef liftNC (f : k →+ R) (g : G → R) : k[G] →+ R :=\n  liftAddHom fun x : G => (AddMonoidHom.mulRight (g x)).comp f\n\n@[simp]\ntheorem liftNC_single (f : k →+ R) (g : G → R) (a : G) (b : k) :\n    liftNC f g (single a b) = f b * g a :=\n  liftAddHom_apply_single _ _ _\n\nend\n\nsection Mul\n\nvariable [Semiring k] [Mul G] [Semiring R]\n\nTarget:\ntheorem liftNC_mul {g_hom : Type*} [FunLike g_hom G R] [MulHomClass g_hom G R]\n    (f : k →+* R) (g : g_hom) (a b : k[G])\n    (h_comm : ∀ {x y}, y ∈ a.support → Commute (f (b x)) (g y)) :\n    liftNC (f : k →+ R) g (a * b) = liftNC (f : k →+ R) g a * liftNC (f : k →+ R) g b :=\n\nProof body:\n","rejected":"by\n  conv_rhs => rw [← sum_single a, ← sum_single b]\n  simp_rw [mul_def, map_finsuppSum, liftNC_single, Finsupp.sum_mul, Finsupp.mul_sum]\n  refine Finset.sum_congr rfl fun y hy => Finset.sum_congr rfl fun x _hx => ?_\n  simp [mul_assoc, (h_comm hy).left_comm]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/MonoidAlgebra","family_id":"liftnc_mul","file_id":"mathlib/Mathlib/Algebra/MonoidAlgebra/Lift.lean","sample_id":"c88d0c7a08db8e0ee629366fac952f34c781e69b50a4902cc54b66d0c91187ae"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6f31eac0331d8119b977de849699fd5e0d1fb2f77898bc029e261ff94821e033","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"59450a4ffd61d7699b103e7b5e0a8e1339913115bd823f4c4a464a5a458f6158","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4793854c2e3af40b9d027f7b421519fea77582f2694be3b8462681fa81f0c2f6","source_sha256":"00cdc131b09c868f94179716d61273d8fdf897ed023d34f30c7085668a3b31ac","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by simpa only [zero_mul, zero_add] using div_add_mod a 0","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.25},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"a9d5f83fd593493db2947f918eb2ac5cb32a0c20121037786a30ed78ae1cde3b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Order.RelClasses\n\nNamespace:\nEuclideanDomain\n\nLocal context:\n/-\nCopyright (c) 2018 Louis Carlin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Louis Carlin, Mario Carneiro\n-/\n/-!\n# Euclidean domains\n\nThis file introduces Euclidean domains and provides the extended Euclidean algorithm. To be precise,\na slightly more general version is provided which is sometimes called a transfinite Euclidean domain\nand differs in the fact that the degree function need not take values in `ℕ` but can take values in\nany well-ordered set. Transfinite Euclidean domains were introduced by Motzkin and examples which\ndon't satisfy the classical notion were provided independently by Hiblot and Nagata.\n\n## Main definitions\n\n* `EuclideanDomain`: Defines Euclidean domain with functions `quotient` and `remainder`. Instances\n  of `Div` and `Mod` are provided, so that one can write `a = b * (a / b) + a % b`.\n* `gcd`: defines the greatest common divisors of two elements of a Euclidean domain.\n* `xgcd`: given two elements `a b : R`, `xgcd a b` defines the pair `(x, y)` such that\n  `x * a + y * b = gcd a b`.\n* `lcm`: defines the lowest common multiple of two elements `a` and `b` of a Euclidean domain as\n  `a * b / (gcd a b)`\n\n## Main statements\n\nSee `Algebra.EuclideanDomain.Basic` for most of the theorems about Euclidean domains,\nincluding Bézout's lemma.\n\nSee `Algebra.EuclideanDomain.Instances` for the fact that `ℤ` is a Euclidean domain,\nas is any field.\n\n## Notation\n\n`≺` denotes the well-founded relation on the Euclidean domain, e.g. in the example of the polynomial\nring over a field, `p ≺ q` for polynomials `p` and `q` if and only if the degree of `p` is less than\nthe degree of `q`.\n\n## Implementation details\n\nInstead of working with a valuation, `EuclideanDomain` is implemented with the existence of a well\nfounded relation `r` on the integral domain `R`, which in the example of `ℤ` would correspond to\nsetting `i ≺ j` for integers `i` and `j` if the absolute value of `i` is smaller than the absolute\nvalue of `j`.\n\n## References\n\n* [Th. Motzkin, *The Euclidean algorithm*][MR32592]\n* [J.-J. Hiblot, *Des anneaux euclidiens dont le plus petit algorithme n'est pas à valeurs finies*]\n  [MR399081]\n* [M. Nagata, *On Euclid algorithm*][MR541021]\n\n\n## Tags\n\nEuclidean domain, transfinite Euclidean domain, Bézout's lemma\n-/\n\n@[expose] public section\n\nuniverse u\n\n/-- A `EuclideanDomain` is a non-trivial commutative ring with a division and a remainder,\n  satisfying `b * (a / b) + a % b = a`.\n  The definition of a Euclidean domain usually includes a valuation function `R → ℕ`.\n  This definition is slightly generalised to include a well-founded relation\n  `r` with the property that `r (a % b) b`, instead of a valuation. -/\n@[wikidata Q867345]\nclass EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R where\n  /-- A division function (denoted `/`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `%` denotes `remainder`. -/\n  protected quotient : R → R → R\n  /-- Division by zero should always give zero by convention. -/\n  protected quotient_zero : ∀ a, quotient a 0 = 0\n  /-- A remainder function (denoted `%`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `/` denotes `quotient`. -/\n  protected remainder : R → R → R\n  /-- The property that links the quotient and remainder functions.\n    This allows us to compute GCDs and LCMs. -/\n  protected quotient_mul_add_remainder_eq : ∀ a b, b * quotient a b + remainder a b = a\n  /-- A well-founded relation on `R`, satisfying `r (a % b) b`.\n    This ensures that the GCD algorithm always terminates. -/\n  protected r : R → R → Prop\n  /-- The relation `r` must be well-founded.\n    This ensures that the GCD algorithm always terminates. -/\n  r_wellFounded : WellFounded r\n  /-- The relation `r` satisfies `r (a % b) b`. -/\n  protected remainder_lt : ∀ (a) {b}, b ≠ 0 → r (remainder a b) b\n  /-- An additional constraint on `r`. -/\n  mul_left_not_lt : ∀ (a) {b}, b ≠ 0 → ¬r (a * b) a\n\n/-\nLean has far more theorems about fields than about Euclidean domains. We thus\nlower the priority of `Euclideandomain.toCommRing`, encouraging typeclass inference\nto try `Field.toCommRing` first. Without this priority-lowering, typeclass inference\nfinds the more inefficient path `Field.toEuclideanDomain.toCommRing` by default. This\npriority change saves over 500G instructions across mathlib. See\nhttps://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/We.20need.20to.20talk.20about.20Euclidean.20Domains/near/594655420\n-/\n-- see Note [lower instance priority]\nattribute [instance 100] EuclideanDomain.toCommRing\n\nnamespace EuclideanDomain\n\nvariable {R : Type u} [EuclideanDomain R]\n\n/-- Abbreviated notation for the well-founded relation `r` in a Euclidean domain. -/\nlocal infixl:50 \" ≺ \" => EuclideanDomain.r\n\nlocal instance wellFoundedRelation : WellFoundedRelation R where\n  rel := EuclideanDomain.r\n  wf := r_wellFounded\n\ninstance isWellFounded : IsWellFounded R (· ≺ ·) where\n  wf := r_wellFounded\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Div R :=\n  ⟨EuclideanDomain.quotient⟩\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Mod R :=\n  ⟨EuclideanDomain.remainder⟩\n\ntheorem div_add_mod (a b : R) : b * (a / b) + a % b = a :=\n  EuclideanDomain.quotient_mul_add_remainder_eq _ _\n\ntheorem mod_add_div (a b : R) : a % b + b * (a / b) = a :=\n  (add_comm _ _).trans (div_add_mod _ _)\n\ntheorem mod_add_div' (m k : R) : m % k + m / k * k = m := by\n  rw [mul_comm]\n  exact mod_add_div _ _\n\ntheorem div_add_mod' (m k : R) : m / k * k + m % k = m := by\n  rw [mul_comm]\n  exact div_add_mod _ _\n\ntheorem mod_lt : ∀ (a) {b : R}, b ≠ 0 → a % b ≺ b :=\n  EuclideanDomain.remainder_lt\n\ntheorem mul_right_not_lt {a : R} (b) (h : a ≠ 0) : ¬a * b ≺ b := by\n  rw [mul_comm]\n  exact mul_left_not_lt b h\n\n@[simp]\n\nTarget:\ntheorem mod_zero (a : R) : a % 0 = a :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_4793854c2e3a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"6193c093694f86b6eb2c5aa1536248f20666edd7902c1a654ace2dafabda826b","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/EuclideanDomain","family_id":"mod_zero","file_id":"mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean","sample_id":"4793854c2e3af40b9d027f7b421519fea77582f2694be3b8462681fa81f0c2f6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bdd86c88ef7416db2f30c34d0d939f5c26ce704f80d0007fe2f340c238437f0f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e8c9dc5f379d2760fb24d04c1fd1b200b12c9f621f4607d9bb2924f7901f72b2","source_sha256":"42683ce9ec3a96f77aab204680a294682617adafa7d756b1f3dbef8d091d30cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine IsPullback.of_forall_isPullback_app fun X => ?_\n  rw [Types.isPullback_iff]\n  refine ⟨congr(($(comp_χ_eq m)).app X), ?_, ?_⟩\n  · simpa using! (mono_iff_injective (m.app X)).mp (inferInstance)\n  · simp only [Functor.const_obj_obj, Functor.sieves_obj, χ_app, Opposite.op_unop,\n      TypeCat.hom_ofHom, TypeCat.Fun.coe_mk, truth_app, Functor.isTerminalConst_from_app,\n      Types.isTerminalPUnit_from_apply, and_true, forall_const]\n    intro p hp\n    simpa [eq_comm] using! congr($(hp).arrows (𝟙 _))","hard_negative":false,"metrics":{"chosen_tokens":123,"rejected_tokens":2,"token_jaccard":0.017241,"token_length_ratio":0.01626},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"aa2527ee65976eebd73c63bb68dd0a3397e360ff95348059ffef0964d55e0f52","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Sites.Closed\npublic import Mathlib.CategoryTheory.Sites.Equivalence\npublic import Mathlib.CategoryTheory.Subobject.Classifier.Defs\npublic import Mathlib.CategoryTheory.Subfunctor.Image\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Edward van de Meent. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Edward van de Meent\n-/\n/-!\n\n# (Elementary) Sheaf Topos\n\nWe define a subobject classifier for categories of sheaves of (large enough) types.\n\n## Main definitions\n\nLet `C` refer to a category with (when relevant) Grothendieck topology `J`.\n\n* `Presheaf.classifier C` is a construction of a subobject classifier in `Cᵒᵖ ⥤ Type (max u v)`.\n* `Sheaf.classifier J` is a construction of a subobject classifier in `Sheaf J (Type (max u v))`.\n* `inferInstance : HasClassifier (Cᵒᵖ ⥤ Type w)` says that `Cᵒᵖ ⥤ Type w` has a subobject\n  classifier if `C` is `w`-essentially small.\n* `inferInstance : HasClassifier (Sheaf J (Type w))` says that `Sheaf J (Type w)` has a\n  subobject classifier if `C` is `w`-essentially small.\n\n## Main results\n\n* Any category of sheaves of types has a subobject classifier if the site is essentially small.\n* As a consequence, (because categories of sheaves are cartesian monoidal and have finite limits,)\n  such categories are Elementary Topoi.\n\n## TODOS:\n\n* generalize `Presheaf.isClosed_χ_app_apply_of` to only assuming `G` is separated\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nnamespace CategoryTheory\n\nvariable {C : Type u} [Category.{v} C]\nopen Limits\n\nsection presheaf\n\nvariable (C) in\n/-- The truth morphism in the category of presheaves. At each component `X : C`, it is the constant\nmap returning `⊤ : Sieve X`. -/\n@[simps]\ndef Presheaf.truth : (Functor.const _).obj PUnit ⟶ Functor.sieves C where\n  app X := ↾fun _ ↦ (⊤ : Sieve X.unop)\n\nvariable {F G : Cᵒᵖ ⥤ Type (max u v)}\n\nset_option backward.defeqAttrib.useBackward true in\n/--\nThe characteristic map of an inclusion of presheaves.\nGiven a monomorphism of sheaves `m : F ⟶ G`, an object X of the site, map an element `x : G(X)`\nto the (closed) sieve on X where `f : Y → X` is in the sieve iff\n  `∃ a ∈ F(Y), G(f)(x) = m_Y(a)`\n-/\n@[simps app]\ndef Presheaf.χ (m : F ⟶ G) : G ⟶ Functor.sieves C where\n  app X := ↾fun x ↦ ⟨fun Y f => ∃ a, G.map f.op x = m.app (.op Y) a, by\n    intro Y Z f ⟨a, ha⟩ g\n    use F.map g.op a\n    simp [ha, NatTrans.naturality_apply]⟩\n\nlemma Presheaf.comp_χ_eq (m : F ⟶ G) : m ≫ Presheaf.χ m =\n    (Functor.isTerminalConst _ Types.isTerminalPUnit).from F ≫ Presheaf.truth C := by\n  ext\n  apply Sieve.ext\n  simp [← NatTrans.naturality_apply]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma Presheaf.isPullback_χ_truth (m : F ⟶ G) [Mono m] :\n    IsPullback m ((Functor.isTerminalConst _ Types.isTerminalPUnit).from F) (χ m) (truth C) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Topos","family_id":"presheaf","file_id":"mathlib/Mathlib/CategoryTheory/Topos/Sheaf.lean","sample_id":"e8c9dc5f379d2760fb24d04c1fd1b200b12c9f621f4607d9bb2924f7901f72b2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8ea2eb314aa329962f79f410c370fc03bc8d9d4238b02a2baace82f999396bf2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f9ac29b8a38906cebe8f0ae27b17348c7dcfa9da287ce6b53ff919e6dfcd5785","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7b5b722d1bc16ae3899d329eba58a7b2d1599243b0ea9acc8a97cda675c28cc8","source_sha256":"34049d18cb594f0a184df593dbd6d4fba0ade793ef81827c21fb1d80f135b5c5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by simp [left_distrib, *]","hard_negative":true,"metrics":{"chosen_tokens":7,"rejected_tokens":2,"token_jaccard":0.125,"token_length_ratio":0.285714},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"aa627b854ccae9aab3d94d37d054ab7ff50bde06c5a52a58fdb592a5a786876f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Basic\npublic import Mathlib.Algebra.Order.Ring.Defs\npublic import Mathlib.Data.Tree.Basic\npublic import Mathlib.Tactic.NormNum.Core\npublic import Mathlib.Util.SynthesizeUsing\n\nNamespace:\nMathlib.Tactic.CancelDenoms\n\nLocal context:\n/-\nCopyright (c) 2020 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n-/\npublic meta import Mathlib.Data.Tree.Basic\npublic meta import Mathlib.Algebra.Group.Nat.Defs\n\n/-!\n# A tactic for canceling numeric denominators\n\nThis file defines tactics that cancel numeric denominators from field Expressions.\n\nAs an example, we want to transform a comparison `5*(a/3 + b/4) < c/3` into the equivalent\n`5*(4*a + 3*b) < 4*c`.\n\n## Implementation notes\n\nThe tooling here was originally written for `linarith`, not intended as an interactive tactic.\nThe interactive version has been split off because it is sometimes convenient to use on its own.\nThere are likely some rough edges to it.\n\nImproving this tactic would be a good project for someone interested in learning tactic programming.\n-/\n\npublic meta section\n\nopen Lean Parser Tactic Mathlib Meta NormNum Qq\n\ninitialize registerTraceClass `CancelDenoms\n\nnamespace Mathlib.Tactic.CancelDenoms\n\n/-! ### Lemmas used in the procedure -/\n\ntheorem mul_subst {α} [CommRing α] {n1 n2 k e1 e2 t1 t2 : α}\n    (h1 : n1 * e1 = t1) (h2 : n2 * e2 = t2) (h3 : n1 * n2 = k) : k * (e1 * e2) = t1 * t2 := by\n  rw [← h3, mul_comm n1, mul_assoc n2, ← mul_assoc n1, h1,\n      ← mul_assoc n2, mul_comm n2, mul_assoc, h2]\n\ntheorem div_subst {α} [Field α] {n1 n2 k e1 e2 t1 : α}\n    (h1 : n1 * e1 = t1) (h2 : n2 / e2 = 1) (h3 : n1 * n2 = k) : k * (e1 / e2) = t1 := by\n  rw [← h3, mul_assoc, mul_div_left_comm, h2, ← mul_assoc, h1, mul_comm, one_mul]\n\ntheorem cancel_factors_eq_div {α} [Field α] {n e e' : α}\n    (h : n * e = e') (h2 : n ≠ 0) : e = e' / n :=\n  eq_div_of_mul_eq h2 <| by rwa [mul_comm] at h\n\nTarget:\ntheorem add_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :\n    n * (e1 + e2) = t1 + t2 :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_7b5b722d1bc1","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"68eabacc3e463870b47a1289e8507d28484f649756e74ba239131c2a3e597108","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/CancelDenoms","family_id":"add_subst","file_id":"mathlib/Mathlib/Tactic/CancelDenoms/Core.lean","sample_id":"7b5b722d1bc16ae3899d329eba58a7b2d1599243b0ea9acc8a97cda675c28cc8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5fc5fcc6ed6270b255f9132aa985cf7070d173e497c862b80ffbe80e51562de8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5db3bcdbdea6a829d6f37e0e356b2ab39fb7f30061303a2b8a97d9076fba0b9e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0f82b3064f5d92a16f748b7b934596ecec0fdc7677d78faf22eb29829f183996","source_sha256":"dbc79381fe985a1a450b97d09f056dd0638cc629a12da566daa44070340e962c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [Set.Sized, Set.mem_iUnion, forall_exists_index]\n  exact forall_comm\n\n-- `simp` normal form is `sized_iUnion`.","hard_negative":false,"metrics":{"chosen_tokens":27,"rejected_tokens":34,"token_jaccard":0.95,"token_length_ratio":1.259259},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"aa9ab3899a840f64b1d4246d99ddd70e0994c2514a99a39f70a20f39033fa0a2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Order.Interval.Finset.Nat\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nNamespace:\nSet\n\nLocal context:\n/-\nCopyright (c) 2021 Bhavik Mehta, Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Alena Gusakov, Yaël Dillies\n-/\n/-!\n# `r`-sets and slice\n\nThis file defines the `r`-th slice of a set family and provides a way to say that a set family is\nmade of `r`-sets.\n\nAn `r`-set is a finset of cardinality `r` (aka of *size* `r`). The `r`-th slice of a set family is\nthe set family made of its `r`-sets.\n\n## Main declarations\n\n* `Set.Sized`: `A.Sized r` means that `A` only contains `r`-sets.\n* `Finset.slice`: `A.slice r` is the set of `r`-sets in `A`.\n\n## Notation\n\n`A # r` is notation for `A.slice r` in scope `finset_family`.\n-/\n\n@[expose] public section\n\n\nopen Finset Nat\n\nvariable {α : Type*} {ι : Sort*} {κ : ι → Sort*}\n\nnamespace Set\n\nvariable {A B : Set (Finset α)} {s : Finset α} {r : ℕ}\n\n/-! ### Families of `r`-sets -/\n\n\n/-- `Sized r A` means that every Finset in `A` has size `r`. -/\ndef Sized (r : ℕ) (A : Set (Finset α)) : Prop := ∀ ⦃x⦄, x ∈ A → #x = r\n\ntheorem Sized.mono (h : A ⊆ B) (hB : B.Sized r) : A.Sized r := fun _x hx => hB <| h hx\n\n@[simp] lemma sized_empty : (∅ : Set (Finset α)).Sized r := by simp [Sized]\n@[simp] lemma sized_singleton : ({s} : Set (Finset α)).Sized r ↔ #s = r := by simp [Sized]\n\ntheorem sized_union : (A ∪ B).Sized r ↔ A.Sized r ∧ B.Sized r :=\n  ⟨fun hA => ⟨hA.mono subset_union_left, hA.mono subset_union_right⟩, fun hA _x hx =>\n    hx.elim (fun h => hA.1 h) fun h => hA.2 h⟩\n\nalias ⟨_, sized.union⟩ := sized_union\n\n--TODO: A `forall_iUnion` lemma would be handy here.\n@[simp]\n\nTarget:\ntheorem sized_iUnion {f : ι → Set (Finset α)} : (⋃ i, f i).Sized r ↔ ∀ i, (f i).Sized r :=\n\nProof body:\n","rejected":"```lean\nby\n  simp_rw [Set.Sized, Set.mem_iUnion, forall_exists_index]\n  exact forall_comm\n\n-- `simp` normal form is `sized_iUnion`.\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finset","family_id":"sized_iunion","file_id":"mathlib/Mathlib/Data/Finset/Slice.lean","sample_id":"0f82b3064f5d92a16f748b7b934596ecec0fdc7677d78faf22eb29829f183996"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8cf4a63a26dc0e5b281fd2da3d3e4050e7d2b711edcf9188ce1a11ef8a2b0753","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"07ea5168f4a5f0cbe84ebdfb15ab18e8497cb981fc02bf407291fbbaa4492325","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4814cbf7f3508d036f5e5a6a7155650097bbec6785ba9dace923119e89bbbe59","source_sha256":"bc1107b51b023f15ba6b116e921d72825e5fd28811d110becc324cfa9ebc2fb7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold mkFinCons\n  exact coe_mk (v := Fin.cons y (N.subtype ∘ b)) _ _\n\n/-- Let `b` be a basis for a submodule `N ≤ O`. If `y ∈ O` is linear independent of `N`\nand `y` and `N` together span the whole of `O`, then there is a basis for `O`\nwhose basis vectors are given by `Fin.cons y b`. -/\nnoncomputable def mkFinConsOfLE {n : ℕ} {N O : Submodule R M} (y : M) (yO : y ∈ O)\n    (b : Basis (Fin n) R N) (hNO : N ≤ O) (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0)\n    (hsp : ∀ z ∈ O, ∃ c : R, z + c • y ∈ N) : Basis (Fin (n + 1)) R O :=\n  mkFinCons ⟨y, yO⟩ (b.map (Submodule.comapSubtypeEquivOfLe hNO).symm)\n    (fun c x hc hx => hli c x (Submodule.mem_comap.mp hc) (congr_arg ((↑) : O → M) hx))\n    fun z => hsp z z.2","hard_negative":true,"metrics":{"chosen_tokens":262,"rejected_tokens":3,"token_jaccard":0.022472,"token_length_ratio":0.01145},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"ab59a70b514264ba51cc2251127873feceec15be7021f74105cd74dc15d24c4f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Basis.Basic\npublic import Mathlib.LinearAlgebra.Pi\n\nNamespace:\nModule.Basis\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Kevin H. Wilson\n-/\n/-!\n# Bases indexed by `Fin`\n-/\n\n@[expose] public section\n\nassert_not_exists Ordinal\n\nnoncomputable section\n\nuniverse u\n\nopen Function Set Submodule Finsupp\n\nvariable {ι : Type*} {ι' : Type*} {R : Type*} {R₂ : Type*} {M : Type*} {M' : Type*}\n\nnamespace Module\n\nopen LinearMap\n\nvariable {v : ι → M}\nvariable [Ring R] [CommRing R₂] [AddCommGroup M]\nvariable [Module R M] [Module R₂ M]\nvariable {x y : M}\nvariable (b : Basis ι R M)\n\nnamespace Basis\n\nsection Fin\n\n/-- Let `b` be a basis for a submodule `N` of `M`. If `y : M` is linear independent of `N`\nand `y` and `N` together span the whole of `M`, then there is a basis for `M`\nwhose basis vectors are given by `Fin.cons y b`. -/\nnoncomputable def mkFinCons {n : ℕ} {N : Submodule R M} (y : M) (b : Basis (Fin n) R N)\n    (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :\n    Basis (Fin (n + 1)) R M :=\n  have span_b : N = Submodule.span R (Set.range (N.subtype ∘ b)) := by\n    rw [Set.range_comp, Submodule.span_image, b.span_eq, Submodule.map_subtype_top]\n  Basis.mk (v := Fin.cons y (N.subtype ∘ b))\n    ((b.linearIndependent.map' N.subtype (Submodule.ker_subtype _)).finCons' _ _\n      (by\n        intro c x hx hc\n        rw [← span_b] at hx\n        exact hli c x hx hc))\n    fun x _ => by simpa [Submodule.mem_span_insert', span_b] using hsp x\n\n@[simp]\n\nTarget:\ntheorem coe_mkFinCons {n : ℕ} {N : Submodule R M} (y : M) (b : Basis (Fin n) R N)\n    (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :\n    (mkFinCons y b hli hsp : Fin (n + 1) → M) = Fin.cons y ((↑) ∘ b) :=\n\nProof body:\n","rejected":"by\n  exact coe_mkFinCons","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"4a7764f3ab31d965bf33dd6357092b657777d0754e02c92947f328cd62b225c8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Basis","family_id":"coe_mkfincons","file_id":"mathlib/Mathlib/LinearAlgebra/Basis/Fin.lean","sample_id":"4814cbf7f3508d036f5e5a6a7155650097bbec6785ba9dace923119e89bbbe59"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3e9d48f580176762fa7a5e0ebfd29db0eb21f956c84ea97eca970101095c5878","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"57ea28a7b08d8f885b958267bc30048df940f453f6ddd0117c4636da80751a9c","source_sha256":"f6612dc0cd9222234e5394f61b2340672945324a405e24dff4457d1abcc92db2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  letI : MeasurableSpace (ℕ × Ω) := 𝓕.predictable\n  letI : MeasurableSpace Ω := 𝓕 n\n  let X m := (Function.curry (h𝓕.approx m) (n + 1))\n  refine ⟨(fun m ↦ SimpleFunc.mk (X m) ?_ ?_), (fun ω ↦ h𝓕.tendsto_approx ⟨(n + 1), ω⟩)⟩\n  · intro s\n    rw [(by aesop : X m ⁻¹' {s} = {ω | (n + 1, ω) ∈ h𝓕.approx m ⁻¹' {s}})]\n    apply measurableSet_prodMk_add_one_of_predictable\n    apply (h𝓕.approx m).measurableSet_fiber\n  · apply (h𝓕.approx m).finite_range.subset\n    rw [Set.range_subset_iff]\n    aesop","hard_negative":false,"metrics":{"chosen_tokens":154,"rejected_tokens":2,"token_jaccard":0.035088,"token_length_ratio":0.012987},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"ab7fc8aeab3490a6ee50dd3a6194778463d48d84a5275842966070cc00bf2ff4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Process.Adapted\n\nNamespace:\nMeasureTheory.IsStronglyPredictable\n\nLocal context:\n/-\nCopyright (c) 2025 Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kexing Ying\n-/\n/-!\n# Predictable σ-algebra\n\nThis file defines the predictable σ-algebra associated to a filtration, as well as the\nnotion of predictable processes. We prove that predictable processes are progressively measurable\nand adapted. We also give an equivalent characterization of predictability for discrete processes.\n\n## Main definitions\n\n* `Filtration.predictable` : The predictable σ-algebra associated to a filtration.\n* `IsStronglyPredictable` : A process is predictable if it is measurable with respect to the\n  predictable σ-algebra.\n\n## Main results\n\n* `IsStronglyPredictable.isStronglyProgressive` : A predictable process is progressively measurable.\n* `IsStronglyPredictable.iff_measurable_add_one` : `u` is a discrete predictable process iff\n  `u (n + 1)` is `𝓕 n`-measurable and `u 0` is `𝓕 0`-measurable.\n\n## Tags\n\npredictable, previsible\n\n-/\n\n@[expose] public section\n\nopen Filter Order TopologicalSpace\n\nopen scoped MeasureTheory NNReal ENNReal Topology\n\nnamespace MeasureTheory\n\nvariable {Ω ι : Type*} {m : MeasurableSpace Ω} {E : Type*}\n\nsection\n\nnamespace Filtration\n\n/-- Given a filtration `𝓕`, the predictable σ-algebra is the σ-algebra on `ι × Ω` generated by\nsets of the form `(t, ∞) × A` for `t ∈ ι` and `A ∈ 𝓕 t` and `{⊥} × A` for `A ∈ 𝓕 ⊥`. -/\n@[implicit_reducible]\ndef predictable [Preorder ι] [OrderBot ι] (𝓕 : Filtration ι m) : MeasurableSpace (ι × Ω) :=\n  MeasurableSpace.generateFrom <|\n    {s | ∃ A, MeasurableSet[𝓕 ⊥] A ∧ s = {⊥} ×ˢ A} ∪\n    {s | ∃ i A, MeasurableSet[𝓕 i] A ∧ s = Set.Ioi i ×ˢ A}\n\nend Filtration\n\nlemma measurableSet_predictable_singleton_bot_prod [LinearOrder ι] [OrderBot ι]\n    {𝓕 : Filtration ι m} {s : Set Ω} (hs : MeasurableSet[𝓕 ⊥] s) :\n    MeasurableSet[𝓕.predictable] <| {⊥} ×ˢ s :=\n  MeasurableSpace.measurableSet_generateFrom <| Or.inl ⟨s, hs, rfl⟩\n\nlemma measurableSet_predictable_Ioi_prod [LinearOrder ι] [OrderBot ι]\n    {𝓕 : Filtration ι m} {i : ι} {s : Set Ω} (hs : MeasurableSet[𝓕 i] s) :\n    MeasurableSet[𝓕.predictable] <| Set.Ioi i ×ˢ s :=\n  MeasurableSpace.measurableSet_generateFrom <| Or.inr ⟨i, s, hs, rfl⟩\n\n/-- Sets of the form `(i, j] × A` for any `A ∈ 𝓕 i` are measurable with respect to the predictable\nσ-algebra. -/\nlemma measurableSet_predictable_Ioc_prod [LinearOrder ι] [OrderBot ι]\n    {𝓕 : Filtration ι m} (i j : ι) {s : Set Ω} (hs : MeasurableSet[𝓕 i] s) :\n    MeasurableSet[𝓕.predictable] <| Set.Ioc i j ×ˢ s := by\n  obtain hij | hij := le_total j i\n  · simp [hij]\n  · rw [← Set.Ioi_sdiff_Ioi, (by simp : (Set.Ioi i \\ Set.Ioi j) ×ˢ s\n      = Set.Ioi i ×ˢ (s \\ s) ∪ (Set.Ioi i \\ Set.Ioi j) ×ˢ s), ← Set.prod_sdiff_prod]\n    exact (measurableSet_predictable_Ioi_prod hs).diff\n      (measurableSet_predictable_Ioi_prod <| 𝓕.mono hij _ hs)\n\nlemma measurableSet_predictable_singleton_prod\n    {𝓕 : Filtration ℕ m} {n : ℕ} {s : Set Ω} (hs : MeasurableSet[𝓕 n] s) :\n    MeasurableSet[𝓕.predictable] <| {n + 1} ×ˢ s := by\n  rw [(_ : {n + 1} = Set.Ioc n (n + 1))]\n  · exact measurableSet_predictable_Ioc_prod _ _ hs\n  · ext m\n    simp only [Set.mem_singleton_iff, Set.mem_Ioc]\n    lia\n\nlemma measurableSet_prodMk_add_one_of_predictable {𝓕 : Filtration ℕ m} {s : Set (ℕ × Ω)}\n    (hs : MeasurableSet[𝓕.predictable] s) (n : ℕ) :\n    MeasurableSet[𝓕 n] {ω | (n + 1, ω) ∈ s} := by\n  rw [(by aesop : {ω | (n + 1, ω) ∈ s} = (Prod.mk (α := Set.singleton (n + 1)) (β := Ω)\n      ⟨n + 1, rfl⟩) ⁻¹' ((fun (p : Set.singleton (n + 1) × Ω) ↦ ((p.1 : ℕ), p.2)) ⁻¹' s))]\n  refine measurableSet_preimage (mβ := Subtype.instMeasurableSpace.prod (𝓕 n))\n    measurable_prodMk_left <| measurableSet_preimage ?_ hs\n  rw [measurable_iff_comap_le, MeasurableSpace.comap_le_iff_le_map]\n  refine MeasurableSpace.generateFrom_le ?_\n  rintro - (⟨A, hA, rfl⟩ | ⟨i, A, hA, rfl⟩)\n  · rw [MeasurableSpace.map_def,\n      (_ : (fun (p : Set.singleton (n + 1) × Ω) ↦ ((p.1 : ℕ), p.2)) ⁻¹' ({⊥} ×ˢ A) = ∅)]\n    · simp\n    · ext p\n      simp only [Nat.bot_eq_zero, Set.mem_preimage, Set.mem_prod, Set.mem_singleton_iff,\n        Set.mem_empty_iff_false, iff_false, not_and]\n      exact fun hp1 ↦ False.elim <| Nat.succ_ne_zero n (hp1 ▸ p.1.2.symm)\n  · rw [MeasurableSpace.map_def]\n    obtain hni | hin := lt_or_ge n i\n    · rw [(_ : (fun (p : Set.singleton (n + 1) × Ω) ↦ ((p.1 : ℕ), p.2)) ⁻¹' (Set.Ioi i ×ˢ A) = ∅)]\n      · simp\n      · ext p\n        simp only [Set.mem_preimage, Set.mem_prod, Set.mem_Ioi, Set.mem_empty_iff_false,\n          iff_false, not_and]\n        rw [p.1.2]\n        grind\n    · rw [(_ : (fun (p : Set.singleton (n + 1) × Ω) ↦ ((p.1 : ℕ), p.2)) ⁻¹' (Set.Ioi i ×ˢ A)\n          = {⟨n + 1, rfl⟩} ×ˢ A)]\n      · exact MeasurableSet.prod (MeasurableSet.of_subtype_image trivial) (𝓕.mono hin _ hA)\n      · ext p\n        simp only [Set.mem_preimage, Set.mem_prod, Set.mem_Ioi, Set.mem_singleton_iff,\n          and_congr_left_iff]\n        intro hp2\n        rw [p.1.2]\n        exact ⟨fun _ ↦ by aesop, fun _ ↦ lt_add_one_iff.2 hin⟩\n\nlemma measurableSpace_le_predictable_of_measurableSet [Preorder ι] [OrderBot ι]\n    {𝓕 : Filtration ι m} {m' : MeasurableSpace (ι × Ω)}\n    (hm'bot : ∀ A, MeasurableSet[𝓕 ⊥] A → MeasurableSet[m'] ({⊥} ×ˢ A))\n    (hm' : ∀ i A, MeasurableSet[𝓕 i] A → MeasurableSet[m'] ((Set.Ioi i) ×ˢ A)) :\n    𝓕.predictable ≤ m' := by\n  refine MeasurableSpace.generateFrom_le ?_\n  rintro - (⟨A, hA, rfl⟩ | ⟨i, A, hA, rfl⟩)\n  · exact hm'bot A hA\n  · exact hm' i A hA\n\n/-- The inclusion map from [0,i] × Ω with the subtype × 𝓕 i σ-algebra) to ι × Ω with the\npredictable σ-algebra is measurable -/\nlemma measurable_inclusion_predictable [LinearOrder ι] [OrderBot ι] [MeasurableSpace ι]\n    [TopologicalSpace ι] [OpensMeasurableSpace ι] [OrderClosedTopology ι] {𝓕 : Filtration ι m} {i} :\n    @Measurable (Set.Iic i × Ω) (ι × Ω) (Subtype.instMeasurableSpace.prod (𝓕 i)) 𝓕.predictable\n    fun x ↦ ⟨x.1.val, x.2⟩ := by\n  rw [measurable_iff_comap_le]\n  refine MeasurableSpace.comap_le_iff_le_map.2 <|\n    measurableSpace_le_predictable_of_measurableSet ?_ ?_\n  · intros A hA\n    simp only [MeasurableSpace.map_def,\n      (by aesop : (fun (p : Set.Iic i × Ω) ↦ ((p.1 : ι), p.2)) ⁻¹' ({⊥} ×ˢ A) = {⊥} ×ˢ A)]\n    exact (measurableSet_singleton _).prod <| 𝓕.mono bot_le _ hA\n  · intros j A hA\n    simp only [MeasurableSpace.map_def]\n    obtain hji | hij := le_total j i\n    · rw [(by grind : (fun (p : Set.Iic i × Ω) ↦ ((p.1 : ι), p.2)) ⁻¹' Set.Ioi j ×ˢ A\n        = (Subtype.val ⁻¹' (Set.Ioc j i)) ×ˢ A)]\n      exact (measurable_subtype_coe measurableSet_Ioc).prod (𝓕.mono hji _ hA)\n    · simp [(by grind : (fun (p : Set.Iic i × Ω) ↦ ((p.1 : ι), p.2)) ⁻¹' Set.Ioi j ×ˢ A = ∅)]\n\nend\n\nvariable [TopologicalSpace E]\n\n/-- A process is said to be predictable if it is measurable with respect to the predictable\nσ-algebra. -/\ndef IsStronglyPredictable [Preorder ι] [OrderBot ι] (𝓕 : Filtration ι m) (u : ι → Ω → E) :=\n  StronglyMeasurable[𝓕.predictable] <| Function.uncurry u\n\nnamespace IsStronglyPredictable\n\nopen Filtration\n\nvariable [LinearOrder ι] [OrderBot ι] [MeasurableSpace ι] [TopologicalSpace ι]\n    [OpensMeasurableSpace ι] [OrderClosedTopology ι]\n\n/-- A predictable process is progressively measurable. -/\nlemma isStronglyProgressive {𝓕 : Filtration ι m} {u : ι → Ω → E} (h𝓕 : IsStronglyPredictable 𝓕 u) :\n    IsStronglyProgressive 𝓕 u := by\n  intro i\n  letI : MeasurableSpace (ι × Ω) := 𝓕.predictable\n  letI : MeasurableSpace (Set.Iic i × Ω) := Subtype.instMeasurableSpace.prod (𝓕 i)\n  let X m (x : Set.Iic i × Ω) := h𝓕.approx m ⟨x.1, x.2⟩\n  refine ⟨fun m ↦ SimpleFunc.mk (X m) ?_ ?_, ?_⟩\n  · exact fun e ↦ measurable_inclusion_predictable <| (h𝓕.approx m).measurableSet_fiber e\n  · exact Set.Finite.subset (h𝓕.approx m).finite_range (by grind)\n  · exact fun n ↦ by apply h𝓕.tendsto_approx\n\n/-- A predictable process is adapted. -/\nlemma stronglyAdapted {𝓕 : Filtration ι m} {u : ι → Ω → E} (h𝓕 : IsStronglyPredictable 𝓕 u) :\n    StronglyAdapted 𝓕 u :=\n  h𝓕.isStronglyProgressive.stronglyAdapted\n\nsection Discrete\n\n/-- If `u` is a discrete predictable process, then `u (n + 1)` is `𝓕 n`-measurable. -/\n\nTarget:\nlemma measurable_add_one {𝓕 : Filtration ℕ m} {u : ℕ → Ω → E}\n    (h𝓕 : IsStronglyPredictable 𝓕 u) (n : ℕ) : StronglyMeasurable[𝓕 n] (u (n + 1)) :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Process","family_id":"measurable_add_one","file_id":"mathlib/Mathlib/Probability/Process/Predictable.lean","sample_id":"57ea28a7b08d8f885b958267bc30048df940f453f6ddd0117c4636da80751a9c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e68fac77a9fc516d7c10bdaed9dcefeaf4db7121c9ed0829fb3a283ec7ee3c03","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"735bf5dce5b321e535055fa4c86fa49603a8028a22626fad8900aee15b3514d6","source_sha256":"0688c3fec656e5c98eb2a67e2bb9366dc426159747d3f41fd3b47b03b8868e64","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [range, List.range_succ, ← coe_add, Multiset.add_comm, range, coe_singleton, singleton_add]","hard_negative":false,"metrics":{"chosen_tokens":22,"rejected_tokens":3,"token_jaccard":0.058824,"token_length_ratio":0.136364},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"ac198000550733419f064fb751817a48cb38b4c8103814152f676aca57562dca","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Multiset.UnionInter\n\nNamespace:\nMultiset\n\nLocal context:\n/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-! # `Multiset.range n` gives `{0, 1, ..., n-1}` as a multiset. -/\n\n@[expose] public section\n\nassert_not_exists Monoid\n\nopen List Nat\n\nnamespace Multiset\n\n-- range\n/-- `range n` is the multiset lifted from the list `range n`,\n  that is, the set `{0, 1, ..., n-1}`. -/\ndef range (n : ℕ) : Multiset ℕ :=\n  List.range n\n\ntheorem coe_range (n : ℕ) : ↑(List.range n) = range n :=\n  rfl\n\n@[simp]\ntheorem range_zero : range 0 = 0 :=\n  rfl\n\n@[simp]\n\nTarget:\ntheorem range_succ (n : ℕ) : range (succ n) = n ::ₘ range n :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Multiset","family_id":"range_succ","file_id":"mathlib/Mathlib/Data/Multiset/Range.lean","sample_id":"735bf5dce5b321e535055fa4c86fa49603a8028a22626fad8900aee15b3514d6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"42ee086aa0660c3672751c6ea4362a4dc2000a25c82c17c805dafacc7e49fe69","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cd37bf158342299f6da986fa2bd70e6d1e0dce8b07e6b8db5b9fc40dd9c7db15","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"141687aaa229bc38944346a4f56eb62c28baec7cf1b9f600772bfcea49a063a4","source_sha256":"37e5cf98a4004b5dd772a5a2594f4a59dd8c3a858390b7b2963d50078d06b866","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have eq₁ : _ = (π₁ i, j, k).base := PreservesPullback.iso_hom_fst (forget C) _ _\n  have eq₂ : _ = (π₂ i, j, k).base := PreservesPullback.iso_hom_snd (forget C) _ _\n  rw [← eq₁, ← eq₂, TopCat.coe_comp, Set.image_comp, TopCat.coe_comp, Set.preimage_comp,\n    Set.image_preimage_eq]\n  · simp only [forget_obj, forget_map, TopCat.pullback_snd_image_fst_preimage]\n  rw [← TopCat.epi_iff_surjective]\n  infer_instance","hard_negative":true,"metrics":{"chosen_tokens":105,"rejected_tokens":5,"token_jaccard":0.045455,"token_length_ratio":0.047619},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"ac479104f0653b9f118303bb920662f773988851ef38eb8260cedea96f63a646","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Gluing\npublic import Mathlib.Geometry.RingedSpace.OpenImmersion\npublic import Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits\n\nNamespace:\nAlgebraicGeometry.PresheafedSpace.GlueData\n\nLocal context:\n/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Gluing structured spaces\n\nGiven a family of gluing data of structured spaces (presheafed spaces, sheafed spaces, or locally\nringed spaces), we may glue them together.\n\nThe construction should be \"sealed\" and considered as a black box, while only using the API\nprovided.\n\n## Main definitions\n\n* `AlgebraicGeometry.PresheafedSpace.GlueData`: A structure containing the family of gluing data.\n* `CategoryTheory.GlueData.glued`: The glued presheafed space.\n  This is defined as the multicoequalizer of `∐ V i j ⇉ ∐ U i`, so that the general colimit API\n  can be used.\n* `CategoryTheory.GlueData.ι`: The immersion `ι i : U i ⟶ glued` for each `i : J`.\n\n## Main results\n\n* `AlgebraicGeometry.PresheafedSpace.GlueData.ιIsOpenImmersion`: The map `ι i : U i ⟶ glued`\n  is an open immersion for each `i : J`.\n* `AlgebraicGeometry.PresheafedSpace.GlueData.ι_jointly_surjective` : The underlying maps of\n  `ι i : U i ⟶ glued` are jointly surjective.\n* `AlgebraicGeometry.PresheafedSpace.GlueData.vPullbackConeIsLimit` : `V i j` is the pullback\n  (intersection) of `U i` and `U j` over the glued space.\n\nAnalogous results are also provided for `SheafedSpace` and `LocallyRingedSpace`.\n\n## Implementation details\n\nAlmost the whole file is dedicated to showing that `ι i` is an open immersion. The fact that\nthis is an open embedding of topological spaces follows from `Mathlib/Topology/Gluing.lean`, and it\nremains to construct `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_X, ι i '' U)` for each `U ⊆ U i`.\nSince `Γ(𝒪_X, ι i '' U)` is the limit of `diagram_over_open`, the components of the structure\nsheaves of the spaces in the gluing diagram, we need to construct a map\n`ιInvApp_π_app : Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_V, U_V)` for each `V` in the gluing diagram.\n\nWe will refer to ![this diagram](https://i.imgur.com/P0phrwr.png) in the following docstrings.\nThe `X` is the glued space, and the dotted arrow is a partial inverse guaranteed by the fact\nthat it is an open immersion. The map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_{U_j}, _)` is given by the composition\nof the red arrows, and the map `Γ(𝒪_{U_i}, U) ⟶ Γ(𝒪_{V_{jk}}, _)` is given by the composition of the\nblue arrows. To lift this into a map from `Γ(𝒪_X, ι i '' U)`, we also need to show that these\ncommute with the maps in the diagram (the green arrows), which is just a lengthy diagram-chasing.\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen TopologicalSpace CategoryTheory Opposite Topology\n\nopen CategoryTheory.Limits AlgebraicGeometry.PresheafedSpace\n\nopen AlgebraicGeometry.PresheafedSpace.IsOpenImmersion\n\nopen CategoryTheory.GlueData\n\nnamespace AlgebraicGeometry\n\nuniverse v u\n\nvariable (C : Type u) [Category.{v} C]\n\nnamespace PresheafedSpace\n\n/-- A family of gluing data consists of\n1. An index type `J`\n2. A presheafed space `U i` for each `i : J`.\n3. A presheafed space `V i j` for each `i j : J`.\n   (Note that this is `J × J → PresheafedSpace C` rather than `J → J → PresheafedSpace C` to\n   connect to the limits library more easily.)\n4. An open immersion `f i j : V i j ⟶ U i` for each `i j : J`.\n5. A transition map `t i j : V i j ⟶ V j i` for each `i j : J`.\n\nsuch that\n6. `f i i` is an isomorphism.\n7. `t i i` is the identity.\n8. `V i j ×[U i] V i k ⟶ V i j ⟶ V j i` factors through `V j k ×[U j] V j i ⟶ V j i` via some\n   `t' : V i j ×[U i] V i k ⟶ V j k ×[U j] V j i`.\n9. `t' i j k ≫ t' j k i ≫ t' k i j = 𝟙 _`.\n\nWe can then glue the spaces `U i` together by identifying `V i j` with `V j i`, such\nthat the `U i`'s are open subspaces of the glued space.\n-/\nstructure GlueData extends CategoryTheory.GlueData (PresheafedSpace.{v, u, v} C) where\n  f_open : ∀ i j, IsOpenImmersion (f i j)\n\nattribute [instance] GlueData.f_open\n\nnamespace GlueData\n\nvariable {C}\nvariable (D : GlueData.{v, u} C)\n\nlocal notation \"𝖣\" => D.toGlueData\n\nlocal notation \"π₁ \" i \", \" j \", \" k => pullback.fst (D.f i j) (D.f i k)\n\nlocal notation \"π₂ \" i \", \" j \", \" k => pullback.snd (D.f i j) (D.f i k)\n\nset_option quotPrecheck false\nlocal notation \"π₁⁻¹ \" i \", \" j \", \" k =>\n  (PresheafedSpace.IsOpenImmersion.pullbackFstOfRight (D.f i j) (D.f i k)).invApp\n\nset_option quotPrecheck false\nlocal notation \"π₂⁻¹ \" i \", \" j \", \" k =>\n  (PresheafedSpace.IsOpenImmersion.pullbackSndOfLeft (D.f i j) (D.f i k)).invApp\n\n/-- The glue data of topological spaces associated to a family of glue data of PresheafedSpaces. -/\nabbrev toTopGlueData : TopCat.GlueData :=\n  { f_open := fun i j => (D.f_open i j).base_open\n    toGlueData := 𝖣.mapGlueData (forget C) }\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem ι_isOpenEmbedding [HasLimits C] (i : D.J) : IsOpenEmbedding (𝖣.ι i).base := by\n  rw [← show _ = (𝖣.ι i).base from 𝖣.ι_gluedIso_inv (PresheafedSpace.forget _) _, TopCat.coe_comp]\n  exact (TopCat.homeoOfIso (𝖣.gluedIso (PresheafedSpace.forget _)).symm).isOpenEmbedding.comp\n      (D.toTopGlueData.ι_isOpenEmbedding i)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem pullback_base (i j k : D.J) (S : Set (D.V (i, j)).carrier) :\n    (π₂ i, j, k) '' (π₁ i, j, k) ⁻¹' S = D.f i k ⁻¹' D.f i j '' S :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_141687aaa229","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"562437279b8862124dae3df3a30096b846c56d73ea89694590bac7368768b816","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/RingedSpace","family_id":"pullback_base","file_id":"mathlib/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean","sample_id":"141687aaa229bc38944346a4f56eb62c28baec7cf1b9f600772bfcea49a063a4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"95de6bb494dcefbe243626eb340736b29bf3b291e155dac01267e899f1bb69c2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c031c9432732d05cd237126cbcac6a473645115eab86e0eb5195fb77c4512d50","source_sha256":"afb2352b0e8eda21bd88c47cda5f9b0a4c162f3f78eedf36a33613392805b346","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [of_eq, Prefunctor.comp_assoc, Prefunctor.comp_assoc, Functor.toPrefunctor_comp]\n  dsimp [lift]\n  rw [Quotient.lift_spec, Paths.lift_spec, Quiver.Symmetrify.lift_spec]","hard_negative":false,"metrics":{"chosen_tokens":37,"rejected_tokens":3,"token_jaccard":0.05,"token_length_ratio":0.081081},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"ac55e5156f89a8dc337214916a9aaa262e95b5a44d813f906847a2139202dc29","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Groupoid\npublic import Mathlib.CategoryTheory.PathCategory.Basic\n\nNamespace:\nQuiver.FreeGroupoid\n\nLocal context:\n/-\nCopyright (c) 2022 Rémi Bottinelli. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémi Bottinelli\n-/\n/-!\n# Free groupoid on a quiver\n\nThis file defines the free groupoid on a quiver, the lifting of a prefunctor to its unique\nextension as a functor from the free groupoid, and proves uniqueness of this extension.\n\n## Main results\n\nGiven the type `V` and a quiver instance on `V`:\n\n- `Quiver.FreeGroupoid V`: a type synonym for `V`.\n- `Quiver.FreeGroupoid.instGroupoid`: the `Groupoid` instance on `Quiver.FreeGroupoid V`.\n- `lift`: the lifting of a prefunctor from `V` to `V'` where `V'` is a groupoid, to a functor.\n  `Quiver.FreeGroupoid V ⥤ V'`.\n- `lift_spec` and `lift_unique`: the proofs that, respectively, `lift` indeed is a lifting\n  and is the unique one.\n\n## Implementation notes\n\nThe free groupoid is first defined by symmetrifying the quiver, taking the induced path category\nand finally quotienting by the reducibility relation.\n\n-/\n\n@[expose] public section\n\nopen Set Function\n\nnamespace Quiver\n\nopen CategoryTheory\n\nuniverse u v u' v' u'' v''\n\nvariable {V : Type u} [Quiver.{v} V]\n\n/-- Shorthand for the \"forward\" arrow corresponding to `f` in `paths <| symmetrify V` -/\nabbrev Hom.toPosPath {X Y : V} (f : X ⟶ Y) :\n    (CategoryTheory.Paths.categoryPaths <| Quiver.Symmetrify V).Hom X Y :=\n  f.toPos.toPath\n\n/-- Shorthand for the \"forward\" arrow corresponding to `f` in `paths <| symmetrify V` -/\nabbrev Hom.toNegPath {X Y : V} (f : X ⟶ Y) :\n    (CategoryTheory.Paths.categoryPaths <| Quiver.Symmetrify V).Hom Y X :=\n  f.toNeg.toPath\n\n/-- The \"reduction\" relation -/\ninductive FreeGroupoid.redStep : HomRel (Paths (Quiver.Symmetrify V))\n  | step (X Z : Quiver.Symmetrify V) (f : X ⟶ Z) :\n    redStep (𝟙 ((Paths.of (Quiver.Symmetrify V)).obj X)) (f.toPath ≫ (Quiver.reverse f).toPath)\n\n/-- The underlying vertices of the free groupoid -/\nprotected def FreeGroupoid (V) [Q : Quiver V] :=\n  CategoryTheory.Quotient (@FreeGroupoid.redStep V Q)\n\nnamespace FreeGroupoid\n\nopen Quiver\n\ninstance {V} [Quiver V] [Nonempty V] : Nonempty (Quiver.FreeGroupoid V) := by\n  inhabit V; exact ⟨⟨@default V _⟩⟩\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem congr_reverse {X Y : Paths <| Quiver.Symmetrify V} (p q : X ⟶ Y) :\n    HomRel.CompClosure redStep p q → HomRel.CompClosure redStep p.reverse q.reverse := by\n  rintro ⟨_, _, XW, _, _, WY, _, _, f⟩\n  have : HomRel.CompClosure redStep (WY.reverse ≫ 𝟙 _ ≫ XW.reverse)\n      (WY.reverse ≫ (f.toPath ≫ (Quiver.reverse f).toPath) ≫ XW.reverse) := by\n    constructor\n    constructor\n  simpa only [CategoryStruct.comp, CategoryStruct.id, Quiver.Path.reverse, Quiver.Path.nil_comp,\n    Quiver.Path.reverse_comp, Quiver.reverse_reverse, Quiver.Path.reverse_toPath,\n    Quiver.Path.comp_assoc] using this\n\nopen Relation in\ntheorem congr_comp_reverse {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :\n    Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (p ≫ p.reverse) =\n      Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (𝟙 X) := by\n  apply Quot.eqvGen_sound\n  induction p with\n  | nil => apply EqvGen.refl\n  | cons q f ih =>\n    simp only [Quiver.Path.reverse]\n    fapply EqvGen.trans\n    -- Porting note: dot notation for `Quiver.Path.*` and `Quiver.Hom.*` not working\n    · exact q ≫ Quiver.Path.reverse q\n    · apply EqvGen.symm\n      apply EqvGen.rel\n      have : HomRel.CompClosure redStep (q ≫ 𝟙 _ ≫ Quiver.Path.reverse q)\n          (q ≫ (Quiver.Hom.toPath f ≫ Quiver.Hom.toPath (Quiver.reverse f)) ≫\n            Quiver.Path.reverse q) := by\n        apply HomRel.CompClosure.intro\n        apply redStep.step\n      simp only [Category.assoc, Category.id_comp] at this ⊢\n      -- Porting note: `simp` cannot see how `Quiver.Path.comp_assoc` is relevant, so change to\n      -- category notation\n      change HomRel.CompClosure redStep (q ≫ Quiver.Path.reverse q)\n        (Quiver.Path.cons q f ≫ (Quiver.Hom.toPath (Quiver.reverse f)) ≫ (Quiver.Path.reverse q))\n      simp only [← Category.assoc] at this ⊢\n      exact this\n    · exact ih\n\ntheorem congr_reverse_comp {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :\n    Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (p.reverse ≫ p) =\n      Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (𝟙 Y) := by\n  nth_rw 2 [← Quiver.Path.reverse_reverse p]\n  apply congr_comp_reverse\n\ninstance : Category (Quiver.FreeGroupoid V) :=\n  Quotient.category redStep\n\n/-- The inverse of an arrow in the free groupoid -/\ndef quotInv {X Y : Quiver.FreeGroupoid V} (f : X ⟶ Y) : Y ⟶ X :=\n  Quot.liftOn f (fun pp => Quot.mk _ <| pp.reverse) fun pp qq con =>\n    Quot.sound <| congr_reverse pp qq con\n\ninstance instGroupoid : Groupoid (Quiver.FreeGroupoid V) where\n  inv := quotInv\n  inv_comp p := Quot.inductionOn p fun pp => congr_reverse_comp pp\n  comp_inv p := Quot.inductionOn p fun pp => congr_comp_reverse pp\n\n/-- The inclusion of the quiver on `V` to the underlying quiver on `FreeGroupoid V` -/\ndef of (V) [Quiver V] : V ⥤q Quiver.FreeGroupoid V where\n  obj X := ⟨X⟩\n  map f := Quot.mk _ f.toPosPath\n\ntheorem of_eq :\n    of V = (Quiver.Symmetrify.of ⋙q (Paths.of (Quiver.Symmetrify V))).comp\n      (Quotient.functor <| @redStep V _).toPrefunctor := rfl\n\nsection UniversalProperty\n\nvariable {V' : Type u'} [Groupoid V']\n\n/-- The lift of a prefunctor to a groupoid, to a functor from `FreeGroupoid V` -/\ndef lift (φ : V ⥤q V') : Quiver.FreeGroupoid V ⥤ V' :=\n  CategoryTheory.Quotient.lift _ (Paths.lift <| Quiver.Symmetrify.lift φ) <| by\n    rintro _ _ _ _ ⟨X, Y, f⟩\n    -- Porting note: `simp` does not work, so manually `rewrite`\n    erw [Paths.lift_nil, Paths.lift_cons, Quiver.Path.comp_nil, Paths.lift_toPath,\n      Quiver.Symmetrify.lift_reverse]\n    symm\n    apply Groupoid.comp_inv\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem lift_spec (φ : V ⥤q V') : of V ⋙q (lift φ).toPrefunctor = φ :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Groupoid","family_id":"lift_spec","file_id":"mathlib/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean","sample_id":"c031c9432732d05cd237126cbcac6a473645115eab86e0eb5195fb77c4512d50"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2f0e35fce05624abb1b88e07b05ba59ec3da85bd198bf90cf2c93b8503cfca8a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"93a0ad9cdbe52633a2a6476e78bca93ac766df3668d5e36441bd4a7ce8340870","source_sha256":"15d1aa45ed96e60a321174c4449c6369f20349d72b788c380b869a678d1f4952","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let g : X ⟶ (⨆ i, U i : Y.Opens) := IsOpenImmersion.lift (Scheme.Opens.ι _) f (by simpa using H)\n  rw [← IsOpenImmersion.lift_fac (⨆ i, U i).ι f (by simpa using H)]\n  apply MorphismProperty.RespectsRight.postcomp (Q := @IsOpenImmersion) _ inferInstance\n  rw [iff_of_iSup_eq_top (P := P) (U := fun i : ι ↦ (⨆ i, U i).ι ⁻¹ᵁ U i)]\n  · intro i\n    have heq : g ⁻¹ᵁ (⨆ i, U i).ι ⁻¹ᵁ U i = f ⁻¹ᵁ U i := by\n      change (g ≫ (⨆ i, U i).ι) ⁻¹ᵁ U i = _\n      simp [g]\n    let e : Arrow.mk (g ∣_ (⨆ i, U i).ι ⁻¹ᵁ U i) ≅ Arrow.mk (f ∣_ U i) :=\n        Arrow.isoMk (X.isoOfEq heq) (Scheme.Opens.isoOfLE (le_iSup U i)) <| by\n      simp [← CategoryTheory.cancel_mono (U i).ι, g]\n    rw [P.arrow_mk_iso_iff e]\n    exact hf i\n  apply (⨆ i, U i).ι.image_injective\n  dsimp\n  rw [Scheme.Hom.image_iSup, Scheme.Hom.image_top_eq_opensRange, Scheme.Opens.opensRange_ι]\n  simp [Scheme.Hom.image_preimage_eq_opensRange_inf, le_iSup U]","hard_negative":false,"metrics":{"chosen_tokens":294,"rejected_tokens":2,"token_jaccard":0.013158,"token_length_ratio":0.006803},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"ac6f33e286ce6690c295765e6cf8d3c77022dd6a5df88c1ea7b3dea3524a33e7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Limits\npublic import Mathlib.CategoryTheory.MorphismProperty.Local\npublic import Mathlib.Data.List.TFAE\n\nNamespace:\nAlgebraicGeometry.IsZariskiLocalAtTarget\n\nLocal context:\n/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Properties of morphisms between Schemes\n\nWe provide the basic framework for talking about properties of morphisms between Schemes.\n\nA `MorphismProperty Scheme` is a predicate on morphisms between schemes. For properties local at\nthe target, its behaviour is entirely determined by its definition on morphisms into affine schemes,\nwhich we call an `AffineTargetMorphismProperty`. In this file, we provide API lemmas for properties\nlocal at the target, and special support for those properties whose `AffineTargetMorphismProperty`\ntakes on a simpler form. We also provide API lemmas for properties local at the source.\nThe main interfaces of the API are the typeclasses `IsZariskiLocalAtTarget`,\n`IsZariskiLocalAtSource` and `HasAffineProperty`, which we describe in detail below.\n\n## `IsZariskiLocalAtTarget`\n\n- `AlgebraicGeometry.IsZariskiLocalAtTarget`: We say that `IsZariskiLocalAtTarget P` for\n  `P : MorphismProperty Scheme` if\n  1. `P` respects isomorphisms.\n  2. `P` holds for `f ∣_ U` for an open cover `U` of `Y` if and only if `P` holds for `f`.\n\nFor a morphism property `P` local at the target and `f : X ⟶ Y`, we provide these API lemmas:\n\n- `AlgebraicGeometry.IsZariskiLocalAtTarget.of_isPullback`:\n    `P` is preserved under pullback along open immersions.\n- `AlgebraicGeometry.IsZariskiLocalAtTarget.restrict`:\n    `P f → P (f ∣_ U)` for an open `U` of `Y`.\n- `AlgebraicGeometry.IsZariskiLocalAtTarget.iff_of_iSup_eq_top`:\n    `P f ↔ ∀ i, P (f ∣_ U i)` for a family `U` of open sets covering `Y`.\n- `AlgebraicGeometry.IsZariskiLocalAtTarget.iff_of_openCover`:\n    `P f ↔ ∀ i, P (𝒰.pullbackHom f i)` for `𝒰 : Y.OpenCover`.\n\n## `IsZariskiLocalAtSource`\n\n- `AlgebraicGeometry.IsZariskiLocalAtSource`: We say that `IsZariskiLocalAtSource P` for\n  `P : MorphismProperty Scheme` if\n  1. `P` respects isomorphisms.\n  2. `P` holds for `𝒰.f i ≫ f` for an open cover `𝒰` of `X` iff `P` holds for `f : X ⟶ Y`.\n\nFor a morphism property `P` local at the source and `f : X ⟶ Y`, we provide these API lemmas:\n\n- `AlgebraicGeometry.IsZariskiLocalAtSource.comp`:\n    `P` is preserved under composition with open immersions at the source.\n- `AlgebraicGeometry.IsZariskiLocalAtSource.iff_of_iSup_eq_top`:\n    `P f ↔ ∀ i, P ((U i).ι ≫ f)` for a family `U` of open sets covering `X`.\n- `AlgebraicGeometry.IsZariskiLocalAtSource.iff_of_openCover`:\n    `P f ↔ ∀ i, P (𝒰.f i ≫ f)` for `𝒰 : X.OpenCover`.\n- `AlgebraicGeometry.IsZariskiLocalAtSource.of_isOpenImmersion`: If `P` contains identities then `P`\n    holds for open immersions.\n\n## `AffineTargetMorphismProperty`\n\n- `AlgebraicGeometry.AffineTargetMorphismProperty`:\n    The type of predicates on `f : X ⟶ Y` with `Y` affine.\n- `AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal`: We say that `P.IsLocal` if `P`\n    satisfies the assumptions of the affine communication lemma\n    (`AlgebraicGeometry.of_affine_open_cover`). That is,\n    1. `P` respects isomorphisms.\n    2. If `P` holds for `f : X ⟶ Y`, then `P` holds for `f ∣_ Y.basicOpen r` for any\n      global section `r`.\n    3. If `P` holds for `f ∣_ Y.basicOpen r` for all `r` in a spanning set of the global sections,\n      then `P` holds for `f`.\n\n## `HasAffineProperty`\n\n- `AlgebraicGeometry.HasAffineProperty`:\n  `HasAffineProperty P Q` is a type class asserting that `P` is local at the target,\n  and over affine schemes, it is equivalent to `Q : AffineTargetMorphismProperty`.\n\nFor `HasAffineProperty P Q` and `f : X ⟶ Y`, we provide these API lemmas:\n\n- `AlgebraicGeometry.HasAffineProperty.of_isPullback`:\n    `P` is preserved under pullback along open immersions from affine schemes.\n- `AlgebraicGeometry.HasAffineProperty.restrict`:\n    `P f → Q (f ∣_ U)` for affine `U` of `Y`.\n- `AlgebraicGeometry.HasAffineProperty.iff_of_iSup_eq_top`:\n    `P f ↔ ∀ i, Q (f ∣_ U i)` for a family `U` of affine open sets covering `Y`.\n- `AlgebraicGeometry.HasAffineProperty.iff_of_openCover`:\n    `P f ↔ ∀ i, Q (𝒰.pullbackHom f i)` for affine open covers `𝒰` of `Y`.\n- `AlgebraicGeometry.HasAffineProperty.isStableUnderBaseChange`:\n    If `Q` is stable under affine base change, then `P` is stable under arbitrary base change.\n\n## Implementation details\n\nThe properties `IsZariskiLocalAtTarget` and `IsZariskiLocalAtSource` are defined as abbreviations\nfor the respective local property of morphism properties defined generally for categories equipped\nwith a `Precoverage`.\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\nopen TopologicalSpace CategoryTheory CategoryTheory.Limits Opposite\n\nnoncomputable section\n\nnamespace AlgebraicGeometry\n\n/-- A property is Zariski-local at target if it is local at target in the Zariski topology. -/\nabbrev IsZariskiLocalAtTarget (P : MorphismProperty Scheme.{u}) :=\n  P.IsLocalAtTarget Scheme.zariskiPrecoverage\n\nnamespace IsZariskiLocalAtTarget\n\n/--\n`P` is local at the target if\n1. `P` respects isomorphisms.\n2. If `P` holds for `f : X ⟶ Y`, then `P` holds for `f ∣_ U` for any `U`.\n3. If `P` holds for `f ∣_ U` for an open cover `U` of `Y`, then `P` holds for `f`.\n-/\nprotected lemma mk' {P : MorphismProperty Scheme} [P.RespectsIso]\n    (restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens), P f → P (f ∣_ U))\n    (of_sSup_eq_top :\n      ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), iSup U = ⊤ →\n        (∀ i, P (f ∣_ U i)) → P f) :\n    IsZariskiLocalAtTarget P := by\n  refine .mk_of_iff_of_zeroHypercover fun {X Y} f 𝒰 ↦ ?_\n  refine ⟨fun hf i ↦ (P.arrow_mk_iso_iff (morphismRestrictOpensRange _ _)).mp (restrict _ _ hf),\n    fun h ↦ ?_⟩\n  refine of_sSup_eq_top f _ (Scheme.OpenCover.iSup_opensRange <| .ulift 𝒰) ?_\n  exact fun i ↦ (P.arrow_mk_iso_iff (morphismRestrictOpensRange f _)).mpr (h _)\n\nvariable {P : MorphismProperty Scheme.{u}} [IsZariskiLocalAtTarget P]\n  {X Y : Scheme.{u}} {f : X ⟶ Y} (𝒰 : Y.OpenCover)\n\nlemma of_isPullback {UX UY : Scheme.{u}} {iY : UY ⟶ Y} [IsOpenImmersion iY]\n    {iX : UX ⟶ X} {f' : UX ⟶ UY} (h : IsPullback iX f' f iY) (H : P f) : P f' :=\n  MorphismProperty.IsLocalAtTarget.of_isPullback (Y.affineCover.add iY) .none h H\n\ntheorem restrict (hf : P f) (U : Y.Opens) : P (f ∣_ U) :=\n  of_isPullback (isPullback_morphismRestrict f U).flip hf\n\nlemma of_iSup_eq_top {ι} (U : ι → Y.Opens) (hU : iSup U = ⊤)\n    (H : ∀ i, P (f ∣_ U i)) : P f := by\n  refine (P.iff_of_zeroHypercover_target\n    (Y.openCoverOfIsOpenCover (s := Set.range U) Subtype.val (by ext; simp [← hU]))).mpr fun i ↦ ?_\n  obtain ⟨_, i, rfl⟩ := i\n  refine (P.arrow_mk_iso_iff (morphismRestrictOpensRange f _)).mp ?_\n  change P (f ∣_ (U i).ι.opensRange)\n  rw [Scheme.Opens.opensRange_ι]\n  exact H i\n\ntheorem iff_of_iSup_eq_top {ι} (U : ι → Y.Opens) (hU : iSup U = ⊤) :\n    P f ↔ ∀ i, P (f ∣_ U i) :=\n  ⟨fun H _ ↦ restrict H _, of_iSup_eq_top U hU⟩\n\nlemma of_openCover (H : ∀ i, P (𝒰.pullbackHom f i)) : P f := by\n  apply of_iSup_eq_top (fun i ↦ (𝒰.f i).opensRange) 𝒰.iSup_opensRange\n  exact fun i ↦ (P.arrow_mk_iso_iff (morphismRestrictOpensRange f _)).mpr (H i)\n\ntheorem iff_of_openCover (𝒰 : Y.OpenCover) :\n    P f ↔ ∀ i, P (𝒰.pullbackHom f i) :=\n  ⟨fun H _ ↦ of_isPullback (.of_hasPullback _ _) H, of_openCover _⟩\n\nset_option backward.defeqAttrib.useBackward true in\n\nTarget:\nlemma of_range_subset_iSup [P.RespectsRight @IsOpenImmersion] {ι : Type*} (U : ι → Y.Opens)\n    (H : Set.range f ⊆ (⨆ i, U i : Y.Opens)) (hf : ∀ i, P (f ∣_ U i)) : P f :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Morphisms","family_id":"of_range_subset_isup","file_id":"mathlib/Mathlib/AlgebraicGeometry/Morphisms/Basic.lean","sample_id":"93a0ad9cdbe52633a2a6476e78bca93ac766df3668d5e36441bd4a7ce8340870"},"split":"train"}
{"binding":{"chosen_proof_sha256":"af8967d58c125507dfdd8a870d8e2d61acb151c1a146b922c5673f800fa162b8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"89069437bed98817c329262f0d45f97272b726fdfd39addabd0aab53182ba049","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2f8dba92a8be5eab58edb05807e69424a9fcd6a33e1126c48358e2bed2791279","source_sha256":"46ad986a41048710426d3296a8284bc983e91e0a000d6a497db675e512528c15","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [mem_disjSum, or_iff_right]\n  · simp only [inr.injEq, exists_eq_right]\n  rintro ⟨a, _, ha⟩\n  exact inl_ne_inr ha","hard_negative":false,"metrics":{"chosen_tokens":28,"rejected_tokens":35,"token_jaccard":0.916667,"token_length_ratio":1.25},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"ac79195f2a25646b1e73b23d0f80d4270cb41a4cbf19d36e68f55e31d17751a8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.Multiset\n\nNamespace:\nMultiset\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Disjoint sum of multisets\n\nThis file defines the disjoint sum of two multisets as `Multiset (α ⊕ β)`. Beware not to confuse\nwith the `Multiset.sum` operation which computes the additive sum.\n\n## Main declarations\n\n* `Multiset.disjSum`: `s.disjSum t` is the disjoint sum of `s` and `t`.\n-/\n\n@[expose] public section\n\n\nopen Sum\n\nnamespace Multiset\n\nvariable {α β γ : Type*} (s : Multiset α) (t : Multiset β)\n\n/-- Disjoint sum of multisets. -/\ndef disjSum : Multiset (α ⊕ β) :=\n  s.map inl + t.map inr\n\n@[simp]\ntheorem zero_disjSum : (0 : Multiset α).disjSum t = t.map inr :=\n  Multiset.zero_add _\n\n@[simp]\ntheorem disjSum_zero : s.disjSum (0 : Multiset β) = s.map inl :=\n  Multiset.add_zero _\n\n@[simp]\ntheorem card_disjSum : Multiset.card (s.disjSum t) = Multiset.card s + Multiset.card t := by\n  rw [disjSum, card_add, card_map, card_map]\n\nvariable {s t} {s₁ s₂ : Multiset α} {t₁ t₂ : Multiset β} {a : α} {b : β} {x : α ⊕ β}\n\ntheorem mem_disjSum : x ∈ s.disjSum t ↔ (∃ a, a ∈ s ∧ inl a = x) ∨ ∃ b, b ∈ t ∧ inr b = x := by\n  simp_rw [disjSum, mem_add, mem_map]\n\n@[simp]\ntheorem inl_mem_disjSum : inl a ∈ s.disjSum t ↔ a ∈ s := by\n  rw [mem_disjSum, or_iff_left]\n  · simp only [inl.injEq, exists_eq_right]\n  rintro ⟨b, _, hb⟩\n  exact inr_ne_inl hb\n\n@[simp]\n\nTarget:\ntheorem inr_mem_disjSum : inr b ∈ s.disjSum t ↔ b ∈ t :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [mem_disjSum, or_iff_right]\n  · simp only [inr.injEq, exists_eq_right]\n  rintro ⟨a, _, ha⟩\n  exact inl_ne_inr ha\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Multiset","family_id":"inr_mem_disjsum","file_id":"mathlib/Mathlib/Data/Multiset/Sum.lean","sample_id":"2f8dba92a8be5eab58edb05807e69424a9fcd6a33e1126c48358e2bed2791279"},"split":"train"}
{"binding":{"chosen_proof_sha256":"55840963317e3a0d749bfa4ab4253cf6959feaf799aad1f633d62acc28d49b26","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a1e462f08764c6d42623fb769d4457124c492b34cbacfef092d7509dfdd31de7","source_sha256":"02df2dcc4ba0253814e33afb4b0bcd9e1c8a127adf2c967cb9666cee9d9d5264","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by valid) : F.map' i i = 𝟙 _ := F.map_id _","hard_negative":false,"metrics":{"chosen_tokens":17,"rejected_tokens":2,"token_jaccard":0.071429,"token_length_ratio":0.117647},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"ac84000199581a56459e1f2d1dfd752af7c78c3d9cd0dafb411bf97dae304fd3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Nat.Defs\npublic import Mathlib.CategoryTheory.Category.Preorder\npublic import Mathlib.CategoryTheory.Comma.Arrow\npublic import Mathlib.Data.Fintype.Basic\npublic import Mathlib.Tactic.FinCases\npublic import Mathlib.Tactic.SuppressCompilation\n\nNamespace:\nCategoryTheory.ComposableArrows\n\nLocal context:\n/-\nCopyright (c) 2023 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Composable arrows\n\nIf `C` is a category, the type of `n`-simplices in the nerve of `C` identifies\nto the type of functors `Fin (n + 1) ⥤ C`, which can be thought of as families of `n` composable\narrows in `C`. In this file, we introduce and study this category `ComposableArrows C n`\nof `n` composable arrows in `C`.\n\nIf `F : ComposableArrows C n`, we define `F.left` as the leftmost object, `F.right` as the\nrightmost object, and `F.hom : F.left ⟶ F.right` is the canonical map.\n\nThe most significant definition in this file is the constructor\n`F.precomp f : ComposableArrows C (n + 1)` for `F : ComposableArrows C n` and `f : X ⟶ F.left`:\n\"it shifts `F` towards the right and inserts `f` on the left\". This `precomp` has\ngood definitional properties.\n\nIn the namespace `CategoryTheory.ComposableArrows`, we provide constructors\nlike `mk₁ f`, `mk₂ f g`, `mk₃ f g h` for `ComposableArrows C n` for small `n`.\n\nTODO (@joelriou):\n* construct some elements in `ComposableArrows m (Fin (n + 1))` for small `n`\n  the precomposition with which shall induce functors\n  `ComposableArrows C n ⥤ ComposableArrows C m` which correspond to simplicial operations\n  (specifically faces) with good definitional properties (this might be necessary for\n  up to `n = 7` in order to formalize spectral sequences following Verdier)\n\n-/\n\nset_option backward.defeqAttrib.useBackward true\n\n@[expose] public section\n\nset_option backward.privateInPublic true\n\n/-!\nNew `simprocs` that run even in `dsimp` have caused breakages in this file.\n\n(e.g. `dsimp` can now simplify `2 + 3` to `5`)\n\nFor now, we just turn off the offending simprocs in this file.\n\n*However*, hopefully it is possible to refactor the material here so that no disabling of\nsimprocs is needed.\n\nSee issue https://github.com/leanprover-community/mathlib4/issues/27382.\n-/\nattribute [-simp] Fin.reduceFinMk\n\nnamespace CategoryTheory\n\nopen Category\n\nvariable (C : Type*) [Category* C]\n\n/-- `ComposableArrows C n` is the type of functors `Fin (n + 1) ⥤ C`. -/\nabbrev ComposableArrows (n : ℕ) := Fin (n + 1) ⥤ C\n\nnamespace ComposableArrows\n\nvariable {C} {n m : ℕ}\nvariable (F G : ComposableArrows C n)\n\n-- We do not yet replace `omega` with `lia` here, as it is measurably slower.\n/-- A wrapper for `omega` which prefaces it with some quick and useful attempts -/\nmacro \"valid\" : tactic =>\n  `(tactic| first | assumption | apply zero_le | apply le_rfl | transitivity <;> assumption | omega)\n\n/-- The `i`th object (with `i : ℕ` such that `i ≤ n`) of `F : ComposableArrows C n`. -/\n@[simp]\nabbrev obj' (i : ℕ) (hi : i ≤ n := by valid) : C := F.obj ⟨i, by lia⟩\n\n/-- The map `F.obj' i ⟶ F.obj' j` when `F : ComposableArrows C n`, and `i` and `j`\nare natural numbers such that `i ≤ j ≤ n`. -/\n@[simp]\nabbrev map' (i j : ℕ) (hij : i ≤ j := by valid) (hjn : j ≤ n := by valid) :\n    F.obj ⟨i, by lia⟩ ⟶ F.obj ⟨j, by lia⟩ :=\n  F.map (homOfLE (by simp only [Fin.mk_le_mk]; valid))\n\nTarget:\nlemma map'_self (i : ℕ) (hi : i ≤ n :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/ComposableArrows","family_id":"map'_self","file_id":"mathlib/Mathlib/CategoryTheory/ComposableArrows/Basic.lean","sample_id":"a1e462f08764c6d42623fb769d4457124c492b34cbacfef092d7509dfdd31de7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"02be6bd02ac451c737251c72bcfefcd2bda23393af373ca35cd3c4b0bb93ffd2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a09da6d7ec6d1cfa34656994fd49c4f73c6c9af7b3c929a8c052ae4721cb3d4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4dda6d8eed8aa9990b9ce99d74443691206268bc0aae2f322f2853e45e3b954a","source_sha256":"a7aa42b03cd6d38aac48f2dc40ba1b38bc0a247c798efe96de58375304c838ee","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [coe_inv, inv_def, normSq_coe, inv_one, ofReal_one, mul_one]","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.2},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"aca33954fc274c6e79fc7d53eabeccec17ec0aca0f15cd5938a87616decde659","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic\npublic import Mathlib.Analysis.Normed.Field.UnitBall\n\nNamespace:\nCircle\n\nLocal context:\n/-\nCopyright (c) 2021 Heather Macbeth. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Heather Macbeth\n-/\n/-!\n# The circle\n\nThis file defines `Circle` to be the metric sphere (`Metric.sphere`) in `ℂ` centred at `0` of\nradius `1`.  We equip it with the following structure:\n\n* a submonoid of `ℂ`\n* a group\n* a topological group\n\nWe furthermore define `Circle.exp` to be the natural map `fun t ↦ exp (t * I)` from `ℝ` to\n`Circle`, and show that this map is a group homomorphism.\n\nWe define two additive characters onto the circle:\n* `Real.fourierChar`: The character `fun x ↦ exp ((2 * π * x) * I)` (for which we introduce the\n  notation `𝐞` in the scope `FourierTransform`). This uses the analyst convention that there is a\n  `2 * π` in the exponent.\n* `Real.probChar`: The character `fun x ↦ exp (x * I)`, which uses the probabilist convention that\n  there is no `2 * π` in the exponent.\n\n## Implementation notes\n\nBecause later (in `Geometry.Manifold.Instances.Sphere`) one wants to equip the circle with a smooth\nmanifold structure borrowed from `Metric.sphere`, the underlying set is\n`{z : ℂ | abs (z - 0) = 1}`.  This prevents certain algebraic facts from working definitionally --\nfor example, the circle is not defeq to `{z : ℂ | abs z = 1}`, which is the kernel of `Complex.abs`\nconsidered as a homomorphism from `ℂ` to `ℝ`, nor is it defeq to `{z : ℂ | normSq z = 1}`, which\nis the kernel of the homomorphism `Complex.normSq` from `ℂ` to `ℝ`.\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen Complex Function Metric ComplexConjugate\n\n/-- The unit circle in `ℂ`. -/\ndef Circle : Type := Submonoid.unitSphere ℂ\nderiving TopologicalSpace\n\nnamespace Circle\nvariable {x y : Circle}\n\ninstance instCoeOut : CoeOut Circle ℂ := subtypeCoe\n\ninstance instCommGroup : CommGroup Circle := inferInstanceAs <| CommGroup (sphere _ _)\ninstance : HasDistribNeg Circle := inferInstanceAs <| HasDistribNeg (sphere _ _)\ninstance : ContinuousNeg Circle := inferInstanceAs <| ContinuousNeg (sphere _ _)\ninstance instMetricSpace : MetricSpace Circle := inferInstanceAs <| MetricSpace (sphere _ _)\n\n@[ext] lemma ext : (x : ℂ) = y → x = y := Subtype.ext\n\nlemma coe_injective : Injective ((↑) : Circle → ℂ) := fun _ _ ↦ ext\n\n-- Not simp because `SetLike.coe_eq_coe` already proves it\nlemma coe_inj : (x : ℂ) = y ↔ x = y := coe_injective.eq_iff\n\nlemma norm_coe (z : Circle) : ‖(z : ℂ)‖ = 1 := mem_sphere_zero_iff_norm.1 z.2\n\n@[simp] lemma normSq_coe (z : Circle) : normSq z = 1 := by simp [normSq_eq_norm_sq]\n@[simp] lemma coe_ne_zero (z : Circle) : (z : ℂ) ≠ 0 := ne_zero_of_mem_unit_sphere z\n@[simp, norm_cast] lemma coe_one : ↑(1 : Circle) = (1 : ℂ) := rfl\n-- Not simp because `OneMemClass.coe_eq_one` already proves it\n@[norm_cast] lemma coe_eq_one : (x : ℂ) = 1 ↔ x = 1 := by rw [← coe_inj, coe_one]\n@[simp, norm_cast] lemma coe_mul (z w : Circle) : ↑(z * w) = (z : ℂ) * w := rfl\n@[simp, norm_cast] lemma coe_inv (z : Circle) : ↑z⁻¹ = (z : ℂ)⁻¹ := rfl\n\nTarget:\nlemma coe_inv_eq_conj (z : Circle) : ↑z⁻¹ = conj (z : ℂ) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_4dda6d8eed8a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"209e5553c0e8f45f8f144b7c10b0380a9f59fb90fca9d1700761562c9f3a5e68","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Complex","family_id":"coe_inv_eq_conj","file_id":"mathlib/Mathlib/Analysis/Complex/Circle.lean","sample_id":"4dda6d8eed8aa9990b9ce99d74443691206268bc0aae2f322f2853e45e3b954a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1fd411efd0adcfd9a0a62aeb48dc422a81d7841cde6fbd3cbad674180785c839","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"324ea35ce6179c7986048fb9f8e478d89d1f28abd4d7eccf06b0337eaf7ec28a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bdf9d42bef05adc3f26afabef77136e641ea973a1c33f1f631c7b584f6c81787","source_sha256":"ef9864ce066b7696b62174e7a3b3a449882e3ab320bb2e19ef0e02cda6607291","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp +instances only [r_iff_exists, Subtype.exists, exists_prop, OreLocalization.oreEqv,\n    smul_eq_mul, Submonoid.mk_smul]\n  constructor\n  · rintro ⟨u, hu, e⟩\n    exact ⟨_, mul_mem hu x.2.2, u * y.2, by rw [mul_assoc, mul_assoc, ← e], mul_right_comm _ _ _⟩\n  · rintro ⟨u, hu, v, e₁, e₂⟩\n    exact ⟨u, hu, by rw [← mul_assoc, e₂, mul_right_comm, ← e₁, mul_assoc, mul_comm y.1]⟩","hard_negative":false,"metrics":{"chosen_tokens":112,"rejected_tokens":117,"token_jaccard":0.913043,"token_length_ratio":1.044643},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"aca4e195617fff08a2cc2c2d5da5f511add2ecf2ca787d6964f105706d4ee584","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\npublic import Mathlib.Algebra.Group.Submonoid.Operations\npublic import Mathlib.Algebra.Regular.Basic\npublic import Mathlib.GroupTheory.Congruence.Hom\npublic import Mathlib.GroupTheory.OreLocalization.Basic\n\nNamespace:\nLocalization\n\nLocal context:\n/-\nCopyright (c) 2019 Amelia Livingston. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Amelia Livingston\n-/\n/-!\n# Localizations of commutative monoids\n\nLocalizing a commutative ring at one of its submonoids does not rely on the ring's addition, so\nwe can generalize localizations to commutative monoids.\n\nWe characterize the localization of a commutative monoid `M` at a submonoid `S` up to\nisomorphism; that is, a commutative monoid `N` is the localization of `M` at `S` iff we can find a\nmonoid homomorphism `f : M →* N` satisfying 3 properties:\n1. For all `y ∈ S`, `f y` is a unit;\n2. For all `z : N`, there exists `(x, y) : M × S` such that `z * f y = f x`;\n3. For all `x, y : M` such that `f x = f y`, there exists `c ∈ S` such that `x * c = y * c`.\n   (The converse is a consequence of 1.)\n\nGiven such a localization map `f : M →* N`, we can define the surjection\n`Submonoid.LocalizationMap.mk'` sending `(x, y) : M × S` to `f x * (f y)⁻¹`. Mapping properties\nof the localization (e.g. extending a map from `M → P` to `N` if the image of `S` is contained in\nthe units) are treated in a later file `Mathlib.GroupTheory.MonoidLocalization.Maps`.\n\nWe also define the quotient of `M × S` by the unique congruence relation (equivalence relation\npreserving a binary operation) `r` such that for any other congruence relation `s` on `M × S`\nsatisfying '`∀ y ∈ S`, `(1, 1) ∼ (y, y)` under `s`', we have that `(x₁, y₁) ∼ (x₂, y₂)` by `s`\nwhenever `(x₁, y₁) ∼ (x₂, y₂)` by `r`. We show this relation is equivalent to the standard\nlocalization relation.\nThis defines the localization as a quotient type, `Localization`, but the majority of\nsubsequent lemmas in the file are given in terms of localizations up to isomorphism, using maps\nwhich satisfy the characteristic predicate.\n\nThe Grothendieck group construction corresponds to localizing at the top submonoid, namely making\nevery element invertible.\n\n## Implementation notes\n\nIn maths it is natural to reason up to isomorphism, but in Lean we cannot naturally `rewrite` one\nstructure with an isomorphic one; one way around this is to isolate a predicate characterizing\na structure up to isomorphism, and reason about things that satisfy the predicate.\n\nThe infimum form of the localization congruence relation is chosen as 'canonical' here, since it\nshortens some proofs.\n\nTo reason about the localization as a quotient type, use `mk_eq_monoidOf_mk'` and associated\nlemmas. These show the quotient map `mk : M → S → Localization S` equals the\nsurjection `LocalizationMap.mk'` induced by the map\n`Localization.monoidOf : Submonoid.LocalizationMap S (Localization S)` (where `of` establishes the\nlocalization as a quotient type satisfies the characteristic predicate). The lemma\n`mk_eq_monoidOf_mk'` hence gives you access to the results in the rest of the file, which are about\nthe `LocalizationMap.mk'` induced by any localization map.\n\n## TODO\n\n* Show that the localization at the top monoid is a group.\n* Generalise to (nonempty) subsemigroups.\n* If we acquire more bundlings, we can make `Localization.mkOrderEmbedding` be an ordered monoid\n  embedding.\n\n## Tags\nlocalization, monoid localization, quotient monoid, congruence relation, characteristic predicate,\ncommutative monoid, grothendieck group\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero Ring\n\nopen Function\nnamespace AddSubmonoid\n\nvariable {M : Type*} [AddCommMonoid M] (S : AddSubmonoid M) (N : Type*) [AddCommMonoid N]\n\nvariable {N} in\n/-- A predicate characterizing homomorphisms between additive monoids `M` and `N` that form a\ncommutative triangle with the canonical map from `M` to its localization at `S` and\nsome isomorphism between `N` and the localization. -/\nstructure IsLocalizationMap (S : AddSubmonoid M) (f : M → N) where\n  map_addUnits (y : S) : IsAddUnit (f y)\n  surj (z : N) : ∃ x : M × S, z + f x.2 = f x.1\n  exists_of_eq {x y} : f x = f y → ∃ c : S, c + x = c + y\n\n/-- The type of AddMonoid homomorphisms satisfying the characteristic predicate: if `f : M →+ N`\nsatisfies this predicate, then `N` is isomorphic to the localization of `M` at `S`. -/\nstructure LocalizationMap extends M →ₙ+ N where\n  isLocalizationMap : IsLocalizationMap S toFun\n\n/-- The additive homomorphism underlying a `LocalizationMap` of `AddCommMonoid`s. -/\nadd_decl_doc LocalizationMap.toAddHom\n\nend AddSubmonoid\n\nsection CommMonoid\n\nvariable {M : Type*} [CommMonoid M] (S : Submonoid M) (N : Type*) [CommMonoid N] {P : Type*}\n  [CommMonoid P]\n\nnamespace Submonoid\n\nvariable {N} in\n/-- A predicate characterizing homomorphisms between monoids `M` and `N` that form a\ncommutative triangle with the canonical map from `M` to its localization at `S` and\nsome isomorphism between `N` and the localization. -/\n@[to_additive (attr := mk_iff)]\nstructure IsLocalizationMap (S : Submonoid M) (f : M → N) where\n  map_units (y : S) : IsUnit (f y)\n  surj (z : N) : ∃ x : M × S, z * f x.2 = f x.1\n  exists_of_eq {x y} : f x = f y → ∃ c : S, c * x = c * y\n\n/-- The type of monoid homomorphisms satisfying the characteristic predicate: if `f : M →* N`\nsatisfies this predicate, then `N` is isomorphic to the localization of `M` at `S`. -/\n@[to_additive] structure LocalizationMap extends M →ₙ* N where\n  isLocalizationMap : IsLocalizationMap S toFun\n\n/-- The multiplicative homomorphism underlying a `LocalizationMap`. -/\nadd_decl_doc LocalizationMap.toMulHom\n\nend Submonoid\n\nnamespace Localization\n\n/- Ensure that `@[to_additive]` uses the right namespace before the definition of `Localization`. -/\ninsert_to_additive_translation Localization AddLocalization\n\n/-- The congruence relation on `M × S`, `M` a `CommMonoid` and `S` a submonoid of `M`, whose\nquotient is the localization of `M` at `S`, defined as the unique congruence relation on\n`M × S` such that for any other congruence relation `s` on `M × S` where for all `y ∈ S`,\n`(1, 1) ∼ (y, y)` under `s`, we have that `(x₁, y₁) ∼ (x₂, y₂)` by `r` implies\n`(x₁, y₁) ∼ (x₂, y₂)` by `s`. -/\n@[to_additive\n/-- The congruence relation on `M × S`, `M` an `AddCommMonoid` and `S` an `AddSubmonoid` of `M`,\nwhose quotient is the localization of `M` at `S`, defined as the unique congruence relation on\n`M × S` such that for any other congruence relation `s` on `M × S` where for all `y ∈ S`,\n`(0, 0) ∼ (y, y)` under `s`, we have that `(x₁, y₁) ∼ (x₂, y₂)` by `r` implies\n`(x₁, y₁) ∼ (x₂, y₂)` by `s`. -/]\ndef r (S : Submonoid M) : Con (M × S) :=\n  sInf { c | ∀ y : S, c 1 (y, y) }\n\n/-- An alternate form of the congruence relation on `M × S`, `M` a `CommMonoid` and `S` a\nsubmonoid of `M`, whose quotient is the localization of `M` at `S`. -/\n@[to_additive\n/-- An alternate form of the congruence relation on `M × S`, `M` a `CommMonoid` and `S` a\nsubmonoid of `M`, whose quotient is the localization of `M` at `S`. -/]\ndef r' : Con (M × S) := by\n  -- note we multiply by `c` on the left so that we can later generalize to `•`\n  refine\n    { r := fun a b : M × S ↦ ∃ c : S, ↑c * (↑b.2 * a.1) = c * (a.2 * b.1)\n      iseqv := ⟨fun a ↦ ⟨1, rfl⟩, fun ⟨c, hc⟩ ↦ ⟨c, hc.symm⟩, ?_⟩\n      mul' := ?_ }\n  · rintro a b c ⟨t₁, ht₁⟩ ⟨t₂, ht₂⟩\n    use t₂ * t₁ * b.2\n    simp only [Submonoid.coe_mul]\n    calc\n      (t₂ * t₁ * b.2 : M) * (c.2 * a.1) = t₂ * c.2 * (t₁ * (b.2 * a.1)) := by ac_rfl\n      _ = t₁ * a.2 * (t₂ * (c.2 * b.1)) := by rw [ht₁]; ac_rfl\n      _ = t₂ * t₁ * b.2 * (a.2 * c.1) := by rw [ht₂]; ac_rfl\n  · rintro a b c d ⟨t₁, ht₁⟩ ⟨t₂, ht₂⟩\n    use t₂ * t₁\n    calc\n      (t₂ * t₁ : M) * (b.2 * d.2 * (a.1 * c.1)) = t₂ * (d.2 * c.1) * (t₁ * (b.2 * a.1)) := by ac_rfl\n      _ = (t₂ * t₁ : M) * (a.2 * c.2 * (b.1 * d.1)) := by rw [ht₁, ht₂]; ac_rfl\n\n/-- The congruence relation used to localize a `CommMonoid` at a submonoid can be expressed\nequivalently as an infimum (see `Localization.r`) or explicitly\n(see `Localization.r'`). -/\n@[to_additive\n/-- The additive congruence relation used to localize an `AddCommMonoid` at a submonoid can be\nexpressed equivalently as an infimum (see `AddLocalization.r`) or explicitly\n(see `AddLocalization.r'`). -/]\ntheorem r_eq_r' : r S = r' S :=\n  le_antisymm (sInf_le fun _ ↦ ⟨1, by simp⟩) <|\n    le_sInf fun b H ⟨p, q⟩ ⟨x, y⟩ ⟨t, ht⟩ ↦ by\n      rw [← one_mul (p, q), ← one_mul (x, y)]\n      refine b.trans (b.mul (H (t * y)) (b.refl _)) ?_\n      convert! b.symm (b.mul (H (t * q)) (b.refl (x, y))) using 1\n      dsimp only [Prod.mk_mul_mk, Submonoid.coe_mul] at ht ⊢\n      simp_rw [mul_assoc, ht, mul_comm y q]\n\nvariable {S}\n\n@[to_additive]\ntheorem r_iff_exists {x y : M × S} : r S x y ↔ ∃ c : S, ↑c * (↑y.2 * x.1) = c * (x.2 * y.1) := by\n  simp only [r_eq_r' S, r', Con.rel_mk]\n\n@[to_additive]\n\nTarget:\ntheorem r_iff_oreEqv_r {x y : M × S} : r S x y ↔ (OreLocalization.oreEqv S M).r x y :=\n\nProof body:\n","rejected":"by\n  simp +instances only [r_iff_exists, Subtype.exists, exists_prop, OreLocalization.oreEqv,\n    smul_eq_mul, Submonoid.mk_smul]\n  constructor\n  · rintro ⟨u, hu, e⟩\n    exact ⟨_, mul_mem hu x.2.2, u * y.2, by rw [mul_assoc, mul_assoc, ← e], mul_right_comm _ _ _⟩\n  · rintro ⟨u, hu, v, e₁, e₂⟩\n    exact ⟨u, hu, by rw [← mul_assoc, e₂, mul_right_comm, ← e₁, mul_assoc, mul_comm y.1]⟩\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/MonoidLocalization","family_id":"r_iff_oreeqv_r","file_id":"mathlib/Mathlib/GroupTheory/MonoidLocalization/Basic.lean","sample_id":"bdf9d42bef05adc3f26afabef77136e641ea973a1c33f1f631c7b584f6c81787"},"split":"train"}
{"binding":{"chosen_proof_sha256":"22008102f7d34b93c9ef351ad767f156bb7824bb62e00233f134ea27d1c44cf6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2aad5e99e9d08105118c7fe3a6e9502304ab85de3fce7e613074176387085c34","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f1be1ab398e8e6f5f51fea76b3b1c755d097ca53fd2fb60f4b3eb6a00e32ab92","source_sha256":"2ecbfb606a976f2e47686b1a1c471e54098f1481b48ac0c349b703cde7d54204","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [eq_geometrically, GeometricallyReduced.eq_geometrically,\n    GeometricallyIrreducible.eq_geometrically, ← geometrically_inf]\n  eta_expand\n  simp [isIntegral_iff_irreducibleSpace_and_isReduced, and_comm]","hard_negative":true,"metrics":{"chosen_tokens":23,"rejected_tokens":2,"token_jaccard":0.0625,"token_length_ratio":0.086957},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"aca994ad1a8871b19352d41d1bb2ae6b4010df66d27f04fed4ce0fb24d39136b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Geometrically.Reduced\npublic import Mathlib.AlgebraicGeometry.Geometrically.Irreducible\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2026 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Geometrically Integral Schemes\n\n## Main results\n- `AlgebraicGeometry.GeometricallyIntegral`:\n  We say that morphism `f : X ⟶ Y` is geometrically integral if for all `Spec K ⟶ Y` with `K`\n  a field, `X ×[Y] Spec K` is integral.\n  We also provide the fact that this is stable under base change (`by infer_instance`)\n- `GeometricallyIntegral.iff_geometricallyIntegral_fiber`:\n  A scheme is geometrically integral over `S` iff the fibers of all\n  `s : S` are geometrically integral.\n- `AlgebraicGeometry.GeometricallyIntegral.isIntegral_of_isLocallyNoetherian`:\n  If `X` is geometrically integral, flat, and universally open (e.g. when over a field),\n  over an integral locally noetherian scheme, then `X` is also integral.\n- `AlgebraicGeometry.GeometricallyIntegral.isIntegral_of_subsingleton`:\n  If `X` is geometrically integral over a field, then it is integral.\n-/\n\npublic section\n\nopen CategoryTheory MorphismProperty Limits\n\nnamespace AlgebraicGeometry\n\nvariable {X Y Z S : Scheme} (f : X ⟶ S) (g : Y ⟶ S)\n\n/-- We say that morphism `f : X ⟶ Y` is geometrically integral if for all `Spec K ⟶ Y` with `K`\na field, `X ×[Y] Spec K` is integral. -/\n@[mk_iff]\nclass GeometricallyIntegral (f : X ⟶ Y) : Prop where\n  geometrically_isIntegral : geometrically IsIntegral f\n\nlemma GeometricallyIntegral.eq_geometrically :\n    @GeometricallyIntegral = geometrically IsIntegral := by\n  ext; exact geometricallyIntegral_iff _\n\nTarget:\nlemma GeometricallyIntegral.eq_geometricallyReduced_inf_geometricallyIrreducible :\n    @GeometricallyIntegral =\n      (@GeometricallyReduced ⊓ @GeometricallyIrreducible : MorphismProperty Scheme) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_f1be1ab398e8","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"511f381c93065221d7065f1c7b1a341ca5020a824aff117afff08123d34556f4","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Geometrically","family_id":"geometricallyintegral","file_id":"mathlib/Mathlib/AlgebraicGeometry/Geometrically/Integral.lean","sample_id":"f1be1ab398e8e6f5f51fea76b3b1c755d097ca53fd2fb60f4b3eb6a00e32ab92"},"split":"train"}
{"binding":{"chosen_proof_sha256":"69cb17d2f43d6924d494a8be87b9a52df302f4a4d1c3a416993a289830e67b47","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2c9653db60a3ddae63cedc1cc9e2ba6e8d6ce8800e7284483f64df67132c5de6","source_sha256":"368f0cb56789c53c6820ea0e367b83289d651593fc19d224d3364287d2adc7a4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases hR : R = 0\n  · simp_all\n  have H : ContinuousOn (circleAverage f c) (Set.Ioc 0 |R|) := by\n    refine (h₁f.2.mono ?_).circleAverage (fun z hz ↦ hz.1.le)\n    intro x hx\n    rw [closure_ball _ (by aesop), mem_closedBall_iff_norm]\n    exact hx.2\n  rw [← circleAverage_abs_radius]\n  apply H.eq_of_eqOn_Ioo (by aesop)\n  · intro r hr\n    apply HarmonicOnNhd.circleAverage_eq\n    · apply h₁f.1.mono\n      rw [abs_of_pos hr.1]\n      exact closedBall_subset_ball hr.2","hard_negative":false,"metrics":{"chosen_tokens":114,"rejected_tokens":2,"token_jaccard":0.035714,"token_length_ratio":0.017544},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"acc4f3b35322e7c127477ee14383aac88fba3c495c8c1fccde10ba670ca99791","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Complex.Harmonic.Analytic\npublic import Mathlib.Analysis.Complex.MeanValue\npublic import Mathlib.Analysis.InnerProductSpace.Harmonic.HarmonicContOnCl\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Stefan Kebekus. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stefan Kebekus\n-/\n/-!\n# The Mean Value Property of Harmonic Functions on the Complex Plane\n-/\n\npublic section\n\nopen InnerProductSpace Metric Real\n\nvariable {f : ℂ → ℝ} {c : ℂ} {R : ℝ}\n\n/--\nThe **Mean Value Property** of harmonic functions: If `f : ℂ → ℝ` is harmonic in a neighborhood of a\nclosed disc of radius `R` and center `c`, then the circle average `circleAverage f c R` equals\n`f c`.\n-/\ntheorem HarmonicOnNhd.circleAverage_eq (hf : HarmonicOnNhd f (closedBall c |R|)) :\n    circleAverage f c R = f c := by\n  obtain ⟨e, h₁e, h₂e⟩ := (isCompact_closedBall c |R|).exists_thickening_subset_open\n    (isOpen_setOf_harmonicAt f) hf\n  rw [thickening_closedBall h₁e (abs_nonneg R)] at h₂e\n  obtain ⟨F, h₁F, h₂F⟩ := InnerProductSpace.HarmonicOnNhd.exists_analyticOnNhd_ball_re_eq h₂e\n  have h₃F : DifferentiableOn ℂ F (closure (ball c |R|)) := by\n    intro x hx\n    apply (h₁F x _).differentiableWithinAt\n    grind [mem_ball, mem_closedBall.1 (closure_ball_subset_closedBall hx)]\n  have h₄F : Set.EqOn (Complex.reCLM ∘ F) f (sphere c |R|) :=\n    fun x hx ↦ h₂F (sphere_subset_ball (lt_add_of_pos_left |R| h₁e) hx)\n  rw [← circleAverage_congr_sphere h₄F, Complex.reCLM.circleAverage_comp_comm,\n    h₃F.diffContOnCl.circleAverage]\n  · apply h₂F\n    simp [mem_ball, dist_self, add_pos_of_pos_of_nonneg h₁e (abs_nonneg R)]\n  · apply (h₁F.continuousOn.mono (fun _ _ ↦ by simp_all [dist_eq_norm])).circleIntegrable'\n\n/--\nThe **Mean Value Property** of harmonic functions: If `f : ℂ → ℝ` is harmonic on a disc of radius\n`|R|` and center `c` and continuous on its closure, then the circle average `circleAverage f c R`\nequals `f c`.\n-/\n\nTarget:\ntheorem HarmonicContOnCl.circleAverage_eq {f : ℂ → ℝ} {c : ℂ} {R : ℝ}\n    (h₁f : HarmonicContOnCl f (ball c |R|)) :\n    circleAverage f c R = f c :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Complex","family_id":"harmoniccontoncl","file_id":"mathlib/Mathlib/Analysis/Complex/Harmonic/MeanValue.lean","sample_id":"2c9653db60a3ddae63cedc1cc9e2ba6e8d6ce8800e7284483f64df67132c5de6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"73ba32450f7eaa32ac31b9569cb16af401e9b0fc7b4daddcdd761838e6b70efb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2299db0e612a8842de7ca07c9ac6d1044626542cd35b496219419b3ec83f8ec1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8138a062ed7aa5e0b5f610f4aeff3a70100ca75f6fdd0cf8c052c2cf58487e47","source_sha256":"04666e1a54002d0f7b310bb68c623ecf47eda2066cc741dda3dda2f5fc6f4c64","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  letI : TopologicalSpace Y := ⊥; haveI := discreteTopology_bot Y\n  exact (isCompact_range hf.continuous).finite_of_discrete","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":5,"token_jaccard":0.15,"token_length_ratio":0.238095},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"ace213d4ac65d212b0bda6500924bda111f27ebb2415ba025192577707662db2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Indicator\npublic import Mathlib.Topology.Connected.LocallyConnected\npublic import Mathlib.Topology.Sets.Closeds\n\nNamespace:\nIsLocallyConstant\n\nLocal context:\n/-\nCopyright (c) 2021 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\n/-!\n# Locally constant functions\n\nThis file sets up the theory of locally constant function from a topological space to a type.\n\n## Main definitions and constructions\n\n* `IsLocallyConstant f` : a map `f : X → Y` where `X` is a topological space is locally\n                            constant if every set in `Y` has an open preimage.\n* `LocallyConstant X Y` : the type of locally constant maps from `X` to `Y`\n* `LocallyConstant.map` : push-forward of locally constant maps\n* `LocallyConstant.comap` : pull-back of locally constant maps\n-/\n\n@[expose] public section\n\nvariable {X Y Z α : Type*} [TopologicalSpace X]\n\nopen Set Filter\nopen scoped Topology\n\n/-- A function between topological spaces is locally constant if the preimage of any set is open. -/\ndef IsLocallyConstant (f : X → Y) : Prop :=\n  ∀ s : Set Y, IsOpen (f ⁻¹' s)\n\nnamespace IsLocallyConstant\n\nopen List in\nprotected theorem tfae (f : X → Y) :\n    TFAE [IsLocallyConstant f,\n      ∀ x, ∀ᶠ x' in 𝓝 x, f x' = f x,\n      ∀ x, IsOpen { x' | f x' = f x },\n      ∀ y, IsOpen (f ⁻¹' {y}),\n      ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x] := by\n  tfae_have 1 → 4 := fun h y => h {y}\n  tfae_have 4 → 3 := fun h x => h (f x)\n  tfae_have 3 → 2 := fun h x => IsOpen.mem_nhds (h x) rfl\n  tfae_have 2 → 5\n  | h, x => by\n    rcases mem_nhds_iff.1 (h x) with ⟨U, eq, hU, hx⟩\n    exact ⟨U, hU, hx, eq⟩\n  tfae_have 5 → 1\n  | h, s => by\n    refine isOpen_iff_forall_mem_open.2 fun x hx ↦ ?_\n    rcases h x with ⟨U, hU, hxU, eq⟩\n    exact ⟨U, fun x' hx' => mem_preimage.2 <| (eq x' hx').symm ▸ hx, hU, hxU⟩\n  tfae_finish\n\n@[nontriviality]\ntheorem of_discrete [DiscreteTopology X] (f : X → Y) : IsLocallyConstant f := fun _ =>\n  isOpen_discrete _\n\ntheorem isOpen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsOpen { x | f x = y } :=\n  hf {y}\n\ntheorem isClosed_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClosed { x | f x = y } :=\n  ⟨hf {y}ᶜ⟩\n\ntheorem isClopen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClopen { x | f x = y } :=\n  ⟨isClosed_fiber hf _, isOpen_fiber hf _⟩\n\ntheorem iff_exists_open (f : X → Y) :\n    IsLocallyConstant f ↔ ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x :=\n  (IsLocallyConstant.tfae f).out 0 4\n\ntheorem iff_eventually_eq (f : X → Y) : IsLocallyConstant f ↔ ∀ x, ∀ᶠ y in 𝓝 x, f y = f x :=\n  (IsLocallyConstant.tfae f).out 0 1\n\ntheorem exists_open {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x :=\n  (iff_exists_open f).1 hf x\n\nprotected theorem eventually_eq {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    ∀ᶠ y in 𝓝 x, f y = f x :=\n  (iff_eventually_eq f).1 hf x\n\ntheorem iff_isOpen_fiber_apply {f : X → Y} : IsLocallyConstant f ↔ ∀ x, IsOpen (f ⁻¹' {f x}) :=\n  (IsLocallyConstant.tfae f).out 0 2\n\ntheorem iff_isOpen_fiber {f : X → Y} : IsLocallyConstant f ↔ ∀ y, IsOpen (f ⁻¹' {y}) :=\n  (IsLocallyConstant.tfae f).out 0 3\n\nprotected theorem continuous [TopologicalSpace Y] {f : X → Y} (hf : IsLocallyConstant f) :\n    Continuous f :=\n  ⟨fun _ _ => hf _⟩\n\ntheorem iff_continuous {_ : TopologicalSpace Y} [DiscreteTopology Y] (f : X → Y) :\n    IsLocallyConstant f ↔ Continuous f :=\n  ⟨IsLocallyConstant.continuous, fun h s => h.isOpen_preimage s (isOpen_discrete _)⟩\n\ntheorem of_constant (f : X → Y) (h : ∀ x y, f x = f y) : IsLocallyConstant f :=\n  (iff_eventually_eq f).2 fun _ => Eventually.of_forall fun _ => h _ _\n\nprotected theorem const (y : Y) : IsLocallyConstant (Function.const X y) :=\n  of_constant _ fun _ _ => rfl\n\nprotected theorem comp {f : X → Y} (hf : IsLocallyConstant f) (g : Y → Z) :\n    IsLocallyConstant (g ∘ f) := fun s => by\n  rw [Set.preimage_comp]\n  exact hf _\n\ntheorem prodMk {Y'} {f : X → Y} {f' : X → Y'} (hf : IsLocallyConstant f)\n    (hf' : IsLocallyConstant f') : IsLocallyConstant fun x => (f x, f' x) :=\n  (iff_eventually_eq _).2 fun x =>\n    (hf.eventually_eq x).mp <| (hf'.eventually_eq x).mono fun _ hf' hf => Prod.ext hf hf'\n\ntheorem comp₂ {Y₁ Y₂ Z : Type*} {f : X → Y₁} {g : X → Y₂} (hf : IsLocallyConstant f)\n    (hg : IsLocallyConstant g) (h : Y₁ → Y₂ → Z) : IsLocallyConstant fun x => h (f x) (g x) :=\n  (hf.prodMk hg).comp fun x : Y₁ × Y₂ => h x.1 x.2\n\ntheorem comp_continuous [TopologicalSpace Y] {g : Y → Z} {f : X → Y} (hg : IsLocallyConstant g)\n    (hf : Continuous f) : IsLocallyConstant (g ∘ f) := fun s => by\n  rw [Set.preimage_comp]\n  exact hf.isOpen_preimage _ (hg _)\n\n/-- A locally constant function is constant on any preconnected set. -/\ntheorem apply_eq_of_isPreconnected {f : X → Y} (hf : IsLocallyConstant f) {s : Set X}\n    (hs : IsPreconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by\n  let U := f ⁻¹' {f y}\n  suffices x ∉ Uᶜ from Classical.not_not.1 this\n  intro hxV\n  specialize hs U Uᶜ (hf {f y}) (hf {f y}ᶜ) _ ⟨y, ⟨hy, rfl⟩⟩ ⟨x, ⟨hx, hxV⟩⟩\n  · simp only [union_compl_self, subset_univ]\n  · simp only [inter_empty, Set.not_nonempty_empty, inter_compl_self] at hs\n\ntheorem apply_eq_of_preconnectedSpace [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f)\n    (x y : X) : f x = f y :=\n  hf.apply_eq_of_isPreconnected isPreconnected_univ trivial trivial\n\ntheorem eq_const [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    f = Function.const X (f x) :=\n  funext fun y => hf.apply_eq_of_preconnectedSpace y x\n\ntheorem exists_eq_const [PreconnectedSpace X] [Nonempty Y] {f : X → Y} (hf : IsLocallyConstant f) :\n    ∃ y, f = Function.const X y := by\n  rcases isEmpty_or_nonempty X with h | h\n  · exact ⟨Classical.arbitrary Y, funext <| h.elim⟩\n  · exact ⟨f (Classical.arbitrary X), hf.eq_const _⟩\n\ntheorem iff_is_const [PreconnectedSpace X] {f : X → Y} : IsLocallyConstant f ↔ ∀ x y, f x = f y :=\n  ⟨fun h _ _ => h.apply_eq_of_isPreconnected isPreconnected_univ trivial trivial, of_constant _⟩\n\nTarget:\ntheorem range_finite [CompactSpace X] {f : X → Y} (hf : IsLocallyConstant f) :\n    (Set.range f).Finite :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_8138a062ed7a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"9513c60c0e0eec4da6f7664da9fb896178298d8a8b4a18c57fa777604db848cd","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/LocallyConstant","family_id":"range_finite","file_id":"mathlib/Mathlib/Topology/LocallyConstant/Basic.lean","sample_id":"8138a062ed7aa5e0b5f610f4aeff3a70100ca75f6fdd0cf8c052c2cf58487e47"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d409d608f76c2016efc5fdd42930cab75c91e811c1cdb05b5a33c8b551b1384f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"63d4c6f7bdfaad78a31356f5390bcb2b6c5c4366f0f8ad12785fcd4685b1c7e7","source_sha256":"231219a77cbb3d5335e2c2c8fee21d9a8c54aa585e63a8a73c88281c1837f6a8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro i\n  have h_eq_iUnion : {ω | τ ω ≤ i} = ⋃ k ≤ i, {ω | τ ω = k} := by\n    ext ω\n    simp only [Set.mem_setOf_eq, Set.mem_iUnion, exists_prop]\n    cases τ ω with\n    | top => simp\n    | coe a => norm_cast; simp\n  rw [h_eq_iUnion]\n  refine MeasurableSet.biUnion (Set.to_countable _) fun k hk => ?_\n  exact f.mono hk _ (hτ k)","hard_negative":false,"metrics":{"chosen_tokens":91,"rejected_tokens":3,"token_jaccard":0.06,"token_length_ratio":0.032967},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"ad1c17c702b77db041118e3cfca2b85bcf6d16e78c9915a76ea8c91bee70941c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Process.Adapted\npublic import Mathlib.MeasureTheory.Constructions.BorelSpace.WithTop\npublic import Mathlib.Data.ENat.Lattice\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2021 Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kexing Ying, Rémy Degenne\n-/\n/-!\n# Stopping times, stopped processes and stopped values\n\nDefinition and properties of stopping times.\n\n## Main definitions\n\n* `MeasureTheory.IsStoppingTime`: a stopping time with respect to some filtration `f` on a\n  measurable space `Ω` is a function `τ : Ω → WithTop ι` such that for all `i : ι`,\n  the preimage of `{j | j ≤ i}` along `τ` is `f i`-measurable\n* `MeasureTheory.IsStoppingTime.measurableSpace`: the σ-algebra associated with a stopping time\n\n## Main results\n\n* `IsStronglyProgressive.stoppedProcess`: the stopped process of a progressively measurable process\n  is progressively measurable.\n* `memLp_stoppedProcess`: if a process belongs to `ℒp` at every time in `ℕ`, then its stopped\n  process belongs to `ℒp` as well.\n\n## Implementation notes\n\nFor a filtration on a type `ι`, we define stopping times as functions from the measurable space `Ω`\nto `WithTop ι`, which allows stopping times that can take an infinite value, represented by\n`⊤ : WithTop ι`.\n\nThis means that if we have a process `X : ι → Ω → β` and a stopping time `τ : Ω → WithTop ι`, then\nto consider the value of `X` at the stopping time `τ ω`, we need to write `X (τ ω).untopA ω`,\nin which `(τ ω).untopA` is the value of `τ ω` in `ι` if `τ ω ≠ ⊤` and some arbitrary value if\n`τ ω = ⊤`.\n\nWhile indexing would be more convenient if we defined stopping times as functions from `Ω` to `ι`,\nthis would prevent us from using stopping times as in standard mathematical literature, where a\ntypical example of stopping time is the first time an event occurs, which may never happen.\nConsider for example the first time a coin lands heads when flipping it infinitely many times:\nthis is almost surely finite, but possibly infinite. We could also not use a function `Ω → ι` with\narbitrary value for the infinite case, because this would be incompatible with the stopping time\nproperty.\n\n## Tags\n\nstopping time, stochastic process\n\n-/\n\n@[expose] public section\n\nopen Filter Order TopologicalSpace WithTop\n\nopen scoped MeasureTheory NNReal ENNReal Topology\n\nnamespace MeasureTheory\n\nvariable {Ω β ι : Type*} {m : MeasurableSpace Ω}\n\n/-! ### Stopping times -/\n\n\n/-- A stopping time with respect to some filtration `f` is a function\n`τ` such that for all `i`, the preimage of `{j | j ≤ i}` along `τ` is measurable\nwith respect to `f i`.\n\nIntuitively, the stopping time `τ` describes some stopping rule such that at time\n`i`, we may determine it with the information we have at time `i`. -/\ndef IsStoppingTime [Preorder ι] (f : Filtration ι m) (τ : Ω → WithTop ι) :=\n  ∀ i : ι, MeasurableSet[f i] <| {ω | τ ω ≤ i}\n\ntheorem isStoppingTime_const [Preorder ι] (f : Filtration ι m) (i : ι) :\n    IsStoppingTime f fun _ => i := fun j => by simp only [MeasurableSet.const]\n\nsection MeasurableSet\n\nsection Preorder\n\nvariable [Preorder ι] {f : Filtration ι m} {τ : Ω → WithTop ι}\n\nprotected theorem IsStoppingTime.measurableSet_le (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω ≤ i} :=\n  hτ i\n\ntheorem IsStoppingTime.measurableSet_lt_of_pred [PredOrder ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω < i} := by\n  by_cases hi_min : IsMin i\n  · suffices {ω : Ω | τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i)\n    ext1 ω\n    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]\n    rw [isMin_iff_forall_not_lt] at hi_min\n    cases τ ω with\n    | top => simp\n    | coe t => exact mod_cast hi_min t\n  have : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iic (pred i : ι) := by\n    ext ω\n    push _ ∈ _\n    cases τ ω with\n    | top => simp\n    | coe t =>\n      simp only [coe_lt_coe, coe_le_coe]\n      rw [le_pred_iff_of_not_isMin hi_min]\n  rw [this]\n  exact f.mono (pred_le i) _ (hτ.measurableSet_le <| pred i)\n\nend Preorder\n\nsection CountableStoppingTime\n\nnamespace IsStoppingTime\n\nvariable [PartialOrder ι] {τ : Ω → WithTop ι} {f : Filtration ι m}\n\nprotected theorem measurableSet_eq_of_countable_range (hτ : IsStoppingTime f τ)\n    (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω | τ ω = i} := by\n  have : {ω | τ ω = i} = {ω | τ ω ≤ i} \\ ⋃ (j ∈ Set.range τ) (_ : j < i), {ω | τ ω ≤ j} := by\n    ext1 a\n    simp only [Set.mem_setOf_eq, Set.mem_range, Set.iUnion_exists, Set.iUnion_iUnion_eq',\n      Set.mem_sdiff, Set.mem_iUnion, exists_prop, not_exists, not_and]\n    constructor <;> intro h\n    · simp only [h, lt_iff_le_not_ge, le_refl, and_imp, imp_self, imp_true_iff, and_self_iff]\n    · exact h.1.eq_or_lt.resolve_right fun h_lt => h.2 a h_lt le_rfl\n  rw [this]\n  refine (hτ.measurableSet_le i).diff ?_\n  refine MeasurableSet.biUnion h_countable fun j _ => ?_\n  classical\n  rw [Set.iUnion_eq_if]\n  split_ifs with hji\n  · lift j to ι using (ne_top_of_lt hji)\n    exact f.mono (mod_cast hji.le) _ (hτ.measurableSet_le j)\n  · exact @MeasurableSet.empty _ (f i)\n\nprotected theorem measurableSet_eq_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω = i} :=\n  hτ.measurableSet_eq_of_countable_range (Set.to_countable _) i\n\nprotected theorem measurableSet_lt_of_countable_range (hτ : IsStoppingTime f τ)\n    (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω | τ ω < i} := by\n  have : {ω | τ ω < i} = {ω | τ ω ≤ i} \\ {ω | τ ω = i} := by ext1 ω; simp [lt_iff_le_and_ne]\n  rw [this]\n  exact (hτ.measurableSet_le i).diff (hτ.measurableSet_eq_of_countable_range h_countable i)\n\nprotected theorem measurableSet_lt_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω < i} :=\n  hτ.measurableSet_lt_of_countable_range (Set.to_countable _) i\n\nprotected theorem measurableSet_ge_of_countable_range {ι} [LinearOrder ι] {τ : Ω → WithTop ι}\n    {f : Filtration ι m} (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) :\n    MeasurableSet[f i] {ω | i ≤ τ ω} := by\n  have : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ := by\n    ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt]\n  rw [this]\n  exact (hτ.measurableSet_lt_of_countable_range h_countable i).compl\n\nprotected theorem measurableSet_ge_of_countable {ι} [LinearOrder ι] {τ : Ω → WithTop ι}\n    {f : Filtration ι m} [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | i ≤ τ ω} :=\n  hτ.measurableSet_ge_of_countable_range (Set.to_countable _) i\n\nend IsStoppingTime\n\nend CountableStoppingTime\n\nsection LinearOrder\n\nvariable [LinearOrder ι] {f : Filtration ι m} {τ : Ω → WithTop ι}\n\ntheorem IsStoppingTime.measurableSet_gt (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | i < τ ω} := by\n  have : {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ := by\n    ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_le]\n  rw [this]\n  exact (hτ.measurableSet_le i).compl\n\nsection TopologicalSpace\n\nvariable [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι]\n\n/-- Auxiliary lemma for `MeasureTheory.IsStoppingTime.measurableSet_lt`. -/\ntheorem IsStoppingTime.measurableSet_lt_of_isLUB (hτ : IsStoppingTime f τ) (i : ι)\n    (h_lub : IsLUB (Set.Iio i) i) : MeasurableSet[f i] {ω | τ ω < i} := by\n  by_cases hi_min : IsMin i\n  · suffices {ω | τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i)\n    ext1 ω\n    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]\n    cases τ ω with\n    | top => simp\n    | coe t => norm_cast; exact isMin_iff_forall_not_lt.mp hi_min t\n  obtain ⟨seq, -, -, h_tendsto, h_bound⟩ :\n      ∃ seq : ℕ → ι, Monotone seq ∧ (∀ j, seq j ≤ i) ∧ Tendsto seq atTop (𝓝 i) ∧ ∀ j, seq j < i :=\n    h_lub.exists_seq_monotone_tendsto (not_isMin_iff.mp hi_min)\n  have h_Iio_eq_Union : Set.Iio (i : WithTop ι) = ⋃ j, {k : WithTop ι | k ≤ seq j} := by\n    ext1 k\n    push _ ∈ _\n    refine ⟨fun hk_lt_i => ?_, fun h_exists_k_le_seq => ?_⟩\n    · rw [tendsto_atTop'] at h_tendsto\n      cases k with\n      | top => simp at hk_lt_i\n      | coe k =>\n        norm_cast at hk_lt_i ⊢\n        have h_nhds : Set.Ici k ∈ 𝓝 i :=\n          mem_nhds_iff.mpr ⟨Set.Ioi k, Set.Ioi_subset_Ici le_rfl, isOpen_Ioi, hk_lt_i⟩\n        obtain ⟨a, ha⟩ : ∃ a : ℕ, ∀ b : ℕ, b ≥ a → k ≤ seq b := h_tendsto (Set.Ici k) h_nhds\n        exact ⟨a, ha a le_rfl⟩\n    · obtain ⟨j, hk_seq_j⟩ := h_exists_k_le_seq\n      exact hk_seq_j.trans_lt (mod_cast h_bound j)\n  have h_lt_eq_preimage : {ω | τ ω < i} = τ ⁻¹' Set.Iio i := by\n    ext1 ω; push _ ∈ _; rfl\n  rw [h_lt_eq_preimage, h_Iio_eq_Union]\n  simp only [Set.preimage_iUnion, Set.preimage_setOf_eq]\n  exact MeasurableSet.iUnion fun n => f.mono (h_bound n).le _ (hτ.measurableSet_le (seq n))\n\ntheorem IsStoppingTime.measurableSet_lt (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω < i} := by\n  obtain ⟨i', hi'_lub⟩ : ∃ i', IsLUB (Set.Iio i) i' := exists_lub_Iio i\n  rcases lub_Iio_eq_self_or_Iio_eq_Iic i hi'_lub with hi'_eq_i | h_Iio_eq_Iic\n  · rw [← hi'_eq_i] at hi'_lub ⊢\n    exact hτ.measurableSet_lt_of_isLUB i' hi'_lub\n  · have h_lt_eq_preimage : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iio i := rfl\n    have h_Iio_eq_Iic' : Set.Iio (i : WithTop ι) = Set.Iic (i' : WithTop ι) := by\n      rw [← image_coe_Iio, ← image_coe_Iic, h_Iio_eq_Iic]\n    rw [h_lt_eq_preimage, h_Iio_eq_Iic']\n    exact f.mono (le_of_isLUB_Iio i hi'_lub) _ (hτ.measurableSet_le i')\n\ntheorem IsStoppingTime.measurableSet_ge (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | i ≤ τ ω} := by\n  have : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ := by\n    ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt]\n  rw [this]\n  exact (hτ.measurableSet_lt i).compl\n\ntheorem IsStoppingTime.measurableSet_eq (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω = i} := by\n  have : {ω | τ ω = i} = {ω | τ ω ≤ i} ∩ {ω | τ ω ≥ i} := by\n    ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_inter_iff, le_antisymm_iff]\n  rw [this]\n  exact (hτ.measurableSet_le i).inter (hτ.measurableSet_ge i)\n\ntheorem IsStoppingTime.measurableSet_eq_le (hτ : IsStoppingTime f τ) {i j : ι} (hle : i ≤ j) :\n    MeasurableSet[f j] {ω | τ ω = i} :=\n  f.mono hle _ <| hτ.measurableSet_eq i\n\ntheorem IsStoppingTime.measurableSet_lt_le (hτ : IsStoppingTime f τ) {i j : ι} (hle : i ≤ j) :\n    MeasurableSet[f j] {ω | τ ω < i} :=\n  f.mono hle _ <| hτ.measurableSet_lt i\n\nend TopologicalSpace\n\nend LinearOrder\n\nsection Countable\n\nTarget:\ntheorem isStoppingTime_of_measurableSet_eq [Preorder ι] [Countable ι] {f : Filtration ι m}\n    {τ : Ω → WithTop ι} (hτ : ∀ i, MeasurableSet[f i] {ω | τ ω = i}) : IsStoppingTime f τ :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Process","family_id":"isstoppingtime_of_measurableset_eq","file_id":"mathlib/Mathlib/Probability/Process/Stopping.lean","sample_id":"63d4c6f7bdfaad78a31356f5390bcb2b6c5c4366f0f8ad12785fcd4685b1c7e7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5cfc8f87ea330fcf416830673402289862153b1294b56a5e4d961dfa30463860","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ae5b4930d39b46c6599bc819ee08d84d74c21f7814376e5d0bf28dd0aa423e48","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a931f0b3f033fc32c017444ea0eb1a5592318ecdbcdc093591ad2758f58c4528","source_sha256":"dc37d7e189f3a4d32e4029e0df16c9eeb5ca8b81845b68289b84bb8e5dd20ee8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h => .of_comp (f := ({·} : β → Compacts β)) ?_, map_injective hf⟩\n  simp_rw [Function.comp_def, ← map_singleton hf]\n  exact h.comp singleton_injective","hard_negative":false,"metrics":{"chosen_tokens":43,"rejected_tokens":48,"token_jaccard":0.897436,"token_length_ratio":1.116279},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"adb86cc875c6377ada0cf61e174f3ca4c0b8f1e8c91bdac97aab8eb6b33796e2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Sets.Closeds\npublic import Mathlib.Topology.QuasiSeparated\n\nNamespace:\nTopologicalSpace.Compacts\n\nLocal context:\n/-\nCopyright (c) 2020 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Yaël Dillies\n-/\n/-!\n# Compact sets\n\nWe define a few types of compact sets in a topological space.\n\n## Main Definitions\n\nFor a topological space `α`,\n* `TopologicalSpace.Compacts α`: The type of compact sets.\n* `TopologicalSpace.NonemptyCompacts α`: The type of non-empty compact sets.\n* `TopologicalSpace.PositiveCompacts α`: The type of compact sets with non-empty interior.\n* `TopologicalSpace.CompactOpens α`: The type of compact open sets. This is a central object in the\n  study of spectral spaces.\n-/\n\n@[expose] public section\n\n\nopen Set\n\nvariable {α β γ : Type*} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]\n\nnamespace TopologicalSpace\n\n/-! ### Compact sets -/\n\n/-- The type of compact sets of a topological space. -/\nstructure Compacts (α : Type*) [TopologicalSpace α] where\n  /-- the carrier set, i.e. the points in this set -/\n  carrier : Set α\n  isCompact' : IsCompact carrier\n\nnamespace Compacts\n\ninstance : SetLike (Compacts α) α where\n  coe := Compacts.carrier\n  coe_injective s t h := by cases s; cases t; congr\n\ninstance : PartialOrder (Compacts α) := .ofSetLike (Compacts α) α\n\n/-- See Note [custom simps projection]. -/\ndef Simps.coe (s : Compacts α) : Set α := s\n\ninitialize_simps_projections Compacts (carrier → coe, as_prefix coe)\n\nprotected theorem isCompact (s : Compacts α) : IsCompact (s : Set α) :=\n  s.isCompact'\n\ninstance (K : Compacts α) : CompactSpace K :=\n  isCompact_iff_compactSpace.1 K.isCompact\n\n/-- Reinterpret a compact as a closed set. -/\n@[simps]\ndef toCloseds [T2Space α] (s : Compacts α) : Closeds α :=\n  ⟨s, s.isCompact.isClosed⟩\n\n@[simp]\ntheorem mem_toCloseds [T2Space α] {x : α} {s : Compacts α} :\n    x ∈ s.toCloseds ↔ x ∈ s :=\n  Iff.rfl\n\ntheorem toCloseds_injective [T2Space α] : Function.Injective (toCloseds (α := α)) :=\n  .of_comp (f := SetLike.coe) SetLike.coe_injective\n\ninstance : CanLift (Set α) (Compacts α) (↑) IsCompact where prf K hK := ⟨⟨K, hK⟩, rfl⟩\n\n@[ext]\nprotected theorem ext {s t : Compacts α} (h : (s : Set α) = t) : s = t :=\n  SetLike.ext' h\n\n@[simp]\ntheorem coe_mk (s : Set α) (h) : (mk s h : Set α) = s :=\n  rfl\n\n@[simp]\ntheorem carrier_eq_coe (s : Compacts α) : s.carrier = s :=\n  rfl\n\ninstance : Max (Compacts α) :=\n  ⟨fun s t => ⟨s ∪ t, s.isCompact.union t.isCompact⟩⟩\n\ninstance [T2Space α] : Min (Compacts α) :=\n  ⟨fun s t => ⟨s ∩ t, s.isCompact.inter t.isCompact⟩⟩\n\ninstance [CompactSpace α] : Top (Compacts α) :=\n  ⟨⟨univ, isCompact_univ⟩⟩\n\ninstance : Bot (Compacts α) :=\n  ⟨⟨∅, isCompact_empty⟩⟩\n\ninstance : SemilatticeSup (Compacts α) :=\n  fast_instance% SetLike.coe_injective.semilatticeSup _ .rfl .rfl fun _ _ ↦ rfl\n\ninstance [T2Space α] : DistribLattice (Compacts α) :=\n  fast_instance% SetLike.coe_injective.distribLattice _ .rfl .rfl (fun _ _ ↦ rfl) fun _ _ ↦ rfl\n\ninstance : OrderBot (Compacts α) :=\n  fast_instance% OrderBot.lift ((↑) : _ → Set α) (fun _ _ => id) rfl\n\ninstance [CompactSpace α] : BoundedOrder (Compacts α) :=\n  fast_instance% BoundedOrder.lift ((↑) : _ → Set α) (fun _ _ => id) rfl rfl\n\n/-- The type of compact sets is inhabited, with default element the empty set. -/\ninstance : Inhabited (Compacts α) := ⟨⊥⟩\n\ninstance [IsEmpty α] : Unique (Compacts α) where\n  uniq _ := Compacts.ext (Subsingleton.elim _ _)\n\n@[simp]\ntheorem coe_sup (s t : Compacts α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t :=\n  rfl\n\n@[simp]\ntheorem coe_inf [T2Space α] (s t : Compacts α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t :=\n  rfl\n\n@[simp]\ntheorem coe_top [CompactSpace α] : (↑(⊤ : Compacts α) : Set α) = univ :=\n  rfl\n\n@[simp]\ntheorem coe_bot : (↑(⊥ : Compacts α) : Set α) = ∅ :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_eq_empty {s : Compacts α} : (s : Set α) = ∅ ↔ s = ⊥ :=\n  SetLike.coe_injective.eq_iff' rfl\n\n@[simp]\ntheorem coe_nonempty {s : Compacts α} : (s : Set α).Nonempty ↔ s ≠ ⊥ :=\n  nonempty_iff_ne_empty.trans coe_eq_empty.not\n\n@[simp]\ntheorem coe_finset_sup {ι : Type*} {s : Finset ι} {f : ι → Compacts α} :\n    (↑(s.sup f) : Set α) = s.sup fun i => ↑(f i) := by\n  refine Finset.cons_induction_on s rfl fun a s _ h => ?_\n  simp_rw [Finset.sup_cons, coe_sup, sup_eq_union]\n  congr\n\n@[simps]\ninstance : Singleton α (Compacts α) where\n  singleton x := ⟨{x}, isCompact_singleton⟩\n\n@[simp]\ntheorem mem_singleton (x y : α) : x ∈ ({y} : Compacts α) ↔ x = y :=\n  Iff.rfl\n\n@[simp]\ntheorem toCloseds_singleton [T2Space α] (x : α) : toCloseds {x} = {x} :=\n  rfl\n\ntheorem singleton_injective : Function.Injective ({·} : α → Compacts α) :=\n  .of_comp (f := SetLike.coe) Set.singleton_injective\n\n@[simp]\ntheorem singleton_inj {x y : α} : ({x} : Compacts α) = {y} ↔ x = y :=\n  singleton_injective.eq_iff\n\ninstance [Nonempty α] : Nontrivial (Compacts α) := by\n  constructor\n  obtain ⟨x⟩ := ‹Nonempty α›\n  exact ⟨⊥, {x}, ne_of_apply_ne SetLike.coe (Set.empty_ne_singleton x)⟩\n\n@[simp]\ntheorem subsingleton_iff : Subsingleton (Compacts α) ↔ IsEmpty α := by\n  refine ⟨fun h => ?_, fun _ => inferInstance⟩\n  contrapose! h\n  infer_instance\n\n@[simp]\ntheorem nontrivial_iff : Nontrivial (Compacts α) ↔ Nonempty α := by\n  rw [← not_subsingleton_iff_nontrivial, subsingleton_iff, not_isEmpty_iff]\n\n/-- The image of a compact set under a continuous function. -/\nprotected def map (f : α → β) (hf : Continuous f) (K : Compacts α) : Compacts β :=\n  ⟨f '' K.1, K.2.image hf⟩\n\n@[simp, norm_cast]\ntheorem coe_map {f : α → β} (hf : Continuous f) (s : Compacts α) : (s.map f hf : Set β) = f '' s :=\n  rfl\n\n@[simp]\ntheorem map_id (K : Compacts α) : K.map id continuous_id = K :=\n  Compacts.ext <| Set.image_id _\n\ntheorem map_comp (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g) (K : Compacts α) :\n    K.map (f ∘ g) (hf.comp hg) = (K.map g hg).map f hf :=\n  Compacts.ext <| Set.image_comp _ _ _\n\ntheorem map_injective {f : α → β} (hf : Continuous f) (hf' : Function.Injective f) :\n    Function.Injective (Compacts.map f hf) :=\n  .of_comp (f := SetLike.coe) <| hf'.image_injective.comp SetLike.coe_injective\n\n@[simp]\ntheorem map_singleton {f : α → β} (hf : Continuous f) (x : α) : Compacts.map f hf {x} = {f x} :=\n  Compacts.ext Set.image_singleton\n\n@[simp]\n\nTarget:\ntheorem map_injective_iff {f : α → β} (hf : Continuous f) :\n    Function.Injective (Compacts.map f hf) ↔ Function.Injective f :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  refine ⟨fun h => .of_comp (f := ({·} : β → Compacts β)) ?_, map_injective hf⟩\n  simp_rw [Function.comp_def, ← map_singleton hf]\n  exact h.comp singleton_injective","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Sets","family_id":"map_injective_iff","file_id":"mathlib/Mathlib/Topology/Sets/Compacts.lean","sample_id":"a931f0b3f033fc32c017444ea0eb1a5592318ecdbcdc093591ad2758f58c4528"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d110cfbf729fa1ead7d2f77345ddd3124e402db97adfe146f40e24d032256864","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ebbba5759c869c22b0a2b53d7b28f55271b278697a38f30616e920a0848b78b1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"099a68aecd4c932bd014347b7efcc8259e886e53c407723e803e298e292ed9b6","source_sha256":"e14d2dddf8bf8e732d384fcfc239ee89a73c0a738a4da1d1d8efcb0c46627981","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := RingHom.toMorphismProperty_respectsIso_iff.mp RingHom.isIntegral_respectsIso\n  rw [HasAffineProperty.iff_of_isAffine (P := @IsIntegralHom), and_iff_right]\n  exacts [MorphismProperty.arrow_mk_iso_iff (RingHom.toMorphismProperty RingHom.IsIntegral)\n    (arrowIsoΓSpecOfIsAffine φ).symm, inferInstance]","hard_negative":true,"metrics":{"chosen_tokens":49,"rejected_tokens":3,"token_jaccard":0.030303,"token_length_ratio":0.061224},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"ae43b96cf08f0c8939f7150d82c133ffbe1d64219c741cad04c9ffe1f8f2fd23","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.Separated\npublic import Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed\npublic import Mathlib.RingTheory.RingHom.Integral\n\nNamespace:\nAlgebraicGeometry.IsIntegralHom\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Integral morphisms of schemes\n\nA morphism of schemes `f : X ⟶ Y` is integral if the preimage\nof an arbitrary affine open subset of `Y` is affine and the induced ring map is integral.\n\nIt is equivalent to ask only that `Y` is covered by affine opens whose preimage is affine\nand the induced ring map is integral.\n\n-/\n\npublic section\n\nuniverse v u\n\nopen CategoryTheory TopologicalSpace Opposite MorphismProperty\n\nnamespace AlgebraicGeometry\n\n/-- A morphism of schemes `X ⟶ Y` is integral if the preimage of any affine open subset of `Y` is\naffine and the induced ring hom on sections is integral. -/\n@[mk_iff]\nclass IsIntegralHom {X Y : Scheme} (f : X ⟶ Y) : Prop extends IsAffineHom f where\n  isIntegral_app (f) (U : Y.Opens) (hU : IsAffineOpen U) : (f.app U).hom.IsIntegral\n\nalias Scheme.Hom.isIntegral_app := IsIntegralHom.isIntegral_app\n\nnamespace IsIntegralHom\n\nvariable {X Y Z S : Scheme.{u}}\n\ninstance hasAffineProperty : HasAffineProperty @IsIntegralHom\n    fun X _ f _ ↦ IsAffine X ∧ RingHom.IsIntegral (f.app ⊤).hom := by\n  change HasAffineProperty @IsIntegralHom (affineAnd RingHom.IsIntegral)\n  rw [HasAffineProperty.affineAnd_iff _ RingHom.isIntegral_respectsIso\n    RingHom.isIntegral_isStableUnderBaseChange.localizationPreserves.away\n    RingHom.isIntegral_ofLocalizationSpan]\n  simp [isIntegralHom_iff]\n\ninstance : IsStableUnderComposition @IsIntegralHom :=\n  HasAffineProperty.affineAnd_isStableUnderComposition (Q := RingHom.IsIntegral) hasAffineProperty\n    RingHom.isIntegral_stableUnderComposition\n\ninstance : IsStableUnderBaseChange @IsIntegralHom :=\n  HasAffineProperty.affineAnd_isStableUnderBaseChange (Q := RingHom.IsIntegral) hasAffineProperty\n    RingHom.isIntegral_respectsIso RingHom.isIntegral_isStableUnderBaseChange\n\ninstance (priority := low) (f : X ⟶ Y) [IsClosedImmersion f] : IsIntegralHom f where\n  isIntegral_app U hU := (RingHom.Finite.of_surjective _ (f.app_surjective U hU)).to_isIntegral\n\ninstance : IsMultiplicative @IsIntegralHom where\n  id_mem _ := inferInstance\n\ninstance (f : X ⟶ Y) (g : Y ⟶ Z) [IsIntegralHom f] [IsIntegralHom g] : IsIntegralHom (f ≫ g) :=\n  MorphismProperty.comp_mem _ _ _ ‹_› ‹_›\n\ninstance (f : X ⟶ S) (g : Y ⟶ S) [IsIntegralHom g] : IsIntegralHom (Limits.pullback.fst f g) :=\n  MorphismProperty.pullback_fst f g inferInstance\n\ninstance (f : X ⟶ S) (g : Y ⟶ S) [IsIntegralHom f] : IsIntegralHom (Limits.pullback.snd f g) :=\n  MorphismProperty.pullback_snd f g inferInstance\n\ninstance (f : X ⟶ Y) (V : Y.Opens) [IsIntegralHom f] : IsIntegralHom (f ∣_ V) :=\n  IsZariskiLocalAtTarget.restrict ‹_› V\n\ninstance : MorphismProperty.HasOfPostcompProperty @IsIntegralHom @IsSeparated :=\n  MorphismProperty.hasOfPostcompProperty_iff_le_diagonal.mpr\n    fun _ _ _ _ ↦ inferInstanceAs (IsIntegralHom _)\n\nlemma of_comp (f : X ⟶ Y) (g : Y ⟶ Z) [IsIntegralHom (f ≫ g)] [IsSeparated g] :\n    IsIntegralHom f := MorphismProperty.of_postcomp _ _ g ‹_› ‹_›\n\nlemma comp_iff {f : X ⟶ Y} {g : Y ⟶ Z} [IsIntegralHom g] :\n    IsIntegralHom (f ≫ g) ↔ IsIntegralHom f :=\n  ⟨fun _ ↦ .of_comp f g, fun _ ↦ inferInstance⟩\n\nTarget:\nlemma SpecMap_iff {R S : CommRingCat} {φ : R ⟶ S} :\n    IsIntegralHom (Spec.map φ) ↔ φ.hom.IsIntegral :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_099a68aecd4c","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f85e18ecc80b27190dbeafa6808df25a09153b6e2171842ef242f03abc507a40","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Morphisms","family_id":"specmap_iff","file_id":"mathlib/Mathlib/AlgebraicGeometry/Morphisms/Integral.lean","sample_id":"099a68aecd4c932bd014347b7efcc8259e886e53c407723e803e298e292ed9b6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8b4eba4a7d76ab6115b8794619b18b2fd26643e7d931d0fcde4744d29b34df3e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3eb24946d7afb6ed4e9945af45b2ccb512ce0450e76dfc41dd46ff1f9072c845","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"40d4ba4e788dd7fbc85ecb01598bd57d742b372ed1f0e1d36e21ca910229b975","source_sha256":"6ea826f0633d3930f7872110f571f16f76059ec3668e205ef02a1089b9b31873","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n    rw [← card_image_of_injective s' Subtype.coe_injective]\n    convert! ht (s'.image fun z => z.1) using 1\n    apply congr_arg\n    ext y\n    simp","hard_negative":true,"metrics":{"chosen_tokens":32,"rejected_tokens":3,"token_jaccard":0.034483,"token_length_ratio":0.09375},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"aed6d954183d7f2b7be63285f408ee1fb6d9c57f1aab635927b150d35c5b39f6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Basic\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Data.Set.Finite.Basic\n\nNamespace:\nHallMarriageTheorem\n\nLocal context:\n/-\nCopyright (c) 2021 Alena Gusakov, Bhavik Mehta, Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alena Gusakov, Bhavik Mehta, Kyle Miller\n-/\n/-!\n# Hall's Marriage Theorem for finite index types\n\nThis module proves the basic form of Hall's theorem.\nIn contrast to the theorem described in `Combinatorics.Hall.Basic`, this\nversion requires that the indexed family `t : ι → Finset α` have `ι` be finite.\nThe `Combinatorics.Hall.Basic` module applies a compactness argument to this version\nto remove the `Finite` constraint on `ι`.\n\nThe modules are split like this since the generalized statement\ndepends on the topology and category theory libraries, but the finite\ncase in this module has few dependencies.\n\nA description of this formalization is in [Gusakov2021].\n\n## Main statements\n\n* `Finset.all_card_le_biUnion_card_iff_existsInjective'` is Hall's theorem with\n  a finite index set.  This is elsewhere generalized to\n  `Finset.all_card_le_biUnion_card_iff_existsInjective`.\n\n## Tags\n\nHall's Marriage Theorem, indexed families\n-/\n\npublic section\n\n\nopen Finset\n\nuniverse u v\n\nnamespace HallMarriageTheorem\n\nvariable {ι : Type u} {α : Type v} [DecidableEq α] {t : ι → Finset α}\n\nsection Fintype\n\nvariable [Fintype ι]\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem hall_cond_of_erase {x : ι} (a : α)\n    (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t))\n    (s' : Finset { x' : ι | x' ≠ x }) : #s' ≤ #(s'.biUnion fun x' => (t x').erase a) := by\n  haveI := Classical.decEq ι\n  specialize ha (s'.image fun z => z.1)\n  rw [image_nonempty, Finset.card_image_of_injective s' Subtype.coe_injective] at ha\n  by_cases! he : s'.Nonempty\n  · have ha' : #s' < #(s'.biUnion fun x => t x) := by\n      convert! ha he fun h => by simpa [← h] using mem_univ x using 2\n      ext x\n      simp only [mem_image, mem_biUnion, SetCoe.exists, exists_and_right,\n        exists_eq_right]\n    rw [← erase_biUnion]\n    by_cases hb : a ∈ s'.biUnion fun x => t x\n    · rw [card_erase_of_mem hb]\n      exact Nat.le_sub_one_of_lt ha'\n    · rw [erase_eq_of_notMem hb]\n      exact Nat.le_of_lt ha'\n  · subst s'\n    simp\n\n/-- First case of the inductive step: assuming that\n`∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)`\nand that the statement of **Hall's Marriage Theorem** is true for all\n`ι'` of cardinality ≤ `n`, then it is true for `ι` of cardinality `n + 1`.\n-/\ntheorem hall_hard_inductive_step_A {n : ℕ} (hn : Fintype.card ι = n + 1)\n    (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))\n    (ih :\n      ∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),\n        Fintype.card ι' ≤ n →\n          (∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →\n            ∃ f : ι' → α, Function.Injective f ∧ ∀ x, f x ∈ t' x)\n    (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t)) :\n    ∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by\n  haveI : Nonempty ι := Fintype.card_pos_iff.mp (hn.symm ▸ Nat.succ_pos _)\n  haveI := Classical.decEq ι\n  -- Choose an arbitrary element `x : ι` and `y : t x`.\n  let x := Classical.arbitrary ι\n  have tx_ne : (t x).Nonempty := by\n    rw [← Finset.card_pos]\n    calc\n      0 < 1 := Nat.one_pos\n      _ ≤ #(.biUnion {x} t) := ht {x}\n      _ = (t x).card := by rw [Finset.singleton_biUnion]\n  choose y hy using tx_ne\n  -- Restrict to everything except `x` and `y`.\n  let ι' := { x' : ι | x' ≠ x }\n  let t' : ι' → Finset α := fun x' => (t x').erase y\n  have card_ι' : Fintype.card ι' = n :=\n    calc\n      Fintype.card ι' = Fintype.card ι - 1 := Set.card_ne_eq _\n      _ = n := by rw [hn, Nat.add_succ_sub_one, add_zero]\n  rcases ih t' card_ι'.le (hall_cond_of_erase y ha) with ⟨f', hfinj, hfr⟩\n  -- Extend the resulting function.\n  refine ⟨fun z => if h : z = x then y else f' ⟨z, h⟩, ?_, ?_⟩\n  · rintro z₁ z₂\n    have key : ∀ {x}, y ≠ f' x := by\n      intro x h\n      simpa [t', ← h] using hfr x\n    by_cases h₁ : z₁ = x <;> by_cases h₂ : z₂ = x <;>\n      simp [h₁, h₂, hfinj.eq_iff, key, key.symm]\n  · intro z\n    simp only\n    split_ifs with hz\n    · rwa [hz]\n    · specialize hfr ⟨z, hz⟩\n      rw [mem_erase] at hfr\n      exact hfr.2\n\nTarget:\ntheorem hall_cond_of_restrict {ι : Type u} {t : ι → Finset α} {s : Finset ι}\n    (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t)) (s' : Finset (s : Set ι)) :\n    #s' ≤ #(s'.biUnion fun a' => t a') :=\n\nProof body:\n","rejected":"by\n  exact hall_cond_of_restrict","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"282836e62cc7826b4d35289f8c2a38c8213b53d0f6dd968e4198fe188b33193e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Hall","family_id":"hall_cond_of_restrict","file_id":"mathlib/Mathlib/Combinatorics/Hall/Finite.lean","sample_id":"40d4ba4e788dd7fbc85ecb01598bd57d742b372ed1f0e1d36e21ca910229b975"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b54a12dda5879cff4ee4a1d4e164d0430717d0b1ed9dc7072b52b58f4f7965cc","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cc46ae0cd6fbf8b5c482692a262adbee52fa600011e9053140abc53478aa5ddd","source_sha256":"3cc5f24e166336d305cafd0a6e88d1acf7bd4033e91b083c3ca5b2a6bcc3c9bc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine eq_bot_iff.2 (le_trans jacobson_bot_polynomial_le_sInf_map_maximal ?_)\n  refine fun f hf => (Submodule.mem_bot R[X]).2 <| Polynomial.ext fun n =>\n    Trans.trans (?_ : coeff f n = 0) (coeff_zero n).symm\n  suffices f.coeff n ∈ Ideal.jacobson ⊥ by rwa [h, Submodule.mem_bot] at this\n  exact mem_sInf.2 fun j hj => (mem_map_C_iff.1 ((mem_sInf.1 hf) ⟨j, ⟨hj.2, rfl⟩⟩)) n","hard_negative":false,"metrics":{"chosen_tokens":108,"rejected_tokens":2,"token_jaccard":0.037736,"token_length_ratio":0.018519},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"aef03cd48d0139ad7f0a3eeaddbeb26c9213a004649c509b56a1681e4541ad96","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Jacobson.Ideal\npublic import Mathlib.RingTheory.Polynomial.Quotient\n\nNamespace:\nIdeal\n\nLocal context:\n/-\nCopyright (c) 2020 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\n/-!\n# Jacobson radical of polynomial ring\n\n-/\n\npublic section\n\nnamespace Ideal\n\nsection Polynomial\n\nopen Polynomial\n\nvariable {R : Type*} [CommRing R]\n\ntheorem jacobson_bot_polynomial_le_sInf_map_maximal :\n    jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by\n  refine le_sInf fun J => exists_imp.2 fun j hj => ?_\n  haveI : j.IsMaximal := hj.1\n  refine Trans.trans (jacobson_mono bot_le) (le_of_eq ?_ : J.jacobson ≤ J)\n  suffices t : (⊥ : Ideal (Polynomial (R ⧸ j))).jacobson = ⊥ by\n    rw [← hj.2, jacobson_eq_iff_jacobson_quotient_eq_bot]\n    replace t := congr_arg (map (polynomialQuotientEquivQuotientPolynomial j).toRingHom) t\n    rwa [map_jacobson_of_bijective _, map_bot] at t\n    exact RingEquiv.bijective (polynomialQuotientEquivQuotientPolynomial j)\n  refine eq_bot_iff.2 fun f hf => ?_\n  have r1 : (X : (R ⧸ j)[X]) ≠ 0 := ne_of_apply_ne (coeff · 1) <| by simp\n  simpa [r1] using eq_C_of_degree_eq_zero (degree_eq_zero_of_isUnit ((mem_jacobson_bot.1 hf) X))\n\nTarget:\ntheorem jacobson_bot_polynomial_of_jacobson_bot (h : jacobson (⊥ : Ideal R) = ⊥) :\n    jacobson (⊥ : Ideal R[X]) = ⊥ :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Jacobson","family_id":"jacobson_bot_polynomial_of_jacobson_bot","file_id":"mathlib/Mathlib/RingTheory/Jacobson/Polynomial.lean","sample_id":"cc46ae0cd6fbf8b5c482692a262adbee52fa600011e9053140abc53478aa5ddd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9a1a0b80645445024d8bb550e3e8810e677e51b85aec8657ff2f9f66fe77107a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"33d6750ddf63884a3db77c1ac0c0a14e4b5da0e25cd340d2ef99ffb5417edbe1","source_sha256":"70cf8c7f92181f101a609663b51cd6a85c94239ea2fd1fac014099a79ca682e9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isProperMap_iff_isClosedMap_and_compact_fibers]\n  refine ⟨f.continuous, f.isClosedMap, fun y ↦ ?_⟩\n  have := compactSpace_of_universallyClosed (pullback.snd f (Y.fromSpecResidueField y))\n  rw [← Scheme.range_fromSpecResidueField, ← Scheme.Pullback.range_fst]\n  exact isCompact_range (Scheme.Hom.continuous _)","hard_negative":true,"metrics":{"chosen_tokens":60,"rejected_tokens":8,"token_jaccard":0.076923,"token_length_ratio":0.133333},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"af03e155d216c8fb668e10c22cdfe0eaf90435cb353ddbfbc5398e15ecf92e42","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion\npublic import Mathlib.AlgebraicGeometry.PullbackCarrier\npublic import Mathlib.Topology.LocalAtTarget\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Universally closed morphism\n\nA morphism of schemes `f : X ⟶ Y` is universally closed if `X ×[Y] Y' ⟶ Y'` is a closed map\nfor all base change `Y' ⟶ Y`.\nThis implies that `f` is topologically proper (`AlgebraicGeometry.Scheme.Hom.isProperMap`).\n\nWe show that being universally closed is local at the target, and is stable under compositions and\nbase changes.\n\n-/\n\npublic section\n\n\nnoncomputable section\n\nopen CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace\n\nuniverse v u\n\nnamespace AlgebraicGeometry\n\nvariable {X Y : Scheme.{u}} (f : X ⟶ Y)\n\nopen CategoryTheory.MorphismProperty\n\n/-- A morphism of schemes `f : X ⟶ Y` is universally closed if the base change `X ×[Y] Y' ⟶ Y'`\nalong any morphism `Y' ⟶ Y` is (topologically) a closed map.\n-/\n@[mk_iff]\nclass UniversallyClosed (f : X ⟶ Y) : Prop where\n  universally_isClosedMap : universally (topologically @IsClosedMap) f\n\n@[deprecated (since := \"2026-01-20\")]\nalias UniversallyClosed.out := UniversallyClosed.universally_isClosedMap\n\nlemma Scheme.Hom.isClosedMap {X Y : Scheme} (f : X ⟶ Y) [UniversallyClosed f] :\n    IsClosedMap f := UniversallyClosed.universally_isClosedMap _ _ _ IsPullback.of_id_snd\n\ntheorem universallyClosed_eq : @UniversallyClosed = universally (topologically @IsClosedMap) := by\n  ext X Y f; rw [universallyClosed_iff]\n\ninstance (priority := 900) [IsClosedImmersion f] : UniversallyClosed f := by\n  rw [universallyClosed_eq]\n  intro X' Y' i₁ i₂ f' hf\n  have hf' : IsClosedImmersion f' :=\n    MorphismProperty.of_isPullback hf.flip inferInstance\n  exact f'.isClosedEmbedding.isClosedMap\n\ntheorem universallyClosed_respectsIso : RespectsIso @UniversallyClosed :=\n  universallyClosed_eq.symm ▸ universally_respectsIso (topologically @IsClosedMap)\n\ninstance universallyClosed_isStableUnderBaseChange : IsStableUnderBaseChange @UniversallyClosed :=\n  universallyClosed_eq.symm ▸ universally_isStableUnderBaseChange (topologically @IsClosedMap)\n\ninstance isClosedMap_isStableUnderComposition :\n    IsStableUnderComposition (topologically @IsClosedMap) where\n  comp_mem f g hf hg := IsClosedMap.comp (f := f) (g := g) hg hf\n\ninstance universallyClosed_isStableUnderComposition :\n    IsStableUnderComposition @UniversallyClosed := by\n  rw [universallyClosed_eq]\n  infer_instance\n\nlemma UniversallyClosed.of_comp_surjective {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)\n    [UniversallyClosed (f ≫ g)] [Surjective f] : UniversallyClosed g := by\n  constructor\n  intro X' Y' i₁ i₂ f' H\n  have := UniversallyClosed.universally_isClosedMap _ _ _\n    ((IsPullback.of_hasPullback i₁ f).paste_horiz H)\n  exact IsClosedMap.of_comp_surjective (MorphismProperty.pullback_fst (P := @Surjective) _ _ ‹_›).1\n    (Scheme.Hom.continuous _) this\n\ninstance universallyClosedTypeComp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)\n    [hf : UniversallyClosed f] [hg : UniversallyClosed g] : UniversallyClosed (f ≫ g) :=\n  comp_mem _ _ _ hf hg\n\ninstance : MorphismProperty.IsMultiplicative @UniversallyClosed where\n  id_mem _ := inferInstance\n\ninstance universallyClosed_fst {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hg : UniversallyClosed g] :\n    UniversallyClosed (pullback.fst f g) :=\n  MorphismProperty.pullback_fst f g hg\n\ninstance universallyClosed_snd {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hf : UniversallyClosed f] :\n    UniversallyClosed (pullback.snd f g) :=\n  MorphismProperty.pullback_snd f g hf\n\ninstance universallyClosed_isZariskiLocalAtTarget : IsZariskiLocalAtTarget @UniversallyClosed := by\n  rw [universallyClosed_eq]\n  apply universally_isZariskiLocalAtTarget\n  intro X Y f ι U hU H\n  simp_rw [topologically, morphismRestrict_base] at H\n  exact hU.isClosedMap_iff_restrictPreimage.mpr H\n\ninstance (f : X ⟶ Y) (V : Y.Opens) [UniversallyClosed f] : UniversallyClosed (f ∣_ V) :=\n  IsZariskiLocalAtTarget.restrict ‹_› V\n\nopen Scheme.Pullback _root_.PrimeSpectrum MvPolynomial in\n/-- If `X` is universally closed over a field, then `X` is quasi-compact. -/\nlemma compactSpace_of_universallyClosed\n    {K} [Field K] (f : X ⟶ Spec (.of K)) [UniversallyClosed f] : CompactSpace X := by\n  classical\n  let 𝒰 : X.OpenCover := X.affineCover\n  let U (i : 𝒰.I₀) : X.Opens := (𝒰.f i).opensRange\n  let T : Scheme := Spec (.of <| MvPolynomial 𝒰.I₀ K)\n  let q : T ⟶ Spec (.of K) := Spec.map (CommRingCat.ofHom MvPolynomial.C)\n  let Ti (i : 𝒰.I₀) : T.Opens := basicOpen (MvPolynomial.X i)\n  let fT : pullback f q ⟶ T := pullback.snd f q\n  let p : pullback f q ⟶ X := pullback.fst f q\n  let Z : Set (pullback f q :) := (⨆ i, fT ⁻¹ᵁ (Ti i) ⊓ p ⁻¹ᵁ (U i) : (pullback f q).Opens)ᶜ\n  have hZ : IsClosed Z := by\n    simp only [Z, isClosed_compl_iff, Opens.coe_iSup, Opens.coe_inf, Opens.map_coe]\n    exact isOpen_iUnion fun i ↦ (fT.continuous.1 _ (Ti i).2).inter (p.continuous.1 _ (U i).2)\n  let Zc : T.Opens := ⟨(fT '' Z)ᶜ, (fT.isClosedMap _ hZ).isOpen_compl⟩\n  let ψ : MvPolynomial 𝒰.I₀ K →ₐ[K] K := MvPolynomial.aeval (fun _ ↦ 1)\n  let t : T := Spec.map (CommRingCat.ofHom ψ.toRingHom) default\n  have ht (i : 𝒰.I₀) : t ∈ Ti i := show ψ (.X i) ≠ 0 by simp [ψ]\n  have htZc : t ∈ Zc := by\n    intro ⟨z, hz, hzt⟩\n    suffices ∃ i, fT z ∈ Ti i ∧ p z ∈ U i from hz (by simpa)\n    exact ⟨𝒰.idx (p z), hzt ▸ ht _, by simpa [U] using 𝒰.covers (p z)⟩\n  obtain ⟨U', ⟨g, rfl⟩, htU', hU'le⟩ := Opens.isBasis_iff_nbhd.mp isBasis_basic_opens htZc\n  let σ : Finset 𝒰.I₀ := MvPolynomial.vars g\n  let φ : MvPolynomial 𝒰.I₀ K →+* MvPolynomial 𝒰.I₀ K :=\n    (MvPolynomial.aeval fun i : 𝒰.I₀ ↦ if i ∈ σ then MvPolynomial.X i else 0).toRingHom\n  let t' : T := Spec.map (CommRingCat.ofHom φ) t\n  have ht'g : t' ∈ PrimeSpectrum.basicOpen g :=\n    show φ g ∉ t.asIdeal from (show φ g = g from aeval_ite_mem_eq_self g subset_rfl).symm ▸ htU'\n  have h : t' ∉ fT '' Z := hU'le ht'g\n  suffices ⋃ i ∈ σ, (U i).1 = Set.univ from\n    ⟨this ▸ Finset.isCompact_biUnion _ fun i _ ↦ isCompact_range (𝒰.f i).continuous⟩\n  rw [Set.iUnion₂_eq_univ_iff]\n  contrapose! h\n  obtain ⟨x, hx⟩ := h\n  obtain ⟨z, rfl, hzr⟩ := exists_preimage_pullback x t' (Subsingleton.elim (f x) (q t'))\n  suffices ∀ i, t ∈ (Ti i).comap ⟨_, continuous_comap φ⟩ → p z ∉ U i from\n    ⟨z, by simpa [Z, p, fT, hzr], hzr⟩\n  intro i hi₁ hi₂\n  rw [comap_basicOpen, show φ (.X i) = 0 by simpa [φ] using (hx i · hi₂), basicOpen_zero] at hi₁\n  cases hi₁\n\nset_option backward.isDefEq.respectTransparency false in\n@[stacks 04XU]\n\nTarget:\nlemma Scheme.Hom.isProperMap (f : X ⟶ Y) [UniversallyClosed f] : IsProperMap f :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"ec2f02bf97c18a5bded120090c52d3efb2036960ae95fe7f5b72ffed08280a37","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Morphisms","family_id":"scheme","file_id":"mathlib/Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean","sample_id":"33d6750ddf63884a3db77c1ac0c0a14e4b5da0e25cd340d2ef99ffb5417edbe1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2139989fd770c5936c4b547ad3ae043d8606b002f3ccc3b366913c36130fefc8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"89245e196e8f1c3866de6d9410eff906de2a4f7ec9bd4db4b755d68b297fae0f","source_sha256":"7cfbfc972c134864031de87e3b5157be2d7af83e375a2c0e1cc9ff7db598678d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  filter_upwards [rnDeriv_compProd h_ac ν] with p hp\n  simp_rw [hp, ENNReal.toReal_mul]\n  by_cases h_zero1 : ((∂μ/∂ν) p.1).toReal = 0\n  · simp [h_zero1]\n  by_cases h_zero2 : ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal = 0\n  · simp [h_zero2]\n  simp [log_mul h_zero1 h_zero2]","hard_negative":false,"metrics":{"chosen_tokens":76,"rejected_tokens":3,"token_jaccard":0.027027,"token_length_ratio":0.039474},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"af113fa099ea8205eb6ac77f9dd39c64b916806def9ab73576cc9cd0439e095b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.InformationTheory.KullbackLeibler.Basic\npublic import Mathlib.Probability.Kernel.Composition.MeasureCompProd\npublic import Mathlib.Probability.Notation\nimport Mathlib.Probability.Kernel.Composition.IntegralCompProd\nimport Mathlib.Probability.Kernel.Composition.RadonNikodym\n\nNamespace:\nInformationTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne, Lorenzo Luccioli\n-/\n/-!\n# Chain rule for the Kullback-Leibler divergence\n\nSuppose that we have two finite joint measures on a product `𝓧 × 𝓨`, which can be decomposed as\n`μ ⊗ₘ κ` and `ν ⊗ₘ η`, where `μ` and `ν` are measures on `𝓧` and `κ` and `η` are Markov kernels\nfrom `𝓧` to `𝓨`. Then we can express the Kullback-Leibler divergence between these two joint\nmeasures as a sum of `klDiv μ ν` and the conditional Kullback-Leibler divergence between the kernels\n`κ` and `η`, averaged over `μ`. The resulting equality is most often written as\n`klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + μ[fun x ↦ klDiv (κ x) (η x)]`.\n\nHere we first prove the following version:\n`klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)`.\nThis version avoids the issue of measurability of the function `x ↦ klDiv (κ x) (η x)`, which is not\nalways guaranteed, and thus holds for all measurable spaces `𝓧` and `𝓨`, without any assumptions.\n\n## Main statements\n\n* `klDiv_compProd_eq_add`: `klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)`\n* `klDiv_compProd_left`: `klDiv (μ ⊗ₘ κ) (ν ⊗ₘ κ) = klDiv μ ν`\n\n## Proof\n\nThe main ingredient is the chain rule for Radon-Nikodym derivatives:\n`∂(μ ⊗ₘ κ)/∂(ν ⊗ₘ η) = ∂μ/∂ν * ∂(μ ⊗ₘ κ)/∂(μ ⊗ₘ η)`.\nThen, omitting edge cases, the Kullback-Leibler divergence is an integral of a logarithm of the\nderivative on the left, which decomposes into a sum of two integrals of logarithms.\nWe now give a more detailed outline of the proof.\n\nThe Kullback-Leibler divergence `klDiv μ ν` is defined with an if-then-else statement:\nif the measures are absolutely continuous (`μ ≪ ν`) and the log-likelihood ratio `llr μ ν` is\nintegrable, then it is defined as `∫ x, llr μ ν x ∂μ + ν.real univ - μ.real univ`, otherwise\nit is defined to be `∞`.\n\nWe first deal with the case in which absolute continuity does not hold. The main observation is\nthat `μ ⊗ₘ κ ≪ ν ⊗ₘ η ↔ μ ≪ ν ∧ μ ⊗ₘ κ ≪ μ ⊗ₘ η`, which means that if one of the two sides of the\nKL equality is infinite because of lack of absolute continuity, then the other side is also infinite\nfor the same reason.\n\nThen, we deal with the case in which absolute continuity holds but integrability does not. Again,\nwe can show a similar equivalence for integrability, which allows us to conclude that both sides\nare infinite.\n`Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)` is equivalent to\n`Integrable (llr μ ν) μ ∧ Integrable (llr (μ ⊗ₘ κ) (μ ⊗ₘ η)) (μ ⊗ₘ κ)`.\nThe proof of this equivalence relies on the convexity of the function `x ↦ x * log x`.\n\nFinally, we prove the equality in the case in which both absolute continuity and integrability hold.\nIn that case, `klDiv μ ν = ∫ x, llr μ ν x ∂μ + ν.real univ - μ.real univ` and similarly for\nthe other terms. It is easy to see that it suffices to prove the equality of the integrals parts.\nFinally, the computation for the integral of the log-likelihood ratio is as follows:\n```\n∫ p, llr (μ ⊗ₘ κ) (ν ⊗ₘ η) p ∂(μ ⊗ₘ κ)\n_ = ∫ p, ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal * log ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal ∂(ν ⊗ₘ η)\n_ = ∫ p, ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal *\n    (log ((∂μ/∂ν) p.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal) ∂(ν ⊗ₘ η)\n_ = ∫ p, (log ((∂μ/∂ν) p.1).toReal + log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal) ∂(μ ⊗ₘ κ)\n_ = ∫ p, log ((∂μ/∂ν) p.1).toReal ∂(μ ⊗ₘ κ) + ∫ p, log ((∂μ ⊗ₘ κ/∂μ ⊗ₘ η) p).toReal ∂(μ ⊗ₘ κ)\n_ = ∫ a, llr μ ν a ∂μ + ∫ p, llr (μ ⊗ₘ κ) (μ ⊗ₘ η) p ∂(μ ⊗ₘ κ)\n```\n\n## TODO\n\nAdd a version of the chain rule for the integral form of the contional KL divergence, i.e.\n`μ[fun x ↦ klDiv (κ x) (η x)]`.\n\n-/\n\npublic section\n\nopen Real MeasureTheory Set ProbabilityTheory\nopen scoped ENNReal\n\nnamespace InformationTheory\n\nvariable {𝓧 𝓨 : Type*} {m𝓧 : MeasurableSpace 𝓧} {m𝓨 : MeasurableSpace 𝓨}\n  {μ ν : Measure 𝓧} {κ η : Kernel 𝓧 𝓨}\n  [IsFiniteMeasure μ] [IsFiniteMeasure ν] [IsMarkovKernel κ] [IsMarkovKernel η]\n\n/-- If the log-likelihood ration between two composition-products is integrable, then so is the\nlog-likelihood ratio between the two measures on the first space. -/\nlemma integrable_llr_of_integrable_llr_compProd\n    (h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η) (h_int : Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)) :\n    Integrable (llr μ ν) μ := by\n  have ⟨hμν_ac, hκη_ac⟩ := Measure.absolutelyContinuous_compProd_iff.mp h_ac\n  rw [← integrable_rnDeriv_mul_log_iff h_ac] at h_int\n  replace h_int := convexOn_mul_log.integrable_apply_rnDeriv_of_integrable_compProd\n    continuous_mul_log.stronglyMeasurable continuous_mul_log.continuousWithinAt h_int hκη_ac\n  exact (integrable_rnDeriv_mul_log_iff hμν_ac).mp h_int\n\nTarget:\nlemma rnDeriv_compProd_mul_log_eq_mul_add (h_ac : μ ⊗ₘ κ ≪ μ ⊗ₘ η) :\n    ∀ᵐ p ∂(ν ⊗ₘ η), ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal * log ((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal =\n      (((∂μ ⊗ₘ κ/∂ν ⊗ₘ η) p).toReal * (log ((∂μ/∂ν) p.1).toReal +\n        log ((∂(μ ⊗ₘ κ)/∂(μ ⊗ₘ η)) p).toReal)) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"InformationTheory/KullbackLeibler","family_id":"rnderiv_compprod_mul_log_eq_mul_add","file_id":"mathlib/Mathlib/InformationTheory/KullbackLeibler/ChainRule.lean","sample_id":"89245e196e8f1c3866de6d9410eff906de2a4f7ec9bd4db4b755d68b297fae0f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a3707e1c83ef260ac94852111d4f8128b1e6f031babe4d0a83c426fc4ebac9e6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9944f4492e2edc232793a5f7e57320d19cd2217d990686c0e9b0e255481105bf","source_sha256":"0a9373d9f782e07a7cdfc4c29959b20ea9b224cdd8ac732a23d6364e81e86934","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [Part.fix]\n  rw [assert_neg h']","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.25},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"af2390a997ed2e52c9f5ae2d97cad750fc86e8c5d2fa402d8e0d9a483df496a3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Part\npublic import Mathlib.Data.Nat.Find\npublic import Mathlib.Data.Nat.Upto\npublic import Mathlib.Data.Stream.Defs\n\nNamespace:\nPart\n\nLocal context:\n/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\n/-!\n# Fixed point\n\nThis module defines a generic `fix` operator for defining recursive\ncomputations that are not necessarily well-founded or productive.\nAn instance is defined for `Part`.\n\n## Main definition\n\n* class `Fix`\n* `Part.fix`\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\nvariable {α : Type*} {β : α → Type*}\n\n/-- `Fix α` provides a `fix` operator to define recursive computation\nvia the fixed point of function of type `α → α`. -/\nclass Fix (α : Type*) where\n  /-- `fix f` represents the computation of a fixed point for `f`. -/\n  fix : (α → α) → α\n\nnamespace Part\n\nopen Part Nat Nat.Upto\n\nsection Basic\n\nvariable (f : (∀ a, Part (β a)) → (∀ a, Part (β a)))\n\n/-- A series of successive, finite approximation of the fixed point of `f`, defined by\n`approx f n = f^[n] ⊥`. The limit of this chain is the fixed point of `f`. -/\ndef Fix.approx : Stream' (∀ a, Part (β a))\n  | 0 => ⊥\n  | Nat.succ i => f (Fix.approx i)\n\n/-- loop body for finding the fixed point of `f` -/\ndef fixAux {p : ℕ → Prop} (i : Nat.Upto p) (g : ∀ j : Nat.Upto p, i < j → ∀ a, Part (β a)) :\n    ∀ a, Part (β a) :=\n  f fun x : α => (assert ¬p i.val) fun h : ¬p i.val => g (i.succ h) (Nat.lt_succ_self _) x\n\n/-- The least fixed point of `f`.\n\nIf `f` is a continuous function (according to complete partial orders),\nit satisfies the equations:\n\n  1. `fix f = f (fix f)`          (is a fixed point)\n  2. `∀ X, f X ≤ X → fix f ≤ X`   (least fixed point)\n-/\nprotected def fix (x : α) : Part (β x) :=\n  (Part.assert (∃ i, (Fix.approx f i x).Dom)) fun h =>\n    WellFounded.fix.{1} (Nat.Upto.wf h) (fixAux f) Nat.Upto.zero x\n\nopen Classical in\nprotected theorem fix_def {x : α} (h' : ∃ i, (Fix.approx f i x).Dom) :\n    Part.fix f x = Fix.approx f (Nat.succ (Nat.find h')) x := by\n  let p := fun i : ℕ => (Fix.approx f i x).Dom\n  have : p (Nat.find h') := Nat.find_spec h'\n  generalize hk : Nat.find h' = k\n  replace hk : Nat.find h' = k + (@Upto.zero p).val := hk\n  rw [hk] at this\n  revert hk\n  dsimp [Part.fix]; rw [assert_pos h']; revert this\n  generalize Upto.zero = z; intro _this hk\n  suffices ∀ x' hwf,\n    WellFounded.fix hwf (fixAux f) z x' = Fix.approx f (succ k) x'\n    from this _ _\n  induction k generalizing z with\n  | zero =>\n    intro x' _\n    rw [Fix.approx, WellFounded.fix_eq, fixAux]\n    congr\n    ext x : 1\n    rw [assert_neg]\n    · rfl\n    · rw [Nat.zero_add] at _this\n      simpa only [not_not, Coe]\n  | succ n n_ih =>\n    intro x' _\n    rw [Fix.approx, WellFounded.fix_eq, fixAux]\n    congr\n    ext : 1\n    have hh : ¬(Fix.approx f z.val x).Dom := by\n      apply Nat.find_min h'\n      lia\n    rw [succ_add_eq_add_succ] at _this hk\n    rw [assert_pos hh, n_ih (Upto.succ z hh) _this hk]\n\nTarget:\ntheorem fix_def' {x : α} (h' : ¬∃ i, (Fix.approx f i x).Dom) : Part.fix f x = none :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control","family_id":"fix_def","file_id":"mathlib/Mathlib/Control/Fix.lean","sample_id":"9944f4492e2edc232793a5f7e57320d19cd2217d990686c0e9b0e255481105bf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"760a46c45735c2286ac991019ea42b32b81c6e8ac4991d80d094da0aa71c32a8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5ec0b0e55183ff152ce79ab5d2c02cf74a8634036c497bc83a0db9d263f074b7","source_sha256":"158ce289cc94776d7ace3f5f8788d7841da0114f64a958ffadba75342e4312aa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [destruct]\n  cases f0 : s.1 0 <;> intro h\n  · contradiction\n  · apply Subtype.ext\n    funext n\n    induction n with\n    | zero => injection h with h'; rwa [h'] at f0\n    | succ n IH => exact s.2 IH","hard_negative":false,"metrics":{"chosen_tokens":53,"rejected_tokens":2,"token_jaccard":0.026316,"token_length_ratio":0.037736},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"af3d53f9f2332874af61370edf7d012bb72e84e32ebed15fd47719bcfc5b1d67","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.Find\npublic import Mathlib.Data.Stream.Init\npublic import Mathlib.Logic.Relator\npublic import Mathlib.Tactic.Common\npublic import Batteries.Tactic.Lint.Simp\n\nNamespace:\nComputation\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Coinductive formalization of unbounded computations.\n\nThis file provides a `Computation` type where `Computation α` is the type of\nunbounded computations returning `α`.\n-/\n\n@[expose] public section\n\nopen Function\n\nuniverse u v w\n\n/-\ncoinductive Computation (α : Type u) : Type u\n| pure : α → Computation α\n| think : Computation α → Computation α\n-/\n/-- `Computation α` is the type of unbounded computations returning `α`.\n  An element of `Computation α` is an infinite sequence of `Option α` such\n  that if `f n = some a` for some `n` then it is constantly `some a` after that. -/\ndef Computation (α : Type u) : Type u :=\n  { f : Stream' (Option α) // ∀ ⦃n a⦄, f n = some a → f (n + 1) = some a }\n\nnamespace Computation\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n-- constructors\n/-- `pure a` is the computation that immediately terminates with result `a`. -/\ndef pure (a : α) : Computation α :=\n  ⟨Stream'.const (some a), fun _ _ => id⟩\n\ninstance : CoeTC α (Computation α) :=\n  ⟨pure⟩\n\n-- note [use has_coe_t]\n/-- `think c` is the computation that delays for one \"tick\" and then performs\n  computation `c`. -/\ndef think (c : Computation α) : Computation α :=\n  ⟨Stream'.cons none c.1, fun n a h => by\n    rcases n with - | n\n    · contradiction\n    · exact c.2 h⟩\n\n/-- `thinkN c n` is the computation that delays for `n` ticks and then performs\n  computation `c`. -/\ndef thinkN (c : Computation α) : ℕ → Computation α\n  | 0 => c\n  | n + 1 => think (thinkN c n)\n\n-- check for immediate result\n/-- `head c` is the first step of computation, either `some a` if `c = pure a`\n  or `none` if `c = think c'`. -/\ndef head (c : Computation α) : Option α :=\n  c.1.head\n\n-- one step of computation\n/-- `tail c` is the remainder of computation, either `c` if `c = pure a`\n  or `c'` if `c = think c'`. -/\ndef tail (c : Computation α) : Computation α :=\n  ⟨c.1.tail, fun _ _ h => c.2 h⟩\n\n/-- `empty α` is the computation that never returns, an infinite sequence of\n  `think`s. -/\ndef empty (α) : Computation α :=\n  ⟨Stream'.const none, fun _ _ => id⟩\n\ninstance : Inhabited (Computation α) :=\n  ⟨empty _⟩\n\n/-- `runFor c n` evaluates `c` for `n` steps and returns the result, or `none`\n  if it did not terminate after `n` steps. -/\ndef runFor : Computation α → ℕ → Option α :=\n  Subtype.val\n\n/-- `destruct c` is the destructor for `Computation α` as a coinductive type.\n  It returns `inl a` if `c = pure a` and `inr c'` if `c = think c'`. -/\ndef destruct (c : Computation α) : α ⊕ (Computation α) :=\n  match c.1 0 with\n  | none => Sum.inr (tail c)\n  | some a => Sum.inl a\n\n/-- `run c` is an unsound meta function that runs `c` to completion, possibly\n  resulting in an infinite loop in the VM. -/\nunsafe def run : Computation α → α\n  | c =>\n    match destruct c with\n    | Sum.inl a => a\n    | Sum.inr ca => run ca\n\nTarget:\ntheorem destruct_eq_pure {s : Computation α} {a : α} : destruct s = Sum.inl a → s = pure a :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Seq","family_id":"destruct_eq_pure","file_id":"mathlib/Mathlib/Data/Seq/Computation.lean","sample_id":"5ec0b0e55183ff152ce79ab5d2c02cf74a8634036c497bc83a0db9d263f074b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1a29d536dd839cb2ac804ef83ab5abb79ba3d4ec297a0fa4ca91c3477c2aaaa0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"45ea66a4976169e006a11c1610afe29ec4c0c4a950d061222eb9e544fe4ac87d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a793e6cc882c9887527edba47d8bc663c406cd6c0a53d249021c37e71fed657c","source_sha256":"0558464e45d5ed116146fb004c344db832eef9939fa1268196e1a1a60376c0f3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine OreLocalization.ind fun x s ↦ ?_\n  obtain ⟨i, j, hne, heq⟩ := Finite.exists_ne_map_eq_of_infinite (α := ℕ) (s ^ · • x)\n  wlog! hlt : j < i generalizing i j\n  · exact this j i hne.symm heq.symm (hne.lt_of_le hlt)\n  use s ^ (i - (j + 1)) • x\n  rw [oreDiv_eq_iff]\n  refine ⟨s ^ j, (s ^ (j + 1)).1, ?_, ?_⟩\n  · change s ^ j • x = s ^ (j + 1) • s ^ (i - (j + 1)) • x\n    rw [← mul_smul, ← pow_add, Nat.add_sub_cancel' hlt, heq]\n  · simp_rw [SubmonoidClass.coe_pow, OneMemClass.coe_one, mul_one, pow_succ]","hard_negative":false,"metrics":{"chosen_tokens":165,"rejected_tokens":170,"token_jaccard":0.938462,"token_length_ratio":1.030303},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"af4a092b8d38ca7b20fe1444e35c85799e62230915f1f587f15eba9863de4975","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Pigeonhole\npublic import Mathlib.GroupTheory.OreLocalization.Basic\npublic import Mathlib.SetTheory.Cardinal.Arithmetic\n\nNamespace:\nOreLocalization\n\nLocal context:\n/-\nCopyright (c) 2024 Jz Pan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jz Pan\n-/\n/-!\n\n# Cardinality of Ore localizations\n\nThis file contains some results on cardinality of Ore localizations.\n\n## TODO\n\n- Prove or disprove `OreLocalization.cardinalMk_le_lift_cardinalMk_of_commute`\n  with `Commute` assumption removed.\n\n-/\n\npublic section\n\nuniverse u v\n\nopen Cardinal Function\n\nnamespace OreLocalization\n\nvariable {R : Type u} [Monoid R] (S : Submonoid R) [OreLocalization.OreSet S]\n  (X : Type v) [MulAction R X]\n\n@[to_additive]\ntheorem oreDiv_one_surjective_of_finite_left [Finite S] :\n    Surjective (fun x ↦ x /ₒ (1 : ↥S) : X → OreLocalization S X) := by\n  refine OreLocalization.ind fun x s ↦ ?_\n  obtain ⟨i, j, hne, heq⟩ := Finite.exists_ne_map_eq_of_infinite (α := ℕ) (s ^ ·)\n  wlog! hlt : j < i generalizing i j\n  · exact this j i hne.symm heq.symm (hne.lt_of_le hlt)\n  use s ^ (i - (j + 1)) • x\n  rw [oreDiv_eq_iff]\n  refine ⟨s ^ j, (s ^ (j + 1)).1, ?_, ?_⟩\n  · change s ^ j • x = s ^ (j + 1) • s ^ (i - (j + 1)) • x\n    rw [← mul_smul, ← pow_add, Nat.add_sub_cancel' hlt, heq]\n  · simp_rw [SubmonoidClass.coe_pow, OneMemClass.coe_one, mul_one, pow_succ]\n\n@[to_additive]\n\nTarget:\ntheorem oreDiv_one_surjective_of_finite_right [Finite X] :\n    Surjective (fun x ↦ x /ₒ (1 : ↥S) : X → OreLocalization S X) :=\n\nProof body:\n","rejected":"by\n  refine OreLocalization.ind fun x s ↦ ?_\n  obtain ⟨i, j, hne, heq⟩ := Finite.exists_ne_map_eq_of_infinite (α := ℕ) (s ^ · • x)\n  wlog! hlt : j < i generalizing i j\n  · exact this j i hne.symm heq.symm (hne.lt_of_le hlt)\n  use s ^ (i - (j + 1)) • x\n  rw [oreDiv_eq_iff]\n  refine ⟨s ^ j, (s ^ (j + 1)).1, ?_, ?_⟩\n  · change s ^ j • x = s ^ (j + 1) • s ^ (i - (j + 1)) • x\n    rw [← mul_smul, ← pow_add, Nat.add_sub_cancel' hlt, heq]\n  · simp_rw [SubmonoidClass.coe_pow, OneMemClass.coe_one, mul_one, pow_succ]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/OreLocalization","family_id":"orediv_one_surjective_of_finite_right","file_id":"mathlib/Mathlib/GroupTheory/OreLocalization/Cardinality.lean","sample_id":"a793e6cc882c9887527edba47d8bc663c406cd6c0a53d249021c37e71fed657c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f0fe75d1087a41898b65fcdcdb5116eac360500295f3e4e2a2c616668dd8129d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7fe45320e9d9e2a8740d56b6ea8a40e7b3a70cdd082e9f5d7517045e42a33ea2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5ae786a0f2c5c57019f4c1e4edc7524aa65574003fae2570dc839740f449cb98","source_sha256":"afb2352b0e8eda21bd88c47cda5f9b0a4c162f3f78eedf36a33613392805b346","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  nth_rw 2 [← Quiver.Path.reverse_reverse p]\n  apply congr_comp_reverse","hard_negative":false,"metrics":{"chosen_tokens":14,"rejected_tokens":19,"token_jaccard":0.722222,"token_length_ratio":1.357143},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"af82112c37087d1a8fb10a343478e315ae9f1146a8ffe6aa3dc998f132c4d15b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Groupoid\npublic import Mathlib.CategoryTheory.PathCategory.Basic\n\nNamespace:\nQuiver.FreeGroupoid\n\nLocal context:\n/-\nCopyright (c) 2022 Rémi Bottinelli. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémi Bottinelli\n-/\n/-!\n# Free groupoid on a quiver\n\nThis file defines the free groupoid on a quiver, the lifting of a prefunctor to its unique\nextension as a functor from the free groupoid, and proves uniqueness of this extension.\n\n## Main results\n\nGiven the type `V` and a quiver instance on `V`:\n\n- `Quiver.FreeGroupoid V`: a type synonym for `V`.\n- `Quiver.FreeGroupoid.instGroupoid`: the `Groupoid` instance on `Quiver.FreeGroupoid V`.\n- `lift`: the lifting of a prefunctor from `V` to `V'` where `V'` is a groupoid, to a functor.\n  `Quiver.FreeGroupoid V ⥤ V'`.\n- `lift_spec` and `lift_unique`: the proofs that, respectively, `lift` indeed is a lifting\n  and is the unique one.\n\n## Implementation notes\n\nThe free groupoid is first defined by symmetrifying the quiver, taking the induced path category\nand finally quotienting by the reducibility relation.\n\n-/\n\n@[expose] public section\n\nopen Set Function\n\nnamespace Quiver\n\nopen CategoryTheory\n\nuniverse u v u' v' u'' v''\n\nvariable {V : Type u} [Quiver.{v} V]\n\n/-- Shorthand for the \"forward\" arrow corresponding to `f` in `paths <| symmetrify V` -/\nabbrev Hom.toPosPath {X Y : V} (f : X ⟶ Y) :\n    (CategoryTheory.Paths.categoryPaths <| Quiver.Symmetrify V).Hom X Y :=\n  f.toPos.toPath\n\n/-- Shorthand for the \"forward\" arrow corresponding to `f` in `paths <| symmetrify V` -/\nabbrev Hom.toNegPath {X Y : V} (f : X ⟶ Y) :\n    (CategoryTheory.Paths.categoryPaths <| Quiver.Symmetrify V).Hom Y X :=\n  f.toNeg.toPath\n\n/-- The \"reduction\" relation -/\ninductive FreeGroupoid.redStep : HomRel (Paths (Quiver.Symmetrify V))\n  | step (X Z : Quiver.Symmetrify V) (f : X ⟶ Z) :\n    redStep (𝟙 ((Paths.of (Quiver.Symmetrify V)).obj X)) (f.toPath ≫ (Quiver.reverse f).toPath)\n\n/-- The underlying vertices of the free groupoid -/\nprotected def FreeGroupoid (V) [Q : Quiver V] :=\n  CategoryTheory.Quotient (@FreeGroupoid.redStep V Q)\n\nnamespace FreeGroupoid\n\nopen Quiver\n\ninstance {V} [Quiver V] [Nonempty V] : Nonempty (Quiver.FreeGroupoid V) := by\n  inhabit V; exact ⟨⟨@default V _⟩⟩\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem congr_reverse {X Y : Paths <| Quiver.Symmetrify V} (p q : X ⟶ Y) :\n    HomRel.CompClosure redStep p q → HomRel.CompClosure redStep p.reverse q.reverse := by\n  rintro ⟨_, _, XW, _, _, WY, _, _, f⟩\n  have : HomRel.CompClosure redStep (WY.reverse ≫ 𝟙 _ ≫ XW.reverse)\n      (WY.reverse ≫ (f.toPath ≫ (Quiver.reverse f).toPath) ≫ XW.reverse) := by\n    constructor\n    constructor\n  simpa only [CategoryStruct.comp, CategoryStruct.id, Quiver.Path.reverse, Quiver.Path.nil_comp,\n    Quiver.Path.reverse_comp, Quiver.reverse_reverse, Quiver.Path.reverse_toPath,\n    Quiver.Path.comp_assoc] using this\n\nopen Relation in\ntheorem congr_comp_reverse {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :\n    Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (p ≫ p.reverse) =\n      Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (𝟙 X) := by\n  apply Quot.eqvGen_sound\n  induction p with\n  | nil => apply EqvGen.refl\n  | cons q f ih =>\n    simp only [Quiver.Path.reverse]\n    fapply EqvGen.trans\n    -- Porting note: dot notation for `Quiver.Path.*` and `Quiver.Hom.*` not working\n    · exact q ≫ Quiver.Path.reverse q\n    · apply EqvGen.symm\n      apply EqvGen.rel\n      have : HomRel.CompClosure redStep (q ≫ 𝟙 _ ≫ Quiver.Path.reverse q)\n          (q ≫ (Quiver.Hom.toPath f ≫ Quiver.Hom.toPath (Quiver.reverse f)) ≫\n            Quiver.Path.reverse q) := by\n        apply HomRel.CompClosure.intro\n        apply redStep.step\n      simp only [Category.assoc, Category.id_comp] at this ⊢\n      -- Porting note: `simp` cannot see how `Quiver.Path.comp_assoc` is relevant, so change to\n      -- category notation\n      change HomRel.CompClosure redStep (q ≫ Quiver.Path.reverse q)\n        (Quiver.Path.cons q f ≫ (Quiver.Hom.toPath (Quiver.reverse f)) ≫ (Quiver.Path.reverse q))\n      simp only [← Category.assoc] at this ⊢\n      exact this\n    · exact ih\n\nTarget:\ntheorem congr_reverse_comp {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :\n    Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (p.reverse ≫ p) =\n      Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (𝟙 Y) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  nth_rw 2 [← Quiver.Path.reverse_reverse p]\n  apply congr_comp_reverse","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Groupoid","family_id":"congr_reverse_comp","file_id":"mathlib/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean","sample_id":"5ae786a0f2c5c57019f4c1e4edc7524aa65574003fae2570dc839740f449cb98"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5c5e46221d8b229ee79bbd41b4c1bdba3112695c6156a1e23176053e1c54427e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"23be217614af4f9bf2fdcf78fa95677947589c258c37fac0c3f5c94e2fa5861b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cb00d28e03305d658f5fd091ef6af272ded07e63d8901ab1f776e0190fa772a7","source_sha256":"5b291100fb3630420e5a4ec2ae128eaf5dffa0b8e5840aaf136af3ede6f696c3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by subst h; rfl","hard_negative":true,"metrics":{"chosen_tokens":5,"rejected_tokens":3,"token_jaccard":0.142857,"token_length_ratio":0.6},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"af915a31132bd8857f6299a643f4c99837855fb061e101d6ba87c1815a1a9a94","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.EqToHom\npublic import Mathlib.CategoryTheory.Functor.Const\npublic import Mathlib.CategoryTheory.Opposites\npublic import Mathlib.Data.Prod.Basic\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2017 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Kim Morrison\n-/\n/-!\n# Cartesian products of categories\n\nWe define the category instance on `C × D` when `C` and `D` are categories.\n\nWe define:\n* `sectL C Z` : the functor `C ⥤ C × D` given by `X ↦ ⟨X, Z⟩`\n* `sectR Z D` : the functor `D ⥤ C × D` given by `Y ↦ ⟨Z, Y⟩`\n* `fst`       : the functor `⟨X, Y⟩ ↦ X`\n* `snd`       : the functor `⟨X, Y⟩ ↦ Y`\n* `swap`      : the functor `C × D ⥤ D × C` given by `⟨X, Y⟩ ↦ ⟨Y, X⟩`\n    (and the fact that this is an equivalence)\n\nWe further define `evaluation : C ⥤ (C ⥤ D) ⥤ D` and `evaluationUncurried : C × (C ⥤ D) ⥤ D`,\nand products of functors and natural transformations, written `F.prod G` and `α.prod β`.\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Functor\n\n-- declare the `v`'s first; see `CategoryTheory.Category` for an explanation\nuniverse v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄\n\nsection\n\nvariable (C : Type u₁) [CategoryStruct.{v₁} C] (D : Type u₂) [CategoryStruct.{v₂} D]\n\n/-- `CategoryStruct.prod C D` gives the Cartesian product of two `CategoryStruct`'s. -/\n@[simps id_fst id_snd comp_fst comp_snd Hom]\ninstance prod : CategoryStruct.{max v₁ v₂} (C × D) where\n  Hom X Y := (X.1 ⟶ Y.1) × (X.2 ⟶ Y.2)\n  id X := ⟨𝟙 X.1, 𝟙 X.2⟩\n  comp f g := (f.1 ≫ g.1, f.2 ≫ g.2)\n\nvariable {C D}\n\nnamespace Prod\n\n@[ext]\nlemma hom_ext {X Y : C × D} {f g : X ⟶ Y} (h₁ : f.1 = g.1) (h₂ : f.2 = g.2) : f = g :=\n  Prod.ext h₁ h₂\n\n/-- Construct a morphism in a product category by giving its constituent components.\nThis constructor should be preferred over `Prod.mk`, because Lean infers better the\nsource and target of the resulting morphism. -/\nabbrev mkHom {X₁ X₂ : C} {Y₁ Y₂ : D} (f : X₁ ⟶ X₂) (g : Y₁ ⟶ Y₂) : (X₁, Y₁) ⟶ (X₂, Y₂) :=\n  ⟨f, g⟩\n\n@[inherit_doc Prod.mkHom]\nscoped infixr:70 \" ×ₘ \" => Prod.mkHom\n\n/-- Analogue of `Prod.mk.injEq` in this setting. -/\nlemma mkHom_eq {X₁ X₂ : C} {Y₁ Y₂ : D} (f f' : X₁ ⟶ X₂) (g g' : Y₁ ⟶ Y₂) :\n    f ×ₘ g = f' ×ₘ g' ↔ (f = f' ∧ g = g') :=\n  Prod.hom_ext_iff\n\nend Prod\n\nopen Prod\n\n/-! Two rfl lemmas that cannot be generated by `@[simps]`. -/\n\n@[simp]\ntheorem prod_id (X : C) (Y : D) : 𝟙 (X, Y) = 𝟙 X ×ₘ 𝟙 Y :=\n  rfl\n\ntheorem prod_id' (X : C) (Y : D) : 𝟙 (X, Y) = (𝟙 X, 𝟙 Y) :=\n  rfl\n\n@[simp]\ntheorem prod_comp {P Q R : C} {S T U : D} (f : (P, S) ⟶ (Q, T)) (g : (Q, T) ⟶ (R, U)) :\n    f ≫ g = f.1 ≫ g.1 ×ₘ f.2 ≫ g.2 :=\n  rfl\n\n/-! Two `eqToHom` lemmas about projections. -/\n\n@[simp]\ntheorem eqToHom_fst {X Y : C × D} (h : X = Y) :\n    (eqToHom h).1 = eqToHom (h ▸ rfl) := by subst h; rfl\n\n@[simp]\n\nTarget:\ntheorem eqToHom_snd {X Y : C × D} (h : X = Y) :\n    (eqToHom h).2 = eqToHom (h ▸ rfl) :=\n\nProof body:\n","rejected":"by\n  exact eqToHom_snd","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"7d357bb85d9cbc2292a0c338bcaeb556cef462b3a65e91a472fb92f95d8d62df","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Products","family_id":"eqtohom_snd","file_id":"mathlib/Mathlib/CategoryTheory/Products/Basic.lean","sample_id":"cb00d28e03305d658f5fd091ef6af272ded07e63d8901ab1f776e0190fa772a7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1656a7f84780a26168e97da85997b830e60c4a9c21bd50667f35238ce2172d26","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0e4b1666a2df18b8f82460264078354bcb621eb24caecc149f02733eb9e19be1","source_sha256":"04666e1a54002d0f7b310bb68c623ecf47eda2066cc741dda3dda2f5fc6f4c64","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let U := f ⁻¹' {f y}\n  suffices x ∉ Uᶜ from Classical.not_not.1 this\n  intro hxV\n  specialize hs U Uᶜ (hf {f y}) (hf {f y}ᶜ) _ ⟨y, ⟨hy, rfl⟩⟩ ⟨x, ⟨hx, hxV⟩⟩\n  · simp only [union_compl_self, subset_univ]\n  · simp only [inter_empty, Set.not_nonempty_empty, inter_compl_self] at hs","hard_negative":false,"metrics":{"chosen_tokens":87,"rejected_tokens":2,"token_jaccard":0.020833,"token_length_ratio":0.022989},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"afbc40eeb3212726ba09fd48d70f1a1557e8a415303ff43d2a4348eed39482b4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Indicator\npublic import Mathlib.Topology.Connected.LocallyConnected\npublic import Mathlib.Topology.Sets.Closeds\n\nNamespace:\nIsLocallyConstant\n\nLocal context:\n/-\nCopyright (c) 2021 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\n/-!\n# Locally constant functions\n\nThis file sets up the theory of locally constant function from a topological space to a type.\n\n## Main definitions and constructions\n\n* `IsLocallyConstant f` : a map `f : X → Y` where `X` is a topological space is locally\n                            constant if every set in `Y` has an open preimage.\n* `LocallyConstant X Y` : the type of locally constant maps from `X` to `Y`\n* `LocallyConstant.map` : push-forward of locally constant maps\n* `LocallyConstant.comap` : pull-back of locally constant maps\n-/\n\n@[expose] public section\n\nvariable {X Y Z α : Type*} [TopologicalSpace X]\n\nopen Set Filter\nopen scoped Topology\n\n/-- A function between topological spaces is locally constant if the preimage of any set is open. -/\ndef IsLocallyConstant (f : X → Y) : Prop :=\n  ∀ s : Set Y, IsOpen (f ⁻¹' s)\n\nnamespace IsLocallyConstant\n\nopen List in\nprotected theorem tfae (f : X → Y) :\n    TFAE [IsLocallyConstant f,\n      ∀ x, ∀ᶠ x' in 𝓝 x, f x' = f x,\n      ∀ x, IsOpen { x' | f x' = f x },\n      ∀ y, IsOpen (f ⁻¹' {y}),\n      ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x] := by\n  tfae_have 1 → 4 := fun h y => h {y}\n  tfae_have 4 → 3 := fun h x => h (f x)\n  tfae_have 3 → 2 := fun h x => IsOpen.mem_nhds (h x) rfl\n  tfae_have 2 → 5\n  | h, x => by\n    rcases mem_nhds_iff.1 (h x) with ⟨U, eq, hU, hx⟩\n    exact ⟨U, hU, hx, eq⟩\n  tfae_have 5 → 1\n  | h, s => by\n    refine isOpen_iff_forall_mem_open.2 fun x hx ↦ ?_\n    rcases h x with ⟨U, hU, hxU, eq⟩\n    exact ⟨U, fun x' hx' => mem_preimage.2 <| (eq x' hx').symm ▸ hx, hU, hxU⟩\n  tfae_finish\n\n@[nontriviality]\ntheorem of_discrete [DiscreteTopology X] (f : X → Y) : IsLocallyConstant f := fun _ =>\n  isOpen_discrete _\n\ntheorem isOpen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsOpen { x | f x = y } :=\n  hf {y}\n\ntheorem isClosed_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClosed { x | f x = y } :=\n  ⟨hf {y}ᶜ⟩\n\ntheorem isClopen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClopen { x | f x = y } :=\n  ⟨isClosed_fiber hf _, isOpen_fiber hf _⟩\n\ntheorem iff_exists_open (f : X → Y) :\n    IsLocallyConstant f ↔ ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x :=\n  (IsLocallyConstant.tfae f).out 0 4\n\ntheorem iff_eventually_eq (f : X → Y) : IsLocallyConstant f ↔ ∀ x, ∀ᶠ y in 𝓝 x, f y = f x :=\n  (IsLocallyConstant.tfae f).out 0 1\n\ntheorem exists_open {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x :=\n  (iff_exists_open f).1 hf x\n\nprotected theorem eventually_eq {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    ∀ᶠ y in 𝓝 x, f y = f x :=\n  (iff_eventually_eq f).1 hf x\n\ntheorem iff_isOpen_fiber_apply {f : X → Y} : IsLocallyConstant f ↔ ∀ x, IsOpen (f ⁻¹' {f x}) :=\n  (IsLocallyConstant.tfae f).out 0 2\n\ntheorem iff_isOpen_fiber {f : X → Y} : IsLocallyConstant f ↔ ∀ y, IsOpen (f ⁻¹' {y}) :=\n  (IsLocallyConstant.tfae f).out 0 3\n\nprotected theorem continuous [TopologicalSpace Y] {f : X → Y} (hf : IsLocallyConstant f) :\n    Continuous f :=\n  ⟨fun _ _ => hf _⟩\n\ntheorem iff_continuous {_ : TopologicalSpace Y} [DiscreteTopology Y] (f : X → Y) :\n    IsLocallyConstant f ↔ Continuous f :=\n  ⟨IsLocallyConstant.continuous, fun h s => h.isOpen_preimage s (isOpen_discrete _)⟩\n\ntheorem of_constant (f : X → Y) (h : ∀ x y, f x = f y) : IsLocallyConstant f :=\n  (iff_eventually_eq f).2 fun _ => Eventually.of_forall fun _ => h _ _\n\nprotected theorem const (y : Y) : IsLocallyConstant (Function.const X y) :=\n  of_constant _ fun _ _ => rfl\n\nprotected theorem comp {f : X → Y} (hf : IsLocallyConstant f) (g : Y → Z) :\n    IsLocallyConstant (g ∘ f) := fun s => by\n  rw [Set.preimage_comp]\n  exact hf _\n\ntheorem prodMk {Y'} {f : X → Y} {f' : X → Y'} (hf : IsLocallyConstant f)\n    (hf' : IsLocallyConstant f') : IsLocallyConstant fun x => (f x, f' x) :=\n  (iff_eventually_eq _).2 fun x =>\n    (hf.eventually_eq x).mp <| (hf'.eventually_eq x).mono fun _ hf' hf => Prod.ext hf hf'\n\ntheorem comp₂ {Y₁ Y₂ Z : Type*} {f : X → Y₁} {g : X → Y₂} (hf : IsLocallyConstant f)\n    (hg : IsLocallyConstant g) (h : Y₁ → Y₂ → Z) : IsLocallyConstant fun x => h (f x) (g x) :=\n  (hf.prodMk hg).comp fun x : Y₁ × Y₂ => h x.1 x.2\n\ntheorem comp_continuous [TopologicalSpace Y] {g : Y → Z} {f : X → Y} (hg : IsLocallyConstant g)\n    (hf : Continuous f) : IsLocallyConstant (g ∘ f) := fun s => by\n  rw [Set.preimage_comp]\n  exact hf.isOpen_preimage _ (hg _)\n\n/-- A locally constant function is constant on any preconnected set. -/\n\nTarget:\ntheorem apply_eq_of_isPreconnected {f : X → Y} (hf : IsLocallyConstant f) {s : Set X}\n    (hs : IsPreconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) : f x = f y :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/LocallyConstant","family_id":"apply_eq_of_ispreconnected","file_id":"mathlib/Mathlib/Topology/LocallyConstant/Basic.lean","sample_id":"0e4b1666a2df18b8f82460264078354bcb621eb24caecc149f02733eb9e19be1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5af7d1d4d55c3a73fbff3849f778662494382e96b2f978d7cb9fc32bd567bdc1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"17c8f688ffbfe3f08997e2e441397e109ef966a6e5472eeef0ec671a2f11c5b3","source_sha256":"23d7dd6c4328a8c8adae4c7811bfae5c38bb2e82b7410307d7aa82730380d65f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro f\n  obtain ⟨⟨f⟩, rfl⟩ := (bijective_leftHomotopyClassToHom L X Y).2 f\n  exact ⟨f, rfl⟩","hard_negative":false,"metrics":{"chosen_tokens":27,"rejected_tokens":2,"token_jaccard":0.052632,"token_length_ratio":0.074074},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"afcb29c71a84c8b9c87ac9be774228de3f63418389ead84e0787370e04cf4f29","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy\n\nNamespace:\nHomotopicalAlgebra\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The fundamental lemma of homotopical algebra\n\nLet `C` be a model category. Let `L : C ⥤ H` be a localization functor\nwith respect to weak equivalences in `C`. We obtain the fundamental\nlemma of homotopical algebra: if `X` is cofibrant and `Y` fibrant,\nthe map `(X ⟶ Y) → (L.obj X ⟶ L.obj Y)` identifies `L.obj X ⟶ L.obj Y`\nto the quotient of `X ⟶ Y` by the homotopy relation (in this case,\nthe left and right homotopy relations coincide).\n\n## References\n* [Daniel G. Quillen, Homotopical algebra, I.1][Quillen1967]\n\n-/\n\n@[expose] public section\n\nopen CategoryTheory Limits\n\nnamespace HomotopicalAlgebra\n\nvariable {C : Type*} [Category* C] [ModelCategory C] {H : Type*} [Category* H]\n  (L : C ⥤ H) [L.IsLocalization (weakEquivalences _)]\n  {X Y : C}\n\n/-- The map `LeftHomotopyClass X Y → (L.obj X ⟶ L.obj Y)` when `L` is\na localization functor with respect to `weakEquivalences C`. -/\ndef leftHomotopyClassToHom : LeftHomotopyClass X Y → (L.obj X ⟶ L.obj Y) :=\n  Quot.lift L.map (fun _ _ h ↦ h.factorsThroughLocalization.map_eq _)\n\n@[simp]\nlemma leftHomotopyClassToHom_mk (f : X ⟶ Y) :\n    leftHomotopyClassToHom L (.mk f) = L.map f := rfl\n\n/-- The map `RightHomotopyClass X Y → (L.obj X ⟶ L.obj Y)` when `L` is\na localization functor with respect to `weakEquivalences C`. -/\ndef rightHomotopyClassToHom : RightHomotopyClass X Y → (L.obj X ⟶ L.obj Y) :=\n  Quot.lift L.map (fun _ _ h ↦ h.factorsThroughLocalization.map_eq _)\n\n@[simp]\nlemma rightHomotopyClassToHom_mk (f : X ⟶ Y) :\n    rightHomotopyClassToHom L (.mk f) = L.map f := rfl\n\nvariable (X Y)\nlemma bijective_leftHomotopyClassToHom_iff_bijective_rightHomotopyClassToHom\n    [IsCofibrant X] [IsFibrant Y] :\n    Function.Bijective (leftHomotopyClassToHom L : LeftHomotopyClass X Y → _) ↔\n    Function.Bijective (rightHomotopyClassToHom L : RightHomotopyClass X Y → _) := by\n  have : (leftHomotopyClassToHom L : LeftHomotopyClass X Y → _) =\n      rightHomotopyClassToHom L ∘ leftHomotopyClassEquivRightHomotopyClass := by\n    ext f\n    obtain ⟨f, rfl⟩ := f.mk_surjective\n    simp\n  simp [this]\n\nsection\n\nvariable [IsCofibrant X] [IsFibrant Y]\n\nlemma bijective_rightHomotopyClassToHom :\n    Function.Bijective (rightHomotopyClassToHom L : RightHomotopyClass X Y → _) := by\n  wlog _ : IsCofibrant Y generalizing Y\n  · obtain ⟨Y', _, p, _, _⟩ := CofibrantObject.HoCat.exists_resolution Y\n    have _ : IsFibrant Y' := isFibrant_of_fibration p\n    have hY' := this Y' inferInstance\n    simp only [← bijective_leftHomotopyClassToHom_iff_bijective_rightHomotopyClassToHom] at hY' ⊢\n    have := Localization.inverts L (weakEquivalences _) p\n      (by rwa [← weakEquivalence_iff])\n    rw [← Function.Bijective.of_comp_iff _\n      (LeftHomotopyClass.postcomp_bijective_of_fibration_of_weakEquivalence _ p)]\n    convert! (Iso.homCongr (Iso.refl (L.obj X)) (asIso (L.map p))).bijective.comp hY'\n    ext f\n    obtain ⟨f, rfl⟩ := f.mk_surjective\n    simp\n  wlog _ : IsFibrant X generalizing X\n  · obtain ⟨X', i, _, _, _⟩ : ∃ (X' : C) (i : X ⟶ X'), Cofibration i ∧ WeakEquivalence i ∧\n        IsFibrant X' :=\n      ⟨_, FibrantObject.HoCat.iResolutionObj X, inferInstance, inferInstance, inferInstance⟩\n    have _ := isCofibrant_of_cofibration i\n    have hX' := this X' inferInstance\n    have := Localization.inverts L (weakEquivalences _) i\n      (by rwa [← weakEquivalence_iff])\n    rw [← Function.Bijective.of_comp_iff _\n      (RightHomotopyClass.precomp_bijective_of_cofibration_of_weakEquivalence Y i)]\n    convert! (Iso.homCongr (asIso (L.map i)) (Iso.refl (L.obj Y))).symm.bijective.comp hX'\n    ext f\n    obtain ⟨f, rfl⟩ := f.mk_surjective\n    simp\n  let E := Localization.uniq BifibrantObject.toHoCat (BifibrantObject.ι ⋙ L) (weakEquivalences _)\n  let e : BifibrantObject.toHoCat ⋙ E.functor ≅ BifibrantObject.ι ⋙ L :=\n    Localization.compUniqFunctor BifibrantObject.toHoCat (BifibrantObject.ι ⋙ L)\n      (weakEquivalences _)\n  have : rightHomotopyClassToHom L =\n      (BifibrantObject.HoCat.homEquivRight.trans (E.fullyFaithfulFunctor.homEquiv.trans\n        (Iso.homCongr (e.app (.mk X)) (e.app (.mk Y))))) := by\n    ext f\n    obtain ⟨f, rfl⟩ := RightHomotopyClass.mk_surjective f\n    exact (NatIso.naturality_1 e (BifibrantObject.homMk f)).symm\n  rw [this]\n  exact Equiv.bijective _\n\nlemma bijective_leftHomotopyClassToHom :\n    Function.Bijective (leftHomotopyClassToHom L : LeftHomotopyClass X Y → _) := by\n  rw [bijective_leftHomotopyClassToHom_iff_bijective_rightHomotopyClassToHom]\n  exact bijective_rightHomotopyClassToHom L X Y\n\nTarget:\nlemma map_surjective_of_isLocalization :\n    Function.Surjective (L.map : (X ⟶ Y) → _) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/ModelCategory","family_id":"map_surjective_of_islocalization","file_id":"mathlib/Mathlib/AlgebraicTopology/ModelCategory/FundamentalLemma.lean","sample_id":"17c8f688ffbfe3f08997e2e441397e109ef966a6e5472eeef0ec671a2f11c5b3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5137effd66997cbfcff7aea301f3e52b8dc7efd3fad26dd29be5d337e1b39c36","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"70992bd7daf4c7526afc1418802c19b19b9ccb0aaae0bf013a2ed9af653b15f8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0830875cbcc8e3b2fa3c485a8dfb76651d4e973c898e771f3c9e1024b82d5806","source_sha256":"1727a112351b769f97d17a6ca3e660167f05fda3d2180a8ca1a53c6a4fe0c013","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp","hard_negative":true,"metrics":{"chosen_tokens":16,"rejected_tokens":5,"token_jaccard":0.111111,"token_length_ratio":0.3125},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"b000f4d05fa2d070326b4a65a06a6bf5289161500c96fcbda8ca83540b2a9324","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Basic\npublic import Mathlib.Algebra.Order.AbsoluteValue.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Absolute values and the integers\n\nThis file contains some results on absolute values applied to integers.\n\n## Main results\n\n* `AbsoluteValue.map_units_int`: an absolute value sends all units of `ℤ` to `1`\n-/\n\npublic section\n\nvariable {R S : Type*} [Ring R] [CommRing S] [LinearOrder S] [IsStrictOrderedRing S]\n\n@[simp]\n\nTarget:\ntheorem AbsoluteValue.map_units_int (abv : AbsoluteValue ℤ S) (x : ℤˣ) : abv x = 1 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_0830875cbcc8","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"0e0464254a41f8d70a8a2c2073ed364198414fe763c551396cd8b4f9523dc7d3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Int","family_id":"absolutevalue","file_id":"mathlib/Mathlib/Data/Int/AbsoluteValue.lean","sample_id":"0830875cbcc8e3b2fa3c485a8dfb76651d4e973c898e771f3c9e1024b82d5806"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f11fe3945a9851f302dfa7c95843f91ea71198e392581ee0ee3ba699adf230d7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9757f7a78bda71d8e7517609a4d26326ae1122e116d5aaf449c4d98b5651cdcf","source_sha256":"b7c2e561541c6c2f2a91b3c7b8f53597d9188114f47fe81a231097a061f20b60","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [toPrincipalIdeal]; exact Units.ext_iff","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.166667,"token_length_ratio":0.272727},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"b00d899fb071b143c8beba00a5f12b533ece721404e20704e853074e6f9eb2e2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.Ideal.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# The ideal class group\n\nThis file defines the ideal class group `ClassGroup R` of fractional ideals of `R`\ninside its field of fractions.\n\n## Main definitions\n\n- `toPrincipalIdeal` sends an invertible `x : K` to an invertible fractional ideal\n- `ClassGroup` is the quotient of invertible fractional ideals modulo `toPrincipalIdeal.range`\n- `ClassGroup.mk0` sends a nonzero integral ideal in a Dedekind domain to its class\n\n## Main results\n- `ClassGroup.mk0_eq_mk0_iff` shows the equivalence with the \"classical\" definition,\n  where `I ~ J` iff `x I = y J` for `x y ≠ (0 : R)`\n- `ClassGroup.mk0_eq_quotientMk` identifies `ClassGroup.mk0` with the canonical quotient\n  projection on `(FractionalIdeal R⁰ (FractionRing R))ˣ`\n\n## Implementation details\n\nThe definition of `ClassGroup R` involves `FractionRing R`. However, the API should be completely\nidentical no matter the choice of field of fractions for `R`.\n-/\n\n@[expose] public noncomputable section\n\n\nvariable {R K : Type*} [CommRing R] [Field K] [Algebra R K] [IsFractionRing R K]\n\nopen scoped nonZeroDivisors\n\nopen IsLocalization IsFractionRing FractionalIdeal Units\n\nsection\n\nvariable (R K)\n\n/-- `toPrincipalIdeal R K x` sends `x ≠ 0 : K` to the fractional `R`-ideal generated by `x` -/\nirreducible_def toPrincipalIdeal : Kˣ →* (FractionalIdeal R⁰ K)ˣ :=\n  { toFun := fun x =>\n      ⟨spanSingleton _ x, spanSingleton _ x⁻¹, by\n        simp only [spanSingleton_one, Units.mul_inv', spanSingleton_mul_spanSingleton], by\n        simp only [spanSingleton_one, Units.inv_mul', spanSingleton_mul_spanSingleton]⟩\n    map_mul' := fun x y =>\n      ext (by simp only [Units.val_mul, spanSingleton_mul_spanSingleton])\n    map_one' := ext (by simp only [spanSingleton_one, Units.val_one]) }\n\nvariable {R K}\n\n@[simp]\ntheorem coe_toPrincipalIdeal (x : Kˣ) :\n    (toPrincipalIdeal R K x : FractionalIdeal R⁰ K) = spanSingleton _ (x : K) := by\n  simp only [toPrincipalIdeal]; rfl\n\n@[simp]\n\nTarget:\ntheorem toPrincipalIdeal_eq_iff {I : (FractionalIdeal R⁰ K)ˣ} {x : Kˣ} :\n    toPrincipalIdeal R K x = I ↔ spanSingleton R⁰ (x : K) = I :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/ClassGroup","family_id":"toprincipalideal_eq_iff","file_id":"mathlib/Mathlib/RingTheory/ClassGroup/Basic.lean","sample_id":"9757f7a78bda71d8e7517609a4d26326ae1122e116d5aaf449c4d98b5651cdcf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"36b9bfe8ecf433634e45203e8b895f570c57f0adf9b70b7ca8ace15a79e21d3e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"be90198cd5d1c08c4c9d1cf652ee121e5725615dc5e07c134efc8db6ae98a159","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bcfecfd0c0cb2071cab40553cc6bdd48d75c67302392e5f99abe6ae29aa62d18","source_sha256":"c10b46784e847fb243b6f808f2d97c20493e59f01e570b069afc8a2b315200ff","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  simp [single, ← Finsupp.smul_single]","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":15,"token_jaccard":0.785714,"token_length_ratio":1.363636},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"b00ef8a1028ff3b4d037b60a1ad4f00701c8039c95a90f0fa6fe4e5573615f11","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GroupWithZero.Action.TransferInstance\npublic import Mathlib.Algebra.Module.Defs\npublic import Mathlib.Data.Finsupp.Basic\npublic import Mathlib.Data.Finsupp.SMulWithZero\n\nNamespace:\nMonoidAlgebra\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Yury Kudryashov, Kim Morrison\n-/\n/-!\n# Monoid algebras\n\nWhen the domain of a `Finsupp` has a multiplicative or additive structure, we can define\na convolution product. To mathematicians this structure is known as the \"monoid algebra\",\ni.e. the finite formal linear combinations over a given semiring of elements of a monoid `M`.\nThe \"group ring\" `ℤ[G]` or the \"group algebra\" `k[G]` are typical uses.\n\nIn fact the construction of the \"monoid algebra\" makes sense when `M` is not even a monoid, but\nmerely a magma, i.e., when `M` carries a multiplication which is not required to satisfy any\nconditions at all. In this case the construction yields a not-necessarily-unital,\nnot-necessarily-associative algebra but it is still adjoint to the forgetful functor from such\nalgebras to magmas, and we prove this as `MonoidAlgebra.liftMagma`.\n\nIn this file we define `MonoidAlgebra R M := M →₀ R`, and `AddMonoidAlgebra R M`\nin the same way, and then define the convolution product on these.\n\nWhen the domain is additive, this is used to define polynomials:\n```\nPolynomial R := AddMonoidAlgebra R ℕ\nMvPolynomial σ α := AddMonoidAlgebra R (σ →₀ ℕ)\n```\nNote: `Polynomial R` is currently a wrapper around `AddMonoidAlgebra R ℕ` and not defeq to it.\nThere is ongoing work to make it defeq.\nSee https://github.com/leanprover-community/mathlib4/pull/25273\n\nWhen the domain is multiplicative, e.g. a group, this will be used to define the group ring.\n\n## Notation\n\nWe introduce the notation `R[M]` for both `MonoidAlgebra R M` and `AddMonoidAlgebra R M`.\nThe notations are scoped to their respective namespaces, and which one `R[M]` resolves to therefore\ndepends on which of the two namespaces is open.\n\n## TODO\n\nUse `coeff`/`ofCoeff` more widely. See\nhttps://github.com/leanprover-community/mathlib4/pull/36746\nhttps://github.com/leanprover-community/mathlib4/pull/25273\n-/\n\n@[expose] public section\n\nassert_not_exists NonUnitalAlgHom AlgEquiv\n\nnoncomputable section\n\nopen Finsupp hiding single\n\nvariable {R S G M N O ι : Type*}\n\n/-- The monoid algebra over a semiring `R` generated by the monoid `M`.\n\nIt is the type of finite formal `R`-linear combinations of terms of `M`,\nendowed with the convolution product. -/\n@[to_additive (relevant_arg := M)\n/-- The additive monoid algebra over a semiring `R` generated by the additive monoid `M`.\n\nIt is the type of finite formal `R`-linear combinations of terms of `M`,\nendowed with the convolution product. -/, to_additive_dont_translate]\ndef MonoidAlgebra (R M : Type*) [Semiring R] : Type _ := M →₀ R\n\nnamespace AddMonoidAlgebra\n\n@[inherit_doc AddMonoidAlgebra]\nscoped syntax:max (priority := high) term noWs \"[\" term \"]\" : term\n\nmacro_rules | `($R[$M]) => `(AddMonoidAlgebra $R $M)\n\n/-- Unexpander for `AddMonoidAlgebra`. -/\n@[scoped app_unexpander AddMonoidAlgebra]\nmeta def unexpander : Lean.PrettyPrinter.Unexpander\n  | `($_ $R $M) => `($R[$M])\n  | _ => throw ()\n\nend AddMonoidAlgebra\n\nnamespace MonoidAlgebra\n\n@[inherit_doc MonoidAlgebra]\nscoped syntax:max (priority := high) term noWs \"[\" term \"]\" : term\n\nmacro_rules | `($R[$M]) => `(MonoidAlgebra $R $M)\n\n/-- Unexpander for `MonoidAlgebra`. -/\n@[scoped app_unexpander MonoidAlgebra]\nmeta def unexpander : Lean.PrettyPrinter.Unexpander\n  | `($_ $R $M) => `($R[$M])\n  | _ => throw ()\n\nsection Semiring\nvariable [Semiring R] {x y : R[M]} {r r₁ r₂ : R} {m m' m₁ m₂ : M}\n\n/-- Construct an element of the monoid algebra `R[M]` from its coefficients `M →₀ R`. -/\n@[to_additive\n/-- Construct an element of the additive monoid algebra `R[M]` from its coefficients `M →₀ R`. -/]\ndef ofCoeff (x : M →₀ R) : R[M] := x\n\n/-- The coefficients `M →₀ R` of an element of the monoid algebra `R[M]`. -/\n@[to_additive\n/-- The coefficients `M →₀ R` of an element of the additive monoid algebra `R[M]`. -/]\ndef coeff (x : R[M]) : M →₀ R := x\n\n@[to_additive (attr := simp)] lemma coeff_ofCoeff (x : M →₀ R) : coeff (ofCoeff x) = x := rfl\n@[to_additive (attr := simp)] lemma ofCoeff_coeff (x : R[M]) : ofCoeff x.coeff = x := rfl\n\n/-- `MonoidAlgebra.coeff` as an equiv. -/\n@[to_additive (attr := simps apply symm_apply)\n/-- `AddMonoidAlgebra.coeff` as an equiv. -/]\ndef coeffEquiv : R[M] ≃ (M →₀ R) where\n  toFun := coeff\n  invFun := ofCoeff\n  left_inv _ := rfl\n  right_inv _ := rfl\n\n@[to_additive] lemma «forall» {P : R[M] → Prop} : (∀ p, P p) ↔ ∀ q, P (ofCoeff q) :=\n  coeffEquiv.forall_congr_left\n\n@[to_additive] lemma «exists» {P : R[M] → Prop} : (∃ p, P p) ↔ ∃ q, P (ofCoeff q) :=\n  coeffEquiv.exists_congr_left\n\n@[to_additive]\nlemma coeff_injective : (coeff : R[M] → M →₀ R).Injective := coeffEquiv.injective\n\n@[to_additive]\nlemma ofCoeff_injective : (ofCoeff : (M →₀ R) → R[M]).Injective := coeffEquiv.symm.injective\n\n@[to_additive (attr := simp)]\nlemma coeff_inj : x.coeff = y.coeff ↔ x = y := coeff_injective.eq_iff\n\n@[to_additive]\nlemma ofCoeff_inj {x y : M →₀ R} : ofCoeff x = ofCoeff y ↔ x = y := ofCoeff_injective.eq_iff\n\n@[to_additive] instance inhabited : Inhabited R[M] :=\n  inferInstanceAs <| Inhabited <| M →₀ R\n\n@[to_additive] instance nontrivial [Nontrivial R] [Nonempty M] : Nontrivial R[M] :=\n  inferInstanceAs <| Nontrivial <| M →₀ R\n\n@[to_additive] instance unique [Subsingleton R] : Unique R[M] :=\n  inferInstanceAs <| Unique <| M →₀ R\n\n@[to_additive] instance instDecidableEq [DecidableEq R] [DecidableEq M] : DecidableEq R[M] :=\n  inferInstanceAs <| DecidableEq <| M →₀ R\n\n-- TODO: this instance abuses definitional equality with `Finsupp.mapRange`\n@[to_additive] instance addCommMonoid : AddCommMonoid R[M] :=\n  fast_instance% { (inferInstance : AddCommMonoid <| M →₀ R) with\n    nsmul n x := x.mapRange (n • ·) (smul_zero _) }\n\n@[to_additive] instance instIsCancelAdd [IsCancelAdd R] : IsCancelAdd R[M] :=\n  inferInstanceAs <| IsCancelAdd <| M →₀ R\n\n-- TODO: Replace this with `coeff`. See https://github.com/leanprover-community/mathlib4/pull/36746\n#adaptation_note /-- Since nightly-2026-03-22,\nthis is needed or we get errors in UniversalFactorizationRing.lean -/\nset_option backward.inferInstanceAs.wrap false in\n@[to_additive] instance instCoeFun : CoeFun R[M] fun _ ↦ M → R :=\n  inferInstanceAs <| CoeFun (M →₀ R) fun _ ↦ M → R\n\n/-- A copy of `Finsupp.ext` for `MonoidAlgebra`. -/\n@[to_additive (attr := ext) /-- A copy of `Finsupp.ext` for `AddMonoidAlgebra`. -/]\nlemma ext ⦃f g : R[M]⦄ (hfg : ∀ m, f m = g m) : f = g := Finsupp.ext hfg\n\n/-- `MonoidAlgebra.coeff` as an `AddEquiv`. -/\n@[to_additive (attr := simps! apply symm_apply)\n/-- `AddMonoidAlgebra.coeff` as an `AddEquiv`. -/]\ndef coeffAddEquiv : R[M] ≃+ (M →₀ R) := coeffEquiv.addEquiv\n\n@[to_additive (attr := simp)] lemma coeff_zero : coeff (0 : R[M]) = 0 := rfl\n@[to_additive (attr := simp)] lemma ofCoeff_zero : (ofCoeff 0 : R[M]) = 0 := rfl\n@[to_additive (attr := simp)] lemma coeff_eq_zero : coeff x = 0 ↔ x = 0 := coeff_inj\n@[to_additive (attr := simp)] lemma ofCoeff_eq_zero {x : M →₀ R} : ofCoeff x = 0 ↔ x = 0 :=\n  ofCoeff_inj\n\n@[to_additive (attr := simp)]\nlemma coeff_add (x y : R[M]) : coeff (x + y) = coeff x + coeff y := rfl\n\n@[to_additive (attr := simp)]\nlemma ofCoeff_add (x y : M →₀ R) : ofCoeff (x + y) = ofCoeff x + ofCoeff y := rfl\n\n@[to_additive (attr := simp)]\nlemma coeff_sum (s : Finset ι) (f : ι → R[M]) :\n    coeff (∑ i ∈ s, f i) = ∑ i ∈ s, coeff (f i) := map_sum coeffAddEquiv ..\n\n@[to_additive (attr := simp)]\nlemma ofCoeff_sum (s : Finset ι) (f : ι → M →₀ R) :\n    ofCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, ofCoeff (f i) := map_sum coeffAddEquiv.symm ..\n\n@[to_additive (attr := simp)]\nlemma coeff_finsuppSum [AddCommMonoid N] (f : ι →₀ N) (g : ι → N → R[M]) :\n    coeff (f.sum g) = f.sum (fun i n ↦ coeff (g i n)) := map_finsuppSum coeffAddEquiv ..\n\n@[to_additive (attr := simp)]\nlemma ofCoeff_finsuppSum [AddCommMonoid N] (f : ι →₀ N) (g : ι → N → M →₀ R) :\n    ofCoeff (f.sum g) = f.sum (fun i n ↦ ofCoeff (g i n)) := map_finsuppSum coeffAddEquiv.symm ..\n\n-- TODO: This definition is very leaky, and we later have frequent problems conflating the two\n-- versions of `single`. Perhaps someone wants to try making this a `def` rather than an `abbrev`?\n-- In Mathlib 3 this was locally reducible.\n/-- `MonoidAlgebra.single m r` for `m : M`, `r : R` is the element `rm : R[M]`. -/\n@[to_additive\n/-- `AddMonoidAlgebra.single m r` for `m : M`, `r : R` is the element `rm : R[M]`. -/]\nabbrev single (m : M) (r : R) : R[M] := Finsupp.single m r\n\n/-- Remove a term from an element of the monoid algebra. -/\n@[to_additive /-- Remove a term from an element of the additive monoid algebra. -/]\ndef erase (m : M) (x : R[M]) : R[M] := .ofCoeff <| .erase m x.coeff\n\n@[to_additive (attr := simp)]\nlemma coeff_erase (m : M) (x : R[M]) : (x.erase m).coeff = x.coeff.erase m := rfl\n\n@[to_additive (attr := simp)]\nlemma ofCoeff_erase (m : M) (x : M →₀ R) : ofCoeff (x.erase m) = (ofCoeff x).erase m := rfl\n\n@[to_additive (attr := simp)]\nlemma erase_zero (m : M) : erase m (0 : R[M]) = 0 := by simp [erase]\n\n@[to_additive (attr := simp)]\nlemma erase_single (m : M) (r : R) : erase m (single m r) = 0 := by\n  simp [erase, ofCoeff, coeff]; rfl\n\n/-- Replace the `m`-th coefficient of an element `x` of the monoid algebra by a given value `r : R`.\nIf `r = 0`, this is equal to `x.erase m`. -/\n@[to_additive\n/-- Replace the `m`-th coefficient of an element `x` of the monoid algebra by a given value `r : R`.\nIf `r = 0`, this is equal to `x.erase m`. -/]\ndef update (m : M) (r : R) (x : R[M]) : R[M] :=\n  ofCoeff (x.coeff.update m r)\n\n@[to_additive (attr := simp)]\nlemma coeff_update (m : M) (r : R) (x : R[M]) :\n    (x.update m r).coeff = x.coeff.update m r := rfl\n\n@[to_additive (attr := simp)]\nlemma ofCoeff_update (m : M) (r : R) (x : M →₀ R) :\n    ofCoeff (x.update m r) = (ofCoeff x).update m r := rfl\n\nsection SMul\n\n/-! ### Basic scalar multiplication instances\n\nThis section collects instances needed for the algebraic structure of `Polynomial`,\nwhich is defined in terms of `MonoidAlgebra`.\nFurther results on scalar multiplication can be found in\n`Mathlib/Algebra/MonoidAlgebra/Module.lean`.\n-/\n\nvariable {A : Type*} [SMulZeroClass A R]\n\n-- TODO: this instance abuses definitional equality with `Finsupp.mapRange`\n@[to_additive (dont_translate := A) smulZeroClass]\ninstance smulZeroClass : SMulZeroClass A R[M] :=\n  fast_instance% { (inferInstance : SMulZeroClass A (M →₀ R)) with\n    smul a x := x.mapRange (a • ·) (smul_zero _) }\n\nsection\n-- Ensure that the different smul instances do not create a diamond.\nexample : (smulZeroClass (A := ℕ) (R := R) (M := M)).toSMul = addCommMonoid.toNSMul := by\n  with_reducible_and_instances rfl\n\n-- Ensure that smul has good defeq properties\nprivate local instance {α} [Monoid M] [SMul M α] : SMul Mˣ α where smul m a := (m : M) • a\nexample [Monoid A] (a : Units A) (x : R[M]) :\n    a • x = (a : A) • x := by\n  with_reducible_and_instances rfl\nend\n\n@[to_additive (dont_translate := A) (attr := simp) coeff_smul]\nlemma coeff_smul (a : A) (x : R[M]) : coeff (a • x) = a • coeff x := rfl\n\n@[to_additive (dont_translate := A) (attr := simp) ofCoeff_smul]\nlemma ofCoeff_smul (a : A) (x : M →₀ R) : ofCoeff (a • x) = a • ofCoeff x := rfl\n\n@[to_additive (attr := simp) (dont_translate := A) smul_apply]\nlemma smul_apply (a : A) (x : R[M]) (m : M) : (a • x) m = a • x m := rfl\n\n@[to_additive (attr := simp) (dont_translate := A) smul_single]\n\nTarget:\nlemma smul_single (a : A) (m : M) (r : R) : a • single m r = single m (a • r) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  ext\n  simp [single, ← Finsupp.smul_single]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/MonoidAlgebra","family_id":"smul_single","file_id":"mathlib/Mathlib/Algebra/MonoidAlgebra/Defs.lean","sample_id":"bcfecfd0c0cb2071cab40553cc6bdd48d75c67302392e5f99abe6ae29aa62d18"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4a140838d46443a223247967ef703140b4be6ea2808c8d4f93f75968349bd809","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b579425a25a44dbf03d7aec5fcfc9f633b6db243ba8c3f42fbd8c2796e7b54f2","source_sha256":"38b81cd46858f243db931b63ae095bb873edc08d47b08e77c1c254a8ae6a5653","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := h.out\n  simp [this]\n\n/- evalCast -/","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.2},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"b0108481ac663956f3a661c3f8f9327eac7375b084f845f516b5b57355b39b63","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Defs\npublic import Mathlib.Algebra.Algebra.Basic\npublic import Mathlib.Tactic.Ring.RingNF\n\nNamespace:\nMathlib.Tactic.Algebra\n\nLocal context:\n/-\nCopyright (c) 2025 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Arend Mellendijk\n-/\n/-! # Lemmas for the `algebra` tactic.\n-/\n\n@[expose] public section\n\nopen Mathlib.Meta.NormNum\n\nnamespace Mathlib.Tactic.Algebra\n\nsection ring\n\nvariable {R A : Type*} [CommRing R] [CommRing A] [Algebra R A]\n\n/- evalCast -/\ntheorem isInt_negOfNat_eq {a : A} {lit : ℕ} (h : IsInt a (Int.negOfNat lit)) :\n    a = algebraMap R A (Int.rawCast (Int.negOfNat lit) + 0 : R) + 0 := by\n  simp [h.out]\n\nend ring\n\nsection semifield\n\nvariable {R A : Type*} [Semifield R] [Semifield A] [Algebra R A]\n\n/- evalCast -/\ntheorem isNNRat_eq_rawCast {a : A} {n d : ℕ} (h : IsNNRat a n d) :\n    a = algebraMap R A (NNRat.rawCast n d + 0 : R) + 0 := by\n  simp [Mathlib.Tactic.Ring.cast_nnrat h]\n\nend semifield\n\nsection field\n\nvariable {R A : Type*} [Field R] [Field A] [Algebra R A]\n\n/- evalCast -/\ntheorem isRat_eq_rawCast {a : A} {n d : ℕ} (h : IsRat a (.negOfNat n) d) :\n    a = algebraMap R A (Rat.rawCast (.negOfNat n) d + 0 : R) + 0 := by\n  simp [Mathlib.Tactic.Ring.cast_rat h]\n\nend field\n\nvariable {R A : Type*} [sR : CommSemiring R] [sA : CommSemiring A] [sAlg : Algebra R A]\n\n/- evalCast -/\n\nTarget:\ntheorem isNat_zero_eq {a : A} (h : IsNat a 0) : a = 0 :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Algebra","family_id":"isnat_zero_eq","file_id":"mathlib/Mathlib/Tactic/Algebra/Lemmas.lean","sample_id":"b579425a25a44dbf03d7aec5fcfc9f633b6db243ba8c3f42fbd8c2796e7b54f2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"73aa3a7c73d07e3e5e0d2c9a0dc0fab61b3371c9b2e729b99cd088422d654f85","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ba99dd8f6392afa104bfc6ac9136b281cb75f223a9c829f9749723664ecb6a1c","source_sha256":"135835936222dd833aca95a55105deb9bdc869adf3601290cb91b6f312b8a295","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases Finite.exists_equiv_fin β with ⟨n, ⟨e⟩⟩\n  classical exact .of_equiv (Set.range (e ∘ f)) (Equiv.ofInjective _ (e.injective.comp H)).symm\n\n-- see Note [lower instance priority]","hard_negative":false,"metrics":{"chosen_tokens":53,"rejected_tokens":2,"token_jaccard":0.027027,"token_length_ratio":0.037736},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"b021b760ce8bf23cb553f9595796c2bc8b7145bc493171c3dcb1f2fb40b785c4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Card\npublic import Mathlib.Data.List.NodupEquivFin\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Equivalences between `Fintype`, `Fin` and `Finite`\n\nThis file defines the bijection between a `Fintype α` and `Fin (Fintype.card α)`, and uses this to\nrelate `Fintype` with `Finite`. From that we can derive properties of `Finite` and `Infinite`,\nand show some instances of `Infinite`.\n\n## Main declarations\n\n* `Fintype.truncEquivFin`: A fintype `α` is computably equivalent to `Fin (card α)`. The\n  `Trunc`-free, noncomputable version is `Fintype.equivFin`.\n* `Fintype.truncEquivOfCardEq` `Fintype.equivOfCardEq`: Two fintypes of same cardinality are\n  equivalent. See above.\n* `Fin.equiv_iff_eq`: `Fin m ≃ Fin n` iff `m = n`.\n* `Infinite.natEmbedding`: An embedding of `ℕ` into an infinite type.\n\nTypes which have an injection from/a surjection to an `Infinite` type are themselves `Infinite`.\nSee `Infinite.of_injective` and `Infinite.of_surjective`.\n\n## Instances\n\nWe provide `Infinite` instances for\n* specific types: `ℕ`, `ℤ`, `String`\n* type constructors: `Multiset α`, `List α`\n\n-/\n\n@[expose] public section\n\nassert_not_exists Monoid\n\nopen Function\n\nuniverse u v\n\nvariable {α β γ : Type*}\n\nopen Finset\n\nnamespace Fintype\n\n/-- There is (computably) an equivalence between `α` and `Fin (card α)`.\n\nSince it is not unique and depends on which permutation\nof the universe list is used, the equivalence is wrapped in `Trunc` to\npreserve computability.\n\nSee `Fintype.equivFin` for the noncomputable version,\nand `Fintype.truncEquivFinOfCardEq` and `Fintype.equivFinOfCardEq`\nfor an equiv `α ≃ Fin n` given `Fintype.card α = n`.\n\nSee `Fintype.truncFinBijection` for a version without `[DecidableEq α]`.\n-/\ndef truncEquivFin (α) [DecidableEq α] [Fintype α] : Trunc (α ≃ Fin (card α)) := by\n  unfold card Finset.card\n  exact\n    Quot.recOnSubsingleton\n      (motive := fun s : Multiset α =>\n        (∀ x : α, x ∈ s) → s.Nodup → Trunc (α ≃ Fin (Multiset.card s)))\n      univ.val\n      (fun l (h : ∀ x : α, x ∈ l) (nd : l.Nodup) => Trunc.mk (nd.getEquivOfForallMemList _ h).symm)\n      mem_univ_val univ.2\n\n/-- There is (noncomputably) an equivalence between `α` and `Fin (card α)`.\n\nSee `Fintype.truncEquivFin` for the computable version,\nand `Fintype.truncEquivFinOfCardEq` and `Fintype.equivFinOfCardEq`\nfor an equiv `α ≃ Fin n` given `Fintype.card α = n`.\n-/\nnoncomputable def equivFin (α) [Fintype α] : α ≃ Fin (card α) :=\n  letI := Classical.decEq α\n  (truncEquivFin α).out\n\n/-- There is (computably) a bijection between `Fin (card α)` and `α`.\n\nSince it is not unique and depends on which permutation\nof the universe list is used, the bijection is wrapped in `Trunc` to\npreserve computability.\n\nSee `Fintype.truncEquivFin` for a version that gives an equivalence\ngiven `[DecidableEq α]`.\n-/\ndef truncFinBijection (α) [Fintype α] : Trunc { f : Fin (card α) → α // Bijective f } := by\n  unfold card Finset.card\n  refine\n    Quot.recOnSubsingleton\n      (motive := fun s : Multiset α =>\n        (∀ x : α, x ∈ s) → s.Nodup → Trunc {f : Fin (Multiset.card s) → α // Bijective f})\n      univ.val\n      (fun l (h : ∀ x : α, x ∈ l) (nd : l.Nodup) => Trunc.mk (nd.getBijectionOfForallMemList _ h))\n      mem_univ_val univ.2\n\nend Fintype\n\nnamespace Fintype\n\nsection\n\nvariable [Fintype α] [Fintype β]\n\n/-- If the cardinality of `α` is `n`, there is computably a bijection between `α` and `Fin n`.\n\nSee `Fintype.equivFinOfCardEq` for the noncomputable definition,\nand `Fintype.truncEquivFin` and `Fintype.equivFin` for the bijection `α ≃ Fin (card α)`.\n-/\ndef truncEquivFinOfCardEq [DecidableEq α] {n : ℕ} (h : Fintype.card α = n) : Trunc (α ≃ Fin n) :=\n  (truncEquivFin α).map fun e => e.trans (finCongr h)\n\n/-- If the cardinality of `α` is `n`, there is noncomputably a bijection between `α` and `Fin n`.\n\nSee `Fintype.truncEquivFinOfCardEq` for the computable definition,\nand `Fintype.truncEquivFin` and `Fintype.equivFin` for the bijection `α ≃ Fin (card α)`.\n-/\nnoncomputable def equivFinOfCardEq {n : ℕ} (h : Fintype.card α = n) : α ≃ Fin n :=\n  letI := Classical.decEq α\n  (truncEquivFinOfCardEq h).out\n\n/-- Two `Fintype`s with the same cardinality are (computably) in bijection.\n\nSee `Fintype.equivOfCardEq` for the noncomputable version,\nand `Fintype.truncEquivFinOfCardEq` and `Fintype.equivFinOfCardEq` for\nthe specialization to `Fin`.\n-/\ndef truncEquivOfCardEq [DecidableEq α] [DecidableEq β] (h : card α = card β) : Trunc (α ≃ β) :=\n  (truncEquivFinOfCardEq h).bind fun e => (truncEquivFin β).map fun e' => e.trans e'.symm\n\n/-- Two `Fintype`s with the same cardinality are (noncomputably) in bijection.\n\nSee `Fintype.truncEquivOfCardEq` for the computable version,\nand `Fintype.truncEquivFinOfCardEq` and `Fintype.equivFinOfCardEq` for\nthe specialization to `Fin`.\n-/\nnoncomputable def equivOfCardEq (h : card α = card β) : α ≃ β := by\n  letI := Classical.decEq α\n  letI := Classical.decEq β\n  exact (truncEquivOfCardEq h).out\n\nend\n\ntheorem card_eq {α β} [_F : Fintype α] [_G : Fintype β] : card α = card β ↔ Nonempty (α ≃ β) :=\n  ⟨fun h =>\n    haveI := Classical.propDecidable\n    (truncEquivOfCardEq h).nonempty,\n    fun ⟨f⟩ => card_congr f⟩\n\nend Fintype\n\n/-!\n### Relation to `Finite`\n\nIn this section we prove that `α : Type*` is `Finite` if and only if `Fintype α` is nonempty.\n-/\n\nprotected theorem Fintype.finite {α : Type*} (_inst : Fintype α) : Finite α :=\n  ⟨Fintype.equivFin α⟩\n\nset_option linter.unusedFintypeInType false in\n/-- For efficiency reasons, we want `Finite` instances to have higher\npriority than ones coming from `Fintype` instances. -/\ninstance (priority := 900) Finite.of_fintype (α : Type*) [Fintype α] : Finite α :=\n  Fintype.finite ‹_›\n\ntheorem finite_iff_nonempty_fintype (α : Type*) : Finite α ↔ Nonempty (Fintype α) :=\n  ⟨fun _ => nonempty_fintype α, fun ⟨_⟩ => inferInstance⟩\n\n/-- Noncomputably get a `Fintype` instance from a `Finite` instance. This is not an\ninstance because we want `Fintype` instances to be useful for computations. -/\n@[instance_reducible]\nnoncomputable def Fintype.ofFinite (α : Type*) [Finite α] : Fintype α :=\n  (nonempty_fintype α).some\n\nTarget:\ntheorem Finite.of_injective {α β : Sort*} [Finite β] (f : α → β) (H : Injective f) : Finite α :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Fintype","family_id":"finite","file_id":"mathlib/Mathlib/Data/Fintype/EquivFin.lean","sample_id":"ba99dd8f6392afa104bfc6ac9136b281cb75f223a9c829f9749723664ecb6a1c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cd750d60c8401fd18ff81d5b091050a985f85f02e4e58110a29f7aecd6f9f0ac","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f168373cfd1f31b36e20d360cfbb18b1a6084a7395ef31c4c3eb93947132babd","source_sha256":"586377b5167eecc243d7b879009165834566ed22063836fb061931738feab6a2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [card_eq_card_quotient_mul_card_subgroup s, @dvd_mul_right ℕ]\n\nvariable {H : Type*} [Group H]","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":8,"token_jaccard":0.086957,"token_length_ratio":0.380952},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"b061d0f7784ceb23a4dd3a7f223126f171f34c40ace73f6afdd43410473d0006","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.Coset.Basic\npublic import Mathlib.SetTheory.Cardinal.Finite\n\nNamespace:\nSubgroup\n\nLocal context:\n/-\nCopyright (c) 2018 Mitchell Rowett. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mitchell Rowett, Kim Morrison\n-/\n/-!\n# Lagrange's theorem: the order of a subgroup divides the order of the group.\n\n* `Subgroup.card_subgroup_dvd_card`: Lagrange's theorem (for multiplicative groups);\n  there is an analogous version for additive groups\n\n-/\n\npublic section\n\nassert_not_exists Field\n\nopen scoped Pointwise\n\nvariable {α : Type*} [Group α] {s : Subgroup α}\n\nnamespace QuotientGroup\n\n@[to_additive]\ninstance fintype [Fintype α] (s : Subgroup α) [DecidableRel (leftRel s).r] : Fintype (α ⧸ s) :=\n  Quotient.fintype (leftRel s)\n\n@[to_additive]\ninstance (priority := 100) finite [Finite α] : Finite (α ⧸ s) :=\n  Quotient.finite _\n\n@[to_additive]\ninstance fintypeQuotientRightRel [Fintype (α ⧸ s)] :\n    Fintype (Quotient (QuotientGroup.rightRel s)) :=\n  .ofEquiv (α ⧸ s) (QuotientGroup.quotientRightRelEquivQuotientLeftRel s).symm\n\nvariable (s) in\n@[to_additive]\nlemma card_quotient_rightRel [Fintype (α ⧸ s)] :\n    Fintype.card (Quotient (QuotientGroup.rightRel s)) = Fintype.card (α ⧸ s) :=\n  Fintype.ofEquiv_card (QuotientGroup.quotientRightRelEquivQuotientLeftRel s).symm\n\nend QuotientGroup\n\nnamespace Subgroup\n\n@[to_additive AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup]\ntheorem card_eq_card_quotient_mul_card_subgroup (s : Subgroup α) :\n    Nat.card α = Nat.card (α ⧸ s) * Nat.card s := by\n  rw [← Nat.card_prod]; exact Nat.card_congr Subgroup.groupEquivQuotientProdSubgroup\n\n@[to_additive]\nlemma card_mul_eq_card_subgroup_mul_card_quotient (s : Subgroup α) (t : Set α) :\n    Nat.card (t * s : Set α) = Nat.card s * Nat.card (t.image (↑) : Set (α ⧸ s)) := by\n  rw [← Nat.card_prod, Nat.card_congr]\n  apply Equiv.trans _ (QuotientGroup.preimageMkEquivSubgroupProdSet _ _)\n  rw [QuotientGroup.preimage_image_mk]\n  convert! Equiv.refl ↑(t * s)\n  aesop (add simp [Set.mem_mul])\n\n/-- **Lagrange's Theorem**: The order of a subgroup divides the order of its ambient group. -/\n@[to_additive (attr := wikidata Q505798) /-- **Lagrange's Theorem**: The order of an additive\nsubgroup divides the order of its ambient additive group. -/]\ntheorem card_subgroup_dvd_card (s : Subgroup α) : Nat.card s ∣ Nat.card α := by\n  classical simp [card_eq_card_quotient_mul_card_subgroup s, @dvd_mul_left ℕ]\n\n@[to_additive]\n\nTarget:\ntheorem card_quotient_dvd_card (s : Subgroup α) : Nat.card (α ⧸ s) ∣ Nat.card α :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"507fabdaaf3c0886cf347528bd5cb83a491b6979b19e352c734731a872a2f8f9","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Coset","family_id":"card_quotient_dvd_card","file_id":"mathlib/Mathlib/GroupTheory/Coset/Card.lean","sample_id":"f168373cfd1f31b36e20d360cfbb18b1a6084a7395ef31c4c3eb93947132babd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c1ea19ea49c645e697b6d5466464113860db944a733b14a16a41636e7325b4ec","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3065ddea640d150a719f7ca546348a960da065f54e8ab9a9d7476b502ae8f9af","source_sha256":"8015c56b849258c13cb728b7c24ca8097da450018f8c58e78e02913505a6b7f7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨x, rfl⟩ := x.mk_surjective\n  obtain ⟨y, rfl⟩ := y.mk_surjective\n  obtain rfl : x = y := h\n  rfl","hard_negative":true,"metrics":{"chosen_tokens":30,"rejected_tokens":8,"token_jaccard":0.166667,"token_length_ratio":0.266667},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"b071e4f739699b10dc7463c73093a908e8f23fb1beb679191c8ba6b563cff1b7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.SimplicialObject.Basic\npublic import Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal\npublic import Mathlib.AlgebraicTopology.SimplicialSet.CompStruct\npublic import Mathlib.AlgebraicTopology.SimplexCategory.Truncated\npublic import Mathlib.CategoryTheory.Category.ReflQuiv\npublic import Mathlib.Combinatorics.Quiver.ReflQuiver\npublic import Mathlib.AlgebraicTopology.SimplicialSet.Monoidal\npublic import Mathlib.CategoryTheory.Category.Cat.Terminal\n\nNamespace:\nSSet.Truncated.HomotopyCategory\n\nLocal context:\n/-\nCopyright (c) 2024 Mario Carneiro and Emily Riehl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Emily Riehl, Joël Riou\n-/\n/-!\n\n# The homotopy category of a simplicial set\n\nThe homotopy category of a simplicial set is defined as a quotient of the free category on its\nunderlying reflexive quiver (equivalently its one truncation). The quotient imposes an additional\nhom relation on this free category, asserting that `f ≫ g = h` whenever `f`, `g`, and `h` are\nrespectively the 2nd, 0th, and 1st faces of a 2-simplex.\n\nIn fact, the associated functor\n\n`SSet.hoFunctor : SSet.{u} ⥤ Cat.{u, u} := SSet.truncation 2 ⋙ SSet.hoFunctor₂`\n\nis defined by first restricting from simplicial sets to 2-truncated simplicial sets (throwing away\nthe data that is not used for the construction of the homotopy category) and then composing with an\nanalogously defined `SSet.hoFunctor₂ : SSet.Truncated.{u} 2 ⥤ Cat.{u,u}` implemented relative to\nthe syntax of the 2-truncated simplex category.\n\nIn the file `Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean` we show the functor\n`SSet.hoFunctor` to be left adjoint to the nerve by providing an analogous decomposition of the\nnerve functor, made by possible by the fact that nerves of categories are 2-coskeletal, and then\ncomposing a pair of adjunctions, which factor through the category of 2-truncated simplicial sets.\n-/\n\n@[expose] public section\n\nnamespace SSet\nopen CategoryTheory Category Limits Functor Opposite Simplicial Nerve\nopen SimplexCategory.Truncated SimplicialObject.Truncated\n\nuniverse v u\n\n/-- A 2-truncated simplicial set `S` has an underlying refl quiver with `S _⦋0⦌₂` as its underlying\ntype. -/\ndef OneTruncation₂ (S : SSet.Truncated 2) := S _⦋0⦌₂\n\nnamespace OneTruncation₂\n\n/-- A 2-truncated simplicial set `S` has an underlying refl quiver `SSet.OneTruncation₂ S`. -/\n@[simps -isSimp]\ninstance reflQuiver (S : SSet.Truncated 2) : ReflQuiver (OneTruncation₂ S) where\n  Hom := Truncated.Edge\n  id := Truncated.Edge.id\n\n@[ext]\nlemma hom_ext\n    {S : SSet.Truncated 2} {x y : OneTruncation₂ S} {f g : x ⟶ y}\n    (h : f.edge = g.edge) : f = g :=\n  Truncated.Edge.ext h\n\n/-- The prefunctor on refl quivers `OneTruncation₂` induced by a morphism\nof `2`-truncated simplicial sets. -/\n@[simps]\ndef map {S T : SSet.Truncated 2} (f : S ⟶ T) :\n    OneTruncation₂ S ⥤rq OneTruncation₂ T where\n  obj x := f.app _ x\n  map e := e.map f\n  map_id x := by ext; simp [← NatTrans.naturality_apply, reflQuiver_id]\n\nend OneTruncation₂\n\n/-- The functor that carries a 2-truncated simplicial set to its underlying refl quiver. -/\n@[simps]\ndef oneTruncation₂ : SSet.Truncated.{u} 2 ⥤ ReflQuiv.{u, u} where\n  obj S := ReflQuiv.of (OneTruncation₂ S)\n  map f := OneTruncation₂.map f\n\nnamespace OneTruncation₂\n\n@[simp]\nlemma homOfEq_edge\n    {X : SSet.Truncated.{u} 2} {x₁ y₁ x₂ y₂ : OneTruncation₂ X}\n    (f : x₁ ⟶ y₁) (hx : x₁ = x₂) (hy : y₁ = y₂) :\n    (Quiver.homOfEq f hx hy).edge = f.edge := by\n  subst hx hy\n  rfl\n\nsection\nvariable {C : Type u} [Category.{v} C]\n\n/-- An equivalence between the type of objects underlying a category and the type of 0-simplices in\nthe 2-truncated nerve. -/\n@[simps! -isSimp]\ndef nerveEquiv : OneTruncation₂ ((SSet.truncation 2).obj (nerve C)) ≃ C :=\n  CategoryTheory.nerveEquiv\n\n/-- A hom equivalence over the function `OneTruncation₂.nerveEquiv`. -/\ndef nerveHomEquiv {X Y : OneTruncation₂ ((SSet.truncation 2).obj (nerve C))} :\n    (X ⟶ Y) ≃ (nerveEquiv X ⟶ nerveEquiv Y) :=\n  nerve.homEquiv\n\nlemma nerveHomEquiv_apply {X Y : OneTruncation₂ ((SSet.truncation 2).obj (nerve C))}\n    (f : X ⟶ Y) :\n    nerveHomEquiv f = eqToHom (congr_arg ComposableArrows.left f.src_eq.symm) ≫\n      f.edge.hom ≫ eqToHom (congr_arg ComposableArrows.left f.tgt_eq) :=\n  rfl\n\n@[simp]\nlemma nerveHomEquiv_id (X : OneTruncation₂ ((SSet.truncation 2).obj (nerve C))) :\n    nerveHomEquiv (𝟙rq X) = 𝟙 _ :=\n  nerve.homEquiv_id _\n\n/-- The refl quiver underlying a nerve is isomorphic to the refl quiver underlying the category. -/\ndef ofNerve₂ (C : Type u) [Category.{u} C] :\n    ReflQuiv.of (OneTruncation₂ ((truncation 2).obj (nerve C))) ≅ ReflQuiv.of C :=\n  ReflQuiv.isoOfEquiv.{u, u} OneTruncation₂.nerveEquiv\n    (fun _ _ ↦ OneTruncation₂.nerveHomEquiv) nerveHomEquiv_id\n\nlemma nerve_hom_ext {X : (SSet.Truncated 2)} {C : Type u} [Category.{u} C]\n    {F G : X ⟶ ((truncation 2).obj (nerve C))}\n    (h : OneTruncation₂.map F = OneTruncation₂.map G) : F = G :=\n  SSet.Truncated.IsStrictSegal.hom_ext (fun f ↦ by\n    obtain ⟨x₀, x₁, f, rfl⟩ := Truncated.Edge.exists_of_simplex f\n    simpa using congr_arg Truncated.Edge.edge (ReflPrefunctor.congr_hom h f))\n\nend\nend OneTruncation₂\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The refl quiver underlying a nerve is naturally isomorphic to the refl quiver underlying the\ncategory. -/\n@[simps! hom_app_obj hom_app_map inv_app_obj_obj inv_app_obj_map inv_app_map]\ndef OneTruncation₂.ofNerve₂.natIso :\n    nerveFunctor₂.{u, u} ⋙ SSet.oneTruncation₂ ≅ ReflQuiv.forget :=\n  NatIso.ofComponents (fun C => OneTruncation₂.ofNerve₂ C)\n    (fun F ↦ ReflPrefunctor.ext (by cat_disch) (fun x y f ↦ by\n      obtain ⟨f, rfl, rfl⟩ := f\n      dsimp [ofNerve₂, ReflQuiv.isoOfEquiv, ReflQuiv.isoOfQuivIso,\n        Quiv.isoOfEquiv, nerveHomEquiv_apply]\n      simp only [comp_id, id_comp]\n      rfl))\n\nset_option backward.privateInPublic true in\nprivate lemma map_map_of_eq.{w} {C : Type u} [Category.{v} C] (V : Cᵒᵖ ⥤ Type w) {X Y Z : C}\n    {α : X ⟶ Y} {β : Y ⟶ Z} {γ : X ⟶ Z} {φ} :\n    α ≫ β = γ → V.map α.op (V.map β.op φ) = V.map γ.op φ := by\n  rintro rfl\n  simp\n\nnamespace Truncated\n\n/-- The map that picks up the initial vertex of a 2-simplex, as a morphism in the 2-truncated\nsimplex category. -/\ndef ι0₂ : ⦋0⦌₂ ⟶ ⦋2⦌₂ := δ₂ (n := 0) 1 ≫ δ₂ (n := 1) 1\n\n/-- The map that picks up the middle vertex of a 2-simplex, as a morphism in the 2-truncated\nsimplex category. -/\ndef ι1₂ : ⦋0⦌₂ ⟶ ⦋2⦌₂ := δ₂ (n := 0) 0 ≫ δ₂ (n := 1) 2\n\n/-- The map that picks up the final vertex of a 2-simplex, as a morphism in the 2-truncated\nsimplex category. -/\ndef ι2₂ : ⦋0⦌₂ ⟶ ⦋2⦌₂ := δ₂ (n := 0) 0 ≫ δ₂ (n := 1) 1\n\n/-- The initial vertex of a 2-simplex in a 2-truncated simplicial set. -/\ndef ev0₂ {V : SSet.Truncated 2} (φ : V _⦋2⦌₂) : OneTruncation₂ V := V.map ι0₂.op φ\n\n/-- The middle vertex of a 2-simplex in a 2-truncated simplicial set. -/\ndef ev1₂ {V : SSet.Truncated 2} (φ : V _⦋2⦌₂) : OneTruncation₂ V := V.map ι1₂.op φ\n\n/-- The final vertex of a 2-simplex in a 2-truncated simplicial set. -/\ndef ev2₂ {V : SSet.Truncated 2} (φ : V _⦋2⦌₂) : OneTruncation₂ V := V.map ι2₂.op φ\n\n/-- The 0th face of a 2-simplex, as a morphism in the 2-truncated simplex category. -/\ndef δ0₂ : ⦋1⦌₂ ⟶ ⦋2⦌₂ := δ₂ (n := 1) 0\n\n/-- The 1st face of a 2-simplex, as a morphism in the 2-truncated simplex category. -/\ndef δ1₂ : ⦋1⦌₂ ⟶ ⦋2⦌₂ := δ₂ (n := 1) 1\n\n/-- The 2nd face of a 2-simplex, as a morphism in the 2-truncated simplex category. -/\ndef δ2₂ : ⦋1⦌₂ ⟶ ⦋2⦌₂ := δ₂ (n := 1) 2\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- The arrow in the ReflQuiver `OneTruncation₂ V` of a 2-truncated simplicial set arising from the\n0th face of a 2-simplex. -/\ndef ev12₂ {V : SSet.Truncated 2} (φ : V _⦋2⦌₂) : ev1₂ φ ⟶ ev2₂ φ :=\n  ⟨V.map δ0₂.op φ,\n    map_map_of_eq V (InducedCategory.hom_ext\n      (SimplexCategory.δ_comp_δ (i := 0) (j := 1) (by decide)).symm),\n    map_map_of_eq V rfl⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- The arrow in the ReflQuiver `OneTruncation₂ V` of a 2-truncated simplicial set arising from the\n1st face of a 2-simplex. -/\ndef ev02₂ {V : SSet.Truncated 2} (φ : V _⦋2⦌₂) : ev0₂ φ ⟶ ev2₂ φ :=\n  ⟨V.map δ1₂.op φ, map_map_of_eq V rfl, map_map_of_eq V rfl⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- The arrow in the ReflQuiver `OneTruncation₂ V` of a 2-truncated simplicial set arising from the\n2nd face of a 2-simplex. -/\ndef ev01₂ {V : SSet.Truncated 2} (φ : V _⦋2⦌₂) : ev0₂ φ ⟶ ev1₂ φ :=\n  ⟨V.map δ2₂.op φ,\n    map_map_of_eq V (InducedCategory.hom_ext (SimplexCategory.δ_comp_δ (j := 1) le_rfl)),\n    map_map_of_eq V rfl⟩\n\nend Truncated\n\nnamespace OneTruncation₂\n\nvariable (V : SSet.Truncated.{u} 2)\n\n/-- The 2-simplices in a 2-truncated simplicial set `V` generate a hom relation on the free\ncategory on the underlying refl quiver of `V`. -/\ninductive HoRel₂ : HomRel (Cat.FreeRefl (OneTruncation₂ V)) where\n  | of_compStruct {x₀ x₁ x₂ : V _⦋0⦌₂} {e₀₁ : Truncated.Edge x₀ x₁}\n    {e₁₂ : Truncated.Edge x₁ x₂} {e₀₂ : Truncated.Edge x₀ x₂}\n    (h : Truncated.Edge.CompStruct e₀₁ e₁₂ e₀₂) :\n    HoRel₂\n      ((Cat.FreeRefl.quotientFunctor (OneTruncation₂ V)).map\n        (Quiver.Hom.toPath e₀₁ ≫ Quiver.Hom.toPath e₁₂))\n      ((Cat.FreeRefl.quotientFunctor (OneTruncation₂ V)).map (Quiver.Hom.toPath e₀₂))\n\nend OneTruncation₂\n\nnamespace Truncated\n\nvariable (V W : SSet.Truncated.{u} 2)\n\n/-- The type underlying the homotopy category of a 2-truncated simplicial set `V`. -/\ndef HomotopyCategory : Type u :=\n  Quotient (OneTruncation₂.HoRel₂ V)\n  deriving Category.{u}\n\nnamespace HomotopyCategory\n\n/-- A canonical functor from the free category on the refl quiver underlying a 2-truncated\nsimplicial set `V` to its homotopy category. -/\ndef quotientFunctor :\n    Cat.FreeRefl (OneTruncation₂ V) ⥤ V.HomotopyCategory :=\n  Quotient.functor _\n\ninstance : (quotientFunctor V).Full :=\n  Quotient.full_functor _\n\nvariable {V}\n\n/-- Constructor for objects of the homotopy category of a `2`-truncated simplicial set. -/\ndef mk (x : V _⦋0⦌₂) : V.HomotopyCategory :=\n  (quotientFunctor V).obj (.mk x)\n\nlemma mk_surjective : Function.Surjective (mk (V := V)) := by\n  rintro ⟨⟨x⟩⟩\n  exact ⟨x, rfl⟩\n\nTarget:\nlemma ext {x y : V.HomotopyCategory} (h : x.as.as = y.as.as) : x = y :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"23efcc095954488a8537405ce81963fcfbb3152d5a7ed3e257f2eff7539a2f84","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplicialSet","family_id":"ext","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplicialSet/HomotopyCat.lean","sample_id":"3065ddea640d150a719f7ca546348a960da065f54e8ab9a9d7476b502ae8f9af"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d110cfbf729fa1ead7d2f77345ddd3124e402db97adfe146f40e24d032256864","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"099a68aecd4c932bd014347b7efcc8259e886e53c407723e803e298e292ed9b6","source_sha256":"e14d2dddf8bf8e732d384fcfc239ee89a73c0a738a4da1d1d8efcb0c46627981","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := RingHom.toMorphismProperty_respectsIso_iff.mp RingHom.isIntegral_respectsIso\n  rw [HasAffineProperty.iff_of_isAffine (P := @IsIntegralHom), and_iff_right]\n  exacts [MorphismProperty.arrow_mk_iso_iff (RingHom.toMorphismProperty RingHom.IsIntegral)\n    (arrowIsoΓSpecOfIsAffine φ).symm, inferInstance]","hard_negative":false,"metrics":{"chosen_tokens":49,"rejected_tokens":2,"token_jaccard":0.03125,"token_length_ratio":0.040816},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"b0ffaeadda617610b297d58de5e49a2ea9cee929a1d0e410bf34dfbe8c370857","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.Separated\npublic import Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed\npublic import Mathlib.RingTheory.RingHom.Integral\n\nNamespace:\nAlgebraicGeometry.IsIntegralHom\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Integral morphisms of schemes\n\nA morphism of schemes `f : X ⟶ Y` is integral if the preimage\nof an arbitrary affine open subset of `Y` is affine and the induced ring map is integral.\n\nIt is equivalent to ask only that `Y` is covered by affine opens whose preimage is affine\nand the induced ring map is integral.\n\n-/\n\npublic section\n\nuniverse v u\n\nopen CategoryTheory TopologicalSpace Opposite MorphismProperty\n\nnamespace AlgebraicGeometry\n\n/-- A morphism of schemes `X ⟶ Y` is integral if the preimage of any affine open subset of `Y` is\naffine and the induced ring hom on sections is integral. -/\n@[mk_iff]\nclass IsIntegralHom {X Y : Scheme} (f : X ⟶ Y) : Prop extends IsAffineHom f where\n  isIntegral_app (f) (U : Y.Opens) (hU : IsAffineOpen U) : (f.app U).hom.IsIntegral\n\nalias Scheme.Hom.isIntegral_app := IsIntegralHom.isIntegral_app\n\nnamespace IsIntegralHom\n\nvariable {X Y Z S : Scheme.{u}}\n\ninstance hasAffineProperty : HasAffineProperty @IsIntegralHom\n    fun X _ f _ ↦ IsAffine X ∧ RingHom.IsIntegral (f.app ⊤).hom := by\n  change HasAffineProperty @IsIntegralHom (affineAnd RingHom.IsIntegral)\n  rw [HasAffineProperty.affineAnd_iff _ RingHom.isIntegral_respectsIso\n    RingHom.isIntegral_isStableUnderBaseChange.localizationPreserves.away\n    RingHom.isIntegral_ofLocalizationSpan]\n  simp [isIntegralHom_iff]\n\ninstance : IsStableUnderComposition @IsIntegralHom :=\n  HasAffineProperty.affineAnd_isStableUnderComposition (Q := RingHom.IsIntegral) hasAffineProperty\n    RingHom.isIntegral_stableUnderComposition\n\ninstance : IsStableUnderBaseChange @IsIntegralHom :=\n  HasAffineProperty.affineAnd_isStableUnderBaseChange (Q := RingHom.IsIntegral) hasAffineProperty\n    RingHom.isIntegral_respectsIso RingHom.isIntegral_isStableUnderBaseChange\n\ninstance (priority := low) (f : X ⟶ Y) [IsClosedImmersion f] : IsIntegralHom f where\n  isIntegral_app U hU := (RingHom.Finite.of_surjective _ (f.app_surjective U hU)).to_isIntegral\n\ninstance : IsMultiplicative @IsIntegralHom where\n  id_mem _ := inferInstance\n\ninstance (f : X ⟶ Y) (g : Y ⟶ Z) [IsIntegralHom f] [IsIntegralHom g] : IsIntegralHom (f ≫ g) :=\n  MorphismProperty.comp_mem _ _ _ ‹_› ‹_›\n\ninstance (f : X ⟶ S) (g : Y ⟶ S) [IsIntegralHom g] : IsIntegralHom (Limits.pullback.fst f g) :=\n  MorphismProperty.pullback_fst f g inferInstance\n\ninstance (f : X ⟶ S) (g : Y ⟶ S) [IsIntegralHom f] : IsIntegralHom (Limits.pullback.snd f g) :=\n  MorphismProperty.pullback_snd f g inferInstance\n\ninstance (f : X ⟶ Y) (V : Y.Opens) [IsIntegralHom f] : IsIntegralHom (f ∣_ V) :=\n  IsZariskiLocalAtTarget.restrict ‹_› V\n\ninstance : MorphismProperty.HasOfPostcompProperty @IsIntegralHom @IsSeparated :=\n  MorphismProperty.hasOfPostcompProperty_iff_le_diagonal.mpr\n    fun _ _ _ _ ↦ inferInstanceAs (IsIntegralHom _)\n\nlemma of_comp (f : X ⟶ Y) (g : Y ⟶ Z) [IsIntegralHom (f ≫ g)] [IsSeparated g] :\n    IsIntegralHom f := MorphismProperty.of_postcomp _ _ g ‹_› ‹_›\n\nlemma comp_iff {f : X ⟶ Y} {g : Y ⟶ Z} [IsIntegralHom g] :\n    IsIntegralHom (f ≫ g) ↔ IsIntegralHom f :=\n  ⟨fun _ ↦ .of_comp f g, fun _ ↦ inferInstance⟩\n\nTarget:\nlemma SpecMap_iff {R S : CommRingCat} {φ : R ⟶ S} :\n    IsIntegralHom (Spec.map φ) ↔ φ.hom.IsIntegral :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Morphisms","family_id":"specmap_iff","file_id":"mathlib/Mathlib/AlgebraicGeometry/Morphisms/Integral.lean","sample_id":"099a68aecd4c932bd014347b7efcc8259e886e53c407723e803e298e292ed9b6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0f3c5c923c48eece1af3a7c60f1fa94f43ad2e6d58c10a4a8ef6046b414d8933","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d0df3140da7fa364792a7ce4f3fa0fb068f28c3068cdfb76bb78b78e0d7a5c60","source_sha256":"d5350ee23560ecfe7472ddc39d464fb66256d2af6cef5b34446a54a647e9b358","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain rfl | hs := s.eq_empty_or_nonempty\n  · rw [smul_set_empty, Real.sSup_empty, Real.sInf_empty, smul_zero]\n  obtain rfl | ha' := ha.eq_or_lt\n  · rw [zero_smul_set hs, zero_smul]\n    exact csSup_singleton 0\n  by_cases h : BddBelow s\n  · exact ((OrderIso.smulRightDual ℝ ha').map_csInf' hs h).symm\n  · rw [Real.sSup_of_not_bddAbove (mt (bddAbove_smul_iff_of_neg ha').1 h),\n        Real.sInf_of_not_bddBelow h, smul_zero]","hard_negative":false,"metrics":{"chosen_tokens":89,"rejected_tokens":3,"token_jaccard":0.045455,"token_length_ratio":0.033708},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"b12757db08e9c7fb991df5ce809ac076468d3b8e98f206d0acbb21bf3fa3a3a9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set\npublic import Mathlib.Algebra.Order.Archimedean.Real.Basic\npublic import Mathlib.Algebra.Order.Module.Pointwise\npublic import Mathlib.Order.ConditionallyCompleteLattice.Indexed\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Eric Wieser\n-/\n/-!\n# Pointwise operations on sets of reals\n\nThis file relates `sInf (a • s)`/`sSup (a • s)` with `a • sInf s`/`a • sSup s` for `s : Set ℝ`.\n\nFrom these, it relates `⨅ i, a • f i` / `⨆ i, a • f i` with `a • (⨅ i, f i)` / `a • (⨆ i, f i)`,\nand provides lemmas about distributing `*` over `⨅` and `⨆`.\n\n## TODO\n\nThis is true more generally for conditionally complete linear order whose default value is `0`. We\ndon't have those yet.\n-/\n\npublic section\n\nassert_not_exists Finset\n\nopen Set\n\nopen scoped Pointwise\n\nvariable {ι : Sort*} {α : Type*} [Field α] [LinearOrder α] [IsStrictOrderedRing α]\n\nsection MulActionWithZero\n\nvariable [MulActionWithZero α ℝ] [IsOrderedModule α ℝ] {a : α}\n\ntheorem Real.sInf_smul_of_nonneg (ha : 0 ≤ a) (s : Set ℝ) : sInf (a • s) = a • sInf s := by\n  obtain rfl | hs := s.eq_empty_or_nonempty\n  · rw [smul_set_empty, Real.sInf_empty, smul_zero]\n  obtain rfl | ha' := ha.eq_or_lt\n  · rw [zero_smul_set hs, zero_smul]\n    exact csInf_singleton 0\n  by_cases h : BddBelow s\n  · exact ((OrderIso.smulRight ha').map_csInf' hs h).symm\n  · rw [Real.sInf_of_not_bddBelow (mt (bddBelow_smul_iff_of_pos ha').1 h),\n        Real.sInf_of_not_bddBelow h, smul_zero]\n\ntheorem Real.smul_iInf_of_nonneg (ha : 0 ≤ a) (f : ι → ℝ) : (a • ⨅ i, f i) = ⨅ i, a • f i :=\n  (Real.sInf_smul_of_nonneg ha _).symm.trans <| congr_arg sInf <| (range_comp _ _).symm\n\ntheorem Real.sSup_smul_of_nonneg (ha : 0 ≤ a) (s : Set ℝ) : sSup (a • s) = a • sSup s := by\n  obtain rfl | hs := s.eq_empty_or_nonempty\n  · rw [smul_set_empty, Real.sSup_empty, smul_zero]\n  obtain rfl | ha' := ha.eq_or_lt\n  · rw [zero_smul_set hs, zero_smul]\n    exact csSup_singleton 0\n  by_cases h : BddAbove s\n  · exact ((OrderIso.smulRight ha').map_csSup' hs h).symm\n  · rw [Real.sSup_of_not_bddAbove (mt (bddAbove_smul_iff_of_pos ha').1 h),\n        Real.sSup_of_not_bddAbove h, smul_zero]\n\ntheorem Real.smul_iSup_of_nonneg (ha : 0 ≤ a) (f : ι → ℝ) : (a • ⨆ i, f i) = ⨆ i, a • f i :=\n  (Real.sSup_smul_of_nonneg ha _).symm.trans <| congr_arg sSup <| (range_comp _ _).symm\n\nend MulActionWithZero\n\nsection Module\n\nvariable [Module α ℝ] [IsOrderedModule α ℝ] {a : α}\n\ntheorem Real.sInf_smul_of_nonpos (ha : a ≤ 0) (s : Set ℝ) : sInf (a • s) = a • sSup s := by\n  obtain rfl | hs := s.eq_empty_or_nonempty\n  · rw [smul_set_empty, Real.sInf_empty, Real.sSup_empty, smul_zero]\n  obtain rfl | ha' := ha.eq_or_lt\n  · rw [zero_smul_set hs, zero_smul]\n    exact csInf_singleton 0\n  by_cases h : BddAbove s\n  · exact ((OrderIso.smulRightDual ℝ ha').map_csSup' hs h).symm\n  · rw [Real.sInf_of_not_bddBelow (mt (bddBelow_smul_iff_of_neg ha').1 h),\n        Real.sSup_of_not_bddAbove h, smul_zero]\n\ntheorem Real.smul_iSup_of_nonpos (ha : a ≤ 0) (f : ι → ℝ) : (a • ⨆ i, f i) = ⨅ i, a • f i :=\n  (Real.sInf_smul_of_nonpos ha _).symm.trans <| congr_arg sInf <| (range_comp _ _).symm\n\nTarget:\ntheorem Real.sSup_smul_of_nonpos (ha : a ≤ 0) (s : Set ℝ) : sSup (a • s) = a • sInf s :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Real","family_id":"real","file_id":"mathlib/Mathlib/Data/Real/Pointwise.lean","sample_id":"d0df3140da7fa364792a7ce4f3fa0fb068f28c3068cdfb76bb78b78e0d7a5c60"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9c89a7a96cac539d3af0940d447c59a7152af477d925ba54bdc0d7f1377d3647","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7bc5d50d79474e2eb00b8d1589e6b3098d673cf91d3b5c0556d07b768e75e3bc","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"70071016d6f956ee807fdebd39b7e38611eccc815e374ddf4ad147924380a172","source_sha256":"ebde9039546abaab19fae84dcc9dee6e17f2d20f60817bbfa41ae3bdde9f8a83","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext; simp [mem_coe, mem_extended_iff]","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":2,"token_jaccard":0.1,"token_length_ratio":0.222222},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"b1c31560e411c4c18d0b7e0ad40f4b33df7fdb657a43b98ab2be4d26ea30ba9f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas\n\nNamespace:\nFractionalIdeal\n\nLocal context:\n/-\nCopyright (c) 2024 James Sundstrom. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: James Sundstrom, Xavier Roblot\n-/\n/-!\n# Extension of fractional ideals\n\nThis file defines the extension of a fractional ideal along a ring homomorphism.\n\n## Main definitions\n\n* `FractionalIdeal.extended`: Let `A` and `B` be commutative rings with respective localizations\n  `IsLocalization M K` and `IsLocalization N L`. Let `f : A →+* B` be a ring homomorphism with\n  `hf : M ≤ Submonoid.comap f N`. If `I : FractionalIdeal M K`, then the extension of `I` along\n  `f` is `extended L hf I : FractionalIdeal N L`.\n* `FractionalIdeal.extendedHom'`: The ring homomorphism version of `FractionalIdeal.extended`.\n* `FractionalIdeal.extendedHom`: For `A ⊆ B` an extension of domains, the ring homomorphism that\n  sends a fractional ideal of `A` to a fractional ideal of `B`.\n\n## Main results\n\n* `FractionalIdeal.extendedHom_injective`: the map `FractionalIdeal.extendedHom` is injective.\n* `FractionalIdeal.extended_extended`: extending fractional ideals is compatible with composition\n  of ring homomorphisms.\n* `FractionalIdeal.extendedHom'_comp`: the homomorphisms induced by extension of fractional\n  ideals compose in towers.\n* `Ideal.map_algebraMap_injective`: For `A ⊆ B` an extension of Dedekind domains, the map that\n  sends an ideal `I` of `A` to `I·B` is injective.\n\n## Tags\n\nfractional ideal, fractional ideals, extended, extension\n-/\n\n@[expose] public section\n\nopen IsLocalization FractionalIdeal Module Submodule\n\nnamespace FractionalIdeal\n\nsection RingHom\n\nvariable {A : Type*} [CommRing A] {B : Type*} [CommRing B] {f : A →+* B}\nvariable {K : Type*} {M : Submonoid A} [CommRing K] [Algebra A K] [IsLocalization M K]\nvariable (L : Type*) {N : Submonoid B} [CommRing L] [Algebra B L] [IsLocalization N L]\nvariable (hf : M ≤ Submonoid.comap f N)\nvariable (I : FractionalIdeal M K) (J : FractionalIdeal M K)\n\n/-- Given commutative rings `A` and `B` with respective localizations `IsLocalization M K` and\n`IsLocalization N L`, and a ring homomorphism `f : A →+* B` satisfying `M ≤ Submonoid.comap f N`, a\nfractional ideal `I` of `A` can be extended along `f` to a fractional ideal of `B`. -/\ndef extended (I : FractionalIdeal M K) : FractionalIdeal N L where\n  val := span B <| (IsLocalization.map (S := K) L f hf) '' I\n  property := by\n    have ⟨a, ha, frac⟩ := I.isFractional\n    refine ⟨f a, hf ha, fun b hb ↦ ?_⟩\n    refine span_induction (fun x hx ↦ ?_) ⟨0, by simp⟩\n      (fun x y _ _ hx hy ↦ smul_add (f a) x y ▸ isInteger_add hx hy) (fun b c _ hc ↦ ?_) hb\n    · rcases hx with ⟨k, kI, rfl⟩\n      obtain ⟨c, hc⟩ := frac k kI\n      exact ⟨f c, by simp [← IsLocalization.map_smul, ← hc]⟩\n    · rw [← smul_assoc, smul_eq_mul, mul_comm (f a), ← smul_eq_mul, smul_assoc]\n      exact isInteger_smul hc\n\nlocal notation \"map_f\" => (IsLocalization.map (S := K) L f hf)\n\nlemma mem_extended_iff (x : L) : x ∈ I.extended L hf ↔ x ∈ span B (map_f '' I) := by\n  constructor <;> { intro hx; simpa }\n\n@[simp]\n\nTarget:\nlemma coe_extended_eq_span : I.extended L hf = span B (map_f '' I) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_70071016d6f9","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"94a470b8317c21461ef0556c5f357f4049d9a43f6115a46158f4519a68161ff3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/FractionalIdeal","family_id":"coe_extended_eq_span","file_id":"mathlib/Mathlib/RingTheory/FractionalIdeal/Extended.lean","sample_id":"70071016d6f956ee807fdebd39b7e38611eccc815e374ddf4ad147924380a172"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2e876053dd524313a985f03953e4b880c7de87fe925bf70214a79832334fbbdd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2478a8e6254a32b82717f4f3721b328383f977ee01f8637439d9aea70831141b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3b9738526fb66d231f0474bd9878a45063a3e22be465efe04b4fbbe9d62f823a","source_sha256":"3c917efc6600c0893eb5c055755a108b9a20ae48ebe963f7f5db8b06582c587d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨F, hF⟩ := hc.exists_eq_range (hn.image _)\n  simp only [Set.ext_iff, Quotient.forall, mem_image, mem_range] at hF\n  simp_rw [Quotient.eq_mk_iff_out] at hF\n  have hF' : ∀ n : ℕ, (F n).out ∈ K := by\n    intro n\n    obtain ⟨P, hP1, hP2⟩ := (hF (F n).out).2 ⟨n, Setoid.refl _⟩\n    -- Porting note: fix hP2 because `Quotient.out (Quotient.mk' x) ≈ a` was not simplified\n    -- to `x ≈ a` in hF\n    replace hP2 := Setoid.trans (Setoid.symm (Quotient.mk_out P)) hP2\n    exact (hp.is_equiv_invariant_of_fg fg _ _ hP2).1 hP1\n  choose P hPK hP hFP using fun (N : K) (n : ℕ) => jep N N.2 (F (n + 1)).out (hF' _)\n  let G : ℕ → K := @Nat.rec (fun _ => K) ⟨(F 0).out, hF' 0⟩ fun n N => ⟨P N n, hPK N n⟩\n  let f : ∀ (i j : ℕ), i ≤ j → (G i).val ↪[L] (G j).val :=\n    DirectedSystem.natLERec fun n => (hP _ n).some\n  refine ⟨Bundled.of (@DirectLimit L _ _ (fun n ↦ (G n).val) _ f _ _), ?_, ?_⟩\n  · exact DirectLimit.cg _ (fun n => (fg _ (G n).2).cg)\n  · refine (age_directLimit (fun n ↦ (G n).val) f).trans\n      (subset_antisymm (iUnion_subset fun n N hN => hp (G n).val (G n).2 hN) fun N KN => ?_)\n    have : Quotient.out (Quotient.mk' N) ≈ N := Quotient.eq_mk_iff_out.mp rfl\n    obtain ⟨n, ⟨e⟩⟩ := (hF N).1 ⟨N, KN, this⟩\n    refine mem_iUnion_of_mem n ⟨fg _ KN, ⟨Embedding.comp ?_ e.symm.toEmbedding⟩⟩\n    rcases n with - | n\n    · dsimp [G]; exact Embedding.refl _ _\n    · dsimp [G]; exact (hFP _ n).some","hard_negative":false,"metrics":{"chosen_tokens":480,"rejected_tokens":487,"token_jaccard":0.99187,"token_length_ratio":1.014583},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"b2167ae7201e8d27f946fc899bcbe77c5ad5fe1f5fb62477c3012366789c7791","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.ModelTheory.FinitelyGenerated\npublic import Mathlib.ModelTheory.PartialEquiv\npublic import Mathlib.ModelTheory.Bundled\npublic import Mathlib.Algebra.Order.Archimedean.Basic\n\nNamespace:\nFirstOrder.Language\n\nLocal context:\n/-\nCopyright (c) 2022 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson, Gabin Kolly\n-/\n/-!\n# Fraïssé Classes and Fraïssé Limits\n\nThis file pertains to the ages of countable first-order structures. The age of a structure is the\nclass of all finitely-generated structures that embed into it.\n\nOf particular interest are Fraïssé classes, which are exactly the ages of countable\nultrahomogeneous structures. To each is associated a unique (up to nonunique isomorphism)\nFraïssé limit - the countable ultrahomogeneous structure with that age.\n\n## Main Definitions\n\n- `FirstOrder.Language.age` is the class of finitely-generated structures that embed into a\n  particular structure.\n- A class `K` is `FirstOrder.Language.Hereditary` when all finitely-generated\n  structures that embed into structures in `K` are also in `K`.\n- A class `K` has `FirstOrder.Language.JointEmbedding` when for every `M`, `N` in\n  `K`, there is another structure in `K` into which both `M` and `N` embed.\n- A class `K` has `FirstOrder.Language.Amalgamation` when for any pair of embeddings\n  of a structure `M` in `K` into other structures in `K`, those two structures can be embedded into\n  a fourth structure in `K` such that the resulting square of embeddings commutes.\n- `FirstOrder.Language.IsFraisse` indicates that a class is nonempty, essentially countable,\n  and satisfies the hereditary, joint embedding, and amalgamation properties.\n- `FirstOrder.Language.IsFraisseLimit` indicates that a structure is a Fraïssé limit for a given\n  class.\n\n## Main Results\n\n- We show that the age of any structure is isomorphism-invariant and satisfies the hereditary and\n  joint-embedding properties.\n- `FirstOrder.Language.age.countable_quotient` shows that the age of any countable structure is\n  essentially countable.\n- `FirstOrder.Language.exists_countable_is_age_of_iff` gives necessary and sufficient conditions\n  for a class to be the age of a countable structure in a language with countably many functions.\n- `FirstOrder.Language.IsFraisseLimit.nonempty_equiv` shows that any class which is Fraïssé has\n  at most one Fraïssé limit up to equivalence.\n- `FirstOrder.Language.empty.isFraisseLimit_of_countable_infinite` shows that any countably infinite\n  structure in the empty language is a Fraïssé limit of the class of finite structures.\n- `FirstOrder.Language.empty.isFraisse_finite` shows that the class of finite structures in the\n  empty language is Fraïssé.\n\n## Implementation Notes\n\n- Classes of structures are formalized with `Set (Bundled L.Structure)`.\n- Some results pertain to countable limit structures, others to countably-generated limit\n  structures. In the case of a language with countably many function symbols, these are equivalent.\n\n## References\n\n- [W. Hodges, *A Shorter Model Theory*][Hodges97]\n- [K. Tent, M. Ziegler, *A Course in Model Theory*][Tent_Ziegler]\n\n## TODO\n\n- Show existence of Fraïssé limits\n\n-/\n\n@[expose] public section\n\n\nuniverse u v w w'\n\nopen scoped FirstOrder\n\nopen Set CategoryTheory\n\nnamespace FirstOrder\n\nnamespace Language\n\nopen Structure Substructure\n\nvariable (L : Language.{u, v})\n\n/-! ### The Age of a Structure and Fraïssé Classes -/\n\n\n/-- The age of a structure `M` is the class of finitely-generated structures that embed into it. -/\ndef age (M : Type w) [L.Structure M] : Set (Bundled.{w} L.Structure) :=\n  {N | Structure.FG L N ∧ Nonempty (N ↪[L] M)}\n\nvariable {L}\nvariable (K : Set (Bundled.{w} L.Structure))\n\n/-- A class `K` has the hereditary property when all finitely-generated structures that embed into\n  structures in `K` are also in `K`. -/\ndef Hereditary : Prop :=\n  ∀ M : Bundled.{w} L.Structure, M ∈ K → L.age M ⊆ K\n\n/-- A class `K` has the joint embedding property when for every `M`, `N` in `K`, there is another\n  structure in `K` into which both `M` and `N` embed. -/\ndef JointEmbedding : Prop :=\n  DirectedOn (fun M N : Bundled.{w} L.Structure => Nonempty (M ↪[L] N)) K\n\n/-- A class `K` has the amalgamation property when for any pair of embeddings of a structure `M` in\n  `K` into other structures in `K`, those two structures can be embedded into a fourth structure in\n  `K` such that the resulting square of embeddings commutes. -/\ndef Amalgamation : Prop :=\n  ∀ (M N P : Bundled.{w} L.Structure) (MN : M ↪[L] N) (MP : M ↪[L] P),\n    M ∈ K → N ∈ K → P ∈ K → ∃ (Q : Bundled.{w} L.Structure) (NQ : N ↪[L] Q) (PQ : P ↪[L] Q),\n      Q ∈ K ∧ NQ.comp MN = PQ.comp MP\n\n/-- A Fraïssé class is a nonempty, essentially countable class of structures satisfying the\nhereditary, joint embedding, and amalgamation properties. -/\nclass IsFraisse : Prop where\n  is_nonempty : K.Nonempty\n  FG : ∀ M : Bundled.{w} L.Structure, M ∈ K → Structure.FG L M\n  is_essentially_countable : (Quotient.mk' '' K).Countable\n  hereditary : Hereditary K\n  jointEmbedding : JointEmbedding K\n  amalgamation : Amalgamation K\n\nvariable {K} (L) (M : Type w) [Structure L M]\n\ntheorem age.is_equiv_invariant (N P : Bundled.{w} L.Structure) (h : Nonempty (N ≃[L] P)) :\n    N ∈ L.age M ↔ P ∈ L.age M :=\n  and_congr h.some.fg_iff\n    ⟨Nonempty.map fun x => Embedding.comp x h.some.symm.toEmbedding,\n      Nonempty.map fun x => Embedding.comp x h.some.toEmbedding⟩\n\nvariable {L} {M} {N : Type w} [Structure L N]\n\ntheorem Embedding.age_subset_age (MN : M ↪[L] N) : L.age M ⊆ L.age N := fun _ =>\n  And.imp_right (Nonempty.map MN.comp)\n\ntheorem Equiv.age_eq_age (MN : M ≃[L] N) : L.age M = L.age N :=\n  le_antisymm MN.toEmbedding.age_subset_age MN.symm.toEmbedding.age_subset_age\n\ntheorem Structure.FG.mem_age_of_equiv {M N : Bundled L.Structure} (h : Structure.FG L M)\n    (MN : Nonempty (M ≃[L] N)) : N ∈ L.age M :=\n  ⟨MN.some.fg_iff.1 h, ⟨MN.some.symm.toEmbedding⟩⟩\n\ntheorem Hereditary.is_equiv_invariant_of_fg (h : Hereditary K)\n    (fg : ∀ M : Bundled.{w} L.Structure, M ∈ K → Structure.FG L M) (M N : Bundled.{w} L.Structure)\n    (hn : Nonempty (M ≃[L] N)) : M ∈ K ↔ N ∈ K :=\n  ⟨fun MK => h M MK ((fg M MK).mem_age_of_equiv hn),\n   fun NK => h N NK ((fg N NK).mem_age_of_equiv ⟨hn.some.symm⟩)⟩\n\ntheorem IsFraisse.is_equiv_invariant [h : IsFraisse K] {M N : Bundled.{w} L.Structure}\n    (hn : Nonempty (M ≃[L] N)) : M ∈ K ↔ N ∈ K :=\n  h.hereditary.is_equiv_invariant_of_fg h.FG M N hn\n\nvariable (M)\n\ntheorem age.nonempty : (L.age M).Nonempty :=\n  ⟨Bundled.of (Substructure.closure L (∅ : Set M)),\n    (fg_iff_structure_fg _).1 (fg_closure Set.finite_empty), ⟨Substructure.subtype _⟩⟩\n\ntheorem age.hereditary : Hereditary (L.age M) := fun _ hN _ hP => hN.2.some.age_subset_age hP\n\ntheorem age.jointEmbedding : JointEmbedding (L.age M) := fun _ hN _ hP =>\n  ⟨Bundled.of (↥(hN.2.some.toHom.range ⊔ hP.2.some.toHom.range)),\n    ⟨(fg_iff_structure_fg _).1 ((hN.1.range hN.2.some.toHom).sup (hP.1.range hP.2.some.toHom)),\n      ⟨Substructure.subtype _⟩⟩,\n    ⟨Embedding.comp (inclusion le_sup_left) hN.2.some.equivRange.toEmbedding⟩,\n    ⟨Embedding.comp (inclusion le_sup_right) hP.2.some.equivRange.toEmbedding⟩⟩\n\nvariable {M} in\ntheorem age.fg_substructure {S : L.Substructure M} (fg : S.FG) : Bundled.mk S ∈ L.age M := by\n  exact ⟨(Substructure.fg_iff_structure_fg _).1 fg, ⟨subtype _⟩⟩\n\n/-- Any class in the age of a structure has a representative which is a finitely generated\nsubstructure. -/\ntheorem age.has_representative_as_substructure :\n    ∀ C ∈ Quotient.mk' '' L.age M, ∃ V : {V : L.Substructure M // FG V},\n      ⟦Bundled.mk V⟧ = C := by\n  rintro _ ⟨N, ⟨N_fg, ⟨N_incl⟩⟩, N_eq⟩\n  refine N_eq.symm ▸ ⟨⟨N_incl.toHom.range, ?_⟩, Quotient.sound ⟨N_incl.equivRange.symm⟩⟩\n  exact FG.range N_fg (Embedding.toHom N_incl)\n\n/-- The age of a countable structure is essentially countable (has countably many isomorphism\nclasses). -/\ntheorem age.countable_quotient [h : Countable M] : (Quotient.mk' '' L.age M).Countable := by\n  classical\n  refine (congr_arg _ (Set.ext <| Quotient.forall.2 fun N => ?_)).mp\n    (countable_range fun s : Finset M => ⟦⟨closure L (s : Set M), inferInstance⟩⟧)\n  constructor\n  · rintro ⟨s, hs⟩\n    use Bundled.of (closure L (s : Set M))\n    exact ⟨⟨(fg_iff_structure_fg _).1 (fg_closure s.finite_toSet), ⟨Substructure.subtype _⟩⟩, hs⟩\n  · simp only [mem_range, Quotient.eq]\n    rintro ⟨P, ⟨⟨s, hs⟩, ⟨PM⟩⟩, hP2⟩\n    refine ⟨s.image PM, Setoid.trans (b := P) ?_ <| Quotient.exact hP2⟩\n    rw [← Embedding.coe_toHom, Finset.coe_image, closure_image PM.toHom, hs, ← Hom.range_eq_map]\n    exact ⟨PM.equivRange.symm⟩\n\nset_option backward.isDefEq.respectTransparency false in\n-- This is not a simp-lemma because it does not apply to itself.\n/-- The age of a direct limit of structures is the union of the ages of the structures. -/\ntheorem age_directLimit {ι : Type w} [Preorder ι] [IsDirectedOrder ι] [Nonempty ι]\n    (G : ι → Type max w w') [∀ i, L.Structure (G i)] (f : ∀ i j, i ≤ j → G i ↪[L] G j)\n    [DirectedSystem G fun i j h => f i j h] : L.age (DirectLimit G f) = ⋃ i : ι, L.age (G i) := by\n  classical\n  ext M\n  simp only [mem_iUnion]\n  constructor\n  · rintro ⟨Mfg, ⟨e⟩⟩\n    obtain ⟨s, hs⟩ := Mfg.range e.toHom\n    let out := @Quotient.out _ (DirectLimit.setoid G f)\n    obtain ⟨i, hi⟩ := Finset.exists_le (s.image (Sigma.fst ∘ out))\n    have e' := (DirectLimit.of L ι G f i).equivRange.symm.toEmbedding\n    refine ⟨i, Mfg, ⟨e'.comp ((Substructure.inclusion ?_).comp e.equivRange.toEmbedding)⟩⟩\n    rw [← hs, closure_le]\n    intro x hx\n    refine ⟨f (out x).1 i (hi (out x).1 (Finset.mem_image_of_mem _ hx)) (out x).2, ?_⟩\n    rw [Embedding.coe_toHom, DirectLimit.of_apply, @Quotient.mk_eq_iff_out _ (_),\n      DirectLimit.equiv_iff G f (le_refl _) (hi (out x).1 (Finset.mem_image_of_mem _ hx)),\n      DirectedSystem.map_self]\n  · rintro ⟨i, Mfg, ⟨e⟩⟩\n    exact ⟨Mfg, ⟨Embedding.comp (DirectLimit.of L ι G f i) e⟩⟩\n\n/-- Sufficient conditions for a class to be the age of a countably-generated structure. -/\n\nTarget:\ntheorem exists_cg_is_age_of (hn : K.Nonempty)\n    (hc : (Quotient.mk' '' K).Countable)\n    (fg : ∀ M : Bundled.{w} L.Structure, M ∈ K → Structure.FG L M) (hp : Hereditary K)\n    (jep : JointEmbedding K) : ∃ M : Bundled.{w} L.Structure, Structure.CG L M ∧ L.age M = K :=\n\nProof body:\n","rejected":"```lean\nby\n  obtain ⟨F, hF⟩ := hc.exists_eq_range (hn.image _)\n  simp only [Set.ext_iff, Quotient.forall, mem_image, mem_range] at hF\n  simp_rw [Quotient.eq_mk_iff_out] at hF\n  have hF' : ∀ n : ℕ, (F n).out ∈ K := by\n    intro n\n    obtain ⟨P, hP1, hP2⟩ := (hF (F n).out).2 ⟨n, Setoid.refl _⟩\n    -- Porting note: fix hP2 because `Quotient.out (Quotient.mk' x) ≈ a` was not simplified\n    -- to `x ≈ a` in hF\n    replace hP2 := Setoid.trans (Setoid.symm (Quotient.mk_out P)) hP2\n    exact (hp.is_equiv_invariant_of_fg fg _ _ hP2).1 hP1\n  choose P hPK hP hFP using fun (N : K) (n : ℕ) => jep N N.2 (F (n + 1)).out (hF' _)\n  let G : ℕ → K := @Nat.rec (fun _ => K) ⟨(F 0).out, hF' 0⟩ fun n N => ⟨P N n, hPK N n⟩\n  let f : ∀ (i j : ℕ), i ≤ j → (G i).val ↪[L] (G j).val :=\n    DirectedSystem.natLERec fun n => (hP _ n).some\n  refine ⟨Bundled.of (@DirectLimit L _ _ (fun n ↦ (G n).val) _ f _ _), ?_, ?_⟩\n  · exact DirectLimit.cg _ (fun n => (fg _ (G n).2).cg)\n  · refine (age_directLimit (fun n ↦ (G n).val) f).trans\n      (subset_antisymm (iUnion_subset fun n N hN => hp (G n).val (G n).2 hN) fun N KN => ?_)\n    have : Quotient.out (Quotient.mk' N) ≈ N := Quotient.eq_mk_iff_out.mp rfl\n    obtain ⟨n, ⟨e⟩⟩ := (hF N).1 ⟨N, KN, this⟩\n    refine mem_iUnion_of_mem n ⟨fg _ KN, ⟨Embedding.comp ?_ e.symm.toEmbedding⟩⟩\n    rcases n with - | n\n    · dsimp [G]; exact Embedding.refl _ _\n    · dsimp [G]; exact (hFP _ n).some\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"ModelTheory","family_id":"exists_cg_is_age_of","file_id":"mathlib/Mathlib/ModelTheory/Fraisse.lean","sample_id":"3b9738526fb66d231f0474bd9878a45063a3e22be465efe04b4fbbe9d62f823a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b1c7cebb6805403539cf95bc62a5c22e029b7da315a1e90239408f66439637c6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d7f033ea05a77516a16a1eb1a4d807d0521fcc849b70bb12f74c3c1a0dde6968","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f97d07cceec981c168c5f2eca445cd4e95e1981b7df2ab5f563b675d89d0e28f","source_sha256":"bd4b2b01d950a3b9c8fd78b82f11b816abe9ec16bf6d55a504ac2e051e9cf9fe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [superpolynomialDecay_iff_abs_tendsto_zero] at hf ⊢\n  refine fun z =>\n    tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds (hf z)\n      (Eventually.of_forall fun x => abs_nonneg _) (hfg.mono fun x hx => ?_)\n  calc\n    |k x ^ z * g x| = |k x ^ z| * |g x| := abs_mul (k x ^ z) (g x)\n    _ ≤ |k x ^ z| * |f x| := by gcongr _ * ?_; exact hx\n    _ = |k x ^ z * f x| := (abs_mul (k x ^ z) (f x)).symm","hard_negative":true,"metrics":{"chosen_tokens":123,"rejected_tokens":5,"token_jaccard":0.069767,"token_length_ratio":0.04065},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"b23e569f379f81a586c80741403aca03ad3607a6b3ce607fa27fabc277a3aed4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Eval.Defs\npublic import Mathlib.Analysis.Asymptotics.Lemmas\n\nNamespace:\nAsymptotics\n\nLocal context:\n/-\nCopyright (c) 2021 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\n/-!\n# Super-Polynomial Function Decay\n\nThis file defines a predicate `Asymptotics.SuperpolynomialDecay f` for a function satisfying\none of the following equivalent definitions (the definition is in terms of the first condition):\n\n* `x ^ n * f` tends to `𝓝 0` for all (or sufficiently large) naturals `n`\n* `|x ^ n * f|` tends to `𝓝 0` for all naturals `n` (`superpolynomialDecay_iff_abs_tendsto_zero`)\n* `|x ^ n * f|` is bounded for all naturals `n` (`superpolynomialDecay_iff_abs_isBoundedUnder`)\n* `f` is `o(x ^ c)` for all integers `c` (`superpolynomialDecay_iff_isLittleO`)\n* `f` is `O(x ^ c)` for all integers `c` (`superpolynomialDecay_iff_isBigO`)\n\nThese conditions are all equivalent to conditions in terms of polynomials, replacing `x ^ c` with\n  `p(x)` or `p(x)⁻¹` as appropriate, since asymptotically `p(x)` behaves like `X ^ p.natDegree`.\nThese further equivalences are not proven in mathlib but would be good future projects.\n\nThe definition of superpolynomial decay for `f : α → β` is relative to a parameter `k : α → β`.\nSuper-polynomial decay then means `f x` decays faster than `(k x) ^ c` for all integers `c`.\nEquivalently `f x` decays faster than `p.eval (k x)` for all polynomials `p : β[X]`.\nThe definition is also relative to a filter `l : Filter α` where the decay rate is compared.\n\nWhen the map `k` is given by `n ↦ ↑n : ℕ → ℝ` this defines negligible functions:\nhttps://en.wikipedia.org/wiki/Negligible_function\n\nWhen the map `k` is given by `(r₁,...,rₙ) ↦ r₁*...*rₙ : ℝⁿ → ℝ` this is equivalent\n  to the definition of rapidly decreasing functions given here:\nhttps://ncatlab.org/nlab/show/rapidly+decreasing+function\n\n## Main statements\n\n* `SuperpolynomialDecay.polynomial_mul` says that if `f(x)` is negligible,\n    then so is `p(x) * f(x)` for any polynomial `p`.\n* `superpolynomialDecay_iff_zpow_tendsto_zero` gives an equivalence between definitions in terms\n    of decaying faster than `k(x) ^ n` for all naturals `n` or `k(x) ^ c` for all integer `c`.\n-/\n\n@[expose] public section\n\n\nnamespace Asymptotics\n\nopen Topology Polynomial\n\nopen Filter\n\n/-- `f` has superpolynomial decay in parameter `k` along filter `l` if\n  `k ^ n * f` tends to zero at `l` for all naturals `n` -/\ndef SuperpolynomialDecay {α β : Type*} [TopologicalSpace β] [CommSemiring β] (l : Filter α)\n    (k : α → β) (f : α → β) :=\n  ∀ n : ℕ, Tendsto (fun a : α => k a ^ n * f a) l (𝓝 0)\n\nvariable {α β : Type*} {l : Filter α} {k : α → β} {f g g' : α → β}\n\nsection CommSemiring\n\nvariable [TopologicalSpace β] [CommSemiring β]\n\ntheorem SuperpolynomialDecay.congr' (hf : SuperpolynomialDecay l k f) (hfg : f =ᶠ[l] g) :\n    SuperpolynomialDecay l k g := fun z =>\n  (hf z).congr' (EventuallyEq.mul (EventuallyEq.refl l _) hfg)\n\ntheorem SuperpolynomialDecay.congr (hf : SuperpolynomialDecay l k f) (hfg : ∀ x, f x = g x) :\n    SuperpolynomialDecay l k g := fun z =>\n  (hf z).congr fun x => (congr_arg fun a => k x ^ z * a) <| hfg x\n\n@[simp]\ntheorem superpolynomialDecay_zero (l : Filter α) (k : α → β) : SuperpolynomialDecay l k 0 :=\n  fun z => by simpa only [Pi.zero_apply, mul_zero] using tendsto_const_nhds\n\ntheorem SuperpolynomialDecay.add [ContinuousAdd β] (hf : SuperpolynomialDecay l k f)\n    (hg : SuperpolynomialDecay l k g) : SuperpolynomialDecay l k (f + g) := fun z => by\n  simpa only [mul_add, add_zero, Pi.add_apply] using (hf z).add (hg z)\n\ntheorem SuperpolynomialDecay.mul [ContinuousMul β] (hf : SuperpolynomialDecay l k f)\n    (hg : SuperpolynomialDecay l k g) : SuperpolynomialDecay l k (f * g) := fun z => by\n  simpa only [mul_assoc, one_mul, mul_zero, pow_zero] using! (hf z).mul (hg 0)\n\ntheorem SuperpolynomialDecay.mul_const [ContinuousMul β] (hf : SuperpolynomialDecay l k f) (c : β) :\n    SuperpolynomialDecay l k fun n => f n * c := fun z => by\n  simpa only [← mul_assoc, zero_mul] using Tendsto.mul_const c (hf z)\n\ntheorem SuperpolynomialDecay.const_mul [ContinuousMul β] (hf : SuperpolynomialDecay l k f) (c : β) :\n    SuperpolynomialDecay l k fun n => c * f n :=\n  (hf.mul_const c).congr fun _ => mul_comm _ _\n\ntheorem SuperpolynomialDecay.param_mul (hf : SuperpolynomialDecay l k f) :\n    SuperpolynomialDecay l k (k * f) := fun z =>\n  tendsto_nhds.2 fun s hs hs0 =>\n    l.sets_of_superset ((tendsto_nhds.1 (hf <| z + 1)) s hs hs0) fun x hx => by\n      simpa only [Set.mem_preimage, Pi.mul_apply, ← mul_assoc, ← pow_succ] using hx\n\ntheorem SuperpolynomialDecay.mul_param (hf : SuperpolynomialDecay l k f) :\n    SuperpolynomialDecay l k (f * k) :=\n  hf.param_mul.congr fun _ => mul_comm _ _\n\ntheorem SuperpolynomialDecay.param_pow_mul (hf : SuperpolynomialDecay l k f) (n : ℕ) :\n    SuperpolynomialDecay l k (k ^ n * f) := by\n  induction n with\n  | zero => simpa only [one_mul, pow_zero] using hf\n  | succ n hn => simpa only [pow_succ', mul_assoc] using hn.param_mul\n\ntheorem SuperpolynomialDecay.mul_param_pow (hf : SuperpolynomialDecay l k f) (n : ℕ) :\n    SuperpolynomialDecay l k (f * k ^ n) :=\n  (hf.param_pow_mul n).congr fun _ => mul_comm _ _\n\ntheorem SuperpolynomialDecay.polynomial_mul [ContinuousAdd β] [ContinuousMul β]\n    (hf : SuperpolynomialDecay l k f) (p : β[X]) :\n    SuperpolynomialDecay l k fun x => (p.eval <| k x) * f x :=\n  Polynomial.induction_on' p (fun p q hp hq => by simpa [add_mul] using! hp.add hq) fun n c => by\n    simpa [mul_assoc] using! (hf.param_pow_mul n).const_mul c\n\ntheorem SuperpolynomialDecay.mul_polynomial [ContinuousAdd β] [ContinuousMul β]\n    (hf : SuperpolynomialDecay l k f) (p : β[X]) :\n    SuperpolynomialDecay l k fun x => f x * (p.eval <| k x) :=\n  (hf.polynomial_mul p).congr fun _ => mul_comm _ _\n\nend CommSemiring\n\nsection OrderedCommSemiring\n\nvariable [TopologicalSpace β] [CommSemiring β] [PartialOrder β] [IsOrderedRing β] [OrderTopology β]\n\ntheorem SuperpolynomialDecay.trans_eventuallyLE (hk : 0 ≤ᶠ[l] k) (hg : SuperpolynomialDecay l k g)\n    (hg' : SuperpolynomialDecay l k g') (hfg : g ≤ᶠ[l] f) (hfg' : f ≤ᶠ[l] g') :\n    SuperpolynomialDecay l k f := fun z =>\n  tendsto_of_tendsto_of_tendsto_of_le_of_le' (hg z) (hg' z)\n    (by filter_upwards [hfg, hk] with x hx (hx' : 0 ≤ k x) using by gcongr)\n    (by filter_upwards [hfg', hk] with x hx (hx' : 0 ≤ k x) using by gcongr)\n\nend OrderedCommSemiring\n\nsection LinearOrderedCommRing\n\nvariable [TopologicalSpace β] [CommRing β] [LinearOrder β] [IsStrictOrderedRing β] [OrderTopology β]\nvariable (l k f)\n\ntheorem superpolynomialDecay_iff_abs_tendsto_zero :\n    SuperpolynomialDecay l k f ↔ ∀ n : ℕ, Tendsto (fun a : α => |k a ^ n * f a|) l (𝓝 0) :=\n  ⟨fun h z => (tendsto_zero_iff_abs_tendsto_zero _).1 (h z), fun h z =>\n    (tendsto_zero_iff_abs_tendsto_zero _).2 (h z)⟩\n\ntheorem superpolynomialDecay_iff_superpolynomialDecay_abs :\n    SuperpolynomialDecay l k f ↔ SuperpolynomialDecay l (fun a => |k a|) fun a => |f a| :=\n  (superpolynomialDecay_iff_abs_tendsto_zero l k f).trans\n    (by simp_rw [SuperpolynomialDecay, abs_mul, abs_pow])\n\nvariable {l k f}\n\nTarget:\ntheorem SuperpolynomialDecay.trans_eventually_abs_le (hf : SuperpolynomialDecay l k f)\n    (hfg : abs ∘ g ≤ᶠ[l] abs ∘ f) : SuperpolynomialDecay l k g :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_f97d07cceec9","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"8be3c6327935bda09fd2da3c8a57759bf4c7a29302587a1221e704619ddeee44","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Asymptotics","family_id":"superpolynomialdecay","file_id":"mathlib/Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean","sample_id":"f97d07cceec981c168c5f2eca445cd4e95e1981b7df2ab5f563b675d89d0e28f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f8689006584708e8f5f2ea5a3447138e87630e033d57129a1fcb9fca25aae75e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1b257d253a1d99e1ebaccfa098b639ce8bc9949faf2e55ad6ca74bb6e26b343e","source_sha256":"d05b0d2ddc5c920ffdbae14bb142f5937424e23790dd69cbc6cb4ae7e530bb6a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let h := MorphismProperty.factorizationData (trivialCofibrations C) (fibrations C)\n      (terminal.from X.obj)\n  have := isCofibrant_of_cofibration h.i\n  have : IsFibrant h.Z := by\n    rw [isFibrant_iff_of_isTerminal h.p terminalIsTerminal]\n    infer_instance\n  exact ⟨BifibrantObject.mk h.Z, homMk h.i, inferInstanceAs (Cofibration h.i),\n    inferInstanceAs (WeakEquivalence h.i)⟩\n\n/-- Given `X : CofibrantObject C`, this is a choice of bifibrant resolution of `X`. -/\nnoncomputable def bifibrantResolutionObj (X : CofibrantObject C) :\n    BifibrantObject C :=\n  (exists_bifibrant X).choose\n\n/-- Given `X : CofibrantObject C`, this is a trivial cofibration\nfrom `X` to a choice of bifibrant resolution. -/\nnoncomputable def iBifibrantResolutionObj (X : CofibrantObject C) :\n    X ⟶ BifibrantObject.ιCofibrantObject.obj (bifibrantResolutionObj X) :=\n  (exists_bifibrant X).choose_spec.choose","hard_negative":false,"metrics":{"chosen_tokens":180,"rejected_tokens":5,"token_jaccard":0.030303,"token_length_ratio":0.027778},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"b25216169f7611bacc187d47990993c081d5decbab23d3dd2b33e2d3648cabe3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy\npublic import Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy\npublic import Mathlib.CategoryTheory.Localization.CalculusOfFractions.OfAdjunction\npublic import Mathlib.CategoryTheory.Quotient.LocallySmall\n\nNamespace:\nHomotopicalAlgebra.CofibrantObject\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The homotopy category of bifibrant objects\n\nWe construct the homotopy category `BifibrantObject.HoCat C` of bifibrant\nobjects in a model category `C` and show that the functor\n`BifibrantObject.toHoCat : BifibrantObject C ⥤ BifibrantObject.HoCat C`\nis a localization functor with respect to weak equivalences.\nWe also show that certain localizer morphisms are localized weak equivalences,\nwhich can be understood by saying that we obtain the same localized\ncategory (up to equivalence) by inverting weak equivalences in `C`,\n`CofibrantObject C`, `FibrantObject C` or `BifibrantObject C`.\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nopen CategoryTheory Limits\n\nnamespace HomotopicalAlgebra\n\nvariable {C : Type u} [Category.{v} C] [ModelCategory C]\n\nnamespace BifibrantObject\n\nvariable (C) in\n/-- The homotopy relation on the category of bifibrant objects. -/\ndef homRel : HomRel (BifibrantObject C) :=\n  fun _ _ f g ↦ RightHomotopyRel f.hom g.hom\n\nlemma homRel_iff_rightHomotopyRel {X Y : BifibrantObject C} {f g : X ⟶ Y} :\n    homRel C f g ↔ RightHomotopyRel f.hom g.hom := Iff.rfl\n\nlemma homRel_iff_leftHomotopyRel {X Y : BifibrantObject C} {f g : X ⟶ Y} :\n    homRel C f g ↔ LeftHomotopyRel f.hom g.hom := by\n  rw [homRel_iff_rightHomotopyRel, leftHomotopyRel_iff_rightHomotopyRel]\n\ninstance : HomRel.IsStableUnderPostcomp (homRel C) where\n  comp_right _ h := h.postcomp _\n\ninstance : HomRel.IsStableUnderPrecomp (homRel C) where\n  comp_left _ _ _ h := h.precomp _\n\ninstance : Congruence (homRel C) where\n  equivalence :=\n    { refl _ := .refl _\n      symm h := .symm h\n      trans h₁ h₂ := .trans h₁ h₂ }\n\nvariable (C) in\n/-- The homotopy category of bifibrant objects. -/\nabbrev HoCat := Quotient (BifibrantObject.homRel C)\n\n/-- The quotient functor from the category of bifibrant objects to its\nhomotopy category. -/\ndef toHoCat : BifibrantObject C ⥤ HoCat C := Quotient.functor _\n\nlemma toHoCat_obj_surjective : Function.Surjective (toHoCat (C := C)).obj :=\n  fun ⟨_⟩ ↦ ⟨_, rfl⟩\n\ninstance : Functor.Full (toHoCat (C := C)) := by dsimp [toHoCat]; infer_instance\n\nlemma toHoCat_map_eq {X Y : BifibrantObject C} {f g : X ⟶ Y}\n    (h : homRel C f g) :\n    toHoCat.map f = toHoCat.map g :=\n  CategoryTheory.Quotient.sound _ h\n\nlemma toHoCat_map_eq_iff {X Y : BifibrantObject C} (f g : X ⟶ Y) :\n    toHoCat.map f = toHoCat.map g ↔ homRel C f g :=\n  Quotient.functor_map_eq_iff _ _ _\n\ninstance [LocallySmall.{w} C] : LocallySmall.{w} (HoCat C) := by\n  dsimp [HoCat]\n  infer_instance\n\nsection\n\nvariable {D : Type*} [Category* D]\n\nlemma inverts_iff_factors (F : BifibrantObject C ⥤ D) :\n    (weakEquivalences _).IsInvertedBy F ↔\n    ∀ ⦃K L : BifibrantObject C⦄ (f g : K ⟶ L),\n      homRel C f g → F.map f = F.map g := by\n  refine ⟨fun H K L f g h ↦ ?_, fun h X Y f hf ↦ ?_⟩\n  · obtain ⟨P, _, ⟨h⟩⟩ := h.exists_very_good_pathObject\n    have := isCofibrant_of_cofibration P.ι\n    have : IsIso (F.map (homMk P.ι)) := H _ (by\n      rw [← weakEquivalence_iff, weakEquivalence_iff_of_objectProperty]\n      exact inferInstanceAs (WeakEquivalence P.ι))\n    simp only [show f = homMk h.h ≫ homMk P.p₀ by cat_disch,\n      show g = homMk h.h ≫ homMk P.p₁ by cat_disch, Functor.map_comp]\n    congr 1\n    simp [← cancel_epi (F.map (homMk P.ι)), ← Functor.map_comp]\n  · rw [← weakEquivalence_iff, weakEquivalence_iff_of_objectProperty] at hf\n    obtain ⟨g', h₁, h₂⟩ := RightHomotopyClass.whitehead f.hom\n    refine ⟨F.map (homMk g'), ?_, ?_⟩\n    all_goals\n      rw [← F.map_comp, ← F.map_id]\n      apply h\n      assumption\n\n/-- The strict universal property of the localization with respect\nto weak equivalences for the quotient functor\n`toHoCat : BifibrantObject C ⥤ BifibrantObject.HoCat C`. -/\ndef strictUniversalPropertyFixedTargetToHoCat :\n    Localization.StrictUniversalPropertyFixedTarget\n      toHoCat (weakEquivalences (BifibrantObject C)) D where\n  inverts := by\n    rw [inverts_iff_factors]\n    intro K L f g h\n    exact CategoryTheory.Quotient.sound _ h\n  lift F hF := CategoryTheory.Quotient.lift _ F\n    (by rwa [inverts_iff_factors] at hF)\n  fac F hF := rfl\n  uniq _ _ h := Quotient.lift_unique' _ _ _ h\n\nend\n\ninstance : toHoCat.IsLocalization (weakEquivalences (BifibrantObject C)) :=\n  .mk' _ _ strictUniversalPropertyFixedTargetToHoCat\n    strictUniversalPropertyFixedTargetToHoCat\n\ninstance {X Y : BifibrantObject C} (f : X ⟶ Y) [hf : WeakEquivalence f] :\n    IsIso (toHoCat.map f) :=\n  Localization.inverts toHoCat (weakEquivalences _) f (by rwa [weakEquivalence_iff] at hf)\n\nsection\n\nvariable {X Y : C} [IsCofibrant X] [IsCofibrant Y] [IsFibrant X] [IsFibrant Y]\n\n/-- Right homotopy classes of maps between bifibrant objects identify\nto morphisms in the homotopy category `BifibrantObject.HoCat`. -/\ndef HoCat.homEquivRight :\n    RightHomotopyClass X Y ≃ (toHoCat.obj (mk X) ⟶ toHoCat.obj (mk Y)) where\n  toFun := Quot.lift (fun f ↦ toHoCat.map (homMk f)) (fun _ _ h ↦ by rwa [toHoCat_map_eq_iff])\n  invFun := Quot.lift (fun f ↦ .mk f.hom) (fun _ _ h ↦ by\n    simpa [RightHomotopyClass.mk_eq_mk_iff] using! h)\n  left_inv := by rintro ⟨f⟩; rfl\n  right_inv := by rintro ⟨f⟩; rfl\n\n@[simp]\nlemma HoCat.homEquivRight_apply (f : X ⟶ Y) :\n    HoCat.homEquivRight (.mk f) = toHoCat.map (homMk f) := rfl\n\n@[simp]\nlemma HoCat.homEquivRight_symm_apply (f : X ⟶ Y) :\n    HoCat.homEquivRight.symm (toHoCat.map (homMk f)) = .mk f := rfl\n\n/-- Left homotopy classes of maps between bifibrant objects identify\nto morphisms in the homotopy category `BifibrantObject.HoCat`. -/\ndef HoCat.homEquivLeft :\n    LeftHomotopyClass X Y ≃ (toHoCat.obj (mk X) ⟶ toHoCat.obj (mk Y)) :=\n  leftHomotopyClassEquivRightHomotopyClass.trans HoCat.homEquivRight\n\n@[simp]\nlemma HoCat.homEquivLeft_apply (f : X ⟶ Y) :\n    HoCat.homEquivLeft (.mk f) = toHoCat.map (homMk f) := by\n  simp [homEquivLeft]\n\n@[simp]\nlemma HoCat.homEquivLeft_symm_apply (f : X ⟶ Y) :\n    HoCat.homEquivRight.symm (toHoCat.map (homMk f)) = .mk f := rfl\n\nend\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The inclusion functor `BifibrantObject.HoCat C ⥤ FibrantObject.HoCat C`. -/\ndef HoCat.ιFibrantObject : HoCat C ⥤ FibrantObject.HoCat C :=\n  CategoryTheory.Quotient.lift _\n    (BifibrantObject.ιFibrantObject ⋙ FibrantObject.toHoCat) (fun _ _ _ _ h ↦ by\n      simpa [FibrantObject.toHoCat_map_eq_iff, FibrantObject.homRel_iff_leftHomotopyRel,\n        homRel_iff_leftHomotopyRel] using h)\n\n@[simp]\nlemma HoCat.ιFibrantObject_obj (X : BifibrantObject C) :\n    HoCat.ιFibrantObject.obj (toHoCat.obj X) =\n      FibrantObject.toHoCat.obj (BifibrantObject.ιFibrantObject.obj X) :=\n  rfl\n\n@[simp]\nlemma HoCat.ιFibrantObject_map_toHoCat_map {X Y : BifibrantObject C} (f : X ⟶ Y) :\n    HoCat.ιFibrantObject.map (toHoCat.map f) =\n      FibrantObject.toHoCat.map (FibrantObject.homMk f.hom) :=\n  rfl\n\n/-- The isomorphism `toHoCat ⋙ HoCat.ιFibrantObject ≅ ιFibrantObject ⋙ FibrantObject.toHoCat`\nbetween functors `BifibrantObject C ⥤ FibrantObject.HoCat C`. -/\ndef toHoCatCompιFibrantObject :\n    toHoCat (C := C) ⋙ HoCat.ιFibrantObject ≅\n      ιFibrantObject ⋙ FibrantObject.toHoCat := Iso.refl _\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The inclusion functor `BifibrantObject.HoCat C ⥤ CofibrantObject.HoCat C`. -/\ndef HoCat.ιCofibrantObject : HoCat C ⥤ CofibrantObject.HoCat C :=\n  CategoryTheory.Quotient.lift _\n    (BifibrantObject.ιCofibrantObject ⋙ CofibrantObject.toHoCat) (fun _ _ _ _ h ↦ by\n      simpa [CofibrantObject.toHoCat_map_eq_iff])\n\n@[simp]\nlemma HoCat.ιCofibrantObject_obj (X : BifibrantObject C) :\n    HoCat.ιCofibrantObject.obj (toHoCat.obj X) =\n      CofibrantObject.toHoCat.obj (BifibrantObject.ιCofibrantObject.obj X) :=\n  rfl\n\n@[simp]\nlemma HoCat.ιCofibrantObject_map_toHoCat_map {X Y : BifibrantObject C} (f : X ⟶ Y) :\n    HoCat.ιCofibrantObject.map (toHoCat.map f) =\n      CofibrantObject.toHoCat.map (CofibrantObject.homMk f.hom) :=\n  rfl\n\n/-- The isomorphism\n`toHoCat ⋙ HoCat.ιCofibrantObject ≅ ιCofibrantObject ⋙ CofibrantObject.toHoCat`\nbetween functors `BifibrantObject C ⥤ CofibrantObject.HoCat C`. -/\ndef toHoCatCompιCofibrantObject :\n    toHoCat (C := C) ⋙ HoCat.ιCofibrantObject ≅\n      ιCofibrantObject ⋙ CofibrantObject.toHoCat := Iso.refl _\n\nend BifibrantObject\n\nnamespace CofibrantObject\n\nTarget:\nlemma exists_bifibrant (X : CofibrantObject C) :\n    ∃ (Y : BifibrantObject C) (i : X ⟶ BifibrantObject.ιCofibrantObject.obj Y),\n      Cofibration (ι.map i) ∧ WeakEquivalence (ι.map i) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/ModelCategory","family_id":"exists_bifibrant","file_id":"mathlib/Mathlib/AlgebraicTopology/ModelCategory/BifibrantObjectHomotopy.lean","sample_id":"1b257d253a1d99e1ebaccfa098b639ce8bc9949faf2e55ad6ca74bb6e26b343e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9d077355c8aafccdf2f765766657b2aef7013299b24845a61f82fd2eda23f5d5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"eeee9288bac943d40417c3f56fb23df234a9e4e2137314ce3ccd6a1779c20989","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"471ab45e244c5046c4d6db5db33762cae30b42d81f33c3c388fbdb7ee1366b37","source_sha256":"bdc962e9d41e4ed417e034e5bc54ca34f64e30878dd13535c17ebf9d0ca4959d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  zify\n  convert! Rat.num_mul_num_eq_num_mul_gcd q₁ q₂ <;> norm_cast","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":22,"token_jaccard":0.875,"token_length_ratio":1.466667},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"b2a6a27982231f34c0259c3bc5ec4ed4b798124e625a892f70896f8acbf51030","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Rat\npublic import Mathlib.Algebra.Group.Indicator\npublic import Mathlib.Algebra.Order.Field.Rat\npublic import Mathlib.Data.Rat.Lemmas\npublic import Mathlib.Tactic.Zify\n\nNamespace:\nNNRat\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Field and action structures on the nonnegative rationals\n\nThis file provides additional results about `NNRat` that cannot live in earlier files due to import\ncycles.\n-/\n\n@[expose] public section\n\nopen Function\nopen scoped NNRat\n\nnamespace NNRat\nvariable {α : Type*} {q : ℚ≥0}\n\n@[simp, norm_cast]\nlemma coe_indicator (s : Set α) (f : α → ℚ≥0) (a : α) :\n    ((s.indicator f a : ℚ≥0) : ℚ) = s.indicator (fun x ↦ ↑(f x)) a :=\n  map_indicator coeHom _ _ _\n\nend NNRat\n\nopen NNRat\n\nnamespace Rat\n\nvariable {p q : ℚ}\n\nlemma toNNRat_inv (q : ℚ) : toNNRat q⁻¹ = (toNNRat q)⁻¹ := by\n  obtain hq | hq := le_total q 0\n  · rw [toNNRat_eq_zero.mpr hq, inv_zero, toNNRat_eq_zero.mpr (inv_nonpos.mpr hq)]\n  · nth_rw 1 [← Rat.coe_toNNRat q hq]\n    rw [← coe_inv, toNNRat_coe]\n\nlemma toNNRat_div (hp : 0 ≤ p) : toNNRat (p / q) = toNNRat p / toNNRat q := by\n  rw [div_eq_mul_inv, div_eq_mul_inv, ← toNNRat_inv, ← toNNRat_mul hp]\n\nlemma toNNRat_div' (hq : 0 ≤ q) : toNNRat (p / q) = toNNRat p / toNNRat q := by\n  rw [div_eq_inv_mul, div_eq_inv_mul, toNNRat_mul (inv_nonneg.2 hq), toNNRat_inv]\n\nend Rat\n\n/-! ### Numerator and denominator -/\n\nnamespace NNRat\n\nvariable {q : ℚ≥0}\n\n/-- A recursor for nonnegative rationals in terms of numerators and denominators. -/\nprotected def rec {α : ℚ≥0 → Sort*} (h : ∀ m n : ℕ, α (m / n)) (q : ℚ≥0) : α q := by\n  rw [← num_div_den q]; apply h\n\ntheorem mul_num (q₁ q₂ : ℚ≥0) :\n    (q₁ * q₂).num = q₁.num * q₂.num / Nat.gcd (q₁.num * q₂.num) (q₁.den * q₂.den) := by\n  zify\n  convert! Rat.mul_num q₁ q₂ <;> norm_cast\n\ntheorem mul_den (q₁ q₂ : ℚ≥0) :\n    (q₁ * q₂).den = q₁.den * q₂.den / Nat.gcd (q₁.num * q₂.num) (q₁.den * q₂.den) := by\n  convert! Rat.mul_den q₁ q₂\n  norm_cast\n\n/-- A version of `NNRat.mul_den` without division. -/\ntheorem den_mul_den_eq_den_mul_gcd (q₁ q₂ : ℚ≥0) :\n    q₁.den * q₂.den = (q₁ * q₂).den * ((q₁.num * q₂.num).gcd (q₁.den * q₂.den)) := by\n  convert! Rat.den_mul_den_eq_den_mul_gcd q₁ q₂\n  norm_cast\n\n/-- A version of `NNRat.mul_num` without division. -/\n\nTarget:\ntheorem num_mul_num_eq_num_mul_gcd (q₁ q₂ : ℚ≥0) :\n    q₁.num * q₂.num = (q₁ * q₂).num * ((q₁.num * q₂.num).gcd (q₁.den * q₂.den)) :=\n\nProof body:\n","rejected":"```lean\nby\n  zify\n  convert! Rat.num_mul_num_eq_num_mul_gcd q₁ q₂ <;> norm_cast\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/NNRat","family_id":"num_mul_num_eq_num_mul_gcd","file_id":"mathlib/Mathlib/Data/NNRat/Lemmas.lean","sample_id":"471ab45e244c5046c4d6db5db33762cae30b42d81f33c3c388fbdb7ee1366b37"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6932312c5f36cec3a14fb1d6fd3d6a1837663ad65e9f1ee82ae3d44d58f2d03c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"062acd11f87739fb15d069f3d86b913dd7cb13fa854b155bc79d97daea43f761","source_sha256":"c3981253356b7afc328e1095904e6a5b84d7b6867b9673136639d9cfe765100e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let _ : Nontrivial N := by simp [← Module.supportDim_ne_bot_iff_nontrivial R, dim]\n  rw [PrimeSpectrum.zeroLocus_eq_singleton]\n  apply le_antisymm\n  · intro p hp\n    by_contra nmem\n    push _ ∈ _ at nmem\n    have : p < ⟨maximalIdeal R, IsMaximal.isPrime' (maximalIdeal R)⟩ :=\n      lt_of_le_of_ne (IsLocalRing.le_maximalIdeal IsPrime.ne_top') nmem\n    have : Module.supportDim R N > 0 := by\n      simp only [Module.supportDim, gt_iff_lt, Order.krullDim_pos_iff, Subtype.exists,\n        Subtype.mk_lt_mk, exists_prop]\n      use p\n      simpa [hp] using! ⟨_, IsLocalRing.closedPoint_mem_support R N, this⟩\n    exact (ne_of_lt this) dim.symm\n  · simpa using! IsLocalRing.closedPoint_mem_support R N","hard_negative":false,"metrics":{"chosen_tokens":135,"rejected_tokens":2,"token_jaccard":0.015152,"token_length_ratio":0.014815},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"b2aee270782a042c5aede4eaf0e2a53d9ce091da448f0e09807b4935dfef650c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.KrullDimension.NonZeroDivisors\npublic import Mathlib.RingTheory.Spectrum.Prime.Module\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Nailin Guan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Nailin Guan\n-/\n/-!\n\n# Krull Dimension of Module\n\nIn this file we define `Module.supportDim R M` for an `R`-module `M` as\nthe krull dimension of its support. It is equal to the krull dimension of `R / Ann M` when\n`M` is finitely generated.\n\n-/\n\n@[expose] public section\n\nvariable (R : Type*) [CommRing R]\n\nvariable (M : Type*) [AddCommGroup M] [Module R M] (N : Type*) [AddCommGroup N] [Module R N]\n\nnamespace Module\n\nopen Order\n\n/-- The krull dimension of module, defined as `krullDim` of its support. -/\nnoncomputable def supportDim : WithBot ℕ∞ :=\n  krullDim (Module.support R M)\n\n@[nontriviality]\nlemma supportDim_eq_bot_of_subsingleton [Subsingleton M] : supportDim R M = ⊥ := by\n  simpa [supportDim, support_eq_empty_iff]\n\nlemma supportDim_ne_bot_of_nontrivial [Nontrivial M] : supportDim R M ≠ ⊥ := by\n  have : Nonempty (Module.support R M) := nonempty_support_of_nontrivial.to_subtype\n  simp [supportDim]\n\nlemma supportDim_eq_bot_iff_subsingleton : supportDim R M = ⊥ ↔ Subsingleton M := by\n  simp [supportDim, krullDim_eq_bot_iff, support_eq_empty_iff]\n\nlemma supportDim_ne_bot_iff_nontrivial : supportDim R M ≠ ⊥ ↔ Nontrivial M := by\n  simp [supportDim, krullDim_eq_bot_iff, support_eq_empty_iff, not_subsingleton_iff_nontrivial]\n\nlemma supportDim_eq_ringKrullDim_quotient_annihilator [Module.Finite R M] :\n    supportDim R M = ringKrullDim (R ⧸ annihilator R M) := by\n  simp only [supportDim]\n  rw [support_eq_zeroLocus, ringKrullDim_quotient]\n\nlemma supportDim_self_eq_ringKrullDim : supportDim R R = ringKrullDim R := by\n  have : annihilator R R = ⊥ :=\n    annihilator_eq_bot.mpr ((faithfulSMul_iff_algebraMap_injective R R).mpr fun {a₁ a₂} a ↦ a)\n  rw [supportDim_eq_ringKrullDim_quotient_annihilator, this]\n  exact (RingEquiv.ringKrullDim (RingEquiv.quotientBot R))\n\nlemma supportDim_le_ringKrullDim : supportDim R M ≤ ringKrullDim R :=\n  krullDim_le_of_strictMono (fun a ↦ a) fun {_ _} lt ↦ lt\n\nvariable {R M N}\n\nlemma supportDim_quotient_eq_ringKrullDim (I : Ideal R) :\n    supportDim R (R ⧸ I) = ringKrullDim (R ⧸ I) := by\n  rw [supportDim_eq_ringKrullDim_quotient_annihilator, Ideal.annihilator_quotient]\n\nlemma supportDim_le_of_injective (f : M →ₗ[R] N) (h : Function.Injective f) :\n    supportDim R M ≤ supportDim R N :=\n  krullDim_le_of_strictMono (fun a ↦ ⟨a.1, Module.support_subset_of_injective f h a.2⟩)\n    (fun {_ _} lt ↦ lt)\n\nlemma supportDim_le_of_surjective (f : M →ₗ[R] N) (h : Function.Surjective f) :\n    supportDim R N ≤ supportDim R M :=\n  krullDim_le_of_strictMono (fun a ↦ ⟨a.1, Module.support_subset_of_surjective f h a.2⟩)\n    (fun {_ _} lt ↦ lt)\n\nlemma supportDim_eq_of_equiv (e : M ≃ₗ[R] N) :\n    supportDim R M = supportDim R N :=\n  le_antisymm (supportDim_le_of_injective e e.injective)\n    (supportDim_le_of_surjective e e.surjective)\n\nend Module\n\nopen Ideal IsLocalRing\n\nTarget:\nlemma support_of_supportDim_eq_zero [IsLocalRing R]\n    (dim : Module.supportDim R N = 0) :\n    Module.support R N = PrimeSpectrum.zeroLocus (maximalIdeal R) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/KrullDimension","family_id":"support_of_supportdim_eq_zero","file_id":"mathlib/Mathlib/RingTheory/KrullDimension/Module.lean","sample_id":"062acd11f87739fb15d069f3d86b913dd7cb13fa854b155bc79d97daea43f761"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e13f4e0fbff74fe329154bb27fe6656b950cc8d302959791efca88cd4237a5ab","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"17969d4630b562cbde413c26c21cf5e30805ec3aa06850d6c27cc2864964be90","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"727321c81fb49fb44e787715ddf3c0aafbedfa69ef549affbe7e565a44f0be51","source_sha256":"6c1d4f9fced239f6f8c0c8785a4325f383edabba30747fe08c8262d3408cd815","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h => ?_, fun ⟨h₁, h₂⟩ => Equiv.antisymm h₁ h₂⟩\n  obtain - | ⟨h₁, h₂⟩ := h\n  · simp [Lists'.Subset.refl]\n  · exact ⟨h₁, h₂⟩","hard_negative":true,"metrics":{"chosen_tokens":56,"rejected_tokens":5,"token_jaccard":0.103448,"token_length_ratio":0.089286},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"b2c0f7cd9dbe977cf68e6e075ae305d92f51eeb43fd63c04bcecacea4c5ac5ba","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.Monoid.NatCast\npublic import Mathlib.Algebra.Ring.Nat\npublic import Mathlib.Data.Sigma.Basic\npublic import Batteries.Tactic.Lint.TypeClass\n\nNamespace:\nLists\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# A computable model of ZFA without infinity\n\nIn this file we define finite hereditary lists. This is useful for calculations in naive set theory.\n\nWe distinguish two kinds of ZFA lists:\n* Atoms. Directly correspond to an element of the original type.\n* Proper ZFA lists. Can be thought of (but are not implemented) as a list of ZFA lists (not\n  necessarily proper).\n\nFor example, `Lists ℕ` contains stuff like `23`, `[]`, `[37]`, `[1, [[2], 3], 4]`.\n\n## Implementation note\n\nAs we want to be able to append both atoms and proper ZFA lists to proper ZFA lists, it's handy that\natoms and proper ZFA lists belong to the same type, even though atoms of `α` could be modelled as\n`α` directly. But we don't want to be able to append anything to atoms.\n\nThis calls for a two-step definition of ZFA lists:\n* First, define ZFA prelists as atoms and proper ZFA prelists. Those proper ZFA prelists are defined\n  by inductive appending of (not necessarily proper) ZFA lists.\n* Second, define ZFA lists by rubbing out the distinction between atoms and proper lists.\n\n## Main declarations\n\n* `Lists' α false`: Atoms as ZFA prelists. Basically a copy of `α`.\n* `Lists' α true`: Proper ZFA prelists. Defined inductively from the empty ZFA prelist\n  (`Lists'.nil`) and from appending a ZFA prelist to a proper ZFA prelist (`Lists'.cons a l`).\n* `Lists α`: ZFA lists. Sum of the atoms and proper ZFA prelists.\n* `Finsets α`: ZFA sets. Defined as `Lists` quotiented by `Lists.Equiv`, the extensional\n  equivalence.\n-/\n\n@[expose] public section\n\n\nvariable {α : Type*}\n\n/-- Prelists, helper type to define `Lists`. `Lists' α false` are the \"atoms\", a copy of `α`.\n`Lists' α true` are the \"proper\" ZFA prelists, inductively defined from the empty ZFA prelist and\nfrom appending a ZFA prelist to a proper ZFA prelist. It is made so that you can't append anything\nto an atom while having only one appending function for appending both atoms and proper ZFA prelists\nto a proper ZFA prelist. -/\ninductive Lists'.{u} (α : Type u) : Bool → Type u\n  | atom : α → Lists' α false\n  | nil : Lists' α true\n  | cons' {b} : Lists' α b → Lists' α true → Lists' α true\n  deriving DecidableEq\ncompile_inductive% Lists'\n\n/-- Hereditarily finite list, aka ZFA list. A ZFA list is either an \"atom\" (`b = false`),\ncorresponding to an element of `α`, or a \"proper\" ZFA list, inductively defined from the empty ZFA\nlist and from appending a ZFA list to a proper ZFA list. -/\ndef Lists (α : Type*) :=\n  Σ b, Lists' α b\n\nnamespace Lists'\n\ninstance [Inhabited α] : ∀ b, Inhabited (Lists' α b)\n  | true => ⟨nil⟩\n  | false => ⟨atom default⟩\n\n/-- Appending a ZFA list to a proper ZFA prelist. -/\ndef cons : Lists α → Lists' α true → Lists' α true\n  | ⟨_, a⟩, l => cons' a l\n\n/-- Converts a ZFA prelist to a `List` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp]\ndef toList : ∀ {b}, Lists' α b → List (Lists α)\n  | _, atom _ => []\n  | _, nil => []\n  | _, cons' a l => ⟨_, a⟩ :: l.toList\n\n@[simp]\ntheorem toList_cons (a : Lists α) (l) : toList (cons a l) = a :: l.toList := rfl\n\n/-- Converts a `List` of ZFA lists to a proper ZFA prelist. -/\n@[simp]\ndef ofList : List (Lists α) → Lists' α true\n  | [] => nil\n  | a :: l => cons a (ofList l)\n\n@[simp]\ntheorem to_ofList (l : List (Lists α)) : toList (ofList l) = l := by induction l <;> simp [*]\n\n@[simp]\ntheorem of_toList : ∀ l : Lists' α true, ofList (toList l) = l :=\n  suffices ∀ (b) (h : true = b) (l : Lists' α b),\n      let l' : Lists' α true := h ▸ l\n      ofList (toList l') = l'\n    from this _ rfl\n  fun b h l => by\n    induction l with\n    | atom => cases h\n    | nil => simp\n    | cons' b a _ IH => simpa [cons] using IH rfl\n\n/-- Recursion/induction principle for `Lists'.ofList`. -/\n@[elab_as_elim]\ndef recOfList {motive : Lists' α true → Sort*} (ofList : ∀ l, motive (ofList l)) : ∀ l, motive l :=\n  fun l ↦ cast (by simp) <| ofList (l.toList)\n\nend Lists'\n\nmutual\n  /-- Equivalence of ZFA lists. Defined inductively. -/\n  inductive Lists.Equiv : Lists α → Lists α → Prop\n    | refl (l) : Lists.Equiv l l\n    | antisymm {l₁ l₂ : Lists' α true} :\n      Lists'.Subset l₁ l₂ → Lists'.Subset l₂ l₁ → Lists.Equiv ⟨_, l₁⟩ ⟨_, l₂⟩\n\n  /-- Subset relation for ZFA lists. Defined inductively. -/\n  inductive Lists'.Subset : Lists' α true → Lists' α true → Prop\n    | nil {l} : Lists'.Subset Lists'.nil l\n    | cons {a a' l l'} :\n      Lists.Equiv a a' →\n        a' ∈ Lists'.toList l' → Lists'.Subset l l' → Lists'.Subset (Lists'.cons a l) l'\nend\n\nlocal infixl:50 \" ~ \" => Lists.Equiv\n\nnamespace Lists'\n\ninstance : HasSubset (Lists' α true) :=\n  ⟨Lists'.Subset⟩\n\n/-- ZFA prelist membership. A ZFA list is in a ZFA prelist if some element of this ZFA prelist is\nequivalent as a ZFA list to this ZFA list. -/\ninstance {b} : Membership (Lists α) (Lists' α b) :=\n  ⟨fun l a => ∃ a' ∈ l.toList, a ~ a'⟩\n\ntheorem mem_def {b a} {l : Lists' α b} : a ∈ l ↔ ∃ a' ∈ l.toList, a ~ a' :=\n  Iff.rfl\n\n@[simp]\ntheorem mem_cons {a y l} : a ∈ @cons α y l ↔ a ~ y ∨ a ∈ l := by\n  simp [mem_def, or_and_right, exists_or]\n\ntheorem cons_subset {a} {l₁ l₂ : Lists' α true} : Lists'.cons a l₁ ⊆ l₂ ↔ a ∈ l₂ ∧ l₁ ⊆ l₂ := by\n  refine ⟨fun h => ?_, fun ⟨⟨a', m, e⟩, s⟩ => Subset.cons e m s⟩\n  generalize h' : Lists'.cons a l₁ = l₁' at h\n  obtain - | @⟨a', _, _, _, e, m, s⟩ := h\n  · cases a\n    cases h'\n  cases a; cases a'; cases h'; exact ⟨⟨_, m, e⟩, s⟩\n\ntheorem ofList_subset {l₁ l₂ : List (Lists α)} (h : l₁ ⊆ l₂) :\n    Lists'.ofList l₁ ⊆ Lists'.ofList l₂ := by\n  induction l₁ with\n  | nil => exact Subset.nil\n  | cons _ _ l₁_ih =>\n    refine Subset.cons (Lists.Equiv.refl _) ?_ (l₁_ih (List.subset_of_cons_subset h))\n    simp only [List.cons_subset] at h; simp [h]\n\n@[refl]\ntheorem Subset.refl {l : Lists' α true} : l ⊆ l := by\n  rw [← Lists'.of_toList l]; exact ofList_subset (List.Subset.refl _)\n\ntheorem subset_nil {l : Lists' α true} : l ⊆ Lists'.nil → l = Lists'.nil := by\n  rw [← of_toList l]\n  induction toList l <;> intro h\n  · rfl\n  · rcases cons_subset.1 h with ⟨⟨_, ⟨⟩, _⟩, _⟩\n\ntheorem mem_of_subset' {a} : ∀ {l₁ l₂ : Lists' α true} (_ : l₁ ⊆ l₂) (_ : a ∈ l₁.toList), a ∈ l₂\n  | nil, _, Lists'.Subset.nil, h => by cases h\n  | cons' a0 l0, l₂, s, h => by\n    obtain - | ⟨e, m, s⟩ := s\n    simp only [toList, Sigma.eta, List.mem_cons] at h\n    rcases h with (rfl | h)\n    · exact ⟨_, m, e⟩\n    · exact mem_of_subset' s h\n\ntheorem subset_def {l₁ l₂ : Lists' α true} : l₁ ⊆ l₂ ↔ ∀ a ∈ l₁.toList, a ∈ l₂ :=\n  ⟨fun H _ => mem_of_subset' H, fun H => by\n    induction l₁ using recOfList with | _ l₁\n    induction l₁ with\n    | nil => exact Subset.nil\n    | cons h t t_ih =>\n      simp only [to_ofList, ofList, toList_cons, List.mem_cons, forall_eq_or_imp] at *\n      exact cons_subset.2 ⟨H.1, t_ih H.2⟩⟩\n\nend Lists'\n\nnamespace Lists\n\n/-- Sends `a : α` to the corresponding atom in `Lists α`. -/\n@[match_pattern]\ndef atom (a : α) : Lists α :=\n  ⟨_, Lists'.atom a⟩\n\n/-- Converts a proper ZFA prelist to a ZFA list. -/\n@[match_pattern]\ndef of' (l : Lists' α true) : Lists α :=\n  ⟨_, l⟩\n\n/-- Converts a ZFA list to a `List` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp]\ndef toList : Lists α → List (Lists α)\n  | ⟨_, l⟩ => l.toList\n\n/-- Predicate stating that a ZFA list is proper. -/\ndef IsList (l : Lists α) : Prop :=\n  l.1\n\n/-- Converts a `List` of ZFA lists to a ZFA list. -/\ndef ofList (l : List (Lists α)) : Lists α :=\n  of' (Lists'.ofList l)\n\ntheorem isList_toList (l : List (Lists α)) : IsList (ofList l) :=\n  Eq.refl _\n\ntheorem to_ofList (l : List (Lists α)) : toList (ofList l) = l := by simp [ofList, of']\n\ntheorem of_toList : ∀ {l : Lists α}, IsList l → ofList (toList l) = l\n  | ⟨true, l⟩, _ => by simp_all [ofList, of']\n\ninstance : Inhabited (Lists α) :=\n  ⟨of' Lists'.nil⟩\n\ninstance [DecidableEq α] : DecidableEq (Lists α) := inferInstanceAs <| DecidableEq (Sigma _)\n\ninstance [SizeOf α] : SizeOf (Lists α) := inferInstanceAs <| SizeOf (Sigma _)\n\n/-- A recursion principle for pairs of ZFA lists and proper ZFA prelists. -/\ndef inductionMut (C : Lists α → Sort*) (D : Lists' α true → Sort*)\n    (C0 : ∀ a, C (atom a)) (C1 : ∀ l, D l → C (of' l))\n    (D0 : D Lists'.nil) (D1 : ∀ a l, C a → D l → D (Lists'.cons a l)) :\n    PProd (∀ l, C l) (∀ l, D l) := by\n  suffices ∀ {b} (l : Lists' α b),\n      PProd (C ⟨_, l⟩)\n        (match b, l with\n        | true, l => D l\n        | false, _ => PUnit)\n    by exact ⟨fun ⟨b, l⟩ => (this _).1, fun l => (this l).2⟩\n  intro b l\n  induction l with\n  | atom => exact ⟨C0 _, ⟨⟩⟩\n  | nil => exact ⟨C1 _ D0, D0⟩\n  | cons' a l IH₁ IH =>\n    have : D (Lists'.cons' a l) := D1 ⟨_, _⟩ _ IH₁.1 IH.2\n    exact ⟨C1 _ this, this⟩\n\n/-- Membership of ZFA list. A ZFA list belongs to a proper ZFA list if it belongs to the latter as a\nproper ZFA prelist. An atom has no members. -/\ndef mem (a : Lists α) : Lists α → Prop\n  | ⟨false, _⟩ => False\n  | ⟨_, l⟩ => a ∈ l\n\ninstance : Membership (Lists α) (Lists α) where\n  mem ls l := mem l ls\n\nTarget:\ntheorem isList_of_mem {a : Lists α} : ∀ {l : Lists α}, a ∈ l → IsList l\n  | ⟨_, Lists'.nil⟩, _ => rfl\n  | ⟨_, Lists'.cons' _ _⟩, _ => rfl\n\ntheorem Equiv.antisymm_iff {l₁ l₂ : Lists' α true} : of' l₁ ~ of' l₂ ↔ l₁ ⊆ l₂ ∧ l₂ ⊆ l₁ :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_727321c81fb4","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"493cd1762115e34cafdf59884185112811c261b16b38124fc5fc44f684c749e1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory","family_id":"islist_of_mem","file_id":"mathlib/Mathlib/SetTheory/Lists.lean","sample_id":"727321c81fb49fb44e787715ddf3c0aafbedfa69ef549affbe7e565a44f0be51"},"split":"train"}
{"binding":{"chosen_proof_sha256":"882f9cd65c324f45860677174f52bfb0dee89c309337831b30132904e328a1fb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5ad3f4c1074336dbfd17b6d6041c87d9308d36a6123c6ad702c726184bd0c5fe","source_sha256":"aac9c42310c3c68ba48257d401225da11673aa0693f3a499d296961d7c43ce42","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  lift s to Finset α using hs\n  classical\n  rw [stabilizer_coe_finset, mem_stabilizer_finset_iff_smul_finset_subset, ← Finset.coe_smul_finset,\n    Finset.coe_subset]","hard_negative":false,"metrics":{"chosen_tokens":24,"rejected_tokens":3,"token_jaccard":0.047619,"token_length_ratio":0.125},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"b2dbb29721b7dbdee80ab92d474c45c927c68299a45f9c7cd9dabdeeb7e58c36","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Pointwise.Finset\npublic import Mathlib.GroupTheory.QuotientGroup.Defs\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\n\nNamespace:\nMulAction\n\nLocal context:\n/-\nCopyright (c) 2023 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Stabilizer of a set under a pointwise action\n\nThis file characterises the stabilizer of a set/finset under the pointwise action of a group.\n-/\n\npublic section\n\nopen Function MulOpposite Set\nopen scoped Pointwise\n\nnamespace MulAction\nvariable {G H α : Type*}\n\n/-! ### Stabilizer of a set -/\n\nsection Set\nsection Group\nvariable [Group G] [Group H] [MulAction G α] {a : G} {s t : Set α}\n\n@[to_additive (attr := simp)]\nlemma stabilizer_empty : stabilizer G (∅ : Set α) = ⊤ :=\n  Subgroup.coe_eq_univ.1 <| eq_univ_of_forall fun _a ↦ smul_set_empty\n\n@[to_additive (attr := simp)]\nlemma stabilizer_univ : stabilizer G (Set.univ : Set α) = ⊤ := by\n  ext\n  simp\n\n@[to_additive (attr := simp)]\nlemma stabilizer_singleton (b : α) : stabilizer G ({b} : Set α) = stabilizer G b := by ext; simp\n\n@[to_additive]\nlemma mem_stabilizer_set {s : Set α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s := by\n  refine mem_stabilizer_iff.trans ⟨fun h b ↦ ?_, fun h ↦ ?_⟩\n  · rw [← (smul_mem_smul_set_iff : a • b ∈ _ ↔ _), h]\n  simp_rw [Set.ext_iff, mem_smul_set_iff_inv_smul_mem]\n  exact ((MulAction.toPerm a).forall_congr' <| by simp [Iff.comm]).1 h\n\n@[to_additive]\nlemma map_stabilizer_le (f : G →* H) (s : Set G) :\n    (stabilizer G s).map f ≤ stabilizer H (f '' s) := by\n  rintro a\n  simp only [Subgroup.mem_map, mem_stabilizer_iff, forall_exists_index, and_imp]\n  rintro a ha rfl\n  rw [← image_smul_distrib, ha]\n\n@[to_additive (attr := simp)]\nlemma stabilizer_mul_self (s : Set G) : (stabilizer G s : Set G) * s = s := by\n  ext\n  refine ⟨?_, fun h ↦ ⟨_, (stabilizer G s).one_mem, _, h, one_mul _⟩⟩\n  rintro ⟨a, ha, b, hb, rfl⟩\n  rw [← mem_stabilizer_iff.1 ha]\n  exact smul_mem_smul_set hb\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_apply₂ {f : Set α → Set α → Set α}\n    (hf : ∀ a : G, a • f s t = f (a • s) (a • t)) :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (f s t) := by aesop (add simp [SetLike.le_def])\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_union :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s ∪ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_union\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_inter :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s ∩ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_inter\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_sdiff :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s \\ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_sdiff\n\n@[to_additive]\nlemma stabilizer_union_eq_left (hdisj : Disjoint s t) (hstab : stabilizer G s ≤ stabilizer G t)\n    (hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G t) :\n    stabilizer G (s ∪ t) = stabilizer G s := by\n  refine le_antisymm ?_ ?_\n  · calc\n      stabilizer G (s ∪ t)\n        ≤ stabilizer G (s ∪ t) ⊓ stabilizer G t := by simpa\n      _ ≤ stabilizer G ((s ∪ t) \\ t) := stabilizer_inf_stabilizer_le_stabilizer_sdiff\n      _ = stabilizer G s := by rw [union_sdiff_cancel_right]; simpa [← disjoint_iff_inter_eq_empty]\n  · calc\n      stabilizer G s\n        ≤ stabilizer G s ⊓ stabilizer G t := by simpa\n      _ ≤ stabilizer G (s ∪ t) := stabilizer_inf_stabilizer_le_stabilizer_union\n\n@[to_additive]\nlemma stabilizer_union_eq_right (hdisj : Disjoint s t) (hstab : stabilizer G t ≤ stabilizer G s)\n    (hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G s) :\n    stabilizer G (s ∪ t) = stabilizer G t := by\n  rw [union_comm, stabilizer_union_eq_left hdisj.symm hstab (union_comm .. ▸ hstab_union)]\n\nvariable {s : Set G}\n\nopen scoped RightActions in\n@[to_additive]\nlemma op_smul_set_stabilizer_subset (ha : a ∈ s) : (stabilizer G s : Set G) <• a ⊆ s :=\n  smul_set_subset_iff.2 fun b hb ↦ by rw [← hb]; exact smul_mem_smul_set ha\n\n@[to_additive]\nlemma stabilizer_subset_div_right (ha : a ∈ s) : ↑(stabilizer G s) ⊆ s / {a} := fun b hb ↦\n  ⟨_, by rwa [← smul_eq_mul, mem_stabilizer_set.1 hb], _, mem_singleton _, mul_div_cancel_right _ _⟩\n\n@[to_additive]\nlemma stabilizer_finite (hs₀ : s.Nonempty) (hs : s.Finite) : (stabilizer G s : Set G).Finite := by\n  obtain ⟨a, ha⟩ := hs₀\n  exact (hs.div <| finite_singleton _).subset <| stabilizer_subset_div_right ha\n\nend Group\n\nsection CommGroup\nvariable [CommGroup G] {s t : Set G} {a : G}\n\n@[to_additive]\nlemma smul_set_stabilizer_subset (ha : a ∈ s) : a • (stabilizer G s : Set G) ⊆ s := by\n  simpa using op_smul_set_stabilizer_subset ha\n\nend CommGroup\nend Set\n\nvariable [Group G] [Group H] [MulAction G α] {a : G}\n\n/-! ### Stabilizer of a subgroup -/\n\nsection Subgroup\n\n-- TODO: Is there a lemma that could unify the following three very similar lemmas?\n\n@[to_additive (attr := simp)]\nlemma stabilizer_subgroup (s : Subgroup G) : stabilizer G (s : Set G) = s := by\n  simp_rw [SetLike.ext_iff, mem_stabilizer_set]\n  refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_left ha⟩\n  simpa only [smul_eq_mul, SetLike.mem_coe, mul_one] using (h 1).2 s.one_mem\n\n@[to_additive (attr := simp)]\nlemma stabilizer_op_subgroup (s : Subgroup G) : stabilizer Gᵐᵒᵖ (s : Set G) = s.op := by\n  simp_rw [SetLike.ext_iff, mem_stabilizer_set]\n  simp only [smul_eq_mul_unop, SetLike.mem_coe, Subgroup.mem_op, «forall», unop_op]\n  refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_right ha⟩\n  simpa only [op_smul_eq_mul, SetLike.mem_coe, one_mul] using (h 1).2 s.one_mem\n\n@[to_additive (attr := simp)]\nlemma stabilizer_subgroup_op (s : Subgroup Gᵐᵒᵖ) : stabilizer G (s : Set Gᵐᵒᵖ) = s.unop := by\n  simp_rw [SetLike.ext_iff, mem_stabilizer_set]\n  refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_right ha⟩\n  have : 1 * MulOpposite.op a ∈ s := (h 1).2 s.one_mem\n  simpa only [op_smul_eq_mul, SetLike.mem_coe, one_mul] using! this\n\nend Subgroup\n\n/-! ### Stabilizer of a finset -/\n\nsection Finset\nvariable [DecidableEq α]\n\n@[to_additive (attr := simp, norm_cast)]\nlemma stabilizer_coe_finset (s : Finset α) : stabilizer G (s : Set α) = stabilizer G s := by\n  ext; simp [← Finset.coe_inj]\n\n@[to_additive (attr := simp)]\nlemma stabilizer_finset_empty : stabilizer G (∅ : Finset α) = ⊤ :=\n  Subgroup.coe_eq_univ.1 <| eq_univ_of_forall Finset.smul_finset_empty\n\n@[to_additive (attr := simp)]\nlemma stabilizer_finset_univ [Fintype α] : stabilizer G (Finset.univ : Finset α) = ⊤ := by\n  ext\n  simp\n\n@[to_additive (attr := simp)]\nlemma stabilizer_finset_singleton (b : α) : stabilizer G ({b} : Finset α) = stabilizer G b := by\n  ext; simp\n\n@[to_additive]\nlemma mem_stabilizer_finset {s : Finset α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s := by\n  simp_rw [← stabilizer_coe_finset, mem_stabilizer_set, Finset.mem_coe]\n\n@[to_additive]\nlemma mem_stabilizer_finset_iff_subset_smul_finset {s : Finset α} :\n    a ∈ stabilizer G s ↔ s ⊆ a • s := by\n  rw [mem_stabilizer_iff, Finset.subset_iff_eq_of_card_le (Finset.card_smul_finset _ _).le, eq_comm]\n\n@[to_additive]\nlemma mem_stabilizer_finset_iff_smul_finset_subset {s : Finset α} :\n    a ∈ stabilizer G s ↔ a • s ⊆ s := by\n  rw [mem_stabilizer_iff, Finset.subset_iff_eq_of_card_le (Finset.card_smul_finset _ _).ge]\n\n@[to_additive]\nlemma mem_stabilizer_finset' {s : Finset α} : a ∈ stabilizer G s ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ s := by\n  rw [← Subgroup.inv_mem_iff, mem_stabilizer_finset_iff_subset_smul_finset]\n  simp_rw [← Finset.mem_inv_smul_finset_iff, Finset.subset_iff]\n\nend Finset\n\n/-! ### Stabilizer of a finite set -/\n\nvariable {s : Set α}\n\n@[to_additive]\nlemma mem_stabilizer_set_iff_subset_smul_set {s : Set α} (hs : s.Finite) :\n    a ∈ stabilizer G s ↔ s ⊆ a • s := by\n  lift s to Finset α using hs\n  classical\n  rw [stabilizer_coe_finset, mem_stabilizer_finset_iff_subset_smul_finset, ← Finset.coe_smul_finset,\n    Finset.coe_subset]\n\n@[to_additive]\n\nTarget:\nlemma mem_stabilizer_set_iff_smul_set_subset {s : Set α} (hs : s.Finite) :\n    a ∈ stabilizer G s ↔ a • s ⊆ s :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Pointwise","family_id":"mem_stabilizer_set_iff_smul_set_subset","file_id":"mathlib/Mathlib/Algebra/Pointwise/Stabilizer.lean","sample_id":"5ad3f4c1074336dbfd17b6d6041c87d9308d36a6123c6ad702c726184bd0c5fe"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5a49187c184286dd02eeb6ca07b11443ba36cdee6d43f2c64edd8e661703bcce","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"262a69776ad7e9ea0f6337da04d3acff30a162c8be6e90db51ab03458c4bc192","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"394f27a561e1c227c4d6a3e91cad146754404323a259cba6efddc68715f8ebd8","source_sha256":"c0e3fa9405ddf4fdd99cd3aa6c9ff7258178f89496e31a48d0f8f121f656258a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [mem_Icc, insertNth_le_iff, le_insertNth_iff, and_assoc, @and_left_comm (x ≤ q₂ i)]","hard_negative":true,"metrics":{"chosen_tokens":22,"rejected_tokens":2,"token_jaccard":0.05,"token_length_ratio":0.090909},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"b3018b782d5d1976cf0542376bd5284262da5c3e89f6fbd44e33f097a49f9964","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.VecNotation\npublic import Mathlib.Logic.Equiv.Fin.Basic\npublic import Mathlib.Order.Fin.Basic\npublic import Mathlib.Order.PiLex\npublic import Mathlib.Order.Interval.Set.Defs\n\nNamespace:\nFin\n\nLocal context:\n/-\nCopyright (c) 2019 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes\n-/\n/-!\n# Order properties on tuples\n-/\n\n@[expose] public section\n\nassert_not_exists Monoid\n\nopen Function Set\n\nnamespace Fin\nvariable {m n : ℕ} {α : Fin (n + 1) → Type*} (x : α 0) (q : ∀ i, α i) (p : ∀ i : Fin n, α i.succ)\n  (i : Fin n) (y : α i.succ) (z : α 0)\n\nlemma pi_lex_lt_cons_cons {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ}\n    (s : ∀ {i : Fin n.succ}, α i → α i → Prop) :\n    Pi.Lex (· < ·) (@s) (Fin.cons x₀ x) (Fin.cons y₀ y) ↔\n      s x₀ y₀ ∨ x₀ = y₀ ∧ Pi.Lex (· < ·) (@fun i : Fin n ↦ @s i.succ) x y := by\n  simp_rw [Pi.Lex, Fin.exists_fin_succ, Fin.cons_succ, Fin.cons_zero, Fin.forall_iff_succ]\n  simp [and_assoc, exists_and_left]\n\nvariable [∀ i, Preorder (α i)]\n\nTarget:\nlemma insertNth_mem_Icc {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)}\n    {q₁ q₂ : ∀ j, α j} :\n    i.insertNth x p ∈ Icc q₁ q₂ ↔\n      x ∈ Icc (q₁ i) (q₂ i) ∧ p ∈ Icc (fun j ↦ q₁ (i.succAbove j)) fun j ↦ q₂ (i.succAbove j) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_394f27a561e1","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"15aa71346e634dec7a8f214ce48684ce23f853180708f069c88dda272cc8ec98","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Fin","family_id":"insertnth_mem_icc","file_id":"mathlib/Mathlib/Order/Fin/Tuple.lean","sample_id":"394f27a561e1c227c4d6a3e91cad146754404323a259cba6efddc68715f8ebd8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d4714964e668b31f0718a64704bfc3c462347f6aac0cd63c4a504fd6654a6944","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4f54315a32700ee5c1139f169b34823e5e3a9893b9b5fa8693e6b412a2e8b57f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bcfe71f4669f39454a991e49045a95b381eb505965239b851eaa93006f10ef03","source_sha256":"3ea4a54ee767d2f1814d8b90accb092fa9ec4a1fd88519664221a22cfd80fa82","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  subst s_card\n  simp [basis_apply]\n\nvariable (s : powersetCard I m) (t : powersetCard I n)","hard_negative":true,"metrics":{"chosen_tokens":22,"rejected_tokens":3,"token_jaccard":0.052632,"token_length_ratio":0.136364},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"b33fced4ee87b0cce33d82f8999a74500691a74fd6ef581ea5a39be643d94dd0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.ExteriorAlgebra.Grading\npublic import Mathlib.LinearAlgebra.ExteriorPower.Basis\n\nNamespace:\nExteriorAlgebra\n\nLocal context:\n/-\nCopyright (c) 2026 Daniel Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Morrison\n-/\n/-!\n# Basis for `ExteriorAlgebra`\n-/\n\n@[expose] public section\n\nnamespace ExteriorAlgebra\n\nopen Module Set Set.powersetCard exteriorPower\n\nvariable {R M : Type*} {m n : ℕ} {I : Type*} [LinearOrder I] [CommRing R]\n  [AddCommGroup M] [Module R M] (b : Module.Basis I R M)\n\n/-- The direct sum decomposition of the exterior algebra from the graded algebra structure. -/\ninstance : DirectSum.Decomposition (fun n ↦ ⋀[R]^n M) :=\n  GradedRing.toDecomposition (self := ExteriorAlgebra.gradedAlgebra R M)\n\n/-- If `b` is a basis of `M` (indexed by a linearly ordered type), the basis of the exterior\nalgebra of `M` formed by the `n`-fold exterior products of elements of `b` for each `n`. -/\nnoncomputable def _root_.Module.Basis.ExteriorAlgebra : Basis (Finset I) R (ExteriorAlgebra R M) :=\n  .reindex\n    ((DirectSum.Decomposition.isInternal (fun n => ⋀[R]^n M)).collectedBasis b.exteriorPower)\n    Set.powersetCard.prodEquiv\n\nlemma basis_apply (s : Finset I) :\n    b.ExteriorAlgebra s = ιMulti_family R s.card b (prodEquiv.symm s).2 := by\n  simp [Basis.ExteriorAlgebra]\n\nTarget:\nlemma basis_apply_ofCard {s : Finset I} (s_card : s.card = n) :\n    b.ExteriorAlgebra s = ιMulti_family R n b (ofCard s_card) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_bcfe71f4669f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"b918206dd12ac3d3424feb43ed65c3eb6b3495c71471b5a1957a553db184b17f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/ExteriorAlgebra","family_id":"basis_apply_ofcard","file_id":"mathlib/Mathlib/LinearAlgebra/ExteriorAlgebra/Basis.lean","sample_id":"bcfe71f4669f39454a991e49045a95b381eb505965239b851eaa93006f10ef03"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bc0a4d32839ad3481e8623af1efc75f8290857aa2e5f1191a4c4a6c87919cec5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b84e567048f9867d2afcbc469aa5345705444fa465f18ef54146f3abb4b505ef","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0e6ca74f069156e403ca37b6a55058b4c1f3d343ff3c45b25117782dc63111e3","source_sha256":"cd2a9f6a6ac4ffdabcb0fdd545e558d6cc8573a01d198e082fa0d6a20deaa548","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply fun s ↦ (residue_ne_zero_iff_isUnit s).mp\n  rw [map_aeval_eq_aeval_map (ψ := residue S) (φ := residue R) rfl,\n    ← derivative_map, minpoly_map_residue hadj]\n  exact (Algebra.IsSeparable.isSeparable _ _).aeval_derivative_ne_zero (minpoly.aeval _ _)","hard_negative":false,"metrics":{"chosen_tokens":53,"rejected_tokens":60,"token_jaccard":0.944444,"token_length_ratio":1.132075},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"b40d10cdca82b64d5ef018944718daf379b4b52cb46125b34be96a46b28244da","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.IsAdjoinRoot\npublic import Mathlib.RingTheory.LocalRing.Quotient\npublic import Mathlib.RingTheory.Smooth.Flat\npublic import Mathlib.RingTheory.Unramified.LocalRing\n\nNamespace:\nIsLocalRing\n\nLocal context:\n/-\nCopyright (c) 2026 University of Washington Math AI Lab. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bianca Viray, Bryan Boehnke, Grant Yang, George Peykanu, Tianshuo Wang\n-/\n\n/-!\n# Étale extensions of local rings\n\nWe prove that a finite étale extension of local rings is monogenic (generated by a single element),\nand that the derivative of the minimal polynomial evaluated at the generator is a unit.\nThese are parts 1 and 2 of Lemma 3.2 of [arXiv:2503.07846](https://arxiv.org/abs/2503.07846).\n\n## Main results\n\n* `IsLocalRing.exists_adjoin_eq_top`: a finite étale extension of local rings is generated by a\n  single element (Lemma 3.2, part 1).\n* `IsLocalRing.isUnit_aeval_derivative_minpoly_of_adjoin_eq_top`: if `R → S` is étale and\n  `R[β] = S`, then `f'(β)` is a unit, where `f = minpoly R β` (Lemma 3.2, part 2).\n\n## Key intermediate results\n\n* `IsLocalRing.adjoin_residue_eq_top_iff_adjoin_eq_top`: `β` generates `S` over `R` iff\n  `β mod m_S` generates `S/m_S` over `R/m_R`.\n* `IsLocalRing.finrank_eq_finrank_residueField`: for finite étale extensions of local rings,\n  `finrank R S = finrank (ResidueField R) (ResidueField S)`.\n* `IsLocalRing.minpoly_map_residue`: the minimal polynomial of `β` over `R` maps to the\n  minimal polynomial of `β mod m_S` over the residue field.\n\n## Future work\n\nThe following results from [arXiv:2503.07846](https://arxiv.org/abs/2503.07846) (formalized at [uw-math-ai/monogenic-extensions](https://github.com/uw-math-ai/monogenic-extensions)) are planned for\nfuture PRs:\n\n* **Converse**: If `S ≅ R[X]/(f)` with `f` monic and `f'(root)` a unit, then `R → S` is étale.\n* **Lemma 3.1** (partial étale case): If `R` and `S` are local integral domains with `R`\n  integrally closed, `S` a UFD, `R → S` finite and injective, and there exists a height-one\n  prime `q ⊆ S` such that `R/(q ∩ R) → S/q` is étale, then `S ≅ R[X]/(f)` for some monic `f`.\n\n## Tags\n\nétale, monogenic, local ring, minimal polynomial, residue field\n-/\n\n@[expose] public section\n\nnamespace IsLocalRing\n\nvariable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]\n    [IsLocalRing S] [IsLocalRing R] [Module.Finite R S] [FaithfulSMul R S]\n\nopen Polynomial IsLocalRing Algebra\n\n/-- When `β` generates `S` over `R`, the residue `β₀ = β mod m_S`\ngenerates `S/m_S` over `R/m_R`. -/\nlemma adjoin_residue_eq_top_iff_adjoin_eq_top [Algebra.FormallyUnramified R S] (β : S) :\n    Algebra.adjoin (ResidueField R) {residue S β} = ⊤ ↔ Algebra.adjoin R {β} = ⊤ := by\n  constructor\n  · intro hβ\n    refine eq_top_iff.mpr <| Submodule.le_of_le_smul_of_le_jacobson_bot\n      (Module.finite_def.mp inferInstance) (IsLocalRing.maximalIdeal_le_jacobson ⊥)\n      (?_ : ⊤ ≤ (adjoin R {β}).toSubmodule ⊔ maximalIdeal R • ⊤)\n    intro s _\n    rw [adjoin_singleton_eq_range_aeval, AlgHom.range_eq_top] at hβ\n    obtain ⟨p, hp⟩ := hβ (residue S s)\n    obtain ⟨q, rfl⟩ := Polynomial.map_surjective _ residue_surjective p\n    rw [Ideal.smul_top_eq_map]\n    refine Submodule.mem_sup.mpr ⟨aeval β q, ?_, s - aeval β q, ?_, by ring⟩\n    · rw [adjoin_singleton_eq_range_aeval]; exact ⟨q, rfl⟩\n    · rw [Algebra.FormallyUnramified.map_maximalIdeal, Submodule.restrictScalars_mem,\n        ← Ideal.Quotient.eq]\n      -- def eq abuse since IsLocalRing.residue needs to be an abbrev\n      rw [← map_aeval_eq_aeval_map (ψ := residue S) (φ := residue R) rfl] at hp\n      exact hp.symm\n  · intro hβ_gen\n    rw [Algebra.adjoin_singleton_eq_range_aeval, AlgHom.range_eq_top] at *\n    intro x\n    obtain ⟨s, rfl⟩ := residue_surjective (R := S) x\n    obtain ⟨p, rfl⟩ := hβ_gen s\n    exact ⟨p.map (residue R), by\n      rw [← map_aeval_eq_aeval_map (ψ := residue S) (φ := residue R) rfl p β]⟩\n\n/-- A finite étale extension of local rings is generated by a single element.\nThis is Lemma 3.2, part 1 of [arXiv:2503.07846](https://arxiv.org/abs/2503.07846).\nThe proof lifts a primitive element of the residue field extension via Nakayama's lemma. -/\ntheorem exists_adjoin_eq_top [Algebra.FormallyUnramified R S] :\n    ∃ β : S, Algebra.adjoin R {β} = ⊤ := by\n  obtain ⟨β₀, hβ₀⟩ := Field.exists_primitive_element (ResidueField R) (ResidueField S)\n  obtain ⟨β, hβ⟩ := residue_surjective (R := S) β₀\n  refine ⟨β, adjoin_residue_eq_top_iff_adjoin_eq_top β |>.mp ?_⟩\n  rw [hβ,\n    ← IntermediateField.adjoin_simple_toSubalgebra_of_isAlgebraic (IsAlgebraic.of_finite _ _),\n    hβ₀, IntermediateField.top_toSubalgebra]\n\n/-- For finite étale extensions of local rings,\n`finrank R S = finrank (ResidueField R) (ResidueField S)`. -/\nlemma finrank_eq_finrank_residueField [Algebra.Etale R S] :\n    Module.finrank R S =\n      Module.finrank (ResidueField R) (ResidueField S) := by\n  have : Module.Free R S := Module.free_of_flat_of_isLocalRing\n  have e := AddEquiv.toLinearEquiv (R := R ⧸ maximalIdeal R) (Ideal.quotEquivOfEq <|\n      Algebra.FormallyUnramified.map_maximalIdeal (R := R) (S := S)).toAddEquiv\n    ?_\n  · rw [← finrank_quotient_map (R := R) (S := S)]\n    exact e.finrank_eq -- again IsLocalRing.residue should be abbrev\n  · intro r x\n    obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective r\n    obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x\n    simp only [RingEquiv.toAddEquiv_eq_coe]; rfl\n\n/-- For a monogenic étale extension of local rings, the minimal polynomial of `β`\nmaps to the minimal polynomial of `β mod m_S` over the residue field. -/\nlemma minpoly_map_residue [Algebra.Etale R S]\n    {β : S} (hadj : Algebra.adjoin R {β} = ⊤) :\n    (minpoly R β).map (residue R) = minpoly (ResidueField R) (residue S β) := by\n  have h := minpoly.monic <| Algebra.IsIntegral.isIntegral (R := R) β\n  -- Both monic, same degree, divisibility ⟹ equal\n  refine eq_of_monic_of_dvd_of_natDegree_le\n    (minpoly.monic <| Algebra.IsIntegral.isIntegral <| residue S β)\n    (h.map _) (minpoly.dvd (ResidueField R) (residue S β) ?_) ?_\n  · rw [← map_aeval_eq_aeval_map (ψ := residue S) (φ := residue R) rfl]\n    simp\n  · have : Module.Free R S := Module.free_of_flat_of_isLocalRing\n    have hβ₀ := (adjoin_residue_eq_top_iff_adjoin_eq_top β).mpr hadj\n    rw [h.natDegree_map _,\n      ← (IsAdjoinRootMonic.mkOfAdjoinEqTop' hadj).finrank,\n      finrank_eq_finrank_residueField,\n      (IsAdjoinRootMonic.mkOfAdjoinEqTop' hβ₀).finrank]\n\n/-- If `R → S` is étale and `R[β] = S`, then `f'(β)` is a unit in `S`,\nwhere `f = minpoly R β`. The proof reduces to separability of the\nresidue field extension via `minpoly_map_residue`. -/\n\nTarget:\nlemma isUnit_aeval_derivative_minpoly_of_adjoin_eq_top\n    [Algebra.Etale R S] {β : S}\n    (hadj : Algebra.adjoin R {β} = ⊤) :\n    IsUnit (aeval β (minpoly R β).derivative) :=\n\nProof body:\n","rejected":"```lean\nby\n  apply fun s ↦ (residue_ne_zero_iff_isUnit s).mp\n  rw [map_aeval_eq_aeval_map (ψ := residue S) (φ := residue R) rfl,\n    ← derivative_map, minpoly_map_residue hadj]\n  exact (Algebra.IsSeparable.isSeparable _ _).aeval_derivative_ne_zero (minpoly.aeval _ _)\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/LocalRing","family_id":"isunit_aeval_derivative_minpoly_of_adjoin_eq_top","file_id":"mathlib/Mathlib/RingTheory/LocalRing/Etale.lean","sample_id":"0e6ca74f069156e403ca37b6a55058b4c1f3d343ff3c45b25117782dc63111e3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"afd4fe6929e62277a458a32fdf80154f965674b8199603407ac172cf56934b1a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7b702b3f54a49b24809e03e0a017a431793ec768bf4e679fe9f430ed6393e5c4","source_sha256":"704b186061db0b8a3733d8cc4c73447096dc6713be40f982a89e46b27c980656","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine\n    le_trans ?_ <| ((Subobject.pullback (π(π₁, prod.map F.f (𝟙 _) ≫ G.F))).monotone F.le).trans <|\n    le_trans ?_ <| ((Subobject.pullback (prod.map F.f (𝟙 Z.tgt))).monotone G.le).trans ?_\n    <;> simp [← Subobject.pullback_comp]","hard_negative":false,"metrics":{"chosen_tokens":90,"rejected_tokens":3,"token_jaccard":0.027778,"token_length_ratio":0.033333},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"b4385880b93569abe7193d668dcaab35826fcd95d3f5e94b230c4f558fc8ab8c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Subobject.Basic\n\nNamespace:\nCategoryTheory.Dial\n\nLocal context:\n/-\nCopyright (c) 2024 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Dialectica category\n\nWe define the category `Dial` of the Dialectica interpretation, after [dialectica1989].\n\n## Background\n\nDialectica categories are important models of linear type theory. They satisfy most of the\ndistinctions that linear logic was meant to introduce and many models do not satisfy, like the\nindependence of constants. Many linear type theories are being used at the\nmoment--[nLab] describes some of them: for quantum systems, for effects in programming, for linear\ndependent types. In particular, dialectica categories are connected to polynomial functors, being a\nslightly more sophisticated version of polynomial types, as discussed, for instance, in Moss and\nvon Glehn's [*Dialectica models of type theory*]. As such they are related to the polynomial\nconstructions being [developed][Poly] by Awodey, Riehl, and Hazratpour. For the non-dependent\nversion developed here several applications are known to Petri Nets, small cardinals\nin Set Theory, state in imperative programming, and others, see [Dialectica Categories].\n\n## References\n\n* [Valeria de Paiva, The Dialectica Categories.][dialectica1989]\n  ([pdf](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-213.pdf))\n\n[nLab]: https://ncatlab.org/nlab/show/linear+type+theory\n[*Dialectica models of type theory*]: https://arxiv.org/abs/2105.00283\n[Poly]: https://github.com/sinhp/Poly\n[Dialectica Categories]: https://github.com/vcvpaiva/DialecticaCategories\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nnamespace CategoryTheory\n\nopen Limits\n\nuniverse v u\nvariable {C : Type u} [Category.{v} C] [HasFiniteProducts C] [HasPullbacks C]\n\nvariable (C) in\n/-- The Dialectica category. An object of the category is a triple `⟨U, X, α ⊆ U × X⟩`,\nand a morphism from `⟨U, X, α⟩` to `⟨V, Y, β⟩` is a pair `(f : U ⟶ V, F : U ⨯ Y ⟶ X)` such that\n`{(u,y) | α(u, F(u, y))} ⊆ {(u,y) | β(f(u), y)}`. The subset `α` is actually encoded as an element\nof `Subobject (U × X)`, and the above inequality is expressed using pullbacks. -/\nstructure Dial where\n  /-- The source object -/\n  src : C\n  /-- The target object -/\n  tgt : C\n  /-- A subobject of `src ⨯ tgt`, interpreted as a relation -/\n  rel : Subobject (src ⨯ tgt)\n\nnamespace Dial\n\nlocal notation \"π₁\" => prod.fst\nlocal notation \"π₂\" => prod.snd\nlocal notation \"π(\" a \", \" b \")\" => prod.lift a b\n\n/-- A morphism in the `Dial C` category from `⟨U, X, α⟩` to `⟨V, Y, β⟩` is a pair\n`(f : U ⟶ V, F : U ⨯ Y ⟶ X)` such that `{(u,y) | α(u, F(u, y))} ≤ {(u,y) | β(f(u), y)}`. -/\n@[ext] structure Hom (X Y : Dial C) where\n  /-- Maps the sources -/\n  f : X.src ⟶ Y.src\n  /-- Maps the targets (contravariantly) -/\n  F : X.src ⨯ Y.tgt ⟶ X.tgt\n  /-- This says `{(u, y) | α(u, F(u, y))} ⊆ {(u, y) | β(f(u), y)}` using subobject pullbacks -/\n  le :\n    (Subobject.pullback π(π₁, F)).obj X.rel ≤\n    (Subobject.pullback (prod.map f (𝟙 _))).obj Y.rel\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem comp_le_lemma {X Y Z : Dial C} (F : Dial.Hom X Y) (G : Dial.Hom Y Z) :\n    (Subobject.pullback π(π₁, π(π₁, prod.map F.f (𝟙 _) ≫ G.F) ≫ F.F)).obj X.rel ≤\n    (Subobject.pullback (prod.map (F.f ≫ G.f) (𝟙 Z.tgt))).obj Z.rel :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Dialectica","family_id":"comp_le_lemma","file_id":"mathlib/Mathlib/CategoryTheory/Dialectica/Basic.lean","sample_id":"7b702b3f54a49b24809e03e0a017a431793ec768bf4e679fe9f430ed6393e5c4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a76192e154cb0c91a316e618a2d1566a7760c095bf189498c16a175bdc2813bb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"84988f78a63dd6457f39f9a79617a3487d4fd48238e4f573cd4146da7395b4a6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e4f8810a3c97776fcd6674079eb5c88b07ed5c16b257876471689d8a0da4bfcc","source_sha256":"57eda5dc13e802c426fc6aca07052f2fd43d6d64cc1f0f72ed78d6e2a265c4fe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  generalize h' : [] = L'\n  intro h\n  rcases h with - | ⟨L₁, L₂⟩\n  simp at h'","hard_negative":false,"metrics":{"chosen_tokens":25,"rejected_tokens":29,"token_jaccard":0.846154,"token_length_ratio":1.16},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"b4b40aff18267f3f1d44d3d36c4be6a1f0bb62d7ecd52295691185c91c44ff46","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Pi.Basic\npublic import Mathlib.Algebra.Group.Subgroup.Ker\npublic import Mathlib.Data.List.Chain\npublic import Mathlib.Algebra.Group.Int.Defs\npublic import Mathlib.Algebra.BigOperators.Group.List.Basic\npublic import Mathlib.Algebra.Group.Nat.Defs\npublic import Mathlib.Tactic.CrossRefAttribute\n\nNamespace:\nFreeGroup.Red\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n/-!\n# Free groups\n\nThis file defines free groups over a type. Furthermore, it is shown that the free group construction\nis an instance of a monad. For the result that `FreeGroup` is the left adjoint to the forgetful\nfunctor from groups to types, see `Mathlib/Algebra/Category/Grp/Adjunctions.lean`.\n\n## Main definitions\n\n* `FreeGroup`/`FreeAddGroup`: the free group (resp. free additive group) associated to a type\n  `α` defined as the words over `a : α × Bool` modulo the relation `a * x * x⁻¹ * b = a * b`.\n* `FreeGroup.mk`/`FreeAddGroup.mk`: the canonical quotient map `List (α × Bool) → FreeGroup α`.\n* `FreeGroup.of`/`FreeAddGroup.of`: the canonical injection `α → FreeGroup α`.\n* `FreeGroup.lift f`/`FreeAddGroup.lift`: the canonical group homomorphism `FreeGroup α →* G`\n  given a group `G` and a function `f : α → G`.\n\n## Main statements\n\n* `FreeGroup.Red.church_rosser`/`FreeAddGroup.Red.church_rosser`: The Church-Rosser theorem for word\n  reduction (also known as Newman's diamond lemma).\n* `FreeGroup.freeGroupUnitEquivInt`: The free group over the one-point type\n  is isomorphic to the integers.\n* The free group construction is an instance of a monad.\n\n## Implementation details\n\nFirst we introduce the one step reduction relation `FreeGroup.Red.Step`:\n`w * x * x⁻¹ * v   ~>   w * v`, its reflexive transitive closure `FreeGroup.Red.trans`\nand prove that its join is an equivalence relation. Then we introduce `FreeGroup α` as a quotient\nover `FreeGroup.Red.Step`.\n\nFor the additive version we introduce the same relation under a different name so that we can\ndistinguish the quotient types more easily.\n\n\n## Tags\n\nfree group, Newman's diamond lemma, Church-Rosser theorem\n-/\n\n@[expose] public section\n\nopen Relation\nopen scoped List\n\nuniverse u v w\n\nvariable {α : Type u}\n\nattribute [local simp] List.append_eq_has_append\n\n/- Ensure that `@[to_additive]` uses the right namespace before the definition of `FreeGroup`. -/\ninsert_to_additive_translation FreeGroup FreeAddGroup\n\n/-- Reduction step for the additive free group relation: `w + x + (-x) + v ~> w + v` -/\ninductive FreeAddGroup.Red.Step : List (α × Bool) → List (α × Bool) → Prop\n  | not {L₁ L₂ x b} : FreeAddGroup.Red.Step (L₁ ++ (x, b) :: (x, not b) :: L₂) (L₁ ++ L₂)\n\nattribute [simp] FreeAddGroup.Red.Step.not\n\n/-- Reduction step for the multiplicative free group relation: `w * x * x⁻¹ * v ~> w * v` -/\n@[to_additive]\ninductive FreeGroup.Red.Step : List (α × Bool) → List (α × Bool) → Prop\n  | not {L₁ L₂ x b} : FreeGroup.Red.Step (L₁ ++ (x, b) :: (x, not b) :: L₂) (L₁ ++ L₂)\n\nattribute [simp] FreeGroup.Red.Step.not\n\nnamespace FreeGroup\n\nvariable {L L₁ L₂ L₃ L₄ : List (α × Bool)}\n\n/-- Reflexive-transitive closure of `Red.Step` -/\n@[to_additive /-- Reflexive-transitive closure of `Red.Step` -/]\ndef Red : List (α × Bool) → List (α × Bool) → Prop :=\n  ReflTransGen Red.Step\n\n@[to_additive (attr := refl)]\ntheorem Red.refl : Red L L :=\n  ReflTransGen.refl\n\n@[to_additive (attr := trans)]\ntheorem Red.trans : Red L₁ L₂ → Red L₂ L₃ → Red L₁ L₃ :=\n  ReflTransGen.trans\n\nnamespace Red\n\n/-- Predicate asserting that the word `w₁` can be reduced to `w₂` in one step, i.e. there are words\n`w₃ w₄` and letter `x` such that `w₁ = w₃xx⁻¹w₄` and `w₂ = w₃w₄` -/\n@[to_additive /-- Predicate asserting that the word `w₁` can be reduced to `w₂` in one step, i.e.\nthere are words `w₃ w₄` and letter `x` such that `w₁ = w₃ + x + (-x) + w₄` and `w₂ = w₃w₄` -/]\ntheorem Step.length : ∀ {L₁ L₂ : List (α × Bool)}, Step L₁ L₂ → L₂.length + 2 = L₁.length\n  | _, _, @Red.Step.not _ L1 L2 x b => by rw [List.length_append, List.length_append]; rfl\n\n@[to_additive (attr := simp)]\ntheorem Step.not_rev {x b} : Step (L₁ ++ (x, !b) :: (x, b) :: L₂) (L₁ ++ L₂) := by\n  cases b <;> exact Step.not\n\n@[to_additive (attr := simp)]\ntheorem Step.cons_not {x b} : Red.Step ((x, b) :: (x, !b) :: L) L :=\n  @Step.not _ [] _ _ _\n\n@[to_additive (attr := simp)]\ntheorem Step.cons_not_rev {x b} : Red.Step ((x, !b) :: (x, b) :: L) L :=\n  @Red.Step.not_rev _ [] _ _ _\n\n@[to_additive]\ntheorem Step.append_left : ∀ {L₁ L₂ L₃ : List (α × Bool)}, Step L₂ L₃ → Step (L₁ ++ L₂) (L₁ ++ L₃)\n  | _, _, _, Red.Step.not => by rw [← List.append_assoc, ← List.append_assoc]; constructor\n\n@[to_additive]\ntheorem Step.cons {x} (H : Red.Step L₁ L₂) : Red.Step (x :: L₁) (x :: L₂) :=\n  @Step.append_left _ [x] _ _ H\n\n@[to_additive]\n\nTarget:\ntheorem Step.append_right : ∀ {L₁ L₂ L₃ : List (α × Bool)}, Step L₁ L₂ → Step (L₁ ++ L₃) (L₂ ++ L₃)\n  | _, _, _, Red.Step.not => by simp\n\n@[to_additive]\ntheorem not_step_nil : ¬Step [] L :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  generalize h' : [] = L'\n  intro h\n  rcases h with - | ⟨L₁, L₂⟩\n  simp at h'","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/FreeGroup","family_id":"step","file_id":"mathlib/Mathlib/GroupTheory/FreeGroup/Basic.lean","sample_id":"e4f8810a3c97776fcd6674079eb5c88b07ed5c16b257876471689d8a0da4bfcc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0378e91a096e555db6c2dcff0f9d43073fa97735128ddc7daf75d45b02b151ac","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f6367fab7503d969f0bae44dba3d63ca6fddd75a99ad5b0e2f65909eedb272b7","source_sha256":"2dd9b3d55f05fb237e2d926d68cfafdde7ae626a66f9d4e360c17c788bc0eada","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply le_antisymm\n  · refine iSup₂_le fun i hi y hy => ⟨DFinsupp.single i ⟨y, hy⟩, ?_⟩\n    rw [AddMonoidHom.comp_apply, filterAddMonoidHom_apply, filter_single_pos _ _ hi]\n    exact sumAddHom_single _ _ _\n  · rintro x ⟨v, rfl⟩\n    refine dfinsuppSumAddHom_mem _ _ _ fun i _ => ?_\n    refine AddSubmonoid.mem_iSup_of_mem i ?_\n    by_cases hp : p i\n    · simp [hp]\n    · simp [hp]","hard_negative":false,"metrics":{"chosen_tokens":87,"rejected_tokens":2,"token_jaccard":0.022727,"token_length_ratio":0.022989},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"b4f42aa2a088b0bf5f539e1b4a4e51526b5286faff376ae83408575ada8a9552","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Submonoid.BigOperators\npublic import Mathlib.Algebra.Group.Submonoid.Membership\npublic import Mathlib.Data.DFinsupp.BigOperators\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Kenny Lau\n-/\n/-!\n# `DFinsupp` and submonoids\n\nThis file mainly concerns the interaction between submonoids and products/sums of `DFinsupp`s.\n\n## Main results\n\n* `AddSubmonoid.mem_iSup_iff_exists_dfinsupp`: elements of the supremum of additive commutative\n  monoids can be given by taking finite sums of elements of each monoid.\n* `AddSubmonoid.mem_bsupr_iff_exists_dfinsupp`: elements of the supremum of additive commutative\n  monoids can be given by taking finite sums of elements of each monoid.\n-/\n\npublic section\n\n\nuniverse u u₁ u₂ v v₁ v₂ v₃ w x y l\n\nvariable {ι : Type u} {γ : Type w} {β : ι → Type v} {β₁ : ι → Type v₁} {β₂ : ι → Type v₂}\n\nopen DFinsupp\n\nvariable [DecidableEq ι]\n\n@[to_additive]\ntheorem dfinsuppProd_mem [∀ i, Zero (β i)] [∀ (i) (x : β i), Decidable (x ≠ 0)]\n    [CommMonoid γ] {S : Type*} [SetLike S γ] [SubmonoidClass S γ]\n    (s : S) (f : Π₀ i, β i) (g : ∀ i, β i → γ)\n    (h : ∀ c, f c ≠ 0 → g c (f c) ∈ s) : f.prod g ∈ s :=\n  prod_mem fun _ hi => h _ <| mem_support_iff.1 hi\n\ntheorem dfinsuppSumAddHom_mem [∀ i, AddZeroClass (β i)] [AddCommMonoid γ] {S : Type*}\n    [SetLike S γ] [AddSubmonoidClass S γ] (s : S) (f : Π₀ i, β i) (g : ∀ i, β i →+ γ)\n    (h : ∀ c, f c ≠ 0 → g c (f c) ∈ s) : DFinsupp.sumAddHom g f ∈ s := by\n  classical\n    rw [DFinsupp.sumAddHom_apply]\n    exact dfinsuppSum_mem s f (g ·) h\n\n/-- The supremum of a family of commutative additive submonoids is equal to the range of\n`DFinsupp.sumAddHom`; that is, every element in the `iSup` can be produced from taking a finite\nnumber of non-zero elements of `S i`, coercing them to `γ`, and summing them. -/\ntheorem AddSubmonoid.iSup_eq_mrange_dfinsuppSumAddHom\n    [AddCommMonoid γ] (S : ι → AddSubmonoid γ) :\n    iSup S = AddMonoidHom.mrange (DFinsupp.sumAddHom fun i => (S i).subtype) := by\n  apply le_antisymm\n  · apply iSup_le _\n    intro i y hy\n    exact ⟨DFinsupp.single i ⟨y, hy⟩, DFinsupp.sumAddHom_single _ _ _⟩\n  · rintro x ⟨v, rfl⟩\n    exact dfinsuppSumAddHom_mem _ v _ fun i _ => (le_iSup S i : S i ≤ _) (v i).prop\n\n/-- The bounded supremum of a family of commutative additive submonoids is equal to the range of\n`DFinsupp.sumAddHom` composed with `DFinsupp.filterAddMonoidHom`; that is, every element in the\nbounded `iSup` can be produced from taking a finite number of non-zero elements from the `S i` that\nsatisfy `p i`, coercing them to `γ`, and summing them. -/\n\nTarget:\ntheorem AddSubmonoid.bsupr_eq_mrange_dfinsuppSumAddHom (p : ι → Prop) [DecidablePred p]\n    [AddCommMonoid γ] (S : ι → AddSubmonoid γ) :\n    ⨆ (i) (_ : p i), S i =\n      AddMonoidHom.mrange ((sumAddHom fun i => (S i).subtype).comp (filterAddMonoidHom _ p)) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/DFinsupp","family_id":"addsubmonoid","file_id":"mathlib/Mathlib/Data/DFinsupp/Submonoid.lean","sample_id":"f6367fab7503d969f0bae44dba3d63ca6fddd75a99ad5b0e2f65909eedb272b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7d2f77b8c1bb155392db6b9e50f050fefd652d8cabc59188758f6cc33ac80619","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"682d48ad8b9e90576e74171f482cee42e861a91ce0c174d5f3522cee18da37e1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bb469c3613f92d8cabf0255e6c012999e3bdcfb6d61f1c8b6b5d3a6dc8c383d5","source_sha256":"3ee6cc92b0085948fb0c591bbd98433993abf27d506b4cc259fc13467d3e2a06","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by convert! aeval_algebraMap_apply A x p","hard_negative":true,"metrics":{"chosen_tokens":7,"rejected_tokens":5,"token_jaccard":0.090909,"token_length_ratio":0.714286},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"b4f5286e1bbd5caa5a11829f2cd53e1f0836e087eb59ae8a07171a56c061f3dd","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Subalgebra.Tower\npublic import Mathlib.Algebra.MvPolynomial.Eval\n\nNamespace:\nSubalgebra\n\nLocal context:\n/-\nCopyright (c) 2022 Yuyang Zhao. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yuyang Zhao\n-/\n/-!\n# Algebra towers for multivariate polynomial\n\nThis file proves some basic results about the algebra tower structure for the type\n`MvPolynomial σ R`.\n\nThis structure itself is provided elsewhere as `MvPolynomial.isScalarTower`\n\nWhen you update this file, you can also try to make a corresponding update in\n`RingTheory.Polynomial.Tower`.\n-/\n\npublic section\n\n\nvariable (R A B : Type*) {σ : Type*}\n\nnamespace MvPolynomial\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\nvariable [Algebra R A] [Algebra A B] [Algebra R B]\nvariable [IsScalarTower R A B]\nvariable {R B}\n\ntheorem aeval_map_algebraMap (x : σ → B) (p : MvPolynomial σ R) :\n    aeval x (map (algebraMap R A) p) = aeval x p := by\n  rw [aeval_def, aeval_def, eval₂_map, IsScalarTower.algebraMap_eq R A B]\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\nvariable [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B]\nvariable {R A}\n\ntheorem aeval_algebraMap_apply (x : σ → A) (p : MvPolynomial σ R) :\n    aeval (algebraMap A B ∘ x) p = algebraMap A B (MvPolynomial.aeval x p) := by\n  rw [aeval_def, aeval_def, ← coe_eval₂Hom, ← coe_eval₂Hom, map_eval₂Hom, ←\n    IsScalarTower.algebraMap_eq, Function.comp_def]\n\n@[simp]\nlemma aeval_C_comp_left {ι : Type*} (f : σ → A) (p : MvPolynomial σ R) :\n    aeval (C (σ := ι) ∘ f) p = C (aeval f p) :=\n  aeval_algebraMap_apply ..\n\nlemma aeval_algebraMap_eq_zero_iff [IsDomain A] [Module.IsTorsionFree A B] [Nontrivial B]\n    (x : σ → A) (p : MvPolynomial σ R) : aeval (algebraMap A B ∘ x) p = 0 ↔ aeval x p = 0 := by\n  rw [aeval_algebraMap_apply, Algebra.algebraMap_eq_smul_one, smul_eq_zero,\n    iff_false_intro (one_ne_zero' B), or_false]\n\ntheorem aeval_algebraMap_eq_zero_iff_of_injective {x : σ → A} {p : MvPolynomial σ R}\n    (h : Function.Injective (algebraMap A B)) :\n    aeval (algebraMap A B ∘ x) p = 0 ↔ aeval x p = 0 := by\n  rw [aeval_algebraMap_apply, ← (algebraMap A B).map_zero, h.eq_iff]\n\nend CommSemiring\n\nend MvPolynomial\n\nnamespace Subalgebra\n\nopen MvPolynomial\n\nsection CommSemiring\n\nvariable {R A} [CommSemiring R] [CommSemiring A] [Algebra R A]\n\n@[simp]\n\nTarget:\ntheorem mvPolynomial_aeval_coe (S : Subalgebra R A) (x : σ → S) (p : MvPolynomial σ R) :\n    aeval (fun i => (x i : A)) p = aeval x p :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_bb469c3613f9","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c873ee2aca5ae636fdca5c53d085bd704c25b4cd75dc9004e3f1047a9a94e084","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/MvPolynomial","family_id":"mvpolynomial_aeval_coe","file_id":"mathlib/Mathlib/RingTheory/MvPolynomial/Tower.lean","sample_id":"bb469c3613f92d8cabf0255e6c012999e3bdcfb6d61f1c8b6b5d3a6dc8c383d5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dca549e925c3a05e8b8bc1baae8e33cdf145c6f1d35ee1dc1c15d213d724ea07","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e912fa5d231de1681e276afa664abc4b41eaddb661771201b381c52309cdb1ea","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6aa245c658f8d36bf9da386e1af0245093910d695a91f4bad772966128f1d7d5","source_sha256":"5a992f718439ad7f2b978ed86a7b3a95db162d52a4ab577079eb77f016af349f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by cat_disch) : f = g :=\n  Hom.ext w","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":2,"token_jaccard":0.076923,"token_length_ratio":0.166667},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"b50f3b67cdc1d8179ac9a024c59b10168a6dfd0f45e9933589d3ce251054c185","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Shapes.IsTerminal\npublic import Mathlib.CategoryTheory.Functor.EpiMono\n\nNamespace:\nCategoryTheory.Endofunctor.Coalgebra\n\nLocal context:\n/-\nCopyright (c) 2022 Joseph Hua. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison, Bhavik Mehta, Johan Commelin, Reid Barton, Robert Y. Lewis, Joseph Hua\n-/\n/-!\n\n# Algebras of endofunctors\n\nThis file defines (co)algebras of an endofunctor, and provides the category instance for them.\nIt also defines the forgetful functor from the category of (co)algebras. It is shown that the\nstructure map of the initial algebra of an endofunctor is an isomorphism. Furthermore, it is shown\nthat for an adjunction `F ⊣ G` the category of algebras over `F` is equivalent to the category of\ncoalgebras over `G`.\n\n## TODO\n\n* Prove that if the countable infinite product over the powers of the endofunctor exists, then\n  algebras over the endofunctor coincide with algebras over the free monad on the endofunctor.\n-/\n\n@[expose] public section\n\n\nuniverse v u\n\nnamespace CategoryTheory\n\nnamespace Endofunctor\n\nvariable {C : Type u} [Category.{v} C]\n\n/-- An algebra of an endofunctor; `str` stands for \"structure morphism\" -/\nstructure Algebra (F : C ⥤ C) where\n  /-- carrier of the algebra -/\n  a : C\n  /-- structure morphism of the algebra -/\n  str : F.obj a ⟶ a\n\ninstance [Inhabited C] : Inhabited (Algebra (𝟭 C)) :=\n  ⟨⟨default, 𝟙 _⟩⟩\n\nnamespace Algebra\n\nvariable {F : C ⥤ C} (A : Algebra F) {A₀ A₁ A₂ : Algebra F}\n\n/-\n```\n        str\n   F A₀ -----> A₀\n    |          |\nF f |          | f\n    V          V\n   F A₁ -----> A₁\n        str\n```\n-/\n/-- A morphism between algebras of endofunctor `F` -/\n@[ext]\nstructure Hom (A₀ A₁ : Algebra F) where\n  /-- underlying morphism between the carriers -/\n  f : A₀.1 ⟶ A₁.1\n  /-- compatibility condition -/\n  h : F.map f ≫ A₁.str = A₀.str ≫ f := by cat_disch\n\nattribute [reassoc (attr := simp)] Hom.h\n\nnamespace Hom\n\n/-- The identity morphism of an algebra of endofunctor `F` -/\ndef id : Hom A A where f := 𝟙 _\n\ninstance : Inhabited (Hom A A) :=\n  ⟨{ f := 𝟙 _ }⟩\n\n/-- The composition of morphisms between algebras of endofunctor `F` -/\ndef comp (f : Hom A₀ A₁) (g : Hom A₁ A₂) : Hom A₀ A₂ where f := f.1 ≫ g.1\n\nend Hom\n\ninstance (F : C ⥤ C) : CategoryStruct (Algebra F) where\n  Hom := Hom\n  id := Hom.id\n  comp := @Hom.comp _ _ _\n\n@[ext]\nlemma ext {A B : Algebra F} {f g : A ⟶ B} (w : f.f = g.f := by cat_disch) : f = g :=\n  Hom.ext w\n\n@[simp]\ntheorem id_eq_id : Algebra.Hom.id A = 𝟙 A :=\n  rfl\n\n@[simp]\ntheorem id_f : (𝟙 _ : A ⟶ A).1 = 𝟙 A.1 :=\n  rfl\n\nvariable (f : A₀ ⟶ A₁) (g : A₁ ⟶ A₂)\n\n@[simp]\ntheorem comp_eq_comp : Algebra.Hom.comp f g = f ≫ g :=\n  rfl\n\n@[simp]\ntheorem comp_f : (f ≫ g).1 = f.1 ≫ g.1 :=\n  rfl\n\n/-- Algebras of an endofunctor `F` form a category -/\ninstance (F : C ⥤ C) : Category (Algebra F) := { }\n\n/-- To construct an isomorphism of algebras, it suffices to give an isomorphism of the As which\ncommutes with the structure morphisms.\n-/\n@[simps!]\ndef isoMk (h : A₀.1 ≅ A₁.1) (w : F.map h.hom ≫ A₁.str = A₀.str ≫ h.hom := by cat_disch) :\n    A₀ ≅ A₁ where\n  hom := { f := h.hom }\n  inv :=\n    { f := h.inv\n      h := by\n        rw [h.eq_comp_inv, Category.assoc, ← w, ← Functor.map_comp_assoc]\n        simp }\n\n/-- The forgetful functor from the category of algebras, forgetting the algebraic structure. -/\n@[simps]\ndef forget (F : C ⥤ C) : Algebra F ⥤ C where\n  obj A := A.1\n  map := Hom.f\n\n/-- An algebra morphism with an underlying isomorphism hom in `C` is an algebra isomorphism. -/\ntheorem iso_of_iso (f : A₀ ⟶ A₁) [IsIso f.1] : IsIso f :=\n  ⟨⟨{ f := inv f.1\n      h := by simp }, by cat_disch, by cat_disch⟩⟩\n\ninstance forget_reflects_iso : (forget F).ReflectsIsomorphisms where reflects := iso_of_iso\n\ninstance forget_faithful : (forget F).Faithful := { }\n\n/-- An algebra morphism with an underlying epimorphism hom in `C` is an algebra epimorphism. -/\ntheorem epi_of_epi {X Y : Algebra F} (f : X ⟶ Y) [h : Epi f.1] : Epi f :=\n  (forget F).epi_of_epi_map h\n\n/-- An algebra morphism with an underlying monomorphism hom in `C` is an algebra monomorphism. -/\ntheorem mono_of_mono {X Y : Algebra F} (f : X ⟶ Y) [h : Mono f.1] : Mono f :=\n  (forget F).mono_of_mono_map h\n\n/-- From a natural transformation `α : G → F` we get a functor from\nalgebras of `F` to algebras of `G`.\n-/\n@[simps]\ndef functorOfNatTrans {F G : C ⥤ C} (α : G ⟶ F) : Algebra F ⥤ Algebra G where\n  obj A :=\n    { a := A.1\n      str := α.app _ ≫ A.str }\n  map f := { f := f.1 }\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The identity transformation induces the identity endofunctor on the category of algebras. -/\n@[simps!]\ndef functorOfNatTransId : functorOfNatTrans (𝟙 F) ≅ 𝟭 _ :=\n  NatIso.ofComponents fun X => isoMk (Iso.refl _)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- A composition of natural transformations gives the composition of corresponding functors. -/\n@[simps!]\ndef functorOfNatTransComp {F₀ F₁ F₂ : C ⥤ C} (α : F₀ ⟶ F₁) (β : F₁ ⟶ F₂) :\n    functorOfNatTrans (α ≫ β) ≅ functorOfNatTrans β ⋙ functorOfNatTrans α :=\n  NatIso.ofComponents fun X => isoMk (Iso.refl _)\n\n/--\nIf `α` and `β` are two equal natural transformations, then the functors of algebras induced by them\nare isomorphic.\nWe define it like this as opposed to using `eq_to_iso` so that the components are nicer to prove\nlemmas about.\n-/\n@[simps!]\ndef functorOfNatTransEq {F G : C ⥤ C} {α β : F ⟶ G} (h : α = β) :\n    functorOfNatTrans α ≅ functorOfNatTrans β :=\n  NatIso.ofComponents fun X => isoMk (Iso.refl _)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- Naturally isomorphic endofunctors give equivalent categories of algebras.\nFurthermore, they are equivalent as categories over `C`, that is,\nwe have `equiv_of_nat_iso h ⋙ forget = forget`.\n-/\n@[simps]\ndef equivOfNatIso {F G : C ⥤ C} (α : F ≅ G) : Algebra F ≌ Algebra G where\n  functor := functorOfNatTrans α.inv\n  inverse := functorOfNatTrans α.hom\n  unitIso := functorOfNatTransId.symm ≪≫ functorOfNatTransEq (by simp) ≪≫ functorOfNatTransComp _ _\n  counitIso :=\n    (functorOfNatTransComp _ _).symm ≪≫ functorOfNatTransEq (by simp) ≪≫ functorOfNatTransId\n\nnamespace Initial\n\nvariable {A : Algebra F} (h : Limits.IsInitial A)\n/-- The inverse of the structure map of an initial algebra -/\n@[simp]\ndef strInv : A.1 ⟶ F.obj A.1 :=\n  (h.to ⟨F.obj A.a, F.map A.str⟩).f\n\ntheorem left_inv' :\n    ⟨strInv h ≫ A.str, by rw [← Category.assoc, F.map_comp, strInv, ← Hom.h]⟩ = 𝟙 A :=\n  Limits.IsInitial.hom_ext h _ (𝟙 A)\n\ntheorem left_inv : strInv h ≫ A.str = 𝟙 _ :=\n  congr_arg Hom.f (left_inv' h)\n\ntheorem right_inv : A.str ≫ strInv h = 𝟙 _ := by\n  rw [strInv, ← (h.to ⟨F.obj A.1, F.map A.str⟩).h, ← F.map_id, ← F.map_comp]\n  congr\n  exact left_inv h\n\n/-- The structure map of the initial algebra is an isomorphism,\nhence endofunctors preserve their initial algebras\n-/\ntheorem str_isIso (h : Limits.IsInitial A) : IsIso A.str :=\n  { out := ⟨strInv h, right_inv _, left_inv _⟩ }\n\nend Initial\n\nend Algebra\n\n/-- A coalgebra of an endofunctor; `str` stands for \"structure morphism\" -/\nstructure Coalgebra (F : C ⥤ C) where\n  /-- carrier of the coalgebra -/\n  V : C\n  /-- structure morphism of the coalgebra -/\n  str : V ⟶ F.obj V\n\ninstance [Inhabited C] : Inhabited (Coalgebra (𝟭 C)) :=\n  ⟨⟨default, 𝟙 _⟩⟩\n\nnamespace Coalgebra\n\nvariable {F : C ⥤ C} (V : Coalgebra F) {V₀ V₁ V₂ : Coalgebra F}\n\n/-\n```\n        str\n    V₀ -----> F V₀\n    |          |\n  f |          | F f\n    V          V\n    V₁ -----> F V₁\n        str\n```\n-/\n/-- A morphism between coalgebras of an endofunctor `F` -/\n@[ext]\nstructure Hom (V₀ V₁ : Coalgebra F) where\n  /-- underlying morphism between two carriers -/\n  f : V₀.1 ⟶ V₁.1\n  /-- compatibility condition -/\n  h : V₀.str ≫ F.map f = f ≫ V₁.str := by cat_disch\n\nattribute [reassoc (attr := simp)] Hom.h\n\nnamespace Hom\n\n/-- The identity morphism of an algebra of endofunctor `F` -/\ndef id : Hom V V where f := 𝟙 _\n\ninstance : Inhabited (Hom V V) :=\n  ⟨{ f := 𝟙 _ }⟩\n\n/-- The composition of morphisms between algebras of endofunctor `F` -/\ndef comp (f : Hom V₀ V₁) (g : Hom V₁ V₂) : Hom V₀ V₂ where f := f.1 ≫ g.1\n\nend Hom\n\ninstance (F : C ⥤ C) : CategoryStruct (Coalgebra F) where\n  Hom := Hom\n  id := Hom.id\n  comp := @Hom.comp _ _ _\n\n@[ext]\n\nTarget:\nlemma ext {A B : Coalgebra F} {f g : A ⟶ B} (w : f.f = g.f :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_6aa245c658f8","rejection":{"accepted":false,"failure_type":"unknown_tactic","output_sha256":"b2f6e75651ce1c07631343cb324f01a6566eee8a4fd1e94b8a4566aa4689da09","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Endofunctor","family_id":"ext","file_id":"mathlib/Mathlib/CategoryTheory/Endofunctor/Algebra.lean","sample_id":"6aa245c658f8d36bf9da386e1af0245093910d695a91f4bad772966128f1d7d5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6db23e1003596f7373b150a65cdc1843f9cceb615009a5e13fae5265b46dd136","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f362ce03318e89ad5084cd4764e967b35a3cc6ed1f7b8e8ce93da29b6d4f234b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"70ee751da2d1168fe9d7a11059342b3a3f15c210604e29d540ce2dd98fb9afcf","source_sha256":"1d77f6aaa753df530e96edf4819026a453d32ae6fc473ac307ea722c26a6c807","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro L₁ L₂ hflat\n  apply Subtype.ext\n  exact h L₁.val L₂.val L₁.prop L₂.prop hflat","hard_negative":false,"metrics":{"chosen_tokens":30,"rejected_tokens":35,"token_jaccard":0.777778,"token_length_ratio":1.166667},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"b51623415e585f54efe78cebe5e355db0aff9ecb355dd22a469048cf37136b98","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Basic\n\nNamespace:\nInformationTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Elazar Gershuni. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Elazar Gershuni\n-/\n/-!\n# Uniquely Decodable Codes\n\nThis file defines uniquely decodable codes and proves basic properties.\n\n## Main definitions\n\n* `UniquelyDecodable`: A set of codewords is uniquely decodable if distinct concatenations\n  of codewords yield distinct strings.\n\n## Main results\n\n* `UniquelyDecodable.epsilon_not_mem`: Uniquely decodable codes cannot contain the empty\n  string.\n* `UniquelyDecodable.flatten_injective`: The flatten function is injective on lists of\n  codewords from a uniquely decodable code.\n-/\n\n@[expose] public section\n\nnamespace InformationTheory\n\nvariable {α : Type*}\n\n/-- A set of lists is uniquely decodable if distinct concatenations yield distinct strings. -/\ndef UniquelyDecodable (S : Set (List α)) : Prop :=\n  ∀ (L₁ L₂ : List (List α)),\n    (∀ w ∈ L₁, w ∈ S) → (∀ w ∈ L₂, w ∈ S) →\n    L₁.flatten = L₂.flatten → L₁ = L₂\n\nvariable {S : Set (List α)}\n\n/-- If a code is uniquely decodable, it does not contain the empty string.\n\nThe empty string can be \"decoded\" as either zero or two copies of itself,\nviolating unique decodability. -/\nlemma UniquelyDecodable.epsilon_not_mem\n    (h : UniquelyDecodable S) :\n    [] ∉ S := by\n  simpa using h [[]] [[], []]\n\nTarget:\nlemma UniquelyDecodable.flatten_injective (h : UniquelyDecodable S) :\n    Function.Injective (fun (L : {L : List (List α) // ∀ x ∈ L, x ∈ S}) => L.val.flatten) :=\n\nProof body:\n","rejected":"by\n  intro L₁ L₂ hflat\n  apply Subtype.ext\n  exact h L₁.val L₂.val L₁.prop L₂.prop hflat\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"InformationTheory/Coding","family_id":"uniquelydecodable","file_id":"mathlib/Mathlib/InformationTheory/Coding/UniquelyDecodable.lean","sample_id":"70ee751da2d1168fe9d7a11059342b3a3f15c210604e29d540ce2dd98fb9afcf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5668706c687646eb1a543fe99da9ca45de8dce6aa203fed4f6e41809be74067a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8ec2135d00a06c0bdde5ffa67ec3d4e80d4d037337441a05011662b1b800eac1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f7c063a4e5099ef9d392d9d1440be2d9799586ee41a1ffac00a85cda728bea8f","source_sha256":"abc84df69b88508535f62fd151f496b3587000aca07e5e3a2f098ce3ffb6b87d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos\n  have hk₀' : (0 : ℝ) < k := Int.cast_pos.mpr hk₀\n  have hden : ((j / k : ℚ).den : ℤ) ≤ k := by\n    convert! le_of_dvd hk₀ (Rat.den_dvd j k)\n    exact Rat.intCast_div_eq_divInt _ _\n  refine ⟨j / k, ?_, Nat.cast_le.mp (hden.trans hk₁)⟩\n  rw [← div_div, le_div_iff₀ (Nat.cast_pos.mpr <| Rat.pos _ : (0 : ℝ) < _)]\n  refine (mul_le_mul_of_nonneg_left (Int.cast_le.mpr hden : _ ≤ (k : ℝ)) (abs_nonneg _)).trans ?_\n  rwa [← abs_of_pos hk₀', Rat.cast_div, Rat.cast_intCast, Rat.cast_intCast, ← abs_mul, sub_mul,\n    div_mul_cancel₀ _ hk₀'.ne', mul_comm]","hard_negative":true,"metrics":{"chosen_tokens":192,"rejected_tokens":3,"token_jaccard":0.029851,"token_length_ratio":0.015625},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"b51625a5a6878b6ddc5559abc8a5e2e2f40bb6fb285adc67f84356ef2fb1fcf0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.Real.Irrational\npublic import Mathlib.RingTheory.Coprime.Lemmas\npublic import Mathlib.RingTheory.Int.Basic\npublic import Mathlib.Tactic.Basic\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2022 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Geißer, Michael Stoll\n-/\n/-!\n# Diophantine Approximation\n\nThe first part of this file gives proofs of various versions of\n**Dirichlet's approximation theorem** and its important consequence that when $\\xi$ is an\nirrational real number, then there are infinitely many rationals $x/y$ (in lowest terms)\nsuch that\n$$\\left|\\xi - \\frac{x}{y}\\right| < \\frac{1}{y^2} \\,.$$\nThe proof is based on the pigeonhole principle.\n\nThe second part of the file gives a proof of **Legendre's Theorem** on rational approximation,\nwhich states that if $\\xi$ is a real number and $x/y$ is a rational number such that\n$$\\left|\\xi - \\frac{x}{y}\\right| < \\frac{1}{2y^2} \\,,$$\nthen $x/y$ must be a convergent of the continued fraction expansion of $\\xi$.\n\n## Main statements\n\nThe main results are three variants of Dirichlet's approximation theorem:\n* `Real.exists_int_int_abs_mul_sub_le`, which states that for all real `ξ` and natural `0 < n`,\n  there are integers `j` and `k` with `0 < k ≤ n` and `|k*ξ - j| ≤ 1/(n+1)`,\n* `Real.exists_nat_abs_mul_sub_round_le`, which replaces `j` by `round(k*ξ)` and uses\n  a natural number `k`,\n* `Real.exists_rat_abs_sub_le_and_den_le`, which says that there is a rational number `q`\n  satisfying `|ξ - q| ≤ 1/((n+1)*q.den)` and `q.den ≤ n`,\n\nand\n* `Real.infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational`, which states that\n  for irrational `ξ`, the set `{q : ℚ | |ξ - q| < 1/q.den^2}` is infinite.\n\nWe also show a converse,\n* `Rat.finite_rat_abs_sub_lt_one_div_den_sq`, which states that the set above is finite\n  when `ξ` is a rational number.\n\nBoth statements are combined to give an equivalence,\n`Real.infinite_rat_abs_sub_lt_one_div_den_sq_iff_irrational`.\n\nThere are two versions of Legendre's Theorem. One, `Real.exists_rat_eq_convergent`, uses\n`Real.convergent`, a simple recursive definition of the convergents that is also defined\nin this file, whereas the other, `Real.exists_convs_eq_rat` defined in the file\n`Mathlib/NumberTheory/DiophantineApproximation/ContinuedFractions.lean`, uses\n`GenContFract.convs` of `GenContFract.of ξ`.\n\n## Implementation notes\n\nWe use the namespace `Real` for the results on real numbers and `Rat` for the results\non rational numbers. We introduce a secondary namespace `Real.ContfracLegendre`\nto separate off a definition and some technical auxiliary lemmas used in the proof\nof Legendre's Theorem. For remarks on the proof of Legendre's Theorem, see below.\n\n## References\n\n<https://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem>\n<https://de.wikipedia.org/wiki/Kettenbruch> (The German Wikipedia page on continued\nfractions is much more extensive than the English one.)\n\n## Tags\n\nDiophantine approximation, Dirichlet's approximation theorem, continued fraction\n-/\n\n@[expose] public section\n\n\nnamespace Real\n\nsection Dirichlet\n\n/-!\n### Dirichlet's approximation theorem\n\nWe show that for any real number `ξ` and positive natural `n`, there is a fraction `q`\nsuch that `q.den ≤ n` and `|ξ - q| ≤ 1/((n+1)*q.den)`.\n-/\n\n\nopen Finset Int\n\n/-- *Dirichlet's approximation theorem:*\nFor any real number `ξ` and positive natural `n`, there are integers `j` and `k`,\nwith `0 < k ≤ n` and `|k*ξ - j| ≤ 1/(n+1)`.\n\nSee also `Real.exists_nat_abs_mul_sub_round_le`. -/\ntheorem exists_int_int_abs_mul_sub_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :\n    ∃ j k : ℤ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - j| ≤ 1 / (n + 1) := by\n  let f : ℤ → ℤ := fun m => ⌊fract (ξ * m) * (n + 1)⌋\n  have hn : 0 < (n : ℝ) + 1 := mod_cast Nat.succ_pos _\n  have hfu := fun m : ℤ => mul_lt_of_lt_one_left hn <| fract_lt_one (ξ * ↑m)\n  conv in |_| ≤ _ => rw [mul_comm, le_div_iff₀ hn, ← abs_of_pos hn, ← abs_mul]\n  let D := Icc (0 : ℤ) n\n  by_cases! H : ∃ m ∈ D, f m = n\n  · obtain ⟨m, hm, hf⟩ := H\n    have hf' : ((n : ℤ) : ℝ) ≤ fract (ξ * m) * (n + 1) := hf ▸ floor_le (fract (ξ * m) * (n + 1))\n    have hm₀ : 0 < m := by\n      have hf₀ : f 0 = 0 := by\n        simp only [f, cast_zero, mul_zero, fract_zero, zero_mul, floor_zero]\n      refine Ne.lt_of_le (fun h => n_pos.ne ?_) (mem_Icc.mp hm).1\n      exact mod_cast hf₀.symm.trans (h.symm ▸ hf : f 0 = n)\n    refine ⟨⌊ξ * m⌋ + 1, m, hm₀, (mem_Icc.mp hm).2, ?_⟩\n    rw [cast_add, ← sub_sub, sub_mul, cast_one, one_mul, abs_le]\n    refine\n      ⟨le_sub_iff_add_le.mpr ?_, sub_le_iff_le_add.mpr <| le_of_lt <| (hfu m).trans <| lt_one_add _⟩\n    simpa only [neg_add_cancel_comm_assoc] using! hf'\n  · have hD : #(Ico (0 : ℤ) n) < #D := by rw [card_Icc, card_Ico]; exact lt_add_one n\n    have hfu' : ∀ m, f m ≤ n := fun m => lt_add_one_iff.mp (floor_lt.mpr (mod_cast hfu m))\n    have hwd : ∀ m : ℤ, m ∈ D → f m ∈ Ico (0 : ℤ) n := fun x hx =>\n      mem_Ico.mpr\n        ⟨floor_nonneg.mpr (mul_nonneg (fract_nonneg (ξ * x)) hn.le), Ne.lt_of_le (H x hx) (hfu' x)⟩\n    obtain ⟨x, hx, y, hy, x_lt_y, hxy⟩ : ∃ x ∈ D, ∃ y ∈ D, x < y ∧ f x = f y := by\n      obtain ⟨x, hx, y, hy, x_ne_y, hxy⟩ := exists_ne_map_eq_of_card_lt_of_maps_to hD hwd\n      rcases lt_trichotomy x y with (h | h | h)\n      exacts [⟨x, hx, y, hy, h, hxy⟩, False.elim (x_ne_y h), ⟨y, hy, x, hx, h, hxy.symm⟩]\n    refine\n      ⟨⌊ξ * y⌋ - ⌊ξ * x⌋, y - x, sub_pos_of_lt x_lt_y,\n        sub_le_iff_le_add.mpr <| le_add_of_le_of_nonneg (mem_Icc.mp hy).2 (mem_Icc.mp hx).1, ?_⟩\n    convert_to! |fract (ξ * y) * (n + 1) - fract (ξ * x) * (n + 1)| ≤ 1\n    · congr; push_cast; simp only [fract]; ring\n    exact (abs_sub_lt_one_of_floor_eq_floor hxy.symm).le\n\n/-- *Dirichlet's approximation theorem:*\nFor any real number `ξ` and positive natural `n`, there is a natural number `k`,\nwith `0 < k ≤ n` such that `|k*ξ - round(k*ξ)| ≤ 1/(n+1)`.\n-/\ntheorem exists_nat_abs_mul_sub_round_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :\n    ∃ k : ℕ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - round (↑k * ξ)| ≤ 1 / (n + 1) := by\n  obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos\n  have hk := toNat_of_nonneg hk₀.le\n  rw [← hk] at hk₀ hk₁ h\n  exact ⟨k.toNat, natCast_pos.mp hk₀, Nat.cast_le.mp hk₁, (round_le (↑k.toNat * ξ) j).trans h⟩\n\n/-- *Dirichlet's approximation theorem:*\nFor any real number `ξ` and positive natural `n`, there is a fraction `q`\nsuch that `q.den ≤ n` and `|ξ - q| ≤ 1/((n+1)*q.den)`.\n\nSee also `AddCircle.exists_norm_nsmul_le`. -/\n\nTarget:\ntheorem exists_rat_abs_sub_le_and_den_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :\n    ∃ q : ℚ, |ξ - q| ≤ 1 / ((n + 1) * q.den) ∧ q.den ≤ n :=\n\nProof body:\n","rejected":"by\n  exact exists_rat_abs_sub_le_and_den_le","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"d52c7cdd2bd404deb9a1bc208ae6e61ce682e00c89dc18d04785079eb47b1656","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/DiophantineApproximation","family_id":"exists_rat_abs_sub_le_and_den_le","file_id":"mathlib/Mathlib/NumberTheory/DiophantineApproximation/Basic.lean","sample_id":"f7c063a4e5099ef9d392d9d1440be2d9799586ee41a1ffac00a85cda728bea8f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"31b27ce5f99a3bf78bf18a3e524b1460389b6e7ac4458283ab4e0eee4a7b4805","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1c5375cf4fad0c1095d3773e5d3eececd8a87bfceddd1a86fb9b46d851ed3a4b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"767863c4f273ffd20ab08a8048315c1504cb395ceb5b970fd18d5025077b58cc","source_sha256":"f31e0ea5c5dcaaef4fe7a595be40e4dcc670dcec12bcae3dfce625de497aea8b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold unfolds; rw [corec_eq]","hard_negative":true,"metrics":{"chosen_tokens":8,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.625},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"b531b8837ebdb8b4f5a462c898794a4490f023239ae9bfe90637257df017c8a6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Stream.Defs\npublic import Mathlib.Logic.Function.Basic\npublic import Mathlib.Data.Nat.Basic\npublic import Mathlib.Tactic.Common\n\nNamespace:\nStream'\n\nLocal context:\n/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Streams a.k.a. infinite lists a.k.a. infinite sequences\n-/\n\n@[expose] public section\n\nopen Nat Function Option\n\nnamespace Stream'\n\nuniverse u v w\nvariable {α : Type u} {β : Type v} {δ : Type w}\nvariable (m n : ℕ) (x y : List α) (a b : Stream' α)\n\ninstance [Inhabited α] : Inhabited (Stream' α) :=\n  ⟨Stream'.const default⟩\n\n@[simp] protected theorem eta (s : Stream' α) : head s :: tail s = s :=\n  funext fun i => by cases i <;> rfl\n\n/-- Alias for `Stream'.eta` to match `List` API. -/\nalias cons_head_tail := Stream'.eta\n\n@[ext]\nprotected theorem ext {s₁ s₂ : Stream' α} : (∀ n, get s₁ n = get s₂ n) → s₁ = s₂ :=\n  fun h => funext h\n\n@[simp]\ntheorem get_zero_cons (a : α) (s : Stream' α) : get (a::s) 0 = a :=\n  rfl\n\n@[simp]\ntheorem head_cons (a : α) (s : Stream' α) : head (a::s) = a :=\n  rfl\n\n@[simp]\ntheorem tail_cons (a : α) (s : Stream' α) : tail (a::s) = s :=\n  rfl\n\n@[simp]\ntheorem get_drop (n m : ℕ) (s : Stream' α) : get (drop m s) n = get s (m + n) := by\n  rw [Nat.add_comm]\n  rfl\n\ntheorem tail_eq_drop (s : Stream' α) : tail s = drop 1 s :=\n  rfl\n\n@[simp]\ntheorem drop_drop (n m : ℕ) (s : Stream' α) : drop n (drop m s) = drop (m + n) s := by\n  ext; simp [Nat.add_assoc]\n\n@[simp] theorem get_tail {n : ℕ} {s : Stream' α} : s.tail.get n = s.get (n + 1) := rfl\n\n@[simp] theorem tail_drop' {i : ℕ} {s : Stream' α} : tail (drop i s) = s.drop (i + 1) := by\n  ext; simp [Nat.add_comm, Nat.add_left_comm]\n\n@[simp] theorem drop_tail' {i : ℕ} {s : Stream' α} : drop i (tail s) = s.drop (i + 1) := rfl\n\ntheorem tail_drop (n : ℕ) (s : Stream' α) : tail (drop n s) = drop n (tail s) := by simp\n\ntheorem get_succ (n : ℕ) (s : Stream' α) : get s (succ n) = get (tail s) n :=\n  rfl\n\n@[simp]\ntheorem get_succ_cons (n : ℕ) (s : Stream' α) (x : α) : get (x :: s) n.succ = get s n :=\n  rfl\n\n@[simp] lemma get_cons_append_zero {a : α} {x : List α} {s : Stream' α} :\n    (a :: x ++ₛ s).get 0 = a := rfl\n\n@[simp] lemma append_eq_cons {a : α} {as : Stream' α} : [a] ++ₛ as = a :: as := rfl\n\n@[simp] theorem drop_zero {s : Stream' α} : s.drop 0 = s := rfl\n\ntheorem drop_succ (n : ℕ) (s : Stream' α) : drop (succ n) s = drop n (tail s) :=\n  rfl\n\ntheorem head_drop (a : Stream' α) (n : ℕ) : (a.drop n).head = a.get n := by simp\n\ntheorem cons_injective2 : Function.Injective2 (cons : α → Stream' α → Stream' α) := fun x y s t h =>\n  ⟨by rw [← get_zero_cons x s, h, get_zero_cons],\n    Stream'.ext fun n => by rw [← get_succ_cons n _ x, h, get_succ_cons]⟩\n\ntheorem cons_injective_left (s : Stream' α) : Function.Injective fun x => cons x s :=\n  cons_injective2.left _\n\ntheorem cons_injective_right (x : α) : Function.Injective (cons x) :=\n  cons_injective2.right _\n\ntheorem all_def (p : α → Prop) (s : Stream' α) : All p s = ∀ n, p (get s n) :=\n  rfl\n\ntheorem any_def (p : α → Prop) (s : Stream' α) : Any p s = ∃ n, p (get s n) :=\n  rfl\n\n@[simp]\ntheorem mem_cons (a : α) (s : Stream' α) : a ∈ a::s :=\n  Exists.intro 0 rfl\n\ntheorem mem_cons_of_mem {a : α} {s : Stream' α} (b : α) : a ∈ s → a ∈ b::s := fun ⟨n, h⟩ =>\n  Exists.intro (succ n) (by rw [get_succ, tail_cons, h])\n\ntheorem eq_or_mem_of_mem_cons {a b : α} {s : Stream' α} : (a ∈ b::s) → a = b ∨ a ∈ s :=\n    fun ⟨n, h⟩ => by\n  rcases n with - | n'\n  · left\n    exact h\n  · right\n    rw [get_succ, tail_cons] at h\n    exact ⟨n', h⟩\n\ntheorem mem_of_get_eq {n : ℕ} {s : Stream' α} {a : α} : a = get s n → a ∈ s := fun h =>\n  Exists.intro n h\n\ntheorem mem_iff_exists_get_eq {s : Stream' α} {a : α} : a ∈ s ↔ ∃ n, a = s.get n where\n  mp := by simp [Membership.mem, any_def]\n  mpr h := mem_of_get_eq h.choose_spec\n\nsection Map\n\nvariable (f : α → β)\n\ntheorem drop_map (n : ℕ) (s : Stream' α) : drop n (map f s) = map f (drop n s) :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem get_map (n : ℕ) (s : Stream' α) : get (map f s) n = f (get s n) :=\n  rfl\n\ntheorem tail_map (s : Stream' α) : tail (map f s) = map f (tail s) := rfl\n\n@[simp]\ntheorem head_map (s : Stream' α) : head (map f s) = f (head s) :=\n  rfl\n\ntheorem map_eq (s : Stream' α) : map f s = f (head s)::map f (tail s) := by\n  rw [← Stream'.eta (map f s), tail_map, head_map]\n\ntheorem map_cons (a : α) (s : Stream' α) : map f (a::s) = f a::map f s := by\n  rw [← Stream'.eta (map f (a::s)), map_eq]; rfl\n\n@[simp]\ntheorem map_id (s : Stream' α) : map id s = s :=\n  rfl\n\n@[simp]\ntheorem map_map (g : β → δ) (f : α → β) (s : Stream' α) : map g (map f s) = map (g ∘ f) s :=\n  rfl\n\n@[simp]\ntheorem map_tail (s : Stream' α) : map f (tail s) = tail (map f s) :=\n  rfl\n\ntheorem mem_map {a : α} {s : Stream' α} : a ∈ s → f a ∈ map f s := fun ⟨n, h⟩ =>\n  Exists.intro n (by rw [get_map, h])\n\ntheorem exists_of_mem_map {f} {b : β} {s : Stream' α} : b ∈ map f s → ∃ a, a ∈ s ∧ f a = b :=\n  fun ⟨n, h⟩ => ⟨get s n, ⟨n, rfl⟩, h.symm⟩\n\nend Map\n\nsection Zip\n\nvariable (f : α → β → δ)\n\ntheorem drop_zip (n : ℕ) (s₁ : Stream' α) (s₂ : Stream' β) :\n    drop n (zip f s₁ s₂) = zip f (drop n s₁) (drop n s₂) :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem get_zip (n : ℕ) (s₁ : Stream' α) (s₂ : Stream' β) :\n    get (zip f s₁ s₂) n = f (get s₁ n) (get s₂ n) :=\n  rfl\n\ntheorem head_zip (s₁ : Stream' α) (s₂ : Stream' β) : head (zip f s₁ s₂) = f (head s₁) (head s₂) :=\n  rfl\n\ntheorem tail_zip (s₁ : Stream' α) (s₂ : Stream' β) :\n    tail (zip f s₁ s₂) = zip f (tail s₁) (tail s₂) :=\n  rfl\n\ntheorem zip_eq (s₁ : Stream' α) (s₂ : Stream' β) :\n    zip f s₁ s₂ = f (head s₁) (head s₂)::zip f (tail s₁) (tail s₂) := by\n  rw [← Stream'.eta (zip f s₁ s₂)]; rfl\n\n@[simp]\ntheorem get_enum (s : Stream' α) (n : ℕ) : get (enum s) n = (n, s.get n) :=\n  rfl\n\ntheorem enum_eq_zip (s : Stream' α) : enum s = zip Prod.mk nats s :=\n  rfl\n\nend Zip\n\n@[simp]\ntheorem mem_const (a : α) : a ∈ const a :=\n  Exists.intro 0 rfl\n\ntheorem const_eq (a : α) : const a = a::const a := by\n  apply Stream'.ext; intro n\n  cases n <;> rfl\n\n@[simp]\ntheorem tail_const (a : α) : tail (const a) = const a :=\n  suffices tail (a::const a) = const a by rwa [← const_eq] at this\n  rfl\n\n@[simp]\ntheorem map_const (f : α → β) (a : α) : map f (const a) = const (f a) :=\n  rfl\n\n@[simp]\ntheorem get_const (n : ℕ) (a : α) : get (const a) n = a :=\n  rfl\n\n@[simp]\ntheorem drop_const (n : ℕ) (a : α) : drop n (const a) = const a :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem head_iterate (f : α → α) (a : α) : head (iterate f a) = a :=\n  rfl\n\ntheorem get_succ_iterate' (n : ℕ) (f : α → α) (a : α) :\n    get (iterate f a) (succ n) = f (get (iterate f a) n) := rfl\n\ntheorem tail_iterate (f : α → α) (a : α) : tail (iterate f a) = iterate f (f a) := by\n  ext n\n  rw [get_tail]\n  induction n with\n  | zero => rfl\n  | succ n ih => rw [get_succ_iterate', ih, get_succ_iterate']\n\ntheorem iterate_eq (f : α → α) (a : α) : iterate f a = a::iterate f (f a) := by\n  rw [← Stream'.eta (iterate f a)]\n  rw [tail_iterate]; rfl\n\n@[simp]\ntheorem get_zero_iterate (f : α → α) (a : α) : get (iterate f a) 0 = a :=\n  rfl\n\ntheorem get_succ_iterate (n : ℕ) (f : α → α) (a : α) :\n    get (iterate f a) (succ n) = get (iterate f (f a)) n := by rw [get_succ, tail_iterate]\n\nsection Bisim\n\nvariable (R : Stream' α → Stream' α → Prop)\n\n/-- equivalence relation -/\nlocal infixl:50 \" ~ \" => R\n\n/-- Streams `s₁` and `s₂` are defined to be bisimulations if\ntheir heads are equal and tails are bisimulations. -/\ndef IsBisimulation :=\n  ∀ ⦃s₁ s₂⦄, s₁ ~ s₂ →\n      head s₁ = head s₂ ∧ tail s₁ ~ tail s₂\n\ntheorem get_of_bisim (bisim : IsBisimulation R) {s₁ s₂} :\n    ∀ n, s₁ ~ s₂ → get s₁ n = get s₂ n ∧ drop (n + 1) s₁ ~ drop (n + 1) s₂\n  | 0, h => bisim h\n  | n + 1, h =>\n    match bisim h with\n    | ⟨_, trel⟩ => get_of_bisim bisim n trel\n\n-- If two streams are bisimilar, then they are equal\ntheorem eq_of_bisim (bisim : IsBisimulation R) {s₁ s₂} : s₁ ~ s₂ → s₁ = s₂ := fun r =>\n  Stream'.ext fun n => And.left (get_of_bisim R bisim n r)\n\nend Bisim\n\ntheorem bisim_simple (s₁ s₂ : Stream' α) :\n    head s₁ = head s₂ → s₁ = tail s₁ → s₂ = tail s₂ → s₁ = s₂ := fun hh ht₁ ht₂ =>\n  eq_of_bisim (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂)\n    (fun s₁ s₂ ⟨h₁, h₂, h₃⟩ => by grind)\n    (And.intro hh (And.intro ht₁ ht₂))\n\ntheorem coinduction {s₁ s₂ : Stream' α} :\n    head s₁ = head s₂ →\n      (∀ (β : Type u) (fr : Stream' α → β),\n      fr s₁ = fr s₂ → fr (tail s₁) = fr (tail s₂)) → s₁ = s₂ :=\n  fun hh ht =>\n  eq_of_bisim\n    (fun s₁ s₂ =>\n      head s₁ = head s₂ ∧\n        ∀ (β : Type u) (fr : Stream' α → β), fr s₁ = fr s₂ → fr (tail s₁) = fr (tail s₂))\n    (fun s₁ s₂ h =>\n      have h₁ : head s₁ = head s₂ := And.left h\n      have h₂ : head (tail s₁) = head (tail s₂) := And.right h α (@head α) h₁\n      have h₃ :\n        ∀ (β : Type u) (fr : Stream' α → β),\n          fr (tail s₁) = fr (tail s₂) → fr (tail (tail s₁)) = fr (tail (tail s₂)) :=\n        fun β fr => And.right h β fun s => fr (tail s)\n      And.intro h₁ (And.intro h₂ h₃))\n    (And.intro hh ht)\n\n@[simp]\ntheorem iterate_id (a : α) : iterate id a = const a :=\n  coinduction rfl fun β fr ch => by rw [tail_iterate, tail_const]; exact ch\n\ntheorem map_iterate (f : α → α) (a : α) : iterate f (f a) = map f (iterate f a) := by\n  funext n\n  induction n with\n  | zero => rfl\n  | succ n ih =>\n    unfold map iterate get\n    rw [map, get] at ih\n    rw [iterate]\n    exact congrArg f ih\n\nsection Corec\n\ntheorem corec_def (f : α → β) (g : α → α) (a : α) : corec f g a = map f (iterate g a) :=\n  rfl\n\ntheorem corec_eq (f : α → β) (g : α → α) (a : α) : corec f g a = f a :: corec f g (g a) := by\n  rw [corec_def, map_eq, head_iterate, tail_iterate]; rfl\n\ntheorem corec_id_id_eq_const (a : α) : corec id id a = const a := by\n  rw [corec_def, map_id, iterate_id]\n\ntheorem corec_id_f_eq_iterate (f : α → α) (a : α) : corec id f a = iterate f a :=\n  rfl\n\nend Corec\n\nsection Corec'\n\ntheorem corec'_eq (f : α → β × α) (a : α) : corec' f a = (f a).1 :: corec' f (f a).2 :=\n  corec_eq _ _ _\n\nend Corec'\n\nTarget:\ntheorem unfolds_eq (g : α → β) (f : α → α) (a : α) : unfolds g f a = g a :: unfolds g f (f a) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_767863c4f273","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"88df87fef37fa62737ab5369eec4a3819ceb6e8eea2deb54603aa2b7cb0ddb22","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Stream","family_id":"unfolds_eq","file_id":"mathlib/Mathlib/Data/Stream/Init.lean","sample_id":"767863c4f273ffd20ab08a8048315c1504cb395ceb5b970fd18d5025077b58cc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"909daa16cd181b78a203d073bc4331c1f30425ac8b8431620e4cd7f332d4a635","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"21cb92a17308c1ecba7cf2fdd675eece83d81c8e124af2d07bc4b2ba8c377a43","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a13d8df10b3157064561e6f5629998f83e9b759c9ef6010480db3810fec50f71","source_sha256":"3b79245b81821b44a91daf15c8bd7da3bce2a30e3ee9d29e2db6a7d0e0195196","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · intro H\n    induction H using Submodule.span_induction with\n    | mem a ha =>\n      obtain ⟨a, ha, rfl⟩ := ha\n      exact ⟨C a, fun i ↦ by rw [coeff_C]; aesop, aeval_C _ _⟩\n    | zero => exact ⟨0, by simp, aeval_zero _⟩\n    | add a b ha hb ha' hb' =>\n      obtain ⟨a, ha, ha'⟩ := ha'\n      obtain ⟨b, hb, hb'⟩ := hb'\n      exact ⟨a + b, fun i ↦ by simpa using add_mem (ha i) (hb i), by simp [ha', hb']⟩\n    | smul a b hb hb' =>\n      obtain ⟨b', hb, hb'⟩ := hb'\n      have ⟨p, hp⟩ := adjoin_eq_exists_aeval R x a\n      refine ⟨p * b', fun i ↦ ?_, by simp [hp, hb']⟩\n      rw [coeff_mul]\n      exact sum_mem fun i hi ↦ Ideal.mul_mem_left _ _ (hb _)\n  · rintro ⟨p, hp, hp'⟩\n    have : y = ∑ i ∈ p.support, p.coeff i • ⟨_, (X ^ i).aeval_mem_adjoin_singleton _ x⟩ := by\n      trans ∑ i ∈ p.support, ⟨_, (C (p.coeff i) * X ^ i).aeval_mem_adjoin_singleton _ x⟩\n      · ext1\n        simp only [AddSubmonoidClass.coe_finsetSum, ← map_sum, ← hp', ← as_sum_support_C_mul_X_pow]\n      · congr with i\n        simp [Algebra.smul_def]\n    simp_rw [this, Algebra.smul_def]\n    exact sum_mem fun i _ ↦ Ideal.mul_mem_right _ _ (Ideal.mem_map_of_mem _ (hp i))","hard_negative":true,"metrics":{"chosen_tokens":312,"rejected_tokens":2,"token_jaccard":0.010638,"token_length_ratio":0.00641},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"b556bf1c369b40dfd91e7a88f69331bc08aa6133313376a903aca4569f09a89b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.RingDivision\npublic import Mathlib.RingTheory.Adjoin.Polynomial.Basic\npublic import Mathlib.RingTheory.Ideal.Maps\n\nNamespace:\nAlgebra\n\nLocal context:\n/-\nCopyright (c) 2019 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n/-!\n# Ideals in polynomial rings\n-/\n\npublic section\n\nnoncomputable section\n\nopen Polynomial\n\nopen Finset\n\nuniverse u v w\n\nnamespace Polynomial\n\nvariable {R : Type*} [CommRing R] {a : R}\n\ntheorem mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero {b : R[X]} {P : R[X][X]} :\n    P ∈ Ideal.span {C (X - C a), X - C b} ↔ (P.eval b).eval a = 0 := by\n  rw [Ideal.mem_span_pair]\n  constructor <;> intro h\n  · rcases h with ⟨_, _, rfl⟩\n    simp\n  · rcases dvd_iff_isRoot.mpr h with ⟨p, hp⟩\n    rcases @X_sub_C_dvd_sub_C_eval _ b _ P with ⟨q, hq⟩\n    exact ⟨C p, q, by rw [mul_comm, mul_comm q, eq_add_of_sub_eq' hq, hp, C_mul]⟩\n\ntheorem ker_evalRingHom (x : R) : RingHom.ker (evalRingHom x) = Ideal.span {X - C x} := by\n  ext y\n  simp [Ideal.mem_span_singleton, dvd_iff_isRoot, RingHom.mem_ker]\n\n@[simp]\ntheorem ker_modByMonicHom {q : R[X]} (hq : q.Monic) :\n    LinearMap.ker (Polynomial.modByMonicHom q) = (Ideal.span {q}).restrictScalars R :=\n  Submodule.ext fun _ => (mem_ker_modByMonic hq).trans Ideal.mem_span_singleton.symm\n\n@[simp]\nlemma ker_constantCoeff : RingHom.ker constantCoeff = .span {(X : R[X])} := by\n  refine le_antisymm (fun p hp ↦ ?_) (by simp [Ideal.span_le])\n  simp only [RingHom.mem_ker, constantCoeff_apply, ← Polynomial.X_dvd_iff] at hp\n  rwa [Ideal.mem_span_singleton]\n\nend Polynomial\n\nnamespace Algebra\n\nvariable {R S : Type*}\n\nTarget:\nlemma mem_ideal_map_adjoin [CommSemiring R] [Semiring S] [Algebra R S] (x : S) (I : Ideal R)\n    {y : R[x]} :\n    y ∈ I.map (algebraMap R (R[x])) ↔\n      ∃ p : R[X], (∀ i, p.coeff i ∈ I) ∧ Polynomial.aeval x p = y :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_a13d8df10b31","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"50dd4c9cd7569858f6c7ce052cff48d4a7509c4568bf47a765575af3b4e78faa","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Polynomial","family_id":"mem_ideal_map_adjoin","file_id":"mathlib/Mathlib/RingTheory/Polynomial/Ideal.lean","sample_id":"a13d8df10b3157064561e6f5629998f83e9b759c9ef6010480db3810fec50f71"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6ed251e98bcc09fc840097a1704c71dc97a2385a1bbef8d0c694a48f689a68b3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b482550037179e53cdaa84a61ee77776411a6edd944e1331a4115712f6b6c73c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0d7956b6ebdcf8c47b68d95a35dc56b9f93a664feaf120a85ef061f559b47042","source_sha256":"1513994cab9129a1b43fa5d8ae4d7aeabdb9cc958d905bbce24e5e0da36801ec","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [OneSkeletonGraph_isLink, closedCell_zero_eq_singleton, closedCell_zero_eq_singleton,\n    singleton_union]","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":15,"token_jaccard":0.666667,"token_length_ratio":1.363636},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"b55f2f4072dda5af039bd4a80814b1c4213415d26f25ff5dbd5ae22610658c45","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.CWComplex.Classical.Finite\npublic import Mathlib.Combinatorics.Graph.Basic\n\nNamespace:\nTopology.CWComplex.OneSkeletonGraph\n\nLocal context:\n/-\nCopyright (c) 2026 Jun Kwon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jun Kwon\n-/\n/-!\n# 1-skeletons of CW complexes as graphs\n\nIn this file we define the 1-skeleton of a CW complex as a graph.\n\n## Main definitions\n* `CWComplex.OneSkeletonGraph`: the 1-skeleton of a CW complex as a graph.\n\n-/\n\npublic section\n\nopen Metric Set Graph\n\nnamespace Topology\n\nvariable {X : Type*} [TopologicalSpace X]\n\n/-- The 1-skeleton of a CW complex as a graph. -/\n@[expose, simps]\ndef CWComplex.OneSkeletonGraph (C : Set X) [CWComplex C] : Graph (cell C 0) (cell C 1) where\n  vertexSet := univ\n  edgeSet := univ\n  IsLink e x y := cellFrontier 1 e = closedCell 0 x ∪ closedCell 0 y\n  isLink_symm := by grind [symm_def]\n  eq_or_eq_of_isLink_of_isLink e x y z w h1 h2 := by\n    simp_rw [closedCell_zero_eq_singleton] at h1 h2\n    rw [h1] at h2\n    simp only [(RelCWComplex.injective_map_zero C).eq_iff, union_singleton, pair_eq_pair_iff] at h2\n    tauto\n  left_mem_of_isLink _ _ _ _ := mem_univ _\n  edge_mem_iff_exists_isLink e := by\n    simp only [mem_univ, true_iff]\n    exact exists_cellFrontier_one_eq e\n\nnamespace CWComplex.OneSkeletonGraph\n\nvariable {C : Set X} [CWComplex C]\n\nTarget:\nlemma isLink_iff_pair (e : cell C 1) (x y : cell C 0) :\n    (OneSkeletonGraph C).IsLink e x y ↔ cellFrontier 1 e = {map 0 x ![], map 0 y ![]} :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [OneSkeletonGraph_isLink, closedCell_zero_eq_singleton, closedCell_zero_eq_singleton,\n    singleton_union]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/CWComplex","family_id":"islink_iff_pair","file_id":"mathlib/Mathlib/Topology/CWComplex/Classical/Graph.lean","sample_id":"0d7956b6ebdcf8c47b68d95a35dc56b9f93a664feaf120a85ef061f559b47042"},"split":"train"}
{"binding":{"chosen_proof_sha256":"75765532e9f719b9017c00ea0609cd6aeeb9a4a2bf7c2d14d50fca0b0687a396","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"19c41640b1c800a77df051f9d2a4e841800bf0cd1e162f71aed963a11a6e9795","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e5781a280c95429bef6967a38411ca2235131ecb05d9745326c5155130e9a4ab","source_sha256":"d5ae8a6ac85e02d23384eb0f0f9e5cb6a39c74c5455bdde189d5500807ce86cf","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    obtain ⟨x, hx⟩ := h.mul_inv_cancel ha\n    simpa [← mul_assoc, h.mul_comm, hx] using congr_arg (x * ·) hb\n  mul_right_cancel_of_ne_zero ha _ _ hb := by\n    obtain ⟨x, hx⟩ := h.mul_inv_cancel ha\n    simpa [mul_assoc, hx] using congr_arg (· * x) hb\n  exists_pair_ne := h.exists_pair_ne","hard_negative":true,"metrics":{"chosen_tokens":68,"rejected_tokens":2,"token_jaccard":0.034483,"token_length_ratio":0.029412},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"b5625bde5ab6dcbbae7aa99f0a65ed3c6ff80518560623dfbd6b94761f593242","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Defs\npublic import Mathlib.Tactic.Common\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2014 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro\n-/\n/-!\n# `IsField` predicate\n\nPredicate on a (semi)ring that it is a (semi)field, i.e. that the multiplication is\ncommutative, that it has more than one element and that all non-zero elements have a\nmultiplicative inverse. In contrast to `Field`, which contains the data of a function associating\nto an element of the field its multiplicative inverse, this predicate only assumes the existence\nand can therefore more easily be used to e.g. transfer along ring isomorphisms.\n-/\n\n@[expose] public section\n\nuniverse u\n\nsection IsField\n\n/-- A predicate to express that a (semi)ring is a (semi)field.\n\nThis is mainly useful because such a predicate does not contain data,\nand can therefore be easily transported along ring isomorphisms.\nAdditionally, this is useful when trying to prove that\na particular ring structure extends to a (semi)field. -/\nstructure IsField (R : Type u) [Semiring R] : Prop where\n  /-- For a semiring to be a field, it must have two distinct elements. -/\n  exists_pair_ne : ∃ x y : R, x ≠ y\n  /-- Fields are commutative. -/\n  mul_comm : ∀ x y : R, x * y = y * x\n  /-- Nonzero elements have multiplicative inverses. -/\n  mul_inv_cancel : ∀ {a : R}, a ≠ 0 → ∃ b, a * b = 1\n\n/-- Transferring from `Semifield` to `IsField`. -/\ntheorem Semifield.toIsField (R : Type u) [Semifield R] : IsField R where\n  __ := ‹Semifield R›\n  mul_inv_cancel {a} ha := ⟨a⁻¹, mul_inv_cancel₀ ha⟩\n\n/-- Transferring from `Field` to `IsField`. -/\ntheorem Field.toIsField (R : Type u) [Field R] : IsField R :=\n  Semifield.toIsField _\n\n@[simp]\ntheorem IsField.nontrivial {R : Type u} [Semiring R] (h : IsField R) : Nontrivial R :=\n  ⟨h.exists_pair_ne⟩\n\nTarget:\nlemma IsField.isDomain {R : Type u} [Semiring R] (h : IsField R) : IsDomain R where\n  mul_left_cancel_of_ne_zero ha _ _ hb :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_e5781a280c95","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"5d2938203b0b8c1eb2df6767b3693e8328df755d04270f35f82858cd253cbade","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Field","family_id":"isfield","file_id":"mathlib/Mathlib/Algebra/Field/IsField.lean","sample_id":"e5781a280c95429bef6967a38411ca2235131ecb05d9745326c5155130e9a4ab"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8a67a77843a75ba583cb43ac0e535963aaf96ee59db334d6516f871f6e4da8bd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9f9a70e78a68c5afe844cbabb734e8a42046f9373491df56a598c7ff77b730eb","source_sha256":"bdcd2022da453373ffbdfe5bdbc273db0a9656383ede2bdc6575e68ee03c9aa0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← @isBounded_univ, isBounded_induced, image_univ]","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":2,"token_jaccard":0.090909,"token_length_ratio":0.181818},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"b5666fe90493a29321e4e941036eb0c188e8ec20bb1f4a9028fed1533765d6f7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.TypeTags.Basic\npublic import Mathlib.Topology.Bornology.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Bornology structure on products and subtypes\n\nIn this file we define `Bornology` and `BoundedSpace` instances on `α × β`, `Π i, X i`, and\n`{x // p x}`. We also prove basic lemmas about `Bornology.cobounded` and `Bornology.IsBounded`\non these types.\n-/\n\npublic section\n\n\nopen Set Filter Bornology Function\n\nopen Filter\n\nvariable {α β ι : Type*} {X : ι → Type*} [Bornology α] [Bornology β]\n  [∀ i, Bornology (X i)]\n\ninstance Prod.instBornology : Bornology (α × β) where\n  cobounded := (cobounded α).coprod (cobounded β)\n  le_cofinite :=\n    @coprod_cofinite α β ▸ coprod_mono ‹Bornology α›.le_cofinite ‹Bornology β›.le_cofinite\n\ninstance Pi.instBornology : Bornology (∀ i, X i) where\n  cobounded := Filter.coprodᵢ fun i => cobounded (X i)\n  le_cofinite := iSup_le fun _ ↦ (comap_mono (Bornology.le_cofinite _)).trans (comap_cofinite_le _)\n\n/-- Inverse image of a bornology. -/\nabbrev Bornology.induced {α β : Type*} [Bornology β] (f : α → β) : Bornology α where\n  cobounded := comap f (cobounded β)\n  le_cofinite := (comap_mono (Bornology.le_cofinite β)).trans (comap_cofinite_le _)\n\ninstance {p : α → Prop} : Bornology (Subtype p) :=\n  Bornology.induced (Subtype.val : Subtype p → α)\n\nnamespace Bornology\n\n/-!\n### Bounded sets in `α × β`\n-/\n\n\ntheorem cobounded_prod : cobounded (α × β) = (cobounded α).coprod (cobounded β) :=\n  rfl\n\ntheorem isBounded_image_fst_and_snd {s : Set (α × β)} :\n    IsBounded (Prod.fst '' s) ∧ IsBounded (Prod.snd '' s) ↔ IsBounded s :=\n  compl_mem_coprod.symm\n\nlemma IsBounded.image_fst {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.fst '' s) :=\n  (isBounded_image_fst_and_snd.2 hs).1\n\nlemma IsBounded.image_snd {s : Set (α × β)} (hs : IsBounded s) : IsBounded (Prod.snd '' s) :=\n  (isBounded_image_fst_and_snd.2 hs).2\n\nvariable {s : Set α} {t : Set β} {S : ∀ i, Set (X i)}\n\ntheorem IsBounded.fst_of_prod (h : IsBounded (s ×ˢ t)) (ht : t.Nonempty) : IsBounded s :=\n  fst_image_prod s ht ▸ h.image_fst\n\ntheorem IsBounded.snd_of_prod (h : IsBounded (s ×ˢ t)) (hs : s.Nonempty) : IsBounded t :=\n  snd_image_prod hs t ▸ h.image_snd\n\ntheorem IsBounded.prod (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s ×ˢ t) :=\n  isBounded_image_fst_and_snd.1\n    ⟨hs.subset <| fst_image_prod_subset _ _, ht.subset <| snd_image_prod_subset _ _⟩\n\ntheorem isBounded_prod_of_nonempty (hne : Set.Nonempty (s ×ˢ t)) :\n    IsBounded (s ×ˢ t) ↔ IsBounded s ∧ IsBounded t :=\n  ⟨fun h => ⟨h.fst_of_prod hne.snd, h.snd_of_prod hne.fst⟩, fun h => h.1.prod h.2⟩\n\ntheorem isBounded_prod : IsBounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ IsBounded s ∧ IsBounded t := by\n  rcases s.eq_empty_or_nonempty with (rfl | hs); · simp\n  rcases t.eq_empty_or_nonempty with (rfl | ht); · simp\n  simp only [hs.ne_empty, ht.ne_empty, isBounded_prod_of_nonempty (hs.prod ht), false_or]\n\ntheorem isBounded_prod_self : IsBounded (s ×ˢ s) ↔ IsBounded s := by\n  rcases s.eq_empty_or_nonempty with (rfl | hs); · simp\n  exact (isBounded_prod_of_nonempty (hs.prod hs)).trans and_self_iff\n\n/-!\n### Bounded sets in `Π i, X i`\n-/\n\n\ntheorem cobounded_pi : cobounded (∀ i, X i) = Filter.coprodᵢ fun i => cobounded (X i) :=\n  rfl\n\ntheorem forall_isBounded_image_eval_iff {s : Set (∀ i, X i)} :\n    (∀ i, IsBounded (eval i '' s)) ↔ IsBounded s :=\n  compl_mem_coprodᵢ.symm\n\nlemma IsBounded.image_eval {s : Set (∀ i, X i)} (hs : IsBounded s) (i : ι) :\n    IsBounded (eval i '' s) :=\n  forall_isBounded_image_eval_iff.2 hs i\n\ntheorem IsBounded.pi (h : ∀ i, IsBounded (S i)) : IsBounded (pi univ S) :=\n  forall_isBounded_image_eval_iff.1 fun i => (h i).subset eval_image_univ_pi_subset\n\ntheorem isBounded_pi_of_nonempty (hne : (pi univ S).Nonempty) :\n    IsBounded (pi univ S) ↔ ∀ i, IsBounded (S i) :=\n  ⟨fun H i => @eval_image_univ_pi _ _ _ i hne ▸ forall_isBounded_image_eval_iff.2 H i, IsBounded.pi⟩\n\ntheorem isBounded_pi : IsBounded (pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ i, IsBounded (S i) := by\n  by_cases hne : ∃ i, S i = ∅\n  · simp [hne, univ_pi_eq_empty_iff.2 hne]\n  · simp only [hne, false_or]\n    simp only [not_exists, ← nonempty_iff_ne_empty, ← univ_pi_nonempty_iff] at hne\n    exact isBounded_pi_of_nonempty hne\n\n/-!\n### Bounded sets in `{x // p x}`\n-/\n\n\ntheorem isBounded_induced {α β : Type*} [Bornology β] {f : α → β} {s : Set α} :\n    @IsBounded α (Bornology.induced f) s ↔ IsBounded (f '' s) :=\n  compl_mem_comap\n\ntheorem isBounded_image_subtype_val {p : α → Prop} {s : Set { x // p x }} :\n    IsBounded (Subtype.val '' s) ↔ IsBounded s :=\n  isBounded_induced.symm\n\nend Bornology\n\n/-!\n### Bounded spaces\n-/\n\n\nopen Bornology\n\ninstance [BoundedSpace α] [BoundedSpace β] : BoundedSpace (α × β) := by\n  simp [← cobounded_eq_bot_iff, cobounded_prod]\n\ninstance [∀ i, BoundedSpace (X i)] : BoundedSpace (∀ i, X i) := by\n  simp [← cobounded_eq_bot_iff, cobounded_pi]\n\nTarget:\ntheorem boundedSpace_induced_iff {α β : Type*} [Bornology β] {f : α → β} :\n    @BoundedSpace α (Bornology.induced f) ↔ IsBounded (range f) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Bornology","family_id":"boundedspace_induced_iff","file_id":"mathlib/Mathlib/Topology/Bornology/Constructions.lean","sample_id":"9f9a70e78a68c5afe844cbabb734e8a42046f9373491df56a598c7ff77b730eb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2fe6742b90ce6b6b787aeae843d8b7476e8ec34c6e8c54b97436e93c09e1ac78","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9edf83ad0b7479676e136f7279c23e835e46a349748bddabb519ed9ff699acc7","source_sha256":"06fcefd9d2e8fefa05b9d4bdca9149cb45a5aceb7ddd5cd1ef4e639ced848f8f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : IsNoetherianRing A := Algebra.FiniteType.isNoetherianRing R A\n  rw [finite_iff_isArtinianRing, isArtinianRing_iff_isNoetherianRing_krullDimLE_zero,\n    and_iff_right this]","hard_negative":false,"metrics":{"chosen_tokens":22,"rejected_tokens":3,"token_jaccard":0.047619,"token_length_ratio":0.136364},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"b56b454402599c5959b0ec7f09ea09ade70068ddd6f350240aca62598ff80d5c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.HopkinsLevitzki\npublic import Mathlib.RingTheory.Jacobson.Ring\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu, Andrew Yang\n-/\n/-!\n# Artinian rings over Jacobson rings\n\n## Main results\n- `Module.finite_iff_isArtinianRing`: If `A` is a finite type algebra over an Artinian ring `R`,\n  then `A` is finite over `R` if and only if `A` is an Artinian ring.\n\n-/\n\npublic section\n\nvariable (R A : Type*) [CommRing R] [CommRing A] [Algebra R A] [Algebra.FiniteType R A]\n\nattribute [local instance] IsArtinianRing.fieldOfSubtypeIsMaximal in\nlemma Module.finite_of_isSemisimpleRing [IsJacobsonRing R] [IsSemisimpleRing A] :\n    Module.Finite R A :=\n  have (I : MaximalSpectrum A) := finite_of_finite_type_of_isJacobsonRing R (A ⧸ I.asIdeal)\n  .equiv ((IsArtinianRing.equivPi A).restrictScalars R).toLinearEquiv.symm\n\n/-- If `A` is a finite type algebra over `R`, then `A` is an Artinian ring and `R` is Jacobson\nimplies `A` is finite over `R`. -/\n/- If made an instance, causes timeouts synthesizing `FaithfulSMul R I.ResidueField` at\n`Ideal.algebraMap_residueField_eq_zero` and `Ideal.ker_algebraMap_residueField` during\nsimpNF linting. -/\nlemma Module.finite_of_isArtinianRing [IsJacobsonRing R] [IsArtinianRing A] :\n    Module.Finite R A :=\n  have := finite_of_isSemisimpleRing R (A ⧸ Ring.jacobson A)\n  IsSemiprimaryRing.finite_of_isArtinian R A A\n\n/-- If `A` is a finite type algebra over an Artinian ring `R`,\nthen `A` is finite over `R` if and only if `A` is an Artinian ring. -/\nlemma Module.finite_iff_isArtinianRing [IsArtinianRing R] :\n    Module.Finite R A ↔ IsArtinianRing A :=\n  ⟨isArtinian_of_tower _ ∘ ((IsArtinianRing.tfae R A).out 0 2).mp,\n    fun _ ↦ finite_of_isArtinianRing R A⟩\n\n/-- If `A` is a finite type algebra over an Artinian ring `R`,\nthen `A` is finite over `R` if and only if `dim A = 0`. -/\n\nTarget:\nlemma Module.finite_iff_krullDimLE_zero [IsArtinianRing R] :\n    Module.Finite R A ↔ Ring.KrullDimLE 0 A :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Jacobson","family_id":"module","file_id":"mathlib/Mathlib/RingTheory/Jacobson/Artinian.lean","sample_id":"9edf83ad0b7479676e136f7279c23e835e46a349748bddabb519ed9ff699acc7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ca01e87830c1462b512c1bad10dd02834fa50907bb2a15cdffba13757dee4a07","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"32a4aa28ca53ec5f375880fb329a6c92a05a17c1c9d5e8c664a77a4bde982211","source_sha256":"dbbf637f35822ee42ab01ded669b703900488e4b298d437d2dd114913838068f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine mt ((injective_iff_map_eq_zero _).mp bS.algebraMap_injective _) ?_\n  simp only [Finset.prod_eq_zero_iff, not_exists]\n  rintro x ⟨hx, rfl⟩\n  exact finsetApprox.zero_notMem bS adm hx","hard_negative":false,"metrics":{"chosen_tokens":40,"rejected_tokens":2,"token_jaccard":0.066667,"token_length_ratio":0.05},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"b5789e37ebbdbe9370f3f392555d742b65c21135ed23aedf208525c81079b22c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Pow.Real\npublic import Mathlib.LinearAlgebra.FreeModule.PID\npublic import Mathlib.LinearAlgebra.Matrix.AbsoluteValue\npublic import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue\npublic import Mathlib.RingTheory.ClassGroup.Basic\npublic import Mathlib.RingTheory.DedekindDomain.IntegralClosure\npublic import Mathlib.RingTheory.Norm.Basic\n\nNamespace:\nClassGroup\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Class numbers of global fields\nIn this file, we use the notion of \"admissible absolute value\" to prove\nfiniteness of the class group for number fields and function fields.\n\n## Main definitions\n- `ClassGroup.fintypeOfAdmissibleOfAlgebraic`: if `R` has an admissible absolute value,\n  its integral closure has a finite class group\n-/\n\n@[expose] public section\n\nopen Module Ring\nopen scoped nonZeroDivisors\n\nnamespace ClassGroup\nsection EuclideanDomain\n\nvariable {R S : Type*} (K L : Type*) [EuclideanDomain R] [CommRing S] [IsDomain S]\nvariable [Field K] [Field L]\nvariable [Algebra R K] [IsFractionRing R K]\nvariable [Algebra K L] [FiniteDimensional K L] [Algebra.IsSeparable K L]\nvariable [algRL : Algebra R L] [IsScalarTower R K L]\nvariable [Algebra R S] [Algebra S L]\nvariable [ist : IsScalarTower R S L]\nvariable (abv : AbsoluteValue R ℤ)\nvariable {ι : Type*} [DecidableEq ι] [Fintype ι] (bS : Basis ι R S)\n\n/-- If `b` is an `R`-basis of `S` of cardinality `n`, then `normBound abv b` is an integer\nsuch that for every `R`-integral element `a : S` with coordinates `≤ y`,\nwe have `Algebra.norm a ≤ normBound abv b * y ^ n`. (See also `norm_le` and `norm_lt`). -/\nnoncomputable def normBound : ℤ :=\n  let n := Fintype.card ι\n  let i : ι := Nonempty.some bS.index_nonempty\n  let m : ℤ :=\n    Finset.max'\n      (Finset.univ.image fun ijk : ι × ι × ι =>\n        abv (Algebra.leftMulMatrix bS (bS ijk.1) ijk.2.1 ijk.2.2))\n      ⟨_, Finset.mem_image.mpr ⟨⟨i, i, i⟩, Finset.mem_univ _, rfl⟩⟩\n  Nat.factorial n • (n • m) ^ n\n\ntheorem normBound_pos : 0 < normBound abv bS := by\n  obtain ⟨i, j, k, hijk⟩ : ∃ i j k, Algebra.leftMulMatrix bS (bS i) j k ≠ 0 := by\n    by_contra! h\n    obtain ⟨i⟩ := bS.index_nonempty\n    apply bS.ne_zero i\n    apply\n      (injective_iff_map_eq_zero (Algebra.leftMulMatrix bS)).mp (Algebra.leftMulMatrix_injective bS)\n    ext j k\n    simp [h]\n  simp only [normBound, Algebra.smul_def, eq_natCast]\n  apply mul_pos (Int.natCast_pos.mpr (Nat.factorial_pos _))\n  refine pow_pos (mul_pos (Int.natCast_pos.mpr (Fintype.card_pos_iff.mpr ⟨i⟩)) ?_) _\n  refine lt_of_lt_of_le (abv.pos hijk) (Finset.le_max' _ _ ?_)\n  exact Finset.mem_image.mpr ⟨⟨i, j, k⟩, Finset.mem_univ _, rfl⟩\n\n/-- If the `R`-integral element `a : S` has coordinates `≤ y` with respect to some basis `b`,\nits norm is less than `normBound abv b * y ^ dim S`. -/\ntheorem norm_le (a : S) {y : ℤ} (hy : ∀ k, abv (bS.repr a k) ≤ y) :\n    abv (Algebra.norm R a) ≤ normBound abv bS * y ^ Fintype.card ι := by\n  conv_lhs => rw [← bS.sum_repr a]\n  rw [Algebra.norm_apply, ← LinearMap.det_toMatrix bS]\n  simp only [map_sum, map_smul, map_sum, map_smul,\n    normBound, smul_mul_assoc, ← mul_pow]\n  convert! Matrix.det_sum_smul_le Finset.univ _ hy using 3\n  · rw [Finset.card_univ, smul_mul_assoc, mul_comm]\n  · intro i j k\n    apply Finset.le_max'\n    exact Finset.mem_image.mpr ⟨⟨i, j, k⟩, Finset.mem_univ _, rfl⟩\n\n/-- If the `R`-integral element `a : S` has coordinates `< y` with respect to some basis `b`,\nits norm is strictly less than `normBound abv b * y ^ dim S`. -/\ntheorem norm_lt {T : Type*} [Ring T] [LinearOrder T] [IsStrictOrderedRing T] (a : S) {y : T}\n    (hy : ∀ k, (abv (bS.repr a k) : T) < y) :\n    (abv (Algebra.norm R a) : T) < normBound abv bS * y ^ Fintype.card ι := by\n  obtain ⟨i⟩ := bS.index_nonempty\n  have him : (Finset.univ.image fun k => abv (bS.repr a k)).Nonempty :=\n    ⟨_, Finset.mem_image.mpr ⟨i, Finset.mem_univ _, rfl⟩⟩\n  set y' : ℤ := Finset.max' _ him with y'_def\n  have hy' : ∀ k, abv (bS.repr a k) ≤ y' := by\n    intro k\n    exact @Finset.le_max' ℤ _ _ _ (Finset.mem_image.mpr ⟨k, Finset.mem_univ _, rfl⟩)\n  have : (y' : T) < y := by\n    rw [y'_def,\n      ← Finset.max'_image (show Monotone (_ : ℤ → T) from fun x y h => Int.cast_le.mpr h)\n          _ (him.image _)]\n    apply (Finset.max'_lt_iff _ (him.image _)).mpr\n    simp only [Finset.mem_image]\n    rintro _ ⟨x, ⟨k, -, rfl⟩, rfl⟩\n    exact hy k\n  have y'_nonneg : 0 ≤ y' := le_trans (abv.nonneg _) (hy' i)\n  apply (Int.cast_le.mpr (norm_le abv bS a hy')).trans_lt\n  simp only [Int.cast_mul, Int.cast_pow]\n  apply mul_lt_mul' le_rfl\n  · exact pow_lt_pow_left₀ this (by positivity) (@Fintype.card_ne_zero _ _ ⟨i⟩)\n  · positivity\n  · exact Int.cast_pos.mpr (normBound_pos abv bS)\n\n\n/-- A nonzero ideal has an element of minimal norm. -/\ntheorem exists_min (I : (Ideal S)⁰) :\n    ∃ b ∈ (I : Ideal S),\n      b ≠ 0 ∧ ∀ c ∈ (I : Ideal S), abv (Algebra.norm R c) < abv (Algebra.norm R b) → c =\n      (0 : S) := by\n  obtain ⟨_, ⟨b, b_mem, b_ne_zero, rfl⟩, min⟩ := @Int.exists_least_of_bdd\n      (fun a => ∃ b ∈ (I : Ideal S), b ≠ (0 : S) ∧ abv (Algebra.norm R b) = a)\n    (by\n      use 0\n      rintro _ ⟨b, _, _, rfl⟩\n      apply abv.nonneg)\n    (by\n      obtain ⟨b, b_mem, b_ne_zero⟩ := (I : Ideal S).ne_bot_iff.mp (nonZeroDivisors.coe_ne_zero I)\n      exact ⟨_, ⟨b, b_mem, b_ne_zero, rfl⟩⟩)\n  refine ⟨b, b_mem, b_ne_zero, ?_⟩\n  intro c hc lt\n  contrapose! lt with c_ne_zero\n  exact min _ ⟨c, hc, c_ne_zero, rfl⟩\n\nsection IsAdmissible\n\nvariable {abv}\nvariable (adm : abv.IsAdmissible)\n\n/-- If we have a large enough set of elements in `R^ι`, then there will be a pair\nwhose remainders are close together. We'll show that all sets of cardinality\nat least `cardM bS adm` elements satisfy this condition.\n\nThe value of `cardM` is not at all optimal: for specific choices of `R`,\nthe minimum cardinality can be exponentially smaller.\n-/\nnoncomputable def cardM : ℕ :=\n  adm.card (normBound abv bS ^ (-1 / Fintype.card ι : ℝ)) ^ Fintype.card ι\n\nvariable [Infinite R]\n\n/-- In the following results, we need a large set of distinct elements of `R`. -/\nnoncomputable def distinctElems : Fin (cardM bS adm).succ ↪ R :=\n  Fin.valEmbedding.trans (Infinite.natEmbedding R)\n\nvariable [DecidableEq R]\n\n/-- `finsetApprox` is a finite set such that each fractional ideal in the integral closure\ncontains an element close to `finsetApprox`. -/\nnoncomputable def finsetApprox : Finset R :=\n  (Finset.univ.image fun xy : _ × _ => distinctElems bS adm xy.1 - distinctElems bS adm xy.2).erase\n    0\n\ntheorem finsetApprox.zero_notMem : (0 : R) ∉ finsetApprox bS adm :=\n  Finset.notMem_erase _ _\n\n@[simp]\ntheorem mem_finsetApprox {x : R} :\n    x ∈ finsetApprox bS adm ↔ ∃ i j, i ≠ j ∧ distinctElems bS adm i - distinctElems bS adm j =\n    x := by\n  simp only [finsetApprox, Finset.mem_erase, Finset.mem_image]\n  constructor\n  · rintro ⟨hx, ⟨i, j⟩, _, rfl⟩\n    refine ⟨i, j, ?_, rfl⟩\n    rintro rfl\n    simp at hx\n  · rintro ⟨i, j, hij, rfl⟩\n    refine ⟨?_, ⟨i, j⟩, Finset.mem_univ _, rfl⟩\n    rw [Ne, sub_eq_zero]\n    exact fun h => hij ((distinctElems bS adm).injective h)\n\nsection Real\n\nopen Real\n\nattribute [-instance] Real.decidableEq\n\n/-- We can approximate `a / b : L` with `q / r`, where `r` has finitely many options for `L`. -/\ntheorem exists_mem_finsetApprox (a : S) {b} (hb : b ≠ (0 : R)) :\n    ∃ q : S,\n      ∃ r ∈ finsetApprox bS adm, abv (Algebra.norm R (r • a - b • q)) <\n      abv (Algebra.norm R (algebraMap R S b)) := by\n  have dim_pos := Fintype.card_pos_iff.mpr bS.index_nonempty\n  set ε : ℝ := normBound abv bS ^ (-1 / Fintype.card ι : ℝ) with ε_eq\n  have hε : 0 < ε := Real.rpow_pos_of_pos (Int.cast_pos.mpr (normBound_pos abv bS)) _\n  have ε_le : (normBound abv bS : ℝ) * (abv b • ε) ^ (Fintype.card ι : ℝ)\n                ≤ abv b ^ (Fintype.card ι : ℝ) := by\n    have := normBound_pos abv bS\n    have := abv.nonneg b\n    rw [ε_eq, Algebra.smul_def, eq_intCast, mul_rpow, ← rpow_mul, div_mul_cancel₀, rpow_neg_one,\n      mul_left_comm, mul_inv_cancel₀, mul_one, rpow_natCast] <;>\n      try norm_cast; lia\n    · exact Int.cast_nonneg this\n    · linarith\n  set μ : Fin (cardM bS adm).succ ↪ R := distinctElems bS adm\n  let s : ι →₀ R := bS.repr a\n  have s_eq : ∀ i, s i = bS.repr a i := fun i => rfl\n  let qs : Fin (cardM bS adm).succ → ι → R := fun j i => μ j * s i / b\n  let rs : Fin (cardM bS adm).succ → ι → R := fun j i => μ j * s i % b\n  have r_eq : ∀ j i, rs j i = μ j * s i % b := fun i j => rfl\n  have μ_eq : ∀ i j, μ j * s i = b * qs j i + rs j i := by\n    intro i j\n    rw [r_eq, EuclideanDomain.div_add_mod]\n  have μ_mul_a_eq : ∀ j, μ j • a = b • ∑ i, qs j i • bS i + ∑ i, rs j i • bS i := by\n    intro j\n    rw [← bS.sum_repr a]\n    simp only [μ, qs, rs, Finset.smul_sum, ← Finset.sum_add_distrib]\n    refine Finset.sum_congr rfl fun i _ => ?_\n    rw [← s_eq, ← mul_smul, μ_eq, add_smul, mul_smul, ← μ_eq]\n  obtain ⟨j, k, j_ne_k, hjk⟩ := adm.exists_approx hε hb fun j i => μ j * s i\n  have hjk' : ∀ i, (abv (rs k i - rs j i) : ℝ) < abv b • ε := by simpa only [r_eq] using hjk\n  let q := ∑ i, (qs k i - qs j i) • bS i\n  set r := μ k - μ j with r_eq\n  refine ⟨q, r, (mem_finsetApprox bS adm).mpr ?_, ?_⟩\n  · exact ⟨k, j, j_ne_k.symm, rfl⟩\n  have : r • a - b • q = ∑ x : ι, (rs k x • bS x - rs j x • bS x) := by\n    simp only [q, r_eq, sub_smul, μ_mul_a_eq, Finset.smul_sum, ← Finset.sum_add_distrib,\n      ← Finset.sum_sub_distrib, smul_sub]\n    refine Finset.sum_congr rfl fun x _ => ?_\n    ring\n  rw [this, Algebra.norm_algebraMap_of_basis bS, abv.map_pow]\n  refine Int.cast_lt.mp ((norm_lt abv bS _ fun i => lt_of_le_of_lt ?_ (hjk' i)).trans_le ?_)\n  · apply le_of_eq\n    congr\n    simp_rw [map_sum, map_sub, map_smul, Finset.sum_apply',\n      Finsupp.sub_apply, Finsupp.smul_apply, Finset.sum_sub_distrib, Basis.repr_self_apply,\n      smul_eq_mul, mul_boole, Finset.sum_ite_eq', Finset.mem_univ, if_true]\n  · exact mod_cast ε_le\n\n/-- We can approximate `a / b : L` with `q / r`, where `r` has finitely many options for `L`. -/\ntheorem exists_mem_finset_approx' [Algebra.IsAlgebraic R S] (a : S) {b : S} (hb : b ≠ 0) :\n    ∃ q : S,\n      ∃ r ∈ finsetApprox bS adm, abv (Algebra.norm R (r • a - q * b)) < abv (Algebra.norm R b) := by\n  obtain ⟨a', b', hb', h⟩ := Algebra.IsAlgebraic.exists_smul_eq_mul R a hb\n  obtain ⟨q, r, hr, hqr⟩ := exists_mem_finsetApprox bS adm a' hb'\n  refine ⟨q, r, hr, ?_⟩\n  refine\n    lt_of_mul_lt_mul_left ?_ (show 0 ≤ abv (Algebra.norm R (algebraMap R S b')) from abv.nonneg _)\n  refine\n    lt_of_le_of_lt (le_of_eq ?_)\n      (mul_lt_mul hqr le_rfl (abv.pos ((Algebra.norm_ne_zero_iff_of_basis bS).mpr hb))\n        (abv.nonneg _))\n  rw [← abv.map_mul, ← map_mul, ← abv.map_mul, ← map_mul, ← Algebra.smul_def,\n    smul_sub b', sub_mul, smul_comm, h, mul_comm b a', Algebra.smul_mul_assoc r a' b,\n    Algebra.smul_mul_assoc b' q b]\n\nend Real\n\nTarget:\ntheorem prod_finsetApprox_ne_zero : algebraMap R S (∏ m ∈ finsetApprox bS adm, m) ≠ 0 :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/ClassNumber","family_id":"prod_finsetapprox_ne_zero","file_id":"mathlib/Mathlib/NumberTheory/ClassNumber/Finite.lean","sample_id":"32a4aa28ca53ec5f375880fb329a6c92a05a17c1c9d5e8c664a77a4bde982211"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a3b30f98b08bb268a88816f205f3d07c5c34faaa5cf59c628e0cdbe5b5a6a78c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"10a45c3e9911e5b7b3fc6816e3e6db463f07fc1907b25aacc0bdf965063786b1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e87fe6db087248fc861a35d6fd880f370abae53489bf00891ff2b411146d68fa","source_sha256":"00cdc131b09c868f94179716d61273d8fdf897ed023d34f30c7085668a3b31ac","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [mul_comm]\n  exact mul_left_not_lt b h","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":2,"token_jaccard":0.1,"token_length_ratio":0.222222},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"b58a35579fec5583888a353a943f4675d0443910cd36bb092fb95a28a678d0bf","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Order.RelClasses\n\nNamespace:\nEuclideanDomain\n\nLocal context:\n/-\nCopyright (c) 2018 Louis Carlin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Louis Carlin, Mario Carneiro\n-/\n/-!\n# Euclidean domains\n\nThis file introduces Euclidean domains and provides the extended Euclidean algorithm. To be precise,\na slightly more general version is provided which is sometimes called a transfinite Euclidean domain\nand differs in the fact that the degree function need not take values in `ℕ` but can take values in\nany well-ordered set. Transfinite Euclidean domains were introduced by Motzkin and examples which\ndon't satisfy the classical notion were provided independently by Hiblot and Nagata.\n\n## Main definitions\n\n* `EuclideanDomain`: Defines Euclidean domain with functions `quotient` and `remainder`. Instances\n  of `Div` and `Mod` are provided, so that one can write `a = b * (a / b) + a % b`.\n* `gcd`: defines the greatest common divisors of two elements of a Euclidean domain.\n* `xgcd`: given two elements `a b : R`, `xgcd a b` defines the pair `(x, y)` such that\n  `x * a + y * b = gcd a b`.\n* `lcm`: defines the lowest common multiple of two elements `a` and `b` of a Euclidean domain as\n  `a * b / (gcd a b)`\n\n## Main statements\n\nSee `Algebra.EuclideanDomain.Basic` for most of the theorems about Euclidean domains,\nincluding Bézout's lemma.\n\nSee `Algebra.EuclideanDomain.Instances` for the fact that `ℤ` is a Euclidean domain,\nas is any field.\n\n## Notation\n\n`≺` denotes the well-founded relation on the Euclidean domain, e.g. in the example of the polynomial\nring over a field, `p ≺ q` for polynomials `p` and `q` if and only if the degree of `p` is less than\nthe degree of `q`.\n\n## Implementation details\n\nInstead of working with a valuation, `EuclideanDomain` is implemented with the existence of a well\nfounded relation `r` on the integral domain `R`, which in the example of `ℤ` would correspond to\nsetting `i ≺ j` for integers `i` and `j` if the absolute value of `i` is smaller than the absolute\nvalue of `j`.\n\n## References\n\n* [Th. Motzkin, *The Euclidean algorithm*][MR32592]\n* [J.-J. Hiblot, *Des anneaux euclidiens dont le plus petit algorithme n'est pas à valeurs finies*]\n  [MR399081]\n* [M. Nagata, *On Euclid algorithm*][MR541021]\n\n\n## Tags\n\nEuclidean domain, transfinite Euclidean domain, Bézout's lemma\n-/\n\n@[expose] public section\n\nuniverse u\n\n/-- A `EuclideanDomain` is a non-trivial commutative ring with a division and a remainder,\n  satisfying `b * (a / b) + a % b = a`.\n  The definition of a Euclidean domain usually includes a valuation function `R → ℕ`.\n  This definition is slightly generalised to include a well-founded relation\n  `r` with the property that `r (a % b) b`, instead of a valuation. -/\n@[wikidata Q867345]\nclass EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R where\n  /-- A division function (denoted `/`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `%` denotes `remainder`. -/\n  protected quotient : R → R → R\n  /-- Division by zero should always give zero by convention. -/\n  protected quotient_zero : ∀ a, quotient a 0 = 0\n  /-- A remainder function (denoted `%`) on `R`.\n    This satisfies the property `b * (a / b) + a % b = a`, where `/` denotes `quotient`. -/\n  protected remainder : R → R → R\n  /-- The property that links the quotient and remainder functions.\n    This allows us to compute GCDs and LCMs. -/\n  protected quotient_mul_add_remainder_eq : ∀ a b, b * quotient a b + remainder a b = a\n  /-- A well-founded relation on `R`, satisfying `r (a % b) b`.\n    This ensures that the GCD algorithm always terminates. -/\n  protected r : R → R → Prop\n  /-- The relation `r` must be well-founded.\n    This ensures that the GCD algorithm always terminates. -/\n  r_wellFounded : WellFounded r\n  /-- The relation `r` satisfies `r (a % b) b`. -/\n  protected remainder_lt : ∀ (a) {b}, b ≠ 0 → r (remainder a b) b\n  /-- An additional constraint on `r`. -/\n  mul_left_not_lt : ∀ (a) {b}, b ≠ 0 → ¬r (a * b) a\n\n/-\nLean has far more theorems about fields than about Euclidean domains. We thus\nlower the priority of `Euclideandomain.toCommRing`, encouraging typeclass inference\nto try `Field.toCommRing` first. Without this priority-lowering, typeclass inference\nfinds the more inefficient path `Field.toEuclideanDomain.toCommRing` by default. This\npriority change saves over 500G instructions across mathlib. See\nhttps://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/We.20need.20to.20talk.20about.20Euclidean.20Domains/near/594655420\n-/\n-- see Note [lower instance priority]\nattribute [instance 100] EuclideanDomain.toCommRing\n\nnamespace EuclideanDomain\n\nvariable {R : Type u} [EuclideanDomain R]\n\n/-- Abbreviated notation for the well-founded relation `r` in a Euclidean domain. -/\nlocal infixl:50 \" ≺ \" => EuclideanDomain.r\n\nlocal instance wellFoundedRelation : WellFoundedRelation R where\n  rel := EuclideanDomain.r\n  wf := r_wellFounded\n\ninstance isWellFounded : IsWellFounded R (· ≺ ·) where\n  wf := r_wellFounded\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Div R :=\n  ⟨EuclideanDomain.quotient⟩\n\n-- see Note [lower instance priority]\ninstance (priority := 70) : Mod R :=\n  ⟨EuclideanDomain.remainder⟩\n\ntheorem div_add_mod (a b : R) : b * (a / b) + a % b = a :=\n  EuclideanDomain.quotient_mul_add_remainder_eq _ _\n\ntheorem mod_add_div (a b : R) : a % b + b * (a / b) = a :=\n  (add_comm _ _).trans (div_add_mod _ _)\n\ntheorem mod_add_div' (m k : R) : m % k + m / k * k = m := by\n  rw [mul_comm]\n  exact mod_add_div _ _\n\ntheorem div_add_mod' (m k : R) : m / k * k + m % k = m := by\n  rw [mul_comm]\n  exact div_add_mod _ _\n\ntheorem mod_lt : ∀ (a) {b : R}, b ≠ 0 → a % b ≺ b :=\n  EuclideanDomain.remainder_lt\n\nTarget:\ntheorem mul_right_not_lt {a : R} (b) (h : a ≠ 0) : ¬a * b ≺ b :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_e87fe6db0872","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"5b45c37cfb1c4d5e9d2bf2812af809dbb59514f5517dd2d6f4d802304d1c3b6f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/EuclideanDomain","family_id":"mul_right_not_lt","file_id":"mathlib/Mathlib/Algebra/EuclideanDomain/Defs.lean","sample_id":"e87fe6db087248fc861a35d6fd880f370abae53489bf00891ff2b411146d68fa"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a6d320117b4f0031ed5a5af663cf8c7397e183bff5fdfb69cb2aa2ba76b6c4c2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4966b3585ecb4af08a122a7051ce365cbfa0c8a70a6162ab9acb0ed1c2e560de","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a81f25208ce01c9c1e94e1a7e16ac4440ac59daf394e99a5ec3de20f35dc63a8","source_sha256":"22d84e531512fb453c5c22eb12c263f109619421771ae337215e1ec65fb0d46f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← mul_right_inj (x : M), mul_inv_cancel, mul_leftInvEquiv_symm]","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":22,"token_jaccard":0.875,"token_length_ratio":1.466667},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"b5ae2932e37eb0ec40caff371064a0bd2054e1255c2405d7d5eb4e81b0d4f709","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Submonoid.Pointwise\n\nNamespace:\nSubmonoid\n\nLocal context:\n/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Submonoid of inverses\n\nGiven a submonoid `N` of a monoid `M`, we define the submonoid `N.leftInv` as the submonoid of\nleft inverses of `N`. When `M` is commutative, we may define `fromCommLeftInv : N.leftInv →* N`\nsince the inverses are unique. When `N ≤ IsUnit.Submonoid M`, this is precisely\nthe pointwise inverse of `N`, and we may define `leftInvEquiv : S.leftInv ≃* S`.\n\nFor the pointwise inverse of submonoids of groups, please refer to the file\n`Mathlib/Algebra/Group/Submonoid/Pointwise.lean`.\n\n`N.leftInv` is distinct from `N.units`, which is the subgroup of `Mˣ` containing all units that are\nin `N`. See the implementation notes of `Mathlib/Algebra/Group/Submonoid/Units.lean` for more\ndetails on related constructions.\n\n## TODO\n\nDefine the submonoid of right inverses and two-sided inverses.\nSee the comments of https://github.com/leanprover-community/mathlib4/pull/10679 for a possible\nimplementation.\n-/\n\n@[expose] public section\n\n\nvariable {M : Type*}\n\nnamespace Submonoid\n\n@[to_additive]\nnoncomputable instance [Monoid M] : Group (IsUnit.submonoid M) :=\n  { (inferInstance : Monoid (IsUnit.submonoid M)) with\n    inv := fun x ↦ ⟨x.prop.unit⁻¹.val, x.prop.unit⁻¹.isUnit⟩\n    inv_mul_cancel := fun x ↦\n      Subtype.ext ((Units.val_mul x.prop.unit⁻¹ _).trans x.prop.unit.inv_val) }\n\n@[to_additive]\nnoncomputable instance [CommMonoid M] : CommGroup (IsUnit.submonoid M) :=\n  { (inferInstance : Group (IsUnit.submonoid M)) with\n    mul_comm := fun a b ↦ by convert! mul_comm a b }\n\n@[to_additive]\ntheorem _root_.IsUnit.submonoid.coe_inv [Monoid M] (x : IsUnit.submonoid M) :\n    ↑x⁻¹ = (↑x.prop.unit⁻¹ : M) :=\n  rfl\n\n@[deprecated (since := \"2026-05-24\")]\nalias _root_.AddSubmonoid.IsUnit.Submonoid.coe_neg := IsAddUnit.addSubmonoid.coe_neg\n@[to_additive existing, deprecated (since := \"2026-05-24\")]\nalias IsUnit.Submonoid.coe_inv := IsUnit.submonoid.coe_inv\n\nsection Monoid\n\nvariable [Monoid M] (S : Submonoid M)\n\n/-- `S.leftInv` is the submonoid containing all the left inverses of `S`. -/\n@[to_additive\n/-- `S.leftNeg` is the additive submonoid containing all the left additive inverses of `S`. -/]\ndef leftInv : Submonoid M where\n  carrier := { x : M | ∃ y : S, x * y = 1 }\n  one_mem' := ⟨1, mul_one 1⟩\n  mul_mem' := fun {a} _b ⟨a', ha⟩ ⟨b', hb⟩ ↦\n    ⟨b' * a', by simp only [coe_mul, ← mul_assoc, mul_assoc a, hb, mul_one, ha]⟩\n\n@[to_additive]\ntheorem leftInv_leftInv_le : S.leftInv.leftInv ≤ S := by\n  rintro x ⟨⟨y, z, h₁⟩, h₂ : x * y = 1⟩\n  convert! z.prop\n  rw [← mul_one x, ← h₁, ← mul_assoc, h₂, one_mul]\n\n@[to_additive]\ntheorem unit_mem_leftInv (x : Mˣ) (hx : (x : M) ∈ S) : ((x⁻¹ :) : M) ∈ S.leftInv :=\n  ⟨⟨x, hx⟩, x.inv_val⟩\n\n@[to_additive]\ntheorem leftInv_leftInv_eq (hS : S ≤ IsUnit.submonoid M) : S.leftInv.leftInv = S := by\n  refine le_antisymm S.leftInv_leftInv_le ?_\n  intro x hx\n  have : x = ((hS hx).unit⁻¹⁻¹ : Mˣ) := by\n    rw [inv_inv (hS hx).unit]\n    rfl\n  rw [this]\n  exact S.leftInv.unit_mem_leftInv _ (S.unit_mem_leftInv _ hx)\n\n/-- The function from `S.leftInv` to `S` sending an element to its right inverse in `S`.\nThis is a `MonoidHom` when `M` is commutative. -/\n@[to_additive\n/-- The function from `S.leftAdd` to `S` sending an element to its right additive\ninverse in `S`. This is an `AddMonoidHom` when `M` is commutative. -/]\nnoncomputable def fromLeftInv : S.leftInv → S := fun x ↦ x.prop.choose\n\n@[to_additive (attr := simp)]\ntheorem mul_fromLeftInv (x : S.leftInv) : (x : M) * S.fromLeftInv x = 1 :=\n  x.prop.choose_spec\n\n@[to_additive (attr := simp)]\ntheorem fromLeftInv_one : S.fromLeftInv 1 = 1 :=\n  (one_mul _).symm.trans (Subtype.ext <| S.mul_fromLeftInv 1)\n\nend Monoid\n\nsection CommMonoid\n\nvariable [CommMonoid M] (S : Submonoid M)\n\n@[to_additive (attr := simp)]\ntheorem fromLeftInv_mul (x : S.leftInv) : (S.fromLeftInv x : M) * x = 1 := by\n  rw [mul_comm, mul_fromLeftInv]\n\n@[to_additive]\ntheorem leftInv_le_isUnit : S.leftInv ≤ IsUnit.submonoid M := fun x ⟨y, hx⟩ ↦\n  ⟨⟨x, y, hx, mul_comm x y ▸ hx⟩, rfl⟩\n\n@[to_additive]\ntheorem fromLeftInv_eq_iff (a : S.leftInv) (b : M) :\n    (S.fromLeftInv a : M) = b ↔ (a : M) * b = 1 := by\n  rw [← IsUnit.mul_right_inj (leftInv_le_isUnit _ a.prop), S.mul_fromLeftInv, eq_comm]\n\n/-- The `MonoidHom` from `S.leftInv` to `S` sending an element to its right inverse in `S`. -/\n@[to_additive (attr := simps) /-- The `AddMonoidHom` from `S.leftNeg` to `S` sending an element to\nits right additive inverse in  `S`. -/]\nnoncomputable def fromCommLeftInv : S.leftInv →* S where\n  toFun := S.fromLeftInv\n  map_one' := S.fromLeftInv_one\n  map_mul' x y :=\n    Subtype.ext <| by\n      rw [fromLeftInv_eq_iff, mul_comm x, Submonoid.coe_mul, Submonoid.coe_mul, mul_assoc, ←\n        mul_assoc (x : M), mul_fromLeftInv, one_mul, mul_fromLeftInv]\n\nvariable (hS : S ≤ IsUnit.submonoid M)\n\n/-- The submonoid of pointwise inverse of `S` is `MulEquiv` to `S`. -/\n@[to_additive (attr := simps apply) /-- The additive submonoid of pointwise additive inverse of `S`\nis `AddEquiv` to `S`. -/]\nnoncomputable def leftInvEquiv : S.leftInv ≃* S :=\n  { S.fromCommLeftInv with\n    invFun := fun x ↦ ⟨↑(hS x.2).unit⁻¹, x, by simp⟩\n    left_inv := by\n      intro x\n      ext\n      simp [← Units.mul_eq_one_iff_inv_eq]\n    right_inv := by\n      rintro ⟨x, hx⟩\n      ext\n      simp [fromLeftInv_eq_iff] }\n\n@[to_additive (attr := simp)]\ntheorem fromLeftInv_leftInvEquiv_symm (x : S) : S.fromLeftInv ((S.leftInvEquiv hS).symm x) = x :=\n  (S.leftInvEquiv hS).right_inv x\n\n@[to_additive (attr := simp)]\ntheorem leftInvEquiv_symm_fromLeftInv (x : S.leftInv) :\n    (S.leftInvEquiv hS).symm (S.fromLeftInv x) = x :=\n  (S.leftInvEquiv hS).left_inv x\n\n@[to_additive]\ntheorem leftInvEquiv_mul (x : S.leftInv) : (S.leftInvEquiv hS x : M) * x = 1 := by\n  simpa only [leftInvEquiv_apply, fromCommLeftInv] using fromLeftInv_mul S x\n\n@[to_additive]\ntheorem mul_leftInvEquiv (x : S.leftInv) : (x : M) * S.leftInvEquiv hS x = 1 := by\n  simp only [leftInvEquiv_apply, fromCommLeftInv, mul_fromLeftInv]\n\n@[to_additive (attr := simp)]\ntheorem leftInvEquiv_symm_mul (x : S) : ((S.leftInvEquiv hS).symm x : M) * x = 1 := by\n  convert! S.mul_leftInvEquiv hS ((S.leftInvEquiv hS).symm x)\n  simp\n\n@[to_additive (attr := simp)]\ntheorem mul_leftInvEquiv_symm (x : S) : (x : M) * (S.leftInvEquiv hS).symm x = 1 := by\n  convert! S.leftInvEquiv_mul hS ((S.leftInvEquiv hS).symm x)\n  simp\n\nend CommMonoid\n\nsection Group\n\nvariable [Group M] (S : Submonoid M)\n\nopen scoped Pointwise\n\n@[to_additive]\ntheorem leftInv_eq_inv : S.leftInv = S⁻¹ :=\n  Submonoid.ext fun _ ↦\n    ⟨fun h ↦ Submonoid.mem_inv.mpr ((inv_eq_of_mul_eq_one_right h.choose_spec).symm ▸\n      h.choose.prop),\n      fun h ↦ ⟨⟨_, h⟩, mul_inv_cancel _⟩⟩\n\n@[to_additive (attr := simp)]\ntheorem fromLeftInv_eq_inv (x : S.leftInv) : (S.fromLeftInv x : M) = (x : M)⁻¹ := by\n  rw [← mul_right_inj (x : M), mul_inv_cancel, mul_fromLeftInv]\n\nend Group\n\nsection CommGroup\n\nvariable [CommGroup M] (S : Submonoid M) (hS : S ≤ IsUnit.submonoid M)\n\n@[to_additive (attr := simp)]\n\nTarget:\ntheorem leftInvEquiv_symm_eq_inv (x : S) : ((S.leftInvEquiv hS).symm x : M) = (x : M)⁻¹ :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [← mul_right_inj (x : M), mul_inv_cancel, mul_leftInvEquiv_symm]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Submonoid","family_id":"leftinvequiv_symm_eq_inv","file_id":"mathlib/Mathlib/GroupTheory/Submonoid/Inverses.lean","sample_id":"a81f25208ce01c9c1e94e1a7e16ac4440ac59daf394e99a5ec3de20f35dc63a8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ef657d6d5fc4b7d8887ac9017ba9ce06e5bcee28dc51338cd8d6ad2502a8eccb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"88998f6d1d9f42ffd0de1fc1703302506033691bb0a25ab3af2d1f4d8ddd6fc6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4b6a657545abaaf7eebda573d07dd91e0d6097e9312e9cce6c12048585c3e4e7","source_sha256":"ffda45eef0509a041f24079788fb5aa3f3f4500975a78975ccc64409c3de362b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [← union, disjoint_union_right, disjoint_iUnion_right]\n  exact ⟨disjointBase n i , fun _ _ ↦ disjoint_openCell_of_ne (by lia)⟩","hard_negative":true,"metrics":{"chosen_tokens":26,"rejected_tokens":3,"token_jaccard":0.086957,"token_length_ratio":0.115385},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"b5b644305a4bc30f47ab5d2ab05b81300e09cb52741c70993b7b243b6e0fc12d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.CWComplex.Classical.Finite\npublic import Mathlib.Analysis.Normed.Module.RCLike.Real\n\nNamespace:\nTopology\n\nLocal context:\n/-\nCopyright (c) 2025 Floris van Doorn and Hannah Scholz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Hannah Scholz\n-/\n/-!\n# Subcomplexes\n\nIn this file we discuss subcomplexes of CW complexes.\nThe definition of subcomplexes is in the file `Mathlib/Topology/CWComplex/Classical/Basic.lean`.\n\n## Main results\n* `RelCWComplex.Subcomplex.instRelCWComplex`: a subcomplex of a (relative) CW complex is again a\n  (relative) CW complex.\n\n## References\n* [K. Jänich, *Topology*][Janich1984]\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Metric Set\n\nnamespace Topology\n\nvariable {X : Type*} [t : TopologicalSpace X] {C D : Set X}\n\n@[alias_in CWComplex.Subcomplex]\nlemma RelCWComplex.Subcomplex.closedCell_subset_of_mem [T2Space X] [RelCWComplex C D]\n    (E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) :\n    closedCell n i ⊆ E := by\n  rw [← closure_openCell_eq_closedCell, E.closed.closure_subset_iff, ← E.union]\n  apply subset_union_of_subset_right\n  exact subset_iUnion_of_subset n\n    (subset_iUnion (fun (j : ↑(E.I n)) ↦ openCell (C := C) n j) ⟨i, hi⟩)\n\n@[alias_in CWComplex.Subcomplex]\nlemma RelCWComplex.Subcomplex.openCell_subset_of_mem [T2Space X] [RelCWComplex C D]\n    (E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) :\n    openCell n i ⊆ E :=\n  (openCell_subset_closedCell n i).trans (closedCell_subset_of_mem E hi)\n\n@[alias_in CWComplex.Subcomplex]\nlemma RelCWComplex.Subcomplex.cellFrontier_subset_of_mem [T2Space X] [RelCWComplex C D]\n    (E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) :\n    cellFrontier n i ⊆ E :=\n  (cellFrontier_subset_closedCell n i).trans (closedCell_subset_of_mem E hi)\n\n/-- A subcomplex is the union of its closed cells and its base. -/\nlemma RelCWComplex.Subcomplex.union_closedCell [T2Space X] [RelCWComplex C D] (E : Subcomplex C) :\n    D ∪ ⋃ (n : ℕ) (j : E.I n), closedCell (C := C) n j = E := by\n  apply subset_antisymm\n  · apply union_subset E.base_subset\n    exact iUnion₂_subset fun n i ↦ closedCell_subset_of_mem E i.2\n  · rw [← E.union]\n    apply union_subset_union_right\n    apply iUnion₂_mono fun n i ↦ ?_\n    exact openCell_subset_closedCell (C := C) n i\n\n/-- A subcomplex is the union of its closed cells. -/\nlemma CWComplex.Subcomplex.union_closedCell [T2Space X] [CWComplex C] (E : Subcomplex C) :\n    ⋃ (n : ℕ) (j : E.I n), closedCell (C := C) n j = E :=\n  (empty_union _).symm.trans (RelCWComplex.Subcomplex.union_closedCell E)\n\n@[alias_in CWComplex.Subcomplex]\n\nTarget:\nlemma RelCWComplex.Subcomplex.disjoint_openCell_subcomplex_of_not_mem [RelCWComplex C D]\n    (E : Subcomplex C) {n : ℕ} {i : cell C n} (h : i ∉ E.I n) : Disjoint (openCell n i) E :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_4b6a657545ab","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"308561b0193d05667e50dbbdaeaf05078510337353c6133486b5e57abe419859","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/CWComplex","family_id":"relcwcomplex","file_id":"mathlib/Mathlib/Topology/CWComplex/Classical/Subcomplex.lean","sample_id":"4b6a657545abaaf7eebda573d07dd91e0d6097e9312e9cce6c12048585c3e4e7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ff1ae62e27377baaefea54f50e782d5e8647a507a59b2f78b75ee566cf2b6bd5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5c3559b3cf2ee9a241bb81b7e23213b68dd9f8307cc2252bb99a753e0c04c076","source_sha256":"73ebc2e72f7678b20eb20bcb8222b112e9c86c209dc6aa3492fcf8a265aa9fa7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [iff_comp_injective]\n  intro B _ _ I hI f₁ f₂ e\n  ext x\n  have : f₁ x - f₂ x ∈ I := by\n    simpa [Ideal.Quotient.mk_eq_mk_iff_sub_mem] using AlgHom.congr_fun e x\n  have := Polynomial.eval_add_of_sq_eq_zero ((minpoly K x).map (algebraMap K B)) (f₂ x)\n    (f₁ x - f₂ x) (show (f₁ x - f₂ x) ^ 2 ∈ ⊥ from hI ▸ Ideal.pow_mem_pow this 2)\n  simp only [add_sub_cancel, eval_map_algebraMap, aeval_algHom_apply, minpoly.aeval, map_zero,\n    derivative_map, zero_add] at this\n  rwa [eq_comm, ((isUnit_iff_ne_zero.mpr\n    ((Algebra.IsSeparable.isSeparable K x).aeval_derivative_ne_zero\n      (minpoly.aeval K x))).map f₂).mul_right_eq_zero, sub_eq_zero] at this\n\nvariable [FormallyUnramified K A] [EssFiniteType K A]\nvariable [FormallyUnramified K L] [EssFiniteType K L]","hard_negative":false,"metrics":{"chosen_tokens":187,"rejected_tokens":2,"token_jaccard":0.013889,"token_length_ratio":0.010695},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"b60c025fc5f730573cd95f50e625985832e27f5c94a9dcaac7a7cc889fcd27b3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.PurelyInseparable.Basic\npublic import Mathlib.RingTheory.Artinian.Ring\npublic import Mathlib.RingTheory.Unramified.Finite\npublic import Mathlib.RingTheory.Unramified.Locus\n\nNamespace:\nAlgebra.FormallyUnramified\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Unramified algebras over fields\n\n## Main results\n\nLet `K` be a field, `A` be a `K`-algebra and `L` be a field extension of `K`.\n\n- `Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_isLocalRing`:\n    If `A` is `K`-unramified and `K` is alg-closed, then `K = A`.\n- `Algebra.FormallyUnramified.isReduced_of_field`:\n    If `A` is `K`-unramified then `A` is reduced.\n- `Algebra.FormallyUnramified.iff_isSeparable`:\n    `L` is unramified over `K` iff `L` is separable over `K`.\n\n## References\n\n- [B. Iversen, *Generic Local Structure of the Morphisms in Commutative Algebra*][iversen]\n\n-/\n\npublic section\n\nopen Algebra Module Polynomial\nopen scoped TensorProduct\n\nuniverse u\n\nvariable (K A L : Type*) [Field K] [Field L] [CommRing A] [Algebra K A] [Algebra K L]\n\nnamespace Algebra.FormallyUnramified\n\nTarget:\ntheorem of_isSeparable [Algebra.IsSeparable K L] : FormallyUnramified K L :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Unramified","family_id":"of_isseparable","file_id":"mathlib/Mathlib/RingTheory/Unramified/Field.lean","sample_id":"5c3559b3cf2ee9a241bb81b7e23213b68dd9f8307cc2252bb99a753e0c04c076"},"split":"train"}
{"binding":{"chosen_proof_sha256":"faeb8949323faf61ca5ff806cf60d2c823a2170b0852dfc5d54df3a865ffc0fd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8befb9059d33074287276d870632867722349ec1e96ce117c820c3b37806ed7c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"988fd8cb4d8e17e38f2a76981b98bfba51705b13e2c6dc4822f9d73a946a1e06","source_sha256":"8ca4ffc48e91301292a0b84aa8e139dd34c405da224b4628cd66cbb6fa0e45d8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [zariskisMainProperty_iff, ← top_le_iff, SetLike.le_def,\n    Submonoid.mem_powers_iff, mem_integralClosure_iff]","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":23,"token_jaccard":0.777778,"token_length_ratio":1.277778},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"b62d138266ed1e5f9934e927550d5f8bdfc5b66548d6ed304421b760723ef6d6","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Algebraic.StronglyTranscendental\npublic import Mathlib.RingTheory.Conductor\npublic import Mathlib.RingTheory.Ideal.Quotient.Nilpotent\npublic import Mathlib.RingTheory.IntegralClosure.GoingDown\npublic import Mathlib.RingTheory.Polynomial.IsIntegral\npublic import Mathlib.RingTheory.QuasiFinite.Polynomial\npublic import Mathlib.Algebra.Algebra.Shrink\n\nNamespace:\nAlgebra\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Algebraic Zariski's Main Theorem\n\nThe statement of Zariski's main theorem is the following:\nGiven a finite type `R`-algebra `S`, and `p` a prime of `S` such that `S` is quasi-finite at `R`,\nthen there exists a `f ∉ p` such that `S[1/f]` is isomorphic to `R'[1/f]` where `R'` is the integral\nclosure of `R` in `S`.\n\nWe follow https://stacks.math.columbia.edu/tag/00PI and proceed in the following steps\n\n1. `Algebra.ZariskisMainProperty.of_adjoin_eq_top`:\n  The case where `S = R[X]/I`.\n  The key is `Polynomial.not_ker_le_map_C_of_surjective_of_quasiFiniteAt`\n  which shows that there exists some `g ∈ I` such that some coefficient `gᵢ ∉ p`.\n  Then one basically takes `f = gᵢ` and `g` becomes monic in `R[1/gᵢ][X]` up to some minor technical\n  issues, and then `S[1/gᵢ]` is basically integral over `R[1/gᵢ]`.\n2. `Algebra.ZariskisMainProperty.of_algHom_polynomial`:\n  The case where `S` is finite over `R⟨x⟩` for some `x : S`.\n  The following key results are first established:\n  - `isStronglyTranscendental_mk_radical_conductor`:\n    Let `𝔣` be the conductor of `x` (i.e. the largest `S`-ideal in `R⟨x⟩`).\n    `x` as an element of `S/√𝔣` is strongly transcendental over `R`.\n  - `Algebra.not_quasiFiniteAt_of_stronglyTranscendental`:\n    If `S` is reduced, then `x : S` is not strongly transcendental over `R`.\n    One first reduces to when `R ⊆ S` are domains, and then to when `R` is integrally closed.\n    A going down theorem is now available, which could be applied to\n    `Polynomial.map_under_lt_comap_of_quasiFiniteAt`:`(p ∩ R)[X] < p ∩ R<x>` to get a contradiction.\n\n  The second result applied to `S/√𝔣` together with the first result implies that\n  `p` does not contain `𝔣`.\n  The claim then follows from `Localization.localRingHom_bijective_of_not_conductor_le`.\n3. `Algebra.ZariskisMainProperty.of_algHom_mvPolynomial`:\n  The case where `S` is finite over `R⟨x₁,...,xₙ⟩`. This is proved using induction on `n`.\n\n## Main definition and results\n- `Algebra.ZariskisMainProperty`:\n  We say that an `R` algebra `S` satisfies the Zariski's main property at a prime `p` of `S`\n  if there exists `r ∉ p` in the integral closure `S'` of `R` in `S`, such that `S'[1/r] = S[1/r]`.\n- `Algebra.ZariskisMainProperty.of_finiteType`:\n  If `S` is finite type over `R` and quasi-finite at `p`, then `ZariskisMainProperty` holds.\n- `Algebra.QuasiFiniteAt.exists_fg_and_exists_notMem_and_awayMap_bijective`:\n  If `S` is finite type over `R` and quasi-finite at `p`,\n  then there exists a subalgebra `S'` of `R` that is finitely generated as an `R`-module,\n  and some `r ∈ S'` such that `r ∉ p` and `S'[1/r] = S[1/r]`.\n-/\n\n@[expose] public section\n\nvariable {R S T : Type*} [CommRing R] [CommRing S] [Algebra R S] [CommRing T] [Algebra R T]\n\nopen scoped TensorProduct nonZeroDivisors\n\nopen Polynomial\n\nnamespace Algebra\n\nvariable (R) in\n/-- We say that an `R` algebra `S` satisfies the Zariski's main property at a prime `p` of `S`\nif there exists `r ∉ p` in the integral closure `S'` of `R` in `S`, such that `S'[1/r] = S[1/r]`. -/\ndef ZariskisMainProperty (p : Ideal S) : Prop :=\n  ∃ r : integralClosure R S, r.1 ∉ p ∧ Function.Bijective\n    (Localization.awayMap (integralClosure R S).val.toRingHom r)\n\nlemma zariskisMainProperty_iff {p : Ideal S} :\n    ZariskisMainProperty R p ↔ ∃ r ∉ p, IsIntegral R r ∧ ∀ x, ∃ m, IsIntegral R (r ^ m * x) := by\n  simp only [ZariskisMainProperty, Subtype.exists, ← exists_prop, @exists_comm (_ ∉ p)]\n  refine exists₃_congr fun r hr hrp ↦ ?_\n  rw [Function.Bijective, and_iff_right\n    (by exact IsLocalization.map_injective_of_injective _ _ _ Subtype.val_injective),\n    Localization.awayMap_surjective_iff]\n  simp [mem_integralClosure_iff]\n\nlemma zariskisMainProperty_iff' {p : Ideal S} :\n    ZariskisMainProperty R p ↔ ∃ r ∉ p, ∀ x, ∃ m, IsIntegral R (r ^ m * x) := by\n  refine zariskisMainProperty_iff.trans (exists_congr fun r ↦ and_congr_right fun hrp ↦\n    and_iff_right_of_imp fun H ↦ ?_)\n  obtain ⟨n, hn⟩ := H r\n  rw [← pow_succ] at hn\n  exact (IsIntegral.pow_iff (by simp)).mp hn\n\nTarget:\nlemma zariskisMainProperty_iff_exists_saturation_eq_top {p : Ideal S} :\n    ZariskisMainProperty R p ↔ ∃ r ∉ p, ∃ h : IsIntegral R r,\n      (integralClosure R S).saturation (.powers r) (by simpa [Submonoid.powers_le]) = ⊤ :=\n\nProof body:\n","rejected":"by\n  simp [zariskisMainProperty_iff, ← top_le_iff, SetLike.le_def,\n    Submonoid.mem_powers_iff, mem_integralClosure_iff]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory","family_id":"zariskismainproperty_iff_exists_saturation_eq_top","file_id":"mathlib/Mathlib/RingTheory/ZariskisMainTheorem.lean","sample_id":"988fd8cb4d8e17e38f2a76981b98bfba51705b13e2c6dc4822f9d73a946a1e06"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0378e91a096e555db6c2dcff0f9d43073fa97735128ddc7daf75d45b02b151ac","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"90b3c016e6178b281419875d8a8be9e9c3cb336bf5ef639055b41756038381dc","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f6367fab7503d969f0bae44dba3d63ca6fddd75a99ad5b0e2f65909eedb272b7","source_sha256":"2dd9b3d55f05fb237e2d926d68cfafdde7ae626a66f9d4e360c17c788bc0eada","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply le_antisymm\n  · refine iSup₂_le fun i hi y hy => ⟨DFinsupp.single i ⟨y, hy⟩, ?_⟩\n    rw [AddMonoidHom.comp_apply, filterAddMonoidHom_apply, filter_single_pos _ _ hi]\n    exact sumAddHom_single _ _ _\n  · rintro x ⟨v, rfl⟩\n    refine dfinsuppSumAddHom_mem _ _ _ fun i _ => ?_\n    refine AddSubmonoid.mem_iSup_of_mem i ?_\n    by_cases hp : p i\n    · simp [hp]\n    · simp [hp]","hard_negative":true,"metrics":{"chosen_tokens":87,"rejected_tokens":3,"token_jaccard":0.045455,"token_length_ratio":0.034483},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"b64e1f3a32fcaeabda1692f23578b2639372c35373a6c15da73564591d014eab","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Submonoid.BigOperators\npublic import Mathlib.Algebra.Group.Submonoid.Membership\npublic import Mathlib.Data.DFinsupp.BigOperators\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Kenny Lau\n-/\n/-!\n# `DFinsupp` and submonoids\n\nThis file mainly concerns the interaction between submonoids and products/sums of `DFinsupp`s.\n\n## Main results\n\n* `AddSubmonoid.mem_iSup_iff_exists_dfinsupp`: elements of the supremum of additive commutative\n  monoids can be given by taking finite sums of elements of each monoid.\n* `AddSubmonoid.mem_bsupr_iff_exists_dfinsupp`: elements of the supremum of additive commutative\n  monoids can be given by taking finite sums of elements of each monoid.\n-/\n\npublic section\n\n\nuniverse u u₁ u₂ v v₁ v₂ v₃ w x y l\n\nvariable {ι : Type u} {γ : Type w} {β : ι → Type v} {β₁ : ι → Type v₁} {β₂ : ι → Type v₂}\n\nopen DFinsupp\n\nvariable [DecidableEq ι]\n\n@[to_additive]\ntheorem dfinsuppProd_mem [∀ i, Zero (β i)] [∀ (i) (x : β i), Decidable (x ≠ 0)]\n    [CommMonoid γ] {S : Type*} [SetLike S γ] [SubmonoidClass S γ]\n    (s : S) (f : Π₀ i, β i) (g : ∀ i, β i → γ)\n    (h : ∀ c, f c ≠ 0 → g c (f c) ∈ s) : f.prod g ∈ s :=\n  prod_mem fun _ hi => h _ <| mem_support_iff.1 hi\n\ntheorem dfinsuppSumAddHom_mem [∀ i, AddZeroClass (β i)] [AddCommMonoid γ] {S : Type*}\n    [SetLike S γ] [AddSubmonoidClass S γ] (s : S) (f : Π₀ i, β i) (g : ∀ i, β i →+ γ)\n    (h : ∀ c, f c ≠ 0 → g c (f c) ∈ s) : DFinsupp.sumAddHom g f ∈ s := by\n  classical\n    rw [DFinsupp.sumAddHom_apply]\n    exact dfinsuppSum_mem s f (g ·) h\n\n/-- The supremum of a family of commutative additive submonoids is equal to the range of\n`DFinsupp.sumAddHom`; that is, every element in the `iSup` can be produced from taking a finite\nnumber of non-zero elements of `S i`, coercing them to `γ`, and summing them. -/\ntheorem AddSubmonoid.iSup_eq_mrange_dfinsuppSumAddHom\n    [AddCommMonoid γ] (S : ι → AddSubmonoid γ) :\n    iSup S = AddMonoidHom.mrange (DFinsupp.sumAddHom fun i => (S i).subtype) := by\n  apply le_antisymm\n  · apply iSup_le _\n    intro i y hy\n    exact ⟨DFinsupp.single i ⟨y, hy⟩, DFinsupp.sumAddHom_single _ _ _⟩\n  · rintro x ⟨v, rfl⟩\n    exact dfinsuppSumAddHom_mem _ v _ fun i _ => (le_iSup S i : S i ≤ _) (v i).prop\n\n/-- The bounded supremum of a family of commutative additive submonoids is equal to the range of\n`DFinsupp.sumAddHom` composed with `DFinsupp.filterAddMonoidHom`; that is, every element in the\nbounded `iSup` can be produced from taking a finite number of non-zero elements from the `S i` that\nsatisfy `p i`, coercing them to `γ`, and summing them. -/\n\nTarget:\ntheorem AddSubmonoid.bsupr_eq_mrange_dfinsuppSumAddHom (p : ι → Prop) [DecidablePred p]\n    [AddCommMonoid γ] (S : ι → AddSubmonoid γ) :\n    ⨆ (i) (_ : p i), S i =\n      AddMonoidHom.mrange ((sumAddHom fun i => (S i).subtype).comp (filterAddMonoidHom _ p)) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_f6367fab7503","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"a2726fc91447bd8d1d4fb8a3ff11f09e2f1bc98dbbe62658502108d99dd3b80a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/DFinsupp","family_id":"addsubmonoid","file_id":"mathlib/Mathlib/Data/DFinsupp/Submonoid.lean","sample_id":"f6367fab7503d969f0bae44dba3d63ca6fddd75a99ad5b0e2f65909eedb272b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ace5857b04711e3f6e5ec35f7db5255eacbf0110e349e471daaf3e18ca693ed4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"81a2ff22c892d0579ef9c1611ce99af9ab34db9e64c6ba19b54102e66d3272a3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"22290c022264f0f12d52570adb279eb6ca130b8e7ce84838db022df1840fc31a","source_sha256":"f47eb7e9445cb948d9f1cba9a87c2cadc067f856113f1f8e2a417357cd103064","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases! hn : n = 0\n  · rw [pow_eq_top_iff.mpr (Or.inr hn)]\n    exact isUnit_of_subsingleton _\n  exact (isUnit_mk_pow_iff_notMem I hn).mpr hx","hard_negative":false,"metrics":{"chosen_tokens":33,"rejected_tokens":37,"token_jaccard":0.892857,"token_length_ratio":1.121212},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"b6563503c39397490182642400e447a3ec41931e5c453d3b1ccf73ea252b0b59","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Ideal.Quotient.Operations\npublic import Mathlib.RingTheory.Nilpotent.Lemmas\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Nilpotent elements in quotient rings\n-/\n\npublic section\n\ntheorem Ideal.isRadical_iff_quotient_reduced {R : Type*} [CommRing R] (I : Ideal R) :\n    I.IsRadical ↔ IsReduced (R ⧸ I) := by\n  conv_lhs => rw [← @Ideal.mk_ker R _ I]\n  exact RingHom.ker_isRadical_iff_reduced_of_surjective Quotient.mk_surjective\n\nvariable {S : Type*} [CommRing S] (I : Ideal S)\n\n/-- Let `P` be a property on ideals. If `P` holds for square-zero ideals, and if\n  `P I → P (J ⧸ I) → P J`, then `P` holds for all nilpotent ideals. -/\ntheorem Ideal.IsNilpotent.induction_on (hI : IsNilpotent I)\n    {P : ∀ ⦃S : Type _⦄ [CommRing S], Ideal S → Prop}\n    (h₁ : ∀ ⦃S : Type _⦄ [CommRing S], ∀ I : Ideal S, I ^ 2 = ⊥ → P I)\n    (h₂ : ∀ ⦃S : Type _⦄ [CommRing S], ∀ I J : Ideal S, I ≤ J → P I →\n      P (J.map (Ideal.Quotient.mk I)) → P J) :\n    P I := by\n  obtain ⟨n, hI : I ^ n = ⊥⟩ := hI\n  induction n using Nat.strong_induction_on generalizing S with | _ n H\n  by_cases hI' : I = ⊥\n  · subst hI'\n    apply h₁\n    rw [← Ideal.zero_eq_bot, zero_pow two_ne_zero]\n  rcases n with - | n\n  · rw [pow_zero, Ideal.one_eq_top] at hI\n    haveI := subsingleton_of_bot_eq_top hI.symm\n    exact (hI' (Subsingleton.elim _ _)).elim\n  rcases n with - | n\n  · rw [pow_one] at hI\n    exact (hI' hI).elim\n  apply h₂ (I ^ 2) _ (Ideal.pow_le_self two_ne_zero)\n  · apply H n.succ _ (I ^ 2)\n    · rw [← pow_mul, eq_bot_iff, ← hI, Nat.succ_eq_add_one]\n      apply Ideal.pow_le_pow_right (by lia)\n    · exact n.succ.lt_succ_self\n  · apply h₁\n    rw [← Ideal.map_pow, Ideal.map_quotient_self]\n\ntheorem IsNilpotent.isUnit_quotient_mk_iff {R : Type*} [CommRing R] {I : Ideal R}\n    (hI : IsNilpotent I) {x : R} : IsUnit (Ideal.Quotient.mk I x) ↔ IsUnit x := by\n  refine ⟨?_, fun h => h.map <| Ideal.Quotient.mk I⟩\n  revert x\n  apply Ideal.IsNilpotent.induction_on (S := R) I hI <;> clear hI I\n  swap\n  · introv e h₁ h₂ h₃\n    apply h₁\n    apply h₂\n    exact\n      h₃.map\n        ((DoubleQuot.quotQuotEquivQuotSup I J).trans\n              (Ideal.quotEquivOfEq (sup_eq_right.mpr e))).symm.toRingHom\n  · introv e H\n    obtain ⟨y, hy⟩ := Ideal.Quotient.mk_surjective (↑H.unit⁻¹ : S ⧸ I)\n    have : Ideal.Quotient.mk I (x * y) = Ideal.Quotient.mk I 1 := by\n      rw [map_one, map_mul, hy, IsUnit.mul_val_inv]\n    rw [Ideal.Quotient.eq] at this\n    have : (x * y - 1) ^ 2 = 0 := by\n      rw [← Ideal.mem_bot, ← e]\n      exact Ideal.pow_mem_pow this _\n    have : x * (y * (2 - x * y)) = 1 := by\n      rw [eq_comm, ← sub_eq_zero, ← this]\n      ring\n    exact .of_mul_eq_one _ this\n\ntheorem Ideal.Quotient.isUnit_mk_pow_iff_isUnit_mk {x : S} {n : ℕ} (hn : n ≠ 0) :\n    IsUnit (Ideal.Quotient.mk (I ^ n) x) ↔ IsUnit (Ideal.Quotient.mk I x) := by\n  rw [← IsNilpotent.isUnit_quotient_mk_iff (I := Ideal.map (Ideal.Quotient.mk (I ^ n)) I)]\n  · rw [← isUnit_map_iff (DoubleQuot.quotQuotEquivQuotOfLE (Ideal.pow_le_self hn))]\n    rfl\n  · use n\n    simp [← Ideal.map_pow]\n\ntheorem Ideal.Quotient.isUnit_mk_pow_iff_notMem [I.IsMaximal] {n : ℕ} (hn : n ≠ 0) {x : S} :\n    IsUnit (mk (I ^ n) x) ↔ x ∉ I := by\n  let := Ideal.Quotient.field I\n  rw [isUnit_mk_pow_iff_isUnit_mk I hn, isUnit_iff_ne_zero]\n  exact Ideal.Quotient.eq_zero_iff_mem.not\n\nTarget:\ntheorem Ideal.Quotient.isUnit_mk_pow_of_notMem [I.IsMaximal] {n : ℕ} {x : S} (hx : x ∉ I) :\n    IsUnit (mk (I ^ n) x) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  by_cases! hn : n = 0\n  · rw [pow_eq_top_iff.mpr (Or.inr hn)]\n    exact isUnit_of_subsingleton _\n  exact (isUnit_mk_pow_iff_notMem I hn).mpr hx","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Ideal","family_id":"ideal","file_id":"mathlib/Mathlib/RingTheory/Ideal/Quotient/Nilpotent.lean","sample_id":"22290c022264f0f12d52570adb279eb6ca130b8e7ce84838db022df1840fc31a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ea63d2911a78e27e3d0efd6769a583eaec219edad30fe20857b3d050d16b7158","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d05263c7bf55a58e8c8de9c28c2d1ec2455cd4bbdc429607ef41551255875eaa","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f611f13c4698d3e99009d189350bca949401d584539fd12a5756b6e5b77a146b","source_sha256":"546544902b60193a288cef69b3c4f5d20597fa5d3b2853c4808d2cc29f4c6640","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  letI := (IntermediateField.inclusion H.le).toAlgebra\n  have : IsScalarTower F E L := .of_algebraMap_eq' rfl\n  refine lt_of_le_of_ne ?_ ?_\n  · exact Module.finrank_top_le_finrank_of_isScalarTower _ _ _\n  · exact .symm (mt (eq_of_le_of_finrank_eq' H.le) H.ne)","hard_negative":true,"metrics":{"chosen_tokens":53,"rejected_tokens":2,"token_jaccard":0.03125,"token_length_ratio":0.037736},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"b66988e7eb714cfcdb8c43abf10b1551f3f24909939445c93c9743c83053eccb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.IntermediateField.Basic\npublic import Mathlib.FieldTheory.Minpoly.Basic\npublic import Mathlib.FieldTheory.Tower\npublic import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition\npublic import Mathlib.RingTheory.Algebraic.Integral\n\nNamespace:\nIntermediateField\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Results on finite dimensionality and algebraicity of intermediate fields.\n-/\n\n@[expose] public section\n\nopen Module\n\nvariable {K L : Type*} [Field K] [Field L] [Algebra K L]\n  {S : IntermediateField K L}\n\ntheorem IntermediateField.coe_isIntegral_iff {R : Type*} [CommRing R] [Algebra R K] [Algebra R L]\n    [IsScalarTower R K L] {x : S} : IsIntegral R (x : L) ↔ IsIntegral R x :=\n  isIntegral_algHom_iff (S.val.restrictScalars R) Subtype.val_injective\n\n/-- Turn an algebraic subalgebra into an intermediate field, `Subalgebra.IsAlgebraic` version. -/\ndef Subalgebra.IsAlgebraic.toIntermediateField {S : Subalgebra K L} (hS : S.IsAlgebraic) :\n    IntermediateField K L where\n  toSubalgebra := S\n  inv_mem' x hx := Algebra.adjoin_le_iff.mpr\n    (Set.singleton_subset_iff.mpr hx) (hS x hx).isIntegral.inv_mem_adjoin\n\n/-- Turn an algebraic subalgebra into an intermediate field, `Algebra.IsAlgebraic` version. -/\nabbrev Algebra.IsAlgebraic.toIntermediateField (S : Subalgebra K L) [Algebra.IsAlgebraic K S] :\n    IntermediateField K L := (S.isAlgebraic_iff.mpr ‹_›).toIntermediateField\n\nnamespace IntermediateField\n\ninstance isAlgebraic_tower_bot [Algebra.IsAlgebraic K L] : Algebra.IsAlgebraic K S :=\n  Algebra.IsAlgebraic.of_injective S.val S.val.injective\n\ninstance isAlgebraic_tower_top [Algebra.IsAlgebraic K L] : Algebra.IsAlgebraic S L :=\n  Algebra.IsAlgebraic.tower_top (K := K) S\n\nsection FiniteDimensional\n\nvariable (F E : IntermediateField K L)\n\ninstance finiteDimensional_left [FiniteDimensional K L] : FiniteDimensional K F := .left K F L\ninstance finiteDimensional_right [FiniteDimensional K L] : FiniteDimensional F L := .right K F L\n\n@[simp]\ntheorem rank_eq_rank_subalgebra : Module.rank K F.toSubalgebra = Module.rank K F :=\n  rfl\n\n@[simp]\ntheorem finrank_eq_finrank_subalgebra : finrank K F.toSubalgebra = finrank K F :=\n  rfl\n\nvariable {F} {E}\n\n/-- If `F ≤ E` are two intermediate fields of `L / K` such that `[E : K] ≤ [F : K]` are finite,\nthen `F = E`. -/\ntheorem eq_of_le_of_finrank_le [hfin : FiniteDimensional K E] (h_le : F ≤ E)\n    (h_finrank : finrank K E ≤ finrank K F) : F = E :=\n  haveI : Module.Finite K E.toSubalgebra := hfin\n  toSubalgebra_injective <| Subalgebra.eq_of_le_of_finrank_le h_le h_finrank\n\n/-- If `F ≤ E` are two intermediate fields of `L / K` such that `[F : K] = [E : K]` are finite,\nthen `F = E`. -/\ntheorem eq_of_le_of_finrank_eq [FiniteDimensional K E] (h_le : F ≤ E)\n    (h_finrank : finrank K F = finrank K E) : F = E :=\n  eq_of_le_of_finrank_le h_le h_finrank.ge\n\n/-- If `F ≤ E` are two intermediate fields of `L / K` such that `[E : K]` is finite,\nthen `F = E` iff `[F : K] = [E : K]`. -/\ntheorem eq_iff_finrank_eq_of_le [FiniteDimensional K E] (h_le : F ≤ E) :\n    F = E ↔ finrank K F = finrank K E :=\n  ⟨fun h ↦ by rw [h], eq_of_le_of_finrank_eq h_le⟩\n\n-- If `F ≤ E` are two intermediate fields of a finite extension `L / K` such that\n-- `[L : F] ≤ [L : E]`, then `F = E`. Marked as private since it's a direct corollary of\n-- `eq_of_le_of_finrank_le'` (the `FiniteDimensional K L` implies `FiniteDimensional F L`\n-- automatically by typeclass resolution).\nprivate theorem eq_of_le_of_finrank_le'' [FiniteDimensional K L] (h_le : F ≤ E)\n    (h_finrank : finrank F L ≤ finrank E L) : F = E := by\n  apply eq_of_le_of_finrank_le h_le\n  have h1 := finrank_mul_finrank K F L\n  have h2 := finrank_mul_finrank K E L\n  have h3 : 0 < finrank E L := finrank_pos\n  nlinarith\n\n/-- If `F ≤ E` are two intermediate fields of `L / K` such that `[L : F] ≤ [L : E]` are finite,\nthen `F = E`. -/\ntheorem eq_of_le_of_finrank_le' [FiniteDimensional F L] (h_le : F ≤ E)\n    (h_finrank : finrank F L ≤ finrank E L) : F = E := by\n  refine le_antisymm h_le (fun l hl ↦ ?_)\n  rwa [← mem_extendScalars (le_refl F), eq_of_le_of_finrank_le''\n    ((extendScalars_le_extendScalars_iff (le_refl F) h_le).2 h_le) h_finrank, mem_extendScalars]\n\n/-- If `F ≤ E` are two intermediate fields of `L / K` such that `[L : F] = [L : E]` are finite,\nthen `F = E`. -/\ntheorem eq_of_le_of_finrank_eq' [FiniteDimensional F L] (h_le : F ≤ E)\n    (h_finrank : finrank F L = finrank E L) : F = E :=\n  eq_of_le_of_finrank_le' h_le h_finrank.le\n\n/-- If `F ≤ E` are two intermediate fields of `L / K` such that `[L : F]` is finite,\nthen `F = E` iff `[L : F] = [L : E]`. -/\ntheorem eq_iff_finrank_eq_of_le' [FiniteDimensional F L] (h_le : F ≤ E) :\n    F = E ↔ finrank F L = finrank E L :=\n  ⟨fun h ↦ by rw [h], eq_of_le_of_finrank_eq' h_le⟩\n\nTarget:\nlemma finrank_lt_of_gt [FiniteDimensional F L] (H : F < E) :\n    Module.finrank E L < Module.finrank F L :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_f611f13c4698","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"aacfb5302f375a74a50816b0dbc1a8cef7fd24c09b08f05ba6286e386e1b7630","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/IntermediateField","family_id":"finrank_lt_of_gt","file_id":"mathlib/Mathlib/FieldTheory/IntermediateField/Algebraic.lean","sample_id":"f611f13c4698d3e99009d189350bca949401d584539fd12a5756b6e5b77a146b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"08218aabda46401fe50c2dc9de3ff5011d981b5e7f148f3d58bd8c99abd66739","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"631dd733f33921d58948a0df45ceabb189d48a1055bf4f8fc988734c4839b6cb","source_sha256":"a532f8917e86d9e21f6417bd7287cabd2ab46e6ff2388a4cee138b7afaddc6d0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [simplyConnectedSpace_iff, equiv_punit_iff_unique,\n    FundamentalGroupoid.nonempty_iff X, and_congr_right_iff, Nonempty.forall]\n  intros\n  exact ⟨fun h _ _ => h _ _, fun h _ _ => h _ _⟩","hard_negative":false,"metrics":{"chosen_tokens":40,"rejected_tokens":2,"token_jaccard":0.041667,"token_length_ratio":0.05},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"b66c6d427bca768a114f217572f7bbc1097ca2977d6b33a362cfaa9b12d8c09c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps\npublic import Mathlib.Topology.Homotopy.Contractible\npublic import Mathlib.CategoryTheory.PUnit\npublic import Mathlib.AlgebraicTopology.FundamentalGroupoid.PUnit\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Praneeth Kolichala. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Praneeth Kolichala\n-/\n/-!\n# Simply connected spaces\nThis file defines simply connected spaces.\nA topological space is simply connected if its fundamental groupoid is equivalent to `Unit`.\n\nWe also define the corresponding predicate for sets.\n\n## Main theorems\n  - `simply_connected_iff_unique_homotopic` - A space is simply connected if and only if it is\n    nonempty and there is a unique path up to homotopy between any two points\n\n  - `SimplyConnectedSpace.ofContractible` - A contractible space is simply connected\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen CategoryTheory\nopen scoped ContinuousMap\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n\n/-- A simply connected space is one whose fundamental groupoid is equivalent to `Discrete Unit` -/\n@[mk_iff]\nclass SimplyConnectedSpace (X : Type*) [TopologicalSpace X] : Prop where\n  equiv_unit : Nonempty (FundamentalGroupoid X ≌ Discrete Unit)\n\n@[deprecated (since := \"2026-01-08\")]\nalias simply_connected_def := simplyConnectedSpace_iff\n\nTarget:\ntheorem simply_connected_iff_unique_homotopic (X : Type*) [TopologicalSpace X] :\n    SimplyConnectedSpace X ↔\n      Nonempty X ∧ ∀ x y : X, Nonempty (Unique (Path.Homotopic.Quotient x y)) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/FundamentalGroupoid","family_id":"simply_connected_iff_unique_homotopic","file_id":"mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean","sample_id":"631dd733f33921d58948a0df45ceabb189d48a1055bf4f8fc988734c4839b6cb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"31a2975efba554cce04bfa852066668eb27b98da82a9d33652d228e97ecec653","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7e8c17610a3baa53fbdd954115a7d7f9ea7c655e0a0e548683cb4072c1194356","source_sha256":"75f644a2aa9ce33f4e3a9b3af6b61387bae9fdea7ce3dbf0ef1ec4c5bea1cda5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [removeFactor, natDegree_divByMonic _ (monic_X_sub_C _), natDegree_map, natDegree_X_sub_C]","hard_negative":false,"metrics":{"chosen_tokens":16,"rejected_tokens":2,"token_jaccard":0.071429,"token_length_ratio":0.125},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"b66f439d309434f1998e9d144d7db1dc2ab1be527ace70537efb2deb489d8777","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharP.Algebra\npublic import Mathlib.FieldTheory.SplittingField.IsSplittingField\npublic import Mathlib.RingTheory.Algebraic.Basic\n\nNamespace:\nPolynomial\n\nLocal context:\n/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Splitting fields\n\nIn this file we prove the existence and uniqueness of splitting fields.\n\n## Main definitions\n\n* `Polynomial.SplittingField f`: A fixed splitting field of the polynomial `f`.\n\n## Main statements\n\n* `Polynomial.IsSplittingField.algEquiv`: Every splitting field of a polynomial `f` is isomorphic\n  to `SplittingField f` and thus, being a splitting field is unique up to isomorphism.\n\n## Implementation details\nWe construct a `SplittingFieldAux` without worrying about whether the instances satisfy nice\ndefinitional equalities. Then the actual `SplittingField` is defined to be a quotient of a\n`MvPolynomial` ring by the kernel of the obvious map into `SplittingFieldAux`. Because the\nactual `SplittingField` will be a quotient of a `MvPolynomial`, it has nice instances on it.\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nuniverse u v w\n\nvariable {F : Type u} {K : Type v} {L : Type w}\n\nnamespace Polynomial\n\nvariable [Field K] [Field L] [Field F]\n\nopen Polynomial\n\nsection SplittingField\n\nopen Classical in\n/-- Non-computably choose an irreducible factor from a polynomial. -/\ndef factor (f : K[X]) : K[X] :=\n  if H : ∃ g, Irreducible g ∧ g ∣ f then Classical.choose H else X\n\ntheorem irreducible_factor (f : K[X]) : Irreducible (factor f) := by\n  rw [factor]\n  split_ifs with H\n  · exact (Classical.choose_spec H).1\n  · exact irreducible_X\n\n/-- See note [fact non-instances]. -/\ntheorem fact_irreducible_factor (f : K[X]) : Fact (Irreducible (factor f)) :=\n  ⟨irreducible_factor f⟩\n\nattribute [local instance] fact_irreducible_factor\n\ntheorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f := by\n  by_cases hf2 : f = 0; · rw [hf2]; exact dvd_zero _\n  rw [factor, dif_pos (WfDvdMonoid.exists_irreducible_factor hf1 hf2)]\n  exact (Classical.choose_spec <| WfDvdMonoid.exists_irreducible_factor hf1 hf2).2\n\ntheorem factor_dvd_of_degree_ne_zero {f : K[X]} (hf : f.degree ≠ 0) : factor f ∣ f :=\n  factor_dvd_of_not_isUnit (mt degree_eq_zero_of_isUnit hf)\n\ntheorem factor_dvd_of_natDegree_ne_zero {f : K[X]} (hf : f.natDegree ≠ 0) : factor f ∣ f :=\n  factor_dvd_of_degree_ne_zero (mt natDegree_eq_of_degree_eq_some hf)\n\nlemma isCoprime_iff_aeval_ne_zero (f g : K[X]) : IsCoprime f g ↔ ∀ {A : Type v} [CommRing A]\n    [IsDomain A] [Algebra K A] (a : A), aeval a f ≠ 0 ∨ aeval a g ≠ 0 := by\n  refine ⟨fun h => aeval_ne_zero_of_isCoprime h, fun h => isCoprime_of_dvd _ _ ?_ fun x hx _ => ?_⟩\n  · replace h := @h K _ _ _ 0\n    contrapose! h\n    rw [h.left, h.right, map_zero, and_self]\n  · rintro ⟨_, rfl⟩ ⟨_, rfl⟩\n    replace h := not_and_or.mpr <| h <| AdjoinRoot.root x.factor\n    simp only [AdjoinRoot.aeval_eq, AdjoinRoot.mk_eq_zero,\n      dvd_mul_of_dvd_left <| factor_dvd_of_not_isUnit hx, true_and, not_true] at h\n\n/-- Divide a polynomial f by `X - C r` where `r` is a root of `f` in a bigger field extension. -/\ndef removeFactor (f : K[X]) : Polynomial (AdjoinRoot <| factor f) :=\n  map (AdjoinRoot.of f.factor) f /ₘ (X - C (AdjoinRoot.root f.factor))\n\ntheorem X_sub_C_mul_removeFactor (f : K[X]) (hf : f.natDegree ≠ 0) :\n    (X - C (AdjoinRoot.root f.factor)) * f.removeFactor = map (AdjoinRoot.of f.factor) f := by\n  let ⟨g, hg⟩ := factor_dvd_of_natDegree_ne_zero hf\n  apply (mul_divByMonic_eq_iff_isRoot\n    (R := AdjoinRoot f.factor) (a := AdjoinRoot.root f.factor)).mpr\n  rw [IsRoot.def, eval_map, hg, eval₂_mul, ← hg, AdjoinRoot.eval₂_root, zero_mul]\n\nTarget:\ntheorem natDegree_removeFactor (f : K[X]) : f.removeFactor.natDegree = f.natDegree - 1 :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/SplittingField","family_id":"natdegree_removefactor","file_id":"mathlib/Mathlib/FieldTheory/SplittingField/Construction.lean","sample_id":"7e8c17610a3baa53fbdd954115a7d7f9ea7c655e0a0e548683cb4072c1194356"},"split":"train"}
{"binding":{"chosen_proof_sha256":"768478b669f1b0c48360ff5cb94ef6db06e9b1b6432c6287e4928bb1cb2132ad","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2ad48a41750f9ee7416f3ed47f9d4bdc606c25a12a288f7161359dab4138e50a","source_sha256":"a9f42fb2a650677b7f31f134a9fdb63cf294a003ee4e030eb51993a29af692d7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [eq_comm, ← ltb_cons_addChar' c]\n  simp","hard_negative":false,"metrics":{"chosen_tokens":10,"rejected_tokens":2,"token_jaccard":0.090909,"token_length_ratio":0.2},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"b689f922ce9b0cddf688b98239699931002cdf4999808b36a301f5f63d48158c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.String.Lemmas\npublic import Mathlib.Data.List.Lex\npublic import Mathlib.Data.Char\npublic import Mathlib.Algebra.Order.Group.Nat\nimport all Init.Data.String.Iterator  -- for unfolding `Iterator.curr`\nimport all Init.Data.Ord.String  -- for unfolding `String.compare`\n\nNamespace:\nString\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Strings\n\nSupplementary theorems about the `String` type.\n-/\n\n@[expose] public section\n\nnamespace String\n\n/-- `<` on string iterators. This coincides with `<` on strings as lists. -/\ndef ltb (s₁ s₂ : Legacy.Iterator) : Bool :=\n  if s₂.hasNext then\n    if s₁.hasNext then\n      if s₁.curr = s₂.curr then\n        ltb s₁.next s₂.next\n      else s₁.curr < s₂.curr\n    else true\n  else false\n\n/-- Induction on `String.ltb`. -/\n@[no_expose] def ltb.inductionOn.{u} {motive : Legacy.Iterator → Legacy.Iterator → Sort u}\n    (it₁ it₂ : Legacy.Iterator)\n    (ind : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      i₁.get s₁ = i₂.get s₂ →\n        motive (Legacy.Iterator.next ⟨s₁, i₁⟩) (Legacy.Iterator.next ⟨s₂, i₂⟩) →\n          motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (eq : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      ¬ i₁.get s₁ = i₂.get s₂ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (base₁ : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → ¬ Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (base₂ : ∀ s₁ s₂ i₁ i₂, ¬ Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩) :\n    motive it₁ it₂ :=\n  if h₂ : it₂.hasNext then\n    if h₁ : it₁.hasNext then\n      if heq : it₁.curr = it₂.curr then\n        ind it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq (inductionOn it₁.next it₂.next ind eq base₁ base₂)\n      else eq it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq\n    else base₁ it₁.s it₂.s it₁.i it₂.i h₂ h₁\n  else base₂ it₁.s it₂.s it₁.i it₂.i h₂\n\ntheorem ltb_cons_addChar' (c : Char) (s₁ s₂ : Legacy.Iterator) :\n    ltb ⟨ofList (c :: s₁.s.toList), s₁.i + c⟩ ⟨ofList (c :: s₂.s.toList), s₂.i + c⟩ =\n      ltb s₁ s₂ := by\n  fun_induction ltb s₁ s₂ with\n  | case1 s₁ s₂ h₁ h₂ h ih =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_pos (by simpa using h₂), if_pos, ← ih]\n    · simp only [Legacy.Iterator.next, Pos.Raw.next, get_cons_addChar, ofList_toList]\n      congr 2 <;> apply Pos.Raw.add_char_right_comm\n    · simpa only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList] using h\n  | case2 s₁ s₂ h₁ h₂ h =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_pos (by simpa using h₂), if_neg]\n    · simp only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList, decide_eq_decide]\n    · simpa only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList] using h\n  | case3 s₁ s₂ h₁ h₂ =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_neg (by simpa using h₂)]\n  | case4 s₁ s₂ h₁ =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, if_neg (by simpa using h₁)]\n\nTarget:\ntheorem ltb_cons_addChar (c : Char) (cs₁ cs₂ : List Char) (i₁ i₂ : Pos.Raw) :\n    ltb ⟨ofList (c :: cs₁), i₁ + c⟩ ⟨ofList (c :: cs₂), i₂ + c⟩ =\n      ltb ⟨ofList cs₁, i₁⟩ ⟨ofList cs₂, i₂⟩ :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/String","family_id":"ltb_cons_addchar","file_id":"mathlib/Mathlib/Data/String/Basic.lean","sample_id":"2ad48a41750f9ee7416f3ed47f9d4bdc606c25a12a288f7161359dab4138e50a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bbb068c1b254f9a62b1fe49e6c1cd52f49bd7cbde7d3a811059b2617b8439517","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7887373d14e0ce8b2a10aa48b0884c760c0da2834c67ad161196e05577bc27fe","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c87ad3ae24029059963e03348c139139f0d135dcb45808f3d2427d4e578980f8","source_sha256":"50e4a4d798ba2dd2872d163b6a5aeb55d8cf128d153e35dcc8c630f4eadcbabe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h0 : v πᵥ ≠ 0 := by simpa using uniformizingPolynomial_ne_zero hle\n  induction f using RatFunc.induction_on with\n  | f p q hq =>\n    use (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {p})).factors -\n      (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {q})).factors\n    simp only [map_div₀, map_pow, zpow_sub₀ h0, zpow_natCast,\n      valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle hq,\n      valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle\n        (p := p) (by aesop)]","hard_negative":true,"metrics":{"chosen_tokens":116,"rejected_tokens":3,"token_jaccard":0.018868,"token_length_ratio":0.025862},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"b699dca0b8715f1070646516ab65da3e81f9db24bec979dd721ddf6ae5b705dc","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.Finite.Valuation\npublic import Mathlib.NumberTheory.FunctionField\npublic import Mathlib.RingTheory.Valuation.Discrete.Basic\n\nNamespace:\nRatFunc\n\nLocal context:\n/-\nCopyright (c) 2025 María Inés de Frutos-Fernández & Xavier Généreux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: María Inés de Frutos-Fernández, Xavier Généreux\n-/\n/-!\n# Ostrowski's theorem for `K(X)`\n\nThis file proves Ostrowski's theorem for the field of rational functions `K(X)`, where `K` is any\nfield: if `v` is a discrete valuation on `K(X)` which is trivial on elements of `K`, then `v` is\nequivalent to either the `I`-adic valuation for some `I : HeightOneSpectrum K[X]`, or to the\nvaluation at infinity `FunctionField.inftyValuation K`.\n\n## Main results\n- `RatFunc.valuation_isEquiv_infty_or_adic`: Ostrowski's theorem for `K(X)`.\n-/\n\n@[expose] public noncomputable section\n\n\nopen Multiplicative WithZero\n\nvariable {K Γ : Type*} [Field K] [LinearOrderedCommGroupWithZero Γ] {v : Valuation (RatFunc K) Γ}\n\nnamespace RatFunc\n\nsection Infinity\n\nopen Polynomial Valuation\n\nlemma valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X {f : RatFunc K}\n    [v.IsTrivialOn K] (hlt : 1 < v X) (hf : f ≠ 0) : v f = v RatFunc.X ^ f.intDegree := by\n  induction f using RatFunc.induction_on with\n  | f p q hq =>\n    rw [intDegree_div (by grind only) (by grind only), v.map_div, zpow_sub₀ (ne_zero_of_lt hlt)]\n    simp_rw [intDegree_polynomial, zpow_natCast, ← coePolynomial_eq_algebraMap]\n    have hp : p ≠ 0 := by contrapose hf; simp [hf]\n    rw [valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X _ hlt hp,\n      valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X _ hlt hq]\n\nvariable [DecidableEq (RatFunc K)]\n\nlemma valuation_isEquiv_inftyValuation_of_one_lt_valuation_X [v.IsTrivialOn K] (hlt : 1 < v X) :\n    v.IsEquiv (inftyValuation K) := by\n  refine isEquiv_iff_val_lt_one.mpr fun {f} ↦ ?_\n  rcases eq_or_ne f 0 with rfl | hf\n  · simp\n  · have hlt' : 1 < inftyValuation K X := by simp [← exp_zero]\n    rw [valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt hf,\n      valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt' hf]\n    grind [one_le_zpow_iff_right₀]\n\nend Infinity\n\nopen IsDedekindDomain HeightOneSpectrum Set Valuation Polynomial\n\nlemma setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty [v.IsNontrivial] [v.IsTrivialOn K]\n    (hle : v RatFunc.X ≤ 1) : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty := by\n  obtain ⟨w, h0, h1⟩ := IsNontrivial.exists_lt_one (v := v)\n  induction w using RatFunc.induction_on with\n  | f p q =>\n    simp only [ne_eq, _root_.div_eq_zero_iff, FaithfulSMul.algebraMap_eq_zero_iff, not_or,\n      map_div₀] at *\n    have hor : ¬v ↑p = 1 ∨ ¬v ↑q = 1 := by rw [← not_and_or]; aesop\n    suffices ∀ r : K[X], v (↑r) ≠ 1 → r ≠ 0 → {p : K[X] | v ↑p < 1 ∧ ¬p = 0}.Nonempty by\n      exact Or.elim hor (fun hp ↦ this p hp h0.1) (fun hq ↦ this q hq h0.2)\n    exact fun r hr hr0 ↦ ⟨r, lt_iff_le_and_ne.mpr\n      ⟨Polynomial.valuation_le_one_of_valuation_X_le_one _ hle r, hr⟩, hr0⟩\n\nprivate lemma one_le_valuation_factor (hne : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty) {a b : K[X]}\n    (hab : v ↑(a * b) < 1 ∧ a ≠ 0 ∧ b ≠ 0) (hπᵥ : degree_lt_wf.min _ hne = a * b)\n    (hb : ¬IsUnit b) : 1 ≤ v ↑a := by\n  set πᵥ := degree_lt_wf.min _ hne\n  have hda : a.degree < πᵥ.degree := by\n    have hbpos := degree_pos_of_ne_zero_of_nonunit hab.2.2 hb\n    simp_rw [hπᵥ, degree_mul, degree_eq_natDegree hab.2.1, degree_eq_natDegree hab.2.2] at hbpos ⊢\n    norm_cast\n    simpa using hbpos\n  have hlea := imp_not_comm.mp (degree_lt_wf.not_lt_min _) hda\n  grind\n\nlemma irreducible_min_polynomial_valuation_lt_one_and_ne_zero [v.IsTrivialOn K]\n    (hne : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty) :\n    Irreducible (degree_lt_wf.min {p : K[X] | v p < 1 ∧ p ≠ 0} hne) := by\n  set πᵥ := degree_lt_wf.min _ hne\n  have hπᵥ : v πᵥ < 1 ∧ πᵥ ≠ 0 := degree_lt_wf.min_mem _ hne\n  refine irreducible_iff.mpr ⟨?_, fun a b hab ↦ ?_⟩\n  · simp only [Polynomial.isUnit_iff, isUnit_iff_ne_zero]\n    intro ⟨a, ha0, ha⟩\n    rw [← ha, coePolynomial, algebraMap_C, ← algebraMap_eq_C] at hπᵥ\n    grind\n  · by_contra! H\n    simp only [hab, ne_eq, mul_eq_zero, not_or] at hπᵥ\n    have hva := one_le_valuation_factor hne hπᵥ hab H.2\n    simp only [mul_comm a b, @and_comm (¬a = 0)] at hπᵥ hab\n    have := Right.one_le_mul (one_le_valuation_factor hne hπᵥ hab H.1) hva\n    simp only [coePolynomial_eq_algebraMap, map_mul] at hπᵥ this\n    grind\n\nsection valuation_X_le_one\n\nvariable [v.IsNontrivial] [v.IsTrivialOn K] (hle : v RatFunc.X ≤ 1)\n\n/-- A uniformizing element for the valuation `v`, as a polynomial in `K[X]`. -/\nabbrev uniformizingPolynomial : K[X] :=\n  WellFounded.min degree_lt_wf _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)\n\n@[inherit_doc]\nlocal notation \"πᵥ\" => uniformizingPolynomial hle\n\nlemma uniformizingPolynomial_ne_zero : πᵥ ≠ 0 := by\n  have := degree_lt_wf.min_mem _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)\n  simp_all [uniformizingPolynomial]\n\nlemma valuation_uniformizingPolynomial_lt_one : v πᵥ < 1 := by\n  simpa using! (degree_lt_wf.min_mem _\n    (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)).1\n\nopen Ideal in\n/-- The maximal ideal of `K[X]` generated by the `uniformizingPolynomial` for `v`. -/\ndef valuationIdeal : HeightOneSpectrum K[X] where\n  asIdeal := Submodule.span K[X] {πᵥ}\n  isPrime := IsMaximal.isPrime (PrincipalIdealRing.isMaximal_of_irreducible\n    (irreducible_min_polynomial_valuation_lt_one_and_ne_zero\n      (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)))\n  ne_bot := by simpa using uniformizingPolynomial_ne_zero hle\n\n@[inherit_doc]\nlocal notation \"Pᵥ\" => RatFunc.valuationIdeal hle\n\nsection Associates\n\nopen EuclideanDomain in\nlemma valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one {p : K[X]}\n    (hp : p ≠ 0) :\n    v (algebraMap K[X] (RatFunc K) p) = v (πᵥ ^ ((Associates.mk (Pᵥ).asIdeal).count\n      (Associates.mk (Ideal.span {p})).factors)) := by\n  set π := πᵥ\n  have hne := setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle\n  have hπirr : Irreducible π := irreducible_min_polynomial_valuation_lt_one_and_ne_zero hne\n  obtain ⟨k, q, hnq, heq⟩ := WfDvdMonoid.max_power_factor hp hπirr\n  have hπ : π ∈ _ := degree_lt_wf.min_mem _ hne\n  simp only [ne_eq, mem_setOf] at hπ\n  nth_rw 1 [heq]\n  simp only [map_mul, map_pow]\n  suffices v (algebraMap K[X] (RatFunc K) q) = 1 by\n    simp only [this, mul_one]\n    congr\n    exact (Ideal.count_associates_eq (irreducible_iff_prime.mp hπirr) hnq heq).symm\n  rw [← mod_add_div q π, map_add]\n  rw [← mod_eq_zero] at hnq\n  suffices v (algebraMap K[X] (RatFunc K) (q % π)) = 1 ∧\n      v (algebraMap K[X] (RatFunc K) (π * (q / π))) < 1 by\n    obtain ⟨h₁, h₂⟩ := this\n    rw [← h₁] at h₂ ⊢\n    exact Valuation.map_add_eq_of_lt_left _ h₂\n  constructor\n  · rw [← coePolynomial_eq_algebraMap]\n    have hnπ : q % π ∉ {p : K[X] | v ↑p < 1 ∧ p ≠ 0} :=\n      imp_not_comm.mp (degree_lt_wf.not_lt_min _) (EuclideanDomain.remainder_lt q hπ.2)\n    have := Polynomial.valuation_le_one_of_valuation_X_le_one _ hle (q % π)\n    grind\n  · simpa only [map_mul, ← coePolynomial_eq_algebraMap]\n      using mul_lt_one_of_lt_of_le hπ.1 <| (q / π).valuation_le_one_of_valuation_X_le_one _ hle\n\nTarget:\nlemma exists_zpow_uniformizingPolynomial {f : RatFunc K} (hf : f ≠ 0) :\n    ∃ (z : ℤ), v f = v πᵥ ^ z :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_c87ad3ae2402","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c3c4f4aadecd04077ed58cc7d8706d04245d9a690e48822b76699b5cb154be57","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/RatFunc","family_id":"exists_zpow_uniformizingpolynomial","file_id":"mathlib/Mathlib/NumberTheory/RatFunc/Ostrowski.lean","sample_id":"c87ad3ae24029059963e03348c139139f0d135dcb45808f3d2427d4e578980f8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2b60f1251ce380c3c88892d3140d40e19399072a863e960ad4ea10cd3795106f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9a258eabbca29a5c3771ee588ea1b0dec5082cf5ef32ba82797313921b1c69f8","source_sha256":"3f2d058c8c184ba4c5b0f27fc93d3aa940869e09cbde2b006e4860db27ae9682","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases exists_pair_ne α with ⟨x, y, hxy⟩\n  cases lt_or_gt_of_ne hxy <;> exact ⟨_, _, ‹_›⟩","hard_negative":true,"metrics":{"chosen_tokens":28,"rejected_tokens":8,"token_jaccard":0.038462,"token_length_ratio":0.285714},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"b6aa1915879309ab5bbf519805ba77ed2bd9bd2cbbd01fe32ad90a30fa7a0ece","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Prod.Basic\npublic import Mathlib.Logic.Function.Basic\npublic import Mathlib.Logic.Nontrivial.Defs\npublic import Mathlib.Logic.Unique\npublic import Mathlib.Order.Defs.LinearOrder\nimport Mathlib.Tactic.Attr.Register\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Nontrivial types\n\nResults about `Nontrivial`.\n-/\n\n@[expose] public section\n\nvariable {α : Type*} {β : Type*}\n\n-- `x` and `y` are explicit here, as they are often needed to guide typechecking of `h`.\ntheorem nontrivial_of_lt [Preorder α] (x y : α) (h : x < y) : Nontrivial α :=\n  ⟨⟨x, y, ne_of_lt h⟩⟩\n\nTarget:\ntheorem exists_pair_lt (α : Type*) [Nontrivial α] [LinearOrder α] : ∃ x y : α, x < y :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"d19e3b2ffed3bc088683f6ee7555b22a069dfda563d38856803dcb2388a43300","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Nontrivial","family_id":"exists_pair_lt","file_id":"mathlib/Mathlib/Logic/Nontrivial/Basic.lean","sample_id":"9a258eabbca29a5c3771ee588ea1b0dec5082cf5ef32ba82797313921b1c69f8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8269b551fa1e3baafeb590db545d184ccf387498c0689c5533fadd958fae33b7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a2717c01644181afb92dd63d96aa9ca9ce454ffe813c7c0184be6728f9e504e2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"56dfb71d32b180653fe5db01d6101ad74566b666462b27c7889ed2d552e18dc1","source_sha256":"2330b04bbf9f5616db3c6c30d466b364ee306b290c33249fea3b2f667c7433f7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let adj := mapPullbackAdj g\n  let U : Over Z := Over.mk (φ₁ ≫ snd f g)\n  let φ₁' : U ⟶ (pullback g).obj (Over.mk f) := Over.homMk φ₁\n  let φ₂' : U ⟶ (pullback g).obj (Over.mk f) := Over.homMk φ₂ (by simpa using! h₂.symm)\n  have : φ₁' = φ₂' := by\n    apply (adj.homEquiv U _).symm.injective\n    apply (Over.forget X).map_injective\n    simpa using! h₁\n  exact congr_arg CommaMorphism.left this","hard_negative":true,"metrics":{"chosen_tokens":126,"rejected_tokens":3,"token_jaccard":0.044444,"token_length_ratio":0.02381},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"b6b5b67a640c29ca60017a12982da20c1225a7e0c371e7c210aba43c10e4a96d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Comma.Over.Pullback\npublic import Mathlib.CategoryTheory.Monoidal.Cartesian.Basic\npublic import Mathlib.CategoryTheory.Adjunction.Unique\npublic import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Defs\n\nNamespace:\nCategoryTheory.ChosenPullbacksAlong\n\nLocal context:\n/-\nCopyright (c) 2025 Sina Hazratpour. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sina Hazratpour\n-/\n/-!\n# Chosen pullbacks along a morphism\n\n## Main declarations\n\n- `ChosenPullbacksAlong` : For a morphism `f : Y ⟶ X` in `C`, the type class\n  `ChosenPullbacksAlong f` provides the data of a pullback functor `Over X ⥤ Over Y`\n  as a right adjoint to `Over.map f`.\n\n## Main results\n\n- We prove that `ChosenPullbacksAlong` has good closure properties: isos have chosen pullbacks,\n  and composition of morphisms with chosen pullbacks have chosen pullbacks.\n\n- We prove that chosen pullbacks yield usual pullbacks: `ChosenPullbacksAlong.isPullback`\n  proves that for morphisms `f` and `g` with the same codomain, the object\n  `ChosenPullbacksAlong.pullbackObj f g` together with morphisms\n  `ChosenPullbacksAlong.fst f g` and `ChosenPullbacksAlong.snd f g` form a pullback square\n  over `f` and `g`.\n\n- We prove that in cartesian monoidal categories, morphisms to the terminal tensor unit and\n  the product projections have chosen pullbacks.\n\n-/\n\n@[expose] public section\n\nuniverse v₁ v₂ u₁ u₂\n\nnamespace CategoryTheory\n\nopen Category Limits CartesianMonoidalCategory MonoidalCategory Over\n\nvariable {C : Type u₁} [Category.{v₁} C]\n\n/-- A functorial choice of pullbacks along a morphism `f : Y ⟶ X` in `C` given by a functor\n`Over X ⥤ Over Y` which is a right adjoint to the functor `Over.map f`. -/\nclass ChosenPullbacksAlong {Y X : C} (f : Y ⟶ X) where\n  /-- The pullback functor along `f`. -/\n  pullback : Over X ⥤ Over Y\n  /-- The adjunction between `Over.map f` and `pullback f`. -/\n  mapPullbackAdj (f) : Over.map f ⊣ pullback\n\nvariable (C) in\n/-- A category has chosen pullbacks if every morphism has a chosen pullback. -/\nabbrev ChosenPullbacks := Π {X Y : C} (f : Y ⟶ X), ChosenPullbacksAlong f\n\nnamespace ChosenPullbacksAlong\n\n/-- Relating the existing noncomputable `HasPullbacksAlong` typeclass to `ChosenPullbacksAlong`. -/\n@[simps, implicit_reducible]\nnoncomputable def ofHasPullbacksAlong {Y X : C} (f : Y ⟶ X) [HasPullbacksAlong f] :\n    ChosenPullbacksAlong f where\n  pullback := Over.pullback f\n  mapPullbackAdj := Over.mapPullbackAdj f\n\n/-- The identity morphism has a functorial choice of pullbacks. -/\n@[implicit_reducible]\ndef id (X : C) : ChosenPullbacksAlong (𝟙 X) where\n  pullback := 𝟭 _\n  mapPullbackAdj := (Adjunction.id).ofNatIsoLeft (Over.mapId _).symm\n\n/-- Any chosen pullback functor of the identity morphism is naturally isomorphic to the identity\nfunctor. -/\ndef pullbackId (X : C) [ChosenPullbacksAlong (𝟙 X)] :\n    pullback (𝟙 X) ≅ 𝟭 (Over X) :=\n  (mapPullbackAdj (𝟙 X)).rightAdjointUniq (id X).mapPullbackAdj\n\n@[reassoc (attr := simp)]\ntheorem unit_pullbackId_hom_app (X : C) [ChosenPullbacksAlong (𝟙 X)] (Y : Over X) :\n    (mapPullbackAdj (𝟙 X)).unit.app Y ≫ (pullbackId X).hom.app ((Over.map (𝟙 X)).obj Y) =\n      (id X).mapPullbackAdj.unit.app Y := by\n  rw [pullbackId, Adjunction.unit_rightAdjointUniq_hom_app]\n\n@[reassoc (attr := simp)]\ntheorem unit_pullbackId_hom (X : C) [ChosenPullbacksAlong (𝟙 X)] :\n    (mapPullbackAdj (𝟙 X)).unit ≫ (Over.map (𝟙 X)).whiskerLeft (pullbackId X).hom =\n      (id X).mapPullbackAdj.unit := by\n  rw [pullbackId, Adjunction.unit_rightAdjointUniq_hom]\n\n@[reassoc (attr := simp)]\ntheorem pullbackId_hom_counit (X : C) [ChosenPullbacksAlong (𝟙 X)] :\n    Functor.whiskerRight (pullbackId X).hom (Over.map (𝟙 X)) ≫ (id X).mapPullbackAdj.counit =\n      (mapPullbackAdj (𝟙 X)).counit := by\n  have := Adjunction.rightAdjointUniq_hom_counit (mapPullbackAdj (𝟙 X)) (id X).mapPullbackAdj\n  rw [pullbackId, Adjunction.rightAdjointUniq_hom_counit]\n\nset_option backward.defeqAttrib.useBackward true in\n/-- Every isomorphism has a functorial choice of pullbacks. -/\n@[simps, implicit_reducible]\ndef iso {Y X : C} (f : Y ≅ X) : ChosenPullbacksAlong f.hom where\n  pullback.obj Z := Over.mk (Z.hom ≫ f.inv)\n  pullback.map {Y Z} g := Over.homMk (g.left)\n  mapPullbackAdj.unit.app T := Over.homMk (𝟙 T.left)\n  mapPullbackAdj.counit.app U := Over.homMk (𝟙 _)\n\n/-- The inverse of an isomorphism has a functorial choice of pullbacks. -/\n@[simps!, implicit_reducible]\ndef isoInv {Y X : C} (f : Y ≅ X) : ChosenPullbacksAlong f.inv := iso f.symm\n\n/-- The composition of morphisms with chosen pullbacks has a chosen pullback. -/\n@[implicit_reducible]\ndef comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z)\n    [ChosenPullbacksAlong f] [ChosenPullbacksAlong g] : ChosenPullbacksAlong (f ≫ g) where\n  pullback := pullback g ⋙ pullback f\n  mapPullbackAdj := ((mapPullbackAdj f).comp (mapPullbackAdj g)).ofNatIsoLeft\n    (Over.mapComp f g).symm\n\n/-- Any chosen pullback of a composite of morphisms is naturally isomorphic to the composition of\nchosen pullback functors. -/\ndef pullbackComp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z)\n    [ChosenPullbacksAlong f] [ChosenPullbacksAlong g] [ChosenPullbacksAlong (f ≫ g)] :\n    pullback (f ≫ g) ≅ pullback g ⋙ pullback f :=\n  Adjunction.rightAdjointUniq (mapPullbackAdj (f ≫ g)) ((comp f g).mapPullbackAdj)\n\n@[reassoc (attr := simp)]\ntheorem unit_pullbackComp_hom {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z)\n    [ChosenPullbacksAlong f] [ChosenPullbacksAlong g] [ChosenPullbacksAlong (f ≫ g)] :\n    (mapPullbackAdj (f ≫ g)).unit ≫ (Over.map (f ≫ g)).whiskerLeft (pullbackComp f g).hom =\n      (comp f g).mapPullbackAdj.unit := by\n  rw [pullbackComp, Adjunction.unit_rightAdjointUniq_hom]\n\n@[reassoc (attr := simp)]\ntheorem pullbackComp_hom_counit {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z)\n    [ChosenPullbacksAlong f] [ChosenPullbacksAlong g] [ChosenPullbacksAlong (f ≫ g)] :\n    Functor.whiskerRight (pullbackComp f g).hom (Over.map (f ≫ g)) ≫\n      (comp f g).mapPullbackAdj.counit =\n      (mapPullbackAdj (f ≫ g)).counit := by\n  rw [pullbackComp, Adjunction.rightAdjointUniq_hom_counit]\n\nset_option backward.defeqAttrib.useBackward true in\n/-- In cartesian monoidal categories, any morphism to the terminal tensor unit has a functorial\nchoice of pullbacks. -/\n@[instance_reducible, simps]\ndef cartesianMonoidalCategoryToUnit [CartesianMonoidalCategory C] {X : C} (f : X ⟶ 𝟙_ C) :\n    ChosenPullbacksAlong f where\n  pullback.obj Y := Over.mk (snd Y.left X)\n  pullback.map {Y Z} g := Over.homMk (g.left ▷ X)\n  mapPullbackAdj.unit.app T := Over.homMk (lift (𝟙 _) (T.hom))\n  mapPullbackAdj.counit.app U := Over.homMk (fst _ _)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- In cartesian monoidal categories, the first product projections `fst` have a functorial choice\nof pullbacks. -/\n@[simps, implicit_reducible]\ndef cartesianMonoidalCategoryFst [CartesianMonoidalCategory C] (X Y : C) :\n    ChosenPullbacksAlong (fst X Y : X ⊗ Y ⟶ X) where\n  pullback.obj Z := Over.mk (Z.hom ▷ Y)\n  pullback.map g := Over.homMk (g.left ▷ Y)\n  mapPullbackAdj.unit.app T := Over.homMk (lift (𝟙 _) (T.hom ≫ snd _ _))\n  mapPullbackAdj.counit.app U := Over.homMk (fst _ _)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- In cartesian monoidal categories, the second product projections `snd` have a functorial choice\nof pullbacks. -/\n@[simps, implicit_reducible]\ndef cartesianMonoidalCategorySnd [CartesianMonoidalCategory C] (X Y : C) :\n    ChosenPullbacksAlong (snd X Y : X ⊗ Y ⟶ Y) where\n  pullback.obj Z := Over.mk (X ◁ Z.hom)\n  pullback.map g := Over.homMk (X ◁ g.left)\n  mapPullbackAdj.unit.app T := Over.homMk (lift (T.hom ≫ fst _ _) (𝟙 _))\n  mapPullbackAdj.counit.app U := Over.homMk (snd _ _)\n\nsection PullbackFromChosenPullbacksAlongs\n\nvariable {Y Z X : C} (f : Y ⟶ X) (g : Z ⟶ X) [ChosenPullbacksAlong g]\n\n/-- The underlying object of the chosen pullback along `g` of `f`. -/\nabbrev pullbackObj : C := ((pullback g).obj (Over.mk f)).left\n\n/-- A morphism in `Over X` from the chosen pullback along `g` of `f` to `Over.mk f`. -/\nabbrev fst' : (Over.map g).obj ((pullback g).obj (Over.mk f)) ⟶ Over.mk f :=\n  (mapPullbackAdj g).counit.app <| Over.mk f\n\n/-- The first projection from the chosen pullback along `g` of `f` to the domain of `f`. -/\nabbrev fst : pullbackObj f g ⟶ Y := fst' f g |>.left\n\ntheorem fst'_left : (fst' f g).left = fst f g := rfl\n\n/-- The second projection from the chosen pullback along `g` of `f` to the domain of `g`. -/\nabbrev snd : pullbackObj f g ⟶ Z := (pullback g).obj (Over.mk f) |>.hom\n\n/-- A morphism in `Over X` from the chosen pullback along `g` of `f` to `Over.mk g`. -/\nabbrev snd' : (Over.map g).obj ((pullback g).obj (Over.mk f)) ⟶ (Over.mk g) :=\n  Over.homMk (snd f g)\n\ntheorem snd'_left : (snd' f g).left = snd f g := rfl\n\nvariable {f g}\n\n@[reassoc]\ntheorem condition : fst f g ≫ f = snd f g ≫ g :=\n  Over.w (fst' f g)\n\nvariable (f g) in\n@[ext]\n\nTarget:\ntheorem hom_ext {W : C} {φ₁ φ₂ : W ⟶ pullbackObj f g} (h₁ : φ₁ ≫ fst _ _ = φ₂ ≫ fst _ _)\n    (h₂ : φ₁ ≫ snd _ _ = φ₂ ≫ snd _ _) :\n    φ₁ = φ₂ :=\n\nProof body:\n","rejected":"by\n  exact hom_ext","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"9b2785e9207717371a0fe5b97ec4bf30758e3e39f285a35903c6e16d7905009e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/LocallyCartesianClosed","family_id":"hom_ext","file_id":"mathlib/Mathlib/CategoryTheory/LocallyCartesianClosed/ChosenPullbacksAlong.lean","sample_id":"56dfb71d32b180653fe5db01d6101ad74566b666462b27c7889ed2d552e18dc1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"018854a18bc4ddad72fb2634603cc3211f19d9c9b481d90eec379823ad5a6440","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8e6d776a112cd89de7aa1f1a126b48dfead5089fd7abc542320a1969d23d73fc","source_sha256":"f589366b76253f55a5f00af110f0088b3f55c67577d56940b18639c5f1a2a3b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [← zsmul_eq_mul, ofAdd_zsmul, map_zpow]","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":2,"token_jaccard":0.090909,"token_length_ratio":0.181818},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"b6d24590cd0e0007427823199b16c7dae66ad56fbff561f6172b968aede15488","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.Order.SemiconjSup\n\nNamespace:\nCircleDeg1Lift\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Translation number of a monotone real map that commutes with `x ↦ x + 1`\n\nLet `f : ℝ → ℝ` be a monotone map such that `f (x + 1) = f x + 1` for all `x`. Then the limit\n$$\n  \\tau(f)=\\lim_{n\\to\\infty}{f^n(x)-x}{n}\n$$\nexists and does not depend on `x`. This number is called the *translation number* of `f`.\nDifferent authors use different notation for this number: `τ`, `ρ`, `rot`, etc\n\nIn this file we define a structure `CircleDeg1Lift` for bundled maps with these properties, define\ntranslation number of `f : CircleDeg1Lift`, prove some estimates relating `f^n(x)-x` to `τ(f)`. In\ncase of a continuous map `f` we also prove that `f` admits a point `x` such that `f^n(x)=x+m` if and\nonly if `τ(f)=m/n`.\n\nMaps of this type naturally appear as lifts of orientation-preserving circle homeomorphisms. More\nprecisely, let `f` be an orientation-preserving homeomorphism of the circle $S^1=ℝ/ℤ$, and\nconsider a real number `a` such that\n`⟦a⟧ = f 0`, where `⟦⟧` means the natural projection `ℝ → ℝ/ℤ`. Then there exists a unique\ncontinuous function `F : ℝ → ℝ` such that `F 0 = a` and `⟦F x⟧ = f ⟦x⟧` for all `x` (this fact is\nnot formalized yet). This function is strictly monotone, continuous, and satisfies\n`F (x + 1) = F x + 1`. The number `⟦τ F⟧ : ℝ / ℤ` is called the *rotation number* of `f`.\nIt does not depend on the choice of `a`.\n\n## Main definitions\n\n* `CircleDeg1Lift`: a monotone map `f : ℝ → ℝ` such that `f (x + 1) = f x + 1` for all `x`;\n  the type `CircleDeg1Lift` is equipped with `Lattice` and `Monoid` structures; the\n  multiplication is given by composition: `(f * g) x = f (g x)`.\n* `CircleDeg1Lift.translationNumber`: translation number of `f : CircleDeg1Lift`.\n\n## Main statements\n\nWe prove the following properties of `CircleDeg1Lift.translationNumber`.\n\n* `CircleDeg1Lift.translationNumber_eq_of_dist_bounded`: if the distance between `(f^n) 0`\n  and `(g^n) 0` is bounded from above uniformly in `n : ℕ`, then `f` and `g` have equal\n  translation numbers.\n\n* `CircleDeg1Lift.translationNumber_eq_of_semiconjBy`: if two `CircleDeg1Lift` maps `f`, `g`\n  are semiconjugate by a `CircleDeg1Lift` map, then `τ f = τ g`.\n\n* `CircleDeg1Lift.translationNumber_units_inv`: if `f` is an invertible `CircleDeg1Lift` map\n  (equivalently, `f` is a lift of an orientation-preserving circle homeomorphism), then\n  the translation number of `f⁻¹` is the negative of the translation number of `f`.\n\n* `CircleDeg1Lift.translationNumber_mul_of_commute`: if `f` and `g` commute, then\n  `τ (f * g) = τ f + τ g`.\n\n* `CircleDeg1Lift.translationNumber_eq_rat_iff`: the translation number of `f` is equal to\n  a rational number `m / n` if and only if `(f^n) x = x + m` for some `x`.\n\n* `CircleDeg1Lift.semiconj_of_bijective_of_translationNumber_eq`: if `f` and `g` are two\n  bijective `CircleDeg1Lift` maps and their translation numbers are equal, then these\n  maps are semiconjugate to each other.\n\n* `CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq`: let `f₁` and `f₂` be\n  two actions of a group `G` on the circle by degree 1 maps (formally, `f₁` and `f₂` are two\n  homomorphisms from `G →* CircleDeg1Lift`). If the translation numbers of `f₁ g` and `f₂ g` are\n  equal to each other for all `g : G`, then these two actions are semiconjugate by some\n  `F : CircleDeg1Lift`. This is a version of Proposition 5.4 from [Étienne Ghys, Groupes\n  d'homéomorphismes du cercle et cohomologie bornée][ghys87:groupes].\n\n## Notation\n\nWe use a local notation `τ` for the translation number of `f : CircleDeg1Lift`.\n\n## Implementation notes\n\nWe define the translation number of `f : CircleDeg1Lift` to be the limit of the sequence\n`(f ^ (2 ^ n)) 0 / (2 ^ n)`, then prove that `((f ^ n) x - x) / n` tends to this number for any `x`.\nThis way it is much easier to prove that the limit exists and basic properties of the limit.\n\nWe define translation number for a wider class of maps `f : ℝ → ℝ` instead of lifts of orientation\npreserving circle homeomorphisms for two reasons:\n\n* non-strictly monotone circle self-maps with discontinuities naturally appear as Poincaré maps\n  for some flows on the two-torus (e.g., one can take a constant flow and glue in a few Cherry\n  cells);\n* definition and some basic properties still work for this class.\n\n## References\n\n* [Étienne Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée][ghys87:groupes]\n\n## TODO\n\nHere are some short-term goals.\n\n* Introduce a structure or a typeclass for lifts of circle homeomorphisms. We use\n  `Units CircleDeg1Lift` for now, but it's better to have a dedicated type (or a typeclass?).\n\n* Prove that the `SemiconjBy` relation on circle homeomorphisms is an equivalence relation.\n\n* Introduce `ConditionallyCompleteLattice` structure, use it in the proof of\n  `CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq`.\n\n* Prove that the orbits of the irrational rotation are dense in the circle. Deduce that a\n  homeomorphism with an irrational rotation is semiconjugate to the corresponding irrational\n  translation by a continuous `CircleDeg1Lift`.\n\n## Tags\n\ncircle homeomorphism, rotation number\n-/\n\n@[expose] public section\n\nopen Filter Set Int Topology\nopen Function hiding Commute\n\n/-!\n### Definition and monoid structure\n-/\n\n/-- A lift of a monotone degree one map `S¹ → S¹`. -/\nstructure CircleDeg1Lift : Type extends ℝ →o ℝ where\n  map_add_one' : ∀ x, toFun (x + 1) = toFun x + 1\n\nnamespace CircleDeg1Lift\n\ninstance : FunLike CircleDeg1Lift ℝ ℝ where\n  coe f := f.toFun\n  coe_injective | ⟨⟨_, _⟩, _⟩, ⟨⟨_, _⟩, _⟩, rfl => rfl\n\ninstance : OrderHomClass CircleDeg1Lift ℝ ℝ where\n  map_rel f _ _ h := f.monotone' h\n\n@[simp] theorem coe_mk (f h) : ⇑(mk f h) = f := rfl\n\nvariable (f g : CircleDeg1Lift)\n\n@[simp] theorem coe_toOrderHom : ⇑f.toOrderHom = f := rfl\n\nprotected theorem monotone : Monotone f := f.monotone'\n\n@[gcongr, mono] theorem mono {x y} (h : x ≤ y) : f x ≤ f y := f.monotone h\n\ntheorem strictMono_iff_injective : StrictMono f ↔ Injective f :=\n  f.monotone.strictMono_iff_injective\n\n@[simp]\ntheorem map_add_one : ∀ x, f (x + 1) = f x + 1 :=\n  f.map_add_one'\n\n@[simp]\ntheorem map_one_add (x : ℝ) : f (1 + x) = 1 + f x := by rw [add_comm, map_add_one, add_comm 1]\n\n@[ext]\ntheorem ext ⦃f g : CircleDeg1Lift⦄ (h : ∀ x, f x = g x) : f = g :=\n  DFunLike.ext f g h\n\ninstance : Monoid CircleDeg1Lift where\n  mul f g :=\n    { toOrderHom := f.1.comp g.1\n      map_add_one' := fun x => by simp [map_add_one] }\n  one := ⟨.id, fun _ => rfl⟩\n  mul_one _ := rfl\n  one_mul _ := rfl\n  mul_assoc _ _ _ := DFunLike.coe_injective rfl\n\ninstance : Inhabited CircleDeg1Lift := ⟨1⟩\n\n@[simp]\ntheorem coe_mul : ⇑(f * g) = f ∘ g :=\n  rfl\n\ntheorem mul_apply (x) : (f * g) x = f (g x) :=\n  rfl\n\n@[simp]\ntheorem coe_one : ⇑(1 : CircleDeg1Lift) = id :=\n  rfl\n\ninstance unitsHasCoeToFun : CoeFun CircleDeg1Liftˣ fun _ => ℝ → ℝ :=\n  ⟨fun f => ⇑(f : CircleDeg1Lift)⟩\n\n@[simp]\ntheorem units_inv_apply_apply (f : CircleDeg1Liftˣ) (x : ℝ) :\n    (f⁻¹ : CircleDeg1Liftˣ) (f x) = x := by simp only [← mul_apply, f.inv_mul, coe_one, id]\n\n@[simp]\ntheorem units_apply_inv_apply (f : CircleDeg1Liftˣ) (x : ℝ) :\n    f ((f⁻¹ : CircleDeg1Liftˣ) x) = x := by simp only [← mul_apply, f.mul_inv, coe_one, id]\n\n/-- If a lift of a circle map is bijective, then it is an order automorphism of the line. -/\ndef toOrderIso : CircleDeg1Liftˣ →* ℝ ≃o ℝ where\n  toFun f :=\n    { toFun := f\n      invFun := ⇑f⁻¹\n      left_inv := units_inv_apply_apply f\n      right_inv := units_apply_inv_apply f\n      map_rel_iff' := ⟨fun h => by simpa using mono (↑f⁻¹) h, mono f⟩ }\n  map_one' := rfl\n  map_mul' _ _ := rfl\n\n@[simp]\ntheorem coe_toOrderIso (f : CircleDeg1Liftˣ) : ⇑(toOrderIso f) = f :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderIso_symm (f : CircleDeg1Liftˣ) :\n    ⇑(toOrderIso f).symm = (f⁻¹ : CircleDeg1Liftˣ) :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderIso_inv (f : CircleDeg1Liftˣ) : ⇑(toOrderIso f)⁻¹ = (f⁻¹ : CircleDeg1Liftˣ) :=\n  rfl\n\ntheorem isUnit_iff_bijective {f : CircleDeg1Lift} : IsUnit f ↔ Bijective f :=\n  ⟨fun ⟨u, h⟩ => h ▸ (toOrderIso u).bijective, fun h =>\n    Units.isUnit\n      { val := f\n        inv :=\n          { toFun := (Equiv.ofBijective f h).symm\n            monotone' := fun x y hxy =>\n              (f.strictMono_iff_injective.2 h.1).le_iff_le.1\n                (by simp only [Equiv.ofBijective_apply_symm_apply f h, hxy])\n            map_add_one' := fun x =>\n              h.1 <| by simp only [Equiv.ofBijective_apply_symm_apply f, f.map_add_one] }\n        val_inv := ext <| Equiv.ofBijective_apply_symm_apply f h\n        inv_val := ext <| Equiv.ofBijective_symm_apply_apply f h }⟩\n\ntheorem coe_pow : ∀ n : ℕ, ⇑(f ^ n) = f^[n]\n  | 0 => rfl\n  | n + 1 => by\n    simp [coe_pow n, pow_succ]\n\ntheorem semiconjBy_iff_semiconj {f g₁ g₂ : CircleDeg1Lift} :\n    SemiconjBy f g₁ g₂ ↔ Semiconj f g₁ g₂ :=\n  CircleDeg1Lift.ext_iff\n\ntheorem commute_iff_commute {f g : CircleDeg1Lift} : Commute f g ↔ Function.Commute f g :=\n  CircleDeg1Lift.ext_iff\n\n/-!\n### Translate by a constant\n-/\n\n\n/-- The map `y ↦ x + y` as a `CircleDeg1Lift`. More precisely, we define a homomorphism from\n`Multiplicative ℝ` to `CircleDeg1Liftˣ`, so the translation by `x` is\n`translation (Multiplicative.ofAdd x)`. -/\ndef translate : Multiplicative ℝ →* CircleDeg1Liftˣ := MonoidHom.toHomUnits <|\n  { toFun x := ⟨⟨fun y => x.toAdd + y, add_right_mono⟩, fun _ => (add_assoc ..).symm⟩\n    map_one' := ext zero_add\n    map_mul' _ _ := ext <| add_assoc _ _ }\n\n@[simp]\ntheorem translate_apply (x y : ℝ) : translate (Multiplicative.ofAdd x) y = x + y :=\n  rfl\n\n@[simp]\ntheorem translate_inv_apply (x y : ℝ) : (translate <| Multiplicative.ofAdd x)⁻¹ y = -x + y :=\n  rfl\n\n@[simp]\n\nTarget:\ntheorem translate_zpow (x : ℝ) (n : ℤ) :\n    translate (Multiplicative.ofAdd x) ^ n = translate (Multiplicative.ofAdd <| ↑n * x) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/Circle","family_id":"translate_zpow","file_id":"mathlib/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean","sample_id":"8e6d776a112cd89de7aa1f1a126b48dfead5089fd7abc542320a1969d23d73fc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a139748a86235708cf34b19e320c41795f248777d1dc57cebe0b73665a686a07","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"59f057c902f53c00cf6f43ba76e1799f474e433421931bfc687b36d55befea24","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9844a70ca9aa095c0c679f810bb6057aadb712cb3e2289c8ec909fdd876bd5a3","source_sha256":"6ea826f0633d3930f7872110f571f16f76059ec3668e205ef02a1089b9b31873","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · exact HallMarriageTheorem.hall_hard_inductive\n  · rintro ⟨f, hf₁, hf₂⟩ s\n    rw [← card_image_of_injective s hf₁]\n    apply card_le_card\n    intro\n    rw [mem_image, mem_biUnion]\n    rintro ⟨x, hx, rfl⟩\n    exact ⟨x, hx, hf₂ x⟩","hard_negative":true,"metrics":{"chosen_tokens":54,"rejected_tokens":2,"token_jaccard":0.033333,"token_length_ratio":0.037037},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"b6f163953875c77810db4e7f37cd191b8b3e17842f477ba1a0309bbdaa18cd09","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Basic\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Data.Set.Finite.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Alena Gusakov, Bhavik Mehta, Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alena Gusakov, Bhavik Mehta, Kyle Miller\n-/\n/-!\n# Hall's Marriage Theorem for finite index types\n\nThis module proves the basic form of Hall's theorem.\nIn contrast to the theorem described in `Combinatorics.Hall.Basic`, this\nversion requires that the indexed family `t : ι → Finset α` have `ι` be finite.\nThe `Combinatorics.Hall.Basic` module applies a compactness argument to this version\nto remove the `Finite` constraint on `ι`.\n\nThe modules are split like this since the generalized statement\ndepends on the topology and category theory libraries, but the finite\ncase in this module has few dependencies.\n\nA description of this formalization is in [Gusakov2021].\n\n## Main statements\n\n* `Finset.all_card_le_biUnion_card_iff_existsInjective'` is Hall's theorem with\n  a finite index set.  This is elsewhere generalized to\n  `Finset.all_card_le_biUnion_card_iff_existsInjective`.\n\n## Tags\n\nHall's Marriage Theorem, indexed families\n-/\n\npublic section\n\n\nopen Finset\n\nuniverse u v\n\nnamespace HallMarriageTheorem\n\nvariable {ι : Type u} {α : Type v} [DecidableEq α] {t : ι → Finset α}\n\nsection Fintype\n\nvariable [Fintype ι]\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem hall_cond_of_erase {x : ι} (a : α)\n    (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t))\n    (s' : Finset { x' : ι | x' ≠ x }) : #s' ≤ #(s'.biUnion fun x' => (t x').erase a) := by\n  haveI := Classical.decEq ι\n  specialize ha (s'.image fun z => z.1)\n  rw [image_nonempty, Finset.card_image_of_injective s' Subtype.coe_injective] at ha\n  by_cases! he : s'.Nonempty\n  · have ha' : #s' < #(s'.biUnion fun x => t x) := by\n      convert! ha he fun h => by simpa [← h] using mem_univ x using 2\n      ext x\n      simp only [mem_image, mem_biUnion, SetCoe.exists, exists_and_right,\n        exists_eq_right]\n    rw [← erase_biUnion]\n    by_cases hb : a ∈ s'.biUnion fun x => t x\n    · rw [card_erase_of_mem hb]\n      exact Nat.le_sub_one_of_lt ha'\n    · rw [erase_eq_of_notMem hb]\n      exact Nat.le_of_lt ha'\n  · subst s'\n    simp\n\n/-- First case of the inductive step: assuming that\n`∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)`\nand that the statement of **Hall's Marriage Theorem** is true for all\n`ι'` of cardinality ≤ `n`, then it is true for `ι` of cardinality `n + 1`.\n-/\ntheorem hall_hard_inductive_step_A {n : ℕ} (hn : Fintype.card ι = n + 1)\n    (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))\n    (ih :\n      ∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),\n        Fintype.card ι' ≤ n →\n          (∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →\n            ∃ f : ι' → α, Function.Injective f ∧ ∀ x, f x ∈ t' x)\n    (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t)) :\n    ∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by\n  haveI : Nonempty ι := Fintype.card_pos_iff.mp (hn.symm ▸ Nat.succ_pos _)\n  haveI := Classical.decEq ι\n  -- Choose an arbitrary element `x : ι` and `y : t x`.\n  let x := Classical.arbitrary ι\n  have tx_ne : (t x).Nonempty := by\n    rw [← Finset.card_pos]\n    calc\n      0 < 1 := Nat.one_pos\n      _ ≤ #(.biUnion {x} t) := ht {x}\n      _ = (t x).card := by rw [Finset.singleton_biUnion]\n  choose y hy using tx_ne\n  -- Restrict to everything except `x` and `y`.\n  let ι' := { x' : ι | x' ≠ x }\n  let t' : ι' → Finset α := fun x' => (t x').erase y\n  have card_ι' : Fintype.card ι' = n :=\n    calc\n      Fintype.card ι' = Fintype.card ι - 1 := Set.card_ne_eq _\n      _ = n := by rw [hn, Nat.add_succ_sub_one, add_zero]\n  rcases ih t' card_ι'.le (hall_cond_of_erase y ha) with ⟨f', hfinj, hfr⟩\n  -- Extend the resulting function.\n  refine ⟨fun z => if h : z = x then y else f' ⟨z, h⟩, ?_, ?_⟩\n  · rintro z₁ z₂\n    have key : ∀ {x}, y ≠ f' x := by\n      intro x h\n      simpa [t', ← h] using hfr x\n    by_cases h₁ : z₁ = x <;> by_cases h₂ : z₂ = x <;>\n      simp [h₁, h₂, hfinj.eq_iff, key, key.symm]\n  · intro z\n    simp only\n    split_ifs with hz\n    · rwa [hz]\n    · specialize hfr ⟨z, hz⟩\n      rw [mem_erase] at hfr\n      exact hfr.2\n\ntheorem hall_cond_of_restrict {ι : Type u} {t : ι → Finset α} {s : Finset ι}\n    (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t)) (s' : Finset (s : Set ι)) :\n    #s' ≤ #(s'.biUnion fun a' => t a') := by\n  classical\n    rw [← card_image_of_injective s' Subtype.coe_injective]\n    convert! ht (s'.image fun z => z.1) using 1\n    apply congr_arg\n    ext y\n    simp\n\ntheorem hall_cond_of_compl {ι : Type u} {t : ι → Finset α} {s : Finset ι}\n    (hus : #s = #(s.biUnion t)) (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))\n    (s' : Finset (sᶜ : Set ι)) : #s' ≤ #(s'.biUnion fun x' => t x' \\ s.biUnion t) := by\n  haveI := Classical.decEq ι\n  have disj : Disjoint s (s'.image fun z => z.1) := by\n    simp only [disjoint_left, not_exists, mem_image, SetCoe.exists, exists_and_right,\n      exists_eq_right]\n    intro x hx hc _\n    exact absurd hx hc\n  have : #s' = #(s ∪ s'.image fun z => z.1) - #s := by\n    simp [disj, card_image_of_injective _ Subtype.coe_injective, Nat.add_sub_cancel_left]\n  rw [this, hus]\n  refine (Nat.sub_le_sub_right (ht _) _).trans ?_\n  rw [← card_sdiff_of_subset]\n  · gcongr\n    intro t\n    simp only [mem_biUnion, mem_sdiff, not_exists, mem_image, and_imp, mem_union,\n      exists_imp]\n    rintro x (hx | ⟨x', hx', rfl⟩) rat hs\n    · exact False.elim <| (hs x) <| And.intro hx rat\n    · use x', hx', rat, hs\n  · apply biUnion_subset_biUnion_of_subset_left\n    apply subset_union_left\n\n/-- Second case of the inductive step: assuming that\n`∃ (s : Finset ι), s ≠ univ → #s = #(s.biUnion t)`\nand that the statement of **Hall's Marriage Theorem** is true for all\n`ι'` of cardinality ≤ `n`, then it is true for `ι` of cardinality `n + 1`.\n-/\ntheorem hall_hard_inductive_step_B {n : ℕ} (hn : Fintype.card ι = n + 1)\n    (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))\n    (ih :\n      ∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),\n        Fintype.card ι' ≤ n →\n          (∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →\n            ∃ f : ι' → α, Function.Injective f ∧ ∀ x, f x ∈ t' x)\n    (s : Finset ι) (hs : s.Nonempty) (hns : s ≠ univ) (hus : #s = #(s.biUnion t)) :\n    ∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by\n  haveI := Classical.decEq ι\n  -- Restrict to `s`\n  rw [Nat.add_one] at hn\n  have card_ι'_le : Fintype.card s ≤ n := by\n    apply Nat.le_of_lt_succ\n    calc\n      Fintype.card s = #s := Fintype.card_coe _\n      _ < Fintype.card ι := (card_lt_iff_ne_univ _).mpr hns\n      _ = n.succ := hn\n  let t' : s → Finset α := fun x' => t x'\n  rcases ih t' card_ι'_le (hall_cond_of_restrict ht) with ⟨f', hf', hsf'⟩\n  -- Restrict to `sᶜ` in the domain and `(s.biUnion t)ᶜ` in the codomain.\n  set ι'' := (s : Set ι)ᶜ\n  let t'' : ι'' → Finset α := fun a'' => t a'' \\ s.biUnion t\n  have card_ι''_le : Fintype.card ι'' ≤ n := by\n    simp_rw [ι'', ← Nat.lt_succ_iff, ← hn, ← Finset.coe_compl, coe_sort_coe]\n    rwa [Fintype.card_coe, card_compl_lt_iff_nonempty]\n  rcases ih t'' card_ι''_le (hall_cond_of_compl hus ht) with ⟨f'', hf'', hsf''⟩\n  -- Put them together\n  have f''_notMem_biUnion : ∀ (x'') (hx'' : x'' ∉ s), f'' ⟨x'', hx''⟩ ∉ s.biUnion t := by\n    intro x'' hx''\n    have h := hsf'' ⟨x'', hx''⟩\n    rw [mem_sdiff] at h\n    exact h.2\n  have im_disj :\n      ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : x'' ∉ s), f' ⟨x', hx'⟩ ≠ f'' ⟨x'', hx''⟩ := by\n    grind\n  refine ⟨fun x => if h : x ∈ s then f' ⟨x, h⟩ else f'' ⟨x, h⟩, ?_, ?_⟩\n  · refine hf'.dite _ hf'' (@fun x x' => im_disj x x' _ _)\n  · intro x\n    simp only\n    split_ifs with h\n    · exact hsf' ⟨x, h⟩\n    · exact sdiff_subset (hsf'' ⟨x, h⟩)\n\nend Fintype\n\nvariable [Finite ι]\n\n/-- Here we combine the two inductive steps into a full strong induction proof,\ncompleting the proof the harder direction of **Hall's Marriage Theorem**.\n-/\ntheorem hall_hard_inductive (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t)) :\n    ∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by\n  cases nonempty_fintype ι\n  generalize hn : Fintype.card ι = m\n  induction m using Nat.strongRecOn generalizing ι with | ind n ih => _\n  rcases n with (_ | n)\n  · rw [Fintype.card_eq_zero_iff] at hn\n    exact ⟨isEmptyElim, isEmptyElim, isEmptyElim⟩\n  · have ih' : ∀ (ι' : Type u) [Fintype ι'] (t' : ι' → Finset α), Fintype.card ι' ≤ n →\n        (∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →\n        ∃ f : ι' → α, Function.Injective f ∧ ∀ x, f x ∈ t' x := by\n      intro ι' _ _ hι' ht'\n      exact ih _ (Nat.lt_succ_of_le hι') ht' _ rfl\n    by_cases! h : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t)\n    · refine hall_hard_inductive_step_A hn ht (@fun ι' => ih' ι') h\n    · rcases h with ⟨s, sne, snu, sle⟩\n      exact hall_hard_inductive_step_B hn ht (@fun ι' => ih' ι')\n        s sne snu (Nat.le_antisymm (ht _) sle)\n\nend HallMarriageTheorem\n\n/-- This is the version of **Hall's Marriage Theorem** in terms of indexed\nfamilies of finite sets `t : ι → Finset α` with `ι` finite.\nIt states that there is a set of distinct representatives if and only\nif every union of `k` of the sets has at least `k` elements.\n\nSee `Finset.all_card_le_biUnion_card_iff_exists_injective` for a version\nwhere the `Finite ι` constraint is removed.\n-/\n\nTarget:\ntheorem Finset.all_card_le_biUnion_card_iff_existsInjective' {ι α : Type*} [Finite ι]\n    [DecidableEq α] (t : ι → Finset α) :\n    (∀ s : Finset ι, #s ≤ #(s.biUnion t)) ↔\n      ∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_9844a70ca9aa","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"42f7ec6e02aae223df7362714fd72ad4d1d2ff79f28a8be2874dcfc3f883c847","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Hall","family_id":"finset","file_id":"mathlib/Mathlib/Combinatorics/Hall/Finite.lean","sample_id":"9844a70ca9aa095c0c679f810bb6057aadb712cb3e2289c8ec909fdd876bd5a3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b1c7cebb6805403539cf95bc62a5c22e029b7da315a1e90239408f66439637c6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"08841f49db68791c6da3b198f7e39c3a02b070c837643d3e23c4d5b852e89aa6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f97d07cceec981c168c5f2eca445cd4e95e1981b7df2ab5f563b675d89d0e28f","source_sha256":"bd4b2b01d950a3b9c8fd78b82f11b816abe9ec16bf6d55a504ac2e051e9cf9fe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [superpolynomialDecay_iff_abs_tendsto_zero] at hf ⊢\n  refine fun z =>\n    tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds (hf z)\n      (Eventually.of_forall fun x => abs_nonneg _) (hfg.mono fun x hx => ?_)\n  calc\n    |k x ^ z * g x| = |k x ^ z| * |g x| := abs_mul (k x ^ z) (g x)\n    _ ≤ |k x ^ z| * |f x| := by gcongr _ * ?_; exact hx\n    _ = |k x ^ z * f x| := (abs_mul (k x ^ z) (f x)).symm","hard_negative":false,"metrics":{"chosen_tokens":123,"rejected_tokens":128,"token_jaccard":0.911111,"token_length_ratio":1.04065},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"b6f9172b44bf3d8ec067a6d06f65ceb3613e81037a297137dddb5cb860f7b98e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Eval.Defs\npublic import Mathlib.Analysis.Asymptotics.Lemmas\n\nNamespace:\nAsymptotics\n\nLocal context:\n/-\nCopyright (c) 2021 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\n/-!\n# Super-Polynomial Function Decay\n\nThis file defines a predicate `Asymptotics.SuperpolynomialDecay f` for a function satisfying\none of the following equivalent definitions (the definition is in terms of the first condition):\n\n* `x ^ n * f` tends to `𝓝 0` for all (or sufficiently large) naturals `n`\n* `|x ^ n * f|` tends to `𝓝 0` for all naturals `n` (`superpolynomialDecay_iff_abs_tendsto_zero`)\n* `|x ^ n * f|` is bounded for all naturals `n` (`superpolynomialDecay_iff_abs_isBoundedUnder`)\n* `f` is `o(x ^ c)` for all integers `c` (`superpolynomialDecay_iff_isLittleO`)\n* `f` is `O(x ^ c)` for all integers `c` (`superpolynomialDecay_iff_isBigO`)\n\nThese conditions are all equivalent to conditions in terms of polynomials, replacing `x ^ c` with\n  `p(x)` or `p(x)⁻¹` as appropriate, since asymptotically `p(x)` behaves like `X ^ p.natDegree`.\nThese further equivalences are not proven in mathlib but would be good future projects.\n\nThe definition of superpolynomial decay for `f : α → β` is relative to a parameter `k : α → β`.\nSuper-polynomial decay then means `f x` decays faster than `(k x) ^ c` for all integers `c`.\nEquivalently `f x` decays faster than `p.eval (k x)` for all polynomials `p : β[X]`.\nThe definition is also relative to a filter `l : Filter α` where the decay rate is compared.\n\nWhen the map `k` is given by `n ↦ ↑n : ℕ → ℝ` this defines negligible functions:\nhttps://en.wikipedia.org/wiki/Negligible_function\n\nWhen the map `k` is given by `(r₁,...,rₙ) ↦ r₁*...*rₙ : ℝⁿ → ℝ` this is equivalent\n  to the definition of rapidly decreasing functions given here:\nhttps://ncatlab.org/nlab/show/rapidly+decreasing+function\n\n## Main statements\n\n* `SuperpolynomialDecay.polynomial_mul` says that if `f(x)` is negligible,\n    then so is `p(x) * f(x)` for any polynomial `p`.\n* `superpolynomialDecay_iff_zpow_tendsto_zero` gives an equivalence between definitions in terms\n    of decaying faster than `k(x) ^ n` for all naturals `n` or `k(x) ^ c` for all integer `c`.\n-/\n\n@[expose] public section\n\n\nnamespace Asymptotics\n\nopen Topology Polynomial\n\nopen Filter\n\n/-- `f` has superpolynomial decay in parameter `k` along filter `l` if\n  `k ^ n * f` tends to zero at `l` for all naturals `n` -/\ndef SuperpolynomialDecay {α β : Type*} [TopologicalSpace β] [CommSemiring β] (l : Filter α)\n    (k : α → β) (f : α → β) :=\n  ∀ n : ℕ, Tendsto (fun a : α => k a ^ n * f a) l (𝓝 0)\n\nvariable {α β : Type*} {l : Filter α} {k : α → β} {f g g' : α → β}\n\nsection CommSemiring\n\nvariable [TopologicalSpace β] [CommSemiring β]\n\ntheorem SuperpolynomialDecay.congr' (hf : SuperpolynomialDecay l k f) (hfg : f =ᶠ[l] g) :\n    SuperpolynomialDecay l k g := fun z =>\n  (hf z).congr' (EventuallyEq.mul (EventuallyEq.refl l _) hfg)\n\ntheorem SuperpolynomialDecay.congr (hf : SuperpolynomialDecay l k f) (hfg : ∀ x, f x = g x) :\n    SuperpolynomialDecay l k g := fun z =>\n  (hf z).congr fun x => (congr_arg fun a => k x ^ z * a) <| hfg x\n\n@[simp]\ntheorem superpolynomialDecay_zero (l : Filter α) (k : α → β) : SuperpolynomialDecay l k 0 :=\n  fun z => by simpa only [Pi.zero_apply, mul_zero] using tendsto_const_nhds\n\ntheorem SuperpolynomialDecay.add [ContinuousAdd β] (hf : SuperpolynomialDecay l k f)\n    (hg : SuperpolynomialDecay l k g) : SuperpolynomialDecay l k (f + g) := fun z => by\n  simpa only [mul_add, add_zero, Pi.add_apply] using (hf z).add (hg z)\n\ntheorem SuperpolynomialDecay.mul [ContinuousMul β] (hf : SuperpolynomialDecay l k f)\n    (hg : SuperpolynomialDecay l k g) : SuperpolynomialDecay l k (f * g) := fun z => by\n  simpa only [mul_assoc, one_mul, mul_zero, pow_zero] using! (hf z).mul (hg 0)\n\ntheorem SuperpolynomialDecay.mul_const [ContinuousMul β] (hf : SuperpolynomialDecay l k f) (c : β) :\n    SuperpolynomialDecay l k fun n => f n * c := fun z => by\n  simpa only [← mul_assoc, zero_mul] using Tendsto.mul_const c (hf z)\n\ntheorem SuperpolynomialDecay.const_mul [ContinuousMul β] (hf : SuperpolynomialDecay l k f) (c : β) :\n    SuperpolynomialDecay l k fun n => c * f n :=\n  (hf.mul_const c).congr fun _ => mul_comm _ _\n\ntheorem SuperpolynomialDecay.param_mul (hf : SuperpolynomialDecay l k f) :\n    SuperpolynomialDecay l k (k * f) := fun z =>\n  tendsto_nhds.2 fun s hs hs0 =>\n    l.sets_of_superset ((tendsto_nhds.1 (hf <| z + 1)) s hs hs0) fun x hx => by\n      simpa only [Set.mem_preimage, Pi.mul_apply, ← mul_assoc, ← pow_succ] using hx\n\ntheorem SuperpolynomialDecay.mul_param (hf : SuperpolynomialDecay l k f) :\n    SuperpolynomialDecay l k (f * k) :=\n  hf.param_mul.congr fun _ => mul_comm _ _\n\ntheorem SuperpolynomialDecay.param_pow_mul (hf : SuperpolynomialDecay l k f) (n : ℕ) :\n    SuperpolynomialDecay l k (k ^ n * f) := by\n  induction n with\n  | zero => simpa only [one_mul, pow_zero] using hf\n  | succ n hn => simpa only [pow_succ', mul_assoc] using hn.param_mul\n\ntheorem SuperpolynomialDecay.mul_param_pow (hf : SuperpolynomialDecay l k f) (n : ℕ) :\n    SuperpolynomialDecay l k (f * k ^ n) :=\n  (hf.param_pow_mul n).congr fun _ => mul_comm _ _\n\ntheorem SuperpolynomialDecay.polynomial_mul [ContinuousAdd β] [ContinuousMul β]\n    (hf : SuperpolynomialDecay l k f) (p : β[X]) :\n    SuperpolynomialDecay l k fun x => (p.eval <| k x) * f x :=\n  Polynomial.induction_on' p (fun p q hp hq => by simpa [add_mul] using! hp.add hq) fun n c => by\n    simpa [mul_assoc] using! (hf.param_pow_mul n).const_mul c\n\ntheorem SuperpolynomialDecay.mul_polynomial [ContinuousAdd β] [ContinuousMul β]\n    (hf : SuperpolynomialDecay l k f) (p : β[X]) :\n    SuperpolynomialDecay l k fun x => f x * (p.eval <| k x) :=\n  (hf.polynomial_mul p).congr fun _ => mul_comm _ _\n\nend CommSemiring\n\nsection OrderedCommSemiring\n\nvariable [TopologicalSpace β] [CommSemiring β] [PartialOrder β] [IsOrderedRing β] [OrderTopology β]\n\ntheorem SuperpolynomialDecay.trans_eventuallyLE (hk : 0 ≤ᶠ[l] k) (hg : SuperpolynomialDecay l k g)\n    (hg' : SuperpolynomialDecay l k g') (hfg : g ≤ᶠ[l] f) (hfg' : f ≤ᶠ[l] g') :\n    SuperpolynomialDecay l k f := fun z =>\n  tendsto_of_tendsto_of_tendsto_of_le_of_le' (hg z) (hg' z)\n    (by filter_upwards [hfg, hk] with x hx (hx' : 0 ≤ k x) using by gcongr)\n    (by filter_upwards [hfg', hk] with x hx (hx' : 0 ≤ k x) using by gcongr)\n\nend OrderedCommSemiring\n\nsection LinearOrderedCommRing\n\nvariable [TopologicalSpace β] [CommRing β] [LinearOrder β] [IsStrictOrderedRing β] [OrderTopology β]\nvariable (l k f)\n\ntheorem superpolynomialDecay_iff_abs_tendsto_zero :\n    SuperpolynomialDecay l k f ↔ ∀ n : ℕ, Tendsto (fun a : α => |k a ^ n * f a|) l (𝓝 0) :=\n  ⟨fun h z => (tendsto_zero_iff_abs_tendsto_zero _).1 (h z), fun h z =>\n    (tendsto_zero_iff_abs_tendsto_zero _).2 (h z)⟩\n\ntheorem superpolynomialDecay_iff_superpolynomialDecay_abs :\n    SuperpolynomialDecay l k f ↔ SuperpolynomialDecay l (fun a => |k a|) fun a => |f a| :=\n  (superpolynomialDecay_iff_abs_tendsto_zero l k f).trans\n    (by simp_rw [SuperpolynomialDecay, abs_mul, abs_pow])\n\nvariable {l k f}\n\nTarget:\ntheorem SuperpolynomialDecay.trans_eventually_abs_le (hf : SuperpolynomialDecay l k f)\n    (hfg : abs ∘ g ≤ᶠ[l] abs ∘ f) : SuperpolynomialDecay l k g :=\n\nProof body:\n","rejected":"by\n  rw [superpolynomialDecay_iff_abs_tendsto_zero] at hf ⊢\n  refine fun z =>\n    tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds (hf z)\n      (Eventually.of_forall fun x => abs_nonneg _) (hfg.mono fun x hx => ?_)\n  calc\n    |k x ^ z * g x| = |k x ^ z| * |g x| := abs_mul (k x ^ z) (g x)\n    _ ≤ |k x ^ z| * |f x| := by gcongr _ * ?_; exact hx\n    _ = |k x ^ z * f x| := (abs_mul (k x ^ z) (f x)).symm\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Asymptotics","family_id":"superpolynomialdecay","file_id":"mathlib/Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean","sample_id":"f97d07cceec981c168c5f2eca445cd4e95e1981b7df2ab5f563b675d89d0e28f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1c6409609dd76697d622025294c8bc5145bcfe414ffbcf9cd053bd8d9c3b4785","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"986cb62814f148ed04d4c7dd4be579d36d7daa6ad59549d192831b6fd602cdb2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"de181494cf6df2f46264b7108f8b5fe8ade01b1f3c5c162ddf30528ffc055308","source_sha256":"e262008d04b45974f36575d7d297716d0a028e3928a5aa24f21cf62385c66b81","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let w'' : V₂.op ⋙ eB.op.functor ≅ eT.op.functor ⋙ V₃.op := NatIso.op w'\n  have : (w ≫ₕ w'.hom).op = (w.op ≫ᵥ w''.hom) := by ext; simp [w'']\n  rw [← guitartExact_op_iff, ← guitartExact_op_iff w,\n    ← vComp_iff_of_equivalences _ _ _ w'', this]\n  rfl","hard_negative":true,"metrics":{"chosen_tokens":80,"rejected_tokens":3,"token_jaccard":0.02439,"token_length_ratio":0.0375},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"b718e42fc4cf14eb28d7d82f80e60f7351aac8a778e673c1c1699aa7669d1468","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.GuitartExact.Opposite\n\nNamespace:\nCategoryTheory.TwoSquare.GuitartExact\n\nLocal context:\n/-\nCopyright (c) 2026 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Horizontal composition of Guitart exact squares\n\nIn this file, we show that the horizontal composition of Guitart exact squares\nis Guitart exact.\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Category\n\nvariable {C₁ C₂ C₃ D₁ D₂ D₃ : Type*} [Category* C₁] [Category* C₂] [Category* C₃]\n  [Category* D₁] [Category* D₂] [Category* D₃]\n\nnamespace TwoSquare\n\nsection WhiskerHorizontal\n\nvariable {T : C₁ ⥤ D₁} {L : C₁ ⥤ C₂} {R : D₁ ⥤ D₂} {B : C₂ ⥤ D₂} (w : TwoSquare T L R B)\n  {T' : C₁ ⥤ D₁} {B' : C₂ ⥤ D₂}\n\n/-- Given `w : TwoSquare T L R B`, one may obtain a 2-square `TwoSquare T' L R B'` if we\nprovide natural transformations `α : T ⟶ T'` and `β : B' ⟶ B`. -/\n@[simps!]\ndef whiskerHorizontal (α : T' ⟶ T) (β : B ⟶ B') :\n    TwoSquare T' L R B' :=\n  (w.whiskerTop α).whiskerBottom β\n\nnamespace GuitartExact\n\nset_option backward.defeqAttrib.useBackward true in\n/-- A 2-square stays Guitart exact if we replace the top and bottom functors\nby isomorphic functors. See also `whiskerHorizontal_iff`. -/\nlemma whiskerHorizontal [w.GuitartExact] (α : T ≅ T') (β : B ≅ B') :\n    (w.whiskerHorizontal α.inv β.hom).GuitartExact := by\n  rw [guitartExact_iff_final]\n  intro X₂\n  let e : costructuredArrowRightwards (w.whiskerHorizontal α.inv β.hom) X₂ ≅\n      w.costructuredArrowRightwards X₂ ⋙ (CostructuredArrow.mapIso (β.app X₂)).functor :=\n    NatIso.ofComponents (fun f ↦ CostructuredArrow.isoMk (α.symm.app f.left))\n  rw [Functor.final_natIso_iff e]\n  infer_instance\n\n/-- A 2-square is Guitart exact iff it is so after replacing the top and bottom functors by\nisomorphic functors. -/\n@[simp]\nlemma whiskerHorizontal_iff (α : T ≅ T') (β : B ≅ B') :\n    (w.whiskerHorizontal α.inv β.hom).GuitartExact ↔ w.GuitartExact := by\n  rw [← guitartExact_op_iff, ← w.guitartExact_op_iff,\n    ← whiskerVertical_iff w.op (NatIso.op α.symm) (NatIso.op β.symm)]\n  rfl\n\ninstance [w.GuitartExact] (α : T' ⟶ T) (β : B ⟶ B')\n    [IsIso α] [IsIso β] : (w.whiskerHorizontal α β).GuitartExact :=\n  whiskerHorizontal w (asIso α).symm (asIso β)\n\nend GuitartExact\n\nend WhiskerHorizontal\n\nsection HorizontalComposition\n\nvariable {V₁ : C₁ ⥤ D₁} {T₁ : C₁ ⥤ C₂} {B₁ : D₁ ⥤ D₂} {V₂ : C₂ ⥤ D₂}\n  (w : TwoSquare T₁ V₁ V₂ B₁)\n  {T₂ : C₂ ⥤ C₃} {B₂ : D₂ ⥤ D₃} {V₃ : C₃ ⥤ D₃}\n  (w' : TwoSquare T₂ V₂ V₃ B₂)\n\n/-- The horizontal composition of 2-squares. (Variant where we allow the replacement of\nthe horizontal compositions by isomorphic functors.) -/\n@[simps!]\ndef hComp' {T₁₂ : C₁ ⥤ C₃} {B₁₂ : D₁ ⥤ D₃} (eT : T₁ ⋙ T₂ ≅ T₁₂) (eB : B₁ ⋙ B₂ ≅ B₁₂) :\n    TwoSquare T₁₂ V₁ V₃ B₁₂ :=\n  (w ≫ₕ w').whiskerHorizontal eT.inv eB.hom\n\nnamespace GuitartExact\n\nset_option backward.defeqAttrib.useBackward true in\ninstance hComp [w.GuitartExact] [w'.GuitartExact] :\n    (w ≫ₕ w').GuitartExact := by\n  rw [← guitartExact_op_iff]\n  have : (w ≫ₕ w').op = w.op ≫ᵥ w'.op := by ext; simp\n  rw [this]\n  exact inferInstanceAs (w.op ≫ᵥ w'.op).GuitartExact\n\ninstance hComp' {T₁₂ : C₁ ⥤ C₃} {B₁₂ : D₁ ⥤ D₃} (eT : T₁ ⋙ T₂ ≅ T₁₂) (eB : B₁ ⋙ B₂ ≅ B₁₂)\n    [w.GuitartExact] [w'.GuitartExact] :\n    (w.hComp' w' eT eB).GuitartExact := by\n  dsimp only [TwoSquare.hComp']\n  infer_instance\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The canonical isomorphism between\n`w.costructuredArrowRightwards Y₁ ⋙ w'.costructuredArrowRightwards (B₁.obj Y₁)` and\n`(w ≫ₕ w').costructuredArrowRightwards Y₁`. -/\ndef costructuredArrowRightwardsComp (Y₁ : D₁) :\n    w.costructuredArrowRightwards Y₁ ⋙ w'.costructuredArrowRightwards (B₁.obj Y₁) ≅\n      (w ≫ₕ w').costructuredArrowRightwards Y₁ :=\n  NatIso.ofComponents (fun _ => CostructuredArrow.isoMk (Iso.refl _))\n\nlemma of_hComp [B₁.EssSurj] [w.GuitartExact] [(w ≫ₕ w').GuitartExact] :\n    w'.GuitartExact := by\n  rw [guitartExact_iff_final]\n  intro Y₂\n  rw [costructuredArrowRightwards_final_iff_of_iso _ (B₁.objObjPreimageIso Y₂).symm]\n  have : (w.costructuredArrowRightwards (B₁.objPreimage Y₂) ⋙\n      w'.costructuredArrowRightwards (B₁.obj (B₁.objPreimage Y₂))).Final :=\n    (Functor.final_of_natIso (costructuredArrowRightwardsComp w w' _).symm :)\n  exact Functor.final_of_final_comp (w.costructuredArrowRightwards (B₁.objPreimage Y₂)) _\n\nlemma of_hComp' {T₁₂ : C₁ ⥤ C₃} {B₁₂ : D₁ ⥤ D₃} (eT : T₁ ⋙ T₂ ≅ T₁₂) (eB : B₁ ⋙ B₂ ≅ B₁₂)\n    [B₁.EssSurj] [w.GuitartExact] [h : (w.hComp' w' eT eB).GuitartExact] :\n    w'.GuitartExact := by\n  dsimp [TwoSquare.hComp'] at h\n  rw [whiskerHorizontal_iff] at h\n  exact of_hComp w w'\n\nlemma hComp_iff_of_essSurj [B₁.EssSurj] [w.GuitartExact] :\n    (w ≫ₕ w').GuitartExact ↔ w'.GuitartExact :=\n  ⟨fun _ ↦ of_hComp w w', fun _ ↦ inferInstance⟩\n\nlemma hComp'_iff_of_essSurj\n    {T₁₂ : C₁ ⥤ C₃} {B₁₂ : D₁ ⥤ D₃} (eT : T₁ ⋙ T₂ ≅ T₁₂) (eB : B₁ ⋙ B₂ ≅ B₁₂)\n    [B₁.EssSurj] [w.GuitartExact] :\n    (w.hComp' w' eT eB).GuitartExact ↔ w'.GuitartExact :=\n  ⟨fun _ ↦ of_hComp' w w' eT eB, fun _ ↦ inferInstance⟩\n\nset_option backward.defeqAttrib.useBackward true in\n\nTarget:\nlemma hComp_iff_of_equivalences (eT : C₂ ≌ C₃) (eB : D₂ ≌ D₃)\n    (w' : eT.functor ⋙ V₃ ≅ V₂ ⋙ eB.functor) :\n    (w ≫ₕ w'.hom).GuitartExact ↔ w.GuitartExact :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_de181494cf6d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d997171ef5a3e87589788677fd6325b625cd8c2a7c475773ee3eb52a5adcbff1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/GuitartExact","family_id":"hcomp_iff_of_equivalences","file_id":"mathlib/Mathlib/CategoryTheory/GuitartExact/HorizontalComposition.lean","sample_id":"de181494cf6df2f46264b7108f8b5fe8ade01b1f3c5c162ddf30528ffc055308"},"split":"train"}
{"binding":{"chosen_proof_sha256":"85d4d418bf899b74281aa7eeb44015fb0b192ec7ec0ce946edf50b0cafb9b526","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"fbd4866a79e5cecb09bbdac5f941dd35a0fa3615af86da3a07086d351d137826","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ed1374ebe28e9b99f5e81874724199c0edfb80ce0f4bc23069e0f6178e331a7a","source_sha256":"4c2fd0b515ef4b06865711215377bd2bbde8a660c25b8d72d68a97c859a1494a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  simp [map, hw]","hard_negative":true,"metrics":{"chosen_tokens":8,"rejected_tokens":3,"token_jaccard":0.1,"token_length_ratio":0.375},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"b72c6c3b16b380e125223b6b73cc28fa128d0cda88996f80396a8052c1bf35ca","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.LocallyFinsupp\npublic import Mathlib.Topology.Spectral.Basic\n\nNamespace:\nFunction.locallyFinsupp\n\nLocal context:\n/-\nCopyright (c) 2026 Raphael Douglas Giles. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Raphael Douglas Giles\n-/\n/-!\n# Pushforward of functions with locally finite support\n\nIn this file we define the notion of the pushforward of a function with locally finite support\nbetween prespectral spaces along a spectral map. This is used for defining the (proper) pushforward\nof algebraic cycles in algebraic geometry.\n\n## Main declarations\n\n- `Function.locallyFinsupp.map`: If `f : X → Y` is a spectral map between spectral spaces and\n  `c : X → R` is locally of finite support, the pushforward of `c` along `f` at `y : Y` is\n  `∑ᶠ x ∈ f ⁻¹' {y}, c x * w x`, where `w : X → R` is a weight function.\n\n## Notes\n\nIn the case of algebraic cycles, the weight function used in `Function.locallyFinsupp.map` will be\nspecialized to the degree of the residue field extension\n(see https://stacks.math.columbia.edu/tag/02R4).\n-/\n\n@[expose] public section\n\nopen Set Order Topology TopologicalSpace\n\nvariable {X Y R : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n  {f : X → Y} (hf : IsSpectralMap f) (w : X → R)\n\nnamespace Function.locallyFinsupp\n\nvariable [Semiring R] {W : Set Y} (hW : IsOpen W) (c : Function.locallyFinsupp X R)\n  [PrespectralSpace Y]\n\nvariable (f) in\n/--\nThe pushforward of a function `c` of locally finite support by a spectral map with respect to a\nweight function `w`.\n-/\nnoncomputable\ndef map (hf : IsSpectralMap f) (c : locallyFinsupp X R) : Function.locallyFinsupp Y R where\n  toFun z := ∑ᶠ x ∈ f ⁻¹' {z}, c x * w x\n  supportWithinDomain' := by simp\n  supportLocallyFiniteWithinDomain' y _ := by\n    obtain ⟨U, hU⟩ := (PrespectralSpace.isTopologicalBasis (X := Y)).exists_subset_of_mem_open\n      (by simp : y ∈ ⊤) (by simp)\n    refine ⟨U, IsOpen.mem_nhds hU.1.1 hU.2.1, ?_⟩\n    suffices h : (U ∩ {z | (f ⁻¹' {z} ∩ support ⇑c).Nonempty}).Finite by\n      refine h.subset (inter_subset_inter_right U fun y hy ↦ ?_)\n      obtain ⟨x, (hx : f x = y), h'⟩ := exists_ne_zero_of_finsum_mem_ne_zero hy\n      use x\n      grind [mem_support]\n    suffices (f ⁻¹' (U ∩ {z | (f ⁻¹' {z} ∩ c.support).Nonempty}) ∩ c.support).Finite from\n      (this.image f).subset (fun a ha ↦ by grind [Set.Nonempty])\n    exact (c.locallyFiniteSupport.finite_inter_support_of_isCompact <| hf.2 hU.1.1 hU.1.2).subset\n      (by simp; grind)\n\n@[simp]\nlemma map_apply (hf : IsSpectralMap f) (c : locallyFinsupp X R) (y : Y) :\n    map f w hf c y = ∑ᶠ x ∈ f ⁻¹' {y}, c x * w x := rfl\n\nlemma support_map_subset_of_forall_mem (s : Set X) (t : Set Y) (hc : c.support ⊆ s)\n    (h : ∀ x : X, x ∈ s → w x ≠ 0 → f x ∈ t) : (map f w hf c).support ⊆ t := by\n  intro y hy\n  obtain ⟨x, (rfl : f x = y), h'⟩ := exists_ne_zero_of_finsum_mem_ne_zero hy\n  grind [mem_support]\n\n@[simp]\n\nTarget:\nlemma map_id [PrespectralSpace X] (hw : ∀ z : X, w z = 1) :\n    map id w isSpectralMap_id c = c :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_ed1374ebe28e","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"bfa530c85cedc13583d1953d314416650ae95521403b5ad879a6ec23bcbd1666","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/LocallyFinsupp","family_id":"map_id","file_id":"mathlib/Mathlib/Topology/LocallyFinsupp/Pushforward.lean","sample_id":"ed1374ebe28e9b99f5e81874724199c0edfb80ce0f4bc23069e0f6178e331a7a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"97f9e88375230bfbd4354802c179b8f0d71d6327ad14ee730bd5022e5fe5beee","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ea9df1cb32d99e6e86eeba3223e23a8702301d811768bfe8abc2f5b4fa039194","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2a418ca608341e871713a18e1c34bfa17c3846a9b9d6a3f1d347ce4be3076bae","source_sha256":"610b55cc6b034881bfda1b6609f87ef5fc28aecf81f7b61240acfb496bc220e4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [center_eq_bot, Algebra.mem_bot]\n  simp [eq_comm]","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":20,"token_jaccard":0.846154,"token_length_ratio":1.538462},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"b73f437f00185e944294a17517b200429bd355dc1530840c326a1c8e2791e423","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Central.Defs\nimport Mathlib.Algebra.Module.Torsion.Field\n\nNamespace:\nAlgebra.IsCentral\n\nLocal context:\n/-\nCopyright (c) 2024 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Jujian Zhang, Yunzhou Xie\n-/\n/-!\n# Central Algebras\n\nIn this file, we prove some basic results about central algebras over a commutative ring.\n\n## Main results\n\n- `Algebra.IsCentral.center_eq_bot`: the center of a central algebra over `K` is equal to `K`.\n- `Algebra.IsCentral.self`: a commutative ring is a central algebra over itself.\n- `Algebra.IsCentral.baseField_essentially_unique`: Let `D/K/k` be a tower of scalars where\n  `K` and `k` are fields. If `D` is a nontrivial central algebra over `k`, `K` is isomorphic to `k`.\n-/\n\npublic section\n\nuniverse u v\n\nnamespace Algebra.IsCentral\n\nvariable (K : Type u) [CommSemiring K] (D D' : Type v) [Semiring D] [Algebra K D]\n  [h : IsCentral K D] [Semiring D'] [Algebra K D']\n\n@[simp]\nlemma center_eq_bot : Subalgebra.center K D = ⊥ := eq_bot_iff.2 IsCentral.out\n\nvariable {D} in\n\nTarget:\nlemma mem_center_iff {x : D} : x ∈ Subalgebra.center K D ↔ ∃ (a : K), x = algebraMap K D a :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [center_eq_bot, Algebra.mem_bot]\n  simp [eq_comm]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Central","family_id":"mem_center_iff","file_id":"mathlib/Mathlib/Algebra/Central/Basic.lean","sample_id":"2a418ca608341e871713a18e1c34bfa17c3846a9b9d6a3f1d347ce4be3076bae"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ebb83eb68b98b09837a3cd4cf60703741f66ac05dee2a408489caf38cd22110c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9fd02cd1275b92cba5b435d3686fa58fec1fc7b2cad79d852c9e074db6daabbc","source_sha256":"b1ceba138fc84cb225f35cbdb2dcc1af4a948959a8326e3efc95adc26f7cbe28","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold rieszContentAux\n  gcongr\n  apply rieszContentAux_image_nonempty","hard_negative":true,"metrics":{"chosen_tokens":6,"rejected_tokens":8,"token_jaccard":0.083333,"token_length_ratio":1.333333},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"b76d0e0bdc98443cf611a21967f1f245b28de17051d6f6fc551b1be902847cd6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Measure.Content\npublic import Mathlib.Topology.ContinuousMap.CompactlySupported\npublic import Mathlib.Topology.PartitionOfUnity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Jesse Reimann. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jesse Reimann, Kalle Kytölä\n-/\n/-!\n# Riesz–Markov–Kakutani representation theorem\n\nThis file prepares technical definitions and results for the Riesz-Markov-Kakutani representation\ntheorem on a locally compact T2 space `X`. As a special case, the statements about linear\nfunctionals on bounded continuous functions follows. Actual theorems, depending on the\nlinearity (`ℝ`, `ℝ≥0` or `ℂ`), are proven in separate files\n(`Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Real.lean`,\n`Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/NNReal.lean`...)\n\nTo make use of the existing API, the measure is constructed from a content `λ` on the\ncompact subsets of a locally compact space X, rather than the usual construction of open sets in the\nliterature.\n\n## References\n\n* [Walter Rudin, Real and Complex Analysis.][Rud87]\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen scoped BoundedContinuousFunction NNReal ENNReal\nopen Set Function TopologicalSpace CompactlySupported CompactlySupportedContinuousMap\n  MeasureTheory\n\nvariable {X : Type*} [TopologicalSpace X]\nvariable (Λ : C_c(X, ℝ≥0) →ₗ[ℝ≥0] ℝ≥0)\n\n/-! ### Construction of the content: -/\n\nsection Monotone\n\nlemma CompactlySupportedContinuousMap.monotone_of_nnreal : Monotone Λ := by\n  intro f₁ f₂ h\n  obtain ⟨g, hg⟩ := CompactlySupportedContinuousMap.exists_add_of_le h\n  rw [← hg]\n  simp\n\nend Monotone\n\n/-- Given a positive linear functional `Λ` on continuous compactly supported functions on `X`\nwith values in `ℝ≥0`, for `K ⊆ X` compact define `λ(K) = inf {Λf | 1≤f on K}`.\nWhen `X` is a locally compact T2 space, this will be shown to be a\ncontent, and will be shown to agree with the Riesz measure on the compact subsets `K ⊆ X`. -/\ndef rieszContentAux : Compacts X → ℝ≥0 := fun K =>\n  sInf (Λ '' { f : C_c(X, ℝ≥0) | ∀ x ∈ K, (1 : ℝ≥0) ≤ f x })\n\nsection RieszMonotone\n\nvariable [T2Space X] [LocallyCompactSpace X]\n\n/-- For any compact subset `K ⊆ X`, there exist some compactly supported continuous nonnegative\nfunctions `f` on `X` such that `f ≥ 1` on `K`. -/\ntheorem rieszContentAux_image_nonempty (K : Compacts X) :\n    (Λ '' { f : C_c(X, ℝ≥0) | ∀ x ∈ K, (1 : ℝ≥0) ≤ f x }).Nonempty := by\n  rw [image_nonempty]\n  obtain ⟨V, hVcp, hKsubintV⟩ := exists_compact_superset K.2\n  have hIsCompact_closure_interior : IsCompact (closure (interior V)) := by\n    apply IsCompact.of_isClosed_subset hVcp isClosed_closure\n    nth_rw 2 [← closure_eq_iff_isClosed.mpr (IsCompact.isClosed hVcp)]\n    exact closure_mono interior_subset\n  obtain ⟨f, hsuppfsubV, hfeq1onK, hfinicc⟩ :=\n    exists_tsupport_one_of_isOpen_isClosed isOpen_interior hIsCompact_closure_interior\n      (IsCompact.isClosed K.2) hKsubintV\n  have hfHasCompactSupport : HasCompactSupport f :=\n    IsCompact.of_isClosed_subset hVcp (isClosed_tsupport f)\n      (Set.Subset.trans hsuppfsubV interior_subset)\n  use nnrealPart ⟨f, hfHasCompactSupport⟩\n  intro x hx\n  apply le_of_eq\n  simp only [nnrealPart_apply, CompactlySupportedContinuousMap.coe_mk]\n  rw [← Real.toNNReal_one, Real.toNNReal_eq_toNNReal_iff (zero_le_one' ℝ) (hfinicc x).1]\n  exact hfeq1onK.symm hx\n\n/-- Riesz content `λ` (associated with a positive linear functional `Λ`) is\nmonotone: if `K₁ ⊆ K₂` are compact subsets in `X`, then `λ(K₁) ≤ λ(K₂)`. -/\n\nTarget:\ntheorem rieszContentAux_mono {K₁ K₂ : Compacts X} (h : K₁ ≤ K₂) :\n    rieszContentAux Λ K₁ ≤ rieszContentAux Λ K₂ :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"d329a38231e339bade0e65e9fa058419fb0d1ee37f77c506c8627201d3b718ee","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Integral","family_id":"rieszcontentaux_mono","file_id":"mathlib/Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Basic.lean","sample_id":"9fd02cd1275b92cba5b435d3686fa58fec1fc7b2cad79d852c9e074db6daabbc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3c1f07b625c847d49cc43a89c05d3139e535c28f43f07276611fd8f7f1db78de","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e9b755e8d18b77c79dddaaf9d3ff11dacc42cf2890d5bd630e35c18555caf641","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cb9967aa5d54ae74d6b7c334cf672946e1ac267085a38bfbf8b9d07e6fea6e09","source_sha256":"5baad0a48c840bf6cedcfdfce26a9261ba521811fb4fc226af3afd123575d446","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by trunc) (hb : b.len ≤ n := by trunc)\n    (hc : c.len ≤ n := by trunc) :\n    tr (f ≫ g) = tr f ≫ tr g :=\n  rfl","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":5,"token_jaccard":0.086957,"token_length_ratio":0.119048},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"b774febead84c3f3fd39087cf7c31957af20ee8e9aa3307afcf0801a9b0c4db2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Category.Preorder\npublic import Mathlib.CategoryTheory.Opposites\npublic import Mathlib.Order.Fin.Basic\npublic import Mathlib.Util.Superscript\n\nNamespace:\nSimplexCategory.Truncated\n\nLocal context:\n/-\nCopyright (c) 2020 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Kim Morrison, Adam Topaz\n-/\n/-! # The simplex category\n\nWe construct a skeletal model of the simplex category, with an object `⦋n⦌` for each `n : ℕ`, and\nmorphisms `⦋n⦌ ⟶ ⦋m⦌` identify to monotone maps from `Fin (n + 1)` to `Fin (m + 1)`.\n\nIn `Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean`, we show that this category\nis equivalent to `NonemptyFinLinOrd`.\n\n## Remarks\n\nWe provide the following functions to work with these objects:\n1. `SimplexCategory.mk` creates an object of `SimplexCategory` out of a natural number.\n  Use the notation `⦋n⦌` in the `Simplicial` locale.\n2. `SimplexCategory.len` gives the \"length\" of an object of `SimplexCategory`, as a natural.\n3. `SimplexCategory.Hom.mk` makes a morphism out of a monotone map between `Fin`'s.\n4. `SimplexCategory.Hom.toOrderHom` gives the underlying monotone map associated to a\n  term of `SimplexCategory.Hom`.\n\n## Notation\n\n* `⦋n⦌` denotes the `n`-dimensional simplex. This notation is available with\n  `open Simplicial`.\n* `⦋m⦌ₙ` denotes the `m`-dimensional simplex in the `n`-truncated simplex category.\n  The truncation proof `p : m ≤ n` can also be provided using the syntax `⦋m, p⦌ₙ`.\n  This notation is available with `open SimplexCategory.Truncated`.\n-/\n\n@[expose] public section\n\nuniverse v\n\nopen CategoryTheory\n\n/-- The simplex category:\n* for each `n : ℕ`, there is an object `⦋n⦌`;\n* morphisms `⦋n⦌ ⟶ ⦋m⦌` are monotone functions `Fin (n+1) → Fin (m+1)`\n-/\n@[ext]\nstructure SimplexCategory : Type where\n  /-- Constructor `ℕ → SimplexCategory`. -/\n  mk ::\n  /-- The length of an object in `SimplexCategory` -/\n  len : ℕ\n\nnamespace SimplexCategory\n\n/-- the `n`-dimensional simplex can be denoted `⦋n⦌` -/\nscoped[Simplicial] notation \"⦋\" n \"⦌\" => SimplexCategory.mk n\n\nopen Simplicial\n\ntheorem len_mk (n : ℕ) : ⦋n⦌.len = n := rfl\n\n@[simp]\ntheorem mk_len (n : SimplexCategory) : ⦋n.len⦌ = n :=\n  rfl\n\n/-- Morphisms in the `SimplexCategory`. -/\nprotected def Hom (a b : SimplexCategory) :=\n  Fin (a.len + 1) →o Fin (b.len + 1)\n\nnamespace Hom\n\n/-- Make a morphism in `SimplexCategory` from a monotone map of `Fin`'s. -/\ndef mk {a b : SimplexCategory} (f : Fin (a.len + 1) →o Fin (b.len + 1)) : SimplexCategory.Hom a b :=\n  f\n\n/-- Recover the monotone map from a morphism in the simplex category. -/\ndef toOrderHom {a b : SimplexCategory} (f : SimplexCategory.Hom a b) :\n    Fin (a.len + 1) →o Fin (b.len + 1) :=\n  f\n\ntheorem ext' {a b : SimplexCategory} (f g : SimplexCategory.Hom a b) :\n    f.toOrderHom = g.toOrderHom → f = g :=\n  id\n\n@[simp]\ntheorem mk_toOrderHom {a b : SimplexCategory} (f : SimplexCategory.Hom a b) : mk f.toOrderHom = f :=\n  rfl\n\n@[simp]\ntheorem toOrderHom_mk {a b : SimplexCategory} (f : Fin (a.len + 1) →o Fin (b.len + 1)) :\n    (mk f).toOrderHom = f :=\n  rfl\n\ntheorem mk_toOrderHom_apply {a b : SimplexCategory} (f : Fin (a.len + 1) →o Fin (b.len + 1))\n    (i : Fin (a.len + 1)) : (mk f).toOrderHom i = f i :=\n  rfl\n\n/-- Identity morphisms of `SimplexCategory`. -/\n@[simp]\ndef id (a : SimplexCategory) : SimplexCategory.Hom a a :=\n  mk OrderHom.id\n\n/-- Composition of morphisms of `SimplexCategory`. -/\n@[simp]\ndef comp {a b c : SimplexCategory} (f : SimplexCategory.Hom b c) (g : SimplexCategory.Hom a b) :\n    SimplexCategory.Hom a c :=\n  mk <| f.toOrderHom.comp g.toOrderHom\n\nend Hom\n\nattribute [irreducible] SimplexCategory.Hom\n\ninstance smallCategory : SmallCategory.{0} SimplexCategory where\n  Hom n m := SimplexCategory.Hom n m\n  id _ := SimplexCategory.Hom.id _\n  comp f g := SimplexCategory.Hom.comp g f\n\n@[simp]\nlemma id_toOrderHom (a : SimplexCategory) :\n    Hom.toOrderHom (𝟙 a) = OrderHom.id := rfl\n\n@[simp]\nlemma comp_toOrderHom {a b c : SimplexCategory} (f : a ⟶ b) (g : b ⟶ c) :\n    (f ≫ g).toOrderHom = g.toOrderHom.comp f.toOrderHom := rfl\n\n@[ext]\ntheorem Hom.ext {a b : SimplexCategory} (f g : a ⟶ b) :\n    f.toOrderHom = g.toOrderHom → f = g :=\n  Hom.ext' _ _\n\n/-- Homs in `SimplexCategory` are equivalent to order-preserving functions of finite linear\norders. -/\ndef homEquivOrderHom {a b : SimplexCategory} :\n    (a ⟶ b) ≃ (Fin (a.len + 1) →o Fin (b.len + 1)) where\n  toFun := Hom.toOrderHom\n  invFun := Hom.mk\n\n/-- Homs in `SimplexCategory` are equivalent to functors between finite linear orders. -/\ndef homEquivFunctor {a b : SimplexCategory} :\n    (a ⟶ b) ≃ (Fin (a.len + 1) ⥤ Fin (b.len + 1)) :=\n  SimplexCategory.homEquivOrderHom.trans OrderHom.equivFunctor\n\n/-- The truncated simplex category. -/\nabbrev Truncated (n : ℕ) :=\n  ObjectProperty.FullSubcategory fun a : SimplexCategory => a.len ≤ n\n\nnamespace Truncated\n\ninstance {n} : Inhabited (Truncated n) :=\n  ⟨⟨⦋0⦌, by simp⟩⟩\n\n/-- The fully faithful inclusion of the truncated simplex category into the usual\nsimplex category.\n-/\nabbrev inclusion (n : ℕ) : SimplexCategory.Truncated n ⥤ SimplexCategory :=\n  ObjectProperty.ι _\n\n/-- A proof that the full subcategory inclusion is fully faithful -/\nnoncomputable def inclusion.fullyFaithful (n : ℕ) :\n    (inclusion n : Truncated n ⥤ _).op.FullyFaithful :=\n  Functor.FullyFaithful.ofFullyFaithful _\n\n@[ext]\ntheorem Hom.ext {n} {a b : Truncated n} (f g : a ⟶ b)\n    (h : f.hom.toOrderHom = g.hom.toOrderHom) : f = g :=\n  ObjectProperty.hom_ext _ (SimplexCategory.Hom.ext _ _ h)\n\n/-- A quick attempt to prove that `⦋m⦌` is `n`-truncated (`⦋m⦌.len ≤ n`). -/\nscoped macro \"trunc\" : tactic =>\n  `(tactic| first | assumption | dsimp only [SimplexCategory.len_mk] <;> lia)\n\nopen Mathlib.Tactic (subscriptTerm) in\n/-- For `m ≤ n`, `⦋m⦌ₙ` is the `m`-dimensional simplex in `Truncated n`. The\nproof `p : m ≤ n` can also be provided using the syntax `⦋m, p⦌ₙ`. -/\nscoped syntax:max (name := mkNotation)\n  \"⦋\" term (\",\" term)? \"⦌\" noWs subscriptTerm : term\nscoped macro_rules\n  | `(⦋$m:term⦌$n:subscript) =>\n    `((⟨SimplexCategory.mk $m, by first | trunc |\n      fail \"Failed to prove truncation property. Try writing `⦋m, by ...⦌ₙ`.\"⟩ :\n      SimplexCategory.Truncated $n))\n  | `(⦋$m:term, $p:term⦌$n:subscript) =>\n    `((⟨SimplexCategory.mk $m, $p⟩ : SimplexCategory.Truncated $n))\n\n/-- Make a morphism in `Truncated n` from a morphism in `SimplexCategory`. This\nis equivalent to `@id (⦋a⦌ₙ ⟶ ⦋b⦌ₙ) f`. -/\nabbrev Hom.tr {n : ℕ} {a b : SimplexCategory} (f : a ⟶ b)\n    (ha : a.len ≤ n := by trunc) (hb : b.len ≤ n := by trunc) :\n    (⟨a, ha⟩ : Truncated n) ⟶ ⟨b, hb⟩ :=\n  ObjectProperty.homMk f\n\n@[simp]\nlemma Hom.tr_id {n : ℕ} (a : SimplexCategory) (ha : a.len ≤ n := by trunc) :\n    Hom.tr (𝟙 a) ha = 𝟙 _ := rfl\n\n@[reassoc]\n\nTarget:\nlemma Hom.tr_comp {n : ℕ} {a b c : SimplexCategory} (f : a ⟶ b) (g : b ⟶ c)\n    (ha : a.len ≤ n :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_cb9967aa5d54","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"04b6c5833103f468e67ace4601ecc004ea5ab4cae53e2bfcbfc47efe49c8e5d1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplexCategory","family_id":"hom","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplexCategory/Defs.lean","sample_id":"cb9967aa5d54ae74d6b7c334cf672946e1ac267085a38bfbf8b9d07e6fea6e09"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a4055cfe4983bb68dd9d6d5e363dc43c3ee390bf3b7daa2b26130a168ae26243","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c6bc706d166e6b8c0ec0ad881ad47670431feaffc163f29325cfeefd84fa16cb","source_sha256":"b5f291a07b438990d3572a453453e2d3f7dd16b400e7d0a6bac2c42be601d0ed","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases eq_or_ne p 1 with rfl | hp\n  · simp\n  · rcases eq_or_ne n 0 with rfl | hn\n    · simp\n    · simp [← padicValNat_eq_emultiplicity_of_ne_one, *]","hard_negative":false,"metrics":{"chosen_tokens":30,"rejected_tokens":5,"token_jaccard":0.041667,"token_length_ratio":0.166667},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"b79d7015d103273fd5f3d35400c28c04300ba0da8ecbe6f4d59b1284b2679796","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.MaxPowDiv\npublic import Mathlib.RingTheory.Multiplicity\npublic import Mathlib.Data.Nat.Factors\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis, Matthew Robert Ballard\n-/\n/-!\n# `p`-adic Valuation\n\nThis file defines the `p`-adic valuation on `ℕ`, `ℤ`, and `ℚ`.\n\nThe `p`-adic valuation on `ℚ` is the difference of the multiplicities of `p` in the numerator and\ndenominator of `q`. This function obeys the standard properties of a valuation, with the appropriate\nassumptions on `p`. The `p`-adic valuations on `ℕ` and `ℤ` agree with that on `ℚ`.\n\nThe valuation induces a norm on `ℚ`. This norm is defined in\n`Mathlib/NumberTheory/Padics/PadicNorm.lean`.\n-/\n\n@[expose] public section\n\nassert_not_exists Field\n\nuniverse u\n\nopen Nat\n\nvariable {p : ℕ}\n\ntheorem padicValNat_eq_emultiplicity_of_ne_one (hp : p ≠ 1) {n : ℕ} (hn : n ≠ 0) :\n    padicValNat p n = emultiplicity p n := by\n  rw [eq_comm, emultiplicity_eq_coe, pow_dvd_iff_le_padicValNat hp hn,\n    pow_dvd_iff_le_padicValNat hp hn]\n  simp\n\n@[simp]\n\nTarget:\ntheorem Nat.toNat_emultiplicity (p n : ℕ) : (emultiplicity p n).toNat = padicValNat p n :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Padics","family_id":"nat","file_id":"mathlib/Mathlib/NumberTheory/Padics/PadicVal/Defs.lean","sample_id":"c6bc706d166e6b8c0ec0ad881ad47670431feaffc163f29325cfeefd84fa16cb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"de9ab835bffa0c1a69cfad4d6364d7e5552c21f89d63ae535eda5295f7d4c8b1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3a6d8b994748d4e07066dd4b842c5e18a4a267de9cc4efed1e2d52e8ad6a24cb","source_sha256":"354c54d863521e4c2e2cb8eb45baaec320a74608e472a8d10bc34f45771b2911","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨reg.1.of_flat_of_isBaseChange hf, ?_⟩\n  rw [← Ideal.map_ofList]\n  exact ((hf.map_smul_top_ne_top_iff_of_faithfullyFlat R M _).mpr reg.2.symm).symm","hard_negative":false,"metrics":{"chosen_tokens":40,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.075},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"b7ff13dd209914bcbf221e15d0f8369a53eecd92df7333c9a68b2b3af283a676","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Flat.FaithfullyFlat.Basic\npublic import Mathlib.RingTheory.Flat.Localization\npublic import Mathlib.RingTheory.Regular.RegularSequence\n\nNamespace:\nRingTheory.Sequence\n\nLocal context:\n/-\nCopyright (c) 2025 Yongle Hu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yongle Hu, Nailin Guan\n-/\n/-!\n# `RingTheory.Sequence.IsWeaklyRegular` is stable under flat base change\n\n## Main results\n* `RingTheory.Sequence.IsWeaklyRegular.of_flat_of_isBaseChange`: Let `R` be a commutative ring,\n  `M` be an `R`-module, `S` be a flat `R`-algebra, `N` be the base change of `M` to `S`.\n  If `[r₁, …, rₙ]` is a weakly regular `M`-sequence, then its image in `N` is a weakly regular\n  `N`-sequence.\n-/\n\npublic section\n\nnamespace RingTheory.Sequence\n\nopen Module\n\nvariable {R S M N : Type*} [CommRing R] [CommRing S] [Algebra R S]\n  [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower R S N]\n\n/-- Let `R` be a commutative ring, `M` be an `R`-module, `S` be a flat `R`-algebra, `N` be the base\n  change of `M` to `S`. If `[r₁, …, rₙ]` is a weakly regular `M`-sequence, then its image in `N` is\n  a weakly regular `N`-sequence. -/\ntheorem IsWeaklyRegular.of_flat_of_isBaseChange [Flat R S] {f : M →ₗ[R] N} (hf : IsBaseChange S f)\n    {rs : List R} (reg : IsWeaklyRegular M rs) : IsWeaklyRegular N (rs.map (algebraMap R S)) := by\n  induction rs generalizing M N with\n  | nil => simp\n  | cons x _ ih =>\n    simp only [List.map_cons, isWeaklyRegular_cons_iff] at reg ⊢\n    have e := (QuotSMulTop.algebraMapTensorEquivTensorQuotSMulTop x M S).symm ≪≫ₗ\n      QuotSMulTop.congr ((algebraMap R S) x) hf.equiv\n    have hg : IsBaseChange S <|\n        e.toLinearMap.restrictScalars R ∘ₗ TensorProduct.mk R S (QuotSMulTop x M) 1 :=\n      IsBaseChange.of_equiv e (fun _ ↦ by simp)\n    exact ⟨reg.1.of_flat_of_isBaseChange hf, ih hg reg.2⟩\n\ntheorem IsWeaklyRegular.of_flat [Flat R S] {rs : List R} (reg : IsWeaklyRegular R rs) :\n    IsWeaklyRegular S (rs.map (algebraMap R S)) :=\n  reg.of_flat_of_isBaseChange (IsBaseChange.linearMap R S)\n\nvariable (S) (T : Submonoid R) [IsLocalization T S]\n\ntheorem IsWeaklyRegular.of_isLocalizedModule (f : M →ₗ[R] N) [IsLocalizedModule T f]\n    {rs : List R} (reg : IsWeaklyRegular M rs) : IsWeaklyRegular N (rs.map (algebraMap R S)) :=\n  have : Flat R S := IsLocalization.flat S T\n  reg.of_flat_of_isBaseChange (IsLocalizedModule.isBaseChange T S f)\n\ninclude T in\ntheorem IsWeaklyRegular.of_isLocalization {rs : List R} (reg : IsWeaklyRegular R rs) :\n    IsWeaklyRegular S (rs.map (algebraMap R S)) :=\n  reg.of_isLocalizedModule S T (Algebra.linearMap R S)\n\nvariable (p : Ideal R) [p.IsPrime] [IsLocalization.AtPrime S p]\n\ntheorem IsWeaklyRegular.isRegular_of_isLocalizedModule_of_mem\n    [Nontrivial N] [Module.Finite S N] (f : M →ₗ[R] N) [IsLocalizedModule.AtPrime p f]\n    {rs : List R} (reg : IsWeaklyRegular M rs) (mem : ∀ r ∈ rs, r ∈ p) :\n    IsRegular N (rs.map (algebraMap R S)) := by\n  have : IsLocalRing S := IsLocalization.AtPrime.isLocalRing S p\n  refine (IsLocalRing.isRegular_iff_isWeaklyRegular_of_subset_maximalIdeal (fun _ hr ↦ ?_)).mpr <|\n    reg.of_isLocalizedModule S p.primeCompl f\n  rcases List.mem_map.mp hr with ⟨r, hr, eq⟩\n  simpa only [← eq, IsLocalization.AtPrime.to_map_mem_maximal_iff S p] using mem r hr\n\ntheorem IsWeaklyRegular.isRegular_of_isLocalization_of_mem\n    {rs : List R} (reg : IsWeaklyRegular R rs) (mem : ∀ r ∈ rs, r ∈ p) :\n    IsRegular S (rs.map (algebraMap R S)) :=\n  have : Nontrivial S := IsLocalization.AtPrime.nontrivial S p\n  reg.isRegular_of_isLocalizedModule_of_mem S p (Algebra.linearMap R S) mem\n\nvariable {S} [FaithfullyFlat R S]\n\n/-- Let `R` be a commutative ring, `M` be an `R`-module, `S` be a faithfully flat `R`-algebra,\n  `N` be the base change of `M` to `S`. If `[r₁, …, rₙ]` is a regular `M`-sequence, then its image\n  in `N` is a regular `N`-sequence. -/\n\nTarget:\ntheorem IsRegular.of_faithfullyFlat_of_isBaseChange {f : M →ₗ[R] N} (hf : IsBaseChange S f)\n    {rs : List R} (reg : IsRegular M rs) : IsRegular N (rs.map (algebraMap R S)) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Regular","family_id":"isregular","file_id":"mathlib/Mathlib/RingTheory/Regular/Flat.lean","sample_id":"3a6d8b994748d4e07066dd4b842c5e18a4a267de9cc4efed1e2d52e8ad6a24cb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f144c9feaed254a6570ff3fe42b4f3dd16c597e6f392545a7ebcf73899e7bd04","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"94202eba3f813b725b97a5df43ca9261065f4c2ad32c2cd56e50b36e5c530beb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c5c582457298af65c0d36d7c0a352351e64ed55551e90cd77c5b767876e6c965","source_sha256":"cc48552fd983736c871bbaff8e8b09090c080c9d6073d135463b6637bbb51f7b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical exact computable_iff_re_compl_re","hard_negative":false,"metrics":{"chosen_tokens":4,"rejected_tokens":11,"token_jaccard":0.666667,"token_length_ratio":2.75},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"b81d208dce261020ed2eb76033189af03c4c0320bd5b6066ba7c9fb6a07263cc","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Computability.PartrecCode\n\nNamespace:\nComputablePred\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Computable and Recursively Enumerable Predicates\n\nThis file defines computable (`ComputablePred`) and recursively enumerable (`REPred`)\npredicates. It also provides basic closure properties and Post's theorem on the\nequivalence of recursive, r.e., and co-r.e. sets.\n-/\n\n@[expose] public section\n\nopen List (Vector)\nopen Encodable Denumerable\n\nnamespace Nat.Partrec\n\nopen Computable Part\n\ntheorem merge' {f g} (hf : Nat.Partrec f) (hg : Nat.Partrec g) :\n    ∃ h, Nat.Partrec h ∧\n      ∀ a, (∀ x ∈ h a, x ∈ f a ∨ x ∈ g a) ∧ ((h a).Dom ↔ (f a).Dom ∨ (g a).Dom) := by\n  obtain ⟨cf, rfl⟩ := Code.exists_code.1 hf\n  obtain ⟨cg, rfl⟩ := Code.exists_code.1 hg\n  have : Nat.Partrec fun n => Nat.rfindOpt fun k => cf.evaln k n <|> cg.evaln k n :=\n    Partrec.nat_iff.1\n      (Partrec.rfindOpt <|\n        Primrec.option_orElse.to_comp.comp\n          (Code.primrec_evaln.to_comp.comp <| (snd.pair (const cf)).pair fst)\n          (Code.primrec_evaln.to_comp.comp <| (snd.pair (const cg)).pair fst))\n  refine ⟨_, this, fun n => ?_⟩\n  have : ∀ x ∈ rfindOpt fun k ↦ Code.evaln k cf n <|> Code.evaln k cg n,\n      x ∈ Code.eval cf n ∨ x ∈ Code.eval cg n := by\n    intro x h\n    obtain ⟨k, e⟩ := Nat.rfindOpt_spec h\n    rw [Option.mem_def, Option.orElse_eq_some, ← Option.mem_def, ← Option.mem_def] at e\n    obtain e | ⟨-, e⟩ := e <;> simp [Code.evaln_sound e]\n  refine ⟨this, fun h ↦ (this _ ⟨h, rfl⟩).imp Exists.fst Exists.fst, fun h ↦ ?_⟩\n  rw [Nat.rfindOpt_dom]\n  simp only [dom_iff_mem, Code.evaln_complete, Option.mem_def] at h\n  obtain ⟨x, k, e⟩ | ⟨x, k, e⟩ := h\n  · exact ⟨k, x, by simp [e]⟩\n  · refine ⟨k, ?_⟩\n    rcases cf.evaln k n with - | y\n    · exact ⟨x, by simp [e]⟩\n    · exact ⟨y, by simp⟩\n\nend Nat.Partrec\n\nnamespace Partrec\n\nvariable {α : Type*} {β : Type*} {γ : Type*} {σ : Type*}\nvariable [Primcodable α] [Primcodable β] [Primcodable γ] [Primcodable σ]\n\nopen Computable Part\n\nopen Nat.Partrec (Code)\n\nopen Nat.Partrec.Code\n\ntheorem merge' {f g : α →. σ} (hf : Partrec f) (hg : Partrec g) :\n    ∃ k : α →. σ,\n      Partrec k ∧ ∀ a, (∀ x ∈ k a, x ∈ f a ∨ x ∈ g a) ∧ ((k a).Dom ↔ (f a).Dom ∨ (g a).Dom) := by\n  let ⟨k, hk, H⟩ := Nat.Partrec.merge' (bind_decode₂_iff.1 hf) (bind_decode₂_iff.1 hg)\n  let k' (a : α) := (k (encode a)).bind fun n => (decode (α := σ) n : Part σ)\n  refine\n    ⟨k', ((nat_iff.2 hk).comp Computable.encode).bind (Computable.decode.ofOption.comp snd).to₂,\n      fun a => ?_⟩\n  have : ∀ x ∈ k' a, x ∈ f a ∨ x ∈ g a := by\n    intro x h'\n    simp only [k', mem_coe, mem_bind_iff, Option.mem_def] at h'\n    obtain ⟨n, hn, hx⟩ := h'\n    have := (H _).1 _ hn\n    simp only [decode₂_encode, coe_some, bind_some, mem_map_iff] at this\n    obtain ⟨a', ha, rfl⟩ | ⟨a', ha, rfl⟩ := this <;> simp only [encodek, Option.some_inj] at hx <;>\n      rw [hx] at ha\n    · exact Or.inl ha\n    · exact Or.inr ha\n  refine ⟨this, ⟨fun h => (this _ ⟨h, rfl⟩).imp Exists.fst Exists.fst, ?_⟩⟩\n  intro h\n  rw [bind_dom]\n  have hk : (k (encode a)).Dom :=\n    (H _).2.2 (by simpa only [encodek₂, bind_some, coe_some] using! h)\n  exists hk\n  simp only [mem_map_iff, mem_coe, mem_bind_iff, Option.mem_def] at H\n  obtain ⟨a', _, y, _, e⟩ | ⟨a', _, y, _, e⟩ := (H _).1 _ ⟨hk, rfl⟩ <;>\n    simp only [e.symm, encodek, coe_some, some_dom]\n\ntheorem merge {f g : α →. σ} (hf : Partrec f) (hg : Partrec g)\n    (H : ∀ (a), ∀ x ∈ f a, ∀ y ∈ g a, x = y) :\n    ∃ k : α →. σ, Partrec k ∧ ∀ a x, x ∈ k a ↔ x ∈ f a ∨ x ∈ g a :=\n  let ⟨k, hk, K⟩ := merge' hf hg\n  ⟨k, hk, fun a x =>\n    ⟨(K _).1 _, fun h => by\n      have : (k a).Dom := (K _).2.2 (h.imp Exists.fst Exists.fst)\n      refine ⟨this, ?_⟩\n      rcases h with h | h <;> rcases (K _).1 _ ⟨this, rfl⟩ with h' | h'\n      · exact mem_unique h' h\n      · exact (H _ _ h _ h').symm\n      · exact H _ _ h' _ h\n      · exact mem_unique h' h⟩⟩\n\ntheorem cond {c : α → Bool} {f : α →. σ} {g : α →. σ} (hc : Computable c) (hf : Partrec f)\n    (hg : Partrec g) : Partrec fun a => cond (c a) (f a) (g a) :=\n  let ⟨cf, ef⟩ := exists_code.1 hf\n  let ⟨cg, eg⟩ := exists_code.1 hg\n  ((eval_part.comp (Computable.cond hc (const cf) (const cg)) Computable.encode).bind\n    ((@Computable.decode σ _).comp snd).ofOption.to₂).of_eq\n    fun a => by cases c a <;> simp [ef, eg, encodek]\n\nnonrec theorem sumCasesOn {f : α → β ⊕ γ} {g : α → β →. σ} {h : α → γ →. σ} (hf : Computable f)\n    (hg : Partrec₂ g) (hh : Partrec₂ h) : @Partrec _ σ _ _ fun a => Sum.casesOn (f a) (g a) (h a) :=\n  option_some_iff.1 <|\n    (cond (sumCasesOn hf (const true).to₂ (const false).to₂)\n          (sumCasesOn_left hf (option_some_iff.2 hg).to₂ (const Option.none).to₂)\n          (sumCasesOn_right hf (const Option.none).to₂ (option_some_iff.2 hh).to₂)).of_eq\n      fun a => by cases f a <;> simp only [Bool.cond_true, Bool.cond_false]\n\nend Partrec\n\n/-- A computable predicate is one whose indicator function is computable. -/\ndef ComputablePred {α} [Primcodable α] (p : α → Prop) :=\n  ∃ (_ : DecidablePred p), Computable fun a => decide (p a)\n\nsection decide\n\nvariable {α} [Primcodable α]\n\nprotected lemma ComputablePred.decide {p : α → Prop} [DecidablePred p] (hp : ComputablePred p) :\n    Computable (fun a => decide (p a)) := by\n  convert! hp.choose_spec\n\nlemma Computable.computablePred {p : α → Prop} [DecidablePred p]\n    (hp : Computable (fun a => decide (p a))) : ComputablePred p :=\n  ⟨inferInstance, hp⟩\n\nlemma computablePred_iff_computable_decide {p : α → Prop} [DecidablePred p] :\n    ComputablePred p ↔ Computable (fun a => decide (p a)) where\n  mp := ComputablePred.decide\n  mpr := Computable.computablePred\n\nlemma PrimrecPred.computablePred {α} [Primcodable α] {p : α → Prop} :\n    (hp : PrimrecPred p) → ComputablePred p\n  | ⟨_, hp⟩ => hp.to_comp.computablePred\n\nend decide\n\n/-- A recursively enumerable predicate is one which is the domain of a computable partial function.\n-/\ndef REPred {α} [Primcodable α] (p : α → Prop) :=\n  Partrec fun a => Part.assert (p a) fun _ => Part.some ()\n\ntheorem REPred.of_eq {α} [Primcodable α] {p q : α → Prop} (hp : REPred p) (H : ∀ a, p a ↔ q a) :\n    REPred q :=\n  (funext fun a => propext (H a) : p = q) ▸ hp\n\ntheorem Partrec.dom_re {α β} [Primcodable α] [Primcodable β] {f : α →. β} (h : Partrec f) :\n    REPred fun a => (f a).Dom :=\n  (h.map (Computable.const ()).to₂).of_eq fun n => Part.ext fun _ => by simp [Part.dom_iff_mem]\n\ntheorem ComputablePred.of_eq {α} [Primcodable α] {p q : α → Prop} (hp : ComputablePred p)\n    (H : ∀ a, p a ↔ q a) : ComputablePred q :=\n  (funext fun a => propext (H a) : p = q) ▸ hp\n\nnamespace Computable\n\n/-- If `P` is computable, and if for every `x` there exists an `n` such that `P x n` holds,\nthen the function mapping `x` to the minimal such `n` (using `Nat.find`) is computable.\nThis formally bridges `Partrec.rfind` with total unbounded search. -/\nlemma find {α : Type*} [Primcodable α] {P : α → ℕ → Prop} [DecidableRel P]\n    (hP_comp : ComputablePred (fun p : α × ℕ => P p.1 p.2)) (hP_ex : ∀ x, ∃ n, P x n) :\n    Computable (fun x => Nat.find (hP_ex x)) := by\n  have h : Partrec (fun x ↦ Nat.rfind fun n => Part.some (decide (P x n))) :=\n    Partrec.rfind hP_comp.decide.partrec\n  refine h.of_eq_tot fun x ↦ ?_\n  simp +contextual [Nat.find_spec]\n\nend Computable\n\nnamespace ComputablePred\n\nvariable {α : Type*} [Primcodable α]\n\nopen Nat.Partrec (Code)\n\nopen Nat.Partrec.Code Computable\n\ntheorem computable_iff {p : α → Prop} :\n    ComputablePred p ↔ ∃ f : α → Bool, Computable f ∧ p = fun a => (f a : Prop) :=\n  ⟨fun ⟨_, h⟩ => ⟨_, h, funext fun _ => propext (Bool.decide_iff _).symm⟩, by\n    rintro ⟨f, h, rfl⟩; exact ⟨by infer_instance, by simpa using h⟩⟩\n\nprotected theorem not {p : α → Prop} :\n    (hp : ComputablePred p) → ComputablePred fun a => ¬p a\n  | ⟨_, hp⟩ => Computable.computablePred <| Primrec.not.to_comp.comp hp |>.of_eq <| by simp\n\n/-- The computable functions are closed under if-then-else definitions\nwith computable predicates. -/\ntheorem ite {f₁ f₂ : ℕ → ℕ} (hf₁ : Computable f₁) (hf₂ : Computable f₂)\n    {c : ℕ → Prop} [DecidablePred c] (hc : ComputablePred c) :\n    Computable fun k ↦ if c k then f₁ k else f₂ k := by\n  simpa [Bool.cond_decide] using hc.decide.cond hf₁ hf₂\n\ntheorem to_re {p : α → Prop} (hp : ComputablePred p) : REPred p := by\n  obtain ⟨f, hf, rfl⟩ := computable_iff.1 hp\n  unfold REPred\n  dsimp only []\n  refine\n    (Partrec.cond hf (Decidable.Partrec.const' (Part.some ())) Partrec.none).of_eq fun n =>\n      Part.ext fun a => ?_\n  cases a; cases f n <;> simp\n\n-- Post's theorem on the equivalence of r.e., co-r.e. sets and\n-- computable sets. The assumption that p is decidable is required\n-- unless we assume Markov's principle or LEM.\nset_option linter.unusedDecidableInType false in\ntheorem computable_iff_re_compl_re {p : α → Prop} [DecidablePred p] :\n    ComputablePred p ↔ REPred p ∧ REPred fun a => ¬p a :=\n  ⟨fun h => ⟨h.to_re, h.not.to_re⟩, fun ⟨h₁, h₂⟩ =>\n    ⟨‹_›, by\n      obtain ⟨k, pk, hk⟩ :=\n        Partrec.merge (h₁.map (Computable.const true).to₂) (h₂.map (Computable.const false).to₂)\n        (by\n          intro a x hx y hy\n          simp only [Part.mem_map_iff, Part.mem_assert_iff, Part.mem_some_iff, exists_prop,\n            and_true, exists_const] at hx hy\n          cases hy.1 hx.1)\n      refine Partrec.of_eq pk fun n => Part.eq_some_iff.2 ?_\n      rw [hk]\n      simp only [Part.mem_map_iff, Part.mem_assert_iff, Part.mem_some_iff, exists_prop, and_true,\n        true_eq_decide_iff, and_self, exists_const, false_eq_decide_iff]\n      apply Decidable.em⟩⟩\n\nTarget:\ntheorem computable_iff_re_compl_re' {p : α → Prop} :\n    ComputablePred p ↔ REPred p ∧ REPred fun a => ¬p a :=\n\nProof body:\n","rejected":"```lean\nby\n  classical exact computable_iff_re_compl_re\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability","family_id":"computable_iff_re_compl_re","file_id":"mathlib/Mathlib/Computability/RE.lean","sample_id":"c5c582457298af65c0d36d7c0a352351e64ed55551e90cd77c5b767876e6c965"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c785440b2e793b1a2afd030ccba707286c555ec39b3feca32e7d614cb913da7e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"541065cf7e2b0c48add8ba4df93b11dbe3388e1bea47ee835a3a06d99127e75a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7734458b70658c76e232f731e838552dc282d9587d72ef9af109c7d77dc4c28f","source_sha256":"56d11af83a40725621595d46ae5793c5903e2fe6ca0b04b01b11f907274c0c49","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let ⟨z, hz⟩ := (F.map (algebraMap R ℤ_[p])).evalSubFactor x y\n  simp only [Polynomial.eval_map_algebraMap] at hz\n  calc\n    ‖F.aeval x - F.aeval y‖ = ‖z‖ * ‖x - y‖ := by simp [hz]\n    _ ≤ 1 * ‖x - y‖ := by gcongr; apply PadicInt.norm_le_one\n    _ = ‖x - y‖ := by simp","hard_negative":true,"metrics":{"chosen_tokens":90,"rejected_tokens":2,"token_jaccard":0.02381,"token_length_ratio":0.022222},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"b82fa13b5408bbe18e684f3c2a0674479166b6c3f6b00b4e9df065c636d48285","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Identities\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.NumberTheory.Padics.PadicIntegers\npublic import Mathlib.Topology.Algebra.Polynomial\npublic import Mathlib.Topology.MetricSpace.CauSeqFilter\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n-/\n/-!\n# Hensel's lemma on `ℤ_p`\n\nThis file proves Hensel's lemma on `ℤ_p`, roughly following Keith Conrad's writeup:\n<http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/hensel.pdf>\n\nHensel's lemma gives a simple condition for the existence of a root of a polynomial.\n\nThe proof and motivation are described in the paper\n[R. Y. Lewis, *A formal proof of Hensel's lemma over the p-adic integers*][lewis2019].\n\n## References\n\n* <http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/hensel.pdf>\n* [R. Y. Lewis, *A formal proof of Hensel's lemma over the p-adic integers*][lewis2019]\n* <https://en.wikipedia.org/wiki/Hensel%27s_lemma>\n\n## Tags\n\np-adic, p adic, padic, p-adic integer\n-/\n\npublic section\n\n\nnoncomputable section\n\nopen Topology\n\n-- We begin with some general lemmas that are used below in the computation.\n\nTarget:\ntheorem padic_polynomial_dist {p : ℕ} [Fact p.Prime] {R : Type*} [CommSemiring R] [Algebra R ℤ_[p]]\n    (F : Polynomial R) (x y : ℤ_[p]) :\n    ‖F.aeval x - F.aeval y‖ ≤ ‖x - y‖ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_7734458b7065","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"1bad10184d63d65f42ce81a74a1abb9dc755a9ce7e098270e7f99299d9735c30","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Padics","family_id":"padic_polynomial_dist","file_id":"mathlib/Mathlib/NumberTheory/Padics/Hensel.lean","sample_id":"7734458b70658c76e232f731e838552dc282d9587d72ef9af109c7d77dc4c28f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"86252bc5f575ff25cd53bd8cf62a6e1c8e93248c81bafebff2879ba682468c22","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"baab1cc1fa864e751c82e0367d07242c8d6bc220c02f497937b6fb933c82b1c4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"24bec75d4c990cfaa562f6c11f1e1ed4950885ceeb40f6408a0260da05969bf6","source_sha256":"155f9e8471ee4b30d28b3dbc51f2958ab69f6c864e7fa3081250ca89e1bfbb0f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply toFinsupp_injective\n  ext x\n  simp_rw [toFinsupp_equivMapDomain, Finsupp.equivMapDomain_apply, toFinsupp_mapDomain,\n    Finsupp.mapDomain_equiv_apply]","hard_negative":true,"metrics":{"chosen_tokens":19,"rejected_tokens":3,"token_jaccard":0.058824,"token_length_ratio":0.157895},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"b8c4a9c305c0d37ff4b15303c8d780a8ac3603404b40b49e1c711ec5d1bfd217","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.SkewMonoidAlgebra.Basic\npublic import Mathlib.Algebra.Module.BigOperators\npublic import Mathlib.Algebra.Algebra.Equiv\n\nNamespace:\nSkewMonoidAlgebra\n\nLocal context:\n/-\nCopyright (c) 2025 Xavier Généreux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: María Inés de Frutos Fernández, Xavier Généreux\n-/\n/-!\n# Lemmas about different kinds of \"lifts\" to `SkewMonoidAlgebra`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nnamespace SkewMonoidAlgebra\n\nvariable {k G H : Type*}\n\nsection lift\n\nvariable [CommSemiring k] [Monoid G] [Monoid H]\nvariable {A B : Type*} [Semiring A] [Algebra k A] [Semiring B] [Algebra k B]\n\n/-- `liftNCRingHom` as an `AlgHom`, for when `f` is an `AlgHom` -/\ndef liftNCAlgHom [MulSemiringAction G A] [SMulCommClass G k A] (f : A →ₐ[k] B)\n    (g : G →* B) (h_comm : ∀ {x y}, (f (y • x)) * g y = (g y) * (f x)) :\n    SkewMonoidAlgebra A G →ₐ[k] B where\n  __ := liftNCRingHom (f : A →+* B) g h_comm\n  commutes' := by simp [liftNCRingHom]\n\n/- Hypotheses needed for `k`-algebra homomorphism from `SkewMonoidAlgebra k G`-/\nvariable [MulSemiringAction G k] [SMulCommClass G k k]\n\nvariable (k G A)\n\n/-- Any monoid homomorphism `G →* A` can be lifted to an algebra homomorphism\n  `SkewMonoidAlgebra k G →ₐ[k] A`. -/\ndef lift : (G →* A) ≃ (AlgHom k (SkewMonoidAlgebra k G) A) where\n  invFun f := (f : SkewMonoidAlgebra k G →* A).comp (of k G)\n  toFun F := by\n    apply liftNCAlgHom (Algebra.ofId k A) F\n    simp_rw [show ∀ (g : G) (r : k), g • r = r by\n        exact fun _ _ ↦ smul_algebraMap _ (algebraMap k k _)]\n    exact Algebra.commutes _ _\n  left_inv f := by\n    ext\n    simp [liftNCAlgHom, liftNCRingHom]\n  right_inv F := by\n    ext\n    simp [liftNCAlgHom, liftNCRingHom]\n\nvariable {k G A}\n\ntheorem lift_apply' (F : G →* A) (f : SkewMonoidAlgebra k G) :\n    lift k G A F f = f.sum fun a b ↦ algebraMap k A b * F a := rfl\n\ntheorem lift_apply (F : G →* A) (f : SkewMonoidAlgebra k G) :\n    lift k G A F f = f.sum fun a b ↦ b • F a := by simp [lift_apply', Algebra.smul_def]\n\ntheorem lift_def (F : G →* A) : (lift k G A F : SkewMonoidAlgebra k G → A) =\n    liftNC ((algebraMap k A : k →+* A) : k →+ A) F := rfl\n\n@[simp]\ntheorem lift_symm_apply (F : AlgHom k (SkewMonoidAlgebra k G) A) (x : G) :\n    (lift k G A).symm F x = F (single x 1) := rfl\n\ntheorem lift_of (F : G →* A) (x) : lift k G A F (of k G x) = F x := by\n  rw [of_apply, ← lift_symm_apply, Equiv.symm_apply_apply]\n\n@[simp]\ntheorem lift_single (F : G →* A) (a b) : lift k G A F (single a b) = b • F a := by\n  rw [lift_def, liftNC_single, Algebra.smul_def, AddMonoidHom.coe_coe]\n\ntheorem lift_unique' (F : AlgHom k (SkewMonoidAlgebra k G) A) :\n    F = lift k G A ((F : SkewMonoidAlgebra k G →* A).comp (of k G)) :=\n  ((lift k G A).apply_symm_apply F).symm\n\n/-- Decomposition of a `k`-algebra homomorphism from `SkewMonoidAlgebra k G` by\n  its values on `F (single a 1)`. -/\ntheorem lift_unique (F : AlgHom k (SkewMonoidAlgebra k G) A)\n    (f : SkewMonoidAlgebra k G) : F f = f.sum fun a b ↦ b • F (single a 1) := by\n  conv_lhs =>\n    rw [lift_unique' F]\n    simp [lift_apply]\n\n/-- If `f : G → H` is a multiplicative homomorphism between two monoids, then\n`mapDomain f` is an algebra homomorphism between their monoid algebras. -/\n@[simps!]\ndef mapDomainAlgHom (k A : Type*) [CommSemiring k] [Semiring A] [Algebra k A] {H F : Type*}\n    [Monoid H] [FunLike F G H] [MonoidHomClass F G H] [MulSemiringAction G A]\n    [MulSemiringAction H A] [SMulCommClass G k A] [SMulCommClass H k A] {f : F}\n    (hf : ∀ (a : G) (x : A), a • x = (f a) • x) :\n    SkewMonoidAlgebra A G →ₐ[k] SkewMonoidAlgebra A H where\n  __ := mapDomainRingHom hf\n  commutes' := by simp [mapDomainRingHom]\n\nend lift\n\nsection equivMapDomain\n\nvariable [AddCommMonoid k]\n\n/-- Given `f : G ≃ H`, we can map `l : SkewMonoidAlgebra k G` to\n`equivMapDomain f l : SkewMonoidAlgebra k H` (computably) by mapping the support forwards\nand the function backwards. -/\ndef equivMapDomain (f : G ≃ H) (l : SkewMonoidAlgebra k G) : SkewMonoidAlgebra k H where\n  toFinsupp := ⟨l.support.map f.toEmbedding, fun a ↦ l.coeff (f.symm a), by simp⟩\n\n@[simp]\ntheorem coeff_equivMapDomain (f : G ≃ H) (l : SkewMonoidAlgebra k G) (b : H) :\n    (equivMapDomain f l).coeff b = l.coeff (f.symm b) :=\n  rfl\n\nlemma toFinsupp_equivMapDomain (f : G ≃ H) (l : SkewMonoidAlgebra k G) :\n    (equivMapDomain f l).toFinsupp = Finsupp.equivMapDomain f l.toFinsupp := rfl\n\nTarget:\ntheorem equivMapDomain_eq_mapDomain (f : G ≃ H) (l : SkewMonoidAlgebra k G) :\n    equivMapDomain f l = mapDomain f l :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_24bec75d4c99","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"916a87c2160d1adf9e6e74cbb5d23af3e5484db541dd1f01a62bee5a45aa9bf1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/SkewMonoidAlgebra","family_id":"equivmapdomain_eq_mapdomain","file_id":"mathlib/Mathlib/Algebra/SkewMonoidAlgebra/Lift.lean","sample_id":"24bec75d4c990cfaa562f6c11f1e1ed4950885ceeb40f6408a0260da05969bf6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"73fbc60842c9edf00faf984e5736df9245f2ebd275179f441c2133f339aab829","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"46dbe9e477ddbb7606c3f29bdeda37a0f7a2816233719f84ab33136a865943b2","source_sha256":"148d7eba9bc7c4474f48190ef655e8ff7d2748109cb095b07744bfb109f40298","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases H : ∃ s : Finset S, Nonempty (Basis s R S)\n  · rw [trace_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some]\n  · simp [trace_eq_zero_of_not_exists_basis R H, finrank_eq_zero_of_not_exists_basis_finset H]","hard_negative":false,"metrics":{"chosen_tokens":44,"rejected_tokens":5,"token_jaccard":0.033333,"token_length_ratio":0.113636},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"b8dac7a9c3bfc09124cc7900fa12053fcd575d1ac69d45cd874a7f41c6c748d7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas\npublic import Mathlib.LinearAlgebra.Matrix.BilinearForm\npublic import Mathlib.LinearAlgebra.Trace\n\nNamespace:\nAlgebra\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Trace for (finite) ring extensions.\n\nSuppose we have an `R`-algebra `S` with a finite basis. For each `s : S`,\nthe trace of the linear map given by multiplying by `s` gives information about\nthe roots of the minimal polynomial of `s` over `R`.\n\n## Main definitions\n\n* `Algebra.trace R S x`: the trace of an element `s` of an `R`-algebra `S`\n* `Algebra.traceForm R S`: bilinear form sending `x`, `y` to the trace of `x * y`\n* `Algebra.traceMatrix R b`: the matrix whose `(i j)`-th element is the trace of `b i * b j`.\n\n## Main results\n\n* `trace_algebraMap_of_basis`, `trace_algebraMap`: if `x : K`, then `Tr_{L/K} x = [L : K] x`\n* `trace_trace_of_basis`, `trace_trace`: `Tr_{L/K} (Tr_{F/L} x) = Tr_{F/K} x`\n\n## Implementation notes\n\nTypically, the trace is defined specifically for finite field extensions.\nThe definition is as general as possible and the assumption that the extension is finite\nis added to the lemmas as needed.\n\nWe only define the trace for left multiplication (`Algebra.leftMulMatrix`,\ni.e. `LinearMap.mulLeft`).\nFor now, the definitions assume `S` is commutative, so the choice doesn't matter anyway.\n\n## References\n\n* https://en.wikipedia.org/wiki/Field_trace\n\n-/\n\n@[expose] public section\n\n\nuniverse w\n\nvariable {R S T : Type*} [CommRing R] [CommRing S] [CommRing T]\nvariable [Algebra R S] [Algebra R T]\nvariable {ι : Type w} [Fintype ι]\n\nopen Module\n\nopen LinearMap (BilinForm)\nopen LinearMap\n\nopen Matrix\n\nopen scoped Matrix\n\nnamespace Algebra\n\nvariable (R S)\n\n/-- The trace of an element `s` of an `R`-algebra is the trace of `(s * ·)`,\nas an `R`-linear map. -/\n@[stacks 0BIF \"Trace\"]\nnoncomputable def trace : S →ₗ[R] R :=\n  (LinearMap.trace R S).comp (lmul R S).toLinearMap\n\nvariable {S}\n\n-- Not a `simp` lemma since there are more interesting ways to rewrite `trace R S x`,\n-- for example `trace_trace`\ntheorem trace_apply (x) : trace R S x = LinearMap.trace R S (lmul R S x) :=\n  rfl\n\ntheorem trace_eq_zero_of_not_exists_basis (h : ¬∃ s : Finset S, Nonempty (Basis s R S)) :\n    trace R S = 0 := by ext s; simp [trace_apply, LinearMap.trace, h]\n\nvariable {R}\n\n-- Can't be a `simp` lemma because it depends on a choice of basis\ntheorem trace_eq_matrix_trace [DecidableEq ι] (b : Basis ι R S) (s : S) :\n    trace R S s = Matrix.trace (Algebra.leftMulMatrix b s) := by\n  rw [trace_apply, LinearMap.trace_eq_matrix_trace _ b, ← toMatrix_lmul_eq]; rfl\n\n/-- If `x` is in the base field `K`, then the trace is `[L : K] * x`. -/\ntheorem trace_algebraMap_of_basis (b : Basis ι R S) (x : R) :\n    trace R S (algebraMap R S x) = Fintype.card ι • x := by\n  haveI := Classical.decEq ι\n  rw [trace_apply, LinearMap.trace_eq_matrix_trace R b, Matrix.trace]\n  convert! Finset.sum_const x\n  simp [-coe_lmul_eq_mul]\n\n\n/-- The trace map from `R` to itself is the identity map. -/\n@[simp] theorem trace_self : trace R R = LinearMap.id := by\n  ext; simpa using trace_algebraMap_of_basis (.singleton (Fin 1) R) 1\n\ntheorem trace_self_apply (a) : trace R R a = a := by simp\n\n/-- If `x` is in the base field `K`, then the trace is `[L : K] * x`.\n\n(If `L` is not finite-dimensional over `K`, then `trace` and `finrank` return `0`.)\n-/\n@[simp]\n\nTarget:\ntheorem trace_algebraMap [StrongRankCondition R] [Module.Free R S] (x : R) :\n    trace R S (algebraMap R S x) = finrank R S • x :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Trace","family_id":"trace_algebramap","file_id":"mathlib/Mathlib/RingTheory/Trace/Defs.lean","sample_id":"46dbe9e477ddbb7606c3f29bdeda37a0f7a2816233719f84ab33136a865943b2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cf1663d691843ffb2d1c18b62f9a504b9533198687278010c9c58b7251f1dcfb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8b1fcb5d375f7d0c76ac8b5dc5be9785fa4e3630fe7326a206b1b90b1962024a","source_sha256":"5efd5852b126afb746be3b592d500b1af5de2649a35029fd57ec7bbb27770fcb","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp only [LiftR, LiftR']\n  apply exists_iff_exists_of_mono F (f' rr _ _) (g' rr _ _)\n  · ext i ⟨x, _⟩ : 2\n    cases i <;> rfl\n  · intros\n    simp +unfoldPartialApp only [map_map, TypeVec.comp]\n    apply iff_of_eq -- Switch to `eq` so we can use `ext`\n    congr <;> ext i ⟨x, _⟩ <;> cases i <;> rfl","hard_negative":false,"metrics":{"chosen_tokens":88,"rejected_tokens":2,"token_jaccard":0.02,"token_length_ratio":0.022727},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"b941c1372f365304d19704d50fb8b535a936c642d675e131b99e199262bfcb80","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.Fin2\npublic import Mathlib.Data.TypeVec\npublic import Mathlib.Logic.Equiv.Defs\n\nNamespace:\nMvFunctor\n\nLocal context:\n/-\nCopyright (c) 2018 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Mario Carneiro, Simon Hudon\n-/\n/-!\n\n# Functors between the category of tuples of types, and the category Type\n\nFeatures:\n\n* `MvFunctor n` : the type class of multivariate functors\n* `f <$$> x`    : notation for map\n\n-/\n\n@[expose] public section\n\n\nuniverse u v w\n\nopen MvFunctor\n\n/-- Multivariate functors, i.e. functor between the category of type vectors\nand the category of Type -/\nclass MvFunctor {n : ℕ} (F : TypeVec n → Type*) where\n  /-- Multivariate map, if `f : α ⟹ β` and `x : F α` then `f <$$> x : F β`. -/\n  map : ∀ {α β : TypeVec n}, α ⟹ β → F α → F β\n\n/-- Multivariate map, if `f : α ⟹ β` and `x : F α` then `f <$$> x : F β` -/\nscoped[MvFunctor] infixr:100 \" <$$> \" => MvFunctor.map\n\nvariable {n : ℕ}\n\nnamespace MvFunctor\n\nvariable {α β : TypeVec.{u} n} {F : TypeVec.{u} n → Type v} [MvFunctor F]\n\n/-- predicate lifting over multivariate functors -/\ndef LiftP {α : TypeVec n} (P : ∀ i, α i → Prop) (x : F α) : Prop :=\n  ∃ u : F (fun i => Subtype (P i)), (fun i => @Subtype.val _ (P i)) <$$> u = x\n\n/-- relational lifting over multivariate functors -/\ndef LiftR {α : TypeVec n} (R : ∀ ⦃i⦄, α i → α i → Prop) (x y : F α) : Prop :=\n  ∃ u : F (fun i => { p : α i × α i // R p.fst p.snd }),\n    (fun i (t : { p : α i × α i // R p.fst p.snd }) => t.val.fst) <$$> u = x ∧\n      (fun i (t : { p : α i × α i // R p.fst p.snd }) => t.val.snd) <$$> u = y\n\n/-- given `x : F α` and a projection `i` of type vector `α`, `supp x i` is the set\nof `α.i` contained in `x` -/\ndef supp {α : TypeVec n} (x : F α) (i : Fin2 n) : Set (α i) :=\n  { y : α i | ∀ ⦃P⦄, LiftP P x → P i y }\n\ntheorem of_mem_supp {α : TypeVec n} {x : F α} {P : ∀ ⦃i⦄, α i → Prop} (h : LiftP P x) (i : Fin2 n) :\n    ∀ y ∈ supp x i, P y := fun _y hy => hy h\n\nend MvFunctor\n\n\n\n/-- laws for `MvFunctor` -/\nclass LawfulMvFunctor {n : ℕ} (F : TypeVec n → Type*) [MvFunctor F] : Prop where\n  /-- `map` preserved identities, i.e., maps identity on `α` to identity on `F α` -/\n  id_map : ∀ {α : TypeVec n} (x : F α), TypeVec.id <$$> x = x\n  /-- `map` preserves compositions -/\n  comp_map :\n    ∀ {α β γ : TypeVec n} (g : α ⟹ β) (h : β ⟹ γ) (x : F α), (h ⊚ g) <$$> x = h <$$> g <$$> x\n\nopen Nat TypeVec\n\nnamespace MvFunctor\n\nexport LawfulMvFunctor (comp_map)\n\nopen LawfulMvFunctor\n\nvariable {α β γ : TypeVec.{u} n}\nvariable {F : TypeVec.{u} n → Type v} [MvFunctor F]\nvariable (P : α ⟹ «repeat» n Prop) (R : α ⊗ α ⟹ «repeat» n Prop)\n\n/-- adapt `MvFunctor.LiftP` to accept predicates as arrows -/\ndef LiftP' : F α → Prop :=\n  MvFunctor.LiftP fun i x => ofRepeat <| P i x\n\n\n/-- adapt `MvFunctor.LiftR` to accept relations as arrows -/\ndef LiftR' : F α → F α → Prop :=\n  MvFunctor.LiftR @fun i x y => ofRepeat <| R i <| TypeVec.prod.mk _ x y\n\nvariable [LawfulMvFunctor F]\n\n@[simp]\ntheorem id_map (x : F α) : TypeVec.id <$$> x = x :=\n  LawfulMvFunctor.id_map x\n\n@[simp]\ntheorem id_map' (x : F α) : (fun _i a => a) <$$> x = x :=\n  id_map x\n\ntheorem map_map (g : α ⟹ β) (h : β ⟹ γ) (x : F α) : h <$$> g <$$> x = (h ⊚ g) <$$> x :=\n  Eq.symm <| comp_map _ _ _\n\nsection LiftP'\n\nvariable (F) in\ntheorem exists_iff_exists_of_mono {P : F α → Prop} {q : F β → Prop}\n    (f : α ⟹ β) (g : β ⟹ α)\n    (h₀ : f ⊚ g = TypeVec.id)\n    (h₁ : ∀ u : F α, P u ↔ q (f <$$> u)) :\n    (∃ u : F α, P u) ↔ ∃ u : F β, q u := by\n  constructor <;> rintro ⟨u, h₂⟩\n  · refine ⟨f <$$> u, ?_⟩\n    apply (h₁ u).mp h₂\n  · refine ⟨g <$$> u, ?_⟩\n    rw [h₁]\n    simp only [MvFunctor.map_map, h₀, LawfulMvFunctor.id_map, h₂]\n\ntheorem LiftP_def (x : F α) : LiftP' P x ↔ ∃ u : F (Subtype_ P), subtypeVal P <$$> u = x :=\n  exists_iff_exists_of_mono F _ _ (toSubtype_of_subtype P) (by simp [MvFunctor.map_map])\n\ntheorem LiftR_def (x y : F α) :\n    LiftR' R x y ↔\n      ∃ u : F (Subtype_ R),\n        (TypeVec.prod.fst ⊚ subtypeVal R) <$$> u = x ∧\n          (TypeVec.prod.snd ⊚ subtypeVal R) <$$> u = y :=\n  exists_iff_exists_of_mono _ _ _ (toSubtype'_of_subtype' R) (by\n    simp only [map_map, comp_assoc, subtypeVal_toSubtype']\n    simp +unfoldPartialApp [comp])\n\nend LiftP'\n\nend MvFunctor\n\nnamespace MvFunctor\n\nsection LiftPLastPredIff\n\nvariable {F : TypeVec.{u} (n + 1) → Type*} [MvFunctor F] [LawfulMvFunctor F] {α : TypeVec.{u} n}\n\nvariable {β : Type u}\nvariable (pp : β → Prop)\n\nprivate def f :\n    ∀ n α,\n      (fun i : Fin2 (n + 1) => { p_1 // ofRepeat (PredLast' α pp i p_1) }) ⟹ fun i : Fin2 (n + 1) =>\n        { p_1 : (α ::: β) i // PredLast α pp p_1 }\n  | _, α, Fin2.fs i, x =>\n    ⟨x.val, cast (by grind [PredLast]) x.property⟩\n  | _, _, Fin2.fz, x => ⟨x.val, x.property⟩\n\nprivate def g :\n    ∀ n α,\n      (fun i : Fin2 (n + 1) => { p_1 : (α ::: β) i // PredLast α pp p_1 }) ⟹ fun i : Fin2 (n + 1) =>\n        { p_1 // ofRepeat (PredLast' α pp i p_1) }\n  | _, α, Fin2.fs i, x =>\n    ⟨x.val, cast (by simp only [PredLast]; erw [const_iff_true]) x.property⟩\n  | _, _, Fin2.fz, x => ⟨x.val, x.property⟩\n\ntheorem LiftP_PredLast_iff {β} (P : β → Prop) (x : F (α ::: β)) :\n    LiftP' (PredLast' _ P) x ↔ LiftP (PredLast _ P) x := by\n  dsimp only [LiftP, LiftP']\n  apply exists_iff_exists_of_mono F (f _ n α) (g _ n α)\n  · ext i ⟨x, _⟩\n    cases i <;> rfl\n  · intros\n    rw [MvFunctor.map_map]\n    dsimp +unfoldPartialApp [(· ⊚ ·)]\n    suffices (fun i => Subtype.val) = (fun i x => (MvFunctor.f P n α i x).val) by rw [this]\n    ext i ⟨x, _⟩\n    cases i <;> rfl\n\nvariable (rr : β → β → Prop)\n\nset_option backward.isDefEq.respectTransparency false in\nprivate def f' :\n    ∀ n α,\n      (fun i : Fin2 (n + 1) =>\n          { p_1 : _ × _ // ofRepeat (RelLast' α rr i (TypeVec.prod.mk _ p_1.fst p_1.snd)) }) ⟹\n        fun i : Fin2 (n + 1) => { p_1 : (α ::: β) i × _ // RelLast α rr p_1.fst p_1.snd }\n  | _, α, Fin2.fs i, x =>\n    ⟨x.val, cast (by simp only [RelLast]; erw [repeatEq_iff_eq]) x.property⟩\n  | _, _, Fin2.fz, x => ⟨x.val, x.property⟩\n\nset_option backward.isDefEq.respectTransparency false in\nprivate def g' :\n    ∀ n α,\n      (fun i : Fin2 (n + 1) => { p_1 : (α ::: β) i × _ // RelLast α rr p_1.fst p_1.snd }) ⟹\n        fun i : Fin2 (n + 1) =>\n        { p_1 : _ × _ // ofRepeat (RelLast' α rr i (TypeVec.prod.mk _ p_1.1 p_1.2)) }\n  | _, α, Fin2.fs i, x =>\n    ⟨x.val, cast (by simp only [RelLast]; erw [repeatEq_iff_eq]) x.property⟩\n  | _, _, Fin2.fz, x => ⟨x.val, x.property⟩\n\nTarget:\ntheorem LiftR_RelLast_iff (x y : F (α ::: β)) :\n    LiftR' (RelLast' _ rr) x y ↔ LiftR (RelLast _ rr) x y :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control/Functor","family_id":"liftr_rellast_iff","file_id":"mathlib/Mathlib/Control/Functor/Multivariate.lean","sample_id":"8b1fcb5d375f7d0c76ac8b5dc5be9785fa4e3630fe7326a206b1b90b1962024a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"be5d4d87008e815276be22f67ac5016b50a8ac860e36c4ba227914947bb79f69","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9e9aa61e2c3db4106475af3ddd5d69cd0b728c021823d1df2d1d07e5583ba8a1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9e0d38f023447d9d9affb59a11c3c96b5f0f51e0a6c3e310742ed531f794b9ce","source_sha256":"ebf0fa9c73e678969fdd70bcd10ba88fbd15d277d2467b36c6ff110685cacf37","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext <;> simp; ext <;> simp <;> (rw [← Category.assoc]; congr 1; simp)","hard_negative":true,"metrics":{"chosen_tokens":29,"rejected_tokens":2,"token_jaccard":0.055556,"token_length_ratio":0.068966},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"b991e633bebf6edb4a973f8fb08af9ee000d38f9a597157f3174592c6fbcc5f3","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Subobject.Lattice\npublic import Mathlib.CategoryTheory.Monoidal.Braided.Basic\npublic import Mathlib.CategoryTheory.Dialectica.Basic\n\nNamespace:\nCategoryTheory.Dial\n\nLocal context:\n/-\nCopyright (c) 2024 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# The Dialectica category is symmetric monoidal\n\nWe show that the category `Dial` has a symmetric monoidal category structure.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nnamespace CategoryTheory\n\nopen MonoidalCategory Limits\n\nuniverse v u\nvariable {C : Type u} [Category.{v} C] [HasFiniteProducts C] [HasPullbacks C]\n\nnamespace Dial\n\nlocal notation \"π₁\" => prod.fst\nlocal notation \"π₂\" => prod.snd\nlocal notation \"π(\" a \", \" b \")\" => prod.lift a b\n\n/-- The object `X ⊗ Y` in the `Dial C` category just tuples the left and right components. -/\n@[simps] def tensorObjImpl (X Y : Dial C) : Dial C where\n  src := X.src ⨯ Y.src\n  tgt := X.tgt ⨯ Y.tgt\n  rel :=\n    (Subobject.pullback (prod.map π₁ π₁)).obj X.rel ⊓\n    (Subobject.pullback (prod.map π₂ π₂)).obj Y.rel\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The functorial action of `X ⊗ Y` in `Dial C`. -/\n@[simps] def tensorHomImpl {X₁ X₂ Y₁ Y₂ : Dial C} (f : X₁ ⟶ X₂) (g : Y₁ ⟶ Y₂) :\n    tensorObjImpl X₁ Y₁ ⟶ tensorObjImpl X₂ Y₂ where\n  f := prod.map f.f g.f\n  F := π(prod.map π₁ π₁ ≫ f.F, prod.map π₂ π₂ ≫ g.F)\n  le := by\n    simp only [tensorObjImpl, Subobject.inf_pullback]\n    apply inf_le_inf <;> rw [← Subobject.pullback_comp, ← Subobject.pullback_comp]\n    · have := (Subobject.pullback (prod.map π₁ π₁ :\n        (X₁.src ⨯ Y₁.src) ⨯ X₂.tgt ⨯ Y₂.tgt ⟶ _)).monotone (Hom.le f)\n      rw [← Subobject.pullback_comp, ← Subobject.pullback_comp] at this\n      convert! this using 3 <;> simp\n    · have := (Subobject.pullback (prod.map π₂ π₂ :\n        (X₁.src ⨯ Y₁.src) ⨯ X₂.tgt ⨯ Y₂.tgt ⟶ _)).monotone (Hom.le g)\n      rw [← Subobject.pullback_comp, ← Subobject.pullback_comp] at this\n      convert! this using 3 <;> simp\n\n/-- The unit for the tensor `X ⊗ Y` in `Dial C`. -/\n@[simps] def tensorUnitImpl : Dial C := { src := ⊤_ _, tgt := ⊤_ _, rel := ⊤ }\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- Left unit cancellation `1 ⊗ X ≅ X` in `Dial C`. -/\n@[simps!] def leftUnitorImpl (X : Dial C) : tensorObjImpl tensorUnitImpl X ≅ X :=\n  isoMk (Limits.prod.leftUnitor _) (Limits.prod.leftUnitor _) <| by simp [Subobject.pullback_top]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- Right unit cancellation `X ⊗ 1 ≅ X` in `Dial C`. -/\n@[simps!] def rightUnitorImpl (X : Dial C) : tensorObjImpl X tensorUnitImpl ≅ X :=\n  isoMk (Limits.prod.rightUnitor _) (Limits.prod.rightUnitor _) <| by simp [Subobject.pullback_top]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The associator for tensor, `(X ⊗ Y) ⊗ Z ≅ X ⊗ (Y ⊗ Z)` in `Dial C`. -/\n@[simps!]\ndef associatorImpl (X Y Z : Dial C) :\n    tensorObjImpl (tensorObjImpl X Y) Z ≅ tensorObjImpl X (tensorObjImpl Y Z) :=\n  isoMk (prod.associator ..) (prod.associator ..) <| by\n    simp [Subobject.inf_pullback, ← Subobject.pullback_comp, inf_assoc]\n\n@[simps!]\ninstance : MonoidalCategoryStruct (Dial C) where\n  tensorUnit := tensorUnitImpl\n  tensorObj := tensorObjImpl\n  whiskerLeft X _ _ f := tensorHomImpl (𝟙 X) f\n  whiskerRight f Y := tensorHomImpl f (𝟙 Y)\n  tensorHom := tensorHomImpl\n  leftUnitor := leftUnitorImpl\n  rightUnitor := rightUnitorImpl\n  associator := associatorImpl\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\ntheorem id_tensorHom_id (X₁ X₂ : Dial C) : (𝟙 X₁ ⊗ₘ 𝟙 X₂ : _ ⟶ _) = 𝟙 (X₁ ⊗ X₂ : Dial C) := by\n  cat_disch\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n-- TODO: fix the non-terminal simp\nset_option linter.flexible false in\n\nTarget:\ntheorem tensorHom_comp_tensorHom {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : Dial C}\n    (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁) (g₂ : Y₂ ⟶ Z₂) :\n    (f₁ ⊗ₘ f₂) ≫ (g₁ ⊗ₘ g₂) = (f₁ ≫ g₁) ⊗ₘ (f₂ ≫ g₂) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_9e0d38f02344","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"8bd4050442c3bf81f26fc58474f5899953cfe9f977479513cb5426a40fc6b434","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Dialectica","family_id":"tensorhom_comp_tensorhom","file_id":"mathlib/Mathlib/CategoryTheory/Dialectica/Monoidal.lean","sample_id":"9e0d38f023447d9d9affb59a11c3c96b5f0f51e0a6c3e310742ed531f794b9ce"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ec2ccc87cb1b0580e84707f89c57da330dcbae57aa0cfba902f545e3a992cb11","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704302e78285e7c22e2b57a7665d68d1ff79e275868a3f817e12057d1bc284f7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d885a32647abb9166c82b5317470ebda2b4ddd48a31b71c34892848d64e4f116","source_sha256":"50e4a4d798ba2dd2872d163b6a5aeb55d8cf128d153e35dcc8c630f4eadcbabe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_contra h\n  exact absurd Valuation.IsEquiv.refl (h ▸ adicValuation_not_isEquiv_infty_valuation p)","hard_negative":true,"metrics":{"chosen_tokens":16,"rejected_tokens":3,"token_jaccard":0.133333,"token_length_ratio":0.1875},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"b99af4c1e895c6e0995a2ce65a9c6cbbc5ccd50cac1d1ed1965da32543179bb0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.Finite.Valuation\npublic import Mathlib.NumberTheory.FunctionField\npublic import Mathlib.RingTheory.Valuation.Discrete.Basic\n\nNamespace:\nRatFunc\n\nLocal context:\n/-\nCopyright (c) 2025 María Inés de Frutos-Fernández & Xavier Généreux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: María Inés de Frutos-Fernández, Xavier Généreux\n-/\n/-!\n# Ostrowski's theorem for `K(X)`\n\nThis file proves Ostrowski's theorem for the field of rational functions `K(X)`, where `K` is any\nfield: if `v` is a discrete valuation on `K(X)` which is trivial on elements of `K`, then `v` is\nequivalent to either the `I`-adic valuation for some `I : HeightOneSpectrum K[X]`, or to the\nvaluation at infinity `FunctionField.inftyValuation K`.\n\n## Main results\n- `RatFunc.valuation_isEquiv_infty_or_adic`: Ostrowski's theorem for `K(X)`.\n-/\n\n@[expose] public noncomputable section\n\n\nopen Multiplicative WithZero\n\nvariable {K Γ : Type*} [Field K] [LinearOrderedCommGroupWithZero Γ] {v : Valuation (RatFunc K) Γ}\n\nnamespace RatFunc\n\nsection Infinity\n\nopen Polynomial Valuation\n\nlemma valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X {f : RatFunc K}\n    [v.IsTrivialOn K] (hlt : 1 < v X) (hf : f ≠ 0) : v f = v RatFunc.X ^ f.intDegree := by\n  induction f using RatFunc.induction_on with\n  | f p q hq =>\n    rw [intDegree_div (by grind only) (by grind only), v.map_div, zpow_sub₀ (ne_zero_of_lt hlt)]\n    simp_rw [intDegree_polynomial, zpow_natCast, ← coePolynomial_eq_algebraMap]\n    have hp : p ≠ 0 := by contrapose hf; simp [hf]\n    rw [valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X _ hlt hp,\n      valuation_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X _ hlt hq]\n\nvariable [DecidableEq (RatFunc K)]\n\nlemma valuation_isEquiv_inftyValuation_of_one_lt_valuation_X [v.IsTrivialOn K] (hlt : 1 < v X) :\n    v.IsEquiv (inftyValuation K) := by\n  refine isEquiv_iff_val_lt_one.mpr fun {f} ↦ ?_\n  rcases eq_or_ne f 0 with rfl | hf\n  · simp\n  · have hlt' : 1 < inftyValuation K X := by simp [← exp_zero]\n    rw [valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt hf,\n      valuation_eq_valuation_X_zpow_intDegree_of_one_lt_valuation_X hlt' hf]\n    grind [one_le_zpow_iff_right₀]\n\nend Infinity\n\nopen IsDedekindDomain HeightOneSpectrum Set Valuation Polynomial\n\nlemma setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty [v.IsNontrivial] [v.IsTrivialOn K]\n    (hle : v RatFunc.X ≤ 1) : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty := by\n  obtain ⟨w, h0, h1⟩ := IsNontrivial.exists_lt_one (v := v)\n  induction w using RatFunc.induction_on with\n  | f p q =>\n    simp only [ne_eq, _root_.div_eq_zero_iff, FaithfulSMul.algebraMap_eq_zero_iff, not_or,\n      map_div₀] at *\n    have hor : ¬v ↑p = 1 ∨ ¬v ↑q = 1 := by rw [← not_and_or]; aesop\n    suffices ∀ r : K[X], v (↑r) ≠ 1 → r ≠ 0 → {p : K[X] | v ↑p < 1 ∧ ¬p = 0}.Nonempty by\n      exact Or.elim hor (fun hp ↦ this p hp h0.1) (fun hq ↦ this q hq h0.2)\n    exact fun r hr hr0 ↦ ⟨r, lt_iff_le_and_ne.mpr\n      ⟨Polynomial.valuation_le_one_of_valuation_X_le_one _ hle r, hr⟩, hr0⟩\n\nprivate lemma one_le_valuation_factor (hne : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty) {a b : K[X]}\n    (hab : v ↑(a * b) < 1 ∧ a ≠ 0 ∧ b ≠ 0) (hπᵥ : degree_lt_wf.min _ hne = a * b)\n    (hb : ¬IsUnit b) : 1 ≤ v ↑a := by\n  set πᵥ := degree_lt_wf.min _ hne\n  have hda : a.degree < πᵥ.degree := by\n    have hbpos := degree_pos_of_ne_zero_of_nonunit hab.2.2 hb\n    simp_rw [hπᵥ, degree_mul, degree_eq_natDegree hab.2.1, degree_eq_natDegree hab.2.2] at hbpos ⊢\n    norm_cast\n    simpa using hbpos\n  have hlea := imp_not_comm.mp (degree_lt_wf.not_lt_min _) hda\n  grind\n\nlemma irreducible_min_polynomial_valuation_lt_one_and_ne_zero [v.IsTrivialOn K]\n    (hne : {p : K[X] | v p < 1 ∧ p ≠ 0}.Nonempty) :\n    Irreducible (degree_lt_wf.min {p : K[X] | v p < 1 ∧ p ≠ 0} hne) := by\n  set πᵥ := degree_lt_wf.min _ hne\n  have hπᵥ : v πᵥ < 1 ∧ πᵥ ≠ 0 := degree_lt_wf.min_mem _ hne\n  refine irreducible_iff.mpr ⟨?_, fun a b hab ↦ ?_⟩\n  · simp only [Polynomial.isUnit_iff, isUnit_iff_ne_zero]\n    intro ⟨a, ha0, ha⟩\n    rw [← ha, coePolynomial, algebraMap_C, ← algebraMap_eq_C] at hπᵥ\n    grind\n  · by_contra! H\n    simp only [hab, ne_eq, mul_eq_zero, not_or] at hπᵥ\n    have hva := one_le_valuation_factor hne hπᵥ hab H.2\n    simp only [mul_comm a b, @and_comm (¬a = 0)] at hπᵥ hab\n    have := Right.one_le_mul (one_le_valuation_factor hne hπᵥ hab H.1) hva\n    simp only [coePolynomial_eq_algebraMap, map_mul] at hπᵥ this\n    grind\n\nsection valuation_X_le_one\n\nvariable [v.IsNontrivial] [v.IsTrivialOn K] (hle : v RatFunc.X ≤ 1)\n\n/-- A uniformizing element for the valuation `v`, as a polynomial in `K[X]`. -/\nabbrev uniformizingPolynomial : K[X] :=\n  WellFounded.min degree_lt_wf _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)\n\n@[inherit_doc]\nlocal notation \"πᵥ\" => uniformizingPolynomial hle\n\nlemma uniformizingPolynomial_ne_zero : πᵥ ≠ 0 := by\n  have := degree_lt_wf.min_mem _ (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)\n  simp_all [uniformizingPolynomial]\n\nlemma valuation_uniformizingPolynomial_lt_one : v πᵥ < 1 := by\n  simpa using! (degree_lt_wf.min_mem _\n    (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)).1\n\nopen Ideal in\n/-- The maximal ideal of `K[X]` generated by the `uniformizingPolynomial` for `v`. -/\ndef valuationIdeal : HeightOneSpectrum K[X] where\n  asIdeal := Submodule.span K[X] {πᵥ}\n  isPrime := IsMaximal.isPrime (PrincipalIdealRing.isMaximal_of_irreducible\n    (irreducible_min_polynomial_valuation_lt_one_and_ne_zero\n      (setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle)))\n  ne_bot := by simpa using uniformizingPolynomial_ne_zero hle\n\n@[inherit_doc]\nlocal notation \"Pᵥ\" => RatFunc.valuationIdeal hle\n\nsection Associates\n\nopen EuclideanDomain in\nlemma valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one {p : K[X]}\n    (hp : p ≠ 0) :\n    v (algebraMap K[X] (RatFunc K) p) = v (πᵥ ^ ((Associates.mk (Pᵥ).asIdeal).count\n      (Associates.mk (Ideal.span {p})).factors)) := by\n  set π := πᵥ\n  have hne := setOf_polynomial_valuation_lt_one_and_ne_zero_nonempty hle\n  have hπirr : Irreducible π := irreducible_min_polynomial_valuation_lt_one_and_ne_zero hne\n  obtain ⟨k, q, hnq, heq⟩ := WfDvdMonoid.max_power_factor hp hπirr\n  have hπ : π ∈ _ := degree_lt_wf.min_mem _ hne\n  simp only [ne_eq, mem_setOf] at hπ\n  nth_rw 1 [heq]\n  simp only [map_mul, map_pow]\n  suffices v (algebraMap K[X] (RatFunc K) q) = 1 by\n    simp only [this, mul_one]\n    congr\n    exact (Ideal.count_associates_eq (irreducible_iff_prime.mp hπirr) hnq heq).symm\n  rw [← mod_add_div q π, map_add]\n  rw [← mod_eq_zero] at hnq\n  suffices v (algebraMap K[X] (RatFunc K) (q % π)) = 1 ∧\n      v (algebraMap K[X] (RatFunc K) (π * (q / π))) < 1 by\n    obtain ⟨h₁, h₂⟩ := this\n    rw [← h₁] at h₂ ⊢\n    exact Valuation.map_add_eq_of_lt_left _ h₂\n  constructor\n  · rw [← coePolynomial_eq_algebraMap]\n    have hnπ : q % π ∉ {p : K[X] | v ↑p < 1 ∧ p ≠ 0} :=\n      imp_not_comm.mp (degree_lt_wf.not_lt_min _) (EuclideanDomain.remainder_lt q hπ.2)\n    have := Polynomial.valuation_le_one_of_valuation_X_le_one _ hle (q % π)\n    grind\n  · simpa only [map_mul, ← coePolynomial_eq_algebraMap]\n      using mul_lt_one_of_lt_of_le hπ.1 <| (q / π).valuation_le_one_of_valuation_X_le_one _ hle\n\nlemma exists_zpow_uniformizingPolynomial {f : RatFunc K} (hf : f ≠ 0) :\n    ∃ (z : ℤ), v f = v πᵥ ^ z := by\n  have h0 : v πᵥ ≠ 0 := by simpa using uniformizingPolynomial_ne_zero hle\n  induction f using RatFunc.induction_on with\n  | f p q hq =>\n    use (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {p})).factors -\n      (Associates.mk (Pᵥ).asIdeal).count (Associates.mk (Ideal.span {q})).factors\n    simp only [map_div₀, map_pow, zpow_sub₀ h0, zpow_natCast,\n      valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle hq,\n      valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle\n        (p := p) (by aesop)]\n\nlemma uniformizingPolynomial_isUniformizer [hv : IsRankOneDiscrete v] :\n    v.IsUniformizer πᵥ := by\n  have h0 : v πᵥ ≠ 0 := by simpa using uniformizingPolynomial_ne_zero hle\n  rw [IsUniformizer, ← hv.valueGroup_genLTOne_eq_generator, ← h0.isUnit.unit_spec, Units.val_inj]\n  apply LinearOrderedCommGroup.Subgroup.genLTOne_unique\n  · rw [← Units.val_lt_val, h0.isUnit.unit_spec, Units.val_one]\n    exact valuation_uniformizingPolynomial_lt_one hle\n  · ext γ\n    simp only [coePolynomial_eq_algebraMap, MonoidWithZeroHom.mem_valueGroup_iff_of_comm, ne_eq,\n      map_eq_zero, Subgroup.mem_zpowers_iff]\n    refine ⟨fun ⟨k, hk⟩ ↦ ?_, fun ⟨a, ha, b, hab⟩ ↦ ?_⟩\n    · use 1, one_ne_zero, πᵥ ^ k\n      simp only [← Units.val_inj, Units.val_zpow_eq_zpow_val] at hk\n      simp [← hk]\n    · obtain ⟨ka, hka⟩ := exists_zpow_uniformizingPolynomial hle ha\n      obtain ⟨kb, hkb⟩ := exists_zpow_uniformizingPolynomial hle (f := b) (by aesop)\n      rw [MonoidWithZeroHom.coe_ofClass, hka, hkb] at hab\n      use kb - ka\n      have : v ↑πᵥ ^ ka ≠ 0 := zpow_ne_zero _ h0\n      simp [zpow_sub, ← Units.val_inj, ← coePolynomial_eq_algebraMap, field, ← hab]\n\nlemma valuation_isEquiv_valuationIdeal_adic_of_valuation_X_le_one [IsRankOneDiscrete v] :\n    v.IsEquiv ((Pᵥ).valuation (RatFunc K)) := by\n  rw [isEquiv_iff_val_le_one]\n  intro f\n  rcases eq_or_ne f 0 with rfl | hf0\n  · simp\n  · induction f using RatFunc.induction_on with\n    | f p q hq0 =>\n      have hp0 : p ≠ 0 := by simp_all\n      set pi := πᵥ with hpi_def\n      have hpi : v.IsUniformizer (pi : RatFunc K) := uniformizingPolynomial_isUniformizer hle\n      simp only [map_div₀, valuation_of_algebraMap, intValuation_def, exp_neg, if_neg hp0,\n        if_neg hq0, div_inv_eq_mul]\n      rw [valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle hp0,\n        valuation_eq_valuation_uniformizingPolynomial_pow_of_valuation_X_le_one hle hq0]\n      simp_all [div_le_one₀, inv_mul_le_one₀,\n        (pow_le_pow_iff_right_of_lt_one₀ (by simp_all) (IsRankOneDiscrete.generator_lt_one v))]\n\nend Associates\n\nend valuation_X_le_one\n\nlemma adicValuation_not_isEquiv_infty_valuation [DecidableEq (RatFunc K)]\n    (p : IsDedekindDomain.HeightOneSpectrum K[X]) :\n    ¬ (p.valuation (RatFunc K)).IsEquiv (inftyValuation K) := by\n  simp only [isEquiv_iff_val_le_one]\n  push Not\n  refine ⟨X, .inl ⟨p.valuation_le_one _, ?_⟩⟩\n  rw [inftyValuation.X, ← log_lt_iff_lt_exp one_ne_zero, log_one]\n  exact zero_lt_one\n\nTarget:\nlemma adicValuation_ne_inftyValuation [DecidableEq (RatFunc K)]\n   (p : IsDedekindDomain.HeightOneSpectrum K[X]) :\n    p.valuation (RatFunc K) ≠ inftyValuation K :=\n\nProof body:\n","rejected":"by\n  exact adicValuation_ne_inftyValuation","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"de869dc7a0655d59fd0eb33feb975f83bd34fd14d0f3eb1d872c1845a26a0f2d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/RatFunc","family_id":"adicvaluation_ne_inftyvaluation","file_id":"mathlib/Mathlib/NumberTheory/RatFunc/Ostrowski.lean","sample_id":"d885a32647abb9166c82b5317470ebda2b4ddd48a31b71c34892848d64e4f116"},"split":"train"}
{"binding":{"chosen_proof_sha256":"078c3ba86f93f2f5c0b7638f81f2db59b5ef54f0de3d73360e15d4f2f0ce975a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5f55f4262f31682a3e12d8625b11abf778bb7c39ed9ebef5aef18cf9e8a8d94d","source_sha256":"5719c6ab1b79cb92451b632f8631c4fdd8e58d363dc71c3722b36068c3f61537","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [normThreshold_zero]\n  calc a / (1 - √2)\n  _ ≤ 0 := div_nonpos_of_nonneg_of_nonpos ha_pos.le (by simp)\n  _ < a := ha_pos","hard_negative":false,"metrics":{"chosen_tokens":32,"rejected_tokens":3,"token_jaccard":0.038462,"token_length_ratio":0.09375},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"b9b2b746a4ee88a63ebcba3858f8858eb66a34ff85967d99c5e71504fba8ab8d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecificLimits.ArithmeticGeometric\npublic import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap\npublic import Mathlib.MeasureTheory.Function.L1Space.Integrable\n\nNamespace:\nProbabilityTheory.Fernique\n\nLocal context:\n/-\nCopyright (c) 2025 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n-/\n/-!\n# Fernique's theorem for rotation-invariant measures\n\nLet `μ` be a finite measure on a second-countable normed space `E` such that the product measure\n`μ.prod μ` on `E × E` is invariant by rotation of angle `-π/4`.\nThen there exists a constant `C > 0` such that the function `x ↦ exp (C * ‖x‖ ^ 2)` is integrable\nwith respect to `μ`.\n\n## Sketch of the proof\n\nThe main case of the proof is for `μ` a probability measure such that there exists a positive\n`a : ℝ` such that `2⁻¹ < μ {x | ‖x‖ ≤ a} < 1`. If `μ` is a probability measure and `a` does not\nexist then we can show that there is a ball with finite radius of measure 1, and the result is true\nfor `C = 1` (for example), since `x ↦ exp (‖x‖ ^ 2)` is almost surely bounded.\nWe then choose such an `a`.\n\nIn order to show the existence of `C` such that `x ↦ exp (C * ‖x‖ ^ 2)` is integrable, we prove as\nintermediate result that for `a, c` with `2⁻¹ < c ≤ μ {x | ‖x‖ ≤ a}`,\nthe integral `∫⁻ x, exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2) ∂μ` is bounded by a finite quantity\n(`logRatio c` is a multiple of `log (c / (1 - c))`). We can then take `C = logRatio c * a⁻¹ ^ 2`.\n\nWe now turn to the proof of the intermediate result.\n\nFirst in `measure_le_mul_measure_gt_le_of_map_rotation_eq_self` we prove that if a measure `μ` is\nsuch that `μ.prod μ` is invariant by rotation of angle `-π/4` then\n`μ {x | ‖x‖ ≤ a} * μ {x | b < ‖x‖} ≤ μ {x | (b - a) / √2 < ‖x‖} ^ 2`.\nThe rotation invariance is used only through that inequality.\n\nWe define a sequence of thresholds `t n` inductively by `t 0 = a` and `t (n + 1) = √2 * t n + a`.\nThey are chosen such that the invariance by rotation gives\n`μ {x | ‖x‖ ≤ a} * μ {x | t (n + 1) < ‖x‖} ≤ μ {x | t n < ‖x‖} ^ 2`.\nThanks to that inequality we can show that `μ {x | t n < ‖x‖}` decreases fast with `n`:\nfor `mₐ = μ {x | ‖x‖ ≤ a}`, `μ {x | t n < ‖x‖} ≤ mₐ * exp (- log (mₐ / (1 - mₐ)) * 2 ^ n)`.\n\nWe cut the space into annuli `{x | t n < ‖x‖ ≤ t n + 1}` and bound the integral separately on\neach annulus. On that set the function `exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2)` is bounded by\n`exp (logRatio c * a⁻¹ ^ 2 * t (n + 1) ^ 2)`, which is in turn less than\n`exp (2⁻¹ * log (c / (1 - c)) * 2 ^ n)` (from the definition of the threshold `t` and `logRatio c`).\nThe measure of the annulus is bounded by `μ {x | t n < ‖x‖}`, for which we derived an upper bound\nabove. The function gets exponentially large, but `μ {x | t n < ‖x‖}` decreases even faster, so the\nintegral is bounded by a quantity of the form `exp (- u * 2 ^ n)` for `u>0`.\nSumming over all annuli (over `n`) gives a finite value for the integral.\n\n## Main statements\n\n* `lintegral_exp_mul_sq_norm_le_of_map_rotation_eq_self`: for `μ` a probability measure\n  whose product with itself is invariant by rotation and for `a, c` with\n  `2⁻¹ < c ≤ μ {x | ‖x‖ ≤ a}`, the integral `∫⁻ x, exp (logRatio c * a⁻¹ ^ 2 * ‖x‖ ^ 2) ∂μ`\n  is bounded by a quantity that does not depend on `a`.\n* `exists_integrable_exp_sq_of_map_rotation_eq_self`: Fernique's theorem for finite measures\n  whose product is invariant by rotation.\n\n## References\n\n* [Xavier Fernique, *Intégrabilité des vecteurs gaussiens*][fernique1970integrabilite]\n* [Martin Hairer, *An introduction to stochastic PDEs*][hairer2009introduction]\n\n## TODO\n\nFrom the intermediate result `lintegral_exp_mul_sq_norm_le_of_map_rotation_eq_self`,\nwe can deduce bounds on all the moments of the measure `μ` as function of powers of\nthe first moment.\n\n-/\n\n@[expose] public section\n\nopen MeasureTheory ProbabilityTheory Complex NormedSpace Filter\nopen scoped ENNReal NNReal Real Topology\n\nsection Aux\n\nlemma StrictMono.exists_between_of_tendsto_atTop {β : Type*} [LinearOrder β] {t : ℕ → β}\n    (ht_mono : StrictMono t) (ht_tendsto : Tendsto t atTop atTop) {x : β} (hx : t 0 < x) :\n    ∃ n, t n < x ∧ x ≤ t (n + 1) := by\n  have h : ∃ n, x ≤ t n := by\n    simp only [tendsto_atTop_atTop_iff_of_monotone ht_mono.monotone] at ht_tendsto\n    exact ht_tendsto x\n  have h' m := Nat.find_min h (m := m)\n  simp only [not_le] at h'\n  exact ⟨Nat.find h - 1, h' _ (by simp [hx]), by simp [Nat.find_spec h, hx]⟩\n\nend Aux\n\nnamespace ProbabilityTheory\n\nvariable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ℝ E]\n\n/-- The rotation in `E × E` with angle `θ`, as a continuous linear map. -/\nnoncomputable\ndef _root_.ContinuousLinearMap.rotation (θ : ℝ) : E × E →L[ℝ] E × E where\n  toFun := fun x ↦ (Real.cos θ • x.1 + Real.sin θ • x.2, - Real.sin θ • x.1 + Real.cos θ • x.2)\n  map_add' x y := by\n    simp only [Prod.fst_add, smul_add, Prod.snd_add, neg_smul, Prod.mk_add_mk]\n    abel_nf\n  map_smul' c x := by simp [smul_comm c]\n\nlemma _root_.ContinuousLinearMap.rotation_apply (θ : ℝ) (x : E × E) :\n    ContinuousLinearMap.rotation θ x\n     = (Real.cos θ • x.1 + Real.sin θ • x.2, -Real.sin θ • x.1 + Real.cos θ • x.2) := rfl\n\nvariable [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E] {μ : Measure E} {a : ℝ}\n\n/-- If a measure `μ` is such that `μ.prod μ` is invariant by rotation of angle `-π/4` then\n`μ {x | ‖x‖ ≤ a} * μ {x | b < ‖x‖} ≤ μ {x | (b - a) / √2 < ‖x‖} ^ 2`. -/\nlemma measure_le_mul_measure_gt_le_of_map_rotation_eq_self [SFinite μ]\n    (h : (μ.prod μ).map (ContinuousLinearMap.rotation (-(π / 4))) = μ.prod μ)\n    (a b : ℝ) :\n    μ {x | ‖x‖ ≤ a} * μ {x | b < ‖x‖} ≤ μ {x | (b - a) / √2 < ‖x‖} ^ 2 := by\n  calc μ {x | ‖x‖ ≤ a} * μ {x | b < ‖x‖}\n  _ = (μ.prod μ) ({x | ‖x‖ ≤ a} ×ˢ {y | b < ‖y‖}) := by rw [Measure.prod_prod]\n    -- This is the measure of two bands in the plane (draw a picture!)\n  _ = (μ.prod μ) {p | ‖p.1‖ ≤ a ∧ b < ‖p.2‖} := rfl\n  _ = ((μ.prod μ).map (ContinuousLinearMap.rotation (-(π / 4)))) {p | ‖p.1‖ ≤ a ∧ b < ‖p.2‖} := by\n    -- We can rotate the bands since `μ.prod μ` is invariant under rotation\n    rw [h]\n  _ = (μ.prod μ) {p | ‖p.1 - p.2‖ / √2 ≤ a ∧ b < ‖p.1 + p.2‖ / √2} := by\n    rw [Measure.map_apply (by fun_prop)]\n    swap\n    · refine MeasurableSet.inter ?_ ?_\n      · change MeasurableSet {p : E × E | ‖p.1‖ ≤ a}\n        exact measurableSet_le (by fun_prop) (by fun_prop)\n      · change MeasurableSet {p : E × E | b < ‖p.2‖}\n        exact measurableSet_lt (by fun_prop) (by fun_prop)\n    congr 1\n    simp only [Set.preimage_setOf_eq, ContinuousLinearMap.rotation_apply, Real.cos_neg,\n      Real.cos_pi_div_four, Real.sin_neg, Real.sin_pi_div_four, neg_smul, neg_neg]\n    have h_twos : ‖2⁻¹ * √2‖ = (√2)⁻¹ := by\n      simp only [norm_mul, norm_inv, Real.norm_ofNat, Real.norm_eq_abs]\n      rw [abs_of_nonneg (by positivity)]\n      nth_rw 1 [← Real.sq_sqrt (by simp : (0 : ℝ) ≤ 2)]\n      rw [pow_two, mul_inv, mul_assoc, inv_mul_cancel₀ (by positivity), mul_one]\n    congr! with p\n    · rw [← sub_eq_add_neg, ← smul_sub, norm_smul, div_eq_inv_mul, div_eq_inv_mul, h_twos]\n    · rw [← smul_add, norm_smul, div_eq_inv_mul, div_eq_inv_mul, h_twos]\n  _ ≤ (μ.prod μ) {p | (b - a) / √2 < ‖p.1‖ ∧ (b - a) / √2 < ‖p.2‖} := by\n    -- The rotated bands are contained in quadrants.\n    refine measure_mono fun p ↦ ?_\n    simp only [Set.mem_setOf_eq, and_imp]\n    intro hp1 hp2\n    suffices (b - a) / √2 < min ‖p.1‖ ‖p.2‖ from lt_min_iff.mp this\n    calc (b - a) / √2\n    _ < (‖p.1 + p.2‖ - ‖p.1 - p.2‖) / 2 := by\n      suffices b - a < ‖p.1 + p.2‖ / √2 - ‖p.1 - p.2‖ / √2 by\n        calc (b - a) / √2 < (‖p.1 + p.2‖ / √2 - ‖p.1 - p.2‖ / √2) / √2 := by gcongr\n        _ = (‖p.1 + p.2‖ - ‖p.1 - p.2‖) / 2 := by\n          field_simp; rw [Real.sq_sqrt (by positivity)]; ring\n      calc b - a < ‖p.1 + p.2‖ / √2 - a := by gcongr\n      _ ≤ ‖p.1 + p.2‖ / √2 - ‖p.1 - p.2‖ / √2 := by gcongr\n    _ ≤ min ‖p.1‖ ‖p.2‖ := by\n      have := norm_add_sub_norm_sub_le_two_mul_min p.1 p.2\n      linarith\n  _ = (μ.prod μ) ({x | (b - a) / √2 < ‖x‖} ×ˢ {y | (b - a) / √2 < ‖y‖}) := rfl\n  _ ≤ μ {x | (b - a) / √2 < ‖x‖} ^ 2 := by rw [Measure.prod_prod, pow_two]\n\nnamespace Fernique\n\n/-- A sequence of real thresholds that will be used to cut the space into annuli.\nChosen such that for a rotation invariant measure, an application of lemma\n`measure_le_mul_measure_gt_le_of_map_rotation_eq_self` gives\n`μ {x | ‖x‖ ≤ a} * μ {x | normThreshold a (n + 1) < ‖x‖} ≤ μ {x | normThreshold a n < ‖x‖} ^ 2`. -/\nnoncomputable def normThreshold (a : ℝ) : ℕ → ℝ := arithGeom √2 a a\n\nlemma normThreshold_zero : normThreshold a 0 = a := rfl\n\nlemma normThreshold_add_one (n : ℕ) : normThreshold a (n + 1) = √2 * normThreshold a n + a := rfl\n\nlemma measure_le_mul_measure_gt_normThreshold_le_of_map_rotation_eq_self [SFinite μ]\n    (h_rot : (μ.prod μ).map (ContinuousLinearMap.rotation (-(π / 4))) = μ.prod μ) (a : ℝ) (n : ℕ) :\n    μ {x | ‖x‖ ≤ a} * μ {x | normThreshold a (n + 1) < ‖x‖}\n      ≤ μ {x | normThreshold a n < ‖x‖} ^ 2 := by\n  convert! measure_le_mul_measure_gt_le_of_map_rotation_eq_self h_rot _ _\n  simp [normThreshold_add_one]\n\nTarget:\nlemma lt_normThreshold_zero (ha_pos : 0 < a) : a / (1 - √2) < normThreshold a 0 :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Distributions","family_id":"lt_normthreshold_zero","file_id":"mathlib/Mathlib/Probability/Distributions/Fernique.lean","sample_id":"5f55f4262f31682a3e12d8625b11abf778bb7c39ed9ebef5aef18cf9e8a8d94d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ea338c8c6deda19356f39b2f9dc6bfce64075bf47da045c1a31eb2b19f1fe070","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5bfe544231c20197189c9a0b8a98816ff308792447dd9ac8dab546df3eb7f12a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7c30455a13ea31c0fdc5ee48caccc8329851df034a9e411c0a2bb1d61bbf2a76","source_sha256":"dc37d7e189f3a4d32e4029e0df16c9eeb5ca8b81845b68289b84bb8e5dd20ee8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h => ?_, fun _ => inferInstance⟩\n  contrapose! h\n  infer_instance","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":25,"token_jaccard":0.875,"token_length_ratio":1.388889},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"b9fa642f52de2754fde4a9b671585515fa8d7d2d66365b41a5e19d88ff075baa","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Sets.Closeds\npublic import Mathlib.Topology.QuasiSeparated\n\nNamespace:\nTopologicalSpace.Compacts\n\nLocal context:\n/-\nCopyright (c) 2020 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Yaël Dillies\n-/\n/-!\n# Compact sets\n\nWe define a few types of compact sets in a topological space.\n\n## Main Definitions\n\nFor a topological space `α`,\n* `TopologicalSpace.Compacts α`: The type of compact sets.\n* `TopologicalSpace.NonemptyCompacts α`: The type of non-empty compact sets.\n* `TopologicalSpace.PositiveCompacts α`: The type of compact sets with non-empty interior.\n* `TopologicalSpace.CompactOpens α`: The type of compact open sets. This is a central object in the\n  study of spectral spaces.\n-/\n\n@[expose] public section\n\n\nopen Set\n\nvariable {α β γ : Type*} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]\n\nnamespace TopologicalSpace\n\n/-! ### Compact sets -/\n\n/-- The type of compact sets of a topological space. -/\nstructure Compacts (α : Type*) [TopologicalSpace α] where\n  /-- the carrier set, i.e. the points in this set -/\n  carrier : Set α\n  isCompact' : IsCompact carrier\n\nnamespace Compacts\n\ninstance : SetLike (Compacts α) α where\n  coe := Compacts.carrier\n  coe_injective s t h := by cases s; cases t; congr\n\ninstance : PartialOrder (Compacts α) := .ofSetLike (Compacts α) α\n\n/-- See Note [custom simps projection]. -/\ndef Simps.coe (s : Compacts α) : Set α := s\n\ninitialize_simps_projections Compacts (carrier → coe, as_prefix coe)\n\nprotected theorem isCompact (s : Compacts α) : IsCompact (s : Set α) :=\n  s.isCompact'\n\ninstance (K : Compacts α) : CompactSpace K :=\n  isCompact_iff_compactSpace.1 K.isCompact\n\n/-- Reinterpret a compact as a closed set. -/\n@[simps]\ndef toCloseds [T2Space α] (s : Compacts α) : Closeds α :=\n  ⟨s, s.isCompact.isClosed⟩\n\n@[simp]\ntheorem mem_toCloseds [T2Space α] {x : α} {s : Compacts α} :\n    x ∈ s.toCloseds ↔ x ∈ s :=\n  Iff.rfl\n\ntheorem toCloseds_injective [T2Space α] : Function.Injective (toCloseds (α := α)) :=\n  .of_comp (f := SetLike.coe) SetLike.coe_injective\n\ninstance : CanLift (Set α) (Compacts α) (↑) IsCompact where prf K hK := ⟨⟨K, hK⟩, rfl⟩\n\n@[ext]\nprotected theorem ext {s t : Compacts α} (h : (s : Set α) = t) : s = t :=\n  SetLike.ext' h\n\n@[simp]\ntheorem coe_mk (s : Set α) (h) : (mk s h : Set α) = s :=\n  rfl\n\n@[simp]\ntheorem carrier_eq_coe (s : Compacts α) : s.carrier = s :=\n  rfl\n\ninstance : Max (Compacts α) :=\n  ⟨fun s t => ⟨s ∪ t, s.isCompact.union t.isCompact⟩⟩\n\ninstance [T2Space α] : Min (Compacts α) :=\n  ⟨fun s t => ⟨s ∩ t, s.isCompact.inter t.isCompact⟩⟩\n\ninstance [CompactSpace α] : Top (Compacts α) :=\n  ⟨⟨univ, isCompact_univ⟩⟩\n\ninstance : Bot (Compacts α) :=\n  ⟨⟨∅, isCompact_empty⟩⟩\n\ninstance : SemilatticeSup (Compacts α) :=\n  fast_instance% SetLike.coe_injective.semilatticeSup _ .rfl .rfl fun _ _ ↦ rfl\n\ninstance [T2Space α] : DistribLattice (Compacts α) :=\n  fast_instance% SetLike.coe_injective.distribLattice _ .rfl .rfl (fun _ _ ↦ rfl) fun _ _ ↦ rfl\n\ninstance : OrderBot (Compacts α) :=\n  fast_instance% OrderBot.lift ((↑) : _ → Set α) (fun _ _ => id) rfl\n\ninstance [CompactSpace α] : BoundedOrder (Compacts α) :=\n  fast_instance% BoundedOrder.lift ((↑) : _ → Set α) (fun _ _ => id) rfl rfl\n\n/-- The type of compact sets is inhabited, with default element the empty set. -/\ninstance : Inhabited (Compacts α) := ⟨⊥⟩\n\ninstance [IsEmpty α] : Unique (Compacts α) where\n  uniq _ := Compacts.ext (Subsingleton.elim _ _)\n\n@[simp]\ntheorem coe_sup (s t : Compacts α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t :=\n  rfl\n\n@[simp]\ntheorem coe_inf [T2Space α] (s t : Compacts α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t :=\n  rfl\n\n@[simp]\ntheorem coe_top [CompactSpace α] : (↑(⊤ : Compacts α) : Set α) = univ :=\n  rfl\n\n@[simp]\ntheorem coe_bot : (↑(⊥ : Compacts α) : Set α) = ∅ :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_eq_empty {s : Compacts α} : (s : Set α) = ∅ ↔ s = ⊥ :=\n  SetLike.coe_injective.eq_iff' rfl\n\n@[simp]\ntheorem coe_nonempty {s : Compacts α} : (s : Set α).Nonempty ↔ s ≠ ⊥ :=\n  nonempty_iff_ne_empty.trans coe_eq_empty.not\n\n@[simp]\ntheorem coe_finset_sup {ι : Type*} {s : Finset ι} {f : ι → Compacts α} :\n    (↑(s.sup f) : Set α) = s.sup fun i => ↑(f i) := by\n  refine Finset.cons_induction_on s rfl fun a s _ h => ?_\n  simp_rw [Finset.sup_cons, coe_sup, sup_eq_union]\n  congr\n\n@[simps]\ninstance : Singleton α (Compacts α) where\n  singleton x := ⟨{x}, isCompact_singleton⟩\n\n@[simp]\ntheorem mem_singleton (x y : α) : x ∈ ({y} : Compacts α) ↔ x = y :=\n  Iff.rfl\n\n@[simp]\ntheorem toCloseds_singleton [T2Space α] (x : α) : toCloseds {x} = {x} :=\n  rfl\n\ntheorem singleton_injective : Function.Injective ({·} : α → Compacts α) :=\n  .of_comp (f := SetLike.coe) Set.singleton_injective\n\n@[simp]\ntheorem singleton_inj {x y : α} : ({x} : Compacts α) = {y} ↔ x = y :=\n  singleton_injective.eq_iff\n\ninstance [Nonempty α] : Nontrivial (Compacts α) := by\n  constructor\n  obtain ⟨x⟩ := ‹Nonempty α›\n  exact ⟨⊥, {x}, ne_of_apply_ne SetLike.coe (Set.empty_ne_singleton x)⟩\n\n@[simp]\n\nTarget:\ntheorem subsingleton_iff : Subsingleton (Compacts α) ↔ IsEmpty α :=\n\nProof body:\n","rejected":"```lean\nby\n  refine ⟨fun h => ?_, fun _ => inferInstance⟩\n  contrapose! h\n  infer_instance\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Sets","family_id":"subsingleton_iff","file_id":"mathlib/Mathlib/Topology/Sets/Compacts.lean","sample_id":"7c30455a13ea31c0fdc5ee48caccc8329851df034a9e411c0a2bb1d61bbf2a76"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f30223a701d84b8cf4910174a2ebd64c60a3db6a916ec5748abb240a8d7ca7c4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"43783c386dd24adc229bb59f8ae27239586f2fb44a8bc33452052e849aeb7226","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"766f3502274c6a836f440a41e6af60197e8da15e11cc99e0e09ed8fb000b1581","source_sha256":"a1ecfd868e0d49b8667b2cdba350ee4f2ceb12fbe0b51094508e312f262dbf5e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using (toInt_le_toInt val j i).not","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":16,"token_jaccard":0.6875,"token_length_ratio":1.454545},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"ba1984bf89c8eafef1003d1127aea540aa84ac6d65bfcfea3a2bdb6643f6aec5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.List.Pairwise\npublic import Batteries.Tactic.GeneralizeProofs\npublic import Mathlib.Tactic.Order.CollectFacts\n\nNamespace:\nMathlib.Tactic.Order.ToInt\n\nLocal context:\n/-\nCopyright (c) 2025 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\npublic meta import Mathlib.Util.AtomM\npublic meta import Mathlib.Util.Qq\npublic meta import Std.Data.HashMap.AdditionalOperations\n\n/-!\n# Translating linear orders to ℤ\n\nIn this file we implement the translation of a problem in any linearly ordered type to a problem in\n`ℤ`. This allows us to use the `lia` tactic to solve it.\n\nWhile the core algorithm of the `order` tactic is complete for the theory of linear orders in the\nsignature (`<`, `≤`),\nit becomes incomplete in the signature with lattice operations `⊓` and `⊔`. With these operations,\nthe problem becomes NP-hard, and the idea is to reuse a smart and efficient procedure, such as\n`lia`.\n\n## TODO\n\nMigrate to `grind` when it is ready.\n-/\n\npublic meta section\n\nnamespace Mathlib.Tactic.Order.ToInt\n\nvariable {α : Type*} [LinearOrder α] {n : ℕ} (val : Fin n → α)\n\n/-- The main theorem asserting the existence of a translation.\nWe use `Classical.choose` to turn this into a value for use in the `order` tactic,\nsee `toInt`.\n-/\ntheorem exists_translation : ∃ tr : Fin n → ℤ, ∀ i j, val i ≤ val j ↔ tr i ≤ tr j := by\n  let li := List.ofFn val\n  let sli := li.mergeSort\n  have (i : Fin n) : ∃ j : Fin sli.length, sli[j] = val i := by\n    apply List.get_of_mem\n    rw [List.Perm.mem_iff (List.mergeSort_perm _ _)]\n    simp [li]\n  use fun i ↦ (this i).choose\n  intro i j\n  simp only [Fin.getElem_fin, Int.ofNat_le]\n  by_cases h_eq : val i = val j\n  · simp [h_eq]\n  generalize_proofs _ hi hj\n  rw [← hi.choose_spec, ← hj.choose_spec] at h_eq\n  conv_lhs => rw [← hi.choose_spec, ← hj.choose_spec]\n  have := li.pairwise_mergeSort (le := fun a b ↦ decide (a ≤ b))\n      (fun a b c ↦ by simpa using le_trans) (by simpa using le_total)\n  rw [List.pairwise_iff_get] at this\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · contrapose! h\n    exact lt_of_le_of_ne (by simpa using (this hj.choose hi.choose (by simpa)))\n      (fun h ↦ h_eq (h.symm))\n  · simpa using this hi.choose hj.choose (by apply lt_of_le_of_ne h; contrapose h_eq; simp [h_eq])\n\n/-- Auxiliary definition used by the `order` tactic to transfer facts in a linear order to `ℤ`. -/\nnoncomputable def toInt (k : Fin n) : ℤ :=\n  (exists_translation val).choose k\n\nvariable (i j k : Fin n)\n\ntheorem toInt_le_toInt : toInt val i ≤ toInt val j ↔ val i ≤ val j := by\n  simp [toInt, (exists_translation val).choose_spec]\n\nTarget:\ntheorem toInt_lt_toInt : toInt val i < toInt val j ↔ val i < val j :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  simpa using (toInt_le_toInt val j i).not","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Order","family_id":"toint_lt_toint","file_id":"mathlib/Mathlib/Tactic/Order/ToInt.lean","sample_id":"766f3502274c6a836f440a41e6af60197e8da15e11cc99e0e09ed8fb000b1581"},"split":"train"}
{"binding":{"chosen_proof_sha256":"90db882f63b60882a9cc2eecf39977196f2e1c4d889d00ccb138dc84cbbbe60a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4ed39d9450f827e035b841d5a8626fbdbc747601015b7cb760148514ffb7149a","source_sha256":"9a532fef3a07582dfc854d1c03bc13ebe1db0f3d339ef6b3416a8baedc6e0ea2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [trop_add_def, trop_eq_iff_eq_untrop]\n  simp [min_eq_iff]","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":8,"token_jaccard":0.066667,"token_length_ratio":0.727273},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"ba33d24dc0116f27d944d1e8497aec7bfb137f3caf6d1693c47877b5e98f2f98","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.AddGroupWithTop\npublic import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax\npublic import Mathlib.Algebra.Order.Monoid.Unbundled.Pow\npublic import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Order.Hom.Basic\n\nNamespace:\nTropical\n\nLocal context:\n/-\nCopyright (c) 2021 Yakov Pechersky. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yakov Pechersky\n-/\n/-!\n\n# Tropical algebraic structures\n\nThis file defines algebraic structures of the (min-)tropical numbers, up to the tropical semiring.\nSome basic lemmas about conversion from the base type `R` to `Tropical R` are provided, as\nwell as the expected implementations of tropical addition and tropical multiplication.\n\n## Main declarations\n\n* `Tropical R`: The type synonym of the tropical interpretation of `R`.\n    If `[LinearOrder R]`, then addition on `R` is via `min`.\n* `Semiring (Tropical R)`: A `LinearOrderedAddCommMonoidWithTop R`\n    induces a `Semiring (Tropical R)`. If one solely has `[LinearOrderedAddCommMonoid R]`,\n    then the \"tropicalization of `R`\" would be `Tropical (WithTop R)`.\n\n## Implementation notes\n\nThe tropical structure relies on `Top` and `min`. For the max-tropical numbers, use\n`OrderDual R`.\n\nInspiration was drawn from the implementation of `Additive`/`Multiplicative`/`Opposite`,\nwhere a type synonym is created with some barebones API, and quickly made irreducible.\n\nAlgebraic structures are provided with as few typeclass assumptions as possible, even though\nmost references rely on `Semiring (Tropical R)` for building up the whole theory.\n\n## References followed\n\n* https://arxiv.org/pdf/math/0408099.pdf\n* https://www.mathenjeans.fr/sites/default/files/sujets/tropical_geometry_-_casagrande.pdf\n\n-/\n\n@[expose] public section\n\nassert_not_exists Nat.instMulOneClass\n\nuniverse u v\n\nvariable (R : Type u)\n\n/-- The tropicalization of a type `R`. -/\ndef Tropical : Type u :=\n  R\n\nvariable {R}\n\nnamespace Tropical\n\n/-- Reinterpret `x : R` as an element of `Tropical R`.\nSee `Tropical.tropEquiv` for the equivalence.\n-/\ndef trop : R → Tropical R :=\n  id\n\n/-- Reinterpret `x : Tropical R` as an element of `R`.\nSee `Tropical.tropEquiv` for the equivalence. -/\n@[pp_nodot]\ndef untrop : Tropical R → R :=\n  id\n\ntheorem trop_injective : Function.Injective (trop : R → Tropical R) := fun _ _ => id\n\ntheorem untrop_injective : Function.Injective (untrop : Tropical R → R) := fun _ _ => id\n\n@[simp]\ntheorem trop_inj_iff (x y : R) : trop x = trop y ↔ x = y :=\n  Iff.rfl\n\n@[simp]\ntheorem untrop_inj_iff (x y : Tropical R) : untrop x = untrop y ↔ x = y :=\n  Iff.rfl\n\n@[simp]\ntheorem trop_untrop (x : Tropical R) : trop (untrop x) = x :=\n  rfl\n\n@[simp]\ntheorem untrop_trop (x : R) : untrop (trop x) = x :=\n  rfl\n\nattribute [irreducible] Tropical\n\ntheorem leftInverse_trop : Function.LeftInverse (trop : R → Tropical R) untrop :=\n  trop_untrop\n\ntheorem rightInverse_trop : Function.RightInverse (trop : R → Tropical R) untrop :=\n  untrop_trop\n\n/-- Reinterpret `x : R` as an element of `Tropical R`.\nSee `Tropical.tropOrderIso` for the order-preserving equivalence. -/\ndef tropEquiv : R ≃ Tropical R where\n  toFun := trop\n  invFun := untrop\n  left_inv := untrop_trop\n  right_inv := trop_untrop\n\n@[simp]\ntheorem tropEquiv_coe_fn : (tropEquiv : R → Tropical R) = trop :=\n  rfl\n\n@[simp]\ntheorem tropEquiv_symm_coe_fn : (tropEquiv.symm : Tropical R → R) = untrop :=\n  rfl\n\ntheorem trop_eq_iff_eq_untrop {x : R} {y} : trop x = y ↔ x = untrop y :=\n  tropEquiv.apply_eq_iff_eq_symm_apply\n\ntheorem untrop_eq_iff_eq_trop {x} {y : R} : untrop x = y ↔ x = trop y :=\n  tropEquiv.symm.apply_eq_iff_eq_symm_apply\n\ntheorem injective_trop : Function.Injective (trop : R → Tropical R) :=\n  tropEquiv.injective\n\ntheorem injective_untrop : Function.Injective (untrop : Tropical R → R) :=\n  tropEquiv.symm.injective\n\ntheorem surjective_trop : Function.Surjective (trop : R → Tropical R) :=\n  tropEquiv.surjective\n\ntheorem surjective_untrop : Function.Surjective (untrop : Tropical R → R) :=\n  tropEquiv.symm.surjective\n\ninstance [Inhabited R] : Inhabited (Tropical R) :=\n  ⟨trop default⟩\n\n/-- Recursing on an `x' : Tropical R` is the same as recursing on an `x : R` reinterpreted\nas a term of `Tropical R` via `trop x`. -/\n@[simp]\ndef tropRec {F : Tropical R → Sort v} (h : ∀ X, F (trop X)) : ∀ X, F X := fun X => h (untrop X)\n\ninstance [DecidableEq R] : DecidableEq (Tropical R) := fun _ _ =>\n  decidable_of_iff _ injective_untrop.eq_iff\n\nsection Order\n\ninstance instLETropical [LE R] : LE (Tropical R) where le x y := untrop x ≤ untrop y\n\n@[simp]\ntheorem untrop_le_iff [LE R] {x y : Tropical R} : untrop x ≤ untrop y ↔ x ≤ y :=\n  Iff.rfl\n\ninstance decidableLE [LE R] [DecidableLE R] : DecidableLE (Tropical R) := fun x y =>\n  ‹DecidableLE R› (untrop x) (untrop y)\n\ninstance instLTTropical [LT R] : LT (Tropical R) where lt x y := untrop x < untrop y\n\n@[simp]\ntheorem untrop_lt_iff [LT R] {x y : Tropical R} : untrop x < untrop y ↔ x < y :=\n  Iff.rfl\n\ninstance decidableLT [LT R] [DecidableLT R] : DecidableLT (Tropical R) := fun x y =>\n  ‹DecidableLT R› (untrop x) (untrop y)\n\ninstance instPreorderTropical [Preorder R] : Preorder (Tropical R) :=\n  { instLETropical, instLTTropical with\n    le_refl := fun x => le_refl (untrop x)\n    le_trans := fun _ _ _ h h' => le_trans (α := R) h h'\n    lt_iff_le_not_ge := fun _ _ => lt_iff_le_not_ge (α := R) }\n\n/-- Reinterpret `x : R` as an element of `Tropical R`, preserving the order. -/\ndef tropOrderIso [Preorder R] : R ≃o Tropical R :=\n  { tropEquiv with map_rel_iff' := untrop_le_iff }\n\n@[simp]\ntheorem tropOrderIso_coe_fn [Preorder R] : (tropOrderIso : R → Tropical R) = trop :=\n  rfl\n\n@[simp]\ntheorem tropOrderIso_symm_coe_fn [Preorder R] : (tropOrderIso.symm : Tropical R → R) = untrop :=\n  rfl\n\ntheorem trop_monotone [Preorder R] : Monotone (trop : R → Tropical R) := fun _ _ => id\n\ntheorem untrop_monotone [Preorder R] : Monotone (untrop : Tropical R → R) := fun _ _ => id\n\ninstance instPartialOrderTropical [PartialOrder R] : PartialOrder (Tropical R) :=\n  { instPreorderTropical with le_antisymm := fun _ _ h h' => untrop_injective (le_antisymm h h') }\n\ninstance instZeroTropical [Top R] : Zero (Tropical R) :=\n  ⟨trop ⊤⟩\n\ninstance instTopTropical [Top R] : Top (Tropical R) :=\n  ⟨0⟩\n\n@[simp]\ntheorem untrop_zero [Top R] : untrop (0 : Tropical R) = ⊤ :=\n  rfl\n\n@[simp]\ntheorem trop_top [Top R] : trop (⊤ : R) = 0 :=\n  rfl\n\n@[simp]\ntheorem trop_coe_ne_zero (x : R) : trop (x : WithTop R) ≠ 0 :=\n  nofun\n\n@[simp]\ntheorem zero_ne_trop_coe (x : R) : 0 ≠ (trop x : Tropical (WithTop R)) :=\n  nofun\n\n@[simp]\ntheorem le_zero [LE R] [OrderTop R] (x : Tropical R) : x ≤ 0 :=\n  le_top (α := R)\n\ninstance [LE R] [OrderTop R] : OrderTop (Tropical R) :=\n  { instTopTropical with le_top := fun _ => le_top (α := R) }\n\nvariable [LinearOrder R]\n\n/-- Tropical addition is the minimum of two underlying elements of `R`. -/\ninstance : Add (Tropical R) :=\n  ⟨fun x y => trop (min (untrop x) (untrop y))⟩\n\ninstance instAddCommSemigroupTropical : AddCommSemigroup (Tropical R) where\n  add_assoc _ _ _ := untrop_injective (min_assoc _ _ _)\n  add_comm _ _ := untrop_injective (min_comm _ _)\n\n@[simp]\ntheorem untrop_add (x y : Tropical R) : untrop (x + y) = min (untrop x) (untrop y) :=\n  rfl\n\n@[simp]\ntheorem trop_min (x y : R) : trop (min x y) = trop x + trop y :=\n  rfl\n\n@[simp]\ntheorem trop_inf (x y : R) : trop (x ⊓ y) = trop x + trop y :=\n  rfl\n\ntheorem trop_add_def (x y : Tropical R) : x + y = trop (min (untrop x) (untrop y)) :=\n  rfl\n\ninstance instLinearOrderTropical : LinearOrder (Tropical R) :=\n  { instPartialOrderTropical with\n    le_total := fun a b => le_total (untrop a) (untrop b)\n    toDecidableLE := Tropical.decidableLE\n    toDecidableEq := Tropical.instDecidableEq\n    toDecidableLT := Tropical.decidableLT\n    max := fun a b => trop (max (untrop a) (untrop b))\n    max_def := fun a b => untrop_injective (by\n      simp only [max_def, untrop_le_iff, untrop_trop]; split_ifs <;> simp)\n    min := (· + ·)\n    min_def := fun a b => untrop_injective (by\n      simp only [untrop_add, min_def, untrop_le_iff]; split_ifs <;> simp) }\n\n@[simp]\ntheorem untrop_sup (x y : Tropical R) : untrop (x ⊔ y) = untrop x ⊔ untrop y :=\n  rfl\n\n@[simp]\ntheorem untrop_max (x y : Tropical R) : untrop (max x y) = max (untrop x) (untrop y) :=\n  rfl\n\n@[simp]\ntheorem min_eq_add : (min : Tropical R → Tropical R → Tropical R) = (· + ·) :=\n  rfl\n\n@[simp]\ntheorem inf_eq_add : ((· ⊓ ·) : Tropical R → Tropical R → Tropical R) = (· + ·) :=\n  rfl\n\ntheorem trop_max_def (x y : Tropical R) : max x y = trop (max (untrop x) (untrop y)) :=\n  rfl\n\ntheorem trop_sup_def (x y : Tropical R) : x ⊔ y = trop (untrop x ⊔ untrop y) :=\n  rfl\n\n@[simp]\ntheorem add_eq_left ⦃x y : Tropical R⦄ (h : x ≤ y) : x + y = x :=\n  untrop_injective (by simpa using h)\n\n@[simp]\ntheorem add_eq_right ⦃x y : Tropical R⦄ (h : y ≤ x) : x + y = y :=\n  untrop_injective (by simpa using h)\n\ntheorem add_eq_left_iff {x y : Tropical R} : x + y = x ↔ x ≤ y := by\n  rw [trop_add_def, trop_eq_iff_eq_untrop, ← untrop_le_iff, min_eq_left_iff]\n\ntheorem add_eq_right_iff {x y : Tropical R} : x + y = y ↔ y ≤ x := by\n  rw [trop_add_def, trop_eq_iff_eq_untrop, ← untrop_le_iff, min_eq_right_iff]\n\ntheorem add_self (x : Tropical R) : x + x = x :=\n  untrop_injective (min_eq_right le_rfl)\n\nTarget:\ntheorem add_eq_iff {x y z : Tropical R} : x + y = z ↔ x = z ∧ x ≤ y ∨ y = z ∧ y ≤ x :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"be90f8317b7681069fbf6dcb01f7547f3e588d84187009adc6ff5af03651bf7c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Tropical","family_id":"add_eq_iff","file_id":"mathlib/Mathlib/Algebra/Tropical/Basic.lean","sample_id":"4ed39d9450f827e035b841d5a8626fbdbc747601015b7cb760148514ffb7149a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e0cdaf3a4b43a4f8fc2adc6c6eee964e632fc1aa70e6d8ebca268efd1191ebba","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3c58404f57528d0f09544fa0d6422c11989243f1a83384b09f5595af7aef1431","source_sha256":"59e5e41190d4901dfcf9839eb63e1593f695e949ec1d2d7f0996228faa2acac0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Presieve.EffectiveEpimorphic.iff_forall_isSheafFor_yoneda] at hR\n  rw [← isSheafFor_comp_uliftFunctor_iff]\n  refine Presieve.isSheafFor_iso (F ⋙ uliftFunctor.{v}).uliftYonedaReprXIso ?_\n  dsimp only [uliftYoneda, Functor.comp_obj, Functor.whiskeringRight_obj_obj]\n  rw [isSheafFor_comp_uliftFunctor_iff]\n  exact hR _","hard_negative":false,"metrics":{"chosen_tokens":53,"rejected_tokens":3,"token_jaccard":0.090909,"token_length_ratio":0.056604},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"ba8621d3d67f773bc682b52493cdd35150c112ea226d68d336d2c9a3e05d07e7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Sites.SheafOfTypes\npublic import Mathlib.CategoryTheory.EffectiveEpi.Basic\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2023 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\n/-!\n\n# Effective epimorphic sieves\n\nWe define the notion of effective epimorphic (pre)sieves and provide some API for relating the\nnotion with the notions of effective epimorphism and effective epimorphic family.\n\nMore precisely, if `f` is a morphism, then `f` is an effective epi if and only if the sieve\nit generates is effective epimorphic; see `CategoryTheory.Sieve.effectiveEpimorphic_singleton`.\nThe analogous statement for a family of morphisms is in the theorem\n`CategoryTheory.Sieve.effectiveEpimorphic_family`.\n\n-/\n\nuniverse w v u\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Limits\n\nvariable {C : Type u} [Category.{v} C]\n\n/-- A sieve is effective epimorphic if the associated cocone is a colimit cocone. -/\ndef Sieve.EffectiveEpimorphic {X : C} (S : Sieve X) : Prop :=\n  Nonempty (IsColimit (S : Presieve X).cocone)\n\n/-- A presieve is effective epimorphic if the cocone associated to the sieve it generates\nis a colimit cocone. -/\nabbrev Presieve.EffectiveEpimorphic {X : C} (S : Presieve X) : Prop :=\n  (Sieve.generate S).EffectiveEpimorphic\n\n/--\nThe sieve of morphisms which factor through a given morphism `f`.\nThis is equal to `Sieve.generate (Presieve.singleton f)`, but has\nmore convenient definitional properties.\n-/\ndef Sieve.generateSingleton {X Y : C} (f : Y ⟶ X) : Sieve X where\n  arrows Z g := ∃ (e : Z ⟶ Y), e ≫ f = g\n  downward_closed := by\n    rintro W Z g ⟨e, rfl⟩ q\n    exact ⟨q ≫ e, by simp⟩\n\nlemma Sieve.generateSingleton_eq {X Y : C} (f : Y ⟶ X) :\n    Sieve.generate (Presieve.singleton f) = Sieve.generateSingleton f := by\n  ext Z g\n  constructor\n  · rintro ⟨W, i, p, ⟨⟩, rfl⟩\n    exact ⟨i, rfl⟩\n  · rintro ⟨g, h⟩\n    exact ⟨Y, g, f, ⟨⟩, h⟩\n\nlemma Sieve.EffectiveEpimorphic.iff_forall_isSheafFor_yoneda {X : C} (S : Sieve X) :\n    S.EffectiveEpimorphic ↔ ∀ Y, S.arrows.IsSheafFor (yoneda.obj Y) :=\n  S.forallYonedaIsSheaf_iff_colimit.symm\n\nlemma Presieve.EffectiveEpimorphic.iff_forall_isSheafFor_yoneda {X : C} (R : Presieve X) :\n    R.EffectiveEpimorphic ↔ ∀ Y, R.IsSheafFor (yoneda.obj Y) := by\n  simp_rw [Presieve.isSheafFor_iff_generate R,\n    Presieve.EffectiveEpimorphic, Sieve.EffectiveEpimorphic.iff_forall_isSheafFor_yoneda]\n\nset_option backward.defeqAttrib.useBackward true in\n\nTarget:\nlemma Presieve.EffectiveEpimorphic.isSheafFor_of_isRepresentable {X : C} {R : Presieve X}\n    (hR : R.EffectiveEpimorphic) (F : Cᵒᵖ ⥤ Type w) [F.IsRepresentable] :\n    R.IsSheafFor F :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Sites","family_id":"presieve","file_id":"mathlib/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean","sample_id":"3c58404f57528d0f09544fa0d6422c11989243f1a83384b09f5595af7aef1431"},"split":"train"}
{"binding":{"chosen_proof_sha256":"94092e3c9ae7aafc8bfd4b060b03f1f7bf7ca38bd35810157e3b4c1cdb58864b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bd9359a574dcaca31acf15cd4ed4c49c4962b2ea1c2de798364f401e03500718","source_sha256":"5b5f327a6479776df18ad9f770e834a46924f37fc00aca2b6d876cc1986c4651","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext1\n  apply map_C\n\n-- mixing the two monad structures","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":8,"token_jaccard":0.0625,"token_length_ratio":0.727273},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"ba8c7f42ae5f0b2e50502f2385a7de1e35a2b25cba009ee491e5fd8a02c6676f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.Rename\npublic import Mathlib.Algebra.MvPolynomial.Variables\n\nNamespace:\nMvPolynomial\n\nLocal context:\n/-\nCopyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Robert Y. Lewis\n-/\n/-!\n\n# Monad operations on `MvPolynomial`\n\nThis file defines two monadic operations on `MvPolynomial`. Given `p : MvPolynomial σ R`,\n\n* `MvPolynomial.bind₁` and `MvPolynomial.join₁` operate on the variable type `σ`.\n* `MvPolynomial.bind₂` and `MvPolynomial.join₂` operate on the coefficient type `R`.\n\n- `MvPolynomial.bind₁ f φ` with `f : σ → MvPolynomial τ R` and `φ : MvPolynomial σ R`,\n  is the polynomial `φ(f 1, ..., f i, ...) : MvPolynomial τ R`.\n- `MvPolynomial.join₁ φ` with `φ : MvPolynomial (MvPolynomial σ R) R` collapses `φ` to\n  a `MvPolynomial σ R`, by evaluating `φ` under the map `X f ↦ f` for `f : MvPolynomial σ R`.\n  In other words, if you have a polynomial `φ` in a set of variables indexed by a polynomial ring,\n  you evaluate the polynomial in these indexing polynomials.\n- `MvPolynomial.bind₂ f φ` with `f : R →+* MvPolynomial σ S` and `φ : MvPolynomial σ R`\n  is the `MvPolynomial σ S` obtained from `φ` by mapping the coefficients of `φ` through `f`\n  and considering the resulting polynomial as polynomial expression in `MvPolynomial σ R`.\n- `MvPolynomial.join₂ φ` with `φ : MvPolynomial σ (MvPolynomial σ R)` collapses `φ` to\n  a `MvPolynomial σ R`, by considering `φ` as polynomial expression in `MvPolynomial σ R`.\n\nThese operations themselves have algebraic structure: `MvPolynomial.bind₁`\nand `MvPolynomial.join₁` are algebra homs and\n`MvPolynomial.bind₂` and `MvPolynomial.join₂` are ring homs.\n\nThey interact in convenient ways with `MvPolynomial.rename`, `MvPolynomial.map`,\n`MvPolynomial.vars`, and other polynomial operations.\nIndeed, `MvPolynomial.rename` is the \"map\" operation for the (`bind₁`, `join₁`) pair,\nwhereas `MvPolynomial.map` is the \"map\" operation for the other pair.\n\n## Implementation notes\n\nWe add a `LawfulMonad` instance for the (`bind₁`, `join₁`) pair.\nThe second pair cannot be instantiated as a `Monad`,\nsince it is not a monad in `Type` but in `CommRingCat` (or rather `CommSemiRingCat`).\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nnamespace MvPolynomial\n\nopen Finsupp\n\nvariable {σ : Type*} {τ : Type*}\nvariable {R S T : Type*} [CommSemiring R] [CommSemiring S] [CommSemiring T]\n\n/--\n`bind₁` is the \"left-hand side\" bind operation on `MvPolynomial`, operating on the variable type.\nGiven a polynomial `p : MvPolynomial σ R` and a map `f : σ → MvPolynomial τ R` taking variables\nin `p` to polynomials in the variable type `τ`, `bind₁ f p` replaces each variable in `p` with\nits value under `f`, producing a new polynomial in `τ`. The coefficient type remains the same.\nThis operation is an algebra hom.\n-/\ndef bind₁ (f : σ → MvPolynomial τ R) : MvPolynomial σ R →ₐ[R] MvPolynomial τ R :=\n  aeval f\n\n/-- `bind₂` is the \"right-hand side\" bind operation on `MvPolynomial`,\noperating on the coefficient type.\nGiven a polynomial `p : MvPolynomial σ R` and\na map `f : R → MvPolynomial σ S` taking coefficients in `p` to polynomials over a new ring `S`,\n`bind₂ f p` replaces each coefficient in `p` with its value under `f`,\nproducing a new polynomial over `S`.\nThe variable type remains the same. This operation is a ring hom.\n-/\ndef bind₂ (f : R →+* MvPolynomial σ S) : MvPolynomial σ R →+* MvPolynomial σ S :=\n  eval₂Hom f X\n\n/--\n`join₁` is the monadic join operation corresponding to `MvPolynomial.bind₁`. Given a polynomial `p`\nwith coefficients in `R` whose variables are polynomials in `σ` with coefficients in `R`,\n`join₁ p` collapses `p` to a polynomial with variables in `σ` and coefficients in `R`.\nThis operation is an algebra hom.\n-/\ndef join₁ : MvPolynomial (MvPolynomial σ R) R →ₐ[R] MvPolynomial σ R :=\n  aeval id\n\n/--\n`join₂` is the monadic join operation corresponding to `MvPolynomial.bind₂`. Given a polynomial `p`\nwith variables in `σ` whose coefficients are polynomials in `σ` with coefficients in `R`,\n`join₂ p` collapses `p` to a polynomial with variables in `σ` and coefficients in `R`.\nThis operation is a ring hom.\n-/\ndef join₂ : MvPolynomial σ (MvPolynomial σ R) →+* MvPolynomial σ R :=\n  eval₂Hom (RingHom.id _) X\n\n@[simp]\ntheorem aeval_eq_bind₁ (f : σ → MvPolynomial τ R) : aeval f = bind₁ f :=\n  rfl\n\n@[simp]\ntheorem eval₂Hom_C_eq_bind₁ (f : σ → MvPolynomial τ R) : eval₂Hom C f = bind₁ f :=\n  rfl\n\n@[simp]\ntheorem eval₂Hom_eq_bind₂ (f : R →+* MvPolynomial σ S) : eval₂Hom f X = bind₂ f :=\n  rfl\n\nsection\n\nvariable (σ R)\n\n@[simp]\ntheorem aeval_id_eq_join₁ : aeval id = @join₁ σ R _ :=\n  rfl\n\ntheorem eval₂Hom_C_id_eq_join₁ (φ : MvPolynomial (MvPolynomial σ R) R) :\n    eval₂Hom C id φ = join₁ φ :=\n  rfl\n\n@[simp]\ntheorem eval₂Hom_id_X_eq_join₂ : eval₂Hom (RingHom.id _) X = @join₂ σ R _ :=\n  rfl\n\nend\n\n-- In this file, we don't want to use these simp lemmas,\n-- because we first need to show how these new definitions interact\n-- and the proofs fall back on unfolding the definitions and call simp afterwards\nattribute [-simp]\n  aeval_eq_bind₁ eval₂Hom_C_eq_bind₁ eval₂Hom_eq_bind₂ aeval_id_eq_join₁ eval₂Hom_id_X_eq_join₂\n\n@[simp]\ntheorem bind₁_X_right (f : σ → MvPolynomial τ R) (i : σ) : bind₁ f (X i) = f i :=\n  aeval_X f i\n\n@[simp]\ntheorem bind₂_X_right (f : R →+* MvPolynomial σ S) (i : σ) : bind₂ f (X i) = X i :=\n  eval₂Hom_X' f X i\n\n@[simp]\ntheorem bind₁_X_left : bind₁ (X : σ → MvPolynomial σ R) = AlgHom.id R _ := by\n  ext1 i\n  simp\n\nvariable (f : σ → MvPolynomial τ R)\n\ntheorem bind₁_C_right (f : σ → MvPolynomial τ R) (x) : bind₁ f (C x) = C x := algHom_C _ _\n\n@[simp]\ntheorem bind₂_C_right (f : R →+* MvPolynomial σ S) (r : R) : bind₂ f (C r) = f r :=\n  eval₂Hom_C f X r\n\n@[simp]\ntheorem bind₂_C_left : bind₂ (C : R →+* MvPolynomial σ R) = RingHom.id _ := by ext : 2 <;> simp\n\n@[simp]\ntheorem bind₂_comp_C (f : R →+* MvPolynomial σ S) : (bind₂ f).comp C = f :=\n  RingHom.ext <| bind₂_C_right _\n\n@[simp]\ntheorem join₂_map (f : R →+* MvPolynomial σ S) (φ : MvPolynomial σ R) :\n    join₂ (map f φ) = bind₂ f φ := by simp only [join₂, bind₂, eval₂Hom_map_hom, RingHom.id_comp]\n\n@[simp]\ntheorem join₂_comp_map (f : R →+* MvPolynomial σ S) : join₂.comp (map f) = bind₂ f :=\n  RingHom.ext <| join₂_map _\n\ntheorem aeval_id_rename (f : σ → MvPolynomial τ R) (p : MvPolynomial σ R) :\n    aeval id (rename f p) = aeval f p := by rw [aeval_rename, Function.id_comp]\n\n@[simp]\ntheorem join₁_rename (f : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) :\n    join₁ (rename f φ) = bind₁ f φ :=\n  aeval_id_rename _ _\n\n@[simp]\ntheorem bind₁_id : bind₁ (@id (MvPolynomial σ R)) = join₁ :=\n  rfl\n\n@[simp]\ntheorem bind₂_id : bind₂ (RingHom.id (MvPolynomial σ R)) = join₂ :=\n  rfl\n\ntheorem bind₁_bind₁ {υ : Type*} (f : σ → MvPolynomial τ R) (g : τ → MvPolynomial υ R)\n    (φ : MvPolynomial σ R) : (bind₁ g) (bind₁ f φ) = bind₁ (fun i => bind₁ g (f i)) φ := by\n  simp [bind₁, ← comp_aeval]\n\ntheorem bind₁_comp_bind₁ {υ : Type*} (f : σ → MvPolynomial τ R) (g : τ → MvPolynomial υ R) :\n    (bind₁ g).comp (bind₁ f) = bind₁ fun i => bind₁ g (f i) := by\n  ext1\n  apply bind₁_bind₁\n\ntheorem bind₂_comp_bind₂ (f : R →+* MvPolynomial σ S) (g : S →+* MvPolynomial σ T) :\n    (bind₂ g).comp (bind₂ f) = bind₂ ((bind₂ g).comp f) := by ext : 2 <;> simp\n\ntheorem bind₂_bind₂ (f : R →+* MvPolynomial σ S) (g : S →+* MvPolynomial σ T)\n    (φ : MvPolynomial σ R) : (bind₂ g) (bind₂ f φ) = bind₂ ((bind₂ g).comp f) φ :=\n  RingHom.congr_fun (bind₂_comp_bind₂ f g) φ\n\ntheorem rename_comp_bind₁ {υ : Type*} (f : σ → MvPolynomial τ R) (g : τ → υ) :\n    (rename g).comp (bind₁ f) = bind₁ fun i => rename g <| f i := by\n  ext1 i\n  simp\n\ntheorem rename_bind₁ {υ : Type*} (f : σ → MvPolynomial τ R) (g : τ → υ) (φ : MvPolynomial σ R) :\n    rename g (bind₁ f φ) = bind₁ (fun i => rename g <| f i) φ :=\n  AlgHom.congr_fun (rename_comp_bind₁ f g) φ\n\ntheorem map_bind₂ (f : R →+* MvPolynomial σ S) (g : S →+* T) (φ : MvPolynomial σ R) :\n    map g (bind₂ f φ) = bind₂ ((map g).comp f) φ := by\n  simp only [bind₂, eval₂_comp_right, coe_eval₂Hom, eval₂_map]\n  congr 1 with : 1\n  simp only [Function.comp_apply, map_X]\n\ntheorem bind₁_comp_rename {υ : Type*} (f : τ → MvPolynomial υ R) (g : σ → τ) :\n    (bind₁ f).comp (rename g) = bind₁ (f ∘ g) := by\n  ext1 i\n  simp\n\ntheorem bind₁_rename {υ : Type*} (f : τ → MvPolynomial υ R) (g : σ → τ) (φ : MvPolynomial σ R) :\n    bind₁ f (rename g φ) = bind₁ (f ∘ g) φ :=\n  AlgHom.congr_fun (bind₁_comp_rename f g) φ\n\ntheorem bind₂_map (f : S →+* MvPolynomial σ T) (g : R →+* S) (φ : MvPolynomial σ R) :\n    bind₂ f (map g φ) = bind₂ (f.comp g) φ := by simp [bind₂]\n\n@[simp]\n\nTarget:\ntheorem map_comp_C (f : R →+* S) : (map f).comp (C : R →+* MvPolynomial σ R) = C.comp f :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"a973c13228eede58ab54b527611548b7f5b3fa375abd09aaa93bb492d70fa0ff","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/MvPolynomial","family_id":"map_comp_c","file_id":"mathlib/Mathlib/Algebra/MvPolynomial/Monad.lean","sample_id":"bd9359a574dcaca31acf15cd4ed4c49c4962b2ea1c2de798364f401e03500718"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2e876053dd524313a985f03953e4b880c7de87fe925bf70214a79832334fbbdd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"fc2d6d9e9f6a1af7e7b952f5f1068ccb99e235fcd9e7771fd11ac13083136f6e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3b9738526fb66d231f0474bd9878a45063a3e22be465efe04b4fbbe9d62f823a","source_sha256":"3c917efc6600c0893eb5c055755a108b9a20ae48ebe963f7f5db8b06582c587d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨F, hF⟩ := hc.exists_eq_range (hn.image _)\n  simp only [Set.ext_iff, Quotient.forall, mem_image, mem_range] at hF\n  simp_rw [Quotient.eq_mk_iff_out] at hF\n  have hF' : ∀ n : ℕ, (F n).out ∈ K := by\n    intro n\n    obtain ⟨P, hP1, hP2⟩ := (hF (F n).out).2 ⟨n, Setoid.refl _⟩\n    -- Porting note: fix hP2 because `Quotient.out (Quotient.mk' x) ≈ a` was not simplified\n    -- to `x ≈ a` in hF\n    replace hP2 := Setoid.trans (Setoid.symm (Quotient.mk_out P)) hP2\n    exact (hp.is_equiv_invariant_of_fg fg _ _ hP2).1 hP1\n  choose P hPK hP hFP using fun (N : K) (n : ℕ) => jep N N.2 (F (n + 1)).out (hF' _)\n  let G : ℕ → K := @Nat.rec (fun _ => K) ⟨(F 0).out, hF' 0⟩ fun n N => ⟨P N n, hPK N n⟩\n  let f : ∀ (i j : ℕ), i ≤ j → (G i).val ↪[L] (G j).val :=\n    DirectedSystem.natLERec fun n => (hP _ n).some\n  refine ⟨Bundled.of (@DirectLimit L _ _ (fun n ↦ (G n).val) _ f _ _), ?_, ?_⟩\n  · exact DirectLimit.cg _ (fun n => (fg _ (G n).2).cg)\n  · refine (age_directLimit (fun n ↦ (G n).val) f).trans\n      (subset_antisymm (iUnion_subset fun n N hN => hp (G n).val (G n).2 hN) fun N KN => ?_)\n    have : Quotient.out (Quotient.mk' N) ≈ N := Quotient.eq_mk_iff_out.mp rfl\n    obtain ⟨n, ⟨e⟩⟩ := (hF N).1 ⟨N, KN, this⟩\n    refine mem_iUnion_of_mem n ⟨fg _ KN, ⟨Embedding.comp ?_ e.symm.toEmbedding⟩⟩\n    rcases n with - | n\n    · dsimp [G]; exact Embedding.refl _ _\n    · dsimp [G]; exact (hFP _ n).some","hard_negative":true,"metrics":{"chosen_tokens":480,"rejected_tokens":2,"token_jaccard":0.00813,"token_length_ratio":0.004167},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"babbafd82167d7331ee873a548c626c87490edf0232c96aced02e84e9ec17d3e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.ModelTheory.FinitelyGenerated\npublic import Mathlib.ModelTheory.PartialEquiv\npublic import Mathlib.ModelTheory.Bundled\npublic import Mathlib.Algebra.Order.Archimedean.Basic\n\nNamespace:\nFirstOrder.Language\n\nLocal context:\n/-\nCopyright (c) 2022 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson, Gabin Kolly\n-/\n/-!\n# Fraïssé Classes and Fraïssé Limits\n\nThis file pertains to the ages of countable first-order structures. The age of a structure is the\nclass of all finitely-generated structures that embed into it.\n\nOf particular interest are Fraïssé classes, which are exactly the ages of countable\nultrahomogeneous structures. To each is associated a unique (up to nonunique isomorphism)\nFraïssé limit - the countable ultrahomogeneous structure with that age.\n\n## Main Definitions\n\n- `FirstOrder.Language.age` is the class of finitely-generated structures that embed into a\n  particular structure.\n- A class `K` is `FirstOrder.Language.Hereditary` when all finitely-generated\n  structures that embed into structures in `K` are also in `K`.\n- A class `K` has `FirstOrder.Language.JointEmbedding` when for every `M`, `N` in\n  `K`, there is another structure in `K` into which both `M` and `N` embed.\n- A class `K` has `FirstOrder.Language.Amalgamation` when for any pair of embeddings\n  of a structure `M` in `K` into other structures in `K`, those two structures can be embedded into\n  a fourth structure in `K` such that the resulting square of embeddings commutes.\n- `FirstOrder.Language.IsFraisse` indicates that a class is nonempty, essentially countable,\n  and satisfies the hereditary, joint embedding, and amalgamation properties.\n- `FirstOrder.Language.IsFraisseLimit` indicates that a structure is a Fraïssé limit for a given\n  class.\n\n## Main Results\n\n- We show that the age of any structure is isomorphism-invariant and satisfies the hereditary and\n  joint-embedding properties.\n- `FirstOrder.Language.age.countable_quotient` shows that the age of any countable structure is\n  essentially countable.\n- `FirstOrder.Language.exists_countable_is_age_of_iff` gives necessary and sufficient conditions\n  for a class to be the age of a countable structure in a language with countably many functions.\n- `FirstOrder.Language.IsFraisseLimit.nonempty_equiv` shows that any class which is Fraïssé has\n  at most one Fraïssé limit up to equivalence.\n- `FirstOrder.Language.empty.isFraisseLimit_of_countable_infinite` shows that any countably infinite\n  structure in the empty language is a Fraïssé limit of the class of finite structures.\n- `FirstOrder.Language.empty.isFraisse_finite` shows that the class of finite structures in the\n  empty language is Fraïssé.\n\n## Implementation Notes\n\n- Classes of structures are formalized with `Set (Bundled L.Structure)`.\n- Some results pertain to countable limit structures, others to countably-generated limit\n  structures. In the case of a language with countably many function symbols, these are equivalent.\n\n## References\n\n- [W. Hodges, *A Shorter Model Theory*][Hodges97]\n- [K. Tent, M. Ziegler, *A Course in Model Theory*][Tent_Ziegler]\n\n## TODO\n\n- Show existence of Fraïssé limits\n\n-/\n\n@[expose] public section\n\n\nuniverse u v w w'\n\nopen scoped FirstOrder\n\nopen Set CategoryTheory\n\nnamespace FirstOrder\n\nnamespace Language\n\nopen Structure Substructure\n\nvariable (L : Language.{u, v})\n\n/-! ### The Age of a Structure and Fraïssé Classes -/\n\n\n/-- The age of a structure `M` is the class of finitely-generated structures that embed into it. -/\ndef age (M : Type w) [L.Structure M] : Set (Bundled.{w} L.Structure) :=\n  {N | Structure.FG L N ∧ Nonempty (N ↪[L] M)}\n\nvariable {L}\nvariable (K : Set (Bundled.{w} L.Structure))\n\n/-- A class `K` has the hereditary property when all finitely-generated structures that embed into\n  structures in `K` are also in `K`. -/\ndef Hereditary : Prop :=\n  ∀ M : Bundled.{w} L.Structure, M ∈ K → L.age M ⊆ K\n\n/-- A class `K` has the joint embedding property when for every `M`, `N` in `K`, there is another\n  structure in `K` into which both `M` and `N` embed. -/\ndef JointEmbedding : Prop :=\n  DirectedOn (fun M N : Bundled.{w} L.Structure => Nonempty (M ↪[L] N)) K\n\n/-- A class `K` has the amalgamation property when for any pair of embeddings of a structure `M` in\n  `K` into other structures in `K`, those two structures can be embedded into a fourth structure in\n  `K` such that the resulting square of embeddings commutes. -/\ndef Amalgamation : Prop :=\n  ∀ (M N P : Bundled.{w} L.Structure) (MN : M ↪[L] N) (MP : M ↪[L] P),\n    M ∈ K → N ∈ K → P ∈ K → ∃ (Q : Bundled.{w} L.Structure) (NQ : N ↪[L] Q) (PQ : P ↪[L] Q),\n      Q ∈ K ∧ NQ.comp MN = PQ.comp MP\n\n/-- A Fraïssé class is a nonempty, essentially countable class of structures satisfying the\nhereditary, joint embedding, and amalgamation properties. -/\nclass IsFraisse : Prop where\n  is_nonempty : K.Nonempty\n  FG : ∀ M : Bundled.{w} L.Structure, M ∈ K → Structure.FG L M\n  is_essentially_countable : (Quotient.mk' '' K).Countable\n  hereditary : Hereditary K\n  jointEmbedding : JointEmbedding K\n  amalgamation : Amalgamation K\n\nvariable {K} (L) (M : Type w) [Structure L M]\n\ntheorem age.is_equiv_invariant (N P : Bundled.{w} L.Structure) (h : Nonempty (N ≃[L] P)) :\n    N ∈ L.age M ↔ P ∈ L.age M :=\n  and_congr h.some.fg_iff\n    ⟨Nonempty.map fun x => Embedding.comp x h.some.symm.toEmbedding,\n      Nonempty.map fun x => Embedding.comp x h.some.toEmbedding⟩\n\nvariable {L} {M} {N : Type w} [Structure L N]\n\ntheorem Embedding.age_subset_age (MN : M ↪[L] N) : L.age M ⊆ L.age N := fun _ =>\n  And.imp_right (Nonempty.map MN.comp)\n\ntheorem Equiv.age_eq_age (MN : M ≃[L] N) : L.age M = L.age N :=\n  le_antisymm MN.toEmbedding.age_subset_age MN.symm.toEmbedding.age_subset_age\n\ntheorem Structure.FG.mem_age_of_equiv {M N : Bundled L.Structure} (h : Structure.FG L M)\n    (MN : Nonempty (M ≃[L] N)) : N ∈ L.age M :=\n  ⟨MN.some.fg_iff.1 h, ⟨MN.some.symm.toEmbedding⟩⟩\n\ntheorem Hereditary.is_equiv_invariant_of_fg (h : Hereditary K)\n    (fg : ∀ M : Bundled.{w} L.Structure, M ∈ K → Structure.FG L M) (M N : Bundled.{w} L.Structure)\n    (hn : Nonempty (M ≃[L] N)) : M ∈ K ↔ N ∈ K :=\n  ⟨fun MK => h M MK ((fg M MK).mem_age_of_equiv hn),\n   fun NK => h N NK ((fg N NK).mem_age_of_equiv ⟨hn.some.symm⟩)⟩\n\ntheorem IsFraisse.is_equiv_invariant [h : IsFraisse K] {M N : Bundled.{w} L.Structure}\n    (hn : Nonempty (M ≃[L] N)) : M ∈ K ↔ N ∈ K :=\n  h.hereditary.is_equiv_invariant_of_fg h.FG M N hn\n\nvariable (M)\n\ntheorem age.nonempty : (L.age M).Nonempty :=\n  ⟨Bundled.of (Substructure.closure L (∅ : Set M)),\n    (fg_iff_structure_fg _).1 (fg_closure Set.finite_empty), ⟨Substructure.subtype _⟩⟩\n\ntheorem age.hereditary : Hereditary (L.age M) := fun _ hN _ hP => hN.2.some.age_subset_age hP\n\ntheorem age.jointEmbedding : JointEmbedding (L.age M) := fun _ hN _ hP =>\n  ⟨Bundled.of (↥(hN.2.some.toHom.range ⊔ hP.2.some.toHom.range)),\n    ⟨(fg_iff_structure_fg _).1 ((hN.1.range hN.2.some.toHom).sup (hP.1.range hP.2.some.toHom)),\n      ⟨Substructure.subtype _⟩⟩,\n    ⟨Embedding.comp (inclusion le_sup_left) hN.2.some.equivRange.toEmbedding⟩,\n    ⟨Embedding.comp (inclusion le_sup_right) hP.2.some.equivRange.toEmbedding⟩⟩\n\nvariable {M} in\ntheorem age.fg_substructure {S : L.Substructure M} (fg : S.FG) : Bundled.mk S ∈ L.age M := by\n  exact ⟨(Substructure.fg_iff_structure_fg _).1 fg, ⟨subtype _⟩⟩\n\n/-- Any class in the age of a structure has a representative which is a finitely generated\nsubstructure. -/\ntheorem age.has_representative_as_substructure :\n    ∀ C ∈ Quotient.mk' '' L.age M, ∃ V : {V : L.Substructure M // FG V},\n      ⟦Bundled.mk V⟧ = C := by\n  rintro _ ⟨N, ⟨N_fg, ⟨N_incl⟩⟩, N_eq⟩\n  refine N_eq.symm ▸ ⟨⟨N_incl.toHom.range, ?_⟩, Quotient.sound ⟨N_incl.equivRange.symm⟩⟩\n  exact FG.range N_fg (Embedding.toHom N_incl)\n\n/-- The age of a countable structure is essentially countable (has countably many isomorphism\nclasses). -/\ntheorem age.countable_quotient [h : Countable M] : (Quotient.mk' '' L.age M).Countable := by\n  classical\n  refine (congr_arg _ (Set.ext <| Quotient.forall.2 fun N => ?_)).mp\n    (countable_range fun s : Finset M => ⟦⟨closure L (s : Set M), inferInstance⟩⟧)\n  constructor\n  · rintro ⟨s, hs⟩\n    use Bundled.of (closure L (s : Set M))\n    exact ⟨⟨(fg_iff_structure_fg _).1 (fg_closure s.finite_toSet), ⟨Substructure.subtype _⟩⟩, hs⟩\n  · simp only [mem_range, Quotient.eq]\n    rintro ⟨P, ⟨⟨s, hs⟩, ⟨PM⟩⟩, hP2⟩\n    refine ⟨s.image PM, Setoid.trans (b := P) ?_ <| Quotient.exact hP2⟩\n    rw [← Embedding.coe_toHom, Finset.coe_image, closure_image PM.toHom, hs, ← Hom.range_eq_map]\n    exact ⟨PM.equivRange.symm⟩\n\nset_option backward.isDefEq.respectTransparency false in\n-- This is not a simp-lemma because it does not apply to itself.\n/-- The age of a direct limit of structures is the union of the ages of the structures. -/\ntheorem age_directLimit {ι : Type w} [Preorder ι] [IsDirectedOrder ι] [Nonempty ι]\n    (G : ι → Type max w w') [∀ i, L.Structure (G i)] (f : ∀ i j, i ≤ j → G i ↪[L] G j)\n    [DirectedSystem G fun i j h => f i j h] : L.age (DirectLimit G f) = ⋃ i : ι, L.age (G i) := by\n  classical\n  ext M\n  simp only [mem_iUnion]\n  constructor\n  · rintro ⟨Mfg, ⟨e⟩⟩\n    obtain ⟨s, hs⟩ := Mfg.range e.toHom\n    let out := @Quotient.out _ (DirectLimit.setoid G f)\n    obtain ⟨i, hi⟩ := Finset.exists_le (s.image (Sigma.fst ∘ out))\n    have e' := (DirectLimit.of L ι G f i).equivRange.symm.toEmbedding\n    refine ⟨i, Mfg, ⟨e'.comp ((Substructure.inclusion ?_).comp e.equivRange.toEmbedding)⟩⟩\n    rw [← hs, closure_le]\n    intro x hx\n    refine ⟨f (out x).1 i (hi (out x).1 (Finset.mem_image_of_mem _ hx)) (out x).2, ?_⟩\n    rw [Embedding.coe_toHom, DirectLimit.of_apply, @Quotient.mk_eq_iff_out _ (_),\n      DirectLimit.equiv_iff G f (le_refl _) (hi (out x).1 (Finset.mem_image_of_mem _ hx)),\n      DirectedSystem.map_self]\n  · rintro ⟨i, Mfg, ⟨e⟩⟩\n    exact ⟨Mfg, ⟨Embedding.comp (DirectLimit.of L ι G f i) e⟩⟩\n\n/-- Sufficient conditions for a class to be the age of a countably-generated structure. -/\n\nTarget:\ntheorem exists_cg_is_age_of (hn : K.Nonempty)\n    (hc : (Quotient.mk' '' K).Countable)\n    (fg : ∀ M : Bundled.{w} L.Structure, M ∈ K → Structure.FG L M) (hp : Hereditary K)\n    (jep : JointEmbedding K) : ∃ M : Bundled.{w} L.Structure, Structure.CG L M ∧ L.age M = K :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_3b9738526fb6","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"186804f81a80cd23fad2ab389897abc2b41bd66a6b84eebbd01e80703c0fe2e6","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"ModelTheory","family_id":"exists_cg_is_age_of","file_id":"mathlib/Mathlib/ModelTheory/Fraisse.lean","sample_id":"3b9738526fb66d231f0474bd9878a45063a3e22be465efe04b4fbbe9d62f823a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"faf6bc027aba61a3300a23f2d313e28567ba4f3a90cdcf2560518c60032b7b01","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6ae87415fab9e244b6175127e9ef4c306de7de201cffbebb4a8e68f34e4df20b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bb2be5794bc2bef47d57ee2509f8e94d0f35879b91907a617a92bba8e4b1af69","source_sha256":"b6f10e56b5f036623bcf6641814f24797b2faa600c67007e65fd922ddca8f380","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  let ι := Module.Free.ChooseBasisIndex R S\n  let b : Module.Basis ι R S := Module.Free.chooseBasis R S\n  rw [trace_eq_matrix_trace b, trace_eq_matrix_trace (basisQuotient b), AddMonoidHom.map_trace]\n  congr 1\n  ext i j\n  simp only [leftMulMatrix_apply, coe_lmul_eq_mul, LinearMap.toMatrix_apply,\n    basisQuotient_apply, LinearMap.mul_apply', Matrix.map_apply, ← map_mul,\n    basisQuotient_repr]","hard_negative":true,"metrics":{"chosen_tokens":75,"rejected_tokens":3,"token_jaccard":0.022222,"token_length_ratio":0.04},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"baf5be089a505559b27c1b7048d7d84f877d57647bd2a87ab88a7e598c8b726d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.Dvr\npublic import Mathlib.RingTheory.IntegralClosure.IntegralRestrict\npublic import Mathlib.RingTheory.LocalRing.Quotient\npublic import Mathlib.RingTheory.Localization.AtPrime.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Riccardo Brasca. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang, Riccardo Brasca\n-/\n/-!\n\nWe gather results about the relations between the trace map on `B → A` and the trace map on\nquotients and localizations.\n\n## Main Results\n\n* `Algebra.trace_quotient_eq_of_isDedekindDomain` : The trace map on `B → A` coincides with the\n  trace map on `B⧸pB → A⧸p`.\n\n-/\n\npublic section\n\nvariable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]\n\nopen IsLocalRing FiniteDimensional Module Submodule IsLocalization.AtPrime\n\nsection IsLocalRing\n\nlocal notation \"p\" => maximalIdeal R\nlocal notation \"pS\" => Ideal.map (algebraMap R S) p\n\nvariable [Module.Free R S] [Module.Finite R S]\n\nattribute [local instance] Ideal.Quotient.field\n\nTarget:\nlemma Algebra.trace_quotient_mk [IsLocalRing R] (x : S) :\n    Algebra.trace (R ⧸ p) (S ⧸ pS) (Ideal.Quotient.mk pS x) =\n      Ideal.Quotient.mk p (Algebra.trace R S x) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_bb2be5794bc2","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"4643ca815847487ea26f9ea8a49e064039017395f5b7511ba2d77bbddd81251a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Trace","family_id":"algebra","file_id":"mathlib/Mathlib/RingTheory/Trace/Quotient.lean","sample_id":"bb2be5794bc2bef47d57ee2509f8e94d0f35879b91907a617a92bba8e4b1af69"},"split":"train"}
{"binding":{"chosen_proof_sha256":"077e58b8ebfdb691115134cf203f73b4ed07755ad550ac73f5ae401c07fbed75","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3eed304e6af690a7ca4f95af9814fe112874a0b0efb4f7d091c7be3ce7bffbfa","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bf8d6d5204ce084429052d3f687b1b4007057ecf29fbdc62aab2b93e39615bf4","source_sha256":"f37e97c4ba70607031e3a483123b1b4fa94a20ae8b550c03293099ec4ccab941","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine Measure.ext_of_charFunDual ?_\n  ext L\n  simp_rw [charFunDual_map, charFunDual_prod, charFunDual_eq_of_forall_strongDual_eq_zero hμ,\n    ← Complex.exp_add]\n  rw [← add_div, ← add_div, ← neg_add, ← neg_add]\n  congr 3\n  norm_cast\n  have h1 : (L.comp (.rotation θ)).comp (.inl ℝ E E)\n      = Real.cos θ • L.comp (.inl ℝ E E) - Real.sin θ • L.comp (.inr ℝ E E) := by\n    ext x\n    simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.inl_apply,\n      ContinuousLinearMap.rotation_apply, smul_zero, add_zero]\n    rw [← L.comp_inl_add_comp_inr]\n    simp [-neg_smul, sub_eq_add_neg]\n  have h2 : (L.comp (.rotation θ)).comp (.inr ℝ E E)\n      = Real.sin θ • L.comp (.inl ℝ E E) + Real.cos θ • L.comp (.inr ℝ E E) := by\n    ext x\n    simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.inr_apply,\n      ContinuousLinearMap.rotation_apply, smul_zero, zero_add, add_apply, smul_apply,\n      ContinuousLinearMap.inl_apply, smul_eq_mul]\n    rw [← L.comp_inl_add_comp_inr]\n    simp\n  rw [h1, h2]\n  simp only [FunLike.coe_sub, FunLike.coe_smul,\n    FunLike.coe_add]\n  rw [variance_sub, variance_smul, variance_add, variance_smul, variance_smul, covariance_smul_left,\n    covariance_smul_right, variance_smul, covariance_smul_left, covariance_smul_right]\n  · have h := Real.cos_sq_add_sin_sq θ\n    grind\n  all_goals exact (memLp_dual _ _ _ (by simp)).const_smul _","hard_negative":true,"metrics":{"chosen_tokens":294,"rejected_tokens":2,"token_jaccard":0.012195,"token_length_ratio":0.006803},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"bb0130ecbc03cebd9bfba730d07b889eef7c5fee21e086371a61ad0340cfe66c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Distributions.Fernique\npublic import Mathlib.Probability.Distributions.Gaussian.Basic\n\nNamespace:\nProbabilityTheory.IsGaussian\n\nLocal context:\n/-\nCopyright (c) 2025 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n-/\n/-!\n# Fernique's theorem for Gaussian measures\n\nWe show that the product of two identical Gaussian measures is invariant under rotation.\nWe then deduce Fernique's theorem, which states that for a Gaussian measure `μ`, there exists\n`C > 0` such that the function `x ↦ exp (C * ‖x‖ ^ 2)` is integrable with respect to `μ`.\nAs a consequence, a Gaussian measure has finite moments of all orders.\n\n## Main statements\n\n* `IsGaussian.exists_integrable_exp_sq`: **Fernique's theorem**. For a Gaussian measure on a\n  second-countable normed space, there exists `C > 0` such that the function\n  `x ↦ exp (C * ‖x‖ ^ 2)` is integrable.\n* `IsGaussian.memLp_id`: a Gaussian measure in a second-countable Banach space has finite moments\n  of all orders.\n\n## References\n\n* [Martin Hairer, *An introduction to stochastic PDEs*][hairer2009introduction]\n\n-/\n\npublic section\n\nopen MeasureTheory ProbabilityTheory Complex\nopen scoped ENNReal NNReal Real Topology\n\nnamespace ProbabilityTheory.IsGaussian\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E]\n  {μ : Measure E} [IsGaussian μ]\n\nsection Rotation\n\n/-- Characteristic function of a centered Gaussian measure.\nFor a Gaussian measure, the hypothesis `∀ L : StrongDual ℝ E, μ[L] = 0` is equivalent to the simpler\n`μ[id] = 0`, but at this point we don't know yet that `μ` has a first moment so we can't use it.\nSee `charFunDual_eq_of_integral_eq_zero` -/\nlemma charFunDual_eq_of_forall_strongDual_eq_zero (hμ : ∀ L : StrongDual ℝ E, μ[L] = 0)\n    (L : StrongDual ℝ E) :\n    charFunDual μ L = exp (- Var[L; μ] / 2) := by\n  simp [charFunDual_eq L, integral_complex_ofReal, hμ L, neg_div]\n\n/-- For a centered Gaussian measure `μ`, the product measure `μ.prod μ` is invariant under rotation.\nThe hypothesis `∀ L : StrongDual ℝ E, μ[L] = 0` is equivalent to the simpler\n`μ[id] = 0`, but at this point we don't know yet that `μ` has a first moment so we can't use it.\nSee `map_rotation_eq_self`. -/\n\nTarget:\nlemma map_rotation_eq_self_of_forall_strongDual_eq_zero\n    [SecondCountableTopology E] [CompleteSpace E]\n    (hμ : ∀ L : StrongDual ℝ E, μ[L] = 0) (θ : ℝ) :\n    (μ.prod μ).map (ContinuousLinearMap.rotation θ) = μ.prod μ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_bf8d6d5204ce","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"1371d5e65e7a8aebb3663c8a104dac65b000417f0973bf58769265992e607958","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Distributions","family_id":"map_rotation_eq_self_of_forall_strongdual_eq_zero","file_id":"mathlib/Mathlib/Probability/Distributions/Gaussian/Fernique.lean","sample_id":"bf8d6d5204ce084429052d3f687b1b4007057ecf29fbdc62aab2b93e39615bf4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0fd4fc28ba9a30a85e46c5d1d3ba799c62c0383a953832f15efb88028c8307ee","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"da1265a9cce4836c0845ed096f6cb5e87eaef2399cb62892b9ca5f1918463db5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a75eaccf6d18b5c2eb5e6918fb406ae54ba35c64367a2abe5f21d18d5b2dac31","source_sha256":"2c2357bdce988b5b8902bab0c6b1d4a599e97cee77b05d9e7215f22e881d81ce","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI := Fintype.ofFinite α\n  induction φ with simp only [BoundedFormula.Realize]\n  | equal t₁ t₂ =>\n    rcases term_realize_eq_add_dotProduct t₁ with ⟨k₁, u₁, ht₁⟩\n    rcases term_realize_eq_add_dotProduct t₂ with ⟨k₂, u₂, ht₂⟩\n    convert! Nat.isSemilinearSet_setOf_mulVec_eq ![k₁] ![k₂] (.of ![u₁]) (.of ![u₂])\n    simp [ht₁, ht₂]\n  | rel f => nomatch f\n  | falsum => exact .empty\n  | imp _ _ ih₁ ih₂ =>\n    convert! (ih₂.compl.inter ih₁).compl using 1\n    simp [setOf_inter_eq_sep, imp_iff_not_or, compl_setOf]\n  | @all n φ ih =>\n    let e := (Equiv.sumAssoc α (Fin n) (Fin 1)).trans (Equiv.sumCongr (.refl α) finSumFinEquiv)\n    rw [← isSemilinearSet_image_iff (LinearEquiv.funCongrLeft ℕ ℕ e)] at ih\n    convert! ih.compl.proj.compl using 1\n    simp_rw [compl_setOf, not_exists, Fin.forall_fin_succ_pi, Fin.forall_fin_zero_pi,\n      mem_compl_iff, mem_image, not_not, ← LinearEquiv.eq_symm_apply, LinearEquiv.funCongrLeft_symm,\n      exists_eq_right, mem_setOf, LinearEquiv.funCongrLeft_apply, LinearMap.funLeft,\n      LinearMap.coe_mk, AddHom.coe_mk]\n    congr! 4\n    ext i\n    cases i using Fin.lastCases <;> simp [e]","hard_negative":true,"metrics":{"chosen_tokens":270,"rejected_tokens":2,"token_jaccard":0.010101,"token_length_ratio":0.007407},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"bb79219269b47c8edd64208cda5bd1f1747975ee4ba5807eb914825a7ed14faf","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.ModelTheory.Arithmetic.Presburger.Basic\npublic import Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic\npublic import Mathlib.ModelTheory.Definability\nimport Mathlib.Algebra.Group.Submonoid.Finsupp\nimport Mathlib.LinearAlgebra.Matrix.Notation\n\nNamespace:\nFirstOrder.Language.presburger\n\nLocal context:\n/-\nCopyright (c) 2025 Dexin Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dexin Zhang\n-/\n/-!\n# Presburger definability and semilinear sets\n\nThis file formalizes the classical result that Presburger definable sets are the same as semilinear\nsets. As an application of this result, we show that the graph of multiplication is not Presburger\ndefinable.\n\n## Main Results\n\n- `presburger.definable_iff_isSemilinearSet`: a set is Presburger definable in `ℕ` if and only if it\n  is semilinear.\n- `presburger.definable₁_iff_ultimately_periodic`: in the 1-dimensional case, a set is Presburger\n  arithmetic definable in `ℕ` if and only if it is ultimately periodic, i.e. periodic after some\n  number `k`.\n- `presburger.mul_not_definable`: the graph of multiplication is not Presburger definable in `ℕ`.\n\n## References\n\n* [Seymour Ginsburg and Edwin H. Spanier, *Bounded ALGOL-Like Languages*][ginsburg1964]\n* [Seymour Ginsburg and Edwin H. Spanier, *Semigroups, Presburger Formulas, and\n  Languages*][ginsburg1966]\n* [Samuel Eilenberg and M. P. Schützenberger, *Rational Sets in Commutative Monoids*][eilenberg1969]\n-/\n\npublic section\n\nvariable {α : Type*} {s : Set (α → ℕ)} {A : Set ℕ}\n\nopen Set FirstOrder Language\n\ntheorem IsLinearSet.definable [Finite α] (hs : IsLinearSet s) : A.Definable presburger s := by\n  rw [isLinearSet_iff] at hs\n  rcases hs with ⟨v, t, rfl⟩\n  refine ⟨Formula.iExs t (Formula.iInf fun i : α =>\n    (Term.var (Sum.inl i)).equal\n      (Term.varsToConstants\n        ((v i : presburger.Term _) + presburger.sum Finset.univ fun x : t =>\n          x.1 i • Term.var (Sum.inr (Sum.inr x))))), ?_⟩\n  ext x\n  simp only [mem_vadd_set, SetLike.mem_coe, AddSubmonoid.mem_closure_finset', Finset.univ_eq_attach,\n    nsmul_eq_mul, vadd_eq_add, ↓existsAndEq, true_and, mem_setOf_eq, Formula.realize_iExs,\n    Formula.realize_iInf, Formula.realize_equal, Term.realize_var, Sum.elim_inl,\n    Term.realize_varsToConstants, coe_con, presburger.realize_add, presburger.realize_natCast,\n    Nat.cast_id, presburger.realize_sum, presburger.realize_nsmul, Sum.elim_inr, smul_eq_mul]\n  congr! with a\n  simp_rw [Eq.comm (b := x), fun x : t => mul_comm (a x : α → ℕ) x, funext_iff]\n  congr! 1 with i\n  simp\n\ntheorem IsSemilinearSet.definable [Finite α] (hs : IsSemilinearSet s) :\n    A.Definable presburger s := by\n  rw [isSemilinearSet_iff] at hs\n  rcases hs with ⟨S, hS, rfl⟩\n  choose φ hφ using fun s : S => (hS s.1 s.2).definable\n  refine ⟨Formula.iSup φ, ?_⟩\n  ext x\n  have := fun s hs x => Set.ext_iff.1 (hφ ⟨s, hs⟩).symm x\n  simp only [mem_setOf_eq] at this\n  simp [this]\n\nnamespace FirstOrder.Language.presburger\n\nset_option backward.isDefEq.respectTransparency false in\nlemma term_realize_eq_add_dotProduct [Fintype α] (t : presburger[[A]].Term α) :\n    ∃ (k : ℕ) (u : α → ℕ), ∀ (v : α → ℕ), t.realize v = k + u ⬝ᵥ v := by\n  classical\n  induction t with simp only [Term.realize]\n  | var i =>\n    exact ⟨0, Pi.single i 1, by simp⟩\n  | @func l f ts ih =>\n    cases f with\n    | inl f =>\n      choose k u ih using ih\n      cases f with\n      | zero =>\n        refine ⟨0, 0, fun v => ?_⟩\n        rw [withConstants_funMap_sumInl]\n        simp\n      | one =>\n        refine ⟨1, 0, fun v => ?_⟩\n        rw [withConstants_funMap_sumInl]\n        simp [ih]\n      | add =>\n        refine ⟨k 0 + k 1, u 0 + u 1, fun v => ?_⟩\n        rw [withConstants_funMap_sumInl, add_dotProduct, add_left_comm, add_assoc, add_left_comm,\n          ← add_assoc]\n        simp [ih]\n    | inr f =>\n      cases l with\n      | zero =>\n        refine ⟨f, 0, fun v => ?_⟩\n        rw [withConstants_funMap_sumInr, zero_dotProduct, add_zero]\n        rfl\n      | succ => nomatch f\n\nvariable [Finite α]\n\nTarget:\nlemma isSemilinearSet_boundedFormula_realize {n} (φ : presburger[[A]].BoundedFormula α n) :\n    IsSemilinearSet {v : α ⊕ Fin n → ℕ | φ.Realize (v ∘ Sum.inl) (v ∘ Sum.inr)} :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_a75eaccf6d18","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"5014888971e89aa8983f1efe16af9621e1da499311453b86b4f77297386d5a70","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"ModelTheory/Arithmetic","family_id":"issemilinearset_boundedformula_realize","file_id":"mathlib/Mathlib/ModelTheory/Arithmetic/Presburger/Definability.lean","sample_id":"a75eaccf6d18b5c2eb5e6918fb406ae54ba35c64367a2abe5f21d18d5b2dac31"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eb4847d4ec18aeaffa781e4a9bdcbca4898991cdee673fe5996e2fd2c98cb1bf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cb8c4de23317146f19f32df2485e1f047d44025176348b35088a93dfd2c0286c","source_sha256":"e4f7eab8ce40520bcf25316e25287e36bf06b7a538fd15f40c2755c88337db7b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Function.Injective] at hv\n  push Not at hv\n  rcases hv with ⟨i₁, i₂, heq, hne⟩\n  exact f.map_eq_zero_of_eq v heq hne\n\n/-!\n### Algebraic structure inherited from `MultilinearMap`\n\n`AlternatingMap` carries the same `AddCommMonoid`, `AddCommGroup`, and `Module` structure\nas `MultilinearMap`\n-/","hard_negative":false,"metrics":{"chosen_tokens":72,"rejected_tokens":2,"token_jaccard":0.022222,"token_length_ratio":0.027778},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"bbd08c8041043ff9aab35fab5c7558026f3ac43ef4bcdb9bcd4b34c2e95e31c2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.Perm.Sign\npublic import Mathlib.LinearAlgebra.LinearIndependent.Defs\npublic import Mathlib.LinearAlgebra.Multilinear.Basis\n\nNamespace:\nAlternatingMap\n\nLocal context:\n/-\nCopyright (c) 2020 Zhangir Azerbayev. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser, Zhangir Azerbayev\n-/\n/-!\n# Alternating Maps\n\nWe construct the bundled function `AlternatingMap`, which extends `MultilinearMap` with all the\narguments of the same type.\n\n## Main definitions\n* `AlternatingMap R M N ι` is the space of `R`-linear alternating maps from `ι → M` to `N`.\n* `f.map_eq_zero_of_eq` expresses that `f` is zero when two inputs are equal.\n* `f.map_swap` expresses that `f` is negated when two inputs are swapped.\n* `f.map_perm` expresses how `f` varies by a sign change under a permutation of its inputs.\n* An `AddCommMonoid`, `AddCommGroup`, and `Module` structure over `AlternatingMap`s that\n  matches the definitions over `MultilinearMap`s.\n* `AlternatingMap.domDomCongr`, for permuting the elements within a family.\n* `MultilinearMap.alternatization`, which makes an alternating map out of a non-alternating one.\n* `AlternatingMap.curryLeft`, for binding the leftmost argument of an alternating map indexed\n  by `Fin n.succ`.\n\n## Implementation notes\n`AlternatingMap` is defined in terms of `map_eq_zero_of_eq`, as this is easier to work with than\nusing `map_swap` as a definition, and does not require `Neg N`.\n\n`AlternatingMap`s are provided with a coercion to `MultilinearMap`, along with a set of\n`norm_cast` lemmas that act on the algebraic structure:\n\n* `AlternatingMap.coe_add`\n* `AlternatingMap.coe_zero`\n* `AlternatingMap.coe_sub`\n* `AlternatingMap.coe_neg`\n* `AlternatingMap.coe_smul`\n-/\n\n@[expose] public section\n\nopen Module\n\n-- semiring / add_comm_monoid\n\nvariable {R : Type*} [Semiring R]\nvariable {M : Type*} [AddCommMonoid M] [Module R M]\nvariable {N : Type*} [AddCommMonoid N] [Module R N]\nvariable {P : Type*} [AddCommMonoid P] [Module R P]\n\n-- semiring / add_comm_group\n\nvariable {M' : Type*} [AddCommGroup M'] [Module R M']\nvariable {N' : Type*} [AddCommGroup N'] [Module R N']\nvariable {ι ι' ι'' : Type*}\n\nsection\n\nvariable (R M N ι)\n\n/-- An alternating map from `ι → M` to `N`, denoted `M [⋀^ι]→ₗ[R] N`,\nis a multilinear map that vanishes when two of its arguments are equal. -/\nstructure AlternatingMap extends MultilinearMap R (fun _ : ι => M) N where\n  /-- The map is alternating: if `v` has two equal coordinates, then `f v = 0`. -/\n  map_eq_zero_of_eq' : ∀ (v : ι → M) (i j : ι), v i = v j → i ≠ j → toFun v = 0\n\n@[inherit_doc]\nnotation M \" [⋀^\" ι \"]→ₗ[\" R \"] \" N:100 => AlternatingMap R M N ι\n\nend\n\n/-- The multilinear map associated to an alternating map -/\nadd_decl_doc AlternatingMap.toMultilinearMap\n\nnamespace AlternatingMap\n\nvariable (f f' : M [⋀^ι]→ₗ[R] N)\nvariable (g g₂ : M [⋀^ι]→ₗ[R] N')\nvariable (g' : M' [⋀^ι]→ₗ[R] N')\nvariable (v : ι → M) (v' : ι → M')\n\nopen Function\n\n/-! Basic coercion simp lemmas, largely copied from `RingHom` and `MultilinearMap` -/\n\n\nsection Coercions\n\ninstance instFunLike : FunLike (M [⋀^ι]→ₗ[R] N) (ι → M) N where\n  coe f := f.toFun\n  coe_injective f g h := by\n    rcases f with ⟨⟨_, _, _⟩, _⟩\n    rcases g with ⟨⟨_, _, _⟩, _⟩\n    congr\n\ninitialize_simps_projections AlternatingMap (toFun → apply)\n\n@[simp]\ntheorem toFun_eq_coe : f.toFun = f :=\n  rfl\n\n@[simp]\ntheorem coe_mk (f : MultilinearMap R (fun _ : ι => M) N) (h) :\n    ⇑(⟨f, h⟩ : M [⋀^ι]→ₗ[R] N) = f :=\n  rfl\n\nprotected theorem congr_fun {f g : M [⋀^ι]→ₗ[R] N} (h : f = g) (x : ι → M) : f x = g x :=\n  congr_arg (fun h : M [⋀^ι]→ₗ[R] N => h x) h\n\nprotected theorem congr_arg (f : M [⋀^ι]→ₗ[R] N) {x y : ι → M} (h : x = y) : f x = f y :=\n  congr_arg (fun x : ι → M => f x) h\n\ntheorem coe_injective : Injective ((↑) : M [⋀^ι]→ₗ[R] N → (ι → M) → N) :=\n  DFunLike.coe_injective\n\n@[norm_cast]\ntheorem coe_inj {f g : M [⋀^ι]→ₗ[R] N} : (f : (ι → M) → N) = g ↔ f = g :=\n  coe_injective.eq_iff\n\n@[ext]\ntheorem ext {f f' : M [⋀^ι]→ₗ[R] N} (H : ∀ x, f x = f' x) : f = f' :=\n  DFunLike.ext _ _ H\n\nattribute [coe] AlternatingMap.toMultilinearMap\n\ninstance instCoe : Coe (M [⋀^ι]→ₗ[R] N) (MultilinearMap R (fun _ : ι => M) N) :=\n  ⟨fun x => x.toMultilinearMap⟩\n\n@[simp, norm_cast]\ntheorem coe_multilinearMap : ⇑(f : MultilinearMap R (fun _ : ι => M) N) = f :=\n  rfl\n\ntheorem coe_multilinearMap_injective :\n    Function.Injective ((↑) : M [⋀^ι]→ₗ[R] N → MultilinearMap R (fun _ : ι => M) N) :=\n  fun _ _ h => ext <| MultilinearMap.congr_fun h\n\ntheorem coe_multilinearMap_mk (f : (ι → M) → N) (h₁ h₂ h₃) :\n    ((⟨⟨f, h₁, h₂⟩, h₃⟩ : M [⋀^ι]→ₗ[R] N) : MultilinearMap R (fun _ : ι => M) N) =\n      ⟨f, @h₁, @h₂⟩ := by\n  simp\n\nend Coercions\n\n/-!\n### Simp-normal forms of the structure fields\n\nThese are expressed in terms of `⇑f` instead of `f.toFun`.\n-/\n\n\n@[simp]\ntheorem map_update_add [DecidableEq ι] (i : ι) (x y : M) :\n    f (update v i (x + y)) = f (update v i x) + f (update v i y) :=\n  f.map_update_add' v i x y\n\n@[simp]\ntheorem map_update_sub [DecidableEq ι] (i : ι) (x y : M') :\n    g' (update v' i (x - y)) = g' (update v' i x) - g' (update v' i y) :=\n  g'.toMultilinearMap.map_update_sub v' i x y\n\n@[simp]\ntheorem map_update_neg [DecidableEq ι] (i : ι) (x : M') :\n    g' (update v' i (-x)) = -g' (update v' i x) :=\n  g'.toMultilinearMap.map_update_neg v' i x\n\n@[simp]\ntheorem map_update_smul [DecidableEq ι] (i : ι) (r : R) (x : M) :\n    f (update v i (r • x)) = r • f (update v i x) :=\n  f.map_update_smul' v i r x\n\n-- Cannot be @[simp] because `i` and `j` cannot be inferred by `simp`.\ntheorem map_eq_zero_of_eq (v : ι → M) {i j : ι} (h : v i = v j) (hij : i ≠ j) : f v = 0 :=\n  f.map_eq_zero_of_eq' v i j h hij\n\ntheorem map_coord_zero {m : ι → M} (i : ι) (h : m i = 0) : f m = 0 :=\n  f.toMultilinearMap.map_coord_zero i h\n\n@[simp]\ntheorem map_update_zero [DecidableEq ι] (m : ι → M) (i : ι) : f (update m i 0) = 0 :=\n  f.toMultilinearMap.map_update_zero m i\n\n@[simp]\ntheorem map_zero [Nonempty ι] : f 0 = 0 :=\n  f.toMultilinearMap.map_zero\n\nTarget:\ntheorem map_eq_zero_of_not_injective (v : ι → M) (hv : ¬Function.Injective v) : f v = 0 :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Alternating","family_id":"map_eq_zero_of_not_injective","file_id":"mathlib/Mathlib/LinearAlgebra/Alternating/Basic.lean","sample_id":"cb8c4de23317146f19f32df2485e1f047d44025176348b35088a93dfd2c0286c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0e429534a623f3db5dd3586328da30692723c2a4071b25aa658c931a2d458bab","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a6146b935412bf1cd41ece91f946c0daf117bb2ef301024873391e1ed5ceafb2","source_sha256":"42f43c83c3b42913128aa30933a73363c485cf2bc69fc613214486004b504ca0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [commutator_eq_closure G, ← image_commutatorSet_closureCommutatorRepresentatives, ←\n    MonoidHom.map_closure, ← commutator_eq_closure]\n  exact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _))\n\nvariable [Finite (commutatorSet G)]","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":2,"token_jaccard":0.038462,"token_length_ratio":0.047619},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"bbe0c936d5dd335ffed48a312d30820c94b95e37b50ffa71159130c1e048121a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Subgroup.Finite\npublic import Mathlib.GroupTheory.Commutator.Basic\npublic import Mathlib.GroupTheory.Rank\npublic import Mathlib.GroupTheory.Index\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jordan Brown, Thomas Browning, Patrick Lutz\n-/\n/-!\nThe commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n\npublic section\n\nvariable {G : Type*} [Group G]\n\nnamespace Subgroup\n\n/-- The commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n@[to_additive /-- The commutator of a finite direct product is contained in the direct product of\nthe commutators. -/]\ntheorem commutator_pi_pi_of_finite {η : Type*} [Finite η] {Gs : η → Type*} [∀ i, Group (Gs i)]\n    (H K : ∀ i, Subgroup (Gs i)) : ⁅Subgroup.pi Set.univ H, Subgroup.pi Set.univ K⁆ =\n    Subgroup.pi Set.univ fun i => ⁅H i, K i⁆ := by\n  classical\n    apply le_antisymm (commutator_pi_pi_le H K)\n    rw [pi_le_iff]\n    intro i hi\n    rw [map_commutator]\n    apply commutator_mono <;>\n      · rw [le_pi_iff]\n        intro j _hj\n        rintro _ ⟨x, hx, rfl⟩\n        by_cases h : j = i\n        · subst h\n          simpa using hx\n        · simp [h, one_mem]\n\nvariable [Finite (commutatorSet G)]\n\ninstance : Group.FG (_root_.commutator G) := by\n  rw [commutator_eq_closure]; apply Group.closure_finite_fg\n\nvariable (G) in\nlemma rank_commutator_le_card : Group.rank (_root_.commutator G) ≤ Nat.card (commutatorSet G) := by\n  rw [Subgroup.rank_congr (commutator_eq_closure G)]\n  apply Subgroup.rank_closure_finite_le_nat_card\n\nvariable [Group.FG G]\n\ninstance finiteIndex_center : FiniteIndex (center G) := by\n  obtain ⟨S, -, hS⟩ := Group.rank_spec G\n  exact ⟨mt (Finite.card_eq_zero_of_embedding (quotientCenterEmbedding hS)) Finite.card_pos.ne'⟩\n\nvariable (G) in\nlemma index_center_le_pow : (center G).index ≤ Nat.card (commutatorSet G) ^ Group.rank G := by\n  obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G\n  rw [← hS1, ← Fintype.card_coe, ← Nat.card_eq_fintype_card, ← Finset.coe_sort_coe, ← Nat.card_fun]\n  exact Finite.card_le_of_embedding (quotientCenterEmbedding hS2)\n\nend Subgroup\n\nsection commutatorRepresentatives\n\nopen Subgroup\n\nlemma card_commutatorSet_closureCommutatorRepresentatives :\n    Nat.card (commutatorSet (closureCommutatorRepresentatives G)) = Nat.card (commutatorSet G) := by\n  rw [← image_commutatorSet_closureCommutatorRepresentatives G]\n  exact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _))\n\nTarget:\nlemma card_commutator_closureCommutatorRepresentatives :\n    Nat.card (commutator (closureCommutatorRepresentatives G)) = Nat.card (commutator G) :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Commutator","family_id":"card_commutator_closurecommutatorrepresentatives","file_id":"mathlib/Mathlib/GroupTheory/Commutator/Finite.lean","sample_id":"a6146b935412bf1cd41ece91f946c0daf117bb2ef301024873391e1ed5ceafb2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"72e566f5060686e112fed795e07f0f9fff6e410e86e784e0d5fb3063e2798458","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"80e10db2175365293b28a61fb2baadedb77fa1adfc56606eaf56f02d6e239f96","source_sha256":"5b12ef03a7b761162934544e836024f413f1638f86d711b3735f9f49b0617ce9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  nontriviality α using Subsingleton.elim a 0\n  obtain rfl | h := eq_or_ne a 0\n  · rw [normUnit_zero, zero_mul, normUnit_zero]\n  · rw [normUnit_mul h (Units.ne_zero _), normUnit_coe_units, mul_inv_eq_one]","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":3,"token_jaccard":0.03125,"token_length_ratio":0.071429},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"bc5f9da74c02a2b11357755fc031c6cb6ef4973068a7f5f70e8aae67afe8827c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.Associated\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Jens Wagemaker\n-/\n/-!\n# Monoids with normalization functions, `gcd`, and `lcm`\n\nThis file defines extra structures on `CommMonoidWithZero`s.\n\n## Main Definitions\n\n* `NormalizationMonoid`\n* `GCDMonoid`\n* `NormalizedGCDMonoid`\n* `gcdMonoidOfGCD`, `gcdMonoidOfExistsGCD`, `normalizedGCDMonoidOfGCD`,\n  `normalizedGCDMonoidOfExistsGCD`\n* `gcdMonoidOfLCM`, `gcdMonoidOfExistsLCM`, `normalizedGCDMonoidOfLCM`,\n  `normalizedGCDMonoidOfExistsLCM`\n\nFor the `NormalizedGCDMonoid` instances on `ℕ` and `ℤ`, see `Mathlib/Algebra/GCDMonoid/Nat.lean`.\n\n## Implementation Notes\n\n* `NormalizationMonoid` is defined by assigning to each element a `normUnit` such that multiplying\n  by that unit normalizes the monoid, and `normalize` is an idempotent monoid homomorphism. This\n  definition as currently implemented does casework on `0`.\n\n* `GCDMonoid` contains the definitions of `gcd` and `lcm` with the usual properties. They are\n  both determined up to a unit.\n\n* `NormalizedGCDMonoid` extends `NormalizationMonoid`, so the `gcd` and `lcm` are always\n  normalized. This makes `gcd`s of polynomials easier to work with, but excludes Euclidean domains,\n  and monoids without zero.\n\n* `gcdMonoidOfGCD` and `normalizedGCDMonoidOfGCD` noncomputably construct a `GCDMonoid`\n  (resp. `NormalizedGCDMonoid`) structure just from the `gcd` and its properties.\n\n* `gcdMonoidOfExistsGCD` and `normalizedGCDMonoidOfExistsGCD` noncomputably construct a\n  `GCDMonoid` (resp. `NormalizedGCDMonoid`) structure just from a proof that any two elements\n  have a (not necessarily normalized) `gcd`.\n\n* `gcdMonoidOfLCM` and `normalizedGCDMonoidOfLCM` noncomputably construct a `GCDMonoid`\n  (resp. `NormalizedGCDMonoid`) structure just from the `lcm` and its properties.\n\n* `gcdMonoidOfExistsLCM` and `normalizedGCDMonoidOfExistsLCM` noncomputably construct a\n  `GCDMonoid` (resp. `NormalizedGCDMonoid`) structure just from a proof that any two elements\n  have a (not necessarily normalized) `lcm`.\n\n## TODO\n\n* Port GCD facts about nats, definition of coprime\n* Generalize normalization monoids to commutative (cancellative) monoids with or without zero\n\n## Tags\n\ndivisibility, gcd, lcm, normalize\n-/\n\n@[expose] public section\n\n\nvariable {α : Type*}\n\n/-- Normalization monoid: multiplying with `normUnit` gives a normal form for associated\nelements. -/\nclass NormalizationMonoid (α : Type*) [CommMonoidWithZero α] where\n  /-- `normUnit` assigns to each element of the monoid a unit of the monoid. -/\n  normUnit : α → αˣ\n  /-- The proposition that `normUnit` maps `0` to the identity. -/\n  normUnit_zero : normUnit 0 = 1\n  /-- The proposition that `normUnit` respects multiplication of non-zero elements. -/\n  normUnit_mul : ∀ {a b}, a ≠ 0 → b ≠ 0 → normUnit (a * b) = normUnit a * normUnit b\n  /-- The proposition that `normUnit` maps units to their inverses. -/\n  normUnit_coe_units : ∀ u : αˣ, normUnit u = u⁻¹\n\nexport NormalizationMonoid (normUnit normUnit_zero normUnit_mul normUnit_coe_units)\n\nattribute [simp] normUnit_coe_units normUnit_zero normUnit_mul\n\nsection NormalizationMonoid\n\nvariable [CommMonoidWithZero α] [NormalizationMonoid α]\n\n@[simp]\ntheorem normUnit_one : normUnit (1 : α) = 1 :=\n  normUnit_coe_units 1\n\n/-- Chooses an element of each associate class, by multiplying by `normUnit` -/\ndef normalize : α →*₀ α where\n  toFun x := x * normUnit x\n  map_zero' := by\n    simp only [normUnit_zero]\n    exact mul_one (0 : α)\n  map_one' := by rw [normUnit_one, one_mul]; rfl\n  map_mul' x y :=\n    (by_cases fun hx : x = 0 => by rw [hx, zero_mul, zero_mul, zero_mul]) fun hx =>\n      (by_cases fun hy : y = 0 => by rw [hy, mul_zero, zero_mul, mul_zero]) fun hy => by\n        simp only [normUnit_mul hx hy, Units.val_mul]; simp only [mul_assoc, mul_left_comm y]\n\ntheorem associated_normalize (x : α) : Associated x (normalize x) :=\n  ⟨_, rfl⟩\n\ntheorem normalize_associated (x : α) : Associated (normalize x) x :=\n  (associated_normalize _).symm\n\ntheorem associated_normalize_iff {x y : α} : Associated x (normalize y) ↔ Associated x y :=\n  ⟨fun h => h.trans (normalize_associated y), fun h => h.trans (associated_normalize y)⟩\n\ntheorem normalize_associated_iff {x y : α} : Associated (normalize x) y ↔ Associated x y :=\n  ⟨fun h => (associated_normalize _).trans h, fun h => (normalize_associated _).trans h⟩\n\ntheorem Associates.mk_normalize (x : α) : Associates.mk (normalize x) = Associates.mk x :=\n  Associates.mk_eq_mk_iff_associated.2 (normalize_associated _)\n\ntheorem normalize_apply (x : α) : normalize x = x * normUnit x :=\n  rfl\n\ntheorem normalize_zero : normalize (0 : α) = 0 :=\n  normalize.map_zero\n\ntheorem normalize_one : normalize (1 : α) = 1 :=\n  normalize.map_one\n\ntheorem normalize_coe_units (u : αˣ) : normalize (u : α) = 1 := by simp [normalize_apply]\n\n@[simp]\ntheorem normalize_eq_zero {x : α} : normalize x = 0 ↔ x = 0 :=\n  ⟨fun hx => (associated_zero_iff_eq_zero x).1 <| hx ▸ associated_normalize _, by\n    rintro rfl; exact normalize_zero⟩\n\ntheorem normalize_eq_one {x : α} : normalize x = 1 ↔ IsUnit x :=\n  ⟨fun hx => isUnit_iff_exists_inv.2 ⟨_, hx⟩, fun ⟨u, hu⟩ => hu ▸ normalize_coe_units u⟩\n\n@[simp]\n\nTarget:\ntheorem normUnit_mul_normUnit (a : α) : normUnit (a * normUnit a) = 1 :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/GCDMonoid","family_id":"normunit_mul_normunit","file_id":"mathlib/Mathlib/Algebra/GCDMonoid/Basic.lean","sample_id":"80e10db2175365293b28a61fb2baadedb77fa1adfc56606eaf56f02d6e239f96"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6d7fe2ba7791e82259c7c7dac10b1de23d8b4e925a2fa271084af7b3f600bee5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a83c5a1c18e47c4bbeb57a883865f74ecb181d20c5d98912e304792cdc597370","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3b9bf8a1b9ae9ed64027deecfa9b2b28510490bbc0204132c8e750b77e3c634e","source_sha256":"75f644a2aa9ce33f4e3a9b3af6b61387bae9fdea7ce3dbf0ef1ec4c5bea1cda5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases hf2 : f = 0; · rw [hf2]; exact dvd_zero _\n  rw [factor, dif_pos (WfDvdMonoid.exists_irreducible_factor hf1 hf2)]\n  exact (Classical.choose_spec <| WfDvdMonoid.exists_irreducible_factor hf1 hf2).2","hard_negative":true,"metrics":{"chosen_tokens":45,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.066667},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"bc608ad4d5a66b18b53b40c1b888e5aa05ac5994a98ac6f7db04cf7380ef70ab","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharP.Algebra\npublic import Mathlib.FieldTheory.SplittingField.IsSplittingField\npublic import Mathlib.RingTheory.Algebraic.Basic\n\nNamespace:\nPolynomial\n\nLocal context:\n/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Splitting fields\n\nIn this file we prove the existence and uniqueness of splitting fields.\n\n## Main definitions\n\n* `Polynomial.SplittingField f`: A fixed splitting field of the polynomial `f`.\n\n## Main statements\n\n* `Polynomial.IsSplittingField.algEquiv`: Every splitting field of a polynomial `f` is isomorphic\n  to `SplittingField f` and thus, being a splitting field is unique up to isomorphism.\n\n## Implementation details\nWe construct a `SplittingFieldAux` without worrying about whether the instances satisfy nice\ndefinitional equalities. Then the actual `SplittingField` is defined to be a quotient of a\n`MvPolynomial` ring by the kernel of the obvious map into `SplittingFieldAux`. Because the\nactual `SplittingField` will be a quotient of a `MvPolynomial`, it has nice instances on it.\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nuniverse u v w\n\nvariable {F : Type u} {K : Type v} {L : Type w}\n\nnamespace Polynomial\n\nvariable [Field K] [Field L] [Field F]\n\nopen Polynomial\n\nsection SplittingField\n\nopen Classical in\n/-- Non-computably choose an irreducible factor from a polynomial. -/\ndef factor (f : K[X]) : K[X] :=\n  if H : ∃ g, Irreducible g ∧ g ∣ f then Classical.choose H else X\n\ntheorem irreducible_factor (f : K[X]) : Irreducible (factor f) := by\n  rw [factor]\n  split_ifs with H\n  · exact (Classical.choose_spec H).1\n  · exact irreducible_X\n\n/-- See note [fact non-instances]. -/\ntheorem fact_irreducible_factor (f : K[X]) : Fact (Irreducible (factor f)) :=\n  ⟨irreducible_factor f⟩\n\nattribute [local instance] fact_irreducible_factor\n\nTarget:\ntheorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f :=\n\nProof body:\n","rejected":"by\n  exact factor_dvd_of_not_isUnit","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"8d31474c75b26fc5f278031a6533a2fa4b1d763ed175e5ac03bd90c29cf3b730","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/SplittingField","family_id":"factor_dvd_of_not_isunit","file_id":"mathlib/Mathlib/FieldTheory/SplittingField/Construction.lean","sample_id":"3b9bf8a1b9ae9ed64027deecfa9b2b28510490bbc0204132c8e750b77e3c634e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4f2754f3c1709f7213213c711ba2777cb527cb9fa5e9c5f811685e4d0a510864","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"365ae057ef87f87f7e7b270aae3a76feaf2c96f2449347291717e82ad544989d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2ba6bee601213da996a0d324b595d9dabc6794f2b60ae27ff16595b74c0086ad","source_sha256":"1cf64ac04a5f1f9a7c585c7ed1d6a4cf445109aff4a46ab4c7d2a0b61c1da479","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← IsLocalization.mk'_one (M := K[X]⁰) (FractionRing K[X]) p, ← mk_coe_def, Submonoid.coe_one]","hard_negative":true,"metrics":{"chosen_tokens":32,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.15625},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"bc7671ac1f486d74cdb9990ce7055b7c8579c96e4a860d4aec3212d83bcac40f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Basic\npublic import Mathlib.Algebra.Ring.NonZeroDivisors\npublic import Mathlib.RingTheory.Localization.FractionRing\n\nNamespace:\nRatFunc\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# The field of rational functions\n\nFiles in this folder define the field `K⟮X⟯` of rational functions over a field `K`, show it\nis the field of fractions of `K[X]` and provide the main results concerning it. This file contains\nthe basic definition.\n\nFor connections with Laurent Series, see `Mathlib/RingTheory/LaurentSeries.lean`.\n\n## Main definitions\nWe provide a set of recursion and induction principles:\n- `RatFunc.liftOn`: define a function by mapping a fraction of polynomials `p/q` to `f p q`,\n  if `f` is well-defined in the sense that `p/q = p'/q' → f p q = f p' q'`.\n- `RatFunc.liftOn'`: define a function by mapping a fraction of polynomials `p/q` to `f p q`,\n  if `f` is well-defined in the sense that `f (a * p) (a * q) = f p' q'`.\n- `RatFunc.induction_on`: if `P` holds on `p / q` for all polynomials `p q`, then `P` holds on all\n  rational functions\n\n## Implementation notes\n\nTo provide good API encapsulation and speed up unification problems,\n`RatFunc` is defined as a structure, and all operations are `@[irreducible] def`s\n\nWe need a couple of maps to set up the `Field` and `IsFractionRing` structure,\nnamely `RatFunc.ofFractionRing`, `RatFunc.toFractionRing`, `RatFunc.mk` and\n`RatFunc.toFractionRingRingEquiv`.\nAll these maps get `simp`ed to bundled morphisms like `algebraMap K[X] K⟮X⟯`\nand `IsLocalization.algEquiv`.\n\nThere are separate lifts and maps of homomorphisms, to provide routes of lifting even when\nthe codomain is not a field or even an integral domain.\n\n## References\n\n* [Kleiman, *Misconceptions about $K_X$*][kleiman1979]\n* https://freedommathdance.blogspot.com/2012/11/misconceptions-about-kx.html\n* https://stacks.math.columbia.edu/tag/01X1\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen scoped nonZeroDivisors Polynomial\n\nuniverse u v\n\nvariable (K : Type u)\n\n/-- `RatFunc K` is `K(X)`, the field of rational functions over `K`.\n\nThe inclusion of polynomials into `RatFunc` is `algebraMap K[X] K⟮X⟯`,\nthe maps between `K⟮X⟯` and another field of fractions of `K[X]`,\nespecially `FractionRing K[X]`, are given by `IsLocalization.algEquiv`.\n-/\nstructure RatFunc [CommRing K] : Type u where ofFractionRing ::\n/-- the coercion to the fraction ring of the polynomial ring -/\n  toFractionRing : FractionRing K[X]\n\n@[inherit_doc] scoped[RatFunc] notation:9000 R \"⟮X⟯\" => RatFunc R\n\nnamespace RatFunc\n\nsection CommRing\n\nvariable {K}\nvariable [CommRing K]\n\nsection Rec\n\n/-! ### Constructing `RatFunc`s and their induction principles -/\n\ntheorem ofFractionRing_injective : Function.Injective (ofFractionRing : _ → K⟮X⟯) :=\n  fun _ _ => ofFractionRing.inj\n\ntheorem toFractionRing_injective : Function.Injective (toFractionRing : _ → FractionRing K[X])\n  | ⟨x⟩, ⟨y⟩, xy => by subst xy; rfl\n\n@[simp] lemma toFractionRing_inj {x y : K⟮X⟯} :\n    toFractionRing x = toFractionRing y ↔ x = y :=\n  toFractionRing_injective.eq_iff\n\n/-- Non-dependent recursion principle for `K⟮X⟯`:\nTo construct a term of `P : Sort*` out of `x : K⟮X⟯`,\nit suffices to provide a constructor `f : Π (p q : K[X]), P`\nand a proof that `f p q = f p' q'` for all `p q p' q'` such that `q' * p = q * p'` where\nboth `q` and `q'` are not zero divisors, stated as `q ∉ K[X]⁰`, `q' ∉ K[X]⁰`.\n\nIf considering `K` as an integral domain, this is the same as saying that\nwe construct a value of `P` for such elements of `K⟮X⟯` by setting\n`liftOn (p / q) f _ = f p q`.\n\nWhen `[IsDomain K]`, one can use `RatFunc.liftOn'`, which has the stronger requirement\nof `∀ {p q a : K[X]} (hq : q ≠ 0) (ha : a ≠ 0), f (a * p) (a * q) = f p q)`.\n-/\nprotected irreducible_def liftOn {P : Sort v} (x : K⟮X⟯) (f : K[X] → K[X] → P)\n    (H : ∀ {p q p' q'} (_hq : q ∈ K[X]⁰) (_hq' : q' ∈ K[X]⁰), q' * p = q * p' → f p q = f p' q') :\n    P :=\n  Localization.liftOn (toFractionRing x) (fun p q => f p q) fun {_ _ q q'} h =>\n    H q.2 q'.2 (let ⟨⟨_, _⟩, mul_eq⟩ := Localization.r_iff_exists.mp h\n      mul_cancel_left_coe_nonZeroDivisors.mp mul_eq)\n\ntheorem liftOn_ofFractionRing_mk {P : Sort v} (n : K[X]) (d : K[X]⁰) (f : K[X] → K[X] → P)\n    (H : ∀ {p q p' q'} (_hq : q ∈ K[X]⁰) (_hq' : q' ∈ K[X]⁰), q' * p = q * p' → f p q = f p' q') :\n    RatFunc.liftOn (ofFractionRing (Localization.mk n d)) f @H = f n d := by\n  rw [RatFunc.liftOn]\n  exact Localization.liftOn_mk _ _ _ _\n\ntheorem liftOn_condition_of_liftOn'_condition {P : Sort v} {f : K[X] → K[X] → P}\n    (H : ∀ {p q a} (_ : q ≠ 0) (_ha : a ≠ 0), f (a * p) (a * q) = f p q) ⦃p q p' q' : K[X]⦄\n    (hq : q ≠ 0) (hq' : q' ≠ 0) (h : q' * p = q * p') : f p q = f p' q' :=\n  calc\n    f p q = f (q' * p) (q' * q) := (H hq hq').symm\n    _ = f (q * p') (q * q') := by rw [h, mul_comm q']\n    _ = f p' q' := H hq' hq\n\nsection IsDomain\n\nvariable [IsDomain K]\n\n/-- `RatFunc.mk (p q : K[X])` is `p / q` as a rational function.\n\nIf `q = 0`, then `mk` returns 0.\n\nThis is an auxiliary definition used to define an `Algebra` structure on `RatFunc`;\nthe `simp` normal form of `mk p q` is `algebraMap _ _ p / algebraMap _ _ q`.\n-/\nprotected irreducible_def mk (p q : K[X]) : K⟮X⟯ :=\n  ofFractionRing (algebraMap _ _ p / algebraMap _ _ q)\n\ntheorem mk_eq_div' (p q : K[X]) :\n    RatFunc.mk p q = ofFractionRing (algebraMap _ _ p / algebraMap _ _ q) := by rw [RatFunc.mk]\n\ntheorem mk_zero (p : K[X]) : RatFunc.mk p 0 = ofFractionRing (0 : FractionRing K[X]) := by\n  rw [mk_eq_div', map_zero, div_zero]\n\ntheorem mk_coe_def (p : K[X]) (q : K[X]⁰) :\n    RatFunc.mk p q = ofFractionRing (IsLocalization.mk' _ p q) := by\n  simp only [mk_eq_div', ← Localization.mk_eq_mk', FractionRing.mk_eq_div]\n\ntheorem mk_def_of_mem (p : K[X]) {q} (hq : q ∈ K[X]⁰) :\n    RatFunc.mk p q = ofFractionRing (IsLocalization.mk' (FractionRing K[X]) p ⟨q, hq⟩) := by\n  simp only [← mk_coe_def]\n\ntheorem mk_def_of_ne (p : K[X]) {q : K[X]} (hq : q ≠ 0) :\n    RatFunc.mk p q =\n      ofFractionRing (IsLocalization.mk' (FractionRing K[X]) p\n        ⟨q, mem_nonZeroDivisors_iff_ne_zero.mpr hq⟩) :=\n  mk_def_of_mem p _\n\ntheorem mk_eq_localization_mk (p : K[X]) {q : K[X]} (hq : q ≠ 0) :\n    RatFunc.mk p q =\n      ofFractionRing (Localization.mk p ⟨q, mem_nonZeroDivisors_iff_ne_zero.mpr hq⟩) := by\n  rw [mk_def_of_ne _ hq, Localization.mk_eq_mk']\n\nTarget:\ntheorem mk_one' (p : K[X]) :\n    RatFunc.mk p 1 = ofFractionRing (algebraMap _ _ p) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_2ba6bee60121","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"bc391f7444b504cd9d191cc20dda70df6909688d088f573ebc8144353ac8c3d0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/RatFunc","family_id":"mk_one","file_id":"mathlib/Mathlib/FieldTheory/RatFunc/Defs.lean","sample_id":"2ba6bee601213da996a0d324b595d9dabc6794f2b60ae27ff16595b74c0086ad"},"split":"train"}
{"binding":{"chosen_proof_sha256":"05a30ac01794d34e26abf0bb89eb49ff18aa58a6481d8190121651504b0314f8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b28f2d047f3f02792ce315aca654c65c6b846562c70579c8ee832efe4c79dcf3","source_sha256":"b5f291a07b438990d3572a453453e2d3f7dd16b400e7d0a6bac2c42be601d0ed","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases eq_or_ne n 0 with rfl | hn₀; · simp\n  rcases eq_or_ne p 1 with rfl | hp₁; · simp\n  simpa [*] using pow_dvd_iff_le_padicValNat (k := 1) hp₁ hn₀ |>.symm |>.not","hard_negative":false,"metrics":{"chosen_tokens":48,"rejected_tokens":2,"token_jaccard":0.03125,"token_length_ratio":0.041667},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"bc849a97d22dfe72805687731b79141d216761d9e734b17525b0c9f768097bf7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.MaxPowDiv\npublic import Mathlib.RingTheory.Multiplicity\npublic import Mathlib.Data.Nat.Factors\n\nNamespace:\npadicValNat\n\nLocal context:\n/-\nCopyright (c) 2018 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis, Matthew Robert Ballard\n-/\n/-!\n# `p`-adic Valuation\n\nThis file defines the `p`-adic valuation on `ℕ`, `ℤ`, and `ℚ`.\n\nThe `p`-adic valuation on `ℚ` is the difference of the multiplicities of `p` in the numerator and\ndenominator of `q`. This function obeys the standard properties of a valuation, with the appropriate\nassumptions on `p`. The `p`-adic valuations on `ℕ` and `ℤ` agree with that on `ℚ`.\n\nThe valuation induces a norm on `ℚ`. This norm is defined in\n`Mathlib/NumberTheory/Padics/PadicNorm.lean`.\n-/\n\n@[expose] public section\n\nassert_not_exists Field\n\nuniverse u\n\nopen Nat\n\nvariable {p : ℕ}\n\ntheorem padicValNat_eq_emultiplicity_of_ne_one (hp : p ≠ 1) {n : ℕ} (hn : n ≠ 0) :\n    padicValNat p n = emultiplicity p n := by\n  rw [eq_comm, emultiplicity_eq_coe, pow_dvd_iff_le_padicValNat hp hn,\n    pow_dvd_iff_le_padicValNat hp hn]\n  simp\n\n@[simp]\ntheorem Nat.toNat_emultiplicity (p n : ℕ) : (emultiplicity p n).toNat = padicValNat p n := by\n  rcases eq_or_ne p 1 with rfl | hp\n  · simp\n  · rcases eq_or_ne n 0 with rfl | hn\n    · simp\n    · simp [← padicValNat_eq_emultiplicity_of_ne_one, *]\n\ntheorem padicValNat_def' {n : ℕ} (hp : p ≠ 1) (hn : n ≠ 0) :\n    padicValNat p n = multiplicity p n :=\n  .symm <| multiplicity_eq_of_emultiplicity_eq_some <| .symm <|\n    padicValNat_eq_emultiplicity_of_ne_one hp hn\n\n/-- A simplification of `padicValNat` when one input is prime, by analogy with\n`padicValRat_def`. -/\ntheorem padicValNat_def [hp : Fact p.Prime] {n : ℕ} (hn : n ≠ 0) :\n    padicValNat p n = multiplicity p n :=\n  padicValNat_def' hp.out.ne_one hn\n\n/-- A simplification of `padicValNat` when one input is prime, by analogy with\n`padicValRat_def`. -/\ntheorem padicValNat_eq_emultiplicity [hp : Fact p.Prime] {n : ℕ} (hn : n ≠ 0) :\n    padicValNat p n = emultiplicity p n :=\n  padicValNat_eq_emultiplicity_of_ne_one hp.out.ne_one hn\n\nnamespace padicValNat\n\n@[deprecated (since := \"2026-03-15\")]\nalias maxPowDiv_eq_emultiplicity := padicValNat_eq_emultiplicity\n\n@[deprecated (since := \"2026-03-15\")]\nalias maxPowDiv_eq_multiplicity := padicValNat_def'\n\n@[deprecated padicValNat_zero_right (since := \"2026-03-15\")]\nprotected theorem zero : padicValNat p 0 = 0 := padicValNat_zero_right p\n\n@[deprecated padicValNat_one_right (since := \"2026-03-15\")]\nprotected theorem one : padicValNat p 1 = 0 := padicValNat_one_right p\n\n@[simp]\n\nTarget:\ntheorem eq_zero_iff {n : ℕ} : padicValNat p n = 0 ↔ p = 1 ∨ n = 0 ∨ ¬p ∣ n :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Padics","family_id":"eq_zero_iff","file_id":"mathlib/Mathlib/NumberTheory/Padics/PadicVal/Defs.lean","sample_id":"b28f2d047f3f02792ce315aca654c65c6b846562c70579c8ee832efe4c79dcf3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e064b140ea88edff320b5c2da73ccde8cd1103d2a90b6e31a432655ee413c425","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"049aeda8e9db27c47b238863ebb28fa2fd875d2c8748ec65c678cd547fc5bd2e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0329a22051730a8fdf389196512cdc9c856c4238f5a4d017fdd705ece3a68b4f","source_sha256":"a4af3b558cb27ae34b791ce56691a1ff72f8a093d6a0881a9bb6c593dcee4522","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Semigroup.mem_center_iff]\n  exact Iff.rfl","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":16,"token_jaccard":0.785714,"token_length_ratio":1.333333},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"bca5f429ade416255804bd341308e92162e9bc5de5739947c3f4417f57265e2a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Submonoid.Operations\npublic import Mathlib.GroupTheory.Subsemigroup.Center\n\nNamespace:\nSubmonoid\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Centers of monoids\n\n## Main definitions\n\n* `Submonoid.center`: the center of a monoid\n* `AddSubmonoid.center`: the center of an additive monoid\n\nWe provide `Subgroup.center`, `AddSubgroup.center`, `Subsemiring.center`, and `Subring.center` in\nother files.\n-/\n\n@[expose] public section\n\n-- Guard against import creep\nassert_not_exists Finset\n\nnamespace Submonoid\n\nsection MulOneClass\n\nvariable (M : Type*) [MulOneClass M]\n\n/-- The center of a multiplication with unit `M` is the set of elements that commute with everything\nin `M` -/\n@[to_additive\n/-- The center of an addition with zero `M` is the set of elements that commute with everything in\n`M` -/]\ndef center : Submonoid M where\n  carrier := Set.center M\n  one_mem' := Set.one_mem_center\n  mul_mem' := Set.mul_mem_center\n\n@[to_additive]\ntheorem coe_center : ↑(center M) = Set.center M :=\n  rfl\n\n@[to_additive (attr := simp) AddSubmonoid.center_toAddSubsemigroup]\ntheorem center_toSubsemigroup : (center M).toSubsemigroup = Subsemigroup.center M :=\n  rfl\n\ninstance {M α : Type*} [Monoid M] [MulAction M α] :\n    SMulCommClass ↥(Submonoid.center M) M α where\n  smul_comm c r v := by\n    have := Semigroup.mem_center_iff.1 c.2\n    simp_rw [Submonoid.smul_def, smul_smul, this]\n\ninstance {M α : Type*} [Monoid M] [MulAction M α] :\n    SMulCommClass M (Submonoid.center M) α :=\n  SMulCommClass.symm (Submonoid.center M) M α\n\nvariable {M}\n\n/-- The center of a multiplication with unit is commutative and associative.\n\nThis is not an instance as it forms a non-defeq diamond with `Submonoid.toMonoid` in the `npow`\nfield. -/\n@[to_additive /-- The center of an addition with zero is commutative and associative. -/]\nabbrev center.commMonoid' : CommMonoid (center M) :=\n  { (center M).toMulOneClass, Subsemigroup.center.commSemigroup with }\n\n@[to_additive]\nprotected theorem center_prod {N : Type*} [MulOneClass N] :\n    center (M × N) = prod (center M) (center N) :=\n  SetLike.coe_injective Set.center_prod\n\n@[to_additive]\nprotected theorem center_pi {ι : Type*} {M : ι → Type*} [Π i, MulOneClass (M i)] :\n    center (Π i, M i) = pi .univ fun i ↦ center (M i) :=\n  SetLike.coe_injective Set.center_pi\n\nend MulOneClass\n\nsection Monoid\n\nvariable {M} [Monoid M]\n\n/-- The center of a monoid is commutative. -/\n@[to_additive]\ninstance center.commMonoid : CommMonoid (center M) :=\n  { (center M).toMonoid, Subsemigroup.center.commSemigroup with }\n\n-- no instance diamond, unlike the primed version\nexample : center.commMonoid.toMonoid = Submonoid.toMonoid (center M) := by\n  with_reducible_and_instances rfl\n\n@[to_additive]\n\nTarget:\ntheorem mem_center_iff {z : M} : z ∈ center M ↔ ∀ g, g * z = z * g :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [← Semigroup.mem_center_iff]\n  exact Iff.rfl","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Submonoid","family_id":"mem_center_iff","file_id":"mathlib/Mathlib/GroupTheory/Submonoid/Center.lean","sample_id":"0329a22051730a8fdf389196512cdc9c856c4238f5a4d017fdd705ece3a68b4f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ab45fc24fe8d81ce6bc79891c129b27df70dce3fefc1109e3aecbeb52c81794e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0ccbfff2e32bc19e9550d2514637413d97b726466dde45498d6c8f3060c8d074","source_sha256":"18fda008e7b40ec7b81fa0fd9f7c55475c44e029c9af90e5639e54fa98674982","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext; simp [transvection.apply]","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":8,"token_jaccard":0.066667,"token_length_ratio":0.888889},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"bcb47f6a7eea62ba1dfe11277d6e7a300459db2548c614fc36ddd3244a07f846","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup\npublic import Mathlib.LinearAlgebra.Charpoly.BaseChange\npublic import Mathlib.LinearAlgebra.Dual.BaseChange\npublic import Mathlib.LinearAlgebra.Dual.Lemmas\npublic import Mathlib.LinearAlgebra.FixedSubmodule\n\nNamespace:\nLinearMap.transvection\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir\n-/\n\n/-!\n# Transvections in a module\n\n* When `f : Module.Dual R V` and `v : V`,\n  `LinearMap.transvection f v` is the linear map given by `x ↦ x + f x • v`,\n\n* `LinearMap.transvection.det` shows that the determinant of\n  `LinearMap.transvection f v` is equal to `1 + f v`.\n\n* If, moreover, `f v = 0`, then `LinearEquiv.transvection` shows that it is\n  a linear equivalence.\n\n* `LinearMap.transvections R V`: the set of transvections.\n\n* `LinearEquiv.dilatransvections R V`: the set of linear equivalences\n  whose associated linear map is of the form `LinearMap.transvection f v`.\n\n* `LinearEquiv.transvection.det` shows that it has determinant `1`.\n\n## Note on terminology\n\nIn the mathematical literature, linear maps of the form `LinearMap.transvection f v`\nare only called “transvections” when `f v = 0`. Otherwise, they are sometimes\ncalled “dilations” (especially if `f v ≠ -1`).\n\nThe definition is almost the same as that of `Module.preReflection f v`,\nup to a sign change, which are interesting when `f v = 2`, because they give “reflections”.\n\n-/\n\n@[expose] public section\n\nnamespace LinearMap\n\nopen Module\n\nvariable {R V : Type*} [Semiring R] [AddCommMonoid V] [Module R V]\n\n/-- The transvection associated with a linear form `f` and a vector `v`.\n\nNB. In mathematics, these linear maps are only called “transvections” when `f v = 0`.\nSee also `Module.preReflection` for a similar definition, up to a sign. -/\ndef transvection (f : Dual R V) (v : V) : V →ₗ[R] V where\n  toFun x := x + f x • v\n  map_add' x y := by simp [add_add_add_comm, add_smul]\n  map_smul' r x := by simp [smul_eq_mul, smul_add, mul_smul]\n\nnamespace transvection\n\nopen Submodule LinearMap\n\ntheorem apply (f : Dual R V) (v x : V) :\n    transvection f v x = x + f x • v :=\n  rfl\n\ntheorem comp_of_left_eq_apply {f : Dual R V} {v w : V} {x : V} (hw : f w = 0) :\n    transvection f v (transvection f w x) = transvection f (v + w) x := by\n  simp [transvection, map_add, hw, add_assoc]\n\ntheorem comp_of_left_eq {f : Dual R V} {v w : V} (hw : f w = 0) :\n    (transvection f v) ∘ₗ (transvection f w) = transvection f (v + w) := by\n  ext; simp [comp_of_left_eq_apply hw]\n\ntheorem comp_of_right_eq_apply {f g : Dual R V} {v : V} {x : V} (hf : f v = 0) :\n    (transvection f v) (transvection g v x) = transvection (f + g) v x := by\n  simp [transvection, map_add, hf, add_smul, add_assoc]\n\ntheorem comp_of_right_eq {f g : Dual R V} {v : V} (hf : f v = 0) :\n    (transvection f v) ∘ₗ (transvection g v) = transvection (f + g) v := by\n  ext; simp [comp_of_right_eq_apply hf]\n\n@[simp]\ntheorem of_left_eq_zero (v : V) :\n    transvection (0 : Dual R V) v = id := by\n  ext\n  simp [transvection]\n\n@[simp]\ntheorem of_right_eq_zero (f : Dual R V) :\n    transvection f 0 = id := by\n  ext\n  simp [transvection]\n\ntheorem comp_smul_smul {f : Dual R V} {v : V} {r s : R} :\n    transvection f (r • v) ∘ₗ transvection f (s • v) =\n      transvection f ((r + s + s * f v * r) • v) := by\n  ext x\n  simp only [LinearMap.comp_apply, apply, map_add, map_smul, add_assoc]\n  simp only [smul_add, ← mul_smul, ← add_smul, ← mul_add (f x), mul_assoc]\n\ntheorem eq_id_of_finrank_le_one\n    {R V : Type*} [CommSemiring R] [AddCommMonoid V] [Module R V]\n    [Free R V] [Module.Finite R V] [StrongRankCondition R]\n    {f : Dual R V} {v : V} (hfv : f v = 0) (h1 : finrank R V ≤ 1) :\n    transvection f v = id := by\n  interval_cases h : finrank R V\n  · have : Subsingleton V := (finrank_eq_zero_iff_of_free R V).mp h\n    simp [Subsingleton.eq_zero v]\n  · let b := finBasis R V\n    ext x\n    suffices f x • v = 0 by\n      simp [apply, this]\n    let i : Fin (finrank R V) := ⟨0, by simp [h]⟩\n    suffices ∀ x, x = b.repr x i • (b i) by\n      rw [this v, map_smul, smul_eq_mul, mul_comm] at hfv\n      rw [this x, this v, map_smul, smul_eq_mul, ← mul_smul, mul_assoc, hfv, mul_zero, zero_smul]\n    intro x\n    have : x = ∑ i, b.repr x i • b i := (b.sum_equivFun x).symm\n    rwa [Finset.sum_eq_single_of_mem i (Finset.mem_univ i) (by grind)] at this\n\nTarget:\ntheorem congr {W : Type*} [AddCommMonoid W] [Module R W]\n    (f : Dual R V) (v : V) (e : V ≃ₗ[R] W) :\n    e ∘ₗ (transvection f v) ∘ₗ e.symm = transvection (f ∘ₗ e.symm) (e v) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"6d2f6b138ae4587fe51c1217fff3c7de8aa90926a5011edd654f0604a9ecb3c2","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Transvection","family_id":"congr","file_id":"mathlib/Mathlib/LinearAlgebra/Transvection/Basic.lean","sample_id":"0ccbfff2e32bc19e9550d2514637413d97b726466dde45498d6c8f3060c8d074"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c1ea19ea49c645e697b6d5466464113860db944a733b14a16a41636e7325b4ec","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ef6b60ad5c0052ed74848bbc4c25cc9c35936112de2cf79b96b02c1f692ceaba","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3065ddea640d150a719f7ca546348a960da065f54e8ab9a9d7476b502ae8f9af","source_sha256":"8015c56b849258c13cb728b7c24ca8097da450018f8c58e78e02913505a6b7f7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨x, rfl⟩ := x.mk_surjective\n  obtain ⟨y, rfl⟩ := y.mk_surjective\n  obtain rfl : x = y := h\n  rfl","hard_negative":false,"metrics":{"chosen_tokens":30,"rejected_tokens":35,"token_jaccard":0.777778,"token_length_ratio":1.166667},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"bce75a3a5cd6933a3f1d3b90e8aea812f77f1c801138699125b60340ba43b96d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.SimplicialObject.Basic\npublic import Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal\npublic import Mathlib.AlgebraicTopology.SimplicialSet.CompStruct\npublic import Mathlib.AlgebraicTopology.SimplexCategory.Truncated\npublic import Mathlib.CategoryTheory.Category.ReflQuiv\npublic import Mathlib.Combinatorics.Quiver.ReflQuiver\npublic import Mathlib.AlgebraicTopology.SimplicialSet.Monoidal\npublic import Mathlib.CategoryTheory.Category.Cat.Terminal\n\nNamespace:\nSSet.Truncated.HomotopyCategory\n\nLocal context:\n/-\nCopyright (c) 2024 Mario Carneiro and Emily Riehl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Emily Riehl, Joël Riou\n-/\n/-!\n\n# The homotopy category of a simplicial set\n\nThe homotopy category of a simplicial set is defined as a quotient of the free category on its\nunderlying reflexive quiver (equivalently its one truncation). The quotient imposes an additional\nhom relation on this free category, asserting that `f ≫ g = h` whenever `f`, `g`, and `h` are\nrespectively the 2nd, 0th, and 1st faces of a 2-simplex.\n\nIn fact, the associated functor\n\n`SSet.hoFunctor : SSet.{u} ⥤ Cat.{u, u} := SSet.truncation 2 ⋙ SSet.hoFunctor₂`\n\nis defined by first restricting from simplicial sets to 2-truncated simplicial sets (throwing away\nthe data that is not used for the construction of the homotopy category) and then composing with an\nanalogously defined `SSet.hoFunctor₂ : SSet.Truncated.{u} 2 ⥤ Cat.{u,u}` implemented relative to\nthe syntax of the 2-truncated simplex category.\n\nIn the file `Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean` we show the functor\n`SSet.hoFunctor` to be left adjoint to the nerve by providing an analogous decomposition of the\nnerve functor, made by possible by the fact that nerves of categories are 2-coskeletal, and then\ncomposing a pair of adjunctions, which factor through the category of 2-truncated simplicial sets.\n-/\n\n@[expose] public section\n\nnamespace SSet\nopen CategoryTheory Category Limits Functor Opposite Simplicial Nerve\nopen SimplexCategory.Truncated SimplicialObject.Truncated\n\nuniverse v u\n\n/-- A 2-truncated simplicial set `S` has an underlying refl quiver with `S _⦋0⦌₂` as its underlying\ntype. -/\ndef OneTruncation₂ (S : SSet.Truncated 2) := S _⦋0⦌₂\n\nnamespace OneTruncation₂\n\n/-- A 2-truncated simplicial set `S` has an underlying refl quiver `SSet.OneTruncation₂ S`. -/\n@[simps -isSimp]\ninstance reflQuiver (S : SSet.Truncated 2) : ReflQuiver (OneTruncation₂ S) where\n  Hom := Truncated.Edge\n  id := Truncated.Edge.id\n\n@[ext]\nlemma hom_ext\n    {S : SSet.Truncated 2} {x y : OneTruncation₂ S} {f g : x ⟶ y}\n    (h : f.edge = g.edge) : f = g :=\n  Truncated.Edge.ext h\n\n/-- The prefunctor on refl quivers `OneTruncation₂` induced by a morphism\nof `2`-truncated simplicial sets. -/\n@[simps]\ndef map {S T : SSet.Truncated 2} (f : S ⟶ T) :\n    OneTruncation₂ S ⥤rq OneTruncation₂ T where\n  obj x := f.app _ x\n  map e := e.map f\n  map_id x := by ext; simp [← NatTrans.naturality_apply, reflQuiver_id]\n\nend OneTruncation₂\n\n/-- The functor that carries a 2-truncated simplicial set to its underlying refl quiver. -/\n@[simps]\ndef oneTruncation₂ : SSet.Truncated.{u} 2 ⥤ ReflQuiv.{u, u} where\n  obj S := ReflQuiv.of (OneTruncation₂ S)\n  map f := OneTruncation₂.map f\n\nnamespace OneTruncation₂\n\n@[simp]\nlemma homOfEq_edge\n    {X : SSet.Truncated.{u} 2} {x₁ y₁ x₂ y₂ : OneTruncation₂ X}\n    (f : x₁ ⟶ y₁) (hx : x₁ = x₂) (hy : y₁ = y₂) :\n    (Quiver.homOfEq f hx hy).edge = f.edge := by\n  subst hx hy\n  rfl\n\nsection\nvariable {C : Type u} [Category.{v} C]\n\n/-- An equivalence between the type of objects underlying a category and the type of 0-simplices in\nthe 2-truncated nerve. -/\n@[simps! -isSimp]\ndef nerveEquiv : OneTruncation₂ ((SSet.truncation 2).obj (nerve C)) ≃ C :=\n  CategoryTheory.nerveEquiv\n\n/-- A hom equivalence over the function `OneTruncation₂.nerveEquiv`. -/\ndef nerveHomEquiv {X Y : OneTruncation₂ ((SSet.truncation 2).obj (nerve C))} :\n    (X ⟶ Y) ≃ (nerveEquiv X ⟶ nerveEquiv Y) :=\n  nerve.homEquiv\n\nlemma nerveHomEquiv_apply {X Y : OneTruncation₂ ((SSet.truncation 2).obj (nerve C))}\n    (f : X ⟶ Y) :\n    nerveHomEquiv f = eqToHom (congr_arg ComposableArrows.left f.src_eq.symm) ≫\n      f.edge.hom ≫ eqToHom (congr_arg ComposableArrows.left f.tgt_eq) :=\n  rfl\n\n@[simp]\nlemma nerveHomEquiv_id (X : OneTruncation₂ ((SSet.truncation 2).obj (nerve C))) :\n    nerveHomEquiv (𝟙rq X) = 𝟙 _ :=\n  nerve.homEquiv_id _\n\n/-- The refl quiver underlying a nerve is isomorphic to the refl quiver underlying the category. -/\ndef ofNerve₂ (C : Type u) [Category.{u} C] :\n    ReflQuiv.of (OneTruncation₂ ((truncation 2).obj (nerve C))) ≅ ReflQuiv.of C :=\n  ReflQuiv.isoOfEquiv.{u, u} OneTruncation₂.nerveEquiv\n    (fun _ _ ↦ OneTruncation₂.nerveHomEquiv) nerveHomEquiv_id\n\nlemma nerve_hom_ext {X : (SSet.Truncated 2)} {C : Type u} [Category.{u} C]\n    {F G : X ⟶ ((truncation 2).obj (nerve C))}\n    (h : OneTruncation₂.map F = OneTruncation₂.map G) : F = G :=\n  SSet.Truncated.IsStrictSegal.hom_ext (fun f ↦ by\n    obtain ⟨x₀, x₁, f, rfl⟩ := Truncated.Edge.exists_of_simplex f\n    simpa using congr_arg Truncated.Edge.edge (ReflPrefunctor.congr_hom h f))\n\nend\nend OneTruncation₂\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The refl quiver underlying a nerve is naturally isomorphic to the refl quiver underlying the\ncategory. -/\n@[simps! hom_app_obj hom_app_map inv_app_obj_obj inv_app_obj_map inv_app_map]\ndef OneTruncation₂.ofNerve₂.natIso :\n    nerveFunctor₂.{u, u} ⋙ SSet.oneTruncation₂ ≅ ReflQuiv.forget :=\n  NatIso.ofComponents (fun C => OneTruncation₂.ofNerve₂ C)\n    (fun F ↦ ReflPrefunctor.ext (by cat_disch) (fun x y f ↦ by\n      obtain ⟨f, rfl, rfl⟩ := f\n      dsimp [ofNerve₂, ReflQuiv.isoOfEquiv, ReflQuiv.isoOfQuivIso,\n        Quiv.isoOfEquiv, nerveHomEquiv_apply]\n      simp only [comp_id, id_comp]\n      rfl))\n\nset_option backward.privateInPublic true in\nprivate lemma map_map_of_eq.{w} {C : Type u} [Category.{v} C] (V : Cᵒᵖ ⥤ Type w) {X Y Z : C}\n    {α : X ⟶ Y} {β : Y ⟶ Z} {γ : X ⟶ Z} {φ} :\n    α ≫ β = γ → V.map α.op (V.map β.op φ) = V.map γ.op φ := by\n  rintro rfl\n  simp\n\nnamespace Truncated\n\n/-- The map that picks up the initial vertex of a 2-simplex, as a morphism in the 2-truncated\nsimplex category. -/\ndef ι0₂ : ⦋0⦌₂ ⟶ ⦋2⦌₂ := δ₂ (n := 0) 1 ≫ δ₂ (n := 1) 1\n\n/-- The map that picks up the middle vertex of a 2-simplex, as a morphism in the 2-truncated\nsimplex category. -/\ndef ι1₂ : ⦋0⦌₂ ⟶ ⦋2⦌₂ := δ₂ (n := 0) 0 ≫ δ₂ (n := 1) 2\n\n/-- The map that picks up the final vertex of a 2-simplex, as a morphism in the 2-truncated\nsimplex category. -/\ndef ι2₂ : ⦋0⦌₂ ⟶ ⦋2⦌₂ := δ₂ (n := 0) 0 ≫ δ₂ (n := 1) 1\n\n/-- The initial vertex of a 2-simplex in a 2-truncated simplicial set. -/\ndef ev0₂ {V : SSet.Truncated 2} (φ : V _⦋2⦌₂) : OneTruncation₂ V := V.map ι0₂.op φ\n\n/-- The middle vertex of a 2-simplex in a 2-truncated simplicial set. -/\ndef ev1₂ {V : SSet.Truncated 2} (φ : V _⦋2⦌₂) : OneTruncation₂ V := V.map ι1₂.op φ\n\n/-- The final vertex of a 2-simplex in a 2-truncated simplicial set. -/\ndef ev2₂ {V : SSet.Truncated 2} (φ : V _⦋2⦌₂) : OneTruncation₂ V := V.map ι2₂.op φ\n\n/-- The 0th face of a 2-simplex, as a morphism in the 2-truncated simplex category. -/\ndef δ0₂ : ⦋1⦌₂ ⟶ ⦋2⦌₂ := δ₂ (n := 1) 0\n\n/-- The 1st face of a 2-simplex, as a morphism in the 2-truncated simplex category. -/\ndef δ1₂ : ⦋1⦌₂ ⟶ ⦋2⦌₂ := δ₂ (n := 1) 1\n\n/-- The 2nd face of a 2-simplex, as a morphism in the 2-truncated simplex category. -/\ndef δ2₂ : ⦋1⦌₂ ⟶ ⦋2⦌₂ := δ₂ (n := 1) 2\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- The arrow in the ReflQuiver `OneTruncation₂ V` of a 2-truncated simplicial set arising from the\n0th face of a 2-simplex. -/\ndef ev12₂ {V : SSet.Truncated 2} (φ : V _⦋2⦌₂) : ev1₂ φ ⟶ ev2₂ φ :=\n  ⟨V.map δ0₂.op φ,\n    map_map_of_eq V (InducedCategory.hom_ext\n      (SimplexCategory.δ_comp_δ (i := 0) (j := 1) (by decide)).symm),\n    map_map_of_eq V rfl⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- The arrow in the ReflQuiver `OneTruncation₂ V` of a 2-truncated simplicial set arising from the\n1st face of a 2-simplex. -/\ndef ev02₂ {V : SSet.Truncated 2} (φ : V _⦋2⦌₂) : ev0₂ φ ⟶ ev2₂ φ :=\n  ⟨V.map δ1₂.op φ, map_map_of_eq V rfl, map_map_of_eq V rfl⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- The arrow in the ReflQuiver `OneTruncation₂ V` of a 2-truncated simplicial set arising from the\n2nd face of a 2-simplex. -/\ndef ev01₂ {V : SSet.Truncated 2} (φ : V _⦋2⦌₂) : ev0₂ φ ⟶ ev1₂ φ :=\n  ⟨V.map δ2₂.op φ,\n    map_map_of_eq V (InducedCategory.hom_ext (SimplexCategory.δ_comp_δ (j := 1) le_rfl)),\n    map_map_of_eq V rfl⟩\n\nend Truncated\n\nnamespace OneTruncation₂\n\nvariable (V : SSet.Truncated.{u} 2)\n\n/-- The 2-simplices in a 2-truncated simplicial set `V` generate a hom relation on the free\ncategory on the underlying refl quiver of `V`. -/\ninductive HoRel₂ : HomRel (Cat.FreeRefl (OneTruncation₂ V)) where\n  | of_compStruct {x₀ x₁ x₂ : V _⦋0⦌₂} {e₀₁ : Truncated.Edge x₀ x₁}\n    {e₁₂ : Truncated.Edge x₁ x₂} {e₀₂ : Truncated.Edge x₀ x₂}\n    (h : Truncated.Edge.CompStruct e₀₁ e₁₂ e₀₂) :\n    HoRel₂\n      ((Cat.FreeRefl.quotientFunctor (OneTruncation₂ V)).map\n        (Quiver.Hom.toPath e₀₁ ≫ Quiver.Hom.toPath e₁₂))\n      ((Cat.FreeRefl.quotientFunctor (OneTruncation₂ V)).map (Quiver.Hom.toPath e₀₂))\n\nend OneTruncation₂\n\nnamespace Truncated\n\nvariable (V W : SSet.Truncated.{u} 2)\n\n/-- The type underlying the homotopy category of a 2-truncated simplicial set `V`. -/\ndef HomotopyCategory : Type u :=\n  Quotient (OneTruncation₂.HoRel₂ V)\n  deriving Category.{u}\n\nnamespace HomotopyCategory\n\n/-- A canonical functor from the free category on the refl quiver underlying a 2-truncated\nsimplicial set `V` to its homotopy category. -/\ndef quotientFunctor :\n    Cat.FreeRefl (OneTruncation₂ V) ⥤ V.HomotopyCategory :=\n  Quotient.functor _\n\ninstance : (quotientFunctor V).Full :=\n  Quotient.full_functor _\n\nvariable {V}\n\n/-- Constructor for objects of the homotopy category of a `2`-truncated simplicial set. -/\ndef mk (x : V _⦋0⦌₂) : V.HomotopyCategory :=\n  (quotientFunctor V).obj (.mk x)\n\nlemma mk_surjective : Function.Surjective (mk (V := V)) := by\n  rintro ⟨⟨x⟩⟩\n  exact ⟨x, rfl⟩\n\nTarget:\nlemma ext {x y : V.HomotopyCategory} (h : x.as.as = y.as.as) : x = y :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  obtain ⟨x, rfl⟩ := x.mk_surjective\n  obtain ⟨y, rfl⟩ := y.mk_surjective\n  obtain rfl : x = y := h\n  rfl","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/SimplicialSet","family_id":"ext","file_id":"mathlib/Mathlib/AlgebraicTopology/SimplicialSet/HomotopyCat.lean","sample_id":"3065ddea640d150a719f7ca546348a960da065f54e8ab9a9d7476b502ae8f9af"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cb2db471013d02747d266722ac3b318ad74596861a15ff8911bcdfb53d010405","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7730bf9984af8e1d55b7e5d3174ec963362e837effdd837bd031f3763c30060e","source_sha256":"a09b362ac29d7043b68fab55b278603330bcba1220d1dd2dd405ff015cd23f6a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [GradedAlgebra.proj_apply, decompose_symm_of, Equiv.apply_symm_apply]","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.153846},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"bcfefc018fa695ae83ee562279b82fb0022637279091aee12ebc6749d3be166c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.DirectSum.Algebra\npublic import Mathlib.Algebra.DirectSum.Decomposition\npublic import Mathlib.Algebra.DirectSum.Internal\npublic import Mathlib.Algebra.DirectSum.Ring\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser, Kevin Buzzard, Jujian Zhang, Fangming Li\n-/\n/-!\n# Internally-graded rings and algebras\n\nThis file defines the typeclass `GradedAlgebra 𝒜`, for working with an algebra `A` that is\ninternally graded by a collection of submodules `𝒜 : ι → Submodule R A`.\nSee the docstring of that typeclass for more information.\n\n## Main definitions\n\n* `GradedRing 𝒜`: the typeclass, which is a combination of `SetLike.GradedMonoid`, and\n  `DirectSum.Decomposition 𝒜`.\n* `GradedAlgebra 𝒜`: A convenience alias for `GradedRing` when `𝒜` is a family of submodules.\n* `DirectSum.decomposeRingEquiv 𝒜 : A ≃ₐ[R] ⨁ i, 𝒜 i`, a more bundled version of\n  `DirectSum.decompose 𝒜`.\n* `DirectSum.decomposeAlgEquiv 𝒜 : A ≃ₐ[R] ⨁ i, 𝒜 i`, a more bundled version of\n  `DirectSum.decompose 𝒜`.\n* `GradedAlgebra.proj 𝒜 i` is the linear map from `A` to its degree `i : ι` component, such that\n  `proj 𝒜 i x = decompose 𝒜 x i`.\n\n## Implementation notes\n\nFor now, we do not have internally-graded semirings and internally-graded rings; these can be\nrepresented with `𝒜 : ι → Submodule ℕ A` and `𝒜 : ι → Submodule ℤ A` respectively, since all\n`Semiring`s are ℕ-algebras via `Semiring.toNatAlgebra`, and all `Ring`s are `ℤ`-algebras via\n`Ring.toIntAlgebra`.\n\n## Tags\n\ngraded algebra, graded ring, graded semiring, decomposition\n-/\n\n@[expose] public section\n\n\nopen DirectSum\n\nvariable {ι R A σ : Type*}\n\nsection GradedRing\n\nvariable [DecidableEq ι] [AddMonoid ι] [CommSemiring R] [Semiring A] [Algebra R A]\nvariable [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ)\n\nopen DirectSum\n\n/-- An internally-graded `R`-algebra `A` is one that can be decomposed into a collection\nof `Submodule R A`s indexed by `ι` such that the canonical map `A → ⨁ i, 𝒜 i` is bijective and\nrespects multiplication, i.e. the product of an element of degree `i` and an element of degree `j`\nis an element of degree `i + j`.\n\nNote that the fact that `A` is internally-graded, `GradedAlgebra 𝒜`, implies an externally-graded\nalgebra structure `DirectSum.GAlgebra R (fun i ↦ ↥(𝒜 i))`, which in turn makes available an\n`Algebra R (⨁ i, 𝒜 i)` instance.\n-/\nclass GradedRing (𝒜 : ι → σ) extends SetLike.GradedMonoid 𝒜, DirectSum.Decomposition 𝒜\n\nvariable [GradedRing 𝒜]\n\nnamespace DirectSum\n\n/-- If `A` is graded by `ι` with degree `i` component `𝒜 i`, then it is isomorphic as\na ring to a direct sum of components. -/\ndef decomposeRingEquiv : A ≃+* ⨁ i, 𝒜 i :=\n  RingEquiv.symm\n    { (decomposeAddEquiv 𝒜).symm with\n      map_mul' := (coeRingHom 𝒜).map_mul }\n\n@[simp]\ntheorem decompose_one : decompose 𝒜 (1 : A) = 1 :=\n  map_one (decomposeRingEquiv 𝒜)\n\n@[simp]\ntheorem decompose_symm_one : (decompose 𝒜).symm 1 = (1 : A) :=\n  map_one (decomposeRingEquiv 𝒜).symm\n\n@[simp]\ntheorem decompose_mul (x y : A) : decompose 𝒜 (x * y) = decompose 𝒜 x * decompose 𝒜 y :=\n  map_mul (decomposeRingEquiv 𝒜) x y\n\n@[simp]\ntheorem decompose_symm_mul (x y : ⨁ i, 𝒜 i) :\n    (decompose 𝒜).symm (x * y) = (decompose 𝒜).symm x * (decompose 𝒜).symm y :=\n  map_mul (decomposeRingEquiv 𝒜).symm x y\n\nend DirectSum\n\n/-- The projection maps of a graded ring -/\ndef GradedRing.proj (i : ι) : A →+ A :=\n  (AddSubmonoidClass.subtype (𝒜 i)).comp <|\n    (DFinsupp.evalAddMonoidHom i).comp <|\n      RingHom.toAddMonoidHom <| RingEquiv.toRingHom <| DirectSum.decomposeRingEquiv 𝒜\n\n@[simp]\ntheorem GradedRing.proj_apply (i : ι) (r : A) :\n    GradedRing.proj 𝒜 i r = (decompose 𝒜 r : ⨁ i, 𝒜 i) i :=\n  rfl\n\ntheorem GradedRing.proj_recompose (a : ⨁ i, 𝒜 i) (i : ι) :\n    GradedRing.proj 𝒜 i ((decompose 𝒜).symm a) = (decompose 𝒜).symm (DirectSum.of _ i (a i)) := by\n  rw [GradedRing.proj_apply, decompose_symm_of, Equiv.apply_symm_apply]\n\ntheorem GradedRing.mem_support_iff [∀ (i) (x : 𝒜 i), Decidable (x ≠ 0)] (r : A) (i : ι) :\n    i ∈ (decompose 𝒜 r).support ↔ GradedRing.proj 𝒜 i r ≠ 0 :=\n  DFinsupp.mem_support_iff.trans ZeroMemClass.coe_eq_zero.not.symm\n\nend GradedRing\n\nsection AddCancelMonoid\n\nopen DirectSum\n\nvariable [DecidableEq ι] [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ)\nvariable {i j : ι}\n\nnamespace DirectSum\n\ntheorem coe_decompose_mul_add_of_left_mem [AddLeftCancelMonoid ι] [GradedRing 𝒜] {a b : A}\n    (a_mem : a ∈ 𝒜 i) : (decompose 𝒜 (a * b) (i + j) : A) = a * decompose 𝒜 b j := by\n  lift a to 𝒜 i using a_mem\n  rw [decompose_mul, decompose_coe, coe_of_mul_apply_add]\n\ntheorem coe_decompose_mul_add_of_right_mem [AddRightCancelMonoid ι] [GradedRing 𝒜] {a b : A}\n    (b_mem : b ∈ 𝒜 j) : (decompose 𝒜 (a * b) (i + j) : A) = decompose 𝒜 a i * b := by\n  lift b to 𝒜 j using b_mem\n  rw [decompose_mul, decompose_coe, coe_mul_of_apply_add]\n\ntheorem decompose_mul_add_left [AddLeftCancelMonoid ι] [GradedRing 𝒜] (a : 𝒜 i) {b : A} :\n    decompose 𝒜 (↑a * b) (i + j) =\n      @GradedMonoid.GMul.mul ι (fun i => 𝒜 i) _ _ _ _ a (decompose 𝒜 b j) :=\n  Subtype.ext <| coe_decompose_mul_add_of_left_mem 𝒜 a.2\n\ntheorem decompose_mul_add_right [AddRightCancelMonoid ι] [GradedRing 𝒜] {a : A} (b : 𝒜 j) :\n    decompose 𝒜 (a * ↑b) (i + j) =\n      @GradedMonoid.GMul.mul ι (fun i => 𝒜 i) _ _ _ _ (decompose 𝒜 a i) b :=\n  Subtype.ext <| coe_decompose_mul_add_of_right_mem 𝒜 b.2\n\ntheorem coe_decompose_mul_of_left_mem_zero [AddMonoid ι] [GradedRing 𝒜] {a b : A}\n    (a_mem : a ∈ 𝒜 0) : (decompose 𝒜 (a * b) j : A) = a * decompose 𝒜 b j := by\n  lift a to 𝒜 0 using a_mem\n  rw [decompose_mul, decompose_coe, coe_of_mul_apply_of_mem_zero]\n\ntheorem coe_decompose_mul_of_right_mem_zero [AddMonoid ι] [GradedRing 𝒜] {a b : A}\n    (b_mem : b ∈ 𝒜 0) : (decompose 𝒜 (a * b) i : A) = decompose 𝒜 a i * b := by\n  lift b to 𝒜 0 using b_mem\n  rw [decompose_mul, decompose_coe, coe_mul_of_apply_of_mem_zero]\n\nend DirectSum\n\nend AddCancelMonoid\n\nsection GradedAlgebra\n\nvariable [DecidableEq ι] [AddMonoid ι] [CommSemiring R] [Semiring A] [Algebra R A]\nvariable (𝒜 : ι → Submodule R A)\n\n/-- A special case of `GradedRing` with `σ = Submodule R A`. This is useful both because it\ncan avoid typeclass search, and because it provides a more concise name. -/\nabbrev GradedAlgebra :=\n  GradedRing 𝒜\n\n/-- A helper to construct a `GradedAlgebra` when the `SetLike.GradedMonoid` structure is already\navailable. This makes the `left_inv` condition easier to prove, and phrases the `right_inv`\ncondition in a way that allows custom `@[ext]` lemmas to apply.\n\nSee note [reducible non-instances]. -/\nabbrev GradedAlgebra.ofAlgHom [SetLike.GradedMonoid 𝒜] (decompose : A →ₐ[R] ⨁ i, 𝒜 i)\n    (right_inv : (DirectSum.coeAlgHom 𝒜).comp decompose = AlgHom.id R A)\n    (left_inv : ∀ i (x : 𝒜 i), decompose (x : A) = DirectSum.of (fun i => ↥(𝒜 i)) i x) :\n    GradedAlgebra 𝒜 where\n  decompose' := decompose\n  left_inv := AlgHom.congr_fun right_inv\n  right_inv := by\n    suffices decompose.comp (DirectSum.coeAlgHom 𝒜) = AlgHom.id _ _ from AlgHom.congr_fun this\n    ext i x : 2\n    exact (decompose.congr_arg <| DirectSum.coeAlgHom_of _ _ _).trans (left_inv i x)\n\ninstance (R₀ : Type*) [CommSemiring R₀] [Algebra R₀ R] [Algebra R₀ A] [IsScalarTower R₀ R A]\n    [i : GradedAlgebra 𝒜] : GradedAlgebra (𝒜 · |>.restrictScalars R₀) := { i with }\n\nvariable [GradedAlgebra 𝒜]\n\nnamespace DirectSum\n\n/-- If `A` is graded by `ι` with degree `i` component `𝒜 i`, then it is isomorphic as\nan algebra to a direct sum of components. -/\n-- We have to write the `@[simps]` lemmas by hand to see through the\n-- `AlgEquiv.symm (decomposeAddEquiv 𝒜).symm`.\ndef decomposeAlgEquiv : A ≃ₐ[R] ⨁ i, 𝒜 i :=\n  AlgEquiv.symm\n    { (decomposeAddEquiv 𝒜).symm with\n      map_mul' := map_mul (coeAlgHom 𝒜)\n      commutes' := (coeAlgHom 𝒜).commutes }\n\n@[simp]\nlemma decomposeAlgEquiv_apply (a : A) :\n    decomposeAlgEquiv 𝒜 a = decompose 𝒜 a := rfl\n\n@[simp]\nlemma decomposeAlgEquiv_symm_apply (a : ⨁ i, 𝒜 i) :\n    (decomposeAlgEquiv 𝒜).symm a = (decompose 𝒜).symm a := rfl\n\n@[simp]\nlemma decompose_algebraMap (r : R) :\n    decompose 𝒜 (algebraMap R A r) = algebraMap R (⨁ i, 𝒜 i) r :=\n  (decomposeAlgEquiv 𝒜).commutes r\n\n@[simp]\nlemma decompose_symm_algebraMap (r : R) :\n    (decompose 𝒜).symm (algebraMap R (⨁ i, 𝒜 i) r) = algebraMap R A r :=\n  (decomposeAlgEquiv 𝒜).symm.commutes r\n\nend DirectSum\n\nopen DirectSum\n\n/-- The projection maps of graded algebra -/\ndef GradedAlgebra.proj (𝒜 : ι → Submodule R A) [GradedAlgebra 𝒜] (i : ι) : A →ₗ[R] A :=\n  (𝒜 i).subtype.comp <| (DFinsupp.lapply i).comp <| (decomposeAlgEquiv 𝒜).toAlgHom.toLinearMap\n\n@[simp]\ntheorem GradedAlgebra.proj_apply (i : ι) (r : A) :\n    GradedAlgebra.proj 𝒜 i r = (decompose 𝒜 r : ⨁ i, 𝒜 i) i :=\n  rfl\n\nTarget:\ntheorem GradedAlgebra.proj_recompose (a : ⨁ i, 𝒜 i) (i : ι) :\n    GradedAlgebra.proj 𝒜 i ((decompose 𝒜).symm a) = (decompose 𝒜).symm (of _ i (a i)) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/GradedAlgebra","family_id":"gradedalgebra","file_id":"mathlib/Mathlib/RingTheory/GradedAlgebra/Basic.lean","sample_id":"7730bf9984af8e1d55b7e5d3174ec963362e837effdd837bd031f3763c30060e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"930122cf7815506e3d0570b55467812e4caec3bf1c598928f36877dea55601fa","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"636a9fcc3e8a95e1374f41b3672c8c890f7576b9c9fa7d5f0caae9002e2a7ade","source_sha256":"d6fe9940fe5118171b24b843fb7690a27e16198b1d58f5f56373a58891dc6904","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only [h.comp_eq] using bijOn_fixedPoints_comp f g","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":2,"token_jaccard":0.076923,"token_length_ratio":0.166667},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"bd959678112e1e5907efe9d8f175cd78349bc5212734c961e8bb653c333d14f5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.End\npublic import Mathlib.Data.Set.Function\npublic import Mathlib.Dynamics.FixedPoints.Defs\n\nNamespace:\nFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Fixed points of a self-map\n\nWe prove some simple lemmas about `IsFixedPt` and `∘`, `iterate`, and `Semiconj`.\n\n## Tags\n\nfixed point\n-/\n\npublic section\n\nopen Equiv\n\nuniverse u v\n\nvariable {α β : Type*} {f fa g : α → α} {x : α} {fb : β → β} {e : Perm α}\n\nnamespace Function\n\nopen Function (Commute)\n\nnamespace IsFixedPt\n\n/-- If `x` is a fixed point of `f` and `g`, then it is a fixed point of `f ∘ g`. -/\nprotected theorem comp (hf : IsFixedPt f x) (hg : IsFixedPt g x) : IsFixedPt (f ∘ g) x :=\n  calc\n    f (g x) = f x := congr_arg f hg\n    _ = x := hf\n\n/-- If `x` is a fixed point of `f`, then it is a fixed point of `f^[n]`. -/\nprotected theorem iterate (hf : IsFixedPt f x) (n : ℕ) : IsFixedPt f^[n] x :=\n  iterate_fixed hf n\n\n/-- If `x` is a fixed point of `f ∘ g` and `g`, then it is a fixed point of `f`. -/\ntheorem left_of_comp (hfg : IsFixedPt (f ∘ g) x) (hg : IsFixedPt g x) : IsFixedPt f x :=\n  calc\n    f x = f (g x) := congr_arg f hg.symm\n    _ = x := hfg\n\n/-- If `x` is a fixed point of `f` and `g` is a left inverse of `f`, then `x` is a fixed\npoint of `g`. -/\ntheorem to_leftInverse (hf : IsFixedPt f x) (h : LeftInverse g f) : IsFixedPt g x :=\n  calc\n    g x = g (f x) := congr_arg g hf.symm\n    _ = x := h x\n\n/-- If `g` (semi)conjugates `fa` to `fb`, then it sends fixed points of `fa` to fixed points\nof `fb`. -/\nprotected theorem map {x : α} (hx : IsFixedPt fa x) {g : α → β} (h : Semiconj g fa fb) :\n    IsFixedPt fb (g x) :=\n  calc\n    fb (g x) = g (fa x) := (h.eq x).symm\n    _ = g x := congr_arg g hx\n\nprotected theorem apply {x : α} (hx : IsFixedPt f x) : IsFixedPt f (f x) := by convert! hx\n\ntheorem preimage_iterate {s : Set α} (h : IsFixedPt (Set.preimage f) s) (n : ℕ) :\n    IsFixedPt (Set.preimage f^[n]) s := by\n  rw [Set.preimage_iterate_eq]\n  exact h.iterate n\n\nlemma image_iterate {s : Set α} (h : IsFixedPt (Set.image f) s) (n : ℕ) :\n    IsFixedPt (Set.image f^[n]) s :=\n  Set.image_iterate_eq ▸ h.iterate n\n\nprotected theorem equiv_symm (h : IsFixedPt e x) : IsFixedPt e.symm x :=\n  h.to_leftInverse e.leftInverse_symm\n\n@[simp]\ntheorem equiv_symm_iff : IsFixedPt e.symm x ↔ IsFixedPt e x :=\n  ⟨fun h ↦ e.symm_symm ▸ h.equiv_symm, .equiv_symm⟩\n\nprotected theorem perm_inv (h : IsFixedPt e x) : IsFixedPt (⇑e⁻¹) x :=\n  h.equiv_symm\n\nprotected theorem perm_pow (h : IsFixedPt e x) (n : ℕ) : IsFixedPt (⇑(e ^ n)) x := h.iterate _\n\nprotected theorem perm_zpow (h : IsFixedPt e x) : ∀ n : ℤ, IsFixedPt (⇑(e ^ n)) x\n  | Int.ofNat _ => h.perm_pow _\n  | Int.negSucc n => (h.perm_pow <| n + 1).perm_inv\n\nend IsFixedPt\n\n@[simp]\ntheorem fixedPoints_symm : fixedPoints e.symm = fixedPoints e := by\n  simp [Set.ext_iff]\n\n@[simp]\ntheorem Injective.isFixedPt_apply_iff (hf : Injective f) {x : α} :\n    IsFixedPt f (f x) ↔ IsFixedPt f x :=\n  ⟨fun h => hf h.eq, IsFixedPt.apply⟩\n\n/-- If `g` semiconjugates `fa` to `fb`, then it sends fixed points of `fa` to fixed points\nof `fb`. -/\ntheorem Semiconj.mapsTo_fixedPoints {g : α → β} (h : Semiconj g fa fb) :\n    Set.MapsTo g (fixedPoints fa) (fixedPoints fb) := fun _ hx => hx.map h\n\n/-- Any two maps `f : α → β` and `g : β → α` are inverse of each other on the sets of fixed points\nof `f ∘ g` and `g ∘ f`, respectively. -/\ntheorem invOn_fixedPoints_comp (f : α → β) (g : β → α) :\n    Set.InvOn f g (fixedPoints <| f ∘ g) (fixedPoints <| g ∘ f) :=\n  ⟨fun _ => id, fun _ => id⟩\n\n/-- Any map `f` sends fixed points of `g ∘ f` to fixed points of `f ∘ g`. -/\ntheorem mapsTo_fixedPoints_comp (f : α → β) (g : β → α) :\n    Set.MapsTo f (fixedPoints <| g ∘ f) (fixedPoints <| f ∘ g) := fun _ hx => hx.map fun _ => rfl\n\n/-- Given two maps `f : α → β` and `g : β → α`, `g` is a bijective map between the fixed points\nof `f ∘ g` and the fixed points of `g ∘ f`. The inverse map is `f`, see `invOn_fixedPoints_comp`. -/\ntheorem bijOn_fixedPoints_comp (f : α → β) (g : β → α) :\n    Set.BijOn g (fixedPoints <| f ∘ g) (fixedPoints <| g ∘ f) :=\n  (invOn_fixedPoints_comp f g).bijOn (mapsTo_fixedPoints_comp g f) (mapsTo_fixedPoints_comp f g)\n\n/-- If self-maps `f` and `g` commute, then they are inverse of each other on the set of fixed points\nof `f ∘ g`. This is a particular case of `Function.invOn_fixedPoints_comp`. -/\ntheorem Commute.invOn_fixedPoints_comp (h : Commute f g) :\n    Set.InvOn f g (fixedPoints <| f ∘ g) (fixedPoints <| f ∘ g) := by\n  simpa only [h.comp_eq] using Function.invOn_fixedPoints_comp f g\n\n/-- If self-maps `f` and `g` commute, then `f` is bijective on the set of fixed points of `f ∘ g`.\nThis is a particular case of `Function.bijOn_fixedPoints_comp`. -/\ntheorem Commute.left_bijOn_fixedPoints_comp (h : Commute f g) :\n    Set.BijOn f (fixedPoints <| f ∘ g) (fixedPoints <| f ∘ g) := by\n  simpa only [h.comp_eq] using bijOn_fixedPoints_comp g f\n\n/-- If self-maps `f` and `g` commute, then `g` is bijective on the set of fixed points of `f ∘ g`.\nThis is a particular case of `Function.bijOn_fixedPoints_comp`. -/\n\nTarget:\ntheorem Commute.right_bijOn_fixedPoints_comp (h : Commute f g) :\n    Set.BijOn g (fixedPoints <| f ∘ g) (fixedPoints <| f ∘ g) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/FixedPoints","family_id":"commute","file_id":"mathlib/Mathlib/Dynamics/FixedPoints/Basic.lean","sample_id":"636a9fcc3e8a95e1374f41b3672c8c890f7576b9c9fa7d5f0caae9002e2a7ade"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b276acfbf4317cef3c8ddc23d0eda431624fd9c228809a52c5678c7b8d9bd426","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"02daabd48e88e8b51fc3ee4d773fb0f3f96c121b52e8a9bd621f98c16855be86","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"79c5185e6f28d9b4a517c1d0991b691ef5318525bdf3a4fee785bfeb02cb287e","source_sha256":"b645158f5be7fdfcbd42911af4e66fd4a79189395cdeb22443a3f6445ef45b71","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ENNReal.eq_sub_of_add_eq hg_fin ?_\n  rw [← lintegral_add_right' _ hg]\n  exact lintegral_congr_ae (h_le.mono fun x hx => tsub_add_cancel_of_le hx)","hard_negative":false,"metrics":{"chosen_tokens":28,"rejected_tokens":35,"token_jaccard":0.925926,"token_length_ratio":1.25},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"bda2c8ba66549de33f1c14eee7675f3d997f91806e754da90cfd4f5f14ef2889","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Integral.Lebesgue.Add\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Johannes Hölzl\n-/\n/-!\n# Subtraction of Lebesgue integrals\n\nIn this file we first show that Lebesgue integrals can be subtracted with the expected results –\n`∫⁻ f - ∫⁻ g ≤ ∫⁻ (f - g)`, with equality if `g ≤ f` almost everywhere. Then we prove variants of\nthe monotone convergence theorem that use this subtraction in their proofs.\n-/\n\npublic section\n\nopen Filter ENNReal Topology\n\nnamespace MeasureTheory\n\nvariable {α β : Type*} [MeasurableSpace α] {μ : Measure α}\n\nTarget:\ntheorem lintegral_sub' {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hg_fin : ∫⁻ a, g a ∂μ ≠ ∞)\n    (h_le : g ≤ᵐ[μ] f) : ∫⁻ a, f a - g a ∂μ = ∫⁻ a, f a ∂μ - ∫⁻ a, g a ∂μ :=\n\nProof body:\n","rejected":"```lean\nby\n  refine ENNReal.eq_sub_of_add_eq hg_fin ?_\n  rw [← lintegral_add_right' _ hg]\n  exact lintegral_congr_ae (h_le.mono fun x hx => tsub_add_cancel_of_le hx)\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Integral","family_id":"lintegral_sub","file_id":"mathlib/Mathlib/MeasureTheory/Integral/Lebesgue/Sub.lean","sample_id":"79c5185e6f28d9b4a517c1d0991b691ef5318525bdf3a4fee785bfeb02cb287e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"158b158a558370a66bb2762fb6673634c41f0d235bb36659662b90d5aa0cb875","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"335fd55413b4f96955767217c73d17f78f377c305db1438e043702f98d245c46","source_sha256":"059da84dda9e7cf638670a1c074b01de35e7641bc88d590bf894e2e5fc5223e6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  contrapose! +distrib\n  rw [Set.infinite_image2 hfs hft]\n  grind only [Set.Infinite.nonempty]","hard_negative":false,"metrics":{"chosen_tokens":22,"rejected_tokens":2,"token_jaccard":0.055556,"token_length_ratio":0.090909},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"bdbf38fa8875d55cfe3efa84fe278244e878cd5831329cde87918a0006064b0e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Basic\npublic import Mathlib.Data.Fintype.Prod\npublic import Mathlib.Data.Fintype.Pi\npublic import Mathlib.Algebra.Order.Group.Multiset\npublic import Mathlib.Data.ULift\npublic import Mathlib.Data.Set.NAry\n\nNamespace:\nSet\n\nLocal context:\n/-\nCopyright (c) 2022 Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kyle Miller\n-/\n/-!\n# Finiteness of products\n-/\n\npublic section\n\nassert_not_exists IsOrderedRing MonoidWithZero\n\nvariable {α β : Type*}\n\nnamespace Finite\n\ninstance [Finite α] [Finite β] : Finite (α × β) := by\n  haveI := Fintype.ofFinite α\n  haveI := Fintype.ofFinite β\n  infer_instance\n\ninstance {α β : Sort*} [Finite α] [Finite β] : Finite (PProd α β) :=\n  of_equiv _ Equiv.pprodEquivProdPLift.symm\n\ntheorem prod_left (β) [Finite (α × β)] [Nonempty β] : Finite α :=\n  of_surjective (Prod.fst : α × β → α) Prod.fst_surjective\n\ntheorem prod_right (α) [Finite (α × β)] [Nonempty α] : Finite β :=\n  of_surjective (Prod.snd : α × β → β) Prod.snd_surjective\n\nend Finite\n\nlemma Prod.finite_iff [Nonempty α] [Nonempty β] : Finite (α × β) ↔ Finite α ∧ Finite β where\n  mp _ := ⟨.prod_left β, .prod_right α⟩\n  mpr | ⟨_, _⟩ => inferInstance\n\ninstance Pi.finite {α : Sort*} {β : α → Sort*} [Finite α] [∀ a, Finite (β a)] :\n    Finite (∀ a, β a) := by\n  classical\n  haveI := Fintype.ofFinite (PLift α)\n  haveI := fun a => Fintype.ofFinite (PLift (β a))\n  exact\n    Finite.of_equiv (∀ a : PLift α, PLift (β (Equiv.plift a)))\n      (Equiv.piCongr Equiv.plift fun _ => Equiv.plift)\n\ninstance Function.Embedding.finite {α β : Sort*} [Finite β] : Finite (α ↪ β) := by\n  rcases isEmpty_or_nonempty (α ↪ β) with _ | h\n  · infer_instance\n  · refine h.elim fun f => ?_\n    haveI : Finite α := Finite.of_injective _ f.injective\n    exact Finite.of_injective _ DFunLike.coe_injective\n\ninstance Equiv.finite_right {α β : Sort*} [Finite β] : Finite (α ≃ β) :=\n  Finite.of_injective Equiv.toEmbedding fun e₁ e₂ h => Equiv.ext <| by\n    convert! DFunLike.congr_fun h using 0\n\ninstance Equiv.finite_left {α β : Sort*} [Finite α] : Finite (α ≃ β) :=\n  Finite.of_equiv _ ⟨Equiv.symm, Equiv.symm, Equiv.symm_symm, Equiv.symm_symm⟩\n\n@[to_additive]\ninstance MulEquiv.finite_left {α β : Type*} [Mul α] [Mul β] [Finite α] : Finite (α ≃* β) :=\n  Finite.of_injective toEquiv toEquiv_injective\n\n@[to_additive]\ninstance MulEquiv.finite_right {α β : Type*} [Mul α] [Mul β] [Finite β] : Finite (α ≃* β) :=\n  Finite.of_injective toEquiv toEquiv_injective\n\nopen Set Function\n\nvariable {γ : Type*}\n\nnamespace Set\n\n/-! ### Fintype instances\n\nEvery instance here should have a corresponding `Set.Finite` constructor in the next section.\n-/\n\nsection FintypeInstances\n\ninstance fintypeProd (s : Set α) (t : Set β) [Fintype s] [Fintype t] :\n    Fintype (s ×ˢ t : Set (α × β)) :=\n  Fintype.ofFinset (s.toFinset ×ˢ t.toFinset) <| by simp\n\ninstance fintypeOffDiag (s : Set α) [Fintype s] : Fintype s.offDiag :=\n  Fintype.ofFinset s.toFinset.offDiag <| by simp\n\n/-- `image2 f s t` is `Fintype` if `s` and `t` are. -/\ninstance fintypeImage2 [DecidableEq γ] (f : α → β → γ) (s : Set α) (t : Set β) [hs : Fintype s]\n    [ht : Fintype t] : Fintype (image2 f s t : Set γ) := by\n  rw [← image_prod]\n  apply Set.fintypeImage\n\nend FintypeInstances\n\nend Set\n\n/-! ### Finite instances\n\nThere is seemingly some overlap between the following instances and the `Fintype` instances\nin `Data.Set.Finite`. While every `Fintype` instance gives a `Finite` instance, those\ninstances that depend on `Fintype` or `Decidable` instances need an additional `Finite` instance\nto be able to generally apply.\n\nSome set instances do not appear here since they are consequences of others, for example\n`Subtype.Finite` for subsets of a finite type.\n-/\n\n\nnamespace Finite.Set\n\ninstance finite_prod (s : Set α) (t : Set β) [Finite s] [Finite t] :\n    Finite (s ×ˢ t : Set (α × β)) :=\n  Finite.of_equiv _ (Equiv.Set.prod s t).symm\n\ninstance finite_image2 (f : α → β → γ) (s : Set α) (t : Set β) [Finite s] [Finite t] :\n    Finite (image2 f s t : Set γ) := by\n  rw [← image_prod]\n  infer_instance\n\nend Finite.Set\n\nnamespace Set\n\n/-! ### Constructors for `Set.Finite`\n\nEvery constructor here should have a corresponding `Fintype` instance in the previous section\n(or in the `Fintype` module).\n\nThe implementation of these constructors ideally should be no more than `Set.toFinite`,\nafter possibly setting up some `Fintype` and classical `Decidable` instances.\n-/\n\n\nsection SetFiniteConstructors\n\nsection Prod\n\nvariable {s : Set α} {t : Set β}\n\nprotected theorem Finite.prod (hs : s.Finite) (ht : t.Finite) : (s ×ˢ t : Set (α × β)).Finite := by\n  have := hs.to_subtype\n  have := ht.to_subtype\n  apply toFinite\n\ntheorem Finite.of_prod_left (h : (s ×ˢ t : Set (α × β)).Finite) : t.Nonempty → s.Finite :=\n  fun ⟨b, hb⟩ => (h.image Prod.fst).subset fun a ha => ⟨(a, b), ⟨ha, hb⟩, rfl⟩\n\ntheorem Finite.of_prod_right (h : (s ×ˢ t : Set (α × β)).Finite) : s.Nonempty → t.Finite :=\n  fun ⟨a, ha⟩ => (h.image Prod.snd).subset fun b hb => ⟨(a, b), ⟨ha, hb⟩, rfl⟩\n\nprotected theorem Infinite.prod_left (hs : s.Infinite) (ht : t.Nonempty) : (s ×ˢ t).Infinite :=\n  fun h => hs <| h.of_prod_left ht\n\nprotected theorem Infinite.prod_right (ht : t.Infinite) (hs : s.Nonempty) : (s ×ˢ t).Infinite :=\n  fun h => ht <| h.of_prod_right hs\n\nprotected theorem infinite_prod :\n    (s ×ˢ t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty := by\n  refine ⟨fun h => ?_, ?_⟩\n  · simp_rw [Set.Infinite, @and_comm ¬_, ← Classical.not_imp]\n    by_contra!\n    exact h ((this.1 h.nonempty.snd).prod <| this.2 h.nonempty.fst)\n  · rintro (h | h)\n    · exact h.1.prod_left h.2\n    · exact h.1.prod_right h.2\n\ntheorem finite_prod : (s ×ˢ t).Finite ↔ (s.Finite ∨ t = ∅) ∧ (t.Finite ∨ s = ∅) := by\n  contrapose! +distrib; exact Set.infinite_prod\n\nprotected theorem Finite.offDiag {s : Set α} (hs : s.Finite) : s.offDiag.Finite :=\n  (hs.prod hs).subset s.offDiag_subset_prod\n\nprotected theorem Finite.image2 (f : α → β → γ) (hs : s.Finite) (ht : t.Finite) :\n    (image2 f s t).Finite := by\n  have := hs.to_subtype\n  have := ht.to_subtype\n  apply toFinite\n\nend Prod\n\nend SetFiniteConstructors\n\n/-! ### Properties -/\n\ntheorem Finite.toFinset_prod {s : Set α} {t : Set β} (hs : s.Finite) (ht : t.Finite) :\n    hs.toFinset ×ˢ ht.toFinset = (hs.prod ht).toFinset :=\n  Finset.ext <| by simp\n\ntheorem Finite.toFinset_offDiag {s : Set α} (hs : s.Finite) :\n    hs.offDiag.toFinset = hs.toFinset.offDiag :=\n  Finset.ext <| by simp\n\ntheorem finite_image_fst_and_snd_iff {s : Set (α × β)} :\n    (Prod.fst '' s).Finite ∧ (Prod.snd '' s).Finite ↔ s.Finite :=\n  ⟨fun h => (h.1.prod h.2).subset fun _ h => ⟨mem_image_of_mem _ h, mem_image_of_mem _ h⟩,\n    fun h => ⟨h.image _, h.image _⟩⟩\n\n/-! ### Infinite sets -/\n\nvariable {s t : Set α}\n\nsection Image2\n\nvariable {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β}\n\nprotected theorem Infinite.image2_left (hs : s.Infinite) (hb : b ∈ t)\n    (hf : InjOn (fun a => f a b) s) : (image2 f s t).Infinite :=\n  (hs.image hf).mono <| image_subset_image2_left hb\n\nprotected theorem Infinite.image2_right (ht : t.Infinite) (ha : a ∈ s) (hf : InjOn (f a) t) :\n    (image2 f s t).Infinite :=\n  (ht.image hf).mono <| image_subset_image2_right ha\n\ntheorem infinite_image2 (hfs : ∀ b ∈ t, InjOn (fun a => f a b) s) (hft : ∀ a ∈ s, InjOn (f a) t) :\n    (image2 f s t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty := by\n  refine ⟨fun h => Set.infinite_prod.1 ?_, ?_⟩\n  · rw [← image_uncurry_prod] at h\n    exact h.of_image _\n  · rintro (⟨hs, b, hb⟩ | ⟨ht, a, ha⟩)\n    · exact hs.image2_left hb (hfs _ hb)\n    · exact ht.image2_right ha (hft _ ha)\n\nTarget:\nlemma finite_image2 (hfs : ∀ b ∈ t, InjOn (f · b) s) (hft : ∀ a ∈ s, InjOn (f a) t) :\n    (image2 f s t).Finite ↔ s.Finite ∧ t.Finite ∨ s = ∅ ∨ t = ∅ :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finite","family_id":"finite_image","file_id":"mathlib/Mathlib/Data/Finite/Prod.lean","sample_id":"335fd55413b4f96955767217c73d17f78f377c305db1438e043702f98d245c46"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fcb2f3f3485ce6a70b3afb69269f67e2beb686789a095f74d88d7d21aa424579","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c5ea797717dbad7344a1184169096320fb9fd213b8cc1cea9f3c4ca95cfbc75d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"502371bcc584859610c49145091eb701ac3e0cd90f5beb5f633781b964813dd7","source_sha256":"522c0739e8f7535bf0f954aff339b473dbd0f9e903c0b5a9a4fff87d391d66b3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext S; simp only [smul_def, _root_.zero_smul, zero_def, Pi.zero_apply]","hard_negative":false,"metrics":{"chosen_tokens":19,"rejected_tokens":24,"token_jaccard":0.8,"token_length_ratio":1.263158},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"bde33bd6c990ddf351ecc0a02c5bb46922d5374460e512cf6e8bf84b6059d2d8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorProduct.RightExactness\npublic import Mathlib.RingTheory.Congruence.Hom\npublic import Mathlib.RingTheory.FiniteType\npublic import Mathlib.RingTheory.TensorProduct.DirectLimitFG\n\nNamespace:\nPolynomialLaw\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir & María-Inés de Frutos-Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir & María-Inés de Frutos-Fernández\n-/\n/-! # Polynomial laws on modules\n\nLet `M` and `N` be a modules over a commutative ring `R`.\nA polynomial law `f : PolynomialLaw R M N`, with notation `f : M →ₚₗₗ[R] N`,\nis a “law” that assigns a natural map `PolynomialLaw.toFun' f S : S ⊗[R] M → S ⊗[R] N`\nfor every `R`-algebra `S`.\n\nFor type-theoretic reasons, if `R : Type u`, then the definition of the polynomial map `f`\nis restricted to `R`-algebras `S` such that `S : Type u`.\nUsing the fact that a module is the direct limit of its finitely generated submodules, that a\nfinitely generated subalgebra is a quotient of a polynomial ring in the universe `u`, plus\nthe commutation of tensor products with direct limits, we extend the functor\nto all `R`-algebras.\n\nThe two fields involving the definition of `PolynomialLaw`,\n`PolynomialLaw.toFun'` and `PolynomialLaw.isCompat'` are primed.\nThey are superseded by their universe-polymorphic counterparts,\nthe definition `PolynomialLaw.toFun` and the lemma `PolynomialLaw.isCompat`\nwhich should be used once the theory is properly stated.\n\nFor constructions of general definitions of `PolynomialLaw`\nat a universe-polymorphic level, one needs to lift\nelements in a tensor product to smaller universes.\nFor this, one can make use of\n`PolynomialLaw.exists_lift` or `PolynomialLaw.exists_lift'`,\nor establish appropriate generalizations.\n\n## Main definitions/lemmas\n\n* Instance : `Module R (M →ₚₗ[R] N)` shows that polynomial laws form an `R`-module.\n\n* `PolynomialLaw.ground f` is the map `M → N` corresponding to `PolynomialLaw.toFun' f R` under\n  the isomorphisms `R ⊗[R] M ≃ₗ[R] M`, and similarly for `N`.\n\nIn further works, we construct the coefficients of a polynomial law and show the relation with\npolynomials (when the module `M` is free and finite).\n\n## Implementation notes\n\nIn the literature, the theory is written for commutative rings, but this implementation\nonly assumes `R` is a commutative semiring.\n\n## References\n\n* [Roby, Norbert. 1963. «Lois polynomes et lois formelles en théorie des modules».\n  Annales scientifiques de l’École Normale Supérieure 80 (3): 213‑348](Roby-1963)\n\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nnoncomputable section PolynomialLaw\n\nopen scoped TensorProduct\n\nopen LinearMap TensorProduct AlgHom RingCon\n\n/-- A polynomial law `M →ₚₗ[R] N` between `R`-modules is a functorial family of maps\n`S ⊗[R] M → S ⊗[R] N`, for all `R`-algebras `S`.\n\nFor universe reasons, `S` has to be restricted to the same universe as `R`. -/\n@[ext]\nstructure PolynomialLaw (R : Type u) [CommSemiring R]\n    (M : Type*) [AddCommMonoid M] [Module R M] (N : Type*) [AddCommMonoid N] [Module R N] where\n  /-- The functions `S ⊗[R] M → S ⊗[R] N` underlying a polynomial law -/\n  toFun' (S : Type u) [CommSemiring S] [Algebra R S] : S ⊗[R] M → S ⊗[R] N\n  /-- The compatibility relations between the functions underlying a polynomial law -/\n  isCompat' {S : Type u} [CommSemiring S] [Algebra R S]\n    {S' : Type u} [CommSemiring S'] [Algebra R S'] (φ : S →ₐ[R] S') :\n    φ.toLinearMap.rTensor N ∘ toFun' S = toFun' S' ∘ φ.toLinearMap.rTensor M := by aesop\n\n/-- `M →ₚₗ[R] N` is the type of `R`-polynomial laws from `M` to `N`. -/\nnotation:25 M \" →ₚₗ[\" R:25 \"] \" N:0 => PolynomialLaw R M N\n\n@[local simp]\ntheorem PolynomialLaw.isCompat_apply'\n    {R : Type u} [CommSemiring R] {M : Type*} [AddCommMonoid M] [Module R M]\n    {N : Type*} [AddCommMonoid N] [Module R N] {f : M →ₚₗ[R] N}\n    {S : Type u} [CommSemiring S] [Algebra R S] {S' : Type u} [CommSemiring S'] [Algebra R S']\n    (φ : S →ₐ[R] S') (x : S ⊗[R] M) :\n    (φ.toLinearMap.rTensor N) ((f.toFun' S) x) = (f.toFun' S') (φ.toLinearMap.rTensor M x) := by\n  simpa only using! congr_fun (f.isCompat' φ) x\n\nattribute [local simp] PolynomialLaw.isCompat_apply'\n\nnamespace PolynomialLaw\n\nsection Module\n\nsection CommSemiring\n\nvariable {R : Type u} [CommSemiring R] {M : Type*} [AddCommMonoid M] [Module R M]\n  {N : Type*} [AddCommMonoid N] [Module R N] (r a b : R) (f g : M →ₚₗ[R] N)\n\ninstance : Zero (M →ₚₗ[R] N) := ⟨{ toFun' _ := 0 }⟩\n\n@[simp]\ntheorem zero_def (S : Type u) [CommSemiring S] [Algebra R S] :\n    (0 : PolynomialLaw R M N).toFun' S = 0 := rfl\n\ninstance : Inhabited (PolynomialLaw R M N) := ⟨Zero.zero⟩\n\n/-- The identity as a polynomial law -/\ndef id : M →ₚₗ[R] M where\n  toFun' S _ _ := _root_.id\n\ntheorem id_apply' {S : Type u} [CommSemiring S] [Algebra R S] :\n    (id : M →ₚₗ[R] M).toFun' S = _root_.id := rfl\n\n/-- The sum of two polynomial laws -/\nnoncomputable def add : M →ₚₗ[R] N where\n  toFun' S _ _ := f.toFun' S + g.toFun' S\n\ninstance : Add (PolynomialLaw R M N) := ⟨add⟩\n\n@[simp]\ntheorem add_def (S : Type u) [CommSemiring S] [Algebra R S] :\n    (f + g).toFun' S = f.toFun' S + g.toFun' S := rfl\n\ntheorem add_def_apply (S : Type u) [CommSemiring S] [Algebra R S] (m : S ⊗[R] M) :\n    (f + g).toFun' S m = f.toFun' S m + g.toFun' S m := rfl\n\n/-- External multiplication of a `f : M →ₚₗ[R] N` by `r : R` -/\ndef smul : M →ₚₗ[R] N where\n  toFun' S _ _ := r • f.toFun' S\n\ninstance : SMul R (M →ₚₗ[R] N) := ⟨smul⟩\n\n@[simp]\ntheorem smul_def (S : Type u) [CommSemiring S] [Algebra R S] :\n    (r • f).toFun' S = r • f.toFun' S := rfl\n\ntheorem smul_def_apply (S : Type u) [CommSemiring S] [Algebra R S] (m : S ⊗[R] M) :\n    (r • f).toFun' S m = r • f.toFun' S m := rfl\n\ntheorem add_smul : (a + b) • f = a • f + b • f := by\n  ext; simp only [add_def, smul_def, _root_.add_smul]\n\nTarget:\ntheorem zero_smul : (0 : R) • f = 0 :=\n\nProof body:\n","rejected":"by\n  ext S; simp only [smul_def, _root_.zero_smul, zero_def, Pi.zero_apply]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/PolynomialLaw","family_id":"zero_smul","file_id":"mathlib/Mathlib/RingTheory/PolynomialLaw/Basic.lean","sample_id":"502371bcc584859610c49145091eb701ac3e0cd90f5beb5f633781b964813dd7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"10363aa97aac5a8e4a6ed13d0ebf67594ed188c89c81a730878f4693d1b889fa","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cd6445d62ea0160fdadc6b7b9535f61ac5bf89d79260f9ccdf0cb0c344df5551","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"19c9230dd9532a6df572a3d4cf4ed8cfd2fc18850e231e933699018c9fe60632","source_sha256":"18fda008e7b40ec7b81fa0fd9f7c55475c44e029c9af90e5639e54fa98674982","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x\n  simp only [LinearMap.comp_apply, apply, map_add, map_smul, add_assoc]\n  simp only [smul_add, ← mul_smul, ← add_smul, ← mul_add (f x), mul_assoc]","hard_negative":true,"metrics":{"chosen_tokens":38,"rejected_tokens":2,"token_jaccard":0.04,"token_length_ratio":0.052632},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"be0aef3bab710dee073eaf0a6e8f15f08539a0c444548aa1adb21017809dd5c4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup\npublic import Mathlib.LinearAlgebra.Charpoly.BaseChange\npublic import Mathlib.LinearAlgebra.Dual.BaseChange\npublic import Mathlib.LinearAlgebra.Dual.Lemmas\npublic import Mathlib.LinearAlgebra.FixedSubmodule\n\nNamespace:\nLinearMap.transvection\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir\n-/\n\n/-!\n# Transvections in a module\n\n* When `f : Module.Dual R V` and `v : V`,\n  `LinearMap.transvection f v` is the linear map given by `x ↦ x + f x • v`,\n\n* `LinearMap.transvection.det` shows that the determinant of\n  `LinearMap.transvection f v` is equal to `1 + f v`.\n\n* If, moreover, `f v = 0`, then `LinearEquiv.transvection` shows that it is\n  a linear equivalence.\n\n* `LinearMap.transvections R V`: the set of transvections.\n\n* `LinearEquiv.dilatransvections R V`: the set of linear equivalences\n  whose associated linear map is of the form `LinearMap.transvection f v`.\n\n* `LinearEquiv.transvection.det` shows that it has determinant `1`.\n\n## Note on terminology\n\nIn the mathematical literature, linear maps of the form `LinearMap.transvection f v`\nare only called “transvections” when `f v = 0`. Otherwise, they are sometimes\ncalled “dilations” (especially if `f v ≠ -1`).\n\nThe definition is almost the same as that of `Module.preReflection f v`,\nup to a sign change, which are interesting when `f v = 2`, because they give “reflections”.\n\n-/\n\n@[expose] public section\n\nnamespace LinearMap\n\nopen Module\n\nvariable {R V : Type*} [Semiring R] [AddCommMonoid V] [Module R V]\n\n/-- The transvection associated with a linear form `f` and a vector `v`.\n\nNB. In mathematics, these linear maps are only called “transvections” when `f v = 0`.\nSee also `Module.preReflection` for a similar definition, up to a sign. -/\ndef transvection (f : Dual R V) (v : V) : V →ₗ[R] V where\n  toFun x := x + f x • v\n  map_add' x y := by simp [add_add_add_comm, add_smul]\n  map_smul' r x := by simp [smul_eq_mul, smul_add, mul_smul]\n\nnamespace transvection\n\nopen Submodule LinearMap\n\ntheorem apply (f : Dual R V) (v x : V) :\n    transvection f v x = x + f x • v :=\n  rfl\n\ntheorem comp_of_left_eq_apply {f : Dual R V} {v w : V} {x : V} (hw : f w = 0) :\n    transvection f v (transvection f w x) = transvection f (v + w) x := by\n  simp [transvection, map_add, hw, add_assoc]\n\ntheorem comp_of_left_eq {f : Dual R V} {v w : V} (hw : f w = 0) :\n    (transvection f v) ∘ₗ (transvection f w) = transvection f (v + w) := by\n  ext; simp [comp_of_left_eq_apply hw]\n\ntheorem comp_of_right_eq_apply {f g : Dual R V} {v : V} {x : V} (hf : f v = 0) :\n    (transvection f v) (transvection g v x) = transvection (f + g) v x := by\n  simp [transvection, map_add, hf, add_smul, add_assoc]\n\ntheorem comp_of_right_eq {f g : Dual R V} {v : V} (hf : f v = 0) :\n    (transvection f v) ∘ₗ (transvection g v) = transvection (f + g) v := by\n  ext; simp [comp_of_right_eq_apply hf]\n\n@[simp]\ntheorem of_left_eq_zero (v : V) :\n    transvection (0 : Dual R V) v = id := by\n  ext\n  simp [transvection]\n\n@[simp]\ntheorem of_right_eq_zero (f : Dual R V) :\n    transvection f 0 = id := by\n  ext\n  simp [transvection]\n\nTarget:\ntheorem comp_smul_smul {f : Dual R V} {v : V} {r s : R} :\n    transvection f (r • v) ∘ₗ transvection f (s • v) =\n      transvection f ((r + s + s * f v * r) • v) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_19c9230dd953","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"385719a1adfad89cc12f79a73ff05551050d39afbfee5702224580fb5780ec35","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Transvection","family_id":"comp_smul_smul","file_id":"mathlib/Mathlib/LinearAlgebra/Transvection/Basic.lean","sample_id":"19c9230dd9532a6df572a3d4cf4ed8cfd2fc18850e231e933699018c9fe60632"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b1b2588f46af06b1f0c76c618d2cd12402ece06fbea01ebed4a3cb2260d14513","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e38015f0095dc11e18da07e2408293619a33379a8de7991100acdb86ccbefd14","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"345ae0fc64c4223b685a8043b5b363bbd84f8c383d47f5ccfa20bde13d5436d1","source_sha256":"b281bb44b709ce523fcb2c42bb757b0bde9d86d54aacb866640c7f77cfffa6bd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [mapRange_def]\n  refine (Finset.prod_subset support_mk_subset ?_).trans ?_\n  · intro i h1 h2\n    simp only [mem_support_toFun, ne_eq] at h1\n    simp only [Finset.coe_sort_coe, mem_support_toFun, mk_apply, ne_eq, h1, not_false_iff,\n      dite_eq_ite, ite_true, not_not] at h2\n    simp [h2, h0]\n  · refine Finset.prod_congr rfl ?_\n    intro i h1\n    simp only [mem_support_toFun, ne_eq] at h1\n    simp [h1]","hard_negative":true,"metrics":{"chosen_tokens":87,"rejected_tokens":3,"token_jaccard":0.027027,"token_length_ratio":0.034483},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"be160afeb02fcd319494f44f81b0452ef860b378300a5795751baaa26516d585","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.GroupWithZero.Action\npublic import Mathlib.Data.DFinsupp.Ext\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Sigma\n\nNamespace:\nDFinsupp\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Kenny Lau\n-/\n/-!\n# Dependent functions with finite support\n\nFor a non-dependent version see `Mathlib/Data/Finsupp/Defs.lean`.\n\n## Notation\n\nThis file introduces the notation `Π₀ a, β a` as notation for `DFinsupp β`, mirroring the `α →₀ β`\nnotation used for `Finsupp`. This works for nested binders too, with `Π₀ a b, γ a b` as notation\nfor `DFinsupp (fun a ↦ DFinsupp (γ a))`.\n\n## Implementation notes\n\nThe support is internally represented (in the primed `DFinsupp.support'`) as a `Multiset` that\nrepresents a superset of the true support of the function, quotiented by the always-true relation so\nthat this does not impact equality. This approach has computational benefits over storing a\n`Finset`; it allows us to add together two finitely-supported functions without\nhaving to evaluate the resulting function to recompute its support (which would required\ndecidability of `b = 0` for `b : β i`).\n\nThe true support of the function can still be recovered with `DFinsupp.support`; but these\ndecidability obligations are now postponed to when the support is actually needed. As a consequence,\nthere are two ways to sum a `DFinsupp`: with `DFinsupp.sum` which works over an arbitrary function\nbut requires recomputation of the support and therefore a `Decidable` argument; and with\n`DFinsupp.sumAddHom` which requires an additive morphism, using its properties to show that\nsumming over a superset of the support is sufficient.\n\n`Finsupp` takes an altogether different approach here; it uses `Classical.Decidable` and declares\nthe `Add` instance as noncomputable. This design difference is independent of the fact that\n`DFinsupp` is dependently-typed and `Finsupp` is not; in future, we may want to align these two\ndefinitions, or introduce two more definitions for the other combinations of decisions.\n-/\n\n@[expose] public section\n\nuniverse u u₁ u₂ v v₁ v₂ v₃ w x y l\n\nvariable {ι : Type u} {γ : Type w} {β : ι → Type v} {β₁ : ι → Type v₁} {β₂ : ι → Type v₂}\n\nnamespace DFinsupp\n\nsection Algebra\n\n/-- Evaluation at a point is an `AddMonoidHom`. This is the finitely-supported version of\n`Pi.evalAddMonoidHom`. -/\ndef evalAddMonoidHom [∀ i, AddZeroClass (β i)] (i : ι) : (Π₀ i, β i) →+ β i :=\n  (Pi.evalAddMonoidHom β i).comp coeFnAddMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_finsetSum {α} [∀ i, AddCommMonoid (β i)] (s : Finset α) (g : α → Π₀ i, β i) :\n    ⇑(∑ a ∈ s, g a) = ∑ a ∈ s, ⇑(g a) :=\n  map_sum coeFnAddMonoidHom g s\n\n@[deprecated (since := \"2026-04-08\")] alias coe_finset_sum := coe_finsetSum\n\n@[simp]\ntheorem finsetSum_apply {α} [∀ i, AddCommMonoid (β i)] (s : Finset α) (g : α → Π₀ i, β i) (i : ι) :\n    (∑ a ∈ s, g a) i = ∑ a ∈ s, g a i :=\n  map_sum (evalAddMonoidHom i) g s\n\n@[deprecated (since := \"2026-04-08\")] alias finset_sum_apply := finsetSum_apply\n\nend Algebra\n\nsection ProdAndSum\n\nvariable [DecidableEq ι]\n\n/-- `DFinsupp.prod f g` is the product of `g i (f i)` over the support of `f`. -/\n@[to_additive /-- `sum f g` is the sum of `g i (f i)` over the support of `f`. -/]\ndef prod [∀ i, Zero (β i)] [∀ (i) (x : β i), Decidable (x ≠ 0)] [CommMonoid γ] (f : Π₀ i, β i)\n    (g : ∀ i, β i → γ) : γ :=\n  ∏ i ∈ f.support, g i (f i)\n\n@[to_additive]\ntheorem prod_of_support_subset [∀ i, Zero (β i)]\n    [∀ (i) (x : β i), Decidable (x ≠ 0)] [CommMonoid γ]\n    {f : Π₀ i, β i} {g : (i : ι) → β i → γ} {s : Finset ι}\n    (hs : f.support ⊆ s) (map_zero : ∀ i ∈ s, g i 0 = 1) :\n    f.prod g = ∏ i ∈ s, g i (f i) := by\n  simp only [DFinsupp.prod]\n  apply Finset.prod_subset hs\n  intro i hi hi'\n  simp only [DFinsupp.mem_support_toFun, ne_eq, not_not] at hi'\n  rw [hi', map_zero]\n  exact hi\n\n/-- The product over two dfinsupps agree if the functions agree and are well-behaved within the\nshared support. -/\n@[to_additive (attr := gcongr only)\n/-- The sum over two dfinsupps agree if the functions agree and are well-behaved within the\nshared support. -/]\ntheorem prod_congr_of_eq_on_union\n    [∀ i, Zero (β i)] [∀ (i) (x : β i), Decidable (x ≠ 0)] [CommMonoid γ]\n    {f1 f2 : Π₀ i, β i} {g1 g2 : (i : ι) → β i → γ}\n    (h : ∀ x ∈ f1.support ∪ f2.support, g1 x (f1 x) = g2 x (f2 x))\n    (h1 : ∀ x ∈ f1.support ∪ f2.support, g1 x 0 = 1)\n    (h2 : ∀ x ∈ f1.support ∪ f2.support, g2 x 0 = 1) :\n    f1.prod g1 = f2.prod g2 := by\n  rw [prod_of_support_subset Finset.subset_union_left h1,\n    prod_of_support_subset Finset.subset_union_right h2]\n  exact Finset.prod_congr rfl h\n\n@[to_additive (attr := simp)]\ntheorem _root_.map_dfinsuppProd\n    {R S H : Type*} [∀ i, Zero (β i)] [∀ (i) (x : β i), Decidable (x ≠ 0)]\n    [CommMonoid R] [CommMonoid S] [FunLike H R S] [MonoidHomClass H R S] (h : H) (f : Π₀ i, β i)\n    (g : ∀ i, β i → R) : h (f.prod g) = f.prod fun a b => h (g a b) :=\n  map_prod _ _ _\n\n@[to_additive]\n\nTarget:\ntheorem prod_mapRange_index {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [∀ i, Zero (β₁ i)]\n    [∀ i, Zero (β₂ i)] [∀ (i) (x : β₁ i), Decidable (x ≠ 0)] [∀ (i) (x : β₂ i), Decidable (x ≠ 0)]\n    [CommMonoid γ] {f : ∀ i, β₁ i → β₂ i} {hf : ∀ i, f i 0 = 0} {g : Π₀ i, β₁ i} {h : ∀ i, β₂ i → γ}\n    (h0 : ∀ i, h i 0 = 1) : (mapRange f hf g).prod h = g.prod fun i b => h i (f i b) :=\n\nProof body:\n","rejected":"by\n  exact prod_mapRange_index","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"a2f14ecb13921862f3b89a5fe7809bb51283b0f5e72623843fbb2784b0a193fb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/DFinsupp","family_id":"prod_maprange_index","file_id":"mathlib/Mathlib/Data/DFinsupp/BigOperators.lean","sample_id":"345ae0fc64c4223b685a8043b5b363bbd84f8c383d47f5ccfa20bde13d5436d1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"313c9cf4c1b5eea4156e212ff571b19fbbfa824c1fd6f777992a3ed4cc19b0b1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"eec7b09ae0be76f3b02bd63e3447ae8dcf2f50c68c89e36daa2ab7e9a7a447a8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dd0a6c86eaf74b4453f65b352e9a02cbc6a9bc8e7367f1c0ef0f634a91f60ff2","source_sha256":"44d6e0a4a761ac34c9ab6bb892d46ec9566fb71d10c48050289a8c26b4be1e40","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases hq : 1 < q\n  · rw [← pow_zero q, ofPowerSeries_apply_pow hq, PowerSeries.coeff_zero_eq_constantCoeff]\n  · simp [ofPowerSeries, dif_neg hq]","hard_negative":true,"metrics":{"chosen_tokens":29,"rejected_tokens":3,"token_jaccard":0.043478,"token_length_ratio":0.103448},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"be28e710c68e328f83374b91e821216ae8bc4f3204687b111a044ca5e583f76e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.ArithmeticFunction.Defs\npublic import Mathlib.Order.Northcott\npublic import Mathlib.RingTheory.PowerSeries.Basic\npublic import Mathlib.RingTheory.PowerSeries.PiTopology\npublic import Mathlib.RingTheory.PowerSeries.Substitution\n\nNamespace:\nArithmeticFunction\n\nLocal context:\n/-\nCopyright (c) 2026 Thomas Browning. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Thomas Browning\n-/\n/-!\n# Construction of L-functions\n\nThis file constructs L-functions as formal Dirichlet series.\n\n## Main definitions\n\n* `ArithmeticFunction.ofPowerSeries q f`: L-function `f(q⁻ˢ)` obtained from a power series `f(T)`.\n* `ArithmeticFunction.eulerProduct f`: the Euler product of a family `f i` of Dirichlet series.\n\n## Implementation notes\n\nWe take the following route from polynomials to L-functions:\n* Starting from a polynomial in `T`, `PowerSeries.invOfUnit` gives the reciporical power series.\n* `ofPowerSeries` gives the local Euler factor as a formal Dirichlet series on powers of `q`.\n* `eulerProduct` gives the L-function as the formal product of these local Euler factors.\n* `LSeries` gives the L-function as an analytic function on the right half-plane of convergence.\n\nFor example, the Riemann zeta function `ζ(s)` corresponds to taking `1 - T` at each prime `p`.\n\nFor context, here is a diagram of the possible routes from polynomials to L-functions:\n```\n                   T=q⁻ˢ                     s ∈ ℂ\n[polynomials in T] ----> [polynomials in q⁻ˢ] ----> [analytic function in s]\n          |                           |                           |\n          | (reciprocal)              | (reciprocal)              | (reciprocal)\n          v         T=q⁻ˢ             V          s ∈ ℂ            V\n[power series in T] ----> [power series in q⁻ˢ] ----> [analytic function in s] (the Euler factor)\n          |                           |                           |\n          | (product)                 | (product)                 | (product)\n          v                 T=q⁻ˢ     V               s ∈ ℂ       V\n[multivariate power series] ----> [Dirichlet series] ----> [L-function in s] (the Euler product)\n```\n-/\n\n@[expose] public section\n\nnamespace ArithmeticFunction\n\nsection PowerSeries\n\nvariable {R : Type*}\n\nsection CommSemiring\n\nvariable [CommSemiring R]\n\n/-- The arithmetic function corresponding to the Dirichlet series `f(q⁻ˢ)`.\nFor example, if `f = 1 + X + X² + ...` and `q = p`, then `f(q⁻ˢ) = 1 + p⁻ˢ + p⁻²ˢ + ...`.\n\nIf `q ≤ 1` then `k ↦ q ^ k` is not injective, so we use the junk value `f.constantCoeff`. -/\nnoncomputable def ofPowerSeries (q : ℕ) : PowerSeries R →ₐ[R] ArithmeticFunction R where\n  toFun f := if hq : 1 < q then\n    ⟨Function.extend (q ^ ·) (f.coeff ·) 0, by simp [Nat.ne_zero_of_lt hq]⟩ else\n      algebraMap R (ArithmeticFunction R) f.constantCoeff\n  map_zero' := by ext; split_ifs <;> simp [Function.extend]\n  -- note that `ofPowerSeries.map_one'` relies on the junk value `f.constantCoeff`.\n  map_one' := by\n    ext n\n    split_ifs with hq\n    · by_cases hn : ∃ k, q ^ k = n\n      · obtain ⟨a, rfl⟩ := hn\n        simp [(Nat.pow_right_injective hq).extend_apply, one_apply, hq.ne']\n      · simp [hn, one_apply_ne (fun H ↦ hn ⟨0, H.symm⟩)]\n    · simp\n  map_add' f g := by\n    ext n\n    split_ifs with hq\n    · by_cases h : ∃ a, q ^ a = n\n      · obtain ⟨a, rfl⟩ := h\n        simp [(Nat.pow_right_injective hq).extend_apply]\n      · simp [h]\n    · by_cases hn : n = 1 <;> simp [hn]\n  map_mul' f g := by\n    ext n\n    split_ifs with hq\n    · simp_rw [mul_apply, coe_mk]\n      by_cases hn : ∃ a, q ^ a = n\n      · obtain ⟨k, rfl⟩ := hn\n        rw [(Nat.pow_right_injective hq).extend_apply]\n        have hs : (Finset.antidiagonal k).map (.prodMap ⟨fun k ↦ q ^ k, Nat.pow_right_injective hq⟩\n            ⟨fun k ↦ q ^ k, Nat.pow_right_injective hq⟩) ⊆ (q ^ k).divisorsAntidiagonal :=\n          Nat.antidiagonal_map_subset_divisorsAntidiagonal_pow hq k\n        rw [PowerSeries.coeff_mul k f g, ← Finset.sum_subset hs]\n        · simp [(Nat.pow_right_injective hq).extend_apply]\n        · intro (a, b) hab h\n          by_cases ha : ∃ i, q ^ i = a\n          · by_cases hb : ∃ j, q ^ j = b\n            · obtain ⟨i, rfl⟩ := ha\n              obtain ⟨j, rfl⟩ := hb\n              rw [Nat.mem_divisorsAntidiagonal, ← pow_add, Nat.pow_right_inj hq] at hab\n              simp_rw [Finset.mem_map, not_exists, not_and, Finset.mem_antidiagonal] at h\n              simpa using h (i, j) hab.1\n            · rwa [mul_comm, Function.extend_apply', Pi.zero_apply, zero_mul]\n          · rwa [Function.extend_apply', Pi.zero_apply, zero_mul]\n      · rw [Function.extend_apply' _ _ _ hn, Pi.zero_apply, Finset.sum_eq_zero]\n        intro (a, b) hk\n        obtain ⟨hab, -⟩ := Nat.mem_divisorsAntidiagonal.mp hk\n        by_cases ha : ∃ i, q ^ i = a\n        · by_cases hb : ∃ j, q ^ j = b\n          · obtain ⟨i, rfl⟩ := ha\n            obtain ⟨j, rfl⟩ := hb\n            rw [← pow_add] at hab\n            exact (hn ⟨i + j, hab⟩).elim\n          · rwa [mul_comm, Function.extend_apply', Pi.zero_apply, zero_mul]\n        · rwa [Function.extend_apply', Pi.zero_apply, zero_mul]\n    · simp\n  commutes' x := by\n    ext n\n    split_ifs with hq\n    · simp only [Algebra.algebraMap_eq_smul_one, coe_mk]\n      by_cases hn : ∃ k, q ^ k = n\n      · obtain ⟨k, rfl⟩ := hn\n        simp [(Nat.pow_right_injective hq).extend_apply, one_apply, hq.ne']\n      · rw [Function.extend_apply' _ _ _ hn, Pi.zero_apply, smul_map, one_apply_ne, smul_zero]\n        contrapose hn\n        exact ⟨0, by simp [hn]⟩\n    · simp\n\ntheorem ofPowerSeries_apply {q : ℕ} (hq : 1 < q) (f : PowerSeries R) (n : ℕ) :\n    ofPowerSeries q f n = Function.extend (q ^ ·) (f.coeff ·) 0 n := by\n  simp [ofPowerSeries, dif_pos hq]\n\ntheorem ofPowerSeries_apply_pow {q : ℕ} (hq : 1 < q) (f : PowerSeries R) (k : ℕ) :\n    ofPowerSeries q f (q ^ k) = f.coeff k := by\n  rw [ofPowerSeries_apply hq, (Nat.pow_right_injective hq).extend_apply]\n\ntheorem ofPowerSeries_apply_zero (q : ℕ) (f : PowerSeries R) : ofPowerSeries q f 0 = 0 := by\n  simp\n\n@[simp]\n-- note that `ofPowerSeries_apply_one` relies on the junk value `f.constantCoeff`.\n\nTarget:\ntheorem ofPowerSeries_apply_one (q : ℕ) (f : PowerSeries R) :\n    ofPowerSeries q f 1 = f.constantCoeff :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_dd0a6c86eaf7","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"8967cd195d564f7c2cd89655430989b937bc29d2583c3008fd2c583048117b91","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/ArithmeticFunction","family_id":"ofpowerseries_apply_one","file_id":"mathlib/Mathlib/NumberTheory/ArithmeticFunction/LFunction.lean","sample_id":"dd0a6c86eaf74b4453f65b352e9a02cbc6a9bc8e7367f1c0ef0f634a91f60ff2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7d031507a20582e22ca4729c3a20ae04f876f5f252c02c99897df94087378755","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e4a8552d1a248e6c893e3831be7b2e444a8b37e155eaad0a98d06f94e3457532","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"623cf51d346b229ca13fed1da60cd6df5ba3c3809b92103ef6f67007cec38024","source_sha256":"b7ad320c59fef5b242b6e2dd8c96baf9488a66a587c50e32c8af235ebfc77c21","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext j\n  rw [assoc, assoc, enrichedComp_π, id_comp, tensorHom_def', assoc,\n    ← MonoidalCategory.whiskerLeft_comp_assoc, enrichedId_π,\n    whisker_exchange_assoc, MonoidalCategory.whiskerRight_id, assoc, assoc,\n    Iso.inv_hom_id_assoc]\n  dsimp\n  rw [e_comp_id, comp_id]","hard_negative":false,"metrics":{"chosen_tokens":47,"rejected_tokens":52,"token_jaccard":0.857143,"token_length_ratio":1.106383},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"becaac8453286340912807970a3cd8b7a6c70279ed811cfba300a0128763f6a5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Monoidal.FunctorCategory\npublic import Mathlib.CategoryTheory.Enriched.Ordinary.Basic\npublic import Mathlib.CategoryTheory.Functor.Category\npublic import Mathlib.CategoryTheory.Limits.Shapes.End\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Functor categories are enriched\n\nIf `C` is a `V`-enriched ordinary category, then `J ⥤ C` is also\nboth a `V`-enriched ordinary category and a `J ⥤ V`-enriched\nordinary category, provided `C` has suitable limits.\n\nWe first define the `V`-enriched structure on `J ⥤ C` by saying\nthat if `F₁` and `F₂` are in `J ⥤ C`, then `enrichedHom V F₁ F₂ : V`\nis a suitable limit involving `F₁.obj j ⟶[V] F₂.obj j` for all `j : C`.\nThe `J ⥤ V` object of morphisms `functorEnrichedHom V F₁ F₂ : J ⥤ V`\nis defined by sending `j : J` to the previously defined `enrichedHom`\nfor the \"restriction\" of `F₁` and `F₂` to the category `Under j`.\nThe definition `isLimitConeFunctorEnrichedHom` shows that\n`enriched V F₁ F₂` is the limit of the functor `functorEnrichedHom V F₁ F₂`.\n\n-/\n\nset_option backward.defeqAttrib.useBackward true\n\n@[expose] public section\n\nuniverse v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄\n\nnamespace CategoryTheory.Enriched.FunctorCategory\n\nopen Category MonoidalCategory Limits Functor\n\nvariable (V : Type u₁) [Category.{v₁} V] [MonoidalCategory V]\n  {C : Type u₂} [Category.{v₂} C] {J : Type u₃} [Category.{v₃} J]\n  {K : Type u₄} [Category.{v₄} K] [EnrichedOrdinaryCategory V C]\n\nvariable (F₁ F₂ F₃ F₄ : J ⥤ C)\n\n/-- Given two functors `F₁` and `F₂` from a category `J` to a `V`-enriched\nordinary category `C`, this is the diagram `Jᵒᵖ ⥤ J ⥤ V` whose end shall be\nthe `V`-morphisms in `J ⥤ V` from `F₁` to `F₂`. -/\n@[simps!]\ndef diagram : Jᵒᵖ ⥤ J ⥤ V := F₁.op ⋙ eHomFunctor V C ⋙ (whiskeringLeft J C V).obj F₂\n\n/-- The condition that the end `diagram V F₁ F₂` exists, see `enrichedHom`. -/\nabbrev HasEnrichedHom := HasEnd (diagram V F₁ F₂)\n\nsection\n\nvariable [HasEnrichedHom V F₁ F₂]\n\n/-- The `V`-enriched hom from `F₁` to `F₂` when `F₁` and `F₂` are functors `J ⥤ C`\nand `C` is a `V`-enriched category. -/\nnoncomputable abbrev enrichedHom : V := end_ (diagram V F₁ F₂)\n\n/-- The projection `enrichedHom V F₁ F₂ ⟶ F₁.obj j ⟶[V] F₂.obj j` in the category `V`\nfor any `j : J` when `F₁` and `F₂` are functors `J ⥤ C` and `C` is a `V`-enriched category. -/\nnoncomputable abbrev enrichedHomπ (j : J) : enrichedHom V F₁ F₂ ⟶ F₁.obj j ⟶[V] F₂.obj j :=\n  end_.π _ j\n\n@[reassoc]\nlemma enrichedHom_condition {i j : J} (f : i ⟶ j) :\n    enrichedHomπ V F₁ F₂ i ≫ eHomWhiskerLeft V (F₁.obj i) (F₂.map f) =\n    enrichedHomπ V F₁ F₂ j ≫ eHomWhiskerRight V (F₁.map f) (F₂.obj j) :=\n  end_.condition (diagram V F₁ F₂) f\n\n@[reassoc]\nlemma enrichedHom_condition' {i j : J} (f : i ⟶ j) :\n    enrichedHomπ V F₁ F₂ i ≫ (ρ_ _).inv ≫\n      _ ◁ (eHomEquiv V) (F₂.map f) ≫ eComp V _ _ _ =\n    enrichedHomπ V F₁ F₂ j ≫ (λ_ _).inv ≫\n      (eHomEquiv V) (F₁.map f) ▷ _ ≫ eComp V _ _ _ :=\n  end_.condition (diagram V F₁ F₂) f\n\nvariable {F₁ F₂}\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- Given functors `F₁` and `F₂` in `J ⥤ C`, where `C` is a `V`-enriched ordinary category,\nthis is the bijection `(F₁ ⟶ F₂) ≃ (𝟙_ V ⟶ enrichedHom V F₁ F₂)`. -/\nnoncomputable def homEquiv : (F₁ ⟶ F₂) ≃ (𝟙_ V ⟶ enrichedHom V F₁ F₂) where\n  toFun τ := end_.lift (fun j ↦ eHomEquiv V (τ.app j)) (fun i j f ↦ by\n    trans eHomEquiv V (τ.app i ≫ F₂.map f)\n    · dsimp\n      simp only [eHomEquiv_comp, tensorHom_def_assoc, MonoidalCategory.whiskerRight_id,\n        ← unitors_equal, assoc, Iso.inv_hom_id_assoc, eHomWhiskerLeft]\n    · dsimp\n      simp only [← NatTrans.naturality, eHomEquiv_comp, tensorHom_def', id_whiskerLeft,\n        assoc, Iso.inv_hom_id_assoc, eHomWhiskerRight])\n  invFun g :=\n    { app := fun j ↦ (eHomEquiv V).symm (g ≫ end_.π _ j)\n      naturality := fun i j f ↦ (eHomEquiv V).injective (by\n        simp only [eHomEquiv_comp, Equiv.apply_symm_apply, Iso.cancel_iso_inv_left]\n        conv_rhs =>\n          rw [tensorHom_def_assoc, MonoidalCategory.whiskerRight_id_assoc, assoc,\n            enrichedHom_condition' V F₁ F₂ f]\n        conv_lhs =>\n          rw [tensorHom_def'_assoc, MonoidalCategory.whiskerLeft_comp_assoc,\n            id_whiskerLeft_assoc, id_whiskerLeft_assoc, Iso.inv_hom_id_assoc, unitors_equal]) }\n  left_inv τ := by aesop\n  right_inv g := by aesop\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma homEquiv_apply_π (τ : F₁ ⟶ F₂) (j : J) :\n    homEquiv V τ ≫ enrichedHomπ V _ _ j = eHomEquiv V (τ.app j) := by\n  simp [homEquiv]\n\nend\n\nsection\n\nvariable [HasEnrichedHom V F₁ F₁]\n\n/-- The identity for the `V`-enrichment of the category `J ⥤ C`. -/\nnoncomputable def enrichedId : 𝟙_ V ⟶ enrichedHom V F₁ F₁ := homEquiv _ (𝟙 F₁)\n\n@[reassoc (attr := simp)]\nlemma enrichedId_π (j : J) : enrichedId V F₁ ≫ end_.π _ j = eId V (F₁.obj j) := by\n  simp [enrichedId]\n\n@[simp]\nlemma homEquiv_id : homEquiv V (𝟙 F₁) = enrichedId V F₁ := rfl\n\nend\n\nsection\n\nvariable [HasEnrichedHom V F₁ F₂] [HasEnrichedHom V F₂ F₃] [HasEnrichedHom V F₁ F₃]\n\n/-- The composition for the `V`-enrichment of the category `J ⥤ C`. -/\nnoncomputable def enrichedComp : enrichedHom V F₁ F₂ ⊗ enrichedHom V F₂ F₃ ⟶ enrichedHom V F₁ F₃ :=\n  end_.lift (fun j ↦ (end_.π _ j ⊗ₘ end_.π _ j) ≫ eComp V _ _ _) (fun i j f ↦ by\n    dsimp\n    trans (end_.π (diagram V F₁ F₂) i ⊗ₘ end_.π (diagram V F₂ F₃) j) ≫\n      (ρ_ _).inv ▷ _ ≫ (_ ◁ (eHomEquiv V (F₂.map f))) ▷ _ ≫ eComp V _ (F₂.obj i) _ ▷ _ ≫\n        eComp V _ (F₂.obj j) _\n    · have := end_.condition (diagram V F₂ F₃) f\n      dsimp [eHomWhiskerLeft, eHomWhiskerRight] at this ⊢\n      conv_lhs => rw [assoc, tensorHom_def_assoc]\n      conv_rhs =>\n        rw [tensorHom_def_assoc, whisker_assoc_assoc, e_assoc,\n          triangle_assoc_comp_right_inv_assoc, ← MonoidalCategory.whiskerLeft_comp_assoc,\n          ← MonoidalCategory.whiskerLeft_comp_assoc, ← MonoidalCategory.whiskerLeft_comp_assoc,\n          assoc, assoc, ← this, MonoidalCategory.whiskerLeft_comp_assoc,\n          MonoidalCategory.whiskerLeft_comp_assoc, MonoidalCategory.whiskerLeft_comp_assoc,\n          ← e_assoc, whiskerLeft_rightUnitor_inv_assoc, associator_inv_naturality_right_assoc,\n          Iso.hom_inv_id_assoc, whisker_exchange_assoc, MonoidalCategory.whiskerRight_id_assoc,\n          Iso.inv_hom_id_assoc]\n    · have := end_.condition (diagram V F₁ F₂) f\n      dsimp [eHomWhiskerLeft, eHomWhiskerRight] at this ⊢\n      conv_lhs =>\n        rw [tensorHom_def'_assoc, ← comp_whiskerRight_assoc,\n          ← comp_whiskerRight_assoc, ← comp_whiskerRight_assoc,\n          assoc, assoc, this, comp_whiskerRight_assoc, comp_whiskerRight_assoc,\n          comp_whiskerRight_assoc, leftUnitor_inv_whiskerRight_assoc,\n          ← associator_inv_naturality_left_assoc, ← e_assoc',\n          Iso.inv_hom_id_assoc, ← whisker_exchange_assoc, id_whiskerLeft_assoc,\n          Iso.inv_hom_id_assoc]\n      conv_rhs => rw [assoc, tensorHom_def'_assoc])\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma enrichedComp_π (j : J) :\n    enrichedComp V F₁ F₂ F₃ ≫ end_.π _ j =\n      (end_.π (diagram V F₁ F₂) j ⊗ₘ end_.π (diagram V F₂ F₃) j) ≫ eComp V _ _ _ := by\n  simp [enrichedComp]\n\nvariable {F₁ F₂ F₃}\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc]\nlemma homEquiv_comp (f : F₁ ⟶ F₂) (g : F₂ ⟶ F₃) :\n    (homEquiv V) (f ≫ g) = (λ_ (𝟙_ V)).inv ≫ ((homEquiv V) f ⊗ₘ (homEquiv V) g) ≫\n    enrichedComp V F₁ F₂ F₃ := by\n  ext j\n  simp only [homEquiv_apply_π, NatTrans.comp_app, eHomEquiv_comp, assoc,\n    enrichedComp_π, Functor.op_obj, tensorHom_comp_tensorHom_assoc]\n\nend\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma enriched_id_comp [HasEnrichedHom V F₁ F₁] [HasEnrichedHom V F₁ F₂] :\n    (λ_ (enrichedHom V F₁ F₂)).inv ≫ enrichedId V F₁ ▷ enrichedHom V F₁ F₂ ≫\n      enrichedComp V F₁ F₁ F₂ = 𝟙 _ := by\n  ext j\n  rw [assoc, assoc, enrichedComp_π, id_comp, tensorHom_def, assoc,\n    ← comp_whiskerRight_assoc, enrichedId_π, ← whisker_exchange_assoc,\n    id_whiskerLeft, assoc, assoc, Iso.inv_hom_id_assoc]\n  dsimp\n  rw [e_id_comp, comp_id]\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\n\nTarget:\nlemma enriched_comp_id [HasEnrichedHom V F₁ F₂] [HasEnrichedHom V F₂ F₂] :\n    (ρ_ (enrichedHom V F₁ F₂)).inv ≫ enrichedHom V F₁ F₂ ◁ enrichedId V F₂ ≫\n      enrichedComp V F₁ F₂ F₂ = 𝟙 _ :=\n\nProof body:\n","rejected":"by\n  ext j\n  rw [assoc, assoc, enrichedComp_π, id_comp, tensorHom_def', assoc,\n    ← MonoidalCategory.whiskerLeft_comp_assoc, enrichedId_π,\n    whisker_exchange_assoc, MonoidalCategory.whiskerRight_id, assoc, assoc,\n    Iso.inv_hom_id_assoc]\n  dsimp\n  rw [e_comp_id, comp_id]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Enriched","family_id":"enriched_comp_id","file_id":"mathlib/Mathlib/CategoryTheory/Enriched/FunctorCategory.lean","sample_id":"623cf51d346b229ca13fed1da60cd6df5ba3c3809b92103ef6f67007cec38024"},"split":"train"}
{"binding":{"chosen_proof_sha256":"12e284d615394feaae0bf91b6d2469b44129b24d28aa683dddb8dd6ad90c1ad2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"733568c1cfdc13c4513e891277fa1d4790f8cbee6c14171a33b4c0dcbbf16802","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"141f039ee25caf569389190297e8e170fe2b5f957ffbed666e13fe2928910cf7","source_sha256":"31a5542d1ac83b668eb669f80f15da1c85ba792b90d38c379ac11ec99cc62c14","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  grw [hv.apply_sum_le, ← cbiSup_eq_of_forall (by grind)]\n  simp","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.0625,"token_length_ratio":0.2},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"beddc8e185731c21f3582ee3d00edaeef1a441f87c87ebe50cf5d59df05aa720","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Homogenize\npublic import Mathlib.NumberTheory.Height.Basic\nimport Mathlib.Algebra.Order.Ring.IsNonarchimedean\nimport Mathlib.Data.Fintype.Order\nimport all Mathlib.NumberTheory.Height.Basic\n\nNamespace:\nIsNonarchimedean\n\nLocal context:\n/-\nCopyright (c) 2026 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Height bounds for linear and polynomial maps\n\nWe prove an upper bound for the height of the image of a tuple under a linear map.\n\nWe also prove upper and lower bounds for the height of `fun i ↦ eval P i x`, where `P` is a family\nof homogeneous polynomials over the field `K` of the same degree `N` and `x : ι → K`\nwith `ι` finite.\n-/\n\npublic section\n\nsection aux\n\nprivate lemma Height.iSup_fun_eq_max (f : Fin 2 → ℝ) : iSup f = max (f 0) (f 1) := by\n  rw [show f = ![f 0, f 1] from List.ofFn_inj.mp rfl]\n  exact (max_eq_iSup ..).symm\n\nnamespace IsNonarchimedean\n\nvariable {R α β F : Type*} [CommRing R] [AddCommMonoid β] [FunLike F β ℝ] [NonnegHomClass F β ℝ]\n    [ZeroHomClass F β ℝ] {v : F} {l : α → β}\n\n-- NOTE: The following cannot be moved to Mathlib.Algebra.Order.Ring.IsNonarchimedean,\n--       because it needs the target to be the reals (to have the default value zero\n--       for empty iSups), which are not known there.\n/-- The ultrametric triangle inequality for finite sums. -/\nlemma apply_sum_le (hv : IsNonarchimedean v) {s : Finset α} :\n    v (∑ i ∈ s, l i) ≤ ⨆ i : s, v (l i) := by\n  classical\n  induction s using Finset.induction with\n  | empty => simp\n  | insert a s ha ih =>\n    rw [Finset.sum_insert ha]\n    grw [hv .., ih]\n    refine max_le ?_ ?_\n    · exact Finite.le_ciSup_of_le ⟨_, s.mem_insert_self a⟩ le_rfl\n    · rcases isEmpty_or_nonempty s with hs | hs\n      · simpa using Real.iSup_nonneg_of_nonnegHomClass v _\n      exact ciSup_le fun i ↦ Finite.le_ciSup_of_le (⟨i.val, Finset.mem_insert_of_mem i.prop⟩) le_rfl\n\n/-- The ultrametric triangle inequality for finite sums. -/\n\nTarget:\nlemma apply_sum_univ_le [Fintype α] (hv : IsNonarchimedean v) :\n    v (∑ i, l i) ≤ ⨆ i, v (l i) :=\n\nProof body:\n","rejected":"by\n  exact apply_sum_univ_le","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"9cafc93385394096ca1d871f33c8bf36ee8a696d3a6c946fbc1d2e1eb529be0b","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Height","family_id":"apply_sum_univ_le","file_id":"mathlib/Mathlib/NumberTheory/Height/MvPolynomial.lean","sample_id":"141f039ee25caf569389190297e8e170fe2b5f957ffbed666e13fe2928910cf7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9863212e81e0182629bd15d927cf3f594339d8438f634ee6f333a33c3de5d6eb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0b9613c0f8f4ed0424d4b806c7d2c4c3dd5532324d866b160d25e480beb77d5e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b3ad2032874ba6a0b6f50397e2b035a5757e486348575d6dc25170f7fefc69ec","source_sha256":"5f13df63ef1a33fb7d6444e769a7e9b9cf3ab09860a26a91c473157d6239794f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  conv_rhs => simp [← h.commute_iff, Commute.symm_iff (a := e), commute_iff_eq, ← mem_commutant_iff]","hard_negative":true,"metrics":{"chosen_tokens":24,"rejected_tokens":3,"token_jaccard":0.045455,"token_length_ratio":0.125},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"bf69b8068a4f044aeb110178a284673006e47a724a6526e5270a56d6407d30d0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.CStarAlgebra.Classes\npublic import Mathlib.Analysis.InnerProductSpace.Adjoint\n\nNamespace:\nVonNeumannAlgebra\n\nLocal context:\n/-\nCopyright (c) 2022 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Von Neumann algebras\n\nWe give the \"abstract\" and \"concrete\" definitions of a von Neumann algebra.\nWe still have a major project ahead of us to show the equivalence between these definitions!\n\nAn abstract von Neumann algebra `WStarAlgebra M` is a C⋆ algebra with a Banach space predual,\nper Sakai (1971).\n\nA concrete von Neumann algebra `VonNeumannAlgebra H` (where `H` is a Hilbert space)\nis a \\*-closed subalgebra of bounded operators on `H` which is equal to its double commutant.\n\nWe'll also need to prove the von Neumann double commutant theorem,\nthat the concrete definition is equivalent to a \\*-closed subalgebra which is weakly closed.\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\n/-- Sakai's definition of a von Neumann algebra as a C⋆ algebra with a Banach space predual.\n\nSo that we can unambiguously talk about these \"abstract\" von Neumann algebras\nin parallel with the \"concrete\" ones (weakly closed \\*-subalgebras of B(H)),\nwe name this definition `WStarAlgebra`.\n\nNote that for now we only assert the mere existence of predual, rather than picking one.\nThis may later prove problematic, and need to be revisited.\nPicking one may cause problems with definitional unification of different instances.\nOne the other hand, not picking one means that the weak-\\* topology\n(which depends on a choice of predual) must be defined using the choice,\nand we may be unhappy with the resulting opaqueness of the definition.\n-/\nclass WStarAlgebra (M : Type u) [CStarAlgebra M] : Prop where\n  /-- There is a Banach space `X` whose dual is isometrically (conjugate-linearly) isomorphic\n  to the `WStarAlgebra`. -/\n  exists_predual :\n    ∃ (X : Type u) (_ : NormedAddCommGroup X) (_ : NormedSpace ℂ X) (_ : CompleteSpace X),\n      Nonempty (StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M)\n\n-- TODO: Without this, `VonNeumannAlgebra` times out. Why?\n/-- The double commutant definition of a von Neumann algebra,\nas a \\*-closed subalgebra of bounded operators on a Hilbert space,\nwhich is equal to its double commutant.\n\nNote that this definition is parameterised by the Hilbert space\non which the algebra faithfully acts, as is standard in the literature.\nSee `WStarAlgebra` for the abstract notion (a C⋆-algebra with Banach space predual).\n\nNote this is a bundled structure, parameterised by the Hilbert space `H`,\nrather than a typeclass on the type of elements.\nThus we can't say that the bounded operators `H →L[ℂ] H` form a `VonNeumannAlgebra`\n(although we will later construct the instance `WStarAlgebra (H →L[ℂ] H)`),\nand instead will use `⊤ : VonNeumannAlgebra H`.\n-/\nstructure VonNeumannAlgebra (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H]\n    [CompleteSpace H] extends StarSubalgebra ℂ (H →L[ℂ] H) where\n  /-- The double commutant (a.k.a. centralizer) of a `VonNeumannAlgebra` is itself. -/\n  centralizer_centralizer' : Set.centralizer (Set.centralizer carrier) = carrier\n\n/-- Consider a von Neumann algebra acting on a Hilbert space `H` as a \\*-subalgebra of `H →L[ℂ] H`.\n(That is, we forget that it is equal to its double commutant\nor equivalently that it is closed in the weak and strong operator topologies.)\n-/\nadd_decl_doc VonNeumannAlgebra.toStarSubalgebra\n\nnamespace VonNeumannAlgebra\n\nvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n\ninstance instSetLike : SetLike (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  coe S := S.carrier\n  coe_injective S T h := by obtain ⟨⟨⟨⟨⟨⟨_, _⟩, _⟩, _⟩, _⟩, _⟩, _⟩ := S; cases T; congr\n\ninstance : PartialOrder (VonNeumannAlgebra H) := .ofSetLike (VonNeumannAlgebra H) (H →L[ℂ] H)\n\nnoncomputable instance instStarMemClass : StarMemClass (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  star_mem {s} := s.star_mem'\n\ninstance instSubringClass : SubringClass (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  one_mem {s} := s.one_mem'\n  zero_mem {s} := s.zero_mem'\n  neg_mem {s} a ha := show -a ∈ s.toStarSubalgebra from neg_mem ha\n\n@[simp]\ntheorem mem_carrier {S : VonNeumannAlgebra H} {x : H →L[ℂ] H} :\n    x ∈ S.toStarSubalgebra ↔ x ∈ (S : Set (H →L[ℂ] H)) :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toStarSubalgebra (S : VonNeumannAlgebra H) :\n    (S.toStarSubalgebra : Set (H →L[ℂ] H)) = S :=\n  rfl\n\n@[simp]\ntheorem coe_mk (S : StarSubalgebra ℂ (H →L[ℂ] H)) (h) :\n    ((⟨S, h⟩ : VonNeumannAlgebra H) : Set (H →L[ℂ] H)) = S :=\n  rfl\n\n@[ext]\ntheorem ext {S T : VonNeumannAlgebra H} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem centralizer_centralizer (S : VonNeumannAlgebra H) :\n    Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S :=\n  S.centralizer_centralizer'\n\n/-- The centralizer of a `VonNeumannAlgebra`, as a `VonNeumannAlgebra`. -/\nnoncomputable def commutant (S : VonNeumannAlgebra H) : VonNeumannAlgebra H where\n  toStarSubalgebra := StarSubalgebra.centralizer ℂ (S : Set (H →L[ℂ] H))\n  centralizer_centralizer' := by simp\n\n@[simp]\ntheorem coe_commutant (S : VonNeumannAlgebra H) :\n    ↑S.commutant = Set.centralizer (S : Set (H →L[ℂ] H)) := by\n  simp [commutant]\n\n@[simp]\ntheorem mem_commutant_iff {S : VonNeumannAlgebra H} {z : H →L[ℂ] H} :\n    z ∈ S.commutant ↔ ∀ g ∈ S, g * z = z * g := by\n  rw [← SetLike.mem_coe, coe_commutant]\n  rfl\n\n@[simp]\ntheorem commutant_commutant (S : VonNeumannAlgebra H) : S.commutant.commutant = S :=\n  SetLike.coe_injective <| by simp\n\nopen ContinuousLinearMap in\n/-- An idempotent is an element in a von Neumann algebra if and only if\nits range and kernel are invariant under the commutant. -/\n\nTarget:\ntheorem IsIdempotentElem.mem_iff {e : H →L[ℂ] H} (h : IsIdempotentElem e)\n    (S : VonNeumannAlgebra H) :\n    e ∈ S ↔ ∀ y ∈ S.commutant,\n      e.range ∈ Module.End.invtSubmodule y ∧ e.ker ∈ Module.End.invtSubmodule y :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_b3ad2032874b","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"55112e9037835c620d6ba0fb1a4edc0edec6cac84a7f722d5b0e6f70ca22063c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/VonNeumannAlgebra","family_id":"isidempotentelem","file_id":"mathlib/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean","sample_id":"b3ad2032874ba6a0b6f50397e2b035a5757e486348575d6dc25170f7fefc69ec"},"split":"train"}
{"binding":{"chosen_proof_sha256":"63ce66eb5d2d85e6c3aa93e37342e57afc36c526d7b717542f12e151e6f9e034","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"947f4998f81845289fe2e622281075006b1acfc80588d06808ad972dd3f50b69","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ed0bc1a8156a91f3c77b4c93e0e33c2312cab82d14a2bd0d77876ca7788349af","source_sha256":"1926121bd5a0473c64fcd7320e73eb6e8a92720aeaf7b72deff3f44f28315a03","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let := Localization.AtPrime.algebraOfLiesOver (p.under R) p\n  rw [isUnramifiedAt_iff_map_eq R (p.under R) p, and_iff_right,\n    Ideal.IsDedekindDomain.ramificationIdx_eq_one_iff hp Ideal.map_comap_le]\n  have : Finite (R ⧸ p.under R) :=\n    Ideal.finiteQuotientOfFreeOfNeBot _ (mt Ideal.eq_bot_of_comap_eq_bot hp)\n  have : Finite ((p.under R).ResidueField) := IsLocalization.finite _\n    (nonZeroDivisors (R ⧸ p.under R))\n  infer_instance","hard_negative":false,"metrics":{"chosen_tokens":92,"rejected_tokens":97,"token_jaccard":0.9,"token_length_ratio":1.054348},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"bf89e9579f627d977ad17dcb48950dbbcc62b5d75150dfcd9ccff5f24f810488","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.RamificationInertia.Basic\npublic import Mathlib.RingTheory.LocalRing.ResidueField.Instances\npublic import Mathlib.RingTheory.Unramified.LocalRing\npublic import Mathlib.LinearAlgebra.FreeModule.IdealQuotient\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Unramified and ramification index\n\nWe connect `Ideal.ramificationIdx` to the commutative algebra notion predicate of `IsUnramifiedAt`.\n\n## Main result\n- `Algebra.isUnramifiedAt_iff_of_isDedekindDomain`:\n  Let `R` be a domain of characteristic 0, finite rank over `ℤ`, `S ⊇ R` be a Dedekind domain\n  that is a finite `R`-algebra. Let `p` be a prime of `S`, then `p` is unramified iff `e(p) = 1`.\n\n-/\n\npublic section\n\nvariable {R S T : Type*} [CommRing R] [CommRing S] [CommRing T]\nvariable [Algebra R S] [Algebra S T] [Algebra R T] [IsScalarTower R S T]\n\nlocal notation3 \"e(\" P \"|\" R \")\" =>\n  Ideal.ramificationIdx (Ideal.under R P) P\n\nopen IsLocalRing Algebra\n\nlemma Ideal.ramificationIdx_eq_one_of_isUnramifiedAt\n    {p : Ideal S} [p.IsPrime] [IsNoetherianRing S] [IsUnramifiedAt R p]\n    (hp : p ≠ ⊥) [IsDomain S] [EssFiniteType R S] :\n    e(p|R) = 1 :=\n  let := Localization.AtPrime.algebraOfLiesOver (p.under R) p\n  (Ideal.ramificationIdx_eq_one_of_map_localization Ideal.map_comap_le hp\n    p.primeCompl_le_nonZeroDivisors\n    ((isUnramifiedAt_iff_map_eq R (p.under R) p).mp ‹_›).2)\n\nvariable (R) in\nlemma IsUnramifiedAt.of_liesOver_of_ne_bot\n    (p : Ideal S) (P : Ideal T) [P.LiesOver p] [p.IsPrime] [P.IsPrime]\n    [IsUnramifiedAt R P] [EssFiniteType R S] [EssFiniteType R T]\n    [IsDedekindDomain S] (hP₁ : P.primeCompl ≤ nonZeroDivisors T) (hP₂ : p ≠ ⊥ → P ≠ ⊥) :\n    IsUnramifiedAt R p := by\n  let p₀ : Ideal R := p.under R\n  have : P.LiesOver p₀ := .trans P p p₀\n  let := Localization.AtPrime.algebraOfLiesOver p₀ p\n  let := Localization.AtPrime.algebraOfLiesOver p P\n  let := Localization.AtPrime.algebraOfLiesOver p₀ P\n  have hp₀ : p₀ = P.under R := Ideal.LiesOver.over\n  have : EssFiniteType S T := .of_comp R S T\n  have := Algebra.EssFiniteType.isNoetherianRing S T\n  rw [isUnramifiedAt_iff_map_eq R p₀ p]\n  have ⟨h₁, h₂⟩ := (isUnramifiedAt_iff_map_eq R p₀ P).mp ‹_›\n  refine ⟨Algebra.isSeparable_tower_bot_of_isSeparable _ _ P.ResidueField, ?_⟩\n  by_cases hp : p = ⊥\n  · have : p₀.map (algebraMap R S) = p := by\n      subst hp\n      exact le_bot_iff.mp (Ideal.map_comap_le)\n    rw [IsScalarTower.algebraMap_eq _ S, ← Ideal.map_map, this,\n      Localization.AtPrime.map_eq_maximalIdeal]\n  rw [← Ideal.IsDedekindDomain.ramificationIdx_eq_one_iff hp Ideal.map_comap_le,\n    ← not_ne_iff, Ideal.ramificationIdx_ne_one_iff Ideal.map_comap_le]\n  intro H\n  have := Ideal.ramificationIdx_eq_one_of_map_localization\n    (hp₀ ▸ Ideal.map_comap_le) (hP₂ hp) hP₁ h₂\n  rw [← not_ne_iff, Ideal.ramificationIdx_ne_one_iff (hp₀ ▸ Ideal.map_comap_le)] at this\n  replace H := Ideal.map_mono (f := algebraMap S T) H\n  rw [Ideal.map_map, ← IsScalarTower.algebraMap_eq, Ideal.map_pow] at H\n  refine this (H.trans (Ideal.pow_right_mono ?_ _))\n  exact Ideal.map_le_iff_le_comap.mpr Ideal.LiesOver.over.le\n\nvariable (R) in\n/--\nUp to technical conditions, If `T/S/R` is a tower of algebras, `P` is a prime of `T` unramified\nin `R`, then `P ∩ S` (as a prime of `S`) is also unramified in `R`.\n-/\nlemma Algebra.IsUnramifiedAt.of_liesOver\n    (p : Ideal S) (P : Ideal T) [P.LiesOver p] [p.IsPrime] [P.IsPrime]\n    [IsUnramifiedAt R P] [EssFiniteType R S] [EssFiniteType R T]\n    [IsDedekindDomain S] [IsDomain T] [Module.IsTorsionFree S T] : IsUnramifiedAt R p :=\n  IsUnramifiedAt.of_liesOver_of_ne_bot R p P P.primeCompl_le_nonZeroDivisors\n    (Ideal.ne_bot_of_liesOver_of_ne_bot · P)\n\n/-- Let `R` be a domain of characteristic 0, finite rank over `ℤ`, `S` be a Dedekind domain\nthat is a finite `R`-algebra. Let `p` be a prime of `S`, then `p` is unramified iff `e(p) = 1`. -/\n\nTarget:\nlemma Algebra.isUnramifiedAt_iff_of_isDedekindDomain\n    {p : Ideal S} [p.IsPrime] [IsDedekindDomain S] [EssFiniteType R S] [IsDomain R]\n    [Module.Finite ℤ R] [CharZero R] [Algebra.IsIntegral R S]\n    (hp : p ≠ ⊥) :\n    Algebra.IsUnramifiedAt R p ↔ e(p|R) = 1 :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  let := Localization.AtPrime.algebraOfLiesOver (p.under R) p\n  rw [isUnramifiedAt_iff_map_eq R (p.under R) p, and_iff_right,\n    Ideal.IsDedekindDomain.ramificationIdx_eq_one_iff hp Ideal.map_comap_le]\n  have : Finite (R ⧸ p.under R) :=\n    Ideal.finiteQuotientOfFreeOfNeBot _ (mt Ideal.eq_bot_of_comap_eq_bot hp)\n  have : Finite ((p.under R).ResidueField) := IsLocalization.finite _\n    (nonZeroDivisors (R ⧸ p.under R))\n  infer_instance","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/RamificationInertia","family_id":"algebra","file_id":"mathlib/Mathlib/NumberTheory/RamificationInertia/Unramified.lean","sample_id":"ed0bc1a8156a91f3c77b4c93e0e33c2312cab82d14a2bd0d77876ca7788349af"},"split":"train"}
{"binding":{"chosen_proof_sha256":"99f33d97a3ff079eb01e172cf32d5125fc28f1ebfb2b8c076d51536e340d9ee6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"afc67d04fe81026f8c10733aa377845ea38d7b3b8dae4b1c4ebb2ec01c92084f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dce31b4f3667c3830b421cb76a3fefb37b0de149d8e70ed049af7ee6287ae930","source_sha256":"ab8dcc475894ae6adb2da3524a8462f1faacdbf7c702f22683396e1bbe0456ee","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply tendsto_integral_filter_of_dominated_convergence _ _ _ hf.norm\n  · filter_upwards with v\n    nth_rewrite 2 [show f v = cexp (- (0 : ℝ) * ‖v‖ ^ 2) • f v by simp]\n    apply (Tendsto.cexp _).smul_const\n    exact tendsto_inv_atTop_zero.ofReal.neg.mul_const _\n  · filter_upwards with c using\n      AEStronglyMeasurable.smul (Continuous.aestronglyMeasurable (by fun_prop)) hf.1\n  · filter_upwards [Ici_mem_atTop (0 : ℝ)] with c (hc : 0 ≤ c)\n    filter_upwards with v\n    simp only [ofReal_inv, neg_mul, norm_smul]\n    norm_cast\n    conv_rhs => rw [← one_mul (‖f v‖)]\n    gcongr\n    simp only [norm_eq_abs, abs_exp, exp_le_one_iff, Left.neg_nonpos_iff]\n    positivity\n\nvariable [CompleteSpace E]","hard_negative":true,"metrics":{"chosen_tokens":144,"rejected_tokens":3,"token_jaccard":0.028571,"token_length_ratio":0.020833},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"bfa5b8eace368b97ef5d76e061c3331c0c68cac32282eb03ba5245a9ae1574e7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Integral.PeakFunction\npublic import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2024 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Fourier inversion formula\n\nIn a finite-dimensional real inner product space, we show the Fourier inversion formula, i.e.,\n`𝓕⁻ (𝓕 f) v = f v` if `f` and `𝓕 f` are integrable, and `f` is continuous at `v`. This is proved\nin `MeasureTheory.Integrable.fourier_inversion`. See also `Continuous.fourier_inversion`\ngiving `𝓕⁻ (𝓕 f) = f` under an additional continuity assumption for `f`.\n\nWe use the following proof. A naïve computation gives\n`𝓕⁻ (𝓕 f) v\n= ∫_w exp (2 I π ⟪w, v⟫) 𝓕 f (w) dw\n= ∫_w exp (2 I π ⟪w, v⟫) ∫_x, exp (-2 I π ⟪w, x⟫) f x dx) dw\n= ∫_x (∫_ w, exp (2 I π ⟪w, v - x⟫ dw) f x dx `\n\nHowever, the Fubini step does not make sense for lack of integrability, and the middle integral\n`∫_ w, exp (2 I π ⟪w, v - x⟫ dw` (which one would like to be a Dirac at `v - x`) is not defined.\nTo gain integrability, one multiplies with a Gaussian function `exp (-c⁻¹ ‖w‖^2)`, with a large\n(but finite) `c`. As this function converges pointwise to `1` when `c → ∞`, we get\n`∫_w exp (2 I π ⟪w, v⟫) 𝓕 f (w) dw = lim_c ∫_w exp (-c⁻¹ ‖w‖^2 + 2 I π ⟪w, v⟫) 𝓕 f (w) dw`.\nOne can perform Fubini on the right-hand side for fixed `c`, writing the integral as\n`∫_x (∫_w exp (-c⁻¹‖w‖^2 + 2 I π ⟪w, v - x⟫ dw)) f x dx`.\nThe middle factor is the Fourier transform of a more and more flat function\n(converging to the constant `1`), hence it becomes more and more concentrated, around the\npoint `v`. (Morally, it converges to the Dirac at `v`). Moreover, it has integral one.\nTherefore, multiplying by `f` and integrating, one gets a term converging to `f v` as `c → ∞`.\nSince it also converges to `𝓕⁻ (𝓕 f) v`, this proves the result.\n\nTo check the concentration property of the middle factor and the fact that it has integral one, we\nrely on the explicit computation of the Fourier transform of Gaussians.\n-/\n\npublic section\n\nopen Filter MeasureTheory Complex Module Metric Real Bornology\n\nopen scoped Topology FourierTransform RealInnerProductSpace Complex\n\nvariable {V E : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]\n  [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V]\n  [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → E}\n\nnamespace Real\n\nTarget:\nlemma tendsto_integral_cexp_sq_smul (hf : Integrable f) :\n    Tendsto (fun (c : ℝ) ↦ (∫ v : V, cexp (- c⁻¹ * ‖v‖ ^ 2) • f v))\n      atTop (𝓝 (∫ v : V, f v)) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_dce31b4f3667","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"510d9da003c113862690226ac0151db9fe7da08152bd3adf43d17651e352bfc3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Fourier","family_id":"tendsto_integral_cexp_sq_smul","file_id":"mathlib/Mathlib/Analysis/Fourier/Inversion.lean","sample_id":"dce31b4f3667c3830b421cb76a3fefb37b0de149d8e70ed049af7ee6287ae930"},"split":"train"}
{"binding":{"chosen_proof_sha256":"63cd494fa84f771bdfe24b254aba32aae9ae2b28ee1e64d13bc4951087609f7b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"64b0828b72ceff37d3d9baf47b05b8e5a0d086c4eb08d7061602bb711debc8cf","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3d37610ea90fd4d8210d4ae3653f0e831fc85b4617585bafc5fe04742a37ccee","source_sha256":"54cfbd06b4a139f3904600d55743484d7ae132ed17d8577ed2bfda5db188e8e8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction γ using Quotient.ind with | mk γ =>\n  simpa [← mk_trans, ← mk_symm, ← mk_refl, eq] using Homotopic.trans_symm γ","hard_negative":true,"metrics":{"chosen_tokens":30,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.066667},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"bfad231e82430113335d411ff8a08f5d5bc98e82e3dc570bc59196f27c7a72a1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Groupoid.Grpd.Basic\npublic import Mathlib.Topology.Category.TopCat.Basic\npublic import Mathlib.Topology.Homotopy.Path\npublic import Mathlib.Data.Set.Subsingleton\n\nNamespace:\nPath.Homotopic.Quotient\n\nLocal context:\n/-\nCopyright (c) 2021 Shing Tak Lam. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Shing Tak Lam\n-/\n/-!\n# Fundamental groupoid of a space\n\nGiven a topological space `X`, we can define the fundamental groupoid of `X` to be the category with\nobjects being points of `X`, and morphisms `x ⟶ y` being paths from `x` to `y`, quotiented by\nhomotopy equivalence. With this, the fundamental group of `X` based at `x` is just the automorphism\ngroup of `x`.\n-/\n\n@[expose] public section\n\nopen CategoryTheory\n\nuniverse u\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\nvariable {x₀ x₁ : X}\n\nnoncomputable section\n\nopen unitInterval\n\nnamespace Path\n\nnamespace Homotopy\n\nsection\n\n/-- Auxiliary function for `reflTransSymm`. -/\ndef reflTransSymmAux (x : I × I) : ℝ :=\n  if (x.2 : ℝ) ≤ 1 / 2 then x.1 * 2 * x.2 else x.1 * (2 - 2 * x.2)\n\n@[continuity, fun_prop]\ntheorem continuous_reflTransSymmAux : Continuous reflTransSymmAux :=\n  continuous_if_le (by fun_prop) (by fun_prop) (by fun_prop) (by fun_prop) (by grind)\n\ntheorem reflTransSymmAux_mem_I (x : I × I) : reflTransSymmAux x ∈ I := by\n  dsimp only [reflTransSymmAux]\n  split_ifs\n  · constructor\n    · apply mul_nonneg <;> grind\n    · rw [mul_assoc]\n      apply mul_le_one₀ <;> grind\n  · constructor\n    · apply mul_nonneg <;> grind\n    · apply mul_le_one₀ <;> grind\n\n/-- For any path `p` from `x₀` to `x₁`, we have a homotopy from the constant path based at `x₀` to\n  `p.trans p.symm`. -/\ndef reflTransSymm (p : Path x₀ x₁) : Homotopy (Path.refl x₀) (p.trans p.symm) where\n  toFun x := p ⟨reflTransSymmAux x, reflTransSymmAux_mem_I x⟩\n  continuous_toFun := by fun_prop\n  map_zero_left := by simp [reflTransSymmAux]\n  map_one_left x := by\n    simp only [reflTransSymmAux, Path.trans]\n    cases le_or_gt (x : ℝ) 2⁻¹ with\n    | inl hx => simp [hx, ← extend_apply]\n    | inr hx =>\n      have : p.extend (2 - 2 * ↑x) = p.extend (1 - (2 * ↑x - 1)) := by ring_nf\n      simpa [hx.not_ge, ← extend_apply]\n  prop' t := by norm_num [reflTransSymmAux]\n\n/-- For any path `p` from `x₀` to `x₁`, we have a homotopy from the constant path based at `x₁` to\n  `p.symm.trans p`. -/\ndef reflSymmTrans (p : Path x₀ x₁) : Homotopy (Path.refl x₁) (p.symm.trans p) :=\n  (reflTransSymm p.symm).cast rfl <| congr_arg _ (Path.symm_symm _)\n\nend\n\nsection TransRefl\n\n/-- Auxiliary function for `trans_refl_reparam`. -/\ndef transReflReparamAux (t : I) : ℝ :=\n  if (t : ℝ) ≤ 1 / 2 then 2 * t else 1\n\n@[continuity, fun_prop]\ntheorem continuous_transReflReparamAux : Continuous transReflReparamAux :=\n  continuous_if_le (by fun_prop) (by fun_prop) (by fun_prop) (by fun_prop) (by grind)\n\ntheorem transReflReparamAux_mem_I (t : I) : transReflReparamAux t ∈ I := by\n  unfold transReflReparamAux\n  split_ifs <;> constructor <;> linarith [unitInterval.le_one t, unitInterval.nonneg t]\n\ntheorem transReflReparamAux_zero : transReflReparamAux 0 = 0 := by\n  norm_num [transReflReparamAux]\n\ntheorem transReflReparamAux_one : transReflReparamAux 1 = 1 := by\n  norm_num [transReflReparamAux]\n\ntheorem trans_refl_reparam (p : Path x₀ x₁) :\n    p.trans (Path.refl x₁) =\n      p.reparam (fun t => ⟨transReflReparamAux t, transReflReparamAux_mem_I t⟩) (by fun_prop)\n        (Subtype.ext transReflReparamAux_zero) (Subtype.ext transReflReparamAux_one) := by\n  ext\n  unfold transReflReparamAux\n  simp only [coe_reparam]\n  grind\n\n/-- For any path `p` from `x₀` to `x₁`, we have a homotopy from `p.trans (Path.refl x₁)` to `p`. -/\ndef transRefl (p : Path x₀ x₁) : Homotopy (p.trans (Path.refl x₁)) p :=\n  ((Homotopy.reparam p (fun t => ⟨transReflReparamAux t, transReflReparamAux_mem_I t⟩)\n          (by fun_prop) (Subtype.ext transReflReparamAux_zero)\n          (Subtype.ext transReflReparamAux_one)).cast\n      rfl (trans_refl_reparam p).symm).symm\n\n/-- For any path `p` from `x₀` to `x₁`, we have a homotopy from `(Path.refl x₀).trans p` to `p`. -/\ndef reflTrans (p : Path x₀ x₁) : Homotopy ((Path.refl x₀).trans p) p :=\n  (transRefl p.symm).symm₂.cast (by simp) (by simp)\n\nend TransRefl\n\nsection Assoc\n\n/-- Auxiliary function for `trans_assoc_reparam`. -/\ndef transAssocReparamAux (t : I) : ℝ :=\n  if (t : ℝ) ≤ 1 / 4 then 2 * t else if (t : ℝ) ≤ 1 / 2 then t + 1 / 4 else 1 / 2 * (t + 1)\n\n@[continuity, fun_prop]\ntheorem continuous_transAssocReparamAux : Continuous transAssocReparamAux :=\n  continuous_if_le (by fun_prop) (by fun_prop) (by fun_prop)\n    (continuous_if_le (by fun_prop) (by fun_prop) (by fun_prop) (by fun_prop)\n      (by grind)).continuousOn (by grind)\n\ntheorem transAssocReparamAux_mem_I (t : I) : transAssocReparamAux t ∈ I := by\n  unfold transAssocReparamAux\n  split_ifs <;> constructor <;> linarith [unitInterval.le_one t, unitInterval.nonneg t]\n\ntheorem transAssocReparamAux_zero : transAssocReparamAux 0 = 0 := by\n  norm_num [transAssocReparamAux]\n\ntheorem transAssocReparamAux_one : transAssocReparamAux 1 = 1 := by\n  norm_num [transAssocReparamAux]\n\ntheorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :\n    (p.trans q).trans r =\n      (p.trans (q.trans r)).reparam\n        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by fun_prop)\n        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by\n  ext x\n  simp only [transAssocReparamAux, Path.trans_apply, Function.comp_apply, Path.coe_reparam]\n  split_ifs\n  iterate 12 grind\n  · linarith\n  · linarith\n  · grind\n\n/-- For paths `p q r`, we have a homotopy from `(p.trans q).trans r` to `p.trans (q.trans r)`. -/\ndef transAssoc {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :\n    Homotopy ((p.trans q).trans r) (p.trans (q.trans r)) :=\n  ((Homotopy.reparam (p.trans (q.trans r))\n          (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by fun_prop)\n          (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one)).cast\n      rfl (trans_assoc_reparam p q r).symm).symm\n\nend Assoc\n\nend Homotopy\n\nnamespace Homotopic\n\ntheorem refl_trans (p : Path x₀ x₁) :\n    ((Path.refl x₀).trans p).Homotopic p :=\n  ⟨Homotopy.reflTrans p⟩\n\ntheorem trans_refl (p : Path x₀ x₁) :\n    (p.trans (Path.refl x₁)).Homotopic p :=\n  ⟨Homotopy.transRefl p⟩\n\ntheorem trans_symm (p : Path x₀ x₁) :\n    (p.trans p.symm).Homotopic (Path.refl x₀) :=\n  ⟨(Homotopy.reflTransSymm p).symm⟩\n\ntheorem symm_trans (p : Path x₀ x₁) :\n    (p.symm.trans p).Homotopic (Path.refl x₁) :=\n  ⟨(Homotopy.reflSymmTrans p).symm⟩\n\ntheorem trans_assoc {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :\n    ((p.trans q).trans r).Homotopic (p.trans (q.trans r)) :=\n  ⟨Homotopy.transAssoc p q r⟩\n\nnamespace Quotient\n\n@[simp, grind =]\ntheorem refl_trans (γ : Homotopic.Quotient x₀ x₁) :\n    trans (refl x₀) γ = γ := by\n  induction γ using Quotient.ind with | mk γ =>\n  simpa [← mk_trans, ← mk_refl, eq] using Homotopic.refl_trans γ\n\n@[simp, grind =]\ntheorem trans_refl (γ : Homotopic.Quotient x₀ x₁) :\n    trans γ (refl x₁) = γ := by\n  induction γ using Quotient.ind with | mk γ =>\n  simpa [← mk_trans, ← mk_refl, eq] using Homotopic.trans_refl γ\n\n@[simp, grind =]\n\nTarget:\ntheorem trans_symm (γ : Homotopic.Quotient x₀ x₁) :\n    trans γ (symm γ) = refl x₀ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_3d37610ea90f","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e91437c0d2c6b48ba87952c8f66e370e39310c3c2fec5f9fa0210d7b9d00a0f3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/FundamentalGroupoid","family_id":"trans_symm","file_id":"mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean","sample_id":"3d37610ea90fd4d8210d4ae3653f0e831fc85b4617585bafc5fe04742a37ccee"},"split":"train"}
{"binding":{"chosen_proof_sha256":"105decf31a73d8f03c28fa8bbcf536ea59b795c7d4d6c1c8dd7fade7fd71c4b3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"69ddf01226811bd3e646477572e8227879a86ffb5007988759dbb84b2d53d0f8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2ecf221e60a1de5319a947fdeadd18115cd6ed88d23420eaeb85a40fe0314ea2","source_sha256":"46b1b7b1e57c2ee22dceb33ca3f6863e9e50ed75099a163a5900686dabb9e572","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨K * ε * K, mul_pos (mul_pos K0 ε0) K0, fun {a b} ha hb h => ?_⟩\n  have a0 := K0.trans_le ha\n  have b0 := K0.trans_le hb\n  rw [inv_sub_inv' ((abv_pos abv).1 a0) ((abv_pos abv).1 b0), abv_mul abv, abv_mul abv, abv_inv abv,\n    abv_inv abv, abv_sub abv]\n  refine lt_of_mul_lt_mul_left (lt_of_mul_lt_mul_right ?_ b0.le) a0.le\n  rw [mul_assoc, inv_mul_cancel_right₀ b0.ne', ← mul_assoc, mul_inv_cancel₀ a0.ne', one_mul]\n  refine h.trans_le ?_\n  gcongr\n  exact mul_nonneg a0.le ε0.le","hard_negative":true,"metrics":{"chosen_tokens":131,"rejected_tokens":3,"token_jaccard":0.037037,"token_length_ratio":0.022901},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"bfae18879023840b877ad962e6b5d163f3e718e1dc726152329984b6b3768b47","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Pi\npublic import Mathlib.Algebra.Order.AbsoluteValue.Basic\npublic import Mathlib.Algebra.Order.Field.Basic\npublic import Mathlib.Algebra.Order.Group.MinMax\npublic import Mathlib.Algebra.Ring.Pi\npublic import Mathlib.Data.Setoid.Basic\npublic import Mathlib.GroupTheory.GroupAction.Ring\npublic import Mathlib.Tactic.GCongr\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Cauchy sequences\n\nA basic theory of Cauchy sequences, used in the construction of the reals and p-adic numbers. Where\napplicable, lemmas that will be reused in other contexts have been stated in extra generality.\nThere are other \"versions\" of Cauchyness in the library, in particular Cauchy filters in topology.\nThis is a concrete implementation that is useful for simplicity and computability reasons.\n\n## Important definitions\n\n* `IsCauSeq`: a predicate that says `f : ℕ → β` is Cauchy.\n* `CauSeq`: the type of Cauchy sequences valued in type `β` with respect to an absolute value\n  function `abv`.\n\n## Tags\n\nsequence, cauchy, abs val, absolute value\n-/\n\n@[expose] public section\n\nassert_not_exists Finset Module Submonoid FloorRing\n\nvariable {α β : Type*}\n\nopen IsAbsoluteValue\n\nsection\n\nvariable [Field α] [LinearOrder α] [IsStrictOrderedRing α] [Ring β]\n  (abv : β → α) [IsAbsoluteValue abv]\n\ntheorem rat_add_continuous_lemma {ε : α} (ε0 : 0 < ε) :\n    ∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv (a₁ - b₁) < δ → abv (a₂ - b₂) < δ →\n      abv (a₁ + a₂ - (b₁ + b₂)) < ε :=\n  ⟨ε / 2, half_pos ε0, fun {a₁ a₂ b₁ b₂} h₁ h₂ => by\n    simpa [add_halves, sub_eq_add_neg, add_comm, add_left_comm, add_assoc] using\n      lt_of_le_of_lt (abv_add abv _ _) (add_lt_add h₁ h₂)⟩\n\ntheorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :\n    ∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →\n      abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε := by\n  have K0 : (0 : α) < max 1 (max K₁ K₂) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)\n  have εK := div_pos (half_pos ε0) K0\n  refine ⟨_, εK, fun {a₁ a₂ b₁ b₂} ha₁ hb₂ h₁ h₂ => ?_⟩\n  replace ha₁ := lt_of_lt_of_le ha₁ (le_trans (le_max_left _ K₂) (le_max_right 1 _))\n  replace hb₂ := lt_of_lt_of_le hb₂ (le_trans (le_max_right K₁ _) (le_max_right 1 _))\n  set M := max 1 (max K₁ K₂)\n  have : abv (a₁ - b₁) * abv b₂ + abv (a₂ - b₂) * abv a₁ < ε / 2 / M * M + ε / 2 / M * M := by\n    gcongr\n  rw [← abv_mul abv, mul_comm, div_mul_cancel₀ _ (ne_of_gt K0), ← abv_mul abv, add_halves] at this\n  simpa [sub_eq_add_neg, mul_add, add_mul, add_left_comm] using\n    lt_of_le_of_lt (abv_add abv _ _) this\n\nTarget:\ntheorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]\n    {ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :\n    ∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_2ecf221e60a1","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"0e6829ec1178c92b549141908bb745c977605ae9db5344e4cb50a2da19ec4586","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Order","family_id":"rat_inv_continuous_lemma","file_id":"mathlib/Mathlib/Algebra/Order/CauSeq/Basic.lean","sample_id":"2ecf221e60a1de5319a947fdeadd18115cd6ed88d23420eaeb85a40fe0314ea2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4a41798036ff76446dc750a37ef284895b5d38963c6bb7b381d67845e6108343","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"da422d22caf26f3e5568440ff4f4ce39cf647f59520cb7416b45b8017ec51f47","source_sha256":"aa1352e2143e704eafbedf6a7be68b6abc2920999f9fb6e7acf0e05da957ef08","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : JacobsonSpace ↥X := LocallyOfFiniteType.jacobsonSpace (f ≫ i)\n  refine ext_of_fromSpecResidueField_eq f g i (S ∩ closedPoints X) ?_ ?_ H'\n  · rwa [dense_iff_closure_eq, JacobsonSpace.closure_inter_closedPoints_eq_closure hS,\n      ← dense_iff_closure_eq]\n  · intro x ⟨hxS, hx⟩\n    rw [← cancel_epi (Spec.map (residueFieldIsoBase (f ≫ i) x hx).hom)]\n    refine ext_of_apply_closedPoint_eq i ?_ ?_ (by simpa using H x hxS hx) <;>\n      simp only [Category.assoc, ← SpecMap_residueFieldIsoBase_inv (f ≫ i) x hx, ← Spec.map_comp,\n        Iso.inv_hom_id, Spec.map_id, ← H']","hard_negative":false,"metrics":{"chosen_tokens":126,"rejected_tokens":2,"token_jaccard":0.016129,"token_length_ratio":0.015873},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"bfd210fe30b2ff65a38babfcb4d8ba3ea9daf294d32826be8d14276131d922b7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.Finite\npublic import Mathlib.FieldTheory.IsAlgClosed.Basic\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2026 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Schemes over algebraically closed fields\n\nWe show that if `X` is locally of finite type over an algebraically closed field `k`,\nthen the closed points of `X` are in bijection with the `k`-points of `X`.\nSee `AlgebraicGeometry.pointEquivClosedPoint`.\n\n-/\n\n@[expose] public noncomputable section\n\nopen CategoryTheory\n\nnamespace AlgebraicGeometry\n\nuniverse u\n\nvariable {X Y : Scheme.{u}} {K : Type u} [Field K] [IsAlgClosed K]\n    (f : X ⟶ Spec (.of K)) [LocallyOfFiniteType f] (x : X) (hx : IsClosed {x})\n\n/-- If `X` is a locally of finite type `k`-scheme and `k` is algebraically closed, then\nthe residue field of any closed point of `x` is isomorphic to `k`. -/\ndef residueFieldIsoBase : X.residueField x ≅ .of K :=\n  letI : IsIso (Spec.preimage (X.fromSpecResidueField x ≫ f)) := by\n    have : IsFinite (X.fromSpecResidueField x ≫ f) := by\n      rw [isClosed_singleton_iff_isClosedImmersion] at hx\n      rw [isFinite_iff_locallyOfFiniteType_of_jacobsonSpace]\n      infer_instance\n    rw [ConcreteCategory.isIso_iff_bijective]\n    refine IsAlgClosed.ringHom_bijective_of_isIntegral _ ?_\n    rw [← IsIntegralHom.SpecMap_iff, Spec.map_preimage]\n    infer_instance\n  (asIso (Spec.preimage (X.fromSpecResidueField x ≫ f))).symm\n\n@[simp, reassoc]\nlemma SpecMap_residueFieldIsoBase_inv :\n    Spec.map (residueFieldIsoBase f x hx).inv = X.fromSpecResidueField x ≫ f :=\n  Spec.map_preimage _\n\n/-- If `k` is algebraically closed, this is the `k`-point of `X` associated to a closed point. -/\nnoncomputable\ndef pointOfClosedPoint : Spec (.of K) ⟶ X :=\n  Spec.map (residueFieldIsoBase f x hx).hom ≫ X.fromSpecResidueField x\n\n@[reassoc (attr := simp)]\nlemma pointOfClosedPoint_comp : pointOfClosedPoint f x hx ≫ f = 𝟙 _ := by\n  simp [pointOfClosedPoint, ← SpecMap_residueFieldIsoBase_inv, ← Spec.map_comp]\n\n@[simp]\nlemma pointOfClosedPoint_apply (a : _) : pointOfClosedPoint f x hx a = x := by\n  simp [pointOfClosedPoint]\n\nset_option backward.defeqAttrib.useBackward true in\n/-- If `k` is algebraically closed,\nthen the closed points of `X` are in bijection with the `k`-points of `X`. -/\n@[simps]\ndef pointEquivClosedPoint :\n    {p : Spec (.of K) ⟶ X // p ≫ f = 𝟙 _} ≃ closedPoints X where\n  toFun p := ⟨p.1 (IsLocalRing.closedPoint K), by\n    have := isClosedImmersion_of_comp_eq_id _ _ p.2\n    have := p.1.isClosedEmbedding.isClosed_range\n    rwa [Set.range_eq_singleton] at this\n    exact fun x ↦ congr(p.1 $(Subsingleton.elim _ _))⟩\n  invFun x := ⟨pointOfClosedPoint f x.1 x.2, pointOfClosedPoint_comp f x.1 x.2⟩\n  left_inv p := by\n    ext\n    refine ((Scheme.SpecToEquivOfField _ _).symm_apply_eq (x := ⟨_, _⟩)).mpr ?_\n    rw [Scheme.SpecToEquivOfField_eq_iff]\n    dsimp [Scheme.SpecToEquivOfField]\n    simp only [Category.id_comp, exists_const]\n    generalize_proofs _ h\n    refine (Category.comp_id _).symm.trans (((residueFieldIsoBase f _ h).eq_inv_comp).mp ?_)\n    rw [← Spec.map_injective.eq_iff]\n    simp only [Spec.map_id, Spec.map_comp, SpecMap_residueFieldIsoBase_inv]\n    rw [reassoc_of% Scheme.descResidueField_stalkClosedPointTo_fromSpecResidueField, p.2]\n  right_inv x := by simp\n\nlemma ext_of_apply_closedPoint_eq\n    {f g : Spec (.of K) ⟶ X} (h : X ⟶ Spec (.of K))\n    [LocallyOfFiniteType h]\n    (hf : f ≫ h = 𝟙 _) (hg : g ≫ h = 𝟙 _)\n    (H : f (IsLocalRing.closedPoint K) = g (IsLocalRing.closedPoint K)) : f = g :=\n  congr($((pointEquivClosedPoint h).injective (a₁ := ⟨f, hf⟩) (a₂ := ⟨g, hg⟩) (Subtype.ext H)).1)\n\n/-- Let `X` and `Y` be locally of finite type `K`-schemes with `K` algebraically closed and `Y`\nseparated over `K`. Suppose `X` is reduced, then two `K`-morphisms `f g : X ⟶ Y` are equal if\nthey are equal on the closed points of a dense locally closed subset of `X`. -/\n\nTarget:\nlemma ext_of_apply_eq {f g : X ⟶ Y} (i : Y ⟶ Spec (.of K)) [IsSeparated i] [LocallyOfFiniteType i]\n    [IsReduced X] [LocallyOfFiniteType (f ≫ i)]\n    (S : Set X) (hS : IsLocallyClosed S) (hS' : Dense S)\n    (H : ∀ x ∈ S, IsClosed {x} → f x = g x)\n    (H' : f ≫ i = g ≫ i) : f = g :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/AlgClosed","family_id":"ext_of_apply_eq","file_id":"mathlib/Mathlib/AlgebraicGeometry/AlgClosed/Basic.lean","sample_id":"da422d22caf26f3e5568440ff4f4ce39cf647f59520cb7416b45b8017ec51f47"},"split":"train"}
{"binding":{"chosen_proof_sha256":"06f47103349d9ae4b5fa9c701936477e3308e59432a0c09cdb18eb8a9e2de388","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6cb7857e8c992cbf8974094de8e796c632b14f4a4f590b360fe38825b0471692","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ea19c96b1833a290f3ad4a2c8868771d2cf2aee1459ba1b113ef3ef0cf9c4a4c","source_sha256":"0333a24d4ec160f28f2400f09ad2bc2ec0955c0c7bd92de1261f4a4e19fe5936","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  contrapose!; exact parts_eq_empty_iff","hard_negative":true,"metrics":{"chosen_tokens":6,"rejected_tokens":2,"token_jaccard":0.142857,"token_length_ratio":0.333333},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"bfe88c55f540db35856d776df6cb318428f25f275c95f3e3cfea5e9dcf913a0f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\npublic import Mathlib.Data.Finset.Pairwise\npublic import Mathlib.Data.Finset.Preimage\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Data.Setoid.Basic\npublic import Mathlib.Order.Atoms\npublic import Mathlib.Order.SupIndep\n\nNamespace:\nFinpartition\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Finite partitions\n\nIn this file, we define finite partitions. A finpartition of `a : α` is a finite set of pairwise\ndisjoint parts `parts : Finset α` which does not contain `⊥` and whose supremum is `a`.\n\nFinpartitions of a finset are at the heart of Szemerédi's regularity lemma. They are also studied\npurely order theoretically in Sperner theory.\n\n## Constructions\n\nWe provide many ways to build finpartitions:\n* `Finpartition.ofErase`: Builds a finpartition by erasing `⊥` for you.\n* `Finpartition.ofSubset`: Builds a finpartition from a subset of the parts of a previous\n  finpartition.\n* `Finpartition.empty`: The empty finpartition of `⊥`.\n* `Finpartition.indiscrete`: The indiscrete, aka trivial, aka pure, finpartition made of a single\n  part.\n* `Finpartition.discrete`: The discrete finpartition of `s : Finset α` made of singletons.\n* `Finpartition.toSubtype`: Turns a finpartition of a type to one of a subtype.\n* `Finpartition.bind`: Puts together the finpartitions of the parts of a finpartition into a new\n  finpartition.\n* `Finpartition.extend`: Extends a finpartition of `a` to a finpartition of `a ⊔ b` by adding `b`\n  as a new part.\n* `Finpartition.extendOfLE`: Extends a finpartition of `a` to a finpartition of `b` when `a ≤ b`,\n  by adding `b \\ a` as a new part (if nonempty).\n* `Finpartition.restrict`: Restricts a finpartition of `a` to `b` where `b ≤ a` by intersecting\n  each part with `b`.\n* `Finpartition.ofPairwiseDisjoint`: Builds a finpartition from a finset `parts` of pairwise\n  disjoint elements.\n* `Finpartition.combine`: Combines a family of partitions of pairwise disjoint elements into a\n  partition of their sup.\n* `Finpartition.ofExistsUnique`: Builds a finpartition from a collection of parts such that each\n  element is in exactly one part.\n* `Finpartition.ofSetoid`: With `Fintype α`, constructs the finpartition of `univ : Finset α`\n  induced by the equivalence classes of `s : Setoid α`.\n* `Finpartition.atomise`: Makes a finpartition of `s : Finset α` by breaking `s` along all finsets\n  in `F : Finset (Finset α)`. Two elements of `s` belong to the same part iff they belong to the\n  same elements of `F`.\n\n`Finpartition.indiscrete` and `Finpartition.bind` together form the monadic structure of\n`Finpartition`.\n\n## Implementation notes\n\nForbidding `⊥` as a part follows mathematical tradition and is a pragmatic choice concerning\noperations on `Finpartition`. Not caring about `⊥` being a part or not breaks extensionality (it's\nnot because the parts of `P` and the parts of `Q` have the same elements that `P = Q`). Enforcing\n`⊥` to be a part makes `Finpartition.bind` uglier and doesn't rid us of the need of\n`Finpartition.ofErase`.\n\n## TODO\n\nThe order is the wrong way around to make `Finpartition a` a graded order. Is it bad to depart from\nthe literature and turn the order around?\n\nThe specialisation to `Finset α` could be generalised to atomistic orders.\n-/\n\n@[expose] public section\n\n\nopen Finset Function\n\nvariable {α : Type*}\n\n/-- A finite partition of `a : α` is a pairwise disjoint finite set of elements whose supremum is\n`a`. We forbid `⊥` as a part. -/\n@[ext]\nstructure Finpartition [Lattice α] [OrderBot α] (a : α) where\n  /-- The elements of the finite partition of `a` -/\n  parts : Finset α\n  /-- The partition is supremum-independent -/\n  protected supIndep : parts.SupIndep id\n  /-- The supremum of the partition is `a` -/\n  sup_parts : parts.sup id = a\n  /-- No element of the partition is bottom -/\n  bot_notMem : ⊥ ∉ parts\n  deriving DecidableEq\n\nnamespace Finpartition\n\nsection Lattice\n\nvariable [Lattice α] [OrderBot α]\n\n/-- A `Finpartition` constructor which does not insist on `⊥` not being a part. -/\n@[simps]\ndef ofErase [DecidableEq α] {a : α} (parts : Finset α) (sup_indep : parts.SupIndep id)\n    (sup_parts : parts.sup id = a) : Finpartition a where\n  parts := parts.erase ⊥\n  supIndep := sup_indep.subset (erase_subset _ _)\n  sup_parts := (sup_erase_bot _).trans sup_parts\n  bot_notMem := notMem_erase _ _\n\n/-- A `Finpartition` constructor from a bigger existing finpartition. -/\n@[simps]\ndef ofSubset {a b : α} (P : Finpartition a) {parts : Finset α} (subset : parts ⊆ P.parts)\n    (sup_parts : parts.sup id = b) : Finpartition b :=\n  { parts := parts\n    supIndep := P.supIndep.subset subset\n    sup_parts := sup_parts\n    bot_notMem := fun h ↦ P.bot_notMem (subset h) }\n\nlemma sum_ofSubset_eq_sum {a b : α} (P : Finpartition a) {parts : Finset α}\n    (subset : parts ⊆ P.parts) (sup_parts : parts.sup id = b)\n    {X : Type*} [AddCommMonoid X] (f : α → X) (hf : ∀ p ∈ P.parts, p ∉ parts → f p = 0) :\n    ∑ p ∈ (P.ofSubset subset sup_parts).parts, f p = ∑ p ∈ P.parts, f p :=\n  Finset.sum_subset subset hf\n\n/-- Changes the type of a finpartition to an equal one. -/\n@[simps]\ndef copy {a b : α} (P : Finpartition a) (h : a = b) : Finpartition b where\n  parts := P.parts\n  supIndep := P.supIndep\n  sup_parts := h ▸ P.sup_parts\n  bot_notMem := P.bot_notMem\n\n/-- Transfer a finpartition over an order isomorphism. -/\ndef map {β : Type*} [Lattice β] [OrderBot β] {a : α} (e : α ≃o β) (P : Finpartition a) :\n    Finpartition (e a) where\n  parts := P.parts.map e\n  supIndep u hu _ hb hbu _ hx hxu := by\n    rw [← map_symm_subset] at hu\n    simp only [mem_map_equiv] at hb\n    have := P.supIndep hu hb (by simp [hbu]) (map_rel e.symm hx) ?_\n    · rw [← e.symm.map_bot] at this\n      exact e.symm.map_rel_iff.mp this\n    · convert! e.symm.map_rel_iff.mpr hxu\n      rw [map_finset_sup, sup_map]\n      rfl\n  sup_parts := by simp [← P.sup_parts]\n  bot_notMem := by\n    rw [mem_map_equiv]\n    convert! P.bot_notMem\n    exact e.symm.map_bot\n\n@[simp]\ntheorem parts_map {β : Type*} [Lattice β] [OrderBot β] {a : α} {e : α ≃o β} {P : Finpartition a} :\n    (P.map e).parts = P.parts.map e := rfl\n\nvariable (α)\n\n/-- The empty finpartition. -/\n@[simps]\nprotected def empty : Finpartition (⊥ : α) where\n  parts := ∅\n  supIndep := supIndep_empty _\n  sup_parts := Finset.sup_empty\n  bot_notMem := notMem_empty ⊥\n\ninstance : Inhabited (Finpartition (⊥ : α)) :=\n  ⟨Finpartition.empty α⟩\n\n@[simp]\ntheorem default_eq_empty : (default : Finpartition (⊥ : α)) = Finpartition.empty α :=\n  rfl\n\nvariable {α} {a : α}\n\n/-- The finpartition in one part, aka indiscrete finpartition. -/\n@[simps]\ndef indiscrete (ha : a ≠ ⊥) : Finpartition a where\n  parts := {a}\n  supIndep := supIndep_singleton _ _\n  sup_parts := Finset.sup_singleton\n  bot_notMem h := ha (mem_singleton.1 h).symm\n\nvariable (P : Finpartition a)\n\nprotected theorem le {b : α} (hb : b ∈ P.parts) : b ≤ a :=\n  (le_sup hb).trans P.sup_parts.le\n\ntheorem ne_bot {b : α} (hb : b ∈ P.parts) : b ≠ ⊥ := by\n  intro h\n  refine P.bot_notMem (?_)\n  rw [h] at hb\n  exact hb\n\nprotected theorem disjoint : (P.parts : Set α).PairwiseDisjoint id :=\n  P.supIndep.pairwiseDisjoint\n\nvariable {P}\n\n@[simp]\ntheorem parts_eq_empty_iff : P.parts = ∅ ↔ a = ⊥ := by\n  simp_rw [← P.sup_parts]\n  refine ⟨fun h ↦ ?_, fun h ↦ eq_empty_iff_forall_notMem.2 fun b hb ↦ P.bot_notMem ?_⟩\n  · rw [h]\n    exact Finset.sup_empty\n  · rwa [← le_bot_iff.1 ((le_sup hb).trans h.le)]\n\n@[simp]\n\nTarget:\ntheorem parts_nonempty_iff : P.parts.Nonempty ↔ a ≠ ⊥ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_ea19c96b1833","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"1f1f15d8bb54b35d8fbae03733e059798ff7fab941495223e03a5e90c270d8a2","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Partition","family_id":"parts_nonempty_iff","file_id":"mathlib/Mathlib/Order/Partition/Finpartition.lean","sample_id":"ea19c96b1833a290f3ad4a2c8868771d2cf2aee1459ba1b113ef3ef0cf9c4a4c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"db758460f3feb84a60fe6730f2e7a0ca8f4614d60686027e4a4aa0bdb93eeac5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"56824650124ef37b3559baee31b1277720d3a9f018ce7db07f8712e4b6c90072","source_sha256":"ba819d6726990bcb515ef042ef549f9e2a90b75f672af9b55d2f61118b410281","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases hμ : SFinite μ\n  swap; · rw [compProd_of_not_sfinite _ _ hμ]; simp\n  by_cases hν : SFinite ν\n  swap; · rw [compProd_of_not_sfinite _ _ hν]; simp\n  let s := κ.mutuallySingularSet η\n  have hs : MeasurableSet s := Kernel.measurableSet_mutuallySingularSet κ η\n  symm\n  refine ⟨s, hs, ?_⟩\n  rw [compProd_apply hs, compProd_apply hs.compl]\n  have h_eq a : Prod.mk a ⁻¹' s = Kernel.mutuallySingularSetSlice κ η a := rfl\n  have h1 a : η a (Prod.mk a ⁻¹' s) = 0 := by rw [h_eq, Kernel.measure_mutuallySingularSetSlice]\n  have h2 : ∀ᵐ a ∂μ, κ a (Prod.mk a ⁻¹' s)ᶜ = 0 := by\n    filter_upwards [hκη] with a ha\n    rwa [h_eq, ← Kernel.withDensity_rnDeriv_eq_zero_iff_measure_eq_zero κ η a,\n      Kernel.withDensity_rnDeriv_eq_zero_iff_mutuallySingular]\n  simp [h1, lintegral_congr_ae h2]","hard_negative":false,"metrics":{"chosen_tokens":184,"rejected_tokens":3,"token_jaccard":0.030769,"token_length_ratio":0.016304},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"c0233fccdadf7eb2d3c3b685c0af797c8080e40510af741c1b6a8535e507692d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Kernel.Composition.MeasureCompProd\npublic import Mathlib.Probability.Kernel.RadonNikodym\n\nNamespace:\nMeasureTheory.Measure\n\nLocal context:\n/-\nCopyright (c) 2025 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne, Lorenzo Luccioli\n-/\n/-!\n# Absolute continuity of the composition of measures and kernels\n\nThis file contains some results about the absolute continuity of the composition of measures and\nkernels which use an assumption `CountableOrCountablyGenerated α β` on the measurable spaces.\n\nResults that hold without that assumption are in files about the definitions of compositions and\nproducts, like `Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean` and\n`Mathlib/Probability/Kernel/Composition/MeasureComp.lean`.\n\nThe assumption ensures the measurability of the sets where two kernels are absolutely continuous\nor mutually singular.\n\n## Main statements\n\n* `absolutelyContinuous_compProd_iff'`: `μ ⊗ₘ κ ≪ ν ⊗ₘ η ↔ μ ≪ ν ∧ ∀ᵐ a ∂μ, κ a ≪ η a`.\n\n-/\n\npublic section\n\nopen ProbabilityTheory Filter\n\nopen scoped ENNReal\n\nnamespace MeasureTheory.Measure\n\nvariable {α β : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}\n  {μ ν : Measure α} {κ η : Kernel α β} [IsFiniteKernel κ] [IsFiniteKernel η]\n  [MeasurableSpace.CountableOrCountablyGenerated α β]\n\nTarget:\nlemma MutuallySingular.compProd_of_right (μ ν : Measure α) (hκη : ∀ᵐ a ∂μ, κ a ⟂ₘ η a) :\n    μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Kernel","family_id":"mutuallysingular","file_id":"mathlib/Mathlib/Probability/Kernel/Composition/AbsolutelyContinuous.lean","sample_id":"56824650124ef37b3559baee31b1277720d3a9f018ce7db07f8712e4b6c90072"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1917f8da7209b00ed86e3ac8e7571325a8372a2d3f50420181edba9e230a5abb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c126f860a1889f0eed701887869d359f44f2980da9414820754ce91fa0c2db26","source_sha256":"6d14a53c89ceedf8fdc211fa474370698532792d974cd7aecfc721711170cc1d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_all [SProd.sprod, product, mem_flatMap, mem_map, Prod.ext_iff, and_left_comm]","hard_negative":false,"metrics":{"chosen_tokens":19,"rejected_tokens":2,"token_jaccard":0.066667,"token_length_ratio":0.105263},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"c03c7b3cc128fe980190764f2d5ba4b5183c6575284ad99c6faa8c4f22a08d19","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.List.Defs\npublic import Mathlib.Tactic.Attr.Core\npublic import Mathlib.Tactic.Common\n\nNamespace:\nList\n\nLocal context:\n/-\nCopyright (c) 2015 Leonardo de Moura. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura, Mario Carneiro\n-/\n/-!\n# Lists in product and sigma types\n\nThis file proves basic properties of `List.product` and `List.sigma`, which are list constructions\nliving in `Prod` and `Sigma` types respectively. Their definitions can be found in\n[`Data.List.Defs`](./defs). Beware, this is not about `List.prod`, the multiplicative product.\n-/\n\npublic section\n\n\nvariable {α β : Type*}\n\nnamespace List\n\n/-! ### product -/\n\n\n@[simp]\ntheorem nil_product (l : List β) : (@nil α) ×ˢ l = [] :=\n  rfl\n\n@[simp]\ntheorem product_cons (a : α) (l₁ : List α) (l₂ : List β) :\n    (a :: l₁) ×ˢ l₂ = map (fun b => (a, b)) l₂ ++ (l₁ ×ˢ l₂) :=\n  rfl\n\n@[simp]\ntheorem product_nil : ∀ l : List α, l ×ˢ (@nil β) = []\n  | [] => rfl\n  | _ :: l => by simp [product_cons, product_nil l]\n\n@[simp]\n\nTarget:\ntheorem mem_product {l₁ : List α} {l₂ : List β} {a : α} {b : β} :\n    (a, b) ∈ l₁ ×ˢ l₂ ↔ a ∈ l₁ ∧ b ∈ l₂ :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/List","family_id":"mem_product","file_id":"mathlib/Mathlib/Data/List/ProdSigma.lean","sample_id":"c126f860a1889f0eed701887869d359f44f2980da9414820754ce91fa0c2db26"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4de448a7d71906907793d1dc7af096bf74aa03a276d6cf774b8ce324536dc08e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ad39024358756a3c192f53ac5f381a4640cbba052fc6a25e4685ee5e6418e078","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e762d0b76da2ca7a3ba42e4fe5df7c426f0bb2153a670147e34342a186a3e955","source_sha256":"4db60ea328ce3789c20719560845d3c8bdc3720bf72ad08c23d3cd9a57d627b0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine tendsto_atTop_of_eventually_const (i₀ := Nat.ceil (‖f x‖₊ : ℝ) + 1) fun n hn => ?_\n  rw [Set.indicator_of_notMem]\n  simp only [not_le, Set.mem_setOf_eq]\n  refine lt_of_le_of_lt (Nat.le_ceil _) ?_\n  refine lt_of_lt_of_le (lt_add_one _) ?_\n  norm_cast\n\nvariable {p : ℝ≥0∞}","hard_negative":true,"metrics":{"chosen_tokens":71,"rejected_tokens":2,"token_jaccard":0.020833,"token_length_ratio":0.028169},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"c074818763fcc463b6a6dd997c6e3044fec551c379f014baca056b3128895e14","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Function.ConvergenceInMeasure\npublic import Mathlib.MeasureTheory.Function.L1Space.Integrable\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2022 Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kexing Ying\n-/\n/-!\n# Uniform integrability\n\nThis file contains the definitions for uniform integrability (both in the measure theory sense\nas well as the probability theory sense). This file also contains the Vitali convergence theorem\nwhich establishes a relation between uniform integrability, convergence in measure and\nLp convergence.\n\nUniform integrability plays a vital role in the theory of martingales and most notably is used to\nformulate the martingale convergence theorem.\n\n## Main definitions\n\n* `MeasureTheory.UnifIntegrable`: uniform integrability in the measure theory sense.\n  In particular, a sequence of functions `f` is uniformly integrable if for all `ε > 0`, there\n  exists some `δ > 0` such that for all sets `s` of smaller measure than `δ`, the Lp-norm of\n  `f i` restricted to `s` is smaller than `ε` for all `i`.\n* `MeasureTheory.UniformIntegrable`: uniform integrability in the probability theory sense.\n  In particular, a sequence of measurable functions `f` is uniformly integrable in the\n  probability theory sense if it is uniformly integrable in the measure theory sense and\n  has uniformly bounded Lp-norm.\n\n## Main results\n\n* `MeasureTheory.unifIntegrable_finite`: a finite sequence of Lp functions is uniformly\n  integrable.\n* `MeasureTheory.tendsto_Lp_finite_of_tendsto_ae`: a sequence of Lp functions which is uniformly\n  integrable converges in Lp if they converge almost everywhere.\n* `MeasureTheory.tendstoInMeasure_iff_tendsto_Lp_finite`: Vitali convergence theorem:\n  a sequence of Lp functions converges in Lp if and only if it is uniformly integrable\n  and converges in measure.\n\n## Tags\nuniformly integrable, uniformly absolutely continuous integral, Vitali convergence theorem\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen scoped MeasureTheory NNReal ENNReal Topology\n\nnamespace MeasureTheory\n\nopen Set Filter TopologicalSpace\n\nvariable {α β ι : Type*} {m : MeasurableSpace α} {μ : Measure α} [NormedAddCommGroup β]\n\n/-- Uniform integrability in the measure theory sense.\n\nA sequence of functions `f` is said to be uniformly integrable if for all `ε > 0`, there exists\nsome `δ > 0` such that for all sets `s` with measure less than `δ`, the Lp-norm of `f i`\nrestricted to `s` is less than `ε`.\n\nUniform integrability is also known as uniformly absolutely continuous integrals. -/\ndef UnifIntegrable {_ : MeasurableSpace α} (f : ι → α → β) (p : ℝ≥0∞) (μ : Measure α) : Prop :=\n  ∀ ⦃ε : ℝ⦄ (_ : 0 < ε), ∃ (δ : ℝ) (_ : 0 < δ), ∀ i s,\n    MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε\n\n/-- In probability theory, a family of measurable functions is uniformly integrable if it is\nuniformly integrable in the measure theory sense and is uniformly bounded. -/\ndef UniformIntegrable {_ : MeasurableSpace α} (f : ι → α → β) (p : ℝ≥0∞) (μ : Measure α) : Prop :=\n  (∀ i, AEStronglyMeasurable (f i) μ) ∧ UnifIntegrable f p μ ∧ ∃ C : ℝ≥0, ∀ i, eLpNorm (f i) p μ ≤ C\n\nnamespace UniformIntegrable\n\nprotected theorem aestronglyMeasurable {f : ι → α → β} {p : ℝ≥0∞} (hf : UniformIntegrable f p μ)\n    (i : ι) : AEStronglyMeasurable (f i) μ :=\n  hf.1 i\n\nprotected theorem unifIntegrable {f : ι → α → β} {p : ℝ≥0∞} (hf : UniformIntegrable f p μ) :\n    UnifIntegrable f p μ :=\n  hf.2.1\n\nprotected theorem memLp {f : ι → α → β} {p : ℝ≥0∞} (hf : UniformIntegrable f p μ) (i : ι) :\n    MemLp (f i) p μ :=\n  ⟨hf.1 i,\n    let ⟨_, _, hC⟩ := hf.2\n    lt_of_le_of_lt (hC i) ENNReal.coe_lt_top⟩\n\nend UniformIntegrable\n\nsection UnifIntegrable\n\n/-! ### `UnifIntegrable`\n\nThis section deals with uniform integrability in the measure theory sense. -/\n\n\nnamespace UnifIntegrable\n\nvariable {f g : ι → α → β} {p : ℝ≥0∞}\n\nprotected theorem add (hf : UnifIntegrable f p μ) (hg : UnifIntegrable g p μ) (hp : 1 ≤ p)\n    (hf_meas : ∀ i, AEStronglyMeasurable (f i) μ) (hg_meas : ∀ i, AEStronglyMeasurable (g i) μ) :\n    UnifIntegrable (f + g) p μ := by\n  intro ε hε\n  have hε2 : 0 < ε / 2 := half_pos hε\n  obtain ⟨δ₁, hδ₁_pos, hfδ₁⟩ := hf hε2\n  obtain ⟨δ₂, hδ₂_pos, hgδ₂⟩ := hg hε2\n  refine ⟨min δ₁ δ₂, lt_min hδ₁_pos hδ₂_pos, fun i s hs hμs => ?_⟩\n  simp_rw [Pi.add_apply, Set.indicator_add']\n  refine (eLpNorm_add_le ((hf_meas i).indicator hs) ((hg_meas i).indicator hs) hp).trans ?_\n  have hε_halves : ENNReal.ofReal ε = ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2) := by\n    rw [← ENNReal.ofReal_add hε2.le hε2.le, add_halves]\n  rw [hε_halves]\n  exact add_le_add (hfδ₁ i s hs (hμs.trans (ENNReal.ofReal_le_ofReal (min_le_left _ _))))\n    (hgδ₂ i s hs (hμs.trans (ENNReal.ofReal_le_ofReal (min_le_right _ _))))\n\nprotected theorem neg (hf : UnifIntegrable f p μ) : UnifIntegrable (-f) p μ := by\n  simp_rw [UnifIntegrable, Pi.neg_apply, Set.indicator_neg', eLpNorm_neg]\n  exact hf\n\nprotected theorem sub (hf : UnifIntegrable f p μ) (hg : UnifIntegrable g p μ) (hp : 1 ≤ p)\n    (hf_meas : ∀ i, AEStronglyMeasurable (f i) μ) (hg_meas : ∀ i, AEStronglyMeasurable (g i) μ) :\n    UnifIntegrable (f - g) p μ := by\n  rw [sub_eq_add_neg]\n  exact hf.add hg.neg hp hf_meas fun i => (hg_meas i).neg\n\nprotected theorem ae_eq (hf : UnifIntegrable f p μ) (hfg : ∀ n, f n =ᵐ[μ] g n) :\n    UnifIntegrable g p μ := by\n  classical\n  intro ε hε\n  obtain ⟨δ, hδ_pos, hfδ⟩ := hf hε\n  refine ⟨δ, hδ_pos, fun n s hs hμs => (le_of_eq <| eLpNorm_congr_ae ?_).trans (hfδ n s hs hμs)⟩\n  filter_upwards [hfg n] with x hx\n  simp_rw [Set.indicator_apply, hx]\n\n/-- Uniform integrability is preserved by restriction of the functions to a set. -/\nprotected theorem indicator (hf : UnifIntegrable f p μ) (E : Set α) :\n    UnifIntegrable (fun i => E.indicator (f i)) p μ := fun ε hε ↦ by\n  obtain ⟨δ, hδ_pos, hε⟩ := hf hε\n  refine ⟨δ, hδ_pos, fun i s hs hμs ↦ ?_⟩\n  calc\n    eLpNorm (s.indicator (E.indicator (f i))) p μ\n      = eLpNorm (E.indicator (s.indicator (f i))) p μ := by\n      simp only [indicator_indicator, inter_comm]\n    _ ≤ eLpNorm (s.indicator (f i)) p μ := eLpNorm_indicator_le _\n    _ ≤ ENNReal.ofReal ε := hε _ _ hs hμs\n\n/-- Uniform integrability is preserved by restriction of the measure to a set. -/\nprotected theorem restrict (hf : UnifIntegrable f p μ) (E : Set α) :\n    UnifIntegrable f p (μ.restrict E) := fun ε hε ↦ by\n  obtain ⟨δ, hδ_pos, hδε⟩ := hf hε\n  refine ⟨δ, hδ_pos, fun i s hs hμs ↦ ?_⟩\n  rw [μ.restrict_apply hs, ← measure_toMeasurable] at hμs\n  calc\n    eLpNorm (indicator s (f i)) p (μ.restrict E) = eLpNorm (f i) p (μ.restrict (s ∩ E)) := by\n      rw [eLpNorm_indicator_eq_eLpNorm_restrict hs, μ.restrict_restrict hs]\n    _ ≤ eLpNorm (f i) p (μ.restrict (toMeasurable μ (s ∩ E))) :=\n      eLpNorm_mono_measure _ <| Measure.restrict_mono (subset_toMeasurable _ _) le_rfl\n    _ = eLpNorm (indicator (toMeasurable μ (s ∩ E)) (f i)) p μ :=\n      (eLpNorm_indicator_eq_eLpNorm_restrict (measurableSet_toMeasurable _ _)).symm\n    _ ≤ ENNReal.ofReal ε := hδε i _ (measurableSet_toMeasurable _ _) hμs\n\nend UnifIntegrable\n\ntheorem unifIntegrable_zero_meas [MeasurableSpace α] {p : ℝ≥0∞} {f : ι → α → β} :\n    UnifIntegrable f p (0 : Measure α) :=\n  fun ε _ => ⟨1, one_pos, fun i s _ _ => by simp⟩\n\ntheorem unifIntegrable_congr_ae {p : ℝ≥0∞} {f g : ι → α → β} (hfg : ∀ n, f n =ᵐ[μ] g n) :\n    UnifIntegrable f p μ ↔ UnifIntegrable g p μ :=\n  ⟨fun hf => hf.ae_eq hfg, fun hg => hg.ae_eq fun n => (hfg n).symm⟩\n\nTarget:\ntheorem tendsto_indicator_ge (f : α → β) (x : α) :\n    Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_e762d0b76da2","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"1b6a170d9c37790b9de218cacd60bf88b8968bde11daeaf89f7fb76c8b183eaf","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Function","family_id":"tendsto_indicator_ge","file_id":"mathlib/Mathlib/MeasureTheory/Function/UniformIntegrable.lean","sample_id":"e762d0b76da2ca7a3ba42e4fe5df7c426f0bb2153a670147e34342a186a3e955"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3dbc0fabaa57897ff1e49ce5f7faadddfa1be34ceeabe2d7c9fe2b87f820ba31","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"30a1d731d06c0889b5d0dec83922f821938cb24eb2b822a5d40965e02cea292c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3087fe77077a4804fd4adc39f3b2f67cf5b7a70da11ffe0c839d1c6a058f86e4","source_sha256":"326a725d48085820c8ea061cea860b42ee76be144d5f7dbc1de5f2d04ba00c0a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine .mk _ Etale.respectsIso fun R S T _ _ _ _ _ h h' ↦ ?_\n  rw [etale_algebraMap] at h' ⊢\n  rw [faithfullyFlat_algebraMap_iff] at h\n  exact .of_etale_tensorProduct_of_faithfullyFlat S","hard_negative":true,"metrics":{"chosen_tokens":39,"rejected_tokens":5,"token_jaccard":0.115385,"token_length_ratio":0.128205},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"c0902f6804ee78ea9803d579c38617cb2162f0303a43f92782a6dd9d88e69ee6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.RingHom.Etale\npublic import Mathlib.RingTheory.Finiteness.Descent\npublic import Mathlib.RingTheory.Extension.Cotangent.BaseChange\n\nNamespace:\nRingHom\n\nLocal context:\n/-\nCopyright (c) 2026 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten\n-/\n/-!\n# Etale descends along faithfully flat ring maps\n\nIn this file we show that smooth, unramified and étale algebras descend along faithfully flat\nbase change.\n\n## Main results\n\n- `Algebra.Smooth.of_smooth_tensorProduct_of_faithfullyFlat`: Smooth descends.\n- `Algebra.Unramified.of_smooth_tensorProduct_of_faithfullyFlat`: Unramified descends.\n- `Algebra.Etale.of_etale_tensorProduct_of_faithfullyFlat`: Etale descends.\n\nWe also provide the corresponding `RingHom.CodescendsAlong` lemmas.\n\n## TODOs\n\n- The lemma `Algebra.FormallySmooth.of_formallySmooth_tensorProduct_of_faithfullyFlat` has an\n  additional `Algebra.FinitePresentation` assumption, because the proof uses that a flat module\n  of finite presentation is projective and the former descends. This also holds without\n  the finite presentation assumption, but requires showing that projectivity descends\n  along faithfully flat base change, which is due to Raynaud and Gruson\n  (see https://stacks.math.columbia.edu/tag/058B).\n-/\n\npublic section\n\nopen TensorProduct\n\nnamespace Algebra\n\nvariable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]\nvariable (T : Type*) [CommRing T] [Algebra R T] [Module.FaithfullyFlat R T]\n\nlemma FormallyUnramified.of_formallyUnramified_tensorProduct_of_faithfullyFlat\n    [FormallyUnramified T (T ⊗[R] S)] :\n    FormallyUnramified R S := by\n  constructor\n  let _ : Algebra S (T ⊗[R] S) := TensorProduct.rightAlgebra\n  have : Subsingleton (T ⊗[R] Ω[S⁄R]) :=\n    (KaehlerDifferential.tensorKaehlerEquivBase R T S (T ⊗[R] S)).subsingleton\n  exact Module.FaithfullyFlat.lTensor_reflects_triviality R T _\n\n/-- Formally smooth algebras descend along faithfully flat base change. See the TODO\nin the module docstring. -/\nproof_wanted FormallySmooth.of_formallySmooth_tensorProduct_of_faithfullyFlat\n    {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]\n    (T : Type*) [CommRing T] [Algebra R T] [Module.FaithfullyFlat R T]\n    [FormallySmooth T (T ⊗[R] S)] :\n    FormallySmooth R S\n\nlemma Smooth.of_smooth_tensorProduct_of_faithfullyFlat [Smooth T (T ⊗[R] S)] :\n    Smooth R S := by\n  have : Algebra.FinitePresentation R S := .of_finitePresentation_tensorProduct_of_faithfullyFlat T\n  refine ⟨?_, .of_finitePresentation_tensorProduct_of_faithfullyFlat T⟩\n  rw [formallySmooth_iff]\n  constructor\n  · let _ : Algebra T (S ⊗[R] T) := TensorProduct.rightAlgebra\n    let e : S ⊗[R] T ≃ₐ[T] T ⊗[R] S :=\n      .ofRingEquiv (f := TensorProduct.comm R S T) <| by simp [RingHom.algebraMap_toAlgebra]\n    have : FormallySmooth T (S ⊗[R] T) := .of_equiv e.symm\n    let e' : (S ⊗[R] T) ⊗[S] Ω[S⁄R] ≃ₗ[S ⊗[R] T] Ω[S ⊗[R] T⁄T] :=\n      KaehlerDifferential.tensorKaehlerEquiv R T S (S ⊗[R] T)\n    have : Module.Flat (S ⊗[R] T) ((S ⊗[R] T) ⊗[S] Ω[S⁄R]) := .of_linearEquiv e'\n    have : Module.Flat S Ω[S⁄R] := Module.Flat.of_flat_tensorProduct _ _ (S ⊗[R] T)\n    exact Module.Flat.projective_of_finitePresentation\n  · have : Subsingleton (T ⊗[R] H1Cotangent R S) := (tensorH1CotangentOfFlat R S T).subsingleton\n    exact Module.FaithfullyFlat.lTensor_reflects_triviality R T (H1Cotangent R S)\n\nlemma Unramified.of_unramified_tensorProduct_of_faithfullyFlat [Unramified T (T ⊗[R] S)] :\n    Unramified R S :=\n  ⟨.of_formallyUnramified_tensorProduct_of_faithfullyFlat T,\n    .of_finiteType_tensorProduct_of_faithfullyFlat T⟩\n\nlemma Etale.of_etale_tensorProduct_of_faithfullyFlat [Etale T (T ⊗[R] S)] :\n    Etale R S := by\n  rw [Etale.iff_formallyUnramified_and_smooth]\n  exact ⟨.of_formallyUnramified_tensorProduct_of_faithfullyFlat T,\n    .of_smooth_tensorProduct_of_faithfullyFlat T⟩\n\nend Algebra\n\nnamespace RingHom\n\nlemma Smooth.codescendsAlong_faithfullyFlat : CodescendsAlong Smooth FaithfullyFlat := by\n  refine .mk _ Smooth.respectsIso fun R S T _ _ _ _ _ h h' ↦ ?_\n  rw [smooth_algebraMap] at h' ⊢\n  rw [faithfullyFlat_algebraMap_iff] at h\n  exact .of_smooth_tensorProduct_of_faithfullyFlat S\n\nlemma FormallyUnramified.codescendsAlong_faithfullyFlat :\n    CodescendsAlong FormallyUnramified FaithfullyFlat := by\n  refine .mk _ FormallyUnramified.respectsIso fun R S T _ _ _ _ _ h h' ↦ ?_\n  rw [formallyUnramified_algebraMap] at h' ⊢\n  rw [faithfullyFlat_algebraMap_iff] at h\n  exact .of_formallyUnramified_tensorProduct_of_faithfullyFlat S\n\nTarget:\nlemma Etale.codescendsAlong_faithfullyFlat : CodescendsAlong Etale FaithfullyFlat :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_3087fe77077a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"569be107d7730fe97c5e0e6ca16e9b4cdf8170260586047481181b658fe1b9a8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Etale","family_id":"etale","file_id":"mathlib/Mathlib/RingTheory/Etale/Descent.lean","sample_id":"3087fe77077a4804fd4adc39f3b2f67cf5b7a70da11ffe0c839d1c6a058f86e4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d74b30d157e4d96b787ba5699f29a97d3d77e2405ded9e8ea9925e08645a5660","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"597f39dd347d0f59b6dc44976880e8d5522789c1118ad5494acb64511ad6ee2f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"aad755a0feeacee18eda311e51c7bdd7c4de3325eadaa0977eddb936e76e9ea5","source_sha256":"006922c150073d26eccce5bef3353d7015b01e87320ecec635275017a27177d4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [coeff_eq_ncoeff, ncoeff_eq_zero_of_lt_order A (-n - 1) x]\n  lia\n\n/-- Given an endomorphism-valued function on integers satisfying a pointwise bounded-pole condition,\nwe produce a vertex operator. -/\nnoncomputable def of_coeff (f : ℤ → Module.End R V)\n    (hf : ∀ x, BddBelow (Function.support fun y ↦ f y x)) : VertexOperator R V :=\n  HVertexOperator.of_coeff f fun x ↦ (BddBelow.isWF (hf x)).isPWO","hard_negative":true,"metrics":{"chosen_tokens":99,"rejected_tokens":3,"token_jaccard":0.016667,"token_length_ratio":0.030303},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"c0e9428ec1ddde99a716b52c49659a919237fed57fb053a67457e8cd0c023804","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Vertex.HVertexOperator\npublic import Mathlib.Data.Int.Interval\n\nNamespace:\nVertexOperator\n\nLocal context:\n/-\nCopyright (c) 2024 Scott Carnahan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Carnahan\n-/\n/-!\n# Vertex operators\nIn this file we introduce vertex operators as linear maps to Laurent series.\n\n## Definitions\n* `VertexOperator` is an `R`-linear map from an `R`-module `V` to `LaurentSeries V`.\n* `VertexOperator.ncoeff` is the coefficient of a vertex operator under normalized indexing.\n\n## TODO\n* `HasseDerivative` : A divided-power derivative.\n* `Locality` : A weak form of commutativity.\n* `Residue products` : A family of products on `VertexOperator R V` parametrized by integers.\n\n## References\n* [G. Mason, *Vertex rings and Pierce bundles*][mason2017]\n* [A. Matsuo, K. Nagatomo, *On axioms for a vertex algebra and locality of quantum\n  fields*][matsuo1997]\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nvariable {R V : Type*} [CommRing R] [AddCommGroup V] [Module R V]\n\n/-- A vertex operator over a commutative ring `R` is an `R`-linear map from an `R`-module `V` to\nLaurent series with coefficients in `V`.  We write this as a specialization of the heterogeneous\ncase. -/\nabbrev VertexOperator (R : Type*) (V : Type*) [CommRing R] [AddCommGroup V]\n    [Module R V] := HVertexOperator ℤ R V V\n\nnamespace VertexOperator\n\nopen HVertexOperator\n\n@[ext]\ntheorem ext (A B : VertexOperator R V) (h : ∀ v : V, A v = B v) :\n    A = B := LinearMap.ext h\n\n/-- The coefficient of a vertex operator under normalized indexing. -/\ndef ncoeff : VertexOperator R V →ₗ[R] ℤ → Module.End R V where\n  toFun A n := HVertexOperator.coeff A (-n - 1)\n  map_add' _ _ := by ext; simp\n  map_smul' _ _ := by ext; simp\n\ntheorem ncoeff_apply (A : VertexOperator R V) (n : ℤ) : ncoeff A n = coeff A (-n - 1) :=\n  rfl\n\n/-- In the literature, the `n`th normalized coefficient of a vertex operator `A` is written as\neither `Aₙ` or `A(n)`. -/\nscoped[VertexOperator] notation A \"[[\" n \"]]\" => ncoeff A n\n\n@[simp]\ntheorem coeff_eq_ncoeff (A : VertexOperator R V)\n    (n : ℤ) : HVertexOperator.coeff A n = A[[-n - 1]] := by\n  rw [ncoeff_apply, neg_sub, Int.sub_neg, add_sub_cancel_left]\n\ntheorem ncoeff_eq_zero_of_lt_order (A : VertexOperator R V) (n : ℤ) (x : V)\n    (h : -n - 1 < HahnSeries.order ((HahnModule.of R).symm (A x))) : (A[[n]]) x = 0 := by\n  simp only [ncoeff, HVertexOperator.coeff, LinearMap.coe_mk, AddHom.coe_mk]\n  exact HahnSeries.coeff_eq_zero_of_lt_order h\n\nTarget:\ntheorem coeff_eq_zero_of_lt_order (A : VertexOperator R V) (n : ℤ) (x : V)\n    (h : n < HahnSeries.order ((HahnModule.of R).symm (A x))) : coeff A n x = 0 :=\n\nProof body:\n","rejected":"by\n  exact coeff_eq_zero_of_lt_order","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"7b18f3f54dd5bfb9411f5e773f1653d7dd05981d964f1ec06c8c8d595b85ff58","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Vertex","family_id":"coeff_eq_zero_of_lt_order","file_id":"mathlib/Mathlib/Algebra/Vertex/VertexOperator.lean","sample_id":"aad755a0feeacee18eda311e51c7bdd7c4de3325eadaa0977eddb936e76e9ea5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"909daa16cd181b78a203d073bc4331c1f30425ac8b8431620e4cd7f332d4a635","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a13d8df10b3157064561e6f5629998f83e9b759c9ef6010480db3810fec50f71","source_sha256":"3b79245b81821b44a91daf15c8bd7da3bce2a30e3ee9d29e2db6a7d0e0195196","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · intro H\n    induction H using Submodule.span_induction with\n    | mem a ha =>\n      obtain ⟨a, ha, rfl⟩ := ha\n      exact ⟨C a, fun i ↦ by rw [coeff_C]; aesop, aeval_C _ _⟩\n    | zero => exact ⟨0, by simp, aeval_zero _⟩\n    | add a b ha hb ha' hb' =>\n      obtain ⟨a, ha, ha'⟩ := ha'\n      obtain ⟨b, hb, hb'⟩ := hb'\n      exact ⟨a + b, fun i ↦ by simpa using add_mem (ha i) (hb i), by simp [ha', hb']⟩\n    | smul a b hb hb' =>\n      obtain ⟨b', hb, hb'⟩ := hb'\n      have ⟨p, hp⟩ := adjoin_eq_exists_aeval R x a\n      refine ⟨p * b', fun i ↦ ?_, by simp [hp, hb']⟩\n      rw [coeff_mul]\n      exact sum_mem fun i hi ↦ Ideal.mul_mem_left _ _ (hb _)\n  · rintro ⟨p, hp, hp'⟩\n    have : y = ∑ i ∈ p.support, p.coeff i • ⟨_, (X ^ i).aeval_mem_adjoin_singleton _ x⟩ := by\n      trans ∑ i ∈ p.support, ⟨_, (C (p.coeff i) * X ^ i).aeval_mem_adjoin_singleton _ x⟩\n      · ext1\n        simp only [AddSubmonoidClass.coe_finsetSum, ← map_sum, ← hp', ← as_sum_support_C_mul_X_pow]\n      · congr with i\n        simp [Algebra.smul_def]\n    simp_rw [this, Algebra.smul_def]\n    exact sum_mem fun i _ ↦ Ideal.mul_mem_right _ _ (Ideal.mem_map_of_mem _ (hp i))","hard_negative":false,"metrics":{"chosen_tokens":312,"rejected_tokens":2,"token_jaccard":0.010638,"token_length_ratio":0.00641},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"c15a49c61392328c75b7c9291a6394da75cffb394894910940bcbc2164652b44","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.RingDivision\npublic import Mathlib.RingTheory.Adjoin.Polynomial.Basic\npublic import Mathlib.RingTheory.Ideal.Maps\n\nNamespace:\nAlgebra\n\nLocal context:\n/-\nCopyright (c) 2019 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n/-!\n# Ideals in polynomial rings\n-/\n\npublic section\n\nnoncomputable section\n\nopen Polynomial\n\nopen Finset\n\nuniverse u v w\n\nnamespace Polynomial\n\nvariable {R : Type*} [CommRing R] {a : R}\n\ntheorem mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero {b : R[X]} {P : R[X][X]} :\n    P ∈ Ideal.span {C (X - C a), X - C b} ↔ (P.eval b).eval a = 0 := by\n  rw [Ideal.mem_span_pair]\n  constructor <;> intro h\n  · rcases h with ⟨_, _, rfl⟩\n    simp\n  · rcases dvd_iff_isRoot.mpr h with ⟨p, hp⟩\n    rcases @X_sub_C_dvd_sub_C_eval _ b _ P with ⟨q, hq⟩\n    exact ⟨C p, q, by rw [mul_comm, mul_comm q, eq_add_of_sub_eq' hq, hp, C_mul]⟩\n\ntheorem ker_evalRingHom (x : R) : RingHom.ker (evalRingHom x) = Ideal.span {X - C x} := by\n  ext y\n  simp [Ideal.mem_span_singleton, dvd_iff_isRoot, RingHom.mem_ker]\n\n@[simp]\ntheorem ker_modByMonicHom {q : R[X]} (hq : q.Monic) :\n    LinearMap.ker (Polynomial.modByMonicHom q) = (Ideal.span {q}).restrictScalars R :=\n  Submodule.ext fun _ => (mem_ker_modByMonic hq).trans Ideal.mem_span_singleton.symm\n\n@[simp]\nlemma ker_constantCoeff : RingHom.ker constantCoeff = .span {(X : R[X])} := by\n  refine le_antisymm (fun p hp ↦ ?_) (by simp [Ideal.span_le])\n  simp only [RingHom.mem_ker, constantCoeff_apply, ← Polynomial.X_dvd_iff] at hp\n  rwa [Ideal.mem_span_singleton]\n\nend Polynomial\n\nnamespace Algebra\n\nvariable {R S : Type*}\n\nTarget:\nlemma mem_ideal_map_adjoin [CommSemiring R] [Semiring S] [Algebra R S] (x : S) (I : Ideal R)\n    {y : R[x]} :\n    y ∈ I.map (algebraMap R (R[x])) ↔\n      ∃ p : R[X], (∀ i, p.coeff i ∈ I) ∧ Polynomial.aeval x p = y :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Polynomial","family_id":"mem_ideal_map_adjoin","file_id":"mathlib/Mathlib/RingTheory/Polynomial/Ideal.lean","sample_id":"a13d8df10b3157064561e6f5629998f83e9b759c9ef6010480db3810fec50f71"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ebb5a2a7a52df61da4119d7165c562d64bfda64efb43081a7067ed8d5938555e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"108b8bdc62a57e8b5f470c5fc576e999d28b7edd2245678df50c535869adb4d7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"40b2cc76633014d1e5dccc9d2cd9aa43dc95f8c9291dfcfb08d96f928e0a655e","source_sha256":"3b1ad71a44a0ba95fe172fff4adb35c1c3d0ab3dacc0cacca37d6024c51fd12b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [toComplex_def₂, toComplex_def₂]\n  exact congr_arg₂ _ rfl (Int.cast_neg _)","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":2,"token_jaccard":0.058824,"token_length_ratio":0.1},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"c1679ab85d22b5c2037b5a20a6175948b8b39f116f32c6d6f9dea6e8b729454f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Archimedean.Real.Basic\npublic import Mathlib.Data.Complex.Basic\npublic import Mathlib.Data.Nat.Prime.Basic\npublic import Mathlib.NumberTheory.Zsqrtd.Basic\n\nNamespace:\nGaussianInt\n\nLocal context:\n/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Gaussian integers\n\nThe Gaussian integers are complex integer, complex numbers whose real and imaginary parts are both\nintegers.\n\n## Main definitions\n\nThe Euclidean domain structure on `ℤ[i]` is defined in this file.\n\nThe homomorphism `GaussianInt.toComplex` into the complex numbers is also defined in this file.\n\n## See also\n\nSee `NumberTheory.Zsqrtd.QuadraticReciprocity` for:\n* `prime_iff_mod_four_eq_three_of_nat_prime`:\n  A prime natural number is prime in `ℤ[i]` if and only if it is `3` mod `4`\n\n## Notation\n\nThis file uses the local notation `ℤ[i]` for `GaussianInt`\n\n## Implementation notes\n\nGaussian integers are implemented using the more general definition `Zsqrtd`, the type of integers\nadjoined a square root of `d`, in this case `-1`. The definition is reducible, so that properties\nand definitions about `Zsqrtd` can easily be used.\n-/\n\n@[expose] public section\n\n\nopen Zsqrtd Complex\n\nopen scoped ComplexConjugate\n\n/-- The Gaussian integers, defined as `ℤ√(-1)`. -/\nabbrev GaussianInt : Type :=\n  Zsqrtd (-1)\n\nlocal notation \"ℤ[i]\" => GaussianInt\n\nnamespace GaussianInt\n\ninstance : Repr ℤ[i] :=\n  ⟨fun x _ => \"⟨\" ++ repr x.re ++ \", \" ++ repr x.im ++ \"⟩\"⟩\n\ninstance instCommRing : CommRing ℤ[i] :=\n  Zsqrtd.commRing\n\nsection\n\nattribute [-instance] Complex.instField -- Avoid making things noncomputable unnecessarily.\n\n/-- The embedding of the Gaussian integers into the complex numbers, as a ring homomorphism. -/\ndef toComplex : ℤ[i] →+* ℂ :=\n  Zsqrtd.lift ⟨I, by simp⟩\n\nend\n\ninstance : Coe ℤ[i] ℂ :=\n  ⟨toComplex⟩\n\ntheorem toComplex_def (x : ℤ[i]) : (x : ℂ) = x.re + x.im * I :=\n  rfl\n\ntheorem toComplex_def' (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ) = x + y * I := by simp [toComplex_def]\n\ntheorem toComplex_def₂ (x : ℤ[i]) : (x : ℂ) = ⟨x.re, x.im⟩ := by\n  apply Complex.ext <;> simp [toComplex_def]\n\n@[simp]\ntheorem intCast_re (x : ℤ[i]) : ((x.re : ℤ) : ℝ) = (x : ℂ).re := by simp [toComplex_def]\n\n@[simp]\ntheorem intCast_im (x : ℤ[i]) : ((x.im : ℤ) : ℝ) = (x : ℂ).im := by simp [toComplex_def]\n\n@[simp]\ntheorem re_toComplex (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).re = x := by simp [toComplex_def]\n\n@[simp]\ntheorem im_toComplex (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).im = y := by simp [toComplex_def]\n\ntheorem toComplex_add (x y : ℤ[i]) : ((x + y : ℤ[i]) : ℂ) = x + y :=\n  toComplex.map_add _ _\n\ntheorem toComplex_mul (x y : ℤ[i]) : ((x * y : ℤ[i]) : ℂ) = x * y :=\n  toComplex.map_mul _ _\n\ntheorem toComplex_one : ((1 : ℤ[i]) : ℂ) = 1 :=\n  toComplex.map_one\n\ntheorem toComplex_zero : ((0 : ℤ[i]) : ℂ) = 0 :=\n  toComplex.map_zero\n\ntheorem toComplex_neg (x : ℤ[i]) : ((-x : ℤ[i]) : ℂ) = -x :=\n  toComplex.map_neg _\n\ntheorem toComplex_sub (x y : ℤ[i]) : ((x - y : ℤ[i]) : ℂ) = x - y :=\n  toComplex.map_sub _ _\n\n@[simp]\n\nTarget:\ntheorem toComplex_star (x : ℤ[i]) : ((star x : ℤ[i]) : ℂ) = conj (x : ℂ) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_40b2cc766330","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"0649046269b815a6c7b72f826b9aa9322615538f6cc6b95b9f60978f422ca0c0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Zsqrtd","family_id":"tocomplex_star","file_id":"mathlib/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean","sample_id":"40b2cc76633014d1e5dccc9d2cd9aa43dc95f8c9291dfcfb08d96f928e0a655e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cb2db471013d02747d266722ac3b318ad74596861a15ff8911bcdfb53d010405","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5dcc120b10457a5e5ed4d89893c537fd3c31acd149d3efc4972e246ac9853a3a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7730bf9984af8e1d55b7e5d3174ec963362e837effdd837bd031f3763c30060e","source_sha256":"a09b362ac29d7043b68fab55b278603330bcba1220d1dd2dd405ff015cd23f6a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [GradedAlgebra.proj_apply, decompose_symm_of, Equiv.apply_symm_apply]","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.153846},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"c16a22330d974c47587ef3e38f585c505ad72e1987e9c353922f65a170f94f23","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.DirectSum.Algebra\npublic import Mathlib.Algebra.DirectSum.Decomposition\npublic import Mathlib.Algebra.DirectSum.Internal\npublic import Mathlib.Algebra.DirectSum.Ring\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser, Kevin Buzzard, Jujian Zhang, Fangming Li\n-/\n/-!\n# Internally-graded rings and algebras\n\nThis file defines the typeclass `GradedAlgebra 𝒜`, for working with an algebra `A` that is\ninternally graded by a collection of submodules `𝒜 : ι → Submodule R A`.\nSee the docstring of that typeclass for more information.\n\n## Main definitions\n\n* `GradedRing 𝒜`: the typeclass, which is a combination of `SetLike.GradedMonoid`, and\n  `DirectSum.Decomposition 𝒜`.\n* `GradedAlgebra 𝒜`: A convenience alias for `GradedRing` when `𝒜` is a family of submodules.\n* `DirectSum.decomposeRingEquiv 𝒜 : A ≃ₐ[R] ⨁ i, 𝒜 i`, a more bundled version of\n  `DirectSum.decompose 𝒜`.\n* `DirectSum.decomposeAlgEquiv 𝒜 : A ≃ₐ[R] ⨁ i, 𝒜 i`, a more bundled version of\n  `DirectSum.decompose 𝒜`.\n* `GradedAlgebra.proj 𝒜 i` is the linear map from `A` to its degree `i : ι` component, such that\n  `proj 𝒜 i x = decompose 𝒜 x i`.\n\n## Implementation notes\n\nFor now, we do not have internally-graded semirings and internally-graded rings; these can be\nrepresented with `𝒜 : ι → Submodule ℕ A` and `𝒜 : ι → Submodule ℤ A` respectively, since all\n`Semiring`s are ℕ-algebras via `Semiring.toNatAlgebra`, and all `Ring`s are `ℤ`-algebras via\n`Ring.toIntAlgebra`.\n\n## Tags\n\ngraded algebra, graded ring, graded semiring, decomposition\n-/\n\n@[expose] public section\n\n\nopen DirectSum\n\nvariable {ι R A σ : Type*}\n\nsection GradedRing\n\nvariable [DecidableEq ι] [AddMonoid ι] [CommSemiring R] [Semiring A] [Algebra R A]\nvariable [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ)\n\nopen DirectSum\n\n/-- An internally-graded `R`-algebra `A` is one that can be decomposed into a collection\nof `Submodule R A`s indexed by `ι` such that the canonical map `A → ⨁ i, 𝒜 i` is bijective and\nrespects multiplication, i.e. the product of an element of degree `i` and an element of degree `j`\nis an element of degree `i + j`.\n\nNote that the fact that `A` is internally-graded, `GradedAlgebra 𝒜`, implies an externally-graded\nalgebra structure `DirectSum.GAlgebra R (fun i ↦ ↥(𝒜 i))`, which in turn makes available an\n`Algebra R (⨁ i, 𝒜 i)` instance.\n-/\nclass GradedRing (𝒜 : ι → σ) extends SetLike.GradedMonoid 𝒜, DirectSum.Decomposition 𝒜\n\nvariable [GradedRing 𝒜]\n\nnamespace DirectSum\n\n/-- If `A` is graded by `ι` with degree `i` component `𝒜 i`, then it is isomorphic as\na ring to a direct sum of components. -/\ndef decomposeRingEquiv : A ≃+* ⨁ i, 𝒜 i :=\n  RingEquiv.symm\n    { (decomposeAddEquiv 𝒜).symm with\n      map_mul' := (coeRingHom 𝒜).map_mul }\n\n@[simp]\ntheorem decompose_one : decompose 𝒜 (1 : A) = 1 :=\n  map_one (decomposeRingEquiv 𝒜)\n\n@[simp]\ntheorem decompose_symm_one : (decompose 𝒜).symm 1 = (1 : A) :=\n  map_one (decomposeRingEquiv 𝒜).symm\n\n@[simp]\ntheorem decompose_mul (x y : A) : decompose 𝒜 (x * y) = decompose 𝒜 x * decompose 𝒜 y :=\n  map_mul (decomposeRingEquiv 𝒜) x y\n\n@[simp]\ntheorem decompose_symm_mul (x y : ⨁ i, 𝒜 i) :\n    (decompose 𝒜).symm (x * y) = (decompose 𝒜).symm x * (decompose 𝒜).symm y :=\n  map_mul (decomposeRingEquiv 𝒜).symm x y\n\nend DirectSum\n\n/-- The projection maps of a graded ring -/\ndef GradedRing.proj (i : ι) : A →+ A :=\n  (AddSubmonoidClass.subtype (𝒜 i)).comp <|\n    (DFinsupp.evalAddMonoidHom i).comp <|\n      RingHom.toAddMonoidHom <| RingEquiv.toRingHom <| DirectSum.decomposeRingEquiv 𝒜\n\n@[simp]\ntheorem GradedRing.proj_apply (i : ι) (r : A) :\n    GradedRing.proj 𝒜 i r = (decompose 𝒜 r : ⨁ i, 𝒜 i) i :=\n  rfl\n\ntheorem GradedRing.proj_recompose (a : ⨁ i, 𝒜 i) (i : ι) :\n    GradedRing.proj 𝒜 i ((decompose 𝒜).symm a) = (decompose 𝒜).symm (DirectSum.of _ i (a i)) := by\n  rw [GradedRing.proj_apply, decompose_symm_of, Equiv.apply_symm_apply]\n\ntheorem GradedRing.mem_support_iff [∀ (i) (x : 𝒜 i), Decidable (x ≠ 0)] (r : A) (i : ι) :\n    i ∈ (decompose 𝒜 r).support ↔ GradedRing.proj 𝒜 i r ≠ 0 :=\n  DFinsupp.mem_support_iff.trans ZeroMemClass.coe_eq_zero.not.symm\n\nend GradedRing\n\nsection AddCancelMonoid\n\nopen DirectSum\n\nvariable [DecidableEq ι] [Semiring A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : ι → σ)\nvariable {i j : ι}\n\nnamespace DirectSum\n\ntheorem coe_decompose_mul_add_of_left_mem [AddLeftCancelMonoid ι] [GradedRing 𝒜] {a b : A}\n    (a_mem : a ∈ 𝒜 i) : (decompose 𝒜 (a * b) (i + j) : A) = a * decompose 𝒜 b j := by\n  lift a to 𝒜 i using a_mem\n  rw [decompose_mul, decompose_coe, coe_of_mul_apply_add]\n\ntheorem coe_decompose_mul_add_of_right_mem [AddRightCancelMonoid ι] [GradedRing 𝒜] {a b : A}\n    (b_mem : b ∈ 𝒜 j) : (decompose 𝒜 (a * b) (i + j) : A) = decompose 𝒜 a i * b := by\n  lift b to 𝒜 j using b_mem\n  rw [decompose_mul, decompose_coe, coe_mul_of_apply_add]\n\ntheorem decompose_mul_add_left [AddLeftCancelMonoid ι] [GradedRing 𝒜] (a : 𝒜 i) {b : A} :\n    decompose 𝒜 (↑a * b) (i + j) =\n      @GradedMonoid.GMul.mul ι (fun i => 𝒜 i) _ _ _ _ a (decompose 𝒜 b j) :=\n  Subtype.ext <| coe_decompose_mul_add_of_left_mem 𝒜 a.2\n\ntheorem decompose_mul_add_right [AddRightCancelMonoid ι] [GradedRing 𝒜] {a : A} (b : 𝒜 j) :\n    decompose 𝒜 (a * ↑b) (i + j) =\n      @GradedMonoid.GMul.mul ι (fun i => 𝒜 i) _ _ _ _ (decompose 𝒜 a i) b :=\n  Subtype.ext <| coe_decompose_mul_add_of_right_mem 𝒜 b.2\n\ntheorem coe_decompose_mul_of_left_mem_zero [AddMonoid ι] [GradedRing 𝒜] {a b : A}\n    (a_mem : a ∈ 𝒜 0) : (decompose 𝒜 (a * b) j : A) = a * decompose 𝒜 b j := by\n  lift a to 𝒜 0 using a_mem\n  rw [decompose_mul, decompose_coe, coe_of_mul_apply_of_mem_zero]\n\ntheorem coe_decompose_mul_of_right_mem_zero [AddMonoid ι] [GradedRing 𝒜] {a b : A}\n    (b_mem : b ∈ 𝒜 0) : (decompose 𝒜 (a * b) i : A) = decompose 𝒜 a i * b := by\n  lift b to 𝒜 0 using b_mem\n  rw [decompose_mul, decompose_coe, coe_mul_of_apply_of_mem_zero]\n\nend DirectSum\n\nend AddCancelMonoid\n\nsection GradedAlgebra\n\nvariable [DecidableEq ι] [AddMonoid ι] [CommSemiring R] [Semiring A] [Algebra R A]\nvariable (𝒜 : ι → Submodule R A)\n\n/-- A special case of `GradedRing` with `σ = Submodule R A`. This is useful both because it\ncan avoid typeclass search, and because it provides a more concise name. -/\nabbrev GradedAlgebra :=\n  GradedRing 𝒜\n\n/-- A helper to construct a `GradedAlgebra` when the `SetLike.GradedMonoid` structure is already\navailable. This makes the `left_inv` condition easier to prove, and phrases the `right_inv`\ncondition in a way that allows custom `@[ext]` lemmas to apply.\n\nSee note [reducible non-instances]. -/\nabbrev GradedAlgebra.ofAlgHom [SetLike.GradedMonoid 𝒜] (decompose : A →ₐ[R] ⨁ i, 𝒜 i)\n    (right_inv : (DirectSum.coeAlgHom 𝒜).comp decompose = AlgHom.id R A)\n    (left_inv : ∀ i (x : 𝒜 i), decompose (x : A) = DirectSum.of (fun i => ↥(𝒜 i)) i x) :\n    GradedAlgebra 𝒜 where\n  decompose' := decompose\n  left_inv := AlgHom.congr_fun right_inv\n  right_inv := by\n    suffices decompose.comp (DirectSum.coeAlgHom 𝒜) = AlgHom.id _ _ from AlgHom.congr_fun this\n    ext i x : 2\n    exact (decompose.congr_arg <| DirectSum.coeAlgHom_of _ _ _).trans (left_inv i x)\n\ninstance (R₀ : Type*) [CommSemiring R₀] [Algebra R₀ R] [Algebra R₀ A] [IsScalarTower R₀ R A]\n    [i : GradedAlgebra 𝒜] : GradedAlgebra (𝒜 · |>.restrictScalars R₀) := { i with }\n\nvariable [GradedAlgebra 𝒜]\n\nnamespace DirectSum\n\n/-- If `A` is graded by `ι` with degree `i` component `𝒜 i`, then it is isomorphic as\nan algebra to a direct sum of components. -/\n-- We have to write the `@[simps]` lemmas by hand to see through the\n-- `AlgEquiv.symm (decomposeAddEquiv 𝒜).symm`.\ndef decomposeAlgEquiv : A ≃ₐ[R] ⨁ i, 𝒜 i :=\n  AlgEquiv.symm\n    { (decomposeAddEquiv 𝒜).symm with\n      map_mul' := map_mul (coeAlgHom 𝒜)\n      commutes' := (coeAlgHom 𝒜).commutes }\n\n@[simp]\nlemma decomposeAlgEquiv_apply (a : A) :\n    decomposeAlgEquiv 𝒜 a = decompose 𝒜 a := rfl\n\n@[simp]\nlemma decomposeAlgEquiv_symm_apply (a : ⨁ i, 𝒜 i) :\n    (decomposeAlgEquiv 𝒜).symm a = (decompose 𝒜).symm a := rfl\n\n@[simp]\nlemma decompose_algebraMap (r : R) :\n    decompose 𝒜 (algebraMap R A r) = algebraMap R (⨁ i, 𝒜 i) r :=\n  (decomposeAlgEquiv 𝒜).commutes r\n\n@[simp]\nlemma decompose_symm_algebraMap (r : R) :\n    (decompose 𝒜).symm (algebraMap R (⨁ i, 𝒜 i) r) = algebraMap R A r :=\n  (decomposeAlgEquiv 𝒜).symm.commutes r\n\nend DirectSum\n\nopen DirectSum\n\n/-- The projection maps of graded algebra -/\ndef GradedAlgebra.proj (𝒜 : ι → Submodule R A) [GradedAlgebra 𝒜] (i : ι) : A →ₗ[R] A :=\n  (𝒜 i).subtype.comp <| (DFinsupp.lapply i).comp <| (decomposeAlgEquiv 𝒜).toAlgHom.toLinearMap\n\n@[simp]\ntheorem GradedAlgebra.proj_apply (i : ι) (r : A) :\n    GradedAlgebra.proj 𝒜 i r = (decompose 𝒜 r : ⨁ i, 𝒜 i) i :=\n  rfl\n\nTarget:\ntheorem GradedAlgebra.proj_recompose (a : ⨁ i, 𝒜 i) (i : ι) :\n    GradedAlgebra.proj 𝒜 i ((decompose 𝒜).symm a) = (decompose 𝒜).symm (of _ i (a i)) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_7730bf9984af","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"a4389f0ef08c32b3f06bc1d4370eed33ac3ff832db59e2ecf1897fb9550abcd1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/GradedAlgebra","family_id":"gradedalgebra","file_id":"mathlib/Mathlib/RingTheory/GradedAlgebra/Basic.lean","sample_id":"7730bf9984af8e1d55b7e5d3174ec963362e837effdd837bd031f3763c30060e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"98bbccba21050a97745f14c454bcd7300ae6caac0c9cdef2d7b7857a2d6f0324","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"44e7cadea26a9e80398866fe2e0034ed41855a81055034197f4c41ef5e24dcbf","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"28dab043652a1525adfd5ddc52e6243324bae2d9c4d94e5b8f18a41fa2a7d11f","source_sha256":"ca6f378e1485ca426ed934f409d878b070ba40b0fbd38c240a94cbf961d2f64f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← totallyDisconnectedSpace_subtype_iff]\n  have : Countable s := hs\n  infer_instance","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":7,"token_jaccard":0.117647,"token_length_ratio":0.538462},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"c2535c5a88682902dce7ca90b630ba7885842731491d5583b8873a630a719c79","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.GDelta.MetrizableSpace\npublic import Mathlib.Topology.Separation.CompletelyRegular\npublic import Mathlib.Topology.Separation.Profinite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Bryan Wang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Geoffrey Irving, Bryan Wang, Oliver Nash\n-/\n/-!\n# Further separation lemmas\n-/\n\npublic section\n\nvariable {X : Type*}\n\nnamespace CompletelyRegularSpace\n\nvariable [TopologicalSpace X] [T35Space X]\n\ntheorem totallySeparatedSpace_of_cardinalMk_lt_continuum (h : Cardinal.mk X < Cardinal.continuum) :\n    TotallySeparatedSpace X :=\n  totallySeparatedSpace_of_t0_of_basis_clopen <|\n    CompletelyRegularSpace.isTopologicalBasis_clopens_of_cardinalMk_lt_continuum h\n\ninstance [Countable X] : TotallySeparatedSpace X :=\n  totallySeparatedSpace_of_cardinalMk_lt_continuum <|\n    (Cardinal.mk_le_aleph0_iff.mpr inferInstance).trans_lt Cardinal.aleph0_lt_continuum\n\nprotected lemma _root_.Set.Countable.totallySeparatedSpace {s : Set X} (h : s.Countable) :\n    TotallySeparatedSpace s :=\n  have : _root_.Countable s := h\n  inferInstanceAs (TotallySeparatedSpace s)\n\nend CompletelyRegularSpace\n\n/-- Countable subsets of metric spaces are totally disconnected. -/\n\nTarget:\ntheorem Set.Countable.isTotallyDisconnected [MetricSpace X] {s : Set X} (hs : s.Countable) :\n    IsTotallyDisconnected s :=\n\nProof body:\n","rejected":"by\n  exact Set.Countable.isTotallyDisconnected","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"10e73213d1dd8472f9126b7f7950ef68dd1835640f7557eca13341f677964830","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Separation","family_id":"set","file_id":"mathlib/Mathlib/Topology/Separation/Lemmas.lean","sample_id":"28dab043652a1525adfd5ddc52e6243324bae2d9c4d94e5b8f18a41fa2a7d11f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3f5dfa9a05050a7e8f6e2944c4416e1a85109aff6c3b31b2844a70ac199a91e0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"48fc0507f8a383ca9960ad8374aaee847a8e1c263971c58c39917ac5808cf145","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"44d2da225f20cd3b6e6276fb821a5a22fe9ec474c9711a39a7cbc160460c406f","source_sha256":"8d5705aabd3ec1513ff008c85f46f0fb487e906338de45986cd81d4559495758","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨C, D, rfl⟩ := hNM\n  exact hI.of_delete.of_contract","hard_negative":false,"metrics":{"chosen_tokens":17,"rejected_tokens":22,"token_jaccard":0.75,"token_length_ratio":1.294118},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"c293062afcc42ecc7b5bd7d2cb04e3f1a23c877abe8cb170f292472cbef648f4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Matroid.Minor.Contract\n\nNamespace:\nMatroid\n\nLocal context:\n/-\nCopyright (c) 2025 Peter Nelson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Peter Nelson\n-/\n/-!\n# Matroid Minors\n\nA matroid `N = M ／ C ＼ D` obtained from a matroid `M` by a contraction then a delete,\n(or equivalently, by any number of contractions/deletions in any order) is a *minor* of `M`.\nThis gives a partial order on `Matroid α` that is ubiquitous in matroid theory,\nand interacts nicely with duality and linear representations.\n\nAlthough we provide a `PartialOrder` instance on `Matroid α` corresponding to the minor order,\nwe do not use the `M ≤ N` / `N < M` notation directly,\ninstead writing `N ≤m M` and `N <m M` for more convenient dot notation.\n\n## Main Declarations\n\n* `Matroid.IsMinor N M`, written `N ≤m M`, means that `N = M ／ C ＼ D` for some\n  subset `C` and `D` of `M.E`.\n* `Matroid.IsStrictMinor N M`, written `N <m M`, means that `N = M ／ C ＼ D`\n  for some subsets `C` and `D` of `M.E` that are not both nonempty.\n* `Matroid.IsMinor.exists_eq_contract_delete_disjoint` : we can choose `C` and `D` disjoint.\n\n-/\n\n@[expose] public section\n\nnamespace Matroid\n\nopen Set\n\nsection Minor\n\nvariable {α : Type*} {M M' N : Matroid α} {e f : α} {I C D : Set α}\n\n/-! ### Minors -/\n\n/-- `N` is a minor of `M` if `N = M ／ C ＼ D` for some `C` and `D`.\nThe definition itself does not require `C` and `D` to be disjoint,\nor even to be subsets of the ground set. See `Matroid.IsMinor.exists_eq_contract_delete_disjoint`\nfor the fact that we can choose `C` and `D` with these properties. -/\ndef IsMinor (N M : Matroid α) : Prop := ∃ C D, N = M ／ C ＼ D\n\n/-- `≤m` denotes the minor relation on matroids. -/\ninfixl:50 \" ≤m \" => Matroid.IsMinor\n\n@[simp]\nlemma contract_delete_isMinor (M : Matroid α) (C D : Set α) : M ／ C ＼ D ≤m M :=\n  ⟨C, D, rfl⟩\n\nlemma IsMinor.exists_eq_contract_delete_disjoint (h : N ≤m M) :\n    ∃ (C D : Set α), C ⊆ M.E ∧ D ⊆ M.E ∧ Disjoint C D ∧ N = M ／ C ＼ D := by\n  obtain ⟨C, D, rfl⟩ := h\n  exact ⟨C ∩ M.E, (D ∩ M.E) \\ C, inter_subset_right, sdiff_subset.trans inter_subset_right,\n    disjoint_sdiff_right.mono_left inter_subset_left,\n    by simp [delete_eq_delete_iff, inter_assoc, inter_sdiff_assoc]⟩\n\n/-- `N` is a strict minor of `M` if `N` is a minor of `M` and `N ≠ M`.\nEquivalently, `N` is obtained from `M` by deleting/contracting subsets of the ground set\nthat are not both empty. -/\ndef IsStrictMinor (N M : Matroid α) : Prop := N ≤m M ∧ ¬ M ≤m N\n\n/-- `<m` denotes the strict minor relation on matroids. -/\ninfixl:50 \" <m \" => Matroid.IsStrictMinor\n\nlemma IsMinor.subset (h : N ≤m M) : N.E ⊆ M.E := by\n  obtain ⟨C, D, rfl⟩ := h\n  exact sdiff_subset.trans sdiff_subset\n\nlemma IsMinor.refl {M : Matroid α} : M ≤m M := ⟨∅, ∅, by simp⟩\n\nlemma IsMinor.trans {M₁ M₂ M₃ : Matroid α} (h : M₁ ≤m M₂) (h' : M₂ ≤m M₃) : M₁ ≤m M₃ := by\n  obtain ⟨C₁, D₁, rfl⟩ := h\n  obtain ⟨C₂, D₂, rfl⟩ := h'\n  exact ⟨C₂ ∪ C₁ \\ D₂, D₂ ∪ D₁, by rw [contract_delete_contract_delete']⟩\n\nlemma IsMinor.eq_of_ground_subset (h : N ≤m M) (hE : M.E ⊆ N.E) : M = N := by\n  obtain ⟨C, D, rfl⟩ := h\n  rw [delete_ground, contract_ground, subset_sdiff, subset_sdiff] at hE\n  rw [← contract_inter_ground_eq, hE.1.2.symm.inter_eq, contract_empty, ← delete_inter_ground_eq,\n    hE.2.symm.inter_eq, delete_empty]\n\nlemma IsMinor.antisymm (h : N ≤m M) (h' : M ≤m N) : N = M :=\n  h'.eq_of_ground_subset h.subset\n\n/-- The minor order is a `PartialOrder` on `Matroid α`.\nWe prefer the spelling `N ≤m M` over `N ≤ M` for the dot notation. -/\ninstance (α : Type*) : PartialOrder (Matroid α) where\n  le N M := N ≤m M\n  lt N M := N <m M\n  le_refl _ := IsMinor.refl\n  le_trans _ _ _ := IsMinor.trans\n  le_antisymm _ _ := IsMinor.antisymm\n\nlemma IsMinor.le (h : N ≤m M) : N ≤ M := h\n\nlemma IsStrictMinor.lt (h : N <m M) : N < M := h\n\n@[simp]\nlemma le_eq_isMinor : (fun M M' : Matroid α ↦ M ≤ M') = Matroid.IsMinor := rfl\n\n@[simp]\nlemma lt_eq_isStrictMinor : (fun M M' : Matroid α ↦ M < M') = Matroid.IsStrictMinor := rfl\n\nlemma isStrictMinor_iff_isMinor_ne : N <m M ↔ N ≤m M ∧ N ≠ M :=\n  lt_iff_le_and_ne (α := Matroid α)\n\nlemma IsStrictMinor.ne (h : N <m M) : N ≠ M :=\n  h.lt.ne\n\nlemma isStrictMinor_irrefl (M : Matroid α) : ¬ (M <m M) :=\n  lt_irrefl M\n\nlemma IsStrictMinor.isMinor (h : N <m M) : N ≤m M :=\n  h.lt.le\n\nlemma IsStrictMinor.not_isMinor (h : N <m M) : ¬ (M ≤m N) :=\n  h.lt.not_ge\n\nlemma IsStrictMinor.ssubset (h : N <m M) : N.E ⊂ M.E :=\n  h.isMinor.subset.ssubset_of_ne (fun hE ↦ h.ne (h.isMinor.eq_of_ground_subset hE.symm.subset).symm)\n\nlemma isStrictMinor_iff_isMinor_ssubset : N <m M ↔ N ≤m M ∧ N.E ⊂ M.E :=\n  ⟨fun h ↦ ⟨h.isMinor, h.ssubset⟩, fun ⟨h, hss⟩ ↦ ⟨h, fun h' ↦ hss.ne <| by rw [h'.antisymm h]⟩⟩\n\nlemma IsStrictMinor.trans_isMinor (h : N <m M) (h' : M ≤m M') : N <m M' :=\n  h.lt.trans_le h'\n\nlemma IsMinor.trans_isStrictMinor (h : N ≤m M) (h' : M <m M') : N <m M' :=\n  h.le.trans_lt h'\n\nlemma IsStrictMinor.trans (h : N <m M) (h' : M <m M') : N <m M' :=\n  h.lt.trans h'\n\nTarget:\nlemma Indep.of_isMinor (hI : N.Indep I) (hNM : N ≤m M) : M.Indep I :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  obtain ⟨C, D, rfl⟩ := hNM\n  exact hI.of_delete.of_contract","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Matroid","family_id":"indep","file_id":"mathlib/Mathlib/Combinatorics/Matroid/Minor/Order.lean","sample_id":"44d2da225f20cd3b6e6276fb821a5a22fe9ec474c9711a39a7cbc160460c406f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e849d95b5db02f9c868ba8b0e417961eeea08fc774f074fe05a938dadf630d6e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bf6d1403eb4460ab7069e5f743251c32cc0683a5c043812b0396ebf8e760f9d6","source_sha256":"af794759eaa35d9b988712c78093779e987eaf10b925193a6e559d007441b0b2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_contra!\n  exact hA (hP.subobject_eq_top this)","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.181818},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"c29a91f5e5643b2586363f3479a54d015712abed8aa03ed3cb577f2b5415a34f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.ExtremalEpi\npublic import Mathlib.CategoryTheory.Generator.Basic\npublic import Mathlib.CategoryTheory.Limits.Presentation\n\nNamespace:\nCategoryTheory.ObjectProperty.IsStrongGenerator\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Strong generators\n\nIf `P : ObjectProperty C`, we say that `P` is a strong generator if it is a\ngenerator (in the sense that `IsSeparating P` holds) such that for any\nproper subobject `A ⊂ X`, there exists a morphism `G ⟶ X` which does not factor\nthrough `A` from an object satisfying `P`.\n\nThe main result is the lemma `isStrongGenerator_iff_exists_extremalEpi` which\nsays that if `P` is `w`-small, `C` is locally `w`-small and\nhas coproducts of size `w`, then `P` is a strong generator iff any\nobject of `C` is the target of an extremal epimorphism from a coproduct of\nobjects satisfying `P`.\n\nWe also show that if any object in `C` is a colimit of objects in `S`,\nthen `S` is a strong generator.\n\n## References\n* [Adámek, J. and Rosický, J., *Locally presentable and accessible categories*][Adamek_Rosicky_1994]\n\n-/\n\n@[expose] public section\n\nuniverse w' w v u\n\nnamespace CategoryTheory\n\nopen Limits\n\n\nnamespace ObjectProperty\n\nvariable {C : Type u} [Category.{v} C] (P : ObjectProperty C)\n\n/-- A property `P : ObjectProperty C` is a strong generator\nif it is separating and for any proper subobject `A ⊂ X`, there exists\na morphism `G ⟶ X` which does not factor through `A` from an object\nsuch that `P G` holds. -/\ndef IsStrongGenerator : Prop :=\n  P.IsSeparating ∧ ∀ ⦃X : C⦄ (A : Subobject X),\n    (∀ (G : C) (_ : P G) (f : G ⟶ X), Subobject.Factors A f) → A = ⊤\n\nvariable {P}\n\nlemma isStrongGenerator_iff :\n    P.IsStrongGenerator ↔ P.IsSeparating ∧\n      ∀ ⦃X Y : C⦄ (i : X ⟶ Y) [Mono i],\n        (∀ (G : C) (_ : P G), Function.Surjective (fun (f : G ⟶ X) ↦ f ≫ i)) → IsIso i := by\n  refine ⟨fun ⟨hS₁, hS₂⟩ ↦ ⟨hS₁, fun X Y i _ h ↦ ?_⟩,\n    fun ⟨hS₁, hS₂⟩ ↦ ⟨hS₁, fun X A hA ↦ ?_⟩⟩\n  · rw [Subobject.isIso_iff_mk_eq_top]\n    refine hS₂ _ (fun G hG g ↦ ?_)\n    rw [Subobject.mk_factors_iff]\n    exact h G hG g\n  · rw [← Subobject.isIso_arrow_iff_eq_top]\n    exact hS₂ A.arrow (fun G hG g ↦ ⟨_, Subobject.factorThru_arrow _ _ (hA G hG g)⟩)\n\nnamespace IsStrongGenerator\n\nsection\n\nvariable (hP : P.IsStrongGenerator)\n\ninclude hP\n\nlemma isSeparating : P.IsSeparating := hP.1\n\nlemma subobject_eq_top {X : C} {A : Subobject X}\n    (hA : ∀ (G : C) (_ : P G) (f : G ⟶ X), Subobject.Factors A f) :\n    A = ⊤ :=\n  hP.2 _ hA\n\nlemma isIso_of_mono ⦃X Y : C⦄ (i : X ⟶ Y) [Mono i]\n    (hi : ∀ (G : C) (_ : P G), Function.Surjective (fun (f : G ⟶ X) ↦ f ≫ i)) : IsIso i :=\n  (isStrongGenerator_iff.1 hP).2 i hi\n\nTarget:\nlemma exists_of_subobject_ne_top {X : C} {A : Subobject X} (hA : A ≠ ⊤) :\n    ∃ (G : C) (_ : P G) (f : G ⟶ X), ¬ Subobject.Factors A f :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Generator","family_id":"exists_of_subobject_ne_top","file_id":"mathlib/Mathlib/CategoryTheory/Generator/StrongGenerator.lean","sample_id":"bf6d1403eb4460ab7069e5f743251c32cc0683a5c043812b0396ebf8e760f9d6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1c8bfaba84928466d869f23a6dc4de48708f2d6673fcc79573e01b4de03f6a29","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c4f806797736ab64809b45b40862e20cdbb12085780d491537012af759d12650","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"30e2dcca73b76516afafa57bf58fa1f528e0a923278fa6441a9972d75d07dd33","source_sha256":"5446f542892910a2397a84cd0f024393cbca50d14ecc1fa829e9751db035abd3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [normal_iff] at h ⊢\n  intro x; specialize h (f.symm x)\n  rw [← f.apply_symm_apply x, minpoly.algEquiv_eq, ← f.toAlgHom.comp_algebraMap, ← map_map]\n  exact ⟨h.1.map f, h.2.map _⟩","hard_negative":true,"metrics":{"chosen_tokens":57,"rejected_tokens":5,"token_jaccard":0.088235,"token_length_ratio":0.087719},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"c29aaf23e73dbf05cb8a8d4bc4e81042e68e1ade4ebefe830adc5e8de3197e15","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Splits\npublic import Mathlib.FieldTheory.Galois.Notation\npublic import Mathlib.FieldTheory.IntermediateField.Basic\npublic import Mathlib.FieldTheory.Minpoly.Field\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Thomas Browning, Patrick Lutz\n-/\n/-!\n# Normal field extensions\n\nIn this file we define normal field extensions.\n\n## Main Definitions\n\n- `Normal F K` where `K` is a field extension of `F`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Polynomial IsScalarTower\n\nvariable (F K : Type*) [Field F] [Field K] [Algebra F K]\n\n/-- Typeclass for normal field extensions: an algebraic extension of fields `K/F` is *normal*\nif the minimal polynomial of every element `x` in `K` splits in `K`, i.e. every `F`-conjugate\nof `x` is in `K`. -/\n@[stacks 09HM]\nclass Normal : Prop extends Algebra.IsAlgebraic F K where\n  splits' (x : K) : Splits ((minpoly F x).map (algebraMap F K))\n\nvariable {F K}\n\ntheorem Normal.isIntegral (_ : Normal F K) (x : K) : IsIntegral F x :=\n  Algebra.IsIntegral.isIntegral x\n\ntheorem Normal.splits (_ : Normal F K) (x : K) : Splits ((minpoly F x).map (algebraMap F K)) :=\n  Normal.splits' x\n\ntheorem normal_iff :\n    Normal F K ↔ ∀ x : K, IsIntegral F x ∧ Splits ((minpoly F x).map (algebraMap F K)) :=\n  ⟨fun h x => ⟨h.isIntegral x, h.splits x⟩, fun h =>\n    { isAlgebraic := fun x => (h x).1.isAlgebraic\n      splits' := fun x => (h x).2 }⟩\n\ntheorem Normal.out :\n    Normal F K → ∀ x : K, IsIntegral F x ∧ Splits ((minpoly F x).map (algebraMap F K)) :=\n  normal_iff.1\n\nvariable (F K)\n\ninstance normal_self : Normal F F where\n  isAlgebraic := fun _ => isIntegral_algebraMap.isAlgebraic\n  splits' := fun x => (minpoly.eq_X_sub_C' x).symm ▸ by simp\n\nsection NormalTower\n\nvariable (E : Type*) [Field E] [Algebra F E] [Algebra K E] [IsScalarTower F K E]\n\n@[stacks 09HN]\ntheorem Normal.tower_top_of_normal [h : Normal F E] : Normal K E :=\n  normal_iff.2 fun x => by\n    obtain ⟨hx, hhx⟩ := h.out x\n    rw [algebraMap_eq F K E, ← map_map] at hhx\n    exact ⟨hx.tower_top, hhx.of_dvd (map_ne_zero (map_ne_zero (minpoly.ne_zero hx)))\n      ((map_dvd_map' _).mpr (minpoly.dvd_map_of_isScalarTower F K x))⟩\n\ninstance IntermediateField.normal (K : IntermediateField F E) [Normal F E] : Normal K E :=\n  Normal.tower_top_of_normal F K E\n\ntheorem AlgHom.normal_bijective [h : Normal F E] (ϕ : E →ₐ[F] K) : Function.Bijective ϕ :=\n  h.toIsAlgebraic.bijective_of_isScalarTower' ϕ\n\nvariable {E F}\nvariable {E' : Type*} [Field E'] [Algebra F E']\n\nTarget:\ntheorem Normal.of_algEquiv [h : Normal F E] (f : E ≃ₐ[F] E') : Normal F E' :=\n\nProof body:\n","rejected":"by\n  exact Normal.of_algEquiv","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"cbdd080a81701a668239d4e5a7d1758d4c6b0bd6ffaa7b91509a9b2085371949","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Normal","family_id":"normal","file_id":"mathlib/Mathlib/FieldTheory/Normal/Defs.lean","sample_id":"30e2dcca73b76516afafa57bf58fa1f528e0a923278fa6441a9972d75d07dd33"},"split":"train"}
{"binding":{"chosen_proof_sha256":"56b4e98d034de1994cb22dfefff34d5e35c25783ae7eef0d3fe3f45cbf3ff033","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8b929f3d1885371d6b55ef0cdc5e96bf8ff28a4292aeef57779d2add21f395ef","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"27226b4cef484b3c92a34e5c76b5413440a1f9efdabd65d7d5bb3fbb80618898","source_sha256":"d5cffaaa7c7b308256b47b38d296e80bf7d53217db8c0a040e827e6ffc87b03a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using mul_vle_mul_left ((vle_total 0 1).resolve_right not_vle_one_zero) x","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":22,"token_jaccard":0.866667,"token_length_ratio":1.466667},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"c2a8a70eccaabbb4f95b3eff60005b1aeef1f60fff0eabebbb61a5b4498607fb","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Valuation.Basic\npublic import Mathlib.Data.NNReal.Defs\npublic import Mathlib.Topology.Defs.Filter\npublic import Mathlib.Order.Filter.Bases.Basic\n\nNamespace:\nValuativeRel\n\nLocal context:\n/-\nCopyright (c) 2025 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Liu, Adam Topaz\n-/\n/-!\n\n# Valuative Relations\n\nIn this file we introduce a class called `ValuativeRel R` for a ring `R`.\nThis bundles a relation `vle : R → R → Prop` on `R` which mimics a\npreorder on `R` arising from a valuation.\nWe introduce the notation `x ≤ᵥ y` for this relation.\n\nRecall that the equivalence class of a valuation is *completely* characterized by\nsuch a preorder. Thus, we can think of `ValuativeRel R` as a way of\nsaying that `R` is endowed with an equivalence class of valuations.\n\n## Main Definitions\n\n- `ValuativeRel R` endows a semiring `R` with a relation \"arising from a valuation\". When `R` is a\n  ring, this is equivalent to fixing an equivalence class of valuations on `R`.\n  Use the notation `x ≤ᵥ y` for this relation.\n- `ValuativeRel.valuation R` is the \"canonical\" valuation associated to `ValuativeRel R`,\n  taking values in `ValuativeRel.ValueGroupWithZero R`.\n- Given a valuation `v` on `R` and an instance `[ValuativeRel R]`, writing `[v.Compatible]`\n  ensures that the relation `x ≤ᵥ y` is equivalent to `v x ≤ v y`. Note that\n  it is possible to have `[v.Compatible]` and `[w.Compatible]` for two different valuations on `R`.\n- Given `[ValuativeRel A]`, `[ValuativeRel B]` and `[Algebra A B]`, the class\n  `[ValuativeExtension A B]` ensures that the algebra map `A → B` is compatible with the valuations\n  on `A` and `B`. For example, this can be used to talk about extensions of valued fields.\n\n\n## Remark\n\nThe last two axioms in `ValuativeRel`, namely `vle_mul_cancel` and `not_vle_one_zero`, are\nused to ensure that we have a well-behaved valuation taking values in a *value group* (with zero).\nIn principle, it should be possible to drop these two axioms and obtain a value monoid,\nhowever, such a value monoid would not necessarily embed into an ordered abelian group with zero.\nSimilarly, without these axioms, the support of the valuation need not be a prime ideal.\nWe have thus opted to include these two axioms and obtain a `ValueGroupWithZero` associated to\na `ValuativeRel` in order to best align with the literature about valuations on commutative rings.\n\nFuture work could refactor `ValuativeRel` by dropping the `vle_mul_cancel` and `not_vle_one_zero`\naxioms, opting to make these mixins instead.\n\n## Projects\n\nThe `ValuativeRel` class should eventually replace the existing `Valued` typeclass.\nOnce such a refactor happens, `ValuativeRel` could be renamed to `Valued`.\n\n## TODO\nSplit this file. For instance, the universal properties of `ValueGroupWithZero` and definition of\n`IsRankLeOne` could be separated out.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\n/-- The class `[ValuativeRel R]` class introduces an operator `x ≤ᵥ y : Prop` for `x y : R`\nwhich is the natural relation arising from (the equivalence class of) a valuation on `R` when `R`\nis a ring. More precisely, if `v` is a valuation on `R` then the associated relation is\n`x ≤ᵥ y ↔ v x ≤ v y`. Use this class to talk about the case where `R` is equipped\nwith an equivalence class of valuations. -/\n@[ext]\nclass ValuativeRel (R : Type*) [Semiring R] where\n  /-- The valuation less-equal operator arising from `ValuativeRel`. -/\n  vle : R → R → Prop\n  vle_total (x y) : vle x y ∨ vle y x\n  vle_trans {z y x} : vle x y → vle y z → vle x z\n  vle_add {x y z} : vle x z → vle y z → vle (x + y) z\n  mul_vle_mul_left {x y} (h : vle x y) (z) : vle (x * z) (y * z)\n  vle_mul_cancel {x y z} : ¬ vle z 0 → vle (x * z) (y * z) → vle x y\n  not_vle_one_zero : ¬ vle 1 0\n  vle_mul_comm {x y} : vle (x * y) (y * x)\n\n@[inherit_doc] infix:50 \" ≤ᵥ \" => ValuativeRel.vle\n\nmacro_rules | `($a ≤ᵥ $b) => `(binrel% ValuativeRel.vle $a $b)\n\nnamespace Valuation\n\nvariable {R Γ : Type*} [Ring R] [LinearOrderedCommMonoidWithZero Γ]\n  (v : Valuation R Γ)\n\n/-- We say that a valuation `v` is `Compatible` if the relation `x ≤ᵥ y`\nis equivalent to `v x ≤ v y`. -/\nclass Compatible [ValuativeRel R] where\n  vle_iff_le (x y : R) : x ≤ᵥ y ↔ v x ≤ v y\n\nend Valuation\n\n/-- A preorder on a ring is said to be \"valuative\" if it agrees with the\nvaluative relation. -/\nclass ValuativePreorder (R : Type*) [Semiring R] [ValuativeRel R] [Preorder R] where\n  vle_iff_le (x y : R) : x ≤ᵥ y ↔ x ≤ y\n\nnamespace ValuativeRel\n\n@[deprecated (since := \"2025-12-20\")] alias Rel := vle\n@[deprecated (since := \"2025-12-20\")] alias rel_total := vle_total\n@[deprecated (since := \"2025-12-20\")] alias rel_trans := vle_trans\n@[deprecated (since := \"2025-12-20\")] alias rel_add := vle_add\n@[deprecated (since := \"2025-12-20\")] alias rel_mul_right := mul_vle_mul_left\n@[deprecated (since := \"2025-12-20\")] alias rel_mul_cancel := vle_mul_cancel\n@[deprecated (since := \"2025-12-20\")] alias not_rel_one_zero := not_vle_one_zero\n\nvariable {R : Type*} [Semiring R] [ValuativeRel R] {x x' y y' z : R}\n\n/-- The valuation less-than relation, defined as `x <ᵥ y ↔ ¬ y ≤ᵥ x`. -/\ndef vlt (x y : R) : Prop := ¬ y ≤ᵥ x\n\n@[deprecated (since := \"2025-12-20\")] alias SRel := vlt\n\n@[inherit_doc] infix:50 \" <ᵥ \" => ValuativeRel.vlt\n\nmacro_rules | `($a <ᵥ $b) => `(binrel% ValuativeRel.vlt $a $b)\n\n/-- The valuation equals relation, defined as `x =ᵥ y ↔ x ≤ᵥ y ∧ y ≤ᵥ x`. -/\ndef veq : R → R → Prop := AntisymmRel (· ≤ᵥ ·)\n\n@[inherit_doc] infix:50 \" =ᵥ \" => ValuativeRel.veq\n\nlemma veq_mul_comm (x y : R) : x * y =ᵥ y * x := ⟨vle_mul_comm, vle_mul_comm⟩\n\nmacro_rules | `($a =ᵥ $b) => `(binrel% ValuativeRel.veq $a $b)\n\n@[simp, grind =] lemma not_vle : ¬ x ≤ᵥ y ↔ y <ᵥ x := .rfl\n@[simp, grind =] lemma not_vlt : ¬ x <ᵥ y ↔ y ≤ᵥ x := not_vle.not_left\nlemma veq_def : x =ᵥ y ↔ x ≤ᵥ y ∧ y ≤ᵥ x := .rfl\n\n@[deprecated not_vle (since := \"2025-12-20\")]\nlemma srel_iff : x <ᵥ y ↔ ¬ y ≤ᵥ x := Iff.rfl\n\n@[deprecated (since := \"2025-12-20\")] alias not_srel_iff := not_vlt\n\nprotected alias ⟨_, vle.not_vlt⟩ := not_vlt\nprotected alias ⟨_, vlt.not_vle⟩ := not_vle\n\nlemma veq_comm : x =ᵥ y ↔ y =ᵥ x := antisymmRel_comm\n@[symm] protected alias ⟨veq.symm, _⟩ := veq_comm\n\ninstance : @Std.Symm R (· =ᵥ ·) where\n  symm _ _ := veq.symm\n\nlemma vle_of_veq (h : x =ᵥ y) : x ≤ᵥ y := h.1\nlemma vge_of_veq (h : x =ᵥ y) : y ≤ᵥ x := h.2\n\nprotected alias veq.vle := vle_of_veq\nprotected alias veq.vge := vge_of_veq\n\nlemma not_vlt_of_veq (h : x =ᵥ y) : ¬ x <ᵥ y := h.vge.not_vlt\nlemma not_vgt_of_veq (h : x =ᵥ y) : ¬ y <ᵥ x := h.vle.not_vlt\n\nprotected alias veq.not_vlt := not_vlt_of_veq\nprotected alias veq.not_vgt := not_vgt_of_veq\n\n@[simp, refl] lemma vle_refl (x : R) : x ≤ᵥ x := or_self_iff.1 <| vle_total x x\nlemma vle_rfl : x ≤ᵥ x := vle_refl x\n\n@[deprecated (since := \"2025-12-20\")] alias rel_refl := vle_refl\n@[deprecated (since := \"2025-12-20\")] alias rel_rfl := vle_rfl\n\nprotected alias vle.refl := vle_refl\nprotected alias vle.rfl := vle_rfl\n\ninstance : @Std.Refl R (· ≤ᵥ ·) where\n  refl _ := vle_rfl\n\n@[deprecated (since := \"2025-12-20\")] protected alias Rel.refl := vle.refl\n@[deprecated (since := \"2025-12-20\")] protected alias Rel.rfl := vle.rfl\n\n@[simp, refl] lemma veq_refl (x : R) : x =ᵥ x := AntisymmRel.rfl\nlemma veq_rfl : x =ᵥ x := veq_refl x\n\nprotected alias veq.refl := veq_refl\nprotected alias veq.rfl := veq_rfl\n\ninstance : @Std.Refl R (· =ᵥ ·) where\n  refl _ := veq_rfl\n\n@[simp]\n\nTarget:\ntheorem zero_vle (x : R) : 0 ≤ᵥ x :=\n\nProof body:\n","rejected":"```lean\nby\n  simpa using mul_vle_mul_left ((vle_total 0 1).resolve_right not_vle_one_zero) x\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Valuation","family_id":"zero_vle","file_id":"mathlib/Mathlib/RingTheory/Valuation/ValuativeRel/Basic.lean","sample_id":"27226b4cef484b3c92a34e5c76b5413440a1f9efdabd65d7d5bb3fbb80618898"},"split":"train"}
{"binding":{"chosen_proof_sha256":"335297b55a581bb6fe6dcb38f704182637c77f8f85b5c081f025c29aaa5ebb61","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c092795e5fd0e22d540c502fd1b4fd017f05beade7817be0597a69f2c004e7d6","source_sha256":"c0e3fa9405ddf4fdd99cd3aa6c9ff7258178f89496e31a48d0f8f121f656258a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Option.some_inj]\n  simp [finSuccAboveOrderIso, finSuccAboveEquiv, OrderIso.symm]","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":5,"token_jaccard":0.055556,"token_length_ratio":0.277778},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"c2eb304b244fc4f70332e2a4d38b94ad25cbfa3bf2eec28009a1011b9eff7bb1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.VecNotation\npublic import Mathlib.Logic.Equiv.Fin.Basic\npublic import Mathlib.Order.Fin.Basic\npublic import Mathlib.Order.PiLex\npublic import Mathlib.Order.Interval.Set.Defs\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes\n-/\n/-!\n# Order properties on tuples\n-/\n\n@[expose] public section\n\nassert_not_exists Monoid\n\nopen Function Set\n\nnamespace Fin\nvariable {m n : ℕ} {α : Fin (n + 1) → Type*} (x : α 0) (q : ∀ i, α i) (p : ∀ i : Fin n, α i.succ)\n  (i : Fin n) (y : α i.succ) (z : α 0)\n\nlemma pi_lex_lt_cons_cons {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ}\n    (s : ∀ {i : Fin n.succ}, α i → α i → Prop) :\n    Pi.Lex (· < ·) (@s) (Fin.cons x₀ x) (Fin.cons y₀ y) ↔\n      s x₀ y₀ ∨ x₀ = y₀ ∧ Pi.Lex (· < ·) (@fun i : Fin n ↦ @s i.succ) x y := by\n  simp_rw [Pi.Lex, Fin.exists_fin_succ, Fin.cons_succ, Fin.cons_zero, Fin.forall_iff_succ]\n  simp [and_assoc, exists_and_left]\n\nvariable [∀ i, Preorder (α i)]\n\nlemma insertNth_mem_Icc {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)}\n    {q₁ q₂ : ∀ j, α j} :\n    i.insertNth x p ∈ Icc q₁ q₂ ↔\n      x ∈ Icc (q₁ i) (q₂ i) ∧ p ∈ Icc (fun j ↦ q₁ (i.succAbove j)) fun j ↦ q₂ (i.succAbove j) := by\n  simp only [mem_Icc, insertNth_le_iff, le_insertNth_iff, and_assoc, @and_left_comm (x ≤ q₂ i)]\n\nlemma preimage_insertNth_Icc_of_mem {i : Fin (n + 1)} {x : α i} {q₁ q₂ : ∀ j, α j}\n    (hx : x ∈ Icc (q₁ i) (q₂ i)) :\n    i.insertNth x ⁻¹' Icc q₁ q₂ = Icc (fun j ↦ q₁ (i.succAbove j)) fun j ↦ q₂ (i.succAbove j) :=\n  Set.ext fun p ↦ by simp only [mem_preimage, insertNth_mem_Icc, hx, true_and]\n\nlemma preimage_insertNth_Icc_of_notMem {i : Fin (n + 1)} {x : α i} {q₁ q₂ : ∀ j, α j}\n    (hx : x ∉ Icc (q₁ i) (q₂ i)) : i.insertNth x ⁻¹' Icc q₁ q₂ = ∅ :=\n  Set.ext fun p ↦ by\n    simp only [mem_preimage, insertNth_mem_Icc, hx, false_and, mem_empty_iff_false]\n\nend Fin\n\nopen Fin Matrix\n\nvariable {α : Type*}\n\nopen scoped Relator in\nlemma liftFun_vecCons {n : ℕ} (r : α → α → Prop) [IsTrans α r] {f : Fin (n + 1) → α} {a : α} :\n    ((· < ·) ⇒ r) (vecCons a f) (vecCons a f) ↔ r a (f 0) ∧ ((· < ·) ⇒ r) f f := by\n  simp only [liftFun_iff_succ r, forall_iff_succ, cons_val_succ, cons_val_zero, ← succ_castSucc,\n    castSucc_zero]\n\nvariable [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α}\n\n@[simp] lemma strictMono_vecCons : StrictMono (vecCons a f) ↔ a < f 0 ∧ StrictMono f :=\n  liftFun_vecCons (· < ·)\n\n@[simp]\nlemma monotone_vecCons : Monotone (vecCons a f) ↔ a ≤ f 0 ∧ Monotone f := by\n  simpa only [monotone_iff_forall_lt] using! @liftFun_vecCons α n (· ≤ ·) _ f a\n\n@[simp] lemma monotone_vecEmpty : Monotone ![a]\n  | ⟨0, _⟩, ⟨0, _⟩, _ => le_refl _\n\n@[simp] lemma strictMono_vecEmpty : StrictMono ![a]\n  | ⟨0, _⟩, ⟨0, _⟩, h => (irrefl _ h).elim\n\n@[simp] lemma strictAnti_vecCons : StrictAnti (vecCons a f) ↔ f 0 < a ∧ StrictAnti f :=\n  liftFun_vecCons (· > ·)\n\n@[simp] lemma antitone_vecCons : Antitone (vecCons a f) ↔ f 0 ≤ a ∧ Antitone f :=\n  monotone_vecCons (α := αᵒᵈ)\n\n@[simp] lemma antitone_vecEmpty : Antitone (vecCons a vecEmpty)\n  | ⟨0, _⟩, ⟨0, _⟩, _ => le_rfl\n\n@[simp] lemma strictAnti_vecEmpty : StrictAnti (vecCons a vecEmpty)\n  | ⟨0, _⟩, ⟨0, _⟩, h => (irrefl _ h).elim\n\nlemma StrictMono.vecCons (hf : StrictMono f) (ha : a < f 0) : StrictMono (vecCons a f) :=\n  strictMono_vecCons.2 ⟨ha, hf⟩\n\nlemma StrictAnti.vecCons (hf : StrictAnti f) (ha : f 0 < a) : StrictAnti (vecCons a f) :=\n  strictAnti_vecCons.2 ⟨ha, hf⟩\n\nlemma Monotone.vecCons (hf : Monotone f) (ha : a ≤ f 0) : Monotone (vecCons a f) :=\n  monotone_vecCons.2 ⟨ha, hf⟩\n\nlemma Antitone.vecCons (hf : Antitone f) (ha : f 0 ≤ a) : Antitone (vecCons a f) :=\n  antitone_vecCons.2 ⟨ha, hf⟩\n\nexample : Monotone ![1, 2, 2, 3] := by decide\n\n\nvariable {n : ℕ}\n\n/-- `Π i : Fin 2, α i` is order equivalent to `α 0 × α 1`. See also `OrderIso.finTwoArrowEquiv`\nfor a non-dependent version. -/\ndef OrderIso.piFinTwoIso (α : Fin 2 → Type*) [∀ i, Preorder (α i)] : (∀ i, α i) ≃o α 0 × α 1 where\n  toEquiv := piFinTwoEquiv α\n  map_rel_iff' := Iff.symm Fin.forall_fin_two\n\n/-- The space of functions `Fin 2 → α` is order equivalent to `α × α`. See also\n`OrderIso.piFinTwoIso`. -/\ndef OrderIso.finTwoArrowIso (α : Type*) [Preorder α] : (Fin 2 → α) ≃o α × α :=\n  { OrderIso.piFinTwoIso fun _ => α with toEquiv := finTwoArrowEquiv α }\n\nnamespace Fin\n\n/-- Order isomorphism between tuples of length `n + 1` and pairs of an element and a tuple of length\n`n` given by separating out the first element of the tuple.\n\nThis is `Fin.cons` as an `OrderIso`. -/\n@[simps!, simps toEquiv]\ndef consOrderIso (α : Fin (n + 1) → Type*) [∀ i, LE (α i)] :\n    α 0 × (∀ i, α (succ i)) ≃o ∀ i, α i where\n  toEquiv := consEquiv α\n  map_rel_iff' := forall_iff_succ\n\n/-- Order isomorphism between tuples of length `n + 1` and pairs of an element and a tuple of length\n`n` given by separating out the last element of the tuple.\n\nThis is `Fin.snoc` as an `OrderIso`. -/\n@[simps!, simps toEquiv]\ndef snocOrderIso (α : Fin (n + 1) → Type*) [∀ i, LE (α i)] :\n    α (last n) × (∀ i, α (castSucc i)) ≃o ∀ i, α i where\n  toEquiv := snocEquiv α\n  map_rel_iff' := by simp [Pi.le_def, Prod.le_def, forall_iff_castSucc]\n\n/-- Order isomorphism between tuples of length `n + 1` and pairs of an element and a tuple of length\n`n` given by separating out the `p`-th element of the tuple.\n\nThis is `Fin.insertNth` as an `OrderIso`. -/\n@[simps!, simps toEquiv]\ndef insertNthOrderIso (α : Fin (n + 1) → Type*) [∀ i, LE (α i)] (p : Fin (n + 1)) :\n    α p × (∀ i, α (p.succAbove i)) ≃o ∀ i, α i where\n  toEquiv := insertNthEquiv α p\n  map_rel_iff' := by simp [Pi.le_def, Prod.le_def, p.forall_iff_succAbove]\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp] lemma insertNthOrderIso_zero (α : Fin (n + 1) → Type*) [∀ i, LE (α i)] :\n    insertNthOrderIso α 0 = consOrderIso α := by ext; simp [insertNthOrderIso]\n\n/-- Note this lemma can only be written about non-dependent tuples as `insertNth (last n) = snoc` is\nnot a definitional equality. -/\n@[simp] lemma insertNthOrderIso_last (n : ℕ) (α : Type*) [LE α] :\n    insertNthOrderIso (fun _ ↦ α) (last n) = snocOrderIso (fun _ ↦ α) := by ext; simp\n\nend Fin\n\n/-- `Fin.succAbove` as an order isomorphism between `Fin n` and `{x : Fin (n + 1) // x ≠ p}`. -/\ndef finSuccAboveOrderIso (p : Fin (n + 1)) : Fin n ≃o { x : Fin (n + 1) // x ≠ p } where\n  __ := finSuccAboveEquiv p\n  map_rel_iff' := p.succAboveOrderEmb.map_rel_iff'\n\nlemma finSuccAboveOrderIso_apply (p : Fin (n + 1)) (i : Fin n) :\n    finSuccAboveOrderIso p i = ⟨p.succAbove i, p.succAbove_ne i⟩ := rfl\n\nTarget:\nlemma finSuccAboveOrderIso_symm_apply_last (x : { x : Fin (n + 1) // x ≠ Fin.last n }) :\n    (finSuccAboveOrderIso (Fin.last n)).symm x = Fin.castLT x.1 (Fin.val_lt_last x.2) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Fin","family_id":"finsuccaboveorderiso_symm_apply_last","file_id":"mathlib/Mathlib/Order/Fin/Tuple.lean","sample_id":"c092795e5fd0e22d540c502fd1b4fd017f05beade7817be0597a69f2c004e7d6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e483053df1cef9a9899f381185253ce0593370fe3c8c56da6f87d63a4270c28a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e538998df3d05f93bc2e2817a045d944fa2ac3da368487ea91017aaa253c2833","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"81608c9debe142f31aa7efa37f974504e0e8f8ceff0f0397335ad80e4df5060e","source_sha256":"29f6ed4519c38f3059df6084770133fafb9cbcb638c0482c74b9751c669ac0d9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [localization]\n  simp only [Localization.liftNatTrans_app, Functor.id_obj, Functor.whiskerRight_app,\n    NatTrans.naturality, Functor.comp_map, Functor.id_map, Iso.hom_inv_id_app_assoc]\n\ninclude W","hard_negative":false,"metrics":{"chosen_tokens":38,"rejected_tokens":45,"token_jaccard":0.916667,"token_length_ratio":1.184211},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"c300e8616e64cb5278f68df964d7d2bd474d20aa3d8f5f2b4a17d9c29e3a4a54","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Center.Preadditive\npublic import Mathlib.CategoryTheory.Localization.Predicate\npublic import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor\n\nNamespace:\nCategoryTheory.CatCenter\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Localization of the center of a category\n\nGiven a localization functor `L : C ⥤ D` with respect to `W : MorphismProperty C`,\nwe define a localization map `CatCenter C → CatCenter D` for the centers\nof these categories. In case `L` is an additive functor between preadditive\ncategories, we promote this to a ring morphism `CatCenter C →+* CatCenter D`.\n\n-/\n\n@[expose] public section\n\nuniverse w v₁ v₂ u₁ u₂\n\nnamespace CategoryTheory\n\nvariable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]\n  (r s : CatCenter C) (L : C ⥤ D) (W : MorphismProperty C) [L.IsLocalization W]\n\nnamespace CatCenter\n\n/-- Given `r : CatCenter C` and `L : C ⥤ D` a localization functor with respect\nto `W : MorphismProperty D`, this is the induced element in `CatCenter D`\nobtained by localization. -/\nnoncomputable def localization : CatCenter D :=\n  Localization.liftNatTrans L W L L (𝟭 D) (𝟭 D) (Functor.whiskerRight r L)\n\n@[simp]\n\nTarget:\nlemma localization_app (X : C) :\n    (r.localization L W).app (L.obj X) = L.map (r.app X) :=\n\nProof body:\n","rejected":"```lean\nby\n  dsimp [localization]\n  simp only [Localization.liftNatTrans_app, Functor.id_obj, Functor.whiskerRight_app,\n    NatTrans.naturality, Functor.comp_map, Functor.id_map, Iso.hom_inv_id_app_assoc]\n\ninclude W\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Center","family_id":"localization_app","file_id":"mathlib/Mathlib/CategoryTheory/Center/Localization.lean","sample_id":"81608c9debe142f31aa7efa37f974504e0e8f8ceff0f0397335ad80e4df5060e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"65fb0bf51735f2fd54b4e2c9d7d6b383f1f337560f0bc27999124bd7b667a991","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ec42e30620eff4d5664c510c36fb816ac3d16fcdbca2be2229523be37286abc9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"99fd3b1c7dbbd31133ecfe7e89f6d8fdbe8581f23efe8e63f4413b7eb18ed016","source_sha256":"cc5ba9fe8b6562ffdfbb46bc6d44a21f83997fd70ba7584887f499c1806af956","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hA_T_nonneg : ∀ i j, 0 ≤ Aᵀ i j := fun i j => by\n    simpa [Matrix.transpose_apply] using hA.nonneg j i\n  refine ⟨hA_T_nonneg, ?_⟩\n  intro i j\n  letI : Quiver n := toQuiver A\n  obtain ⟨p, hp_pos⟩ := hA.connected j i\n  cases p with\n  | nil =>\n    simp at hp_pos\n  | @cons b _ q e =>\n    let qT := transposePath (A := A) (q.cons e)\n    letI : Quiver n := toQuiver Aᵀ\n    use qT\n    simp [qT, transposePath, Quiver.Path.length_comp, Quiver.Path.length_toPath]","hard_negative":true,"metrics":{"chosen_tokens":121,"rejected_tokens":2,"token_jaccard":0.016949,"token_length_ratio":0.016529},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"c383ae8416b8f5f30fd609b398bcfaa20389d9329f1d39efd5609f11ada07fd4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Quiver.ConnectedComponent\npublic import Mathlib.Combinatorics.Quiver.Path.Vertices\npublic import Mathlib.Data.Matrix.Mul\n\nNamespace:\nMatrix\n\nLocal context:\n/-\nCopyright (c) 2025 Matteo Cipollina. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Matteo Cipollina\n-/\n/-!\n# Irreducibility and primitivity of nonnegative matrices\n\nThis file develops a graph-theoretic interface for studying the properties of nonnegative square\nmatrices.\n\nWe associate a directed graph (quiver) with a matrix `A`, where an edge `i ⟶ j` exists if and only\nif the entry `A i j` is strictly positive. This allows translating algebraic properties of the\nmatrix (like powers) into graph-theoretic properties of its quiver (like the existence of paths).\n\n## Main definitions\n\n* `Matrix.toQuiver A`: The quiver associated with a matrix `A`, where an edge `i ⟶ j` exists if\n  `0 < A i j`.\n* `Matrix.IsIrreducible A`: A matrix `A` is defined as irreducible if it is entrywise nonnegative\n  and its associated quiver `toQuiver A` is strongly connected. The theorem\n  `Matrix.isIrreducible_iff_exists_pow_pos` proves this graph-theoretic definition is equivalent\n  to the algebraic one in seneta2006 (Def 1.6, p.18): for every pair of indices `(i, j)`, there\n  exists a positive integer `k` such that `(A ^ k) i j > 0`.\n* `Matrix.IsPrimitive A`: A matrix `A` is primitive if it is nonnegative and some power `A ^ k`\n  is strictly positive (all entries are `> 0`), (seneta2006, Definition 1.1, p.14).\n\n## Main results\n\n* `Matrix.pow_apply_pos_iff_nonempty_path`: Establishes the link between matrix powers and graph\n  theory:\n  `(A ^ k) i j > 0` if and only if there is a path of length `k` from `i` to `j` in `toQuiver A`.\n* `Matrix.isIrreducible_iff_exists_pow_pos`: Shows the equivalence between the graph-theoretic\n  definition of irreducibility (strong connectivity) and the algebraic one (existence of a\n  positive entry in some power).\n* `Matrix.IsPrimitive.to_IsIrreducible`: Proves that a primitive matrix is also irreducible\n  (Seneta, p.14).\n* `Matrix.IsIrreducible.transpose`: Shows that the irreducibility property is preserved under\n  transposition.\n\n## Implementation notes\n\nThroughout we work over a linearly ordered ring `R`. Some results require stronger assumptions,\nlike `PosMulStrictMono R` or `Nontrivial R`. Some statements expand matrix powers and thus require\n`[DecidableEq n]` to reason about finite sums.\n\n## TODO\n\nRefactor to use digraphs instead of quivers. A prerequisite for this refactor\nis paths in digraphs.\n\n## References\n\n* [E. Seneta, *Non-negative Matrices and Markov Chains*][seneta2006]\n\n## Tags\n\nmatrix, nonnegative, positive, power, quiver, graph, irreducible, primitive, perron-frobenius\n-/\n\n@[expose] public section\nnamespace Matrix\n\nopen Quiver Quiver.Path\n\nvariable {n R : Type*} [Ring R] [LinearOrder R]\n\n/-- The directed graph (quiver) associated with a matrix `A`,\nwith an edge `i ⟶ j` iff `0 < A i j`. -/\n@[implicit_reducible]\ndef toQuiver (A : Matrix n n R) : Quiver n :=\n  ⟨fun i j => PLift (0 < A i j)⟩\n\n/-- A matrix `A` is irreducible if it is entrywise nonnegative and\nits quiver of positive entries (`toQuiver A`) is strongly connected. -/\n@[mk_iff] structure IsIrreducible (A : Matrix n n R) : Prop where\n  nonneg (i j : n) : 0 ≤ A i j\n  connected : @IsSStronglyConnected n (toQuiver A)\n\n/-- A matrix `A` is primitive if it is entrywise nonnegative\nand some positive power has all entries strictly positive. -/\n@[mk_iff] structure IsPrimitive [Fintype n] [DecidableEq n] (A : Matrix n n R) : Prop where\n  nonneg (i j : n) : 0 ≤ A i j\n  exists_pos_pow : ∃ k > 0, ∀ i j, 0 < (A ^ k) i j\n\nvariable {A : Matrix n n R}\n\n/-- If `A` is irreducible and `n` is non-trivial then every row has a positive entry. -/\nlemma IsIrreducible.exists_pos [Nontrivial n]\n    (h_irr : IsIrreducible A) (i : n) :\n    ∃ j, 0 < A i j := by\n  letI : Quiver n := toQuiver A\n  by_contra h_row\n  have no_out : ∀ j : n, IsEmpty (i ⟶ j) :=\n    fun j => ⟨fun e => h_row ⟨j, e.down⟩⟩\n  obtain ⟨j, hij⟩ := exists_pair_ne n\n  obtain ⟨p, hp_pos⟩ := h_irr.connected i j\n  have h_le : 1 ≤ p.length := Nat.succ_le_of_lt hp_pos\n  have ⟨v, p₁, p₂, _hp_eq, hp₁_len⟩ := p.exists_eq_comp_of_le_length (n := 1) h_le\n  have hlen_ne : p₁.length ≠ 0 := by simp [hp₁_len]\n  obtain ⟨c, p', e, rfl⟩ := (Quiver.Path.length_ne_zero_iff_eq_cons (p := p₁)).1 (by lia)\n  obtain ⟨rfl⟩ : i = c := Quiver.Path.eq_of_length_zero p' (by simp_all)\n  exact (no_out _).false e\n\n/--\nFor a matrix `A` with nonnegative entries, the `(i, j)`-entry of the `k`-th power `A ^ k`\nis strictly positive if and only if there exists a path of length `k` from `i` to `j` in the\nquiver associated to `A` via `toQuiver`. -/\ntheorem pow_apply_pos_iff_nonempty_path\n    [Fintype n] [IsOrderedRing R] [PosMulStrictMono R] [Nontrivial R] [DecidableEq n]\n    (hA : ∀ i j, 0 ≤ A i j) (k : ℕ) (i j : n) :\n    letI := toQuiver A\n    0 < (A ^ k) i j ↔ Nonempty {p : Path i j // p.length = k} := by\n  letI := toQuiver A\n  induction k generalizing i j with\n  | zero =>\n    refine ⟨fun h_pos ↦ ?_, fun ⟨p, hp⟩ ↦ ?_⟩\n    · rcases eq_or_ne i j with rfl | h_eq\n      · exact ⟨⟨Quiver.Path.nil, rfl⟩⟩\n      · simp_all\n    · simp [Quiver.Path.eq_of_length_zero p hp]\n  | succ m ih =>\n    rw [pow_succ, mul_apply]\n    constructor\n    · intro h_pos\n      obtain ⟨l, hl_mem, hl_pos⟩ :\n          ∃ l ∈ (Finset.univ : Finset n), 0 < (A ^ m) i l * A l j := by\n        simpa [Finset.sum_pos_iff_of_nonneg\n                 (fun x _ => mul_nonneg (pow_apply_nonneg hA m i x) (hA x j))]\n          using h_pos\n      have hAm_nonneg : 0 ≤ (A ^ m) i l := pow_apply_nonneg hA m i l\n      have hA_nonneg' : 0 ≤ A l j := hA l j\n      have h_Am : 0 < (A ^ m) i l := by by_contra! h; simp [le_antisymm h hAm_nonneg] at hl_pos\n      have h_A : 0 < A l j := by by_contra! h; simp [le_antisymm h hA_nonneg'] at hl_pos\n      obtain ⟨⟨p, rfl⟩⟩ := (ih i l).mp h_Am\n      exact ⟨p.cons (PLift.up h_A), by simp⟩\n    · rintro ⟨p, hp_len⟩\n      cases p with\n      | nil => simp [Quiver.Path.length] at hp_len\n      | @cons b _ q e =>\n        simp only [Quiver.Path.length_cons, Nat.succ.injEq] at hp_len\n        have h_Am_pos : 0 < (A ^ m) i b := (ih i b).mpr ⟨q, hp_len⟩\n        let h_A_pos := e\n        have h_prod : 0 < (A ^ m) i b * A b j := mul_pos h_Am_pos h_A_pos.down\n        exact\n          (Finset.sum_pos_iff_of_nonneg\n            (fun x _ => mul_nonneg (pow_apply_nonneg hA m i x) (hA x j))).2\n            ⟨b, Finset.mem_univ b, h_prod⟩\n\n/-- Irreducibility of a nonnegative matrix `A` is equivalent to entrywise positivity of some\npower: between any two indices `i, j` there exists a positive integer `k` such that the\n`(i, j)`-entry of `A ^ k` is strictly positive. -/\ntheorem isIrreducible_iff_exists_pow_pos\n    [Fintype n] [IsOrderedRing R] [PosMulStrictMono R] [Nontrivial R] [DecidableEq n]\n    (hA : ∀ i j, 0 ≤ A i j) :\n    IsIrreducible A ↔ ∀ i j, ∃ k > 0, 0 < (A ^ k) i j := by\n  letI : Quiver n := toQuiver A\n  constructor\n  · intro h_irr i j\n    obtain ⟨p, hp_len⟩ := h_irr.2 i j\n    refine ⟨p.length, hp_len, ?_⟩\n    have : Nonempty {q : Path i j // q.length = p.length} := ⟨⟨p, rfl⟩⟩\n    have hpos :=\n      (pow_apply_pos_iff_nonempty_path (A := A) hA p.length i j).2 this\n    simpa using hpos\n  · intro h_exists\n    constructor\n    · exact hA\n    · intro i j\n      obtain ⟨k, hk_pos, hk_entry⟩ := h_exists i j\n      obtain ⟨⟨p, rfl⟩⟩ :=\n        (pow_apply_pos_iff_nonempty_path (A := A) hA k i j).mp hk_entry\n      exact ⟨p, hk_pos⟩\n\n/-- If a nonnegative square matrix `A` is primitive, then `A` is irreducible. -/\ntheorem IsPrimitive.isIrreducible\n    [Fintype n] [IsOrderedRing R] [PosMulStrictMono R] [Nontrivial R] [DecidableEq n]\n    (h_prim : IsPrimitive A) : IsIrreducible A := by\n  obtain ⟨h_nonneg, k, hk_pos, hk_all⟩ := h_prim\n  rw [isIrreducible_iff_exists_pow_pos h_nonneg]\n  aesop\n\n/-! ## Transposition -/\n\n/-- Reverse a path in `toQuiver A` to a path in `toQuiver Aᵀ`, swapping endpoints. -/\ndef transposePath {i j : n} (p : @Quiver.Path n A.toQuiver i j) :\n    @Quiver.Path n Aᵀ.toQuiver j i := by\n  letI : Quiver n := toQuiver A\n  induction p with\n  | nil =>\n    exact (@Quiver.Path.nil _ (toQuiver Aᵀ) _)\n  | @cons b c q e ih =>\n    have eT : 0 < (Aᵀ) c b := by\n      simpa [Matrix.transpose_apply] using e.down\n    exact (@Quiver.Path.comp n (toQuiver Aᵀ) c b i (@Quiver.Hom.toPath n (toQuiver Aᵀ) c b\n      (PLift.up eT)) ih)\n\n/-- Irreducibility is invariant under transpose. -/\n\nTarget:\ntheorem IsIrreducible.transpose (hA : IsIrreducible A) : IsIrreducible Aᵀ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_99fd3b1c7dbb","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b88570995ea7a47ee5c63db757dd22f09fa8725a6f4313168317e911e33af73a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Matrix","family_id":"isirreducible","file_id":"mathlib/Mathlib/LinearAlgebra/Matrix/Irreducible/Defs.lean","sample_id":"99fd3b1c7dbbd31133ecfe7e89f6d8fdbe8581f23efe8e63f4413b7eb18ed016"},"split":"train"}
{"binding":{"chosen_proof_sha256":"01a9a5bebe8404a63c3e4dd6c8ed6a63ca7ab9f3512e93006d4e0432218d9a02","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"301632548b2cf18455f3303dcaa3cf36384b8e48dcb9c37dd4c33d16977a202d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d99fd6fb9456101e1d28de19e801f40e15b75e0d01ea3c483f56e794b32e60bf","source_sha256":"2ce531595998b973f239f8a83dbac8a4626afb2088d76e997a149cdb9eae73d2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let ⟨⟨κ, b⟩⟩ := Module.Free.exists_basis (R := R) (M := M)\n  rw [(equivFreeAlgebra b).toLinearEquiv.rank_eq, FreeAlgebra.rank_eq, mk_list_eq_sum_pow,\n    Basis.mk_eq_rank'' b]","hard_negative":true,"metrics":{"chosen_tokens":47,"rejected_tokens":3,"token_jaccard":0.074074,"token_length_ratio":0.06383},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"c3d97e0e62d3c6d29f11d3ab8b0293bdcd548cacc679d04ca3bd997fd9321abf","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorAlgebra.Basic\npublic import Mathlib.LinearAlgebra.FreeAlgebra\n\nNamespace:\nTensorAlgebra\n\nLocal context:\n/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# A basis for `TensorAlgebra R M`\n\n## Main definitions\n\n* `TensorAlgebra.equivMonoidAlgebra b : TensorAlgebra R M ≃ₐ[R] FreeAlgebra R κ`:\n  the isomorphism given by a basis `b : Basis κ R M`.\n* `Basis.tensorAlgebra b : Basis (FreeMonoid κ) R (TensorAlgebra R M)`:\n  the basis on the tensor algebra given by a basis `b : Basis κ R M`.\n\n## Main results\n\n* `TensorAlgebra.instFreeModule`: the tensor algebra over `M` is free when `M` is\n* `TensorAlgebra.rank_eq`\n\n-/\n\n@[expose] public section\n\nopen Module\n\nnamespace TensorAlgebra\n\nuniverse uκ uR uM\nvariable {κ : Type uκ} {R : Type uR} {M : Type uM}\n\nsection CommSemiring\nvariable [CommSemiring R] [AddCommMonoid M] [Module R M]\n\n/-- A basis provides an algebra isomorphism with the free algebra, replacing each basis vector\nwith its index. -/\nnoncomputable def equivFreeAlgebra (b : Basis κ R M) :\n    TensorAlgebra R M ≃ₐ[R] FreeAlgebra R κ :=\n  AlgEquiv.ofAlgHom\n    (TensorAlgebra.lift _ (Finsupp.linearCombination _ (FreeAlgebra.ι _) ∘ₗ b.repr.toLinearMap))\n    (FreeAlgebra.lift _ (ι R ∘ b))\n    (by ext; simp)\n    (hom_ext <| b.ext fun i => by simp)\n\n@[simp]\nlemma equivFreeAlgebra_ι_apply (b : Basis κ R M) (i : κ) :\n    equivFreeAlgebra b (ι R (b i)) = FreeAlgebra.ι R i :=\n  (TensorAlgebra.lift_ι_apply _ _).trans <| by simp\n\n@[simp]\nlemma equivFreeAlgebra_symm_ι (b : Basis κ R M) (i : κ) :\n    (equivFreeAlgebra b).symm (FreeAlgebra.ι R i) = ι R (b i) :=\n  (equivFreeAlgebra b).toEquiv.symm_apply_eq.mpr <| equivFreeAlgebra_ι_apply b i |>.symm\n\n/-- A basis on `M` can be lifted to a basis on `TensorAlgebra R M` -/\n@[simps! repr_apply]\nnoncomputable def _root_.Module.Basis.tensorAlgebra (b : Basis κ R M) :\n    Basis (FreeMonoid κ) R (TensorAlgebra R M) :=\n  (FreeAlgebra.basisFreeMonoid R κ).map <| (equivFreeAlgebra b).symm.toLinearEquiv\n\n/-- `TensorAlgebra R M` is free when `M` is. -/\ninstance instModuleFree [Module.Free R M] : Module.Free R (TensorAlgebra R M) :=\n  let ⟨⟨_κ, b⟩⟩ := Module.Free.exists_basis (R := R) (M := M)\n  .of_basis b.tensorAlgebra\n\n/-- The `TensorAlgebra` of a free module over a commutative semiring with no zero-divisors has\nno zero-divisors. -/\ninstance instNoZeroDivisors [NoZeroDivisors R] [Module.Free R M] :\n    NoZeroDivisors (TensorAlgebra R M) :=\n  have ⟨⟨_, b⟩⟩ := ‹Module.Free R M›\n  (equivFreeAlgebra b).toMulEquiv.noZeroDivisors\n\nend CommSemiring\n\nsection CommRing\nvariable [CommRing R] [AddCommGroup M] [Module R M]\n\n/-- The `TensorAlgebra` of a free module over an integral domain is a domain. -/\ninstance instIsDomain [IsDomain R] [Module.Free R M] : IsDomain (TensorAlgebra R M) :=\n  NoZeroDivisors.to_isDomain _\n\nattribute [pp_with_univ] Cardinal.lift\n\nopen Cardinal in\n\nTarget:\nlemma rank_eq [Nontrivial R] [Module.Free R M] :\n    Module.rank R (TensorAlgebra R M) = Cardinal.lift.{uR} (sum fun n ↦ Module.rank R M ^ n) :=\n\nProof body:\n","rejected":"by\n  exact rank_eq","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"ee50573d078d5041c122f54fe08b9d8601c9649d459f6a3a579cf71cc6658164","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/TensorAlgebra","family_id":"rank_eq","file_id":"mathlib/Mathlib/LinearAlgebra/TensorAlgebra/Basis.lean","sample_id":"d99fd6fb9456101e1d28de19e801f40e15b75e0d01ea3c483f56e794b32e60bf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f6142b8f4680c491cc9a84d46312116399269b0b324f299edcee5c0f95b61ddc","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ebebed96f1f59661bd31143f834b43b5b9941049affa74d994da26eb1385a91c","source_sha256":"edb52479240700af7f6c572a1b6c456a3b7b18b85aa47a4adb83e507587b6553","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro x y hxy\n  have := M.isUnit_algebraMap_end_of_le_basicOpen _ hle\n  rw [Module.End.isUnit_iff] at this\n  exact this.injective hxy","hard_negative":false,"metrics":{"chosen_tokens":27,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.074074},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"c3e8ce175e8d37cbe89b0260163757984727c7119f61167ae676d43220174807","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Category.ModuleCat.Localization\npublic import Mathlib.Algebra.Category.ModuleCat.Sheaf.Quasicoherent\npublic import Mathlib.AlgebraicGeometry.AffineScheme\npublic import Mathlib.AlgebraicGeometry.Modules.Sheaf\n\nNamespace:\nAlgebraicGeometry.Scheme.Modules\n\nLocal context:\n/-\nCopyright (c) 2024 Weihong Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Johan Commelin, Amelia Livingston, Sophie Morel, Jujian Zhang, Weihong Xu,\n  Andrew Yang, Brian Nugent\n-/\n/-!\n\n# Construction of M^~\n\nGiven any commutative ring `R` and `R`-module `M`, we construct the sheaf `M^~` of `𝒪_SpecR`-modules\nsuch that `M^~(U)` is the set of dependent functions that are locally fractions.\n\n## Main definitions\n* `AlgebraicGeometry.tilde` : `M^~` as a sheaf of `𝒪_{Spec R}`-modules.\n* `AlgebraicGeometry.tilde.adjunction` : `~` is left adjoint to taking global sections.\n\n-/\n\n@[expose] public noncomputable section\n\nuniverse u\n\nopen TopCat AlgebraicGeometry TopologicalSpace CategoryTheory Opposite\n\nvariable {R : CommRingCat.{u}} (M : ModuleCat.{u} R)\n\nnamespace AlgebraicGeometry\n\nopen _root_.PrimeSpectrum\n\n/-- The forgetful functor from `𝒪_{Spec R}` modules to sheaves of `R`-modules. -/\ndef modulesSpecToSheaf :\n    (Spec R).Modules ⥤ TopCat.Sheaf (ModuleCat R) (Spec R) :=\n  SheafOfModules.forgetToSheafModuleCat (Spec R).ringCatSheaf (.op ⊤)\n    (Limits.initialOpOfTerminal Limits.isTerminalTop) ⋙\n  sheafCompose _ (ModuleCat.restrictScalars (Scheme.ΓSpecIso R).inv.hom)\n\n/-- The global section functor for `𝒪_{Spec R}` modules -/\nnoncomputable\ndef moduleSpecΓFunctor : (Spec (.of R)).Modules ⥤ ModuleCat R :=\n  modulesSpecToSheaf ⋙ TopCat.Sheaf.forget _ _ ⋙ (evaluation _ _).obj (.op ⊤)\n\nset_option backward.isDefEq.respectTransparency false in\nopen PrimeSpectrum in\n/-- The forgetful functor from `𝒪_{Spec R}` modules to sheaves of `R`-modules is fully faithful. -/\ndef SpecModulesToSheafFullyFaithful : (modulesSpecToSheaf (R := R)).FullyFaithful where\n  preimage {M N} f := ⟨fun U ↦ ModuleCat.ofHom ⟨(f.1.app U).hom.toAddHom, by\n    intro t m\n    apply TopCat.Presheaf.IsSheaf.section_ext (modulesSpecToSheaf.obj N).2\n    intro x hxU\n    obtain ⟨a, ⟨_, ⟨r, rfl⟩, rfl⟩, hxr, hrU : basicOpen _ ≤ _⟩ :=\n      PrimeSpectrum.isBasis_basic_opens.exists_subset_of_mem_open hxU U.unop.2\n    refine ⟨_, hrU, hxr, ?_⟩\n    refine Eq.trans ?_ (N.val.map_smul (homOfLE hrU).op t _).symm\n    change N.1.map (homOfLE hrU).op (f.1.app _ _) = _ • N.1.map (homOfLE hrU).op (f.1.app _ _)\n    have (x : _) :\n        f.1.app _ (M.1.map (homOfLE hrU).op _) = N.1.map (homOfLE hrU).op (f.1.app _ x) :=\n      congr($(f.1.naturality (homOfLE hrU).op).hom x)\n    rw [← this, ← this, M.val.map_smul]\n    generalize (Spec R).ringCatSheaf.obj.map (homOfLE hrU).op t = t\n    letI := Module.compHom (R := Γ(Spec R, basicOpen r)) Γ(M, basicOpen r)\n      (algebraMap R Γ(Spec R, basicOpen r))\n    haveI : IsScalarTower R Γ(Spec R, basicOpen r) Γ(M, basicOpen r) :=\n      .of_algebraMap_smul fun _ _ ↦ rfl\n    letI := Module.compHom Γ(N, basicOpen r) (algebraMap R Γ(Spec R, basicOpen r))\n    haveI : IsScalarTower R Γ(Spec R, basicOpen r) Γ(N, basicOpen r) :=\n      .of_algebraMap_smul fun _ _ ↦ rfl\n    exact (IsLocalization.linearMap_compatibleSMul (.powers (M := R) r)\n      Γ(Spec R, basicOpen r) Γ(M, basicOpen r) Γ(N, basicOpen r)).map_smul\n      (f.hom.app _).hom _ _⟩, fun i ↦ by ext x; exact congr($(f.1.naturality i).hom x)⟩\n  map_preimage f := rfl\n  preimage_map f := rfl\n\ninstance : (modulesSpecToSheaf (R := R)).Faithful := SpecModulesToSheafFullyFaithful.faithful\n\ninstance : (modulesSpecToSheaf (R := R)).Full := SpecModulesToSheafFullyFaithful.full\n\nnamespace Scheme.Modules\n\nvariable {M : (Spec R).Modules} {U V : (Spec R).Opens}\n\ninstance : Module R Γ(M, U) :=\n  inferInstanceAs <| Module R ((modulesSpecToSheaf.obj M).obj.obj (.op U))\n\ninstance : IsScalarTower R Γ(Spec R, U) Γ(M, U) :=\n  IsScalarTower.of_compHom R Γ(Spec R, U) Γ(M, U)\n\nlemma smul_Spec_def (r : R) (x : Γ(M, U)) :\n    r • x = ((Spec R).presheaf.map U.leTop.op) ((Scheme.ΓSpecIso R).inv r) • x :=\n  rfl\n\n@[simp]\nlemma map_smul_Spec (hUV : .op V ⟶ .op U) (f : R) (x : Γ(M, V)) :\n    dsimp% M.presheaf.map hUV (f • x) = f • M.presheaf.map hUV x :=\n  ((modulesSpecToSheaf.obj M).obj.map hUV).hom.map_smul f x\n\nlemma isUnit_algebraMap_end_of_le_basicOpen (f : R) (hf : U ≤ PrimeSpectrum.basicOpen f) :\n    IsUnit (algebraMap R (Module.End R Γ(M, U)) f) := by\n  rw [Module.End.isUnit_iff]\n  have : ⇑((algebraMap R (Module.End ↑R ↑Γ(M, U))) f) =\n      algebraMap (Γ(Spec R, U)) (Module.End Γ(Spec R, U) Γ(M, U))\n        (((Spec R).presheaf.map (homOfLE hf).op) <| algebraMap R _ f) :=\n    rfl\n  rw [this, ← Module.End.isUnit_iff]\n  exact ((IsLocalization.Away.algebraMap_isUnit _).map _).map _\n\nTarget:\nlemma isSMulRegular_of_le_basicOpen {f : R} (hle : U ≤ PrimeSpectrum.basicOpen f) :\n    IsSMulRegular Γ(M, U) f :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Modules","family_id":"issmulregular_of_le_basicopen","file_id":"mathlib/Mathlib/AlgebraicGeometry/Modules/Tilde.lean","sample_id":"ebebed96f1f59661bd31143f834b43b5b9941049affa74d994da26eb1385a91c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a14f9a077bdd5b6f8dbacb2f648d28154c7ace639c1cada2afcd1dc3728bb20d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"36f6292b2bea01ba2949cab99df0942afa3d01db01b2c1e30655e18e62dad252","source_sha256":"e94c5e4561a46cbb726026eb6950cfdf4d135e8e5c0b4f8f268b42813d064fee","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  intro Z ⟨g, gc⟩ ⟨h, hc⟩ e\n  apply_fun InducedCategory.Hom.hom at e\n  obtain rfl : g = h := ConcreteCategory.hom_ext _ _ fun y ↦ by\n    rw [← (h₁ y).choose_spec]\n    simpa using congr(($e).base.hom (h₁ y).choose)\n  refine SheafedSpace.hom_stalk_ext ⟨g, gc⟩ ⟨g, hc⟩ rfl fun y ↦ ?_\n  rw [← (h₁ y).choose_spec, ← cancel_mono (f.hom.stalkMap (h₁ y).choose), stalkCongr_hom,\n    stalkSpecializes_refl, Category.id_comp, ← PresheafedSpace.stalkMap.comp f.hom ⟨g, gc⟩,\n    ← PresheafedSpace.stalkMap.comp f.hom ⟨g, hc⟩]\n  congr 1","hard_negative":false,"metrics":{"chosen_tokens":158,"rejected_tokens":2,"token_jaccard":0.017241,"token_length_ratio":0.012658},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"c435151482dde8e322d7017ef7aefc216329eb87cb05d3f63dfd75eb7ede5df5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits\npublic import Mathlib.Geometry.RingedSpace.Stalks\npublic import Mathlib.Topology.Sheaves.Functors\n\nNamespace:\nAlgebraicGeometry.SheafedSpace\n\nLocal context:\n/-\nCopyright (c) 2019 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Sheafed spaces\n\nIntroduces the category of topological spaces equipped with a sheaf (taking values in an\narbitrary target category `C`).\n\nWe further describe how to apply functors and natural transformations to the values of the\npresheaves.\n-/\n\n@[expose] public section\n\nopen CategoryTheory TopCat TopologicalSpace Opposite CategoryTheory.Limits CategoryTheory.Category\n  CategoryTheory.Functor Topology\n\nuniverse u v w' w\n\nvariable (C : Type u) [Category.{v} C]\n\n\n-- We could enable the following line:\n-- attribute [local aesop safe cases (rule_sets := [CategoryTheory])] Opposite\n-- but may need\n-- https://github.com/leanprover-community/aesop/issues/59\n\nnamespace AlgebraicGeometry\n\n/-- A `SheafedSpace C` is a topological space equipped with a sheaf of `C`s. -/\nstructure SheafedSpace extends PresheafedSpace C where\n  /-- A sheafed space is a presheafed space which happens to be a sheaf. -/\n  IsSheaf : presheaf.IsSheaf\n\nvariable {C}\n\nnamespace SheafedSpace\n\ninstance coeCarrier : CoeOut (SheafedSpace C) TopCat where coe X := X.carrier\n\ninstance coeSort : CoeSort (SheafedSpace C) Type* where\n  coe X := X.1\n\n/-- Extract the `sheaf C (X : Top)` from a `SheafedSpace C`. -/\ndef sheaf (X : SheafedSpace C) : Sheaf C (X : TopCat) :=\n  ⟨X.presheaf, X.IsSheaf⟩\n\n/-- Not `@[simp]` since it already reduces to `carrier = carrier`. -/\ntheorem mk_coe (carrier) (presheaf) (h) :\n    (({ carrier\n        presheaf\n        IsSheaf := h } : SheafedSpace C) : TopCat) = carrier :=\n  rfl\n\ninstance (X : SheafedSpace C) : TopologicalSpace X :=\n  X.carrier.str\n\n/-- The trivial `unit`-valued sheaf on any topological space. -/\ndef unit (X : TopCat) : SheafedSpace (Discrete Unit) :=\n  { @PresheafedSpace.const (Discrete Unit) _ X ⟨⟨⟩⟩ with IsSheaf := Presheaf.isSheaf_unit _ }\n\ninstance : Inhabited (SheafedSpace (Discrete Unit)) :=\n  ⟨unit (TopCat.of PEmpty)⟩\n\ninstance : Category (SheafedSpace C) :=\n  inferInstanceAs <| Category (InducedCategory (PresheafedSpace C) SheafedSpace.toPresheafedSpace)\n\n@[ext (iff := false)]\ntheorem ext {X Y : SheafedSpace C} (α β : X ⟶ Y) (w : α.hom.base = β.hom.base)\n    (h : α.hom.c ≫ whiskerRight (eqToHom (by rw [w])) _ = β.hom.c) : α = β :=\n  InducedCategory.hom_ext (PresheafedSpace.ext _ _ w h)\n\n/-- Constructor for isomorphisms in the category `SheafedSpace C`. -/\n@[simps]\ndef isoMk {X Y : SheafedSpace C} (e : X.toPresheafedSpace ≅ Y.toPresheafedSpace) : X ≅ Y where\n  hom := InducedCategory.homMk e.hom\n  inv := InducedCategory.homMk e.inv\n  hom_inv_id := InducedCategory.hom_ext e.hom_inv_id\n  inv_hom_id := InducedCategory.hom_ext e.inv_hom_id\n\n/-- Forgetting the sheaf condition is a functor from `SheafedSpace C` to `PresheafedSpace C`. -/\n@[simps! obj map]\ndef forgetToPresheafedSpace : SheafedSpace C ⥤ PresheafedSpace C :=\n  inducedFunctor _\n-- The `Full, Faithful` instances should be constructed by a deriving handler.\n-- https://github.com/leanprover-community/mathlib4/issues/380\n\n/-- The functor `forgetToPresheafedSpace : SheafedSpace C ⥤ PresheafedSpace C`\nis fully faithful. -/\ndef fullyFaithfulForgetToPresheafedSpace :\n    (forgetToPresheafedSpace (C := C)).FullyFaithful where\n  preimage f := InducedCategory.homMk f\n\n@[simp]\nlemma fullyFaithfulForgetToPresheafedSpace_preimage_hom {X Y : SheafedSpace C}\n    (f : forgetToPresheafedSpace.obj X ⟶ forgetToPresheafedSpace.obj Y) :\n    (fullyFaithfulForgetToPresheafedSpace.preimage f).hom = f := rfl\n\ninstance forgetToPresheafedSpace_full : (forgetToPresheafedSpace (C := C)).Full :=\n  fullyFaithfulForgetToPresheafedSpace.full\n\ninstance forgetToPresheafedSpace_faithful : (forgetToPresheafedSpace (C := C)).Faithful :=\n  fullyFaithfulForgetToPresheafedSpace.faithful\n\ninstance is_presheafedSpace_iso {X Y : SheafedSpace C} (f : X ⟶ Y) [IsIso f] :\n    IsIso f.hom :=\n  SheafedSpace.forgetToPresheafedSpace.map_isIso f\n\nsection\n\nattribute [local simp] id comp\n\n@[simp]\ntheorem id_hom (X : SheafedSpace C) : (𝟙 X : X ⟶ X).hom = 𝟙 X.toPresheafedSpace :=\n  rfl\n\n@[simp]\ntheorem id_hom_base (X : SheafedSpace C) : (𝟙 X : X ⟶ X).hom.base = 𝟙 (X : TopCat) :=\n  rfl\n\ntheorem id_hom_c (X : SheafedSpace C) :\n    (𝟙 X : X ⟶ X).hom.c = eqToHom (Presheaf.Pushforward.id_eq X.presheaf).symm :=\n  rfl\n\ntheorem id_hom_c_app (X : SheafedSpace C) (U) :\n    (𝟙 X : X ⟶ X).hom.c.app U = 𝟙 _ := rfl\n\n@[simp]\ntheorem comp_hom_base {X Y Z : SheafedSpace C} (f : X ⟶ Y) (g : Y ⟶ Z) :\n    (f ≫ g).hom.base = f.hom.base ≫ g.hom.base :=\n  rfl\n\n@[simp]\ntheorem comp_hom_c_app {X Y Z : SheafedSpace C} (α : X ⟶ Y) (β : Y ⟶ Z) (U) :\n    (α ≫ β).hom.c.app U =\n      β.hom.c.app U ≫ α.hom.c.app (op ((Opens.map β.hom.base).obj (unop U))) :=\n  rfl\n\ntheorem comp_hom_c_app' {X Y Z : SheafedSpace C} (α : X ⟶ Y) (β : Y ⟶ Z) (U) :\n    (α ≫ β).hom.c.app (op U) =\n      β.hom.c.app (op U) ≫ α.hom.c.app (op ((Opens.map β.hom.base).obj U)) :=\n  rfl\n\ntheorem congr_hom_app {X Y : SheafedSpace C} {α β : X ⟶ Y} (h : α = β) (U) :\n    α.hom.c.app U = β.hom.c.app U ≫ X.presheaf.map (eqToHom (by subst h; rfl)) :=\n  (PresheafedSpace.congr_app (by rw [h]) U)\n\n@[deprecated (since := \"2025-12-18\")] alias id_base := id_hom_base\n@[deprecated (since := \"2025-12-18\")] alias id_c := id_hom_c\n@[deprecated (since := \"2025-12-18\")] alias id_c_app := id_hom_c_app\n@[deprecated (since := \"2025-12-18\")] alias comp_base := comp_hom_base\n@[deprecated (since := \"2025-12-18\")] alias comp_c_app := comp_hom_c_app\n@[deprecated (since := \"2025-12-18\")] alias comp_c_app' := comp_hom_c_app'\n@[deprecated (since := \"2025-12-18\")] alias congr_app := congr_hom_app\n\nvariable (C)\n\n/-- The forgetful functor from `SheafedSpace` to `Top`. -/\ndef forget : SheafedSpace C ⥤ TopCat where\n  obj X := (X : TopCat)\n  map {_ _} f := f.hom.base\n\nend\n\nopen TopCat.Presheaf\n\n/-- The restriction of a sheafed space along an open embedding into the space.\n-/\ndef restrict {U : TopCat} (X : SheafedSpace C) {f : U ⟶ (X : TopCat)} (h : IsOpenEmbedding f) :\n    SheafedSpace C :=\n  { X.toPresheafedSpace.restrict h with IsSheaf := isSheaf_of_isOpenEmbedding h X.IsSheaf }\n\n/-- The map from the restriction of a presheafed space.\n-/\n@[simps!]\ndef ofRestrict {U : TopCat} (X : SheafedSpace C) {f : U ⟶ (X : TopCat)}\n    (h : IsOpenEmbedding f) : X.restrict h ⟶ X :=\n  InducedCategory.homMk (X.toPresheafedSpace.ofRestrict h)\n\n/-- The restriction of a sheafed space `X` to the top subspace is isomorphic to `X` itself.\n-/\n@[simps! hom inv]\ndef restrictTopIso (X : SheafedSpace C) : X.restrict (Opens.isOpenEmbedding ⊤) ≅ X :=\n  isoMk (X.toPresheafedSpace.restrictTopIso)\n\n/-- The global sections, notated Gamma.\n-/\ndef Γ : (SheafedSpace C)ᵒᵖ ⥤ C :=\n  forgetToPresheafedSpace.op ⋙ PresheafedSpace.Γ\n\ntheorem Γ_def : (Γ : _ ⥤ C) = forgetToPresheafedSpace.op ⋙ PresheafedSpace.Γ :=\n  rfl\n\n@[simp]\ntheorem Γ_obj (X : (SheafedSpace C)ᵒᵖ) : Γ.obj X = (unop X).presheaf.obj (op ⊤) :=\n  rfl\n\ntheorem Γ_obj_op (X : SheafedSpace C) : Γ.obj (op X) = X.presheaf.obj (op ⊤) :=\n  rfl\n\n@[simp]\ntheorem Γ_map {X Y : (SheafedSpace C)ᵒᵖ} (f : X ⟶ Y) : Γ.map f = f.unop.hom.c.app (op ⊤) :=\n  rfl\n\ntheorem Γ_map_op {X Y : SheafedSpace C} (f : X ⟶ Y) : Γ.map f.op = f.hom.c.app (op ⊤) :=\n  rfl\n\nnoncomputable instance (J : Type w) [Category.{w'} J] [Small.{v} J] [HasLimitsOfShape Jᵒᵖ C] :\n    CreatesColimitsOfShape J (forgetToPresheafedSpace : SheafedSpace.{_, _, v} C ⥤ _) :=\n  ⟨fun {K} =>\n    createsColimitOfFullyFaithfulOfIso\n      ⟨(PresheafedSpace.colimitCocone (K ⋙ forgetToPresheafedSpace)).pt,\n        limit_isSheaf _ fun j ↦ Sheaf.pushforward_sheaf_of_sheaf _ (K.obj (unop j)).2⟩\n      (colimit.isoColimitCocone ⟨_, PresheafedSpace.colimitCoconeIsColimit _⟩).symm⟩\n\nnoncomputable instance [HasLimits C] :\n    CreatesColimits (forgetToPresheafedSpace : SheafedSpace C ⥤ _) where\n\ninstance (J : Type w) [Category.{w'} J] [Small.{v} J] [HasLimitsOfShape Jᵒᵖ C] :\n    HasColimitsOfShape J (SheafedSpace.{_, _, v} C) :=\n  hasColimitsOfShape_of_hasColimitsOfShape_createsColimitsOfShape forgetToPresheafedSpace\n\ninstance [HasLimits C] : HasColimits.{v} (SheafedSpace C) where\n\ninstance (J : Type w) [Category.{w'} J] [Small.{v} J] [HasLimitsOfShape Jᵒᵖ C] :\n    PreservesColimitsOfShape J (forget.{_, _, v} C) :=\n  Limits.comp_preservesColimitsOfShape forgetToPresheafedSpace (PresheafedSpace.forget C)\n\nnoncomputable instance [HasLimits C] : PreservesColimits (forget.{_, _, v} C) where\n\nsection ConcreteCategory\n\nvariable {FC : C → C → Type*} {CC : C → Type v} [∀ X Y, FunLike (FC X Y) (CC X) (CC Y)]\nvariable [instCC : ConcreteCategory.{v} C FC] [HasColimits C] [HasLimits C]\nvariable [PreservesLimits (CategoryTheory.forget C)]\nvariable [PreservesFilteredColimits (CategoryTheory.forget C)]\nvariable [(CategoryTheory.forget C).ReflectsIsomorphisms]\n\nattribute [local ext] DFunLike.ext in\ninclude instCC in\nlemma hom_stalk_ext {X Y : SheafedSpace C} (f g : X ⟶ Y) (h : f.hom.base = g.hom.base)\n    (h' : ∀ x, f.hom.stalkMap x = (Y.presheaf.stalkCongr (h ▸ rfl)).hom ≫ g.hom.stalkMap x) :\n    f = g := by\n  obtain ⟨f, fc⟩ := f\n  obtain ⟨g, gc⟩ := g\n  obtain rfl : f = g := h\n  congr\n  ext U s\n  refine section_ext X.sheaf _ _ _ fun x hx ↦\n    show X.presheaf.germ _ x _ _ = X.presheaf.germ _ x _ _ from ?_\n  erw [← PresheafedSpace.stalkMap_germ_apply ⟨f, fc⟩, ← PresheafedSpace.stalkMap_germ_apply ⟨f, gc⟩]\n  simp [h']\n\nattribute [local ext] DFunLike.ext in\ninclude instCC in\nlemma mono_of_base_injective_of_stalk_epi {X Y : SheafedSpace C} (f : X ⟶ Y)\n    (h₁ : Function.Injective f.hom.base)\n    (h₂ : ∀ x, Epi (f.hom.stalkMap x)) : Mono f := by\n  constructor\n  intro Z ⟨g, gc⟩ ⟨h, hc⟩ e\n  obtain rfl : g = h := ConcreteCategory.hom_ext _ _ fun x ↦ h₁ congr(($e).hom.base x)\n  refine SheafedSpace.hom_stalk_ext ⟨g, gc⟩ ⟨g, hc⟩ rfl fun x ↦ ?_\n  rw [← cancel_epi (f.hom.stalkMap (g x)), stalkCongr_hom, stalkSpecializes_refl, Category.id_comp,\n    ← PresheafedSpace.stalkMap.comp ⟨g, gc⟩ f.hom, ← PresheafedSpace.stalkMap.comp ⟨g, hc⟩ f.hom]\n  replace e := congr_arg InducedCategory.Hom.hom e\n  congr 1\n\nattribute [local ext] DFunLike.ext in\ninclude instCC in\n\nTarget:\nlemma epi_of_base_surjective_of_stalk_mono {X Y : SheafedSpace C} (f : X ⟶ Y)\n    (h₁ : Function.Surjective f.hom.base)\n    (h₂ : ∀ x, Mono (f.hom.stalkMap x)) : Epi f :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/RingedSpace","family_id":"epi_of_base_surjective_of_stalk_mono","file_id":"mathlib/Mathlib/Geometry/RingedSpace/SheafedSpace.lean","sample_id":"36f6292b2bea01ba2949cab99df0942afa3d01db01b2c1e30655e18e62dad252"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c1390084cb2580392cbfe88281f49b6df9fd83f38d46b0fc57029acaa73ff3f3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a8f650db8d1b0bf326b19579d79bab9b82094108f000482fb9502388fd0f7567","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f378af96f4b655711415908ce319014735dc5bb779d91dee4115105bae787859","source_sha256":"36745b2e6bef135edaaf082a7696e9ea3c46b2f2bf1d19afbff0b004485d93b6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← span_span_of_tower R, finrank_span_eq_finrank]","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":14,"token_jaccard":0.642857,"token_length_ratio":1.555556},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"c4561dcb2df0eedef243beba495a3027f05dd4a823dd427707c6293ffbc8c6fe","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Epi\npublic import Mathlib.LinearAlgebra.Dimension.StrongRankCondition\npublic import Mathlib.LinearAlgebra.Finsupp.LinearCombination\npublic import Mathlib.LinearAlgebra.Span.Basic\npublic import Mathlib.RingTheory.Flat.Basic\npublic import Mathlib.Combinatorics.Matroid.Init\npublic import Mathlib.Data.Nat.Totient\npublic import Mathlib.Data.Sym.Sym2\npublic import Mathlib.LinearAlgebra.FreeModule.PID\npublic import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition\npublic import Mathlib.Tactic.NormNum.GCD\npublic import Mathlib.Tactic.Positivity\n\nNamespace:\nSubmodule\n\nLocal context:\n/-\nCopyright (c) 2026 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# The interaction of linear span and tensor product for mixed scalars.\n-/\n\n@[expose] public section\n\nopen Function TensorProduct\n\nnamespace Submodule\n\nvariable {R : Type*} (A : Type*) {M : Type*}\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [Algebra R A]\n  [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n  (p : Submodule R M)\n\n/-- If `A` is an `R`-algebra and `p` is an `R`-submodule of an `A`-module `M`, this is the natural\nsurjection `A ⊗[R] p → span A p`.\n\nSee also `Submodule.tensorEquivSpan`. -/\ndef tensorToSpan : A ⊗[R] p →ₗ[A] span A (p : Set M) :=\n  AlgebraTensorModule.lift\n    { toFun a := a • p.inclusionSpan A\n      map_add' a b := add_smul a b _\n      map_smul' a b := smul_assoc a b _ }\n\n@[simp] lemma tensorToSpan_apply_tmul (a : A) (x : p) :\n    p.tensorToSpan A (a ⊗ₜ x) = a • (x : M) :=\n  rfl\n\nlemma surjective_tensorToSpan : Surjective (p.tensorToSpan A) := by\n  intro v\n  obtain ⟨f, hf⟩ := (Finsupp.mem_span_iff_linearCombination _ _ _).mp v.property\n  use f.sum fun x a ↦ a ⊗ₜ x\n  rw [map_finsuppSum, Subtype.ext_iff, ← Submodule.subtype_apply, map_finsuppSum]\n  simpa using! hf\n\nvariable [Algebra.IsEpi R A] [Module.Flat R A]\n\nopen Module.Flat LinearMap in\nlemma injective_tensorToSpan : Injective (p.tensorToSpan A) := by\n  let f : A ⊗[R] (span A (p : Set M)) →ₗ[A] span A (p : Set M) :=\n    AlgebraTensorModule.lift <| (restrictScalarsₗ R A _ _ A) ∘ₗ lsmul A (span A (p : Set M))\n  let g : A ⊗[R] p →ₗ[R] A ⊗[R] span A (p : Set M) := (p.inclusionSpan A).lTensor A\n  have hf : Injective f := Algebra.injective_lift_lsmul R A _\n  have hg : Injective g := lTensor_preserves_injective_linearMap _ (p.injective_inclusionSpan A)\n  have : p.tensorToSpan A = f.restrictScalars R ∘ₗ g := by ext; simp [tensorToSpan, f, g]\n  rw [← LinearMap.coe_restrictScalars R, this, coe_comp]\n  exact hf.comp hg\n\n/-- If `A` is a flat epi `R`-algebra and `p` is an `R`-submodule of an `A`-module `M` then the\nnatural surjection from `A ⊗[R] p` to `span A p` is an equivalence. -/\nnoncomputable def tensorEquivSpan : A ⊗[R] p ≃ₗ[A] span A (p : Set M) :=\n  .ofBijective (p.tensorToSpan A) ⟨p.injective_tensorToSpan A, p.surjective_tensorToSpan A⟩\n\n@[simp] lemma tensorEquivSpan_apply_tmul (a : A) (x : p) :\n    p.tensorEquivSpan A (a ⊗ₜ x) = a • (x : M) :=\n  rfl\n\nvariable (R) in\n/-- If `A` is a flat epi `R`-algebra and `s` is a subset of an `A`-module `M` then the natural\nsurjection from `A ⊗[R] span R s` to `span A s` is an equivalence. -/\nnoncomputable def tensorSpanEquivSpan (s : Set M) : A ⊗[R] span R s ≃ₗ[A] span A s :=\n  ((span R s).tensorEquivSpan A).trans <| .ofEq _ _ <| span_span_of_tower R A s\n\n@[simp] lemma coe_tensorSpanEquivSpan_apply_tmul {s : Set M} (a : A) (x : span R s) :\n    tensorSpanEquivSpan R A s (a ⊗ₜ x) = a • (x : M) :=\n  rfl\n\nend CommSemiring\n\nsection CommRing\n\nopen Module\n\nvariable [CommRing R] [CommRing A] [Nontrivial A]\n  [Algebra R A] [Algebra.IsEpi R A] [Module.Flat R A]\n  [AddCommGroup M] [Module R M] [Module A M] [IsScalarTower R A M]\n  (p : Submodule R M) [Free R p] [Module.Finite R p]\n\n@[simp] lemma finrank_span_eq_finrank :\n    finrank A (span A (p : Set M)) = finrank R p := by\n  rcases subsingleton_or_nontrivial R; · simp [Algebra.subsingleton R A]\n  let ι := Free.ChooseBasisIndex R p\n  let b₁ : Basis ι R p := Free.chooseBasis R p\n  let b₂ : Basis ι A (span A (p : Set M)) := (b₁.baseChange A).map <| p.tensorEquivSpan A\n  rw [finrank_eq_card_basis b₁, finrank_eq_card_basis b₂]\n\nvariable (R) in\n\nTarget:\nlemma finrank_span_eq_finrank_span [IsPrincipalIdealRing R] [IsDomain R] [IsTorsionFree R M]\n    (s : Set M) [Module.Finite R (span R s)] :\n    finrank A (span A s) = finrank R (span R s) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  rw [← span_span_of_tower R, finrank_span_eq_finrank]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Span","family_id":"finrank_span_eq_finrank_span","file_id":"mathlib/Mathlib/LinearAlgebra/Span/TensorProduct.lean","sample_id":"f378af96f4b655711415908ce319014735dc5bb779d91dee4115105bae787859"},"split":"train"}
{"binding":{"chosen_proof_sha256":"99a954ce5f0b66b4d53e9b41552229a51be60d1da33548a2621f7dcf8fed671d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"bb92462adba4b538731ab9f0c55d878b2d44620576e2e8c4eb4477e0f7e42b0d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4fa9f7ffaee479f67dcc3f60683ec20977b8a4ed853209875ba4ba0d88e8525f","source_sha256":"fb00166246598a433f1c5ccafeed443b215760ffb2773f535ee79e2155bf0bd1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  use 1, S.one_mem\n  intro b hb\n  rw [one_smul]\n  obtain ⟨b', b'_mem, rfl⟩ := mem_one.mp (h hb)\n  exact Set.mem_range_self b'","hard_negative":false,"metrics":{"chosen_tokens":35,"rejected_tokens":39,"token_jaccard":0.90625,"token_length_ratio":1.114286},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"c4c6088d2958298470367aebf187ef566d21f267e6f55bcc5ba761f960a9f79e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Localization.Integer\npublic import Mathlib.RingTheory.Localization.Submodule\n\nNamespace:\nFractionalIdeal\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen, Filippo A. E. Nuccio\n-/\n/-!\n# Fractional ideals\n\nThis file defines fractional ideals of an integral domain and proves basic facts about them.\n\n## Main definitions\nLet `S` be a submonoid of an integral domain `R` and `P` the localization of `R` at `S`.\n* `IsFractional` defines which `R`-submodules of `P` are fractional ideals\n* `FractionalIdeal S P` is the type of fractional ideals in `P`\n* a coercion `coeIdeal : Ideal R → FractionalIdeal S P`\n* `CommSemiring (FractionalIdeal S P)` instance:\n  the typical ideal operations generalized to fractional ideals\n* `Lattice (FractionalIdeal S P)` instance\n\n## Main statements\n\n  * the `MulLeftMono` and `MulRightMono` instances state that ideal multiplication is monotone\n  * `mul_div_self_cancel_iff` states that `1 / I` is the inverse of `I` if one exists\n\n## Implementation notes\n\nFractional ideals are considered equal when they contain the same elements,\nindependent of the denominator `a : R` such that `a I ⊆ R`.\nThus, we define `FractionalIdeal` to be the subtype of the predicate `IsFractional`,\ninstead of having `FractionalIdeal` be a structure of which `a` is a field.\n\nMost definitions in this file specialize operations from submodules to fractional ideals,\nproving that the result of this operation is fractional if the input is fractional.\nExceptions to this rule are defining `(+) := (⊔)` and `⊥ := 0`,\nin order to reuse their respective proof terms.\nWe can still use `simp` to show `↑I + ↑J = ↑(I + J)` and `↑⊥ = ↑0`.\n\nMany results in fact do not need that `P` is a localization, only that `P` is an\n`R`-algebra. We omit the `IsLocalization` parameter whenever this is practical.\nSimilarly, we don't assume that the localization is a field until we need it to\ndefine ideal quotients. When this assumption is needed, we replace `S` with `R⁰`,\nmaking the localization a field.\n\n## References\n\n  * https://en.wikipedia.org/wiki/Fractional_ideal\n\n## Tags\n\nfractional ideal, fractional ideals, invertible ideal\n-/\n\n@[expose] public section\n\n\nopen IsLocalization Pointwise nonZeroDivisors\n\nsection Defs\n\nvariable {R : Type*} [CommRing R] {S : Submonoid R} {P : Type*} [CommRing P]\nvariable [Algebra R P]\nvariable (S)\n\n/-- A submodule `I` is a fractional ideal with respect to a submonoid `S`\nif `a I ⊆ R` for some `a ∈ S`. -/\ndef IsFractional (I : Submodule R P) :=\n  ∃ a ∈ S, ∀ b ∈ I, IsInteger R (a • b)\n\nvariable (P)\n\n/-- The fractional ideals of a domain `R` are ideals of `R` divided by some `a ∈ R`.\n\nMore precisely, let `P` be a localization of `R` at some submonoid `S`,\nthen a fractional ideal `I ⊆ P` is an `R`-submodule of `P`,\nsuch that there is an `a ∈ S` with `a I ⊆ R`.\n-/\n@[wikidata Q1497184]\ndef FractionalIdeal :=\n  { I : Submodule R P // IsFractional S I }\n\nend Defs\n\nnamespace FractionalIdeal\n\nopen Set Submodule\n\nvariable {R : Type*} [CommRing R] {S : Submonoid R} {P : Type*} [CommRing P]\nvariable [Algebra R P]\n\n/-- Map a fractional ideal `I` to a submodule by forgetting that `∃ a, a I ⊆ R`.\n\nThis implements the coercion `FractionalIdeal S P → Submodule R P`.\n-/\n@[coe]\ndef coeToSubmodule (I : FractionalIdeal S P) : Submodule R P :=\n  I.val\n\n/-- Map a fractional ideal `I` to a submodule by forgetting that `∃ a, a I ⊆ R`.\n\nThis coercion is typically called `coeToSubmodule` in lemma names\n(or `coe` when the coercion is clear from the context),\nnot to be confused with `IsLocalization.coeSubmodule : Ideal R → Submodule R P`\n(which we use to define `coe : Ideal R → FractionalIdeal S P`).\n-/\ninstance : CoeOut (FractionalIdeal S P) (Submodule R P) :=\n  ⟨coeToSubmodule⟩\n\nprotected theorem isFractional (I : FractionalIdeal S P) : IsFractional S (I : Submodule R P) :=\n  I.prop\n\n/-- An element of `S` such that `I.den • I = I.num`, see `FractionalIdeal.num` and\n`FractionalIdeal.den_mul_self_eq_num`. -/\nnoncomputable def den (I : FractionalIdeal S P) : S :=\n  ⟨I.2.choose, I.2.choose_spec.1⟩\n\n/-- An ideal of `R` such that `I.den • I = I.num`, see `FractionalIdeal.den` and\n`FractionalIdeal.den_mul_self_eq_num`. -/\nnoncomputable def num (I : FractionalIdeal S P) : Ideal R :=\n  (I.den • (I : Submodule R P)).comap (Algebra.linearMap R P)\n\ntheorem den_mul_self_eq_num (I : FractionalIdeal S P) :\n    I.den • (I : Submodule R P) = Submodule.map (Algebra.linearMap R P) I.num := by\n  rw [den, num, Submodule.map_comap_eq]\n  refine (inf_of_le_right ?_).symm\n  rintro _ ⟨a, ha, rfl⟩\n  exact I.2.choose_spec.2 a ha\n\n/-- The linear equivalence between the fractional ideal `I` and the integral ideal `I.num`\ndefined by mapping `x` to `I.den • x`, assuming scalar multiplication by `I.den` is injective. -/\nnoncomputable abbrev equivNumOfIsSMulRegular [FaithfulSMul R P] {I : FractionalIdeal S P}\n    (reg : IsSMulRegular P I.den) : I ≃ₗ[R] I.num := by\n  refine LinearEquiv.trans\n    (LinearEquiv.ofBijective ((DistribSMul.toLinearMap R P I.den).restrict fun _ hx ↦ ?_)\n      ⟨fun _ _ hxy ↦ ?_, fun ⟨y, hy⟩ ↦ ?_⟩)\n    (Submodule.equivMapOfInjective (Algebra.linearMap R P)\n      (FaithfulSMul.algebraMap_injective R P) (num I)).symm\n  · rw [← den_mul_self_eq_num]\n    exact Submodule.smul_mem_pointwise_smul _ _ _ hx\n  · simpa [LinearMap.restrict_apply, reg.eq_iff] using hxy\n  · rw [← den_mul_self_eq_num] at hy\n    obtain ⟨x, hx, hxy⟩ := hy\n    exact ⟨⟨x, hx⟩, by simp_rw [LinearMap.restrict_apply, Subtype.ext_iff, ← hxy]; rfl⟩\n\n/-- The linear equivalence between the fractional ideal `I` and the integral ideal `I.num`\ndefined by mapping `x` to `I.den • x`. -/\nnoncomputable def equivNum [IsDomain R] [Module.IsTorsionFree R P] [Nontrivial P]\n    {I : FractionalIdeal S P} (h_nz : (I.den : R) ≠ 0) : I ≃ₗ[R] I.num :=\n  equivNumOfIsSMulRegular (smul_right_injective P h_nz)\n\n/-- The linear equivalence between the fractional ideal `I` in a faithful localization\nand the integral ideal `I.num`. -/\nnoncomputable def equivNumOfIsLocalization [FaithfulSMul R P] [IsLocalization S P]\n    (I : FractionalIdeal S P) : I ≃ₗ[R] I.num :=\n  equivNumOfIsSMulRegular (smul_bijective ..).1\n\nsection SetLike\n\ninstance : SetLike (FractionalIdeal S P) P where\n  coe I := ↑(I : Submodule R P)\n  coe_injective := SetLike.coe_injective.comp Subtype.coe_injective\n\ninstance : PartialOrder (FractionalIdeal S P) := .ofSetLike (FractionalIdeal S P) P\n\n@[simp]\ntheorem mem_coe {I : FractionalIdeal S P} {x : P} : x ∈ (I : Submodule R P) ↔ x ∈ I :=\n  Iff.rfl\n\n/-- Partially-applied version of `FractionalIdeal.ext`. -/\ntheorem coe_ext {I J : FractionalIdeal S P} : (I : Submodule R P) = (J : Submodule R P) → I = J :=\n  Subtype.ext\n\n/-- Partially-applied version of `FractionalIdeal.ext_iff`. -/\ntheorem coe_ext_iff {I J : FractionalIdeal S P} :\n    I = J ↔ (I : Submodule R P) = (J : Submodule R P) :=\n  Subtype.ext_iff\n\n@[ext]\ntheorem ext {I J : FractionalIdeal S P} : (∀ x, x ∈ I ↔ x ∈ J) → I = J :=\n  SetLike.ext\n\n@[simp]\n theorem equivNum_apply [IsDomain R] [Module.IsTorsionFree R P] [Nontrivial P]\n    {I : FractionalIdeal S P} (h_nz : (I.den : R) ≠ 0) (x : I) :\n    algebraMap R P (equivNum h_nz x) = I.den • x := by\n  change Algebra.linearMap R P _ = _\n  rw [equivNum, LinearEquiv.trans_apply, LinearEquiv.ofBijective_apply, LinearMap.restrict_apply,\n    Submodule.map_equivMapOfInjective_symm_apply, Subtype.coe_mk,\n    DistribSMul.toLinearMap_apply]\n\n/-- Copy of a `FractionalIdeal` with a new underlying set equal to the old one.\nUseful to fix definitional equalities. -/\nprotected def copy (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : FractionalIdeal S P :=\n  ⟨Submodule.copy p s hs, by\n    convert! p.isFractional\n    ext\n    simp only [hs]\n    rfl⟩\n\n@[simp]\ntheorem coe_copy (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : ↑(p.copy s hs) = s :=\n  rfl\n\ntheorem coe_eq (p : FractionalIdeal S P) (s : Set P) (hs : s = ↑p) : p.copy s hs = p :=\n  SetLike.coe_injective hs\n\nend SetLike\n\nlemma zero_mem (I : FractionalIdeal S P) : 0 ∈ I := I.coeToSubmodule.zero_mem\n\n@[simp]\ntheorem val_eq_coe (I : FractionalIdeal S P) : I.val = I :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_mk (I : Submodule R P) (hI : IsFractional S I) :\n    coeToSubmodule ⟨I, hI⟩ = I :=\n  rfl\n\ntheorem coeToSet_coeToSubmodule (I : FractionalIdeal S P) :\n    ((I : Submodule R P) : Set P) = I :=\n  rfl\n\n/-! Transfer instances from `Submodule R P` to `FractionalIdeal S P`. -/\n\ninstance (I : FractionalIdeal S P) : Module R I :=\n  Submodule.module (I : Submodule R P)\n\ntheorem coeToSubmodule_injective :\n    Function.Injective (fun (I : FractionalIdeal S P) ↦ (I : Submodule R P)) :=\n  Subtype.coe_injective\n\ntheorem coeToSubmodule_inj {I J : FractionalIdeal S P} : (I : Submodule R P) = J ↔ I = J :=\n  coeToSubmodule_injective.eq_iff\n\nTarget:\ntheorem isFractional_of_le_one (I : Submodule R P) (h : I ≤ 1) : IsFractional S I :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  use 1, S.one_mem\n  intro b hb\n  rw [one_smul]\n  obtain ⟨b', b'_mem, rfl⟩ := mem_one.mp (h hb)\n  exact Set.mem_range_self b'","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/FractionalIdeal","family_id":"isfractional_of_le_one","file_id":"mathlib/Mathlib/RingTheory/FractionalIdeal/Basic.lean","sample_id":"4fa9f7ffaee479f67dcc3f60683ec20977b8a4ed853209875ba4ba0d88e8525f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"50b6a5e16d3844921370e0938982952c322ae0fcb0f107c81a94eedb5977aaa4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b6c7165c0bd788e389a9bc6e6e492d5bb7a7c434702deaec5881955900d53932","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"70a4170ae953a085b2ee61b70a1cefea9bde7f6d81dae6658409edfa420b8c2e","source_sha256":"d0f59f260198c8e37d3bb0bd17106b617a04bc52ddfffa04abdcd1e14bbcbdf5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← lintegral_sum_measure, h.sum_restrict_of_ac hν]","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":17,"token_jaccard":0.6875,"token_length_ratio":1.416667},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"c544df3040aa49a9fe64a495ebb56bf8731013c66f929a22697d190bca352a6f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Group.Action\npublic import Mathlib.MeasureTheory.Group.Pointwise\npublic import Mathlib.MeasureTheory.Integral.Lebesgue.Map\npublic import Mathlib.MeasureTheory.Integral.Bochner.Set\n\nNamespace:\nMeasureTheory.IsFundamentalDomain\n\nLocal context:\n/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth\n-/\n/-!\n# Fundamental domain of a group action\n\nA set `s` is said to be a *fundamental domain* of an action of a group `G` on a measurable space `α`\nwith respect to a measure `μ` if\n\n* `s` is a measurable set;\n\n* the sets `g • s` over all `g : G` cover almost all points of the whole space;\n\n* the sets `g • s`, are pairwise a.e. disjoint, i.e., `μ (g₁ • s ∩ g₂ • s) = 0` whenever `g₁ ≠ g₂`;\n  we require this for `g₂ = 1` in the definition, then deduce it for any two `g₁ ≠ g₂`.\n\nIn this file we prove that in case of a countable group `G` and a measure-preserving action, any two\nfundamental domains have the same measure, and for a `G`-invariant function, its integrals over any\ntwo fundamental domains are equal to each other.\n\nWe also generate additive versions of all theorems in this file using the `to_additive` attribute.\n\n* We define the `HasFundamentalDomain` typeclass, in particular to be able to define the `covolume`\n  of a quotient of `α` by a group `G`, which under reasonable conditions does not depend on the\n  choice of fundamental domain.\n\n* We define the `QuotientMeasureEqMeasurePreimage` typeclass to describe a situation in which a\n  measure `μ` on `α ⧸ G` can be computed by taking a measure `ν` on `α` of the intersection of the\n  pullback with a fundamental domain.\n\n## Main declarations\n\n* `MeasureTheory.IsFundamentalDomain`: Predicate for a set to be a fundamental domain of the\n  action of a group\n* `MeasureTheory.fundamentalFrontier`: Fundamental frontier of a set under the action of a group.\n  Elements of `s` that belong to some other translate of `s`.\n* `MeasureTheory.fundamentalInterior`: Fundamental interior of a set under the action of a group.\n  Elements of `s` that do not belong to any other translate of `s`.\n-/\n\n@[expose] public section\n\n\nopen scoped ENNReal Pointwise Topology NNReal ENNReal MeasureTheory\n\nopen MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace Filter\n\nnamespace MeasureTheory\n\n/-- A measurable set `s` is a *fundamental domain* for an additive action of an additive group `G`\non a measurable space `α` with respect to a measure `μ` if the sets `g +ᵥ s`, `g : G`, are pairwise\na.e. disjoint and cover the whole space. -/\nstructure IsAddFundamentalDomain (G : Type*) {α : Type*} [Zero G] [VAdd G α] [MeasurableSpace α]\n    (s : Set α) (μ : Measure α := by volume_tac) : Prop where\n  protected nullMeasurableSet : NullMeasurableSet s μ\n  protected ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g +ᵥ x ∈ s\n  protected aedisjoint : Pairwise <| (AEDisjoint μ on fun g : G => g +ᵥ s)\n\n/-- A measurable set `s` is a *fundamental domain* for an action of a group `G` on a measurable\nspace `α` with respect to a measure `μ` if the sets `g • s`, `g : G`, are pairwise a.e. disjoint and\ncover the whole space. -/\n@[to_additive IsAddFundamentalDomain]\nstructure IsFundamentalDomain (G : Type*) {α : Type*} [One G] [SMul G α] [MeasurableSpace α]\n    (s : Set α) (μ : Measure α := by volume_tac) : Prop where\n  protected nullMeasurableSet : NullMeasurableSet s μ\n  protected ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g • x ∈ s\n  protected aedisjoint : Pairwise <| (AEDisjoint μ on fun g : G => g • s)\n\nvariable {G H α β E : Type*}\n\nnamespace IsFundamentalDomain\n\nvariable [Group G] [Group H] [MulAction G α] [MeasurableSpace α] [MulAction H β] [MeasurableSpace β]\n  [NormedAddCommGroup E] {s t : Set α} {μ : Measure α}\n\n/-- If for each `x : α`, exactly one of `g • x`, `g : G`, belongs to a measurable set `s`, then `s`\nis a fundamental domain for the action of `G` on `α`. -/\n@[to_additive /-- If for each `x : α`, exactly one of `g +ᵥ x`, `g : G`, belongs to a measurable set\n`s`, then `s` is a fundamental domain for the additive action of `G` on `α`. -/]\ntheorem mk' (h_meas : NullMeasurableSet s μ) (h_exists : ∀ x : α, ∃! g : G, g • x ∈ s) :\n    IsFundamentalDomain G s μ where\n  nullMeasurableSet := h_meas\n  ae_covers := Eventually.of_forall fun x => (h_exists x).exists\n  aedisjoint a b hab := Disjoint.aedisjoint <| disjoint_left.2 fun x hxa hxb => by\n    rw [mem_smul_set_iff_inv_smul_mem] at hxa hxb\n    exact hab (inv_injective <| (h_exists x).unique hxa hxb)\n\n/-- For `s` to be a fundamental domain, it's enough to check\n`MeasureTheory.AEDisjoint (g • s) s` for `g ≠ 1`. -/\n@[to_additive /-- For `s` to be a fundamental domain, it's enough to check\n  `MeasureTheory.AEDisjoint (g +ᵥ s) s` for `g ≠ 0`. -/]\ntheorem mk'' (h_meas : NullMeasurableSet s μ) (h_ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g • x ∈ s)\n    (h_ae_disjoint : ∀ g, g ≠ (1 : G) → AEDisjoint μ (g • s) s)\n    (h_qmp : ∀ g : G, QuasiMeasurePreserving ((g • ·) : α → α) μ μ) :\n    IsFundamentalDomain G s μ where\n  nullMeasurableSet := h_meas\n  ae_covers := h_ae_covers\n  aedisjoint := pairwise_aedisjoint_of_aedisjoint_forall_ne_one h_ae_disjoint h_qmp\n\n/-- If a measurable space has a finite measure `μ` and a countable group `G` acts\nquasi-measure-preservingly, then to show that a set `s` is a fundamental domain, it is sufficient\nto check that its translates `g • s` are (almost) disjoint and that the sum `∑' g, μ (g • s)` is\nsufficiently large. -/\n@[to_additive\n  /-- If a measurable space has a finite measure `μ` and a countable additive group `G` acts\n  quasi-measure-preservingly, then to show that a set `s` is a fundamental domain, it is sufficient\n  to check that its translates `g +ᵥ s` are (almost) disjoint and that the sum `∑' g, μ (g +ᵥ s)` is\n  sufficiently large. -/]\ntheorem mk_of_measure_univ_le [IsFiniteMeasure μ] [Countable G] (h_meas : NullMeasurableSet s μ)\n    (h_ae_disjoint : ∀ g ≠ (1 : G), AEDisjoint μ (g • s) s)\n    (h_qmp : ∀ g : G, QuasiMeasurePreserving (g • · : α → α) μ μ)\n    (h_measure_univ_le : μ (univ : Set α) ≤ ∑' g : G, μ (g • s)) : IsFundamentalDomain G s μ :=\n  have aedisjoint : Pairwise (AEDisjoint μ on fun g : G => g • s) :=\n    pairwise_aedisjoint_of_aedisjoint_forall_ne_one h_ae_disjoint h_qmp\n  { nullMeasurableSet := h_meas\n    aedisjoint\n    ae_covers := by\n      replace h_meas : ∀ g : G, NullMeasurableSet (g • s) μ := fun g => by\n        rw [← inv_inv g, ← preimage_smul]; exact h_meas.preimage (h_qmp g⁻¹)\n      have h_meas' : NullMeasurableSet {a | ∃ g : G, g • a ∈ s} μ := by\n        rw [← iUnion_smul_eq_setOf_exists]; exact .iUnion h_meas\n      rw [ae_iff_measure_eq h_meas', ← iUnion_smul_eq_setOf_exists]\n      refine le_antisymm (measure_mono <| subset_univ _) ?_\n      rw [measure_iUnion₀ aedisjoint h_meas]\n      exact h_measure_univ_le }\n\n@[to_additive]\ntheorem iUnion_smul_ae_eq (h : IsFundamentalDomain G s μ) : ⋃ g : G, g • s =ᵐ[μ] univ :=\n  eventuallyEq_univ.2 <| h.ae_covers.mono fun _ ⟨g, hg⟩ =>\n    mem_iUnion.2 ⟨g⁻¹, _, hg, inv_smul_smul _ _⟩\n\n@[to_additive]\ntheorem measure_ne_zero [Countable G] [SMulInvariantMeasure G α μ]\n    (hμ : μ ≠ 0) (h : IsFundamentalDomain G s μ) : μ s ≠ 0 := by\n  have hc := measure_univ_pos.mpr hμ\n  contrapose! hc\n  rw [← measure_congr h.iUnion_smul_ae_eq]\n  refine le_trans (measure_iUnion_le _) ?_\n  simp_rw [measure_smul, hc, tsum_zero, le_refl]\n\n@[to_additive]\ntheorem mono (h : IsFundamentalDomain G s μ) {ν : Measure α} (hle : ν ≪ μ) :\n    IsFundamentalDomain G s ν :=\n  ⟨h.1.mono_ac hle, hle h.2, h.aedisjoint.mono fun _ _ h => hle h⟩\n\n@[to_additive]\ntheorem preimage_of_equiv {ν : Measure β} (h : IsFundamentalDomain G s μ) {f : β → α}\n    (hf : QuasiMeasurePreserving f ν μ) {e : G → H} (he : Bijective e)\n    (hef : ∀ g, Semiconj f (e g • ·) (g • ·)) : IsFundamentalDomain H (f ⁻¹' s) ν where\n  nullMeasurableSet := h.nullMeasurableSet.preimage hf\n  ae_covers := (hf.ae h.ae_covers).mono fun x ⟨g, hg⟩ => ⟨e g, by rwa [mem_preimage, hef g x]⟩\n  aedisjoint a b hab := by\n    lift e to G ≃ H using he\n    have : (e.symm a⁻¹)⁻¹ ≠ (e.symm b⁻¹)⁻¹ := by simp [hab]\n    have := (h.aedisjoint this).preimage hf\n    simp only [Semiconj] at hef\n    simpa only [onFun, ← preimage_smul_inv, preimage_preimage, ← hef, e.apply_symm_apply, inv_inv]\n      using this\n\n@[to_additive]\ntheorem image_of_equiv {ν : Measure β} (h : IsFundamentalDomain G s μ) (f : α ≃ β)\n    (hf : QuasiMeasurePreserving f.symm ν μ) (e : H ≃ G)\n    (hef : ∀ g, Semiconj f (e g • ·) (g • ·)) : IsFundamentalDomain H (f '' s) ν := by\n  rw [f.image_eq_preimage_symm]\n  refine h.preimage_of_equiv hf e.symm.bijective fun g x => ?_\n  rcases f.surjective x with ⟨x, rfl⟩\n  rw [← hef _ _, f.symm_apply_apply, f.symm_apply_apply, e.apply_symm_apply]\n\n@[to_additive]\ntheorem pairwise_aedisjoint_of_ac {ν} (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) :\n    Pairwise fun g₁ g₂ : G => AEDisjoint ν (g₁ • s) (g₂ • s) :=\n  h.aedisjoint.mono fun _ _ H => hν H\n\n@[to_additive]\ntheorem smul_of_comm {G' : Type*} [Group G'] [MulAction G' α]\n    [MeasurableConstSMul G' α] [SMulInvariantMeasure G' α μ] [SMulCommClass G' G α]\n    (h : IsFundamentalDomain G s μ) (g : G') : IsFundamentalDomain G (g • s) μ :=\n  h.image_of_equiv (MulAction.toPerm g) (measurePreserving_smul _ _).quasiMeasurePreserving\n    (Equiv.refl _) <| smul_comm g\n\nvariable [MeasurableConstSMul G α] [SMulInvariantMeasure G α μ]\n\n@[to_additive]\ntheorem nullMeasurableSet_smul (h : IsFundamentalDomain G s μ) (g : G) :\n    NullMeasurableSet (g • s) μ :=\n  h.nullMeasurableSet.smul g\n\n@[to_additive]\ntheorem restrict_restrict (h : IsFundamentalDomain G s μ) (g : G) (t : Set α) :\n    (μ.restrict t).restrict (g • s) = μ.restrict (g • s ∩ t) :=\n  restrict_restrict₀ ((h.nullMeasurableSet_smul g).mono restrict_le_self)\n\n@[to_additive]\ntheorem smul (h : IsFundamentalDomain G s μ) (g : G) : IsFundamentalDomain G (g • s) μ :=\n  h.image_of_equiv (MulAction.toPerm g) (measurePreserving_smul _ _).quasiMeasurePreserving\n    ⟨fun g' => g⁻¹ * g' * g, fun g' => g * g' * g⁻¹, fun g' => by simp [mul_assoc], fun g' => by\n      simp [mul_assoc]⟩\n    fun g' x => by simp [smul_smul, mul_assoc]\n\nvariable [Countable G] {ν : Measure α}\n\n@[to_additive]\ntheorem sum_restrict_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) :\n    (sum fun g : G => ν.restrict (g • s)) = ν := by\n  rw [← restrict_iUnion_ae (h.aedisjoint.mono fun i j h => hν h) fun g =>\n      (h.nullMeasurableSet_smul g).mono_ac hν,\n    restrict_congr_set (hν h.iUnion_smul_ae_eq), restrict_univ]\n\n@[to_additive]\n\nTarget:\ntheorem lintegral_eq_tsum_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) (f : α → ℝ≥0∞) :\n    ∫⁻ x, f x ∂ν = ∑' g : G, ∫⁻ x in g • s, f x ∂ν :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  rw [← lintegral_sum_measure, h.sum_restrict_of_ac hν]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Group","family_id":"lintegral_eq_tsum_of_ac","file_id":"mathlib/Mathlib/MeasureTheory/Group/FundamentalDomain.lean","sample_id":"70a4170ae953a085b2ee61b70a1cefea9bde7f6d81dae6658409edfa420b8c2e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3816114db1c077f621bfeb2d3b1b98b7b671ab19074e0d158d715632b3696809","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9584eeedec12f1270f773e97e07803ac6bace4ed2fabdce1f1ae60147a31ccc0","source_sha256":"351582fbc976816e329c33b0cb83e6e30d58c72551427bea563b172b7814b639","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro x y hxy\n  let xy : G.obj (pullback π π) := (PreservesPullback.iso G π π).inv <|\n    (TopCat.pullbackIsoProdSubtype (G.map π) (G.map π)).inv ⟨(x, y), hxy⟩\n  have ha' := congr_fun ha xy\n  dsimp at ha'\n  have h₁ : ∀ y, G.map (pullback.fst _ _) ((PreservesPullback.iso G π π).inv y) =\n      pullback.fst (G.map π) (G.map π) y := by\n    simp only [← PreservesPullback.iso_inv_fst]; intro y; rfl\n  have h₂ : ∀ y, G.map (pullback.snd _ _) ((PreservesPullback.iso G π π).inv y) =\n      pullback.snd (G.map π) (G.map π) y := by\n    simp only [← PreservesPullback.iso_inv_snd]; intro y; rfl\n  rw [h₁, h₂, TopCat.pullbackIsoProdSubtype_inv_fst_apply,\n    TopCat.pullbackIsoProdSubtype_inv_snd_apply] at ha'\n  simpa using ha'","hard_negative":true,"metrics":{"chosen_tokens":211,"rejected_tokens":8,"token_jaccard":0.068966,"token_length_ratio":0.037915},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"c55a0667ce6320a7d8c219565f1f502e4fb0077c0ef37bea82fdc312238df32d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Preserves.Opposites\npublic import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison\npublic import Mathlib.Condensed.Basic\npublic import Mathlib.Topology.Category.TopCat.Yoneda\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# The functor from topological spaces to condensed sets\n\nThis file builds on the API from the file `TopCat.Yoneda`. If the forgetful functor to `TopCat` has\nnice properties, like preserving pullbacks and finite coproducts, then this Yoneda presheaf\nsatisfies the sheaf condition for the regular and extensive topologies respectively.\n\nWe apply this API to `CompHaus` and define the functor\n`topCatToCondensedSet : TopCat.{u + 1} ⥤ CondensedSet.{u}`.\n\n-/\n\n@[expose] public section\n\nuniverse w w' v u\n\nopen CategoryTheory Opposite Limits regularTopology ContinuousMap Topology\n\nvariable {C : Type u} [Category.{v} C] (G : C ⥤ TopCat.{w})\n  (X : Type w') [TopologicalSpace X]\n\n/--\nAn auxiliary lemma to that allows us to use `IsQuotientMap.lift` in the proof of\n`equalizerCondition_yonedaPresheaf`.\n-/\n\nTarget:\ntheorem factorsThrough_of_pullbackCondition {Z B : C} {π : Z ⟶ B} [HasPullback π π]\n    [PreservesLimit (cospan π π) G]\n    {a : C(G.obj Z, X)}\n    (ha : a ∘ (G.map (pullback.fst _ _)) = a ∘ (G.map (pullback.snd π π))) :\n    Function.FactorsThrough a (G.map π) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"0b618490eacd78b02138e83eb99423e432e36ab7122ae0c28c40876a46f14a1a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed","family_id":"factorsthrough_of_pullbackcondition","file_id":"mathlib/Mathlib/Condensed/TopComparison.lean","sample_id":"9584eeedec12f1270f773e97e07803ac6bace4ed2fabdce1f1ae60147a31ccc0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"be2b6932dda4e0b3c56c94860ddf8d756e2f7715633872f9e4a77499f7decf69","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0700ede7788b015baae01a8e4aa3cd674212af76354c06e79abb4f1dca42ce86","source_sha256":"7fb54be1edb74454dd74d15ac42b7739b2a4aa95bc5649191f339f2298801887","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [cmpUsing, Ordering.ite_eq_gt_distrib, if_false_right, and_true, if_false_left,\n    and_iff_right_iff_imp, reduceCtorEq]\n  exact fun hba hab ↦ (irrefl a) (_root_.trans hab hba)","hard_negative":false,"metrics":{"chosen_tokens":36,"rejected_tokens":3,"token_jaccard":0.074074,"token_length_ratio":0.083333},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"c5a023fbc19f95476990a981d665d46620209eb34c022a1e6a0f22a28eb17604","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Ordering.Basic\npublic import Mathlib.Order.Defs.Unbundled\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Some `Ordering` lemmas\n-/\n\npublic section\n\nuniverse u\n\nnamespace Ordering\n\n@[simp]\ntheorem ite_eq_lt_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.lt) = if c then a = Ordering.lt else b = Ordering.lt := by\n  by_cases c <;> simp [*]\n\n@[simp]\ntheorem ite_eq_eq_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.eq) = if c then a = Ordering.eq else b = Ordering.eq := by\n  by_cases c <;> simp [*]\n\n@[simp]\ntheorem ite_eq_gt_distrib (c : Prop) [Decidable c] (a b : Ordering) :\n    ((if c then a else b) = Ordering.gt) = if c then a = Ordering.gt else b = Ordering.gt := by\n  by_cases c <;> simp [*]\n\n@[simp]\nlemma dthen_eq_then (o₁ o₂ : Ordering) : o₁.dthen (fun _ => o₂) = o₁.then o₂ := by\n  cases o₁ <;> rfl\n\nend Ordering\n\nsection\n\nvariable {α : Type u} {lt : α → α → Prop} [DecidableRel lt]\n\nattribute [local simp] cmpUsing\n\n@[simp]\ntheorem cmpUsing_eq_lt (a b : α) : (cmpUsing lt a b = Ordering.lt) = lt a b := by\n  simp only [cmpUsing, Ordering.ite_eq_lt_distrib, ite_self, if_false_right, and_true, reduceCtorEq]\n\n@[simp]\n\nTarget:\ntheorem cmpUsing_eq_gt [IsStrictOrder α lt] (a b : α) : cmpUsing lt a b = Ordering.gt ↔ lt b a :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Ordering","family_id":"cmpusing_eq_gt","file_id":"mathlib/Mathlib/Data/Ordering/Lemmas.lean","sample_id":"0700ede7788b015baae01a8e4aa3cd674212af76354c06e79abb4f1dca42ce86"},"split":"train"}
{"binding":{"chosen_proof_sha256":"197871194b11ccd2f7b68c20098508f4fb8d58251817640d3a8e7c9a33767c9c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"56c83df4fcbbc0fb6cb913621257dbef79183303e8f7b47c11db70c5826d5d2a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"41bafb9f17936ff03795282fab01c1edc2ec5df90a2dc068b3d884811a5b8b02","source_sha256":"ced30280014822262039ce197c9e4ae821d789623b770f4b0e6844d4852109a3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h _ _ ↦ by rw [h], fun h ↦ ?_⟩\n  rw [DFunLike.ext'_iff]\n  apply Function.Surjective.injective_comp_right mkAlgHom_surjective\n  simpa [← AlgHom.coe_comp] using MvPolynomial.algHom_ext fun ⟨n, m⟩ ↦ h n m","hard_negative":false,"metrics":{"chosen_tokens":54,"rejected_tokens":58,"token_jaccard":0.909091,"token_length_ratio":1.074074},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"c5cee8610374bccd198e38f2c84561f0425d3c89b6e6c0dbec3968cf6c6cff7f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.Eval\npublic import Mathlib.Algebra.RingQuot\npublic import Mathlib.RingTheory.DividedPowers.Basic\n\nNamespace:\nDividedPowerAlgebra\n\nLocal context:\n/-\nCopyright (c) 2026 Antoine Chambert-Loir & María Inés de Frutos—Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir, María Inés de Frutos—Fernández\n-/\n/-!\n# The universal divided power algebra\n\nLet `R` be a (commutative) semiring and `M` be an `R`-module. In this file we define `Γ_R(M)`,\nthe universal divided power algebra of `M`, as the ring quotient of the polynomial ring\nin the variables `ℕ × M` by the relation `DividedPowerAlgebra.Rel`.\n\n`DividedPowerAlgebra R M` satisfies a weak universal property for morphisms to rings with\ndivided powers (`DividedPowerAlgebra.lift`).\n\n## Main definitions\n\n* `DividedPowerAlgebra.Rel`: the type coding the basic relations that will give rise to the\n  divided power algebra.\n\n* `DividedPowerAlgebra R M`: the universal divided power algebra of the `R`-module `M`,\n  defined as `RingQuot` of `DividedPowerAlgebra.Rel R M`.\n\n* `DividedPowerAlgebra.dp R n m`: for `n : ℕ` and `m : M`, this is the equivalence class of\n  `MvPolynomial.X (⟨n, m⟩)` in `DividedPowerAlgebra R M`.\n\n  When that algebra is endowed with its canonical divided power structure (to be defined),\n  the image of `MvPolynomial.X (n, m)`, for any `n : ℕ` and `m : M`, is equal to\n  the `n`th divided power of the image of `m`.\n\n  The API will be setup so that it is never (never say never…) necessary to lift to `MvPolynomial`.\n\n* `DividedPowerAlgebra.lift`: the weak universal property of `DividedPowerAlgebra R M`.\n\n* `DividedPowerAlgebra.map`: the functoriality map between divided power algebras\n  associated with a linear map of the underlying modules.\n  Given an `R`-algebra `S`, an `S`-module `N` and an `R`-linear map `f : M →ₗ[R] N`,\n  this is the map `DividedPowerAlgebra R M →ₐ[R] DividedPowerAlgebra S N`\n  sending `dp R n m` to `dp S n (f m)`.\n\n## References\n\n* [P. Berthelot (1974), *Cohomologie cristalline des schémas de\n  caractéristique $p$ > 0*][Berthelot-1974]\n\n* [P. Berthelot and A. Ogus (1978), *Notes on crystalline\n  cohomology*][BerthelotOgus-1978]\n\n* [N. Roby (1963), *Lois polynomes et lois formelles en théorie des\n  modules*][Roby-1963]\n\n* [N. Roby (1965), *Les algèbres à puissances dividées*][Roby-1965]\n\n## TODO\n\n* Show in upcoming files that `DividedPowerAlgebra R M` has divided powers.\n\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Finset Ideal MvPolynomial RingQuot\n\nvariable (R M : Type*) [CommSemiring R] [AddCommMonoid M] [Module R M]\n\nnamespace DividedPowerAlgebra\n\n/-- The type coding the basic relations that will give rise to the divided power algebra.\n  The class of `MvPolynomial.X (n, a)` will be equal to `dpow n a`, for `a ∈ M`. -/\ninductive Rel : MvPolynomial (ℕ × M) R → MvPolynomial (ℕ × M) R → Prop\n  | rfl_zero : Rel 0 0 -- Needed for technical reasons.\n  | zero {a : M} : Rel (X (0, a)) 1\n  | smul {r : R} {n : ℕ} {a : M} : Rel (X (n, r • a)) (r ^ n • X (n, a))\n  | mul {m n : ℕ} {a : M} : Rel (X (m, a) * X (n, a)) (Nat.choose (m + n) m • X (m + n, a))\n  | add {n : ℕ} {a b : M} :\n    Rel (X (n, a + b)) ((Finset.antidiagonal n).sum fun k ↦ X (k.1, a) * X (k.2, b))\n\n/-- The ideal of `MvPolynomial (ℕ × M) R` generated by `Rel`. -/\ndef RelI : Ideal (MvPolynomial (ℕ × M) R) := ofRel (DividedPowerAlgebra.Rel R M)\n\nend DividedPowerAlgebra\n\n/-- The divided power algebra of a module M is defined as the ring quotient of the polynomial ring\n  in the variables `ℕ × M` by the ring relation defined by `DividedPowerAlgebra.Rel`.\n  We will later show that that `DividedPowerAlgebra R M` has divided powers.\n  It satisfies a weak universal property for morphisms to rings with divided powers. -/\nabbrev DividedPowerAlgebra := RingQuot (DividedPowerAlgebra.Rel R M)\n\nnamespace DividedPowerAlgebra\n\nopen MvPolynomial\n\nvariable {R M}\n\nlemma mkAlgHom_surjective : Function.Surjective (mkAlgHom R (Rel R M)) :=\n  RingQuot.mkAlgHom_surjective _ _\n\nlemma mkAlgHom_C (a : R) :\n    mkAlgHom R (Rel R M) (C a) = algebraMap R (DividedPowerAlgebra R M) a := by\n  rw [← MvPolynomial.algebraMap_eq, AlgHom.commutes]\n\nlemma mkRingHom_C (a : R) :\n    mkRingHom (Rel R M) (C a) = algebraMap R (DividedPowerAlgebra R M) a := by\n  rw [← mkAlgHom_C, mkAlgHom, AlgHom.coe_mk]\n\nvariable (R) in\n/-- `dp R n m` is the equivalence class of `X (⟨n, m⟩)` in `DividedPowerAlgebra R M`. -/\ndef dp (n : ℕ) (m : M) : DividedPowerAlgebra R M := mkAlgHom R (Rel R M) (X ⟨n, m⟩)\n\ntheorem dp_def (n : ℕ) (m : M) :\n  dp R n m = mkAlgHom R (Rel R M) (X ⟨n, m⟩) := rfl\n\nprotected theorem induction_on' {P : DividedPowerAlgebra R M → Prop} (f : DividedPowerAlgebra R M)\n    (h_C : ∀ a, P (mkAlgHom R (Rel R M) (C a))) (h_add : ∀ f g, P f → P g → P (f + g))\n    (h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)) : P f := by\n  obtain ⟨F, hf⟩ := RingQuot.mkRingHom_surjective (DividedPowerAlgebra.Rel R M) f\n  rw [← hf]\n  induction F using MvPolynomial.induction_on generalizing f with\n  | C a =>\n      convert! h_C a using 1\n      rw [mkAlgHom, AlgHom.coe_mk]\n  | add g1 g2 hg1 hg2 =>\n      rw [map_add]\n      exact h_add _ _ (hg1 ((mkRingHom (Rel R M)) g1) rfl) (hg2 ((mkRingHom (Rel R M)) g2) rfl)\n  | mul_X g nm h =>\n      have h' : (mkRingHom (Rel R M)) (X nm) = dp R nm.1 nm.2 := by\n        simp only [dp_def, Prod.mk.eta, mkAlgHom, AlgHom.coe_mk]\n      rw [_root_.map_mul, h']\n      exact h_dp _ _ _ (h (mkRingHom (Rel R M) g) rfl)\n\n@[elab_as_elim]\nprotected theorem induction_on {P : DividedPowerAlgebra R M → Prop} (f : DividedPowerAlgebra R M)\n    (C : ∀ a, P (algebraMap R _ a)) (add : ∀ f g, P f → P g → P (f + g))\n    (dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)) : P f :=\n  DividedPowerAlgebra.induction_on' f (fun a ↦ by rw [mkAlgHom_C]; exact C a) add dp\n\ntheorem dp_eq_mkRingHom (n : ℕ) (m : M) :\n    dp R n m = mkRingHom (Rel R M) (X (⟨n, m⟩)) := by\n  simp [dp, mkRingHom, mkAlgHom]\n\ntheorem dp_zero {m : M} : dp R 0 m = 1 := by\n  rw [dp_def, ← map_one (mkAlgHom R (Rel R M))]\n  exact RingQuot.mkAlgHom_rel R Rel.zero\n\ntheorem dp_smul {r : R} {n : ℕ} {m : M} : dp R n (r • m) = r ^ n • dp R n m := by\n  rw [dp_def, dp_def, ← map_smul]\n  exact mkAlgHom_rel R Rel.smul\n\ntheorem dp_null {n : ℕ} : dp R n (0 : M) = if n = 0 then 1 else 0 := by\n  cases Nat.eq_zero_or_pos n with\n  | inl hn =>\n    rw [if_pos hn, hn, dp_zero]\n  | inr hn =>\n    rw [if_neg (ne_of_gt hn), ← zero_smul R (0 : M), dp_smul]\n    rw [zero_pow (Nat.pos_iff_ne_zero.mp hn), zero_smul]\n\ntheorem dp_null_of_ne_zero {n : ℕ} (hn : n ≠ 0) : dp R n (0 : M) = 0 := by\n  rw [dp_null, if_neg hn]\n\ntheorem dp_mul {n p : ℕ} {m : M} :\n    dp R n m * dp R p m = (n + p).choose n • dp R (n + p) m := by\n  simp only [dp_def, ← _root_.map_mul, ← map_nsmul]\n  exact mkAlgHom_rel R Rel.mul\n\ntheorem dp_add {n : ℕ} {x y : M} :\n    dp R n (x + y) = (antidiagonal n).sum fun k ↦ dp R k.1 x * dp R k.2 y := by\n  simp only [dp_def]\n  rw [mkAlgHom_rel (A := MvPolynomial (ℕ × M) R) R Rel.add, map_sum,\n    Finset.sum_congr rfl (fun k _ ↦ by rw [_root_.map_mul])]\n\ntheorem dp_sum {ι : Type*} [DecidableEq ι] (s : Finset ι) (q : ℕ) (x : ι → M) :\n    dp R q (s.sum x) =\n      (Finset.sym s q).sum fun k ↦ s.prod fun i ↦ dp R (Multiset.count i k) (x i) :=\n  DividedPowers.dpow_sum' (I := ⊤) _ (fun _ ↦ dp_zero)\n    (fun _ _ ↦ dp_add) dp_null_of_ne_zero (fun _ _ ↦ trivial)\n\ntheorem dp_sum_smul {ι : Type*} [DecidableEq ι] (s : Finset ι) (q : ℕ) (a : ι → R) (x : ι → M) :\n    dp R q (s.sum fun i ↦ a i • x i) =\n      (Finset.sym s q).sum fun k ↦\n        (s.prod fun i ↦ a i ^ Multiset.count i k) •\n          s.prod fun i ↦ dp R (Multiset.count i k) (x i) := by\n  simp_rw [dp_sum, dp_smul, Algebra.smul_def, map_prod, ← Finset.prod_mul_distrib]\n\nopen Nat in\nlemma prod_dp {ι : Type*} {s : Finset ι} {n : ι → ℕ} {m : M} :\n    ∏ i ∈ s, (dp R (n i) m) = (Nat.multinomial s n) * dp R (s.sum n) m := by\n  classical\n  induction s using Finset.induction with\n  | empty =>\n    simp only [prod_empty, multinomial_empty, cast_one, sum_empty, one_mul, dp_zero]\n  | insert _ _ hi hrec =>\n    rw [prod_insert hi, hrec, ← mul_assoc, mul_comm (dp R (n _) m),\n      mul_assoc, dp_mul, ← sum_insert hi, nsmul_eq_mul, ← mul_assoc]\n    congr 1\n    rw [multinomial_insert hi, mul_comm, cast_mul, sum_insert hi]\n\nopen scoped Nat\n\ntheorem natFactorial_mul_dp_eq (n : ℕ) (x : M) :\n    n ! * dp R n x = (dp R 1 x) ^ n := by\n  induction n with\n  | zero => simp [dp_zero]\n  | succ n h =>\n    rw [pow_succ, ← h, mul_assoc, dp_mul, nsmul_eq_mul, ← mul_assoc, ← Nat.cast_mul]\n    simp [mul_comm _ (n + 1), Nat.factorial_succ]\n\nvariable (R M) in\n/-- The canonical linear map `M →ₗ[R] DividedPowerAlgebra R M`. -/\ndef embed : M →ₗ[R] DividedPowerAlgebra R M where\n  toFun m := dp R 1 m\n  map_add' _ _ := by simp [dp_add, Nat.antidiagonal_succ, dp_zero, add_comm]\n  map_smul' _ _ := by simp [dp_smul, pow_one, RingHom.id_apply]\n\ntheorem embed_def (m : M) : embed R M m = dp R 1 m := rfl\n\nTarget:\ntheorem algHom_ext_iff {A : Type*} [CommSemiring A] [Algebra R A]\n    {f g : DividedPowerAlgebra R M →ₐ[R] A} :\n    f = g ↔ ∀ n m, f (dp R n m) = g (dp R n m) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  refine ⟨fun h _ _ ↦ by rw [h], fun h ↦ ?_⟩\n  rw [DFunLike.ext'_iff]\n  apply Function.Surjective.injective_comp_right mkAlgHom_surjective\n  simpa [← AlgHom.coe_comp] using MvPolynomial.algHom_ext fun ⟨n, m⟩ ↦ h n m","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/DividedPowerAlgebra","family_id":"alghom_ext_iff","file_id":"mathlib/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean","sample_id":"41bafb9f17936ff03795282fab01c1edc2ec5df90a2dc068b3d884811a5b8b02"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7333d1a239a893fe44382cfe648dc46c728566269ad1ec5413c976d55e86e914","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9fe351be3c3d262f5458bdcaa60348aa5ed6ae16e7dd76f937e22c5914fe4df1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dabe8f52856a803fb511e285a4cf7be82ea7a0120495bcf033465a0c9f492e10","source_sha256":"3ebf52bd4f56640ff82f3663b47b89c207fb7b51b21d5bfbbe3cfb7edd34865e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa only [Ordinal.liftPrincipalSeg_coe, lift_ord, lift_succ, ord_le, Order.succ_le_iff] using!\n    le_of_lt (Ordinal.liftPrincipalSeg.{u, u + 1}.lt_top (Order.succ c).ord)","hard_negative":true,"metrics":{"chosen_tokens":44,"rejected_tokens":3,"token_jaccard":0.032258,"token_length_ratio":0.068182},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"c5fde899450b8eb48b93cd0927762d7b2ee700ea0a4c3dcba306ea1eac1e48a0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.SetTheory.Ordinal.Basic\n\nNamespace:\nCardinal\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Floris van Doorn\n-/\n/-!\n# Universal ordinal and cardinal\n\n`Cardinal.univ` is the cardinality of the cardinals themselves. Likewise, `Ordinal.univ` is the\norder type of the ordinals. These are related via `Cardinal.univ.ord = Ordinal.univ` and\n`Ordinal.univ.card = Cardinal.univ`.\n\nThe cardinal `Cardinal.univ` is strongly inaccessible. This reflects the fact that in ZFC, the\ncardinals form a proper class. See `IsInaccessible.univ` for a proof.\n\n## Implementation notes\n\nWe actually define `Cardinal.univ` as the cardinality of `Ordinal`, rather than that of `Cardinal`.\nThis makes the basic API easier to set up. See `Cardinal.mk_cardinal` for a proof that\n`Cardinal.univ = #Cardinal`.\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nset_option linter.checkUnivs false in\nopen Ordinal in\n-- intended to be used with explicit universe parameters\n/-- The ordinal `univ.{u, v}` is the order type of `Ordinal.{u}` or `Cardinal.{u}`, as an element of\n`Ordinal.{v}` (when `u < v`). -/\n@[pp_with_univ, nolint checkUnivs]\ndef Ordinal.univ : Ordinal.{max (u + 1) v} :=\n  lift.{v, u + 1} (typeLT Ordinal)\n\nset_option linter.checkUnivs false in\nopen Cardinal in\n-- intended to be used with explicit universe parameters\n/-- The cardinal `univ.{u, v}` is the cardinality of `Ordinal.{u}` or `Cardinal.{u}`, as an element\nof `Cardinal.{v}` (when `u < v`). -/\n@[pp_with_univ, nolint checkUnivs]\ndef Cardinal.univ : Cardinal.{max (u + 1) v} :=\n  lift.{v, u + 1} #Ordinal\n\n/-! ### Universal ordinal -/\n\nnamespace Ordinal\n\n@[simp]\ntheorem type_lt_ordinal : typeLT Ordinal = univ.{u, u + 1} :=\n  (lift_id _).symm\n\n@[deprecated type_lt_ordinal (since := \"2026-03-20\")]\ntheorem univ_id : univ.{u, u + 1} = typeLT Ordinal :=\n  lift_id _\n\n@[simp]\ntheorem lift_univ : lift.{w} univ.{u, v} = univ.{u, max v w} :=\n  lift_lift _\n\ntheorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=\n  congr_fun lift_umax _\n\n/-- Principal segment version of the lift operation on ordinals, embedding `Ordinal.{u}` in\n`Ordinal.{v}` as a principal segment when `u < v`. -/\ndef liftPrincipalSeg : Ordinal.{u} <i Ordinal.{max (u + 1) v} :=\n  ⟨liftInitialSeg.{max (u + 1) v, u}, univ.{u, v}, by\n    refine fun b => inductionOn b ?_; intro β s _\n    rw [univ, ← lift_umax]; constructor <;> intro h\n    · obtain ⟨a, e⟩ := h\n      rw [← e]\n      refine inductionOn a ?_\n      intro α r _\n      exact lift_type_lt.{u, u + 1, max (u + 1) v}.2 ⟨typein r⟩\n    · rw [← lift_id (type s)] at h ⊢\n      obtain ⟨f⟩ := lift_type_lt.{_,_,v}.1 h\n      obtain ⟨f, a, hf⟩ := f\n      exists a\n      induction a using inductionOn with | type α r\n      refine lift_type_eq.{u, max (u + 1) v, max (u + 1) v}.2\n        ⟨(RelIso.ofSurjective (RelEmbedding.ofMonotone ?_ ?_) ?_).symm⟩\n      · exact fun b => enum r ⟨f b, (hf _).1 ⟨_, rfl⟩⟩\n      · refine fun a b h => (typein_lt_typein r).1 ?_\n        rw [typein_enum, typein_enum]\n        exact f.map_rel_iff.2 h\n      · intro a'\n        obtain ⟨b, e⟩ := (hf _).2 (typein_lt_type _ a')\n        exists b\n        simp only [RelEmbedding.ofMonotone_coe]\n        simp [e]⟩\n\n@[simp]\ntheorem liftPrincipalSeg_coe :\n    (liftPrincipalSeg.{u, v} : Ordinal → Ordinal) = lift.{max (u + 1) v} :=\n  rfl\n\n@[simp]\ntheorem liftPrincipalSeg_top : (liftPrincipalSeg.{u, v}).top = univ.{u, v} :=\n  rfl\n\n@[deprecated liftPrincipalSeg_top (since := \"2026-03-20\")]\ntheorem liftPrincipalSeg_top' : liftPrincipalSeg.{u, u + 1}.top = typeLT Ordinal := by\n  simp\n\n@[simp]\ntheorem card_univ : card univ.{u, v} = Cardinal.univ.{u, v} :=\n  rfl\n\nend Ordinal\n\n/-! ### Universal cardinal -/\n\nnamespace Cardinal\n\n@[simp]\ntheorem mk_ordinal : #Ordinal = univ.{u, u + 1} :=\n  (lift_id _).symm\n\n@[deprecated mk_ordinal (since := \"2026-04-22\")]\ntheorem univ_id : univ.{u, u + 1} = #Ordinal :=\n  lift_id _\n\n@[simp]\ntheorem lift_univ : lift.{w} univ.{u, v} = univ.{u, max v w} :=\n  lift_lift _\n\ntheorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=\n  congr_fun lift_umax _\n\nTarget:\ntheorem lift_lt_univ (c : Cardinal) : lift.{u + 1, u} c < univ.{u, u + 1} :=\n\nProof body:\n","rejected":"by\n  exact lift_lt_univ","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"dd1078bf964280d36c3d51095d2a3f4371dcbe8fe09875bd684346014d99989b","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Ordinal","family_id":"lift_lt_univ","file_id":"mathlib/Mathlib/SetTheory/Ordinal/Univ.lean","sample_id":"dabe8f52856a803fb511e285a4cf7be82ea7a0120495bcf033465a0c9f492e10"},"split":"train"}
{"binding":{"chosen_proof_sha256":"768478b669f1b0c48360ff5cb94ef6db06e9b1b6432c6287e4928bb1cb2132ad","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b506c66ce42848c4a4ad0e476f485d2b07d3f08b618fa2c4247e2085d2d34222","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2ad48a41750f9ee7416f3ed47f9d4bdc606c25a12a288f7161359dab4138e50a","source_sha256":"a9f42fb2a650677b7f31f134a9fdb63cf294a003ee4e030eb51993a29af692d7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [eq_comm, ← ltb_cons_addChar' c]\n  simp","hard_negative":true,"metrics":{"chosen_tokens":10,"rejected_tokens":5,"token_jaccard":0.071429,"token_length_ratio":0.5},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"c6187b5e167e246c8b350825a72b760f2cbbe887323447151567e861c05e2737","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.String.Lemmas\npublic import Mathlib.Data.List.Lex\npublic import Mathlib.Data.Char\npublic import Mathlib.Algebra.Order.Group.Nat\nimport all Init.Data.String.Iterator  -- for unfolding `Iterator.curr`\nimport all Init.Data.Ord.String  -- for unfolding `String.compare`\n\nNamespace:\nString\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Strings\n\nSupplementary theorems about the `String` type.\n-/\n\n@[expose] public section\n\nnamespace String\n\n/-- `<` on string iterators. This coincides with `<` on strings as lists. -/\ndef ltb (s₁ s₂ : Legacy.Iterator) : Bool :=\n  if s₂.hasNext then\n    if s₁.hasNext then\n      if s₁.curr = s₂.curr then\n        ltb s₁.next s₂.next\n      else s₁.curr < s₂.curr\n    else true\n  else false\n\n/-- Induction on `String.ltb`. -/\n@[no_expose] def ltb.inductionOn.{u} {motive : Legacy.Iterator → Legacy.Iterator → Sort u}\n    (it₁ it₂ : Legacy.Iterator)\n    (ind : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      i₁.get s₁ = i₂.get s₂ →\n        motive (Legacy.Iterator.next ⟨s₁, i₁⟩) (Legacy.Iterator.next ⟨s₂, i₂⟩) →\n          motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (eq : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      ¬ i₁.get s₁ = i₂.get s₂ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (base₁ : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → ¬ Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (base₂ : ∀ s₁ s₂ i₁ i₂, ¬ Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩) :\n    motive it₁ it₂ :=\n  if h₂ : it₂.hasNext then\n    if h₁ : it₁.hasNext then\n      if heq : it₁.curr = it₂.curr then\n        ind it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq (inductionOn it₁.next it₂.next ind eq base₁ base₂)\n      else eq it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq\n    else base₁ it₁.s it₂.s it₁.i it₂.i h₂ h₁\n  else base₂ it₁.s it₂.s it₁.i it₂.i h₂\n\ntheorem ltb_cons_addChar' (c : Char) (s₁ s₂ : Legacy.Iterator) :\n    ltb ⟨ofList (c :: s₁.s.toList), s₁.i + c⟩ ⟨ofList (c :: s₂.s.toList), s₂.i + c⟩ =\n      ltb s₁ s₂ := by\n  fun_induction ltb s₁ s₂ with\n  | case1 s₁ s₂ h₁ h₂ h ih =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_pos (by simpa using h₂), if_pos, ← ih]\n    · simp only [Legacy.Iterator.next, Pos.Raw.next, get_cons_addChar, ofList_toList]\n      congr 2 <;> apply Pos.Raw.add_char_right_comm\n    · simpa only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList] using h\n  | case2 s₁ s₂ h₁ h₂ h =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_pos (by simpa using h₂), if_neg]\n    · simp only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList, decide_eq_decide]\n    · simpa only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList] using h\n  | case3 s₁ s₂ h₁ h₂ =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_neg (by simpa using h₂)]\n  | case4 s₁ s₂ h₁ =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, if_neg (by simpa using h₁)]\n\nTarget:\ntheorem ltb_cons_addChar (c : Char) (cs₁ cs₂ : List Char) (i₁ i₂ : Pos.Raw) :\n    ltb ⟨ofList (c :: cs₁), i₁ + c⟩ ⟨ofList (c :: cs₂), i₂ + c⟩ =\n      ltb ⟨ofList cs₁, i₁⟩ ⟨ofList cs₂, i₂⟩ :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_2ad48a41750f","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c112f948655427f7e212f06151c0a653e2a7a34517ec7d039ee6fde2a4b16326","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/String","family_id":"ltb_cons_addchar","file_id":"mathlib/Mathlib/Data/String/Basic.lean","sample_id":"2ad48a41750f9ee7416f3ed47f9d4bdc606c25a12a288f7161359dab4138e50a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"18e5d95a99a28ddbd089bbdf5652dbad7467c92bf9a278d6d749571d6703ac97","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6019d53cd390889e024e24ee9eef9da9a2c6ea21be7042d8d912c199aa84c2f6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8c92966f8a3dcfb46f7baa7761df8474960f52637455e799bcbf00ab2a68364d","source_sha256":"a4df2964a0a0195a94022ab24e77297c35474e0a98e2a52eb57d2203822fac2d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [descFactorial_eq_prod_range, factorial_eq_prod_range_add_one]\n  simp only [cast_prod]\n  nth_rw 2 [← card_range n]\n  rw [pow_card_mul_prod]\n  refine prod_congr rfl ?_\n  intro x hx\n  rw [← tsub_add_eq_tsub_tsub_swap,\n    Nat.cast_sub <| Nat.le_trans (Nat.add_one_le_iff.mpr (List.mem_range.mp hx)) h,\n    CharP.cast_eq_zero, zero_sub, cast_succ, neg_add_rev, mul_add, neg_mul, one_mul,\n    mul_one, add_comm]","hard_negative":false,"metrics":{"chosen_tokens":81,"rejected_tokens":88,"token_jaccard":0.960784,"token_length_ratio":1.08642},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"c6528826cec183f7207e67d577463e9fb38e134b4532f1ff420bc68845aedfe8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.Factorial.BigOperators\npublic import Mathlib.Data.ZMod.Basic\n\nNamespace:\nZMod\n\nLocal context:\n/-\nCopyright (c) 2023 Moritz Firsching. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Moritz Firsching\n-/\n/-!\n# Facts about factorials in ZMod\n\nWe collect facts about factorials in context of modular arithmetic.\n\n## Main statements\n\n* `ZMod.cast_descFactorial`: For natural numbers `n` and `p`, where `n` is less than or equal to `p`\n  the descending factorial of `(p - 1)` taken `n` times modulo `p` equals `(-1) ^ n * n!`.\n\n## See also\n\nFor the prime case and involving `factorial` rather than `descFactorial`, see Wilson's theorem:\n* `Nat.prime_iff_fac_equiv_neg_one`\n\n-/\n\npublic section\n\nassert_not_exists TwoSidedIdeal\n\nopen Finset Nat\n\nnamespace ZMod\n\nTarget:\ntheorem cast_descFactorial {n p : ℕ} (h : n ≤ p) :\n    (descFactorial (p - 1) n : ZMod p) = (-1) ^ n * n ! :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [descFactorial_eq_prod_range, factorial_eq_prod_range_add_one]\n  simp only [cast_prod]\n  nth_rw 2 [← card_range n]\n  rw [pow_card_mul_prod]\n  refine prod_congr rfl ?_\n  intro x hx\n  rw [← tsub_add_eq_tsub_tsub_swap,\n    Nat.cast_sub <| Nat.le_trans (Nat.add_one_le_iff.mpr (List.mem_range.mp hx)) h,\n    CharP.cast_eq_zero, zero_sub, cast_succ, neg_add_rev, mul_add, neg_mul, one_mul,\n    mul_one, add_comm]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/ZMod","family_id":"cast_descfactorial","file_id":"mathlib/Mathlib/Data/ZMod/Factorial.lean","sample_id":"8c92966f8a3dcfb46f7baa7761df8474960f52637455e799bcbf00ab2a68364d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3628ed809b6ca5e99b1c81051989cfa58d947fe9238f4a04564fff2b92ccbda9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0b655641f0fbcc76ef7401981695bc0221f56804bae473d2205adf3e718b621a","source_sha256":"b36923632397960a715546258f21a9493171dd1ee07ef68d71942c93bbc53c1e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨W, hW, hWl, hWr, h⟩ := h\n  have e₁ := isDominant_hom_iff_isDominant_restrict_hom f W hW hWl\n  have e₂ := isDominant_hom_iff_isDominant_restrict_hom g W hW hWr\n  dsimp only [restrict_domain, restrict_hom] at ⊢ e₁ e₂ h\n  rw [e₁, h, ← e₂]","hard_negative":true,"metrics":{"chosen_tokens":58,"rejected_tokens":8,"token_jaccard":0.09375,"token_length_ratio":0.137931},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"c6cf8abfbcde9d0c3809197297df66581ef505eaf88921dcf253c59169be4837","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Birational.RationalMap\n\nNamespace:\nAlgebraicGeometry.Scheme.PartialMap\n\nLocal context:\n/-\nCopyright (c) 2026 Justus Springer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Justus Springer\n-/\n/-!\n\n# Dominant rational maps\n\nThis file defines `RationalMap.IsDominant` and establishes its connection to\n`IsDominant` on the underlying partial maps.\n\n## Main definition\n\n- `Scheme.RationalMap.IsDominant`: a rational map is dominant if some (equivalently, any)\n  representative partial map has dominant underlying morphism.\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen CategoryTheory\n\nnamespace AlgebraicGeometry\n\nvariable {X Y : Scheme.{u}}\n\nnamespace Scheme\n\nnamespace PartialMap\n\nset_option backward.defeqAttrib.useBackward true in\n/-- Restricting a dominant partial map to a dense open yields a dominant partial map. -/\nlemma isDominant_restrict_hom (f : X.PartialMap Y) [IsDominant f.hom] (U : X.Opens)\n    (hU : Dense (U : Set X)) (hU' : U ≤ f.domain) : IsDominant (f.restrict U hU hU').hom := by\n  dsimp only [restrict_domain, restrict_hom]\n  have : IsDominant (X.homOfLE hU') := Opens.isDominant_homOfLE hU hU'\n  rwa [IsDominant.comp_iff]\n\n/-- If a restriction of `f` is dominant, then `f` is dominant. -/\nlemma isDominant_hom_of_isDominant_restrict_hom (f : X.PartialMap Y) (U : X.Opens)\n    (hU : Dense (U : Set X)) (hU' : U ≤ f.domain) [H : IsDominant (f.restrict U hU hU').hom] :\n    IsDominant f.hom :=\n  IsDominant.of_comp (X.homOfLE hU') f.hom (H := H)\n\n/-- `f.hom` is dominant iff any restriction of `f` is. -/\nlemma isDominant_hom_iff_isDominant_restrict_hom (f : X.PartialMap Y) (U : X.Opens)\n    (hU : Dense (U : Set X)) (hU' : U ≤ f.domain) :\n    IsDominant f.hom ↔ IsDominant (f.restrict U hU hU').hom :=\n  ⟨fun _ ↦ f.isDominant_restrict_hom U hU hU',\n    fun _ ↦ f.isDominant_hom_of_isDominant_restrict_hom U hU hU'⟩\n\nset_option backward.defeqAttrib.useBackward true in\n/-- Dominance of the underlying morphism is invariant under equivalence of partial maps. -/\n\nTarget:\nlemma isDominant_hom_iff_of_equiv (f g : X.PartialMap Y) (h : f.equiv g) :\n    IsDominant f.hom ↔ IsDominant g.hom :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"08fe55745aee9ead66f239113c1f502f1051027dadf5aeb75b805ea75370f88f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Birational","family_id":"isdominant_hom_iff_of_equiv","file_id":"mathlib/Mathlib/AlgebraicGeometry/Birational/Dominant.lean","sample_id":"0b655641f0fbcc76ef7401981695bc0221f56804bae473d2205adf3e718b621a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"aedbd85f784d006a3b8e0a1c284ce353d4c9554e1c007c5246c151cde67cde07","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b463641116e31fe41b30a3ddc8948cc4710e80378eb58481fecf2b2ecfdbe470","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a3caeca3aca5af45210b187259a35456fb858a420f6704c94186f1b14ec00a25","source_sha256":"898a445ef5a6164c4919dd80610d649e8321c8a06f3edd7c9d6402f5554964c9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  exact ⟨fun h ↦ ⟨⟨@effectiveEpiFamilyStructOfEffectiveEpiDesc _ _ _ _ X π _ h _ _ (fun g ↦\n    (FinitaryPreExtensive.isIso_sigmaDesc_fst (fun a ↦ Sigma.ι X a) g inferInstance).epi_of_iso)⟩⟩,\n    fun _ ↦ inferInstance⟩\n\nvariable {D : Type*} [Category* D] [FinitaryPreExtensive D]\nvariable (F : C ⥤ D) [PreservesFiniteCoproducts F]","hard_negative":false,"metrics":{"chosen_tokens":80,"rejected_tokens":85,"token_jaccard":0.904762,"token_length_ratio":1.0625},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"c6d45450d1ab0aaf2f040125f71b0f33e9dba281c490657d38512ae8c6546d2c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.EffectiveEpi.Preserves\npublic import Mathlib.CategoryTheory.EffectiveEpi.Coproduct\npublic import Mathlib.CategoryTheory.Extensive\npublic import Mathlib.CategoryTheory.Limits.Preserves.Finite\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# Preserving and reflecting effective epis on extensive categories\n\nWe prove that a functor between `FinitaryPreExtensive` categories preserves (resp. reflects) finite\neffective epi families if it preserves (resp. reflects) effective epis.\n-/\n\npublic section\n\nnamespace CategoryTheory\n\nopen Limits\n\nvariable {C : Type*} [Category* C] [FinitaryPreExtensive C]\n\nTarget:\ntheorem effectiveEpi_desc_iff_effectiveEpiFamily {α : Type} [Finite α]\n    {B : C} (X : α → C) (π : (a : α) → X a ⟶ B) :\n    EffectiveEpi (Sigma.desc π) ↔ EffectiveEpiFamily X π :=\n\nProof body:\n","rejected":"by\n  exact ⟨fun h ↦ ⟨⟨@effectiveEpiFamilyStructOfEffectiveEpiDesc _ _ _ _ X π _ h _ _ (fun g ↦\n    (FinitaryPreExtensive.isIso_sigmaDesc_fst (fun a ↦ Sigma.ι X a) g inferInstance).epi_of_iso)⟩⟩,\n    fun _ ↦ inferInstance⟩\n\nvariable {D : Type*} [Category* D] [FinitaryPreExtensive D]\nvariable (F : C ⥤ D) [PreservesFiniteCoproducts F]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/EffectiveEpi","family_id":"effectiveepi_desc_iff_effectiveepifamily","file_id":"mathlib/Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean","sample_id":"a3caeca3aca5af45210b187259a35456fb858a420f6704c94186f1b14ec00a25"},"split":"train"}
{"binding":{"chosen_proof_sha256":"83fc6e2c4e62e7f25a2d9e2ffddabb7d5875dc78574f118790a50383327ea178","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"30d6aae9bfaa67acea75dfbb91ae0fa520b66296064e76c798d32752d9029399","source_sha256":"d2377d09b51f8862ecd1e9604b616865bc29923d42ea22ca4ee787791f41d5e7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨n, hn : ∀ m, n ≤ m → LinearMap.ker (f ^ n) = LinearMap.ker (f ^ m)⟩ :=\n    monotone_stabilizes_iff_noetherian.mpr inferInstance f.iterateKer\n  refine eventually_atTop.mpr ⟨n, fun m hm ↦ disjoint_iff.mpr ?_⟩\n  rw [← hn _ hm, Submodule.eq_bot_iff]\n  rintro - ⟨hx, ⟨x, rfl⟩⟩\n  apply pow_map_zero_of_le hm\n  replace hx : x ∈ LinearMap.ker (f ^ (n + m)) := by\n    simpa [f.pow_apply n, f.pow_apply m, ← f.pow_apply (n + m), ← iterate_add_apply] using hx\n  rwa [← hn _ (n.le_add_right m)] at hx","hard_negative":true,"metrics":{"chosen_tokens":142,"rejected_tokens":8,"token_jaccard":0.05,"token_length_ratio":0.056338},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"c702dddbd722a5f62b3a894af81033a47a71d2864dcd1dcd0fecb0ed476ae025","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Filter.AtTopBot.Basic\npublic import Mathlib.RingTheory.Finiteness.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Kevin Buzzard\n-/\n/-!\n# Noetherian rings and modules\n\nThe following are equivalent for a module M over a ring R:\n1. Every increasing chain of submodules M₁ ⊆ M₂ ⊆ M₃ ⊆ ⋯ eventually stabilises.\n2. Every submodule is finitely generated.\n\nA module satisfying these equivalent conditions is said to be a *Noetherian* R-module.\nA ring is a *Noetherian ring* if it is Noetherian as a module over itself.\n\n(Note that we do not assume yet that our rings are commutative,\nso perhaps this should be called \"left-Noetherian\".\nTo avoid cumbersome names once we specialize to the commutative case,\nwe don't make this explicit in the declaration names.)\n\n## Main definitions\n\nLet `R` be a ring and let `M` and `P` be `R`-modules. Let `N` be an `R`-submodule of `M`.\n\n* `IsNoetherian R M` is the proposition that `M` is a Noetherian `R`-module. It is a class,\n  implemented as the predicate that all `R`-submodules of `M` are finitely generated.\n\n## Main statements\n\n* `isNoetherian_iff` is the theorem that an R-module M is Noetherian iff `>` is well-founded on\n  `Submodule R M`.\n\nNote that the Hilbert basis theorem, that if a commutative ring R is Noetherian then so is R[X],\nis proved in `RingTheory.Polynomial`.\n\n## References\n\n* [M. F. Atiyah and I. G. Macdonald, *Introduction to commutative algebra*][atiyah-macdonald]\n* [P. Samuel, *Algebraic Theory of Numbers*][samuel1967]\n\n## Tags\n\nNoetherian, noetherian, Noetherian ring, Noetherian module, noetherian ring, noetherian module\n\n-/\n\n@[expose] public section\n\nassert_not_exists Finsupp.linearCombination Matrix Pi.basis\n\nopen Set Pointwise\n\n/-- `IsNoetherian R M` is the proposition that `M` is a Noetherian `R`-module,\nimplemented as the predicate that all `R`-submodules of `M` are finitely generated.\n-/\n-- TODO: should this be renamed to `Noetherian`?\nclass IsNoetherian (R M) [Semiring R] [AddCommMonoid M] [Module R M] : Prop where\n  noetherian : ∀ s : Submodule R M, s.FG\n\nattribute [inherit_doc IsNoetherian] IsNoetherian.noetherian\n\nsection\n\nvariable {R : Type*} {M : Type*} {P : Type*}\nvariable [Semiring R] [AddCommMonoid M] [AddCommMonoid P]\nvariable [Module R M] [Module R P]\n\nopen IsNoetherian\n\n/-- An R-module is Noetherian iff all its submodules are finitely-generated. -/\ntheorem isNoetherian_def : IsNoetherian R M ↔ ∀ s : Submodule R M, s.FG :=\n  ⟨fun h => h.noetherian, IsNoetherian.mk⟩\n\ntheorem isNoetherian_submodule {N : Submodule R M} :\n    IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG := by\n  refine ⟨fun ⟨hn⟩ => fun s hs =>\n    have : s ≤ LinearMap.range N.subtype := N.range_subtype.symm ▸ hs\n    Submodule.map_comap_eq_self this ▸ (hn _).map _,\n    fun h => ⟨fun s => ?_⟩⟩\n  specialize h (s.map N.subtype) (Submodule.map_subtype_le N s)\n  exact Submodule.fg_of_fg_map_injective N.subtype Subtype.val_injective h\n\ntheorem isNoetherian_submodule_left {N : Submodule R M} :\n    IsNoetherian R N ↔ ∀ s : Submodule R M, (N ⊓ s).FG :=\n  isNoetherian_submodule.trans ⟨fun H _ => H _ inf_le_left, fun H _ hs => inf_of_le_right hs ▸ H _⟩\n\ntheorem isNoetherian_submodule_right {N : Submodule R M} :\n    IsNoetherian R N ↔ ∀ s : Submodule R M, (s ⊓ N).FG :=\n  isNoetherian_submodule.trans ⟨fun H _ => H _ inf_le_right, fun H _ hs => inf_of_le_left hs ▸ H _⟩\n\ninstance isNoetherian_submodule' [IsNoetherian R M] (N : Submodule R M) : IsNoetherian R N :=\n  isNoetherian_submodule.2 fun _ _ => IsNoetherian.noetherian _\n\ntheorem isNoetherian_of_le {s t : Submodule R M} [ht : IsNoetherian R t] (h : s ≤ t) :\n    IsNoetherian R s :=\n  isNoetherian_submodule.mpr fun _ hs' => isNoetherian_submodule.mp ht _ (le_trans hs' h)\n\nend\n\nopen IsNoetherian Submodule Function\n\nsection\n\nuniverse w\n\nvariable {R M P : Type*} {N : Type w} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N]\n  [Module R N] [AddCommMonoid P] [Module R P]\n\ntheorem isNoetherian_iff' : IsNoetherian R M ↔ WellFoundedGT (Submodule R M) := by\n  refine .trans ?_ ((CompleteLattice.wellFoundedGT_characterisations <| Submodule R M).out 0 3).symm\n  exact\n    ⟨fun ⟨h⟩ => fun k => (fg_iff_compact k).mp (h k), fun h =>\n      ⟨fun k => (fg_iff_compact k).mpr (h k)⟩⟩\n\ntheorem isNoetherian_iff :\n    IsNoetherian R M ↔ WellFounded ((· > ·) : Submodule R M → Submodule R M → Prop) := by\n  rw [isNoetherian_iff', ← isWellFounded_iff]\n\nalias ⟨IsNoetherian.wf, _⟩ := isNoetherian_iff\n\nalias ⟨IsNoetherian.wellFoundedGT, isNoetherian_mk⟩ := isNoetherian_iff'\n\ninstance wellFoundedGT [h : IsNoetherian R M] : WellFoundedGT (Submodule R M) :=\n  h.wellFoundedGT\n\ntheorem isNoetherian_iff_fg_wellFounded :\n    IsNoetherian R M ↔ WellFoundedGT { N : Submodule R M // N.FG } := by\n  let α := { N : Submodule R M // N.FG }\n  constructor\n  · intro H\n    let f : α ↪o Submodule R M := OrderEmbedding.subtype _\n    exact OrderEmbedding.wellFoundedLT f.dual\n  · intro H\n    constructor\n    intro N\n    obtain ⟨⟨N₀, h₁⟩, e : N₀ ≤ N, h₂⟩ :=\n      WellFounded.has_min H.wf { N' : α | N'.1 ≤ N } ⟨⟨⊥, Submodule.fg_bot⟩, @bot_le _ _ _ N⟩\n    convert! h₁\n    refine (e.antisymm ?_).symm\n    by_contra h₃\n    obtain ⟨x, hx₁ : x ∈ N, hx₂ : x ∉ N₀⟩ := Set.not_subset.mp h₃\n    apply hx₂\n    rw [eq_of_le_of_not_lt (le_sup_right : N₀ ≤ _) (h₂\n      ⟨_, Submodule.FG.sup ⟨{x}, by rw [Finset.coe_singleton]⟩ h₁⟩ <|\n      sup_le ((Submodule.span_singleton_le_iff_mem _ _).mpr hx₁) e)]\n    exact (le_sup_left : R ∙ x ≤ _) (Submodule.mem_span_singleton_self _)\n\n/-- A module is Noetherian iff every nonempty set of submodules has a maximal submodule among them.\n-/\ntheorem set_has_maximal_iff_noetherian :\n    (∀ a : Set <| Submodule R M, a.Nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬M' < I) ↔ IsNoetherian R M := by\n  rw [isNoetherian_iff, WellFounded.wellFounded_iff_has_min]\n\n/-- A module is Noetherian iff every increasing chain of submodules stabilizes. -/\ntheorem monotone_stabilizes_iff_noetherian :\n    (∀ f : ℕ →o Submodule R M, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsNoetherian R M := by\n  rw [isNoetherian_iff', wellFoundedGT_iff_monotone_chain_condition]\n\nvariable [IsNoetherian R M]\n\nopen Filter\n/-- For an endomorphism of a Noetherian module, any sufficiently large iterate has disjoint kernel\nand range. -/\n\nTarget:\ntheorem Module.End.eventually_disjoint_ker_pow_range_pow (f : End R M) :\n    ∀ᶠ n in atTop, Disjoint (LinearMap.ker (f ^ n)) (LinearMap.range (f ^ n)) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"f558a8bc29b2dda1c8db968c2e595d63e502ecf0d40a46e52a1cc5d336b9c083","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Noetherian","family_id":"module","file_id":"mathlib/Mathlib/RingTheory/Noetherian/Defs.lean","sample_id":"30d6aae9bfaa67acea75dfbb91ae0fa520b66296064e76c798d32752d9029399"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fc005df75a037e9e437ea57ce191d036c65586027b4016497e6ed6b62c40e9f3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"eb5890534d5da8e62ba89ed4014fd0be99dd1d4b563a4215ecc1fa91874f5077","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9f7fcfe93c14fb87931e79553027be989fd1cf1216f6e4592f76a5cfd75311b1","source_sha256":"5efd5852b126afb746be3b592d500b1af5de2649a35029fd57ec7bbb27770fcb","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp only [LiftP, LiftP']\n  apply exists_iff_exists_of_mono F (f _ n α) (g _ n α)\n  · ext i ⟨x, _⟩\n    cases i <;> rfl\n  · intros\n    rw [MvFunctor.map_map]\n    dsimp +unfoldPartialApp [(· ⊚ ·)]\n    suffices (fun i => Subtype.val) = (fun i x => (MvFunctor.f P n α i x).val) by rw [this]\n    ext i ⟨x, _⟩\n    cases i <;> rfl\n\nvariable (rr : β → β → Prop)","hard_negative":true,"metrics":{"chosen_tokens":111,"rejected_tokens":5,"token_jaccard":0.037037,"token_length_ratio":0.045045},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"c73ca49f94f9c68cc0d5ff46f43be466703c44aad78feba9649fbedafefb41e9","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.Fin2\npublic import Mathlib.Data.TypeVec\npublic import Mathlib.Logic.Equiv.Defs\n\nNamespace:\nMvFunctor\n\nLocal context:\n/-\nCopyright (c) 2018 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Mario Carneiro, Simon Hudon\n-/\n/-!\n\n# Functors between the category of tuples of types, and the category Type\n\nFeatures:\n\n* `MvFunctor n` : the type class of multivariate functors\n* `f <$$> x`    : notation for map\n\n-/\n\n@[expose] public section\n\n\nuniverse u v w\n\nopen MvFunctor\n\n/-- Multivariate functors, i.e. functor between the category of type vectors\nand the category of Type -/\nclass MvFunctor {n : ℕ} (F : TypeVec n → Type*) where\n  /-- Multivariate map, if `f : α ⟹ β` and `x : F α` then `f <$$> x : F β`. -/\n  map : ∀ {α β : TypeVec n}, α ⟹ β → F α → F β\n\n/-- Multivariate map, if `f : α ⟹ β` and `x : F α` then `f <$$> x : F β` -/\nscoped[MvFunctor] infixr:100 \" <$$> \" => MvFunctor.map\n\nvariable {n : ℕ}\n\nnamespace MvFunctor\n\nvariable {α β : TypeVec.{u} n} {F : TypeVec.{u} n → Type v} [MvFunctor F]\n\n/-- predicate lifting over multivariate functors -/\ndef LiftP {α : TypeVec n} (P : ∀ i, α i → Prop) (x : F α) : Prop :=\n  ∃ u : F (fun i => Subtype (P i)), (fun i => @Subtype.val _ (P i)) <$$> u = x\n\n/-- relational lifting over multivariate functors -/\ndef LiftR {α : TypeVec n} (R : ∀ ⦃i⦄, α i → α i → Prop) (x y : F α) : Prop :=\n  ∃ u : F (fun i => { p : α i × α i // R p.fst p.snd }),\n    (fun i (t : { p : α i × α i // R p.fst p.snd }) => t.val.fst) <$$> u = x ∧\n      (fun i (t : { p : α i × α i // R p.fst p.snd }) => t.val.snd) <$$> u = y\n\n/-- given `x : F α` and a projection `i` of type vector `α`, `supp x i` is the set\nof `α.i` contained in `x` -/\ndef supp {α : TypeVec n} (x : F α) (i : Fin2 n) : Set (α i) :=\n  { y : α i | ∀ ⦃P⦄, LiftP P x → P i y }\n\ntheorem of_mem_supp {α : TypeVec n} {x : F α} {P : ∀ ⦃i⦄, α i → Prop} (h : LiftP P x) (i : Fin2 n) :\n    ∀ y ∈ supp x i, P y := fun _y hy => hy h\n\nend MvFunctor\n\n\n\n/-- laws for `MvFunctor` -/\nclass LawfulMvFunctor {n : ℕ} (F : TypeVec n → Type*) [MvFunctor F] : Prop where\n  /-- `map` preserved identities, i.e., maps identity on `α` to identity on `F α` -/\n  id_map : ∀ {α : TypeVec n} (x : F α), TypeVec.id <$$> x = x\n  /-- `map` preserves compositions -/\n  comp_map :\n    ∀ {α β γ : TypeVec n} (g : α ⟹ β) (h : β ⟹ γ) (x : F α), (h ⊚ g) <$$> x = h <$$> g <$$> x\n\nopen Nat TypeVec\n\nnamespace MvFunctor\n\nexport LawfulMvFunctor (comp_map)\n\nopen LawfulMvFunctor\n\nvariable {α β γ : TypeVec.{u} n}\nvariable {F : TypeVec.{u} n → Type v} [MvFunctor F]\nvariable (P : α ⟹ «repeat» n Prop) (R : α ⊗ α ⟹ «repeat» n Prop)\n\n/-- adapt `MvFunctor.LiftP` to accept predicates as arrows -/\ndef LiftP' : F α → Prop :=\n  MvFunctor.LiftP fun i x => ofRepeat <| P i x\n\n\n/-- adapt `MvFunctor.LiftR` to accept relations as arrows -/\ndef LiftR' : F α → F α → Prop :=\n  MvFunctor.LiftR @fun i x y => ofRepeat <| R i <| TypeVec.prod.mk _ x y\n\nvariable [LawfulMvFunctor F]\n\n@[simp]\ntheorem id_map (x : F α) : TypeVec.id <$$> x = x :=\n  LawfulMvFunctor.id_map x\n\n@[simp]\ntheorem id_map' (x : F α) : (fun _i a => a) <$$> x = x :=\n  id_map x\n\ntheorem map_map (g : α ⟹ β) (h : β ⟹ γ) (x : F α) : h <$$> g <$$> x = (h ⊚ g) <$$> x :=\n  Eq.symm <| comp_map _ _ _\n\nsection LiftP'\n\nvariable (F) in\ntheorem exists_iff_exists_of_mono {P : F α → Prop} {q : F β → Prop}\n    (f : α ⟹ β) (g : β ⟹ α)\n    (h₀ : f ⊚ g = TypeVec.id)\n    (h₁ : ∀ u : F α, P u ↔ q (f <$$> u)) :\n    (∃ u : F α, P u) ↔ ∃ u : F β, q u := by\n  constructor <;> rintro ⟨u, h₂⟩\n  · refine ⟨f <$$> u, ?_⟩\n    apply (h₁ u).mp h₂\n  · refine ⟨g <$$> u, ?_⟩\n    rw [h₁]\n    simp only [MvFunctor.map_map, h₀, LawfulMvFunctor.id_map, h₂]\n\ntheorem LiftP_def (x : F α) : LiftP' P x ↔ ∃ u : F (Subtype_ P), subtypeVal P <$$> u = x :=\n  exists_iff_exists_of_mono F _ _ (toSubtype_of_subtype P) (by simp [MvFunctor.map_map])\n\ntheorem LiftR_def (x y : F α) :\n    LiftR' R x y ↔\n      ∃ u : F (Subtype_ R),\n        (TypeVec.prod.fst ⊚ subtypeVal R) <$$> u = x ∧\n          (TypeVec.prod.snd ⊚ subtypeVal R) <$$> u = y :=\n  exists_iff_exists_of_mono _ _ _ (toSubtype'_of_subtype' R) (by\n    simp only [map_map, comp_assoc, subtypeVal_toSubtype']\n    simp +unfoldPartialApp [comp])\n\nend LiftP'\n\nend MvFunctor\n\nnamespace MvFunctor\n\nsection LiftPLastPredIff\n\nvariable {F : TypeVec.{u} (n + 1) → Type*} [MvFunctor F] [LawfulMvFunctor F] {α : TypeVec.{u} n}\n\nvariable {β : Type u}\nvariable (pp : β → Prop)\n\nprivate def f :\n    ∀ n α,\n      (fun i : Fin2 (n + 1) => { p_1 // ofRepeat (PredLast' α pp i p_1) }) ⟹ fun i : Fin2 (n + 1) =>\n        { p_1 : (α ::: β) i // PredLast α pp p_1 }\n  | _, α, Fin2.fs i, x =>\n    ⟨x.val, cast (by grind [PredLast]) x.property⟩\n  | _, _, Fin2.fz, x => ⟨x.val, x.property⟩\n\nprivate def g :\n    ∀ n α,\n      (fun i : Fin2 (n + 1) => { p_1 : (α ::: β) i // PredLast α pp p_1 }) ⟹ fun i : Fin2 (n + 1) =>\n        { p_1 // ofRepeat (PredLast' α pp i p_1) }\n  | _, α, Fin2.fs i, x =>\n    ⟨x.val, cast (by simp only [PredLast]; erw [const_iff_true]) x.property⟩\n  | _, _, Fin2.fz, x => ⟨x.val, x.property⟩\n\nTarget:\ntheorem LiftP_PredLast_iff {β} (P : β → Prop) (x : F (α ::: β)) :\n    LiftP' (PredLast' _ P) x ↔ LiftP (PredLast _ P) x :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_9f7fcfe93c14","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"30dd9fdd086dd9549026b8bf8a622901404ac2baa283626e4fafd465303a6cef","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control/Functor","family_id":"liftp_predlast_iff","file_id":"mathlib/Mathlib/Control/Functor/Multivariate.lean","sample_id":"9f7fcfe93c14fb87931e79553027be989fd1cf1216f6e4592f76a5cfd75311b1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a956e0393455f70ac3f0634d07a69fa1a653d99212eafe809fbf92b858839bc8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a95656ffd38538e1dc9aa485eecac2357b1c568e62017ed23bbaa4a78e647d87","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a1347b35a1da940b0cf7bb9fa19ec8fb67a45c4aec52bd9f946f2c2f04d77b83","source_sha256":"33b47846d8ad6e41763ee1834aea3dbb18288085daf7f82ed174fd1d15b2a25b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · induction fs generalizing t with\n    | nil =>\n      simp only [sequence, mem_pure, imp_self, forall₂_nil_left_iff, exists_eq_left, Set.pure_def,\n        singleton_subset_iff, traverse_nil]\n    | cons b fs ih =>\n      intro ht\n      rcases mem_seq_iff.1 ht with ⟨u, hu, v, hv, ht⟩\n      rcases mem_map_iff_exists_image.1 hu with ⟨w, hw, hwu⟩\n      rcases ih v hv with ⟨us, hus, hu⟩\n      exact ⟨w::us, Forall₂.cons hw hus, (Set.seq_mono hwu hu).trans ht⟩\n  · rintro ⟨us, hus, hs⟩\n    exact mem_of_superset (mem_traverse _ _ hus) hs","hard_negative":true,"metrics":{"chosen_tokens":127,"rejected_tokens":2,"token_jaccard":0.016393,"token_length_ratio":0.015748},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"c741a92a9b54e4bf90f0ef20c42e6e02337cbeca59644b8e832659f0ab5427d5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Control.Traversable.Instances\npublic import Mathlib.Order.Filter.Map\n\nNamespace:\nFilter\n\nLocal context:\n/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Properties of `Traversable.traverse` on `List`s and `Filter`s\n\nIn this file we prove basic properties (monotonicity, membership)\nfor `Traversable.traverse f l`, where `f : β → Filter α` and `l : List β`.\n-/\n\npublic section\n\nopen Set List\n\nnamespace Filter\n\nuniverse u\n\nvariable {α β γ : Type u} {f : β → Filter α} {s : γ → Set α}\n\ntheorem sequence_mono : ∀ as bs : List (Filter α), Forall₂ (· ≤ ·) as bs → sequence as ≤ sequence bs\n  | [], [], Forall₂.nil => le_rfl\n  | _::as, _::bs, Forall₂.cons h hs => seq_mono (map_mono h) (sequence_mono as bs hs)\n\nTarget:\ntheorem mem_traverse :\n    ∀ (fs : List β) (us : List γ),\n      Forall₂ (fun b c => s c ∈ f b) fs us → traverse s us ∈ traverse f fs\n  | [], [], Forall₂.nil => mem_pure.2 <| mem_singleton _\n  | _::fs, _::us, Forall₂.cons h hs => seq_mem_seq (image_mem_map h) (mem_traverse fs us hs)\n\n-- TODO: add a `Filter.HasBasis` statement\ntheorem mem_traverse_iff (fs : List β) (t : Set (List α)) :\n    t ∈ traverse f fs ↔\n      ∃ us : List (Set α), Forall₂ (fun b (s : Set α) => s ∈ f b) fs us ∧ sequence us ⊆ t :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_a1347b35a1da","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"047acb989ff296fd86a71da279282c412282b3ce5deba21d79152cb3ac5ba2e4","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Filter","family_id":"mem_traverse","file_id":"mathlib/Mathlib/Order/Filter/ListTraverse.lean","sample_id":"a1347b35a1da940b0cf7bb9fa19ec8fb67a45c4aec52bd9f946f2c2f04d77b83"},"split":"train"}
{"binding":{"chosen_proof_sha256":"665f9b24ed35095c9258c237fa3353b33a4af46379e1ca75f0d890346defc187","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"df00ef8b543a3838a8dcfd15337dcf5374c33e276a9693b751546ead28ce4bfb","source_sha256":"351582fbc976816e329c33b0cb83e6e30d58c72551427bea563b172b7814b639","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply EqualizerCondition.mk\n  intro Z B π _ _\n  refine ⟨fun a b h ↦ ?_, fun ⟨a, ha⟩ ↦ ?_⟩\n  · simp only [yonedaPresheaf, comp, Quiver.Hom.unop_op, TypeCat.Fun.coe_mk,\n      Set.coe_setOf, mapToEqualizer, Set.mem_setOf_eq, ConcreteCategory.hom_ofHom, Subtype.mk.injEq,\n      mk.injEq] at h\n    simp only [yonedaPresheaf, unop_op]\n    ext x\n    obtain ⟨y, hy⟩ := (hq Z B π).surjective x\n    rw [← hy]\n    exact congr_fun h y\n  · simp only [yonedaPresheaf, comp, Quiver.Hom.unop_op, ConcreteCategory.hom_ofHom,\n      TypeCat.Fun.coe_mk, mk.injEq, Set.mem_setOf_eq] at ha\n    simp only [yonedaPresheaf, comp, Quiver.Hom.unop_op, TypeCat.Fun.coe_mk,\n      Set.coe_setOf, mapToEqualizer, Set.mem_setOf_eq, ConcreteCategory.hom_ofHom, Subtype.mk.injEq]\n    simp only [yonedaPresheaf, unop_op] at a\n    refine ⟨(hq Z B π).lift a (factorsThrough_of_pullbackCondition G X ha), ?_⟩\n    congr 1\n    exact DFunLike.ext'_iff.mp ((hq Z B π).lift_comp a (factorsThrough_of_pullbackCondition G X ha))\n\n/--\nIf `G` preserves finite coproducts (which is the case when `C` is `CompHaus`, `Profinite` or\n`Stonean`), then `yonedaPresheaf` preserves finite products, which is required to be a sheaf for\nthe extensive topology.\n-/\nnoncomputable instance [PreservesFiniteCoproducts G] :\n    PreservesFiniteProducts (yonedaPresheaf G X) :=\n  have := preservesFiniteProducts_op G\n  ⟨fun _ ↦ comp_preservesLimitsOfShape G.op (yonedaPresheaf' X)⟩","hard_negative":false,"metrics":{"chosen_tokens":325,"rejected_tokens":2,"token_jaccard":0.009615,"token_length_ratio":0.006154},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"c778555600e29517ab9267992523e6f71e122438ad968160ffaecc3bc669b9cf","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Preserves.Opposites\npublic import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison\npublic import Mathlib.Condensed.Basic\npublic import Mathlib.Topology.Category.TopCat.Yoneda\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# The functor from topological spaces to condensed sets\n\nThis file builds on the API from the file `TopCat.Yoneda`. If the forgetful functor to `TopCat` has\nnice properties, like preserving pullbacks and finite coproducts, then this Yoneda presheaf\nsatisfies the sheaf condition for the regular and extensive topologies respectively.\n\nWe apply this API to `CompHaus` and define the functor\n`topCatToCondensedSet : TopCat.{u + 1} ⥤ CondensedSet.{u}`.\n\n-/\n\n@[expose] public section\n\nuniverse w w' v u\n\nopen CategoryTheory Opposite Limits regularTopology ContinuousMap Topology\n\nvariable {C : Type u} [Category.{v} C] (G : C ⥤ TopCat.{w})\n  (X : Type w') [TopologicalSpace X]\n\n/--\nAn auxiliary lemma to that allows us to use `IsQuotientMap.lift` in the proof of\n`equalizerCondition_yonedaPresheaf`.\n-/\ntheorem factorsThrough_of_pullbackCondition {Z B : C} {π : Z ⟶ B} [HasPullback π π]\n    [PreservesLimit (cospan π π) G]\n    {a : C(G.obj Z, X)}\n    (ha : a ∘ (G.map (pullback.fst _ _)) = a ∘ (G.map (pullback.snd π π))) :\n    Function.FactorsThrough a (G.map π) := by\n  intro x y hxy\n  let xy : G.obj (pullback π π) := (PreservesPullback.iso G π π).inv <|\n    (TopCat.pullbackIsoProdSubtype (G.map π) (G.map π)).inv ⟨(x, y), hxy⟩\n  have ha' := congr_fun ha xy\n  dsimp at ha'\n  have h₁ : ∀ y, G.map (pullback.fst _ _) ((PreservesPullback.iso G π π).inv y) =\n      pullback.fst (G.map π) (G.map π) y := by\n    simp only [← PreservesPullback.iso_inv_fst]; intro y; rfl\n  have h₂ : ∀ y, G.map (pullback.snd _ _) ((PreservesPullback.iso G π π).inv y) =\n      pullback.snd (G.map π) (G.map π) y := by\n    simp only [← PreservesPullback.iso_inv_snd]; intro y; rfl\n  rw [h₁, h₂, TopCat.pullbackIsoProdSubtype_inv_fst_apply,\n    TopCat.pullbackIsoProdSubtype_inv_snd_apply] at ha'\n  simpa using ha'\n\nset_option backward.isDefEq.respectTransparency false in\n/--\nIf `G` preserves the relevant pullbacks and every effective epi in `C` is a quotient map (which is\nthe case when `C` is `CompHaus` or `Profinite`), then `yonedaPresheaf` satisfies the equalizer\ncondition which is required to be a sheaf for the regular topology.\n-/\n\nTarget:\ntheorem equalizerCondition_yonedaPresheaf\n    [∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π], PreservesLimit (cospan π π) G]\n    (hq : ∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π], IsQuotientMap (G.map π)) :\n      EqualizerCondition (yonedaPresheaf G X) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed","family_id":"equalizercondition_yonedapresheaf","file_id":"mathlib/Mathlib/Condensed/TopComparison.lean","sample_id":"df00ef8b543a3838a8dcfd15337dcf5374c33e276a9693b751546ead28ce4bfb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"81f6811c61bda2ac7e16fe95d39238e8cafe9933dcb6310481ee06482505a998","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"936168df007e15fb85b6f7d1e2a4ec6ccdb182c4008e862b6d8de8b6dd6237e3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9ea116b7daa0ff9aa81b54eb159a02ebd9756dce658d8964cedd444ceb6ca675","source_sha256":"957fb86c605e5aefaf558e2e21f21bba2c9e2fd312cbcb361e91615272ce22b4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [fiber, fiberι, Scheme.Pullback.range_fst, Scheme.range_fromSpecResidueField]","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":5,"token_jaccard":0.133333,"token_length_ratio":0.277778},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"c7c27c1930ea9b6da3dad4d064a11eefa4ea90d775f39ed99eb36b0ed22c4f84","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.PullbackCarrier\npublic import Mathlib.RingTheory.LocalRing.ResidueField.Fiber\npublic import Mathlib.RingTheory.Spectrum.Prime.Jacobson\npublic import Mathlib.AlgebraicGeometry.Morphisms.Affine\npublic import Mathlib.AlgebraicGeometry.Morphisms.FiniteType\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Scheme-theoretic fiber\n\n## Main result\n- `AlgebraicGeometry.Scheme.Hom.fiber`: `f.fiber y` is the scheme-theoretic fiber of `f` at `y`.\n- `AlgebraicGeometry.Scheme.Hom.fiberHomeo`: `f.fiber y` is homeomorphic to `f ⁻¹' {y}`.\n- `AlgebraicGeometry.Scheme.Hom.finite_preimage`: Finite morphisms have finite fibers.\n- `AlgebraicGeometry.Scheme.Hom.discrete_fiber`: Finite morphisms have discrete fibers.\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nnoncomputable section\n\nopen CategoryTheory Limits\n\nnamespace AlgebraicGeometry\n\nvariable {X Y : Scheme.{u}}\n\n/-- `f.fiber y` is the scheme-theoretic fiber of `f` at `y`. -/\ndef Scheme.Hom.fiber (f : X ⟶ Y) (y : Y) : Scheme := pullback f (Y.fromSpecResidueField y)\n\n/-- `f.fiberι y : f.fiber y ⟶ X` is the embedding of the scheme-theoretic fiber into `X`. -/\ndef Scheme.Hom.fiberι (f : X ⟶ Y) (y : Y) : f.fiber y ⟶ X := pullback.fst _ _\n\ninstance (f : X ⟶ Y) (y : Y) : (f.fiber y).CanonicallyOver X where hom := f.fiberι y\n\n/-- The canonical map from the scheme-theoretic fiber to the residue field. -/\ndef Scheme.Hom.fiberToSpecResidueField (f : X ⟶ Y) (y : Y) :\n    f.fiber y ⟶ Spec (Y.residueField y) :=\n  pullback.snd _ _\n\n@[reassoc]\nlemma Scheme.Hom.fiber_fac (f : X ⟶ Y) (y : Y) :\n    f.fiberι y ≫ f = f.fiberToSpecResidueField y ≫ Y.fromSpecResidueField y :=\n  pullback.condition\n\n/-- The fiber of `f` at `y` is naturally a `κ(y)`-scheme. -/\n@[reducible] def Scheme.Hom.fiberOverSpecResidueField\n    (f : X ⟶ Y) (y : Y) : (f.fiber y).Over (Spec (Y.residueField y)) where\n  hom := f.fiberToSpecResidueField y\n\nlemma Scheme.Hom.fiberToSpecResidueField_apply (f : X ⟶ Y) (y : Y) (x : f.fiber y) :\n    f.fiberToSpecResidueField y x = IsLocalRing.closedPoint (Y.residueField y) :=\n  Subsingleton.elim (α := PrimeSpectrum _) _ _\n\nset_option backward.isDefEq.respectTransparency false in\nlemma isPullback_fiberToSpecResidueField_of_isPullback {P X Y Z : Scheme.{u}} {fst : P ⟶ X}\n    {snd : P ⟶ Y} {f : X ⟶ Z} {g : Y ⟶ Z} (h : IsPullback fst snd f g) (y : Y) :\n    IsPullback (pullback.map _ _ _ _ fst (Spec.map (g.residueFieldMap y)) g h.w.symm (by simp))\n      (snd.fiberToSpecResidueField y)\n      (f.fiberToSpecResidueField (g y))\n      (Spec.map (g.residueFieldMap y)) := by\n  refine .of_right (h₁₂ := pullback.fst _ _) ?_ ?_\n      (IsPullback.of_hasPullback f (Z.fromSpecResidueField (g y)))\n  · simpa using! (IsPullback.of_hasPullback _ _).paste_horiz h\n  · simp [Scheme.Hom.fiberToSpecResidueField]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The morphism from the fiber of `Spec S ⟶ Spec R` at some prime `p` to `Spec κ(p)`\nis isomorphic to the map induced by `κ(p) ⟶ κ(p) ⊗[R] S`. -/\nnoncomputable def Spec.fiberToSpecResidueFieldIso (R S : Type u) [CommRing R] [CommRing S]\n    [Algebra R S] (p : PrimeSpectrum R) :\n    Arrow.mk ((Spec.map (CommRingCat.ofHom <| algebraMap R S)).fiberToSpecResidueField p) ≅\n      Arrow.mk (Spec.map <| CommRingCat.ofHom <|\n        algebraMap p.asIdeal.ResidueField (p.asIdeal.Fiber S)) := by\n  refine Arrow.isoMk' _ _\n    (pullbackSymmetry _ _ ≪≫ ?_ ≪≫ pullbackSpecIso R p.asIdeal.ResidueField S) ?_ ?_\n  · refine pullback.congrHom\n      (Scheme.Spec.map_residueFieldIso_inv_eq_fromSpecResidueField (.of R) p).symm rfl ≪≫ ?_\n    refine asIso <| pullback.map _ _ _ _ (Spec.map <| (Scheme.Spec.residueFieldIso (.of R) _).inv)\n      (𝟙 _) (𝟙 _) (by simp) (by simp)\n  · exact Scheme.Spec.mapIso (Scheme.Spec.residueFieldIso (.of R) _).symm.op\n  · cat_disch\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma Scheme.Hom.range_fiberι (f : X ⟶ Y) (y : Y) :\n    Set.range (f.fiberι y) = f ⁻¹' {y} :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_9ea116b7daa0","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"a98d6c6c6a65f5bb99e7c6e1ecfc81c0f48acb301906f82b1bb3c855493b0266","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry","family_id":"scheme","file_id":"mathlib/Mathlib/AlgebraicGeometry/Fiber.lean","sample_id":"9ea116b7daa0ff9aa81b54eb159a02ebd9756dce658d8964cedd444ceb6ca675"},"split":"train"}
{"binding":{"chosen_proof_sha256":"06d455a8b5ee3bf3c5458bfc466011b102068e1311619ad2681ed0ee2fe0c4c3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"73e2230058ad9af7f90bd906974327f3dff5d73f4d9ae73835dc4eb5a665aa63","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"900df9f57575ae9ae5d97a81a010ab1e20e766cd1a88e2d731f34af9d7eca3f8","source_sha256":"afbafd08519a79137e714f52e70a195d46a2c795926cef003418bc25da380e6c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases hβ : Nonempty β\n  · refine small_of_surjective (f := Function.invFun g ∘ f) (fun b => ?_)\n    obtain ⟨a, ha⟩ := h b\n    exact ⟨a, by rw [Function.comp_apply, ha, Function.leftInverse_invFun hg]⟩\n  · simp only [not_nonempty_iff] at hβ\n    infer_instance\n\n/-!\nWe don't define `Countable.toSmall` in this file, to keep imports to `Logic` to a minimum.\n-/","hard_negative":false,"metrics":{"chosen_tokens":93,"rejected_tokens":98,"token_jaccard":0.935484,"token_length_ratio":1.053763},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"c7de665b3ae280a1442fefd5b27719ebc4fdc4395a5fad5a6901eaaad899e7ec","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Logic.Small.Defs\npublic import Mathlib.Logic.Equiv.Set\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Instances and theorems for `Small`.\n\nIn particular we prove `small_of_injective` and `small_of_surjective`.\n-/\n\npublic section\n\nassert_not_exists Countable\n\nuniverse u w v v'\n\n-- TODO(timotree3): lower the priority on this instance?\n-- This instance applies to every synthesis problem of the form `Small ↥s` for some set `s`,\n-- but we have lots of instances of `Small` for specific set constructions.\ninstance small_subtype (α : Type v) [Small.{w} α] (P : α → Prop) : Small.{w} { x // P x } :=\n  small_map (equivShrink α).subtypeEquivOfSubtype'\n\ntheorem small_of_injective {α : Type v} {β : Type w} [Small.{u} β] {f : α → β}\n    (hf : Function.Injective f) : Small.{u} α :=\n  small_map (Equiv.ofInjective f hf)\n\ntheorem small_of_surjective {α : Type v} {β : Type w} [Small.{u} α] {f : α → β}\n    (hf : Function.Surjective f) : Small.{u} β :=\n  small_of_injective (Function.injective_surjInv hf)\n\ninstance (priority := 100) small_subsingleton (α : Type v) [Subsingleton α] : Small.{w} α := by\n  rcases isEmpty_or_nonempty α with ⟨⟩\n  · apply small_map (Equiv.equivPEmpty α)\n  · apply small_map Equiv.punitOfNonemptyOfSubsingleton\n\n/-- This can be seen as a version of `small_of_surjective` in which the function `f` doesn't\nactually land in `β` but in some larger type `γ` related to `β` via an injective function `g`.\n-/\n\nTarget:\ntheorem small_of_injective_of_exists {α : Type v} {β : Type w} {γ : Type v'} [Small.{u} α]\n    (f : α → γ) {g : β → γ} (hg : Function.Injective g) (h : ∀ b : β, ∃ a : α, f a = g b) :\n    Small.{u} β :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  by_cases hβ : Nonempty β\n  · refine small_of_surjective (f := Function.invFun g ∘ f) (fun b => ?_)\n    obtain ⟨a, ha⟩ := h b\n    exact ⟨a, by rw [Function.comp_apply, ha, Function.leftInverse_invFun hg]⟩\n  · simp only [not_nonempty_iff] at hβ\n    infer_instance\n\n/-!\nWe don't define `Countable.toSmall` in this file, to keep imports to `Logic` to a minimum.\n-/","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Small","family_id":"small_of_injective_of_exists","file_id":"mathlib/Mathlib/Logic/Small/Basic.lean","sample_id":"900df9f57575ae9ae5d97a81a010ab1e20e766cd1a88e2d731f34af9d7eca3f8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"153be3b3963b7b4e295fef95384346f85509c50a1d50e691a207cccee178ce99","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"826b1e14ead13499112deffa7b08d1cf2421e03fedbc37a2d7a8c6e2018ea7dd","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fffbd570a9bc968d3091c7ef16d1e7aecf0edf42a3c5fce61027b019b7b6b538","source_sha256":"16dd8d8cb25a5fc3397ffc119122019b2cf6ddea68dae9b29de54500250b117b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · rw [Presheaf.isLocallySurjective_iff_whisker_forget (J := extensiveTopology C)]\n    exact fun h _ ↦\n      surjective_of_isLocallySurjective_sheaf_of_types (Functor.whiskerRight f (forget D)) h\n  · exact fun a ↦\n      Presheaf.isLocallySurjective_of_surjective _ _ (fun _ ↦ a _)","hard_negative":true,"metrics":{"chosen_tokens":49,"rejected_tokens":2,"token_jaccard":0.034483,"token_length_ratio":0.040816},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"c82929fdeb2f3bcfb24c97fd8bb674e0b7d958c7475d18f7738af5e1e78ae8f2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology\npublic import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison\npublic import Mathlib.CategoryTheory.Sites.LocallySurjective\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# Locally surjective morphisms of coherent sheaves\n\nThis file characterises locally surjective morphisms of presheaves for the coherent, regular\nand extensive topologies.\n\n## Main results\n\n* `regularTopology.isLocallySurjective_iff` A morphism of presheaves `f : F ⟶ G` is locally\n  surjective for the regular topology iff for every object `X` of `C`, and every `y : G(X)`, there\n  is an effective epimorphism `φ : X' ⟶ X` and an `x : F(X)` such that `f_{X'}(x) = G(φ)(y)`.\n\n* `coherentTopology.isLocallySurjective_iff` a morphism of sheaves for the coherent topology on a\n  preregular finitary extensive category is locally surjective if and only if it is\n  locally surjective for the regular topology.\n\n* `extensiveTopology.isLocallySurjective_iff` a morphism of sheaves for the extensive topology on a\n  finitary extensive category is locally surjective iff it is objectwise surjective.\n-/\n\npublic section\n\nuniverse w\n\nopen CategoryTheory Sheaf Limits Opposite\n\nnamespace CategoryTheory\n\nvariable {C : Type*} (D : Type*) [Category* C] [Category* D] {FD : D → D → Type*} {CD : D → Type w}\n  [∀ X Y, FunLike (FD X Y) (CD X) (CD Y)] [ConcreteCategory.{w} D FD]\n\nlemma regularTopology.isLocallySurjective_iff [Preregular C] {F G : Cᵒᵖ ⥤ D} (f : F ⟶ G) :\n    Presheaf.IsLocallySurjective (regularTopology C) f ↔\n      ∀ (X : C) (y : ToType (G.obj ⟨X⟩)), (∃ (X' : C) (φ : X' ⟶ X) (_ : EffectiveEpi φ)\n        (x : ToType (F.obj ⟨X'⟩)),\n        f.app ⟨X'⟩ x = G.map ⟨φ⟩ y) := by\n  constructor\n  · intro ⟨h⟩ X y\n    specialize h y\n    rw [regularTopology.mem_sieves_iff_hasEffectiveEpi] at h\n    obtain ⟨X', π, h, h'⟩ := h\n    exact ⟨X', π, h, h'⟩\n  · intro h\n    refine ⟨fun y ↦ ?_⟩\n    obtain ⟨X', π, h, h'⟩ := h _ y\n    rw [regularTopology.mem_sieves_iff_hasEffectiveEpi]\n    exact ⟨X', π, h, h'⟩\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\nlemma extensiveTopology.surjective_of_isLocallySurjective_sheaf_of_types [FinitaryPreExtensive C]\n    {F G : Cᵒᵖ ⥤ Type w} (f : F ⟶ G) [PreservesFiniteProducts F] [PreservesFiniteProducts G]\n      (h : Presheaf.IsLocallySurjective (extensiveTopology C) f) {X : C} :\n        Function.Surjective (f.app (op X)) := by\n  intro x\n  replace h := h.1 x\n  rw [mem_sieves_iff_contains_colimit_cofan] at h\n  obtain ⟨α, _, Y, π, h, h'⟩ := h\n  let y : (a : α) → (F.obj ⟨Y a⟩) := fun a ↦ (h' a).choose\n  let _ : Fintype α := Fintype.ofFinite _\n  let ht := (Types.productLimitCone (fun a ↦ F.obj ⟨Y a⟩)).isLimit\n  let ht' := (Functor.Initial.isLimitWhiskerEquiv (Discrete.opposite α).inverse\n    (Cocone.op (Cofan.mk X π))).symm h.some.op\n  let i : ((a : α) → (F.obj ⟨Y a⟩)) ≅ (F.obj ⟨X⟩) :=\n    ht.conePointsIsoOfNatIso (isLimitOfPreserves F ht')\n      (Discrete.natIso (fun _ ↦ (Iso.refl (F.obj ⟨_⟩))))\n  refine ⟨i.hom y, ?_⟩\n  apply Concrete.isLimit_ext _ (isLimitOfPreserves G ht')\n  intro ⟨a⟩\n  simp only [Functor.comp_obj, Discrete.opposite_inverse_obj, Functor.op_obj, Discrete.functor_obj,\n    Functor.mapCone_pt, Cone.whisker_pt, Cocone.op_pt, Cofan.mk_pt, Functor.const_obj_obj,\n    Functor.mapCone_π_app, Cone.whisker_π, Cocone.op_π, Functor.whiskerLeft_app, NatTrans.op_app,\n    Cofan.mk_ι_app]\n  rw [← (h' a).choose_spec, ← NatTrans.naturality_apply (φ := f)]\n  simp only [IsLimit.conePointsIsoOfNatIso_hom, ← comp_apply, i]\n  erw [IsLimit.map_π]\n  rfl\n\nTarget:\nlemma extensiveTopology.presheafIsLocallySurjective_iff [FinitaryPreExtensive C] {F G : Cᵒᵖ ⥤ D}\n    (f : F ⟶ G) [PreservesFiniteProducts F] [PreservesFiniteProducts G]\n      [PreservesFiniteProducts (forget D)] : Presheaf.IsLocallySurjective (extensiveTopology C) f ↔\n        ∀ (X : C), Function.Surjective (f.app (op X)) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_fffbd570a9bc","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"35d8298951273c49ae6c129d7bea7a08db796bef45304d744145b2f187cd8783","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Sites","family_id":"extensivetopology","file_id":"mathlib/Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean","sample_id":"fffbd570a9bc968d3091c7ef16d1e7aecf0edf42a3c5fce61027b019b7b6b538"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0e10a34c52b1aa6caddc8087552eba2fdbebe27ef42d67264d88c7c0b6ced53f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b4314e3aa68947f9c2ef21b50c145bfc861cf80dd2c7e4d01d1f88506a70358d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b69b8cf11d81b81760b7e2d01790b8cb08e792859b8026925d68c9b27e00c6b1","source_sha256":"0a590cb1cf4bee7c2ccf453bd99af14d58eb8e9c3d2e696c45f4fcbc49319d23","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction H with\n  | basic h =>\n    cases h\n    · exact Or.inr ⟨Set.univ, cond.univ_mem, by simpa⟩\n    · exact Or.inl (by assumption)\n  | univ => exact Or.inl cond.univ_mem\n  | @inter T1 T2 _ _ h1 h2 =>\n    rcases h1 with (h | ⟨Q, hQ, rfl⟩) <;> rcases h2 with (i | ⟨R, hR, rfl⟩)\n    · exact Or.inl (cond.inter_mem h i)\n    · exact\n        Or.inr ⟨T1 ∩ R, cond.inter_mem h hR, by simp only [← Set.inter_assoc, Set.inter_comm _ A]⟩\n    · exact Or.inr ⟨Q ∩ T2, cond.inter_mem hQ i, by simp only [Set.inter_assoc]⟩\n    · exact\n        Or.inr\n          ⟨Q ∩ R, cond.inter_mem hQ hR, by\n            ext x\n            constructor <;> simp +contextual⟩","hard_negative":false,"metrics":{"chosen_tokens":183,"rejected_tokens":188,"token_jaccard":0.920635,"token_length_ratio":1.027322},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"c879fef9c8a937258494a90e703ff9764d8ec8964ab99417bba3d74fe9175804","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Insert\npublic import Mathlib.Data.Set.Lattice\n\nNamespace:\nFiniteInter\n\nLocal context:\n/-\nCopyright (c) 2020 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\n/-!\n# Constructions involving sets of sets.\n\n## Finite Intersections\n\nWe define a structure `FiniteInter` which asserts that a set `S` of subsets of `α` is\nclosed under finite intersections.\n\nWe define `finiteInterClosure` which, given a set `S` of subsets of `α`, is the smallest\nset of subsets of `α` which is closed under finite intersections.\n\n`finiteInterClosure S` is endowed with a term of type `FiniteInter` using\n`finiteInterClosure_finiteInter`.\n\n-/\n\npublic section\n\n\nvariable {α : Type*} (S : Set (Set α))\n\n/-- A structure encapsulating the fact that a set of sets is closed under finite intersection. -/\nstructure FiniteInter : Prop where\n  /-- `univ_mem` states that `Set.univ` is in `S`. -/\n  univ_mem : Set.univ ∈ S\n  /-- `inter_mem` states that any two intersections of sets in `S` is also in `S`. -/\n  inter_mem : ∀ ⦃s⦄, s ∈ S → ∀ ⦃t⦄, t ∈ S → s ∩ t ∈ S\n\nnamespace FiniteInter\n\n/-- The smallest set of sets containing `S` which is closed under finite intersections. -/\ninductive finiteInterClosure : Set (Set α)\n  | basic {s} : s ∈ S → finiteInterClosure s\n  | univ : finiteInterClosure Set.univ\n  | inter {s t} : finiteInterClosure s → finiteInterClosure t → finiteInterClosure (s ∩ t)\n\ntheorem finiteInterClosure_finiteInter : FiniteInter (finiteInterClosure S) :=\n  { univ_mem := finiteInterClosure.univ\n    inter_mem := fun _ h _ => finiteInterClosure.inter h }\n\nvariable {S}\n\ntheorem finiteInter_mem (cond : FiniteInter S) (F : Finset (Set α)) :\n    ↑F ⊆ S → ⋂₀ (↑F : Set (Set α)) ∈ S := by\n  classical\n    refine Finset.induction_on F (fun _ => ?_) ?_\n    · simp [cond.univ_mem]\n    · intro a s _ h1 h2\n      suffices a ∩ ⋂₀ ↑s ∈ S by simpa\n      exact\n        cond.inter_mem (h2 (Finset.mem_insert_self a s))\n          (h1 fun x hx => h2 <| Finset.mem_insert_of_mem hx)\n\nTarget:\ntheorem finiteInterClosure_insert {A : Set α} (cond : FiniteInter S) (P)\n    (H : P ∈ finiteInterClosure (insert A S)) : P ∈ S ∨ ∃ Q ∈ S, P = A ∩ Q :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  induction H with\n  | basic h =>\n    cases h\n    · exact Or.inr ⟨Set.univ, cond.univ_mem, by simpa⟩\n    · exact Or.inl (by assumption)\n  | univ => exact Or.inl cond.univ_mem\n  | @inter T1 T2 _ _ h1 h2 =>\n    rcases h1 with (h | ⟨Q, hQ, rfl⟩) <;> rcases h2 with (i | ⟨R, hR, rfl⟩)\n    · exact Or.inl (cond.inter_mem h i)\n    · exact\n        Or.inr ⟨T1 ∩ R, cond.inter_mem h hR, by simp only [← Set.inter_assoc, Set.inter_comm _ A]⟩\n    · exact Or.inr ⟨Q ∩ T2, cond.inter_mem hQ i, by simp only [Set.inter_assoc]⟩\n    · exact\n        Or.inr\n          ⟨Q ∩ R, cond.inter_mem hQ hR, by\n            ext x\n            constructor <;> simp +contextual⟩","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Set","family_id":"finiteinterclosure_insert","file_id":"mathlib/Mathlib/Data/Set/Constructions.lean","sample_id":"b69b8cf11d81b81760b7e2d01790b8cb08e792859b8026925d68c9b27e00c6b1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c975c088fa8efb7c0cc15ea499939f263cd071ef49bfcc3a4f706c833af7cab0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"77ab52f257e20e46da01266890c31363fd41c9a271f0a1f1ab995fa8f340ab6b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e7de025d50f83819819386b2f880ed49a8eebee597e932160cded70716ac05cb","source_sha256":"394c3d8954491754c0a03071ce7aa65ddc93c4783581cf0b4d6869361547187f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  tfae_have 1 → 2 := fun _ ↦ inferInstance\n  tfae_have 2 → 3 := fun _ ↦ IsAtomic.exists_atom _\n  tfae_have 3 → 1 := fun ⟨I, hI⟩ ↦ by\n    have ⟨_, h⟩ := isSimpleRing_iff_isTwoSided_imp.mp ‹IsSimpleRing R›\n    simp_rw [← isFullyInvariant_iff_isTwoSided] at h\n    have := isSimpleModule_iff_isAtom.mpr hI\n    obtain eq | eq := h _ (.isotypicComponent R R I)\n    · exact (hI.bot_lt.not_ge <| (le_sSup <| by exact ⟨.refl ..⟩).trans_eq eq).elim\n    exact .congr (.symm <| .trans (.ofEq _ _ eq) Submodule.topEquiv)\n  tfae_finish","hard_negative":true,"metrics":{"chosen_tokens":127,"rejected_tokens":3,"token_jaccard":0.034483,"token_length_ratio":0.023622},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"c88115c5b63f5725066c994334b90b1701b1f854fee64cc0c2a074f154903d5f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.FreeModule.Finite.Basic\npublic import Mathlib.RingTheory.FiniteLength\npublic import Mathlib.RingTheory.SimpleModule.Isotypic\npublic import Mathlib.RingTheory.SimpleRing.Congr\npublic import Mathlib.RingTheory.SimpleRing.Matrix\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu\n-/\n/-!\n# Wedderburn–Artin Theorem\n\n## Main results\n\n* `IsSimpleRing.tfae`: a simple ring is semisimple iff it is Artinian,\n  iff it has a minimal left ideal.\n\n* `isSimpleRing_isArtinianRing_iff`: a ring is simple Artinian iff it is semisimple, isotypic,\n  and nontrivial.\n\n* `IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite`: a simple Artinian algebra is\n  isomorphic to a (finite-dimensional) matrix algebra over a division algebra. The division\n  algebra is the opposite of the endomorphism algebra of a simple (i.e., minimal) left ideal.\n\n* `IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite`: a semisimple algebra is\n  isomorphic to a finite direct product of matrix algebras over division algebras. The division\n  algebras are the opposites of the endomorphism algebras of the simple (i.e., minimal)\n  left ideals.\n\n* `IsSimpleRing.exists_algEquiv_matrix_divisionRing_finite`,\n  `IsSemisimpleRing.exists_algEquiv_pi_matrix_divisionRing_finite`:\n  if the simple Artinian / semisimple algebra is finite as a module over a base ring, then the\n  division algebra(s) are also finite over the same ring.\n  If the base ring is an algebraically closed field, the only finite-dimensional division algebra\n  over it is itself, and we obtain `IsSimpleRing.exists_algEquiv_matrix_of_isAlgClosed` and\n  `IsSemisimpleRing.exists_algEquiv_pi_matrix_of_isAlgClosed` (in a later file).\n\n-/\n\npublic section\n\nuniverse u\nvariable (R₀ : Type*) {R : Type u} [CommSemiring R₀] [Ring R] [Algebra R₀ R]\n\n/-- A simple ring is semisimple iff it is Artinian, iff it has a minimal left ideal. -/\n\nTarget:\ntheorem IsSimpleRing.tfae [IsSimpleRing R] : List.TFAE\n    [IsSemisimpleRing R, IsArtinianRing R, ∃ I : Ideal R, IsAtom I] :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_e7de025d50f8","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"3ca9173408122d90bbd2d1cc256b0eecafe21120d00538b390272d6224118338","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/SimpleModule","family_id":"issimplering","file_id":"mathlib/Mathlib/RingTheory/SimpleModule/WedderburnArtin.lean","sample_id":"e7de025d50f83819819386b2f880ed49a8eebee597e932160cded70716ac05cb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"475307141c3a1cbf5e416ce7878a4f3fd1a2e5cb9833e295c816f2bfea6746a8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"607d86a3d00d2a167cfec5e1d61d8780b434f24a964e4e150ae6ac47af27ee0f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8718a7560479506834cd432f1d1bbbcdc14349fe8b10b51da3d79c17a3f32f95","source_sha256":"f0e2b6563fb6bf505db894994fffc54566c37c1de08ac8bba92d16e8dd3bcc5d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [Fintype.subtype_card, Finset.card_univ]","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":5,"token_jaccard":0.153846,"token_length_ratio":0.454545},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"c8c487a8893b0f0f627c1f9bcf93a2d0484260b7b2481fcc661667fd70d28c92","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Powerset\npublic import Mathlib.Data.Fintype.EquivFin\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# fintype instance for `Set α`, when `α` is a fintype\n-/\n\npublic section\n\n\nvariable {α : Type*}\n\nopen Finset\n\ninstance Finset.fintype [Fintype α] : Fintype (Finset α) :=\n  ⟨univ.powerset, fun _ => Finset.mem_powerset.2 (Finset.subset_univ _)⟩\n\n@[simp]\ntheorem Fintype.card_finset [Fintype α] : Fintype.card (Finset α) = 2 ^ Fintype.card α :=\n  Finset.card_powerset Finset.univ\n\nnamespace Finset\nvariable [Fintype α] {s : Finset α} {k : ℕ}\n\n@[simp] lemma powerset_univ : (univ : Finset α).powerset = univ :=\n  coe_injective <| by simp [-coe_eq_univ]\n\nlemma filter_subset_univ [DecidableEq α] (s : Finset α) :\n    ({t | t ⊆ s} : Finset _) = powerset s := by ext; simp\n\n@[simp] lemma powerset_eq_univ : s.powerset = univ ↔ s = univ := by\n  rw [← Finset.powerset_univ, powerset_inj]\n\nlemma mem_powersetCard_univ : s ∈ powersetCard k (univ : Finset α) ↔ #s = k :=\n  mem_powersetCard.trans <| and_iff_right <| subset_univ _\n\nvariable (α)\n\n@[simp] lemma univ_filter_card_eq (k : ℕ) :\n    ({s | #s = k} : Finset (Finset α)) = univ.powersetCard k := by ext; simp\n\nend Finset\n\n@[simp]\n\nTarget:\ntheorem Fintype.card_finset_len [Fintype α] (k : ℕ) :\n    Fintype.card { s : Finset α // #s = k } = Nat.choose (Fintype.card α) k :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_8718a7560479","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d550a9dd5c7d512b8a62f6a897276d41ffa03f3b78bea00e7968cf43d1720249","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Fintype","family_id":"fintype","file_id":"mathlib/Mathlib/Data/Fintype/Powerset.lean","sample_id":"8718a7560479506834cd432f1d1bbbcdc14349fe8b10b51da3d79c17a3f32f95"},"split":"train"}
{"binding":{"chosen_proof_sha256":"340281fbf0a45d34c3a34d2f4a61a24c355dc2f03e29d91afa10e3fb584bf2fd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3b594b8efd7138c1edf9cf4beb8848fba944bf82946b75971c431aea36b7d8df","source_sha256":"34c15bf85e4926f0e35f0d5bc6422a5c85336bdcbed113f50f3b65102f3e6cc1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro a b h\n  simp only [LT.lt, Finsupp.lex_def]\n  simp only [ofLex_toLex, Nat.lt_eq]\n  use a\n  constructor\n  · intro d hd\n    simp only [Finsupp.single_eq_of_ne hd.ne, Finsupp.single_eq_of_ne (hd.trans h).ne]\n  · simp [h.ne']","hard_negative":false,"metrics":{"chosen_tokens":61,"rejected_tokens":2,"token_jaccard":0.033333,"token_length_ratio":0.032787},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"c8de96fd1d3b2bea39328b9aef7fa962b0ff7d4e775ef9ca767eda0f64f1d7e2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finsupp.Order\npublic import Mathlib.Data.DFinsupp.Lex\npublic import Mathlib.Data.Finsupp.ToDFinsupp\n\nNamespace:\nFinsupp\n\nLocal context:\n/-\nCopyright (c) 2022 Damiano Testa. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Damiano Testa\n-/\n/-!\n# Lexicographic order on finitely supported functions\n\nThis file defines the lexicographic order on `Finsupp`.\n-/\n\n@[expose] public section\n\n\nvariable {α N : Type*}\n\nnamespace Finsupp\n\nsection NHasZero\n\nvariable [Zero N]\n\n/-- `Finsupp.Lex r s` is the lexicographic relation on `α →₀ N`, where `α` is ordered by `r`,\nand `N` is ordered by `s`.\n\nThe type synonym `Lex (α →₀ N)` has an order given by `Finsupp.Lex (· < ·) (· < ·)`.\n-/\nprotected def Lex (r : α → α → Prop) (s : N → N → Prop) (x y : α →₀ N) : Prop :=\n  Pi.Lex r s x y\n\ntheorem _root_.Pi.lex_eq_finsupp_lex {r : α → α → Prop} {s : N → N → Prop} (a b : α →₀ N) :\n    Pi.Lex r s a b = Finsupp.Lex r s a b :=\n  rfl\n\ntheorem lex_def {r : α → α → Prop} {s : N → N → Prop} {a b : α →₀ N} :\n    Finsupp.Lex r s a b ↔ ∃ j, (∀ d, r d j → a d = b d) ∧ s (a j) (b j) :=\n  .rfl\n\ntheorem lex_eq_invImage_dfinsupp_lex (r : α → α → Prop) (s : N → N → Prop) :\n    Finsupp.Lex r s = InvImage (DFinsupp.Lex r fun _ ↦ s) toDFinsupp :=\n  rfl\n\ninstance [LT α] [LT N] : LT (Lex (α →₀ N)) :=\n  ⟨fun f g ↦ Finsupp.Lex (· < ·) (· < ·) (ofLex f) (ofLex g)⟩\n\ninstance [LT α] [LT N] : LT (Colex (α →₀ N)) :=\n  ⟨fun f g ↦ Finsupp.Lex (· > ·) (· < ·) (ofColex f) (ofColex g)⟩\n\ntheorem Lex.lt_iff [LT α] [LT N] {a b : Lex (α →₀ N)} :\n    a < b ↔ ∃ i, (∀ j, j < i → a j = b j) ∧ a i < b i :=\n  .rfl\n\n@[deprecated (since := \"2025-11-29\")]\nalias lex_lt_iff := Lex.lt_iff\n\ntheorem Colex.lt_iff [LT α] [LT N] {a b : Colex (α →₀ N)} :\n    a < b ↔ ∃ i, (∀ j, i < j → a j = b j) ∧ a i < b i :=\n  .rfl\n\ntheorem lex_lt_of_lt_of_preorder [Preorder N] (r) [IsStrictOrder α r] {x y : α →₀ N} (hlt : x < y) :\n    ∃ i, (∀ j, r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i :=\n  DFinsupp.lex_lt_of_lt_of_preorder r (id hlt : x.toDFinsupp < y.toDFinsupp)\n\ntheorem lex_lt_of_lt [PartialOrder N] (r) [IsStrictOrder α r] {x y : α →₀ N} (hlt : x < y) :\n    Pi.Lex r (· < ·) x y :=\n  DFinsupp.lex_lt_of_lt r (id hlt : x.toDFinsupp < y.toDFinsupp)\n\ntheorem lex_iff_of_unique [Unique α] [LT N] {r} [Std.Irrefl r] {x y : α →₀ N} :\n    Finsupp.Lex r (· < ·) x y ↔ x default < y default :=\n  Pi.lex_iff_of_unique\n\ntheorem Lex.lt_iff_of_unique [Unique α] [LT N] [Preorder α] {x y : Lex (α →₀ N)} :\n    x < y ↔ x default < y default :=\n  lex_iff_of_unique\n\n@[deprecated (since := \"2025-11-29\")]\nalias lex_lt_iff_of_unique := Lex.lt_iff_of_unique\n\ntheorem Colex.lt_iff_of_unique [Unique α] [LT N] [Preorder α] {x y : Colex (α →₀ N)} :\n    x < y ↔ x default < y default :=\n  Lex.lt_iff_of_unique (α := αᵒᵈ)\n\nvariable [LinearOrder α]\n\ninstance Lex.isStrictOrder [PartialOrder N] : IsStrictOrder (Lex (α →₀ N)) (· < ·) where\n  irrefl _ := lt_irrefl (α := Lex (α → N)) _\n  trans _ _ _ := lt_trans (α := Lex (α → N))\n\ninstance Colex.isStrictOrder [PartialOrder N] : IsStrictOrder (Colex (α →₀ N)) (· < ·) :=\n  Lex.isStrictOrder (α := αᵒᵈ)\n\n/-- The partial order on `Finsupp`s obtained by the lexicographic ordering.\nSee `Finsupp.Lex.linearOrder` for a proof that this partial order is in fact linear. -/\ninstance Lex.partialOrder [PartialOrder N] : PartialOrder (Lex (α →₀ N)) where\n  lt := (· < ·)\n  le x y := ⇑(ofLex x) = ⇑(ofLex y) ∨ x < y\n  __ := PartialOrder.lift (fun x : Lex (α →₀ N) ↦ toLex (⇑(ofLex x)))\n    (DFunLike.coe_injective (F := Finsupp α N))\n\n/-- The partial order on `Finsupp`s obtained by the colexicographic ordering.\nSee `Finsupp.Colex.linearOrder` for a proof that this partial order is in fact linear. -/\ninstance Colex.partialOrder [PartialOrder N] : PartialOrder (Colex (α →₀ N)) where\n  lt := (· < ·)\n  le x y := ⇑(ofColex x) = ⇑(ofColex y) ∨ x < y\n  __ := PartialOrder.lift (fun x : Colex (α →₀ N) ↦ toColex (⇑(ofColex x)))\n    (DFunLike.coe_injective (F := Finsupp α N))\n\n/-- The linear order on `Finsupp`s obtained by the lexicographic ordering. -/\ninstance Lex.linearOrder [LinearOrder N] : LinearOrder (Lex (α →₀ N)) where\n  __ := Lex.partialOrder\n  __ := LinearOrder.lift' (toLex ∘ toDFinsupp ∘ ofLex) finsuppEquivDFinsupp.injective\n\n/-- The linear order on `Finsupp`s obtained by the colexicographic ordering. -/\ninstance Colex.linearOrder [LinearOrder N] : LinearOrder (Colex (α →₀ N)) where\n  lt := (· < ·)\n  le := (· ≤ ·)\n  __ := LinearOrder.lift' (toColex ∘ toDFinsupp ∘ ofColex) finsuppEquivDFinsupp.injective\n\ntheorem Lex.le_iff_of_unique [Unique α] [PartialOrder N] {x y : Lex (α →₀ N)} :\n    x ≤ y ↔ x default ≤ y default :=\n  Pi.lex_le_iff_of_unique\n\n@[deprecated (since := \"2025-11-29\")]\nalias lex_le_iff_of_unique := Lex.le_iff_of_unique\n\ntheorem Colex.le_iff_of_unique [Unique α] [PartialOrder N] {x y : Colex (α →₀ N)} :\n    x ≤ y ↔ x default ≤ y default :=\n  Lex.le_iff_of_unique (α := αᵒᵈ)\n\nTarget:\ntheorem Lex.single_strictAnti : StrictAnti fun (a : α) ↦ toLex (single a 1) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finsupp","family_id":"lex","file_id":"mathlib/Mathlib/Data/Finsupp/Lex.lean","sample_id":"3b594b8efd7138c1edf9cf4beb8848fba944bf82946b75971c431aea36b7d8df"},"split":"train"}
{"binding":{"chosen_proof_sha256":"36b9bfe8ecf433634e45203e8b895f570c57f0adf9b70b7ca8ace15a79e21d3e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2a9d151a1126ceeaa73613f006206f069249394e06348f7d322ca4450f8ab3d8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bcfecfd0c0cb2071cab40553cc6bdd48d75c67302392e5f99abe6ae29aa62d18","source_sha256":"c10b46784e847fb243b6f808f2d97c20493e59f01e570b069afc8a2b315200ff","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  simp [single, ← Finsupp.smul_single]","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.166667,"token_length_ratio":0.272727},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"c9850ca9de24f6253853a2069672e47cf4845bc7220e42116361236e439fcd50","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GroupWithZero.Action.TransferInstance\npublic import Mathlib.Algebra.Module.Defs\npublic import Mathlib.Data.Finsupp.Basic\npublic import Mathlib.Data.Finsupp.SMulWithZero\n\nNamespace:\nMonoidAlgebra\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Yury Kudryashov, Kim Morrison\n-/\n/-!\n# Monoid algebras\n\nWhen the domain of a `Finsupp` has a multiplicative or additive structure, we can define\na convolution product. To mathematicians this structure is known as the \"monoid algebra\",\ni.e. the finite formal linear combinations over a given semiring of elements of a monoid `M`.\nThe \"group ring\" `ℤ[G]` or the \"group algebra\" `k[G]` are typical uses.\n\nIn fact the construction of the \"monoid algebra\" makes sense when `M` is not even a monoid, but\nmerely a magma, i.e., when `M` carries a multiplication which is not required to satisfy any\nconditions at all. In this case the construction yields a not-necessarily-unital,\nnot-necessarily-associative algebra but it is still adjoint to the forgetful functor from such\nalgebras to magmas, and we prove this as `MonoidAlgebra.liftMagma`.\n\nIn this file we define `MonoidAlgebra R M := M →₀ R`, and `AddMonoidAlgebra R M`\nin the same way, and then define the convolution product on these.\n\nWhen the domain is additive, this is used to define polynomials:\n```\nPolynomial R := AddMonoidAlgebra R ℕ\nMvPolynomial σ α := AddMonoidAlgebra R (σ →₀ ℕ)\n```\nNote: `Polynomial R` is currently a wrapper around `AddMonoidAlgebra R ℕ` and not defeq to it.\nThere is ongoing work to make it defeq.\nSee https://github.com/leanprover-community/mathlib4/pull/25273\n\nWhen the domain is multiplicative, e.g. a group, this will be used to define the group ring.\n\n## Notation\n\nWe introduce the notation `R[M]` for both `MonoidAlgebra R M` and `AddMonoidAlgebra R M`.\nThe notations are scoped to their respective namespaces, and which one `R[M]` resolves to therefore\ndepends on which of the two namespaces is open.\n\n## TODO\n\nUse `coeff`/`ofCoeff` more widely. See\nhttps://github.com/leanprover-community/mathlib4/pull/36746\nhttps://github.com/leanprover-community/mathlib4/pull/25273\n-/\n\n@[expose] public section\n\nassert_not_exists NonUnitalAlgHom AlgEquiv\n\nnoncomputable section\n\nopen Finsupp hiding single\n\nvariable {R S G M N O ι : Type*}\n\n/-- The monoid algebra over a semiring `R` generated by the monoid `M`.\n\nIt is the type of finite formal `R`-linear combinations of terms of `M`,\nendowed with the convolution product. -/\n@[to_additive (relevant_arg := M)\n/-- The additive monoid algebra over a semiring `R` generated by the additive monoid `M`.\n\nIt is the type of finite formal `R`-linear combinations of terms of `M`,\nendowed with the convolution product. -/, to_additive_dont_translate]\ndef MonoidAlgebra (R M : Type*) [Semiring R] : Type _ := M →₀ R\n\nnamespace AddMonoidAlgebra\n\n@[inherit_doc AddMonoidAlgebra]\nscoped syntax:max (priority := high) term noWs \"[\" term \"]\" : term\n\nmacro_rules | `($R[$M]) => `(AddMonoidAlgebra $R $M)\n\n/-- Unexpander for `AddMonoidAlgebra`. -/\n@[scoped app_unexpander AddMonoidAlgebra]\nmeta def unexpander : Lean.PrettyPrinter.Unexpander\n  | `($_ $R $M) => `($R[$M])\n  | _ => throw ()\n\nend AddMonoidAlgebra\n\nnamespace MonoidAlgebra\n\n@[inherit_doc MonoidAlgebra]\nscoped syntax:max (priority := high) term noWs \"[\" term \"]\" : term\n\nmacro_rules | `($R[$M]) => `(MonoidAlgebra $R $M)\n\n/-- Unexpander for `MonoidAlgebra`. -/\n@[scoped app_unexpander MonoidAlgebra]\nmeta def unexpander : Lean.PrettyPrinter.Unexpander\n  | `($_ $R $M) => `($R[$M])\n  | _ => throw ()\n\nsection Semiring\nvariable [Semiring R] {x y : R[M]} {r r₁ r₂ : R} {m m' m₁ m₂ : M}\n\n/-- Construct an element of the monoid algebra `R[M]` from its coefficients `M →₀ R`. -/\n@[to_additive\n/-- Construct an element of the additive monoid algebra `R[M]` from its coefficients `M →₀ R`. -/]\ndef ofCoeff (x : M →₀ R) : R[M] := x\n\n/-- The coefficients `M →₀ R` of an element of the monoid algebra `R[M]`. -/\n@[to_additive\n/-- The coefficients `M →₀ R` of an element of the additive monoid algebra `R[M]`. -/]\ndef coeff (x : R[M]) : M →₀ R := x\n\n@[to_additive (attr := simp)] lemma coeff_ofCoeff (x : M →₀ R) : coeff (ofCoeff x) = x := rfl\n@[to_additive (attr := simp)] lemma ofCoeff_coeff (x : R[M]) : ofCoeff x.coeff = x := rfl\n\n/-- `MonoidAlgebra.coeff` as an equiv. -/\n@[to_additive (attr := simps apply symm_apply)\n/-- `AddMonoidAlgebra.coeff` as an equiv. -/]\ndef coeffEquiv : R[M] ≃ (M →₀ R) where\n  toFun := coeff\n  invFun := ofCoeff\n  left_inv _ := rfl\n  right_inv _ := rfl\n\n@[to_additive] lemma «forall» {P : R[M] → Prop} : (∀ p, P p) ↔ ∀ q, P (ofCoeff q) :=\n  coeffEquiv.forall_congr_left\n\n@[to_additive] lemma «exists» {P : R[M] → Prop} : (∃ p, P p) ↔ ∃ q, P (ofCoeff q) :=\n  coeffEquiv.exists_congr_left\n\n@[to_additive]\nlemma coeff_injective : (coeff : R[M] → M →₀ R).Injective := coeffEquiv.injective\n\n@[to_additive]\nlemma ofCoeff_injective : (ofCoeff : (M →₀ R) → R[M]).Injective := coeffEquiv.symm.injective\n\n@[to_additive (attr := simp)]\nlemma coeff_inj : x.coeff = y.coeff ↔ x = y := coeff_injective.eq_iff\n\n@[to_additive]\nlemma ofCoeff_inj {x y : M →₀ R} : ofCoeff x = ofCoeff y ↔ x = y := ofCoeff_injective.eq_iff\n\n@[to_additive] instance inhabited : Inhabited R[M] :=\n  inferInstanceAs <| Inhabited <| M →₀ R\n\n@[to_additive] instance nontrivial [Nontrivial R] [Nonempty M] : Nontrivial R[M] :=\n  inferInstanceAs <| Nontrivial <| M →₀ R\n\n@[to_additive] instance unique [Subsingleton R] : Unique R[M] :=\n  inferInstanceAs <| Unique <| M →₀ R\n\n@[to_additive] instance instDecidableEq [DecidableEq R] [DecidableEq M] : DecidableEq R[M] :=\n  inferInstanceAs <| DecidableEq <| M →₀ R\n\n-- TODO: this instance abuses definitional equality with `Finsupp.mapRange`\n@[to_additive] instance addCommMonoid : AddCommMonoid R[M] :=\n  fast_instance% { (inferInstance : AddCommMonoid <| M →₀ R) with\n    nsmul n x := x.mapRange (n • ·) (smul_zero _) }\n\n@[to_additive] instance instIsCancelAdd [IsCancelAdd R] : IsCancelAdd R[M] :=\n  inferInstanceAs <| IsCancelAdd <| M →₀ R\n\n-- TODO: Replace this with `coeff`. See https://github.com/leanprover-community/mathlib4/pull/36746\n#adaptation_note /-- Since nightly-2026-03-22,\nthis is needed or we get errors in UniversalFactorizationRing.lean -/\nset_option backward.inferInstanceAs.wrap false in\n@[to_additive] instance instCoeFun : CoeFun R[M] fun _ ↦ M → R :=\n  inferInstanceAs <| CoeFun (M →₀ R) fun _ ↦ M → R\n\n/-- A copy of `Finsupp.ext` for `MonoidAlgebra`. -/\n@[to_additive (attr := ext) /-- A copy of `Finsupp.ext` for `AddMonoidAlgebra`. -/]\nlemma ext ⦃f g : R[M]⦄ (hfg : ∀ m, f m = g m) : f = g := Finsupp.ext hfg\n\n/-- `MonoidAlgebra.coeff` as an `AddEquiv`. -/\n@[to_additive (attr := simps! apply symm_apply)\n/-- `AddMonoidAlgebra.coeff` as an `AddEquiv`. -/]\ndef coeffAddEquiv : R[M] ≃+ (M →₀ R) := coeffEquiv.addEquiv\n\n@[to_additive (attr := simp)] lemma coeff_zero : coeff (0 : R[M]) = 0 := rfl\n@[to_additive (attr := simp)] lemma ofCoeff_zero : (ofCoeff 0 : R[M]) = 0 := rfl\n@[to_additive (attr := simp)] lemma coeff_eq_zero : coeff x = 0 ↔ x = 0 := coeff_inj\n@[to_additive (attr := simp)] lemma ofCoeff_eq_zero {x : M →₀ R} : ofCoeff x = 0 ↔ x = 0 :=\n  ofCoeff_inj\n\n@[to_additive (attr := simp)]\nlemma coeff_add (x y : R[M]) : coeff (x + y) = coeff x + coeff y := rfl\n\n@[to_additive (attr := simp)]\nlemma ofCoeff_add (x y : M →₀ R) : ofCoeff (x + y) = ofCoeff x + ofCoeff y := rfl\n\n@[to_additive (attr := simp)]\nlemma coeff_sum (s : Finset ι) (f : ι → R[M]) :\n    coeff (∑ i ∈ s, f i) = ∑ i ∈ s, coeff (f i) := map_sum coeffAddEquiv ..\n\n@[to_additive (attr := simp)]\nlemma ofCoeff_sum (s : Finset ι) (f : ι → M →₀ R) :\n    ofCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, ofCoeff (f i) := map_sum coeffAddEquiv.symm ..\n\n@[to_additive (attr := simp)]\nlemma coeff_finsuppSum [AddCommMonoid N] (f : ι →₀ N) (g : ι → N → R[M]) :\n    coeff (f.sum g) = f.sum (fun i n ↦ coeff (g i n)) := map_finsuppSum coeffAddEquiv ..\n\n@[to_additive (attr := simp)]\nlemma ofCoeff_finsuppSum [AddCommMonoid N] (f : ι →₀ N) (g : ι → N → M →₀ R) :\n    ofCoeff (f.sum g) = f.sum (fun i n ↦ ofCoeff (g i n)) := map_finsuppSum coeffAddEquiv.symm ..\n\n-- TODO: This definition is very leaky, and we later have frequent problems conflating the two\n-- versions of `single`. Perhaps someone wants to try making this a `def` rather than an `abbrev`?\n-- In Mathlib 3 this was locally reducible.\n/-- `MonoidAlgebra.single m r` for `m : M`, `r : R` is the element `rm : R[M]`. -/\n@[to_additive\n/-- `AddMonoidAlgebra.single m r` for `m : M`, `r : R` is the element `rm : R[M]`. -/]\nabbrev single (m : M) (r : R) : R[M] := Finsupp.single m r\n\n/-- Remove a term from an element of the monoid algebra. -/\n@[to_additive /-- Remove a term from an element of the additive monoid algebra. -/]\ndef erase (m : M) (x : R[M]) : R[M] := .ofCoeff <| .erase m x.coeff\n\n@[to_additive (attr := simp)]\nlemma coeff_erase (m : M) (x : R[M]) : (x.erase m).coeff = x.coeff.erase m := rfl\n\n@[to_additive (attr := simp)]\nlemma ofCoeff_erase (m : M) (x : M →₀ R) : ofCoeff (x.erase m) = (ofCoeff x).erase m := rfl\n\n@[to_additive (attr := simp)]\nlemma erase_zero (m : M) : erase m (0 : R[M]) = 0 := by simp [erase]\n\n@[to_additive (attr := simp)]\nlemma erase_single (m : M) (r : R) : erase m (single m r) = 0 := by\n  simp [erase, ofCoeff, coeff]; rfl\n\n/-- Replace the `m`-th coefficient of an element `x` of the monoid algebra by a given value `r : R`.\nIf `r = 0`, this is equal to `x.erase m`. -/\n@[to_additive\n/-- Replace the `m`-th coefficient of an element `x` of the monoid algebra by a given value `r : R`.\nIf `r = 0`, this is equal to `x.erase m`. -/]\ndef update (m : M) (r : R) (x : R[M]) : R[M] :=\n  ofCoeff (x.coeff.update m r)\n\n@[to_additive (attr := simp)]\nlemma coeff_update (m : M) (r : R) (x : R[M]) :\n    (x.update m r).coeff = x.coeff.update m r := rfl\n\n@[to_additive (attr := simp)]\nlemma ofCoeff_update (m : M) (r : R) (x : M →₀ R) :\n    ofCoeff (x.update m r) = (ofCoeff x).update m r := rfl\n\nsection SMul\n\n/-! ### Basic scalar multiplication instances\n\nThis section collects instances needed for the algebraic structure of `Polynomial`,\nwhich is defined in terms of `MonoidAlgebra`.\nFurther results on scalar multiplication can be found in\n`Mathlib/Algebra/MonoidAlgebra/Module.lean`.\n-/\n\nvariable {A : Type*} [SMulZeroClass A R]\n\n-- TODO: this instance abuses definitional equality with `Finsupp.mapRange`\n@[to_additive (dont_translate := A) smulZeroClass]\ninstance smulZeroClass : SMulZeroClass A R[M] :=\n  fast_instance% { (inferInstance : SMulZeroClass A (M →₀ R)) with\n    smul a x := x.mapRange (a • ·) (smul_zero _) }\n\nsection\n-- Ensure that the different smul instances do not create a diamond.\nexample : (smulZeroClass (A := ℕ) (R := R) (M := M)).toSMul = addCommMonoid.toNSMul := by\n  with_reducible_and_instances rfl\n\n-- Ensure that smul has good defeq properties\nprivate local instance {α} [Monoid M] [SMul M α] : SMul Mˣ α where smul m a := (m : M) • a\nexample [Monoid A] (a : Units A) (x : R[M]) :\n    a • x = (a : A) • x := by\n  with_reducible_and_instances rfl\nend\n\n@[to_additive (dont_translate := A) (attr := simp) coeff_smul]\nlemma coeff_smul (a : A) (x : R[M]) : coeff (a • x) = a • coeff x := rfl\n\n@[to_additive (dont_translate := A) (attr := simp) ofCoeff_smul]\nlemma ofCoeff_smul (a : A) (x : M →₀ R) : ofCoeff (a • x) = a • ofCoeff x := rfl\n\n@[to_additive (attr := simp) (dont_translate := A) smul_apply]\nlemma smul_apply (a : A) (x : R[M]) (m : M) : (a • x) m = a • x m := rfl\n\n@[to_additive (attr := simp) (dont_translate := A) smul_single]\n\nTarget:\nlemma smul_single (a : A) (m : M) (r : R) : a • single m r = single m (a • r) :=\n\nProof body:\n","rejected":"by\n  exact smul_single","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"cc05767c2f159dbd97091dc9144cf3af5d7c6e0540f926bdf1aa9cee1bf9a0e1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/MonoidAlgebra","family_id":"smul_single","file_id":"mathlib/Mathlib/Algebra/MonoidAlgebra/Defs.lean","sample_id":"bcfecfd0c0cb2071cab40553cc6bdd48d75c67302392e5f99abe6ae29aa62d18"},"split":"train"}
{"binding":{"chosen_proof_sha256":"63ac599b07e7436f21a63f1b73d4a7f54c73a40c1b4a50f48b5455d9ae2693f2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a4a38b92b5df1cd680b4f1767b59c251ab23b001962fb810e16c3c22929494cf","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9614d072a9105eea2ab6f57e5011e3855902130db93743053ebaa70a564b484b","source_sha256":"a4ded0c37af8d7d4bd1dca4a7e378a31082efe29498bab67ebba55f071776588","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext j\n  dsimp\n  rw [mapBifunctorLeftUnitor_inv_apply, mapBifunctorLeftUnitor_inv_apply, assoc, assoc,\n    ι_mapBifunctorMapMap]\n  dsimp\n  rw [Functor.map_id, NatTrans.id_app, id_comp, ← NatTrans.naturality_assoc,\n    ← NatTrans.naturality_assoc]\n  rfl","hard_negative":true,"metrics":{"chosen_tokens":41,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.121951},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"c9bb09ec8112b748dc3c91abcccff58534697f8e7fe93d1c5e04e506aea549b5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.GradedObject.Associator\npublic import Mathlib.CategoryTheory.GradedObject.Single\n\nNamespace:\nCategoryTheory.GradedObject\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The left and right unitors\n\nGiven a bifunctor `F : C ⥤ D ⥤ D`, an object `X : C` such that `F.obj X ≅ 𝟭 D` and a\nmap `p : I × J → J` such that `hp : ∀ (j : J), p ⟨0, j⟩ = j`,\nwe define an isomorphism of `J`-graded objects for any `Y : GradedObject J D`.\n`mapBifunctorLeftUnitor F X e p hp Y : mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`.\nUnder similar assumptions, we also obtain a right unitor isomorphism\n`mapBifunctorMapObj F p X ((single₀ I).obj Y) ≅ X`. Finally,\nthe lemma `mapBifunctor_triangle` promotes a triangle identity involving functors\nto a triangle identity for the induced functors on graded objects.\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Category Limits\n\nnamespace GradedObject\n\nsection LeftUnitor\n\nvariable {C D I J : Type*} [Category* C] [Category* D]\n  [Zero I] [DecidableEq I] [HasInitial C]\n  (F : C ⥤ D ⥤ D) (X : C) (e : F.obj X ≅ 𝟭 D)\n  [∀ (Y : D), PreservesColimit (Functor.empty.{0} C) (F.flip.obj Y)]\n  (p : I × J → J) (hp : ∀ (j : J), p ⟨0, j⟩ = j)\n  (Y Y' : GradedObject J D) (φ : Y ⟶ Y')\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D` and `Y : GradedObject J D`,\nthis is the isomorphism `((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a ≅ Y a.2`\nwhen `a : I × J` is such that `a.1 = 0`. -/\n@[simps!]\nnoncomputable def mapBifunctorObjSingle₀ObjIso (a : I × J) (ha : a.1 = 0) :\n    ((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a ≅ Y a.2 :=\n  (F.mapIso (singleObjApplyIsoOfEq _ X _ ha)).app _ ≪≫ e.app (Y a.2)\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C` and `Y : GradedObject J D`,\n`((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a` is an initial object\nwhen `a : I × J` is such that `a.1 ≠ 0`. -/\nnoncomputable def mapBifunctorObjSingle₀ObjIsInitial (a : I × J) (ha : a.1 ≠ 0) :\n    IsInitial (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a) :=\n  IsInitial.isInitialObj (F.flip.obj (Y a.2)) _ (isInitialSingleObjApply _ _ _ ha)\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D`, `Y : GradedObject J D` and\n`p : I × J → J` such that `p ⟨0, j⟩ = j` for all `j`,\nthis is the (colimit) cofan which shall be used to construct the isomorphism\n`mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`, see `mapBifunctorLeftUnitor`. -/\nnoncomputable def mapBifunctorLeftUnitorCofan (hp : ∀ (j : J), p ⟨0, j⟩ = j) (Y) (j : J) :\n    (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y).CofanMapObjFun p j :=\n  CofanMapObjFun.mk _ _ _ (Y j) (fun a ha =>\n    if ha : a.1 = 0 then\n      (mapBifunctorObjSingle₀ObjIso F X e Y a ha).hom ≫ eqToHom (by aesop)\n    else\n      (mapBifunctorObjSingle₀ObjIsInitial F X Y a ha).to _)\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp, reassoc]\nlemma mapBifunctorLeftUnitorCofan_inj (j : J) :\n    (mapBifunctorLeftUnitorCofan F X e p hp Y j).inj ⟨⟨0, j⟩, hp j⟩ =\n      (F.map (singleObjApplyIso (0 : I) X).hom).app (Y j) ≫ e.hom.app (Y j) := by\n  simp [mapBifunctorLeftUnitorCofan]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The cofan `mapBifunctorLeftUnitorCofan F X e p hp Y j` is a colimit. -/\nnoncomputable def mapBifunctorLeftUnitorCofanIsColimit (j : J) :\n    IsColimit (mapBifunctorLeftUnitorCofan F X e p hp Y j) :=\n  Cofan.IsColimit.mk _\n    (fun s => e.inv.app (Y j) ≫\n      (F.map (singleObjApplyIso (0 : I) X).inv).app (Y j) ≫ s.inj ⟨⟨0, j⟩, hp j⟩)\n    (fun s => by\n      rintro ⟨⟨i, j'⟩, h⟩\n      by_cases hi : i = 0\n      · subst hi\n        simp only [Set.mem_preimage, hp, Set.mem_singleton_iff] at h\n        subst h\n        simp\n      · apply IsInitial.hom_ext\n        exact mapBifunctorObjSingle₀ObjIsInitial _ _ _ _ hi)\n    (fun s m hm => by simp [← hm ⟨⟨0, j⟩, hp j⟩])\n\ninclude e hp in\nlemma mapBifunctorLeftUnitor_hasMap :\n    HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y) p :=\n  CofanMapObjFun.hasMap _ _ _ (mapBifunctorLeftUnitorCofanIsColimit F X e p hp Y)\n\nvariable [HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y) p]\n  [HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y') p]\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D`, `Y : GradedObject J D` and\n`p : I × J → J` such that `p ⟨0, j⟩ = j` for all `j`,\nthis is the left unitor isomorphism `mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`. -/\nnoncomputable def mapBifunctorLeftUnitor : mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y :=\n  isoMk _ _ (fun j => (CofanMapObjFun.iso\n    (mapBifunctorLeftUnitorCofanIsColimit F X e p hp Y j)).symm)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma ι_mapBifunctorLeftUnitor_hom_apply (j : J) :\n    ιMapBifunctorMapObj F p ((single₀ I).obj X) Y 0 j j (hp j) ≫\n      (mapBifunctorLeftUnitor F X e p hp Y).hom j =\n      (F.map (singleObjApplyIso (0 : I) X).hom).app _ ≫ e.hom.app (Y j) := by\n  dsimp [mapBifunctorLeftUnitor]\n  erw [CofanMapObjFun.ιMapObj_iso_inv]\n  rw [mapBifunctorLeftUnitorCofan_inj]\n\nlemma mapBifunctorLeftUnitor_inv_apply (j : J) :\n    (mapBifunctorLeftUnitor F X e p hp Y).inv j =\n      e.inv.app (Y j) ≫ (F.map (singleObjApplyIso (0 : I) X).inv).app (Y j) ≫\n      ιMapBifunctorMapObj F p ((single₀ I).obj X) Y 0 j j (hp j) := rfl\n\nvariable {Y Y'}\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc]\n\nTarget:\nlemma mapBifunctorLeftUnitor_inv_naturality :\n    φ ≫ (mapBifunctorLeftUnitor F X e p hp Y').inv =\n      (mapBifunctorLeftUnitor F X e p hp Y).inv ≫ mapBifunctorMapMap F p (𝟙 _) φ :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_9614d072a910","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"182eea63fb00055802188d40d115deecf013ebd54a204df7d6d56a61235cdbdf","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/GradedObject","family_id":"mapbifunctorleftunitor_inv_naturality","file_id":"mathlib/Mathlib/CategoryTheory/GradedObject/Unitor.lean","sample_id":"9614d072a9105eea2ab6f57e5011e3855902130db93743053ebaa70a564b484b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0e11e10bbaaaed2dc873b84e165eba0bc5466c4e31acb825aeb49b92a5fe94d8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0da4ea72451dc03589d0ca83202fa7e907d0d6cd1c5ec72a025a542439f2033e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0996acbca200c0dccec01a46d4e81205ed46c98fd950cb9f91af86182dcfbe7d","source_sha256":"5630df9717d5ebdb1ddaef0ed587220f1abeb4c5c8759131e52bd9025a112e08","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  -- Let `Z` be a minimal set of prime ideals such that their product is contained in `J`.\n  obtain ⟨Z₀, hZ₀⟩ := exists_primeSpectrum_prod_le_and_ne_bot_of_domain hNF hI0\n  obtain ⟨Z, ⟨hZI, hprodZ⟩, h_eraseZ⟩ :=\n    wellFounded_lt.has_min\n      {Z | (Z.map asIdeal).prod ≤ I ∧ (Z.map asIdeal).prod ≠ ⊥}\n      ⟨Z₀, hZ₀.1, hZ₀.2⟩\n  obtain ⟨_, hPZ', hPM⟩ := hM.isPrime.multiset_prod_le.mp (hZI.trans hIM)\n  -- Then in fact there is a `P ∈ Z` with `P ≤ M`.\n  obtain ⟨P, hPZ, rfl⟩ := Multiset.mem_map.mp hPZ'\n  classical\n    have := Multiset.map_erase asIdeal (fun _ _ => PrimeSpectrum.ext) P Z\n    obtain ⟨hP0, hZP0⟩ : P.asIdeal ≠ ⊥ ∧ ((Z.erase P).map asIdeal).prod ≠ ⊥ := by\n      rwa [Ne, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←\n        this] at hprodZ\n    -- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.\n    have hPM' := (P.isPrime.isMaximal hP0).eq_of_le hM.ne_top hPM\n    subst hPM'\n    -- By minimality of `Z`, erasing `P` from `Z` is exactly what we need.\n    refine ⟨Z.erase P, ?_, ?_⟩\n    · convert! hZI\n      rw [this, Multiset.cons_erase hPZ']\n    · refine fun h => h_eraseZ (Z.erase P) ⟨h, ?_⟩ (Multiset.erase_lt.mpr hPZ)\n      exact hZP0","hard_negative":true,"metrics":{"chosen_tokens":346,"rejected_tokens":2,"token_jaccard":0.008197,"token_length_ratio":0.00578},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"c9f23e7ff9469079c9e017a90c3dff1e9d3c1b111f58a2032fe9bb800321eb1f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Subalgebra.Pointwise\npublic import Mathlib.RingTheory.DedekindDomain.Basic\npublic import Mathlib.RingTheory.FractionalIdeal.Inverse\npublic import Mathlib.RingTheory.Spectrum.Prime.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Kenji Nakagawa. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio\n-/\n/-!\n# Dedekind domains and invertible ideals\n\nIn this file, we show a ring is a Dedekind domain iff all fractional ideals are invertible,\nand prove instances such as the unique factorization of ideals.\nFurther results on the structure of ideals in a Dedekind domain are found in\n`Mathlib/RingTheory/DedekindDomain/Ideal/Lemmas.lean`.\n\n## Main definitions\n\n- `IsDedekindDomainInv` alternatively defines a Dedekind domain as an integral domain where\n  every nonzero fractional ideal is invertible.\n- `isDedekindDomainInv_iff` shows that this does not depend on the choice of field of\n  fractions.\n\n## Main results:\n\n- `isDedekindDomain_iff_isDedekindDomainInv`\n- `Ideal.uniqueFactorizationMonoid`\n\n## Implementation notes\n\nThe definitions that involve a field of fractions choose a canonical field of fractions,\nbut are independent of that choice. The `..._iff` lemmas express this independence.\n\nOften, definitions assume that Dedekind domains are not fields. We found it more practical\nto add a `(h : ¬ IsField A)` assumption whenever this is explicitly needed.\n\n## References\n\n* [D. Marcus, *Number Fields*][marcus1977number]\n* [J.W.S. Cassels, A. Fröhlich, *Algebraic Number Theory*][cassels1967algebraic]\n* [J. Neukirch, *Algebraic Number Theory*][Neukirch1992]\n\n## Tags\n\ndedekind domain, dedekind ring\n-/\n\n@[expose] public section\n\nvariable (R A K : Type*) [CommRing R] [CommRing A] [Field K]\n\nopen scoped nonZeroDivisors Polynomial\n\nsection Inverse\n\nsection IsDedekindDomainInv\n\nvariable [IsDomain A]\n/-- A Dedekind domain is an integral domain such that every fractional ideal has an inverse.\n\nThis is equivalent to `IsDedekindDomain`.\nIn particular we provide a `CommGroupWithZero` instance,\nassuming `IsDedekindDomain A`, which implies `IsDedekindDomainInv`. For **integral** domain,\n`IsDedekindDomain`(`Inv`) implies only `Ideal.isCancelMulZero`.\n-/\ndef IsDedekindDomainInv : Prop :=\n  ∀ I ≠ (⊥ : FractionalIdeal A⁰ (FractionRing A)), I * I⁻¹ = 1\n\nopen FractionalIdeal\n\nvariable {R A K}\n\ntheorem isDedekindDomainInv_iff [Algebra A K] [IsFractionRing A K] :\n    IsDedekindDomainInv A ↔ ∀ I ≠ (⊥ : FractionalIdeal A⁰ K), I * I⁻¹ = 1 := by\n  let h : FractionalIdeal A⁰ (FractionRing A) ≃+* FractionalIdeal A⁰ K :=\n    FractionalIdeal.mapEquiv (FractionRing.algEquiv A K)\n  refine h.toEquiv.forall_congr (fun {x} => ?_)\n  rw [← h.toEquiv.apply_eq_iff_eq]\n  simp [h]\n\ntheorem FractionalIdeal.adjoinIntegral_eq_one_of_isUnit [Algebra A K] [IsFractionRing A K] (x : K)\n    (hx : IsIntegral A x) (hI : IsUnit (adjoinIntegral A⁰ x hx)) : adjoinIntegral A⁰ x hx = 1 := by\n  set I := adjoinIntegral A⁰ x hx\n  have mul_self : IsIdempotentElem I := by\n    apply coeToSubmodule_injective\n    simp only [coe_mul, adjoinIntegral_coe, I]\n    rw [(Algebra.adjoin A {x}).isIdempotentElem_toSubmodule]\n  convert! congr_arg (· * I⁻¹) mul_self <;>\n    simp only [(mul_inv_cancel_iff_isUnit K).mpr hI, mul_assoc, mul_one]\n\nnamespace IsDedekindDomainInv\n\nvariable [Algebra A K] [IsFractionRing A K] (h : IsDedekindDomainInv A) {I J : FractionalIdeal A⁰ K}\ninclude h\n\n/-- `IsDedekindDomainInv A` implies that fractional ideals over it form a commutative group with\nzero. -/\nnoncomputable abbrev commGroupWithZero : CommGroupWithZero (FractionalIdeal A⁰ K) where\n  inv_zero := inv_zero' _\n  mul_inv_cancel := isDedekindDomainInv_iff.mp h\n  div_eq_mul_inv I J := by\n    obtain rfl | hJ := eq_or_ne J 0\n    · simp [inv_zero']\n    refine le_antisymm ?_ ((FractionalIdeal.le_div_iff_mul_le hJ).2 ?_)\n    · suffices I / J * J ≤ I by\n        simpa [mul_assoc, isDedekindDomainInv_iff.mp h _ hJ] using mul_left_mono (a := J⁻¹) this\n      simp [FractionalIdeal.mul_le, mem_div_iff_of_ne_zero hJ]\n    · rw [mul_assoc, mul_comm _ J, isDedekindDomainInv_iff.mp h _ hJ, mul_one]\n\ntheorem isNoetherianRing : IsNoetherianRing A := by\n  let := h.commGroupWithZero (K := FractionRing A)\n  refine isNoetherianRing_iff.mpr ⟨fun I : Ideal A => ?_⟩\n  by_cases hI : I = ⊥\n  · rw [hI]; apply Submodule.fg_bot\n  have hI : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI\n  exact I.fg_of_isUnit (IsFractionRing.injective A (FractionRing A)) hI.isUnit\n\ntheorem integrallyClosed : IsIntegrallyClosed A := by\n  let := h.commGroupWithZero (K := FractionRing A)\n  -- It suffices to show that for integral `x`,\n  -- `A[x]` (which is a fractional ideal) is in fact equal to `A`.\n  refine (isIntegrallyClosed_iff (FractionRing A)).mpr (fun {x hx} => ?_)\n  rw [← Set.mem_range, ← Algebra.mem_bot, ← Subalgebra.mem_toSubmodule, Algebra.toSubmodule_bot,\n    Submodule.one_eq_span, ← coe_spanSingleton A⁰ (1 : FractionRing A), spanSingleton_one, ←\n    FractionalIdeal.adjoinIntegral_eq_one_of_isUnit x hx (Ne.isUnit _)]\n  · exact mem_adjoinIntegral_self A⁰ x hx\n  · exact fun h => one_ne_zero (eq_zero_iff.mp h 1 (Algebra.adjoin A {x}).one_mem)\n\nopen Ring\n\ntheorem dimensionLEOne : DimensionLEOne A := by\n  -- We're going to show that `P` is maximal because any (maximal) ideal `M`\n  -- that is strictly larger would be `⊤`.\n  let := h.commGroupWithZero (K := FractionRing A)\n  constructor\n  rintro P P_ne hP\n  refine Ideal.isMaximal_def.mpr ⟨hP.ne_top, fun M hM => ?_⟩\n  -- We may assume `P` and `M` (as fractional ideals) are nonzero.\n  have P'_ne : (P : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr P_ne\n  have M'_ne : (M : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hM.ne_bot\n  -- In particular, we'll show `M⁻¹ * P ≤ P`\n  suffices (M⁻¹ : FractionalIdeal A⁰ (FractionRing A)) * P ≤ P by\n    rw [eq_top_iff, ← coeIdeal_le_coeIdeal (FractionRing A), coeIdeal_top]\n    calc\n      (1 : FractionalIdeal A⁰ (FractionRing A)) = (↑M)⁻¹ * P * ((↑P)⁻¹ * M) := by\n        simp [mul_assoc, *]\n      _ ≤ P * ((↑P)⁻¹ * M) := by gcongr\n      _ = M := by simp [*]\n  -- Suppose we have `x ∈ M⁻¹ * P`, then in fact `x = algebraMap _ _ y` for some `y`.\n  intro x hx\n  have le_one : (M⁻¹ : FractionalIdeal A⁰ (FractionRing A)) * P ≤ 1 := by\n    rw [← inv_mul_cancel₀ M'_ne]; gcongr\n  obtain ⟨y, _hy, rfl⟩ := (mem_coeIdeal _).mp (le_one hx)\n  -- Since `M` is strictly greater than `P`, let `z ∈ M \\ P`.\n  obtain ⟨z, hzM, hzp⟩ := SetLike.exists_of_lt hM\n  -- We have `z * y ∈ M * (M⁻¹ * P) = P`.\n  have zy_mem := mul_mem_mul (mem_coeIdeal_of_mem A⁰ hzM) hx\n  rw [← map_mul, ← mul_assoc, mul_inv_cancel₀ M'_ne, one_mul] at zy_mem\n  obtain ⟨zy, hzy, zy_eq⟩ := (mem_coeIdeal A⁰).mp zy_mem\n  rw [IsFractionRing.injective A (FractionRing A) zy_eq] at hzy\n  -- But `P` is a prime ideal, so `z ∉ P` implies `y ∈ P`, as desired.\n  exact mem_coeIdeal_of_mem A⁰ (Or.resolve_left (hP.mem_or_mem hzy) hzp)\n\n/-- Showing one side of the equivalence between the definitions\n`IsDedekindDomainInv` and `IsDedekindDomain` of Dedekind domains. -/\ntheorem isDedekindDomain : IsDedekindDomain A :=\n  { h.isNoetherianRing, h.dimensionLEOne, h.integrallyClosed with }\n\nend IsDedekindDomainInv\n\nend IsDedekindDomainInv\n\nvariable [Algebra A K] [IsFractionRing A K]\n\nvariable {A K}\n\ntheorem FractionalIdeal.one_mem_inv_coe_ideal [IsDomain A] {I : Ideal A} (hI : I ≠ ⊥) :\n    (1 : K) ∈ (I : FractionalIdeal A⁰ K)⁻¹ := by\n  rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI)]\n  intro y hy\n  rw [one_mul]\n  exact coeIdeal_le_one hy\n\n@[deprecated (since := \"2026-04-16\")]\nalias one_mem_inv_coe_ideal := FractionalIdeal.one_mem_inv_coe_ideal\n\n/-- Specialization of `exists_primeSpectrum_prod_le_and_ne_bot_of_domain` to Dedekind domains:\nLet `I : Ideal A` be a nonzero ideal, where `A` is a Dedekind domain that is not a field.\nThen `exists_primeSpectrum_prod_le_and_ne_bot_of_domain` states we can find a product of prime\nideals that is contained within `I`. This lemma extends that result by making the product minimal:\nlet `M` be a maximal ideal that contains `I`, then the product including `M` is contained within `I`\nand the product excluding `M` is not contained within `I`. -/\n\nTarget:\ntheorem PrimeSpectrum.exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A]\n    (hNF : ¬IsField A) {I M : Ideal A} (hI0 : I ≠ ⊥) (hIM : I ≤ M) [hM : M.IsMaximal] :\n    ∃ Z : Multiset (PrimeSpectrum A),\n      (M ::ₘ Z.map asIdeal).prod ≤ I ∧\n        ¬Multiset.prod (Z.map asIdeal) ≤ I :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_0996acbca200","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"04744208de7d7bdddb9dac48ae8cf6a515aca5ab180f1139e812531dac867fc4","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/DedekindDomain","family_id":"primespectrum","file_id":"mathlib/Mathlib/RingTheory/DedekindDomain/Ideal/Basic.lean","sample_id":"0996acbca200c0dccec01a46d4e81205ed46c98fd950cb9f91af86182dcfbe7d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1bc941a2b3a47d131821876175410f2047d53e35df24a12c023e660701f0a1b1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f74f3c30220b04da1fe4f1b1519530cecb6ea35343f0fc2bc9e9549863b219a6","source_sha256":"1c1559946be132288407174c056f2f416039575ec7e1421e4402d1878b97979a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← isLowerSet_compl, ← coe_compl] at h𝒜\n  have := h𝒜.card_inter_le_finset hℬ\n  rwa [card_compl, Fintype.card_finset, tsub_mul, le_tsub_iff_le_tsub, ← mul_tsub, ←\n    card_sdiff_of_subset inter_subset_right, sdiff_inter_self_right, sdiff_compl,\n    _root_.inf_comm] at this\n  · grw [inter_subset_right]\n  · grw [← Fintype.card_finset, card_le_univ]","hard_negative":false,"metrics":{"chosen_tokens":64,"rejected_tokens":3,"token_jaccard":0.028571,"token_length_ratio":0.046875},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"ca2f03f376599b0ab7731ddc7d924f56fca241f65fcc7aaf9ec8ee85d6d874c6","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Ring.Canonical\npublic import Mathlib.Algebra.Order.Ring.Nat\npublic import Mathlib.Combinatorics.SetFamily.Compression.Down\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Order.UpperLower.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Harris-Kleitman inequality\n\nThis file proves the Harris-Kleitman inequality. This relates `#𝒜 * #ℬ` and\n`2 ^ card α * #(𝒜 ∩ ℬ)` where `𝒜` and `ℬ` are upward- or downcard-closed finite families of\nfinsets. This can be interpreted as saying that any two lower sets (resp. any two upper sets)\ncorrelate in the uniform measure.\n\n## Main declarations\n\n* `IsLowerSet.le_card_inter_finset`: One form of the Harris-Kleitman inequality.\n\n## References\n\n* [D. J. Kleitman, *Families of non-disjoint subsets*][kleitman1966]\n-/\n\npublic section\n\n\nopen Finset\n\nvariable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}\n\ntheorem IsLowerSet.nonMemberSubfamily (h : IsLowerSet (𝒜 : Set (Finset α))) :\n    IsLowerSet (𝒜.nonMemberSubfamily a : Set (Finset α)) := fun s t hts => by\n  simp_rw [mem_coe, mem_nonMemberSubfamily]\n  exact And.imp (h hts) (mt <| @hts _)\n\ntheorem IsLowerSet.memberSubfamily (h : IsLowerSet (𝒜 : Set (Finset α))) :\n    IsLowerSet (𝒜.memberSubfamily a : Set (Finset α)) := by\n  rintro s t hts\n  simp_rw [mem_coe, mem_memberSubfamily]\n  exact And.imp (h <| insert_subset_insert _ hts) (mt <| @hts _)\n\ntheorem IsLowerSet.memberSubfamily_subset_nonMemberSubfamily (h : IsLowerSet (𝒜 : Set (Finset α))) :\n    𝒜.memberSubfamily a ⊆ 𝒜.nonMemberSubfamily a := fun s => by\n  rw [mem_memberSubfamily, mem_nonMemberSubfamily]\n  exact And.imp_left (h <| subset_insert _ _)\n\n/-- **Harris-Kleitman inequality**: Any two lower sets of finsets correlate. -/\ntheorem IsLowerSet.le_card_inter_finset' (h𝒜 : IsLowerSet (𝒜 : Set (Finset α)))\n    (hℬ : IsLowerSet (ℬ : Set (Finset α))) (h𝒜s : ∀ t ∈ 𝒜, t ⊆ s) (hℬs : ∀ t ∈ ℬ, t ⊆ s) :\n    #𝒜 * #ℬ ≤ 2 ^ #s * #(𝒜 ∩ ℬ) := by\n  induction s using Finset.induction generalizing 𝒜 ℬ with\n  | empty =>\n    simp_rw [subset_empty, ← subset_singleton_iff', subset_singleton_iff] at h𝒜s hℬs\n    obtain rfl | rfl := h𝒜s\n    · simp only [card_empty, zero_mul, empty_inter, mul_zero, le_refl]\n    obtain rfl | rfl := hℬs\n    · simp\n    · simp only [card_empty, pow_zero, inter_singleton_of_mem, mem_singleton, card_singleton,\n        le_refl]\n  | insert a s hs ih =>\n  rw [card_insert_of_notMem hs, ← card_memberSubfamily_add_card_nonMemberSubfamily a 𝒜, ←\n    card_memberSubfamily_add_card_nonMemberSubfamily a ℬ, add_mul, mul_add, mul_add,\n    add_comm (_ * _), add_add_add_comm]\n  grw [mul_add_mul_le_mul_add_mul\n    (card_le_card h𝒜.memberSubfamily_subset_nonMemberSubfamily) <|\n      card_le_card hℬ.memberSubfamily_subset_nonMemberSubfamily, ← two_mul, pow_succ', mul_assoc]\n  have h₀ : ∀ 𝒞 : Finset (Finset α), (∀ t ∈ 𝒞, t ⊆ insert a s) →\n      ∀ t ∈ 𝒞.nonMemberSubfamily a, t ⊆ s := by\n    rintro 𝒞 h𝒞 t ht\n    rw [mem_nonMemberSubfamily] at ht\n    exact (subset_insert_iff_of_notMem ht.2).1 (h𝒞 _ ht.1)\n  have h₁ : ∀ 𝒞 : Finset (Finset α), (∀ t ∈ 𝒞, t ⊆ insert a s) →\n      ∀ t ∈ 𝒞.memberSubfamily a, t ⊆ s := by\n    rintro 𝒞 h𝒞 t ht\n    rw [mem_memberSubfamily] at ht\n    exact (subset_insert_iff_of_notMem ht.2).1 ((subset_insert _ _).trans <| h𝒞 _ ht.1)\n  gcongr\n  refine (add_le_add (ih h𝒜.memberSubfamily hℬ.memberSubfamily (h₁ _ h𝒜s) <| h₁ _ hℬs) <|\n    ih h𝒜.nonMemberSubfamily hℬ.nonMemberSubfamily (h₀ _ h𝒜s) <| h₀ _ hℬs).trans_eq ?_\n  rw [← mul_add, ← memberSubfamily_inter, ← nonMemberSubfamily_inter,\n    card_memberSubfamily_add_card_nonMemberSubfamily]\n\nvariable [Fintype α]\n\n/-- **Harris-Kleitman inequality**: Any two lower sets of finsets correlate. -/\ntheorem IsLowerSet.le_card_inter_finset (h𝒜 : IsLowerSet (𝒜 : Set (Finset α)))\n    (hℬ : IsLowerSet (ℬ : Set (Finset α))) : #𝒜 * #ℬ ≤ 2 ^ Fintype.card α * #(𝒜 ∩ ℬ) :=\nh𝒜.le_card_inter_finset' hℬ (fun _ _ => subset_univ _) fun _ _ => subset_univ _\n\n/-- **Harris-Kleitman inequality**: Upper sets and lower sets of finsets anticorrelate. -/\ntheorem IsUpperSet.card_inter_le_finset (h𝒜 : IsUpperSet (𝒜 : Set (Finset α)))\n    (hℬ : IsLowerSet (ℬ : Set (Finset α))) :\n    2 ^ Fintype.card α * #(𝒜 ∩ ℬ) ≤ #𝒜 * #ℬ := by\n  rw [← isLowerSet_compl, ← coe_compl] at h𝒜\n  have := h𝒜.le_card_inter_finset hℬ\n  rwa [card_compl, Fintype.card_finset, tsub_mul, tsub_le_iff_tsub_le, ← mul_tsub, ←\n    card_sdiff_of_subset inter_subset_right, sdiff_inter_self_right, sdiff_compl,\n    _root_.inf_comm] at this\n\n/-- **Harris-Kleitman inequality**: Lower sets and upper sets of finsets anticorrelate. -/\ntheorem IsLowerSet.card_inter_le_finset (h𝒜 : IsLowerSet (𝒜 : Set (Finset α)))\n    (hℬ : IsUpperSet (ℬ : Set (Finset α))) :\n    2 ^ Fintype.card α * #(𝒜 ∩ ℬ) ≤ #𝒜 * #ℬ := by\n  rw [inter_comm, mul_comm #𝒜]\n  exact hℬ.card_inter_le_finset h𝒜\n\n/-- **Harris-Kleitman inequality**: Any two upper sets of finsets correlate. -/\n\nTarget:\ntheorem IsUpperSet.le_card_inter_finset (h𝒜 : IsUpperSet (𝒜 : Set (Finset α)))\n    (hℬ : IsUpperSet (ℬ : Set (Finset α))) :\n    #𝒜 * #ℬ ≤ 2 ^ Fintype.card α * #(𝒜 ∩ ℬ) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/SetFamily","family_id":"isupperset","file_id":"mathlib/Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean","sample_id":"f74f3c30220b04da1fe4f1b1519530cecb6ea35343f0fc2bc9e9549863b219a6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7f54226990013dd5c5400459b377a357ecf590ee9b93a656a04179c0ea205d15","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4e7c68a0ae1706e7fbfa84a0b8e103da95b624a1edb0fcbc393efbb9763f79d8","source_sha256":"d54536321e31ab6ecb143f6ff47411119fb9708f05bec728ae60cd91a7b3224f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isUnit_iff_coprime]\n  exact (Prime.dvd_iff_not_coprime <| prime_of_mem_primeFactors h).mp <| dvd_of_mem_primeFactors h","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":8,"token_jaccard":0.043478,"token_length_ratio":0.380952},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"ca60b76884b829a0c1f9e3277150bb48b5858066fdc3a96eca24ec9598f1ff64","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Associated\npublic import Mathlib.Data.ZMod.Basic\npublic import Mathlib.RingTheory.Coprime.Lemmas\n\nNamespace:\nZMod\n\nLocal context:\n/-\nCopyright (c) 2023 Moritz Firsching. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Moritz Firsching, Ashvni Narayanan, Michael Stoll\n-/\n/-!\n# Lemmas about units in `ZMod`.\n-/\n\n@[expose] public section\n\nassert_not_exists TwoSidedIdeal\n\nnamespace ZMod\n\nvariable {n m : ℕ}\n/-- `unitsMap` is a group homomorphism that maps units of `ZMod m` to units of `ZMod n` when `n`\ndivides `m`. -/\ndef unitsMap (hm : n ∣ m) : (ZMod m)ˣ →* (ZMod n)ˣ := Units.map (castHom hm (ZMod n))\n\nlemma unitsMap_def (hm : n ∣ m) : unitsMap hm = Units.map (castHom hm (ZMod n)) := rfl\n\nlemma unitsMap_comp {d : ℕ} (hm : n ∣ m) (hd : m ∣ d) :\n    (unitsMap hm).comp (unitsMap hd) = unitsMap (dvd_trans hm hd) := by\n  simp only [unitsMap_def]\n  rw [← Units.map_comp]\n  exact congr_arg Units.map <| congr_arg RingHom.toMonoidHom <| castHom_comp hm hd\n\n@[simp]\nlemma unitsMap_self (n : ℕ) : unitsMap (dvd_refl n) = MonoidHom.id _ := by\n  simp [unitsMap, castHom_self]\n\n/-- `unitsMap_val` shows that coercing from `(ZMod m)ˣ` to `ZMod n` gives the same result\nwhen going via `(ZMod n)ˣ` and `ZMod m`. -/\nlemma unitsMap_val (h : n ∣ m) (a : (ZMod m)ˣ) :\n    ↑(unitsMap h a) = ((a : ZMod m).cast : ZMod n) := rfl\n\nlemma isUnit_cast_of_dvd (hm : n ∣ m) (a : Units (ZMod m)) : IsUnit (cast (a : ZMod m) : ZMod n) :=\n  Units.isUnit (unitsMap hm a)\ntheorem unitsMap_surjective [hm : NeZero m] (h : n ∣ m) :\n    Function.Surjective (unitsMap h) := by\n  suffices ∀ x : ℕ, x.Coprime n → ∃ k : ℕ, (x + k * n).Coprime m by\n    intro x\n    have ⟨k, hk⟩ := this x.val.val (val_coe_unit_coprime x)\n    refine ⟨unitOfCoprime _ hk, Units.ext ?_⟩\n    have : NeZero n := ⟨fun hn ↦ hm.out (eq_zero_of_zero_dvd (hn ▸ h))⟩\n    simp [unitsMap_def, -castHom_apply]\n  intro x hx\n  let ps : Finset ℕ := {p ∈ m.primeFactors | ¬p ∣ x}\n  use ps.prod id\n  apply Nat.coprime_of_dvd\n  intro p pp hp hpn\n  by_cases hpx : p ∣ x\n  · have h := Nat.dvd_sub hp hpx\n    rw [add_comm, Nat.add_sub_cancel] at h\n    rcases pp.dvd_mul.mp h with h | h\n    · have ⟨q, hq, hq'⟩ := (pp.prime.dvd_finsetProd_iff id).mp h\n      rw [Finset.mem_filter, Nat.mem_primeFactors,\n        ← (Nat.prime_dvd_prime_iff_eq pp hq.1.1).mp hq'] at hq\n      exact hq.2 hpx\n    · exact Nat.Prime.not_coprime_iff_dvd.mpr ⟨p, pp, hpx, h⟩ hx\n  · have pps : p ∈ ps := Finset.mem_filter.mpr ⟨Nat.mem_primeFactors.mpr ⟨pp, hpn, hm.out⟩, hpx⟩\n    have h := Nat.dvd_sub hp ((Finset.dvd_prod_of_mem id pps).mul_right n)\n    rw [Nat.add_sub_cancel] at h\n    contradiction\n\n-- This needs `Nat.primeFactors`, so cannot go into `Mathlib/Data/ZMod/Basic.lean`.\nopen Nat in\n\nTarget:\nlemma not_isUnit_of_mem_primeFactors {n p : ℕ} (h : p ∈ n.primeFactors) :\n    ¬ IsUnit (p : ZMod n) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"d795702a186429c0c2a7edc8a9baa750dc174c3b70c4d0d1eecd3d5c261fa5c5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/ZMod","family_id":"not_isunit_of_mem_primefactors","file_id":"mathlib/Mathlib/Data/ZMod/Units.lean","sample_id":"4e7c68a0ae1706e7fbfa84a0b8e103da95b624a1edb0fcbc393efbb9763f79d8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dad9f4770c642653b3a83e01d76fcc6253ccf22ac06ee8b959ff773f2af21b7e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b9481162d2a39b870dac6d7f3cc6ca153438f1e04d0e98096b1c95aa686092e9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5574e1cf4d2a2166ca8dd385d8105f10751922211dcf708a3a1a0c7976bbd7d3","source_sha256":"e70fb56bf1193d4c52139a9e4e22d574aabafc8e5d21aec95ae03e5150f51760","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  exact\n    { map_smul := fun f r => by\n        simp only [← NatIso.naturality_1 e (r • f), F.map_smul, Linear.smul_comp,\n          NatTrans.naturality, Linear.comp_smul, Iso.inv_hom_id_app_assoc] }","hard_negative":false,"metrics":{"chosen_tokens":45,"rejected_tokens":50,"token_jaccard":0.885714,"token_length_ratio":1.111111},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"ca708eb0155b07a93ca35e3418adf0ea7e9ddef41db818b606b22f9f3fa606ca","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor\npublic import Mathlib.CategoryTheory.Linear.Basic\npublic import Mathlib.Algebra.Module.LinearMap.Rat\n\nNamespace:\nCategoryTheory.Functor\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Linear Functors\n\nAn additive functor between two `R`-linear categories is called *linear*\nif the induced map on hom types is a morphism of `R`-modules.\n\n## Implementation details\n\n`Functor.Linear` is a `Prop`-valued class, defined by saying that\nfor every two objects `X` and `Y`, the map\n`F.map : (X ⟶ Y) → (F.obj X ⟶ F.obj Y)` is a morphism of `R`-modules.\n\n-/\n\n@[expose] public section\n\n\nnamespace CategoryTheory\n\nvariable (R : Type*) [Semiring R] {C D : Type*} [Category* C] [Category* D]\n  [Preadditive C] [Preadditive D] [CategoryTheory.Linear R C] [CategoryTheory.Linear R D]\n  (F : C ⥤ D)\n\n/-- An additive functor `F` is `R`-linear provided `F.map` is an `R`-module morphism. -/\nclass Functor.Linear : Prop where\n  /-- the functor induces a linear map on morphisms -/\n  map_smul : ∀ {X Y : C} (f : X ⟶ Y) (r : R), F.map (r • f) = r • F.map f := by cat_disch\n\nlemma Functor.linear_iff (F : C ⥤ D) :\n    Functor.Linear R F ↔ ∀ (X : C) (r : R), F.map (r • 𝟙 X) = r • 𝟙 (F.obj X) := by\n  constructor\n  · intro h X r\n    rw [h.map_smul, F.map_id]\n  · refine fun h => ⟨fun {X Y} f r => ?_⟩\n    have : r • f = (r • 𝟙 X) ≫ f := by simp\n    rw [this, F.map_comp, h, Linear.smul_comp, Category.id_comp]\n\nsection Linear\n\nnamespace Functor\n\nsection\n\nvariable {R} [Linear R F]\n\n@[simp]\ntheorem map_smul {X Y : C} (r : R) (f : X ⟶ Y) : F.map (r • f) = r • F.map f :=\n  Functor.Linear.map_smul _ _\n\n@[simp]\ntheorem map_units_smul {X Y : C} (r : Rˣ) (f : X ⟶ Y) : F.map (r • f) = r • F.map f := by\n  apply map_smul\n\ninstance : Linear R (𝟭 C) where\n\nsection\n\nvariable {E : Type*} [Category* E] [Preadditive E] [CategoryTheory.Linear R E] (G : D ⥤ E)\n\ninstance [Linear R G] : Linear R (F ⋙ G) where\n\nset_option backward.isDefEq.respectTransparency false in\nlemma linear_of_full_essSurj_comp [F.Full] [F.EssSurj] [Functor.Linear R (F ⋙ G)] :\n    Functor.Linear R G := by\n  refine ⟨fun {X Y} f r ↦ ?_⟩\n  obtain ⟨X', Y', eX, eY, f', rfl⟩ :\n      ∃ (X' Y' : C) (eX : F.obj X' ≅ X) (eY : F.obj Y' ≅ Y)\n        (f' : X' ⟶ Y'), f = eX.inv ≫ F.map f' ≫ eY.hom := by\n    obtain ⟨f', hf'⟩ :=\n      F.map_surjective ((F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv)\n    exact ⟨_, _, F.objObjPreimageIso X, F.objObjPreimageIso Y, f', by cat_disch⟩\n  simpa only [comp_map, map_smul, Linear.smul_comp, Linear.comp_smul, ← G.map_comp]\n    using G.map eX.inv ≫= ((F ⋙ G).map_smul r f') =≫ G.map eY.hom\n\nlemma linear_comp_iff_of_full_of_essSurj [F.Full] [F.EssSurj] :\n    Functor.Linear R (F ⋙ G) ↔ Functor.Linear R G :=\n  ⟨fun _ ↦ linear_of_full_essSurj_comp F G, fun _ ↦ inferInstance⟩\n\nend\n\nvariable (R) [F.Additive]\n\n/-- `F.mapLinearMap` is an `R`-linear map whose underlying function is `F.map`. -/\n@[simps]\ndef mapLinearMap {X Y : C} : (X ⟶ Y) →ₗ[R] F.obj X ⟶ F.obj Y :=\n  { F.mapAddHom with map_smul' := fun r f => F.map_smul r f }\n\ntheorem coe_mapLinearMap {X Y : C} : ⇑(F.mapLinearMap R : (X ⟶ Y) →ₗ[R] _) = F.map := rfl\n\nend\n\nvariable {F} in\n\nTarget:\nlemma linear_of_iso {G : C ⥤ D} (e : F ≅ G) [F.Linear R] : G.Linear R :=\n\nProof body:\n","rejected":"by\n  exact\n    { map_smul := fun f r => by\n        simp only [← NatIso.naturality_1 e (r • f), F.map_smul, Linear.smul_comp,\n          NatTrans.naturality, Linear.comp_smul, Iso.inv_hom_id_app_assoc] }\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Linear","family_id":"linear_of_iso","file_id":"mathlib/Mathlib/CategoryTheory/Linear/LinearFunctor.lean","sample_id":"5574e1cf4d2a2166ca8dd385d8105f10751922211dcf708a3a1a0c7976bbd7d3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a57d660deed4ffd6ef566ef7d9ca6c07159f27d62499692c71e882d53f3bfa34","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"211d293f722fe1130c34a732f32615615b914ff3488d05c85b24b7fea5b7d189","source_sha256":"6515bdb4a68678b27490ec2b3332f6d7b1e529489ff9d844898ec1eda5637ec5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  grind [= lt_iff_le_not_ge]","hard_negative":false,"metrics":{"chosen_tokens":6,"rejected_tokens":2,"token_jaccard":0.142857,"token_length_ratio":0.333333},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"ca849763710d72eb71dfd88a8dd1897311af4d1f754e1be7d576616db52150f6","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Tactic.Alias\npublic import Batteries.Tactic.Trans\npublic import Mathlib.Tactic.ExtendDoc\npublic import Mathlib.Tactic.ToDual\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Orders\n\nDefines classes for preorders and partial orders\nand proves some basic lemmas about them.\n\nWe also define covering relations on a preorder.\nWe say that `b` *covers* `a` if `a < b` and there is no element in between.\nWe say that `b` *weakly covers* `a` if `a ≤ b` and there is no element between `a` and `b`.\nIn a partial order this is equivalent to `a ⋖ b ∨ a = b`,\nin a preorder this is equivalent to `a ⋖ b ∨ (a ≤ b ∧ b ≤ a)`\n\n## Notation\n\n* `a ⋖ b` means that `b` covers `a`.\n* `a ⩿ b` means that `b` weakly covers `a`.\n-/\n\n@[expose] public section\n\nvariable {α : Type*}\n\nsection Preorder\n\n/-!\n### Definition of `Preorder` and lemmas about types with a `Preorder`\n-/\n\n/--\nA preorder is a reflexive, transitive relation `≤`.\nIn a preorder, `a < b` means `a ≤ b ∧ ¬b ≤ a`, and `<` is defined this way by default.\nYou can override this definition to set a better def-eq.\n-/\nclass Preorder (α : Type*) extends LE α, LT α where\n  protected le_refl : ∀ a : α, a ≤ a\n  protected le_trans : ∀ a b c : α, a ≤ b → b ≤ c → a ≤ c\n  lt := fun a b => a ≤ b ∧ ¬b ≤ a\n  protected lt_iff_le_not_ge : ∀ a b : α, a < b ↔ a ≤ b ∧ ¬b ≤ a := by intros; rfl\n\nattribute [to_dual self (reorder := le_trans (a c, 4 5), lt_iff_le_not_ge (a b))] Preorder.mk\n\ninstance [Preorder α] : Std.LawfulOrderLT α where\n  lt_iff := Preorder.lt_iff_le_not_ge\n\ninstance [Preorder α] : Std.IsPreorder α where\n  le_refl := Preorder.le_refl\n  le_trans := Preorder.le_trans\n\nvariable [Preorder α] {a b c : α}\n\n/-- The relation `≤` on a preorder is reflexive. -/\n@[refl] lemma le_refl : ∀ a : α, a ≤ a := Preorder.le_refl\n\n/-- A version of `le_refl` where the argument is implicit -/\nlemma le_rfl : a ≤ a := le_refl a\n\n/-- The relation `≤` on a preorder is transitive. -/\nlemma le_trans : a ≤ b → b ≤ c → a ≤ c := Preorder.le_trans _ _ _\n\n@[to_dual existing le_trans]\nlemma ge_trans : b ≤ a → c ≤ b → c ≤ a := flip le_trans\n\n@[to_dual self]\nlemma lt_iff_le_not_ge : a < b ↔ a ≤ b ∧ ¬b ≤ a := Preorder.lt_iff_le_not_ge _ _\n\n@[to_dual self]\nlemma lt_of_le_not_ge (hab : a ≤ b) (hba : ¬ b ≤ a) : a < b := lt_iff_le_not_ge.2 ⟨hab, hba⟩\n\n@[to_dual ge_of_eq] lemma le_of_eq (hab : a = b) : a ≤ b := by rw [hab]\n@[to_dual self] lemma le_of_lt (hab : a < b) : a ≤ b := (lt_iff_le_not_ge.1 hab).1\n@[to_dual self] lemma not_le_of_gt (hab : a < b) : ¬ b ≤ a := (lt_iff_le_not_ge.1 hab).2\n@[to_dual self] lemma not_lt_of_ge (hab : a ≤ b) : ¬ b < a := imp_not_comm.1 not_le_of_gt hab\n\n@[to_dual self] alias LT.lt.not_ge := not_le_of_gt\n@[to_dual self] alias LE.le.not_gt := not_lt_of_ge\n\nlemma lt_irrefl (a : α) : ¬a < a := fun h ↦ not_le_of_gt h le_rfl\n\n@[to_dual lt_of_lt_of_le']\nlemma lt_of_lt_of_le (hab : a < b) (hbc : b ≤ c) : a < c :=\n  lt_of_le_not_ge (le_trans (le_of_lt hab) hbc) fun hca ↦ not_le_of_gt hab (le_trans hbc hca)\n\n@[to_dual lt_of_le_of_lt']\nlemma lt_of_le_of_lt (hab : a ≤ b) (hbc : b < c) : a < c :=\n  lt_of_le_not_ge (le_trans hab (le_of_lt hbc)) fun hca ↦ not_le_of_gt hbc (le_trans hca hab)\n\n@[to_dual gt_trans]\nlemma lt_trans : a < b → b < c → a < c := fun h₁ h₂ => lt_of_lt_of_le h₁ (le_of_lt h₂)\n\n@[to_dual ne_of_gt]\nlemma ne_of_lt (h : a < b) : a ≠ b := fun he => absurd h (he ▸ lt_irrefl a)\n@[to_dual self]\nlemma lt_asymm (h : a < b) : ¬b < a := fun h1 : b < a => lt_irrefl a (lt_trans h h1)\n\n@[to_dual self] alias not_lt_of_gt := lt_asymm\n\n@[to_dual le_of_lt_or_eq']\nlemma le_of_lt_or_eq (h : a < b ∨ a = b) : a ≤ b := h.elim le_of_lt le_of_eq\n@[to_dual le_of_eq_or_lt']\nlemma le_of_eq_or_lt (h : a = b ∨ a < b) : a ≤ b := h.elim le_of_eq le_of_lt\n\n@[to_dual self]\n\nTarget:\nlemma lt_iff_gt_iff_le_iff_ge : (a < b ↔ b < a) ↔ (a ≤ b ↔ b ≤ a) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Defs","family_id":"lt_iff_gt_iff_le_iff_ge","file_id":"mathlib/Mathlib/Order/Defs/PartialOrder.lean","sample_id":"211d293f722fe1130c34a732f32615615b914ff3488d05c85b24b7fea5b7d189"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c849c5df0799eac828ab19895ffb8d4649eaf0e6c519838aa60e6b56e2b819cc","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a6cc210366796b93c3087a5747f28bf3396e3f7d848298d6e55848e06cc71e8d","source_sha256":"279951b6d00a5d584aef7e70a8e7410771dfbbc000b64122d58d64bf59f2a387","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have a := hP.card_parts_eq_average ht\n  lia","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":8,"token_jaccard":0.230769,"token_length_ratio":0.888889},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"cab24ce71c642e3daea44cf8385b2e40724005a42ad988d3fb3657d624aa9d43","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Ring.Nat\npublic import Mathlib.Data.Set.Equitable\npublic import Mathlib.Logic.Equiv.Fin.Basic\npublic import Mathlib.Order.Partition.Finpartition\npublic import Mathlib.Tactic.ApplyFun\n\nNamespace:\nFinpartition\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Finite equipartitions\n\nThis file defines finite equipartitions, the partitions whose parts all are the same size up to a\ndifference of `1`.\n\n## Main declarations\n\n* `Finpartition.IsEquipartition`: Predicate for a `Finpartition` to be an equipartition.\n* `Finpartition.IsEquipartition.exists_partPreservingEquiv`: part-preserving enumeration of a finset\n  equipped with an equipartition. Indices of elements in the same part are congruent modulo\n  the number of parts.\n-/\n\n@[expose] public section\n\n\nopen Finset Fintype\n\nnamespace Finpartition\n\nvariable {α : Type*} [DecidableEq α] {s t : Finset α} (P : Finpartition s)\n\n/-- An equipartition is a partition whose parts are all the same size, up to a difference of `1`. -/\ndef IsEquipartition : Prop :=\n  (P.parts : Set (Finset α)).EquitableOn card\n\ntheorem isEquipartition_iff_card_parts_eq_average :\n    P.IsEquipartition ↔\n      ∀ a : Finset α, a ∈ P.parts → #a = #s / #P.parts ∨ #a = #s / #P.parts + 1 := by\n  simp_rw [IsEquipartition, Finset.equitableOn_iff, P.sum_card_parts]\n\nvariable {P}\n\nlemma not_isEquipartition :\n    ¬P.IsEquipartition ↔ ∃ a ∈ P.parts, ∃ b ∈ P.parts, #b + 1 < #a := Set.not_equitableOn\n\ntheorem _root_.Set.Subsingleton.isEquipartition (h : (P.parts : Set (Finset α)).Subsingleton) :\n    P.IsEquipartition :=\n  Set.Subsingleton.equitableOn h _\n\ntheorem IsEquipartition.card_parts_eq_average (hP : P.IsEquipartition) (ht : t ∈ P.parts) :\n    #t = #s / #P.parts ∨ #t = #s / #P.parts + 1 :=\n  P.isEquipartition_iff_card_parts_eq_average.1 hP _ ht\n\nTarget:\ntheorem IsEquipartition.card_part_eq_average_iff (hP : P.IsEquipartition) (ht : t ∈ P.parts) :\n    #t = #s / #P.parts ↔ #t ≠ #s / #P.parts + 1 :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e673fcbab29f876fe999052c806b5be7379cd98ce3e5ad9dd5d054f0dc61d4b7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Partition","family_id":"isequipartition","file_id":"mathlib/Mathlib/Order/Partition/Equipartition.lean","sample_id":"a6cc210366796b93c3087a5747f28bf3396e3f7d848298d6e55848e06cc71e8d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9117fb79a21990bf1e7a81654768056689d985c1af2446daa090c5cc9faf03cd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c145cc4ee5288f2772f52444fc547753bf938e2f4ca8d06f9588c20e083de47","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3890101cc6d52342ce69e19715aee46bb30db6af8bbb140e8a5fac3718968504","source_sha256":"80f83e637655104ff57f368934350815480bb54de691c8a5aefe35bab1dd7af7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases eq_or_lt_of_le hj with h | h\n  · rw [h]\n  · exact T.row_weak h cell","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":3,"token_jaccard":0.117647,"token_length_ratio":0.15},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"cb48cd8588f65b0adf6dc28eaa4652aeec5058066f18552e861a17a481a925ca","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Young.YoungDiagram\n\nNamespace:\nSemistandardYoungTableau\n\nLocal context:\n/-\nCopyright (c) 2022 Jake Levinson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jake Levinson\n-/\n/-!\n# Semistandard Young tableaux\n\nA semistandard Young tableau is a filling of a Young diagram by natural numbers, such that\nthe entries are weakly increasing left-to-right along rows (i.e. for fixed `i`), and\nstrictly-increasing top-to-bottom along columns (i.e. for fixed `j`).\n\nAn example of an SSYT of shape `μ = [4, 2, 1]` is:\n\n```text\n0 0 0 2\n1 1\n2\n```\n\nWe represent a semistandard Young tableau as a function `ℕ → ℕ → ℕ`, which is required to be zero\nfor all pairs `(i, j) ∉ μ` and to satisfy the row-weak and column-strict conditions on `μ`.\n\n\n## Main definitions\n\n- `SemistandardYoungTableau (μ : YoungDiagram)`: semistandard Young tableaux of shape `μ`. There is\n  a `coe` instance such that `T i j` is value of the `(i, j)` entry of the semistandard Young\n  tableau `T`.\n- `SemistandardYoungTableau.highestWeight (μ : YoungDiagram)`: the semistandard Young tableau whose\n  `i`th row consists entirely of `i`s, for each `i`.\n\n## Tags\n\nSemistandard Young tableau\n\n## References\n\n<https://en.wikipedia.org/wiki/Young_tableau>\n\n-/\n\n@[expose] public section\n\n\n/-- A semistandard Young tableau is a filling of the cells of a Young diagram by natural\nnumbers, such that the entries in each row are weakly increasing (left to right), and the entries\nin each column are strictly increasing (top to bottom).\n\nHere, a semistandard Young tableau is represented as an unrestricted function `ℕ → ℕ → ℕ` that, for\nreasons of extensionality, is required to vanish outside `μ`. -/\nstructure SemistandardYoungTableau (μ : YoungDiagram) where\n  /-- `entry i j` is value of the `(i, j)` entry of the SSYT `μ`. -/\n  entry : ℕ → ℕ → ℕ\n  /-- The entries in each row are weakly increasing (left to right). -/\n  row_weak' : ∀ {i j1 j2 : ℕ}, j1 < j2 → (i, j2) ∈ μ → entry i j1 ≤ entry i j2\n  /-- The entries in each column are strictly increasing (top to bottom). -/\n  col_strict' : ∀ {i1 i2 j : ℕ}, i1 < i2 → (i2, j) ∈ μ → entry i1 j < entry i2 j\n  /-- `entry` is required to be zero for all pairs `(i, j) ∉ μ`. -/\n  zeros' : ∀ {i j}, (i, j) ∉ μ → entry i j = 0\n\nnamespace SemistandardYoungTableau\n\ninstance instFunLike {μ : YoungDiagram} : FunLike (SemistandardYoungTableau μ) ℕ (ℕ → ℕ) where\n  coe := SemistandardYoungTableau.entry\n  coe_injective T T' h := by\n    cases T\n    cases T'\n    congr\n\n@[simp]\ntheorem to_fun_eq_coe {μ : YoungDiagram} {T : SemistandardYoungTableau μ} :\n    T.entry = (T : ℕ → ℕ → ℕ) :=\n  rfl\n\n@[ext]\ntheorem ext {μ : YoungDiagram} {T T' : SemistandardYoungTableau μ} (h : ∀ i j, T i j = T' i j) :\n    T = T' :=\n  DFunLike.ext T T' fun _ ↦ by\n    funext\n    apply h\n\n/-- Copy of an `SemistandardYoungTableau μ` with a new `entry` equal to the old one. Useful to fix\ndefinitional equalities. -/\nprotected def copy {μ : YoungDiagram} (T : SemistandardYoungTableau μ) (entry' : ℕ → ℕ → ℕ)\n    (h : entry' = T) : SemistandardYoungTableau μ where\n  entry := entry'\n  row_weak' := h.symm ▸ T.row_weak'\n  col_strict' := h.symm ▸ T.col_strict'\n  zeros' := h.symm ▸ T.zeros'\n\n@[simp]\ntheorem coe_copy {μ : YoungDiagram} (T : SemistandardYoungTableau μ) (entry' : ℕ → ℕ → ℕ)\n    (h : entry' = T) : ⇑(T.copy entry' h) = entry' :=\n  rfl\n\ntheorem copy_eq {μ : YoungDiagram} (T : SemistandardYoungTableau μ) (entry' : ℕ → ℕ → ℕ)\n    (h : entry' = T) : T.copy entry' h = T :=\n  DFunLike.ext' h\n\ntheorem row_weak {μ : YoungDiagram} (T : SemistandardYoungTableau μ) {i j1 j2 : ℕ} (hj : j1 < j2)\n    (hcell : (i, j2) ∈ μ) : T i j1 ≤ T i j2 :=\n  T.row_weak' hj hcell\n\ntheorem col_strict {μ : YoungDiagram} (T : SemistandardYoungTableau μ) {i1 i2 j : ℕ} (hi : i1 < i2)\n    (hcell : (i2, j) ∈ μ) : T i1 j < T i2 j :=\n  T.col_strict' hi hcell\n\ntheorem zeros {μ : YoungDiagram} (T : SemistandardYoungTableau μ) {i j : ℕ}\n    (not_cell : (i, j) ∉ μ) : T i j = 0 :=\n  T.zeros' not_cell\n\nTarget:\ntheorem row_weak_of_le {μ : YoungDiagram} (T : SemistandardYoungTableau μ) {i j1 j2 : ℕ}\n    (hj : j1 ≤ j2) (cell : (i, j2) ∈ μ) : T i j1 ≤ T i j2 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_3890101cc6d5","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f2d3a59ad39d43066788a15c26f338dae70973baa17e3a38be83a649860ca131","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Young","family_id":"row_weak_of_le","file_id":"mathlib/Mathlib/Combinatorics/Young/SemistandardTableau.lean","sample_id":"3890101cc6d52342ce69e19715aee46bb30db6af8bbb140e8a5fac3718968504"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b2e750b3c5b8740fb444ea49e43055beae89260e7b9e80c29b949ec289033978","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"844d818def83ae69c024a7d8514dd9d3788fc2851d4e2b66f661cd8ecb827d5f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f0978345095eddbdb23e72c7878138dd691ea3f6ff7d3034237e21db83bc2237","source_sha256":"155f9e8471ee4b30d28b3dbc51f2958ab69f6c864e7fa3081250ca89e1bfbb0f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [of_apply, ← lift_symm_apply, Equiv.symm_apply_apply]","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":19,"token_jaccard":0.846154,"token_length_ratio":1.583333},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"cb65422dbddb8cffab822914ba5a7754cd65f22c5ebf7606bdf7508dc0620f62","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.SkewMonoidAlgebra.Basic\npublic import Mathlib.Algebra.Module.BigOperators\npublic import Mathlib.Algebra.Algebra.Equiv\n\nNamespace:\nSkewMonoidAlgebra\n\nLocal context:\n/-\nCopyright (c) 2025 Xavier Généreux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: María Inés de Frutos Fernández, Xavier Généreux\n-/\n/-!\n# Lemmas about different kinds of \"lifts\" to `SkewMonoidAlgebra`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nnamespace SkewMonoidAlgebra\n\nvariable {k G H : Type*}\n\nsection lift\n\nvariable [CommSemiring k] [Monoid G] [Monoid H]\nvariable {A B : Type*} [Semiring A] [Algebra k A] [Semiring B] [Algebra k B]\n\n/-- `liftNCRingHom` as an `AlgHom`, for when `f` is an `AlgHom` -/\ndef liftNCAlgHom [MulSemiringAction G A] [SMulCommClass G k A] (f : A →ₐ[k] B)\n    (g : G →* B) (h_comm : ∀ {x y}, (f (y • x)) * g y = (g y) * (f x)) :\n    SkewMonoidAlgebra A G →ₐ[k] B where\n  __ := liftNCRingHom (f : A →+* B) g h_comm\n  commutes' := by simp [liftNCRingHom]\n\n/- Hypotheses needed for `k`-algebra homomorphism from `SkewMonoidAlgebra k G`-/\nvariable [MulSemiringAction G k] [SMulCommClass G k k]\n\nvariable (k G A)\n\n/-- Any monoid homomorphism `G →* A` can be lifted to an algebra homomorphism\n  `SkewMonoidAlgebra k G →ₐ[k] A`. -/\ndef lift : (G →* A) ≃ (AlgHom k (SkewMonoidAlgebra k G) A) where\n  invFun f := (f : SkewMonoidAlgebra k G →* A).comp (of k G)\n  toFun F := by\n    apply liftNCAlgHom (Algebra.ofId k A) F\n    simp_rw [show ∀ (g : G) (r : k), g • r = r by\n        exact fun _ _ ↦ smul_algebraMap _ (algebraMap k k _)]\n    exact Algebra.commutes _ _\n  left_inv f := by\n    ext\n    simp [liftNCAlgHom, liftNCRingHom]\n  right_inv F := by\n    ext\n    simp [liftNCAlgHom, liftNCRingHom]\n\nvariable {k G A}\n\ntheorem lift_apply' (F : G →* A) (f : SkewMonoidAlgebra k G) :\n    lift k G A F f = f.sum fun a b ↦ algebraMap k A b * F a := rfl\n\ntheorem lift_apply (F : G →* A) (f : SkewMonoidAlgebra k G) :\n    lift k G A F f = f.sum fun a b ↦ b • F a := by simp [lift_apply', Algebra.smul_def]\n\ntheorem lift_def (F : G →* A) : (lift k G A F : SkewMonoidAlgebra k G → A) =\n    liftNC ((algebraMap k A : k →+* A) : k →+ A) F := rfl\n\n@[simp]\ntheorem lift_symm_apply (F : AlgHom k (SkewMonoidAlgebra k G) A) (x : G) :\n    (lift k G A).symm F x = F (single x 1) := rfl\n\nTarget:\ntheorem lift_of (F : G →* A) (x) : lift k G A F (of k G x) = F x :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [of_apply, ← lift_symm_apply, Equiv.symm_apply_apply]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/SkewMonoidAlgebra","family_id":"lift_of","file_id":"mathlib/Mathlib/Algebra/SkewMonoidAlgebra/Lift.lean","sample_id":"f0978345095eddbdb23e72c7878138dd691ea3f6ff7d3034237e21db83bc2237"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eac7599f7adbe4156b1359f0aff3dffecc04db5fb77ecf8bedd140ea7081aceb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9b4df564c00a9e22a952e465016b91d0c329c338835f66cf4e99551b7d229b42","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7ee01c83c10c22668c1fdc800b54b271e64edd823972fd056c65d61e5211e5df","source_sha256":"bbfec10092a00bc78b9af51067ad47929a5d5df48b980b858f3f6f7bbf3e54e6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction h\n  simp","hard_negative":true,"metrics":{"chosen_tokens":4,"rejected_tokens":5,"token_jaccard":0.125,"token_length_ratio":1.25},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"cba2716e5f78bacc62dd24a91b8d9f17c8bca4b7df7da04f305c8b4873d97de8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Skeleton\npublic import Mathlib.CategoryTheory.Subobject.MonoOver\npublic import Mathlib.CategoryTheory.Skeletal\npublic import Mathlib.CategoryTheory.ConcreteCategory.Basic\npublic import Mathlib.Tactic.ApplyFun\npublic import Mathlib.Tactic.CategoryTheory.Elementwise\npublic import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic\n\nNamespace:\nCategoryTheory.Subobject\n\nLocal context:\n/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Kim Morrison\n-/\n/-!\n# Subobjects\n\nWe define `Subobject X` as the quotient (by isomorphisms) of\n`MonoOver X := {f : Over X // Mono f.hom}`.\n\nHere `MonoOver X` is a thin category (a pair of objects has at most one morphism between them),\nso we can think of it as a preorder. However as it is not skeletal, it is not a partial order.\n\nThere is a coercion from `Subobject X` back to the ambient category `C`\n(using choice to pick a representative), and for `P : Subobject X`,\n`P.arrow : (P : C) ⟶ X` is the inclusion morphism.\n\nWe provide\n* `def pullback [HasPullbacks C] (f : X ⟶ Y) : Subobject Y ⥤ Subobject X`\n* `def map (f : X ⟶ Y) [Mono f] : Subobject X ⥤ Subobject Y`\n* `def «exists_» [HasImages C] (f : X ⟶ Y) : Subobject X ⥤ Subobject Y`\n\nand prove their basic properties and relationships.\nThese are all easy consequences of the earlier development\nof the corresponding functors for `MonoOver`.\n\nThe subobjects of `X` form a preorder making them into a category. We have `X ≤ Y` if and only if\n`X.arrow` factors through `Y.arrow`: see `ofLE`/`ofLEMk`/`ofMkLE`/`ofMkLEMk` and\n`le_of_comm`. Similarly, to show that two subobjects are equal, we can supply an isomorphism between\nthe underlying objects that commutes with the arrows (`eq_of_comm`).\n\nSee also\n\n* `CategoryTheory.Subobject.factorThru` :\n  an API describing factorization of morphisms through subobjects.\n* `CategoryTheory.Subobject.lattice` :\n  the lattice structures on subobjects.\n\n## Notes\n\nThis development originally appeared in Bhavik Mehta's \"Topos theory for Lean\" repository,\nand was ported to mathlib by Kim Morrison.\n\n### Implementation note\n\nCurrently we describe `pullback`, `map`, etc., as functors.\nIt may be better to just say that they are monotone functions,\nand even avoid using categorical language entirely when describing `Subobject X`.\n(It's worth keeping this in mind in future use; it should be a relatively easy change here\nif it looks preferable.)\n\n### Relation to pseudoelements\n\nThere is a separate development of pseudoelements in `CategoryTheory.Abelian.Pseudoelements`,\nas a quotient (but not by isomorphism) of `Over X`.\n\nWhen a morphism `f` has an image, the image represents the same pseudoelement.\nIn a category with images `Pseudoelements X` could be constructed as a quotient of `MonoOver X`.\nIn fact, in an abelian category (I'm not sure in what generality beyond that),\n`Pseudoelements X` agrees with `Subobject X`, but we haven't developed this in mathlib yet.\n\n-/\n\n@[expose] public section\n\n\nuniverse w' w v₁ v₂ v₃ u₁ u₂ u₃\n\nnoncomputable section\n\nnamespace CategoryTheory\n\nopen CategoryTheory CategoryTheory.Category CategoryTheory.Limits\n\nvariable {C : Type u₁} [Category.{v₁} C] {X Y Z : C}\nvariable {D : Type u₂} [Category.{v₂} D]\n\n/-!\nWe now construct the subobject lattice for `X : C`,\nas the quotient by isomorphisms of `MonoOver X`.\n\nSince `MonoOver X` is a thin category, we use `ThinSkeleton` to take the quotient.\n\nEssentially all the structure defined above on `MonoOver X` descends to `Subobject X`,\nwith morphisms becoming inequalities, and isomorphisms becoming equations.\n-/\n\n\n/-- The category of subobjects of `X : C`, defined as isomorphism classes of monomorphisms into `X`.\n-/\ndef Subobject (X : C) :=\n  ThinSkeleton (MonoOver X)\n\ninstance (X : C) : PartialOrder (Subobject X) :=\n  inferInstanceAs <| PartialOrder (ThinSkeleton (MonoOver X))\n\nnamespace Subobject\n\nlemma skeletal (X : C) : Skeletal (Subobject X) := ThinSkeleton.skeletal\n\n/-- Convenience constructor for a subobject. -/\ndef mk {X A : C} (f : A ⟶ X) [Mono f] : Subobject X :=\n  (toThinSkeleton _).obj (MonoOver.mk f)\n\nsection\n\nattribute [local ext] CategoryTheory.Comma\n\nprotected theorem ind {X : C} (p : Subobject X → Prop)\n    (h : ∀ ⦃A : C⦄ (f : A ⟶ X) [Mono f], p (Subobject.mk f)) (P : Subobject X) : p P := by\n  induction P using Quotient.inductionOn' with | _ a\n  exact h a.arrow\n\nprotected theorem ind₂ {X : C} (p : Subobject X → Subobject X → Prop)\n    (h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [Mono f] [Mono g],\n      p (Subobject.mk f) (Subobject.mk g))\n    (P Q : Subobject X) : p P Q := by\n  induction P, Q using Quotient.inductionOn₂' with | _ a b\n  exact h a.arrow b.arrow\n\nend\n\n/-- Declare a function on subobjects of `X` by specifying a function on monomorphisms with\ncodomain `X`. -/\nprotected def lift {α : Sort*} {X : C} (F : ∀ ⦃A : C⦄ (f : A ⟶ X) [Mono f], α)\n    (h :\n      ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [Mono f] [Mono g] (i : A ≅ B),\n        i.hom ≫ g = f → F f = F g) :\n    Subobject X → α := fun P =>\n  Quotient.liftOn' P (fun m => F m.arrow) fun m n ⟨i⟩ =>\n    h m.arrow n.arrow ((MonoOver.forget X ⋙ Over.forget X).mapIso i) (Over.w i.hom.hom)\n\n@[simp]\nprotected theorem lift_mk {α : Sort*} {X : C} (F : ∀ ⦃A : C⦄ (f : A ⟶ X) [Mono f], α) {h A}\n    (f : A ⟶ X) [Mono f] : Subobject.lift F h (Subobject.mk f) = F f :=\n  rfl\n\n/-- The category of subobjects is equivalent to the `MonoOver` category. It is more convenient to\nuse the former due to the partial order instance, but oftentimes it is easier to define structures\non the latter. -/\nnoncomputable def equivMonoOver (X : C) : Subobject X ≌ MonoOver X :=\n  ThinSkeleton.equivalence _\n\n/-- Use choice to pick a representative `MonoOver X` for each `Subobject X`.\n-/\nnoncomputable def representative {X : C} : Subobject X ⥤ MonoOver X :=\n  (equivMonoOver X).functor\n\ninstance : (representative (X := X)).IsEquivalence :=\n  (equivMonoOver X).isEquivalence_functor\n\n/-- Starting with `A : MonoOver X`, we can take its equivalence class in `Subobject X`\nthen pick an arbitrary representative using `representative.obj`.\nThis is isomorphic (in `MonoOver X`) to the original `A`.\n-/\nnoncomputable def representativeIso {X : C} (A : MonoOver X) :\n    representative.obj ((toThinSkeleton _).obj A) ≅ A :=\n  (equivMonoOver X).counitIso.app A\n\n@[simp]\nlemma thinSkeleton_mk_representative_eq_self {X : C} (A : Subobject X) :\n    ThinSkeleton.mk (representative.obj A) = A :=\n  Subobject.skeletal _ ⟨((equivMonoOver X).unitIso.app _).symm⟩\n\n/-- Use choice to pick a representative underlying object in `C` for any `Subobject X`.\n\nPrefer to use the coercion `P : C` rather than explicitly writing `underlying.obj P`.\n-/\nnoncomputable def underlying {X : C} : Subobject X ⥤ C :=\n  representative ⋙ MonoOver.forget _ ⋙ Over.forget _\n\ninstance : CoeOut (Subobject X) C where coe Y := underlying.obj Y\n\n/-- If we construct a `Subobject Y` from an explicit `f : X ⟶ Y` with `[Mono f]`,\nthen pick an arbitrary choice of underlying object `(Subobject.mk f : C)` back in `C`,\nit is isomorphic (in `C`) to the original `X`.\n-/\nnoncomputable def underlyingIso {X Y : C} (f : X ⟶ Y) [Mono f] : (Subobject.mk f : C) ≅ X :=\n  (MonoOver.forget _ ⋙ Over.forget _).mapIso (representativeIso (MonoOver.mk f))\n\n/-- The morphism in `C` from the arbitrarily chosen underlying object to the ambient object.\n-/\nnoncomputable def arrow {X : C} (Y : Subobject X) : (Y : C) ⟶ X :=\n  (representative.obj Y).obj.hom\n\ninstance arrow_mono {X : C} (Y : Subobject X) : Mono Y.arrow :=\n  (representative.obj Y).property\n\n@[simp]\n\nTarget:\ntheorem arrow_congr {A : C} (X Y : Subobject A) (h : X = Y) :\n    eqToHom (congr_arg (fun X : Subobject A => (X : C)) h) ≫ Y.arrow = X.arrow :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_7ee01c83c10c","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d8c382de9011809aa3641b268fe3df78957613522a7747f9ff1f6ff44def4c83","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Subobject","family_id":"arrow_congr","file_id":"mathlib/Mathlib/CategoryTheory/Subobject/Basic.lean","sample_id":"7ee01c83c10c22668c1fdc800b54b271e64edd823972fd056c65d61e5211e5df"},"split":"train"}
{"binding":{"chosen_proof_sha256":"47e39958d92c203177453f8d36c36c3e273fa3ce6ffb582eb81c95f2e2f8425a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"33918059ac643ca50e9d2e3d7ccb2d32565782d0b03a2650b0af67e33602c28c","source_sha256":"1c7994cb953bf15a3e2a75bf5a9f75792705603cc1fc781f9e07bc585e941697","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    apply isTopologicalBasis_of_isOpen_of_nhds (fun U hU ↦ hU.1) <| fun x U hx hU ↦ ?_\n    obtain ⟨V, ⟨hoV, hcV⟩, hfx, hVf⟩ : ∃ V ∈ {V | IsOpen V ∧ IsCompact V}, f x ∈ V ∧ V ⊆ f '' U :=\n      (PrespectralSpace.isTopologicalBasis (X := Y)).isOpen_iff.mp\n        (hf.isOpen_iff_image_isOpen.mp hU) (f x) ⟨x, hx, rfl⟩\n    refine ⟨f ⁻¹' V, ⟨hoV.preimage hf.continuous, ?_⟩, ⟨hfx, fun y hy ↦ ?_⟩⟩\n    · exact hf.toIsInducing.isCompact_preimage' hcV <| Set.SurjOn.subset_range hVf\n    · exact hf.injective.mem_set_image.mp (hVf hy)","hard_negative":true,"metrics":{"chosen_tokens":155,"rejected_tokens":8,"token_jaccard":0.044776,"token_length_ratio":0.051613},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"cbb74a9cdad6c5031bff3a892debaef1846c14087078e2f6cefcb8da25201d78","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Ideal\npublic import Mathlib.Topology.Sets.Compacts\npublic import Mathlib.Topology.Sets.OpenCover\npublic import Mathlib.Topology.Spectral.Hom\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Prespectral spaces\n\nIn this file, we define prespectral spaces as spaces whose lattice of compact opens forms a basis.\n\n-/\n\n@[expose] public section\n\nopen TopologicalSpace Topology\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n\n/-- A space is prespectral if the lattice of compact opens forms a basis. -/\n@[stacks 08YG \"The last condition for spectral spaces\", mk_iff]\nclass PrespectralSpace (X : Type*) [TopologicalSpace X] : Prop where\n  isTopologicalBasis : IsTopologicalBasis { U : Set X | IsOpen U ∧ IsCompact U }\n\n/-- A space is prespectral if it has a basis consisting of compact opens. -/\nlemma PrespectralSpace.of_isTopologicalBasis {B : Set (Set X)}\n    (basis : IsTopologicalBasis B) (isCompact_basis : ∀ U ∈ B, IsCompact U) :\n    PrespectralSpace X where\n  isTopologicalBasis := basis.of_isOpen_of_subset (fun _ h ↦ h.1)\n    fun s hs ↦ ⟨basis.isOpen hs, isCompact_basis s hs⟩\n\n/-- A space is prespectral if it has a basis consisting of compact opens.\nThis is the variant with an indexed basis instead. -/\nlemma PrespectralSpace.of_isTopologicalBasis' {ι : Type*} {b : ι → Set X}\n    (basis : IsTopologicalBasis (Set.range b)) (isCompact_basis : ∀ i, IsCompact (b i)) :\n    PrespectralSpace X :=\n  .of_isTopologicalBasis basis (by simp_all)\n\ninstance (priority := low) [NoetherianSpace X] : PrespectralSpace X :=\n  .of_isTopologicalBasis isTopologicalBasis_opens fun _ _ ↦ NoetherianSpace.isCompact _\n\ninstance (priority := low) [PrespectralSpace X] : LocallyCompactSpace X where\n  local_compact_nhds _ _ hn :=\n    have ⟨V, ⟨hV₁, hV₂⟩, hxV, hVn⟩ := PrespectralSpace.isTopologicalBasis.mem_nhds_iff.mp hn\n    ⟨V, hV₁.mem_nhds hxV, hVn, hV₂⟩\n\nopen PrespectralSpace in\ninstance (priority := low) [T2Space X] [PrespectralSpace X] : TotallySeparatedSpace X :=\n  totallySeparatedSpace_iff_exists_isClopen.mpr fun _ _ hxy ↦\n    have ⟨U, ⟨hU₁, hU₂⟩, hxU, hyU⟩ :=\n      isTopologicalBasis.exists_subset_of_mem_open hxy isClosed_singleton.isOpen_compl\n    ⟨U, ⟨hU₂.isClosed, hU₁⟩, hxU, fun h ↦ hyU h rfl⟩\n\nlemma PrespectralSpace.of_isOpenCover\n    {ι : Type*} {U : ι → Opens X} (hU : IsOpenCover U) [∀ i, PrespectralSpace (U i)] :\n    PrespectralSpace X := by\n  refine .of_isTopologicalBasis (hU.isTopologicalBasis fun i ↦ isTopologicalBasis) ?_\n  simp only [Set.mem_iUnion, Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp,\n    forall_comm (α := Set _), forall_apply_eq_imp_iff₂]\n  exact fun i V hV hV' ↦ hV'.image continuous_subtype_val\n\nlemma PrespectralSpace.of_isInducing [PrespectralSpace Y]\n    (f : X → Y) (hf : IsInducing f) (hf' : IsSpectralMap f) : PrespectralSpace X :=\n  .of_isTopologicalBasis (PrespectralSpace.isTopologicalBasis.isInducing hf) (by\n    simp only [Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp]\n    rintro _ U h₁ h₂ rfl\n    exact hf'.isCompact_preimage_of_isOpen h₁ h₂)\n\nlemma PrespectralSpace.of_isClosedEmbedding [PrespectralSpace Y]\n    (f : X → Y) (hf : IsClosedEmbedding f) : PrespectralSpace X :=\n  .of_isInducing f hf.isInducing hf.isProperMap.isSpectralMap\n\n/-- Let `f : X → Y` be an open embedding of topological spaces.\nIf `Y` is a prespectral space (i.e., the quasi-compact opens of `Y` form a basis),\nthen `X` is also a prespectral space. -/\n\nTarget:\nlemma Topology.IsOpenEmbedding.prespectralSpace [PrespectralSpace Y]\n    {f : X → Y} (hf : IsOpenEmbedding f) :\n    PrespectralSpace X where\n  isTopologicalBasis :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"0c2c1e8d788ac4d0ef9ab5f5a9df5f6694f98c38be6f917b8ec2f35ef4a7a7d5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Spectral","family_id":"topology","file_id":"mathlib/Mathlib/Topology/Spectral/Prespectral.lean","sample_id":"33918059ac643ca50e9d2e3d7ccb2d32565782d0b03a2650b0af67e33602c28c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"75765532e9f719b9017c00ea0609cd6aeeb9a4a2bf7c2d14d50fca0b0687a396","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e5781a280c95429bef6967a38411ca2235131ecb05d9745326c5155130e9a4ab","source_sha256":"d5ae8a6ac85e02d23384eb0f0f9e5cb6a39c74c5455bdde189d5500807ce86cf","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    obtain ⟨x, hx⟩ := h.mul_inv_cancel ha\n    simpa [← mul_assoc, h.mul_comm, hx] using congr_arg (x * ·) hb\n  mul_right_cancel_of_ne_zero ha _ _ hb := by\n    obtain ⟨x, hx⟩ := h.mul_inv_cancel ha\n    simpa [mul_assoc, hx] using congr_arg (· * x) hb\n  exists_pair_ne := h.exists_pair_ne","hard_negative":false,"metrics":{"chosen_tokens":68,"rejected_tokens":2,"token_jaccard":0.034483,"token_length_ratio":0.029412},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"cbe64824b82ac0aee7773986f305cf77de35014f915e5ad47004cd4e851cde46","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Defs\npublic import Mathlib.Tactic.Common\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2014 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro\n-/\n/-!\n# `IsField` predicate\n\nPredicate on a (semi)ring that it is a (semi)field, i.e. that the multiplication is\ncommutative, that it has more than one element and that all non-zero elements have a\nmultiplicative inverse. In contrast to `Field`, which contains the data of a function associating\nto an element of the field its multiplicative inverse, this predicate only assumes the existence\nand can therefore more easily be used to e.g. transfer along ring isomorphisms.\n-/\n\n@[expose] public section\n\nuniverse u\n\nsection IsField\n\n/-- A predicate to express that a (semi)ring is a (semi)field.\n\nThis is mainly useful because such a predicate does not contain data,\nand can therefore be easily transported along ring isomorphisms.\nAdditionally, this is useful when trying to prove that\na particular ring structure extends to a (semi)field. -/\nstructure IsField (R : Type u) [Semiring R] : Prop where\n  /-- For a semiring to be a field, it must have two distinct elements. -/\n  exists_pair_ne : ∃ x y : R, x ≠ y\n  /-- Fields are commutative. -/\n  mul_comm : ∀ x y : R, x * y = y * x\n  /-- Nonzero elements have multiplicative inverses. -/\n  mul_inv_cancel : ∀ {a : R}, a ≠ 0 → ∃ b, a * b = 1\n\n/-- Transferring from `Semifield` to `IsField`. -/\ntheorem Semifield.toIsField (R : Type u) [Semifield R] : IsField R where\n  __ := ‹Semifield R›\n  mul_inv_cancel {a} ha := ⟨a⁻¹, mul_inv_cancel₀ ha⟩\n\n/-- Transferring from `Field` to `IsField`. -/\ntheorem Field.toIsField (R : Type u) [Field R] : IsField R :=\n  Semifield.toIsField _\n\n@[simp]\ntheorem IsField.nontrivial {R : Type u} [Semiring R] (h : IsField R) : Nontrivial R :=\n  ⟨h.exists_pair_ne⟩\n\nTarget:\nlemma IsField.isDomain {R : Type u} [Semiring R] (h : IsField R) : IsDomain R where\n  mul_left_cancel_of_ne_zero ha _ _ hb :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Field","family_id":"isfield","file_id":"mathlib/Mathlib/Algebra/Field/IsField.lean","sample_id":"e5781a280c95429bef6967a38411ca2235131ecb05d9745326c5155130e9a4ab"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9740f164ce7086b555af4b7554cbcc58f11573780fea0b6812b9fc64a3703b2b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"27c7b061eb2df69ed4438c60ac705043bf7b1c99567810c2e2cbfe95dec0ba49","source_sha256":"c27083b8f8cde0437299c837bd1f7a4c8446e30ca2254817bcb2958bf94cb26d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply MonoidHom.eq_of_eqOn_denseM (closure_range_of _)\n  grind [Set.eqOn_range]","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":2,"token_jaccard":0.066667,"token_length_ratio":0.133333},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"cbf36a4d3ec570060cd79be70f581da7037b98227b0973a506144a4e539ba798","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.FreeMonoid.Basic\npublic import Mathlib.Algebra.Group.Submonoid.Operations\npublic import Mathlib.GroupTheory.Congruence.Hom\n\nNamespace:\nPresentedMonoid\n\nLocal context:\n/-\nCopyright (c) 2024 Hannah Fechtner. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Hannah Fechtner\n-/\n/-!\n# Defining a monoid given by generators and relations\n\nGiven relations `rels` on the free monoid on a type `α`, this file constructs the monoid\ngiven by generators `x : α` and relations `rels`.\n\n## Main definitions\n\n* `PresentedMonoid rels`: the quotient of the free monoid on a type `α` by the closure of one-step\n  reductions (arising from a binary relation on free monoid elements `rels`).\n* `PresentedMonoid.of`: The canonical map from `α` to a presented monoid with generators `α`.\n* `PresentedMonoid.lift f`: the canonical monoid homomorphism `PresentedMonoid rels → M`, given\n  a function `f : α → G` from a type `α` to a monoid `M` which satisfies the relations `rels`.\n\n## Tags\n\ngenerators, relations, monoid presentations\n-/\n\n@[expose] public section\n\nvariable {α : Type*}\n\n/-- Given a set of relations, `rels`, over a type `α`, `PresentedMonoid` constructs the monoid with\ngenerators `x : α` and relations `rels` as a quotient of a congruence structure over rels. -/\n@[to_additive /-- Given a set of relations, `rels`, over a type `α`, `PresentedAddMonoid` constructs\nthe monoid with generators `x : α` and relations `rels` as a quotient of an AddCon structure over\nrels -/]\ndef PresentedMonoid (rels : FreeMonoid α → FreeMonoid α → Prop) := (conGen rels).Quotient\n\nnamespace PresentedMonoid\n\nopen Set Submonoid\n\n@[to_additive]\ninstance {rels : FreeMonoid α → FreeMonoid α → Prop} : Monoid (PresentedMonoid rels) :=\n  inferInstanceAs <| Monoid (conGen rels).Quotient\n\n/-- The quotient map from the free monoid on `α` to the presented monoid with the same generators\nand the given relations `rels`. -/\n@[to_additive /-- The quotient map from the free additive monoid on `α` to the presented additive\nmonoid with the same generators and the given relations `rels` -/]\ndef mk (rels : FreeMonoid α → FreeMonoid α → Prop) : FreeMonoid α →* PresentedMonoid rels where\n  toFun := Quotient.mk (conGen rels).toSetoid\n  map_one' := rfl\n  map_mul' := fun _ _ => rfl\n\n/-- `of` is the canonical map from `α` to a presented monoid with generators `x : α`. The term `x`\nis mapped to the equivalence class of the image of `x` in `FreeMonoid α`. -/\n@[to_additive\n/-- `of` is the canonical map from `α` to a presented additive monoid with generators `x : α`. The\nterm `x` is mapped to the equivalence class of the image of `x` in `FreeAddMonoid α`. -/]\ndef of (rels : FreeMonoid α → FreeMonoid α → Prop) (x : α) : PresentedMonoid rels :=\n  mk rels (.of x)\n\nsection inductionOn\n\nvariable {α₁ α₂ α₃ : Type*} {rels₁ : FreeMonoid α₁ → FreeMonoid α₁ → Prop}\n  {rels₂ : FreeMonoid α₂ → FreeMonoid α₂ → Prop} {rels₃ : FreeMonoid α₃ → FreeMonoid α₃ → Prop}\n\nlocal notation \"P₁\" => PresentedMonoid rels₁\nlocal notation \"P₂\" => PresentedMonoid rels₂\nlocal notation \"P₃\" => PresentedMonoid rels₃\n\n@[to_additive (attr := elab_as_elim), induction_eliminator]\nprotected theorem inductionOn {δ : P₁ → Prop} (q : P₁) (h : ∀ a, δ (mk rels₁ a)) : δ q :=\n  Quotient.ind h q\n\n@[to_additive (attr := elab_as_elim)]\nprotected theorem inductionOn₂ {δ : P₁ → P₂ → Prop} (q₁ : P₁) (q₂ : P₂)\n    (h : ∀ a b, δ (mk rels₁ a) (mk rels₂ b)) : δ q₁ q₂ :=\n  Quotient.inductionOn₂ q₁ q₂ h\n\n@[to_additive (attr := elab_as_elim)]\nprotected theorem inductionOn₃ {δ : P₁ → P₂ → P₃ → Prop} (q₁ : P₁)\n    (q₂ : P₂) (q₃ : P₃) (h : ∀ a b c, δ (mk rels₁ a) (mk rels₂ b) (mk rels₃ c)) :\n    δ q₁ q₂ q₃ :=\n  Quotient.inductionOn₃ q₁ q₂ q₃ h\n\nend inductionOn\n\nvariable {α : Type*} {rels : FreeMonoid α → FreeMonoid α → Prop} {x y : FreeMonoid α}\n\nlemma mk_eq_mk_iff : mk rels x = mk rels y ↔ conGen rels x y := Quotient.eq\n\nlemma mk_eq_mk_of_rel (h : rels x y) : mk rels x = mk rels y := mk_eq_mk_iff.2 (.of _ _ h)\n\n/-- The generators of a presented monoid generate the presented monoid. That is, the submonoid\nclosure of the set of generators equals `⊤`. -/\n@[to_additive (attr := simp) /-- The generators of a presented additive monoid generate the\npresented additive monoid. That is, the additive submonoid closure of the set of generators equals\n`⊤`. -/]\ntheorem closure_range_of (rels : FreeMonoid α → FreeMonoid α → Prop) :\n    Submonoid.closure (Set.range (of rels)) = ⊤ := by\n  rw [Submonoid.eq_top_iff']\n  intro x\n  induction x with | _ a\n  induction a with\n  | one => exact Submonoid.one_mem _\n  | of x => exact subset_closure <| by simp [range, of]\n  | mul x y hx hy => exact Submonoid.mul_mem _ hx hy\n\n@[to_additive]\ntheorem surjective_mk {rels : FreeMonoid α → FreeMonoid α → Prop} :\n    Function.Surjective (mk rels) := fun x ↦ PresentedMonoid.inductionOn x fun a ↦ .intro a rfl\n\nsection ToMonoid\nvariable {α M : Type*} [Monoid M] (f : α → M)\nvariable {rels : FreeMonoid α → FreeMonoid α → Prop}\nvariable (h : ∀ a b : FreeMonoid α, rels a b → FreeMonoid.lift f a = FreeMonoid.lift f b)\n\n/-- The extension of a map `f : α → M` that satisfies the given relations to a monoid homomorphism\nfrom `PresentedMonoid rels → M`. -/\n@[to_additive /-- The extension of a map `f : α → M` that satisfies the given relations to an\nadditive-monoid homomorphism from `PresentedAddMonoid rels → M` -/]\ndef lift : PresentedMonoid rels →* M :=\n  Con.lift _ (FreeMonoid.lift f) (Con.conGen_le.2 h)\n\n@[to_additive]\ntheorem toMonoid.unique (g : MonoidHom (conGen rels).Quotient M)\n    (hg : ∀ a : α, g (of rels a) = f a) : g = lift f h :=\n  Con.lift_unique (Con.conGen_le.2 h) g (FreeMonoid.hom_eq hg)\n\n@[to_additive (attr := simp)]\ntheorem lift_of {x : α} : lift f h (of rels x) = f x := rfl\n\nend ToMonoid\n\n@[to_additive (attr := ext)]\n\nTarget:\ntheorem ext {M : Type*} [Monoid M] (rels : FreeMonoid α → FreeMonoid α → Prop)\n    {φ ψ : PresentedMonoid rels →* M} (hx : ∀ (x : α), φ (.of rels x) = ψ (.of rels x)) :\n    φ = ψ :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/PresentedMonoid","family_id":"ext","file_id":"mathlib/Mathlib/Algebra/PresentedMonoid/Basic.lean","sample_id":"27c7b061eb2df69ed4438c60ac705043bf7b1c99567810c2e2cbfe95dec0ba49"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3973b528cf86a6a3105014eb45539f7af155ea83fbda961dfd591ee49e04f82a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"398caa0d1490961bc830406e9179967f9ea5b8ac82e7b5f45e04d7160d1b39c6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a1d25de80740477526c09d1872d0447c8000fb20b6d0ca785a16aeb63ebdfb95","source_sha256":"6295fc893f54a37fd351974b0ff78fe129cc329f6cbb3ba221c06301535ec6ad","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨p₁, p₂, n, e⟩ := P.cond\n  have : aeval x P.f.derivative ∣ aeval x P.g ^ n :=\n    ⟨_, by simpa [h.1] using congr(aeval x $e.symm)⟩\n  exact isUnit_of_dvd_unit this (.pow _ h.2)","hard_negative":true,"metrics":{"chosen_tokens":67,"rejected_tokens":5,"token_jaccard":0.071429,"token_length_ratio":0.074627},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"cbf37a365117d20c230c642370ca887a60fcf7d1f04bfe3f494844283a516d00","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Bivariate\npublic import Mathlib.Algebra.Polynomial.Taylor\npublic import Mathlib.RingTheory.Etale.Basic\npublic import Mathlib.RingTheory.Extension.Presentation.Submersive\npublic import Mathlib.RingTheory.Ideal.IdempotentFG\n\nNamespace:\nStandardEtalePair\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Standard etale maps\n\n## Main definitions\n- `StandardEtalePair`:\n  A pair `f g : R[X]` such that `f` is monic and `f'` is invertible in `R[X][1/g]`.\n- `StandardEtalePair`: The standard etale algebra corresponding to a `StandardEtalePair`.\n- `StandardEtalePair.equivPolynomialQuotient`   : `P.Ring ≃ R[X][Y]/⟨f, Yg-1⟩`\n- `StandardEtalePair.equivAwayAdjoinRoot`       : `P.Ring ≃ (R[X]/f)[1/g]`\n- `StandardEtalePair.equivAwayQuotient`         : `P.Ring ≃ R[X][1/g]/f`\n- `StandardEtalePair.equivMvPolynomialQuotient` : `P.Ring ≃ R[X, Y]/⟨f, Yg-1⟩`\n- `StandardEtalePair.homEquiv`:\n  Maps out of `P.Ring` corresponds to `x` such that `f(x) = 0` and `g(x)` is invertible.\n- We also provide the instance that `P.Ring` is etale over `R`.\n\n- `Algebra.IsStandardEtale`: The class of standard etale algebras.\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen Polynomial\n\nopen scoped Bivariate\n\nnoncomputable section\n\nvariable {R S T : Type*} [CommRing R] [CommRing S] [CommRing T] [Algebra R S] [Algebra R T]\n\nvariable (R) in\n/-- A `StandardEtalePair R` is a pair `f g : R[X]` such that `f` is monic,\nand `f'` is invertible in `R[X][1/g]/f`. -/\nstructure StandardEtalePair : Type _ where\n  /-- The monic polynomial to be quotiented out in a standard etale algebra. -/\n  f : R[X]\n  monic_f : f.Monic\n  /-- The polynomial to be localized away from in a standard etale algebra. -/\n  g : R[X]\n  cond : ∃ p₁ p₂ n, derivative f * p₁ + f * p₂ = g ^ n\n\nvariable (P : StandardEtalePair R)\n\n/-- The standard etale algebra `R[X][Y]/⟨f, Yg-1⟩` associated to a `StandardEtalePair R`.\nAlso see\n`equivPolynomialQuotient   : P.Ring ≃ R[X][Y]/⟨f, Yg-1⟩`\n`equivAwayAdjoinRoot       : P.Ring ≃ (R[X]/f)[1/g]`\n`equivAwayQuotient         : P.Ring ≃ R[X][1/g]/f`\n`equivMvPolynomialQuotient : P.Ring ≃ R[X, Y]/⟨f, Yg-1⟩` -/\nprotected def StandardEtalePair.Ring := R[X][Y] ⧸ Ideal.span {C P.f, Y * C P.g - 1}\n  deriving CommRing, Algebra R\n\nnamespace StandardEtalePair\n\n/-- The `X` in the standard etale algebra `R[X][Y]/⟨f, Yg-1⟩`. -/\nprotected def X : P.Ring := Ideal.Quotient.mk _ (C .X)\n\n/-- There is a map from a standard etale algebra `R[X][Y]/⟨f, Yg-1⟩` to `S` sending `X` to `x` iff\n`f(x) = 0` and `g(x)` is invertible. Also see `StandardEtalePair.homEquiv`. -/\ndef HasMap (x : S) : Prop :=\n  aeval x P.f = 0 ∧ IsUnit (aeval x P.g)\n\n/-- The map `R[X][Y]/⟨f, Yg-1⟩ →ₐ[R] S` sending `X` to `x`, given `P.HasMap x`. -/\ndef lift (x : S) (h : P.HasMap x) : P.Ring →ₐ[R] S :=\n  Ideal.Quotient.liftₐ _ (aevalAeval x ↑(h.2.unit⁻¹))\n    (Ideal.span_le (I := RingHom.ker _).mpr (by simp [Set.pair_subset_iff, h.1]))\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\nlemma lift_X (x : S) (h : P.HasMap x) : P.lift x h P.X = x := by\n  simp [lift, StandardEtalePair.Ring, StandardEtalePair.X]\n\nvariable {P} in\nlemma HasMap.map {x : S} (h : P.HasMap x) (f : S →ₐ[R] T) : P.HasMap (f x) :=\n  ⟨by simp [aeval_algHom, h.1], by simpa [aeval_algHom] using h.2.map f⟩\n\nTarget:\nlemma HasMap.isUnit_derivative_f {x : S} (h : P.HasMap x) :\n    IsUnit (P.f.derivative.aeval x) :=\n\nProof body:\n","rejected":"by\n  exact HasMap.isUnit_derivative_f","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"cd23a21601a9aa4baddde82532dfc4648b79abbfa00258058961f43b8f1ce4be","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Etale","family_id":"hasmap","file_id":"mathlib/Mathlib/RingTheory/Etale/StandardEtale.lean","sample_id":"a1d25de80740477526c09d1872d0447c8000fb20b6d0ca785a16aeb63ebdfb95"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2c32995a07b48e16dcaafc728301779b447c776a69f801e281fcde5579068d01","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fce701ee6ff6a6606b14ec7394e9ebfdd374d824dea16656e00b6156e86509d7","source_sha256":"7da6267f9f5d264ecd59201d6b4195a7718cac6faa70ee30f8ae303009b58a72","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h ↦ fun _ _ _ ↦ h _ _, fun h ↦ fun i₁ i₂ ↦ ?_⟩\n  by_cases hi : i₁ = i₂\n  · simp [hi]\n  · exact h hi","hard_negative":true,"metrics":{"chosen_tokens":44,"rejected_tokens":8,"token_jaccard":0.074074,"token_length_ratio":0.181818},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"cc44fd93977c7d3235055dcd84e0a8fe0a6f7d1728394206c3293625a71c3ff4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.MetricSpace.Pseudo.Defs\npublic import Mathlib.Topology.MetricSpace.Isometry\npublic import Mathlib.Topology.MetricSpace.Dilation\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Jovan Gerbscheid. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jovan Gerbscheid, Newell Jensen\n-/\n/-!\n# Congruences\n\nThis file defines `Congruent`, i.e., the equivalence between indexed families of points in a metric\nspace where all corresponding pairwise distances are the same. The motivating example are\ntriangles in the plane.\n\n## Implementation notes\n\nAfter considering two possible approaches to defining congruence — either based on equal pairwise\ndistances or the existence of an isometric equivalence — we have opted for the broader concept of\nequal pairwise distances. This notion is commonly employed in the literature across various metric\nspaces that lack an isometric equivalence.\n\nFor more details see the [Zulip discussion](https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Euclidean.20Geometry).\n\n## Notation\n\n* `v₁ ≅ v₂`: for `Congruent v₁ v₂`.\n-/\n\n@[expose] public section\n\nvariable {ι ι' : Type*} {P₁ P₂ P₃ P₄ : Type*} {v₁ : ι → P₁} {v₂ : ι → P₂} {v₃ : ι → P₃}\n\nsection PseudoEMetricSpace\n\nvariable [PseudoEMetricSpace P₁] [PseudoEMetricSpace P₂]\nvariable [PseudoEMetricSpace P₃] [PseudoEMetricSpace P₄]\n\n/-- A congruence between indexed sets of vertices v₁ and v₂.\nUse `open scoped Congruent` to access the `v₁ ≅ v₂` notation. -/\ndef Congruent (v₁ : ι → P₁) (v₂ : ι → P₂) : Prop :=\n  ∀ i₁ i₂, edist (v₁ i₁) (v₁ i₂) = edist (v₂ i₁) (v₂ i₂)\n\n@[inherit_doc]\nscoped[Congruent] infixl:25 \" ≅ \" => Congruent\n\n/-- Congruence holds if and only if all extended distances are the same. -/\nlemma congruent_iff_edist_eq :\n    Congruent v₁ v₂ ↔ ∀ i₁ i₂, edist (v₁ i₁) (v₁ i₂) = edist (v₂ i₁) (v₂ i₂) :=\n  Iff.rfl\n\n/-- Congruence holds if and only if all extended distances between points with different\nindices are the same. -/\n\nTarget:\nlemma congruent_iff_pairwise_edist_eq :\n    Congruent v₁ v₂ ↔ Pairwise fun i₁ i₂ ↦ edist (v₁ i₁) (v₁ i₂) = edist (v₂ i₁) (v₂ i₂) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"031b929041c799bda630701bb9bc049f8022549482549716daab2fdba835cbaa","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/MetricSpace","family_id":"congruent_iff_pairwise_edist_eq","file_id":"mathlib/Mathlib/Topology/MetricSpace/Congruence.lean","sample_id":"fce701ee6ff6a6606b14ec7394e9ebfdd374d824dea16656e00b6156e86509d7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3df867611d307b2d4c52a5aeae1c5c93f72412c5fca86e32a6445afd51c4bda0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fe45775e1693cac44dede504e39345f1c5dfd23b430122a30f0b82d3494c2fdd","source_sha256":"2aaaf1d6e3bf8bcc38fe6595b30037cc7d4b213d51f17e0ec2ab4dd92ba3ffe4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  trans fun n ↦ (n / 3 - 2) * ↑((n - 3) / 6) * exp (-4 * √(log ↑((n - 3) / 6)))\n  · simp_rw [sq]\n    refine (IsBigO.mul ?_ ?_).mul ?_\n    · trans fun n ↦ n / 3\n      · simp_rw [div_eq_inv_mul]\n        exact (isBigO_refl ..).const_mul_right (by simp)\n      refine IsLittleO.right_isBigO_sub ?_\n      simpa [div_eq_inv_mul, Function.comp_def] using\n        .atTop_of_const_mul₀ zero_lt_three (by simp [tendsto_natCast_atTop_atTop])\n    · rw [IsBigO_def]\n      refine ⟨12, ?_⟩\n      simp only [IsBigOWith, norm_natCast, eventually_atTop]\n      exact ⟨15, fun x hx ↦ by norm_cast; lia⟩\n    · rw [isBigO_exp_comp_exp_comp]\n      refine ⟨0, ?_⟩\n      simp only [neg_mul, eventually_map, Pi.sub_apply, sub_neg_eq_add, neg_add_le_iff_le_add,\n        add_zero, eventually_atTop]\n      refine ⟨9, fun x hx ↦ ?_⟩\n      gcongr\n      · simp\n        lia\n      · lia\n  · refine .of_norm_eventuallyLE ?_\n    filter_upwards [eventually_ge_atTop 6] with n hn\n    have : (0 : ℝ) ≤ n / 3 - 2 := by rify at hn; linarith\n    simpa [neg_mul, abs_mul, abs_of_nonneg this] using ruzsaSzemerediNumberNat_lower_bound n","hard_negative":false,"metrics":{"chosen_tokens":244,"rejected_tokens":2,"token_jaccard":0.011236,"token_length_ratio":0.008197},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"cc45799f765756cd0ef42f1997f90a05b37faa6314599e8be0c2b5c2142bddcb","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Additive.AP.Three.Behrend\npublic import Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite\npublic import Mathlib.Tactic.Rify\npublic import Mathlib.Tactic.Qify\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# The Ruzsa-Szemerédi problem\n\nThis file proves the lower bound of the Ruzsa-Szemerédi problem. The problem is to find the maximum\nnumber of edges that a graph on `n` vertices can have if all edges belong to at most one triangle.\n\nThe lower bound comes from turning the big 3AP-free set from Behrend's construction into a graph\nthat has the property that every triangle gives a (possibly trivial) arithmetic progression on the\noriginal set.\n\n## Main declarations\n\n* `ruzsaSzemerediNumberNat n`: Maximum number of edges a graph on `n` vertices can have such that\n  each edge belongs to exactly one triangle.\n* `ruzsaSzemerediNumberNat_asymptotic_lower_bound`: There exists a graph with `n` vertices and\n  `Ω((n ^ 2 * exp (-4 * √(log n))))` edges such that each edge belongs to exactly one triangle.\n-/\n\n@[expose] public section\n\nopen Finset Nat Real SimpleGraph Sum3 SimpleGraph.TripartiteFromTriangles\nopen Fintype (card)\nopen scoped Pointwise\n\nvariable {α β : Type*}\n\n/-! ### The Ruzsa-Szemerédi number -/\n\nsection ruzsaSzemerediNumber\nvariable [DecidableEq α] [DecidableEq β] [Fintype α] [Fintype β] {G H : SimpleGraph α}\n\nvariable (α) in\n/-- The **Ruzsa-Szemerédi number** of a fintype is the maximum number of edges a locally linear\ngraph on that type can have.\n\nIn other words, `ruzsaSzemerediNumber α` is the maximum number of edges a graph on `α` can have such\nthat each edge belongs to exactly one triangle. -/\nnoncomputable def ruzsaSzemerediNumber : ℕ := by\n  classical\n  exact Nat.findGreatest (fun m ↦ ∃ (G : SimpleGraph α) (_ : DecidableRel G.Adj),\n    #(G.cliqueFinset 3) = m ∧ G.LocallyLinear) ((card α).choose 3)\n\nopen scoped Classical in\nlemma ruzsaSzemerediNumber_le : ruzsaSzemerediNumber α ≤ (card α).choose 3 := Nat.findGreatest_le _\n\nlemma ruzsaSzemerediNumber_spec :\n    ∃ (G : SimpleGraph α) (_ : DecidableRel G.Adj),\n      #(G.cliqueFinset 3) = ruzsaSzemerediNumber α ∧ G.LocallyLinear := by\n  classical\n  exact @Nat.findGreatest_spec _\n    (fun m ↦ ∃ (G : SimpleGraph α) (_ : DecidableRel G.Adj),\n      #(G.cliqueFinset 3) = m ∧ G.LocallyLinear) _ _ (Nat.zero_le _)\n    ⟨⊥, inferInstance, by simp, locallyLinear_bot⟩\n\nvariable {m n : ℕ}\n\nlemma SimpleGraph.LocallyLinear.le_ruzsaSzemerediNumber [DecidableRel G.Adj]\n    (hG : G.LocallyLinear) : #(G.cliqueFinset 3) ≤ ruzsaSzemerediNumber α := by\n  classical\n  exact le_findGreatest card_cliqueFinset_le ⟨G, inferInstance, by congr, hG⟩\n\nlemma ruzsaSzemerediNumber_mono (f : α ↪ β) : ruzsaSzemerediNumber α ≤ ruzsaSzemerediNumber β := by\n  classical\n  refine findGreatest_mono ?_ (choose_mono _ <| Fintype.card_le_of_embedding f)\n  rintro n ⟨G, _, rfl, hG⟩\n  refine ⟨G.map f, inferInstance, ?_, hG.map _⟩\n  rw [← card_map ⟨map f, Finset.map_injective _⟩, ← cliqueFinset_map G f]\n  decide\n\nlemma ruzsaSzemerediNumber_congr (e : α ≃ β) : ruzsaSzemerediNumber α = ruzsaSzemerediNumber β :=\n  (ruzsaSzemerediNumber_mono (e : α ↪ β)).antisymm <| ruzsaSzemerediNumber_mono e.symm\n\n/-- The `n`-th **Ruzsa-Szemerédi number** is the maximum number of edges a locally linear graph on\n`n` vertices can have.\n\nIn other words, `ruzsaSzemerediNumberNat n` is the maximum number of edges a graph on `n` vertices\ncan have such that each edge belongs to exactly one triangle. -/\nnoncomputable def ruzsaSzemerediNumberNat (n : ℕ) : ℕ := ruzsaSzemerediNumber (Fin n)\n\n@[simp]\nlemma ruzsaSzemerediNumberNat_card : ruzsaSzemerediNumberNat (card α) = ruzsaSzemerediNumber α :=\n  ruzsaSzemerediNumber_congr (Fintype.equivFin _).symm\n\n@[gcongr]\nlemma ruzsaSzemerediNumberNat_mono : Monotone ruzsaSzemerediNumberNat := fun _m _n h =>\n  ruzsaSzemerediNumber_mono (Fin.castLEEmb h)\n\nlemma ruzsaSzemerediNumberNat_le : ruzsaSzemerediNumberNat n ≤ n.choose 3 :=\n  ruzsaSzemerediNumber_le.trans_eq <| by rw [Fintype.card_fin]\n\n@[simp] lemma ruzsaSzemerediNumberNat_zero : ruzsaSzemerediNumberNat 0 = 0 :=\n  le_zero_iff.1 ruzsaSzemerediNumberNat_le\n\n@[simp] lemma ruzsaSzemerediNumberNat_one : ruzsaSzemerediNumberNat 1 = 0 :=\n  le_zero_iff.1 ruzsaSzemerediNumberNat_le\n\n@[simp] lemma ruzsaSzemerediNumberNat_two : ruzsaSzemerediNumberNat 2 = 0 :=\n  le_zero_iff.1 ruzsaSzemerediNumberNat_le\n\nend ruzsaSzemerediNumber\n\n/-! ### The Ruzsa-Szemerédi construction -/\n\nsection RuzsaSzemeredi\nvariable [Fintype α] [CommRing α] {s : Finset α} {x : α × α × α}\n\n/-- The triangle indices for the Ruzsa-Szemerédi construction. -/\nprivate def triangleIndices (s : Finset α) : Finset (α × α × α) :=\n  (univ ×ˢ s).map\n    ⟨fun xa ↦ (xa.1, xa.1 + xa.2, xa.1 + 2 * xa.2), by\n      rintro ⟨x, a⟩ ⟨y, b⟩ h\n      simp only [Prod.ext_iff] at h\n      obtain rfl := h.1\n      obtain rfl := add_right_injective _ h.2.1\n      rfl⟩\n\n@[simp]\nprivate lemma mem_triangleIndices :\n    x ∈ triangleIndices s ↔ ∃ y, ∃ a ∈ s, (y, y + a, y + 2 * a) = x := by simp [triangleIndices]\n\n@[simp]\nprivate lemma card_triangleIndices : #(triangleIndices s) = card α * #s := by\n  simp [triangleIndices]\n\nprivate lemma noAccidental (hs : ThreeAPFree (s : Set α)) :\n    NoAccidental (triangleIndices s : Finset (α × α × α)) where\n  eq_or_eq_or_eq := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ _ _ d a ha rfl rfl rfl b' b hb rfl rfl h₁ d' c hc rfl h₂ rfl\n    have : a + c = b + b := by linear_combination h₁.symm - h₂.symm\n    obtain rfl := hs ha hb hc this\n    simp_all\n\nvariable [Fact <| IsUnit (2 : α)]\n\nprivate instance : ExplicitDisjoint (triangleIndices s : Finset (α × α × α)) where\n  inj₀ := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ x a ha rfl rfl rfl y b hb rfl h₁ h₂\n    linear_combination 2 * h₁.symm - h₂.symm\n  inj₁ := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ x a ha rfl rfl rfl y b hb rfl rfl h\n    simpa [(Fact.out (p := IsUnit (2 : α))).mul_right_inj, eq_comm] using h\n  inj₂ := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ x a ha rfl rfl rfl y b hb rfl h rfl\n    simpa [(Fact.out (p := IsUnit (2 : α))).mul_right_inj, eq_comm] using h\n\nprivate lemma locallyLinear (hs : ThreeAPFree (s : Set α)) :\n    (graph <| triangleIndices s).LocallyLinear :=\n  haveI := noAccidental hs; TripartiteFromTriangles.locallyLinear _\n\nprivate lemma card_edgeFinset (hs : ThreeAPFree (s : Set α)) [DecidableEq α] :\n    #(graph <| triangleIndices s).edgeFinset = 3 * card α * #s := by\n  haveI := noAccidental hs\n  rw [(locallyLinear hs).card_edgeFinset, card_triangles, card_triangleIndices, mul_assoc]\n\nend RuzsaSzemeredi\n\nvariable (α) [Fintype α] [DecidableEq α] [CommRing α] [Fact <| IsUnit (2 : α)]\n\nlemma addRothNumber_le_ruzsaSzemerediNumber :\n    card α * addRothNumber (univ : Finset α) ≤ ruzsaSzemerediNumber (Sum α (Sum α α)) := by\n  obtain ⟨s, -, hscard, hs⟩ := addRothNumber_spec (univ : Finset α)\n  haveI := noAccidental hs\n  rw [← hscard, ← card_triangleIndices, ← card_triangles]\n  exact (locallyLinear hs).le_ruzsaSzemerediNumber\n\nlemma rothNumberNat_le_ruzsaSzemerediNumberNat (n : ℕ) :\n    (2 * n + 1) * rothNumberNat n ≤ ruzsaSzemerediNumberNat (6 * n + 3) := by\n  let α := Fin (2 * n + 1)\n  have : Nat.Coprime 2 (2 * n + 1) := by simp\n  haveI : Fact (IsUnit (2 : Fin (2 * n + 1))) := ⟨by simpa\n    using! (ZMod.unitOfCoprime 2 this).isUnit⟩\n  open scoped Fin.CommRing in\n  calc\n    (2 * n + 1) * rothNumberNat n\n    _ = Fintype.card α * addRothNumber (Iio (n : α)) := by\n      rw [Fin.addRothNumber_eq_rothNumberNat le_rfl, Fintype.card_fin]\n    _ ≤ Fintype.card α * addRothNumber (univ : Finset α) := by\n      gcongr; exact subset_univ _\n    _ ≤ ruzsaSzemerediNumber (Sum α (Sum α α)) := addRothNumber_le_ruzsaSzemerediNumber _\n    _ = ruzsaSzemerediNumberNat (6 * n + 3) := by\n      simp_rw [← ruzsaSzemerediNumberNat_card, Fintype.card_sum, α, Fintype.card_fin]\n      ring_nf\n\n/-- Lower bound on the **Ruzsa-Szemerédi problem** in terms of 3AP-free sets.\n\nIf there exists a 3AP-free subset of `[1, ..., (n - 3) / 6]` of size `m`, then there exists a graph\nwith `n` vertices and `(n / 3 - 2) * m` edges such that each edge belongs to exactly one triangle.\n-/\ntheorem rothNumberNat_le_ruzsaSzemerediNumberNat' :\n    ∀ n : ℕ, (n / 3 - 2 : ℝ) * rothNumberNat ((n - 3) / 6) ≤ ruzsaSzemerediNumberNat n\n  | 0 => by simp\n  | 1 => by simp\n  | 2 => by simp\n  | n + 3 => by\n    calc\n      _ ≤ (↑(2 * (n / 6) + 1) : ℝ) * rothNumberNat (n / 6) :=\n        mul_le_mul_of_nonneg_right ?_ (Nat.cast_nonneg _)\n      _ ≤ (ruzsaSzemerediNumberNat (6 * (n / 6) + 3) : ℝ) := ?_\n      _ ≤ _ := by grw [Nat.mul_div_le]\n    · simp only [cast_add, cast_ofNat, cast_mul, cast_one, tsub_le_iff_right]\n      rw [← div_add_one (three_ne_zero' ℝ), ← le_sub_iff_add_le, div_le_iff₀ (zero_lt_three' ℝ),\n        add_assoc, add_sub_assoc, add_mul, mul_right_comm, add_sub_cancel_left]\n      norm_cast\n      rw [← mul_add_one]\n      exact (Nat.lt_mul_div_succ _ <| by simp).le\n    · norm_cast\n      exact rothNumberNat_le_ruzsaSzemerediNumberNat _\n\n/-- Explicit lower bound on the **Ruzsa-Szemerédi problem**.\n\nThere exists a graph with `n` vertices and\n`(n / 3 - 2) * (n - 3) / 6 * exp (-4 * √(log ((n - 3) / 6)))` edges such that each edge belongs\nto exactly one triangle. -/\ntheorem ruzsaSzemerediNumberNat_lower_bound (n : ℕ) :\n    (n / 3 - 2 : ℝ) * ↑((n - 3) / 6) * exp (-4 * √(log ↑((n - 3) / 6))) ≤\n      ruzsaSzemerediNumberNat n := by\n  rw [mul_assoc]\n  obtain hn | hn := le_total (n / 3 - 2 : ℝ) 0\n  · exact (mul_nonpos_of_nonpos_of_nonneg hn <| by positivity).trans (Nat.cast_nonneg _)\n  exact\n    (mul_le_mul_of_nonneg_left Behrend.roth_lower_bound hn).trans\n      (rothNumberNat_le_ruzsaSzemerediNumberNat' _)\n\nopen Asymptotics Filter\n\n/-- Asymptotic lower bound on the **Ruzsa-Szemerédi problem**.\n\nThere exists a graph with `n` vertices and `Ω((n ^ 2 * exp (-4 * √(log n))))` edges such that\neach edge belongs to exactly one triangle. -/\n\nTarget:\ntheorem ruzsaSzemerediNumberNat_asymptotic_lower_bound :\n    (fun n ↦ n ^ 2 * exp (-4 * √(log n)) : ℕ → ℝ) =O[atTop]\n     fun n ↦ (ruzsaSzemerediNumberNat n : ℝ) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Extremal","family_id":"ruzsaszemeredinumbernat_asymptotic_lower_bound","file_id":"mathlib/Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean","sample_id":"fe45775e1693cac44dede504e39345f1c5dfd23b430122a30f0b82d3494c2fdd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ad208d86006761f39c52958101642c0dd5169f84bd3c5136aba050921aea4aea","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2ba4cc62e244d5056d28140a5aecb6bb0931d3b1d3d603a2cbd5639b8008e4b7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0659aa452ee5b4142dc99d160a8e44415976c14c3bb5194948dedcb030d02219","source_sha256":"dd8040e4f8746a8f8bfe236176883e5dd7af73f9152b910f7d596480c7e29802","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let ⟨g, hg⟩ := (ContinuousLinearMap.id 𝕜 S).exist_extension_of_finiteDimensional_range\n  exact ⟨g, DFunLike.congr_fun hg.symm⟩","hard_negative":true,"metrics":{"chosen_tokens":28,"rejected_tokens":2,"token_jaccard":0.047619,"token_length_ratio":0.071429},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"cc72e41710d4dbab7d78826c54d92bb37e4225ca12595beba6a667772e6d5780","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Convex.Cone.Extension\npublic import Mathlib.Analysis.LocallyConvex.AbsConvexOpen\npublic import Mathlib.Analysis.LocallyConvex.WeakDual\npublic import Mathlib.Analysis.Normed.Module.RCLike.Extend\npublic import Mathlib.Topology.Algebra.Module.FiniteDimension\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2026 Yongxi Lin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yongxi Lin\n-/\n/-!\n# Hahn-Banach theorem for polynormable spaces\n\nIn this file, we prove the analytic Hahn-Banach theorem for polynormable spaces over a field\nsatisfying `IsRCLikeNormedField`. For any continuous linear functional on a subspace, we can extend\nit to the entire space. Note that we cannot use `LocallyConvexSpace` because an\n`IsRCLikeNormedField` has no order structure.\n\nWe prove\n* `Module.Dual.exists_continuous_extension_of_le_seminorm`: Hahn-Banach theorem for linear\n  functionals dominated by a continuous seminorm on polynormable spaces over a field satisfying\n  `IsRCLikeNormedField`.\n* `StrongDual.exists_extension`: Hahn-Banach theorem for continuous linear functionals on\n  polynormable spaces over fields satisfying `IsRCLikeNormedField`.\n\n-/\n\npublic section\n\nopen Module Topology RCLike\n\nopen scoped ComplexConjugate\n\nvariable {𝕜 E : Type*} [AddCommGroup E]\n\ntheorem Module.Dual.exists_extension_of_le_seminorm_real [Module ℝ E]\n    (S : Subspace ℝ E) (f : Dual ℝ S)\n    {p : Seminorm ℝ E} (hp : ∀ x, f x ≤ p x) :\n    ∃ g : Dual ℝ E, (∀ x : S, g x = f x) ∧ ∀ x, |g x| ≤ p x := by\n  obtain ⟨g, hg, hl⟩ := by\n    refine exists_extension_of_le_sublinear ⟨S, f⟩ p (fun _ hc _ => ?_) ?_ hp\n    · simp [map_smul_eq_mul, abs_of_nonneg hc.le]\n    · exact fun x y => map_add_le_add p x y\n  exact ⟨g, hg, p.abs_le_of_le hl⟩\n\nvariable [NormedField 𝕜] [IsRCLikeNormedField 𝕜]\n\ntheorem Module.Dual.exists_extension_of_le_seminorm [Module 𝕜 E] (S : Submodule 𝕜 E) (f : Dual 𝕜 S)\n    {p : Seminorm 𝕜 E} (hp : ∀ x, ‖f x‖ ≤ p x) :\n    ∃ g : Dual 𝕜 E, (∀ x : S, g x = f x) ∧ ∀ x, ‖g x‖ ≤ p x := by\n  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜\n  letI : Module ℝ E := .restrictScalars ℝ 𝕜 E\n  letI : IsScalarTower ℝ 𝕜 E := .restrictScalars _ _ _\n  let fr : Dual ℝ S := reLm.comp (f.restrictScalars ℝ)\n  obtain ⟨g, (hg : ∀ x : S, g x = fr x), hgp⟩ :=\n    fr.exists_extension_of_le_seminorm_real (S.restrictScalars ℝ) (p := p.restrictScalars ℝ)\n      fun x ↦ (re_le_norm (f x)).trans (hp x)\n  refine ⟨g.extendRCLike, fun x ↦ ?_, fun x ↦ ?_⟩\n  · rw [g.extendRCLike_apply, ← Submodule.coe_smul, hg, hg]\n    simp [fr, mul_comm I]\n  · apply norm_extendRCLike_le_seminorm\n    exact hgp\n\nvariable [TopologicalSpace E]\n\n/-- **Hahn-Banach theorem** for linear functionals dominated by a continuous seminorm on\npolynormable spaces over `ℝ`. -/\ntheorem Module.Dual.exists_continuous_extension_of_le_seminorm_real [IsTopologicalAddGroup E]\n    [Module ℝ E] [ContinuousSMul ℝ E] [PolynormableSpace ℝ E] (S : Subspace ℝ E) (f : Dual ℝ S)\n    {p : Seminorm ℝ E} (hp_cont : Continuous p) (hp : ∀ x, f x ≤ p x) :\n    ∃ g : StrongDual ℝ E, (∀ x : S, g x = f x) ∧ ∀ x, |g x| ≤ p x := by\n  obtain ⟨g, hg, hl⟩ := f.exists_extension_of_le_seminorm_real S hp\n  exact ⟨⟨g, (PolynormableSpace.withSeminorms ℝ E).continuous_real_rng g\n    ⟨{⟨p, hp_cont⟩}, 1, fun x ↦ by simpa using (le_abs_self _).trans (hl x)⟩⟩, hg, hl⟩\n\nvariable [Module 𝕜 E] [PolynormableSpace 𝕜 E]\n\n/-- **Hahn-Banach theorem** for linear functionals dominated by a continuous seminorm on\npolynormable spaces over fields satisfying `IsRCLikeNormedField`. -/\ntheorem Module.Dual.exists_continuous_extension_of_le_seminorm (S : Submodule 𝕜 E) (f : Dual 𝕜 S)\n    {p : Seminorm 𝕜 E} (hp_cont : Continuous p) (hp : ∀ x, ‖f x‖ ≤ p x) :\n    ∃ g : StrongDual 𝕜 E, (∀ x : S, g x = f x) ∧ ∀ x, ‖g x‖ ≤ p x := by\n  obtain ⟨g, hg, hle⟩ := Dual.exists_extension_of_le_seminorm S f hp\n  refine ⟨⟨g, (PolynormableSpace.withSeminorms 𝕜 E).continuous_normedSpace_rng 𝕜 g ?_⟩, hg, hle⟩\n  exact ⟨{⟨p, hp_cont⟩}, 1, by simpa⟩\n\n/-- **Hahn-Banach theorem** for continuous linear functionals on polynormable spaces over a field\nsatisfying `IsRCLikeNormedField`. -/\ntheorem StrongDual.exists_extension {𝕜} [NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜]\n    [Module 𝕜 E] [PolynormableSpace 𝕜 E] (S : Submodule 𝕜 E) (f : StrongDual 𝕜 S) :\n    ∃ g : StrongDual 𝕜 E, ∀ x : S, g x = f x := by\n  obtain ⟨q, hq_cont, hq⟩ := Seminorm.exists_le_comp_of_isInducing (f := S.subtype)\n    (p := f.toSeminorm) f.continuous.norm IsInducing.subtypeVal\n  obtain ⟨g, hg, _⟩ := Dual.exists_continuous_extension_of_le_seminorm S f.toLinearMap hq_cont hq\n  exact ⟨g, hg⟩\n\nvariable {F : Type*} [AddCommGroup F] [TopologicalSpace F] [IsTopologicalAddGroup F] [Module 𝕜 F]\n  [ContinuousSMul 𝕜 F] [T2Space F]\n\n/-- Corollary of the polynormable **Hahn-Banach theorem**: if `f : S → F` is a continuous\nlinear map with finite-dimensional range, then `f` extends to a continuous linear map on the whole\nspace. -/\nlemma ContinuousLinearMap.exist_extension_of_finiteDimensional_range {S : Submodule 𝕜 E}\n    (f : S →L[𝕜] F) [FiniteDimensional 𝕜 f.range] :\n    ∃ g : E →L[𝕜] F, f = g.comp S.subtypeL := by\n  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜\n  let b := Module.finBasis 𝕜 f.range\n  let e := b.equivFunL\n  let fi := fun i ↦ (LinearMap.toContinuousLinearMap (b.coord i)).comp\n    (f.codRestrict _ <| LinearMap.mem_range_self _)\n  choose gi hgf using fun i ↦ StrongDual.exists_extension S (fi i)\n  use f.range.subtypeL.comp <| e.symm.toContinuousLinearMap.comp (.pi gi)\n  ext x\n  simp [fi, e, hgf]\n\n/-- A finite-dimensional submodule of a polynormable space over a field satisfying\n`IsRCLikeNormedField` is `Submodule.ClosedComplemented`. -/\n\nTarget:\nlemma Submodule.ClosedComplemented.of_finiteDimensional [PolynormableSpace 𝕜 F] (S : Submodule 𝕜 F)\n    [FiniteDimensional 𝕜 S] : S.ClosedComplemented :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_0659aa452ee5","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e38e17756d1bf500147fbd2c190bca3685edd85c96cf9e93c947b51a3b512ac8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/LocallyConvex","family_id":"submodule","file_id":"mathlib/Mathlib/Analysis/LocallyConvex/HahnBanach.lean","sample_id":"0659aa452ee5b4142dc99d160a8e44415976c14c3bb5194948dedcb030d02219"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e68fac77a9fc516d7c10bdaed9dcefeaf4db7121c9ed0829fb3a283ec7ee3c03","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"76b26d55952be09aac10b04e67c37a508dbe1e70e8aa7c1db85d1639fb9ae756","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"735bf5dce5b321e535055fa4c86fa49603a8028a22626fad8900aee15b3514d6","source_sha256":"0688c3fec656e5c98eb2a67e2bb9366dc426159747d3f41fd3b47b03b8868e64","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [range, List.range_succ, ← coe_add, Multiset.add_comm, range, coe_singleton, singleton_add]","hard_negative":true,"metrics":{"chosen_tokens":22,"rejected_tokens":3,"token_jaccard":0.058824,"token_length_ratio":0.136364},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"cc9e0ff6aea75594c951dfcf7023055b33c737de4b94c289c4f588b2e698ad3c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Multiset.UnionInter\n\nNamespace:\nMultiset\n\nLocal context:\n/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-! # `Multiset.range n` gives `{0, 1, ..., n-1}` as a multiset. -/\n\n@[expose] public section\n\nassert_not_exists Monoid\n\nopen List Nat\n\nnamespace Multiset\n\n-- range\n/-- `range n` is the multiset lifted from the list `range n`,\n  that is, the set `{0, 1, ..., n-1}`. -/\ndef range (n : ℕ) : Multiset ℕ :=\n  List.range n\n\ntheorem coe_range (n : ℕ) : ↑(List.range n) = range n :=\n  rfl\n\n@[simp]\ntheorem range_zero : range 0 = 0 :=\n  rfl\n\n@[simp]\n\nTarget:\ntheorem range_succ (n : ℕ) : range (succ n) = n ::ₘ range n :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_735bf5dce5b3","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"a2ab4480e8be77064093d36caadbc361d6e761483be451486e25a98119941bce","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Multiset","family_id":"range_succ","file_id":"mathlib/Mathlib/Data/Multiset/Range.lean","sample_id":"735bf5dce5b321e535055fa4c86fa49603a8028a22626fad8900aee15b3514d6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"313c9cf4c1b5eea4156e212ff571b19fbbfa824c1fd6f777992a3ed4cc19b0b1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dd0a6c86eaf74b4453f65b352e9a02cbc6a9bc8e7367f1c0ef0f634a91f60ff2","source_sha256":"44d6e0a4a761ac34c9ab6bb892d46ec9566fb71d10c48050289a8c26b4be1e40","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases hq : 1 < q\n  · rw [← pow_zero q, ofPowerSeries_apply_pow hq, PowerSeries.coeff_zero_eq_constantCoeff]\n  · simp [ofPowerSeries, dif_neg hq]","hard_negative":false,"metrics":{"chosen_tokens":29,"rejected_tokens":2,"token_jaccard":0.045455,"token_length_ratio":0.068966},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"ccd847e98947eacff7b6f74af58894f0896a00a7f05df35134e482f54fd07323","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.ArithmeticFunction.Defs\npublic import Mathlib.Order.Northcott\npublic import Mathlib.RingTheory.PowerSeries.Basic\npublic import Mathlib.RingTheory.PowerSeries.PiTopology\npublic import Mathlib.RingTheory.PowerSeries.Substitution\n\nNamespace:\nArithmeticFunction\n\nLocal context:\n/-\nCopyright (c) 2026 Thomas Browning. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Thomas Browning\n-/\n/-!\n# Construction of L-functions\n\nThis file constructs L-functions as formal Dirichlet series.\n\n## Main definitions\n\n* `ArithmeticFunction.ofPowerSeries q f`: L-function `f(q⁻ˢ)` obtained from a power series `f(T)`.\n* `ArithmeticFunction.eulerProduct f`: the Euler product of a family `f i` of Dirichlet series.\n\n## Implementation notes\n\nWe take the following route from polynomials to L-functions:\n* Starting from a polynomial in `T`, `PowerSeries.invOfUnit` gives the reciporical power series.\n* `ofPowerSeries` gives the local Euler factor as a formal Dirichlet series on powers of `q`.\n* `eulerProduct` gives the L-function as the formal product of these local Euler factors.\n* `LSeries` gives the L-function as an analytic function on the right half-plane of convergence.\n\nFor example, the Riemann zeta function `ζ(s)` corresponds to taking `1 - T` at each prime `p`.\n\nFor context, here is a diagram of the possible routes from polynomials to L-functions:\n```\n                   T=q⁻ˢ                     s ∈ ℂ\n[polynomials in T] ----> [polynomials in q⁻ˢ] ----> [analytic function in s]\n          |                           |                           |\n          | (reciprocal)              | (reciprocal)              | (reciprocal)\n          v         T=q⁻ˢ             V          s ∈ ℂ            V\n[power series in T] ----> [power series in q⁻ˢ] ----> [analytic function in s] (the Euler factor)\n          |                           |                           |\n          | (product)                 | (product)                 | (product)\n          v                 T=q⁻ˢ     V               s ∈ ℂ       V\n[multivariate power series] ----> [Dirichlet series] ----> [L-function in s] (the Euler product)\n```\n-/\n\n@[expose] public section\n\nnamespace ArithmeticFunction\n\nsection PowerSeries\n\nvariable {R : Type*}\n\nsection CommSemiring\n\nvariable [CommSemiring R]\n\n/-- The arithmetic function corresponding to the Dirichlet series `f(q⁻ˢ)`.\nFor example, if `f = 1 + X + X² + ...` and `q = p`, then `f(q⁻ˢ) = 1 + p⁻ˢ + p⁻²ˢ + ...`.\n\nIf `q ≤ 1` then `k ↦ q ^ k` is not injective, so we use the junk value `f.constantCoeff`. -/\nnoncomputable def ofPowerSeries (q : ℕ) : PowerSeries R →ₐ[R] ArithmeticFunction R where\n  toFun f := if hq : 1 < q then\n    ⟨Function.extend (q ^ ·) (f.coeff ·) 0, by simp [Nat.ne_zero_of_lt hq]⟩ else\n      algebraMap R (ArithmeticFunction R) f.constantCoeff\n  map_zero' := by ext; split_ifs <;> simp [Function.extend]\n  -- note that `ofPowerSeries.map_one'` relies on the junk value `f.constantCoeff`.\n  map_one' := by\n    ext n\n    split_ifs with hq\n    · by_cases hn : ∃ k, q ^ k = n\n      · obtain ⟨a, rfl⟩ := hn\n        simp [(Nat.pow_right_injective hq).extend_apply, one_apply, hq.ne']\n      · simp [hn, one_apply_ne (fun H ↦ hn ⟨0, H.symm⟩)]\n    · simp\n  map_add' f g := by\n    ext n\n    split_ifs with hq\n    · by_cases h : ∃ a, q ^ a = n\n      · obtain ⟨a, rfl⟩ := h\n        simp [(Nat.pow_right_injective hq).extend_apply]\n      · simp [h]\n    · by_cases hn : n = 1 <;> simp [hn]\n  map_mul' f g := by\n    ext n\n    split_ifs with hq\n    · simp_rw [mul_apply, coe_mk]\n      by_cases hn : ∃ a, q ^ a = n\n      · obtain ⟨k, rfl⟩ := hn\n        rw [(Nat.pow_right_injective hq).extend_apply]\n        have hs : (Finset.antidiagonal k).map (.prodMap ⟨fun k ↦ q ^ k, Nat.pow_right_injective hq⟩\n            ⟨fun k ↦ q ^ k, Nat.pow_right_injective hq⟩) ⊆ (q ^ k).divisorsAntidiagonal :=\n          Nat.antidiagonal_map_subset_divisorsAntidiagonal_pow hq k\n        rw [PowerSeries.coeff_mul k f g, ← Finset.sum_subset hs]\n        · simp [(Nat.pow_right_injective hq).extend_apply]\n        · intro (a, b) hab h\n          by_cases ha : ∃ i, q ^ i = a\n          · by_cases hb : ∃ j, q ^ j = b\n            · obtain ⟨i, rfl⟩ := ha\n              obtain ⟨j, rfl⟩ := hb\n              rw [Nat.mem_divisorsAntidiagonal, ← pow_add, Nat.pow_right_inj hq] at hab\n              simp_rw [Finset.mem_map, not_exists, not_and, Finset.mem_antidiagonal] at h\n              simpa using h (i, j) hab.1\n            · rwa [mul_comm, Function.extend_apply', Pi.zero_apply, zero_mul]\n          · rwa [Function.extend_apply', Pi.zero_apply, zero_mul]\n      · rw [Function.extend_apply' _ _ _ hn, Pi.zero_apply, Finset.sum_eq_zero]\n        intro (a, b) hk\n        obtain ⟨hab, -⟩ := Nat.mem_divisorsAntidiagonal.mp hk\n        by_cases ha : ∃ i, q ^ i = a\n        · by_cases hb : ∃ j, q ^ j = b\n          · obtain ⟨i, rfl⟩ := ha\n            obtain ⟨j, rfl⟩ := hb\n            rw [← pow_add] at hab\n            exact (hn ⟨i + j, hab⟩).elim\n          · rwa [mul_comm, Function.extend_apply', Pi.zero_apply, zero_mul]\n        · rwa [Function.extend_apply', Pi.zero_apply, zero_mul]\n    · simp\n  commutes' x := by\n    ext n\n    split_ifs with hq\n    · simp only [Algebra.algebraMap_eq_smul_one, coe_mk]\n      by_cases hn : ∃ k, q ^ k = n\n      · obtain ⟨k, rfl⟩ := hn\n        simp [(Nat.pow_right_injective hq).extend_apply, one_apply, hq.ne']\n      · rw [Function.extend_apply' _ _ _ hn, Pi.zero_apply, smul_map, one_apply_ne, smul_zero]\n        contrapose hn\n        exact ⟨0, by simp [hn]⟩\n    · simp\n\ntheorem ofPowerSeries_apply {q : ℕ} (hq : 1 < q) (f : PowerSeries R) (n : ℕ) :\n    ofPowerSeries q f n = Function.extend (q ^ ·) (f.coeff ·) 0 n := by\n  simp [ofPowerSeries, dif_pos hq]\n\ntheorem ofPowerSeries_apply_pow {q : ℕ} (hq : 1 < q) (f : PowerSeries R) (k : ℕ) :\n    ofPowerSeries q f (q ^ k) = f.coeff k := by\n  rw [ofPowerSeries_apply hq, (Nat.pow_right_injective hq).extend_apply]\n\ntheorem ofPowerSeries_apply_zero (q : ℕ) (f : PowerSeries R) : ofPowerSeries q f 0 = 0 := by\n  simp\n\n@[simp]\n-- note that `ofPowerSeries_apply_one` relies on the junk value `f.constantCoeff`.\n\nTarget:\ntheorem ofPowerSeries_apply_one (q : ℕ) (f : PowerSeries R) :\n    ofPowerSeries q f 1 = f.constantCoeff :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/ArithmeticFunction","family_id":"ofpowerseries_apply_one","file_id":"mathlib/Mathlib/NumberTheory/ArithmeticFunction/LFunction.lean","sample_id":"dd0a6c86eaf74b4453f65b352e9a02cbc6a9bc8e7367f1c0ef0f634a91f60ff2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"536352e5ddbb900c41d2280e8e59f7791e63d85f699a6f8a27e8c2d97392c7b3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b9b65ec13f7e30e41eeaa6338022c9efbc66d8f0394c7b483c53252b38ab9a8f","source_sha256":"07a36e2d42950767413a6ab43b32d6264e65f7c71e39cd053b21a81cb6699e71","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let S : Set (Ultrafilter M) := ⋂ n, { U | ∀ᶠ m in U, m ∈ FP (a.drop n) }\n  have h := exists_idempotent_in_compact_subsemigroup ?_ S ?_ ?_ ?_\n  · rcases h with ⟨U, hU, U_idem⟩\n    refine ⟨U, U_idem, ?_⟩\n    convert! Set.mem_iInter.mp hU 0\n  · exact Ultrafilter.continuous_mul_left\n  · apply IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed\n    · intro n U hU\n      filter_upwards [hU]\n      rw [← Stream'.drop_drop, ← Stream'.tail_eq_drop]\n      exact FP.tail _\n    · intro n\n      exact ⟨pure _, mem_pure.mpr <| FP.head _⟩\n    · exact (ultrafilter_isClosed_basic _).isCompact\n    · intro n\n      apply ultrafilter_isClosed_basic\n  · exact IsClosed.isCompact (isClosed_iInter fun i => ultrafilter_isClosed_basic _)\n  · intro U hU V hV\n    rw [Set.mem_iInter] at *\n    intro n\n    rw [Set.mem_setOf_eq, Ultrafilter.eventually_mul]\n    filter_upwards [hU n] with m hm\n    obtain ⟨n', hn⟩ := FP.mul hm\n    filter_upwards [hV (n' + n)] with m' hm'\n    apply hn\n    simpa only [Stream'.drop_drop, add_comm] using hm'","hard_negative":true,"metrics":{"chosen_tokens":224,"rejected_tokens":8,"token_jaccard":0.043478,"token_length_ratio":0.035714},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"cce7eae1f3a0b2e14810860437387f7d1427fc35ab4282f05ace605284f4690c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Stream.Init\npublic import Mathlib.Topology.Algebra.Semigroup\npublic import Mathlib.Topology.Compactification.StoneCech\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nNamespace:\nHindman\n\nLocal context:\n/-\nCopyright (c) 2021 David Wärn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David Wärn\n-/\n/-!\n# Hindman's theorem on finite sums\n\nWe prove Hindman's theorem on finite sums, using idempotent ultrafilters.\n\nGiven an infinite sequence `a₀, a₁, a₂, …` of positive integers, the set `FS(a₀, …)` is the set\nof positive integers that can be expressed as a finite sum of `aᵢ`'s, without repetition. Hindman's\ntheorem asserts that whenever the positive integers are finitely colored, there exists a sequence\n`a₀, a₁, a₂, …` such that `FS(a₀, …)` is monochromatic. There is also a stronger version, saying\nthat whenever a set of the form `FS(a₀, …)` is finitely colored, there exists a sequence\n`b₀, b₁, b₂, …` such that `FS(b₀, …)` is monochromatic and contained in `FS(a₀, …)`. We prove both\nthese versions for a general semigroup `M` instead of `ℕ+` since it is no harder, although this\nspecial case implies the general case.\n\nThe idea of the proof is to extend the addition `(+) : M → M → M` to addition `(+) : βM → βM → βM`\non the space `βM` of ultrafilters on `M`. One can prove that if `U` is an _idempotent_ ultrafilter,\ni.e. `U + U = U`, then any `U`-large subset of `M` contains some set `FS(a₀, …)` (see\n`exists_FS_of_large`). And with the help of a general topological argument one can show that any set\nof the form `FS(a₀, …)` is `U`-large according to some idempotent ultrafilter `U` (see\n`exists_idempotent_ultrafilter_le_FS`). This is enough to prove the theorem since in any finite\npartition of a `U`-large set, one of the parts is `U`-large.\n\n## Main results\n\n- `FS_partition_regular`: the strong form of Hindman's theorem\n- `exists_FS_of_finite_cover`: the weak form of Hindman's theorem\n\n## Tags\n\nRamsey theory, ultrafilter\n\n-/\n\n@[expose] public section\n\n\nopen Filter\n\n/-- Multiplication of ultrafilters given by `∀ᶠ m in U*V, p m ↔ ∀ᶠ m in U, ∀ᶠ m' in V, p (m*m')`. -/\n@[to_additive (attr := implicit_reducible)\n/-- Addition of ultrafilters given by `∀ᶠ m in U+V, p m ↔ ∀ᶠ m in U, ∀ᶠ m' in V, p (m+m')`. -/]\ndef Ultrafilter.mul {M} [Mul M] : Mul (Ultrafilter M) where mul U V := (· * ·) <$> U <*> V\n\nattribute [local instance] Ultrafilter.mul Ultrafilter.add\n\n/-- We could have taken this as the definition of `U * V`, but then we would have to prove that it\ndefines an ultrafilter. -/\n@[to_additive]\ntheorem Ultrafilter.eventually_mul {M} [Mul M] (U V : Ultrafilter M) (p : M → Prop) :\n    (∀ᶠ m in ↑(U * V), p m) ↔ ∀ᶠ m in U, ∀ᶠ m' in V, p (m * m') :=\n  Iff.rfl\n\n/-- Semigroup structure on `Ultrafilter M` induced by a semigroup structure on `M`. -/\n@[to_additive (attr := implicit_reducible)\n/-- Additive semigroup structure on `Ultrafilter M` induced by an additive semigroup\nstructure on `M`. -/]\ndef Ultrafilter.semigroup {M} [Semigroup M] : Semigroup (Ultrafilter M) :=\n  { Ultrafilter.mul with\n    mul_assoc := fun U V W =>\n      Ultrafilter.coe_inj.mp <|\n        Filter.ext' fun p => by simp [Ultrafilter.eventually_mul, mul_assoc] }\n\nattribute [local instance] Ultrafilter.semigroup Ultrafilter.addSemigroup\n\n-- We don't prove `continuous_mul_right`, because in general it is false!\n@[to_additive]\ntheorem Ultrafilter.continuous_mul_left {M} [Mul M] (V : Ultrafilter M) :\n    Continuous (· * V) :=\n  ultrafilterBasis_is_basis.continuous_iff.2 <| Set.forall_mem_range.mpr fun s ↦\n    ultrafilter_isOpen_basic { m : M | ∀ᶠ m' in V, m * m' ∈ s }\n\nnamespace Hindman\n\n/-- `FS a` is the set of finite sums in `a`, i.e. `m ∈ FS a` if `m` is the sum of a nonempty\nsubsequence of `a`. We give a direct inductive definition instead of talking about subsequences. -/\ninductive FS {M} [AddSemigroup M] : Stream' M → Set M\n  | head' (a : Stream' M) : FS a a.head\n  | tail' (a : Stream' M) (m : M) (h : FS a.tail m) : FS a m\n  | cons' (a : Stream' M) (m : M) (h : FS a.tail m) : FS a (a.head + m)\n\n/-- `FP a` is the set of finite products in `a`, i.e. `m ∈ FP a` if `m` is the product of a nonempty\nsubsequence of `a`. We give a direct inductive definition instead of talking about subsequences. -/\n@[to_additive FS]\ninductive FP {M} [Semigroup M] : Stream' M → Set M\n  | head' (a : Stream' M) : FP a a.head\n  | tail' (a : Stream' M) (m : M) (h : FP a.tail m) : FP a m\n  | cons' (a : Stream' M) (m : M) (h : FP a.tail m) : FP a (a.head * m)\n\nsection Aliases\n\n/-! Since the constructors for `FS` and `FP` cheat using the `Set M = M → Prop` defeq,\nwe provide match patterns that preserve the defeq correctly in their type. -/\n\nvariable {M} [Semigroup M] (a : Stream' M) (m : M) (h : FP a.tail m)\n\nset_option linter.defProp false in\n/-- Constructor for `FP`. This is the preferred spelling over `FP.head'`. -/\n@[to_additive (attr := match_pattern, nolint defLemma)\n  /-- Constructor for `FS`. This is the preferred spelling over `FS.head'`. -/]\nabbrev FP.head : a.head ∈ FP a := FP.head' a\nset_option linter.defProp false in\n/-- Constructor for `FP`. This is the preferred spelling over `FP.tail'`. -/\n@[to_additive (attr := match_pattern, nolint defLemma)\n  /-- Constructor for `FS`. This is the preferred spelling over `FS.tail'`. -/]\nabbrev FP.tail : m ∈ FP a := FP.tail' a m h\nset_option linter.defProp false in\n/-- Constructor for `FP`. This is the preferred spelling over `FP.cons'`. -/\n@[to_additive (attr := match_pattern, nolint defLemma)\n  /-- Constructor for `FS`. This is the preferred spelling over `FS.cons'`. -/]\nabbrev FP.cons : a.head * m ∈ FP a := FP.cons' a m h\n\nend Aliases\n\n/-- If `m` and `m'` are finite products in `M`, then so is `m * m'`, provided that `m'` is obtained\nfrom a subsequence of `M` starting sufficiently late. -/\n@[to_additive /-- If `m` and `m'` are finite sums in `M`, then so is `m + m'`, provided that `m'`\nis obtained from a subsequence of `M` starting sufficiently late. -/]\ntheorem FP.mul {M} [Semigroup M] {a : Stream' M} {m : M} (hm : m ∈ FP a) :\n    ∃ n, ∀ m' ∈ FP (a.drop n), m * m' ∈ FP a := by\n  induction hm with\n  | head' a => exact ⟨1, fun m hm => FP.cons a m hm⟩\n  | tail' a m _ ih =>\n    obtain ⟨n, hn⟩ := ih\n    use n + 1\n    intro m' hm'\n    exact FP.tail _ _ (hn _ hm')\n  | cons' a m _ ih =>\n    obtain ⟨n, hn⟩ := ih\n    use n + 1\n    intro m' hm'\n    rw [mul_assoc]\n    exact FP.cons _ _ (hn _ hm')\n\n@[to_additive exists_idempotent_ultrafilter_le_FS]\n\nTarget:\ntheorem exists_idempotent_ultrafilter_le_FP {M} [Semigroup M] (a : Stream' M) :\n    ∃ U : Ultrafilter M, U * U = U ∧ ∀ᶠ m in U, m ∈ FP a :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"bd63b81ec55a775fe7c4fab35a1d24822d3e6e52f7c6720988fe9127c817cec7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics","family_id":"exists_idempotent_ultrafilter_le_fp","file_id":"mathlib/Mathlib/Combinatorics/Hindman.lean","sample_id":"b9b65ec13f7e30e41eeaa6338022c9efbc66d8f0394c7b483c53252b38ab9a8f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7fb3cc1775db12a3f434a99b04b7c0a0f79496406a0eb39713c9f5d39c18a786","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b9d6520e23cf7ef508708d905f3af667f91b01c1501bdfb215036dc08f82f360","source_sha256":"3e56b0bca06aab56ed706da176a00f9bb677f8dddf1dc55819d7e75bf8119e45","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [sInter_eq_biInter]\n  exact MeasurableSet.biInter hs h","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.166667,"token_length_ratio":0.272727},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"ccf52a6698a535699465c5ffdbbd533262745c8cbda86f2fe73f1b9258540848","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Countable\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\npublic import Mathlib.Tactic.CrossRefAttribute\npublic import Mathlib.Tactic.FunProp.Attr\npublic import Mathlib.Tactic.Measurability\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro\n-/\n/-!\n# Measurable spaces and measurable functions\n\nThis file defines measurable spaces and measurable functions.\n\nA measurable space is a set equipped with a σ-algebra, a collection of\nsubsets closed under complementation and countable union. A function\nbetween measurable spaces is measurable if the preimage of each\nmeasurable subset is measurable.\n\nσ-algebras on a fixed set `α` form a complete lattice. Here we order\nσ-algebras by writing `m₁ ≤ m₂` if every set which is `m₁`-measurable is\nalso `m₂`-measurable (that is, `m₁` is a subset of `m₂`). In particular, any\ncollection of subsets of `α` generates a smallest σ-algebra which\ncontains all of them.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Measurable_space>\n* <https://en.wikipedia.org/wiki/Sigma-algebra>\n* <https://en.wikipedia.org/wiki/Dynkin_system>\n\n## Tags\n\nmeasurable space, σ-algebra, measurable function\n-/\n\n@[expose] public section\n\nassert_not_exists Covariant MonoidWithZero\n\nopen Set Encodable Function Equiv\n\nvariable {α β γ δ δ' : Type*} {ι : Sort*} {s t u : Set α}\n\n/-- A measurable space is a space equipped with a σ-algebra. -/\n@[class] structure MeasurableSpace (α : Type*) where\n  /-- Predicate saying that a given set is measurable. Use `MeasurableSet` in the root namespace\n  instead. -/\n  MeasurableSet' : Set α → Prop\n  /-- The empty set is a measurable set. Use `MeasurableSet.empty` instead. -/\n  measurableSet_empty : MeasurableSet' ∅\n  /-- The complement of a measurable set is a measurable set. Use `MeasurableSet.compl` instead. -/\n  measurableSet_compl : ∀ s, MeasurableSet' s → MeasurableSet' sᶜ\n  /-- The union of a sequence of measurable sets is a measurable set. Use a more general\n  `MeasurableSet.iUnion` instead. -/\n  measurableSet_iUnion : ∀ f : ℕ → Set α, (∀ i, MeasurableSet' (f i)) → MeasurableSet' (⋃ i, f i)\n\ninstance [h : MeasurableSpace α] : MeasurableSpace αᵒᵈ := h\n\n/-- `MeasurableSet s` means that `s` is measurable (in the ambient measure space on `α`) -/\ndef MeasurableSet [MeasurableSpace α] (s : Set α) : Prop :=\n  ‹MeasurableSpace α›.MeasurableSet' s\n\n/-- Notation for `MeasurableSet` with respect to a non-standard σ-algebra. -/\nscoped[MeasureTheory] notation \"MeasurableSet[\" m \"]\" => @MeasurableSet _ m\n\nopen MeasureTheory\n\nsection\n\nopen scoped symmDiff\n\n@[simp, measurability]\ntheorem MeasurableSet.empty [MeasurableSpace α] : MeasurableSet (∅ : Set α) :=\n  MeasurableSpace.measurableSet_empty _\n\nvariable {m : MeasurableSpace α}\n\n@[measurability]\nprotected theorem MeasurableSet.compl : MeasurableSet s → MeasurableSet sᶜ :=\n  MeasurableSpace.measurableSet_compl _ s\n\nprotected theorem MeasurableSet.of_compl (h : MeasurableSet sᶜ) : MeasurableSet s :=\n  compl_compl s ▸ h.compl\n\n@[simp]\ntheorem MeasurableSet.compl_iff : MeasurableSet sᶜ ↔ MeasurableSet s :=\n  ⟨.of_compl, .compl⟩\n\n@[simp, measurability]\nprotected theorem MeasurableSet.univ : MeasurableSet (univ : Set α) :=\n  .of_compl <| by simp\n\n@[nontriviality, measurability]\ntheorem Subsingleton.measurableSet [Subsingleton α] {s : Set α} : MeasurableSet s :=\n  Subsingleton.set_cases MeasurableSet.empty MeasurableSet.univ s\n\ntheorem MeasurableSet.congr {s t : Set α} (hs : MeasurableSet s) (h : s = t) : MeasurableSet t := by\n  rwa [← h]\n\n@[measurability]\nprotected theorem MeasurableSet.iUnion [Countable ι] ⦃f : ι → Set α⦄\n    (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋃ b, f b) := by\n  cases isEmpty_or_nonempty ι\n  · simp\n  · rcases exists_surjective_nat ι with ⟨e, he⟩\n    rw [← iUnion_congr_of_surjective _ he (fun _ => rfl)]\n    exact m.measurableSet_iUnion _ fun _ => h _\n\nprotected theorem MeasurableSet.biUnion {f : β → Set α} {s : Set β} (hs : s.Countable)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) := by\n  rw [biUnion_eq_iUnion]\n  have := hs.to_subtype\n  exact MeasurableSet.iUnion (by simpa using h)\n\ntheorem Set.Finite.measurableSet_biUnion {f : β → Set α} {s : Set β} (hs : s.Finite)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) :=\n  .biUnion hs.countable h\n\ntheorem Finset.measurableSet_biUnion {f : β → Set α} (s : Finset β)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) :=\n  s.finite_toSet.measurableSet_biUnion h\n\nprotected theorem MeasurableSet.sUnion {s : Set (Set α)} (hs : s.Countable)\n    (h : ∀ t ∈ s, MeasurableSet t) : MeasurableSet (⋃₀ s) := by\n  rw [sUnion_eq_biUnion]\n  exact .biUnion hs h\n\ntheorem Set.Finite.measurableSet_sUnion {s : Set (Set α)} (hs : s.Finite)\n    (h : ∀ t ∈ s, MeasurableSet t) : MeasurableSet (⋃₀ s) :=\n  MeasurableSet.sUnion hs.countable h\n\n@[measurability]\ntheorem MeasurableSet.iInter [Countable ι] {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) :\n    MeasurableSet (⋂ b, f b) :=\n  .of_compl <| by rw [compl_iInter]; exact .iUnion fun b => (h b).compl\n\ntheorem MeasurableSet.biInter {f : β → Set α} {s : Set β} (hs : s.Countable)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) :=\n  .of_compl <| by rw [compl_iInter₂]; exact .biUnion hs fun b hb => (h b hb).compl\n\ntheorem Set.Finite.measurableSet_biInter {f : β → Set α} {s : Set β} (hs : s.Finite)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) :=\n  .biInter hs.countable h\n\ntheorem Finset.measurableSet_biInter {f : β → Set α} (s : Finset β)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) :=\n  s.finite_toSet.measurableSet_biInter h\n\nTarget:\ntheorem MeasurableSet.sInter {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s, MeasurableSet t) :\n    MeasurableSet (⋂₀ s) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/MeasurableSpace","family_id":"measurableset","file_id":"mathlib/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","sample_id":"b9d6520e23cf7ef508708d905f3af667f91b01c1501bdfb215036dc08f82f360"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bc537854a12e226bc8123bdf85ce0d801d68aa77f9f581d84fafad195b8a7d94","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7f4e05d647d8234808511f2d65c328365b205678685e9a78a56123a55f2d6582","source_sha256":"e4f58af6d4a9683ac7a3c2d77d21907a98d376b8c60bad21d7bf0c07390bc700","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨n, f, hf, P, rfl⟩ := P.exists_fin_rename\n  rw [IsNilpotent.map_iff (rename_injective _ hf), MvPolynomial.isNilpotent_iff_of_fintype]\n  lift f to Fin n ↪ σ using hf\n  refine ⟨fun H i ↦ ?_, fun H i ↦ by simpa using H (i.embDomain f)⟩\n  by_cases H : i ∈ Set.range (Finsupp.embDomain f)\n  · aesop\n  · rw [coeff_rename_eq_zero] <;> aesop (add simp Finsupp.embDomain_eq_mapDomain)","hard_negative":false,"metrics":{"chosen_tokens":97,"rejected_tokens":2,"token_jaccard":0.018519,"token_length_ratio":0.020619},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"cd392e11cc293aa5b7c5da907bb1c680b05e01190835d9268c4577a534dfcc04","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.MvPolynomial.Homogeneous\npublic import Mathlib.RingTheory.Polynomial.Nilpotent\n\nNamespace:\nMvPolynomial\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Nilpotents and units in multivariate polynomial rings\n\nWe prove that\n- `MvPolynomial.isNilpotent_iff`:\n  A multivariate polynomial is nilpotent iff all its coefficients are.\n- `MvPolynomial.isUnit_iff`:\n  A multivariate polynomial is invertible iff its constant term is invertible\n  and its other coefficients are nilpotent.\n-/\n\npublic section\n\nnamespace MvPolynomial\n\nvariable {σ R : Type*} [CommRing R] {P : MvPolynomial σ R}\n\n-- Subsumed by `isNilpotent_iff` below.\nprivate theorem isNilpotent_iff_of_fintype [Finite σ] :\n    IsNilpotent P ↔ ∀ i, IsNilpotent (P.coeff i) := by\n  classical\n  -- Note: including `Fintype.ofFinite σ` in the entire context interferes with the `rw` below.\n  refine have := Fintype.ofFinite σ; Fintype.induction_empty_option ?_ ?_ ?_ σ P\n  · intro α β _ e h₁ P\n    rw [← IsNilpotent.map_iff (rename_injective _ e.symm.injective), h₁,\n      (Finsupp.equivCongrLeft e).forall_congr_left]\n    simp [Finsupp.equivMapDomain_eq_mapDomain, coeff_rename_mapDomain _ e.symm.injective]\n  · simp [Unique.forall_iff, ← IsNilpotent.map_iff (isEmptyRingEquiv R PEmpty).injective,\n      -isEmptyRingEquiv_apply, isEmptyRingEquiv_eq_coeff_zero]\n  · intro α _ H P\n    obtain ⟨P, rfl⟩ := (optionEquivLeft _ _).symm.surjective P\n    simp [IsNilpotent.map_iff (optionEquivLeft _ _).symm.injective,\n      Polynomial.isNilpotent_iff, H, Finsupp.optionEquiv.forall_congr_left,\n      ← optionEquivLeft_coeff_some_coeff_none, Finsupp.coe_update]\n\nTarget:\ntheorem isNilpotent_iff : IsNilpotent P ↔ ∀ i, IsNilpotent (P.coeff i) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/MvPolynomial","family_id":"isnilpotent_iff","file_id":"mathlib/Mathlib/Algebra/MvPolynomial/Nilpotent.lean","sample_id":"7f4e05d647d8234808511f2d65c328365b205678685e9a78a56123a55f2d6582"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6c44e0bc1aac850175730ade0c94468670de6fadad8303d7beb2df51cfcf9e9e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"574a2405d1c8dc937c54d23ab5023c4ffb7c0dd44de753fb1b3b583eb8956080","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"94c18dd81e6ce7afdea3016639b32636d0e629ff55fd5405a516ea90027133c6","source_sha256":"ba493885a2be935359e1e232e1bbaeb68ad80f382975839ead7d3190a1fce95f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases a <;> cases b\n  exacts [map_subset_map.2 (hf₁ _ _), disjSum_mono (hg₁ _ _) (hg₂ _ _), Subset.rfl,\n    map_subset_map.2 (hf₂ _ _)]","hard_negative":true,"metrics":{"chosen_tokens":48,"rejected_tokens":2,"token_jaccard":0.04,"token_length_ratio":0.041667},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"cd403baa002656716ed358530a85bcc3e2f41fef11339397c783310126f60b91","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Sum\npublic import Mathlib.Data.Sum.Order\npublic import Mathlib.Order.Interval.Finset.Defs\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Finite intervals in a disjoint union\n\nThis file provides the `LocallyFiniteOrder` instance for the disjoint sum and linear sum of two\norders and calculates the cardinality of their finite intervals.\n-/\n\n@[expose] public section\n\n\nopen Function Sum\n\nnamespace Finset\n\nvariable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}\n\nsection SumLift₂\n\nvariable (f f₁ g₁ : α₁ → β₁ → Finset γ₁) (g f₂ g₂ : α₂ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁` and `α₂ → β₂ → Finset γ₂` to a map\n`α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to `Alternative` functors if we can\nmake sure to keep computability and universe polymorphism. -/\n@[simp]\ndef sumLift₂ : ∀ (_ : α₁ ⊕ α₂) (_ : β₁ ⊕ β₂), Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f a b).map Embedding.inl\n  | inl _, inr _ => ∅\n  | inr _, inl _ => ∅\n  | inr a, inr b => (g a b).map Embedding.inr\n\nvariable {f f₁ g₁ g f₂ g₂} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\ntheorem mem_sumLift₂ :\n    c ∈ sumLift₂ f g a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨\n        ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g a₂ b₂ := by\n  constructor\n  · rcases a with a | a <;> rcases b with b | b\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr ⟨a, b, c, rfl, rfl, rfl, hc⟩\n  · rintro (⟨a, b, c, rfl, rfl, rfl, h⟩ | ⟨a, b, c, rfl, rfl, rfl, h⟩) <;> exact mem_map_of_mem _ h\n\ntheorem inl_mem_sumLift₂ {c₁ : γ₁} :\n    inl c₁ ∈ sumLift₂ f g a b ↔ ∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f a₁ b₁ := by\n  rw [mem_sumLift₂, or_iff_left]\n  · simp only [inl.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inl_ne_inr h\n\ntheorem inr_mem_sumLift₂ {c₂ : γ₂} :\n    inr c₂ ∈ sumLift₂ f g a b ↔ ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ g a₂ b₂ := by\n  rw [mem_sumLift₂, or_iff_right]\n  · simp only [inr.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inr_ne_inl h\n\ntheorem sumLift₂_eq_empty :\n    sumLift₂ f g a b = ∅ ↔\n      (∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f a₁ b₁ = ∅) ∧\n        ∀ a₂ b₂, a = inr a₂ → b = inr b₂ → g a₂ b₂ = ∅ := by\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · constructor <;>\n    · rintro a b rfl rfl\n      exact map_eq_empty.1 h\n  cases a <;> cases b\n  · exact map_eq_empty.2 (h.1 _ _ rfl rfl)\n  · rfl\n  · rfl\n  · exact map_eq_empty.2 (h.2 _ _ rfl rfl)\n\ntheorem sumLift₂_nonempty :\n    (sumLift₂ f g a b).Nonempty ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ (f a₁ b₁).Nonempty) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ (g a₂ b₂).Nonempty := by\n  simp only [nonempty_iff_ne_empty, Ne, sumLift₂_eq_empty, not_and_or, not_forall, exists_prop]\n\ntheorem sumLift₂_mono (h₁ : ∀ a b, f₁ a b ⊆ g₁ a b) (h₂ : ∀ a b, f₂ a b ⊆ g₂ a b) :\n    ∀ a b, sumLift₂ f₁ f₂ a b ⊆ sumLift₂ g₁ g₂ a b\n  | inl _, inl _ => map_subset_map.2 (h₁ _ _)\n  | inl _, inr _ => Subset.rfl\n  | inr _, inl _ => Subset.rfl\n  | inr _, inr _ => map_subset_map.2 (h₂ _ _)\n\nend SumLift₂\n\nsection SumLexLift\nvariable (f₁ f₁' : α₁ → β₁ → Finset γ₁) (f₂ f₂' : α₂ → β₂ → Finset γ₂)\n  (g₁ g₁' : α₁ → β₂ → Finset γ₁) (g₂ g₂' : α₁ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁`, `α₂ → β₂ → Finset γ₂`, `α₁ → β₂ → Finset γ₁`,\n`α₂ → β₂ → Finset γ₂`  to a map `α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to\nalternative monads if we can make sure to keep computability and universe polymorphism. -/\ndef sumLexLift : α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f₁ a b).map Embedding.inl\n  | inl a, inr b => (g₁ a b).disjSum (g₂ a b)\n  | inr _, inl _ => ∅\n  | inr a, inr b => (f₂ a b).map ⟨_, inr_injective⟩\n\n@[simp]\nlemma sumLexLift_inl_inl (a : α₁) (b : β₁) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inl b) = (f₁ a b).map Embedding.inl := rfl\n\n@[simp]\nlemma sumLexLift_inl_inr (a : α₁) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b) = (g₁ a b).disjSum (g₂ a b) := rfl\n\n@[simp]\nlemma sumLexLift_inr_inl (a : α₂) (b : β₁) : sumLexLift f₁ f₂ g₁ g₂ (inr a) (inl b) = ∅ := rfl\n\n@[simp]\nlemma sumLexLift_inr_inr (a : α₂) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inr a) (inr b) = (f₂ a b).map ⟨_, inr_injective⟩ := rfl\n\nvariable {f₁ g₁ f₂ g₂ f₁' g₁' f₂' g₂'} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\nlemma mem_sumLexLift :\n    c ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        (∃ a₁ b₂ c₁, a = inl a₁ ∧ b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨\n          (∃ a₁ b₂ c₂, a = inl a₁ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n            ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂ := by\n  constructor\n  · obtain a | a := a <;> obtain b | b := b\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (mem_disjSum.1 h).elim ?_ ?_\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inr <| Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr (Or.inr <| Or.inr <| ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n  · rintro (⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩ |\n      ⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact mem_map_of_mem _ hc\n    · exact inl_mem_disjSum.2 hc\n    · exact inr_mem_disjSum.2 hc\n    · exact mem_map_of_mem _ hc\n\nlemma inl_mem_sumLexLift {c₁ : γ₁} :\n    inl c₁ ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        ∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ c₁ ∈ g₁ a₁ b₂ := by\n  simp [mem_sumLexLift]\n\nlemma inr_mem_sumLexLift {c₂ : γ₂} :\n    inr c₂ ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ f₂ a₂ b₂ := by\n  simp [mem_sumLexLift]\n\nTarget:\nlemma sumLexLift_mono (hf₁ : ∀ a b, f₁ a b ⊆ f₁' a b) (hf₂ : ∀ a b, f₂ a b ⊆ f₂' a b)\n    (hg₁ : ∀ a b, g₁ a b ⊆ g₁' a b) (hg₂ : ∀ a b, g₂ a b ⊆ g₂' a b) (a : α₁ ⊕ α₂)\n    (b : β₁ ⊕ β₂) : sumLexLift f₁ f₂ g₁ g₂ a b ⊆ sumLexLift f₁' f₂' g₁' g₂' a b :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_94c18dd81e6c","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"77f09a26f72d82ea92f0f47d714b281fbaa42ef06b5115ebd558ac0504204c8d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sum","family_id":"sumlexlift_mono","file_id":"mathlib/Mathlib/Data/Sum/Interval.lean","sample_id":"94c18dd81e6ce7afdea3016639b32636d0e629ff55fd5405a516ea90027133c6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d46e9dc9cd151de5455487d42f3fd890f31162d28298263c77fbfdfebe8161e5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cf0e248544ee9d6ed6e902aaa1a7849b4ce61aeb489bff40d94a5eee7d981eac","source_sha256":"158ce289cc94776d7ace3f5f8788d7841da0114f64a958ffadba75342e4312aa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [destruct]\n  rcases f0 : s.1 0 with - | a' <;> intro h\n  · injection h with h'\n    rw [← h']\n    obtain ⟨f, al⟩ := s\n    apply Subtype.ext\n    dsimp [think, tail]\n    rw [← f0]\n    exact (Stream'.eta f).symm\n  · contradiction","hard_negative":true,"metrics":{"chosen_tokens":65,"rejected_tokens":8,"token_jaccard":0.061224,"token_length_ratio":0.123077},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"cd771895d12affa0e554e40e004b0aaa333822760bfc2cbe14bee6ad65404848","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.Find\npublic import Mathlib.Data.Stream.Init\npublic import Mathlib.Logic.Relator\npublic import Mathlib.Tactic.Common\npublic import Batteries.Tactic.Lint.Simp\n\nNamespace:\nComputation\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Coinductive formalization of unbounded computations.\n\nThis file provides a `Computation` type where `Computation α` is the type of\nunbounded computations returning `α`.\n-/\n\n@[expose] public section\n\nopen Function\n\nuniverse u v w\n\n/-\ncoinductive Computation (α : Type u) : Type u\n| pure : α → Computation α\n| think : Computation α → Computation α\n-/\n/-- `Computation α` is the type of unbounded computations returning `α`.\n  An element of `Computation α` is an infinite sequence of `Option α` such\n  that if `f n = some a` for some `n` then it is constantly `some a` after that. -/\ndef Computation (α : Type u) : Type u :=\n  { f : Stream' (Option α) // ∀ ⦃n a⦄, f n = some a → f (n + 1) = some a }\n\nnamespace Computation\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n-- constructors\n/-- `pure a` is the computation that immediately terminates with result `a`. -/\ndef pure (a : α) : Computation α :=\n  ⟨Stream'.const (some a), fun _ _ => id⟩\n\ninstance : CoeTC α (Computation α) :=\n  ⟨pure⟩\n\n-- note [use has_coe_t]\n/-- `think c` is the computation that delays for one \"tick\" and then performs\n  computation `c`. -/\ndef think (c : Computation α) : Computation α :=\n  ⟨Stream'.cons none c.1, fun n a h => by\n    rcases n with - | n\n    · contradiction\n    · exact c.2 h⟩\n\n/-- `thinkN c n` is the computation that delays for `n` ticks and then performs\n  computation `c`. -/\ndef thinkN (c : Computation α) : ℕ → Computation α\n  | 0 => c\n  | n + 1 => think (thinkN c n)\n\n-- check for immediate result\n/-- `head c` is the first step of computation, either `some a` if `c = pure a`\n  or `none` if `c = think c'`. -/\ndef head (c : Computation α) : Option α :=\n  c.1.head\n\n-- one step of computation\n/-- `tail c` is the remainder of computation, either `c` if `c = pure a`\n  or `c'` if `c = think c'`. -/\ndef tail (c : Computation α) : Computation α :=\n  ⟨c.1.tail, fun _ _ h => c.2 h⟩\n\n/-- `empty α` is the computation that never returns, an infinite sequence of\n  `think`s. -/\ndef empty (α) : Computation α :=\n  ⟨Stream'.const none, fun _ _ => id⟩\n\ninstance : Inhabited (Computation α) :=\n  ⟨empty _⟩\n\n/-- `runFor c n` evaluates `c` for `n` steps and returns the result, or `none`\n  if it did not terminate after `n` steps. -/\ndef runFor : Computation α → ℕ → Option α :=\n  Subtype.val\n\n/-- `destruct c` is the destructor for `Computation α` as a coinductive type.\n  It returns `inl a` if `c = pure a` and `inr c'` if `c = think c'`. -/\ndef destruct (c : Computation α) : α ⊕ (Computation α) :=\n  match c.1 0 with\n  | none => Sum.inr (tail c)\n  | some a => Sum.inl a\n\n/-- `run c` is an unsound meta function that runs `c` to completion, possibly\n  resulting in an infinite loop in the VM. -/\nunsafe def run : Computation α → α\n  | c =>\n    match destruct c with\n    | Sum.inl a => a\n    | Sum.inr ca => run ca\n\ntheorem destruct_eq_pure {s : Computation α} {a : α} : destruct s = Sum.inl a → s = pure a := by\n  dsimp [destruct]\n  cases f0 : s.1 0 <;> intro h\n  · contradiction\n  · apply Subtype.ext\n    funext n\n    induction n with\n    | zero => injection h with h'; rwa [h'] at f0\n    | succ n IH => exact s.2 IH\n\nTarget:\ntheorem destruct_eq_think {s : Computation α} {s'} : destruct s = Sum.inr s' → s = think s' :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"4aa82f11dee5f5777fe3a740f383d8350ea75c726104aa3512cf6ee7fa4a8736","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Seq","family_id":"destruct_eq_think","file_id":"mathlib/Mathlib/Data/Seq/Computation.lean","sample_id":"cf0e248544ee9d6ed6e902aaa1a7849b4ce61aeb489bff40d94a5eee7d981eac"},"split":"train"}
{"binding":{"chosen_proof_sha256":"18e5d95a99a28ddbd089bbdf5652dbad7467c92bf9a278d6d749571d6703ac97","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9573f0056018f8a6c39ac9e88631ce207c46ee22450533263d10a4425cb31589","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8c92966f8a3dcfb46f7baa7761df8474960f52637455e799bcbf00ab2a68364d","source_sha256":"a4df2964a0a0195a94022ab24e77297c35474e0a98e2a52eb57d2203822fac2d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [descFactorial_eq_prod_range, factorial_eq_prod_range_add_one]\n  simp only [cast_prod]\n  nth_rw 2 [← card_range n]\n  rw [pow_card_mul_prod]\n  refine prod_congr rfl ?_\n  intro x hx\n  rw [← tsub_add_eq_tsub_tsub_swap,\n    Nat.cast_sub <| Nat.le_trans (Nat.add_one_le_iff.mpr (List.mem_range.mp hx)) h,\n    CharP.cast_eq_zero, zero_sub, cast_succ, neg_add_rev, mul_add, neg_mul, one_mul,\n    mul_one, add_comm]","hard_negative":true,"metrics":{"chosen_tokens":81,"rejected_tokens":2,"token_jaccard":0.02,"token_length_ratio":0.024691},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"cd7851abec6e7c9cc68e1f33bdd22b706441b83f005d483353e1865197461853","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.Factorial.BigOperators\npublic import Mathlib.Data.ZMod.Basic\n\nNamespace:\nZMod\n\nLocal context:\n/-\nCopyright (c) 2023 Moritz Firsching. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Moritz Firsching\n-/\n/-!\n# Facts about factorials in ZMod\n\nWe collect facts about factorials in context of modular arithmetic.\n\n## Main statements\n\n* `ZMod.cast_descFactorial`: For natural numbers `n` and `p`, where `n` is less than or equal to `p`\n  the descending factorial of `(p - 1)` taken `n` times modulo `p` equals `(-1) ^ n * n!`.\n\n## See also\n\nFor the prime case and involving `factorial` rather than `descFactorial`, see Wilson's theorem:\n* `Nat.prime_iff_fac_equiv_neg_one`\n\n-/\n\npublic section\n\nassert_not_exists TwoSidedIdeal\n\nopen Finset Nat\n\nnamespace ZMod\n\nTarget:\ntheorem cast_descFactorial {n p : ℕ} (h : n ≤ p) :\n    (descFactorial (p - 1) n : ZMod p) = (-1) ^ n * n ! :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_8c92966f8a3d","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"cbfe661bb5791854e7b78f9b74d42cd81e7958aecf672755678555a51a7c5086","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/ZMod","family_id":"cast_descfactorial","file_id":"mathlib/Mathlib/Data/ZMod/Factorial.lean","sample_id":"8c92966f8a3dcfb46f7baa7761df8474960f52637455e799bcbf00ab2a68364d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"048280cfb8f07df3e6d2a0a1b3323465e6476c927989956effb4e1caeb51283e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"abf95977d4892fdcad911aa6c3299b9ebb1a438848e8d7c360677f9b9af439b6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c32bfd18b1eacc9948f2069f737c203f16059abd126861ca53c4f4fff6855765","source_sha256":"d1ba22e22a26245161ec2703bf29aa4e6e9c0ade6d5458aee3e70435922bd387","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨x, w₁, w₂⟩ := x\n  obtain ⟨⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩, -⟩ := h\n  simp [Prod.toFun', Prod.invFun', h₁, h₂]","hard_negative":true,"metrics":{"chosen_tokens":57,"rejected_tokens":2,"token_jaccard":0.041667,"token_length_ratio":0.035088},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"ce18d57bf784b5d6ce27012eca5bfb287ccdb371c8b527f8c278fba6a47c99ad","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.FiberBundle.Basic\n\nNamespace:\nBundle.Trivialization\n\nLocal context:\n/-\nCopyright (c) 2022 Nicolò Cavalleri. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn\n-/\n/-!\n# Standard constructions on fiber bundles\n\nThis file contains several standard constructions on fiber bundles:\n\n* `Bundle.Trivial.fiberBundle 𝕜 B F`: the trivial fiber bundle with model fiber `F` over the base\n  `B`\n\n* `FiberBundle.prod`: for fiber bundles `E₁` and `E₂` over a common base, a fiber bundle structure\n  on their fiberwise product `E₁ ×ᵇ E₂` (the notation stands for `fun x ↦ E₁ x × E₂ x`).\n\n* `FiberBundle.pullback`: for a fiber bundle `E` over `B`, a fiber bundle structure on its\n  pullback `f *ᵖ E` by a map `f : B' → B` (the notation is a type synonym for `E ∘ f`).\n\n## Tags\n\nfiber bundle, fibre bundle, fiberwise product, pullback\n\n-/\n\n@[expose] public section\n\nopen Bundle Filter Set TopologicalSpace Topology\n\n/-! ### The trivial bundle -/\n\nnamespace Bundle\n\nnamespace Trivial\n\nvariable (B : Type*) (F : Type*)\n\n-- TODO: use `TotalSpace.toProd`\ninstance topologicalSpace [t₁ : TopologicalSpace B]\n    [t₂ : TopologicalSpace F] : TopologicalSpace (TotalSpace F (Trivial B F)) :=\n  induced TotalSpace.proj t₁ ⊓ induced (TotalSpace.trivialSnd B F) t₂\n\nvariable [TopologicalSpace B] [TopologicalSpace F]\n\ntheorem isInducing_toProd : IsInducing (TotalSpace.toProd B F) :=\n  ⟨by simp only [instTopologicalSpaceProd, induced_inf, induced_compose]; rfl⟩\n\n/-- Homeomorphism between the total space of the trivial bundle and the Cartesian product. -/\n@[simps!]\ndef homeomorphProd : TotalSpace F (Trivial B F) ≃ₜ B × F :=\n  (TotalSpace.toProd _ _).toHomeomorphOfIsInducing (isInducing_toProd B F)\n\n/-- Local trivialization for trivial bundle. -/\n@[simps!]\ndef trivialization : Trivialization F (π F (Bundle.Trivial B F)) where\n  toOpenPartialHomeomorph := (homeomorphProd B F).toOpenPartialHomeomorph\n  baseSet := univ\n  open_baseSet := isOpen_univ\n  source_eq := rfl\n  target_eq := univ_prod_univ.symm\n  proj_toFun _ _ := rfl\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp] lemma trivialization_symm_apply [Zero F] (b : B) (f : F) :\n    (trivialization B F).symm b f = f := by\n  simp [trivialization, homeomorphProd, TotalSpace.toProd, Trivialization.symm,\n    Pretrivialization.symm, Trivialization.toPretrivialization]\n\n@[simp] lemma toOpenPartialHomeomorph_trivialization_symm_apply (v : B × F) :\n    (trivialization B F).toOpenPartialHomeomorph.symm v = ⟨v.1, v.2⟩ := rfl\n\n/-- Fiber bundle instance on the trivial bundle. -/\n@[simps] instance fiberBundle : FiberBundle F (Bundle.Trivial B F) where\n  trivializationAtlas' := {trivialization B F}\n  trivializationAt' _ := trivialization B F\n  mem_baseSet_trivializationAt' := mem_univ\n  trivialization_mem_atlas' _ := mem_singleton _\n  totalSpaceMk_isInducing' _ := (homeomorphProd B F).symm.isInducing.comp\n    (isInducing_const_prod.2 .id)\n\ntheorem eq_trivialization (e : Trivialization F (π F (Bundle.Trivial B F)))\n    [i : MemTrivializationAtlas e] : e = trivialization B F := i.out\n\nend Trivial\n\nend Bundle\n\n/-! ### Fibrewise product of two bundles -/\n\n\nsection Prod\n\nvariable {B : Type*}\n\nsection Defs\n\nvariable (F₁ : Type*) (E₁ : B → Type*) (F₂ : Type*) (E₂ : B → Type*)\nvariable [TopologicalSpace (TotalSpace F₁ E₁)] [TopologicalSpace (TotalSpace F₂ E₂)]\n\n/-- Equip the total space of the fiberwise product of two fiber bundles `E₁`, `E₂` with\nthe induced topology from the diagonal embedding into `TotalSpace F₁ E₁ × TotalSpace F₂ E₂`. -/\ninstance FiberBundle.Prod.topologicalSpace : TopologicalSpace (TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂)) :=\n  TopologicalSpace.induced\n    (fun p ↦ ((⟨p.1, p.2.1⟩ : TotalSpace F₁ E₁), (⟨p.1, p.2.2⟩ : TotalSpace F₂ E₂)))\n    inferInstance\n\n/-- The diagonal map from the total space of the fiberwise product of two fiber bundles\n`E₁`, `E₂` into `TotalSpace F₁ E₁ × TotalSpace F₂ E₂` is an inducing map. -/\ntheorem FiberBundle.Prod.isInducing_diag :\n    IsInducing (fun p ↦ (⟨p.1, p.2.1⟩, ⟨p.1, p.2.2⟩) :\n      TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂) :=\n  ⟨rfl⟩\n\nend Defs\n\nopen FiberBundle\n\nvariable [TopologicalSpace B] (F₁ : Type*) [TopologicalSpace F₁] (E₁ : B → Type*)\n  [TopologicalSpace (TotalSpace F₁ E₁)] (F₂ : Type*) [TopologicalSpace F₂] (E₂ : B → Type*)\n  [TopologicalSpace (TotalSpace F₂ E₂)]\n\nnamespace Bundle.Trivialization\n\nvariable {F₁ E₁ F₂ E₂}\nvariable (e₁ : Trivialization F₁ (π F₁ E₁)) (e₂ : Trivialization F₂ (π F₂ E₂))\n\n/-- Given trivializations `e₁`, `e₂` for fiber bundles `E₁`, `E₂` over a base `B`, the forward\nfunction for the construction `Trivialization.prod`, the induced\ntrivialization for the fiberwise product of `E₁` and `E₂`. -/\ndef Prod.toFun' : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) → B × F₁ × F₂ :=\n  fun p ↦ ⟨p.1, (e₁ ⟨p.1, p.2.1⟩).2, (e₂ ⟨p.1, p.2.2⟩).2⟩\n\nvariable {e₁ e₂}\n\ntheorem Prod.continuous_to_fun : ContinuousOn (Prod.toFun' e₁ e₂)\n    (π (F₁ × F₂) (E₁ ×ᵇ E₂) ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) := by\n  let f₁ : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) → TotalSpace F₁ E₁ × TotalSpace F₂ E₂ :=\n    fun p ↦ ((⟨p.1, p.2.1⟩ : TotalSpace F₁ E₁), (⟨p.1, p.2.2⟩ : TotalSpace F₂ E₂))\n  let f₂ : TotalSpace F₁ E₁ × TotalSpace F₂ E₂ → (B × F₁) × B × F₂ := fun p ↦ ⟨e₁ p.1, e₂ p.2⟩\n  let f₃ : (B × F₁) × B × F₂ → B × F₁ × F₂ := fun p ↦ ⟨p.1.1, p.1.2, p.2.2⟩\n  have hf₁ : Continuous f₁ := (Prod.isInducing_diag F₁ E₁ F₂ E₂).continuous\n  have hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) :=\n    e₁.toOpenPartialHomeomorph.continuousOn.prodMap e₂.toOpenPartialHomeomorph.continuousOn\n  have hf₃ : Continuous f₃ := by fun_prop\n  refine ((hf₃.comp_continuousOn hf₂).comp hf₁.continuousOn ?_).congr ?_\n  · rw [e₁.source_eq, e₂.source_eq]\n    exact mapsTo_preimage _ _\n  rintro ⟨b, v₁, v₂⟩ ⟨hb₁, _⟩\n  simp only [f₁, f₂, f₃, Prod.toFun', Prod.mk_inj, Function.comp_apply, and_true]\n  rw [e₁.coe_fst]\n  rw [e₁.source_eq, mem_preimage]\n  exact hb₁\n\nvariable (e₁ e₂) [∀ x, Zero (E₁ x)] [∀ x, Zero (E₂ x)]\n\n/-- Given trivializations `e₁`, `e₂` for fiber bundles `E₁`, `E₂` over a base `B`, the inverse\nfunction for the construction `Trivialization.prod`, the induced\ntrivialization for the fiberwise product of `E₁` and `E₂`. -/\nnoncomputable def Prod.invFun' (p : B × F₁ × F₂) : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂) :=\n  ⟨p.1, e₁.symm p.1 p.2.1, e₂.symm p.1 p.2.2⟩\n\nvariable {e₁ e₂}\n\ntheorem Prod.left_inv {x : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂)}\n    (h : x ∈ π (F₁ × F₂) (E₁ ×ᵇ E₂) ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) :\n    Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x := by\n  obtain ⟨x, v₁, v₂⟩ := x\n  obtain ⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩ := h\n  simp [Prod.toFun', Prod.invFun', h₁, h₂]\n\nTarget:\ntheorem Prod.right_inv {x : B × F₁ × F₂}\n    (h : x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ (univ : Set (F₁ × F₂))) :\n    Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_c32bfd18b1ea","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"474a9197a473e01e9da66dedf86759145ab4d652b370ff79ea221942ac3b23f8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/FiberBundle","family_id":"prod","file_id":"mathlib/Mathlib/Topology/FiberBundle/Constructions.lean","sample_id":"c32bfd18b1eacc9948f2069f737c203f16059abd126861ca53c4f4fff6855765"},"split":"train"}
{"binding":{"chosen_proof_sha256":"61c7560cab56bf0e5d3f545a1a32c8f50733fcb31780bdea8dc6b7263c7d7328","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"35877c6cf79fa66db90e5991ad29f2b3f093444f285a8e92079a5bb760e5281a","source_sha256":"d1220062a6e285767ee3caf437f2e0e411175bb2084db2baf1a505199a4fb98e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI : NumberField K := @NumberField.mk _ _ _ (IsCyclotomicExtension.finiteDimensional {n} ℚ K)\n  have H₁ : (aeval (hζ.powerBasis ℚ).gen) (X - 1 : ℤ[X]) = (hζ.subOnePowerBasis ℚ).gen := by simp\n  have H₂ : (aeval (hζ.subOnePowerBasis ℚ).gen) (X + 1 : ℤ[X]) = (hζ.powerBasis ℚ).gen := by simp\n  refine discr_eq_discr_of_toMatrix_coeff_isIntegral _ (fun i j => toMatrix_isIntegral H₁ ?_ ?_ _ _)\n    fun i j => toMatrix_isIntegral H₂ ?_ ?_ _ _\n  · exact hζ.isIntegral (NeZero.pos _)\n  · refine minpoly.isIntegrallyClosed_eq_field_fractions' (K := ℚ) (hζ.isIntegral (NeZero.pos _))\n  · exact (hζ.isIntegral (NeZero.pos _)).sub isIntegral_one\n  · refine minpoly.isIntegrallyClosed_eq_field_fractions' (K := ℚ) ?_\n    exact (hζ.isIntegral (NeZero.pos _)).sub isIntegral_one","hard_negative":false,"metrics":{"chosen_tokens":211,"rejected_tokens":2,"token_jaccard":0.018182,"token_length_ratio":0.009479},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"ce7a38e3725b3b10c30c20e9024a506bbfe445ed86ef0a9b30452f3aad988fc4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots\npublic import Mathlib.RingTheory.DedekindDomain.Dvr\npublic import Mathlib.NumberTheory.NumberField.Discriminant.Defs\n\nNamespace:\nIsPrimitiveRoot\n\nLocal context:\n/-\nCopyright (c) 2022 Riccardo Brasca. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Riccardo Brasca\n-/\n/-!\n# Discriminant of cyclotomic fields\nWe compute the discriminant of a `p ^ n`-th cyclotomic extension.\n\n## Main results\n* `IsCyclotomicExtension.discr_odd_prime` : if `p` is an odd prime such that\n  `IsCyclotomicExtension {p} K L` and `Irreducible (cyclotomic p K)`, then\n  `discr K (hζ.powerBasis K).basis = (-1) ^ ((p - 1) / 2) * p ^ (p - 2)` for any\n  `hζ : IsPrimitiveRoot ζ p`.\n\n-/\n\npublic section\n\n\nuniverse u v\n\nopen Algebra Polynomial Nat IsPrimitiveRoot PowerBasis\n\nopen scoped Polynomial Cyclotomic\n\nnamespace IsPrimitiveRoot\n\nvariable {n : ℕ} [NeZero n] {K : Type u} [Field K] [CharZero K] {ζ : K}\nvariable [ce : IsCyclotomicExtension {n} ℚ K]\n\n/-- The discriminant of the power basis given by a primitive root of unity `ζ` is the same as the\ndiscriminant of the power basis given by `ζ - 1`. -/\n\nTarget:\ntheorem discr_zeta_eq_discr_zeta_sub_one (hζ : IsPrimitiveRoot ζ n) :\n    discr ℚ (hζ.powerBasis ℚ).basis = discr ℚ (hζ.subOnePowerBasis ℚ).basis :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Cyclotomic","family_id":"discr_zeta_eq_discr_zeta_sub_one","file_id":"mathlib/Mathlib/NumberTheory/Cyclotomic/Discriminant.lean","sample_id":"35877c6cf79fa66db90e5991ad29f2b3f093444f285a8e92079a5bb760e5281a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e6c90bca07c73d1bcb9479a5b66cf4f9a197038e47590f39f131d2b92e478d48","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7b81765a8031ad51037f4259b92e28a5ea92d5e155ac346ebd82828d508f44f4","source_sha256":"ebea26ad83d186da19f8d5aec5a4532b5e10776580381e96c50d87a2ed0ca5b4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_contra! h\n  refine ((hfin.biUnion (t := Set.Ici) (by simpa using h)).subset (fun b hb ↦ ?_)).not_infinite\n    (ha.sdiff (Set.finite_singleton a))\n  obtain ⟨x, hax, hxb⟩ := ((show a ≤ b from hb.1).lt_of_ne (Ne.symm hb.2)).exists_covby_le\n  exact Set.mem_biUnion hax hxb","hard_negative":false,"metrics":{"chosen_tokens":87,"rejected_tokens":3,"token_jaccard":0.065217,"token_length_ratio":0.034483},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"ce888ea52d396902f79c533a8004261dbad78ff57a47cc6a608bb9daff08ad0c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Lattice\npublic import Mathlib.Order.Atoms\npublic import Mathlib.Order.Interval.Finset.Defs\npublic import Mathlib.Order.Preorder.Finite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Atoms, Coatoms, Simple Lattices, and Finiteness\n\nThis module contains some results on atoms and simple lattices in the finite context.\n\n## Main results\n* `Finite.to_isAtomic`, `Finite.to_isCoatomic`: Finite partial orders with bottom resp. top\n  are atomic resp. coatomic.\n\n-/\n\npublic section\n\n\nvariable {α β : Type*}\n\nnamespace IsSimpleOrder\n\nvariable [LE α] [BoundedOrder α] [IsSimpleOrder α]\n\nsection DecidableEq\n\n/-- It is important that `IsSimpleOrder` is the last type-class argument of this instance,\nso that type-class inference fails quickly if it doesn't apply.\n\nNote that as of 2025-08-13, this is false. Could someone investigate? -/\nscoped instance (priority := 200) [DecidableEq α] : Fintype α :=\n  Fintype.ofEquiv Bool equivBool.symm\n\nend DecidableEq\n\nscoped instance (priority := 200) : Finite α := by classical infer_instance\n\nend IsSimpleOrder\n\nnamespace Fintype\n\nnamespace IsSimpleOrder\n\nopen scoped _root_.IsSimpleOrder\n\nvariable [LE α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α]\n\ntheorem univ : (Finset.univ : Finset α) = {⊤, ⊥} := by\n  ext\n  simpa using (eq_bot_or_eq_top _).symm\n\ntheorem card : Fintype.card α = 2 :=\n  (Fintype.ofEquiv_card _).trans Fintype.card_bool\n\nend IsSimpleOrder\n\nend Fintype\n\nnamespace Bool\n\ninstance : IsSimpleOrder Bool :=\n  ⟨fun a => by\n    rw [← Finset.mem_singleton, Or.comm, ← Finset.mem_insert, top_eq_true, bot_eq_false, ←\n      Fintype.univ_bool]\n    apply Finset.mem_univ⟩\n\nend Bool\n\nsection Fintype\n\nopen Finset\n\n-- see Note [lower instance priority]\ninstance (priority := 100) Finite.to_isCoatomic [PartialOrder α] [OrderTop α] [Finite α] :\n    IsCoatomic α :=\n  IsStronglyCoatomic.toIsCoatomic α\n\n-- see Note [lower instance priority]\ninstance (priority := 100) Finite.to_isAtomic [PartialOrder α] [OrderBot α] [Finite α] :\n    IsAtomic α :=\n  isCoatomic_dual_iff_isAtomic.mp Finite.to_isCoatomic\n\nend Fintype\n\nsection LocallyFinite\n\nvariable [Preorder α] [LocallyFiniteOrder α]\n\ninstance : IsStronglyAtomic α where\n  exists_covBy_le_of_lt a b hab := by\n    obtain ⟨x, hx, hxmin⟩ := (LocallyFiniteOrder.finsetIoc a b).exists_minimal\n      ⟨b, by simpa [LocallyFiniteOrder.finset_mem_Ioc]⟩\n    simp only [LocallyFiniteOrder.finset_mem_Ioc] at hx hxmin\n    exact ⟨x, ⟨hx.1, fun c hac hcx ↦ hcx.not_ge <| hxmin ⟨hac, hcx.le.trans hx.2⟩ hcx.le⟩, hx.2⟩\n\ninstance : IsStronglyCoatomic α := by\n  rw [← isStronglyAtomic_dual_iff_is_stronglyCoatomic]; infer_instance\n\nend LocallyFinite\n\nsection IsStronglyAtomic\n\nvariable [PartialOrder α] {a : α}\n\nTarget:\ntheorem exists_covby_infinite_Ici_of_infinite_Ici [IsStronglyAtomic α]\n    (ha : (Set.Ici a).Infinite) (hfin : {x | a ⋖ x}.Finite) :\n    ∃ b, a ⋖ b ∧ (Set.Ici b).Infinite :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Atoms","family_id":"exists_covby_infinite_ici_of_infinite_ici","file_id":"mathlib/Mathlib/Order/Atoms/Finite.lean","sample_id":"7b81765a8031ad51037f4259b92e28a5ea92d5e155ac346ebd82828d508f44f4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"be03b33260f8a567b6bd346d2bc5d2f454ac795779b31a7801a8bdb4b9414537","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d6518471441772f8d5a85981c2aec5b7d14ed2edb7d53a8663368eea04cc9127","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ca893de6923284d5337d25661aaf5c8d1221b14895d8ede2751221d59a9b92ee","source_sha256":"ddf0a2dfe4174f459cd3668baa0e6f85d5935c17878e2b7aa6d667af48528ee1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [Functor.HasPointwiseLeftKanExtension]\n  simp only [hasPointwiseLeftKanExtensionAt_iff w]\n  refine ⟨fun h X₄ ↦ ?_, fun h _ ↦ h _⟩\n  rw [← Functor.hasPointwiseLeftKanExtensionAt_iff_of_iso _ _ (B.objObjPreimageIso X₄)]\n  apply h","hard_negative":true,"metrics":{"chosen_tokens":48,"rejected_tokens":3,"token_jaccard":0.03125,"token_length_ratio":0.0625},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"ce8d21f199f85c383f439f9c69463bff5a840c053d1da105f9efc2e311419f69","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.GuitartExact.Basic\npublic import Mathlib.CategoryTheory.Functor.KanExtension.Adjunction\n\nNamespace:\nCategoryTheory.TwoSquare\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Guitart exact squares and Kan extensions\n\nGiven a Guitart exact square `w : T ⋙ R ⟶ L ⋙ B`,\n```\n     T\n  C₁ ⥤ C₂\nL |     | R\n  v     v\n  C₃ ⥤ C₄\n     B\n```\nwe show that an extension `F' : C₄ ⥤ D` of `F : C₂ ⥤ D` along `R`\nis a pointwise left Kan extension at `B.obj X₃` iff\nthe composition `T ⋙ F'` is a pointwise left Kan extension at `X₃`\nof `B ⋙ F'`.\n\nWhen suitable (pointwise) left Kan extensions exist, we also show that\nthe natural transformation of functors `(C₂ ⥤ D) ⥤ C₃ ⥤ D`\n`(whiskeringLeft C₁ C₂ D).obj T ⋙ L.lan ⟶ R.lan ⋙ (whiskeringLeft C₃ C₄ D).obj B`\ninduced by a Guitart exact square `w` is an isomorphism.\n\n## References\n\n* https://ncatlab.org/nlab/show/exact+square\n\n-/\n\n@[expose] public section\n\nuniverse v₁ v₂ v₃ v₄ v₅ u₁ u₂ u₃ u₄ u₅\n\nnamespace CategoryTheory\n\nopen Limits\n\nvariable {C₁ : Type u₁} {C₂ : Type u₂} {C₃ : Type u₃} {C₄ : Type u₄} {D : Type u₅}\n  [Category.{v₁} C₁] [Category.{v₂} C₂] [Category.{v₃} C₃] [Category.{v₄} C₄]\n  [Category.{v₅} D]\n\nnamespace Functor.LeftExtension\n\nvariable {T : C₁ ⥤ C₂} {L : C₁ ⥤ C₃} {R : C₂ ⥤ C₄} {B : C₃ ⥤ C₄}\n  {F : C₂ ⥤ D} (E : R.LeftExtension F)\n\n/-- Given a square `w : TwoSquare T L R B` (consisting of a natural transformation\n`T ⋙ R ⟶ L ⋙ B`), this is the obvious map `R.LeftExtension F → L.LeftExtension (T ⋙ F)`\nobtained by the precomposition with `B` and the postcomposition with `w`. -/\nabbrev compTwoSquare (w : TwoSquare T L R B) : L.LeftExtension (T ⋙ F) :=\n  LeftExtension.mk (B ⋙ E.right)\n    (whiskerLeft _ E.hom ≫ (associator _ _ _).inv ≫\n      whiskerRight w.natTrans _ ≫ (associator _ _ _).hom)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- If `w : TwoSquare T L R B` is a Guitart exact square, and `E` is a left extension\nof `F` along `R`, then `E` is a pointwise left Kan extension of `F` along `R` at\n`B.obj X₃` iff `E.compTwoSquare w` is a pointwise left Kan extension\nof `T ⋙ F` along `L` at `X₃`. -/\nnoncomputable def isPointwiseLeftKanExtensionAtCompTwoSquareEquiv\n    (w : TwoSquare T L R B) (X₃ : C₃) [Final (w.costructuredArrowRightwards X₃)] :\n    (E.compTwoSquare w).IsPointwiseLeftKanExtensionAt X₃ ≃\n      E.IsPointwiseLeftKanExtensionAt (B.obj X₃) := by\n  refine Equiv.trans ?_ (Final.isColimitWhiskerEquiv (w.costructuredArrowRightwards X₃) _)\n  exact IsColimit.equivIsoColimit (Cocone.ext (Iso.refl _))\n\nlemma nonempty_isPointwiseLeftKanExtensionAt_compTwoSquare_iff\n    (w : TwoSquare T L R B) (X₃ : C₃) [Final (w.costructuredArrowRightwards X₃)] :\n    Nonempty ((E.compTwoSquare w).IsPointwiseLeftKanExtensionAt X₃) ↔\n      Nonempty (E.IsPointwiseLeftKanExtensionAt (B.obj X₃)) :=\n  (E.isPointwiseLeftKanExtensionAtCompTwoSquareEquiv w _).nonempty_congr\n\nvariable {E} in\n/-- If `w : TwoSquare T L R B` is a Guitart exact square, and `E` is a pointwise\nleft Kan extension of `F` along `R`, then `E.compTwoSquare w` is a pointwise left\nKan extension of `T ⋙ F` along `L`. -/\nnoncomputable def IsPointwiseLeftKanExtension.compTwoSquare\n    (h : E.IsPointwiseLeftKanExtension) (w : TwoSquare T L R B) [w.GuitartExact] :\n    (E.compTwoSquare w).IsPointwiseLeftKanExtension :=\n  fun X₃ ↦ (E.isPointwiseLeftKanExtensionAtCompTwoSquareEquiv w X₃).symm (h _)\n\n/-- If `w : TwoSquare T L R B` is a Guitart exact square, with `B` essentially surjective,\nand `E` is a left extension of `F` along `R`, then `E` is a pointwise\nleft Kan extension of `F` along `R` provided `E.compTwoSquare w` is a pointwise left\nKan extension of `T ⋙ F` along `L`. -/\nnoncomputable def isPointwiseLeftKanExtensionOfCompTwoSquare\n    (w : TwoSquare T L R B) [w.GuitartExact] [B.EssSurj]\n    (h : (E.compTwoSquare w).IsPointwiseLeftKanExtension) :\n    E.IsPointwiseLeftKanExtension :=\n  fun X₄ ↦ E.isPointwiseLeftKanExtensionAtOfIso'\n    (E.isPointwiseLeftKanExtensionAtCompTwoSquareEquiv w _ (h (B.objPreimage X₄)))\n    (B.objObjPreimageIso X₄)\n\n/-- If `w : TwoSquare T L R B` is a Guitart exact square, with `B` essentially surjective,\nand `E` is a left extension of `F` along `R`, then `E` is a pointwise left Kan extension\nof `F` along `R` iff `E.compTwoSquare w` is a pointwise left Kan extension\nof `T ⋙ F` along `L`. -/\nnoncomputable def isPointwiseLeftKanExtensionEquivOfGuitartExact\n    (w : TwoSquare T L R B) [w.GuitartExact] [B.EssSurj] :\n    (E.compTwoSquare w).IsPointwiseLeftKanExtension ≃\n      E.IsPointwiseLeftKanExtension where\n  toFun h := E.isPointwiseLeftKanExtensionOfCompTwoSquare w h\n  invFun h := h.compTwoSquare w\n  left_inv _ := by subsingleton\n  right_inv _ := by subsingleton\n\nend Functor.LeftExtension\n\nnamespace TwoSquare\n\nvariable {T : C₁ ⥤ C₂} {L : C₁ ⥤ C₃} {R : C₂ ⥤ C₄} {B : C₃ ⥤ C₄}\n   (w : TwoSquare T L R B)\n\ninclude w\n\nlemma hasPointwiseLeftKanExtensionAt_iff\n    (F : C₂ ⥤ D) (X₃ : C₃) [(w.costructuredArrowRightwards X₃).Final] :\n    L.HasPointwiseLeftKanExtensionAt (T ⋙ F) X₃ ↔\n      R.HasPointwiseLeftKanExtensionAt F (B.obj X₃) := by\n  dsimp [Functor.HasPointwiseLeftKanExtensionAt]\n  rw [← Functor.Final.hasColimit_comp_iff (w.costructuredArrowRightwards X₃)]\n  rfl\n\nTarget:\nlemma hasPointwiseLeftKanExtension_iff [w.GuitartExact] [B.EssSurj] (F : C₂ ⥤ D) :\n    L.HasPointwiseLeftKanExtension (T ⋙ F) ↔\n      R.HasPointwiseLeftKanExtension F :=\n\nProof body:\n","rejected":"by\n  exact hasPointwiseLeftKanExtension_iff","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"8e23c2ef8cdba39a2b013f98def7ae5ef3ccb286e07d5856e354cda107382d40","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/GuitartExact","family_id":"haspointwiseleftkanextension_iff","file_id":"mathlib/Mathlib/CategoryTheory/GuitartExact/KanExtension.lean","sample_id":"ca893de6923284d5337d25661aaf5c8d1221b14895d8ede2751221d59a9b92ee"},"split":"train"}
{"binding":{"chosen_proof_sha256":"65fb0bf51735f2fd54b4e2c9d7d6b383f1f337560f0bc27999124bd7b667a991","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"18824e2814fecae627026c691166e00b0e8120c14be4aedda0d23db167d80b57","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"99fd3b1c7dbbd31133ecfe7e89f6d8fdbe8581f23efe8e63f4413b7eb18ed016","source_sha256":"cc5ba9fe8b6562ffdfbb46bc6d44a21f83997fd70ba7584887f499c1806af956","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hA_T_nonneg : ∀ i j, 0 ≤ Aᵀ i j := fun i j => by\n    simpa [Matrix.transpose_apply] using hA.nonneg j i\n  refine ⟨hA_T_nonneg, ?_⟩\n  intro i j\n  letI : Quiver n := toQuiver A\n  obtain ⟨p, hp_pos⟩ := hA.connected j i\n  cases p with\n  | nil =>\n    simp at hp_pos\n  | @cons b _ q e =>\n    let qT := transposePath (A := A) (q.cons e)\n    letI : Quiver n := toQuiver Aᵀ\n    use qT\n    simp [qT, transposePath, Quiver.Path.length_comp, Quiver.Path.length_toPath]","hard_negative":false,"metrics":{"chosen_tokens":121,"rejected_tokens":128,"token_jaccard":0.966667,"token_length_ratio":1.057851},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"ce941dfe552bd2e8cf780933e590251e38b8c5afb4f4c974fa52823f3d70ae90","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Quiver.ConnectedComponent\npublic import Mathlib.Combinatorics.Quiver.Path.Vertices\npublic import Mathlib.Data.Matrix.Mul\n\nNamespace:\nMatrix\n\nLocal context:\n/-\nCopyright (c) 2025 Matteo Cipollina. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Matteo Cipollina\n-/\n/-!\n# Irreducibility and primitivity of nonnegative matrices\n\nThis file develops a graph-theoretic interface for studying the properties of nonnegative square\nmatrices.\n\nWe associate a directed graph (quiver) with a matrix `A`, where an edge `i ⟶ j` exists if and only\nif the entry `A i j` is strictly positive. This allows translating algebraic properties of the\nmatrix (like powers) into graph-theoretic properties of its quiver (like the existence of paths).\n\n## Main definitions\n\n* `Matrix.toQuiver A`: The quiver associated with a matrix `A`, where an edge `i ⟶ j` exists if\n  `0 < A i j`.\n* `Matrix.IsIrreducible A`: A matrix `A` is defined as irreducible if it is entrywise nonnegative\n  and its associated quiver `toQuiver A` is strongly connected. The theorem\n  `Matrix.isIrreducible_iff_exists_pow_pos` proves this graph-theoretic definition is equivalent\n  to the algebraic one in seneta2006 (Def 1.6, p.18): for every pair of indices `(i, j)`, there\n  exists a positive integer `k` such that `(A ^ k) i j > 0`.\n* `Matrix.IsPrimitive A`: A matrix `A` is primitive if it is nonnegative and some power `A ^ k`\n  is strictly positive (all entries are `> 0`), (seneta2006, Definition 1.1, p.14).\n\n## Main results\n\n* `Matrix.pow_apply_pos_iff_nonempty_path`: Establishes the link between matrix powers and graph\n  theory:\n  `(A ^ k) i j > 0` if and only if there is a path of length `k` from `i` to `j` in `toQuiver A`.\n* `Matrix.isIrreducible_iff_exists_pow_pos`: Shows the equivalence between the graph-theoretic\n  definition of irreducibility (strong connectivity) and the algebraic one (existence of a\n  positive entry in some power).\n* `Matrix.IsPrimitive.to_IsIrreducible`: Proves that a primitive matrix is also irreducible\n  (Seneta, p.14).\n* `Matrix.IsIrreducible.transpose`: Shows that the irreducibility property is preserved under\n  transposition.\n\n## Implementation notes\n\nThroughout we work over a linearly ordered ring `R`. Some results require stronger assumptions,\nlike `PosMulStrictMono R` or `Nontrivial R`. Some statements expand matrix powers and thus require\n`[DecidableEq n]` to reason about finite sums.\n\n## TODO\n\nRefactor to use digraphs instead of quivers. A prerequisite for this refactor\nis paths in digraphs.\n\n## References\n\n* [E. Seneta, *Non-negative Matrices and Markov Chains*][seneta2006]\n\n## Tags\n\nmatrix, nonnegative, positive, power, quiver, graph, irreducible, primitive, perron-frobenius\n-/\n\n@[expose] public section\nnamespace Matrix\n\nopen Quiver Quiver.Path\n\nvariable {n R : Type*} [Ring R] [LinearOrder R]\n\n/-- The directed graph (quiver) associated with a matrix `A`,\nwith an edge `i ⟶ j` iff `0 < A i j`. -/\n@[implicit_reducible]\ndef toQuiver (A : Matrix n n R) : Quiver n :=\n  ⟨fun i j => PLift (0 < A i j)⟩\n\n/-- A matrix `A` is irreducible if it is entrywise nonnegative and\nits quiver of positive entries (`toQuiver A`) is strongly connected. -/\n@[mk_iff] structure IsIrreducible (A : Matrix n n R) : Prop where\n  nonneg (i j : n) : 0 ≤ A i j\n  connected : @IsSStronglyConnected n (toQuiver A)\n\n/-- A matrix `A` is primitive if it is entrywise nonnegative\nand some positive power has all entries strictly positive. -/\n@[mk_iff] structure IsPrimitive [Fintype n] [DecidableEq n] (A : Matrix n n R) : Prop where\n  nonneg (i j : n) : 0 ≤ A i j\n  exists_pos_pow : ∃ k > 0, ∀ i j, 0 < (A ^ k) i j\n\nvariable {A : Matrix n n R}\n\n/-- If `A` is irreducible and `n` is non-trivial then every row has a positive entry. -/\nlemma IsIrreducible.exists_pos [Nontrivial n]\n    (h_irr : IsIrreducible A) (i : n) :\n    ∃ j, 0 < A i j := by\n  letI : Quiver n := toQuiver A\n  by_contra h_row\n  have no_out : ∀ j : n, IsEmpty (i ⟶ j) :=\n    fun j => ⟨fun e => h_row ⟨j, e.down⟩⟩\n  obtain ⟨j, hij⟩ := exists_pair_ne n\n  obtain ⟨p, hp_pos⟩ := h_irr.connected i j\n  have h_le : 1 ≤ p.length := Nat.succ_le_of_lt hp_pos\n  have ⟨v, p₁, p₂, _hp_eq, hp₁_len⟩ := p.exists_eq_comp_of_le_length (n := 1) h_le\n  have hlen_ne : p₁.length ≠ 0 := by simp [hp₁_len]\n  obtain ⟨c, p', e, rfl⟩ := (Quiver.Path.length_ne_zero_iff_eq_cons (p := p₁)).1 (by lia)\n  obtain ⟨rfl⟩ : i = c := Quiver.Path.eq_of_length_zero p' (by simp_all)\n  exact (no_out _).false e\n\n/--\nFor a matrix `A` with nonnegative entries, the `(i, j)`-entry of the `k`-th power `A ^ k`\nis strictly positive if and only if there exists a path of length `k` from `i` to `j` in the\nquiver associated to `A` via `toQuiver`. -/\ntheorem pow_apply_pos_iff_nonempty_path\n    [Fintype n] [IsOrderedRing R] [PosMulStrictMono R] [Nontrivial R] [DecidableEq n]\n    (hA : ∀ i j, 0 ≤ A i j) (k : ℕ) (i j : n) :\n    letI := toQuiver A\n    0 < (A ^ k) i j ↔ Nonempty {p : Path i j // p.length = k} := by\n  letI := toQuiver A\n  induction k generalizing i j with\n  | zero =>\n    refine ⟨fun h_pos ↦ ?_, fun ⟨p, hp⟩ ↦ ?_⟩\n    · rcases eq_or_ne i j with rfl | h_eq\n      · exact ⟨⟨Quiver.Path.nil, rfl⟩⟩\n      · simp_all\n    · simp [Quiver.Path.eq_of_length_zero p hp]\n  | succ m ih =>\n    rw [pow_succ, mul_apply]\n    constructor\n    · intro h_pos\n      obtain ⟨l, hl_mem, hl_pos⟩ :\n          ∃ l ∈ (Finset.univ : Finset n), 0 < (A ^ m) i l * A l j := by\n        simpa [Finset.sum_pos_iff_of_nonneg\n                 (fun x _ => mul_nonneg (pow_apply_nonneg hA m i x) (hA x j))]\n          using h_pos\n      have hAm_nonneg : 0 ≤ (A ^ m) i l := pow_apply_nonneg hA m i l\n      have hA_nonneg' : 0 ≤ A l j := hA l j\n      have h_Am : 0 < (A ^ m) i l := by by_contra! h; simp [le_antisymm h hAm_nonneg] at hl_pos\n      have h_A : 0 < A l j := by by_contra! h; simp [le_antisymm h hA_nonneg'] at hl_pos\n      obtain ⟨⟨p, rfl⟩⟩ := (ih i l).mp h_Am\n      exact ⟨p.cons (PLift.up h_A), by simp⟩\n    · rintro ⟨p, hp_len⟩\n      cases p with\n      | nil => simp [Quiver.Path.length] at hp_len\n      | @cons b _ q e =>\n        simp only [Quiver.Path.length_cons, Nat.succ.injEq] at hp_len\n        have h_Am_pos : 0 < (A ^ m) i b := (ih i b).mpr ⟨q, hp_len⟩\n        let h_A_pos := e\n        have h_prod : 0 < (A ^ m) i b * A b j := mul_pos h_Am_pos h_A_pos.down\n        exact\n          (Finset.sum_pos_iff_of_nonneg\n            (fun x _ => mul_nonneg (pow_apply_nonneg hA m i x) (hA x j))).2\n            ⟨b, Finset.mem_univ b, h_prod⟩\n\n/-- Irreducibility of a nonnegative matrix `A` is equivalent to entrywise positivity of some\npower: between any two indices `i, j` there exists a positive integer `k` such that the\n`(i, j)`-entry of `A ^ k` is strictly positive. -/\ntheorem isIrreducible_iff_exists_pow_pos\n    [Fintype n] [IsOrderedRing R] [PosMulStrictMono R] [Nontrivial R] [DecidableEq n]\n    (hA : ∀ i j, 0 ≤ A i j) :\n    IsIrreducible A ↔ ∀ i j, ∃ k > 0, 0 < (A ^ k) i j := by\n  letI : Quiver n := toQuiver A\n  constructor\n  · intro h_irr i j\n    obtain ⟨p, hp_len⟩ := h_irr.2 i j\n    refine ⟨p.length, hp_len, ?_⟩\n    have : Nonempty {q : Path i j // q.length = p.length} := ⟨⟨p, rfl⟩⟩\n    have hpos :=\n      (pow_apply_pos_iff_nonempty_path (A := A) hA p.length i j).2 this\n    simpa using hpos\n  · intro h_exists\n    constructor\n    · exact hA\n    · intro i j\n      obtain ⟨k, hk_pos, hk_entry⟩ := h_exists i j\n      obtain ⟨⟨p, rfl⟩⟩ :=\n        (pow_apply_pos_iff_nonempty_path (A := A) hA k i j).mp hk_entry\n      exact ⟨p, hk_pos⟩\n\n/-- If a nonnegative square matrix `A` is primitive, then `A` is irreducible. -/\ntheorem IsPrimitive.isIrreducible\n    [Fintype n] [IsOrderedRing R] [PosMulStrictMono R] [Nontrivial R] [DecidableEq n]\n    (h_prim : IsPrimitive A) : IsIrreducible A := by\n  obtain ⟨h_nonneg, k, hk_pos, hk_all⟩ := h_prim\n  rw [isIrreducible_iff_exists_pow_pos h_nonneg]\n  aesop\n\n/-! ## Transposition -/\n\n/-- Reverse a path in `toQuiver A` to a path in `toQuiver Aᵀ`, swapping endpoints. -/\ndef transposePath {i j : n} (p : @Quiver.Path n A.toQuiver i j) :\n    @Quiver.Path n Aᵀ.toQuiver j i := by\n  letI : Quiver n := toQuiver A\n  induction p with\n  | nil =>\n    exact (@Quiver.Path.nil _ (toQuiver Aᵀ) _)\n  | @cons b c q e ih =>\n    have eT : 0 < (Aᵀ) c b := by\n      simpa [Matrix.transpose_apply] using e.down\n    exact (@Quiver.Path.comp n (toQuiver Aᵀ) c b i (@Quiver.Hom.toPath n (toQuiver Aᵀ) c b\n      (PLift.up eT)) ih)\n\n/-- Irreducibility is invariant under transpose. -/\n\nTarget:\ntheorem IsIrreducible.transpose (hA : IsIrreducible A) : IsIrreducible Aᵀ :=\n\nProof body:\n","rejected":"```lean\nby\n  have hA_T_nonneg : ∀ i j, 0 ≤ Aᵀ i j := fun i j => by\n    simpa [Matrix.transpose_apply] using hA.nonneg j i\n  refine ⟨hA_T_nonneg, ?_⟩\n  intro i j\n  letI : Quiver n := toQuiver A\n  obtain ⟨p, hp_pos⟩ := hA.connected j i\n  cases p with\n  | nil =>\n    simp at hp_pos\n  | @cons b _ q e =>\n    let qT := transposePath (A := A) (q.cons e)\n    letI : Quiver n := toQuiver Aᵀ\n    use qT\n    simp [qT, transposePath, Quiver.Path.length_comp, Quiver.Path.length_toPath]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Matrix","family_id":"isirreducible","file_id":"mathlib/Mathlib/LinearAlgebra/Matrix/Irreducible/Defs.lean","sample_id":"99fd3b1c7dbbd31133ecfe7e89f6d8fdbe8581f23efe8e63f4413b7eb18ed016"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d4687275514802a11968138ca35e457b8f90609ca53ed1eef831fff19e4d31f5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"10f041559ce44c16bf9a3dea6dd166b47f6a6825476f1e9d77fa0db23316ead5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fd18d30e8b2b216d023535e089915b87e05c113a9e9d53b5e82b2353ec29f830","source_sha256":"586377b5167eecc243d7b879009165834566ed22063836fb061931738feab6a2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Nat.card_prod]; exact Nat.card_congr Subgroup.groupEquivQuotientProdSubgroup","hard_negative":true,"metrics":{"chosen_tokens":16,"rejected_tokens":3,"token_jaccard":0.142857,"token_length_ratio":0.1875},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"cede6e08569eefb2e5d27e82ad8d580693e24f8f7a423093b57e2ea1433a3e82","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.Coset.Basic\npublic import Mathlib.SetTheory.Cardinal.Finite\n\nNamespace:\nSubgroup\n\nLocal context:\n/-\nCopyright (c) 2018 Mitchell Rowett. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mitchell Rowett, Kim Morrison\n-/\n/-!\n# Lagrange's theorem: the order of a subgroup divides the order of the group.\n\n* `Subgroup.card_subgroup_dvd_card`: Lagrange's theorem (for multiplicative groups);\n  there is an analogous version for additive groups\n\n-/\n\npublic section\n\nassert_not_exists Field\n\nopen scoped Pointwise\n\nvariable {α : Type*} [Group α] {s : Subgroup α}\n\nnamespace QuotientGroup\n\n@[to_additive]\ninstance fintype [Fintype α] (s : Subgroup α) [DecidableRel (leftRel s).r] : Fintype (α ⧸ s) :=\n  Quotient.fintype (leftRel s)\n\n@[to_additive]\ninstance (priority := 100) finite [Finite α] : Finite (α ⧸ s) :=\n  Quotient.finite _\n\n@[to_additive]\ninstance fintypeQuotientRightRel [Fintype (α ⧸ s)] :\n    Fintype (Quotient (QuotientGroup.rightRel s)) :=\n  .ofEquiv (α ⧸ s) (QuotientGroup.quotientRightRelEquivQuotientLeftRel s).symm\n\nvariable (s) in\n@[to_additive]\nlemma card_quotient_rightRel [Fintype (α ⧸ s)] :\n    Fintype.card (Quotient (QuotientGroup.rightRel s)) = Fintype.card (α ⧸ s) :=\n  Fintype.ofEquiv_card (QuotientGroup.quotientRightRelEquivQuotientLeftRel s).symm\n\nend QuotientGroup\n\nnamespace Subgroup\n\n@[to_additive AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup]\n\nTarget:\ntheorem card_eq_card_quotient_mul_card_subgroup (s : Subgroup α) :\n    Nat.card α = Nat.card (α ⧸ s) * Nat.card s :=\n\nProof body:\n","rejected":"by\n  exact card_eq_card_quotient_mul_card_subgroup","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"18fe0ae9451f67cb096147aff3a02ef108c959d6377d57d3f2e8059e3a0c8099","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Coset","family_id":"card_eq_card_quotient_mul_card_subgroup","file_id":"mathlib/Mathlib/GroupTheory/Coset/Card.lean","sample_id":"fd18d30e8b2b216d023535e089915b87e05c113a9e9d53b5e82b2353ec29f830"},"split":"train"}
{"binding":{"chosen_proof_sha256":"22533728b6f9b3f9588a530b9a97fe6bf76b4cb9f09b93a8510eba1da1bd326d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f2f0591f91cec44d93c77b4c160bc5278ddc40a39f907725229958f5faa85552","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1709493a1e9d915fafb30165d87390b1f52391762f60fb9606bb3e33e9e6a8da","source_sha256":"42683ce9ec3a96f77aab204680a294682617adafa7d756b1f3dbef8d091d30cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  apply Sieve.ext\n  simp [← NatTrans.naturality_apply]","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":5,"token_jaccard":0.142857,"token_length_ratio":0.384615},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"cf19d9803bca85939da30674e45760400c2439a32773c52dcff974a2cc528f32","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Sites.Closed\npublic import Mathlib.CategoryTheory.Sites.Equivalence\npublic import Mathlib.CategoryTheory.Subobject.Classifier.Defs\npublic import Mathlib.CategoryTheory.Subfunctor.Image\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Edward van de Meent. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Edward van de Meent\n-/\n/-!\n\n# (Elementary) Sheaf Topos\n\nWe define a subobject classifier for categories of sheaves of (large enough) types.\n\n## Main definitions\n\nLet `C` refer to a category with (when relevant) Grothendieck topology `J`.\n\n* `Presheaf.classifier C` is a construction of a subobject classifier in `Cᵒᵖ ⥤ Type (max u v)`.\n* `Sheaf.classifier J` is a construction of a subobject classifier in `Sheaf J (Type (max u v))`.\n* `inferInstance : HasClassifier (Cᵒᵖ ⥤ Type w)` says that `Cᵒᵖ ⥤ Type w` has a subobject\n  classifier if `C` is `w`-essentially small.\n* `inferInstance : HasClassifier (Sheaf J (Type w))` says that `Sheaf J (Type w)` has a\n  subobject classifier if `C` is `w`-essentially small.\n\n## Main results\n\n* Any category of sheaves of types has a subobject classifier if the site is essentially small.\n* As a consequence, (because categories of sheaves are cartesian monoidal and have finite limits,)\n  such categories are Elementary Topoi.\n\n## TODOS:\n\n* generalize `Presheaf.isClosed_χ_app_apply_of` to only assuming `G` is separated\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nnamespace CategoryTheory\n\nvariable {C : Type u} [Category.{v} C]\nopen Limits\n\nsection presheaf\n\nvariable (C) in\n/-- The truth morphism in the category of presheaves. At each component `X : C`, it is the constant\nmap returning `⊤ : Sieve X`. -/\n@[simps]\ndef Presheaf.truth : (Functor.const _).obj PUnit ⟶ Functor.sieves C where\n  app X := ↾fun _ ↦ (⊤ : Sieve X.unop)\n\nvariable {F G : Cᵒᵖ ⥤ Type (max u v)}\n\nset_option backward.defeqAttrib.useBackward true in\n/--\nThe characteristic map of an inclusion of presheaves.\nGiven a monomorphism of sheaves `m : F ⟶ G`, an object X of the site, map an element `x : G(X)`\nto the (closed) sieve on X where `f : Y → X` is in the sieve iff\n  `∃ a ∈ F(Y), G(f)(x) = m_Y(a)`\n-/\n@[simps app]\ndef Presheaf.χ (m : F ⟶ G) : G ⟶ Functor.sieves C where\n  app X := ↾fun x ↦ ⟨fun Y f => ∃ a, G.map f.op x = m.app (.op Y) a, by\n    intro Y Z f ⟨a, ha⟩ g\n    use F.map g.op a\n    simp [ha, NatTrans.naturality_apply]⟩\n\nTarget:\nlemma Presheaf.comp_χ_eq (m : F ⟶ G) : m ≫ Presheaf.χ m =\n    (Functor.isTerminalConst _ Types.isTerminalPUnit).from F ≫ Presheaf.truth C :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_1709493a1e9d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c9aecd76d1904089c04dd55c0b3c964bb31800176af21a9721ef85a9d8077676","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Topos","family_id":"presheaf","file_id":"mathlib/Mathlib/CategoryTheory/Topos/Sheaf.lean","sample_id":"1709493a1e9d915fafb30165d87390b1f52391762f60fb9606bb3e33e9e6a8da"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b8f5229735b933ede0144ab9a48b8b3042b761eb7777cebb1d91a92fa7760495","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5b8e99c563bfea9792268681baf844de6f8a2486d53f29093429574a59b4d8c5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ad36c8eeea38b3617b2fa01b60e158adc224ea844614163c66de06f8b4d2fee4","source_sha256":"42f43c83c3b42913128aa30933a73363c485cf2bc69fc613214486004b504ca0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Subgroup.rank_congr (commutator_eq_closure G)]\n  apply Subgroup.rank_closure_finite_le_nat_card\n\nvariable [Group.FG G]","hard_negative":true,"metrics":{"chosen_tokens":22,"rejected_tokens":3,"token_jaccard":0.055556,"token_length_ratio":0.136364},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"cf3e44883d43be77c53fc13b16e5bddc74e4ba9e43444737b57354098a262f18","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Subgroup.Finite\npublic import Mathlib.GroupTheory.Commutator.Basic\npublic import Mathlib.GroupTheory.Rank\npublic import Mathlib.GroupTheory.Index\n\nNamespace:\nSubgroup\n\nLocal context:\n/-\nCopyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jordan Brown, Thomas Browning, Patrick Lutz\n-/\n/-!\nThe commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n\npublic section\n\nvariable {G : Type*} [Group G]\n\nnamespace Subgroup\n\n/-- The commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n@[to_additive /-- The commutator of a finite direct product is contained in the direct product of\nthe commutators. -/]\ntheorem commutator_pi_pi_of_finite {η : Type*} [Finite η] {Gs : η → Type*} [∀ i, Group (Gs i)]\n    (H K : ∀ i, Subgroup (Gs i)) : ⁅Subgroup.pi Set.univ H, Subgroup.pi Set.univ K⁆ =\n    Subgroup.pi Set.univ fun i => ⁅H i, K i⁆ := by\n  classical\n    apply le_antisymm (commutator_pi_pi_le H K)\n    rw [pi_le_iff]\n    intro i hi\n    rw [map_commutator]\n    apply commutator_mono <;>\n      · rw [le_pi_iff]\n        intro j _hj\n        rintro _ ⟨x, hx, rfl⟩\n        by_cases h : j = i\n        · subst h\n          simpa using hx\n        · simp [h, one_mem]\n\nvariable [Finite (commutatorSet G)]\n\ninstance : Group.FG (_root_.commutator G) := by\n  rw [commutator_eq_closure]; apply Group.closure_finite_fg\n\nvariable (G) in\n\nTarget:\nlemma rank_commutator_le_card : Group.rank (_root_.commutator G) ≤ Nat.card (commutatorSet G) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_ad36c8eeea38","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"6f13cb3ba9bd2d6aabf75f296fbf6369e0ed1a189fe06594efdf541f02a80582","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Commutator","family_id":"rank_commutator_le_card","file_id":"mathlib/Mathlib/GroupTheory/Commutator/Finite.lean","sample_id":"ad36c8eeea38b3617b2fa01b60e158adc224ea844614163c66de06f8b4d2fee4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"988485a8c756571de67c1e7554cc8678471929e0b79e0159e23042969b87deb3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"696e93fb10ae259c09e9614ab7179b8e76e457a174ccaa766c9f948e29436e3e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5dc82f08b0f93626082032ca9aa64a8e948617dc0aa5acf4d820aa7f216565eb","source_sha256":"8bbd9fce2d15ad8ea50e5683616b40f12621dcf6a628ad12261a1d432f9bccf6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction p with\n  | nil => rfl\n  | cons _ _ ih => dsimp [pathToList] at *; rw [ih]","hard_negative":true,"metrics":{"chosen_tokens":25,"rejected_tokens":2,"token_jaccard":0.05,"token_length_ratio":0.08},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"cf51ac8748508eda0ec8f466879e24a6f552127d5a7568e28f6b6b8ad2c914fb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Quiver.Cast\npublic import Mathlib.Combinatorics.Quiver.Symmetric\n\nNamespace:\nQuiver.SingleObj\n\nLocal context:\n/-\nCopyright (c) 2023 Antoine Labelle. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Labelle\n-/\n/-!\n# Single-object quiver\n\nSingle object quiver with a given arrows type.\n\n## Main definitions\n\nGiven a type `α`, `SingleObj α` is the `Unit` type, whose single object is called `star α`, with\n`Quiver` structure such that `star α ⟶ star α` is the type `α`.\nAn element `x : α` can be reinterpreted as an element of `star α ⟶ star α` using\n`toHom`.\nMore generally, a list of elements of `a` can be reinterpreted as a path from `star α` to\nitself using `pathEquivList`.\n-/\n\n@[expose] public section\n\nnamespace Quiver\n\n/-- Type tag on `Unit` used to define single-object quivers. -/\n@[nolint unusedArguments]\ndef SingleObj (_ : Type*) : Type :=\n  Unit\nderiving Unique\n\nnamespace SingleObj\n\nvariable (α β γ : Type*)\n\ninstance : Quiver (SingleObj α) :=\n  ⟨fun _ _ => α⟩\n\n/-- The single object in `SingleObj α`. -/\ndef star : SingleObj α := default\n\nvariable {α β γ}\n\nlemma ext {x y : SingleObj α} : x = y := Unit.ext x y\n\n-- See note [reducible non-instances]\n/-- Equip `SingleObj α` with a reverse operation. -/\nabbrev hasReverse (rev : α → α) : HasReverse (SingleObj α) := ⟨rev⟩\n\n-- See note [reducible non-instances]\n/-- Equip `SingleObj α` with an involutive reverse operation. -/\nabbrev hasInvolutiveReverse (rev : α → α) (h : Function.Involutive rev) :\n    HasInvolutiveReverse (SingleObj α) where\n  toHasReverse := hasReverse rev\n  inv' := h\n\n/-- The type of arrows from `star α` to itself is equivalent to the original type `α`. -/\n@[simps!]\ndef toHom : α ≃ (star α ⟶ star α) :=\n  Equiv.refl _\n\n/-- Prefunctors between two `SingleObj` quivers correspond to functions between the corresponding\narrows types.\n-/\n@[simps]\ndef toPrefunctor : (α → β) ≃ SingleObj α ⥤q SingleObj β where\n  toFun f := ⟨id, f⟩\n  invFun f a := f.map (toHom a)\n\ntheorem toPrefunctor_id : toPrefunctor id = 𝟭q (SingleObj α) :=\n  rfl\n\n@[simp]\ntheorem toPrefunctor_symm_id : toPrefunctor.symm (𝟭q (SingleObj α)) = id :=\n  rfl\n\ntheorem toPrefunctor_comp (f : α → β) (g : β → γ) :\n    toPrefunctor (g ∘ f) = toPrefunctor f ⋙q toPrefunctor g :=\n  rfl\n\n@[simp]\ntheorem toPrefunctor_symm_comp (f : SingleObj α ⥤q SingleObj β) (g : SingleObj β ⥤q SingleObj γ) :\n    toPrefunctor.symm (f ⋙q g) = toPrefunctor.symm g ∘ toPrefunctor.symm f := by\n  simp only [Equiv.symm_apply_eq, toPrefunctor_comp, Equiv.apply_symm_apply]\n\n/-- Auxiliary definition for `quiver.SingleObj.pathEquivList`.\nConverts a path in the quiver `single_obj α` into a list of elements of type `a`.\n-/\ndef pathToList : ∀ {x : SingleObj α}, Path (star α) x → List α\n  | _, Path.nil => []\n  | _, Path.cons p a => a :: pathToList p\n\n/-- Auxiliary definition for `quiver.SingleObj.pathEquivList`.\nConverts a list of elements of type `α` into a path in the quiver `SingleObj α`.\n-/\n@[simp]\ndef listToPath : List α → Path (star α) (star α)\n  | [] => Path.nil\n  | a :: l => (listToPath l).cons a\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem listToPath_pathToList {x : SingleObj α} (p : Path (star α) x) :\n    listToPath (pathToList p) = p.cast rfl ext :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_5dc82f08b0f9","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"9b399d96551725639bb50a3682deb56ddde2efadef5345d3be00177463911dd5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Quiver","family_id":"listtopath_pathtolist","file_id":"mathlib/Mathlib/Combinatorics/Quiver/SingleObj.lean","sample_id":"5dc82f08b0f93626082032ca9aa64a8e948617dc0aa5acf4d820aa7f216565eb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fd734d0795994b2df39d22bec1fc6dc26c9a299bef650b0d07eb3f85b3d6c929","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5bf0e0681c55eec8bc5b1c759d43192dd7b331a9558e5d94174d45141766d782","source_sha256":"9414b3a4a4547da0b65ca4fdfb12cdf19f09a370ed358f728d19141cdf7fb313","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [logHeight_eq_log_mulHeight]\n  exact log_nonneg <| x.one_le_mulHeight","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":2,"token_jaccard":0.076923,"token_length_ratio":0.166667},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"cf870cd02521a00895607acc736e1275673d729a4904343175a46a011008525b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Projectivization.Basic\npublic import Mathlib.NumberTheory.Height.Basic\n\nNamespace:\nProjectivization\n\nLocal context:\n/-\nCopyright (c) 2026 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Heights of points in projective space\n\nWe define the multiplicative (`Projectivization.mulHeight`) and the logarithmic\n(`Projectivization.logHeight`) height of a point in a (finite-dimensional) projective space\nover a field that has a `Height.AdmissibleAbsValues` instance.\n\nThe height is defined to be the height of any representative tuple; it does not depend\non which representative is chosen.\n-/\n\npublic section\n\nnamespace Projectivization\n\nopen Height AdmissibleAbsValues Real\n\nvariable {K : Type*} [Field K] [AdmissibleAbsValues K] {ι : Type*} [Finite ι]\n\nprivate lemma mulHeight_aux (a b : { v : ι → K // v ≠ 0 }) (t : K) (h : a.val = t • b.val) :\n    mulHeight a.val = mulHeight b.val :=\n  have ht : t ≠ 0 := by\n    contrapose! h\n    simpa [h] using a.prop\n  h ▸ mulHeight_smul_eq_mulHeight _ ht\n\nprivate lemma logHeight_aux (a b : { v : ι → K // v ≠ 0 }) (t : K) (h : a.val = t • b.val) :\n    logHeight a.val = logHeight b.val :=\n  congrArg log <| mod_cast mulHeight_aux a b t h\n\n-- We do not expose the bodies of these definitions so that we can keep the \"_aux\" lemmas\n-- above private.\n\n/-- The multiplicative height of a point on a finite-dimensional projective space over `K`\nwith a given basis. -/\nnoncomputable def mulHeight (x : Projectivization K (ι → K)) : ℝ :=\n  x.lift (fun r ↦ Height.mulHeight r.val) mulHeight_aux\n\n/-- The logarithmic height of a point on a finite-dimensional projective space over `K`\nwith a given basis. -/\nnoncomputable def logHeight (x : Projectivization K (ι → K)) : ℝ :=\n  x.lift (fun r ↦ Height.logHeight r.val) logHeight_aux\n\nlemma mulHeight_mk {x : ι → K} (hx : x ≠ 0) : mulHeight (mk K x hx) = Height.mulHeight x := by\n  rfl\n\nlemma logHeight_mk {x : ι → K} (hx : x ≠ 0) : logHeight (mk K x hx) = Height.logHeight x := by\n  rfl\n\nlemma logHeight_eq_log_mulHeight (x : Projectivization K (ι → K)) :\n    logHeight x = log (mulHeight x) := by\n  rw [← x.mk_rep, mulHeight_mk, logHeight_mk, Height.logHeight]\n\nlemma one_le_mulHeight (x : Projectivization K (ι → K)) : 1 ≤ mulHeight x := by\n  rw [← x.mk_rep, mulHeight_mk]\n  exact Height.one_le_mulHeight _\n\nlemma mulHeight_pos (x : Projectivization K (ι → K)) : 0 < mulHeight x :=\n  zero_lt_one.trans_le <| one_le_mulHeight x\n\nlemma mulHeight_ne_zero (x : Projectivization K (ι → K)) : mulHeight x ≠ 0 :=\n  (mulHeight_pos x).ne'\n\nTarget:\nlemma logHeight_nonneg (x : Projectivization K (ι → K)) : 0 ≤ logHeight x :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Height","family_id":"logheight_nonneg","file_id":"mathlib/Mathlib/NumberTheory/Height/Projectivization.lean","sample_id":"5bf0e0681c55eec8bc5b1c759d43192dd7b331a9558e5d94174d45141766d782"},"split":"train"}
{"binding":{"chosen_proof_sha256":"397dccc7f0b407dbb3cdb395970fa551240d91f8fe8d8b6664c36ca222b24937","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a6605051476f011bb510bc68d28ab612266a9c7614f07e64b7362c57ddba7a03","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"890928cea67490b486f404ea924568c9df0ae3523215f6af6df2d816e57e58f7","source_sha256":"c729c093a684b2a7b5066e0c93a4ca1c797418af44bab051680547cf260ff85c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [h]\n\n-- used in the `nlinarith` normalization steps. The `_` argument is for uniformity.","hard_negative":true,"metrics":{"chosen_tokens":25,"rejected_tokens":5,"token_jaccard":0.086957,"token_length_ratio":0.2},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"cfe2fd00e12037051da5cef940133085a0d98147a2d931e18f262705d3afb6f0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Ineq\npublic import Mathlib.Data.Nat.Cast.Order.Ring\n\nNamespace:\nMathlib.Tactic.Linarith\n\nLocal context:\n/-\nCopyright (c) 2020 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n-/\npublic meta import Batteries.Tactic.Lint.Basic\npublic meta import Mathlib.Data.Ineq\npublic meta import Mathlib.Tactic.ToAdditive\n\n/-!\n# Lemmas for `linarith`.\n\nThose in the `Linarith` namespace should stay here.\n\nThose outside the `Linarith` namespace may be deleted as they are ported to mathlib4.\n-/\n\npublic meta section\n\nnamespace Mathlib.Tactic.Linarith\n\nuniverse u\ntheorem lt_irrefl {α : Type u} [Preorder α] {a : α} : ¬a < a := _root_.lt_irrefl a\n\ntheorem eq_of_eq_of_eq {α} [Semiring α] {a b : α} (ha : a = 0) (hb : b = 0) : a + b = 0 := by\n  simp [*]\n\nsection Semiring\nvariable {α : Type u} [Semiring α] [PartialOrder α]\n\ntheorem zero_lt_one [IsStrictOrderedRing α] : (0:α) < 1 :=\n  _root_.zero_lt_one\n\ntheorem le_of_eq_of_le {a b : α} (ha : a = 0) (hb : b ≤ 0) : a + b ≤ 0 := by\n  simp [*]\n\ntheorem lt_of_eq_of_lt {a b : α} (ha : a = 0) (hb : b < 0) : a + b < 0 := by\n  simp [*]\n\ntheorem le_of_le_of_eq {a b : α} (ha : a ≤ 0) (hb : b = 0) : a + b ≤ 0 := by\n  simp [*]\n\ntheorem lt_of_lt_of_eq {a b : α} (ha : a < 0) (hb : b = 0) : a + b < 0 := by\n  simp [*]\n\ntheorem add_nonpos [IsOrderedRing α] {a b : α} (ha : a ≤ 0) (hb : b ≤ 0) :\n    a + b ≤ 0 :=\n  _root_.add_nonpos ha hb\n\ntheorem add_lt_of_le_of_neg [IsStrictOrderedRing α] {a b c : α} (hbc : b ≤ c)\n    (ha : a < 0) : b + a < c :=\n  _root_.add_lt_of_le_of_neg hbc ha\n\ntheorem add_lt_of_neg_of_le [IsStrictOrderedRing α] {a b c : α} (ha : a < 0)\n    (hbc : b ≤ c) : a + b < c :=\n  _root_.add_lt_of_neg_of_le ha hbc\n\ntheorem add_neg [IsStrictOrderedRing α] {a b : α} (ha : a < 0)\n    (hb : b < 0) : a + b < 0 :=\n  _root_.add_neg ha hb\n\nvariable (α) in\nlemma natCast_nonneg [IsOrderedRing α] (n : ℕ) : (0 : α) ≤ n := Nat.cast_nonneg n\n\n-- used alongside `mul_neg` and `mul_nonpos`, so has the same argument pattern for uniformity\n@[nolint unusedArguments]\ntheorem mul_eq [IsOrderedRing α] {a b : α} (ha : a = 0) (_ : 0 < b) : b * a = 0 := by\n  simp [*]\n\nend Semiring\n\nsection Ring\nvariable {α : Type u} [Ring α] [PartialOrder α]\n\ntheorem mul_neg [IsStrictOrderedRing α] {a b : α} (ha : a < 0) (hb : 0 < b) : b * a < 0 :=\n  have : (-b)*a > 0 := mul_pos_of_neg_of_neg (neg_neg_of_pos hb) ha\n  neg_of_neg_pos (by simpa)\n\ntheorem mul_nonpos [IsOrderedRing α] {a b : α} (ha : a ≤ 0) (hb : 0 < b) : b * a ≤ 0 :=\n  have : (-b)*a ≥ 0 := mul_nonneg_of_nonpos_of_nonpos (le_of_lt (neg_neg_of_pos hb)) ha\n  by simpa\n\ntheorem sub_nonpos_of_le [IsOrderedRing α] {a b : α} : a ≤ b → a - b ≤ 0 :=\n  _root_.sub_nonpos_of_le\n\ntheorem sub_neg_of_lt [IsOrderedRing α] {a b : α} : a < b → a - b < 0 :=\n  _root_.sub_neg_of_lt\n\nend Ring\n\n/-- Finds the name of a multiplicative lemma corresponding to an inequality strength. -/\ndef _root_.Mathlib.Ineq.toConstMulName : Ineq → Lean.Name\n  | .lt => ``mul_neg\n  | .le => ``mul_nonpos\n  | .eq => ``mul_eq\n\nlemma eq_of_not_lt_of_not_gt {α} [LinearOrder α] (a b : α) (h1 : ¬ a < b) (h2 : ¬ b < a) : a = b :=\n  le_antisymm (le_of_not_gt h2) (le_of_not_gt h1)\n\n-- used in the `nlinarith` normalization steps. The `_` argument is for uniformity.\n@[nolint unusedArguments]\n\nTarget:\nlemma mul_zero_eq {α} {R : α → α → Prop} [Semiring α] {a b : α} (_ : R a 0) (h : b = 0) :\n    a * b = 0 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_890928cea674","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"4455e949a0b4c849734e0fb188fdc1f6908f82e6b0858689d50ae2c0ebc0c78d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Linarith","family_id":"mul_zero_eq","file_id":"mathlib/Mathlib/Tactic/Linarith/Lemmas.lean","sample_id":"890928cea67490b486f404ea924568c9df0ae3523215f6af6df2d816e57e58f7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"82023a5a0373446dea510bd79a74d7b917c1be62462ff0113fe4a4cf3e1520cd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"87399eed90a7a9d71500444beb6af36405b8a0740724037a49d189fcab7e9413","source_sha256":"49729ab7657bf5b96e3883707dabc942ba77c2ab9a085e0f2d0959061ea37dde","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by simp) :\n    ∃ b : B, b ∉ s := by\n  simpa [-SetLike.coe_set_eq, SetLike.ext'_iff, h_top, Set.ne_univ_iff_exists_notMem] using hs","hard_negative":false,"metrics":{"chosen_tokens":33,"rejected_tokens":5,"token_jaccard":0.035714,"token_length_ratio":0.151515},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"cfea20f0aa41937481e1d8180b262b43f8a562bfb860f605bf16e66117fc9a29","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Tactic.Monotonicity.Attr\npublic import Mathlib.Tactic.SetLike\npublic import Mathlib.Data.Set.Basic\n\nNamespace:\nSetLike\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Typeclass for types with a set-like extensionality property\n\nThe `Membership` typeclass is used to let terms of a type have elements.\nMany instances of `Membership` have a set-like extensionality property:\nthings are equal iff they have the same elements.  The `SetLike`\ntypeclass provides a unified interface to define a `Membership` that is\nextensional in this way.\n\nThe main use of `SetLike` is for algebraic subobjects (such as\n`Submonoid` and `Submodule`), whose non-proof data consists only of a\ncarrier set.  In such a situation, the projection to the carrier set\nis injective.\n\nIn general, a type `A` is `SetLike` with elements of type `B` if it\nhas an injective map to `Set B`.  This module provides standard\nboilerplate for every `SetLike`: a `coe_sort`, a `coe` to set,\nand various extensionality and simp lemmas. The order induced by set inclusion is\ncalled `PartialOrder.ofSetlike`: this is not an instance for flexibility in choosing orders.\nThe class `IsConcreteLE` abstractly states the order is equal to that induced by set inclusion;\nan instance is automatically available when defining a `PartialOrder` as\n`.ofSetLike (MySubobject X) X`.\n\nA typical subobject should be declared as:\n```\nstructure MySubobject (X : Type*) [ObjectTypeclass X] where\n  (carrier : Set X)\n  (op_mem' : ∀ {x : X}, x ∈ carrier → sorry ∈ carrier)\n\nnamespace MySubobject\n\nvariable {X : Type*} [ObjectTypeclass X] {x : X}\n\ninstance : SetLike (MySubobject X) X :=\n  ⟨MySubobject.carrier, fun p q h => by cases p; cases q; congr!⟩\n\ninstance : PartialOrder (MySubobject X) := .ofSetLike (MySubobject X) X\n\n@[simp] lemma mem_carrier {p : MySubobject X} : x ∈ p.carrier ↔ x ∈ (p : Set X) := Iff.rfl\n\n@[ext] theorem ext {p q : MySubobject X} (h : ∀ x, x ∈ p ↔ x ∈ q) : p = q := SetLike.ext h\n\n/-- Copy of a `MySubobject` with a new `carrier` equal to the old one. Useful to fix definitional\nequalities. See Note [range copy pattern]. -/\nprotected def copy (p : MySubobject X) (s : Set X) (hs : s = ↑p) : MySubobject X :=\n  { carrier := s\n    op_mem' := hs.symm ▸ p.op_mem' }\n\n@[simp] lemma coe_copy (p : MySubobject X) (s : Set X) (hs : s = ↑p) :\n  (p.copy s hs : Set X) = s := rfl\n\nlemma copy_eq (p : MySubobject X) (s : Set X) (hs : s = ↑p) : p.copy s hs = p :=\n  SetLike.coe_injective hs\n\nend MySubobject\n```\n\nAn alternative to `SetLike` could have been an extensional `Membership` typeclass:\n```\nclass ExtMembership (α : out_param <| Type u) (β : Type v) extends Membership α β where\n  (ext_iff : ∀ {s t : β}, s = t ↔ ∀ (x : α), x ∈ s ↔ x ∈ t)\n```\nWhile this is equivalent, `SetLike` conveniently uses a carrier set projection directly.\n\n## Tags\n\nsubobjects\n-/\n\n@[expose] public section\n\nassert_not_exists RelIso\n\n/-- A class to indicate that there is a canonical injection between `A` and `Set B`.\n\nThis has the effect of giving terms of `A` elements of type `B` (through a `Membership`\ninstance) and a compatible coercion to `Type*` as a subtype.\n\nNote: if `SetLike.coe` is a projection, implementers should create a simp lemma such as\n```\n@[simp] lemma mem_carrier {p : MySubobject X} : x ∈ p.carrier ↔ x ∈ (p : Set X) := Iff.rfl\n```\nto normalize terms.\n\nIf you declare an unbundled subclass of `SetLike`, for example:\n```\nclass MulMemClass (S : Type*) (M : Type*) [Mul M] [SetLike S M] where\n  ...\n```\nThen you should *not* repeat the `outParam` declaration so `SetLike` will supply the value instead.\nThis ensures your subclass will not have issues with synthesis of the `[Mul M]` parameter starting\nbefore the value of `M` is known.\n-/\n@[notation_class* carrier Simps.findCoercionArgs]\nclass SetLike (A : Type*) (B : outParam Type*) where\n  /-- The coercion from a term of a `SetLike` to its corresponding `Set`. -/\n  protected coe : A → Set B\n  /-- The coercion from a term of a `SetLike` to its corresponding `Set` is injective. -/\n  coe_injective : Function.Injective coe\n\nattribute [coe] SetLike.coe\n\nnamespace SetLike\n\nvariable {A : Type*} {B : Type*} [i : SetLike A B]\n\n@[deprecated (since := \"2026-06-04\")] alias coe_injective' := coe_injective\n\ninstance : CoeTC A (Set B) where coe := SetLike.coe\n\ninstance (priority := 100) instMembership : Membership B A :=\n  ⟨fun p x => x ∈ (p : Set B)⟩\n\ninstance (priority := 100) : CoeSort A (Type _) :=\n  ⟨fun p => { x : B // x ∈ p }⟩\n\nsection Delab\nopen Lean PrettyPrinter.Delaborator SubExpr\n\n/-- For terms that match the `CoeSort` instance's body, pretty print as `↥S`\nrather than as `{ x // x ∈ S }`. The discriminating feature is that membership\nuses the `SetLike.instMembership` instance. -/\n@[app_delab Subtype]\nmeta def delabSubtypeSetLike : Delab := whenPPOption getPPNotation do\n  let #[_, .lam n _ body _] := (← getExpr).getAppArgs | failure\n  guard <| body.isAppOf ``Membership.mem\n  let #[_, _, inst, _, .bvar 0] := body.getAppArgs | failure\n  guard <| inst.isAppOfArity ``instMembership 3\n  let S ← withAppArg <| withBindingBody n <| withNaryArg 3 delab\n  `(↥$S)\n\nend Delab\n\nvariable (p q : A)\n\n@[simp, norm_cast]\ntheorem coe_sort_coe : ((p : Set B) : Type _) = p :=\n  rfl\n\nvariable {p q}\n\nprotected theorem «exists» {q : p → Prop} : (∃ x, q x) ↔ ∃ (x : B) (h : x ∈ p), q ⟨x, ‹_›⟩ :=\n  SetCoe.exists\n\nprotected theorem «forall» {q : p → Prop} : (∀ x, q x) ↔ ∀ (x : B) (h : x ∈ p), q ⟨x, ‹_›⟩ :=\n  SetCoe.forall\n\n@[simp, norm_cast]\ntheorem coe_set_eq : (p : Set B) = q ↔ p = q :=\n  coe_injective.eq_iff\n\n@[norm_cast] lemma coe_ne_coe : (p : Set B) ≠ q ↔ p ≠ q := coe_injective.ne_iff\n\ntheorem ext' (h : (p : Set B) = q) : p = q :=\n  coe_injective h\n\ntheorem ext'_iff : p = q ↔ (p : Set B) = q :=\n  coe_set_eq.symm\n\n/-- Note: implementers of `SetLike` must copy this lemma in order to tag it with `@[ext]`. -/\ntheorem ext (h : ∀ x, x ∈ p ↔ x ∈ q) : p = q :=\n  coe_injective <| Set.ext h\n\ntheorem ext_iff : p = q ↔ ∀ x, x ∈ p ↔ x ∈ q :=\n  coe_injective.eq_iff.symm.trans Set.ext_iff\n\n@[simp, push]\ntheorem mem_coe {x : B} : x ∈ (p : Set B) ↔ x ∈ p :=\n  Iff.rfl\n\n@[simp, norm_cast]\ntheorem coe_eq_coe {x y : p} : (x : B) = y ↔ x = y :=\n  Subtype.ext_iff.symm\n\n@[simp]\ntheorem coe_mem (x : p) : (x : B) ∈ p :=\n  x.2\n\n@[aesop 5% (rule_sets := [SetLike!])]\nlemma mem_of_subset {s : Set B} (hp : s ⊆ p) {x : B} (hx : x ∈ s) : x ∈ p := hp hx\n\n@[simp]\nprotected theorem eta (x : p) (hx : (x : B) ∈ p) : (⟨x, hx⟩ : p) = x := rfl\n\n@[simp] lemma setOf_mem_eq (a : A) : {b | b ∈ a} = a := rfl\n\n@[nontriviality]\nlemma mem_of_subsingleton [Subsingleton B] (S : A) [h : Nonempty S] {b : B} : b ∈ S := by\n  obtain ⟨s, hs⟩ := nonempty_subtype.mp h\n  simpa [Subsingleton.elim b s]\n\n/-- If `s` is a proper element of a `SetLike` structure (i.e., `s ≠ ⊤`) and the top element\ncoerces to the universal set, then there exists an element not in `s`. -/\n\nTarget:\nlemma exists_not_mem_of_ne_top [LE A] [OrderTop A] (s : A) (hs : s ≠ ⊤)\n    (h_top : ((⊤ : A) : Set B) = Set.univ :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/SetLike","family_id":"exists_not_mem_of_ne_top","file_id":"mathlib/Mathlib/Data/SetLike/Basic.lean","sample_id":"87399eed90a7a9d71500444beb6af36405b8a0740724037a49d189fcab7e9413"},"split":"train"}
{"binding":{"chosen_proof_sha256":"447b3ac2404a7fdae48d1d41ba080e9c5434ee1fd77cdc32e3d7e6b39eb7aa1d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b38b14c96b38afbba7c3259cf4c4e7b4c145e1b2ac6c153a770feeac3a7076d5","source_sha256":"ffc9175aacac138c9dfb496e1bcda0b6ab6a54d2ace871f4d5aff79b2c6d448a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h ↦ ?_, fun h ↦ h.units_smul w⟩\n  convert! h.units_smul (fun i ↦ (w i)⁻¹)\n  simp [funext_iff]","hard_negative":false,"metrics":{"chosen_tokens":37,"rejected_tokens":5,"token_jaccard":0.074074,"token_length_ratio":0.135135},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"cfed9ede98240b9168b8094a80f8a6085c721c3883c577a35fa0cd3eeab34840","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Fin\npublic import Mathlib.LinearAlgebra.LinearIndependent.Defs\npublic import Mathlib.Logic.Equiv.Fin.Rotate\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen\n-/\n/-!\n# Linear independence\n\nThis file collects basic consequences of linear (in)dependence and includes specialized tests for\nspecific families of vectors.\n\n## Main statements\n\nWe prove several specialized tests for linear independence of families of vectors and of sets of\nvectors.\n\n* `linearIndependent_empty_type`: a family indexed by an empty type is linearly independent;\n* `linearIndependent_unique_iff`: if `ι` is a singleton, then `LinearIndependent K v` is\n  equivalent to `v default ≠ 0`;\n* `linearIndependent_sum`: type-specific test for linear independence of families of vector\n  fields;\n* `linearIndependent_singleton`: linear independence tests for set operations.\n\nIn many cases we additionally provide dot-style operations (e.g., `LinearIndependent.union`) to\nmake the linear independence tests usable as `hv.insert ha` etc.\n\n## TODO\n\nRework proofs to hold in semirings, by avoiding the path through\n`ker (Finsupp.linearCombination R v) = ⊥`.\n\n## Tags\n\nlinearly dependent, linear dependence, linearly independent, linear independence\n\n-/\n\npublic section\n\nassert_not_exists Cardinal\n\nnoncomputable section\n\nopen Function Set Submodule\n\nuniverse u' u\n\nvariable {ι : Type u'} {ι' : Type*} {R : Type*} {K : Type*} {s : Set ι}\nvariable {M : Type*} {M' : Type*} {V : Type u}\n\nsection Semiring\n\n\nvariable {v : ι → M}\nvariable [Semiring R] [AddCommMonoid M] [AddCommMonoid M']\nvariable [Module R M] [Module R M']\nvariable (R) (v)\n\nvariable {R v}\n\n/-- A set of linearly independent vectors in a module `M` over a semiring `K` is also linearly\nindependent over a subring `R` of `K`.\n\nSee also `LinearIndependent.restrict_scalars'` for a version with more convenient typeclass\nassumptions.\n\nTODO : `LinearIndepOn` version. -/\ntheorem LinearIndependent.restrict_scalars [Semiring K] [SMulWithZero R K] [Module K M]\n    [IsScalarTower R K M] (hinj : Injective fun r : R ↦ r • (1 : K))\n    (li : LinearIndependent K v) : LinearIndependent R v := by\n  intro x y hxy\n  let f := fun r : R => r • (1 : K)\n  have := @li (x.mapRange f (by simp [f])) (y.mapRange f (by simp [f])) ?_\n  · ext i\n    exact hinj congr($this i)\n  simpa [Finsupp.linearCombination, f, Finsupp.sum_mapRange_index]\n\nvariable (R) in\ntheorem LinearIndependent.restrict_scalars' [Semiring K] [SMulWithZero R K] [Module K M]\n    [IsScalarTower R K M] [FaithfulSMul R K] [IsScalarTower R K K] {v : ι → M}\n    (li : LinearIndependent K v) : LinearIndependent R v :=\n  restrict_scalars ((faithfulSMul_iff_injective_smul_one R K).mp inferInstance) li\n\n/-- If `v` is an injective family of vectors such that `f ∘ v` is linearly independent, then `v`\n    spans a submodule disjoint from the kernel of `f`.\nTODO : `LinearIndepOn` version. -/\ntheorem Submodule.range_ker_disjoint {f : M →ₗ[R] M'}\n    (hv : LinearIndependent R (f ∘ v)) :\n    Disjoint (span R (range v)) (LinearMap.ker f) := by\n  rw [LinearIndependent, Finsupp.linearCombination_linear_comp] at hv\n  rw [disjoint_iff_inf_le, ← Set.image_univ, Finsupp.span_image_eq_map_linearCombination,\n    map_inf_eq_map_inf_comap, (LinearMap.ker_comp _ _).symm.trans\n      (LinearMap.ker_eq_bot_of_injective hv), inf_bot_eq, map_bot]\n\n/-- If `M / R` and `M' / R'` are modules, `i : R' → R` is a map, `j : M →+ M'` is a monoid map,\nsuch that they are both injective, and compatible with the scalar\nmultiplications on `M` and `M'`, then `j` sends linearly independent families of vectors to\nlinearly independent families of vectors. As a special case, taking `R = R'`\nit is `LinearIndependent.map_injOn`.\nTODO : `LinearIndepOn` version. -/\ntheorem LinearIndependent.map_of_injective_injectiveₛ {R' M' : Type*}\n    [Semiring R'] [AddCommMonoid M'] [Module R' M'] (hv : LinearIndependent R v)\n    (i : R' → R) (j : M →+ M') (hi : Injective i) (hj : Injective j)\n    (hc : ∀ (r : R') (m : M), j (i r • m) = r • j m) : LinearIndependent R' (j ∘ v) := by\n  rw [linearIndependent_iff'ₛ] at hv ⊢\n  intro S r₁ r₂ H s hs\n  simp_rw [comp_apply, ← hc, ← map_sum] at H\n  exact hi <| hv _ _ _ (hj H) s hs\n\n/-- If `M / R` and `M' / R'` are modules, `i : R → R'` is a surjective map,\nand `j : M →+ M'` is an injective monoid map, such that the scalar multiplications\non `M` and `M'` are compatible, then `j` sends linearly independent families\nof vectors to linearly independent families of vectors. As a special case, taking `R = R'`\nit is `LinearIndependent.map_injOn`.\nTODO : `LinearIndepOn` version. -/\ntheorem LinearIndependent.map_of_surjective_injectiveₛ {R' M' : Type*}\n    [Semiring R'] [AddCommMonoid M'] [Module R' M'] (hv : LinearIndependent R v)\n    (i : R → R') (j : M →+ M') (hi : Surjective i) (hj : Injective j)\n    (hc : ∀ (r : R) (m : M), j (r • m) = i r • j m) : LinearIndependent R' (j ∘ v) := by\n  obtain ⟨i', hi'⟩ := hi.hasRightInverse\n  refine hv.map_of_injective_injectiveₛ i' j (fun _ _ h ↦ ?_) hj fun r m ↦ ?_\n  · apply_fun i at h\n    rwa [hi', hi'] at h\n  rw [hc (i' r) m, hi']\n\n/-- If a linear map is injective on the span of a family of linearly independent vectors, then\nthe family stays linearly independent after composing with the linear map.\nSee `LinearIndependent.map` for the version with `Set.InjOn` replaced by `Disjoint`\nwhen working over a ring. -/\ntheorem LinearIndependent.map_injOn (hv : LinearIndependent R v) (f : M →ₗ[R] M')\n    (hf_inj : Set.InjOn f (span R (Set.range v))) : LinearIndependent R (f ∘ v) :=\n  (f.linearIndependent_iff_of_injOn hf_inj).mpr hv\n\ntheorem LinearIndepOn.map_injOn (hv : LinearIndepOn R v s) (f : M →ₗ[R] M')\n    (hf_inj : Set.InjOn f (span R (v '' s))) : LinearIndepOn R (f ∘ v) s :=\n  (f.linearIndepOn_iff_of_injOn hf_inj).mpr hv\n\ntheorem LinearIndepOn.comp_of_image {s : Set ι'} {f : ι' → ι} (h : LinearIndepOn R v (f '' s))\n    (hf : InjOn f s) : LinearIndepOn R (v ∘ f) s :=\n  LinearIndependent.comp h _ (Equiv.Set.imageOfInjOn _ _ hf).injective\n\ntheorem LinearIndepOn.image_of_comp (f : ι → ι') (g : ι' → M) (hs : LinearIndepOn R (g ∘ f) s) :\n    LinearIndepOn R g (f '' s) := by\n  nontriviality R\n  have : InjOn f s := injOn_iff_injective.2 hs.injective.of_comp\n  exact (linearIndependent_equiv' (Equiv.Set.imageOfInjOn f s this) rfl).1 hs\n\ntheorem LinearIndepOn.id_image (hs : LinearIndepOn R v s) : LinearIndepOn R id (v '' s) :=\n  LinearIndepOn.image_of_comp v id hs\n\ntheorem LinearIndepOn_iff_linearIndepOn_image_injOn [Nontrivial R] :\n    LinearIndepOn R v s ↔ LinearIndepOn R id (v '' s) ∧ InjOn v s :=\n  ⟨fun h ↦ ⟨h.id_image, h.injOn⟩, fun h ↦ (linearIndepOn_iff_image h.2).2 h.1⟩\n\ntheorem linearIndepOn_congr {w : ι → M} (h : EqOn v w s) :\n    LinearIndepOn R v s ↔ LinearIndepOn R w s := by\n  rw [LinearIndepOn, LinearIndepOn]\n  convert! Iff.rfl using 2\n  ext x\n  exact h.symm x.2\n\ntheorem LinearIndepOn.congr {w : ι → M} (hli : LinearIndepOn R v s) (h : EqOn v w s) :\n    LinearIndepOn R w s :=\n  (linearIndepOn_congr h).1 hli\n\ntheorem LinearIndependent.group_smul {G : Type*} [hG : Group G] [MulAction G R]\n    [SMul G M] [IsScalarTower G R M] [SMulCommClass G R M] {v : ι → M}\n    (hv : LinearIndependent R v) (w : ι → G) : LinearIndependent R (w • v) := by\n  rw [linearIndependent_iff''ₛ] at hv ⊢\n  intro s g₁ g₂ hgs hsum i\n  refine (Group.isUnit (w i)).smul_left_cancel.mp ?_\n  refine hv s (fun i ↦ w i • g₁ i) (fun i ↦ w i • g₂ i) (fun i hi ↦ ?_) ?_ i\n  · simp_rw [hgs i hi]\n  · simpa only [smul_assoc, smul_comm] using! hsum\n\n@[simp]\ntheorem LinearIndependent.group_smul_iff {G : Type*} [hG : Group G] [MulAction G R]\n    [MulAction G M] [IsScalarTower G R M] [SMulCommClass G R M] (v : ι → M) (w : ι → G) :\n    LinearIndependent R (w • v) ↔ LinearIndependent R v := by\n  refine ⟨fun h ↦ ?_, fun h ↦ h.group_smul w⟩\n  convert! h.group_smul (fun i ↦ (w i)⁻¹)\n  simp [funext_iff]\n\n-- This lemma cannot be proved with `LinearIndependent.group_smul` since the action of\n-- `Rˣ` on `R` is not commutative.\ntheorem LinearIndependent.units_smul {v : ι → M} (hv : LinearIndependent R v) (w : ι → Rˣ) :\n    LinearIndependent R (w • v) := by\n  rw [linearIndependent_iff''ₛ] at hv ⊢\n  intro s g₁ g₂ hgs hsum i\n  rw [← (w i).mul_left_inj]\n  refine hv s (fun i ↦ g₁ i • w i) (fun i ↦ g₂ i • w i) (fun i hi ↦ ?_) ?_ i\n  · simp_rw [hgs i hi]\n  · simpa only [smul_eq_mul, mul_smul, Pi.smul_apply'] using! hsum\n\n@[simp]\n\nTarget:\ntheorem LinearIndependent.units_smul_iff (v : ι → M) (w : ι → Rˣ) :\n    LinearIndependent R (w • v) ↔ LinearIndependent R v :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/LinearIndependent","family_id":"linearindependent","file_id":"mathlib/Mathlib/LinearAlgebra/LinearIndependent/Basic.lean","sample_id":"b38b14c96b38afbba7c3259cf4c4e7b4c145e1b2ac6c153a770feeac3a7076d5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f95ac83c92042b8eca33c773ee62f7012472f34ee0a3ebc546b077931f47c21d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"83bbcd0a12ed90d64932bbbf23b8435e8984a966305ad698b755df700c2893b8","source_sha256":"e6501306e0c166f31aa2d54b2053e75c750b45815aae03787ae7a4d5aa54d0d7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction hs using Finset.Nonempty.cons_induction with\n  | singleton i => exact ⟨i, by simp⟩\n  | @cons i s hi hs ih =>\n    obtain ⟨j, hj⟩ := ih\n    by_cases hji : f j ≤ f i\n    · refine ⟨i, mem_cons_self .., ?_⟩\n      simp only [mem_cons, forall_eq_or_imp, imp_self, true_and]\n      exact fun k hk hik ↦ _root_.trans (hj.2 hk <| _root_.trans hji hik) hji\n    · exact ⟨j, mem_cons_of_mem hj.1, by simpa [hji] using hj.2⟩","hard_negative":false,"metrics":{"chosen_tokens":112,"rejected_tokens":3,"token_jaccard":0.033898,"token_length_ratio":0.026786},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"d011c227b390a33e811b59dcea5a67f95c597792296f1995078b0a288432c0ab","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Basic\npublic import Mathlib.Order.Minimal\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2025 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Finite preorders and finite sets in a preorder\n\nThis file shows that non-empty finite sets in a preorder have minimal/maximal elements, and\ncontrapositively that non-empty sets without minimal or maximal elements are infinite.\n-/\n\npublic section\n\nvariable {ι α β : Type*}\n\nnamespace Finset\nsection IsTrans\nvariable [LE α] [IsTrans α LE.le] {s : Finset α} {a : α}\n\nTarget:\nlemma exists_maximalFor (f : ι → α) (s : Finset ι) (hs : s.Nonempty) :\n    ∃ i, MaximalFor (· ∈ s) f i :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Preorder","family_id":"exists_maximalfor","file_id":"mathlib/Mathlib/Order/Preorder/Finite.lean","sample_id":"83bbcd0a12ed90d64932bbbf23b8435e8984a966305ad698b755df700c2893b8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b54a12dda5879cff4ee4a1d4e164d0430717d0b1ed9dc7072b52b58f4f7965cc","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4d5c5b1473328f3a599163c0715d3d92ebde8fae4e6d7f49111909f106081cda","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cc46ae0cd6fbf8b5c482692a262adbee52fa600011e9053140abc53478aa5ddd","source_sha256":"3cc5f24e166336d305cafd0a6e88d1acf7bd4033e91b083c3ca5b2a6bcc3c9bc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine eq_bot_iff.2 (le_trans jacobson_bot_polynomial_le_sInf_map_maximal ?_)\n  refine fun f hf => (Submodule.mem_bot R[X]).2 <| Polynomial.ext fun n =>\n    Trans.trans (?_ : coeff f n = 0) (coeff_zero n).symm\n  suffices f.coeff n ∈ Ideal.jacobson ⊥ by rwa [h, Submodule.mem_bot] at this\n  exact mem_sInf.2 fun j hj => (mem_map_C_iff.1 ((mem_sInf.1 hf) ⟨j, ⟨hj.2, rfl⟩⟩)) n","hard_negative":true,"metrics":{"chosen_tokens":108,"rejected_tokens":5,"token_jaccard":0.054545,"token_length_ratio":0.046296},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"d01ab5ddd99060fda5739d8326110c4d4208c6c2bcf1064f3881d797d645d384","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Jacobson.Ideal\npublic import Mathlib.RingTheory.Polynomial.Quotient\n\nNamespace:\nIdeal\n\nLocal context:\n/-\nCopyright (c) 2020 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\n/-!\n# Jacobson radical of polynomial ring\n\n-/\n\npublic section\n\nnamespace Ideal\n\nsection Polynomial\n\nopen Polynomial\n\nvariable {R : Type*} [CommRing R]\n\ntheorem jacobson_bot_polynomial_le_sInf_map_maximal :\n    jacobson (⊥ : Ideal R[X]) ≤ sInf (map (C : R →+* R[X]) '' { J : Ideal R | J.IsMaximal }) := by\n  refine le_sInf fun J => exists_imp.2 fun j hj => ?_\n  haveI : j.IsMaximal := hj.1\n  refine Trans.trans (jacobson_mono bot_le) (le_of_eq ?_ : J.jacobson ≤ J)\n  suffices t : (⊥ : Ideal (Polynomial (R ⧸ j))).jacobson = ⊥ by\n    rw [← hj.2, jacobson_eq_iff_jacobson_quotient_eq_bot]\n    replace t := congr_arg (map (polynomialQuotientEquivQuotientPolynomial j).toRingHom) t\n    rwa [map_jacobson_of_bijective _, map_bot] at t\n    exact RingEquiv.bijective (polynomialQuotientEquivQuotientPolynomial j)\n  refine eq_bot_iff.2 fun f hf => ?_\n  have r1 : (X : (R ⧸ j)[X]) ≠ 0 := ne_of_apply_ne (coeff · 1) <| by simp\n  simpa [r1] using eq_C_of_degree_eq_zero (degree_eq_zero_of_isUnit ((mem_jacobson_bot.1 hf) X))\n\nTarget:\ntheorem jacobson_bot_polynomial_of_jacobson_bot (h : jacobson (⊥ : Ideal R) = ⊥) :\n    jacobson (⊥ : Ideal R[X]) = ⊥ :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_cc46ae0cd6fb","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d71932f558fa7ea1f725c7fb319fea207849c002ae7b2bd316384f6527b37b3e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Jacobson","family_id":"jacobson_bot_polynomial_of_jacobson_bot","file_id":"mathlib/Mathlib/RingTheory/Jacobson/Polynomial.lean","sample_id":"cc46ae0cd6fbf8b5c482692a262adbee52fa600011e9053140abc53478aa5ddd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"76fdca8112ad5b228e441ce2f0fc6042386b4c4cdeafde082a85932a16c29411","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"77ec50a1d8bd4d1d6217637dbd7b2d32354cdbf181fa20f2a6595526bf717d39","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8ceba723b8cfca101595c85d0f256daa590d35c274c45371de9577614b0b3c30","source_sha256":"ebea26ad83d186da19f8d5aec5a4532b5e10776580381e96c50d87a2ed0ca5b4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [← toDual_covBy_toDual_iff (α := α)] at hfin ⊢\n  exact exists_covby_infinite_Ici_of_infinite_Ici (α := αᵒᵈ) ha hfin","hard_negative":true,"metrics":{"chosen_tokens":25,"rejected_tokens":2,"token_jaccard":0.052632,"token_length_ratio":0.08},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"d0900b3fbb59b835695f4a3de4eca213b7dab8d35e453a5858af0319e610148f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Lattice\npublic import Mathlib.Order.Atoms\npublic import Mathlib.Order.Interval.Finset.Defs\npublic import Mathlib.Order.Preorder.Finite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Atoms, Coatoms, Simple Lattices, and Finiteness\n\nThis module contains some results on atoms and simple lattices in the finite context.\n\n## Main results\n* `Finite.to_isAtomic`, `Finite.to_isCoatomic`: Finite partial orders with bottom resp. top\n  are atomic resp. coatomic.\n\n-/\n\npublic section\n\n\nvariable {α β : Type*}\n\nnamespace IsSimpleOrder\n\nvariable [LE α] [BoundedOrder α] [IsSimpleOrder α]\n\nsection DecidableEq\n\n/-- It is important that `IsSimpleOrder` is the last type-class argument of this instance,\nso that type-class inference fails quickly if it doesn't apply.\n\nNote that as of 2025-08-13, this is false. Could someone investigate? -/\nscoped instance (priority := 200) [DecidableEq α] : Fintype α :=\n  Fintype.ofEquiv Bool equivBool.symm\n\nend DecidableEq\n\nscoped instance (priority := 200) : Finite α := by classical infer_instance\n\nend IsSimpleOrder\n\nnamespace Fintype\n\nnamespace IsSimpleOrder\n\nopen scoped _root_.IsSimpleOrder\n\nvariable [LE α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α]\n\ntheorem univ : (Finset.univ : Finset α) = {⊤, ⊥} := by\n  ext\n  simpa using (eq_bot_or_eq_top _).symm\n\ntheorem card : Fintype.card α = 2 :=\n  (Fintype.ofEquiv_card _).trans Fintype.card_bool\n\nend IsSimpleOrder\n\nend Fintype\n\nnamespace Bool\n\ninstance : IsSimpleOrder Bool :=\n  ⟨fun a => by\n    rw [← Finset.mem_singleton, Or.comm, ← Finset.mem_insert, top_eq_true, bot_eq_false, ←\n      Fintype.univ_bool]\n    apply Finset.mem_univ⟩\n\nend Bool\n\nsection Fintype\n\nopen Finset\n\n-- see Note [lower instance priority]\ninstance (priority := 100) Finite.to_isCoatomic [PartialOrder α] [OrderTop α] [Finite α] :\n    IsCoatomic α :=\n  IsStronglyCoatomic.toIsCoatomic α\n\n-- see Note [lower instance priority]\ninstance (priority := 100) Finite.to_isAtomic [PartialOrder α] [OrderBot α] [Finite α] :\n    IsAtomic α :=\n  isCoatomic_dual_iff_isAtomic.mp Finite.to_isCoatomic\n\nend Fintype\n\nsection LocallyFinite\n\nvariable [Preorder α] [LocallyFiniteOrder α]\n\ninstance : IsStronglyAtomic α where\n  exists_covBy_le_of_lt a b hab := by\n    obtain ⟨x, hx, hxmin⟩ := (LocallyFiniteOrder.finsetIoc a b).exists_minimal\n      ⟨b, by simpa [LocallyFiniteOrder.finset_mem_Ioc]⟩\n    simp only [LocallyFiniteOrder.finset_mem_Ioc] at hx hxmin\n    exact ⟨x, ⟨hx.1, fun c hac hcx ↦ hcx.not_ge <| hxmin ⟨hac, hcx.le.trans hx.2⟩ hcx.le⟩, hx.2⟩\n\ninstance : IsStronglyCoatomic α := by\n  rw [← isStronglyAtomic_dual_iff_is_stronglyCoatomic]; infer_instance\n\nend LocallyFinite\n\nsection IsStronglyAtomic\n\nvariable [PartialOrder α] {a : α}\n\ntheorem exists_covby_infinite_Ici_of_infinite_Ici [IsStronglyAtomic α]\n    (ha : (Set.Ici a).Infinite) (hfin : {x | a ⋖ x}.Finite) :\n    ∃ b, a ⋖ b ∧ (Set.Ici b).Infinite := by\n  by_contra! h\n  refine ((hfin.biUnion (t := Set.Ici) (by simpa using h)).subset (fun b hb ↦ ?_)).not_infinite\n    (ha.sdiff (Set.finite_singleton a))\n  obtain ⟨x, hax, hxb⟩ := ((show a ≤ b from hb.1).lt_of_ne (Ne.symm hb.2)).exists_covby_le\n  exact Set.mem_biUnion hax hxb\n\nTarget:\ntheorem exists_covby_infinite_Iic_of_infinite_Iic [IsStronglyCoatomic α]\n    (ha : (Set.Iic a).Infinite) (hfin : {x | x ⋖ a}.Finite) :\n    ∃ b, b ⋖ a ∧ (Set.Iic b).Infinite :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_8ceba723b8cf","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"502959d4e5e30853a8d25f720ed13e8bf00f0a4de75bdd5ac71520f4cdce2aa0","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Atoms","family_id":"exists_covby_infinite_iic_of_infinite_iic","file_id":"mathlib/Mathlib/Order/Atoms/Finite.lean","sample_id":"8ceba723b8cfca101595c85d0f256daa590d35c274c45371de9577614b0b3c30"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d7e6aa18e95f9b73927d19d13b21d8b699befd36d25410b98b9a340426e1233b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"edb8b3e01bb6880ee0bbdea1e1488404f9654431b8c621f67e732f4c167e5480","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2f91e46e3433b86a1e3699a32e11802a7183cc285fe3179f472be8008081dcf8","source_sha256":"76ff543811de607fc5619ca4ca6c9c5878d83cf85372b02e848f8ebf54663e8c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain rfl | hn := eq_or_ne n 0\n  · simpa [ssubset_iff_subset_not_subset, hX₁, -Finset.subset_singleton_iff]\n      using! hX.not_subset_singleton\n  refine (pow_subset_pow_right hX₁ <| n.le_add_right _).ssubset_of_ne ?_\n  contrapose hXclosure with hXn\n  rw [← closure_pow (mod_cast hX₁) hn]\n  wlog hn₁ : n = 1\n  · simp +contextual only [pow_one] at this\n    replace hXn d : X ^ (n + d) = X ^ n := by\n      induction d with\n      | zero => rw [add_zero]\n      | succ d hd =>\n        rw [pow_add, pow_one] at hXn\n        rw [← add_assoc, pow_add, pow_one, hd, ← hXn]\n    exact mod_cast this (one_mem_pow hX₁) (hX.pow hn) one_ne_zero\n      (by simp [hXn, ← pow_mul, mul_two]) (by simp)\n  subst hn₁\n  simp only [ne_eq, one_ne_zero, not_false_eq_true, Nat.reduceAdd, pow_one] at *\n  let Xgp : Subgroup G :=\n  { carrier := X\n    mul_mem' := fun {x y} hx hy ↦ by\n      norm_cast at *\n      simpa [← hXn, ← sq] using! mul_mem_mul hx hy\n    one_mem' := hX₁\n    inv_mem' := fun {x} hx ↦ by\n      norm_cast at *\n      have : x • X ⊆ X := by\n        simpa [← hXn, add_assoc, ← sq] using! smul_finset_subset_mul (t := X) hx\n      have : x • X = X := eq_of_subset_of_card_le this (card_smul_finset ..).ge\n      rw [← eq_inv_smul_iff] at this\n      rw [this]\n      simpa [mem_inv_smul_finset_iff] }\n  exact subset_closure.antisymm <| (closure_le Xgp).2 subset_rfl","hard_negative":false,"metrics":{"chosen_tokens":307,"rejected_tokens":311,"token_jaccard":0.973451,"token_length_ratio":1.013029},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"d0bd2e53ff72f35796e504d06461a6c94f36297019b8d5c2709badaec0ef406c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Pointwise.Finset\npublic import Mathlib.Algebra.Group.Subgroup.Pointwise\npublic import Mathlib.Data.Nat.SuccPred\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2024 Yaël Dillies, Patrick Luo, Eric Rodriguez. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Patrick Luo, Eric Rodriguez\n-/\n/-!\n# Linear lower bound on the growth of a generating set\n\nThis file proves that the growth of a set generating an infinite group is at least linear.\n-/\n\npublic section\n\nopen Subgroup\nopen scoped Pointwise\n\nnamespace Finset\nvariable {G : Type*} [Group G] [DecidableEq G] {X : Finset G} {n : ℕ}\n\n@[to_additive]\n\nTarget:\nlemma pow_ssubset_pow_succ_of_pow_ne_closure (hX₁ : (1 : G) ∈ X) (hX : X.Nontrivial)\n    (hXclosure : (X ^ n : Set G) ≠ closure (X : Set G)) : X ^ n ⊂ X ^ (n + 1) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  obtain rfl | hn := eq_or_ne n 0\n  · simpa [ssubset_iff_subset_not_subset, hX₁, -Finset.subset_singleton_iff]\n      using! hX.not_subset_singleton\n  refine (pow_subset_pow_right hX₁ <| n.le_add_right _).ssubset_of_ne ?_\n  contrapose hXclosure with hXn\n  rw [← closure_pow (mod_cast hX₁) hn]\n  wlog hn₁ : n = 1\n  · simp +contextual only [pow_one] at this\n    replace hXn d : X ^ (n + d) = X ^ n := by\n      induction d with\n      | zero => rw [add_zero]\n      | succ d hd =>\n        rw [pow_add, pow_one] at hXn\n        rw [← add_assoc, pow_add, pow_one, hd, ← hXn]\n    exact mod_cast this (one_mem_pow hX₁) (hX.pow hn) one_ne_zero\n      (by simp [hXn, ← pow_mul, mul_two]) (by simp)\n  subst hn₁\n  simp only [ne_eq, one_ne_zero, not_false_eq_true, Nat.reduceAdd, pow_one] at *\n  let Xgp : Subgroup G :=\n  { carrier := X\n    mul_mem' := fun {x y} hx hy ↦ by\n      norm_cast at *\n      simpa [← hXn, ← sq] using! mul_mem_mul hx hy\n    one_mem' := hX₁\n    inv_mem' := fun {x} hx ↦ by\n      norm_cast at *\n      have : x • X ⊆ X := by\n        simpa [← hXn, add_assoc, ← sq] using! smul_finset_subset_mul (t := X) hx\n      have : x • X = X := eq_of_subset_of_card_le this (card_smul_finset ..).ge\n      rw [← eq_inv_smul_iff] at this\n      rw [this]\n      simpa [mem_inv_smul_finset_iff] }\n  exact subset_closure.antisymm <| (closure_le Xgp).2 subset_rfl","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/Group","family_id":"pow_ssubset_pow_succ_of_pow_ne_closure","file_id":"mathlib/Mathlib/Geometry/Group/Growth/LinearLowerBound.lean","sample_id":"2f91e46e3433b86a1e3699a32e11802a7183cc285fe3179f472be8008081dcf8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ad8808e6ef06989ebffa9f2a94c9e16fc819a971dc924b479659fd09f73634ab","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d3e808dd9addf2387a7c10bc55a218ee8c461d0d226d0c690471da0895ea82f7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"92326efcf9c32173b0a7545e06842af3e1db8144b2ac5b58b5ccfdd3ce10f509","source_sha256":"016de02d162ccc90d858e39e45ebc4da0ae14e5643087f6444b25344546eb84f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [Ideal.eq_top_iff_one, ← Subtype.range_val (s := s), ← Set.range_comp,\n    Finsupp.mem_ideal_span_range_iff_exists_finsupp] at hs' ⊢\n  choose f hf using hs'\n  have : Fintype s := hs.fintype\n  refine ⟨Finsupp.equivFunOnFinite.symm fun i x ↦ f x i, ?_⟩\n  ext i\n  simpa [Finsupp.sum_fintype] using hf i","hard_negative":false,"metrics":{"chosen_tokens":72,"rejected_tokens":79,"token_jaccard":0.958333,"token_length_ratio":1.097222},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"d0c009f86406e39f1e5fdab5b942aa26a244155b09535cbd10a7bd2b06b080f0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.Immersion\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2024 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# `Π Rᵢ`-Points of Schemes\n\nWe show that the canonical map `X(Π Rᵢ) ⟶ Π X(Rᵢ)` (`AlgebraicGeometry.pointsPi`)\nis injective and surjective under various assumptions.\n\n-/\n\n@[expose] public section\n\nopen CategoryTheory Limits PrimeSpectrum\n\nnamespace AlgebraicGeometry\n\nuniverse u v\n\nvariable {ι : Type u} (R : ι → CommRingCat.{u})\n\nTarget:\nlemma Ideal.span_eq_top_of_span_image_evalRingHom\n    {ι} {R : ι → Type*} [∀ i, CommRing (R i)] (s : Set (Π i, R i))\n    (hs : s.Finite) (hs' : ∀ i, Ideal.span (Pi.evalRingHom (R ·) i '' s) = ⊤) :\n    Ideal.span s = ⊤ :=\n\nProof body:\n","rejected":"```lean\nby\n  simp only [Ideal.eq_top_iff_one, ← Subtype.range_val (s := s), ← Set.range_comp,\n    Finsupp.mem_ideal_span_range_iff_exists_finsupp] at hs' ⊢\n  choose f hf using hs'\n  have : Fintype s := hs.fintype\n  refine ⟨Finsupp.equivFunOnFinite.symm fun i x ↦ f x i, ?_⟩\n  ext i\n  simpa [Finsupp.sum_fintype] using hf i\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry","family_id":"ideal","file_id":"mathlib/Mathlib/AlgebraicGeometry/PointsPi.lean","sample_id":"92326efcf9c32173b0a7545e06842af3e1db8144b2ac5b58b5ccfdd3ce10f509"},"split":"train"}
{"binding":{"chosen_proof_sha256":"861cfb34c8727a0d4d437e7097497ea9d3f14804aa184568a47ee591868cf5bf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"56d9c52271c4e055cf68149536f81a5fb94327a8ac0ac8e10a3169f8ff8090ab","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"87030b969f81b483b5bdfa5c63449397f06437d95959ae3498323790baf68ce1","source_sha256":"cd917cebf3f69c14dd2596343577ddfa07783b1b73bd4e0e91a20b1a1be30f59","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Equiv.eq_symm_comp (e := Quiver.symmetrifyStar (φ.obj u))]\n  ext ⟨v, f | g⟩ <;>\n    -- Porting note (https://github.com/leanprover-community/mathlib4/issues/10745): was `simp [Quiver.symmetrifyStar]`\n    simp only [Quiver.symmetrifyStar, Function.comp_apply] <;>\n    erw [Equiv.sigmaSumDistrib_apply, Equiv.sigmaSumDistrib_apply] <;>\n    simp","hard_negative":false,"metrics":{"chosen_tokens":96,"rejected_tokens":101,"token_jaccard":0.928571,"token_length_ratio":1.052083},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"d0d544deba5fd970886613891ca4e0505917097b86d24cb21751d3588db98f3d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Quiver.Cast\npublic import Mathlib.Combinatorics.Quiver.Symmetric\npublic import Mathlib.Data.Sigma.Basic\npublic import Mathlib.Data.Sum.Basic\npublic import Mathlib.Logic.Equiv.Sum\npublic import Mathlib.Tactic.Common\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Labelle, Rémi Bottinelli\n-/\n/-!\n# Covering\n\nThis file defines coverings of quivers as prefunctors that are bijective on the\nso-called stars and costars at each vertex of the domain.\n\n## Main definitions\n\n* `Quiver.Star u` is the type of all arrows with source `u`;\n* `Quiver.Costar u` is the type of all arrows with target `u`;\n* `Prefunctor.star φ u` is the obvious function `star u → star (φ.obj u)`;\n* `Prefunctor.costar φ u` is the obvious function `costar u → costar (φ.obj u)`;\n* `Prefunctor.IsCovering φ` means that `φ.star u` and `φ.costar u` are bijections for all `u`;\n* `Quiver.PathStar u` is the type of all paths with source `u`;\n* `Prefunctor.pathStar u` is the obvious function `PathStar u → PathStar (φ.obj u)`.\n\n## Main statements\n\n* `Prefunctor.IsCovering.pathStar_bijective` states that if `φ` is a covering,\n  then `φ.pathStar u` is a bijection for all `u`.\n  In other words, every path in the codomain of `φ` lifts uniquely to its domain.\n\n## TODO\n\nClean up the namespaces by renaming `Prefunctor` to `Quiver.Prefunctor`.\n\n## Tags\n\nCover, covering, quiver, path, lift\n-/\n\n@[expose] public section\n\n\nopen Function Quiver\n\nuniverse u v w\n\nvariable {U : Type _} [Quiver.{u} U] {V : Type _} [Quiver.{v} V] (φ : U ⥤q V) {W : Type _}\n  [Quiver.{w} W] (ψ : V ⥤q W)\n\n/-- The `Quiver.Star` at a vertex is the collection of arrows whose source is the vertex.\nThe type `Quiver.Star u` is defined to be `Σ (v : U), (u ⟶ v)`. -/\nabbrev Quiver.Star (u : U) :=\n  Σ v : U, u ⟶ v\n\n/-- Constructor for `Quiver.Star`. Defined to be `Sigma.mk`. -/\nprotected abbrev Quiver.Star.mk {u v : U} (f : u ⟶ v) : Quiver.Star u :=\n  ⟨_, f⟩\n\n/-- The `Quiver.Costar` at a vertex is the collection of arrows whose target is the vertex.\nThe type `Quiver.Costar v` is defined to be `Σ (u : U), (u ⟶ v)`. -/\nabbrev Quiver.Costar (v : U) :=\n  Σ u : U, u ⟶ v\n\n/-- Constructor for `Quiver.Costar`. Defined to be `Sigma.mk`. -/\nprotected abbrev Quiver.Costar.mk {u v : U} (f : u ⟶ v) : Quiver.Costar v :=\n  ⟨_, f⟩\n\n/-- A prefunctor induces a map of `Quiver.Star` at every vertex. -/\n@[simps]\ndef Prefunctor.star (u : U) : Quiver.Star u → Quiver.Star (φ.obj u) := fun F =>\n  Quiver.Star.mk (φ.map F.2)\n\n/-- A prefunctor induces a map of `Quiver.Costar` at every vertex. -/\n@[simps]\ndef Prefunctor.costar (u : U) : Quiver.Costar u → Quiver.Costar (φ.obj u) := fun F =>\n  Quiver.Costar.mk (φ.map F.2)\n\n@[simp]\ntheorem Prefunctor.star_apply {u v : U} (e : u ⟶ v) :\n    φ.star u (Quiver.Star.mk e) = Quiver.Star.mk (φ.map e) :=\n  rfl\n\n@[simp]\ntheorem Prefunctor.costar_apply {u v : U} (e : u ⟶ v) :\n    φ.costar v (Quiver.Costar.mk e) = Quiver.Costar.mk (φ.map e) :=\n  rfl\n\ntheorem Prefunctor.star_comp (u : U) : (φ ⋙q ψ).star u = ψ.star (φ.obj u) ∘ φ.star u :=\n  rfl\n\ntheorem Prefunctor.costar_comp (u : U) : (φ ⋙q ψ).costar u = ψ.costar (φ.obj u) ∘ φ.costar u :=\n  rfl\n\n/-- A prefunctor is a covering of quivers if it defines bijections on all stars and costars. -/\nprotected structure Prefunctor.IsCovering : Prop where\n  star_bijective : ∀ u, Bijective (φ.star u)\n  costar_bijective : ∀ u, Bijective (φ.costar u)\n\n@[simp]\ntheorem Prefunctor.IsCovering.map_injective (hφ : φ.IsCovering) {u v : U} :\n    Injective fun f : u ⟶ v => φ.map f := by\n  rintro f g he\n  have : φ.star u (Quiver.Star.mk f) = φ.star u (Quiver.Star.mk g) := by simpa using he\n  simpa using (hφ.star_bijective u).left this\n\ntheorem Prefunctor.IsCovering.comp (hφ : φ.IsCovering) (hψ : ψ.IsCovering) : (φ ⋙q ψ).IsCovering :=\n  ⟨fun _ => (hψ.star_bijective _).comp (hφ.star_bijective _),\n   fun _ => (hψ.costar_bijective _).comp (hφ.costar_bijective _)⟩\n\ntheorem Prefunctor.IsCovering.of_comp_right (hψ : ψ.IsCovering) (hφψ : (φ ⋙q ψ).IsCovering) :\n    φ.IsCovering :=\n  ⟨fun _ => (Bijective.of_comp_iff' (hψ.star_bijective _) _).mp (hφψ.star_bijective _),\n   fun _ => (Bijective.of_comp_iff' (hψ.costar_bijective _) _).mp (hφψ.costar_bijective _)⟩\n\ntheorem Prefunctor.IsCovering.of_comp_left (hφ : φ.IsCovering) (hφψ : (φ ⋙q ψ).IsCovering)\n    (φsur : Surjective φ.obj) : ψ.IsCovering := by\n  refine ⟨fun v => ?_, fun v => ?_⟩ <;> obtain ⟨u, rfl⟩ := φsur v\n  exacts [(Bijective.of_comp_iff _ (hφ.star_bijective u)).mp (hφψ.star_bijective u),\n    (Bijective.of_comp_iff _ (hφ.costar_bijective u)).mp (hφψ.costar_bijective u)]\n\n/-- The star of the symmetrification of a quiver at a vertex `u` is equivalent to the sum of the\nstar and the costar at `u` in the original quiver. -/\ndef Quiver.symmetrifyStar (u : U) :\n    Quiver.Star (Symmetrify.of.obj u) ≃ Quiver.Star u ⊕ Quiver.Costar u :=\n  Equiv.sigmaSumDistrib _ _\n\n/-- The costar of the symmetrification of a quiver at a vertex `u` is equivalent to the sum of the\ncostar and the star at `u` in the original quiver. -/\ndef Quiver.symmetrifyCostar (u : U) :\n    Quiver.Costar (Symmetrify.of.obj u) ≃ Quiver.Costar u ⊕ Quiver.Star u :=\n  Equiv.sigmaSumDistrib _ _\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem Prefunctor.symmetrifyStar (u : U) :\n    φ.symmetrify.star u =\n      (Quiver.symmetrifyStar _).symm ∘ Sum.map (φ.star u) (φ.costar u) ∘\n        Quiver.symmetrifyStar u :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  rw [Equiv.eq_symm_comp (e := Quiver.symmetrifyStar (φ.obj u))]\n  ext ⟨v, f | g⟩ <;>\n    -- Porting note (https://github.com/leanprover-community/mathlib4/issues/10745): was `simp [Quiver.symmetrifyStar]`\n    simp only [Quiver.symmetrifyStar, Function.comp_apply] <;>\n    erw [Equiv.sigmaSumDistrib_apply, Equiv.sigmaSumDistrib_apply] <;>\n    simp","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Quiver","family_id":"prefunctor","file_id":"mathlib/Mathlib/Combinatorics/Quiver/Covering.lean","sample_id":"87030b969f81b483b5bdfa5c63449397f06437d95959ae3498323790baf68ce1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"332fb89312684e7786baadfdf197587e5cdba892591555a524aedd43ed4113c6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"83d409c2d59eb7e379de832485f191b9f404335b11deaf98d29f5102389d2dda","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e31fb78612321e73c41c554f6e35840ea8a109f4f7b6f9fddcff3cf0f701ebac","source_sha256":"810b2eb230c1a433d88acadf7472080648a9bb39145bc2d8989cf9f5cfc8a13b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction n with\n  | zero =>\n    simp\n  | succ n IH =>\n    simp_rw [iteratedLineDerivOp_succ_left, IH, lineDerivOp_add]","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":3,"token_jaccard":0.055556,"token_length_ratio":0.142857},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"d0de58649d2aad3af2998f087a2f18f59d3026a4239f3a82d55d83ed1a32e255","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Equiv.Defs\npublic import Mathlib.Data.Fin.Tuple.Basic\npublic import Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic\npublic import Mathlib.Analysis.InnerProductSpace.CanonicalTensor\n\nNamespace:\nLineDeriv\n\nLocal context:\n/-\nCopyright (c) 2025 Moritz Doll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Moritz Doll\n-/\n/-! # Type classes for derivatives and the Laplacian\n\nIn this file we define notation type classes for line derivatives, also known as partial\nderivatives, and for the Laplacian.\n\nMoreover, we provide type-classes that encode the linear structure.\nWe also define the iterated line derivative and prove elementary properties.\nWe define a Laplacian based on the sum of second derivatives formula and prove that the Laplacian\nthus defined is independent of the choice of basis.\n\nCurrently, this type class is only used by Schwartz functions. Future uses include derivatives on\ntest functions, distributions, tempered distributions, and Sobolev spaces (and other generalized\nfunction spaces).\n-/\n\n@[expose] public noncomputable section\n\nuniverse u' u v w\n\nvariable {ι ι' 𝕜 R V E F V₁ V₂ V₃ : Type*}\n\n/-! ## Line derivative -/\n\nopen Fin\n\n/--\nThe notation typeclass for the line derivative.\n-/\nclass LineDeriv (V : Type u) (E : Type v) (F : outParam (Type w)) where\n  /-- `∂_{v} f` is the line derivative of `f` in direction `v`. The meaning of this notation is\n  type-dependent. -/\n  lineDerivOp : V → E → F\n\nnamespace LineDeriv\n\n@[inherit_doc] scoped notation \"∂_{\" v \"}\" => LineDeriv.lineDerivOp v\n\nvariable {V E : Type*} [LineDeriv V E E]\n\n/-- `∂^{m} f` is the iterated line derivative of `f`, where `m` is a finite number of (different)\ndirections. -/\ndef iteratedLineDerivOp {n : ℕ} : (Fin n → V) → E → E :=\n  Nat.recOn n (fun _ ↦ id) (fun _ rec y ↦ LineDeriv.lineDerivOp (y 0) ∘ rec (tail y))\n\n@[inherit_doc] scoped notation \"∂^{\" v \"}\" => LineDeriv.iteratedLineDerivOp v\n\n@[simp]\ntheorem iteratedLineDerivOp_fin_zero (m : Fin 0 → V) (f : E) : ∂^{m} f = f :=\n  rfl\n\n@[simp]\ntheorem iteratedLineDerivOp_one (m : Fin 1 → V) (f : E) : ∂^{m} f = ∂_{m 0} f :=\n  rfl\n\ntheorem iteratedLineDerivOp_succ_left {n : ℕ} (m : Fin (n + 1) → V) (f : E) :\n    ∂^{m} f = ∂_{m 0} (∂^{tail m} f) :=\n  rfl\n\ntheorem iteratedLineDerivOp_succ_right {n : ℕ} (m : Fin (n + 1) → V) (f : E) :\n    ∂^{m} f = ∂^{init m} (∂_{m (last n)} f) := by\n  induction n with\n  | zero => rfl\n  -- The proof is `∂^{n + 2} = ∂ ∂^{n + 1} = ∂ ∂^n ∂ = ∂^{n+1} ∂`\n  | succ n IH =>\n    have hmzero : init m 0 = m 0 := by simp only [init_def, castSucc_zero]\n    have hmtail : tail m (last n) = m (last n.succ) := by\n      simp only [tail_def, succ_last]\n    calc\n      _ = ∂_{m 0} (∂^{tail m} f) := iteratedLineDerivOp_succ_left _ _\n      _ = ∂_{m 0} (∂^{init <| tail m} (∂_{tail m <| last n} f)) := by\n        congr 1\n        exact IH _\n      _ = _ := by\n        rw [hmtail, iteratedLineDerivOp_succ_left, hmzero, tail_init_eq_init_tail]\n\n@[simp]\ntheorem iteratedLineDerivOp_const_eq_iter_lineDerivOp (n : ℕ) (y : V) (f : E) :\n    ∂^{fun (_ : Fin n) ↦ y} f = ∂_{y}^[n] f := by\n  induction n with\n  | zero => rfl\n  | succ n IH =>\n    rw [iteratedLineDerivOp_succ_left, Function.iterate_succ_apply']\n    congr\n\nend LineDeriv\n\nopen LineDeriv\n\n/--\nThe line derivative is additive, `∂_{v} (x + y) = ∂_{v} x + ∂_{v} y` for all `x y : E`\nand `∂_{v + w} x = ∂_{v} x + ∂_{w} y` for all `v w : V`.\n\nNote that `lineDeriv` on functions is not additive.\n-/\nclass LineDerivAdd (V : Type u) (E : Type v) (F : outParam (Type w))\n    [AddCommGroup V] [AddCommGroup E] [AddCommGroup F] [LineDeriv V E F] where\n  lineDerivOp_add (v : V) (x y : E) : ∂_{v} (x + y) = ∂_{v} x + ∂_{v} y\n  lineDerivOp_left_add (v w : V) (x : E) : ∂_{v + w} x = ∂_{v} x + ∂_{w} x\n\n/--\nThe line derivative commutes with scalar multiplication, `∂_{v} (r • x) = r • ∂_{v} x` for all\n`r : R` and `x : E`.\n-/\nclass LineDerivSMul (R : Type*) (V : Type u) (E : Type v) (F : outParam (Type w))\n    [SMul R E] [SMul R F] [LineDeriv V E F] where\n  lineDerivOp_smul (v : V) (r : R) (x : E) : ∂_{v} (r • x) = r • ∂_{v} x\n\n/--\nThe line derivative commutes with scalar multiplication, `∂_{r • v} x = r • ∂_{v} x` for all\n`r : R` and `v : V`.\n-/\nclass LineDerivLeftSMul (R : Type*) (V : Type u) (E : Type v) (F : outParam (Type w))\n    [SMul R V] [SMul R F] [LineDeriv V E F] where\n  lineDerivOp_left_smul (r : R) (v : V) (x : E) : ∂_{r • v} x = r • ∂_{v} x\n\n/--\nThe line derivative is continuous.\n-/\nclass ContinuousLineDeriv (V : Type u) (E : Type v) (F : outParam (Type w))\n    [TopologicalSpace E] [TopologicalSpace F] [LineDeriv V E F] where\n  continuous_lineDerivOp (v : V) : Continuous (∂_{v} : E → F)\n\nattribute [fun_prop] ContinuousLineDeriv.continuous_lineDerivOp\n\nnamespace LineDeriv\n\nexport LineDerivAdd (lineDerivOp_add)\nexport LineDerivAdd (lineDerivOp_left_add)\nexport LineDerivSMul (lineDerivOp_smul)\nexport LineDerivLeftSMul (lineDerivOp_left_smul)\nexport ContinuousLineDeriv (continuous_lineDerivOp)\n\nsection lineDerivOp\n\nvariable [AddCommGroup V] [AddCommGroup E] [AddCommGroup F] [LineDeriv V E F] [LineDerivAdd V E F]\n\n@[simp]\ntheorem lineDerivOp_zero (v : V) : ∂_{v} (0 : E) = 0 :=\n  map_zero (AddMonoidHom.mk' ∂_{v} (lineDerivOp_add v))\n\n@[simp]\ntheorem lineDerivOp_neg (v : V) (x : E) : ∂_{v} (-x) = - ∂_{v} x :=\n  map_neg (AddMonoidHom.mk' ∂_{v} (lineDerivOp_add v)) x\n\n@[simp]\ntheorem lineDerivOp_sum (v : V) (f : ι → E) (s : Finset ι) :\n    ∂_{v} (∑ i ∈ s, f i) = ∑ i ∈ s, ∂_{v} (f i) :=\n  map_sum (AddMonoidHom.mk' ∂_{v} (lineDerivOp_add v)) f s\n\n@[simp]\ntheorem lineDerivOp_left_zero (x : E) : ∂_{(0 : V)} x = 0 :=\n  map_zero (AddMonoidHom.mk' (∂_{·} x) (lineDerivOp_left_add · · x))\n\n@[simp]\ntheorem lineDerivOp_left_neg (v : V) (x : E) : ∂_{-v} x = - ∂_{v} x :=\n  map_neg (AddMonoidHom.mk' (∂_{·} x) (lineDerivOp_left_add · · x)) v\n\n@[simp]\ntheorem lineDerivOp_left_sum (f : ι → V) (x : E) (s : Finset ι) :\n    ∂_{∑ i ∈ s, f i} x = ∑ i ∈ s, ∂_{f i} x :=\n  map_sum (AddMonoidHom.mk' (∂_{·} x) (lineDerivOp_left_add · · x)) f s\n\nend lineDerivOp\n\nsection lineDerivOpCLM\n\nvariable [Ring R] [AddCommGroup E] [Module R E] [AddCommGroup F] [Module R F]\n  [TopologicalSpace E] [TopologicalSpace F] [AddCommGroup V]\n  [LineDeriv V E F] [LineDerivAdd V E F] [LineDerivSMul R V E F] [ContinuousLineDeriv V E F]\n\nvariable (R E) in\n/-- The line derivative as a continuous linear map. -/\ndef lineDerivOpCLM (m : V) : E →L[R] F where\n  toFun := ∂_{m}\n  map_add' := lineDerivOp_add m\n  map_smul' := lineDerivOp_smul m\n\n@[simp]\ntheorem lineDerivOpCLM_apply (m : V) (x : E) :\n    lineDerivOpCLM R E m x = ∂_{m} x := rfl\n\nend lineDerivOpCLM\n\nsection iteratedLineDerivOp\n\nvariable [LineDeriv V E E]\nvariable {n : ℕ} (m : Fin n → V)\n\nsection add\n\nvariable [AddCommGroup V] [AddCommGroup E] [LineDerivAdd V E E]\n\nTarget:\ntheorem iteratedLineDerivOp_add (x y : E) :\n    ∂^{m} (x + y) = ∂^{m} x + ∂^{m} y :=\n\nProof body:\n","rejected":"by\n  exact iteratedLineDerivOp_add","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"be6270b8b9c29643af7aec79a9a7154679bbf9611d0230444ab04410c1ec3035","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Distribution","family_id":"iteratedlinederivop_add","file_id":"mathlib/Mathlib/Analysis/Distribution/DerivNotation.lean","sample_id":"e31fb78612321e73c41c554f6e35840ea8a109f4f7b6f9fddcff3cf0f701ebac"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4fe5d002e1432ce371569b334eb570afebb4011df3c1ab6e1eb29382c5b0d1c6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4c4c80ad0a7010c105ce75354ba15cb0853afffeb7444c56c493b383d3d52f04","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5d67e5e7b9950556ea8bd8cdfa3767dbe10a6562717cb86050afc39cdaf6a4b7","source_sha256":"85d2f87939059f796ffb510eeb3d6af5b97a8cdc2c5223d679b28800dce394e2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI := Nonempty.map f ‹_›\n  haveI := Nonempty.map g ‹_›\n  exact (map_injective.and map_surjective).trans and_and_and_comm","hard_negative":true,"metrics":{"chosen_tokens":29,"rejected_tokens":5,"token_jaccard":0.142857,"token_length_ratio":0.172414},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"d10c35ffced81b833bd78f200acf5fbaeff258aff5a748e1be8590b21440b575","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Lean.PrettyPrinter.Delaborator.Builtins\npublic import Mathlib.Logic.Function.Defs\npublic import Mathlib.Logic.Function.Iterate\npublic import Mathlib.Tactic.Inhabit\npublic import Batteries.Tactic.Trans\nimport Mathlib.Tactic.Attr.Register\n\nNamespace:\nProd\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Extra facts about `Prod`\n\nThis file proves various simple lemmas about `Prod`.\nIt also defines better delaborators for product projections.\n-/\n\n@[expose] public section\n\nvariable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}\n\nnamespace Prod\n\nlemma swap_eq_iff_eq_swap {x : α × β} {y : β × α} : x.swap = y ↔ x = y.swap := by grind\n\ndef mk.injArrow {x₁ : α} {y₁ : β} {x₂ : α} {y₂ : β} :\n    (x₁, y₁) = (x₂, y₂) → ∀ ⦃P : Sort*⦄, (x₁ = x₂ → y₁ = y₂ → P) → P := by\n  intros h P w\n  cases h\n  exact w rfl rfl\n\n@[simp]\ntheorem mk.eta : ∀ {p : α × β}, (p.1, p.2) = p\n  | (_, _) => rfl\n\ntheorem forall' {p : α → β → Prop} : (∀ x : α × β, p x.1 x.2) ↔ ∀ a b, p a b :=\n  Prod.forall\n\ntheorem exists' {p : α → β → Prop} : (∃ x : α × β, p x.1 x.2) ↔ ∃ a b, p a b :=\n  Prod.exists\n\n@[simp]\ntheorem snd_comp_mk (x : α) : Prod.snd ∘ (Prod.mk x : β → α × β) = id :=\n  rfl\n\n@[simp]\ntheorem fst_comp_mk (x : α) : Prod.fst ∘ (Prod.mk x : β → α × β) = Function.const β x :=\n  rfl\n\nattribute [mfld_simps] map_apply\n\n-- This was previously a `simp` lemma, but no longer is on the basis that it destructures the pair.\n--  See `map_apply`, `map_fst`, and `map_snd` for slightly weaker lemmas in the `simp` set.\ntheorem map_apply' (f : α → γ) (g : β → δ) (p : α × β) : map f g p = (f p.1, g p.2) :=\n  rfl\n\ntheorem map_fst' (f : α → γ) (g : β → δ) : Prod.fst ∘ map f g = f ∘ Prod.fst :=\n  funext <| map_fst f g\n\ntheorem map_snd' (f : α → γ) (g : β → δ) : Prod.snd ∘ map f g = g ∘ Prod.snd :=\n  funext <| map_snd f g\n\ntheorem mk_inj {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) = (a₂, b₂) ↔ a₁ = a₂ ∧ b₁ = b₂ := by simp\n\ntheorem mk_right_injective {α β : Type*} (a : α) : (mk a : β → α × β).Injective := by\n  intro b₁ b₂ h\n  simpa only [true_and, Prod.mk_inj, eq_self_iff_true] using h\n\ntheorem mk_left_injective {α β : Type*} (b : β) : (fun a ↦ mk a b : α → α × β).Injective := by\n  intro b₁ b₂ h\n  simpa only [and_true, eq_self_iff_true, mk_inj] using h\n\nlemma mk_right_inj {a : α} {b₁ b₂ : β} : (a, b₁) = (a, b₂) ↔ b₁ = b₂ :=\n    (mk_right_injective _).eq_iff\n\nlemma mk_left_inj {a₁ a₂ : α} {b : β} : (a₁, b) = (a₂, b) ↔ a₁ = a₂ := (mk_left_injective _).eq_iff\n\ntheorem map_def {f : α → γ} {g : β → δ} : Prod.map f g = fun p : α × β ↦ (f p.1, g p.2) :=\n  funext fun p ↦ Prod.ext (map_fst f g p) (map_snd f g p)\n\ntheorem id_prod : (fun p : α × β ↦ (p.1, p.2)) = id :=\n  rfl\n\n@[simp]\ntheorem map_iterate (f : α → α) (g : β → β) (n : ℕ) :\n    (Prod.map f g)^[n] = Prod.map f^[n] g^[n] := by induction n <;> simp [*, Prod.map_comp_map]\n\ntheorem fst_surjective [h : Nonempty β] : Function.Surjective (@fst α β) :=\n  fun x ↦ h.elim fun y ↦ ⟨⟨x, y⟩, rfl⟩\n\ntheorem snd_surjective [h : Nonempty α] : Function.Surjective (@snd α β) :=\n  fun y ↦ h.elim fun x ↦ ⟨⟨x, y⟩, rfl⟩\n\ntheorem fst_injective [Subsingleton β] : Function.Injective (@fst α β) :=\n  fun _ _ h ↦ Prod.ext h (Subsingleton.elim _ _)\n\ntheorem snd_injective [Subsingleton α] : Function.Injective (@snd α β) :=\n  fun _ _ h ↦ Prod.ext (Subsingleton.elim _ _) h\n\n@[simp]\ntheorem swap_leftInverse : Function.LeftInverse (@swap α β) swap :=\n  swap_swap\n\n@[simp]\ntheorem swap_rightInverse : Function.RightInverse (@swap α β) swap :=\n  swap_swap\n\ntheorem swap_injective : Function.Injective (@swap α β) :=\n  swap_leftInverse.injective\n\ntheorem swap_surjective : Function.Surjective (@swap α β) :=\n  swap_leftInverse.surjective\n\ntheorem swap_bijective : Function.Bijective (@swap α β) :=\n  ⟨swap_injective, swap_surjective⟩\n\ntheorem _root_.Function.Semiconj.swap_map (f : α → α) (g : β → β) :\n    Function.Semiconj swap (map f g) (map g f) :=\n  Function.semiconj_iff_comp_eq.2 (map_comp_swap g f).symm\n\ntheorem eq_iff_fst_eq_snd_eq : ∀ {p q : α × β}, p = q ↔ p.1 = q.1 ∧ p.2 = q.2\n  | ⟨p₁, p₂⟩, ⟨q₁, q₂⟩ => by simp\n\ntheorem fst_eq_iff : ∀ {p : α × β} {x : α}, p.1 = x ↔ p = (x, p.2)\n  | ⟨a, b⟩, x => by simp\n\ntheorem snd_eq_iff : ∀ {p : α × β} {x : β}, p.2 = x ↔ p = (p.1, x)\n  | ⟨a, b⟩, x => by simp\n\nvariable {r : α → α → Prop} {s : β → β → Prop} {x y : α × β}\n\nlemma lex_iff : Prod.Lex r s x y ↔ r x.1 y.1 ∨ x.1 = y.1 ∧ s x.2 y.2 := lex_def\n\ninstance Lex.decidable [DecidableEq α]\n    (r : α → α → Prop) (s : β → β → Prop) [DecidableRel r] [DecidableRel s] :\n    DecidableRel (Prod.Lex r s) :=\n  fun _ _ ↦ decidable_of_decidable_of_iff lex_def.symm\n\n@[refl]\ntheorem Lex.refl_left (r : α → α → Prop) (s : β → β → Prop) [Std.Refl r] : ∀ x, Prod.Lex r s x x\n  | (_, _) => Lex.left _ _ (refl _)\n\ninstance {r : α → α → Prop} {s : β → β → Prop} [Std.Refl r] : Std.Refl (Prod.Lex r s) :=\n  ⟨Lex.refl_left _ _⟩\n\n@[refl]\ntheorem Lex.refl_right (r : α → α → Prop) (s : β → β → Prop) [Std.Refl s] : ∀ x, Prod.Lex r s x x\n  | (_, _) => Lex.right _ (refl _)\n\ninstance {r : α → α → Prop} {s : β → β → Prop} [Std.Refl s] : Std.Refl (Prod.Lex r s) :=\n  ⟨Lex.refl_right _ _⟩\n\ninstance [Std.Irrefl r] [Std.Irrefl s] : Std.Irrefl (Prod.Lex r s) :=\n  ⟨by rintro ⟨i, a⟩ (⟨_, _, h⟩ | ⟨_, h⟩) <;> exact irrefl _ h⟩\n\nset_option linter.style.whitespace false in -- manual alignment is not recognised\n@[trans]\ntheorem Lex.trans {r : α → α → Prop} {s : β → β → Prop} [IsTrans α r] [IsTrans β s] :\n    ∀ {x y z : α × β}, Prod.Lex r s x y → Prod.Lex r s y z → Prod.Lex r s x z\n  | (_, _), (_, _), (_, _), left  _ _ hxy₁, left  _ _ hyz₁ => left  _ _ (_root_.trans hxy₁ hyz₁)\n  | (_, _), (_, _), (_, _), left  _ _ hxy₁, right _ _      => left  _ _ hxy₁\n  | (_, _), (_, _), (_, _), right _ _,      left  _ _ hyz₁ => left  _ _ hyz₁\n  | (_, _), (_, _), (_, _), right _ hxy₂,   right _ hyz₂   => right _ (_root_.trans hxy₂ hyz₂)\n\ninstance {r : α → α → Prop} {s : β → β → Prop} [IsTrans α r] [IsTrans β s] :\n    IsTrans (α × β) (Prod.Lex r s) :=\n  ⟨fun _ _ _ ↦ Lex.trans⟩\n\ninstance {r : α → α → Prop} {s : β → β → Prop} [IsStrictOrder α r] [Std.Antisymm s] :\n    Std.Antisymm (Prod.Lex r s) :=\n  ⟨fun x₁ x₂ h₁₂ h₂₁ ↦\n    match x₁, x₂, h₁₂, h₂₁ with\n    | (a, _), (_, _), .left  _ _ hr₁, .left  _ _ hr₂ => (irrefl a (_root_.trans hr₁ hr₂)).elim\n    | (_, _), (_, _), .left  _ _ hr₁, .right _ _     => (irrefl _ hr₁).elim\n    | (_, _), (_, _), .right _ _,     .left  _ _ hr₂ => (irrefl _ hr₂).elim\n    | (_, _), (_, _), .right _ hs₁,   .right _ hs₂   => antisymm hs₁ hs₂ ▸ rfl⟩\n\ninstance total_left {r : α → α → Prop} {s : β → β → Prop} [Std.Total r] :\n    Std.Total (Prod.Lex r s) :=\n  ⟨fun ⟨a₁, _⟩ ⟨a₂, _⟩ ↦ (Std.Total.total a₁ a₂).imp (Lex.left _ _) (Lex.left _ _)⟩\n\ninstance total_right {r : α → α → Prop} {s : β → β → Prop} [Std.Trichotomous r] [Std.Total s] :\n    Std.Total (Prod.Lex r s) :=\n  ⟨fun ⟨i, a⟩ ⟨j, b⟩ ↦ by\n    obtain hij | rfl | hji := trichotomous_of r i j\n    · exact Or.inl (.left _ _ hij)\n    · exact (total_of s a b).imp (.right _) (.right _)\n    · exact Or.inr (.left _ _ hji) ⟩\n\ninstance trichotomous [Std.Trichotomous r] [Std.Trichotomous s] :\n    Std.Trichotomous (Prod.Lex r s) :=\n  Std.trichotomous_of_rel_or_eq_or_rel_swap <| by\n    intro ⟨i, a⟩ ⟨j, b⟩\n    obtain hij | rfl | hji := trichotomous_of r i j\n    { exact Or.inl (Lex.left _ _ hij) }\n    { exact (trichotomous_of (s) a b).imp3 (Lex.right _) (congr_arg _) (Lex.right _) }\n    { exact Or.inr (Or.inr <| Lex.left _ _ hji) }\n\ninstance [Std.Asymm r] [Std.Asymm s] :\n    Std.Asymm (Prod.Lex r s) where\n  asymm\n  | (_a₁, _a₂), (_b₁, _b₂), .left _ _ h₁, .left _ _ h₂ => Std.Asymm.asymm _ _ h₂ h₁\n  | (_a₁, _a₂), (_, _b₂), .left _ _ h₁, .right _ _ => Std.Asymm.asymm _ _ h₁ h₁\n  | (_a₁, _a₂), (_, _b₂), .right _ _, .left _ _ h₂ => Std.Asymm.asymm _ _ h₂ h₂\n  | (_a₁, _a₂), (_, _b₂), .right _ h₁, .right _ h₂ => Std.Asymm.asymm _ _ h₁ h₂\n\nend Prod\n\nopen Prod\n\nnamespace Function\n\nvariable {f : α → γ} {g : β → δ} {f₁ : α → β} {g₁ : γ → δ} {f₂ : β → α} {g₂ : δ → γ}\n\ntheorem Injective.prodMap (hf : Injective f) (hg : Injective g) : Injective (map f g) :=\n  fun _ _ h ↦ Prod.ext (hf <| congr_arg Prod.fst h) (hg <| congr_arg Prod.snd h)\n\ntheorem Surjective.prodMap (hf : Surjective f) (hg : Surjective g) : Surjective (map f g) :=\n  fun p ↦\n  let ⟨x, hx⟩ := hf p.1\n  let ⟨y, hy⟩ := hg p.2\n  ⟨(x, y), Prod.ext hx hy⟩\n\ntheorem Bijective.prodMap (hf : Bijective f) (hg : Bijective g) : Bijective (map f g) :=\n  ⟨hf.1.prodMap hg.1, hf.2.prodMap hg.2⟩\n\ntheorem LeftInverse.prodMap (hf : LeftInverse f₁ f₂) (hg : LeftInverse g₁ g₂) :\n    LeftInverse (map f₁ g₁) (map f₂ g₂) :=\n  fun a ↦ by rw [Prod.map_map, hf.comp_eq_id, hg.comp_eq_id, map_id, id]\n\ntheorem RightInverse.prodMap :\n    RightInverse f₁ f₂ → RightInverse g₁ g₂ → RightInverse (map f₁ g₁) (map f₂ g₂) :=\n  LeftInverse.prodMap\n\ntheorem Involutive.prodMap {f : α → α} {g : β → β} :\n    Involutive f → Involutive g → Involutive (map f g) :=\n  LeftInverse.prodMap\n\nend Function\n\nnamespace Prod\n\nopen Function\n\n@[simp]\ntheorem map_injective [Nonempty α] [Nonempty β] {f : α → γ} {g : β → δ} :\n    Injective (map f g) ↔ Injective f ∧ Injective g :=\n  ⟨fun h =>\n    ⟨fun a₁ a₂ ha => by\n      inhabit β\n      injection\n        @h (a₁, default) (a₂, default) (congr_arg (fun c : γ => Prod.mk c (g default)) ha :),\n      fun b₁ b₂ hb => by\n      inhabit α\n      injection @h (default, b₁) (default, b₂) (congr_arg (Prod.mk (f default)) hb :)⟩,\n    fun h => h.1.prodMap h.2⟩\n\n@[simp]\ntheorem map_surjective [Nonempty γ] [Nonempty δ] {f : α → γ} {g : β → δ} :\n    Surjective (map f g) ↔ Surjective f ∧ Surjective g :=\n  ⟨fun h =>\n    ⟨fun c => by\n      inhabit δ\n      obtain ⟨⟨a, b⟩, h⟩ := h (c, default)\n      exact ⟨a, congr_arg Prod.fst h⟩,\n      fun d => by\n      inhabit γ\n      obtain ⟨⟨a, b⟩, h⟩ := h (default, d)\n      exact ⟨b, congr_arg Prod.snd h⟩⟩,\n    fun h => h.1.prodMap h.2⟩\n\n@[simp]\n\nTarget:\ntheorem map_bijective [Nonempty α] [Nonempty β] {f : α → γ} {g : β → δ} :\n    Bijective (map f g) ↔ Bijective f ∧ Bijective g :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_5d67e5e7b995","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"71810c68c130487596915437a09ad4fb2412e91d3ed6d930db9ef7155227d74c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Prod","family_id":"map_bijective","file_id":"mathlib/Mathlib/Data/Prod/Basic.lean","sample_id":"5d67e5e7b9950556ea8bd8cdfa3767dbe10a6562717cb86050afc39cdaf6a4b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eac7599f7adbe4156b1359f0aff3dffecc04db5fb77ecf8bedd140ea7081aceb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7ee01c83c10c22668c1fdc800b54b271e64edd823972fd056c65d61e5211e5df","source_sha256":"bbfec10092a00bc78b9af51067ad47929a5d5df48b980b858f3f6f7bbf3e54e6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction h\n  simp","hard_negative":false,"metrics":{"chosen_tokens":4,"rejected_tokens":2,"token_jaccard":0.2,"token_length_ratio":0.5},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"d123bcc2d59669e3c0585bf169b6b41065bfe28292f95739cd0d9475e9a936bc","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Skeleton\npublic import Mathlib.CategoryTheory.Subobject.MonoOver\npublic import Mathlib.CategoryTheory.Skeletal\npublic import Mathlib.CategoryTheory.ConcreteCategory.Basic\npublic import Mathlib.Tactic.ApplyFun\npublic import Mathlib.Tactic.CategoryTheory.Elementwise\npublic import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic\n\nNamespace:\nCategoryTheory.Subobject\n\nLocal context:\n/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Kim Morrison\n-/\n/-!\n# Subobjects\n\nWe define `Subobject X` as the quotient (by isomorphisms) of\n`MonoOver X := {f : Over X // Mono f.hom}`.\n\nHere `MonoOver X` is a thin category (a pair of objects has at most one morphism between them),\nso we can think of it as a preorder. However as it is not skeletal, it is not a partial order.\n\nThere is a coercion from `Subobject X` back to the ambient category `C`\n(using choice to pick a representative), and for `P : Subobject X`,\n`P.arrow : (P : C) ⟶ X` is the inclusion morphism.\n\nWe provide\n* `def pullback [HasPullbacks C] (f : X ⟶ Y) : Subobject Y ⥤ Subobject X`\n* `def map (f : X ⟶ Y) [Mono f] : Subobject X ⥤ Subobject Y`\n* `def «exists_» [HasImages C] (f : X ⟶ Y) : Subobject X ⥤ Subobject Y`\n\nand prove their basic properties and relationships.\nThese are all easy consequences of the earlier development\nof the corresponding functors for `MonoOver`.\n\nThe subobjects of `X` form a preorder making them into a category. We have `X ≤ Y` if and only if\n`X.arrow` factors through `Y.arrow`: see `ofLE`/`ofLEMk`/`ofMkLE`/`ofMkLEMk` and\n`le_of_comm`. Similarly, to show that two subobjects are equal, we can supply an isomorphism between\nthe underlying objects that commutes with the arrows (`eq_of_comm`).\n\nSee also\n\n* `CategoryTheory.Subobject.factorThru` :\n  an API describing factorization of morphisms through subobjects.\n* `CategoryTheory.Subobject.lattice` :\n  the lattice structures on subobjects.\n\n## Notes\n\nThis development originally appeared in Bhavik Mehta's \"Topos theory for Lean\" repository,\nand was ported to mathlib by Kim Morrison.\n\n### Implementation note\n\nCurrently we describe `pullback`, `map`, etc., as functors.\nIt may be better to just say that they are monotone functions,\nand even avoid using categorical language entirely when describing `Subobject X`.\n(It's worth keeping this in mind in future use; it should be a relatively easy change here\nif it looks preferable.)\n\n### Relation to pseudoelements\n\nThere is a separate development of pseudoelements in `CategoryTheory.Abelian.Pseudoelements`,\nas a quotient (but not by isomorphism) of `Over X`.\n\nWhen a morphism `f` has an image, the image represents the same pseudoelement.\nIn a category with images `Pseudoelements X` could be constructed as a quotient of `MonoOver X`.\nIn fact, in an abelian category (I'm not sure in what generality beyond that),\n`Pseudoelements X` agrees with `Subobject X`, but we haven't developed this in mathlib yet.\n\n-/\n\n@[expose] public section\n\n\nuniverse w' w v₁ v₂ v₃ u₁ u₂ u₃\n\nnoncomputable section\n\nnamespace CategoryTheory\n\nopen CategoryTheory CategoryTheory.Category CategoryTheory.Limits\n\nvariable {C : Type u₁} [Category.{v₁} C] {X Y Z : C}\nvariable {D : Type u₂} [Category.{v₂} D]\n\n/-!\nWe now construct the subobject lattice for `X : C`,\nas the quotient by isomorphisms of `MonoOver X`.\n\nSince `MonoOver X` is a thin category, we use `ThinSkeleton` to take the quotient.\n\nEssentially all the structure defined above on `MonoOver X` descends to `Subobject X`,\nwith morphisms becoming inequalities, and isomorphisms becoming equations.\n-/\n\n\n/-- The category of subobjects of `X : C`, defined as isomorphism classes of monomorphisms into `X`.\n-/\ndef Subobject (X : C) :=\n  ThinSkeleton (MonoOver X)\n\ninstance (X : C) : PartialOrder (Subobject X) :=\n  inferInstanceAs <| PartialOrder (ThinSkeleton (MonoOver X))\n\nnamespace Subobject\n\nlemma skeletal (X : C) : Skeletal (Subobject X) := ThinSkeleton.skeletal\n\n/-- Convenience constructor for a subobject. -/\ndef mk {X A : C} (f : A ⟶ X) [Mono f] : Subobject X :=\n  (toThinSkeleton _).obj (MonoOver.mk f)\n\nsection\n\nattribute [local ext] CategoryTheory.Comma\n\nprotected theorem ind {X : C} (p : Subobject X → Prop)\n    (h : ∀ ⦃A : C⦄ (f : A ⟶ X) [Mono f], p (Subobject.mk f)) (P : Subobject X) : p P := by\n  induction P using Quotient.inductionOn' with | _ a\n  exact h a.arrow\n\nprotected theorem ind₂ {X : C} (p : Subobject X → Subobject X → Prop)\n    (h : ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [Mono f] [Mono g],\n      p (Subobject.mk f) (Subobject.mk g))\n    (P Q : Subobject X) : p P Q := by\n  induction P, Q using Quotient.inductionOn₂' with | _ a b\n  exact h a.arrow b.arrow\n\nend\n\n/-- Declare a function on subobjects of `X` by specifying a function on monomorphisms with\ncodomain `X`. -/\nprotected def lift {α : Sort*} {X : C} (F : ∀ ⦃A : C⦄ (f : A ⟶ X) [Mono f], α)\n    (h :\n      ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [Mono f] [Mono g] (i : A ≅ B),\n        i.hom ≫ g = f → F f = F g) :\n    Subobject X → α := fun P =>\n  Quotient.liftOn' P (fun m => F m.arrow) fun m n ⟨i⟩ =>\n    h m.arrow n.arrow ((MonoOver.forget X ⋙ Over.forget X).mapIso i) (Over.w i.hom.hom)\n\n@[simp]\nprotected theorem lift_mk {α : Sort*} {X : C} (F : ∀ ⦃A : C⦄ (f : A ⟶ X) [Mono f], α) {h A}\n    (f : A ⟶ X) [Mono f] : Subobject.lift F h (Subobject.mk f) = F f :=\n  rfl\n\n/-- The category of subobjects is equivalent to the `MonoOver` category. It is more convenient to\nuse the former due to the partial order instance, but oftentimes it is easier to define structures\non the latter. -/\nnoncomputable def equivMonoOver (X : C) : Subobject X ≌ MonoOver X :=\n  ThinSkeleton.equivalence _\n\n/-- Use choice to pick a representative `MonoOver X` for each `Subobject X`.\n-/\nnoncomputable def representative {X : C} : Subobject X ⥤ MonoOver X :=\n  (equivMonoOver X).functor\n\ninstance : (representative (X := X)).IsEquivalence :=\n  (equivMonoOver X).isEquivalence_functor\n\n/-- Starting with `A : MonoOver X`, we can take its equivalence class in `Subobject X`\nthen pick an arbitrary representative using `representative.obj`.\nThis is isomorphic (in `MonoOver X`) to the original `A`.\n-/\nnoncomputable def representativeIso {X : C} (A : MonoOver X) :\n    representative.obj ((toThinSkeleton _).obj A) ≅ A :=\n  (equivMonoOver X).counitIso.app A\n\n@[simp]\nlemma thinSkeleton_mk_representative_eq_self {X : C} (A : Subobject X) :\n    ThinSkeleton.mk (representative.obj A) = A :=\n  Subobject.skeletal _ ⟨((equivMonoOver X).unitIso.app _).symm⟩\n\n/-- Use choice to pick a representative underlying object in `C` for any `Subobject X`.\n\nPrefer to use the coercion `P : C` rather than explicitly writing `underlying.obj P`.\n-/\nnoncomputable def underlying {X : C} : Subobject X ⥤ C :=\n  representative ⋙ MonoOver.forget _ ⋙ Over.forget _\n\ninstance : CoeOut (Subobject X) C where coe Y := underlying.obj Y\n\n/-- If we construct a `Subobject Y` from an explicit `f : X ⟶ Y` with `[Mono f]`,\nthen pick an arbitrary choice of underlying object `(Subobject.mk f : C)` back in `C`,\nit is isomorphic (in `C`) to the original `X`.\n-/\nnoncomputable def underlyingIso {X Y : C} (f : X ⟶ Y) [Mono f] : (Subobject.mk f : C) ≅ X :=\n  (MonoOver.forget _ ⋙ Over.forget _).mapIso (representativeIso (MonoOver.mk f))\n\n/-- The morphism in `C` from the arbitrarily chosen underlying object to the ambient object.\n-/\nnoncomputable def arrow {X : C} (Y : Subobject X) : (Y : C) ⟶ X :=\n  (representative.obj Y).obj.hom\n\ninstance arrow_mono {X : C} (Y : Subobject X) : Mono Y.arrow :=\n  (representative.obj Y).property\n\n@[simp]\n\nTarget:\ntheorem arrow_congr {A : C} (X Y : Subobject A) (h : X = Y) :\n    eqToHom (congr_arg (fun X : Subobject A => (X : C)) h) ≫ Y.arrow = X.arrow :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Subobject","family_id":"arrow_congr","file_id":"mathlib/Mathlib/CategoryTheory/Subobject/Basic.lean","sample_id":"7ee01c83c10c22668c1fdc800b54b271e64edd823972fd056c65d61e5211e5df"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3df867611d307b2d4c52a5aeae1c5c93f72412c5fca86e32a6445afd51c4bda0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"617a91f1c4527ca920c01a59a90105084ba9b7d9266cd812029ecf72841e2d5b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fe45775e1693cac44dede504e39345f1c5dfd23b430122a30f0b82d3494c2fdd","source_sha256":"2aaaf1d6e3bf8bcc38fe6595b30037cc7d4b213d51f17e0ec2ab4dd92ba3ffe4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  trans fun n ↦ (n / 3 - 2) * ↑((n - 3) / 6) * exp (-4 * √(log ↑((n - 3) / 6)))\n  · simp_rw [sq]\n    refine (IsBigO.mul ?_ ?_).mul ?_\n    · trans fun n ↦ n / 3\n      · simp_rw [div_eq_inv_mul]\n        exact (isBigO_refl ..).const_mul_right (by simp)\n      refine IsLittleO.right_isBigO_sub ?_\n      simpa [div_eq_inv_mul, Function.comp_def] using\n        .atTop_of_const_mul₀ zero_lt_three (by simp [tendsto_natCast_atTop_atTop])\n    · rw [IsBigO_def]\n      refine ⟨12, ?_⟩\n      simp only [IsBigOWith, norm_natCast, eventually_atTop]\n      exact ⟨15, fun x hx ↦ by norm_cast; lia⟩\n    · rw [isBigO_exp_comp_exp_comp]\n      refine ⟨0, ?_⟩\n      simp only [neg_mul, eventually_map, Pi.sub_apply, sub_neg_eq_add, neg_add_le_iff_le_add,\n        add_zero, eventually_atTop]\n      refine ⟨9, fun x hx ↦ ?_⟩\n      gcongr\n      · simp\n        lia\n      · lia\n  · refine .of_norm_eventuallyLE ?_\n    filter_upwards [eventually_ge_atTop 6] with n hn\n    have : (0 : ℝ) ≤ n / 3 - 2 := by rify at hn; linarith\n    simpa [neg_mul, abs_mul, abs_of_nonneg this] using ruzsaSzemerediNumberNat_lower_bound n","hard_negative":true,"metrics":{"chosen_tokens":244,"rejected_tokens":2,"token_jaccard":0.011236,"token_length_ratio":0.008197},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"d16739fd19f6911b194a15bfe2ea6158c45bfacdc0559b514070c8eb24af3277","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Combinatorics.Additive.AP.Three.Behrend\npublic import Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite\npublic import Mathlib.Tactic.Rify\npublic import Mathlib.Tactic.Qify\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# The Ruzsa-Szemerédi problem\n\nThis file proves the lower bound of the Ruzsa-Szemerédi problem. The problem is to find the maximum\nnumber of edges that a graph on `n` vertices can have if all edges belong to at most one triangle.\n\nThe lower bound comes from turning the big 3AP-free set from Behrend's construction into a graph\nthat has the property that every triangle gives a (possibly trivial) arithmetic progression on the\noriginal set.\n\n## Main declarations\n\n* `ruzsaSzemerediNumberNat n`: Maximum number of edges a graph on `n` vertices can have such that\n  each edge belongs to exactly one triangle.\n* `ruzsaSzemerediNumberNat_asymptotic_lower_bound`: There exists a graph with `n` vertices and\n  `Ω((n ^ 2 * exp (-4 * √(log n))))` edges such that each edge belongs to exactly one triangle.\n-/\n\n@[expose] public section\n\nopen Finset Nat Real SimpleGraph Sum3 SimpleGraph.TripartiteFromTriangles\nopen Fintype (card)\nopen scoped Pointwise\n\nvariable {α β : Type*}\n\n/-! ### The Ruzsa-Szemerédi number -/\n\nsection ruzsaSzemerediNumber\nvariable [DecidableEq α] [DecidableEq β] [Fintype α] [Fintype β] {G H : SimpleGraph α}\n\nvariable (α) in\n/-- The **Ruzsa-Szemerédi number** of a fintype is the maximum number of edges a locally linear\ngraph on that type can have.\n\nIn other words, `ruzsaSzemerediNumber α` is the maximum number of edges a graph on `α` can have such\nthat each edge belongs to exactly one triangle. -/\nnoncomputable def ruzsaSzemerediNumber : ℕ := by\n  classical\n  exact Nat.findGreatest (fun m ↦ ∃ (G : SimpleGraph α) (_ : DecidableRel G.Adj),\n    #(G.cliqueFinset 3) = m ∧ G.LocallyLinear) ((card α).choose 3)\n\nopen scoped Classical in\nlemma ruzsaSzemerediNumber_le : ruzsaSzemerediNumber α ≤ (card α).choose 3 := Nat.findGreatest_le _\n\nlemma ruzsaSzemerediNumber_spec :\n    ∃ (G : SimpleGraph α) (_ : DecidableRel G.Adj),\n      #(G.cliqueFinset 3) = ruzsaSzemerediNumber α ∧ G.LocallyLinear := by\n  classical\n  exact @Nat.findGreatest_spec _\n    (fun m ↦ ∃ (G : SimpleGraph α) (_ : DecidableRel G.Adj),\n      #(G.cliqueFinset 3) = m ∧ G.LocallyLinear) _ _ (Nat.zero_le _)\n    ⟨⊥, inferInstance, by simp, locallyLinear_bot⟩\n\nvariable {m n : ℕ}\n\nlemma SimpleGraph.LocallyLinear.le_ruzsaSzemerediNumber [DecidableRel G.Adj]\n    (hG : G.LocallyLinear) : #(G.cliqueFinset 3) ≤ ruzsaSzemerediNumber α := by\n  classical\n  exact le_findGreatest card_cliqueFinset_le ⟨G, inferInstance, by congr, hG⟩\n\nlemma ruzsaSzemerediNumber_mono (f : α ↪ β) : ruzsaSzemerediNumber α ≤ ruzsaSzemerediNumber β := by\n  classical\n  refine findGreatest_mono ?_ (choose_mono _ <| Fintype.card_le_of_embedding f)\n  rintro n ⟨G, _, rfl, hG⟩\n  refine ⟨G.map f, inferInstance, ?_, hG.map _⟩\n  rw [← card_map ⟨map f, Finset.map_injective _⟩, ← cliqueFinset_map G f]\n  decide\n\nlemma ruzsaSzemerediNumber_congr (e : α ≃ β) : ruzsaSzemerediNumber α = ruzsaSzemerediNumber β :=\n  (ruzsaSzemerediNumber_mono (e : α ↪ β)).antisymm <| ruzsaSzemerediNumber_mono e.symm\n\n/-- The `n`-th **Ruzsa-Szemerédi number** is the maximum number of edges a locally linear graph on\n`n` vertices can have.\n\nIn other words, `ruzsaSzemerediNumberNat n` is the maximum number of edges a graph on `n` vertices\ncan have such that each edge belongs to exactly one triangle. -/\nnoncomputable def ruzsaSzemerediNumberNat (n : ℕ) : ℕ := ruzsaSzemerediNumber (Fin n)\n\n@[simp]\nlemma ruzsaSzemerediNumberNat_card : ruzsaSzemerediNumberNat (card α) = ruzsaSzemerediNumber α :=\n  ruzsaSzemerediNumber_congr (Fintype.equivFin _).symm\n\n@[gcongr]\nlemma ruzsaSzemerediNumberNat_mono : Monotone ruzsaSzemerediNumberNat := fun _m _n h =>\n  ruzsaSzemerediNumber_mono (Fin.castLEEmb h)\n\nlemma ruzsaSzemerediNumberNat_le : ruzsaSzemerediNumberNat n ≤ n.choose 3 :=\n  ruzsaSzemerediNumber_le.trans_eq <| by rw [Fintype.card_fin]\n\n@[simp] lemma ruzsaSzemerediNumberNat_zero : ruzsaSzemerediNumberNat 0 = 0 :=\n  le_zero_iff.1 ruzsaSzemerediNumberNat_le\n\n@[simp] lemma ruzsaSzemerediNumberNat_one : ruzsaSzemerediNumberNat 1 = 0 :=\n  le_zero_iff.1 ruzsaSzemerediNumberNat_le\n\n@[simp] lemma ruzsaSzemerediNumberNat_two : ruzsaSzemerediNumberNat 2 = 0 :=\n  le_zero_iff.1 ruzsaSzemerediNumberNat_le\n\nend ruzsaSzemerediNumber\n\n/-! ### The Ruzsa-Szemerédi construction -/\n\nsection RuzsaSzemeredi\nvariable [Fintype α] [CommRing α] {s : Finset α} {x : α × α × α}\n\n/-- The triangle indices for the Ruzsa-Szemerédi construction. -/\nprivate def triangleIndices (s : Finset α) : Finset (α × α × α) :=\n  (univ ×ˢ s).map\n    ⟨fun xa ↦ (xa.1, xa.1 + xa.2, xa.1 + 2 * xa.2), by\n      rintro ⟨x, a⟩ ⟨y, b⟩ h\n      simp only [Prod.ext_iff] at h\n      obtain rfl := h.1\n      obtain rfl := add_right_injective _ h.2.1\n      rfl⟩\n\n@[simp]\nprivate lemma mem_triangleIndices :\n    x ∈ triangleIndices s ↔ ∃ y, ∃ a ∈ s, (y, y + a, y + 2 * a) = x := by simp [triangleIndices]\n\n@[simp]\nprivate lemma card_triangleIndices : #(triangleIndices s) = card α * #s := by\n  simp [triangleIndices]\n\nprivate lemma noAccidental (hs : ThreeAPFree (s : Set α)) :\n    NoAccidental (triangleIndices s : Finset (α × α × α)) where\n  eq_or_eq_or_eq := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ _ _ d a ha rfl rfl rfl b' b hb rfl rfl h₁ d' c hc rfl h₂ rfl\n    have : a + c = b + b := by linear_combination h₁.symm - h₂.symm\n    obtain rfl := hs ha hb hc this\n    simp_all\n\nvariable [Fact <| IsUnit (2 : α)]\n\nprivate instance : ExplicitDisjoint (triangleIndices s : Finset (α × α × α)) where\n  inj₀ := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ x a ha rfl rfl rfl y b hb rfl h₁ h₂\n    linear_combination 2 * h₁.symm - h₂.symm\n  inj₁ := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ x a ha rfl rfl rfl y b hb rfl rfl h\n    simpa [(Fact.out (p := IsUnit (2 : α))).mul_right_inj, eq_comm] using h\n  inj₂ := by\n    simp only [mem_triangleIndices, Prod.mk_inj, forall_exists_index, and_imp]\n    rintro _ _ _ _ x a ha rfl rfl rfl y b hb rfl h rfl\n    simpa [(Fact.out (p := IsUnit (2 : α))).mul_right_inj, eq_comm] using h\n\nprivate lemma locallyLinear (hs : ThreeAPFree (s : Set α)) :\n    (graph <| triangleIndices s).LocallyLinear :=\n  haveI := noAccidental hs; TripartiteFromTriangles.locallyLinear _\n\nprivate lemma card_edgeFinset (hs : ThreeAPFree (s : Set α)) [DecidableEq α] :\n    #(graph <| triangleIndices s).edgeFinset = 3 * card α * #s := by\n  haveI := noAccidental hs\n  rw [(locallyLinear hs).card_edgeFinset, card_triangles, card_triangleIndices, mul_assoc]\n\nend RuzsaSzemeredi\n\nvariable (α) [Fintype α] [DecidableEq α] [CommRing α] [Fact <| IsUnit (2 : α)]\n\nlemma addRothNumber_le_ruzsaSzemerediNumber :\n    card α * addRothNumber (univ : Finset α) ≤ ruzsaSzemerediNumber (Sum α (Sum α α)) := by\n  obtain ⟨s, -, hscard, hs⟩ := addRothNumber_spec (univ : Finset α)\n  haveI := noAccidental hs\n  rw [← hscard, ← card_triangleIndices, ← card_triangles]\n  exact (locallyLinear hs).le_ruzsaSzemerediNumber\n\nlemma rothNumberNat_le_ruzsaSzemerediNumberNat (n : ℕ) :\n    (2 * n + 1) * rothNumberNat n ≤ ruzsaSzemerediNumberNat (6 * n + 3) := by\n  let α := Fin (2 * n + 1)\n  have : Nat.Coprime 2 (2 * n + 1) := by simp\n  haveI : Fact (IsUnit (2 : Fin (2 * n + 1))) := ⟨by simpa\n    using! (ZMod.unitOfCoprime 2 this).isUnit⟩\n  open scoped Fin.CommRing in\n  calc\n    (2 * n + 1) * rothNumberNat n\n    _ = Fintype.card α * addRothNumber (Iio (n : α)) := by\n      rw [Fin.addRothNumber_eq_rothNumberNat le_rfl, Fintype.card_fin]\n    _ ≤ Fintype.card α * addRothNumber (univ : Finset α) := by\n      gcongr; exact subset_univ _\n    _ ≤ ruzsaSzemerediNumber (Sum α (Sum α α)) := addRothNumber_le_ruzsaSzemerediNumber _\n    _ = ruzsaSzemerediNumberNat (6 * n + 3) := by\n      simp_rw [← ruzsaSzemerediNumberNat_card, Fintype.card_sum, α, Fintype.card_fin]\n      ring_nf\n\n/-- Lower bound on the **Ruzsa-Szemerédi problem** in terms of 3AP-free sets.\n\nIf there exists a 3AP-free subset of `[1, ..., (n - 3) / 6]` of size `m`, then there exists a graph\nwith `n` vertices and `(n / 3 - 2) * m` edges such that each edge belongs to exactly one triangle.\n-/\ntheorem rothNumberNat_le_ruzsaSzemerediNumberNat' :\n    ∀ n : ℕ, (n / 3 - 2 : ℝ) * rothNumberNat ((n - 3) / 6) ≤ ruzsaSzemerediNumberNat n\n  | 0 => by simp\n  | 1 => by simp\n  | 2 => by simp\n  | n + 3 => by\n    calc\n      _ ≤ (↑(2 * (n / 6) + 1) : ℝ) * rothNumberNat (n / 6) :=\n        mul_le_mul_of_nonneg_right ?_ (Nat.cast_nonneg _)\n      _ ≤ (ruzsaSzemerediNumberNat (6 * (n / 6) + 3) : ℝ) := ?_\n      _ ≤ _ := by grw [Nat.mul_div_le]\n    · simp only [cast_add, cast_ofNat, cast_mul, cast_one, tsub_le_iff_right]\n      rw [← div_add_one (three_ne_zero' ℝ), ← le_sub_iff_add_le, div_le_iff₀ (zero_lt_three' ℝ),\n        add_assoc, add_sub_assoc, add_mul, mul_right_comm, add_sub_cancel_left]\n      norm_cast\n      rw [← mul_add_one]\n      exact (Nat.lt_mul_div_succ _ <| by simp).le\n    · norm_cast\n      exact rothNumberNat_le_ruzsaSzemerediNumberNat _\n\n/-- Explicit lower bound on the **Ruzsa-Szemerédi problem**.\n\nThere exists a graph with `n` vertices and\n`(n / 3 - 2) * (n - 3) / 6 * exp (-4 * √(log ((n - 3) / 6)))` edges such that each edge belongs\nto exactly one triangle. -/\ntheorem ruzsaSzemerediNumberNat_lower_bound (n : ℕ) :\n    (n / 3 - 2 : ℝ) * ↑((n - 3) / 6) * exp (-4 * √(log ↑((n - 3) / 6))) ≤\n      ruzsaSzemerediNumberNat n := by\n  rw [mul_assoc]\n  obtain hn | hn := le_total (n / 3 - 2 : ℝ) 0\n  · exact (mul_nonpos_of_nonpos_of_nonneg hn <| by positivity).trans (Nat.cast_nonneg _)\n  exact\n    (mul_le_mul_of_nonneg_left Behrend.roth_lower_bound hn).trans\n      (rothNumberNat_le_ruzsaSzemerediNumberNat' _)\n\nopen Asymptotics Filter\n\n/-- Asymptotic lower bound on the **Ruzsa-Szemerédi problem**.\n\nThere exists a graph with `n` vertices and `Ω((n ^ 2 * exp (-4 * √(log n))))` edges such that\neach edge belongs to exactly one triangle. -/\n\nTarget:\ntheorem ruzsaSzemerediNumberNat_asymptotic_lower_bound :\n    (fun n ↦ n ^ 2 * exp (-4 * √(log n)) : ℕ → ℝ) =O[atTop]\n     fun n ↦ (ruzsaSzemerediNumberNat n : ℝ) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_fe45775e1693","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"fd0405182fc02c5f97a49773c8d0449bde4e69caa5dfb8fc0778e16e360b202b","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Extremal","family_id":"ruzsaszemeredinumbernat_asymptotic_lower_bound","file_id":"mathlib/Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean","sample_id":"fe45775e1693cac44dede504e39345f1c5dfd23b430122a30f0b82d3494c2fdd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"281c14df73dbd3d58dd1e8053eb70fe2ed484fdf0d2d7eb3b862de0b5a2b7009","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"18dd8b95b77ff3462addfa3b45bcf71a82dbc0f0d284cf7b3ea4a8c80b7ffa63","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e9c86bc71f86c9474c5b0e70638d93a26490f015f3c3db0caf5be6d93809c4ae","source_sha256":"76bf26908fe5bb3f22e0b1de59c408f5480271bc0d25bef1d614882cce4d1085","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : Nonempty J := IsFiltered.nonempty\n  let hc' := Under.isColimitLiftCocone D s c (p ≫ f) h hc\n  obtain ⟨j, q, hq⟩ := exists_hom_of_isColimit (X := Under.mk p) hc' (Under.homMk f rfl)\n  use j, q.right, Under.w q, congr($(hq).right)","hard_negative":false,"metrics":{"chosen_tokens":72,"rejected_tokens":76,"token_jaccard":0.930233,"token_length_ratio":1.055556},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"d1ecee38d2a85c3a90b941b171a2e9a6e6f400286791d07e3cfc50fdae8abf60","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Filtered\npublic import Mathlib.CategoryTheory.Limits.Preserves.Filtered\npublic import Mathlib.CategoryTheory.Limits.Types.Filtered\npublic import Mathlib.CategoryTheory.MorphismProperty.Basic\npublic import Mathlib.CategoryTheory.Presentable.Basic\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Finitely Presentable Objects\n\nWe define finitely presentable objects as a synonym for `ℵ₀`-presentable objects,\nand link this definition with the preservation of filtered colimits.\n\n-/\n\n@[expose] public section\n\n\nuniverse w v' v u' u\n\nnamespace CategoryTheory\n\nopen Limits Opposite Cardinal\n\nvariable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]\n\nattribute [local instance] fact_isRegular_aleph0\n\n/-- A functor `F : C ⥤ D` is finitely accessible if it is `ℵ₀`-accessible.\nEquivalently, it preserves all filtered colimits.\nSee `CategoryTheory.Functor.IsFinitelyAccessible_iff_preservesFilteredColimits`. -/\nabbrev Functor.IsFinitelyAccessible (F : C ⥤ D) : Prop := IsCardinalAccessible.{w} F ℵ₀\n\nlemma Functor.IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize {F : C ⥤ D} :\n    IsFinitelyAccessible.{w} F ↔ PreservesFilteredColimitsOfSize.{w, w} F := by\n  refine ⟨fun ⟨H⟩ ↦ ⟨?_⟩, fun ⟨H⟩ ↦ ⟨?_⟩⟩ <;>\n    simp only [isCardinalFiltered_aleph0_iff] at * <;>\n    exact H\n\nlemma Functor.isFinitelyAccessible_iff_preservesFilteredColimits {F : C ⥤ D} :\n    IsFinitelyAccessible.{v'} F ↔ PreservesFilteredColimits F :=\n  IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize\n\n/-- An object `X` is finitely presentable if `Hom(X, -)` preserves all filtered colimits. -/\nabbrev IsFinitelyPresentable (X : C) : Prop :=\n  IsCardinalPresentable.{w} X ℵ₀\n\nvariable (C) in\n/-- `IsFinitelyPresentable` as an `ObjectProperty` on `C`. This is sometimes called \"compact\". -/\ndef ObjectProperty.isFinitelyPresentable : ObjectProperty C := fun X ↦ IsFinitelyPresentable.{w} X\n\nlemma ObjectProperty.isFinitelyPresentable_eq_isCardinalPresentable :\n    isFinitelyPresentable.{w} C = isCardinalPresentable.{w} C ℵ₀ :=\n  rfl\n\nvariable (C) in\n/-- A morphism `f : X ⟶ Y` is finitely presentable if it is so as an object of `Under X`. -/\ndef MorphismProperty.isFinitelyPresentable : MorphismProperty C :=\n  fun _ _ f ↦ ObjectProperty.isFinitelyPresentable.{w} _ (CategoryTheory.Under.mk f)\n\nlemma isFinitelyPresentable_iff_preservesFilteredColimitsOfSize {X : C} :\n    IsFinitelyPresentable.{w} X ↔ PreservesFilteredColimitsOfSize.{w, w} (coyoneda.obj (op X)) :=\n  Functor.IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize\n\nlemma isFinitelyPresentable_iff_preservesFilteredColimits {X : C} :\n    IsFinitelyPresentable.{v} X ↔ PreservesFilteredColimits (coyoneda.obj (op X)) :=\n  Functor.IsFinitelyAccessible_iff_preservesFilteredColimitsOfSize\n\ninstance (X : C) [IsFinitelyPresentable.{w} X] :\n    PreservesFilteredColimitsOfSize.{w, w} (coyoneda.obj (op X)) := by\n  rw [← isFinitelyPresentable_iff_preservesFilteredColimitsOfSize]\n  infer_instance\n\ninstance (X : (ObjectProperty.isFinitelyPresentable.{w} C).FullSubcategory) :\n    IsFinitelyPresentable.{w} ((ObjectProperty.isFinitelyPresentable.{w} C).ι.obj X) :=\n  X.property\n\nlemma IsFinitelyPresentable.exists_hom_of_isColimit {J : Type w} [SmallCategory J] [IsFiltered J]\n    {D : J ⥤ C} {c : Cocone D} (hc : IsColimit c) {X : C} [IsFinitelyPresentable.{w} X]\n    (f : X ⟶ c.pt) :\n    ∃ (j : J) (p : X ⟶ D.obj j), p ≫ c.ι.app j = f :=\n  Types.jointly_surjective_of_isColimit (isColimitOfPreserves (coyoneda.obj (op X)) hc) f\n\nlemma IsFinitelyPresentable.exists_eq_of_isColimit {J : Type w} [SmallCategory J] [IsFiltered J]\n    {D : J ⥤ C} {c : Cocone D} (hc : IsColimit c) {X : C} [IsFinitelyPresentable.{w} X]\n    {i j : J} (f : X ⟶ D.obj i) (g : X ⟶ D.obj j) (h : f ≫ c.ι.app i = g ≫ c.ι.app j) :\n    ∃ (k : J) (u : i ⟶ k) (v : j ⟶ k), f ≫ D.map u = g ≫ D.map v :=\n  (Types.FilteredColimit.isColimit_eq_iff _ (isColimitOfPreserves (coyoneda.obj (op X)) hc)).mp h\n\nTarget:\nlemma IsFinitelyPresentable.exists_hom_of_isColimit_under\n    {J : Type w} [SmallCategory J] [IsFiltered J] {D : J ⥤ C} {c : Cocone D} (hc : IsColimit c)\n    {X A : C} (p : X ⟶ A) (s : (Functor.const J).obj X ⟶ D)\n    [IsFinitelyPresentable.{w} (Under.mk p)]\n    (f : A ⟶ c.pt) (h : ∀ (j : J), s.app j ≫ c.ι.app j = p ≫ f) :\n    ∃ (j : J) (q : A ⟶ D.obj j), p ≫ q = s.app j ∧ q ≫ c.ι.app j = f :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  have : Nonempty J := IsFiltered.nonempty\n  let hc' := Under.isColimitLiftCocone D s c (p ≫ f) h hc\n  obtain ⟨j, q, hq⟩ := exists_hom_of_isColimit (X := Under.mk p) hc' (Under.homMk f rfl)\n  use j, q.right, Under.w q, congr($(hq).right)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Presentable","family_id":"isfinitelypresentable","file_id":"mathlib/Mathlib/CategoryTheory/Presentable/Finite.lean","sample_id":"e9c86bc71f86c9474c5b0e70638d93a26490f015f3c3db0caf5be6d93809c4ae"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a76192e154cb0c91a316e618a2d1566a7760c095bf189498c16a175bdc2813bb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9151830826aceec6eb28e4eaa5e54cc3c03de4a81418553096c5e52d8af2dcfb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e4f8810a3c97776fcd6674079eb5c88b07ed5c16b257876471689d8a0da4bfcc","source_sha256":"57eda5dc13e802c426fc6aca07052f2fd43d6d64cc1f0f72ed78d6e2a265c4fe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  generalize h' : [] = L'\n  intro h\n  rcases h with - | ⟨L₁, L₂⟩\n  simp at h'","hard_negative":true,"metrics":{"chosen_tokens":25,"rejected_tokens":5,"token_jaccard":0.038462,"token_length_ratio":0.2},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"d1f30df358af3dede6967181b7731f79ed9c80fbf3352adca0bd723b44c14a8c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Pi.Basic\npublic import Mathlib.Algebra.Group.Subgroup.Ker\npublic import Mathlib.Data.List.Chain\npublic import Mathlib.Algebra.Group.Int.Defs\npublic import Mathlib.Algebra.BigOperators.Group.List.Basic\npublic import Mathlib.Algebra.Group.Nat.Defs\npublic import Mathlib.Tactic.CrossRefAttribute\n\nNamespace:\nFreeGroup.Red\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n/-!\n# Free groups\n\nThis file defines free groups over a type. Furthermore, it is shown that the free group construction\nis an instance of a monad. For the result that `FreeGroup` is the left adjoint to the forgetful\nfunctor from groups to types, see `Mathlib/Algebra/Category/Grp/Adjunctions.lean`.\n\n## Main definitions\n\n* `FreeGroup`/`FreeAddGroup`: the free group (resp. free additive group) associated to a type\n  `α` defined as the words over `a : α × Bool` modulo the relation `a * x * x⁻¹ * b = a * b`.\n* `FreeGroup.mk`/`FreeAddGroup.mk`: the canonical quotient map `List (α × Bool) → FreeGroup α`.\n* `FreeGroup.of`/`FreeAddGroup.of`: the canonical injection `α → FreeGroup α`.\n* `FreeGroup.lift f`/`FreeAddGroup.lift`: the canonical group homomorphism `FreeGroup α →* G`\n  given a group `G` and a function `f : α → G`.\n\n## Main statements\n\n* `FreeGroup.Red.church_rosser`/`FreeAddGroup.Red.church_rosser`: The Church-Rosser theorem for word\n  reduction (also known as Newman's diamond lemma).\n* `FreeGroup.freeGroupUnitEquivInt`: The free group over the one-point type\n  is isomorphic to the integers.\n* The free group construction is an instance of a monad.\n\n## Implementation details\n\nFirst we introduce the one step reduction relation `FreeGroup.Red.Step`:\n`w * x * x⁻¹ * v   ~>   w * v`, its reflexive transitive closure `FreeGroup.Red.trans`\nand prove that its join is an equivalence relation. Then we introduce `FreeGroup α` as a quotient\nover `FreeGroup.Red.Step`.\n\nFor the additive version we introduce the same relation under a different name so that we can\ndistinguish the quotient types more easily.\n\n\n## Tags\n\nfree group, Newman's diamond lemma, Church-Rosser theorem\n-/\n\n@[expose] public section\n\nopen Relation\nopen scoped List\n\nuniverse u v w\n\nvariable {α : Type u}\n\nattribute [local simp] List.append_eq_has_append\n\n/- Ensure that `@[to_additive]` uses the right namespace before the definition of `FreeGroup`. -/\ninsert_to_additive_translation FreeGroup FreeAddGroup\n\n/-- Reduction step for the additive free group relation: `w + x + (-x) + v ~> w + v` -/\ninductive FreeAddGroup.Red.Step : List (α × Bool) → List (α × Bool) → Prop\n  | not {L₁ L₂ x b} : FreeAddGroup.Red.Step (L₁ ++ (x, b) :: (x, not b) :: L₂) (L₁ ++ L₂)\n\nattribute [simp] FreeAddGroup.Red.Step.not\n\n/-- Reduction step for the multiplicative free group relation: `w * x * x⁻¹ * v ~> w * v` -/\n@[to_additive]\ninductive FreeGroup.Red.Step : List (α × Bool) → List (α × Bool) → Prop\n  | not {L₁ L₂ x b} : FreeGroup.Red.Step (L₁ ++ (x, b) :: (x, not b) :: L₂) (L₁ ++ L₂)\n\nattribute [simp] FreeGroup.Red.Step.not\n\nnamespace FreeGroup\n\nvariable {L L₁ L₂ L₃ L₄ : List (α × Bool)}\n\n/-- Reflexive-transitive closure of `Red.Step` -/\n@[to_additive /-- Reflexive-transitive closure of `Red.Step` -/]\ndef Red : List (α × Bool) → List (α × Bool) → Prop :=\n  ReflTransGen Red.Step\n\n@[to_additive (attr := refl)]\ntheorem Red.refl : Red L L :=\n  ReflTransGen.refl\n\n@[to_additive (attr := trans)]\ntheorem Red.trans : Red L₁ L₂ → Red L₂ L₃ → Red L₁ L₃ :=\n  ReflTransGen.trans\n\nnamespace Red\n\n/-- Predicate asserting that the word `w₁` can be reduced to `w₂` in one step, i.e. there are words\n`w₃ w₄` and letter `x` such that `w₁ = w₃xx⁻¹w₄` and `w₂ = w₃w₄` -/\n@[to_additive /-- Predicate asserting that the word `w₁` can be reduced to `w₂` in one step, i.e.\nthere are words `w₃ w₄` and letter `x` such that `w₁ = w₃ + x + (-x) + w₄` and `w₂ = w₃w₄` -/]\ntheorem Step.length : ∀ {L₁ L₂ : List (α × Bool)}, Step L₁ L₂ → L₂.length + 2 = L₁.length\n  | _, _, @Red.Step.not _ L1 L2 x b => by rw [List.length_append, List.length_append]; rfl\n\n@[to_additive (attr := simp)]\ntheorem Step.not_rev {x b} : Step (L₁ ++ (x, !b) :: (x, b) :: L₂) (L₁ ++ L₂) := by\n  cases b <;> exact Step.not\n\n@[to_additive (attr := simp)]\ntheorem Step.cons_not {x b} : Red.Step ((x, b) :: (x, !b) :: L) L :=\n  @Step.not _ [] _ _ _\n\n@[to_additive (attr := simp)]\ntheorem Step.cons_not_rev {x b} : Red.Step ((x, !b) :: (x, b) :: L) L :=\n  @Red.Step.not_rev _ [] _ _ _\n\n@[to_additive]\ntheorem Step.append_left : ∀ {L₁ L₂ L₃ : List (α × Bool)}, Step L₂ L₃ → Step (L₁ ++ L₂) (L₁ ++ L₃)\n  | _, _, _, Red.Step.not => by rw [← List.append_assoc, ← List.append_assoc]; constructor\n\n@[to_additive]\ntheorem Step.cons {x} (H : Red.Step L₁ L₂) : Red.Step (x :: L₁) (x :: L₂) :=\n  @Step.append_left _ [x] _ _ H\n\n@[to_additive]\n\nTarget:\ntheorem Step.append_right : ∀ {L₁ L₂ L₃ : List (α × Bool)}, Step L₁ L₂ → Step (L₁ ++ L₃) (L₂ ++ L₃)\n  | _, _, _, Red.Step.not => by simp\n\n@[to_additive]\ntheorem not_step_nil : ¬Step [] L :=\n\nProof body:\n","rejected":"by\n  exact Step.append_right","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"b72b9cec413a2cde1dcf1bc5ef406dcde1093df9810a38bef1e24456eca49488","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/FreeGroup","family_id":"step","file_id":"mathlib/Mathlib/GroupTheory/FreeGroup/Basic.lean","sample_id":"e4f8810a3c97776fcd6674079eb5c88b07ed5c16b257876471689d8a0da4bfcc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"12eef625fe3216831384e04422da6efdbadfd13d594c33e0c21289712e29f94e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3a3c329ac7ef7bd97b6db71b43a7efe1eaaa001c53c4b487b813a42f94dd4a0c","source_sha256":"d7136b89194da5b89655bfe0eeb14ceccd917f2ca80d6a836d008fec03fc59c2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  rw [monomial_def]\n  exact Pi.single_eq_same _ _","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.166667},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"d22bb922cbc5ab42ea810114dfc460eedb61a594b273b5b6722fab3704763f32","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Antidiag.Finsupp\npublic import Mathlib.Data.Finsupp.Weight\npublic import Mathlib.LinearAlgebra.Pi\npublic import Mathlib.Algebra.MvPolynomial.Basic\npublic import Mathlib.Tactic.NormNum\n\nNamespace:\nMvPowerSeries\n\nLocal context:\n/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Kenny Lau\n-/\n/-!\n# Formal (multivariate) power series\n\nThis file defines multivariate formal power series\nand develops the basic properties of these objects.\n\nA formal power series is to a polynomial like an infinite sum is to a finite sum.\n\nWe provide the natural inclusion from multivariate polynomials to multivariate formal power series.\n\n## Main definitions\n\n- `MvPowerSeries.C`: constant power series\n\n- `MvPowerSeries.X`: the indeterminates\n\n- `MvPowerSeries.coeff`, `MvPowerSeries.constantCoeff`:\n  the coefficients of a `MvPowerSeries`, its constant coefficient\n\n- `MvPowerSeries.monomial`: the monomials\n\n- `MvPowerSeries.coeff_mul`: computes the coefficients of the product of two `MvPowerSeries`\n\n- `MvPowerSeries.coeff_prod` : computes the coefficients of products of `MvPowerSeries`\n\n- `MvPowerSeries.coeff_pow` : computes the coefficients of powers of a `MvPowerSeries`\n\n- `MvPowerSeries.coeff_eq_zero_of_constantCoeff_nilpotent`: if the constant coefficient\n  of a `MvPowerSeries` is nilpotent, then some coefficients of its powers are automatically zero\n\n- `MvPowerSeries.map`: apply a `RingHom` to the coefficients of a `MvPowerSeries` (as a `RingHom`).\n\n- `MvPowerSeries.X_pow_dvd_iff`, `MvPowerSeries.X_dvd_iff`: equivalent\n  conditions for (a power of) an indeterminate to divide a `MvPowerSeries`\n\n- `MvPolynomial.toMvPowerSeries`: the canonical coercion from `MvPolynomial` to `MvPowerSeries`\n\n\n## Note\n\nThis file sets up the (semi)ring structure on multivariate power series:\nadditional results are in:\n* `Mathlib/RingTheory/MvPowerSeries/Inverse.lean` : invertibility,\n  formal power series over a local ring form a local ring;\n* `Mathlib/RingTheory/MvPowerSeries/Trunc.lean`: truncation of power series.\n\nIn `Mathlib/RingTheory/PowerSeries/Basic.lean`, formal power series in one variable\nwill be obtained as a particular case, defined by\n  `PowerSeries R := MvPowerSeries Unit R`.\nSee that file for a specific description.\n\n## Implementation notes\n\nIn this file we define multivariate formal power series with\nvariables indexed by `σ` and coefficients in `R` as\n`MvPowerSeries σ R := (σ →₀ ℕ) → R`.\nUnfortunately there is not yet enough API to show that they are the completion\nof the ring of multivariate polynomials. However, we provide most of the infrastructure\nthat is needed to do this. Once I-adic completion (topological or algebraic) is available\nit should not be hard to fill in the details.\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen Finset (antidiagonal mem_antidiagonal)\n\n/-- Multivariate formal power series, where `σ` is the index set of the variables\nand `R` is the coefficient ring. -/\ndef MvPowerSeries (σ : Type*) (R : Type*) :=\n  (σ →₀ ℕ) → R\n\nnamespace MvPowerSeries\n\nopen Finsupp\n\nvariable {σ R : Type*}\n\ninstance [Inhabited R] : Inhabited (MvPowerSeries σ R) :=\n  inferInstanceAs <| Inhabited ((σ →₀ ℕ) → R)\n\ninstance [Zero R] : Zero (MvPowerSeries σ R) :=\n  inferInstanceAs <| Zero ((σ →₀ ℕ) → R)\n\ninstance [AddMonoid R] : AddMonoid (MvPowerSeries σ R) :=\n  inferInstanceAs <| AddMonoid ((σ →₀ ℕ) → R)\n\ninstance [AddGroup R] : AddGroup (MvPowerSeries σ R) :=\n  inferInstanceAs <| AddGroup ((σ →₀ ℕ) → R)\n\ninstance [AddCommMonoid R] : AddCommMonoid (MvPowerSeries σ R) :=\n  inferInstanceAs <| AddCommMonoid ((σ →₀ ℕ) → R)\n\ninstance [AddCommGroup R] : AddCommGroup (MvPowerSeries σ R) :=\n  inferInstanceAs <| AddCommGroup ((σ →₀ ℕ) → R)\n\ninstance [Nontrivial R] : Nontrivial (MvPowerSeries σ R) :=\n  inferInstanceAs <| Nontrivial ((σ →₀ ℕ) → R)\n\ninstance {A} [Semiring R] [AddCommMonoid A] [Module R A] : Module R (MvPowerSeries σ A) :=\n  inferInstanceAs <| Module R ((σ →₀ ℕ) → A)\n\ninstance {A S} [Semiring R] [Semiring S] [AddCommMonoid A] [Module R A] [Module S A] [SMul R S]\n    [IsScalarTower R S A] : IsScalarTower R S (MvPowerSeries σ A) :=\n  inferInstanceAs <| IsScalarTower R S ((σ →₀ ℕ) → A)\n\nsection Semiring\n\nvariable [Semiring R]\n\n/-- The `n`th monomial as multivariate formal power series:\n  it is defined as the `R`-linear map from `R` to the semiring\n  of multivariate formal power series associating to each `a`\n  the map sending `n : σ →₀ ℕ` to the value `a`\n  and sending all other `x : σ →₀ ℕ` different from `n` to `0`. -/\ndef monomial (n : σ →₀ ℕ) : R →ₗ[R] MvPowerSeries σ R :=\n  letI := Classical.decEq σ\n  LinearMap.single R (fun _ ↦ R) n\n\n/-- The `n`th coefficient of a multivariate formal power series. -/\ndef coeff (n : σ →₀ ℕ) : MvPowerSeries σ R →ₗ[R] R :=\n  LinearMap.proj n\n\ntheorem coeff_apply (f : MvPowerSeries σ R) (d : σ →₀ ℕ) : coeff d f = f d :=\n  rfl\n\n/-- Two multivariate formal power series are equal if all their coefficients are equal. -/\n@[ext]\ntheorem ext {φ ψ : MvPowerSeries σ R} (h : ∀ n : σ →₀ ℕ, coeff n φ = coeff n ψ) : φ = ψ :=\n  funext h\n\n/-- Two multivariate formal power series are equal\nif and only if all their coefficients are equal. -/\nadd_decl_doc MvPowerSeries.ext_iff\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem monomial_def [DecidableEq σ] (n : σ →₀ ℕ) :\n    monomial n = LinearMap.single R (fun _ ↦ R) n := by\n  rw [monomial]\n  -- unify the `Decidable` arguments\n  convert! rfl\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem coeff_monomial [DecidableEq σ] (m n : σ →₀ ℕ) (a : R) :\n    coeff m (monomial n a) = if m = n then a else 0 := by\n  dsimp only [coeff, MvPowerSeries]\n  rw [monomial_def, LinearMap.proj_apply (i := m), LinearMap.single_apply, Pi.single_apply]\n\n@[simp]\n\nTarget:\ntheorem coeff_monomial_same (n : σ →₀ ℕ) (a : R) : coeff n (monomial n a) = a :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/MvPowerSeries","family_id":"coeff_monomial_same","file_id":"mathlib/Mathlib/RingTheory/MvPowerSeries/Basic.lean","sample_id":"3a3c329ac7ef7bd97b6db71b43a7efe1eaaa001c53c4b487b813a42f94dd4a0c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c3ee73588e36d5e89a2703d1667734bc0b58f87dcefce1c40a3828ad044e5637","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"925f0d2e6688d51b9087e7f5c0acec178d9cb8d6f92572758a5fcf70179d3744","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"812ef1c18921780c46e19c5df601fc11ffc84bd9969d3bb154d739df4d388862","source_sha256":"345dde00cb75d3f9bc3f5909b7969856d423edbbc6d0a61a7a141d42b1e88b8d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [ringKrullDim, ringKrullDim]\n  apply Order.krullDim_le_of_krullDim_preimage_le' (PrimeSpectrum.comap C) ?_ (fun p ↦ ?_)\n  · exact fun {a b} h ↦ Ideal.comap_mono h\n  · rw [show C = (algebraMap R (Polynomial R)) from rfl, Order.krullDim_eq_of_orderIso\n      (PrimeSpectrum.preimageOrderIsoFiber R (Polynomial R) p), ← ringKrullDim,\n      ← ringKrullDim_eq_of_ringEquiv (polyEquivTensor R (p.asIdeal.ResidueField)).toRingEquiv,\n      ← Ring.krullDimLE_iff]\n    infer_instance\n\nvariable {R : Type*} [CommRing R] [IsNoetherianRing R]","hard_negative":false,"metrics":{"chosen_tokens":111,"rejected_tokens":115,"token_jaccard":0.946429,"token_length_ratio":1.036036},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"d265dd4b7438b1716ec0c609c1f979f3ee417741d6c699c7cf76fe83ebb9d76e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.FieldDivision\npublic import Mathlib.RingTheory.KrullDimension.PID\npublic import Mathlib.RingTheory.LocalRing.ResidueField.Fiber\npublic import Mathlib.RingTheory.Ideal.KrullsHeightTheorem\npublic import Mathlib.RingTheory.KrullDimension.NonZeroDivisors\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Jingting Wang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jingting Wang, Sihan Su, Yi Song, Christian Merten\n-/\n/-!\n# Krull dimension of polynomial ring\n\nThis file proves properties of the Krull dimension of the polynomial ring over a commutative ring\n\n## Main results\n\n* `Polynomial.ringKrullDim_le`: the Krull dimension of the polynomial ring over a commutative ring\n  `R` is less than `2 * (ringKrullDim R) + 1`.\n\nFor noetherian rings:\n* `Polynomial.ringKrullDim_of_isNoetherianRing`: the Krull dimension of `R[X]` is `dim R + 1`.\n* `MvPolynomial.ringKrullDim_of_isNoetherianRing`: the Krull dimension of `R[X₁, ..., Xₙ]` is\n  `dim R + n`.\n-/\n\npublic section\n\nTarget:\ntheorem Polynomial.ringKrullDim_le {R : Type*} [CommRing R] :\n    ringKrullDim (Polynomial R) ≤ 2 * (ringKrullDim R) + 1 :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [ringKrullDim, ringKrullDim]\n  apply Order.krullDim_le_of_krullDim_preimage_le' (PrimeSpectrum.comap C) ?_ (fun p ↦ ?_)\n  · exact fun {a b} h ↦ Ideal.comap_mono h\n  · rw [show C = (algebraMap R (Polynomial R)) from rfl, Order.krullDim_eq_of_orderIso\n      (PrimeSpectrum.preimageOrderIsoFiber R (Polynomial R) p), ← ringKrullDim,\n      ← ringKrullDim_eq_of_ringEquiv (polyEquivTensor R (p.asIdeal.ResidueField)).toRingEquiv,\n      ← Ring.krullDimLE_iff]\n    infer_instance\n\nvariable {R : Type*} [CommRing R] [IsNoetherianRing R]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/KrullDimension","family_id":"polynomial","file_id":"mathlib/Mathlib/RingTheory/KrullDimension/Polynomial.lean","sample_id":"812ef1c18921780c46e19c5df601fc11ffc84bd9969d3bb154d739df4d388862"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7cabac5c0510d840ec215a55b2c97b8bd4ff4bb3ba7ca644c381e4c621c7c4ab","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cd50c93d72cf5c91929bb13dde2673ec356e0e8765b54ae8e9771032cf42ce8c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b76866134268ce8c1e930e844ff2aaa363a473acc169ff6b4db1a719899e0bd6","source_sha256":"354c54d863521e4c2e2cb8eb45baaec320a74608e472a8d10bc34f45771b2911","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction rs generalizing M N with\n  | nil => simp\n  | cons x _ ih =>\n    simp only [List.map_cons, isWeaklyRegular_cons_iff] at reg ⊢\n    have e := (QuotSMulTop.algebraMapTensorEquivTensorQuotSMulTop x M S).symm ≪≫ₗ\n      QuotSMulTop.congr ((algebraMap R S) x) hf.equiv\n    have hg : IsBaseChange S <|\n        e.toLinearMap.restrictScalars R ∘ₗ TensorProduct.mk R S (QuotSMulTop x M) 1 :=\n      IsBaseChange.of_equiv e (fun _ ↦ by simp)\n    exact ⟨reg.1.of_flat_of_isBaseChange hf, ih hg reg.2⟩","hard_negative":true,"metrics":{"chosen_tokens":112,"rejected_tokens":5,"token_jaccard":0.047619,"token_length_ratio":0.044643},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"d29e74057896feebd1a9c1fff232618ebcc965167d08999b3e817092c724277d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Flat.FaithfullyFlat.Basic\npublic import Mathlib.RingTheory.Flat.Localization\npublic import Mathlib.RingTheory.Regular.RegularSequence\n\nNamespace:\nRingTheory.Sequence\n\nLocal context:\n/-\nCopyright (c) 2025 Yongle Hu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yongle Hu, Nailin Guan\n-/\n/-!\n# `RingTheory.Sequence.IsWeaklyRegular` is stable under flat base change\n\n## Main results\n* `RingTheory.Sequence.IsWeaklyRegular.of_flat_of_isBaseChange`: Let `R` be a commutative ring,\n  `M` be an `R`-module, `S` be a flat `R`-algebra, `N` be the base change of `M` to `S`.\n  If `[r₁, …, rₙ]` is a weakly regular `M`-sequence, then its image in `N` is a weakly regular\n  `N`-sequence.\n-/\n\npublic section\n\nnamespace RingTheory.Sequence\n\nopen Module\n\nvariable {R S M N : Type*} [CommRing R] [CommRing S] [Algebra R S]\n  [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower R S N]\n\n/-- Let `R` be a commutative ring, `M` be an `R`-module, `S` be a flat `R`-algebra, `N` be the base\n  change of `M` to `S`. If `[r₁, …, rₙ]` is a weakly regular `M`-sequence, then its image in `N` is\n  a weakly regular `N`-sequence. -/\n\nTarget:\ntheorem IsWeaklyRegular.of_flat_of_isBaseChange [Flat R S] {f : M →ₗ[R] N} (hf : IsBaseChange S f)\n    {rs : List R} (reg : IsWeaklyRegular M rs) : IsWeaklyRegular N (rs.map (algebraMap R S)) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_b76866134268","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"ab723cea329cf8c87a6348c867aa4f26160a5c2d28f040fc58edf0f34b0d425f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Regular","family_id":"isweaklyregular","file_id":"mathlib/Mathlib/RingTheory/Regular/Flat.lean","sample_id":"b76866134268ce8c1e930e844ff2aaa363a473acc169ff6b4db1a719899e0bd6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"62c5d9390a895014879439c4bd46fd00c9783796de9ef2a06a744f2c74871aea","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0f39df25f86b79636ec8efd74a8a03a810910ef6427851948542bfbbe91d6d31","source_sha256":"0e2737adb7e8846513bd14b3c89ab969bcc977f3e683f90a3402f41d55ebbfc2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply_fun ((Sigma.desc fun a ↦ pullback.fst g₁ (Sigma.ι X a)) ≫ ·) using\n    (fun a b ↦ (cancel_epi _).mp)\n  ext a\n  simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_ι_app]\n  rw [← Category.assoc, pullback.condition]\n  simp only [Category.assoc, colimit.ι_desc, Cofan.mk_ι_app]\n  apply_fun ((Sigma.desc fun a ↦ pullback.fst (pullback.fst _ _ ≫ g₂) (Sigma.ι X a)) ≫ ·)\n    using (fun a b ↦ (cancel_epi _).mp)\n  ext b\n  simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_ι_app]\n  simp only [← Category.assoc]\n  rw [(Category.assoc _ _ g₂), pullback.condition]\n  simp only [Category.assoc, colimit.ι_desc, Cofan.mk_ι_app]\n  rw [← Category.assoc]\n  apply h\n  apply_fun (pullback.fst g₁ (Sigma.ι X a) ≫ ·) at hg\n  rw [← Category.assoc, pullback.condition] at hg\n  simp only [Category.assoc, colimit.ι_desc, Cofan.mk_ι_app] at hg\n  apply_fun ((Sigma.ι (fun a ↦ pullback _ _) b) ≫ (Sigma.desc fun a ↦\n    pullback.fst (pullback.fst _ _ ≫ g₂) (Sigma.ι X a)) ≫ ·) at hg\n  simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_ι_app] at hg\n  simp only [← Category.assoc] at hg\n  rw [(Category.assoc _ _ g₂), pullback.condition] at hg\n  simpa using hg","hard_negative":true,"metrics":{"chosen_tokens":358,"rejected_tokens":8,"token_jaccard":0.018519,"token_length_ratio":0.022346},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"d3374488ee30b4ea292b548c3eb7ced006637d64263dbf0c17596d355882cadd","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.EffectiveEpi.Basic\npublic import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback\npublic import Mathlib.Tactic.ApplyFun\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2023 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# Effective epimorphic families and coproducts\n\nThis file proves that an effective epimorphic family induces an effective epi from the coproduct if\nthe coproduct exists, and the converse under some more conditions on the coproduct (that it\ninteracts well with pullbacks).\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Limits\n\nvariable {C : Type*} [Category* C]\n\nset_option backward.isDefEq.respectTransparency false in\n/--\nGiven an `EffectiveEpiFamily X π` and a corresponding coproduct cocone, the family descends to an\n`EffectiveEpi` from the coproduct.\n-/\nnoncomputable\ndef effectiveEpiStructIsColimitDescOfEffectiveEpiFamily {B : C} {α : Type*} (X : α → C)\n    (c : Cofan X) (hc : IsColimit c) (π : (a : α) → (X a ⟶ B)) [EffectiveEpiFamily X π] :\n    EffectiveEpiStruct (hc.desc (Cofan.mk B π)) where\n  desc e h := EffectiveEpiFamily.desc X π (fun a ↦ c.ι.app ⟨a⟩ ≫ e) (fun a₁ a₂ g₁ g₂ hg ↦ by\n    simp only [← Category.assoc]\n    exact h (g₁ ≫ c.ι.app ⟨a₁⟩) (g₂ ≫ c.ι.app ⟨a₂⟩) (by simpa))\n  fac e h := hc.hom_ext (fun ⟨j⟩ ↦ (by simp))\n  uniq e _ m hm := EffectiveEpiFamily.uniq X π (fun a ↦ c.ι.app ⟨a⟩ ≫ e)\n      (fun _ _ _ _ hg ↦ (by simp [← hm, reassoc_of% hg])) m (fun _ ↦ (by simp [← hm]))\n\nset_option backward.defeqAttrib.useBackward true in\ninstance {B : C} {α : Type*} (X : α → C) (π : (a : α) → (X a ⟶ B)) [HasCoproduct X]\n    [EffectiveEpiFamily X π] : EffectiveEpi (Sigma.desc π) := by\n  let e := effectiveEpiStructIsColimitDescOfEffectiveEpiFamily X _ (coproductIsCoproduct _) π\n  simp only [Cofan.mk_pt, coproductIsCoproduct, colimit.cocone_x, IsColimit.ofIsoColimit_desc,\n    Cocone.ext_inv_hom, Iso.refl_inv, colimit.isColimit_desc, Category.id_comp] at e\n  exact ⟨⟨e⟩⟩\n\nexample {B : C} {α : Type*} (X : α → C) (π : (a : α) → (X a ⟶ B)) [EffectiveEpiFamily X π]\n    [HasCoproduct X] : Epi (Sigma.desc π) := inferInstance\n\nset_option backward.isDefEq.respectTransparency false in\n/--\nThis is an auxiliary lemma used twice in the definition of  `EffectiveEpiFamilyOfEffectiveEpiDesc`.\nIt is the `h` hypothesis of `EffectiveEpi.desc` and `EffectiveEpi.fac`.\n-/\n\nTarget:\ntheorem effectiveEpiFamilyStructOfEffectiveEpiDesc_aux {B : C} {α : Type*} {X : α → C}\n    {π : (a : α) → X a ⟶ B} [HasCoproduct X]\n    [∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)]\n    [∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a ↦ pullback g (Sigma.ι X a)]\n    [∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a ↦ pullback.fst g (Sigma.ι X a))]\n    {W : C} {e : (a : α) → X a ⟶ W} (h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂),\n      g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂) {Z : C}\n    {g₁ g₂ : Z ⟶ ∐ fun b ↦ X b} (hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π) :\n    g₁ ≫ Sigma.desc e = g₂ ≫ Sigma.desc e :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"0c50d6b1cc0f95d6accb96a7306cfcff78bb1f3be19aa9d1439567e546878b55","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/EffectiveEpi","family_id":"effectiveepifamilystructofeffectiveepidesc_aux","file_id":"mathlib/Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean","sample_id":"0f39df25f86b79636ec8efd74a8a03a810910ef6427851948542bfbbe91d6d31"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d69154e0d9492b7c81193d7944884dcb10496ed41d6438469a8fe12e15c51a76","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6a8521660d4ef846f6eee10f1b6a7111248906a80446120366f799acf8507e31","source_sha256":"6c1d4f9fced239f6f8c0c8785a4325f383edabba30747fe08c8262d3408cd815","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h => ?_, fun ⟨⟨a', m, e⟩, s⟩ => Subset.cons e m s⟩\n  generalize h' : Lists'.cons a l₁ = l₁' at h\n  obtain - | @⟨a', _, _, _, e, m, s⟩ := h\n  · cases a\n    cases h'\n  cases a; cases a'; cases h'; exact ⟨⟨_, m, e⟩, s⟩","hard_negative":false,"metrics":{"chosen_tokens":91,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.032967},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"d385ff832830447ab50576842da49a6cf4f36facfb3f9dac411653dbe7b2aa5e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.Monoid.NatCast\npublic import Mathlib.Algebra.Ring.Nat\npublic import Mathlib.Data.Sigma.Basic\npublic import Batteries.Tactic.Lint.TypeClass\n\nNamespace:\nLists'\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# A computable model of ZFA without infinity\n\nIn this file we define finite hereditary lists. This is useful for calculations in naive set theory.\n\nWe distinguish two kinds of ZFA lists:\n* Atoms. Directly correspond to an element of the original type.\n* Proper ZFA lists. Can be thought of (but are not implemented) as a list of ZFA lists (not\n  necessarily proper).\n\nFor example, `Lists ℕ` contains stuff like `23`, `[]`, `[37]`, `[1, [[2], 3], 4]`.\n\n## Implementation note\n\nAs we want to be able to append both atoms and proper ZFA lists to proper ZFA lists, it's handy that\natoms and proper ZFA lists belong to the same type, even though atoms of `α` could be modelled as\n`α` directly. But we don't want to be able to append anything to atoms.\n\nThis calls for a two-step definition of ZFA lists:\n* First, define ZFA prelists as atoms and proper ZFA prelists. Those proper ZFA prelists are defined\n  by inductive appending of (not necessarily proper) ZFA lists.\n* Second, define ZFA lists by rubbing out the distinction between atoms and proper lists.\n\n## Main declarations\n\n* `Lists' α false`: Atoms as ZFA prelists. Basically a copy of `α`.\n* `Lists' α true`: Proper ZFA prelists. Defined inductively from the empty ZFA prelist\n  (`Lists'.nil`) and from appending a ZFA prelist to a proper ZFA prelist (`Lists'.cons a l`).\n* `Lists α`: ZFA lists. Sum of the atoms and proper ZFA prelists.\n* `Finsets α`: ZFA sets. Defined as `Lists` quotiented by `Lists.Equiv`, the extensional\n  equivalence.\n-/\n\n@[expose] public section\n\n\nvariable {α : Type*}\n\n/-- Prelists, helper type to define `Lists`. `Lists' α false` are the \"atoms\", a copy of `α`.\n`Lists' α true` are the \"proper\" ZFA prelists, inductively defined from the empty ZFA prelist and\nfrom appending a ZFA prelist to a proper ZFA prelist. It is made so that you can't append anything\nto an atom while having only one appending function for appending both atoms and proper ZFA prelists\nto a proper ZFA prelist. -/\ninductive Lists'.{u} (α : Type u) : Bool → Type u\n  | atom : α → Lists' α false\n  | nil : Lists' α true\n  | cons' {b} : Lists' α b → Lists' α true → Lists' α true\n  deriving DecidableEq\ncompile_inductive% Lists'\n\n/-- Hereditarily finite list, aka ZFA list. A ZFA list is either an \"atom\" (`b = false`),\ncorresponding to an element of `α`, or a \"proper\" ZFA list, inductively defined from the empty ZFA\nlist and from appending a ZFA list to a proper ZFA list. -/\ndef Lists (α : Type*) :=\n  Σ b, Lists' α b\n\nnamespace Lists'\n\ninstance [Inhabited α] : ∀ b, Inhabited (Lists' α b)\n  | true => ⟨nil⟩\n  | false => ⟨atom default⟩\n\n/-- Appending a ZFA list to a proper ZFA prelist. -/\ndef cons : Lists α → Lists' α true → Lists' α true\n  | ⟨_, a⟩, l => cons' a l\n\n/-- Converts a ZFA prelist to a `List` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp]\ndef toList : ∀ {b}, Lists' α b → List (Lists α)\n  | _, atom _ => []\n  | _, nil => []\n  | _, cons' a l => ⟨_, a⟩ :: l.toList\n\n@[simp]\ntheorem toList_cons (a : Lists α) (l) : toList (cons a l) = a :: l.toList := rfl\n\n/-- Converts a `List` of ZFA lists to a proper ZFA prelist. -/\n@[simp]\ndef ofList : List (Lists α) → Lists' α true\n  | [] => nil\n  | a :: l => cons a (ofList l)\n\n@[simp]\ntheorem to_ofList (l : List (Lists α)) : toList (ofList l) = l := by induction l <;> simp [*]\n\n@[simp]\ntheorem of_toList : ∀ l : Lists' α true, ofList (toList l) = l :=\n  suffices ∀ (b) (h : true = b) (l : Lists' α b),\n      let l' : Lists' α true := h ▸ l\n      ofList (toList l') = l'\n    from this _ rfl\n  fun b h l => by\n    induction l with\n    | atom => cases h\n    | nil => simp\n    | cons' b a _ IH => simpa [cons] using IH rfl\n\n/-- Recursion/induction principle for `Lists'.ofList`. -/\n@[elab_as_elim]\ndef recOfList {motive : Lists' α true → Sort*} (ofList : ∀ l, motive (ofList l)) : ∀ l, motive l :=\n  fun l ↦ cast (by simp) <| ofList (l.toList)\n\nend Lists'\n\nmutual\n  /-- Equivalence of ZFA lists. Defined inductively. -/\n  inductive Lists.Equiv : Lists α → Lists α → Prop\n    | refl (l) : Lists.Equiv l l\n    | antisymm {l₁ l₂ : Lists' α true} :\n      Lists'.Subset l₁ l₂ → Lists'.Subset l₂ l₁ → Lists.Equiv ⟨_, l₁⟩ ⟨_, l₂⟩\n\n  /-- Subset relation for ZFA lists. Defined inductively. -/\n  inductive Lists'.Subset : Lists' α true → Lists' α true → Prop\n    | nil {l} : Lists'.Subset Lists'.nil l\n    | cons {a a' l l'} :\n      Lists.Equiv a a' →\n        a' ∈ Lists'.toList l' → Lists'.Subset l l' → Lists'.Subset (Lists'.cons a l) l'\nend\n\nlocal infixl:50 \" ~ \" => Lists.Equiv\n\nnamespace Lists'\n\ninstance : HasSubset (Lists' α true) :=\n  ⟨Lists'.Subset⟩\n\n/-- ZFA prelist membership. A ZFA list is in a ZFA prelist if some element of this ZFA prelist is\nequivalent as a ZFA list to this ZFA list. -/\ninstance {b} : Membership (Lists α) (Lists' α b) :=\n  ⟨fun l a => ∃ a' ∈ l.toList, a ~ a'⟩\n\ntheorem mem_def {b a} {l : Lists' α b} : a ∈ l ↔ ∃ a' ∈ l.toList, a ~ a' :=\n  Iff.rfl\n\n@[simp]\ntheorem mem_cons {a y l} : a ∈ @cons α y l ↔ a ~ y ∨ a ∈ l := by\n  simp [mem_def, or_and_right, exists_or]\n\nTarget:\ntheorem cons_subset {a} {l₁ l₂ : Lists' α true} : Lists'.cons a l₁ ⊆ l₂ ↔ a ∈ l₂ ∧ l₁ ⊆ l₂ :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory","family_id":"cons_subset","file_id":"mathlib/Mathlib/SetTheory/Lists.lean","sample_id":"6a8521660d4ef846f6eee10f1b6a7111248906a80446120366f799acf8507e31"},"split":"train"}
{"binding":{"chosen_proof_sha256":"89a379c0af7b46c3f249bf357aea100510d92246e204cc5b623f3d9530166237","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4c75218715d8106b241c827488496f008634ee5b9e0f5a4864174b7282c10c31","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"58255fb9a08e3ea4853832b4b2e6754cd8d2484fe722ffff05ba10373b5631a3","source_sha256":"4b6a49b0497716f9a81e32d511b44e7e3917e3a9d925e53fe50fefb2d06728f2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← LieSubalgebra.toSubmodule_inj, range_toSubmodule, LieIdeal.toLieSubalgebra_toSubmodule,\n   ker_toSubmodule]\n  exact LinearMap.range_inl R L₁ L₂","hard_negative":true,"metrics":{"chosen_tokens":25,"rejected_tokens":2,"token_jaccard":0.047619,"token_length_ratio":0.08},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"d393e29c6c0b4b23685331b0849da0bb026d2c492afec5165fc29ee5c53ce18c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Lie.Ideal\n\nNamespace:\nLieHom\n\nLocal context:\n/-\nCopyright (c) 2026 Leonid Ryvkin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonid Ryvkin\n-/\n/-! ### Products of Lie algebras\n\nThis file defines the Lie algebra structure the Product of two Lie algebras\n\n## Main definitions\n\n- products in the domain:\n  - `LieHom.fst` The first projection of a product is a Lie algebra map.\n  - `LieHom.snd` The second projection of a product is a Lie algebra map.\n  - `LieHom.prod_ext` Split equality of Lie algebra homomorphisms from a product into Lie algebra\n  homomorphism over each component,\n- products in the codomain:\n  - `LieHom.inl` The left injection into a product is a Lie algebra map.\n  - `LieHom.inr` The right injection into a product is a Lie algebra map.\n  - `LieHom.prod` The prod of two Lie algebra homomorphisms is a Lie algebra homomorphism.\n- products in both domain and codomain:\n  - `LieHom.prodMap` the `Prod.map` of two Lie algebra homomorphisms is a Lie algebra homomorphism.\n\n## Todo: Extend to further functionality from LinearMap.prod e.g.\n - Lie Equivalences related to products\n - Lie Submodule statements\n\n-/\n\n@[expose] public section\n\nvariable {R L₁ L₂ L L₃ L₄ L₅ L₆ : Type*}\n  [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂]\n  [LieRing L] [LieAlgebra R L] [LieRing L₃] [LieAlgebra R L₃] [LieRing L₄] [LieAlgebra R L₄]\n  [LieRing L₅] [LieAlgebra R L₅] [LieRing L₆] [LieAlgebra R L₆]\n\nnamespace LieAlgebra.Prod\n\ninstance instLieRing : LieRing (L₁ × L₂) where\n  bracket x y := ⟨⁅x.1, y.1⁆, ⁅x.2, y.2⁆⟩\n  add_lie := by simp\n  lie_add := by simp\n  lie_self := by simp\n  leibniz_lie := by simp\n\n@[simp]\ntheorem bracket_apply (x y : L₁ × L₂) : ⁅x, y⁆ = ⟨⁅x.1, y.1⁆, ⁅x.2, y.2⁆⟩ := rfl\n\ninstance instLieAlgebra : LieAlgebra R (L₁ × L₂) where\n  lie_smul _ _ _ := by simp\n\nend LieAlgebra.Prod\n\nnamespace LieHom\n\nsection\nvariable (R L₁ L₂)\n\n/-- The first projection of a product is a Lie algebra map. -/\ndef fst : L₁ × L₂ →ₗ⁅R⁆ L₁ where\n  toLinearMap := LinearMap.fst R L₁ L₂\n  map_lie' := by simp\n\n/-- The second projection of a product is a Lie algebra map. -/\ndef snd : L₁ × L₂ →ₗ⁅R⁆ L₂ where\n  toLinearMap := LinearMap.snd R L₁ L₂\n  map_lie' := by simp\n\n/-- The left injection into a product is a Lie algebra map. -/\ndef inl : L₁ →ₗ⁅R⁆ L₁ × L₂ where\n  toLinearMap := LinearMap.inl R L₁ L₂\n  map_lie' := by simp\n\n/-- The right injection into a product is a Lie algebra map. -/\ndef inr : L₂ →ₗ⁅R⁆ L₁ × L₂ where\n  toLinearMap := LinearMap.inr R L₁ L₂\n  map_lie' := by simp\n\nend\n\n@[simp] theorem fst_apply (x : L₁ × L₂) : fst R L₁ L₂ x = x.1 := rfl\n\n@[simp] theorem snd_apply (x : L₁ × L₂) : snd R L₁ L₂ x = x.2 := rfl\n\n@[simp, norm_cast] lemma coe_fst : ⇑(fst R L₁ L₂) = Prod.fst := rfl\n\n@[simp, norm_cast] lemma coe_snd : ⇑(snd R L₁ L₂) = Prod.snd := rfl\n\ntheorem fst_surjective : Function.Surjective (fst R L₁ L₂) := fun x => ⟨(x, 0), rfl⟩\n\ntheorem snd_surjective : Function.Surjective (snd R L₁ L₂) := fun x => ⟨(0, x), rfl⟩\n\n/-- The prod of two Lie algebra homomorphisms is a Lie algebra homomorphism. -/\n@[simps!]\ndef prod (f : L →ₗ⁅R⁆ L₁) (g : L →ₗ⁅R⁆ L₂) : L →ₗ⁅R⁆ L₁ × L₂ where\n  toLinearMap := LinearMap.prod f g\n  map_lie' := by simp\n\ntheorem coe_prod (f : L →ₗ⁅R⁆ L₁) (g : L →ₗ⁅R⁆ L₂) : ⇑(f.prod g) = Function.prod f g :=\n  rfl\n\n@[simp]\ntheorem fst_prod (f : L →ₗ⁅R⁆ L₁) (g : L →ₗ⁅R⁆ L₂) : (fst R L₁ L₂).comp (prod f g) = f := rfl\n\n@[simp]\ntheorem snd_prod (f : L →ₗ⁅R⁆ L₁) (g : L →ₗ⁅R⁆ L₂) : (snd R L₁ L₂).comp (prod f g) = g := rfl\n\n@[simp]\ntheorem pair_fst_snd : prod (fst R L₁ L₂) (snd R L₁ L₂) = LieHom.id := rfl\n\ntheorem prod_comp (f : L₁ →ₗ⁅R⁆ L₂) (g : L₁ →ₗ⁅R⁆ L)\n    (h : L →ₗ⁅R⁆ L₁) : (f.prod g).comp h = (f.comp h).prod (g.comp h) :=\n  rfl\n\ntheorem inl_apply (x : L₁) : inl R L₁ L₂ x = (x, 0) := rfl\n\ntheorem inr_apply (x : L₂) : inr R L₁ L₂ x = (0, x) := rfl\n\n@[simp] theorem coe_inl : (inl R L₁ L₂ : L₁ → L₁ × L₂) = fun x => (x, 0) := rfl\n\n@[simp] theorem coe_inr : (inr R L₁ L₂ : L₂ → L₁ × L₂) = Prod.mk 0 := rfl\n\ntheorem inl_injective : Function.Injective (inl R L₁ L₂) := fun _ => by simp\n\ntheorem inr_injective : Function.Injective (inr R L₁ L₂) := fun _ => by simp\n\nsection\nvariable (R L₁ L₂)\n\nTarget:\ntheorem range_inl : range (inl R L₁ L₂) = ker (snd R L₁ L₂) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_58255fb9a08e","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b7c3ccdfdf8c112fc71a01a46e82a77657cc7884027865ef065fab75887a7add","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Lie","family_id":"range_inl","file_id":"mathlib/Mathlib/Algebra/Lie/Prod.lean","sample_id":"58255fb9a08e3ea4853832b4b2e6754cd8d2484fe722ffff05ba10373b5631a3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7222ee9e22761482f65e7ecb6d7dbb6c39c2c6b75d4bfee6e89b7661c7ff185a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"609454c90f9d78f7e8599fb9bd8862114435a2dded9aae5b46a4c1c57aa273f2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9a1583a6f100445b753a9ba00ceefda2c0d6b67968367089199c06ca3f1c14f0","source_sha256":"1232470fae437db186c8cbd5a8bd94565b5cb80ee4c956cf59b078515170c352","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine le_antisymm ?_ subset_span\n  let s' : Submodule R M :=\n    { carrier := s\n      add_mem' := add_mem\n      zero_mem' := zero_mem _\n      smul_mem' := SMulMemClass.smul_mem }\n  exact span_le (p := s') |>.mpr le_rfl","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":47,"token_jaccard":0.894737,"token_length_ratio":1.119048},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"d3b95664075e20d1bac6be66c8af4bb5cdc630adb7ee2543f4841301e23f927c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Submodule.Lattice\npublic import Mathlib.Algebra.Group.Pointwise.Set.Basic\n\nNamespace:\nSubmodule\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,\n  Heather Macbeth\n-/\n/-!\n# The span of a set of vectors, as a submodule\n\n* `Submodule.span s` is defined to be the smallest submodule containing the set `s`.\n\n## Notation\n\n* We introduce the notation `R ∙ v` for the span of a singleton, `Submodule.span R {v}`.  This is\n  `\\span`, not the same as the scalar multiplication `•`/`\\bub`.\n\n-/\n\n@[expose] public section\n\nassert_not_exists Field\n\nvariable {R R₂ K M M₂ V S : Type*}\n\nnamespace Submodule\n\nopen Function Set\n\nopen scoped Pointwise\n\nsection AddCommMonoid\n\nvariable [Semiring R] [AddCommMonoid M] [Module R M]\nvariable {x : M} (p p' : Submodule R M)\nvariable [Semiring R₂] {σ₁₂ : R →+* R₂}\nvariable [AddCommMonoid M₂] [Module R₂ M₂]\nvariable {F : Type*} [FunLike F M M₂] [SemilinearMapClass F σ₁₂ M M₂]\n\nsection\n\nvariable (R) in\n/-- The span of a set `s ⊆ M` is the smallest submodule of M that contains `s`. -/\ndef span (s : Set M) : Submodule R M :=\n  sInf { p | s ⊆ p }\n\n/-- An `R`-submodule of `M` is principal if it is generated by one element. -/\n@[mk_iff, wikidata Q44382]\nclass IsPrincipal (S : Submodule R M) : Prop where\n  principal (S) : ∃ a, S = span R {a}\n\ninstance (x : R) : (span R {x}).IsPrincipal := ⟨x, rfl⟩\n\nnamespace IsPrincipal\n\n/-- `generator I`, if `I` is a principal submodule, is an `x ∈ M` such that `span R {x} = I` -/\nnoncomputable def generator (S : Submodule R M) [S.IsPrincipal] : M :=\n  Classical.choose (principal S)\n\ntheorem span_singleton_generator (S : Submodule R M) [S.IsPrincipal] : span R {generator S} = S :=\n  (Classical.choose_spec (principal S)).symm\n\nend IsPrincipal\n\nend\n\nvariable {s t : Set M}\n\ntheorem mem_span : x ∈ span R s ↔ ∀ p : Submodule R M, s ⊆ p → x ∈ p :=\n  mem_iInter₂\n\n@[simp, aesop safe 20 (rule_sets := [SetLike])]\ntheorem subset_span : s ⊆ span R s := fun _ h => mem_span.2 fun _ hp => hp h\n\n@[aesop 80% (rule_sets := [SetLike])]\ntheorem mem_span_of_mem {s : Set M} {x : M} (hx : x ∈ s) : x ∈ span R s := subset_span hx\n\ntheorem span_le {p} : span R s ≤ p ↔ s ⊆ p :=\n  ⟨Subset.trans subset_span, fun ss _ h => mem_span.1 h _ ss⟩\n\n@[gcongr] theorem span_mono (h : s ⊆ t) : span R s ≤ span R t :=\n  span_le.2 <| Subset.trans h subset_span\n\ntheorem span_monotone : Monotone (span R : Set M → Submodule R M) := fun _ _ => span_mono\n\ntheorem span_eq_of_le (h₁ : s ⊆ p) (h₂ : p ≤ span R s) : span R s = p :=\n  le_antisymm (span_le.2 h₁) h₂\n\ntheorem span_eq : span R (p : Set M) = p :=\n  span_eq_of_le _ (Subset.refl _) subset_span\n\ntheorem span_eq_span (hs : s ⊆ span R t) (ht : t ⊆ span R s) : span R s = span R t :=\n  le_antisymm (span_le.2 hs) (span_le.2 ht)\n\n/-- A version of `Submodule.span_eq` for subobjects closed under addition and scalar multiplication\nand containing zero. In general, this should not be used directly, but can be used to quickly\ngenerate proofs for specific types of subobjects. -/\n\nTarget:\nlemma coe_span_eq_self [SetLike S M] [AddSubmonoidClass S M] [SMulMemClass S R M] (s : S) :\n    (span R (s : Set M) : Set M) = s :=\n\nProof body:\n","rejected":"by\n  refine le_antisymm ?_ subset_span\n  let s' : Submodule R M :=\n    { carrier := s\n      add_mem' := add_mem\n      zero_mem' := zero_mem _\n      smul_mem' := SMulMemClass.smul_mem }\n  exact span_le (p := s') |>.mpr le_rfl\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Span","family_id":"coe_span_eq_self","file_id":"mathlib/Mathlib/LinearAlgebra/Span/Defs.lean","sample_id":"9a1583a6f100445b753a9ba00ceefda2c0d6b67968367089199c06ca3f1c14f0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"06d455a8b5ee3bf3c5458bfc466011b102068e1311619ad2681ed0ee2fe0c4c3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"900df9f57575ae9ae5d97a81a010ab1e20e766cd1a88e2d731f34af9d7eca3f8","source_sha256":"afbafd08519a79137e714f52e70a195d46a2c795926cef003418bc25da380e6c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases hβ : Nonempty β\n  · refine small_of_surjective (f := Function.invFun g ∘ f) (fun b => ?_)\n    obtain ⟨a, ha⟩ := h b\n    exact ⟨a, by rw [Function.comp_apply, ha, Function.leftInverse_invFun hg]⟩\n  · simp only [not_nonempty_iff] at hβ\n    infer_instance\n\n/-!\nWe don't define `Countable.toSmall` in this file, to keep imports to `Logic` to a minimum.\n-/","hard_negative":true,"metrics":{"chosen_tokens":93,"rejected_tokens":8,"token_jaccard":0.048387,"token_length_ratio":0.086022},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"d3dfc8db459b49e584bac58e3b8322e97023a97b80b599e146daa006fccdc23b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Logic.Small.Defs\npublic import Mathlib.Logic.Equiv.Set\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Instances and theorems for `Small`.\n\nIn particular we prove `small_of_injective` and `small_of_surjective`.\n-/\n\npublic section\n\nassert_not_exists Countable\n\nuniverse u w v v'\n\n-- TODO(timotree3): lower the priority on this instance?\n-- This instance applies to every synthesis problem of the form `Small ↥s` for some set `s`,\n-- but we have lots of instances of `Small` for specific set constructions.\ninstance small_subtype (α : Type v) [Small.{w} α] (P : α → Prop) : Small.{w} { x // P x } :=\n  small_map (equivShrink α).subtypeEquivOfSubtype'\n\ntheorem small_of_injective {α : Type v} {β : Type w} [Small.{u} β] {f : α → β}\n    (hf : Function.Injective f) : Small.{u} α :=\n  small_map (Equiv.ofInjective f hf)\n\ntheorem small_of_surjective {α : Type v} {β : Type w} [Small.{u} α] {f : α → β}\n    (hf : Function.Surjective f) : Small.{u} β :=\n  small_of_injective (Function.injective_surjInv hf)\n\ninstance (priority := 100) small_subsingleton (α : Type v) [Subsingleton α] : Small.{w} α := by\n  rcases isEmpty_or_nonempty α with ⟨⟩\n  · apply small_map (Equiv.equivPEmpty α)\n  · apply small_map Equiv.punitOfNonemptyOfSubsingleton\n\n/-- This can be seen as a version of `small_of_surjective` in which the function `f` doesn't\nactually land in `β` but in some larger type `γ` related to `β` via an injective function `g`.\n-/\n\nTarget:\ntheorem small_of_injective_of_exists {α : Type v} {β : Type w} {γ : Type v'} [Small.{u} α]\n    (f : α → γ) {g : β → γ} (hg : Function.Injective g) (h : ∀ b : β, ∃ a : α, f a = g b) :\n    Small.{u} β :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"5cc6f3952d89cc33c8cd31014488ff0fb61ac61709db41d6027ebcf3da2238e3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Small","family_id":"small_of_injective_of_exists","file_id":"mathlib/Mathlib/Logic/Small/Basic.lean","sample_id":"900df9f57575ae9ae5d97a81a010ab1e20e766cd1a88e2d731f34af9d7eca3f8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7c1a95bd387fa99509421bd9a6c435e484eea5c2a9222b6585afb56f5e4cfe03","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e1b8702d9134a47fb4aaaee0e4e491b5e1c2b4c9e2d4f18f9d53c13ea906985c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fb8d70162809714734e39e1803545fb5adac8b0c13fe3dded4a257dc63c28f16","source_sha256":"79526dcfa55da9c5f7685aed6cead7ebb3b44c830f8cd589df3db791b7c151e7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply tendsto_nhdsNE_of_tendsto_atTop _ _ h_neg\n  convert! h_pos","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":14,"token_jaccard":0.615385,"token_length_ratio":1.555556},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"d40c0c30ea25630886387b3b2bb3583ea46a1ae7e6c487a896e03ed10264df82","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Algebra.Order.Field\npublic import Mathlib.Topology.Maps.Basic\npublic import Mathlib.Analysis.Asymptotics.Lemmas\n\nNamespace:\nTactic.ComputeAsymptotics\n\nLocal context:\n/-\nCopyright (c) 2026 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\n/-!\n# Conversion lemmas\n\nThe main procedure of the `compute_asymptotics` tactic is able to compute limits of functions at\n`atTop` filter. This file contains lemmas we use to reduce other asymptotic goals to\nthe case `Tendsto f atTop l`.\n\n## Main theorems\n\nThis file contains the following lemmas:\n* `tendsto_nhdsGT_of_tendsto_atTop` for `Tendsto f (𝓝[>] c) l`\n* `tendsto_nhdsLT_of_tendsto_atTop` for `Tendsto f (𝓝[<] c) l`\n* `tendsto_nhdsNE_of_tendsto_atTop` for `Tendsto f (𝓝[≠] c) l`\n* `isBigO_of_div_tendsto_atTop` and `isBigO_of_div_tendsto_atBot` for `f =O[l] g`\n\nWe also use lemmas from other files:\n* `tendsto_comp_neg_atTop_iff` for `Tendsto f atBot l`\n* `IsLittleO.of_tendsto_div_atBot` and `IsLittleO.of_tendsto_div_atTop` for `f =o[l] g`\n* `isEquivalent_of_tendsto_one` for `f ∼ g`\n-/\n\npublic section\n\nopen Filter Topology Asymptotics\n\nnamespace Tactic.ComputeAsymptotics\n\nvariable {α 𝕜 : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [TopologicalSpace 𝕜]\n  [OrderTopology 𝕜] {l : Filter α} (f : 𝕜 → α) (c : 𝕜)\n\ntheorem tendsto_nhdsGT_of_tendsto_atTop (h : Tendsto (fun x ↦ f (c + x⁻¹)) atTop l) :\n    Tendsto f (𝓝[>] c) l := by\n  simpa [← Function.comp_def, Tendsto, ← Filter.map_map] using h\n\ntheorem tendsto_nhdsLT_of_tendsto_atTop (h : Tendsto (fun x ↦ f (c - x⁻¹)) atTop l) :\n    Tendsto f (𝓝[<] c) l := by\n  convert_to Tendsto (f ∘ (fun x ↦ c + x) ∘ Neg.neg ∘ Inv.inv) atTop l at h\n  · ext\n    simp [AddGroupWithOne.sub_eq_add_neg]\n  simpa [Tendsto, ← Filter.map_map] using h\n\ntheorem tendsto_nhdsNE_of_tendsto_atTop (h_neg : Tendsto (fun x ↦ f (c - x⁻¹)) atTop l)\n    (h_pos : Tendsto (fun x ↦ f (c + x⁻¹)) atTop l) :\n    Tendsto f (𝓝[≠] c) l := by\n  simpa [Tendsto, ← nhdsLT_sup_nhdsGT] using\n    ⟨tendsto_nhdsLT_of_tendsto_atTop _ _ h_neg, tendsto_nhdsGT_of_tendsto_atTop _ _ h_pos⟩\n\nTarget:\ntheorem tendsto_nhdsNE_of_tendsto_atTop_nhds_of_eq [TopologicalSpace α] {a b : α}\n    (h_neg : Tendsto (fun x ↦ f (c - x⁻¹)) atTop (𝓝 a))\n    (h_pos : Tendsto (fun x ↦ f (c + x⁻¹)) atTop (𝓝 b)) (h_eq : a = b) :\n    Tendsto f (𝓝[≠] c) (𝓝 a) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  apply tendsto_nhdsNE_of_tendsto_atTop _ _ h_neg\n  convert! h_pos","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/ComputeAsymptotics","family_id":"tendsto_nhdsne_of_tendsto_attop_nhds_of_eq","file_id":"mathlib/Mathlib/Tactic/ComputeAsymptotics/Lemmas.lean","sample_id":"fb8d70162809714734e39e1803545fb5adac8b0c13fe3dded4a257dc63c28f16"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a4055cfe4983bb68dd9d6d5e363dc43c3ee390bf3b7daa2b26130a168ae26243","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c6bc706d166e6b8c0ec0ad881ad47670431feaffc163f29325cfeefd84fa16cb","source_sha256":"b5f291a07b438990d3572a453453e2d3f7dd16b400e7d0a6bac2c42be601d0ed","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases eq_or_ne p 1 with rfl | hp\n  · simp\n  · rcases eq_or_ne n 0 with rfl | hn\n    · simp\n    · simp [← padicValNat_eq_emultiplicity_of_ne_one, *]","hard_negative":true,"metrics":{"chosen_tokens":30,"rejected_tokens":8,"token_jaccard":0.038462,"token_length_ratio":0.266667},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"d41edaa9b05f5914bdf1616ef914c8e6f79d4cdbae0c12a60c0adbae466af048","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.MaxPowDiv\npublic import Mathlib.RingTheory.Multiplicity\npublic import Mathlib.Data.Nat.Factors\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis, Matthew Robert Ballard\n-/\n/-!\n# `p`-adic Valuation\n\nThis file defines the `p`-adic valuation on `ℕ`, `ℤ`, and `ℚ`.\n\nThe `p`-adic valuation on `ℚ` is the difference of the multiplicities of `p` in the numerator and\ndenominator of `q`. This function obeys the standard properties of a valuation, with the appropriate\nassumptions on `p`. The `p`-adic valuations on `ℕ` and `ℤ` agree with that on `ℚ`.\n\nThe valuation induces a norm on `ℚ`. This norm is defined in\n`Mathlib/NumberTheory/Padics/PadicNorm.lean`.\n-/\n\n@[expose] public section\n\nassert_not_exists Field\n\nuniverse u\n\nopen Nat\n\nvariable {p : ℕ}\n\ntheorem padicValNat_eq_emultiplicity_of_ne_one (hp : p ≠ 1) {n : ℕ} (hn : n ≠ 0) :\n    padicValNat p n = emultiplicity p n := by\n  rw [eq_comm, emultiplicity_eq_coe, pow_dvd_iff_le_padicValNat hp hn,\n    pow_dvd_iff_le_padicValNat hp hn]\n  simp\n\n@[simp]\n\nTarget:\ntheorem Nat.toNat_emultiplicity (p n : ℕ) : (emultiplicity p n).toNat = padicValNat p n :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"0726dbe9508ffb5b8389710ae5a2d03a256cb9660ad31da536e957962632166a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Padics","family_id":"nat","file_id":"mathlib/Mathlib/NumberTheory/Padics/PadicVal/Defs.lean","sample_id":"c6bc706d166e6b8c0ec0ad881ad47670431feaffc163f29325cfeefd84fa16cb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5c5e46221d8b229ee79bbd41b4c1bdba3112695c6156a1e23176053e1c54427e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"036367c9e9da4688a5df880210fa2dfa4c3a6a61527ecf77f086ca65e67f7b28","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cb00d28e03305d658f5fd091ef6af272ded07e63d8901ab1f776e0190fa772a7","source_sha256":"5b291100fb3630420e5a4ec2ae128eaf5dffa0b8e5840aaf136af3ede6f696c3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by subst h; rfl","hard_negative":false,"metrics":{"chosen_tokens":5,"rejected_tokens":9,"token_jaccard":0.555556,"token_length_ratio":1.8},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"d459a671e4eb43b7e5112088509b0c6314c9b458b4b0b461df798feaef4b3627","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.EqToHom\npublic import Mathlib.CategoryTheory.Functor.Const\npublic import Mathlib.CategoryTheory.Opposites\npublic import Mathlib.Data.Prod.Basic\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2017 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Kim Morrison\n-/\n/-!\n# Cartesian products of categories\n\nWe define the category instance on `C × D` when `C` and `D` are categories.\n\nWe define:\n* `sectL C Z` : the functor `C ⥤ C × D` given by `X ↦ ⟨X, Z⟩`\n* `sectR Z D` : the functor `D ⥤ C × D` given by `Y ↦ ⟨Z, Y⟩`\n* `fst`       : the functor `⟨X, Y⟩ ↦ X`\n* `snd`       : the functor `⟨X, Y⟩ ↦ Y`\n* `swap`      : the functor `C × D ⥤ D × C` given by `⟨X, Y⟩ ↦ ⟨Y, X⟩`\n    (and the fact that this is an equivalence)\n\nWe further define `evaluation : C ⥤ (C ⥤ D) ⥤ D` and `evaluationUncurried : C × (C ⥤ D) ⥤ D`,\nand products of functors and natural transformations, written `F.prod G` and `α.prod β`.\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Functor\n\n-- declare the `v`'s first; see `CategoryTheory.Category` for an explanation\nuniverse v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄\n\nsection\n\nvariable (C : Type u₁) [CategoryStruct.{v₁} C] (D : Type u₂) [CategoryStruct.{v₂} D]\n\n/-- `CategoryStruct.prod C D` gives the Cartesian product of two `CategoryStruct`'s. -/\n@[simps id_fst id_snd comp_fst comp_snd Hom]\ninstance prod : CategoryStruct.{max v₁ v₂} (C × D) where\n  Hom X Y := (X.1 ⟶ Y.1) × (X.2 ⟶ Y.2)\n  id X := ⟨𝟙 X.1, 𝟙 X.2⟩\n  comp f g := (f.1 ≫ g.1, f.2 ≫ g.2)\n\nvariable {C D}\n\nnamespace Prod\n\n@[ext]\nlemma hom_ext {X Y : C × D} {f g : X ⟶ Y} (h₁ : f.1 = g.1) (h₂ : f.2 = g.2) : f = g :=\n  Prod.ext h₁ h₂\n\n/-- Construct a morphism in a product category by giving its constituent components.\nThis constructor should be preferred over `Prod.mk`, because Lean infers better the\nsource and target of the resulting morphism. -/\nabbrev mkHom {X₁ X₂ : C} {Y₁ Y₂ : D} (f : X₁ ⟶ X₂) (g : Y₁ ⟶ Y₂) : (X₁, Y₁) ⟶ (X₂, Y₂) :=\n  ⟨f, g⟩\n\n@[inherit_doc Prod.mkHom]\nscoped infixr:70 \" ×ₘ \" => Prod.mkHom\n\n/-- Analogue of `Prod.mk.injEq` in this setting. -/\nlemma mkHom_eq {X₁ X₂ : C} {Y₁ Y₂ : D} (f f' : X₁ ⟶ X₂) (g g' : Y₁ ⟶ Y₂) :\n    f ×ₘ g = f' ×ₘ g' ↔ (f = f' ∧ g = g') :=\n  Prod.hom_ext_iff\n\nend Prod\n\nopen Prod\n\n/-! Two rfl lemmas that cannot be generated by `@[simps]`. -/\n\n@[simp]\ntheorem prod_id (X : C) (Y : D) : 𝟙 (X, Y) = 𝟙 X ×ₘ 𝟙 Y :=\n  rfl\n\ntheorem prod_id' (X : C) (Y : D) : 𝟙 (X, Y) = (𝟙 X, 𝟙 Y) :=\n  rfl\n\n@[simp]\ntheorem prod_comp {P Q R : C} {S T U : D} (f : (P, S) ⟶ (Q, T)) (g : (Q, T) ⟶ (R, U)) :\n    f ≫ g = f.1 ≫ g.1 ×ₘ f.2 ≫ g.2 :=\n  rfl\n\n/-! Two `eqToHom` lemmas about projections. -/\n\n@[simp]\ntheorem eqToHom_fst {X Y : C × D} (h : X = Y) :\n    (eqToHom h).1 = eqToHom (h ▸ rfl) := by subst h; rfl\n\n@[simp]\n\nTarget:\ntheorem eqToHom_snd {X Y : C × D} (h : X = Y) :\n    (eqToHom h).2 = eqToHom (h ▸ rfl) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby subst h; rfl","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Products","family_id":"eqtohom_snd","file_id":"mathlib/Mathlib/CategoryTheory/Products/Basic.lean","sample_id":"cb00d28e03305d658f5fd091ef6af272ded07e63d8901ab1f776e0190fa772a7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0be2d8fff9f4e0668c3d36339336ba1af35cd1dd0eb1d09d433511d22d232ffa","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"43e80cf9725299057626e124ada0c7fa7af7a66a7d45d1cc2cc1eef0f94a344c","source_sha256":"7077627c568c3d5d4d50c5ab694c1de5a4aa92663f9998f759f2dc7e9ceb07d6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [← h₀.ordConnected_iff_of_bdd (OrderBot.bddBelow I) h₂, ordConnected_iff_disjoint_Ioo_empty]","hard_negative":false,"metrics":{"chosen_tokens":19,"rejected_tokens":5,"token_jaccard":0.047619,"token_length_ratio":0.263158},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"d46182572fdd73679b7647720c8c82b9495822213fea1abd1fb8668c2af124c0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Int.ConditionallyCompleteOrder\npublic import Mathlib.Data.Int.Interval\npublic import Mathlib.Data.Int.SuccPred\npublic import Mathlib.Order.Lattice.Nat\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Bhavik Mehta, Oliver Nash\n-/\n/-!\n# Order-connected subsets of linear orders\n\nIn this file we provide some results about order-connected subsets of linear orders, together with\nsome convenience lemmas for characterising closed intervals in certain concrete types such as `ℤ`,\n`ℕ`, and `Fin n`.\n\n## Main results:\n* `Set.ordConnected_iff_disjoint_Ioo_empty`: a characterisation of `Set.OrdConnected` for\n  locally-finite linear orders.\n* `Set.Nonempty.ordConnected_iff_of_bdd`: a characterisation of closed intervals for locally-finite\n  conditionally complete linear orders.\n* `Set.Nonempty.ordConnected_iff_of_bdd'`: a characterisation of closed intervals for\n  locally-finite complete linear orders (convenient for `Fin n`).\n* `Set.Nonempty.eq_Icc_iff_nat`: characterisation of closed intervals for `ℕ`.\n* `Set.Nonempty.eq_Icc_iff_int`: characterisation of closed intervals for `ℤ`.\n-/\n\npublic section\n\nvariable {α : Type*} {I : Set α}\n\nlemma Set.Nonempty.ordConnected_iff_of_bdd\n    [ConditionallyCompleteLinearOrder α] [LocallyFiniteOrder α]\n    (h₀ : I.Nonempty) (h₁ : BddBelow I) (h₂ : BddAbove I) :\n    I.OrdConnected ↔ I = Icc (sInf I) (sSup I) :=\n  have h₄ : I.Finite := h₁.finite_of_bddAbove h₂\n  ⟨fun _ ↦ le_antisymm (subset_Icc_csInf_csSup h₁ h₂)\n    (I.Icc_subset (h₀.csInf_mem h₄) (h₀.csSup_mem h₄)), fun h₃ ↦ h₃ ▸ ordConnected_Icc⟩\n\n/-- A version of `Set.Nonempty.ordConnected_iff_of_bdd` for complete linear orders, such as `Fin n`,\nin which the explicit boundedness hypotheses are not necessary. -/\nlemma Set.Nonempty.ordConnected_iff_of_bdd' [ConditionallyCompleteLinearOrder α]\n    [OrderTop α] [OrderBot α] [LocallyFiniteOrder α]\n    (h₀ : I.Nonempty) :\n    I.OrdConnected ↔ I = Icc (sInf I) (sSup I) :=\n  h₀.ordConnected_iff_of_bdd (OrderBot.bddBelow I) (OrderTop.bddAbove I)\n\n/- TODO The `LocallyFiniteOrder` assumption here is probably too strong (e.g., it rules out `ℝ`\nfor which this result holds). However at the time of writing it is not clear what weaker\nassumption(s) should replace it. -/\nlemma Set.ordConnected_iff_disjoint_Ioo_empty [LinearOrder α] [LocallyFiniteOrder α] :\n    I.OrdConnected ↔ ∀ᵉ (x ∈ I) (y ∈ I), Disjoint (Ioo x y) I → Ioo x y = ∅ := by\n  simp_rw [← Set.subset_compl_iff_disjoint_right]\n  refine ⟨fun h' x hx y hy hxy ↦ ?_, fun h' ↦ ordConnected_of_Ioo fun x hx y hy hxy z hz ↦ ?_⟩\n  · suffices ∀ z, x < z → y ≤ z by ext z; simpa using this z\n    intro z hz\n    suffices z ∉ Ioo x y by simp_all\n    exact fun contra ↦ hxy contra <| h'.out hx hy <| mem_Icc_of_Ioo contra\n  · by_contra hz'\n    obtain ⟨x', hx', hx''⟩ :=\n      ((finite_Icc x z).inter_of_right I).exists_le_maximal ⟨hx, le_refl _, hz.1.le⟩\n    have hxz : x' < z := lt_of_le_of_ne hx''.1.2.2 (ne_of_mem_of_not_mem hx''.1.1 hz')\n    obtain ⟨y', hy', hy''⟩ :=\n      ((finite_Icc z y).inter_of_right I).exists_le_minimal ⟨hy, hz.2.le, le_refl _⟩\n    have hzy : z < y' := lt_of_le_of_ne' hy''.1.2.1 (ne_of_mem_of_not_mem hy''.1.1 hz')\n    have h₃ : Ioc x' z ⊆ Iᶜ := fun t ht ht' ↦ hx''.not_gt (⟨ht', le_trans hx' ht.1.le, ht.2⟩) ht.1\n    have h₄ : Ico z y' ⊆ Iᶜ := fun t ht ht' ↦ hy''.not_lt (⟨ht', ht.1, le_trans ht.2.le hy'⟩) ht.2\n    have h₅ : Ioo x' y' ⊆ Iᶜ := by\n      simp only [← Ioc_union_Ico_eq_Ioo hxz hzy, union_subset_iff, and_true, h₃, h₄]\n    exact eq_empty_iff_forall_notMem.1 (h' x' hx''.prop.1 y' hy''.prop.1 h₅) z ⟨hxz, hzy⟩\n\nTarget:\nlemma Set.Nonempty.eq_Icc_iff_nat {I : Set ℕ}\n    (h₀ : I.Nonempty) (h₂ : BddAbove I) :\n    I = Icc (sInf I) (sSup I) ↔ ∀ᵉ (x ∈ I) (y ∈ I), Disjoint (Ioo x y) I → y ≤ x + 1 :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Interval","family_id":"set","file_id":"mathlib/Mathlib/Order/Interval/Set/OrdConnectedLinear.lean","sample_id":"43e80cf9725299057626e124ada0c7fa7af7a66a7d45d1cc2cc1eef0f94a344c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"36cd5f448af1bf1ecf44142699671ba22a07e7da26e9b39c0449f7bdf315a9b7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dfed06d389dc5d68e96360f58a96e6261e6d908c48220f135bdf99a31b4e44ba","source_sha256":"a852fe8639a8e4d7780824b945071eaf8d714c077a80f3e3c0dcb22ec20d5088","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨P, φ, hsurj⟩ := H\n  obtain ⟨g, rfl⟩ := h\n  obtain ⟨a, ha⟩ := hsurj (algebraMap _ _ g)\n  have : IsLocalization.Away (f * g) (Localization.Away (φ a)) :=\n    ha ▸ .mul' (Localization.Away f) _ _ _\n  have : IsStandardEtale R (Localization.Away a) := .of_isLocalizationAway a\n  exact .mk _ (IsLocalization.Away.mapₐ_surjective_of_surjective\n    (Aₚ := Localization.Away a) (Bₚ := Localization.Away (φ a)) a hsurj)","hard_negative":false,"metrics":{"chosen_tokens":116,"rejected_tokens":3,"token_jaccard":0.04878,"token_length_ratio":0.025862},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"d47bbe6b333a43389bef7fcd7969edc55b4298ae1b53f63bc61c1e6155609e47","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Etale.StandardEtale\npublic import Mathlib.RingTheory.LocalRing.ResidueField.Instances\npublic import Mathlib.RingTheory.RingHom.StandardSmooth\npublic import Mathlib.RingTheory.Unramified.LocalRing\npublic import Mathlib.RingTheory.ZariskisMainTheorem\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Local structure of unramified algebras\n\nIn this file, we will prove that if `S` is a finite type `R`-algebra unramified at `Q`, then\nthere exists `f ∉ Q` and a standard etale algebra `A` over `R` that surjects onto `S[1/f]`.\nGeometrically, this says that unramified morphisms locally are closed subsets of etale covers.\n\nAs a corollary, we also obtain results about the local structure of etale and smooth algebras.\n\n## Main definition and results\n- `HasStandardEtaleSurjectionOn`: The predicate\n  \"there exists a standard etale algebra `A` over `R` that surjects onto `S[1/f]`\".\n- `Algebra.IsUnramifiedAt.exists_hasStandardEtaleSurjectionOn`:\n  If `S` is a finite type `R`-algebra that is unramified at a prime `p`, then\n  there exists a standard etale algebra over `R` that surjects onto `S[1/f]` for some `f ∉ p`.\n- `Algebra.IsEtaleAt.exists_isStandardEtale`:\n  If `S` is a finitely presented `R`-algebra that is etale at a prime `p`, then\n  `S[1/f]` is standard etale for some `f ∉ p`.\n- `Algebra.IsSmoothAt.exists_isStandardEtale_mvPolynomial`:\n  If `S` is a finitely presented `R`-algebra that is smooth at a prime `p`, then\n  there exists some `f ∉ p` such that `S[1/f]` is `R`-isomorphic to a standard etale algebra\n  over `R[x₁,...,xₙ]`.\n\n-/\n\n@[expose] public section\n\nopen Polynomial TensorProduct Algebra\n\nopen scoped nonZeroDivisors\n\nvariable {R A S : Type*} [CommRing R] [CommRing A] [CommRing S] [Algebra R S] [Algebra R A]\n\nvariable (R) in\n/-- The predicate \"there exists a standard etale algebra `A` over `R` that surjects onto `S[1/f]`\".\nWe shall show if `S` is `R`-unramified at `Q` then there exists `f ∉ Q` satisfying it. -/\ndef HasStandardEtaleSurjectionOn (f : S) : Prop :=\n  ∃ (P : StandardEtalePair R) (φ : P.Ring →ₐ[R] Localization.Away f), Function.Surjective φ\n\nlemma HasStandardEtaleSurjectionOn.mk [IsStandardEtale R A]\n    {Sf : Type*} [CommRing Sf] [Algebra R Sf] [Algebra S Sf] [IsScalarTower R S Sf]\n    {f : S} [IsLocalization.Away f Sf] (φ : A →ₐ[R] Sf) (H : Function.Surjective φ) :\n    HasStandardEtaleSurjectionOn R f :=\n  let P : StandardEtalePresentation R A := Nonempty.some inferInstance\n  ⟨P.P, (((IsLocalization.algEquiv (.powers f) (Localization.Away f) Sf).restrictScalars R)\n    |>.symm.toAlgHom).comp (φ.comp P.equivRing.symm.toAlgHom), by simpa⟩\n\nTarget:\nlemma HasStandardEtaleSurjectionOn.of_dvd\n    {f g : S} (H : HasStandardEtaleSurjectionOn R f) (h : f ∣ g) :\n    HasStandardEtaleSurjectionOn R g :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Unramified","family_id":"hasstandardetalesurjectionon","file_id":"mathlib/Mathlib/RingTheory/Unramified/LocalStructure.lean","sample_id":"dfed06d389dc5d68e96360f58a96e6261e6d908c48220f135bdf99a31b4e44ba"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2038792152bb22c0a5ae377d16d2f16e94b117cceae2db86d9ede3e4d520885b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"59fa1cfe920c44354539a08a73d5bce71392689b4372837276159b664aba9fb8","source_sha256":"5a8a601bded85c9c17f5a633b68ceab704a71b9b3614d423fc312dd608794122","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← cancel_epi ((shiftFunctorZero _ A).inv.app X), Iso.inv_hom_id_app,\n    pullbackShiftFunctorZero_inv_app, assoc, Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app]\n  rfl","hard_negative":false,"metrics":{"chosen_tokens":35,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.057143},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"d4a3ec71c9964cbc3e870b7f0fc4112c9d1033af41e984ab29f31064950b455e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Shift.Adjunction\npublic import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2023 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The pullback of a shift by a monoid morphism\n\nGiven a shift by a monoid `B` on a category `C` and a monoid morphism `φ : A →+ B`,\nwe define a shift by `A` on a category `PullbackShift C φ` which is a type synonym for `C`.\n\nIf `F : C ⥤ D` is a functor between categories equipped with shifts by `B`, we define\na type synonym `PullbackShift.functor F φ` for `F`. When `F` has a `CommShift` structure\nby `B`, we define a pulled back `CommShift` structure by `A` on `PullbackShift.functor F φ`.\n\nSimilarly, if `τ` is a natural transformation between functors `F,G : C ⥤ D`, we define\na type synonym\n`PullbackShift.natTrans τ φ : PullbackShift.functor F φ ⟶ PullbackShift.functor G φ`.\nWhen `τ` has a `CommShift` structure by `B` (i.e. is compatible with `CommShift` structures\non `F` and `G`), we define a pulled back `CommShift` structure by `A` on\n`PullbackShift.natTrans τ φ`.\n\nFinally, if we have an adjunction `F ⊣ G` (with `G : D ⥤ C`), we define a type synonym\n`PullbackShift.adjunction adj φ : PullbackShift.functor F φ ⊣ PullbackShift.functor G φ`\nand we show that, if `adj` is compatible with `CommShift` structures\non `F` and `G`, then `PullbackShift.adjunction adj φ` is also compatible with the pulled back\n`CommShift` structures.\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Limits Category\n\nvariable (C : Type*) [Category* C] {A B : Type*} [AddMonoid A] [AddMonoid B]\n\n/-- The category `PullbackShift C φ` is equipped with a shift such that for all `a`,\nthe shift functor by `a` is `shiftFunctor C (φ a)`. -/\n@[nolint unusedArguments]\ndef PullbackShift [HasShift C B] (_ : A →+ B) := C\nderiving Category\n\nattribute [local instance] endofunctorMonoidalCategory\n\nvariable [HasShift C B] (φ : A →+ B)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The shift on `PullbackShift C φ` is obtained by precomposing the shift on `C` with\nthe monoidal functor `Discrete.addMonoidalFunctor φ : Discrete A ⥤ Discrete B`. -/\ninstance : HasShift (PullbackShift C φ) A where\n  shift := Discrete.addMonoidalFunctor φ ⋙ shiftMonoidalFunctor C B\n\ninstance [HasZeroObject C] : HasZeroObject (PullbackShift C φ) :=\n  inferInstanceAs <| HasZeroObject C\n\ninstance [Preadditive C] : Preadditive (PullbackShift C φ) :=\n  inferInstanceAs <| Preadditive C\n\ninstance [Preadditive C] (a : A) [(shiftFunctor C (φ a)).Additive] :\n    (shiftFunctor (PullbackShift C φ) a).Additive :=\n  inferInstanceAs (shiftFunctor C (φ a)).Additive\n\n/-- When `b = φ a`, this is the canonical\nisomorphism `shiftFunctor (PullbackShift C φ) a ≅ shiftFunctor C b`. -/\ndef pullbackShiftIso (a : A) (b : B) (h : b = φ a) :\n    shiftFunctor (PullbackShift C φ) a ≅ shiftFunctor C b := eqToIso (by subst h; rfl)\n\nvariable {C}\nvariable (X : PullbackShift C φ) (a₁ a₂ a₃ : A) (h : a₁ + a₂ = a₃) (b₁ b₂ b₃ : B)\n  (h₁ : b₁ = φ a₁) (h₂ : b₂ = φ a₂) (h₃ : b₃ = φ a₃)\n\nset_option backward.defeqAttrib.useBackward true in\nlemma pullbackShiftFunctorZero_inv_app :\n    (shiftFunctorZero _ A).inv.app X =\n      (shiftFunctorZero C B).inv.app X ≫ (pullbackShiftIso C φ 0 0 (by simp)).inv.app X := by\n  change (shiftFunctorZero C B).inv.app X ≫ _ = _\n  dsimp [Discrete.eqToHom, Discrete.addMonoidalFunctor_ε]\n  congr 2\n  apply eqToHom_map\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma pullbackShiftFunctorZero_hom_app :\n    (shiftFunctorZero _ A).hom.app X =\n      (pullbackShiftIso C φ 0 0 (by simp)).hom.app X ≫ (shiftFunctorZero C B).hom.app X :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Shift","family_id":"pullbackshiftfunctorzero_hom_app","file_id":"mathlib/Mathlib/CategoryTheory/Shift/Pullback.lean","sample_id":"59fa1cfe920c44354539a08a73d5bce71392689b4372837276159b664aba9fb8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1246a108b01c281615de5c0ac2e3b8d9e0aa2fea03820a7bb20910f1f044ae33","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4180d4b77b71dc3fb2a1ccac790bfafb68b3d5a563a499f05b950658539e98e7","source_sha256":"5f13df63ef1a33fb7d6444e769a7e9b9cf3ab09860a26a91c473157d6239794f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [he.isIdempotentElem.mem_iff, he.isIdempotentElem.range_mem_invtSubmodule_iff,\n    he.isIdempotentElem.ker_mem_invtSubmodule_iff, forall_and, and_iff_left_iff_imp, ← mul_def]\n  intro h x hx\n  simpa [he.isSelfAdjoint.star_eq] using! congr(star $(h _ (star_mem hx)))","hard_negative":false,"metrics":{"chosen_tokens":55,"rejected_tokens":5,"token_jaccard":0.058824,"token_length_ratio":0.090909},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"d4ccb20caa622b1c8645dcd0cc2fc309acbef6558cc35c991cc51017a1fd8fa7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.CStarAlgebra.Classes\npublic import Mathlib.Analysis.InnerProductSpace.Adjoint\n\nNamespace:\nVonNeumannAlgebra\n\nLocal context:\n/-\nCopyright (c) 2022 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Von Neumann algebras\n\nWe give the \"abstract\" and \"concrete\" definitions of a von Neumann algebra.\nWe still have a major project ahead of us to show the equivalence between these definitions!\n\nAn abstract von Neumann algebra `WStarAlgebra M` is a C⋆ algebra with a Banach space predual,\nper Sakai (1971).\n\nA concrete von Neumann algebra `VonNeumannAlgebra H` (where `H` is a Hilbert space)\nis a \\*-closed subalgebra of bounded operators on `H` which is equal to its double commutant.\n\nWe'll also need to prove the von Neumann double commutant theorem,\nthat the concrete definition is equivalent to a \\*-closed subalgebra which is weakly closed.\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\n/-- Sakai's definition of a von Neumann algebra as a C⋆ algebra with a Banach space predual.\n\nSo that we can unambiguously talk about these \"abstract\" von Neumann algebras\nin parallel with the \"concrete\" ones (weakly closed \\*-subalgebras of B(H)),\nwe name this definition `WStarAlgebra`.\n\nNote that for now we only assert the mere existence of predual, rather than picking one.\nThis may later prove problematic, and need to be revisited.\nPicking one may cause problems with definitional unification of different instances.\nOne the other hand, not picking one means that the weak-\\* topology\n(which depends on a choice of predual) must be defined using the choice,\nand we may be unhappy with the resulting opaqueness of the definition.\n-/\nclass WStarAlgebra (M : Type u) [CStarAlgebra M] : Prop where\n  /-- There is a Banach space `X` whose dual is isometrically (conjugate-linearly) isomorphic\n  to the `WStarAlgebra`. -/\n  exists_predual :\n    ∃ (X : Type u) (_ : NormedAddCommGroup X) (_ : NormedSpace ℂ X) (_ : CompleteSpace X),\n      Nonempty (StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M)\n\n-- TODO: Without this, `VonNeumannAlgebra` times out. Why?\n/-- The double commutant definition of a von Neumann algebra,\nas a \\*-closed subalgebra of bounded operators on a Hilbert space,\nwhich is equal to its double commutant.\n\nNote that this definition is parameterised by the Hilbert space\non which the algebra faithfully acts, as is standard in the literature.\nSee `WStarAlgebra` for the abstract notion (a C⋆-algebra with Banach space predual).\n\nNote this is a bundled structure, parameterised by the Hilbert space `H`,\nrather than a typeclass on the type of elements.\nThus we can't say that the bounded operators `H →L[ℂ] H` form a `VonNeumannAlgebra`\n(although we will later construct the instance `WStarAlgebra (H →L[ℂ] H)`),\nand instead will use `⊤ : VonNeumannAlgebra H`.\n-/\nstructure VonNeumannAlgebra (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H]\n    [CompleteSpace H] extends StarSubalgebra ℂ (H →L[ℂ] H) where\n  /-- The double commutant (a.k.a. centralizer) of a `VonNeumannAlgebra` is itself. -/\n  centralizer_centralizer' : Set.centralizer (Set.centralizer carrier) = carrier\n\n/-- Consider a von Neumann algebra acting on a Hilbert space `H` as a \\*-subalgebra of `H →L[ℂ] H`.\n(That is, we forget that it is equal to its double commutant\nor equivalently that it is closed in the weak and strong operator topologies.)\n-/\nadd_decl_doc VonNeumannAlgebra.toStarSubalgebra\n\nnamespace VonNeumannAlgebra\n\nvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n\ninstance instSetLike : SetLike (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  coe S := S.carrier\n  coe_injective S T h := by obtain ⟨⟨⟨⟨⟨⟨_, _⟩, _⟩, _⟩, _⟩, _⟩, _⟩ := S; cases T; congr\n\ninstance : PartialOrder (VonNeumannAlgebra H) := .ofSetLike (VonNeumannAlgebra H) (H →L[ℂ] H)\n\nnoncomputable instance instStarMemClass : StarMemClass (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  star_mem {s} := s.star_mem'\n\ninstance instSubringClass : SubringClass (VonNeumannAlgebra H) (H →L[ℂ] H) where\n  add_mem {s} := s.add_mem'\n  mul_mem {s} := s.mul_mem'\n  one_mem {s} := s.one_mem'\n  zero_mem {s} := s.zero_mem'\n  neg_mem {s} a ha := show -a ∈ s.toStarSubalgebra from neg_mem ha\n\n@[simp]\ntheorem mem_carrier {S : VonNeumannAlgebra H} {x : H →L[ℂ] H} :\n    x ∈ S.toStarSubalgebra ↔ x ∈ (S : Set (H →L[ℂ] H)) :=\n  Iff.rfl\n\n@[simp]\ntheorem coe_toStarSubalgebra (S : VonNeumannAlgebra H) :\n    (S.toStarSubalgebra : Set (H →L[ℂ] H)) = S :=\n  rfl\n\n@[simp]\ntheorem coe_mk (S : StarSubalgebra ℂ (H →L[ℂ] H)) (h) :\n    ((⟨S, h⟩ : VonNeumannAlgebra H) : Set (H →L[ℂ] H)) = S :=\n  rfl\n\n@[ext]\ntheorem ext {S T : VonNeumannAlgebra H} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n\n@[simp]\ntheorem centralizer_centralizer (S : VonNeumannAlgebra H) :\n    Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S :=\n  S.centralizer_centralizer'\n\n/-- The centralizer of a `VonNeumannAlgebra`, as a `VonNeumannAlgebra`. -/\nnoncomputable def commutant (S : VonNeumannAlgebra H) : VonNeumannAlgebra H where\n  toStarSubalgebra := StarSubalgebra.centralizer ℂ (S : Set (H →L[ℂ] H))\n  centralizer_centralizer' := by simp\n\n@[simp]\ntheorem coe_commutant (S : VonNeumannAlgebra H) :\n    ↑S.commutant = Set.centralizer (S : Set (H →L[ℂ] H)) := by\n  simp [commutant]\n\n@[simp]\ntheorem mem_commutant_iff {S : VonNeumannAlgebra H} {z : H →L[ℂ] H} :\n    z ∈ S.commutant ↔ ∀ g ∈ S, g * z = z * g := by\n  rw [← SetLike.mem_coe, coe_commutant]\n  rfl\n\n@[simp]\ntheorem commutant_commutant (S : VonNeumannAlgebra H) : S.commutant.commutant = S :=\n  SetLike.coe_injective <| by simp\n\nopen ContinuousLinearMap in\n/-- An idempotent is an element in a von Neumann algebra if and only if\nits range and kernel are invariant under the commutant. -/\ntheorem IsIdempotentElem.mem_iff {e : H →L[ℂ] H} (h : IsIdempotentElem e)\n    (S : VonNeumannAlgebra H) :\n    e ∈ S ↔ ∀ y ∈ S.commutant,\n      e.range ∈ Module.End.invtSubmodule y ∧ e.ker ∈ Module.End.invtSubmodule y := by\n  conv_rhs => simp [← h.commute_iff, Commute.symm_iff (a := e), commute_iff_eq, ← mem_commutant_iff]\n\nopen VonNeumannAlgebra ContinuousLinearMap in\n/-- A star projection is an element in a von Neumann algebra if and only if\nits range is invariant under the commutant. -/\n\nTarget:\ntheorem IsStarProjection.mem_iff {e : H →L[ℂ] H} (he : IsStarProjection e)\n    (S : VonNeumannAlgebra H) :\n    e ∈ S ↔ ∀ y ∈ S.commutant, e.range ∈ Module.End.invtSubmodule y :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/VonNeumannAlgebra","family_id":"isstarprojection","file_id":"mathlib/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean","sample_id":"4180d4b77b71dc3fb2a1ccac790bfafb68b3d5a563a499f05b950658539e98e7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ba260adf630a9d5dff8237b3bea2f33b1eff285cacd7e89777abb74bff12da31","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3ac2a51e138f18231efcaee4ea66d9527d30cacc059832c5ae4dbc98d07e43ec","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"eb0438819632308cf7faaf08d1c110925f195fcd644c416f40abd06f69c1666f","source_sha256":"da03aef6be316c0f1e4d46eb4e8f0617ba6def9cf6bbead1497df151ed7bf988","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases x; rfl\n\nnonrec def ExceptT.callCC {ε} [MonadCont m] {α β : Type _}\n    (f : Label α (ExceptT ε m) β → ExceptT ε m α) : ExceptT ε m α :=\n  ExceptT.mk (callCC fun x : Label _ m β => ExceptT.run <| f (ExceptT.mkLabel x))","hard_negative":false,"metrics":{"chosen_tokens":73,"rejected_tokens":78,"token_jaccard":0.897436,"token_length_ratio":1.068493},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"d4cf6823d0940896a28819abf16677f73619f7e6a2ca45b62597e673e005b9d4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Control.Monad.Basic\npublic import Mathlib.Control.Monad.Writer\npublic import Mathlib.Control.Lawful\npublic import Batteries.Tactic.Congr\npublic import Batteries.Lean.Except\nimport all Init.Control.Option  -- for unfolding `Option.lift`\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\n/-!\n# Continuation Monad\n\nMonad encapsulating continuation passing programming style, similar to\nHaskell's `Cont`, `ContT` and `MonadCont`:\n<https://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Cont.html>\n<https://hackage.haskell.org/package/transformers-0.6.2.0/docs/Control-Monad-Trans-Cont.html>\n-/\n\n@[expose] public section\n\nuniverse u v w u₀ u₁ v₀ v₁\n\nstructure MonadCont.Label (α : Type w) (m : Type u → Type v) (β : Type u) where\n  apply : α → m β\n\nabbrev MonadCont.goto {α β} {m : Type u → Type v} (f : MonadCont.Label α m β) (x : α) :=\n  f.apply x\n\nclass MonadCont (m : Type u → Type v) where\n  callCC : ∀ {α β}, (MonadCont.Label α m β → m α) → m α\n\nopen MonadCont\n\nclass LawfulMonadCont (m : Type u → Type v) [Monad m] [MonadCont m] : Prop\n    extends LawfulMonad m where\n  callCC_bind_right {α ω γ} (cmd : m α) (next : Label ω m γ → α → m ω) :\n    (callCC fun f => cmd >>= next f) = cmd >>= fun x => callCC fun f => next f x\n  callCC_bind_left {α} (β) (x : α) (dead : Label α m β → β → m α) :\n    (callCC fun f : Label α m β => goto f x >>= dead f) = pure x\n  callCC_dummy {α β} (dummy : m α) : (callCC fun _ : Label α m β => dummy) = dummy\n\nexport LawfulMonadCont (callCC_bind_right callCC_bind_left callCC_dummy)\n\ndef ContT (r : Type u) (m : Type u → Type v) (α : Type w) :=\n  (α → m r) → m r\n\nabbrev Cont (r : Type u) (α : Type w) :=\n  ContT r Id α\n\nnamespace ContT\n\nexport MonadCont (Label goto)\n\nvariable {r : Type u} {m : Type u → Type v} {α β : Type w}\n\n/-- Build a `ContT` from a function taking a continuation callback. -/\ndef mk (f : (α → m r) → m r) : ContT r m α := f\n\n/-- Run a `ContT` with a provided callback. -/\ndef run (x : ContT r m α) : (α → m r) → m r := x\n\ndef map (f : m r → m r) (x : ContT r m α) : ContT r m α :=\n  f ∘ x\n\ntheorem run_contT_map_contT (f : m r → m r) (x : ContT r m α) : run (map f x) = f ∘ run x :=\n  rfl\n\ndef withContT (f : (β → m r) → α → m r) (x : ContT r m α) : ContT r m β := fun g => x <| f g\n\ntheorem run_withContT (f : (β → m r) → α → m r) (x : ContT r m α) :\n    run (withContT f x) = run x ∘ f :=\n  rfl\n\n@[ext]\nprotected theorem ext {x y : ContT r m α} (h : ∀ f, x.run f = y.run f) : x = y := by\n  unfold ContT; ext; apply h\n\ninstance : Monad (ContT r m) where\n  pure x f := f x\n  bind x f g := x fun i => f i g\n\n@[simp]\ntheorem run_mk (f : (α → m r) → m r) (k : α → m r) : (.mk f : ContT r m α).run k = f k := rfl\n\n@[simp]\ntheorem run_pure (a : α) (k : α → m r) : (pure a : ContT r m α).run k = k a := rfl\n\n@[simp]\ntheorem run_bind (x : ContT r m α) (f : α → ContT r m β) (k : β → m r) :\n    (x >>= f).run k = x.run fun x => (f x).run k := rfl\n\n@[simp]\ntheorem run_map (f : α → β) (x : ContT r m α) (k : β → m r) :\n    (f <$> x).run k = x.run (k ∘ f) := rfl\n\n@[simp]\ntheorem run_seq (f : ContT r m (α → β)) (x : ContT r m α) (k : β → m r) :\n    (f <*> x).run k = f.run fun f => x.run (k ∘ f) := rfl\n\n@[simp]\ntheorem run_seqLeft (x : ContT r m α) (y : ContT r m β) (k : α → m r) :\n    (x <* y).run k = x.run fun x => y.run fun _ => k x := rfl\n\n@[simp]\ntheorem run_seqRight (x : ContT r m α) (y : ContT r m β) (k : β → m r) :\n    (x *> y).run k = x.run fun _ => y.run k := rfl\n\ninstance : LawfulMonad (ContT r m) := LawfulMonad.mk'\n  (id_map := by intros; rfl)\n  (pure_bind := by intros; ext; rfl)\n  (bind_assoc := by intros; ext; rfl)\n\ninstance [Monad m] : MonadLift m (ContT r m) where\n  monadLift x := .mk fun k => x >>= k\n\n@[simp]\ntheorem run_monadLift [Monad m] {α} (x : m α) (k : α → m r) :\n    (monadLift x : ContT r m α).run k = x >>= k := rfl\n\ntheorem monadLift_bind [Monad m] [LawfulMonad m] {α β} (x : m α) (f : α → m β) :\n    (monadLift (x >>= f) : ContT r m β) = monadLift x >>= monadLift ∘ f := by\n  ext\n  simp only [bind_assoc, run_bind, run_monadLift, Function.comp_apply]\n\ninstance : MonadCont (ContT r m) where\n  callCC f := .mk fun k => f ⟨fun x => .mk fun _ => k x⟩ k\n\n@[simp]\ntheorem run_callCC (f : Label α (ContT r m) β → ContT r m α) (k : α → m r) :\n    (callCC f).run k = (f ⟨fun x => .mk fun _ => k x⟩).run k := rfl\n\ninstance : LawfulMonadCont (ContT r m) where\n  callCC_bind_right := by intros; ext; rfl\n  callCC_bind_left := by intros; ext; rfl\n  callCC_dummy := by intros; ext; rfl\n\n/-- Note that `tryCatch` does not have correct behavior in this monad:\n```\ndef foo : ContT Bool (Except String) Bool := do\n  let x ← try\n    pure true\n  catch _ =>\n    return false\n  throw s!\"oh no {x}\"\n#eval foo.run pure\n-- `Except.ok false`, no error\n```\nHere, the `throwError` is being run inside the `try`.\nSee [Zulip](https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/MonadExcept.20in.20the.20ContT.20monad/near/375341221)\nfor further discussion.\n-/\ninstance (ε) [MonadExceptOf ε m] : MonadExceptOf ε (ContT r m) where\n  throw e := .mk fun _ => throw e\n  tryCatch act h := .mk fun k => tryCatch (act.run k) fun e => (h e).run k\n\n@[simp]\ntheorem run_throw {ε} [MonadExceptOf ε m]\n    (e : ε) (f : α → m r) :\n    (throw e : ContT r m α).run f = throw e := rfl\n\n@[simp]\ntheorem run_tryCatch {ε} [MonadExceptOf ε m]\n    (act : ContT r m α) (h : ε → ContT r m α) (f : α → m r) :\n    (tryCatch act h : ContT r m α).run f = tryCatch (act.run f) fun e => (h e).run f := rfl\n\nend ContT\n\nvariable {m : Type u → Type v}\n\nsection\nvariable [Monad m]\n\ndef ExceptT.mkLabel {α β ε} : Label (Except.{u, u} ε α) m β → Label α (ExceptT ε m) β\n  | ⟨f⟩ => ⟨fun a => monadLift <| f (Except.ok a)⟩\n\nTarget:\ntheorem ExceptT.goto_mkLabel {α β ε : Type _} (x : Label (Except.{u, u} ε α) m β) (i : α) :\n    goto (ExceptT.mkLabel x) i = ExceptT.mk (Except.ok <$> goto x (Except.ok i)) :=\n\nProof body:\n","rejected":"by\n  cases x; rfl\n\nnonrec def ExceptT.callCC {ε} [MonadCont m] {α β : Type _}\n    (f : Label α (ExceptT ε m) β → ExceptT ε m α) : ExceptT ε m α :=\n  ExceptT.mk (callCC fun x : Label _ m β => ExceptT.run <| f (ExceptT.mkLabel x))\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Control/Monad","family_id":"exceptt","file_id":"mathlib/Mathlib/Control/Monad/Cont.lean","sample_id":"eb0438819632308cf7faaf08d1c110925f195fcd644c416f40abd06f69c1666f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"164db8212cd6dbe524941c096782dd0230af417452ad9fde38bafe58fdc70918","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4855e611094d50cafe6a8152849ab586521a79c6cc0f085db2e12cea390127a5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"470ba23482b167ed1d27d1364fdd8b69790cc7acc5d9ae4345731e59f6a235df","source_sha256":"6a1fd2aa47744142955f801b29fb06cc46e0d439ea80c9cccba97a19a60fbbe9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext\n  apply IsIntegralClosure.algebraMap_injective B A L\n  simp","hard_negative":true,"metrics":{"chosen_tokens":10,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.3},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"d4e4f820f136d6acab55ac283ac4deed85cf08700c8a5bb108d41d35b8a2c4e2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.IntegralClosure\npublic import Mathlib.RingTheory.RingHom.Finite\npublic import Mathlib.RingTheory.Localization.LocalizationLocalization\npublic import Mathlib.RingTheory.Localization.NormTrace\npublic import Mathlib.RingTheory.Norm.Transitivity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Restriction of various maps between fields to integrally closed subrings.\n\nIn this file, we assume `A` is an integrally closed domain; `K` is the fraction ring of `A`;\n`L` is a finite extension of `K`; `B` is the integral closure of `A` in `L`.\nWe call this the AKLB setup.\n\n## Main definitions\n- `galRestrict`: The restriction `Aut(L/K) → Aut(B/A)` as an `MulEquiv` in an AKLB setup.\n- `Algebra.intTrace`: The trace map of a finite extension of integrally closed domains `B/A` is\n  defined to be the restriction of the trace map of `Frac(B)/Frac(A)`.\n- `Algebra.intNorm`: The norm map of a finite extension of integrally closed domains `B/A` is\n  defined to be the restriction of the norm map of `Frac(B)/Frac(A)`.\n\n-/\n\n@[expose] public section\n\nopen Module nonZeroDivisors\n\nvariable (A K L L₂ L₃ B B₂ B₃ : Type*)\nvariable [CommRing A] [CommRing B] [CommRing B₂] [CommRing B₃]\nvariable [Algebra A B] [Algebra A B₂] [Algebra A B₃]\nvariable [Field K] [Field L] [Field L₂] [Field L₃]\nvariable [Algebra A K] [IsFractionRing A K]\nvariable [Algebra K L] [Algebra A L] [IsScalarTower A K L]\nvariable [Algebra K L₂] [Algebra A L₂] [IsScalarTower A K L₂]\nvariable [Algebra K L₃] [Algebra A L₃] [IsScalarTower A K L₃]\nvariable [Algebra B L] [IsScalarTower A B L] [IsIntegralClosure B A L]\nvariable [Algebra B₂ L₂] [IsScalarTower A B₂ L₂] [IsIntegralClosure B₂ A L₂]\nvariable [Algebra B₃ L₃] [IsScalarTower A B₃ L₃] [IsIntegralClosure B₃ A L₃]\n\nsection galois\n\nsection galRestrict'\nvariable {K L L₂ L₃}\nomit [IsFractionRing A K]\n\n/-- A generalization of `galRestrictHom` beyond endomorphisms. -/\nnoncomputable\ndef galRestrict' (f : L →ₐ[K] L₂) : (B →ₐ[A] B₂) :=\n  (IsIntegralClosure.equiv A (integralClosure A L₂) L₂ B₂).toAlgHom.comp\n      (((f.restrictScalars A).comp (IsScalarTower.toAlgHom A B L)).codRestrict\n        (integralClosure A L₂) (fun x ↦ IsIntegral.map _ (IsIntegralClosure.isIntegral A L x)))\n\n@[simp]\nlemma algebraMap_galRestrict'_apply (σ : L →ₐ[K] L₂) (x : B) :\n    algebraMap B₂ L₂ (galRestrict' A B B₂ σ x) = σ (algebraMap B L x) := by\n  simp [galRestrict', galRestrict', Subalgebra.algebraMap_eq]\n\n@[simp]\n\nTarget:\ntheorem galRestrict'_id : galRestrict' A B B (.id K L) = .id A B :=\n\nProof body:\n","rejected":"by\n  exact galRestrict'_id","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"266766fa865e14c002a877f43b0593744227e7a7688118f7e65aa14c319fad13","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/IntegralClosure","family_id":"galrestrict'_id","file_id":"mathlib/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean","sample_id":"470ba23482b167ed1d27d1364fdd8b69790cc7acc5d9ae4345731e59f6a235df"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0e429534a623f3db5dd3586328da30692723c2a4071b25aa658c931a2d458bab","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3a2ce68ac4ec49658754ad114bcf7a919e8e245e21ce98150b79ee6bdc8ff022","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a6146b935412bf1cd41ece91f946c0daf117bb2ef301024873391e1ed5ceafb2","source_sha256":"42f43c83c3b42913128aa30933a73363c485cf2bc69fc613214486004b504ca0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [commutator_eq_closure G, ← image_commutatorSet_closureCommutatorRepresentatives, ←\n    MonoidHom.map_closure, ← commutator_eq_closure]\n  exact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _))\n\nvariable [Finite (commutatorSet G)]","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":5,"token_jaccard":0.111111,"token_length_ratio":0.119048},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"d505c97f243cf4a62744d479cee6088c4bf9e53420dc377e5b80b663bbde82ba","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Subgroup.Finite\npublic import Mathlib.GroupTheory.Commutator.Basic\npublic import Mathlib.GroupTheory.Rank\npublic import Mathlib.GroupTheory.Index\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jordan Brown, Thomas Browning, Patrick Lutz\n-/\n/-!\nThe commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n\npublic section\n\nvariable {G : Type*} [Group G]\n\nnamespace Subgroup\n\n/-- The commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n@[to_additive /-- The commutator of a finite direct product is contained in the direct product of\nthe commutators. -/]\ntheorem commutator_pi_pi_of_finite {η : Type*} [Finite η] {Gs : η → Type*} [∀ i, Group (Gs i)]\n    (H K : ∀ i, Subgroup (Gs i)) : ⁅Subgroup.pi Set.univ H, Subgroup.pi Set.univ K⁆ =\n    Subgroup.pi Set.univ fun i => ⁅H i, K i⁆ := by\n  classical\n    apply le_antisymm (commutator_pi_pi_le H K)\n    rw [pi_le_iff]\n    intro i hi\n    rw [map_commutator]\n    apply commutator_mono <;>\n      · rw [le_pi_iff]\n        intro j _hj\n        rintro _ ⟨x, hx, rfl⟩\n        by_cases h : j = i\n        · subst h\n          simpa using hx\n        · simp [h, one_mem]\n\nvariable [Finite (commutatorSet G)]\n\ninstance : Group.FG (_root_.commutator G) := by\n  rw [commutator_eq_closure]; apply Group.closure_finite_fg\n\nvariable (G) in\nlemma rank_commutator_le_card : Group.rank (_root_.commutator G) ≤ Nat.card (commutatorSet G) := by\n  rw [Subgroup.rank_congr (commutator_eq_closure G)]\n  apply Subgroup.rank_closure_finite_le_nat_card\n\nvariable [Group.FG G]\n\ninstance finiteIndex_center : FiniteIndex (center G) := by\n  obtain ⟨S, -, hS⟩ := Group.rank_spec G\n  exact ⟨mt (Finite.card_eq_zero_of_embedding (quotientCenterEmbedding hS)) Finite.card_pos.ne'⟩\n\nvariable (G) in\nlemma index_center_le_pow : (center G).index ≤ Nat.card (commutatorSet G) ^ Group.rank G := by\n  obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G\n  rw [← hS1, ← Fintype.card_coe, ← Nat.card_eq_fintype_card, ← Finset.coe_sort_coe, ← Nat.card_fun]\n  exact Finite.card_le_of_embedding (quotientCenterEmbedding hS2)\n\nend Subgroup\n\nsection commutatorRepresentatives\n\nopen Subgroup\n\nlemma card_commutatorSet_closureCommutatorRepresentatives :\n    Nat.card (commutatorSet (closureCommutatorRepresentatives G)) = Nat.card (commutatorSet G) := by\n  rw [← image_commutatorSet_closureCommutatorRepresentatives G]\n  exact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _))\n\nTarget:\nlemma card_commutator_closureCommutatorRepresentatives :\n    Nat.card (commutator (closureCommutatorRepresentatives G)) = Nat.card (commutator G) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_a6146b935412","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"2eddd0377156620e39225547bbddf6495e1b63b123aa59fff372655776e15e11","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Commutator","family_id":"card_commutator_closurecommutatorrepresentatives","file_id":"mathlib/Mathlib/GroupTheory/Commutator/Finite.lean","sample_id":"a6146b935412bf1cd41ece91f946c0daf117bb2ef301024873391e1ed5ceafb2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bbbd68780f20b0315d7a36f5a0356a461835a08a2fef3351d5866ce71a89c492","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"37064592f892514616569c3cb0eac4f45c5514046251695c1a052fcd11d2dc27","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"550eea517b6f57ead3df265df6cf73d2563e67f3ae71b6c49159805a9ce8003d","source_sha256":"a7f1c7b70d08168190ea72793a69601025e762dd93aea5040d82a7e6ab550627","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_pow] at hxy ⊢\n  rw [hxy]","hard_negative":false,"metrics":{"chosen_tokens":23,"rejected_tokens":28,"token_jaccard":0.772727,"token_length_ratio":1.217391},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"d510268167f816313ec88d4c6465146cc7a8c348db9faa500803269afd3eced3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Submodule.Map\npublic import Mathlib.Algebra.Polynomial.Eval.Defs\npublic import Mathlib.RingTheory.Ideal.Quotient.Defs\npublic import Mathlib.Algebra.Module.Submodule.RestrictScalars\npublic import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic\n\nNamespace:\nSModEq\n\nLocal context:\n/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n/-!\n# modular equivalence for submodule\n-/\n\n@[expose] public section\n\n\nopen Submodule\n\nopen Polynomial\n\nvariable {R : Type*} [Ring R]\nvariable {S : Type*} [Ring S]\nvariable {A : Type*} [CommRing A]\nvariable {M : Type*} [AddCommGroup M] [Module R M] [Module S M] (U U₁ U₂ : Submodule R M)\nvariable {x x₁ x₂ y y₁ y₂ z z₁ z₂ : M}\nvariable {N : Type*} [AddCommGroup N] [Module R N] (V V₁ V₂ : Submodule R N)\n\n/-- A predicate saying two elements of a module are equivalent modulo a submodule. -/\ndef SModEq (x y : M) : Prop :=\n  (Submodule.Quotient.mk x : M ⧸ U) = Submodule.Quotient.mk y\n\n@[inherit_doc] notation:50 x \" ≡ \" y \" [SMOD \" N \"]\" => SModEq N x y\n\nvariable {U U₁ U₂}\n\nprotected theorem SModEq.def :\n    x ≡ y [SMOD U] ↔ (Submodule.Quotient.mk x : M ⧸ U) = Submodule.Quotient.mk y :=\n  Iff.rfl\n\nnamespace SModEq\n\ntheorem sub_mem : x ≡ y [SMOD U] ↔ x - y ∈ U := by rw [SModEq.def, Submodule.Quotient.eq]\n\n@[simp]\ntheorem top : x ≡ y [SMOD (⊤ : Submodule R M)] :=\n  (Submodule.Quotient.eq ⊤).2 mem_top\n\n@[simp]\ntheorem bot : x ≡ y [SMOD (⊥ : Submodule R M)] ↔ x = y := by\n  rw [SModEq.def, Submodule.Quotient.eq, mem_bot, sub_eq_zero]\n\n@[gcongr, mono]\ntheorem mono (HU : U₁ ≤ U₂) (hxy : x ≡ y [SMOD U₁]) : x ≡ y [SMOD U₂] :=\n  (Submodule.Quotient.eq U₂).2 <| HU <| (Submodule.Quotient.eq U₁).1 hxy\n\nlemma of_toAddSubgroup_le {U : Submodule R M} {V : Submodule S M}\n    (h : U.toAddSubgroup ≤ V.toAddSubgroup) {x y : M} (hxy : x ≡ y [SMOD U]) : x ≡ y [SMOD V] := by\n  simp only [SModEq, Submodule.Quotient.eq] at hxy ⊢\n  exact h hxy\n\n@[refl, simp]\nprotected theorem refl (x : M) : x ≡ x [SMOD U] :=\n  @rfl _ _\n\nprotected theorem rfl : x ≡ x [SMOD U] :=\n  SModEq.refl _\n\ninstance : Std.Refl (SModEq U) :=\n  ⟨SModEq.refl⟩\n\n@[symm]\nnonrec theorem symm (hxy : x ≡ y [SMOD U]) : y ≡ x [SMOD U] :=\n  hxy.symm\n\ntheorem comm : x ≡ y [SMOD U] ↔ y ≡ x [SMOD U] := ⟨symm, symm⟩\n\n@[trans]\nnonrec theorem trans (hxy : x ≡ y [SMOD U]) (hyz : y ≡ z [SMOD U]) : x ≡ z [SMOD U] :=\n  hxy.trans hyz\n\ninstance instTrans : Trans (SModEq U) (SModEq U) (SModEq U) where\n  trans := trans\n\n@[gcongr]\ntheorem add (hxy₁ : x₁ ≡ y₁ [SMOD U]) (hxy₂ : x₂ ≡ y₂ [SMOD U]) : x₁ + x₂ ≡ y₁ + y₂ [SMOD U] := by\n  rw [SModEq.def] at hxy₁ hxy₂ ⊢\n  simp_rw [Quotient.mk_add, hxy₁, hxy₂]\n\n@[gcongr]\ntheorem sum {ι} {s : Finset ι} {x y : ι → M}\n    (hxy : ∀ i ∈ s, x i ≡ y i [SMOD U]) : ∑ i ∈ s, x i ≡ ∑ i ∈ s, y i [SMOD U] := by\n  classical\n  induction s using Finset.cons_induction with\n  | empty => simp [SModEq.rfl]\n  | cons i s _ ih =>\n    grw [Finset.sum_cons, Finset.sum_cons, hxy i (Finset.mem_cons_self i s),\n      ih (fun j hj ↦ hxy j (Finset.mem_cons_of_mem hj))]\n\n@[gcongr]\ntheorem smul (hxy : x ≡ y [SMOD U]) (c : R) : c • x ≡ c • y [SMOD U] := by\n  rw [SModEq.def] at hxy ⊢\n  simp_rw [Quotient.mk_smul, hxy]\n\n@[gcongr]\nlemma nsmul (hxy : x ≡ y [SMOD U]) (n : ℕ) : n • x ≡ n • y [SMOD U] := by\n  rw [SModEq.def] at hxy ⊢\n  simp_rw [Quotient.mk_smul, hxy]\n\n@[gcongr]\nlemma zsmul (hxy : x ≡ y [SMOD U]) (n : ℤ) : n • x ≡ n • y [SMOD U] := by\n  rw [SModEq.def] at hxy ⊢\n  simp_rw [Quotient.mk_smul, hxy]\n\n@[gcongr]\ntheorem mul {I : Ideal A} {x₁ x₂ y₁ y₂ : A} (hxy₁ : x₁ ≡ y₁ [SMOD I])\n    (hxy₂ : x₂ ≡ y₂ [SMOD I]) : x₁ * x₂ ≡ y₁ * y₂ [SMOD I] := by\n  simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_mul] at hxy₁ hxy₂ ⊢\n  rw [hxy₁, hxy₂]\n\n@[gcongr]\ntheorem prod {I : Ideal A} {ι} {s : Finset ι} {x y : ι → A}\n    (hxy : ∀ i ∈ s, x i ≡ y i [SMOD I]) : ∏ i ∈ s, x i ≡ ∏ i ∈ s, y i [SMOD I] := by\n  classical\n  induction s using Finset.cons_induction with\n  | empty => simp [SModEq.rfl]\n  | cons i s _ ih =>\n    grw [Finset.prod_cons, Finset.prod_cons, hxy i (Finset.mem_cons_self i s),\n      ih (fun j hj ↦ hxy j (Finset.mem_cons_of_mem hj))]\n\n@[gcongr]\n\nTarget:\nlemma pow {I : Ideal A} {x y : A} (n : ℕ) (hxy : x ≡ y [SMOD I]) :\n    x ^ n ≡ y ^ n [SMOD I] :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_pow] at hxy ⊢\n  rw [hxy]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/SModEq","family_id":"pow","file_id":"mathlib/Mathlib/LinearAlgebra/SModEq/Basic.lean","sample_id":"550eea517b6f57ead3df265df6cf73d2563e67f3ae71b6c49159805a9ce8003d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0726e99880ae39e0c58cb38c38f992cfa6af39d47cf8ffe5ea36831ca442aef8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c379f0ea095080b6167d0f938b9ad3f7690e51b74a1939fe2962960f954fb874","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"06e54fa94e0402146a1056a8e38c4177d4d7f6c2eb72f119267197e2349465f8","source_sha256":"1c7994cb953bf15a3e2a75bf5a9f75792705603cc1fc781f9e07bc585e941697","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨Us, hUs, heq⟩ := TopologicalSpace.Opens.isBasis_iff_cover.mp\n    (PrespectralSpace.isBasis_opens X) ⟨f ⁻¹' U, hU.preimage hfc⟩\n  obtain ⟨t, ht⟩ := by\n    refine hc.elim_finite_subcover (fun s : Us ↦ f '' s.1) (fun s ↦ h _ s.1.2) (fun x hx ↦ ?_)\n    obtain ⟨x, rfl⟩ := hs hx\n    obtain ⟨i, hi, hx⟩ := mem_sSup.mp <| by rwa [← heq]\n    exact Set.mem_iUnion.mpr ⟨⟨i, hi⟩, x, hx, rfl⟩\n  refine ⟨⨆ s ∈ t, s.1, ?_, ?_⟩\n  · simp only [iSup_mk, carrier_eq_coe, coe_mk]\n    exact t.finite_toSet.isCompact_biUnion fun i _ ↦ hUs i.2\n  · simp only [iSup_mk, carrier_eq_coe, Set.iUnion_coe_set, coe_mk, Set.image_iUnion]\n    convert_to ⋃ i ∈ t, f '' i.1 = U\n    · simp\n    · refine subset_antisymm (fun x ↦ ?_) ht\n      simp_rw [Set.mem_iUnion]\n      rintro ⟨i, hi, x, hx, rfl⟩\n      have := heq ▸ mem_sSup.mpr ⟨i.1, i.2, hx⟩\n      exact this","hard_negative":true,"metrics":{"chosen_tokens":252,"rejected_tokens":5,"token_jaccard":0.037037,"token_length_ratio":0.019841},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"d5277a335d09abbbbedda64e08211f91c1ec3d35a94c8249310fec3959835981","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Ideal\npublic import Mathlib.Topology.Sets.Compacts\npublic import Mathlib.Topology.Sets.OpenCover\npublic import Mathlib.Topology.Spectral.Hom\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Prespectral spaces\n\nIn this file, we define prespectral spaces as spaces whose lattice of compact opens forms a basis.\n\n-/\n\n@[expose] public section\n\nopen TopologicalSpace Topology\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n\n/-- A space is prespectral if the lattice of compact opens forms a basis. -/\n@[stacks 08YG \"The last condition for spectral spaces\", mk_iff]\nclass PrespectralSpace (X : Type*) [TopologicalSpace X] : Prop where\n  isTopologicalBasis : IsTopologicalBasis { U : Set X | IsOpen U ∧ IsCompact U }\n\n/-- A space is prespectral if it has a basis consisting of compact opens. -/\nlemma PrespectralSpace.of_isTopologicalBasis {B : Set (Set X)}\n    (basis : IsTopologicalBasis B) (isCompact_basis : ∀ U ∈ B, IsCompact U) :\n    PrespectralSpace X where\n  isTopologicalBasis := basis.of_isOpen_of_subset (fun _ h ↦ h.1)\n    fun s hs ↦ ⟨basis.isOpen hs, isCompact_basis s hs⟩\n\n/-- A space is prespectral if it has a basis consisting of compact opens.\nThis is the variant with an indexed basis instead. -/\nlemma PrespectralSpace.of_isTopologicalBasis' {ι : Type*} {b : ι → Set X}\n    (basis : IsTopologicalBasis (Set.range b)) (isCompact_basis : ∀ i, IsCompact (b i)) :\n    PrespectralSpace X :=\n  .of_isTopologicalBasis basis (by simp_all)\n\ninstance (priority := low) [NoetherianSpace X] : PrespectralSpace X :=\n  .of_isTopologicalBasis isTopologicalBasis_opens fun _ _ ↦ NoetherianSpace.isCompact _\n\ninstance (priority := low) [PrespectralSpace X] : LocallyCompactSpace X where\n  local_compact_nhds _ _ hn :=\n    have ⟨V, ⟨hV₁, hV₂⟩, hxV, hVn⟩ := PrespectralSpace.isTopologicalBasis.mem_nhds_iff.mp hn\n    ⟨V, hV₁.mem_nhds hxV, hVn, hV₂⟩\n\nopen PrespectralSpace in\ninstance (priority := low) [T2Space X] [PrespectralSpace X] : TotallySeparatedSpace X :=\n  totallySeparatedSpace_iff_exists_isClopen.mpr fun _ _ hxy ↦\n    have ⟨U, ⟨hU₁, hU₂⟩, hxU, hyU⟩ :=\n      isTopologicalBasis.exists_subset_of_mem_open hxy isClosed_singleton.isOpen_compl\n    ⟨U, ⟨hU₂.isClosed, hU₁⟩, hxU, fun h ↦ hyU h rfl⟩\n\nlemma PrespectralSpace.of_isOpenCover\n    {ι : Type*} {U : ι → Opens X} (hU : IsOpenCover U) [∀ i, PrespectralSpace (U i)] :\n    PrespectralSpace X := by\n  refine .of_isTopologicalBasis (hU.isTopologicalBasis fun i ↦ isTopologicalBasis) ?_\n  simp only [Set.mem_iUnion, Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp,\n    forall_comm (α := Set _), forall_apply_eq_imp_iff₂]\n  exact fun i V hV hV' ↦ hV'.image continuous_subtype_val\n\nlemma PrespectralSpace.of_isInducing [PrespectralSpace Y]\n    (f : X → Y) (hf : IsInducing f) (hf' : IsSpectralMap f) : PrespectralSpace X :=\n  .of_isTopologicalBasis (PrespectralSpace.isTopologicalBasis.isInducing hf) (by\n    simp only [Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp]\n    rintro _ U h₁ h₂ rfl\n    exact hf'.isCompact_preimage_of_isOpen h₁ h₂)\n\nlemma PrespectralSpace.of_isClosedEmbedding [PrespectralSpace Y]\n    (f : X → Y) (hf : IsClosedEmbedding f) : PrespectralSpace X :=\n  .of_isInducing f hf.isInducing hf.isProperMap.isSpectralMap\n\n/-- Let `f : X → Y` be an open embedding of topological spaces.\nIf `Y` is a prespectral space (i.e., the quasi-compact opens of `Y` form a basis),\nthen `X` is also a prespectral space. -/\nlemma Topology.IsOpenEmbedding.prespectralSpace [PrespectralSpace Y]\n    {f : X → Y} (hf : IsOpenEmbedding f) :\n    PrespectralSpace X where\n  isTopologicalBasis := by\n    apply isTopologicalBasis_of_isOpen_of_nhds (fun U hU ↦ hU.1) <| fun x U hx hU ↦ ?_\n    obtain ⟨V, ⟨hoV, hcV⟩, hfx, hVf⟩ : ∃ V ∈ {V | IsOpen V ∧ IsCompact V}, f x ∈ V ∧ V ⊆ f '' U :=\n      (PrespectralSpace.isTopologicalBasis (X := Y)).isOpen_iff.mp\n        (hf.isOpen_iff_image_isOpen.mp hU) (f x) ⟨x, hx, rfl⟩\n    refine ⟨f ⁻¹' V, ⟨hoV.preimage hf.continuous, ?_⟩, ⟨hfx, fun y hy ↦ ?_⟩⟩\n    · exact hf.toIsInducing.isCompact_preimage' hcV <| Set.SurjOn.subset_range hVf\n    · exact hf.injective.mem_set_image.mp (hVf hy)\n\ninstance PrespectralSpace.sigma {ι : Type*} (X : ι → Type*) [∀ i, TopologicalSpace (X i)]\n    [∀ i, PrespectralSpace (X i)] : PrespectralSpace (Σ i, X i) :=\n  .of_isTopologicalBasis (IsTopologicalBasis.sigma fun i ↦ isTopologicalBasis) fun U hU ↦ by\n    simp_rw [Set.mem_iUnion] at hU\n    obtain ⟨i, V, hV, rfl⟩ := hU\n    exact hV.2.image continuous_sigmaMk\n\nvariable (X) in\nlemma PrespectralSpace.isBasis_opens [PrespectralSpace X] :\n    TopologicalSpace.Opens.IsBasis { U : Opens X | IsCompact (U : Set X) } := by\n  dsimp only [TopologicalSpace.Opens.IsBasis]\n  convert! isTopologicalBasis (X := X)\n  ext s\n  exact ⟨fun ⟨V, hV, heq⟩ ↦ heq ▸ ⟨V.2, hV⟩, fun h ↦ ⟨⟨s, h.1⟩, h.2, rfl⟩⟩\n\n/-- In a prespectral space, the lattice of opens is determined by its lattice of compact opens. -/\ndef PrespectralSpace.opensEquiv [PrespectralSpace X] :\n    Opens X ≃o Order.Ideal (CompactOpens X) where\n  toFun U := ⟨⟨{ V | (V : Set X) ⊆ U }, fun U₁ U₂ h₁ h₂ ↦ subset_trans (α := Set X) h₁ h₂⟩,\n    ⟨⊥, by simp⟩, fun U₁ h₁ U₂ h₂ ↦ ⟨U₁ ⊔ U₂, by aesop, le_sup_left, le_sup_right⟩⟩\n  invFun I := ⨆ U ∈ I, U.toOpens\n  left_inv U := by\n    apply le_antisymm\n    · simp only [iSup_le_iff]\n      exact fun _ ↦ id\n    · intro x hxU\n      obtain ⟨V, ⟨h₁, h₂⟩, hxV, hVU⟩ := isTopologicalBasis.exists_subset_of_mem_open hxU U.2\n      simp only [Opens.mem_iSup]\n      exact ⟨⟨⟨_, h₂⟩, h₁⟩, hVU, hxV⟩\n  right_inv I := by\n    ext U\n    dsimp\n    change U.toOpens ≤ _ ↔ _\n    refine ⟨fun H ↦ ?_, fun h ↦ le_iSup₂ (f := fun U (h : U ∈ I) ↦ U.toOpens) U h⟩\n    simp only [← SetLike.coe_subset_coe, Opens.iSup_mk, Opens.carrier_eq_coe, Opens.coe_mk] at H\n    obtain ⟨s, hsI, hs, hU⟩ := U.isCompact.elim_finite_subcover_image (fun U _ ↦ U.2) H\n    exact I.lower (a := hs.toFinset.sup fun i ↦ i) (by simpa [← SetLike.coe_subset_coe]) (by simpa)\n  map_rel_iff' {U V} := by\n    change (∀ (W : CompactOpens X), (W : Set X) ⊆ U → (W : Set X) ⊆ V) ↔ U ≤ V\n    refine ⟨?_, fun H W ↦ (le_trans · H)⟩\n    intro H x hxU\n    obtain ⟨W, ⟨h₁, h₂⟩, hxW, hWU⟩ := isTopologicalBasis.exists_subset_of_mem_open hxU U.2\n    exact H ⟨⟨W, h₂⟩, h₁⟩ hWU hxW\n\nopen TopologicalSpace Opens in\n/-- If `X` has a basis of compact opens and `f : X → S` is open, every\ncompact open of `S` is the image of a compact open of `X`. -/\n\nTarget:\nlemma IsOpenMap.exists_opens_image_eq_of_prespectralSpace [PrespectralSpace X] {f : X → Y}\n    (hfc : Continuous f) (h : IsOpenMap f) {U : Set Y} (hs : U ⊆ Set.range f) (hU : IsOpen U)\n    (hc : IsCompact U) : ∃ (V : Opens X), IsCompact V.1 ∧ f '' V = U :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_06e54fa94e04","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"e0d8c79aeba6d4b88d11e7e5a6ae32701f2c0a5e59919f7a512a5316382a34ec","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Spectral","family_id":"isopenmap","file_id":"mathlib/Mathlib/Topology/Spectral/Prespectral.lean","sample_id":"06e54fa94e0402146a1056a8e38c4177d4d7f6c2eb72f119267197e2349465f8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5a49665e0c5d2151828a184678146c90b5369415677dd342dff1e4392eda48b8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"83ec59d8b8358fb8a455e8f01681be42384d6a03b957fc77c05e2d58d2ad540d","source_sha256":"c729c093a684b2a7b5066e0c93a4ca1c797418af44bab051680547cf260ff85c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [*]","hard_negative":false,"metrics":{"chosen_tokens":5,"rejected_tokens":3,"token_jaccard":0.142857,"token_length_ratio":0.6},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"d5579f7c9e9f306460c26c6fb40c36a2d9e61093d6ec0e271b3fd19b232b2843","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Ineq\npublic import Mathlib.Data.Nat.Cast.Order.Ring\n\nNamespace:\nMathlib.Tactic.Linarith\n\nLocal context:\n/-\nCopyright (c) 2020 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n-/\npublic meta import Batteries.Tactic.Lint.Basic\npublic meta import Mathlib.Data.Ineq\npublic meta import Mathlib.Tactic.ToAdditive\n\n/-!\n# Lemmas for `linarith`.\n\nThose in the `Linarith` namespace should stay here.\n\nThose outside the `Linarith` namespace may be deleted as they are ported to mathlib4.\n-/\n\npublic meta section\n\nnamespace Mathlib.Tactic.Linarith\n\nuniverse u\ntheorem lt_irrefl {α : Type u} [Preorder α] {a : α} : ¬a < a := _root_.lt_irrefl a\n\nTarget:\ntheorem eq_of_eq_of_eq {α} [Semiring α] {a b : α} (ha : a = 0) (hb : b = 0) : a + b = 0 :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Linarith","family_id":"eq_of_eq_of_eq","file_id":"mathlib/Mathlib/Tactic/Linarith/Lemmas.lean","sample_id":"83ec59d8b8358fb8a455e8f01681be42384d6a03b957fc77c05e2d58d2ad540d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2851538f3a7c40f88dfdee204d38577a3a636017ca196f0ed156206cf7eba4b2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8bbc6f2b397a8cc463b8503d4ac88ef3657456a084f7c9bea02c0c20011fd255","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"06f9768559f2fe8088fc3798603424c8e8c34edf39453ccd8c0d887cb067503b","source_sha256":"06f9836a8093746b062f55af45a77b834aa17f6fb321985aabfd940c6ffdaecf","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction hx using span_induction with\n  | mem x h => obtain ⟨i, rfl⟩ := h; cases j <;> simp [h]\n  | zero => simp\n  | add u v _ _ hu hv => simp [hu, hv]\n  | smul t u _ hu => simp [hu]","hard_negative":true,"metrics":{"chosen_tokens":59,"rejected_tokens":3,"token_jaccard":0.027027,"token_length_ratio":0.050847},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"d5a6e83da85dcaee0dd3b77ab61f0b5839c52c0af2c7c3c2b717b30993c5ff14","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Lie.Matrix\npublic import Mathlib.Algebra.Lie.OfAssociative\npublic import Mathlib.Algebra.Lie.Weights.Basic\npublic import Mathlib.LinearAlgebra.Eigenspace.Matrix\npublic import Mathlib.LinearAlgebra.LinearIndependent.BaseChange\npublic import Mathlib.LinearAlgebra.RootSystem.CartanMatrix\n\nNamespace:\nRootPairing.GeckConstruction\n\nLocal context:\n/-\nCopyright (c) 2025 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Geck's construction of a Lie algebra associated to a root system\n\nThis file contains an implementation of Geck's construction of a semisimple Lie algebra from a\nreduced crystallographic root system. It follows [Geck](Geck2017) quite closely.\n\n## Main definitions:\n* `RootPairing.GeckConstruction.lieAlgebra`: the Geck construction of the Lie algebra associated to\n  a root system with distinguished base.\n* `RootPairing.GeckConstruction.cartanSubalgebra`: a distinguished subalgebra corresponding to a\n  Cartan subalgebra of the Geck construction.\n* `RootPairing.GeckConstruction.cartanSubalgebra_le_lieAlgebra`: the distinguished subalgebra is\n  contained in the Geck construction.\n\n## Alternative approaches\n\nThere are at least three ways to construct a Lie algebra from a root system:\n1. As a quotient of a free Lie algebra, using the Serre relations\n2. Directly defining the Lie bracket on $H ⊕ K^∣Φ|$\n3. The Geck construction\n\nWe comment on these as follows:\n1. This construction takes just a matrix as input. It yields a semisimple Lie algebra iff the\n   matrix is a Cartan matrix but it is quite a lot of work to prove this. On the other hand, it also\n   allows construction of Kac-Moody Lie algebras. It has been implemented as `Matrix.ToLieAlgebra`\n   but as of May 2025, almost nothing has been proved about it in Mathlib.\n2. This construction takes a root system with base as input, together with sufficient additional\n   data to determine a collection of extraspecial pairs of roots. The additional data for the\n   extraspecial pairs is required to pin down certain signs when defining the Lie bracket. (These\n   signs can be interpreted as a set-theoretic splitting of Tits's extension of the Weyl group by\n   an elementary 2-group of order $2^l$ where $l$ is the rank.)\n3. This construction takes a root system with base as input and is implemented here.\n\nThere seems to be no known construction of a Lie algebra from a root system without first choosing\na base: https://mathoverflow.net/questions/495434/\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Function Set Submodule\nopen scoped Matrix\n\nattribute [local simp] Matrix.mul_apply Matrix.one_apply Matrix.diagonal_apply\n\nnamespace RootPairing.GeckConstruction\n\nvariable {ι R M N : Type*} [CommRing R]\n  [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]\n  {P : RootPairing ι R M N} [P.IsCrystallographic] {b : P.Base}\n\n/-- Part of an `sl₂` triple used in Geck's construction of a Lie algebra from a root system. -/\ndef h (i : b.support) :\n    Matrix (b.support ⊕ ι) (b.support ⊕ ι) R :=\n  open scoped Classical in\n  .fromBlocks 0 0 0 (.diagonal (P.pairingIn ℤ · i))\n\nlemma h_def [DecidableEq ι] (i : b.support) :\n    h i = .fromBlocks 0 0 0 (.diagonal (P.pairingIn ℤ · i)) := by\n  ext (j | j) (k | k) <;> simp [h, Matrix.diagonal_apply]\n\nlemma h_eq_diagonal [DecidableEq ι] (i : b.support) :\n    h i = .diagonal (Sum.elim 0 (P.pairingIn ℤ · i)) := by\n  ext (j | j) (k | k) <;> simp [h, Matrix.diagonal_apply]\n\nvariable (b) in\nlemma linearIndependent_h [Finite ι] [CharZero R] [IsDomain R] [P.IsRootSystem] :\n    LinearIndependent R (h (b := b)) := by\n  classical\n  have : Matrix.diagLinearMap (b.support ⊕ ι) R R ∘ h =\n      Sum.elimZeroLeft ∘ fun i : b.support ↦ algebraMap ℤ R ∘ (P.pairingIn ℤ · i) := by\n    ext; rw [comp_apply, h_def]; aesop\n  apply LinearIndependent.of_comp (Matrix.diagLinearMap _ _ _)\n  rw [this, LinearMap.linearIndependent_iff_of_injOn _ Sum.elim_injective'.injOn,\n    linearIndependent_algebraMap_comp_iff]\n  suffices LinearIndependent ℤ (fun i j : b.support ↦ P.pairingIn ℤ j i) from\n    this.of_linearIndependent_subset b.support\n  apply b.cartanMatrix.transpose.linearIndependent_rows_of_det_ne_zero\n  rw [Matrix.det_transpose, ← Matrix.nondegenerate_iff_det_ne_zero]\n  exact b.cartanMatrix_nondegenerate\n\nlemma span_range_h_le_range_diagonal [DecidableEq ι] :\n    span R (range h) ≤ LinearMap.range (Matrix.diagonalLinearMap (b.support ⊕ ι) R R) := by\n  rw [span_le]\n  rintro - ⟨i, rfl⟩\n  rw [h_eq_diagonal]\n  exact LinearMap.mem_range_self _ _\n\nopen Matrix in\n@[simp] lemma diagonal_elim_mem_span_h_iff [DecidableEq ι] {d : ι → R} :\n    diagonal (Sum.elim 0 d) ∈ span R (range <| h (b := b)) ↔\n      d ∈ span R (range <| fun (i : b.support) j ↦ (P.pairingIn ℤ j i : R)) := by\n  let g : Matrix ι ι R →ₗ[R] Matrix (b.support ⊕ ι) (b.support ⊕ ι) R :=\n    { toFun := .fromBlocks 0 0 0\n      map_add' x y := by ext (i | i) (j | j) <;> simp\n      map_smul' t x := by ext (i | i) (j | j) <;> simp }\n  have h₀ : Injective (g ∘ diagonalLinearMap ι R R) := fun _ _ hd ↦ funext <| by simpa [g] using hd\n  have h₁ {d : ι → R} : diagonal (Sum.elim 0 d) = g (diagonalLinearMap ι R R d) := by\n    ext (i | i) (j | j) <;> simp [g]\n  have h₂ : range h = g '' (diagonalLinearMap ι R R ''\n    (range <| fun (i : b.support) j ↦ (P.pairingIn ℤ j i : R))) := by ext; simp [g, h_def]\n  simp_rw [h₁, h₂, span_image, ← map_comp, ← comp_apply (f := g), mem_map, LinearMap.coe_comp,\n    h₀.eq_iff, exists_eq_right]\n\nTarget:\nlemma apply_sum_inl_eq_zero_of_mem_span_h\n    (i : b.support) (j : b.support ⊕ ι) {x : Matrix (b.support ⊕ ι) (b.support ⊕ ι) R}\n    (hx : x ∈ span R (range h)) :\n    x j (Sum.inl i) = 0 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_06f9768559f2","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"af2478621552e6fd0d65aa2949052d08477f0c4af0da82df09dd1b51e8daa920","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/RootSystem","family_id":"apply_sum_inl_eq_zero_of_mem_span_h","file_id":"mathlib/Mathlib/LinearAlgebra/RootSystem/GeckConstruction/Basic.lean","sample_id":"06f9768559f2fe8088fc3798603424c8e8c34edf39453ccd8c0d887cb067503b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e79e7e0dc18cc849a7cc458c4f413b9fd00b3758b5b3041d108cd5fc7c5dd925","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"05a81ca678f684a0e635006515f905d4791271bb1386b472a3d4d7bf0d040e70","source_sha256":"94fce5005b3452fffd0935ed1c8380f811db0b64c02673ffec8176805d9749dd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun _ ↦ ?_, fun _ ↦ inferInstance⟩\n  let e : G.inv ⋙ G ⋙ F ≅ F := (associator _ _ _).symm ≪≫\n    isoWhiskerRight (G.asEquivalence.counitIso) _ ≪≫ F.leftUnitor\n  exact of_iso e","hard_negative":false,"metrics":{"chosen_tokens":54,"rejected_tokens":3,"token_jaccard":0.09375,"token_length_ratio":0.055556},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"d6189f58a46af849cf19871b1782fa0855d7b13edb85caa98302eab34fdebc41","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Functor.KanExtension.DenseAt\npublic import Mathlib.CategoryTheory.Limits.Presheaf\npublic import Mathlib.CategoryTheory.Generator.StrongGenerator\n\nNamespace:\nCategoryTheory.Functor\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Dense functors\n\nA functor `F : C ⥤ D` is dense (`F.IsDense`) if `𝟭 D` is a pointwise\nleft Kan extension of `F` along itself, i.e. any `Y : D` is the\ncolimit of all `F.obj X` for all morphisms `F.obj X ⟶ Y` (which\nis the condition `F.DenseAt Y`).\nWhen `F` is full, we show that this\nis equivalent to saying that the restricted Yoneda functor\n`D ⥤ Cᵒᵖ ⥤ Type _` is fully faithful (see the lemma\n`Functor.isDense_iff_fullyFaithful_restrictedULiftYoneda`).\n\nWe also show that the range of a dense functor is a strong\ngenerator (see `Functor.isStrongGenerator_of_isDense`).\n\n## References\n\n* https://ncatlab.org/nlab/show/dense+subcategory\n\n-/\n\n@[expose] public section\n\nuniverse w v₁ v₂ v₃ u₁ u₂ u₃\n\nnamespace CategoryTheory\n\nopen Limits Opposite Presheaf ConcreteCategory\n\nvariable {C : Type u₁} {D : Type u₂} [Category.{v₁} C] [Category.{v₂} D]\n  {C' : Type u₃} [Category.{v₃} C']\n\nnamespace Functor\n\n/-- A functor `F : C ⥤ D` is dense if any `Y : D` is a canonical colimit\nrelatively to `F`. -/\nclass IsDense (F : C ⥤ D) : Prop where\n  isDenseAt (F) (Y : D) : F.isDenseAt Y\n\n/-- This is a choice of structure `F.DenseAt Y` when `F : C ⥤ D`\nis dense, and `Y : D`. -/\nnoncomputable def denseAt (F : C ⥤ D) [F.IsDense] (Y : D) : F.DenseAt Y :=\n  (IsDense.isDenseAt F Y).some\n\nlemma isDense_iff_nonempty_isPointwiseLeftKanExtension (F : C ⥤ D) :\n    F.IsDense ↔\n      Nonempty ((LeftExtension.mk _ (rightUnitor F).inv).IsPointwiseLeftKanExtension) :=\n  ⟨fun _ ↦ ⟨fun _ ↦ F.denseAt _⟩, fun ⟨h⟩ ↦ ⟨fun _ ↦ ⟨h _⟩⟩⟩\n\ninstance (F : C ⥤ D) [F.IsDense] : Functor.IsLeftKanExtension (𝟭 D) (Functor.rightUnitor F).inv :=\n  ((Functor.isDense_iff_nonempty_isPointwiseLeftKanExtension F).mp ‹_›).some.isLeftKanExtension\n\ninstance (F : C ⥤ D) [F.IsDense] : F.HasPointwiseLeftKanExtension F :=\n  fun X ↦ (Functor.IsDense.isDenseAt F X).some.hasPointwiseLeftKanExtensionAt\n\nlemma IsDense.of_iso {F G : C ⥤ D} (e : F ≅ G) [F.IsDense] :\n    G.IsDense where\n  isDenseAt Y := by\n    rw [← Functor.congr_isDenseAt e]\n    exact ⟨F.denseAt Y⟩\n\nlemma IsDense.iff_of_iso {F G : C ⥤ D} (e : F ≅ G) :\n    F.IsDense ↔ G.IsDense :=\n  ⟨fun _ ↦ of_iso e, fun _ ↦ of_iso e.symm⟩\n\nvariable (F : C ⥤ D)\n\ninstance (G : C' ⥤ C) [F.IsDense] [G.IsEquivalence] :\n    (G ⋙ F).IsDense where\n  isDenseAt Y := ⟨(F.denseAt Y).precompOfFinal G⟩\n\nTarget:\nlemma IsDense.comp_left_iff_of_isEquivalence (G : C' ⥤ C) [G.IsEquivalence] :\n    (G ⋙ F).IsDense ↔ F.IsDense :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Functor","family_id":"isdense","file_id":"mathlib/Mathlib/CategoryTheory/Functor/KanExtension/Dense.lean","sample_id":"05a81ca678f684a0e635006515f905d4791271bb1386b472a3d4d7bf0d040e70"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c165c49ef010a17e120cfdcd1a6cc6afaacc95eed29ab44ee041da167fa437e8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"bc6b5ae320fe350ef6424ba0b4da9753b031045ff502bf96126cd0bccf9f88c4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c0fbb14cf7d5c8d77a74ea9851f8a6d43a29d0444f221fd7a52f0350a1b23692","source_sha256":"e0ea8313840496a509375804bd2710d8b2f93f0c2a8c8922e53c600d1e07dd2a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  -- this follows from a covering argument using the sets satisfying `ρ a ≤ ν a`.\n  apply ENNReal.le_of_forall_pos_le_add fun ε εpos _ => ?_\n  obtain ⟨U, sU, U_open, νU⟩ : ∃ (U : Set α), s ⊆ U ∧ IsOpen U ∧ ν U ≤ ν s + ε :=\n    exists_isOpen_le_add s ν (ENNReal.coe_pos.2 εpos).ne'\n  let f : α → Set (Set α) := fun _ => {a | ρ a ≤ ν a ∧ a ⊆ U}\n  have h : v.FineSubfamilyOn f s := by\n    apply v.fineSubfamilyOn_of_frequently f s fun x hx => ?_\n    have :=\n      (hs x hx).and_eventually\n        ((v.eventually_filterAt_mem_setsAt x).and\n          (v.eventually_filterAt_subset_of_nhds (U_open.mem_nhds (sU hx))))\n    apply Frequently.mono this\n    rintro a ⟨ρa, _, aU⟩\n    exact ⟨ρa, aU⟩\n  haveI : Encodable h.index := h.index_countable.toEncodable\n  calc\n    ρ s ≤ ∑' x : h.index, ρ (h.covering x) := h.measure_le_tsum_of_absolutelyContinuous hρ\n    _ ≤ ∑' x : h.index, ν (h.covering x) := ENNReal.tsum_le_tsum fun x => (h.covering_mem x.2).1\n    _ = ν (⋃ x : h.index, h.covering x) := by\n      rw [measure_iUnion h.covering_disjoint_subtype fun i => h.measurableSet_u i.2]\n    _ ≤ ν U := (measure_mono (iUnion_subset fun i => (h.covering_mem i.2).2))\n    _ ≤ ν s + ε := νU","hard_negative":true,"metrics":{"chosen_tokens":324,"rejected_tokens":3,"token_jaccard":0.020619,"token_length_ratio":0.009259},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"d66dcb7709be2eb89578ad64882996984efa2422571e7721296777ed54536487","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Covering.VitaliFamily\npublic import Mathlib.MeasureTheory.Function.AEMeasurableOrder\npublic import Mathlib.MeasureTheory.Integral.Average\npublic import Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue\npublic import Mathlib.MeasureTheory.Measure.Regular\n\nNamespace:\nVitaliFamily\n\nLocal context:\n/-\nCopyright (c) 2021 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Differentiation of measures\n\nOn a second countable metric space with a measure `μ`, consider a Vitali family (i.e., for each `x`\none has a family of sets shrinking to `x`, with a good behavior with respect to covering theorems).\nConsider also another measure `ρ`. Then, for almost every `x`, the ratio `ρ a / μ a` converges when\n`a` shrinks to `x` along the Vitali family, towards the Radon-Nikodym derivative of `ρ` with\nrespect to `μ`. This is the main theorem on differentiation of measures.\n\nThis theorem is proved in this file, under the name `VitaliFamily.ae_tendsto_rnDeriv`. Note that,\nalmost surely, `μ a` is eventually positive and finite (see\n`VitaliFamily.ae_eventually_measure_pos` and `VitaliFamily.eventually_measure_lt_top`), so the\nratio really makes sense.\n\nFor concrete applications, one needs concrete instances of Vitali families, as provided for instance\nby `Besicovitch.vitaliFamily` (for balls) or by `Vitali.vitaliFamily` (for doubling measures).\n\nSpecific applications to Lebesgue density points and the Lebesgue differentiation theorem are also\nderived:\n* `VitaliFamily.ae_tendsto_measure_inter_div` states that, for almost every point `x ∈ s`,\n  then `μ (s ∩ a) / μ a` tends to `1` as `a` shrinks to `x` along a Vitali family.\n* `VitaliFamily.ae_tendsto_average_norm_sub` states that, for almost every point `x`, then the\n  average of `y ↦ ‖f y - f x‖` on `a` tends to `0` as `a` shrinks to `x` along a Vitali family.\n\n## Sketch of proof\n\nLet `v` be a Vitali family for `μ`. Assume for simplicity that `ρ` is absolutely continuous with\nrespect to `μ`, as the case of a singular measure is easier.\n\nIt is easy to see that a set `s` on which `liminf ρ a / μ a < q` satisfies `ρ s ≤ q * μ s`, by using\na disjoint subcovering provided by the definition of Vitali families. Similarly for the limsup.\nIt follows that a set on which `ρ a / μ a` oscillates has measure `0`, and therefore that\n`ρ a / μ a` converges almost surely (`VitaliFamily.ae_tendsto_div`). Moreover, on a set where the\nlimit is close to a constant `c`, one gets `ρ s ∼ c μ s`, using again a covering lemma as above.\nIt follows that `ρ` is equal to `μ.withDensity (v.limRatio ρ x)`, where `v.limRatio ρ x` is the\nlimit of `ρ a / μ a` at `x` (which is well defined almost everywhere). By uniqueness of the\nRadon-Nikodym derivative, one gets `v.limRatio ρ x = ρ.rnDeriv μ x` almost everywhere, completing\nthe proof.\n\nThere is a difficulty in this sketch: this argument works well when `v.limRatio ρ` is measurable,\nbut there is no guarantee that this is the case, especially if one doesn't make further assumptions\non the Vitali family. We use an indirect argument to show that `v.limRatio ρ` is always\nalmost everywhere measurable, again based on the disjoint subcovering argument\n(see `VitaliFamily.exists_measurable_supersets_limRatio`), and then proceed as sketched above\nbut replacing `v.limRatio ρ` by a measurable version called `v.limRatioMeas ρ`.\n\n## Counterexample\n\nThe standing assumption in this file is that spaces are second countable. Without this assumption,\nmeasures may be zero locally but nonzero globally, which is not compatible with differentiation\ntheory (which deduces global information from local one). Here is an example displaying this\nbehavior.\n\nDefine a measure `μ` by `μ s = 0` if `s` is covered by countably many balls of radius `1`,\nand `μ s = ∞` otherwise. This is indeed a countably additive measure, which is moreover\nlocally finite and doubling at small scales. It vanishes on every ball of radius `1`, so all the\nquantities in differentiation theory (defined as ratios of measures as the radius tends to zero)\nmake no sense. However, the measure is not globally zero if the space is big enough.\n\n## References\n\n* [Herbert Federer, Geometric Measure Theory, Chapter 2.9][Federer1996]\n-/\n\n@[expose] public section\n\nopen MeasureTheory Metric Set Filter TopologicalSpace MeasureTheory.Measure\n\nopen scoped Filter ENNReal MeasureTheory NNReal Topology\n\nvariable {α : Type*} [PseudoMetricSpace α] {m0 : MeasurableSpace α} {μ : Measure α}\n  (v : VitaliFamily μ)\n  {E : Type*} [NormedAddCommGroup E]\n\nnamespace VitaliFamily\n\n/-- The limit along a Vitali family of `ρ a / μ a` where it makes sense, and garbage otherwise.\nDo *not* use this definition: it is only a temporary device to show that this ratio tends almost\neverywhere to the Radon-Nikodym derivative. -/\nnoncomputable def limRatio (ρ : Measure α) (x : α) : ℝ≥0∞ :=\n  limUnder (v.filterAt x) fun a => ρ a / μ a\n\n/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive\nmeasure. (This is a nontrivial result, following from the covering property of Vitali families). -/\ntheorem ae_eventually_measure_pos [SecondCountableTopology α] :\n    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by\n  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs\n  simp -zeta only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs\n  change μ s = 0\n  let f : α → Set (Set α) := fun _ => {a | μ a = 0}\n  have h : v.FineSubfamilyOn f s := by\n    intro x hx ε εpos\n    rw [hs] at hx\n    simp only [frequently_filterAt_iff, gt_iff_lt, mem_setOf_eq] at hx\n    rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩\n    exact ⟨a, ⟨a_sets, μa⟩, ax⟩\n  refine le_antisymm ?_ bot_le\n  calc\n    μ s ≤ ∑' x : h.index, μ (h.covering x) := h.measure_le_tsum\n    _ = ∑' x : h.index, 0 := by congr; ext1 x; exact h.covering_mem x.2\n    _ = 0 := by simp only [tsum_zero]\n\n/-- For every point `x`, sufficiently small sets in a Vitali family around `x` have finite measure.\n(This is a trivial result, following from the fact that the measure is locally finite). -/\ntheorem eventually_measure_lt_top [IsLocallyFiniteMeasure μ] (x : α) :\n    ∀ᶠ a in v.filterAt x, μ a < ∞ :=\n  (μ.finiteAt_nhds x).eventually.filter_mono inf_le_left\n\n/-- If two measures `ρ` and `ν` have, at every point of a set `s`, arbitrarily small sets in a\nVitali family satisfying `ρ a ≤ ν a`, then `ρ s ≤ ν s` if `ρ ≪ μ`. -/\n\nTarget:\ntheorem measure_le_of_frequently_le [SecondCountableTopology α] [BorelSpace α] {ρ : Measure α}\n    (ν : Measure α) [IsLocallyFiniteMeasure ν] (hρ : ρ ≪ μ) (s : Set α)\n    (hs : ∀ x ∈ s, ∃ᶠ a in v.filterAt x, ρ a ≤ ν a) : ρ s ≤ ν s :=\n\nProof body:\n","rejected":"by\n  exact measure_le_of_frequently_le","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"d1a9cbad5c1ec41b19af4e0eabb3ae5601ad6a18569cbef2d6111783fc2137eb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Covering","family_id":"measure_le_of_frequently_le","file_id":"mathlib/Mathlib/MeasureTheory/Covering/Differentiation.lean","sample_id":"c0fbb14cf7d5c8d77a74ea9851f8a6d43a29d0444f221fd7a52f0350a1b23692"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6bc6459c3657e743f6b65ab2f495aaf259c49629a6dfb3dcfa555bfa29fc68bb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b6ed2b9a81e9433eac524aab243c09c5cb4d432e96ecd8f31ce797043909f1bd","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"56582689395b3fd4d53bdd9aa4d7c0b70282e024f7b7737ed578d8946fb979bf","source_sha256":"bc065e5d26147920810f7e88f906504ecfcce0ffd1da2e0b44dc4df7256292f1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hf : Monotone fun n => ⨅ i ≥ n, f i := fun n m h => biInf_mono fun i => h.trans\n  rw [← Monotone.iSup_nat_add hf k]\n  · simp_rw [iInf_ge_eq_iInf_nat_add, ← Nat.add_assoc]\n\n-- Not `@[simp]` since the subterm `?f (i + ?k)` produces an ugly higher-order unification problem.\n-- (Although the `simpNF` linter does not complain.)\n-- See: https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/complete_lattice.20and.20has_sup/near/316497982","hard_negative":true,"metrics":{"chosen_tokens":143,"rejected_tokens":5,"token_jaccard":0.02439,"token_length_ratio":0.034965},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"d6db77a1d0ef97448f0a8576c54de9b71ec2c97c961bc819b20b5f95561df51b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Bool.Set\npublic import Mathlib.Data.Nat.Set\npublic import Mathlib.Order.CompleteLattice.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Theory of complete lattices\n\nThis file contains results on complete lattices that need more theory to develop.\n\n## Naming conventions\n\nIn lemma names,\n* `sSup` is called `sSup`\n* `sInf` is called `sInf`\n* `⨆ i, s i` is called `iSup`\n* `⨅ i, s i` is called `iInf`\n* `⨆ i j, s i j` is called `iSup₂`. This is an `iSup` inside an `iSup`.\n* `⨅ i j, s i j` is called `iInf₂`. This is an `iInf` inside an `iInf`.\n* `⨆ i ∈ s, t i` is called `biSup` for \"bounded `iSup`\". This is the special case of `iSup₂`\n  where `j : i ∈ s`.\n* `⨅ i ∈ s, t i` is called `biInf` for \"bounded `iInf`\". This is the special case of `iInf₂`\n  where `j : i ∈ s`.\n\n## Notation\n\n* `⨆ i, f i` : `iSup f`, the supremum of the range of `f`;\n* `⨅ i, f i` : `iInf f`, the infimum of the range of `f`.\n-/\n\npublic section\n\nopen Function OrderDual Set\n\nvariable {α β γ : Type*} {ι ι' : Sort*} {κ : ι → Sort*} {κ' : ι' → Sort*}\n\nopen OrderDual\n\nsection\n\nvariable [CompleteLattice α] {f g s : ι → α} {a b : α}\n\n/-!\n### `iSup` and `iInf` under `Bool`\n-/\n\n@[to_dual]\ntheorem iSup_bool_eq {f : Bool → α} : ⨆ b : Bool, f b = f true ⊔ f false := by\n  rw [iSup, Bool.range_eq, sSup_pair, sup_comm]\n\n@[to_dual]\ntheorem sup_eq_iSup (x y : α) : x ⊔ y = ⨆ b : Bool, cond b x y := by\n  rw [iSup_bool_eq, Bool.cond_true, Bool.cond_false]\n\n/-!\n### `iSup` and `iInf` under `ℕ`\n-/\n\n@[to_dual]\ntheorem iSup_ge_eq_iSup_nat_add (u : ℕ → α) (n : ℕ) : ⨆ i ≥ n, u i = ⨆ i, u (i + n) := by\n  apply le_antisymm <;> simp only [iSup_le_iff]\n  · refine fun i hi => le_sSup ⟨i - n, ?_⟩\n    dsimp only\n    rw [Nat.sub_add_cancel hi]\n  · exact fun i => le_sSup ⟨i + n, iSup_pos (Nat.le_add_left _ _)⟩\n\n-- `to_dual` cannot translate between `Monotone` and `Antitone`.\ntheorem Monotone.iSup_nat_add {f : ℕ → α} (hf : Monotone f) (k : ℕ) : ⨆ n, f (n + k) = ⨆ n, f n :=\n  le_antisymm (iSup_le fun i => le_iSup _ (i + k)) <| iSup_mono fun i => hf <| Nat.le_add_right i k\n\ntheorem Antitone.iInf_nat_add {f : ℕ → α} (hf : Antitone f) (k : ℕ) : ⨅ n, f (n + k) = ⨅ n, f n :=\n  hf.dual_right.iSup_nat_add k\n\n-- Not `@[simp]` since the subterm `?f (i + ?k)` produces an ugly higher-order unification problem.\n-- (Although the `simpNF` linter does not complain.)\n-- See: https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/complete_lattice.20and.20has_sup/near/316497982\n\nTarget:\ntheorem iSup_iInf_ge_nat_add (f : ℕ → α) (k : ℕ) :\n    ⨆ n, ⨅ i ≥ n, f (i + k) = ⨆ n, ⨅ i ≥ n, f i :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_56582689395b","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"05d49dd2c544ee0b85a30592f3bd7ba0ed052e1d08f0005e5e6516189ab25d35","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/CompleteLattice","family_id":"isup_iinf_ge_nat_add","file_id":"mathlib/Mathlib/Order/CompleteLattice/Lemmas.lean","sample_id":"56582689395b3fd4d53bdd9aa4d7c0b70282e024f7b7737ed578d8946fb979bf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5403bbdd6b8db64d0404aea07e7fcb0b3cc956807fed1c2cfdeeee9bac43ff23","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a3a4139f0052b0a32136e52937242d5f55eef48c36e8634d6c5292950c12886e","source_sha256":"e6e1b45f74734f944289eb9bfce978e134476af3b67ff11012d6e865d086c8c1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  simp [h.neg_to_eq, div_nonneg]","hard_negative":false,"metrics":{"chosen_tokens":10,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.3},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"d6dc05f4c00067c0306a3dcfa5493a5f4324c0752440de05c941dd06fe682b26","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Floor.Semiring\npublic import Mathlib.Data.NNRat.Order\npublic import Mathlib.Data.Rat.Floor\n\nNamespace:\nMathlib.Meta.NormNum\n\nLocal context:\n/-\nCopyright (c) 2024 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\npublic meta import Mathlib.Data.Rat.Floor\n\n/-!\n# Floor Function for Non-negative Rational Numbers\n\n## Summary\n\nWe define the `FloorSemiring` instance on `ℚ≥0`, and relate its operators to `NNRat.cast`.\n\nNote that we cannot talk about `Int.fract`, which currently only works for rings.\n\n## Tags\n\nnnrat, rationals, ℚ≥0, floor\n-/\n\n@[expose] public section\n\nassert_not_exists Finset\n\nnamespace NNRat\n\ninstance : FloorSemiring ℚ≥0 where\n  floor q := ⌊q.val⌋₊\n  ceil q := ⌈q.val⌉₊\n  floor_of_neg h := by simpa using h.trans zero_lt_one\n  gc_floor {a n} h := by rw [← NNRat.coe_le_coe, Nat.le_floor_iff] <;> norm_cast\n  gc_ceil {a b} := by rw [← NNRat.coe_le_coe, Nat.ceil_le]; norm_cast\n\n@[simp, norm_cast]\ntheorem floor_coe (q : ℚ≥0) : ⌊(q : ℚ)⌋₊ = ⌊q⌋₊ := rfl\n\n@[simp, norm_cast]\ntheorem ceil_coe (q : ℚ≥0) : ⌈(q : ℚ)⌉₊ = ⌈q⌉₊ := rfl\n\n@[simp, norm_cast]\ntheorem coe_floor (q : ℚ≥0) : ↑⌊q⌋₊ = ⌊(q : ℚ)⌋ := Int.natCast_floor_eq_floor q.coe_nonneg\n\n@[simp, norm_cast]\ntheorem coe_ceil (q : ℚ≥0) : ↑⌈q⌉₊ = ⌈(q : ℚ)⌉ := Int.natCast_ceil_eq_ceil q.coe_nonneg\n\nprotected theorem floor_def (q : ℚ≥0) : ⌊q⌋₊ = q.num / q.den := by\n  rw [← Int.natCast_inj, NNRat.coe_floor, Rat.floor_def', Int.natCast_ediv, den_coe, num_coe]\n\nsection Semifield\n\nvariable {K} [Semifield K] [LinearOrder K] [IsStrictOrderedRing K] [FloorSemiring K]\n\n@[simp, norm_cast]\ntheorem floor_cast (x : ℚ≥0) : ⌊(x : K)⌋₊ = ⌊x⌋₊ :=\n  (Nat.floor_eq_iff x.cast_nonneg).2 (mod_cast (Nat.floor_eq_iff x.cast_nonneg).1 (Eq.refl ⌊x⌋₊))\n\n@[simp, norm_cast]\ntheorem ceil_cast (x : ℚ≥0) : ⌈(x : K)⌉₊ = ⌈x⌉₊ := by\n  obtain rfl | hx := eq_or_ne x 0\n  · simp\n  · refine (Nat.ceil_eq_iff ?_).2 (mod_cast (Nat.ceil_eq_iff ?_).1 (Eq.refl ⌈x⌉₊)) <;> simpa\n\nend Semifield\n\nsection Field\n\nvariable {K} [Field K] [LinearOrder K] [IsStrictOrderedRing K] [FloorRing K]\n\n@[simp, norm_cast]\ntheorem intFloor_cast (x : ℚ≥0) : ⌊(x : K)⌋ = ⌊(x : ℚ)⌋ := by\n  rw [Int.floor_eq_iff, ← coe_floor]\n  norm_cast\n  norm_cast\n  rw [Nat.cast_add_one, ← Nat.floor_eq_iff zero_le]\n\n@[simp, norm_cast]\ntheorem intCeil_cast (x : ℚ≥0) : ⌈(x : K)⌉ = ⌈(x : ℚ)⌉ := by\n  rw [Int.ceil_eq_iff, ← coe_ceil, sub_lt_iff_lt_add]\n  constructor\n  · exact_mod_cast NNRat.cast_strictMono <| Nat.ceil_lt_add_one zero_le\n  · rw [Int.cast_natCast, NNRat.cast_le_natCast]\n    exact Nat.le_ceil _\n\nend Field\n\n@[norm_cast]\ntheorem floor_natCast_div_natCast (n d : ℕ) : ⌊(↑n / ↑d : ℚ≥0)⌋₊ = n / d :=\n  Rat.natFloor_natCast_div_natCast n d\n\nend NNRat\n\nnamespace Mathlib.Meta.NormNum\n\nopen Qq\n\n/-!\n### `norm_num` extension for `Nat.ceil`\n-/\n\ntheorem IsNat.natCeil {R : Type*} [Semiring R] [LinearOrder R] [IsStrictOrderedRing R]\n    [FloorSemiring R] (r : R) (m : ℕ) : IsNat r m → IsNat (⌈r⌉₊) m := by\n  rintro ⟨⟨⟩⟩\n  exact ⟨by simp⟩\n\ntheorem IsInt.natCeil {R : Type*} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] [FloorSemiring R]\n    (r : R) (m : ℕ) : IsInt r (.negOfNat m) → IsNat (⌈r⌉₊) 0 := by\n  rintro ⟨⟨⟩⟩\n  exact ⟨by simp⟩\n\ntheorem IsNNRat.natCeil {R : Type*} [Semifield R] [LinearOrder R] [IsStrictOrderedRing R]\n    [FloorSemiring R] (r : R) (n d : ℕ) (h : IsNNRat r n d) (res : ℕ)\n    (hres : ⌈(n / d : ℚ≥0)⌉₊ = res) : IsNat ⌈r⌉₊ res := by\n  constructor\n  rw [← hres, h.to_eq rfl rfl, ← @NNRat.ceil_cast R]\n  simp\n\nTarget:\ntheorem IsRat.natCeil {R : Type*} [Field R] [LinearOrder R] [IsStrictOrderedRing R]\n    [FloorSemiring R] (r : R) (n d : ℕ) (h : IsRat r (.negOfNat n) d) : IsNat ⌈r⌉₊ 0 :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/NNRat","family_id":"israt","file_id":"mathlib/Mathlib/Data/NNRat/Floor.lean","sample_id":"a3a4139f0052b0a32136e52937242d5f55eef48c36e8634d6c5292950c12886e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3816114db1c077f621bfeb2d3b1b98b7b671ab19074e0d158d715632b3696809","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9584eeedec12f1270f773e97e07803ac6bace4ed2fabdce1f1ae60147a31ccc0","source_sha256":"351582fbc976816e329c33b0cb83e6e30d58c72551427bea563b172b7814b639","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro x y hxy\n  let xy : G.obj (pullback π π) := (PreservesPullback.iso G π π).inv <|\n    (TopCat.pullbackIsoProdSubtype (G.map π) (G.map π)).inv ⟨(x, y), hxy⟩\n  have ha' := congr_fun ha xy\n  dsimp at ha'\n  have h₁ : ∀ y, G.map (pullback.fst _ _) ((PreservesPullback.iso G π π).inv y) =\n      pullback.fst (G.map π) (G.map π) y := by\n    simp only [← PreservesPullback.iso_inv_fst]; intro y; rfl\n  have h₂ : ∀ y, G.map (pullback.snd _ _) ((PreservesPullback.iso G π π).inv y) =\n      pullback.snd (G.map π) (G.map π) y := by\n    simp only [← PreservesPullback.iso_inv_snd]; intro y; rfl\n  rw [h₁, h₂, TopCat.pullbackIsoProdSubtype_inv_fst_apply,\n    TopCat.pullbackIsoProdSubtype_inv_snd_apply] at ha'\n  simpa using ha'","hard_negative":false,"metrics":{"chosen_tokens":211,"rejected_tokens":5,"token_jaccard":0.034483,"token_length_ratio":0.023697},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"d6e121eb35031ac83c85f775b6057b146539bacf320891cbf3da106951c09f84","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Preserves.Opposites\npublic import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison\npublic import Mathlib.Condensed.Basic\npublic import Mathlib.Topology.Category.TopCat.Yoneda\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# The functor from topological spaces to condensed sets\n\nThis file builds on the API from the file `TopCat.Yoneda`. If the forgetful functor to `TopCat` has\nnice properties, like preserving pullbacks and finite coproducts, then this Yoneda presheaf\nsatisfies the sheaf condition for the regular and extensive topologies respectively.\n\nWe apply this API to `CompHaus` and define the functor\n`topCatToCondensedSet : TopCat.{u + 1} ⥤ CondensedSet.{u}`.\n\n-/\n\n@[expose] public section\n\nuniverse w w' v u\n\nopen CategoryTheory Opposite Limits regularTopology ContinuousMap Topology\n\nvariable {C : Type u} [Category.{v} C] (G : C ⥤ TopCat.{w})\n  (X : Type w') [TopologicalSpace X]\n\n/--\nAn auxiliary lemma to that allows us to use `IsQuotientMap.lift` in the proof of\n`equalizerCondition_yonedaPresheaf`.\n-/\n\nTarget:\ntheorem factorsThrough_of_pullbackCondition {Z B : C} {π : Z ⟶ B} [HasPullback π π]\n    [PreservesLimit (cospan π π) G]\n    {a : C(G.obj Z, X)}\n    (ha : a ∘ (G.map (pullback.fst _ _)) = a ∘ (G.map (pullback.snd π π))) :\n    Function.FactorsThrough a (G.map π) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed","family_id":"factorsthrough_of_pullbackcondition","file_id":"mathlib/Mathlib/Condensed/TopComparison.lean","sample_id":"9584eeedec12f1270f773e97e07803ac6bace4ed2fabdce1f1ae60147a31ccc0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7098d4ede92516c6303541ad23377423cb7f62e04005055a6e4b359a31869d3c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2b9acd22df8b3e0b46d29d9433ec7adfa97c3ca8da662c2868b5f6dac8fb61d2","source_sha256":"1c0dcf6a3ce55bec7a88672314ea6ae6741a6599903854de40abb691f935e16b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    intro ⟨h1, h2⟩; use y.drop x.length, h2\n    nth_rw 1 [← List.take_append_drop x.length y]\n    simpa [-List.take_append_drop, List.prefix_iff_eq_take, hl] using h1\n  mpr := by simp +contextual [pullSub]","hard_negative":true,"metrics":{"chosen_tokens":53,"rejected_tokens":8,"token_jaccard":0.054054,"token_length_ratio":0.150943},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"d6e38178d798945ba66f5aab8461267cf1e02c8eb17d233755c0b057250e3009","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.CompleteLattice.SetLike\n\nNamespace:\nDescriptive.Tree\n\nLocal context:\n/-\nCopyright (c) 2024 Sven Manthe. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sven Manthe\n-/\n/-!\n# Trees in the sense of descriptive set theory\n\nThis file defines trees of depth `ω` in the sense of descriptive set theory as sets of finite\nsequences that are stable under taking prefixes.\n\n## Main declarations\n\n* `tree A`: a (possibly infinite) tree of depth at most `ω` with nodes in `A`\n-/\n\n@[expose] public section\n\nnamespace Descriptive\n\n/-- A tree is a set of finite sequences, implemented as `List A`, that is stable under\n  taking prefixes. For the definition we use the equivalent property `x ++ [a] ∈ T → x ∈ T`,\n  which is more convenient to check. We define `tree A` as a complete sublattice of\n  `Set (List A)`, which coerces to the type of trees on `A`. -/\ndef tree (A : Type*) : CompleteSublattice (Set (List A)) :=\n  CompleteSublattice.mk' {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T}\n    (by rintro S hS x a ⟨t, ht, hx⟩; use t, ht, hS ht hx)\n    (by rintro S hS x a h T hT; exact hS hT <| h T hT)\n\n@[simps!] instance (A : Type*) : SetLike (tree A) (List A) := SetLike.instSubtypeSet\n\nexample (A : Type*) : PartialOrder (tree A) := inferInstance\n\nnamespace Tree\nvariable {A : Type*} {S T : tree A}\n\nlemma mem_of_append {x y : List A} (h : x ++ y ∈ T) : x ∈ T := by\n  induction y generalizing x with\n  | nil => simpa using h\n  | cons y ys ih => exact T.prop (ih (by simpa))\n\nlemma mem_of_prefix {x y : List A} (h' : x <+: y) (h : y ∈ T) : x ∈ T := by\n  obtain ⟨_, rfl⟩ := h'; exact mem_of_append h\n\ninstance : Trans List.IsPrefix (fun x (T : tree A) ↦ x ∈ T) (fun x T ↦ x ∈ T) where\n  trans := mem_of_prefix\n\nlemma singleton_mem (T : tree A) {a : A} {x : List A} (h : a :: x ∈ T) : [a] ∈ T :=\n  mem_of_prefix ⟨x, rfl⟩ h\n\n@[simp] lemma tree_eq_bot : T = ⊥ ↔ [] ∉ T where\n  mp := by rintro rfl; simp\n  mpr h := by ext x; simpa using fun h' ↦ h <| mem_of_prefix x.nil_prefix h'\n\nlemma take_mem {n : ℕ} (x : T) : x.val.take n ∈ T :=\n  mem_of_prefix (x.val.take_prefix n) x.prop\n\n/-- A variant of `List.take` internally to a tree -/\n@[simps] def take (n : ℕ) (x : T) : T := ⟨x.val.take n, take_mem x⟩\n\n@[simp] lemma take_take (m n : ℕ) (x : T) : take m (take n x) = take (m ⊓ n) x := by\n  simp [Subtype.ext_iff, List.take_take]\n\n@[simp] lemma take_eq_take {x : T} {m n : ℕ} :\n    take m x = take n x ↔ m ⊓ x.val.length = n ⊓ x.val.length := by simp [Subtype.ext_iff]\n\n-- ### `subAt`\n\nvariable (T) (x y : List A)\n\n/-- The residual tree obtained by regarding the node x as new root -/\ndef subAt : tree A :=\n  ⟨(x ++ ·)⁻¹' T, fun _ a _ ↦ mem_of_append (y := [a]) (by rwa [List.append_assoc])⟩\n\n@[simp] lemma mem_subAt : y ∈ subAt T x ↔ x ++ y ∈ T := Iff.rfl\n\n@[simp] lemma subAt_nil : subAt T [] = T := rfl\n\n@[simp] lemma subAt_append : subAt (subAt T x) y = subAt T (x ++ y) := by ext; simp\n\n@[gcongr] lemma subAt_mono (h : S ≤ T) : subAt S x ≤ subAt T x :=\n  Set.preimage_mono h\n\n/-- A variant of `List.drop` that takes values in `subAt` -/\n@[simps] def drop (n : ℕ) (x : T) : subAt T (Tree.take n x).val :=\n  ⟨x.val.drop n, by simp⟩\n\n-- ### `pullSub`\n\n/-- Adjoint of `subAt`, given by pasting x before the root of T. Explicitly,\n  elements are prefixes of x or x with an element of T appended -/\ndef pullSub : tree A where\n  val := { y | y.take x.length <+: x ∧ y.drop x.length ∈ T }\n  property := fun y a ⟨h1, h2⟩ ↦\n    ⟨((y.prefix_append [a]).take x.length).trans h1,\n    mem_of_prefix ((y.prefix_append [a]).drop x.length) h2⟩\n\nvariable {T x y}\n\nlemma mem_pullSub_short (hl : y.length ≤ x.length) : y ∈ pullSub T x ↔ y <+: x ∧ [] ∈ T := by\n  simp [pullSub, List.take_of_length_le hl, List.drop_eq_nil_iff.mpr hl]\n\nTarget:\nlemma mem_pullSub_long (hl : x.length ≤ y.length) : y ∈ pullSub T x ↔ ∃ z ∈ T, y = x ++ z where\n  mp :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"ea0de1a29252f2d6a594ccb340c5d91e082e190c949187667db7765671193290","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Descriptive","family_id":"mem_pullsub_long","file_id":"mathlib/Mathlib/SetTheory/Descriptive/Tree.lean","sample_id":"2b9acd22df8b3e0b46d29d9433ec7adfa97c3ca8da662c2868b5f6dac8fb61d2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"05c1e20ccd52d20617d06c19bb3d65322091abd072a38ef854172815fc9654f9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b6f984148eb97ed2790857133a651dc0f8e9044fcb08153a43a976f8c1a53100","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"15e83a19ac7a4ffb734f70f90cc5cd72e116c7c5b68a279718e9f6a477143d88","source_sha256":"129f2f3aa776146d58955f965fc598f57556bb308d25e687562d21d0c89b2ff2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨?_, fun h ↦ (isCyclic_of_card_le_two h).isMulCommutative⟩\n  classical\n  rintro ⟨⟨h⟩⟩\n  rw [← not_lt, ← Set.ncard_univ, Set.two_lt_ncard_iff]\n  rintro ⟨a, b, c, _, _, _, hab, hac, hbc⟩\n  apply hbc\n  simp_rw [Perm.ext_iff] at h\n  simpa [swap_apply_of_ne_of_ne hab hac] using h (swap a b) (swap b c) a","hard_negative":true,"metrics":{"chosen_tokens":86,"rejected_tokens":3,"token_jaccard":0.02381,"token_length_ratio":0.034884},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"d6f5a7bebf6480677ec74e26742a1f4ca268ad683f839a13ff91de346a6f2b85","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Perm\npublic import Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic\npublic import Mathlib.SetTheory.Cardinal.Finite\n\nNamespace:\nEquiv.Perm\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir\n-/\n/-! # Properties of `Equiv.Perm` on `Finite` types\n\nLet `α` be a `Finite` type.\n\n* `Nat.card_perm`: cardinality of `Equiv.Perm α`.\n\n* `Equiv.Perm.isCyclic_of_card_le_two`: if `Nat.card α ≤ 2`,\n  then `Equiv.Perm α` is cyclic.\n\n* `Equiv.Perm.isCyclic_iff_card_le_two`: `Equiv.Perm α` is cyclic iff `Nat.card α ≤ 2`.\n\n* `Equiv.Perm.isMulCommutative_iff_card_le_two`: `Equiv.Perm α` is commutative iff `Nat.card α ≤ 2`.\n\n-/\n\npublic section\n\nassert_not_exists Field\n\nopen Equiv Nat\n\nvariable {α : Type*} [Finite α]\n\nnamespace Nat\n\ntheorem card_perm : Nat.card (Perm α) = (Nat.card α)! := by\n  classical\n  have := Fintype.ofFinite α\n  rw [card_eq_fintype_card, card_eq_fintype_card, Fintype.card_perm]\n\nend Nat\n\nnamespace Equiv.Perm\n\ntheorem isCyclic_of_card_le_two (hα : Nat.card α ≤ 2) :\n    IsCyclic (Perm α) := by\n  apply isCyclic_of_card_dvd_prime (p := 2)\n  simpa [card_perm] using factorial_dvd_factorial hα\n\nTarget:\ntheorem isMulCommutative_iff_card_le_two :\n    IsMulCommutative (Perm α) ↔ Nat.card α ≤ 2 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_15e83a19ac7a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"792ce9c247b7905e4fbbc8ebeea7a2f08bb351b4fb5cf55e27a87c8b4a9ab80b","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finite","family_id":"ismulcommutative_iff_card_le_two","file_id":"mathlib/Mathlib/Data/Finite/Perm.lean","sample_id":"15e83a19ac7a4ffb734f70f90cc5cd72e116c7c5b68a279718e9f6a477143d88"},"split":"train"}
{"binding":{"chosen_proof_sha256":"084e758470934f76ebf17be847788f3818be1118ff79efcd27a52969227aebb9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2357e78341a07f48e92532f87e0bfabcf4fba0a7567069cbcd02ca8502dd6972","source_sha256":"4b34e7f92543da53c331f629ae697e47be9c96439c6558dd62e8cb4d9bb21d9b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h₁ : x⁻¹ ≠ 1 := by rwa [inv_eq_one_div, Ne, div_eq_iff_mul_eq hx0, one_mul]\n  have h₂ : x⁻¹ - 1 ≠ 0 := mt sub_eq_zero.1 h₁\n  have h₃ : x - 1 ≠ 0 := mt sub_eq_zero.1 hx1\n  have h₄ : x * (x ^ n)⁻¹ = (x ^ n)⁻¹ * x :=\n    Nat.recOn n (by simp) fun n h => by\n      rw [pow_succ', mul_inv_rev, ← mul_assoc, h, mul_assoc, mul_inv_cancel₀ hx0, mul_assoc,\n        inv_mul_cancel₀ hx0]\n  rw [geom_sum_eq h₁, div_eq_iff_mul_eq h₂, ← mul_right_inj' h₃, ← mul_assoc, ← mul_assoc,\n    mul_inv_cancel₀ h₃]\n  simp only [inv_pow, sub_eq_add_neg, mul_add, one_mul, mul_neg, add_mul, mul_inv_cancel₀ hx0,\n    neg_mul, mul_assoc, mul_one, add_comm, neg_add_rev, neg_neg, h₄, add_left_comm]\n  rw [add_comm _ (-x), add_assoc, add_assoc _ _ 1]","hard_negative":false,"metrics":{"chosen_tokens":194,"rejected_tokens":5,"token_jaccard":0.029851,"token_length_ratio":0.025773},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"d6fc59f1efd36b12d3fe2695991cc2ed6bac84539e65aabead8f59c272da08b2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Basic\npublic import Mathlib.Algebra.Ring.GeomSum\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2019 Neil Strickland. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Neil Strickland\n-/\n/-!\n# Partial sums of geometric series in a field\n\nThis file determines the values of the geometric series $\\sum_{i=0}^{n-1} x^i$ and\n$\\sum_{i=0}^{n-1} x^i y^{n-1-i}$ and variants thereof.\n\n## Main statements\n\n* `geom_sum_Ico` proves that $\\sum_{i=m}^{n-1} x^i=\\frac{x^n-x^m}{x-1}$ in a division ring.\n* `geom_sum₂_Ico` proves that $\\sum_{i=m}^{n-1} x^iy^{n - 1 - i}=\\frac{x^n-y^{n-m}x^m}{x-y}$\n  in a field.\n\nSeveral variants are recorded, generalising in particular to the case of a division ring in\nwhich `x` and `y` commute.\n-/\n\npublic section\n\nassert_not_exists IsOrderedRing\n\nvariable {R K : Type*}\n\nopen Finset MulOpposite\n\nsection DivisionRing\nvariable [DivisionRing K] {x y : K}\n\nprotected theorem Commute.geom_sum₂ (h' : Commute x y) (h : x ≠ y)\n    (n : ℕ) : ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ n) / (x - y) := by\n  have : x - y ≠ 0 := by simp_all [sub_eq_iff_eq_add]\n  rw [← h'.geom_sum₂_mul, mul_div_cancel_right₀ _ this]\n\ntheorem geom_sum_eq (h : x ≠ 1) (n : ℕ) : ∑ i ∈ range n, x ^ i = (x ^ n - 1) / (x - 1) := by\n  have : x - 1 ≠ 0 := by simp_all [sub_eq_iff_eq_add]\n  rw [← geom_sum_mul, mul_div_cancel_right₀ _ this]\n\nprotected theorem Commute.geom_sum₂_Ico (h : Commute x y) (hxy : x ≠ y) {m n : ℕ} (hmn : m ≤ n) :\n    ∑ i ∈ Finset.Ico m n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ (n - m) * x ^ m) / (x - y) := by\n  have : x - y ≠ 0 := by simp_all [sub_eq_iff_eq_add]\n  rw [← h.geom_sum₂_Ico_mul hmn, mul_div_cancel_right₀ _ this]\n\nlemma geom_sum_Ico (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) :\n    ∑ i ∈ Finset.Ico m n, x ^ i = (x ^ n - x ^ m) / (x - 1) := by\n  simp only [sum_Ico_eq_sub _ hmn, geom_sum_eq hx, div_sub_div_same, sub_sub_sub_cancel_right]\n\nlemma geom_sum_Ico' (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) :\n    ∑ i ∈ Finset.Ico m n, x ^ i = (x ^ m - x ^ n) / (1 - x) := by\n  simpa [geom_sum_Ico hx hmn] using neg_div_neg_eq (x ^ m - x ^ n) (1 - x)\n\nTarget:\nlemma geom_sum_inv (hx1 : x ≠ 1) (hx0 : x ≠ 0) (n : ℕ) :\n    ∑ i ∈ range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Field","family_id":"geom_sum_inv","file_id":"mathlib/Mathlib/Algebra/Field/GeomSum.lean","sample_id":"2357e78341a07f48e92532f87e0bfabcf4fba0a7567069cbcd02ca8502dd6972"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c1390084cb2580392cbfe88281f49b6df9fd83f38d46b0fc57029acaa73ff3f3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f378af96f4b655711415908ce319014735dc5bb779d91dee4115105bae787859","source_sha256":"36745b2e6bef135edaaf082a7696e9ea3c46b2f2bf1d19afbff0b004485d93b6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← span_span_of_tower R, finrank_span_eq_finrank]","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":8,"token_jaccard":0.066667,"token_length_ratio":0.888889},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"d72f0646f153fca669402859e8812380009159611cdc44d39118c88132681fc2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Epi\npublic import Mathlib.LinearAlgebra.Dimension.StrongRankCondition\npublic import Mathlib.LinearAlgebra.Finsupp.LinearCombination\npublic import Mathlib.LinearAlgebra.Span.Basic\npublic import Mathlib.RingTheory.Flat.Basic\npublic import Mathlib.Combinatorics.Matroid.Init\npublic import Mathlib.Data.Nat.Totient\npublic import Mathlib.Data.Sym.Sym2\npublic import Mathlib.LinearAlgebra.FreeModule.PID\npublic import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition\npublic import Mathlib.Tactic.NormNum.GCD\npublic import Mathlib.Tactic.Positivity\n\nNamespace:\nSubmodule\n\nLocal context:\n/-\nCopyright (c) 2026 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# The interaction of linear span and tensor product for mixed scalars.\n-/\n\n@[expose] public section\n\nopen Function TensorProduct\n\nnamespace Submodule\n\nvariable {R : Type*} (A : Type*) {M : Type*}\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [Algebra R A]\n  [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n  (p : Submodule R M)\n\n/-- If `A` is an `R`-algebra and `p` is an `R`-submodule of an `A`-module `M`, this is the natural\nsurjection `A ⊗[R] p → span A p`.\n\nSee also `Submodule.tensorEquivSpan`. -/\ndef tensorToSpan : A ⊗[R] p →ₗ[A] span A (p : Set M) :=\n  AlgebraTensorModule.lift\n    { toFun a := a • p.inclusionSpan A\n      map_add' a b := add_smul a b _\n      map_smul' a b := smul_assoc a b _ }\n\n@[simp] lemma tensorToSpan_apply_tmul (a : A) (x : p) :\n    p.tensorToSpan A (a ⊗ₜ x) = a • (x : M) :=\n  rfl\n\nlemma surjective_tensorToSpan : Surjective (p.tensorToSpan A) := by\n  intro v\n  obtain ⟨f, hf⟩ := (Finsupp.mem_span_iff_linearCombination _ _ _).mp v.property\n  use f.sum fun x a ↦ a ⊗ₜ x\n  rw [map_finsuppSum, Subtype.ext_iff, ← Submodule.subtype_apply, map_finsuppSum]\n  simpa using! hf\n\nvariable [Algebra.IsEpi R A] [Module.Flat R A]\n\nopen Module.Flat LinearMap in\nlemma injective_tensorToSpan : Injective (p.tensorToSpan A) := by\n  let f : A ⊗[R] (span A (p : Set M)) →ₗ[A] span A (p : Set M) :=\n    AlgebraTensorModule.lift <| (restrictScalarsₗ R A _ _ A) ∘ₗ lsmul A (span A (p : Set M))\n  let g : A ⊗[R] p →ₗ[R] A ⊗[R] span A (p : Set M) := (p.inclusionSpan A).lTensor A\n  have hf : Injective f := Algebra.injective_lift_lsmul R A _\n  have hg : Injective g := lTensor_preserves_injective_linearMap _ (p.injective_inclusionSpan A)\n  have : p.tensorToSpan A = f.restrictScalars R ∘ₗ g := by ext; simp [tensorToSpan, f, g]\n  rw [← LinearMap.coe_restrictScalars R, this, coe_comp]\n  exact hf.comp hg\n\n/-- If `A` is a flat epi `R`-algebra and `p` is an `R`-submodule of an `A`-module `M` then the\nnatural surjection from `A ⊗[R] p` to `span A p` is an equivalence. -/\nnoncomputable def tensorEquivSpan : A ⊗[R] p ≃ₗ[A] span A (p : Set M) :=\n  .ofBijective (p.tensorToSpan A) ⟨p.injective_tensorToSpan A, p.surjective_tensorToSpan A⟩\n\n@[simp] lemma tensorEquivSpan_apply_tmul (a : A) (x : p) :\n    p.tensorEquivSpan A (a ⊗ₜ x) = a • (x : M) :=\n  rfl\n\nvariable (R) in\n/-- If `A` is a flat epi `R`-algebra and `s` is a subset of an `A`-module `M` then the natural\nsurjection from `A ⊗[R] span R s` to `span A s` is an equivalence. -/\nnoncomputable def tensorSpanEquivSpan (s : Set M) : A ⊗[R] span R s ≃ₗ[A] span A s :=\n  ((span R s).tensorEquivSpan A).trans <| .ofEq _ _ <| span_span_of_tower R A s\n\n@[simp] lemma coe_tensorSpanEquivSpan_apply_tmul {s : Set M} (a : A) (x : span R s) :\n    tensorSpanEquivSpan R A s (a ⊗ₜ x) = a • (x : M) :=\n  rfl\n\nend CommSemiring\n\nsection CommRing\n\nopen Module\n\nvariable [CommRing R] [CommRing A] [Nontrivial A]\n  [Algebra R A] [Algebra.IsEpi R A] [Module.Flat R A]\n  [AddCommGroup M] [Module R M] [Module A M] [IsScalarTower R A M]\n  (p : Submodule R M) [Free R p] [Module.Finite R p]\n\n@[simp] lemma finrank_span_eq_finrank :\n    finrank A (span A (p : Set M)) = finrank R p := by\n  rcases subsingleton_or_nontrivial R; · simp [Algebra.subsingleton R A]\n  let ι := Free.ChooseBasisIndex R p\n  let b₁ : Basis ι R p := Free.chooseBasis R p\n  let b₂ : Basis ι A (span A (p : Set M)) := (b₁.baseChange A).map <| p.tensorEquivSpan A\n  rw [finrank_eq_card_basis b₁, finrank_eq_card_basis b₂]\n\nvariable (R) in\n\nTarget:\nlemma finrank_span_eq_finrank_span [IsPrincipalIdealRing R] [IsDomain R] [IsTorsionFree R M]\n    (s : Set M) [Module.Finite R (span R s)] :\n    finrank A (span A s) = finrank R (span R s) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"d7b290445319af70fe27fcdc7fa8d5a53873f01da9d97260db5f6cec71fa2cd2","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Span","family_id":"finrank_span_eq_finrank_span","file_id":"mathlib/Mathlib/LinearAlgebra/Span/TensorProduct.lean","sample_id":"f378af96f4b655711415908ce319014735dc5bb779d91dee4115105bae787859"},"split":"train"}
{"binding":{"chosen_proof_sha256":"536352e5ddbb900c41d2280e8e59f7791e63d85f699a6f8a27e8c2d97392c7b3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b9b65ec13f7e30e41eeaa6338022c9efbc66d8f0394c7b483c53252b38ab9a8f","source_sha256":"07a36e2d42950767413a6ab43b32d6264e65f7c71e39cd053b21a81cb6699e71","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let S : Set (Ultrafilter M) := ⋂ n, { U | ∀ᶠ m in U, m ∈ FP (a.drop n) }\n  have h := exists_idempotent_in_compact_subsemigroup ?_ S ?_ ?_ ?_\n  · rcases h with ⟨U, hU, U_idem⟩\n    refine ⟨U, U_idem, ?_⟩\n    convert! Set.mem_iInter.mp hU 0\n  · exact Ultrafilter.continuous_mul_left\n  · apply IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed\n    · intro n U hU\n      filter_upwards [hU]\n      rw [← Stream'.drop_drop, ← Stream'.tail_eq_drop]\n      exact FP.tail _\n    · intro n\n      exact ⟨pure _, mem_pure.mpr <| FP.head _⟩\n    · exact (ultrafilter_isClosed_basic _).isCompact\n    · intro n\n      apply ultrafilter_isClosed_basic\n  · exact IsClosed.isCompact (isClosed_iInter fun i => ultrafilter_isClosed_basic _)\n  · intro U hU V hV\n    rw [Set.mem_iInter] at *\n    intro n\n    rw [Set.mem_setOf_eq, Ultrafilter.eventually_mul]\n    filter_upwards [hU n] with m hm\n    obtain ⟨n', hn⟩ := FP.mul hm\n    filter_upwards [hV (n' + n)] with m' hm'\n    apply hn\n    simpa only [Stream'.drop_drop, add_comm] using hm'","hard_negative":false,"metrics":{"chosen_tokens":224,"rejected_tokens":5,"token_jaccard":0.032967,"token_length_ratio":0.022321},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"d76b9b34356b855c1a406a2693056e9986d986aa828b02dd3b6549022fea3c34","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Stream.Init\npublic import Mathlib.Topology.Algebra.Semigroup\npublic import Mathlib.Topology.Compactification.StoneCech\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nNamespace:\nHindman\n\nLocal context:\n/-\nCopyright (c) 2021 David Wärn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David Wärn\n-/\n/-!\n# Hindman's theorem on finite sums\n\nWe prove Hindman's theorem on finite sums, using idempotent ultrafilters.\n\nGiven an infinite sequence `a₀, a₁, a₂, …` of positive integers, the set `FS(a₀, …)` is the set\nof positive integers that can be expressed as a finite sum of `aᵢ`'s, without repetition. Hindman's\ntheorem asserts that whenever the positive integers are finitely colored, there exists a sequence\n`a₀, a₁, a₂, …` such that `FS(a₀, …)` is monochromatic. There is also a stronger version, saying\nthat whenever a set of the form `FS(a₀, …)` is finitely colored, there exists a sequence\n`b₀, b₁, b₂, …` such that `FS(b₀, …)` is monochromatic and contained in `FS(a₀, …)`. We prove both\nthese versions for a general semigroup `M` instead of `ℕ+` since it is no harder, although this\nspecial case implies the general case.\n\nThe idea of the proof is to extend the addition `(+) : M → M → M` to addition `(+) : βM → βM → βM`\non the space `βM` of ultrafilters on `M`. One can prove that if `U` is an _idempotent_ ultrafilter,\ni.e. `U + U = U`, then any `U`-large subset of `M` contains some set `FS(a₀, …)` (see\n`exists_FS_of_large`). And with the help of a general topological argument one can show that any set\nof the form `FS(a₀, …)` is `U`-large according to some idempotent ultrafilter `U` (see\n`exists_idempotent_ultrafilter_le_FS`). This is enough to prove the theorem since in any finite\npartition of a `U`-large set, one of the parts is `U`-large.\n\n## Main results\n\n- `FS_partition_regular`: the strong form of Hindman's theorem\n- `exists_FS_of_finite_cover`: the weak form of Hindman's theorem\n\n## Tags\n\nRamsey theory, ultrafilter\n\n-/\n\n@[expose] public section\n\n\nopen Filter\n\n/-- Multiplication of ultrafilters given by `∀ᶠ m in U*V, p m ↔ ∀ᶠ m in U, ∀ᶠ m' in V, p (m*m')`. -/\n@[to_additive (attr := implicit_reducible)\n/-- Addition of ultrafilters given by `∀ᶠ m in U+V, p m ↔ ∀ᶠ m in U, ∀ᶠ m' in V, p (m+m')`. -/]\ndef Ultrafilter.mul {M} [Mul M] : Mul (Ultrafilter M) where mul U V := (· * ·) <$> U <*> V\n\nattribute [local instance] Ultrafilter.mul Ultrafilter.add\n\n/-- We could have taken this as the definition of `U * V`, but then we would have to prove that it\ndefines an ultrafilter. -/\n@[to_additive]\ntheorem Ultrafilter.eventually_mul {M} [Mul M] (U V : Ultrafilter M) (p : M → Prop) :\n    (∀ᶠ m in ↑(U * V), p m) ↔ ∀ᶠ m in U, ∀ᶠ m' in V, p (m * m') :=\n  Iff.rfl\n\n/-- Semigroup structure on `Ultrafilter M` induced by a semigroup structure on `M`. -/\n@[to_additive (attr := implicit_reducible)\n/-- Additive semigroup structure on `Ultrafilter M` induced by an additive semigroup\nstructure on `M`. -/]\ndef Ultrafilter.semigroup {M} [Semigroup M] : Semigroup (Ultrafilter M) :=\n  { Ultrafilter.mul with\n    mul_assoc := fun U V W =>\n      Ultrafilter.coe_inj.mp <|\n        Filter.ext' fun p => by simp [Ultrafilter.eventually_mul, mul_assoc] }\n\nattribute [local instance] Ultrafilter.semigroup Ultrafilter.addSemigroup\n\n-- We don't prove `continuous_mul_right`, because in general it is false!\n@[to_additive]\ntheorem Ultrafilter.continuous_mul_left {M} [Mul M] (V : Ultrafilter M) :\n    Continuous (· * V) :=\n  ultrafilterBasis_is_basis.continuous_iff.2 <| Set.forall_mem_range.mpr fun s ↦\n    ultrafilter_isOpen_basic { m : M | ∀ᶠ m' in V, m * m' ∈ s }\n\nnamespace Hindman\n\n/-- `FS a` is the set of finite sums in `a`, i.e. `m ∈ FS a` if `m` is the sum of a nonempty\nsubsequence of `a`. We give a direct inductive definition instead of talking about subsequences. -/\ninductive FS {M} [AddSemigroup M] : Stream' M → Set M\n  | head' (a : Stream' M) : FS a a.head\n  | tail' (a : Stream' M) (m : M) (h : FS a.tail m) : FS a m\n  | cons' (a : Stream' M) (m : M) (h : FS a.tail m) : FS a (a.head + m)\n\n/-- `FP a` is the set of finite products in `a`, i.e. `m ∈ FP a` if `m` is the product of a nonempty\nsubsequence of `a`. We give a direct inductive definition instead of talking about subsequences. -/\n@[to_additive FS]\ninductive FP {M} [Semigroup M] : Stream' M → Set M\n  | head' (a : Stream' M) : FP a a.head\n  | tail' (a : Stream' M) (m : M) (h : FP a.tail m) : FP a m\n  | cons' (a : Stream' M) (m : M) (h : FP a.tail m) : FP a (a.head * m)\n\nsection Aliases\n\n/-! Since the constructors for `FS` and `FP` cheat using the `Set M = M → Prop` defeq,\nwe provide match patterns that preserve the defeq correctly in their type. -/\n\nvariable {M} [Semigroup M] (a : Stream' M) (m : M) (h : FP a.tail m)\n\nset_option linter.defProp false in\n/-- Constructor for `FP`. This is the preferred spelling over `FP.head'`. -/\n@[to_additive (attr := match_pattern, nolint defLemma)\n  /-- Constructor for `FS`. This is the preferred spelling over `FS.head'`. -/]\nabbrev FP.head : a.head ∈ FP a := FP.head' a\nset_option linter.defProp false in\n/-- Constructor for `FP`. This is the preferred spelling over `FP.tail'`. -/\n@[to_additive (attr := match_pattern, nolint defLemma)\n  /-- Constructor for `FS`. This is the preferred spelling over `FS.tail'`. -/]\nabbrev FP.tail : m ∈ FP a := FP.tail' a m h\nset_option linter.defProp false in\n/-- Constructor for `FP`. This is the preferred spelling over `FP.cons'`. -/\n@[to_additive (attr := match_pattern, nolint defLemma)\n  /-- Constructor for `FS`. This is the preferred spelling over `FS.cons'`. -/]\nabbrev FP.cons : a.head * m ∈ FP a := FP.cons' a m h\n\nend Aliases\n\n/-- If `m` and `m'` are finite products in `M`, then so is `m * m'`, provided that `m'` is obtained\nfrom a subsequence of `M` starting sufficiently late. -/\n@[to_additive /-- If `m` and `m'` are finite sums in `M`, then so is `m + m'`, provided that `m'`\nis obtained from a subsequence of `M` starting sufficiently late. -/]\ntheorem FP.mul {M} [Semigroup M] {a : Stream' M} {m : M} (hm : m ∈ FP a) :\n    ∃ n, ∀ m' ∈ FP (a.drop n), m * m' ∈ FP a := by\n  induction hm with\n  | head' a => exact ⟨1, fun m hm => FP.cons a m hm⟩\n  | tail' a m _ ih =>\n    obtain ⟨n, hn⟩ := ih\n    use n + 1\n    intro m' hm'\n    exact FP.tail _ _ (hn _ hm')\n  | cons' a m _ ih =>\n    obtain ⟨n, hn⟩ := ih\n    use n + 1\n    intro m' hm'\n    rw [mul_assoc]\n    exact FP.cons _ _ (hn _ hm')\n\n@[to_additive exists_idempotent_ultrafilter_le_FS]\n\nTarget:\ntheorem exists_idempotent_ultrafilter_le_FP {M} [Semigroup M] (a : Stream' M) :\n    ∃ U : Ultrafilter M, U * U = U ∧ ∀ᶠ m in U, m ∈ FP a :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics","family_id":"exists_idempotent_ultrafilter_le_fp","file_id":"mathlib/Mathlib/Combinatorics/Hindman.lean","sample_id":"b9b65ec13f7e30e41eeaa6338022c9efbc66d8f0394c7b483c53252b38ab9a8f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"158b158a558370a66bb2762fb6673634c41f0d235bb36659662b90d5aa0cb875","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8612ceda65694614a4d71f561ffbd968d004c5a966d3aa2d3fae0e9f179ac15b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"335fd55413b4f96955767217c73d17f78f377c305db1438e043702f98d245c46","source_sha256":"059da84dda9e7cf638670a1c074b01de35e7641bc88d590bf894e2e5fc5223e6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  contrapose! +distrib\n  rw [Set.infinite_image2 hfs hft]\n  grind only [Set.Infinite.nonempty]","hard_negative":true,"metrics":{"chosen_tokens":22,"rejected_tokens":5,"token_jaccard":0.1,"token_length_ratio":0.227273},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"d76bc76c1ad27c830388eaa7e43f163dd05fff942a8ab139ea6ceacb0356630d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Basic\npublic import Mathlib.Data.Fintype.Prod\npublic import Mathlib.Data.Fintype.Pi\npublic import Mathlib.Algebra.Order.Group.Multiset\npublic import Mathlib.Data.ULift\npublic import Mathlib.Data.Set.NAry\n\nNamespace:\nSet\n\nLocal context:\n/-\nCopyright (c) 2022 Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kyle Miller\n-/\n/-!\n# Finiteness of products\n-/\n\npublic section\n\nassert_not_exists IsOrderedRing MonoidWithZero\n\nvariable {α β : Type*}\n\nnamespace Finite\n\ninstance [Finite α] [Finite β] : Finite (α × β) := by\n  haveI := Fintype.ofFinite α\n  haveI := Fintype.ofFinite β\n  infer_instance\n\ninstance {α β : Sort*} [Finite α] [Finite β] : Finite (PProd α β) :=\n  of_equiv _ Equiv.pprodEquivProdPLift.symm\n\ntheorem prod_left (β) [Finite (α × β)] [Nonempty β] : Finite α :=\n  of_surjective (Prod.fst : α × β → α) Prod.fst_surjective\n\ntheorem prod_right (α) [Finite (α × β)] [Nonempty α] : Finite β :=\n  of_surjective (Prod.snd : α × β → β) Prod.snd_surjective\n\nend Finite\n\nlemma Prod.finite_iff [Nonempty α] [Nonempty β] : Finite (α × β) ↔ Finite α ∧ Finite β where\n  mp _ := ⟨.prod_left β, .prod_right α⟩\n  mpr | ⟨_, _⟩ => inferInstance\n\ninstance Pi.finite {α : Sort*} {β : α → Sort*} [Finite α] [∀ a, Finite (β a)] :\n    Finite (∀ a, β a) := by\n  classical\n  haveI := Fintype.ofFinite (PLift α)\n  haveI := fun a => Fintype.ofFinite (PLift (β a))\n  exact\n    Finite.of_equiv (∀ a : PLift α, PLift (β (Equiv.plift a)))\n      (Equiv.piCongr Equiv.plift fun _ => Equiv.plift)\n\ninstance Function.Embedding.finite {α β : Sort*} [Finite β] : Finite (α ↪ β) := by\n  rcases isEmpty_or_nonempty (α ↪ β) with _ | h\n  · infer_instance\n  · refine h.elim fun f => ?_\n    haveI : Finite α := Finite.of_injective _ f.injective\n    exact Finite.of_injective _ DFunLike.coe_injective\n\ninstance Equiv.finite_right {α β : Sort*} [Finite β] : Finite (α ≃ β) :=\n  Finite.of_injective Equiv.toEmbedding fun e₁ e₂ h => Equiv.ext <| by\n    convert! DFunLike.congr_fun h using 0\n\ninstance Equiv.finite_left {α β : Sort*} [Finite α] : Finite (α ≃ β) :=\n  Finite.of_equiv _ ⟨Equiv.symm, Equiv.symm, Equiv.symm_symm, Equiv.symm_symm⟩\n\n@[to_additive]\ninstance MulEquiv.finite_left {α β : Type*} [Mul α] [Mul β] [Finite α] : Finite (α ≃* β) :=\n  Finite.of_injective toEquiv toEquiv_injective\n\n@[to_additive]\ninstance MulEquiv.finite_right {α β : Type*} [Mul α] [Mul β] [Finite β] : Finite (α ≃* β) :=\n  Finite.of_injective toEquiv toEquiv_injective\n\nopen Set Function\n\nvariable {γ : Type*}\n\nnamespace Set\n\n/-! ### Fintype instances\n\nEvery instance here should have a corresponding `Set.Finite` constructor in the next section.\n-/\n\nsection FintypeInstances\n\ninstance fintypeProd (s : Set α) (t : Set β) [Fintype s] [Fintype t] :\n    Fintype (s ×ˢ t : Set (α × β)) :=\n  Fintype.ofFinset (s.toFinset ×ˢ t.toFinset) <| by simp\n\ninstance fintypeOffDiag (s : Set α) [Fintype s] : Fintype s.offDiag :=\n  Fintype.ofFinset s.toFinset.offDiag <| by simp\n\n/-- `image2 f s t` is `Fintype` if `s` and `t` are. -/\ninstance fintypeImage2 [DecidableEq γ] (f : α → β → γ) (s : Set α) (t : Set β) [hs : Fintype s]\n    [ht : Fintype t] : Fintype (image2 f s t : Set γ) := by\n  rw [← image_prod]\n  apply Set.fintypeImage\n\nend FintypeInstances\n\nend Set\n\n/-! ### Finite instances\n\nThere is seemingly some overlap between the following instances and the `Fintype` instances\nin `Data.Set.Finite`. While every `Fintype` instance gives a `Finite` instance, those\ninstances that depend on `Fintype` or `Decidable` instances need an additional `Finite` instance\nto be able to generally apply.\n\nSome set instances do not appear here since they are consequences of others, for example\n`Subtype.Finite` for subsets of a finite type.\n-/\n\n\nnamespace Finite.Set\n\ninstance finite_prod (s : Set α) (t : Set β) [Finite s] [Finite t] :\n    Finite (s ×ˢ t : Set (α × β)) :=\n  Finite.of_equiv _ (Equiv.Set.prod s t).symm\n\ninstance finite_image2 (f : α → β → γ) (s : Set α) (t : Set β) [Finite s] [Finite t] :\n    Finite (image2 f s t : Set γ) := by\n  rw [← image_prod]\n  infer_instance\n\nend Finite.Set\n\nnamespace Set\n\n/-! ### Constructors for `Set.Finite`\n\nEvery constructor here should have a corresponding `Fintype` instance in the previous section\n(or in the `Fintype` module).\n\nThe implementation of these constructors ideally should be no more than `Set.toFinite`,\nafter possibly setting up some `Fintype` and classical `Decidable` instances.\n-/\n\n\nsection SetFiniteConstructors\n\nsection Prod\n\nvariable {s : Set α} {t : Set β}\n\nprotected theorem Finite.prod (hs : s.Finite) (ht : t.Finite) : (s ×ˢ t : Set (α × β)).Finite := by\n  have := hs.to_subtype\n  have := ht.to_subtype\n  apply toFinite\n\ntheorem Finite.of_prod_left (h : (s ×ˢ t : Set (α × β)).Finite) : t.Nonempty → s.Finite :=\n  fun ⟨b, hb⟩ => (h.image Prod.fst).subset fun a ha => ⟨(a, b), ⟨ha, hb⟩, rfl⟩\n\ntheorem Finite.of_prod_right (h : (s ×ˢ t : Set (α × β)).Finite) : s.Nonempty → t.Finite :=\n  fun ⟨a, ha⟩ => (h.image Prod.snd).subset fun b hb => ⟨(a, b), ⟨ha, hb⟩, rfl⟩\n\nprotected theorem Infinite.prod_left (hs : s.Infinite) (ht : t.Nonempty) : (s ×ˢ t).Infinite :=\n  fun h => hs <| h.of_prod_left ht\n\nprotected theorem Infinite.prod_right (ht : t.Infinite) (hs : s.Nonempty) : (s ×ˢ t).Infinite :=\n  fun h => ht <| h.of_prod_right hs\n\nprotected theorem infinite_prod :\n    (s ×ˢ t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty := by\n  refine ⟨fun h => ?_, ?_⟩\n  · simp_rw [Set.Infinite, @and_comm ¬_, ← Classical.not_imp]\n    by_contra!\n    exact h ((this.1 h.nonempty.snd).prod <| this.2 h.nonempty.fst)\n  · rintro (h | h)\n    · exact h.1.prod_left h.2\n    · exact h.1.prod_right h.2\n\ntheorem finite_prod : (s ×ˢ t).Finite ↔ (s.Finite ∨ t = ∅) ∧ (t.Finite ∨ s = ∅) := by\n  contrapose! +distrib; exact Set.infinite_prod\n\nprotected theorem Finite.offDiag {s : Set α} (hs : s.Finite) : s.offDiag.Finite :=\n  (hs.prod hs).subset s.offDiag_subset_prod\n\nprotected theorem Finite.image2 (f : α → β → γ) (hs : s.Finite) (ht : t.Finite) :\n    (image2 f s t).Finite := by\n  have := hs.to_subtype\n  have := ht.to_subtype\n  apply toFinite\n\nend Prod\n\nend SetFiniteConstructors\n\n/-! ### Properties -/\n\ntheorem Finite.toFinset_prod {s : Set α} {t : Set β} (hs : s.Finite) (ht : t.Finite) :\n    hs.toFinset ×ˢ ht.toFinset = (hs.prod ht).toFinset :=\n  Finset.ext <| by simp\n\ntheorem Finite.toFinset_offDiag {s : Set α} (hs : s.Finite) :\n    hs.offDiag.toFinset = hs.toFinset.offDiag :=\n  Finset.ext <| by simp\n\ntheorem finite_image_fst_and_snd_iff {s : Set (α × β)} :\n    (Prod.fst '' s).Finite ∧ (Prod.snd '' s).Finite ↔ s.Finite :=\n  ⟨fun h => (h.1.prod h.2).subset fun _ h => ⟨mem_image_of_mem _ h, mem_image_of_mem _ h⟩,\n    fun h => ⟨h.image _, h.image _⟩⟩\n\n/-! ### Infinite sets -/\n\nvariable {s t : Set α}\n\nsection Image2\n\nvariable {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β}\n\nprotected theorem Infinite.image2_left (hs : s.Infinite) (hb : b ∈ t)\n    (hf : InjOn (fun a => f a b) s) : (image2 f s t).Infinite :=\n  (hs.image hf).mono <| image_subset_image2_left hb\n\nprotected theorem Infinite.image2_right (ht : t.Infinite) (ha : a ∈ s) (hf : InjOn (f a) t) :\n    (image2 f s t).Infinite :=\n  (ht.image hf).mono <| image_subset_image2_right ha\n\ntheorem infinite_image2 (hfs : ∀ b ∈ t, InjOn (fun a => f a b) s) (hft : ∀ a ∈ s, InjOn (f a) t) :\n    (image2 f s t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty := by\n  refine ⟨fun h => Set.infinite_prod.1 ?_, ?_⟩\n  · rw [← image_uncurry_prod] at h\n    exact h.of_image _\n  · rintro (⟨hs, b, hb⟩ | ⟨ht, a, ha⟩)\n    · exact hs.image2_left hb (hfs _ hb)\n    · exact ht.image2_right ha (hft _ ha)\n\nTarget:\nlemma finite_image2 (hfs : ∀ b ∈ t, InjOn (f · b) s) (hft : ∀ a ∈ s, InjOn (f a) t) :\n    (image2 f s t).Finite ↔ s.Finite ∧ t.Finite ∨ s = ∅ ∨ t = ∅ :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_335fd55413b4","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"cfdd3ec414294a277a56cc8a1108925e669261b4ef1f757484c85f9793598d15","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finite","family_id":"finite_image","file_id":"mathlib/Mathlib/Data/Finite/Prod.lean","sample_id":"335fd55413b4f96955767217c73d17f78f377c305db1438e043702f98d245c46"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ac66c6e721cc952e221200b69cfe94cb7c90e16c08e83be7b74b05ee3c9aa39f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"00dd289d0a927f45e5d74bbf587cb38856e9d5b6466ba5bfff5b0d1d26ea5e31","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b7150f0dc88e25f3382430b108a0281da820a234916871dd325c0904c8b12aaf","source_sha256":"dbf8a3d3c567c2c63c51c166e27c3e89875f40118b51843cf1e9965b53ac82aa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by aesop\n\nvariable [Fintype ι]","hard_negative":true,"metrics":{"chosen_tokens":7,"rejected_tokens":3,"token_jaccard":0.111111,"token_length_ratio":0.428571},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"d79c2ed443f68b76e27ed1e6bc7fe16a2d20dee638685f1b7d7d9a92e1ba3ead","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Sigma\npublic import Mathlib.Data.Fintype.OfMap\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# fintype instances for sigma types\n-/\n\npublic section\n\n\nopen Function\n\nopen Nat\n\nuniverse u v\n\nvariable {ι α : Type*} {κ : ι → Type*} [Π i, Fintype (κ i)]\n\nopen Finset\n\nlemma Set.biUnion_finsetSigma_univ (s : Finset ι) (f : Sigma κ → Set α) :\n    ⋃ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij = ⋃ i ∈ s, ⋃ j, f ⟨i, j⟩ := by aesop\n\nlemma Set.biUnion_finsetSigma_univ' (s : Finset ι) (f : Π i, κ i → Set α) :\n    ⋃ i ∈ s, ⋃ j, f i j = ⋃ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij.1 ij.2 := by aesop\n\nlemma Set.biInter_finsetSigma_univ (s : Finset ι) (f : Sigma κ → Set α) :\n    ⋂ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij = ⋂ i ∈ s, ⋂ j, f ⟨i, j⟩ := by aesop\n\nattribute [local simp] Sigma.forall in\n\nTarget:\nlemma Set.biInter_finsetSigma_univ' (s : Finset ι) (f : Π i, κ i → Set α) :\n    ⋂ i ∈ s, ⋂ j, f i j = ⋂ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij.1 ij.2 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_b7150f0dc88e","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"ba2429c9d797e8900081ef5cd5fec30396a39367bf1bada1f4c8e3620fe8156c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Fintype","family_id":"set","file_id":"mathlib/Mathlib/Data/Fintype/Sigma.lean","sample_id":"b7150f0dc88e25f3382430b108a0281da820a234916871dd325c0904c8b12aaf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"021873745f79b607afe2480d1a3ec9b92a92bc4d1a53d9616a37575a2e4fbf1f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"fd18fa607c6f84a743396868ee103846e47d38db70873f4de5b55770b1ff7293","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"88184057873a0a559cc6d5510520e38470290a3d57980287af7d4cf570e8c138","source_sha256":"554524b5a9af2e4b62f0d83e3222b642d629dcdc418afd3afc6a8cda3e670128","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rw [← top_sdiff', ← top_sdiff', map_sdiff, map_top]","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":3,"token_jaccard":0.090909,"token_length_ratio":0.230769},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"d7ac4155f5abb850288d89d59118ae06d499891ce24c34e66843a8d1fe85af1c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Hom.BoundedLattice\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Heyting algebra morphisms\n\nA Heyting homomorphism between two Heyting algebras is a bounded lattice homomorphism that preserves\nHeyting implication.\n\nWe use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to\nbe satisfied by itself and all stricter types.\n\n## Types of morphisms\n\n* `HeytingHom`: Heyting homomorphisms.\n* `CoheytingHom`: Co-Heyting homomorphisms.\n* `BiheytingHom`: Bi-Heyting homomorphisms.\n\n## Typeclasses\n\n* `HeytingHomClass`\n* `CoheytingHomClass`\n* `BiheytingHomClass`\n-/\n\n@[expose] public section\n\n\nopen Function\n\nvariable {F α β γ δ : Type*}\n\n/-- The type of Heyting homomorphisms from `α` to `β`. Bounded lattice homomorphisms that preserve\nHeyting implication. -/\nstructure HeytingHom (α β : Type*) [HeytingAlgebra α] [HeytingAlgebra β] extends\n  LatticeHom α β where\n  /-- The proposition that a Heyting homomorphism preserves the bottom element. -/\n  protected map_bot' : toFun ⊥ = ⊥\n  /-- The proposition that a Heyting homomorphism preserves the Heyting implication. -/\n  protected map_himp' : ∀ a b, toFun (a ⇨ b) = toFun a ⇨ toFun b\n\n/-- The type of co-Heyting homomorphisms from `α` to `β`. Bounded lattice homomorphisms that\npreserve difference. -/\nstructure CoheytingHom (α β : Type*) [CoheytingAlgebra α] [CoheytingAlgebra β] extends\n  LatticeHom α β where\n  /-- The proposition that a co-Heyting homomorphism preserves the top element. -/\n  protected map_top' : toFun ⊤ = ⊤\n  /-- The proposition that a co-Heyting homomorphism preserves the difference operation. -/\n  protected map_sdiff' : ∀ a b, toFun (a \\ b) = toFun a \\ toFun b\n\n/-- The type of bi-Heyting homomorphisms from `α` to `β`. Bounded lattice homomorphisms that\npreserve Heyting implication and difference. -/\nstructure BiheytingHom (α β : Type*) [BiheytingAlgebra α] [BiheytingAlgebra β] extends\n  LatticeHom α β where\n  /-- The proposition that a bi-Heyting homomorphism preserves the Heyting implication. -/\n  protected map_himp' : ∀ a b, toFun (a ⇨ b) = toFun a ⇨ toFun b\n  /-- The proposition that a bi-Heyting homomorphism preserves the difference operation. -/\n  protected map_sdiff' : ∀ a b, toFun (a \\ b) = toFun a \\ toFun b\n\n/-- `HeytingHomClass F α β` states that `F` is a type of Heyting homomorphisms.\n\nYou should extend this class when you extend `HeytingHom`. -/\nclass HeytingHomClass (F α β : Type*) [HeytingAlgebra α] [HeytingAlgebra β] [FunLike F α β] : Prop\n    extends LatticeHomClass F α β where\n  /-- The proposition that a Heyting homomorphism preserves the bottom element. -/\n  map_bot (f : F) : f ⊥ = ⊥\n  /-- The proposition that a Heyting homomorphism preserves the Heyting implication. -/\n  map_himp (f : F) : ∀ a b, f (a ⇨ b) = f a ⇨ f b\n\n/-- `CoheytingHomClass F α β` states that `F` is a type of co-Heyting homomorphisms.\n\nYou should extend this class when you extend `CoheytingHom`. -/\nclass CoheytingHomClass (F α β : Type*) [CoheytingAlgebra α] [CoheytingAlgebra β] [FunLike F α β] :\n    Prop\n  extends LatticeHomClass F α β where\n  /-- The proposition that a co-Heyting homomorphism preserves the top element. -/\n  map_top (f : F) : f ⊤ = ⊤\n  /-- The proposition that a co-Heyting homomorphism preserves the difference operation. -/\n  map_sdiff (f : F) : ∀ a b, f (a \\ b) = f a \\ f b\n\n/-- `BiheytingHomClass F α β` states that `F` is a type of bi-Heyting homomorphisms.\n\nYou should extend this class when you extend `BiheytingHom`. -/\nclass BiheytingHomClass (F α β : Type*) [BiheytingAlgebra α] [BiheytingAlgebra β] [FunLike F α β] :\n    Prop\n  extends LatticeHomClass F α β where\n  /-- The proposition that a bi-Heyting homomorphism preserves the Heyting implication. -/\n  map_himp (f : F) : ∀ a b, f (a ⇨ b) = f a ⇨ f b\n  /-- The proposition that a bi-Heyting homomorphism preserves the difference operation. -/\n  map_sdiff (f : F) : ∀ a b, f (a \\ b) = f a \\ f b\n\nexport HeytingHomClass (map_himp)\n\nexport CoheytingHomClass (map_sdiff)\n\nattribute [simp] map_himp map_sdiff\n\nsection Hom\n\nvariable [FunLike F α β]\n\n/-! This section passes in some instances implicitly. See note [implicit instance arguments] -/\n\n-- See note [lower instance priority]\ninstance (priority := 100) HeytingHomClass.toBoundedLatticeHomClass [HeytingAlgebra α]\n    {_ : HeytingAlgebra β} [HeytingHomClass F α β] : BoundedLatticeHomClass F α β :=\n  { ‹HeytingHomClass F α β› with\n    map_top := fun f => by rw [← @himp_self α _ ⊥, ← himp_self, map_himp] }\n\n-- See note [lower instance priority]\ninstance (priority := 100) CoheytingHomClass.toBoundedLatticeHomClass [CoheytingAlgebra α]\n    {_ : CoheytingAlgebra β} [CoheytingHomClass F α β] : BoundedLatticeHomClass F α β :=\n  { ‹CoheytingHomClass F α β› with\n    map_bot := fun f => by rw [← @sdiff_self α _ ⊤, ← sdiff_self, map_sdiff] }\n\n-- See note [lower instance priority]\ninstance (priority := 100) BiheytingHomClass.toHeytingHomClass [BiheytingAlgebra α]\n    {_ : BiheytingAlgebra β} [BiheytingHomClass F α β] : HeytingHomClass F α β :=\n  { ‹BiheytingHomClass F α β› with\n    map_bot := fun f => by rw [← @sdiff_self α _ ⊤, ← sdiff_self, BiheytingHomClass.map_sdiff] }\n\n-- See note [lower instance priority]\ninstance (priority := 100) BiheytingHomClass.toCoheytingHomClass [BiheytingAlgebra α]\n    {_ : BiheytingAlgebra β} [BiheytingHomClass F α β] : CoheytingHomClass F α β :=\n  { ‹BiheytingHomClass F α β› with\n    map_top := fun f => by rw [← @himp_self α _ ⊥, ← himp_self, map_himp] }\n\nend Hom\n\nsection Equiv\n\nvariable [EquivLike F α β]\n\n-- See note [lower instance priority]\ninstance (priority := 100) OrderIsoClass.toHeytingHomClass [HeytingAlgebra α]\n    {_ : HeytingAlgebra β} [OrderIsoClass F α β] : HeytingHomClass F α β :=\n  { OrderIsoClass.toBoundedLatticeHomClass with\n    map_himp := fun f a b =>\n      eq_of_forall_le_iff fun c => by\n        simp only [← map_inv_le_iff, le_himp_iff]\n        rw [← OrderIsoClass.map_le_map_iff f]\n        simp }\n\n-- See note [lower instance priority]\ninstance (priority := 100) OrderIsoClass.toCoheytingHomClass [CoheytingAlgebra α]\n    {_ : CoheytingAlgebra β} [OrderIsoClass F α β] : CoheytingHomClass F α β :=\n  { OrderIsoClass.toBoundedLatticeHomClass with\n    map_sdiff := fun f a b =>\n      eq_of_forall_ge_iff fun c => by\n        simp only [← le_map_inv_iff, sdiff_le_iff]\n        rw [← OrderIsoClass.map_le_map_iff f]\n        simp }\n\n-- See note [lower instance priority]\ninstance (priority := 100) OrderIsoClass.toBiheytingHomClass [BiheytingAlgebra α]\n    {_ : BiheytingAlgebra β} [OrderIsoClass F α β] : BiheytingHomClass F α β :=\n  { OrderIsoClass.toLatticeHomClass with\n    map_himp := fun f a b =>\n      eq_of_forall_le_iff fun c => by\n        simp only [← map_inv_le_iff, le_himp_iff]\n        rw [← OrderIsoClass.map_le_map_iff f]\n        simp\n    map_sdiff := fun f a b =>\n      eq_of_forall_ge_iff fun c => by\n        simp only [← le_map_inv_iff, sdiff_le_iff]\n        rw [← OrderIsoClass.map_le_map_iff f]\n        simp }\n\nend Equiv\n\nvariable [FunLike F α β]\n\ninstance BoundedLatticeHomClass.toBiheytingHomClass [BooleanAlgebra α] [BooleanAlgebra β]\n    [BoundedLatticeHomClass F α β] : BiheytingHomClass F α β :=\n  { ‹BoundedLatticeHomClass F α β› with\n    map_himp := fun f a b => by rw [himp_eq, himp_eq, map_sup, (isCompl_compl.map _).compl_eq]\n    map_sdiff := fun f a b => by rw [sdiff_eq, sdiff_eq, map_inf, (isCompl_compl.map _).compl_eq] }\n\nsection HeytingAlgebra\n\nopen scoped symmDiff\n\nvariable [HeytingAlgebra α] [HeytingAlgebra β] [HeytingHomClass F α β] (f : F)\n\n@[simp]\ntheorem map_compl (a : α) : f aᶜ = (f a)ᶜ := by rw [← himp_bot, ← himp_bot, map_himp, map_bot]\n\n@[simp]\ntheorem map_bihimp (a b : α) : f (a ⇔ b) = f a ⇔ f b := by simp_rw [bihimp, map_inf, map_himp]\n\nend HeytingAlgebra\n\nsection CoheytingAlgebra\n\nopen scoped symmDiff\n\nvariable [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingHomClass F α β] (f : F)\n\n@[simp]\n\nTarget:\ntheorem map_hnot (a : α) : f (￢a) = ￢f a :=\n\nProof body:\n","rejected":"by\n  exact map_hnot","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"a27d149a9beb24a22ef2df2790ecc2d675ac796a059b5d3f801a349847061103","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Heyting","family_id":"map_hnot","file_id":"mathlib/Mathlib/Order/Heyting/Hom.lean","sample_id":"88184057873a0a559cc6d5510520e38470290a3d57980287af7d4cf570e8c138"},"split":"train"}
{"binding":{"chosen_proof_sha256":"de67f9b6451b5db3ba017889f07c8f6879cb98f3fc2736a86445d62c14d1c664","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ccd1b6ebbfc7925ea5a53c372ff24897d1bab4d6c043d828ade3166fe263bae3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ba8a1ee7158309c944c89fb3fff36b668ca7628b981cba7184536316677b8d1f","source_sha256":"94021f194f9dbe2b3ff10c0fe9e60d21f920f7ec47e18de0732170ff2d7ff47e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Squarefree, not_forall]\n  exact ⟨0, by simp⟩","hard_negative":true,"metrics":{"chosen_tokens":14,"rejected_tokens":3,"token_jaccard":0.153846,"token_length_ratio":0.214286},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"d7c860fc6318f5c13edf1a3b94d79a08a4f56f7b42ebc19e035e475bfda2fe8e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Coprime.Lemmas\npublic import Mathlib.RingTheory.Nilpotent.Basic\npublic import Mathlib.RingTheory.UniqueFactorizationDomain.GCDMonoid\npublic import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\n/-!\n# Squarefree elements of monoids\nAn element of a monoid is squarefree when it is not divisible by any squares\nexcept the squares of units.\n\nResults about squarefree natural numbers are proved in `Data.Nat.Squarefree`.\n\n## Main Definitions\n- `Squarefree r` indicates that `r` is only divisible by `x * x` if `x` is a unit.\n\n## Main Results\n- `multiplicity.squarefree_iff_emultiplicity_le_one`: `x` is `Squarefree` iff for every `y`, either\n  `emultiplicity y x ≤ 1` or `IsUnit y`.\n- `UniqueFactorizationMonoid.squarefree_iff_nodup_factors`: A nonzero element `x` of a unique\n  factorization monoid is squarefree iff `factors x` has no duplicate factors.\n\n## Tags\nsquarefree, multiplicity\n\n-/\n\n@[expose] public section\n\n\nvariable {R : Type*}\n\n/-- An element of a monoid is squarefree if the only squares that\n  divide it are the squares of units. -/\ndef Squarefree [Monoid R] (r : R) : Prop :=\n  ∀ x : R, x * x ∣ r → IsUnit x\n\ntheorem IsRelPrime.of_squarefree_mul [CommMonoid R] {m n : R} (h : Squarefree (m * n)) :\n    IsRelPrime m n := fun c hca hcb ↦ h c (mul_dvd_mul hca hcb)\n\n@[simp]\ntheorem IsUnit.squarefree [CommMonoid R] {x : R} (h : IsUnit x) : Squarefree x := fun _ hdvd =>\n  isUnit_of_mul_isUnit_left (isUnit_of_dvd_unit hdvd h)\n\ntheorem squarefree_one [CommMonoid R] : Squarefree (1 : R) :=\n  isUnit_one.squarefree\n\n@[simp]\n\nTarget:\ntheorem not_squarefree_zero [MonoidWithZero R] [Nontrivial R] : ¬Squarefree (0 : R) :=\n\nProof body:\n","rejected":"by\n  exact not_squarefree_zero","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"11f7d20af14dcfb0a445b11b6308d279e981fd0bf57915cbfdabf4bdec103a6e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Squarefree","family_id":"not_squarefree_zero","file_id":"mathlib/Mathlib/Algebra/Squarefree/Basic.lean","sample_id":"ba8a1ee7158309c944c89fb3fff36b668ca7628b981cba7184536316677b8d1f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"61f4608135cc80db3b144a8f0b385332ee17b58f1c351db640ceccbb2a9d7cc8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"149a9c772317371fc2b6f906fcbdef5bbd1de66741d24c40b626de4d07eb07f8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6a4d757b91c5500aa6ea19c37733ffcdf411c7c5b0b8869100541b0d79adff6c","source_sha256":"47c01594d211b3c54399310f0cb3b9f706ed5bf9e27bc1387c290b2a49698c96","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x; grind [eq_one_or_toFreeMonoid x]","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":13,"token_jaccard":0.666667,"token_length_ratio":1.444444},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"d7d9b5951413722646760d813fd90bca977a651fd25b09e061471c9614065fb6","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.FreeMonoid.Basic\npublic import Mathlib.Algebra.Free\npublic import Mathlib.Algebra.Group.WithOne.Basic\npublic import Mathlib.Algebra.Group.Units.Basic\npublic import Mathlib.Data.Set.Operations\nimport Mathlib.Data.Set.Insert\n\nNamespace:\nFreeSemigroup\n\nLocal context:\n/-\nCopyright (c) 2026 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\n/-!\n# Relation between the free semigroup and the free monoid\n\nWe provide some constructions relating the free semigroup and the free monoid on the same type.\n\n## Main definitions\n* `FreeSemigroup.toFreeMonoid`: the natural embedding of the free semigroup into the free monoid.\n* `FreeMonoid.equivWithOneFreeSemigroup`: the free monoid is isomorphic to the free semigroup\n  with a `1` added.\n-/\n\npublic section\n\nvariable {α : Type*}\n\nnamespace FreeSemigroup\n\nopen FreeMonoid\n\n/--\nThe natural embedding of the free semigroup into the free monoid.\nThis is injective (`FreeSemigroup.toFreeMonoid_injective`), and its image\nconsists of all non-`1` elements of the free monoid (`FreeSemigroup.eq_one_or_toFreeMonoid`).\n-/\n@[expose, to_additive /-- The natural embedding of the free additive semigroup into the\nfree additive monoid. This is injective (`FreeAddSemigroup.toFreeAddMonoid_injective`), and its\nimage consists of all non-`0` elements of the free additive monoid\n(`FreeAddSemigroup.eq_zero_or_toFreeAddMonoid`). -/]\ndef toFreeMonoid : FreeSemigroup α →ₙ* FreeMonoid α :=\n  lift FreeMonoid.of\n\n@[to_additive (attr := simp, grind =)]\nlemma toFreeMonoid_of (x : α) : toFreeMonoid (.of x) = .of x := rfl\n\n@[to_additive]\nlemma toFreeMonoid_mk_eq_cons (x : α) (xs : List α) :\n    toFreeMonoid ⟨x, xs⟩ = FreeMonoid.ofList (x :: xs) := by\n  suffices ∀ x : FreeMonoid α, (xs.map FreeMonoid.of).foldl (· * ·) x = x * ofList xs by\n    simpa [← List.foldl_map, lift_mk_eq_foldl, toFreeMonoid, lift] using this (FreeMonoid.of x)\n  induction xs with grind [ofList_nil, ofList_cons]\n\n@[to_additive (attr := grind .)]\nlemma toFreeMonoid_injective : Function.Injective (@toFreeMonoid α) := by\n  rintro ⟨x, xs⟩ ⟨y, ys⟩ h\n  simp only [toFreeMonoid_mk_eq_cons, Equiv.apply_eq_iff_eq] at h\n  simpa using h\n\n@[to_additive (attr := simp, grind .)]\nlemma toFreeMonoid_ne_one (x : FreeSemigroup α) : toFreeMonoid x ≠ 1 := by\n  induction x with simp\n\n@[to_additive]\nlemma eq_one_or_toFreeMonoid (x : FreeMonoid α) : x = 1 ∨ ∃ y, toFreeMonoid y = x :=\n  x.inductionOn' (by simp) <| by\n    rintro b _ (rfl | ⟨y, rfl⟩)\n    · exact Or.inr ⟨of b, by simp⟩\n    · exact Or.inr ⟨of b * y, by simp⟩\n\n@[to_additive (attr := simp)]\n\nTarget:\nlemma range_toFreeMonoid : Set.range (@toFreeMonoid α) = {1}ᶜ :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  ext x; grind [eq_one_or_toFreeMonoid x]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/FreeMonoid","family_id":"range_tofreemonoid","file_id":"mathlib/Mathlib/Algebra/FreeMonoid/FreeSemigroup.lean","sample_id":"6a4d757b91c5500aa6ea19c37733ffcdf411c7c5b0b8869100541b0d79adff6c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c3ee73588e36d5e89a2703d1667734bc0b58f87dcefce1c40a3828ad044e5637","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4b54c5b2c0a2b38c02774c2f2d109212cba9b9850dfbb54da563993b14988633","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"812ef1c18921780c46e19c5df601fc11ffc84bd9969d3bb154d739df4d388862","source_sha256":"345dde00cb75d3f9bc3f5909b7969856d423edbbc6d0a61a7a141d42b1e88b8d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [ringKrullDim, ringKrullDim]\n  apply Order.krullDim_le_of_krullDim_preimage_le' (PrimeSpectrum.comap C) ?_ (fun p ↦ ?_)\n  · exact fun {a b} h ↦ Ideal.comap_mono h\n  · rw [show C = (algebraMap R (Polynomial R)) from rfl, Order.krullDim_eq_of_orderIso\n      (PrimeSpectrum.preimageOrderIsoFiber R (Polynomial R) p), ← ringKrullDim,\n      ← ringKrullDim_eq_of_ringEquiv (polyEquivTensor R (p.asIdeal.ResidueField)).toRingEquiv,\n      ← Ring.krullDimLE_iff]\n    infer_instance\n\nvariable {R : Type*} [CommRing R] [IsNoetherianRing R]","hard_negative":true,"metrics":{"chosen_tokens":111,"rejected_tokens":5,"token_jaccard":0.074074,"token_length_ratio":0.045045},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"d7def9b2d1fdb8891a271096c6e19d91fc62aef814052678c16faef4dc1ad72b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.FieldDivision\npublic import Mathlib.RingTheory.KrullDimension.PID\npublic import Mathlib.RingTheory.LocalRing.ResidueField.Fiber\npublic import Mathlib.RingTheory.Ideal.KrullsHeightTheorem\npublic import Mathlib.RingTheory.KrullDimension.NonZeroDivisors\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Jingting Wang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jingting Wang, Sihan Su, Yi Song, Christian Merten\n-/\n/-!\n# Krull dimension of polynomial ring\n\nThis file proves properties of the Krull dimension of the polynomial ring over a commutative ring\n\n## Main results\n\n* `Polynomial.ringKrullDim_le`: the Krull dimension of the polynomial ring over a commutative ring\n  `R` is less than `2 * (ringKrullDim R) + 1`.\n\nFor noetherian rings:\n* `Polynomial.ringKrullDim_of_isNoetherianRing`: the Krull dimension of `R[X]` is `dim R + 1`.\n* `MvPolynomial.ringKrullDim_of_isNoetherianRing`: the Krull dimension of `R[X₁, ..., Xₙ]` is\n  `dim R + n`.\n-/\n\npublic section\n\nTarget:\ntheorem Polynomial.ringKrullDim_le {R : Type*} [CommRing R] :\n    ringKrullDim (Polynomial R) ≤ 2 * (ringKrullDim R) + 1 :=\n\nProof body:\n","rejected":"by\n  exact Polynomial.ringKrullDim_le","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"05f8c76094614921265b6d75219be5df87c573ab92ef1fa2980fca8354e5f3ee","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/KrullDimension","family_id":"polynomial","file_id":"mathlib/Mathlib/RingTheory/KrullDimension/Polynomial.lean","sample_id":"812ef1c18921780c46e19c5df601fc11ffc84bd9969d3bb154d739df4d388862"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0e69e1ffdf18797894f9d6a7bd81718318be487366506848fac73b20c0a23f03","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"78f61c99b43e2dea3859f08d8e1ca140e6f02fb952817c79e7649e7f8aa0fc3c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e1cca0717031d58aaaf848a8d08234c2dff2f8f85f108568fd9abbcb33e5e463","source_sha256":"71fe80c3d870cb012c2842689740577b7ac73128332425c67807885eb1ca905f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [internallyProjective_iff_tensor_condition]\n  refine ⟨fun h A B e he S g ↦ ?_, fun h A B e he S g ↦ ?_⟩\n  · specialize h e S ((β_ _ _).hom ≫ g)\n    obtain ⟨S', π, hπ, g', hh⟩ := h\n    refine ⟨S', π, hπ, (β_ _ _).inv ≫ g', ?_⟩\n    simp [← hh]\n  · specialize h e S ((β_ _ _).inv ≫ g)\n    obtain ⟨S', π, hπ, g', hh⟩ := h\n    refine ⟨S', π, hπ, (β_ _ _).hom ≫ g', ?_⟩\n    simp [← hh]","hard_negative":true,"metrics":{"chosen_tokens":151,"rejected_tokens":3,"token_jaccard":0.052632,"token_length_ratio":0.019868},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"d7e671c167c5a85dacc2edad6bc9a8c9461032af9d93adbb384fa08bbe5f61cb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Preadditive.Projective.Internal\npublic import Mathlib.Condensed.Light.Epi\npublic import Mathlib.Condensed.Light.Functors\npublic import Mathlib.Condensed.Light.Monoidal\n\nNamespace:\nLightCondensed\n\nLocal context:\n/-\nCopyright (c) 2025 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# Characterization of internal projectivity in light condensed modules\n\nThis file gives an explicit condition on light condensed modules over a ring `R` to be internally\nprojective, namely the following:\n\n`internallyProjective_iff_tensor_condition`: `P : LightCondMod R` is internally projective if and\nonly if, for all `A B : LightCondMod R`, for all epimorphisms `e : A ⟶ B`, for all\n`S : LightProfinite` and all morphisms `g : P ⊗ R[S] ⟶ B`, there exists a `S' : LightProfinite`\nwith a surjection `π : S' ⟶ S` and a morphism `g' : P ⊗ R[S'] ⟶ A`, making the diagram\n```\nP ⊗ R[S'] --> A\n  |           |\n  v           v\nP ⊗ R[S]  --> B\n```\ncommute.\n\nWe also provide the analogous characterization with the tensor product commuted the other way around\n(see `internallyProjective_iff_tensor_condition'`), and the special cases when `P` is the free\ncondensed module on a condensed set (`free_internallyProjective_iff_tensor_condition`,\n`free_internallyProjective_iff_tensor_condition'`) and when `P` is the free condensed module on a\nlight profinite set (`free_lightProfinite_internallyProjective_iff_tensor_condition`/\n`free_lightProfinite_internallyProjective_iff_tensor_condition'`).\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen CategoryTheory Category MonoidalCategory Functor Monoidal LaxMonoidal OplaxMonoidal\n\nvariable (R : Type u) [CommRing R]\n\nnamespace LightCondensed\n\n/--\nThe `S`-valued points of the internal hom `A ⟶[LightCondMod R] B` are in bijection with\nmorphisms `A ⊗ R[S] ⟶ B`.\n-/\nnoncomputable def ihomPoints (A B : LightCondMod.{u} R) (S : LightProfinite) :\n    (A ⟶[LightCondMod R] B).obj.obj ⟨S⟩ ≃ ((A ⊗ ((free R).obj S.toCondensed)) ⟶ B) :=\n  (((freeForgetAdjunction R).homEquiv _ _).trans\n    (coherentTopology _).yonedaEquiv).symm.trans\n      ((ihom.adjunction A).homEquiv _ _).symm\n\nlemma ihomPoints_apply (A B : LightCondMod.{u} R) (S : LightProfinite)\n    (x : (A ⟶[LightCondMod R] B).obj.obj ⟨S⟩) :\n    ihomPoints R A B S x = (MonoidalClosed.uncurry (((freeForgetAdjunction R).homEquiv _ _).symm\n      ((coherentTopology LightProfinite.{u}).yonedaEquiv.symm x))) :=\n  rfl\n\nlemma ihomPoints_symm_apply (A B : LightCondMod.{u} R) (S : LightProfinite)\n    (x : (A ⊗ ((free R).obj S.toCondensed)) ⟶ B) :\n    (ihomPoints R A B S).symm x = (coherentTopology LightProfinite.{u}).yonedaEquiv\n      ((freeForgetAdjunction R).homEquiv _ _ (MonoidalClosed.curry x)) :=\n  rfl\n\nset_option backward.isDefEq.respectTransparency false in\nlemma ihom_map_val_app (A B P : LightCondMod.{u} R) (S : LightProfinite) (e : A ⟶ B)\n    (x : (P ⟶[LightCondMod R] A).obj.obj ⟨S⟩) :\n    (((ihom P).map e).hom.app ⟨S⟩) x = (ihomPoints R P B S).symm (ihomPoints R P A S x ≫ e) := by\n  apply (ihomPoints R P B S).injective\n  simp only [ihomPoints_apply, ← MonoidalClosed.uncurry_natural_right,\n    ← Adjunction.homEquiv_naturality_right_symm, Equiv.apply_symm_apply]\n  congr\n  apply (coherentTopology LightProfinite.{u}).yonedaEquiv.injective\n  simp [dsimp% GrothendieckTopology.yonedaEquiv_comp]\n\nset_option backward.isDefEq.respectTransparency false in\nlemma ihomPoints_symm_comp (B P : LightCondMod.{u} R) (S S' : LightProfinite) (π : S ⟶ S')\n    (f : P ⊗ (free R).obj S'.toCondensed ⟶ B) :\n    (ihomPoints R P B S).symm (P ◁ (free R).map (lightProfiniteToLightCondSet.map π) ≫ f) =\n      ((P ⟶[LightCondMod R] B).obj.map π.op) ((ihomPoints R P B S').symm f) := by\n  simpa [ihomPoints_symm_apply, MonoidalClosed.curry_natural_left, Adjunction.homEquiv_apply] using!\n    (GrothendieckTopology.yonedaEquiv_naturality _ _ _).symm\n\nset_option backward.defeqAttrib.useBackward true in\n/--\n`P : LightCondMod R` is internally projective if and\nonly if, for all `A B : LightCondMod R`, for all epimorphisms `e : A ⟶ B`, for all\n`S : LightProfinite` and all morphisms `g : P ⊗ R[S] ⟶ B`, there exists a `S' : LightProfinite`\nwith a surjection `π : S' ⟶ S` and a morphism `g' : P ⊗ R[S'] ⟶ A`, making the diagram\n```\nP ⊗ R[S'] --> A\n  |           |\n  v           v\nP ⊗ R[S]  --> B\n```\ncommute.\n-/\nlemma internallyProjective_iff_tensor_condition (P : LightCondMod R) : InternallyProjective P ↔\n    ∀ {A B : LightCondMod R} (e : A ⟶ B) [Epi e],\n      (∀ (S : LightProfinite) (g : P ⊗ (free R).obj S.toCondensed ⟶ B), ∃ (S' : LightProfinite)\n        (π : S' ⟶ S) (_ : Function.Surjective π) (g' : P ⊗ (free R).obj S'.toCondensed ⟶ A),\n          (P ◁ ((lightProfiniteToLightCondSet ⋙ free R).map π)) ≫ g = g' ≫ e) := by\n  refine ⟨fun ⟨h⟩ A B e he S g ↦ ?_, fun h ↦ ⟨⟨fun {A B} e he ↦ ?_⟩⟩⟩\n  · have hh := h.1 e\n    rw [LightCondMod.epi_iff_locallySurjective_on_lightProfinite] at hh\n    specialize hh S ((ihomPoints R P B S).symm g)\n    obtain ⟨S', π, hπ, g', hh⟩ := hh\n    refine ⟨S', π, hπ, (ihomPoints _ _ _ _) g', ?_⟩\n    rw [ihom_map_val_app] at hh\n    apply (ihomPoints R P B S').symm.injective\n    rw [hh]\n    exact ihomPoints_symm_comp R B P S' S π g\n  · rw [LightCondMod.epi_iff_locallySurjective_on_lightProfinite]\n    intro S g\n    specialize h e S ((ihomPoints _ _ _ _) g)\n    obtain ⟨S', π, hπ, g', hh⟩ := h\n    refine ⟨S', π, hπ, (ihomPoints _ _ _ _).symm g', ?_⟩\n    rw [ihom_map_val_app]\n    have := ihomPoints_symm_comp R B P S' S π ((ihomPoints R P B S) g)\n    dsimp at hh\n    rw [hh] at this\n    simp [this, Quiver.Hom.op]\n\nset_option backward.defeqAttrib.useBackward true in\n/--\n`P : LightCondMod R` is internally projective if and\nonly if, for all `A B : LightCondMod R`, for all epimorphisms `e : A ⟶ B`, for all\n`S : LightProfinite` and all morphisms `g : R[S] ⊗ P ⟶ B`, there exists a `S' : LightProfinite`\nwith a surjection `π : S' ⟶ S` and a morphism `g' : R[S'] ⊗ P ⟶ A`, making the diagram\n```\nR[S'] ⊗ P --> A\n  |           |\n  v           v\nR[S] ⊗ P  --> B\n```\ncommute.\n-/\n\nTarget:\nlemma internallyProjective_iff_tensor_condition' (P : LightCondMod R) : InternallyProjective P ↔\n    ∀ {A B : LightCondMod R} (e : A ⟶ B) [Epi e],\n      (∀ (S : LightProfinite) (g : (free R).obj S.toCondensed ⊗ P ⟶ B), ∃ (S' : LightProfinite)\n        (π : S' ⟶ S) (_ : Function.Surjective π) (g' : (free R).obj S'.toCondensed ⊗ P ⟶ A),\n          (((lightProfiniteToLightCondSet ⋙ free R).map π) ▷ P) ≫ g = g' ≫ e) :=\n\nProof body:\n","rejected":"by\n  exact internallyProjective_iff_tensor_condition","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"c096491035f5ce3dddee76453d5b0f89fb32e29fd0957800c7d092760c8155ad","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed/Light","family_id":"internallyprojective_iff_tensor_condition","file_id":"mathlib/Mathlib/Condensed/Light/InternallyProjective.lean","sample_id":"e1cca0717031d58aaaf848a8d08234c2dff2f8f85f108568fd9abbcb33e5e463"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3f65d3f4792abc1e42ea24793b4621de9bedda45327f9e9eb8d4aee686a05437","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"45bf8f219b5d1572a7b7bfc82220b21da6d9c0dadf354c8db833eeedf98c284d","source_sha256":"5f15c2a4e2cafa011e3dad0d984da70295687e57a0186d3fe7fe49952e8b2ae6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← LinearOrderedCommGroupWithZero.wellFoundedOn_setOf_ge_gt_iff_nonempty_discrete_of_ne_zero\n    one_ne_zero, ← Set.wellFoundedOn_range]\n  classical\n  refine ⟨fun h ↦ (h.mapsTo Subtype.val ?_).mono' (by simp), fun h ↦ (h.mapsTo ?_ ?_).mono' ?_⟩\n  · rintro ⟨_, x, rfl⟩\n    simp only [← Subtype.coe_le_coe, OneMemClass.coe_one, Set.mem_setOf_eq, Set.mem_range,\n      Function.comp_apply]\n    intro hx\n    obtain ⟨y, rfl⟩ := hv.exists_of_le_one hx\n    exact ⟨y, by simp⟩\n  · exact fun x ↦ if hx : x ∈ MonoidHom.mrange v then ⟨x, hx⟩ else 1\n  · intro\n    simp only [Set.mem_range, Function.comp_apply, MonoidHom.mem_mrange, Set.mem_setOf_eq,\n      forall_exists_index]\n    rintro x rfl\n    simp [← Subtype.coe_le_coe, hv.map_le_one]\n  · simp [Function.onFun]","hard_negative":false,"metrics":{"chosen_tokens":173,"rejected_tokens":3,"token_jaccard":0.032258,"token_length_ratio":0.017341},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"d7e693be6455f4408655cb63864e00ad1a5880554a0d577ef0843ac416529a48","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.ArchimedeanDensely\npublic import Mathlib.RingTheory.Valuation.ValuationRing\n\nNamespace:\nValuation.Integers\n\nLocal context:\n/-\nCopyright (c) 2024 Yakov Pechersky. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yakov Pechersky\n-/\n/-!\n# Ring of integers under a given valuation in a multiplicatively archimedean codomain\n\n-/\n\npublic section\n\nsection Field\n\nvariable {F Γ₀ O : Type*} [Field F] [LinearOrderedCommGroupWithZero Γ₀]\n  [CommRing O] [Algebra O F] {v : Valuation F Γ₀}\n\ninstance MonoidWithZeroHom.instLinearOrderedCommGroupWithZeroMrange (v : F →*₀ Γ₀) :\n    LinearOrderedCommGroupWithZero (MonoidHom.mrange v) where\n  bot := ⟨⊥, by simp [bot_eq_zero]⟩\n  bot_le a := by simp [bot_eq_zero, ← Subtype.coe_le_coe]\n  isBot_zero a := by simp [← Subtype.coe_le_coe]\n  mul_lt_mul_of_pos_left := by\n    simp only [← Subtype.coe_lt_coe, val_mrange_zero, Submonoid.coe_mul, Subtype.forall,\n      MonoidHom.mem_mrange, forall_exists_index, forall_apply_eq_imp_iff]\n    rintro a ha b c hbc\n    gcongr\n\ninstance Valuation.instLinearOrderedCommGroupWithZeroMrange :\n    LinearOrderedCommGroupWithZero (MonoidHom.mrange v) :=\n  inferInstanceAs (LinearOrderedCommGroupWithZero (MonoidHom.mrange (.ofClass v : F →*₀ Γ₀)))\n\nnamespace Valuation.Integers\n\nopen scoped Function in\nlemma wfDvdMonoid_iff_wellFounded_gt_on_v (hv : Integers v O) :\n    WfDvdMonoid O ↔ WellFounded ((· > ·) on (v ∘ algebraMap O F)) := by\n  refine ⟨fun _ ↦ wellFounded_dvdNotUnit.mono ?_, fun h ↦ ⟨h.mono ?_⟩⟩ <;>\n  simp [Function.onFun, hv.dvdNotUnit_iff_lt]\n\nopen scoped Function WithZero in\n\nTarget:\nlemma wellFounded_gt_on_v_iff_discrete_mrange [Nontrivial (MonoidHom.mrange v)ˣ]\n    (hv : Integers v O) :\n    WellFounded ((· > ·) on (v ∘ algebraMap O F)) ↔\n      Nonempty (MonoidHom.mrange v ≃*o ℤᵐ⁰) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Valuation","family_id":"wellfounded_gt_on_v_iff_discrete_mrange","file_id":"mathlib/Mathlib/RingTheory/Valuation/Archimedean.lean","sample_id":"45bf8f219b5d1572a7b7bfc82220b21da6d9c0dadf354c8db833eeedf98c284d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"974123677968aef5cec49d697aa3b1eb6faa233c3680735631e7ed936621e266","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5b17a6a744755a2e6fb4017210906255fc2a4dac573058efd5f854676fc92bca","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"aab605044ac1a0cc179875c4807508cccde8f4829eb26975d9537a751ad57521","source_sha256":"174951d99036d6995b4ac6d486de807bedb58f340aa902f02502a7216515bf63","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [integral_fintype .of_finite]\n  congr with x\n  rw [measureReal_def]\n  congr 2\n  exact PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _)","hard_negative":true,"metrics":{"chosen_tokens":28,"rejected_tokens":3,"token_jaccard":0.090909,"token_length_ratio":0.107143},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"d84b95db5a15767d1f9b7be3e5461acb24cd0a857587283ce9cafda591ad3157","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.ProbabilityMassFunction.Basic\npublic import Mathlib.Probability.ProbabilityMassFunction.Constructions\npublic import Mathlib.MeasureTheory.Integral.Bochner.SumMeasure\n\nNamespace:\nPMF\n\nLocal context:\n/-\nCopyright (c) 2023 Joachim Breitner. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joachim Breitner\n-/\n/-!\n# Integrals with a measure derived from probability mass functions.\n\nThis file connects `PMF` with `integral`. The main result is that the integral (i.e. the expected\nvalue) with regard to a measure derived from a `PMF` is a sum weighted by the `PMF`.\n\nIt also provides the expected value for specific probability mass functions.\n-/\n\npublic section\n\nnamespace PMF\n\nopen MeasureTheory NNReal ENNReal TopologicalSpace\n\nsection General\n\nvariable {α : Type*} [MeasurableSpace α] [MeasurableSingletonClass α]\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]\n\ntheorem integral_eq_tsum (p : PMF α) (f : α → E) (hf : Integrable f p.toMeasure) :\n    ∫ a, f a ∂(p.toMeasure) = ∑' a, (p a).toReal • f a := calc\n  _ = ∫ a in p.support, f a ∂(p.toMeasure) := by rw [restrict_toMeasure_support p]\n  _ = ∑' (a : support p), (p.toMeasure {a.val}).toReal • f a := by\n    apply setIntegral_countable f p.support_countable\n    rwa [IntegrableOn, restrict_toMeasure_support p]\n  _ = ∑' (a : support p), (p a).toReal • f a := by\n    congr with x; congr 2\n    apply PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _)\n  _ = ∑' a, (p a).toReal • f a :=\n    tsum_subtype_eq_of_support_subset <| calc\n      (fun a ↦ (p a).toReal • f a).support ⊆ (fun a ↦ (p a).toReal).support :=\n        Function.support_smul_subset_left _ _\n      _ ⊆ support p := fun x h1 h2 => h1 (by simp [h2])\n\nTarget:\ntheorem integral_eq_sum [Fintype α] (p : PMF α) (f : α → E) :\n    ∫ a, f a ∂(p.toMeasure) = ∑ a, (p a).toReal • f a :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_aab605044ac1","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"16b6d9253327d213169b422bdddba1982f944f595a03a61524e43c858fa8a25d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/ProbabilityMassFunction","family_id":"integral_eq_sum","file_id":"mathlib/Mathlib/Probability/ProbabilityMassFunction/Integrals.lean","sample_id":"aab605044ac1a0cc179875c4807508cccde8f4829eb26975d9537a751ad57521"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8cf4a63a26dc0e5b281fd2da3d3e4050e7d2b711edcf9188ce1a11ef8a2b0753","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"21e6b6874ff31a22dfe9207b446fa97b3b77e9fdcdd69af5c1b2e58ae8c599a9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4814cbf7f3508d036f5e5a6a7155650097bbec6785ba9dace923119e89bbbe59","source_sha256":"bc1107b51b023f15ba6b116e921d72825e5fd28811d110becc324cfa9ebc2fb7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold mkFinCons\n  exact coe_mk (v := Fin.cons y (N.subtype ∘ b)) _ _\n\n/-- Let `b` be a basis for a submodule `N ≤ O`. If `y ∈ O` is linear independent of `N`\nand `y` and `N` together span the whole of `O`, then there is a basis for `O`\nwhose basis vectors are given by `Fin.cons y b`. -/\nnoncomputable def mkFinConsOfLE {n : ℕ} {N O : Submodule R M} (y : M) (yO : y ∈ O)\n    (b : Basis (Fin n) R N) (hNO : N ≤ O) (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0)\n    (hsp : ∀ z ∈ O, ∃ c : R, z + c • y ∈ N) : Basis (Fin (n + 1)) R O :=\n  mkFinCons ⟨y, yO⟩ (b.map (Submodule.comapSubtypeEquivOfLe hNO).symm)\n    (fun c x hc hx => hli c x (Submodule.mem_comap.mp hc) (congr_arg ((↑) : O → M) hx))\n    fun z => hsp z z.2","hard_negative":false,"metrics":{"chosen_tokens":262,"rejected_tokens":266,"token_jaccard":0.967033,"token_length_ratio":1.015267},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"d87158316733db7657f547ef8bf00e6a0812c6a62fcd3580f0eda65e9517a730","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Basis.Basic\npublic import Mathlib.LinearAlgebra.Pi\n\nNamespace:\nModule.Basis\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Kevin H. Wilson\n-/\n/-!\n# Bases indexed by `Fin`\n-/\n\n@[expose] public section\n\nassert_not_exists Ordinal\n\nnoncomputable section\n\nuniverse u\n\nopen Function Set Submodule Finsupp\n\nvariable {ι : Type*} {ι' : Type*} {R : Type*} {R₂ : Type*} {M : Type*} {M' : Type*}\n\nnamespace Module\n\nopen LinearMap\n\nvariable {v : ι → M}\nvariable [Ring R] [CommRing R₂] [AddCommGroup M]\nvariable [Module R M] [Module R₂ M]\nvariable {x y : M}\nvariable (b : Basis ι R M)\n\nnamespace Basis\n\nsection Fin\n\n/-- Let `b` be a basis for a submodule `N` of `M`. If `y : M` is linear independent of `N`\nand `y` and `N` together span the whole of `M`, then there is a basis for `M`\nwhose basis vectors are given by `Fin.cons y b`. -/\nnoncomputable def mkFinCons {n : ℕ} {N : Submodule R M} (y : M) (b : Basis (Fin n) R N)\n    (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :\n    Basis (Fin (n + 1)) R M :=\n  have span_b : N = Submodule.span R (Set.range (N.subtype ∘ b)) := by\n    rw [Set.range_comp, Submodule.span_image, b.span_eq, Submodule.map_subtype_top]\n  Basis.mk (v := Fin.cons y (N.subtype ∘ b))\n    ((b.linearIndependent.map' N.subtype (Submodule.ker_subtype _)).finCons' _ _\n      (by\n        intro c x hx hc\n        rw [← span_b] at hx\n        exact hli c x hx hc))\n    fun x _ => by simpa [Submodule.mem_span_insert', span_b] using hsp x\n\n@[simp]\n\nTarget:\ntheorem coe_mkFinCons {n : ℕ} {N : Submodule R M} (y : M) (b : Basis (Fin n) R N)\n    (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :\n    (mkFinCons y b hli hsp : Fin (n + 1) → M) = Fin.cons y ((↑) ∘ b) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  unfold mkFinCons\n  exact coe_mk (v := Fin.cons y (N.subtype ∘ b)) _ _\n\n/-- Let `b` be a basis for a submodule `N ≤ O`. If `y ∈ O` is linear independent of `N`\nand `y` and `N` together span the whole of `O`, then there is a basis for `O`\nwhose basis vectors are given by `Fin.cons y b`. -/\nnoncomputable def mkFinConsOfLE {n : ℕ} {N O : Submodule R M} (y : M) (yO : y ∈ O)\n    (b : Basis (Fin n) R N) (hNO : N ≤ O) (hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0)\n    (hsp : ∀ z ∈ O, ∃ c : R, z + c • y ∈ N) : Basis (Fin (n + 1)) R O :=\n  mkFinCons ⟨y, yO⟩ (b.map (Submodule.comapSubtypeEquivOfLe hNO).symm)\n    (fun c x hc hx => hli c x (Submodule.mem_comap.mp hc) (congr_arg ((↑) : O → M) hx))\n    fun z => hsp z z.2","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Basis","family_id":"coe_mkfincons","file_id":"mathlib/Mathlib/LinearAlgebra/Basis/Fin.lean","sample_id":"4814cbf7f3508d036f5e5a6a7155650097bbec6785ba9dace923119e89bbbe59"},"split":"train"}
{"binding":{"chosen_proof_sha256":"82dcb0744887ee0b2b5e66ebed4e1d1db29ac81ed20eb128a4882cf8f95fc26d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f073ff6b550aeff5c73e243e84ebf2d91d1d169b0b3b2c4fe873846d588b6590","source_sha256":"3ee6cc92b0085948fb0c591bbd98433993abf27d506b4cc259fc13467d3e2a06","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [aeval_def, aeval_def, ← coe_eval₂Hom, ← coe_eval₂Hom, map_eval₂Hom, ←\n    IsScalarTower.algebraMap_eq, Function.comp_def]","hard_negative":false,"metrics":{"chosen_tokens":30,"rejected_tokens":3,"token_jaccard":0.055556,"token_length_ratio":0.1},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"d8736f7c9f6736ca56971777ae615b67b5447ec1c57f7895c97411c72a95b597","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Subalgebra.Tower\npublic import Mathlib.Algebra.MvPolynomial.Eval\n\nNamespace:\nMvPolynomial\n\nLocal context:\n/-\nCopyright (c) 2022 Yuyang Zhao. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yuyang Zhao\n-/\n/-!\n# Algebra towers for multivariate polynomial\n\nThis file proves some basic results about the algebra tower structure for the type\n`MvPolynomial σ R`.\n\nThis structure itself is provided elsewhere as `MvPolynomial.isScalarTower`\n\nWhen you update this file, you can also try to make a corresponding update in\n`RingTheory.Polynomial.Tower`.\n-/\n\npublic section\n\n\nvariable (R A B : Type*) {σ : Type*}\n\nnamespace MvPolynomial\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\nvariable [Algebra R A] [Algebra A B] [Algebra R B]\nvariable [IsScalarTower R A B]\nvariable {R B}\n\ntheorem aeval_map_algebraMap (x : σ → B) (p : MvPolynomial σ R) :\n    aeval x (map (algebraMap R A) p) = aeval x p := by\n  rw [aeval_def, aeval_def, eval₂_map, IsScalarTower.algebraMap_eq R A B]\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\nvariable [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B]\nvariable {R A}\n\nTarget:\ntheorem aeval_algebraMap_apply (x : σ → A) (p : MvPolynomial σ R) :\n    aeval (algebraMap A B ∘ x) p = algebraMap A B (MvPolynomial.aeval x p) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/MvPolynomial","family_id":"aeval_algebramap_apply","file_id":"mathlib/Mathlib/RingTheory/MvPolynomial/Tower.lean","sample_id":"f073ff6b550aeff5c73e243e84ebf2d91d1d169b0b3b2c4fe873846d588b6590"},"split":"train"}
{"binding":{"chosen_proof_sha256":"05b7a960b09bb29c3bdb02159f0748a158edd33fe39806e604a0595973e7a4e9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"87930c51804421981909023af82af6c0ef02d38b6a61f89d18e4440f826827c3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"861b7d8a1fe1c1ac90ef7cb56d31d4772a1a7fac4746b22305f0316ae345cc5c","source_sha256":"2c2357bdce988b5b8902bab0c6b1d4a599e97cee77b05d9e7215f22e881d81ce","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isSemilinearSet_iff] at hs\n  rcases hs with ⟨S, hS, rfl⟩\n  choose φ hφ using fun s : S => (hS s.1 s.2).definable\n  refine ⟨Formula.iSup φ, ?_⟩\n  ext x\n  have := fun s hs x => Set.ext_iff.1 (hφ ⟨s, hs⟩).symm x\n  simp only [mem_setOf_eq] at this\n  simp [this]","hard_negative":true,"metrics":{"chosen_tokens":85,"rejected_tokens":3,"token_jaccard":0.021277,"token_length_ratio":0.035294},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"d8a2f212b600d258da347dfa0539f57b51f68742221c8cf664b02d02eebc8e9f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.ModelTheory.Arithmetic.Presburger.Basic\npublic import Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic\npublic import Mathlib.ModelTheory.Definability\nimport Mathlib.Algebra.Group.Submonoid.Finsupp\nimport Mathlib.LinearAlgebra.Matrix.Notation\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Dexin Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dexin Zhang\n-/\n/-!\n# Presburger definability and semilinear sets\n\nThis file formalizes the classical result that Presburger definable sets are the same as semilinear\nsets. As an application of this result, we show that the graph of multiplication is not Presburger\ndefinable.\n\n## Main Results\n\n- `presburger.definable_iff_isSemilinearSet`: a set is Presburger definable in `ℕ` if and only if it\n  is semilinear.\n- `presburger.definable₁_iff_ultimately_periodic`: in the 1-dimensional case, a set is Presburger\n  arithmetic definable in `ℕ` if and only if it is ultimately periodic, i.e. periodic after some\n  number `k`.\n- `presburger.mul_not_definable`: the graph of multiplication is not Presburger definable in `ℕ`.\n\n## References\n\n* [Seymour Ginsburg and Edwin H. Spanier, *Bounded ALGOL-Like Languages*][ginsburg1964]\n* [Seymour Ginsburg and Edwin H. Spanier, *Semigroups, Presburger Formulas, and\n  Languages*][ginsburg1966]\n* [Samuel Eilenberg and M. P. Schützenberger, *Rational Sets in Commutative Monoids*][eilenberg1969]\n-/\n\npublic section\n\nvariable {α : Type*} {s : Set (α → ℕ)} {A : Set ℕ}\n\nopen Set FirstOrder Language\n\ntheorem IsLinearSet.definable [Finite α] (hs : IsLinearSet s) : A.Definable presburger s := by\n  rw [isLinearSet_iff] at hs\n  rcases hs with ⟨v, t, rfl⟩\n  refine ⟨Formula.iExs t (Formula.iInf fun i : α =>\n    (Term.var (Sum.inl i)).equal\n      (Term.varsToConstants\n        ((v i : presburger.Term _) + presburger.sum Finset.univ fun x : t =>\n          x.1 i • Term.var (Sum.inr (Sum.inr x))))), ?_⟩\n  ext x\n  simp only [mem_vadd_set, SetLike.mem_coe, AddSubmonoid.mem_closure_finset', Finset.univ_eq_attach,\n    nsmul_eq_mul, vadd_eq_add, ↓existsAndEq, true_and, mem_setOf_eq, Formula.realize_iExs,\n    Formula.realize_iInf, Formula.realize_equal, Term.realize_var, Sum.elim_inl,\n    Term.realize_varsToConstants, coe_con, presburger.realize_add, presburger.realize_natCast,\n    Nat.cast_id, presburger.realize_sum, presburger.realize_nsmul, Sum.elim_inr, smul_eq_mul]\n  congr! with a\n  simp_rw [Eq.comm (b := x), fun x : t => mul_comm (a x : α → ℕ) x, funext_iff]\n  congr! 1 with i\n  simp\n\nTarget:\ntheorem IsSemilinearSet.definable [Finite α] (hs : IsSemilinearSet s) :\n    A.Definable presburger s :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_861b7d8a1fe1","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"2a94c432c40aaa0aba8cb92a60d95b10abb52590df992e99cc2868658811769a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"ModelTheory/Arithmetic","family_id":"issemilinearset","file_id":"mathlib/Mathlib/ModelTheory/Arithmetic/Presburger/Definability.lean","sample_id":"861b7d8a1fe1c1ac90ef7cb56d31d4772a1a7fac4746b22305f0316ae345cc5c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"63ac599b07e7436f21a63f1b73d4a7f54c73a40c1b4a50f48b5455d9ae2693f2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9614d072a9105eea2ab6f57e5011e3855902130db93743053ebaa70a564b484b","source_sha256":"a4ded0c37af8d7d4bd1dca4a7e378a31082efe29498bab67ebba55f071776588","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext j\n  dsimp\n  rw [mapBifunctorLeftUnitor_inv_apply, mapBifunctorLeftUnitor_inv_apply, assoc, assoc,\n    ι_mapBifunctorMapMap]\n  dsimp\n  rw [Functor.map_id, NatTrans.id_app, id_comp, ← NatTrans.naturality_assoc,\n    ← NatTrans.naturality_assoc]\n  rfl","hard_negative":false,"metrics":{"chosen_tokens":41,"rejected_tokens":2,"token_jaccard":0.045455,"token_length_ratio":0.04878},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"d8a350d3c1398f256d304df6a3c403c53734ff00717d729efa9f689d743e7d01","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.GradedObject.Associator\npublic import Mathlib.CategoryTheory.GradedObject.Single\n\nNamespace:\nCategoryTheory.GradedObject\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The left and right unitors\n\nGiven a bifunctor `F : C ⥤ D ⥤ D`, an object `X : C` such that `F.obj X ≅ 𝟭 D` and a\nmap `p : I × J → J` such that `hp : ∀ (j : J), p ⟨0, j⟩ = j`,\nwe define an isomorphism of `J`-graded objects for any `Y : GradedObject J D`.\n`mapBifunctorLeftUnitor F X e p hp Y : mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`.\nUnder similar assumptions, we also obtain a right unitor isomorphism\n`mapBifunctorMapObj F p X ((single₀ I).obj Y) ≅ X`. Finally,\nthe lemma `mapBifunctor_triangle` promotes a triangle identity involving functors\nto a triangle identity for the induced functors on graded objects.\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Category Limits\n\nnamespace GradedObject\n\nsection LeftUnitor\n\nvariable {C D I J : Type*} [Category* C] [Category* D]\n  [Zero I] [DecidableEq I] [HasInitial C]\n  (F : C ⥤ D ⥤ D) (X : C) (e : F.obj X ≅ 𝟭 D)\n  [∀ (Y : D), PreservesColimit (Functor.empty.{0} C) (F.flip.obj Y)]\n  (p : I × J → J) (hp : ∀ (j : J), p ⟨0, j⟩ = j)\n  (Y Y' : GradedObject J D) (φ : Y ⟶ Y')\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D` and `Y : GradedObject J D`,\nthis is the isomorphism `((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a ≅ Y a.2`\nwhen `a : I × J` is such that `a.1 = 0`. -/\n@[simps!]\nnoncomputable def mapBifunctorObjSingle₀ObjIso (a : I × J) (ha : a.1 = 0) :\n    ((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a ≅ Y a.2 :=\n  (F.mapIso (singleObjApplyIsoOfEq _ X _ ha)).app _ ≪≫ e.app (Y a.2)\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C` and `Y : GradedObject J D`,\n`((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a` is an initial object\nwhen `a : I × J` is such that `a.1 ≠ 0`. -/\nnoncomputable def mapBifunctorObjSingle₀ObjIsInitial (a : I × J) (ha : a.1 ≠ 0) :\n    IsInitial (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a) :=\n  IsInitial.isInitialObj (F.flip.obj (Y a.2)) _ (isInitialSingleObjApply _ _ _ ha)\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D`, `Y : GradedObject J D` and\n`p : I × J → J` such that `p ⟨0, j⟩ = j` for all `j`,\nthis is the (colimit) cofan which shall be used to construct the isomorphism\n`mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`, see `mapBifunctorLeftUnitor`. -/\nnoncomputable def mapBifunctorLeftUnitorCofan (hp : ∀ (j : J), p ⟨0, j⟩ = j) (Y) (j : J) :\n    (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y).CofanMapObjFun p j :=\n  CofanMapObjFun.mk _ _ _ (Y j) (fun a ha =>\n    if ha : a.1 = 0 then\n      (mapBifunctorObjSingle₀ObjIso F X e Y a ha).hom ≫ eqToHom (by aesop)\n    else\n      (mapBifunctorObjSingle₀ObjIsInitial F X Y a ha).to _)\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp, reassoc]\nlemma mapBifunctorLeftUnitorCofan_inj (j : J) :\n    (mapBifunctorLeftUnitorCofan F X e p hp Y j).inj ⟨⟨0, j⟩, hp j⟩ =\n      (F.map (singleObjApplyIso (0 : I) X).hom).app (Y j) ≫ e.hom.app (Y j) := by\n  simp [mapBifunctorLeftUnitorCofan]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The cofan `mapBifunctorLeftUnitorCofan F X e p hp Y j` is a colimit. -/\nnoncomputable def mapBifunctorLeftUnitorCofanIsColimit (j : J) :\n    IsColimit (mapBifunctorLeftUnitorCofan F X e p hp Y j) :=\n  Cofan.IsColimit.mk _\n    (fun s => e.inv.app (Y j) ≫\n      (F.map (singleObjApplyIso (0 : I) X).inv).app (Y j) ≫ s.inj ⟨⟨0, j⟩, hp j⟩)\n    (fun s => by\n      rintro ⟨⟨i, j'⟩, h⟩\n      by_cases hi : i = 0\n      · subst hi\n        simp only [Set.mem_preimage, hp, Set.mem_singleton_iff] at h\n        subst h\n        simp\n      · apply IsInitial.hom_ext\n        exact mapBifunctorObjSingle₀ObjIsInitial _ _ _ _ hi)\n    (fun s m hm => by simp [← hm ⟨⟨0, j⟩, hp j⟩])\n\ninclude e hp in\nlemma mapBifunctorLeftUnitor_hasMap :\n    HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y) p :=\n  CofanMapObjFun.hasMap _ _ _ (mapBifunctorLeftUnitorCofanIsColimit F X e p hp Y)\n\nvariable [HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y) p]\n  [HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y') p]\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D`, `Y : GradedObject J D` and\n`p : I × J → J` such that `p ⟨0, j⟩ = j` for all `j`,\nthis is the left unitor isomorphism `mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`. -/\nnoncomputable def mapBifunctorLeftUnitor : mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y :=\n  isoMk _ _ (fun j => (CofanMapObjFun.iso\n    (mapBifunctorLeftUnitorCofanIsColimit F X e p hp Y j)).symm)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma ι_mapBifunctorLeftUnitor_hom_apply (j : J) :\n    ιMapBifunctorMapObj F p ((single₀ I).obj X) Y 0 j j (hp j) ≫\n      (mapBifunctorLeftUnitor F X e p hp Y).hom j =\n      (F.map (singleObjApplyIso (0 : I) X).hom).app _ ≫ e.hom.app (Y j) := by\n  dsimp [mapBifunctorLeftUnitor]\n  erw [CofanMapObjFun.ιMapObj_iso_inv]\n  rw [mapBifunctorLeftUnitorCofan_inj]\n\nlemma mapBifunctorLeftUnitor_inv_apply (j : J) :\n    (mapBifunctorLeftUnitor F X e p hp Y).inv j =\n      e.inv.app (Y j) ≫ (F.map (singleObjApplyIso (0 : I) X).inv).app (Y j) ≫\n      ιMapBifunctorMapObj F p ((single₀ I).obj X) Y 0 j j (hp j) := rfl\n\nvariable {Y Y'}\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc]\n\nTarget:\nlemma mapBifunctorLeftUnitor_inv_naturality :\n    φ ≫ (mapBifunctorLeftUnitor F X e p hp Y').inv =\n      (mapBifunctorLeftUnitor F X e p hp Y).inv ≫ mapBifunctorMapMap F p (𝟙 _) φ :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/GradedObject","family_id":"mapbifunctorleftunitor_inv_naturality","file_id":"mathlib/Mathlib/CategoryTheory/GradedObject/Unitor.lean","sample_id":"9614d072a9105eea2ab6f57e5011e3855902130db93743053ebaa70a564b484b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b9120dc0c9841f5cc0966a2e218d4adcc087ef91568585fb5a852de6e1d93e10","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d306a3f4f2440500e38cd155adf08ac6a9da3781a1fac0d1e531b3e51b76d9e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3e937a169766b398011457da21839b47e8a3e949c8dec6560a9e40da808f1d10","source_sha256":"afb2352b0e8eda21bd88c47cda5f9b0a4c162f3f78eedf36a33613392805b346","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply Quotient.lift_unique\n  apply Paths.lift_unique\n  fapply @Quiver.Symmetrify.lift_unique _ _ _ _ _ _ _ _ _\n  · rw [← Functor.toPrefunctor_comp]\n    exact hΦ\n  · rintro X Y f\n    simp only [← Functor.toPrefunctor_comp, Prefunctor.comp_map, Paths.of_map]\n    change Φ.map (Groupoid.inv ((Quotient.functor redStep).toPrefunctor.map f.toPath)) =\n      Groupoid.inv (Φ.map ((Quotient.functor redStep).toPrefunctor.map f.toPath))\n    have := Functor.map_inv Φ ((Quotient.functor redStep).toPrefunctor.map f.toPath)\n    convert! this <;> simp only [Groupoid.inv_eq_inv]","hard_negative":true,"metrics":{"chosen_tokens":139,"rejected_tokens":2,"token_jaccard":0.018868,"token_length_ratio":0.014388},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"d8d9cc740d7ada3478fc0d6809d3ff3117b1a7c000f3e4ef02aef562b4cfa75d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Groupoid\npublic import Mathlib.CategoryTheory.PathCategory.Basic\n\nNamespace:\nQuiver.FreeGroupoid\n\nLocal context:\n/-\nCopyright (c) 2022 Rémi Bottinelli. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémi Bottinelli\n-/\n/-!\n# Free groupoid on a quiver\n\nThis file defines the free groupoid on a quiver, the lifting of a prefunctor to its unique\nextension as a functor from the free groupoid, and proves uniqueness of this extension.\n\n## Main results\n\nGiven the type `V` and a quiver instance on `V`:\n\n- `Quiver.FreeGroupoid V`: a type synonym for `V`.\n- `Quiver.FreeGroupoid.instGroupoid`: the `Groupoid` instance on `Quiver.FreeGroupoid V`.\n- `lift`: the lifting of a prefunctor from `V` to `V'` where `V'` is a groupoid, to a functor.\n  `Quiver.FreeGroupoid V ⥤ V'`.\n- `lift_spec` and `lift_unique`: the proofs that, respectively, `lift` indeed is a lifting\n  and is the unique one.\n\n## Implementation notes\n\nThe free groupoid is first defined by symmetrifying the quiver, taking the induced path category\nand finally quotienting by the reducibility relation.\n\n-/\n\n@[expose] public section\n\nopen Set Function\n\nnamespace Quiver\n\nopen CategoryTheory\n\nuniverse u v u' v' u'' v''\n\nvariable {V : Type u} [Quiver.{v} V]\n\n/-- Shorthand for the \"forward\" arrow corresponding to `f` in `paths <| symmetrify V` -/\nabbrev Hom.toPosPath {X Y : V} (f : X ⟶ Y) :\n    (CategoryTheory.Paths.categoryPaths <| Quiver.Symmetrify V).Hom X Y :=\n  f.toPos.toPath\n\n/-- Shorthand for the \"forward\" arrow corresponding to `f` in `paths <| symmetrify V` -/\nabbrev Hom.toNegPath {X Y : V} (f : X ⟶ Y) :\n    (CategoryTheory.Paths.categoryPaths <| Quiver.Symmetrify V).Hom Y X :=\n  f.toNeg.toPath\n\n/-- The \"reduction\" relation -/\ninductive FreeGroupoid.redStep : HomRel (Paths (Quiver.Symmetrify V))\n  | step (X Z : Quiver.Symmetrify V) (f : X ⟶ Z) :\n    redStep (𝟙 ((Paths.of (Quiver.Symmetrify V)).obj X)) (f.toPath ≫ (Quiver.reverse f).toPath)\n\n/-- The underlying vertices of the free groupoid -/\nprotected def FreeGroupoid (V) [Q : Quiver V] :=\n  CategoryTheory.Quotient (@FreeGroupoid.redStep V Q)\n\nnamespace FreeGroupoid\n\nopen Quiver\n\ninstance {V} [Quiver V] [Nonempty V] : Nonempty (Quiver.FreeGroupoid V) := by\n  inhabit V; exact ⟨⟨@default V _⟩⟩\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem congr_reverse {X Y : Paths <| Quiver.Symmetrify V} (p q : X ⟶ Y) :\n    HomRel.CompClosure redStep p q → HomRel.CompClosure redStep p.reverse q.reverse := by\n  rintro ⟨_, _, XW, _, _, WY, _, _, f⟩\n  have : HomRel.CompClosure redStep (WY.reverse ≫ 𝟙 _ ≫ XW.reverse)\n      (WY.reverse ≫ (f.toPath ≫ (Quiver.reverse f).toPath) ≫ XW.reverse) := by\n    constructor\n    constructor\n  simpa only [CategoryStruct.comp, CategoryStruct.id, Quiver.Path.reverse, Quiver.Path.nil_comp,\n    Quiver.Path.reverse_comp, Quiver.reverse_reverse, Quiver.Path.reverse_toPath,\n    Quiver.Path.comp_assoc] using this\n\nopen Relation in\ntheorem congr_comp_reverse {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :\n    Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (p ≫ p.reverse) =\n      Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (𝟙 X) := by\n  apply Quot.eqvGen_sound\n  induction p with\n  | nil => apply EqvGen.refl\n  | cons q f ih =>\n    simp only [Quiver.Path.reverse]\n    fapply EqvGen.trans\n    -- Porting note: dot notation for `Quiver.Path.*` and `Quiver.Hom.*` not working\n    · exact q ≫ Quiver.Path.reverse q\n    · apply EqvGen.symm\n      apply EqvGen.rel\n      have : HomRel.CompClosure redStep (q ≫ 𝟙 _ ≫ Quiver.Path.reverse q)\n          (q ≫ (Quiver.Hom.toPath f ≫ Quiver.Hom.toPath (Quiver.reverse f)) ≫\n            Quiver.Path.reverse q) := by\n        apply HomRel.CompClosure.intro\n        apply redStep.step\n      simp only [Category.assoc, Category.id_comp] at this ⊢\n      -- Porting note: `simp` cannot see how `Quiver.Path.comp_assoc` is relevant, so change to\n      -- category notation\n      change HomRel.CompClosure redStep (q ≫ Quiver.Path.reverse q)\n        (Quiver.Path.cons q f ≫ (Quiver.Hom.toPath (Quiver.reverse f)) ≫ (Quiver.Path.reverse q))\n      simp only [← Category.assoc] at this ⊢\n      exact this\n    · exact ih\n\ntheorem congr_reverse_comp {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :\n    Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (p.reverse ≫ p) =\n      Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (𝟙 Y) := by\n  nth_rw 2 [← Quiver.Path.reverse_reverse p]\n  apply congr_comp_reverse\n\ninstance : Category (Quiver.FreeGroupoid V) :=\n  Quotient.category redStep\n\n/-- The inverse of an arrow in the free groupoid -/\ndef quotInv {X Y : Quiver.FreeGroupoid V} (f : X ⟶ Y) : Y ⟶ X :=\n  Quot.liftOn f (fun pp => Quot.mk _ <| pp.reverse) fun pp qq con =>\n    Quot.sound <| congr_reverse pp qq con\n\ninstance instGroupoid : Groupoid (Quiver.FreeGroupoid V) where\n  inv := quotInv\n  inv_comp p := Quot.inductionOn p fun pp => congr_reverse_comp pp\n  comp_inv p := Quot.inductionOn p fun pp => congr_comp_reverse pp\n\n/-- The inclusion of the quiver on `V` to the underlying quiver on `FreeGroupoid V` -/\ndef of (V) [Quiver V] : V ⥤q Quiver.FreeGroupoid V where\n  obj X := ⟨X⟩\n  map f := Quot.mk _ f.toPosPath\n\ntheorem of_eq :\n    of V = (Quiver.Symmetrify.of ⋙q (Paths.of (Quiver.Symmetrify V))).comp\n      (Quotient.functor <| @redStep V _).toPrefunctor := rfl\n\nsection UniversalProperty\n\nvariable {V' : Type u'} [Groupoid V']\n\n/-- The lift of a prefunctor to a groupoid, to a functor from `FreeGroupoid V` -/\ndef lift (φ : V ⥤q V') : Quiver.FreeGroupoid V ⥤ V' :=\n  CategoryTheory.Quotient.lift _ (Paths.lift <| Quiver.Symmetrify.lift φ) <| by\n    rintro _ _ _ _ ⟨X, Y, f⟩\n    -- Porting note: `simp` does not work, so manually `rewrite`\n    erw [Paths.lift_nil, Paths.lift_cons, Quiver.Path.comp_nil, Paths.lift_toPath,\n      Quiver.Symmetrify.lift_reverse]\n    symm\n    apply Groupoid.comp_inv\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem lift_spec (φ : V ⥤q V') : of V ⋙q (lift φ).toPrefunctor = φ := by\n  rw [of_eq, Prefunctor.comp_assoc, Prefunctor.comp_assoc, Functor.toPrefunctor_comp]\n  dsimp [lift]\n  rw [Quotient.lift_spec, Paths.lift_spec, Quiver.Symmetrify.lift_spec]\n\nTarget:\ntheorem lift_unique (φ : V ⥤q V') (Φ : Quiver.FreeGroupoid V ⥤ V')\n    (hΦ : of V ⋙q Φ.toPrefunctor = φ) : Φ = lift φ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_3e937a169766","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"4dc30413705b6c0ac90a7eb0a2fa52fe2b255819df0dad627d99a6b42f858190","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Groupoid","family_id":"lift_unique","file_id":"mathlib/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean","sample_id":"3e937a169766b398011457da21839b47e8a3e949c8dec6560a9e40da808f1d10"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6ec31665b33411506cda64640e21f6774b047c685e35cff9b518e65219ef78f0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a2717c01644181afb92dd63d96aa9ca9ce454ffe813c7c0184be6728f9e504e2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"96fed0cd3e0fdbf1ac8825b9ea3ac8717dd91342a9fea198dd325c9dbb8823cc","source_sha256":"7d9672c8f14a833f3923b39b4803102d55499036394ed799a1e56e11ac046c23","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine c.isColimit.hom_ext (fun j ↦ ?_)\n  dsimp\n  induction j using SuccOrder.limitRecOn with\n  | isMin j hj =>\n    obtain rfl := hj.eq_bot\n    simpa [← cancel_epi c.isoBot.inv] using h₀\n  | succ j hj hj' =>\n    apply (c.attachCells j hj).hom_ext\n    · simpa using hj'\n    · intro i\n      simpa only [Category.assoc, Cells.ι] using h ({ hj := hj, k := i, .. })\n  | isSuccLimit j hj hj' =>\n    exact (c.F.isColimitOfIsWellOrderContinuous j hj).hom_ext\n      (fun ⟨k, hk⟩ ↦ by simpa using hj' k hk)","hard_negative":true,"metrics":{"chosen_tokens":129,"rejected_tokens":3,"token_jaccard":0.050847,"token_length_ratio":0.023256},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"d947f5e436279310eacea7956ed43d6233726b3a2a0bd95276b6c19f914ad5b1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.RelativeCellComplex.AttachCells\npublic import Mathlib.CategoryTheory.MorphismProperty.TransfiniteComposition\n\nNamespace:\nHomotopicalAlgebra.RelativeCellComplex\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Relative cell complexes\n\nIn this file, we define a structure `RelativeCellComplex` which expresses\nthat a morphism `f : X ⟶ Y` is a transfinite composition of morphisms,\nall of which consist in attaching cells. Here, we allow a different\nfamily of authorized cells at each step. For example, (relative)\nCW-complexes are defined in the file `Mathlib/Topology/CWComplex/Abstract/Basic.lean`\nby requiring that at the `n`th step, we attach `n`-disks along their\nboundaries.\n\nThis structure `RelativeCellComplex` is also used in the\nformalization of the small object argument,\nsee the file `Mathlib/CategoryTheory/SmallObject/IsCardinalForSmallObjectArgument.lean`.\n\n## References\n* https://ncatlab.org/nlab/show/small+object+argument\n\n-/\n\n@[expose] public section\n\nuniverse w w' t v u\n\nopen CategoryTheory\n\nnamespace HomotopicalAlgebra\n\nvariable {C : Type u} [Category.{v} C]\n  {J : Type w'} [LinearOrder J] [OrderBot J] [SuccOrder J] [WellFoundedLT J]\n  {α : J → Type t} {A B : (j : J) → α j → C}\n  (basicCell : (j : J) → (i : α j) → A j i ⟶ B j i) {X Y : C} (f : X ⟶ Y)\n\n/-- Let `J` be a well-ordered type. Assume that for each `j : J`, we\nhave a family `basicCell j` of morphisms. A relative cell complex\nis a morphism `f : X ⟶ Y` which is a transfinite composition of morphisms\nin such a way that at the step `j : J`, we attach cells in the family `basicCell j`. -/\nstructure RelativeCellComplex\n    extends TransfiniteCompositionOfShape J f where\n  /-- If `j` is not the maximum element, `F.obj (Order.succ j)` is obtained\n  from `F.obj j` by attaching cells in the family of morphisms `basicCell j`. -/\n  attachCells (j : J) (hj : ¬ IsMax j) :\n    AttachCells.{w} (basicCell j) (F.map (homOfLE (Order.le_succ j)))\n\nnamespace RelativeCellComplex\n\nvariable {basicCell f} (c : RelativeCellComplex basicCell f)\n\n/-- The index type of cells in a relative cell complex. -/\nstructure Cells where\n  /-- the step where the cell is added -/\n  j : J\n  hj : ¬ IsMax j\n  /-- the index of the cell -/\n  k : (c.attachCells j hj).ι\n\nvariable {c} in\n/-- Given a cell `γ` in a relative cell complex, this is the corresponding\nindex in the family of morphisms `basicCell γ.j`. -/\ndef Cells.i (γ : Cells c) : α γ.j := (c.attachCells γ.j γ.hj).π γ.k\n\nvariable {c} in\n/-- The inclusion of a cell. -/\ndef Cells.ι (γ : Cells c) : B γ.j γ.i ⟶ Y :=\n  (c.attachCells γ.j γ.hj).cell γ.k ≫ c.incl.app (Order.succ γ.j)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma hom_ext {Z : C} {φ₁ φ₂ : Y ⟶ Z} (h₀ : f ≫ φ₁ = f ≫ φ₂)\n    (h : ∀ (γ : Cells c), γ.ι ≫ φ₁ = γ.ι ≫ φ₂) :\n    φ₁ = φ₂ :=\n\nProof body:\n","rejected":"by\n  exact hom_ext","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"f1cd0e069abadb8a5f95105164c9d75a38a4e9e481983e6bbe2893f443685f83","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/RelativeCellComplex","family_id":"hom_ext","file_id":"mathlib/Mathlib/AlgebraicTopology/RelativeCellComplex/Basic.lean","sample_id":"96fed0cd3e0fdbf1ac8825b9ea3ac8717dd91342a9fea198dd325c9dbb8823cc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"da195f4abae7ab0a2c1c536bf1f2974db5c532e362792146fd16a70b703b9aa9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7b6515db8de5c99b6ea544300d4f908ec793c9bb1f46381312ffcb7d47fb0013","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f4c589a7f4818f438b23af657c2db224655a94001f25229b505dee753deb8fe1","source_sha256":"cc48552fd983736c871bbaff8e8b09090c080c9d6073d135463b6637bbb51f7b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨cf, rfl⟩ := Code.exists_code.1 hf\n  obtain ⟨cg, rfl⟩ := Code.exists_code.1 hg\n  have : Nat.Partrec fun n => Nat.rfindOpt fun k => cf.evaln k n <|> cg.evaln k n :=\n    Partrec.nat_iff.1\n      (Partrec.rfindOpt <|\n        Primrec.option_orElse.to_comp.comp\n          (Code.primrec_evaln.to_comp.comp <| (snd.pair (const cf)).pair fst)\n          (Code.primrec_evaln.to_comp.comp <| (snd.pair (const cg)).pair fst))\n  refine ⟨_, this, fun n => ?_⟩\n  have : ∀ x ∈ rfindOpt fun k ↦ Code.evaln k cf n <|> Code.evaln k cg n,\n      x ∈ Code.eval cf n ∨ x ∈ Code.eval cg n := by\n    intro x h\n    obtain ⟨k, e⟩ := Nat.rfindOpt_spec h\n    rw [Option.mem_def, Option.orElse_eq_some, ← Option.mem_def, ← Option.mem_def] at e\n    obtain e | ⟨-, e⟩ := e <;> simp [Code.evaln_sound e]\n  refine ⟨this, fun h ↦ (this _ ⟨h, rfl⟩).imp Exists.fst Exists.fst, fun h ↦ ?_⟩\n  rw [Nat.rfindOpt_dom]\n  simp only [dom_iff_mem, Code.evaln_complete, Option.mem_def] at h\n  obtain ⟨x, k, e⟩ | ⟨x, k, e⟩ := h\n  · exact ⟨k, x, by simp [e]⟩\n  · refine ⟨k, ?_⟩\n    rcases cf.evaln k n with - | y\n    · exact ⟨x, by simp [e]⟩\n    · exact ⟨y, by simp⟩","hard_negative":false,"metrics":{"chosen_tokens":350,"rejected_tokens":355,"token_jaccard":0.950617,"token_length_ratio":1.014286},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"d9858a395ec41d8db259717a9d6ffb63ec259591c7f15029b954ac3127578e25","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Computability.PartrecCode\n\nNamespace:\nNat.Partrec\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Computable and Recursively Enumerable Predicates\n\nThis file defines computable (`ComputablePred`) and recursively enumerable (`REPred`)\npredicates. It also provides basic closure properties and Post's theorem on the\nequivalence of recursive, r.e., and co-r.e. sets.\n-/\n\n@[expose] public section\n\nopen List (Vector)\nopen Encodable Denumerable\n\nnamespace Nat.Partrec\n\nopen Computable Part\n\nTarget:\ntheorem merge' {f g} (hf : Nat.Partrec f) (hg : Nat.Partrec g) :\n    ∃ h, Nat.Partrec h ∧\n      ∀ a, (∀ x ∈ h a, x ∈ f a ∨ x ∈ g a) ∧ ((h a).Dom ↔ (f a).Dom ∨ (g a).Dom) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  obtain ⟨cf, rfl⟩ := Code.exists_code.1 hf\n  obtain ⟨cg, rfl⟩ := Code.exists_code.1 hg\n  have : Nat.Partrec fun n => Nat.rfindOpt fun k => cf.evaln k n <|> cg.evaln k n :=\n    Partrec.nat_iff.1\n      (Partrec.rfindOpt <|\n        Primrec.option_orElse.to_comp.comp\n          (Code.primrec_evaln.to_comp.comp <| (snd.pair (const cf)).pair fst)\n          (Code.primrec_evaln.to_comp.comp <| (snd.pair (const cg)).pair fst))\n  refine ⟨_, this, fun n => ?_⟩\n  have : ∀ x ∈ rfindOpt fun k ↦ Code.evaln k cf n <|> Code.evaln k cg n,\n      x ∈ Code.eval cf n ∨ x ∈ Code.eval cg n := by\n    intro x h\n    obtain ⟨k, e⟩ := Nat.rfindOpt_spec h\n    rw [Option.mem_def, Option.orElse_eq_some, ← Option.mem_def, ← Option.mem_def] at e\n    obtain e | ⟨-, e⟩ := e <;> simp [Code.evaln_sound e]\n  refine ⟨this, fun h ↦ (this _ ⟨h, rfl⟩).imp Exists.fst Exists.fst, fun h ↦ ?_⟩\n  rw [Nat.rfindOpt_dom]\n  simp only [dom_iff_mem, Code.evaln_complete, Option.mem_def] at h\n  obtain ⟨x, k, e⟩ | ⟨x, k, e⟩ := h\n  · exact ⟨k, x, by simp [e]⟩\n  · refine ⟨k, ?_⟩\n    rcases cf.evaln k n with - | y\n    · exact ⟨x, by simp [e]⟩\n    · exact ⟨y, by simp⟩","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability","family_id":"merge","file_id":"mathlib/Mathlib/Computability/RE.lean","sample_id":"f4c589a7f4818f438b23af657c2db224655a94001f25229b505dee753deb8fe1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"507db4142d79d356282c11878756b5338f2f435787c1651165c4326a0009c857","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ec54f6d939d178c00cf294ecb4fb203e2e64557208f1e05332abec7888fc767b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2eaa4c3a7a8979e42c8b78cd304c273792da5d766a00dd97d6cfbc8a6d8171e8","source_sha256":"3b1ad71a44a0ba95fe172fff4adb35c1c3d0ab3dacc0cacca37d6024c51fd12b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases x; rw [Zsqrtd.norm, normSq]; simp","hard_negative":true,"metrics":{"chosen_tokens":14,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.214286},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"da399269a39da501ee63f70f7235812cefcc4cb5c6f055a8e3890b7d093cc3dc","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Archimedean.Real.Basic\npublic import Mathlib.Data.Complex.Basic\npublic import Mathlib.Data.Nat.Prime.Basic\npublic import Mathlib.NumberTheory.Zsqrtd.Basic\n\nNamespace:\nGaussianInt\n\nLocal context:\n/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Gaussian integers\n\nThe Gaussian integers are complex integer, complex numbers whose real and imaginary parts are both\nintegers.\n\n## Main definitions\n\nThe Euclidean domain structure on `ℤ[i]` is defined in this file.\n\nThe homomorphism `GaussianInt.toComplex` into the complex numbers is also defined in this file.\n\n## See also\n\nSee `NumberTheory.Zsqrtd.QuadraticReciprocity` for:\n* `prime_iff_mod_four_eq_three_of_nat_prime`:\n  A prime natural number is prime in `ℤ[i]` if and only if it is `3` mod `4`\n\n## Notation\n\nThis file uses the local notation `ℤ[i]` for `GaussianInt`\n\n## Implementation notes\n\nGaussian integers are implemented using the more general definition `Zsqrtd`, the type of integers\nadjoined a square root of `d`, in this case `-1`. The definition is reducible, so that properties\nand definitions about `Zsqrtd` can easily be used.\n-/\n\n@[expose] public section\n\n\nopen Zsqrtd Complex\n\nopen scoped ComplexConjugate\n\n/-- The Gaussian integers, defined as `ℤ√(-1)`. -/\nabbrev GaussianInt : Type :=\n  Zsqrtd (-1)\n\nlocal notation \"ℤ[i]\" => GaussianInt\n\nnamespace GaussianInt\n\ninstance : Repr ℤ[i] :=\n  ⟨fun x _ => \"⟨\" ++ repr x.re ++ \", \" ++ repr x.im ++ \"⟩\"⟩\n\ninstance instCommRing : CommRing ℤ[i] :=\n  Zsqrtd.commRing\n\nsection\n\nattribute [-instance] Complex.instField -- Avoid making things noncomputable unnecessarily.\n\n/-- The embedding of the Gaussian integers into the complex numbers, as a ring homomorphism. -/\ndef toComplex : ℤ[i] →+* ℂ :=\n  Zsqrtd.lift ⟨I, by simp⟩\n\nend\n\ninstance : Coe ℤ[i] ℂ :=\n  ⟨toComplex⟩\n\ntheorem toComplex_def (x : ℤ[i]) : (x : ℂ) = x.re + x.im * I :=\n  rfl\n\ntheorem toComplex_def' (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ) = x + y * I := by simp [toComplex_def]\n\ntheorem toComplex_def₂ (x : ℤ[i]) : (x : ℂ) = ⟨x.re, x.im⟩ := by\n  apply Complex.ext <;> simp [toComplex_def]\n\n@[simp]\ntheorem intCast_re (x : ℤ[i]) : ((x.re : ℤ) : ℝ) = (x : ℂ).re := by simp [toComplex_def]\n\n@[simp]\ntheorem intCast_im (x : ℤ[i]) : ((x.im : ℤ) : ℝ) = (x : ℂ).im := by simp [toComplex_def]\n\n@[simp]\ntheorem re_toComplex (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).re = x := by simp [toComplex_def]\n\n@[simp]\ntheorem im_toComplex (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).im = y := by simp [toComplex_def]\n\ntheorem toComplex_add (x y : ℤ[i]) : ((x + y : ℤ[i]) : ℂ) = x + y :=\n  toComplex.map_add _ _\n\ntheorem toComplex_mul (x y : ℤ[i]) : ((x * y : ℤ[i]) : ℂ) = x * y :=\n  toComplex.map_mul _ _\n\ntheorem toComplex_one : ((1 : ℤ[i]) : ℂ) = 1 :=\n  toComplex.map_one\n\ntheorem toComplex_zero : ((0 : ℤ[i]) : ℂ) = 0 :=\n  toComplex.map_zero\n\ntheorem toComplex_neg (x : ℤ[i]) : ((-x : ℤ[i]) : ℂ) = -x :=\n  toComplex.map_neg _\n\ntheorem toComplex_sub (x y : ℤ[i]) : ((x - y : ℤ[i]) : ℂ) = x - y :=\n  toComplex.map_sub _ _\n\n@[simp]\ntheorem toComplex_star (x : ℤ[i]) : ((star x : ℤ[i]) : ℂ) = conj (x : ℂ) := by\n  rw [toComplex_def₂, toComplex_def₂]\n  exact congr_arg₂ _ rfl (Int.cast_neg _)\n\n@[simp]\ntheorem toComplex_inj {x y : ℤ[i]} : (x : ℂ) = y ↔ x = y := by\n  cases x; cases y; simp [toComplex_def₂]\n\nlemma toComplex_injective : Function.Injective GaussianInt.toComplex :=\n  fun ⦃_ _⦄ ↦ toComplex_inj.mp\n\n@[simp]\ntheorem toComplex_eq_zero {x : ℤ[i]} : (x : ℂ) = 0 ↔ x = 0 := by\n  rw [← toComplex_zero, toComplex_inj]\n\n@[simp]\ntheorem intCast_real_norm (x : ℤ[i]) : (x.norm : ℝ) = Complex.normSq (x : ℂ) := by\n  rw [Zsqrtd.norm, normSq]; simp\n\n@[simp]\n\nTarget:\ntheorem intCast_complex_norm (x : ℤ[i]) : (x.norm : ℂ) = Complex.normSq (x : ℂ) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_2eaa4c3a7a89","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"8edab4bc7b7599fbfb0f4dbf116b372c1d60f04dd638f87e3843688cd37edf09","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Zsqrtd","family_id":"intcast_complex_norm","file_id":"mathlib/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean","sample_id":"2eaa4c3a7a8979e42c8b78cd304c273792da5d766a00dd97d6cfbc8a6d8171e8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"73fbc60842c9edf00faf984e5736df9245f2ebd275179f441c2133f339aab829","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"46dbe9e477ddbb7606c3f29bdeda37a0f7a2816233719f84ab33136a865943b2","source_sha256":"148d7eba9bc7c4474f48190ef655e8ff7d2748109cb095b07744bfb109f40298","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases H : ∃ s : Finset S, Nonempty (Basis s R S)\n  · rw [trace_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some]\n  · simp [trace_eq_zero_of_not_exists_basis R H, finrank_eq_zero_of_not_exists_basis_finset H]","hard_negative":true,"metrics":{"chosen_tokens":44,"rejected_tokens":8,"token_jaccard":0.064516,"token_length_ratio":0.181818},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"da39f368ee49de12e46d00531694af219415e6aa74eb3875c9915cb7f3c5b8cf","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas\npublic import Mathlib.LinearAlgebra.Matrix.BilinearForm\npublic import Mathlib.LinearAlgebra.Trace\n\nNamespace:\nAlgebra\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Trace for (finite) ring extensions.\n\nSuppose we have an `R`-algebra `S` with a finite basis. For each `s : S`,\nthe trace of the linear map given by multiplying by `s` gives information about\nthe roots of the minimal polynomial of `s` over `R`.\n\n## Main definitions\n\n* `Algebra.trace R S x`: the trace of an element `s` of an `R`-algebra `S`\n* `Algebra.traceForm R S`: bilinear form sending `x`, `y` to the trace of `x * y`\n* `Algebra.traceMatrix R b`: the matrix whose `(i j)`-th element is the trace of `b i * b j`.\n\n## Main results\n\n* `trace_algebraMap_of_basis`, `trace_algebraMap`: if `x : K`, then `Tr_{L/K} x = [L : K] x`\n* `trace_trace_of_basis`, `trace_trace`: `Tr_{L/K} (Tr_{F/L} x) = Tr_{F/K} x`\n\n## Implementation notes\n\nTypically, the trace is defined specifically for finite field extensions.\nThe definition is as general as possible and the assumption that the extension is finite\nis added to the lemmas as needed.\n\nWe only define the trace for left multiplication (`Algebra.leftMulMatrix`,\ni.e. `LinearMap.mulLeft`).\nFor now, the definitions assume `S` is commutative, so the choice doesn't matter anyway.\n\n## References\n\n* https://en.wikipedia.org/wiki/Field_trace\n\n-/\n\n@[expose] public section\n\n\nuniverse w\n\nvariable {R S T : Type*} [CommRing R] [CommRing S] [CommRing T]\nvariable [Algebra R S] [Algebra R T]\nvariable {ι : Type w} [Fintype ι]\n\nopen Module\n\nopen LinearMap (BilinForm)\nopen LinearMap\n\nopen Matrix\n\nopen scoped Matrix\n\nnamespace Algebra\n\nvariable (R S)\n\n/-- The trace of an element `s` of an `R`-algebra is the trace of `(s * ·)`,\nas an `R`-linear map. -/\n@[stacks 0BIF \"Trace\"]\nnoncomputable def trace : S →ₗ[R] R :=\n  (LinearMap.trace R S).comp (lmul R S).toLinearMap\n\nvariable {S}\n\n-- Not a `simp` lemma since there are more interesting ways to rewrite `trace R S x`,\n-- for example `trace_trace`\ntheorem trace_apply (x) : trace R S x = LinearMap.trace R S (lmul R S x) :=\n  rfl\n\ntheorem trace_eq_zero_of_not_exists_basis (h : ¬∃ s : Finset S, Nonempty (Basis s R S)) :\n    trace R S = 0 := by ext s; simp [trace_apply, LinearMap.trace, h]\n\nvariable {R}\n\n-- Can't be a `simp` lemma because it depends on a choice of basis\ntheorem trace_eq_matrix_trace [DecidableEq ι] (b : Basis ι R S) (s : S) :\n    trace R S s = Matrix.trace (Algebra.leftMulMatrix b s) := by\n  rw [trace_apply, LinearMap.trace_eq_matrix_trace _ b, ← toMatrix_lmul_eq]; rfl\n\n/-- If `x` is in the base field `K`, then the trace is `[L : K] * x`. -/\ntheorem trace_algebraMap_of_basis (b : Basis ι R S) (x : R) :\n    trace R S (algebraMap R S x) = Fintype.card ι • x := by\n  haveI := Classical.decEq ι\n  rw [trace_apply, LinearMap.trace_eq_matrix_trace R b, Matrix.trace]\n  convert! Finset.sum_const x\n  simp [-coe_lmul_eq_mul]\n\n\n/-- The trace map from `R` to itself is the identity map. -/\n@[simp] theorem trace_self : trace R R = LinearMap.id := by\n  ext; simpa using trace_algebraMap_of_basis (.singleton (Fin 1) R) 1\n\ntheorem trace_self_apply (a) : trace R R a = a := by simp\n\n/-- If `x` is in the base field `K`, then the trace is `[L : K] * x`.\n\n(If `L` is not finite-dimensional over `K`, then `trace` and `finrank` return `0`.)\n-/\n@[simp]\n\nTarget:\ntheorem trace_algebraMap [StrongRankCondition R] [Module.Free R S] (x : R) :\n    trace R S (algebraMap R S x) = finrank R S • x :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b10e727d1b894d0dc5840f098008a9a689fb8692fea504e91c338123141c6a0a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Trace","family_id":"trace_algebramap","file_id":"mathlib/Mathlib/RingTheory/Trace/Defs.lean","sample_id":"46dbe9e477ddbb7606c3f29bdeda37a0f7a2816233719f84ab33136a865943b2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7727f8d1a6315f3170b243f902f0e2ff87bac72c93a1e273349cdbafdd693264","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ca2fb92b9a27f6c7dccff0a61d2e985ccf8e34e6dd61d51c2352528f0acd6ec2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1205eb9e1bd541c4f97a6b6d682d3d9f05b7c621a6ffc31f4b5fc185468fec86","source_sha256":"ae512c5fcdaa1f75b5ee08453eb6bee821b508070e712fcb1cd05ec73381f2d7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [homEquiv]\n  ext j\n  simp [← uncurry_natural_left]","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.25},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"da52b4cdaad8444c0bdc25def01d49962168f4ad613971ad6a49b091f0376dc9","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Monoidal.Closed.Enrichment\npublic import Mathlib.CategoryTheory.Enriched.FunctorCategory\n\nNamespace:\nCategoryTheory.MonoidalClosed.FunctorCategory\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Functor categories are monoidal closed\n\nLet `C` be a monoidal closed category. Let `J` be a category. In this file,\nwe obtain that the category `J ⥤ C` is monoidal closed if `C` has suitable\nlimits.\n\n-/\n\n@[expose] public section\n\nuniverse v₁ v₂ u₁ u₂\n\nnamespace CategoryTheory\n\nopen Category Limits MonoidalCategory\n\nnamespace MonoidalClosed\n\nnamespace FunctorCategory\n\nopen Enriched.FunctorCategory\n\nvariable {C : Type u₁} [Category.{v₁} C] [MonoidalCategory C] [MonoidalClosed C]\n  {J : Type u₂} [Category.{v₂} J]\n  [∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂]\n\nattribute [local simp] enrichedCategorySelf_hom\n\nsection\n\nvariable {F₁ F₂ F₂' F₃ F₃' : J ⥤ C}\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The bijection `(F₁ ⊗ F₂ ⟶ F₃) ≃ (F₂ ⟶ functorEnrichedHom C F₁ F₃)` when `F₁`, `F₂`\nand `F₃` are functors `J ⥤ C`, and `C` is monoidal closed. -/\nnoncomputable def homEquiv : (F₁ ⊗ F₂ ⟶ F₃) ≃ (F₂ ⟶ functorEnrichedHom C F₁ F₃) where\n  toFun f :=\n    { app j := end_.lift (fun k ↦ F₂.map k.hom ≫ curry (f.app k.right))\n        (fun k₁ k₂ φ ↦ by\n          dsimp\n          simp only [enrichedOrdinaryCategorySelf_eHomWhiskerLeft, Category.assoc,\n            enrichedOrdinaryCategorySelf_eHomWhiskerRight]\n          rw [← curry_natural_left_assoc, ← curry_natural_left_assoc,\n            ← curry_natural_right, curry_pre_app, Category.assoc,\n            ← f.naturality φ.right, Monoidal.tensorObj_map, tensorHom_def_assoc,\n            ← Under.w φ, Functor.map_comp, MonoidalCategory.whiskerLeft_comp_assoc,\n            whisker_exchange_assoc]) }\n  invFun g :=\n    { app j := uncurry (g.app j ≫ enrichedHomπ C _ _ (Under.mk (𝟙 j)))\n      naturality j j' φ := by\n        dsimp\n        rw [← uncurry_natural_right, tensorHom_def'_assoc, ← uncurry_pre_app,\n          ← uncurry_natural_left, Category.assoc, Category.assoc,\n          NatTrans.naturality_assoc, functorEnrichedHom_map,\n          end_.lift_π_assoc, enrichedOrdinaryCategorySelf_eHomWhiskerRight]\n        dsimp\n        rw [pre_id, NatTrans.id_app, enrichedOrdinaryCategorySelf_eHomWhiskerLeft,\n          Functor.map_id, Category.comp_id, Category.comp_id]\n        congr 2\n        rw [← enrichedOrdinaryCategorySelf_eHomWhiskerRight,\n          ← enrichedOrdinaryCategorySelf_eHomWhiskerLeft]\n        let α : Under.mk (𝟙 j) ⟶ (Under.map φ).obj (Under.mk (𝟙 j')) := Under.homMk φ\n        exact (enrichedHom_condition C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃) α).symm }\n  left_inv f := by cat_disch\n  right_inv g := by\n    ext j\n    dsimp\n    ext k\n    -- this following list was obtained by\n    -- `simp? [enrichedOrdinaryCategorySelf_eHomWhiskerLeft, Under.map, Comma.mapLeft]`\n    simp only [diagram_obj_obj, Functor.comp_obj, Under.forget_obj, enrichedCategorySelf_hom,\n      curry_uncurry, NatTrans.naturality_assoc, functorEnrichedHom_obj, functorEnrichedHom_map,\n      Under.map, Comma.mapLeft, Functor.const_obj_obj, Functor.id_obj, Discrete.natTrans_app,\n      StructuredArrow.left_eq_id, end_.lift_π, Under.mk_right, Under.mk_hom, Iso.refl_inv,\n      NatTrans.id_app, enrichedOrdinaryCategorySelf_eHomWhiskerRight, pre_id, Iso.refl_hom,\n      enrichedOrdinaryCategorySelf_eHomWhiskerLeft, Functor.map_id, Category.comp_id]\n    congr\n    simp\n\nTarget:\nlemma homEquiv_naturality_two_symm (f₂ : F₂ ⟶ F₂') (g : F₂' ⟶ functorEnrichedHom C F₁ F₃) :\n    homEquiv.symm (f₂ ≫ g) = F₁ ◁ f₂ ≫ homEquiv.symm g :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_1205eb9e1bd5","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"b72b318e26b4e21a4fa5bf047eb1636e011f725c15ff6bdb5bfd29d95c349169","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Monoidal","family_id":"homequiv_naturality_two_symm","file_id":"mathlib/Mathlib/CategoryTheory/Monoidal/Closed/FunctorCategory/Basic.lean","sample_id":"1205eb9e1bd541c4f97a6b6d682d3d9f05b7c621a6ffc31f4b5fc185468fec86"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9342ddf1a3670b64a91f411236ca705a4f712a410fef1d38cc662b4389a061cb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"83c5bc19a8475efdecefba11a168fc6bb576240b8897c80b4ded959f3ff7248c","source_sha256":"42f43c83c3b42913128aa30933a73363c485cf2bc69fc613214486004b504ca0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← image_commutatorSet_closureCommutatorRepresentatives G]\n  exact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _))","hard_negative":false,"metrics":{"chosen_tokens":24,"rejected_tokens":3,"token_jaccard":0.105263,"token_length_ratio":0.125},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"daa470165dbee0295d2c5556c9eb3b972c9bb4964873bde1292449fc02bf7054","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Subgroup.Finite\npublic import Mathlib.GroupTheory.Commutator.Basic\npublic import Mathlib.GroupTheory.Rank\npublic import Mathlib.GroupTheory.Index\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jordan Brown, Thomas Browning, Patrick Lutz\n-/\n/-!\nThe commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n\npublic section\n\nvariable {G : Type*} [Group G]\n\nnamespace Subgroup\n\n/-- The commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n@[to_additive /-- The commutator of a finite direct product is contained in the direct product of\nthe commutators. -/]\ntheorem commutator_pi_pi_of_finite {η : Type*} [Finite η] {Gs : η → Type*} [∀ i, Group (Gs i)]\n    (H K : ∀ i, Subgroup (Gs i)) : ⁅Subgroup.pi Set.univ H, Subgroup.pi Set.univ K⁆ =\n    Subgroup.pi Set.univ fun i => ⁅H i, K i⁆ := by\n  classical\n    apply le_antisymm (commutator_pi_pi_le H K)\n    rw [pi_le_iff]\n    intro i hi\n    rw [map_commutator]\n    apply commutator_mono <;>\n      · rw [le_pi_iff]\n        intro j _hj\n        rintro _ ⟨x, hx, rfl⟩\n        by_cases h : j = i\n        · subst h\n          simpa using hx\n        · simp [h, one_mem]\n\nvariable [Finite (commutatorSet G)]\n\ninstance : Group.FG (_root_.commutator G) := by\n  rw [commutator_eq_closure]; apply Group.closure_finite_fg\n\nvariable (G) in\nlemma rank_commutator_le_card : Group.rank (_root_.commutator G) ≤ Nat.card (commutatorSet G) := by\n  rw [Subgroup.rank_congr (commutator_eq_closure G)]\n  apply Subgroup.rank_closure_finite_le_nat_card\n\nvariable [Group.FG G]\n\ninstance finiteIndex_center : FiniteIndex (center G) := by\n  obtain ⟨S, -, hS⟩ := Group.rank_spec G\n  exact ⟨mt (Finite.card_eq_zero_of_embedding (quotientCenterEmbedding hS)) Finite.card_pos.ne'⟩\n\nvariable (G) in\nlemma index_center_le_pow : (center G).index ≤ Nat.card (commutatorSet G) ^ Group.rank G := by\n  obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G\n  rw [← hS1, ← Fintype.card_coe, ← Nat.card_eq_fintype_card, ← Finset.coe_sort_coe, ← Nat.card_fun]\n  exact Finite.card_le_of_embedding (quotientCenterEmbedding hS2)\n\nend Subgroup\n\nsection commutatorRepresentatives\n\nopen Subgroup\n\nTarget:\nlemma card_commutatorSet_closureCommutatorRepresentatives :\n    Nat.card (commutatorSet (closureCommutatorRepresentatives G)) = Nat.card (commutatorSet G) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Commutator","family_id":"card_commutatorset_closurecommutatorrepresentatives","file_id":"mathlib/Mathlib/GroupTheory/Commutator/Finite.lean","sample_id":"83c5bc19a8475efdecefba11a168fc6bb576240b8897c80b4ded959f3ff7248c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"50b6a5e16d3844921370e0938982952c322ae0fcb0f107c81a94eedb5977aaa4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"70a4170ae953a085b2ee61b70a1cefea9bde7f6d81dae6658409edfa420b8c2e","source_sha256":"d0f59f260198c8e37d3bb0bd17106b617a04bc52ddfffa04abdcd1e14bbcbdf5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← lintegral_sum_measure, h.sum_restrict_of_ac hν]","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":8,"token_jaccard":0.058824,"token_length_ratio":0.666667},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"dac8859800f1833a7520afb0d9b70478e26970cb634411885de31c2d1dfd65f5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Group.Action\npublic import Mathlib.MeasureTheory.Group.Pointwise\npublic import Mathlib.MeasureTheory.Integral.Lebesgue.Map\npublic import Mathlib.MeasureTheory.Integral.Bochner.Set\n\nNamespace:\nMeasureTheory.IsFundamentalDomain\n\nLocal context:\n/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth\n-/\n/-!\n# Fundamental domain of a group action\n\nA set `s` is said to be a *fundamental domain* of an action of a group `G` on a measurable space `α`\nwith respect to a measure `μ` if\n\n* `s` is a measurable set;\n\n* the sets `g • s` over all `g : G` cover almost all points of the whole space;\n\n* the sets `g • s`, are pairwise a.e. disjoint, i.e., `μ (g₁ • s ∩ g₂ • s) = 0` whenever `g₁ ≠ g₂`;\n  we require this for `g₂ = 1` in the definition, then deduce it for any two `g₁ ≠ g₂`.\n\nIn this file we prove that in case of a countable group `G` and a measure-preserving action, any two\nfundamental domains have the same measure, and for a `G`-invariant function, its integrals over any\ntwo fundamental domains are equal to each other.\n\nWe also generate additive versions of all theorems in this file using the `to_additive` attribute.\n\n* We define the `HasFundamentalDomain` typeclass, in particular to be able to define the `covolume`\n  of a quotient of `α` by a group `G`, which under reasonable conditions does not depend on the\n  choice of fundamental domain.\n\n* We define the `QuotientMeasureEqMeasurePreimage` typeclass to describe a situation in which a\n  measure `μ` on `α ⧸ G` can be computed by taking a measure `ν` on `α` of the intersection of the\n  pullback with a fundamental domain.\n\n## Main declarations\n\n* `MeasureTheory.IsFundamentalDomain`: Predicate for a set to be a fundamental domain of the\n  action of a group\n* `MeasureTheory.fundamentalFrontier`: Fundamental frontier of a set under the action of a group.\n  Elements of `s` that belong to some other translate of `s`.\n* `MeasureTheory.fundamentalInterior`: Fundamental interior of a set under the action of a group.\n  Elements of `s` that do not belong to any other translate of `s`.\n-/\n\n@[expose] public section\n\n\nopen scoped ENNReal Pointwise Topology NNReal ENNReal MeasureTheory\n\nopen MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace Filter\n\nnamespace MeasureTheory\n\n/-- A measurable set `s` is a *fundamental domain* for an additive action of an additive group `G`\non a measurable space `α` with respect to a measure `μ` if the sets `g +ᵥ s`, `g : G`, are pairwise\na.e. disjoint and cover the whole space. -/\nstructure IsAddFundamentalDomain (G : Type*) {α : Type*} [Zero G] [VAdd G α] [MeasurableSpace α]\n    (s : Set α) (μ : Measure α := by volume_tac) : Prop where\n  protected nullMeasurableSet : NullMeasurableSet s μ\n  protected ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g +ᵥ x ∈ s\n  protected aedisjoint : Pairwise <| (AEDisjoint μ on fun g : G => g +ᵥ s)\n\n/-- A measurable set `s` is a *fundamental domain* for an action of a group `G` on a measurable\nspace `α` with respect to a measure `μ` if the sets `g • s`, `g : G`, are pairwise a.e. disjoint and\ncover the whole space. -/\n@[to_additive IsAddFundamentalDomain]\nstructure IsFundamentalDomain (G : Type*) {α : Type*} [One G] [SMul G α] [MeasurableSpace α]\n    (s : Set α) (μ : Measure α := by volume_tac) : Prop where\n  protected nullMeasurableSet : NullMeasurableSet s μ\n  protected ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g • x ∈ s\n  protected aedisjoint : Pairwise <| (AEDisjoint μ on fun g : G => g • s)\n\nvariable {G H α β E : Type*}\n\nnamespace IsFundamentalDomain\n\nvariable [Group G] [Group H] [MulAction G α] [MeasurableSpace α] [MulAction H β] [MeasurableSpace β]\n  [NormedAddCommGroup E] {s t : Set α} {μ : Measure α}\n\n/-- If for each `x : α`, exactly one of `g • x`, `g : G`, belongs to a measurable set `s`, then `s`\nis a fundamental domain for the action of `G` on `α`. -/\n@[to_additive /-- If for each `x : α`, exactly one of `g +ᵥ x`, `g : G`, belongs to a measurable set\n`s`, then `s` is a fundamental domain for the additive action of `G` on `α`. -/]\ntheorem mk' (h_meas : NullMeasurableSet s μ) (h_exists : ∀ x : α, ∃! g : G, g • x ∈ s) :\n    IsFundamentalDomain G s μ where\n  nullMeasurableSet := h_meas\n  ae_covers := Eventually.of_forall fun x => (h_exists x).exists\n  aedisjoint a b hab := Disjoint.aedisjoint <| disjoint_left.2 fun x hxa hxb => by\n    rw [mem_smul_set_iff_inv_smul_mem] at hxa hxb\n    exact hab (inv_injective <| (h_exists x).unique hxa hxb)\n\n/-- For `s` to be a fundamental domain, it's enough to check\n`MeasureTheory.AEDisjoint (g • s) s` for `g ≠ 1`. -/\n@[to_additive /-- For `s` to be a fundamental domain, it's enough to check\n  `MeasureTheory.AEDisjoint (g +ᵥ s) s` for `g ≠ 0`. -/]\ntheorem mk'' (h_meas : NullMeasurableSet s μ) (h_ae_covers : ∀ᵐ x ∂μ, ∃ g : G, g • x ∈ s)\n    (h_ae_disjoint : ∀ g, g ≠ (1 : G) → AEDisjoint μ (g • s) s)\n    (h_qmp : ∀ g : G, QuasiMeasurePreserving ((g • ·) : α → α) μ μ) :\n    IsFundamentalDomain G s μ where\n  nullMeasurableSet := h_meas\n  ae_covers := h_ae_covers\n  aedisjoint := pairwise_aedisjoint_of_aedisjoint_forall_ne_one h_ae_disjoint h_qmp\n\n/-- If a measurable space has a finite measure `μ` and a countable group `G` acts\nquasi-measure-preservingly, then to show that a set `s` is a fundamental domain, it is sufficient\nto check that its translates `g • s` are (almost) disjoint and that the sum `∑' g, μ (g • s)` is\nsufficiently large. -/\n@[to_additive\n  /-- If a measurable space has a finite measure `μ` and a countable additive group `G` acts\n  quasi-measure-preservingly, then to show that a set `s` is a fundamental domain, it is sufficient\n  to check that its translates `g +ᵥ s` are (almost) disjoint and that the sum `∑' g, μ (g +ᵥ s)` is\n  sufficiently large. -/]\ntheorem mk_of_measure_univ_le [IsFiniteMeasure μ] [Countable G] (h_meas : NullMeasurableSet s μ)\n    (h_ae_disjoint : ∀ g ≠ (1 : G), AEDisjoint μ (g • s) s)\n    (h_qmp : ∀ g : G, QuasiMeasurePreserving (g • · : α → α) μ μ)\n    (h_measure_univ_le : μ (univ : Set α) ≤ ∑' g : G, μ (g • s)) : IsFundamentalDomain G s μ :=\n  have aedisjoint : Pairwise (AEDisjoint μ on fun g : G => g • s) :=\n    pairwise_aedisjoint_of_aedisjoint_forall_ne_one h_ae_disjoint h_qmp\n  { nullMeasurableSet := h_meas\n    aedisjoint\n    ae_covers := by\n      replace h_meas : ∀ g : G, NullMeasurableSet (g • s) μ := fun g => by\n        rw [← inv_inv g, ← preimage_smul]; exact h_meas.preimage (h_qmp g⁻¹)\n      have h_meas' : NullMeasurableSet {a | ∃ g : G, g • a ∈ s} μ := by\n        rw [← iUnion_smul_eq_setOf_exists]; exact .iUnion h_meas\n      rw [ae_iff_measure_eq h_meas', ← iUnion_smul_eq_setOf_exists]\n      refine le_antisymm (measure_mono <| subset_univ _) ?_\n      rw [measure_iUnion₀ aedisjoint h_meas]\n      exact h_measure_univ_le }\n\n@[to_additive]\ntheorem iUnion_smul_ae_eq (h : IsFundamentalDomain G s μ) : ⋃ g : G, g • s =ᵐ[μ] univ :=\n  eventuallyEq_univ.2 <| h.ae_covers.mono fun _ ⟨g, hg⟩ =>\n    mem_iUnion.2 ⟨g⁻¹, _, hg, inv_smul_smul _ _⟩\n\n@[to_additive]\ntheorem measure_ne_zero [Countable G] [SMulInvariantMeasure G α μ]\n    (hμ : μ ≠ 0) (h : IsFundamentalDomain G s μ) : μ s ≠ 0 := by\n  have hc := measure_univ_pos.mpr hμ\n  contrapose! hc\n  rw [← measure_congr h.iUnion_smul_ae_eq]\n  refine le_trans (measure_iUnion_le _) ?_\n  simp_rw [measure_smul, hc, tsum_zero, le_refl]\n\n@[to_additive]\ntheorem mono (h : IsFundamentalDomain G s μ) {ν : Measure α} (hle : ν ≪ μ) :\n    IsFundamentalDomain G s ν :=\n  ⟨h.1.mono_ac hle, hle h.2, h.aedisjoint.mono fun _ _ h => hle h⟩\n\n@[to_additive]\ntheorem preimage_of_equiv {ν : Measure β} (h : IsFundamentalDomain G s μ) {f : β → α}\n    (hf : QuasiMeasurePreserving f ν μ) {e : G → H} (he : Bijective e)\n    (hef : ∀ g, Semiconj f (e g • ·) (g • ·)) : IsFundamentalDomain H (f ⁻¹' s) ν where\n  nullMeasurableSet := h.nullMeasurableSet.preimage hf\n  ae_covers := (hf.ae h.ae_covers).mono fun x ⟨g, hg⟩ => ⟨e g, by rwa [mem_preimage, hef g x]⟩\n  aedisjoint a b hab := by\n    lift e to G ≃ H using he\n    have : (e.symm a⁻¹)⁻¹ ≠ (e.symm b⁻¹)⁻¹ := by simp [hab]\n    have := (h.aedisjoint this).preimage hf\n    simp only [Semiconj] at hef\n    simpa only [onFun, ← preimage_smul_inv, preimage_preimage, ← hef, e.apply_symm_apply, inv_inv]\n      using this\n\n@[to_additive]\ntheorem image_of_equiv {ν : Measure β} (h : IsFundamentalDomain G s μ) (f : α ≃ β)\n    (hf : QuasiMeasurePreserving f.symm ν μ) (e : H ≃ G)\n    (hef : ∀ g, Semiconj f (e g • ·) (g • ·)) : IsFundamentalDomain H (f '' s) ν := by\n  rw [f.image_eq_preimage_symm]\n  refine h.preimage_of_equiv hf e.symm.bijective fun g x => ?_\n  rcases f.surjective x with ⟨x, rfl⟩\n  rw [← hef _ _, f.symm_apply_apply, f.symm_apply_apply, e.apply_symm_apply]\n\n@[to_additive]\ntheorem pairwise_aedisjoint_of_ac {ν} (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) :\n    Pairwise fun g₁ g₂ : G => AEDisjoint ν (g₁ • s) (g₂ • s) :=\n  h.aedisjoint.mono fun _ _ H => hν H\n\n@[to_additive]\ntheorem smul_of_comm {G' : Type*} [Group G'] [MulAction G' α]\n    [MeasurableConstSMul G' α] [SMulInvariantMeasure G' α μ] [SMulCommClass G' G α]\n    (h : IsFundamentalDomain G s μ) (g : G') : IsFundamentalDomain G (g • s) μ :=\n  h.image_of_equiv (MulAction.toPerm g) (measurePreserving_smul _ _).quasiMeasurePreserving\n    (Equiv.refl _) <| smul_comm g\n\nvariable [MeasurableConstSMul G α] [SMulInvariantMeasure G α μ]\n\n@[to_additive]\ntheorem nullMeasurableSet_smul (h : IsFundamentalDomain G s μ) (g : G) :\n    NullMeasurableSet (g • s) μ :=\n  h.nullMeasurableSet.smul g\n\n@[to_additive]\ntheorem restrict_restrict (h : IsFundamentalDomain G s μ) (g : G) (t : Set α) :\n    (μ.restrict t).restrict (g • s) = μ.restrict (g • s ∩ t) :=\n  restrict_restrict₀ ((h.nullMeasurableSet_smul g).mono restrict_le_self)\n\n@[to_additive]\ntheorem smul (h : IsFundamentalDomain G s μ) (g : G) : IsFundamentalDomain G (g • s) μ :=\n  h.image_of_equiv (MulAction.toPerm g) (measurePreserving_smul _ _).quasiMeasurePreserving\n    ⟨fun g' => g⁻¹ * g' * g, fun g' => g * g' * g⁻¹, fun g' => by simp [mul_assoc], fun g' => by\n      simp [mul_assoc]⟩\n    fun g' x => by simp [smul_smul, mul_assoc]\n\nvariable [Countable G] {ν : Measure α}\n\n@[to_additive]\ntheorem sum_restrict_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) :\n    (sum fun g : G => ν.restrict (g • s)) = ν := by\n  rw [← restrict_iUnion_ae (h.aedisjoint.mono fun i j h => hν h) fun g =>\n      (h.nullMeasurableSet_smul g).mono_ac hν,\n    restrict_congr_set (hν h.iUnion_smul_ae_eq), restrict_univ]\n\n@[to_additive]\n\nTarget:\ntheorem lintegral_eq_tsum_of_ac (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) (f : α → ℝ≥0∞) :\n    ∫⁻ x, f x ∂ν = ∑' g : G, ∫⁻ x in g • s, f x ∂ν :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e4d8209d607dde102bdf2fc0b5f89a5283191c9a08be3c51adfeeb599f60b0f3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Group","family_id":"lintegral_eq_tsum_of_ac","file_id":"mathlib/Mathlib/MeasureTheory/Group/FundamentalDomain.lean","sample_id":"70a4170ae953a085b2ee61b70a1cefea9bde7f6d81dae6658409edfa420b8c2e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3fa4e6a02899a81962f514978e80ca0d79529b0b1a773bcbf2dd1c22e7f2b19e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4d0c2d672852344cf952109d7ab97db7597c3d965c93d98077539794d96bcb44","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6402417a187a534989eab3a2c52dac736091b114295a76d321088d940f0ef62a","source_sha256":"d30cbd6919379ea815bed52dd0199b189327488f1457353c63e7bcff05fa2d66","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Order.covBy_iff_add_one_eq, Order.covBy_iff_add_one_eq]\n  exact Int.natCast_inj","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":20,"token_jaccard":0.6875,"token_length_ratio":1.333333},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"daf82d8b75c71f3e108cb7c038daf2ca7b6db7b7c8b85eb7d14c7b84b7f5bcba","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Ring.Int\npublic import Mathlib.Data.Nat.SuccPred\n\nNamespace:\nInt\n\nLocal context:\n/-\nCopyright (c) 2021 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Successors and predecessors of integers\n\nIn this file, we show that `ℤ` is both an archimedean `SuccOrder` and an archimedean `PredOrder`.\n-/\n\npublic section\n\n\nopen Function Order\n\nnamespace Int\n\ninstance instSuccOrder : SuccOrder ℤ :=\n  { SuccOrder.ofSuccLeIff succ fun {_ _} => Iff.rfl with succ := succ }\n\ninstance instSuccAddOrder : SuccAddOrder ℤ := ⟨fun _ => rfl⟩\n\ninstance instPredOrder : PredOrder ℤ where\n  pred := pred\n  pred_le _ := (sub_one_lt_of_le le_rfl).le\n  min_of_le_pred ha := ((sub_one_lt_of_le le_rfl).not_ge ha).elim\n  le_pred_of_lt {_ _} := le_sub_one_of_lt\n\ninstance instPredSubOrder : PredSubOrder ℤ := ⟨fun _ => rfl⟩\n\n@[simp]\ntheorem succ_eq_succ : Order.succ = succ :=\n  rfl\n\n@[simp]\ntheorem pred_eq_pred : Order.pred = pred :=\n  rfl\n\ninstance : IsSuccArchimedean ℤ :=\n  ⟨fun {a b} h =>\n    ⟨(b - a).toNat, by rw [succ_iterate, toNat_sub_of_le h, ← add_sub_assoc, add_sub_cancel_left]⟩⟩\n\ninstance : IsPredArchimedean ℤ :=\n  ⟨fun {a b} h =>\n    ⟨(b - a).toNat, by rw [pred_iterate, toNat_sub_of_le h, sub_sub_cancel]⟩⟩\n\n/-! ### Covering relation -/\n\n\n@[simp, norm_cast]\n\nTarget:\ntheorem natCast_covBy {a b : ℕ} : (a : ℤ) ⋖ b ↔ a ⋖ b :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  rw [Order.covBy_iff_add_one_eq, Order.covBy_iff_add_one_eq]\n  exact Int.natCast_inj","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Int","family_id":"natcast_covby","file_id":"mathlib/Mathlib/Data/Int/SuccPred.lean","sample_id":"6402417a187a534989eab3a2c52dac736091b114295a76d321088d940f0ef62a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8f3a85b33714251a1a29f92568064949152f5e84c72c953dd09287cb0b6f1455","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"309747378fad33151e6310726ff0174aa0d05877efaa42bd524d8d5cf49ca2f9","source_sha256":"d9780449d2613fd42914cf0df8a7f4380debc01adc843be01bb83642da361938","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [NormNum.IsNat.to_eq h rfl]\n  exact Nat.cast_nonneg' n","hard_negative":true,"metrics":{"chosen_tokens":16,"rejected_tokens":8,"token_jaccard":0.05,"token_length_ratio":0.5},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"db11a052999061de3591ca82b17722b270bf5ede64bef6b5f18ad58356889b2a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Invertible\npublic import Mathlib.Algebra.Order.Ring.Cast\npublic import Mathlib.Tactic.HaveI\npublic import Mathlib.Tactic.NormNum.Core\n\nNamespace:\nMathlib.Meta.Positivity\n\nLocal context:\n/-\nCopyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Heather Macbeth, Yaël Dillies\n-/\npublic meta import Mathlib.Control.Basic\n\n/-!\n## `positivity` core functionality\n\nThis file sets up the `positivity` tactic and the `@[positivity]` attribute,\nwhich allow for plugging in new positivity functionality around a positivity-based driver.\nThe actual behavior is in `@[positivity]`-tagged definitions in `Tactic.Positivity.Basic`\nand elsewhere.\n-/\n\npublic meta section\n\nopen Lean\nopen Lean.Meta Qq Lean.Elab Term\n\n/-- A definition of type `PositivityExt` tagged `@[positivity t]` extends the `positivity` tactic.\nThe term (with underscores) `t` indicates which expressions this extension accepts.\nAn extension will be given an expression `e : α`, together with hypotheses\n`[Zero α] [PartialOrder α]` and attempts to prove `e > 0`, `e ≥ 0`, or `e ≠ 0`.\n\nWhen `Positivity.core` calls this extension on an expression `e`, it does not guarantee that `e`\nmatches `t` perfectly: validate the form of the expression (using e.g.\n`match_expr (← withReducible (whnf e))`) before building a proof. See also the\n`let .app ... ← withReducible (whnf e) | throwError ...` lines in the example below.\n\nAn extension can call `Mathlib.Meta.Positivity.core` to recursively solve subgoals.\n\nExample:\n```lean\n@[positivity ite _ _ _] def evalIte : PositivityExt where eval {u α} zα pα e := do\n  let .app (.app (.app (.app f (p : Q(Prop))) (_ : Q(Decidable $p))) (a : Q($α))) (b : Q($α))\n    ← withReducible (whnf e) | throwError \"not ite\"\n  haveI' : $e =Q ite $p $a $b := ⟨⟩\n  guard <| ← withDefault <| withNewMCtxDepth <| isDefEq f q(ite (α := $α))\n  let ra ← core zα pα a; let rb ← core zα pα b\n  ...\n```\n-/\nsyntax (name := positivity) \"positivity \" term,+ : attr\n\nlemma ne_of_ne_of_eq' {α : Sort*} {a c b : α} (hab : (a : α) ≠ c) (hbc : a = b) : b ≠ c := hbc ▸ hab\n\nnamespace Mathlib.Meta.Positivity\n\nvariable {u : Level} {α : Q(Type u)} (zα : Q(Zero $α))\n\n/-- The result of `positivity` running on an expression `e` of type `α`. -/\ninductive Strictness (e : Q($α)) : Option Q(PartialOrder $α) → Type where\n  | positive {pα : Q(PartialOrder $α)} (pf : Q(0 < $e)) : Strictness e pα\n  | nonnegative {pα : Q(PartialOrder $α)} (pf : Q(0 ≤ $e)) : Strictness e pα\n  | nonzero {pα?} (pf : Q($e ≠ 0)) : Strictness e pα?\n  | none {pα?} : Strictness e pα?\n\n/-- Gives a generic description of the `positivity` result. -/\ndef Strictness.toString {e pα?} : Strictness zα e pα? → String\n  | positive _ => \"positive\"\n  | nonnegative _ => \"nonnegative\"\n  | nonzero _ => \"nonzero\"\n  | none => \"none\"\n\n/-- Extract a proof that `e` is positive, if possible, from `Strictness` information about `e`. -/\ndef Strictness.toPositive {e pα} : Strictness zα e (some pα) → Option Q(0 < $e)\n  | .positive pf => some pf\n  | _ => .none\n\n/-- Extract a proof that `e` is nonnegative, if possible, from `Strictness` information about `e`.\n-/\ndef Strictness.toNonneg {e pα} : Strictness zα e (some pα) → Option Q(0 ≤ $e)\n  | .positive pf => some q(le_of_lt $pf)\n  | .nonnegative pf => some pf\n  | _ => .none\n\n/-- Extract a proof that `e` is nonzero, if possible, from `Strictness` information about `e`. -/\ndef Strictness.toNonzero {e pα?} : Strictness zα e pα? → Option Q($e ≠ 0)\n  | .positive pf => some q(ne_of_gt $pf)\n  | .nonzero pf => some pf\n  | _ => .none\n\n/-- An extension for `positivity`. -/\nstructure PositivityExt where\n  /-- Attempts to prove an expression `e : α` is `>0`, `≥0`, or `≠0`. -/\n  eval {u : Level} {α : Q(Type u)} (zα : Q(Zero $α)) (pα? : Option Q(PartialOrder $α)) (e : Q($α)) :\n    MetaM (Strictness zα e pα?)\n\n/-- Read a `positivity` extension from a declaration of the right type. -/\ndef mkPositivityExt (n : Name) : ImportM PositivityExt := do\n  let { env, opts, .. } ← read\n  IO.ofExcept <| unsafe env.evalConstCheck PositivityExt opts ``PositivityExt n\n\n/-- Each `positivity` extension is labelled with a collection of patterns\nwhich determine the expressions to which it should be applied. -/\nabbrev Entry := Array (Array DiscrTree.Key) × Name\n\n/-- Environment extensions for `positivity` declarations -/\ninitialize positivityExt : PersistentEnvExtension Entry (Entry × PositivityExt)\n    (List Entry × DiscrTree PositivityExt) ←\n  -- we only need this to deduplicate entries in the DiscrTree\n  have : BEq PositivityExt := ⟨fun _ _ => false⟩\n  let insert kss v dt := kss.foldl (fun dt ks => dt.insertKeyValue ks v) dt\n  registerPersistentEnvExtension {\n    mkInitial := pure ([], {})\n    addImportedFn := fun s => do\n      let dt ← s.foldlM (init := {}) fun dt s => s.foldlM (init := dt) fun dt (kss, n) => do\n        pure (insert kss (← mkPositivityExt n) dt)\n      pure ([], dt)\n    addEntryFn := fun (entries, s) ((kss, n), ext) => ((kss, n) :: entries, insert kss ext s)\n    exportEntriesFn := fun s => s.1.reverse.toArray\n  }\n\ninitialize registerBuiltinAttribute {\n  name := `positivity\n  descr := \"adds a positivity extension\"\n  applicationTime := .afterCompilation\n  add := fun declName stx kind => match stx with\n    | `(attr| positivity $es,*) => do\n      ensureAttrDeclIsMeta `positivity declName kind\n      unless kind == AttributeKind.global do\n        throwError \"invalid attribute 'positivity', must be global\"\n      let env ← getEnv\n      unless (env.getModuleIdxFor? declName).isNone do\n        throwError \"invalid attribute 'positivity', declaration is in an imported module\"\n      if (IR.getSorryDep env declName).isSome then return -- ignore in progress definitions\n      let ext ← mkPositivityExt declName\n      let keys ← MetaM.run' <| es.getElems.mapM fun stx => do\n        let e ← TermElabM.run' <| withSaveInfoContext <| withAutoBoundImplicit <|\n          withReader ({ · with ignoreTCFailures := true }) do\n            let e ← elabTerm stx none\n            let (_, _, e) ← lambdaMetaTelescope (← mkLambdaFVars (← getLCtx).getFVars e)\n            return e\n        DiscrTree.mkPath e\n      setEnv <| positivityExt.addEntry env ((keys, declName), ext)\n      -- TODO: track what `[positivity]` decls are actually used at use sites\n      recordExtraRevUseOfCurrentModule\n    | _ => throwUnsupportedSyntax\n}\n\nvariable {A : Type*} {e : A}\n\nlemma pos_of_isNat {n : ℕ} [Semiring A] [PartialOrder A] [IsOrderedRing A] [Nontrivial A]\n    (h : NormNum.IsNat e n) (w : Nat.ble 1 n = true) : 0 < (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  apply Nat.cast_pos.2\n  simpa using! w\n\nlemma pos_of_isNat' {n : ℕ}\n    [AddMonoidWithOne A] [PartialOrder A] [AddLeftMono A] [ZeroLEOneClass A] [h'' : NeZero (1 : A)]\n    (h : NormNum.IsNat e n) (w : Nat.ble 1 n = true) : 0 < (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  apply Nat.cast_pos'.2\n  simpa using! w\n\nlemma nonneg_of_isNat {n : ℕ} [Semiring A] [PartialOrder A] [IsOrderedRing A]\n    (h : NormNum.IsNat e n) : 0 ≤ (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  exact Nat.cast_nonneg n\n\nTarget:\nlemma nonneg_of_isNat' {n : ℕ}\n    [AddMonoidWithOne A] [PartialOrder A] [AddLeftMono A] [ZeroLEOneClass A]\n    (h : NormNum.IsNat e n) : 0 ≤ (e : A) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"c4daf9cc9bd1142b4482128155f37d52c62741d70069740638b790eab762f2b2","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Positivity","family_id":"nonneg_of_isnat","file_id":"mathlib/Mathlib/Tactic/Positivity/Core.lean","sample_id":"309747378fad33151e6310726ff0174aa0d05877efaa42bd524d8d5cf49ca2f9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"63a4845b3fb713f383d6aefe7ca814d2f2062db02f00a1f2f03bbbb49cf53c38","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2b78fa8807ec50bfc8ab5c93ca7eeaa86e8f4d73fc76cde03273374bc7683948","source_sha256":"e433c304165198b6af36e2c9a124b401b67821948088461657a4c34ad256bb4a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  replace h (a m) : Q₁ (a ⊗ₜ m) = Q₂ (a ⊗ₜ m) := by\n    rw [← mul_one a, ← smul_eq_mul, ← smul_tmul', QuadraticMap.map_smul, QuadraticMap.map_smul, h]\n  ext x\n  induction x with\n  | tmul => simp [h]\n  | zero => simp\n  | add x y hx hy =>\n    have : Q₁.polarBilin = Q₂.polarBilin := by\n      ext\n      dsimp [polar]\n      rw [← TensorProduct.tmul_add, h, h, h]\n    replace := congr($this x y)\n    dsimp [polar] at this\n    linear_combination (norm := module) this + hx + hy","hard_negative":true,"metrics":{"chosen_tokens":129,"rejected_tokens":8,"token_jaccard":0.071429,"token_length_ratio":0.062016},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"db4a05b23ed844b81836f43fb8fbd24f75cf0303c1f87b67f90d1da7953eeedb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.BilinearForm.TensorProduct\npublic import Mathlib.LinearAlgebra.QuadraticForm.Basic\npublic import Mathlib.Tactic.LinearCombination\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# The quadratic form on a tensor product\n\n## Main definitions\n\n* `QuadraticForm.tensorDistrib (Q₁ ⊗ₜ Q₂)`: the quadratic form on `M₁ ⊗ M₂` constructed by applying\n  `Q₁` on `M₁` and `Q₂` on `M₂`. This construction is not available in characteristic two.\n\n-/\n\n@[expose] public section\n\nuniverse uR uA uM₁ uM₂ uN₁ uN₂\n\nvariable {R : Type uR} {A : Type uA} {M₁ : Type uM₁} {M₂ : Type uM₂} {N₁ : Type uN₁} {N₂ : Type uN₂}\n\nopen LinearMap (BilinMap BilinForm)\nopen TensorProduct QuadraticMap\n\nsection CommRing\nvariable [CommRing R] [CommRing A]\nvariable [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup N₁] [AddCommGroup N₂]\nvariable [Algebra R A] [Module R M₁] [Module A M₁] [Module R N₁] [Module A N₁]\nvariable [SMulCommClass R A M₁] [IsScalarTower R A M₁] [IsScalarTower R A N₁]\nvariable [Module R M₂] [Module R N₂]\n\nsection InvertibleTwo\nvariable [Invertible (2 : R)]\n\nnamespace QuadraticMap\n\nvariable (R A) in\n/-- The tensor product of two quadratic maps injects into quadratic maps on tensor products.\n\nNote this is heterobasic; the quadratic map on the left can take values in a module over a larger\nring than the one on the right. -/\ndef tensorDistrib :\n    QuadraticMap A M₁ N₁ ⊗[R] QuadraticMap R M₂ N₂ →ₗ[A] QuadraticMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=\n  letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm\n  -- while `letI`s would produce a better term than `let`, they would make this already-slow\n  -- definition even slower.\n  let toQ := BilinMap.toQuadraticMapLinearMap A A (M₁ ⊗[R] M₂)\n  let tmulB := BilinMap.tensorDistrib R A (M₁ := M₁) (M₂ := M₂)\n  let toB := AlgebraTensorModule.map\n      (QuadraticMap.associated : QuadraticMap A M₁ N₁ →ₗ[A] BilinMap A M₁ N₁)\n      (QuadraticMap.associated : QuadraticMap R M₂ N₂ →ₗ[R] BilinMap R M₂ N₂)\n  toQ ∘ₗ tmulB ∘ₗ toB\n\n@[simp]\ntheorem tensorDistrib_tmul (Q₁ : QuadraticMap A M₁ N₁) (Q₂ : QuadraticMap R M₂ N₂) (m₁ : M₁)\n    (m₂ : M₂) : tensorDistrib R A (Q₁ ⊗ₜ Q₂) (m₁ ⊗ₜ m₂) = Q₁ m₁ ⊗ₜ Q₂ m₂ :=\n  letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm\n  (BilinMap.tensorDistrib_tmul _ _ _ _ _ _).trans <| congr_arg₂ _\n    (associated_eq_self_apply _ _ _) (associated_eq_self_apply _ _ _)\n\n/-- The tensor product of two quadratic maps, a shorthand for dot notation. -/\nprotected abbrev tmul (Q₁ : QuadraticMap A M₁ N₁)\n    (Q₂ : QuadraticMap R M₂ N₂) : QuadraticMap A (M₁ ⊗[R] M₂) (N₁ ⊗[R] N₂) :=\n  tensorDistrib R A (Q₁ ⊗ₜ[R] Q₂)\n\ntheorem associated_tmul [Invertible (2 : A)]\n    (Q₁ : QuadraticMap A M₁ N₁) (Q₂ : QuadraticMap R M₂ N₂) :\n    (Q₁.tmul Q₂).associated = Q₁.associated.tmul Q₂.associated := by\n  letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm\n  rw [QuadraticMap.tmul, BilinMap.tmul]\n  have : Subsingleton (Invertible (2 : A)) := inferInstance\n  convert!\n    associated_left_inverse A\n      (LinearMap.BilinMap.tmul_isSymm (QuadraticMap.associated_isSymm A Q₁)\n        (QuadraticMap.associated_isSymm R Q₂))\n\nend QuadraticMap\n\nnamespace QuadraticForm\n\nvariable (R A) in\n/-- The tensor product of two quadratic forms injects into quadratic forms on tensor products.\n\nNote this is heterobasic; the quadratic form on the left can take values in a larger ring than\nthe one on the right. -/\ndef tensorDistrib :\n    QuadraticForm A M₁ ⊗[R] QuadraticForm R M₂ →ₗ[A] QuadraticForm A (M₁ ⊗[R] M₂) :=\n  (AlgebraTensorModule.rid R A A).congrQuadraticMap.toLinearMap ∘ₗ QuadraticMap.tensorDistrib R A\n\n-- TODO: make the RHS `MulOpposite.op (Q₂ m₂) • Q₁ m₁` so that this has a nicer defeq for\n-- `R = A` of `Q₁ m₁ * Q₂ m₂`.\n@[simp]\ntheorem tensorDistrib_tmul (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) (m₁ : M₁) (m₂ : M₂) :\n    tensorDistrib R A (Q₁ ⊗ₜ Q₂) (m₁ ⊗ₜ m₂) = Q₂ m₂ • Q₁ m₁ :=\n  letI : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm\n  (LinearMap.BilinForm.tensorDistrib_tmul _ _ _ _ _ _ _ _).trans <| congr_arg₂ _\n    (associated_eq_self_apply _ _ _) (associated_eq_self_apply _ _ _)\n\n/-- The tensor product of two quadratic forms, a shorthand for dot notation. -/\nprotected abbrev tmul (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) :\n    QuadraticForm A (M₁ ⊗[R] M₂) :=\n  tensorDistrib R A (Q₁ ⊗ₜ[R] Q₂)\n\ntheorem associated_tmul [Invertible (2 : A)] (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) :\n    (Q₁.tmul Q₂).associated = BilinForm.tmul Q₁.associated Q₂.associated := by\n  rw [BilinForm.tmul, BilinForm.tensorDistrib, LinearMap.comp_apply, ← BilinMap.tmul,\n    ← QuadraticMap.associated_tmul Q₁ Q₂, LinearEquiv.coe_coe, LinearEquiv.congrRight₂_apply]\n  ext : 6\n  simp [associated_apply]\n  rfl\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem polarBilin_tmul [Invertible (2 : A)] (Q₁ : QuadraticForm A M₁) (Q₂ : QuadraticForm R M₂) :\n    polarBilin (Q₁.tmul Q₂) = ⅟(2 : A) • BilinForm.tmul (polarBilin Q₁) (polarBilin Q₂) := by\n  simp_rw [← two_nsmul_associated A, ← two_nsmul_associated R, BilinForm.tmul, tmul_smul,\n    ← smul_tmul', map_nsmul, associated_tmul]\n  rw [smul_comm (_ : A) (_ : ℕ), ← smul_assoc, two_smul _ (_ : A), invOf_two_add_invOf_two,\n    one_smul]\n\nvariable (A) in\n/-- The base change of a quadratic form. -/\nprotected def baseChange (Q : QuadraticForm R M₂) : QuadraticForm A (A ⊗[R] M₂) :=\n  QuadraticForm.tmul (R := R) (A := A) (M₁ := A) (M₂ := M₂) (QuadraticMap.sq (R := A)) Q\n\n@[simp]\ntheorem baseChange_tmul (Q : QuadraticForm R M₂) (a : A) (m₂ : M₂) :\n    Q.baseChange A (a ⊗ₜ m₂) = Q m₂ • (a * a) :=\n  tensorDistrib_tmul _ _ _ _\n\ntheorem associated_baseChange [Invertible (2 : A)] (Q : QuadraticForm R M₂) :\n    associated (R := A) (Q.baseChange A) = BilinForm.baseChange A (associated (R := R) Q) := by\n  dsimp only [QuadraticForm.baseChange, LinearMap.baseChange]\n  rw [associated_tmul (QuadraticMap.sq (R := A)) Q, associated_sq]\n  exact rfl\n\ntheorem polarBilin_baseChange [Invertible (2 : A)] (Q : QuadraticForm R M₂) :\n    polarBilin (Q.baseChange A) = BilinForm.baseChange A (polarBilin Q) := by\n  rw [QuadraticForm.baseChange, BilinForm.baseChange, polarBilin_tmul, BilinForm.tmul,\n    ← map_smul, smul_tmul', ← two_nsmul_associated R, coe_associatedHom, associated_sq,\n    smul_comm, ← smul_assoc, two_smul, invOf_two_add_invOf_two, one_smul]\n\nend QuadraticForm\n\nend InvertibleTwo\n\nset_option backward.defeqAttrib.useBackward true in\n/-- If two quadratic maps from `A ⊗[R] M₂` agree on elements of the form `1 ⊗ m`, they are equal.\n\nIn other words, if a base change exists for a quadratic map, it is unique.\n\nNote that unlike `QuadraticForm.baseChange`, this does not need `Invertible (2 : R)`. -/\n@[ext]\n\nTarget:\ntheorem baseChange_ext ⦃Q₁ Q₂ : QuadraticMap A (A ⊗[R] M₂) N₁⦄\n    (h : ∀ m, Q₁ (1 ⊗ₜ m) = Q₂ (1 ⊗ₜ m)) :\n    Q₁ = Q₂ :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"8d9a5c1cf44686dc0e3b6d87703a49db13127a36fee0054aad54515bebc20d8a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/QuadraticForm","family_id":"basechange_ext","file_id":"mathlib/Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean","sample_id":"2b78fa8807ec50bfc8ab5c93ca7eeaa86e8f4d73fc76cde03273374bc7683948"},"split":"train"}
{"binding":{"chosen_proof_sha256":"105da6d71d1fe288f69e471a4b967a3a862bded59f3b72ae1eccbacf108cf793","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"316e9edc7144d415a7e46a6a2a53e203a71e0c03a7d0b39706e889dc27b42fb3","source_sha256":"23217497e3239734dae1b907161ba142454fe3246530e95a20731c7a2dcfd18b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  -- Again, it is a \"left inverse equals right inverse\" argument in the convolution monoid.\n  apply left_inv_eq_right_inv\n    (M := Conv A (A ⊗ A))\n    (a := Δ[A])\n  · rw [Conv.mul_eq, Conv.one_eq]\n    simp only [comp_whiskerRight, tensor_whiskerLeft, MonObj.tensorObj.mul_def, Category.assoc,\n      MonObj.tensorObj.one_def]\n    simp only [tensorμ]\n    simp only [Category.assoc, Iso.inv_hom_id_assoc]\n    exact antipode_comul₁ A\n  · rw [Conv.mul_eq, Conv.one_eq]\n    simp only [whiskerLeft_comp, tensor_whiskerLeft, Category.assoc, Iso.inv_hom_id_assoc,\n      MonObj.tensorObj.mul_def, MonObj.tensorObj.one_def]\n    simp only [tensorμ]\n    simp only [Category.assoc, Iso.inv_hom_id_assoc]\n    exact antipode_comul₂ A","hard_negative":false,"metrics":{"chosen_tokens":156,"rejected_tokens":2,"token_jaccard":0.018519,"token_length_ratio":0.012821},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"db8dd6781955bd68b0bb8039cde37bd6b19e6fff9fe4e80dc5b956676165a089","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Monoidal.Bimon_\npublic import Mathlib.CategoryTheory.Monoidal.Conv\n\nNamespace:\nCategoryTheory.HopfObj\n\nLocal context:\n/-\nCopyright (c) 2024 Lean FRO LLC. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# The category of Hopf monoids in a braided monoidal category.\n\n\n## TODO\n\n* Show that in a Cartesian monoidal category Hopf monoids are exactly group objects.\n* Show that `Hopf (ModuleCat R) ≌ HopfAlgCat R`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nuniverse v₁ v₂ u₁ u₂ u\n\nopen CategoryTheory MonoidalCategory\n\nnamespace CategoryTheory\nvariable {C : Type u₁} [Category.{v₁} C] [MonoidalCategory.{v₁} C] [BraidedCategory C]\n\nopen scoped MonObj ComonObj\n\n/--\nA Hopf monoid in a braided category `C` is a bimonoid object in `C` equipped with an antipode.\n-/\nclass HopfObj (X : C) extends BimonObj X where\n  /-- The antipode is an endomorphism of the underlying object of the Hopf monoid. -/\n  antipode : X ⟶ X\n  antipode_left (X) : Δ ≫ antipode ▷ X ≫ μ = ε ≫ η := by cat_disch\n  antipode_right (X) : Δ ≫ X ◁ antipode ≫ μ = ε ≫ η := by cat_disch\n\nnamespace HopfObj\n\n@[inherit_doc] scoped notation \"𝒮\" => HopfObj.antipode\n@[inherit_doc] scoped notation \"𝒮[\" M \"]\" => HopfObj.antipode (X := M)\n\nattribute [reassoc (attr := simp)] antipode_left antipode_right\n\n\nend HopfObj\n\nvariable (C)\n\n/--\nA Hopf monoid in a braided category `C` is a bimonoid object in `C` equipped with an antipode.\n-/\nstructure Hopf where\n  /-- The underlying object in the ambient monoidal category -/\n  X : C\n  [hopf : HopfObj X]\n\nattribute [instance] Hopf.hopf\n\nnamespace Hopf\n\nvariable {C}\n\n/-- A Hopf monoid is a bimonoid. -/\ndef toBimon (A : Hopf C) : Bimon C := .mk' A.X\n\n/--\nMorphisms of Hopf monoids are just morphisms of the underlying bimonoids.\nIn fact they automatically intertwine the antipodes, proved below.\n-/\ninstance : Category (Hopf C) :=\n  inferInstanceAs <| Category (InducedCategory (Bimon C) Hopf.toBimon)\n\nend Hopf\n\nnamespace HopfObj\n\nvariable {C}\n\n/-- Morphisms of Hopf monoids intertwine the antipodes. -/\ntheorem hom_antipode {A B : C} [HopfObj A] [HopfObj B] (f : A ⟶ B) [IsBimonHom f] :\n    f ≫ 𝒮 = 𝒮 ≫ f := by\n  -- We show these elements are equal by exhibiting an element in the convolution algebra\n  -- between `A` (as a comonoid) and `B` (as a monoid),\n  -- such that the LHS is a left inverse, and the RHS is a right inverse.\n  apply left_inv_eq_right_inv\n    (M := Conv A B)\n    (a := f)\n  · rw [Conv.mul_eq, Conv.one_eq]\n    simp only [comp_whiskerRight, Category.assoc]\n    slice_lhs 3 4 =>\n      rw [← whisker_exchange]\n    slice_lhs 2 3 =>\n      rw [← tensorHom_def]\n    slice_lhs 1 2 =>\n      rw [← IsComonHom.hom_comul f]\n    slice_lhs 2 4 =>\n      rw [antipode_left]\n    slice_lhs 1 2 =>\n      rw [IsComonHom.hom_counit]\n  · rw [Conv.mul_eq, Conv.one_eq]\n    simp only [whiskerLeft_comp, Category.assoc]\n    slice_lhs 2 3 =>\n      rw [← whisker_exchange]\n    slice_lhs 3 4 =>\n      rw [← tensorHom_def]\n    slice_lhs 3 4 =>\n      rw [← IsMonHom.mul_hom]\n    slice_lhs 1 3 =>\n      rw [antipode_right]\n    slice_lhs 2 3 =>\n      rw [IsMonHom.one_hom]\n\n@[reassoc (attr := simp)]\ntheorem one_antipode (A : C) [HopfObj A] : η[A] ≫ 𝒮[A] = η[A] := by\n  have := (rfl : η[A] ≫ Δ[A] ≫ (𝒮[A] ▷ A) ≫ μ[A] = _)\n  conv at this =>\n    rhs\n    rw [antipode_left]\n  rw [Bimon.one_comul_assoc, tensorHom_def_assoc, unitors_inv_equal,\n    ← rightUnitor_inv_naturality_assoc, whisker_exchange_assoc, ← rightUnitor_inv_naturality_assoc,\n    rightUnitor_inv_naturality_assoc] at this\n  simpa\n\n@[reassoc (attr := simp)]\ntheorem antipode_counit (A : C) [HopfObj A] : 𝒮[A] ≫ ε[A] = ε[A] := by\n  have := (rfl : Δ[A] ≫ (𝒮[A] ▷ A) ≫ μ[A] ≫ ε[A] = _)\n  conv at this =>\n    rhs\n    rw [antipode_left_assoc]\n  rw [Bimon.mul_counit, tensorHom_def', Category.assoc, ← whisker_exchange_assoc] at this\n  simpa [unitors_equal]\n\n/-!\n## The antipode is an antihomomorphism with respect to both the monoid and comonoid structures.\n-/\n\ntheorem antipode_comul₁ (A : C) [HopfObj A] :\n    Δ[A] ≫\n      𝒮[A] ▷ A ≫\n      Δ[A] ▷ A ≫\n      (α_ A A A).hom ≫\n      A ◁ A ◁ Δ[A] ≫\n      A ◁ (α_ A A A).inv ≫\n      A ◁ (β_ A A).hom ▷ A ≫\n      A ◁ (α_ A A A).hom ≫\n      (α_ A A (A ⊗ A)).inv ≫\n      (μ[A] ⊗ₘ μ[A]) =\n    ε[A] ≫ (λ_ (𝟙_ C)).inv ≫ (η[A] ⊗ₘ η[A]) := by\n  slice_lhs 3 5 =>\n    rw [← associator_naturality_right, ← Category.assoc, ← tensorHom_def]\n  slice_lhs 3 9 =>\n    rw [Bimon.compatibility]\n  slice_lhs 1 3 =>\n    rw [antipode_left]\n  simp [MonObj.tensorObj.one_def]\n\n/--\nAuxiliary calculation for `antipode_comul`.\nThis calculation calls for some ASCII art out of This Week's Finds.\n\n```\n   |   |\n   n   n\n  | \\ / |\n  |  /  |\n  | / \\ |\n  | | S S\n  | | \\ /\n  | |  /\n  | | / \\\n  \\ / \\ /\n   v   v\n    \\ /\n     v\n     |\n```\n\nWe move the left antipode up through the crossing,\nthe right antipode down through the crossing,\nthe right multiplication down across the strand,\nreassociate the comultiplications,\nthen use `antipode_right` then `antipode_left` to simplify.\n-/\ntheorem antipode_comul₂ (A : C) [HopfObj A] :\n    Δ[A] ≫\n      Δ[A] ▷ A ≫\n      (α_ A A A).hom ≫\n      A ◁ A ◁ Δ[A] ≫\n      A ◁ A ◁ (β_ A A).hom ≫\n      A ◁ A ◁ (𝒮[A] ⊗ₘ 𝒮[A]) ≫\n      A ◁ (α_ A A A).inv ≫\n      A ◁ (β_ A A).hom ▷ A ≫\n      A ◁ (α_ A A A).hom ≫\n      (α_ A A (A ⊗ A)).inv ≫\n      (μ[A] ⊗ₘ μ[A]) =\n    ε[A] ≫ (λ_ (𝟙_ C)).inv ≫ (η[A] ⊗ₘ η[A]) := by\n  -- We should write a version of `slice_lhs` that zooms through whiskerings.\n  slice_lhs 6 6 =>\n    simp only [tensorHom_def', whiskerLeft_comp]\n  slice_lhs 7 8 =>\n    rw [← whiskerLeft_comp, associator_inv_naturality_middle, whiskerLeft_comp]\n  slice_lhs 8 9 =>\n    rw [← whiskerLeft_comp, ← comp_whiskerRight, BraidedCategory.braiding_naturality_right,\n      comp_whiskerRight, whiskerLeft_comp]\n  slice_lhs 9 10 =>\n    rw [← whiskerLeft_comp, associator_naturality_left, whiskerLeft_comp]\n  slice_lhs 5 6 =>\n    rw [← whiskerLeft_comp, ← whiskerLeft_comp, ← BraidedCategory.braiding_naturality_left,\n      whiskerLeft_comp, whiskerLeft_comp]\n  slice_lhs 11 12 =>\n    rw [tensorHom_def', ← Category.assoc, ← associator_inv_naturality_right]\n  slice_lhs 10 11 =>\n    rw [← whiskerLeft_comp, ← whisker_exchange, whiskerLeft_comp]\n  slice_lhs 6 10 =>\n    simp only [← whiskerLeft_comp]\n    rw [← BraidedCategory.hexagon_reverse_assoc, Iso.inv_hom_id_assoc,\n      ← BraidedCategory.braiding_naturality_left]\n    simp only [whiskerLeft_comp]\n  rw [ComonObj.comul_assoc_flip_assoc, Iso.inv_hom_id_assoc]\n  slice_lhs 2 3 =>\n    simp only [← whiskerLeft_comp]\n    rw [ComonObj.comul_assoc]\n    simp only [whiskerLeft_comp]\n  slice_lhs 3 7 =>\n    simp only [← whiskerLeft_comp]\n    rw [← associator_naturality_middle_assoc, Iso.hom_inv_id_assoc]\n    simp only [← comp_whiskerRight]\n    rw [antipode_right]\n    simp only [comp_whiskerRight]\n    simp only [whiskerLeft_comp]\n  slice_lhs 2 3 =>\n    simp only [← whiskerLeft_comp]\n    rw [ComonObj.counit_comul]\n    simp only [whiskerLeft_comp]\n  slice_lhs 3 4 =>\n    simp only [← whiskerLeft_comp]\n    rw [BraidedCategory.braiding_naturality_left]\n    simp only [whiskerLeft_comp]\n  slice_lhs 4 5 =>\n    simp only [← whiskerLeft_comp]\n    rw [whisker_exchange]\n    simp only [whiskerLeft_comp]\n  slice_lhs 5 7 =>\n    rw [associator_inv_naturality_right_assoc, whisker_exchange]\n  simp only [braiding_tensorUnit_left,\n    whiskerLeft_comp, whiskerLeft_rightUnitor_inv,\n    whiskerRight_id, whiskerLeft_rightUnitor, Category.assoc, Iso.hom_inv_id_assoc,\n    Iso.inv_hom_id_assoc, whiskerLeft_inv_hom_assoc, antipode_right_assoc]\n  rw [rightUnitor_inv_naturality_assoc, tensorHom_def]\n  monoidal\n\nTarget:\ntheorem antipode_comul (A : C) [HopfObj A] :\n    𝒮[A] ≫ Δ[A] = Δ[A] ≫ (β_ _ _).hom ≫ (𝒮[A] ⊗ₘ 𝒮[A]) :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Monoidal","family_id":"antipode_comul","file_id":"mathlib/Mathlib/CategoryTheory/Monoidal/Hopf_.lean","sample_id":"316e9edc7144d415a7e46a6a2a53e203a71e0c03a7d0b39706e889dc27b42fb3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"06f4331fcea43b6ff267c69e7aee0a119d3182db477558ac433ba49db7222320","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"818b600996d16659e53d7447c449dd582739374ebc0426ce42dc34ce1bd8ca1e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"23f66eb623e6a66dd8750aa398395aca87f38f1c676ee283cf181c130d4fe931","source_sha256":"cc2a1bcce9a68ef1d53580ab6c1e41e421aac9f362e958d30242edc06eefee5f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let K := Algebra.adjoin F (Set.range x)\n  suffices #E = #K by simp [K, this, ← lift_mk_eq'.2 ⟨hx.1.aevalEquiv.toEquiv⟩]\n  haveI : Algebra.IsAlgebraic K E := hx.isAlgebraic\n  refine le_antisymm ?_ (mk_le_of_injective Subtype.val_injective)\n  haveI : Infinite K := hx.1.aevalEquiv.infinite_iff.1 inferInstance\n  simpa only [sup_eq_left.2 (aleph0_le_mk K)] using Algebra.IsAlgebraic.cardinalMk_le_max K E","hard_negative":false,"metrics":{"chosen_tokens":96,"rejected_tokens":101,"token_jaccard":0.927273,"token_length_ratio":1.052083},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"db9ce4472b197c521db30f011e5d720dc0be8db8e4611e2bc2a5753d5070d5e5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.IntermediateField.Adjoin.Basic\npublic import Mathlib.FieldTheory.MvRatFunc.Rank\npublic import Mathlib.RingTheory.Algebraic.Cardinality\npublic import Mathlib.RingTheory.AlgebraicIndependent.Adjoin\npublic import Mathlib.RingTheory.AlgebraicIndependent.Transcendental\npublic import Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Cardinality of a transcendence basis\n\nThis file concerns the cardinality of a transcendence basis.\n\n## References\n\n* [Stacks: Transcendence](https://stacks.math.columbia.edu/tag/030D)\n\n## Tags\ntranscendence basis, transcendence degree, transcendence\n\n-/\n\npublic section\n\nnoncomputable section\n\nopen Function Set Subalgebra MvPolynomial Algebra\n\nuniverse u v w\n\nopen AlgebraicIndependent\n\nopen Cardinal\n\nTarget:\ntheorem IsTranscendenceBasis.lift_cardinalMk_eq_max_lift\n    {F : Type u} {E : Type v} [CommRing F] [Nontrivial F] [CommRing E] [IsDomain E] [Algebra F E]\n    {ι : Type w} {x : ι → E} [Nonempty ι] (hx : IsTranscendenceBasis F x) :\n    lift.{max u w} #E = lift.{max v w} #F ⊔ lift.{max u v} #ι ⊔ ℵ₀ :=\n\nProof body:\n","rejected":"by\n  let K := Algebra.adjoin F (Set.range x)\n  suffices #E = #K by simp [K, this, ← lift_mk_eq'.2 ⟨hx.1.aevalEquiv.toEquiv⟩]\n  haveI : Algebra.IsAlgebraic K E := hx.isAlgebraic\n  refine le_antisymm ?_ (mk_le_of_injective Subtype.val_injective)\n  haveI : Infinite K := hx.1.aevalEquiv.infinite_iff.1 inferInstance\n  simpa only [sup_eq_left.2 (aleph0_le_mk K)] using Algebra.IsAlgebraic.cardinalMk_le_max K E\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/AlgebraicIndependent","family_id":"istranscendencebasis","file_id":"mathlib/Mathlib/RingTheory/AlgebraicIndependent/RankAndCardinality.lean","sample_id":"23f66eb623e6a66dd8750aa398395aca87f38f1c676ee283cf181c130d4fe931"},"split":"train"}
{"binding":{"chosen_proof_sha256":"42771a2147cb91bca0f9e576050654f9c7f45762679c01f645ed998108481986","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5eafb8d4f210e0305decd4f2fa8d9481c3b2ad15eb87b03d6f08b5fbabeaece5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e2222f0bad75d2e7deb1fd25fdb3ceb2a09b7f0046dfa22182f45b77e039767c","source_sha256":"ec4c2033e9d1a1249c37ea9aa929a55ae2775b2733369f392ca904414b15e64c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    simpa using hκ.irreducible hs <| Std.lt_of_lt_of_le hsp (hφ s)","hard_negative":true,"metrics":{"chosen_tokens":19,"rejected_tokens":5,"token_jaccard":0.1,"token_length_ratio":0.263158},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"dbba5c7c59a953d24dab2804161e961263d55484c906e11436c82d4350505efc","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Kernel.Composition.Comp\n\nNamespace:\nProbabilityTheory.Kernel\n\nLocal context:\n/-\nCopyright (c) 2025 Bryan Wang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bryan Wang\n-/\n/-!\n# Irreducibility of kernels\n\nA kernel `κ : Kernel α α` is `φ`-irreducible, for a given measure `φ` on `α`,\nif for every measurable set `A` with positive measure under `φ`, and for every `a : α`,\nthere exists a positive integer `n` such that we have `(κ ^ n) a A > 0`.\n\nWhen the kernel `κ` is the transition kernel of a Markov chain,\nthis precisely means that the Markov chain is `φ`-irreducible,\nthat is, there is a positive probability of reaching any\n(`φ`-positive measure) set of states from any other state within a finite number of steps.\n\n## Main definitions\n\n* `ProbabilityTheory.Kernel.IsIrreducible`:\n  irreducibility of a given kernel with respect to a measure `φ`.\n\n## Main statements\n\n* `isIrreducible_of_le_measure`: If a kernel `κ` is irreducible with respect to a measure `φ₂`,\n  then it is also irreducible with respect to any measure `φ₁` with `φ₁ ≤ φ₂`.\n\n## References\n\n* [Meyn, S.P. and Tweedie, R.L., *Markov Chains and Stochastic Stability*][meyntweedie1993]\n* [C Robert, G Casella, *Monte Carlo Statistical Methods*][robertcasella2004]\n\n-/\n\npublic section\n\nopen MeasureTheory\n\nopen scoped MeasureTheory ENNReal ProbabilityTheory\n\nnamespace ProbabilityTheory\n\nvariable {α β : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}\n\nnamespace Kernel\n\n/-- A kernel `κ : Kernel α α` is `φ`-irreducible (w.r.t. a given measure `φ` on `α`),\nif for every measurable set `A` with positive measure under `φ`,\nand for every `a : α`, there exists an integer `n` such that `(κ ^ n) a A > 0`.\nRef. *Meyn-Tweedie* Proposition 4.2.1(ii), page 89 -/\n@[mk_iff]\nclass IsIrreducible (φ : Measure α) (κ : Kernel α α) : Prop where\n  irreducible ⦃A⦄ (hA : MeasurableSet A) (hφA : φ A > 0) a :\n    ∃ (n : ℕ), (κ ^ n) a A > 0\n\ninstance {φ : Measure α} [Subsingleton α] :\n    IsIrreducible φ Kernel.id where\n  irreducible s hs hsp a := by\n    use 1;\n    have ha : a ∈ s :=\n      Subsingleton.mem_iff_nonempty.mpr\n        <| MeasureTheory.nonempty_of_measure_ne_zero (μ := φ) (ne_of_lt hsp).symm\n    simp [id_apply, ha]\n\ninstance {c : ℝ≥0∞} {φ : Measure α} {κ : Kernel α α} [hκ : IsIrreducible φ κ] :\n    IsIrreducible (c • φ) κ where\n  irreducible s hs hsp := by\n    simpa using hκ.irreducible hs <| by simp_all\n\nTarget:\nlemma isIrreducible_of_le_measure {φ₁ φ₂ : Measure α} (hφ : φ₁ ≤ φ₂)\n    {κ : Kernel α α} [hκ : IsIrreducible φ₂ κ] :\n    IsIrreducible φ₁ κ where\n  irreducible s hs hsp :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_e2222f0bad75","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"5cf2e967e2a16fa94f4a10281bcae4682f524618156a8067a2fdf0620fa527bf","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Kernel","family_id":"isirreducible_of_le_measure","file_id":"mathlib/Mathlib/Probability/Kernel/Irreducible.lean","sample_id":"e2222f0bad75d2e7deb1fd25fdb3ceb2a09b7f0046dfa22182f45b77e039767c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ffc956ff392a03eea39dc40bd447e1d225bcdcdd1b88caef9ee391f6b7a9878c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1bdf66f02f800f9f6bc8f802f4eb7010531ea0fed644e6fb319377ffa1bd244a","source_sha256":"250e0bf39b2fceb88cc44db25e786e52b9f1629e4d745089c27db7ba4ae4e60e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [card_of_finrank k V h]","hard_negative":true,"metrics":{"chosen_tokens":8,"rejected_tokens":8,"token_jaccard":0.071429,"token_length_ratio":1.0},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"dbbaae38c6c20bfb3aafb81c546ccd10f3b5565e38b0f1cf96ba0a103f4a7d42","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.GeomSum\npublic import Mathlib.Algebra.GroupWithZero.Units.Fintype\npublic import Mathlib.Data.Finite.Sum\npublic import Mathlib.GroupTheory.GroupAction.Quotient\npublic import Mathlib.LinearAlgebra.Projectivization.Basic\n\nNamespace:\nProjectivization\n\nLocal context:\n/-\nCopyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Judith Ludwig, Christian Merten\n-/\n/-!\n# Cardinality of projective spaces\n\nWe compute the cardinality of `ℙ k V` if `k` is a finite field.\n\n-/\n\n@[expose] public section\n\nnamespace Projectivization\n\nopen scoped LinearAlgebra.Projectivization\n\nsection\n\nvariable (k V : Type*) [DivisionRing k] [AddCommGroup V] [Module k V]\n\n/-- `ℙ k V` is equivalent to the quotient of the non-zero elements of `V` by `kˣ`. -/\ndef equivQuotientOrbitRel : ℙ k V ≃ Quotient (MulAction.orbitRel kˣ { v : V // v ≠ 0 }) :=\n  Quotient.congr (Equiv.refl _) (fun x y ↦ (Units.orbitRel_nonZero_iff k V x y).symm)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The non-zero elements of `V` are equivalent to the product of `ℙ k V` with the units of `k`. -/\nnoncomputable def nonZeroEquivProjectivizationProdUnits : { v : V // v ≠ 0 } ≃ ℙ k V × kˣ :=\n  let e := MulAction.selfEquivOrbitsQuotientProd <| fun b ↦ by\n    rw [(Units.nonZeroSubMul k V).stabilizer_of_subMul,\n      Module.stabilizer_units_eq_bot_of_ne_zero k b.property]\n  e.trans (Equiv.prodCongrLeft (fun _ ↦ (equivQuotientOrbitRel k V).symm))\n\ninstance isEmpty_of_subsingleton [Subsingleton V] : IsEmpty (ℙ k V) := by\n  have : IsEmpty { v : V // v ≠ 0 } := ⟨fun v ↦ v.2 (Subsingleton.elim v.1 0)⟩\n  simpa using (nonZeroEquivProjectivizationProdUnits k V).symm.isEmpty\n\n/-- If `V` is a finite `k`-module and `k` is finite, `ℙ k V` is finite. -/\ninstance finite_of_finite [Finite V] : Finite (ℙ k V) :=\n  have : Finite (ℙ k V × kˣ) := Finite.of_equiv _ (nonZeroEquivProjectivizationProdUnits k V)\n  Finite.prod_left kˣ\n\nlemma finite_iff_of_finite [Finite k] : Finite (ℙ k V) ↔ Finite V := by\n  classical\n  refine ⟨fun h ↦ ?_, fun h ↦ inferInstance⟩\n  let e := nonZeroEquivProjectivizationProdUnits k V\n  have : Finite { v : V // v ≠ 0 } := Finite.of_equiv _ e.symm\n  let eq : { v : V // v ≠ 0 } ⊕ Unit ≃ V :=\n    ⟨(Sum.elim Subtype.val (fun _ ↦ 0)), fun v ↦ if h : v = 0 then Sum.inr () else Sum.inl ⟨v, h⟩,\n      by intro x; aesop, by intro x; aesop⟩\n  exact Finite.of_equiv _ eq\n\n/-- Fraction free cardinality formula for the points of `ℙ k V` if `k` and `V` are finite\n(for silly reasons the formula also holds when `k` and `V` are infinite).\nSee `Projectivization.card'` and `Projectivization.card''` for other spellings of the formula. -/\nlemma card : Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1) := by\n  nontriviality V\n  cases finite_or_infinite k with\n  | inr h =>\n    have : Infinite V := Module.Free.infinite k V\n    simp\n  | inl h =>\n  cases finite_or_infinite V with\n  | inr h =>\n    have := not_iff_not.mpr (finite_iff_of_finite k V)\n    push Not at this\n    have : Infinite (ℙ k V) := by rwa [this]\n    simp\n  | inl h =>\n  classical\n  haveI : Fintype V := Fintype.ofFinite V\n  haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)\n  haveI : Fintype k := Fintype.ofFinite k\n  have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp\n  simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]\n  rw [Fintype.card_congr (nonZeroEquivProjectivizationProdUnits k V), Fintype.card_prod]\n\n/-- Cardinality formula for the points of `ℙ k V` if `k` and `V` are finite with less\nnatural subtraction. -/\nlemma card' [Finite V] : Nat.card V = Nat.card (ℙ k V) * (Nat.card k - 1) + 1 := by\n  rw [← card k V]\n  have : Nat.card V > 0 := Nat.card_pos\n  lia\n\nend\n\nvariable (k V : Type*) [Field k] [AddCommGroup V] [Module k V]\n\n/-- Cardinality formula for the points of `ℙ k V` if `k` and `V` are finite expressed\nas a fraction. -/\nlemma card'' [Finite k] : Nat.card (ℙ k V) = (Nat.card V - 1) / (Nat.card k - 1) := by\n  have : 1 < Nat.card k := Finite.one_lt_card\n  rw [card k, Nat.mul_div_cancel]\n  lia\n\nlemma card_of_finrank [Finite k] {n : ℕ} (h : Module.finrank k V = n) :\n    Nat.card (ℙ k V) = ∑ i ∈ Finset.range n, Nat.card k ^ i := by\n  wlog hf : Finite V\n  · have : Infinite (ℙ k V) := by\n      contrapose! hf\n      rwa [finite_iff_of_finite] at hf\n    have : n = 0 := by\n      rw [← h]\n      apply Module.finrank_of_not_finite\n      contrapose hf\n      simpa using Module.finite_of_finite k\n    simp [this]\n  have : 1 < Nat.card k := Finite.one_lt_card\n  refine Nat.mul_right_cancel (m := Nat.card k - 1) (by lia) ?_\n  let e : V ≃ₗ[k] (Fin n → k) := LinearEquiv.ofFinrankEq _ _ (by simpa)\n  have hc : Nat.card V = Nat.card k ^ n := by simp [Nat.card_congr e.toEquiv, Nat.card_fun]\n  zify\n  conv_rhs => rw [Int.natCast_sub this.le, Int.natCast_one, geom_sum_mul]\n  rw [← Int.natCast_mul, ← card k V, hc]\n  simp\n\nTarget:\nlemma card_of_finrank_two [Finite k] (h : Module.finrank k V = 2) :\n    Nat.card (ℙ k V) = Nat.card k + 1 :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"2cc64911b870965fa06b50d4633de8b8db6a00126ef1bc7a0c851098d7f2b369","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Projectivization","family_id":"card_of_finrank_two","file_id":"mathlib/Mathlib/LinearAlgebra/Projectivization/Cardinality.lean","sample_id":"1bdf66f02f800f9f6bc8f802f4eb7010531ea0fed644e6fb319377ffa1bd244a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"603b10c8e8d8264cdfe3ea1be9c171071aba17b8e3411dbc942a8512b4e99c56","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"119ff96b176e6f80d0102930408c2c230e7d2b42ec62a08736a7ae0ef94a8be7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"57919ee9fe831200d4f5697405ab93efe8a6295ef8776bc2c7eff0fee53e76ad","source_sha256":"b81d767611ae69fe62afcbf3affb9ca909a694d39d19ca39528661161a4334bc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI := Fintype.ofFinite ι\n  haveI := fun i => Fintype.ofFinite (κ i)\n  classical\n  refine (Basis.piTensorProduct b).dualBasis.ext (fun p ↦ ?_)\n  refine (Basis.piTensorProduct b).ext (fun q ↦ ?_)\n  simp [Finsupp.single_apply, Fintype.prod_ite_zero, ← funext_iff]","hard_negative":false,"metrics":{"chosen_tokens":67,"rejected_tokens":72,"token_jaccard":0.871795,"token_length_ratio":1.074627},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"dc0ed27cb68b81ab0b7599fb0db7fd3b7effde970d34ae0d96f1004177b2bbff","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Dual.Basis\npublic import Mathlib.LinearAlgebra.FreeModule.Finite.Basic\npublic import Mathlib.LinearAlgebra.PiTensorProduct.Basis\n\nNamespace:\nPiTensorProduct\n\nLocal context:\n/-\nCopyright (c) 2025 Daniel Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Morrison, Sophie Morel\n-/\n/-!\n# Tensor products of dual spaces\n\n## Main definitions\n\n* `PiTensorProduct.dualDistrib`: The canonical linear map from `⨂[R] i, Dual R (M i)` to\n  `Dual R (⨂[R] i, M i)`, sending `⨂ₜ[R] i, f i` to the composition of\n  `PiTensorProduct.map f` with the linear equivalence `⨂[R] i, R →ₗ R` given by multiplication.\n\n* `PiTensorProduct.dualDistribEquiv`: A linear equivalence between `⨂[R] i, Dual R (M i)`\n  and `Dual R (⨂[R] i, M i)` when all `M i` are finite free modules. If\n  `f : (i : ι) → Dual R (M i)`, then this equivalence sends `⨂ₜ[R] i, f i` to the composition of\n  `PiTensorProduct.map f` with the natural isomorphism `⨂[R] i, R ≃ R` given by multiplication.\n-/\n\n@[expose] public section\n\nnamespace PiTensorProduct\n\nopen PiTensorProduct LinearMap Module TensorProduct\n\nvariable {ι : Type*}\n\nsection SemiRing\n\nvariable {R : Type*} {M : ι → Type*} [CommSemiring R] [Π i, AddCommMonoid (M i)]\n  [Π i, Module R (M i)]\n\n/-- The canonical linear map from `⨂[R] i, Dual R (M i)` to `Dual R (⨂[R] i, M i)`,\nsending `⨂ₜ[R] i, f i` to the composition of `PiTensorProduct.map f` with\nthe linear equivalence `⨂[R] i, R →ₗ R` given by multiplication. -/\nnoncomputable def dualDistrib [Finite ι] : (⨂[R] i, Dual R (M i)) →ₗ[R] Dual R (⨂[R] i, M i) :=\n  haveI := Fintype.ofFinite ι\n  (LinearMap.compRight _ (constantBaseRingEquiv ι R).toLinearMap) ∘ₗ piTensorHomMap\n\n@[simp]\ntheorem dualDistrib_apply [Fintype ι] (f : Π i, Dual R (M i)) (m : Π i, M i) :\n    dualDistrib (⨂ₜ[R] i, f i) (⨂ₜ[R] i, m i) = ∏ i, (f i) (m i) := by\n  rw [dualDistrib, Subsingleton.elim (Fintype.ofFinite ι) ‹_›]\n  simp\n\nend SemiRing\n\nsection Ring\n\nvariable {R : Type*} {κ : ι → Type*} {M : ι → Type*} [CommRing R] [Π i, AddCommGroup (M i)]\n  [Π i, Module R (M i)]\n\nopen Classical in\n/-- An inverse to `PiTensorProduct.dualDistrib` given bases. -/\nnoncomputable def dualDistribInvOfBasis [Finite ι] [∀ i, Finite (κ i)]\n    (b : Π i, Basis (κ i) R (M i)) :\n    Dual R (⨂[R] i, M i) →ₗ[R] ⨂[R] i, Dual R (M i) :=\n  haveI := Fintype.ofFinite ι\n  haveI := fun i => Fintype.ofFinite (κ i)\n  ∑ p : (Π i, κ i), (ringLmapEquivSelf R ℕ _).symm (⨂ₜ[R] i, (b i).dualBasis (p i)) ∘ₗ\n    (applyₗ (⨂ₜ[R] i, b i (p i)))\n\nopen Classical in\n@[simp]\ntheorem dualDistribInvOfBasis_apply [Fintype ι] [∀ i, Fintype (κ i)] (b : Π i, Basis (κ i) R (M i))\n    (f : Dual R (⨂[R] i, M i)) : dualDistribInvOfBasis b f =\n    ∑ p : (Π i, κ i), f (⨂ₜ[R] i, b i (p i)) • (⨂ₜ[R] i, (b i).dualBasis (p i)) := by\n  simp only [dualDistribInvOfBasis, Basis.coe_dualBasis, ringLmapEquivSelf_symm_apply, coe_sum,\n    coe_comp, coe_smulRight, End.one_apply, Finset.sum_apply, Function.comp_apply,\n    applyₗ_apply_apply]\n  convert! rfl\n\nTarget:\ntheorem dualDistrib_dualDistribInvOfBasis_left_inverse [Finite ι] [∀ i, Finite (κ i)]\n    (b : Π i, Basis (κ i) R (M i)) :\n    (dualDistrib) ∘ₗ (dualDistribInvOfBasis b) = LinearMap.id :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  haveI := Fintype.ofFinite ι\n  haveI := fun i => Fintype.ofFinite (κ i)\n  classical\n  refine (Basis.piTensorProduct b).dualBasis.ext (fun p ↦ ?_)\n  refine (Basis.piTensorProduct b).ext (fun q ↦ ?_)\n  simp [Finsupp.single_apply, Fintype.prod_ite_zero, ← funext_iff]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/PiTensorProduct","family_id":"dualdistrib_dualdistribinvofbasis_left_inverse","file_id":"mathlib/Mathlib/LinearAlgebra/PiTensorProduct/Dual.lean","sample_id":"57919ee9fe831200d4f5697405ab93efe8a6295ef8776bc2c7eff0fee53e76ad"},"split":"train"}
{"binding":{"chosen_proof_sha256":"894118c14dc5e14117bef012c677ec8f7a21a4cdd6ea74ed8d6415838166eb42","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3f8d2316a45406574dc026647834f24f20ab2b7e88800df83732567cc670b4eb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a4e28351bd9a0b41f4d5ba3c08dc4215d63a0d8b4b1907903fe2be3c09e51fcd","source_sha256":"82faa569879adae1105c3c846784886561d45f7b1b0c67c1d5374fc07267c247","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases hE : CompleteSpace E; swap\n  · simp [integral, hE, hasSum_zero]\n  rw [integral_tsum (fun i ↦ (hF_int i).1)]\n  · exact (hF_sum.of_norm_bounded fun i ↦ norm_integral_le_integral_norm _).hasSum\n  have (i : ι) : ∫⁻ a, ‖F i a‖ₑ ∂μ = ‖∫ a, ‖F i a‖ ∂μ‖ₑ := by\n    dsimp [enorm]\n    rw [lintegral_coe_eq_integral _ (hF_int i).norm, coe_nnreal_eq, coe_nnnorm,\n      Real.norm_of_nonneg (integral_nonneg (fun a ↦ norm_nonneg (F i a)))]\n    simp only [coe_nnnorm]\n  rw [funext this]\n  exact ENNReal.tsum_coe_ne_top_iff_summable.2 <| NNReal.summable_coe.1 hF_sum.abs","hard_negative":false,"metrics":{"chosen_tokens":144,"rejected_tokens":149,"token_jaccard":0.941176,"token_length_ratio":1.034722},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"dc4cc39a6d31e4ef9e7f2a32ed51faa859690c022d0202bd32c864cbc59099c0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable\npublic import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic\nimport Mathlib.Topology.Algebra.IsUniformGroup.Order\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2019 Zhouhang Zhou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Zhouhang Zhou, Yury Kudryashov, Patrick Massot, Louis (Yiyang) Liu\n-/\n/-!\n# The dominated convergence theorem\n\nThis file collects various results related to the Lebesgue dominated convergence theorem\nfor the Bochner integral.\n\n## Main results\n- `MeasureTheory.tendsto_integral_of_dominated_convergence`:\n  the Lebesgue dominated convergence theorem for the Bochner integral\n- `MeasureTheory.hasSum_integral_of_dominated_convergence`:\n  the Lebesgue dominated convergence theorem for series\n- `MeasureTheory.integral_tsum`, `MeasureTheory.integral_tsum_of_summable_integral_norm`:\n  the integral and `tsum`s commute, if the norms of the functions form a summable series\n- `intervalIntegral.hasSum_integral_of_dominated_convergence`: the Lebesgue dominated convergence\n  theorem for parametric interval integrals\n- `intervalIntegral.continuous_of_dominated_interval`: continuity of the interval integral\n  w.r.t. a parameter\n- `intervalIntegral.continuous_primitive` and friends: primitives of interval integrable\n  measurable functions are continuous\n\n-/\n\npublic section\n\nopen MeasureTheory Metric\n\n/-!\n## The Lebesgue dominated convergence theorem for the Bochner integral\n-/\nsection DominatedConvergenceTheorem\n\nopen Set Filter TopologicalSpace ENNReal\nopen scoped Topology Interval\n\nnamespace MeasureTheory\n\nvariable {α E G : Type*}\n  [NormedAddCommGroup E] [NormedSpace ℝ E]\n  [NormedAddCommGroup G] [NormedSpace ℝ G]\n  {m : MeasurableSpace α} {μ : Measure α}\n\n/-- **Lebesgue dominated convergence theorem** provides sufficient conditions under which almost\n  everywhere convergence of a sequence of functions implies the convergence of their integrals.\n  We could weaken the condition `bound_integrable` to require `HasFiniteIntegral bound μ` instead\n  (i.e. not requiring that `bound` is measurable), but in all applications proving integrability\n  is easier. -/\ntheorem tendsto_integral_of_dominated_convergence {F : ℕ → α → G} {f : α → G} (bound : α → ℝ)\n    (F_measurable : ∀ n, AEStronglyMeasurable (F n) μ) (bound_integrable : Integrable bound μ)\n    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)\n    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :\n    Tendsto (fun n => ∫ a, F n a ∂μ) atTop (𝓝 <| ∫ a, f a ∂μ) := by\n  simp only [integral_eq_setToFun]\n  exact tendsto_setToFun_of_dominated_convergence (dominatedFinMeasAdditive_weightedSMul μ)\n    bound F_measurable bound_integrable h_bound h_lim\n\n/-- Lebesgue dominated convergence theorem for filters with a countable basis -/\ntheorem tendsto_integral_filter_of_dominated_convergence {ι} {l : Filter ι} [l.IsCountablyGenerated]\n    {F : ι → α → G} {f : α → G} (bound : α → ℝ) (hF_meas : ∀ᶠ n in l, AEStronglyMeasurable (F n) μ)\n    (h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (bound_integrable : Integrable bound μ)\n    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a))) :\n    Tendsto (fun n => ∫ a, F n a ∂μ) l (𝓝 <| ∫ a, f a ∂μ) := by\n  simp only [integral_eq_setToFun]\n  exact tendsto_setToFun_filter_of_dominated_convergence (dominatedFinMeasAdditive_weightedSMul μ)\n    bound hF_meas h_bound bound_integrable h_lim\n\n/-- Lebesgue dominated convergence theorem for series. -/\ntheorem hasSum_integral_of_dominated_convergence {ι} [Countable ι] {F : ι → α → G} {f : α → G}\n    (bound : ι → α → ℝ) (hF_meas : ∀ n, AEStronglyMeasurable (F n) μ)\n    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound n a)\n    (bound_summable : ∀ᵐ a ∂μ, Summable fun n => bound n a)\n    (bound_integrable : Integrable (fun a => ∑' n, bound n a) μ)\n    (h_lim : ∀ᵐ a ∂μ, HasSum (fun n => F n a) (f a)) :\n    HasSum (fun n => ∫ a, F n a ∂μ) (∫ a, f a ∂μ) := by\n  simp only [integral_eq_setToFun]\n  exact hasSum_setToFun_of_dominated_convergence _ bound hF_meas h_bound bound_summable\n    bound_integrable h_lim\n\ntheorem integral_tsum {ι} [Countable ι] {f : ι → α → G} (hf : ∀ i, AEStronglyMeasurable (f i) μ)\n    (hf' : ∑' i, ∫⁻ a : α, ‖f i a‖ₑ ∂μ ≠ ∞) :\n    ∫ a, ∑' i, f i a ∂μ = ∑' i, ∫ a, f i a ∂μ := by\n  by_cases hG : CompleteSpace G; swap\n  · simp [integral, hG]\n  simp only [integral_eq_setToFun]\n  exact setToFun_tsum _ hf hf'\n\nTarget:\nlemma hasSum_integral_of_summable_integral_norm {ι} [Countable ι] {F : ι → α → E}\n    (hF_int : ∀ i : ι, Integrable (F i) μ) (hF_sum : Summable fun i ↦ ∫ a, ‖F i a‖ ∂μ) :\n    HasSum (∫ a, F · a ∂μ) (∫ a, (∑' i, F i a) ∂μ) :=\n\nProof body:\n","rejected":"by\n  by_cases hE : CompleteSpace E; swap\n  · simp [integral, hE, hasSum_zero]\n  rw [integral_tsum (fun i ↦ (hF_int i).1)]\n  · exact (hF_sum.of_norm_bounded fun i ↦ norm_integral_le_integral_norm _).hasSum\n  have (i : ι) : ∫⁻ a, ‖F i a‖ₑ ∂μ = ‖∫ a, ‖F i a‖ ∂μ‖ₑ := by\n    dsimp [enorm]\n    rw [lintegral_coe_eq_integral _ (hF_int i).norm, coe_nnreal_eq, coe_nnnorm,\n      Real.norm_of_nonneg (integral_nonneg (fun a ↦ norm_nonneg (F i a)))]\n    simp only [coe_nnnorm]\n  rw [funext this]\n  exact ENNReal.tsum_coe_ne_top_iff_summable.2 <| NNReal.summable_coe.1 hF_sum.abs\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Integral","family_id":"hassum_integral_of_summable_integral_norm","file_id":"mathlib/Mathlib/MeasureTheory/Integral/DominatedConvergence.lean","sample_id":"a4e28351bd9a0b41f4d5ba3c08dc4215d63a0d8b4b1907903fe2be3c09e51fcd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d46e9dc9cd151de5455487d42f3fd890f31162d28298263c77fbfdfebe8161e5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cf0e248544ee9d6ed6e902aaa1a7849b4ce61aeb489bff40d94a5eee7d981eac","source_sha256":"158ce289cc94776d7ace3f5f8788d7841da0114f64a958ffadba75342e4312aa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [destruct]\n  rcases f0 : s.1 0 with - | a' <;> intro h\n  · injection h with h'\n    rw [← h']\n    obtain ⟨f, al⟩ := s\n    apply Subtype.ext\n    dsimp [think, tail]\n    rw [← f0]\n    exact (Stream'.eta f).symm\n  · contradiction","hard_negative":false,"metrics":{"chosen_tokens":65,"rejected_tokens":5,"token_jaccard":0.041667,"token_length_ratio":0.076923},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"dc517ef0a9a5c9cd6925f53b5dc447ac7ed56163db9ef3b9325a5e586dbc7fe8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.Find\npublic import Mathlib.Data.Stream.Init\npublic import Mathlib.Logic.Relator\npublic import Mathlib.Tactic.Common\npublic import Batteries.Tactic.Lint.Simp\n\nNamespace:\nComputation\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Coinductive formalization of unbounded computations.\n\nThis file provides a `Computation` type where `Computation α` is the type of\nunbounded computations returning `α`.\n-/\n\n@[expose] public section\n\nopen Function\n\nuniverse u v w\n\n/-\ncoinductive Computation (α : Type u) : Type u\n| pure : α → Computation α\n| think : Computation α → Computation α\n-/\n/-- `Computation α` is the type of unbounded computations returning `α`.\n  An element of `Computation α` is an infinite sequence of `Option α` such\n  that if `f n = some a` for some `n` then it is constantly `some a` after that. -/\ndef Computation (α : Type u) : Type u :=\n  { f : Stream' (Option α) // ∀ ⦃n a⦄, f n = some a → f (n + 1) = some a }\n\nnamespace Computation\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n-- constructors\n/-- `pure a` is the computation that immediately terminates with result `a`. -/\ndef pure (a : α) : Computation α :=\n  ⟨Stream'.const (some a), fun _ _ => id⟩\n\ninstance : CoeTC α (Computation α) :=\n  ⟨pure⟩\n\n-- note [use has_coe_t]\n/-- `think c` is the computation that delays for one \"tick\" and then performs\n  computation `c`. -/\ndef think (c : Computation α) : Computation α :=\n  ⟨Stream'.cons none c.1, fun n a h => by\n    rcases n with - | n\n    · contradiction\n    · exact c.2 h⟩\n\n/-- `thinkN c n` is the computation that delays for `n` ticks and then performs\n  computation `c`. -/\ndef thinkN (c : Computation α) : ℕ → Computation α\n  | 0 => c\n  | n + 1 => think (thinkN c n)\n\n-- check for immediate result\n/-- `head c` is the first step of computation, either `some a` if `c = pure a`\n  or `none` if `c = think c'`. -/\ndef head (c : Computation α) : Option α :=\n  c.1.head\n\n-- one step of computation\n/-- `tail c` is the remainder of computation, either `c` if `c = pure a`\n  or `c'` if `c = think c'`. -/\ndef tail (c : Computation α) : Computation α :=\n  ⟨c.1.tail, fun _ _ h => c.2 h⟩\n\n/-- `empty α` is the computation that never returns, an infinite sequence of\n  `think`s. -/\ndef empty (α) : Computation α :=\n  ⟨Stream'.const none, fun _ _ => id⟩\n\ninstance : Inhabited (Computation α) :=\n  ⟨empty _⟩\n\n/-- `runFor c n` evaluates `c` for `n` steps and returns the result, or `none`\n  if it did not terminate after `n` steps. -/\ndef runFor : Computation α → ℕ → Option α :=\n  Subtype.val\n\n/-- `destruct c` is the destructor for `Computation α` as a coinductive type.\n  It returns `inl a` if `c = pure a` and `inr c'` if `c = think c'`. -/\ndef destruct (c : Computation α) : α ⊕ (Computation α) :=\n  match c.1 0 with\n  | none => Sum.inr (tail c)\n  | some a => Sum.inl a\n\n/-- `run c` is an unsound meta function that runs `c` to completion, possibly\n  resulting in an infinite loop in the VM. -/\nunsafe def run : Computation α → α\n  | c =>\n    match destruct c with\n    | Sum.inl a => a\n    | Sum.inr ca => run ca\n\ntheorem destruct_eq_pure {s : Computation α} {a : α} : destruct s = Sum.inl a → s = pure a := by\n  dsimp [destruct]\n  cases f0 : s.1 0 <;> intro h\n  · contradiction\n  · apply Subtype.ext\n    funext n\n    induction n with\n    | zero => injection h with h'; rwa [h'] at f0\n    | succ n IH => exact s.2 IH\n\nTarget:\ntheorem destruct_eq_think {s : Computation α} {s'} : destruct s = Sum.inr s' → s = think s' :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Seq","family_id":"destruct_eq_think","file_id":"mathlib/Mathlib/Data/Seq/Computation.lean","sample_id":"cf0e248544ee9d6ed6e902aaa1a7849b4ce61aeb489bff40d94a5eee7d981eac"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f9aa1f2b0c3ec5667a46cb932e6917f3da85ddecbe4e4201b9fff6f973d74358","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ca4d4f9887ee3a892e6506dc6023e4d2417fcaf8ede1273a6e1bb19013900ef6","source_sha256":"575fbe654cd51b260a28c1b494f58357a3030804869742bdef7a7dc326096822","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨a, b, hij⟩ | ⟨a, b, hab⟩ := h\n  · exact Lex.left _ _ (hr _ _ hij)\n  · exact Lex.right _ _ (hs _ _ _ hab)","hard_negative":true,"metrics":{"chosen_tokens":46,"rejected_tokens":8,"token_jaccard":0.071429,"token_length_ratio":0.173913},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"dcc05359c9291761e1844ec50340d7fb715564638a7956ec09a5d91eb1ad8029","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Logic.Function.Defs\npublic import Mathlib.Order.Defs.Unbundled\npublic import Batteries.Logic\n\nNamespace:\nSigma\n\nLocal context:\n/-\nCopyright (c) 2021 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Lexicographic order on a sigma type\n\nThis defines the lexicographical order of two arbitrary relations on a sigma type and proves some\nlemmas about `PSigma.Lex`, which is defined in core Lean.\n\nGiven a relation in the index type and a relation on each summand, the lexicographical order on the\nsigma type relates `a` and `b` if their summands are related or they are in the same summand and\nrelated by the summand's relation.\n\n## See also\n\nRelated files are:\n* `Combinatorics.CoLex`: Colexicographic order on finite sets.\n* `Data.List.Lex`: Lexicographic order on lists.\n* `Data.Sigma.Order`: Lexicographic order on `Σ i, α i` per say.\n* `Data.PSigma.Order`: Lexicographic order on `Σ' i, α i`.\n* `Data.Prod.Lex`: Lexicographic order on `α × β`. Can be thought of as the special case of\n  `Sigma.Lex` where all summands are the same\n-/\n\npublic section\n\n\nnamespace Sigma\n\nvariable {ι : Type*} {α : ι → Type*} {r r₁ r₂ : ι → ι → Prop} {s s₁ s₂ : ∀ i, α i → α i → Prop}\n  {a b : Σ i, α i}\n\n/-- The lexicographical order on a sigma type. It takes in a relation on the index type and a\nrelation for each summand. `a` is related to `b` iff their summands are related or they are in the\nsame summand and are related through the summand's relation. -/\ninductive Lex (r : ι → ι → Prop) (s : ∀ i, α i → α i → Prop) : ∀ _ _ : Σ i, α i, Prop\n  | left {i j : ι} (a : α i) (b : α j) : r i j → Lex r s ⟨i, a⟩ ⟨j, b⟩\n  | right {i : ι} (a b : α i) : s i a b → Lex r s ⟨i, a⟩ ⟨i, b⟩\n\ntheorem lex_iff : Lex r s a b ↔ r a.1 b.1 ∨ ∃ h : a.1 = b.1, s b.1 (h.rec a.2) b.2 := by\n  constructor\n  · rintro (⟨a, b, hij⟩ | ⟨a, b, hab⟩)\n    · exact Or.inl hij\n    · exact Or.inr ⟨rfl, hab⟩\n  · obtain ⟨i, a⟩ := a\n    dsimp only\n    rintro (h | ⟨rfl, h⟩)\n    · exact Lex.left _ _ h\n    · exact Lex.right _ _ h\n\ninstance Lex.decidable (r : ι → ι → Prop) (s : ∀ i, α i → α i → Prop) [DecidableEq ι]\n    [DecidableRel r] [∀ i, DecidableRel (s i)] : DecidableRel (Lex r s) := fun _ _ =>\n  decidable_of_decidable_of_iff lex_iff.symm\n\nTarget:\ntheorem Lex.mono (hr : ∀ a b, r₁ a b → r₂ a b) (hs : ∀ i a b, s₁ i a b → s₂ i a b) {a b : Σ i, α i}\n    (h : Lex r₁ s₁ a b) : Lex r₂ s₂ a b :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"fb5945bfef9761abaa40a4699371e72c7942f202e9d512de263d772c1281314e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sigma","family_id":"lex","file_id":"mathlib/Mathlib/Data/Sigma/Lex.lean","sample_id":"ca4d4f9887ee3a892e6506dc6023e4d2417fcaf8ede1273a6e1bb19013900ef6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dd729798355787e2f50246686d3074ecb0b708500ef8ae38cb56d8d7dbde9c7f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8657a579acb61ab7d8c3c4f5eccef630cadd2b7fd2cc9beb288901f9e0b06266","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"724a8437e52e6581c6973db3a8c052c428831eeb06b3711d37815c86ff48be5a","source_sha256":"77767c60d3dd433d634008abee2943f40d23ac4eab051267efb892f5f5bd8ab6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  induction s using Finset.induction with\n  | empty => simp\n  | insert _ _ _ _ => simp_all [lcm_eq_nat_lcm, Nat.factorization_lcm]","hard_negative":false,"metrics":{"chosen_tokens":28,"rejected_tokens":35,"token_jaccard":0.913043,"token_length_ratio":1.25},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"dd2584b875cfc15da3567ad48bc83d7a6e25e9251521b6467ca181fffc6d6c56","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GCDMonoid.Finset\npublic import Mathlib.Algebra.GCDMonoid.Nat\npublic import Mathlib.Data.Nat.GCD.Basic\npublic import Mathlib.RingTheory.Coprime.Lemmas\npublic import Mathlib.Data.Nat.Factorization.Basic\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2025 Snir Broshi. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Snir Broshi\n-/\n/-!\n# `Finset.lcm` lemmas\n\n## Tags\n\nfinset, lcm, prod, coprime, Rat.den\n-/\n\npublic section\n\nnamespace Finset\n\nvariable {ι α : Type*} [CommMonoidWithZero α] [NormalizedGCDMonoid α]\n\ntheorem lcm_dvd_prod (s : Finset ι) (f : ι → α) : s.lcm f ∣ s.prod f :=\n  lcm_dvd fun _ ↦ dvd_prod_of_mem _\n\ntheorem associated_lcm_prod {s : Finset ι} {f : ι → α} (h : Set.Pairwise s <| IsRelPrime.onFun f) :\n    Associated (s.lcm f) (s.prod f) :=\n  associated_of_dvd_dvd (s.lcm_dvd_prod f) (s.prod_dvd_of_isRelPrime h fun _ ↦ dvd_lcm)\n\ntheorem lcm_eq_prod {s : Finset ι} {f : ι → ℕ} (h : Set.Pairwise s <| Nat.Coprime.onFun f) :\n    s.lcm f = s.prod f := by\n  rw [show Nat.Coprime = IsRelPrime by ext; exact Nat.coprime_iff_isRelPrime] at h\n  exact associated_lcm_prod h |>.eq_of_normalized (normalize_eq _) (normalize_eq _)\n\n/-- An analogue of `Nat.factorization_lcm` for `Finset.lcm`. -/\n\nTarget:\ntheorem factorization_lcm {f : ι → ℕ} {s : Finset ι} (hf : ∀ k ∈ s, f k ≠ 0) (p : ℕ) :\n    (s.lcm f).factorization p = s.sup fun a ↦ (f a).factorization p :=\n\nProof body:\n","rejected":"```lean\nby\n  classical\n  induction s using Finset.induction with\n  | empty => simp\n  | insert _ _ _ _ => simp_all [lcm_eq_nat_lcm, Nat.factorization_lcm]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/GCDMonoid","family_id":"factorization_lcm","file_id":"mathlib/Mathlib/Algebra/GCDMonoid/FinsetLemmas.lean","sample_id":"724a8437e52e6581c6973db3a8c052c428831eeb06b3711d37815c86ff48be5a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a1aa8b6078219bed7946cc7376ea4840b9f886018bc53ce22deefb93a518874b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9eef900fc40300308ff1ecb2aed829f9ba2f08a85e294734879bf55ae56f16ca","source_sha256":"fee0fafbf6e1c4c8ae7d7cbed3ad72a2c79bb54752f320fe911fff838084de11","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  change c = (ZMod.ringEquivCongr hn) (toFun n g)\n  rw [← toFun_unique' n g (ZMod.ringEquivCongr hn.symm c)\n    (fun t ht ↦ by rw [hc t ht, ZMod.ringEquivCongr_val]), ← ZMod.ringEquivCongr_symm hn,\n    RingEquiv.apply_symm_apply]","hard_negative":true,"metrics":{"chosen_tokens":58,"rejected_tokens":8,"token_jaccard":0.028571,"token_length_ratio":0.137931},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"dd8ea58cd440957f0c7eb0093a3d20f7f5edee70c52c4597835ab67943ed0a03","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.Aut\npublic import Mathlib.NumberTheory.Padics.RingHoms\npublic import Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity\npublic import Mathlib.RingTheory.RootsOfUnity.Minpoly\npublic import Mathlib.FieldTheory.KrullTopology\n\nNamespace:\nmodularCyclotomicCharacter\n\nLocal context:\n/-\nCopyright (c) 2023 Hanneke Wiersema. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Hanneke Wiersema, Andrew Yang\n-/\n/-!\n\n# The cyclotomic character\n\nLet `L` be an integral domain and let `n : ℕ+` be a positive integer. If `μₙ` is the\ngroup of `n`th roots of unity in `L` then any field automorphism `g` of `L`\ninduces an automorphism of `μₙ` which, being a cyclic group, must be of\nthe form `ζ ↦ ζ^j` for some integer `j = j(g)`, well-defined in `ZMod d`, with\n`d` the cardinality of `μₙ`. The function `j` is a group homomorphism\n`(L ≃+* L) →* ZMod d`.\n\nFuture work: If `L` is separably closed (e.g. algebraically closed) and `p` is a prime\nnumber such that `p ≠ 0` in `L`, then applying the above construction with\n`n = p^i` (noting that the size of `μₙ` is `p^i`) gives a compatible collection of\ngroup homomorphisms `(L ≃+* L) →* ZMod (p^i)` which glue to give\na group homomorphism `(L ≃+* L) →* ℤₚ`; this is the `p`-adic cyclotomic character.\n\n## Important definitions\n\nLet `L` be an integral domain, `g : L ≃+* L` and `n : ℕ+`. Let `d` be the number of `n`th roots\nof `1` in `L`.\n\n* `modularCyclotomicCharacter L n hn : (L ≃+* L) →* (ZMod n)ˣ` sends `g` to the unique `j` such\n  that `g(ζ)=ζ^j` for all `ζ : rootsOfUnity n L`. Here `hn` is a proof that there\n  are `n` `n`th roots of unity in `L`.\n\n* `cyclotomicCharacter L p : (L ≃+* L) →* ℤ_[p]ˣ` sends `g` to the unique `j` such\n  that `g(ζ) = ζ ^ (j mod pⁱ)` for all `pⁱ`-th roots of unity `ζ`.\n\n  Note: This is defined to be the trivial character if `L` does not have enough roots of unity.\n\n## Implementation note\n\nIn theory this could be set up as some theory about monoids, being a character\non monoid isomorphisms, but under the hypotheses that the `n`-th roots of unity\nare cyclic. The advantage of sticking to integral domains is that finite subgroups\nare guaranteed to be cyclic, so the weaker assumption that there are `n` `n`th\nroots of unity is enough. All the applications I'm aware of are when `L` is a\nfield anyway.\n\nAlthough I don't know whether it's of any use, `modularCyclotomicCharacter'`\nis the general case for integral domains, with target in `(ZMod d)ˣ`\nwhere `d` is the number of `n`th roots of unity in `L`.\n\n## TODO\n\n* Prove the compatibility of `modularCyclotomicCharacter n` and `modularCyclotomicCharacter m`\n  if `n ∣ m`.\n\n## Tags\n\ncyclotomic character\n-/\n\n@[expose] public section\n\nuniverse u\nvariable {L : Type u} [CommRing L] [IsDomain L]\n\n/-\n\n## The mod n theory\n\n-/\n\nvariable (n : ℕ) [NeZero n]\n\ntheorem rootsOfUnity.integer_power_of_ringEquiv (g : L ≃+* L) :\n    ∃ m : ℤ, ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ m : Lˣ) := by\n  obtain ⟨m, hm⟩ := MonoidHom.map_cyclic ((g : L ≃* L).restrictRootsOfUnity n).toMonoidHom\n  exact ⟨m, fun t ↦ Units.ext_iff.1 <| SetCoe.ext_iff.2 <| hm t⟩\n\ntheorem rootsOfUnity.integer_power_of_ringEquiv' (g : L ≃+* L) :\n    ∃ m : ℤ, ∀ t ∈ rootsOfUnity n L, g (t : Lˣ) = (t ^ m : Lˣ) := by\n  simpa using rootsOfUnity.integer_power_of_ringEquiv n g\n\n/-- `modularCyclotomicCharacter_aux g n` is a non-canonical auxiliary integer `j`,\n  only well-defined modulo the number of `n`-th roots of unity in `L`, such that `g(ζ)=ζ^j`\n  for all `n`-th roots of unity `ζ` in `L`. -/\nnoncomputable def modularCyclotomicCharacter.aux (g : L ≃+* L) (n : ℕ) [NeZero n] : ℤ :=\n  (rootsOfUnity.integer_power_of_ringEquiv n g).choose\n\n-- the only thing we know about `modularCyclotomicCharacter_aux g n`\ntheorem modularCyclotomicCharacter.aux_spec (g : L ≃+* L) (n : ℕ) [NeZero n] :\n    ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ (modularCyclotomicCharacter.aux g n) : Lˣ) :=\n  (rootsOfUnity.integer_power_of_ringEquiv n g).choose_spec\n\ntheorem modularCyclotomicCharacter.pow_dvd_aux_pow_sub_aux_pow\n    (g : L ≃+* L) (p : ℕ) [Fact p.Prime] [∀ i, HasEnoughRootsOfUnity L (p ^ i)]\n    {i k : ℕ} (hi : k ≤ i) : (p : ℤ) ^ k ∣ aux g (p ^ i) - aux g (p ^ k) := by\n  obtain ⟨i, rfl⟩ := exists_add_of_le hi\n  obtain ⟨ζ, hζ⟩ := HasEnoughRootsOfUnity.exists_primitiveRoot L (p ^ (k + i))\n  have h := hζ.pow (a := p ^ i) (Nat.pos_of_neZero _) (Nat.pow_add' _ _ _)\n  have h_unit : (h.isUnit NeZero.out).unit =\n      (hζ.isUnit NeZero.out).unit ^ (p ^ i) := by ext; rfl\n  have H₁ := aux_spec g (p ^ (k + i))\n    ⟨_, (hζ.isUnit_unit NeZero.out).mem_rootsOfUnity⟩\n  have H₂ := aux_spec g (p ^ k)\n    ⟨_, (h.isUnit_unit NeZero.out).mem_rootsOfUnity⟩\n  simp only [IsUnit.unit_spec, map_pow] at H₁ H₂\n  rw [H₁, ← Units.val_pow_eq_pow_val, ← Units.ext_iff, h_unit, ← div_eq_one] at H₂\n  simp only [← zpow_natCast, ← zpow_mul, div_eq_mul_inv, ← zpow_sub] at H₂\n  rw [(hζ.isUnit_unit NeZero.out).zpow_eq_one_iff_dvd, mul_comm, ← mul_sub] at H₂\n  conv_lhs at H₂ => rw [Nat.pow_add', Nat.cast_mul]\n  rwa [mul_dvd_mul_iff_left (by simp [NeZero.ne p]), Nat.cast_pow] at H₂\n\n/-- If `g` is a ring automorphism of `L`, and `n : ℕ+`, then\n  `modularCyclotomicCharacter.toFun n g` is the `j : ZMod d` such that `g(ζ)=ζ^j` for all\n  `n`-th roots of unity. Here `d` is the number of `n`th roots of unity in `L`. -/\nnoncomputable def modularCyclotomicCharacter.toFun (n : ℕ) [NeZero n] (g : L ≃+* L) :\n    ZMod (Fintype.card (rootsOfUnity n L)) :=\n  modularCyclotomicCharacter.aux g n\n\nnamespace modularCyclotomicCharacter\n\nlocal notation \"χ₀\" => modularCyclotomicCharacter.toFun\n\n/-- The formula which characterises the output of `modularCyclotomicCharacter g n`. -/\ntheorem toFun_spec (g : L ≃+* L) {n : ℕ} [NeZero n] (t : rootsOfUnity n L) :\n    g (t : Lˣ) = (t ^ (χ₀ n g).val : Lˣ) := by\n  rw [modularCyclotomicCharacter.aux_spec g n t, ← zpow_natCast, modularCyclotomicCharacter.toFun,\n    ZMod.val_intCast, ← Subgroup.coe_zpow]\n  exact Units.ext_iff.1 <| SetCoe.ext_iff.2 <|\n    zpow_eq_zpow_emod _ pow_card_eq_one (G := rootsOfUnity n L)\n\ntheorem toFun_spec' (g : L ≃+* L) {n : ℕ} [NeZero n] {t : Lˣ} (ht : t ∈ rootsOfUnity n L) :\n    g t = t ^ (χ₀ n g).val :=\n  toFun_spec g ⟨t, ht⟩\n\ntheorem toFun_spec'' (g : L ≃+* L) {n : ℕ} [NeZero n] {t : L} (ht : IsPrimitiveRoot t n) :\n    g t = t ^ (χ₀ n g).val :=\n  toFun_spec' g (SetLike.coe_mem ht.toRootsOfUnity)\n\n/-- If g(t)=t^c for all roots of unity, then c=χ(g). -/\ntheorem toFun_unique (g : L ≃+* L) (c : ZMod (Fintype.card (rootsOfUnity n L)))\n    (hc : ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ c.val : Lˣ)) : c = χ₀ n g := by\n  apply IsCyclic.ext Nat.card_eq_fintype_card (fun ζ ↦ ?_)\n  specialize hc ζ\n  suffices ((ζ ^ c.val : Lˣ) : L) = (ζ ^ (χ₀ n g).val : Lˣ) by exact_mod_cast this\n  rw [← toFun_spec g ζ, hc]\n\ntheorem toFun_unique' (g : L ≃+* L) (c : ZMod (Fintype.card (rootsOfUnity n L)))\n    (hc : ∀ t ∈ rootsOfUnity n L, g t = t ^ c.val) : c = χ₀ n g :=\n  toFun_unique n g c (fun ⟨_, ht⟩ ↦ hc _ ht)\n\nlemma id : χ₀ n (RingEquiv.refl L) = 1 := by\n  refine (toFun_unique n (RingEquiv.refl L) 1 <| fun t ↦ ?_).symm\n  have : 1 ≤ Fintype.card { x // x ∈ rootsOfUnity n L } := Fin.size_positive'\n  obtain (h | h) := this.lt_or_eq\n  · have := Fact.mk h\n    simp [ZMod.val_one]\n  · have := Fintype.card_le_one_iff_subsingleton.mp h.ge\n    obtain rfl : t = 1 := Subsingleton.elim t 1\n    simp\n\nlemma comp (g h : L ≃+* L) : χ₀ n (g * h) =\n    χ₀ n g * χ₀ n h := by\n  refine (toFun_unique n (g * h) _ <| fun ζ ↦ ?_).symm\n  change g (h (ζ : Lˣ)) = _\n  rw [toFun_spec, ← Subgroup.coe_pow, toFun_spec, mul_comm, Subgroup.coe_pow, ← pow_mul,\n    ← Subgroup.coe_pow]\n  congr 2\n  norm_cast\n  simp only [pow_eq_pow_iff_modEq, ← ZMod.natCast_eq_natCast_iff,\n    ZMod.natCast_val, Nat.cast_mul, ZMod.cast_mul (m := orderOf ζ) orderOf_dvd_card]\n\nend modularCyclotomicCharacter\n\nvariable (L)\n\n/-- Given a positive integer `n`, `modularCyclotomicCharacter' n` is a\nmultiplicative homomorphism from the automorphisms of a field `L` to `(ℤ/dℤ)ˣ`,\nwhere `d` is the number of `n`-th roots of unity in `L`. It is uniquely\ncharacterised by the property that `g(ζ)=ζ^(modularCyclotomicCharacter n g)`\nfor `g` an automorphism of `L` and `ζ` an `n`th root of unity. -/\nnoncomputable\ndef modularCyclotomicCharacter' (n : ℕ) [NeZero n] :\n    (L ≃+* L) →* (ZMod (Fintype.card { x // x ∈ rootsOfUnity n L }))ˣ := MonoidHom.toHomUnits\n  { toFun := modularCyclotomicCharacter.toFun n\n    map_one' := modularCyclotomicCharacter.id n\n    map_mul' := modularCyclotomicCharacter.comp n }\n\nlemma modularCyclotomicCharacter'.spec' (g : L ≃+* L) {t : Lˣ} (ht : t ∈ rootsOfUnity n L) :\n    g t = t ^ ((modularCyclotomicCharacter' L n g) : ZMod\n      (Fintype.card { x // x ∈ rootsOfUnity n L })).val :=\n  modularCyclotomicCharacter.toFun_spec' g ht\n\nlemma modularCyclotomicCharacter'.unique' (g : L ≃+* L)\n    {c : ZMod (Fintype.card { x // x ∈ rootsOfUnity n L })}\n    (hc : ∀ t ∈ rootsOfUnity n L, g t = t ^ c.val) :\n    c = modularCyclotomicCharacter' L n g :=\n  modularCyclotomicCharacter.toFun_unique' _ _ _ hc\n\n/-- Given a positive integer `n` and a field `L` containing `n` `n`th roots\nof unity, `modularCyclotomicCharacter n` is a multiplicative homomorphism from the\nautomorphisms of `L` to `(ℤ/nℤ)ˣ`. It is uniquely characterised by the property that\n`g(ζ)=ζ^(modularCyclotomicCharacter n g)` for `g` an automorphism of `L` and `ζ` any `n`th root\nof unity. -/\nnoncomputable def modularCyclotomicCharacter {n : ℕ} [NeZero n]\n    (hn : Fintype.card { x // x ∈ rootsOfUnity n L } = n) :\n    (L ≃+* L) →* (ZMod n)ˣ :=\n  (Units.mapEquiv <| (ZMod.ringEquivCongr hn).toMulEquiv).toMonoidHom.comp\n  (modularCyclotomicCharacter' L n)\n\nnamespace modularCyclotomicCharacter\n\nvariable {n : ℕ} [NeZero n] (hn : Fintype.card { x // x ∈ rootsOfUnity n L } = n)\n\nlemma spec (g : L ≃+* L) {t : Lˣ} (ht : t ∈ rootsOfUnity n L) :\n    g t = t ^ ((modularCyclotomicCharacter L hn g) : ZMod n).val := by\n  rw [toFun_spec' g ht]\n  congr 1\n  exact (ZMod.ringEquivCongr_val _ _).symm\n\nTarget:\nlemma unique (g : L ≃+* L) {c : ZMod n} (hc : ∀ t ∈ rootsOfUnity n L, g t = t ^ c.val) :\n    c = modularCyclotomicCharacter L hn g :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e97e24884878fe0740831dff521ece2b8373761f634b9ff757f1b8c7bf91c9e7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Cyclotomic","family_id":"unique","file_id":"mathlib/Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean","sample_id":"9eef900fc40300308ff1ecb2aed829f9ba2f08a85e294734879bf55ae56f16ca"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5e09f885c765d1667ba798f4a88bd6a2385014bfb40b892ea398481ff81d2e8d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"87892b698b5aa0c2157518ba3c76b8916dafacf71da97fb690d2885a5985adff","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"abc072079305390f0bef511c2c60314512d25bfb1b8f705b8e28581cf1707079","source_sha256":"c7fff7c9f2f71a64bbae1f040362fa5f16b7992ea23b8c9d1ff27baa1835cd99","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  -- The key is that φ preserves labels, so edges labelled 's' stay labelled 's'\n  let e : v ⟶ ι s • v := ⟨s, rfl⟩\n  -- φ.map e is an edge from φ.obj v, and its label is preserved\n  have h := (φ.map e).property\n  -- This says: `ι (φ.map e).val • φ.obj v = φ.obj (ι s • v)`\n  -- We need to show `(φ.map e).val = s`\n  have label_eq : (φ.map e).val = s := by\n    -- `φm` says `φ ⋙q labelling = labelling`\n    -- So `(φ ⋙q labelling).map e = labelling.map e`\n    have : (φ ⋙q labelling W ι).map e = (labelling V ι).map e := by\n      rw [φm]\n    simp only [Prefunctor.comp_map, labelling_map] at this\n    exact this\n  rw [label_eq] at h\n  exact h.symm","hard_negative":false,"metrics":{"chosen_tokens":216,"rejected_tokens":221,"token_jaccard":0.96,"token_length_ratio":1.023148},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"dd975c851d378a1fee31d4ddf6ca2bccacb73e1cf08593c553fd6e4c99205214","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Defs\npublic import Mathlib.Algebra.Group.Basic\npublic import Mathlib.Combinatorics.Quiver.Covering\npublic import Mathlib.Combinatorics.Quiver.SingleObj\n\nNamespace:\nQuiver.SchreierGraph\n\nLocal context:\n/-\nCopyright (c) 2026 Runtian Zhou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Runtian Zhou\n-/\n/-!\n# Schreier Graphs\n\nThis module defines Schreier graphs as quivers with labelled edges.\n\nGiven a monoid `M` acting on a type `V` and a map `ι : S → M`, the Schreier graph has\nvertices `V` and a directed edge `x → ι(s) • x` for each `x : V` and `s : S`.\n\n## Main definitions\n\n* `SchreierGraph V ι` - The Schreier graph of an action, with vertices of type `V` and edges\n  labelled by elements of `S` via `ι : S → M`.\n* `SchreierGraph.labelling` - The prefunctor from a Schreier graph to `SingleObj S` that\n  extracts edge labels.\n\n## Main results\n\n* `SchreierGraph.labelling_isCovering` - The labelling prefunctor is a covering when we have\n  a group action.\n\n## Examples\n\n* The (left) **Cayley graph** of a group `M` with generators `ι : S → M` is the Schreier graph\n  where `V = M` and the action is left multiplication.\n\n## Implementation notes\n\nAlthough referred to informally as graphs, Schreier graphs have multiple, directed, labelled\nedges between nodes and so are implemented here as quivers.\n\n## References\n\n* [Y. Vorobets, *Notes on the Schreier graphs of the Grigorchuk group*][Vorobets2012]\n-/\n\n@[expose] public section\n\nnamespace Quiver\n\n/-- A Schreier graph for a monoid `M` acting on `V` with generators `ι : S → M`.\nVertices are elements of `V`, and there is an edge from `x` to `y` for each `s : S`\nsuch that `ι s • x = y`. -/\n@[nolint unusedArguments, ext]\nstructure SchreierGraph (V : Type*) {M : Type*} [SMul M V] {S : Type*} (_ι : S → M) where\n  /-- Wraps a vertex of the acted-upon type into the Schreier graph. -/\n  ofVertex ::\n  /-- The underlying vertex. -/\n  toVertex : V\n\nnamespace SchreierGraph\n\nsection Basic\n\nvariable (V : Type*) {M : Type*} [SMul M V] {S : Type*} (ι : S → M)\n\n/-- Equivalence between the original vertex type and the Schreier graph type. -/\n@[simps]\ndef equiv : V ≃ SchreierGraph V ι where\n  toFun := SchreierGraph.ofVertex\n  invFun := SchreierGraph.toVertex\n  left_inv _ := rfl\n  right_inv _ := rfl\n\n/-- Transport the scalar multiplication to the Schreier graph vertices. -/\ninstance schreierGraphSMul : SMul M (SchreierGraph V ι) where\n  smul x y := ⟨x • y.toVertex⟩\n\n/-- The quiver structure on a Schreier graph. An arrow from `x` to `y` exists when\nthere is an `s : S` such that `(ι s) • x = y`. -/\ninstance schreierGraphQuiver : Quiver (SchreierGraph V ι) where\n  Hom x y := { s : S // (ι s) • x = y }\n\n/-- The labelling of arrows in a Schreier graph by elements of `S`.\nThis is encoded as a prefunctor to `SingleObj S`. -/\n@[simps]\ndef labelling : SchreierGraph V ι ⥤q SingleObj S where\n  obj _ := SingleObj.star S\n  map e := e.val\n\nend Basic\n\nsection MulAction\n\nvariable (V : Type*) {M : Type*} [Monoid M] [MulAction M V] {S : Type*} (ι : S → M)\n\n/-- The monoid acts on the vertices of the Schreier graph. -/\ninstance : MulAction M (SchreierGraph V ι) where\n  one_smul x := by\n    ext\n    exact one_smul M x.toVertex\n  mul_smul a b x := by\n    ext\n    exact mul_smul a b x.toVertex\n\nend MulAction\n\nsection GroupAction\n\n/-!\n### Schreier graphs for group actions\n\nWhen we have a group action, the labelling becomes a covering.\n-/\n\nvariable {V : Type*} {M : Type*} [Group M] [MulAction M V] {S : Type*} (ι : S → M)\n\n/-- The star map of the labelling prefunctor as an equivalence. -/\n@[simps]\ndef labellingStarEquiv (x : SchreierGraph V ι) :\n    Quiver.Star x ≃ Quiver.Star (SingleObj.star S) where\n  toFun := (labelling V ι).star x\n  invFun := fun ⟨_, s⟩ => ⟨ι s • x, s, rfl⟩\n  left_inv := fun ⟨_, _, rfl⟩ => rfl\n  right_inv := fun ⟨_, _⟩ => rfl\n\n/-- The costar map of the labelling prefunctor as an equivalence. -/\n@[simps]\ndef labellingCostarEquiv (x : SchreierGraph V ι) :\n    Quiver.Costar x ≃ Quiver.Costar (SingleObj.star S) where\n  toFun := (labelling V ι).costar x\n  invFun := fun ⟨_, s⟩ => ⟨(ι s)⁻¹ • x, s, by simp⟩\n  left_inv := by\n    rintro ⟨v, s, hs⟩\n    simp only [Prefunctor.costar_apply, labelling_map]\n    have : (ι s)⁻¹ • x = v := by rw [← hs, inv_smul_smul]\n    subst this; rfl\n  right_inv := fun ⟨_, _⟩ => rfl\n\n/-- The labelling prefunctor is a covering for Schreier graphs with group actions. -/\ntheorem labelling_isCovering : (labelling V ι).IsCovering where\n  star_bijective u := (labellingStarEquiv ι u).bijective\n  costar_bijective u := (labellingCostarEquiv ι u).bijective\n\n/-- If a prefunctor between Schreier graphs commutes with the labelling (i.e., labels are\npreserved), then it commutes with the group action. In other words, morphisms that preserve edge\nlabels also preserve the group structure. -/\n\nTarget:\nlemma map_smul_of_comp_labelling_eq {W : Type*} [MulAction M W]\n    (φ : SchreierGraph V ι ⥤q SchreierGraph W ι) (φm : φ ⋙q labelling W ι = labelling V ι)\n    (v : SchreierGraph V ι) (s : S) :\n    φ.obj (ι s • v) = ι s • (φ.obj v) :=\n\nProof body:\n","rejected":"by\n  -- The key is that φ preserves labels, so edges labelled 's' stay labelled 's'\n  let e : v ⟶ ι s • v := ⟨s, rfl⟩\n  -- φ.map e is an edge from φ.obj v, and its label is preserved\n  have h := (φ.map e).property\n  -- This says: `ι (φ.map e).val • φ.obj v = φ.obj (ι s • v)`\n  -- We need to show `(φ.map e).val = s`\n  have label_eq : (φ.map e).val = s := by\n    -- `φm` says `φ ⋙q labelling = labelling`\n    -- So `(φ ⋙q labelling).map e = labelling.map e`\n    have : (φ ⋙q labelling W ι).map e = (labelling V ι).map e := by\n      rw [φm]\n    simp only [Prefunctor.comp_map, labelling_map] at this\n    exact this\n  rw [label_eq] at h\n  exact h.symm\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Quiver","family_id":"map_smul_of_comp_labelling_eq","file_id":"mathlib/Mathlib/Combinatorics/Quiver/Schreier.lean","sample_id":"abc072079305390f0bef511c2c60314512d25bfb1b8f705b8e28581cf1707079"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e9b659a1781e41131bb39de2a0211b972065e3468a0c84eab21865825ba5a5a7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7f463aa725b7762c153926d14972d9db6f40590bb5632a97064d1dc9f7d0cf39","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9c259b24270f51c0a7679a42e8272d4047f65364bf40859a29277a457689ac84","source_sha256":"b5b8fef319876fc584704ec0902863ab8016fe0ba4dee5dd358f86f4688b13cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Rat.isSquare_intCast_iff, ← irrational_sqrt_ratCast_iff_of_nonneg (mod_cast hz),\n    Rat.cast_intCast]","hard_negative":true,"metrics":{"chosen_tokens":19,"rejected_tokens":2,"token_jaccard":0.0625,"token_length_ratio":0.105263},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"ddc547b0ff98548a0b9e86d840355a1165de28e12642e09fb746fc783fc63977","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Rat\npublic import Mathlib.Data.Nat.Prime.Int\npublic import Mathlib.Data.Rat.Sqrt\npublic import Mathlib.Analysis.Real.Sqrt\npublic import Mathlib.RingTheory.Algebraic.Basic\npublic import Mathlib.Tactic.CrossRefAttribute\npublic import Mathlib.Tactic.IntervalCases\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov\n-/\n/-!\n# Irrational real numbers\n\nIn this file we define a predicate `Irrational` on `ℝ`, prove that the `n`-th root of an integer\nnumber is irrational if it is not integer, and that `√(q : ℚ)` is irrational if and only if\n`¬IsSquare q ∧ 0 ≤ q`.\n\nWe also provide dot-style constructors like `Irrational.add_ratCast`, `Irrational.ratCast_sub` etc.\n\nWith the `Decidable` instances in this file, is possible to prove `Irrational √n` using `decide`,\nwhen `n` is a numeric literal or cast;\nbut this only works if you `unseal Nat.sqrt.iter in` before the theorem where you use this proof.\n-/\n\n@[expose] public section\n\n\nopen Rat Real\n\n/-- A real number is irrational if it is not equal to any rational number. -/\n@[wikidata Q607728]\ndef Irrational (x : ℝ) :=\n  x ∉ Set.range ((↑) : ℚ → ℝ)\n\ntheorem irrational_iff_ne_rational (x : ℝ) : Irrational x ↔ ∀ a b : ℤ, b ≠ 0 → x ≠ a / b := by\n  simp [Irrational, Rat.forall, eq_comm]\n\ntheorem Irrational.ne_rational {x : ℝ} (hx : Irrational x) (a b : ℤ) : x ≠ a / b := by\n  rintro rfl; exact hx ⟨a / b, by simp⟩\n\ntheorem exists_rat_of_not_irrational {x : ℝ} (hx : ¬ Irrational x) : ∃ (q : ℚ), x = q := by\n  grind [Irrational]\n\n/-- A transcendental real number is irrational. -/\ntheorem Transcendental.irrational {r : ℝ} (tr : Transcendental ℚ r) : Irrational r := by\n  rintro ⟨a, rfl⟩\n  exact tr (isAlgebraic_algebraMap a)\n\n/-!\n### Irrationality of roots of integer and rational numbers\n-/\n\n\n/-- If `x^n`, `n > 0`, is integer and is not the `n`-th power of an integer, then\n`x` is irrational. -/\ntheorem irrational_nrt_of_notint_nrt {x : ℝ} (n : ℕ) (m : ℤ) (hxr : x ^ n = m)\n    (hv : ¬∃ y : ℤ, x = y) (hnpos : 0 < n) : Irrational x := by\n  rintro ⟨⟨N, D, P, C⟩, rfl⟩\n  rw [← cast_pow] at hxr\n  have c1 : ((D : ℤ) : ℝ) ≠ 0 := by\n    rw [Int.cast_ne_zero, Int.natCast_ne_zero]\n    exact P\n  have c2 : ((D : ℤ) : ℝ) ^ n ≠ 0 := pow_ne_zero _ c1\n  rw [mk_eq_divInt, cast_pow, cast_divInt, div_pow, div_eq_iff_mul_eq c2, ← Int.cast_pow,\n    ← Int.cast_pow, ← Int.cast_mul, Int.cast_inj] at hxr\n  have hdivn : (D : ℤ) ^ n ∣ N ^ n := Dvd.intro_left m hxr\n  rw [← Int.dvd_natAbs, ← Int.natCast_pow, Int.natCast_dvd_natCast, Int.natAbs_pow,\n    Nat.pow_dvd_pow_iff hnpos.ne'] at hdivn\n  obtain rfl : D = 1 := by rw [← Nat.gcd_eq_right hdivn, C.gcd_eq_one]\n  refine hv ⟨N, ?_⟩\n  rw [mk_eq_divInt, Int.ofNat_one, divInt_one, cast_intCast]\n\n/-- If `x^n = m` is an integer and `n` does not divide the `multiplicity p m`, then `x`\nis irrational. -/\ntheorem irrational_nrt_of_n_not_dvd_multiplicity {x : ℝ} (n : ℕ) {m : ℤ} (hm : m ≠ 0) (p : ℕ)\n    [hp : Fact p.Prime] (hxr : x ^ n = m)\n    (hv : multiplicity (p : ℤ) m % n ≠ 0) :\n    Irrational x := by\n  rcases Nat.eq_zero_or_pos n with (rfl | hnpos)\n  · rw [eq_comm, pow_zero, ← Int.cast_one, Int.cast_inj] at hxr\n    simp [hxr, multiplicity_of_one_right (mt isUnit_iff_dvd_one.1\n      (mt Int.natCast_dvd_natCast.1 hp.1.not_dvd_one))] at hv\n  refine irrational_nrt_of_notint_nrt _ _ hxr ?_ hnpos\n  rintro ⟨y, rfl⟩\n  rw [← Int.cast_pow, Int.cast_inj] at hxr\n  subst m\n  have : y ≠ 0 := by rintro rfl; rw [zero_pow hnpos.ne'] at hm; exact hm rfl\n  rw [(Int.finiteMultiplicity_iff.2 ⟨by simp [hp.1.ne_one], this⟩).multiplicity_pow\n    (Nat.prime_iff_prime_int.1 hp.1), Nat.mul_mod_right] at hv\n  exact hv rfl\n\ntheorem irrational_sqrt_of_multiplicity_odd (m : ℤ) (hm : 0 < m) (p : ℕ) [hp : Fact p.Prime]\n    (Hpv : multiplicity (p : ℤ) m % 2 = 1) :\n    Irrational (√m) :=\n  @irrational_nrt_of_n_not_dvd_multiplicity _ 2 _ (Ne.symm (ne_of_lt hm)) p hp\n    (sq_sqrt (Int.cast_nonneg hm.le)) (by rw [Hpv]; exact one_ne_zero)\n\n@[simp] theorem not_irrational_zero : ¬Irrational 0 := not_not_intro ⟨0, Rat.cast_zero⟩\n@[simp] theorem not_irrational_one : ¬Irrational 1 := not_not_intro ⟨1, Rat.cast_one⟩\n\ntheorem irrational_sqrt_ratCast_iff_of_nonneg {q : ℚ} (hq : 0 ≤ q) :\n    Irrational (√q) ↔ ¬IsSquare q := by\n  refine Iff.not (?_ : Exists _ ↔ Exists _)\n  constructor\n  · rintro ⟨y, hy⟩\n    refine ⟨y, Rat.cast_injective (α := ℝ) ?_⟩\n    rw [Rat.cast_mul, hy, mul_self_sqrt (Rat.cast_nonneg.2 hq)]\n  · rintro ⟨q', rfl⟩\n    exact ⟨|q'|, mod_cast (sqrt_mul_self_eq_abs q').symm⟩\n\ntheorem irrational_sqrt_ratCast_iff {q : ℚ} :\n    Irrational (√q) ↔ ¬IsSquare q ∧ 0 ≤ q := by\n  obtain hq | hq := le_or_gt 0 q\n  · simp_rw [irrational_sqrt_ratCast_iff_of_nonneg hq, and_iff_left hq]\n  · rw [sqrt_eq_zero_of_nonpos (Rat.cast_nonpos.2 hq.le)]\n    simp_rw [not_irrational_zero, false_iff, not_and, not_le, hq, implies_true]\n\nTarget:\ntheorem irrational_sqrt_intCast_iff_of_nonneg {z : ℤ} (hz : 0 ≤ z) :\n    Irrational (√z) ↔ ¬IsSquare z :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_9c259b24270f","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e70eacf945b4856ad5fc128f39c26b5629cbb0279a30afdae702c4445eef0b08","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Real","family_id":"irrational_sqrt_intcast_iff_of_nonneg","file_id":"mathlib/Mathlib/NumberTheory/Real/Irrational.lean","sample_id":"9c259b24270f51c0a7679a42e8272d4047f65364bf40859a29277a457689ac84"},"split":"train"}
{"binding":{"chosen_proof_sha256":"679574657f773b5361b65d8bc217854b5dee221b7dde4fd6f07b322f738d2333","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4dbb11ad844b5c9a90a253f6b1ac76bca67a8a2155f9a417c3905c407ea9974c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bf23546e5597548a8a703fba0f21b06b38585a3dc1e1cc164336dbb8b96373a4","source_sha256":"a9cefe337b8a57141453affd06f1d59efc30b22cebabbd270f973d0add5ab80b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    rwa [← map_natCast' _ he, ← map_natCast' _ he, he'.eq_iff, Nat.cast_inj] at h","hard_negative":false,"metrics":{"chosen_tokens":23,"rejected_tokens":30,"token_jaccard":0.888889,"token_length_ratio":1.304348},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"dddc649bdfc68de2bbd52f1dcb780984ea1b5c81c079620c7c277de38638a7d4","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharZero.Defs\npublic import Mathlib.Algebra.Group.Equiv.Defs\npublic import Mathlib.Data.Nat.Cast.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Paul Lezeau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Paul Lezeau\n-/\n/-!\n# Transporting `CharZero` across injective `AddMonoidHom`s\n\nThis file exists in order to avoid adding extra imports to other files in this subdirectory.\n-/\n\npublic section\n\nTarget:\ntheorem CharZero.of_addMonoidHom {M N : Type*} [AddCommMonoidWithOne M] [AddCommMonoidWithOne N]\n    [CharZero M] (e : M →+ N) (he : e 1 = 1) (he' : Function.Injective e) : CharZero N where\n  cast_injective n m h :=\n\nProof body:\n","rejected":"```lean\nby\n    rwa [← map_natCast' _ he, ← map_natCast' _ he, he'.eq_iff, Nat.cast_inj] at h\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/CharZero","family_id":"charzero","file_id":"mathlib/Mathlib/Algebra/CharZero/AddMonoidHom.lean","sample_id":"bf23546e5597548a8a703fba0f21b06b38585a3dc1e1cc164336dbb8b96373a4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"594293ce5d84984c59a5d88c876de5c8200038ce21c7d54ee21a2ba975b3d8b6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"10cc88fb02c4baabbf38e38ccad452ff78588a5790d63f1b1e27909a05a6c964","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d7b302fa097a85ad40628a8309be13365dc90a026d8ed805296461111bd052b1","source_sha256":"ae9de95805258feb351386be6f3451ef917af14e0cd4a5d5b560efd102bade12","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rw [eq_bot_iff, sup_le_iff]; simp","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":2,"token_jaccard":0.1,"token_length_ratio":0.222222},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"de0341c5cb1994fbaddc899d9e596a84446ab57613bf2618401f9685d46720d9","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.BoundedOrder.Basic\npublic import Mathlib.Order.Lattice\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Bounded lattices\n\nThis file contains miscellaneous lemmas about lattices with top or bottom elements.\n\n## Common lattices\n\n* Distributive lattices with a bottom element. Notated by `[DistribLattice α] [OrderBot α]`.\n  It captures the properties of `Disjoint` that are common to `GeneralizedBooleanAlgebra` and\n  `DistribLattice` when `OrderBot`.\n* Bounded and distributive lattice. Notated by `[DistribLattice α] [BoundedOrder α]`.\n  Typical examples include `Prop` and `Set α`.\n-/\n\npublic section\n\nopen Function OrderDual\n\nvariable {α β : Type*}\n\n/-! ### Top, bottom element -/\n\nsection SemilatticeSupTop\n\nvariable [SemilatticeSup α] [OrderTop α]\n\n@[to_dual] theorem top_sup_eq (a : α) : ⊤ ⊔ a = ⊤ := sup_of_le_left le_top\n@[to_dual] theorem sup_top_eq (a : α) : a ⊔ ⊤ = ⊤ := sup_of_le_right le_top\n\nend SemilatticeSupTop\n\nsection SemilatticeSupBot\n\nvariable [SemilatticeSup α] [OrderBot α] {a b : α}\n\n@[to_dual] theorem bot_sup_eq (a : α) : ⊥ ⊔ a = a := sup_of_le_right bot_le\n@[to_dual] theorem sup_bot_eq (a : α) : a ⊔ ⊥ = a := sup_of_le_left bot_le\n\n@[to_dual (attr := simp, grind =)]\n\nTarget:\ntheorem sup_eq_bot_iff : a ⊔ b = ⊥ ↔ a = ⊥ ∧ b = ⊥ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_d7b302fa097a","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"416fbce2dfd5d8e4ba5024b99605f69fc05a5ba1901b3b0ffc49248298e43714","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/BoundedOrder","family_id":"sup_eq_bot_iff","file_id":"mathlib/Mathlib/Order/BoundedOrder/Lattice.lean","sample_id":"d7b302fa097a85ad40628a8309be13365dc90a026d8ed805296461111bd052b1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ce223990efe19a3952beb8c48c08eec52484b0eabebe4090119bef4f0e984746","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"210dd76ae67ac01eff7a888fdcf07f7b44a5b4586ca30f9a119d28a997965242","source_sha256":"b1dde0e82be30e8a9ca87b23c46d04afc92be61818622ba6d4603daa4d4a9f70","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by simp [mk, ← ofCauchy_add]","hard_negative":false,"metrics":{"chosen_tokens":8,"rejected_tokens":2,"token_jaccard":0.111111,"token_length_ratio":0.25},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"de276568fab9f8b33a96d4f775d53e9d0853e082c52d3ca3ecb4b8873af38e9c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.CauSeq.Completion\npublic import Mathlib.Algebra.Order.Ring.Rat\npublic import Mathlib.Data.Rat.Cast.Defs\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Floris van Doorn\n-/\n/-!\n# Real numbers from Cauchy sequences\n\nThis file defines `ℝ` as the type of equivalence classes of Cauchy sequences of rational numbers.\nThis choice is motivated by how easy it is to prove that `ℝ` is a commutative ring, by simply\nlifting everything to `ℚ`.\n\nThe facts that the real numbers are an Archimedean floor ring,\nand a conditionally complete linear order,\nhave been deferred to the file `Mathlib/Data/Real/Archimedean.lean`,\nin order to keep the imports here simple.\n\nThe fact that the real numbers are a (trivial) \\*-ring has similarly been deferred to\n`Mathlib/Data/Real/Star.lean`.\n-/\n\n@[expose] public section\n\n\nassert_not_exists Finset Module Submonoid FloorRing\n\n/-- The type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational\nnumbers. -/\n@[wikidata Q12916, wikidata Q2584477]\nstructure Real where ofCauchy ::\n  /-- The underlying Cauchy completion -/\n  cauchy : CauSeq.Completion.Cauchy (abs : ℚ → ℚ)\n\n@[inherit_doc]\nnotation \"ℝ\" => Real\n\nnamespace CauSeq.Completion\n\n-- this can't go in `Data.Real.CauSeqCompletion` as the structure on `ℚ` isn't available\n@[simp]\ntheorem ofRat_rat {abv : ℚ → ℚ} [IsAbsoluteValue abv] (q : ℚ) :\n    ofRat (q : ℚ) = (q : Cauchy abv) :=\n  rfl\n\nend CauSeq.Completion\n\nnamespace Real\n\nopen CauSeq CauSeq.Completion\n\nvariable {x : ℝ}\n\ntheorem ext_cauchy_iff : ∀ {x y : Real}, x = y ↔ x.cauchy = y.cauchy\n  | ⟨a⟩, ⟨b⟩ => by rw [ofCauchy.injEq]\n\ntheorem ext_cauchy {x y : Real} : x.cauchy = y.cauchy → x = y :=\n  ext_cauchy_iff.2\n\n/-- The real numbers are isomorphic to the quotient of Cauchy sequences on the rationals. -/\ndef equivCauchy : ℝ ≃ CauSeq.Completion.Cauchy (abs : ℚ → ℚ) :=\n  ⟨Real.cauchy, Real.ofCauchy, fun ⟨_⟩ => rfl, fun _ => rfl⟩\n\nset_option backward.privateInPublic true in\n-- irreducible doesn't work for instances: https://github.com/leanprover-community/lean/issues/511\nprivate irreducible_def zero : ℝ :=\n  ⟨0⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def one : ℝ :=\n  ⟨1⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def add : ℝ → ℝ → ℝ\n  | ⟨a⟩, ⟨b⟩ => ⟨a + b⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def neg : ℝ → ℝ\n  | ⟨a⟩ => ⟨-a⟩\n\nset_option backward.privateInPublic true in\nprivate irreducible_def mul : ℝ → ℝ → ℝ\n  | ⟨a⟩, ⟨b⟩ => ⟨a * b⟩\n\nset_option backward.privateInPublic true in\nprivate noncomputable irreducible_def inv' : ℝ → ℝ\n  | ⟨a⟩ => ⟨a⁻¹⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Zero ℝ :=\n  ⟨zero⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : One ℝ :=\n  ⟨one⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Add ℝ :=\n  ⟨add⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Neg ℝ :=\n  ⟨neg⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : Mul ℝ :=\n  ⟨mul⟩\n\ninstance : Sub ℝ :=\n  ⟨fun a b => a + -b⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nnoncomputable instance : Inv ℝ :=\n  ⟨inv'⟩\n\ntheorem ofCauchy_zero : (⟨0⟩ : ℝ) = 0 :=\n  zero_def.symm\n\ntheorem ofCauchy_one : (⟨1⟩ : ℝ) = 1 :=\n  one_def.symm\n\ntheorem ofCauchy_add (a b) : (⟨a + b⟩ : ℝ) = ⟨a⟩ + ⟨b⟩ :=\n  (add_def _ _).symm\n\ntheorem ofCauchy_neg (a) : (⟨-a⟩ : ℝ) = -⟨a⟩ :=\n  (neg_def _).symm\n\ntheorem ofCauchy_sub (a b) : (⟨a - b⟩ : ℝ) = ⟨a⟩ - ⟨b⟩ := by\n  rw [sub_eq_add_neg, ofCauchy_add, ofCauchy_neg]\n  rfl\n\ntheorem ofCauchy_mul (a b) : (⟨a * b⟩ : ℝ) = ⟨a⟩ * ⟨b⟩ :=\n  (mul_def _ _).symm\n\ntheorem ofCauchy_inv {f} : (⟨f⁻¹⟩ : ℝ) = ⟨f⟩⁻¹ :=\n  show _ = inv' _ by rw [inv']\n\ntheorem cauchy_zero : (0 : ℝ).cauchy = 0 :=\n  show zero.cauchy = 0 by rw [zero_def]\n\ntheorem cauchy_one : (1 : ℝ).cauchy = 1 :=\n  show one.cauchy = 1 by rw [one_def]\n\ntheorem cauchy_add : ∀ a b, (a + b : ℝ).cauchy = a.cauchy + b.cauchy\n  | ⟨a⟩, ⟨b⟩ => show (add _ _).cauchy = _ by rw [add_def]\n\ntheorem cauchy_neg : ∀ a, (-a : ℝ).cauchy = -a.cauchy\n  | ⟨a⟩ => show (neg _).cauchy = _ by rw [neg_def]\n\ntheorem cauchy_mul : ∀ a b, (a * b : ℝ).cauchy = a.cauchy * b.cauchy\n  | ⟨a⟩, ⟨b⟩ => show (mul _ _).cauchy = _ by rw [mul_def]\n\ntheorem cauchy_sub : ∀ a b, (a - b : ℝ).cauchy = a.cauchy - b.cauchy\n  | ⟨a⟩, ⟨b⟩ => by\n    rw [sub_eq_add_neg, ← cauchy_neg, ← cauchy_add]\n    rfl\n\ntheorem cauchy_inv : ∀ f, (f⁻¹ : ℝ).cauchy = f.cauchy⁻¹\n  | ⟨f⟩ => show (inv' _).cauchy = _ by rw [inv']\n\ninstance instNatCast : NatCast ℝ where natCast n := ⟨n⟩\ninstance instIntCast : IntCast ℝ where intCast z := ⟨z⟩\ninstance instNNRatCast : NNRatCast ℝ where nnratCast q := ⟨q⟩\ninstance instRatCast : RatCast ℝ where ratCast q := ⟨q⟩\n\nlemma ofCauchy_natCast (n : ℕ) : (⟨n⟩ : ℝ) = n := rfl\nlemma ofCauchy_intCast (z : ℤ) : (⟨z⟩ : ℝ) = z := rfl\nlemma ofCauchy_nnratCast (q : ℚ≥0) : (⟨q⟩ : ℝ) = q := rfl\nlemma ofCauchy_ratCast (q : ℚ) : (⟨q⟩ : ℝ) = q := rfl\n\nlemma cauchy_natCast (n : ℕ) : (n : ℝ).cauchy = n := rfl\nlemma cauchy_intCast (z : ℤ) : (z : ℝ).cauchy = z := rfl\nlemma cauchy_nnratCast (q : ℚ≥0) : (q : ℝ).cauchy = q := rfl\nlemma cauchy_ratCast (q : ℚ) : (q : ℝ).cauchy = q := rfl\n\ninstance commRing : CommRing ℝ where\n  natCast n := ⟨n⟩\n  intCast z := ⟨z⟩\n  npow := @npowRec ℝ ⟨1⟩ ⟨(· * ·)⟩\n  nsmul := @nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩\n  zsmul := @zsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩ ⟨@Neg.neg ℝ _⟩ (@nsmulRec ℝ ⟨0⟩ ⟨(· + ·)⟩)\n  add_zero a := by apply ext_cauchy; simp [cauchy_add, cauchy_zero]\n  zero_add a := by apply ext_cauchy; simp [cauchy_add, cauchy_zero]\n  add_comm a b := by apply ext_cauchy; simp only [cauchy_add, add_comm]\n  add_assoc a b c := by apply ext_cauchy; simp only [cauchy_add, add_assoc]\n  mul_zero a := by apply ext_cauchy; simp [cauchy_mul, cauchy_zero]\n  zero_mul a := by apply ext_cauchy; simp [cauchy_mul, cauchy_zero]\n  mul_one a := by apply ext_cauchy; simp [cauchy_mul, cauchy_one]\n  one_mul a := by apply ext_cauchy; simp [cauchy_mul, cauchy_one]\n  mul_comm a b := by apply ext_cauchy; simp only [cauchy_mul, mul_comm]\n  mul_assoc a b c := by apply ext_cauchy; simp only [cauchy_mul, mul_assoc]\n  left_distrib a b c := by apply ext_cauchy; simp only [cauchy_add, cauchy_mul, mul_add]\n  right_distrib a b c := by apply ext_cauchy; simp only [cauchy_add, cauchy_mul, add_mul]\n  neg_add_cancel a := by apply ext_cauchy; simp [cauchy_add, cauchy_neg, cauchy_zero]\n  natCast_zero := by apply ext_cauchy; simp [cauchy_zero]\n  natCast_succ n := by apply ext_cauchy; simp [cauchy_one, cauchy_add]\n  intCast_negSucc z := by apply ext_cauchy; simp [cauchy_neg, cauchy_natCast]\n\n/-- `Real.equivCauchy` as a ring equivalence. -/\n@[simps]\ndef ringEquivCauchy : ℝ ≃+* CauSeq.Completion.Cauchy (abs : ℚ → ℚ) :=\n  { equivCauchy with\n    toFun := cauchy\n    invFun := ofCauchy\n    map_add' := cauchy_add\n    map_mul' := cauchy_mul }\n\n/-! Extra instances to short-circuit type class resolution.\n\nThese short-circuits have an additional property of ensuring that a computable path is found; if\n`Field ℝ` is found first, then decaying it to these typeclasses would result in a `noncomputable`\nversion of them. -/\n\ninstance instRing : Ring ℝ := by infer_instance\n\ninstance : CommSemiring ℝ := by infer_instance\n\ninstance semiring : Semiring ℝ := by infer_instance\n\ninstance : CommMonoidWithZero ℝ := by infer_instance\n\ninstance : MonoidWithZero ℝ := by infer_instance\n\ninstance : AddCommGroup ℝ := by infer_instance\n\ninstance : AddGroup ℝ := by infer_instance\n\ninstance : AddCommMonoid ℝ := by infer_instance\n\ninstance : AddMonoid ℝ := by infer_instance\n\ninstance : AddLeftCancelSemigroup ℝ := by infer_instance\n\ninstance : AddRightCancelSemigroup ℝ := by infer_instance\n\ninstance : AddCommSemigroup ℝ := by infer_instance\n\ninstance : AddSemigroup ℝ := by infer_instance\n\ninstance : CommMonoid ℝ := by infer_instance\n\ninstance : Monoid ℝ := by infer_instance\n\ninstance : CommSemigroup ℝ := by infer_instance\n\ninstance : Semigroup ℝ := by infer_instance\n\ninstance : Inhabited ℝ :=\n  ⟨0⟩\n\n/-- Make a real number from a Cauchy sequence of rationals (by taking the equivalence class). -/\ndef mk (x : CauSeq ℚ abs) : ℝ :=\n  ⟨CauSeq.Completion.mk x⟩\n\ntheorem mk_eq {f g : CauSeq ℚ abs} : mk f = mk g ↔ f ≈ g :=\n  ext_cauchy_iff.trans CauSeq.Completion.mk_eq\n\nset_option backward.privateInPublic true in\nprivate irreducible_def lt : ℝ → ℝ → Prop\n  | ⟨x⟩, ⟨y⟩ =>\n    (Quotient.liftOn₂ x y (· < ·)) fun _ _ _ _ hf hg =>\n      propext <|\n        ⟨fun h => lt_of_eq_of_lt (Setoid.symm hf) (lt_of_lt_of_eq h hg), fun h =>\n          lt_of_eq_of_lt hf (lt_of_lt_of_eq h (Setoid.symm hg))⟩\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\ninstance : LT ℝ :=\n  ⟨lt⟩\n\ntheorem lt_cauchy {f g} : (⟨⟦f⟧⟩ : ℝ) < ⟨⟦g⟧⟩ ↔ f < g :=\n  show lt _ _ ↔ _ by rw [lt_def]; rfl\n\n@[simp]\ntheorem mk_lt {f g : CauSeq ℚ abs} : mk f < mk g ↔ f < g :=\n  lt_cauchy\n\ntheorem mk_zero : mk 0 = 0 := by rw [← ofCauchy_zero]; rfl\n\ntheorem mk_one : mk 1 = 1 := by rw [← ofCauchy_one]; rfl\n\nTarget:\ntheorem mk_add {f g : CauSeq ℚ abs} : mk (f + g) = mk f + mk g :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Real","family_id":"mk_add","file_id":"mathlib/Mathlib/Data/Real/Basic.lean","sample_id":"210dd76ae67ac01eff7a888fdcf07f7b44a5b4586ca30f9a119d28a997965242"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dbc5a53bc3d6fc4141510bd9f512049166f30bf6d3c576c98bf05b032824182f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"3b4adf8c6ba159c4e6085837800b0118519e92103704fdde5b2a79efe825ccf5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a3ef0314d959f005739e2ea0a1bbd35c4cb66fa53980fc55488bf2d3ac971efc","source_sha256":"0f7f045d7dc25461fbd356aab76e5db8e19b16035129f69e67b31933f14c577d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by cfc_cont_tac)\n    (ha : p a := by cfc_tac) (ha' : ∀ i, q i (a i) := by cfc_tac) :\n    cfc f a = fun i => cfc f (a i) := by\n  ext i\n  let φ := Pi.evalStarAlgHom S A i\n  exact φ.map_cfc f a (by rwa [Pi.spectrum_eq]) (continuous_apply i) ha (ha' i)","hard_negative":true,"metrics":{"chosen_tokens":79,"rejected_tokens":2,"token_jaccard":0.027778,"token_length_ratio":0.025316},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"de2810f02b36b8cbbe37329db682a3a19741ddb7bda1e0dc5f7b2d6a29e66e8f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique\npublic import Mathlib.Algebra.Algebra.Spectrum.Pi\npublic import Mathlib.Algebra.Star.StarAlgHom\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Frédéric Dupuis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Frédéric Dupuis\n-/\n/-! # The continuous functional calculus on product types\n\nThis file contains results about the continuous functional calculus on (indexed) product types.\n\n## Main theorems\n\n+ `cfc_map_pi` and `cfcₙ_map_pi`: given `a : ∀ i, A i`, then `cfc f a = fun i => cfc f (a i)`\n  (and likewise for the non-unital version)\n+ `cfc_map_prod` and `cfcₙ_map_prod`: given `a : A` and `b : B`, then\n  `cfc f (a, b) = (cfc f a, cfc f b)` (and likewise for the non-unital version)\n-/\n\npublic section\n\nsection nonunital_pi\n\nvariable {ι R S : Type*} {A : ι → Type*} [CommSemiring R] [Nontrivial R] [StarRing R]\n  [MetricSpace R]\n  [IsTopologicalSemiring R] [ContinuousStar R] [CommRing S] [Algebra R S]\n  [∀ i, NonUnitalRing (A i)] [∀ i, Module S (A i)] [∀ i, Module R (A i)]\n  [∀ i, IsScalarTower R S (A i)] [∀ i, SMulCommClass R (A i) (A i)]\n  [∀ i, IsScalarTower R (A i) (A i)]\n  [∀ i, StarRing (A i)] [∀ i, TopologicalSpace (A i)] {p : (∀ i, A i) → Prop}\n  {q : (i : ι) → A i → Prop}\n  [NonUnitalContinuousFunctionalCalculus R (∀ i, A i) p]\n  [∀ i, NonUnitalContinuousFunctionalCalculus R (A i) (q i)]\n  [∀ i, ContinuousMapZero.UniqueHom R (A i)]\n\ninclude S in\nlemma cfcₙ_map_pi (f : R → R) (a : ∀ i, A i)\n    (hf : ContinuousOn f (⋃ i, quasispectrum R (a i)) := by cfc_cont_tac)\n    (ha : p a := by cfc_tac) (ha' : ∀ i, q i (a i) := by cfc_tac) :\n    cfcₙ f a = fun i => cfcₙ f (a i) := by\n  by_cases hf₀ : f 0 = 0\n  · ext i\n    have : Nonempty ι := ⟨i⟩\n    let φ := Pi.evalNonUnitalStarAlgHom S A i\n    exact φ.map_cfcₙ f a (by rwa [Pi.quasispectrum_eq]) hf₀ (continuous_apply i) ha (ha' i)\n  · simp only [cfcₙ_apply_of_not_map_zero _ hf₀, Pi.zero_def]\n\nend nonunital_pi\n\nsection nonunital_prod\n\nvariable {A B R S : Type*} [CommSemiring R] [CommRing S] [Nontrivial R] [StarRing R]\n  [MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [Algebra R S] [NonUnitalRing A]\n  [NonUnitalRing B] [Module S A] [Module R A] [Module R B] [Module S B]\n  [SMulCommClass R A A] [SMulCommClass R B B] [IsScalarTower R A A] [IsScalarTower R B B]\n  [StarRing A] [StarRing B] [TopologicalSpace A] [TopologicalSpace B]\n  [IsScalarTower R S A] [IsScalarTower R S B]\n  {pab : A × B → Prop} {pa : A → Prop} {pb : B → Prop}\n  [NonUnitalContinuousFunctionalCalculus R (A × B) pab]\n  [NonUnitalContinuousFunctionalCalculus R A pa]\n  [NonUnitalContinuousFunctionalCalculus R B pb]\n  [ContinuousMapZero.UniqueHom R A] [ContinuousMapZero.UniqueHom R B]\n\ninclude S in\nlemma cfcₙ_map_prod (f : R → R) (a : A) (b : B)\n    (hf : ContinuousOn f (quasispectrum R a ∪ quasispectrum R b) := by cfc_cont_tac)\n    (hab : pab (a, b) := by cfc_tac) (ha : pa a := by cfc_tac) (hb : pb b := by cfc_tac) :\n    cfcₙ f (a, b) = (cfcₙ f a, cfcₙ f b) := by\n  by_cases hf₀ : f 0 = 0\n  case pos =>\n    ext\n    case fst =>\n      let φ := NonUnitalStarAlgHom.fst S A B\n      exact φ.map_cfcₙ f (a, b) (by rwa [Prod.quasispectrum_eq]) hf₀ continuous_fst hab ha\n    case snd =>\n      let φ := NonUnitalStarAlgHom.snd S A B\n      exact φ.map_cfcₙ f (a, b) (by rwa [Prod.quasispectrum_eq]) hf₀ continuous_snd hab hb\n  case neg =>\n    simp [cfcₙ_apply_of_not_map_zero _ hf₀, eqComm]\n\nend nonunital_prod\n\nsection unital_pi\n\nvariable {ι R S : Type*} {A : ι → Type*} [CommSemiring R] [StarRing R] [MetricSpace R]\n  [IsTopologicalSemiring R] [ContinuousStar R] [CommRing S] [Algebra R S]\n  [∀ i, Ring (A i)] [∀ i, Algebra S (A i)] [∀ i, Algebra R (A i)] [∀ i, IsScalarTower R S (A i)]\n  [hinst : IsScalarTower R S (∀ i, A i)]\n  [∀ i, StarRing (A i)] [∀ i, TopologicalSpace (A i)] {p : (∀ i, A i) → Prop}\n  {q : (i : ι) → A i → Prop}\n  [ContinuousFunctionalCalculus R (∀ i, A i) p]\n  [∀ i, ContinuousFunctionalCalculus R (A i) (q i)]\n  [∀ i, ContinuousMap.UniqueHom R (A i)]\n\ninclude S in\n\nTarget:\nlemma cfc_map_pi (f : R → R) (a : ∀ i, A i)\n    (hf : ContinuousOn f (⋃ i, spectrum R (a i)) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_a3ef0314d959","rejection":{"accepted":false,"failure_type":"unknown_tactic","output_sha256":"1720986c071f477950b363d28ec55e272866775b35ed16b16e36bcd7fc1c58ab","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/CStarAlgebra","family_id":"cfc_map_pi","file_id":"mathlib/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Pi.lean","sample_id":"a3ef0314d959f005739e2ea0a1bbd35c4cb66fa53980fc55488bf2d3ac971efc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"97f1a84bedb0ef6995acbe3d016b9163eb9ac17f32bd0890956d8e9bd5498309","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"642f434381ee52651be87c870d541417455012d1f2a106209471684b9d27b5aa","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"567f3c5c6d2031f7fb93f23ec479080605b61b08ad68d9daa949e04606ce6590","source_sha256":"48e76914e12791b3eb841b9933ad6f80e72c4515bc3faaf828abf1cb41030052","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    refine (hB.hasLaw I).congr ?_\n    have : ∀ᵐ ω ∂P, ∀ i : I, B i ω = C i ω := ae_all_iff.2 fun _ ↦ h _\n    filter_upwards [this] with ω hω using funext fun i ↦ (hω i).symm","hard_negative":true,"metrics":{"chosen_tokens":61,"rejected_tokens":3,"token_jaccard":0.025641,"token_length_ratio":0.04918},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"de4876aefdf437d3dd811f28c79bc21208d422ddac10f7b577021c0463c5ce68","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily\npublic import Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Def\npublic import Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic\nimport Mathlib.Probability.Distributions.Gaussian.CharFun\nimport Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic\nimport Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence\nimport Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic\nimport Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence\nimport Mathlib.Probability.Independence.Process.HasIndepIncrements.IsGaussianProcess\n\nNamespace:\nProbabilityTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Etienne Marion. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Etienne Marion\n-/\n/-!\n# Brownian motion\n\nIn this file we define two predicates over stochastic processes `X : ℝ≥0 → Ω → ℝ` given\na probability measure `P : Measure Ω`. `IsPreBrownianReal X P` means that\n`X` is a pre-Brownian motion. It means that it has the law of the Brownian motion, namely that\nits finite dimensional distributions are given by `projectiveFamily`. Then\n`IsBrownianReal X P` means that `X` is a Brownian motion, which means that it is a pre-Brownian\nmotion with almost surely continuous paths.\n\nWe prove that a centered Gaussian process `X` with covariances given by `cov[X s, X t; P] = min s t`\nis a pre-Brownian motion and provide basic invariance properties. We also prove the\nweak Markov property: if `B` is a pre-Brownian motion and `t₀ : ℝ≥0`, then the process\n`t ↦ B (t + t₀) - B t₀` is a pre-Brownian motion independent from `(B t | t ≤ t₀)`.\n\n## Main definitions\n\n* `IsPreBrownianReal X P`: A stochastic process is called pre-Brownian if its finite-dimensional\n  laws are those of the Brownian motion, see `projectiveFamily`.\n* `IsBrownianReal X P`: A stochastic process is called Brownian if its finite-dimensional laws\n  are those of the Brownian motion, see `IsPreBrownianReal`,\n  and if it has almost-surely continuous paths.\n\n## Main statements\n\n* `IsGaussianProcess.isPreBrownianReal_of_covariance`: A centered Gaussian process with the right\n  covariance is a pre-Brownian motion.\n* `HasIndepIncrements.isPreBrownianReal_of_hasLaw`: A stochastic process `X` with independent\n  increments and such that for all `t`, `X t` has law `gaussianReal 0 t` is a pre-Brownian motion.\n* `IsPreBrownianReal.indepFun_shift`: The weak Markov property: If `B` is a pre-Brownian motion,\n  then `B (t₀ + t) - B t₀` is a pre-Brownian motion which is independent from `(B t, t ≤ t₀)`.\n\n## Tags\n\npre-Brownian motion, Brownian motion, Markov property\n\n-/\n\n@[expose] public section\n\nopen MeasureTheory ProbabilityTheory.BrownianReal\nopen scoped ENNReal NNReal Topology\n\nvariable {Ω : Type*} {mΩ : MeasurableSpace Ω} {B X : ℝ≥0 → Ω → ℝ} {P : Measure Ω}\n\nnamespace ProbabilityTheory\n\nsection IsPreBrownianReal\n\n/-! ### Pre-Brownian motion -/\n\n/-- A stochastic process is called **pre-Brownian** if its finite-dimensional laws are those\nof the Brownian motion, see `projectiveFamily`.\n\nNote: we name the constructor `mk'` so as to define later `IsPreBrownianReal.mk`, which to\npre-Brownian motion will associate a continuous modification,\nin a way similar to `AEMeasurable.mk`. -/\nstructure IsPreBrownianReal (X : ℝ≥0 → Ω → ℝ) (P : Measure Ω := by volume_tac) : Prop where\n  mk' ::\n  hasLaw : ∀ I : Finset ℝ≥0, HasLaw (fun ω ↦ I.restrict (X · ω)) (projectiveFamily I) P\n\n/- A modification of a pre-Brownian is pre-Brownian. -/\n\nTarget:\nlemma IsPreBrownianReal.congr {C : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P)\n    (h : ∀ t, B t =ᵐ[P] C t) :\n    IsPreBrownianReal C P where\n  hasLaw I :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_567f3c5c6d20","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"3854c832d12260f8da7ecf4033b7244980b20bc0f12ef341f19858ebeac6583a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/BrownianMotion","family_id":"isprebrownianreal","file_id":"mathlib/Mathlib/Probability/BrownianMotion/Basic.lean","sample_id":"567f3c5c6d2031f7fb93f23ec479080605b61b08ad68d9daa949e04606ce6590"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cb1bd661524d698c211f8037516f3bb09f8383d70d26a48c78df08d165e667a0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8c194e12f2fbe5697fa400d07d4529ac20ba6d8c4893ecec80e27968f828edce","source_sha256":"c35204fe93e22d960b9e0f1d04e2f2f5f2bf73b540a7e7fec0bfcf5bf1349499","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases Int.Prime.dvd_mul hp h with hp2 | hpp\n  · apply Or.intro_left\n    exact le_antisymm (Nat.le_of_dvd zero_lt_two hp2) (Nat.Prime.two_le hp)\n  · apply Or.intro_right\n    rw [sq, Int.natAbs_mul] at hpp\n    exact or_self_iff.mp ((Nat.Prime.dvd_mul hp).mp hpp)","hard_negative":false,"metrics":{"chosen_tokens":67,"rejected_tokens":3,"token_jaccard":0.057143,"token_length_ratio":0.044776},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"de4d9b7748b2272e998ef79a3cffc51621db31412985917960e08b3df64ad832","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.EuclideanDomain.Basic\npublic import Mathlib.Algebra.EuclideanDomain.Int\npublic import Mathlib.Algebra.GCDMonoid.Nat\npublic import Mathlib.Data.Nat.Prime.Int\npublic import Mathlib.RingTheory.PrincipalIdealDomain\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson\n-/\n/-!\n# Divisibility over ℤ\n\nThis file collects results for the integers that use ring theory in their proofs or\ncases of ℤ being examples of structures in ring theory.\n\n## Main statements\n\n* `Int.Prime.dvd_mul'`: A prime number dividing a product in ℤ divides at least one factor.\n* `Int.exists_prime_and_dvd`: Every non-unit integer has a prime divisor.\n* `Int.prime_iff_natAbs_prime`: Primality in ℤ corresponds to primality of its absolute value in ℕ.\n* `Int.span_natAbs`: The principal ideal generated by `a.natAbs` is equal to that of `a`.\n\n## Tags\n\nprime, irreducible, integers, normalization monoid, gcd monoid, greatest common divisor\n-/\n\npublic section\n\nnamespace Int\n\n\ntheorem isCoprime_iff_nat_coprime {a b : ℤ} : IsCoprime a b ↔ Nat.Coprime a.natAbs b.natAbs := by\n  rw [isCoprime_iff_gcd_eq_one, Nat.coprime_iff_gcd_eq_one, gcd_eq_natAbs]\n\n/-- If `gcd a (m * n) ≠ 1`, then `gcd a m ≠ 1` or `gcd a n ≠ 1`. -/\ntheorem gcd_ne_one_iff_gcd_mul_right_ne_one {a : ℤ} {m n : ℕ} :\n    a.gcd (m * n) ≠ 1 ↔ a.gcd m ≠ 1 ∨ a.gcd n ≠ 1 := by\n  simp only [← isCoprime_iff_gcd_eq_one, ← not_and_or, not_iff_not, IsCoprime.mul_right_iff]\n\ntheorem sq_of_gcd_eq_one {a b c : ℤ} (h : Int.gcd a b = 1) (heq : a * b = c ^ 2) :\n    ∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 := by\n  have h' : IsUnit (GCDMonoid.gcd a b) := by\n    rw [← coe_gcd, h, Int.ofNat_one]\n    exact isUnit_one\n  obtain ⟨d, ⟨u, hu⟩⟩ := exists_associated_pow_of_mul_eq_pow h' heq\n  use d\n  rw [← hu]\n  rcases Int.units_eq_one_or u with hu' | hu' <;>\n    · rw [hu']\n      simp\n\ntheorem sq_of_isCoprime {a b c : ℤ} (h : IsCoprime a b) (heq : a * b = c ^ 2) :\n    ∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 :=\n  sq_of_gcd_eq_one (isCoprime_iff_gcd_eq_one.mp h) heq\n\ntheorem natAbs_euclideanDomain_gcd (a b : ℤ) :\n    Int.natAbs (EuclideanDomain.gcd a b) = Int.gcd a b := by\n  apply Nat.dvd_antisymm <;> rw [← Int.natCast_dvd_natCast]\n  · rw [Int.natAbs_dvd]\n    exact Int.dvd_coe_gcd (EuclideanDomain.gcd_dvd_left _ _) (EuclideanDomain.gcd_dvd_right _ _)\n  · rw [Int.dvd_natAbs]\n    exact EuclideanDomain.dvd_gcd (Int.gcd_dvd_left ..) (Int.gcd_dvd_right ..)\n\nend Int\n\ntheorem Int.Prime.dvd_mul {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ m * n) :\n    p ∣ m.natAbs ∨ p ∣ n.natAbs := by\n  rwa [← hp.dvd_mul, ← Int.natAbs_mul, ← Int.natCast_dvd]\n\ntheorem Int.Prime.dvd_mul' {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ m * n) :\n    (p : ℤ) ∣ m ∨ (p : ℤ) ∣ n := by\n  rw [Int.natCast_dvd, Int.natCast_dvd]\n  exact Int.Prime.dvd_mul hp h\n\ntheorem Int.Prime.dvd_pow {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ n ^ k) :\n    p ∣ n.natAbs := by\n  rw [Int.natCast_dvd, Int.natAbs_pow] at h\n  exact hp.dvd_of_dvd_pow h\n\ntheorem Int.Prime.dvd_pow' {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ n ^ k) :\n    (p : ℤ) ∣ n := by\n  rw [Int.natCast_dvd]\n  exact Int.Prime.dvd_pow hp h\n\nTarget:\ntheorem prime_two_or_dvd_of_dvd_two_mul_pow_self_two {m : ℤ} {p : ℕ} (hp : Nat.Prime p)\n    (h : (p : ℤ) ∣ 2 * m ^ 2) : p = 2 ∨ p ∣ Int.natAbs m :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Int","family_id":"prime_two_or_dvd_of_dvd_two_mul_pow_self_two","file_id":"mathlib/Mathlib/RingTheory/Int/Basic.lean","sample_id":"8c194e12f2fbe5697fa400d07d4529ac20ba6d8c4893ecec80e27968f828edce"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7605cac82936a34e51b347b493387fd67f6c60e4f590ba9df4da446fc48c38a4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1e9f94a5050c3f631d6ef4550fc4148994113fabe92f719d1a84b2cac8998b0b","source_sha256":"74b6571c8b4ffa69041159c6a44858bd2f705cc61f45fce1f1c46bf3161b1e36","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let l := hi.lift ((pullback.fst αX f) ≫ h') ((pullback.snd αX f) ≫ g)\n    (by simp only [Category.assoc, hh.toCommSq.w, pullback.condition_assoc, ← H.w])\n  use (pullback αX f), (pullback.fst αX f), l, (pullback.snd αX f)\n  refine ⟨IsPullback.of_hasPullback αX f, ?_, ?_⟩\n  · refine IsPullback.of_right' ?_ hi\n    rw [← H.w]\n    exact IsPullback.paste_horiz (IsPullback.of_hasPullback αX f) hh\n  · refine (H' (pullback.fst αX f) l h' i' (pullback.snd αX f) αX αY αZ\n      (IsPullback.of_hasPullback αX f) ?_\n        hh.toCommSq hi.toCommSq ⟨by simp only [IsPullback.lift_fst, l]⟩).2 ⟨hh, hi⟩\n    · refine IsPullback.of_right' ?_ hi\n      rw [← H.w]\n      exact IsPullback.paste_horiz (IsPullback.of_hasPullback αX f) hh","hard_negative":false,"metrics":{"chosen_tokens":219,"rejected_tokens":2,"token_jaccard":0.019608,"token_length_ratio":0.009132},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"de63d39d0a76042311e71b364e950ecda63311c443e0f3d397ad31c7eb43a223","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Extensive\npublic import Mathlib.CategoryTheory.Limits.Shapes.KernelPair\npublic import Mathlib.CategoryTheory.Limits.Constructions.EpiMono\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang, Jack McKoen\n-/\n/-!\n\n# Adhesive categories\n\n## Main definitions\n- `CategoryTheory.IsPushout.IsVanKampen`: A convenience formulation for a pushout being\n  a van Kampen colimit.\n- `CategoryTheory.Adhesive`: A category is adhesive if it has pushouts and pullbacks along\n  monomorphisms, and such pushouts are van Kampen.\n\n## Main Results\n- `CategoryTheory.Type.adhesive`: The category of `Type` is adhesive.\n- `CategoryTheory.Adhesive.isPullback_of_isPushout_of_mono_left`: In adhesive categories,\n  pushouts along monomorphisms are pullbacks.\n- `CategoryTheory.Adhesive.mono_of_isPushout_of_mono_left`: In adhesive categories,\n  monomorphisms are stable under pushouts.\n- `CategoryTheory.Adhesive.toRegularMonoCategory`: Monomorphisms in adhesive categories are\n  regular (this implies that adhesive categories are balanced).\n- `CategoryTheory.adhesive_functor`: The category `C ⥤ D` is adhesive if `D`\n  has all pullbacks and all pushouts and is adhesive\n\n## References\n- https://ncatlab.org/nlab/show/adhesive+category\n- [Stephen Lack and Paweł Sobociński, Adhesive Categories][adhesive2004]\n\n-/\n\n@[expose] public section\n\n\nnamespace CategoryTheory\n\nopen Limits\n\nuniverse v' u' v u\n\nvariable {J : Type v'} [Category.{u'} J] {C : Type u} [Category.{v} C]\nvariable {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z}\n\n-- This only makes sense when the original diagram is a pushout.\n/-- A convenient formulation for a pushout being a van Kampen colimit. For any commutative cube of\nwhich a van Kampen pushout forms the bottom face and the back faces are pullbacks, the front faces\nare pullbacks if and only if the top face is a pushout. See `IsPushout.isVanKampen_iff` below. -/\n@[nolint unusedArguments]\ndef IsPushout.IsVanKampen (_ : IsPushout f g h i) : Prop :=\n  ∀ ⦃W' X' Y' Z' : C⦄ (f' : W' ⟶ X') (g' : W' ⟶ Y') (h' : X' ⟶ Z') (i' : Y' ⟶ Z') (αW : W' ⟶ W)\n    (αX : X' ⟶ X) (αY : Y' ⟶ Y) (αZ : Z' ⟶ Z) (_ : IsPullback f' αW αX f)\n    (_ : IsPullback g' αW αY g) (_ : CommSq h' αX αZ h) (_ : CommSq i' αY αZ i)\n    (_ : CommSq f' g' h' i'), IsPushout f' g' h' i' ↔ IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i\n\n/-- If a van Kampen pushout forms the bottom face of a commutative \"half-cube\" whose front faces\nare pullbacks, then there exist two back faces which are pullbacks and a top face which is a\npushout. -/\n\nTarget:\nlemma IsPushout.IsVanKampen.exists_cube_filling {H : IsPushout f g h i} (H' : H.IsVanKampen)\n    {X' Y' Z' : C} {h' : X' ⟶ Z'} {i' : Y' ⟶ Z'} {αX : X' ⟶ X} {αY : Y' ⟶ Y} {αZ : Z' ⟶ Z}\n    [HasPullback αX f] (hh : IsPullback h' αX αZ h) (hi : IsPullback i' αY αZ i) :\n    ∃ (W' : C) (f' : W' ⟶ X') (g' : W' ⟶ Y') (αW : W' ⟶ W),\n      IsPullback f' αW αX f ∧ IsPullback g' αW αY g ∧ IsPushout f' g' h' i' :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Adhesive","family_id":"ispushout","file_id":"mathlib/Mathlib/CategoryTheory/Adhesive/Basic.lean","sample_id":"1e9f94a5050c3f631d6ef4550fc4148994113fabe92f719d1a84b2cac8998b0b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0d4d6dccf2c84c2f8c5355bc4fd7dec43e2d86ae85215b172de94088a2f48bcb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"81e1863881d17ff2139ffc9deef5d1a646205eba55871a26235d2346169a0605","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dd8a4792fd8238e946d75aea51e934ebc1fab1139603667bd836f2fb5b0acf87","source_sha256":"62aebf63d41c64c52529eefd998601766828e0e4d42e1874f5417f5225e112d0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext x\n  simp only [spectrum.mem_iff, isUnit_iff_forall_ne_zero, not_forall, sub_apply,\n    Classical.not_not, Set.mem_range,\n    sub_eq_zero, @eq_comm _ (x • 1 : R) _, Set.mem_preimage, Algebra.algebraMap_eq_smul_one,\n    smul_apply, one_apply]","hard_negative":true,"metrics":{"chosen_tokens":50,"rejected_tokens":5,"token_jaccard":0.055556,"token_length_ratio":0.1},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"ded8e25b58d4b16c92e30e3342eee5a9724de00ce89acb5e30a5a60000064d99","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Ring.Units\npublic import Mathlib.Algebra.Algebra.Spectrum.Basic\npublic import Mathlib.Topology.ContinuousMap.Algebra\n\nNamespace:\nContinuousMap\n\nLocal context:\n/-\nCopyright (c) 2022 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\n/-!\n# Units of continuous functions\n\nThis file concerns itself with `C(X, M)ˣ` and `C(X, Mˣ)` when `X` is a topological space\nand `M` has some monoid structure compatible with its topology.\n-/\n\n@[expose] public section\n\n\nvariable {X M R 𝕜 : Type*} [TopologicalSpace X]\n\nnamespace ContinuousMap\n\nsection Monoid\n\nvariable [Monoid M] [TopologicalSpace M] [ContinuousMul M]\n\n/-- Equivalence between continuous maps into the units of a monoid with continuous multiplication\nand the units of the monoid of continuous maps. -/\n-- `simps` generates some lemmas here with LHS not in simp normal form,\n-- so we write them out manually below.\n@[to_additive (attr := simps apply_val_apply symm_apply_apply_val)\n/-- Equivalence between continuous maps into the additive units of an additive monoid with\ncontinuous addition and the additive units of the additive monoid of continuous maps. -/]\ndef unitsLift : C(X, Mˣ) ≃ C(X, M)ˣ where\n  toFun f :=\n    { val := ⟨fun x => f x, Units.continuous_val.comp f.continuous⟩\n      inv := ⟨fun x => ↑(f x)⁻¹, Units.continuous_val.comp (continuous_inv.comp f.continuous)⟩\n      val_inv := ext fun _ => Units.mul_inv _\n      inv_val := ext fun _ => Units.inv_mul _ }\n  invFun f :=\n    { toFun := fun x =>\n        ⟨(f : C(X, M)) x, (↑f⁻¹ : C(X, M)) x,\n          ContinuousMap.congr_fun f.mul_inv x, ContinuousMap.congr_fun f.inv_mul x⟩\n      continuous_toFun := continuous_induced_rng.2 <|\n        (f : C(X, M)).continuous.prodMk <|\n        MulOpposite.continuous_op.comp (↑f⁻¹ : C(X, M)).continuous }\n\n@[to_additive (attr := simp)]\nlemma unitsLift_apply_inv_apply (f : C(X, Mˣ)) (x : X) :\n    (↑(ContinuousMap.unitsLift f)⁻¹ : C(X, M)) x = (f x)⁻¹ :=\n  rfl\n\n@[to_additive (attr := simp)]\nlemma unitsLift_symm_apply_apply_inv' (f : C(X, M)ˣ) (x : X) :\n    (ContinuousMap.unitsLift.symm f x)⁻¹ = (↑f⁻¹ : C(X, M)) x := by\n  rfl\n\nend Monoid\n\nsection NormedRing\n\nvariable [NormedRing R] [CompleteSpace R]\n\ntheorem continuous_isUnit_unit {f : C(X, R)} (h : ∀ x, IsUnit (f x)) :\n    Continuous fun x => (h x).unit := by\n  refine\n    continuous_induced_rng.2\n      (Continuous.prodMk f.continuous\n        (MulOpposite.continuous_op.comp (continuous_iff_continuousAt.mpr fun x => ?_)))\n  have := NormedRing.inverse_continuousAt (h x).unit\n  simp only\n  simp only [← Ring.inverse_unit, IsUnit.unit_spec] at this ⊢\n  exact this.comp (f.continuousAt x)\n\n/-- Construct a continuous map into the group of units of a normed ring from a function into the\nnormed ring and a proof that every element of the range is a unit. -/\n@[simps]\nnoncomputable def unitsOfForallIsUnit {f : C(X, R)} (h : ∀ x, IsUnit (f x)) : C(X, Rˣ) where\n  toFun x := (h x).unit\n  continuous_toFun := continuous_isUnit_unit h\n\ninstance canLift :\n    CanLift C(X, R) C(X, Rˣ) (fun f => ⟨fun x => f x, Units.continuous_val.comp f.continuous⟩)\n      fun f => ∀ x, IsUnit (f x) where\n  prf f h := ⟨unitsOfForallIsUnit h, by ext; rfl⟩\n\ntheorem isUnit_iff_forall_isUnit (f : C(X, R)) : IsUnit f ↔ ∀ x, IsUnit (f x) :=\n  Iff.intro (fun h => fun x => ⟨unitsLift.symm h.unit x, rfl⟩) fun h =>\n    ⟨ContinuousMap.unitsLift (unitsOfForallIsUnit h), by ext; rfl⟩\n\nend NormedRing\n\nsection NormedField\n\nvariable [NormedField 𝕜] [NormedDivisionRing R] [Algebra 𝕜 R] [CompleteSpace R]\n\ntheorem isUnit_iff_forall_ne_zero (f : C(X, R)) : IsUnit f ↔ ∀ x, f x ≠ 0 := by\n  simp_rw [f.isUnit_iff_forall_isUnit, isUnit_iff_ne_zero]\n\nTarget:\ntheorem spectrum_eq_preimage_range (f : C(X, R)) :\n    spectrum 𝕜 f = algebraMap _ _ ⁻¹' Set.range f :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_dd8a4792fd82","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"b384432fc14513cf7b08186780ad391a05726732f7998892c2d5e54436d88b9e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/ContinuousMap","family_id":"spectrum_eq_preimage_range","file_id":"mathlib/Mathlib/Topology/ContinuousMap/Units.lean","sample_id":"dd8a4792fd8238e946d75aea51e934ebc1fab1139603667bd836f2fb5b0acf87"},"split":"train"}
{"binding":{"chosen_proof_sha256":"507db4142d79d356282c11878756b5338f2f435787c1651165c4326a0009c857","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2eaa4c3a7a8979e42c8b78cd304c273792da5d766a00dd97d6cfbc8a6d8171e8","source_sha256":"3b1ad71a44a0ba95fe172fff4adb35c1c3d0ab3dacc0cacca37d6024c51fd12b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases x; rw [Zsqrtd.norm, normSq]; simp","hard_negative":false,"metrics":{"chosen_tokens":14,"rejected_tokens":2,"token_jaccard":0.071429,"token_length_ratio":0.142857},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"dee95716440da9607832256ea4a1a366bba6ab62e6910449c1aa4d48ae3f8afd","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Archimedean.Real.Basic\npublic import Mathlib.Data.Complex.Basic\npublic import Mathlib.Data.Nat.Prime.Basic\npublic import Mathlib.NumberTheory.Zsqrtd.Basic\n\nNamespace:\nGaussianInt\n\nLocal context:\n/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Gaussian integers\n\nThe Gaussian integers are complex integer, complex numbers whose real and imaginary parts are both\nintegers.\n\n## Main definitions\n\nThe Euclidean domain structure on `ℤ[i]` is defined in this file.\n\nThe homomorphism `GaussianInt.toComplex` into the complex numbers is also defined in this file.\n\n## See also\n\nSee `NumberTheory.Zsqrtd.QuadraticReciprocity` for:\n* `prime_iff_mod_four_eq_three_of_nat_prime`:\n  A prime natural number is prime in `ℤ[i]` if and only if it is `3` mod `4`\n\n## Notation\n\nThis file uses the local notation `ℤ[i]` for `GaussianInt`\n\n## Implementation notes\n\nGaussian integers are implemented using the more general definition `Zsqrtd`, the type of integers\nadjoined a square root of `d`, in this case `-1`. The definition is reducible, so that properties\nand definitions about `Zsqrtd` can easily be used.\n-/\n\n@[expose] public section\n\n\nopen Zsqrtd Complex\n\nopen scoped ComplexConjugate\n\n/-- The Gaussian integers, defined as `ℤ√(-1)`. -/\nabbrev GaussianInt : Type :=\n  Zsqrtd (-1)\n\nlocal notation \"ℤ[i]\" => GaussianInt\n\nnamespace GaussianInt\n\ninstance : Repr ℤ[i] :=\n  ⟨fun x _ => \"⟨\" ++ repr x.re ++ \", \" ++ repr x.im ++ \"⟩\"⟩\n\ninstance instCommRing : CommRing ℤ[i] :=\n  Zsqrtd.commRing\n\nsection\n\nattribute [-instance] Complex.instField -- Avoid making things noncomputable unnecessarily.\n\n/-- The embedding of the Gaussian integers into the complex numbers, as a ring homomorphism. -/\ndef toComplex : ℤ[i] →+* ℂ :=\n  Zsqrtd.lift ⟨I, by simp⟩\n\nend\n\ninstance : Coe ℤ[i] ℂ :=\n  ⟨toComplex⟩\n\ntheorem toComplex_def (x : ℤ[i]) : (x : ℂ) = x.re + x.im * I :=\n  rfl\n\ntheorem toComplex_def' (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ) = x + y * I := by simp [toComplex_def]\n\ntheorem toComplex_def₂ (x : ℤ[i]) : (x : ℂ) = ⟨x.re, x.im⟩ := by\n  apply Complex.ext <;> simp [toComplex_def]\n\n@[simp]\ntheorem intCast_re (x : ℤ[i]) : ((x.re : ℤ) : ℝ) = (x : ℂ).re := by simp [toComplex_def]\n\n@[simp]\ntheorem intCast_im (x : ℤ[i]) : ((x.im : ℤ) : ℝ) = (x : ℂ).im := by simp [toComplex_def]\n\n@[simp]\ntheorem re_toComplex (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).re = x := by simp [toComplex_def]\n\n@[simp]\ntheorem im_toComplex (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).im = y := by simp [toComplex_def]\n\ntheorem toComplex_add (x y : ℤ[i]) : ((x + y : ℤ[i]) : ℂ) = x + y :=\n  toComplex.map_add _ _\n\ntheorem toComplex_mul (x y : ℤ[i]) : ((x * y : ℤ[i]) : ℂ) = x * y :=\n  toComplex.map_mul _ _\n\ntheorem toComplex_one : ((1 : ℤ[i]) : ℂ) = 1 :=\n  toComplex.map_one\n\ntheorem toComplex_zero : ((0 : ℤ[i]) : ℂ) = 0 :=\n  toComplex.map_zero\n\ntheorem toComplex_neg (x : ℤ[i]) : ((-x : ℤ[i]) : ℂ) = -x :=\n  toComplex.map_neg _\n\ntheorem toComplex_sub (x y : ℤ[i]) : ((x - y : ℤ[i]) : ℂ) = x - y :=\n  toComplex.map_sub _ _\n\n@[simp]\ntheorem toComplex_star (x : ℤ[i]) : ((star x : ℤ[i]) : ℂ) = conj (x : ℂ) := by\n  rw [toComplex_def₂, toComplex_def₂]\n  exact congr_arg₂ _ rfl (Int.cast_neg _)\n\n@[simp]\ntheorem toComplex_inj {x y : ℤ[i]} : (x : ℂ) = y ↔ x = y := by\n  cases x; cases y; simp [toComplex_def₂]\n\nlemma toComplex_injective : Function.Injective GaussianInt.toComplex :=\n  fun ⦃_ _⦄ ↦ toComplex_inj.mp\n\n@[simp]\ntheorem toComplex_eq_zero {x : ℤ[i]} : (x : ℂ) = 0 ↔ x = 0 := by\n  rw [← toComplex_zero, toComplex_inj]\n\n@[simp]\ntheorem intCast_real_norm (x : ℤ[i]) : (x.norm : ℝ) = Complex.normSq (x : ℂ) := by\n  rw [Zsqrtd.norm, normSq]; simp\n\n@[simp]\n\nTarget:\ntheorem intCast_complex_norm (x : ℤ[i]) : (x.norm : ℂ) = Complex.normSq (x : ℂ) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Zsqrtd","family_id":"intcast_complex_norm","file_id":"mathlib/Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean","sample_id":"2eaa4c3a7a8979e42c8b78cd304c273792da5d766a00dd97d6cfbc8a6d8171e8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8c5d22170da6aa84840f7344a2c31c2097b39c782de19947a86a24798df1b587","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"40a9e698de3c8772948fb6e716fb277d4c72e8e6913a687ba653657faca9f64e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"c360ffd931955f77da36029b2c2c6057afeb7e861c5fc96268f3b26c2470d808","source_sha256":"082955edd445e22305bf58ec1b7c606429333e8cae7ee766f31439d19efb2538","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rintro ⟨f, a, hf⟩ ⟨g, b, hg⟩ rfl\n  congr\n  refine extensional_of_trichotomous_of_irrefl s fun x ↦ ?_\n  rw [← hf, hg]","hard_negative":true,"metrics":{"chosen_tokens":33,"rejected_tokens":3,"token_jaccard":0.037037,"token_length_ratio":0.090909},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"df122cb9182694ca0715ec07d35c16e064fdab2b910354947873b9d81e5d140e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Sum.Order\npublic import Mathlib.Order.Hom.Lex\npublic import Mathlib.Order.RelIso.Set\npublic import Mathlib.Order.UpperLower.Basic\npublic import Mathlib.Order.WellFounded\n\nNamespace:\nPrincipalSeg\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios\n-/\n/-!\n# Initial and principal segments\n\nThis file defines initial and principal segment embeddings. Though these definitions make sense for\narbitrary relations, they're intended for use with well orders.\n\nAn initial segment is simply a lower set, i.e. if `x` belongs to the range, then any `y < x` also\nbelongs to the range. A principal segment is a set of the form `Set.Iio x` for some `x`.\n\nAn initial segment embedding `r ≼i s` is an order embedding `r ↪ s` such that its range is an\ninitial segment. Likewise, a principal segment embedding `r ≺i s` has a principal segment for a\nrange.\n\n## Main definitions\n\n* `InitialSeg r s`: Type of initial segment embeddings of `r` into `s`, denoted by `r ≼i s`.\n* `PrincipalSeg r s`: Type of principal segment embeddings of `r` into `s`, denoted by `r ≺i s`.\n\nThe lemmas `Ordinal.type_le_iff` and `Ordinal.type_lt_iff` tell us that `≼i` corresponds to the `≤`\nrelation on ordinals, while `≺i` corresponds to the `<` relation. This prompts us to think of\n`PrincipalSeg` as a \"strict\" version of `InitialSeg`.\n\n## Notation\n\nThese notations belong to the `InitialSeg` locale.\n\n* `r ≼i s`: the type of initial segment embeddings of `r` into `s`.\n* `r ≺i s`: the type of principal segment embeddings of `r` into `s`.\n* `α ≤i β` is an abbreviation for `(· < ·) ≼i (· < ·)`.\n* `α <i β` is an abbreviation for `(· < ·) ≺i (· < ·)`.\n-/\n\n@[expose] public section\n\n/-! ### Initial segment embeddings -/\n\nuniverse u\n\nvariable {α β γ : Type*} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop}\n\nopen Function\n\n/-- If `r` is a relation on `α` and `s` in a relation on `β`, then `f : r ≼i s` is an order\nembedding whose `Set.range` is a lower set. That is, whenever `b < f a` in `β` then `b` is in the\nrange of `f`. -/\nstructure InitialSeg {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends r ↪r s where\n  /-- The order embedding is an initial segment -/\n  mem_range_of_rel' : ∀ a b, s b (toRelEmbedding a) → b ∈ Set.range toRelEmbedding\n\n@[inherit_doc]\nscoped[InitialSeg] infixl:25 \" ≼i \" => InitialSeg\n\n/-- An `InitialSeg` between the `<` relations of two types. -/\nnotation3:25 α:24 \" ≤i \" β:25 => @InitialSeg α β (· < ·) (· < ·)\n\nnamespace InitialSeg\n\ninstance : Coe (r ≼i s) (r ↪r s) :=\n  ⟨InitialSeg.toRelEmbedding⟩\n\ninstance : FunLike (r ≼i s) α β where\n  coe f := f.toFun\n  coe_injective := by\n    rintro ⟨f, hf⟩ ⟨g, hg⟩ h\n    congr with x\n    exact congr_fun h x\n\ninstance : EmbeddingLike (r ≼i s) α β where\n  injective' f := f.inj'\n\ninstance : RelHomClass (r ≼i s) r s where\n  map_rel f := f.map_rel_iff.2\n\n/-- An initial segment embedding between the `<` relations of two partial orders is an order\nembedding. -/\ndef toOrderEmbedding [PartialOrder α] [PartialOrder β] (f : α ≤i β) : α ↪o β :=\n  f.orderEmbeddingOfLTEmbedding\n\n@[simp]\ntheorem toOrderEmbedding_apply [PartialOrder α] [PartialOrder β] (f : α ≤i β) (x : α) :\n    f.toOrderEmbedding x = f x :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderEmbedding [PartialOrder α] [PartialOrder β] (f : α ≤i β) :\n    (f.toOrderEmbedding : α → β) = f :=\n  rfl\n\ninstance [PartialOrder α] [PartialOrder β] : OrderHomClass (α ≤i β) α β where\n  map_rel f := f.toOrderEmbedding.map_rel_iff.2\n\n@[ext] lemma ext {f g : r ≼i s} (h : ∀ x, f x = g x) : f = g :=\n  DFunLike.ext f g h\n\n@[simp]\ntheorem coe_coe_fn (f : r ≼i s) : ((f : r ↪r s) : α → β) = f :=\n  rfl\n\ntheorem mem_range_of_rel (f : r ≼i s) {a : α} {b : β} : s b (f a) → b ∈ Set.range f :=\n  f.mem_range_of_rel' _ _\n\ntheorem map_rel_iff {a b : α} (f : r ≼i s) : s (f a) (f b) ↔ r a b :=\n  f.map_rel_iff'\n\ntheorem inj (f : r ≼i s) {a b : α} : f a = f b ↔ a = b :=\n  f.toRelEmbedding.inj\n\ntheorem exists_eq_iff_rel (f : r ≼i s) {a : α} {b : β} : s b (f a) ↔ ∃ a', f a' = b ∧ r a' a :=\n  ⟨fun h => by\n    rcases f.mem_range_of_rel h with ⟨a', rfl⟩\n    exact ⟨a', rfl, f.map_rel_iff.1 h⟩,\n    fun ⟨_, e, h⟩ => e ▸ f.map_rel_iff.2 h⟩\n\n/-- A relation isomorphism is an initial segment embedding -/\n@[simps!]\ndef _root_.RelIso.toInitialSeg (f : r ≃r s) : r ≼i s :=\n  ⟨f, by simp⟩\n\n/-- The identity function shows that `≼i` is reflexive -/\n@[refl]\nprotected def refl (r : α → α → Prop) : r ≼i r :=\n  (RelIso.refl r).toInitialSeg\n\ninstance (r : α → α → Prop) : Inhabited (r ≼i r) :=\n  ⟨InitialSeg.refl r⟩\n\n/-- Composition of functions shows that `≼i` is transitive -/\n@[trans]\nprotected def trans (f : r ≼i s) (g : s ≼i t) : r ≼i t :=\n  ⟨f.1.trans g.1, fun a c h => by\n    simp only [RelEmbedding.coe_trans, coe_coe_fn, comp_apply] at h ⊢\n    rcases g.2 _ _ h with ⟨b, rfl⟩; have h := g.map_rel_iff.1 h\n    rcases f.2 _ _ h with ⟨a', rfl⟩; exact ⟨a', rfl⟩⟩\n\n@[simp]\ntheorem refl_apply (x : α) : InitialSeg.refl r x = x :=\n  rfl\n\n@[simp]\ntheorem trans_apply (f : r ≼i s) (g : s ≼i t) (a : α) : (f.trans g) a = g (f a) :=\n  rfl\n\ninstance subsingleton_of_trichotomous_of_irrefl [Std.Trichotomous s] [Std.Irrefl s]\n    [IsWellFounded α r] : Subsingleton (r ≼i s) where\n  allEq f g := by\n    ext a\n    refine IsWellFounded.induction r a fun b IH =>\n      extensional_of_trichotomous_of_irrefl s fun x => ?_\n    rw [f.exists_eq_iff_rel, g.exists_eq_iff_rel]\n    exact exists_congr fun x => and_congr_left fun hx => IH _ hx ▸ Iff.rfl\n\n/-- Given a well order `s`, there is at most one initial segment embedding of `r` into `s`. -/\ninstance [IsWellOrder β s] : Subsingleton (r ≼i s) :=\n  ⟨fun a => have := a.isWellFounded; Subsingleton.elim a⟩\n\nprotected theorem eq [IsWellOrder β s] (f g : r ≼i s) (a) : f a = g a := by\n  rw [Subsingleton.elim f g]\n\ntheorem eq_relIso [IsWellOrder β s] (f : r ≼i s) (g : r ≃r s) (a : α) : g a = f a :=\n  InitialSeg.eq g.toInitialSeg f a\n\n/-- If we have order embeddings between `α` and `β` whose ranges are initial segments, and `β` is a\nwell order, then `α` and `β` are order-isomorphic. -/\ndef antisymm [IsWellOrder β s] (f : r ≼i s) (g : s ≼i r) : r ≃r s :=\n  have := f.toRelEmbedding.isWellOrder\n  ⟨⟨f, g, (f.trans g).eq (InitialSeg.refl _), (g.trans f).eq (InitialSeg.refl _)⟩, f.map_rel_iff'⟩\n\n@[simp]\ntheorem antisymm_toFun [IsWellOrder β s] (f : r ≼i s) (g : s ≼i r) : (antisymm f g : α → β) = f :=\n  rfl\n\n@[simp]\ntheorem antisymm_symm [IsWellOrder α r] [IsWellOrder β s] (f : r ≼i s) (g : s ≼i r) :\n    (antisymm f g).symm = antisymm g f :=\n  RelIso.coe_fn_injective rfl\n\n/-- An initial segment embedding is either an isomorphism, or a principal segment embedding.\n\nSee also `InitialSeg.ltOrEq`. -/\ntheorem eq_or_principal [IsWellOrder β s] (f : r ≼i s) :\n    Surjective f ∨ ∃ b, ∀ x, x ∈ Set.range f ↔ s x b := by\n  apply or_iff_not_imp_right.2\n  intro h b\n  push Not at h\n  apply IsWellFounded.induction s b\n  intro x IH\n  obtain ⟨y, ⟨hy, hs⟩ | ⟨hy, hs⟩⟩ := h x\n  · obtain (rfl | h) := (trichotomous y x).resolve_left hs\n    · exact hy\n    · obtain ⟨z, rfl⟩ := hy\n      exact f.mem_range_of_rel h\n  · obtain ⟨z, rfl⟩ := IH y hs\n    cases hy (Set.mem_range_self z)\n\n/-- Restrict the codomain of an initial segment -/\ndef codRestrict (p : Set β) (f : r ≼i s) (H : ∀ a, f a ∈ p) : r ≼i Subrel s (· ∈ p) :=\n  ⟨RelEmbedding.codRestrict p f H, fun a ⟨b, m⟩ h =>\n    let ⟨a', e⟩ := f.mem_range_of_rel h\n    ⟨a', by subst e; rfl⟩⟩\n\n@[simp]\ntheorem codRestrict_apply (p) (f : r ≼i s) (H a) : codRestrict p f H a = ⟨f a, H a⟩ :=\n  rfl\n\n/-- Initial segment embedding from an empty type. -/\ndef ofIsEmpty (r : α → α → Prop) (s : β → β → Prop) [IsEmpty α] : r ≼i s :=\n  ⟨RelEmbedding.ofIsEmpty r s, isEmptyElim⟩\n\n/-- Initial segment embedding of an order `r` into the disjoint union of `r` and `s`. -/\ndef leAdd (r : α → α → Prop) (s : β → β → Prop) : r ≼i Sum.Lex r s :=\n  ⟨⟨⟨Sum.inl, fun _ _ => Sum.inl.inj⟩, Sum.lex_inl_inl⟩, fun a b => by\n    cases b <;> [exact fun _ => ⟨_, rfl⟩; exact False.elim ∘ Sum.lex_inr_inl]⟩\n\n@[simp]\ntheorem leAdd_apply (r : α → α → Prop) (s : β → β → Prop) (a) : leAdd r s a = Sum.inl a :=\n  rfl\n\nprotected theorem acc (f : r ≼i s) (a : α) : Acc r a ↔ Acc s (f a) :=\n  ⟨by\n    refine fun h => Acc.recOn h fun a _ ha => Acc.intro _ fun b hb => ?_\n    obtain ⟨a', rfl⟩ := f.mem_range_of_rel hb\n    exact ha _ (f.map_rel_iff.mp hb), f.toRelEmbedding.acc a⟩\n\nend InitialSeg\n\n/-! ### Principal segments -/\n\n/-- If `r` is a relation on `α` and `s` in a relation on `β`, then `f : r ≺i s` is an initial\nsegment embedding whose range is `Set.Iio x` for some element `x`. If `β` is a well order, this is\nequivalent to the embedding not being surjective. -/\nstructure PrincipalSeg {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends r ↪r s where\n  /-- The supremum of the principal segment -/\n  top : β\n  /-- The range of the order embedding is the set of elements `b` such that `s b top` -/\n  mem_range_iff_rel' : ∀ b, b ∈ Set.range toRelEmbedding ↔ s b top\n\n@[inherit_doc]\nscoped[InitialSeg] infixl:25 \" ≺i \" => PrincipalSeg\n\n/-- A `PrincipalSeg` between the `<` relations of two types. -/\nnotation3:25 α:24 \" <i \" β:25 => @PrincipalSeg α β (· < ·) (· < ·)\n\nopen scoped InitialSeg\n\nnamespace PrincipalSeg\n\ninstance : CoeOut (r ≺i s) (r ↪r s) :=\n  ⟨PrincipalSeg.toRelEmbedding⟩\n\ninstance : CoeFun (r ≺i s) fun _ => α → β :=\n  ⟨fun f => f⟩\n\nTarget:\ntheorem toRelEmbedding_injective [Std.Irrefl s] [Std.Trichotomous s] :\n    Function.Injective (@toRelEmbedding α β r s) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_c360ffd93195","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"920979c161884480a4bd263a7263c5c0256808166c6779734e0c4c23f9aab373","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order","family_id":"torelembedding_injective","file_id":"mathlib/Mathlib/Order/InitialSeg.lean","sample_id":"c360ffd931955f77da36029b2c2c6057afeb7e861c5fc96268f3b26c2470d808"},"split":"train"}
{"binding":{"chosen_proof_sha256":"58196917b1a24b2be60c95c2b5d60e57c3e3c717c1a775b2f741e63bbfde64ca","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7fc44abc4155c5513e4bbdb5b46fb28fc64e1f85540625b34c97451c5b546fb4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"41dfd38896021be841aea499e4a561d7890d2e4973f5f2dd9b0701af439b41dd","source_sha256":"931482f0b3c940b38e80debe6ca13abb163dc5dd9b614292ad2b1da12781602e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h := lintegral_condCDF ((dirac Unit.unit).prod μ) x\n  simpa only [fst_prod, prod_prod, measure_univ, one_mul, lintegral_dirac] using! h","hard_negative":false,"metrics":{"chosen_tokens":33,"rejected_tokens":40,"token_jaccard":0.928571,"token_length_ratio":1.212121},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"df30f9b5050c3ba2447f63c9d64ae15bb4c8881bcf5e39df8c002ab4476969fd","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Kernel.Disintegration.CondCDF\npublic import Mathlib.Tactic.CrossRefAttribute\n\nNamespace:\nProbabilityTheory\n\nLocal context:\n/-\nCopyright (c) 2023 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n-/\n/-!\n# Cumulative distribution function of a real probability measure\n\nThe cumulative distribution function (cdf) of a probability measure over `ℝ` is a monotone, right\ncontinuous function with limit 0 at -∞ and 1 at +∞, such that `cdf μ x = μ (Iic x)` for all `x : ℝ`.\nTwo probability measures are equal if and only if they have the same cdf.\n\n## Main definitions\n\n* `ProbabilityTheory.cdf μ`: cumulative distribution function of `μ : Measure ℝ`, defined as the\n  conditional cdf (`ProbabilityTheory.condCDF`) of the product measure\n  `(Measure.dirac Unit.unit).prod μ` evaluated at `Unit.unit`.\n\nThe definition could be replaced by the more elementary `cdf μ x = μ.real (Iic x)`, but using\n`condCDF` gives us access to its API, from which most properties of the cdf follow directly.\n\n## Main statements\n\n* `ProbabilityTheory.ofReal_cdf`: for a probability measure `μ` and `x : ℝ`,\n  `ENNReal.ofReal (cdf μ x) = μ (Iic x)`.\n* `MeasureTheory.Measure.ext_of_cdf`: two probability measures are equal if and only if they have\n  the same cdf.\n\n## TODO\n\nThe definition could be extended to a finite measure by rescaling `condCDF`, but it would be nice\nto have more structure on Stieltjes functions first. Right now, if `f` is a Stieltjes function,\n`2 • f` makes no sense. We could define Stieltjes functions as a submodule.\n\nThe definition could be extended to `ℝⁿ`, either by extending the definition of `condCDF`, or by\nusing another construction here.\n-/\n\n@[expose] public section\n\nopen MeasureTheory Measure Set Filter\n\nopen scoped Topology\n\nnamespace ProbabilityTheory\n\n/-- Cumulative distribution function of a real measure. The definition currently makes sense only\nfor probability measures. In that case, it satisfies `cdf μ x = μ.real (Iic x)` (see\n`ProbabilityTheory.cdf_eq_real`). -/\n@[wikidata Q386228]\nnoncomputable\ndef cdf (μ : Measure ℝ) : StieltjesFunction ℝ :=\n  condCDF ((dirac Unit.unit).prod μ) Unit.unit\n\nsection ExplicitMeasureArg\nvariable (μ : Measure ℝ)\n\n/-- The cdf is non-negative. -/\nlemma cdf_nonneg (x : ℝ) : 0 ≤ cdf μ x := condCDF_nonneg _ _ _\n\n/-- The cdf is lower or equal to 1. -/\nlemma cdf_le_one (x : ℝ) : cdf μ x ≤ 1 := condCDF_le_one _ _ _\n\n/-- The cdf is monotone. -/\nlemma monotone_cdf : Monotone (cdf μ) := (condCDF _ _).mono\n\n/-- The cdf tends to 0 at -∞. -/\nlemma tendsto_cdf_atBot : Tendsto (cdf μ) atBot (𝓝 0) := tendsto_condCDF_atBot _ _\n\n/-- The cdf tends to 1 at +∞. -/\nlemma tendsto_cdf_atTop : Tendsto (cdf μ) atTop (𝓝 1) := tendsto_condCDF_atTop _ _\n\nTarget:\nlemma ofReal_cdf [IsProbabilityMeasure μ] (x : ℝ) : ENNReal.ofReal (cdf μ x) = μ (Iic x) :=\n\nProof body:\n","rejected":"```lean\nby\n  have h := lintegral_condCDF ((dirac Unit.unit).prod μ) x\n  simpa only [fst_prod, prod_prod, measure_univ, one_mul, lintegral_dirac] using! h\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability","family_id":"ofreal_cdf","file_id":"mathlib/Mathlib/Probability/CDF.lean","sample_id":"41dfd38896021be841aea499e4a561d7890d2e4973f5f2dd9b0701af439b41dd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"956d0d39593774896edc51480ad38b07e1edca7a7a86a5d842d9b626672e0346","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1985f9bc8f1d271e9024a4e9b57f47f6440b8628042b13c228851b2c53549153","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a21a121517cc485ec8b0fac3320d37a95ffa767d8d52dbb2b3c8baa9ed0c81a3","source_sha256":"d2377d09b51f8862ecd1e9604b616865bc29923d42ea22ca4ee787791f41d5e7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isNoetherian_iff', wellFoundedGT_iff_monotone_chain_condition]\n\nvariable [IsNoetherian R M]","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":20,"token_jaccard":0.846154,"token_length_ratio":1.538462},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"df47fef6b0c1945d58fb1f7f60ca2c1543d491bb058b2c5d8d868bcae2b07163","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Filter.AtTopBot.Basic\npublic import Mathlib.RingTheory.Finiteness.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Kevin Buzzard\n-/\n/-!\n# Noetherian rings and modules\n\nThe following are equivalent for a module M over a ring R:\n1. Every increasing chain of submodules M₁ ⊆ M₂ ⊆ M₃ ⊆ ⋯ eventually stabilises.\n2. Every submodule is finitely generated.\n\nA module satisfying these equivalent conditions is said to be a *Noetherian* R-module.\nA ring is a *Noetherian ring* if it is Noetherian as a module over itself.\n\n(Note that we do not assume yet that our rings are commutative,\nso perhaps this should be called \"left-Noetherian\".\nTo avoid cumbersome names once we specialize to the commutative case,\nwe don't make this explicit in the declaration names.)\n\n## Main definitions\n\nLet `R` be a ring and let `M` and `P` be `R`-modules. Let `N` be an `R`-submodule of `M`.\n\n* `IsNoetherian R M` is the proposition that `M` is a Noetherian `R`-module. It is a class,\n  implemented as the predicate that all `R`-submodules of `M` are finitely generated.\n\n## Main statements\n\n* `isNoetherian_iff` is the theorem that an R-module M is Noetherian iff `>` is well-founded on\n  `Submodule R M`.\n\nNote that the Hilbert basis theorem, that if a commutative ring R is Noetherian then so is R[X],\nis proved in `RingTheory.Polynomial`.\n\n## References\n\n* [M. F. Atiyah and I. G. Macdonald, *Introduction to commutative algebra*][atiyah-macdonald]\n* [P. Samuel, *Algebraic Theory of Numbers*][samuel1967]\n\n## Tags\n\nNoetherian, noetherian, Noetherian ring, Noetherian module, noetherian ring, noetherian module\n\n-/\n\n@[expose] public section\n\nassert_not_exists Finsupp.linearCombination Matrix Pi.basis\n\nopen Set Pointwise\n\n/-- `IsNoetherian R M` is the proposition that `M` is a Noetherian `R`-module,\nimplemented as the predicate that all `R`-submodules of `M` are finitely generated.\n-/\n-- TODO: should this be renamed to `Noetherian`?\nclass IsNoetherian (R M) [Semiring R] [AddCommMonoid M] [Module R M] : Prop where\n  noetherian : ∀ s : Submodule R M, s.FG\n\nattribute [inherit_doc IsNoetherian] IsNoetherian.noetherian\n\nsection\n\nvariable {R : Type*} {M : Type*} {P : Type*}\nvariable [Semiring R] [AddCommMonoid M] [AddCommMonoid P]\nvariable [Module R M] [Module R P]\n\nopen IsNoetherian\n\n/-- An R-module is Noetherian iff all its submodules are finitely-generated. -/\ntheorem isNoetherian_def : IsNoetherian R M ↔ ∀ s : Submodule R M, s.FG :=\n  ⟨fun h => h.noetherian, IsNoetherian.mk⟩\n\ntheorem isNoetherian_submodule {N : Submodule R M} :\n    IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG := by\n  refine ⟨fun ⟨hn⟩ => fun s hs =>\n    have : s ≤ LinearMap.range N.subtype := N.range_subtype.symm ▸ hs\n    Submodule.map_comap_eq_self this ▸ (hn _).map _,\n    fun h => ⟨fun s => ?_⟩⟩\n  specialize h (s.map N.subtype) (Submodule.map_subtype_le N s)\n  exact Submodule.fg_of_fg_map_injective N.subtype Subtype.val_injective h\n\ntheorem isNoetherian_submodule_left {N : Submodule R M} :\n    IsNoetherian R N ↔ ∀ s : Submodule R M, (N ⊓ s).FG :=\n  isNoetherian_submodule.trans ⟨fun H _ => H _ inf_le_left, fun H _ hs => inf_of_le_right hs ▸ H _⟩\n\ntheorem isNoetherian_submodule_right {N : Submodule R M} :\n    IsNoetherian R N ↔ ∀ s : Submodule R M, (s ⊓ N).FG :=\n  isNoetherian_submodule.trans ⟨fun H _ => H _ inf_le_right, fun H _ hs => inf_of_le_left hs ▸ H _⟩\n\ninstance isNoetherian_submodule' [IsNoetherian R M] (N : Submodule R M) : IsNoetherian R N :=\n  isNoetherian_submodule.2 fun _ _ => IsNoetherian.noetherian _\n\ntheorem isNoetherian_of_le {s t : Submodule R M} [ht : IsNoetherian R t] (h : s ≤ t) :\n    IsNoetherian R s :=\n  isNoetherian_submodule.mpr fun _ hs' => isNoetherian_submodule.mp ht _ (le_trans hs' h)\n\nend\n\nopen IsNoetherian Submodule Function\n\nsection\n\nuniverse w\n\nvariable {R M P : Type*} {N : Type w} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N]\n  [Module R N] [AddCommMonoid P] [Module R P]\n\ntheorem isNoetherian_iff' : IsNoetherian R M ↔ WellFoundedGT (Submodule R M) := by\n  refine .trans ?_ ((CompleteLattice.wellFoundedGT_characterisations <| Submodule R M).out 0 3).symm\n  exact\n    ⟨fun ⟨h⟩ => fun k => (fg_iff_compact k).mp (h k), fun h =>\n      ⟨fun k => (fg_iff_compact k).mpr (h k)⟩⟩\n\ntheorem isNoetherian_iff :\n    IsNoetherian R M ↔ WellFounded ((· > ·) : Submodule R M → Submodule R M → Prop) := by\n  rw [isNoetherian_iff', ← isWellFounded_iff]\n\nalias ⟨IsNoetherian.wf, _⟩ := isNoetherian_iff\n\nalias ⟨IsNoetherian.wellFoundedGT, isNoetherian_mk⟩ := isNoetherian_iff'\n\ninstance wellFoundedGT [h : IsNoetherian R M] : WellFoundedGT (Submodule R M) :=\n  h.wellFoundedGT\n\ntheorem isNoetherian_iff_fg_wellFounded :\n    IsNoetherian R M ↔ WellFoundedGT { N : Submodule R M // N.FG } := by\n  let α := { N : Submodule R M // N.FG }\n  constructor\n  · intro H\n    let f : α ↪o Submodule R M := OrderEmbedding.subtype _\n    exact OrderEmbedding.wellFoundedLT f.dual\n  · intro H\n    constructor\n    intro N\n    obtain ⟨⟨N₀, h₁⟩, e : N₀ ≤ N, h₂⟩ :=\n      WellFounded.has_min H.wf { N' : α | N'.1 ≤ N } ⟨⟨⊥, Submodule.fg_bot⟩, @bot_le _ _ _ N⟩\n    convert! h₁\n    refine (e.antisymm ?_).symm\n    by_contra h₃\n    obtain ⟨x, hx₁ : x ∈ N, hx₂ : x ∉ N₀⟩ := Set.not_subset.mp h₃\n    apply hx₂\n    rw [eq_of_le_of_not_lt (le_sup_right : N₀ ≤ _) (h₂\n      ⟨_, Submodule.FG.sup ⟨{x}, by rw [Finset.coe_singleton]⟩ h₁⟩ <|\n      sup_le ((Submodule.span_singleton_le_iff_mem _ _).mpr hx₁) e)]\n    exact (le_sup_left : R ∙ x ≤ _) (Submodule.mem_span_singleton_self _)\n\n/-- A module is Noetherian iff every nonempty set of submodules has a maximal submodule among them.\n-/\ntheorem set_has_maximal_iff_noetherian :\n    (∀ a : Set <| Submodule R M, a.Nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬M' < I) ↔ IsNoetherian R M := by\n  rw [isNoetherian_iff, WellFounded.wellFounded_iff_has_min]\n\n/-- A module is Noetherian iff every increasing chain of submodules stabilizes. -/\n\nTarget:\ntheorem monotone_stabilizes_iff_noetherian :\n    (∀ f : ℕ →o Submodule R M, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsNoetherian R M :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [isNoetherian_iff', wellFoundedGT_iff_monotone_chain_condition]\n\nvariable [IsNoetherian R M]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Noetherian","family_id":"monotone_stabilizes_iff_noetherian","file_id":"mathlib/Mathlib/RingTheory/Noetherian/Defs.lean","sample_id":"a21a121517cc485ec8b0fac3320d37a95ffa767d8d52dbb2b3c8baa9ed0c81a3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1a3ab110d23e9c5a2fcf5c8f847d9eb4eca9aceac610399fa60f5ab4f28d4c1a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cc198a628839d26b84d2d45a4274cbe04c295e1ce79ccc05957e8799a01460b5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a2b4875ff5d193c274945a397a372732cc39a1c971c7cc22bbea2e4368fe786d","source_sha256":"98d8a988714b8411039a52ca8cb94b19a051f27bd0e995b1332bddd6c23da9ea","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨c₁, hc₁, c₂, _, h⟩ := hf b hb\n  refine ⟨c₁, hc₁, ?_⟩\n  filter_upwards [h]\n  exact fun _ H u hu => (H u hu).1","hard_negative":true,"metrics":{"chosen_tokens":49,"rejected_tokens":5,"token_jaccard":0.09375,"token_length_ratio":0.102041},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"dfc398597cdb868d5ca82828afde77fa5c1bdd9ff95b99194ed2ae1329078ef6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecialFunctions.Pow.Real\npublic import Mathlib.Algebra.Order.ToIntervalMod\npublic import Mathlib.Analysis.SpecialFunctions.Log.Base\nimport Mathlib.Algebra.Order.Interval.Set.Group\n\nNamespace:\nAkraBazziRecurrence.GrowsPolynomially\n\nLocal context:\n/-\nCopyright (c) 2023 Frédéric Dupuis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Frédéric Dupuis\n-/\n/-!\n# Akra-Bazzi theorem: the polynomial growth condition\n\nThis file defines and develops an API for the polynomial growth condition that appears in the\nstatement of the Akra-Bazzi theorem: for the theorem to hold, the function `g` must\nsatisfy the condition that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b*n` and `n` for any\nconstant `b ∈ (0,1)`.\n\n## Implementation notes\n\nOur definition requires that the condition hold for any `b ∈ (0,1)`. This is equivalent to requiring\nit only for `b = 1 / 2` (or any other particular value in `(0, 1)`). While this could, in principle,\nmake it harder to prove that a particular function grows polynomially, this issue does not seem to\narise in practice.\n\n-/\n\n@[expose] public section\n\nopen Finset Real Filter Asymptotics\nopen scoped Topology\n\nnamespace AkraBazziRecurrence\n\n/-- The growth condition that the function `g` must satisfy for the Akra-Bazzi theorem to apply.\nIt roughly states that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b * n` and `n`, for any\nconstant `b ∈ (0, 1)`. -/\ndef GrowsPolynomially (f : ℝ → ℝ) : Prop :=\n  ∀ b ∈ Set.Ioo 0 1, ∃ c₁ > 0, ∃ c₂ > 0,\n    ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * (f x)) (c₂ * f x)\n\nnamespace GrowsPolynomially\n\nlemma congr_of_eventuallyEq {f g : ℝ → ℝ} (hfg : f =ᶠ[atTop] g) (hg : GrowsPolynomially g) :\n    GrowsPolynomially f := by\n  intro b hb\n  have hg' := hg b hb\n  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hg'⟩ := hg'\n  refine ⟨c₁, hc₁_mem, c₂, hc₂_mem, ?_⟩\n  filter_upwards [hg', (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hfg, hfg]\n    with x hx₁ hx₂ hx₃\n  intro u hu\n  rw [hx₂ u hu.1, hx₃]\n  exact hx₁ u hu\n\nlemma iff_eventuallyEq {f g : ℝ → ℝ} (h : f =ᶠ[atTop] g) :\n    GrowsPolynomially f ↔ GrowsPolynomially g :=\n  ⟨fun hf => congr_of_eventuallyEq h.symm hf, fun hg => congr_of_eventuallyEq h hg⟩\n\nvariable {f : ℝ → ℝ}\n\nlemma eventually_atTop_le {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ≤ c * f x := by\n  obtain ⟨c₁, _, c₂, hc₂, h⟩ := hf b hb\n  refine ⟨c₂, hc₂, ?_⟩\n  filter_upwards [h]\n  exact fun _ H u hu => (H u hu).2\n\nlemma eventually_atTop_le_nat {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (b * n) n, f u ≤ c * f n := by\n  obtain ⟨c, hc_mem, hc⟩ := hf.eventually_atTop_le hb\n  exact ⟨c, hc_mem, hc.natCast_atTop⟩\n\nTarget:\nlemma eventually_atTop_ge {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :\n    ∃ c > 0, ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, c * f x ≤ f u :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_a2b4875ff5d1","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"3e05d5e56de7a45ff5241391ad8b77d0035c03d48f743271e1e8c99142bdd7c9","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Computability/AkraBazzi","family_id":"eventually_attop_ge","file_id":"mathlib/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean","sample_id":"a2b4875ff5d193c274945a397a372732cc39a1c971c7cc22bbea2e4368fe786d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"72e566f5060686e112fed795e07f0f9fff6e410e86e784e0d5fb3063e2798458","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6c38d9122f13fca6acc952741077e5353e8c6277695e07bb15da53db5e09eafb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"80e10db2175365293b28a61fb2baadedb77fa1adfc56606eaf56f02d6e239f96","source_sha256":"5b12ef03a7b761162934544e836024f413f1638f86d711b3735f9f49b0617ce9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  nontriviality α using Subsingleton.elim a 0\n  obtain rfl | h := eq_or_ne a 0\n  · rw [normUnit_zero, zero_mul, normUnit_zero]\n  · rw [normUnit_mul h (Units.ne_zero _), normUnit_coe_units, mul_inv_eq_one]","hard_negative":true,"metrics":{"chosen_tokens":42,"rejected_tokens":3,"token_jaccard":0.03125,"token_length_ratio":0.071429},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"dfd45f2176b4e9abc1326e7ce855ef0e386472da31f8caf31b4658befa03ee13","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.Associated\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Jens Wagemaker\n-/\n/-!\n# Monoids with normalization functions, `gcd`, and `lcm`\n\nThis file defines extra structures on `CommMonoidWithZero`s.\n\n## Main Definitions\n\n* `NormalizationMonoid`\n* `GCDMonoid`\n* `NormalizedGCDMonoid`\n* `gcdMonoidOfGCD`, `gcdMonoidOfExistsGCD`, `normalizedGCDMonoidOfGCD`,\n  `normalizedGCDMonoidOfExistsGCD`\n* `gcdMonoidOfLCM`, `gcdMonoidOfExistsLCM`, `normalizedGCDMonoidOfLCM`,\n  `normalizedGCDMonoidOfExistsLCM`\n\nFor the `NormalizedGCDMonoid` instances on `ℕ` and `ℤ`, see `Mathlib/Algebra/GCDMonoid/Nat.lean`.\n\n## Implementation Notes\n\n* `NormalizationMonoid` is defined by assigning to each element a `normUnit` such that multiplying\n  by that unit normalizes the monoid, and `normalize` is an idempotent monoid homomorphism. This\n  definition as currently implemented does casework on `0`.\n\n* `GCDMonoid` contains the definitions of `gcd` and `lcm` with the usual properties. They are\n  both determined up to a unit.\n\n* `NormalizedGCDMonoid` extends `NormalizationMonoid`, so the `gcd` and `lcm` are always\n  normalized. This makes `gcd`s of polynomials easier to work with, but excludes Euclidean domains,\n  and monoids without zero.\n\n* `gcdMonoidOfGCD` and `normalizedGCDMonoidOfGCD` noncomputably construct a `GCDMonoid`\n  (resp. `NormalizedGCDMonoid`) structure just from the `gcd` and its properties.\n\n* `gcdMonoidOfExistsGCD` and `normalizedGCDMonoidOfExistsGCD` noncomputably construct a\n  `GCDMonoid` (resp. `NormalizedGCDMonoid`) structure just from a proof that any two elements\n  have a (not necessarily normalized) `gcd`.\n\n* `gcdMonoidOfLCM` and `normalizedGCDMonoidOfLCM` noncomputably construct a `GCDMonoid`\n  (resp. `NormalizedGCDMonoid`) structure just from the `lcm` and its properties.\n\n* `gcdMonoidOfExistsLCM` and `normalizedGCDMonoidOfExistsLCM` noncomputably construct a\n  `GCDMonoid` (resp. `NormalizedGCDMonoid`) structure just from a proof that any two elements\n  have a (not necessarily normalized) `lcm`.\n\n## TODO\n\n* Port GCD facts about nats, definition of coprime\n* Generalize normalization monoids to commutative (cancellative) monoids with or without zero\n\n## Tags\n\ndivisibility, gcd, lcm, normalize\n-/\n\n@[expose] public section\n\n\nvariable {α : Type*}\n\n/-- Normalization monoid: multiplying with `normUnit` gives a normal form for associated\nelements. -/\nclass NormalizationMonoid (α : Type*) [CommMonoidWithZero α] where\n  /-- `normUnit` assigns to each element of the monoid a unit of the monoid. -/\n  normUnit : α → αˣ\n  /-- The proposition that `normUnit` maps `0` to the identity. -/\n  normUnit_zero : normUnit 0 = 1\n  /-- The proposition that `normUnit` respects multiplication of non-zero elements. -/\n  normUnit_mul : ∀ {a b}, a ≠ 0 → b ≠ 0 → normUnit (a * b) = normUnit a * normUnit b\n  /-- The proposition that `normUnit` maps units to their inverses. -/\n  normUnit_coe_units : ∀ u : αˣ, normUnit u = u⁻¹\n\nexport NormalizationMonoid (normUnit normUnit_zero normUnit_mul normUnit_coe_units)\n\nattribute [simp] normUnit_coe_units normUnit_zero normUnit_mul\n\nsection NormalizationMonoid\n\nvariable [CommMonoidWithZero α] [NormalizationMonoid α]\n\n@[simp]\ntheorem normUnit_one : normUnit (1 : α) = 1 :=\n  normUnit_coe_units 1\n\n/-- Chooses an element of each associate class, by multiplying by `normUnit` -/\ndef normalize : α →*₀ α where\n  toFun x := x * normUnit x\n  map_zero' := by\n    simp only [normUnit_zero]\n    exact mul_one (0 : α)\n  map_one' := by rw [normUnit_one, one_mul]; rfl\n  map_mul' x y :=\n    (by_cases fun hx : x = 0 => by rw [hx, zero_mul, zero_mul, zero_mul]) fun hx =>\n      (by_cases fun hy : y = 0 => by rw [hy, mul_zero, zero_mul, mul_zero]) fun hy => by\n        simp only [normUnit_mul hx hy, Units.val_mul]; simp only [mul_assoc, mul_left_comm y]\n\ntheorem associated_normalize (x : α) : Associated x (normalize x) :=\n  ⟨_, rfl⟩\n\ntheorem normalize_associated (x : α) : Associated (normalize x) x :=\n  (associated_normalize _).symm\n\ntheorem associated_normalize_iff {x y : α} : Associated x (normalize y) ↔ Associated x y :=\n  ⟨fun h => h.trans (normalize_associated y), fun h => h.trans (associated_normalize y)⟩\n\ntheorem normalize_associated_iff {x y : α} : Associated (normalize x) y ↔ Associated x y :=\n  ⟨fun h => (associated_normalize _).trans h, fun h => (normalize_associated _).trans h⟩\n\ntheorem Associates.mk_normalize (x : α) : Associates.mk (normalize x) = Associates.mk x :=\n  Associates.mk_eq_mk_iff_associated.2 (normalize_associated _)\n\ntheorem normalize_apply (x : α) : normalize x = x * normUnit x :=\n  rfl\n\ntheorem normalize_zero : normalize (0 : α) = 0 :=\n  normalize.map_zero\n\ntheorem normalize_one : normalize (1 : α) = 1 :=\n  normalize.map_one\n\ntheorem normalize_coe_units (u : αˣ) : normalize (u : α) = 1 := by simp [normalize_apply]\n\n@[simp]\ntheorem normalize_eq_zero {x : α} : normalize x = 0 ↔ x = 0 :=\n  ⟨fun hx => (associated_zero_iff_eq_zero x).1 <| hx ▸ associated_normalize _, by\n    rintro rfl; exact normalize_zero⟩\n\ntheorem normalize_eq_one {x : α} : normalize x = 1 ↔ IsUnit x :=\n  ⟨fun hx => isUnit_iff_exists_inv.2 ⟨_, hx⟩, fun ⟨u, hu⟩ => hu ▸ normalize_coe_units u⟩\n\n@[simp]\n\nTarget:\ntheorem normUnit_mul_normUnit (a : α) : normUnit (a * normUnit a) = 1 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_80e10db21753","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"69cc636998a3d566ccc6f782a3b839e93f4e7c2a4a4d27665ee7d86f7f14de4a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/GCDMonoid","family_id":"normunit_mul_normunit","file_id":"mathlib/Mathlib/Algebra/GCDMonoid/Basic.lean","sample_id":"80e10db2175365293b28a61fb2baadedb77fa1adfc56606eaf56f02d6e239f96"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5af7d1d4d55c3a73fbff3849f778662494382e96b2f978d7cb9fc32bd567bdc1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9dd28da9ecf3c55e13be992634a8a033ad3c7c927f9493226c620253ba629f4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"17c8f688ffbfe3f08997e2e441397e109ef966a6e5472eeef0ec671a2f11c5b3","source_sha256":"23d7dd6c4328a8c8adae4c7811bfae5c38bb2e82b7410307d7aa82730380d65f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro f\n  obtain ⟨⟨f⟩, rfl⟩ := (bijective_leftHomotopyClassToHom L X Y).2 f\n  exact ⟨f, rfl⟩","hard_negative":true,"metrics":{"chosen_tokens":27,"rejected_tokens":2,"token_jaccard":0.052632,"token_length_ratio":0.074074},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"e00c41090846da0caf24584284de55a37a12d8ef5e9ae7133d86b4305cee74ec","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy\n\nNamespace:\nHomotopicalAlgebra\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The fundamental lemma of homotopical algebra\n\nLet `C` be a model category. Let `L : C ⥤ H` be a localization functor\nwith respect to weak equivalences in `C`. We obtain the fundamental\nlemma of homotopical algebra: if `X` is cofibrant and `Y` fibrant,\nthe map `(X ⟶ Y) → (L.obj X ⟶ L.obj Y)` identifies `L.obj X ⟶ L.obj Y`\nto the quotient of `X ⟶ Y` by the homotopy relation (in this case,\nthe left and right homotopy relations coincide).\n\n## References\n* [Daniel G. Quillen, Homotopical algebra, I.1][Quillen1967]\n\n-/\n\n@[expose] public section\n\nopen CategoryTheory Limits\n\nnamespace HomotopicalAlgebra\n\nvariable {C : Type*} [Category* C] [ModelCategory C] {H : Type*} [Category* H]\n  (L : C ⥤ H) [L.IsLocalization (weakEquivalences _)]\n  {X Y : C}\n\n/-- The map `LeftHomotopyClass X Y → (L.obj X ⟶ L.obj Y)` when `L` is\na localization functor with respect to `weakEquivalences C`. -/\ndef leftHomotopyClassToHom : LeftHomotopyClass X Y → (L.obj X ⟶ L.obj Y) :=\n  Quot.lift L.map (fun _ _ h ↦ h.factorsThroughLocalization.map_eq _)\n\n@[simp]\nlemma leftHomotopyClassToHom_mk (f : X ⟶ Y) :\n    leftHomotopyClassToHom L (.mk f) = L.map f := rfl\n\n/-- The map `RightHomotopyClass X Y → (L.obj X ⟶ L.obj Y)` when `L` is\na localization functor with respect to `weakEquivalences C`. -/\ndef rightHomotopyClassToHom : RightHomotopyClass X Y → (L.obj X ⟶ L.obj Y) :=\n  Quot.lift L.map (fun _ _ h ↦ h.factorsThroughLocalization.map_eq _)\n\n@[simp]\nlemma rightHomotopyClassToHom_mk (f : X ⟶ Y) :\n    rightHomotopyClassToHom L (.mk f) = L.map f := rfl\n\nvariable (X Y)\nlemma bijective_leftHomotopyClassToHom_iff_bijective_rightHomotopyClassToHom\n    [IsCofibrant X] [IsFibrant Y] :\n    Function.Bijective (leftHomotopyClassToHom L : LeftHomotopyClass X Y → _) ↔\n    Function.Bijective (rightHomotopyClassToHom L : RightHomotopyClass X Y → _) := by\n  have : (leftHomotopyClassToHom L : LeftHomotopyClass X Y → _) =\n      rightHomotopyClassToHom L ∘ leftHomotopyClassEquivRightHomotopyClass := by\n    ext f\n    obtain ⟨f, rfl⟩ := f.mk_surjective\n    simp\n  simp [this]\n\nsection\n\nvariable [IsCofibrant X] [IsFibrant Y]\n\nlemma bijective_rightHomotopyClassToHom :\n    Function.Bijective (rightHomotopyClassToHom L : RightHomotopyClass X Y → _) := by\n  wlog _ : IsCofibrant Y generalizing Y\n  · obtain ⟨Y', _, p, _, _⟩ := CofibrantObject.HoCat.exists_resolution Y\n    have _ : IsFibrant Y' := isFibrant_of_fibration p\n    have hY' := this Y' inferInstance\n    simp only [← bijective_leftHomotopyClassToHom_iff_bijective_rightHomotopyClassToHom] at hY' ⊢\n    have := Localization.inverts L (weakEquivalences _) p\n      (by rwa [← weakEquivalence_iff])\n    rw [← Function.Bijective.of_comp_iff _\n      (LeftHomotopyClass.postcomp_bijective_of_fibration_of_weakEquivalence _ p)]\n    convert! (Iso.homCongr (Iso.refl (L.obj X)) (asIso (L.map p))).bijective.comp hY'\n    ext f\n    obtain ⟨f, rfl⟩ := f.mk_surjective\n    simp\n  wlog _ : IsFibrant X generalizing X\n  · obtain ⟨X', i, _, _, _⟩ : ∃ (X' : C) (i : X ⟶ X'), Cofibration i ∧ WeakEquivalence i ∧\n        IsFibrant X' :=\n      ⟨_, FibrantObject.HoCat.iResolutionObj X, inferInstance, inferInstance, inferInstance⟩\n    have _ := isCofibrant_of_cofibration i\n    have hX' := this X' inferInstance\n    have := Localization.inverts L (weakEquivalences _) i\n      (by rwa [← weakEquivalence_iff])\n    rw [← Function.Bijective.of_comp_iff _\n      (RightHomotopyClass.precomp_bijective_of_cofibration_of_weakEquivalence Y i)]\n    convert! (Iso.homCongr (asIso (L.map i)) (Iso.refl (L.obj Y))).symm.bijective.comp hX'\n    ext f\n    obtain ⟨f, rfl⟩ := f.mk_surjective\n    simp\n  let E := Localization.uniq BifibrantObject.toHoCat (BifibrantObject.ι ⋙ L) (weakEquivalences _)\n  let e : BifibrantObject.toHoCat ⋙ E.functor ≅ BifibrantObject.ι ⋙ L :=\n    Localization.compUniqFunctor BifibrantObject.toHoCat (BifibrantObject.ι ⋙ L)\n      (weakEquivalences _)\n  have : rightHomotopyClassToHom L =\n      (BifibrantObject.HoCat.homEquivRight.trans (E.fullyFaithfulFunctor.homEquiv.trans\n        (Iso.homCongr (e.app (.mk X)) (e.app (.mk Y))))) := by\n    ext f\n    obtain ⟨f, rfl⟩ := RightHomotopyClass.mk_surjective f\n    exact (NatIso.naturality_1 e (BifibrantObject.homMk f)).symm\n  rw [this]\n  exact Equiv.bijective _\n\nlemma bijective_leftHomotopyClassToHom :\n    Function.Bijective (leftHomotopyClassToHom L : LeftHomotopyClass X Y → _) := by\n  rw [bijective_leftHomotopyClassToHom_iff_bijective_rightHomotopyClassToHom]\n  exact bijective_rightHomotopyClassToHom L X Y\n\nTarget:\nlemma map_surjective_of_isLocalization :\n    Function.Surjective (L.map : (X ⟶ Y) → _) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_17c8f688ffbf","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b0fc25684118d3615137d7395da7edad57c073ae729c00dd9ac826d5b4c1ba03","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/ModelCategory","family_id":"map_surjective_of_islocalization","file_id":"mathlib/Mathlib/AlgebraicTopology/ModelCategory/FundamentalLemma.lean","sample_id":"17c8f688ffbfe3f08997e2e441397e109ef966a6e5472eeef0ec671a2f11c5b3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8f9fde22f4f5e3c6a342c774b51f2e9a3888b3631f985c3a4bfd65b729742429","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bbc6b49baeedf3e55b05cb3f8d67c144875e1981fa10f0ed9325586b2eeede01","source_sha256":"92db59c0cca19f9c28529fd17474c1f344e67bd83f08b4344e7abe51502b9b08","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext : 1\n  simp [derivative']","hard_negative":false,"metrics":{"chosen_tokens":8,"rejected_tokens":3,"token_jaccard":0.1,"token_length_ratio":0.375},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"e01a3736bee50197d3da8d7b02368d6dfe0e69ecf1d6f6b6965ca04d20a1fa70","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.AlgebraMap\npublic import Mathlib.Algebra.Polynomial.Derivative\npublic import Mathlib.Algebra.Polynomial.Module.AEval\npublic import Mathlib.RingTheory.Adjoin.Polynomial.Basic\npublic import Mathlib.RingTheory.Derivation.Basic\n\nNamespace:\nPolynomial\n\nLocal context:\n/-\nCopyright (c) 2023 Kevin Buzzard. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Richard M. Hill\n-/\n/-!\n# Derivations of univariate polynomials\n\nIn this file we prove that an `R`-derivation of `Polynomial R` is determined by its value on `X`.\nWe also provide a constructor `Polynomial.mkDerivation` that\nbuilds a derivation from its value on `X`, and a linear equivalence\n`Polynomial.mkDerivationEquiv` between `A` and `Derivation (Polynomial R) A`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nnamespace Polynomial\n\nsection CommSemiring\n\nvariable {R A : Type*} [CommSemiring R]\n\n/-- `Polynomial.derivative` as a derivation. -/\n@[simps]\ndef derivative' : Derivation R R[X] R[X] where\n  toFun := derivative\n  map_add' _ _ := derivative_add\n  map_smul' := derivative_smul\n  map_one_eq_zero' := derivative_one\n  leibniz' f g := by simp [mul_comm, add_comm, derivative_mul]\n\nvariable [AddCommMonoid A] [Module R A] [Module (Polynomial R) A]\n\n@[simp]\ntheorem derivation_C (D : Derivation R R[X] A) (a : R) : D (C a) = 0 :=\n  D.map_algebraMap a\n\n@[simp]\ntheorem C_smul_derivation_apply (D : Derivation R R[X] A) (a : R) (f : R[X]) :\n    C a • D f = a • D f := by\n  have : C a • D f = D (C a * f) := by simp\n  rw [this, C_mul', D.map_smul]\n\n@[ext]\ntheorem derivation_ext {D₁ D₂ : Derivation R R[X] A} (h : D₁ X = D₂ X) : D₁ = D₂ :=\n  Derivation.ext fun f => Derivation.eqOn_adjoin (Set.eqOn_singleton.2 h) <| by\n    simp only [adjoin_X, Algebra.coe_top, Set.mem_univ]\n\nvariable [IsScalarTower R (Polynomial R) A]\nvariable (R)\n\n/-- The derivation on `R[X]` that takes the value `a` on `X`. -/\ndef mkDerivation : A →ₗ[R] Derivation R R[X] A where\n  toFun := fun a ↦ (LinearMap.toSpanSingleton R[X] A a).compDer derivative'\n  map_add' := fun a b ↦ by ext; simp\n  map_smul' := fun t a ↦ by ext; simp\n\nlemma mkDerivation_apply (a : A) (f : R[X]) :\n    mkDerivation R a f = derivative f • a := by\n  rfl\n\n@[simp]\ntheorem mkDerivation_X (a : A) : mkDerivation R a X = a := by simp [mkDerivation_apply]\n\nTarget:\nlemma mkDerivation_one_eq_derivative' : mkDerivation R (1 : R[X]) = derivative' :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Polynomial","family_id":"mkderivation_one_eq_derivative","file_id":"mathlib/Mathlib/Algebra/Polynomial/Derivation.lean","sample_id":"bbc6b49baeedf3e55b05cb3f8d67c144875e1981fa10f0ed9325586b2eeede01"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c0bd9ef2de4bda624d2c87bea71eb5ca5fdf0a8bf96e99ba0215cfd11acebae5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"94a2116bd60b99beb1018a62dbb0caea14627fadc812ba92de4745a089513b32","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"57e63da4c8d7ec6a478c9fdfc7ae432be0adeebf4eec412381dc418d3098ca3d","source_sha256":"f9292a333dd089609a858e0aaa7af2194ec9cf6135c7c88cedf01931adf88866","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · intro h\n    constructor\n    rintro _ _ f (hf : epimorphisms C f)\n    rw [epi_iff_surjective]\n    rw [← h] at hf\n    exact hf\n  · intro\n    apply le_antisymm (surjective_le_epimorphisms C)\n    intro _ _ f hf\n    have : Epi f := hf\n    change Function.Surjective ((forget C).map f)\n    rw [← epi_iff_surjective]\n    infer_instance","hard_negative":true,"metrics":{"chosen_tokens":68,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.073529},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"e071ab865477f81173b1e02a7fc0f9132d60878de71f612a0cade8e4d7da6b84","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Constructions.EpiMono\npublic import Mathlib.CategoryTheory.Limits.Preserves.Basic\npublic import Mathlib.CategoryTheory.Limits.Shapes.Images\npublic import Mathlib.CategoryTheory.MorphismProperty.Concrete\npublic import Mathlib.CategoryTheory.Types.Basic\n\nNamespace:\nCategoryTheory.ConcreteCategory\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Epi and mono in concrete categories\n\nIn this file, we relate epimorphisms and monomorphisms in a concrete category `C`\nto surjective and injective morphisms, and we show that if `C` has\nstrong epi mono factorizations and is such that `forget C` preserves\nboth epi and mono, then any morphism in `C` can be factored in a\nfunctorial manner as a composition of a surjective morphism followed\nby an injective morphism.\n\n-/\n\n@[expose] public section\n\nuniverse w v v' u u'\n\nnamespace CategoryTheory\n\nvariable {C : Type u} [Category.{v} C] {FC : C → C → Type*} {CC : C → Type w}\nvariable [∀ X Y, FunLike (FC X Y) (CC X) (CC Y)] [ConcreteCategory.{w} C FC]\n\nopen Limits MorphismProperty\n\nnamespace ConcreteCategory\n\nsection\n\ninstance [(forget C).PreservesMonomorphisms] {X Y : C} (f : X ⟶ Y) [Mono f] :\n    Mono (↾f) := Functor.map_mono (forget C) f\n\ninstance [(forget C).PreservesEpimorphisms] {X Y : C} (f : X ⟶ Y) [Epi f] :\n    Epi (↾f) := Functor.map_epi (forget C) f\n\n/-- In any concrete category, injective morphisms are monomorphisms. -/\ntheorem mono_of_injective {X Y : C} (f : X ⟶ Y) (i : Function.Injective f) :\n    Mono f :=\n  (forget C).mono_of_mono_map ((mono_iff_injective ((forget C).map f)).2 i)\n\ninstance forget₂_preservesMonomorphisms (C : Type u) (D : Type u')\n    [Category.{v} C] [Category.{v'} D]\n    {FC : C → C → Type*} {CC : C → Type w}\n    [∀ X Y, FunLike (FC X Y) (CC X) (CC Y)]\n    [ConcreteCategory C FC]\n    {FD : D → D → Type*} {CD : D → Type w}\n    [∀ X Y, FunLike (FD X Y) (CD X) (CD Y)]\n    [ConcreteCategory D FD]\n    [HasForget₂ C D] [(forget C).PreservesMonomorphisms] :\n    (forget₂ C D).PreservesMonomorphisms :=\n  have : (forget₂ C D ⋙ forget D).PreservesMonomorphisms := by\n    simp only [HasForget₂.forget_comp]\n    infer_instance\n  Functor.preservesMonomorphisms_of_preserves_of_reflects _ (forget D)\n\ninstance forget₂_preservesEpimorphisms (C : Type u) (D : Type u')\n    [Category.{v} C] [Category.{v'} D]\n    {FC : C → C → Type*} {CC : C → Type w}\n    [∀ X Y, FunLike (FC X Y) (CC X) (CC Y)]\n    [ConcreteCategory C FC]\n    {FD : D → D → Type*} {CD : D → Type w}\n    [∀ X Y, FunLike (FD X Y) (CD X) (CD Y)]\n    [ConcreteCategory D FD]\n    [HasForget₂ C D] [(forget C).PreservesEpimorphisms] :\n    (forget₂ C D).PreservesEpimorphisms :=\n  have : (forget₂ C D ⋙ forget D).PreservesEpimorphisms := by\n    simp only [HasForget₂.forget_comp]\n    infer_instance\n  Functor.preservesEpimorphisms_of_preserves_of_reflects _ (forget D)\n\nvariable (C)\n\nlemma surjective_le_epimorphisms :\n    MorphismProperty.surjective C ≤ epimorphisms C :=\n  fun _ _ _ hf => (forget C).epi_of_epi_map ((epi_iff_surjective _).2 hf)\n\nlemma injective_le_monomorphisms :\n    MorphismProperty.injective C ≤ monomorphisms C :=\n  fun _ _ _ hf => (forget C).mono_of_mono_map ((mono_iff_injective _).2 hf)\n\nTarget:\nlemma surjective_eq_epimorphisms_iff :\n    MorphismProperty.surjective C = epimorphisms C ↔ (forget C).PreservesEpimorphisms :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_57e63da4c8d7","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"fd4b504bfacd8f24cbfc3d39c84caf099edfcbc8198923639b1a0f7edf00b2c1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/ConcreteCategory","family_id":"surjective_eq_epimorphisms_iff","file_id":"mathlib/Mathlib/CategoryTheory/ConcreteCategory/EpiMono.lean","sample_id":"57e63da4c8d7ec6a478c9fdfc7ae432be0adeebf4eec412381dc418d3098ca3d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"47e39958d92c203177453f8d36c36c3e273fa3ce6ffb582eb81c95f2e2f8425a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"00b449300b9da4fbf1fe53aee201d84aa293839de6b8a6fffb29ea3f6ab07ccb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"33918059ac643ca50e9d2e3d7ccb2d32565782d0b03a2650b0af67e33602c28c","source_sha256":"1c7994cb953bf15a3e2a75bf5a9f75792705603cc1fc781f9e07bc585e941697","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    apply isTopologicalBasis_of_isOpen_of_nhds (fun U hU ↦ hU.1) <| fun x U hx hU ↦ ?_\n    obtain ⟨V, ⟨hoV, hcV⟩, hfx, hVf⟩ : ∃ V ∈ {V | IsOpen V ∧ IsCompact V}, f x ∈ V ∧ V ⊆ f '' U :=\n      (PrespectralSpace.isTopologicalBasis (X := Y)).isOpen_iff.mp\n        (hf.isOpen_iff_image_isOpen.mp hU) (f x) ⟨x, hx, rfl⟩\n    refine ⟨f ⁻¹' V, ⟨hoV.preimage hf.continuous, ?_⟩, ⟨hfx, fun y hy ↦ ?_⟩⟩\n    · exact hf.toIsInducing.isCompact_preimage' hcV <| Set.SurjOn.subset_range hVf\n    · exact hf.injective.mem_set_image.mp (hVf hy)","hard_negative":false,"metrics":{"chosen_tokens":155,"rejected_tokens":160,"token_jaccard":0.940299,"token_length_ratio":1.032258},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"e0be8bd239483da4aed00df8ece624f1be100f52c53f62f9ca053eed3daf8321","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Ideal\npublic import Mathlib.Topology.Sets.Compacts\npublic import Mathlib.Topology.Sets.OpenCover\npublic import Mathlib.Topology.Spectral.Hom\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Prespectral spaces\n\nIn this file, we define prespectral spaces as spaces whose lattice of compact opens forms a basis.\n\n-/\n\n@[expose] public section\n\nopen TopologicalSpace Topology\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n\n/-- A space is prespectral if the lattice of compact opens forms a basis. -/\n@[stacks 08YG \"The last condition for spectral spaces\", mk_iff]\nclass PrespectralSpace (X : Type*) [TopologicalSpace X] : Prop where\n  isTopologicalBasis : IsTopologicalBasis { U : Set X | IsOpen U ∧ IsCompact U }\n\n/-- A space is prespectral if it has a basis consisting of compact opens. -/\nlemma PrespectralSpace.of_isTopologicalBasis {B : Set (Set X)}\n    (basis : IsTopologicalBasis B) (isCompact_basis : ∀ U ∈ B, IsCompact U) :\n    PrespectralSpace X where\n  isTopologicalBasis := basis.of_isOpen_of_subset (fun _ h ↦ h.1)\n    fun s hs ↦ ⟨basis.isOpen hs, isCompact_basis s hs⟩\n\n/-- A space is prespectral if it has a basis consisting of compact opens.\nThis is the variant with an indexed basis instead. -/\nlemma PrespectralSpace.of_isTopologicalBasis' {ι : Type*} {b : ι → Set X}\n    (basis : IsTopologicalBasis (Set.range b)) (isCompact_basis : ∀ i, IsCompact (b i)) :\n    PrespectralSpace X :=\n  .of_isTopologicalBasis basis (by simp_all)\n\ninstance (priority := low) [NoetherianSpace X] : PrespectralSpace X :=\n  .of_isTopologicalBasis isTopologicalBasis_opens fun _ _ ↦ NoetherianSpace.isCompact _\n\ninstance (priority := low) [PrespectralSpace X] : LocallyCompactSpace X where\n  local_compact_nhds _ _ hn :=\n    have ⟨V, ⟨hV₁, hV₂⟩, hxV, hVn⟩ := PrespectralSpace.isTopologicalBasis.mem_nhds_iff.mp hn\n    ⟨V, hV₁.mem_nhds hxV, hVn, hV₂⟩\n\nopen PrespectralSpace in\ninstance (priority := low) [T2Space X] [PrespectralSpace X] : TotallySeparatedSpace X :=\n  totallySeparatedSpace_iff_exists_isClopen.mpr fun _ _ hxy ↦\n    have ⟨U, ⟨hU₁, hU₂⟩, hxU, hyU⟩ :=\n      isTopologicalBasis.exists_subset_of_mem_open hxy isClosed_singleton.isOpen_compl\n    ⟨U, ⟨hU₂.isClosed, hU₁⟩, hxU, fun h ↦ hyU h rfl⟩\n\nlemma PrespectralSpace.of_isOpenCover\n    {ι : Type*} {U : ι → Opens X} (hU : IsOpenCover U) [∀ i, PrespectralSpace (U i)] :\n    PrespectralSpace X := by\n  refine .of_isTopologicalBasis (hU.isTopologicalBasis fun i ↦ isTopologicalBasis) ?_\n  simp only [Set.mem_iUnion, Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp,\n    forall_comm (α := Set _), forall_apply_eq_imp_iff₂]\n  exact fun i V hV hV' ↦ hV'.image continuous_subtype_val\n\nlemma PrespectralSpace.of_isInducing [PrespectralSpace Y]\n    (f : X → Y) (hf : IsInducing f) (hf' : IsSpectralMap f) : PrespectralSpace X :=\n  .of_isTopologicalBasis (PrespectralSpace.isTopologicalBasis.isInducing hf) (by\n    simp only [Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp]\n    rintro _ U h₁ h₂ rfl\n    exact hf'.isCompact_preimage_of_isOpen h₁ h₂)\n\nlemma PrespectralSpace.of_isClosedEmbedding [PrespectralSpace Y]\n    (f : X → Y) (hf : IsClosedEmbedding f) : PrespectralSpace X :=\n  .of_isInducing f hf.isInducing hf.isProperMap.isSpectralMap\n\n/-- Let `f : X → Y` be an open embedding of topological spaces.\nIf `Y` is a prespectral space (i.e., the quasi-compact opens of `Y` form a basis),\nthen `X` is also a prespectral space. -/\n\nTarget:\nlemma Topology.IsOpenEmbedding.prespectralSpace [PrespectralSpace Y]\n    {f : X → Y} (hf : IsOpenEmbedding f) :\n    PrespectralSpace X where\n  isTopologicalBasis :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n    apply isTopologicalBasis_of_isOpen_of_nhds (fun U hU ↦ hU.1) <| fun x U hx hU ↦ ?_\n    obtain ⟨V, ⟨hoV, hcV⟩, hfx, hVf⟩ : ∃ V ∈ {V | IsOpen V ∧ IsCompact V}, f x ∈ V ∧ V ⊆ f '' U :=\n      (PrespectralSpace.isTopologicalBasis (X := Y)).isOpen_iff.mp\n        (hf.isOpen_iff_image_isOpen.mp hU) (f x) ⟨x, hx, rfl⟩\n    refine ⟨f ⁻¹' V, ⟨hoV.preimage hf.continuous, ?_⟩, ⟨hfx, fun y hy ↦ ?_⟩⟩\n    · exact hf.toIsInducing.isCompact_preimage' hcV <| Set.SurjOn.subset_range hVf\n    · exact hf.injective.mem_set_image.mp (hVf hy)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Spectral","family_id":"topology","file_id":"mathlib/Mathlib/Topology/Spectral/Prespectral.lean","sample_id":"33918059ac643ca50e9d2e3d7ccb2d32565782d0b03a2650b0af67e33602c28c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"156fa55c2e6ebaab51efb61e1b6415d519f8bca6eaddaf4eff8b3327fd1b34c8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b9d038f0b31926725a67cabc50ad8dfeb3e914d41076f9b7c6d575a332e9b255","source_sha256":"e2806df73ecfc81f91bdaba3be881cdfeef7117f485c371a23cf9b886b2f1661","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    rcases le_total x 0 with (neg | pos)\n    · exact .inr <| isSquare_of_nonneg (neg_nonneg_of_nonpos neg)\n    · exact .inl <| isSquare_of_nonneg pos\n  exists_isRoot_of_odd_natDegree := exists_isRoot_of_odd_natDegree\n\nvariable [IsRealClosed R]","hard_negative":false,"metrics":{"chosen_tokens":38,"rejected_tokens":2,"token_jaccard":0.037037,"token_length_ratio":0.052632},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"e12d3e799ab9bab09a38aedfa11d3af2c09c7f06dd96e2c42c1ff2bbcfea4b8c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Degree.Domain\npublic import Mathlib.Algebra.Polynomial.Eval.Defs\npublic import Mathlib.Algebra.Ring.Semireal.Defs\npublic import Mathlib.Tactic.LinearCombination\n\nNamespace:\nIsRealClosed\n\nLocal context:\n/-\nCopyright (c) 2025 Artie Khovanov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Artie Khovanov\n-/\n/-!\n# Real Closed Field\n\nA field `R` is real closed if all of the following hold:\n1. `R` is real (that is, `-1` is not a sum of squares in `R`).\n2. for every `x` in `R`, one of `x` or `-x` is a square.\n3. every odd-degree polynomial over `R` has a root in `R`.\n\nA real closed field is an algebraic generalisation of the real numbers.\n\nIn this file we define real closed fields and prove some of their properties.\n\nTODO (Artie Khovanov) : equivalent conditions for a real field to be real closed\nTODO (Artie Khovanov) : real numbers, real algebraic numbers, hyperreals form a real closed field\n\n## Main Definitions\n\n- `IsRealClosed R` is the typeclass saying `R` is a real closed field.\n\n## Tags\n\nreal closed, rcf\n\n-/\n\npublic section\n\nopen Polynomial\n\n/--\nA field `R` is real closed if all of the following hold:\n1. `R` is real (that is, `-1` is not a sum of squares in `R`).\n2. for every `x` in `R`, one of `x` or `-x` is a square.\n3. every odd-degree polynomial over `R` has a root in `R`.\n-/\nclass IsRealClosed (R : Type*) [Field R] : Prop extends IsSemireal R where\n  isSquare_or_isSquare_neg (x : R) : IsSquare x ∨ IsSquare (-x)\n  exists_isRoot_of_odd_natDegree {f : R[X]} (hf : Odd f.natDegree) : ∃ x, f.IsRoot x\n\nattribute [aesop 90% forward] IsRealClosed.isSquare_or_isSquare_neg\n\nnamespace IsRealClosed\n\nuniverse u\n\nvariable {R : Type u} [Field R]\n\nTarget:\ntheorem of_linearOrderedField [LinearOrder R] [IsStrictOrderedRing R]\n    (isSquare_of_nonneg : ∀ {x : R}, 0 ≤ x → IsSquare x)\n    (exists_isRoot_of_odd_natDegree : ∀ {f : R[X]}, Odd f.natDegree → ∃ x, f.IsRoot x) :\n    IsRealClosed R where\n  isSquare_or_isSquare_neg {x} :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/IsRealClosed","family_id":"of_linearorderedfield","file_id":"mathlib/Mathlib/FieldTheory/IsRealClosed/Basic.lean","sample_id":"b9d038f0b31926725a67cabc50ad8dfeb3e914d41076f9b7c6d575a332e9b255"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dc1780cb3fb0b15d1b902233abc1a076e6c3e1b7b64ffb6ddfc98b9761298fce","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"34da45620930cb8bcc150c1331c6a664b3ea40f34777268fd778065e1a422539","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1a6f779a26d6933a1ab93cb5daf06274698653e526a8b75426906873d50dc3c8","source_sha256":"36095e70549b743e6d6e18e94107fa315e72325dc59944c049976dea39a27dfe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine .fromProd (fun a d hd₁ hd₂ c hdc ↦ ?_) (fun b d hd₁ hd₂ c hdc ↦ ?_)\n  · have e1 : (fun b ↦ (f a, g b)) '' d = {f a} ×ˢ (g '' d) := by aesop\n    simp_rw [Prod.map_apply, e1]\n    exact .prod (singleton_nonempty _) (hd₁.image _) isLUB_singleton (hg hd₁ hd₂ hdc)\n  · have e2 : ((fun a ↦ (f a, g b)) '' d) = (f '' d) ×ˢ {g b} := by aesop\n    simp_rw [Prod.map_apply, e2]\n    exact .prod (hd₁.image _) (singleton_nonempty _) (hf hd₁ hd₂ hdc) isLUB_singleton","hard_negative":false,"metrics":{"chosen_tokens":166,"rejected_tokens":171,"token_jaccard":0.92,"token_length_ratio":1.03012},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"e14b0bb81c7ff2020373790c9097fb3dc33ee23b5e9c4c020e650f09436622c3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ScottContinuity\npublic import Mathlib.Order.Bounds.Lattice\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Christopher Hoskin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christopher Hoskin\n-/\n/-!\n# Scott continuity on product spaces\n\n## Main result\n\n- `ScottContinuous_prod_of_ScottContinuous`: A function is Scott continuous on a product space if it\n  is Scott continuous in each variable.\n- `ScottContinuousOn.inf₂`: For complete linear orders, the meet operation is Scott continuous.\n\n-/\n\npublic section\n\nopen Set\n\nvariable {α β γ : Type*}\n\n/-- If  is Scott continuous on a product space if it is Scott continuous and monotone in each\nvariable -/\nlemma ScottContinuousOn.fromProd [Preorder α] [Preorder β] [Preorder γ]\n    {f : α × β → γ} {D : Set (Set (α × β))}\n    (h₁ : ∀ a, ScottContinuousOn ((fun d => Prod.snd '' d) '' D) (fun b => f (a, b)))\n    (h₂ : ∀ b, ScottContinuousOn ((fun d => Prod.fst '' d) '' D) (fun a => f (a, b)))\n    (h₁' : ∀ a, Monotone (fun b => f (a, b))) (h₂' : ∀ b, Monotone (fun a => f (a, b))) :\n    ScottContinuousOn D f := fun d hX hd₁ hd₂ ⟨p1, p2⟩ hdp => by\n  rw [isLUB_congr ((monotone_prod_iff.mpr ⟨h₁', h₂'⟩).upperBounds_image_of_directedOn_prod hd₂),\n    ← iUnion_of_singleton_coe (Prod.fst '' d), iUnion_prod_const, image_iUnion,\n    ← isLUB_iUnion_iff_of_isLUB (fun a => by\n      rw [singleton_prod, image_image f (fun b ↦ (a, b))]\n      exact h₁ _ (mem_image_of_mem (fun d ↦ Prod.snd '' d) hX) (Nonempty.image Prod.snd hd₁)\n        (DirectedOn.snd hd₂) (isLUB_prod.mp hdp).2) _, Set.range]\n  convert!\n    (h₂ _ (mem_image_of_mem (fun d ↦ Prod.fst '' d) hX) (Nonempty.image Prod.fst hd₁)\n      (DirectedOn.fst hd₂) (isLUB_prod.mp hdp).1)\n  ext : 1\n  simp_all only [Subtype.exists, mem_image, Prod.exists,\n    exists_and_right, exists_eq_right, exists_prop, mem_setOf_eq]\n\nlemma ScottContinuous.fromProd {γ : Type*} [Preorder α] [Preorder β] [Preorder γ]\n    {f : α × β → γ} (h₁ : ∀ a, ScottContinuous (fun b => f (a, b)))\n    (h₂ : ∀ b, ScottContinuous (fun a => f (a, b))) : ScottContinuous f := by\n  simp_rw [← scottContinuousOn_univ] at ⊢\n  exact .fromProd (fun a ↦ (h₁ a).scottContinuousOn) (fun b ↦ (h₂ b).scottContinuousOn)\n    (fun a ↦ (h₁ a).monotone) (fun b ↦ (h₂ b).monotone)\n\nTarget:\nlemma ScottContinuous.prod {α' β' : Type*} [Preorder α] [Preorder β] [Preorder α'] [Preorder β']\n    {f : α → α'} {g : β → β'} (hf : ScottContinuous f) (hg : ScottContinuous g) :\n    ScottContinuous (Prod.map f g) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  refine .fromProd (fun a d hd₁ hd₂ c hdc ↦ ?_) (fun b d hd₁ hd₂ c hdc ↦ ?_)\n  · have e1 : (fun b ↦ (f a, g b)) '' d = {f a} ×ˢ (g '' d) := by aesop\n    simp_rw [Prod.map_apply, e1]\n    exact .prod (singleton_nonempty _) (hd₁.image _) isLUB_singleton (hg hd₁ hd₂ hdc)\n  · have e2 : ((fun a ↦ (f a, g b)) '' d) = (f '' d) ×ˢ {g b} := by aesop\n    simp_rw [Prod.map_apply, e2]\n    exact .prod (hd₁.image _) (singleton_nonempty _) (hf hd₁ hd₂ hdc) isLUB_singleton","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/ScottContinuity","family_id":"scottcontinuous","file_id":"mathlib/Mathlib/Order/ScottContinuity/Prod.lean","sample_id":"1a6f779a26d6933a1ab93cb5daf06274698653e526a8b75426906873d50dc3c8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bb1d914f2c85ca879a4ab7ab44641a41b11803eff45090adf0fe5af8fb8479b6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"87b680f48e655bff7c8ab451ec2a00022c0fb4843e94ac946cebd92f485e49fb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"26f5274475c00cabc639fcde1f5b699c28f967e875673ef858d4e214040ea235","source_sha256":"8deb8a50d24f74f7392dcccfc5a64bc688ef77075e655438ef9523a05e0e1481","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction n <;> grind [AllZero, sign]","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":16,"token_jaccard":0.75,"token_length_ratio":1.333333},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"e15ce64fab7abc90c13f02e5624b160ceeaadae59a44a9ad9572cf9237f2e961","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Real.Basic\n\nNamespace:\nTactic.ComputeAsymptotics.UnitMonomial.AllZero\n\nLocal context:\n/-\nCopyright (c) 2026 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\n/-!\n\n# Predicates on monomials\n\nIn this file we define `UnitMonomial`: type to represent monomials without coefficient as a list of\nits exponents.  `[e₁, e₂, ..., eₙ]` corresponds to `basis[0] ^ e₁ * ... * basis[n] ^ eₙ` where\n`basis` is the basis of functions.\n\nThen we define some predicates for these lists:\n1. `FirstNonzeroIsPos li` means that the first non-zero element of the list `li` is positive.\n2. `FirstNonzeroIsNeg li` means that the first non-zero element of the list `li` is negative.\n3. `AllZero li` means that all elements in `li` are zero.\n\nThis trichotomy determines the asymptotic behaviour of a monomial:\n`FirstNonzeroIsPos` means it tends to infinity, `FirstNonzeroIsNeg` means it tends to zero and\n`AllZero` means it tends to a constant.\n-/\n\n@[expose] public section\n\nnamespace Tactic.ComputeAsymptotics\n\n/-- Unit monomial, represented as a list of its exponents. `[e₁, e₂, ..., eₙ]` corresponds to\n`basis[0] ^ e₁ * ... * basis[n] ^ eₙ` where `basis` is the basis of functions. -/\nabbrev UnitMonomial := List ℝ\n\nnamespace UnitMonomial\n\n/-- Type representing a sign of the first non-zero exponent, returned by `sign`. -/\ninductive Sign\n| pos | neg | zero\n\n/-- Sign of the first non-zero exponent of a unit monomial. -/\nnoncomputable def sign : UnitMonomial → Sign\n  | [] => .zero\n  | hd :: tl =>\n    if 0 < hd then\n      .pos\n    else if hd < 0 then\n      .neg\n    else\n      sign tl\n\n/-- Predicate stating that the first non-zero exponent is positive. -/\ndef FirstNonzeroIsPos (m : UnitMonomial) : Prop := m.sign = .pos\n\n/-- Predicate stating that the first non-zero exponent is negative. -/\ndef FirstNonzeroIsNeg (m : UnitMonomial) : Prop := m.sign = .neg\n\n/-- Predicate stating that all exponents are zero. -/\ndef AllZero (m : UnitMonomial) : Prop := m.sign = .zero\n\nnamespace AllZero\n\ntheorem nil : AllZero [] :=\n  rfl\n\n@[simp]\ntheorem cons_iff {hd : ℝ} {tl : UnitMonomial} :\n    AllZero (hd :: tl) ↔ hd = 0 ∧ AllZero tl := by\n  grind [AllZero, sign]\n\ntheorem of_tail {hd : ℝ} {tl : UnitMonomial} (h_hd : hd = 0) (h_tl : AllZero tl) :\n    AllZero (hd :: tl) :=\n  cons_iff.mpr ⟨h_hd, h_tl⟩\n\nTarget:\ntheorem replicate {n : ℕ} : AllZero (List.replicate n 0) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  induction n <;> grind [AllZero, sign]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/ComputeAsymptotics","family_id":"replicate","file_id":"mathlib/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Predicates.lean","sample_id":"26f5274475c00cabc639fcde1f5b699c28f967e875673ef858d4e214040ea235"},"split":"train"}
{"binding":{"chosen_proof_sha256":"81a9d05bf0ed54d4fa50cc7311fe12bd2332adf9f758867f731c0d0d23db9024","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ba8c41a3c9a3ed4626d2c2866e09bd88dd10053f726387e0a0ad28498aa7a5e7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"15f253266f8e1f550df58301525f21097e11cc98ed777b7a946120509154235e","source_sha256":"e01a6cb1d107abe5a5bc2e366e62863bb223a3de261cf05b6f9895b4d0fc06e8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext i\n  by_cases h : i = i₀\n  · subst h\n    simp\n  · rw [subtypeNeLift_of_neq _ _ _ _ h]","hard_negative":true,"metrics":{"chosen_tokens":24,"rejected_tokens":3,"token_jaccard":0.055556,"token_length_ratio":0.125},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"e26a0adc8bd3349d4c3d99cc10637ce74d8bfe19c4b177af8e509a377a51f5ad","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Logic.Equiv.Option\n\nNamespace:\nFunction\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Extending a function from the complement of a singleton\n\nIn this file, we define `Function.subtypeNeLift` which allows to\nextend a (dependent) function defined on the complement of a singleton.\n\n-/\n\n@[expose] public section\n\nnamespace Function\n\nvariable {ι : Type*} [DecidableEq ι] {M : ι → Type*} (i₀ : ι)\n  (f : ∀ (j : { i // i ≠ i₀ }), M j) (x : M i₀)\n\n/-- Given `i₀ : ι` and `x : M i₀`, this is the (dependent) map `(i : ι) → M i`\nwhose value at `i₀` is `x` and which extends a given map on the complement of `{i₀}`. -/\ndef subtypeNeLift (i : ι) : M i :=\n  if h : i = i₀ then by rw [h]; exact x else f ⟨i, h⟩\n\n@[simp]\nlemma subtypeNeLift_self : subtypeNeLift i₀ f x i₀ = x := dif_pos rfl\n\nlemma subtypeNeLift_of_neq (i : ι) (h : i ≠ i₀) :\n    subtypeNeLift i₀ f x i = f ⟨i, h⟩ := dif_neg h\n\n@[simp]\n\nTarget:\nlemma subtypeNeLift_restriction (φ : ∀ i, M i) (i₀ : ι) :\n    subtypeNeLift i₀ (fun i ↦ φ i) (φ i₀) = φ :=\n\nProof body:\n","rejected":"by\n  exact subtypeNeLift_restriction","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"b577b45f916f3ad4d4835b3899fedd07a946a1eba1f6f3d5d5b6a538eb7cefc5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data","family_id":"subtypenelift_restriction","file_id":"mathlib/Mathlib/Data/SubtypeNeLift.lean","sample_id":"15f253266f8e1f550df58301525f21097e11cc98ed777b7a946120509154235e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"49bdfcf01bbb1a8c4c2fb91482006ee7e27f5a1dbdbae3ae54b463c762ee66e9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6af328522e0ae1497586eaa9d33047feb15607dc4be8ec65639a14531857f1f4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"65e0c46ab80804e8a5cd071bf84cbfc579f19ac01cda5e58e506b75975df165e","source_sha256":"35550c7de96df6629396d8c6f9793bd06aaa1489dd49abd1f5374fad7d4fd60d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext X; simp [Equivalence.unit]","hard_negative":true,"metrics":{"chosen_tokens":10,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.3},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"e2e58748dd0e4b5eef5db9c3700472a24b005ba0a194fa7e61b4d7bb49bb0fbf","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Adjunction.Basic\npublic import Mathlib.CategoryTheory.Yoneda\npublic import Mathlib.CategoryTheory.Opposites\n\nNamespace:\nCategoryTheory.Adjunction\n\nLocal context:\n/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Thomas Read, Andrew Yang\n-/\n/-!\n# Opposite adjunctions\n\nThis file contains constructions to relate adjunctions of functors to adjunctions of their\nopposites.\n\n## Tags\nadjunction, opposite, uniqueness\n-/\n\n@[expose] public section\n\n\nopen CategoryTheory\n\nuniverse v₁ v₂ u₁ u₂\n\n-- morphism levels before object levels. See note [category theory universes].\nvariable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]\n\nnamespace CategoryTheory.Adjunction\n\nattribute [local simp] homEquiv_unit homEquiv_counit\n\n/-- If `G` is adjoint to `F` then `F.unop` is adjoint to `G.unop`. -/\n@[simps]\ndef unop {F : Cᵒᵖ ⥤ Dᵒᵖ} {G : Dᵒᵖ ⥤ Cᵒᵖ} (h : G ⊣ F) : F.unop ⊣ G.unop where\n  unit := NatTrans.unop h.counit\n  counit := NatTrans.unop h.unit\n  left_triangle_components _ := Quiver.Hom.op_inj (h.right_triangle_components _)\n  right_triangle_components _ := Quiver.Hom.op_inj (h.left_triangle_components _)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- If `G` is adjoint to `F` then `F.op` is adjoint to `G.op`. -/\n@[simps]\ndef op {F : C ⥤ D} {G : D ⥤ C} (h : G ⊣ F) : F.op ⊣ G.op where\n  unit := NatTrans.op h.counit\n  counit := NatTrans.op h.unit\n  left_triangle_components _ := Quiver.Hom.unop_inj (by simp)\n  right_triangle_components _ := Quiver.Hom.unop_inj (by simp)\n\n/-- If `F` is adjoint to `G.leftOp` then `G` is adjoint to `F.leftOp`. -/\n@[simps]\ndef leftOp {F : C ⥤ Dᵒᵖ} {G : D ⥤ Cᵒᵖ} (a : F ⊣ G.leftOp) : G ⊣ F.leftOp where\n  unit := NatTrans.unop a.counit\n  counit := NatTrans.op a.unit\n  left_triangle_components X := congr($(a.right_triangle_components (.op X)).op)\n  right_triangle_components X := congr($(a.left_triangle_components X.unop).unop)\n\n/-- If `F.rightOp` is adjoint to `G` then `G.rightOp` is adjoint to `F`. -/\n@[simps]\ndef rightOp {F : Cᵒᵖ ⥤ D} {G : Dᵒᵖ ⥤ C} (a : F.rightOp ⊣ G) : G.rightOp ⊣ F where\n  unit := NatTrans.unop a.counit\n  counit := NatTrans.op a.unit\n  left_triangle_components X := congr($(a.right_triangle_components (.op X)).op)\n  right_triangle_components X := congr($(a.left_triangle_components X.unop).unop)\n\nset_option backward.defeqAttrib.useBackward true in\nlemma leftOp_eq {F : C ⥤ Dᵒᵖ} {G : D ⥤ Cᵒᵖ} (a : F ⊣ G.leftOp) :\n    a.leftOp = (opOpEquivalence D).symm.toAdjunction.comp a.op := by\n  ext X; simp [Equivalence.unit]\n\nset_option backward.defeqAttrib.useBackward true in\n\nTarget:\nlemma rightOp_eq {F : Cᵒᵖ ⥤ D} {G : Dᵒᵖ ⥤ C} (a : F.rightOp ⊣ G) :\n    a.rightOp = (opOpEquivalence D).symm.toAdjunction.comp a.op :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_65e0c46ab808","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"bf35ea83ee7e208ca9b5f67529c1890e859879ccf20d1ef5f554c2a0c6abba26","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Adjunction","family_id":"rightop_eq","file_id":"mathlib/Mathlib/CategoryTheory/Adjunction/Opposites.lean","sample_id":"65e0c46ab80804e8a5cd071bf84cbfc579f19ac01cda5e58e506b75975df165e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8f849e0c2e5e9b058d0be7efa5644ffd7d171530c0bf0be894d1f2aff0fde505","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"18167008b708b906a466e4eee73d2d19d1a2245033a09393562894d6c83bfa46","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"76d9a3f6de2535e710efca9f1d5d77672d61eb6404d0c9c6e1be8e32402d6ace","source_sha256":"0ee33967ba2e7eb358f659c9411776ab2218de1f416103e609f36c526927ef27","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [wInner_cWeight_eq_expect, ψ.norm_apply]","hard_negative":true,"metrics":{"chosen_tokens":9,"rejected_tokens":3,"token_jaccard":0.090909,"token_length_ratio":0.333333},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"e30fc623542e5281c71a8fd5476ba50b33642b4886d1f2a8424bdd07ef8fe9d8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Expect\npublic import Mathlib.Algebra.Group.AddChar\npublic import Mathlib.Analysis.RCLike.Inner\n\nNamespace:\nAddChar\n\nLocal context:\n/-\nCopyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Orthogonality of characters of a finite abelian group\n\nThis file proves that characters of a finite abelian group are orthogonal, and in particular that\nthere are at most as many characters as there are elements of the group.\n-/\n\npublic section\n\nopen Finset hiding card\nopen Fintype (card)\nopen Function RCLike\nopen scoped BigOperators ComplexConjugate DirectSum\n\nvariable {G H R : Type*}\n\nnamespace AddChar\nsection AddGroup\nvariable [AddGroup G]\n\nsection Semifield\nvariable [Fintype G] [Semifield R] [CharZero R] {ψ : AddChar G R}\n\nlemma expect_eq_ite (ψ : AddChar G R) : 𝔼 a, ψ a = if ψ = 0 then 1 else 0 := by\n  simp [Fintype.expect_eq_sum_div_card, sum_eq_ite, ite_div]\n\nlemma expect_eq_zero_iff_ne_zero : 𝔼 x, ψ x = 0 ↔ ψ ≠ 0 := by\n  rw [expect_eq_ite, one_ne_zero.ite_eq_right_iff]\n\nlemma expect_ne_zero_iff_eq_zero : 𝔼 x, ψ x ≠ 0 ↔ ψ = 0 := expect_eq_zero_iff_ne_zero.not_left\n\nend Semifield\n\nsection RCLike\nvariable [RCLike R] [Fintype G]\n\nTarget:\nlemma wInner_cWeight_self (ψ : AddChar G R) : ⟪(ψ : G → R), ψ⟫ₙ_[R] = 1 :=\n\nProof body:\n","rejected":"by\n  exact wInner_cWeight_self","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"1209904076d2721414bd2f88050e40d914c83da391e0e4c96722dac7cc9455ce","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Fourier","family_id":"winner_cweight_self","file_id":"mathlib/Mathlib/Analysis/Fourier/FiniteAbelian/Orthogonality.lean","sample_id":"76d9a3f6de2535e710efca9f1d5d77672d61eb6404d0c9c6e1be8e32402d6ace"},"split":"train"}
{"binding":{"chosen_proof_sha256":"42680d60d5e78b64557581d1ab549dd5b4e6ea0d21a689bb03eb5cdec7a02fe8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"026c42c1a18ace69185b5b2383ea579a45b7bd0313b181e0953d0847129aa197","source_sha256":"695353e91eba40e45d9ecd0e58f8b00a56d575b25190f682a1689e29c7316a51","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold logDeriv\n  simp [deriv_algebraMap]","hard_negative":false,"metrics":{"chosen_tokens":7,"rejected_tokens":3,"token_jaccard":0.111111,"token_length_ratio":0.428571},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"e345837be9274fe2df5bab4d0e93a7238089b978ddaadb48545e48e8ce95282b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Derivation.MapCoeffs\npublic import Mathlib.FieldTheory.PrimitiveElement\n\nNamespace:\nDifferential\n\nLocal context:\n/-\nCopyright (c) 2024 Daniel Weber. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Weber\n-/\n/-!\n# Differential Fields\n\nThis file defines the logarithmic derivative `Differential.logDeriv` and proves properties of it.\nThis is defined algebraically, compared to `logDeriv` which is analytical.\n-/\n\n@[expose] public section\n\nnamespace Differential\n\nopen algebraMap Polynomial IntermediateField\n\nvariable {R : Type*} [Field R] [Differential R] (a b : R)\n\n/--\nThe logarithmic derivative of a is a′ / a.\n-/\ndef logDeriv : R := a′ / a\n\n@[simp]\nlemma logDeriv_zero : logDeriv (0 : R) = 0 := by\n  simp [logDeriv]\n\n@[simp]\nlemma logDeriv_one : logDeriv (1 : R) = 0 := by\n  simp [logDeriv]\n\nlemma logDeriv_mul (ha : a ≠ 0) (hb : b ≠ 0) : logDeriv (a * b) = logDeriv a + logDeriv b := by\n  unfold logDeriv\n  simp [field]\n  ring\n\nlemma logDeriv_div (ha : a ≠ 0) (hb : b ≠ 0) : logDeriv (a / b) = logDeriv a - logDeriv b := by\n  unfold logDeriv\n  simp [field, Derivation.leibniz_div]\n\n@[simp]\nlemma logDeriv_pow (n : ℕ) (a : R) : logDeriv (a ^ n) = n * logDeriv a := by\n  induction n with\n  | zero => simp\n  | succ n h2 =>\n    obtain rfl | hb := eq_or_ne a 0\n    · simp\n    · rw [Nat.cast_add, Nat.cast_one, add_mul, one_mul, ← h2, pow_succ, logDeriv_mul] <;>\n      simp [hb]\n\nlemma logDeriv_eq_zero : logDeriv a = 0 ↔ a′ = 0 :=\n  ⟨fun h ↦ by simp only [logDeriv, _root_.div_eq_zero_iff] at h; rcases h with h|h <;> simp [h],\n  fun h ↦ by unfold logDeriv at *; simp [h]⟩\n\nlemma logDeriv_multisetProd {ι : Type*} (s : Multiset ι) {f : ι → R} (h : ∀ x ∈ s, f x ≠ 0) :\n    logDeriv (s.map f).prod = (s.map fun x ↦ logDeriv (f x)).sum := by\n  induction s using Multiset.induction_on\n  · simp\n  · rename_i h₂\n    simp only [Multiset.map_cons, Multiset.sum_cons, Multiset.prod_cons]\n    rw [← h₂]\n    · apply logDeriv_mul\n      · simp [h]\n      · simp_all\n    · simp_all\n\nlemma logDeriv_prod (ι : Type*) (s : Finset ι) (f : ι → R) (h : ∀ x ∈ s, f x ≠ 0) :\n    logDeriv (∏ x ∈ s, f x) = ∑ x ∈ s, logDeriv (f x) := logDeriv_multisetProd _ h\n\nlemma logDeriv_prod_of_eq_zero (ι : Type*) (s : Finset ι) (f : ι → R) (h : ∀ x ∈ s, f x = 0) :\n    logDeriv (∏ x ∈ s, f x) = ∑ x ∈ s, logDeriv (f x) := by\n  unfold logDeriv\n  simp_all\n\nTarget:\nlemma logDeriv_algebraMap {F K : Type*} [Field F] [Field K] [Differential F] [Differential K]\n    [Algebra F K] [DifferentialAlgebra F K]\n    (a : F) : logDeriv (algebraMap F K a) = algebraMap F K (logDeriv a) :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Differential","family_id":"logderiv_algebramap","file_id":"mathlib/Mathlib/FieldTheory/Differential/Basic.lean","sample_id":"026c42c1a18ace69185b5b2383ea579a45b7bd0313b181e0953d0847129aa197"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bdd71d8bba5ab6bc0aadc26f2ebfd16f9c6d0a1ea732936b96c085c594ae8d5f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"08990ef995bd3c79af527f9a2b0631a1d0c6260f5b1442074911893366fcfd66","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4fba185756badbaa3d579461622f7431189517e5fb7071a627d47ad8323f43b0","source_sha256":"1c0dcf6a3ce55bec7a88672314ea6ae6741a6599903854de40abb691f935e16b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [pullSub, List.take_of_length_le hl, List.drop_eq_nil_iff.mpr hl]","hard_negative":true,"metrics":{"chosen_tokens":17,"rejected_tokens":5,"token_jaccard":0.133333,"token_length_ratio":0.294118},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"e3768cd0cb7530009ac746a7429e4255936c69c78baff3cbb08a710a7d696ef2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.CompleteLattice.SetLike\n\nNamespace:\nDescriptive.Tree\n\nLocal context:\n/-\nCopyright (c) 2024 Sven Manthe. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sven Manthe\n-/\n/-!\n# Trees in the sense of descriptive set theory\n\nThis file defines trees of depth `ω` in the sense of descriptive set theory as sets of finite\nsequences that are stable under taking prefixes.\n\n## Main declarations\n\n* `tree A`: a (possibly infinite) tree of depth at most `ω` with nodes in `A`\n-/\n\n@[expose] public section\n\nnamespace Descriptive\n\n/-- A tree is a set of finite sequences, implemented as `List A`, that is stable under\n  taking prefixes. For the definition we use the equivalent property `x ++ [a] ∈ T → x ∈ T`,\n  which is more convenient to check. We define `tree A` as a complete sublattice of\n  `Set (List A)`, which coerces to the type of trees on `A`. -/\ndef tree (A : Type*) : CompleteSublattice (Set (List A)) :=\n  CompleteSublattice.mk' {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T}\n    (by rintro S hS x a ⟨t, ht, hx⟩; use t, ht, hS ht hx)\n    (by rintro S hS x a h T hT; exact hS hT <| h T hT)\n\n@[simps!] instance (A : Type*) : SetLike (tree A) (List A) := SetLike.instSubtypeSet\n\nexample (A : Type*) : PartialOrder (tree A) := inferInstance\n\nnamespace Tree\nvariable {A : Type*} {S T : tree A}\n\nlemma mem_of_append {x y : List A} (h : x ++ y ∈ T) : x ∈ T := by\n  induction y generalizing x with\n  | nil => simpa using h\n  | cons y ys ih => exact T.prop (ih (by simpa))\n\nlemma mem_of_prefix {x y : List A} (h' : x <+: y) (h : y ∈ T) : x ∈ T := by\n  obtain ⟨_, rfl⟩ := h'; exact mem_of_append h\n\ninstance : Trans List.IsPrefix (fun x (T : tree A) ↦ x ∈ T) (fun x T ↦ x ∈ T) where\n  trans := mem_of_prefix\n\nlemma singleton_mem (T : tree A) {a : A} {x : List A} (h : a :: x ∈ T) : [a] ∈ T :=\n  mem_of_prefix ⟨x, rfl⟩ h\n\n@[simp] lemma tree_eq_bot : T = ⊥ ↔ [] ∉ T where\n  mp := by rintro rfl; simp\n  mpr h := by ext x; simpa using fun h' ↦ h <| mem_of_prefix x.nil_prefix h'\n\nlemma take_mem {n : ℕ} (x : T) : x.val.take n ∈ T :=\n  mem_of_prefix (x.val.take_prefix n) x.prop\n\n/-- A variant of `List.take` internally to a tree -/\n@[simps] def take (n : ℕ) (x : T) : T := ⟨x.val.take n, take_mem x⟩\n\n@[simp] lemma take_take (m n : ℕ) (x : T) : take m (take n x) = take (m ⊓ n) x := by\n  simp [Subtype.ext_iff, List.take_take]\n\n@[simp] lemma take_eq_take {x : T} {m n : ℕ} :\n    take m x = take n x ↔ m ⊓ x.val.length = n ⊓ x.val.length := by simp [Subtype.ext_iff]\n\n-- ### `subAt`\n\nvariable (T) (x y : List A)\n\n/-- The residual tree obtained by regarding the node x as new root -/\ndef subAt : tree A :=\n  ⟨(x ++ ·)⁻¹' T, fun _ a _ ↦ mem_of_append (y := [a]) (by rwa [List.append_assoc])⟩\n\n@[simp] lemma mem_subAt : y ∈ subAt T x ↔ x ++ y ∈ T := Iff.rfl\n\n@[simp] lemma subAt_nil : subAt T [] = T := rfl\n\n@[simp] lemma subAt_append : subAt (subAt T x) y = subAt T (x ++ y) := by ext; simp\n\n@[gcongr] lemma subAt_mono (h : S ≤ T) : subAt S x ≤ subAt T x :=\n  Set.preimage_mono h\n\n/-- A variant of `List.drop` that takes values in `subAt` -/\n@[simps] def drop (n : ℕ) (x : T) : subAt T (Tree.take n x).val :=\n  ⟨x.val.drop n, by simp⟩\n\n-- ### `pullSub`\n\n/-- Adjoint of `subAt`, given by pasting x before the root of T. Explicitly,\n  elements are prefixes of x or x with an element of T appended -/\ndef pullSub : tree A where\n  val := { y | y.take x.length <+: x ∧ y.drop x.length ∈ T }\n  property := fun y a ⟨h1, h2⟩ ↦\n    ⟨((y.prefix_append [a]).take x.length).trans h1,\n    mem_of_prefix ((y.prefix_append [a]).drop x.length) h2⟩\n\nvariable {T x y}\n\nTarget:\nlemma mem_pullSub_short (hl : y.length ≤ x.length) : y ∈ pullSub T x ↔ y <+: x ∧ [] ∈ T :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_4fba185756ba","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"eadec439be49bd25fc97ff1d3e8135532a1a5972b123b66e8c35c33e5133e522","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Descriptive","family_id":"mem_pullsub_short","file_id":"mathlib/Mathlib/SetTheory/Descriptive/Tree.lean","sample_id":"4fba185756badbaa3d579461622f7431189517e5fb7071a627d47ad8323f43b0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3a259627f5a5b32fa98a969454c4436a807645c0fdfa0bcf717382246cd616fd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"89ecc5dec26210b39010cde2e335b6d4cf0a4c1bf7eb4d3bd4ba691ff88b6aba","source_sha256":"f31e0ea5c5dcaaef4fe7a595be40e4dcc670dcec12bcae3dfce625de497aea8b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply Stream'.ext; intro n\n  cases n <;> rfl","hard_negative":true,"metrics":{"chosen_tokens":14,"rejected_tokens":8,"token_jaccard":0.055556,"token_length_ratio":0.571429},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"e391990692053e4022446fd8dddf4246db6ca9e99bcaa1dc1f6f7ddb755ffe8f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Stream.Defs\npublic import Mathlib.Logic.Function.Basic\npublic import Mathlib.Data.Nat.Basic\npublic import Mathlib.Tactic.Common\n\nNamespace:\nStream'\n\nLocal context:\n/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\n/-!\n# Streams a.k.a. infinite lists a.k.a. infinite sequences\n-/\n\n@[expose] public section\n\nopen Nat Function Option\n\nnamespace Stream'\n\nuniverse u v w\nvariable {α : Type u} {β : Type v} {δ : Type w}\nvariable (m n : ℕ) (x y : List α) (a b : Stream' α)\n\ninstance [Inhabited α] : Inhabited (Stream' α) :=\n  ⟨Stream'.const default⟩\n\n@[simp] protected theorem eta (s : Stream' α) : head s :: tail s = s :=\n  funext fun i => by cases i <;> rfl\n\n/-- Alias for `Stream'.eta` to match `List` API. -/\nalias cons_head_tail := Stream'.eta\n\n@[ext]\nprotected theorem ext {s₁ s₂ : Stream' α} : (∀ n, get s₁ n = get s₂ n) → s₁ = s₂ :=\n  fun h => funext h\n\n@[simp]\ntheorem get_zero_cons (a : α) (s : Stream' α) : get (a::s) 0 = a :=\n  rfl\n\n@[simp]\ntheorem head_cons (a : α) (s : Stream' α) : head (a::s) = a :=\n  rfl\n\n@[simp]\ntheorem tail_cons (a : α) (s : Stream' α) : tail (a::s) = s :=\n  rfl\n\n@[simp]\ntheorem get_drop (n m : ℕ) (s : Stream' α) : get (drop m s) n = get s (m + n) := by\n  rw [Nat.add_comm]\n  rfl\n\ntheorem tail_eq_drop (s : Stream' α) : tail s = drop 1 s :=\n  rfl\n\n@[simp]\ntheorem drop_drop (n m : ℕ) (s : Stream' α) : drop n (drop m s) = drop (m + n) s := by\n  ext; simp [Nat.add_assoc]\n\n@[simp] theorem get_tail {n : ℕ} {s : Stream' α} : s.tail.get n = s.get (n + 1) := rfl\n\n@[simp] theorem tail_drop' {i : ℕ} {s : Stream' α} : tail (drop i s) = s.drop (i + 1) := by\n  ext; simp [Nat.add_comm, Nat.add_left_comm]\n\n@[simp] theorem drop_tail' {i : ℕ} {s : Stream' α} : drop i (tail s) = s.drop (i + 1) := rfl\n\ntheorem tail_drop (n : ℕ) (s : Stream' α) : tail (drop n s) = drop n (tail s) := by simp\n\ntheorem get_succ (n : ℕ) (s : Stream' α) : get s (succ n) = get (tail s) n :=\n  rfl\n\n@[simp]\ntheorem get_succ_cons (n : ℕ) (s : Stream' α) (x : α) : get (x :: s) n.succ = get s n :=\n  rfl\n\n@[simp] lemma get_cons_append_zero {a : α} {x : List α} {s : Stream' α} :\n    (a :: x ++ₛ s).get 0 = a := rfl\n\n@[simp] lemma append_eq_cons {a : α} {as : Stream' α} : [a] ++ₛ as = a :: as := rfl\n\n@[simp] theorem drop_zero {s : Stream' α} : s.drop 0 = s := rfl\n\ntheorem drop_succ (n : ℕ) (s : Stream' α) : drop (succ n) s = drop n (tail s) :=\n  rfl\n\ntheorem head_drop (a : Stream' α) (n : ℕ) : (a.drop n).head = a.get n := by simp\n\ntheorem cons_injective2 : Function.Injective2 (cons : α → Stream' α → Stream' α) := fun x y s t h =>\n  ⟨by rw [← get_zero_cons x s, h, get_zero_cons],\n    Stream'.ext fun n => by rw [← get_succ_cons n _ x, h, get_succ_cons]⟩\n\ntheorem cons_injective_left (s : Stream' α) : Function.Injective fun x => cons x s :=\n  cons_injective2.left _\n\ntheorem cons_injective_right (x : α) : Function.Injective (cons x) :=\n  cons_injective2.right _\n\ntheorem all_def (p : α → Prop) (s : Stream' α) : All p s = ∀ n, p (get s n) :=\n  rfl\n\ntheorem any_def (p : α → Prop) (s : Stream' α) : Any p s = ∃ n, p (get s n) :=\n  rfl\n\n@[simp]\ntheorem mem_cons (a : α) (s : Stream' α) : a ∈ a::s :=\n  Exists.intro 0 rfl\n\ntheorem mem_cons_of_mem {a : α} {s : Stream' α} (b : α) : a ∈ s → a ∈ b::s := fun ⟨n, h⟩ =>\n  Exists.intro (succ n) (by rw [get_succ, tail_cons, h])\n\ntheorem eq_or_mem_of_mem_cons {a b : α} {s : Stream' α} : (a ∈ b::s) → a = b ∨ a ∈ s :=\n    fun ⟨n, h⟩ => by\n  rcases n with - | n'\n  · left\n    exact h\n  · right\n    rw [get_succ, tail_cons] at h\n    exact ⟨n', h⟩\n\ntheorem mem_of_get_eq {n : ℕ} {s : Stream' α} {a : α} : a = get s n → a ∈ s := fun h =>\n  Exists.intro n h\n\ntheorem mem_iff_exists_get_eq {s : Stream' α} {a : α} : a ∈ s ↔ ∃ n, a = s.get n where\n  mp := by simp [Membership.mem, any_def]\n  mpr h := mem_of_get_eq h.choose_spec\n\nsection Map\n\nvariable (f : α → β)\n\ntheorem drop_map (n : ℕ) (s : Stream' α) : drop n (map f s) = map f (drop n s) :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem get_map (n : ℕ) (s : Stream' α) : get (map f s) n = f (get s n) :=\n  rfl\n\ntheorem tail_map (s : Stream' α) : tail (map f s) = map f (tail s) := rfl\n\n@[simp]\ntheorem head_map (s : Stream' α) : head (map f s) = f (head s) :=\n  rfl\n\ntheorem map_eq (s : Stream' α) : map f s = f (head s)::map f (tail s) := by\n  rw [← Stream'.eta (map f s), tail_map, head_map]\n\ntheorem map_cons (a : α) (s : Stream' α) : map f (a::s) = f a::map f s := by\n  rw [← Stream'.eta (map f (a::s)), map_eq]; rfl\n\n@[simp]\ntheorem map_id (s : Stream' α) : map id s = s :=\n  rfl\n\n@[simp]\ntheorem map_map (g : β → δ) (f : α → β) (s : Stream' α) : map g (map f s) = map (g ∘ f) s :=\n  rfl\n\n@[simp]\ntheorem map_tail (s : Stream' α) : map f (tail s) = tail (map f s) :=\n  rfl\n\ntheorem mem_map {a : α} {s : Stream' α} : a ∈ s → f a ∈ map f s := fun ⟨n, h⟩ =>\n  Exists.intro n (by rw [get_map, h])\n\ntheorem exists_of_mem_map {f} {b : β} {s : Stream' α} : b ∈ map f s → ∃ a, a ∈ s ∧ f a = b :=\n  fun ⟨n, h⟩ => ⟨get s n, ⟨n, rfl⟩, h.symm⟩\n\nend Map\n\nsection Zip\n\nvariable (f : α → β → δ)\n\ntheorem drop_zip (n : ℕ) (s₁ : Stream' α) (s₂ : Stream' β) :\n    drop n (zip f s₁ s₂) = zip f (drop n s₁) (drop n s₂) :=\n  Stream'.ext fun _ => rfl\n\n@[simp]\ntheorem get_zip (n : ℕ) (s₁ : Stream' α) (s₂ : Stream' β) :\n    get (zip f s₁ s₂) n = f (get s₁ n) (get s₂ n) :=\n  rfl\n\ntheorem head_zip (s₁ : Stream' α) (s₂ : Stream' β) : head (zip f s₁ s₂) = f (head s₁) (head s₂) :=\n  rfl\n\ntheorem tail_zip (s₁ : Stream' α) (s₂ : Stream' β) :\n    tail (zip f s₁ s₂) = zip f (tail s₁) (tail s₂) :=\n  rfl\n\ntheorem zip_eq (s₁ : Stream' α) (s₂ : Stream' β) :\n    zip f s₁ s₂ = f (head s₁) (head s₂)::zip f (tail s₁) (tail s₂) := by\n  rw [← Stream'.eta (zip f s₁ s₂)]; rfl\n\n@[simp]\ntheorem get_enum (s : Stream' α) (n : ℕ) : get (enum s) n = (n, s.get n) :=\n  rfl\n\ntheorem enum_eq_zip (s : Stream' α) : enum s = zip Prod.mk nats s :=\n  rfl\n\nend Zip\n\n@[simp]\ntheorem mem_const (a : α) : a ∈ const a :=\n  Exists.intro 0 rfl\n\nTarget:\ntheorem const_eq (a : α) : const a = a::const a :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"875b042262b31d7859253425e5195e587b69724726f762af3d3cd6db1de2cc95","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Stream","family_id":"const_eq","file_id":"mathlib/Mathlib/Data/Stream/Init.lean","sample_id":"89ecc5dec26210b39010cde2e335b6d4cf0a4c1bf7eb4d3bd4ba691ff88b6aba"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a90544dc513a05353e26becaa7c94b77ef02a2abe517eb6e4b51662aba7c3cd3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6c4b77c678413a1803601330d163141e89d2e2dacf1588a9c4dd253dfe7027ba","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5c25dc9430d8645a6917c9b8ba0f572387c246c6713ecceebd925b758bc2e637","source_sha256":"b8e096f7469de9f8271b2d043d71be202bf790e32cee2afb83b7783c6d94bbb7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa [← Basis.coord_apply] using LinearMap.congr_fun (basis_coord R n b s) x","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":24,"token_jaccard":0.863636,"token_length_ratio":1.2},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"e3a52afda624cbcdce425c40ce5ac9fa0c1a62233b604e7709c525156be9b06e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.ExteriorPower.Basic\npublic import Mathlib.LinearAlgebra.ExteriorPower.Pairing\npublic import Mathlib.RingTheory.Finiteness.Subalgebra\npublic import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition\n\nNamespace:\nexteriorPower\n\nLocal context:\n/-\nCopyright (c) 2025 Daniel Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sophie Morel, Daniel Morrison\n-/\n/-!\n# Constructs a basis for exterior powers\n-/\n\n@[expose] public section\n\nvariable {R K M E : Type*} {n : ℕ}\n  [CommRing R] [Field K] [AddCommGroup M] [Module R M] [AddCommGroup E] [Module K E]\n\nnamespace exteriorPower\n\n/-! Finiteness of the exterior power. -/\n\n/-- The `n`th exterior power of a finite module is a finite module. -/\ninstance instFinite [Module.Finite R M] : Module.Finite R (⋀[R]^n M) := by\n  rw [Module.Finite.iff_fg, ExteriorAlgebra.exteriorPower, LinearMap.range_eq_map]\n  exact Submodule.FG.pow (Submodule.FG.map _ Module.Finite.fg_top) n\n\n/-! We construct a basis of `⋀[R]^n M` from a basis of `M`. -/\n\nopen Module Set Set.powersetCard\n\nvariable (R n)\n\n/-- If `b` is a basis of `M` indexed by a linearly ordered type `I` and `s` is a finset of\n`I` of cardinality `n`, then we get a linear form on the `n`th exterior power of `M` by\napplying the `exteriorPower.linearForm` construction to the family of linear forms\ngiven by the coordinates of `b` indexed by elements of `s` (ordered using the given order on\n`I`). -/\nnoncomputable def ιMultiDual {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s : powersetCard I n) : Module.Dual R (⋀[R]^n M) :=\n  pairingDual R M n (ιMulti_family R n b.coord s)\n\n@[simp]\nlemma ιMultiDual_apply_ιMulti {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s : powersetCard I n) (v : Fin n → M) :\n    ιMultiDual R n b s (ιMulti R n v) =\n    (Matrix.of fun i j => b.coord (powersetCard.ofFinEmbEquiv.symm s j) (v i)).det := by\n  simp [ιMultiDual, ιMulti_family, pairingDual_ιMulti_ιMulti]\n\n/-- Let `b` be a basis of `M` indexed by a linearly ordered type `I` and `s` be a finset of `I`\nof cardinality `n`. If we apply the linear form on `⋀[R]^n M` defined by `b` and `s`\nto the exterior product of the `b i` for `i ∈ s`, then we get `1`. -/\nlemma ιMultiDual_apply_diag {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s : powersetCard I n) :\n    ιMultiDual R n b s (ιMulti_family R n b s) = 1 := by\n  rw [ιMulti_family, ιMultiDual_apply_ιMulti]\n  suffices Matrix.of (fun i j => b.coord (powersetCard.ofFinEmbEquiv.symm s j)\n    (b (powersetCard.ofFinEmbEquiv.symm s i))) = 1 by\n    simp_rw [Function.comp_apply, this, Matrix.det_one]\n  ext\n  simp [Matrix.one_apply, Finsupp.single_apply]\n\n/-- Let `b` be a basis of `M` indexed by a linearly ordered type `I` and `s` be a finset of `I`\nof cardinality `n`. Let `t` be a finset of `I` of cardinality `n` such that `s ≠ t`. If we apply\nthe linear form on `⋀[R]^n M` defined by `b` and `s` to the exterior product of the\n`b i` for `i ∈ t`, then we get `0`. -/\nlemma ιMultiDual_apply_nondiag {I : Type*} [LinearOrder I] (b : Basis I R M)\n    (s t : powersetCard I n) (hst : s ≠ t) :\n    ιMultiDual R n b s (ιMulti_family R n b t) = 0 := by\n  rw [ιMulti_family, ιMultiDual_apply_ιMulti]\n  obtain ⟨i, his, hit⟩ := (exists_mem_notMem_iff_ne s t).mp hst\n  obtain ⟨k, rfl⟩ := (mem_range_ofFinEmbEquiv_symm_iff_mem s i).mpr his\n  apply Matrix.det_eq_zero_of_column_eq_zero k\n  simp_rw [Matrix.of_apply, Basis.coord_apply, Function.comp_apply, Basis.repr_self]\n  intro j\n  apply Finsupp.single_eq_of_ne\n  by_contra! h\n  apply hit\n  rw [h, powersetCard.ofFinEmbEquiv_symm_apply, ← powersetCard.mem_coe_iff]\n  exact Finset.orderEmbOfFin_mem t.val t.prop j\n\n/-- If `b` is a basis of `M` (indexed by a linearly ordered type), then the family\n`exteriorPower.ιMulti R n b` of the `n`-fold exterior products of its elements is linearly\nindependent in the `n`th exterior power of `M`. -/\nlemma ιMulti_family_linearIndependent_ofBasis {I : Type*} [LinearOrder I] (b : Basis I R M) :\n    LinearIndependent R (ιMulti_family R n b) :=\n  LinearIndependent.of_pairwise_dual_eq_zero_one _ (fun s ↦ ιMultiDual R n b s)\n    (fun _ _ h => ιMultiDual_apply_nondiag R n b _ _ h)\n    (fun _ => ιMultiDual_apply_diag _ _ _ _)\n\nvariable {R} in\n/-- If `b` is a basis of `M` (indexed by a linearly ordered type), the basis of the `n`th\nexterior power of `M` formed by the `n`-fold exterior products of elements of `b`. -/\nnoncomputable def _root_.Module.Basis.exteriorPower {I : Type*} [LinearOrder I] (b : Basis I R M) :\n    Basis (powersetCard I n) R (⋀[R]^n M) :=\n  Basis.mk (ιMulti_family_linearIndependent_ofBasis _ _ _)\n    (eq_top_iff.mp <| ιMulti_family_span_of_span R b.span_eq)\n\n@[simp]\nlemma coe_basis {I : Type*} [LinearOrder I] (b : Basis I R M) :\n    DFunLike.coe (b.exteriorPower n) = ιMulti_family R n b :=\n  Basis.coe_mk _ _\n\nlemma basis_apply {I : Type*} [LinearOrder I] (b : Basis I R M) (s : powersetCard I n) :\n    b.exteriorPower n s = ιMulti_family R n b s := by\n  rw [coe_basis]\n\n/-- If `b` is a basis of `M` indexed by a linearly ordered type `I` and `B` is the corresponding\nbasis of the `n`th exterior power of `M`, indexed by the set of finsets `s` of `I` of cardinality\n`n`, then the coordinate function of `B` at `s` is the linear form on the `n`th exterior power\ndefined by `b` and `s` in `exteriorPower.ιMultiDual`. -/\nlemma basis_coord {I : Type*} [LinearOrder I] (b : Basis I R M) (s : powersetCard I n) :\n    Basis.coord (b.exteriorPower n) s = ιMultiDual R n b s := by\n  apply LinearMap.ext_on (ιMulti_family_span_of_span R (Basis.span_eq b))\n  rintro x ⟨t, rfl⟩\n  rw [Basis.coord_apply]\n  by_cases! hst : s = t\n  · rw [hst, ιMultiDual_apply_diag, ← basis_apply, Basis.repr_self, Finsupp.single_eq_same]\n  · rw [ιMultiDual_apply_nondiag R n b s t hst, ← basis_apply, Basis.repr_self,\n      Finsupp.single_eq_of_ne hst]\n\nTarget:\nlemma basis_repr_apply {I : Type*} [LinearOrder I] (b : Basis I R M) (x : ⋀[R]^n M)\n    (s : powersetCard I n) :\n    Basis.repr (b.exteriorPower n) x s = ιMultiDual R n b s x :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  simpa [← Basis.coord_apply] using LinearMap.congr_fun (basis_coord R n b s) x","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/ExteriorPower","family_id":"basis_repr_apply","file_id":"mathlib/Mathlib/LinearAlgebra/ExteriorPower/Basis.lean","sample_id":"5c25dc9430d8645a6917c9b8ba0f572387c246c6713ecceebd925b758bc2e637"},"split":"train"}
{"binding":{"chosen_proof_sha256":"55394e3b94e4002c95f3216c1764d8bfdefb9347ecc72f11001d5f75ebad2dcb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"fbcf882d1eaeaa3d8c9140c5df42fa2a0c87136f9f984da508a1d8da2d736f8b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"65f5358ecacae8480f414a0ec307a37a8131f8d46fb796e1817fa646fac75533","source_sha256":"b302dc459101a0b475ce68391aafc25fa9a801436e4b136046216248b891e1f7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rintro x ⟨hf, hg⟩\n  rw [mem_fixedPoints_iff] at *\n  rw [comp_apply, hg, hf]","hard_negative":true,"metrics":{"chosen_tokens":22,"rejected_tokens":3,"token_jaccard":0.058824,"token_length_ratio":0.136364},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"e3c3630c76505294850432fb40bc577b775b580b4872a75922a8777ec74c3f2e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Operations\n\nNamespace:\nFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Fixed points of a self-map\n\nIn this file we define the set `Function.fixedPoints` of fixed points of a function `f : α → α`.\nThe related predicate `IsFixedPt` is defined in `Mathlib.Logic.Function.Defs`.\n\n## Tags\n\nfixed point\n-/\n\n@[expose] public section\n\nnamespace Function\n\nvariable {α : Type*} {x : α} {f g : α → α}\n\n/-- The set of fixed points of a map `f : α → α`. -/\ndef fixedPoints (f : α → α) : Set α :=\n  { x : α | IsFixedPt f x }\n\ninstance fixedPoints.decidable [DecidableEq α] (f : α → α) (x : α) :\n    Decidable (x ∈ fixedPoints f) :=\n  IsFixedPt.decidable\n\n@[simp]\ntheorem mem_fixedPoints : x ∈ fixedPoints f ↔ IsFixedPt f x :=\n  .rfl\n\ntheorem mem_fixedPoints_iff {α : Type*} {f : α → α} {x : α} : x ∈ fixedPoints f ↔ f x = x :=\n  .rfl\n\n@[simp]\ntheorem fixedPoints_id : fixedPoints (@id α) = Set.univ :=\n  Set.ext fun _ => by simpa using isFixedPt_id _\n\n@[simp]\n\nTarget:\ntheorem inter_subset_fixedPoints_comp : fixedPoints f ∩ fixedPoints g ⊆ fixedPoints (f ∘ g) :=\n\nProof body:\n","rejected":"by\n  exact inter_subset_fixedPoints_comp","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"b987ab2babce512194b884122cc8f0fee577a5ea3abbc5c07d67cb59a8ff7971","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/FixedPoints","family_id":"inter_subset_fixedpoints_comp","file_id":"mathlib/Mathlib/Dynamics/FixedPoints/Defs.lean","sample_id":"65f5358ecacae8480f414a0ec307a37a8131f8d46fb796e1817fa646fac75533"},"split":"train"}
{"binding":{"chosen_proof_sha256":"186eee1641eb21a1d1c5d6e7242727e956490464bf744e56de75eb2d692d16ff","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"4a080b9895c1efd55f50ec53f39f68ef545210268ba93f76bc3813b18b45f3eb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"88a199b64f05b0ccb332981cf02b5c961ab80a822fd1dae1175ba359f20a3e82","source_sha256":"f5b68a895eb5e12a4d24474be37696a6e7dfc784571814edb493f00c4110f491","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← singleton_vadd]; exact (h₁ x).add h₂","hard_negative":false,"metrics":{"chosen_tokens":17,"rejected_tokens":22,"token_jaccard":0.8,"token_length_ratio":1.294118},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"e3d42cc7997015f95bb78a442e2c41ed2372b82eaf3c09e2a1d53ae4430b0ccf","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set\npublic import Mathlib.Algebra.Ring.Action.Pointwise.Set\npublic import Mathlib.Topology.Bornology.Basic\n\nNamespace:\nAbsorbent\n\nLocal context:\n/-\nCopyright (c) 2020 Jean Lo, Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jean Lo, Yury Kudryashov\n-/\n/-!\n# Absorption of sets\n\nLet `M` act on `α`, let `A` and `B` be sets in `α`.\nWe say that `A` *absorbs* `B` if for sufficiently large `a : M`, we have `B ⊆ a • A`.\nFormally, \"for sufficiently large `a : M`\" means \"for all but a bounded set of `a`\".\n\nTraditionally, this definition is formulated\nfor the action of a (semi)normed ring on a module over that ring.\n\nWe formulate it in a more general settings for two reasons:\n\n- this way we don't have to depend on metric spaces, normed rings etc;\n- some proofs look nicer with this definition than with something like\n  `∃ r : ℝ, ∀ a : R, r ≤ ‖a‖ → B ⊆ a • A`.\n\nIf `M` is a `GroupWithZero` (e.g., a division ring),\nthe sets absorbing a given set form a filter, see `Filter.absorbing`.\n\n## Implementation notes\n\nFor now, all theorems assume that we deal with (a generalization of) a module over a division ring.\nSome lemmas have multiplicative versions for `MulDistribMulAction`s.\nThey can be added later when someone needs them.\n\n## Keywords\n\nabsorbs, absorbent\n-/\n\n@[expose] public section\n\nassert_not_exists Real\n\nopen Set Bornology Filter\nopen scoped Pointwise\n\nsection Defs\n\nvariable (M : Type*) {α : Type*} [Bornology M] [SMul M α]\n\n/-- A set `s` absorbs another set `t` if `t` is contained in all scalings of `s`\nby all but a bounded set of elements. -/\ndef Absorbs (s t : Set α) : Prop :=\n  ∀ᶠ a in cobounded M, t ⊆ a • s\n\n/-- A set is *absorbent* if it absorbs every singleton. -/\ndef Absorbent (s : Set α) : Prop :=\n  ∀ x, Absorbs M s {x}\n\nend Defs\n\nnamespace Absorbs\n\nsection SMul\n\nvariable {M α : Type*} [Bornology M] [SMul M α] {s s₁ s₂ t t₁ t₂ : Set α} {S T : Set (Set α)}\n\nprotected lemma empty : Absorbs M s ∅ := by simp [Absorbs]\n\nprotected lemma eventually (h : Absorbs M s t) : ∀ᶠ a in cobounded M, t ⊆ a • s := h\n\n@[simp] lemma of_boundedSpace [BoundedSpace M] : Absorbs M s t := by simp [Absorbs]\n\nlemma mono_left (h : Absorbs M s₁ t) (hs : s₁ ⊆ s₂) : Absorbs M s₂ t :=\n  h.mono fun _a ha ↦ ha.trans <| smul_set_mono hs\n\nlemma mono_right (h : Absorbs M s t₁) (ht : t₂ ⊆ t₁) : Absorbs M s t₂ :=\n  h.mono fun _ ↦ ht.trans\n\nlemma mono (h : Absorbs M s₁ t₁) (hs : s₁ ⊆ s₂) (ht : t₂ ⊆ t₁) : Absorbs M s₂ t₂ :=\n  (h.mono_left hs).mono_right ht\n\n@[simp]\nlemma _root_.absorbs_union : Absorbs M s (t₁ ∪ t₂) ↔ Absorbs M s t₁ ∧ Absorbs M s t₂ := by\n  simp [Absorbs]\n\nprotected lemma union (h₁ : Absorbs M s t₁) (h₂ : Absorbs M s t₂) : Absorbs M s (t₁ ∪ t₂) :=\n  absorbs_union.2 ⟨h₁, h₂⟩\n\nlemma _root_.Set.Finite.absorbs_sUnion {T : Set (Set α)} (hT : T.Finite) :\n    Absorbs M s (⋃₀ T) ↔ ∀ t ∈ T, Absorbs M s t := by\n  simp [Absorbs, hT]\n\nprotected lemma sUnion (hT : T.Finite) (hs : ∀ t ∈ T, Absorbs M s t) :\n    Absorbs M s (⋃₀ T) :=\n  hT.absorbs_sUnion.2 hs\n\n@[simp]\nlemma _root_.absorbs_iUnion {ι : Sort*} [Finite ι] {t : ι → Set α} :\n    Absorbs M s (⋃ i, t i) ↔ ∀ i, Absorbs M s (t i) :=\n  (finite_range t).absorbs_sUnion.trans forall_mem_range\n\nprotected alias ⟨_, iUnion⟩ := absorbs_iUnion\n\nlemma _root_.Set.Finite.absorbs_biUnion {ι : Type*} {t : ι → Set α} {I : Set ι} (hI : I.Finite) :\n    Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i) := by\n  simp [Absorbs, hI]\n\nprotected alias ⟨_, biUnion⟩ := Set.Finite.absorbs_biUnion\n\n@[simp]\nlemma _root_.absorbs_biUnion_finset {ι : Type*} {t : ι → Set α} {I : Finset ι} :\n    Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i) :=\n  I.finite_toSet.absorbs_biUnion\n\nprotected alias ⟨_, biUnion_finset⟩ := absorbs_biUnion_finset\n\nend SMul\n\nsection AddZero\n\nvariable {M E : Type*} [Bornology M] {s₁ s₂ t₁ t₂ : Set E}\n\nprotected lemma add [AddZeroClass E] [DistribSMul M E]\n    (h₁ : Absorbs M s₁ t₁) (h₂ : Absorbs M s₂ t₂) : Absorbs M (s₁ + s₂) (t₁ + t₂) :=\n  h₂.mp <| h₁.eventually.mono fun x hx₁ hx₂ ↦ by rw [smul_add]; exact add_subset_add hx₁ hx₂\n\nprotected lemma zero [Zero E] [SMulZeroClass M E] {s : Set E} (hs : 0 ∈ s) : Absorbs M s 0 :=\n  Eventually.of_forall fun _ ↦ zero_subset.2 <| zero_mem_smul_set hs\n\nend AddZero\n\nend Absorbs\n\nsection GroupWithZero\n\nvariable {G₀ α : Type*} [GroupWithZero G₀] [Bornology G₀] [MulAction G₀ α]\n  {s t u : Set α} {S : Set (Set α)}\n\n@[simp]\nprotected lemma Absorbs.univ : Absorbs G₀ univ s :=\n  (eventually_ne_cobounded 0).mono fun a ha ↦ by rw [smul_set_univ₀ ha]; apply subset_univ\n\nlemma absorbs_iff_eventually_cobounded_mapsTo :\n    Absorbs G₀ s t ↔ ∀ᶠ c in cobounded G₀, MapsTo (c⁻¹ • ·) t s :=\n  eventually_congr <| (eventually_ne_cobounded 0).mono fun c hc ↦ by\n    rw [← preimage_smul_inv₀ hc]; rfl\n\nalias ⟨eventually_cobounded_mapsTo, _⟩ := absorbs_iff_eventually_cobounded_mapsTo\n\n@[simp]\nlemma absorbs_inter : Absorbs G₀ (s ∩ t) u ↔ Absorbs G₀ s u ∧ Absorbs G₀ t u := by\n  simp only [absorbs_iff_eventually_cobounded_mapsTo, mapsTo_inter, eventually_and]\n\nprotected lemma Absorbs.inter (hs : Absorbs G₀ s u) (ht : Absorbs G₀ t u) : Absorbs G₀ (s ∩ t) u :=\n  absorbs_inter.2 ⟨hs, ht⟩\n\nvariable (G₀ u) in\n/-- The filter of sets that absorb `u`. -/\ndef Filter.absorbing : Filter α where\n  sets := {s | Absorbs G₀ s u}\n  univ_sets := .univ\n  sets_of_superset h := h.mono_left\n  inter_sets := .inter\n\n@[simp]\nlemma Filter.mem_absorbing : s ∈ absorbing G₀ u ↔ Absorbs G₀ s u := .rfl\n\nlemma Set.Finite.absorbs_sInter (hS : S.Finite) :\n    Absorbs G₀ (⋂₀ S) t ↔ ∀ s ∈ S, Absorbs G₀ s t :=\n  sInter_mem (f := absorbing G₀ t) hS\n\nprotected alias ⟨_, Absorbs.sInter⟩ := Set.Finite.absorbs_sInter\n\n@[simp]\nlemma absorbs_iInter {ι : Sort*} [Finite ι] {s : ι → Set α} :\n    Absorbs G₀ (⋂ i, s i) t ↔ ∀ i, Absorbs G₀ (s i) t :=\n  iInter_mem (f := absorbing G₀ t)\n\nprotected alias ⟨_, Absorbs.iInter⟩ := absorbs_iInter\n\nlemma Set.Finite.absorbs_biInter {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set α} :\n    Absorbs G₀ (⋂ i ∈ I, s i) t ↔ ∀ i ∈ I, Absorbs G₀ (s i) t :=\n  biInter_mem (f := absorbing G₀ t) hI\n\nprotected alias ⟨_, Absorbs.biInter⟩ := Set.Finite.absorbs_biInter\n\n@[simp]\nlemma absorbs_zero_iff [NeBot (cobounded G₀)]\n    {E : Type*} [AddMonoid E] [DistribMulAction G₀ E] {s : Set E} :\n    Absorbs G₀ s 0 ↔ 0 ∈ s := by\n  simp only [absorbs_iff_eventually_cobounded_mapsTo, ← singleton_zero,\n    mapsTo_singleton, smul_zero, eventually_const]\n\nend GroupWithZero\n\nsection AddGroup\n\nvariable {M E : Type*} [Monoid M] [AddGroup E] [DistribMulAction M E] [Bornology M]\n\n@[simp]\nlemma absorbs_neg_neg {s t : Set E} : Absorbs M (-s) (-t) ↔ Absorbs M s t := by simp [Absorbs]\n\nalias ⟨Absorbs.of_neg_neg, Absorbs.neg_neg⟩ := absorbs_neg_neg\n\nlemma Absorbs.sub {s₁ s₂ t₁ t₂ : Set E} (h₁ : Absorbs M s₁ t₁) (h₂ : Absorbs M s₂ t₂) :\n    Absorbs M (s₁ - s₂) (t₁ - t₂) := by\n  simpa only [sub_eq_add_neg] using h₁.add h₂.neg_neg\n\nend AddGroup\n\nnamespace Absorbent\n\nsection SMul\n\nvariable {M α : Type*} [Bornology M] [SMul M α] {s t : Set α}\n\nprotected theorem mono (ht : Absorbent M s) (hsub : s ⊆ t) : Absorbent M t := fun x ↦\n  (ht x).mono_left hsub\n\ntheorem _root_.absorbent_iff_forall_absorbs_singleton : Absorbent M s ↔ ∀ x, Absorbs M s {x} := .rfl\n\nprotected theorem absorbs (hs : Absorbent M s) {x : α} : Absorbs M s {x} := hs x\n\ntheorem absorbs_finite (hs : Absorbent M s) (ht : t.Finite) : Absorbs M s t := by\n  rw [← Set.biUnion_of_singleton t]\n  exact .biUnion ht fun _ _ => hs.absorbs\n\nend SMul\n\nTarget:\ntheorem vadd_absorbs {M E : Type*} [Bornology M] [AddZeroClass E] [DistribSMul M E]\n    {s₁ s₂ t : Set E} {x : E} (h₁ : Absorbent M s₁) (h₂ : Absorbs M s₂ t) :\n    Absorbs M (s₁ + s₂) (x +ᵥ t) :=\n\nProof body:\n","rejected":"by\n  rw [← singleton_vadd]; exact (h₁ x).add h₂\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Bornology","family_id":"vadd_absorbs","file_id":"mathlib/Mathlib/Topology/Bornology/Absorbs.lean","sample_id":"88a199b64f05b0ccb332981cf02b5c961ab80a822fd1dae1175ba359f20a3e82"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7308aa6374fea597c48250db2901872d2efda7ec2b018c03749527c542943e3a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"88dc792ee34d82bdbe14dbea6a9bf9efe530e0a2ea1640a11d02fb687a45c35f","source_sha256":"f5c6a6d95455e9edee0a0f7f8fe9448df43f3ee7944da75da2fff4d714325b9b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨n, hn⟩ := exists_rat_eq_convergent h\n  exact ⟨n, hn.symm ▸ convs_eq_convergent ξ n⟩","hard_negative":false,"metrics":{"chosen_tokens":22,"rejected_tokens":2,"token_jaccard":0.058824,"token_length_ratio":0.090909},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"e4181044c816113275b6b82e51973c7f4d366de6209e1dacd4b7dd7ecb98cdd1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries\npublic import Mathlib.Algebra.ContinuedFractions.Computation.Translations\npublic import Mathlib.NumberTheory.DiophantineApproximation.Basic\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2022 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Geißer, Michael Stoll\n-/\n/-!\n# Diophantine Approximation using continued fractions\n\n## Main statements\n\nThere are two versions of Legendre's Theorem.`Real.exists_rat_eq_convergent`,\ndefined in `Mathlib/NumberTheory/DiophantineApproximation/Basic.lean`, uses `Real.convergent`,\na simple recursive definition of the convergents that is also defined in that file.\nThis file provides `Real.exists_convs_eq_rat`, using `GenContFract.convs` of `GenContFract.of ξ`.\n-/\n\npublic section\n\nsection Convergent\n\nnamespace Real\n\nopen Int\n\n/-!\nOur `convergent`s agree with `GenContFract.convs`.\n-/\n\nopen GenContFract\n\n/-- The `n`th convergent of the `GenContFract.of ξ` agrees with `ξ.convergent n`. -/\ntheorem convs_eq_convergent (ξ : ℝ) (n : ℕ) :\n    (GenContFract.of ξ).convs n = ξ.convergent n := by\n  induction n generalizing ξ with\n  | zero => simp only [zeroth_conv_eq_h, of_h_eq_floor, convergent_zero, Rat.cast_intCast]\n  | succ n ih => rw [convs_succ, ih (fract ξ)⁻¹, convergent_succ, one_div]; norm_cast\n\nend Real\n\nend Convergent\n\nnamespace Real\n\nvariable {ξ : ℝ} {u v : ℤ}\n\n/-- The main result, *Legendre's Theorem* on rational approximation:\nif `ξ` is a real number and `q` is a rational number such that `|ξ - q| < 1/(2*q.den^2)`,\nthen `q` is a convergent of the continued fraction expansion of `ξ`.\nThis is the version using `GenContFract.convs`. -/\n\nTarget:\ntheorem exists_convs_eq_rat {q : ℚ}\n    (h : |ξ - q| < 1 / (2 * (q.den : ℝ) ^ 2)) : ∃ n, (GenContFract.of ξ).convs n = q :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/DiophantineApproximation","family_id":"exists_convs_eq_rat","file_id":"mathlib/Mathlib/NumberTheory/DiophantineApproximation/ContinuedFractions.lean","sample_id":"88dc792ee34d82bdbe14dbea6a9bf9efe530e0a2ea1640a11d02fb687a45c35f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"598e0d2c4be030094c0dbca69a5ab0955fb2fcd00976225f2cdbe9fd5333ec6f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c6ea5e3ef49586d24e451298b05a23e2d4974a2a6682daf18dc5ecd23cd3a512","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f81f8323cb2859a12bdc22124ed1f8f7177b728cd0e3476841f3fae387cfd30b","source_sha256":"5fbd6e0c9aed4ae0af15e49b0a292eb4e252a42e876e277383b0144f2f430dad","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [List.applyId]\n  cases h₃ : List.dlookup x (List.map Prod.toSigma (xs.zip ys)) with\n  | none =>\n    dsimp [Option.getD]\n    rw [h₁.mem_iff]\n  | some val =>\n    have h₂ : ys.Nodup := h₁.nodup_iff.1 h₀\n    replace h₁ : xs.length = ys.length := h₁.length_eq\n    dsimp\n    induction xs generalizing ys with\n    | nil => contradiction\n    | cons x' xs xs_ih =>\n      rcases ys with - | ⟨y, ys⟩\n      · cases h₃\n      dsimp [List.dlookup] at h₃; split_ifs at h₃ with h\n      · rw [Option.some_inj] at h₃\n        subst x'; subst val\n        simp only [List.mem_cons, true_or]\n      · obtain - | ⟨h₀, h₅⟩ := h₀\n        obtain - | ⟨h₂, h₄⟩ := h₂\n        have h₆ := Nat.succ.inj h₁\n        specialize xs_ih h₅ h₃ h₄ h₆\n        simp only [Ne.symm h, xs_ih, List.mem_cons]\n        suffices val ∈ ys by tauto\n        rw [← Option.mem_def, List.mem_dlookup_iff] at h₃\n        · simp only [Prod.toSigma, List.mem_map, Prod.exists] at h₃\n          rcases h₃ with ⟨a, b, h₃, h₄, h₅⟩\n          apply (List.of_mem_zip h₃).2\n        simp only [List.NodupKeys, List.keys, comp_def, Prod.fst_toSigma, List.map_map]\n        rwa [List.map_fst_zip (le_of_eq h₆)]","hard_negative":false,"metrics":{"chosen_tokens":314,"rejected_tokens":321,"token_jaccard":0.979798,"token_length_ratio":1.022293},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"e4630180ab38f365d85471fbc2a7edc17dd870324c6a71664cac901689e57e57","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.MLList.Basic\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.ZeroLEOne\npublic import Mathlib.Data.DFinsupp.Defs\npublic import Mathlib.Data.Finsupp.Defs\npublic import Mathlib.Data.Int.Range\npublic import Mathlib.Data.List.Sigma\npublic import Mathlib.Data.PNat.Notation\npublic import Mathlib.Tactic.Bound.Init\n\nNamespace:\nPlausible.InjectiveFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\npublic meta import Mathlib.Data.Finsupp.ToDFinsupp\npublic meta import Mathlib.Data.Int.Range\npublic meta import Mathlib.Data.List.Sigma\npublic meta import Plausible.Functions\n\n/-!\n## `Plausible`: generators for functions\n\nThis file defines `Sampleable` instances for `ℤ → ℤ` injective functions.\n\nInjective functions are generated by creating a list of numbers and\na permutation of that list. The permutation insures that every input\nis mapped to a unique output. When an input is not found in the list\nthe input itself is used as an output.\n\nInjective functions `f : α → α` could be generated easily instead of\n`ℤ → ℤ` by generating a `List α`, removing duplicates and creating a\npermutation. One has to be careful when generating the domain to make\nit vast enough that, when generating arguments to apply `f` to,\nthey argument should be likely to lie in the domain of `f`. This is\nthe reason that injective functions `f : ℤ → ℤ` are generated by\nfixing the domain to the range `[-2*size .. 2*size]`, with `size`\nthe size parameter of the `gen` monad.\n\nMuch of the machinery provided in this file is applicable to generate\ninjective functions of type `α → α` and new instances should be easy\nto define.\n\nOther classes of functions such as monotone functions can generated using\nsimilar techniques. For monotone functions, generating two lists, sorting them\nand matching them should suffice, with appropriate default values.\nSome care must be taken for shrinking such functions to make sure\ntheir defining property is invariant through shrinking. Injective\nfunctions are an example of how complicated it can get.\n-/\n\n@[expose] public meta section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type v}\n\nnamespace Plausible\n\nnamespace TotalFunction\n\nsection Finsupp\n\nvariable [DecidableEq α]\n\n/--\nThis theorem exists because plausible does not have access to dlookup but\nmathlib has all the theory for it and wants to use it. We probably want to\nbring these two together at some point.\n-/\nprivate theorem apply_eq_dlookup (m : List (Σ _ : α, β)) (y : β) (x : α) :\n     (withDefault m y).apply x = (m.dlookup x).getD y := by\n  dsimp only [apply]\n  congr 1\n  induction m with\n  | nil => simp\n  | cons p m ih =>\n    rcases p with ⟨fst, snd⟩\n    by_cases heq : fst = x\n    · simp [heq]\n    · rw [List.dlookup_cons_ne]\n      · simp [heq, ih]\n      · symm\n        simp [heq]\n\nvariable [Zero β] [DecidableEq β]\n\n/-- Map a `TotalFunction` to one whose default value is zero so that it represents a `Finsupp`. -/\n@[simp]\ndef zeroDefault : TotalFunction α β → TotalFunction α β\n  | .withDefault A _ => .withDefault A 0\n\n/-- The support of a zero default `TotalFunction`. -/\ndef zeroDefaultSupp : TotalFunction α β → Finset α\n  | .withDefault A _ =>\n    List.toFinset <| (A.dedupKeys.filter fun ab => Sigma.snd ab ≠ 0).map Sigma.fst\n\n/-- Create a finitely supported function from a total function by taking the default value to\nzero. -/\ndef applyFinsupp (tf : TotalFunction α β) : α →₀ β where\n  support := zeroDefaultSupp tf\n  toFun := tf.zeroDefault.apply\n  mem_support_toFun := by\n    intro a\n    rcases tf with ⟨A, y⟩\n    simp only [zeroDefaultSupp, List.mem_map, List.mem_filter, exists_and_right,\n      List.mem_toFinset, exists_eq_right, Sigma.exists, Ne, zeroDefault]\n    rw [apply_eq_dlookup]\n    constructor\n    · rintro ⟨od, hval, hod⟩\n      have := List.mem_dlookup (List.nodupKeys_dedupKeys A) hval\n      rw [(_ : List.dlookup a A = od)]\n      · simpa using hod\n      · simpa [List.dlookup_dedupKeys]\n    · intro h\n      use (A.dlookup a).getD (0 : β)\n      rw [← List.dlookup_dedupKeys] at h ⊢\n      simp only [h, ← List.mem_dlookup_iff A.nodupKeys_dedupKeys, not_false_iff, Option.mem_def]\n      cases haA : List.dlookup a A.dedupKeys\n      · simp [haA] at h\n      · simp\n\nvariable [SampleableExt α] [SampleableExt β] [Repr α]\n\ninstance Finsupp.sampleableExt : SampleableExt (α →₀ β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\n-- TODO: support a non-constant codomain type\ninstance DFinsupp.sampleableExt : SampleableExt (Π₀ _ : α, β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp.toDFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\nend Finsupp\nend TotalFunction\n\nopen _root_.List\n\n/-- Data structure specifying a total function using a list of pairs\nand a default value returned when the input is not in the domain of\nthe partial function.\n\n`mapToSelf f` encodes `x ↦ f x` when `x ∈ f` and `x ↦ x`,\ni.e. `x` to itself, otherwise.\n\nWe use `Σ` to encode mappings instead of `×` because we\nrely on the association list API defined in `Mathlib/Data/List/Sigma.lean`.\n-/\ninductive InjectiveFunction (α : Type u) : Type u\n  | mapToSelf (xs : List (Σ _ : α, α)) :\n      xs.map Sigma.fst ~ xs.map Sigma.snd → List.Nodup (xs.map Sigma.snd) → InjectiveFunction α\n\ninstance : Inhabited (InjectiveFunction α) :=\n  ⟨⟨[], List.Perm.nil, List.nodup_nil⟩⟩\n\nnamespace InjectiveFunction\n\n/-- Apply a total function to an argument. -/\ndef apply [DecidableEq α] : InjectiveFunction α → α → α\n  | InjectiveFunction.mapToSelf m _ _, x => (m.dlookup x).getD x\n\n/-- Produce a string for a given `InjectiveFunction`.\nThe output is of the form `[x₀ ↦ f x₀, .. xₙ ↦ f xₙ, x ↦ x]`.\nUnlike for `TotalFunction`, the default value is not a constant\nbut the identity function.\n-/\nprotected def repr [Repr α] : InjectiveFunction α → String\n  | InjectiveFunction.mapToSelf m _ _ => s! \"[{TotalFunction.reprAux m}x ↦ x]\"\n\ninstance (α : Type u) [Repr α] : Repr (InjectiveFunction α) where\n  reprPrec f _p := InjectiveFunction.repr f\n\n/-- Interpret a list of pairs as a total function, defaulting to\nthe identity function when no entries are found for a given function -/\ndef List.applyId [DecidableEq α] (xs : List (α × α)) (x : α) : α :=\n  ((xs.map Prod.toSigma).dlookup x).getD x\n\n@[simp]\ntheorem List.applyId_cons [DecidableEq α] (xs : List (α × α)) (x y z : α) :\n    List.applyId ((y, z)::xs) x = if y = x then z else List.applyId xs x := by\n  simp only [List.applyId, List.dlookup, eq_rec_constant, Prod.toSigma, List.map]\n  split_ifs <;> rfl\n\nopen Function\nopen List\n\nopen Nat\n\ntheorem List.applyId_zip_eq [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs)\n    (h₁ : xs.length = ys.length) (x y : α) (i : ℕ) (h₂ : xs[i]? = some x) :\n    List.applyId.{u} (xs.zip ys) x = y ↔ ys[i]? = some y := by\n  induction xs generalizing ys i with\n  | nil => cases h₂\n  | cons x' xs xs_ih =>\n    cases i\n    · simp only [length_cons, lt_add_iff_pos_left, add_pos_iff, Nat.lt_add_one, or_true,\n        getElem?_eq_getElem, getElem_cons_zero, Option.some.injEq] at h₂\n      subst h₂\n      cases ys\n      · cases h₁\n      · simp\n    · cases ys\n      · cases h₁\n      · obtain - | ⟨h₀, h₁⟩ := h₀\n        simp only [getElem?_cons_succ, zip_cons_cons, applyId_cons] at h₂ ⊢\n        rw [if_neg]\n        · apply xs_ih <;> solve_by_elim [Nat.succ.inj]\n        · apply h₀; apply List.mem_of_getElem? h₂\n\nTarget:\ntheorem applyId_mem_iff [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs) (h₁ : xs ~ ys)\n    (x : α) : List.applyId.{u} (xs.zip ys) x ∈ ys ↔ x ∈ xs :=\n\nProof body:\n","rejected":"```lean\nby\n  simp only [List.applyId]\n  cases h₃ : List.dlookup x (List.map Prod.toSigma (xs.zip ys)) with\n  | none =>\n    dsimp [Option.getD]\n    rw [h₁.mem_iff]\n  | some val =>\n    have h₂ : ys.Nodup := h₁.nodup_iff.1 h₀\n    replace h₁ : xs.length = ys.length := h₁.length_eq\n    dsimp\n    induction xs generalizing ys with\n    | nil => contradiction\n    | cons x' xs xs_ih =>\n      rcases ys with - | ⟨y, ys⟩\n      · cases h₃\n      dsimp [List.dlookup] at h₃; split_ifs at h₃ with h\n      · rw [Option.some_inj] at h₃\n        subst x'; subst val\n        simp only [List.mem_cons, true_or]\n      · obtain - | ⟨h₀, h₅⟩ := h₀\n        obtain - | ⟨h₂, h₄⟩ := h₂\n        have h₆ := Nat.succ.inj h₁\n        specialize xs_ih h₅ h₃ h₄ h₆\n        simp only [Ne.symm h, xs_ih, List.mem_cons]\n        suffices val ∈ ys by tauto\n        rw [← Option.mem_def, List.mem_dlookup_iff] at h₃\n        · simp only [Prod.toSigma, List.mem_map, Prod.exists] at h₃\n          rcases h₃ with ⟨a, b, h₃, h₄, h₅⟩\n          apply (List.of_mem_zip h₃).2\n        simp only [List.NodupKeys, List.keys, comp_def, Prod.fst_toSigma, List.map_map]\n        rwa [List.map_fst_zip (le_of_eq h₆)]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Testing/Plausible","family_id":"applyid_mem_iff","file_id":"mathlib/Mathlib/Testing/Plausible/Functions.lean","sample_id":"f81f8323cb2859a12bdc22124ed1f8f7177b728cd0e3476841f3fae387cfd30b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dbc5a53bc3d6fc4141510bd9f512049166f30bf6d3c576c98bf05b032824182f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9ae8a4625029f9ac2796e24d13902444fe030ae53370fcf79ea19d3b85c1e327","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a3ef0314d959f005739e2ea0a1bbd35c4cb66fa53980fc55488bf2d3ac971efc","source_sha256":"0f7f045d7dc25461fbd356aab76e5db8e19b16035129f69e67b31933f14c577d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by cfc_cont_tac)\n    (ha : p a := by cfc_tac) (ha' : ∀ i, q i (a i) := by cfc_tac) :\n    cfc f a = fun i => cfc f (a i) := by\n  ext i\n  let φ := Pi.evalStarAlgHom S A i\n  exact φ.map_cfc f a (by rwa [Pi.spectrum_eq]) (continuous_apply i) ha (ha' i)","hard_negative":false,"metrics":{"chosen_tokens":79,"rejected_tokens":86,"token_jaccard":0.945946,"token_length_ratio":1.088608},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"e46432044006992c37b35a13124082ee01ca2c2f48e20b314e4fad7c262be8d1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique\npublic import Mathlib.Algebra.Algebra.Spectrum.Pi\npublic import Mathlib.Algebra.Star.StarAlgHom\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Frédéric Dupuis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Frédéric Dupuis\n-/\n/-! # The continuous functional calculus on product types\n\nThis file contains results about the continuous functional calculus on (indexed) product types.\n\n## Main theorems\n\n+ `cfc_map_pi` and `cfcₙ_map_pi`: given `a : ∀ i, A i`, then `cfc f a = fun i => cfc f (a i)`\n  (and likewise for the non-unital version)\n+ `cfc_map_prod` and `cfcₙ_map_prod`: given `a : A` and `b : B`, then\n  `cfc f (a, b) = (cfc f a, cfc f b)` (and likewise for the non-unital version)\n-/\n\npublic section\n\nsection nonunital_pi\n\nvariable {ι R S : Type*} {A : ι → Type*} [CommSemiring R] [Nontrivial R] [StarRing R]\n  [MetricSpace R]\n  [IsTopologicalSemiring R] [ContinuousStar R] [CommRing S] [Algebra R S]\n  [∀ i, NonUnitalRing (A i)] [∀ i, Module S (A i)] [∀ i, Module R (A i)]\n  [∀ i, IsScalarTower R S (A i)] [∀ i, SMulCommClass R (A i) (A i)]\n  [∀ i, IsScalarTower R (A i) (A i)]\n  [∀ i, StarRing (A i)] [∀ i, TopologicalSpace (A i)] {p : (∀ i, A i) → Prop}\n  {q : (i : ι) → A i → Prop}\n  [NonUnitalContinuousFunctionalCalculus R (∀ i, A i) p]\n  [∀ i, NonUnitalContinuousFunctionalCalculus R (A i) (q i)]\n  [∀ i, ContinuousMapZero.UniqueHom R (A i)]\n\ninclude S in\nlemma cfcₙ_map_pi (f : R → R) (a : ∀ i, A i)\n    (hf : ContinuousOn f (⋃ i, quasispectrum R (a i)) := by cfc_cont_tac)\n    (ha : p a := by cfc_tac) (ha' : ∀ i, q i (a i) := by cfc_tac) :\n    cfcₙ f a = fun i => cfcₙ f (a i) := by\n  by_cases hf₀ : f 0 = 0\n  · ext i\n    have : Nonempty ι := ⟨i⟩\n    let φ := Pi.evalNonUnitalStarAlgHom S A i\n    exact φ.map_cfcₙ f a (by rwa [Pi.quasispectrum_eq]) hf₀ (continuous_apply i) ha (ha' i)\n  · simp only [cfcₙ_apply_of_not_map_zero _ hf₀, Pi.zero_def]\n\nend nonunital_pi\n\nsection nonunital_prod\n\nvariable {A B R S : Type*} [CommSemiring R] [CommRing S] [Nontrivial R] [StarRing R]\n  [MetricSpace R] [IsTopologicalSemiring R] [ContinuousStar R] [Algebra R S] [NonUnitalRing A]\n  [NonUnitalRing B] [Module S A] [Module R A] [Module R B] [Module S B]\n  [SMulCommClass R A A] [SMulCommClass R B B] [IsScalarTower R A A] [IsScalarTower R B B]\n  [StarRing A] [StarRing B] [TopologicalSpace A] [TopologicalSpace B]\n  [IsScalarTower R S A] [IsScalarTower R S B]\n  {pab : A × B → Prop} {pa : A → Prop} {pb : B → Prop}\n  [NonUnitalContinuousFunctionalCalculus R (A × B) pab]\n  [NonUnitalContinuousFunctionalCalculus R A pa]\n  [NonUnitalContinuousFunctionalCalculus R B pb]\n  [ContinuousMapZero.UniqueHom R A] [ContinuousMapZero.UniqueHom R B]\n\ninclude S in\nlemma cfcₙ_map_prod (f : R → R) (a : A) (b : B)\n    (hf : ContinuousOn f (quasispectrum R a ∪ quasispectrum R b) := by cfc_cont_tac)\n    (hab : pab (a, b) := by cfc_tac) (ha : pa a := by cfc_tac) (hb : pb b := by cfc_tac) :\n    cfcₙ f (a, b) = (cfcₙ f a, cfcₙ f b) := by\n  by_cases hf₀ : f 0 = 0\n  case pos =>\n    ext\n    case fst =>\n      let φ := NonUnitalStarAlgHom.fst S A B\n      exact φ.map_cfcₙ f (a, b) (by rwa [Prod.quasispectrum_eq]) hf₀ continuous_fst hab ha\n    case snd =>\n      let φ := NonUnitalStarAlgHom.snd S A B\n      exact φ.map_cfcₙ f (a, b) (by rwa [Prod.quasispectrum_eq]) hf₀ continuous_snd hab hb\n  case neg =>\n    simp [cfcₙ_apply_of_not_map_zero _ hf₀, eqComm]\n\nend nonunital_prod\n\nsection unital_pi\n\nvariable {ι R S : Type*} {A : ι → Type*} [CommSemiring R] [StarRing R] [MetricSpace R]\n  [IsTopologicalSemiring R] [ContinuousStar R] [CommRing S] [Algebra R S]\n  [∀ i, Ring (A i)] [∀ i, Algebra S (A i)] [∀ i, Algebra R (A i)] [∀ i, IsScalarTower R S (A i)]\n  [hinst : IsScalarTower R S (∀ i, A i)]\n  [∀ i, StarRing (A i)] [∀ i, TopologicalSpace (A i)] {p : (∀ i, A i) → Prop}\n  {q : (i : ι) → A i → Prop}\n  [ContinuousFunctionalCalculus R (∀ i, A i) p]\n  [∀ i, ContinuousFunctionalCalculus R (A i) (q i)]\n  [∀ i, ContinuousMap.UniqueHom R (A i)]\n\ninclude S in\n\nTarget:\nlemma cfc_map_pi (f : R → R) (a : ∀ i, A i)\n    (hf : ContinuousOn f (⋃ i, spectrum R (a i)) :=\n\nProof body:\n","rejected":"```lean\nby cfc_cont_tac)\n    (ha : p a := by cfc_tac) (ha' : ∀ i, q i (a i) := by cfc_tac) :\n    cfc f a = fun i => cfc f (a i) := by\n  ext i\n  let φ := Pi.evalStarAlgHom S A i\n  exact φ.map_cfc f a (by rwa [Pi.spectrum_eq]) (continuous_apply i) ha (ha' i)\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/CStarAlgebra","family_id":"cfc_map_pi","file_id":"mathlib/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Pi.lean","sample_id":"a3ef0314d959f005739e2ea0a1bbd35c4cb66fa53980fc55488bf2d3ac971efc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"363f3ee37f17de1a17c56efabc2dfd6a0b7c412da4a7733db969356b0041dbda","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a921660930bcd0f4c9bd0e614c1768bb72ffc642b6565ee11d368d38df7c680d","source_sha256":"a4e7292deef4515ad5e54a931a89dcffc81e82a037be0f261e9c4a92f324bcea","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply SetLike.coe_injective\n  simp only [coe_centralizer, Set.centralizer_centralizer_centralizer]\n\nvariable {M} in","hard_negative":false,"metrics":{"chosen_tokens":19,"rejected_tokens":2,"token_jaccard":0.052632,"token_length_ratio":0.105263},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"e46a4a9ccb7d77c5e92d504a7eaa438d7516f165f88e415be977cc1fece7045c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.Subsemigroup.Centralizer\npublic import Mathlib.GroupTheory.Submonoid.Center\n\nNamespace:\nSubmonoid\n\nLocal context:\n/-\nCopyright (c) 2021 Thomas Browning. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Thomas Browning\n-/\n/-!\n# Centralizers of magmas and monoids\n\n## Main definitions\n\n* `Submonoid.centralizer`: the centralizer of a subset of a monoid\n* `AddSubmonoid.centralizer`: the centralizer of a subset of an additive monoid\n\nWe provide `Subgroup.centralizer`, `AddSubgroup.centralizer` in other files.\n-/\n\n@[expose] public section\n\n-- Guard against import creep\nassert_not_exists Finset\n\nvariable {M : Type*} {S T : Set M}\n\nnamespace Submonoid\n\nsection\n\nvariable [Monoid M] (S)\n\n/-- The centralizer of a subset of a monoid `M`. -/\n@[to_additive /-- The centralizer of a subset of an additive monoid. -/]\ndef centralizer : Submonoid M where\n  carrier := S.centralizer\n  one_mem' := S.one_mem_centralizer\n  mul_mem' := Set.mul_mem_centralizer\n\n@[to_additive (attr := simp, norm_cast)]\ntheorem coe_centralizer : ↑(centralizer S) = S.centralizer :=\n  rfl\n\n@[to_additive AddSubmonoid.centralizer_toAddSubsemigroup]\ntheorem centralizer_toSubsemigroup : (centralizer S).toSubsemigroup = Subsemigroup.centralizer S :=\n  rfl\n\nvariable {S}\n\n@[to_additive]\ntheorem mem_centralizer_iff {z : M} : z ∈ centralizer S ↔ ∀ g ∈ S, g * z = z * g :=\n  Iff.rfl\n\n@[to_additive]\ntheorem center_le_centralizer (s) : center M ≤ centralizer s :=\n  s.center_subset_centralizer\n\n@[to_additive]\ninstance decidableMemCentralizer (a) [Decidable <| ∀ b ∈ S, b * a = a * b] :\n    Decidable (a ∈ centralizer S) :=\n  decidable_of_iff' _ mem_centralizer_iff\n\n@[to_additive]\ntheorem centralizer_le (h : S ⊆ T) : centralizer T ≤ centralizer S :=\n  Set.centralizer_subset h\n\n@[to_additive (attr := simp)]\ntheorem centralizer_eq_top_iff_subset {s : Set M} : centralizer s = ⊤ ↔ s ⊆ center M :=\n  SetLike.ext'_iff.trans Set.centralizer_eq_top_iff_subset\n\nvariable (M)\n\n@[to_additive (attr := simp)]\ntheorem centralizer_univ : centralizer Set.univ = center M :=\n  SetLike.ext' (Set.centralizer_univ M)\n\n@[to_additive]\nlemma le_centralizer_centralizer {s : Submonoid M} : s ≤ centralizer (centralizer (s : Set M)) :=\n  Set.subset_centralizer_centralizer\n\n@[to_additive (attr := simp)]\n\nTarget:\nlemma centralizer_centralizer_centralizer {s : Set M} :\n    centralizer s.centralizer.centralizer = centralizer s :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Submonoid","family_id":"centralizer_centralizer_centralizer","file_id":"mathlib/Mathlib/GroupTheory/Submonoid/Centralizer.lean","sample_id":"a921660930bcd0f4c9bd0e614c1768bb72ffc642b6565ee11d368d38df7c680d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ff8a2edf1706af29211010fc997e519b058214e2acf8d5e15ccbda28fa02aa4d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6fc83af12926d67f82e7ea31db06b0a746a421286c20a9485cd6854580deafe5","source_sha256":"c27083b8f8cde0437299c837bd1f7a4c8446e30ca2254817bcb2958bf94cb26d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Submonoid.eq_top_iff']\n  intro x\n  induction x with | _ a\n  induction a with\n  | one => exact Submonoid.one_mem _\n  | of x => exact subset_closure <| by simp [range, of]\n  | mul x y hx hy => exact Submonoid.mul_mem _ hx hy","hard_negative":false,"metrics":{"chosen_tokens":55,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.054545},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"e488dc514a8a3997590112b8da35690178bf8b36f3d4a45abb0db6317c67be6f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.FreeMonoid.Basic\npublic import Mathlib.Algebra.Group.Submonoid.Operations\npublic import Mathlib.GroupTheory.Congruence.Hom\n\nNamespace:\nPresentedMonoid\n\nLocal context:\n/-\nCopyright (c) 2024 Hannah Fechtner. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Hannah Fechtner\n-/\n/-!\n# Defining a monoid given by generators and relations\n\nGiven relations `rels` on the free monoid on a type `α`, this file constructs the monoid\ngiven by generators `x : α` and relations `rels`.\n\n## Main definitions\n\n* `PresentedMonoid rels`: the quotient of the free monoid on a type `α` by the closure of one-step\n  reductions (arising from a binary relation on free monoid elements `rels`).\n* `PresentedMonoid.of`: The canonical map from `α` to a presented monoid with generators `α`.\n* `PresentedMonoid.lift f`: the canonical monoid homomorphism `PresentedMonoid rels → M`, given\n  a function `f : α → G` from a type `α` to a monoid `M` which satisfies the relations `rels`.\n\n## Tags\n\ngenerators, relations, monoid presentations\n-/\n\n@[expose] public section\n\nvariable {α : Type*}\n\n/-- Given a set of relations, `rels`, over a type `α`, `PresentedMonoid` constructs the monoid with\ngenerators `x : α` and relations `rels` as a quotient of a congruence structure over rels. -/\n@[to_additive /-- Given a set of relations, `rels`, over a type `α`, `PresentedAddMonoid` constructs\nthe monoid with generators `x : α` and relations `rels` as a quotient of an AddCon structure over\nrels -/]\ndef PresentedMonoid (rels : FreeMonoid α → FreeMonoid α → Prop) := (conGen rels).Quotient\n\nnamespace PresentedMonoid\n\nopen Set Submonoid\n\n@[to_additive]\ninstance {rels : FreeMonoid α → FreeMonoid α → Prop} : Monoid (PresentedMonoid rels) :=\n  inferInstanceAs <| Monoid (conGen rels).Quotient\n\n/-- The quotient map from the free monoid on `α` to the presented monoid with the same generators\nand the given relations `rels`. -/\n@[to_additive /-- The quotient map from the free additive monoid on `α` to the presented additive\nmonoid with the same generators and the given relations `rels` -/]\ndef mk (rels : FreeMonoid α → FreeMonoid α → Prop) : FreeMonoid α →* PresentedMonoid rels where\n  toFun := Quotient.mk (conGen rels).toSetoid\n  map_one' := rfl\n  map_mul' := fun _ _ => rfl\n\n/-- `of` is the canonical map from `α` to a presented monoid with generators `x : α`. The term `x`\nis mapped to the equivalence class of the image of `x` in `FreeMonoid α`. -/\n@[to_additive\n/-- `of` is the canonical map from `α` to a presented additive monoid with generators `x : α`. The\nterm `x` is mapped to the equivalence class of the image of `x` in `FreeAddMonoid α`. -/]\ndef of (rels : FreeMonoid α → FreeMonoid α → Prop) (x : α) : PresentedMonoid rels :=\n  mk rels (.of x)\n\nsection inductionOn\n\nvariable {α₁ α₂ α₃ : Type*} {rels₁ : FreeMonoid α₁ → FreeMonoid α₁ → Prop}\n  {rels₂ : FreeMonoid α₂ → FreeMonoid α₂ → Prop} {rels₃ : FreeMonoid α₃ → FreeMonoid α₃ → Prop}\n\nlocal notation \"P₁\" => PresentedMonoid rels₁\nlocal notation \"P₂\" => PresentedMonoid rels₂\nlocal notation \"P₃\" => PresentedMonoid rels₃\n\n@[to_additive (attr := elab_as_elim), induction_eliminator]\nprotected theorem inductionOn {δ : P₁ → Prop} (q : P₁) (h : ∀ a, δ (mk rels₁ a)) : δ q :=\n  Quotient.ind h q\n\n@[to_additive (attr := elab_as_elim)]\nprotected theorem inductionOn₂ {δ : P₁ → P₂ → Prop} (q₁ : P₁) (q₂ : P₂)\n    (h : ∀ a b, δ (mk rels₁ a) (mk rels₂ b)) : δ q₁ q₂ :=\n  Quotient.inductionOn₂ q₁ q₂ h\n\n@[to_additive (attr := elab_as_elim)]\nprotected theorem inductionOn₃ {δ : P₁ → P₂ → P₃ → Prop} (q₁ : P₁)\n    (q₂ : P₂) (q₃ : P₃) (h : ∀ a b c, δ (mk rels₁ a) (mk rels₂ b) (mk rels₃ c)) :\n    δ q₁ q₂ q₃ :=\n  Quotient.inductionOn₃ q₁ q₂ q₃ h\n\nend inductionOn\n\nvariable {α : Type*} {rels : FreeMonoid α → FreeMonoid α → Prop} {x y : FreeMonoid α}\n\nlemma mk_eq_mk_iff : mk rels x = mk rels y ↔ conGen rels x y := Quotient.eq\n\nlemma mk_eq_mk_of_rel (h : rels x y) : mk rels x = mk rels y := mk_eq_mk_iff.2 (.of _ _ h)\n\n/-- The generators of a presented monoid generate the presented monoid. That is, the submonoid\nclosure of the set of generators equals `⊤`. -/\n@[to_additive (attr := simp) /-- The generators of a presented additive monoid generate the\npresented additive monoid. That is, the additive submonoid closure of the set of generators equals\n`⊤`. -/]\n\nTarget:\ntheorem closure_range_of (rels : FreeMonoid α → FreeMonoid α → Prop) :\n    Submonoid.closure (Set.range (of rels)) = ⊤ :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/PresentedMonoid","family_id":"closure_range_of","file_id":"mathlib/Mathlib/Algebra/PresentedMonoid/Basic.lean","sample_id":"6fc83af12926d67f82e7ea31db06b0a746a421286c20a9485cd6854580deafe5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"018854a18bc4ddad72fb2634603cc3211f19d9c9b481d90eec379823ad5a6440","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8d1c313ba94921d07ea1a0040f878f446732c6634c73f66233da8fe3c4ca1e5e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8e6d776a112cd89de7aa1f1a126b48dfead5089fd7abc542320a1969d23d73fc","source_sha256":"f589366b76253f55a5f00af110f0088b3f55c67577d56940b18639c5f1a2a3b7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [← zsmul_eq_mul, ofAdd_zsmul, map_zpow]","hard_negative":true,"metrics":{"chosen_tokens":11,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.272727},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"e494692203a4f53c5f513a358ebbee1784d8af94eeadd1fb53bc0832b527095c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.SpecificLimits.Basic\npublic import Mathlib.Order.SemiconjSup\n\nNamespace:\nCircleDeg1Lift\n\nLocal context:\n/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\n/-!\n# Translation number of a monotone real map that commutes with `x ↦ x + 1`\n\nLet `f : ℝ → ℝ` be a monotone map such that `f (x + 1) = f x + 1` for all `x`. Then the limit\n$$\n  \\tau(f)=\\lim_{n\\to\\infty}{f^n(x)-x}{n}\n$$\nexists and does not depend on `x`. This number is called the *translation number* of `f`.\nDifferent authors use different notation for this number: `τ`, `ρ`, `rot`, etc\n\nIn this file we define a structure `CircleDeg1Lift` for bundled maps with these properties, define\ntranslation number of `f : CircleDeg1Lift`, prove some estimates relating `f^n(x)-x` to `τ(f)`. In\ncase of a continuous map `f` we also prove that `f` admits a point `x` such that `f^n(x)=x+m` if and\nonly if `τ(f)=m/n`.\n\nMaps of this type naturally appear as lifts of orientation-preserving circle homeomorphisms. More\nprecisely, let `f` be an orientation-preserving homeomorphism of the circle $S^1=ℝ/ℤ$, and\nconsider a real number `a` such that\n`⟦a⟧ = f 0`, where `⟦⟧` means the natural projection `ℝ → ℝ/ℤ`. Then there exists a unique\ncontinuous function `F : ℝ → ℝ` such that `F 0 = a` and `⟦F x⟧ = f ⟦x⟧` for all `x` (this fact is\nnot formalized yet). This function is strictly monotone, continuous, and satisfies\n`F (x + 1) = F x + 1`. The number `⟦τ F⟧ : ℝ / ℤ` is called the *rotation number* of `f`.\nIt does not depend on the choice of `a`.\n\n## Main definitions\n\n* `CircleDeg1Lift`: a monotone map `f : ℝ → ℝ` such that `f (x + 1) = f x + 1` for all `x`;\n  the type `CircleDeg1Lift` is equipped with `Lattice` and `Monoid` structures; the\n  multiplication is given by composition: `(f * g) x = f (g x)`.\n* `CircleDeg1Lift.translationNumber`: translation number of `f : CircleDeg1Lift`.\n\n## Main statements\n\nWe prove the following properties of `CircleDeg1Lift.translationNumber`.\n\n* `CircleDeg1Lift.translationNumber_eq_of_dist_bounded`: if the distance between `(f^n) 0`\n  and `(g^n) 0` is bounded from above uniformly in `n : ℕ`, then `f` and `g` have equal\n  translation numbers.\n\n* `CircleDeg1Lift.translationNumber_eq_of_semiconjBy`: if two `CircleDeg1Lift` maps `f`, `g`\n  are semiconjugate by a `CircleDeg1Lift` map, then `τ f = τ g`.\n\n* `CircleDeg1Lift.translationNumber_units_inv`: if `f` is an invertible `CircleDeg1Lift` map\n  (equivalently, `f` is a lift of an orientation-preserving circle homeomorphism), then\n  the translation number of `f⁻¹` is the negative of the translation number of `f`.\n\n* `CircleDeg1Lift.translationNumber_mul_of_commute`: if `f` and `g` commute, then\n  `τ (f * g) = τ f + τ g`.\n\n* `CircleDeg1Lift.translationNumber_eq_rat_iff`: the translation number of `f` is equal to\n  a rational number `m / n` if and only if `(f^n) x = x + m` for some `x`.\n\n* `CircleDeg1Lift.semiconj_of_bijective_of_translationNumber_eq`: if `f` and `g` are two\n  bijective `CircleDeg1Lift` maps and their translation numbers are equal, then these\n  maps are semiconjugate to each other.\n\n* `CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq`: let `f₁` and `f₂` be\n  two actions of a group `G` on the circle by degree 1 maps (formally, `f₁` and `f₂` are two\n  homomorphisms from `G →* CircleDeg1Lift`). If the translation numbers of `f₁ g` and `f₂ g` are\n  equal to each other for all `g : G`, then these two actions are semiconjugate by some\n  `F : CircleDeg1Lift`. This is a version of Proposition 5.4 from [Étienne Ghys, Groupes\n  d'homéomorphismes du cercle et cohomologie bornée][ghys87:groupes].\n\n## Notation\n\nWe use a local notation `τ` for the translation number of `f : CircleDeg1Lift`.\n\n## Implementation notes\n\nWe define the translation number of `f : CircleDeg1Lift` to be the limit of the sequence\n`(f ^ (2 ^ n)) 0 / (2 ^ n)`, then prove that `((f ^ n) x - x) / n` tends to this number for any `x`.\nThis way it is much easier to prove that the limit exists and basic properties of the limit.\n\nWe define translation number for a wider class of maps `f : ℝ → ℝ` instead of lifts of orientation\npreserving circle homeomorphisms for two reasons:\n\n* non-strictly monotone circle self-maps with discontinuities naturally appear as Poincaré maps\n  for some flows on the two-torus (e.g., one can take a constant flow and glue in a few Cherry\n  cells);\n* definition and some basic properties still work for this class.\n\n## References\n\n* [Étienne Ghys, Groupes d'homéomorphismes du cercle et cohomologie bornée][ghys87:groupes]\n\n## TODO\n\nHere are some short-term goals.\n\n* Introduce a structure or a typeclass for lifts of circle homeomorphisms. We use\n  `Units CircleDeg1Lift` for now, but it's better to have a dedicated type (or a typeclass?).\n\n* Prove that the `SemiconjBy` relation on circle homeomorphisms is an equivalence relation.\n\n* Introduce `ConditionallyCompleteLattice` structure, use it in the proof of\n  `CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq`.\n\n* Prove that the orbits of the irrational rotation are dense in the circle. Deduce that a\n  homeomorphism with an irrational rotation is semiconjugate to the corresponding irrational\n  translation by a continuous `CircleDeg1Lift`.\n\n## Tags\n\ncircle homeomorphism, rotation number\n-/\n\n@[expose] public section\n\nopen Filter Set Int Topology\nopen Function hiding Commute\n\n/-!\n### Definition and monoid structure\n-/\n\n/-- A lift of a monotone degree one map `S¹ → S¹`. -/\nstructure CircleDeg1Lift : Type extends ℝ →o ℝ where\n  map_add_one' : ∀ x, toFun (x + 1) = toFun x + 1\n\nnamespace CircleDeg1Lift\n\ninstance : FunLike CircleDeg1Lift ℝ ℝ where\n  coe f := f.toFun\n  coe_injective | ⟨⟨_, _⟩, _⟩, ⟨⟨_, _⟩, _⟩, rfl => rfl\n\ninstance : OrderHomClass CircleDeg1Lift ℝ ℝ where\n  map_rel f _ _ h := f.monotone' h\n\n@[simp] theorem coe_mk (f h) : ⇑(mk f h) = f := rfl\n\nvariable (f g : CircleDeg1Lift)\n\n@[simp] theorem coe_toOrderHom : ⇑f.toOrderHom = f := rfl\n\nprotected theorem monotone : Monotone f := f.monotone'\n\n@[gcongr, mono] theorem mono {x y} (h : x ≤ y) : f x ≤ f y := f.monotone h\n\ntheorem strictMono_iff_injective : StrictMono f ↔ Injective f :=\n  f.monotone.strictMono_iff_injective\n\n@[simp]\ntheorem map_add_one : ∀ x, f (x + 1) = f x + 1 :=\n  f.map_add_one'\n\n@[simp]\ntheorem map_one_add (x : ℝ) : f (1 + x) = 1 + f x := by rw [add_comm, map_add_one, add_comm 1]\n\n@[ext]\ntheorem ext ⦃f g : CircleDeg1Lift⦄ (h : ∀ x, f x = g x) : f = g :=\n  DFunLike.ext f g h\n\ninstance : Monoid CircleDeg1Lift where\n  mul f g :=\n    { toOrderHom := f.1.comp g.1\n      map_add_one' := fun x => by simp [map_add_one] }\n  one := ⟨.id, fun _ => rfl⟩\n  mul_one _ := rfl\n  one_mul _ := rfl\n  mul_assoc _ _ _ := DFunLike.coe_injective rfl\n\ninstance : Inhabited CircleDeg1Lift := ⟨1⟩\n\n@[simp]\ntheorem coe_mul : ⇑(f * g) = f ∘ g :=\n  rfl\n\ntheorem mul_apply (x) : (f * g) x = f (g x) :=\n  rfl\n\n@[simp]\ntheorem coe_one : ⇑(1 : CircleDeg1Lift) = id :=\n  rfl\n\ninstance unitsHasCoeToFun : CoeFun CircleDeg1Liftˣ fun _ => ℝ → ℝ :=\n  ⟨fun f => ⇑(f : CircleDeg1Lift)⟩\n\n@[simp]\ntheorem units_inv_apply_apply (f : CircleDeg1Liftˣ) (x : ℝ) :\n    (f⁻¹ : CircleDeg1Liftˣ) (f x) = x := by simp only [← mul_apply, f.inv_mul, coe_one, id]\n\n@[simp]\ntheorem units_apply_inv_apply (f : CircleDeg1Liftˣ) (x : ℝ) :\n    f ((f⁻¹ : CircleDeg1Liftˣ) x) = x := by simp only [← mul_apply, f.mul_inv, coe_one, id]\n\n/-- If a lift of a circle map is bijective, then it is an order automorphism of the line. -/\ndef toOrderIso : CircleDeg1Liftˣ →* ℝ ≃o ℝ where\n  toFun f :=\n    { toFun := f\n      invFun := ⇑f⁻¹\n      left_inv := units_inv_apply_apply f\n      right_inv := units_apply_inv_apply f\n      map_rel_iff' := ⟨fun h => by simpa using mono (↑f⁻¹) h, mono f⟩ }\n  map_one' := rfl\n  map_mul' _ _ := rfl\n\n@[simp]\ntheorem coe_toOrderIso (f : CircleDeg1Liftˣ) : ⇑(toOrderIso f) = f :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderIso_symm (f : CircleDeg1Liftˣ) :\n    ⇑(toOrderIso f).symm = (f⁻¹ : CircleDeg1Liftˣ) :=\n  rfl\n\n@[simp]\ntheorem coe_toOrderIso_inv (f : CircleDeg1Liftˣ) : ⇑(toOrderIso f)⁻¹ = (f⁻¹ : CircleDeg1Liftˣ) :=\n  rfl\n\ntheorem isUnit_iff_bijective {f : CircleDeg1Lift} : IsUnit f ↔ Bijective f :=\n  ⟨fun ⟨u, h⟩ => h ▸ (toOrderIso u).bijective, fun h =>\n    Units.isUnit\n      { val := f\n        inv :=\n          { toFun := (Equiv.ofBijective f h).symm\n            monotone' := fun x y hxy =>\n              (f.strictMono_iff_injective.2 h.1).le_iff_le.1\n                (by simp only [Equiv.ofBijective_apply_symm_apply f h, hxy])\n            map_add_one' := fun x =>\n              h.1 <| by simp only [Equiv.ofBijective_apply_symm_apply f, f.map_add_one] }\n        val_inv := ext <| Equiv.ofBijective_apply_symm_apply f h\n        inv_val := ext <| Equiv.ofBijective_symm_apply_apply f h }⟩\n\ntheorem coe_pow : ∀ n : ℕ, ⇑(f ^ n) = f^[n]\n  | 0 => rfl\n  | n + 1 => by\n    simp [coe_pow n, pow_succ]\n\ntheorem semiconjBy_iff_semiconj {f g₁ g₂ : CircleDeg1Lift} :\n    SemiconjBy f g₁ g₂ ↔ Semiconj f g₁ g₂ :=\n  CircleDeg1Lift.ext_iff\n\ntheorem commute_iff_commute {f g : CircleDeg1Lift} : Commute f g ↔ Function.Commute f g :=\n  CircleDeg1Lift.ext_iff\n\n/-!\n### Translate by a constant\n-/\n\n\n/-- The map `y ↦ x + y` as a `CircleDeg1Lift`. More precisely, we define a homomorphism from\n`Multiplicative ℝ` to `CircleDeg1Liftˣ`, so the translation by `x` is\n`translation (Multiplicative.ofAdd x)`. -/\ndef translate : Multiplicative ℝ →* CircleDeg1Liftˣ := MonoidHom.toHomUnits <|\n  { toFun x := ⟨⟨fun y => x.toAdd + y, add_right_mono⟩, fun _ => (add_assoc ..).symm⟩\n    map_one' := ext zero_add\n    map_mul' _ _ := ext <| add_assoc _ _ }\n\n@[simp]\ntheorem translate_apply (x y : ℝ) : translate (Multiplicative.ofAdd x) y = x + y :=\n  rfl\n\n@[simp]\ntheorem translate_inv_apply (x y : ℝ) : (translate <| Multiplicative.ofAdd x)⁻¹ y = -x + y :=\n  rfl\n\n@[simp]\n\nTarget:\ntheorem translate_zpow (x : ℝ) (n : ℤ) :\n    translate (Multiplicative.ofAdd x) ^ n = translate (Multiplicative.ofAdd <| ↑n * x) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_8e6d776a112c","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d867d33b3eddfdee5b1f3d8cc8f0d4f6eb7c6edfce4ee68dec2eb02bf0e91ebb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Dynamics/Circle","family_id":"translate_zpow","file_id":"mathlib/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean","sample_id":"8e6d776a112cd89de7aa1f1a126b48dfead5089fd7abc542320a1969d23d73fc"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a577b07c0d33a825fd2224d959c4010ad6df89972c66448b93e986876b718d2f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"2b642afd7ce1ed1e1837b4bb707a0533102a2379b1d34750b7d1b4b919d1b131","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"56bb01dbdf1213a5571dfdae532b08e414ed78cc621e447a3cc9e4531d26ca79","source_sha256":"2fde3cadc36a49aaa16d3276079af548a012432072c96911e1dbf6460053e397","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  calc\n    Var[X; μ | m]\n    _ = μ[X ^ 2 - 2 * X * μ[X | m] + μ[X | m] ^ 2 | m] := by rw [condVar, sub_sq]\n    _ =ᵐ[μ] μ[X ^ 2 | m] - 2 * μ[X | m] ^ 2 + μ[X | m] ^ 2 := by\n      have aux₀ : Integrable (X ^ 2) μ := hX.integrable_sq\n      have aux₁ : Integrable (2 * X * μ[X | m]) μ := by\n        rw [mul_assoc]\n        exact (memLp_one_iff_integrable.1 <| hX.condExp.mul hX).const_mul _\n      have aux₂ : Integrable (μ[X | m] ^ 2) μ := hX.condExp.integrable_sq\n      filter_upwards [condExp_add (m := m) (aux₀.sub aux₁) aux₂, condExp_sub (m := m) aux₀ aux₁,\n        condExp_mul_of_stronglyMeasurable_right stronglyMeasurable_condExp aux₁\n          ((hX.integrable one_le_two).const_mul _), condExp_ofNat (m := m) 2 X]\n        with ω hω₀ hω₁ hω₂ hω₃\n      simp [hω₀, hω₁, hω₂, hω₃,\n        condExp_of_stronglyMeasurable hm (stronglyMeasurable_condExp.pow _) aux₂]\n      simp [mul_assoc, sq]\n    _ = μ[X ^ 2 | m] - μ[X | m] ^ 2 := by ring","hard_negative":true,"metrics":{"chosen_tokens":288,"rejected_tokens":3,"token_jaccard":0.031746,"token_length_ratio":0.010417},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"e4d870a0abe6553c1f65d53d619e4619b8c74758a9de10c4b964080427e3f3fb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut\npublic import Mathlib.MeasureTheory.Integral.Average\npublic import Mathlib.Probability.Moments.Variance\n\nNamespace:\nProbabilityTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Conditional variance\n\nThis file defines the variance of a real-valued random variable conditional to a sigma-algebra.\n\n## TODO\n\nDefine the Lebesgue conditional variance. See\n[GibbsMeasure](https://github.com/james18lpc/GibbsMeasure) for a definition of the Lebesgue\nconditional expectation.\n-/\n\n@[expose] public section\n\nopen MeasureTheory Filter\nopen scoped ENNReal\n\nnamespace ProbabilityTheory\nvariable {Ω : Type*} {m₀ m m' : MeasurableSpace Ω} {hm : m ≤ m₀} {X Y : Ω → ℝ} {μ : Measure[m₀] Ω}\n  {s : Set Ω}\n\nvariable (m X μ) in\n/-- Conditional variance of a real-valued random variable. It is defined as `0` if any one of the\nfollowing conditions is true:\n- `m` is not a sub-σ-algebra of `m₀`,\n- `μ` is not σ-finite with respect to `m`,\n- `X - μ[X | m]` is not square-integrable. -/\nnoncomputable def condVar : Ω → ℝ := μ[(X - μ[X | m]) ^ 2 | m]\n\n@[inherit_doc] scoped notation \"Var[\" X \"; \" μ \" | \" m \"]\" => condVar m X μ\n\n/-- Conditional variance of a real-valued random variable. It is defined as `0` if any one of the\nfollowing conditions is true:\n- `m` is not a sub-σ-algebra of `m₀`,\n- `volume` is not σ-finite with respect to `m`,\n- `X - 𝔼[X | m]` is not square-integrable. -/\nscoped notation \"Var[\" f \"|\" m \"]\" => Var[f; MeasureTheory.volume | m]\n\nlemma condVar_of_not_le (hm : ¬m ≤ m₀) : Var[X; μ | m] = 0 := by rw [condVar, condExp_of_not_le hm]\n\nlemma condVar_of_not_sigmaFinite (hμm : ¬SigmaFinite (μ.trim hm)) :\n    Var[X; μ | m] = 0 := by rw [condVar, condExp_of_not_sigmaFinite hm hμm]\n\nopen scoped Classical in\nlemma condVar_of_sigmaFinite [SigmaFinite (μ.trim hm)] :\n    Var[X; μ | m] =\n      if Integrable (fun ω ↦ (X ω - (μ[X | m]) ω) ^ 2) μ then\n        if StronglyMeasurable[m] (fun ω ↦ (X ω - (μ[X | m]) ω) ^ 2) then\n          fun ω ↦ (X ω - (μ[X | m]) ω) ^ 2\n        else aestronglyMeasurable_condExpL1.mk (condExpL1 hm μ fun ω ↦ (X ω - (μ[X | m]) ω) ^ 2)\n      else 0 := condExp_of_sigmaFinite _\n\nlemma condVar_of_stronglyMeasurable [SigmaFinite (μ.trim hm)]\n    (hX : StronglyMeasurable[m] X) (hXint : Integrable ((X - μ[X | m]) ^ 2) μ) :\n    Var[X; μ | m] = fun ω ↦ (X ω - (μ[X | m]) ω) ^ 2 :=\n  condExp_of_stronglyMeasurable _ ((hX.sub stronglyMeasurable_condExp).pow _) hXint\n\nlemma condVar_of_not_integrable (hXint : ¬ Integrable (fun ω ↦ (X ω - (μ[X | m]) ω) ^ 2) μ) :\n    Var[X; μ | m] = 0 := condExp_of_not_integrable hXint\n\n@[simp] lemma condVar_zero : Var[0; μ | m] = 0 := by simp [condVar]\n\n@[simp]\nlemma condVar_const (hm : m ≤ m₀) (c : ℝ) : Var[fun _ ↦ c; μ | m] = 0 := by\n  obtain rfl | hc := eq_or_ne c 0\n  · simp [← Pi.zero_def]\n  by_cases hμm : IsFiniteMeasure μ\n  · simp [condVar, hm]\n  · simp [condVar, condExp_of_not_integrable, integrable_const_iff_isFiniteMeasure hc,\n      integrable_const_iff_isFiniteMeasure <| pow_ne_zero _ hc, hμm, Pi.pow_def]\n\nlemma stronglyMeasurable_condVar : StronglyMeasurable[m] (Var[X; μ | m]) :=\n  stronglyMeasurable_condExp\n\nlemma condVar_congr_ae (h : X =ᵐ[μ] Y) : Var[X; μ | m] =ᵐ[μ] Var[Y; μ | m] :=\n  condExp_congr_ae <| by filter_upwards [h, condExp_congr_ae h] with ω hω hω'; dsimp; rw [hω, hω']\n\nlemma condVar_of_aestronglyMeasurable [hμm : SigmaFinite (μ.trim hm)]\n    (hX : AEStronglyMeasurable[m] X μ) (hXint : Integrable ((X - μ[X | m]) ^ 2) μ) :\n    Var[X; μ | m] =ᵐ[μ] (X - μ[X | m]) ^ 2 :=\n  condExp_of_aestronglyMeasurable' _ ((continuous_pow _).comp_aestronglyMeasurable\n    (hX.sub stronglyMeasurable_condExp.aestronglyMeasurable)) hXint\n\nlemma integrable_condVar : Integrable Var[X; μ | m] μ := integrable_condExp\n\n/-- The integral of the conditional variance `Var[X | m]` over an `m`-measurable set is equal to\nthe integral of `(X - μ[X | m]) ^ 2` on that set. -/\nlemma setIntegral_condVar [SigmaFinite (μ.trim hm)] (hX : Integrable ((X - μ[X | m]) ^ 2) μ)\n    (hs : MeasurableSet[m] s) :\n    ∫ ω in s, (Var[X; μ | m]) ω ∂μ = ∫ ω in s, (X ω - (μ[X | m]) ω) ^ 2 ∂μ :=\n  setIntegral_condExp _ hX hs\n\n-- `(· ^ 2)` is a postfix operator called `_sq` in lemma names, but\n-- `condVar_ae_eq_condExp_sq_sub_condExp_sq` is a bit ridiculous, so we exceptionally denote it by\n-- `sq_` as it were a prefix.\n\nTarget:\nlemma condVar_ae_eq_condExp_sq_sub_sq_condExp (hm : m ≤ m₀) [IsFiniteMeasure μ] (hX : MemLp X 2 μ) :\n    Var[X; μ | m] =ᵐ[μ] μ[X ^ 2 | m] - μ[X | m] ^ 2 :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_56bb01dbdf12","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"b06209a8391beb4631a67bbdd6ee34087b65f550caeebe12c5e1346ac0141193","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability","family_id":"condvar_ae_eq_condexp_sq_sub_sq_condexp","file_id":"mathlib/Mathlib/Probability/CondVar.lean","sample_id":"56bb01dbdf1213a5571dfdae532b08e414ed78cc621e447a3cc9e4531d26ca79"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bdd86c88ef7416db2f30c34d0d939f5c26ce704f80d0007fe2f340c238437f0f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e1a889ef8d34f3c24811588b71ee3a8a194988f1f5be4944a7c1296be8eaac8e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e8c9dc5f379d2760fb24d04c1fd1b200b12c9f621f4607d9bb2924f7901f72b2","source_sha256":"42683ce9ec3a96f77aab204680a294682617adafa7d756b1f3dbef8d091d30cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine IsPullback.of_forall_isPullback_app fun X => ?_\n  rw [Types.isPullback_iff]\n  refine ⟨congr(($(comp_χ_eq m)).app X), ?_, ?_⟩\n  · simpa using! (mono_iff_injective (m.app X)).mp (inferInstance)\n  · simp only [Functor.const_obj_obj, Functor.sieves_obj, χ_app, Opposite.op_unop,\n      TypeCat.hom_ofHom, TypeCat.Fun.coe_mk, truth_app, Functor.isTerminalConst_from_app,\n      Types.isTerminalPUnit_from_apply, and_true, forall_const]\n    intro p hp\n    simpa [eq_comm] using! congr($(hp).arrows (𝟙 _))","hard_negative":true,"metrics":{"chosen_tokens":123,"rejected_tokens":3,"token_jaccard":0.016949,"token_length_ratio":0.02439},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"e4f858d19f4af3774d53a0b1649b4583fad48b7bb78c54f5f26a499051cc8c79","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Sites.Closed\npublic import Mathlib.CategoryTheory.Sites.Equivalence\npublic import Mathlib.CategoryTheory.Subobject.Classifier.Defs\npublic import Mathlib.CategoryTheory.Subfunctor.Image\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Edward van de Meent. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Edward van de Meent\n-/\n/-!\n\n# (Elementary) Sheaf Topos\n\nWe define a subobject classifier for categories of sheaves of (large enough) types.\n\n## Main definitions\n\nLet `C` refer to a category with (when relevant) Grothendieck topology `J`.\n\n* `Presheaf.classifier C` is a construction of a subobject classifier in `Cᵒᵖ ⥤ Type (max u v)`.\n* `Sheaf.classifier J` is a construction of a subobject classifier in `Sheaf J (Type (max u v))`.\n* `inferInstance : HasClassifier (Cᵒᵖ ⥤ Type w)` says that `Cᵒᵖ ⥤ Type w` has a subobject\n  classifier if `C` is `w`-essentially small.\n* `inferInstance : HasClassifier (Sheaf J (Type w))` says that `Sheaf J (Type w)` has a\n  subobject classifier if `C` is `w`-essentially small.\n\n## Main results\n\n* Any category of sheaves of types has a subobject classifier if the site is essentially small.\n* As a consequence, (because categories of sheaves are cartesian monoidal and have finite limits,)\n  such categories are Elementary Topoi.\n\n## TODOS:\n\n* generalize `Presheaf.isClosed_χ_app_apply_of` to only assuming `G` is separated\n\n-/\n\n@[expose] public section\n\nuniverse w v u\n\nnamespace CategoryTheory\n\nvariable {C : Type u} [Category.{v} C]\nopen Limits\n\nsection presheaf\n\nvariable (C) in\n/-- The truth morphism in the category of presheaves. At each component `X : C`, it is the constant\nmap returning `⊤ : Sieve X`. -/\n@[simps]\ndef Presheaf.truth : (Functor.const _).obj PUnit ⟶ Functor.sieves C where\n  app X := ↾fun _ ↦ (⊤ : Sieve X.unop)\n\nvariable {F G : Cᵒᵖ ⥤ Type (max u v)}\n\nset_option backward.defeqAttrib.useBackward true in\n/--\nThe characteristic map of an inclusion of presheaves.\nGiven a monomorphism of sheaves `m : F ⟶ G`, an object X of the site, map an element `x : G(X)`\nto the (closed) sieve on X where `f : Y → X` is in the sieve iff\n  `∃ a ∈ F(Y), G(f)(x) = m_Y(a)`\n-/\n@[simps app]\ndef Presheaf.χ (m : F ⟶ G) : G ⟶ Functor.sieves C where\n  app X := ↾fun x ↦ ⟨fun Y f => ∃ a, G.map f.op x = m.app (.op Y) a, by\n    intro Y Z f ⟨a, ha⟩ g\n    use F.map g.op a\n    simp [ha, NatTrans.naturality_apply]⟩\n\nlemma Presheaf.comp_χ_eq (m : F ⟶ G) : m ≫ Presheaf.χ m =\n    (Functor.isTerminalConst _ Types.isTerminalPUnit).from F ≫ Presheaf.truth C := by\n  ext\n  apply Sieve.ext\n  simp [← NatTrans.naturality_apply]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma Presheaf.isPullback_χ_truth (m : F ⟶ G) [Mono m] :\n    IsPullback m ((Functor.isTerminalConst _ Types.isTerminalPUnit).from F) (χ m) (truth C) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_e8c9dc5f379d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"56d8e37a1910b600afedc60e4ba1afc1830c8bac9ac37b2689b5dfcfc2773de7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Topos","family_id":"presheaf","file_id":"mathlib/Mathlib/CategoryTheory/Topos/Sheaf.lean","sample_id":"e8c9dc5f379d2760fb24d04c1fd1b200b12c9f621f4607d9bb2924f7901f72b2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"407de8aea2c8c096a4b6c72b70c997e3bb00015ef6148a6a35ed2aaaed552283","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"fe26dba9dee59cdbe8ccc661ca4aeb3988f6a3efc29a757d12badd0f01116421","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"99a463e8b538cc1129c05262ca34214f41e74d8600f9e3aa84ec10439902b1f8","source_sha256":"fdcf591465bcad30a0f93e23ea718e1d3a47c3c81447ba4e02561aad0c6cb720","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    obtain ⟨T', e, h'⟩ := hF T hT\n    exact (ShortComplex.exact_iff_of_iso\n      (F.mapShortComplex.mapIso ((shortComplexOfDistTriangleIsoOfIso e hT)))).2 h'","hard_negative":false,"metrics":{"chosen_tokens":36,"rejected_tokens":41,"token_jaccard":0.851852,"token_length_ratio":1.138889},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"e51e995438167d7e6e6ca3840514af9d351377186e954ddd7eb889fb65a45e7a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.ShortComplex.Exact\npublic import Mathlib.CategoryTheory.Shift.ShiftSequence\npublic import Mathlib.CategoryTheory.Triangulated.Functor\npublic import Mathlib.CategoryTheory.Triangulated.Subcategory\npublic import Mathlib.Algebra.Homology.ExactSequence\n\nNamespace:\nCategoryTheory.Functor\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-! # Homological functors\n\nIn this file, given a functor `F : C ⥤ A` from a pretriangulated category to\nan abelian category, we define the type class `F.IsHomological`, which is the property\nthat `F` sends distinguished triangles in `C` to exact sequences in `A`.\n\nIf `F` has been endowed with `[F.ShiftSequence ℤ]`, then we may think\nof the functor `F` as a `H^0`, and then the `H^n` functors are the functors `F.shift n : C ⥤ A`:\nwe have isomorphisms `(F.shift n).obj X ≅ F.obj (X⟦n⟧)`, but through the choice of this\n\"shift sequence\", the user may provide functors with better definitional properties.\n\nGiven a triangle `T` in `C`, we define a connecting homomorphism\n`F.homologySequenceδ T n₀ n₁ h : (F.shift n₀).obj T.obj₃ ⟶ (F.shift n₁).obj T.obj₁`\nunder the assumption `h : n₀ + 1 = n₁`. When `T` is distinguished, this connecting\nhomomorphism is part of a long exact sequence\n`... ⟶ (F.shift n₀).obj T.obj₁ ⟶ (F.shift n₀).obj T.obj₂ ⟶ (F.shift n₀).obj T.obj₃ ⟶ ...`\n\nThe exactness of this long exact sequence is given by three lemmas\n`F.homologySequence_exact₁`, `F.homologySequence_exact₂` and `F.homologySequence_exact₃`.\n\nIf `F` is a homological functor, we define the strictly full triangulated subcategory\n`F.homologicalKernel`: it consists of objects `X : C` such that for all `n : ℤ`,\n`(F.shift n).obj X` (or `F.obj (X⟦n⟧)`) is zero. We show that a morphism `f` in `C`\nbelongs to `F.homologicalKernel.trW` (i.e. the cone of `f` is in this kernel) iff\n`(F.shift n).map f` is an isomorphism for all `n : ℤ`.\n\nNote: depending on the sources, homological functors are sometimes\ncalled cohomological functors, while certain authors use \"cohomological functors\"\nfor \"contravariant\" functors (i.e. functors `Cᵒᵖ ⥤ A`).\n\n## References\n* [Jean-Louis Verdier, *Des catégories dérivées des catégories abéliennes*][verdier1996]\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Category Limits Pretriangulated ZeroObject Preadditive\n\nvariable {C D A : Type*} [Category* C] [HasShift C ℤ]\n  [Category* D] [HasZeroObject D] [HasShift D ℤ] [Preadditive D]\n  [∀ (n : ℤ), (CategoryTheory.shiftFunctor D n).Additive] [Pretriangulated D]\n  [Category* A]\n\nnamespace Functor\n\nvariable (F : C ⥤ A)\n\n/-- The kernel of a homological functor `F : C ⥤ A` is the strictly full\ntriangulated subcategory consisting of objects `X` such that\nfor all `n : ℤ`, `F.obj (X⟦n⟧)` is zero. -/\ndef homologicalKernel : ObjectProperty C :=\n  fun X ↦ ∀ (n : ℤ), IsZero (F.obj (X⟦n⟧))\n\nlemma mem_homologicalKernel_iff [F.ShiftSequence ℤ] (X : C) :\n    F.homologicalKernel X ↔ ∀ (n : ℤ), IsZero ((F.shift n).obj X) := by\n  simp only [← fun (n : ℤ) => Iso.isZero_iff ((F.isoShift n).app X),\n    homologicalKernel, comp_obj]\n\nsection Pretriangulated\n\nvariable [HasZeroObject C] [Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive]\n  [Pretriangulated C] [Abelian A]\n\n/-- A functor from a pretriangulated category to an abelian category is a homological functor\nif it sends distinguished triangles to exact sequences. -/\nclass IsHomological : Prop extends F.PreservesZeroMorphisms where\n  exact (T : Triangle C) (hT : T ∈ distTriang C) :\n    ((shortComplexOfDistTriangle T hT).map F).Exact\n\nlemma map_distinguished_exact [F.IsHomological] (T : Triangle C) (hT : T ∈ distTriang C) :\n    ((shortComplexOfDistTriangle T hT).map F).Exact :=\n  IsHomological.exact _ hT\n\ninstance (L : C ⥤ D) (F : D ⥤ A) [L.CommShift ℤ] [L.IsTriangulated] [F.IsHomological] :\n    (L ⋙ F).IsHomological where\n  exact T hT := F.map_distinguished_exact _ (L.map_distinguished T hT)\n\nTarget:\nlemma IsHomological.mk' [F.PreservesZeroMorphisms]\n    (hF : ∀ (T : Pretriangulated.Triangle C) (hT : T ∈ distTriang C),\n      ∃ (T' : Pretriangulated.Triangle C) (e : T ≅ T'),\n      ((shortComplexOfDistTriangle T' (isomorphic_distinguished _ hT _ e.symm)).map F).Exact) :\n    F.IsHomological where\n  exact T hT :=\n\nProof body:\n","rejected":"by\n    obtain ⟨T', e, h'⟩ := hF T hT\n    exact (ShortComplex.exact_iff_of_iso\n      (F.mapShortComplex.mapIso ((shortComplexOfDistTriangleIsoOfIso e hT)))).2 h'\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Triangulated","family_id":"ishomological","file_id":"mathlib/Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean","sample_id":"99a463e8b538cc1129c05262ca34214f41e74d8600f9e3aa84ec10439902b1f8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fd734d0795994b2df39d22bec1fc6dc26c9a299bef650b0d07eb3f85b3d6c929","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"572db0c3d1eca879ca2f4b84f5b5bc898ed3b1d25d9e872f516e248e3e2ec2c6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5bf0e0681c55eec8bc5b1c759d43192dd7b331a9558e5d94174d45141766d782","source_sha256":"9414b3a4a4547da0b65ca4fdfb12cdf19f09a370ed358f728d19141cdf7fb313","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [logHeight_eq_log_mulHeight]\n  exact log_nonneg <| x.one_le_mulHeight","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":5,"token_jaccard":0.214286,"token_length_ratio":0.416667},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"e5aad54417fb656db3478b2193d4e9aee9a51f8bfb67aa4e3e934540e8578dd6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.Projectivization.Basic\npublic import Mathlib.NumberTheory.Height.Basic\n\nNamespace:\nProjectivization\n\nLocal context:\n/-\nCopyright (c) 2026 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Heights of points in projective space\n\nWe define the multiplicative (`Projectivization.mulHeight`) and the logarithmic\n(`Projectivization.logHeight`) height of a point in a (finite-dimensional) projective space\nover a field that has a `Height.AdmissibleAbsValues` instance.\n\nThe height is defined to be the height of any representative tuple; it does not depend\non which representative is chosen.\n-/\n\npublic section\n\nnamespace Projectivization\n\nopen Height AdmissibleAbsValues Real\n\nvariable {K : Type*} [Field K] [AdmissibleAbsValues K] {ι : Type*} [Finite ι]\n\nprivate lemma mulHeight_aux (a b : { v : ι → K // v ≠ 0 }) (t : K) (h : a.val = t • b.val) :\n    mulHeight a.val = mulHeight b.val :=\n  have ht : t ≠ 0 := by\n    contrapose! h\n    simpa [h] using a.prop\n  h ▸ mulHeight_smul_eq_mulHeight _ ht\n\nprivate lemma logHeight_aux (a b : { v : ι → K // v ≠ 0 }) (t : K) (h : a.val = t • b.val) :\n    logHeight a.val = logHeight b.val :=\n  congrArg log <| mod_cast mulHeight_aux a b t h\n\n-- We do not expose the bodies of these definitions so that we can keep the \"_aux\" lemmas\n-- above private.\n\n/-- The multiplicative height of a point on a finite-dimensional projective space over `K`\nwith a given basis. -/\nnoncomputable def mulHeight (x : Projectivization K (ι → K)) : ℝ :=\n  x.lift (fun r ↦ Height.mulHeight r.val) mulHeight_aux\n\n/-- The logarithmic height of a point on a finite-dimensional projective space over `K`\nwith a given basis. -/\nnoncomputable def logHeight (x : Projectivization K (ι → K)) : ℝ :=\n  x.lift (fun r ↦ Height.logHeight r.val) logHeight_aux\n\nlemma mulHeight_mk {x : ι → K} (hx : x ≠ 0) : mulHeight (mk K x hx) = Height.mulHeight x := by\n  rfl\n\nlemma logHeight_mk {x : ι → K} (hx : x ≠ 0) : logHeight (mk K x hx) = Height.logHeight x := by\n  rfl\n\nlemma logHeight_eq_log_mulHeight (x : Projectivization K (ι → K)) :\n    logHeight x = log (mulHeight x) := by\n  rw [← x.mk_rep, mulHeight_mk, logHeight_mk, Height.logHeight]\n\nlemma one_le_mulHeight (x : Projectivization K (ι → K)) : 1 ≤ mulHeight x := by\n  rw [← x.mk_rep, mulHeight_mk]\n  exact Height.one_le_mulHeight _\n\nlemma mulHeight_pos (x : Projectivization K (ι → K)) : 0 < mulHeight x :=\n  zero_lt_one.trans_le <| one_le_mulHeight x\n\nlemma mulHeight_ne_zero (x : Projectivization K (ι → K)) : mulHeight x ≠ 0 :=\n  (mulHeight_pos x).ne'\n\nTarget:\nlemma logHeight_nonneg (x : Projectivization K (ι → K)) : 0 ≤ logHeight x :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_5bf0e0681c55","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"ec4d7488e3ed492df0db21140eac51abfc3c7f2a2ea21d1a87907f1536a7e2d1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Height","family_id":"logheight_nonneg","file_id":"mathlib/Mathlib/NumberTheory/Height/Projectivization.lean","sample_id":"5bf0e0681c55eec8bc5b1c759d43192dd7b331a9558e5d94174d45141766d782"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a923c894044509b83369abcd9431b69d970849ba97f75437eb89f230df9823e5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"35834d0e98d574f6e1ed88656df05a2d1ef4513396ba99f07b5eb2ab7937c26e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0884596d77461ad97de009c6f8b8d539833c319659cef7be0f8610e17da4299a","source_sha256":"68dce35114cf22d288a51cfbefe877f5725b4487e27909680b1aae1c2b647510","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [IsHausdorff.eq_iff_smodEq (I := 𝔭)]\n  simp only [smul_eq_mul, mul_top]\n  intro n\n  cases n\n  · simp\n  · simp [SModEq, mk_pow_fontaineTheta]","hard_negative":true,"metrics":{"chosen_tokens":32,"rejected_tokens":5,"token_jaccard":0.076923,"token_length_ratio":0.15625},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"e5e378d24a4b511073c55ca348aa6d93dfca0de859222172ab8a6744db594bef","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.AdicCompletion.Functoriality\npublic import Mathlib.RingTheory.AdicCompletion.RingHom\npublic import Mathlib.RingTheory.Perfectoid.Untilt\npublic import Mathlib.RingTheory.WittVector.TeichmullerSeries\n\nNamespace:\nWittVector\n\nLocal context:\n/-\nCopyright (c) 2025 Jiedong Jiang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jiedong Jiang\n-/\n/-!\n# Fontaine's θ map\nIn this file, we define Fontaine's `θ` map, which is a ring\nhomomorphism from the Witt vector `𝕎 R♭` of the tilt of a perfectoid ring `R`\nto `R` itself. Our definition of `θ` does not require that `R` is perfectoid in the first place.\nWe only need `R` to be `p`-adically complete.\n\n## Main Definitions\n* `fontaineTheta` : Fontaine's θ map, which is a ring homomorphism from `𝕎 R♭` to `R`.\n\n## Main Theorems\n* `fontaineTheta_teichmuller` : `θ([x])` is the untilt of `x`.\n* `fontaineTheta_surjective` : Fontaine's θ map is surjective.\n\n## TODO\nEstablish that our definition (explicit construction of `θ mod p ^ n`) agrees with the\ndeformation-theoretic approach via the cotangent complex, as in\n[Bhatt, *Lecture notes for a class on perfectoid spaces*.\nRemark 6.1.7](https://www.math.ias.edu/~bhatt/teaching/mat679w17/lectures.pdf).\n\n## Tags\nFontaine's theta map, perfectoid theory, p-adic Hodge theory\n\n## Reference\n\n* [Fontaine, *Sur Certains Types de Représentations p-Adiques du Groupe de Galois d'un Corps Local;\n  Construction d'un Anneau de Barsotti-Tate*][fontaine1982certains]\n* [Fontaine, *Le corps des périodes p-adiques*][fontaine1994corps]\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen Ideal Quotient PreTilt WittVector\n\nnoncomputable section\n\nvariable {R : Type u} [CommRing R] {p : ℕ} [Fact p.Prime]\n\nlocal notation \"𝕎 \" A:100 => WittVector p A\nlocal notation A \"♭\" => PreTilt A p\nlocal notation3 \"𝔭\" => span {(p : R)}\n\nnamespace WittVector\n\n/-!\n## θ as a ring homomorphism\nLet `𝔭` denote the ideal of `R` generated by the prime number `p`. In this section, we first\ndefine the ring homomorphism `fontaineThetaModPPow : 𝕎 R♭ →+* R ⧸ 𝔭 ^ (n + 1)`.\nThen we show they are compatible with each other and lift to a\nring homomorphism `fontaineTheta : 𝕎 R♭ →+* R`.\n\nTo prove this, we define `fontaineThetaModPPow` as a composition of the following ring\nhomomorphisms.\n\n`𝕎 R♭ --𝕎(Frob^-n)-> 𝕎 R♭ --𝕎(coeff 0)-> 𝕎(R/𝔭) --gh_n-> R/𝔭^(n+1)`\n\nHere, the ring map `gh_n` fits in the following diagram.\n\n```\n𝕎(R)  --ghost_n->   R\n|                   |\nv                   v\n𝕎(R/𝔭) --gh_n-> R/𝔭^(n+1)\n```\n-/\n\ntheorem ker_map_le_ker_mk_comp_ghostComponent (n : ℕ) :\n    RingHom.ker (WittVector.map (Ideal.Quotient.mk 𝔭)) ≤\n    RingHom.ker (((Ideal.Quotient.mk (𝔭 ^ (n + 1)))).comp\n    (WittVector.ghostComponent (p := p) n)) := by\n  intro x\n  simp only [RingHom.mem_ker, map_eq_zero_iff, RingHom.comp_apply]\n  intro h\n  simp only [ghostComponent]\n  apply_fun Ideal.quotEquivOfEq (Ideal.span_singleton_pow _ (n + 1))\n  simp only [RingHom.coe_comp, Function.comp_apply, Pi.evalRingHom_apply, ghostMap_apply,\n    quotEquivOfEq_mk, map_zero]\n  simp only [eq_zero_iff_dvd] at h ⊢\n  exact pow_dvd_ghostComponent_of_dvd_coeff (fun _ _ ↦ h _)\n\n/--\nThe lift ring map `gh_n : 𝕎(R/𝔭) →+* R/𝔭^(n+1)` of the `n`-th ghost component\n`𝕎(R) →+* R` along the surjective ring map `𝕎(R) →+* 𝕎(R/𝔭)`.\n-/\ndef ghostComponentModPPow (n : ℕ) : 𝕎 (R ⧸ 𝔭) →+* R ⧸ 𝔭 ^ (n + 1) :=\n  RingHom.liftOfSurjective (WittVector.map (Ideal.Quotient.mk 𝔭))\n    (map_surjective _ Ideal.Quotient.mk_surjective) ⟨((Ideal.Quotient.mk (𝔭 ^ (n + 1)))).comp\n      (WittVector.ghostComponent n), ker_map_le_ker_mk_comp_ghostComponent n⟩\n\n@[simp]\ntheorem ghostComponentModPPow_map_mk (n : ℕ) (x : 𝕎 R) :\n    ghostComponentModPPow n (WittVector.map (Ideal.Quotient.mk 𝔭) x) =\n    WittVector.ghostComponent n x :=\n  RingHom.liftOfSurjective_comp_apply ..\n\n@[simp]\ntheorem quotEquivOfEq_ghostComponentModPPow (x : 𝕎 (R ⧸ 𝔭)) (h : 𝔭 ^ (0 + 1) = 𝔭) :\n    quotEquivOfEq h (ghostComponentModPPow 0 x) = ghostComponent 0 x := by\n  obtain ⟨y, hy⟩ := map_surjective _ Ideal.Quotient.mk_surjective x\n  simp [← hy, ghostComponent_apply]\n\nvariable [Fact ¬IsUnit (p : R)] [IsAdicComplete (span {(p : R)}) R]\n-- local notation 𝔭 does not work in [IsAdicComplete (span {(p : R)}) R]\n\n@[simp]\ntheorem ghostComponentModPPow_teichmuller_coeff (n : ℕ) (x : R♭) :\n    ghostComponentModPPow n (teichmuller p (PreTilt.coeff n x)) =\n    Ideal.Quotient.mk (𝔭 ^ (n + 1)) x.untilt := by\n  simpa using ghostComponentModPPow_map_mk n\n    (teichmuller p ((((_root_.frobeniusEquiv _ p).symm ^ n) x).untilt))\n\nvariable (R p) in\n/--\nThe Fontaine's theta map modulo `p^(n+1)`.\nIt is the composition of the following ring homomorphisms.\n`𝕎 R♭ --𝕎(Frob^-n)-> 𝕎 R♭ --𝕎(coeff 0)-> 𝕎(R/p) --gh_n-> R/p^(n+1)`\n-/\ndef fontaineThetaModPPow (n : ℕ) : 𝕎 R♭ →+* R ⧸ 𝔭 ^ (n + 1) :=\n  (ghostComponentModPPow n).comp (((WittVector.map (PreTilt.coeff 0))).comp\n    (WittVector.map ((_root_.frobeniusEquiv (R♭) p).symm ^ n : R♭ →+* R♭)))\n\n@[simp]\ntheorem fontaineThetaModPPow_teichmuller (n : ℕ) (x : R♭) :\n    fontaineThetaModPPow R p n (teichmuller p x) = Ideal.Quotient.mk _ x.untilt := by\n  simp [fontaineThetaModPPow]\n\ntheorem factorPowSucc_comp_fontaineThetaModPPow (n : ℕ) :\n    (factorPowSucc _ _).comp (fontaineThetaModPPow R p (n + 1)) = fontaineThetaModPPow R p n := by\n  apply eq_of_apply_teichmuller_eq ((factorPowSucc _ _).comp (fontaineThetaModPPow R p (n + 1)))\n    (fontaineThetaModPPow R p n)\n  · use n + 1\n    have : p = Ideal.Quotient.mk (𝔭 ^ (n + 1)) p := by\n      simp [map_natCast]\n    rw [this, ← map_pow, Ideal.Quotient.eq_zero_iff_mem]\n    exact Ideal.pow_mem_pow (mem_span_singleton_self _) _\n  simp [fontaineThetaModPPow]\n\ntheorem factorPowSucc_fontaineThetaModPPow_eq (n : ℕ) (x : 𝕎 R♭) :\n    factorPowSucc _ _ ((fontaineThetaModPPow R p (n + 1)) x) = fontaineThetaModPPow R p n x := by\n  simp [← factorPowSucc_comp_fontaineThetaModPPow n]\n\nopen IsAdicComplete\n\nvariable (R p) in\n/--\nThe Fontaine's θ map from `𝕎 R♭` to `R`.\nIt is the limit of the ring maps `fontaineThetaModPPow n` from `𝕎 R♭` to `R/p^(n+1)`.\n-/\ndef fontaineTheta : 𝕎 R♭ →+* R :=\n  Order.succ_strictMono.liftRingHom 𝔭 _ (factorPowSucc_comp_fontaineThetaModPPow _)\n\ntheorem mk_pow_fontaineTheta (n : ℕ) (x : 𝕎 R♭) :\n    Ideal.Quotient.mk (𝔭 ^ (n + 1)) (fontaineTheta R p x) = fontaineThetaModPPow R p n x :=\n  Order.succ_strictMono.mk_liftRingHom 𝔭 _ (factorPowSucc_comp_fontaineThetaModPPow _) x\n\ntheorem mk_fontaineTheta (x : 𝕎 R♭) :\n    Ideal.Quotient.mk 𝔭 (fontaineTheta R p x) = PreTilt.coeff 0 (x.coeff 0) := by\n  have := mk_pow_fontaineTheta 0 x\n  simp only [Nat.reduceAdd] at this\n  apply_fun Ideal.quotEquivOfEq (pow_one (p : R) ▸ Ideal.span_singleton_pow (p : R) 1) at this\n  simp only [quotEquivOfEq_mk] at this\n  rw [this]\n  simp [fontaineThetaModPPow, ghostComponent_apply, RingHom.one_def]\n\n@[simp]\n\nTarget:\ntheorem fontaineTheta_teichmuller (x : R♭) : fontaineTheta R p (teichmuller p x) = x.untilt :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_0884596d7746","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"224c0d630c8fcf8814d5ee96e1555f8189450f8d4fe89f89103d87906420ddef","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Perfectoid","family_id":"fontainetheta_teichmuller","file_id":"mathlib/Mathlib/RingTheory/Perfectoid/FontaineTheta.lean","sample_id":"0884596d77461ad97de009c6f8b8d539833c319659cef7be0f8610e17da4299a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"42680d60d5e78b64557581d1ab549dd5b4e6ea0d21a689bb03eb5cdec7a02fe8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d015ef1b5c637ff972dc63934850928ed1b5069c8619f18dfa9cfa9e5bb85ac5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"026c42c1a18ace69185b5b2383ea579a45b7bd0313b181e0953d0847129aa197","source_sha256":"695353e91eba40e45d9ecd0e58f8b00a56d575b25190f682a1689e29c7316a51","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold logDeriv\n  simp [deriv_algebraMap]","hard_negative":true,"metrics":{"chosen_tokens":7,"rejected_tokens":3,"token_jaccard":0.111111,"token_length_ratio":0.428571},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"e5fb78cabcd71bc97c01d07e35922281aee6d077397bee38dbed8e8d8415ddce","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Derivation.MapCoeffs\npublic import Mathlib.FieldTheory.PrimitiveElement\n\nNamespace:\nDifferential\n\nLocal context:\n/-\nCopyright (c) 2024 Daniel Weber. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Weber\n-/\n/-!\n# Differential Fields\n\nThis file defines the logarithmic derivative `Differential.logDeriv` and proves properties of it.\nThis is defined algebraically, compared to `logDeriv` which is analytical.\n-/\n\n@[expose] public section\n\nnamespace Differential\n\nopen algebraMap Polynomial IntermediateField\n\nvariable {R : Type*} [Field R] [Differential R] (a b : R)\n\n/--\nThe logarithmic derivative of a is a′ / a.\n-/\ndef logDeriv : R := a′ / a\n\n@[simp]\nlemma logDeriv_zero : logDeriv (0 : R) = 0 := by\n  simp [logDeriv]\n\n@[simp]\nlemma logDeriv_one : logDeriv (1 : R) = 0 := by\n  simp [logDeriv]\n\nlemma logDeriv_mul (ha : a ≠ 0) (hb : b ≠ 0) : logDeriv (a * b) = logDeriv a + logDeriv b := by\n  unfold logDeriv\n  simp [field]\n  ring\n\nlemma logDeriv_div (ha : a ≠ 0) (hb : b ≠ 0) : logDeriv (a / b) = logDeriv a - logDeriv b := by\n  unfold logDeriv\n  simp [field, Derivation.leibniz_div]\n\n@[simp]\nlemma logDeriv_pow (n : ℕ) (a : R) : logDeriv (a ^ n) = n * logDeriv a := by\n  induction n with\n  | zero => simp\n  | succ n h2 =>\n    obtain rfl | hb := eq_or_ne a 0\n    · simp\n    · rw [Nat.cast_add, Nat.cast_one, add_mul, one_mul, ← h2, pow_succ, logDeriv_mul] <;>\n      simp [hb]\n\nlemma logDeriv_eq_zero : logDeriv a = 0 ↔ a′ = 0 :=\n  ⟨fun h ↦ by simp only [logDeriv, _root_.div_eq_zero_iff] at h; rcases h with h|h <;> simp [h],\n  fun h ↦ by unfold logDeriv at *; simp [h]⟩\n\nlemma logDeriv_multisetProd {ι : Type*} (s : Multiset ι) {f : ι → R} (h : ∀ x ∈ s, f x ≠ 0) :\n    logDeriv (s.map f).prod = (s.map fun x ↦ logDeriv (f x)).sum := by\n  induction s using Multiset.induction_on\n  · simp\n  · rename_i h₂\n    simp only [Multiset.map_cons, Multiset.sum_cons, Multiset.prod_cons]\n    rw [← h₂]\n    · apply logDeriv_mul\n      · simp [h]\n      · simp_all\n    · simp_all\n\nlemma logDeriv_prod (ι : Type*) (s : Finset ι) (f : ι → R) (h : ∀ x ∈ s, f x ≠ 0) :\n    logDeriv (∏ x ∈ s, f x) = ∑ x ∈ s, logDeriv (f x) := logDeriv_multisetProd _ h\n\nlemma logDeriv_prod_of_eq_zero (ι : Type*) (s : Finset ι) (f : ι → R) (h : ∀ x ∈ s, f x = 0) :\n    logDeriv (∏ x ∈ s, f x) = ∑ x ∈ s, logDeriv (f x) := by\n  unfold logDeriv\n  simp_all\n\nTarget:\nlemma logDeriv_algebraMap {F K : Type*} [Field F] [Field K] [Differential F] [Differential K]\n    [Algebra F K] [DifferentialAlgebra F K]\n    (a : F) : logDeriv (algebraMap F K a) = algebraMap F K (logDeriv a) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_026c42c1a18a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"acc443b90d6ae3ebd9c8f957836d90a83a2a8c0e38400fd43a2209b58605cf24","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Differential","family_id":"logderiv_algebramap","file_id":"mathlib/Mathlib/FieldTheory/Differential/Basic.lean","sample_id":"026c42c1a18ace69185b5b2383ea579a45b7bd0313b181e0953d0847129aa197"},"split":"train"}
{"binding":{"chosen_proof_sha256":"efb528cee40199b8a7f10c4af81ecc5b602aa05a4991b3b60a5968591ef7ad2c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2c31379e40cbecc4b70db87b9684b86d81f7fb6fcff5b69f3c8ba705b4e20f03","source_sha256":"7aa6ff736d259829acf4509d7de6faf2fc5cbce6e43d3b6e8a128a0408a44be4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [section_]\n  rw [assoc, dsimp% X.hom.naturality, comp_id]\n  exact ed.s'_comp_ε","hard_negative":true,"metrics":{"chosen_tokens":24,"rejected_tokens":8,"token_jaccard":0.041667,"token_length_ratio":0.333333},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"e607631f47e054ca315a6b6f1e741508f29c9d36923edc13ebcb03447e960826","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Homology.Homotopy\npublic import Mathlib.AlgebraicTopology.AlternatingFaceMapComplex\npublic import Mathlib.AlgebraicTopology.CechNerve\npublic import Mathlib.AlgebraicTopology.SimplicialObject.DeltaZeroIter\npublic import Mathlib.AlgebraicTopology.SimplicialObject.Homotopy\npublic import Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex\n\nNamespace:\nCategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy\n\nLocal context:\n/-\nCopyright (c) 2022 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n\n# Augmented simplicial objects with an extra degeneracy\n\nIn simplicial homotopy theory, in order to prove that the connected components\nof a simplicial set `X` are contractible, it suffices to construct an extra\ndegeneracy as it is defined in *Simplicial Homotopy Theory* by Goerss-Jardine p. 190.\nIt consists of a series of maps `π₀ X → X _⦋0⦌` and `X _⦋n⦌ → X _⦋n+1⦌` which\nbehave formally like an extra degeneracy `σ (-1)`. It can be thought as a datum\nassociated to the augmented simplicial set `X → π₀ X`.\n\nIn this file, we adapt this definition to the case of augmented\nsimplicial objects in any category.\n\n## Main definitions\n\n- the structure `ExtraDegeneracy X` for any `X : SimplicialObject.Augmented C`\n- `ExtraDegeneracy.map`: extra degeneracies are preserved by the application of any\n  functor `C ⥤ D`\n- `SSet.Augmented.StandardSimplex.extraDegeneracy`: the standard `n`-simplex has\n  an extra degeneracy\n- `Arrow.AugmentedCechNerve.extraDegeneracy`: the Čech nerve of a split\n  epimorphism has an extra degeneracy\n- `ExtraDegeneracy.homotopyEquiv`: in the case the category `C` is preadditive,\n  if we have an extra degeneracy on `X : SimplicialObject.Augmented C`, then\n  the augmentation on the alternating face map complex of `X` is a homotopy\n  equivalence.\n- `ExtraDegeneracy.homotopy`: if we have an extra degeneracy `ed` on\n  `X : SimplicialObject.Augmented C` (for any category `C`), then\n  the morphism `X.hom ≫ ed.section_` is homotopic to `𝟙 X.left`.\n\n## References\n* [Paul G. Goerss, John F. Jardine, *Simplicial Homotopy Theory*][goerss-jardine-2009]\n* [M. Barr, J. Kennison, J. and R. Robert,\n  *Contractible simplicial objects*][barr-kennison-robert-2019]\n\n-/\n\nset_option backward.defeqAttrib.useBackward true\n\n@[expose] public section\n\n\nopen CategoryTheory Category SimplicialObject.Augmented Opposite Simplicial\n\nnamespace CategoryTheory\n\nnamespace SimplicialObject\n\nnamespace Augmented\n\nvariable {C : Type*} [Category* C]\n\n/-- The datum of an extra degeneracy is a technical condition on\naugmented simplicial objects. The morphisms `s'` and `s n` of the\nstructure formally behave like extra degeneracies `σ (-1)`. -/\n@[ext]\nstructure ExtraDegeneracy (X : SimplicialObject.Augmented C) where\n  /-- a section of the augmentation in dimension `0` -/\n  s' : X.right ⟶ X.left _⦋0⦌\n  /-- the extra degeneracy -/\n  s : ∀ n : ℕ, X.left _⦋n⦌ ⟶ X.left _⦋n + 1⦌\n  s'_comp_ε : dsimp% s' ≫ X.hom.app (op ⦋0⦌) = 𝟙 X.right := by cat_disch\n  s₀_comp_δ₁ : dsimp% s 0 ≫ X.left.δ 1 = X.hom.app (op ⦋0⦌) ≫ s' := by cat_disch\n  s_comp_δ₀ : ∀ n : ℕ, s n ≫ X.left.δ 0 = 𝟙 _ := by cat_disch\n  s_comp_δ :\n    ∀ (n : ℕ) (i : Fin (n + 2)), s (n + 1) ≫ X.left.δ i.succ = X.left.δ i ≫ s n := by cat_disch\n  s_comp_σ :\n    ∀ (n : ℕ) (i : Fin (n + 1)), s n ≫ X.left.σ i.succ = X.left.σ i ≫ s (n + 1) := by cat_disch\n\nnamespace ExtraDegeneracy\n\nattribute [reassoc] s₀_comp_δ₁ s_comp_δ s_comp_σ\nattribute [reassoc (attr := simp)] s'_comp_ε s_comp_δ₀\n\nattribute [local simp←] Functor.map_comp in\nattribute [local simp] s₀_comp_δ₁ s_comp_δ s_comp_σ in\n/-- If `ed` is an extra degeneracy for `X : SimplicialObject.Augmented C` and\n`F : C ⥤ D` is a functor, then `ed.map F` is an extra degeneracy for the\naugmented simplicial object in `D` obtained by applying `F` to `X`. -/\ndef map {D : Type*} [Category* D] {X : SimplicialObject.Augmented C} (ed : ExtraDegeneracy X)\n    (F : C ⥤ D) : ExtraDegeneracy (((whiskering _ _).obj F).obj X) where\n  s' := F.map ed.s'\n  s n := F.map (ed.s n)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- If `X` and `Y` are isomorphic augmented simplicial objects, then an extra\ndegeneracy for `X` gives also an extra degeneracy for `Y` -/\ndef ofIso {X Y : SimplicialObject.Augmented C} (e : X ≅ Y) (ed : ExtraDegeneracy X) :\n    ExtraDegeneracy Y where\n  s' := (point.mapIso e).inv ≫ ed.s' ≫ (drop.mapIso e).hom.app (op ⦋0⦌)\n  s n := (drop.mapIso e).inv.app (op ⦋n⦌) ≫ ed.s n ≫ (drop.mapIso e).hom.app (op ⦋n + 1⦌)\n  s'_comp_ε := by\n    simpa [dsimp% w₀] using dsimp% (point.mapIso e).inv_hom_id\n  s₀_comp_δ₁ := by\n    simp [← SimplicialObject.δ_naturality, s₀_comp_δ₁_assoc, dsimp% w₀_assoc]\n  s_comp_δ₀ n := by\n    simpa [← SimplicialObject.δ_naturality] using\n      congr_app (drop.mapIso e).inv_hom_id (op ⦋n⦌)\n  s_comp_δ n i := by\n    simp [← SimplicialObject.δ_naturality, s_comp_δ_assoc,\n      ← SimplicialObject.δ_naturality_assoc]\n  s_comp_σ n i := by\n    simp [← SimplicialObject.σ_naturality, s_comp_σ_assoc,\n      ← SimplicialObject.σ_naturality_assoc]\n\nvariable {X : SimplicialObject.Augmented C} (ed : ExtraDegeneracy X)\n\nattribute [local simp←] Functor.map_comp in\n/-- The section of the augmentation that is induced by the extradegeneracy. -/\ndef section_ : (SimplicialObject.const C).obj X.right ⟶ X.left where\n  app n := ed.s' ≫ X.left.map (SimplexCategory.isTerminalZero.from _).op\n\n@[simp]\nlemma section_app_op_mk_zero :\n    ed.section_.app (op ⦋0⦌) = ed.s' := by\n  simp [section_]\n\n@[reassoc (attr := simp)]\n\nTarget:\nlemma section_app_comp_hom_app (n : SimplexCategoryᵒᵖ) :\n    dsimp% ed.section_.app n ≫ X.hom.app n = 𝟙 _ :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"2858a7b4be4e3b02b0efa758201e23ddbddf9aedef914abfc0289048fd745d15","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology","family_id":"section_app_comp_hom_app","file_id":"mathlib/Mathlib/AlgebraicTopology/ExtraDegeneracy.lean","sample_id":"2c31379e40cbecc4b70db87b9684b86d81f7fb6fcff5b69f3c8ba705b4e20f03"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e26bc9c73cc42d866d4dadb94c4f9430e4046c553d052939bf6c8d1e66df3de3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0a6f4bb20aaaaf779ecf54289d8b160260f76ce5009987f9c2d3a2e21c31fec6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bad13452563a2e4433cf2964ac3661b1745a1d26c8fd4dce023319c5196a9b1e","source_sha256":"1c7994cb953bf15a3e2a75bf5a9f75792705603cc1fc781f9e07bc585e941697","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine hK.induction_on ⟨∅, by simp⟩ (fun s t hst ⟨W, Wc, Wo, hKW, hWU⟩ ↦ ?_) ?_ ?_\n  · use W, Wc, Wo, subset_trans hst hKW, hWU\n  · intro s t ⟨W₁, Wc₁, Wo₁, hKW₁, hWU₁⟩ ⟨W₂, Wc₂, Wo₂, hKW₂, hWU₂⟩\n    exact ⟨W₁ ∪ W₂, Wc₁.union Wc₂, Wo₁.union Wo₂, Set.union_subset_union hKW₁ hKW₂,\n      Set.union_subset hWU₁ hWU₂⟩\n  · intro x hx\n    obtain ⟨V, h, hxV, hVU⟩ :=\n      PrespectralSpace.isTopologicalBasis.exists_subset_of_mem_open (hKU hx) hU\n    exact ⟨V, mem_nhdsWithin.mpr ⟨V, h.1, hxV, Set.inter_subset_left⟩, V, h.2, h.1, subset_rfl, hVU⟩","hard_negative":true,"metrics":{"chosen_tokens":182,"rejected_tokens":2,"token_jaccard":0.017857,"token_length_ratio":0.010989},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"e61edf5dfe93d0b8cfd1256929925a82f00b345dfb406a20942a30039c21e23c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Ideal\npublic import Mathlib.Topology.Sets.Compacts\npublic import Mathlib.Topology.Sets.OpenCover\npublic import Mathlib.Topology.Spectral.Hom\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Prespectral spaces\n\nIn this file, we define prespectral spaces as spaces whose lattice of compact opens forms a basis.\n\n-/\n\n@[expose] public section\n\nopen TopologicalSpace Topology\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n\n/-- A space is prespectral if the lattice of compact opens forms a basis. -/\n@[stacks 08YG \"The last condition for spectral spaces\", mk_iff]\nclass PrespectralSpace (X : Type*) [TopologicalSpace X] : Prop where\n  isTopologicalBasis : IsTopologicalBasis { U : Set X | IsOpen U ∧ IsCompact U }\n\n/-- A space is prespectral if it has a basis consisting of compact opens. -/\nlemma PrespectralSpace.of_isTopologicalBasis {B : Set (Set X)}\n    (basis : IsTopologicalBasis B) (isCompact_basis : ∀ U ∈ B, IsCompact U) :\n    PrespectralSpace X where\n  isTopologicalBasis := basis.of_isOpen_of_subset (fun _ h ↦ h.1)\n    fun s hs ↦ ⟨basis.isOpen hs, isCompact_basis s hs⟩\n\n/-- A space is prespectral if it has a basis consisting of compact opens.\nThis is the variant with an indexed basis instead. -/\nlemma PrespectralSpace.of_isTopologicalBasis' {ι : Type*} {b : ι → Set X}\n    (basis : IsTopologicalBasis (Set.range b)) (isCompact_basis : ∀ i, IsCompact (b i)) :\n    PrespectralSpace X :=\n  .of_isTopologicalBasis basis (by simp_all)\n\ninstance (priority := low) [NoetherianSpace X] : PrespectralSpace X :=\n  .of_isTopologicalBasis isTopologicalBasis_opens fun _ _ ↦ NoetherianSpace.isCompact _\n\ninstance (priority := low) [PrespectralSpace X] : LocallyCompactSpace X where\n  local_compact_nhds _ _ hn :=\n    have ⟨V, ⟨hV₁, hV₂⟩, hxV, hVn⟩ := PrespectralSpace.isTopologicalBasis.mem_nhds_iff.mp hn\n    ⟨V, hV₁.mem_nhds hxV, hVn, hV₂⟩\n\nopen PrespectralSpace in\ninstance (priority := low) [T2Space X] [PrespectralSpace X] : TotallySeparatedSpace X :=\n  totallySeparatedSpace_iff_exists_isClopen.mpr fun _ _ hxy ↦\n    have ⟨U, ⟨hU₁, hU₂⟩, hxU, hyU⟩ :=\n      isTopologicalBasis.exists_subset_of_mem_open hxy isClosed_singleton.isOpen_compl\n    ⟨U, ⟨hU₂.isClosed, hU₁⟩, hxU, fun h ↦ hyU h rfl⟩\n\nlemma PrespectralSpace.of_isOpenCover\n    {ι : Type*} {U : ι → Opens X} (hU : IsOpenCover U) [∀ i, PrespectralSpace (U i)] :\n    PrespectralSpace X := by\n  refine .of_isTopologicalBasis (hU.isTopologicalBasis fun i ↦ isTopologicalBasis) ?_\n  simp only [Set.mem_iUnion, Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp,\n    forall_comm (α := Set _), forall_apply_eq_imp_iff₂]\n  exact fun i V hV hV' ↦ hV'.image continuous_subtype_val\n\nlemma PrespectralSpace.of_isInducing [PrespectralSpace Y]\n    (f : X → Y) (hf : IsInducing f) (hf' : IsSpectralMap f) : PrespectralSpace X :=\n  .of_isTopologicalBasis (PrespectralSpace.isTopologicalBasis.isInducing hf) (by\n    simp only [Set.mem_image, Set.mem_setOf_eq, forall_exists_index, and_imp]\n    rintro _ U h₁ h₂ rfl\n    exact hf'.isCompact_preimage_of_isOpen h₁ h₂)\n\nlemma PrespectralSpace.of_isClosedEmbedding [PrespectralSpace Y]\n    (f : X → Y) (hf : IsClosedEmbedding f) : PrespectralSpace X :=\n  .of_isInducing f hf.isInducing hf.isProperMap.isSpectralMap\n\n/-- Let `f : X → Y` be an open embedding of topological spaces.\nIf `Y` is a prespectral space (i.e., the quasi-compact opens of `Y` form a basis),\nthen `X` is also a prespectral space. -/\nlemma Topology.IsOpenEmbedding.prespectralSpace [PrespectralSpace Y]\n    {f : X → Y} (hf : IsOpenEmbedding f) :\n    PrespectralSpace X where\n  isTopologicalBasis := by\n    apply isTopologicalBasis_of_isOpen_of_nhds (fun U hU ↦ hU.1) <| fun x U hx hU ↦ ?_\n    obtain ⟨V, ⟨hoV, hcV⟩, hfx, hVf⟩ : ∃ V ∈ {V | IsOpen V ∧ IsCompact V}, f x ∈ V ∧ V ⊆ f '' U :=\n      (PrespectralSpace.isTopologicalBasis (X := Y)).isOpen_iff.mp\n        (hf.isOpen_iff_image_isOpen.mp hU) (f x) ⟨x, hx, rfl⟩\n    refine ⟨f ⁻¹' V, ⟨hoV.preimage hf.continuous, ?_⟩, ⟨hfx, fun y hy ↦ ?_⟩⟩\n    · exact hf.toIsInducing.isCompact_preimage' hcV <| Set.SurjOn.subset_range hVf\n    · exact hf.injective.mem_set_image.mp (hVf hy)\n\ninstance PrespectralSpace.sigma {ι : Type*} (X : ι → Type*) [∀ i, TopologicalSpace (X i)]\n    [∀ i, PrespectralSpace (X i)] : PrespectralSpace (Σ i, X i) :=\n  .of_isTopologicalBasis (IsTopologicalBasis.sigma fun i ↦ isTopologicalBasis) fun U hU ↦ by\n    simp_rw [Set.mem_iUnion] at hU\n    obtain ⟨i, V, hV, rfl⟩ := hU\n    exact hV.2.image continuous_sigmaMk\n\nvariable (X) in\nlemma PrespectralSpace.isBasis_opens [PrespectralSpace X] :\n    TopologicalSpace.Opens.IsBasis { U : Opens X | IsCompact (U : Set X) } := by\n  dsimp only [TopologicalSpace.Opens.IsBasis]\n  convert! isTopologicalBasis (X := X)\n  ext s\n  exact ⟨fun ⟨V, hV, heq⟩ ↦ heq ▸ ⟨V.2, hV⟩, fun h ↦ ⟨⟨s, h.1⟩, h.2, rfl⟩⟩\n\n/-- In a prespectral space, the lattice of opens is determined by its lattice of compact opens. -/\ndef PrespectralSpace.opensEquiv [PrespectralSpace X] :\n    Opens X ≃o Order.Ideal (CompactOpens X) where\n  toFun U := ⟨⟨{ V | (V : Set X) ⊆ U }, fun U₁ U₂ h₁ h₂ ↦ subset_trans (α := Set X) h₁ h₂⟩,\n    ⟨⊥, by simp⟩, fun U₁ h₁ U₂ h₂ ↦ ⟨U₁ ⊔ U₂, by aesop, le_sup_left, le_sup_right⟩⟩\n  invFun I := ⨆ U ∈ I, U.toOpens\n  left_inv U := by\n    apply le_antisymm\n    · simp only [iSup_le_iff]\n      exact fun _ ↦ id\n    · intro x hxU\n      obtain ⟨V, ⟨h₁, h₂⟩, hxV, hVU⟩ := isTopologicalBasis.exists_subset_of_mem_open hxU U.2\n      simp only [Opens.mem_iSup]\n      exact ⟨⟨⟨_, h₂⟩, h₁⟩, hVU, hxV⟩\n  right_inv I := by\n    ext U\n    dsimp\n    change U.toOpens ≤ _ ↔ _\n    refine ⟨fun H ↦ ?_, fun h ↦ le_iSup₂ (f := fun U (h : U ∈ I) ↦ U.toOpens) U h⟩\n    simp only [← SetLike.coe_subset_coe, Opens.iSup_mk, Opens.carrier_eq_coe, Opens.coe_mk] at H\n    obtain ⟨s, hsI, hs, hU⟩ := U.isCompact.elim_finite_subcover_image (fun U _ ↦ U.2) H\n    exact I.lower (a := hs.toFinset.sup fun i ↦ i) (by simpa [← SetLike.coe_subset_coe]) (by simpa)\n  map_rel_iff' {U V} := by\n    change (∀ (W : CompactOpens X), (W : Set X) ⊆ U → (W : Set X) ⊆ V) ↔ U ≤ V\n    refine ⟨?_, fun H W ↦ (le_trans · H)⟩\n    intro H x hxU\n    obtain ⟨W, ⟨h₁, h₂⟩, hxW, hWU⟩ := isTopologicalBasis.exists_subset_of_mem_open hxU U.2\n    exact H ⟨⟨W, h₂⟩, h₁⟩ hWU hxW\n\nopen TopologicalSpace Opens in\n/-- If `X` has a basis of compact opens and `f : X → S` is open, every\ncompact open of `S` is the image of a compact open of `X`. -/\nlemma IsOpenMap.exists_opens_image_eq_of_prespectralSpace [PrespectralSpace X] {f : X → Y}\n    (hfc : Continuous f) (h : IsOpenMap f) {U : Set Y} (hs : U ⊆ Set.range f) (hU : IsOpen U)\n    (hc : IsCompact U) : ∃ (V : Opens X), IsCompact V.1 ∧ f '' V = U := by\n  obtain ⟨Us, hUs, heq⟩ := TopologicalSpace.Opens.isBasis_iff_cover.mp\n    (PrespectralSpace.isBasis_opens X) ⟨f ⁻¹' U, hU.preimage hfc⟩\n  obtain ⟨t, ht⟩ := by\n    refine hc.elim_finite_subcover (fun s : Us ↦ f '' s.1) (fun s ↦ h _ s.1.2) (fun x hx ↦ ?_)\n    obtain ⟨x, rfl⟩ := hs hx\n    obtain ⟨i, hi, hx⟩ := mem_sSup.mp <| by rwa [← heq]\n    exact Set.mem_iUnion.mpr ⟨⟨i, hi⟩, x, hx, rfl⟩\n  refine ⟨⨆ s ∈ t, s.1, ?_, ?_⟩\n  · simp only [iSup_mk, carrier_eq_coe, coe_mk]\n    exact t.finite_toSet.isCompact_biUnion fun i _ ↦ hUs i.2\n  · simp only [iSup_mk, carrier_eq_coe, Set.iUnion_coe_set, coe_mk, Set.image_iUnion]\n    convert_to ⋃ i ∈ t, f '' i.1 = U\n    · simp\n    · refine subset_antisymm (fun x ↦ ?_) ht\n      simp_rw [Set.mem_iUnion]\n      rintro ⟨i, hi, x, hx, rfl⟩\n      have := heq ▸ mem_sSup.mpr ⟨i.1, i.2, hx⟩\n      exact this\n\nTarget:\nlemma PrespectralSpace.exists_isCompact_and_isOpen_between [PrespectralSpace X] {K U : Set X}\n    (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :\n    ∃ (W : Set X), IsCompact W ∧ IsOpen W ∧ K ⊆ W ∧ W ⊆ U :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_bad13452563a","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"dc4cfc1cd759f5e14e7891398413460ff0d0ad5db91dd302647a141b9a62a36c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Spectral","family_id":"prespectralspace","file_id":"mathlib/Mathlib/Topology/Spectral/Prespectral.lean","sample_id":"bad13452563a2e4433cf2964ac3661b1745a1d26c8fd4dce023319c5196a9b1e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3f00e9105f1d5704619d01594b14b1a2c0b0c68fe25f9bbf22b6c6f972b81c9f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"877aceedde815db9287812ae1efc9e2b30bc7887edece15c54d849073f805d0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a26f956f637c764b2f59cef3d4d51fd447c50773705cdd84c865b39ccbebb619","source_sha256":"009168811e0aa7b7b2aa8bada71ce25e1da7960123a5f42f2e4c63ec9822c044","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [moduleCat_exact_iff_function_exact] at hC ⊢\n  exact rTensor_exact M hC","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.166667},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"e6e2af2afe02914c6d4fb820267857235e4158a0953ed64c9da5f163d9056227","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Flat.Basic\npublic import Mathlib.Algebra.Homology.ShortComplex.ModuleCat\npublic import Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed\n\nNamespace:\nModule.Flat\n\nLocal context:\n/-\nCopyright (c) 2024 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang\n-/\n/-!\n# Tensoring with a flat module is an exact functor\n\nIn this file we prove that tensoring with a flat module is an exact functor.\n\n## Main results\n\n- `Module.Flat.iff_lTensor_preserves_shortComplex_exact`: an `R`-module `M` is flat if and only if\n  for every exact sequence `A ⟶ B ⟶ C`, `M ⊗ A ⟶ M ⊗ B ⟶ M ⊗ C` is also exact.\n\n- `Module.Flat.iff_rTensor_preserves_shortComplex_exact`: an `R`-module `M` is flat if and only if\n  for every short exact sequence `A ⟶ B ⟶ C`, `A ⊗ M ⟶ B ⊗ M ⟶ C ⊗ M` is also exact.\n\n## TODO\n\n- Relate flatness with `Tor`\n\n-/\n\npublic section\n\nuniverse u\n\nopen CategoryTheory MonoidalCategory ShortComplex.ShortExact\n\nnamespace Module.Flat\n\nvariable {R : Type u} [CommRing R] (M : ModuleCat.{u} R)\n\nlemma lTensor_shortComplex_exact [Flat R M] (C : ShortComplex <| ModuleCat R) (hC : C.Exact) :\n    C.map (tensorLeft M) |>.Exact := by\n  rw [moduleCat_exact_iff_function_exact] at hC ⊢\n  exact lTensor_exact M hC\n\nTarget:\nlemma rTensor_shortComplex_exact [Flat R M] (C : ShortComplex <| ModuleCat R) (hC : C.Exact) :\n    C.map (tensorRight M) |>.Exact :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_a26f956f637c","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"ee40da114424fa8d7e761d3980499cd260f0e36b905ac8390918f01ef71cdaf7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Flat","family_id":"rtensor_shortcomplex_exact","file_id":"mathlib/Mathlib/RingTheory/Flat/CategoryTheory.lean","sample_id":"a26f956f637c764b2f59cef3d4d51fd447c50773705cdd84c865b39ccbebb619"},"split":"train"}
{"binding":{"chosen_proof_sha256":"db758460f3feb84a60fe6730f2e7a0ca8f4614d60686027e4a4aa0bdb93eeac5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"762cc97dcff582c8c7abd91dcd8f80b24e2934458abe729819171e73921c3dea","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"56824650124ef37b3559baee31b1277720d3a9f018ce7db07f8712e4b6c90072","source_sha256":"ba819d6726990bcb515ef042ef549f9e2a90b75f672af9b55d2f61118b410281","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases hμ : SFinite μ\n  swap; · rw [compProd_of_not_sfinite _ _ hμ]; simp\n  by_cases hν : SFinite ν\n  swap; · rw [compProd_of_not_sfinite _ _ hν]; simp\n  let s := κ.mutuallySingularSet η\n  have hs : MeasurableSet s := Kernel.measurableSet_mutuallySingularSet κ η\n  symm\n  refine ⟨s, hs, ?_⟩\n  rw [compProd_apply hs, compProd_apply hs.compl]\n  have h_eq a : Prod.mk a ⁻¹' s = Kernel.mutuallySingularSetSlice κ η a := rfl\n  have h1 a : η a (Prod.mk a ⁻¹' s) = 0 := by rw [h_eq, Kernel.measure_mutuallySingularSetSlice]\n  have h2 : ∀ᵐ a ∂μ, κ a (Prod.mk a ⁻¹' s)ᶜ = 0 := by\n    filter_upwards [hκη] with a ha\n    rwa [h_eq, ← Kernel.withDensity_rnDeriv_eq_zero_iff_measure_eq_zero κ η a,\n      Kernel.withDensity_rnDeriv_eq_zero_iff_mutuallySingular]\n  simp [h1, lintegral_congr_ae h2]","hard_negative":true,"metrics":{"chosen_tokens":184,"rejected_tokens":5,"token_jaccard":0.029851,"token_length_ratio":0.027174},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"e715233f7e1b1c656f2acadee5ac2e9ce43cf45b1e6d578e347df0df854e6ad7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Kernel.Composition.MeasureCompProd\npublic import Mathlib.Probability.Kernel.RadonNikodym\n\nNamespace:\nMeasureTheory.Measure\n\nLocal context:\n/-\nCopyright (c) 2025 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne, Lorenzo Luccioli\n-/\n/-!\n# Absolute continuity of the composition of measures and kernels\n\nThis file contains some results about the absolute continuity of the composition of measures and\nkernels which use an assumption `CountableOrCountablyGenerated α β` on the measurable spaces.\n\nResults that hold without that assumption are in files about the definitions of compositions and\nproducts, like `Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean` and\n`Mathlib/Probability/Kernel/Composition/MeasureComp.lean`.\n\nThe assumption ensures the measurability of the sets where two kernels are absolutely continuous\nor mutually singular.\n\n## Main statements\n\n* `absolutelyContinuous_compProd_iff'`: `μ ⊗ₘ κ ≪ ν ⊗ₘ η ↔ μ ≪ ν ∧ ∀ᵐ a ∂μ, κ a ≪ η a`.\n\n-/\n\npublic section\n\nopen ProbabilityTheory Filter\n\nopen scoped ENNReal\n\nnamespace MeasureTheory.Measure\n\nvariable {α β : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}\n  {μ ν : Measure α} {κ η : Kernel α β} [IsFiniteKernel κ] [IsFiniteKernel η]\n  [MeasurableSpace.CountableOrCountablyGenerated α β]\n\nTarget:\nlemma MutuallySingular.compProd_of_right (μ ν : Measure α) (hκη : ∀ᵐ a ∂μ, κ a ⟂ₘ η a) :\n    μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η :=\n\nProof body:\n","rejected":"by\n  exact MutuallySingular.compProd_of_right","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"fcc8f52fe37e395077694ac40e4079267326a6f17df55f31deeb90789c060839","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Kernel","family_id":"mutuallysingular","file_id":"mathlib/Mathlib/Probability/Kernel/Composition/AbsolutelyContinuous.lean","sample_id":"56824650124ef37b3559baee31b1277720d3a9f018ce7db07f8712e4b6c90072"},"split":"train"}
{"binding":{"chosen_proof_sha256":"760a46c45735c2286ac991019ea42b32b81c6e8ac4991d80d094da0aa71c32a8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"95d16f8d87604e31af296cb85c57b5f4bc76b11ab120d1b5d1ddc04c9afe723c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5ec0b0e55183ff152ce79ab5d2c02cf74a8634036c497bc83a0db9d263f074b7","source_sha256":"158ce289cc94776d7ace3f5f8788d7841da0114f64a958ffadba75342e4312aa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [destruct]\n  cases f0 : s.1 0 <;> intro h\n  · contradiction\n  · apply Subtype.ext\n    funext n\n    induction n with\n    | zero => injection h with h'; rwa [h'] at f0\n    | succ n IH => exact s.2 IH","hard_negative":true,"metrics":{"chosen_tokens":53,"rejected_tokens":2,"token_jaccard":0.026316,"token_length_ratio":0.037736},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"e782a773dcd5fc20a642f86e98bc3a424fa41bf6bf0859d6a4a1f5495e1ba3d1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.Find\npublic import Mathlib.Data.Stream.Init\npublic import Mathlib.Logic.Relator\npublic import Mathlib.Tactic.Common\npublic import Batteries.Tactic.Lint.Simp\n\nNamespace:\nComputation\n\nLocal context:\n/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Coinductive formalization of unbounded computations.\n\nThis file provides a `Computation` type where `Computation α` is the type of\nunbounded computations returning `α`.\n-/\n\n@[expose] public section\n\nopen Function\n\nuniverse u v w\n\n/-\ncoinductive Computation (α : Type u) : Type u\n| pure : α → Computation α\n| think : Computation α → Computation α\n-/\n/-- `Computation α` is the type of unbounded computations returning `α`.\n  An element of `Computation α` is an infinite sequence of `Option α` such\n  that if `f n = some a` for some `n` then it is constantly `some a` after that. -/\ndef Computation (α : Type u) : Type u :=\n  { f : Stream' (Option α) // ∀ ⦃n a⦄, f n = some a → f (n + 1) = some a }\n\nnamespace Computation\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n-- constructors\n/-- `pure a` is the computation that immediately terminates with result `a`. -/\ndef pure (a : α) : Computation α :=\n  ⟨Stream'.const (some a), fun _ _ => id⟩\n\ninstance : CoeTC α (Computation α) :=\n  ⟨pure⟩\n\n-- note [use has_coe_t]\n/-- `think c` is the computation that delays for one \"tick\" and then performs\n  computation `c`. -/\ndef think (c : Computation α) : Computation α :=\n  ⟨Stream'.cons none c.1, fun n a h => by\n    rcases n with - | n\n    · contradiction\n    · exact c.2 h⟩\n\n/-- `thinkN c n` is the computation that delays for `n` ticks and then performs\n  computation `c`. -/\ndef thinkN (c : Computation α) : ℕ → Computation α\n  | 0 => c\n  | n + 1 => think (thinkN c n)\n\n-- check for immediate result\n/-- `head c` is the first step of computation, either `some a` if `c = pure a`\n  or `none` if `c = think c'`. -/\ndef head (c : Computation α) : Option α :=\n  c.1.head\n\n-- one step of computation\n/-- `tail c` is the remainder of computation, either `c` if `c = pure a`\n  or `c'` if `c = think c'`. -/\ndef tail (c : Computation α) : Computation α :=\n  ⟨c.1.tail, fun _ _ h => c.2 h⟩\n\n/-- `empty α` is the computation that never returns, an infinite sequence of\n  `think`s. -/\ndef empty (α) : Computation α :=\n  ⟨Stream'.const none, fun _ _ => id⟩\n\ninstance : Inhabited (Computation α) :=\n  ⟨empty _⟩\n\n/-- `runFor c n` evaluates `c` for `n` steps and returns the result, or `none`\n  if it did not terminate after `n` steps. -/\ndef runFor : Computation α → ℕ → Option α :=\n  Subtype.val\n\n/-- `destruct c` is the destructor for `Computation α` as a coinductive type.\n  It returns `inl a` if `c = pure a` and `inr c'` if `c = think c'`. -/\ndef destruct (c : Computation α) : α ⊕ (Computation α) :=\n  match c.1 0 with\n  | none => Sum.inr (tail c)\n  | some a => Sum.inl a\n\n/-- `run c` is an unsound meta function that runs `c` to completion, possibly\n  resulting in an infinite loop in the VM. -/\nunsafe def run : Computation α → α\n  | c =>\n    match destruct c with\n    | Sum.inl a => a\n    | Sum.inr ca => run ca\n\nTarget:\ntheorem destruct_eq_pure {s : Computation α} {a : α} : destruct s = Sum.inl a → s = pure a :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_5ec0b0e55183","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"0156e296883168cf0f8c430a64bac182d96a8ea134a52814107376b65614f5e6","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Seq","family_id":"destruct_eq_pure","file_id":"mathlib/Mathlib/Data/Seq/Computation.lean","sample_id":"5ec0b0e55183ff152ce79ab5d2c02cf74a8634036c497bc83a0db9d263f074b7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8ea2eb314aa329962f79f410c370fc03bc8d9d4238b02a2baace82f999396bf2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7b5b722d1bc16ae3899d329eba58a7b2d1599243b0ea9acc8a97cda675c28cc8","source_sha256":"34049d18cb594f0a184df593dbd6d4fba0ade793ef81827c21fb1d80f135b5c5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by simp [left_distrib, *]","hard_negative":false,"metrics":{"chosen_tokens":7,"rejected_tokens":2,"token_jaccard":0.125,"token_length_ratio":0.285714},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"e790e5919324c47a89e2bf661643ba35f690ac97d93ea74dc72dd30ad283563f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Field.Basic\npublic import Mathlib.Algebra.Order.Ring.Defs\npublic import Mathlib.Data.Tree.Basic\npublic import Mathlib.Tactic.NormNum.Core\npublic import Mathlib.Util.SynthesizeUsing\n\nNamespace:\nMathlib.Tactic.CancelDenoms\n\nLocal context:\n/-\nCopyright (c) 2020 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n-/\npublic meta import Mathlib.Data.Tree.Basic\npublic meta import Mathlib.Algebra.Group.Nat.Defs\n\n/-!\n# A tactic for canceling numeric denominators\n\nThis file defines tactics that cancel numeric denominators from field Expressions.\n\nAs an example, we want to transform a comparison `5*(a/3 + b/4) < c/3` into the equivalent\n`5*(4*a + 3*b) < 4*c`.\n\n## Implementation notes\n\nThe tooling here was originally written for `linarith`, not intended as an interactive tactic.\nThe interactive version has been split off because it is sometimes convenient to use on its own.\nThere are likely some rough edges to it.\n\nImproving this tactic would be a good project for someone interested in learning tactic programming.\n-/\n\npublic meta section\n\nopen Lean Parser Tactic Mathlib Meta NormNum Qq\n\ninitialize registerTraceClass `CancelDenoms\n\nnamespace Mathlib.Tactic.CancelDenoms\n\n/-! ### Lemmas used in the procedure -/\n\ntheorem mul_subst {α} [CommRing α] {n1 n2 k e1 e2 t1 t2 : α}\n    (h1 : n1 * e1 = t1) (h2 : n2 * e2 = t2) (h3 : n1 * n2 = k) : k * (e1 * e2) = t1 * t2 := by\n  rw [← h3, mul_comm n1, mul_assoc n2, ← mul_assoc n1, h1,\n      ← mul_assoc n2, mul_comm n2, mul_assoc, h2]\n\ntheorem div_subst {α} [Field α] {n1 n2 k e1 e2 t1 : α}\n    (h1 : n1 * e1 = t1) (h2 : n2 / e2 = 1) (h3 : n1 * n2 = k) : k * (e1 / e2) = t1 := by\n  rw [← h3, mul_assoc, mul_div_left_comm, h2, ← mul_assoc, h1, mul_comm, one_mul]\n\ntheorem cancel_factors_eq_div {α} [Field α] {n e e' : α}\n    (h : n * e = e') (h2 : n ≠ 0) : e = e' / n :=\n  eq_div_of_mul_eq h2 <| by rwa [mul_comm] at h\n\nTarget:\ntheorem add_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :\n    n * (e1 + e2) = t1 + t2 :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/CancelDenoms","family_id":"add_subst","file_id":"mathlib/Mathlib/Tactic/CancelDenoms/Core.lean","sample_id":"7b5b722d1bc16ae3899d329eba58a7b2d1599243b0ea9acc8a97cda675c28cc8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3dd577e7ce2be81d92ba051dd678c134f2a6bcae0a357014a7b7228cadb929c6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1a64a2331eda0da0fbaef78bea3f8bc0d6cc844d45d3bd42a1481a8c40fbd0be","source_sha256":"76ff543811de607fc5619ca4ca6c9c5878d83cf85372b02e848f8ebf54663e8c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine strictMonoOn_of_lt_add_one ⟨?_⟩ fun n _ _ hn ↦\n    pow_ssubset_pow_succ_of_pow_ne_closure hX₁ hX hn\n  rintro - - n hn m ⟨-, hmn⟩ hm\n  apply hn\n  obtain rfl | hm₀ := m.eq_zero_or_pos\n  · simp [eq_comm (a := (1 : Set _)), coe_set_eq_one, -Set.subset_singleton_iff,\n      hX.coe.not_subset_singleton] at hm\n  · calc (X : Set G) ^ (n - 1)\n    _ = X ^ (n - m) * X ^ (m - 1) := by rw [← pow_add]; congr 1; lia\n    _ = closure (X : Set G) := by rw [hm, Set.pow_mul_subgroupClosure hX.nonempty.to_set]","hard_negative":false,"metrics":{"chosen_tokens":138,"rejected_tokens":2,"token_jaccard":0.016393,"token_length_ratio":0.014493},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"e818a51a5a29fc97a4b7cb7357dec1cf1371b84ae4b09de99cb24e3d096c8715","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Pointwise.Finset\npublic import Mathlib.Algebra.Group.Subgroup.Pointwise\npublic import Mathlib.Data.Nat.SuccPred\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2024 Yaël Dillies, Patrick Luo, Eric Rodriguez. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Patrick Luo, Eric Rodriguez\n-/\n/-!\n# Linear lower bound on the growth of a generating set\n\nThis file proves that the growth of a set generating an infinite group is at least linear.\n-/\n\npublic section\n\nopen Subgroup\nopen scoped Pointwise\n\nnamespace Finset\nvariable {G : Type*} [Group G] [DecidableEq G] {X : Finset G} {n : ℕ}\n\n@[to_additive]\nlemma pow_ssubset_pow_succ_of_pow_ne_closure (hX₁ : (1 : G) ∈ X) (hX : X.Nontrivial)\n    (hXclosure : (X ^ n : Set G) ≠ closure (X : Set G)) : X ^ n ⊂ X ^ (n + 1) := by\n  obtain rfl | hn := eq_or_ne n 0\n  · simpa [ssubset_iff_subset_not_subset, hX₁, -Finset.subset_singleton_iff]\n      using! hX.not_subset_singleton\n  refine (pow_subset_pow_right hX₁ <| n.le_add_right _).ssubset_of_ne ?_\n  contrapose hXclosure with hXn\n  rw [← closure_pow (mod_cast hX₁) hn]\n  wlog hn₁ : n = 1\n  · simp +contextual only [pow_one] at this\n    replace hXn d : X ^ (n + d) = X ^ n := by\n      induction d with\n      | zero => rw [add_zero]\n      | succ d hd =>\n        rw [pow_add, pow_one] at hXn\n        rw [← add_assoc, pow_add, pow_one, hd, ← hXn]\n    exact mod_cast this (one_mem_pow hX₁) (hX.pow hn) one_ne_zero\n      (by simp [hXn, ← pow_mul, mul_two]) (by simp)\n  subst hn₁\n  simp only [ne_eq, one_ne_zero, not_false_eq_true, Nat.reduceAdd, pow_one] at *\n  let Xgp : Subgroup G :=\n  { carrier := X\n    mul_mem' := fun {x y} hx hy ↦ by\n      norm_cast at *\n      simpa [← hXn, ← sq] using! mul_mem_mul hx hy\n    one_mem' := hX₁\n    inv_mem' := fun {x} hx ↦ by\n      norm_cast at *\n      have : x • X ⊆ X := by\n        simpa [← hXn, add_assoc, ← sq] using! smul_finset_subset_mul (t := X) hx\n      have : x • X = X := eq_of_subset_of_card_le this (card_smul_finset ..).ge\n      rw [← eq_inv_smul_iff] at this\n      rw [this]\n      simpa [mem_inv_smul_finset_iff] }\n  exact subset_closure.antisymm <| (closure_le Xgp).2 subset_rfl\n\n@[to_additive]\n\nTarget:\nlemma pow_right_strictMonoOn (hX₁ : 1 ∈ X) (hX : X.Nontrivial) :\n    StrictMonoOn (fun n ↦ X ^ n) {n | (X ^ (n - 1) : Set G) ≠ closure (X : Set G)} :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/Group","family_id":"pow_right_strictmonoon","file_id":"mathlib/Mathlib/Geometry/Group/Growth/LinearLowerBound.lean","sample_id":"1a64a2331eda0da0fbaef78bea3f8bc0d6cc844d45d3bd42a1481a8c40fbd0be"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b276acfbf4317cef3c8ddc23d0eda431624fd9c228809a52c5678c7b8d9bd426","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cee550d49a61e658e839ba188228371f82697399ac94b3e73478d451bf3a111f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"79c5185e6f28d9b4a517c1d0991b691ef5318525bdf3a4fee785bfeb02cb287e","source_sha256":"b645158f5be7fdfcbd42911af4e66fd4a79189395cdeb22443a3f6445ef45b71","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ENNReal.eq_sub_of_add_eq hg_fin ?_\n  rw [← lintegral_add_right' _ hg]\n  exact lintegral_congr_ae (h_le.mono fun x hx => tsub_add_cancel_of_le hx)","hard_negative":true,"metrics":{"chosen_tokens":28,"rejected_tokens":2,"token_jaccard":0.038462,"token_length_ratio":0.071429},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"e86c24faae6ec8491a439691532d09ed4719c1844a3aabe8bbdbd7e17ea70102","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Integral.Lebesgue.Add\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Johannes Hölzl\n-/\n/-!\n# Subtraction of Lebesgue integrals\n\nIn this file we first show that Lebesgue integrals can be subtracted with the expected results –\n`∫⁻ f - ∫⁻ g ≤ ∫⁻ (f - g)`, with equality if `g ≤ f` almost everywhere. Then we prove variants of\nthe monotone convergence theorem that use this subtraction in their proofs.\n-/\n\npublic section\n\nopen Filter ENNReal Topology\n\nnamespace MeasureTheory\n\nvariable {α β : Type*} [MeasurableSpace α] {μ : Measure α}\n\nTarget:\ntheorem lintegral_sub' {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hg_fin : ∫⁻ a, g a ∂μ ≠ ∞)\n    (h_le : g ≤ᵐ[μ] f) : ∫⁻ a, f a - g a ∂μ = ∫⁻ a, f a ∂μ - ∫⁻ a, g a ∂μ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_79c5185e6f28","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b664c36587e0e4eda73b80984757acc7dcc47faf42493ba5e7ab78493230f5d6","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Integral","family_id":"lintegral_sub","file_id":"mathlib/Mathlib/MeasureTheory/Integral/Lebesgue/Sub.lean","sample_id":"79c5185e6f28d9b4a517c1d0991b691ef5318525bdf3a4fee785bfeb02cb287e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bf3037486d0c6b30f0d1754728261534db0d266ca9de3c9d72286f831a1bc900","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"aeb2769c934ebca5ccdab77e64272ed83031848c9f7d2012206430aec7e08fe6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"53e21ad5e315c317c8084e011afafd42127d23df856e46c6d95998155a1cbf26","source_sha256":"610b55cc6b034881bfda1b6609f87ef5fc28aecf81f7b61240acfb496bc220e4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI : IsCentral K D :=\n  { out := fun x ↦ show x ∈ Subalgebra.center k D → _ by\n      simp only [center_eq_bot, mem_bot, Set.mem_range, forall_exists_index]\n      rintro x rfl\n      exact ⟨algebraMap k K x, by simp [algebraMap_eq_smul_one, smul_assoc]⟩ }\n  refine ⟨FaithfulSMul.algebraMap_injective k K, fun x => ?_⟩\n  have H : algebraMap K D x ∈ (Subalgebra.center K D : Set D) := Subalgebra.algebraMap_mem _ _\n  rw [show (Subalgebra.center K D : Set D) = Subalgebra.center k D by rfl] at H\n  simp only [center_eq_bot, coe_bot, Set.mem_range] at H\n  obtain ⟨x', H⟩ := H\n  exact ⟨x', (algebraMap K D).injective <| by simp [← H, algebraMap_eq_smul_one]⟩","hard_negative":true,"metrics":{"chosen_tokens":161,"rejected_tokens":2,"token_jaccard":0.016667,"token_length_ratio":0.012422},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"e8aef76c07f5ec5dcb28123ad7347aaf76c954448e5fe2c80991b1b9e77d6831","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Central.Defs\nimport Mathlib.Algebra.Module.Torsion.Field\n\nNamespace:\nAlgebra.IsCentral\n\nLocal context:\n/-\nCopyright (c) 2024 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Jujian Zhang, Yunzhou Xie\n-/\n/-!\n# Central Algebras\n\nIn this file, we prove some basic results about central algebras over a commutative ring.\n\n## Main results\n\n- `Algebra.IsCentral.center_eq_bot`: the center of a central algebra over `K` is equal to `K`.\n- `Algebra.IsCentral.self`: a commutative ring is a central algebra over itself.\n- `Algebra.IsCentral.baseField_essentially_unique`: Let `D/K/k` be a tower of scalars where\n  `K` and `k` are fields. If `D` is a nontrivial central algebra over `k`, `K` is isomorphic to `k`.\n-/\n\npublic section\n\nuniverse u v\n\nnamespace Algebra.IsCentral\n\nvariable (K : Type u) [CommSemiring K] (D D' : Type v) [Semiring D] [Algebra K D]\n  [h : IsCentral K D] [Semiring D'] [Algebra K D']\n\n@[simp]\nlemma center_eq_bot : Subalgebra.center K D = ⊥ := eq_bot_iff.2 IsCentral.out\n\nvariable {D} in\nlemma mem_center_iff {x : D} : x ∈ Subalgebra.center K D ↔ ∃ (a : K), x = algebraMap K D a := by\n  rw [center_eq_bot, Algebra.mem_bot]\n  simp [eq_comm]\n\ninstance self : IsCentral K K where\n  out x := by simp [Algebra.mem_bot]\n\nTarget:\nlemma baseField_essentially_unique\n    (k K D : Type*) [Field k] [Field K] [Ring D] [Nontrivial D]\n    [Algebra k K] [Algebra K D] [Algebra k D] [IsScalarTower k K D]\n    [IsCentral k D] :\n    Function.Bijective (algebraMap k K) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_53e21ad5e315","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"15f53928d893e815ce916eea7ba03424c15b673c35d96347ff26e047052004e9","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Central","family_id":"basefield_essentially_unique","file_id":"mathlib/Mathlib/Algebra/Central/Basic.lean","sample_id":"53e21ad5e315c317c8084e011afafd42127d23df856e46c6d95998155a1cbf26"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b62df895e5ecbe83fea44bae35972a3514a09e499e02633c6e24648777cc00bf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"69656bd82022794f70a1f8fdd7df61df1e6eb18f7b6001f52e155f26ad69127a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5e07b58c14b8fda2d13e55aab1c1520c2c2eed27aece7742846d0bae8fae4ed6","source_sha256":"5d8f7f707a69d48e10d7817e517c71568b55a688a7dc3473a0fa55b05f45f810","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro contr\n  obtain ⟨a, b, hab, mcfab⟩ := mcf\n  specialize contr ![![a, b], ![b, a]]\n  rw [Fin.sum_univ_two', ← mcfab.left, ← two_nsmul] at contr\n  have sharp :\n    2 • ((ω.tt ![![a, b], ![b, a]]).map (fun _ => f ![a, b])).sum <\n    2 • ((ω.tt ![![a, b], ![b, a]]).map f).sum := by\n    have half_sharp :\n      ((ω.tt ![![a, b], ![b, a]]).map (fun _ => f ![a, b])).sum <\n      ((ω.tt ![![a, b], ![b, a]]).map f).sum := by\n      apply Multiset.sum_lt_sum\n      · intro r rin\n        exact le_of_lt (mcfab.rows_lt_aux hab symmega rin)\n      · obtain ⟨g, _⟩ := valid.contains\n        have : (fun i => g ((Function.swap ![![a, b], ![b, a]]) i)) ∈ ω.tt ![![a, b], ![b, a]] := by\n          simp only [FractionalOperation.tt, Multiset.mem_map]\n          use g\n        exact ⟨_, this, mcfab.rows_lt_aux hab symmega this⟩\n    rw [two_nsmul, two_nsmul]\n    exact add_lt_add half_sharp half_sharp\n  have impos : 2 • (ω.map (fun _ => f ![a, b])).sum < ω.size • 2 • f ![a, b] := by\n    convert! lt_of_lt_of_le sharp contr\n    simp [FractionalOperation.tt, Multiset.map_map]\n  have rhs_swap : ω.size • 2 • f ![a, b] = 2 • ω.size • f ![a, b] := nsmul_left_comm ..\n  have distrib : (ω.map (fun _ => f ![a, b])).sum = ω.size • f ![a, b] := by simp\n  rw [rhs_swap, distrib] at impos\n  exact ne_of_lt impos rfl","hard_negative":false,"metrics":{"chosen_tokens":456,"rejected_tokens":463,"token_jaccard":0.974684,"token_length_ratio":1.015351},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"e8b20cd4e6e4b0022f47598611935f630e4af42f4878fe7afd2cc0b39f68058c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Fin\npublic import Mathlib.Algebra.Order.BigOperators.Group.Multiset\npublic import Mathlib.Data.Fin.VecNotation\npublic import Mathlib.LinearAlgebra.Matrix.Notation\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Martin Dvorak. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Martin Dvorak\n-/\n/-!\n\n# General-Valued Constraint Satisfaction Problems\n\nGeneral-Valued CSP is a very broad class of problems in discrete optimization.\nGeneral-Valued CSP subsumes Min-Cost-Hom (including 3-SAT for example) and Finite-Valued CSP.\n\n## Main definitions\n* `ValuedCSP`: A VCSP template; fixes a domain, a codomain, and allowed cost functions.\n* `ValuedCSP.Term`: One summand in a VCSP instance; calls a concrete function from given template.\n* `ValuedCSP.Term.evalSolution`: An evaluation of the VCSP term for given solution.\n* `ValuedCSP.Instance`: An instance of a VCSP problem over given template.\n* `ValuedCSP.Instance.evalSolution`: An evaluation of the VCSP instance for given solution.\n* `ValuedCSP.Instance.IsOptimumSolution`: Is given solution a minimum of the VCSP instance?\n* `Function.HasMaxCutProperty`: Can given binary function express the Max-Cut problem?\n* `FractionalOperation`: Multiset of operations on given domain of the same arity.\n* `FractionalOperation.IsSymmetricFractionalPolymorphismFor`: Is given fractional operation a\n  symmetric fractional polymorphism for given VCSP template?\n\n## References\n* [D. A. Cohen, M. C. Cooper, P. Creed, P. G. Jeavons, S. Živný,\n  *An Algebraic Theory of Complexity for Discrete Optimisation*][cohen2012]\n\n-/\n\n@[expose] public section\n\n/-- A template for a valued CSP problem over a domain `D` with costs in `C`.\nRegarding `C` we want to support `Bool`, `Nat`, `ENat`, `Int`, `Rat`, `NNRat`,\n`Real`, `NNReal`, `EReal`, `ENNReal`, and tuples made of any of those types. -/\n@[nolint unusedArguments]\nabbrev ValuedCSP (D C : Type*) [AddCommMonoid C] [PartialOrder C] [IsOrderedAddMonoid C] :=\n  Set (Σ (n : ℕ), (Fin n → D) → C) -- Cost functions `D^n → C` for any `n`\n\nvariable {D C : Type*} [AddCommMonoid C] [PartialOrder C] [IsOrderedAddMonoid C]\n\n/-- A term in a valued CSP instance over the template `Γ`. -/\nstructure ValuedCSP.Term (Γ : ValuedCSP D C) (ι : Type*) where\n  /-- Arity of the function -/\n  n : ℕ\n  /-- Which cost function is instantiated -/\n  f : (Fin n → D) → C\n  /-- The cost function comes from the template -/\n  inΓ : ⟨n, f⟩ ∈ Γ\n  /-- Which variables are plugged as arguments to the cost function -/\n  app : Fin n → ι\n\n/-- Evaluation of a `Γ` term `t` for given solution `x`. -/\ndef ValuedCSP.Term.evalSolution {Γ : ValuedCSP D C} {ι : Type*}\n    (t : Γ.Term ι) (x : ι → D) : C :=\n  t.f (x ∘ t.app)\n\n/-- A valued CSP instance over the template `Γ` with variables indexed by `ι`. -/\nabbrev ValuedCSP.Instance (Γ : ValuedCSP D C) (ι : Type*) : Type _ :=\n  Multiset (Γ.Term ι)\n\n/-- Evaluation of a `Γ` instance `I` for given solution `x`. -/\ndef ValuedCSP.Instance.evalSolution {Γ : ValuedCSP D C} {ι : Type*}\n    (I : Γ.Instance ι) (x : ι → D) : C :=\n  (I.map (·.evalSolution x)).sum\n\n/-- Condition for `x` being an optimum solution (min) to given `Γ` instance `I`. -/\ndef ValuedCSP.Instance.IsOptimumSolution {Γ : ValuedCSP D C} {ι : Type*}\n    (I : Γ.Instance ι) (x : ι → D) : Prop :=\n  ∀ y : ι → D, I.evalSolution x ≤ I.evalSolution y\n\n/-- Function `f` has Max-Cut property at labels `a` and `b` when `argmin f` is exactly\n`{ ![a, b], ![b, a] }`. -/\ndef Function.HasMaxCutPropertyAt (f : (Fin 2 → D) → C) (a b : D) : Prop :=\n  f ![a, b] = f ![b, a] ∧\n    ∀ x y : D, f ![a, b] ≤ f ![x, y] ∧ (f ![a, b] = f ![x, y] → a = x ∧ b = y ∨ a = y ∧ b = x)\n\n/-- Function `f` has Max-Cut property at some two non-identical labels. -/\ndef Function.HasMaxCutProperty (f : (Fin 2 → D) → C) : Prop :=\n  ∃ a b : D, a ≠ b ∧ f.HasMaxCutPropertyAt a b\n\n/-- Fractional operation is a finite unordered collection of D^m → D possibly with duplicates. -/\nabbrev FractionalOperation (D : Type*) (m : ℕ) : Type _ :=\n  Multiset ((Fin m → D) → D)\n\nvariable {m : ℕ}\n\n/-- Arity of the \"output\" of the fractional operation. -/\n@[simp]\ndef FractionalOperation.size (ω : FractionalOperation D m) : ℕ := ω.card\n\n/-- Fractional operation is valid iff nonempty. -/\ndef FractionalOperation.IsValid (ω : FractionalOperation D m) : Prop :=\n  ω ≠ ∅\n\n/-- Valid fractional operation contains an operation. -/\nlemma FractionalOperation.IsValid.contains {ω : FractionalOperation D m} (valid : ω.IsValid) :\n    ∃ g : (Fin m → D) → D, g ∈ ω :=\n  Multiset.exists_mem_of_ne_zero valid\n\n/-- Fractional operation applied to a transposed table of values. -/\ndef FractionalOperation.tt {ι : Type*} (ω : FractionalOperation D m) (x : Fin m → ι → D) :\n    Multiset (ι → D) :=\n  ω.map (fun (g : (Fin m → D) → D) (i : ι) => g ((Function.swap x) i))\n\n/-- Cost function admits given fractional operation, i.e., `ω` improves `f` in the `≤` sense. -/\ndef Function.AdmitsFractional {n : ℕ} (f : (Fin n → D) → C) (ω : FractionalOperation D m) : Prop :=\n  ∀ x : (Fin m → (Fin n → D)),\n    m • ((ω.tt x).map f).sum ≤ ω.size • Finset.univ.sum (fun i => f (x i))\n\n/-- Fractional operation is a fractional polymorphism for given VCSP template. -/\ndef FractionalOperation.IsFractionalPolymorphismFor\n    (ω : FractionalOperation D m) (Γ : ValuedCSP D C) : Prop :=\n  ∀ f ∈ Γ, f.snd.AdmitsFractional ω\n\n/-- Fractional operation is symmetric. -/\ndef FractionalOperation.IsSymmetric (ω : FractionalOperation D m) : Prop :=\n  ∀ x y : (Fin m → D), List.Perm (List.ofFn x) (List.ofFn y) → ∀ g ∈ ω, g x = g y\n\n/-- Fractional operation is a symmetric fractional polymorphism for given VCSP template. -/\ndef FractionalOperation.IsSymmetricFractionalPolymorphismFor\n    (ω : FractionalOperation D m) (Γ : ValuedCSP D C) : Prop :=\n  ω.IsFractionalPolymorphismFor Γ ∧ ω.IsSymmetric\n\nlemma Function.HasMaxCutPropertyAt.rows_lt_aux {C : Type*} [PartialOrder C]\n    {f : (Fin 2 → D) → C} {a b : D} (mcf : f.HasMaxCutPropertyAt a b) (hab : a ≠ b)\n    {ω : FractionalOperation D 2} (symmega : ω.IsSymmetric)\n    {r : Fin 2 → D} (rin : r ∈ (ω.tt ![![a, b], ![b, a]])) :\n    f ![a, b] < f r := by\n  rw [FractionalOperation.tt, Multiset.mem_map] at rin\n  rw [show r = ![r 0, r 1] by simp [← List.ofFn_inj]]\n  apply lt_of_le_of_ne (mcf.right (r 0) (r 1)).left\n  intro equ\n  have asymm : r 0 ≠ r 1 := by\n    rcases (mcf.right (r 0) (r 1)).right equ with ⟨ha0, hb1⟩ | ⟨ha1, hb0⟩\n    · rw [ha0, hb1] at hab\n      exact hab\n    · rw [ha1, hb0] at hab\n      exact hab.symm\n  apply asymm\n  obtain ⟨o, in_omega, rfl⟩ := rin\n  change o (fun j => ![![a, b], ![b, a]] j 0) = o (fun j => ![![a, b], ![b, a]] j 1)\n  convert! symmega ![a, b] ![b, a] (by simp [List.Perm.swap]) o in_omega using 2 <;>\n    simp [Matrix.const_fin1_eq]\n\nvariable {C : Type*} [AddCommMonoid C] [PartialOrder C] [IsOrderedCancelAddMonoid C]\n\nTarget:\nlemma Function.HasMaxCutProperty.forbids_commutativeFractionalPolymorphism\n    {f : (Fin 2 → D) → C} (mcf : f.HasMaxCutProperty)\n    {ω : FractionalOperation D 2} (valid : ω.IsValid) (symmega : ω.IsSymmetric) :\n    ¬ f.AdmitsFractional ω :=\n\nProof body:\n","rejected":"```lean\nby\n  intro contr\n  obtain ⟨a, b, hab, mcfab⟩ := mcf\n  specialize contr ![![a, b], ![b, a]]\n  rw [Fin.sum_univ_two', ← mcfab.left, ← two_nsmul] at contr\n  have sharp :\n    2 • ((ω.tt ![![a, b], ![b, a]]).map (fun _ => f ![a, b])).sum <\n    2 • ((ω.tt ![![a, b], ![b, a]]).map f).sum := by\n    have half_sharp :\n      ((ω.tt ![![a, b], ![b, a]]).map (fun _ => f ![a, b])).sum <\n      ((ω.tt ![![a, b], ![b, a]]).map f).sum := by\n      apply Multiset.sum_lt_sum\n      · intro r rin\n        exact le_of_lt (mcfab.rows_lt_aux hab symmega rin)\n      · obtain ⟨g, _⟩ := valid.contains\n        have : (fun i => g ((Function.swap ![![a, b], ![b, a]]) i)) ∈ ω.tt ![![a, b], ![b, a]] := by\n          simp only [FractionalOperation.tt, Multiset.mem_map]\n          use g\n        exact ⟨_, this, mcfab.rows_lt_aux hab symmega this⟩\n    rw [two_nsmul, two_nsmul]\n    exact add_lt_add half_sharp half_sharp\n  have impos : 2 • (ω.map (fun _ => f ![a, b])).sum < ω.size • 2 • f ![a, b] := by\n    convert! lt_of_lt_of_le sharp contr\n    simp [FractionalOperation.tt, Multiset.map_map]\n  have rhs_swap : ω.size • 2 • f ![a, b] = 2 • ω.size • f ![a, b] := nsmul_left_comm ..\n  have distrib : (ω.map (fun _ => f ![a, b])).sum = ω.size • f ![a, b] := by simp\n  rw [rhs_swap, distrib] at impos\n  exact ne_of_lt impos rfl\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Optimization","family_id":"function","file_id":"mathlib/Mathlib/Combinatorics/Optimization/ValuedCSP.lean","sample_id":"5e07b58c14b8fda2d13e55aab1c1520c2c2eed27aece7742846d0bae8fae4ed6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c7cf3138eb7d775d3fb111ddb9cf2f6f27656c166d40a0c93cd557614d2fd3af","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0e1d06e3f61ab839dc7deda0008c234fa9944bd61ed2eed9659122bd46611dc4","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2b323eed78bdd6ac7a8d6ff6b4eecb4cf02033f76acf0f63ce6e3ba7ef4dfad0","source_sha256":"f94f55c4ba2322bc4c471438f5b24643adb8c39a2c5e74b34a6830fdd6741d84","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases xs\n  simp only [reverse, snoc, cons, toList_mk]\n  congr\n  simp [toList, append_def]","hard_negative":true,"metrics":{"chosen_tokens":21,"rejected_tokens":2,"token_jaccard":0.0625,"token_length_ratio":0.095238},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"e906435fb5959a8730c6063ef9f71c1cd977500604fdbfc7e8fb8faa7c60b84a","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Vector.Basic\n\nNamespace:\nList.Vector\n\nLocal context:\n/-\nCopyright (c) 2023 Alex Keizer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alex Keizer\n-/\n/-!\n  This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single\n  element to the back of a vector.\n\n  It provides a collection of lemmas that show how different `Vector` operations reduce when their\n  argument is `snoc xs x`.\n\n  Also, an alternative, reverse, induction principle is added, that breaks down a vector into\n  `snoc xs x` for its inductive case. Effectively doing induction from right-to-left\n-/\n\n@[expose] public section\n\nnamespace List\n\nnamespace Vector\n\nvariable {α β σ φ : Type*} {n : ℕ} {x : α} {s : σ} (xs : Vector α n)\n\n/-- Append a single element to the end of a vector -/\ndef snoc : Vector α n → α → Vector α (n + 1) :=\n  fun xs x => xs ++ x ::ᵥ Vector.nil\n\n/-! ## Simplification lemmas -/\n\nsection Simp\n\nvariable {y : α}\n\n@[simp]\ntheorem snoc_cons : (x ::ᵥ xs).snoc y = x ::ᵥ (xs.snoc y) :=\n  rfl\n\n@[simp]\ntheorem snoc_nil : (nil.snoc x) = x ::ᵥ nil :=\n  rfl\n\n@[simp]\ntheorem reverse_cons : reverse (x ::ᵥ xs) = (reverse xs).snoc x := by\n  cases xs\n  simp only [reverse, cons, toList_mk, List.reverse_cons, snoc]\n  congr\n\n@[simp]\n\nTarget:\ntheorem reverse_snoc : reverse (xs.snoc x) = x ::ᵥ (reverse xs) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_2b323eed78bd","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"ae3ba5e2b838b56637524d15611f7683482c62c5670ee010b2dbed8231cae075","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Vector","family_id":"reverse_snoc","file_id":"mathlib/Mathlib/Data/Vector/Snoc.lean","sample_id":"2b323eed78bdd6ac7a8d6ff6b4eecb4cf02033f76acf0f63ce6e3ba7ef4dfad0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"792cf305c49d49d53ed8828dada761a3db9769ab9dc9ac7edc92e61179edcded","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1d3d2dc551d90c30ba5f909bc69c9cc130fae372a274b0debe56a42386450ace","source_sha256":"287a704d503b237a43f85c1d8d085e3a9c8170677a7fe6bc6d74659961e522de","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · rw [Ne, eq_none_iff', not_not]\n    exact fun h => ⟨o.get h, eq_some_iff.2 (get_mem h)⟩\n  · rintro ⟨x, rfl⟩\n    apply some_ne_none","hard_negative":true,"metrics":{"chosen_tokens":38,"rejected_tokens":8,"token_jaccard":0.028571,"token_length_ratio":0.210526},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"e930901b112a4bd0dc8715a3cdd4be0163fea80879d4d54523dd9a732b395790","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Defs\npublic import Mathlib.Data.Set.Subsingleton\npublic import Mathlib.Logic.Equiv.Defs\n\nNamespace:\nPart\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Jeremy Avigad, Simon Hudon\n-/\n/-!\n# Partial values of a type\n\nThis file defines `Part α`, the partial values of a type.\n`o : Part α` carries a proposition `o.Dom`, its domain, along with a function `get : o.Dom → α`, its\nvalue. The rule is then that every partial value has a value but, to access it, you need to provide\na proof of the domain.\n`Part α` behaves the same as `Option α` except that `o : Option α` is decidably `none` or `some a`\nfor some `a : α`, while the domain of `o : Part α` doesn't have to be decidable. That means you can\ntranslate back and forth between a partial value with a decidable domain and an option, and\n`Option α` and `Part α` are classically equivalent. In general, `Part α` is bigger than `Option α`.\n\n## Main declarations\n`Option`-like declarations:\n* `Part.none`: The partial value whose domain is `False`.\n* `Part.some a`: The partial value whose domain is `True` and whose value is `a`.\n* `Part.ofOption`: Converts an `Option α` to a `Part α` by sending `none` to `none` and `some a` to\n  `some a`.\n* `Part.toOption`: Converts a `Part α` with a decidable domain to an `Option α`.\n* `Part.equivOption`: Classical equivalence between `Part α` and `Option α`.\n\nMonadic structure:\n* `Part.bind`: `o.bind f` has value `(f (o.get _)).get _` (`f o` morally) and is defined when `o`\n  and `f (o.get _)` are defined.\n* `Part.map`: Maps the value and keeps the same domain.\n\nOther:\n* `Part.restrict`: `Part.restrict p o` replaces the domain of `o : Part α` by `p : Prop` so long as\n  `p → o.Dom`.\n* `Part.assert`: `assert p f` appends `p` to the domains of the values of a partial function.\n* `Part.unwrap`: Gets the value of a partial value regardless of its domain. Unsound.\n\n## Notation\nFor `a : α`, `o : Part α`, `a ∈ o` means that `o` is defined and equal to `a`. Formally, it means\n`o.Dom` and `o.get _ = a`.\n-/\n\n@[expose] public section\n\nassert_not_exists RelIso\n\nopen Function\n\n/-- `Part α` is the type of \"partial values\" of type `α`. It\n  is similar to `Option α` except the domain condition can be an\n  arbitrary proposition, not necessarily decidable. -/\nstructure Part.{u} (α : Type u) : Type u where\n  /-- The domain of a partial value -/\n  Dom : Prop\n  /-- Extract a value from a partial value given a proof of `Dom` -/\n  get : Dom → α\n\nnamespace Part\n\nvariable {α : Type*} {β : Type*} {γ : Type*}\n\n/-- Convert a `Part α` with a decidable domain to an option -/\ndef toOption (o : Part α) [Decidable o.Dom] : Option α :=\n  if h : Dom o then some (o.get h) else none\n\n@[simp] lemma toOption_isSome (o : Part α) [Decidable o.Dom] : o.toOption.isSome ↔ o.Dom := by\n  by_cases h : o.Dom <;> simp [h, toOption]\n\n@[simp] lemma toOption_eq_none (o : Part α) [Decidable o.Dom] : o.toOption = none ↔ ¬o.Dom := by\n  by_cases h : o.Dom <;> simp [h, toOption]\n\n/-- `Part` extensionality -/\ntheorem ext' : ∀ {o p : Part α}, (o.Dom ↔ p.Dom) → (∀ h₁ h₂, o.get h₁ = p.get h₂) → o = p\n  | ⟨od, o⟩, ⟨pd, p⟩, H1, H2 => by\n    have t : od = pd := propext H1\n    cases t; rw [show o = p from funext fun p => H2 p p]\n\n/-- `Part` eta expansion -/\n@[simp]\ntheorem eta : ∀ o : Part α, (⟨o.Dom, fun h => o.get h⟩ : Part α) = o\n  | ⟨_, _⟩ => rfl\n\n/-- `a ∈ o` means that `o` is defined and equal to `a` -/\nprotected def Mem (o : Part α) (a : α) : Prop :=\n  ∃ h, o.get h = a\n\ninstance : Membership α (Part α) :=\n  ⟨Part.Mem⟩\n\ntheorem mem_eq (a : α) (o : Part α) : (a ∈ o) = ∃ h, o.get h = a :=\n  rfl\n\ntheorem dom_iff_mem : ∀ {o : Part α}, o.Dom ↔ ∃ y, y ∈ o\n  | ⟨_, f⟩ => ⟨fun h => ⟨f h, h, rfl⟩, fun ⟨_, h, rfl⟩ => h⟩\n\ntheorem get_mem {o : Part α} (h) : get o h ∈ o :=\n  ⟨_, rfl⟩\n\n@[simp]\ntheorem mem_mk_iff {p : Prop} {o : p → α} {a : α} : a ∈ Part.mk p o ↔ ∃ h, o h = a :=\n  Iff.rfl\n\n/-- `Part` extensionality -/\n@[ext]\ntheorem ext {o p : Part α} (H : ∀ a, a ∈ o ↔ a ∈ p) : o = p :=\n  (ext' ⟨fun h => ((H _).1 ⟨h, rfl⟩).fst, fun h => ((H _).2 ⟨h, rfl⟩).fst⟩) fun _ _ =>\n    ((H _).2 ⟨_, rfl⟩).snd\n\n/-- The `none` value in `Part` has a `False` domain and an empty function. -/\ndef none : Part α :=\n  ⟨False, False.rec⟩\n\ninstance : Inhabited (Part α) :=\n  ⟨none⟩\n\n@[simp]\ntheorem notMem_none (a : α) : a ∉ @none α := fun h => h.fst\n\n/-- The `some a` value in `Part` has a `True` domain and the\n  function returns `a`. -/\ndef some (a : α) : Part α :=\n  ⟨True, fun _ => a⟩\n\n@[simp]\ntheorem some_dom (a : α) : (some a).Dom :=\n  trivial\n\ntheorem mem_unique : ∀ {a b : α} {o : Part α}, a ∈ o → b ∈ o → a = b\n  | _, _, ⟨_, _⟩, ⟨_, rfl⟩, ⟨_, rfl⟩ => rfl\n\ntheorem mem_right_unique : ∀ {a : α} {o p : Part α}, a ∈ o → a ∈ p → o = p\n  | _, _, _, ⟨ho, _⟩, ⟨hp, _⟩ => ext' (iff_of_true ho hp) (by simp [*])\n\ntheorem Mem.left_unique : Relator.LeftUnique ((· ∈ ·) : α → Part α → Prop) := fun _ _ _ =>\n  mem_unique\n\ntheorem Mem.right_unique : Relator.RightUnique ((· ∈ ·) : α → Part α → Prop) := fun _ _ _ =>\n  mem_right_unique\n\ntheorem get_eq_of_mem {o : Part α} {a} (h : a ∈ o) (h') : get o h' = a :=\n  mem_unique ⟨_, rfl⟩ h\n\nprotected theorem subsingleton (o : Part α) : Set.Subsingleton { a | a ∈ o } := fun _ ha _ hb =>\n  mem_unique ha hb\n\n@[simp]\ntheorem get_some {a : α} (ha : (some a).Dom) : get (some a) ha = a :=\n  rfl\n\ntheorem mem_some (a : α) : a ∈ some a :=\n  ⟨trivial, rfl⟩\n\n@[simp]\ntheorem mem_some_iff {a b} : b ∈ (some a : Part α) ↔ b = a :=\n  ⟨fun ⟨_, e⟩ => e.symm, fun e => ⟨trivial, e.symm⟩⟩\n\ntheorem eq_some_iff {a : α} {o : Part α} : o = some a ↔ a ∈ o :=\n  ⟨fun e => e.symm ▸ mem_some _, fun ⟨h, e⟩ => e ▸ ext' (iff_true_intro h) fun _ _ => rfl⟩\n\ntheorem eq_none_iff {o : Part α} : o = none ↔ ∀ a, a ∉ o :=\n  ⟨fun e => e.symm ▸ notMem_none, fun h => ext (by simpa)⟩\n\ntheorem eq_none_iff' {o : Part α} : o = none ↔ ¬o.Dom :=\n  ⟨fun e => e.symm ▸ id, fun h => eq_none_iff.2 fun _ h' => h h'.fst⟩\n\n@[simp]\ntheorem not_none_dom : ¬(none : Part α).Dom :=\n  id\n\n@[simp]\ntheorem some_ne_none (x : α) : some x ≠ none := by\n  intro h\n  exact true_ne_false (congr_arg Dom h)\n\n@[simp]\ntheorem none_ne_some (x : α) : none ≠ some x :=\n  (some_ne_none x).symm\n\nTarget:\ntheorem ne_none_iff {o : Part α} : o ≠ none ↔ ∃ x, o = some x :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"ee3ae9ce77e19904f43c1b09679d7113197bbae3a2593580404f180b6e6e593c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data","family_id":"ne_none_iff","file_id":"mathlib/Mathlib/Data/Part.lean","sample_id":"1d3d2dc551d90c30ba5f909bc69c9cc130fae372a274b0debe56a42386450ace"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8beab1ef316edfb8523c7f6d8a094e22bcc81e68fa42d1a455d344f9254b12c0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"be98270d4e641660001524cb4a6a1e9cfae5fb603cff7ae0acf7c6d8fed02aa2","source_sha256":"39dfdad0e4eff3ed2f02c59604c6ff92849183f69003baf79902622224068963","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [MonomialOrder.lex_lt_iff, Finsupp.Lex.lt_iff_of_unique, ofLex_toLex]","hard_negative":false,"metrics":{"chosen_tokens":16,"rejected_tokens":2,"token_jaccard":0.071429,"token_length_ratio":0.125},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"e969e5b3c8d4e056026dca3539a6d386dbc9578f424d6fc68de98525c8b54ae1","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finsupp.Lex\npublic import Mathlib.Data.Finsupp.WellFounded\npublic import Mathlib.Data.List.TFAE\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Antoine Chambert-Loir. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir\n-/\n/-! # Monomial orders\n\n## Monomial orders\n\nA *monomial order* is well ordering relation on a type of the form `σ →₀ ℕ` which\nis compatible with addition and for which `0` is the smallest element.\nSince several monomial orders may have to be used simultaneously, one cannot\nget them as instances.\n\nIn this formalization, they are presented as a structure `MonomialOrder` which encapsulates\n`MonomialOrder.toSyn`, an additive and monotone isomorphism to a linearly ordered cancellative\nadditive commutative monoid.\nThe entry `MonomialOrder.wf` asserts that `MonomialOrder.syn` is well founded.\n\nThe terminology comes from commutative algebra and algebraic geometry, especially Gröbner bases,\nwhere `c : σ →₀ ℕ` are exponents of monomials.\n\nGiven a monomial order `m : MonomialOrder σ`, we provide the notation\n`c ≼[m] d` and `c ≺[m] d` to compare `c d : σ →₀ ℕ` with respect to `m`.\nIt is activated using `open scoped MonomialOrder`.\n\n## Examples\n\nCommutative algebra defines many monomial orders, with different usefulness ranges.\nIn this file, we provide the basic example of lexicographic ordering.\nFor the graded lexicographic ordering, see `Mathlib/Data/Finsupp/MonomialOrder/DegLex.lean`\n\n* `MonomialOrder.lex` : the lexicographic ordering on `σ →₀ ℕ`.\n  For this, `σ` needs to be embedded with an ordering relation which satisfies `WellFoundedGT σ`.\n  (This last property is automatic when `σ` is finite).\n\nThe type synonym is `Lex (σ →₀ ℕ)` and the two lemmas `MonomialOrder.lex_le_iff`\nand `MonomialOrder.lex_lt_iff` rewrite the ordering as comparisons in the type `Lex (σ →₀ ℕ)`.\n\n## References\n\n* [Cox, Little and O'Shea, *Ideals, varieties, and algorithms*][coxlittleoshea1997]\n* [Becker and Weispfenning, *Gröbner bases*][Becker-Weispfenning1993]\n\n## Note\n\nIn algebraic geometry, when the finitely many variables are indexed by integers,\nit is customary to order them using the opposite order : `MvPolynomial.X 0 > MvPolynomial.X 1 > … `\n\n-/\n\n@[expose] public section\n\n/-- Monomial orders : equivalence of `σ →₀ ℕ` with a well-ordered type -/\nstructure MonomialOrder (σ : Type*) where\n  /-- The synonym type -/\n  syn : Type*\n  /-- `syn` is an additive commutative monoid -/\n  acm : AddCommMonoid syn := by infer_instance\n  /-- `syn` is linearly ordered -/\n  lo : LinearOrder syn := by infer_instance\n  /-- `syn` is a linearly ordered cancellative additive commutative monoid -/\n  iocam : IsOrderedCancelAddMonoid syn := by infer_instance\n  /-- the additive equivalence from `σ →₀ ℕ` to `syn` -/\n  toSyn : (σ →₀ ℕ) ≃+ syn\n  /-- `toSyn` is monotone -/\n  toSyn_monotone : Monotone toSyn\n  /-- `syn` is a well ordering -/\n  wf : WellFoundedLT syn := by infer_instance\n\nattribute [instance] MonomialOrder.acm MonomialOrder.lo MonomialOrder.iocam MonomialOrder.wf\n\nnamespace MonomialOrder\n\nvariable {σ : Type*} (m : MonomialOrder σ)\n\nlemma le_add_right (a b : σ →₀ ℕ) :\n    m.toSyn a ≤ m.toSyn a + m.toSyn b := by\n  rw [← map_add]\n  exact m.toSyn_monotone le_self_add\n\ninstance orderBot : OrderBot (m.syn) where\n  bot := 0\n  bot_le a := by\n    have := m.le_add_right 0 (m.toSyn.symm a)\n    simpa [map_add, zero_add]\n\n@[simp]\ntheorem bot_eq_zero : (⊥ : m.syn) = 0 := rfl\n\n@[simp]\nlemma zero_le (a : m.syn) : 0 ≤ a := bot_le\n\ntheorem eq_zero_iff {a : m.syn} : a = 0 ↔ a ≤ 0 := eq_bot_iff\n\nlemma toSyn_eq_zero_iff (a : σ →₀ ℕ) :\n    m.toSyn a = 0 ↔ a = 0 := AddEquiv.map_eq_zero_iff m.toSyn\n\nlemma toSyn_lt_iff_ne_zero {a : m.syn} :\n    0 < a ↔ a ≠ 0 := bot_lt_iff_ne_bot\n\nlemma toSyn_strictMono : StrictMono (m.toSyn) := by\n  apply m.toSyn_monotone.strictMono_of_injective m.toSyn.injective\n\n/-- Given a monomial order, notation for the corresponding strict order relation on `σ →₀ ℕ` -/\nscoped\nnotation:50 c \" ≺[\" m:25 \"] \" d:50 => (MonomialOrder.toSyn m c < MonomialOrder.toSyn m d)\n\n/-- Given a monomial order, notation for the corresponding order relation on `σ →₀ ℕ` -/\nscoped\nnotation:50 c \" ≼[\" m:25 \"] \" d:50 => (MonomialOrder.toSyn m c ≤ MonomialOrder.toSyn m d)\n\nend MonomialOrder\n\nsection Lex\n\nopen Finsupp\n\nopen scoped MonomialOrder\n\n-- The linear order on `Finsupp`s obtained by the lexicographic ordering. -/\nnoncomputable instance {α N : Type*} [LinearOrder α]\n    [AddCommMonoid N] [PartialOrder N] [IsOrderedCancelAddMonoid N] :\n    IsOrderedCancelAddMonoid (Lex (α →₀ N)) where\n  le_of_add_le_add_left a b c h := by simpa only [add_le_add_iff_left] using h\n  add_le_add_left a b h c := by simpa using h\n\n/-- for the lexicographic ordering, X 0 * X 1 < X 0 ^ 2 -/\nexample : toLex (Finsupp.single 0 2) > toLex (Finsupp.single 0 1 + Finsupp.single 1 1) := by\n  use 0; simp\n\n/-- for the lexicographic ordering, X 1 < X 0 -/\nexample : toLex (Finsupp.single 1 1) < toLex (Finsupp.single 0 1) := by\n  use 0; simp\n\n/-- for the lexicographic ordering, X 1 < X 0 ^ 2 -/\nexample : toLex (Finsupp.single 1 1) < toLex (Finsupp.single 0 2) := by\n  use 0; simp\n\nvariable {σ : Type*} [LinearOrder σ]\n\n/-- The lexicographic order on `σ →₀ ℕ`, as a `MonomialOrder` -/\nnoncomputable def MonomialOrder.lex [WellFoundedGT σ] :\n    MonomialOrder σ where\n  syn := Lex (σ →₀ ℕ)\n  toSyn :=\n  { toEquiv := toLex\n    map_add' := toLex_add }\n  toSyn_monotone := Finsupp.toLex_monotone\n\ntheorem MonomialOrder.lex_le_iff [WellFoundedGT σ] {c d : σ →₀ ℕ} :\n    c ≼[lex] d ↔ toLex c ≤ toLex d := Iff.rfl\n\ntheorem MonomialOrder.lex_lt_iff [WellFoundedGT σ] {c d : σ →₀ ℕ} :\n    c ≺[lex] d ↔ toLex c < toLex d := Iff.rfl\n\nTarget:\ntheorem MonomialOrder.lex_lt_iff_of_unique [Unique σ] {c d : σ →₀ ℕ} :\n    c ≺[lex] d ↔ c default < d default :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finsupp","family_id":"monomialorder","file_id":"mathlib/Mathlib/Data/Finsupp/MonomialOrder.lean","sample_id":"be98270d4e641660001524cb4a6a1e9cfae5fb603cff7ae0acf7c6d8fed02aa2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1bc941a2b3a47d131821876175410f2047d53e35df24a12c023e660701f0a1b1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a86870c89a4a8513a9ca5e86eb6ee5debb67a1914500d7a9ac2268bfb0d90827","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f74f3c30220b04da1fe4f1b1519530cecb6ea35343f0fc2bc9e9549863b219a6","source_sha256":"1c1559946be132288407174c056f2f416039575ec7e1421e4402d1878b97979a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← isLowerSet_compl, ← coe_compl] at h𝒜\n  have := h𝒜.card_inter_le_finset hℬ\n  rwa [card_compl, Fintype.card_finset, tsub_mul, le_tsub_iff_le_tsub, ← mul_tsub, ←\n    card_sdiff_of_subset inter_subset_right, sdiff_inter_self_right, sdiff_compl,\n    _root_.inf_comm] at this\n  · grw [inter_subset_right]\n  · grw [← Fintype.card_finset, card_le_univ]","hard_negative":true,"metrics":{"chosen_tokens":64,"rejected_tokens":3,"token_jaccard":0.028571,"token_length_ratio":0.046875},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"e99392e2f755e5bc88a364a126b3392d4a22d19725d62099c672c7a962194dee","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Ring.Canonical\npublic import Mathlib.Algebra.Order.Ring.Nat\npublic import Mathlib.Combinatorics.SetFamily.Compression.Down\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Order.UpperLower.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Harris-Kleitman inequality\n\nThis file proves the Harris-Kleitman inequality. This relates `#𝒜 * #ℬ` and\n`2 ^ card α * #(𝒜 ∩ ℬ)` where `𝒜` and `ℬ` are upward- or downcard-closed finite families of\nfinsets. This can be interpreted as saying that any two lower sets (resp. any two upper sets)\ncorrelate in the uniform measure.\n\n## Main declarations\n\n* `IsLowerSet.le_card_inter_finset`: One form of the Harris-Kleitman inequality.\n\n## References\n\n* [D. J. Kleitman, *Families of non-disjoint subsets*][kleitman1966]\n-/\n\npublic section\n\n\nopen Finset\n\nvariable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}\n\ntheorem IsLowerSet.nonMemberSubfamily (h : IsLowerSet (𝒜 : Set (Finset α))) :\n    IsLowerSet (𝒜.nonMemberSubfamily a : Set (Finset α)) := fun s t hts => by\n  simp_rw [mem_coe, mem_nonMemberSubfamily]\n  exact And.imp (h hts) (mt <| @hts _)\n\ntheorem IsLowerSet.memberSubfamily (h : IsLowerSet (𝒜 : Set (Finset α))) :\n    IsLowerSet (𝒜.memberSubfamily a : Set (Finset α)) := by\n  rintro s t hts\n  simp_rw [mem_coe, mem_memberSubfamily]\n  exact And.imp (h <| insert_subset_insert _ hts) (mt <| @hts _)\n\ntheorem IsLowerSet.memberSubfamily_subset_nonMemberSubfamily (h : IsLowerSet (𝒜 : Set (Finset α))) :\n    𝒜.memberSubfamily a ⊆ 𝒜.nonMemberSubfamily a := fun s => by\n  rw [mem_memberSubfamily, mem_nonMemberSubfamily]\n  exact And.imp_left (h <| subset_insert _ _)\n\n/-- **Harris-Kleitman inequality**: Any two lower sets of finsets correlate. -/\ntheorem IsLowerSet.le_card_inter_finset' (h𝒜 : IsLowerSet (𝒜 : Set (Finset α)))\n    (hℬ : IsLowerSet (ℬ : Set (Finset α))) (h𝒜s : ∀ t ∈ 𝒜, t ⊆ s) (hℬs : ∀ t ∈ ℬ, t ⊆ s) :\n    #𝒜 * #ℬ ≤ 2 ^ #s * #(𝒜 ∩ ℬ) := by\n  induction s using Finset.induction generalizing 𝒜 ℬ with\n  | empty =>\n    simp_rw [subset_empty, ← subset_singleton_iff', subset_singleton_iff] at h𝒜s hℬs\n    obtain rfl | rfl := h𝒜s\n    · simp only [card_empty, zero_mul, empty_inter, mul_zero, le_refl]\n    obtain rfl | rfl := hℬs\n    · simp\n    · simp only [card_empty, pow_zero, inter_singleton_of_mem, mem_singleton, card_singleton,\n        le_refl]\n  | insert a s hs ih =>\n  rw [card_insert_of_notMem hs, ← card_memberSubfamily_add_card_nonMemberSubfamily a 𝒜, ←\n    card_memberSubfamily_add_card_nonMemberSubfamily a ℬ, add_mul, mul_add, mul_add,\n    add_comm (_ * _), add_add_add_comm]\n  grw [mul_add_mul_le_mul_add_mul\n    (card_le_card h𝒜.memberSubfamily_subset_nonMemberSubfamily) <|\n      card_le_card hℬ.memberSubfamily_subset_nonMemberSubfamily, ← two_mul, pow_succ', mul_assoc]\n  have h₀ : ∀ 𝒞 : Finset (Finset α), (∀ t ∈ 𝒞, t ⊆ insert a s) →\n      ∀ t ∈ 𝒞.nonMemberSubfamily a, t ⊆ s := by\n    rintro 𝒞 h𝒞 t ht\n    rw [mem_nonMemberSubfamily] at ht\n    exact (subset_insert_iff_of_notMem ht.2).1 (h𝒞 _ ht.1)\n  have h₁ : ∀ 𝒞 : Finset (Finset α), (∀ t ∈ 𝒞, t ⊆ insert a s) →\n      ∀ t ∈ 𝒞.memberSubfamily a, t ⊆ s := by\n    rintro 𝒞 h𝒞 t ht\n    rw [mem_memberSubfamily] at ht\n    exact (subset_insert_iff_of_notMem ht.2).1 ((subset_insert _ _).trans <| h𝒞 _ ht.1)\n  gcongr\n  refine (add_le_add (ih h𝒜.memberSubfamily hℬ.memberSubfamily (h₁ _ h𝒜s) <| h₁ _ hℬs) <|\n    ih h𝒜.nonMemberSubfamily hℬ.nonMemberSubfamily (h₀ _ h𝒜s) <| h₀ _ hℬs).trans_eq ?_\n  rw [← mul_add, ← memberSubfamily_inter, ← nonMemberSubfamily_inter,\n    card_memberSubfamily_add_card_nonMemberSubfamily]\n\nvariable [Fintype α]\n\n/-- **Harris-Kleitman inequality**: Any two lower sets of finsets correlate. -/\ntheorem IsLowerSet.le_card_inter_finset (h𝒜 : IsLowerSet (𝒜 : Set (Finset α)))\n    (hℬ : IsLowerSet (ℬ : Set (Finset α))) : #𝒜 * #ℬ ≤ 2 ^ Fintype.card α * #(𝒜 ∩ ℬ) :=\nh𝒜.le_card_inter_finset' hℬ (fun _ _ => subset_univ _) fun _ _ => subset_univ _\n\n/-- **Harris-Kleitman inequality**: Upper sets and lower sets of finsets anticorrelate. -/\ntheorem IsUpperSet.card_inter_le_finset (h𝒜 : IsUpperSet (𝒜 : Set (Finset α)))\n    (hℬ : IsLowerSet (ℬ : Set (Finset α))) :\n    2 ^ Fintype.card α * #(𝒜 ∩ ℬ) ≤ #𝒜 * #ℬ := by\n  rw [← isLowerSet_compl, ← coe_compl] at h𝒜\n  have := h𝒜.le_card_inter_finset hℬ\n  rwa [card_compl, Fintype.card_finset, tsub_mul, tsub_le_iff_tsub_le, ← mul_tsub, ←\n    card_sdiff_of_subset inter_subset_right, sdiff_inter_self_right, sdiff_compl,\n    _root_.inf_comm] at this\n\n/-- **Harris-Kleitman inequality**: Lower sets and upper sets of finsets anticorrelate. -/\ntheorem IsLowerSet.card_inter_le_finset (h𝒜 : IsLowerSet (𝒜 : Set (Finset α)))\n    (hℬ : IsUpperSet (ℬ : Set (Finset α))) :\n    2 ^ Fintype.card α * #(𝒜 ∩ ℬ) ≤ #𝒜 * #ℬ := by\n  rw [inter_comm, mul_comm #𝒜]\n  exact hℬ.card_inter_le_finset h𝒜\n\n/-- **Harris-Kleitman inequality**: Any two upper sets of finsets correlate. -/\n\nTarget:\ntheorem IsUpperSet.le_card_inter_finset (h𝒜 : IsUpperSet (𝒜 : Set (Finset α)))\n    (hℬ : IsUpperSet (ℬ : Set (Finset α))) :\n    #𝒜 * #ℬ ≤ 2 ^ Fintype.card α * #(𝒜 ∩ ℬ) :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_f74f3c30220b","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"05eb0ee50ebd46da9d8572e1496eec440c3164e19b700dfd4a014905dc6b877f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/SetFamily","family_id":"isupperset","file_id":"mathlib/Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean","sample_id":"f74f3c30220b04da1fe4f1b1519530cecb6ea35343f0fc2bc9e9549863b219a6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ff8a2edf1706af29211010fc997e519b058214e2acf8d5e15ccbda28fa02aa4d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5eeb3348906625369fd323ad89c763297184762c26c7f0ae15b04e2241c764ab","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6fc83af12926d67f82e7ea31db06b0a746a421286c20a9485cd6854580deafe5","source_sha256":"c27083b8f8cde0437299c837bd1f7a4c8446e30ca2254817bcb2958bf94cb26d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Submonoid.eq_top_iff']\n  intro x\n  induction x with | _ a\n  induction a with\n  | one => exact Submonoid.one_mem _\n  | of x => exact subset_closure <| by simp [range, of]\n  | mul x y hx hy => exact Submonoid.mul_mem _ hx hy","hard_negative":true,"metrics":{"chosen_tokens":55,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.054545},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"e9c117ba684ad29d1b46171d33c2246f35280032df588a1e6ff6caf43b0da2c6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.FreeMonoid.Basic\npublic import Mathlib.Algebra.Group.Submonoid.Operations\npublic import Mathlib.GroupTheory.Congruence.Hom\n\nNamespace:\nPresentedMonoid\n\nLocal context:\n/-\nCopyright (c) 2024 Hannah Fechtner. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Hannah Fechtner\n-/\n/-!\n# Defining a monoid given by generators and relations\n\nGiven relations `rels` on the free monoid on a type `α`, this file constructs the monoid\ngiven by generators `x : α` and relations `rels`.\n\n## Main definitions\n\n* `PresentedMonoid rels`: the quotient of the free monoid on a type `α` by the closure of one-step\n  reductions (arising from a binary relation on free monoid elements `rels`).\n* `PresentedMonoid.of`: The canonical map from `α` to a presented monoid with generators `α`.\n* `PresentedMonoid.lift f`: the canonical monoid homomorphism `PresentedMonoid rels → M`, given\n  a function `f : α → G` from a type `α` to a monoid `M` which satisfies the relations `rels`.\n\n## Tags\n\ngenerators, relations, monoid presentations\n-/\n\n@[expose] public section\n\nvariable {α : Type*}\n\n/-- Given a set of relations, `rels`, over a type `α`, `PresentedMonoid` constructs the monoid with\ngenerators `x : α` and relations `rels` as a quotient of a congruence structure over rels. -/\n@[to_additive /-- Given a set of relations, `rels`, over a type `α`, `PresentedAddMonoid` constructs\nthe monoid with generators `x : α` and relations `rels` as a quotient of an AddCon structure over\nrels -/]\ndef PresentedMonoid (rels : FreeMonoid α → FreeMonoid α → Prop) := (conGen rels).Quotient\n\nnamespace PresentedMonoid\n\nopen Set Submonoid\n\n@[to_additive]\ninstance {rels : FreeMonoid α → FreeMonoid α → Prop} : Monoid (PresentedMonoid rels) :=\n  inferInstanceAs <| Monoid (conGen rels).Quotient\n\n/-- The quotient map from the free monoid on `α` to the presented monoid with the same generators\nand the given relations `rels`. -/\n@[to_additive /-- The quotient map from the free additive monoid on `α` to the presented additive\nmonoid with the same generators and the given relations `rels` -/]\ndef mk (rels : FreeMonoid α → FreeMonoid α → Prop) : FreeMonoid α →* PresentedMonoid rels where\n  toFun := Quotient.mk (conGen rels).toSetoid\n  map_one' := rfl\n  map_mul' := fun _ _ => rfl\n\n/-- `of` is the canonical map from `α` to a presented monoid with generators `x : α`. The term `x`\nis mapped to the equivalence class of the image of `x` in `FreeMonoid α`. -/\n@[to_additive\n/-- `of` is the canonical map from `α` to a presented additive monoid with generators `x : α`. The\nterm `x` is mapped to the equivalence class of the image of `x` in `FreeAddMonoid α`. -/]\ndef of (rels : FreeMonoid α → FreeMonoid α → Prop) (x : α) : PresentedMonoid rels :=\n  mk rels (.of x)\n\nsection inductionOn\n\nvariable {α₁ α₂ α₃ : Type*} {rels₁ : FreeMonoid α₁ → FreeMonoid α₁ → Prop}\n  {rels₂ : FreeMonoid α₂ → FreeMonoid α₂ → Prop} {rels₃ : FreeMonoid α₃ → FreeMonoid α₃ → Prop}\n\nlocal notation \"P₁\" => PresentedMonoid rels₁\nlocal notation \"P₂\" => PresentedMonoid rels₂\nlocal notation \"P₃\" => PresentedMonoid rels₃\n\n@[to_additive (attr := elab_as_elim), induction_eliminator]\nprotected theorem inductionOn {δ : P₁ → Prop} (q : P₁) (h : ∀ a, δ (mk rels₁ a)) : δ q :=\n  Quotient.ind h q\n\n@[to_additive (attr := elab_as_elim)]\nprotected theorem inductionOn₂ {δ : P₁ → P₂ → Prop} (q₁ : P₁) (q₂ : P₂)\n    (h : ∀ a b, δ (mk rels₁ a) (mk rels₂ b)) : δ q₁ q₂ :=\n  Quotient.inductionOn₂ q₁ q₂ h\n\n@[to_additive (attr := elab_as_elim)]\nprotected theorem inductionOn₃ {δ : P₁ → P₂ → P₃ → Prop} (q₁ : P₁)\n    (q₂ : P₂) (q₃ : P₃) (h : ∀ a b c, δ (mk rels₁ a) (mk rels₂ b) (mk rels₃ c)) :\n    δ q₁ q₂ q₃ :=\n  Quotient.inductionOn₃ q₁ q₂ q₃ h\n\nend inductionOn\n\nvariable {α : Type*} {rels : FreeMonoid α → FreeMonoid α → Prop} {x y : FreeMonoid α}\n\nlemma mk_eq_mk_iff : mk rels x = mk rels y ↔ conGen rels x y := Quotient.eq\n\nlemma mk_eq_mk_of_rel (h : rels x y) : mk rels x = mk rels y := mk_eq_mk_iff.2 (.of _ _ h)\n\n/-- The generators of a presented monoid generate the presented monoid. That is, the submonoid\nclosure of the set of generators equals `⊤`. -/\n@[to_additive (attr := simp) /-- The generators of a presented additive monoid generate the\npresented additive monoid. That is, the additive submonoid closure of the set of generators equals\n`⊤`. -/]\n\nTarget:\ntheorem closure_range_of (rels : FreeMonoid α → FreeMonoid α → Prop) :\n    Submonoid.closure (Set.range (of rels)) = ⊤ :=\n\nProof body:\n","rejected":"by\n  exact closure_range_of","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"ef9721e5fd819ac80aba9290f5eba2aed2ccdc6e8e5af40c53a0a2c41903b05b","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/PresentedMonoid","family_id":"closure_range_of","file_id":"mathlib/Mathlib/Algebra/PresentedMonoid/Basic.lean","sample_id":"6fc83af12926d67f82e7ea31db06b0a746a421286c20a9485cd6854580deafe5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d6d98f4113b14782373b73a8bb3f9c78ad64b883beec8e3deb78731e5a49b554","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7e250c2a0a366c6dc511bcc2bdae4c79d5eb964d8cab919edc4393813525710c","source_sha256":"a9f42fb2a650677b7f31f134a9fdb63cf294a003ee4e030eb51993a29af692d7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [← toList_inj]","hard_negative":false,"metrics":{"chosen_tokens":6,"rejected_tokens":2,"token_jaccard":0.142857,"token_length_ratio":0.333333},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"e9fe65010e0644eeaaa8c7e18b44a4f6ae20fe20f8665c7dfc56b0edc86bd478","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.String.Lemmas\npublic import Mathlib.Data.List.Lex\npublic import Mathlib.Data.Char\npublic import Mathlib.Algebra.Order.Group.Nat\nimport all Init.Data.String.Iterator  -- for unfolding `Iterator.curr`\nimport all Init.Data.Ord.String  -- for unfolding `String.compare`\n\nNamespace:\nString\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Strings\n\nSupplementary theorems about the `String` type.\n-/\n\n@[expose] public section\n\nnamespace String\n\n/-- `<` on string iterators. This coincides with `<` on strings as lists. -/\ndef ltb (s₁ s₂ : Legacy.Iterator) : Bool :=\n  if s₂.hasNext then\n    if s₁.hasNext then\n      if s₁.curr = s₂.curr then\n        ltb s₁.next s₂.next\n      else s₁.curr < s₂.curr\n    else true\n  else false\n\n/-- Induction on `String.ltb`. -/\n@[no_expose] def ltb.inductionOn.{u} {motive : Legacy.Iterator → Legacy.Iterator → Sort u}\n    (it₁ it₂ : Legacy.Iterator)\n    (ind : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      i₁.get s₁ = i₂.get s₂ →\n        motive (Legacy.Iterator.next ⟨s₁, i₁⟩) (Legacy.Iterator.next ⟨s₂, i₂⟩) →\n          motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (eq : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      ¬ i₁.get s₁ = i₂.get s₂ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (base₁ : ∀ s₁ s₂ i₁ i₂, Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → ¬ Legacy.Iterator.hasNext ⟨s₁, i₁⟩ →\n      motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)\n    (base₂ : ∀ s₁ s₂ i₁ i₂, ¬ Legacy.Iterator.hasNext ⟨s₂, i₂⟩ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩) :\n    motive it₁ it₂ :=\n  if h₂ : it₂.hasNext then\n    if h₁ : it₁.hasNext then\n      if heq : it₁.curr = it₂.curr then\n        ind it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq (inductionOn it₁.next it₂.next ind eq base₁ base₂)\n      else eq it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq\n    else base₁ it₁.s it₂.s it₁.i it₂.i h₂ h₁\n  else base₂ it₁.s it₂.s it₁.i it₂.i h₂\n\ntheorem ltb_cons_addChar' (c : Char) (s₁ s₂ : Legacy.Iterator) :\n    ltb ⟨ofList (c :: s₁.s.toList), s₁.i + c⟩ ⟨ofList (c :: s₂.s.toList), s₂.i + c⟩ =\n      ltb s₁ s₂ := by\n  fun_induction ltb s₁ s₂ with\n  | case1 s₁ s₂ h₁ h₂ h ih =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_pos (by simpa using h₂), if_pos, ← ih]\n    · simp only [Legacy.Iterator.next, Pos.Raw.next, get_cons_addChar, ofList_toList]\n      congr 2 <;> apply Pos.Raw.add_char_right_comm\n    · simpa only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList] using h\n  | case2 s₁ s₂ h₁ h₂ h =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_pos (by simpa using h₂), if_neg]\n    · simp only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList, decide_eq_decide]\n    · simpa only [Legacy.Iterator.curr, get_cons_addChar, ofList_toList] using h\n  | case3 s₁ s₂ h₁ h₂ =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, Legacy.Iterator.hasNext_cons_addChar,\n      if_pos (by simpa using h₁), if_neg (by simpa using h₂)]\n  | case4 s₁ s₂ h₁ =>\n    rw [ltb, Legacy.Iterator.hasNext_cons_addChar, if_neg (by simpa using h₁)]\n\ntheorem ltb_cons_addChar (c : Char) (cs₁ cs₂ : List Char) (i₁ i₂ : Pos.Raw) :\n    ltb ⟨ofList (c :: cs₁), i₁ + c⟩ ⟨ofList (c :: cs₂), i₂ + c⟩ =\n      ltb ⟨ofList cs₁, i₁⟩ ⟨ofList cs₂, i₂⟩ := by\n  rw [eq_comm, ← ltb_cons_addChar' c]\n  simp\n\ntheorem lt_iff_toList_lt {s₁ s₂ : String} : s₁ < s₂ ↔ s₁.toList < s₂.toList :=\n  Iff.rfl\n\n@[simp]\ntheorem lt_iff_ltb {s₁ s₂ : String} :\n    s₁ < s₂ ↔ ltb (String.Legacy.iter s₁) (String.Legacy.iter s₂) := by\n  rw [Iff.comm]\n  obtain ⟨s₁, rfl⟩ := s₁.exists_eq_ofList\n  obtain ⟨s₂, rfl⟩ := s₂.exists_eq_ofList\n  simp only [lt_iff_toList_lt, String.Legacy.iter, String.Legacy.mkIterator, String.toList_ofList]\n  induction s₁ generalizing s₂ <;> cases s₂\n  · unfold ltb; decide\n  · rename_i c₂ cs₂; apply iff_of_true\n    · unfold ltb\n      simp [Legacy.Iterator.hasNext, Char.utf8Size_pos]\n    · apply List.nil_lt_cons\n  · rename_i c₁ cs₁ ih; apply iff_of_false\n    · unfold ltb\n      simp [Legacy.Iterator.hasNext]\n    · apply not_lt_of_gt; apply List.nil_lt_cons\n  · rename_i c₁ cs₁ ih c₂ cs₂; unfold ltb\n    simp only [Legacy.Iterator.hasNext, Pos.Raw.byteIdx_zero, rawEndPos_ofList, utf8Len_cons,\n      add_pos_iff, Char.utf8Size_pos, or_true, decide_true, ↓reduceIte, Legacy.Iterator.curr,\n      Pos.Raw.get, String.toList_ofList, Pos.Raw.utf8GetAux, Legacy.Iterator.next, Pos.Raw.next,\n      Bool.ite_eq_true_distrib, decide_eq_true_eq]\n    split_ifs with h\n    · subst c₂\n      suffices ltb ⟨ofList (c₁ :: cs₁), (0 : Pos.Raw) + c₁⟩\n          ⟨ofList (c₁ :: cs₂), (0 : Pos.Raw) + c₁⟩ =\n            ltb ⟨ofList cs₁, 0⟩ ⟨ofList cs₂, 0⟩ by\n        rw [this]; exact (ih cs₂).trans List.lex_cons_iff.symm\n      apply ltb_cons_addChar\n    · refine ⟨List.Lex.rel, fun e ↦ ?_⟩\n      cases e <;> rename_i h'\n      · assumption\n      · contradiction\n\n@[deprecated \"Use the new String API\" (since := \"2026-04-01\")]\ntheorem toList_nonempty :\n    ∀ {s : String}, s ≠ \"\" → s.toList = String.Legacy.front s :: (String.Legacy.drop s 1).toList\n  | s, h => by\n    obtain ⟨l, rfl⟩ := s.exists_eq_ofList\n    match l with\n    | [] => simp at h\n    | c::cs => simp [Legacy.front, Pos.Raw.get, Pos.Raw.utf8GetAux]\n\n@[simp]\ntheorem head_empty : \"\".toList.head! = default :=\n  rfl\n\nprivate theorem le_iff_not_lt {s₁ s₂ : String} : s₁ ≤ s₂ ↔ ¬ s₂ < s₁ :=\n  Iff.rfl\n\ntheorem le_iff_toList_le {s₁ s₂ : String} : s₁ ≤ s₂ ↔ s₁.toList ≤ s₂.toList := by\n  rw [String.le_iff_not_lt, lt_iff_toList_lt, not_lt]\n\ninstance : LinearOrder String where\n  le_refl _ := le_iff_toList_le.mpr le_rfl\n  le_trans a b c := by\n    simp only [le_iff_toList_le]\n    apply le_trans\n  lt_iff_le_not_ge a b := by\n    simp only [lt_iff_toList_lt, le_iff_toList_le, lt_iff_le_not_ge]\n  le_antisymm a b := by\n    simp only [le_iff_toList_le, ← toList_inj]\n    apply le_antisymm\n  le_total a b := by\n    simp only [le_iff_toList_le]\n    apply le_total\n  toDecidableLE := inferInstance\n  toDecidableEq := inferInstance\n  toDecidableLT := String.decidableLT\n  compare_eq_compareOfLessAndEq a b := by simp [Ord.compare, String.compare]\n\nTarget:\ntheorem ofList_eq {l : List Char} {s : String} : ofList l = s ↔ l = s.toList :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/String","family_id":"oflist_eq","file_id":"mathlib/Mathlib/Data/String/Basic.lean","sample_id":"7e250c2a0a366c6dc511bcc2bdae4c79d5eb964d8cab919edc4393813525710c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c860446eb6ebd47f56bc80449291da30bdf73e713b21c6c028e07a7ce9d17e8f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"06b9a04bef694021e6ebb3add5b89c19bc6e1db1dc192efb15fa2fd734811214","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"624cb99c0cbd1997bbdc0b1f1c47bd9e2e1caf360a81585dc2e8556116e02b79","source_sha256":"6f89eec31c4c30c0e54882861b32ee0325842f53b6b6cd8f4cb71a0f6148c838","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n      rw [Iso.conj_apply, Iso.eq_inv_comp]; simp [f.hom.comm]","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":4,"token_jaccard":0.2,"token_length_ratio":0.2},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"ea125adff66c8fba3e1bff37bad43de207334652656d02a93a4396d92a566b11","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Category.Grp.Basic\npublic import Mathlib.Algebra.Ring.PUnit\npublic import Mathlib.CategoryTheory.Adjunction.Limits\npublic import Mathlib.CategoryTheory.Conj\npublic import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic\npublic import Mathlib.CategoryTheory.Limits.Preserves.Basic\npublic import Mathlib.CategoryTheory.SingleObj\npublic import Mathlib.Tactic.ApplyFun\n\nNamespace:\nAction\n\nLocal context:\n/-\nCopyright (c) 2020 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# `Action V G`, the category of actions of a monoid `G` inside some category `V`.\n\nThe prototypical example is `V = ModuleCat R`,\nwhere `Action (ModuleCat R) G` is the category of `R`-linear representations of `G`.\n\nWe check `Action V G ≌ (CategoryTheory.SingleObj G ⥤ V)`,\nand construct the restriction functors\n`res {G H} [Monoid G] [Monoid H] (f : G →* H) : Action V H ⥤ Action V G`.\n-/\n\n@[expose] public section\n\n\nuniverse u v\n\nopen CategoryTheory Limits\n\nvariable (V : Type*) [Category* V]\n\n-- Note: this is _not_ a categorical action of `G` on `V`.\n/-- An `Action V G` represents a bundled action of\nthe monoid `G` on an object of some category `V`.\n\nAs an example, when `V = ModuleCat R`, this is an `R`-linear representation of `G`,\nwhile when `V = Type` this is a `G`-action.\n-/\nstructure Action (G : Type*) [Monoid G] where\n  /-- The object this action acts on -/\n  V : V\n  /-- The underlying monoid homomorphism of this action -/\n  ρ : G →* End V\n\nnamespace Action\n\nvariable {V}\n\ntheorem ρ_one {G : Type*} [Monoid G] (A : Action V G) : A.ρ 1 = 𝟙 A.V := by simp\n\n/-- When a group acts, we can lift the action to the group of automorphisms. -/\n@[simps]\ndef ρAut {G : Type*} [Group G] (A : Action V G) : G →* Aut A.V where\n  toFun g :=\n    { hom := A.ρ g\n      inv := A.ρ (g⁻¹ : G)\n      hom_inv_id := (A.ρ.map_mul (g⁻¹ : G) g).symm.trans (by rw [inv_mul_cancel, ρ_one])\n      inv_hom_id := (A.ρ.map_mul g (g⁻¹ : G)).symm.trans (by rw [mul_inv_cancel, ρ_one]) }\n  map_one' := Aut.ext A.ρ.map_one\n  map_mul' x y := Aut.ext (A.ρ.map_mul x y)\n\nvariable (G : Type*) [Monoid G]\n\nsection\n\n/-- The action defined by sending every monoid element to the identity. -/\n@[simps]\ndef trivial (X : V) : Action V G := { V := X, ρ := 1 }\n\ninstance inhabited' : Inhabited (Action Type* G) :=\n  ⟨⟨PUnit, 1⟩⟩\n\ninstance : Inhabited (Action AddCommGrpCat G) :=\n  ⟨trivial G <| AddCommGrpCat.of PUnit⟩\n\nend\n\nvariable {G}\n\n/-- A homomorphism of `Action V G`s is a morphism between the underlying objects,\ncommuting with the action of `G`.\n-/\n@[ext]\nstructure Hom (M N : Action V G) where\n  /-- The morphism between the underlying objects of this action -/\n  hom : M.V ⟶ N.V\n  comm : ∀ g : G, M.ρ g ≫ hom = hom ≫ N.ρ g := by cat_disch\n\nnamespace Hom\n\nattribute [reassoc] comm\nattribute [local simp] comm comm_assoc\n\n/-- The identity morphism on an `Action V G`. -/\n@[simps]\ndef id (M : Action V G) : Action.Hom M M where hom := 𝟙 M.V\n\ninstance (M : Action V G) : Inhabited (Action.Hom M M) :=\n  ⟨id M⟩\n\n/-- The composition of two `Action V G` homomorphisms is the composition of the underlying maps.\n-/\n@[simps]\ndef comp {M N K : Action V G} (p : Action.Hom M N) (q : Action.Hom N K) : Action.Hom M K where\n  hom := p.hom ≫ q.hom\n\nend Hom\n\ninstance : Category (Action V G) where\n  Hom M N := Hom M N\n  id M := Hom.id M\n  comp f g := Hom.comp f g\n\nlemma hom_injective {M N : Action V G} : Function.Injective (Hom.hom : (M ⟶ N) → (M.V ⟶ N.V)) :=\n  fun _ _ ↦ Hom.ext\n\n@[ext]\nlemma hom_ext {M N : Action V G} (φ₁ φ₂ : M ⟶ N) (h : φ₁.hom = φ₂.hom) : φ₁ = φ₂ :=\n  Hom.ext h\n\n@[simp]\ntheorem id_hom (M : Action V G) : (𝟙 M : Hom M M).hom = 𝟙 M.V :=\n  rfl\n\n@[simp, reassoc]\ntheorem comp_hom {M N K : Action V G} (f : M ⟶ N) (g : N ⟶ K) :\n    (f ≫ g : Hom M K).hom = f.hom ≫ g.hom :=\n  rfl\n\n@[reassoc (attr := simp)]\ntheorem hom_inv_hom {M N : Action V G} (f : M ≅ N) :\n    f.hom.hom ≫ f.inv.hom = 𝟙 M.V := by\n  rw [← comp_hom, Iso.hom_inv_id, id_hom]\n\n@[reassoc (attr := simp)]\ntheorem inv_hom_hom {M N : Action V G} (f : M ≅ N) :\n    f.inv.hom ≫ f.hom.hom = 𝟙 N.V := by\n  rw [← comp_hom, Iso.inv_hom_id, id_hom]\n\n/-- Construct an isomorphism of `G` actions/representations\nfrom an isomorphism of the underlying objects,\nwhere the forward direction commutes with the group action. -/\n@[simps]\ndef mkIso {M N : Action V G} (f : M.V ≅ N.V)\n    (comm : ∀ g : G, M.ρ g ≫ f.hom = f.hom ≫ N.ρ g := by cat_disch) : M ≅ N where\n  hom :=\n    { hom := f.hom\n      comm := comm }\n  inv :=\n    { hom := f.inv\n      comm := fun g => by have w := comm g =≫ f.inv; simp at w; simp [w] }\n\ninstance (priority := 100) isIso_of_hom_isIso {M N : Action V G} (f : M ⟶ N) [IsIso f.hom] :\n    IsIso f := (mkIso (asIso f.hom) f.comm).isIso_hom\n\ninstance isIso_hom_mk {M N : Action V G} (f : M.V ⟶ N.V) [IsIso f] (w) :\n    @IsIso _ _ M N (Hom.mk f w) :=\n  (mkIso (asIso f) w).isIso_hom\n\ninstance {M N : Action V G} (f : M ≅ N) : IsIso f.hom.hom where\n  out := ⟨f.inv.hom, by simp⟩\n\ninstance {M N : Action V G} (f : M ≅ N) : IsIso f.inv.hom where\n  out := ⟨f.hom.hom, by simp⟩\n\nnamespace FunctorCategoryEquivalence\n\n/-- Auxiliary definition for `functorCategoryEquivalence`. -/\n@[simps]\ndef functor : Action V G ⥤ SingleObj G ⥤ V where\n  obj M :=\n    { obj := fun _ => M.V\n      map := fun g => M.ρ g\n      map_id := fun _ => M.ρ.map_one\n      map_comp := fun g h => M.ρ.map_mul h g }\n  map f :=\n    { app := fun _ => f.hom\n      naturality := fun _ _ g => f.comm g }\n\n/-- Auxiliary definition for `functorCategoryEquivalence`. -/\n@[simps]\ndef inverse : (SingleObj G ⥤ V) ⥤ Action V G where\n  obj F :=\n    { V := F.obj PUnit.unit\n      ρ :=\n        { toFun := fun g => F.map g\n          map_one' := F.map_id PUnit.unit\n          map_mul' := fun g h => F.map_comp h g } }\n  map f :=\n    { hom := f.app PUnit.unit\n      comm := fun g => f.naturality g }\n\nset_option backward.defeqAttrib.useBackward true in\n/-- Auxiliary definition for `functorCategoryEquivalence`. -/\n@[simps!]\ndef unitIso : 𝟭 (Action V G) ≅ functor ⋙ inverse :=\n  NatIso.ofComponents fun M => mkIso (Iso.refl _)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Auxiliary definition for `functorCategoryEquivalence`. -/\n@[simps!]\ndef counitIso : inverse ⋙ functor ≅ 𝟭 (SingleObj G ⥤ V) :=\n  NatIso.ofComponents fun M => NatIso.ofComponents fun _ => Iso.refl _\n\nend FunctorCategoryEquivalence\n\nsection\n\nopen FunctorCategoryEquivalence\n\nvariable (V G)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The category of actions of `G` in the category `V`\nis equivalent to the functor category `SingleObj G ⥤ V`.\n-/\n@[simps]\ndef functorCategoryEquivalence : Action V G ≌ SingleObj G ⥤ V where\n  functor := functor\n  inverse := inverse\n  unitIso := unitIso\n  counitIso := counitIso\n\ninstance : (FunctorCategoryEquivalence.functor (V := V) (G := G)).IsEquivalence :=\n  (functorCategoryEquivalence V G).isEquivalence_functor\n\ninstance : (FunctorCategoryEquivalence.inverse (V := V) (G := G)).IsEquivalence :=\n  (functorCategoryEquivalence V G).isEquivalence_inverse\n\nend\n\nsection Forget\n\nvariable (V G)\n\n/-- (implementation) The forgetful functor from bundled actions to the underlying objects.\n\nUse the `CategoryTheory.forget` API provided by the `ConcreteCategory` instance below,\nrather than using this directly.\n-/\n@[simps]\ndef forget : Action V G ⥤ V where\n  obj M := M.V\n  map f := f.hom\n\ninstance : (forget V G).Faithful where map_injective w := Hom.ext w\n\n/-- The type of `V`-morphisms that can be lifted back to morphisms in the category `Action`. -/\nabbrev HomSubtype {FV : V → V → Type*} {CV : V → Type*} [∀ X Y, FunLike (FV X Y) (CV X) (CV Y)]\n    [ConcreteCategory V FV] (M N : Action V G) :=\n  { f : FV M.V N.V // ∀ g : G,\n      f ∘ ConcreteCategory.hom (M.ρ g) = ConcreteCategory.hom (N.ρ g) ∘ f }\n\ninstance {FV : V → V → Type*} {CV : V → Type*} [∀ X Y, FunLike (FV X Y) (CV X) (CV Y)]\n    [ConcreteCategory V FV] (M N : Action V G) :\n    FunLike (HomSubtype V G M N) (CV M.V) (CV N.V) where\n  coe f := f.1\n  coe_injective _ _ h := Subtype.ext (DFunLike.coe_injective h)\n\ninstance {FV : V → V → Type*} {CV : V → Type*} [∀ X Y, FunLike (FV X Y) (CV X) (CV Y)]\n    [ConcreteCategory V FV] : ConcreteCategory (Action V G) (HomSubtype V G) where\n  hom f := ⟨ConcreteCategory.hom (C := V) f.1, fun g => by\n    ext\n    simpa using CategoryTheory.congr_fun (f.2 g) _⟩\n  ofHom f := ⟨ConcreteCategory.ofHom (C := V) f, fun g => ConcreteCategory.ext_apply fun x => by\n    simpa [ConcreteCategory.hom_ofHom] using congr_fun (f.2 g) x⟩\n  hom_ofHom _ := by dsimp; ext; simp [ConcreteCategory.hom_ofHom]\n  ofHom_hom _ := by ext; simp [ConcreteCategory.ofHom_hom]\n  id_apply := ConcreteCategory.id_apply (C := V)\n  comp_apply _ _ := ConcreteCategory.comp_apply (C := V) _ _\n\ninstance hasForgetToV {FV : V → V → Type*} {CV : V → Type*} [∀ X Y, FunLike (FV X Y) (CV X) (CV Y)]\n    [ConcreteCategory V FV] : HasForget₂ (Action V G) V where forget₂ := forget V G\n\n/-- The forgetful functor is intertwined by `functorCategoryEquivalence` with\nevaluation at `PUnit.star`. -/\ndef functorCategoryEquivalenceCompEvaluation :\n    (functorCategoryEquivalence V G).functor ⋙ (evaluation _ _).obj PUnit.unit ≅ forget V G :=\n  Iso.refl _\n\nnoncomputable instance preservesLimits_forget [HasLimits V] :\n    PreservesLimits (forget V G) :=\n  Limits.preservesLimits_of_natIso (Action.functorCategoryEquivalenceCompEvaluation V G)\n\nnoncomputable instance preservesColimits_forget [HasColimits V] :\n    PreservesColimits (forget V G) :=\n  preservesColimits_of_natIso (Action.functorCategoryEquivalenceCompEvaluation V G)\n\n-- TODO construct categorical images?\nend Forget\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\ntheorem Iso.conj_ρ {M N : Action V G} (f : M ≅ N) (g : G) :\n    N.ρ g = ((forget V G).mapIso f).conj (M.ρ g) :=\n\nProof body:\n","rejected":"by\n  exact Iso.","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"0526e519c502c7f3af5c8269e0ea6c9f1216ce6bcc71827b8163d76243295809","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Action","family_id":"iso","file_id":"mathlib/Mathlib/CategoryTheory/Action/Basic.lean","sample_id":"624cb99c0cbd1997bbdc0b1f1c47bd9e2e1caf360a81585dc2e8556116e02b79"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2ec97858c3d58fdf4fe63a79d350f35bc6aeb51e4c5540c62fc1b50ae3db70dc","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"24fdd48bb31b77f033a3807b5d2ae5cf241f8ce52dab77c38b258e50fa0319c6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0a540970c97c25da7889f6fd2aaf6aa1ff05b98f178319841aaf0767908b62e7","source_sha256":"a1ecfd868e0d49b8667b2cdba350ee4f2ceb12fbe0b51094508e312f262dbf5e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [toInt, (exists_translation val).choose_spec]","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":2,"token_jaccard":0.076923,"token_length_ratio":0.166667},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"ea4c8e3ad74f39e0018bb681c03c79cff7077f8d8970a0a4f218fe1cbb331ae3","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.List.Pairwise\npublic import Batteries.Tactic.GeneralizeProofs\npublic import Mathlib.Tactic.Order.CollectFacts\n\nNamespace:\nMathlib.Tactic.Order.ToInt\n\nLocal context:\n/-\nCopyright (c) 2025 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\npublic meta import Mathlib.Util.AtomM\npublic meta import Mathlib.Util.Qq\npublic meta import Std.Data.HashMap.AdditionalOperations\n\n/-!\n# Translating linear orders to ℤ\n\nIn this file we implement the translation of a problem in any linearly ordered type to a problem in\n`ℤ`. This allows us to use the `lia` tactic to solve it.\n\nWhile the core algorithm of the `order` tactic is complete for the theory of linear orders in the\nsignature (`<`, `≤`),\nit becomes incomplete in the signature with lattice operations `⊓` and `⊔`. With these operations,\nthe problem becomes NP-hard, and the idea is to reuse a smart and efficient procedure, such as\n`lia`.\n\n## TODO\n\nMigrate to `grind` when it is ready.\n-/\n\npublic meta section\n\nnamespace Mathlib.Tactic.Order.ToInt\n\nvariable {α : Type*} [LinearOrder α] {n : ℕ} (val : Fin n → α)\n\n/-- The main theorem asserting the existence of a translation.\nWe use `Classical.choose` to turn this into a value for use in the `order` tactic,\nsee `toInt`.\n-/\ntheorem exists_translation : ∃ tr : Fin n → ℤ, ∀ i j, val i ≤ val j ↔ tr i ≤ tr j := by\n  let li := List.ofFn val\n  let sli := li.mergeSort\n  have (i : Fin n) : ∃ j : Fin sli.length, sli[j] = val i := by\n    apply List.get_of_mem\n    rw [List.Perm.mem_iff (List.mergeSort_perm _ _)]\n    simp [li]\n  use fun i ↦ (this i).choose\n  intro i j\n  simp only [Fin.getElem_fin, Int.ofNat_le]\n  by_cases h_eq : val i = val j\n  · simp [h_eq]\n  generalize_proofs _ hi hj\n  rw [← hi.choose_spec, ← hj.choose_spec] at h_eq\n  conv_lhs => rw [← hi.choose_spec, ← hj.choose_spec]\n  have := li.pairwise_mergeSort (le := fun a b ↦ decide (a ≤ b))\n      (fun a b c ↦ by simpa using le_trans) (by simpa using le_total)\n  rw [List.pairwise_iff_get] at this\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · contrapose! h\n    exact lt_of_le_of_ne (by simpa using (this hj.choose hi.choose (by simpa)))\n      (fun h ↦ h_eq (h.symm))\n  · simpa using this hi.choose hj.choose (by apply lt_of_le_of_ne h; contrapose h_eq; simp [h_eq])\n\n/-- Auxiliary definition used by the `order` tactic to transfer facts in a linear order to `ℤ`. -/\nnoncomputable def toInt (k : Fin n) : ℤ :=\n  (exists_translation val).choose k\n\nvariable (i j k : Fin n)\n\nTarget:\ntheorem toInt_le_toInt : toInt val i ≤ toInt val j ↔ val i ≤ val j :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_0a540970c97c","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"54c1735b0c0c4233d5f05bdddf4de1dbfcb743828bc92b39fa078c67edcb2894","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Order","family_id":"toint_le_toint","file_id":"mathlib/Mathlib/Tactic/Order/ToInt.lean","sample_id":"0a540970c97c25da7889f6fd2aaf6aa1ff05b98f178319841aaf0767908b62e7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"dad9f4770c642653b3a83e01d76fcc6253ccf22ac06ee8b959ff773f2af21b7e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"590ba2178e0f149ed4668d8c75b51af1b4013027d6ae0eacfc21ea23f67cada1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5574e1cf4d2a2166ca8dd385d8105f10751922211dcf708a3a1a0c7976bbd7d3","source_sha256":"e70fb56bf1193d4c52139a9e4e22d574aabafc8e5d21aec95ae03e5150f51760","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  exact\n    { map_smul := fun f r => by\n        simp only [← NatIso.naturality_1 e (r • f), F.map_smul, Linear.smul_comp,\n          NatTrans.naturality, Linear.comp_smul, Iso.inv_hom_id_app_assoc] }","hard_negative":true,"metrics":{"chosen_tokens":45,"rejected_tokens":5,"token_jaccard":0.090909,"token_length_ratio":0.111111},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"ea6afe74a3927e6d9949dfd87607a8cad111368214e3cd948799ffbc36a5a9ff","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor\npublic import Mathlib.CategoryTheory.Linear.Basic\npublic import Mathlib.Algebra.Module.LinearMap.Rat\n\nNamespace:\nCategoryTheory.Functor\n\nLocal context:\n/-\nCopyright (c) 2021 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Linear Functors\n\nAn additive functor between two `R`-linear categories is called *linear*\nif the induced map on hom types is a morphism of `R`-modules.\n\n## Implementation details\n\n`Functor.Linear` is a `Prop`-valued class, defined by saying that\nfor every two objects `X` and `Y`, the map\n`F.map : (X ⟶ Y) → (F.obj X ⟶ F.obj Y)` is a morphism of `R`-modules.\n\n-/\n\n@[expose] public section\n\n\nnamespace CategoryTheory\n\nvariable (R : Type*) [Semiring R] {C D : Type*} [Category* C] [Category* D]\n  [Preadditive C] [Preadditive D] [CategoryTheory.Linear R C] [CategoryTheory.Linear R D]\n  (F : C ⥤ D)\n\n/-- An additive functor `F` is `R`-linear provided `F.map` is an `R`-module morphism. -/\nclass Functor.Linear : Prop where\n  /-- the functor induces a linear map on morphisms -/\n  map_smul : ∀ {X Y : C} (f : X ⟶ Y) (r : R), F.map (r • f) = r • F.map f := by cat_disch\n\nlemma Functor.linear_iff (F : C ⥤ D) :\n    Functor.Linear R F ↔ ∀ (X : C) (r : R), F.map (r • 𝟙 X) = r • 𝟙 (F.obj X) := by\n  constructor\n  · intro h X r\n    rw [h.map_smul, F.map_id]\n  · refine fun h => ⟨fun {X Y} f r => ?_⟩\n    have : r • f = (r • 𝟙 X) ≫ f := by simp\n    rw [this, F.map_comp, h, Linear.smul_comp, Category.id_comp]\n\nsection Linear\n\nnamespace Functor\n\nsection\n\nvariable {R} [Linear R F]\n\n@[simp]\ntheorem map_smul {X Y : C} (r : R) (f : X ⟶ Y) : F.map (r • f) = r • F.map f :=\n  Functor.Linear.map_smul _ _\n\n@[simp]\ntheorem map_units_smul {X Y : C} (r : Rˣ) (f : X ⟶ Y) : F.map (r • f) = r • F.map f := by\n  apply map_smul\n\ninstance : Linear R (𝟭 C) where\n\nsection\n\nvariable {E : Type*} [Category* E] [Preadditive E] [CategoryTheory.Linear R E] (G : D ⥤ E)\n\ninstance [Linear R G] : Linear R (F ⋙ G) where\n\nset_option backward.isDefEq.respectTransparency false in\nlemma linear_of_full_essSurj_comp [F.Full] [F.EssSurj] [Functor.Linear R (F ⋙ G)] :\n    Functor.Linear R G := by\n  refine ⟨fun {X Y} f r ↦ ?_⟩\n  obtain ⟨X', Y', eX, eY, f', rfl⟩ :\n      ∃ (X' Y' : C) (eX : F.obj X' ≅ X) (eY : F.obj Y' ≅ Y)\n        (f' : X' ⟶ Y'), f = eX.inv ≫ F.map f' ≫ eY.hom := by\n    obtain ⟨f', hf'⟩ :=\n      F.map_surjective ((F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv)\n    exact ⟨_, _, F.objObjPreimageIso X, F.objObjPreimageIso Y, f', by cat_disch⟩\n  simpa only [comp_map, map_smul, Linear.smul_comp, Linear.comp_smul, ← G.map_comp]\n    using G.map eX.inv ≫= ((F ⋙ G).map_smul r f') =≫ G.map eY.hom\n\nlemma linear_comp_iff_of_full_of_essSurj [F.Full] [F.EssSurj] :\n    Functor.Linear R (F ⋙ G) ↔ Functor.Linear R G :=\n  ⟨fun _ ↦ linear_of_full_essSurj_comp F G, fun _ ↦ inferInstance⟩\n\nend\n\nvariable (R) [F.Additive]\n\n/-- `F.mapLinearMap` is an `R`-linear map whose underlying function is `F.map`. -/\n@[simps]\ndef mapLinearMap {X Y : C} : (X ⟶ Y) →ₗ[R] F.obj X ⟶ F.obj Y :=\n  { F.mapAddHom with map_smul' := fun r f => F.map_smul r f }\n\ntheorem coe_mapLinearMap {X Y : C} : ⇑(F.mapLinearMap R : (X ⟶ Y) →ₗ[R] _) = F.map := rfl\n\nend\n\nvariable {F} in\n\nTarget:\nlemma linear_of_iso {G : C ⥤ D} (e : F ≅ G) [F.Linear R] : G.Linear R :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_5574e1cf4d2a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c0db2dab9e0b841e5e0aebb9f70d1ed23e4d24f04ad165c3b475f6b8e1b0ba4c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Linear","family_id":"linear_of_iso","file_id":"mathlib/Mathlib/CategoryTheory/Linear/LinearFunctor.lean","sample_id":"5574e1cf4d2a2166ca8dd385d8105f10751922211dcf708a3a1a0c7976bbd7d3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3b83d6405db8d7b67a22120881e91f0e3ce367fb9b3ca08e1bae760a47811d62","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"101bb0c8bbdc01a3ce620c24a4a892863f7886402f279956b3d5a75cfbb8da92","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ae60c1f3772d8f03a4244e5c6a05fff7f8704657d4e3b6abd61ebd01d5c23d05","source_sha256":"ced30280014822262039ce197c9e4ae821d789623b770f4b0e6844d4852109a3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases Nat.eq_zero_or_pos n with\n  | inl hn =>\n    rw [if_pos hn, hn, dp_zero]\n  | inr hn =>\n    rw [if_neg (ne_of_gt hn), ← zero_smul R (0 : M), dp_smul]\n    rw [zero_pow (Nat.pos_iff_ne_zero.mp hn), zero_smul]","hard_negative":true,"metrics":{"chosen_tokens":57,"rejected_tokens":5,"token_jaccard":0.057143,"token_length_ratio":0.087719},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"eab0006b56dbe1683205adcb590b71172e17a6b15486509630dff9cd92b8dbb5","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.Eval\npublic import Mathlib.Algebra.RingQuot\npublic import Mathlib.RingTheory.DividedPowers.Basic\n\nNamespace:\nDividedPowerAlgebra\n\nLocal context:\n/-\nCopyright (c) 2026 Antoine Chambert-Loir & María Inés de Frutos—Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir, María Inés de Frutos—Fernández\n-/\n/-!\n# The universal divided power algebra\n\nLet `R` be a (commutative) semiring and `M` be an `R`-module. In this file we define `Γ_R(M)`,\nthe universal divided power algebra of `M`, as the ring quotient of the polynomial ring\nin the variables `ℕ × M` by the relation `DividedPowerAlgebra.Rel`.\n\n`DividedPowerAlgebra R M` satisfies a weak universal property for morphisms to rings with\ndivided powers (`DividedPowerAlgebra.lift`).\n\n## Main definitions\n\n* `DividedPowerAlgebra.Rel`: the type coding the basic relations that will give rise to the\n  divided power algebra.\n\n* `DividedPowerAlgebra R M`: the universal divided power algebra of the `R`-module `M`,\n  defined as `RingQuot` of `DividedPowerAlgebra.Rel R M`.\n\n* `DividedPowerAlgebra.dp R n m`: for `n : ℕ` and `m : M`, this is the equivalence class of\n  `MvPolynomial.X (⟨n, m⟩)` in `DividedPowerAlgebra R M`.\n\n  When that algebra is endowed with its canonical divided power structure (to be defined),\n  the image of `MvPolynomial.X (n, m)`, for any `n : ℕ` and `m : M`, is equal to\n  the `n`th divided power of the image of `m`.\n\n  The API will be setup so that it is never (never say never…) necessary to lift to `MvPolynomial`.\n\n* `DividedPowerAlgebra.lift`: the weak universal property of `DividedPowerAlgebra R M`.\n\n* `DividedPowerAlgebra.map`: the functoriality map between divided power algebras\n  associated with a linear map of the underlying modules.\n  Given an `R`-algebra `S`, an `S`-module `N` and an `R`-linear map `f : M →ₗ[R] N`,\n  this is the map `DividedPowerAlgebra R M →ₐ[R] DividedPowerAlgebra S N`\n  sending `dp R n m` to `dp S n (f m)`.\n\n## References\n\n* [P. Berthelot (1974), *Cohomologie cristalline des schémas de\n  caractéristique $p$ > 0*][Berthelot-1974]\n\n* [P. Berthelot and A. Ogus (1978), *Notes on crystalline\n  cohomology*][BerthelotOgus-1978]\n\n* [N. Roby (1963), *Lois polynomes et lois formelles en théorie des\n  modules*][Roby-1963]\n\n* [N. Roby (1965), *Les algèbres à puissances dividées*][Roby-1965]\n\n## TODO\n\n* Show in upcoming files that `DividedPowerAlgebra R M` has divided powers.\n\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Finset Ideal MvPolynomial RingQuot\n\nvariable (R M : Type*) [CommSemiring R] [AddCommMonoid M] [Module R M]\n\nnamespace DividedPowerAlgebra\n\n/-- The type coding the basic relations that will give rise to the divided power algebra.\n  The class of `MvPolynomial.X (n, a)` will be equal to `dpow n a`, for `a ∈ M`. -/\ninductive Rel : MvPolynomial (ℕ × M) R → MvPolynomial (ℕ × M) R → Prop\n  | rfl_zero : Rel 0 0 -- Needed for technical reasons.\n  | zero {a : M} : Rel (X (0, a)) 1\n  | smul {r : R} {n : ℕ} {a : M} : Rel (X (n, r • a)) (r ^ n • X (n, a))\n  | mul {m n : ℕ} {a : M} : Rel (X (m, a) * X (n, a)) (Nat.choose (m + n) m • X (m + n, a))\n  | add {n : ℕ} {a b : M} :\n    Rel (X (n, a + b)) ((Finset.antidiagonal n).sum fun k ↦ X (k.1, a) * X (k.2, b))\n\n/-- The ideal of `MvPolynomial (ℕ × M) R` generated by `Rel`. -/\ndef RelI : Ideal (MvPolynomial (ℕ × M) R) := ofRel (DividedPowerAlgebra.Rel R M)\n\nend DividedPowerAlgebra\n\n/-- The divided power algebra of a module M is defined as the ring quotient of the polynomial ring\n  in the variables `ℕ × M` by the ring relation defined by `DividedPowerAlgebra.Rel`.\n  We will later show that that `DividedPowerAlgebra R M` has divided powers.\n  It satisfies a weak universal property for morphisms to rings with divided powers. -/\nabbrev DividedPowerAlgebra := RingQuot (DividedPowerAlgebra.Rel R M)\n\nnamespace DividedPowerAlgebra\n\nopen MvPolynomial\n\nvariable {R M}\n\nlemma mkAlgHom_surjective : Function.Surjective (mkAlgHom R (Rel R M)) :=\n  RingQuot.mkAlgHom_surjective _ _\n\nlemma mkAlgHom_C (a : R) :\n    mkAlgHom R (Rel R M) (C a) = algebraMap R (DividedPowerAlgebra R M) a := by\n  rw [← MvPolynomial.algebraMap_eq, AlgHom.commutes]\n\nlemma mkRingHom_C (a : R) :\n    mkRingHom (Rel R M) (C a) = algebraMap R (DividedPowerAlgebra R M) a := by\n  rw [← mkAlgHom_C, mkAlgHom, AlgHom.coe_mk]\n\nvariable (R) in\n/-- `dp R n m` is the equivalence class of `X (⟨n, m⟩)` in `DividedPowerAlgebra R M`. -/\ndef dp (n : ℕ) (m : M) : DividedPowerAlgebra R M := mkAlgHom R (Rel R M) (X ⟨n, m⟩)\n\ntheorem dp_def (n : ℕ) (m : M) :\n  dp R n m = mkAlgHom R (Rel R M) (X ⟨n, m⟩) := rfl\n\nprotected theorem induction_on' {P : DividedPowerAlgebra R M → Prop} (f : DividedPowerAlgebra R M)\n    (h_C : ∀ a, P (mkAlgHom R (Rel R M) (C a))) (h_add : ∀ f g, P f → P g → P (f + g))\n    (h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)) : P f := by\n  obtain ⟨F, hf⟩ := RingQuot.mkRingHom_surjective (DividedPowerAlgebra.Rel R M) f\n  rw [← hf]\n  induction F using MvPolynomial.induction_on generalizing f with\n  | C a =>\n      convert! h_C a using 1\n      rw [mkAlgHom, AlgHom.coe_mk]\n  | add g1 g2 hg1 hg2 =>\n      rw [map_add]\n      exact h_add _ _ (hg1 ((mkRingHom (Rel R M)) g1) rfl) (hg2 ((mkRingHom (Rel R M)) g2) rfl)\n  | mul_X g nm h =>\n      have h' : (mkRingHom (Rel R M)) (X nm) = dp R nm.1 nm.2 := by\n        simp only [dp_def, Prod.mk.eta, mkAlgHom, AlgHom.coe_mk]\n      rw [_root_.map_mul, h']\n      exact h_dp _ _ _ (h (mkRingHom (Rel R M) g) rfl)\n\n@[elab_as_elim]\nprotected theorem induction_on {P : DividedPowerAlgebra R M → Prop} (f : DividedPowerAlgebra R M)\n    (C : ∀ a, P (algebraMap R _ a)) (add : ∀ f g, P f → P g → P (f + g))\n    (dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)) : P f :=\n  DividedPowerAlgebra.induction_on' f (fun a ↦ by rw [mkAlgHom_C]; exact C a) add dp\n\ntheorem dp_eq_mkRingHom (n : ℕ) (m : M) :\n    dp R n m = mkRingHom (Rel R M) (X (⟨n, m⟩)) := by\n  simp [dp, mkRingHom, mkAlgHom]\n\ntheorem dp_zero {m : M} : dp R 0 m = 1 := by\n  rw [dp_def, ← map_one (mkAlgHom R (Rel R M))]\n  exact RingQuot.mkAlgHom_rel R Rel.zero\n\ntheorem dp_smul {r : R} {n : ℕ} {m : M} : dp R n (r • m) = r ^ n • dp R n m := by\n  rw [dp_def, dp_def, ← map_smul]\n  exact mkAlgHom_rel R Rel.smul\n\nTarget:\ntheorem dp_null {n : ℕ} : dp R n (0 : M) = if n = 0 then 1 else 0 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_ae60c1f3772d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f3de0e0817445d24df80e1e4ea3eca7b938ff0b999f9be43813ca53f1e15c930","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/DividedPowerAlgebra","family_id":"dp_null","file_id":"mathlib/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean","sample_id":"ae60c1f3772d8f03a4244e5c6a05fff7f8704657d4e3b6abd61ebd01d5c23d05"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5cffe96cad332b72c0ae2797275016448548d2c1b107d88e2f58540117e63993","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"501b3d3640125c7dca7fdd0dca6da79f794dca628efb68f4fea9ebb8533db64a","source_sha256":"f4774116e10dc728faae25beafedcc16dba2b4511270606e92d6263e552fccf8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    obtain ⟨a', m⟩ := w.mem_essImage α\n    obtain ⟨m'⟩ := m\n    exact ⟨a', ⟨(Iso.toEquiv m').symm.trans Equiv.ulift⟩⟩","hard_negative":false,"metrics":{"chosen_tokens":38,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.052632},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"eae35e9bc0abc7429d87da5104dbfce2911fadec5427c67c0f16daca079649aa","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.EssentialImage\npublic import Mathlib.CategoryTheory.Types.Basic\npublic import Mathlib.Logic.UnivLE\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# Universe inequalities and essential surjectivity of `uliftFunctor`.\n\nWe show `UnivLE.{max u v, v} ↔ EssSurj (uliftFunctor.{u, v} : Type v ⥤ Type max u v)`.\n-/\n\n@[expose] public section\n\nopen CategoryTheory\n\nuniverse u v\n\nnoncomputable section\n\nTarget:\ntheorem UnivLE.ofEssSurj (w : (uliftFunctor.{u, v} : Type v ⥤ Type max u v).EssSurj) :\n    UnivLE.{max u v, v} where\n  small α :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory","family_id":"univle","file_id":"mathlib/Mathlib/CategoryTheory/UnivLE.lean","sample_id":"501b3d3640125c7dca7fdd0dca6da79f794dca628efb68f4fea9ebb8533db64a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a7995287f4a56e7a4f65ad1427f3f2d4f2887e143a6beb10ba5ae796bad00236","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b1bc28c81cb7045d821825b4ece0233164026129ddc5a05e462819a0fc04d775","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"cc3728dc6e72e50fd255371dff3e5f651d3ce2d9c638da326516235746fae11d","source_sha256":"02c39b43c769989fbb2fff40a558624b26a5d6a7cd12a1509d9e9ee7ee0339ba","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← subsingleton_iff_zero_eq_one, OreLocalization.one_def,\n    OreLocalization.zero_def, oreDiv_eq_iff]\n  simp","hard_negative":true,"metrics":{"chosen_tokens":17,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.176471},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"eb205cf6009981287765aa26ad6a37fad36d5e1855fa0cdb006e6a4789458974","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Submonoid.DistribMulAction\npublic import Mathlib.GroupTheory.OreLocalization.Basic\npublic import Mathlib.Algebra.GroupWithZero.Defs\n\nNamespace:\nOreLocalization\n\nLocal context:\n/-\nCopyright (c) 2022 Jakob von Raumer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jakob von Raumer, Kevin Klinge, Andrew Yang\n-/\n/-!\n\n# Localization over left Ore sets.\n\nThis file proves results on the localization of rings (monoids with zeros) over a left Ore set.\n\n## References\n\n* <https://ncatlab.org/nlab/show/Ore+localization>\n* [Zoran Škoda, *Noncommutative localization in noncommutative geometry*][skoda2006]\n\n\n## Tags\nlocalization, Ore, non-commutative\n\n-/\n\n@[expose] public section\n\nassert_not_exists RelIso\n\nuniverse u\n\nnamespace OreLocalization\n\nsection MonoidWithZero\n\nvariable {R : Type*} [MonoidWithZero R] {S : Submonoid R} [OreSet S]\n\n@[simp]\ntheorem zero_oreDiv' (s : S) : (0 : R) /ₒ s = 0 := by\n  rw [OreLocalization.zero_def, oreDiv_eq_iff]\n  exact ⟨s, 1, by simp [Submonoid.smul_def]⟩\n\ninstance : MonoidWithZero R[S⁻¹] where\n  zero_mul x := by\n    induction x using OreLocalization.ind with | _ r s\n    rw [OreLocalization.zero_def, oreDiv_mul_char 0 r 1 s 0 1 (by simp), zero_mul, one_mul]\n  mul_zero x := by\n    induction x using OreLocalization.ind with | _ r s\n    rw [OreLocalization.zero_def, mul_div_one, mul_zero, zero_oreDiv', zero_oreDiv']\n\nTarget:\ntheorem subsingleton_iff :\n    Subsingleton R[S⁻¹] ↔ 0 ∈ S :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_cc3728dc6e72","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"8315a55c10b2ffead848a9c09be4c3e85b84a1b56269bde6ea574d30cbedf58e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/OreLocalization","family_id":"subsingleton_iff","file_id":"mathlib/Mathlib/RingTheory/OreLocalization/Basic.lean","sample_id":"cc3728dc6e72e50fd255371dff3e5f651d3ce2d9c638da326516235746fae11d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cde0cb2c54480a9a9ba9dd0ed55105a1ff6f413d177d544464139dccab437478","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a94a4e3a9c0eb0f38e2f97aea7bedf8bb571bea3024fe648b328ac2bab07bd36","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dfc0baea2eabd164611e1d1c93cf31cf2ae01a14f1445c0b6a91023c66e00ae4","source_sha256":"5d27fa86937e3368de02fb4ec1d31ef3389387a61446fa516bf2a6e60873be5c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  simp_rw [← AlgHom.coe_toRingHom, ← coeff_map, subProdXSubC, Polynomial.map_sub,\n    Polynomial.map_prod, Polynomial.map_sub, map_X, map_C, toSplittingField,\n    AlgHom.coe_toRingHom, MvPolynomial.aeval_X, dif_pos h,\n    ← (finEquivRoots (Monics.splits_finsetProd h)).symm.prod_comp, Equiv.apply_symm_apply]\n  rw [Finset.prod_coe_sort (f := fun x : _ × ℕ ↦ X - C x.1), (Multiset.toEnumFinset _)\n    |>.prod_eq_multiset_prod, ← Function.comp_def (X - C ·) Prod.fst, ← Multiset.map_map,\n    Multiset.map_toEnumFinset_fst, map_map, AlgHom.comp_algebraMap]\n  conv in map _ _ => rw [Splits.eq_prod_roots (Monics.splits_finsetProd h)]\n  rw [leadingCoeff_map, f.2, map_one, C_1, one_mul, sub_self, coeff_zero]\n\nvariable (k)","hard_negative":false,"metrics":{"chosen_tokens":165,"rejected_tokens":169,"token_jaccard":0.961538,"token_length_ratio":1.024242},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"eb3fec4fd73da96382b484712105da259587b9234422075ee99083db774f6dbb","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharP.Algebra\npublic import Mathlib.Data.Multiset.Fintype\npublic import Mathlib.FieldTheory.IsAlgClosed.Basic\npublic import Mathlib.FieldTheory.SplittingField.Construction\n\nNamespace:\nAlgebraicClosure\n\nLocal context:\n/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n/-!\n# Algebraic Closure\n\nIn this file we construct the algebraic closure of a field\n\n## Main Definitions\n\n- `AlgebraicClosure k` is an algebraic closure of `k` (in the same universe).\n  It is constructed by taking the polynomial ring generated by indeterminates\n  $X_{f,1}, \\dots, X_{f,\\deg f}$ corresponding to roots of monic irreducible\n  polynomials `f` with coefficients in `k`, and quotienting out by a maximal\n  ideal containing every $f - \\prod_i (X - X_{f,i})$. The proof follows\n  https://kconrad.math.uconn.edu/blurbs/galoistheory/algclosureshorter.pdf.\n\n## Tags\n\nalgebraic closure, algebraically closed\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nnoncomputable section\n\nopen Polynomial\n\nvariable (k : Type u) [Field k]\n\nnamespace AlgebraicClosure\n\n/-- The subtype of monic polynomials. -/\ndef Monics : Type u := {f : k[X] // f.Monic}\n\n/-- `Vars k` provides `n` variables $X_{f,1}, \\dots, X_{f,n}$ for each monic polynomial\n`f : k[X]` of degree `n`. -/\ndef Vars : Type u := Σ f : Monics k, Fin f.1.natDegree\n\nvariable {k} in\n/-- Given a monic polynomial `f : k[X]`,\n`subProdXSubC f` is the polynomial $f - \\prod_i (X - X_{f,i})$. -/\ndef subProdXSubC (f : Monics k) : (MvPolynomial (Vars k) k)[X] :=\n  f.1.map (algebraMap _ _) - ∏ i : Fin f.1.natDegree, (X - C (MvPolynomial.X ⟨f, i⟩))\n\n/-- The span of all coefficients of `subProdXSubC f` as `f` ranges all polynomials in `k[X]`. -/\ndef spanCoeffs : Ideal (MvPolynomial (Vars k) k) :=\n  Ideal.span <| Set.range fun fn : Monics k × ℕ ↦ (subProdXSubC fn.1).coeff fn.2\n\nvariable {k}\n\n/-- If a monic polynomial `f : k[X]` splits in `K`,\nthen it has as many roots (counting multiplicity) as its degree. -/\ndef finEquivRoots {K} [Field K] [DecidableEq K] {i : k →+* K} {f : Monics k}\n    (hf : (f.1.map i).Splits) : Fin f.1.natDegree ≃ (f.1.map i).roots.toEnumFinset :=\n  .symm <| Finset.equivFinOfCardEq <| by\n    rwa [splits_iff_card_roots,\n      ← Multiset.card_toEnumFinset, f.2.natDegree_map] at hf\n\nlemma Monics.splits_finsetProd {s : Finset (Monics k)} {f : Monics k} (hf : f ∈ s) :\n    (f.1.map (algebraMap k (SplittingField (∏ f ∈ s, f.1)))).Splits :=\n  (splits_prod_iff fun j _ ↦ map_ne_zero j.2.ne_zero).mp\n    (by simpa [Polynomial.map_prod] using SplittingField.splits (∏ f ∈ s, f.1)) f hf\n\nopen Classical in\n/-- Given a finite set of monic polynomials, construct an algebra homomorphism\nto the splitting field of the product of the polynomials\nsending indeterminates $X_{f_i}$ to the distinct roots of `f`. -/\ndef toSplittingField (s : Finset (Monics k)) :\n    MvPolynomial (Vars k) k →ₐ[k] SplittingField (∏ f ∈ s, f.1) :=\n  MvPolynomial.aeval fun fi ↦\n    if hf : fi.1 ∈ s then (finEquivRoots (Monics.splits_finsetProd hf) fi.2).1.1 else 37\n\nTarget:\ntheorem toSplittingField_coeff {s : Finset (Monics k)} {f} (h : f ∈ s) (n) :\n    toSplittingField s ((subProdXSubC f).coeff n) = 0 :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  classical\n  simp_rw [← AlgHom.coe_toRingHom, ← coeff_map, subProdXSubC, Polynomial.map_sub,\n    Polynomial.map_prod, Polynomial.map_sub, map_X, map_C, toSplittingField,\n    AlgHom.coe_toRingHom, MvPolynomial.aeval_X, dif_pos h,\n    ← (finEquivRoots (Monics.splits_finsetProd h)).symm.prod_comp, Equiv.apply_symm_apply]\n  rw [Finset.prod_coe_sort (f := fun x : _ × ℕ ↦ X - C x.1), (Multiset.toEnumFinset _)\n    |>.prod_eq_multiset_prod, ← Function.comp_def (X - C ·) Prod.fst, ← Multiset.map_map,\n    Multiset.map_toEnumFinset_fst, map_map, AlgHom.comp_algebraMap]\n  conv in map _ _ => rw [Splits.eq_prod_roots (Monics.splits_finsetProd h)]\n  rw [leadingCoeff_map, f.2, map_one, C_1, one_mul, sub_self, coeff_zero]\n\nvariable (k)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/IsAlgClosed","family_id":"tosplittingfield_coeff","file_id":"mathlib/Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean","sample_id":"dfc0baea2eabd164611e1d1c93cf31cf2ae01a14f1445c0b6a91023c66e00ae4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"851de4796bc656b1357eef23d637285576683be0e867f23ab28b2999ef31f864","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a95312207d0271d9516315b3dd214e2726915b26938f24e5ee114b92566a3ab3","source_sha256":"aa35c22bbec7297b71adc1698506626520835012db568036d1b21d7492380df0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  letI := (Sᵀ.toNormedAddCommGroup hS.transpose)\n  letI := (Sᵀ.toInnerProductSpace hS.transpose.posSemidef)\n  ext i j\n  rw [LDL.lowerInv, Basis.coePiBasisFun.toMatrix_eq_transpose, coe_gramSchmidtBasis]\n  rfl\n\nnoncomputable instance LDL.invertibleLowerInv : Invertible (LDL.lowerInv hS) := by\n  rw [LDL.lowerInv_eq_gramSchmidtBasis]\n  haveI :=\n    Basis.invertibleToMatrix (Pi.basisFun 𝕜 n)\n      (@gramSchmidtBasis 𝕜 (n → 𝕜) _ (Sᵀ.toNormedAddCommGroup hS.transpose)\n        (Sᵀ.toInnerProductSpace hS.transpose.posSemidef) n _ _ _ (Pi.basisFun 𝕜 n))\n  infer_instance","hard_negative":false,"metrics":{"chosen_tokens":119,"rejected_tokens":2,"token_jaccard":0.022222,"token_length_ratio":0.016807},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"eb4778c5ebae8996440892b5fa966fd54c78548577b3f955c595d04f699d89ab","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho\npublic import Mathlib.Analysis.Matrix.PosDef\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Alexander Bentkamp. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alexander Bentkamp\n-/\n/-! # LDL decomposition\n\nThis file proves the LDL-decomposition of matrices: Any positive definite matrix `S` can be\ndecomposed as `S = LDLᴴ` where `L` is a lower-triangular matrix and `D` is a diagonal matrix.\n\n## Main definitions\n\n* `LDL.lower` is the lower triangular matrix `L`.\n* `LDL.lowerInv` is the inverse of the lower triangular matrix `L`.\n* `LDL.diag` is the diagonal matrix `D`.\n\n## Main result\n\n* `LDL.lower_conj_diag` states that any positive definite matrix can be decomposed as `LDLᴴ`.\n\n## TODO\n\n* Prove that `LDL.lower` is lower triangular from `LDL.lowerInv_triangular`.\n\n-/\n\n@[expose] public section\n\nopen Module\n\nvariable {𝕜 : Type*} [RCLike 𝕜]\nvariable {n : Type*} [LinearOrder n] [WellFoundedLT n] [LocallyFiniteOrderBot n]\n\nsection set_options\n\nset_option quotPrecheck false\nlocal notation \"⟪\" x \", \" y \"⟫ₑ\" => inner 𝕜 (WithLp.toLp 2 x) (WithLp.toLp 2 y)\n\nopen Matrix InnerProductSpace\n\nopen scoped ComplexOrder\n\nvariable {S : Matrix n n 𝕜} [Fintype n] (hS : S.PosDef)\n\n/-- The inverse of the lower triangular matrix `L` of the LDL-decomposition. It is obtained by\napplying Gram-Schmidt-Orthogonalization w.r.t. the inner product induced by `Sᵀ` on the standard\nbasis vectors `Pi.basisFun`. -/\nnoncomputable def LDL.lowerInv : Matrix n n 𝕜 :=\n  @gramSchmidt 𝕜 (n → 𝕜) _ (Sᵀ.toNormedAddCommGroup hS.transpose)\n    (Sᵀ.toInnerProductSpace hS.transpose.posSemidef) n _ _ _ (Pi.basisFun 𝕜 n)\n\nTarget:\ntheorem LDL.lowerInv_eq_gramSchmidtBasis :\n    LDL.lowerInv hS =\n      ((Pi.basisFun 𝕜 n).toMatrix\n          (@gramSchmidtBasis 𝕜 (n → 𝕜) _ (Sᵀ.toNormedAddCommGroup hS.transpose)\n            (Sᵀ.toInnerProductSpace hS.transpose.posSemidef) n _ _ _ (Pi.basisFun 𝕜 n)))ᵀ :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Matrix","family_id":"ldl","file_id":"mathlib/Mathlib/Analysis/Matrix/LDL.lean","sample_id":"a95312207d0271d9516315b3dd214e2726915b26938f24e5ee114b92566a3ab3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"077e58b8ebfdb691115134cf203f73b4ed07755ad550ac73f5ae401c07fbed75","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e48c355b13d78c993332691198cc2ae1e2231f2b135a0dc55526ebee8e9f8c98","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bf8d6d5204ce084429052d3f687b1b4007057ecf29fbdc62aab2b93e39615bf4","source_sha256":"f37e97c4ba70607031e3a483123b1b4fa94a20ae8b550c03293099ec4ccab941","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine Measure.ext_of_charFunDual ?_\n  ext L\n  simp_rw [charFunDual_map, charFunDual_prod, charFunDual_eq_of_forall_strongDual_eq_zero hμ,\n    ← Complex.exp_add]\n  rw [← add_div, ← add_div, ← neg_add, ← neg_add]\n  congr 3\n  norm_cast\n  have h1 : (L.comp (.rotation θ)).comp (.inl ℝ E E)\n      = Real.cos θ • L.comp (.inl ℝ E E) - Real.sin θ • L.comp (.inr ℝ E E) := by\n    ext x\n    simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.inl_apply,\n      ContinuousLinearMap.rotation_apply, smul_zero, add_zero]\n    rw [← L.comp_inl_add_comp_inr]\n    simp [-neg_smul, sub_eq_add_neg]\n  have h2 : (L.comp (.rotation θ)).comp (.inr ℝ E E)\n      = Real.sin θ • L.comp (.inl ℝ E E) + Real.cos θ • L.comp (.inr ℝ E E) := by\n    ext x\n    simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.inr_apply,\n      ContinuousLinearMap.rotation_apply, smul_zero, zero_add, add_apply, smul_apply,\n      ContinuousLinearMap.inl_apply, smul_eq_mul]\n    rw [← L.comp_inl_add_comp_inr]\n    simp\n  rw [h1, h2]\n  simp only [FunLike.coe_sub, FunLike.coe_smul,\n    FunLike.coe_add]\n  rw [variance_sub, variance_smul, variance_add, variance_smul, variance_smul, covariance_smul_left,\n    covariance_smul_right, variance_smul, covariance_smul_left, covariance_smul_right]\n  · have h := Real.cos_sq_add_sin_sq θ\n    grind\n  all_goals exact (memLp_dual _ _ _ (by simp)).const_smul _","hard_negative":false,"metrics":{"chosen_tokens":294,"rejected_tokens":301,"token_jaccard":0.975904,"token_length_ratio":1.02381},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"eb6215e7a354d143f296d4919f5f9a4989098c8f631b655d1865a5f118552539","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Distributions.Fernique\npublic import Mathlib.Probability.Distributions.Gaussian.Basic\n\nNamespace:\nProbabilityTheory.IsGaussian\n\nLocal context:\n/-\nCopyright (c) 2025 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n-/\n/-!\n# Fernique's theorem for Gaussian measures\n\nWe show that the product of two identical Gaussian measures is invariant under rotation.\nWe then deduce Fernique's theorem, which states that for a Gaussian measure `μ`, there exists\n`C > 0` such that the function `x ↦ exp (C * ‖x‖ ^ 2)` is integrable with respect to `μ`.\nAs a consequence, a Gaussian measure has finite moments of all orders.\n\n## Main statements\n\n* `IsGaussian.exists_integrable_exp_sq`: **Fernique's theorem**. For a Gaussian measure on a\n  second-countable normed space, there exists `C > 0` such that the function\n  `x ↦ exp (C * ‖x‖ ^ 2)` is integrable.\n* `IsGaussian.memLp_id`: a Gaussian measure in a second-countable Banach space has finite moments\n  of all orders.\n\n## References\n\n* [Martin Hairer, *An introduction to stochastic PDEs*][hairer2009introduction]\n\n-/\n\npublic section\n\nopen MeasureTheory ProbabilityTheory Complex\nopen scoped ENNReal NNReal Real Topology\n\nnamespace ProbabilityTheory.IsGaussian\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E]\n  {μ : Measure E} [IsGaussian μ]\n\nsection Rotation\n\n/-- Characteristic function of a centered Gaussian measure.\nFor a Gaussian measure, the hypothesis `∀ L : StrongDual ℝ E, μ[L] = 0` is equivalent to the simpler\n`μ[id] = 0`, but at this point we don't know yet that `μ` has a first moment so we can't use it.\nSee `charFunDual_eq_of_integral_eq_zero` -/\nlemma charFunDual_eq_of_forall_strongDual_eq_zero (hμ : ∀ L : StrongDual ℝ E, μ[L] = 0)\n    (L : StrongDual ℝ E) :\n    charFunDual μ L = exp (- Var[L; μ] / 2) := by\n  simp [charFunDual_eq L, integral_complex_ofReal, hμ L, neg_div]\n\n/-- For a centered Gaussian measure `μ`, the product measure `μ.prod μ` is invariant under rotation.\nThe hypothesis `∀ L : StrongDual ℝ E, μ[L] = 0` is equivalent to the simpler\n`μ[id] = 0`, but at this point we don't know yet that `μ` has a first moment so we can't use it.\nSee `map_rotation_eq_self`. -/\n\nTarget:\nlemma map_rotation_eq_self_of_forall_strongDual_eq_zero\n    [SecondCountableTopology E] [CompleteSpace E]\n    (hμ : ∀ L : StrongDual ℝ E, μ[L] = 0) (θ : ℝ) :\n    (μ.prod μ).map (ContinuousLinearMap.rotation θ) = μ.prod μ :=\n\nProof body:\n","rejected":"```lean\nby\n  refine Measure.ext_of_charFunDual ?_\n  ext L\n  simp_rw [charFunDual_map, charFunDual_prod, charFunDual_eq_of_forall_strongDual_eq_zero hμ,\n    ← Complex.exp_add]\n  rw [← add_div, ← add_div, ← neg_add, ← neg_add]\n  congr 3\n  norm_cast\n  have h1 : (L.comp (.rotation θ)).comp (.inl ℝ E E)\n      = Real.cos θ • L.comp (.inl ℝ E E) - Real.sin θ • L.comp (.inr ℝ E E) := by\n    ext x\n    simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.inl_apply,\n      ContinuousLinearMap.rotation_apply, smul_zero, add_zero]\n    rw [← L.comp_inl_add_comp_inr]\n    simp [-neg_smul, sub_eq_add_neg]\n  have h2 : (L.comp (.rotation θ)).comp (.inr ℝ E E)\n      = Real.sin θ • L.comp (.inl ℝ E E) + Real.cos θ • L.comp (.inr ℝ E E) := by\n    ext x\n    simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.inr_apply,\n      ContinuousLinearMap.rotation_apply, smul_zero, zero_add, add_apply, smul_apply,\n      ContinuousLinearMap.inl_apply, smul_eq_mul]\n    rw [← L.comp_inl_add_comp_inr]\n    simp\n  rw [h1, h2]\n  simp only [FunLike.coe_sub, FunLike.coe_smul,\n    FunLike.coe_add]\n  rw [variance_sub, variance_smul, variance_add, variance_smul, variance_smul, covariance_smul_left,\n    covariance_smul_right, variance_smul, covariance_smul_left, covariance_smul_right]\n  · have h := Real.cos_sq_add_sin_sq θ\n    grind\n  all_goals exact (memLp_dual _ _ _ (by simp)).const_smul _\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Distributions","family_id":"map_rotation_eq_self_of_forall_strongdual_eq_zero","file_id":"mathlib/Mathlib/Probability/Distributions/Gaussian/Fernique.lean","sample_id":"bf8d6d5204ce084429052d3f687b1b4007057ecf29fbdc62aab2b93e39615bf4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d3371ad1ae3b9f3bec6391019a5fb1c62af2682915c2a167e06751160d75c560","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f5c3570decc7db710ef7c794c68d9c0b62e8e6967dc4c3bfe5c338cfbcf456b8","source_sha256":"f6bedc69ab0e67353ba4b185fe6aec08e361441cb856f5f803e100c5cd160384","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hb' : (0 : ℝ) < ↑(abs b) := Int.cast_pos.mpr (abs_pos.mpr hb)\n  have hbε : 0 < abs b • ε := by\n    rw [Algebra.smul_def]\n    exact mul_pos hb' hε\n  have hfloor : ∀ i, 0 ≤ floor ((A i % b : ℤ) / abs b • ε : ℝ) :=\n    fun _ ↦ floor_nonneg.mpr (div_nonneg (cast_nonneg (emod_nonneg _ hb)) hbε.le)\n  refine ⟨fun i ↦ ⟨natAbs (floor ((A i % b : ℤ) / abs b • ε : ℝ)), ?_⟩, ?_⟩\n  · rw [← ofNat_lt, natAbs_of_nonneg (hfloor i), floor_lt, Algebra.smul_def, eq_intCast, ← div_div]\n    apply lt_of_lt_of_le _ (Nat.le_ceil _)\n    gcongr\n    rw [div_lt_one hb', cast_lt]\n    exact Int.emod_lt_abs _ hb\n  intro i₀ i₁ hi\n  have hi : (⌊↑(A i₀ % b) / abs b • ε⌋.natAbs : ℤ) = ⌊↑(A i₁ % b) / abs b • ε⌋.natAbs :=\n    congr_arg ((↑) : ℕ → ℤ) (Fin.mk_eq_mk.mp hi)\n  rw [natAbs_of_nonneg (hfloor i₀), natAbs_of_nonneg (hfloor i₁)] at hi\n  have hi := abs_sub_lt_one_of_floor_eq_floor hi\n  rw [abs_sub_comm, ← sub_div, abs_div, abs_of_nonneg hbε.le, div_lt_iff₀ hbε, one_mul] at hi\n  rwa [Int.cast_abs, Int.cast_sub]\n\n/-- `abs : ℤ → ℤ` is an admissible absolute value. -/\nnoncomputable def absIsAdmissible : IsAdmissible AbsoluteValue.abs :=\n  { AbsoluteValue.abs_isEuclidean with\n    card := fun ε ↦ ⌈1 / ε⌉₊\n    exists_partition' := fun n _ hε _ hb ↦ exists_partition_int n hε hb }\n\nnoncomputable instance : Inhabited (IsAdmissible AbsoluteValue.abs) :=\n  ⟨absIsAdmissible⟩","hard_negative":false,"metrics":{"chosen_tokens":376,"rejected_tokens":5,"token_jaccard":0.02521,"token_length_ratio":0.013298},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"eb82db3972614fd1a12feaf2694e4d6711a28ff6374c54141a44eff3f057b946","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Basic\npublic import Mathlib.Algebra.Order.Archimedean.Real.Basic\npublic import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue\n\nNamespace:\nAbsoluteValue\n\nLocal context:\n/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n/-!\n# Admissible absolute value on the integers\nThis file defines an admissible absolute value `AbsoluteValue.absIsAdmissible`\nwhich we use to show the class number of the ring of integers of a number field\nis finite.\n\n## Main results\n\n* `AbsoluteValue.absIsAdmissible` shows the \"standard\" absolute value on `ℤ`,\n  mapping negative `x` to `-x`, is admissible.\n-/\n\n@[expose] public section\n\n\nnamespace AbsoluteValue\n\nopen Int\n\n/-- We can partition a finite family into `partition_card ε` sets, such that the remainders\nin each set are close together. -/\n\nTarget:\ntheorem exists_partition_int (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : ℤ} (hb : b ≠ 0) (A : Fin n → ℤ) :\n    ∃ t : Fin n → Fin ⌈1 / ε⌉₊,\n    ∀ i₀ i₁, t i₀ = t i₁ → ↑(abs (A i₁ % b - A i₀ % b)) < abs b • ε :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/ClassNumber","family_id":"exists_partition_int","file_id":"mathlib/Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean","sample_id":"f5c3570decc7db710ef7c794c68d9c0b62e8e6967dc4c3bfe5c338cfbcf456b8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e483053df1cef9a9899f381185253ce0593370fe3c8c56da6f87d63a4270c28a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c12cbb39a416f92d3d0c274fc8bf118a1a15c55c3fcf584f88dc89020d197fff","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"81608c9debe142f31aa7efa37f974504e0e8f8ceff0f0397335ad80e4df5060e","source_sha256":"29f6ed4519c38f3059df6084770133fafb9cbcb638c0482c74b9751c669ac0d9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [localization]\n  simp only [Localization.liftNatTrans_app, Functor.id_obj, Functor.whiskerRight_app,\n    NatTrans.naturality, Functor.comp_map, Functor.id_map, Iso.hom_inv_id_app_assoc]\n\ninclude W","hard_negative":true,"metrics":{"chosen_tokens":38,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.052632},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"eb86e79b2950c3380aa89f2ee59348bb338b32d1c84232b24246fbe1b64042f6","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Center.Preadditive\npublic import Mathlib.CategoryTheory.Localization.Predicate\npublic import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor\n\nNamespace:\nCategoryTheory.CatCenter\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Localization of the center of a category\n\nGiven a localization functor `L : C ⥤ D` with respect to `W : MorphismProperty C`,\nwe define a localization map `CatCenter C → CatCenter D` for the centers\nof these categories. In case `L` is an additive functor between preadditive\ncategories, we promote this to a ring morphism `CatCenter C →+* CatCenter D`.\n\n-/\n\n@[expose] public section\n\nuniverse w v₁ v₂ u₁ u₂\n\nnamespace CategoryTheory\n\nvariable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]\n  (r s : CatCenter C) (L : C ⥤ D) (W : MorphismProperty C) [L.IsLocalization W]\n\nnamespace CatCenter\n\n/-- Given `r : CatCenter C` and `L : C ⥤ D` a localization functor with respect\nto `W : MorphismProperty D`, this is the induced element in `CatCenter D`\nobtained by localization. -/\nnoncomputable def localization : CatCenter D :=\n  Localization.liftNatTrans L W L L (𝟭 D) (𝟭 D) (Functor.whiskerRight r L)\n\n@[simp]\n\nTarget:\nlemma localization_app (X : C) :\n    (r.localization L W).app (L.obj X) = L.map (r.app X) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_81608c9debe1","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b2623d2650365934678f7940e494b7457db9b2f160a04704689df2836048eb34","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Center","family_id":"localization_app","file_id":"mathlib/Mathlib/CategoryTheory/Center/Localization.lean","sample_id":"81608c9debe142f31aa7efa37f974504e0e8f8ceff0f0397335ad80e4df5060e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3dbc0fabaa57897ff1e49ce5f7faadddfa1be34ceeabe2d7c9fe2b87f820ba31","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3087fe77077a4804fd4adc39f3b2f67cf5b7a70da11ffe0c839d1c6a058f86e4","source_sha256":"326a725d48085820c8ea061cea860b42ee76be144d5f7dbc1de5f2d04ba00c0a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine .mk _ Etale.respectsIso fun R S T _ _ _ _ _ h h' ↦ ?_\n  rw [etale_algebraMap] at h' ⊢\n  rw [faithfullyFlat_algebraMap_iff] at h\n  exact .of_etale_tensorProduct_of_faithfullyFlat S","hard_negative":false,"metrics":{"chosen_tokens":39,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.051282},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"eb93870a44729af10ae612e457be11de757a19bf8b0c561a89d091ea074a365d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.RingHom.Etale\npublic import Mathlib.RingTheory.Finiteness.Descent\npublic import Mathlib.RingTheory.Extension.Cotangent.BaseChange\n\nNamespace:\nRingHom\n\nLocal context:\n/-\nCopyright (c) 2026 Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten\n-/\n/-!\n# Etale descends along faithfully flat ring maps\n\nIn this file we show that smooth, unramified and étale algebras descend along faithfully flat\nbase change.\n\n## Main results\n\n- `Algebra.Smooth.of_smooth_tensorProduct_of_faithfullyFlat`: Smooth descends.\n- `Algebra.Unramified.of_smooth_tensorProduct_of_faithfullyFlat`: Unramified descends.\n- `Algebra.Etale.of_etale_tensorProduct_of_faithfullyFlat`: Etale descends.\n\nWe also provide the corresponding `RingHom.CodescendsAlong` lemmas.\n\n## TODOs\n\n- The lemma `Algebra.FormallySmooth.of_formallySmooth_tensorProduct_of_faithfullyFlat` has an\n  additional `Algebra.FinitePresentation` assumption, because the proof uses that a flat module\n  of finite presentation is projective and the former descends. This also holds without\n  the finite presentation assumption, but requires showing that projectivity descends\n  along faithfully flat base change, which is due to Raynaud and Gruson\n  (see https://stacks.math.columbia.edu/tag/058B).\n-/\n\npublic section\n\nopen TensorProduct\n\nnamespace Algebra\n\nvariable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]\nvariable (T : Type*) [CommRing T] [Algebra R T] [Module.FaithfullyFlat R T]\n\nlemma FormallyUnramified.of_formallyUnramified_tensorProduct_of_faithfullyFlat\n    [FormallyUnramified T (T ⊗[R] S)] :\n    FormallyUnramified R S := by\n  constructor\n  let _ : Algebra S (T ⊗[R] S) := TensorProduct.rightAlgebra\n  have : Subsingleton (T ⊗[R] Ω[S⁄R]) :=\n    (KaehlerDifferential.tensorKaehlerEquivBase R T S (T ⊗[R] S)).subsingleton\n  exact Module.FaithfullyFlat.lTensor_reflects_triviality R T _\n\n/-- Formally smooth algebras descend along faithfully flat base change. See the TODO\nin the module docstring. -/\nproof_wanted FormallySmooth.of_formallySmooth_tensorProduct_of_faithfullyFlat\n    {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]\n    (T : Type*) [CommRing T] [Algebra R T] [Module.FaithfullyFlat R T]\n    [FormallySmooth T (T ⊗[R] S)] :\n    FormallySmooth R S\n\nlemma Smooth.of_smooth_tensorProduct_of_faithfullyFlat [Smooth T (T ⊗[R] S)] :\n    Smooth R S := by\n  have : Algebra.FinitePresentation R S := .of_finitePresentation_tensorProduct_of_faithfullyFlat T\n  refine ⟨?_, .of_finitePresentation_tensorProduct_of_faithfullyFlat T⟩\n  rw [formallySmooth_iff]\n  constructor\n  · let _ : Algebra T (S ⊗[R] T) := TensorProduct.rightAlgebra\n    let e : S ⊗[R] T ≃ₐ[T] T ⊗[R] S :=\n      .ofRingEquiv (f := TensorProduct.comm R S T) <| by simp [RingHom.algebraMap_toAlgebra]\n    have : FormallySmooth T (S ⊗[R] T) := .of_equiv e.symm\n    let e' : (S ⊗[R] T) ⊗[S] Ω[S⁄R] ≃ₗ[S ⊗[R] T] Ω[S ⊗[R] T⁄T] :=\n      KaehlerDifferential.tensorKaehlerEquiv R T S (S ⊗[R] T)\n    have : Module.Flat (S ⊗[R] T) ((S ⊗[R] T) ⊗[S] Ω[S⁄R]) := .of_linearEquiv e'\n    have : Module.Flat S Ω[S⁄R] := Module.Flat.of_flat_tensorProduct _ _ (S ⊗[R] T)\n    exact Module.Flat.projective_of_finitePresentation\n  · have : Subsingleton (T ⊗[R] H1Cotangent R S) := (tensorH1CotangentOfFlat R S T).subsingleton\n    exact Module.FaithfullyFlat.lTensor_reflects_triviality R T (H1Cotangent R S)\n\nlemma Unramified.of_unramified_tensorProduct_of_faithfullyFlat [Unramified T (T ⊗[R] S)] :\n    Unramified R S :=\n  ⟨.of_formallyUnramified_tensorProduct_of_faithfullyFlat T,\n    .of_finiteType_tensorProduct_of_faithfullyFlat T⟩\n\nlemma Etale.of_etale_tensorProduct_of_faithfullyFlat [Etale T (T ⊗[R] S)] :\n    Etale R S := by\n  rw [Etale.iff_formallyUnramified_and_smooth]\n  exact ⟨.of_formallyUnramified_tensorProduct_of_faithfullyFlat T,\n    .of_smooth_tensorProduct_of_faithfullyFlat T⟩\n\nend Algebra\n\nnamespace RingHom\n\nlemma Smooth.codescendsAlong_faithfullyFlat : CodescendsAlong Smooth FaithfullyFlat := by\n  refine .mk _ Smooth.respectsIso fun R S T _ _ _ _ _ h h' ↦ ?_\n  rw [smooth_algebraMap] at h' ⊢\n  rw [faithfullyFlat_algebraMap_iff] at h\n  exact .of_smooth_tensorProduct_of_faithfullyFlat S\n\nlemma FormallyUnramified.codescendsAlong_faithfullyFlat :\n    CodescendsAlong FormallyUnramified FaithfullyFlat := by\n  refine .mk _ FormallyUnramified.respectsIso fun R S T _ _ _ _ _ h h' ↦ ?_\n  rw [formallyUnramified_algebraMap] at h' ⊢\n  rw [faithfullyFlat_algebraMap_iff] at h\n  exact .of_formallyUnramified_tensorProduct_of_faithfullyFlat S\n\nTarget:\nlemma Etale.codescendsAlong_faithfullyFlat : CodescendsAlong Etale FaithfullyFlat :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Etale","family_id":"etale","file_id":"mathlib/Mathlib/RingTheory/Etale/Descent.lean","sample_id":"3087fe77077a4804fd4adc39f3b2f67cf5b7a70da11ffe0c839d1c6a058f86e4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"105decf31a73d8f03c28fa8bbcf536ea59b795c7d4d6c1c8dd7fade7fd71c4b3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2ecf221e60a1de5319a947fdeadd18115cd6ed88d23420eaeb85a40fe0314ea2","source_sha256":"46b1b7b1e57c2ee22dceb33ca3f6863e9e50ed75099a163a5900686dabb9e572","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨K * ε * K, mul_pos (mul_pos K0 ε0) K0, fun {a b} ha hb h => ?_⟩\n  have a0 := K0.trans_le ha\n  have b0 := K0.trans_le hb\n  rw [inv_sub_inv' ((abv_pos abv).1 a0) ((abv_pos abv).1 b0), abv_mul abv, abv_mul abv, abv_inv abv,\n    abv_inv abv, abv_sub abv]\n  refine lt_of_mul_lt_mul_left (lt_of_mul_lt_mul_right ?_ b0.le) a0.le\n  rw [mul_assoc, inv_mul_cancel_right₀ b0.ne', ← mul_assoc, mul_inv_cancel₀ a0.ne', one_mul]\n  refine h.trans_le ?_\n  gcongr\n  exact mul_nonneg a0.le ε0.le","hard_negative":false,"metrics":{"chosen_tokens":131,"rejected_tokens":3,"token_jaccard":0.037037,"token_length_ratio":0.022901},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"ebe7d5c250a824506d86b6ec2b7ce46c08b6730d51e2ea6fb6a628ec3b75bd64","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Pi\npublic import Mathlib.Algebra.Order.AbsoluteValue.Basic\npublic import Mathlib.Algebra.Order.Field.Basic\npublic import Mathlib.Algebra.Order.Group.MinMax\npublic import Mathlib.Algebra.Ring.Pi\npublic import Mathlib.Data.Setoid.Basic\npublic import Mathlib.GroupTheory.GroupAction.Ring\npublic import Mathlib.Tactic.GCongr\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Cauchy sequences\n\nA basic theory of Cauchy sequences, used in the construction of the reals and p-adic numbers. Where\napplicable, lemmas that will be reused in other contexts have been stated in extra generality.\nThere are other \"versions\" of Cauchyness in the library, in particular Cauchy filters in topology.\nThis is a concrete implementation that is useful for simplicity and computability reasons.\n\n## Important definitions\n\n* `IsCauSeq`: a predicate that says `f : ℕ → β` is Cauchy.\n* `CauSeq`: the type of Cauchy sequences valued in type `β` with respect to an absolute value\n  function `abv`.\n\n## Tags\n\nsequence, cauchy, abs val, absolute value\n-/\n\n@[expose] public section\n\nassert_not_exists Finset Module Submonoid FloorRing\n\nvariable {α β : Type*}\n\nopen IsAbsoluteValue\n\nsection\n\nvariable [Field α] [LinearOrder α] [IsStrictOrderedRing α] [Ring β]\n  (abv : β → α) [IsAbsoluteValue abv]\n\ntheorem rat_add_continuous_lemma {ε : α} (ε0 : 0 < ε) :\n    ∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv (a₁ - b₁) < δ → abv (a₂ - b₂) < δ →\n      abv (a₁ + a₂ - (b₁ + b₂)) < ε :=\n  ⟨ε / 2, half_pos ε0, fun {a₁ a₂ b₁ b₂} h₁ h₂ => by\n    simpa [add_halves, sub_eq_add_neg, add_comm, add_left_comm, add_assoc] using\n      lt_of_le_of_lt (abv_add abv _ _) (add_lt_add h₁ h₂)⟩\n\ntheorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :\n    ∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →\n      abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε := by\n  have K0 : (0 : α) < max 1 (max K₁ K₂) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)\n  have εK := div_pos (half_pos ε0) K0\n  refine ⟨_, εK, fun {a₁ a₂ b₁ b₂} ha₁ hb₂ h₁ h₂ => ?_⟩\n  replace ha₁ := lt_of_lt_of_le ha₁ (le_trans (le_max_left _ K₂) (le_max_right 1 _))\n  replace hb₂ := lt_of_lt_of_le hb₂ (le_trans (le_max_right K₁ _) (le_max_right 1 _))\n  set M := max 1 (max K₁ K₂)\n  have : abv (a₁ - b₁) * abv b₂ + abv (a₂ - b₂) * abv a₁ < ε / 2 / M * M + ε / 2 / M * M := by\n    gcongr\n  rw [← abv_mul abv, mul_comm, div_mul_cancel₀ _ (ne_of_gt K0), ← abv_mul abv, add_halves] at this\n  simpa [sub_eq_add_neg, mul_add, add_mul, add_left_comm] using\n    lt_of_le_of_lt (abv_add abv _ _) this\n\nTarget:\ntheorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]\n    {ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :\n    ∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Order","family_id":"rat_inv_continuous_lemma","file_id":"mathlib/Mathlib/Algebra/Order/CauSeq/Basic.lean","sample_id":"2ecf221e60a1de5319a947fdeadd18115cd6ed88d23420eaeb85a40fe0314ea2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0bd39f4463d7de80fb25819949e6b30d4389ae1eaeb6979f0ff98507bcbb9bbf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"858a6efb9e7b2421f0ba9ed1937a4f310af7de5c57c401fa7766f953dbd930e5","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f97af12d0eda400dcad66bb72ae73c904310f1b01ffca3c206a7820ced091c83","source_sha256":"a2d058104de73af2ef7925ee923e8edeb8d6ca8dc9be066178103a6d13175051","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cat_disch\n\nvariable {D : Type u'} [𝒟 : Category.{u'} D] (I J : D ⥤ C) (ρ : I ⟶ J) {W : D}","hard_negative":true,"metrics":{"chosen_tokens":39,"rejected_tokens":3,"token_jaccard":0.04,"token_length_ratio":0.076923},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"ec14c38aace76d9a3771dc710abd0d3e27f84d637c896af208c67309c3629161","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Elementwise\npublic import Mathlib.CategoryTheory.EpiMono\npublic import Mathlib.Tactic.PPWithUniv\npublic import Mathlib.Tactic.ToAdditive\n\nNamespace:\nCategoryTheory.FunctorToTypes\n\nLocal context:\n/-\nCopyright (c) 2017 Kim Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Kim Morrison, Johannes Hölzl, Dagur Asgeirsson\n-/\n/-!\n# The category `Type`.\n\nIn this section we define a `LargeCategory` structure on `Type u`, in such a way that it becomes a\n`ConcreteCategory`.\n\n## Implementation\n\nWe define the one-field structure `TypeCat.Fun` to wrap a function between types, and a `FunLike`\ninstance on it. Then we define a one-field structure `TypeCat.Hom` which wraps a `Fun`. The\nmorphisms in the category `Type u` are defined to be `TypeCat.Hom`, and the `FC` parameter of\nthe `ConcreteCategory` instance is `TypeCat.Fun`. `TypeCat.Fun` serves as a layer of separation\nbetween the `FC` parameter of the `ConcreteCategory` instance and bare functions, to avoid defining\na `FunLike` instance on the latter (which would give two non-reducibly defeq coercions from\nmorphisms in `Type` to functions), and the outer nesting `TypeCat.Hom` gives a layer of separation\nbetween morphisms and `FC`, as is done for all concrete categories in mathlib.\n\nTo promote a function to a morphism in this category, we provide the abbreviation `↾f`,\nas well as a corresponding notation `↾f`. (Entered as `\\upr `.)\n\n## Main definitions\n\nWe define `uliftFunctor`, from `Type u` to `Type (max u v)`, and show that it is fully faithful\n(but not, of course, essentially surjective).\n\nWe prove some basic facts about the category `Type`:\n*  epimorphisms are surjections and monomorphisms are injections,\n* `Iso` is both `Iso` and `Equiv` to `Equiv` (at least within a fixed universe),\n* every type level `IsLawfulFunctor` gives a categorical functor `Type ⥤ Type`\n  (the corresponding fact about monads is in `Mathlib/CategoryTheory/Monad/Types.lean`).\n-/\n\n@[expose] public section\n\n-- morphism levels before object levels. See note [category theory universes].\nuniverse v w u u'\n\nnamespace TypeCat\n\n/-- A one-field structure wrapping a function between types. -/\n@[ext]\nstructure Fun (X Y : Type*) where\n  /-- The underlying function. -/\n  toFun : X → Y\n\ninstance instFunLikeFun {X Y : Type*} : FunLike (Fun X Y) X Y where\n  coe f x := f.toFun x\n  coe_injective _ := by aesop\n\ninitialize_simps_projections Fun (toFun → apply)\n\nlemma Fun.mk_apply {X Y : Type*} (f : X → Y) (x : X) : (Fun.mk f) x = f x :=\n  rfl\n\n@[simp]\nlemma Fun.coe_mk {X Y : Type*} (f : X → Y) : (Fun.mk f : X → Y) = f :=\n  rfl\n\n/-- The identity function as a `Fun`. -/\n@[simps! +dsimpLhs]\ndef Fun.id (X : Type*) : Fun X X := Fun.mk _root_.id\n\n/-- Composition of `Fun`s. -/\n@[simps! +dsimpLhs]\ndef Fun.comp {X Y Z : Type*} (f : Fun Y Z) (g : Fun X Y) : Fun X Z := mk (f.toFun ∘ g.toFun)\n\n/-- The equivalence between `Fun`s and functions between types. -/\ndef Fun.homEquiv (X Y : Type u) : (Fun X Y) ≃ (X → Y) where\n  toFun f := f\n  invFun f := ⟨f⟩\n  left_inv := by intro; rfl\n  right_inv := by intro; rfl\n\n/-- The type of morphisms in `Type`. -/\n@[ext]\nstructure Hom (X Y : Type u) where\n  private mk ::\n  /-- The underlying function -/\n  hom' : Fun X Y\n\nend TypeCat\n\nopen TypeCat CategoryTheory\n\nset_option backward.privateInPublic true in\n@[to_additive_do_translate] -- Expressions involving this instance can still be additivized.\ninstance CategoryTheory.types : Category.{u} (Type u) where\n  Hom := Hom\n  id X := .mk <| .id X\n  comp f g := .mk <| g.hom'.comp f.hom'\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/--\nThe concrete category instance on `Type u`.\n\nNote: sometimes one needs to specify explicitly `(CC := fun X ↦ X)` to help typeclass inference.\n-/\ninstance : ConcreteCategory.{u} (Type u) Fun where\n  hom := Hom.hom'\n  ofHom := Hom.mk\n\nexample (X Y : Type u) (f : X ⟶ Y) : (f : X → Y) = (ConcreteCategory.hom f : X → Y) := by\n  with_reducible rfl\n\nexample (X Y : Type u) (f : X ⟶ Y) (x : X) : f x = (f : X → Y) x := by\n  with_reducible rfl\n\nexample (X Y : Type*) (f : Fun X Y) : (f : X → Y) = f := by\n  with_reducible rfl\n\nexample (X Y : Type*) (f : Fun X Y) (x : X) : f x = (f : X → Y) x := by\n  with_reducible rfl\n\nnamespace TypeCat\n\n/-- Turn a morphism in `Type` back into a function. -/\nabbrev Hom.hom {X Y : Type u} (f : Hom X Y) : Fun X Y :=\n  ConcreteCategory.hom (C := Type u) f\n\n/-- Typecheck a function as a morphism in `Type`. -/\nabbrev ofHom {X Y : Type u} (f : X → Y) : X ⟶ Y :=\n  ConcreteCategory.ofHom (Fun.mk f)\n\nend TypeCat\n\nnamespace CategoryTheory\n\n@[inherit_doc]\nscoped notation \"↾\" f:200 => TypeCat.ofHom f\n\nend CategoryTheory\n\nnamespace TypeCat\n\n/-- Use the `ConcreteCategory.hom` projection for `@[simps]` lemmas. -/\ndef Hom.Simps.hom (X Y : Type u) (f : X ⟶ Y) :=\n  ConcreteCategory.hom f\n\ninitialize_simps_projections Hom (hom' → hom)\n\n@[simp]\nlemma Fun.toFun_apply {X Y : Type u} (f : Fun X Y) (x : X) : f.toFun x = f x :=\n  rfl\n\nexample (X : Type u) : CategoryTheory.ToType X = X := by with_reducible rfl\n\n@[simp]\nlemma ofHom_eq {X Y : Type u} (f : X ⟶ Y) : ofHom f = f :=\n  rfl\n\n@[simp high]\nlemma hom_ofHom {X Y : Type u} (f : X → Y) : Hom.hom (ofHom f) = Fun.mk f := rfl\n\n@[simp]\nlemma ofHom_hom {X Y : Type u} (f : X ⟶ Y) : ofHom (Hom.hom f) = f := rfl\n\n@[simp]\nlemma ofHom_apply {X Y : Type u} (f : X → Y) (x : X) :\n    (↾f) x = f x :=\n  rfl\n\n/-- `TypeCat.Hom.hom` bundled as an `Equiv`. -/\ndef homEquiv {X Y : Type u} : (X ⟶ Y) ≃ (X → Y) :=\n  (ConcreteCategory.homEquiv (C := Type u)).trans (Fun.homEquiv _ _)\n\n@[simp]\nlemma homEquiv_apply {X Y : Type u} (f : X ⟶ Y) :\n    homEquiv f = f :=\n  rfl\n\n@[simp]\nlemma homEquiv_symm_apply {X Y : Type u} (f : X → Y) :\n    homEquiv.symm f = ofHom f :=\n  rfl\n\nlemma congr_arg {X Y : Type u} (f : X ⟶ Y) {x x' : X} (h : x = x') : f x = f x' := by\n  rw [h]\n\nend TypeCat\n\nnamespace CategoryTheory\n\ntheorem types_id (X : Type u) : (𝟙 X : _ → _) = id :=\n  rfl\n\ntheorem types_comp {X Y Z : Type u} (f : X ⟶ Y) (g : Y ⟶ Z) :\n    ConcreteCategory.hom (f ≫ g) = g ∘ f :=\n  rfl\n\n@[simp]\nlemma types_id_apply (X : Type u) (x : X) : 𝟙 X x = x :=\n  rfl\n\n@[simp]\nlemma types_comp_apply {X Y Z : Type u} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :\n    (f ≫ g) x = g (f x) :=\n  rfl\n\n@[congr]\nlemma types_congr_hom {X Y : Type u} {f g : X ⟶ Y} (h : f = g) (x : X) : f x = g x :=\n  ConcreteCategory.congr_hom h x\n\n@[deprecated (since := \"2026-02-09\")] alias hom_inv_id_apply := Iso.hom_inv_id_apply\n@[deprecated (since := \"2026-02-09\")] alias inv_hom_id_apply := Iso.inv_hom_id_apply\n@[deprecated (since := \"2026-02-09\")] alias asHom := ofHom\n\nnamespace Functor\n\nvariable {J : Type u} [Category.{v} J]\n\n/-- The sections of a functor `F : J ⥤ Type` are\nthe choices of a point `u j : F.obj j` for each `j`,\nsuch that `F.map f (u j) = u j'` for every morphism `f : j ⟶ j'`.\n\nWe later use these to define limits in `Type` and in many concrete categories.\n-/\ndef sections (F : J ⥤ Type w) : Set (∀ j, F.obj j) :=\n  { u | ∀ {j j'} (f : j ⟶ j'), F.map f (u j) = u j' }\n\n@[simp]\nlemma sections_property {F : J ⥤ Type w} (s : F.sections)\n    {j j' : J} (f : j ⟶ j') : F.map f (s.val j) = s.val j' :=\n  s.property f\n\nlemma sections_ext_iff {F : J ⥤ Type w} {x y : F.sections} : x = y ↔ ∀ j, x.val j = y.val j :=\n  Subtype.ext_iff.trans funext_iff\n\nvariable (J)\n\n/-- The functor which sends a functor to types to its sections. -/\n@[simps]\ndef sectionsFunctor : (J ⥤ Type w) ⥤ Type max u w where\n  obj F := F.sections\n  map {F G} φ := ↾fun x ↦ ⟨fun j => φ.app j (x.1 j), fun {j j'} f =>\n    by simp [← NatTrans.naturality_apply, x.2 f]⟩\n\nend Functor\n\nnamespace FunctorToTypes\n\nvariable {C : Type u} [Category.{v} C] (F G H : C ⥤ Type w) {X Y Z : C}\nvariable (σ : F ⟶ G) (τ : G ⟶ H)\n\nattribute [elementwise nosimp] Functor.map_comp Functor.map_id NatTrans.comp_app\n\n@[deprecated Functor.map_comp_apply (since := \"2026-03-09\")]\ntheorem map_comp_apply (f : X ⟶ Y) (g : Y ⟶ Z) (a : F.obj X) :\n    (F.map (f ≫ g)) a = (F.map g) ((F.map f) a) :=\n  F.map_comp_apply f g a\n\n@[deprecated Functor.map_id_apply (since := \"2026-03-09\")]\ntheorem map_id_apply (a : F.obj X) : (F.map (𝟙 X)) a = a :=\n  F.map_id_apply X a\n\n@[deprecated (since := \"2026-02-09\")] alias naturality := NatTrans.naturality_apply\n\n@[deprecated NatTrans.comp_app_apply (since := \"2026-03-09\")]\ntheorem comp (x : F.obj X) : (σ ≫ τ).app X x = τ.app X (σ.app X x) :=\n  σ.comp_app_apply τ X x\n\nattribute [elementwise (attr := simp)] eqToHom_map_comp\n\n@[deprecated \"Use `elementwise_of% eqToHom_map_comp` instead\" (since := \"2026-02-09\")]\n\nTarget:\ntheorem eqToHom_map_comp_apply (p : X = Y) (q : Y = Z) (x : F.obj X) :\n    F.map (eqToHom q) (F.map (eqToHom p) x) = F.map (eqToHom <| p.trans q) x :=\n\nProof body:\n","rejected":"by\n  exact eqToHom_map_comp_apply","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"65f2a877d7f430bd6963e7a5c69524377ed8f3293c7aca38aae8583b39265536","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Types","family_id":"eqtohom_map_comp_apply","file_id":"mathlib/Mathlib/CategoryTheory/Types/Basic.lean","sample_id":"f97af12d0eda400dcad66bb72ae73c904310f1b01ffca3c206a7820ced091c83"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bdd21497f8131a5f5a6c2c2e48cfeb6b69583b6be9858ce150b32d0a5afe6448","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d0483fc373e622a8e612626bc8876496e04af6c7616a72da81eed27781805d64","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2f1e37c134e39703c11252b834c1e076724924538d8af83dbd0bc1c8dab12e37","source_sha256":"7f41b4373f1155c76bd92e338de143f67cd5443f9a140d26207473ba9237819c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Representation.IntertwiningMap.toLinearMap_apply, d_eq]; rfl","hard_negative":true,"metrics":{"chosen_tokens":14,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.214286},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"ec5067fac237b1d528f0ee34433fbcb8da489bc7edf5fa3afd39b34b0ea8a162","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Category.ModuleCat.Adjunctions\npublic import Mathlib.AlgebraicTopology.ExtraDegeneracy\npublic import Mathlib.CategoryTheory.Abelian.Ext\npublic import Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced\npublic import Mathlib.RepresentationTheory.Rep.Iso\n\nNamespace:\nRep.standardComplex\n\nLocal context:\n/-\nCopyright (c) 2022 Amelia Livingston. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Amelia Livingston\n-/\n/-!\n# The standard and bar resolutions of `k` as a trivial `k`-linear `G`-representation\n\nGiven a commutative ring `k` and a group `G`, this file defines two projective resolutions of `k`\nas a trivial `k`-linear `G`-representation.\n\nThe first one, the standard resolution, has objects `k[Gⁿ⁺¹]` equipped with the diagonal\nrepresentation, and differential defined by `(g₀, ..., gₙ) ↦ ∑ (-1)ⁱ • (g₀, ..., ĝᵢ, ..., gₙ)`.\n\nWe define this as the alternating face map complex associated to an appropriate simplicial\n`k`-linear `G`-representation. This simplicial object is the `linearization` of the simplicial\n`G`-set given by the universal cover of the classifying space of `G`, `EG`. We prove this\nsimplicial `G`-set `EG` is isomorphic to the Čech nerve of the natural arrow of `G`-sets\n`G ⟶ {pt}`.\n\nWe then use this isomorphism to deduce that as a complex of `k`-modules, the standard resolution\nof `k` as a trivial `G`-representation is homotopy equivalent to the complex with `k` at 0 and 0\nelsewhere.\n\nPutting this material together allows us to define `Rep.standardResolution`, the\nstandard projective resolution of `k` as a trivial `k`-linear `G`-representation.\n\nWe then construct the bar resolution. The `n`th object in this complex is the representation on\n`Gⁿ →₀ k[G]` defined pointwise by the left regular representation on `k[G]`. The differentials are\ndefined by sending `(g₀, ..., gₙ)` to\n`g₀·(g₁, ..., gₙ) + ∑ (-1)ʲ⁺¹·(g₀, ..., gⱼgⱼ₊₁, ..., gₙ) + (-1)ⁿ⁺¹·(g₀, ..., gₙ₋₁)` for\n`j = 0, ..., n - 1`.\n\nIn `RepresentationTheory.Rep` we define an isomorphism `Rep.diagonalSuccIsoFree` between\n`k[Gⁿ⁺¹] ≅ (Gⁿ →₀ k[G])` sending `(g₀, ..., gₙ) ↦ g₀·(g₀⁻¹g₁, ..., gₙ₋₁⁻¹gₙ)`.\nWe show that this isomorphism defines a commutative square with the bar resolution differential and\nthe standard resolution differential, and thus conclude that the bar resolution differential\nsquares to zero and that `Rep.diagonalSuccIsoFree` defines an isomorphism between the two\ncomplexes. We carry the exactness properties across this isomorphism to conclude the bar resolution\nis a projective resolution too, in `Rep.barResolution`.\n\nIn `Mathlib/RepresentationTheory/Homological/GroupHomology/Basic.lean` and\n`Mathlib/RepresentationTheory/Homological/GroupCohomology/Basic.lean`, we then use\n`Rep.barResolution` to define the inhomogeneous (co)chains of a representation, useful for\ncomputing group (co)homology.\n\n## Main definitions\n\n * `groupCohomology.resolution.ofMulActionBasis`\n * `classifyingSpaceUniversalCover`\n * `Rep.standardComplex.forget₂ToModuleCatHomotopyEquiv`\n * `Rep.standardResolution`\n\nTODO: There's bad DefEq abuses in `Action` and the way we do `Rep.standardComplex` should be\n  unified with continuous cohomology, therefore we should remove the use of `Action` in `Rep` which\n  would remove all the unification hints in this file.\n-/\n\n@[expose] public noncomputable section\n\nsuppress_compilation\n\nopen CategoryTheory Finsupp\nopen scoped MonoidAlgebra\n\nuniverse u v w\n\nvariable {k G : Type u} [CommRing k] {n : ℕ}\n\nlocal notation \"Gⁿ\" => Fin n → G\n\nset_option quotPrecheck false\nlocal notation \"Gⁿ⁺¹\" => Fin (n + 1) → G\n\nvariable (G)\n\n/-- The simplicial `G`-set sending `[n]` to `Gⁿ⁺¹` equipped with the diagonal action of `G`. -/\n@[simps obj map]\ndef classifyingSpaceUniversalCover [Monoid G] :\n    SimplicialObject (Action (Type u) G) where\n  obj n := Action.ofMulAction G (Fin (n.unop.len + 1) → G)\n  map f :=\n    { hom := ↾fun x => x ∘ f.unop.toOrderHom\n      comm := fun _ => rfl }\n  map_id _ := rfl\n  map_comp _ _ := rfl\n\nnamespace classifyingSpaceUniversalCover\n\nopen CategoryTheory.Limits\n\nvariable [Monoid G]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- When the category is `G`-Set, `cechNerveTerminalFrom` of `G` with the left regular action is\nisomorphic to `EG`, the universal cover of the classifying space of `G` as a simplicial `G`-set. -/\ndef cechNerveTerminalFromIso : cechNerveTerminalFrom (Action.ofMulAction G (G)) ≅\n    classifyingSpaceUniversalCover G :=\n  NatIso.ofComponents (fun _ => limit.isoLimitCone (Action.ofMulActionLimitCone _ _)) fun f => by\n    refine IsLimit.hom_ext (Action.ofMulActionLimitCone.{u, 0} G fun _ => G).2 fun j => ?_\n    dsimp only [cechNerveTerminalFrom, Pi.lift]\n    rw [Category.assoc, limit.isoLimitCone_hom_π, limit.lift_π, Category.assoc]\n    exact (limit.isoLimitCone_hom_π _ _).symm\n\n/-- As a simplicial set, `cechNerveTerminalFrom` of a monoid `G` is isomorphic to the universal\ncover of the classifying space of `G` as a simplicial set. -/\ndef cechNerveTerminalFromIsoCompForget :\n    cechNerveTerminalFrom G ≅ classifyingSpaceUniversalCover G ⋙ forget _ := by\n  refine NatIso.ofComponents (fun _ => Types.productIso _) fun _ => ?_\n  ext : 2\n  exact Matrix.ext fun _ _ => Pi.lift_π_apply (f := fun _ ↦ G) _ _ _\n\nvariable (k)\n\nopen AlgebraicTopology SimplicialObject.Augmented SimplicialObject CategoryTheory.Arrow\n\n/-- The universal cover of the classifying space of `G` as a simplicial set, augmented by the map\nfrom `Fin 1 → G` to the terminal object in `Type u`. -/\ndef compForgetAugmented : SimplicialObject.Augmented (Type u) :=\n  SimplicialObject.augment (classifyingSpaceUniversalCover G ⋙ forget _) (terminal _)\n    (terminal.from _) fun _ _ _ => Subsingleton.elim _ _\n\nset_option backward.defeqAttrib.useBackward true in\n/-- The augmented Čech nerve of the map from `Fin 1 → G` to the terminal object in `Type u` has an\nextra degeneracy. -/\ndef extraDegeneracyAugmentedCechNerve :\n    ExtraDegeneracy (Arrow.mk <| terminal.from G).augmentedCechNerve :=\n  AugmentedCechNerve.extraDegeneracy (Arrow.mk <| terminal.from G)\n    ⟨↾fun _ => (1 : G), by cat_disch⟩\n\n/-- The universal cover of the classifying space of `G` as a simplicial set, augmented by the map\nfrom `Fin 1 → G` to the terminal object in `Type u`, has an extra degeneracy. -/\ndef extraDegeneracyCompForgetAugmented : ExtraDegeneracy (compForgetAugmented G) := by\n  refine\n    ExtraDegeneracy.ofIso (?_ : (Arrow.mk <| terminal.from G).augmentedCechNerve ≅ _)\n      (extraDegeneracyAugmentedCechNerve G)\n  exact\n    Comma.isoMk (CechNerveTerminalFrom.iso G ≪≫ cechNerveTerminalFromIsoCompForget G)\n      (Iso.refl _) (by ext : 1; exact IsTerminal.hom_ext terminalIsTerminal _ _)\n\n/-- The free functor `Type u ⥤ ModuleCat.{u} k` applied to the universal cover of the classifying\nspace of `G` as a simplicial set, augmented by the map from `Fin 1 → G` to the terminal object\nin `Type u`. -/\ndef compForgetAugmented.toModule : SimplicialObject.Augmented (ModuleCat.{u} k) :=\n  ((SimplicialObject.Augmented.whiskering _ _).obj (ModuleCat.free k)).obj (compForgetAugmented G)\n\n/-- If we augment the universal cover of the classifying space of `G` as a simplicial set by the\nmap from `Fin 1 → G` to the terminal object in `Type u`, then apply the free functor\n`Type u ⥤ ModuleCat.{u} k`, the resulting augmented simplicial `k`-module has an extra\ndegeneracy. -/\ndef extraDegeneracyCompForgetAugmentedToModule :\n    ExtraDegeneracy (compForgetAugmented.toModule k G) :=\n  ExtraDegeneracy.map (extraDegeneracyCompForgetAugmented G) (ModuleCat.free k)\n\nend classifyingSpaceUniversalCover\n\nvariable (k)\n\n/-- The standard resolution of `k` as a trivial representation, defined as the alternating\nface map complex of a simplicial `k`-linear `G`-representation. -/\ndef Rep.standardComplex [Monoid G] :=\n  (AlgebraicTopology.alternatingFaceMapComplex (Rep k G)).obj\n    (classifyingSpaceUniversalCover G ⋙ linearization k G)\n\nnamespace Rep.standardComplex\n\nopen classifyingSpaceUniversalCover AlgebraicTopology CategoryTheory.Limits\n\n/-- The `k`-linear map underlying the differential in the standard resolution of `k` as a trivial\n`k`-linear `G`-representation. It sends `(g₀, ..., gₙ) ↦ ∑ (-1)ⁱ • (g₀, ..., ĝᵢ, ..., gₙ)`. -/\ndef d (G : Type u) (n : ℕ) : ((Fin (n + 1) → G) →₀ k) →ₗ[k] (Fin n → G) →₀ k :=\n  Finsupp.lift ((Fin n → G) →₀ k) k (Fin (n + 1) → G) fun g =>\n    (@Finset.univ (Fin (n + 1)) _).sum fun p =>\n      Finsupp.single (g ∘ p.succAbove) ((-1 : k) ^ (p : ℕ))\n\nvariable {k G}\n\n@[simp]\ntheorem d_of {n : ℕ} (c : Fin (n + 1) → G) :\n    d k G n (Finsupp.single c 1) =\n      Finset.univ.sum fun p : Fin (n + 1) =>\n        Finsupp.single (c ∘ p.succAbove) ((-1 : k) ^ (p : ℕ)) := by\n  simp [d]\n\nlemma d_single {n : ℕ} (c : Fin (n + 1) → G) (r : k) :\n    d k G n (Finsupp.single c r) =\n      Finset.univ.sum fun p : Fin (n + 1) =>\n        Finsupp.single (c ∘ p.succAbove) (r * (-1 : k) ^ (p : ℕ)) := by\n  rw [← mul_one r, ← smul_eq_mul, ← smul_single, map_smul, d_of]\n  simp [Finset.smul_sum]\n\nvariable (k G) [Monoid G]\n\n/-- The `n`th object of the standard resolution of `k` is definitionally isomorphic to `k[Gⁿ⁺¹]`\nequipped with the representation induced by the diagonal action of `G`. -/\ndef xIso (n : ℕ) : (standardComplex k G).X n ≅ Rep.ofMulAction k G (Fin (n + 1) → G) :=\n  Iso.refl _\n\ninstance x_projective (G : Type u) [Group G] (n : ℕ) :\n    Projective ((standardComplex k G).X n) := by\n  classical exact inferInstanceAs <| Projective (Rep.diagonal k G (n + 1))\n\nset_option backward.defeqAttrib.useBackward true in\nunif_hint where ⊢ Action.V (Action.ofMulAction G (Fin (n + 1) → G)) ≟ Fin (n + 1) → G in\nset_option backward.isDefEq.respectTransparency false in\n/-- Simpler expression for the differential in the standard resolution of `k` as a\n`G`-representation. It sends `(g₀, ..., gₙ₊₁) ↦ ∑ (-1)ⁱ • (g₀, ..., ĝᵢ, ..., gₙ₊₁)`. -/\ntheorem d_eq (n : ℕ) : ((standardComplex k G).d (n + 1) n).hom.toLinearMap =\n    d k G (n + 1) := by\n  refine Finsupp.lhom_ext' fun (x : Fin (n + 2) → G) => LinearMap.ext_ring ?_\n  simp [standardComplex, Action.ofMulAction_V, SimplicialObject.δ, SimplexCategory.δ,\n    Fin.succAboveOrderEmb, ← Int.cast_smul_eq_zsmul k ((-1) ^ _ : ℤ), ← ofHom_smul, ← ofHom_sum,\n    Representation.IntertwiningMap.coe_toLinearMap, Representation.IntertwiningMap.sum_apply,\n    Representation.IntertwiningMap.smul_apply, (Representation.linearizeMap_single), smul_single,\n    smul_eq_mul, mul_one]\n\nTarget:\nlemma d_apply {n : ℕ} (f : (Fin (n + 1 + 1) → G) →₀ k) :\n    ((standardComplex k G).d (n + 1) n).hom f = d k G (n + 1) f :=\n\nProof body:\n","rejected":"by\n  exact d_apply","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"8d3599e7d736eda53fd168596994dc260fb83738990a5a85a1abb47430d8a6d3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RepresentationTheory/Homological","family_id":"d_apply","file_id":"mathlib/Mathlib/RepresentationTheory/Homological/Resolution.lean","sample_id":"2f1e37c134e39703c11252b834c1e076724924538d8af83dbd0bc1c8dab12e37"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6695cef3757a420cfb80a318d6ef88b082eb4eef77453a026e398af8fe7ac543","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"37f235bb48e86794e4b09884c23d8d2d7e5709381d2c53948b9b73b1480e3fa6","source_sha256":"bc065e5d26147920810f7e88f906504ecfcce0ffd1da2e0b44dc4df7256292f1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [iSup, Bool.range_eq, sSup_pair, sup_comm]","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.230769},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"ec6a1df1373504910d7477ce1d9df77133967f4aea7dd6e0da648d2233a31c49","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Bool.Set\npublic import Mathlib.Data.Nat.Set\npublic import Mathlib.Order.CompleteLattice.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Theory of complete lattices\n\nThis file contains results on complete lattices that need more theory to develop.\n\n## Naming conventions\n\nIn lemma names,\n* `sSup` is called `sSup`\n* `sInf` is called `sInf`\n* `⨆ i, s i` is called `iSup`\n* `⨅ i, s i` is called `iInf`\n* `⨆ i j, s i j` is called `iSup₂`. This is an `iSup` inside an `iSup`.\n* `⨅ i j, s i j` is called `iInf₂`. This is an `iInf` inside an `iInf`.\n* `⨆ i ∈ s, t i` is called `biSup` for \"bounded `iSup`\". This is the special case of `iSup₂`\n  where `j : i ∈ s`.\n* `⨅ i ∈ s, t i` is called `biInf` for \"bounded `iInf`\". This is the special case of `iInf₂`\n  where `j : i ∈ s`.\n\n## Notation\n\n* `⨆ i, f i` : `iSup f`, the supremum of the range of `f`;\n* `⨅ i, f i` : `iInf f`, the infimum of the range of `f`.\n-/\n\npublic section\n\nopen Function OrderDual Set\n\nvariable {α β γ : Type*} {ι ι' : Sort*} {κ : ι → Sort*} {κ' : ι' → Sort*}\n\nopen OrderDual\n\nsection\n\nvariable [CompleteLattice α] {f g s : ι → α} {a b : α}\n\n/-!\n### `iSup` and `iInf` under `Bool`\n-/\n\n@[to_dual]\n\nTarget:\ntheorem iSup_bool_eq {f : Bool → α} : ⨆ b : Bool, f b = f true ⊔ f false :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/CompleteLattice","family_id":"isup_bool_eq","file_id":"mathlib/Mathlib/Order/CompleteLattice/Lemmas.lean","sample_id":"37f235bb48e86794e4b09884c23d8d2d7e5709381d2c53948b9b73b1480e3fa6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8e7840e8c83d8f557e8647b534950399050e6f0e066df7ccc73d874d601c7f9a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4791ed0fbefe25e9f05c1dec233b6ddc5e09c117965a9cb9936eb95639f9e525","source_sha256":"e2806df73ecfc81f91bdaba3be881cdfeef7117f485c371a23cf9b886b2f1661","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases exists_isRoot_of_odd_natDegree (f := X ^ n - C x) (by simp [hn]) with ⟨r, hr⟩\n  exact ⟨r, by linear_combination - (by simpa using hr : r ^ n - x = 0)⟩","hard_negative":true,"metrics":{"chosen_tokens":48,"rejected_tokens":8,"token_jaccard":0.088235,"token_length_ratio":0.166667},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"ec79520bbc966b95c638fbf29e1b7366c17c717c3e587043647c86a6077aa7f4","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Degree.Domain\npublic import Mathlib.Algebra.Polynomial.Eval.Defs\npublic import Mathlib.Algebra.Ring.Semireal.Defs\npublic import Mathlib.Tactic.LinearCombination\n\nNamespace:\nIsRealClosed\n\nLocal context:\n/-\nCopyright (c) 2025 Artie Khovanov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Artie Khovanov\n-/\n/-!\n# Real Closed Field\n\nA field `R` is real closed if all of the following hold:\n1. `R` is real (that is, `-1` is not a sum of squares in `R`).\n2. for every `x` in `R`, one of `x` or `-x` is a square.\n3. every odd-degree polynomial over `R` has a root in `R`.\n\nA real closed field is an algebraic generalisation of the real numbers.\n\nIn this file we define real closed fields and prove some of their properties.\n\nTODO (Artie Khovanov) : equivalent conditions for a real field to be real closed\nTODO (Artie Khovanov) : real numbers, real algebraic numbers, hyperreals form a real closed field\n\n## Main Definitions\n\n- `IsRealClosed R` is the typeclass saying `R` is a real closed field.\n\n## Tags\n\nreal closed, rcf\n\n-/\n\npublic section\n\nopen Polynomial\n\n/--\nA field `R` is real closed if all of the following hold:\n1. `R` is real (that is, `-1` is not a sum of squares in `R`).\n2. for every `x` in `R`, one of `x` or `-x` is a square.\n3. every odd-degree polynomial over `R` has a root in `R`.\n-/\nclass IsRealClosed (R : Type*) [Field R] : Prop extends IsSemireal R where\n  isSquare_or_isSquare_neg (x : R) : IsSquare x ∨ IsSquare (-x)\n  exists_isRoot_of_odd_natDegree {f : R[X]} (hf : Odd f.natDegree) : ∃ x, f.IsRoot x\n\nattribute [aesop 90% forward] IsRealClosed.isSquare_or_isSquare_neg\n\nnamespace IsRealClosed\n\nuniverse u\n\nvariable {R : Type u} [Field R]\n\ntheorem of_linearOrderedField [LinearOrder R] [IsStrictOrderedRing R]\n    (isSquare_of_nonneg : ∀ {x : R}, 0 ≤ x → IsSquare x)\n    (exists_isRoot_of_odd_natDegree : ∀ {f : R[X]}, Odd f.natDegree → ∃ x, f.IsRoot x) :\n    IsRealClosed R where\n  isSquare_or_isSquare_neg {x} := by\n    rcases le_total x 0 with (neg | pos)\n    · exact .inr <| isSquare_of_nonneg (neg_nonneg_of_nonpos neg)\n    · exact .inl <| isSquare_of_nonneg pos\n  exists_isRoot_of_odd_natDegree := exists_isRoot_of_odd_natDegree\n\nvariable [IsRealClosed R]\n\n@[aesop 50%]\ntheorem _root_.IsSquare.of_not_isSquare_neg {x : R} (hx : ¬ IsSquare (-x)) : IsSquare x := by aesop\n\n@[aesop 80%]\ntheorem isSquare_neg_of_not_isSquare {x : R} (hx : ¬ IsSquare x) : IsSquare (-x) := by aesop\n\nTarget:\ntheorem exists_eq_pow_of_odd (x : R) {n : ℕ} (hn : Odd n) : ∃ r, x = r ^ n :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"6b172166f97b5999080c8c0ba1efd10a29e400e20c72f0d46c89a86c80a7d551","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/IsRealClosed","family_id":"exists_eq_pow_of_odd","file_id":"mathlib/Mathlib/FieldTheory/IsRealClosed/Basic.lean","sample_id":"4791ed0fbefe25e9f05c1dec233b6ddc5e09c117965a9cb9936eb95639f9e525"},"split":"train"}
{"binding":{"chosen_proof_sha256":"693a609014e39cacfdbaa86bb0c471973c1d57e2dd7fdea8e3f828ddfc7bc6a5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"535e482a49088acaa3ff943705a11086429404a3e1464c08016c546a7309ccb5","source_sha256":"f83ecd09e0c1dc0009269016250874e7bd19aed3be89d47a83b8fee4b9eb7cc4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    refine RelEmbedding.wellFounded_iff_isEmpty.mpr ⟨fun a => ?_⟩\n    suffices sSup (Set.range a) ∈ Set.range a by\n      obtain ⟨n, hn⟩ := Set.mem_range.mp this\n      have h' : sSup (Set.range a) < a (n + 1) := by\n        change _ > _\n        simp [← hn, a.map_rel_iff]\n      apply lt_irrefl (a (n + 1))\n      apply lt_of_le_of_lt _ h'\n      apply le_sSup\n      apply Set.mem_range_self\n    apply h (Set.range a)\n    · use a 37\n      apply Set.mem_range_self\n    · rintro x ⟨m, hm⟩ y ⟨n, hn⟩\n      use m ⊔ n\n      rw [← hm, ← hn]\n      apply RelHomClass.map_sup a","hard_negative":false,"metrics":{"chosen_tokens":141,"rejected_tokens":2,"token_jaccard":0.016393,"token_length_ratio":0.014184},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"ec80c49cd0f348cdad817d2c0b27f9daff6180144b0ea5be4525821da4b4230c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Atoms\npublic import Mathlib.Order.OrderIsoNat\npublic import Mathlib.Order.RelIso.Set\npublic import Mathlib.Order.SupClosed\npublic import Mathlib.Order.SupIndep\npublic import Mathlib.Order.Zorn\npublic import Mathlib.Data.Finset.Order\npublic import Mathlib.Order.Interval.Set.OrderIso\npublic import Mathlib.Data.Finite.Set\npublic import Mathlib.Tactic.TFAE\n\nNamespace:\nCompleteLattice\n\nLocal context:\n/-\nCopyright (c) 2021 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\n/-!\n# Compactness properties for complete lattices\n\nFor complete lattices, there are numerous equivalent ways to express the fact that the relation `>`\nis well-founded. In this file we define three especially-useful characterisations and provide\nproofs that they are indeed equivalent to well-foundedness.\n\n## Main definitions\n* `CompleteLattice.IsSupClosedCompact`\n* `CompleteLattice.IsSupFiniteCompact`\n* `IsCompactElement`\n* `IsCompactlyGenerated`\n\n## Main results\nThe main result is that the following four conditions are equivalent for a complete lattice:\n* `well_founded (>)`\n* `CompleteLattice.IsSupClosedCompact`\n* `CompleteLattice.IsSupFiniteCompact`\n* `∀ k, IsCompactElement k`\n\nThis is demonstrated by means of the following four lemmas:\n* `CompleteLattice.WellFounded.isSupFiniteCompact`\n* `CompleteLattice.IsSupFiniteCompact.isSupClosedCompact`\n* `CompleteLattice.IsSupClosedCompact.wellFounded`\n* `CompleteLattice.isSupFiniteCompact_iff_all_elements_compact`\n\nWe also show well-founded lattices are compactly generated\n(`CompleteLattice.isCompactlyGenerated_of_wellFounded`).\n\n## References\n- [G. Călugăreanu, *Lattice Concepts of Module Theory*][calugareanu]\n\n## Tags\n\ncomplete lattice, well-founded, compact\n-/\n\n@[expose] public section\n\nopen Set\n/-- An element `k` is compact if any directed set with `LUB` (least upper bound) above\n`k` has already got above `k` at some point in the set.\nSuch an element is also called \"finite\" or \"S-compact\". -/\ndef IsCompactElement {α : Type*} [PartialOrder α] (k : α) :=\n  ∀ (s : Set α) (u : α),\n    s.Nonempty →\n    DirectedOn (· ≤ ·) s →\n    IsLUB s u →\n    k ≤ u →\n    ∃ x ∈ s, k ≤ x\n\nvariable {ι : Sort*} {α : Type*} [CompleteLattice α] {f : ι → α}\n\nnamespace CompleteLattice\n\nvariable (α)\n\n/-- A compactness property for a complete lattice is that any `sup`-closed non-empty subset\ncontains its `sSup`. -/\ndef IsSupClosedCompact : Prop :=\n  ∀ (s : Set α) (_ : s.Nonempty), SupClosed s → sSup s ∈ s\n\n/-- A compactness property for a complete lattice is that any subset has a finite subset with the\nsame `sSup`. -/\ndef IsSupFiniteCompact : Prop :=\n  ∀ s : Set α, ∃ t : Finset α, ↑t ⊆ s ∧ sSup s = t.sup id\n\n/-- An element `k` is compact if and only if any directed set with `sSup` above\n`k` already got above `k` at some point in the set. -/\ntheorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :\n    IsCompactElement k ↔\n      ∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x := by\n  constructor\n  · intro hk s hs hs' h_le\n    exact hk s (sSup s) hs hs' (isLUB_sSup s) h_le\n  · intro h s u hs hs' hu h_le\n    rw [isLUB_iff_sSup_eq] at hu\n    rw [← hu] at h_le\n    exact h s hs hs' h_le\n\n/-- An element `k` of is compact if any set with `sSup`\nabove `k` has a finite subset with `sSup` above `k`. -/\ntheorem isCompactElement_iff_exists_le_sSup_of_le_sSup (k : α) :\n    IsCompactElement k ↔ ∀ s : Set α, k ≤ sSup s → ∃ t : Finset α, ↑t ⊆ s ∧ k ≤ t.sup id := by\n  classical\n    rw [isCompactElement_iff_le_of_directed_sSup_le]\n    constructor\n    · intro hk s hsup\n      -- Consider the set of finite joins of elements of the (plain) set s.\n      let S : Set α := { x | ∃ t : Finset α, ↑t ⊆ s ∧ x = t.sup id }\n      -- S is directed, nonempty, and still has sup above k.\n      have dir_US : DirectedOn (· ≤ ·) S := by\n        rintro x ⟨c, hc⟩ y ⟨d, hd⟩\n        use x ⊔ y\n        constructor\n        · use c ∪ d\n          constructor\n          · simp only [hc.left, hd.left, Set.union_subset_iff, Finset.coe_union, and_self_iff]\n          · simp only [hc.right, hd.right, Finset.sup_union]\n        simp only [and_self_iff, le_sup_left, le_sup_right]\n      have sup_S : sSup s ≤ sSup S := by\n        apply sSup_le_sSup\n        intro x hx\n        use {x}\n        simpa only [and_true, id, Finset.coe_singleton, eq_self_iff_true,\n          Finset.sup_singleton, Set.singleton_subset_iff]\n      have Sne : S.Nonempty := by\n        suffices ⊥ ∈ S from Set.nonempty_of_mem this\n        use ∅\n        simp\n      -- Now apply the defn of compact and finish.\n      obtain ⟨j, ⟨hjS, hjk⟩⟩ := hk S Sne dir_US (le_trans hsup sup_S)\n      obtain ⟨t, ⟨htS, htsup⟩⟩ := hjS\n      use t\n      exact ⟨htS, by rwa [← htsup]⟩\n    · intro hk s hne hdir hsup\n      obtain ⟨t, ht⟩ := hk s hsup\n      -- certainly every element of t is below something in s, since ↑t ⊆ s.\n      have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y := fun x hxt => ⟨x, ht.left hxt, le_rfl⟩\n      obtain ⟨x, ⟨hxs, hsupx⟩⟩ := Finset.sup_le_of_le_directed s hne hdir t t_below_s\n      exact ⟨x, ⟨hxs, le_trans ht.right hsupx⟩⟩\n\ntheorem isCompactElement_iff_exists_le_iSup_of_le_iSup.{u} {α : Type u} [CompleteLattice α]\n    (k : α) : IsCompactElement k ↔\n      ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s := by\n  classical\n    rw [isCompactElement_iff_exists_le_sSup_of_le_sSup]\n    constructor\n    · intro H ι s hs\n      obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs\n      have : ∀ x : t, ∃ i, s i = x := fun x => ht x.prop\n      choose f hf using this\n      refine ⟨Finset.univ.image f, ht'.trans ?_⟩\n      rw [Finset.sup_le_iff]\n      intro b hb\n      rw [← show s (f ⟨b, hb⟩) = id b from hf _]\n      exact Finset.le_sup (Finset.mem_image_of_mem f <| Finset.mem_univ (Subtype.mk b hb))\n    · intro H s hs\n      obtain ⟨t, ht⟩ :=\n        H s Subtype.val\n          (by\n            delta iSup\n            rwa [Subtype.range_coe])\n      refine ⟨t.image Subtype.val, by simp, ht.trans ?_⟩\n      rw [Finset.sup_le_iff]\n      exact fun x hx => @Finset.le_sup _ _ _ _ _ id _ (Finset.mem_image_of_mem Subtype.val hx)\n\ntheorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type*}\n    (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i := by\n  classical\n    rw [isCompactElement_iff_le_of_directed_sSup_le] at hk\n    let g : Finset ι → α := fun s => ⨆ i ∈ s, f i\n    have h1 : DirectedOn (· ≤ ·) (Set.range g) := by\n      rintro - ⟨s, rfl⟩ - ⟨t, rfl⟩\n      exact\n        ⟨g (s ∪ t), ⟨s ∪ t, rfl⟩, iSup_le_iSup_of_subset Finset.subset_union_left,\n          iSup_le_iSup_of_subset Finset.subset_union_right⟩\n    have h2 : k ≤ sSup (Set.range g) :=\n      h.trans\n        (iSup_le fun i =>\n          le_sSup_of_le ⟨{i}, rfl⟩\n            (le_iSup_of_le i (le_iSup_of_le (Finset.mem_singleton_self i) le_rfl)))\n    obtain ⟨-, ⟨s, rfl⟩, hs⟩ := hk (Set.range g) (Set.range_nonempty g) h1 h2\n    exact ⟨s, hs⟩\n\n/-- A compact element `k` has the property that any directed set lying strictly below `k` has\nits `sSup` strictly below `k`. -/\ntheorem IsCompactElement.directed_sSup_lt_of_lt {α : Type*} [CompleteLattice α] {k : α}\n    (hk : IsCompactElement k) {s : Set α} (hemp : s.Nonempty) (hdir : DirectedOn (· ≤ ·) s)\n    (hbelow : ∀ x ∈ s, x < k) : sSup s < k := by\n  rw [isCompactElement_iff_le_of_directed_sSup_le] at hk\n  by_contra h\n  have sSup' : sSup s ≤ k := sSup_le fun s hs => (hbelow s hs).le\n  replace sSup : sSup s = k := eq_iff_le_not_lt.mpr ⟨sSup', h⟩\n  obtain ⟨x, hxs, hkx⟩ := hk s hemp hdir sSup.symm.le\n  obtain hxk := hbelow x hxs\n  exact hxk.ne (hxk.le.antisymm hkx)\n\ntheorem isCompactElement_finsetSup {α β : Type*} [CompleteLattice α] {f : β → α} (s : Finset β)\n    (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) := by\n  classical\n    simp_rw [isCompactElement_iff_le_of_directed_sSup_le] at ⊢ h\n    intro d hemp hdir hsup\n    rw [← Function.id_comp f]\n    rw [← Finset.sup_image]\n    apply Finset.sup_le_of_le_directed d hemp hdir\n    rintro x hx\n    obtain ⟨p, ⟨hps, rfl⟩⟩ := Finset.mem_image.mp hx\n    specialize h p hps\n    specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)\n    simpa only [exists_prop]\n\ntheorem WellFoundedGT.isSupFiniteCompact [WellFoundedGT α] :\n    IsSupFiniteCompact α := fun s => by\n  let S := { x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x }\n  obtain ⟨m, ⟨t, ⟨ht₁, rfl⟩⟩, hm⟩ := wellFounded_gt.has_min S ⟨⊥, ∅, by simp⟩\n  refine ⟨t, ht₁, (sSup_le fun y hy => ?_).antisymm ?_⟩\n  · classical\n    rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))\n        (hm _ ⟨insert y t, by simp [Set.insert_subset_iff, hy, ht₁]⟩)]\n    simp\n  · rw [Finset.sup_id_eq_sSup]\n    exact sSup_le_sSup ht₁\n\ntheorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) :\n    IsSupClosedCompact α := by\n  intro s hne hsc; obtain ⟨t, ht₁, ht₂⟩ := h s; clear h\n  rcases t.eq_empty_or_nonempty with rfl | h\n  · rw [Finset.sup_empty] at ht₂\n    rw [ht₂]\n    simp [eq_singleton_bot_of_sSup_eq_bot_of_nonempty ht₂ hne]\n  · rw [ht₂]\n    exact hsc.finsetSup_mem h ht₁\n\nTarget:\ntheorem IsSupClosedCompact.wellFoundedGT (h : IsSupClosedCompact α) :\n    WellFoundedGT α where\n  wf :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/CompactlyGenerated","family_id":"issupclosedcompact","file_id":"mathlib/Mathlib/Order/CompactlyGenerated/Basic.lean","sample_id":"535e482a49088acaa3ff943705a11086429404a3e1464c08016c546a7309ccb5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ee1a0ff1d116d0272d387712ee44ccfa46821cca224349b1934d9628b210bb5f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f97002eb6f5349cba0b0cba353ad0fc3306516a27d9c160813ec6a889f48cb83","source_sha256":"31a5542d1ac83b668eb669f80f15da1c85ba792b90d38c379ac11ec99cc62c14","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases isEmpty_or_nonempty ι\n  · simp\n  rcases isEmpty_or_nonempty ι'\n  · simp\n  refine ciSup_le fun j ↦ ?_\n  grw [hv.apply_sum_le]\n  simp only [map_mul]\n  have (f : ι → ℝ) : ⨆ i : ↥Finset.univ, f i.val = ⨆ i, f i :=\n    Function.Surjective.iSup_comp (fun i ↦ ⟨⟨i, Finset.mem_univ i⟩, rfl⟩) f\n  rw [this fun i ↦ v (A (j, i)) * v (x i)]\n  refine ciSup_le fun i ↦ ?_\n  gcongr\n  · exact Real.iSup_nonneg_of_nonnegHomClass v _\n  · exact Finite.le_ciSup_of_le (j, i) le_rfl\n  · exact Finite.le_ciSup_of_le i le_rfl","hard_negative":true,"metrics":{"chosen_tokens":136,"rejected_tokens":8,"token_jaccard":0.065574,"token_length_ratio":0.058824},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"ec82a054689e58ee915aee959c5c38192815f472375f1e685c01e34725989459","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Homogenize\npublic import Mathlib.NumberTheory.Height.Basic\nimport Mathlib.Algebra.Order.Ring.IsNonarchimedean\nimport Mathlib.Data.Fintype.Order\nimport all Mathlib.NumberTheory.Height.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2026 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Height bounds for linear and polynomial maps\n\nWe prove an upper bound for the height of the image of a tuple under a linear map.\n\nWe also prove upper and lower bounds for the height of `fun i ↦ eval P i x`, where `P` is a family\nof homogeneous polynomials over the field `K` of the same degree `N` and `x : ι → K`\nwith `ι` finite.\n-/\n\npublic section\n\nsection aux\n\nprivate lemma Height.iSup_fun_eq_max (f : Fin 2 → ℝ) : iSup f = max (f 0) (f 1) := by\n  rw [show f = ![f 0, f 1] from List.ofFn_inj.mp rfl]\n  exact (max_eq_iSup ..).symm\n\nnamespace IsNonarchimedean\n\nvariable {R α β F : Type*} [CommRing R] [AddCommMonoid β] [FunLike F β ℝ] [NonnegHomClass F β ℝ]\n    [ZeroHomClass F β ℝ] {v : F} {l : α → β}\n\n-- NOTE: The following cannot be moved to Mathlib.Algebra.Order.Ring.IsNonarchimedean,\n--       because it needs the target to be the reals (to have the default value zero\n--       for empty iSups), which are not known there.\n/-- The ultrametric triangle inequality for finite sums. -/\nlemma apply_sum_le (hv : IsNonarchimedean v) {s : Finset α} :\n    v (∑ i ∈ s, l i) ≤ ⨆ i : s, v (l i) := by\n  classical\n  induction s using Finset.induction with\n  | empty => simp\n  | insert a s ha ih =>\n    rw [Finset.sum_insert ha]\n    grw [hv .., ih]\n    refine max_le ?_ ?_\n    · exact Finite.le_ciSup_of_le ⟨_, s.mem_insert_self a⟩ le_rfl\n    · rcases isEmpty_or_nonempty s with hs | hs\n      · simpa using Real.iSup_nonneg_of_nonnegHomClass v _\n      exact ciSup_le fun i ↦ Finite.le_ciSup_of_le (⟨i.val, Finset.mem_insert_of_mem i.prop⟩) le_rfl\n\n/-- The ultrametric triangle inequality for finite sums. -/\nlemma apply_sum_univ_le [Fintype α] (hv : IsNonarchimedean v) :\n    v (∑ i, l i) ≤ ⨆ i, v (l i) := by\n  grw [hv.apply_sum_le, ← cbiSup_eq_of_forall (by grind)]\n  simp\n\nend IsNonarchimedean\n\nend aux\n\n/-!\n### Upper bound for the height of the image under a linear map\n-/\n\nvariable {K : Type*} [Field K] {ι ι' : Type*} [Fintype ι] [Finite ι']\n\n-- The \"local\" version of the bound for (archimedean) absolute values.\nlemma AbsoluteValue.iSup_abv_linearMap_apply_le (v : AbsoluteValue K ℝ) (A : ι' × ι → K)\n    (x : ι → K) :\n    ⨆ j, v (∑ i, A (j, i) * x i) ≤ Nat.card ι * (⨆ ji, v (A ji)) * ⨆ i, v (x i) := by\n  rcases isEmpty_or_nonempty ι'\n  · simp\n  refine ciSup_le fun j ↦ ?_\n  grw [v.sum_le]\n  simp only [map_mul]\n  grw [Finset.sum_le_sum (g := fun _ ↦ (⨆ ji, v (A ji)) * ⨆ i, v (x i)) fun i _ ↦ ?h]\n  case h =>\n    gcongr\n    · exact Real.iSup_nonneg_of_nonnegHomClass v _\n    · exact Finite.le_ciSup_of_le (j, i) le_rfl\n    · exact Finite.le_ciSup_of_le i le_rfl\n  rw [Finset.sum_const, nsmul_eq_mul, mul_assoc, Finset.card_univ, Nat.card_eq_fintype_card]\n\n-- The \"local\" version of the bound for nonarchimedean absolute values.\n\nTarget:\nlemma IsNonarchimedean.iSup_abv_linearMap_apply_le {v : AbsoluteValue K ℝ} (hv : IsNonarchimedean v)\n    (A : ι' × ι → K) (x : ι → K) :\n    ⨆ j, v (∑ i, A (j, i) * x i) ≤ (⨆ ji, v (A ji)) * ⨆ i, v (x i) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e7b325d3971bc618b22eaceb8f99c56e33102df74382887a5499a5012a35b981","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Height","family_id":"isnonarchimedean","file_id":"mathlib/Mathlib/NumberTheory/Height/MvPolynomial.lean","sample_id":"f97002eb6f5349cba0b0cba353ad0fc3306516a27d9c160813ec6a889f48cb83"},"split":"train"}
{"binding":{"chosen_proof_sha256":"df3d8d4ac17ee91cfc6c4a0a0a21cbff1216a809c22aaf032985f2bc0080f0a1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"bd5e201734617f1764adce1a82681aaccc105ea9d73b16cc49127c0b30d1739f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"502c601dc704b016dbc827a9cd8efa7a22333a98a499271c49028df90994ce61","source_sha256":"0d4d2f5ac977ed70567b806a9e6298c3776d513de7fc9f5724c6140a65d7d751","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal' (I := ⊤) (f := f) (by simp) hg\n\nvariable [I.IsTwoSided]","hard_negative":true,"metrics":{"chosen_tokens":25,"rejected_tokens":5,"token_jaccard":0.1,"token_length_ratio":0.2},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"eca6970ad9aadcc1afffef302d502587e09e85eda9b0edd13173bcb155a7bfad","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Ideal.Operations\npublic import Mathlib.RingTheory.Ideal.BigOperators\npublic import Mathlib.RingTheory.PowerSeries.Basic\n\nNamespace:\nPowerSeries\n\nLocal context:\n/-\nCopyright (c) 2025 Jz Pan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jz Pan\n-/\n/-!\n\n# Some results on the coefficients of multiplication of two power series\n\n## Main results\n\n- `PowerSeries.coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal`,\n  `PowerSeries.coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal'`:\n  if for all `i ≤ n` (resp. for all `i`), the `i`-th coefficients of power series `f` and `g` are\n  in ideals `I` and `J`, respectively, then for all `i ≤ n` (resp. for all `i`), the `i`-th\n  coefficients of `f * g` are in `I * J`.\n\n- `PowerSeries.coeff_mul_mem_ideal_of_coeff_right_mem_ideal`,\n  `PowerSeries.coeff_mul_mem_ideal_of_coeff_right_mem_ideal'`:\n  if for all `i ≤ n` (resp. for all `i`), the `i`-th coefficients of power series `g` are\n  in ideal `I`, then for all `i ≤ n` (resp. for all `i`), the `i`-th coefficients of `f * g` are\n  in `I`.\n\n- `PowerSeries.coeff_mul_mem_ideal_of_coeff_left_mem_ideal`,\n  `PowerSeries.coeff_mul_mem_ideal_of_coeff_left_mem_ideal'`:\n  if for all `i ≤ n` (resp. for all `i`), the `i`-th coefficients of power series `f` are\n  in ideal `I`, then for all `i ≤ n` (resp. for all `i`), the `i`-th coefficients of `f * g` are\n  in `I`.\n\n-/\n\npublic section\n\nnamespace PowerSeries\n\nvariable {A : Type*} [Semiring A] {I J : Ideal A} {f g : A⟦X⟧} (n : ℕ)\n\ntheorem coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal (hf : ∀ i ≤ n, coeff i f ∈ I)\n    (hg : ∀ i ≤ n, coeff i g ∈ J) : ∀ i ≤ n, coeff i (f * g) ∈ I * J := fun i hi ↦ by\n  rw [coeff_mul]\n  exact Ideal.sum_mem _ fun p hp ↦ Ideal.mul_mem_mul\n    (hf _ ((Finset.antidiagonal.fst_le hp).trans hi))\n    (hg _ ((Finset.antidiagonal.snd_le hp).trans hi))\n\ntheorem coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal' (hf : ∀ i, coeff i f ∈ I)\n    (hg : ∀ i, coeff i g ∈ J) : ∀ i, coeff i (f * g) ∈ I * J :=\n  fun i ↦ coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal i\n    (fun i _ ↦ hf i) (fun i _ ↦ hg i) i le_rfl\n\ntheorem coeff_mul_mem_ideal_of_coeff_right_mem_ideal\n    (hg : ∀ i ≤ n, coeff i g ∈ I) : ∀ i ≤ n, coeff i (f * g) ∈ I := by\n  simpa using coeff_mul_mem_ideal_mul_ideal_of_coeff_mem_ideal (I := ⊤) (f := f) n (by simp) hg\n\nTarget:\ntheorem coeff_mul_mem_ideal_of_coeff_right_mem_ideal'\n    (hg : ∀ i, coeff i g ∈ I) : ∀ i, coeff i (f * g) ∈ I :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_502c601dc704","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"e42d7e59da61f7268c6e9e0b8dca29c058ce195a98cfcea47ca1ebb7c6b9cc8d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/PowerSeries","family_id":"coeff_mul_mem_ideal_of_coeff_right_mem_ideal","file_id":"mathlib/Mathlib/RingTheory/PowerSeries/CoeffMulMem.lean","sample_id":"502c601dc704b016dbc827a9cd8efa7a22333a98a499271c49028df90994ce61"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1791f6bfaef40a4ae8b1dcdf06c489870e6b8a0fe97f09d60381a63c54a541cd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"46398d714eff44fac1bfd9b634185d6379a7e206107aac2457d7bb3c445fb473","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9b3f45e527412ab937dbda0aad67341cd01387028790afa9d9ec5b4d5e084b9c","source_sha256":"171194abb27453412c000e21c620bfe335cc5467bbff820dbe99451d3255de0a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [irreducibleComponents_eq_singleton, Set.mem_singleton_iff] at hZ\n  simp [irreducibleComponentOpen, irreducibleComponents_eq_singleton, hZ]","hard_negative":true,"metrics":{"chosen_tokens":19,"rejected_tokens":5,"token_jaccard":0.125,"token_length_ratio":0.263158},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"ecb6357fd6320f4962ce6c2e526d6444c358f27803cc7ca519595441f5090e28","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion\npublic import Mathlib.AlgebraicGeometry.Noetherian\n\nNamespace:\nAlgebraicGeometry.Scheme\n\nLocal context:\n/-\nCopyright (c) 2026 Thomas Browning, Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Thomas Browning, Andrew Yang\n-/\n/-!\n# Subscheme structure on an irreducible component\n\nWe define the subscheme structure on an irreducible component of a Noetherian scheme. Typically,\none takes the reduced induced subscheme structure, but this will throw away information if the\nirreducible component is not already reduced. Instead, we take the closed subscheme defined by\nthe kernel of the restriction to the complement of the union of the other irreducible components.\nFor example, if `X` is irreducible then this will give back the original scheme `X`.\n\n## Main definition\n* `AlgebraicGeometry.Scheme.irreducibleComponentIdeal`: The ideal sheaf data associated to an\n  irreducible component of a Noetherian scheme.\n* `AlgebraicGeometry.Scheme.irreducibleComponent`: The subscheme structure on an irreducible\n  component of a Noetherian scheme.\n\n## TODO\n\nProve that for affine schemes this subscheme structure is defined by the kernel of the\nlocalization away from the union of the other minimal prime ideals.\n\n-/\n\n@[expose] public section\n\nuniverse u\n\nnamespace AlgebraicGeometry.Scheme\n\nvariable (X : Scheme.{u}) (Z : Set X) (hZ : Z ∈ irreducibleComponents X) [IsNoetherian X]\n\n/-- The complement of the irreducible components unequal to `Z` of a Noetherian scheme. -/\ndef irreducibleComponentOpen : Opens X :=\n  ⟨(⋃₀ (irreducibleComponents X \\ {Z}))ᶜ, by\n    rw [Set.sUnion_eq_biUnion, isOpen_compl_iff]\n    exact TopologicalSpace.NoetherianSpace.finite_irreducibleComponents.sdiff.isClosed_biUnion\n      fun W hW ↦ isClosed_of_mem_irreducibleComponents W hW.1⟩\n\n/-- The ideal sheaf data associated to an irreducible component of a Noetherian scheme. -/\ndef irreducibleComponentIdeal : X.IdealSheafData where\n  __ := (irreducibleComponentOpen X Z).ι.ker\n  supportSet := Z\n  supportSet_eq_iInter_zeroLocus := by\n    rw [← IdealSheafData.coe_support_eq_eq_iInter_zeroLocus, Hom.support_ker, Opens.range_ι]\n    exact (closure_sUnion_irreducibleComponents_sdiff_singleton\n      TopologicalSpace.NoetherianSpace.finite_irreducibleComponents Z hZ).symm\n\ntheorem irreducibleComponentIdeal_def :\n    irreducibleComponentIdeal X Z hZ = (irreducibleComponentOpen X Z).ι.ker := by\n  ext\n  rfl\n\n/-- The subscheme structure on an irreducible component of a Noetherian scheme. -/\nnoncomputable def irreducibleComponent : Scheme :=\n  (X.irreducibleComponentIdeal Z hZ).subscheme\n\n/-- The inclusion from an irreducible component of a Noetherian scheme. -/\nnoncomputable def irreducibleComponentι : X.irreducibleComponent Z hZ ⟶ X :=\n  (X.irreducibleComponentIdeal Z hZ).subschemeι\n\nlemma irreducibleComponentι_apply (x : X.irreducibleComponent Z hZ) :\n    X.irreducibleComponentι Z hZ x = x.1 :=\n  rfl\n\ninstance : IsClosedImmersion (X.irreducibleComponentι Z hZ) :=\n  inferInstanceAs (IsClosedImmersion (X.irreducibleComponentIdeal Z hZ).subschemeι)\n\ninstance : IrreducibleSpace (X.irreducibleComponent Z hZ) :=\n  Subtype.irreducibleSpace hZ.1\n\ninclude hZ in\n\nTarget:\ntheorem irreducibleComponentOpen_eq_top [IrreducibleSpace X] :\n    irreducibleComponentOpen X Z = ⊤ :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_9b3f45e52741","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"4cf93d311610720b133c971c5ed08d8ca386dc7ca2e746028c092b8b7b921294","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/IdealSheaf","family_id":"irreduciblecomponentopen_eq_top","file_id":"mathlib/Mathlib/AlgebraicGeometry/IdealSheaf/IrreducibleComponent.lean","sample_id":"9b3f45e527412ab937dbda0aad67341cd01387028790afa9d9ec5b4d5e084b9c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"536871cc5d7d8fd19b1680026dd5fcc3bae6cf4953adabbd17e9ab3ecd407260","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"225045c52101b712c12567b3371378463073288c9fac6b8c54c0b3ba3281af97","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7fa1587a449af9bd025ff7fcf8e9e6160d9d026e5e59b7be7c90ab324dc63b3b","source_sha256":"ba493885a2be935359e1e232e1bbaeb68ad80f382975839ead7d3190a1fce95f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h ↦ ⟨?_, ?_, ?_⟩, fun h ↦ ?_⟩\n  any_goals rintro a b rfl rfl; exact map_eq_empty.1 h\n  · rintro a b rfl rfl; exact disjSum_eq_empty.1 h\n  cases a <;> cases b\n  · exact map_eq_empty.2 (h.1 _ _ rfl rfl)\n  · simp [h.2.1 _ _ rfl rfl]\n  · rfl\n  · exact map_eq_empty.2 (h.2.2 _ _ rfl rfl)","hard_negative":false,"metrics":{"chosen_tokens":99,"rejected_tokens":104,"token_jaccard":0.861111,"token_length_ratio":1.050505},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"ecc3e599ab5a5e434ba137e69f212d3077cca7f202dcbc8785e0019f6c8c1bbb","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Sum\npublic import Mathlib.Data.Sum.Order\npublic import Mathlib.Order.Interval.Finset.Defs\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Finite intervals in a disjoint union\n\nThis file provides the `LocallyFiniteOrder` instance for the disjoint sum and linear sum of two\norders and calculates the cardinality of their finite intervals.\n-/\n\n@[expose] public section\n\n\nopen Function Sum\n\nnamespace Finset\n\nvariable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}\n\nsection SumLift₂\n\nvariable (f f₁ g₁ : α₁ → β₁ → Finset γ₁) (g f₂ g₂ : α₂ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁` and `α₂ → β₂ → Finset γ₂` to a map\n`α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to `Alternative` functors if we can\nmake sure to keep computability and universe polymorphism. -/\n@[simp]\ndef sumLift₂ : ∀ (_ : α₁ ⊕ α₂) (_ : β₁ ⊕ β₂), Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f a b).map Embedding.inl\n  | inl _, inr _ => ∅\n  | inr _, inl _ => ∅\n  | inr a, inr b => (g a b).map Embedding.inr\n\nvariable {f f₁ g₁ g f₂ g₂} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\ntheorem mem_sumLift₂ :\n    c ∈ sumLift₂ f g a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨\n        ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g a₂ b₂ := by\n  constructor\n  · rcases a with a | a <;> rcases b with b | b\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr ⟨a, b, c, rfl, rfl, rfl, hc⟩\n  · rintro (⟨a, b, c, rfl, rfl, rfl, h⟩ | ⟨a, b, c, rfl, rfl, rfl, h⟩) <;> exact mem_map_of_mem _ h\n\ntheorem inl_mem_sumLift₂ {c₁ : γ₁} :\n    inl c₁ ∈ sumLift₂ f g a b ↔ ∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f a₁ b₁ := by\n  rw [mem_sumLift₂, or_iff_left]\n  · simp only [inl.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inl_ne_inr h\n\ntheorem inr_mem_sumLift₂ {c₂ : γ₂} :\n    inr c₂ ∈ sumLift₂ f g a b ↔ ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ g a₂ b₂ := by\n  rw [mem_sumLift₂, or_iff_right]\n  · simp only [inr.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inr_ne_inl h\n\ntheorem sumLift₂_eq_empty :\n    sumLift₂ f g a b = ∅ ↔\n      (∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f a₁ b₁ = ∅) ∧\n        ∀ a₂ b₂, a = inr a₂ → b = inr b₂ → g a₂ b₂ = ∅ := by\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · constructor <;>\n    · rintro a b rfl rfl\n      exact map_eq_empty.1 h\n  cases a <;> cases b\n  · exact map_eq_empty.2 (h.1 _ _ rfl rfl)\n  · rfl\n  · rfl\n  · exact map_eq_empty.2 (h.2 _ _ rfl rfl)\n\ntheorem sumLift₂_nonempty :\n    (sumLift₂ f g a b).Nonempty ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ (f a₁ b₁).Nonempty) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ (g a₂ b₂).Nonempty := by\n  simp only [nonempty_iff_ne_empty, Ne, sumLift₂_eq_empty, not_and_or, not_forall, exists_prop]\n\ntheorem sumLift₂_mono (h₁ : ∀ a b, f₁ a b ⊆ g₁ a b) (h₂ : ∀ a b, f₂ a b ⊆ g₂ a b) :\n    ∀ a b, sumLift₂ f₁ f₂ a b ⊆ sumLift₂ g₁ g₂ a b\n  | inl _, inl _ => map_subset_map.2 (h₁ _ _)\n  | inl _, inr _ => Subset.rfl\n  | inr _, inl _ => Subset.rfl\n  | inr _, inr _ => map_subset_map.2 (h₂ _ _)\n\nend SumLift₂\n\nsection SumLexLift\nvariable (f₁ f₁' : α₁ → β₁ → Finset γ₁) (f₂ f₂' : α₂ → β₂ → Finset γ₂)\n  (g₁ g₁' : α₁ → β₂ → Finset γ₁) (g₂ g₂' : α₁ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁`, `α₂ → β₂ → Finset γ₂`, `α₁ → β₂ → Finset γ₁`,\n`α₂ → β₂ → Finset γ₂`  to a map `α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to\nalternative monads if we can make sure to keep computability and universe polymorphism. -/\ndef sumLexLift : α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f₁ a b).map Embedding.inl\n  | inl a, inr b => (g₁ a b).disjSum (g₂ a b)\n  | inr _, inl _ => ∅\n  | inr a, inr b => (f₂ a b).map ⟨_, inr_injective⟩\n\n@[simp]\nlemma sumLexLift_inl_inl (a : α₁) (b : β₁) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inl b) = (f₁ a b).map Embedding.inl := rfl\n\n@[simp]\nlemma sumLexLift_inl_inr (a : α₁) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b) = (g₁ a b).disjSum (g₂ a b) := rfl\n\n@[simp]\nlemma sumLexLift_inr_inl (a : α₂) (b : β₁) : sumLexLift f₁ f₂ g₁ g₂ (inr a) (inl b) = ∅ := rfl\n\n@[simp]\nlemma sumLexLift_inr_inr (a : α₂) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inr a) (inr b) = (f₂ a b).map ⟨_, inr_injective⟩ := rfl\n\nvariable {f₁ g₁ f₂ g₂ f₁' g₁' f₂' g₂'} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\nlemma mem_sumLexLift :\n    c ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        (∃ a₁ b₂ c₁, a = inl a₁ ∧ b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨\n          (∃ a₁ b₂ c₂, a = inl a₁ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n            ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂ := by\n  constructor\n  · obtain a | a := a <;> obtain b | b := b\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (mem_disjSum.1 h).elim ?_ ?_\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inr <| Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr (Or.inr <| Or.inr <| ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n  · rintro (⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩ |\n      ⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact mem_map_of_mem _ hc\n    · exact inl_mem_disjSum.2 hc\n    · exact inr_mem_disjSum.2 hc\n    · exact mem_map_of_mem _ hc\n\nlemma inl_mem_sumLexLift {c₁ : γ₁} :\n    inl c₁ ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        ∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ c₁ ∈ g₁ a₁ b₂ := by\n  simp [mem_sumLexLift]\n\nlemma inr_mem_sumLexLift {c₂ : γ₂} :\n    inr c₂ ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ f₂ a₂ b₂ := by\n  simp [mem_sumLexLift]\n\nlemma sumLexLift_mono (hf₁ : ∀ a b, f₁ a b ⊆ f₁' a b) (hf₂ : ∀ a b, f₂ a b ⊆ f₂' a b)\n    (hg₁ : ∀ a b, g₁ a b ⊆ g₁' a b) (hg₂ : ∀ a b, g₂ a b ⊆ g₂' a b) (a : α₁ ⊕ α₂)\n    (b : β₁ ⊕ β₂) : sumLexLift f₁ f₂ g₁ g₂ a b ⊆ sumLexLift f₁' f₂' g₁' g₂' a b := by\n  cases a <;> cases b\n  exacts [map_subset_map.2 (hf₁ _ _), disjSum_mono (hg₁ _ _) (hg₂ _ _), Subset.rfl,\n    map_subset_map.2 (hf₂ _ _)]\n\nTarget:\nlemma sumLexLift_eq_empty :\n    sumLexLift f₁ f₂ g₁ g₂ a b = ∅ ↔\n      (∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f₁ a₁ b₁ = ∅) ∧\n        (∀ a₁ b₂, a = inl a₁ → b = inr b₂ → g₁ a₁ b₂ = ∅ ∧ g₂ a₁ b₂ = ∅) ∧\n          ∀ a₂ b₂, a = inr a₂ → b = inr b₂ → f₂ a₂ b₂ = ∅ :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  refine ⟨fun h ↦ ⟨?_, ?_, ?_⟩, fun h ↦ ?_⟩\n  any_goals rintro a b rfl rfl; exact map_eq_empty.1 h\n  · rintro a b rfl rfl; exact disjSum_eq_empty.1 h\n  cases a <;> cases b\n  · exact map_eq_empty.2 (h.1 _ _ rfl rfl)\n  · simp [h.2.1 _ _ rfl rfl]\n  · rfl\n  · exact map_eq_empty.2 (h.2.2 _ _ rfl rfl)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sum","family_id":"sumlexlift_eq_empty","file_id":"mathlib/Mathlib/Data/Sum/Interval.lean","sample_id":"7fa1587a449af9bd025ff7fcf8e9e6160d9d026e5e59b7be7c90ab324dc63b3b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c0bd9ef2de4bda624d2c87bea71eb5ca5fdf0a8bf96e99ba0215cfd11acebae5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9eff0f1367e2de98b6a2bc05edad8e2501e78c93f1adc88a1096fef708167fd0","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"57e63da4c8d7ec6a478c9fdfc7ae432be0adeebf4eec412381dc418d3098ca3d","source_sha256":"f9292a333dd089609a858e0aaa7af2194ec9cf6135c7c88cedf01931adf88866","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · intro h\n    constructor\n    rintro _ _ f (hf : epimorphisms C f)\n    rw [epi_iff_surjective]\n    rw [← h] at hf\n    exact hf\n  · intro\n    apply le_antisymm (surjective_le_epimorphisms C)\n    intro _ _ f hf\n    have : Epi f := hf\n    change Function.Surjective ((forget C).map f)\n    rw [← epi_iff_surjective]\n    infer_instance","hard_negative":false,"metrics":{"chosen_tokens":68,"rejected_tokens":73,"token_jaccard":0.894737,"token_length_ratio":1.073529},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"ed2d54c72fc21ebe02584e0e8b9df9d408b23d7bcc91801cf22381e32e81ad2d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Limits.Constructions.EpiMono\npublic import Mathlib.CategoryTheory.Limits.Preserves.Basic\npublic import Mathlib.CategoryTheory.Limits.Shapes.Images\npublic import Mathlib.CategoryTheory.MorphismProperty.Concrete\npublic import Mathlib.CategoryTheory.Types.Basic\n\nNamespace:\nCategoryTheory.ConcreteCategory\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Epi and mono in concrete categories\n\nIn this file, we relate epimorphisms and monomorphisms in a concrete category `C`\nto surjective and injective morphisms, and we show that if `C` has\nstrong epi mono factorizations and is such that `forget C` preserves\nboth epi and mono, then any morphism in `C` can be factored in a\nfunctorial manner as a composition of a surjective morphism followed\nby an injective morphism.\n\n-/\n\n@[expose] public section\n\nuniverse w v v' u u'\n\nnamespace CategoryTheory\n\nvariable {C : Type u} [Category.{v} C] {FC : C → C → Type*} {CC : C → Type w}\nvariable [∀ X Y, FunLike (FC X Y) (CC X) (CC Y)] [ConcreteCategory.{w} C FC]\n\nopen Limits MorphismProperty\n\nnamespace ConcreteCategory\n\nsection\n\ninstance [(forget C).PreservesMonomorphisms] {X Y : C} (f : X ⟶ Y) [Mono f] :\n    Mono (↾f) := Functor.map_mono (forget C) f\n\ninstance [(forget C).PreservesEpimorphisms] {X Y : C} (f : X ⟶ Y) [Epi f] :\n    Epi (↾f) := Functor.map_epi (forget C) f\n\n/-- In any concrete category, injective morphisms are monomorphisms. -/\ntheorem mono_of_injective {X Y : C} (f : X ⟶ Y) (i : Function.Injective f) :\n    Mono f :=\n  (forget C).mono_of_mono_map ((mono_iff_injective ((forget C).map f)).2 i)\n\ninstance forget₂_preservesMonomorphisms (C : Type u) (D : Type u')\n    [Category.{v} C] [Category.{v'} D]\n    {FC : C → C → Type*} {CC : C → Type w}\n    [∀ X Y, FunLike (FC X Y) (CC X) (CC Y)]\n    [ConcreteCategory C FC]\n    {FD : D → D → Type*} {CD : D → Type w}\n    [∀ X Y, FunLike (FD X Y) (CD X) (CD Y)]\n    [ConcreteCategory D FD]\n    [HasForget₂ C D] [(forget C).PreservesMonomorphisms] :\n    (forget₂ C D).PreservesMonomorphisms :=\n  have : (forget₂ C D ⋙ forget D).PreservesMonomorphisms := by\n    simp only [HasForget₂.forget_comp]\n    infer_instance\n  Functor.preservesMonomorphisms_of_preserves_of_reflects _ (forget D)\n\ninstance forget₂_preservesEpimorphisms (C : Type u) (D : Type u')\n    [Category.{v} C] [Category.{v'} D]\n    {FC : C → C → Type*} {CC : C → Type w}\n    [∀ X Y, FunLike (FC X Y) (CC X) (CC Y)]\n    [ConcreteCategory C FC]\n    {FD : D → D → Type*} {CD : D → Type w}\n    [∀ X Y, FunLike (FD X Y) (CD X) (CD Y)]\n    [ConcreteCategory D FD]\n    [HasForget₂ C D] [(forget C).PreservesEpimorphisms] :\n    (forget₂ C D).PreservesEpimorphisms :=\n  have : (forget₂ C D ⋙ forget D).PreservesEpimorphisms := by\n    simp only [HasForget₂.forget_comp]\n    infer_instance\n  Functor.preservesEpimorphisms_of_preserves_of_reflects _ (forget D)\n\nvariable (C)\n\nlemma surjective_le_epimorphisms :\n    MorphismProperty.surjective C ≤ epimorphisms C :=\n  fun _ _ _ hf => (forget C).epi_of_epi_map ((epi_iff_surjective _).2 hf)\n\nlemma injective_le_monomorphisms :\n    MorphismProperty.injective C ≤ monomorphisms C :=\n  fun _ _ _ hf => (forget C).mono_of_mono_map ((mono_iff_injective _).2 hf)\n\nTarget:\nlemma surjective_eq_epimorphisms_iff :\n    MorphismProperty.surjective C = epimorphisms C ↔ (forget C).PreservesEpimorphisms :=\n\nProof body:\n","rejected":"by\n  constructor\n  · intro h\n    constructor\n    rintro _ _ f (hf : epimorphisms C f)\n    rw [epi_iff_surjective]\n    rw [← h] at hf\n    exact hf\n  · intro\n    apply le_antisymm (surjective_le_epimorphisms C)\n    intro _ _ f hf\n    have : Epi f := hf\n    change Function.Surjective ((forget C).map f)\n    rw [← epi_iff_surjective]\n    infer_instance\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/ConcreteCategory","family_id":"surjective_eq_epimorphisms_iff","file_id":"mathlib/Mathlib/CategoryTheory/ConcreteCategory/EpiMono.lean","sample_id":"57e63da4c8d7ec6a478c9fdfc7ae432be0adeebf4eec412381dc418d3098ca3d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3fa3b1641e3a223c1b1c2acb29844761f01c0429d5fdec1fe9c7e636fcc33dde","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"717673c42ec9cf536eab52497d443d7e188d9ccddc0b2dc34242dbb48af892e4","source_sha256":"a29151d12db11777bb9d805b9535e76a5c2a8f2d2a791e6ae7bfa4c562e67f6b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  suffices h₁ : n ≤ sInf {m | p (m - n)} by\n    convert! sInf_add h₁\n    simp_rw [Nat.add_sub_cancel_right]\n  obtain ⟨m, hm⟩ := nonempty_of_pos_sInf h\n  refine\n    le_csInf ⟨m + n, ?_⟩ fun b hb ↦\n      le_of_not_gt fun hbn ↦\n        ne_of_mem_of_not_mem ?_ (notMem_of_lt_sInf h) (Nat.sub_eq_zero_of_le hbn.le)\n  · dsimp\n    rwa [Nat.add_sub_cancel_right]\n  · exact hb","hard_negative":false,"metrics":{"chosen_tokens":83,"rejected_tokens":2,"token_jaccard":0.019231,"token_length_ratio":0.024096},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"ed827a658dd554d5a0cec33e2121789c7a69cd2b0b6fb3a8725cbd5322796a02","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ConditionallyCompleteLattice.Finset\npublic import Mathlib.Order.Interval.Finset.Nat\n\nNamespace:\nNat\n\nLocal context:\n/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Floris van Doorn, Gabriel Ebner, Yury Kudryashov\n-/\n/-!\n# Conditionally complete linear order structure on `ℕ`\n\nIn this file we\n\n* define a `ConditionallyCompleteLinearOrderBot` structure on `ℕ`;\n* prove a few lemmas about `iSup`/`iInf`/`Set.iUnion`/`Set.iInter` and natural numbers.\n-/\n\n@[expose] public section\n\nassert_not_exists MonoidWithZero\n\nopen Set\n\nnamespace Nat\n\nopen scoped Classical in\nnoncomputable instance : InfSet ℕ :=\n  ⟨fun s ↦ if h : ∃ n, n ∈ s then @Nat.find (fun n ↦ n ∈ s) _ h else 0⟩\n\nopen scoped Classical in\nnoncomputable instance : SupSet ℕ :=\n  ⟨fun s ↦ if h : ∃ n, ∀ a ∈ s, a ≤ n then @Nat.find (fun n ↦ ∀ a ∈ s, a ≤ n) _ h else 0⟩\n\nopen scoped Classical in\ntheorem sInf_def {s : Set ℕ} (h : s.Nonempty) : sInf s = @Nat.find (fun n ↦ n ∈ s) _ h :=\n  dif_pos _\n\nopen scoped Classical in\ntheorem sSup_def {s : Set ℕ} (h : ∃ n, ∀ a ∈ s, a ≤ n) :\n    sSup s = @Nat.find (fun n ↦ ∀ a ∈ s, a ≤ n) _ h :=\n  dif_pos _\n\ntheorem _root_.Set.Infinite.Nat.sSup_eq_zero {s : Set ℕ} (h : s.Infinite) : sSup s = 0 :=\n  dif_neg fun ⟨n, hn⟩ ↦\n    let ⟨k, hks, hk⟩ := h.exists_gt n\n    (hn k hks).not_gt hk\n\ntheorem sSup_of_not_bddAbove {s : Set ℕ} (h : ¬BddAbove s) : sSup s = 0 :=\n  Set.Infinite.Nat.sSup_eq_zero <| Set.infinite_of_not_bddAbove h\n\nlemma iSup_of_not_bddAbove {ι : Sort*} {f : ι → ℕ} (h : ¬ BddAbove (Set.range f)) :\n    (⨆ i, f i : ℕ) = 0 := Nat.sSup_of_not_bddAbove h\n\n@[simp]\ntheorem sInf_eq_zero {s : Set ℕ} : sInf s = 0 ↔ 0 ∈ s ∨ s = ∅ := by\n  cases eq_empty_or_nonempty s with\n  | inl h => subst h\n             simp only [or_true, InfSet.sInf,\n                        mem_empty_iff_false, exists_false, dif_neg, not_false_iff]\n  | inr h => simp only [h.ne_empty, or_false, Nat.sInf_def, h, Nat.find_eq_zero]\n\n@[simp]\ntheorem sInf_empty : sInf ∅ = 0 := by\n  rw [sInf_eq_zero]\n  right\n  rfl\n\n@[simp]\ntheorem iInf_of_empty {ι : Sort*} [IsEmpty ι] (f : ι → ℕ) : iInf f = 0 := by\n  rw [iInf_of_isEmpty, sInf_empty]\n\n/-- This combines `Nat.iInf_of_empty` with `ciInf_const`. -/\n@[simp]\nlemma iInf_const_zero {ι : Sort*} : ⨅ _ : ι, 0 = 0 :=\n  (isEmpty_or_nonempty ι).elim (fun h ↦ by simp) fun h ↦ sInf_eq_zero.2 <| by simp\n\ntheorem sInf_mem {s : Set ℕ} (h : s.Nonempty) : sInf s ∈ s := by\n  classical\n  rw [Nat.sInf_def h]\n  exact Nat.find_spec h\n\ntheorem notMem_of_lt_sInf {s : Set ℕ} {m : ℕ} (hm : m < sInf s) : m ∉ s := by\n  classical\n  cases eq_empty_or_nonempty s with\n  | inl h => subst h; apply notMem_empty\n  | inr h => rw [Nat.sInf_def h] at hm; exact Nat.find_min h hm\n\nprotected theorem sInf_le {s : Set ℕ} {m : ℕ} (hm : m ∈ s) : sInf s ≤ m := by\n  classical\n  rw [Nat.sInf_def ⟨m, hm⟩]\n  exact Nat.find_min' ⟨m, hm⟩ hm\n\ntheorem nonempty_of_pos_sInf {s : Set ℕ} (h : 0 < sInf s) : s.Nonempty := by\n  by_contra contra\n  rw [Set.not_nonempty_iff_eq_empty] at contra\n  have h' : sInf s ≠ 0 := ne_of_gt h\n  apply h'\n  rw [Nat.sInf_eq_zero]\n  right\n  assumption\n\ntheorem nonempty_of_sInf_eq_succ {s : Set ℕ} {k : ℕ} (h : sInf s = k + 1) : s.Nonempty :=\n  nonempty_of_pos_sInf (h.symm ▸ succ_pos k : sInf s > 0)\n\ntheorem eq_Ici_of_nonempty_of_upward_closed {s : Set ℕ} (hs : s.Nonempty)\n    (hs' : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s) : s = Ici (sInf s) :=\n  ext fun n ↦ ⟨fun H ↦ Nat.sInf_le H, fun H ↦ hs' (sInf s) n H (sInf_mem hs)⟩\n\ntheorem sInf_upward_closed_eq_succ_iff {s : Set ℕ} (hs : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s)\n    (k : ℕ) : sInf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s := by\n  classical\n  constructor\n  · intro H\n    rw [eq_Ici_of_nonempty_of_upward_closed (nonempty_of_sInf_eq_succ _) hs, H, mem_Ici, mem_Ici]\n    · exact ⟨le_rfl, k.not_succ_le_self⟩\n    · exact k\n    · assumption\n  · rintro ⟨H, H'⟩\n    rw [sInf_def (⟨_, H⟩ : s.Nonempty), find_eq_iff]\n    exact ⟨H, fun n hnk hns ↦ H' <| hs n k (Nat.lt_succ_iff.mp hnk) hns⟩\n\n/-- This instance is necessary, otherwise the lattice operations would be derived via\n`ConditionallyCompleteLinearOrderBot` and marked as noncomputable. -/\ninstance : Lattice ℕ :=\n  LinearOrder.toLattice\n\nopen scoped Classical in\nnoncomputable instance : ConditionallyCompleteLinearOrderBot ℕ :=\n  { (inferInstance : OrderBot ℕ), (LinearOrder.toLattice : Lattice ℕ),\n    (inferInstance : LinearOrder ℕ) with\n    isLUB_csSup _ hn hb := sSup_def hb ▸ Nat.isLeast_find hb\n    isGLB_csInf _ hn hb := sInf_def hn ▸ (Nat.isLeast_find hn).isGLB\n    csSup_empty := by\n      simp only [sSup_def, Set.mem_empty_iff_false, forall_const, forall_prop_of_false,\n        not_false_iff, exists_const]\n      apply bot_unique (Nat.find_min' _ _)\n      trivial\n    csSup_of_not_bddAbove := by\n      intro s hs\n      simp only [sSup,\n        mem_empty_iff_false, IsEmpty.forall_iff, forall_const, exists_const, dite_true]\n      rw [dif_neg]\n      · exact le_antisymm (zero_le _) (find_le trivial)\n      · exact hs\n    csInf_of_not_bddBelow := fun s hs ↦ by simp at hs }\n\ntheorem sSup_mem {s : Set ℕ} (h₁ : s.Nonempty) (h₂ : BddAbove s) : sSup s ∈ s :=\n  let ⟨k, hk⟩ := h₂\n  h₁.csSup_mem ((finite_le_nat k).subset hk)\n\ntheorem sInf_add {n : ℕ} {p : ℕ → Prop} (hn : n ≤ sInf { m | p m }) :\n    sInf { m | p (m + n) } + n = sInf { m | p m } := by\n  classical\n  obtain h | ⟨m, hm⟩ := { m | p (m + n) }.eq_empty_or_nonempty\n  · rw [h, Nat.sInf_empty, zero_add]\n    obtain hnp | hnp := hn.eq_or_lt\n    · exact hnp\n    suffices hp : p (sInf { m | p m } - n + n) from (h.subset hp).elim\n    rw [Nat.sub_add_cancel hn]\n    exact csInf_mem (nonempty_of_pos_sInf <| n.zero_le.trans_lt hnp)\n  · have hp : ∃ n, n ∈ { m | p m } := ⟨_, hm⟩\n    rw [Nat.sInf_def ⟨m, hm⟩, Nat.sInf_def hp]\n    rw [Nat.sInf_def hp] at hn\n    exact find_add hn\n\nTarget:\ntheorem sInf_add' {n : ℕ} {p : ℕ → Prop} (h : 0 < sInf { m | p m }) :\n    sInf { m | p m } + n = sInf { m | p (m - n) } :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Lattice","family_id":"sinf_add","file_id":"mathlib/Mathlib/Order/Lattice/Nat.lean","sample_id":"717673c42ec9cf536eab52497d443d7e188d9ccddc0b2dc34242dbb48af892e4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1656a7f84780a26168e97da85997b830e60c4a9c21bd50667f35238ce2172d26","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ef209941b3f109c06162c2f7b35734faae1ce153c38fae1172904a0db76a8205","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0e4b1666a2df18b8f82460264078354bcb621eb24caecc149f02733eb9e19be1","source_sha256":"04666e1a54002d0f7b310bb68c623ecf47eda2066cc741dda3dda2f5fc6f4c64","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let U := f ⁻¹' {f y}\n  suffices x ∉ Uᶜ from Classical.not_not.1 this\n  intro hxV\n  specialize hs U Uᶜ (hf {f y}) (hf {f y}ᶜ) _ ⟨y, ⟨hy, rfl⟩⟩ ⟨x, ⟨hx, hxV⟩⟩\n  · simp only [union_compl_self, subset_univ]\n  · simp only [inter_empty, Set.not_nonempty_empty, inter_compl_self] at hs","hard_negative":true,"metrics":{"chosen_tokens":87,"rejected_tokens":2,"token_jaccard":0.020833,"token_length_ratio":0.022989},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"ed97b22deebdb842b3c5b937fc0062d8ae2ab14856db9da5d82ac0b1d98e6a56","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Notation.Indicator\npublic import Mathlib.Topology.Connected.LocallyConnected\npublic import Mathlib.Topology.Sets.Closeds\n\nNamespace:\nIsLocallyConstant\n\nLocal context:\n/-\nCopyright (c) 2021 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\n/-!\n# Locally constant functions\n\nThis file sets up the theory of locally constant function from a topological space to a type.\n\n## Main definitions and constructions\n\n* `IsLocallyConstant f` : a map `f : X → Y` where `X` is a topological space is locally\n                            constant if every set in `Y` has an open preimage.\n* `LocallyConstant X Y` : the type of locally constant maps from `X` to `Y`\n* `LocallyConstant.map` : push-forward of locally constant maps\n* `LocallyConstant.comap` : pull-back of locally constant maps\n-/\n\n@[expose] public section\n\nvariable {X Y Z α : Type*} [TopologicalSpace X]\n\nopen Set Filter\nopen scoped Topology\n\n/-- A function between topological spaces is locally constant if the preimage of any set is open. -/\ndef IsLocallyConstant (f : X → Y) : Prop :=\n  ∀ s : Set Y, IsOpen (f ⁻¹' s)\n\nnamespace IsLocallyConstant\n\nopen List in\nprotected theorem tfae (f : X → Y) :\n    TFAE [IsLocallyConstant f,\n      ∀ x, ∀ᶠ x' in 𝓝 x, f x' = f x,\n      ∀ x, IsOpen { x' | f x' = f x },\n      ∀ y, IsOpen (f ⁻¹' {y}),\n      ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x] := by\n  tfae_have 1 → 4 := fun h y => h {y}\n  tfae_have 4 → 3 := fun h x => h (f x)\n  tfae_have 3 → 2 := fun h x => IsOpen.mem_nhds (h x) rfl\n  tfae_have 2 → 5\n  | h, x => by\n    rcases mem_nhds_iff.1 (h x) with ⟨U, eq, hU, hx⟩\n    exact ⟨U, hU, hx, eq⟩\n  tfae_have 5 → 1\n  | h, s => by\n    refine isOpen_iff_forall_mem_open.2 fun x hx ↦ ?_\n    rcases h x with ⟨U, hU, hxU, eq⟩\n    exact ⟨U, fun x' hx' => mem_preimage.2 <| (eq x' hx').symm ▸ hx, hU, hxU⟩\n  tfae_finish\n\n@[nontriviality]\ntheorem of_discrete [DiscreteTopology X] (f : X → Y) : IsLocallyConstant f := fun _ =>\n  isOpen_discrete _\n\ntheorem isOpen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsOpen { x | f x = y } :=\n  hf {y}\n\ntheorem isClosed_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClosed { x | f x = y } :=\n  ⟨hf {y}ᶜ⟩\n\ntheorem isClopen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClopen { x | f x = y } :=\n  ⟨isClosed_fiber hf _, isOpen_fiber hf _⟩\n\ntheorem iff_exists_open (f : X → Y) :\n    IsLocallyConstant f ↔ ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x :=\n  (IsLocallyConstant.tfae f).out 0 4\n\ntheorem iff_eventually_eq (f : X → Y) : IsLocallyConstant f ↔ ∀ x, ∀ᶠ y in 𝓝 x, f y = f x :=\n  (IsLocallyConstant.tfae f).out 0 1\n\ntheorem exists_open {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x :=\n  (iff_exists_open f).1 hf x\n\nprotected theorem eventually_eq {f : X → Y} (hf : IsLocallyConstant f) (x : X) :\n    ∀ᶠ y in 𝓝 x, f y = f x :=\n  (iff_eventually_eq f).1 hf x\n\ntheorem iff_isOpen_fiber_apply {f : X → Y} : IsLocallyConstant f ↔ ∀ x, IsOpen (f ⁻¹' {f x}) :=\n  (IsLocallyConstant.tfae f).out 0 2\n\ntheorem iff_isOpen_fiber {f : X → Y} : IsLocallyConstant f ↔ ∀ y, IsOpen (f ⁻¹' {y}) :=\n  (IsLocallyConstant.tfae f).out 0 3\n\nprotected theorem continuous [TopologicalSpace Y] {f : X → Y} (hf : IsLocallyConstant f) :\n    Continuous f :=\n  ⟨fun _ _ => hf _⟩\n\ntheorem iff_continuous {_ : TopologicalSpace Y} [DiscreteTopology Y] (f : X → Y) :\n    IsLocallyConstant f ↔ Continuous f :=\n  ⟨IsLocallyConstant.continuous, fun h s => h.isOpen_preimage s (isOpen_discrete _)⟩\n\ntheorem of_constant (f : X → Y) (h : ∀ x y, f x = f y) : IsLocallyConstant f :=\n  (iff_eventually_eq f).2 fun _ => Eventually.of_forall fun _ => h _ _\n\nprotected theorem const (y : Y) : IsLocallyConstant (Function.const X y) :=\n  of_constant _ fun _ _ => rfl\n\nprotected theorem comp {f : X → Y} (hf : IsLocallyConstant f) (g : Y → Z) :\n    IsLocallyConstant (g ∘ f) := fun s => by\n  rw [Set.preimage_comp]\n  exact hf _\n\ntheorem prodMk {Y'} {f : X → Y} {f' : X → Y'} (hf : IsLocallyConstant f)\n    (hf' : IsLocallyConstant f') : IsLocallyConstant fun x => (f x, f' x) :=\n  (iff_eventually_eq _).2 fun x =>\n    (hf.eventually_eq x).mp <| (hf'.eventually_eq x).mono fun _ hf' hf => Prod.ext hf hf'\n\ntheorem comp₂ {Y₁ Y₂ Z : Type*} {f : X → Y₁} {g : X → Y₂} (hf : IsLocallyConstant f)\n    (hg : IsLocallyConstant g) (h : Y₁ → Y₂ → Z) : IsLocallyConstant fun x => h (f x) (g x) :=\n  (hf.prodMk hg).comp fun x : Y₁ × Y₂ => h x.1 x.2\n\ntheorem comp_continuous [TopologicalSpace Y] {g : Y → Z} {f : X → Y} (hg : IsLocallyConstant g)\n    (hf : Continuous f) : IsLocallyConstant (g ∘ f) := fun s => by\n  rw [Set.preimage_comp]\n  exact hf.isOpen_preimage _ (hg _)\n\n/-- A locally constant function is constant on any preconnected set. -/\n\nTarget:\ntheorem apply_eq_of_isPreconnected {f : X → Y} (hf : IsLocallyConstant f) {s : Set X}\n    (hs : IsPreconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) : f x = f y :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_0e4b1666a2df","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"66e0c5e66c7c99602389f5f8be187515a655cf0d45a469631b313073d226049a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/LocallyConstant","family_id":"apply_eq_of_ispreconnected","file_id":"mathlib/Mathlib/Topology/LocallyConstant/Basic.lean","sample_id":"0e4b1666a2df18b8f82460264078354bcb621eb24caecc149f02733eb9e19be1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d0424b22b522a25f022cebb1346824da37ff580ade98ae91173e442bd2f6ba06","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"dcd5259bf50f78916804d2a00f0af0f07069dde7346adffee5dfd3b5d64479ef","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"093333b00fcbe2feec5384aaaba84c83717794caa26f2c6955f67a6005b04fa4","source_sha256":"6ab13085b4631b0c5e2a6d22147b2e528fc5eed37a5ef3189daeebd872d6998b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply IsPullback.of_right (t := (inf_isPullback ((«exists» f).obj A') B').flip)\n    (p := by simp [frobeniusMorphism])\n  simpa [frobeniusMorphism, IsPullback.lift_fst, ← imageFactorisation_F_m,\n    (isPullback f B').paste_horiz_iff] using\n    (inf_isPullback A' ((Subobject.pullback f).obj B')).flip","hard_negative":false,"metrics":{"chosen_tokens":72,"rejected_tokens":76,"token_jaccard":0.916667,"token_length_ratio":1.055556},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"ed9b6d443a5e9104fd27b9c6d1caa8c73bcbd5de0637891306aded607c82a6a3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.ExtremalEpi\npublic import Mathlib.CategoryTheory.MorphismProperty.Limits\npublic import Mathlib.CategoryTheory.Sites.Coherent.Basic\n\nNamespace:\nCategoryTheory.Regular\n\nLocal context:\n/-\nCopyright (c) 2025 Fernando Chu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Fernando Chu\n-/\n/-!\n# Regular categories\n\nA regular category is a category with finite limits such that each kernel pair has a coequalizer\nand such that regular epimorphisms are stable under pullback.\n\nThese categories provide a good ground to develop the calculus of relations, as well as being the\nsemantics for regular logic.\n\n## Main results\n\n* We show that every regular category has strong epi-mono factorisations, following Theorem 1.11\n  in [Gran2021].\n* We show that every regular category satisfies Frobenius reciprocity. That is, that in their\n  internal language, we have `∃ x, (P(x) ⊓ Q)` iff `(∃ x, P(x)) ⊓ Q`, for a proposition `Q` not\n  depending on `x`.\n\n## Future work\n* Show that every topos is regular\n* Show that regular logic has an interpretation in regular categories\n\n## References\n* [Marino Gran, An Introduction to Regular Categories][Gran2021]\n* <https://ncatlab.org/nlab/show/regular+category>\n-/\n\n@[expose] public section\n\nopen CategoryTheory Limits\n\nuniverse u v\n\nnamespace CategoryTheory\n\nvariable (C : Type u) [Category.{v} C]\n\n/--\nA regular category is a category with finite limits, such that every kernel pair has a coequalizer,\nand such that regular epimorphisms are stable under base change.\n-/\nclass Regular extends HasFiniteLimits C where\n  hasCoequalizer_of_isKernelPair {X Y Z : C} {f : X ⟶ Y} {g₁ g₂ : Z ⟶ X} :\n    IsKernelPair f g₁ g₂ → HasCoequalizer g₁ g₂\n  regularEpiIsStableUnderBaseChange : MorphismProperty.IsStableUnderBaseChange (.regularEpi C)\n\nvariable {C} [Regular C]\n\ninstance {X Y B : C} (f : X ⟶ B) (g : Y ⟶ B) [HasPullback f g] [IsRegularEpi f] :\n    IsRegularEpi (pullback.snd f g) := by\n  apply Regular.regularEpiIsStableUnderBaseChange.of_isPullback (IsPullback.of_hasPullback f g)\n  dsimp [MorphismProperty.regularEpi]\n  infer_instance\n\ninstance {X Y B : C} (f : X ⟶ B) (g : Y ⟶ B) [HasPullback f g] [IsRegularEpi g] :\n    IsRegularEpi (pullback.fst f g) := by\n  apply Regular.regularEpiIsStableUnderBaseChange.of_isPullback (IsPullback.of_hasPullback f g).flip\n  dsimp [MorphismProperty.regularEpi]\n  infer_instance\n\ninstance : Preregular C where\n  exists_fac f g := ⟨_, pullback.snd g f, inferInstance, pullback.fst g f, pullback.condition⟩\n\nvariable {X Y : C} (f : X ⟶ Y)\n\nnamespace Regular\n\nsection StrongEpiMonoFactorisation\n\nlocal instance : HasCoequalizer (pullback.fst f f) (pullback.snd f f) :=\n  Regular.hasCoequalizer_of_isKernelPair <| IsKernelPair.of_hasPullback f\n\nset_option backward.isDefEq.respectTransparency false in\ninstance : Mono (coequalizer.desc f pullback.condition) := by\n  -- It suffices to show that the two projections from the kernel pair are equal:\n  apply (IsKernelPair.of_hasPullback _).mono_of_eq_fst_snd\n  /- We fill in the kernel pair square of `f` as follows:\n  ```\n                  g₁                   fst\n  pullback f f------->pullback e k₁----------> X\n        |                 |                    |\n      g₂|                 |snd                 |e\n        v        fst      v            k₁      v\n  pullback k₂ e------>pullback m m---------->coeq\n        |                                      |\n     snd|                                      |m\n        v              e ≫ m = f               v\n        X------------------------------------->Y\n  ```\n  Where `m`, `e`, `k₁`, `k₂`, `g₁`, `g₂` are defined below, `fst` and `snd` denote the projections\n  in the pullbacks indicated as the source of those morphisms, and `coeq` is the coequalizer of the\n  two projections in from the kernel pair of `f`.\n  -/\n  let m := (coequalizer.desc f pullback.condition)\n  let e := coequalizer.π (pullback.fst f f) (pullback.snd f f)\n  let k₁ := pullback.fst m m\n  let k₂ := pullback.snd m m\n  let d : pullback f f ⟶ (pullback m m) :=\n    pullback.lift (pullback.fst f f ≫ e) (pullback.snd f f ≫ e) (by simp [m, e, pullback.condition])\n  let g₁ : pullback f f ⟶ (pullback e k₁) := pullback.lift (pullback.fst f f) d (by simp [d, k₁])\n  let g₂ : pullback f f ⟶ (pullback k₂ e) := pullback.lift d (pullback.snd f f) (by simp [d, k₂])\n  /-\n  Since the big square, the bottom square, and the top right square above are pullback squares,\n  the top left square is also a pullback square.\n  -/\n  have h : IsPullback g₁ g₂ (pullback.snd e k₁) (pullback.fst k₂ e) := by\n    refine .of_right ?_ (by simp [g₁, g₂]) (.of_hasPullback e k₁)\n    refine .of_bot ?_ ?_ (.paste_horiz (.of_hasPullback k₂ e) (.of_hasPullback m m))\n    · simpa [g₁, g₂, e, m, pullback.lift_fst, pullback.lift_snd] using .of_hasPullback f f\n    · simp [g₁, g₂, k₁, d]\n  /-\n  Since `g₁` is the base change of a regular epi (the map `fst` in the middle row of the diagram\n  above, which itself is a regular epi because it is a base change of the regular epi `e`),\n  it is a regular epi.\n  -/\n  have : IsRegularEpi g₁ := by\n    apply Regular.regularEpiIsStableUnderBaseChange.of_isPullback h.flip\n    dsimp [MorphismProperty.regularEpi]\n    infer_instance\n  -- We precompose with the epimorphism `g₁ ≫ pullback.snd e k₁`, and finish\n  rw [← cancel_epi (g₁ ≫ pullback.snd e k₁)]\n  convert! coequalizer.condition (pullback.fst f f) (pullback.snd f f) using 1\n  all_goals cat_disch\n\nset_option backward.isDefEq.respectTransparency false in\n/--\nIn a regular category, every morphism `f : X ⟶ Y` factors as `e ≫ m`, where `e` is the projection\nmap to the coequalizer of the kernel pair of `f`, and `m` is the canonical map from that\ncoequalizer to `Y`. In particular, `f` factors as a strong epimorphism followed by a monomorphism.\n-/\nnoncomputable def strongEpiMonoFactorisation : StrongEpiMonoFactorisation f where\n  I := coequalizer (pullback.fst f f) (pullback.snd f f)\n  m := coequalizer.desc f pullback.condition\n  e := coequalizer.π (pullback.fst f f) (pullback.snd f f)\n\ninstance : IsRegularEpi (strongEpiMonoFactorisation f).e := by\n  dsimp [strongEpiMonoFactorisation]\n  infer_instance\n\n/--\nIn a regular category, every morphism `f` factors as `e ≫ m`, with `e` a strong epimorphism\nand `m` a monomorphism.\n-/\ninstance hasStrongEpiMonoFactorisations : HasStrongEpiMonoFactorisations C where\n  has_fac f := ⟨strongEpiMonoFactorisation f⟩\n\nset_option backward.isDefEq.respectTransparency false in\n/-- In a regular category, every extremal epimorphism is a regular epimorphism. -/\nnoncomputable def regularEpiOfExtremalEpi [h : ExtremalEpi f] : RegularEpi f :=\n  have := h.isIso (strongEpiMonoFactorisation f).e (strongEpiMonoFactorisation f).m (by simp)\n  RegularEpi.ofArrowIso (Arrow.isoMk (f := .mk (strongEpiMonoFactorisation f).e) (Iso.refl _)\n    (asIso (strongEpiMonoFactorisation f).m)) (IsRegularEpi.getStruct _)\n\ninstance isRegularEpi_of_extremalEpi (f : X ⟶ Y) [ExtremalEpi f] : IsRegularEpi f :=\n  ⟨⟨regularEpiOfExtremalEpi f⟩⟩\n\nend StrongEpiMonoFactorisation\n\nsection Frobenius\n\nopen Subobject\n\nvariable {A B : C} (f : A ⟶ B) (A' : Subobject A) (B' : Subobject B)\n\nset_option backward.isDefEq.respectTransparency false in\n/--\nGiven a morphism `f : A ⟶ B` and subobjects `A' ⟶ A` and `B' ⟶ B`, we have a canonical morphism\n`(A' ⊓ (Subobject.pullback f).obj B') ⟶ ((«exists» f).obj A' ⊓ B')`.\nThis morphism is part of a `StrongEpiMonoFactorisation` of\n`(A' ⊓ (Subobject.pullback f).obj B').arrow ≫ f`, see `frobeniusStrongEpiMonoFactorisation`.\n-/\nnoncomputable def frobeniusMorphism :\n    underlying.obj (A' ⊓ (Subobject.pullback f).obj B') ⟶\n      underlying.obj ((«exists» f).obj A' ⊓ B') :=\n  (inf_isPullback ((«exists» f).obj A') B').flip.lift\n    ((ofLE _ _ (inf_le_right A' ((Subobject.pullback f).obj B'))) ≫ (pullbackπ _ _))\n    ((ofLE _ _ (inf_le_left A' ((Subobject.pullback f).obj B'))) ≫ (imageFactorisation f A').F.e)\n    (by simp [← imageFactorisation_F_m, (isPullback _ _).w])\n\nset_option backward.isDefEq.respectTransparency false in\n\nTarget:\nlemma frobeniusMorphism_isPullback :\n    IsPullback (frobeniusMorphism f A' B')\n      ((ofLE _ _ (inf_le_left A' ((Subobject.pullback f).obj B'))))\n      ((ofLE _ _ (inf_le_left ((«exists» f).obj A') B')))\n      (imageFactorisation _ _).F.e :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  apply IsPullback.of_right (t := (inf_isPullback ((«exists» f).obj A') B').flip)\n    (p := by simp [frobeniusMorphism])\n  simpa [frobeniusMorphism, IsPullback.lift_fst, ← imageFactorisation_F_m,\n    (isPullback f B').paste_horiz_iff] using\n    (inf_isPullback A' ((Subobject.pullback f).obj B')).flip","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/RegularCategory","family_id":"frobeniusmorphism_ispullback","file_id":"mathlib/Mathlib/CategoryTheory/RegularCategory/Basic.lean","sample_id":"093333b00fcbe2feec5384aaaba84c83717794caa26f2c6955f67a6005b04fa4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e1fced99958acc84dd8fa2ad1ee6184c3708ee54a7e8d9105ea3938d1bcfadce","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cb0ad84f124428492c2b06be9a3022ca3cb001bac30fe935b8073004a6ce941b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8ccb2339aba41c643ce9ab6237c697b1faecc53decbb2a2025ef71895c9c0e9f","source_sha256":"cd00025c9f1c99662e498ca3e267a5f89784b2f84527a76f8e7a37bf2514f3a2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  refine ⟨fun h₂ ↦ by simp [isBigO_of_le' (c := 0), h₂], ?_⟩\n  contrapose\n  intro h₁\n  obtain ⟨e, he⟩ := exists_single_le_pos (lt_of_le_of_ne h (h₁ ·.symm))\n  rw [isBigO_iff'']\n  push Not\n  intro a ha\n  simp only [Pi.one_apply, norm_eq_abs, frequently_atTop, abs_one]\n  intro b\n  obtain ⟨c, hc⟩ := eventually_atTop.1\n    (isLittleO_iff.1 (one_isLittleO_logCounting_single (e := e)) ha)\n  let ℓ := 1 + max ‖e‖ (max |b| |c|)\n  have h₁ℓ : c ≤ ℓ := by grind\n  have h₂ℓ : 1 ≤ ℓ := by simp [ℓ]\n  use 1 + ℓ, (show b ≤ 1 + ℓ by grind)\n  calc 1\n    _ ≤ (a * |logCounting (single e 1) ℓ|) := by simpa [h₁ℓ] using hc ℓ\n    _ ≤ (a * |logCounting D ℓ|) := by\n      gcongr\n      · apply logCounting_nonneg (single_pos.2 Int.one_pos).le h₂ℓ\n      · apply logCounting_le he h₂ℓ\n    _ < a * |logCounting D (1 + ℓ)| := by\n      gcongr 2\n      rw [abs_of_nonneg (logCounting_nonneg h h₂ℓ),\n        abs_of_nonneg (logCounting_nonneg h (by grind))]\n      apply logCounting_strictMono he <;> grind","hard_negative":true,"metrics":{"chosen_tokens":264,"rejected_tokens":3,"token_jaccard":0.011765,"token_length_ratio":0.011364},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"edefe588435b2baa98bf0e451b0e25f6c9ce8e9d94ee06703b3d0950abbc8f27","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic\n\nNamespace:\nFunction.locallyFinsuppWithin\n\nLocal context:\n/-\nCopyright (c) 2026 Stefan Kebekus. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stefan Kebekus\n-/\n/-!\n# Asymptotic Behavior of the Logarithmic Counting Function\n\nIf `f` is meromorphic over a field `𝕜`, we show that the logarithmic counting function for the\npoles of `f` is asymptotically bounded if and only if `f` has only removable singularities.  See\nPage 170f of [Lang, *Introduction to Complex Hyperbolic Spaces*][MR886677] for a detailed\ndiscussion.\n\n## Implementation Notes\n\nWe establish the result first for the logarithmic counting function for functions with locally\nfinite support on `𝕜` and then specialize to the setting where the function with locally finite\nsupport is the pole or zero-divisor of a meromorphic function.\n\n## TODO\n\nEstablish the analogous characterization of meromorphic functions with finite set of poles, as\nfunctions whose logarithmic counting function is big-O of `log`.\n-/\n\npublic section\n\nopen Asymptotics Filter Function Real Set\n\nnamespace Function.locallyFinsuppWithin\n\nvariable\n  {E : Type*} [NormedAddCommGroup E]\n\n/-!\n## Logarithmic Counting Functions for Functions with Locally Finite Support\n-/\n\n/--\nQualitative consequence of `logCounting_single_eq_log_sub_const`. The constant function `1 : ℝ → ℝ`\nis little o of the logarithmic counting function attached to `single e`.\n-/\nlemma one_isLittleO_logCounting_single [DecidableEq E] [ProperSpace E] {e : E} :\n    (1 : ℝ → ℝ) =o[atTop] logCounting (single e 1) := by\n  have hΘ : (fun r ↦ log r - log ‖e‖) =Θ[atTop] log :=\n    (IsEquivalent.sub_isLittleO IsEquivalent.refl isLittleO_const_log_atTop).isTheta\n  have h₁ : (1 : ℝ → ℝ) =o[atTop] fun r ↦ log r - log ‖e‖ :=\n    (hΘ.isLittleO_congr_right).2 isLittleO_const_log_atTop\n  refine h₁.congr' EventuallyEq.rfl ?_\n  filter_upwards [eventually_ge_atTop ‖e‖] with r hr\n  simp [logCounting_single_eq_log_sub_const hr]\n\n/--\nA non-negative function with locally finite support is zero if and only if its logarithmic counting\nfunctions is asymptotically bounded.\n-/\n\nTarget:\nlemma zero_iff_logCounting_bounded [ProperSpace E]\n    {D : locallyFinsuppWithin (univ : Set E) ℤ} (h : 0 ≤ D) :\n    D = 0 ↔ logCounting D =O[atTop] (1 : ℝ → ℝ) :=\n\nProof body:\n","rejected":"by\n  exact zero_iff_logCounting_bounded","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"54a81d5d82b0000aef0e5cc67afa757acf92564980c3205b3689f7f278c69bb3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Complex","family_id":"zero_iff_logcounting_bounded","file_id":"mathlib/Mathlib/Analysis/Complex/ValueDistribution/LogCounting/Asymptotic.lean","sample_id":"8ccb2339aba41c643ce9ab6237c697b1faecc53decbb2a2025ef71895c9c0e9f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"911bfcaad6d4d642cf91de9a824237ea9c93ccf06c46a0091532a586f961081e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"61d172a6c1c40adedfc16e781f60f95fac4d1f6ec88c11c9a5b57c8a1f496302","source_sha256":"aac9c42310c3c68ba48257d401225da11673aa0693f3a499d296961d7c43ce42","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [SetLike.ext_iff, mem_stabilizer_set]\n  refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_right ha⟩\n  have : 1 * MulOpposite.op a ∈ s := (h 1).2 s.one_mem\n  simpa only [op_smul_eq_mul, SetLike.mem_coe, one_mul] using! this","hard_negative":false,"metrics":{"chosen_tokens":63,"rejected_tokens":2,"token_jaccard":0.047619,"token_length_ratio":0.031746},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"edfbdcd260d1183aa211d99b518b5a2b79a9b540becd5bdef9fdf38273cb9d34","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Action.Pointwise.Finset\npublic import Mathlib.GroupTheory.QuotientGroup.Defs\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\n\nNamespace:\nMulAction\n\nLocal context:\n/-\nCopyright (c) 2023 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Stabilizer of a set under a pointwise action\n\nThis file characterises the stabilizer of a set/finset under the pointwise action of a group.\n-/\n\npublic section\n\nopen Function MulOpposite Set\nopen scoped Pointwise\n\nnamespace MulAction\nvariable {G H α : Type*}\n\n/-! ### Stabilizer of a set -/\n\nsection Set\nsection Group\nvariable [Group G] [Group H] [MulAction G α] {a : G} {s t : Set α}\n\n@[to_additive (attr := simp)]\nlemma stabilizer_empty : stabilizer G (∅ : Set α) = ⊤ :=\n  Subgroup.coe_eq_univ.1 <| eq_univ_of_forall fun _a ↦ smul_set_empty\n\n@[to_additive (attr := simp)]\nlemma stabilizer_univ : stabilizer G (Set.univ : Set α) = ⊤ := by\n  ext\n  simp\n\n@[to_additive (attr := simp)]\nlemma stabilizer_singleton (b : α) : stabilizer G ({b} : Set α) = stabilizer G b := by ext; simp\n\n@[to_additive]\nlemma mem_stabilizer_set {s : Set α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s := by\n  refine mem_stabilizer_iff.trans ⟨fun h b ↦ ?_, fun h ↦ ?_⟩\n  · rw [← (smul_mem_smul_set_iff : a • b ∈ _ ↔ _), h]\n  simp_rw [Set.ext_iff, mem_smul_set_iff_inv_smul_mem]\n  exact ((MulAction.toPerm a).forall_congr' <| by simp [Iff.comm]).1 h\n\n@[to_additive]\nlemma map_stabilizer_le (f : G →* H) (s : Set G) :\n    (stabilizer G s).map f ≤ stabilizer H (f '' s) := by\n  rintro a\n  simp only [Subgroup.mem_map, mem_stabilizer_iff, forall_exists_index, and_imp]\n  rintro a ha rfl\n  rw [← image_smul_distrib, ha]\n\n@[to_additive (attr := simp)]\nlemma stabilizer_mul_self (s : Set G) : (stabilizer G s : Set G) * s = s := by\n  ext\n  refine ⟨?_, fun h ↦ ⟨_, (stabilizer G s).one_mem, _, h, one_mul _⟩⟩\n  rintro ⟨a, ha, b, hb, rfl⟩\n  rw [← mem_stabilizer_iff.1 ha]\n  exact smul_mem_smul_set hb\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_apply₂ {f : Set α → Set α → Set α}\n    (hf : ∀ a : G, a • f s t = f (a • s) (a • t)) :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (f s t) := by aesop (add simp [SetLike.le_def])\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_union :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s ∪ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_union\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_inter :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s ∩ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_inter\n\n@[to_additive]\nlemma stabilizer_inf_stabilizer_le_stabilizer_sdiff :\n    stabilizer G s ⊓ stabilizer G t ≤ stabilizer G (s \\ t) :=\n  stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_sdiff\n\n@[to_additive]\nlemma stabilizer_union_eq_left (hdisj : Disjoint s t) (hstab : stabilizer G s ≤ stabilizer G t)\n    (hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G t) :\n    stabilizer G (s ∪ t) = stabilizer G s := by\n  refine le_antisymm ?_ ?_\n  · calc\n      stabilizer G (s ∪ t)\n        ≤ stabilizer G (s ∪ t) ⊓ stabilizer G t := by simpa\n      _ ≤ stabilizer G ((s ∪ t) \\ t) := stabilizer_inf_stabilizer_le_stabilizer_sdiff\n      _ = stabilizer G s := by rw [union_sdiff_cancel_right]; simpa [← disjoint_iff_inter_eq_empty]\n  · calc\n      stabilizer G s\n        ≤ stabilizer G s ⊓ stabilizer G t := by simpa\n      _ ≤ stabilizer G (s ∪ t) := stabilizer_inf_stabilizer_le_stabilizer_union\n\n@[to_additive]\nlemma stabilizer_union_eq_right (hdisj : Disjoint s t) (hstab : stabilizer G t ≤ stabilizer G s)\n    (hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G s) :\n    stabilizer G (s ∪ t) = stabilizer G t := by\n  rw [union_comm, stabilizer_union_eq_left hdisj.symm hstab (union_comm .. ▸ hstab_union)]\n\nvariable {s : Set G}\n\nopen scoped RightActions in\n@[to_additive]\nlemma op_smul_set_stabilizer_subset (ha : a ∈ s) : (stabilizer G s : Set G) <• a ⊆ s :=\n  smul_set_subset_iff.2 fun b hb ↦ by rw [← hb]; exact smul_mem_smul_set ha\n\n@[to_additive]\nlemma stabilizer_subset_div_right (ha : a ∈ s) : ↑(stabilizer G s) ⊆ s / {a} := fun b hb ↦\n  ⟨_, by rwa [← smul_eq_mul, mem_stabilizer_set.1 hb], _, mem_singleton _, mul_div_cancel_right _ _⟩\n\n@[to_additive]\nlemma stabilizer_finite (hs₀ : s.Nonempty) (hs : s.Finite) : (stabilizer G s : Set G).Finite := by\n  obtain ⟨a, ha⟩ := hs₀\n  exact (hs.div <| finite_singleton _).subset <| stabilizer_subset_div_right ha\n\nend Group\n\nsection CommGroup\nvariable [CommGroup G] {s t : Set G} {a : G}\n\n@[to_additive]\nlemma smul_set_stabilizer_subset (ha : a ∈ s) : a • (stabilizer G s : Set G) ⊆ s := by\n  simpa using op_smul_set_stabilizer_subset ha\n\nend CommGroup\nend Set\n\nvariable [Group G] [Group H] [MulAction G α] {a : G}\n\n/-! ### Stabilizer of a subgroup -/\n\nsection Subgroup\n\n-- TODO: Is there a lemma that could unify the following three very similar lemmas?\n\n@[to_additive (attr := simp)]\nlemma stabilizer_subgroup (s : Subgroup G) : stabilizer G (s : Set G) = s := by\n  simp_rw [SetLike.ext_iff, mem_stabilizer_set]\n  refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_left ha⟩\n  simpa only [smul_eq_mul, SetLike.mem_coe, mul_one] using (h 1).2 s.one_mem\n\n@[to_additive (attr := simp)]\nlemma stabilizer_op_subgroup (s : Subgroup G) : stabilizer Gᵐᵒᵖ (s : Set G) = s.op := by\n  simp_rw [SetLike.ext_iff, mem_stabilizer_set]\n  simp only [smul_eq_mul_unop, SetLike.mem_coe, Subgroup.mem_op, «forall», unop_op]\n  refine fun a ↦ ⟨fun h ↦ ?_, fun ha b ↦ s.mul_mem_cancel_right ha⟩\n  simpa only [op_smul_eq_mul, SetLike.mem_coe, one_mul] using (h 1).2 s.one_mem\n\n@[to_additive (attr := simp)]\n\nTarget:\nlemma stabilizer_subgroup_op (s : Subgroup Gᵐᵒᵖ) : stabilizer G (s : Set Gᵐᵒᵖ) = s.unop :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/Pointwise","family_id":"stabilizer_subgroup_op","file_id":"mathlib/Mathlib/Algebra/Pointwise/Stabilizer.lean","sample_id":"61d172a6c1c40adedfc16e781f60f95fac4d1f6ec88c11c9a5b57c8a1f496302"},"split":"train"}
{"binding":{"chosen_proof_sha256":"2ec97858c3d58fdf4fe63a79d350f35bc6aeb51e4c5540c62fc1b50ae3db70dc","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e9f1b50cf06579e6c3b06af93b5a67fd43fa3476c16efdef9f68581f80521c64","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0a540970c97c25da7889f6fd2aaf6aa1ff05b98f178319841aaf0767908b62e7","source_sha256":"a1ecfd868e0d49b8667b2cdba350ee4f2ceb12fbe0b51094508e312f262dbf5e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [toInt, (exists_translation val).choose_spec]","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":19,"token_jaccard":0.857143,"token_length_ratio":1.583333},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"ee05fc5cdb91ecf72849f9fbf2d659f99dc88d53d3a5bc55b83c7e85d43a513c","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.List.Pairwise\npublic import Batteries.Tactic.GeneralizeProofs\npublic import Mathlib.Tactic.Order.CollectFacts\n\nNamespace:\nMathlib.Tactic.Order.ToInt\n\nLocal context:\n/-\nCopyright (c) 2025 Vasilii Nesterov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Vasilii Nesterov\n-/\npublic meta import Mathlib.Util.AtomM\npublic meta import Mathlib.Util.Qq\npublic meta import Std.Data.HashMap.AdditionalOperations\n\n/-!\n# Translating linear orders to ℤ\n\nIn this file we implement the translation of a problem in any linearly ordered type to a problem in\n`ℤ`. This allows us to use the `lia` tactic to solve it.\n\nWhile the core algorithm of the `order` tactic is complete for the theory of linear orders in the\nsignature (`<`, `≤`),\nit becomes incomplete in the signature with lattice operations `⊓` and `⊔`. With these operations,\nthe problem becomes NP-hard, and the idea is to reuse a smart and efficient procedure, such as\n`lia`.\n\n## TODO\n\nMigrate to `grind` when it is ready.\n-/\n\npublic meta section\n\nnamespace Mathlib.Tactic.Order.ToInt\n\nvariable {α : Type*} [LinearOrder α] {n : ℕ} (val : Fin n → α)\n\n/-- The main theorem asserting the existence of a translation.\nWe use `Classical.choose` to turn this into a value for use in the `order` tactic,\nsee `toInt`.\n-/\ntheorem exists_translation : ∃ tr : Fin n → ℤ, ∀ i j, val i ≤ val j ↔ tr i ≤ tr j := by\n  let li := List.ofFn val\n  let sli := li.mergeSort\n  have (i : Fin n) : ∃ j : Fin sli.length, sli[j] = val i := by\n    apply List.get_of_mem\n    rw [List.Perm.mem_iff (List.mergeSort_perm _ _)]\n    simp [li]\n  use fun i ↦ (this i).choose\n  intro i j\n  simp only [Fin.getElem_fin, Int.ofNat_le]\n  by_cases h_eq : val i = val j\n  · simp [h_eq]\n  generalize_proofs _ hi hj\n  rw [← hi.choose_spec, ← hj.choose_spec] at h_eq\n  conv_lhs => rw [← hi.choose_spec, ← hj.choose_spec]\n  have := li.pairwise_mergeSort (le := fun a b ↦ decide (a ≤ b))\n      (fun a b c ↦ by simpa using le_trans) (by simpa using le_total)\n  rw [List.pairwise_iff_get] at this\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · contrapose! h\n    exact lt_of_le_of_ne (by simpa using (this hj.choose hi.choose (by simpa)))\n      (fun h ↦ h_eq (h.symm))\n  · simpa using this hi.choose hj.choose (by apply lt_of_le_of_ne h; contrapose h_eq; simp [h_eq])\n\n/-- Auxiliary definition used by the `order` tactic to transfer facts in a linear order to `ℤ`. -/\nnoncomputable def toInt (k : Fin n) : ℤ :=\n  (exists_translation val).choose k\n\nvariable (i j k : Fin n)\n\nTarget:\ntheorem toInt_le_toInt : toInt val i ≤ toInt val j ↔ val i ≤ val j :=\n\nProof body:\n","rejected":"```lean\nby\n  simp [toInt, (exists_translation val).choose_spec]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Order","family_id":"toint_le_toint","file_id":"mathlib/Mathlib/Tactic/Order/ToInt.lean","sample_id":"0a540970c97c25da7889f6fd2aaf6aa1ff05b98f178319841aaf0767908b62e7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5a23d360fd69064fcfa72d36f2b8acb7fa78696675236423d19ef91035657ba8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"92bd547cc8e3c70d9fdff8bf30a6d290d35b3641d5d90d6ba3b22bc018d47a89","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"45bdf565a874c8590b8b9e375ecaaa19d5c42795fb168fac5a8abcf519632618","source_sha256":"7e17e88add6ba664803471632353df2ce7dad3fce436e36f21eedb5e3c301b76","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have _ := t.adj₂.isLeftAdjoint\n  refine ⟨fun h ↦ by rw [← map_rightToLeft_app]; exact G.map_epi _, fun h ↦ ?_⟩\n  rw [epi_rightToLeft_app_iff_epi_map_adj₁_unit_app]\n  simpa using epi_comp (t.adj₂.unit.app (H.obj X)) (H.map (t.adj₂.counit.app X ≫ t.adj₁.unit.app X))","hard_negative":true,"metrics":{"chosen_tokens":85,"rejected_tokens":3,"token_jaccard":0.046512,"token_length_ratio":0.035294},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"ee18c16159a818421101de62b1b8fc5cad2541a64f45cff8d53135b2214a5232","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Adjunction.Opposites\npublic import Mathlib.CategoryTheory.Adjunction.Unique\npublic import Mathlib.CategoryTheory.Monad.Adjunction\n\nNamespace:\nCategoryTheory.Adjunction.Triple\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson, Ben Eltschig\n-/\n/-!\n\n# Adjoint triples\n\nThis file concerns adjoint triples `F ⊣ G ⊣ H` of functors `F H : C ⥤ D`, `G : D ⥤ C`. We first\nprove that `F` is fully faithful iff `H` is, and then prove results about the two special cases\nwhere `G` is fully faithful or `F` and `H` are.\n\n## Main results\n\nAll results are about an adjoint triple `F ⊣ G ⊣ H` where `adj₁ : F ⊣ G` and `adj₂ : G ⊣ H`. We\nbundle the adjunctions in a structure `Triple F G H`.\n* `fullyFaithfulEquiv`: `F` is fully faithful iff `H` is.\n* `rightToLeft`: the canonical natural transformation `H ⟶ F` that exists whenever `G` is fully\n  faithful. This is defined as the preimage of `adj₂.counit ≫ adj₁.unit` under whiskering with `G`,\n  but formulas in terms of the units resp. counits of the adjunctions are also given.\n* `whiskerRight_rightToLeft`: whiskering `rightToLeft : H ⟶ F` with `G` yields\n  `adj₂.counit ≫ adj₁.unit : H ⋙ G ⟶ F ⋙ G`.\n* `epi_rightToLeft_app_iff_epi_map_adj₁_unit_app`: `rightToLeft : H ⟶ F` is epic at `X` iff the\n  image of `adj₁.unit.app X` under `H` is.\n* `epi_rightToLeft_app_iff_epi_map_adj₂_counit_app`: `rightToLeft : H ⟶ F` is epic at `X` iff the\n  image of `adj₂.counit.app X` under `F` is.\n* `epi_rightToLeft_app_iff`: when `H` preserves epimorphisms, `rightToLeft : H ⟶ F` is epic at `X`\n  iff `adj₂.counit ≫ adj₁.unit : H ⋙ G ⟶ F ⋙ G` is.\n* `leftToRight`: the canonical natural transformation `F ⟶ H` that exists whenever `F` and `H` are\n  fully faithful. This is defined in terms of the units of the adjunctions, but a formula in terms\n  of the counits is also given.\n* `whiskerLeft_leftToRight`: whiskering `G` with `leftToRight : F ⟶ H` yields\n  `adj₁.counit ≫ adj₂.unit : G ⋙ F ⟶ G ⋙ H`.\n* `mono_leftToRight_app_iff_mono_adj₂_unit_app`: `leftToRight : F ⟶ H` is monic at `X` iff\n  `adj₂.unit` is monic at `F.obj X`.\n* `mono_leftToRight_app_iff_mono_adj₁_counit_app`: `leftToRight : F ⟶ H` is monic at `X` iff\n  `adj₁.counit` is monic at `H.obj X`.\n* `mono_leftToRight_app_iff`: `leftToRight : F ⟶ H` is componentwise monic iff\n  `adj₁.counit ≫ adj₂.unit : G ⋙ F ⟶ G ⋙ H` is.\n-/\n\n@[expose] public section\n\nopen CategoryTheory Functor\n\nvariable {C D : Type*} [Category* C] [Category* D]\nvariable (F : C ⥤ D) (G : D ⥤ C) (H : C ⥤ D)\n\n/-- Structure containing the two adjunctions of an adjoint triple `F ⊣ G ⊣ H`. -/\nstructure CategoryTheory.Adjunction.Triple where\n  /-- Adjunction `F ⊣ G` of the adjoint triple `F ⊣ G ⊣ H`. -/\n  adj₁ : F ⊣ G\n  /-- Adjunction `G ⊣ H` of the adjoint triple `F ⊣ G ⊣ H`. -/\n  adj₂ : G ⊣ H\n\nnamespace CategoryTheory.Adjunction.Triple\n\nvariable {F G H} (t : Triple F G H)\n\nlemma isIso_unit_iff_isIso_counit : IsIso t.adj₁.unit ↔ IsIso t.adj₂.counit := by\n  let adj : F ⋙ G ⊣ H ⋙ G := t.adj₁.comp t.adj₂\n  constructor\n  · intro h\n    let idAdj : 𝟭 C ⊣ H ⋙ G := adj.ofNatIsoLeft (asIso t.adj₁.unit).symm\n    exact t.adj₂.isIso_counit_of_iso (idAdj.rightAdjointUniq id)\n  · intro h\n    let adjId : F ⋙ G ⊣ 𝟭 C := adj.ofNatIsoRight (asIso t.adj₂.counit)\n    exact t.adj₁.isIso_unit_of_iso (adjId.leftAdjointUniq id)\n\n/--\nGiven an adjoint triple `F ⊣ G ⊣ H`, the left adjoint `F` is fully faithful if and only if the\nright adjoint `H` is fully faithful.\n-/\nnoncomputable def fullyFaithfulEquiv : F.FullyFaithful ≃ H.FullyFaithful where\n  toFun h :=\n    haveI := h.full\n    haveI := h.faithful\n    haveI : IsIso t.adj₂.counit := by\n      rw [← t.isIso_unit_iff_isIso_counit]\n      infer_instance\n    t.adj₂.fullyFaithfulROfIsIsoCounit\n  invFun h :=\n    haveI := h.full\n    haveI := h.faithful\n    haveI : IsIso t.adj₁.unit := by\n      rw [t.isIso_unit_iff_isIso_counit]\n      infer_instance\n    t.adj₁.fullyFaithfulLOfIsIsoUnit\n  left_inv _ := Subsingleton.elim _ _\n  right_inv _ := Subsingleton.elim _ _\n\n/-- The adjoint triple `H.op ⊣ G.op ⊣ F.op` dual to an adjoint triple `F ⊣ G ⊣ H`. -/\n@[simps]\nprotected def op : Triple H.op G.op F.op where\n  adj₁ := t.adj₂.op\n  adj₂ := t.adj₁.op\n\nsection InnerFullyFaithful\n\nvariable [G.Full] [G.Faithful]\n\n/-- The natural transformation `H ⟶ F` that exists for every adjoint triple `F ⊣ G ⊣ H` where `G`\nis fully faithful, given here as the preimage of `adj₂.counit ≫ adj₁.unit : H ⋙ G ⟶ F ⋙ G`\nunder whiskering with `G`. -/\nnoncomputable def rightToLeft : H ⟶ F :=\n  ((FullyFaithful.ofFullyFaithful G).whiskeringRight _).preimage (t.adj₂.counit ≫ t.adj₁.unit)\n\n/-- For an adjoint triple `F ⊣ G ⊣ H` where `G` is fully faithful, whiskering the natural\ntransformation `H ⟶ F` with `G` yields the composition of the counit of the second adjunction with\nthe unit of the first adjunction. -/\n@[simp, reassoc]\nlemma whiskerRight_rightToLeft : whiskerRight t.rightToLeft G = t.adj₂.counit ≫ t.adj₁.unit :=\n  ((FullyFaithful.ofFullyFaithful G).whiskeringRight _).map_preimage _\n\n/-- For an adjoint triple `F ⊣ G ⊣ H` where `G` is fully faithful, the images of the components of\nthe natural transformation `H ⟶ F` under `G` are the components of the composition of counit of the\nsecond adjunction with the unit of the first adjunction. -/\n@[simp, reassoc]\nlemma map_rightToLeft_app (X : C) :\n    G.map (t.rightToLeft.app X) = t.adj₂.counit.app X ≫ t.adj₁.unit.app X :=\n  congr_app t.whiskerRight_rightToLeft X\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The natural transformation `H ⟶ F` for an adjoint triple `F ⊣ G ⊣ H` with `G` fully faithful\nis also equal to the whiskered unit `H ⟶ F ⋙ G ⋙ H` of the first adjunction followed by the\ninverse of the whiskered unit `F ⟶ F ⋙ G ⋙ H` of the second. -/\nlemma rightToLeft_eq_units :\n    t.rightToLeft = H.leftUnitor.inv ≫ whiskerRight t.adj₁.unit H ≫ (Functor.associator _ _ _).hom ≫\n    inv (whiskerLeft F t.adj₂.unit) ≫ F.rightUnitor.hom := by\n  ext X; apply G.map_injective; simp [rightToLeft]\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- The natural transformation `H ⟶ F` for an adjoint triple `F ⊣ G ⊣ H` with `G` fully faithful\nis also equal to the inverse of the whiskered counit `H ⋙ G ⋙ F ⟶ H` of the first adjunction\nfollowed by the whiskered counit `H ⋙ G ⋙ F ⟶ F` of the second. -/\nlemma rightToLeft_eq_counits :\n    t.rightToLeft = H.rightUnitor.inv ≫ inv (whiskerLeft H t.adj₁.counit) ≫\n    (Functor.associator _ _ _).inv ≫ whiskerRight t.adj₂.counit F ≫ F.leftUnitor.hom := by\n  ext X; apply G.map_injective; simp [rightToLeft]\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma adj₁_counit_app_rightToLeft_app (X : C) :\n    t.adj₁.counit.app (H.obj X) ≫ t.rightToLeft.app X = F.map (t.adj₂.counit.app X) :=\n  G.map_injective (by simp [← cancel_epi (t.adj₁.unit.app _)])\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma rightToLeft_app_adj₂_unit_app (X : C) :\n    t.rightToLeft.app X ≫ t.adj₂.unit.app (F.obj X) = H.map (t.adj₁.unit.app X) :=\n  G.map_injective (by simp [← cancel_mono (t.adj₂.counit.app _)])\n\nset_option backward.defeqAttrib.useBackward true in\n/-- For an adjoint triple `F ⊣ G ⊣ H` where `G` is fully faithful, the natural transformation\n`F.op ⟶ H.op` obtained from the dual adjoint triple `H.op ⊣ G.op ⊣ F.op` is dual to the natural\ntransformation `H ⟶ F`. -/\n@[simp]\nlemma op_rightToLeft : t.op.rightToLeft = NatTrans.op t.rightToLeft := by\n  ext\n  rw [rightToLeft_eq_units, rightToLeft_eq_counits]\n  simp\n\nset_option backward.isDefEq.respectTransparency false in\n/-- For an adjoint triple `F ⊣ G ⊣ H` where `G` is fully faithful, the natural transformation\n`H ⟶ F` is epic at `X` iff the image of the unit of the adjunction `F ⊣ G` under `H` is. -/\nlemma epi_rightToLeft_app_iff_epi_map_adj₁_unit_app {X : C} :\n    Epi (t.rightToLeft.app X) ↔ Epi (H.map (t.adj₁.unit.app X)) := by\n  rw [← epi_comp_iff_of_isIso _ (t.adj₂.unit.app (F.obj X)), rightToLeft_app_adj₂_unit_app]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- For an adjoint triple `F ⊣ G ⊣ H` where `G` is fully faithful, the natural transformation\n`H ⟶ F` is epic at `X` iff the image of the counit of the adjunction `G ⊣ H` under `F` is. -/\nlemma epi_rightToLeft_app_iff_epi_map_adj₂_counit_app {X : C} :\n    Epi (t.rightToLeft.app X) ↔ Epi (F.map (t.adj₂.counit.app X)) := by\n  rw [← epi_comp_iff_of_epi (t.adj₁.counit.app (H.obj X)), adj₁_counit_app_rightToLeft_app]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- For an adjoint triple `F ⊣ G ⊣ H` where `G` is fully faithful and `H` preserves epimorphisms\n(which is for example the case if `H` has a further right adjoint), the components of the natural\ntransformation `H ⟶ F` are epic iff the respective components of the natural transformation\n`H ⋙ G ⟶ F ⋙ G` obtained from the units and counits of the adjunctions are. -/\n\nTarget:\nlemma epi_rightToLeft_app_iff [H.PreservesEpimorphisms] {X : C} :\n    Epi (t.rightToLeft.app X) ↔ Epi (t.adj₂.counit.app X ≫ t.adj₁.unit.app X) :=\n\nProof body:\n","rejected":"by\n  exact epi_rightToLeft_app_iff","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"ddc482d0d6fba2edf01bb84f2468dd9fd76e044d18a7ea81fc0872c827bf4c6c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Adjunction","family_id":"epi_righttoleft_app_iff","file_id":"mathlib/Mathlib/CategoryTheory/Adjunction/Triple.lean","sample_id":"45bdf565a874c8590b8b9e375ecaaa19d5c42795fb168fac5a8abcf519632618"},"split":"train"}
{"binding":{"chosen_proof_sha256":"397dccc7f0b407dbb3cdb395970fa551240d91f8fe8d8b6664c36ca222b24937","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"890928cea67490b486f404ea924568c9df0ae3523215f6af6df2d816e57e58f7","source_sha256":"c729c093a684b2a7b5066e0c93a4ca1c797418af44bab051680547cf260ff85c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [h]\n\n-- used in the `nlinarith` normalization steps. The `_` argument is for uniformity.","hard_negative":false,"metrics":{"chosen_tokens":25,"rejected_tokens":2,"token_jaccard":0.047619,"token_length_ratio":0.08},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"ee3bd8c9e361c4b26e6bd1744283cee677f4c49dda674ced0123c2f4cf4cb3a2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Ineq\npublic import Mathlib.Data.Nat.Cast.Order.Ring\n\nNamespace:\nMathlib.Tactic.Linarith\n\nLocal context:\n/-\nCopyright (c) 2020 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n-/\npublic meta import Batteries.Tactic.Lint.Basic\npublic meta import Mathlib.Data.Ineq\npublic meta import Mathlib.Tactic.ToAdditive\n\n/-!\n# Lemmas for `linarith`.\n\nThose in the `Linarith` namespace should stay here.\n\nThose outside the `Linarith` namespace may be deleted as they are ported to mathlib4.\n-/\n\npublic meta section\n\nnamespace Mathlib.Tactic.Linarith\n\nuniverse u\ntheorem lt_irrefl {α : Type u} [Preorder α] {a : α} : ¬a < a := _root_.lt_irrefl a\n\ntheorem eq_of_eq_of_eq {α} [Semiring α] {a b : α} (ha : a = 0) (hb : b = 0) : a + b = 0 := by\n  simp [*]\n\nsection Semiring\nvariable {α : Type u} [Semiring α] [PartialOrder α]\n\ntheorem zero_lt_one [IsStrictOrderedRing α] : (0:α) < 1 :=\n  _root_.zero_lt_one\n\ntheorem le_of_eq_of_le {a b : α} (ha : a = 0) (hb : b ≤ 0) : a + b ≤ 0 := by\n  simp [*]\n\ntheorem lt_of_eq_of_lt {a b : α} (ha : a = 0) (hb : b < 0) : a + b < 0 := by\n  simp [*]\n\ntheorem le_of_le_of_eq {a b : α} (ha : a ≤ 0) (hb : b = 0) : a + b ≤ 0 := by\n  simp [*]\n\ntheorem lt_of_lt_of_eq {a b : α} (ha : a < 0) (hb : b = 0) : a + b < 0 := by\n  simp [*]\n\ntheorem add_nonpos [IsOrderedRing α] {a b : α} (ha : a ≤ 0) (hb : b ≤ 0) :\n    a + b ≤ 0 :=\n  _root_.add_nonpos ha hb\n\ntheorem add_lt_of_le_of_neg [IsStrictOrderedRing α] {a b c : α} (hbc : b ≤ c)\n    (ha : a < 0) : b + a < c :=\n  _root_.add_lt_of_le_of_neg hbc ha\n\ntheorem add_lt_of_neg_of_le [IsStrictOrderedRing α] {a b c : α} (ha : a < 0)\n    (hbc : b ≤ c) : a + b < c :=\n  _root_.add_lt_of_neg_of_le ha hbc\n\ntheorem add_neg [IsStrictOrderedRing α] {a b : α} (ha : a < 0)\n    (hb : b < 0) : a + b < 0 :=\n  _root_.add_neg ha hb\n\nvariable (α) in\nlemma natCast_nonneg [IsOrderedRing α] (n : ℕ) : (0 : α) ≤ n := Nat.cast_nonneg n\n\n-- used alongside `mul_neg` and `mul_nonpos`, so has the same argument pattern for uniformity\n@[nolint unusedArguments]\ntheorem mul_eq [IsOrderedRing α] {a b : α} (ha : a = 0) (_ : 0 < b) : b * a = 0 := by\n  simp [*]\n\nend Semiring\n\nsection Ring\nvariable {α : Type u} [Ring α] [PartialOrder α]\n\ntheorem mul_neg [IsStrictOrderedRing α] {a b : α} (ha : a < 0) (hb : 0 < b) : b * a < 0 :=\n  have : (-b)*a > 0 := mul_pos_of_neg_of_neg (neg_neg_of_pos hb) ha\n  neg_of_neg_pos (by simpa)\n\ntheorem mul_nonpos [IsOrderedRing α] {a b : α} (ha : a ≤ 0) (hb : 0 < b) : b * a ≤ 0 :=\n  have : (-b)*a ≥ 0 := mul_nonneg_of_nonpos_of_nonpos (le_of_lt (neg_neg_of_pos hb)) ha\n  by simpa\n\ntheorem sub_nonpos_of_le [IsOrderedRing α] {a b : α} : a ≤ b → a - b ≤ 0 :=\n  _root_.sub_nonpos_of_le\n\ntheorem sub_neg_of_lt [IsOrderedRing α] {a b : α} : a < b → a - b < 0 :=\n  _root_.sub_neg_of_lt\n\nend Ring\n\n/-- Finds the name of a multiplicative lemma corresponding to an inequality strength. -/\ndef _root_.Mathlib.Ineq.toConstMulName : Ineq → Lean.Name\n  | .lt => ``mul_neg\n  | .le => ``mul_nonpos\n  | .eq => ``mul_eq\n\nlemma eq_of_not_lt_of_not_gt {α} [LinearOrder α] (a b : α) (h1 : ¬ a < b) (h2 : ¬ b < a) : a = b :=\n  le_antisymm (le_of_not_gt h2) (le_of_not_gt h1)\n\n-- used in the `nlinarith` normalization steps. The `_` argument is for uniformity.\n@[nolint unusedArguments]\n\nTarget:\nlemma mul_zero_eq {α} {R : α → α → Prop} [Semiring α] {a b : α} (_ : R a 0) (h : b = 0) :\n    a * b = 0 :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Linarith","family_id":"mul_zero_eq","file_id":"mathlib/Mathlib/Tactic/Linarith/Lemmas.lean","sample_id":"890928cea67490b486f404ea924568c9df0ae3523215f6af6df2d816e57e58f7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"82dcb0744887ee0b2b5e66ebed4e1d1db29ac81ed20eb128a4882cf8f95fc26d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"43907cb29f977ec17d763b2547778aa86bd01db31fe2f305fdcaf6f7d3243fef","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f073ff6b550aeff5c73e243e84ebf2d91d1d169b0b3b2c4fe873846d588b6590","source_sha256":"3ee6cc92b0085948fb0c591bbd98433993abf27d506b4cc259fc13467d3e2a06","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [aeval_def, aeval_def, ← coe_eval₂Hom, ← coe_eval₂Hom, map_eval₂Hom, ←\n    IsScalarTower.algebraMap_eq, Function.comp_def]","hard_negative":true,"metrics":{"chosen_tokens":30,"rejected_tokens":3,"token_jaccard":0.055556,"token_length_ratio":0.1},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"ee48d6f75702c5da7975234bcda47ce6931e015f02521c36145d3c57fd46a5ab","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Subalgebra.Tower\npublic import Mathlib.Algebra.MvPolynomial.Eval\n\nNamespace:\nMvPolynomial\n\nLocal context:\n/-\nCopyright (c) 2022 Yuyang Zhao. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yuyang Zhao\n-/\n/-!\n# Algebra towers for multivariate polynomial\n\nThis file proves some basic results about the algebra tower structure for the type\n`MvPolynomial σ R`.\n\nThis structure itself is provided elsewhere as `MvPolynomial.isScalarTower`\n\nWhen you update this file, you can also try to make a corresponding update in\n`RingTheory.Polynomial.Tower`.\n-/\n\npublic section\n\n\nvariable (R A B : Type*) {σ : Type*}\n\nnamespace MvPolynomial\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\nvariable [Algebra R A] [Algebra A B] [Algebra R B]\nvariable [IsScalarTower R A B]\nvariable {R B}\n\ntheorem aeval_map_algebraMap (x : σ → B) (p : MvPolynomial σ R) :\n    aeval x (map (algebraMap R A) p) = aeval x p := by\n  rw [aeval_def, aeval_def, eval₂_map, IsScalarTower.algebraMap_eq R A B]\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\nvariable [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B]\nvariable {R A}\n\nTarget:\ntheorem aeval_algebraMap_apply (x : σ → A) (p : MvPolynomial σ R) :\n    aeval (algebraMap A B ∘ x) p = algebraMap A B (MvPolynomial.aeval x p) :=\n\nProof body:\n","rejected":"by\n  exact aeval_algebraMap_apply","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"45c36b824df0a71cc3b55591248d7a788dc004061e2e389e93371119f3b6afbb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/MvPolynomial","family_id":"aeval_algebramap_apply","file_id":"mathlib/Mathlib/RingTheory/MvPolynomial/Tower.lean","sample_id":"f073ff6b550aeff5c73e243e84ebf2d91d1d169b0b3b2c4fe873846d588b6590"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a1aa8b6078219bed7946cc7376ea4840b9f886018bc53ce22deefb93a518874b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9eef900fc40300308ff1ecb2aed829f9ba2f08a85e294734879bf55ae56f16ca","source_sha256":"fee0fafbf6e1c4c8ae7d7cbed3ad72a2c79bb54752f320fe911fff838084de11","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  change c = (ZMod.ringEquivCongr hn) (toFun n g)\n  rw [← toFun_unique' n g (ZMod.ringEquivCongr hn.symm c)\n    (fun t ht ↦ by rw [hc t ht, ZMod.ringEquivCongr_val]), ← ZMod.ringEquivCongr_symm hn,\n    RingEquiv.apply_symm_apply]","hard_negative":false,"metrics":{"chosen_tokens":58,"rejected_tokens":5,"token_jaccard":0.030303,"token_length_ratio":0.086207},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"ee6f80b73f01e3f1281a99a35b2ad1fecb2ee59ff7dd0970735351583463d57f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.Aut\npublic import Mathlib.NumberTheory.Padics.RingHoms\npublic import Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity\npublic import Mathlib.RingTheory.RootsOfUnity.Minpoly\npublic import Mathlib.FieldTheory.KrullTopology\n\nNamespace:\nmodularCyclotomicCharacter\n\nLocal context:\n/-\nCopyright (c) 2023 Hanneke Wiersema. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Hanneke Wiersema, Andrew Yang\n-/\n/-!\n\n# The cyclotomic character\n\nLet `L` be an integral domain and let `n : ℕ+` be a positive integer. If `μₙ` is the\ngroup of `n`th roots of unity in `L` then any field automorphism `g` of `L`\ninduces an automorphism of `μₙ` which, being a cyclic group, must be of\nthe form `ζ ↦ ζ^j` for some integer `j = j(g)`, well-defined in `ZMod d`, with\n`d` the cardinality of `μₙ`. The function `j` is a group homomorphism\n`(L ≃+* L) →* ZMod d`.\n\nFuture work: If `L` is separably closed (e.g. algebraically closed) and `p` is a prime\nnumber such that `p ≠ 0` in `L`, then applying the above construction with\n`n = p^i` (noting that the size of `μₙ` is `p^i`) gives a compatible collection of\ngroup homomorphisms `(L ≃+* L) →* ZMod (p^i)` which glue to give\na group homomorphism `(L ≃+* L) →* ℤₚ`; this is the `p`-adic cyclotomic character.\n\n## Important definitions\n\nLet `L` be an integral domain, `g : L ≃+* L` and `n : ℕ+`. Let `d` be the number of `n`th roots\nof `1` in `L`.\n\n* `modularCyclotomicCharacter L n hn : (L ≃+* L) →* (ZMod n)ˣ` sends `g` to the unique `j` such\n  that `g(ζ)=ζ^j` for all `ζ : rootsOfUnity n L`. Here `hn` is a proof that there\n  are `n` `n`th roots of unity in `L`.\n\n* `cyclotomicCharacter L p : (L ≃+* L) →* ℤ_[p]ˣ` sends `g` to the unique `j` such\n  that `g(ζ) = ζ ^ (j mod pⁱ)` for all `pⁱ`-th roots of unity `ζ`.\n\n  Note: This is defined to be the trivial character if `L` does not have enough roots of unity.\n\n## Implementation note\n\nIn theory this could be set up as some theory about monoids, being a character\non monoid isomorphisms, but under the hypotheses that the `n`-th roots of unity\nare cyclic. The advantage of sticking to integral domains is that finite subgroups\nare guaranteed to be cyclic, so the weaker assumption that there are `n` `n`th\nroots of unity is enough. All the applications I'm aware of are when `L` is a\nfield anyway.\n\nAlthough I don't know whether it's of any use, `modularCyclotomicCharacter'`\nis the general case for integral domains, with target in `(ZMod d)ˣ`\nwhere `d` is the number of `n`th roots of unity in `L`.\n\n## TODO\n\n* Prove the compatibility of `modularCyclotomicCharacter n` and `modularCyclotomicCharacter m`\n  if `n ∣ m`.\n\n## Tags\n\ncyclotomic character\n-/\n\n@[expose] public section\n\nuniverse u\nvariable {L : Type u} [CommRing L] [IsDomain L]\n\n/-\n\n## The mod n theory\n\n-/\n\nvariable (n : ℕ) [NeZero n]\n\ntheorem rootsOfUnity.integer_power_of_ringEquiv (g : L ≃+* L) :\n    ∃ m : ℤ, ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ m : Lˣ) := by\n  obtain ⟨m, hm⟩ := MonoidHom.map_cyclic ((g : L ≃* L).restrictRootsOfUnity n).toMonoidHom\n  exact ⟨m, fun t ↦ Units.ext_iff.1 <| SetCoe.ext_iff.2 <| hm t⟩\n\ntheorem rootsOfUnity.integer_power_of_ringEquiv' (g : L ≃+* L) :\n    ∃ m : ℤ, ∀ t ∈ rootsOfUnity n L, g (t : Lˣ) = (t ^ m : Lˣ) := by\n  simpa using rootsOfUnity.integer_power_of_ringEquiv n g\n\n/-- `modularCyclotomicCharacter_aux g n` is a non-canonical auxiliary integer `j`,\n  only well-defined modulo the number of `n`-th roots of unity in `L`, such that `g(ζ)=ζ^j`\n  for all `n`-th roots of unity `ζ` in `L`. -/\nnoncomputable def modularCyclotomicCharacter.aux (g : L ≃+* L) (n : ℕ) [NeZero n] : ℤ :=\n  (rootsOfUnity.integer_power_of_ringEquiv n g).choose\n\n-- the only thing we know about `modularCyclotomicCharacter_aux g n`\ntheorem modularCyclotomicCharacter.aux_spec (g : L ≃+* L) (n : ℕ) [NeZero n] :\n    ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ (modularCyclotomicCharacter.aux g n) : Lˣ) :=\n  (rootsOfUnity.integer_power_of_ringEquiv n g).choose_spec\n\ntheorem modularCyclotomicCharacter.pow_dvd_aux_pow_sub_aux_pow\n    (g : L ≃+* L) (p : ℕ) [Fact p.Prime] [∀ i, HasEnoughRootsOfUnity L (p ^ i)]\n    {i k : ℕ} (hi : k ≤ i) : (p : ℤ) ^ k ∣ aux g (p ^ i) - aux g (p ^ k) := by\n  obtain ⟨i, rfl⟩ := exists_add_of_le hi\n  obtain ⟨ζ, hζ⟩ := HasEnoughRootsOfUnity.exists_primitiveRoot L (p ^ (k + i))\n  have h := hζ.pow (a := p ^ i) (Nat.pos_of_neZero _) (Nat.pow_add' _ _ _)\n  have h_unit : (h.isUnit NeZero.out).unit =\n      (hζ.isUnit NeZero.out).unit ^ (p ^ i) := by ext; rfl\n  have H₁ := aux_spec g (p ^ (k + i))\n    ⟨_, (hζ.isUnit_unit NeZero.out).mem_rootsOfUnity⟩\n  have H₂ := aux_spec g (p ^ k)\n    ⟨_, (h.isUnit_unit NeZero.out).mem_rootsOfUnity⟩\n  simp only [IsUnit.unit_spec, map_pow] at H₁ H₂\n  rw [H₁, ← Units.val_pow_eq_pow_val, ← Units.ext_iff, h_unit, ← div_eq_one] at H₂\n  simp only [← zpow_natCast, ← zpow_mul, div_eq_mul_inv, ← zpow_sub] at H₂\n  rw [(hζ.isUnit_unit NeZero.out).zpow_eq_one_iff_dvd, mul_comm, ← mul_sub] at H₂\n  conv_lhs at H₂ => rw [Nat.pow_add', Nat.cast_mul]\n  rwa [mul_dvd_mul_iff_left (by simp [NeZero.ne p]), Nat.cast_pow] at H₂\n\n/-- If `g` is a ring automorphism of `L`, and `n : ℕ+`, then\n  `modularCyclotomicCharacter.toFun n g` is the `j : ZMod d` such that `g(ζ)=ζ^j` for all\n  `n`-th roots of unity. Here `d` is the number of `n`th roots of unity in `L`. -/\nnoncomputable def modularCyclotomicCharacter.toFun (n : ℕ) [NeZero n] (g : L ≃+* L) :\n    ZMod (Fintype.card (rootsOfUnity n L)) :=\n  modularCyclotomicCharacter.aux g n\n\nnamespace modularCyclotomicCharacter\n\nlocal notation \"χ₀\" => modularCyclotomicCharacter.toFun\n\n/-- The formula which characterises the output of `modularCyclotomicCharacter g n`. -/\ntheorem toFun_spec (g : L ≃+* L) {n : ℕ} [NeZero n] (t : rootsOfUnity n L) :\n    g (t : Lˣ) = (t ^ (χ₀ n g).val : Lˣ) := by\n  rw [modularCyclotomicCharacter.aux_spec g n t, ← zpow_natCast, modularCyclotomicCharacter.toFun,\n    ZMod.val_intCast, ← Subgroup.coe_zpow]\n  exact Units.ext_iff.1 <| SetCoe.ext_iff.2 <|\n    zpow_eq_zpow_emod _ pow_card_eq_one (G := rootsOfUnity n L)\n\ntheorem toFun_spec' (g : L ≃+* L) {n : ℕ} [NeZero n] {t : Lˣ} (ht : t ∈ rootsOfUnity n L) :\n    g t = t ^ (χ₀ n g).val :=\n  toFun_spec g ⟨t, ht⟩\n\ntheorem toFun_spec'' (g : L ≃+* L) {n : ℕ} [NeZero n] {t : L} (ht : IsPrimitiveRoot t n) :\n    g t = t ^ (χ₀ n g).val :=\n  toFun_spec' g (SetLike.coe_mem ht.toRootsOfUnity)\n\n/-- If g(t)=t^c for all roots of unity, then c=χ(g). -/\ntheorem toFun_unique (g : L ≃+* L) (c : ZMod (Fintype.card (rootsOfUnity n L)))\n    (hc : ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ c.val : Lˣ)) : c = χ₀ n g := by\n  apply IsCyclic.ext Nat.card_eq_fintype_card (fun ζ ↦ ?_)\n  specialize hc ζ\n  suffices ((ζ ^ c.val : Lˣ) : L) = (ζ ^ (χ₀ n g).val : Lˣ) by exact_mod_cast this\n  rw [← toFun_spec g ζ, hc]\n\ntheorem toFun_unique' (g : L ≃+* L) (c : ZMod (Fintype.card (rootsOfUnity n L)))\n    (hc : ∀ t ∈ rootsOfUnity n L, g t = t ^ c.val) : c = χ₀ n g :=\n  toFun_unique n g c (fun ⟨_, ht⟩ ↦ hc _ ht)\n\nlemma id : χ₀ n (RingEquiv.refl L) = 1 := by\n  refine (toFun_unique n (RingEquiv.refl L) 1 <| fun t ↦ ?_).symm\n  have : 1 ≤ Fintype.card { x // x ∈ rootsOfUnity n L } := Fin.size_positive'\n  obtain (h | h) := this.lt_or_eq\n  · have := Fact.mk h\n    simp [ZMod.val_one]\n  · have := Fintype.card_le_one_iff_subsingleton.mp h.ge\n    obtain rfl : t = 1 := Subsingleton.elim t 1\n    simp\n\nlemma comp (g h : L ≃+* L) : χ₀ n (g * h) =\n    χ₀ n g * χ₀ n h := by\n  refine (toFun_unique n (g * h) _ <| fun ζ ↦ ?_).symm\n  change g (h (ζ : Lˣ)) = _\n  rw [toFun_spec, ← Subgroup.coe_pow, toFun_spec, mul_comm, Subgroup.coe_pow, ← pow_mul,\n    ← Subgroup.coe_pow]\n  congr 2\n  norm_cast\n  simp only [pow_eq_pow_iff_modEq, ← ZMod.natCast_eq_natCast_iff,\n    ZMod.natCast_val, Nat.cast_mul, ZMod.cast_mul (m := orderOf ζ) orderOf_dvd_card]\n\nend modularCyclotomicCharacter\n\nvariable (L)\n\n/-- Given a positive integer `n`, `modularCyclotomicCharacter' n` is a\nmultiplicative homomorphism from the automorphisms of a field `L` to `(ℤ/dℤ)ˣ`,\nwhere `d` is the number of `n`-th roots of unity in `L`. It is uniquely\ncharacterised by the property that `g(ζ)=ζ^(modularCyclotomicCharacter n g)`\nfor `g` an automorphism of `L` and `ζ` an `n`th root of unity. -/\nnoncomputable\ndef modularCyclotomicCharacter' (n : ℕ) [NeZero n] :\n    (L ≃+* L) →* (ZMod (Fintype.card { x // x ∈ rootsOfUnity n L }))ˣ := MonoidHom.toHomUnits\n  { toFun := modularCyclotomicCharacter.toFun n\n    map_one' := modularCyclotomicCharacter.id n\n    map_mul' := modularCyclotomicCharacter.comp n }\n\nlemma modularCyclotomicCharacter'.spec' (g : L ≃+* L) {t : Lˣ} (ht : t ∈ rootsOfUnity n L) :\n    g t = t ^ ((modularCyclotomicCharacter' L n g) : ZMod\n      (Fintype.card { x // x ∈ rootsOfUnity n L })).val :=\n  modularCyclotomicCharacter.toFun_spec' g ht\n\nlemma modularCyclotomicCharacter'.unique' (g : L ≃+* L)\n    {c : ZMod (Fintype.card { x // x ∈ rootsOfUnity n L })}\n    (hc : ∀ t ∈ rootsOfUnity n L, g t = t ^ c.val) :\n    c = modularCyclotomicCharacter' L n g :=\n  modularCyclotomicCharacter.toFun_unique' _ _ _ hc\n\n/-- Given a positive integer `n` and a field `L` containing `n` `n`th roots\nof unity, `modularCyclotomicCharacter n` is a multiplicative homomorphism from the\nautomorphisms of `L` to `(ℤ/nℤ)ˣ`. It is uniquely characterised by the property that\n`g(ζ)=ζ^(modularCyclotomicCharacter n g)` for `g` an automorphism of `L` and `ζ` any `n`th root\nof unity. -/\nnoncomputable def modularCyclotomicCharacter {n : ℕ} [NeZero n]\n    (hn : Fintype.card { x // x ∈ rootsOfUnity n L } = n) :\n    (L ≃+* L) →* (ZMod n)ˣ :=\n  (Units.mapEquiv <| (ZMod.ringEquivCongr hn).toMulEquiv).toMonoidHom.comp\n  (modularCyclotomicCharacter' L n)\n\nnamespace modularCyclotomicCharacter\n\nvariable {n : ℕ} [NeZero n] (hn : Fintype.card { x // x ∈ rootsOfUnity n L } = n)\n\nlemma spec (g : L ≃+* L) {t : Lˣ} (ht : t ∈ rootsOfUnity n L) :\n    g t = t ^ ((modularCyclotomicCharacter L hn g) : ZMod n).val := by\n  rw [toFun_spec' g ht]\n  congr 1\n  exact (ZMod.ringEquivCongr_val _ _).symm\n\nTarget:\nlemma unique (g : L ≃+* L) {c : ZMod n} (hc : ∀ t ∈ rootsOfUnity n L, g t = t ^ c.val) :\n    c = modularCyclotomicCharacter L hn g :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Cyclotomic","family_id":"unique","file_id":"mathlib/Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean","sample_id":"9eef900fc40300308ff1ecb2aed829f9ba2f08a85e294734879bf55ae56f16ca"},"split":"train"}
{"binding":{"chosen_proof_sha256":"598e0d2c4be030094c0dbca69a5ab0955fb2fcd00976225f2cdbe9fd5333ec6f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0e7d9d80f9ff2362b1a3f4e14dd4db88de36ccbe64b62e3cce1c80a42b4ccb93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f81f8323cb2859a12bdc22124ed1f8f7177b728cd0e3476841f3fae387cfd30b","source_sha256":"5fbd6e0c9aed4ae0af15e49b0a292eb4e252a42e876e277383b0144f2f430dad","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [List.applyId]\n  cases h₃ : List.dlookup x (List.map Prod.toSigma (xs.zip ys)) with\n  | none =>\n    dsimp [Option.getD]\n    rw [h₁.mem_iff]\n  | some val =>\n    have h₂ : ys.Nodup := h₁.nodup_iff.1 h₀\n    replace h₁ : xs.length = ys.length := h₁.length_eq\n    dsimp\n    induction xs generalizing ys with\n    | nil => contradiction\n    | cons x' xs xs_ih =>\n      rcases ys with - | ⟨y, ys⟩\n      · cases h₃\n      dsimp [List.dlookup] at h₃; split_ifs at h₃ with h\n      · rw [Option.some_inj] at h₃\n        subst x'; subst val\n        simp only [List.mem_cons, true_or]\n      · obtain - | ⟨h₀, h₅⟩ := h₀\n        obtain - | ⟨h₂, h₄⟩ := h₂\n        have h₆ := Nat.succ.inj h₁\n        specialize xs_ih h₅ h₃ h₄ h₆\n        simp only [Ne.symm h, xs_ih, List.mem_cons]\n        suffices val ∈ ys by tauto\n        rw [← Option.mem_def, List.mem_dlookup_iff] at h₃\n        · simp only [Prod.toSigma, List.mem_map, Prod.exists] at h₃\n          rcases h₃ with ⟨a, b, h₃, h₄, h₅⟩\n          apply (List.of_mem_zip h₃).2\n        simp only [List.NodupKeys, List.keys, comp_def, Prod.fst_toSigma, List.map_map]\n        rwa [List.map_fst_zip (le_of_eq h₆)]","hard_negative":true,"metrics":{"chosen_tokens":314,"rejected_tokens":2,"token_jaccard":0.010204,"token_length_ratio":0.006369},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"ee724fde98b857ec18cbd839c62dba6a0ea7a6e96c0c5256737ec4874bc8f62b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.MLList.Basic\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.ZeroLEOne\npublic import Mathlib.Data.DFinsupp.Defs\npublic import Mathlib.Data.Finsupp.Defs\npublic import Mathlib.Data.Int.Range\npublic import Mathlib.Data.List.Sigma\npublic import Mathlib.Data.PNat.Notation\npublic import Mathlib.Tactic.Bound.Init\n\nNamespace:\nPlausible.InjectiveFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\npublic meta import Mathlib.Data.Finsupp.ToDFinsupp\npublic meta import Mathlib.Data.Int.Range\npublic meta import Mathlib.Data.List.Sigma\npublic meta import Plausible.Functions\n\n/-!\n## `Plausible`: generators for functions\n\nThis file defines `Sampleable` instances for `ℤ → ℤ` injective functions.\n\nInjective functions are generated by creating a list of numbers and\na permutation of that list. The permutation insures that every input\nis mapped to a unique output. When an input is not found in the list\nthe input itself is used as an output.\n\nInjective functions `f : α → α` could be generated easily instead of\n`ℤ → ℤ` by generating a `List α`, removing duplicates and creating a\npermutation. One has to be careful when generating the domain to make\nit vast enough that, when generating arguments to apply `f` to,\nthey argument should be likely to lie in the domain of `f`. This is\nthe reason that injective functions `f : ℤ → ℤ` are generated by\nfixing the domain to the range `[-2*size .. 2*size]`, with `size`\nthe size parameter of the `gen` monad.\n\nMuch of the machinery provided in this file is applicable to generate\ninjective functions of type `α → α` and new instances should be easy\nto define.\n\nOther classes of functions such as monotone functions can generated using\nsimilar techniques. For monotone functions, generating two lists, sorting them\nand matching them should suffice, with appropriate default values.\nSome care must be taken for shrinking such functions to make sure\ntheir defining property is invariant through shrinking. Injective\nfunctions are an example of how complicated it can get.\n-/\n\n@[expose] public meta section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type v}\n\nnamespace Plausible\n\nnamespace TotalFunction\n\nsection Finsupp\n\nvariable [DecidableEq α]\n\n/--\nThis theorem exists because plausible does not have access to dlookup but\nmathlib has all the theory for it and wants to use it. We probably want to\nbring these two together at some point.\n-/\nprivate theorem apply_eq_dlookup (m : List (Σ _ : α, β)) (y : β) (x : α) :\n     (withDefault m y).apply x = (m.dlookup x).getD y := by\n  dsimp only [apply]\n  congr 1\n  induction m with\n  | nil => simp\n  | cons p m ih =>\n    rcases p with ⟨fst, snd⟩\n    by_cases heq : fst = x\n    · simp [heq]\n    · rw [List.dlookup_cons_ne]\n      · simp [heq, ih]\n      · symm\n        simp [heq]\n\nvariable [Zero β] [DecidableEq β]\n\n/-- Map a `TotalFunction` to one whose default value is zero so that it represents a `Finsupp`. -/\n@[simp]\ndef zeroDefault : TotalFunction α β → TotalFunction α β\n  | .withDefault A _ => .withDefault A 0\n\n/-- The support of a zero default `TotalFunction`. -/\ndef zeroDefaultSupp : TotalFunction α β → Finset α\n  | .withDefault A _ =>\n    List.toFinset <| (A.dedupKeys.filter fun ab => Sigma.snd ab ≠ 0).map Sigma.fst\n\n/-- Create a finitely supported function from a total function by taking the default value to\nzero. -/\ndef applyFinsupp (tf : TotalFunction α β) : α →₀ β where\n  support := zeroDefaultSupp tf\n  toFun := tf.zeroDefault.apply\n  mem_support_toFun := by\n    intro a\n    rcases tf with ⟨A, y⟩\n    simp only [zeroDefaultSupp, List.mem_map, List.mem_filter, exists_and_right,\n      List.mem_toFinset, exists_eq_right, Sigma.exists, Ne, zeroDefault]\n    rw [apply_eq_dlookup]\n    constructor\n    · rintro ⟨od, hval, hod⟩\n      have := List.mem_dlookup (List.nodupKeys_dedupKeys A) hval\n      rw [(_ : List.dlookup a A = od)]\n      · simpa using hod\n      · simpa [List.dlookup_dedupKeys]\n    · intro h\n      use (A.dlookup a).getD (0 : β)\n      rw [← List.dlookup_dedupKeys] at h ⊢\n      simp only [h, ← List.mem_dlookup_iff A.nodupKeys_dedupKeys, not_false_iff, Option.mem_def]\n      cases haA : List.dlookup a A.dedupKeys\n      · simp [haA] at h\n      · simp\n\nvariable [SampleableExt α] [SampleableExt β] [Repr α]\n\ninstance Finsupp.sampleableExt : SampleableExt (α →₀ β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\n-- TODO: support a non-constant codomain type\ninstance DFinsupp.sampleableExt : SampleableExt (Π₀ _ : α, β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp.toDFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\nend Finsupp\nend TotalFunction\n\nopen _root_.List\n\n/-- Data structure specifying a total function using a list of pairs\nand a default value returned when the input is not in the domain of\nthe partial function.\n\n`mapToSelf f` encodes `x ↦ f x` when `x ∈ f` and `x ↦ x`,\ni.e. `x` to itself, otherwise.\n\nWe use `Σ` to encode mappings instead of `×` because we\nrely on the association list API defined in `Mathlib/Data/List/Sigma.lean`.\n-/\ninductive InjectiveFunction (α : Type u) : Type u\n  | mapToSelf (xs : List (Σ _ : α, α)) :\n      xs.map Sigma.fst ~ xs.map Sigma.snd → List.Nodup (xs.map Sigma.snd) → InjectiveFunction α\n\ninstance : Inhabited (InjectiveFunction α) :=\n  ⟨⟨[], List.Perm.nil, List.nodup_nil⟩⟩\n\nnamespace InjectiveFunction\n\n/-- Apply a total function to an argument. -/\ndef apply [DecidableEq α] : InjectiveFunction α → α → α\n  | InjectiveFunction.mapToSelf m _ _, x => (m.dlookup x).getD x\n\n/-- Produce a string for a given `InjectiveFunction`.\nThe output is of the form `[x₀ ↦ f x₀, .. xₙ ↦ f xₙ, x ↦ x]`.\nUnlike for `TotalFunction`, the default value is not a constant\nbut the identity function.\n-/\nprotected def repr [Repr α] : InjectiveFunction α → String\n  | InjectiveFunction.mapToSelf m _ _ => s! \"[{TotalFunction.reprAux m}x ↦ x]\"\n\ninstance (α : Type u) [Repr α] : Repr (InjectiveFunction α) where\n  reprPrec f _p := InjectiveFunction.repr f\n\n/-- Interpret a list of pairs as a total function, defaulting to\nthe identity function when no entries are found for a given function -/\ndef List.applyId [DecidableEq α] (xs : List (α × α)) (x : α) : α :=\n  ((xs.map Prod.toSigma).dlookup x).getD x\n\n@[simp]\ntheorem List.applyId_cons [DecidableEq α] (xs : List (α × α)) (x y z : α) :\n    List.applyId ((y, z)::xs) x = if y = x then z else List.applyId xs x := by\n  simp only [List.applyId, List.dlookup, eq_rec_constant, Prod.toSigma, List.map]\n  split_ifs <;> rfl\n\nopen Function\nopen List\n\nopen Nat\n\ntheorem List.applyId_zip_eq [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs)\n    (h₁ : xs.length = ys.length) (x y : α) (i : ℕ) (h₂ : xs[i]? = some x) :\n    List.applyId.{u} (xs.zip ys) x = y ↔ ys[i]? = some y := by\n  induction xs generalizing ys i with\n  | nil => cases h₂\n  | cons x' xs xs_ih =>\n    cases i\n    · simp only [length_cons, lt_add_iff_pos_left, add_pos_iff, Nat.lt_add_one, or_true,\n        getElem?_eq_getElem, getElem_cons_zero, Option.some.injEq] at h₂\n      subst h₂\n      cases ys\n      · cases h₁\n      · simp\n    · cases ys\n      · cases h₁\n      · obtain - | ⟨h₀, h₁⟩ := h₀\n        simp only [getElem?_cons_succ, zip_cons_cons, applyId_cons] at h₂ ⊢\n        rw [if_neg]\n        · apply xs_ih <;> solve_by_elim [Nat.succ.inj]\n        · apply h₀; apply List.mem_of_getElem? h₂\n\nTarget:\ntheorem applyId_mem_iff [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs) (h₁ : xs ~ ys)\n    (x : α) : List.applyId.{u} (xs.zip ys) x ∈ ys ↔ x ∈ xs :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_f81f8323cb28","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"14357a8b814e64f70e03d6fdfbf8e9aab01ee485782e15c79488bc1a423be472","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Testing/Plausible","family_id":"applyid_mem_iff","file_id":"mathlib/Mathlib/Testing/Plausible/Functions.lean","sample_id":"f81f8323cb2859a12bdc22124ed1f8f7177b728cd0e3476841f3fae387cfd30b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"fb278dc04d580276be7c164bc304823032754bdaf50110401d324e4d50eff42b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"96dbde4f4151db83b5ae77d34dc896b57800591a2988fbcdab30f0f92312396b","source_sha256":"6c1d4f9fced239f6f8c0c8785a4325f383edabba30747fe08c8262d3408cd815","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [mem_def, or_and_right, exists_or]","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.555556},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"eea6608762fd6737734fe12b74ea1be0ddcf199d63508ed0ebc91ba910b00eb3","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.Monoid.NatCast\npublic import Mathlib.Algebra.Ring.Nat\npublic import Mathlib.Data.Sigma.Basic\npublic import Batteries.Tactic.Lint.TypeClass\n\nNamespace:\nLists'\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# A computable model of ZFA without infinity\n\nIn this file we define finite hereditary lists. This is useful for calculations in naive set theory.\n\nWe distinguish two kinds of ZFA lists:\n* Atoms. Directly correspond to an element of the original type.\n* Proper ZFA lists. Can be thought of (but are not implemented) as a list of ZFA lists (not\n  necessarily proper).\n\nFor example, `Lists ℕ` contains stuff like `23`, `[]`, `[37]`, `[1, [[2], 3], 4]`.\n\n## Implementation note\n\nAs we want to be able to append both atoms and proper ZFA lists to proper ZFA lists, it's handy that\natoms and proper ZFA lists belong to the same type, even though atoms of `α` could be modelled as\n`α` directly. But we don't want to be able to append anything to atoms.\n\nThis calls for a two-step definition of ZFA lists:\n* First, define ZFA prelists as atoms and proper ZFA prelists. Those proper ZFA prelists are defined\n  by inductive appending of (not necessarily proper) ZFA lists.\n* Second, define ZFA lists by rubbing out the distinction between atoms and proper lists.\n\n## Main declarations\n\n* `Lists' α false`: Atoms as ZFA prelists. Basically a copy of `α`.\n* `Lists' α true`: Proper ZFA prelists. Defined inductively from the empty ZFA prelist\n  (`Lists'.nil`) and from appending a ZFA prelist to a proper ZFA prelist (`Lists'.cons a l`).\n* `Lists α`: ZFA lists. Sum of the atoms and proper ZFA prelists.\n* `Finsets α`: ZFA sets. Defined as `Lists` quotiented by `Lists.Equiv`, the extensional\n  equivalence.\n-/\n\n@[expose] public section\n\n\nvariable {α : Type*}\n\n/-- Prelists, helper type to define `Lists`. `Lists' α false` are the \"atoms\", a copy of `α`.\n`Lists' α true` are the \"proper\" ZFA prelists, inductively defined from the empty ZFA prelist and\nfrom appending a ZFA prelist to a proper ZFA prelist. It is made so that you can't append anything\nto an atom while having only one appending function for appending both atoms and proper ZFA prelists\nto a proper ZFA prelist. -/\ninductive Lists'.{u} (α : Type u) : Bool → Type u\n  | atom : α → Lists' α false\n  | nil : Lists' α true\n  | cons' {b} : Lists' α b → Lists' α true → Lists' α true\n  deriving DecidableEq\ncompile_inductive% Lists'\n\n/-- Hereditarily finite list, aka ZFA list. A ZFA list is either an \"atom\" (`b = false`),\ncorresponding to an element of `α`, or a \"proper\" ZFA list, inductively defined from the empty ZFA\nlist and from appending a ZFA list to a proper ZFA list. -/\ndef Lists (α : Type*) :=\n  Σ b, Lists' α b\n\nnamespace Lists'\n\ninstance [Inhabited α] : ∀ b, Inhabited (Lists' α b)\n  | true => ⟨nil⟩\n  | false => ⟨atom default⟩\n\n/-- Appending a ZFA list to a proper ZFA prelist. -/\ndef cons : Lists α → Lists' α true → Lists' α true\n  | ⟨_, a⟩, l => cons' a l\n\n/-- Converts a ZFA prelist to a `List` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp]\ndef toList : ∀ {b}, Lists' α b → List (Lists α)\n  | _, atom _ => []\n  | _, nil => []\n  | _, cons' a l => ⟨_, a⟩ :: l.toList\n\n@[simp]\ntheorem toList_cons (a : Lists α) (l) : toList (cons a l) = a :: l.toList := rfl\n\n/-- Converts a `List` of ZFA lists to a proper ZFA prelist. -/\n@[simp]\ndef ofList : List (Lists α) → Lists' α true\n  | [] => nil\n  | a :: l => cons a (ofList l)\n\n@[simp]\ntheorem to_ofList (l : List (Lists α)) : toList (ofList l) = l := by induction l <;> simp [*]\n\n@[simp]\ntheorem of_toList : ∀ l : Lists' α true, ofList (toList l) = l :=\n  suffices ∀ (b) (h : true = b) (l : Lists' α b),\n      let l' : Lists' α true := h ▸ l\n      ofList (toList l') = l'\n    from this _ rfl\n  fun b h l => by\n    induction l with\n    | atom => cases h\n    | nil => simp\n    | cons' b a _ IH => simpa [cons] using IH rfl\n\n/-- Recursion/induction principle for `Lists'.ofList`. -/\n@[elab_as_elim]\ndef recOfList {motive : Lists' α true → Sort*} (ofList : ∀ l, motive (ofList l)) : ∀ l, motive l :=\n  fun l ↦ cast (by simp) <| ofList (l.toList)\n\nend Lists'\n\nmutual\n  /-- Equivalence of ZFA lists. Defined inductively. -/\n  inductive Lists.Equiv : Lists α → Lists α → Prop\n    | refl (l) : Lists.Equiv l l\n    | antisymm {l₁ l₂ : Lists' α true} :\n      Lists'.Subset l₁ l₂ → Lists'.Subset l₂ l₁ → Lists.Equiv ⟨_, l₁⟩ ⟨_, l₂⟩\n\n  /-- Subset relation for ZFA lists. Defined inductively. -/\n  inductive Lists'.Subset : Lists' α true → Lists' α true → Prop\n    | nil {l} : Lists'.Subset Lists'.nil l\n    | cons {a a' l l'} :\n      Lists.Equiv a a' →\n        a' ∈ Lists'.toList l' → Lists'.Subset l l' → Lists'.Subset (Lists'.cons a l) l'\nend\n\nlocal infixl:50 \" ~ \" => Lists.Equiv\n\nnamespace Lists'\n\ninstance : HasSubset (Lists' α true) :=\n  ⟨Lists'.Subset⟩\n\n/-- ZFA prelist membership. A ZFA list is in a ZFA prelist if some element of this ZFA prelist is\nequivalent as a ZFA list to this ZFA list. -/\ninstance {b} : Membership (Lists α) (Lists' α b) :=\n  ⟨fun l a => ∃ a' ∈ l.toList, a ~ a'⟩\n\ntheorem mem_def {b a} {l : Lists' α b} : a ∈ l ↔ ∃ a' ∈ l.toList, a ~ a' :=\n  Iff.rfl\n\n@[simp]\n\nTarget:\ntheorem mem_cons {a y l} : a ∈ @cons α y l ↔ a ~ y ∨ a ∈ l :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory","family_id":"mem_cons","file_id":"mathlib/Mathlib/SetTheory/Lists.lean","sample_id":"96dbde4f4151db83b5ae77d34dc896b57800591a2988fbcdab30f0f92312396b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e064b140ea88edff320b5c2da73ccde8cd1103d2a90b6e31a432655ee413c425","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1030995450c428ffdced06a8e3ef05db04bead2eb83e57105332459e6339ac07","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0329a22051730a8fdf389196512cdc9c856c4238f5a4d017fdd705ece3a68b4f","source_sha256":"a4af3b558cb27ae34b791ce56691a1ff72f8a093d6a0881a9bb6c593dcee4522","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Semigroup.mem_center_iff]\n  exact Iff.rfl","hard_negative":true,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.272727,"token_length_ratio":0.25},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"eeacf16943bd3416a414d44bdd14ada2c7d23bf6a1993811104dc683527df04f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Submonoid.Operations\npublic import Mathlib.GroupTheory.Subsemigroup.Center\n\nNamespace:\nSubmonoid\n\nLocal context:\n/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n/-!\n# Centers of monoids\n\n## Main definitions\n\n* `Submonoid.center`: the center of a monoid\n* `AddSubmonoid.center`: the center of an additive monoid\n\nWe provide `Subgroup.center`, `AddSubgroup.center`, `Subsemiring.center`, and `Subring.center` in\nother files.\n-/\n\n@[expose] public section\n\n-- Guard against import creep\nassert_not_exists Finset\n\nnamespace Submonoid\n\nsection MulOneClass\n\nvariable (M : Type*) [MulOneClass M]\n\n/-- The center of a multiplication with unit `M` is the set of elements that commute with everything\nin `M` -/\n@[to_additive\n/-- The center of an addition with zero `M` is the set of elements that commute with everything in\n`M` -/]\ndef center : Submonoid M where\n  carrier := Set.center M\n  one_mem' := Set.one_mem_center\n  mul_mem' := Set.mul_mem_center\n\n@[to_additive]\ntheorem coe_center : ↑(center M) = Set.center M :=\n  rfl\n\n@[to_additive (attr := simp) AddSubmonoid.center_toAddSubsemigroup]\ntheorem center_toSubsemigroup : (center M).toSubsemigroup = Subsemigroup.center M :=\n  rfl\n\ninstance {M α : Type*} [Monoid M] [MulAction M α] :\n    SMulCommClass ↥(Submonoid.center M) M α where\n  smul_comm c r v := by\n    have := Semigroup.mem_center_iff.1 c.2\n    simp_rw [Submonoid.smul_def, smul_smul, this]\n\ninstance {M α : Type*} [Monoid M] [MulAction M α] :\n    SMulCommClass M (Submonoid.center M) α :=\n  SMulCommClass.symm (Submonoid.center M) M α\n\nvariable {M}\n\n/-- The center of a multiplication with unit is commutative and associative.\n\nThis is not an instance as it forms a non-defeq diamond with `Submonoid.toMonoid` in the `npow`\nfield. -/\n@[to_additive /-- The center of an addition with zero is commutative and associative. -/]\nabbrev center.commMonoid' : CommMonoid (center M) :=\n  { (center M).toMulOneClass, Subsemigroup.center.commSemigroup with }\n\n@[to_additive]\nprotected theorem center_prod {N : Type*} [MulOneClass N] :\n    center (M × N) = prod (center M) (center N) :=\n  SetLike.coe_injective Set.center_prod\n\n@[to_additive]\nprotected theorem center_pi {ι : Type*} {M : ι → Type*} [Π i, MulOneClass (M i)] :\n    center (Π i, M i) = pi .univ fun i ↦ center (M i) :=\n  SetLike.coe_injective Set.center_pi\n\nend MulOneClass\n\nsection Monoid\n\nvariable {M} [Monoid M]\n\n/-- The center of a monoid is commutative. -/\n@[to_additive]\ninstance center.commMonoid : CommMonoid (center M) :=\n  { (center M).toMonoid, Subsemigroup.center.commSemigroup with }\n\n-- no instance diamond, unlike the primed version\nexample : center.commMonoid.toMonoid = Submonoid.toMonoid (center M) := by\n  with_reducible_and_instances rfl\n\n@[to_additive]\n\nTarget:\ntheorem mem_center_iff {z : M} : z ∈ center M ↔ ∀ g, g * z = z * g :=\n\nProof body:\n","rejected":"by\n  exact mem_center_iff","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"c632675d0a6c71ce194c5c1b710cd95e9669b65b8359b345104489d3f8f4d9eb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Submonoid","family_id":"mem_center_iff","file_id":"mathlib/Mathlib/GroupTheory/Submonoid/Center.lean","sample_id":"0329a22051730a8fdf389196512cdc9c856c4238f5a4d017fdd705ece3a68b4f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"228b521f0401eea33d29434cfc850bb07b080dd681e51957a62ad89f2e88ea24","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3edd16c3ca0318c2e5d045f7be510639e5b1ebe6db412620dbe457e127e3ccb4","source_sha256":"d55fd8bd8c5069614d92f45b31915666a5cb1c81ecb8657cfb3d4cc03acc7e68","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rwa [sign_apply, if_neg <| asymm ha, if_pos]","hard_negative":false,"metrics":{"chosen_tokens":13,"rejected_tokens":2,"token_jaccard":0.076923,"token_length_ratio":0.153846},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"eee5014b17e295abd299a5b478463c2fb8bb51e5386eba100ce74c867acf7c30","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GroupWithZero.Defs\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Algebra.Order.Ring.Defs\npublic import Mathlib.Tactic.DeriveFintype  -- shake: keep (deriving handlers not tracked yet)\npublic import Mathlib.Data.Multiset.Defs\npublic import Mathlib.Data.Fintype.Defs\npublic import Mathlib.Algebra.Group.Equiv.Defs\npublic import Mathlib.Algebra.Group.Int.Defs\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Eric Rodriguez. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Rodriguez\n-/\n/-!\n# Sign type\n\nThis file defines the type of signs $\\{-1, 0, 1\\}$ and its basic arithmetic instances.\n-/\n\n@[expose] public section\n\n-- Don't generate unnecessary `sizeOf_spec` lemmas which the `simpNF` linter will complain about.\nset_option genSizeOfSpec false in\n/-- The type of signs. -/\ninductive SignType\n  | zero\n  | neg\n  | pos\n  deriving DecidableEq, Inhabited, Fintype\n\nnamespace SignType\n\ninstance : Zero SignType :=\n  ⟨zero⟩\n\ninstance : One SignType :=\n  ⟨pos⟩\n\ninstance : Neg SignType :=\n  ⟨fun s =>\n    match s with\n    | neg => pos\n    | zero => zero\n    | pos => neg⟩\n\n@[simp]\ntheorem zero_eq_zero : zero = 0 :=\n  rfl\n\n@[simp]\ntheorem neg_eq_neg_one : neg = -1 :=\n  rfl\n\n@[simp]\ntheorem pos_eq_one : pos = 1 :=\n  rfl\n\ntheorem trichotomy (a : SignType) : a = -1 ∨ a = 0 ∨ a = 1 := by\n  cases a <;> simp\n\ninstance : Mul SignType :=\n  ⟨fun x y =>\n    match x with\n    | neg => -y\n    | zero => zero\n    | pos => y⟩\n\n/-- The less-than-or-equal relation on signs. -/\nprotected inductive LE : SignType → SignType → Prop\n  | of_neg (a) : SignType.LE neg a\n  | zero : SignType.LE zero zero\n  | of_pos (a) : SignType.LE a pos\n\ninstance : LE SignType :=\n  ⟨SignType.LE⟩\n\ninstance : DecidableLE SignType := fun a b => by\n  cases a <;> cases b <;> first | exact isTrue (by constructor) | exact isFalse (by rintro ⟨_⟩)\n\n/-- We can define a `Field` instance on `SignType`, but it's not mathematically sensible,\nso we only define the `CommGroupWithZero`. -/\ninstance : CommGroupWithZero SignType where\n  inv := id\n  mul_zero a := by cases a <;> rfl\n  zero_mul a := by cases a <;> rfl\n  mul_one a := by cases a <;> rfl\n  one_mul a := by cases a <;> rfl\n  mul_inv_cancel a ha := by cases a <;> trivial\n  mul_comm := by decide\n  mul_assoc := by decide\n  exists_pair_ne := ⟨0, 1, by rintro ⟨_⟩⟩\n  inv_zero := rfl\n\ninstance : LinearOrder SignType where\n  le_refl a := by cases a <;> constructor\n  le_total := by decide\n  le_antisymm := by decide\n  le_trans := by decide\n  toDecidableLE := instDecidableLE\n\ninstance : BoundedOrder SignType where\n  top := 1\n  le_top := LE.of_pos\n  bot := -1\n  bot_le :=\n    #adaptation_note /-- https://github.com/leanprover/lean4/pull/6053\n    Added `by exact`, but don't understand why it was needed. -/\n    by exact LE.of_neg\n\ninstance : HasDistribNeg SignType where\n  neg_neg := by rintro ⟨_⟩ <;> rfl\n  neg_mul := by rintro ⟨_⟩ ⟨_⟩ <;> rfl\n  mul_neg := by rintro ⟨_⟩ ⟨_⟩ <;> rfl\n\n/-- `SignType` is equivalent to `Fin 3`. -/\ndef fin3Equiv : SignType ≃* Fin 3 where\n  toFun a :=\n    match a with\n    | 0 => ⟨0, by simp⟩\n    | 1 => ⟨1, by simp⟩\n    | -1 => ⟨2, by simp⟩\n  invFun a :=\n    match a with\n    | ⟨0, _⟩ => 0\n    | ⟨1, _⟩ => 1\n    | ⟨2, _⟩ => -1\n  left_inv a := by cases a <;> rfl\n  right_inv a :=\n    match a with\n    | ⟨0, _⟩ => by simp\n    | ⟨1, _⟩ => by simp\n    | ⟨2, _⟩ => by simp\n  map_mul' a b := by\n    cases a <;> cases b <;> rfl\n\nsection CaseBashing\n\ntheorem nonneg_iff {a : SignType} : 0 ≤ a ↔ a = 0 ∨ a = 1 := by decide +revert\n\ntheorem nonneg_iff_ne_neg_one {a : SignType} : 0 ≤ a ↔ a ≠ -1 := by decide +revert\n\ntheorem neg_one_lt_iff {a : SignType} : -1 < a ↔ 0 ≤ a := by decide +revert\n\ntheorem nonpos_iff {a : SignType} : a ≤ 0 ↔ a = -1 ∨ a = 0 := by decide +revert\n\ntheorem nonpos_iff_ne_one {a : SignType} : a ≤ 0 ↔ a ≠ 1 := by decide +revert\n\ntheorem lt_one_iff {a : SignType} : a < 1 ↔ a ≤ 0 := by decide +revert\n\n@[simp]\ntheorem neg_iff {a : SignType} : a < 0 ↔ a = -1 := by decide +revert\n\n@[simp]\ntheorem le_neg_one_iff {a : SignType} : a ≤ -1 ↔ a = -1 :=\n  le_bot_iff\n\n@[simp]\ntheorem pos_iff {a : SignType} : 0 < a ↔ a = 1 := by decide +revert\n\n@[simp]\ntheorem one_le_iff {a : SignType} : 1 ≤ a ↔ a = 1 :=\n  top_le_iff\n\n@[simp]\ntheorem neg_one_le (a : SignType) : -1 ≤ a :=\n  bot_le\n\n@[simp]\ntheorem le_one (a : SignType) : a ≤ 1 :=\n  le_top\n\n@[simp]\ntheorem not_lt_neg_one (a : SignType) : ¬a < -1 :=\n  not_lt_bot\n\n@[simp]\ntheorem not_one_lt (a : SignType) : ¬1 < a :=\n  not_top_lt\n\n@[simp]\ntheorem self_eq_neg_iff {a : SignType} : a = -a ↔ a = 0 := by decide +revert\n\n@[simp]\ntheorem neg_eq_self_iff {a : SignType} : -a = a ↔ a = 0 := by decide +revert\n\n@[simp]\ntheorem neg_eq_zero_iff {a : SignType} : -a = 0 ↔ a = 0 := by decide +revert\n\n@[simp]\ntheorem neg_one_lt_one : (-1 : SignType) < 1 :=\n  bot_lt_top\n\n@[simp]\nprotected theorem neg_le_neg_iff {a b : SignType} : -a ≤ -b ↔ b ≤ a := by decide +revert\n\n@[simp]\nprotected theorem neg_lt_neg_iff {a b : SignType} : -a < -b ↔ b < a := by decide +revert\n\nend CaseBashing\n\nsection cast\n\nvariable {α : Type*} [Zero α] [One α] [Neg α]\n\n/-- Turn a `SignType` into zero, one, or minus one. This is a coercion instance. -/\n@[coe]\ndef cast : SignType → α\n  | zero => 0\n  | pos => 1\n  | neg => -1\n\n/--\nThis can't be a `CoeTail` or `Coe` instance because we don't want it to fire when `SignType` isn't\ninvolved in the coercion (or `CoeHead` or `CoeOut` because of `outParam`s). The only other\nuser-exposed option is `CoeDep` then, which allows us to match on both given and expected type.\n-/\ninstance (s : SignType) : CoeDep SignType s α :=\n  ⟨cast s⟩\n\n/-- Casting out of `SignType` respects composition with functions preserving `0, 1, -1`. -/\nlemma map_cast' {β : Type*} [One β] [Neg β] [Zero β]\n    (f : α → β) (h₁ : f 1 = 1) (h₂ : f 0 = 0) (h₃ : f (-1) = -1) (s : SignType) :\n    f s = s := by\n  cases s <;> simp only [SignType.cast, h₁, h₂, h₃]\n\n/-- Casting out of `SignType` respects composition with suitable bundled homomorphism types. -/\nlemma map_cast {α β F : Type*} [AddGroupWithOne α] [One β] [SubtractionMonoid β]\n    [FunLike F α β] [AddMonoidHomClass F α β] [OneHomClass F α β] (f : F) (s : SignType) :\n    f s = s := by\n  apply map_cast' <;> simp\n\n@[simp]\ntheorem coe_zero : ↑(0 : SignType) = (0 : α) :=\n  rfl\n\n@[simp]\ntheorem coe_one : ↑(1 : SignType) = (1 : α) :=\n  rfl\n\n@[simp]\ntheorem coe_neg_one : ↑(-1 : SignType) = (-1 : α) :=\n  rfl\n\n@[simp, norm_cast]\nlemma coe_neg {α : Type*} [One α] [SubtractionMonoid α] (s : SignType) :\n    (↑(-s) : α) = -↑s := by\n  cases s <;> simp\n\nend cast\n\nend SignType\n\nvariable {α : Type*}\n\nopen SignType\n\nsection Preorder\n\nvariable [Zero α] [Preorder α] [DecidableLT α] {a : α}\n\n/-- The sign of an element is 1 if it's positive, -1 if negative, 0 otherwise. -/\ndef SignType.sign : α →o SignType :=\n  ⟨fun a => if 0 < a then 1 else if a < 0 then -1 else 0, fun a b h => by\n    dsimp\n    split_ifs with h₁ h₂ h₃ h₄ _ _ h₂ h₃ <;> try constructor\n    · cases lt_irrefl 0 (h₁.trans <| h.trans_lt h₃)\n    · cases h₂ (h₁.trans_le h)\n    · cases h₄ (h.trans_lt h₃)⟩\n\ntheorem sign_apply : sign a = ite (0 < a) 1 (ite (a < 0) (-1) 0) :=\n  rfl\n\n@[simp]\ntheorem sign_zero : sign (0 : α) = 0 := by simp [sign_apply]\n\n@[simp]\ntheorem sign_pos (ha : 0 < a) : sign a = 1 := by rwa [sign_apply, if_pos]\n\n@[simp]\n\nTarget:\ntheorem sign_neg (ha : a < 0) : sign a = -1 :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sign","family_id":"sign_neg","file_id":"mathlib/Mathlib/Data/Sign/Defs.lean","sample_id":"3edd16c3ca0318c2e5d045f7be510639e5b1ebe6db412620dbe457e127e3ccb4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"edfba660bad11fda81e3675a3dfc917ea4d8a1ad555ebc02b9f4efe9064c3712","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"eb30e3700a643dea643a5205a242d36b207ad0dab3863713d02015f2c9e83d3a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3b27666f8db369265222b785fd8e88c116b6148890f43b6b80939bde09ac3038","source_sha256":"4c2fd0b515ef4b06865711215377bd2bbde8a660c25b8d72d68a97c859a1494a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro y hy\n  obtain ⟨x, (rfl : f x = y), h'⟩ := exists_ne_zero_of_finsum_mem_ne_zero hy\n  grind [mem_support]","hard_negative":false,"metrics":{"chosen_tokens":26,"rejected_tokens":33,"token_jaccard":0.916667,"token_length_ratio":1.269231},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"ef0ee77312338d5339f167ddcfeeff059c016179f398ea3b6b51ed2763c6cd46","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.LocallyFinsupp\npublic import Mathlib.Topology.Spectral.Basic\n\nNamespace:\nFunction.locallyFinsupp\n\nLocal context:\n/-\nCopyright (c) 2026 Raphael Douglas Giles. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Raphael Douglas Giles\n-/\n/-!\n# Pushforward of functions with locally finite support\n\nIn this file we define the notion of the pushforward of a function with locally finite support\nbetween prespectral spaces along a spectral map. This is used for defining the (proper) pushforward\nof algebraic cycles in algebraic geometry.\n\n## Main declarations\n\n- `Function.locallyFinsupp.map`: If `f : X → Y` is a spectral map between spectral spaces and\n  `c : X → R` is locally of finite support, the pushforward of `c` along `f` at `y : Y` is\n  `∑ᶠ x ∈ f ⁻¹' {y}, c x * w x`, where `w : X → R` is a weight function.\n\n## Notes\n\nIn the case of algebraic cycles, the weight function used in `Function.locallyFinsupp.map` will be\nspecialized to the degree of the residue field extension\n(see https://stacks.math.columbia.edu/tag/02R4).\n-/\n\n@[expose] public section\n\nopen Set Order Topology TopologicalSpace\n\nvariable {X Y R : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n  {f : X → Y} (hf : IsSpectralMap f) (w : X → R)\n\nnamespace Function.locallyFinsupp\n\nvariable [Semiring R] {W : Set Y} (hW : IsOpen W) (c : Function.locallyFinsupp X R)\n  [PrespectralSpace Y]\n\nvariable (f) in\n/--\nThe pushforward of a function `c` of locally finite support by a spectral map with respect to a\nweight function `w`.\n-/\nnoncomputable\ndef map (hf : IsSpectralMap f) (c : locallyFinsupp X R) : Function.locallyFinsupp Y R where\n  toFun z := ∑ᶠ x ∈ f ⁻¹' {z}, c x * w x\n  supportWithinDomain' := by simp\n  supportLocallyFiniteWithinDomain' y _ := by\n    obtain ⟨U, hU⟩ := (PrespectralSpace.isTopologicalBasis (X := Y)).exists_subset_of_mem_open\n      (by simp : y ∈ ⊤) (by simp)\n    refine ⟨U, IsOpen.mem_nhds hU.1.1 hU.2.1, ?_⟩\n    suffices h : (U ∩ {z | (f ⁻¹' {z} ∩ support ⇑c).Nonempty}).Finite by\n      refine h.subset (inter_subset_inter_right U fun y hy ↦ ?_)\n      obtain ⟨x, (hx : f x = y), h'⟩ := exists_ne_zero_of_finsum_mem_ne_zero hy\n      use x\n      grind [mem_support]\n    suffices (f ⁻¹' (U ∩ {z | (f ⁻¹' {z} ∩ c.support).Nonempty}) ∩ c.support).Finite from\n      (this.image f).subset (fun a ha ↦ by grind [Set.Nonempty])\n    exact (c.locallyFiniteSupport.finite_inter_support_of_isCompact <| hf.2 hU.1.1 hU.1.2).subset\n      (by simp; grind)\n\n@[simp]\nlemma map_apply (hf : IsSpectralMap f) (c : locallyFinsupp X R) (y : Y) :\n    map f w hf c y = ∑ᶠ x ∈ f ⁻¹' {y}, c x * w x := rfl\n\nTarget:\nlemma support_map_subset_of_forall_mem (s : Set X) (t : Set Y) (hc : c.support ⊆ s)\n    (h : ∀ x : X, x ∈ s → w x ≠ 0 → f x ∈ t) : (map f w hf c).support ⊆ t :=\n\nProof body:\n","rejected":"```lean\nby\n  intro y hy\n  obtain ⟨x, (rfl : f x = y), h'⟩ := exists_ne_zero_of_finsum_mem_ne_zero hy\n  grind [mem_support]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/LocallyFinsupp","family_id":"support_map_subset_of_forall_mem","file_id":"mathlib/Mathlib/Topology/LocallyFinsupp/Pushforward.lean","sample_id":"3b27666f8db369265222b785fd8e88c116b6148890f43b6b80939bde09ac3038"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f13ac1fd7bb3c4deec76c60552efe11696b97aca72a742edff11d90154604aae","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1de4a2ae584ee9a9376192b0c305c1c4427057eda902e8636079376cebce05ed","source_sha256":"a9ec2e65ed93a7fc2e8ad578160c3cad69b1b5b9ffe2a184226c12a8f804dde0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply elim_of_ne\n  rintro rfl\n  exact hl.notMem hj","hard_negative":true,"metrics":{"chosen_tokens":10,"rejected_tokens":8,"token_jaccard":0.0625,"token_length_ratio":0.8},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"ef159c1a3602bb1c572f6467f3c6ca1440f5f786f4d9d12dd203e1f193b3a54f","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.List.Nodup\npublic import Mathlib.Data.Set.Prod\n\nNamespace:\nList.TProd\n\nLocal context:\n/-\nCopyright (c) 2020 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn\n-/\n/-!\n# Finite products of types\n\nThis file defines the product of types over a list. For `l : List ι` and `α : ι → Type v` we define\n`List.TProd α l = l.foldr (fun i β ↦ α i × β) PUnit`.\nThis type should not be used if `∀ i, α i` or `∀ i ∈ l, α i` can be used instead\n(in the last expression, we could also replace the list `l` by a set or a finset).\nThis type is used as an intermediary between binary products and finitary products.\nThe application of this type is finitary product measures, but it could be used in any\nconstruction/theorem that is easier to define/prove on binary products than on finitary products.\n\n* Once we have the construction on binary products (like binary product measures in\n  `MeasureTheory.prod`), we can easily define a finitary version on the type `TProd l α`\n  by iterating. Properties can also be easily extended from the binary case to the finitary case\n  by iterating.\n* Then we can use the equivalence `List.TProd.piEquivTProd` below (or enhanced versions of it,\n  like a `MeasurableEquiv` for product measures) to get the construction on `∀ i : ι, α i`, at\n  least when assuming `[Fintype ι] [Encodable ι]` (using `Encodable.sortedUniv`).\n  Using `attribute [local instance] Fintype.toEncodable` we can get rid of the argument\n  `[Encodable ι]`.\n\n## Main definitions\n\n* We have the equivalence `TProd.piEquivTProd : (∀ i, α i) ≃ TProd α l`\n  if `l` contains every element of `ι` exactly once.\n* The product of sets is `Set.tprod : (∀ i, Set (α i)) → Set (TProd α l)`.\n-/\n\n@[expose] public section\n\n\nopen List Function\nuniverse u v\nvariable {ι : Type u} {α : ι → Type v} {i j : ι} {l : List ι}\n\nnamespace List\n\nvariable (α) in\n/-- The product of a family of types over a list. -/\nabbrev TProd (l : List ι) : Type v :=\n  l.foldr (fun i β => α i × β) PUnit\n\nnamespace TProd\n\n/-- Turning a function `f : ∀ i, α i` into an element of the iterated product `TProd α l`. -/\nprotected def mk : ∀ (l : List ι) (_f : ∀ i, α i), TProd α l\n  | [] => fun _ => PUnit.unit\n  | i :: is => fun f => (f i, TProd.mk is f)\n\ninstance [∀ i, Inhabited (α i)] : Inhabited (TProd α l) :=\n  ⟨TProd.mk l default⟩\n\n@[simp]\ntheorem fst_mk (i : ι) (l : List ι) (f : ∀ i, α i) : (TProd.mk (i :: l) f).1 = f i :=\n  rfl\n\n@[simp]\ntheorem snd_mk (i : ι) (l : List ι) (f : ∀ i, α i) :\n    (TProd.mk.{u, v} (i :: l) f).2 = TProd.mk.{u, v} l f :=\n  rfl\n\nvariable [DecidableEq ι]\n\n/-- Given an element of the iterated product `l.Prod α`, take a projection into direction `i`.\n  If `i` appears multiple times in `l`, this chooses the first component in direction `i`. -/\nprotected def elim : ∀ {l : List ι} (_ : TProd α l) {i : ι} (_ : i ∈ l), α i\n  | i :: is, v, j, hj =>\n    if hji : j = i then by\n      subst hji\n      exact v.1\n    else TProd.elim v.2 ((List.mem_cons.mp hj).resolve_left hji)\n\n@[simp]\ntheorem elim_self (v : TProd α (i :: l)) : v.elim mem_cons_self = v.1 := by simp [TProd.elim]\n\n@[simp]\ntheorem elim_of_ne (hj : j ∈ i :: l) (hji : j ≠ i) (v : TProd α (i :: l)) :\n    v.elim hj = TProd.elim v.2 ((List.mem_cons.mp hj).resolve_left hji) := by simp [TProd.elim, hji]\n\n@[simp]\n\nTarget:\ntheorem elim_of_mem (hl : (i :: l).Nodup) (hj : j ∈ l) (v : TProd α (i :: l)) :\n    v.elim (mem_cons_of_mem _ hj) = TProd.elim v.2 hj :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b312ad9c187435d65336cfd9c9f55003070b241ed639aad0060f412837b94a1b","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Prod","family_id":"elim_of_mem","file_id":"mathlib/Mathlib/Data/Prod/TProd.lean","sample_id":"1de4a2ae584ee9a9376192b0c305c1c4427057eda902e8636079376cebce05ed"},"split":"train"}
{"binding":{"chosen_proof_sha256":"36cd5f448af1bf1ecf44142699671ba22a07e7da26e9b39c0449f7bdf315a9b7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5e5a3af9a844517e88c0337432636443b0443a851921ceae5bc6c7fa1f90fec9","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dfed06d389dc5d68e96360f58a96e6261e6d908c48220f135bdf99a31b4e44ba","source_sha256":"a852fe8639a8e4d7780824b945071eaf8d714c077a80f3e3c0dcb22ec20d5088","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨P, φ, hsurj⟩ := H\n  obtain ⟨g, rfl⟩ := h\n  obtain ⟨a, ha⟩ := hsurj (algebraMap _ _ g)\n  have : IsLocalization.Away (f * g) (Localization.Away (φ a)) :=\n    ha ▸ .mul' (Localization.Away f) _ _ _\n  have : IsStandardEtale R (Localization.Away a) := .of_isLocalizationAway a\n  exact .mk _ (IsLocalization.Away.mapₐ_surjective_of_surjective\n    (Aₚ := Localization.Away a) (Bₚ := Localization.Away (φ a)) a hsurj)","hard_negative":true,"metrics":{"chosen_tokens":116,"rejected_tokens":5,"token_jaccard":0.071429,"token_length_ratio":0.043103},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"ef33b2d6d5337b62730b97d7c8628f39f6a24004f44f30c771b97036d40c1335","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Etale.StandardEtale\npublic import Mathlib.RingTheory.LocalRing.ResidueField.Instances\npublic import Mathlib.RingTheory.RingHom.StandardSmooth\npublic import Mathlib.RingTheory.Unramified.LocalRing\npublic import Mathlib.RingTheory.ZariskisMainTheorem\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Local structure of unramified algebras\n\nIn this file, we will prove that if `S` is a finite type `R`-algebra unramified at `Q`, then\nthere exists `f ∉ Q` and a standard etale algebra `A` over `R` that surjects onto `S[1/f]`.\nGeometrically, this says that unramified morphisms locally are closed subsets of etale covers.\n\nAs a corollary, we also obtain results about the local structure of etale and smooth algebras.\n\n## Main definition and results\n- `HasStandardEtaleSurjectionOn`: The predicate\n  \"there exists a standard etale algebra `A` over `R` that surjects onto `S[1/f]`\".\n- `Algebra.IsUnramifiedAt.exists_hasStandardEtaleSurjectionOn`:\n  If `S` is a finite type `R`-algebra that is unramified at a prime `p`, then\n  there exists a standard etale algebra over `R` that surjects onto `S[1/f]` for some `f ∉ p`.\n- `Algebra.IsEtaleAt.exists_isStandardEtale`:\n  If `S` is a finitely presented `R`-algebra that is etale at a prime `p`, then\n  `S[1/f]` is standard etale for some `f ∉ p`.\n- `Algebra.IsSmoothAt.exists_isStandardEtale_mvPolynomial`:\n  If `S` is a finitely presented `R`-algebra that is smooth at a prime `p`, then\n  there exists some `f ∉ p` such that `S[1/f]` is `R`-isomorphic to a standard etale algebra\n  over `R[x₁,...,xₙ]`.\n\n-/\n\n@[expose] public section\n\nopen Polynomial TensorProduct Algebra\n\nopen scoped nonZeroDivisors\n\nvariable {R A S : Type*} [CommRing R] [CommRing A] [CommRing S] [Algebra R S] [Algebra R A]\n\nvariable (R) in\n/-- The predicate \"there exists a standard etale algebra `A` over `R` that surjects onto `S[1/f]`\".\nWe shall show if `S` is `R`-unramified at `Q` then there exists `f ∉ Q` satisfying it. -/\ndef HasStandardEtaleSurjectionOn (f : S) : Prop :=\n  ∃ (P : StandardEtalePair R) (φ : P.Ring →ₐ[R] Localization.Away f), Function.Surjective φ\n\nlemma HasStandardEtaleSurjectionOn.mk [IsStandardEtale R A]\n    {Sf : Type*} [CommRing Sf] [Algebra R Sf] [Algebra S Sf] [IsScalarTower R S Sf]\n    {f : S} [IsLocalization.Away f Sf] (φ : A →ₐ[R] Sf) (H : Function.Surjective φ) :\n    HasStandardEtaleSurjectionOn R f :=\n  let P : StandardEtalePresentation R A := Nonempty.some inferInstance\n  ⟨P.P, (((IsLocalization.algEquiv (.powers f) (Localization.Away f) Sf).restrictScalars R)\n    |>.symm.toAlgHom).comp (φ.comp P.equivRing.symm.toAlgHom), by simpa⟩\n\nTarget:\nlemma HasStandardEtaleSurjectionOn.of_dvd\n    {f g : S} (H : HasStandardEtaleSurjectionOn R f) (h : f ∣ g) :\n    HasStandardEtaleSurjectionOn R g :=\n\nProof body:\n","rejected":"by\n  exact HasStandardEtaleSurjectionOn.of_dvd","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"0889cafc10672d8068b2176e3a3eee7572788cd57e50d584d8565308d88a0af7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Unramified","family_id":"hasstandardetalesurjectionon","file_id":"mathlib/Mathlib/RingTheory/Unramified/LocalStructure.lean","sample_id":"dfed06d389dc5d68e96360f58a96e6261e6d908c48220f135bdf99a31b4e44ba"},"split":"train"}
{"binding":{"chosen_proof_sha256":"06f4331fcea43b6ff267c69e7aee0a119d3182db477558ac433ba49db7222320","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9ad8141b680b0c8ac6adb8c9f1a6e8450d259fc0383bdc75fa9af0d259f636fb","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"23f66eb623e6a66dd8750aa398395aca87f38f1c676ee283cf181c130d4fe931","source_sha256":"cc2a1bcce9a68ef1d53580ab6c1e41e421aac9f362e958d30242edc06eefee5f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let K := Algebra.adjoin F (Set.range x)\n  suffices #E = #K by simp [K, this, ← lift_mk_eq'.2 ⟨hx.1.aevalEquiv.toEquiv⟩]\n  haveI : Algebra.IsAlgebraic K E := hx.isAlgebraic\n  refine le_antisymm ?_ (mk_le_of_injective Subtype.val_injective)\n  haveI : Infinite K := hx.1.aevalEquiv.infinite_iff.1 inferInstance\n  simpa only [sup_eq_left.2 (aleph0_le_mk K)] using Algebra.IsAlgebraic.cardinalMk_le_max K E","hard_negative":true,"metrics":{"chosen_tokens":96,"rejected_tokens":5,"token_jaccard":0.037037,"token_length_ratio":0.052083},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"ef540060d6cc529e4a7cc4a5a7a0ee2f8113a24bcdc3a28230a8baacb9aca582","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.IntermediateField.Adjoin.Basic\npublic import Mathlib.FieldTheory.MvRatFunc.Rank\npublic import Mathlib.RingTheory.Algebraic.Cardinality\npublic import Mathlib.RingTheory.AlgebraicIndependent.Adjoin\npublic import Mathlib.RingTheory.AlgebraicIndependent.Transcendental\npublic import Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Cardinality of a transcendence basis\n\nThis file concerns the cardinality of a transcendence basis.\n\n## References\n\n* [Stacks: Transcendence](https://stacks.math.columbia.edu/tag/030D)\n\n## Tags\ntranscendence basis, transcendence degree, transcendence\n\n-/\n\npublic section\n\nnoncomputable section\n\nopen Function Set Subalgebra MvPolynomial Algebra\n\nuniverse u v w\n\nopen AlgebraicIndependent\n\nopen Cardinal\n\nTarget:\ntheorem IsTranscendenceBasis.lift_cardinalMk_eq_max_lift\n    {F : Type u} {E : Type v} [CommRing F] [Nontrivial F] [CommRing E] [IsDomain E] [Algebra F E]\n    {ι : Type w} {x : ι → E} [Nonempty ι] (hx : IsTranscendenceBasis F x) :\n    lift.{max u w} #E = lift.{max v w} #F ⊔ lift.{max u v} #ι ⊔ ℵ₀ :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_23f66eb623e6","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"b8b545dda2e848be3ee8f3769b4e9cdd1a5e03889479bc7f1dec4ad4a0aa04ac","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/AlgebraicIndependent","family_id":"istranscendencebasis","file_id":"mathlib/Mathlib/RingTheory/AlgebraicIndependent/RankAndCardinality.lean","sample_id":"23f66eb623e6a66dd8750aa398395aca87f38f1c676ee283cf181c130d4fe931"},"split":"train"}
{"binding":{"chosen_proof_sha256":"caf7c1208514616ff661b06fa1617859cfde2ba543e0b78c240ce9e176c8198e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"50e5e16cfcfd38d6568b5e49b24fb4dda95d8927c47ec6721aa44339e4393525","source_sha256":"1d77f6aaa753df530e96edf4819026a453d32ae6fc473ac307ea722c26a6c807","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa using h [[]] [[], []]","hard_negative":true,"metrics":{"chosen_tokens":15,"rejected_tokens":8,"token_jaccard":0.076923,"token_length_ratio":0.533333},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"ef82e70677932199ade1f3599e4ab697891a57e01986703e2586193dc2b90756","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Basic\n\nNamespace:\nInformationTheory\n\nLocal context:\n/-\nCopyright (c) 2026 Elazar Gershuni. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Elazar Gershuni\n-/\n/-!\n# Uniquely Decodable Codes\n\nThis file defines uniquely decodable codes and proves basic properties.\n\n## Main definitions\n\n* `UniquelyDecodable`: A set of codewords is uniquely decodable if distinct concatenations\n  of codewords yield distinct strings.\n\n## Main results\n\n* `UniquelyDecodable.epsilon_not_mem`: Uniquely decodable codes cannot contain the empty\n  string.\n* `UniquelyDecodable.flatten_injective`: The flatten function is injective on lists of\n  codewords from a uniquely decodable code.\n-/\n\n@[expose] public section\n\nnamespace InformationTheory\n\nvariable {α : Type*}\n\n/-- A set of lists is uniquely decodable if distinct concatenations yield distinct strings. -/\ndef UniquelyDecodable (S : Set (List α)) : Prop :=\n  ∀ (L₁ L₂ : List (List α)),\n    (∀ w ∈ L₁, w ∈ S) → (∀ w ∈ L₂, w ∈ S) →\n    L₁.flatten = L₂.flatten → L₁ = L₂\n\nvariable {S : Set (List α)}\n\n/-- If a code is uniquely decodable, it does not contain the empty string.\n\nThe empty string can be \"decoded\" as either zero or two copies of itself,\nviolating unique decodability. -/\n\nTarget:\nlemma UniquelyDecodable.epsilon_not_mem\n    (h : UniquelyDecodable S) :\n    [] ∉ S :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"df3051af8ba2e3662905dff00b02f6a2922ea6ff62ed33928904990cd5926a2d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"InformationTheory/Coding","family_id":"uniquelydecodable","file_id":"mathlib/Mathlib/InformationTheory/Coding/UniquelyDecodable.lean","sample_id":"50e5e16cfcfd38d6568b5e49b24fb4dda95d8927c47ec6721aa44339e4393525"},"split":"train"}
{"binding":{"chosen_proof_sha256":"63cd494fa84f771bdfe24b254aba32aae9ae2b28ee1e64d13bc4951087609f7b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3d37610ea90fd4d8210d4ae3653f0e831fc85b4617585bafc5fe04742a37ccee","source_sha256":"54cfbd06b4a139f3904600d55743484d7ae132ed17d8577ed2bfda5db188e8e8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction γ using Quotient.ind with | mk γ =>\n  simpa [← mk_trans, ← mk_symm, ← mk_refl, eq] using Homotopic.trans_symm γ","hard_negative":false,"metrics":{"chosen_tokens":30,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.066667},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"efc83613e376964d676e178b4faeddb5ccad8e011d1756ef0af2c2bb87e8d813","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Groupoid.Grpd.Basic\npublic import Mathlib.Topology.Category.TopCat.Basic\npublic import Mathlib.Topology.Homotopy.Path\npublic import Mathlib.Data.Set.Subsingleton\n\nNamespace:\nPath.Homotopic.Quotient\n\nLocal context:\n/-\nCopyright (c) 2021 Shing Tak Lam. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Shing Tak Lam\n-/\n/-!\n# Fundamental groupoid of a space\n\nGiven a topological space `X`, we can define the fundamental groupoid of `X` to be the category with\nobjects being points of `X`, and morphisms `x ⟶ y` being paths from `x` to `y`, quotiented by\nhomotopy equivalence. With this, the fundamental group of `X` based at `x` is just the automorphism\ngroup of `x`.\n-/\n\n@[expose] public section\n\nopen CategoryTheory\n\nuniverse u\n\nvariable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\nvariable {x₀ x₁ : X}\n\nnoncomputable section\n\nopen unitInterval\n\nnamespace Path\n\nnamespace Homotopy\n\nsection\n\n/-- Auxiliary function for `reflTransSymm`. -/\ndef reflTransSymmAux (x : I × I) : ℝ :=\n  if (x.2 : ℝ) ≤ 1 / 2 then x.1 * 2 * x.2 else x.1 * (2 - 2 * x.2)\n\n@[continuity, fun_prop]\ntheorem continuous_reflTransSymmAux : Continuous reflTransSymmAux :=\n  continuous_if_le (by fun_prop) (by fun_prop) (by fun_prop) (by fun_prop) (by grind)\n\ntheorem reflTransSymmAux_mem_I (x : I × I) : reflTransSymmAux x ∈ I := by\n  dsimp only [reflTransSymmAux]\n  split_ifs\n  · constructor\n    · apply mul_nonneg <;> grind\n    · rw [mul_assoc]\n      apply mul_le_one₀ <;> grind\n  · constructor\n    · apply mul_nonneg <;> grind\n    · apply mul_le_one₀ <;> grind\n\n/-- For any path `p` from `x₀` to `x₁`, we have a homotopy from the constant path based at `x₀` to\n  `p.trans p.symm`. -/\ndef reflTransSymm (p : Path x₀ x₁) : Homotopy (Path.refl x₀) (p.trans p.symm) where\n  toFun x := p ⟨reflTransSymmAux x, reflTransSymmAux_mem_I x⟩\n  continuous_toFun := by fun_prop\n  map_zero_left := by simp [reflTransSymmAux]\n  map_one_left x := by\n    simp only [reflTransSymmAux, Path.trans]\n    cases le_or_gt (x : ℝ) 2⁻¹ with\n    | inl hx => simp [hx, ← extend_apply]\n    | inr hx =>\n      have : p.extend (2 - 2 * ↑x) = p.extend (1 - (2 * ↑x - 1)) := by ring_nf\n      simpa [hx.not_ge, ← extend_apply]\n  prop' t := by norm_num [reflTransSymmAux]\n\n/-- For any path `p` from `x₀` to `x₁`, we have a homotopy from the constant path based at `x₁` to\n  `p.symm.trans p`. -/\ndef reflSymmTrans (p : Path x₀ x₁) : Homotopy (Path.refl x₁) (p.symm.trans p) :=\n  (reflTransSymm p.symm).cast rfl <| congr_arg _ (Path.symm_symm _)\n\nend\n\nsection TransRefl\n\n/-- Auxiliary function for `trans_refl_reparam`. -/\ndef transReflReparamAux (t : I) : ℝ :=\n  if (t : ℝ) ≤ 1 / 2 then 2 * t else 1\n\n@[continuity, fun_prop]\ntheorem continuous_transReflReparamAux : Continuous transReflReparamAux :=\n  continuous_if_le (by fun_prop) (by fun_prop) (by fun_prop) (by fun_prop) (by grind)\n\ntheorem transReflReparamAux_mem_I (t : I) : transReflReparamAux t ∈ I := by\n  unfold transReflReparamAux\n  split_ifs <;> constructor <;> linarith [unitInterval.le_one t, unitInterval.nonneg t]\n\ntheorem transReflReparamAux_zero : transReflReparamAux 0 = 0 := by\n  norm_num [transReflReparamAux]\n\ntheorem transReflReparamAux_one : transReflReparamAux 1 = 1 := by\n  norm_num [transReflReparamAux]\n\ntheorem trans_refl_reparam (p : Path x₀ x₁) :\n    p.trans (Path.refl x₁) =\n      p.reparam (fun t => ⟨transReflReparamAux t, transReflReparamAux_mem_I t⟩) (by fun_prop)\n        (Subtype.ext transReflReparamAux_zero) (Subtype.ext transReflReparamAux_one) := by\n  ext\n  unfold transReflReparamAux\n  simp only [coe_reparam]\n  grind\n\n/-- For any path `p` from `x₀` to `x₁`, we have a homotopy from `p.trans (Path.refl x₁)` to `p`. -/\ndef transRefl (p : Path x₀ x₁) : Homotopy (p.trans (Path.refl x₁)) p :=\n  ((Homotopy.reparam p (fun t => ⟨transReflReparamAux t, transReflReparamAux_mem_I t⟩)\n          (by fun_prop) (Subtype.ext transReflReparamAux_zero)\n          (Subtype.ext transReflReparamAux_one)).cast\n      rfl (trans_refl_reparam p).symm).symm\n\n/-- For any path `p` from `x₀` to `x₁`, we have a homotopy from `(Path.refl x₀).trans p` to `p`. -/\ndef reflTrans (p : Path x₀ x₁) : Homotopy ((Path.refl x₀).trans p) p :=\n  (transRefl p.symm).symm₂.cast (by simp) (by simp)\n\nend TransRefl\n\nsection Assoc\n\n/-- Auxiliary function for `trans_assoc_reparam`. -/\ndef transAssocReparamAux (t : I) : ℝ :=\n  if (t : ℝ) ≤ 1 / 4 then 2 * t else if (t : ℝ) ≤ 1 / 2 then t + 1 / 4 else 1 / 2 * (t + 1)\n\n@[continuity, fun_prop]\ntheorem continuous_transAssocReparamAux : Continuous transAssocReparamAux :=\n  continuous_if_le (by fun_prop) (by fun_prop) (by fun_prop)\n    (continuous_if_le (by fun_prop) (by fun_prop) (by fun_prop) (by fun_prop)\n      (by grind)).continuousOn (by grind)\n\ntheorem transAssocReparamAux_mem_I (t : I) : transAssocReparamAux t ∈ I := by\n  unfold transAssocReparamAux\n  split_ifs <;> constructor <;> linarith [unitInterval.le_one t, unitInterval.nonneg t]\n\ntheorem transAssocReparamAux_zero : transAssocReparamAux 0 = 0 := by\n  norm_num [transAssocReparamAux]\n\ntheorem transAssocReparamAux_one : transAssocReparamAux 1 = 1 := by\n  norm_num [transAssocReparamAux]\n\ntheorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :\n    (p.trans q).trans r =\n      (p.trans (q.trans r)).reparam\n        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by fun_prop)\n        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by\n  ext x\n  simp only [transAssocReparamAux, Path.trans_apply, Function.comp_apply, Path.coe_reparam]\n  split_ifs\n  iterate 12 grind\n  · linarith\n  · linarith\n  · grind\n\n/-- For paths `p q r`, we have a homotopy from `(p.trans q).trans r` to `p.trans (q.trans r)`. -/\ndef transAssoc {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :\n    Homotopy ((p.trans q).trans r) (p.trans (q.trans r)) :=\n  ((Homotopy.reparam (p.trans (q.trans r))\n          (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by fun_prop)\n          (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one)).cast\n      rfl (trans_assoc_reparam p q r).symm).symm\n\nend Assoc\n\nend Homotopy\n\nnamespace Homotopic\n\ntheorem refl_trans (p : Path x₀ x₁) :\n    ((Path.refl x₀).trans p).Homotopic p :=\n  ⟨Homotopy.reflTrans p⟩\n\ntheorem trans_refl (p : Path x₀ x₁) :\n    (p.trans (Path.refl x₁)).Homotopic p :=\n  ⟨Homotopy.transRefl p⟩\n\ntheorem trans_symm (p : Path x₀ x₁) :\n    (p.trans p.symm).Homotopic (Path.refl x₀) :=\n  ⟨(Homotopy.reflTransSymm p).symm⟩\n\ntheorem symm_trans (p : Path x₀ x₁) :\n    (p.symm.trans p).Homotopic (Path.refl x₁) :=\n  ⟨(Homotopy.reflSymmTrans p).symm⟩\n\ntheorem trans_assoc {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :\n    ((p.trans q).trans r).Homotopic (p.trans (q.trans r)) :=\n  ⟨Homotopy.transAssoc p q r⟩\n\nnamespace Quotient\n\n@[simp, grind =]\ntheorem refl_trans (γ : Homotopic.Quotient x₀ x₁) :\n    trans (refl x₀) γ = γ := by\n  induction γ using Quotient.ind with | mk γ =>\n  simpa [← mk_trans, ← mk_refl, eq] using Homotopic.refl_trans γ\n\n@[simp, grind =]\ntheorem trans_refl (γ : Homotopic.Quotient x₀ x₁) :\n    trans γ (refl x₁) = γ := by\n  induction γ using Quotient.ind with | mk γ =>\n  simpa [← mk_trans, ← mk_refl, eq] using Homotopic.trans_refl γ\n\n@[simp, grind =]\n\nTarget:\ntheorem trans_symm (γ : Homotopic.Quotient x₀ x₁) :\n    trans γ (symm γ) = refl x₀ :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicTopology/FundamentalGroupoid","family_id":"trans_symm","file_id":"mathlib/Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean","sample_id":"3d37610ea90fd4d8210d4ae3653f0e831fc85b4617585bafc5fe04742a37ccee"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eff3de0495382d14fb1653bae8a0cc2d9bf54b578da62db37b5ffda72bfdcbdf","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"afd985109f6ef269349dc0bffcdceccc63b3d84c995b597d262f4e063e21c276","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"27c82bde5cab40ddeb3cf4fe4ff5388002b1c6d7395440a1ff8ff9c86f617bfd","source_sha256":"e91ba8652f5a9acc4cc194c5b5cca3581b49fb91d7b9b32974af75019938ff7e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [← le_bot_iff, not_le_iff_exists]","hard_negative":true,"metrics":{"chosen_tokens":8,"rejected_tokens":2,"token_jaccard":0.111111,"token_length_ratio":0.25},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"f00ee8e61073715ee7acb2f5586bbce306ced294cc03c35a440902768205a0d0","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.ObjectProperty.ClosedUnderIsomorphisms\npublic import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory\npublic import Mathlib.Order.CompleteLattice.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# ObjectProperty is a complete lattice\n\n-/\n\npublic section\n\nuniverse v u\n\nnamespace CategoryTheory.ObjectProperty\n\nvariable {C : Type u} [Category.{v} C]\n\nexample : CompleteLattice (ObjectProperty C) := inferInstance\n\nsection\n\nvariable (P Q : ObjectProperty C) (X : C)\n\n@[simp high] lemma prop_inf_iff : (P ⊓ Q) X ↔ P X ∧ Q X := Iff.rfl\n\n@[simp high] lemma prop_sup_iff : (P ⊔ Q) X ↔ P X ∨ Q X := Iff.rfl\n\ninstance nonempty_sup_left [P.Nonempty] : (P ⊔ Q).Nonempty :=\n  nonempty_of_prop (Or.inl P.prop_arbitrary)\n\ninstance nonempty_sup_right [Q.Nonempty] : (P ⊔ Q).Nonempty :=\n  nonempty_of_prop (Or.inr Q.prop_arbitrary)\n\ninstance nonempty_top [Nonempty C] : (⊤ : ObjectProperty C).Nonempty :=\n  nonempty_of_prop (X := Classical.arbitrary C) (by trivial)\n\nlemma isoClosure_sup : (P ⊔ Q).isoClosure = P.isoClosure ⊔ Q.isoClosure := by\n  ext X\n  simp only [prop_sup_iff]\n  constructor\n  · rintro ⟨Y, hY, ⟨e⟩⟩\n    simp only [prop_sup_iff] at hY\n    obtain hY | hY := hY\n    · exact Or.inl ⟨Y, hY, ⟨e⟩⟩\n    · exact Or.inr ⟨Y, hY, ⟨e⟩⟩\n  · rintro (hY | hY)\n    · exact monotone_isoClosure le_sup_left _ hY\n    · exact monotone_isoClosure le_sup_right _ hY\n\ninstance [P.IsClosedUnderIsomorphisms] [Q.IsClosedUnderIsomorphisms] :\n    (P ⊔ Q).IsClosedUnderIsomorphisms := by\n  simp only [isClosedUnderIsomorphisms_iff_isoClosure_eq_self, isoClosure_sup, isoClosure_eq_self]\n\ninstance [P.IsClosedUnderIsomorphisms] [Q.IsClosedUnderIsomorphisms] :\n    IsClosedUnderIsomorphisms (P ⊓ Q) where\n  of_iso e h := ⟨IsClosedUnderIsomorphisms.of_iso e h.1, IsClosedUnderIsomorphisms.of_iso e h.2⟩\n\ninstance : IsClosedUnderIsomorphisms (⊥ : ObjectProperty C) where\n  of_iso _ h := h\n\ninstance : IsClosedUnderIsomorphisms (⊤ : ObjectProperty C) where\n  of_iso := by simp\n\nend\n\nsection\n\nvariable {α : Sort*} (P : α → ObjectProperty C) (X : C)\n\n@[simp high] lemma prop_iSup_iff :\n    (⨆ (a : α), P a) X ↔ ∃ (a : α), P a X := by simp\n\nlemma nonempty_iSup (a : α) [(P a).Nonempty] : (⨆ a, P a).Nonempty :=\n  nonempty_of_prop ((prop_iSup_iff P _).mpr ⟨a, (P a).prop_arbitrary⟩)\n\nlemma isoClosure_iSup :\n    ((⨆ (a : α), P a)).isoClosure = ⨆ (a : α), (P a).isoClosure := by\n  refine le_antisymm ?_ ?_\n  · rintro X ⟨Y, hY, ⟨e⟩⟩\n    simp only [prop_iSup_iff] at hY ⊢\n    obtain ⟨a, hY⟩ := hY\n    exact ⟨a, _, hY, ⟨e⟩⟩\n  · simp only [iSup_le_iff]\n    intro a\n    rw [isoClosure_le_iff]\n    exact (le_iSup P a).trans (le_isoClosure _)\n\ninstance [∀ a, (P a).IsClosedUnderIsomorphisms] :\n    ((⨆ (a : α), P a)).IsClosedUnderIsomorphisms := by\n  simp only [isClosedUnderIsomorphisms_iff_isoClosure_eq_self,\n    isoClosure_iSup, isoClosure_eq_self]\n\nend\n\n@[push]\n\nTarget:\nlemma ne_bot_iff_exists (P : ObjectProperty C) : ¬ P = ⊥ ↔ ∃ X, P X :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_27c82bde5cab","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"84aa0638ac85283a283ba37781cc99c69de4a07e4a8896b2cdc264de01ee694f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/ObjectProperty","family_id":"ne_bot_iff_exists","file_id":"mathlib/Mathlib/CategoryTheory/ObjectProperty/CompleteLattice.lean","sample_id":"27c82bde5cab40ddeb3cf4fe4ff5388002b1c6d7395440a1ff8ff9c86f617bfd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ac66c6e721cc952e221200b69cfe94cb7c90e16c08e83be7b74b05ee3c9aa39f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b7150f0dc88e25f3382430b108a0281da820a234916871dd325c0904c8b12aaf","source_sha256":"dbf8a3d3c567c2c63c51c166e27c3e89875f40118b51843cf1e9965b53ac82aa","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by aesop\n\nvariable [Fintype ι]","hard_negative":false,"metrics":{"chosen_tokens":7,"rejected_tokens":2,"token_jaccard":0.125,"token_length_ratio":0.285714},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"f01d723b7e41cc3f22c75baf7d0baec0f5d98e5a6ff4f1c580478a5b112846c8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Sigma\npublic import Mathlib.Data.Fintype.OfMap\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# fintype instances for sigma types\n-/\n\npublic section\n\n\nopen Function\n\nopen Nat\n\nuniverse u v\n\nvariable {ι α : Type*} {κ : ι → Type*} [Π i, Fintype (κ i)]\n\nopen Finset\n\nlemma Set.biUnion_finsetSigma_univ (s : Finset ι) (f : Sigma κ → Set α) :\n    ⋃ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij = ⋃ i ∈ s, ⋃ j, f ⟨i, j⟩ := by aesop\n\nlemma Set.biUnion_finsetSigma_univ' (s : Finset ι) (f : Π i, κ i → Set α) :\n    ⋃ i ∈ s, ⋃ j, f i j = ⋃ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij.1 ij.2 := by aesop\n\nlemma Set.biInter_finsetSigma_univ (s : Finset ι) (f : Sigma κ → Set α) :\n    ⋂ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij = ⋂ i ∈ s, ⋂ j, f ⟨i, j⟩ := by aesop\n\nattribute [local simp] Sigma.forall in\n\nTarget:\nlemma Set.biInter_finsetSigma_univ' (s : Finset ι) (f : Π i, κ i → Set α) :\n    ⋂ i ∈ s, ⋂ j, f i j = ⋂ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij.1 ij.2 :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Fintype","family_id":"set","file_id":"mathlib/Mathlib/Data/Fintype/Sigma.lean","sample_id":"b7150f0dc88e25f3382430b108a0281da820a234916871dd325c0904c8b12aaf"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ae21492afcd16aacb24a3ebd74e7f36439c097260b4eee6f2b4d2294937fc244","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"0aeef97f4cfea55c0640efb59e6d6792baab329447e04559e5b0bd1794a70844","source_sha256":"139083b5f0b4f5b6fe528c0c9bf16219e4a814c177511d56fdc66704924ae51e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases F; rfl","hard_negative":true,"metrics":{"chosen_tokens":5,"rejected_tokens":8,"token_jaccard":0.090909,"token_length_ratio":1.6},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"f032bb11f67502421fc0445b8fc1e819184d507f9b3c12c8c9e009842d763a79","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Analysis.Filter\npublic import Mathlib.Topology.Bases\npublic import Mathlib.Topology.LocallyFinite\n\nNamespace:\nCtop\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Computational realization of topological spaces (experimental)\n\nThis file provides infrastructure to compute with topological spaces.\n\n## Main declarations\n\n* `Ctop`: Realization of a topology basis.\n* `Ctop.Realizer`: Realization of a topological space. `Ctop` that generates the given topology.\n* `LocallyFinite.Realizer`: Realization of the local finiteness of an indexed family of sets.\n* `Compact.Realizer`: Realization of the compactness of a set.\n-/\n\n@[expose] public section\n\n\nopen Set\n\nopen Filter hiding Realizer\n\nopen Topology\n\n/-- A `Ctop α σ` is a realization of a topology (basis) on `α`,\n  represented by a type `σ` together with operations for the top element and\n  the intersection operation. -/\nstructure Ctop (α σ : Type*) where\n  f : σ → Set α\n  top : α → σ\n  top_mem : ∀ x : α, x ∈ f (top x)\n  inter : ∀ (a b) (x : α), x ∈ f a ∩ f b → σ\n  inter_mem : ∀ a b x h, x ∈ f (inter a b x h)\n  inter_sub : ∀ a b x h, f (inter a b x h) ⊆ f a ∩ f b\n\nvariable {α : Type*} {β : Type*} {σ : Type*} {τ : Type*}\n\ninstance : Inhabited (Ctop α (Set α)) :=\n  ⟨{  f := id\n      top := singleton\n      top_mem := mem_singleton\n      inter := fun s t _ _ ↦ s ∩ t\n      inter_mem := fun _s _t _a ↦ id\n      inter_sub := fun _s _t _a _ha ↦ Subset.rfl }⟩\n\nnamespace Ctop\n\nsection\n\nvariable (F : Ctop α σ)\n\ninstance : CoeFun (Ctop α σ) fun _ ↦ σ → Set α :=\n  ⟨Ctop.f⟩\n\ntheorem coe_mk (f T h₁ I h₂ h₃ a) : (@Ctop.mk α σ f T h₁ I h₂ h₃) a = f a := rfl\n\n/-- Map a Ctop to an equivalent representation type. -/\ndef ofEquiv (E : σ ≃ τ) : Ctop α σ → Ctop α τ\n  | ⟨f, T, h₁, I, h₂, h₃⟩ =>\n    { f := fun a ↦ f (E.symm a)\n      top := fun x ↦ E (T x)\n      top_mem := fun x ↦ by simpa using h₁ x\n      inter := fun a b x h ↦ E (I (E.symm a) (E.symm b) x h)\n      inter_mem := fun a b x h ↦ by simpa using h₂ (E.symm a) (E.symm b) x h\n      inter_sub := fun a b x h ↦ by simpa using h₃ (E.symm a) (E.symm b) x h }\n\n@[simp]\n\nTarget:\ntheorem ofEquiv_val (E : σ ≃ τ) (F : Ctop α σ) (a : τ) : F.ofEquiv E a = F (E.symm a) :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e1bfc47e19243f1e07cf27fe20e5649df380f0cbefb679fcf6ef2a1db3b9a6c6","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Analysis","family_id":"ofequiv_val","file_id":"mathlib/Mathlib/Data/Analysis/Topology.lean","sample_id":"0aeef97f4cfea55c0640efb59e6d6792baab329447e04559e5b0bd1794a70844"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8f849e0c2e5e9b058d0be7efa5644ffd7d171530c0bf0be894d1f2aff0fde505","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6214205efd64002eb318e2ef8f8fc42439596f4d92526bc7d926061ba4591e2d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"76d9a3f6de2535e710efca9f1d5d77672d61eb6404d0c9c6e1be8e32402d6ace","source_sha256":"0ee33967ba2e7eb358f659c9411776ab2218de1f416103e609f36c526927ef27","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [wInner_cWeight_eq_expect, ψ.norm_apply]","hard_negative":false,"metrics":{"chosen_tokens":9,"rejected_tokens":13,"token_jaccard":0.75,"token_length_ratio":1.444444},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"f06aa2b37201de8b83023d4bf449e683ba1ed34caa647b711dd414336217278d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Expect\npublic import Mathlib.Algebra.Group.AddChar\npublic import Mathlib.Analysis.RCLike.Inner\n\nNamespace:\nAddChar\n\nLocal context:\n/-\nCopyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Bhavik Mehta\n-/\n/-!\n# Orthogonality of characters of a finite abelian group\n\nThis file proves that characters of a finite abelian group are orthogonal, and in particular that\nthere are at most as many characters as there are elements of the group.\n-/\n\npublic section\n\nopen Finset hiding card\nopen Fintype (card)\nopen Function RCLike\nopen scoped BigOperators ComplexConjugate DirectSum\n\nvariable {G H R : Type*}\n\nnamespace AddChar\nsection AddGroup\nvariable [AddGroup G]\n\nsection Semifield\nvariable [Fintype G] [Semifield R] [CharZero R] {ψ : AddChar G R}\n\nlemma expect_eq_ite (ψ : AddChar G R) : 𝔼 a, ψ a = if ψ = 0 then 1 else 0 := by\n  simp [Fintype.expect_eq_sum_div_card, sum_eq_ite, ite_div]\n\nlemma expect_eq_zero_iff_ne_zero : 𝔼 x, ψ x = 0 ↔ ψ ≠ 0 := by\n  rw [expect_eq_ite, one_ne_zero.ite_eq_right_iff]\n\nlemma expect_ne_zero_iff_eq_zero : 𝔼 x, ψ x ≠ 0 ↔ ψ = 0 := expect_eq_zero_iff_ne_zero.not_left\n\nend Semifield\n\nsection RCLike\nvariable [RCLike R] [Fintype G]\n\nTarget:\nlemma wInner_cWeight_self (ψ : AddChar G R) : ⟪(ψ : G → R), ψ⟫ₙ_[R] = 1 :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  simp [wInner_cWeight_eq_expect, ψ.norm_apply]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Fourier","family_id":"winner_cweight_self","file_id":"mathlib/Mathlib/Analysis/Fourier/FiniteAbelian/Orthogonality.lean","sample_id":"76d9a3f6de2535e710efca9f1d5d77672d61eb6404d0c9c6e1be8e32402d6ace"},"split":"train"}
{"binding":{"chosen_proof_sha256":"51d70e6a078b791d67a01b436c28657296809a50b25f00afb27df7a2d7b5b11b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"262e7e5adf8e8babe0e63c45b4903dee903b3ee27ba049f91b8e508b8e48e77e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2f4e8f65d6a8377847b7d2fb1bab8531851b8db89761aa4db8d2e360d21ed05e","source_sha256":"0941fd5c915c6b6e1c0d21af55aa642c5a111f5a1f5f8c8078f1d0878568b340","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply SheafOfModules.hom_ext\n  ext U x\n  exact congr($(H U.unop) x)","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":5,"token_jaccard":0.176471,"token_length_ratio":0.25},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"f0734afcf758245bb29ec920018b429d72d44c2c868c31e2f39c330526233171","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Category.ModuleCat.Sheaf.Abelian\npublic import Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits\npublic import Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackContinuous\npublic import Mathlib.AlgebraicGeometry.Modules.Presheaf\npublic import Mathlib.AlgebraicGeometry.OpenImmersion\npublic import Mathlib.CategoryTheory.Bicategory.Adjunction.Adj\npublic import Mathlib.CategoryTheory.Bicategory.Adjunction.Cat\npublic import Mathlib.CategoryTheory.Bicategory.Functor.LocallyDiscrete\n\nNamespace:\nAlgebraicGeometry.Scheme.Modules\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou, Andrew Yang\n-/\n/-!\n# The category of sheaves of modules over a scheme\n\nIn this file, we define the abelian category of sheaves of modules\n`X.Modules` over a scheme `X`, and study its basic functoriality.\n\n-/\n\n@[expose] public section\n\nuniverse t u\n\nopen CategoryTheory Limits TopologicalSpace SheafOfModules Bicategory\n\nnamespace AlgebraicGeometry.Scheme\n\nvariable {X Y Z T : Scheme.{u}}\n\nvariable (X) in\n/-- The category of sheaves of modules over a scheme. -/\ndef Modules := SheafOfModules.{u} X.ringCatSheaf\n\nnamespace Modules\n\n/-- Morphisms between `𝒪ₓ`-modules. Use `Hom.app` to act on sections. -/\ndef Hom (M N : X.Modules) : Type u := SheafOfModules.Hom M N\n\ninstance : Category X.Modules where\n  Hom := Modules.Hom\n  __ := (inferInstance : Category (SheafOfModules.{u} X.ringCatSheaf))\n\nnoncomputable instance : Abelian X.Modules :=\n  inferInstanceAs <| Abelian (SheafOfModules.{u} X.ringCatSheaf)\ninstance : HasLimits X.Modules := inferInstanceAs (HasLimits (SheafOfModules X.ringCatSheaf))\ninstance : HasColimits X.Modules := inferInstanceAs (HasColimits (SheafOfModules X.ringCatSheaf))\n\nsection Functor\n\nvariable (X) in\n/-- The forgetful functor from `𝒪ₓ`-modules to presheaves of modules.\nThis is mostly useful to transport results from (pre)sheaves of modules to `𝒪ₓ`-modules and\nusually shouldn't be used directly when working with actual `𝒪ₓ`-modules. -/\ndef toPresheafOfModules : X.Modules ⥤ X.PresheafOfModules := SheafOfModules.forget _\n\n/-- The forgetful functor from `𝒪ₓ`-modules to presheaves of modules is fully faithful. -/\ndef fullyFaithfulToPresheafOfModules : (Modules.toPresheafOfModules X).FullyFaithful :=\n  SheafOfModules.fullyFaithfulForget _\n\ninstance : (toPresheafOfModules X).Full := fullyFaithfulToPresheafOfModules.full\ninstance : (toPresheafOfModules X).Faithful := fullyFaithfulToPresheafOfModules.faithful\ninstance : (toPresheafOfModules X).IsRightAdjoint :=\n  (PresheafOfModules.sheafificationAdjunction (𝟙 X.ringCatSheaf.obj)).isRightAdjoint\n\nvariable (X) in\n/-- The forgetful functor from `𝒪ₓ`-modules to presheaves of abelian groups. -/\nnoncomputable def toPresheaf : X.Modules ⥤ TopCat.Presheaf Ab X :=\n  toPresheafOfModules X ⋙ PresheafOfModules.toPresheaf _\n\ninstance : (toPresheaf X).Faithful := .comp _ (PresheafOfModules.toPresheaf _)\ninstance : PreservesLimits (toPresheaf X) := comp_preservesLimits _ (PresheafOfModules.toPresheaf _)\ninstance : (toPresheaf X).ReflectsIsomorphisms :=\n  reflectsIsomorphisms_comp _ (PresheafOfModules.toPresheaf _)\n\nend Functor\n\nvariable {M N K : X.Modules} {φ : M ⟶ N} {U V : X.Opens}\n\nsection Presheaf\n\n/-- The underlying abelian presheaf of an `𝒪ₓ`-module. -/\nnoncomputable def presheaf (M : X.Modules) : TopCat.Presheaf Ab X := M.1.presheaf\n\n/-- Notation for sections of a presheaf of module. -/\nscoped[AlgebraicGeometry] notation3 \"Γ(\" M \", \" U \")\" => (Scheme.Modules.presheaf M).obj (.op U)\n\ninstance : Module Γ(X, U) Γ(M, U) := (M.val.obj (.op U)).isModule\n\nvariable (M) in\n@[simp] lemma map_smul (i : U ⟶ V) (r : Γ(X, V)) (x : Γ(M, V)) :\n    M.presheaf.map i.op (r • x) = X.presheaf.map i.op r • M.presheaf.map i.op x :=\n  M.val.map_smul _ _ _\n\n/-- Scalar multiplication as an endomorphism of `Γ(M, U)`. -/\ndef smul : Γ(X, U) →+* End Γ(M, U) :=\n  (M.val.obj (.op U)).smul\n\n@[simp]\nlemma smul_apply (r : Γ(X, U)) (x : Γ(M, U)) : (M.smul r).hom x = r • x := rfl\n\n@[reassoc (attr := simp)]\nlemma map_comp_smul (i : U ⟶ V) (r : Γ(X, V)) :\n    M.smul r ≫ M.presheaf.map i.op = M.presheaf.map i.op ≫ M.smul (X.presheaf.map i.op r) := by\n  ext\n  simp\n\n/-- The underlying map between abelian presheaves of a morphism of `𝒪ₓ`-modules. -/\nnoncomputable def Hom.mapPresheaf (φ : M ⟶ N) : M.presheaf ⟶ N.presheaf :=\n  (toPresheaf X).map φ\n\n/-- The application of a morphism of `𝒪ₓ`-modules to sections. -/\ndef Hom.app (φ : M ⟶ N) (U : X.Opens) : Γ(M, U) ⟶ Γ(N, U) :=\n  (forget₂ _ _).map (φ.val.app (.op U))\n\n@[simp] lemma mapPresheaf_app (φ : M ⟶ N) (U) : φ.mapPresheaf.app U = φ.app U.unop := rfl\n\n@[simp]\nlemma Hom.app_smul (φ : M ⟶ N) (r : Γ(X, U)) (x : Γ(M, U)) :\n    φ.app U (r • x) = r • φ.app U x :=\n  (φ.val.app (.op U)).hom.map_smul r x\n\n@[simp] lemma Hom.add_app (φ ψ : M ⟶ N) : (φ + ψ).app U = φ.app U + ψ.app U := rfl\n@[simp] lemma Hom.sub_app (φ ψ : M ⟶ N) : (φ - ψ).app U = φ.app U - ψ.app U := rfl\n@[simp] lemma Hom.zero_app : (0 : M ⟶ N).app U = 0 := rfl\n@[simp] lemma Hom.id_app (M : X.Modules) : (𝟙 M :).app U = 𝟙 _ := rfl\n@[simp] lemma Hom.comp_app (φ : M ⟶ N) (ψ : N ⟶ K) : (φ ≫ ψ).app U = φ.app U ≫ ψ.app U := rfl\n\n@[ext]\n\nTarget:\nlemma hom_ext (f g : M ⟶ N) (H : ∀ U, f.app U = g.app U) : f = g :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_2f4e8f65d6a8","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1d771dafcc0d7485c8a06e17a181d18253bf8b881f91c382dc5e0382af0f0065","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Modules","family_id":"hom_ext","file_id":"mathlib/Mathlib/AlgebraicGeometry/Modules/Sheaf.lean","sample_id":"2f4e8f65d6a8377847b7d2fb1bab8531851b8db89761aa4db8d2e360d21ed05e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c3e85a55413d2bdbfb981811de0e7022d287d440d63c91a6c4ae1530e90621b9","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8dd69de99772e4b60d7d7c33b72ef5655efe8caa0668569bdbff7dc2b9a6a47c","source_sha256":"da1f22be161c5bc7cbd676252bea6b38bb4aa0497872bcdf52a178b9d4951829","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simpa [beta, unbeta, coprimes] using mod_eq_of_modEq\n    ((chineseRemainderOfFinset (l[·]) (coprimes (l[·])) Finset.univ\n      (by simp [coprimes])\n      (by simpa using Set.pairwise_univ.mpr (pairwise_coprime_coprimes _))).prop i (by simp))\n    (coprimes_lt _ _)","hard_negative":false,"metrics":{"chosen_tokens":67,"rejected_tokens":3,"token_jaccard":0.034483,"token_length_ratio":0.044776},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"f094f0308943b6ea5172b1f1a5a1d6f6bd6d6bed50b7ef1ab70a783fe8e8c2af","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.ModEq\npublic import Mathlib.Data.Nat.ChineseRemainder\npublic import Mathlib.Data.Nat.Prime.Defs\npublic import Mathlib.Data.Nat.Pairing\npublic import Mathlib.Order.Fin.Basic\npublic import Mathlib.Data.Finset.Lattice.Fold\npublic import Mathlib.Data.Fintype.Basic\npublic import Mathlib.Data.Nat.Factorial.Basic\n\nNamespace:\nNat\n\nLocal context:\n/-\nCopyright (c) 2023 Shogo Saito. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Shogo Saito. Adapted for mathlib by Hunter Monroe\n-/\n/-!\n# Gödel's Beta Function Lemma\n\nThis file proves Gödel's Beta Function Lemma, used to prove the First Incompleteness Theorem. It\npermits quantification over finite sequences of natural numbers in formal theories of arithmetic.\nThis Beta Function has no connection with the unrelated Beta Function defined in analysis. Note\nthat `Nat.beta` and `Nat.unbeta` provide similar functionality to `Encodable.encodeList` and\n`Encodable.decodeList`. We define these separately, because it is easier to prove that `Nat.beta`\nand `Nat.unbeta` are arithmetically definable, and this is hard to prove that for\n`Encodable.encodeList` and `Encodable.decodeList` directly. The arithmetic\ndefinability is needed for the proof of the First Incompleteness Theorem.\n\n## Main result\n\n- `beta_unbeta_coe`: Gödel's Beta Function Lemma.\n\n## Implementation note\n\nThis code is a step towards eventually including a proof of Gödel's First Incompleteness Theorem\nand other key results from the repository https://github.com/iehality/lean4-logic.\n\n## References\n\n* [R. Kaye, *Models of Peano arithmetic*][kaye1991]\n* <https://en.wikipedia.org/wiki/G%C3%B6del%27s_%CE%B2_function>\n\n## Tags\n\nGödel, beta function\n-/\n\n@[expose] public section\n\nnamespace Nat\n\nlemma coprime_mul_succ {n m a} (ha : m - n ∣ a) : Coprime (n * a + 1) (m * a + 1) :=\n  Nat.coprime_of_dvd fun p pp hn hm => by\n    have : p ∣ (m - n) * a := by\n      simpa [Nat.succ_sub_succ, ← Nat.mul_sub_right_distrib] using\n        Nat.dvd_sub hm hn\n    have : p ∣ a := by\n      rcases (Nat.Prime.dvd_mul pp).mp this with (hp | hp)\n      · exact Nat.dvd_trans hp ha\n      · exact hp\n    apply pp.ne_one\n    simpa [Nat.add_sub_cancel_left] using Nat.dvd_sub hn (this.mul_left n)\n\nvariable {m : ℕ}\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nprivate def supOfSeq (a : Fin m → ℕ) : ℕ := max m (Finset.sup .univ a) + 1\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nprivate def coprimes (a : Fin m → ℕ) : Fin m → ℕ := fun i => (i + 1) * (supOfSeq a)! + 1\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nlemma coprimes_lt (a : Fin m → ℕ) (i) : a i < coprimes a i := by\n  have h₁ : a i < supOfSeq a :=\n    Nat.lt_add_one_iff.mpr (le_max_of_le_right <| Finset.le_sup (by simp))\n  have h₂ : supOfSeq a ≤ (i + 1) * (supOfSeq a)! + 1 :=\n    le_trans (self_le_factorial _) (le_trans (Nat.le_mul_of_pos_left (supOfSeq a)! (succ_pos i))\n      (le_add_right _ _))\n  simpa only [coprimes] using lt_of_lt_of_le h₁ h₂\n\nopen scoped Function in -- required for scoped `on` notation\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nprivate lemma pairwise_coprime_coprimes (a : Fin m → ℕ) : Pairwise (Coprime on coprimes a) := by\n  intro i j hij\n  wlog! ltij : i < j\n  · exact (this a hij.symm (lt_of_le_of_ne ltij hij.symm)).symm\n  unfold Function.onFun coprimes\n  have hja : j < supOfSeq a := lt_of_lt_of_le j.prop (le_succ_of_le (le_max_left _ _))\n  exact coprime_mul_succ\n    (Nat.dvd_factorial (by lia)\n      (by simpa only [Nat.succ_sub_succ] using le_of_lt (lt_of_le_of_lt (sub_le j i) hja)))\n\n/-- Gödel's Beta Function. This is similar to `(Encodable.decodeList)[i]`, but it is easier to\nprove that it is arithmetically definable. -/\ndef beta (n i : ℕ) : ℕ := n.unpair.1 % ((i + 1) * n.unpair.2 + 1)\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n/-- Inverse of Gödel's Beta Function. This is similar to `Encodable.encodeList`, but it is easier\nto prove that it is arithmetically definable. -/\ndef unbeta (l : List ℕ) : ℕ :=\n  (chineseRemainderOfFinset (ι := Fin l.length) (l[·]) (coprimes (l[·])) Finset.univ\n    (by simp [coprimes])\n    (by simpa using Set.pairwise_univ.mpr (pairwise_coprime_coprimes _)) : ℕ).pair\n  (supOfSeq (m := l.length) (l[·]))!\n\n/-- **Gödel's Beta Function Lemma** -/\n\nTarget:\nlemma beta_unbeta_coe (l : List ℕ) (i : Fin l.length) : beta (unbeta l) i = l[i] :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Godel","family_id":"beta_unbeta_coe","file_id":"mathlib/Mathlib/Logic/Godel/GodelBetaFunction.lean","sample_id":"8dd69de99772e4b60d7d7c33b72ef5655efe8caa0668569bdbff7dc2b9a6a47c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d93c674782daaf325f075697a98c3704ebb23b2b31510b6d8f8e1890f07f7bba","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"405da6d6559020f606063085aeff9021bd5519416af46cf3a4716c5395f6448a","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dac74140b5028292313ba8ba24d22052ad637c133b6e08b981669e5ff7752981","source_sha256":"5e794e43f06e38bed06c0789971ed6287919f753a812f6be7d0a2ba23c851951","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [noZeroSMulDivisors_iff, or_iff_not_imp_left]\n  exact ⟨fun h r hr m eq ↦ h eq hr, fun h r m eq hr ↦ h r hr m eq⟩","hard_negative":false,"metrics":{"chosen_tokens":33,"rejected_tokens":40,"token_jaccard":0.894737,"token_length_ratio":1.212121},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"f0d86fbefe0be20fae7af12a90cb20af9ea9cab28024fe42fb214fe9574c44ae","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Module.Torsion.Free\npublic import Mathlib.Tactic.Contrapose\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2015 Nathaniel Thomas. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen, Yury Kudryashov, Joseph Myers, Heather Macbeth, Kim Morrison, Yaël Dillies\n-/\n/-!\n# `NoZeroSMulDivisors`\n\nThis file defines the `NoZeroSMulDivisors` class, and includes some tests\nfor the vanishing of elements (especially in modules over division rings).\n\n## Usage notes\n\nNote that `NoZeroSMulDivisors` is deprecated in favor of `Module.IsTorsionFree`, which is the\nmathematically correct generalisation to semimodules.\n-/\n\npublic section\n\nassert_not_exists RelIso Multiset Set.indicator Pi.single_smul₀\n\nvariable {R M G : Type*}\n\n/-- `NoZeroSMulDivisors R M` states that a scalar multiple is `0` only if either argument is `0`.\nThis is a version of saying that `M` is torsion free, without assuming `R` is zero-divisor free.\n\nThe main application of `NoZeroSMulDivisors R M`, when `M` is a module,\nis the result `smul_eq_zero`: a scalar multiple is `0` iff either argument is `0`.\n\nIt is a generalization of the `NoZeroDivisors` class to heterogeneous multiplication.\n-/\n@[mk_iff]\nclass NoZeroSMulDivisors (R M : Type*) [Zero R] [Zero M] [SMul R M] : Prop where\n  /-- If scalar multiplication yields zero, either the scalar or the vector was zero. -/\n  eq_zero_or_eq_zero_of_smul_eq_zero : ∀ {c : R} {x : M}, c • x = 0 → c = 0 ∨ x = 0\n\nexport NoZeroSMulDivisors (eq_zero_or_eq_zero_of_smul_eq_zero)\n\n/-- Pullback a `NoZeroSMulDivisors` instance along an injective function. -/\ntheorem Function.Injective.noZeroSMulDivisors {R M N : Type*} [Zero R] [Zero M] [Zero N]\n    [SMul R M] [SMul R N] [NoZeroSMulDivisors R N] (f : M → N) (hf : Function.Injective f)\n    (h0 : f 0 = 0) (hs : ∀ (c : R) (x : M), f (c • x) = c • f x) : NoZeroSMulDivisors R M :=\n  ⟨fun {_ _} h =>\n    Or.imp_right (@hf _ _) <| h0.symm ▸ eq_zero_or_eq_zero_of_smul_eq_zero (by rw [← hs, h, h0])⟩\n\n-- See note [lower instance priority]\ninstance (priority := 100) NoZeroDivisors.toNoZeroSMulDivisors [Zero R] [Mul R]\n    [NoZeroDivisors R] : NoZeroSMulDivisors R R :=\n  ⟨fun {_ _} => eq_zero_or_eq_zero_of_mul_eq_zero⟩\n\ninstance [Semiring R] [IsDomain R] [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] :\n    Module.IsTorsionFree R M where\n  isSMulRegular r hr m₁ m₂ hm := by\n    dsimp at hm\n    rw [← sub_eq_zero, ← smul_sub] at hm\n    simpa [hr.ne_zero, sub_eq_zero] using eq_zero_or_eq_zero_of_smul_eq_zero hm\n\nTarget:\ntheorem noZeroSMulDivisors_iff_right_eq_zero_of_smul [Zero R] [Zero M] [SMul R M] :\n    NoZeroSMulDivisors R M ↔ ∀ r : R, r ≠ 0 → ∀ m : M, r • m = 0 → m = 0 :=\n\nProof body:\n","rejected":"```lean\nby\n  simp_rw [noZeroSMulDivisors_iff, or_iff_not_imp_left]\n  exact ⟨fun h r hr m eq ↦ h eq hr, fun h r m eq hr ↦ h r hr m eq⟩\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/NoZeroSMulDivisors","family_id":"nozerosmuldivisors_iff_right_eq_zero_of_smul","file_id":"mathlib/Mathlib/Algebra/NoZeroSMulDivisors/Defs.lean","sample_id":"dac74140b5028292313ba8ba24d22052ad637c133b6e08b981669e5ff7752981"},"split":"train"}
{"binding":{"chosen_proof_sha256":"24551563a4e7d2141b3625f37d2a2f5a066ec7e2be8f2231f8bc3aa32f433342","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d99aae00c5ca2aaa8d6ff924384016fd0cee5a5a87a4990e7e4a09ef188f1d18","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8d47a007615a250bf24cd20685fe57ad901c62e3e8a8706c9506725d94de99c7","source_sha256":"da76ff62b317f067f0d000725a7dfe7652cd6b413da74f1215e83607e9c9c512","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isUnit_iff_and, and_iff_right ha]","hard_negative":true,"metrics":{"chosen_tokens":8,"rejected_tokens":3,"token_jaccard":0.1,"token_length_ratio":0.375},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"f11659d9565f9873a790ac3a5bf15cef6941f572162a9d0dc34eeab09b5fc53e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Ideal.Maximal\n\nNamespace:\nSubmonoid\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Chris Hughes, Mario Carneiro\n-/\n/-!\n# The set of non-invertible elements of a monoid\n\n## Main definitions\n\n* `nonunits` is the set of non-invertible elements of a monoid.\n\n## Main results\n\n* `exists_max_ideal_of_mem_nonunits`: every element of `nonunits` is contained in a maximal ideal\n-/\n\n@[expose] public section\n\n\nvariable {F α β : Type*} {a b : α}\n\n/-- The set of non-invertible elements of a monoid. -/\ndef nonunits (α : Type*) [Monoid α] : Set α :=\n  { a | ¬IsUnit a }\n\n@[simp]\ntheorem mem_nonunits_iff [Monoid α] : a ∈ nonunits α ↔ ¬IsUnit a :=\n  Iff.rfl\n\ntheorem mul_mem_nonunits_right [CommMonoid α] : b ∈ nonunits α → a * b ∈ nonunits α :=\n  mt isUnit_of_mul_isUnit_right\n\ntheorem mul_mem_nonunits_left [CommMonoid α] : a ∈ nonunits α → a * b ∈ nonunits α :=\n  mt isUnit_of_mul_isUnit_left\n\ntheorem zero_mem_nonunits [MonoidWithZero α] : 0 ∈ nonunits α ↔ (0 : α) ≠ 1 :=\n  not_congr isUnit_zero_iff\n\n@[simp high] -- High priority shortcut lemma\ntheorem one_notMem_nonunits [Monoid α] : (1 : α) ∉ nonunits α :=\n  not_not_intro isUnit_one\n\n@[simp high] -- High priority shortcut lemma\ntheorem map_mem_nonunits_iff [Monoid α] [Monoid β] [FunLike F α β] [MonoidHomClass F α β] (f : F)\n    [IsLocalHom f] (a) : f a ∈ nonunits β ↔ a ∈ nonunits α :=\n  ⟨fun h ha => h <| ha.map f, fun h ha => h <| ha.of_map⟩\n\ntheorem coe_subset_nonunits [Semiring α] {I : Ideal α} (h : I ≠ ⊤) : (I : Set α) ⊆ nonunits α :=\n  fun _x hx hu => h <| I.eq_top_of_isUnit_mem hx hu\n\ntheorem exists_max_ideal_of_mem_nonunits [CommSemiring α] (h : a ∈ nonunits α) :\n    ∃ I : Ideal α, I.IsMaximal ∧ a ∈ I := by\n  have : Ideal.span ({a} : Set α) ≠ ⊤ := by\n    intro H\n    rw [Ideal.span_singleton_eq_top] at H\n    contradiction\n  rcases Ideal.exists_le_maximal _ this with ⟨I, Imax, H⟩\n  use I, Imax\n  apply H\n  apply Ideal.subset_span\n  exact Set.mem_singleton a\n\nnamespace Submonoid\n\nvariable {C : Type*} [SetLike C α]\n\ntheorem inv_mem_of_isUnit [DivisionMonoid α] [SubmonoidClass C α] {S : C} {a : S} (ha : IsUnit a) :\n    (a : α)⁻¹ ∈ S := by\n  obtain ⟨u, rfl⟩ := ha\n  convert! u⁻¹.1.2\n  exact (map_inv ((subtype <| ofClass S).comp <| Units.coeHom S) u).symm\n\nsection Group\n\nvariable [Group α] [SubmonoidClass C α] {S : C} {a : S}\n\ntheorem isUnit_iff : IsUnit a ↔ (a : α)⁻¹ ∈ S where\n  mp := inv_mem_of_isUnit\n  mpr h := ⟨⟨a, ⟨_, h⟩, Subtype.ext (mul_inv_cancel _), Subtype.ext (inv_mul_cancel _)⟩, rfl⟩\n\nprotected theorem mem_nonunits_iff : a ∈ nonunits S ↔ (a : α)⁻¹ ∉ S := by\n  rw [mem_nonunits_iff, isUnit_iff]\n\nend Group\n\nsection GroupWithZero\n\nvariable [GroupWithZero α] [SubmonoidClass C α] {S : C} {a : S}\n\ntheorem isUnit_iff_and : IsUnit a ↔ (a : α) ≠ 0 ∧ (a : α)⁻¹ ∈ S where\n  mp h := ⟨(h.map <| subtype <| ofClass S).ne_zero, inv_mem_of_isUnit h⟩\n  mpr h :=\n    ⟨⟨a, ⟨_, h.2⟩, Subtype.ext (mul_inv_cancel₀ h.1), Subtype.ext (inv_mul_cancel₀ h.1)⟩, rfl⟩\n\nTarget:\ntheorem isUnit_iff_of_ne_zero (ha : (a : α) ≠ 0) : IsUnit a ↔ (a : α)⁻¹ ∈ S :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_8d47a007615a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"c2a96c4d30f845f9980da0b0069a18d844b6bf0e11874b36c83183bc723aa902","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Ideal","family_id":"isunit_iff_of_ne_zero","file_id":"mathlib/Mathlib/RingTheory/Ideal/Nonunits.lean","sample_id":"8d47a007615a250bf24cd20685fe57ad901c62e3e8a8706c9506725d94de99c7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"82c27cb6e99a7d6e45601e4127587eb4a9e18558df87ff23a7cf9764fabc33f4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"12e4ed73b2164c9034e09e737bb93b021152af848a112fdd9ed5bf5c79a07e22","source_sha256":"6a1fd2aa47744142955f801b29fb06cc46e0d439ea80c9cccba97a19a60fbbe9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [galRestrict', galRestrict', Subalgebra.algebraMap_eq]","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":2,"token_jaccard":0.1,"token_length_ratio":0.181818},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"f1388ddceea1419c2c899e93563748b40f8df14dc285d9228d5fd1c1a96db4b9","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.DedekindDomain.IntegralClosure\npublic import Mathlib.RingTheory.RingHom.Finite\npublic import Mathlib.RingTheory.Localization.LocalizationLocalization\npublic import Mathlib.RingTheory.Localization.NormTrace\npublic import Mathlib.RingTheory.Norm.Transitivity\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Restriction of various maps between fields to integrally closed subrings.\n\nIn this file, we assume `A` is an integrally closed domain; `K` is the fraction ring of `A`;\n`L` is a finite extension of `K`; `B` is the integral closure of `A` in `L`.\nWe call this the AKLB setup.\n\n## Main definitions\n- `galRestrict`: The restriction `Aut(L/K) → Aut(B/A)` as an `MulEquiv` in an AKLB setup.\n- `Algebra.intTrace`: The trace map of a finite extension of integrally closed domains `B/A` is\n  defined to be the restriction of the trace map of `Frac(B)/Frac(A)`.\n- `Algebra.intNorm`: The norm map of a finite extension of integrally closed domains `B/A` is\n  defined to be the restriction of the norm map of `Frac(B)/Frac(A)`.\n\n-/\n\n@[expose] public section\n\nopen Module nonZeroDivisors\n\nvariable (A K L L₂ L₃ B B₂ B₃ : Type*)\nvariable [CommRing A] [CommRing B] [CommRing B₂] [CommRing B₃]\nvariable [Algebra A B] [Algebra A B₂] [Algebra A B₃]\nvariable [Field K] [Field L] [Field L₂] [Field L₃]\nvariable [Algebra A K] [IsFractionRing A K]\nvariable [Algebra K L] [Algebra A L] [IsScalarTower A K L]\nvariable [Algebra K L₂] [Algebra A L₂] [IsScalarTower A K L₂]\nvariable [Algebra K L₃] [Algebra A L₃] [IsScalarTower A K L₃]\nvariable [Algebra B L] [IsScalarTower A B L] [IsIntegralClosure B A L]\nvariable [Algebra B₂ L₂] [IsScalarTower A B₂ L₂] [IsIntegralClosure B₂ A L₂]\nvariable [Algebra B₃ L₃] [IsScalarTower A B₃ L₃] [IsIntegralClosure B₃ A L₃]\n\nsection galois\n\nsection galRestrict'\nvariable {K L L₂ L₃}\nomit [IsFractionRing A K]\n\n/-- A generalization of `galRestrictHom` beyond endomorphisms. -/\nnoncomputable\ndef galRestrict' (f : L →ₐ[K] L₂) : (B →ₐ[A] B₂) :=\n  (IsIntegralClosure.equiv A (integralClosure A L₂) L₂ B₂).toAlgHom.comp\n      (((f.restrictScalars A).comp (IsScalarTower.toAlgHom A B L)).codRestrict\n        (integralClosure A L₂) (fun x ↦ IsIntegral.map _ (IsIntegralClosure.isIntegral A L x)))\n\n@[simp]\n\nTarget:\nlemma algebraMap_galRestrict'_apply (σ : L →ₐ[K] L₂) (x : B) :\n    algebraMap B₂ L₂ (galRestrict' A B B₂ σ x) = σ (algebraMap B L x) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/IntegralClosure","family_id":"algebramap_galrestrict'_apply","file_id":"mathlib/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean","sample_id":"12e4ed73b2164c9034e09e737bb93b021152af848a112fdd9ed5bf5c79a07e22"},"split":"train"}
{"binding":{"chosen_proof_sha256":"65f2254f5d37c07b087f477ca102b136468a343c041fb416c4b1c505f1281550","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"fb692754d646fb760ef043b5d79501f41f1d28e50ad2da85d3e2605daffd725b","source_sha256":"a6234c28d344c069679c6c490f57e0cad723a087d9f05f1fbee4f95afdc863a6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← linearIndepOn_univ_iff, ← Finset.coe_univ, show @Finset.univ (Fin 2) _ = {0,1} from rfl,\n    Finset.coe_insert, Finset.coe_singleton, LinearIndepOn.pair_iff _ (by trivial)]\n  simp","hard_negative":false,"metrics":{"chosen_tokens":48,"rejected_tokens":2,"token_jaccard":0.03125,"token_length_ratio":0.041667},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"f17cc3451f130505797ef4551c70e1b478ff8d612a5a0a3b81d5a334c92920da","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fin.Tuple.Reflection\npublic import Mathlib.LinearAlgebra.Dual.Defs\npublic import Mathlib.LinearAlgebra.Finsupp.SumProd\npublic import Mathlib.LinearAlgebra.LinearIndependent.Basic\npublic import Mathlib.LinearAlgebra.Pi\npublic import Mathlib.Logic.Equiv.Fin.Rotate\npublic import Mathlib.Tactic.FinCases\npublic import Mathlib.Tactic.Module\npublic import Mathlib.Tactic.Abel\npublic import Mathlib.Tactic.NormNum.Ineq\nimport Mathlib.Algebra.Module.Torsion.Field\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen\n-/\n/-!\n# Linear independence\n\nThis file collects consequences of linear (in)dependence and includes specialized tests for\nspecific families of vectors, requiring more theory to state.\n\n## Main statements\n\nWe prove several specialized tests for linear independence of families of vectors and of sets of\nvectors.\n\n* `linearIndependent_option`, `linearIndependent_finCons`,\n  `linearIndependent_finSucc`, `linearIndependent_finSnoc`: type-specific tests for linear\n  independence of families of vector fields;\n* `linearIndependent_insert`, `linearIndependent_pair`: linear independence tests for set operations\n\nIn many cases we additionally provide dot-style operations (e.g., `LinearIndependent.union`) to\nmake the linear independence tests usable as `hv.insert ha` etc.\n\nWe also prove that, when working over a division ring,\nany family of vectors includes a linear independent subfamily spanning the same subspace.\n\n## TODO\n\nRework proofs to hold in semirings, by avoiding the path through\n`ker (Finsupp.linearCombination R v) = ⊥`.\n\n## Tags\n\nlinearly dependent, linear dependence, linearly independent, linear independence\n\n-/\n\n@[expose] public section\n\n\nassert_not_exists Cardinal\n\nnoncomputable section\n\nopen Function Module Set Submodule\n\nuniverse u' u\n\nvariable {ι : Type u'} {ι' : Type*} {R : Type*} {K : Type*} {s : Set ι}\nvariable {M : Type*} {M' : Type*} {V : Type u}\n\nsection Semiring\n\n\nvariable {v : ι → M}\nvariable [Semiring R] [AddCommMonoid M] [AddCommMonoid M']\nvariable [Module R M] [Module R M']\nvariable (R) (v)\n\nvariable {R v}\n\n/-- A finite family of vectors `v i` is linear independent iff the linear map that sends\n`c : ι → R` to `∑ i, c i • v i` is injective. -/\ntheorem Fintype.linearIndependent_iff'ₛ [Fintype ι] [DecidableEq ι] :\n    LinearIndependent R v ↔\n      Injective (LinearMap.lsum R (fun _ ↦ R) ℕ fun i ↦ LinearMap.id.smulRight (v i)) := by\n  simp [Fintype.linearIndependent_iffₛ, Injective, funext_iff]\n\nlemma LinearIndependent.pair_iffₛ {x y : M} :\n    LinearIndependent R ![x, y] ↔\n      ∀ (s t s' t' : R), s • x + t • y = s' • x + t' • y → s = s' ∧ t = t' := by\n  simp [Fintype.linearIndependent_iffₛ, Fin.forall_fin_two, ← FinVec.forall_iff]; rfl\n\nlemma LinearIndependent.eq_of_pair {x y : M} (h : LinearIndependent R ![x, y])\n    {s t s' t' : R} (h' : s • x + t • y = s' • x + t' • y) : s = s' ∧ t = t' :=\n  pair_iffₛ.mp h _ _ _ _ h'\n\nlemma LinearIndependent.eq_zero_of_pair' {x y : M} (h : LinearIndependent R ![x, y])\n    {s t : R} (h' : s • x = t • y) : s = 0 ∧ t = 0 := by\n  suffices H : s = 0 ∧ 0 = t from ⟨H.1, H.2.symm⟩\n  exact h.eq_of_pair (by simpa using h')\n\nlemma LinearIndependent.eq_zero_of_pair {x y : M} (h : LinearIndependent R ![x, y])\n    {s t : R} (h' : s • x + t • y = 0) : s = 0 ∧ t = 0 := by\n  replace h := @h (.single 0 s + .single 1 t) 0 ?_\n  · exact ⟨by simpa using congr($h 0), by simpa using congr($h 1)⟩\n  simpa\n\nsection Indexed\n\ntheorem linearIndepOn_iUnion_of_directed {η : Type*} {s : η → Set ι} (hs : Directed (· ⊆ ·) s)\n    (h : ∀ i, LinearIndepOn R v (s i)) : LinearIndepOn R v (⋃ i, s i) := by\n  by_cases hη : Nonempty η\n  · refine linearIndepOn_of_finite (⋃ i, s i) fun t ht ft => ?_\n    rcases finite_subset_iUnion ft ht with ⟨I, fi, hI⟩\n    rcases hs.finset_le fi.toFinset with ⟨i, hi⟩\n    exact (h i).mono (Subset.trans hI <| iUnion₂_subset fun j hj => hi j (fi.mem_toFinset.2 hj))\n  · refine (linearIndepOn_empty R v).mono (t := iUnion (s ·)) ?_\n    rintro _ ⟨_, ⟨i, _⟩, _⟩\n    exact hη ⟨i⟩\n\ntheorem linearIndepOn_sUnion_of_directed {s : Set (Set ι)} (hs : DirectedOn (· ⊆ ·) s)\n    (h : ∀ a ∈ s, LinearIndepOn R v a) : LinearIndepOn R v (⋃₀ s) := by\n  rw [sUnion_eq_iUnion]\n  exact linearIndepOn_iUnion_of_directed hs.directed_val (by simpa using h)\n\ntheorem linearIndepOn_biUnion_of_directed {η} {s : Set η} {t : η → Set ι}\n    (hs : DirectedOn (t ⁻¹'o (· ⊆ ·)) s) (h : ∀ a ∈ s, LinearIndepOn R v (t a)) :\n    LinearIndepOn R v (⋃ a ∈ s, t a) := by\n  rw [biUnion_eq_iUnion]\n  exact linearIndepOn_iUnion_of_directed (directed_comp.2 <| hs.directed_val) (by simpa using h)\n\nend Indexed\n\nsection repr\n\nvariable (ι R M) in\ntheorem iSupIndep_range_lsingle :\n    iSupIndep fun i : ι ↦ LinearMap.range (Finsupp.lsingle (R := R) (M := M) i) := by\n  refine fun i ↦ disjoint_iff_inf_le.mpr ?_\n  rintro x ⟨⟨m, rfl⟩, hm⟩\n  suffices ⨆ j ≠ i, LinearMap.range (Finsupp.lsingle j) ≤ Finsupp.supported M R {i}ᶜ by\n    have := (Finsupp.mem_supported ..).mp (this hm); simp_all\n  refine iSup₂_le fun j ne ↦ ?_\n  rintro _ ⟨m, rfl⟩\n  simp [Finsupp.mem_supported, ne]\n\ntheorem LinearMap.iSupIndep_map (f : M →ₗ[R] M') (inj : Injective f) {m : ι → Submodule R M}\n    (ind : iSupIndep m) : iSupIndep fun i ↦ (m i).map f := by\n  simp_rw [iSupIndep, disjoint_iff_inf_le] at ind ⊢\n  rintro i _ ⟨⟨x, hxi, rfl⟩, hx⟩\n  rw [ind i ⟨hxi, _⟩]; · simp\n  simp_rw [← Submodule.map_iSup] at hx\n  have ⟨y, hy, eq⟩ := hx\n  simpa [← inj eq]\n\nvariable (hv : LinearIndependent R v)\n\n/-- See also `iSupIndep_iff_linearIndependent_of_ne_zero`. -/\ntheorem LinearIndependent.iSupIndep_span_singleton (hv : LinearIndependent R v) :\n    iSupIndep fun i => R ∙ v i := by\n  convert! LinearMap.iSupIndep_map _ hv (iSupIndep_range_lsingle ι R R)\n  ext; simp [mem_span_singleton]\n\nend repr\n\nsection union\n\nopen LinearMap Finsupp\n\ntheorem linearIndependent_inl_union_inr' {v : ι → M} {v' : ι' → M'}\n    (hv : LinearIndependent R v) (hv' : LinearIndependent R v') :\n    LinearIndependent R (Sum.elim (inl R M M' ∘ v) (inr R M M' ∘ v')) := by\n  have : linearCombination R (Sum.elim (inl R M M' ∘ v) (inr R M M' ∘ v')) =\n      .prodMap (linearCombination R v) (linearCombination R v') ∘ₗ\n      (sumFinsuppLEquivProdFinsupp R).toLinearMap := by ext (_ | _) <;> simp\n  rw [LinearIndependent, this]\n  simpa [LinearMap.coe_prodMap] using ⟨hv, hv'⟩\n\ntheorem LinearIndependent.inl_union_inr {s : Set M} {t : Set M'}\n    (hs : LinearIndependent R (fun x => x : s → M))\n    (ht : LinearIndependent R (fun x => x : t → M')) :\n    LinearIndependent R (fun x => x : ↥(inl R M M' '' s ∪ inr R M M' '' t) → M × M') := by\n  nontriviality R\n  let e : s ⊕ t ≃ ↥(inl R M M' '' s ∪ inr R M M' '' t) :=\n    .ofBijective (Sum.elim (fun i ↦ ⟨_, .inl ⟨_, i.2, rfl⟩⟩) fun i ↦ ⟨_, .inr ⟨_, i.2, rfl⟩⟩)\n      ⟨by rintro (_ | _) (_ | _) eq <;> simp [hs.ne_zero, ht.ne_zero] at eq <;> aesop,\n        by rintro ⟨_, ⟨_, _, rfl⟩ | ⟨_, _, rfl⟩⟩ <;> aesop⟩\n  refine (linearIndependent_equiv' e ?_).mp (linearIndependent_inl_union_inr' hs ht)\n  ext (_ | _) <;> rfl\n\nend union\n\nsection Maximal\n\nuniverse v w\n\nvariable (R)\n\n/-- TODO : refactor to use `Maximal`. -/\ntheorem exists_maximal_linearIndepOn' (v : ι → M) :\n    ∃ s : Set ι, (LinearIndepOn R v s) ∧ ∀ t : Set ι, s ⊆ t → (LinearIndepOn R v t) → s = t := by\n  let indep : Set ι → Prop := fun s => LinearIndepOn R v s\n  let X := { I : Set ι // indep I }\n  let r : X → X → Prop := fun I J => I.1 ⊆ J.1\n  have key : ∀ c : Set X, IsChain r c → indep (⋃ (I : X) (_ : I ∈ c), I) := by\n    intro c hc\n    dsimp [indep]\n    rw [linearIndepOn_iffₛ]\n    intro f hfsupp g hgsupp hsum\n    rcases eq_empty_or_nonempty c with (rfl | hn)\n    · rw [show f = 0 by simpa using! hfsupp, show g = 0 by simpa using! hgsupp]\n    haveI : Std.Refl r := ⟨fun _ => Set.Subset.refl _⟩\n    classical\n    obtain ⟨I, _I_mem, hI⟩ : ∃ I ∈ c, (f.support ∪ g.support : Set ι) ⊆ I :=\n      f.support.coe_union _ ▸ hc.directedOn.exists_mem_subset_of_finset_subset_biUnion hn <| by\n        simpa using! And.intro hfsupp hgsupp\n    exact linearIndepOn_iffₛ.mp I.2 f (subset_union_left.trans hI)\n      g (subset_union_right.trans hI) hsum\n  obtain ⟨⟨I, hli : indep I⟩, hmax : ∀ a, r ⟨I, hli⟩ a → r a ⟨I, hli⟩⟩ :=\n    exists_maximal_of_chains_bounded (r := r)\n      (fun c hc => ⟨⟨⋃ I ∈ c, (I : Set ι), key c hc⟩, fun I => Set.subset_biUnion_of_mem⟩)\n      Set.Subset.trans\n  exact ⟨I, hli, fun J hsub hli => Set.Subset.antisymm hsub (hmax ⟨J, hli⟩ hsub)⟩\n\nend Maximal\n\nlemma Submodule.codisjoint_span_image_of_codisjoint (hv : Submodule.span R (Set.range v) = ⊤)\n    {s t : Set ι} (hst : Codisjoint s t) :\n    Codisjoint (Submodule.span R (v '' s)) (Submodule.span R (v '' t)) := by\n  rw [Finsupp.span_image_eq_map_linearCombination, Finsupp.span_image_eq_map_linearCombination]\n  refine Submodule.codisjoint_map ?_ (Finsupp.codisjoint_supported_supported hst)\n  rwa [← LinearMap.range_eq_top, Finsupp.range_linearCombination]\n\nlemma LinearIndependent.isCompl_span_image (h₁ : LinearIndependent R v)\n    (h₂ : Submodule.span R (Set.range v) = ⊤) {s t : Set ι} (hst : IsCompl s t) :\n    IsCompl (Submodule.span R (v '' s)) (Submodule.span R (v '' t)) :=\n  ⟨h₁.disjoint_span_image hst.1, Submodule.codisjoint_span_image_of_codisjoint h₂ hst.2⟩\n\nend Semiring\n\nsection Module\n\nvariable {v : ι → M}\nvariable [Ring R] [AddCommGroup M] [AddCommGroup M']\nvariable [Module R M] [Module R M']\n\n/-- A finite family of vectors `v i` is linear independent iff the linear map that sends\n`c : ι → R` to `∑ i, c i • v i` has the trivial kernel. -/\ntheorem Fintype.linearIndependent_iff' [Fintype ι] [DecidableEq ι] :\n    LinearIndependent R v ↔\n      LinearMap.ker (LinearMap.lsum R (fun _ ↦ R) ℕ fun i ↦ LinearMap.id.smulRight (v i)) = ⊥ := by\n  simp [Fintype.linearIndependent_iff, LinearMap.ker_eq_bot', funext_iff]\n\n/-- `linearIndepOn_pair_iff` is a simpler version over fields. -/\nlemma LinearIndepOn.pair_iff {i j : ι} (f : ι → M) (hij : i ≠ j) :\n    LinearIndepOn R f {i,j} ↔ ∀ c d : R, c • f i + d • f j = 0 → c = 0 ∧ d = 0 := by\n  classical\n  rw [linearIndepOn_iff'']\n  refine ⟨fun h c d hcd ↦ ?_, fun h t g ht hg0 h0 ↦ ?_⟩\n  · specialize h {i, j} (Pi.single i c + Pi.single j d)\n    simpa +contextual [Finset.sum_pair, Pi.single_apply, hij, hij.symm, hcd] using h\n  have ht' : t ⊆ {i, j} := by simpa [← Finset.coe_subset]\n  rw [Finset.sum_subset ht', Finset.sum_pair hij] at h0\n  · obtain ⟨hi0, hj0⟩ := h _ _ h0\n    exact fun k hkt ↦ Or.elim (ht hkt) (fun h ↦ h ▸ hi0) (fun h ↦ h ▸ hj0)\n  simp +contextual [hg0]\n\nsection Pair\n\nvariable {x y : M}\n\n/-- Also see `LinearIndependent.pair_iff'` for a simpler version over fields. -/\n\nTarget:\nlemma LinearIndependent.pair_iff :\n    LinearIndependent R ![x, y] ↔ ∀ (s t : R), s • x + t • y = 0 → s = 0 ∧ t = 0 :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/LinearIndependent","family_id":"linearindependent","file_id":"mathlib/Mathlib/LinearAlgebra/LinearIndependent/Lemmas.lean","sample_id":"fb692754d646fb760ef043b5d79501f41f1d28e50ad2da85d3e2605daffd725b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4acb529aee5f1ece8076fbbdc844587e9d93366665ac8d5f6045ad6127fa4525","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"beebaf6401da7ba6ae02cab75a18db90132912fa7666bc24027b84d9741587a1","source_sha256":"3ebf52bd4f56640ff82f3663b47b89c207fb7b51b21d5bfbbe3cfb7edd34865e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have := lift_lt.{_, max (u + 1) v}.2 (lift_lt_univ c)\n  rw [lift_lift, lift_univ, univ_umax.{u, v}] at this\n  exact this","hard_negative":true,"metrics":{"chosen_tokens":40,"rejected_tokens":8,"token_jaccard":0.09375,"token_length_ratio":0.2},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"f1bbc1ef0c05ab80e553f6410e67f9887290073aed26bf4701718eda3b3a6977","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.SetTheory.Ordinal.Basic\n\nNamespace:\nCardinal\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Floris van Doorn\n-/\n/-!\n# Universal ordinal and cardinal\n\n`Cardinal.univ` is the cardinality of the cardinals themselves. Likewise, `Ordinal.univ` is the\norder type of the ordinals. These are related via `Cardinal.univ.ord = Ordinal.univ` and\n`Ordinal.univ.card = Cardinal.univ`.\n\nThe cardinal `Cardinal.univ` is strongly inaccessible. This reflects the fact that in ZFC, the\ncardinals form a proper class. See `IsInaccessible.univ` for a proof.\n\n## Implementation notes\n\nWe actually define `Cardinal.univ` as the cardinality of `Ordinal`, rather than that of `Cardinal`.\nThis makes the basic API easier to set up. See `Cardinal.mk_cardinal` for a proof that\n`Cardinal.univ = #Cardinal`.\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nset_option linter.checkUnivs false in\nopen Ordinal in\n-- intended to be used with explicit universe parameters\n/-- The ordinal `univ.{u, v}` is the order type of `Ordinal.{u}` or `Cardinal.{u}`, as an element of\n`Ordinal.{v}` (when `u < v`). -/\n@[pp_with_univ, nolint checkUnivs]\ndef Ordinal.univ : Ordinal.{max (u + 1) v} :=\n  lift.{v, u + 1} (typeLT Ordinal)\n\nset_option linter.checkUnivs false in\nopen Cardinal in\n-- intended to be used with explicit universe parameters\n/-- The cardinal `univ.{u, v}` is the cardinality of `Ordinal.{u}` or `Cardinal.{u}`, as an element\nof `Cardinal.{v}` (when `u < v`). -/\n@[pp_with_univ, nolint checkUnivs]\ndef Cardinal.univ : Cardinal.{max (u + 1) v} :=\n  lift.{v, u + 1} #Ordinal\n\n/-! ### Universal ordinal -/\n\nnamespace Ordinal\n\n@[simp]\ntheorem type_lt_ordinal : typeLT Ordinal = univ.{u, u + 1} :=\n  (lift_id _).symm\n\n@[deprecated type_lt_ordinal (since := \"2026-03-20\")]\ntheorem univ_id : univ.{u, u + 1} = typeLT Ordinal :=\n  lift_id _\n\n@[simp]\ntheorem lift_univ : lift.{w} univ.{u, v} = univ.{u, max v w} :=\n  lift_lift _\n\ntheorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=\n  congr_fun lift_umax _\n\n/-- Principal segment version of the lift operation on ordinals, embedding `Ordinal.{u}` in\n`Ordinal.{v}` as a principal segment when `u < v`. -/\ndef liftPrincipalSeg : Ordinal.{u} <i Ordinal.{max (u + 1) v} :=\n  ⟨liftInitialSeg.{max (u + 1) v, u}, univ.{u, v}, by\n    refine fun b => inductionOn b ?_; intro β s _\n    rw [univ, ← lift_umax]; constructor <;> intro h\n    · obtain ⟨a, e⟩ := h\n      rw [← e]\n      refine inductionOn a ?_\n      intro α r _\n      exact lift_type_lt.{u, u + 1, max (u + 1) v}.2 ⟨typein r⟩\n    · rw [← lift_id (type s)] at h ⊢\n      obtain ⟨f⟩ := lift_type_lt.{_,_,v}.1 h\n      obtain ⟨f, a, hf⟩ := f\n      exists a\n      induction a using inductionOn with | type α r\n      refine lift_type_eq.{u, max (u + 1) v, max (u + 1) v}.2\n        ⟨(RelIso.ofSurjective (RelEmbedding.ofMonotone ?_ ?_) ?_).symm⟩\n      · exact fun b => enum r ⟨f b, (hf _).1 ⟨_, rfl⟩⟩\n      · refine fun a b h => (typein_lt_typein r).1 ?_\n        rw [typein_enum, typein_enum]\n        exact f.map_rel_iff.2 h\n      · intro a'\n        obtain ⟨b, e⟩ := (hf _).2 (typein_lt_type _ a')\n        exists b\n        simp only [RelEmbedding.ofMonotone_coe]\n        simp [e]⟩\n\n@[simp]\ntheorem liftPrincipalSeg_coe :\n    (liftPrincipalSeg.{u, v} : Ordinal → Ordinal) = lift.{max (u + 1) v} :=\n  rfl\n\n@[simp]\ntheorem liftPrincipalSeg_top : (liftPrincipalSeg.{u, v}).top = univ.{u, v} :=\n  rfl\n\n@[deprecated liftPrincipalSeg_top (since := \"2026-03-20\")]\ntheorem liftPrincipalSeg_top' : liftPrincipalSeg.{u, u + 1}.top = typeLT Ordinal := by\n  simp\n\n@[simp]\ntheorem card_univ : card univ.{u, v} = Cardinal.univ.{u, v} :=\n  rfl\n\nend Ordinal\n\n/-! ### Universal cardinal -/\n\nnamespace Cardinal\n\n@[simp]\ntheorem mk_ordinal : #Ordinal = univ.{u, u + 1} :=\n  (lift_id _).symm\n\n@[deprecated mk_ordinal (since := \"2026-04-22\")]\ntheorem univ_id : univ.{u, u + 1} = #Ordinal :=\n  lift_id _\n\n@[simp]\ntheorem lift_univ : lift.{w} univ.{u, v} = univ.{u, max v w} :=\n  lift_lift _\n\ntheorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=\n  congr_fun lift_umax _\n\ntheorem lift_lt_univ (c : Cardinal) : lift.{u + 1, u} c < univ.{u, u + 1} := by\n  simpa only [Ordinal.liftPrincipalSeg_coe, lift_ord, lift_succ, ord_le, Order.succ_le_iff] using!\n    le_of_lt (Ordinal.liftPrincipalSeg.{u, u + 1}.lt_top (Order.succ c).ord)\n\nTarget:\ntheorem lift_lt_univ' (c : Cardinal) : lift.{max (u + 1) v, u} c < univ.{u, v} :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"bd943cb64286310513638bec0fd0e80fbfdbb9cffe28b8feea552363e234cf7e","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"SetTheory/Ordinal","family_id":"lift_lt_univ","file_id":"mathlib/Mathlib/SetTheory/Ordinal/Univ.lean","sample_id":"beebaf6401da7ba6ae02cab75a18db90132912fa7666bc24027b84d9741587a1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"187dbb6cbe73173dbea1f8f380afdbd3c2e979ea26a53fbec596e63a5f1c929e","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9f7eb7bcc0d3153a05f5019eda72791728388ea6466ec87e305f6457e18d50ee","source_sha256":"72c080c49c4d252a9088a8037fd0035ad61560d6f34ce2947cf3bd3ceaf7e3f4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by by_cases a = 0 <;> grind [zero_le]","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.071429,"token_length_ratio":0.25},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"f1f91a332ac2afd361de4f1547aaf502447e01923e093e604b7813b593b064cf","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.FieldTheory.Finite.Basic\npublic import Mathlib.RingTheory.Valuation.Basic\n\nNamespace:\nFiniteField\n\nLocal context:\n/-\nCopyright (c) 2026 María Inés de Frutos-Fernández, Xavier Généreux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: María Inés de Frutos-Fernández, Xavier Généreux\n-/\n/-!\n# Valuations on an algebra over a finite field.\n-/\n\npublic section\n\nnamespace FiniteField\n\nopen Valuation\n\nvariable {Fq A Γ : Type*} [Field Fq] [Finite Fq] [Ring A] [Algebra Fq A]\n  [LinearOrderedCommMonoidWithZero Γ] (v : Valuation A Γ)\n\n@[grind =>]\nlemma valuation_algebraMap_eq_one (a : Fq) (ha : a ≠ 0) : v (algebraMap Fq A a) = 1 := by\n  have : Fintype Fq := Fintype.ofFinite Fq\n  have hpow : (v (algebraMap Fq A a)) ^ (Fintype.card Fq - 1) = 1 := by\n    simp [← map_pow, FiniteField.pow_card_sub_one_eq_one a ha]\n  grind [pow_eq_one_iff, → IsPrimePow.two_le, FiniteField.isPrimePow_card]\n\nTarget:\nlemma valuation_algebraMap_le_one (v : Valuation A Γ) (a : Fq) :\n    v (algebraMap Fq A a) ≤ 1 :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Finite","family_id":"valuation_algebramap_le_one","file_id":"mathlib/Mathlib/FieldTheory/Finite/Valuation.lean","sample_id":"9f7eb7bcc0d3153a05f5019eda72791728388ea6466ec87e305f6457e18d50ee"},"split":"train"}
{"binding":{"chosen_proof_sha256":"99f33d97a3ff079eb01e172cf32d5125fc28f1ebfb2b8c076d51536e340d9ee6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dce31b4f3667c3830b421cb76a3fefb37b0de149d8e70ed049af7ee6287ae930","source_sha256":"ab8dcc475894ae6adb2da3524a8462f1faacdbf7c702f22683396e1bbe0456ee","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply tendsto_integral_filter_of_dominated_convergence _ _ _ hf.norm\n  · filter_upwards with v\n    nth_rewrite 2 [show f v = cexp (- (0 : ℝ) * ‖v‖ ^ 2) • f v by simp]\n    apply (Tendsto.cexp _).smul_const\n    exact tendsto_inv_atTop_zero.ofReal.neg.mul_const _\n  · filter_upwards with c using\n      AEStronglyMeasurable.smul (Continuous.aestronglyMeasurable (by fun_prop)) hf.1\n  · filter_upwards [Ici_mem_atTop (0 : ℝ)] with c (hc : 0 ≤ c)\n    filter_upwards with v\n    simp only [ofReal_inv, neg_mul, norm_smul]\n    norm_cast\n    conv_rhs => rw [← one_mul (‖f v‖)]\n    gcongr\n    simp only [norm_eq_abs, abs_exp, exp_le_one_iff, Left.neg_nonpos_iff]\n    positivity\n\nvariable [CompleteSpace E]","hard_negative":false,"metrics":{"chosen_tokens":144,"rejected_tokens":2,"token_jaccard":0.014286,"token_length_ratio":0.013889},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"f24cbde0b979366f70a0bc4794768059d7d873df5d9ce207ea02c4c31ade800a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Integral.PeakFunction\npublic import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2024 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Fourier inversion formula\n\nIn a finite-dimensional real inner product space, we show the Fourier inversion formula, i.e.,\n`𝓕⁻ (𝓕 f) v = f v` if `f` and `𝓕 f` are integrable, and `f` is continuous at `v`. This is proved\nin `MeasureTheory.Integrable.fourier_inversion`. See also `Continuous.fourier_inversion`\ngiving `𝓕⁻ (𝓕 f) = f` under an additional continuity assumption for `f`.\n\nWe use the following proof. A naïve computation gives\n`𝓕⁻ (𝓕 f) v\n= ∫_w exp (2 I π ⟪w, v⟫) 𝓕 f (w) dw\n= ∫_w exp (2 I π ⟪w, v⟫) ∫_x, exp (-2 I π ⟪w, x⟫) f x dx) dw\n= ∫_x (∫_ w, exp (2 I π ⟪w, v - x⟫ dw) f x dx `\n\nHowever, the Fubini step does not make sense for lack of integrability, and the middle integral\n`∫_ w, exp (2 I π ⟪w, v - x⟫ dw` (which one would like to be a Dirac at `v - x`) is not defined.\nTo gain integrability, one multiplies with a Gaussian function `exp (-c⁻¹ ‖w‖^2)`, with a large\n(but finite) `c`. As this function converges pointwise to `1` when `c → ∞`, we get\n`∫_w exp (2 I π ⟪w, v⟫) 𝓕 f (w) dw = lim_c ∫_w exp (-c⁻¹ ‖w‖^2 + 2 I π ⟪w, v⟫) 𝓕 f (w) dw`.\nOne can perform Fubini on the right-hand side for fixed `c`, writing the integral as\n`∫_x (∫_w exp (-c⁻¹‖w‖^2 + 2 I π ⟪w, v - x⟫ dw)) f x dx`.\nThe middle factor is the Fourier transform of a more and more flat function\n(converging to the constant `1`), hence it becomes more and more concentrated, around the\npoint `v`. (Morally, it converges to the Dirac at `v`). Moreover, it has integral one.\nTherefore, multiplying by `f` and integrating, one gets a term converging to `f v` as `c → ∞`.\nSince it also converges to `𝓕⁻ (𝓕 f) v`, this proves the result.\n\nTo check the concentration property of the middle factor and the fact that it has integral one, we\nrely on the explicit computation of the Fourier transform of Gaussians.\n-/\n\npublic section\n\nopen Filter MeasureTheory Complex Module Metric Real Bornology\n\nopen scoped Topology FourierTransform RealInnerProductSpace Complex\n\nvariable {V E : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]\n  [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V]\n  [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → E}\n\nnamespace Real\n\nTarget:\nlemma tendsto_integral_cexp_sq_smul (hf : Integrable f) :\n    Tendsto (fun (c : ℝ) ↦ (∫ v : V, cexp (- c⁻¹ * ‖v‖ ^ 2) • f v))\n      atTop (𝓝 (∫ v : V, f v)) :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Fourier","family_id":"tendsto_integral_cexp_sq_smul","file_id":"mathlib/Mathlib/Analysis/Fourier/Inversion.lean","sample_id":"dce31b4f3667c3830b421cb76a3fefb37b0de149d8e70ed049af7ee6287ae930"},"split":"train"}
{"binding":{"chosen_proof_sha256":"887cd45eb0ad0270c148f43d5c6fb5160031574c8c6c116689e7fb4bdff4ffd8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"00174d518458fef9a309864e49f69be9bd090110f25e5905b3ad9090d83909e4","source_sha256":"0558464e45d5ed116146fb004c344db832eef9939fa1268196e1a1a60376c0f3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases finite_or_infinite X with _ | _\n  · have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_right S X)\n    rwa [lift_umax.{v, u}, lift_id'] at this\n  have key (x : X) (s s' : S) (h : s • x = s' • x) (hc : Commute s s') : x /ₒ s = x /ₒ s' := by\n    rw [oreDiv_eq_iff]\n    refine ⟨s, s'.1, h, ?_⟩\n    · exact_mod_cast hc\n  let i (x : X × S) := x.1 /ₒ x.2\n  have hsurj : Surjective i := Quotient.mk''_surjective\n  have hi := rightInverse_surjInv hsurj\n  let j := (fun x : X × S ↦ (x.1, x.2 • x.1)) ∘ surjInv hsurj\n  suffices Injective j by\n    have := lift_mk_le_lift_mk_of_injective this\n    rwa [lift_umax.{v, u}, lift_id', mk_prod, lift_id, lift_mul, mul_eq_self (by simp)] at this\n  intro\n  grind","hard_negative":true,"metrics":{"chosen_tokens":190,"rejected_tokens":8,"token_jaccard":0.052632,"token_length_ratio":0.042105},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"f288b1217155fde1b29741560b09e8035d82ac315bdd14a0cccdc44f3fc01fd7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Pigeonhole\npublic import Mathlib.GroupTheory.OreLocalization.Basic\npublic import Mathlib.SetTheory.Cardinal.Arithmetic\n\nNamespace:\nOreLocalization\n\nLocal context:\n/-\nCopyright (c) 2024 Jz Pan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jz Pan\n-/\n/-!\n\n# Cardinality of Ore localizations\n\nThis file contains some results on cardinality of Ore localizations.\n\n## TODO\n\n- Prove or disprove `OreLocalization.cardinalMk_le_lift_cardinalMk_of_commute`\n  with `Commute` assumption removed.\n\n-/\n\npublic section\n\nuniverse u v\n\nopen Cardinal Function\n\nnamespace OreLocalization\n\nvariable {R : Type u} [Monoid R] (S : Submonoid R) [OreLocalization.OreSet S]\n  (X : Type v) [MulAction R X]\n\n@[to_additive]\ntheorem oreDiv_one_surjective_of_finite_left [Finite S] :\n    Surjective (fun x ↦ x /ₒ (1 : ↥S) : X → OreLocalization S X) := by\n  refine OreLocalization.ind fun x s ↦ ?_\n  obtain ⟨i, j, hne, heq⟩ := Finite.exists_ne_map_eq_of_infinite (α := ℕ) (s ^ ·)\n  wlog! hlt : j < i generalizing i j\n  · exact this j i hne.symm heq.symm (hne.lt_of_le hlt)\n  use s ^ (i - (j + 1)) • x\n  rw [oreDiv_eq_iff]\n  refine ⟨s ^ j, (s ^ (j + 1)).1, ?_, ?_⟩\n  · change s ^ j • x = s ^ (j + 1) • s ^ (i - (j + 1)) • x\n    rw [← mul_smul, ← pow_add, Nat.add_sub_cancel' hlt, heq]\n  · simp_rw [SubmonoidClass.coe_pow, OneMemClass.coe_one, mul_one, pow_succ]\n\n@[to_additive]\ntheorem oreDiv_one_surjective_of_finite_right [Finite X] :\n    Surjective (fun x ↦ x /ₒ (1 : ↥S) : X → OreLocalization S X) := by\n  refine OreLocalization.ind fun x s ↦ ?_\n  obtain ⟨i, j, hne, heq⟩ := Finite.exists_ne_map_eq_of_infinite (α := ℕ) (s ^ · • x)\n  wlog! hlt : j < i generalizing i j\n  · exact this j i hne.symm heq.symm (hne.lt_of_le hlt)\n  use s ^ (i - (j + 1)) • x\n  rw [oreDiv_eq_iff]\n  refine ⟨s ^ j, (s ^ (j + 1)).1, ?_, ?_⟩\n  · change s ^ j • x = s ^ (j + 1) • s ^ (i - (j + 1)) • x\n    rw [← mul_smul, ← pow_add, Nat.add_sub_cancel' hlt, heq]\n  · simp_rw [SubmonoidClass.coe_pow, OneMemClass.coe_one, mul_one, pow_succ]\n\n@[to_additive]\ntheorem numeratorHom_surjective_of_finite [Finite S] : Surjective (numeratorHom (S := S)) :=\n  oreDiv_one_surjective_of_finite_left S R\n\n@[to_additive]\ntheorem cardinalMk_le_max : #(OreLocalization S X) ≤ max (lift.{v} #S) (lift.{u} #X) := by\n  rcases finite_or_infinite X with _ | _\n  · have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_right S X)\n    rw [lift_umax.{v, u}, lift_id'] at this\n    exact le_max_of_le_right this\n  rcases finite_or_infinite S with _ | _\n  · have := lift_mk_le_lift_mk_of_surjective (oreDiv_one_surjective_of_finite_left S X)\n    rw [lift_umax.{v, u}, lift_id'] at this\n    exact le_max_of_le_right this\n  convert! ←\n    mk_le_of_surjective (show Surjective fun x : X × S ↦ x.1 /ₒ x.2 from Quotient.mk''_surjective)\n  rw [mk_prod, mul_comm]\n  refine mul_eq_max ?_ ?_ <;> simp\n\n@[to_additive]\ntheorem cardinalMk_le : #(OreLocalization S R) ≤ #R := by\n  convert! ← cardinalMk_le_max S R\n  simp_rw [lift_id, max_eq_right_iff, mk_subtype_le]\n\n-- TODO: remove the `Commute` assumption\n@[to_additive]\n\nTarget:\ntheorem cardinalMk_le_lift_cardinalMk_of_commute (hc : ∀ s s' : S, Commute s s') :\n    #(OreLocalization S X) ≤ lift.{u} #X :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"9e5fa086937ddbc936aca1e5073aa97e5282eed6598ca162b5cb0943fe6d74c6","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/OreLocalization","family_id":"cardinalmk_le_lift_cardinalmk_of_commute","file_id":"mathlib/Mathlib/GroupTheory/OreLocalization/Cardinality.lean","sample_id":"00174d518458fef9a309864e49f69be9bd090110f25e5905b3ad9090d83909e4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cd750d60c8401fd18ff81d5b091050a985f85f02e4e58110a29f7aecd6f9f0ac","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"68d3eb64f1e33f838236f5814f4593cf223615ea272cf57c1d4b29e0bdf4e53f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f168373cfd1f31b36e20d360cfbb18b1a6084a7395ef31c4c3eb93947132babd","source_sha256":"586377b5167eecc243d7b879009165834566ed22063836fb061931738feab6a2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [card_eq_card_quotient_mul_card_subgroup s, @dvd_mul_right ℕ]\n\nvariable {H : Type*} [Group H]","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":26,"token_jaccard":0.818182,"token_length_ratio":1.238095},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"f2e1555249406de1da330958165d290725c296f9313c7f4953d084731fce1ca5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.GroupTheory.Coset.Basic\npublic import Mathlib.SetTheory.Cardinal.Finite\n\nNamespace:\nSubgroup\n\nLocal context:\n/-\nCopyright (c) 2018 Mitchell Rowett. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mitchell Rowett, Kim Morrison\n-/\n/-!\n# Lagrange's theorem: the order of a subgroup divides the order of the group.\n\n* `Subgroup.card_subgroup_dvd_card`: Lagrange's theorem (for multiplicative groups);\n  there is an analogous version for additive groups\n\n-/\n\npublic section\n\nassert_not_exists Field\n\nopen scoped Pointwise\n\nvariable {α : Type*} [Group α] {s : Subgroup α}\n\nnamespace QuotientGroup\n\n@[to_additive]\ninstance fintype [Fintype α] (s : Subgroup α) [DecidableRel (leftRel s).r] : Fintype (α ⧸ s) :=\n  Quotient.fintype (leftRel s)\n\n@[to_additive]\ninstance (priority := 100) finite [Finite α] : Finite (α ⧸ s) :=\n  Quotient.finite _\n\n@[to_additive]\ninstance fintypeQuotientRightRel [Fintype (α ⧸ s)] :\n    Fintype (Quotient (QuotientGroup.rightRel s)) :=\n  .ofEquiv (α ⧸ s) (QuotientGroup.quotientRightRelEquivQuotientLeftRel s).symm\n\nvariable (s) in\n@[to_additive]\nlemma card_quotient_rightRel [Fintype (α ⧸ s)] :\n    Fintype.card (Quotient (QuotientGroup.rightRel s)) = Fintype.card (α ⧸ s) :=\n  Fintype.ofEquiv_card (QuotientGroup.quotientRightRelEquivQuotientLeftRel s).symm\n\nend QuotientGroup\n\nnamespace Subgroup\n\n@[to_additive AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup]\ntheorem card_eq_card_quotient_mul_card_subgroup (s : Subgroup α) :\n    Nat.card α = Nat.card (α ⧸ s) * Nat.card s := by\n  rw [← Nat.card_prod]; exact Nat.card_congr Subgroup.groupEquivQuotientProdSubgroup\n\n@[to_additive]\nlemma card_mul_eq_card_subgroup_mul_card_quotient (s : Subgroup α) (t : Set α) :\n    Nat.card (t * s : Set α) = Nat.card s * Nat.card (t.image (↑) : Set (α ⧸ s)) := by\n  rw [← Nat.card_prod, Nat.card_congr]\n  apply Equiv.trans _ (QuotientGroup.preimageMkEquivSubgroupProdSet _ _)\n  rw [QuotientGroup.preimage_image_mk]\n  convert! Equiv.refl ↑(t * s)\n  aesop (add simp [Set.mem_mul])\n\n/-- **Lagrange's Theorem**: The order of a subgroup divides the order of its ambient group. -/\n@[to_additive (attr := wikidata Q505798) /-- **Lagrange's Theorem**: The order of an additive\nsubgroup divides the order of its ambient additive group. -/]\ntheorem card_subgroup_dvd_card (s : Subgroup α) : Nat.card s ∣ Nat.card α := by\n  classical simp [card_eq_card_quotient_mul_card_subgroup s, @dvd_mul_left ℕ]\n\n@[to_additive]\n\nTarget:\ntheorem card_quotient_dvd_card (s : Subgroup α) : Nat.card (α ⧸ s) ∣ Nat.card α :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  simp [card_eq_card_quotient_mul_card_subgroup s, @dvd_mul_right ℕ]\n\nvariable {H : Type*} [Group H]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Coset","family_id":"card_quotient_dvd_card","file_id":"mathlib/Mathlib/GroupTheory/Coset/Card.lean","sample_id":"f168373cfd1f31b36e20d360cfbb18b1a6084a7395ef31c4c3eb93947132babd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6d7fe2ba7791e82259c7c7dac10b1de23d8b4e925a2fa271084af7b3f600bee5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3b9bf8a1b9ae9ed64027deecfa9b2b28510490bbc0204132c8e750b77e3c634e","source_sha256":"75f644a2aa9ce33f4e3a9b3af6b61387bae9fdea7ce3dbf0ef1ec4c5bea1cda5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases hf2 : f = 0; · rw [hf2]; exact dvd_zero _\n  rw [factor, dif_pos (WfDvdMonoid.exists_irreducible_factor hf1 hf2)]\n  exact (Classical.choose_spec <| WfDvdMonoid.exists_irreducible_factor hf1 hf2).2","hard_negative":false,"metrics":{"chosen_tokens":45,"rejected_tokens":3,"token_jaccard":0.066667,"token_length_ratio":0.066667},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"f35a8e7db0623e1e93c90dd1632daf4551e9dceb04b874670bc5a2d449d43c8a","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.CharP.Algebra\npublic import Mathlib.FieldTheory.SplittingField.IsSplittingField\npublic import Mathlib.RingTheory.Algebraic.Basic\n\nNamespace:\nPolynomial\n\nLocal context:\n/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\n/-!\n# Splitting fields\n\nIn this file we prove the existence and uniqueness of splitting fields.\n\n## Main definitions\n\n* `Polynomial.SplittingField f`: A fixed splitting field of the polynomial `f`.\n\n## Main statements\n\n* `Polynomial.IsSplittingField.algEquiv`: Every splitting field of a polynomial `f` is isomorphic\n  to `SplittingField f` and thus, being a splitting field is unique up to isomorphism.\n\n## Implementation details\nWe construct a `SplittingFieldAux` without worrying about whether the instances satisfy nice\ndefinitional equalities. Then the actual `SplittingField` is defined to be a quotient of a\n`MvPolynomial` ring by the kernel of the obvious map into `SplittingFieldAux`. Because the\nactual `SplittingField` will be a quotient of a `MvPolynomial`, it has nice instances on it.\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nuniverse u v w\n\nvariable {F : Type u} {K : Type v} {L : Type w}\n\nnamespace Polynomial\n\nvariable [Field K] [Field L] [Field F]\n\nopen Polynomial\n\nsection SplittingField\n\nopen Classical in\n/-- Non-computably choose an irreducible factor from a polynomial. -/\ndef factor (f : K[X]) : K[X] :=\n  if H : ∃ g, Irreducible g ∧ g ∣ f then Classical.choose H else X\n\ntheorem irreducible_factor (f : K[X]) : Irreducible (factor f) := by\n  rw [factor]\n  split_ifs with H\n  · exact (Classical.choose_spec H).1\n  · exact irreducible_X\n\n/-- See note [fact non-instances]. -/\ntheorem fact_irreducible_factor (f : K[X]) : Fact (Irreducible (factor f)) :=\n  ⟨irreducible_factor f⟩\n\nattribute [local instance] fact_irreducible_factor\n\nTarget:\ntheorem factor_dvd_of_not_isUnit {f : K[X]} (hf1 : ¬IsUnit f) : factor f ∣ f :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/SplittingField","family_id":"factor_dvd_of_not_isunit","file_id":"mathlib/Mathlib/FieldTheory/SplittingField/Construction.lean","sample_id":"3b9bf8a1b9ae9ed64027deecfa9b2b28510490bbc0204132c8e750b77e3c634e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"197871194b11ccd2f7b68c20098508f4fb8d58251817640d3a8e7c9a33767c9c","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7d23803bcb7a0b31219450a0f71c8fc39a61b24e5be9b632d0c838c60ba4d872","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"41bafb9f17936ff03795282fab01c1edc2ec5df90a2dc068b3d884811a5b8b02","source_sha256":"ced30280014822262039ce197c9e4ae821d789623b770f4b0e6844d4852109a3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h _ _ ↦ by rw [h], fun h ↦ ?_⟩\n  rw [DFunLike.ext'_iff]\n  apply Function.Surjective.injective_comp_right mkAlgHom_surjective\n  simpa [← AlgHom.coe_comp] using MvPolynomial.algHom_ext fun ⟨n, m⟩ ↦ h n m","hard_negative":true,"metrics":{"chosen_tokens":54,"rejected_tokens":3,"token_jaccard":0.03125,"token_length_ratio":0.055556},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"f371a38e650cd99b2a7379d936de257bacd66cb57805b5348dcc452b7706fc8e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.MvPolynomial.Eval\npublic import Mathlib.Algebra.RingQuot\npublic import Mathlib.RingTheory.DividedPowers.Basic\n\nNamespace:\nDividedPowerAlgebra\n\nLocal context:\n/-\nCopyright (c) 2026 Antoine Chambert-Loir & María Inés de Frutos—Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir, María Inés de Frutos—Fernández\n-/\n/-!\n# The universal divided power algebra\n\nLet `R` be a (commutative) semiring and `M` be an `R`-module. In this file we define `Γ_R(M)`,\nthe universal divided power algebra of `M`, as the ring quotient of the polynomial ring\nin the variables `ℕ × M` by the relation `DividedPowerAlgebra.Rel`.\n\n`DividedPowerAlgebra R M` satisfies a weak universal property for morphisms to rings with\ndivided powers (`DividedPowerAlgebra.lift`).\n\n## Main definitions\n\n* `DividedPowerAlgebra.Rel`: the type coding the basic relations that will give rise to the\n  divided power algebra.\n\n* `DividedPowerAlgebra R M`: the universal divided power algebra of the `R`-module `M`,\n  defined as `RingQuot` of `DividedPowerAlgebra.Rel R M`.\n\n* `DividedPowerAlgebra.dp R n m`: for `n : ℕ` and `m : M`, this is the equivalence class of\n  `MvPolynomial.X (⟨n, m⟩)` in `DividedPowerAlgebra R M`.\n\n  When that algebra is endowed with its canonical divided power structure (to be defined),\n  the image of `MvPolynomial.X (n, m)`, for any `n : ℕ` and `m : M`, is equal to\n  the `n`th divided power of the image of `m`.\n\n  The API will be setup so that it is never (never say never…) necessary to lift to `MvPolynomial`.\n\n* `DividedPowerAlgebra.lift`: the weak universal property of `DividedPowerAlgebra R M`.\n\n* `DividedPowerAlgebra.map`: the functoriality map between divided power algebras\n  associated with a linear map of the underlying modules.\n  Given an `R`-algebra `S`, an `S`-module `N` and an `R`-linear map `f : M →ₗ[R] N`,\n  this is the map `DividedPowerAlgebra R M →ₐ[R] DividedPowerAlgebra S N`\n  sending `dp R n m` to `dp S n (f m)`.\n\n## References\n\n* [P. Berthelot (1974), *Cohomologie cristalline des schémas de\n  caractéristique $p$ > 0*][Berthelot-1974]\n\n* [P. Berthelot and A. Ogus (1978), *Notes on crystalline\n  cohomology*][BerthelotOgus-1978]\n\n* [N. Roby (1963), *Lois polynomes et lois formelles en théorie des\n  modules*][Roby-1963]\n\n* [N. Roby (1965), *Les algèbres à puissances dividées*][Roby-1965]\n\n## TODO\n\n* Show in upcoming files that `DividedPowerAlgebra R M` has divided powers.\n\n\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Finset Ideal MvPolynomial RingQuot\n\nvariable (R M : Type*) [CommSemiring R] [AddCommMonoid M] [Module R M]\n\nnamespace DividedPowerAlgebra\n\n/-- The type coding the basic relations that will give rise to the divided power algebra.\n  The class of `MvPolynomial.X (n, a)` will be equal to `dpow n a`, for `a ∈ M`. -/\ninductive Rel : MvPolynomial (ℕ × M) R → MvPolynomial (ℕ × M) R → Prop\n  | rfl_zero : Rel 0 0 -- Needed for technical reasons.\n  | zero {a : M} : Rel (X (0, a)) 1\n  | smul {r : R} {n : ℕ} {a : M} : Rel (X (n, r • a)) (r ^ n • X (n, a))\n  | mul {m n : ℕ} {a : M} : Rel (X (m, a) * X (n, a)) (Nat.choose (m + n) m • X (m + n, a))\n  | add {n : ℕ} {a b : M} :\n    Rel (X (n, a + b)) ((Finset.antidiagonal n).sum fun k ↦ X (k.1, a) * X (k.2, b))\n\n/-- The ideal of `MvPolynomial (ℕ × M) R` generated by `Rel`. -/\ndef RelI : Ideal (MvPolynomial (ℕ × M) R) := ofRel (DividedPowerAlgebra.Rel R M)\n\nend DividedPowerAlgebra\n\n/-- The divided power algebra of a module M is defined as the ring quotient of the polynomial ring\n  in the variables `ℕ × M` by the ring relation defined by `DividedPowerAlgebra.Rel`.\n  We will later show that that `DividedPowerAlgebra R M` has divided powers.\n  It satisfies a weak universal property for morphisms to rings with divided powers. -/\nabbrev DividedPowerAlgebra := RingQuot (DividedPowerAlgebra.Rel R M)\n\nnamespace DividedPowerAlgebra\n\nopen MvPolynomial\n\nvariable {R M}\n\nlemma mkAlgHom_surjective : Function.Surjective (mkAlgHom R (Rel R M)) :=\n  RingQuot.mkAlgHom_surjective _ _\n\nlemma mkAlgHom_C (a : R) :\n    mkAlgHom R (Rel R M) (C a) = algebraMap R (DividedPowerAlgebra R M) a := by\n  rw [← MvPolynomial.algebraMap_eq, AlgHom.commutes]\n\nlemma mkRingHom_C (a : R) :\n    mkRingHom (Rel R M) (C a) = algebraMap R (DividedPowerAlgebra R M) a := by\n  rw [← mkAlgHom_C, mkAlgHom, AlgHom.coe_mk]\n\nvariable (R) in\n/-- `dp R n m` is the equivalence class of `X (⟨n, m⟩)` in `DividedPowerAlgebra R M`. -/\ndef dp (n : ℕ) (m : M) : DividedPowerAlgebra R M := mkAlgHom R (Rel R M) (X ⟨n, m⟩)\n\ntheorem dp_def (n : ℕ) (m : M) :\n  dp R n m = mkAlgHom R (Rel R M) (X ⟨n, m⟩) := rfl\n\nprotected theorem induction_on' {P : DividedPowerAlgebra R M → Prop} (f : DividedPowerAlgebra R M)\n    (h_C : ∀ a, P (mkAlgHom R (Rel R M) (C a))) (h_add : ∀ f g, P f → P g → P (f + g))\n    (h_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)) : P f := by\n  obtain ⟨F, hf⟩ := RingQuot.mkRingHom_surjective (DividedPowerAlgebra.Rel R M) f\n  rw [← hf]\n  induction F using MvPolynomial.induction_on generalizing f with\n  | C a =>\n      convert! h_C a using 1\n      rw [mkAlgHom, AlgHom.coe_mk]\n  | add g1 g2 hg1 hg2 =>\n      rw [map_add]\n      exact h_add _ _ (hg1 ((mkRingHom (Rel R M)) g1) rfl) (hg2 ((mkRingHom (Rel R M)) g2) rfl)\n  | mul_X g nm h =>\n      have h' : (mkRingHom (Rel R M)) (X nm) = dp R nm.1 nm.2 := by\n        simp only [dp_def, Prod.mk.eta, mkAlgHom, AlgHom.coe_mk]\n      rw [_root_.map_mul, h']\n      exact h_dp _ _ _ (h (mkRingHom (Rel R M) g) rfl)\n\n@[elab_as_elim]\nprotected theorem induction_on {P : DividedPowerAlgebra R M → Prop} (f : DividedPowerAlgebra R M)\n    (C : ∀ a, P (algebraMap R _ a)) (add : ∀ f g, P f → P g → P (f + g))\n    (dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ) (m : M), P f → P (f * dp R n m)) : P f :=\n  DividedPowerAlgebra.induction_on' f (fun a ↦ by rw [mkAlgHom_C]; exact C a) add dp\n\ntheorem dp_eq_mkRingHom (n : ℕ) (m : M) :\n    dp R n m = mkRingHom (Rel R M) (X (⟨n, m⟩)) := by\n  simp [dp, mkRingHom, mkAlgHom]\n\ntheorem dp_zero {m : M} : dp R 0 m = 1 := by\n  rw [dp_def, ← map_one (mkAlgHom R (Rel R M))]\n  exact RingQuot.mkAlgHom_rel R Rel.zero\n\ntheorem dp_smul {r : R} {n : ℕ} {m : M} : dp R n (r • m) = r ^ n • dp R n m := by\n  rw [dp_def, dp_def, ← map_smul]\n  exact mkAlgHom_rel R Rel.smul\n\ntheorem dp_null {n : ℕ} : dp R n (0 : M) = if n = 0 then 1 else 0 := by\n  cases Nat.eq_zero_or_pos n with\n  | inl hn =>\n    rw [if_pos hn, hn, dp_zero]\n  | inr hn =>\n    rw [if_neg (ne_of_gt hn), ← zero_smul R (0 : M), dp_smul]\n    rw [zero_pow (Nat.pos_iff_ne_zero.mp hn), zero_smul]\n\ntheorem dp_null_of_ne_zero {n : ℕ} (hn : n ≠ 0) : dp R n (0 : M) = 0 := by\n  rw [dp_null, if_neg hn]\n\ntheorem dp_mul {n p : ℕ} {m : M} :\n    dp R n m * dp R p m = (n + p).choose n • dp R (n + p) m := by\n  simp only [dp_def, ← _root_.map_mul, ← map_nsmul]\n  exact mkAlgHom_rel R Rel.mul\n\ntheorem dp_add {n : ℕ} {x y : M} :\n    dp R n (x + y) = (antidiagonal n).sum fun k ↦ dp R k.1 x * dp R k.2 y := by\n  simp only [dp_def]\n  rw [mkAlgHom_rel (A := MvPolynomial (ℕ × M) R) R Rel.add, map_sum,\n    Finset.sum_congr rfl (fun k _ ↦ by rw [_root_.map_mul])]\n\ntheorem dp_sum {ι : Type*} [DecidableEq ι] (s : Finset ι) (q : ℕ) (x : ι → M) :\n    dp R q (s.sum x) =\n      (Finset.sym s q).sum fun k ↦ s.prod fun i ↦ dp R (Multiset.count i k) (x i) :=\n  DividedPowers.dpow_sum' (I := ⊤) _ (fun _ ↦ dp_zero)\n    (fun _ _ ↦ dp_add) dp_null_of_ne_zero (fun _ _ ↦ trivial)\n\ntheorem dp_sum_smul {ι : Type*} [DecidableEq ι] (s : Finset ι) (q : ℕ) (a : ι → R) (x : ι → M) :\n    dp R q (s.sum fun i ↦ a i • x i) =\n      (Finset.sym s q).sum fun k ↦\n        (s.prod fun i ↦ a i ^ Multiset.count i k) •\n          s.prod fun i ↦ dp R (Multiset.count i k) (x i) := by\n  simp_rw [dp_sum, dp_smul, Algebra.smul_def, map_prod, ← Finset.prod_mul_distrib]\n\nopen Nat in\nlemma prod_dp {ι : Type*} {s : Finset ι} {n : ι → ℕ} {m : M} :\n    ∏ i ∈ s, (dp R (n i) m) = (Nat.multinomial s n) * dp R (s.sum n) m := by\n  classical\n  induction s using Finset.induction with\n  | empty =>\n    simp only [prod_empty, multinomial_empty, cast_one, sum_empty, one_mul, dp_zero]\n  | insert _ _ hi hrec =>\n    rw [prod_insert hi, hrec, ← mul_assoc, mul_comm (dp R (n _) m),\n      mul_assoc, dp_mul, ← sum_insert hi, nsmul_eq_mul, ← mul_assoc]\n    congr 1\n    rw [multinomial_insert hi, mul_comm, cast_mul, sum_insert hi]\n\nopen scoped Nat\n\ntheorem natFactorial_mul_dp_eq (n : ℕ) (x : M) :\n    n ! * dp R n x = (dp R 1 x) ^ n := by\n  induction n with\n  | zero => simp [dp_zero]\n  | succ n h =>\n    rw [pow_succ, ← h, mul_assoc, dp_mul, nsmul_eq_mul, ← mul_assoc, ← Nat.cast_mul]\n    simp [mul_comm _ (n + 1), Nat.factorial_succ]\n\nvariable (R M) in\n/-- The canonical linear map `M →ₗ[R] DividedPowerAlgebra R M`. -/\ndef embed : M →ₗ[R] DividedPowerAlgebra R M where\n  toFun m := dp R 1 m\n  map_add' _ _ := by simp [dp_add, Nat.antidiagonal_succ, dp_zero, add_comm]\n  map_smul' _ _ := by simp [dp_smul, pow_one, RingHom.id_apply]\n\ntheorem embed_def (m : M) : embed R M m = dp R 1 m := rfl\n\nTarget:\ntheorem algHom_ext_iff {A : Type*} [CommSemiring A] [Algebra R A]\n    {f g : DividedPowerAlgebra R M →ₐ[R] A} :\n    f = g ↔ ∀ n m, f (dp R n m) = g (dp R n m) :=\n\nProof body:\n","rejected":"by\n  exact algHom_ext_iff","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"4b41f2a593c49deb962ebe4b78ce3bd32d961a187f21987ee4c90767fe7bc73f","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/DividedPowerAlgebra","family_id":"alghom_ext_iff","file_id":"mathlib/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean","sample_id":"41bafb9f17936ff03795282fab01c1edc2ec5df90a2dc068b3d884811a5b8b02"},"split":"train"}
{"binding":{"chosen_proof_sha256":"97f1a84bedb0ef6995acbe3d016b9163eb9ac17f32bd0890956d8e9bd5498309","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"148d4bb481d5fb6dd96abb5cffc2629c9e26450cf546a2d15ee9ec7face5cf4f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"567f3c5c6d2031f7fb93f23ec479080605b61b08ad68d9daa949e04606ce6590","source_sha256":"48e76914e12791b3eb841b9933ad6f80e72c4515bc3faaf828abf1cb41030052","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    refine (hB.hasLaw I).congr ?_\n    have : ∀ᵐ ω ∂P, ∀ i : I, B i ω = C i ω := ae_all_iff.2 fun _ ↦ h _\n    filter_upwards [this] with ω hω using funext fun i ↦ (hω i).symm","hard_negative":false,"metrics":{"chosen_tokens":61,"rejected_tokens":3,"token_jaccard":0.025641,"token_length_ratio":0.04918},"negative_category":"by_exact_placeholder","negative_mode":"placeholder","pair_id":"f40309344a510224d9770d3ea2213bfcb54fa2857bc5d2ab522fb6fcb66ee5ed","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily\npublic import Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Def\npublic import Mathlib.Probability.Independence.Process.HasIndepIncrements.Basic\nimport Mathlib.Probability.Distributions.Gaussian.CharFun\nimport Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic\nimport Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence\nimport Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic\nimport Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence\nimport Mathlib.Probability.Independence.Process.HasIndepIncrements.IsGaussianProcess\n\nNamespace:\nProbabilityTheory\n\nLocal context:\n/-\nCopyright (c) 2025 Etienne Marion. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Etienne Marion\n-/\n/-!\n# Brownian motion\n\nIn this file we define two predicates over stochastic processes `X : ℝ≥0 → Ω → ℝ` given\na probability measure `P : Measure Ω`. `IsPreBrownianReal X P` means that\n`X` is a pre-Brownian motion. It means that it has the law of the Brownian motion, namely that\nits finite dimensional distributions are given by `projectiveFamily`. Then\n`IsBrownianReal X P` means that `X` is a Brownian motion, which means that it is a pre-Brownian\nmotion with almost surely continuous paths.\n\nWe prove that a centered Gaussian process `X` with covariances given by `cov[X s, X t; P] = min s t`\nis a pre-Brownian motion and provide basic invariance properties. We also prove the\nweak Markov property: if `B` is a pre-Brownian motion and `t₀ : ℝ≥0`, then the process\n`t ↦ B (t + t₀) - B t₀` is a pre-Brownian motion independent from `(B t | t ≤ t₀)`.\n\n## Main definitions\n\n* `IsPreBrownianReal X P`: A stochastic process is called pre-Brownian if its finite-dimensional\n  laws are those of the Brownian motion, see `projectiveFamily`.\n* `IsBrownianReal X P`: A stochastic process is called Brownian if its finite-dimensional laws\n  are those of the Brownian motion, see `IsPreBrownianReal`,\n  and if it has almost-surely continuous paths.\n\n## Main statements\n\n* `IsGaussianProcess.isPreBrownianReal_of_covariance`: A centered Gaussian process with the right\n  covariance is a pre-Brownian motion.\n* `HasIndepIncrements.isPreBrownianReal_of_hasLaw`: A stochastic process `X` with independent\n  increments and such that for all `t`, `X t` has law `gaussianReal 0 t` is a pre-Brownian motion.\n* `IsPreBrownianReal.indepFun_shift`: The weak Markov property: If `B` is a pre-Brownian motion,\n  then `B (t₀ + t) - B t₀` is a pre-Brownian motion which is independent from `(B t, t ≤ t₀)`.\n\n## Tags\n\npre-Brownian motion, Brownian motion, Markov property\n\n-/\n\n@[expose] public section\n\nopen MeasureTheory ProbabilityTheory.BrownianReal\nopen scoped ENNReal NNReal Topology\n\nvariable {Ω : Type*} {mΩ : MeasurableSpace Ω} {B X : ℝ≥0 → Ω → ℝ} {P : Measure Ω}\n\nnamespace ProbabilityTheory\n\nsection IsPreBrownianReal\n\n/-! ### Pre-Brownian motion -/\n\n/-- A stochastic process is called **pre-Brownian** if its finite-dimensional laws are those\nof the Brownian motion, see `projectiveFamily`.\n\nNote: we name the constructor `mk'` so as to define later `IsPreBrownianReal.mk`, which to\npre-Brownian motion will associate a continuous modification,\nin a way similar to `AEMeasurable.mk`. -/\nstructure IsPreBrownianReal (X : ℝ≥0 → Ω → ℝ) (P : Measure Ω := by volume_tac) : Prop where\n  mk' ::\n  hasLaw : ∀ I : Finset ℝ≥0, HasLaw (fun ω ↦ I.restrict (X · ω)) (projectiveFamily I) P\n\n/- A modification of a pre-Brownian is pre-Brownian. -/\n\nTarget:\nlemma IsPreBrownianReal.congr {C : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P)\n    (h : ∀ t, B t =ᵐ[P] C t) :\n    IsPreBrownianReal C P where\n  hasLaw I :=\n\nProof body:\n","rejected":"by\n  exact placeholder","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/BrownianMotion","family_id":"isprebrownianreal","file_id":"mathlib/Mathlib/Probability/BrownianMotion/Basic.lean","sample_id":"567f3c5c6d2031f7fb93f23ec479080605b61b08ad68d9daa949e04606ce6590"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5914fc3ed55a874cb691c1f7b7c08b38f9e056106005fb4efe3ad9109dba7f6a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"94baa657612564dcd8266b8f23b88fc4f5d958c0822493bf537bbfa331437520","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"636539d6c0febb4c83b00bdc22993462b52f9a53f319750f1f167f122ce30681","source_sha256":"56dc78571f595294b2d5c06f0925083625898756f6d4c19bdd95afa1a756b177","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply (Ideal.homogeneousCore 𝒜 I).isHomogeneous.isPrime_of_homogeneous_mem_or_mem\n  · exact ne_top_of_le_ne_top h.ne_top (Ideal.toIdeal_homogeneousCore_le 𝒜 I)\n  rintro x y hx hy hxy\n  have H := h.mem_or_mem (Ideal.toIdeal_homogeneousCore_le 𝒜 I hxy)\n  refine H.imp ?_ ?_\n  · exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hx\n  · exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hy","hard_negative":false,"metrics":{"chosen_tokens":66,"rejected_tokens":71,"token_jaccard":0.888889,"token_length_ratio":1.075758},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"f403dd22f2d145b528e5e9f653595d4e75bac7c3a6ad468bfbc1e02f4caa6242","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2022 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang, Eric Wieser\n-/\n/-!\n\nThis file contains a proof that the radical of any homogeneous ideal is a homogeneous ideal\n\n## Main statements\n\n* `Ideal.IsHomogeneous.isPrime_iff`: for any `I : Ideal A`, if `I` is homogeneous, then\n  `I` is prime if and only if `I` is homogeneously prime, i.e. `I ≠ ⊤` and if `x, y` are\n  homogeneous elements such that `x * y ∈ I`, then at least one of `x,y` is in `I`.\n* `Ideal.IsPrime.homogeneousCore`: for any `I : Ideal A`, if `I` is prime, then\n  `I.homogeneous_core 𝒜` (i.e. the largest homogeneous ideal contained in `I`) is also prime.\n* `Ideal.IsHomogeneous.radical`: for any `I : Ideal A`, if `I` is homogeneous, then the\n  radical of `I` is homogeneous as well.\n* `HomogeneousIdeal.radical`: for any `I : HomogeneousIdeal 𝒜`, `I.radical` is the\n  radical of `I` as a `HomogeneousIdeal 𝒜`.\n\n## Implementation details\n\nThroughout this file, the indexing type `ι` of grading is assumed to be a\nlinearly ordered cancellative monoid. This might be stronger than necessary but cancelling\nproperty is strictly necessary; for a counterexample of how `Ideal.IsHomogeneous.isPrime_iff`\nfails for a non-cancellative set see `Counterexamples/HomogeneousPrimeNotPrime.lean`.\n\n## Tags\n\nhomogeneous, radical\n-/\n\n@[expose] public section\n\n\nopen GradedRing DirectSum SetLike Finset\n\nvariable {ι σ A : Type*}\nvariable [CommRing A]\nvariable [AddCommMonoid ι] [LinearOrder ι] [IsOrderedCancelAddMonoid ι]\nvariable [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜]\n\ntheorem Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem {I : Ideal A} (hI : I.IsHomogeneous 𝒜)\n    (I_ne_top : I ≠ ⊤)\n    (homogeneous_mem_or_mem :\n      ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I) :\n    Ideal.IsPrime I :=\n  ⟨I_ne_top, by\n    intro x y hxy\n    by_contra! ⟨rid₁, rid₂⟩\n    classical\n      /-\n        The idea of the proof is the following :\n        since `x * y ∈ I` and `I` homogeneous, then `proj i (x * y) ∈ I` for any `i : ι`.\n        Then consider two sets `{i ∈ x.support | xᵢ ∉ I}` and `{j ∈ y.support | yⱼ ∉ J}`;\n        let `max₁, max₂` be the maximum of the two sets, then `proj (max₁ + max₂) (x * y) ∈ I`.\n        Then, `proj max₁ x ∉ I` and `proj max₂ j ∉ I`\n        but `proj i x ∈ I` for all `max₁ < i` and `proj j y ∈ I` for all `max₂ < j`.\n        `  proj (max₁ + max₂) (x * y)`\n        `= ∑ {(i, j) ∈ supports | i + j = max₁ + max₂}, xᵢ * yⱼ`\n        `= proj max₁ x * proj max₂ y`\n        `  + ∑ {(i, j) ∈ supports \\ {(max₁, max₂)} | i + j = max₁ + max₂}, xᵢ * yⱼ`.\n        This is a contradiction, because both `proj (max₁ + max₂) (x * y) ∈ I` and the sum on the\n        right-hand side is in `I` however `proj max₁ x * proj max₂ y` is not in `I`.\n        -/\n      set set₁ := {i ∈ (decompose 𝒜 x).support | proj 𝒜 i x ∉ I} with set₁_eq\n      set set₂ := {i ∈ (decompose 𝒜 y).support | proj 𝒜 i y ∉ I} with set₂_eq\n      have nonempty :\n        ∀ x : A, x ∉ I → {i ∈ (decompose 𝒜 x).support | proj 𝒜 i x ∉ I}.Nonempty := by\n        intro x hx\n        rw [filter_nonempty_iff]\n        contrapose! hx\n        simp_rw [proj_apply] at hx\n        rw [← sum_support_decompose 𝒜 x]\n        exact Ideal.sum_mem _ hx\n      set max₁ := set₁.max' (nonempty x rid₁)\n      set max₂ := set₂.max' (nonempty y rid₂)\n      have mem_max₁ : max₁ ∈ set₁ := max'_mem set₁ (nonempty x rid₁)\n      have mem_max₂ : max₂ ∈ set₂ := max'_mem set₂ (nonempty y rid₂)\n      replace hxy : proj 𝒜 (max₁ + max₂) (x * y) ∈ I := hI _ hxy\n      have mem_I : proj 𝒜 max₁ x * proj 𝒜 max₂ y ∈ I := by\n        set antidiag :=\n          {z ∈ (decompose 𝒜 x).support ×ˢ (decompose 𝒜 y).support | z.1 + z.2 = max₁ + max₂}\n           with ha\n        have mem_antidiag : (max₁, max₂) ∈ antidiag := by\n          simp only [antidiag, mem_filter, mem_product]\n          exact ⟨⟨mem_of_mem_filter _ mem_max₁, mem_of_mem_filter _ mem_max₂⟩, trivial⟩\n        have eq_add_sum :=\n          calc\n            proj 𝒜 (max₁ + max₂) (x * y) = ∑ ij ∈ antidiag, proj 𝒜 ij.1 x * proj 𝒜 ij.2 y := by\n              simp_rw [ha, proj_apply, DirectSum.decompose_mul, DirectSum.coe_mul_apply 𝒜]\n            _ =\n                proj 𝒜 max₁ x * proj 𝒜 max₂ y +\n                  ∑ ij ∈ antidiag.erase (max₁, max₂), proj 𝒜 ij.1 x * proj 𝒜 ij.2 y :=\n              (add_sum_erase _ _ mem_antidiag).symm\n        rw [eq_sub_of_add_eq eq_add_sum.symm]\n        refine Ideal.sub_mem _ hxy (Ideal.sum_mem _ fun z H => ?_)\n        rcases z with ⟨i, j⟩\n        simp only [antidiag, mem_erase, Prod.mk_inj, Ne, mem_filter, mem_product] at H\n        rcases H with ⟨H₁, ⟨H₂, H₃⟩, H₄⟩\n        have max_lt : max₁ < i ∨ max₂ < j := by\n          convert! le_or_lt_of_add_le_add H₄.ge using 1\n          rw [Ne.le_iff_lt]\n          rintro rfl\n          cases H₁ ⟨rfl, add_left_cancel H₄⟩\n        rcases max_lt with max_lt | max_lt\n        · -- in this case `max₁ < i`, then `xᵢ ∈ I`; for otherwise `i ∈ set₁` then `i ≤ max₁`.\n          have notMem : i ∉ set₁ := fun h =>\n            lt_irrefl _ ((max'_lt_iff set₁ (nonempty x rid₁)).mp max_lt i h)\n          rw [set₁_eq] at notMem\n          simp only [not_and, Classical.not_not, mem_filter] at notMem\n          exact Ideal.mul_mem_right _ I (notMem H₂)\n        · -- in this case `max₂ < j`, then `yⱼ ∈ I`; for otherwise `j ∈ set₂`, then `j ≤ max₂`.\n          have notMem : j ∉ set₂ := fun h =>\n            lt_irrefl _ ((max'_lt_iff set₂ (nonempty y rid₂)).mp max_lt j h)\n          rw [set₂_eq] at notMem\n          simp only [not_and, Classical.not_not, mem_filter] at notMem\n          exact Ideal.mul_mem_left I _ (notMem H₃)\n      have notMem_I : proj 𝒜 max₁ x * proj 𝒜 max₂ y ∉ I := by\n        have neither_mem : proj 𝒜 max₁ x ∉ I ∧ proj 𝒜 max₂ y ∉ I := by\n          rw [mem_filter] at mem_max₁ mem_max₂\n          exact ⟨mem_max₁.2, mem_max₂.2⟩\n        intro _rid\n        rcases homogeneous_mem_or_mem ⟨max₁, SetLike.coe_mem _⟩ ⟨max₂, SetLike.coe_mem _⟩ mem_I\n          with h | h\n        · apply neither_mem.1 h\n        · apply neither_mem.2 h\n      exact notMem_I mem_I⟩\n\ntheorem Ideal.IsHomogeneous.isPrime_iff {I : Ideal A} (h : I.IsHomogeneous 𝒜) :\n    I.IsPrime ↔\n      I ≠ ⊤ ∧\n        ∀ {x y : A},\n          IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I :=\n  ⟨fun HI => ⟨HI.ne_top, fun _ _ hxy => Ideal.IsPrime.mem_or_mem HI hxy⟩,\n    fun ⟨I_ne_top, homogeneous_mem_or_mem⟩ =>\n    h.isPrime_of_homogeneous_mem_or_mem I_ne_top @homogeneous_mem_or_mem⟩\n\nTarget:\ntheorem Ideal.IsPrime.homogeneousCore {I : Ideal A} (h : I.IsPrime) :\n    (I.homogeneousCore 𝒜).toIdeal.IsPrime :=\n\nProof body:\n","rejected":"by\n  apply (Ideal.homogeneousCore 𝒜 I).isHomogeneous.isPrime_of_homogeneous_mem_or_mem\n  · exact ne_top_of_le_ne_top h.ne_top (Ideal.toIdeal_homogeneousCore_le 𝒜 I)\n  rintro x y hx hy hxy\n  have H := h.mem_or_mem (Ideal.toIdeal_homogeneousCore_le 𝒜 I hxy)\n  refine H.imp ?_ ?_\n  · exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hx\n  · exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hy\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/GradedAlgebra","family_id":"ideal","file_id":"mathlib/Mathlib/RingTheory/GradedAlgebra/Radical.lean","sample_id":"636539d6c0febb4c83b00bdc22993462b52f9a53f319750f1f167f122ce30681"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3db3fcbd4e98076f3d0533d22872c1a4a27f5605675d9e65936bcaf6671c864f","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"f1ecebbbb93ce9395668a94b0ca695098896585aeb91c7e3dab1d04b2360f274","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5f10fa5c3e8b99e44987fd6f1820e919de6c733d5031ca673f738ff243d9f672","source_sha256":"36095e70549b743e6d6e18e94107fa315e72325dc59944c049976dea39a27dfe","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [← scottContinuousOn_univ] at ⊢\n  exact .fromProd (fun a ↦ (h₁ a).scottContinuousOn) (fun b ↦ (h₂ b).scottContinuousOn)\n    (fun a ↦ (h₁ a).monotone) (fun b ↦ (h₂ b).monotone)","hard_negative":true,"metrics":{"chosen_tokens":59,"rejected_tokens":3,"token_jaccard":0.086957,"token_length_ratio":0.050847},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"f404585a26f6d125c21efbb18736bf413e4c51bd0e08e6b83622d37dbf533078","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ScottContinuity\npublic import Mathlib.Order.Bounds.Lattice\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Christopher Hoskin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christopher Hoskin\n-/\n/-!\n# Scott continuity on product spaces\n\n## Main result\n\n- `ScottContinuous_prod_of_ScottContinuous`: A function is Scott continuous on a product space if it\n  is Scott continuous in each variable.\n- `ScottContinuousOn.inf₂`: For complete linear orders, the meet operation is Scott continuous.\n\n-/\n\npublic section\n\nopen Set\n\nvariable {α β γ : Type*}\n\n/-- If  is Scott continuous on a product space if it is Scott continuous and monotone in each\nvariable -/\nlemma ScottContinuousOn.fromProd [Preorder α] [Preorder β] [Preorder γ]\n    {f : α × β → γ} {D : Set (Set (α × β))}\n    (h₁ : ∀ a, ScottContinuousOn ((fun d => Prod.snd '' d) '' D) (fun b => f (a, b)))\n    (h₂ : ∀ b, ScottContinuousOn ((fun d => Prod.fst '' d) '' D) (fun a => f (a, b)))\n    (h₁' : ∀ a, Monotone (fun b => f (a, b))) (h₂' : ∀ b, Monotone (fun a => f (a, b))) :\n    ScottContinuousOn D f := fun d hX hd₁ hd₂ ⟨p1, p2⟩ hdp => by\n  rw [isLUB_congr ((monotone_prod_iff.mpr ⟨h₁', h₂'⟩).upperBounds_image_of_directedOn_prod hd₂),\n    ← iUnion_of_singleton_coe (Prod.fst '' d), iUnion_prod_const, image_iUnion,\n    ← isLUB_iUnion_iff_of_isLUB (fun a => by\n      rw [singleton_prod, image_image f (fun b ↦ (a, b))]\n      exact h₁ _ (mem_image_of_mem (fun d ↦ Prod.snd '' d) hX) (Nonempty.image Prod.snd hd₁)\n        (DirectedOn.snd hd₂) (isLUB_prod.mp hdp).2) _, Set.range]\n  convert!\n    (h₂ _ (mem_image_of_mem (fun d ↦ Prod.fst '' d) hX) (Nonempty.image Prod.fst hd₁)\n      (DirectedOn.fst hd₂) (isLUB_prod.mp hdp).1)\n  ext : 1\n  simp_all only [Subtype.exists, mem_image, Prod.exists,\n    exists_and_right, exists_eq_right, exists_prop, mem_setOf_eq]\n\nTarget:\nlemma ScottContinuous.fromProd {γ : Type*} [Preorder α] [Preorder β] [Preorder γ]\n    {f : α × β → γ} (h₁ : ∀ a, ScottContinuous (fun b => f (a, b)))\n    (h₂ : ∀ b, ScottContinuous (fun a => f (a, b))) : ScottContinuous f :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_5f10fa5c3e8b","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"49d62ccef480c865a9ee8a98cb5d10270d13820caec9dc015e12c84e6a648408","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/ScottContinuity","family_id":"scottcontinuous","file_id":"mathlib/Mathlib/Order/ScottContinuity/Prod.lean","sample_id":"5f10fa5c3e8b99e44987fd6f1820e919de6c733d5031ca673f738ff243d9f672"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6695cef3757a420cfb80a318d6ef88b082eb4eef77453a026e398af8fe7ac543","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9f2e69ab8abb2e6cb6d83b900a61296628713ac81f5453a29fe9b5c63247d26f","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"37f235bb48e86794e4b09884c23d8d2d7e5709381d2c53948b9b73b1480e3fa6","source_sha256":"bc065e5d26147920810f7e88f906504ecfcce0ffd1da2e0b44dc4df7256292f1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [iSup, Bool.range_eq, sSup_pair, sup_comm]","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":3,"token_jaccard":0.076923,"token_length_ratio":0.230769},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"f408bc396de89130f3c3f895044a48a5b933e57f1583ef4773ae5898578c7fb7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Bool.Set\npublic import Mathlib.Data.Nat.Set\npublic import Mathlib.Order.CompleteLattice.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\n/-!\n# Theory of complete lattices\n\nThis file contains results on complete lattices that need more theory to develop.\n\n## Naming conventions\n\nIn lemma names,\n* `sSup` is called `sSup`\n* `sInf` is called `sInf`\n* `⨆ i, s i` is called `iSup`\n* `⨅ i, s i` is called `iInf`\n* `⨆ i j, s i j` is called `iSup₂`. This is an `iSup` inside an `iSup`.\n* `⨅ i j, s i j` is called `iInf₂`. This is an `iInf` inside an `iInf`.\n* `⨆ i ∈ s, t i` is called `biSup` for \"bounded `iSup`\". This is the special case of `iSup₂`\n  where `j : i ∈ s`.\n* `⨅ i ∈ s, t i` is called `biInf` for \"bounded `iInf`\". This is the special case of `iInf₂`\n  where `j : i ∈ s`.\n\n## Notation\n\n* `⨆ i, f i` : `iSup f`, the supremum of the range of `f`;\n* `⨅ i, f i` : `iInf f`, the infimum of the range of `f`.\n-/\n\npublic section\n\nopen Function OrderDual Set\n\nvariable {α β γ : Type*} {ι ι' : Sort*} {κ : ι → Sort*} {κ' : ι' → Sort*}\n\nopen OrderDual\n\nsection\n\nvariable [CompleteLattice α] {f g s : ι → α} {a b : α}\n\n/-!\n### `iSup` and `iInf` under `Bool`\n-/\n\n@[to_dual]\n\nTarget:\ntheorem iSup_bool_eq {f : Bool → α} : ⨆ b : Bool, f b = f true ⊔ f false :=\n\nProof body:\n","rejected":"by\n  exact iSup_bool_eq","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"5fedb2df1b8cb8b6969b85a9360e6f03449667df4c301a59b87eb28996a68fdb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/CompleteLattice","family_id":"isup_bool_eq","file_id":"mathlib/Mathlib/Order/CompleteLattice/Lemmas.lean","sample_id":"37f235bb48e86794e4b09884c23d8d2d7e5709381d2c53948b9b73b1480e3fa6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"eaff7377a9ad125a2190653d93c4626decbfd214d6bde989ce3ba7f1259b5118","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b2e47c015eac842014bd0418a98dd81b89c3cda4aaab1b55fb3e23238faff8d7","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b748ffc79ac3338e3482458996caeaad3f245a010466e177b0f35271384d2cf9","source_sha256":"a487b8b8276a05b76f39e70e3e24ea2993bb9cba3cee679a6d47a9f00c0fc13a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  cases nonempty_fintype α\n  simpa using exists_max_image univ f univ_nonempty","hard_negative":true,"metrics":{"chosen_tokens":10,"rejected_tokens":5,"token_jaccard":0.071429,"token_length_ratio":0.5},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"f444fb714b082981328cc58890c8a7a3bb5fe75c9f285f4954682165ce3256f2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Max\npublic import Mathlib.Data.Fintype.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n/-!\n# Lemmas relating fintypes and order/lattice structure.\n-/\n\npublic section\n\n\nopen Function\n\nopen Nat\n\nuniverse u v\n\nvariable {ι α β : Type*}\n\nnamespace Finset\n\nvariable [Fintype α] {s : Finset α}\n\n/-- A special case of `Finset.sup_eq_iSup` that omits the useless `x ∈ univ` binder. -/\ntheorem sup_univ_eq_iSup [CompleteLattice β] (f : α → β) : Finset.univ.sup f = iSup f :=\n  (sup_eq_iSup _ f).trans <| congr_arg _ <| funext fun _ => iSup_pos (mem_univ _)\n\n/-- A special case of `Finset.inf_eq_iInf` that omits the useless `x ∈ univ` binder. -/\ntheorem inf_univ_eq_iInf [CompleteLattice β] (f : α → β) : Finset.univ.inf f = iInf f :=\n  @sup_univ_eq_iSup _ βᵒᵈ _ _ (f : α → βᵒᵈ)\n\n@[simp]\ntheorem fold_inf_univ [SemilatticeInf α] [OrderBot α] (a : α) :\n    (Finset.univ.fold min a fun x => x) = ⊥ :=\n  eq_bot_iff.2 <|\n    ((Finset.fold_op_rel_iff_and <| @le_inf_iff α _).1 le_rfl).2 ⊥ <| Finset.mem_univ _\n\n@[simp]\ntheorem fold_sup_univ [SemilatticeSup α] [OrderTop α] (a : α) :\n    (Finset.univ.fold max a fun x => x) = ⊤ :=\n  @fold_inf_univ αᵒᵈ _ _ _ _\n\nlemma mem_inf [DecidableEq α] {s : Finset ι} {f : ι → Finset α} {a : α} :\n    a ∈ s.inf f ↔ ∀ i ∈ s, a ∈ f i := by induction s using Finset.cons_induction <;> simp [*]\n\nend Finset\n\nopen Finset\n\nTarget:\ntheorem Finite.exists_max [Finite α] [Nonempty α] [LinearOrder β] (f : α → β) :\n    ∃ x₀ : α, ∀ x, f x ≤ f x₀ :=\n\nProof body:\n","rejected":"by\n  exact Finite.exists_max","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"cff0f934a9303e1bb8f27b908850b70b29b9035458347987071d64fa56a8e6dd","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Fintype","family_id":"finite","file_id":"mathlib/Mathlib/Data/Fintype/Lattice.lean","sample_id":"b748ffc79ac3338e3482458996caeaad3f245a010466e177b0f35271384d2cf9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"87c6abbbe5679e2db362b7fc92a4107a665977a26b58a1424f12d41fb84a8bc6","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"247835032373bede62a267228c5e4b2505cd3be8f3492601d0c9ed7061a29c2d","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2e2771fc6ad14eff3a9b98719630e14505321f0c2d2dbcd9d1df71f5b966db93","source_sha256":"695353e91eba40e45d9ecd0e58f8b00a56d575b25190f682a1689e29c7316a51","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  unfold logDeriv\n  simp [field]\n  ring","hard_negative":false,"metrics":{"chosen_tokens":8,"rejected_tokens":15,"token_jaccard":0.8,"token_length_ratio":1.875},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"f487d09facef28f5e8d6ef8378dae74c803863fcb31b5a9cd769649b7419a61e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.RingTheory.Derivation.MapCoeffs\npublic import Mathlib.FieldTheory.PrimitiveElement\n\nNamespace:\nDifferential\n\nLocal context:\n/-\nCopyright (c) 2024 Daniel Weber. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Daniel Weber\n-/\n/-!\n# Differential Fields\n\nThis file defines the logarithmic derivative `Differential.logDeriv` and proves properties of it.\nThis is defined algebraically, compared to `logDeriv` which is analytical.\n-/\n\n@[expose] public section\n\nnamespace Differential\n\nopen algebraMap Polynomial IntermediateField\n\nvariable {R : Type*} [Field R] [Differential R] (a b : R)\n\n/--\nThe logarithmic derivative of a is a′ / a.\n-/\ndef logDeriv : R := a′ / a\n\n@[simp]\nlemma logDeriv_zero : logDeriv (0 : R) = 0 := by\n  simp [logDeriv]\n\n@[simp]\nlemma logDeriv_one : logDeriv (1 : R) = 0 := by\n  simp [logDeriv]\n\nTarget:\nlemma logDeriv_mul (ha : a ≠ 0) (hb : b ≠ 0) : logDeriv (a * b) = logDeriv a + logDeriv b :=\n\nProof body:\n","rejected":"```lean\nby\n  unfold logDeriv\n  simp [field]\n  ring\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/Differential","family_id":"logderiv_mul","file_id":"mathlib/Mathlib/FieldTheory/Differential/Basic.lean","sample_id":"2e2771fc6ad14eff3a9b98719630e14505321f0c2d2dbcd9d1df71f5b966db93"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a2f7906395a8fceee2188be53f90d4ed089166e4d7b933b3ba61e10fb185f1d5","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"aa6161864a0c5f5bc6339fe001dfcdf0b66c0b5bdb3cf1678dcb5f82be308aa0","source_sha256":"3e56b0bca06aab56ed706da176a00f9bb677f8dddf1dc55819d7e75bf8119e45","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  split_ifs with h\n  exacts [hs h, ht h]","hard_negative":false,"metrics":{"chosen_tokens":12,"rejected_tokens":3,"token_jaccard":0.083333,"token_length_ratio":0.25},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"f500c030a1d861d97ff842fe13223759f23266b7cb59c888378b79826ba51fed","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Countable\npublic import Mathlib.Order.ConditionallyCompleteLattice.Basic\npublic import Mathlib.Tactic.CrossRefAttribute\npublic import Mathlib.Tactic.FunProp.Attr\npublic import Mathlib.Tactic.Measurability\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro\n-/\n/-!\n# Measurable spaces and measurable functions\n\nThis file defines measurable spaces and measurable functions.\n\nA measurable space is a set equipped with a σ-algebra, a collection of\nsubsets closed under complementation and countable union. A function\nbetween measurable spaces is measurable if the preimage of each\nmeasurable subset is measurable.\n\nσ-algebras on a fixed set `α` form a complete lattice. Here we order\nσ-algebras by writing `m₁ ≤ m₂` if every set which is `m₁`-measurable is\nalso `m₂`-measurable (that is, `m₁` is a subset of `m₂`). In particular, any\ncollection of subsets of `α` generates a smallest σ-algebra which\ncontains all of them.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Measurable_space>\n* <https://en.wikipedia.org/wiki/Sigma-algebra>\n* <https://en.wikipedia.org/wiki/Dynkin_system>\n\n## Tags\n\nmeasurable space, σ-algebra, measurable function\n-/\n\n@[expose] public section\n\nassert_not_exists Covariant MonoidWithZero\n\nopen Set Encodable Function Equiv\n\nvariable {α β γ δ δ' : Type*} {ι : Sort*} {s t u : Set α}\n\n/-- A measurable space is a space equipped with a σ-algebra. -/\n@[class] structure MeasurableSpace (α : Type*) where\n  /-- Predicate saying that a given set is measurable. Use `MeasurableSet` in the root namespace\n  instead. -/\n  MeasurableSet' : Set α → Prop\n  /-- The empty set is a measurable set. Use `MeasurableSet.empty` instead. -/\n  measurableSet_empty : MeasurableSet' ∅\n  /-- The complement of a measurable set is a measurable set. Use `MeasurableSet.compl` instead. -/\n  measurableSet_compl : ∀ s, MeasurableSet' s → MeasurableSet' sᶜ\n  /-- The union of a sequence of measurable sets is a measurable set. Use a more general\n  `MeasurableSet.iUnion` instead. -/\n  measurableSet_iUnion : ∀ f : ℕ → Set α, (∀ i, MeasurableSet' (f i)) → MeasurableSet' (⋃ i, f i)\n\ninstance [h : MeasurableSpace α] : MeasurableSpace αᵒᵈ := h\n\n/-- `MeasurableSet s` means that `s` is measurable (in the ambient measure space on `α`) -/\ndef MeasurableSet [MeasurableSpace α] (s : Set α) : Prop :=\n  ‹MeasurableSpace α›.MeasurableSet' s\n\n/-- Notation for `MeasurableSet` with respect to a non-standard σ-algebra. -/\nscoped[MeasureTheory] notation \"MeasurableSet[\" m \"]\" => @MeasurableSet _ m\n\nopen MeasureTheory\n\nsection\n\nopen scoped symmDiff\n\n@[simp, measurability]\ntheorem MeasurableSet.empty [MeasurableSpace α] : MeasurableSet (∅ : Set α) :=\n  MeasurableSpace.measurableSet_empty _\n\nvariable {m : MeasurableSpace α}\n\n@[measurability]\nprotected theorem MeasurableSet.compl : MeasurableSet s → MeasurableSet sᶜ :=\n  MeasurableSpace.measurableSet_compl _ s\n\nprotected theorem MeasurableSet.of_compl (h : MeasurableSet sᶜ) : MeasurableSet s :=\n  compl_compl s ▸ h.compl\n\n@[simp]\ntheorem MeasurableSet.compl_iff : MeasurableSet sᶜ ↔ MeasurableSet s :=\n  ⟨.of_compl, .compl⟩\n\n@[simp, measurability]\nprotected theorem MeasurableSet.univ : MeasurableSet (univ : Set α) :=\n  .of_compl <| by simp\n\n@[nontriviality, measurability]\ntheorem Subsingleton.measurableSet [Subsingleton α] {s : Set α} : MeasurableSet s :=\n  Subsingleton.set_cases MeasurableSet.empty MeasurableSet.univ s\n\ntheorem MeasurableSet.congr {s t : Set α} (hs : MeasurableSet s) (h : s = t) : MeasurableSet t := by\n  rwa [← h]\n\n@[measurability]\nprotected theorem MeasurableSet.iUnion [Countable ι] ⦃f : ι → Set α⦄\n    (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋃ b, f b) := by\n  cases isEmpty_or_nonempty ι\n  · simp\n  · rcases exists_surjective_nat ι with ⟨e, he⟩\n    rw [← iUnion_congr_of_surjective _ he (fun _ => rfl)]\n    exact m.measurableSet_iUnion _ fun _ => h _\n\nprotected theorem MeasurableSet.biUnion {f : β → Set α} {s : Set β} (hs : s.Countable)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) := by\n  rw [biUnion_eq_iUnion]\n  have := hs.to_subtype\n  exact MeasurableSet.iUnion (by simpa using h)\n\ntheorem Set.Finite.measurableSet_biUnion {f : β → Set α} {s : Set β} (hs : s.Finite)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) :=\n  .biUnion hs.countable h\n\ntheorem Finset.measurableSet_biUnion {f : β → Set α} (s : Finset β)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋃ b ∈ s, f b) :=\n  s.finite_toSet.measurableSet_biUnion h\n\nprotected theorem MeasurableSet.sUnion {s : Set (Set α)} (hs : s.Countable)\n    (h : ∀ t ∈ s, MeasurableSet t) : MeasurableSet (⋃₀ s) := by\n  rw [sUnion_eq_biUnion]\n  exact .biUnion hs h\n\ntheorem Set.Finite.measurableSet_sUnion {s : Set (Set α)} (hs : s.Finite)\n    (h : ∀ t ∈ s, MeasurableSet t) : MeasurableSet (⋃₀ s) :=\n  MeasurableSet.sUnion hs.countable h\n\n@[measurability]\ntheorem MeasurableSet.iInter [Countable ι] {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) :\n    MeasurableSet (⋂ b, f b) :=\n  .of_compl <| by rw [compl_iInter]; exact .iUnion fun b => (h b).compl\n\ntheorem MeasurableSet.biInter {f : β → Set α} {s : Set β} (hs : s.Countable)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) :=\n  .of_compl <| by rw [compl_iInter₂]; exact .biUnion hs fun b hb => (h b hb).compl\n\ntheorem Set.Finite.measurableSet_biInter {f : β → Set α} {s : Set β} (hs : s.Finite)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) :=\n  .biInter hs.countable h\n\ntheorem Finset.measurableSet_biInter {f : β → Set α} (s : Finset β)\n    (h : ∀ b ∈ s, MeasurableSet (f b)) : MeasurableSet (⋂ b ∈ s, f b) :=\n  s.finite_toSet.measurableSet_biInter h\n\ntheorem MeasurableSet.sInter {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s, MeasurableSet t) :\n    MeasurableSet (⋂₀ s) := by\n  rw [sInter_eq_biInter]\n  exact MeasurableSet.biInter hs h\n\ntheorem Set.Finite.measurableSet_sInter {s : Set (Set α)} (hs : s.Finite)\n    (h : ∀ t ∈ s, MeasurableSet t) : MeasurableSet (⋂₀ s) :=\n  MeasurableSet.sInter hs.countable h\n\n@[simp, measurability]\nprotected theorem MeasurableSet.union {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)\n    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ ∪ s₂) := by\n  rw [union_eq_iUnion]\n  exact .iUnion (Bool.forall_bool.2 ⟨h₂, h₁⟩)\n\n@[simp, measurability]\nprotected theorem MeasurableSet.inter {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)\n    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ ∩ s₂) := by\n  rw [inter_eq_compl_compl_union_compl]\n  exact (h₁.compl.union h₂.compl).compl\n\n@[simp, measurability]\nprotected theorem MeasurableSet.diff {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)\n    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ \\ s₂) :=\n  h₁.inter h₂.compl\n\n@[simp, measurability]\nprotected lemma MeasurableSet.himp {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂) :\n    MeasurableSet (s₁ ⇨ s₂) := by rw [himp_eq]; exact h₂.union h₁.compl\n\n@[simp, measurability]\nprotected theorem MeasurableSet.symmDiff {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)\n    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ ∆ s₂) :=\n  (h₁.diff h₂).union (h₂.diff h₁)\n\n@[simp, measurability]\nprotected lemma MeasurableSet.bihimp {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁)\n    (h₂ : MeasurableSet s₂) : MeasurableSet (s₁ ⇔ s₂) := (h₂.himp h₁).inter (h₁.himp h₂)\n\n@[simp, measurability]\nprotected theorem MeasurableSet.ite {t s₁ s₂ : Set α} (ht : MeasurableSet t)\n    (h₁ : MeasurableSet s₁) (h₂ : MeasurableSet s₂) : MeasurableSet (t.ite s₁ s₂) :=\n  (h₁.inter ht).union (h₂.diff ht)\n\nopen Classical in\n\nTarget:\ntheorem MeasurableSet.ite' {s t : Set α} {p : Prop} (hs : p → MeasurableSet s)\n    (ht : ¬p → MeasurableSet t) : MeasurableSet (ite p s t) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/MeasurableSpace","family_id":"measurableset","file_id":"mathlib/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean","sample_id":"aa6161864a0c5f5bc6339fe001dfcdf0b66c0b5bdb3cf1678dcb5f82be308aa0"},"split":"train"}
{"binding":{"chosen_proof_sha256":"958795982b90c7f479780dff2a7eb96698a38b055a8fc856a14fd71baa43ed6a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c7aee18ddb84fda2619825e8c317978cacad561c4ffc10a3e6cb0840000226dc","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7885120c783d836f7907adab08dd0290a06a8a8cf35af5f387b7e22d740bb408","source_sha256":"4068661203cf4c8a8500ce96a89d4eba2dfb134d6a130d9d8f6fe5c686e7101d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by rw [IsLUB.sSup_eq (h d₁ d₂ (isLUB_iff_sSup_eq.mpr rfl))]\n  mpr h _ d₁ d₂ _ hda := by rw [isLUB_iff_sSup_eq, ← (h d₁ d₂), IsLUB.sSup_eq hda]\n\nalias ⟨ScottContinuous.map_sSup, ScottContinuous.of_map_sSup⟩ :=\n  scottContinuous_iff_map_sSup","hard_negative":true,"metrics":{"chosen_tokens":61,"rejected_tokens":2,"token_jaccard":0.034483,"token_length_ratio":0.032787},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"f576b260c4266e6b0f58023991dff75b58e2813bd47e839418b76429e542a29d","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.ScottContinuity.Prod\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Christopher Hoskin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christopher Hoskin\n-/\n/-!\n\n# Scott continuity on complete lattices\n\n## Main results\n\n- `scottContinuous_iff_map_sSup`: A function is Scott continuous if and only if it commutes with\n  `sSup` on directed sets.\n\n-/\n\npublic section\n\nvariable {α β : Type*}\n\nsection CompleteLattice\n\nvariable [CompleteLattice α] [CompleteLattice β]\n\n/- `f` is Scott continuous if and only if it commutes with `sSup` on directed sets -/\n\nTarget:\nlemma scottContinuous_iff_map_sSup {f : α → β} :\n    ScottContinuous f ↔\n      ∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (· ≤ ·) d → f (sSup d) = sSup (f '' d) where\n  mp h _ d₁ d₂ :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_7885120c783d","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"fe51d2f00845dfceba088f47e1cfe4c166c8d1a699d09446097765153ebc774a","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/ScottContinuity","family_id":"scottcontinuous_iff_map_ssup","file_id":"mathlib/Mathlib/Order/ScottContinuity/Complete.lean","sample_id":"7885120c783d836f7907adab08dd0290a06a8a8cf35af5f387b7e22d740bb408"},"split":"train"}
{"binding":{"chosen_proof_sha256":"aea622174bd7323409020f456ba921f86430f2be7a1ef46c09da2cf27c53cb7b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"79ff1c5746539b95fb7b7bd18831f461eb7a59f72b537902f6b1acccf75b210c","source_sha256":"0722d600b193d93cb9e55becbab169fa607364d0057f53064083850c9da2e6cc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have hh : 0 < Γ.strictWidthInfty := Γ.strictWidthInfty_pos_iff.mpr Fact.out\n  have hΓ : Γ.strictWidthInfty ∈ Γ.strictPeriods := Γ.strictWidthInfty_mem_strictPeriods\n  have hanal := ModularFormClass.analyticAt_cuspFunction_zero f hh hΓ\n  have hper := periodic_comp_ofComplex f hΓ\n  simp_rw [IsZeroAtImInfty, ZeroAtFilter, ← h, ← cuspFunction_apply_zero hh hanal hper]\n  exact (hanal.continuousAt.tendsto.comp (qParam_tendsto_atImInfty hh)).congr\n    (fun τ ↦ SlashInvariantFormClass.eq_cuspFunction f τ hΓ hh.ne')","hard_negative":false,"metrics":{"chosen_tokens":96,"rejected_tokens":2,"token_jaccard":0.021277,"token_length_ratio":0.020833},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"f5b2298cc151431196248c8a08e93525403792c1ed139b53c12093a8e9b79682","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.ModularForms.QExpansion\npublic import Mathlib.NumberTheory.ModularForms.LevelOne.Basic\npublic import Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion\n\nNamespace:\nModularForm\n\nLocal context:\n/-\nCopyright (c) 2026 Chris Birkbeck. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Birkbeck\n-/\n/-!\n# Cusp form submodule and IsCuspForm predicate\n\nThis file defines the inclusion of cusp forms into modular forms as a linear map, the cusp form\nsubmodule of modular forms, and the `IsCuspForm` predicate. It also provides a direct constructor\n`ModularForm.toCuspForm` for building cusp forms from modular forms with vanishing constant\nq-expansion coefficient (for `𝒮ℒ`).\n\n## Main definitions\n\n* `CuspForm.toModularFormₗ`: the inclusion `CuspForm Γ k →ₗ[ℂ] ModularForm Γ k`.\n* `ModularForm.cuspFormSubmodule`: the submodule of `ModularForm Γ k` consisting of cusp forms.\n* `ModularForm.IsCuspForm`: predicate that a modular form lies in the cusp form submodule.\n* `ModularForm.toCuspForm`: builds a `CuspForm 𝒮ℒ k` from a `ModularForm` whose q-expansion\n  has vanishing constant term.\n\n## Main results\n\n* `CuspForm.toModularFormₗ_injective`: the inclusion is injective.\n* `CuspForm.equivCuspFormSubmodule`: `CuspForm Γ k ≃ₗ[ℂ] cuspFormSubmodule Γ k`.\n* `ModularForm.isCuspForm_iff_coeffZero_eq_zero`: for `𝒮ℒ`, `IsCuspForm` is equivalent to the\n  q-expansion having vanishing constant term.\n-/\n\n@[expose] public noncomputable section\n\nopen UpperHalfPlane ModularForm Complex SlashInvariantForm SlashInvariantFormClass\n  ModularFormClass MatrixGroups OnePoint Filter Topology\n\nvariable {Γ : Subgroup (GL (Fin 2) ℝ)} {k : ℤ}\n\nnamespace CuspForm\n\n/-- The inclusion of cusp forms into modular forms, as a ℂ-linear map. -/\ndef toModularFormₗ [Γ.HasDetOne] : CuspForm Γ k →ₗ[ℂ] ModularForm Γ k where\n  toFun := ModularFormClass.modularForm\n  map_add' _ _ := rfl\n  map_smul' _ _ := rfl\n\n@[simp]\nlemma toModularFormₗ_apply [Γ.HasDetOne] (f : CuspForm Γ k) (z : ℍ) :\n    (toModularFormₗ f) z = f z := rfl\n\nlemma toModularFormₗ_eq_coe [Γ.HasDetOne] (f : CuspForm Γ k) :\n    toModularFormₗ f = (f : ModularForm Γ k) := rfl\n\nlemma toModularFormₗ_injective [Γ.HasDetOne] :\n    Function.Injective (toModularFormₗ : CuspForm Γ k → ModularForm Γ k) :=\n  fun _ _ h ↦ DFunLike.ext _ _ fun z ↦ DFunLike.congr_fun h z\n\nend CuspForm\n\nnamespace ModularForm\n\n/-- The submodule of `ModularForm Γ k` consisting of cusp forms, defined as the range of\nthe inclusion `CuspForm.toModularFormₗ`. -/\ndef cuspFormSubmodule (Γ : Subgroup (GL (Fin 2) ℝ)) (k : ℤ) [Γ.HasDetOne] :\n    Submodule ℂ (ModularForm Γ k) :=\n  LinearMap.range CuspForm.toModularFormₗ\n\n/-- A modular form is a cusp form if it lies in the cusp form submodule. -/\ndef IsCuspForm [Γ.HasDetOne] (f : ModularForm Γ k) : Prop :=\n  f ∈ cuspFormSubmodule Γ k\n\n@[simp]\nlemma mem_cuspFormSubmodule_iff [Γ.HasDetOne] {f : ModularForm Γ k} :\n    f ∈ cuspFormSubmodule Γ k ↔ IsCuspForm f := Iff.rfl\n\n/-- The cusp form submodule is linearly equivalent to the type of cusp forms. -/\ndef CuspForm.equivCuspFormSubmodule (Γ : Subgroup (GL (Fin 2) ℝ)) (k : ℤ) [Γ.HasDetOne] :\n    CuspForm Γ k ≃ₗ[ℂ] cuspFormSubmodule Γ k :=\n  LinearEquiv.ofInjective CuspForm.toModularFormₗ CuspForm.toModularFormₗ_injective\n\n/-- The underlying modular form (via `toModularFormₗ`) of a `CuspForm` is itself a cusp form. -/\nlemma CuspForm.isCuspForm_toModularFormₗ {Γ : Subgroup (GL (Fin 2) ℝ)} [Γ.HasDetOne]\n    (f : CuspForm Γ k) : ModularForm.IsCuspForm f.toModularFormₗ := by\n  simp [← mem_cuspFormSubmodule_iff, ModularForm.cuspFormSubmodule]\n\n/-- A modular form is a cusp form if and only if it vanishes at every cusp. This is the\ngeneral characterization valid for any subgroup. -/\nlemma isCuspForm_iff [Γ.HasDetOne] (f : ModularForm Γ k) :\n    IsCuspForm f ↔ ∀ {c}, IsCusp c Γ → c.IsZeroAt f k :=\n  ⟨fun ⟨g, hg⟩ _ ↦ hg ▸ g.zero_at_cusps', fun h ↦ ⟨⟨f, f.holo', h⟩, rfl⟩⟩\n\n/-- A modular form with `valueAtInfty f = 0` is zero at infinity. -/\n\nTarget:\nlemma isZeroAtImInfty_of_valueAtInfty_eq_zero {F : Type*} [FunLike F ℍ ℂ]\n    [DiscreteTopology Γ] [Γ.HasDetPlusMinusOne] [Fact (IsCusp ∞ Γ)] [ModularFormClass F Γ k]\n    (f : F) (h : valueAtInfty f = 0) : IsZeroAtImInfty f :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/ModularForms","family_id":"iszeroatiminfty_of_valueatinfty_eq_zero","file_id":"mathlib/Mathlib/NumberTheory/ModularForms/CuspFormSubmodule.lean","sample_id":"79ff1c5746539b95fb7b7bd18831f461eb7a59f72b537902f6b1acccf75b210c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ef657d6d5fc4b7d8887ac9017ba9ce06e5bcee28dc51338cd8d6ad2502a8eccb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4b6a657545abaaf7eebda573d07dd91e0d6097e9312e9cce6c12048585c3e4e7","source_sha256":"ffda45eef0509a041f24079788fb5aa3f3f4500975a78975ccc64409c3de362b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp_rw [← union, disjoint_union_right, disjoint_iUnion_right]\n  exact ⟨disjointBase n i , fun _ _ ↦ disjoint_openCell_of_ne (by lia)⟩","hard_negative":false,"metrics":{"chosen_tokens":26,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.076923},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"f5e05730d55da25be7402f3dd0ddb95d1b4c02281377512c0749332cda48217f","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.CWComplex.Classical.Finite\npublic import Mathlib.Analysis.Normed.Module.RCLike.Real\n\nNamespace:\nTopology\n\nLocal context:\n/-\nCopyright (c) 2025 Floris van Doorn and Hannah Scholz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Hannah Scholz\n-/\n/-!\n# Subcomplexes\n\nIn this file we discuss subcomplexes of CW complexes.\nThe definition of subcomplexes is in the file `Mathlib/Topology/CWComplex/Classical/Basic.lean`.\n\n## Main results\n* `RelCWComplex.Subcomplex.instRelCWComplex`: a subcomplex of a (relative) CW complex is again a\n  (relative) CW complex.\n\n## References\n* [K. Jänich, *Topology*][Janich1984]\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nopen Metric Set\n\nnamespace Topology\n\nvariable {X : Type*} [t : TopologicalSpace X] {C D : Set X}\n\n@[alias_in CWComplex.Subcomplex]\nlemma RelCWComplex.Subcomplex.closedCell_subset_of_mem [T2Space X] [RelCWComplex C D]\n    (E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) :\n    closedCell n i ⊆ E := by\n  rw [← closure_openCell_eq_closedCell, E.closed.closure_subset_iff, ← E.union]\n  apply subset_union_of_subset_right\n  exact subset_iUnion_of_subset n\n    (subset_iUnion (fun (j : ↑(E.I n)) ↦ openCell (C := C) n j) ⟨i, hi⟩)\n\n@[alias_in CWComplex.Subcomplex]\nlemma RelCWComplex.Subcomplex.openCell_subset_of_mem [T2Space X] [RelCWComplex C D]\n    (E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) :\n    openCell n i ⊆ E :=\n  (openCell_subset_closedCell n i).trans (closedCell_subset_of_mem E hi)\n\n@[alias_in CWComplex.Subcomplex]\nlemma RelCWComplex.Subcomplex.cellFrontier_subset_of_mem [T2Space X] [RelCWComplex C D]\n    (E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) :\n    cellFrontier n i ⊆ E :=\n  (cellFrontier_subset_closedCell n i).trans (closedCell_subset_of_mem E hi)\n\n/-- A subcomplex is the union of its closed cells and its base. -/\nlemma RelCWComplex.Subcomplex.union_closedCell [T2Space X] [RelCWComplex C D] (E : Subcomplex C) :\n    D ∪ ⋃ (n : ℕ) (j : E.I n), closedCell (C := C) n j = E := by\n  apply subset_antisymm\n  · apply union_subset E.base_subset\n    exact iUnion₂_subset fun n i ↦ closedCell_subset_of_mem E i.2\n  · rw [← E.union]\n    apply union_subset_union_right\n    apply iUnion₂_mono fun n i ↦ ?_\n    exact openCell_subset_closedCell (C := C) n i\n\n/-- A subcomplex is the union of its closed cells. -/\nlemma CWComplex.Subcomplex.union_closedCell [T2Space X] [CWComplex C] (E : Subcomplex C) :\n    ⋃ (n : ℕ) (j : E.I n), closedCell (C := C) n j = E :=\n  (empty_union _).symm.trans (RelCWComplex.Subcomplex.union_closedCell E)\n\n@[alias_in CWComplex.Subcomplex]\n\nTarget:\nlemma RelCWComplex.Subcomplex.disjoint_openCell_subcomplex_of_not_mem [RelCWComplex C D]\n    (E : Subcomplex C) {n : ℕ} {i : cell C n} (h : i ∉ E.I n) : Disjoint (openCell n i) E :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/CWComplex","family_id":"relcwcomplex","file_id":"mathlib/Mathlib/Topology/CWComplex/Classical/Subcomplex.lean","sample_id":"4b6a657545abaaf7eebda573d07dd91e0d6097e9312e9cce6c12048585c3e4e7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"54c464d6ce26bdef8908aa10edaaa4627d275c464a5704aa3ef40ec7a4c12203","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d0ac2882ec427829126b8598a0819730b07105bce20b1228031035f70c0b76c9","source_sha256":"0c7310750ac9c48298092f0639863d1202fe8a8e7eda47b7ee8abc6e42090355","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rintro ⟨m, hm⟩\n  have hm' : ∀ a, f a ≤ m := fun a ↦ hm <| Set.mem_range_self _\n  obtain ⟨a₀⟩ := ‹Nonempty α›\n  suffices ∀ b, f a₀ ≤ b → ∃ a, b < f a by\n    obtain ⟨a, ha⟩ : ∃ a, m < f a := this m (hm' a₀)\n    exact ha.not_ge (hm' a)\n  have h : ∀ a, ∃ a', f a < f a' := fun a ↦ (exists_gt a).imp (fun a' h ↦ hf h)\n  apply Succ.rec\n  · exact h a₀\n  rintro b _ ⟨a, hba⟩\n  exact (h a).imp (fun a' ↦ (succ_le_of_lt hba).trans_lt)","hard_negative":false,"metrics":{"chosen_tokens":153,"rejected_tokens":3,"token_jaccard":0.038462,"token_length_ratio":0.019608},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"f5f4402442cd70dd6207fa4eb9bfc671f1d5a766e4ae24cb223939b9f3d96c29","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.SuccPred.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Archimedean successor and predecessor\n\n* `IsSuccArchimedean`: `SuccOrder` where `succ` iterated to an element gives all the greater\n  ones.\n* `IsPredArchimedean`: `PredOrder` where `pred` iterated to an element gives all the smaller\n  ones.\n-/\n\npublic section\n\nvariable {α β : Type*}\n\nopen Order Function\n\n/-- A `SuccOrder` is succ-archimedean if one can go from any two comparable elements by iterating\n`succ` -/\nclass IsSuccArchimedean (α : Type*) [Preorder α] [SuccOrder α] : Prop where\n  /-- If `a ≤ b` then one can get to `a` from `b` by iterating `succ` -/\n  exists_succ_iterate_of_le {a b : α} (h : a ≤ b) : ∃ n, succ^[n] a = b\n\n/-- A `PredOrder` is pred-archimedean if one can go from any two comparable elements by iterating\n`pred` -/\n@[to_dual existing]\nclass IsPredArchimedean (α : Type*) [Preorder α] [PredOrder α] : Prop where\n  /-- If `a ≤ b` then one can get to `b` from `a` by iterating `pred` -/\n  exists_pred_iterate_of_le {a b : α} (h : a ≤ b) : ∃ n, pred^[n] b = a\n\nexport IsSuccArchimedean (exists_succ_iterate_of_le)\nexport IsPredArchimedean (exists_pred_iterate_of_le)\n\nattribute [to_dual existing] exists_succ_iterate_of_le\n\nsection Preorder\n\nvariable [Preorder α]\n\n-- `to_dual` cannot yet reorder arguments of arguments\ninstance [SuccOrder α] [IsSuccArchimedean α] : IsPredArchimedean αᵒᵈ :=\n  ⟨fun {a b} h => by convert! exists_succ_iterate_of_le h.ofDual⟩\n\n@[to_dual existing]\ninstance [PredOrder α] [IsPredArchimedean α] : IsSuccArchimedean αᵒᵈ :=\n  ⟨fun {a b} h => by convert! exists_pred_iterate_of_le h.ofDual⟩\n\nsection SuccOrder\n\nvariable [SuccOrder α] [IsSuccArchimedean α] {a b : α}\n\n@[to_dual]\ntheorem LE.le.exists_succ_iterate (h : a ≤ b) : ∃ n, succ^[n] a = b :=\n  exists_succ_iterate_of_le h\n\n@[to_dual]\ntheorem exists_succ_iterate_iff_le : (∃ n, succ^[n] a = b) ↔ a ≤ b := by\n  refine ⟨?_, exists_succ_iterate_of_le⟩\n  rintro ⟨n, rfl⟩\n  exact id_le_iterate_of_id_le le_succ n a\n\n-- TODO: rename to `Order.succ_rec`?\n/-- Induction principle on a type with a `SuccOrder` for all elements above a given element `m`. -/\n@[to_dual (attr := elab_as_elim) Pred.rec\n/-- Induction principle on a type with a `PredOrder` for all elements below a given element `m`. -/]\ntheorem Succ.rec {m : α} {P : ∀ n, m ≤ n → Prop} (rfl : P m le_rfl)\n    (succ : ∀ n (hmn : m ≤ n), P n hmn → P (succ n) (hmn.trans <| le_succ _)) ⦃n : α⦄\n    (hmn : m ≤ n) : P n hmn := by\n  obtain ⟨n, rfl⟩ := hmn.exists_succ_iterate\n  induction n with\n  | zero => exact rfl\n  | succ n ih =>\n    simp_rw [Function.iterate_succ_apply']\n    exact succ _ (id_le_iterate_of_id_le le_succ n m) (ih _)\n\n@[to_dual Pred.rec_iff]\ntheorem Succ.rec_iff {p : α → Prop} (hsucc : ∀ a, p a ↔ p (succ a)) {a b : α} (h : a ≤ b) :\n    p a ↔ p b := by\n  obtain ⟨n, rfl⟩ := h.exists_succ_iterate\n  exact Iterate.rec (fun b => p a ↔ p b) Iff.rfl (fun c hc => hc.trans (hsucc _)) n\n\n@[to_dual le_total_of_directed]\nlemma le_total_of_codirected {r v₁ v₂ : α} (h₁ : r ≤ v₁) (h₂ : r ≤ v₂) : v₁ ≤ v₂ ∨ v₂ ≤ v₁ := by\n  obtain ⟨n, rfl⟩ := h₁.exists_succ_iterate\n  obtain ⟨m, rfl⟩ := h₂.exists_succ_iterate\n  clear h₁ h₂\n  wlog h : n ≤ m\n  · rw [Or.comm]\n    apply this\n    exact Nat.le_of_not_ge h\n  left\n  obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le h\n  rw [Nat.add_comm, Function.iterate_add, Function.comp_apply]\n  apply Order.le_succ_iterate\n\nend SuccOrder\n\nend Preorder\n\nsection PartialOrder\n\nvariable [PartialOrder α]\n\n@[to_dual (reorder := h₁ h₂) lt_or_le_of_directed]\nlemma lt_or_le_of_codirected [SuccOrder α] [IsSuccArchimedean α] {r v₁ v₂ : α} (h₁ : r ≤ v₁)\n    (h₂ : r ≤ v₂) : v₁ < v₂ ∨ v₂ ≤ v₁ := by\n  rw [Classical.or_iff_not_imp_right]\n  intro nh\n  rcases le_total_of_codirected h₁ h₂ with h | h\n  · apply lt_of_le_of_ne h (ne_of_not_le nh).symm\n  · contradiction\n\n-- `to_dual` cannot yet reorder arguments of arguments\n/--\nThis isn't an instance due to a loop with `LinearOrder`.\n-/\n-- See note [reducible non-instances]\nabbrev IsSuccArchimedean.linearOrder [SuccOrder α] [IsSuccArchimedean α]\n     [DecidableEq α] [DecidableLE α] [DecidableLT α]\n     [IsCodirectedOrder α] : LinearOrder α where\n  le_total a b :=\n    have ⟨c, ha, hb⟩ := directed_of (· ≥ ·) a b\n    le_total_of_codirected ha hb\n  toDecidableEq := inferInstance\n  toDecidableLE := inferInstance\n  toDecidableLT := inferInstance\n\n/--\nThis isn't an instance due to a loop with `LinearOrder`.\n-/\n-- See note [reducible non-instances]\n@[to_dual existing]\nabbrev IsPredArchimedean.linearOrder [PredOrder α] [IsPredArchimedean α]\n     [DecidableEq α] [DecidableLE α] [DecidableLT α]\n     [IsDirectedOrder α] : LinearOrder α :=\n  letI : LinearOrder αᵒᵈ := IsSuccArchimedean.linearOrder\n  inferInstanceAs (LinearOrder αᵒᵈᵒᵈ)\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [LinearOrder α]\n\nsection SuccOrder\nvariable [SuccOrder α]\n\n@[deprecated (since := \"2026-02-05\")] alias succ_max := Order.succ_max\n@[deprecated (since := \"2026-02-05\")] alias succ_min := Order.succ_min\n\n@[deprecated (since := \"2026-02-05\")] alias pred_max := Order.pred_max\n@[deprecated (since := \"2026-02-05\")] alias pred_min := Order.pred_min\n\nvariable [IsSuccArchimedean α] {a b : α}\n\n@[to_dual]\ntheorem exists_succ_iterate_or : (∃ n, succ^[n] a = b) ∨ ∃ n, succ^[n] b = a :=\n  (le_total a b).imp exists_succ_iterate_of_le exists_succ_iterate_of_le\n\n@[to_dual Pred.rec_linear]\ntheorem Succ.rec_linear {p : α → Prop} (hsucc : ∀ a, p a ↔ p (succ a)) (a b : α) : p a ↔ p b :=\n  (le_total a b).elim (Succ.rec_iff hsucc) fun h => (Succ.rec_iff hsucc h).symm\n\nend SuccOrder\n\nend LinearOrder\n\nsection bdd_range\nvariable [Preorder α] [Nonempty α] [Preorder β] {f : α → β}\n\n@[to_dual]\n\nTarget:\nlemma StrictMono.not_bddAbove_range_of_isSuccArchimedean [NoMaxOrder α] [SuccOrder β]\n    [IsSuccArchimedean β] (hf : StrictMono f) : ¬ BddAbove (Set.range f) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/SuccPred","family_id":"strictmono","file_id":"mathlib/Mathlib/Order/SuccPred/Archimedean.lean","sample_id":"d0ac2882ec427829126b8598a0819730b07105bce20b1228031035f70c0b76c9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9eee49c5561e216892ae126cb44a6a373a2be5f508e4256a2d5f533053f0d013","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"21c394e35574aad91bc8d293cafaa5383d2921ea96daa7d1c4c40298b7820907","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3e916153cf2d028f0fe2211091c80661d43ef75952801649ac6582ce0158842e","source_sha256":"1b33d2ba913b39d2ffb044888cc338bb0c7eb620bd058c3099d88a86d55171a5","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [(ofSet s hs).trans_ofSet hs']\n  ext <;> simp [hs'.interior_eq]","hard_negative":true,"metrics":{"chosen_tokens":22,"rejected_tokens":5,"token_jaccard":0.095238,"token_length_ratio":0.227273},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"f5ff602d72fc6513f309e21650df2caab9ea091788674fedcb4d0c450b53094b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.OpenPartialHomeomorph.IsImage\n\nNamespace:\nOpenPartialHomeomorph\n\nLocal context:\n/-\nCopyright (c) 2019 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Partial homeomorphisms: composition\n\n## Main definitions\n\n* `OpenPartialHomeomorph.trans`: the composition of two open partial homeomorphisms\n-/\n\n@[expose] public section\n\nopen Function Set Filter Topology\n\nvariable {X X' : Type*} {Y Y' : Type*} {Z Z' : Type*}\n  [TopologicalSpace X] [TopologicalSpace X'] [TopologicalSpace Y] [TopologicalSpace Y']\n  [TopologicalSpace Z] [TopologicalSpace Z']\n\nnamespace OpenPartialHomeomorph\n\nvariable (e : OpenPartialHomeomorph X Y)\n\n/-!\n## Composition\n\n`trans`: composition of two open partial homeomorphisms\n-/\nsection trans\n\nvariable (e' : OpenPartialHomeomorph Y Z)\n\n/-- Composition of two open partial homeomorphisms when the target of the first and the source of\nthe second coincide. -/\n@[simps! apply symm_apply toPartialHomeomorph, simps! -isSimp source target]\nprotected def trans' (h : e.target = e'.source) : OpenPartialHomeomorph X Z where\n  toPartialEquiv := PartialEquiv.trans' e.toPartialEquiv e'.toPartialEquiv h\n  open_source := e.open_source\n  open_target := e'.open_target\n  continuousOn_toFun := e'.continuousOn.comp e.continuousOn <| h ▸ e.mapsTo\n  continuousOn_invFun := e.continuousOn_symm.comp e'.continuousOn_symm <| h.symm ▸ e'.mapsTo_symm\n\n/-- Composing two open partial homeomorphisms, by restricting to the maximal domain where their\ncomposition is well defined.\nWithin the `Manifold` namespace, there is the notation `e ≫ₕ f` for this. -/\n@[trans]\nprotected def trans : OpenPartialHomeomorph X Z :=\n  OpenPartialHomeomorph.trans' (e.symm.restrOpen e'.source e'.open_source).symm\n    (e'.restrOpen e.target e.open_target) (by simp [inter_comm])\n\n@[simp, mfld_simps]\ntheorem trans_toPartialEquiv :\n    (e.trans e').toPartialEquiv = e.toPartialEquiv.trans e'.toPartialEquiv :=\n  rfl\n\n@[simp, mfld_simps]\ntheorem coe_trans : (e.trans e' : X → Z) = e' ∘ e :=\n  rfl\n\n@[simp, mfld_simps]\ntheorem coe_trans_symm : ((e.trans e').symm : Z → X) = e.symm ∘ e'.symm :=\n  rfl\n\ntheorem trans_apply {x : X} : (e.trans e') x = e' (e x) :=\n  rfl\n\ntheorem trans_symm_eq_symm_trans_symm : (e.trans e').symm = e'.symm.trans e.symm := rfl\n\n/-- This could be considered as a simp lemma, but there are many situations where it makes something\nsimple into something more complicated. -/\ntheorem trans_source : (e.trans e').source = e.source ∩ e ⁻¹' e'.source :=\n  PartialEquiv.trans_source e.toPartialEquiv e'.toPartialEquiv\n\ntheorem trans_source' : (e.trans e').source = e.source ∩ e ⁻¹' (e.target ∩ e'.source) :=\n  PartialEquiv.trans_source' e.toPartialEquiv e'.toPartialEquiv\n\ntheorem trans_source'' : (e.trans e').source = e.symm '' (e.target ∩ e'.source) :=\n  PartialEquiv.trans_source'' e.toPartialEquiv e'.toPartialEquiv\n\ntheorem image_trans_source : e '' (e.trans e').source = e.target ∩ e'.source :=\n  PartialEquiv.image_trans_source e.toPartialEquiv e'.toPartialEquiv\n\ntheorem trans_target : (e.trans e').target = e'.target ∩ e'.symm ⁻¹' e.target :=\n  rfl\n\ntheorem trans_target' : (e.trans e').target = e'.target ∩ e'.symm ⁻¹' (e'.source ∩ e.target) :=\n  trans_source' e'.symm e.symm\n\ntheorem trans_target'' : (e.trans e').target = e' '' (e'.source ∩ e.target) :=\n  trans_source'' e'.symm e.symm\n\ntheorem inv_image_trans_target : e'.symm '' (e.trans e').target = e'.source ∩ e.target :=\n  image_trans_source e'.symm e.symm\n\ntheorem trans_assoc (e'' : OpenPartialHomeomorph Z Z') :\n    (e.trans e').trans e'' = e.trans (e'.trans e'') :=\n  toPartialEquiv_injective <| e.1.trans_assoc _ _\n\n@[simp, mfld_simps]\ntheorem trans_refl : e.trans (OpenPartialHomeomorph.refl Y) = e :=\n  toPartialHomeomorph_injective (PartialHomeomorph.toPartialEquiv_injective e.1.trans_refl)\n\n@[simp, mfld_simps]\ntheorem refl_trans : (OpenPartialHomeomorph.refl X).trans e = e :=\n  toPartialHomeomorph_injective (PartialHomeomorph.toPartialEquiv_injective e.1.refl_trans)\n\ntheorem trans_ofSet {s : Set Y} (hs : IsOpen s) : e.trans (ofSet s hs) = e.restr (e ⁻¹' s) :=\n  OpenPartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) <| by\n    rw [trans_source, restr_source, ofSet_source, ← preimage_interior, hs.interior_eq]\n\ntheorem trans_of_set' {s : Set Y} (hs : IsOpen s) :\n    e.trans (ofSet s hs) = e.restr (e.source ∩ e ⁻¹' s) := by rw [trans_ofSet, restr_source_inter]\n\ntheorem ofSet_trans {s : Set X} (hs : IsOpen s) : (ofSet s hs).trans e = e.restr s :=\n  OpenPartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) <|\n    by simp [hs.interior_eq, inter_comm]\n\ntheorem ofSet_trans' {s : Set X} (hs : IsOpen s) :\n    (ofSet s hs).trans e = e.restr (e.source ∩ s) := by\n  rw [ofSet_trans, restr_source_inter]\n\n@[simp, mfld_simps]\n\nTarget:\ntheorem ofSet_trans_ofSet {s : Set X} (hs : IsOpen s) {s' : Set X} (hs' : IsOpen s') :\n    (ofSet s hs).trans (ofSet s' hs') = ofSet (s ∩ s') (IsOpen.inter hs hs') :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_3e916153cf2d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"1dc5df246dc2fe1cb8c23f9d4a3f03f50a685894a0142280c5bdeaefeb92baf7","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/OpenPartialHomeomorph","family_id":"ofset_trans_ofset","file_id":"mathlib/Mathlib/Topology/OpenPartialHomeomorph/Composition.lean","sample_id":"3e916153cf2d028f0fe2211091c80661d43ef75952801649ac6582ce0158842e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9a8be05ccc00ea06c742ae40a8ee2f89b7edc8fc907070b2e2dab340e52a4333","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e7fe54825ee652c35445a03a95e094b2909563decc1dc0fb57d4423c11f5ece1","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7f88709361e31e7cfe4a2d2240e22f16a88fdfa93df42a71071e525d8a5017f9","source_sha256":"99fe4affc634c570de70587e929805d16958622f49de95b17da87beb4716efa3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  induction t with\n  | nil => rw [traverse, traverse, η.preserves_pure]\n  | node v l r hl hr =>\n    rw [traverse, traverse, η.preserves_seq, η.preserves_seq, η.preserves_map, hl, hr,\n      Function.comp_apply]","hard_negative":false,"metrics":{"chosen_tokens":51,"rejected_tokens":56,"token_jaccard":0.833333,"token_length_ratio":1.098039},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"f630852fb6dbd3b820dd1cac396ae2dbfabb1fc909edd1102d20332890df7e5b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Tree.Basic\npublic import Mathlib.Control.Applicative\npublic import Mathlib.Control.Traversable.Basic\n\nNamespace:\nBinaryTree\n\nLocal context:\n/-\nCopyright (c) 2025 Edward van de Meent. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Edward van de Meent\n-/\n/-!\n# Traversable Binary Tree\n\nProvides a `Traversable` instance for the `Tree` type.\n-/\n\npublic section\n\nuniverse u v w\n\nnamespace BinaryTree\nsection Traverse\nvariable {α β : Type*}\n\ninstance : Traversable BinaryTree where\n  map := map\n  traverse := traverse\n\nlemma comp_traverse\n    {F : Type u → Type v} {G : Type v → Type w} [Applicative F] [Applicative G]\n    [LawfulApplicative G] {β : Type v} {γ : Type u} (f : β → F γ) (g : α → G β)\n    (t : BinaryTree α) : t.traverse (Functor.Comp.mk ∘ (f <$> ·) ∘ g) =\n      Functor.Comp.mk ((·.traverse f) <$> (t.traverse g)) := by\n  induction t with\n  | nil => rw [traverse, traverse, map_pure, traverse]; rfl\n  | node v l r hl hr =>\n    rw [traverse, hl, hr, traverse]\n    simp only [Function.comp_def, Function.comp_apply, Functor.Comp.map_mk, Functor.map_map,\n      Comp.seq_mk, seq_map_assoc, map_seq]\n    rfl\n\nlemma traverse_eq_map_id (f : α → β) (t : BinaryTree α) :\n    t.traverse ((pure : β → Id β) ∘ f) = pure (t.map f) := by\n  induction t with\n  | nil => rw [traverse, map]\n  | node v l r hl hr =>\n    rw [traverse, map, hl, hr, Function.comp_apply, map_pure, pure_seq, map_pure, pure_seq,\n      map_pure]\n\nTarget:\nlemma naturality {F G : Type u → Type*} [Applicative F] [Applicative G] [LawfulApplicative F]\n    [LawfulApplicative G] (η : ApplicativeTransformation F G) {β : Type u} (f : α → F β)\n    (t : BinaryTree α) : η (t.traverse f) = t.traverse (η.app β ∘ f : α → G β) :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  induction t with\n  | nil => rw [traverse, traverse, η.preserves_pure]\n  | node v l r hl hr =>\n    rw [traverse, traverse, η.preserves_seq, η.preserves_seq, η.preserves_map, hl, hr,\n      Function.comp_apply]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Tree","family_id":"naturality","file_id":"mathlib/Mathlib/Data/Tree/Traversable.lean","sample_id":"7f88709361e31e7cfe4a2d2240e22f16a88fdfa93df42a71071e525d8a5017f9"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c8b268100380170538174b78ca7f2045ab5992f487a935451a91e1cbc0d99b37","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"00ef4f5a3e97159007721ad98486c2afdf173186bd65f645a604852c1e31ba9e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"81a0159374a335acfc3a3be57440b1fb6c4f215c8d52062ffeefc8554afc5a68","source_sha256":"da1f22be161c5bc7cbd676252bea6b38bb4aa0497872bcdf52a178b9d4951829","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have h₁ : a i < supOfSeq a :=\n    Nat.lt_add_one_iff.mpr (le_max_of_le_right <| Finset.le_sup (by simp))\n  have h₂ : supOfSeq a ≤ (i + 1) * (supOfSeq a)! + 1 :=\n    le_trans (self_le_factorial _) (le_trans (Nat.le_mul_of_pos_left (supOfSeq a)! (succ_pos i))\n      (le_add_right _ _))\n  simpa only [coprimes] using lt_of_lt_of_le h₁ h₂","hard_negative":true,"metrics":{"chosen_tokens":87,"rejected_tokens":5,"token_jaccard":0.046512,"token_length_ratio":0.057471},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"f63b09ce645bcd94d895f6c1d0e21514ea1db435057cef1188b364df666d428b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Nat.ModEq\npublic import Mathlib.Data.Nat.ChineseRemainder\npublic import Mathlib.Data.Nat.Prime.Defs\npublic import Mathlib.Data.Nat.Pairing\npublic import Mathlib.Order.Fin.Basic\npublic import Mathlib.Data.Finset.Lattice.Fold\npublic import Mathlib.Data.Fintype.Basic\npublic import Mathlib.Data.Nat.Factorial.Basic\n\nNamespace:\nNat\n\nLocal context:\n/-\nCopyright (c) 2023 Shogo Saito. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Shogo Saito. Adapted for mathlib by Hunter Monroe\n-/\n/-!\n# Gödel's Beta Function Lemma\n\nThis file proves Gödel's Beta Function Lemma, used to prove the First Incompleteness Theorem. It\npermits quantification over finite sequences of natural numbers in formal theories of arithmetic.\nThis Beta Function has no connection with the unrelated Beta Function defined in analysis. Note\nthat `Nat.beta` and `Nat.unbeta` provide similar functionality to `Encodable.encodeList` and\n`Encodable.decodeList`. We define these separately, because it is easier to prove that `Nat.beta`\nand `Nat.unbeta` are arithmetically definable, and this is hard to prove that for\n`Encodable.encodeList` and `Encodable.decodeList` directly. The arithmetic\ndefinability is needed for the proof of the First Incompleteness Theorem.\n\n## Main result\n\n- `beta_unbeta_coe`: Gödel's Beta Function Lemma.\n\n## Implementation note\n\nThis code is a step towards eventually including a proof of Gödel's First Incompleteness Theorem\nand other key results from the repository https://github.com/iehality/lean4-logic.\n\n## References\n\n* [R. Kaye, *Models of Peano arithmetic*][kaye1991]\n* <https://en.wikipedia.org/wiki/G%C3%B6del%27s_%CE%B2_function>\n\n## Tags\n\nGödel, beta function\n-/\n\n@[expose] public section\n\nnamespace Nat\n\nlemma coprime_mul_succ {n m a} (ha : m - n ∣ a) : Coprime (n * a + 1) (m * a + 1) :=\n  Nat.coprime_of_dvd fun p pp hn hm => by\n    have : p ∣ (m - n) * a := by\n      simpa [Nat.succ_sub_succ, ← Nat.mul_sub_right_distrib] using\n        Nat.dvd_sub hm hn\n    have : p ∣ a := by\n      rcases (Nat.Prime.dvd_mul pp).mp this with (hp | hp)\n      · exact Nat.dvd_trans hp ha\n      · exact hp\n    apply pp.ne_one\n    simpa [Nat.add_sub_cancel_left] using Nat.dvd_sub hn (this.mul_left n)\n\nvariable {m : ℕ}\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nprivate def supOfSeq (a : Fin m → ℕ) : ℕ := max m (Finset.sup .univ a) + 1\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\nprivate def coprimes (a : Fin m → ℕ) : Fin m → ℕ := fun i => (i + 1) * (supOfSeq a)! + 1\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n\nTarget:\nlemma coprimes_lt (a : Fin m → ℕ) (i) : a i < coprimes a i :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_81a0159374a3","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"be7acb484b09252e54bfb7c47367a9747a31079f3f308761d7ea6b5fefd7a7c8","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Logic/Godel","family_id":"coprimes_lt","file_id":"mathlib/Mathlib/Logic/Godel/GodelBetaFunction.lean","sample_id":"81a0159374a335acfc3a3be57440b1fb6c4f215c8d52062ffeefc8554afc5a68"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5cfc8f87ea330fcf416830673402289862153b1294b56a5e4d961dfa30463860","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a931f0b3f033fc32c017444ea0eb1a5592318ecdbcdc093591ad2758f58c4528","source_sha256":"dc37d7e189f3a4d32e4029e0df16c9eeb5ca8b81845b68289b84bb8e5dd20ee8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h => .of_comp (f := ({·} : β → Compacts β)) ?_, map_injective hf⟩\n  simp_rw [Function.comp_def, ← map_singleton hf]\n  exact h.comp singleton_injective","hard_negative":true,"metrics":{"chosen_tokens":43,"rejected_tokens":8,"token_jaccard":0.076923,"token_length_ratio":0.186047},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"f647bd28ea8d91ee9a617020d27e1c858ce9e61c3023e66a40a4855e44fa2fae","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Sets.Closeds\npublic import Mathlib.Topology.QuasiSeparated\n\nNamespace:\nTopologicalSpace.Compacts\n\nLocal context:\n/-\nCopyright (c) 2020 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Yaël Dillies\n-/\n/-!\n# Compact sets\n\nWe define a few types of compact sets in a topological space.\n\n## Main Definitions\n\nFor a topological space `α`,\n* `TopologicalSpace.Compacts α`: The type of compact sets.\n* `TopologicalSpace.NonemptyCompacts α`: The type of non-empty compact sets.\n* `TopologicalSpace.PositiveCompacts α`: The type of compact sets with non-empty interior.\n* `TopologicalSpace.CompactOpens α`: The type of compact open sets. This is a central object in the\n  study of spectral spaces.\n-/\n\n@[expose] public section\n\n\nopen Set\n\nvariable {α β γ : Type*} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]\n\nnamespace TopologicalSpace\n\n/-! ### Compact sets -/\n\n/-- The type of compact sets of a topological space. -/\nstructure Compacts (α : Type*) [TopologicalSpace α] where\n  /-- the carrier set, i.e. the points in this set -/\n  carrier : Set α\n  isCompact' : IsCompact carrier\n\nnamespace Compacts\n\ninstance : SetLike (Compacts α) α where\n  coe := Compacts.carrier\n  coe_injective s t h := by cases s; cases t; congr\n\ninstance : PartialOrder (Compacts α) := .ofSetLike (Compacts α) α\n\n/-- See Note [custom simps projection]. -/\ndef Simps.coe (s : Compacts α) : Set α := s\n\ninitialize_simps_projections Compacts (carrier → coe, as_prefix coe)\n\nprotected theorem isCompact (s : Compacts α) : IsCompact (s : Set α) :=\n  s.isCompact'\n\ninstance (K : Compacts α) : CompactSpace K :=\n  isCompact_iff_compactSpace.1 K.isCompact\n\n/-- Reinterpret a compact as a closed set. -/\n@[simps]\ndef toCloseds [T2Space α] (s : Compacts α) : Closeds α :=\n  ⟨s, s.isCompact.isClosed⟩\n\n@[simp]\ntheorem mem_toCloseds [T2Space α] {x : α} {s : Compacts α} :\n    x ∈ s.toCloseds ↔ x ∈ s :=\n  Iff.rfl\n\ntheorem toCloseds_injective [T2Space α] : Function.Injective (toCloseds (α := α)) :=\n  .of_comp (f := SetLike.coe) SetLike.coe_injective\n\ninstance : CanLift (Set α) (Compacts α) (↑) IsCompact where prf K hK := ⟨⟨K, hK⟩, rfl⟩\n\n@[ext]\nprotected theorem ext {s t : Compacts α} (h : (s : Set α) = t) : s = t :=\n  SetLike.ext' h\n\n@[simp]\ntheorem coe_mk (s : Set α) (h) : (mk s h : Set α) = s :=\n  rfl\n\n@[simp]\ntheorem carrier_eq_coe (s : Compacts α) : s.carrier = s :=\n  rfl\n\ninstance : Max (Compacts α) :=\n  ⟨fun s t => ⟨s ∪ t, s.isCompact.union t.isCompact⟩⟩\n\ninstance [T2Space α] : Min (Compacts α) :=\n  ⟨fun s t => ⟨s ∩ t, s.isCompact.inter t.isCompact⟩⟩\n\ninstance [CompactSpace α] : Top (Compacts α) :=\n  ⟨⟨univ, isCompact_univ⟩⟩\n\ninstance : Bot (Compacts α) :=\n  ⟨⟨∅, isCompact_empty⟩⟩\n\ninstance : SemilatticeSup (Compacts α) :=\n  fast_instance% SetLike.coe_injective.semilatticeSup _ .rfl .rfl fun _ _ ↦ rfl\n\ninstance [T2Space α] : DistribLattice (Compacts α) :=\n  fast_instance% SetLike.coe_injective.distribLattice _ .rfl .rfl (fun _ _ ↦ rfl) fun _ _ ↦ rfl\n\ninstance : OrderBot (Compacts α) :=\n  fast_instance% OrderBot.lift ((↑) : _ → Set α) (fun _ _ => id) rfl\n\ninstance [CompactSpace α] : BoundedOrder (Compacts α) :=\n  fast_instance% BoundedOrder.lift ((↑) : _ → Set α) (fun _ _ => id) rfl rfl\n\n/-- The type of compact sets is inhabited, with default element the empty set. -/\ninstance : Inhabited (Compacts α) := ⟨⊥⟩\n\ninstance [IsEmpty α] : Unique (Compacts α) where\n  uniq _ := Compacts.ext (Subsingleton.elim _ _)\n\n@[simp]\ntheorem coe_sup (s t : Compacts α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t :=\n  rfl\n\n@[simp]\ntheorem coe_inf [T2Space α] (s t : Compacts α) : (↑(s ⊓ t) : Set α) = ↑s ∩ ↑t :=\n  rfl\n\n@[simp]\ntheorem coe_top [CompactSpace α] : (↑(⊤ : Compacts α) : Set α) = univ :=\n  rfl\n\n@[simp]\ntheorem coe_bot : (↑(⊥ : Compacts α) : Set α) = ∅ :=\n  rfl\n\n@[simp, norm_cast]\ntheorem coe_eq_empty {s : Compacts α} : (s : Set α) = ∅ ↔ s = ⊥ :=\n  SetLike.coe_injective.eq_iff' rfl\n\n@[simp]\ntheorem coe_nonempty {s : Compacts α} : (s : Set α).Nonempty ↔ s ≠ ⊥ :=\n  nonempty_iff_ne_empty.trans coe_eq_empty.not\n\n@[simp]\ntheorem coe_finset_sup {ι : Type*} {s : Finset ι} {f : ι → Compacts α} :\n    (↑(s.sup f) : Set α) = s.sup fun i => ↑(f i) := by\n  refine Finset.cons_induction_on s rfl fun a s _ h => ?_\n  simp_rw [Finset.sup_cons, coe_sup, sup_eq_union]\n  congr\n\n@[simps]\ninstance : Singleton α (Compacts α) where\n  singleton x := ⟨{x}, isCompact_singleton⟩\n\n@[simp]\ntheorem mem_singleton (x y : α) : x ∈ ({y} : Compacts α) ↔ x = y :=\n  Iff.rfl\n\n@[simp]\ntheorem toCloseds_singleton [T2Space α] (x : α) : toCloseds {x} = {x} :=\n  rfl\n\ntheorem singleton_injective : Function.Injective ({·} : α → Compacts α) :=\n  .of_comp (f := SetLike.coe) Set.singleton_injective\n\n@[simp]\ntheorem singleton_inj {x y : α} : ({x} : Compacts α) = {y} ↔ x = y :=\n  singleton_injective.eq_iff\n\ninstance [Nonempty α] : Nontrivial (Compacts α) := by\n  constructor\n  obtain ⟨x⟩ := ‹Nonempty α›\n  exact ⟨⊥, {x}, ne_of_apply_ne SetLike.coe (Set.empty_ne_singleton x)⟩\n\n@[simp]\ntheorem subsingleton_iff : Subsingleton (Compacts α) ↔ IsEmpty α := by\n  refine ⟨fun h => ?_, fun _ => inferInstance⟩\n  contrapose! h\n  infer_instance\n\n@[simp]\ntheorem nontrivial_iff : Nontrivial (Compacts α) ↔ Nonempty α := by\n  rw [← not_subsingleton_iff_nontrivial, subsingleton_iff, not_isEmpty_iff]\n\n/-- The image of a compact set under a continuous function. -/\nprotected def map (f : α → β) (hf : Continuous f) (K : Compacts α) : Compacts β :=\n  ⟨f '' K.1, K.2.image hf⟩\n\n@[simp, norm_cast]\ntheorem coe_map {f : α → β} (hf : Continuous f) (s : Compacts α) : (s.map f hf : Set β) = f '' s :=\n  rfl\n\n@[simp]\ntheorem map_id (K : Compacts α) : K.map id continuous_id = K :=\n  Compacts.ext <| Set.image_id _\n\ntheorem map_comp (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g) (K : Compacts α) :\n    K.map (f ∘ g) (hf.comp hg) = (K.map g hg).map f hf :=\n  Compacts.ext <| Set.image_comp _ _ _\n\ntheorem map_injective {f : α → β} (hf : Continuous f) (hf' : Function.Injective f) :\n    Function.Injective (Compacts.map f hf) :=\n  .of_comp (f := SetLike.coe) <| hf'.image_injective.comp SetLike.coe_injective\n\n@[simp]\ntheorem map_singleton {f : α → β} (hf : Continuous f) (x : α) : Compacts.map f hf {x} = {f x} :=\n  Compacts.ext Set.image_singleton\n\n@[simp]\n\nTarget:\ntheorem map_injective_iff {f : α → β} (hf : Continuous f) :\n    Function.Injective (Compacts.map f hf) ↔ Function.Injective f :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"5020a6c2a642297b0a53c5cde7e8f712789a337cdccc08855de7b987ca248699","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Sets","family_id":"map_injective_iff","file_id":"mathlib/Mathlib/Topology/Sets/Compacts.lean","sample_id":"a931f0b3f033fc32c017444ea0eb1a5592318ecdbcdc093591ad2758f58c4528"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b9120dc0c9841f5cc0966a2e218d4adcc087ef91568585fb5a852de6e1d93e10","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"21dca1957ea463f842223ca3af36fd706d107c71eacaf080c4782485b62711e2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3e937a169766b398011457da21839b47e8a3e949c8dec6560a9e40da808f1d10","source_sha256":"afb2352b0e8eda21bd88c47cda5f9b0a4c162f3f78eedf36a33613392805b346","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply Quotient.lift_unique\n  apply Paths.lift_unique\n  fapply @Quiver.Symmetrify.lift_unique _ _ _ _ _ _ _ _ _\n  · rw [← Functor.toPrefunctor_comp]\n    exact hΦ\n  · rintro X Y f\n    simp only [← Functor.toPrefunctor_comp, Prefunctor.comp_map, Paths.of_map]\n    change Φ.map (Groupoid.inv ((Quotient.functor redStep).toPrefunctor.map f.toPath)) =\n      Groupoid.inv (Φ.map ((Quotient.functor redStep).toPrefunctor.map f.toPath))\n    have := Functor.map_inv Φ ((Quotient.functor redStep).toPrefunctor.map f.toPath)\n    convert! this <;> simp only [Groupoid.inv_eq_inv]","hard_negative":false,"metrics":{"chosen_tokens":139,"rejected_tokens":146,"token_jaccard":0.962963,"token_length_ratio":1.05036},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"f64d8856c2db234d6983bc981b971cde755f38221b23d795ded5b2f4b1e145d0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Groupoid\npublic import Mathlib.CategoryTheory.PathCategory.Basic\n\nNamespace:\nQuiver.FreeGroupoid\n\nLocal context:\n/-\nCopyright (c) 2022 Rémi Bottinelli. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémi Bottinelli\n-/\n/-!\n# Free groupoid on a quiver\n\nThis file defines the free groupoid on a quiver, the lifting of a prefunctor to its unique\nextension as a functor from the free groupoid, and proves uniqueness of this extension.\n\n## Main results\n\nGiven the type `V` and a quiver instance on `V`:\n\n- `Quiver.FreeGroupoid V`: a type synonym for `V`.\n- `Quiver.FreeGroupoid.instGroupoid`: the `Groupoid` instance on `Quiver.FreeGroupoid V`.\n- `lift`: the lifting of a prefunctor from `V` to `V'` where `V'` is a groupoid, to a functor.\n  `Quiver.FreeGroupoid V ⥤ V'`.\n- `lift_spec` and `lift_unique`: the proofs that, respectively, `lift` indeed is a lifting\n  and is the unique one.\n\n## Implementation notes\n\nThe free groupoid is first defined by symmetrifying the quiver, taking the induced path category\nand finally quotienting by the reducibility relation.\n\n-/\n\n@[expose] public section\n\nopen Set Function\n\nnamespace Quiver\n\nopen CategoryTheory\n\nuniverse u v u' v' u'' v''\n\nvariable {V : Type u} [Quiver.{v} V]\n\n/-- Shorthand for the \"forward\" arrow corresponding to `f` in `paths <| symmetrify V` -/\nabbrev Hom.toPosPath {X Y : V} (f : X ⟶ Y) :\n    (CategoryTheory.Paths.categoryPaths <| Quiver.Symmetrify V).Hom X Y :=\n  f.toPos.toPath\n\n/-- Shorthand for the \"forward\" arrow corresponding to `f` in `paths <| symmetrify V` -/\nabbrev Hom.toNegPath {X Y : V} (f : X ⟶ Y) :\n    (CategoryTheory.Paths.categoryPaths <| Quiver.Symmetrify V).Hom Y X :=\n  f.toNeg.toPath\n\n/-- The \"reduction\" relation -/\ninductive FreeGroupoid.redStep : HomRel (Paths (Quiver.Symmetrify V))\n  | step (X Z : Quiver.Symmetrify V) (f : X ⟶ Z) :\n    redStep (𝟙 ((Paths.of (Quiver.Symmetrify V)).obj X)) (f.toPath ≫ (Quiver.reverse f).toPath)\n\n/-- The underlying vertices of the free groupoid -/\nprotected def FreeGroupoid (V) [Q : Quiver V] :=\n  CategoryTheory.Quotient (@FreeGroupoid.redStep V Q)\n\nnamespace FreeGroupoid\n\nopen Quiver\n\ninstance {V} [Quiver V] [Nonempty V] : Nonempty (Quiver.FreeGroupoid V) := by\n  inhabit V; exact ⟨⟨@default V _⟩⟩\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem congr_reverse {X Y : Paths <| Quiver.Symmetrify V} (p q : X ⟶ Y) :\n    HomRel.CompClosure redStep p q → HomRel.CompClosure redStep p.reverse q.reverse := by\n  rintro ⟨_, _, XW, _, _, WY, _, _, f⟩\n  have : HomRel.CompClosure redStep (WY.reverse ≫ 𝟙 _ ≫ XW.reverse)\n      (WY.reverse ≫ (f.toPath ≫ (Quiver.reverse f).toPath) ≫ XW.reverse) := by\n    constructor\n    constructor\n  simpa only [CategoryStruct.comp, CategoryStruct.id, Quiver.Path.reverse, Quiver.Path.nil_comp,\n    Quiver.Path.reverse_comp, Quiver.reverse_reverse, Quiver.Path.reverse_toPath,\n    Quiver.Path.comp_assoc] using this\n\nopen Relation in\ntheorem congr_comp_reverse {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :\n    Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (p ≫ p.reverse) =\n      Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (𝟙 X) := by\n  apply Quot.eqvGen_sound\n  induction p with\n  | nil => apply EqvGen.refl\n  | cons q f ih =>\n    simp only [Quiver.Path.reverse]\n    fapply EqvGen.trans\n    -- Porting note: dot notation for `Quiver.Path.*` and `Quiver.Hom.*` not working\n    · exact q ≫ Quiver.Path.reverse q\n    · apply EqvGen.symm\n      apply EqvGen.rel\n      have : HomRel.CompClosure redStep (q ≫ 𝟙 _ ≫ Quiver.Path.reverse q)\n          (q ≫ (Quiver.Hom.toPath f ≫ Quiver.Hom.toPath (Quiver.reverse f)) ≫\n            Quiver.Path.reverse q) := by\n        apply HomRel.CompClosure.intro\n        apply redStep.step\n      simp only [Category.assoc, Category.id_comp] at this ⊢\n      -- Porting note: `simp` cannot see how `Quiver.Path.comp_assoc` is relevant, so change to\n      -- category notation\n      change HomRel.CompClosure redStep (q ≫ Quiver.Path.reverse q)\n        (Quiver.Path.cons q f ≫ (Quiver.Hom.toPath (Quiver.reverse f)) ≫ (Quiver.Path.reverse q))\n      simp only [← Category.assoc] at this ⊢\n      exact this\n    · exact ih\n\ntheorem congr_reverse_comp {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :\n    Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (p.reverse ≫ p) =\n      Quot.mk (@HomRel.CompClosure _ _ redStep _ _) (𝟙 Y) := by\n  nth_rw 2 [← Quiver.Path.reverse_reverse p]\n  apply congr_comp_reverse\n\ninstance : Category (Quiver.FreeGroupoid V) :=\n  Quotient.category redStep\n\n/-- The inverse of an arrow in the free groupoid -/\ndef quotInv {X Y : Quiver.FreeGroupoid V} (f : X ⟶ Y) : Y ⟶ X :=\n  Quot.liftOn f (fun pp => Quot.mk _ <| pp.reverse) fun pp qq con =>\n    Quot.sound <| congr_reverse pp qq con\n\ninstance instGroupoid : Groupoid (Quiver.FreeGroupoid V) where\n  inv := quotInv\n  inv_comp p := Quot.inductionOn p fun pp => congr_reverse_comp pp\n  comp_inv p := Quot.inductionOn p fun pp => congr_comp_reverse pp\n\n/-- The inclusion of the quiver on `V` to the underlying quiver on `FreeGroupoid V` -/\ndef of (V) [Quiver V] : V ⥤q Quiver.FreeGroupoid V where\n  obj X := ⟨X⟩\n  map f := Quot.mk _ f.toPosPath\n\ntheorem of_eq :\n    of V = (Quiver.Symmetrify.of ⋙q (Paths.of (Quiver.Symmetrify V))).comp\n      (Quotient.functor <| @redStep V _).toPrefunctor := rfl\n\nsection UniversalProperty\n\nvariable {V' : Type u'} [Groupoid V']\n\n/-- The lift of a prefunctor to a groupoid, to a functor from `FreeGroupoid V` -/\ndef lift (φ : V ⥤q V') : Quiver.FreeGroupoid V ⥤ V' :=\n  CategoryTheory.Quotient.lift _ (Paths.lift <| Quiver.Symmetrify.lift φ) <| by\n    rintro _ _ _ _ ⟨X, Y, f⟩\n    -- Porting note: `simp` does not work, so manually `rewrite`\n    erw [Paths.lift_nil, Paths.lift_cons, Quiver.Path.comp_nil, Paths.lift_toPath,\n      Quiver.Symmetrify.lift_reverse]\n    symm\n    apply Groupoid.comp_inv\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem lift_spec (φ : V ⥤q V') : of V ⋙q (lift φ).toPrefunctor = φ := by\n  rw [of_eq, Prefunctor.comp_assoc, Prefunctor.comp_assoc, Functor.toPrefunctor_comp]\n  dsimp [lift]\n  rw [Quotient.lift_spec, Paths.lift_spec, Quiver.Symmetrify.lift_spec]\n\nTarget:\ntheorem lift_unique (φ : V ⥤q V') (Φ : Quiver.FreeGroupoid V ⥤ V')\n    (hΦ : of V ⋙q Φ.toPrefunctor = φ) : Φ = lift φ :=\n\nProof body:\n","rejected":"```lean\nby\n  apply Quotient.lift_unique\n  apply Paths.lift_unique\n  fapply @Quiver.Symmetrify.lift_unique _ _ _ _ _ _ _ _ _\n  · rw [← Functor.toPrefunctor_comp]\n    exact hΦ\n  · rintro X Y f\n    simp only [← Functor.toPrefunctor_comp, Prefunctor.comp_map, Paths.of_map]\n    change Φ.map (Groupoid.inv ((Quotient.functor redStep).toPrefunctor.map f.toPath)) =\n      Groupoid.inv (Φ.map ((Quotient.functor redStep).toPrefunctor.map f.toPath))\n    have := Functor.map_inv Φ ((Quotient.functor redStep).toPrefunctor.map f.toPath)\n    convert! this <;> simp only [Groupoid.inv_eq_inv]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Groupoid","family_id":"lift_unique","file_id":"mathlib/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean","sample_id":"3e937a169766b398011457da21839b47e8a3e949c8dec6560a9e40da808f1d10"},"split":"train"}
{"binding":{"chosen_proof_sha256":"76fa0ee83574746f588fc7899220fecd9e66129f439898c7d8ec00fbf1b68c95","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"8032f9d6a97228a94984a08de206cf6fcbd3e34a1b273e00d216ffacec09fc8b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bfdf8a5735fb83291498154fe55688e7c5eabb3111989c0465481333ca9088fd","source_sha256":"231219a77cbb3d5335e2c2c8fee21d9a8c54aa585e63a8a73c88281c1837f6a8","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨i', hi'_lub⟩ : ∃ i', IsLUB (Set.Iio i) i' := exists_lub_Iio i\n  rcases lub_Iio_eq_self_or_Iio_eq_Iic i hi'_lub with hi'_eq_i | h_Iio_eq_Iic\n  · rw [← hi'_eq_i] at hi'_lub ⊢\n    exact hτ.measurableSet_lt_of_isLUB i' hi'_lub\n  · have h_lt_eq_preimage : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iio i := rfl\n    have h_Iio_eq_Iic' : Set.Iio (i : WithTop ι) = Set.Iic (i' : WithTop ι) := by\n      rw [← image_coe_Iio, ← image_coe_Iic, h_Iio_eq_Iic]\n    rw [h_lt_eq_preimage, h_Iio_eq_Iic']\n    exact f.mono (le_of_isLUB_Iio i hi'_lub) _ (hτ.measurableSet_le i')","hard_negative":true,"metrics":{"chosen_tokens":128,"rejected_tokens":3,"token_jaccard":0.033898,"token_length_ratio":0.023438},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"f67ce98cbdd36221db6089ccf00e575f9144e988a9d64e1f3d0a377f4bdb031c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.Process.Adapted\npublic import Mathlib.MeasureTheory.Constructions.BorelSpace.WithTop\npublic import Mathlib.Data.ENat.Lattice\n\nNamespace:\nMeasureTheory\n\nLocal context:\n/-\nCopyright (c) 2021 Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kexing Ying, Rémy Degenne\n-/\n/-!\n# Stopping times, stopped processes and stopped values\n\nDefinition and properties of stopping times.\n\n## Main definitions\n\n* `MeasureTheory.IsStoppingTime`: a stopping time with respect to some filtration `f` on a\n  measurable space `Ω` is a function `τ : Ω → WithTop ι` such that for all `i : ι`,\n  the preimage of `{j | j ≤ i}` along `τ` is `f i`-measurable\n* `MeasureTheory.IsStoppingTime.measurableSpace`: the σ-algebra associated with a stopping time\n\n## Main results\n\n* `IsStronglyProgressive.stoppedProcess`: the stopped process of a progressively measurable process\n  is progressively measurable.\n* `memLp_stoppedProcess`: if a process belongs to `ℒp` at every time in `ℕ`, then its stopped\n  process belongs to `ℒp` as well.\n\n## Implementation notes\n\nFor a filtration on a type `ι`, we define stopping times as functions from the measurable space `Ω`\nto `WithTop ι`, which allows stopping times that can take an infinite value, represented by\n`⊤ : WithTop ι`.\n\nThis means that if we have a process `X : ι → Ω → β` and a stopping time `τ : Ω → WithTop ι`, then\nto consider the value of `X` at the stopping time `τ ω`, we need to write `X (τ ω).untopA ω`,\nin which `(τ ω).untopA` is the value of `τ ω` in `ι` if `τ ω ≠ ⊤` and some arbitrary value if\n`τ ω = ⊤`.\n\nWhile indexing would be more convenient if we defined stopping times as functions from `Ω` to `ι`,\nthis would prevent us from using stopping times as in standard mathematical literature, where a\ntypical example of stopping time is the first time an event occurs, which may never happen.\nConsider for example the first time a coin lands heads when flipping it infinitely many times:\nthis is almost surely finite, but possibly infinite. We could also not use a function `Ω → ι` with\narbitrary value for the infinite case, because this would be incompatible with the stopping time\nproperty.\n\n## Tags\n\nstopping time, stochastic process\n\n-/\n\n@[expose] public section\n\nopen Filter Order TopologicalSpace WithTop\n\nopen scoped MeasureTheory NNReal ENNReal Topology\n\nnamespace MeasureTheory\n\nvariable {Ω β ι : Type*} {m : MeasurableSpace Ω}\n\n/-! ### Stopping times -/\n\n\n/-- A stopping time with respect to some filtration `f` is a function\n`τ` such that for all `i`, the preimage of `{j | j ≤ i}` along `τ` is measurable\nwith respect to `f i`.\n\nIntuitively, the stopping time `τ` describes some stopping rule such that at time\n`i`, we may determine it with the information we have at time `i`. -/\ndef IsStoppingTime [Preorder ι] (f : Filtration ι m) (τ : Ω → WithTop ι) :=\n  ∀ i : ι, MeasurableSet[f i] <| {ω | τ ω ≤ i}\n\ntheorem isStoppingTime_const [Preorder ι] (f : Filtration ι m) (i : ι) :\n    IsStoppingTime f fun _ => i := fun j => by simp only [MeasurableSet.const]\n\nsection MeasurableSet\n\nsection Preorder\n\nvariable [Preorder ι] {f : Filtration ι m} {τ : Ω → WithTop ι}\n\nprotected theorem IsStoppingTime.measurableSet_le (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω ≤ i} :=\n  hτ i\n\ntheorem IsStoppingTime.measurableSet_lt_of_pred [PredOrder ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω < i} := by\n  by_cases hi_min : IsMin i\n  · suffices {ω : Ω | τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i)\n    ext1 ω\n    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]\n    rw [isMin_iff_forall_not_lt] at hi_min\n    cases τ ω with\n    | top => simp\n    | coe t => exact mod_cast hi_min t\n  have : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iic (pred i : ι) := by\n    ext ω\n    push _ ∈ _\n    cases τ ω with\n    | top => simp\n    | coe t =>\n      simp only [coe_lt_coe, coe_le_coe]\n      rw [le_pred_iff_of_not_isMin hi_min]\n  rw [this]\n  exact f.mono (pred_le i) _ (hτ.measurableSet_le <| pred i)\n\nend Preorder\n\nsection CountableStoppingTime\n\nnamespace IsStoppingTime\n\nvariable [PartialOrder ι] {τ : Ω → WithTop ι} {f : Filtration ι m}\n\nprotected theorem measurableSet_eq_of_countable_range (hτ : IsStoppingTime f τ)\n    (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω | τ ω = i} := by\n  have : {ω | τ ω = i} = {ω | τ ω ≤ i} \\ ⋃ (j ∈ Set.range τ) (_ : j < i), {ω | τ ω ≤ j} := by\n    ext1 a\n    simp only [Set.mem_setOf_eq, Set.mem_range, Set.iUnion_exists, Set.iUnion_iUnion_eq',\n      Set.mem_sdiff, Set.mem_iUnion, exists_prop, not_exists, not_and]\n    constructor <;> intro h\n    · simp only [h, lt_iff_le_not_ge, le_refl, and_imp, imp_self, imp_true_iff, and_self_iff]\n    · exact h.1.eq_or_lt.resolve_right fun h_lt => h.2 a h_lt le_rfl\n  rw [this]\n  refine (hτ.measurableSet_le i).diff ?_\n  refine MeasurableSet.biUnion h_countable fun j _ => ?_\n  classical\n  rw [Set.iUnion_eq_if]\n  split_ifs with hji\n  · lift j to ι using (ne_top_of_lt hji)\n    exact f.mono (mod_cast hji.le) _ (hτ.measurableSet_le j)\n  · exact @MeasurableSet.empty _ (f i)\n\nprotected theorem measurableSet_eq_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω = i} :=\n  hτ.measurableSet_eq_of_countable_range (Set.to_countable _) i\n\nprotected theorem measurableSet_lt_of_countable_range (hτ : IsStoppingTime f τ)\n    (h_countable : (Set.range τ).Countable) (i : ι) : MeasurableSet[f i] {ω | τ ω < i} := by\n  have : {ω | τ ω < i} = {ω | τ ω ≤ i} \\ {ω | τ ω = i} := by ext1 ω; simp [lt_iff_le_and_ne]\n  rw [this]\n  exact (hτ.measurableSet_le i).diff (hτ.measurableSet_eq_of_countable_range h_countable i)\n\nprotected theorem measurableSet_lt_of_countable [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω < i} :=\n  hτ.measurableSet_lt_of_countable_range (Set.to_countable _) i\n\nprotected theorem measurableSet_ge_of_countable_range {ι} [LinearOrder ι] {τ : Ω → WithTop ι}\n    {f : Filtration ι m} (hτ : IsStoppingTime f τ) (h_countable : (Set.range τ).Countable) (i : ι) :\n    MeasurableSet[f i] {ω | i ≤ τ ω} := by\n  have : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ := by\n    ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt]\n  rw [this]\n  exact (hτ.measurableSet_lt_of_countable_range h_countable i).compl\n\nprotected theorem measurableSet_ge_of_countable {ι} [LinearOrder ι] {τ : Ω → WithTop ι}\n    {f : Filtration ι m} [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | i ≤ τ ω} :=\n  hτ.measurableSet_ge_of_countable_range (Set.to_countable _) i\n\nend IsStoppingTime\n\nend CountableStoppingTime\n\nsection LinearOrder\n\nvariable [LinearOrder ι] {f : Filtration ι m} {τ : Ω → WithTop ι}\n\ntheorem IsStoppingTime.measurableSet_gt (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | i < τ ω} := by\n  have : {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ := by\n    ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_le]\n  rw [this]\n  exact (hτ.measurableSet_le i).compl\n\nsection TopologicalSpace\n\nvariable [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι]\n\n/-- Auxiliary lemma for `MeasureTheory.IsStoppingTime.measurableSet_lt`. -/\ntheorem IsStoppingTime.measurableSet_lt_of_isLUB (hτ : IsStoppingTime f τ) (i : ι)\n    (h_lub : IsLUB (Set.Iio i) i) : MeasurableSet[f i] {ω | τ ω < i} := by\n  by_cases hi_min : IsMin i\n  · suffices {ω | τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i)\n    ext1 ω\n    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]\n    cases τ ω with\n    | top => simp\n    | coe t => norm_cast; exact isMin_iff_forall_not_lt.mp hi_min t\n  obtain ⟨seq, -, -, h_tendsto, h_bound⟩ :\n      ∃ seq : ℕ → ι, Monotone seq ∧ (∀ j, seq j ≤ i) ∧ Tendsto seq atTop (𝓝 i) ∧ ∀ j, seq j < i :=\n    h_lub.exists_seq_monotone_tendsto (not_isMin_iff.mp hi_min)\n  have h_Iio_eq_Union : Set.Iio (i : WithTop ι) = ⋃ j, {k : WithTop ι | k ≤ seq j} := by\n    ext1 k\n    push _ ∈ _\n    refine ⟨fun hk_lt_i => ?_, fun h_exists_k_le_seq => ?_⟩\n    · rw [tendsto_atTop'] at h_tendsto\n      cases k with\n      | top => simp at hk_lt_i\n      | coe k =>\n        norm_cast at hk_lt_i ⊢\n        have h_nhds : Set.Ici k ∈ 𝓝 i :=\n          mem_nhds_iff.mpr ⟨Set.Ioi k, Set.Ioi_subset_Ici le_rfl, isOpen_Ioi, hk_lt_i⟩\n        obtain ⟨a, ha⟩ : ∃ a : ℕ, ∀ b : ℕ, b ≥ a → k ≤ seq b := h_tendsto (Set.Ici k) h_nhds\n        exact ⟨a, ha a le_rfl⟩\n    · obtain ⟨j, hk_seq_j⟩ := h_exists_k_le_seq\n      exact hk_seq_j.trans_lt (mod_cast h_bound j)\n  have h_lt_eq_preimage : {ω | τ ω < i} = τ ⁻¹' Set.Iio i := by\n    ext1 ω; push _ ∈ _; rfl\n  rw [h_lt_eq_preimage, h_Iio_eq_Union]\n  simp only [Set.preimage_iUnion, Set.preimage_setOf_eq]\n  exact MeasurableSet.iUnion fun n => f.mono (h_bound n).le _ (hτ.measurableSet_le (seq n))\n\nTarget:\ntheorem IsStoppingTime.measurableSet_lt (hτ : IsStoppingTime f τ) (i : ι) :\n    MeasurableSet[f i] {ω | τ ω < i} :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_bfdf8a5735fb","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"32897803ca433a0d9beec2943b2ed277e1b3d2c1a186fd501a0dcb0483dede80","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability/Process","family_id":"isstoppingtime","file_id":"mathlib/Mathlib/Probability/Process/Stopping.lean","sample_id":"bfdf8a5735fb83291498154fe55688e7c5eabb3111989c0465481333ca9088fd"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b4cbd01d6c035dc37c5d6b8fd96656834089520d56a98eb2574c117f82f08ca7","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ff462f64730661be30ba8210dab7fdd5b52f832ee784082523a3c4e53c389888","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2e83f293bd34554caad077053dfa4b4856eabf4ae072c64cf35623b00558a124","source_sha256":"ea6805b6e04e1a10e4f1800fbf6f645af1b3fbe7b7bc9db5c014508a4b9f528f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp [dens, card_biUnion h, sum_div]","hard_negative":true,"metrics":{"chosen_tokens":10,"rejected_tokens":5,"token_jaccard":0.076923,"token_length_ratio":0.5},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"f6b354b68e1294c29aca8a9bed2ee65c21c785e11708c699b977094fb51f70f2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Ring.Finset\npublic import Mathlib.Algebra.Field.Defs\npublic import Mathlib.Data.Finset.Density\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Daniel Weber\n-/\n/-!\n# Results about big operators with values in a field\n-/\n\npublic section\n\nopen Fintype\n\nvariable {ι K : Type*} [DivisionSemiring K]\n\nlemma Multiset.sum_map_div (s : Multiset ι) (f : ι → K) (a : K) :\n    (s.map (fun x ↦ f x / a)).sum = (s.map f).sum / a := by\n  simp only [div_eq_mul_inv, Multiset.sum_map_mul_right]\n\nlemma Finset.sum_div (s : Finset ι) (f : ι → K) (a : K) :\n    (∑ i ∈ s, f i) / a = ∑ i ∈ s, f i / a := by simp only [div_eq_mul_inv, sum_mul]\n\n-- TODO: Move these to `Algebra.BigOperators.Group.Finset.Basic`, next to the corresponding `card`\n-- lemmas, once `Finset.dens` doesn't depend on `Field` anymore.\nnamespace Finset\nvariable {α β : Type*} [Fintype β]\n\n@[simp]\nlemma dens_disjiUnion (s : Finset α) (t : α → Finset β) (h) :\n    (s.disjiUnion t h).dens = ∑ a ∈ s, (t a).dens := by\n  simp [dens, sum_div]\n\nvariable {s : Finset α} {t : α → Finset β}\n\nTarget:\nlemma dens_biUnion [DecidableEq β] (h : (s : Set α).PairwiseDisjoint t) :\n    (s.biUnion t).dens = ∑ u ∈ s, (t u).dens :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_2e83f293bd34","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"58782bd5f256f976a5b1fc56262c1a3d1c9bea23d0c5866366b912b74bed32fd","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Algebra/BigOperators","family_id":"dens_biunion","file_id":"mathlib/Mathlib/Algebra/BigOperators/Field.lean","sample_id":"2e83f293bd34554caad077053dfa4b4856eabf4ae072c64cf35623b00558a124"},"split":"train"}
{"binding":{"chosen_proof_sha256":"362e79b2088809d4572239c27914a8ae94ced25dc58902293b9373aadce248ad","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"ac907c8dc3bdd145394ac850e70ea474947839ba769ccb329f2bcc257c881732","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"215bc10c17214a80cdf028f691cd9e0cb3d40b6838bce81a23b452ba150e84be","source_sha256":"1b477641415d2c99d052691b438f4aaca539c9f1e08663583ff663a2f6ef7d8c","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  haveI : Fintype ι := Fintype.ofFinite ι\n  rw [fromDirectSumEquiv, ← fromDFinsuppEquiv_single]\n  convert! rfl","hard_negative":true,"metrics":{"chosen_tokens":20,"rejected_tokens":5,"token_jaccard":0.095238,"token_length_ratio":0.25},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"f6f70f9eae1c4405bb11626f2f870afa1905b5920027a873b2f5002de15dc05e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.DirectSum.Module\npublic import Mathlib.LinearAlgebra.Multilinear.DFinsupp\n\nNamespace:\nMultilinearMap\n\nLocal context:\n/-\nCopyright (c) 2024 Sophie Morel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sophie Morel\n-/\n/-!\n# Multilinear maps from direct sums\n\nThis file describes multilinear maps on direct sums.\n\n## Main results\n\n* `MultilinearMap.fromDirectSumEquiv` : If `ι` is a `Fintype`, `κ i` is a family of types\n  indexed by `ι` and we are given an `R`-module `M i j` for every `i : ι` and `j : κ i`, this is\n  the linear equivalence between `Π p : (i : ι) → κ i, MultilinearMap R (fun i ↦ M i (p i)) M'` and\n  `MultilinearMap R (fun i ↦ ⨁ j : κ i, M i j) M'`.\n-/\n\n@[expose] public section\n\nnamespace MultilinearMap\n\nopen DirectSum\n\nvariable {R ι M' : Type*} {κ : ι → Type*} {M : (i : ι) → κ i → Type*}\nvariable [CommSemiring R]\nvariable [∀ i j, AddCommMonoid (M i j)] [∀ i j, Module R (M i j)] [AddCommMonoid M'] [Module R M']\n\n/-- Two multilinear maps from direct sums are equal if they agree on the generators. -/\n@[ext]\ntheorem directSum_ext [Finite ι] [(i : ι) → DecidableEq (κ i)]\n    ⦃f g : MultilinearMap R (fun i ↦ ⨁ j : κ i, M i j) M'⦄\n    (h : ∀ p : (i : ι) → κ i,\n      f.compLinearMap (fun i => DirectSum.lof _ _ _ (p i)) =\n      g.compLinearMap (fun i => DirectSum.lof _ _ _ (p i))) : f = g :=\n  dfinsupp_ext h\n\nvariable [DecidableEq ι]\n\n/-- The linear equivalence between families indexed by `p : Π i : ι, κ i` of multilinear maps\non the `fun i ↦ M i (p i)` and the space of multilinear map on `fun i ↦ ⨁ j : κ i, M i j`. -/\nnoncomputable def fromDirectSumEquiv [Finite ι] :\n    ((p : (i : ι) → κ i) → MultilinearMap R (fun i ↦ M i (p i)) M') ≃ₗ[R]\n    MultilinearMap R (fun i ↦ ⨁ j : κ i, M i j) M' :=\n  haveI : Fintype ι := Fintype.ofFinite ι\n  haveI : (i : ι) → DecidableEq (κ i) := fun i ↦ Classical.typeDecidableEq (κ i)\n  fromDFinsuppEquiv _ _\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\n\nTarget:\ntheorem fromDirectSumEquiv_lof [Finite ι] [(i : ι) → DecidableEq (κ i)]\n    (f : (p : (i : ι) → κ i) → MultilinearMap R (fun i ↦ M i (p i)) M')\n    (p : (i : ι) → κ i) (x : (i : ι) → M i (p i)) :\n    fromDirectSumEquiv f (fun i => lof R _ _ _ (x i)) = f p x :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_215bc10c1721","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"276859439ea45b61dd158f27a79ea1aac44893dafad095803d5112223ef1b8c5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Multilinear","family_id":"fromdirectsumequiv_lof","file_id":"mathlib/Mathlib/LinearAlgebra/Multilinear/DirectSum.lean","sample_id":"215bc10c17214a80cdf028f691cd9e0cb3d40b6838bce81a23b452ba150e84be"},"split":"train"}
{"binding":{"chosen_proof_sha256":"55204f99d585e2efb5ba03ffeb23f7cf47ecd4bbf63bb94951cbe11d0f4291b8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"af47faf185b783519da8aa97ae8dad4e2871ff425196acbb014b04737938526c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a1caf932105bba53888c1df521c9cec9b3321d4d5015df07eac43fb607350537","source_sha256":"d9780449d2613fd42914cf0df8a7f4380debc01adc843be01bb83642da361938","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [NormNum.IsInt.neg_to_eq h rfl]\n  simp only [ne_eq, neg_eq_zero]\n  apply ne_of_gt\n  simpa using! w","hard_negative":true,"metrics":{"chosen_tokens":24,"rejected_tokens":5,"token_jaccard":0.083333,"token_length_ratio":0.208333},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"f7086b2e5dd2f6b88d37c654ae5234053d2595d72bfb179565e082705b726884","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Order.Invertible\npublic import Mathlib.Algebra.Order.Ring.Cast\npublic import Mathlib.Tactic.HaveI\npublic import Mathlib.Tactic.NormNum.Core\n\nNamespace:\nMathlib.Meta.Positivity\n\nLocal context:\n/-\nCopyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Heather Macbeth, Yaël Dillies\n-/\npublic meta import Mathlib.Control.Basic\n\n/-!\n## `positivity` core functionality\n\nThis file sets up the `positivity` tactic and the `@[positivity]` attribute,\nwhich allow for plugging in new positivity functionality around a positivity-based driver.\nThe actual behavior is in `@[positivity]`-tagged definitions in `Tactic.Positivity.Basic`\nand elsewhere.\n-/\n\npublic meta section\n\nopen Lean\nopen Lean.Meta Qq Lean.Elab Term\n\n/-- A definition of type `PositivityExt` tagged `@[positivity t]` extends the `positivity` tactic.\nThe term (with underscores) `t` indicates which expressions this extension accepts.\nAn extension will be given an expression `e : α`, together with hypotheses\n`[Zero α] [PartialOrder α]` and attempts to prove `e > 0`, `e ≥ 0`, or `e ≠ 0`.\n\nWhen `Positivity.core` calls this extension on an expression `e`, it does not guarantee that `e`\nmatches `t` perfectly: validate the form of the expression (using e.g.\n`match_expr (← withReducible (whnf e))`) before building a proof. See also the\n`let .app ... ← withReducible (whnf e) | throwError ...` lines in the example below.\n\nAn extension can call `Mathlib.Meta.Positivity.core` to recursively solve subgoals.\n\nExample:\n```lean\n@[positivity ite _ _ _] def evalIte : PositivityExt where eval {u α} zα pα e := do\n  let .app (.app (.app (.app f (p : Q(Prop))) (_ : Q(Decidable $p))) (a : Q($α))) (b : Q($α))\n    ← withReducible (whnf e) | throwError \"not ite\"\n  haveI' : $e =Q ite $p $a $b := ⟨⟩\n  guard <| ← withDefault <| withNewMCtxDepth <| isDefEq f q(ite (α := $α))\n  let ra ← core zα pα a; let rb ← core zα pα b\n  ...\n```\n-/\nsyntax (name := positivity) \"positivity \" term,+ : attr\n\nlemma ne_of_ne_of_eq' {α : Sort*} {a c b : α} (hab : (a : α) ≠ c) (hbc : a = b) : b ≠ c := hbc ▸ hab\n\nnamespace Mathlib.Meta.Positivity\n\nvariable {u : Level} {α : Q(Type u)} (zα : Q(Zero $α))\n\n/-- The result of `positivity` running on an expression `e` of type `α`. -/\ninductive Strictness (e : Q($α)) : Option Q(PartialOrder $α) → Type where\n  | positive {pα : Q(PartialOrder $α)} (pf : Q(0 < $e)) : Strictness e pα\n  | nonnegative {pα : Q(PartialOrder $α)} (pf : Q(0 ≤ $e)) : Strictness e pα\n  | nonzero {pα?} (pf : Q($e ≠ 0)) : Strictness e pα?\n  | none {pα?} : Strictness e pα?\n\n/-- Gives a generic description of the `positivity` result. -/\ndef Strictness.toString {e pα?} : Strictness zα e pα? → String\n  | positive _ => \"positive\"\n  | nonnegative _ => \"nonnegative\"\n  | nonzero _ => \"nonzero\"\n  | none => \"none\"\n\n/-- Extract a proof that `e` is positive, if possible, from `Strictness` information about `e`. -/\ndef Strictness.toPositive {e pα} : Strictness zα e (some pα) → Option Q(0 < $e)\n  | .positive pf => some pf\n  | _ => .none\n\n/-- Extract a proof that `e` is nonnegative, if possible, from `Strictness` information about `e`.\n-/\ndef Strictness.toNonneg {e pα} : Strictness zα e (some pα) → Option Q(0 ≤ $e)\n  | .positive pf => some q(le_of_lt $pf)\n  | .nonnegative pf => some pf\n  | _ => .none\n\n/-- Extract a proof that `e` is nonzero, if possible, from `Strictness` information about `e`. -/\ndef Strictness.toNonzero {e pα?} : Strictness zα e pα? → Option Q($e ≠ 0)\n  | .positive pf => some q(ne_of_gt $pf)\n  | .nonzero pf => some pf\n  | _ => .none\n\n/-- An extension for `positivity`. -/\nstructure PositivityExt where\n  /-- Attempts to prove an expression `e : α` is `>0`, `≥0`, or `≠0`. -/\n  eval {u : Level} {α : Q(Type u)} (zα : Q(Zero $α)) (pα? : Option Q(PartialOrder $α)) (e : Q($α)) :\n    MetaM (Strictness zα e pα?)\n\n/-- Read a `positivity` extension from a declaration of the right type. -/\ndef mkPositivityExt (n : Name) : ImportM PositivityExt := do\n  let { env, opts, .. } ← read\n  IO.ofExcept <| unsafe env.evalConstCheck PositivityExt opts ``PositivityExt n\n\n/-- Each `positivity` extension is labelled with a collection of patterns\nwhich determine the expressions to which it should be applied. -/\nabbrev Entry := Array (Array DiscrTree.Key) × Name\n\n/-- Environment extensions for `positivity` declarations -/\ninitialize positivityExt : PersistentEnvExtension Entry (Entry × PositivityExt)\n    (List Entry × DiscrTree PositivityExt) ←\n  -- we only need this to deduplicate entries in the DiscrTree\n  have : BEq PositivityExt := ⟨fun _ _ => false⟩\n  let insert kss v dt := kss.foldl (fun dt ks => dt.insertKeyValue ks v) dt\n  registerPersistentEnvExtension {\n    mkInitial := pure ([], {})\n    addImportedFn := fun s => do\n      let dt ← s.foldlM (init := {}) fun dt s => s.foldlM (init := dt) fun dt (kss, n) => do\n        pure (insert kss (← mkPositivityExt n) dt)\n      pure ([], dt)\n    addEntryFn := fun (entries, s) ((kss, n), ext) => ((kss, n) :: entries, insert kss ext s)\n    exportEntriesFn := fun s => s.1.reverse.toArray\n  }\n\ninitialize registerBuiltinAttribute {\n  name := `positivity\n  descr := \"adds a positivity extension\"\n  applicationTime := .afterCompilation\n  add := fun declName stx kind => match stx with\n    | `(attr| positivity $es,*) => do\n      ensureAttrDeclIsMeta `positivity declName kind\n      unless kind == AttributeKind.global do\n        throwError \"invalid attribute 'positivity', must be global\"\n      let env ← getEnv\n      unless (env.getModuleIdxFor? declName).isNone do\n        throwError \"invalid attribute 'positivity', declaration is in an imported module\"\n      if (IR.getSorryDep env declName).isSome then return -- ignore in progress definitions\n      let ext ← mkPositivityExt declName\n      let keys ← MetaM.run' <| es.getElems.mapM fun stx => do\n        let e ← TermElabM.run' <| withSaveInfoContext <| withAutoBoundImplicit <|\n          withReader ({ · with ignoreTCFailures := true }) do\n            let e ← elabTerm stx none\n            let (_, _, e) ← lambdaMetaTelescope (← mkLambdaFVars (← getLCtx).getFVars e)\n            return e\n        DiscrTree.mkPath e\n      setEnv <| positivityExt.addEntry env ((keys, declName), ext)\n      -- TODO: track what `[positivity]` decls are actually used at use sites\n      recordExtraRevUseOfCurrentModule\n    | _ => throwUnsupportedSyntax\n}\n\nvariable {A : Type*} {e : A}\n\nlemma pos_of_isNat {n : ℕ} [Semiring A] [PartialOrder A] [IsOrderedRing A] [Nontrivial A]\n    (h : NormNum.IsNat e n) (w : Nat.ble 1 n = true) : 0 < (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  apply Nat.cast_pos.2\n  simpa using! w\n\nlemma pos_of_isNat' {n : ℕ}\n    [AddMonoidWithOne A] [PartialOrder A] [AddLeftMono A] [ZeroLEOneClass A] [h'' : NeZero (1 : A)]\n    (h : NormNum.IsNat e n) (w : Nat.ble 1 n = true) : 0 < (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  apply Nat.cast_pos'.2\n  simpa using! w\n\nlemma nonneg_of_isNat {n : ℕ} [Semiring A] [PartialOrder A] [IsOrderedRing A]\n    (h : NormNum.IsNat e n) : 0 ≤ (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  exact Nat.cast_nonneg n\n\nlemma nonneg_of_isNat' {n : ℕ}\n    [AddMonoidWithOne A] [PartialOrder A] [AddLeftMono A] [ZeroLEOneClass A]\n    (h : NormNum.IsNat e n) : 0 ≤ (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  exact Nat.cast_nonneg' n\n\nTarget:\nlemma nz_of_isNegNat {n : ℕ} [Ring A] [PartialOrder A] [IsStrictOrderedRing A]\n    (h : NormNum.IsInt e (.negOfNat n)) (w : Nat.ble 1 n = true) : (e : A) ≠ 0 :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_a1caf932105b","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"162755ecab5b0686c9310caa07662a0758e51d8868fe59bc3d64ba85932447e1","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Tactic/Positivity","family_id":"nz_of_isnegnat","file_id":"mathlib/Mathlib/Tactic/Positivity/Core.lean","sample_id":"a1caf932105bba53888c1df521c9cec9b3321d4d5015df07eac43fb607350537"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b1b2588f46af06b1f0c76c618d2cd12402ece06fbea01ebed4a3cb2260d14513","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"698ba750f435c2d8477d0231c1f5e3c7957f821edec8b6e868b1d3243469b612","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"345ae0fc64c4223b685a8043b5b363bbd84f8c383d47f5ccfa20bde13d5436d1","source_sha256":"b281bb44b709ce523fcb2c42bb757b0bde9d86d54aacb866640c7f77cfffa6bd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [mapRange_def]\n  refine (Finset.prod_subset support_mk_subset ?_).trans ?_\n  · intro i h1 h2\n    simp only [mem_support_toFun, ne_eq] at h1\n    simp only [Finset.coe_sort_coe, mem_support_toFun, mk_apply, ne_eq, h1, not_false_iff,\n      dite_eq_ite, ite_true, not_not] at h2\n    simp [h2, h0]\n  · refine Finset.prod_congr rfl ?_\n    intro i h1\n    simp only [mem_support_toFun, ne_eq] at h1\n    simp [h1]","hard_negative":false,"metrics":{"chosen_tokens":87,"rejected_tokens":91,"token_jaccard":0.921053,"token_length_ratio":1.045977},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"f71b769ab34dcb653e577d30813f04560c9cc692b359c17fe772a4d72ccc78d8","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.GroupWithZero.Action\npublic import Mathlib.Data.DFinsupp.Ext\npublic import Mathlib.Algebra.BigOperators.Group.Finset.Sigma\n\nNamespace:\nDFinsupp\n\nLocal context:\n/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Kenny Lau\n-/\n/-!\n# Dependent functions with finite support\n\nFor a non-dependent version see `Mathlib/Data/Finsupp/Defs.lean`.\n\n## Notation\n\nThis file introduces the notation `Π₀ a, β a` as notation for `DFinsupp β`, mirroring the `α →₀ β`\nnotation used for `Finsupp`. This works for nested binders too, with `Π₀ a b, γ a b` as notation\nfor `DFinsupp (fun a ↦ DFinsupp (γ a))`.\n\n## Implementation notes\n\nThe support is internally represented (in the primed `DFinsupp.support'`) as a `Multiset` that\nrepresents a superset of the true support of the function, quotiented by the always-true relation so\nthat this does not impact equality. This approach has computational benefits over storing a\n`Finset`; it allows us to add together two finitely-supported functions without\nhaving to evaluate the resulting function to recompute its support (which would required\ndecidability of `b = 0` for `b : β i`).\n\nThe true support of the function can still be recovered with `DFinsupp.support`; but these\ndecidability obligations are now postponed to when the support is actually needed. As a consequence,\nthere are two ways to sum a `DFinsupp`: with `DFinsupp.sum` which works over an arbitrary function\nbut requires recomputation of the support and therefore a `Decidable` argument; and with\n`DFinsupp.sumAddHom` which requires an additive morphism, using its properties to show that\nsumming over a superset of the support is sufficient.\n\n`Finsupp` takes an altogether different approach here; it uses `Classical.Decidable` and declares\nthe `Add` instance as noncomputable. This design difference is independent of the fact that\n`DFinsupp` is dependently-typed and `Finsupp` is not; in future, we may want to align these two\ndefinitions, or introduce two more definitions for the other combinations of decisions.\n-/\n\n@[expose] public section\n\nuniverse u u₁ u₂ v v₁ v₂ v₃ w x y l\n\nvariable {ι : Type u} {γ : Type w} {β : ι → Type v} {β₁ : ι → Type v₁} {β₂ : ι → Type v₂}\n\nnamespace DFinsupp\n\nsection Algebra\n\n/-- Evaluation at a point is an `AddMonoidHom`. This is the finitely-supported version of\n`Pi.evalAddMonoidHom`. -/\ndef evalAddMonoidHom [∀ i, AddZeroClass (β i)] (i : ι) : (Π₀ i, β i) →+ β i :=\n  (Pi.evalAddMonoidHom β i).comp coeFnAddMonoidHom\n\n@[simp, norm_cast]\ntheorem coe_finsetSum {α} [∀ i, AddCommMonoid (β i)] (s : Finset α) (g : α → Π₀ i, β i) :\n    ⇑(∑ a ∈ s, g a) = ∑ a ∈ s, ⇑(g a) :=\n  map_sum coeFnAddMonoidHom g s\n\n@[deprecated (since := \"2026-04-08\")] alias coe_finset_sum := coe_finsetSum\n\n@[simp]\ntheorem finsetSum_apply {α} [∀ i, AddCommMonoid (β i)] (s : Finset α) (g : α → Π₀ i, β i) (i : ι) :\n    (∑ a ∈ s, g a) i = ∑ a ∈ s, g a i :=\n  map_sum (evalAddMonoidHom i) g s\n\n@[deprecated (since := \"2026-04-08\")] alias finset_sum_apply := finsetSum_apply\n\nend Algebra\n\nsection ProdAndSum\n\nvariable [DecidableEq ι]\n\n/-- `DFinsupp.prod f g` is the product of `g i (f i)` over the support of `f`. -/\n@[to_additive /-- `sum f g` is the sum of `g i (f i)` over the support of `f`. -/]\ndef prod [∀ i, Zero (β i)] [∀ (i) (x : β i), Decidable (x ≠ 0)] [CommMonoid γ] (f : Π₀ i, β i)\n    (g : ∀ i, β i → γ) : γ :=\n  ∏ i ∈ f.support, g i (f i)\n\n@[to_additive]\ntheorem prod_of_support_subset [∀ i, Zero (β i)]\n    [∀ (i) (x : β i), Decidable (x ≠ 0)] [CommMonoid γ]\n    {f : Π₀ i, β i} {g : (i : ι) → β i → γ} {s : Finset ι}\n    (hs : f.support ⊆ s) (map_zero : ∀ i ∈ s, g i 0 = 1) :\n    f.prod g = ∏ i ∈ s, g i (f i) := by\n  simp only [DFinsupp.prod]\n  apply Finset.prod_subset hs\n  intro i hi hi'\n  simp only [DFinsupp.mem_support_toFun, ne_eq, not_not] at hi'\n  rw [hi', map_zero]\n  exact hi\n\n/-- The product over two dfinsupps agree if the functions agree and are well-behaved within the\nshared support. -/\n@[to_additive (attr := gcongr only)\n/-- The sum over two dfinsupps agree if the functions agree and are well-behaved within the\nshared support. -/]\ntheorem prod_congr_of_eq_on_union\n    [∀ i, Zero (β i)] [∀ (i) (x : β i), Decidable (x ≠ 0)] [CommMonoid γ]\n    {f1 f2 : Π₀ i, β i} {g1 g2 : (i : ι) → β i → γ}\n    (h : ∀ x ∈ f1.support ∪ f2.support, g1 x (f1 x) = g2 x (f2 x))\n    (h1 : ∀ x ∈ f1.support ∪ f2.support, g1 x 0 = 1)\n    (h2 : ∀ x ∈ f1.support ∪ f2.support, g2 x 0 = 1) :\n    f1.prod g1 = f2.prod g2 := by\n  rw [prod_of_support_subset Finset.subset_union_left h1,\n    prod_of_support_subset Finset.subset_union_right h2]\n  exact Finset.prod_congr rfl h\n\n@[to_additive (attr := simp)]\ntheorem _root_.map_dfinsuppProd\n    {R S H : Type*} [∀ i, Zero (β i)] [∀ (i) (x : β i), Decidable (x ≠ 0)]\n    [CommMonoid R] [CommMonoid S] [FunLike H R S] [MonoidHomClass H R S] (h : H) (f : Π₀ i, β i)\n    (g : ∀ i, β i → R) : h (f.prod g) = f.prod fun a b => h (g a b) :=\n  map_prod _ _ _\n\n@[to_additive]\n\nTarget:\ntheorem prod_mapRange_index {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [∀ i, Zero (β₁ i)]\n    [∀ i, Zero (β₂ i)] [∀ (i) (x : β₁ i), Decidable (x ≠ 0)] [∀ (i) (x : β₂ i), Decidable (x ≠ 0)]\n    [CommMonoid γ] {f : ∀ i, β₁ i → β₂ i} {hf : ∀ i, f i 0 = 0} {g : Π₀ i, β₁ i} {h : ∀ i, β₂ i → γ}\n    (h0 : ∀ i, h i 0 = 1) : (mapRange f hf g).prod h = g.prod fun i b => h i (f i b) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [mapRange_def]\n  refine (Finset.prod_subset support_mk_subset ?_).trans ?_\n  · intro i h1 h2\n    simp only [mem_support_toFun, ne_eq] at h1\n    simp only [Finset.coe_sort_coe, mem_support_toFun, mk_apply, ne_eq, h1, not_false_iff,\n      dite_eq_ite, ite_true, not_not] at h2\n    simp [h2, h0]\n  · refine Finset.prod_congr rfl ?_\n    intro i h1\n    simp only [mem_support_toFun, ne_eq] at h1\n    simp [h1]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/DFinsupp","family_id":"prod_maprange_index","file_id":"mathlib/Mathlib/Data/DFinsupp/BigOperators.lean","sample_id":"345ae0fc64c4223b685a8043b5b363bbd84f8c383d47f5ccfa20bde13d5436d1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"51d70e6a078b791d67a01b436c28657296809a50b25f00afb27df7a2d7b5b11b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"1288fc6a5cfa445715d48101a0256f0dac3c079cec56dd98e1c13594ce0e8285","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2f4e8f65d6a8377847b7d2fb1bab8531851b8db89761aa4db8d2e360d21ed05e","source_sha256":"0941fd5c915c6b6e1c0d21af55aa642c5a111f5a1f5f8c8078f1d0878568b340","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply SheafOfModules.hom_ext\n  ext U x\n  exact congr($(H U.unop) x)","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":25,"token_jaccard":0.789474,"token_length_ratio":1.25},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"f728e0161850a867694e274da513091deecd760ed626de3fa45e1c94120b2f9b","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Category.ModuleCat.Sheaf.Abelian\npublic import Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits\npublic import Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackContinuous\npublic import Mathlib.AlgebraicGeometry.Modules.Presheaf\npublic import Mathlib.AlgebraicGeometry.OpenImmersion\npublic import Mathlib.CategoryTheory.Bicategory.Adjunction.Adj\npublic import Mathlib.CategoryTheory.Bicategory.Adjunction.Cat\npublic import Mathlib.CategoryTheory.Bicategory.Functor.LocallyDiscrete\n\nNamespace:\nAlgebraicGeometry.Scheme.Modules\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou, Andrew Yang\n-/\n/-!\n# The category of sheaves of modules over a scheme\n\nIn this file, we define the abelian category of sheaves of modules\n`X.Modules` over a scheme `X`, and study its basic functoriality.\n\n-/\n\n@[expose] public section\n\nuniverse t u\n\nopen CategoryTheory Limits TopologicalSpace SheafOfModules Bicategory\n\nnamespace AlgebraicGeometry.Scheme\n\nvariable {X Y Z T : Scheme.{u}}\n\nvariable (X) in\n/-- The category of sheaves of modules over a scheme. -/\ndef Modules := SheafOfModules.{u} X.ringCatSheaf\n\nnamespace Modules\n\n/-- Morphisms between `𝒪ₓ`-modules. Use `Hom.app` to act on sections. -/\ndef Hom (M N : X.Modules) : Type u := SheafOfModules.Hom M N\n\ninstance : Category X.Modules where\n  Hom := Modules.Hom\n  __ := (inferInstance : Category (SheafOfModules.{u} X.ringCatSheaf))\n\nnoncomputable instance : Abelian X.Modules :=\n  inferInstanceAs <| Abelian (SheafOfModules.{u} X.ringCatSheaf)\ninstance : HasLimits X.Modules := inferInstanceAs (HasLimits (SheafOfModules X.ringCatSheaf))\ninstance : HasColimits X.Modules := inferInstanceAs (HasColimits (SheafOfModules X.ringCatSheaf))\n\nsection Functor\n\nvariable (X) in\n/-- The forgetful functor from `𝒪ₓ`-modules to presheaves of modules.\nThis is mostly useful to transport results from (pre)sheaves of modules to `𝒪ₓ`-modules and\nusually shouldn't be used directly when working with actual `𝒪ₓ`-modules. -/\ndef toPresheafOfModules : X.Modules ⥤ X.PresheafOfModules := SheafOfModules.forget _\n\n/-- The forgetful functor from `𝒪ₓ`-modules to presheaves of modules is fully faithful. -/\ndef fullyFaithfulToPresheafOfModules : (Modules.toPresheafOfModules X).FullyFaithful :=\n  SheafOfModules.fullyFaithfulForget _\n\ninstance : (toPresheafOfModules X).Full := fullyFaithfulToPresheafOfModules.full\ninstance : (toPresheafOfModules X).Faithful := fullyFaithfulToPresheafOfModules.faithful\ninstance : (toPresheafOfModules X).IsRightAdjoint :=\n  (PresheafOfModules.sheafificationAdjunction (𝟙 X.ringCatSheaf.obj)).isRightAdjoint\n\nvariable (X) in\n/-- The forgetful functor from `𝒪ₓ`-modules to presheaves of abelian groups. -/\nnoncomputable def toPresheaf : X.Modules ⥤ TopCat.Presheaf Ab X :=\n  toPresheafOfModules X ⋙ PresheafOfModules.toPresheaf _\n\ninstance : (toPresheaf X).Faithful := .comp _ (PresheafOfModules.toPresheaf _)\ninstance : PreservesLimits (toPresheaf X) := comp_preservesLimits _ (PresheafOfModules.toPresheaf _)\ninstance : (toPresheaf X).ReflectsIsomorphisms :=\n  reflectsIsomorphisms_comp _ (PresheafOfModules.toPresheaf _)\n\nend Functor\n\nvariable {M N K : X.Modules} {φ : M ⟶ N} {U V : X.Opens}\n\nsection Presheaf\n\n/-- The underlying abelian presheaf of an `𝒪ₓ`-module. -/\nnoncomputable def presheaf (M : X.Modules) : TopCat.Presheaf Ab X := M.1.presheaf\n\n/-- Notation for sections of a presheaf of module. -/\nscoped[AlgebraicGeometry] notation3 \"Γ(\" M \", \" U \")\" => (Scheme.Modules.presheaf M).obj (.op U)\n\ninstance : Module Γ(X, U) Γ(M, U) := (M.val.obj (.op U)).isModule\n\nvariable (M) in\n@[simp] lemma map_smul (i : U ⟶ V) (r : Γ(X, V)) (x : Γ(M, V)) :\n    M.presheaf.map i.op (r • x) = X.presheaf.map i.op r • M.presheaf.map i.op x :=\n  M.val.map_smul _ _ _\n\n/-- Scalar multiplication as an endomorphism of `Γ(M, U)`. -/\ndef smul : Γ(X, U) →+* End Γ(M, U) :=\n  (M.val.obj (.op U)).smul\n\n@[simp]\nlemma smul_apply (r : Γ(X, U)) (x : Γ(M, U)) : (M.smul r).hom x = r • x := rfl\n\n@[reassoc (attr := simp)]\nlemma map_comp_smul (i : U ⟶ V) (r : Γ(X, V)) :\n    M.smul r ≫ M.presheaf.map i.op = M.presheaf.map i.op ≫ M.smul (X.presheaf.map i.op r) := by\n  ext\n  simp\n\n/-- The underlying map between abelian presheaves of a morphism of `𝒪ₓ`-modules. -/\nnoncomputable def Hom.mapPresheaf (φ : M ⟶ N) : M.presheaf ⟶ N.presheaf :=\n  (toPresheaf X).map φ\n\n/-- The application of a morphism of `𝒪ₓ`-modules to sections. -/\ndef Hom.app (φ : M ⟶ N) (U : X.Opens) : Γ(M, U) ⟶ Γ(N, U) :=\n  (forget₂ _ _).map (φ.val.app (.op U))\n\n@[simp] lemma mapPresheaf_app (φ : M ⟶ N) (U) : φ.mapPresheaf.app U = φ.app U.unop := rfl\n\n@[simp]\nlemma Hom.app_smul (φ : M ⟶ N) (r : Γ(X, U)) (x : Γ(M, U)) :\n    φ.app U (r • x) = r • φ.app U x :=\n  (φ.val.app (.op U)).hom.map_smul r x\n\n@[simp] lemma Hom.add_app (φ ψ : M ⟶ N) : (φ + ψ).app U = φ.app U + ψ.app U := rfl\n@[simp] lemma Hom.sub_app (φ ψ : M ⟶ N) : (φ - ψ).app U = φ.app U - ψ.app U := rfl\n@[simp] lemma Hom.zero_app : (0 : M ⟶ N).app U = 0 := rfl\n@[simp] lemma Hom.id_app (M : X.Modules) : (𝟙 M :).app U = 𝟙 _ := rfl\n@[simp] lemma Hom.comp_app (φ : M ⟶ N) (ψ : N ⟶ K) : (φ ≫ ψ).app U = φ.app U ≫ ψ.app U := rfl\n\n@[ext]\n\nTarget:\nlemma hom_ext (f g : M ⟶ N) (H : ∀ U, f.app U = g.app U) : f = g :=\n\nProof body:\n","rejected":"by\n  apply SheafOfModules.hom_ext\n  ext U x\n  exact congr($(H U.unop) x)\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Modules","family_id":"hom_ext","file_id":"mathlib/Mathlib/AlgebraicGeometry/Modules/Sheaf.lean","sample_id":"2f4e8f65d6a8377847b7d2fb1bab8531851b8db89761aa4db8d2e360d21ed05e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a73fac1f1d110d78557015f3b6ec0716b6ddeed8c82d3ea7e29e1289e23f50bb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b648013aeb700becb315f9a7957067c0708e7ac467886972e3266769c8bb1dbe","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8b9a52eec598c7910eb23c645754ae85116421ba3d320b896639db65e6cbf421","source_sha256":"087c4fdfa1fe64dbc38821c1fa8aea61c46d7fa4bef6ee724aa05ce31b21dbc2","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [preimage_image_mk_eq_iUnion_image, iUnion_subtype, ← image2_mul, ← iUnion_image_right]\n  simp only [SetLike.mem_coe]","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":27,"token_jaccard":0.882353,"token_length_ratio":1.35},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"f72c0006a3a432226cec6afe7b30edfbe71f1e3422c3f63ed8f82f301a5ce327","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Quotient\npublic import Mathlib.Algebra.Group.Action.Opposite\npublic import Mathlib.Algebra.Group.Subgroup.MulOpposite\npublic import Mathlib.GroupTheory.GroupAction.Defs\npublic import Mathlib.Algebra.Group.Pointwise.Set.Basic\n\nNamespace:\nQuotientGroup\n\nLocal context:\n/-\nCopyright (c) 2018 Mitchell Rowett. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mitchell Rowett, Kim Morrison\n-/\n/-!\n# Cosets\n\nThis file develops the basic theory of left and right cosets.\n\nWhen `G` is a group and `a : G`, `s : Set G`, with  `open scoped Pointwise` we can write:\n* the left coset of `s` by `a` as `a • s`\n* the right coset of `s` by `a` as `MulOpposite.op a • s` (or `op a • s` with `open MulOpposite`,\n  or `s <• a` with `open scoped Pointwise RightActions`)\n\nIf instead `G` is an additive group, we can write (with  `open scoped Pointwise` still)\n* the left coset of `s` by `a` as `a +ᵥ s`\n* the right coset of `s` by `a` as `AddOpposite.op a +ᵥ s` (or `op a +ᵥ s` with `open AddOpposite`,\n  or `s <+ᵥ a` with `open scoped Pointwise RightActions`)\n\n## Main definitions\n\n* `QuotientGroup.quotient s`: the quotient type representing the left cosets with respect to a\n  subgroup `s`, for an `AddGroup` this is `QuotientAddGroup.quotient s`.\n* `QuotientGroup.mk`: the canonical map from `α` to `α/s` for a subgroup `s` of `α`, for an\n  `AddGroup` this is `QuotientAddGroup.mk`.\n\n## Notation\n\n* `G ⧸ H` is the quotient of the (additive) group `G` by the (additive) subgroup `H`\n\n## TODO\n\nProperly merge with pointwise actions on sets, by renaming and deduplicating lemmas as appropriate.\n-/\n\n@[expose] public section\n\nassert_not_exists Cardinal\n\nopen Function Set\nopen scoped Pointwise\n\nvariable {α : Type*}\n\n/- Ensure that `@[to_additive]` uses the right namespace. -/\ninsert_to_additive_translation QuotientGroup QuotientAddGroup\n\nnamespace QuotientGroup\n\nvariable [Group α] (s : Subgroup α)\n\n/-- The equivalence relation corresponding to the partition of a group by left cosets\nof a subgroup. -/\n@[to_additive (attr := implicit_reducible)\n  /-- The equivalence relation corresponding to the partition of a group by left cosets\nof a subgroup. -/]\ndef leftRel : Setoid α :=\n  MulAction.orbitRel s.op α\n\nvariable {s} in\n@[to_additive]\ntheorem leftRel_apply {x y : α} : leftRel s x y ↔ x⁻¹ * y ∈ s :=\n  calc\n    (∃ a : s.op, y * MulOpposite.unop a = x) ↔ ∃ a : s, y * a = x :=\n      s.equivOp.symm.exists_congr_left\n    _ ↔ ∃ a : s, x⁻¹ * y = a⁻¹ := by\n      simp only [inv_mul_eq_iff_eq_mul, Subgroup.coe_inv, eq_mul_inv_iff_mul_eq]\n    _ ↔ x⁻¹ * y ∈ s := by simp [exists_inv_mem_iff_exists_mem]\n\n@[to_additive]\ntheorem leftRel_eq : ⇑(leftRel s) = fun x y => x⁻¹ * y ∈ s :=\n  funext₂ <| by\n    simp only [eq_iff_iff]\n    apply leftRel_apply\n\n@[to_additive]\ninstance leftRelDecidable [DecidablePred (· ∈ s)] : DecidableRel (leftRel s).r := fun x y => by\n  rw [leftRel_eq]\n  exact ‹DecidablePred (· ∈ s)› _\n\n/-- `α ⧸ s` is the quotient type representing the left cosets of `s`. If `s` is a normal subgroup,\n`α ⧸ s` is a group -/\n@[to_additive /-- `α ⧸ s` is the quotient type representing the left cosets of `s`. If `s` is a\nnormal subgroup, `α ⧸ s` is a group -/]\ninstance instHasQuotientSubgroup : HasQuotient α (Subgroup α) :=\n  ⟨fun s => Quotient (leftRel s)⟩\n\n@[to_additive]\ninstance [DecidablePred (· ∈ s)] : DecidableEq (α ⧸ s) :=\n  @Quotient.decidableEq _ _ (leftRelDecidable _)\n\n/-- The equivalence relation corresponding to the partition of a group by right cosets of a\nsubgroup. -/\n@[to_additive (attr := implicit_reducible)\n  /-- The equivalence relation corresponding to the partition of a group by right cosets\n  of a subgroup. -/]\ndef rightRel : Setoid α :=\n  MulAction.orbitRel s α\n\nvariable {s} in\n@[to_additive]\ntheorem rightRel_apply {x y : α} : rightRel s x y ↔ y * x⁻¹ ∈ s :=\n  calc\n    (∃ a : s, (a : α) * y = x) ↔ ∃ a : s, y * x⁻¹ = a⁻¹ := by\n      simp only [mul_inv_eq_iff_eq_mul, Subgroup.coe_inv, eq_inv_mul_iff_mul_eq]\n    _ ↔ y * x⁻¹ ∈ s := by simp [exists_inv_mem_iff_exists_mem]\n\n@[to_additive]\ntheorem rightRel_eq : ⇑(rightRel s) = fun x y => y * x⁻¹ ∈ s :=\n  funext₂ <| by\n    simp only [eq_iff_iff]\n    apply rightRel_apply\n\n@[to_additive]\ninstance rightRelDecidable [DecidablePred (· ∈ s)] : DecidableRel (rightRel s).r := fun x y => by\n  rw [rightRel_eq]\n  exact ‹DecidablePred (· ∈ s)› _\n\n/-- Right cosets are in bijection with left cosets. -/\n@[to_additive /-- Right cosets are in bijection with left cosets. -/]\ndef quotientRightRelEquivQuotientLeftRel : Quotient (QuotientGroup.rightRel s) ≃ α ⧸ s where\n  toFun :=\n    Quotient.map' (fun g => g⁻¹) fun a b => by\n      rw [leftRel_apply, rightRel_apply]\n      exact fun h => (congr_arg (· ∈ s) (by simp)).mp (s.inv_mem h)\n  invFun :=\n    Quotient.map' (fun g => g⁻¹) fun a b => by\n      rw [leftRel_apply, rightRel_apply]\n      exact fun h => (congr_arg (· ∈ s) (by simp)).mp (s.inv_mem h)\n  left_inv g :=\n    Quotient.inductionOn' g fun g =>\n      Quotient.sound'\n        (by\n          simp only [inv_inv]\n          exact Quotient.exact' rfl)\n  right_inv g :=\n    Quotient.inductionOn' g fun g =>\n      Quotient.sound'\n        (by\n          simp only [inv_inv]\n          exact Quotient.exact' rfl)\n\nend QuotientGroup\n\nnamespace QuotientGroup\n\nvariable [Group α] {s : Subgroup α}\n\n/-- The canonical map from a group `α` to the quotient `α ⧸ s`. -/\n@[to_additive (attr := coe)\n/-- The canonical map from an `AddGroup` `α` to the quotient `α ⧸ s`. -/]\nabbrev mk (a : α) : α ⧸ s :=\n  Quotient.mk'' a\n\n@[to_additive]\ntheorem mk_surjective : Function.Surjective <| @mk _ _ s :=\n  Quotient.mk''_surjective\n\n@[to_additive (attr := simp)]\nlemma range_mk : range (QuotientGroup.mk (s := s)) = univ := range_eq_univ.mpr mk_surjective\n\n@[to_additive (attr := elab_as_elim)]\ntheorem induction_on {C : α ⧸ s → Prop} (x : α ⧸ s) (H : ∀ z, C (QuotientGroup.mk z)) : C x :=\n  Quotient.inductionOn' x H\n\n@[to_additive]\ninstance : Coe α (α ⧸ s) :=\n  ⟨mk⟩\n\n@[to_additive] alias induction_on' := induction_on\n\n@[to_additive (attr := simp)]\ntheorem quotient_liftOn_mk {β} (f : α → β) (h) (x : α) : Quotient.liftOn' (x : α ⧸ s) f h = f x :=\n  rfl\n\n@[to_additive]\ntheorem forall_mk {C : α ⧸ s → Prop} : (∀ x : α ⧸ s, C x) ↔ ∀ x : α, C x :=\n  mk_surjective.forall\n\n@[to_additive]\ntheorem exists_mk {C : α ⧸ s → Prop} : (∃ x : α ⧸ s, C x) ↔ ∃ x : α, C x :=\n  mk_surjective.exists\n\n@[to_additive]\ninstance (s : Subgroup α) : Inhabited (α ⧸ s) :=\n  ⟨((1 : α) : α ⧸ s)⟩\n\n@[to_additive]\nprotected theorem eq {a b : α} : (a : α ⧸ s) = b ↔ a⁻¹ * b ∈ s :=\n  calc\n    _ ↔ leftRel s a b := Quotient.eq''\n    _ ↔ _ := by rw [leftRel_apply]\n\n@[to_additive]\ntheorem out_eq' (a : α ⧸ s) : mk a.out = a :=\n  Quotient.out_eq' a\n\nvariable (s)\n\n/-- It can be useful to write `obtain ⟨h, H⟩ := mk_out_eq_mul ...`, and then `rw [H]` or\n`simp_rw [H]` or `simp only [H]`. In order for `simp_rw` and `simp only` to work, this lemma is\nstated in terms of an arbitrary `h : s`, rather than the specific `h = g⁻¹ * (mk g).out`. -/\n@[to_additive QuotientAddGroup.mk_out_eq_mul]\ntheorem mk_out_eq_mul (g : α) : ∃ h : s, (mk g : α ⧸ s).out = g * h :=\n  ⟨⟨g⁻¹ * (mk g).out, QuotientGroup.eq.mp (mk g).out_eq'.symm⟩, by rw [mul_inv_cancel_left]⟩\n\nvariable {s} {a b : α}\n\n@[to_additive (attr := simp)]\ntheorem mk_mul_of_mem (a : α) (hb : b ∈ s) : (mk (a * b) : α ⧸ s) = mk a := by\n  rwa [QuotientGroup.eq, mul_inv_rev, inv_mul_cancel_right, s.inv_mem_iff]\n\n@[to_additive]\ntheorem preimage_image_mk (N : Subgroup α) (s : Set α) :\n    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = ⋃ x : N, (· * (x : α)) ⁻¹' s := by\n  ext x\n  simp only [QuotientGroup.eq, SetLike.exists, exists_prop, Set.mem_preimage, Set.mem_iUnion,\n    Set.mem_image]\n  exact\n    ⟨fun ⟨y, hs, hN⟩ => ⟨_, N.inv_mem hN, by simpa using hs⟩, fun ⟨z, hz, hxz⟩ =>\n      ⟨x * z, hxz, by simpa using hz⟩⟩\n\n@[to_additive]\ntheorem preimage_image_mk_eq_iUnion_image (N : Subgroup α) (s : Set α) :\n    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = ⋃ x : N, (· * (x : α)) '' s := by\n  rw [preimage_image_mk, iUnion_congr_of_surjective (·⁻¹) inv_surjective]\n  exact fun x ↦ image_mul_right'\n\n@[to_additive]\n\nTarget:\ntheorem preimage_image_mk_eq_mul (N : Subgroup α) (s : Set α) :\n    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = s * N :=\n\nProof body:\n","rejected":"```lean\nby\n  rw [preimage_image_mk_eq_iUnion_image, iUnion_subtype, ← image2_mul, ← iUnion_image_right]\n  simp only [SetLike.mem_coe]\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Coset","family_id":"preimage_image_mk_eq_mul","file_id":"mathlib/Mathlib/GroupTheory/Coset/Defs.lean","sample_id":"8b9a52eec598c7910eb23c645754ae85116421ba3d320b896639db65e6cbf421"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4fff200b3969b2099ac39b5df9f12ed101d4778e55341e30c66e404ab2007cba","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"a775a92a4cceb4ec9087e1badba45ff9b0ab3bb9b4309c273c6dc757e24b808b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"68d51948bcb20f9102720ae7d20bb860c7d2a1e39fd0c3d403bf97713844dd99","source_sha256":"3ebef0ce1ff4e8804552a350539c21f4e3856ca570ca14d4062fa9367e8ccafd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨k, rfl⟩ := hn\n  rw [pow_add, pow_one, pow_mul, sq]\n  cases s <;> simp","hard_negative":false,"metrics":{"chosen_tokens":25,"rejected_tokens":30,"token_jaccard":0.814815,"token_length_ratio":1.2},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"f72cd6146f1f97818067312ca140822391c175e045472f54d8a22647dd903a49","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.GroupWithZero.Units.Lemmas\npublic import Mathlib.Algebra.Order.BigOperators.Group.Finset\npublic import Mathlib.Algebra.Order.Ring.Cast\npublic import Mathlib.Data.Fintype.BigOperators\npublic import Mathlib.Data.Sign.Defs\n\nNamespace:\nSignType\n\nLocal context:\n/-\nCopyright (c) 2022 Eric Rodriguez. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Rodriguez\n-/\n/-!\n# Sign function\n\nThis file defines the sign function for types with zero and a decidable less-than relation, and\nproves some basic theorems about it.\n-/\n\n@[expose] public section\n\nuniverse u\nvariable {α : Type u}\n\nnamespace SignType\n\n/-- Casting `SignType → ℤ → α` is the same as casting directly `SignType → α`. -/\n@[simp, norm_cast]\nlemma intCast_cast {α : Type*} [AddGroupWithOne α] (s : SignType) : ((s : ℤ) : α) = s :=\n  map_cast' _ Int.cast_one Int.cast_zero (@Int.cast_one α _ ▸ Int.cast_neg 1) _\n\nTarget:\ntheorem pow_odd (s : SignType) {n : ℕ} (hn : Odd n) : s ^ n = s :=\n\nProof body:\n","rejected":"by\n  obtain ⟨k, rfl⟩ := hn\n  rw [pow_add, pow_one, pow_mul, sq]\n  cases s <;> simp\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sign","family_id":"pow_odd","file_id":"mathlib/Mathlib/Data/Sign/Basic.lean","sample_id":"68d51948bcb20f9102720ae7d20bb860c7d2a1e39fd0c3d403bf97713844dd99"},"split":"train"}
{"binding":{"chosen_proof_sha256":"be03b33260f8a567b6bd346d2bc5d2f454ac795779b31a7801a8bdb4b9414537","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6f7dc276bda9e16d09662fbe3919eda2fa7e75341411061da3821af3e089d88c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ca893de6923284d5337d25661aaf5c8d1221b14895d8ede2751221d59a9b92ee","source_sha256":"ddf0a2dfe4174f459cd3668baa0e6f85d5935c17878e2b7aa6d667af48528ee1","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  dsimp [Functor.HasPointwiseLeftKanExtension]\n  simp only [hasPointwiseLeftKanExtensionAt_iff w]\n  refine ⟨fun h X₄ ↦ ?_, fun h _ ↦ h _⟩\n  rw [← Functor.hasPointwiseLeftKanExtensionAt_iff_of_iso _ _ (B.objObjPreimageIso X₄)]\n  apply h","hard_negative":false,"metrics":{"chosen_tokens":48,"rejected_tokens":52,"token_jaccard":0.909091,"token_length_ratio":1.083333},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"f7a96330f9aab66e7a585adc7be5b00b96de55c8199b015937beb6e15fd88bff","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.GuitartExact.Basic\npublic import Mathlib.CategoryTheory.Functor.KanExtension.Adjunction\n\nNamespace:\nCategoryTheory.TwoSquare\n\nLocal context:\n/-\nCopyright (c) 2025 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Guitart exact squares and Kan extensions\n\nGiven a Guitart exact square `w : T ⋙ R ⟶ L ⋙ B`,\n```\n     T\n  C₁ ⥤ C₂\nL |     | R\n  v     v\n  C₃ ⥤ C₄\n     B\n```\nwe show that an extension `F' : C₄ ⥤ D` of `F : C₂ ⥤ D` along `R`\nis a pointwise left Kan extension at `B.obj X₃` iff\nthe composition `T ⋙ F'` is a pointwise left Kan extension at `X₃`\nof `B ⋙ F'`.\n\nWhen suitable (pointwise) left Kan extensions exist, we also show that\nthe natural transformation of functors `(C₂ ⥤ D) ⥤ C₃ ⥤ D`\n`(whiskeringLeft C₁ C₂ D).obj T ⋙ L.lan ⟶ R.lan ⋙ (whiskeringLeft C₃ C₄ D).obj B`\ninduced by a Guitart exact square `w` is an isomorphism.\n\n## References\n\n* https://ncatlab.org/nlab/show/exact+square\n\n-/\n\n@[expose] public section\n\nuniverse v₁ v₂ v₃ v₄ v₅ u₁ u₂ u₃ u₄ u₅\n\nnamespace CategoryTheory\n\nopen Limits\n\nvariable {C₁ : Type u₁} {C₂ : Type u₂} {C₃ : Type u₃} {C₄ : Type u₄} {D : Type u₅}\n  [Category.{v₁} C₁] [Category.{v₂} C₂] [Category.{v₃} C₃] [Category.{v₄} C₄]\n  [Category.{v₅} D]\n\nnamespace Functor.LeftExtension\n\nvariable {T : C₁ ⥤ C₂} {L : C₁ ⥤ C₃} {R : C₂ ⥤ C₄} {B : C₃ ⥤ C₄}\n  {F : C₂ ⥤ D} (E : R.LeftExtension F)\n\n/-- Given a square `w : TwoSquare T L R B` (consisting of a natural transformation\n`T ⋙ R ⟶ L ⋙ B`), this is the obvious map `R.LeftExtension F → L.LeftExtension (T ⋙ F)`\nobtained by the precomposition with `B` and the postcomposition with `w`. -/\nabbrev compTwoSquare (w : TwoSquare T L R B) : L.LeftExtension (T ⋙ F) :=\n  LeftExtension.mk (B ⋙ E.right)\n    (whiskerLeft _ E.hom ≫ (associator _ _ _).inv ≫\n      whiskerRight w.natTrans _ ≫ (associator _ _ _).hom)\n\nset_option backward.defeqAttrib.useBackward true in\n/-- If `w : TwoSquare T L R B` is a Guitart exact square, and `E` is a left extension\nof `F` along `R`, then `E` is a pointwise left Kan extension of `F` along `R` at\n`B.obj X₃` iff `E.compTwoSquare w` is a pointwise left Kan extension\nof `T ⋙ F` along `L` at `X₃`. -/\nnoncomputable def isPointwiseLeftKanExtensionAtCompTwoSquareEquiv\n    (w : TwoSquare T L R B) (X₃ : C₃) [Final (w.costructuredArrowRightwards X₃)] :\n    (E.compTwoSquare w).IsPointwiseLeftKanExtensionAt X₃ ≃\n      E.IsPointwiseLeftKanExtensionAt (B.obj X₃) := by\n  refine Equiv.trans ?_ (Final.isColimitWhiskerEquiv (w.costructuredArrowRightwards X₃) _)\n  exact IsColimit.equivIsoColimit (Cocone.ext (Iso.refl _))\n\nlemma nonempty_isPointwiseLeftKanExtensionAt_compTwoSquare_iff\n    (w : TwoSquare T L R B) (X₃ : C₃) [Final (w.costructuredArrowRightwards X₃)] :\n    Nonempty ((E.compTwoSquare w).IsPointwiseLeftKanExtensionAt X₃) ↔\n      Nonempty (E.IsPointwiseLeftKanExtensionAt (B.obj X₃)) :=\n  (E.isPointwiseLeftKanExtensionAtCompTwoSquareEquiv w _).nonempty_congr\n\nvariable {E} in\n/-- If `w : TwoSquare T L R B` is a Guitart exact square, and `E` is a pointwise\nleft Kan extension of `F` along `R`, then `E.compTwoSquare w` is a pointwise left\nKan extension of `T ⋙ F` along `L`. -/\nnoncomputable def IsPointwiseLeftKanExtension.compTwoSquare\n    (h : E.IsPointwiseLeftKanExtension) (w : TwoSquare T L R B) [w.GuitartExact] :\n    (E.compTwoSquare w).IsPointwiseLeftKanExtension :=\n  fun X₃ ↦ (E.isPointwiseLeftKanExtensionAtCompTwoSquareEquiv w X₃).symm (h _)\n\n/-- If `w : TwoSquare T L R B` is a Guitart exact square, with `B` essentially surjective,\nand `E` is a left extension of `F` along `R`, then `E` is a pointwise\nleft Kan extension of `F` along `R` provided `E.compTwoSquare w` is a pointwise left\nKan extension of `T ⋙ F` along `L`. -/\nnoncomputable def isPointwiseLeftKanExtensionOfCompTwoSquare\n    (w : TwoSquare T L R B) [w.GuitartExact] [B.EssSurj]\n    (h : (E.compTwoSquare w).IsPointwiseLeftKanExtension) :\n    E.IsPointwiseLeftKanExtension :=\n  fun X₄ ↦ E.isPointwiseLeftKanExtensionAtOfIso'\n    (E.isPointwiseLeftKanExtensionAtCompTwoSquareEquiv w _ (h (B.objPreimage X₄)))\n    (B.objObjPreimageIso X₄)\n\n/-- If `w : TwoSquare T L R B` is a Guitart exact square, with `B` essentially surjective,\nand `E` is a left extension of `F` along `R`, then `E` is a pointwise left Kan extension\nof `F` along `R` iff `E.compTwoSquare w` is a pointwise left Kan extension\nof `T ⋙ F` along `L`. -/\nnoncomputable def isPointwiseLeftKanExtensionEquivOfGuitartExact\n    (w : TwoSquare T L R B) [w.GuitartExact] [B.EssSurj] :\n    (E.compTwoSquare w).IsPointwiseLeftKanExtension ≃\n      E.IsPointwiseLeftKanExtension where\n  toFun h := E.isPointwiseLeftKanExtensionOfCompTwoSquare w h\n  invFun h := h.compTwoSquare w\n  left_inv _ := by subsingleton\n  right_inv _ := by subsingleton\n\nend Functor.LeftExtension\n\nnamespace TwoSquare\n\nvariable {T : C₁ ⥤ C₂} {L : C₁ ⥤ C₃} {R : C₂ ⥤ C₄} {B : C₃ ⥤ C₄}\n   (w : TwoSquare T L R B)\n\ninclude w\n\nlemma hasPointwiseLeftKanExtensionAt_iff\n    (F : C₂ ⥤ D) (X₃ : C₃) [(w.costructuredArrowRightwards X₃).Final] :\n    L.HasPointwiseLeftKanExtensionAt (T ⋙ F) X₃ ↔\n      R.HasPointwiseLeftKanExtensionAt F (B.obj X₃) := by\n  dsimp [Functor.HasPointwiseLeftKanExtensionAt]\n  rw [← Functor.Final.hasColimit_comp_iff (w.costructuredArrowRightwards X₃)]\n  rfl\n\nTarget:\nlemma hasPointwiseLeftKanExtension_iff [w.GuitartExact] [B.EssSurj] (F : C₂ ⥤ D) :\n    L.HasPointwiseLeftKanExtension (T ⋙ F) ↔\n      R.HasPointwiseLeftKanExtension F :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  dsimp [Functor.HasPointwiseLeftKanExtension]\n  simp only [hasPointwiseLeftKanExtensionAt_iff w]\n  refine ⟨fun h X₄ ↦ ?_, fun h _ ↦ h _⟩\n  rw [← Functor.hasPointwiseLeftKanExtensionAt_iff_of_iso _ _ (B.objObjPreimageIso X₄)]\n  apply h","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/GuitartExact","family_id":"haspointwiseleftkanextension_iff","file_id":"mathlib/Mathlib/CategoryTheory/GuitartExact/KanExtension.lean","sample_id":"ca893de6923284d5337d25661aaf5c8d1221b14895d8ede2751221d59a9b92ee"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bd9e89042700066e34db2eae2a5bdca2cc788a55ff318687cbca751f9272e192","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"b21d5b016f524bd98133a7e41c460440682067118e72efe5e423d3346f465316","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"25e33e2ad73b99a239389fcab2e1bb17f4373e01742868a9ff04d6e9bc1274a4","source_sha256":"5fbd6e0c9aed4ae0af15e49b0a292eb4e252a42e876e277383b0144f2f430dad","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [List.applyId, List.dlookup, eq_rec_constant, Prod.toSigma, List.map]\n  split_ifs <;> rfl","hard_negative":false,"metrics":{"chosen_tokens":27,"rejected_tokens":32,"token_jaccard":0.791667,"token_length_ratio":1.185185},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"f7b621d6b3eb4f87d3912540dc3f342ad7dae1bd0cac641b35726d6b2420935e","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.MLList.Basic\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.ZeroLEOne\npublic import Mathlib.Data.DFinsupp.Defs\npublic import Mathlib.Data.Finsupp.Defs\npublic import Mathlib.Data.Int.Range\npublic import Mathlib.Data.List.Sigma\npublic import Mathlib.Data.PNat.Notation\npublic import Mathlib.Tactic.Bound.Init\n\nNamespace:\nPlausible.InjectiveFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\npublic meta import Mathlib.Data.Finsupp.ToDFinsupp\npublic meta import Mathlib.Data.Int.Range\npublic meta import Mathlib.Data.List.Sigma\npublic meta import Plausible.Functions\n\n/-!\n## `Plausible`: generators for functions\n\nThis file defines `Sampleable` instances for `ℤ → ℤ` injective functions.\n\nInjective functions are generated by creating a list of numbers and\na permutation of that list. The permutation insures that every input\nis mapped to a unique output. When an input is not found in the list\nthe input itself is used as an output.\n\nInjective functions `f : α → α` could be generated easily instead of\n`ℤ → ℤ` by generating a `List α`, removing duplicates and creating a\npermutation. One has to be careful when generating the domain to make\nit vast enough that, when generating arguments to apply `f` to,\nthey argument should be likely to lie in the domain of `f`. This is\nthe reason that injective functions `f : ℤ → ℤ` are generated by\nfixing the domain to the range `[-2*size .. 2*size]`, with `size`\nthe size parameter of the `gen` monad.\n\nMuch of the machinery provided in this file is applicable to generate\ninjective functions of type `α → α` and new instances should be easy\nto define.\n\nOther classes of functions such as monotone functions can generated using\nsimilar techniques. For monotone functions, generating two lists, sorting them\nand matching them should suffice, with appropriate default values.\nSome care must be taken for shrinking such functions to make sure\ntheir defining property is invariant through shrinking. Injective\nfunctions are an example of how complicated it can get.\n-/\n\n@[expose] public meta section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type v}\n\nnamespace Plausible\n\nnamespace TotalFunction\n\nsection Finsupp\n\nvariable [DecidableEq α]\n\n/--\nThis theorem exists because plausible does not have access to dlookup but\nmathlib has all the theory for it and wants to use it. We probably want to\nbring these two together at some point.\n-/\nprivate theorem apply_eq_dlookup (m : List (Σ _ : α, β)) (y : β) (x : α) :\n     (withDefault m y).apply x = (m.dlookup x).getD y := by\n  dsimp only [apply]\n  congr 1\n  induction m with\n  | nil => simp\n  | cons p m ih =>\n    rcases p with ⟨fst, snd⟩\n    by_cases heq : fst = x\n    · simp [heq]\n    · rw [List.dlookup_cons_ne]\n      · simp [heq, ih]\n      · symm\n        simp [heq]\n\nvariable [Zero β] [DecidableEq β]\n\n/-- Map a `TotalFunction` to one whose default value is zero so that it represents a `Finsupp`. -/\n@[simp]\ndef zeroDefault : TotalFunction α β → TotalFunction α β\n  | .withDefault A _ => .withDefault A 0\n\n/-- The support of a zero default `TotalFunction`. -/\ndef zeroDefaultSupp : TotalFunction α β → Finset α\n  | .withDefault A _ =>\n    List.toFinset <| (A.dedupKeys.filter fun ab => Sigma.snd ab ≠ 0).map Sigma.fst\n\n/-- Create a finitely supported function from a total function by taking the default value to\nzero. -/\ndef applyFinsupp (tf : TotalFunction α β) : α →₀ β where\n  support := zeroDefaultSupp tf\n  toFun := tf.zeroDefault.apply\n  mem_support_toFun := by\n    intro a\n    rcases tf with ⟨A, y⟩\n    simp only [zeroDefaultSupp, List.mem_map, List.mem_filter, exists_and_right,\n      List.mem_toFinset, exists_eq_right, Sigma.exists, Ne, zeroDefault]\n    rw [apply_eq_dlookup]\n    constructor\n    · rintro ⟨od, hval, hod⟩\n      have := List.mem_dlookup (List.nodupKeys_dedupKeys A) hval\n      rw [(_ : List.dlookup a A = od)]\n      · simpa using hod\n      · simpa [List.dlookup_dedupKeys]\n    · intro h\n      use (A.dlookup a).getD (0 : β)\n      rw [← List.dlookup_dedupKeys] at h ⊢\n      simp only [h, ← List.mem_dlookup_iff A.nodupKeys_dedupKeys, not_false_iff, Option.mem_def]\n      cases haA : List.dlookup a A.dedupKeys\n      · simp [haA] at h\n      · simp\n\nvariable [SampleableExt α] [SampleableExt β] [Repr α]\n\ninstance Finsupp.sampleableExt : SampleableExt (α →₀ β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\n-- TODO: support a non-constant codomain type\ninstance DFinsupp.sampleableExt : SampleableExt (Π₀ _ : α, β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp.toDFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\nend Finsupp\nend TotalFunction\n\nopen _root_.List\n\n/-- Data structure specifying a total function using a list of pairs\nand a default value returned when the input is not in the domain of\nthe partial function.\n\n`mapToSelf f` encodes `x ↦ f x` when `x ∈ f` and `x ↦ x`,\ni.e. `x` to itself, otherwise.\n\nWe use `Σ` to encode mappings instead of `×` because we\nrely on the association list API defined in `Mathlib/Data/List/Sigma.lean`.\n-/\ninductive InjectiveFunction (α : Type u) : Type u\n  | mapToSelf (xs : List (Σ _ : α, α)) :\n      xs.map Sigma.fst ~ xs.map Sigma.snd → List.Nodup (xs.map Sigma.snd) → InjectiveFunction α\n\ninstance : Inhabited (InjectiveFunction α) :=\n  ⟨⟨[], List.Perm.nil, List.nodup_nil⟩⟩\n\nnamespace InjectiveFunction\n\n/-- Apply a total function to an argument. -/\ndef apply [DecidableEq α] : InjectiveFunction α → α → α\n  | InjectiveFunction.mapToSelf m _ _, x => (m.dlookup x).getD x\n\n/-- Produce a string for a given `InjectiveFunction`.\nThe output is of the form `[x₀ ↦ f x₀, .. xₙ ↦ f xₙ, x ↦ x]`.\nUnlike for `TotalFunction`, the default value is not a constant\nbut the identity function.\n-/\nprotected def repr [Repr α] : InjectiveFunction α → String\n  | InjectiveFunction.mapToSelf m _ _ => s! \"[{TotalFunction.reprAux m}x ↦ x]\"\n\ninstance (α : Type u) [Repr α] : Repr (InjectiveFunction α) where\n  reprPrec f _p := InjectiveFunction.repr f\n\n/-- Interpret a list of pairs as a total function, defaulting to\nthe identity function when no entries are found for a given function -/\ndef List.applyId [DecidableEq α] (xs : List (α × α)) (x : α) : α :=\n  ((xs.map Prod.toSigma).dlookup x).getD x\n\n@[simp]\n\nTarget:\ntheorem List.applyId_cons [DecidableEq α] (xs : List (α × α)) (x y z : α) :\n    List.applyId ((y, z)::xs) x = if y = x then z else List.applyId xs x :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  simp only [List.applyId, List.dlookup, eq_rec_constant, Prod.toSigma, List.map]\n  split_ifs <;> rfl","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Testing/Plausible","family_id":"list","file_id":"mathlib/Mathlib/Testing/Plausible/Functions.lean","sample_id":"25e33e2ad73b99a239389fcab2e1bb17f4373e01742868a9ff04d6e9bc1274a4"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8debc390e3c9d99e3175632da6d0d7fd484f1961b70777e58cda8dfa8a7eb610","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5f997efe8cebc3ae55d046a4d771e677d58412dbcd0b904a4ae744f3199fc901","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"950388e963953fd7e6eb950b29af28e1f498743a5197639d25cdf72094ea0aae","source_sha256":"127dc67ffbb65bf1977104fb0e1029785baeff1fe0e2b525c55e8f479bde944b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let a := (lfp.comp h).lfp\n  refine (lfp_le _ ?_).antisymm (lfp_le _ (Eq.le ?_))\n  · exact lfp_le _ h.onDiag.map_lfp.le\n  have ha : (lfp ∘ h) a = a := (lfp.comp h).map_lfp\n  calc\n    h a a = h a (h a).lfp := congr_arg (h a) ha.symm\n    _ = (h a).lfp := (h a).map_lfp\n    _ = a := ha","hard_negative":true,"metrics":{"chosen_tokens":105,"rejected_tokens":2,"token_jaccard":0.033333,"token_length_ratio":0.019048},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"f7c1c590285c84f1f83335c16dbe766f0f45022fb2ef6433999fd9a51eb7bda8","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Dynamics.FixedPoints.Basic\npublic import Mathlib.Order.Hom.Order\npublic import Mathlib.Order.BourbakiWitt\n\nNamespace:\nOrderHom\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Kenny Lau, Yury Kudryashov\n-/\n/-!\n# Fixed point construction on complete lattices\n\nThis file sets up the basic theory of fixed points of a monotone function in a complete lattice.\n\n## Main definitions\n\n* `OrderHom.lfp`: The least fixed point of a bundled monotone function.\n* `OrderHom.gfp`: The greatest fixed point of a bundled monotone function.\n* `OrderHom.prevFixed`: The greatest fixed point of a bundled monotone function smaller than or\n  equal to a given element.\n* `OrderHom.nextFixed`: The least fixed point of a bundled monotone function greater than or\n  equal to a given element.\n* `fixedPoints.completeLattice`: The Knaster-Tarski theorem: fixed points of a monotone\n  self-map of a complete lattice form themselves a complete lattice.\n\n## Tags\n\nfixed point, complete lattice, monotone function\n-/\n\n@[expose] public section\n\n\nuniverse u v w\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\nopen Function (fixedPoints IsFixedPt)\n\nnamespace OrderHom\n\nsection Basic\n\nvariable [CompleteLattice α] (f : α →o α)\n\n/-- Least fixed point of a monotone function -/\ndef lfp : (α →o α) →o α where\n  toFun f := sInf { a | f a ≤ a }\n  monotone' _ _ hle := sInf_le_sInf fun a ha => (hle a).trans ha\n\n/-- Greatest fixed point of a monotone function -/\ndef gfp : (α →o α) →o α where\n  toFun f := sSup { a | a ≤ f a }\n  monotone' _ _ hle := sSup_le_sSup fun a ha => le_trans ha (hle a)\n\ntheorem lfp_le {a : α} (h : f a ≤ a) : f.lfp ≤ a :=\n  sInf_le h\n\ntheorem lfp_le_fixed {a : α} (h : f a = a) : f.lfp ≤ a :=\n  f.lfp_le h.le\n\ntheorem le_lfp {a : α} (h : ∀ b, f b ≤ b → a ≤ b) : a ≤ f.lfp :=\n  le_sInf h\n\ntheorem map_le_lfp {a : α} (ha : a ≤ f.lfp) : f a ≤ f.lfp :=\n  f.le_lfp fun _ hb => (f.mono <| le_sInf_iff.1 ha _ hb).trans hb\n\n@[simp]\ntheorem map_lfp : f f.lfp = f.lfp :=\n  have h : f f.lfp ≤ f.lfp := f.map_le_lfp le_rfl\n  h.antisymm <| f.lfp_le <| f.mono h\n\ntheorem isFixedPt_lfp : IsFixedPt f f.lfp :=\n  f.map_lfp\n\ntheorem lfp_le_map {a : α} (ha : f.lfp ≤ a) : f.lfp ≤ f a :=\n  calc\n    f.lfp = f f.lfp := f.map_lfp.symm\n    _ ≤ f a := f.mono ha\n\ntheorem isLeast_lfp_le : IsLeast { a | f a ≤ a } f.lfp :=\n  ⟨f.map_lfp.le, fun _ => f.lfp_le⟩\n\ntheorem isLeast_lfp : IsLeast (fixedPoints f) f.lfp :=\n  ⟨f.isFixedPt_lfp, fun _ => f.lfp_le_fixed⟩\n\ntheorem lfp_induction {p : α → Prop} (step : ∀ a, p a → a ≤ f.lfp → p (f a))\n    (hSup : ∀ s, (∀ a ∈ s, p a) → p (sSup s)) : p f.lfp := by\n  set s := { a | a ≤ f.lfp ∧ p a }\n  specialize hSup s fun a => And.right\n  suffices sSup s = f.lfp from this ▸ hSup\n  have h : sSup s ≤ f.lfp := sSup_le fun b => And.left\n  have hmem : f (sSup s) ∈ s := ⟨f.map_le_lfp h, step _ hSup h⟩\n  exact h.antisymm (f.lfp_le <| le_sSup hmem)\n\ntheorem le_gfp {a : α} (h : a ≤ f a) : a ≤ f.gfp :=\n  le_sSup h\n\ntheorem gfp_le {a : α} (h : ∀ b, b ≤ f b → b ≤ a) : f.gfp ≤ a :=\n  sSup_le h\n\ntheorem isFixedPt_gfp : IsFixedPt f f.gfp :=\n  f.dual.isFixedPt_lfp\n\n@[simp]\ntheorem map_gfp : f f.gfp = f.gfp :=\n  f.dual.map_lfp\n\ntheorem map_le_gfp {a : α} (ha : a ≤ f.gfp) : f a ≤ f.gfp :=\n  f.dual.lfp_le_map ha\n\ntheorem gfp_le_map {a : α} (ha : f.gfp ≤ a) : f.gfp ≤ f a :=\n  f.dual.map_le_lfp ha\n\ntheorem isGreatest_gfp_le : IsGreatest { a | a ≤ f a } f.gfp :=\n  f.dual.isLeast_lfp_le\n\ntheorem isGreatest_gfp : IsGreatest (fixedPoints f) f.gfp :=\n  f.dual.isLeast_lfp\n\ntheorem gfp_induction {p : α → Prop} (step : ∀ a, p a → f.gfp ≤ a → p (f a))\n    (hInf : ∀ s, (∀ a ∈ s, p a) → p (sInf s)) : p f.gfp :=\n  f.dual.lfp_induction step hInf\n\ntheorem lfp_le_gfp : f.lfp ≤ f.gfp :=\n  f.lfp_le_fixed f.isFixedPt_gfp\n\nend Basic\n\nsection Eqn\n\nvariable [CompleteLattice α] [CompleteLattice β] (f : β →o α) (g : α →o β)\n\n-- Rolling rule\ntheorem map_lfp_comp : f (g.comp f).lfp = (f.comp g).lfp :=\n  le_antisymm ((f.comp g).map_lfp ▸ f.mono (lfp_le_fixed _ <| congr_arg g (f.comp g).map_lfp)) <|\n    lfp_le _ (congr_arg f (g.comp f).map_lfp).le\n\ntheorem map_gfp_comp : f (g.comp f).gfp = (f.comp g).gfp :=\n  f.dual.map_lfp_comp g.dual\n\n-- Diagonal rule\n\nTarget:\ntheorem lfp_lfp (h : α →o α →o α) : (lfp.comp h).lfp = h.onDiag.lfp :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_950388e96395","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"4a322d19e4348a86c1a5a35ebda42ac55a661467b647f2033221587fc25f27fb","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order","family_id":"lfp_lfp","file_id":"mathlib/Mathlib/Order/FixedPoints.lean","sample_id":"950388e963953fd7e6eb950b29af28e1f498743a5197639d25cdf72094ea0aae"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7d2f77b8c1bb155392db6b9e50f050fefd652d8cabc59188758f6cc33ac80619","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"bb469c3613f92d8cabf0255e6c012999e3bdcfb6d61f1c8b6b5d3a6dc8c383d5","source_sha256":"3ee6cc92b0085948fb0c591bbd98433993abf27d506b4cc259fc13467d3e2a06","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by convert! aeval_algebraMap_apply A x p","hard_negative":false,"metrics":{"chosen_tokens":7,"rejected_tokens":2,"token_jaccard":0.125,"token_length_ratio":0.285714},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"f7c43f8ec5bcea5f9b6cc654659693f57368d634d9a67c3a9dd47baa7bc0bea5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Algebra.Subalgebra.Tower\npublic import Mathlib.Algebra.MvPolynomial.Eval\n\nNamespace:\nSubalgebra\n\nLocal context:\n/-\nCopyright (c) 2022 Yuyang Zhao. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yuyang Zhao\n-/\n/-!\n# Algebra towers for multivariate polynomial\n\nThis file proves some basic results about the algebra tower structure for the type\n`MvPolynomial σ R`.\n\nThis structure itself is provided elsewhere as `MvPolynomial.isScalarTower`\n\nWhen you update this file, you can also try to make a corresponding update in\n`RingTheory.Polynomial.Tower`.\n-/\n\npublic section\n\n\nvariable (R A B : Type*) {σ : Type*}\n\nnamespace MvPolynomial\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\nvariable [Algebra R A] [Algebra A B] [Algebra R B]\nvariable [IsScalarTower R A B]\nvariable {R B}\n\ntheorem aeval_map_algebraMap (x : σ → B) (p : MvPolynomial σ R) :\n    aeval x (map (algebraMap R A) p) = aeval x p := by\n  rw [aeval_def, aeval_def, eval₂_map, IsScalarTower.algebraMap_eq R A B]\n\nend Semiring\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring A] [CommSemiring B]\nvariable [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B]\nvariable {R A}\n\ntheorem aeval_algebraMap_apply (x : σ → A) (p : MvPolynomial σ R) :\n    aeval (algebraMap A B ∘ x) p = algebraMap A B (MvPolynomial.aeval x p) := by\n  rw [aeval_def, aeval_def, ← coe_eval₂Hom, ← coe_eval₂Hom, map_eval₂Hom, ←\n    IsScalarTower.algebraMap_eq, Function.comp_def]\n\n@[simp]\nlemma aeval_C_comp_left {ι : Type*} (f : σ → A) (p : MvPolynomial σ R) :\n    aeval (C (σ := ι) ∘ f) p = C (aeval f p) :=\n  aeval_algebraMap_apply ..\n\nlemma aeval_algebraMap_eq_zero_iff [IsDomain A] [Module.IsTorsionFree A B] [Nontrivial B]\n    (x : σ → A) (p : MvPolynomial σ R) : aeval (algebraMap A B ∘ x) p = 0 ↔ aeval x p = 0 := by\n  rw [aeval_algebraMap_apply, Algebra.algebraMap_eq_smul_one, smul_eq_zero,\n    iff_false_intro (one_ne_zero' B), or_false]\n\ntheorem aeval_algebraMap_eq_zero_iff_of_injective {x : σ → A} {p : MvPolynomial σ R}\n    (h : Function.Injective (algebraMap A B)) :\n    aeval (algebraMap A B ∘ x) p = 0 ↔ aeval x p = 0 := by\n  rw [aeval_algebraMap_apply, ← (algebraMap A B).map_zero, h.eq_iff]\n\nend CommSemiring\n\nend MvPolynomial\n\nnamespace Subalgebra\n\nopen MvPolynomial\n\nsection CommSemiring\n\nvariable {R A} [CommSemiring R] [CommSemiring A] [Algebra R A]\n\n@[simp]\n\nTarget:\ntheorem mvPolynomial_aeval_coe (S : Subalgebra R A) (x : σ → S) (p : MvPolynomial σ R) :\n    aeval (fun i => (x i : A)) p = aeval x p :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/MvPolynomial","family_id":"mvpolynomial_aeval_coe","file_id":"mathlib/Mathlib/RingTheory/MvPolynomial/Tower.lean","sample_id":"bb469c3613f92d8cabf0255e6c012999e3bdcfb6d61f1c8b6b5d3a6dc8c383d5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"4477bb12e5cc2a166f61746299c4ae29af03f6b7f73cd9a6d3ff697daf2ff447","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"865c2117838a5f9b6bdb0bfba027decbe6b709de52a194b8d1a351aa165f6f08","source_sha256":"42f43c83c3b42913128aa30933a73363c485cf2bc69fc613214486004b504ca0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G\n  rw [← hS1, ← Fintype.card_coe, ← Nat.card_eq_fintype_card, ← Finset.coe_sort_coe, ← Nat.card_fun]\n  exact Finite.card_le_of_embedding (quotientCenterEmbedding hS2)","hard_negative":true,"metrics":{"chosen_tokens":47,"rejected_tokens":8,"token_jaccard":0.057143,"token_length_ratio":0.170213},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"f7e105969fa9d56a65268298a4d23e405deb59bb39ba29d8ae63d2b998dbe0ba","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Subgroup.Finite\npublic import Mathlib.GroupTheory.Commutator.Basic\npublic import Mathlib.GroupTheory.Rank\npublic import Mathlib.GroupTheory.Index\n\nNamespace:\nSubgroup\n\nLocal context:\n/-\nCopyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jordan Brown, Thomas Browning, Patrick Lutz\n-/\n/-!\nThe commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n\npublic section\n\nvariable {G : Type*} [Group G]\n\nnamespace Subgroup\n\n/-- The commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n@[to_additive /-- The commutator of a finite direct product is contained in the direct product of\nthe commutators. -/]\ntheorem commutator_pi_pi_of_finite {η : Type*} [Finite η] {Gs : η → Type*} [∀ i, Group (Gs i)]\n    (H K : ∀ i, Subgroup (Gs i)) : ⁅Subgroup.pi Set.univ H, Subgroup.pi Set.univ K⁆ =\n    Subgroup.pi Set.univ fun i => ⁅H i, K i⁆ := by\n  classical\n    apply le_antisymm (commutator_pi_pi_le H K)\n    rw [pi_le_iff]\n    intro i hi\n    rw [map_commutator]\n    apply commutator_mono <;>\n      · rw [le_pi_iff]\n        intro j _hj\n        rintro _ ⟨x, hx, rfl⟩\n        by_cases h : j = i\n        · subst h\n          simpa using hx\n        · simp [h, one_mem]\n\nvariable [Finite (commutatorSet G)]\n\ninstance : Group.FG (_root_.commutator G) := by\n  rw [commutator_eq_closure]; apply Group.closure_finite_fg\n\nvariable (G) in\nlemma rank_commutator_le_card : Group.rank (_root_.commutator G) ≤ Nat.card (commutatorSet G) := by\n  rw [Subgroup.rank_congr (commutator_eq_closure G)]\n  apply Subgroup.rank_closure_finite_le_nat_card\n\nvariable [Group.FG G]\n\ninstance finiteIndex_center : FiniteIndex (center G) := by\n  obtain ⟨S, -, hS⟩ := Group.rank_spec G\n  exact ⟨mt (Finite.card_eq_zero_of_embedding (quotientCenterEmbedding hS)) Finite.card_pos.ne'⟩\n\nvariable (G) in\n\nTarget:\nlemma index_center_le_pow : (center G).index ≤ Nat.card (commutatorSet G) ^ Group.rank G :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"98fa8cf9cafe448e3703a1d232dbeda941d8919bea439d47bf06547064ed1e1d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Commutator","family_id":"index_center_le_pow","file_id":"mathlib/Mathlib/GroupTheory/Commutator/Finite.lean","sample_id":"865c2117838a5f9b6bdb0bfba027decbe6b709de52a194b8d1a351aa165f6f08"},"split":"train"}
{"binding":{"chosen_proof_sha256":"586f1604059baac7ae4d22c8ed3b6c941cba1beaead4cb818f84462ea38efbce","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"5e149fff8a99e4451822a4357dbc08d6be645f61e746a18516f5e7a6267b5370","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"dce4fa550c062aaa1e0e5d299822fa3b4591439eeb6e8fb744a8c9d9b1a2391e","source_sha256":"ba493885a2be935359e1e232e1bbaeb68ad80f382975839ead7d3190a1fce95f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  simp only [nonempty_iff_ne_empty, Ne, sumLexLift_eq_empty, not_and_or, exists_prop, not_forall]","hard_negative":false,"metrics":{"chosen_tokens":16,"rejected_tokens":21,"token_jaccard":0.705882,"token_length_ratio":1.3125},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"f80340f087667bdd43d67352e1717d4154923556a67335964dbc76814002d0f6","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Finset.Sum\npublic import Mathlib.Data.Sum.Order\npublic import Mathlib.Order.Interval.Finset.Defs\n\nNamespace:\nFinset\n\nLocal context:\n/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\n/-!\n# Finite intervals in a disjoint union\n\nThis file provides the `LocallyFiniteOrder` instance for the disjoint sum and linear sum of two\norders and calculates the cardinality of their finite intervals.\n-/\n\n@[expose] public section\n\n\nopen Function Sum\n\nnamespace Finset\n\nvariable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}\n\nsection SumLift₂\n\nvariable (f f₁ g₁ : α₁ → β₁ → Finset γ₁) (g f₂ g₂ : α₂ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁` and `α₂ → β₂ → Finset γ₂` to a map\n`α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to `Alternative` functors if we can\nmake sure to keep computability and universe polymorphism. -/\n@[simp]\ndef sumLift₂ : ∀ (_ : α₁ ⊕ α₂) (_ : β₁ ⊕ β₂), Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f a b).map Embedding.inl\n  | inl _, inr _ => ∅\n  | inr _, inl _ => ∅\n  | inr a, inr b => (g a b).map Embedding.inr\n\nvariable {f f₁ g₁ g f₂ g₂} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\ntheorem mem_sumLift₂ :\n    c ∈ sumLift₂ f g a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨\n        ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g a₂ b₂ := by\n  constructor\n  · rcases a with a | a <;> rcases b with b | b\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · refine fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLift₂, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr ⟨a, b, c, rfl, rfl, rfl, hc⟩\n  · rintro (⟨a, b, c, rfl, rfl, rfl, h⟩ | ⟨a, b, c, rfl, rfl, rfl, h⟩) <;> exact mem_map_of_mem _ h\n\ntheorem inl_mem_sumLift₂ {c₁ : γ₁} :\n    inl c₁ ∈ sumLift₂ f g a b ↔ ∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f a₁ b₁ := by\n  rw [mem_sumLift₂, or_iff_left]\n  · simp only [inl.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inl_ne_inr h\n\ntheorem inr_mem_sumLift₂ {c₂ : γ₂} :\n    inr c₂ ∈ sumLift₂ f g a b ↔ ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ g a₂ b₂ := by\n  rw [mem_sumLift₂, or_iff_right]\n  · simp only [inr.injEq, exists_and_left, exists_eq_left']\n  rintro ⟨_, _, c₂, _, _, h, _⟩\n  exact inr_ne_inl h\n\ntheorem sumLift₂_eq_empty :\n    sumLift₂ f g a b = ∅ ↔\n      (∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f a₁ b₁ = ∅) ∧\n        ∀ a₂ b₂, a = inr a₂ → b = inr b₂ → g a₂ b₂ = ∅ := by\n  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩\n  · constructor <;>\n    · rintro a b rfl rfl\n      exact map_eq_empty.1 h\n  cases a <;> cases b\n  · exact map_eq_empty.2 (h.1 _ _ rfl rfl)\n  · rfl\n  · rfl\n  · exact map_eq_empty.2 (h.2 _ _ rfl rfl)\n\ntheorem sumLift₂_nonempty :\n    (sumLift₂ f g a b).Nonempty ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ (f a₁ b₁).Nonempty) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ (g a₂ b₂).Nonempty := by\n  simp only [nonempty_iff_ne_empty, Ne, sumLift₂_eq_empty, not_and_or, not_forall, exists_prop]\n\ntheorem sumLift₂_mono (h₁ : ∀ a b, f₁ a b ⊆ g₁ a b) (h₂ : ∀ a b, f₂ a b ⊆ g₂ a b) :\n    ∀ a b, sumLift₂ f₁ f₂ a b ⊆ sumLift₂ g₁ g₂ a b\n  | inl _, inl _ => map_subset_map.2 (h₁ _ _)\n  | inl _, inr _ => Subset.rfl\n  | inr _, inl _ => Subset.rfl\n  | inr _, inr _ => map_subset_map.2 (h₂ _ _)\n\nend SumLift₂\n\nsection SumLexLift\nvariable (f₁ f₁' : α₁ → β₁ → Finset γ₁) (f₂ f₂' : α₂ → β₂ → Finset γ₂)\n  (g₁ g₁' : α₁ → β₂ → Finset γ₁) (g₂ g₂' : α₁ → β₂ → Finset γ₂)\n\n/-- Lifts maps `α₁ → β₁ → Finset γ₁`, `α₂ → β₂ → Finset γ₂`, `α₁ → β₂ → Finset γ₁`,\n`α₂ → β₂ → Finset γ₂`  to a map `α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)`. Could be generalized to\nalternative monads if we can make sure to keep computability and universe polymorphism. -/\ndef sumLexLift : α₁ ⊕ α₂ → β₁ ⊕ β₂ → Finset (γ₁ ⊕ γ₂)\n  | inl a, inl b => (f₁ a b).map Embedding.inl\n  | inl a, inr b => (g₁ a b).disjSum (g₂ a b)\n  | inr _, inl _ => ∅\n  | inr a, inr b => (f₂ a b).map ⟨_, inr_injective⟩\n\n@[simp]\nlemma sumLexLift_inl_inl (a : α₁) (b : β₁) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inl b) = (f₁ a b).map Embedding.inl := rfl\n\n@[simp]\nlemma sumLexLift_inl_inr (a : α₁) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b) = (g₁ a b).disjSum (g₂ a b) := rfl\n\n@[simp]\nlemma sumLexLift_inr_inl (a : α₂) (b : β₁) : sumLexLift f₁ f₂ g₁ g₂ (inr a) (inl b) = ∅ := rfl\n\n@[simp]\nlemma sumLexLift_inr_inr (a : α₂) (b : β₂) :\n    sumLexLift f₁ f₂ g₁ g₂ (inr a) (inr b) = (f₂ a b).map ⟨_, inr_injective⟩ := rfl\n\nvariable {f₁ g₁ f₂ g₂ f₁' g₁' f₂' g₂'} {a : α₁ ⊕ α₂} {b : β₁ ⊕ β₂} {c : γ₁ ⊕ γ₂}\n\nlemma mem_sumLexLift :\n    c ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        (∃ a₁ b₂ c₁, a = inl a₁ ∧ b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨\n          (∃ a₁ b₂ c₂, a = inl a₁ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n            ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂ := by\n  constructor\n  · obtain a | a := a <;> obtain b | b := b\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩\n    · refine fun h ↦ (mem_disjSum.1 h).elim ?_ ?_\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n      · rintro ⟨c, hc, rfl⟩\n        exact Or.inr (Or.inr <| Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact fun h ↦ (notMem_empty _ h).elim\n    · rw [sumLexLift, mem_map]\n      rintro ⟨c, hc, rfl⟩\n      exact Or.inr (Or.inr <| Or.inr <| ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n  · rintro (⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩ |\n      ⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩)\n    · exact mem_map_of_mem _ hc\n    · exact inl_mem_disjSum.2 hc\n    · exact inr_mem_disjSum.2 hc\n    · exact mem_map_of_mem _ hc\n\nlemma inl_mem_sumLexLift {c₁ : γ₁} :\n    inl c₁ ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n        ∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ c₁ ∈ g₁ a₁ b₂ := by\n  simp [mem_sumLexLift]\n\nlemma inr_mem_sumLexLift {c₂ : γ₂} :\n    inr c₂ ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔\n      (∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n        ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ f₂ a₂ b₂ := by\n  simp [mem_sumLexLift]\n\nlemma sumLexLift_mono (hf₁ : ∀ a b, f₁ a b ⊆ f₁' a b) (hf₂ : ∀ a b, f₂ a b ⊆ f₂' a b)\n    (hg₁ : ∀ a b, g₁ a b ⊆ g₁' a b) (hg₂ : ∀ a b, g₂ a b ⊆ g₂' a b) (a : α₁ ⊕ α₂)\n    (b : β₁ ⊕ β₂) : sumLexLift f₁ f₂ g₁ g₂ a b ⊆ sumLexLift f₁' f₂' g₁' g₂' a b := by\n  cases a <;> cases b\n  exacts [map_subset_map.2 (hf₁ _ _), disjSum_mono (hg₁ _ _) (hg₂ _ _), Subset.rfl,\n    map_subset_map.2 (hf₂ _ _)]\n\nlemma sumLexLift_eq_empty :\n    sumLexLift f₁ f₂ g₁ g₂ a b = ∅ ↔\n      (∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f₁ a₁ b₁ = ∅) ∧\n        (∀ a₁ b₂, a = inl a₁ → b = inr b₂ → g₁ a₁ b₂ = ∅ ∧ g₂ a₁ b₂ = ∅) ∧\n          ∀ a₂ b₂, a = inr a₂ → b = inr b₂ → f₂ a₂ b₂ = ∅ := by\n  refine ⟨fun h ↦ ⟨?_, ?_, ?_⟩, fun h ↦ ?_⟩\n  any_goals rintro a b rfl rfl; exact map_eq_empty.1 h\n  · rintro a b rfl rfl; exact disjSum_eq_empty.1 h\n  cases a <;> cases b\n  · exact map_eq_empty.2 (h.1 _ _ rfl rfl)\n  · simp [h.2.1 _ _ rfl rfl]\n  · rfl\n  · exact map_eq_empty.2 (h.2.2 _ _ rfl rfl)\n\nTarget:\nlemma sumLexLift_nonempty :\n    (sumLexLift f₁ f₂ g₁ g₂ a b).Nonempty ↔\n      (∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ (f₁ a₁ b₁).Nonempty) ∨\n        (∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ ((g₁ a₁ b₂).Nonempty ∨ (g₂ a₁ b₂).Nonempty)) ∨\n          ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ (f₂ a₂ b₂).Nonempty :=\n\nProof body:\n","rejected":"by\n  simp only [nonempty_iff_ne_empty, Ne, sumLexLift_eq_empty, not_and_or, exists_prop, not_forall]\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Sum","family_id":"sumlexlift_nonempty","file_id":"mathlib/Mathlib/Data/Sum/Interval.lean","sample_id":"dce4fa550c062aaa1e0e5d299822fa3b4591439eeb6e8fb744a8c9d9b1a2391e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"124731cfc6276098aa30620798da6c884c01af47d833522f931a5e1fb2b79484","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"45556f8bd9370bd1415c420e560973ba88e4c44babfc2defddcf5c8f39aa00bb","source_sha256":"394c3d8954491754c0a03071ce7aa65ddc93c4783581cf0b4d6869361547187f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_algEquiv_matrix_end_mulOpposite R₀ R\n  have := Module.Finite.equiv e.toLinearEquiv\n  classical exact ⟨n, hn, _, _, _, .of_surjective\n    (Matrix.entryLinearMap R₀ _ (0 : Fin n) (0 : Fin n)) fun f ↦ ⟨fun _ _ ↦ f, rfl⟩, ⟨e⟩⟩","hard_negative":false,"metrics":{"chosen_tokens":82,"rejected_tokens":5,"token_jaccard":0.085714,"token_length_ratio":0.060976},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"f815cab88a19f118c9634447de5186f39605f1536ea4e446c458e07f4c650198","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.FreeModule.Finite.Basic\npublic import Mathlib.RingTheory.FiniteLength\npublic import Mathlib.RingTheory.SimpleModule.Isotypic\npublic import Mathlib.RingTheory.SimpleRing.Congr\npublic import Mathlib.RingTheory.SimpleRing.Matrix\n\nNamespace:\nIsSimpleRing\n\nLocal context:\n/-\nCopyright (c) 2025 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu\n-/\n/-!\n# Wedderburn–Artin Theorem\n\n## Main results\n\n* `IsSimpleRing.tfae`: a simple ring is semisimple iff it is Artinian,\n  iff it has a minimal left ideal.\n\n* `isSimpleRing_isArtinianRing_iff`: a ring is simple Artinian iff it is semisimple, isotypic,\n  and nontrivial.\n\n* `IsSimpleRing.exists_algEquiv_matrix_end_mulOpposite`: a simple Artinian algebra is\n  isomorphic to a (finite-dimensional) matrix algebra over a division algebra. The division\n  algebra is the opposite of the endomorphism algebra of a simple (i.e., minimal) left ideal.\n\n* `IsSemisimpleRing.exists_algEquiv_pi_matrix_end_mulOpposite`: a semisimple algebra is\n  isomorphic to a finite direct product of matrix algebras over division algebras. The division\n  algebras are the opposites of the endomorphism algebras of the simple (i.e., minimal)\n  left ideals.\n\n* `IsSimpleRing.exists_algEquiv_matrix_divisionRing_finite`,\n  `IsSemisimpleRing.exists_algEquiv_pi_matrix_divisionRing_finite`:\n  if the simple Artinian / semisimple algebra is finite as a module over a base ring, then the\n  division algebra(s) are also finite over the same ring.\n  If the base ring is an algebraically closed field, the only finite-dimensional division algebra\n  over it is itself, and we obtain `IsSimpleRing.exists_algEquiv_matrix_of_isAlgClosed` and\n  `IsSemisimpleRing.exists_algEquiv_pi_matrix_of_isAlgClosed` (in a later file).\n\n-/\n\npublic section\n\nuniverse u\nvariable (R₀ : Type*) {R : Type u} [CommSemiring R₀] [Ring R] [Algebra R₀ R]\n\n/-- A simple ring is semisimple iff it is Artinian, iff it has a minimal left ideal. -/\ntheorem IsSimpleRing.tfae [IsSimpleRing R] : List.TFAE\n    [IsSemisimpleRing R, IsArtinianRing R, ∃ I : Ideal R, IsAtom I] := by\n  tfae_have 1 → 2 := fun _ ↦ inferInstance\n  tfae_have 2 → 3 := fun _ ↦ IsAtomic.exists_atom _\n  tfae_have 3 → 1 := fun ⟨I, hI⟩ ↦ by\n    have ⟨_, h⟩ := isSimpleRing_iff_isTwoSided_imp.mp ‹IsSimpleRing R›\n    simp_rw [← isFullyInvariant_iff_isTwoSided] at h\n    have := isSimpleModule_iff_isAtom.mpr hI\n    obtain eq | eq := h _ (.isotypicComponent R R I)\n    · exact (hI.bot_lt.not_ge <| (le_sSup <| by exact ⟨.refl ..⟩).trans_eq eq).elim\n    exact .congr (.symm <| .trans (.ofEq _ _ eq) Submodule.topEquiv)\n  tfae_finish\n\ntheorem IsSimpleRing.isSemisimpleRing_iff_isArtinianRing [IsSimpleRing R] :\n    IsSemisimpleRing R ↔ IsArtinianRing R := tfae.out 0 1\n\ntheorem isSimpleRing_isArtinianRing_iff :\n    IsSimpleRing R ∧ IsArtinianRing R ↔ IsSemisimpleRing R ∧ IsIsotypic R R ∧ Nontrivial R := by\n  refine ⟨fun ⟨_, _⟩ ↦ ?_, fun ⟨_, _, _⟩ ↦ ?_⟩\n  on_goal 1 => have := IsSimpleRing.isSemisimpleRing_iff_isArtinianRing.mpr ‹_›\n  all_goals simp_rw [isIsotypic_iff_isFullyInvariant_imp_bot_or_top,\n      isFullyInvariant_iff_isTwoSided, isSimpleRing_iff_isTwoSided_imp] at *\n  · exact ⟨this, by rwa [and_comm]⟩\n  · exact ⟨⟨‹_›, ‹_›⟩, inferInstance⟩\n\nnamespace IsSimpleRing\n\nvariable (R) [IsSimpleRing R] [IsArtinianRing R]\n\ninstance (priority := low) : IsSemisimpleRing R :=\n  (isSimpleRing_isArtinianRing_iff.mp ⟨‹_›, ‹_›⟩).1\n\ntheorem isIsotypic (M) [AddCommGroup M] [Module R M] : IsIsotypic R M :=\n  (isSimpleRing_isArtinianRing_iff.mp ⟨‹_›, ‹_›⟩).2.1.of_self M\n\n/-- The **Wedderburn–Artin Theorem**: an Artinian simple ring is isomorphic to a matrix\nring over the opposite of the endomorphism ring of its simple module. -/\ntheorem exists_ringEquiv_matrix_end_mulOpposite :\n    ∃ (n : ℕ) (_ : NeZero n) (I : Ideal R) (_ : IsSimpleModule R I),\n      Nonempty (R ≃+* Matrix (Fin n) (Fin n) (Module.End R I)ᵐᵒᵖ) := by\n  have ⟨n, hn, S, hS, ⟨e⟩⟩ := (isIsotypic R R).linearEquiv_fun\n  refine ⟨n, hn, S, hS, ⟨.trans (.opOp R) <| .trans (.op ?_) (.symm .mopMatrix)⟩⟩\n  exact .trans (.moduleEndSelf R) <| .trans e.conjRingEquiv (endVecRingEquivMatrixEnd ..)\n\n/-- The **Wedderburn–Artin Theorem**: an Artinian simple ring is isomorphic to a matrix\nring over a division ring. -/\ntheorem exists_ringEquiv_matrix_divisionRing :\n    ∃ (n : ℕ) (_ : NeZero n) (D : Type u) (_ : DivisionRing D),\n      Nonempty (R ≃+* Matrix (Fin n) (Fin n) D) := by\n  have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_ringEquiv_matrix_end_mulOpposite R\n  classical exact ⟨n, hn, _, _, ⟨e⟩⟩\n\n/-- The **Wedderburn–Artin Theorem**, algebra form: an Artinian simple algebra is isomorphic\nto a matrix algebra over the opposite of the endomorphism algebra of its simple module. -/\ntheorem exists_algEquiv_matrix_end_mulOpposite :\n    ∃ (n : ℕ) (_ : NeZero n) (I : Ideal R) (_ : IsSimpleModule R I),\n      Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) (Module.End R I)ᵐᵒᵖ) := by\n  have ⟨n, hn, S, hS, ⟨e⟩⟩ := (isIsotypic R R).linearEquiv_fun\n  refine ⟨n, hn, S, hS, ⟨.trans (.opOp R₀ R) <| .trans (.op ?_) (.symm .mopMatrix)⟩⟩\n  exact .trans (.moduleEndSelf R₀) <| .trans (e.conjAlgEquiv R₀) (endVecAlgEquivMatrixEnd ..)\n\n/-- The **Wedderburn–Artin Theorem**, algebra form: an Artinian simple algebra is isomorphic\nto a matrix algebra over a division algebra. -/\ntheorem exists_algEquiv_matrix_divisionRing :\n    ∃ (n : ℕ) (_ : NeZero n) (D : Type u) (_ : DivisionRing D) (_ : Algebra R₀ D),\n      Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) D) := by\n  have ⟨n, hn, I, _, ⟨e⟩⟩ := exists_algEquiv_matrix_end_mulOpposite R₀ R\n  classical exact ⟨n, hn, _, _, _, ⟨e⟩⟩\n\n/-- The **Wedderburn–Artin Theorem**, algebra form, finite case: a finite Artinian simple algebra is\nisomorphic to a matrix algebra over a finite division algebra. -/\n\nTarget:\ntheorem exists_algEquiv_matrix_divisionRing_finite [Module.Finite R₀ R] :\n    ∃ (n : ℕ) (_ : NeZero n) (D : Type u) (_ : DivisionRing D) (_ : Algebra R₀ D)\n      (_ : Module.Finite R₀ D), Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) D) :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/SimpleModule","family_id":"exists_algequiv_matrix_divisionring_finite","file_id":"mathlib/Mathlib/RingTheory/SimpleModule/WedderburnArtin.lean","sample_id":"45556f8bd9370bd1415c420e560973ba88e4c44babfc2defddcf5c8f39aa00bb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"443eea2a8757775ee8a442c763e06d326c2fd7f65ba73927f6537e2b3fa75567","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"36013d3b030a4beac323ba03c7b15bdae8a250c801f46d872bcc359cdf27248b","source_sha256":"73244aa9467f7c855deb57de598d12f743787aaa6e228fe2bbfaffba906d38dc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [finite_iff_subgroup_quotient]; constructor <;> assumption","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":2,"token_jaccard":0.090909,"token_length_ratio":0.181818},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"f82250c17f89948d17caf20341ed0704ec3639cb2aa7420fd5082651d3394538","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Subgroup.Finite\npublic import Mathlib.Data.Finite.Prod\npublic import Mathlib.GroupTheory.QuotientGroup.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Kevin Buzzard, Patrick Massot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Patrick Massot\n-/\n-- This file is to a certain extent based on `quotient_module.lean` by Johannes Hölzl.\n/-!\n# Deducing finiteness of a group.\n-/\n\n@[expose] public section\n\nopen Function QuotientGroup Subgroup\nopen scoped Pointwise\n\n\nvariable {F G H : Type*} [Group F] [Group G] [Group H] [Fintype F] [Fintype H]\nvariable (f : F →* G) (g : G →* H)\n\nnamespace Group\n\nopen scoped Classical in\n/-- If `F` and `H` are finite such that `ker(G →* H) ≤ im(F →* G)`, then `G` is finite. -/\n@[to_additive (attr := implicit_reducible)\n/-- If `F` and `H` are finite such that `ker(G →+ H) ≤ im(F →+ G)`, then `G` is finite. -/]\nnoncomputable def fintypeOfKerLeRange (h : g.ker ≤ f.range) : Fintype G :=\n  @Fintype.ofEquiv _ _\n    (@instFintypeProd _ _ (Fintype.ofInjective _ <| kerLift_injective g) <|\n      Fintype.ofInjective _ <| inclusion_injective h)\n    groupEquivQuotientProdSubgroup.symm\n\n/-- If `F` and `H` are finite such that `ker(G →* H) = im(F →* G)`, then `G` is finite. -/\n@[to_additive (attr := implicit_reducible)\n/-- If `F` and `H` are finite such that `ker(G →+ H) = im(F →+ G)`, then `G` is finite. -/]\nnoncomputable def fintypeOfKerEqRange (h : g.ker = f.range) : Fintype G :=\n  fintypeOfKerLeRange _ _ h.le\n\n/-- If `ker(G →* H)` and `H` are finite, then `G` is finite. -/\n@[to_additive (attr := implicit_reducible)\n  /-- If `ker(G →+ H)` and `H` are finite, then `G` is finite. -/]\nnoncomputable def fintypeOfKerOfCodom [Fintype g.ker] : Fintype G :=\n  fintypeOfKerLeRange ((topEquiv : _ ≃* G).toMonoidHom.comp <| inclusion le_top) g fun x hx =>\n    ⟨⟨x, hx⟩, rfl⟩\n\n/-- If `F` and `coker(F →* G)` are finite, then `G` is finite. -/\n@[to_additive (attr := implicit_reducible)\n  /-- If `F` and `coker(F →+ G)` are finite, then `G` is finite. -/]\nnoncomputable def fintypeOfDomOfCoker [Normal f.range] [Fintype <| G ⧸ f.range] : Fintype G :=\n  fintypeOfKerLeRange _ (mk' f.range) fun x => (eq_one_iff x).mp\n\nend Group\n\n@[to_additive]\nlemma finite_iff_subgroup_quotient (H : Subgroup G) : Finite G ↔ Finite H ∧ Finite (G ⧸ H) := by\n  rw [(groupEquivQuotientProdSubgroup (s := H)).finite_iff, Prod.finite_iff, and_comm]\n\n@[to_additive]\n\nTarget:\nlemma Finite.of_subgroup_quotient (H : Subgroup G) [Finite H] [Finite (G ⧸ H)] : Finite G :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/QuotientGroup","family_id":"finite","file_id":"mathlib/Mathlib/GroupTheory/QuotientGroup/Finite.lean","sample_id":"36013d3b030a4beac323ba03c7b15bdae8a250c801f46d872bcc359cdf27248b"},"split":"train"}
{"binding":{"chosen_proof_sha256":"a139748a86235708cf34b19e320c41795f248777d1dc57cebe0b73665a686a07","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"39ec33afe785bb2145068683a69a98c599dd9ded25a579902bb0e5911710d977","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9844a70ca9aa095c0c679f810bb6057aadb712cb3e2289c8ec909fdd876bd5a3","source_sha256":"6ea826f0633d3930f7872110f571f16f76059ec3668e205ef02a1089b9b31873","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  constructor\n  · exact HallMarriageTheorem.hall_hard_inductive\n  · rintro ⟨f, hf₁, hf₂⟩ s\n    rw [← card_image_of_injective s hf₁]\n    apply card_le_card\n    intro\n    rw [mem_image, mem_biUnion]\n    rintro ⟨x, hx, rfl⟩\n    exact ⟨x, hx, hf₂ x⟩","hard_negative":false,"metrics":{"chosen_tokens":54,"rejected_tokens":61,"token_jaccard":0.935484,"token_length_ratio":1.12963},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"f86219efb5cb4f802277f4645c3e4b06de51196cbb103a37732db46c7ad96573","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Fintype.Basic\npublic import Mathlib.Data.Fintype.Powerset\npublic import Mathlib.Data.Set.Finite.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Alena Gusakov, Bhavik Mehta, Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alena Gusakov, Bhavik Mehta, Kyle Miller\n-/\n/-!\n# Hall's Marriage Theorem for finite index types\n\nThis module proves the basic form of Hall's theorem.\nIn contrast to the theorem described in `Combinatorics.Hall.Basic`, this\nversion requires that the indexed family `t : ι → Finset α` have `ι` be finite.\nThe `Combinatorics.Hall.Basic` module applies a compactness argument to this version\nto remove the `Finite` constraint on `ι`.\n\nThe modules are split like this since the generalized statement\ndepends on the topology and category theory libraries, but the finite\ncase in this module has few dependencies.\n\nA description of this formalization is in [Gusakov2021].\n\n## Main statements\n\n* `Finset.all_card_le_biUnion_card_iff_existsInjective'` is Hall's theorem with\n  a finite index set.  This is elsewhere generalized to\n  `Finset.all_card_le_biUnion_card_iff_existsInjective`.\n\n## Tags\n\nHall's Marriage Theorem, indexed families\n-/\n\npublic section\n\n\nopen Finset\n\nuniverse u v\n\nnamespace HallMarriageTheorem\n\nvariable {ι : Type u} {α : Type v} [DecidableEq α] {t : ι → Finset α}\n\nsection Fintype\n\nvariable [Fintype ι]\n\nset_option backward.isDefEq.respectTransparency false in\ntheorem hall_cond_of_erase {x : ι} (a : α)\n    (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t))\n    (s' : Finset { x' : ι | x' ≠ x }) : #s' ≤ #(s'.biUnion fun x' => (t x').erase a) := by\n  haveI := Classical.decEq ι\n  specialize ha (s'.image fun z => z.1)\n  rw [image_nonempty, Finset.card_image_of_injective s' Subtype.coe_injective] at ha\n  by_cases! he : s'.Nonempty\n  · have ha' : #s' < #(s'.biUnion fun x => t x) := by\n      convert! ha he fun h => by simpa [← h] using mem_univ x using 2\n      ext x\n      simp only [mem_image, mem_biUnion, SetCoe.exists, exists_and_right,\n        exists_eq_right]\n    rw [← erase_biUnion]\n    by_cases hb : a ∈ s'.biUnion fun x => t x\n    · rw [card_erase_of_mem hb]\n      exact Nat.le_sub_one_of_lt ha'\n    · rw [erase_eq_of_notMem hb]\n      exact Nat.le_of_lt ha'\n  · subst s'\n    simp\n\n/-- First case of the inductive step: assuming that\n`∀ (s : Finset ι), s.Nonempty → s ≠ univ → #s < #(s.biUnion t)`\nand that the statement of **Hall's Marriage Theorem** is true for all\n`ι'` of cardinality ≤ `n`, then it is true for `ι` of cardinality `n + 1`.\n-/\ntheorem hall_hard_inductive_step_A {n : ℕ} (hn : Fintype.card ι = n + 1)\n    (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))\n    (ih :\n      ∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),\n        Fintype.card ι' ≤ n →\n          (∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →\n            ∃ f : ι' → α, Function.Injective f ∧ ∀ x, f x ∈ t' x)\n    (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t)) :\n    ∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by\n  haveI : Nonempty ι := Fintype.card_pos_iff.mp (hn.symm ▸ Nat.succ_pos _)\n  haveI := Classical.decEq ι\n  -- Choose an arbitrary element `x : ι` and `y : t x`.\n  let x := Classical.arbitrary ι\n  have tx_ne : (t x).Nonempty := by\n    rw [← Finset.card_pos]\n    calc\n      0 < 1 := Nat.one_pos\n      _ ≤ #(.biUnion {x} t) := ht {x}\n      _ = (t x).card := by rw [Finset.singleton_biUnion]\n  choose y hy using tx_ne\n  -- Restrict to everything except `x` and `y`.\n  let ι' := { x' : ι | x' ≠ x }\n  let t' : ι' → Finset α := fun x' => (t x').erase y\n  have card_ι' : Fintype.card ι' = n :=\n    calc\n      Fintype.card ι' = Fintype.card ι - 1 := Set.card_ne_eq _\n      _ = n := by rw [hn, Nat.add_succ_sub_one, add_zero]\n  rcases ih t' card_ι'.le (hall_cond_of_erase y ha) with ⟨f', hfinj, hfr⟩\n  -- Extend the resulting function.\n  refine ⟨fun z => if h : z = x then y else f' ⟨z, h⟩, ?_, ?_⟩\n  · rintro z₁ z₂\n    have key : ∀ {x}, y ≠ f' x := by\n      intro x h\n      simpa [t', ← h] using hfr x\n    by_cases h₁ : z₁ = x <;> by_cases h₂ : z₂ = x <;>\n      simp [h₁, h₂, hfinj.eq_iff, key, key.symm]\n  · intro z\n    simp only\n    split_ifs with hz\n    · rwa [hz]\n    · specialize hfr ⟨z, hz⟩\n      rw [mem_erase] at hfr\n      exact hfr.2\n\ntheorem hall_cond_of_restrict {ι : Type u} {t : ι → Finset α} {s : Finset ι}\n    (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t)) (s' : Finset (s : Set ι)) :\n    #s' ≤ #(s'.biUnion fun a' => t a') := by\n  classical\n    rw [← card_image_of_injective s' Subtype.coe_injective]\n    convert! ht (s'.image fun z => z.1) using 1\n    apply congr_arg\n    ext y\n    simp\n\ntheorem hall_cond_of_compl {ι : Type u} {t : ι → Finset α} {s : Finset ι}\n    (hus : #s = #(s.biUnion t)) (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))\n    (s' : Finset (sᶜ : Set ι)) : #s' ≤ #(s'.biUnion fun x' => t x' \\ s.biUnion t) := by\n  haveI := Classical.decEq ι\n  have disj : Disjoint s (s'.image fun z => z.1) := by\n    simp only [disjoint_left, not_exists, mem_image, SetCoe.exists, exists_and_right,\n      exists_eq_right]\n    intro x hx hc _\n    exact absurd hx hc\n  have : #s' = #(s ∪ s'.image fun z => z.1) - #s := by\n    simp [disj, card_image_of_injective _ Subtype.coe_injective, Nat.add_sub_cancel_left]\n  rw [this, hus]\n  refine (Nat.sub_le_sub_right (ht _) _).trans ?_\n  rw [← card_sdiff_of_subset]\n  · gcongr\n    intro t\n    simp only [mem_biUnion, mem_sdiff, not_exists, mem_image, and_imp, mem_union,\n      exists_imp]\n    rintro x (hx | ⟨x', hx', rfl⟩) rat hs\n    · exact False.elim <| (hs x) <| And.intro hx rat\n    · use x', hx', rat, hs\n  · apply biUnion_subset_biUnion_of_subset_left\n    apply subset_union_left\n\n/-- Second case of the inductive step: assuming that\n`∃ (s : Finset ι), s ≠ univ → #s = #(s.biUnion t)`\nand that the statement of **Hall's Marriage Theorem** is true for all\n`ι'` of cardinality ≤ `n`, then it is true for `ι` of cardinality `n + 1`.\n-/\ntheorem hall_hard_inductive_step_B {n : ℕ} (hn : Fintype.card ι = n + 1)\n    (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))\n    (ih :\n      ∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),\n        Fintype.card ι' ≤ n →\n          (∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →\n            ∃ f : ι' → α, Function.Injective f ∧ ∀ x, f x ∈ t' x)\n    (s : Finset ι) (hs : s.Nonempty) (hns : s ≠ univ) (hus : #s = #(s.biUnion t)) :\n    ∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by\n  haveI := Classical.decEq ι\n  -- Restrict to `s`\n  rw [Nat.add_one] at hn\n  have card_ι'_le : Fintype.card s ≤ n := by\n    apply Nat.le_of_lt_succ\n    calc\n      Fintype.card s = #s := Fintype.card_coe _\n      _ < Fintype.card ι := (card_lt_iff_ne_univ _).mpr hns\n      _ = n.succ := hn\n  let t' : s → Finset α := fun x' => t x'\n  rcases ih t' card_ι'_le (hall_cond_of_restrict ht) with ⟨f', hf', hsf'⟩\n  -- Restrict to `sᶜ` in the domain and `(s.biUnion t)ᶜ` in the codomain.\n  set ι'' := (s : Set ι)ᶜ\n  let t'' : ι'' → Finset α := fun a'' => t a'' \\ s.biUnion t\n  have card_ι''_le : Fintype.card ι'' ≤ n := by\n    simp_rw [ι'', ← Nat.lt_succ_iff, ← hn, ← Finset.coe_compl, coe_sort_coe]\n    rwa [Fintype.card_coe, card_compl_lt_iff_nonempty]\n  rcases ih t'' card_ι''_le (hall_cond_of_compl hus ht) with ⟨f'', hf'', hsf''⟩\n  -- Put them together\n  have f''_notMem_biUnion : ∀ (x'') (hx'' : x'' ∉ s), f'' ⟨x'', hx''⟩ ∉ s.biUnion t := by\n    intro x'' hx''\n    have h := hsf'' ⟨x'', hx''⟩\n    rw [mem_sdiff] at h\n    exact h.2\n  have im_disj :\n      ∀ (x' x'' : ι) (hx' : x' ∈ s) (hx'' : x'' ∉ s), f' ⟨x', hx'⟩ ≠ f'' ⟨x'', hx''⟩ := by\n    grind\n  refine ⟨fun x => if h : x ∈ s then f' ⟨x, h⟩ else f'' ⟨x, h⟩, ?_, ?_⟩\n  · refine hf'.dite _ hf'' (@fun x x' => im_disj x x' _ _)\n  · intro x\n    simp only\n    split_ifs with h\n    · exact hsf' ⟨x, h⟩\n    · exact sdiff_subset (hsf'' ⟨x, h⟩)\n\nend Fintype\n\nvariable [Finite ι]\n\n/-- Here we combine the two inductive steps into a full strong induction proof,\ncompleting the proof the harder direction of **Hall's Marriage Theorem**.\n-/\ntheorem hall_hard_inductive (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t)) :\n    ∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by\n  cases nonempty_fintype ι\n  generalize hn : Fintype.card ι = m\n  induction m using Nat.strongRecOn generalizing ι with | ind n ih => _\n  rcases n with (_ | n)\n  · rw [Fintype.card_eq_zero_iff] at hn\n    exact ⟨isEmptyElim, isEmptyElim, isEmptyElim⟩\n  · have ih' : ∀ (ι' : Type u) [Fintype ι'] (t' : ι' → Finset α), Fintype.card ι' ≤ n →\n        (∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →\n        ∃ f : ι' → α, Function.Injective f ∧ ∀ x, f x ∈ t' x := by\n      intro ι' _ _ hι' ht'\n      exact ih _ (Nat.lt_succ_of_le hι') ht' _ rfl\n    by_cases! h : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t)\n    · refine hall_hard_inductive_step_A hn ht (@fun ι' => ih' ι') h\n    · rcases h with ⟨s, sne, snu, sle⟩\n      exact hall_hard_inductive_step_B hn ht (@fun ι' => ih' ι')\n        s sne snu (Nat.le_antisymm (ht _) sle)\n\nend HallMarriageTheorem\n\n/-- This is the version of **Hall's Marriage Theorem** in terms of indexed\nfamilies of finite sets `t : ι → Finset α` with `ι` finite.\nIt states that there is a set of distinct representatives if and only\nif every union of `k` of the sets has at least `k` elements.\n\nSee `Finset.all_card_le_biUnion_card_iff_exists_injective` for a version\nwhere the `Finite ι` constraint is removed.\n-/\n\nTarget:\ntheorem Finset.all_card_le_biUnion_card_iff_existsInjective' {ι α : Type*} [Finite ι]\n    [DecidableEq α] (t : ι → Finset α) :\n    (∀ s : Finset ι, #s ≤ #(s.biUnion t)) ↔\n      ∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x :=\n\nProof body:\n","rejected":"```lean\nby\n  constructor\n  · exact HallMarriageTheorem.hall_hard_inductive\n  · rintro ⟨f, hf₁, hf₂⟩ s\n    rw [← card_image_of_injective s hf₁]\n    apply card_le_card\n    intro\n    rw [mem_image, mem_biUnion]\n    rintro ⟨x, hx, rfl⟩\n    exact ⟨x, hx, hf₂ x⟩\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Combinatorics/Hall","family_id":"finset","file_id":"mathlib/Mathlib/Combinatorics/Hall/Finite.lean","sample_id":"9844a70ca9aa095c0c679f810bb6057aadb712cb3e2289c8ec909fdd876bd5a3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b663b020e08f48e798d48b7ce39be8da0dce1601bdd6b1231b5d9860f49a1922","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9464b08287d95c785301c54058651202fff8047c0ca4e3764eb7ffff3e6775f3","source_sha256":"c6fb2ee19236e152d132e9c3d3ef383faafb8161f5d12e957a193a47621136bd","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by lia) :\n    t.IsGE ((t.truncGT a).obj X) b :=\n  t.isGE_truncGE_obj ..","hard_negative":true,"metrics":{"chosen_tokens":25,"rejected_tokens":8,"token_jaccard":0.157895,"token_length_ratio":0.32},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"f879a15aa1fd44809244ddacb0c5a2137fe57cafd36dc9e1ce36b967a4b6d1e3","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE\n\nNamespace:\nCategoryTheory.Triangulated.TStructure\n\nLocal context:\n/-\nCopyright (c) 2026 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Truncations for a t-structure\n\nLet `t` be a t-structure on a (pre)triangulated category `C`.\nIn this file, for any `n : ℤ`, we introduce the truncation functors\n`t.truncLE n : C ⥤ C` and `t.truncGT n : C ⥤ C`, as variants of the functors\n`t.truncLT n : C ⥤ C` and `t.truncGE n : C ⥤ C` introduced in the file\n`Mathlib/CategoryTheory/Triangulated/TStructure/TruncLTGE.lean`.\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Limits Pretriangulated\n\nvariable {C : Type*} [Category* C] [Preadditive C] [HasZeroObject C] [HasShift C ℤ]\n  [∀ (n : ℤ), (shiftFunctor C n).Additive] [Pretriangulated C]\n\nnamespace Triangulated\n\nnamespace TStructure\n\nvariable (t : TStructure C)\n\n/-- Given a t-structure `t` on a pretriangulated category `C` and `n : ℤ`, this\nis the `≤ n`-truncation functor. See also the natural transformation `truncLEι`. -/\nnoncomputable def truncLE (n : ℤ) : C ⥤ C := t.truncLT (n + 1)\n\ninstance (n : ℤ) : (t.truncLE n).Additive := by\n  dsimp only [truncLE]\n  infer_instance\n\nlemma isLE_truncLE_obj (X : C) (a b : ℤ) (hn : a ≤ b := by lia) :\n    t.IsLE ((t.truncLE a).obj X) b :=\n  t.isLE_truncLT_obj ..\n\ninstance (n : ℤ) (X : C) : t.IsLE ((t.truncLE n).obj X) n :=\n  t.isLE_truncLE_obj ..\n\n/-- Given a t-structure `t` on a pretriangulated category `C` and `n : ℤ`, this\nis the `> n`-truncation functor. See also the natural transformation `truncGTπ`. -/\nnoncomputable def truncGT (n : ℤ) : C ⥤ C := t.truncGE (n + 1)\n\ninstance (n : ℤ) : (t.truncGT n).Additive := by\n  dsimp only [truncGT]\n  infer_instance\n\nTarget:\nlemma isGE_truncGT_obj (X : C) (a b : ℤ) (hn : b ≤ a + 1 :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"9c6a0dc3a4f168e3df602a1d8fa72c34ca98e1f1b7ae8523235e26ce7a412097","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Triangulated","family_id":"isge_truncgt_obj","file_id":"mathlib/Mathlib/CategoryTheory/Triangulated/TStructure/TruncLEGT.lean","sample_id":"9464b08287d95c785301c54058651202fff8047c0ca4e3764eb7ffff3e6775f3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"12e284d615394feaae0bf91b6d2469b44129b24d28aa683dddb8dd6ad90c1ad2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"141f039ee25caf569389190297e8e170fe2b5f957ffbed666e13fe2928910cf7","source_sha256":"31a5542d1ac83b668eb669f80f15da1c85ba792b90d38c379ac11ec99cc62c14","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  grw [hv.apply_sum_le, ← cbiSup_eq_of_forall (by grind)]\n  simp","hard_negative":false,"metrics":{"chosen_tokens":15,"rejected_tokens":3,"token_jaccard":0.0625,"token_length_ratio":0.2},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"f89146c860e936c5e0a0e744a34533baf0266ca5b6a3bd117d28ae143a87c11d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Homogenize\npublic import Mathlib.NumberTheory.Height.Basic\nimport Mathlib.Algebra.Order.Ring.IsNonarchimedean\nimport Mathlib.Data.Fintype.Order\nimport all Mathlib.NumberTheory.Height.Basic\n\nNamespace:\nIsNonarchimedean\n\nLocal context:\n/-\nCopyright (c) 2026 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\n/-!\n# Height bounds for linear and polynomial maps\n\nWe prove an upper bound for the height of the image of a tuple under a linear map.\n\nWe also prove upper and lower bounds for the height of `fun i ↦ eval P i x`, where `P` is a family\nof homogeneous polynomials over the field `K` of the same degree `N` and `x : ι → K`\nwith `ι` finite.\n-/\n\npublic section\n\nsection aux\n\nprivate lemma Height.iSup_fun_eq_max (f : Fin 2 → ℝ) : iSup f = max (f 0) (f 1) := by\n  rw [show f = ![f 0, f 1] from List.ofFn_inj.mp rfl]\n  exact (max_eq_iSup ..).symm\n\nnamespace IsNonarchimedean\n\nvariable {R α β F : Type*} [CommRing R] [AddCommMonoid β] [FunLike F β ℝ] [NonnegHomClass F β ℝ]\n    [ZeroHomClass F β ℝ] {v : F} {l : α → β}\n\n-- NOTE: The following cannot be moved to Mathlib.Algebra.Order.Ring.IsNonarchimedean,\n--       because it needs the target to be the reals (to have the default value zero\n--       for empty iSups), which are not known there.\n/-- The ultrametric triangle inequality for finite sums. -/\nlemma apply_sum_le (hv : IsNonarchimedean v) {s : Finset α} :\n    v (∑ i ∈ s, l i) ≤ ⨆ i : s, v (l i) := by\n  classical\n  induction s using Finset.induction with\n  | empty => simp\n  | insert a s ha ih =>\n    rw [Finset.sum_insert ha]\n    grw [hv .., ih]\n    refine max_le ?_ ?_\n    · exact Finite.le_ciSup_of_le ⟨_, s.mem_insert_self a⟩ le_rfl\n    · rcases isEmpty_or_nonempty s with hs | hs\n      · simpa using Real.iSup_nonneg_of_nonnegHomClass v _\n      exact ciSup_le fun i ↦ Finite.le_ciSup_of_le (⟨i.val, Finset.mem_insert_of_mem i.prop⟩) le_rfl\n\n/-- The ultrametric triangle inequality for finite sums. -/\n\nTarget:\nlemma apply_sum_univ_le [Fintype α] (hv : IsNonarchimedean v) :\n    v (∑ i, l i) ≤ ⨆ i, v (l i) :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/Height","family_id":"apply_sum_univ_le","file_id":"mathlib/Mathlib/NumberTheory/Height/MvPolynomial.lean","sample_id":"141f039ee25caf569389190297e8e170fe2b5f957ffbed666e13fe2928910cf7"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f187198a893dd0403a98bdeb7c5bab4c0cada1bb485a8d70e6ce244cdd6c05b3","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"10754817056ac7fe2728b7d3c57dd409b9884a07365c423b530430a8a51d4d7e","source_sha256":"42a1a9a755579f25de77a56470f5e17c3dfcdee299a3d64bf9907f705df67f49","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Icc_union_Ioc_eq_Icc le_rfl h, VectorMeasure.of_union (by simp)\n    measurableSet_Icc measurableSet_Ioc, Icc_self, hf.vectorMeasure_singleton,\n    hf.vectorMeasure_Ioc h]\n  simp","hard_negative":true,"metrics":{"chosen_tokens":30,"rejected_tokens":8,"token_jaccard":0.037037,"token_length_ratio":0.266667},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"f8d6d9b9b9181fcf10e6e196868d71b213e17a119fd74afa6ff28879397af82e","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Normed.Group.Defs\npublic import Mathlib.MeasureTheory.Measure.Stieltjes\npublic import Mathlib.MeasureTheory.VectorMeasure.Basic\npublic import Mathlib.Topology.EMetricSpace.BoundedVariation\nimport Mathlib.MeasureTheory.VectorMeasure.AddContent\n\nNamespace:\nBoundedVariationOn\n\nLocal context:\n/-\nCopyright (c) 2026 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Vector valued Stieltjes measure associated to a bounded variation function\n\nLet `α` be a dense linear order with compact segments (e.g. `ℝ` or `ℝ≥0`), and `f : α → E` a\nbounded variation function taking values in a complete additive normed group.\nWe associate to `f` a vector measure, called `BoundedVariationOn.vectorMeasure`. It gives\nmass `f.rightLim b - f.leftLim a` to the interval `[a, b]` (with similar formulas for\nother types of intervals).\n\nFor the construction, we define first an additive content on the set semiring of open-closed\nintervals `(a, b]`, mapping this interval to `f.rightLim b - f.rightLim a`. To extend this content\nto the whole sigma-algebra, by general extension theorems, it is enough to show that it is\ndominated by a finite measure. For this, we can use the Stieltjes measure associated to the\nvariation of `f.rightLim`. The extension we get is not exactly the desired vector measure, as we\nneed to tweak things if there is a bot element `a`: the previous vector measure gives to `{a}` the\nmass `0` instead of the desired `f.rightLim a - f a`, so we add a Dirac mass to correct this defect.\n-/\n\n@[expose] public section\n\nopen Filter Set MeasureTheory MeasurableSpace MeasureTheory\nopen scoped symmDiff Topology NNReal ENNReal\n\nvariable {α : Type*} [LinearOrder α] [DenselyOrdered α] [TopologicalSpace α] [OrderTopology α]\n  [SecondCountableTopology α] [CompactIccSpace α] [hα : MeasurableSpace α] [BorelSpace α]\n  {E : Type*} [NormedAddCommGroup E] [CompleteSpace E]\n  {f : α → E} {a b : α}\n\nnamespace BoundedVariationOn\n\n/-- The Stieltjes function associated to a bounded variation function. It is given by\nthe variation of the function `f.rightLim` from a fixed base point.\nUsing right limits ensures the right continuity, which is used to construct Stieltjes measures. -/\n@[simps] noncomputable def stieltjesFunctionRightLim\n    (hf : BoundedVariationOn f univ) (x₀ : α) : StieltjesFunction α where\n  toFun x := variationOnFromTo f.rightLim univ x₀ x\n  mono' := by\n    rw [← monotoneOn_univ]\n    exact variationOnFromTo.monotoneOn hf.rightLim.locallyBoundedVariationOn (mem_univ _)\n  right_continuous' x := hf.continuousWithinAt_variationOnFromTo_rightLim_Ici\n\nopen scoped Classical in\n/-- Auxiliary measure used to construct the vector measure associated to a bounded variation\nfunction. This is *not* the total variation of this measure in general, as we need to adjust things\nwhen there is a bot element by adding a Dirac mass there. -/\nprivate noncomputable def measureAux (hf : BoundedVariationOn f univ) : Measure α :=\n  if h : Nonempty α then (hf.stieltjesFunctionRightLim h.some).measure else 0\n\nprivate instance (hf : BoundedVariationOn f univ) : IsFiniteMeasure hf.measureAux := by\n  by_cases h : Nonempty α; swap\n  · simp only [BoundedVariationOn.measureAux, h, ↓reduceDIte]\n    infer_instance\n  simp only [BoundedVariationOn.measureAux, h, ↓reduceDIte]\n  apply StieltjesFunction.isFiniteMeasure_of_forall_abs_le\n    (C := (eVariationOn f.rightLim univ).toReal) _ (fun x ↦ ?_)\n  exact variationOnFromTo.abs_le_eVariationOn hf.rightLim\n\n/-- Given a bounded variation function `f`, we can construct a vector measure giving\nmass `f.rightLim v - f.rightLim a` to each open-closed interval `(a, b]`. This is *not* the\nmeasure associated to `f` in general, as we may need to adjust things at the bot element if\nthere is one. -/\nprivate lemma exists_vectorMeasure_le_measureAux (hf : BoundedVariationOn f univ) :\n    ∃ m : VectorMeasure α E, (∀ u v, u ≤ v → m (Set.Ioc u v) = f.rightLim v - f.rightLim u) ∧\n      m botSet = 0 ∧ ∀ s, ‖m s‖ₑ ≤ hf.measureAux s := by\n  /- We will apply the general extension theorem\n  `VectorMeasure.exists_extension_of_isSetSemiring_of_le_measure_of_generateFrom`. For this, we\n  need to check that the additive content is bounded by the measure `measureAux`. -/\n  rcases isEmpty_or_nonempty α with h'α | h'α\n  · exact ⟨0, by simp⟩\n  let m := AddContent.onIoc f.rightLim\n  have A : ∀ s ∈ {s | ∃ u v, u ≤ v ∧ s = Ioc u v}, ‖m s‖ₑ ≤ hf.measureAux s := by\n    rintro s ⟨u, v, huv, rfl⟩\n    rw [AddContent.onIoc_apply huv]\n    simp only [BoundedVariationOn.measureAux, h'α, ↓reduceDIte, StieltjesFunction.measure_Ioc,\n      BoundedVariationOn.stieltjesFunctionRightLim_apply]\n    rw [← variationOnFromTo.add hf.rightLim.locallyBoundedVariationOn\n      (mem_univ h'α.some) (mem_univ u) (mem_univ v)]\n    simp only [add_sub_cancel_left, variationOnFromTo, huv, ↓reduceIte, univ_inter]\n    rw [ENNReal.ofReal_toReal]; swap\n    · exact ((eVariationOn.mono _ (subset_univ _)).trans_lt hf.rightLim.lt_top).ne\n    rw [← edist_eq_enorm_sub]\n    exact eVariationOn.edist_le _ (by grind) (by grind)\n  have B : hα = generateFrom {s | ∃ u v, u ≤ v ∧ s = Ioc u v} := by\n    borelize α\n    convert! borel_eq_generateFrom_Ioc_le α using 2\n    grind only\n  rcases VectorMeasure.exists_extension_of_isSetSemiring_of_le_measure_of_generateFrom\n    IsSetSemiring.Ioc A B with ⟨m', hm', h'm'⟩\n  refine ⟨m', fun u v huv ↦ ?_, ?_, h'm'⟩\n  · rw [hm']\n    · exact AddContent.onIoc_apply huv\n    · exact ⟨u, v, huv, rfl⟩\n  · apply enorm_eq_zero.1\n    apply le_bot_iff.1\n    exact (h'm' _).trans (by simp [measureAux, h'α])\n\nopen scoped Classical in\n/-- The vector measure associated to a bounded variation function `f`, giving mass\n`f.rightLim b - f.leftLim a` to closed intervals `[a, b]`, and similarly for other intervals. -/\n@[no_expose] noncomputable def vectorMeasure (hf : BoundedVariationOn f univ) : VectorMeasure α E :=\n  hf.exists_vectorMeasure_le_measureAux.choose +\n  (if h : ∃ x, IsBot x then VectorMeasure.dirac h.choose (f.rightLim h.choose - f h.choose) else 0)\n\nlemma vectorMeasure_Ioc (hf : BoundedVariationOn f univ) (h : a ≤ b) :\n    hf.vectorMeasure (Ioc a b) = f.rightLim b - f.rightLim a := by\n  classical\n  have A : hf.exists_vectorMeasure_le_measureAux.choose (Ioc a b) =\n      f.rightLim b - f.rightLim a :=\n    hf.exists_vectorMeasure_le_measureAux.choose_spec.1 a b h\n  have B : (if hx : ∃ (x : α), IsBot x then VectorMeasure.dirac hx.choose\n      (f.rightLim hx.choose - f hx.choose) else 0) (Ioc a b) = 0 := by\n    by_cases hx : ∃ (x : α), IsBot x\n    · simp only [hx, ↓reduceDIte]\n      rw [VectorMeasure.dirac_apply_of_notMem]\n      simp only [mem_Ioc, not_and_or, not_lt, not_le]\n      exact Or.inl (hx.choose_spec _)\n    · simp [hx]\n  simp [vectorMeasure, A, B]\n\nlemma vectorMeasure_singleton (hf : BoundedVariationOn f univ) :\n    hf.vectorMeasure {a} = f.rightLim a - f.leftLim a := by\n  by_cases ha : IsBot a\n  · have h : ∃ x, IsBot x := ⟨a, ha⟩\n    have heqa : h.choose = a := subsingleton_isBot _ h.choose_spec ha\n    have A : hf.exists_vectorMeasure_le_measureAux.choose {a} = 0 := by\n      rw [← botSet_eq_singleton_of_isBot ha]\n      exact hf.exists_vectorMeasure_le_measureAux.choose_spec.2.1\n    simp only [vectorMeasure, h, ↓reduceDIte, add_apply, A, zero_add]\n    rw [VectorMeasure.dirac_apply_of_mem (MeasurableSet.singleton a)]\n    · simpa only [heqa, sub_right_inj] using (leftLim_eq_of_isBot ha).symm\n    · simp [heqa]\n  obtain ⟨b, hb⟩ : ∃ b, b < a := by simpa only [IsBot, not_forall, not_le] using ha\n  obtain ⟨u, u_mono, u_lt_a, u_lim⟩ :\n      ∃ u : ℕ → α, StrictMono u ∧ (∀ n : ℕ, u n ∈ Ioo b a) ∧ Tendsto u atTop (𝓝 a) :=\n    exists_seq_strictMono_tendsto' hb\n  replace u_lt_a n : u n < a := (u_lt_a n).2\n  have A : {a} = ⋂ n, Ioc (u n) a := by\n    refine Subset.antisymm (fun x hx => by simp [mem_singleton_iff.1 hx, u_lt_a]) fun x hx => ?_\n    replace hx : ∀ (i : ℕ), u i < x ∧ x ≤ a := by simpa using hx\n    have : a ≤ x := le_of_tendsto' u_lim fun n => (hx n).1.le\n    simp [le_antisymm this (hx 0).2]\n  have L1 : Tendsto (fun n ↦ hf.vectorMeasure (Ioc (u n) a)) atTop (𝓝 (hf.vectorMeasure {a})) := by\n    rw [A]\n    apply VectorMeasure.tendsto_vectorMeasure_iInter_atTop_nat ?_ (fun n ↦ measurableSet_Ioc)\n    exact fun m n hmn ↦ Ioc_subset_Ioc_left (u_mono.monotone hmn)\n  have L2 : Tendsto (fun n ↦ hf.vectorMeasure (Ioc (u n) a)) atTop\n      (𝓝 (f.rightLim a - f.leftLim a)) := by\n    simp_rw [hf.vectorMeasure_Ioc (u_lt_a _).le]\n    apply tendsto_const_nhds.sub\n    have : Tendsto u atTop (𝓝[<] a) := tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _\n      u_lim (Eventually.of_forall u_lt_a)\n    convert! (hf.rightLim.tendsto_leftLim a).comp this using 2\n    have : (𝓝[<] a).NeBot := by\n      rw [← mem_closure_iff_nhdsWithin_neBot, closure_Iio' ⟨b, hb⟩]\n      exact self_mem_Iic\n    exact (leftLim_rightLim (hf.tendsto_leftLim _)).symm\n  exact tendsto_nhds_unique L1 L2\n\nTarget:\nlemma vectorMeasure_Icc (hf : BoundedVariationOn f univ) (h : a ≤ b) :\n    hf.vectorMeasure (Icc a b) = f.rightLim b - f.leftLim a :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"df73b8bbb0e4a4d0b09eb285144a2fa3a4a6171e552c5bfa210247f2dc6df676","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/VectorMeasure","family_id":"vectormeasure_icc","file_id":"mathlib/Mathlib/MeasureTheory/VectorMeasure/BoundedVariation.lean","sample_id":"10754817056ac7fe2728b7d3c57dd409b9884a07365c423b530430a8a51d4d7e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3486bfbcb341bbc6602e3e4c5334d8af2187c1a15d03540967ecccbb4983efe2","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"72a5ec820d1616bed6e09b9298c3d6ba25ecc6e9436ace533e8b768bc56b26f2","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9547247b6d603996c4b3fb961825dcb69ba77ace42e69a17a558d84380f3ec34","source_sha256":"a4ded0c37af8d7d4bd1dca4a7e378a31082efe29498bab67ebba55f071776588","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← cancel_mono (mapBifunctorLeftUnitor F X e p hp Y').inv, assoc, assoc, Iso.hom_inv_id,\n    comp_id, mapBifunctorLeftUnitor_inv_naturality, Iso.hom_inv_id_assoc]","hard_negative":true,"metrics":{"chosen_tokens":33,"rejected_tokens":3,"token_jaccard":0.038462,"token_length_ratio":0.090909},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"f8f44157683393ee5042bf128f54df081332a78f92c001cb0f2544184424efac","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.GradedObject.Associator\npublic import Mathlib.CategoryTheory.GradedObject.Single\n\nNamespace:\nCategoryTheory.GradedObject\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# The left and right unitors\n\nGiven a bifunctor `F : C ⥤ D ⥤ D`, an object `X : C` such that `F.obj X ≅ 𝟭 D` and a\nmap `p : I × J → J` such that `hp : ∀ (j : J), p ⟨0, j⟩ = j`,\nwe define an isomorphism of `J`-graded objects for any `Y : GradedObject J D`.\n`mapBifunctorLeftUnitor F X e p hp Y : mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`.\nUnder similar assumptions, we also obtain a right unitor isomorphism\n`mapBifunctorMapObj F p X ((single₀ I).obj Y) ≅ X`. Finally,\nthe lemma `mapBifunctor_triangle` promotes a triangle identity involving functors\nto a triangle identity for the induced functors on graded objects.\n\n-/\n\n@[expose] public section\n\nnamespace CategoryTheory\n\nopen Category Limits\n\nnamespace GradedObject\n\nsection LeftUnitor\n\nvariable {C D I J : Type*} [Category* C] [Category* D]\n  [Zero I] [DecidableEq I] [HasInitial C]\n  (F : C ⥤ D ⥤ D) (X : C) (e : F.obj X ≅ 𝟭 D)\n  [∀ (Y : D), PreservesColimit (Functor.empty.{0} C) (F.flip.obj Y)]\n  (p : I × J → J) (hp : ∀ (j : J), p ⟨0, j⟩ = j)\n  (Y Y' : GradedObject J D) (φ : Y ⟶ Y')\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D` and `Y : GradedObject J D`,\nthis is the isomorphism `((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a ≅ Y a.2`\nwhen `a : I × J` is such that `a.1 = 0`. -/\n@[simps!]\nnoncomputable def mapBifunctorObjSingle₀ObjIso (a : I × J) (ha : a.1 = 0) :\n    ((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a ≅ Y a.2 :=\n  (F.mapIso (singleObjApplyIsoOfEq _ X _ ha)).app _ ≪≫ e.app (Y a.2)\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C` and `Y : GradedObject J D`,\n`((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a` is an initial object\nwhen `a : I × J` is such that `a.1 ≠ 0`. -/\nnoncomputable def mapBifunctorObjSingle₀ObjIsInitial (a : I × J) (ha : a.1 ≠ 0) :\n    IsInitial (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y a) :=\n  IsInitial.isInitialObj (F.flip.obj (Y a.2)) _ (isInitialSingleObjApply _ _ _ ha)\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D`, `Y : GradedObject J D` and\n`p : I × J → J` such that `p ⟨0, j⟩ = j` for all `j`,\nthis is the (colimit) cofan which shall be used to construct the isomorphism\n`mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`, see `mapBifunctorLeftUnitor`. -/\nnoncomputable def mapBifunctorLeftUnitorCofan (hp : ∀ (j : J), p ⟨0, j⟩ = j) (Y) (j : J) :\n    (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y).CofanMapObjFun p j :=\n  CofanMapObjFun.mk _ _ _ (Y j) (fun a ha =>\n    if ha : a.1 = 0 then\n      (mapBifunctorObjSingle₀ObjIso F X e Y a ha).hom ≫ eqToHom (by aesop)\n    else\n      (mapBifunctorObjSingle₀ObjIsInitial F X Y a ha).to _)\n\nset_option backward.defeqAttrib.useBackward true in\n@[simp, reassoc]\nlemma mapBifunctorLeftUnitorCofan_inj (j : J) :\n    (mapBifunctorLeftUnitorCofan F X e p hp Y j).inj ⟨⟨0, j⟩, hp j⟩ =\n      (F.map (singleObjApplyIso (0 : I) X).hom).app (Y j) ≫ e.hom.app (Y j) := by\n  simp [mapBifunctorLeftUnitorCofan]\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The cofan `mapBifunctorLeftUnitorCofan F X e p hp Y j` is a colimit. -/\nnoncomputable def mapBifunctorLeftUnitorCofanIsColimit (j : J) :\n    IsColimit (mapBifunctorLeftUnitorCofan F X e p hp Y j) :=\n  Cofan.IsColimit.mk _\n    (fun s => e.inv.app (Y j) ≫\n      (F.map (singleObjApplyIso (0 : I) X).inv).app (Y j) ≫ s.inj ⟨⟨0, j⟩, hp j⟩)\n    (fun s => by\n      rintro ⟨⟨i, j'⟩, h⟩\n      by_cases hi : i = 0\n      · subst hi\n        simp only [Set.mem_preimage, hp, Set.mem_singleton_iff] at h\n        subst h\n        simp\n      · apply IsInitial.hom_ext\n        exact mapBifunctorObjSingle₀ObjIsInitial _ _ _ _ hi)\n    (fun s m hm => by simp [← hm ⟨⟨0, j⟩, hp j⟩])\n\ninclude e hp in\nlemma mapBifunctorLeftUnitor_hasMap :\n    HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y) p :=\n  CofanMapObjFun.hasMap _ _ _ (mapBifunctorLeftUnitorCofanIsColimit F X e p hp Y)\n\nvariable [HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y) p]\n  [HasMap (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y') p]\n\n/-- Given `F : C ⥤ D ⥤ D`, `X : C`, `e : F.obj X ≅ 𝟭 D`, `Y : GradedObject J D` and\n`p : I × J → J` such that `p ⟨0, j⟩ = j` for all `j`,\nthis is the left unitor isomorphism `mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y`. -/\nnoncomputable def mapBifunctorLeftUnitor : mapBifunctorMapObj F p ((single₀ I).obj X) Y ≅ Y :=\n  isoMk _ _ (fun j => (CofanMapObjFun.iso\n    (mapBifunctorLeftUnitorCofanIsColimit F X e p hp Y j)).symm)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma ι_mapBifunctorLeftUnitor_hom_apply (j : J) :\n    ιMapBifunctorMapObj F p ((single₀ I).obj X) Y 0 j j (hp j) ≫\n      (mapBifunctorLeftUnitor F X e p hp Y).hom j =\n      (F.map (singleObjApplyIso (0 : I) X).hom).app _ ≫ e.hom.app (Y j) := by\n  dsimp [mapBifunctorLeftUnitor]\n  erw [CofanMapObjFun.ιMapObj_iso_inv]\n  rw [mapBifunctorLeftUnitorCofan_inj]\n\nlemma mapBifunctorLeftUnitor_inv_apply (j : J) :\n    (mapBifunctorLeftUnitor F X e p hp Y).inv j =\n      e.inv.app (Y j) ≫ (F.map (singleObjApplyIso (0 : I) X).inv).app (Y j) ≫\n      ιMapBifunctorMapObj F p ((single₀ I).obj X) Y 0 j j (hp j) := rfl\n\nvariable {Y Y'}\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc]\nlemma mapBifunctorLeftUnitor_inv_naturality :\n    φ ≫ (mapBifunctorLeftUnitor F X e p hp Y').inv =\n      (mapBifunctorLeftUnitor F X e p hp Y).inv ≫ mapBifunctorMapMap F p (𝟙 _) φ := by\n  ext j\n  dsimp\n  rw [mapBifunctorLeftUnitor_inv_apply, mapBifunctorLeftUnitor_inv_apply, assoc, assoc,\n    ι_mapBifunctorMapMap]\n  dsimp\n  rw [Functor.map_id, NatTrans.id_app, id_comp, ← NatTrans.naturality_assoc,\n    ← NatTrans.naturality_assoc]\n  rfl\n\n@[reassoc]\n\nTarget:\nlemma mapBifunctorLeftUnitor_naturality :\n    mapBifunctorMapMap F p (𝟙 _) φ ≫ (mapBifunctorLeftUnitor F X e p hp Y').hom =\n      (mapBifunctorLeftUnitor F X e p hp Y).hom ≫ φ :=\n\nProof body:\n","rejected":"by\n  exact mapBifunctorLeftUnitor_naturality","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"33d3809e37dd3a56fc6ca3d07e6c90b503d3768752892ce82eb43fc0bb79dd1c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/GradedObject","family_id":"mapbifunctorleftunitor_naturality","file_id":"mathlib/Mathlib/CategoryTheory/GradedObject/Unitor.lean","sample_id":"9547247b6d603996c4b3fb961825dcb69ba77ace42e69a17a558d84380f3ec34"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8690ef82dc2fef54f22a4e762da7bbf0fe4e6499a977029b3fad0c51781aa987","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"18212381e9e2d5955dd328bc6ac67fa9f7e70c531634d30c6e4c7970fcb22b77","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"29db24e141a0e3e320d003b92675922c58df588b3780d126668224314302b43a","source_sha256":"b77307aadbea1488a93479741afc5f9d1f9e6198ff638b8b6035869012641ebc","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  rw [ofDirectSumEquiv]\n  convert! ofDFinsuppEquiv_symm_single_tprod p x","hard_negative":false,"metrics":{"chosen_tokens":11,"rejected_tokens":18,"token_jaccard":0.846154,"token_length_ratio":1.636364},"negative_category":"markdown_preamble_trailing","negative_mode":"fence","pair_id":"f91f5235496e941fe9cfd5258fb9aa7c4921cff1c1cdc81164fb61f3f0bfc997","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.PiTensorProduct.Basic\npublic import Mathlib.LinearAlgebra.PiTensorProduct.DFinsupp\npublic import Mathlib.Algebra.DirectSum.Module\n\nNamespace:\nPiTensorProduct\n\nLocal context:\n/-\nCopyright (c) 2024 Sophie Morel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sophie Morel, Eric Wieser\n-/\n/-!\n# Tensor products of direct sums\n\nThis file shows that taking `PiTensorProduct`s commutes with taking `DirectSum`s in all arguments.\n\n## Main results\n\n* `ofDirectSumEquiv`: the linear equivalence between a `PiTensorProduct` of `DirectSum`s\n  and the `DirectSum` of the `PiTensorProduct`s.\n-/\n\n@[expose] public section\n\nnamespace PiTensorProduct\n\nopen PiTensorProduct DirectSum TensorProduct\n\nvariable {R ι : Type*} {κ : ι → Type*} {M : (i : ι) → κ i → Type*}\n  [CommSemiring R] [Π i (j : κ i), AddCommMonoid (M i j)] [Π i (j : κ i), Module R (M i j)]\n\nopen scoped Classical in\n/-- The n-ary tensor product distributes over m-ary direct sums. -/\nnoncomputable def ofDirectSumEquiv [Finite ι] :\n    (⨂[R] i, (⨁ j : κ i, M i j)) ≃ₗ[R] ⨁ p : Π i, κ i, ⨂[R] i, M i (p i) :=\n  have : Fintype ι := Fintype.ofFinite ι\n  ofDFinsuppEquiv\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\ntheorem ofDirectSumEquiv_tprod_lof [Fintype ι] [(i : ι) → DecidableEq (κ i)]\n    (p : Π i, κ i) (x : Π i, M i (p i)) :\n    ofDirectSumEquiv (⨂ₜ[R] i, DirectSum.lof R _ _ (p i) (x i)) =\n      DirectSum.lof R _ _ p (⨂ₜ[R] i, x i) := by\n  classical\n  rw [ofDirectSumEquiv]\n  convert! ofDFinsuppEquiv_tprod_single p x\n\nset_option backward.isDefEq.respectTransparency false in\n@[simp]\n\nTarget:\ntheorem ofDirectSumEquiv_symm_lof_tprod [Fintype ι] [(i : ι) → DecidableEq (κ i)]\n    (p : Π i, κ i) (x : Π i, M i (p i)) :\n    ofDirectSumEquiv.symm (DirectSum.lof R _ _ p (tprod R x)) =\n      (⨂ₜ[R] i, DirectSum.lof R _ _ (p i) (x i)) :=\n\nProof body:\n","rejected":"```lean\nby\n  classical\n  rw [ofDirectSumEquiv]\n  convert! ofDFinsuppEquiv_symm_single_tprod p x\n```","rejection":{"accepted":false,"all_failure_types":["markdown_fence","not_exact_by_body"],"failure_type":"markdown_fence","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/PiTensorProduct","family_id":"ofdirectsumequiv_symm_lof_tprod","file_id":"mathlib/Mathlib/LinearAlgebra/PiTensorProduct/DirectSum.lean","sample_id":"29db24e141a0e3e320d003b92675922c58df588b3780d126668224314302b43a"},"split":"train"}
{"binding":{"chosen_proof_sha256":"bb5993596ab6f0322c25fd89be3f755c85b89f6e4d2f8ad57eb17900f2915564","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"9c73f768c6b902af55b2843988f629261dd4578a5cd1d557b1db1ff4ef22ed0b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4aa461ab228ed2d6d40d0634d68fa0428d1672b0b417d2b773b3a8831f202fa2","source_sha256":"522c0739e8f7535bf0f954aff339b473dbd0f9e903c0b5a9a4fff87d391d66b3","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext; simp only [add_def, smul_def, _root_.add_smul]","hard_negative":false,"metrics":{"chosen_tokens":14,"rejected_tokens":2,"token_jaccard":0.071429,"token_length_ratio":0.142857},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorry","pair_id":"f950f96c8666dc6791b0d4b648f84ceed4268882edc1bf156aa7b80594cd5bdd","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.TensorProduct.RightExactness\npublic import Mathlib.RingTheory.Congruence.Hom\npublic import Mathlib.RingTheory.FiniteType\npublic import Mathlib.RingTheory.TensorProduct.DirectLimitFG\n\nNamespace:\nPolynomialLaw\n\nLocal context:\n/-\nCopyright (c) 2025 Antoine Chambert-Loir & María-Inés de Frutos-Fernández. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Chambert-Loir & María-Inés de Frutos-Fernández\n-/\n/-! # Polynomial laws on modules\n\nLet `M` and `N` be a modules over a commutative ring `R`.\nA polynomial law `f : PolynomialLaw R M N`, with notation `f : M →ₚₗₗ[R] N`,\nis a “law” that assigns a natural map `PolynomialLaw.toFun' f S : S ⊗[R] M → S ⊗[R] N`\nfor every `R`-algebra `S`.\n\nFor type-theoretic reasons, if `R : Type u`, then the definition of the polynomial map `f`\nis restricted to `R`-algebras `S` such that `S : Type u`.\nUsing the fact that a module is the direct limit of its finitely generated submodules, that a\nfinitely generated subalgebra is a quotient of a polynomial ring in the universe `u`, plus\nthe commutation of tensor products with direct limits, we extend the functor\nto all `R`-algebras.\n\nThe two fields involving the definition of `PolynomialLaw`,\n`PolynomialLaw.toFun'` and `PolynomialLaw.isCompat'` are primed.\nThey are superseded by their universe-polymorphic counterparts,\nthe definition `PolynomialLaw.toFun` and the lemma `PolynomialLaw.isCompat`\nwhich should be used once the theory is properly stated.\n\nFor constructions of general definitions of `PolynomialLaw`\nat a universe-polymorphic level, one needs to lift\nelements in a tensor product to smaller universes.\nFor this, one can make use of\n`PolynomialLaw.exists_lift` or `PolynomialLaw.exists_lift'`,\nor establish appropriate generalizations.\n\n## Main definitions/lemmas\n\n* Instance : `Module R (M →ₚₗ[R] N)` shows that polynomial laws form an `R`-module.\n\n* `PolynomialLaw.ground f` is the map `M → N` corresponding to `PolynomialLaw.toFun' f R` under\n  the isomorphisms `R ⊗[R] M ≃ₗ[R] M`, and similarly for `N`.\n\nIn further works, we construct the coefficients of a polynomial law and show the relation with\npolynomials (when the module `M` is free and finite).\n\n## Implementation notes\n\nIn the literature, the theory is written for commutative rings, but this implementation\nonly assumes `R` is a commutative semiring.\n\n## References\n\n* [Roby, Norbert. 1963. «Lois polynomes et lois formelles en théorie des modules».\n  Annales scientifiques de l’École Normale Supérieure 80 (3): 213‑348](Roby-1963)\n\n-/\n\n@[expose] public section\n\nuniverse u v w\n\nnoncomputable section PolynomialLaw\n\nopen scoped TensorProduct\n\nopen LinearMap TensorProduct AlgHom RingCon\n\n/-- A polynomial law `M →ₚₗ[R] N` between `R`-modules is a functorial family of maps\n`S ⊗[R] M → S ⊗[R] N`, for all `R`-algebras `S`.\n\nFor universe reasons, `S` has to be restricted to the same universe as `R`. -/\n@[ext]\nstructure PolynomialLaw (R : Type u) [CommSemiring R]\n    (M : Type*) [AddCommMonoid M] [Module R M] (N : Type*) [AddCommMonoid N] [Module R N] where\n  /-- The functions `S ⊗[R] M → S ⊗[R] N` underlying a polynomial law -/\n  toFun' (S : Type u) [CommSemiring S] [Algebra R S] : S ⊗[R] M → S ⊗[R] N\n  /-- The compatibility relations between the functions underlying a polynomial law -/\n  isCompat' {S : Type u} [CommSemiring S] [Algebra R S]\n    {S' : Type u} [CommSemiring S'] [Algebra R S'] (φ : S →ₐ[R] S') :\n    φ.toLinearMap.rTensor N ∘ toFun' S = toFun' S' ∘ φ.toLinearMap.rTensor M := by aesop\n\n/-- `M →ₚₗ[R] N` is the type of `R`-polynomial laws from `M` to `N`. -/\nnotation:25 M \" →ₚₗ[\" R:25 \"] \" N:0 => PolynomialLaw R M N\n\n@[local simp]\ntheorem PolynomialLaw.isCompat_apply'\n    {R : Type u} [CommSemiring R] {M : Type*} [AddCommMonoid M] [Module R M]\n    {N : Type*} [AddCommMonoid N] [Module R N] {f : M →ₚₗ[R] N}\n    {S : Type u} [CommSemiring S] [Algebra R S] {S' : Type u} [CommSemiring S'] [Algebra R S']\n    (φ : S →ₐ[R] S') (x : S ⊗[R] M) :\n    (φ.toLinearMap.rTensor N) ((f.toFun' S) x) = (f.toFun' S') (φ.toLinearMap.rTensor M x) := by\n  simpa only using! congr_fun (f.isCompat' φ) x\n\nattribute [local simp] PolynomialLaw.isCompat_apply'\n\nnamespace PolynomialLaw\n\nsection Module\n\nsection CommSemiring\n\nvariable {R : Type u} [CommSemiring R] {M : Type*} [AddCommMonoid M] [Module R M]\n  {N : Type*} [AddCommMonoid N] [Module R N] (r a b : R) (f g : M →ₚₗ[R] N)\n\ninstance : Zero (M →ₚₗ[R] N) := ⟨{ toFun' _ := 0 }⟩\n\n@[simp]\ntheorem zero_def (S : Type u) [CommSemiring S] [Algebra R S] :\n    (0 : PolynomialLaw R M N).toFun' S = 0 := rfl\n\ninstance : Inhabited (PolynomialLaw R M N) := ⟨Zero.zero⟩\n\n/-- The identity as a polynomial law -/\ndef id : M →ₚₗ[R] M where\n  toFun' S _ _ := _root_.id\n\ntheorem id_apply' {S : Type u} [CommSemiring S] [Algebra R S] :\n    (id : M →ₚₗ[R] M).toFun' S = _root_.id := rfl\n\n/-- The sum of two polynomial laws -/\nnoncomputable def add : M →ₚₗ[R] N where\n  toFun' S _ _ := f.toFun' S + g.toFun' S\n\ninstance : Add (PolynomialLaw R M N) := ⟨add⟩\n\n@[simp]\ntheorem add_def (S : Type u) [CommSemiring S] [Algebra R S] :\n    (f + g).toFun' S = f.toFun' S + g.toFun' S := rfl\n\ntheorem add_def_apply (S : Type u) [CommSemiring S] [Algebra R S] (m : S ⊗[R] M) :\n    (f + g).toFun' S m = f.toFun' S m + g.toFun' S m := rfl\n\n/-- External multiplication of a `f : M →ₚₗ[R] N` by `r : R` -/\ndef smul : M →ₚₗ[R] N where\n  toFun' S _ _ := r • f.toFun' S\n\ninstance : SMul R (M →ₚₗ[R] N) := ⟨smul⟩\n\n@[simp]\ntheorem smul_def (S : Type u) [CommSemiring S] [Algebra R S] :\n    (r • f).toFun' S = r • f.toFun' S := rfl\n\ntheorem smul_def_apply (S : Type u) [CommSemiring S] [Algebra R S] (m : S ⊗[R] M) :\n    (r • f).toFun' S m = r • f.toFun' S m := rfl\n\nTarget:\ntheorem add_smul : (a + b) • f = a • f + b • f :=\n\nProof body:\n","rejected":"by\n  sorry","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/PolynomialLaw","family_id":"add_smul","file_id":"mathlib/Mathlib/RingTheory/PolynomialLaw/Basic.lean","sample_id":"4aa461ab228ed2d6d40d0634d68fa0428d1672b0b417d2b773b3a8831f202fa2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"80d415d211305782b1e1301d4d0be21abb30589a9327984a8f2cfe8e37267cf8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"cc0e6de2bb15058021ed449d45b1ad5b54fd26d1f20c9b4474c3b484da5b80a0","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"d55812070999d9cd69897ef7878e60eedf847e5cca72e88cad19720567719cd3","source_sha256":"ed84729d6b60d1d061a43d8f6649c91b9089118e6fc8675bd3d550d272a00e60","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  have : ∀ᵐ x ∂μ, (f - f') x = 0 := by\n    apply ae_eq_zero_of_integral_contMDiff_smul_eq_zero I (hf.sub hf')\n    intro g g_diff g_supp\n    simp only [Pi.sub_apply, smul_sub]\n    rw [integral_sub, sub_eq_zero]\n    · exact h g g_diff g_supp\n    · exact hf.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp\n    · exact hf'.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp\n  filter_upwards [this] with x hx\n  simpa [sub_eq_zero] using hx","hard_negative":false,"metrics":{"chosen_tokens":84,"rejected_tokens":89,"token_jaccard":0.924528,"token_length_ratio":1.059524},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"f9556318272271ba6b4a30c43408227d4440d15b079b2ac43d1c2181fb010b96","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Geometry.Manifold.PartitionOfUnity\npublic import Mathlib.MeasureTheory.Function.AEEqOfIntegral\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2023 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Functions which vanish as distributions vanish as functions\n\nIn a finite-dimensional normed real vector space endowed with a Borel measure, consider a locally\nintegrable function whose integral against all compactly supported smooth functions vanishes. Then\nthe function is almost everywhere zero.\nThis is proved in `ae_eq_zero_of_integral_contDiff_smul_eq_zero`.\n\nA version for two functions having the same integral when multiplied by smooth compactly supported\nfunctions is also given in `ae_eq_of_integral_contDiff_smul_eq`.\n\nThese are deduced from the same results on finite-dimensional real manifolds, given respectively\nas `ae_eq_zero_of_integral_contMDiff_smul_eq_zero` and `ae_eq_of_integral_contMDiff_smul_eq`.\n-/\n\npublic section\n\nopen MeasureTheory Filter Metric Function Set TopologicalSpace\n\nopen scoped Topology Manifold ContDiff\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]\n  {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]\n\nsection Manifold\n\nvariable {H : Type*} [TopologicalSpace H] (I : ModelWithCorners ℝ E H)\n  {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]\n  [MeasurableSpace M] [BorelSpace M] [T2Space M]\n  {f f' : M → F} {μ : Measure M}\n\n/-- If a locally integrable function `f` on a finite-dimensional real manifold has zero integral\nwhen multiplied by any smooth compactly supported function, then `f` vanishes almost everywhere. -/\ntheorem ae_eq_zero_of_integral_contMDiff_smul_eq_zero [SigmaCompactSpace M]\n    (hf : LocallyIntegrable f μ)\n    (h : ∀ g : M → ℝ, CMDiff ∞ g → HasCompactSupport g → ∫ x, g x • f x ∂μ = 0) :\n    ∀ᵐ x ∂μ, f x = 0 := by\n  -- record topological properties of `M`\n  have := I.locallyCompactSpace\n  have := ChartedSpace.locallyCompactSpace H M\n  have := I.secondCountableTopology\n  have := ChartedSpace.secondCountable_of_sigmaCompact H M\n  let _ : MetricSpace M := TopologicalSpace.metrizableSpaceMetric M\n  -- it suffices to show that the integral of the function vanishes on any compact set `s`\n  apply ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero' hf (fun s hs ↦ Eq.symm ?_)\n  obtain ⟨δ, δpos, hδ⟩ : ∃ δ, 0 < δ ∧ IsCompact (cthickening δ s) := hs.exists_isCompact_cthickening\n  -- choose a sequence of smooth functions `gₙ` equal to `1` on `s` and vanishing outside of the\n  -- `uₙ`-neighborhood of `s`, where `uₙ` tends to zero. Then each integral `∫ gₙ f` vanishes,\n  -- and by dominated convergence these integrals converge to `∫ x in s, f`.\n  obtain ⟨u, -, u_pos, u_lim⟩ : ∃ u, StrictAnti u ∧ (∀ (n : ℕ), u n ∈ Ioo 0 δ)\n    ∧ Tendsto u atTop (𝓝 0) := exists_seq_strictAnti_tendsto' δpos\n  let v : ℕ → Set M := fun n ↦ thickening (u n) s\n  obtain ⟨K, K_compact, vK⟩ : ∃ K, IsCompact K ∧ ∀ n, v n ⊆ K :=\n    ⟨_, hδ, fun n ↦ thickening_subset_cthickening_of_le (u_pos n).2.le _⟩\n  have : ∀ n, ∃ (g : M → ℝ), support g = v n ∧ CMDiff ∞ g ∧ Set.range g ⊆ Set.Icc 0 1\n          ∧ ∀ x ∈ s, g x = 1 := by\n    intro n\n    rcases exists_contMDiff_support_eq_eq_one_iff I isOpen_thickening hs.isClosed\n      (self_subset_thickening (u_pos n).1 s) with ⟨g, g_smooth, g_range, g_supp, hg⟩\n    exact ⟨g, g_supp, g_smooth, g_range, fun x hx ↦ (hg x).1 hx⟩\n  choose g g_supp g_diff g_range hg using this\n  -- main fact: the integral of `∫ gₙ f` tends to `∫ x in s, f`.\n  have L : Tendsto (fun n ↦ ∫ x, g n x • f x ∂μ) atTop (𝓝 (∫ x in s, f x ∂μ)) := by\n    rw [← integral_indicator hs.measurableSet]\n    let bound : M → ℝ := K.indicator (fun x ↦ ‖f x‖)\n    have A : ∀ n, AEStronglyMeasurable (fun x ↦ g n x • f x) μ :=\n      fun n ↦ (g_diff n).continuous.aestronglyMeasurable.smul hf.aestronglyMeasurable\n    have B : Integrable bound μ := by\n      rw [integrable_indicator_iff K_compact.measurableSet]\n      exact (hf.integrableOn_isCompact K_compact).norm\n    have C : ∀ n, ∀ᵐ x ∂μ, ‖g n x • f x‖ ≤ bound x := by\n      intro n\n      filter_upwards with x\n      rw [norm_smul]\n      refine le_indicator_apply (fun _ ↦ ?_) (fun hxK ↦ ?_)\n      · have : ‖g n x‖ ≤ 1 := by\n          have := g_range n (mem_range_self (f := g n) x)\n          rw [Real.norm_of_nonneg this.1]\n          exact this.2\n        exact mul_le_of_le_one_left (norm_nonneg _) this\n      · have : g n x = 0 := by rw [← notMem_support, g_supp]; contrapose hxK; exact vK n hxK\n        simp [this]\n    have D : ∀ᵐ x ∂μ, Tendsto (fun n => g n x • f x) atTop (𝓝 (s.indicator f x)) := by\n      filter_upwards with x\n      by_cases hxs : x ∈ s\n      · have : ∀ n, g n x = 1 := fun n ↦ hg n x hxs\n        simp [this, indicator_of_mem hxs f]\n      · simp_rw [indicator_of_notMem hxs f]\n        apply tendsto_const_nhds.congr'\n        suffices H : ∀ᶠ n in atTop, g n x = 0 by\n          filter_upwards [H] with n hn using by simp [hn]\n        obtain ⟨ε, εpos, hε⟩ : ∃ ε, 0 < ε ∧ x ∉ thickening ε s := by\n          rw [← hs.isClosed.closure_eq, closure_eq_iInter_thickening s] at hxs\n          simpa using hxs\n        filter_upwards [(tendsto_order.1 u_lim).2 _ εpos] with n hn\n        rw [← notMem_support, g_supp]\n        contrapose hε\n        exact thickening_mono hn.le s hε\n    exact tendsto_integral_of_dominated_convergence bound A B C D\n  -- deduce that `∫ x in s, f = 0` as each integral `∫ gₙ f` vanishes by assumption\n  have : ∀ n, ∫ x, g n x • f x ∂μ = 0 := by\n    refine fun n ↦ h _ (g_diff n) ?_\n    apply HasCompactSupport.of_support_subset_isCompact K_compact\n    simpa [g_supp] using vK n\n  simpa [this] using L\n\n@[deprecated (since := \"2025-12-17\")]\nalias ae_eq_zero_of_integral_smooth_smul_eq_zero := ae_eq_zero_of_integral_contMDiff_smul_eq_zero\n\n-- An instance with keys containing `Opens`\ninstance (U : Opens M) : BorelSpace U := inferInstanceAs (BorelSpace (U : Set M))\n\n/-- If a function `f` locally integrable on an open subset `U` of a finite-dimensional real\n  manifold has zero integral when multiplied by any smooth function compactly supported\n  in `U`, then `f` vanishes almost everywhere in `U`. -/\ntheorem IsOpen.ae_eq_zero_of_integral_contMDiff_smul_eq_zero' {U : Set M} (hU : IsOpen U)\n    (hSig : IsSigmaCompact U) (hf : LocallyIntegrableOn f U μ)\n    (h : ∀ g : M → ℝ,\n      CMDiff ∞ g → HasCompactSupport g → tsupport g ⊆ U → ∫ x, g x • f x ∂μ = 0) :\n    ∀ᵐ x ∂μ, x ∈ U → f x = 0 := by\n  have meas_U := hU.measurableSet\n  rw [← ae_restrict_iff' meas_U, ae_restrict_iff_subtype meas_U]\n  let U : Opens M := ⟨U, hU⟩\n  change ∀ᵐ (x : U) ∂_, _\n  haveI : SigmaCompactSpace U := isSigmaCompact_iff_sigmaCompactSpace.mp hSig\n  refine ae_eq_zero_of_integral_contMDiff_smul_eq_zero I ?_ fun g g_smth g_supp ↦ ?_\n  · exact (locallyIntegrable_comap meas_U).mpr hf\n  specialize h (Subtype.val.extend g 0) (g_smth.extend_zero g_supp)\n    (g_supp.extend_zero continuous_subtype_val) ((g_supp.tsupport_extend_zero_subset\n      continuous_subtype_val).trans <| Subtype.coe_image_subset _ _)\n  rw [← setIntegral_eq_integral_of_forall_compl_eq_zero (s := U) fun x hx ↦ ?_] at h\n  · rw [← integral_subtype_comap] at h\n    · simp_rw [Subtype.val_injective.extend_apply] at h; exact h\n    · exact meas_U\n  rw [Function.extend_apply' _ _ _ (mt _ hx)]\n  · apply zero_smul\n  · rintro ⟨x, rfl⟩; exact x.2\n\n@[deprecated (since := \"2025-12-17\")]\nalias IsOpen.ae_eq_zero_of_integral_smooth_smul_eq_zero' :=\n  IsOpen.ae_eq_zero_of_integral_contMDiff_smul_eq_zero'\n\nvariable [SigmaCompactSpace M]\n\ntheorem IsOpen.ae_eq_zero_of_integral_contMDiff_smul_eq_zero {U : Set M} (hU : IsOpen U)\n    (hf : LocallyIntegrableOn f U μ)\n    (h : ∀ g : M → ℝ,\n      CMDiff ∞ g → HasCompactSupport g → tsupport g ⊆ U → ∫ x, g x • f x ∂μ = 0) :\n    ∀ᵐ x ∂μ, x ∈ U → f x = 0 :=\n  haveI := I.locallyCompactSpace\n  haveI := ChartedSpace.locallyCompactSpace H M\n  haveI := hU.locallyCompactSpace\n  haveI := I.secondCountableTopology\n  haveI := ChartedSpace.secondCountable_of_sigmaCompact H M\n  hU.ae_eq_zero_of_integral_contMDiff_smul_eq_zero' _\n    (isSigmaCompact_iff_sigmaCompactSpace.mpr inferInstance) hf h\n\n@[deprecated (since := \"2025-12-17\")]\nalias IsOpen.ae_eq_zero_of_integral_smooth_smul_eq_zero :=\n  IsOpen.ae_eq_zero_of_integral_contMDiff_smul_eq_zero\n\n/-- If two locally integrable functions on a finite-dimensional real manifold have the same integral\nwhen multiplied by any smooth compactly supported function, then they coincide almost everywhere. -/\n\nTarget:\ntheorem ae_eq_of_integral_contMDiff_smul_eq\n    (hf : LocallyIntegrable f μ) (hf' : LocallyIntegrable f' μ) (h : ∀ (g : M → ℝ),\n      CMDiff ∞ g → HasCompactSupport g → ∫ x, g x • f x ∂μ = ∫ x, g x • f' x ∂μ) :\n    ∀ᵐ x ∂μ, f x = f' x :=\n\nProof body:\n","rejected":"by\n  have : ∀ᵐ x ∂μ, (f - f') x = 0 := by\n    apply ae_eq_zero_of_integral_contMDiff_smul_eq_zero I (hf.sub hf')\n    intro g g_diff g_supp\n    simp only [Pi.sub_apply, smul_sub]\n    rw [integral_sub, sub_eq_zero]\n    · exact h g g_diff g_supp\n    · exact hf.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp\n    · exact hf'.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp\n  filter_upwards [this] with x hx\n  simpa [sub_eq_zero] using hx\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/Distribution","family_id":"ae_eq_of_integral_contmdiff_smul_eq","file_id":"mathlib/Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean","sample_id":"d55812070999d9cd69897ef7878e60eedf847e5cca72e88cad19720567719cd3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7bfddf9976579c9b33ef1adb03a12367ce2324df54f03888b73e637b017ab565","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"6ef6ae9c0f9613c7c3706f6b6b19cc3ed2c14562e6ea2548c5e346b1a3fa96cb","source_sha256":"5ec4c4d2769bb0652aea247f81d7ff99e572b3159717c948f413b35f1ff454f9","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← disjoint_iff]\n  exact disjoint_sdiff_iff_le hz hx\n\n-- cf. `IsCompl.left_le_iff` and `IsCompl.right_le_iff`","hard_negative":false,"metrics":{"chosen_tokens":25,"rejected_tokens":5,"token_jaccard":0.095238,"token_length_ratio":0.2},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"f95d36b7806ce5579355ff7f76edd92a24e172206c21302c20059f9be6ca05fa","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.BooleanAlgebra.Defs\npublic import Mathlib.Tactic.GRewrite\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Bryan Gin-ge Chen\n-/\n/-!\n# Basic properties of Boolean algebras\n\nThis file provides some basic definitions, functions as well as lemmas for functions and type\nclasses related to Boolean algebras as defined in `Mathlib/Order/BooleanAlgebra/Defs.lean`.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations>\n* [*Postulates for Boolean Algebras and Generalized Boolean Algebras*, M.H. Stone][Stone1935]\n* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]\n\n## Tags\n\ngeneralized Boolean algebras, Boolean algebras, lattices, sdiff, compl\n\n-/\n\npublic section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type*} {x y z : α}\n\n/-!\n### Generalized Boolean algebras\n\nSome of the lemmas in this section are from:\n\n* [*Lattice Theory: Foundation*, George Grätzer][Gratzer2011]\n* <https://ncatlab.org/nlab/show/relative+complement>\n* <https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf>\n\n-/\n\n-- We might want an `IsCompl_of` predicate (for relative complements) generalizing `IsCompl`,\n-- however we'd need another type class for lattices with bot, and all the API for that.\nsection GeneralizedBooleanAlgebra\n\nvariable [GeneralizedBooleanAlgebra α]\n\n@[simp]\ntheorem sup_inf_sdiff (x y : α) : x ⊓ y ⊔ x \\ y = x :=\n  GeneralizedBooleanAlgebra.sup_inf_sdiff _ _\n\n@[simp]\ntheorem inf_inf_sdiff (x y : α) : x ⊓ y ⊓ x \\ y = ⊥ :=\n  GeneralizedBooleanAlgebra.inf_inf_sdiff _ _\n\n@[simp]\ntheorem sup_sdiff_inf (x y : α) : x \\ y ⊔ x ⊓ y = x := by rw [sup_comm, sup_inf_sdiff]\n\n@[simp]\ntheorem inf_sdiff_inf (x y : α) : x \\ y ⊓ (x ⊓ y) = ⊥ := by rw [inf_comm, inf_inf_sdiff]\n\n-- see Note [lower instance priority]\ninstance (priority := 100) GeneralizedBooleanAlgebra.toOrderBot : OrderBot α where\n  __ := GeneralizedBooleanAlgebra.toBot\n  bot_le a := by\n    rw [← inf_inf_sdiff a a, inf_assoc]\n    exact inf_le_left\n\ntheorem disjoint_inf_sdiff : Disjoint (x ⊓ y) (x \\ y) :=\n  disjoint_iff_inf_le.mpr (inf_inf_sdiff x y).le\n\n-- TODO: in distributive lattices, relative complements are unique when they exist\ntheorem sdiff_unique (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \\ y = z := by\n  conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]\n  rw [sup_comm] at s\n  conv_rhs at i => rw [← inf_inf_sdiff x y, inf_comm]\n  rw [inf_comm] at i\n  exact (eq_of_inf_eq_sup_eq i s).symm\n\n-- Use `sdiff_le`\nprivate theorem sdiff_le' : x \\ y ≤ x :=\n  calc\n    x \\ y ≤ x ⊓ y ⊔ x \\ y := le_sup_right\n    _ = x := sup_inf_sdiff x y\n\nset_option backward.privateInPublic true in\n-- Use `sdiff_sup_self`\nprivate theorem sdiff_sup_self' : y \\ x ⊔ x = y ⊔ x :=\n  calc\n    y \\ x ⊔ x = y \\ x ⊔ (x ⊔ x ⊓ y) := by rw [sup_inf_self]\n    _ = y ⊓ x ⊔ y \\ x ⊔ x := by ac_rfl\n    _ = y ⊔ x := by rw [sup_inf_sdiff]\n\n@[simp]\ntheorem sdiff_inf_sdiff : x \\ y ⊓ y \\ x = ⊥ :=\n  Eq.symm <|\n    calc\n      ⊥ = x ⊓ (y ⊓ x ⊔ y \\ x) ⊓ x \\ y := by rw [← inf_inf_sdiff, sup_inf_sdiff]\n      _ = (x ⊓ (y ⊓ x) ⊔ x ⊓ y \\ x) ⊓ x \\ y := by rw [inf_sup_left]\n      _ = (y ⊓ (x ⊓ x) ⊔ x ⊓ y \\ x) ⊓ x \\ y := by ac_rfl\n      _ = x ⊓ y \\ x ⊓ x \\ y := by\n          rw [inf_idem, inf_sup_right, ← inf_comm x y, inf_inf_sdiff, bot_sup_eq]\n      _ = x ⊓ x \\ y ⊓ y \\ x := by ac_rfl\n      _ = x \\ y ⊓ y \\ x := by rw [inf_of_le_right sdiff_le']\n\ntheorem disjoint_sdiff_sdiff : Disjoint (x \\ y) (y \\ x) :=\n  disjoint_iff_inf_le.mpr sdiff_inf_sdiff.le\n\n@[simp]\ntheorem inf_sdiff_self_right : x ⊓ y \\ x = ⊥ :=\n  calc\n    x ⊓ y \\ x = (x ⊓ y ⊔ x \\ y) ⊓ y \\ x := by rw [sup_inf_sdiff]\n    _ = ⊥ := by rw [inf_sup_right, inf_comm x y, inf_inf_sdiff, sdiff_inf_sdiff, bot_sup_eq]\n\n@[simp]\ntheorem inf_sdiff_self_left : y \\ x ⊓ x = ⊥ := by rw [inf_comm, inf_sdiff_self_right]\n\nset_option backward.privateInPublic true in\nset_option backward.privateInPublic.warn false in\n-- see Note [lower instance priority]\ninstance (priority := 100) GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra :\n    GeneralizedCoheytingAlgebra α where\n  __ := ‹GeneralizedBooleanAlgebra α›\n  __ := GeneralizedBooleanAlgebra.toOrderBot\n  sdiff := (· \\ ·)\n  sdiff_le_iff y x z :=\n    ⟨fun h =>\n      le_of_inf_le_sup_le\n        (le_of_eq\n          (by grind [sdiff_le', inf_of_le_right, inf_eq_right, inf_sdiff_self_right, bot_sup_eq,\n            inf_sup_right]))\n        (calc\n          y ⊔ y \\ x ≤ y \\ x ⊔ x ⊔ z := by\n            grind [sup_of_le_left, sdiff_le', le_sup_left, sdiff_sup_self']\n          _ = x ⊔ z ⊔ y \\ x := by ac_rfl),\n      fun h => le_of_inf_le_sup_le (inf_sdiff_self_left.trans_le bot_le) (calc\n        y \\ x ⊔ x = y ⊔ x := sdiff_sup_self'\n        _ ≤ x ⊔ z ⊔ x := sup_le_sup_right h x\n        _ ≤ z ⊔ x := by rw [sup_assoc, sup_comm, sup_assoc, sup_idem])⟩\n\ntheorem disjoint_sdiff_self_left : Disjoint (y \\ x) x :=\n  disjoint_iff_inf_le.mpr inf_sdiff_self_left.le\n\ntheorem disjoint_sdiff_self_right : Disjoint x (y \\ x) :=\n  disjoint_iff_inf_le.mpr inf_sdiff_self_right.le\n\nlemma le_sdiff : x ≤ y \\ z ↔ x ≤ y ∧ Disjoint x z :=\n  ⟨fun h ↦ ⟨h.trans sdiff_le, disjoint_sdiff_self_left.mono_left h⟩, fun h ↦\n    by rw [← h.2.sdiff_eq_left]; exact sdiff_le_sdiff_right h.1⟩\n\n@[simp] lemma sdiff_eq_left : x \\ y = x ↔ Disjoint x y :=\n  ⟨fun h ↦ disjoint_sdiff_self_left.mono_left h.ge, Disjoint.sdiff_eq_left⟩\n\n/- TODO: we could make an alternative constructor for `GeneralizedBooleanAlgebra` using\n`Disjoint x (y \\ x)` and `x ⊔ (y \\ x) = y` as axioms. -/\ntheorem Disjoint.sdiff_eq_of_sup_eq (hi : Disjoint x z) (hs : x ⊔ z = y) : y \\ x = z :=\n  have h : y ⊓ x = x := inf_eq_right.2 <| le_sup_left.trans hs.le\n  sdiff_unique (by rw [h, hs]) (by rw [h, hi.eq_bot])\n\nprotected theorem Disjoint.sdiff_unique (hd : Disjoint x z) (hz : z ≤ y) (hs : y ≤ x ⊔ z) :\n    y \\ x = z :=\n  sdiff_unique\n    (by\n      rw [← inf_eq_right] at hs\n      rwa [sup_inf_right, inf_sup_right, sup_comm x, inf_sup_self, inf_comm, sup_comm z,\n        hs, sup_eq_left])\n    (by rw [inf_assoc, hd.eq_bot, inf_bot_eq])\n\n-- cf. `IsCompl.disjoint_left_iff` and `IsCompl.disjoint_right_iff`\ntheorem disjoint_sdiff_iff_le (hz : z ≤ y) (hx : x ≤ y) : Disjoint z (y \\ x) ↔ z ≤ x :=\n  ⟨fun H =>\n    le_of_inf_le_sup_le (le_trans H.le_bot bot_le)\n      (by\n        rw [sup_sdiff_cancel_right hx]\n        grw [sdiff_le]\n        rw [sup_eq_right.2 hz]),\n    fun H => disjoint_sdiff_self_right.mono_left H⟩\n\n-- cf. `IsCompl.le_left_iff` and `IsCompl.le_right_iff`\ntheorem le_iff_disjoint_sdiff (hz : z ≤ y) (hx : x ≤ y) : z ≤ x ↔ Disjoint z (y \\ x) :=\n  (disjoint_sdiff_iff_le hz hx).symm\n\n-- cf. `IsCompl.inf_left_eq_bot_iff` and `IsCompl.inf_right_eq_bot_iff`\n\nTarget:\ntheorem inf_sdiff_eq_bot_iff (hz : z ≤ y) (hx : x ≤ y) : z ⊓ y \\ x = ⊥ ↔ z ≤ x :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/BooleanAlgebra","family_id":"inf_sdiff_eq_bot_iff","file_id":"mathlib/Mathlib/Order/BooleanAlgebra/Basic.lean","sample_id":"6ef6ae9c0f9613c7c3706f6b6b19cc3ed2c14562e6ea2548c5e346b1a3fa96cb"},"split":"train"}
{"binding":{"chosen_proof_sha256":"f77254043cd1457d306845776290c0242267d27c173b195a06e4b8478f5ec277","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"3d805ac3a07f605169cd17d09107c30e012f4d0f06e8571fe5f44b26cb0ba912","source_sha256":"1ee321a81f228358638453570a83aa9bc5ea6966d6e7864b764bf262f805e0a4","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [Set.eq_univ_iff_forall]\n  intro x\n  rw [Set.mem_iUnion]\n  exact 𝒰.exists_eq x","hard_negative":false,"metrics":{"chosen_tokens":20,"rejected_tokens":5,"token_jaccard":0.125,"token_length_ratio":0.25},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"f96ab8eeb0849aeb3bcbe3a89eb152030998157212ece81f89771367f1412bb7","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Sites.MorphismProperty\npublic import Mathlib.CategoryTheory.MorphismProperty.Limits\n\nNamespace:\nAlgebraicGeometry.Scheme\n\nLocal context:\n/-\nCopyright (c) 2024 Christian Merten, Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Christian Merten, Andrew Yang\n-/\n/-!\n# Covers of schemes\n\nThis file provides the basic API for covers of schemes. A cover of a scheme `X` with respect to\na morphism property `P` is a jointly surjective indexed family of scheme morphisms with\ntarget `X` all satisfying `P`.\n\n## Implementation details\n\nThe definition on the pullback of a cover along a morphism depends on results that\nare developed later in the import tree. Hence in this file, they have additional assumptions\nthat will be automatically satisfied in later files. The motivation here is that we already\nknow that these assumptions are satisfied for open immersions and hence the cover API for open\nimmersions can be used to deduce these assumptions in the general case.\n\n-/\n\n@[expose] public section\n\n\nnoncomputable section\n\nopen TopologicalSpace CategoryTheory Opposite CategoryTheory.Limits\n\nuniverse v v₁ v₂ u\n\nnamespace AlgebraicGeometry\n\nnamespace Scheme\n\nvariable (K : Precoverage Scheme.{u})\n\n/-- A coverage `K` on `Scheme` is called jointly surjective if every covering family in `K`\nis jointly surjective. -/\nclass JointlySurjective (K : Precoverage Scheme.{u}) : Prop where\n  exists_eq {X : Scheme.{u}} (S : Presieve X) (hS : S ∈ K X) (x : X) :\n    ∃ (Y : Scheme.{u}) (g : Y ⟶ X), S g ∧ x ∈ Set.range g\n\n/-- A cover of `X` in the coverage `K` is a `0`-hypercover for `K`. -/\nabbrev Cover (K : Precoverage Scheme.{u}) := Precoverage.ZeroHypercover.{v} K\n\nvariable {K}\n\nvariable {X Y Z : Scheme.{u}} (𝒰 : X.Cover K) (f : X ⟶ Z) (g : Y ⟶ Z)\nvariable [∀ x, HasPullback (𝒰.f x ≫ f) g]\n\nlemma Cover.exists_eq [JointlySurjective K] (𝒰 : X.Cover K) (x : X) :\n    ∃ i y, 𝒰.f i y = x := by\n  obtain ⟨Y, g, ⟨i⟩, y, hy⟩ := JointlySurjective.exists_eq 𝒰.presieve₀ 𝒰.mem₀ x\n  use i, y\n\n/-- A choice of an index `i` such that `x` is in the range of `𝒰.f i`. -/\ndef Cover.idx [JointlySurjective K] (𝒰 : X.Cover K) (x : X) : 𝒰.I₀ :=\n  (𝒰.exists_eq x).choose\n\nlemma Cover.covers [JointlySurjective K] (𝒰 : X.Cover K) (x : X) :\n    x ∈ Set.range (𝒰.f (𝒰.idx x)) :=\n  (𝒰.exists_eq x).choose_spec\n\nTarget:\ntheorem Cover.iUnion_range [JointlySurjective K] {X : Scheme.{u}} (𝒰 : X.Cover K) :\n    ⋃ i, Set.range (𝒰.f i) = Set.univ :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Cover","family_id":"cover","file_id":"mathlib/Mathlib/AlgebraicGeometry/Cover/MorphismProperty.lean","sample_id":"3d805ac3a07f605169cd17d09107c30e012f4d0f06e8571fe5f44b26cb0ba912"},"split":"train"}
{"binding":{"chosen_proof_sha256":"63ce66eb5d2d85e6c3aa93e37342e57afc36c526d7b717542f12e151e6f9e034","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ed0bc1a8156a91f3c77b4c93e0e33c2312cab82d14a2bd0d77876ca7788349af","source_sha256":"1926121bd5a0473c64fcd7320e73eb6e8a92720aeaf7b72deff3f44f28315a03","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  let := Localization.AtPrime.algebraOfLiesOver (p.under R) p\n  rw [isUnramifiedAt_iff_map_eq R (p.under R) p, and_iff_right,\n    Ideal.IsDedekindDomain.ramificationIdx_eq_one_iff hp Ideal.map_comap_le]\n  have : Finite (R ⧸ p.under R) :=\n    Ideal.finiteQuotientOfFreeOfNeBot _ (mt Ideal.eq_bot_of_comap_eq_bot hp)\n  have : Finite ((p.under R).ResidueField) := IsLocalization.finite _\n    (nonZeroDivisors (R ⧸ p.under R))\n  infer_instance","hard_negative":true,"metrics":{"chosen_tokens":92,"rejected_tokens":8,"token_jaccard":0.102564,"token_length_ratio":0.086957},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"f9d383b5c0fd528f0b5d3274e723ff5bd68c5d117d88d485e7a529456f6f1e58","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.RamificationInertia.Basic\npublic import Mathlib.RingTheory.LocalRing.ResidueField.Instances\npublic import Mathlib.RingTheory.Unramified.LocalRing\npublic import Mathlib.LinearAlgebra.FreeModule.IdealQuotient\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2025 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n\n# Unramified and ramification index\n\nWe connect `Ideal.ramificationIdx` to the commutative algebra notion predicate of `IsUnramifiedAt`.\n\n## Main result\n- `Algebra.isUnramifiedAt_iff_of_isDedekindDomain`:\n  Let `R` be a domain of characteristic 0, finite rank over `ℤ`, `S ⊇ R` be a Dedekind domain\n  that is a finite `R`-algebra. Let `p` be a prime of `S`, then `p` is unramified iff `e(p) = 1`.\n\n-/\n\npublic section\n\nvariable {R S T : Type*} [CommRing R] [CommRing S] [CommRing T]\nvariable [Algebra R S] [Algebra S T] [Algebra R T] [IsScalarTower R S T]\n\nlocal notation3 \"e(\" P \"|\" R \")\" =>\n  Ideal.ramificationIdx (Ideal.under R P) P\n\nopen IsLocalRing Algebra\n\nlemma Ideal.ramificationIdx_eq_one_of_isUnramifiedAt\n    {p : Ideal S} [p.IsPrime] [IsNoetherianRing S] [IsUnramifiedAt R p]\n    (hp : p ≠ ⊥) [IsDomain S] [EssFiniteType R S] :\n    e(p|R) = 1 :=\n  let := Localization.AtPrime.algebraOfLiesOver (p.under R) p\n  (Ideal.ramificationIdx_eq_one_of_map_localization Ideal.map_comap_le hp\n    p.primeCompl_le_nonZeroDivisors\n    ((isUnramifiedAt_iff_map_eq R (p.under R) p).mp ‹_›).2)\n\nvariable (R) in\nlemma IsUnramifiedAt.of_liesOver_of_ne_bot\n    (p : Ideal S) (P : Ideal T) [P.LiesOver p] [p.IsPrime] [P.IsPrime]\n    [IsUnramifiedAt R P] [EssFiniteType R S] [EssFiniteType R T]\n    [IsDedekindDomain S] (hP₁ : P.primeCompl ≤ nonZeroDivisors T) (hP₂ : p ≠ ⊥ → P ≠ ⊥) :\n    IsUnramifiedAt R p := by\n  let p₀ : Ideal R := p.under R\n  have : P.LiesOver p₀ := .trans P p p₀\n  let := Localization.AtPrime.algebraOfLiesOver p₀ p\n  let := Localization.AtPrime.algebraOfLiesOver p P\n  let := Localization.AtPrime.algebraOfLiesOver p₀ P\n  have hp₀ : p₀ = P.under R := Ideal.LiesOver.over\n  have : EssFiniteType S T := .of_comp R S T\n  have := Algebra.EssFiniteType.isNoetherianRing S T\n  rw [isUnramifiedAt_iff_map_eq R p₀ p]\n  have ⟨h₁, h₂⟩ := (isUnramifiedAt_iff_map_eq R p₀ P).mp ‹_›\n  refine ⟨Algebra.isSeparable_tower_bot_of_isSeparable _ _ P.ResidueField, ?_⟩\n  by_cases hp : p = ⊥\n  · have : p₀.map (algebraMap R S) = p := by\n      subst hp\n      exact le_bot_iff.mp (Ideal.map_comap_le)\n    rw [IsScalarTower.algebraMap_eq _ S, ← Ideal.map_map, this,\n      Localization.AtPrime.map_eq_maximalIdeal]\n  rw [← Ideal.IsDedekindDomain.ramificationIdx_eq_one_iff hp Ideal.map_comap_le,\n    ← not_ne_iff, Ideal.ramificationIdx_ne_one_iff Ideal.map_comap_le]\n  intro H\n  have := Ideal.ramificationIdx_eq_one_of_map_localization\n    (hp₀ ▸ Ideal.map_comap_le) (hP₂ hp) hP₁ h₂\n  rw [← not_ne_iff, Ideal.ramificationIdx_ne_one_iff (hp₀ ▸ Ideal.map_comap_le)] at this\n  replace H := Ideal.map_mono (f := algebraMap S T) H\n  rw [Ideal.map_map, ← IsScalarTower.algebraMap_eq, Ideal.map_pow] at H\n  refine this (H.trans (Ideal.pow_right_mono ?_ _))\n  exact Ideal.map_le_iff_le_comap.mpr Ideal.LiesOver.over.le\n\nvariable (R) in\n/--\nUp to technical conditions, If `T/S/R` is a tower of algebras, `P` is a prime of `T` unramified\nin `R`, then `P ∩ S` (as a prime of `S`) is also unramified in `R`.\n-/\nlemma Algebra.IsUnramifiedAt.of_liesOver\n    (p : Ideal S) (P : Ideal T) [P.LiesOver p] [p.IsPrime] [P.IsPrime]\n    [IsUnramifiedAt R P] [EssFiniteType R S] [EssFiniteType R T]\n    [IsDedekindDomain S] [IsDomain T] [Module.IsTorsionFree S T] : IsUnramifiedAt R p :=\n  IsUnramifiedAt.of_liesOver_of_ne_bot R p P P.primeCompl_le_nonZeroDivisors\n    (Ideal.ne_bot_of_liesOver_of_ne_bot · P)\n\n/-- Let `R` be a domain of characteristic 0, finite rank over `ℤ`, `S` be a Dedekind domain\nthat is a finite `R`-algebra. Let `p` be a prime of `S`, then `p` is unramified iff `e(p) = 1`. -/\n\nTarget:\nlemma Algebra.isUnramifiedAt_iff_of_isDedekindDomain\n    {p : Ideal S} [p.IsPrime] [IsDedekindDomain S] [EssFiniteType R S] [IsDomain R]\n    [Module.Finite ℤ R] [CharZero R] [Algebra.IsIntegral R S]\n    (hp : p ≠ ⊥) :\n    Algebra.IsUnramifiedAt R p ↔ e(p|R) = 1 :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"e505ecb6a0cc32d2b092dfe6c26910892cfff5faf1971a88c446a47acb1fecdf","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/RamificationInertia","family_id":"algebra","file_id":"mathlib/Mathlib/NumberTheory/RamificationInertia/Unramified.lean","sample_id":"ed0bc1a8156a91f3c77b4c93e0e33c2312cab82d14a2bd0d77876ca7788349af"},"split":"train"}
{"binding":{"chosen_proof_sha256":"6986b7de3b811139639af9103eba6dc70b8f13e73607506a5de302cee24db96a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7ad080fabb1c3deb7fe535e2a848f748f1683ac91b902c1df85e15c050b287f0","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1e74a857d727dcb19bacc65e1dd39ea3f06dc3533cc251ea326cf338c51e0a18","source_sha256":"6b438bdcb0b80057beec48c7a9a1f234b44b153474b61732effd69c42c628d3a","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [BilinForm.toMatrixAux, Matrix.toBilin'Aux, toLinearMap₂'Aux_toMatrix₂Aux]","hard_negative":true,"metrics":{"chosen_tokens":18,"rejected_tokens":3,"token_jaccard":0.058824,"token_length_ratio":0.166667},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"f9f83d25656b783f0465073ef91e37afcaea7fec031baca77c629bde35485bcc","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.LinearAlgebra.BilinearForm.Properties\npublic import Mathlib.LinearAlgebra.Matrix.SesquilinearForm\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen, Kexing Ying\n-/\n/-!\n# Bilinear form\n\nThis file defines the conversion between bilinear forms and matrices.\n\n## Main definitions\n\n* `Matrix.toBilin` given a basis define a bilinear form\n* `Matrix.toBilin'` define the bilinear form on `n → R`\n* `BilinForm.toMatrix`: calculate the matrix coefficients of a bilinear form\n* `BilinForm.toMatrix'`: calculate the matrix coefficients of a bilinear form on `n → R`\n\n## Notation\n\nIn this file we use the following type variables:\n- `M₁` is a module over the commutative semiring `R₁`,\n- `M₂` is a module over the commutative ring `R₂`.\n\n## Tags\n\nbilinear form, bilin form, BilinearForm, matrix, basis\n\n-/\n\n@[expose] public section\n\nopen LinearMap (BilinForm)\nopen Module\n\nvariable {R₁ : Type*} {M₁ : Type*} [CommSemiring R₁] [AddCommMonoid M₁] [Module R₁ M₁]\nvariable {R₂ : Type*} {M₂ : Type*} [CommRing R₂] [AddCommGroup M₂] [Module R₂ M₂]\n\nsection Matrix\n\nvariable {n o : Type*}\n\nopen Finset LinearMap Matrix\n\nopen Matrix\n\n/-- The map from `Matrix n n R` to bilinear forms on `n → R`.\n\nThis is an auxiliary definition for the equivalence `Matrix.toBilin'`. -/\ndef Matrix.toBilin'Aux [Fintype n] (M : Matrix n n R₁) : BilinForm R₁ (n → R₁) :=\n  Matrix.toLinearMap₂'Aux _ _ M\n\ntheorem Matrix.toBilin'Aux_single [Fintype n] [DecidableEq n] (M : Matrix n n R₁) (i j : n) :\n    M.toBilin'Aux (Pi.single i 1) (Pi.single j 1) = M i j :=\n  Matrix.toLinearMap₂'Aux_single _ _ _ _ _\n\n/-- The linear map from bilinear forms to `Matrix n n R` given an `n`-indexed basis.\n\nThis is an auxiliary definition for the equivalence `Matrix.toBilin'`. -/\ndef LinearMap.BilinForm.toMatrixAux (b : n → M₁) : BilinForm R₁ M₁ →ₗ[R₁] Matrix n n R₁ :=\n  LinearMap.toMatrix₂Aux R₁ b b\n\n@[deprecated (since := \"2026-01-16\")] alias BilinForm.toMatrixAux := LinearMap.BilinForm.toMatrixAux\n\n@[simp]\ntheorem LinearMap.BilinForm.toMatrixAux_apply (B : BilinForm R₁ M₁) (b : n → M₁) (i j : n) :\n    BilinForm.toMatrixAux b B i j = B (b i) (b j) :=\n  LinearMap.toMatrix₂Aux_apply R₁ B _ _ _ _\n\nvariable [Fintype n] [Fintype o]\n\nTarget:\ntheorem LinearMap.toBilin'Aux_toMatrixAux [DecidableEq n] (B₂ : BilinForm R₁ (n → R₁)) :\n    Matrix.toBilin'Aux (BilinForm.toMatrixAux (fun j => Pi.single j 1) B₂) = B₂ :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_1e74a857d727","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"f2da8552e2a46b5625efd7bed00ae51038cf7b37b991293cf847b66f56ead8f3","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Matrix","family_id":"linearmap","file_id":"mathlib/Mathlib/LinearAlgebra/Matrix/BilinearForm.lean","sample_id":"1e74a857d727dcb19bacc65e1dd39ea3f06dc3533cc251ea326cf338c51e0a18"},"split":"train"}
{"binding":{"chosen_proof_sha256":"8e7840e8c83d8f557e8647b534950399050e6f0e066df7ccc73d874d601c7f9a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4791ed0fbefe25e9f05c1dec233b6ddc5e09c117965a9cb9936eb95639f9e525","source_sha256":"e2806df73ecfc81f91bdaba3be881cdfeef7117f485c371a23cf9b886b2f1661","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases exists_isRoot_of_odd_natDegree (f := X ^ n - C x) (by simp [hn]) with ⟨r, hr⟩\n  exact ⟨r, by linear_combination - (by simpa using hr : r ^ n - x = 0)⟩","hard_negative":false,"metrics":{"chosen_tokens":48,"rejected_tokens":5,"token_jaccard":0.060606,"token_length_ratio":0.104167},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"fa04b9c59ef329e4ac9db395c0705f98ebf0ff96a8ec0c463a36aa63ce23fe63","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Degree.Domain\npublic import Mathlib.Algebra.Polynomial.Eval.Defs\npublic import Mathlib.Algebra.Ring.Semireal.Defs\npublic import Mathlib.Tactic.LinearCombination\n\nNamespace:\nIsRealClosed\n\nLocal context:\n/-\nCopyright (c) 2025 Artie Khovanov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Artie Khovanov\n-/\n/-!\n# Real Closed Field\n\nA field `R` is real closed if all of the following hold:\n1. `R` is real (that is, `-1` is not a sum of squares in `R`).\n2. for every `x` in `R`, one of `x` or `-x` is a square.\n3. every odd-degree polynomial over `R` has a root in `R`.\n\nA real closed field is an algebraic generalisation of the real numbers.\n\nIn this file we define real closed fields and prove some of their properties.\n\nTODO (Artie Khovanov) : equivalent conditions for a real field to be real closed\nTODO (Artie Khovanov) : real numbers, real algebraic numbers, hyperreals form a real closed field\n\n## Main Definitions\n\n- `IsRealClosed R` is the typeclass saying `R` is a real closed field.\n\n## Tags\n\nreal closed, rcf\n\n-/\n\npublic section\n\nopen Polynomial\n\n/--\nA field `R` is real closed if all of the following hold:\n1. `R` is real (that is, `-1` is not a sum of squares in `R`).\n2. for every `x` in `R`, one of `x` or `-x` is a square.\n3. every odd-degree polynomial over `R` has a root in `R`.\n-/\nclass IsRealClosed (R : Type*) [Field R] : Prop extends IsSemireal R where\n  isSquare_or_isSquare_neg (x : R) : IsSquare x ∨ IsSquare (-x)\n  exists_isRoot_of_odd_natDegree {f : R[X]} (hf : Odd f.natDegree) : ∃ x, f.IsRoot x\n\nattribute [aesop 90% forward] IsRealClosed.isSquare_or_isSquare_neg\n\nnamespace IsRealClosed\n\nuniverse u\n\nvariable {R : Type u} [Field R]\n\ntheorem of_linearOrderedField [LinearOrder R] [IsStrictOrderedRing R]\n    (isSquare_of_nonneg : ∀ {x : R}, 0 ≤ x → IsSquare x)\n    (exists_isRoot_of_odd_natDegree : ∀ {f : R[X]}, Odd f.natDegree → ∃ x, f.IsRoot x) :\n    IsRealClosed R where\n  isSquare_or_isSquare_neg {x} := by\n    rcases le_total x 0 with (neg | pos)\n    · exact .inr <| isSquare_of_nonneg (neg_nonneg_of_nonpos neg)\n    · exact .inl <| isSquare_of_nonneg pos\n  exists_isRoot_of_odd_natDegree := exists_isRoot_of_odd_natDegree\n\nvariable [IsRealClosed R]\n\n@[aesop 50%]\ntheorem _root_.IsSquare.of_not_isSquare_neg {x : R} (hx : ¬ IsSquare (-x)) : IsSquare x := by aesop\n\n@[aesop 80%]\ntheorem isSquare_neg_of_not_isSquare {x : R} (hx : ¬ IsSquare x) : IsSquare (-x) := by aesop\n\nTarget:\ntheorem exists_eq_pow_of_odd (x : R) {n : ℕ} (hn : Odd n) : ∃ r, x = r ^ n :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/IsRealClosed","family_id":"exists_eq_pow_of_odd","file_id":"mathlib/Mathlib/FieldTheory/IsRealClosed/Basic.lean","sample_id":"4791ed0fbefe25e9f05c1dec233b6ddc5e09c117965a9cb9936eb95639f9e525"},"split":"train"}
{"binding":{"chosen_proof_sha256":"74ef8f872938daf0468322013395261e7267b7edbfb838bf7741225d397b7c55","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"c9d53b75ff94d9ba91ecd885af73f8533c42e1c175a62180c1762b72f137eff3","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"de57e8cad85d8b44648cf10f3d51ce2288da48a2805529ae83b12ceae714db81","source_sha256":"250e0bf39b2fceb88cc44db25e786e52b9f1629e4d745089c27db7ba4ae4e60e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← card k V]\n  have : Nat.card V > 0 := Nat.card_pos\n  lia","hard_negative":false,"metrics":{"chosen_tokens":21,"rejected_tokens":2,"token_jaccard":0.055556,"token_length_ratio":0.095238},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"admit","pair_id":"fa0e82f69be5bf39dcb91769e347f49b9b4144ae4b0fc279abd7fa3a7d1cc1a0","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Ring.GeomSum\npublic import Mathlib.Algebra.GroupWithZero.Units.Fintype\npublic import Mathlib.Data.Finite.Sum\npublic import Mathlib.GroupTheory.GroupAction.Quotient\npublic import Mathlib.LinearAlgebra.Projectivization.Basic\n\nNamespace:\nProjectivization\n\nLocal context:\n/-\nCopyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Judith Ludwig, Christian Merten\n-/\n/-!\n# Cardinality of projective spaces\n\nWe compute the cardinality of `ℙ k V` if `k` is a finite field.\n\n-/\n\n@[expose] public section\n\nnamespace Projectivization\n\nopen scoped LinearAlgebra.Projectivization\n\nsection\n\nvariable (k V : Type*) [DivisionRing k] [AddCommGroup V] [Module k V]\n\n/-- `ℙ k V` is equivalent to the quotient of the non-zero elements of `V` by `kˣ`. -/\ndef equivQuotientOrbitRel : ℙ k V ≃ Quotient (MulAction.orbitRel kˣ { v : V // v ≠ 0 }) :=\n  Quotient.congr (Equiv.refl _) (fun x y ↦ (Units.orbitRel_nonZero_iff k V x y).symm)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The non-zero elements of `V` are equivalent to the product of `ℙ k V` with the units of `k`. -/\nnoncomputable def nonZeroEquivProjectivizationProdUnits : { v : V // v ≠ 0 } ≃ ℙ k V × kˣ :=\n  let e := MulAction.selfEquivOrbitsQuotientProd <| fun b ↦ by\n    rw [(Units.nonZeroSubMul k V).stabilizer_of_subMul,\n      Module.stabilizer_units_eq_bot_of_ne_zero k b.property]\n  e.trans (Equiv.prodCongrLeft (fun _ ↦ (equivQuotientOrbitRel k V).symm))\n\ninstance isEmpty_of_subsingleton [Subsingleton V] : IsEmpty (ℙ k V) := by\n  have : IsEmpty { v : V // v ≠ 0 } := ⟨fun v ↦ v.2 (Subsingleton.elim v.1 0)⟩\n  simpa using (nonZeroEquivProjectivizationProdUnits k V).symm.isEmpty\n\n/-- If `V` is a finite `k`-module and `k` is finite, `ℙ k V` is finite. -/\ninstance finite_of_finite [Finite V] : Finite (ℙ k V) :=\n  have : Finite (ℙ k V × kˣ) := Finite.of_equiv _ (nonZeroEquivProjectivizationProdUnits k V)\n  Finite.prod_left kˣ\n\nlemma finite_iff_of_finite [Finite k] : Finite (ℙ k V) ↔ Finite V := by\n  classical\n  refine ⟨fun h ↦ ?_, fun h ↦ inferInstance⟩\n  let e := nonZeroEquivProjectivizationProdUnits k V\n  have : Finite { v : V // v ≠ 0 } := Finite.of_equiv _ e.symm\n  let eq : { v : V // v ≠ 0 } ⊕ Unit ≃ V :=\n    ⟨(Sum.elim Subtype.val (fun _ ↦ 0)), fun v ↦ if h : v = 0 then Sum.inr () else Sum.inl ⟨v, h⟩,\n      by intro x; aesop, by intro x; aesop⟩\n  exact Finite.of_equiv _ eq\n\n/-- Fraction free cardinality formula for the points of `ℙ k V` if `k` and `V` are finite\n(for silly reasons the formula also holds when `k` and `V` are infinite).\nSee `Projectivization.card'` and `Projectivization.card''` for other spellings of the formula. -/\nlemma card : Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1) := by\n  nontriviality V\n  cases finite_or_infinite k with\n  | inr h =>\n    have : Infinite V := Module.Free.infinite k V\n    simp\n  | inl h =>\n  cases finite_or_infinite V with\n  | inr h =>\n    have := not_iff_not.mpr (finite_iff_of_finite k V)\n    push Not at this\n    have : Infinite (ℙ k V) := by rwa [this]\n    simp\n  | inl h =>\n  classical\n  haveI : Fintype V := Fintype.ofFinite V\n  haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)\n  haveI : Fintype k := Fintype.ofFinite k\n  have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp\n  simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]\n  rw [Fintype.card_congr (nonZeroEquivProjectivizationProdUnits k V), Fintype.card_prod]\n\n/-- Cardinality formula for the points of `ℙ k V` if `k` and `V` are finite with less\nnatural subtraction. -/\n\nTarget:\nlemma card' [Finite V] : Nat.card V = Nat.card (ℙ k V) * (Nat.card k - 1) + 1 :=\n\nProof body:\n","rejected":"by\n  admit","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"LinearAlgebra/Projectivization","family_id":"card","file_id":"mathlib/Mathlib/LinearAlgebra/Projectivization/Cardinality.lean","sample_id":"de57e8cad85d8b44648cf10f3d51ce2288da48a2805529ae83b12ceae714db81"},"split":"train"}
{"binding":{"chosen_proof_sha256":"cb1bd661524d698c211f8037516f3bb09f8383d70d26a48c78df08d165e667a0","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6989f734ba078bd696f50a35920dafc64111695822a6a97d925ee485b4e281dd","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8c194e12f2fbe5697fa400d07d4529ac20ba6d8c4893ecec80e27968f828edce","source_sha256":"c35204fe93e22d960b9e0f1d04e2f2f5f2bf73b540a7e7fec0bfcf5bf1349499","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rcases Int.Prime.dvd_mul hp h with hp2 | hpp\n  · apply Or.intro_left\n    exact le_antisymm (Nat.le_of_dvd zero_lt_two hp2) (Nat.Prime.two_le hp)\n  · apply Or.intro_right\n    rw [sq, Int.natAbs_mul] at hpp\n    exact or_self_iff.mp ((Nat.Prime.dvd_mul hp).mp hpp)","hard_negative":true,"metrics":{"chosen_tokens":67,"rejected_tokens":3,"token_jaccard":0.057143,"token_length_ratio":0.044776},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"fa34a54be47df85918d4d4bb180a89b30bad3e3e2e2dc1112fe459bf66c982ba","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.EuclideanDomain.Basic\npublic import Mathlib.Algebra.EuclideanDomain.Int\npublic import Mathlib.Algebra.GCDMonoid.Nat\npublic import Mathlib.Data.Nat.Prime.Int\npublic import Mathlib.RingTheory.PrincipalIdealDomain\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson\n-/\n/-!\n# Divisibility over ℤ\n\nThis file collects results for the integers that use ring theory in their proofs or\ncases of ℤ being examples of structures in ring theory.\n\n## Main statements\n\n* `Int.Prime.dvd_mul'`: A prime number dividing a product in ℤ divides at least one factor.\n* `Int.exists_prime_and_dvd`: Every non-unit integer has a prime divisor.\n* `Int.prime_iff_natAbs_prime`: Primality in ℤ corresponds to primality of its absolute value in ℕ.\n* `Int.span_natAbs`: The principal ideal generated by `a.natAbs` is equal to that of `a`.\n\n## Tags\n\nprime, irreducible, integers, normalization monoid, gcd monoid, greatest common divisor\n-/\n\npublic section\n\nnamespace Int\n\n\ntheorem isCoprime_iff_nat_coprime {a b : ℤ} : IsCoprime a b ↔ Nat.Coprime a.natAbs b.natAbs := by\n  rw [isCoprime_iff_gcd_eq_one, Nat.coprime_iff_gcd_eq_one, gcd_eq_natAbs]\n\n/-- If `gcd a (m * n) ≠ 1`, then `gcd a m ≠ 1` or `gcd a n ≠ 1`. -/\ntheorem gcd_ne_one_iff_gcd_mul_right_ne_one {a : ℤ} {m n : ℕ} :\n    a.gcd (m * n) ≠ 1 ↔ a.gcd m ≠ 1 ∨ a.gcd n ≠ 1 := by\n  simp only [← isCoprime_iff_gcd_eq_one, ← not_and_or, not_iff_not, IsCoprime.mul_right_iff]\n\ntheorem sq_of_gcd_eq_one {a b c : ℤ} (h : Int.gcd a b = 1) (heq : a * b = c ^ 2) :\n    ∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 := by\n  have h' : IsUnit (GCDMonoid.gcd a b) := by\n    rw [← coe_gcd, h, Int.ofNat_one]\n    exact isUnit_one\n  obtain ⟨d, ⟨u, hu⟩⟩ := exists_associated_pow_of_mul_eq_pow h' heq\n  use d\n  rw [← hu]\n  rcases Int.units_eq_one_or u with hu' | hu' <;>\n    · rw [hu']\n      simp\n\ntheorem sq_of_isCoprime {a b c : ℤ} (h : IsCoprime a b) (heq : a * b = c ^ 2) :\n    ∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 :=\n  sq_of_gcd_eq_one (isCoprime_iff_gcd_eq_one.mp h) heq\n\ntheorem natAbs_euclideanDomain_gcd (a b : ℤ) :\n    Int.natAbs (EuclideanDomain.gcd a b) = Int.gcd a b := by\n  apply Nat.dvd_antisymm <;> rw [← Int.natCast_dvd_natCast]\n  · rw [Int.natAbs_dvd]\n    exact Int.dvd_coe_gcd (EuclideanDomain.gcd_dvd_left _ _) (EuclideanDomain.gcd_dvd_right _ _)\n  · rw [Int.dvd_natAbs]\n    exact EuclideanDomain.dvd_gcd (Int.gcd_dvd_left ..) (Int.gcd_dvd_right ..)\n\nend Int\n\ntheorem Int.Prime.dvd_mul {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ m * n) :\n    p ∣ m.natAbs ∨ p ∣ n.natAbs := by\n  rwa [← hp.dvd_mul, ← Int.natAbs_mul, ← Int.natCast_dvd]\n\ntheorem Int.Prime.dvd_mul' {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ m * n) :\n    (p : ℤ) ∣ m ∨ (p : ℤ) ∣ n := by\n  rw [Int.natCast_dvd, Int.natCast_dvd]\n  exact Int.Prime.dvd_mul hp h\n\ntheorem Int.Prime.dvd_pow {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ n ^ k) :\n    p ∣ n.natAbs := by\n  rw [Int.natCast_dvd, Int.natAbs_pow] at h\n  exact hp.dvd_of_dvd_pow h\n\ntheorem Int.Prime.dvd_pow' {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ n ^ k) :\n    (p : ℤ) ∣ n := by\n  rw [Int.natCast_dvd]\n  exact Int.Prime.dvd_pow hp h\n\nTarget:\ntheorem prime_two_or_dvd_of_dvd_two_mul_pow_self_two {m : ℤ} {p : ℕ} (hp : Nat.Prime p)\n    (h : (p : ℤ) ∣ 2 * m ^ 2) : p = 2 ∨ p ∣ Int.natAbs m :=\n\nProof body:\n","rejected":"by\n  exact prime_two_or_dvd_of_dvd_two_mul_pow_self_two","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"fb19c0294d8772bd586deef5b6b35c6681d085d091a92a77f20d63fe4777abfe","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Int","family_id":"prime_two_or_dvd_of_dvd_two_mul_pow_self_two","file_id":"mathlib/Mathlib/RingTheory/Int/Basic.lean","sample_id":"8c194e12f2fbe5697fa400d07d4529ac20ba6d8c4893ecec80e27968f828edce"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0c3061002fd6080e1b0c2c6f335b3c8a50d21cd02337f3272c2365c7c29242da","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6b2eb75d0e9cee43259678b2a625c571bc47f4aa455916d04f3614e5e4006ff6","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1a320ce7ba94d641553130a5996d892f469c6c053d11bc8e9c32a512ac9b6215","source_sha256":"88c666c6d936856962069aacdb5158224b504be63ef88b462fdb4de48e42b813","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (NeZero.ne n)\n  intro i\n  simpa using (h i).injective.ne (by decide : (0 : Fin 3) ≠ 1)","hard_negative":false,"metrics":{"chosen_tokens":42,"rejected_tokens":46,"token_jaccard":0.90625,"token_length_ratio":1.095238},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"fa473beaef49c74d79e1c1230952f4e790b1b0851b8963da5c4b2d06b8633658","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.Convex.Between\npublic import Mathlib.Algebra.Ring.Defs\npublic import Mathlib.Tactic.Continuity\n\nNamespace:\nPolygon\n\nLocal context:\n/-\nCopyright (c) 2026 A. M. Berns. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: A. M. Berns\n-/\n/-!\n# Polygons\n\nThis file defines polygons in affine spaces.\nFor the special case `n = 3`, an interconversion is provided with `Affine.Triangle`.\n\n## Main definitions\n\n* `Polygon P n`: A polygon with `n` vertices in a type `P`.\n\n-/\n\n@[expose] public section\n\nopen Set\n\n/-- A polygon with `n` vertices in a type `P`. -/\nstructure Polygon (P : Type*) (n : ℕ) where\n  /-- The vertices of the polygon, indexed by `Fin n`. -/\n  vertices : Fin n → P\n\nnamespace Polygon\n\nvariable {R V P : Type*} {n : ℕ}\n\n/-- A coercion to function so that vertices can\nbe written as `poly i` instead of `poly.vertices i` -/\ninstance : CoeFun (Polygon P n) (fun _ => Fin n → P) where\n  coe := Polygon.vertices\n\n/-- A polygon has nondegenerate edges if adjacent vertices are distinct. -/\ndef HasNondegenerateEdges (poly : Polygon P n) : Prop :=\n  ∀ i : Fin n, poly i ≠ poly (finRotate n i)\n\ntheorem HasNondegenerateEdges.two_le [NeZero n] {poly : Polygon P n}\n    (h : poly.HasNondegenerateEdges) : 2 ≤ n := by\n  by_contra! hlt\n  interval_cases n\n  · simp_all only [neZero_zero_iff_false]\n  · exact h 0 (by simp)\n\nvariable [Ring R] [AddCommGroup V] [Module R V] [AddTorsor V P]\n\nvariable (R) in\n/-- The `i`-th edge as an affine map `R →ᵃ[R] P`. -/\ndef edgePath (poly : Polygon P n) (i : Fin n) : R →ᵃ[R] P :=\n  AffineMap.lineMap (poly i) (poly (finRotate n i))\n\nvariable (R) in\n/-- The `i`-th edge as a set of points using an `affineSegment`. -/\ndef edgeSet [PartialOrder R] (poly : Polygon P n) (i : Fin n) : Set P :=\n  affineSegment R (poly i) (poly (finRotate n i))\n\nvariable (R) in\n/-- The `edgeSet` is equivalent to the image of the `edgePath`. -/\ntheorem edgeSet_eq_image_edgePath [PartialOrder R] (poly : Polygon P n) (i : Fin n) :\n    poly.edgeSet R i = poly.edgePath R i '' Icc (0 : R) 1 := rfl\n\nvariable (R) in\n/-- The boundary of the polygon is the union of all its edges. -/\ndef boundary [PartialOrder R] (poly : Polygon P n) : Set P :=\n  ⋃ i, poly.edgeSet R i\n\nvariable (R) in\n/-- A polygon has nondegenerate vertices if any three consecutive vertices\nare affinely independent. -/\ndef HasNondegenerateVertices [NeZero n] (poly : Polygon P n) : Prop :=\n  ∀ i : Fin n, AffineIndependent R ![poly i, poly (i + 1), poly (i + 2)]\n\n/-- Polygons with nondegenerate vertices also have nondegenerate edges. -/\n\nTarget:\ntheorem HasNondegenerateVertices.hasNondegenerateEdges [NeZero n] [Nontrivial R]\n    {poly : Polygon P n}\n    (h : poly.HasNondegenerateVertices R) : poly.HasNondegenerateEdges :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (NeZero.ne n)\n  intro i\n  simpa using (h i).injective.ne (by decide : (0 : Fin 3) ≠ 1)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Geometry/Polygon","family_id":"hasnondegeneratevertices","file_id":"mathlib/Mathlib/Geometry/Polygon/Basic.lean","sample_id":"1a320ce7ba94d641553130a5996d892f469c6c053d11bc8e9c32a512ac9b6215"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d988ca19062348f5a91c0b5ac55ee8cf0c8bfcea69c1997f85ea9823ef01e86a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"21544d2f51f369dc7014292be7e61bf52982b39514987f4c5a1ef51ecf1523b5","source_sha256":"059da84dda9e7cf638670a1c074b01de35e7641bc88d590bf894e2e5fc5223e6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  refine ⟨fun h => Set.infinite_prod.1 ?_, ?_⟩\n  · rw [← image_uncurry_prod] at h\n    exact h.of_image _\n  · rintro (⟨hs, b, hb⟩ | ⟨ht, a, ha⟩)\n    · exact hs.image2_left hb (hfs _ hb)\n    · exact ht.image2_right ha (hft _ ha)","hard_negative":false,"metrics":{"chosen_tokens":71,"rejected_tokens":5,"token_jaccard":0.076923,"token_length_ratio":0.070423},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"fa6ca32851f464013e1652313b9c0014aa86a090c204d59e88d22da24284d297","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Finite.Basic\npublic import Mathlib.Data.Fintype.Prod\npublic import Mathlib.Data.Fintype.Pi\npublic import Mathlib.Algebra.Order.Group.Multiset\npublic import Mathlib.Data.ULift\npublic import Mathlib.Data.Set.NAry\n\nNamespace:\nSet\n\nLocal context:\n/-\nCopyright (c) 2022 Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kyle Miller\n-/\n/-!\n# Finiteness of products\n-/\n\npublic section\n\nassert_not_exists IsOrderedRing MonoidWithZero\n\nvariable {α β : Type*}\n\nnamespace Finite\n\ninstance [Finite α] [Finite β] : Finite (α × β) := by\n  haveI := Fintype.ofFinite α\n  haveI := Fintype.ofFinite β\n  infer_instance\n\ninstance {α β : Sort*} [Finite α] [Finite β] : Finite (PProd α β) :=\n  of_equiv _ Equiv.pprodEquivProdPLift.symm\n\ntheorem prod_left (β) [Finite (α × β)] [Nonempty β] : Finite α :=\n  of_surjective (Prod.fst : α × β → α) Prod.fst_surjective\n\ntheorem prod_right (α) [Finite (α × β)] [Nonempty α] : Finite β :=\n  of_surjective (Prod.snd : α × β → β) Prod.snd_surjective\n\nend Finite\n\nlemma Prod.finite_iff [Nonempty α] [Nonempty β] : Finite (α × β) ↔ Finite α ∧ Finite β where\n  mp _ := ⟨.prod_left β, .prod_right α⟩\n  mpr | ⟨_, _⟩ => inferInstance\n\ninstance Pi.finite {α : Sort*} {β : α → Sort*} [Finite α] [∀ a, Finite (β a)] :\n    Finite (∀ a, β a) := by\n  classical\n  haveI := Fintype.ofFinite (PLift α)\n  haveI := fun a => Fintype.ofFinite (PLift (β a))\n  exact\n    Finite.of_equiv (∀ a : PLift α, PLift (β (Equiv.plift a)))\n      (Equiv.piCongr Equiv.plift fun _ => Equiv.plift)\n\ninstance Function.Embedding.finite {α β : Sort*} [Finite β] : Finite (α ↪ β) := by\n  rcases isEmpty_or_nonempty (α ↪ β) with _ | h\n  · infer_instance\n  · refine h.elim fun f => ?_\n    haveI : Finite α := Finite.of_injective _ f.injective\n    exact Finite.of_injective _ DFunLike.coe_injective\n\ninstance Equiv.finite_right {α β : Sort*} [Finite β] : Finite (α ≃ β) :=\n  Finite.of_injective Equiv.toEmbedding fun e₁ e₂ h => Equiv.ext <| by\n    convert! DFunLike.congr_fun h using 0\n\ninstance Equiv.finite_left {α β : Sort*} [Finite α] : Finite (α ≃ β) :=\n  Finite.of_equiv _ ⟨Equiv.symm, Equiv.symm, Equiv.symm_symm, Equiv.symm_symm⟩\n\n@[to_additive]\ninstance MulEquiv.finite_left {α β : Type*} [Mul α] [Mul β] [Finite α] : Finite (α ≃* β) :=\n  Finite.of_injective toEquiv toEquiv_injective\n\n@[to_additive]\ninstance MulEquiv.finite_right {α β : Type*} [Mul α] [Mul β] [Finite β] : Finite (α ≃* β) :=\n  Finite.of_injective toEquiv toEquiv_injective\n\nopen Set Function\n\nvariable {γ : Type*}\n\nnamespace Set\n\n/-! ### Fintype instances\n\nEvery instance here should have a corresponding `Set.Finite` constructor in the next section.\n-/\n\nsection FintypeInstances\n\ninstance fintypeProd (s : Set α) (t : Set β) [Fintype s] [Fintype t] :\n    Fintype (s ×ˢ t : Set (α × β)) :=\n  Fintype.ofFinset (s.toFinset ×ˢ t.toFinset) <| by simp\n\ninstance fintypeOffDiag (s : Set α) [Fintype s] : Fintype s.offDiag :=\n  Fintype.ofFinset s.toFinset.offDiag <| by simp\n\n/-- `image2 f s t` is `Fintype` if `s` and `t` are. -/\ninstance fintypeImage2 [DecidableEq γ] (f : α → β → γ) (s : Set α) (t : Set β) [hs : Fintype s]\n    [ht : Fintype t] : Fintype (image2 f s t : Set γ) := by\n  rw [← image_prod]\n  apply Set.fintypeImage\n\nend FintypeInstances\n\nend Set\n\n/-! ### Finite instances\n\nThere is seemingly some overlap between the following instances and the `Fintype` instances\nin `Data.Set.Finite`. While every `Fintype` instance gives a `Finite` instance, those\ninstances that depend on `Fintype` or `Decidable` instances need an additional `Finite` instance\nto be able to generally apply.\n\nSome set instances do not appear here since they are consequences of others, for example\n`Subtype.Finite` for subsets of a finite type.\n-/\n\n\nnamespace Finite.Set\n\ninstance finite_prod (s : Set α) (t : Set β) [Finite s] [Finite t] :\n    Finite (s ×ˢ t : Set (α × β)) :=\n  Finite.of_equiv _ (Equiv.Set.prod s t).symm\n\ninstance finite_image2 (f : α → β → γ) (s : Set α) (t : Set β) [Finite s] [Finite t] :\n    Finite (image2 f s t : Set γ) := by\n  rw [← image_prod]\n  infer_instance\n\nend Finite.Set\n\nnamespace Set\n\n/-! ### Constructors for `Set.Finite`\n\nEvery constructor here should have a corresponding `Fintype` instance in the previous section\n(or in the `Fintype` module).\n\nThe implementation of these constructors ideally should be no more than `Set.toFinite`,\nafter possibly setting up some `Fintype` and classical `Decidable` instances.\n-/\n\n\nsection SetFiniteConstructors\n\nsection Prod\n\nvariable {s : Set α} {t : Set β}\n\nprotected theorem Finite.prod (hs : s.Finite) (ht : t.Finite) : (s ×ˢ t : Set (α × β)).Finite := by\n  have := hs.to_subtype\n  have := ht.to_subtype\n  apply toFinite\n\ntheorem Finite.of_prod_left (h : (s ×ˢ t : Set (α × β)).Finite) : t.Nonempty → s.Finite :=\n  fun ⟨b, hb⟩ => (h.image Prod.fst).subset fun a ha => ⟨(a, b), ⟨ha, hb⟩, rfl⟩\n\ntheorem Finite.of_prod_right (h : (s ×ˢ t : Set (α × β)).Finite) : s.Nonempty → t.Finite :=\n  fun ⟨a, ha⟩ => (h.image Prod.snd).subset fun b hb => ⟨(a, b), ⟨ha, hb⟩, rfl⟩\n\nprotected theorem Infinite.prod_left (hs : s.Infinite) (ht : t.Nonempty) : (s ×ˢ t).Infinite :=\n  fun h => hs <| h.of_prod_left ht\n\nprotected theorem Infinite.prod_right (ht : t.Infinite) (hs : s.Nonempty) : (s ×ˢ t).Infinite :=\n  fun h => ht <| h.of_prod_right hs\n\nprotected theorem infinite_prod :\n    (s ×ˢ t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty := by\n  refine ⟨fun h => ?_, ?_⟩\n  · simp_rw [Set.Infinite, @and_comm ¬_, ← Classical.not_imp]\n    by_contra!\n    exact h ((this.1 h.nonempty.snd).prod <| this.2 h.nonempty.fst)\n  · rintro (h | h)\n    · exact h.1.prod_left h.2\n    · exact h.1.prod_right h.2\n\ntheorem finite_prod : (s ×ˢ t).Finite ↔ (s.Finite ∨ t = ∅) ∧ (t.Finite ∨ s = ∅) := by\n  contrapose! +distrib; exact Set.infinite_prod\n\nprotected theorem Finite.offDiag {s : Set α} (hs : s.Finite) : s.offDiag.Finite :=\n  (hs.prod hs).subset s.offDiag_subset_prod\n\nprotected theorem Finite.image2 (f : α → β → γ) (hs : s.Finite) (ht : t.Finite) :\n    (image2 f s t).Finite := by\n  have := hs.to_subtype\n  have := ht.to_subtype\n  apply toFinite\n\nend Prod\n\nend SetFiniteConstructors\n\n/-! ### Properties -/\n\ntheorem Finite.toFinset_prod {s : Set α} {t : Set β} (hs : s.Finite) (ht : t.Finite) :\n    hs.toFinset ×ˢ ht.toFinset = (hs.prod ht).toFinset :=\n  Finset.ext <| by simp\n\ntheorem Finite.toFinset_offDiag {s : Set α} (hs : s.Finite) :\n    hs.offDiag.toFinset = hs.toFinset.offDiag :=\n  Finset.ext <| by simp\n\ntheorem finite_image_fst_and_snd_iff {s : Set (α × β)} :\n    (Prod.fst '' s).Finite ∧ (Prod.snd '' s).Finite ↔ s.Finite :=\n  ⟨fun h => (h.1.prod h.2).subset fun _ h => ⟨mem_image_of_mem _ h, mem_image_of_mem _ h⟩,\n    fun h => ⟨h.image _, h.image _⟩⟩\n\n/-! ### Infinite sets -/\n\nvariable {s t : Set α}\n\nsection Image2\n\nvariable {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β}\n\nprotected theorem Infinite.image2_left (hs : s.Infinite) (hb : b ∈ t)\n    (hf : InjOn (fun a => f a b) s) : (image2 f s t).Infinite :=\n  (hs.image hf).mono <| image_subset_image2_left hb\n\nprotected theorem Infinite.image2_right (ht : t.Infinite) (ha : a ∈ s) (hf : InjOn (f a) t) :\n    (image2 f s t).Infinite :=\n  (ht.image hf).mono <| image_subset_image2_right ha\n\nTarget:\ntheorem infinite_image2 (hfs : ∀ b ∈ t, InjOn (fun a => f a b) s) (hft : ∀ a ∈ s, InjOn (f a) t) :\n    (image2 f s t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/Finite","family_id":"infinite_image","file_id":"mathlib/Mathlib/Data/Finite/Prod.lean","sample_id":"21544d2f51f369dc7014292be7e61bf52982b39514987f4c5a1ef51ecf1523b5"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9427a4321d73c521b7283942cf3246fe882d0cae4ae971903987493e7ed9579b","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"09f801cfefe7ea06610ee0102da6822633ab19e08b2a3f6edb5ea40c65abf274","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"7c3c6fabe63da232039bfcfad352f1cc6f001b2ce6fe12a8d6099283072984c1","source_sha256":"d54536321e31ab6ecb143f6ff47411119fb9708f05bec728ae60cd91a7b3224f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isUnit_iff_exists]\n  exact ⟨n, inv_mul_of_unit _ h, mul_inv_of_unit _ h⟩","hard_negative":true,"metrics":{"chosen_tokens":17,"rejected_tokens":3,"token_jaccard":0.133333,"token_length_ratio":0.176471},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"fa74dd762057b0b0d341e9b48021931b8229ae30377b6655d52f4d2b686ee826","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.BigOperators.Associated\npublic import Mathlib.Data.ZMod.Basic\npublic import Mathlib.RingTheory.Coprime.Lemmas\n\nNamespace:\nZMod\n\nLocal context:\n/-\nCopyright (c) 2023 Moritz Firsching. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Moritz Firsching, Ashvni Narayanan, Michael Stoll\n-/\n/-!\n# Lemmas about units in `ZMod`.\n-/\n\n@[expose] public section\n\nassert_not_exists TwoSidedIdeal\n\nnamespace ZMod\n\nvariable {n m : ℕ}\n/-- `unitsMap` is a group homomorphism that maps units of `ZMod m` to units of `ZMod n` when `n`\ndivides `m`. -/\ndef unitsMap (hm : n ∣ m) : (ZMod m)ˣ →* (ZMod n)ˣ := Units.map (castHom hm (ZMod n))\n\nlemma unitsMap_def (hm : n ∣ m) : unitsMap hm = Units.map (castHom hm (ZMod n)) := rfl\n\nlemma unitsMap_comp {d : ℕ} (hm : n ∣ m) (hd : m ∣ d) :\n    (unitsMap hm).comp (unitsMap hd) = unitsMap (dvd_trans hm hd) := by\n  simp only [unitsMap_def]\n  rw [← Units.map_comp]\n  exact congr_arg Units.map <| congr_arg RingHom.toMonoidHom <| castHom_comp hm hd\n\n@[simp]\nlemma unitsMap_self (n : ℕ) : unitsMap (dvd_refl n) = MonoidHom.id _ := by\n  simp [unitsMap, castHom_self]\n\n/-- `unitsMap_val` shows that coercing from `(ZMod m)ˣ` to `ZMod n` gives the same result\nwhen going via `(ZMod n)ˣ` and `ZMod m`. -/\nlemma unitsMap_val (h : n ∣ m) (a : (ZMod m)ˣ) :\n    ↑(unitsMap h a) = ((a : ZMod m).cast : ZMod n) := rfl\n\nlemma isUnit_cast_of_dvd (hm : n ∣ m) (a : Units (ZMod m)) : IsUnit (cast (a : ZMod m) : ZMod n) :=\n  Units.isUnit (unitsMap hm a)\ntheorem unitsMap_surjective [hm : NeZero m] (h : n ∣ m) :\n    Function.Surjective (unitsMap h) := by\n  suffices ∀ x : ℕ, x.Coprime n → ∃ k : ℕ, (x + k * n).Coprime m by\n    intro x\n    have ⟨k, hk⟩ := this x.val.val (val_coe_unit_coprime x)\n    refine ⟨unitOfCoprime _ hk, Units.ext ?_⟩\n    have : NeZero n := ⟨fun hn ↦ hm.out (eq_zero_of_zero_dvd (hn ▸ h))⟩\n    simp [unitsMap_def, -castHom_apply]\n  intro x hx\n  let ps : Finset ℕ := {p ∈ m.primeFactors | ¬p ∣ x}\n  use ps.prod id\n  apply Nat.coprime_of_dvd\n  intro p pp hp hpn\n  by_cases hpx : p ∣ x\n  · have h := Nat.dvd_sub hp hpx\n    rw [add_comm, Nat.add_sub_cancel] at h\n    rcases pp.dvd_mul.mp h with h | h\n    · have ⟨q, hq, hq'⟩ := (pp.prime.dvd_finsetProd_iff id).mp h\n      rw [Finset.mem_filter, Nat.mem_primeFactors,\n        ← (Nat.prime_dvd_prime_iff_eq pp hq.1.1).mp hq'] at hq\n      exact hq.2 hpx\n    · exact Nat.Prime.not_coprime_iff_dvd.mpr ⟨p, pp, hpx, h⟩ hx\n  · have pps : p ∈ ps := Finset.mem_filter.mpr ⟨Nat.mem_primeFactors.mpr ⟨pp, hpn, hm.out⟩, hpx⟩\n    have h := Nat.dvd_sub hp ((Finset.dvd_prod_of_mem id pps).mul_right n)\n    rw [Nat.add_sub_cancel] at h\n    contradiction\n\n-- This needs `Nat.primeFactors`, so cannot go into `Mathlib/Data/ZMod/Basic.lean`.\nopen Nat in\nlemma not_isUnit_of_mem_primeFactors {n p : ℕ} (h : p ∈ n.primeFactors) :\n    ¬ IsUnit (p : ZMod n) := by\n  rw [isUnit_iff_coprime]\n  exact (Prime.dvd_iff_not_coprime <| prime_of_mem_primeFactors h).mp <| dvd_of_mem_primeFactors h\n\nset_option backward.isDefEq.respectTransparency false in\n/-- Any element of `ZMod N` has the form `u * d` where `u` is a unit and `d` is a divisor of `N`. -/\nlemma eq_unit_mul_divisor {N : ℕ} (a : ZMod N) :\n    ∃ d : ℕ, d ∣ N ∧ ∃ (u : ZMod N), IsUnit u ∧ a = u * d := by\n  rcases eq_or_ne N 0 with rfl | hN\n  -- Silly special case : N = 0. Of no mathematical interest, but true, so let's prove it.\n  · change ℤ at a\n    rcases eq_or_ne a 0 with rfl | ha\n    · refine ⟨0, dvd_zero _, 1, isUnit_one, by rw [Nat.cast_zero, mul_zero]⟩\n    refine ⟨a.natAbs, dvd_zero _, Int.sign a, ?_, (Int.sign_mul_natAbs a).symm⟩\n    rcases lt_or_gt_of_ne ha with h | h\n    · simp only [Int.sign_eq_neg_one_of_neg h, IsUnit.neg_iff, isUnit_one]\n    · simp only [Int.sign_eq_one_of_pos h, isUnit_one]\n  -- now the interesting case\n  have : NeZero N := ⟨hN⟩\n  -- Define `d` as the GCD of a lift of `a` and `N`.\n  let d := a.val.gcd N\n  have hd : d ≠ 0 := Nat.gcd_ne_zero_right hN\n  obtain ⟨a₀, (ha₀ : _ = d * _)⟩ := a.val.gcd_dvd_left N\n  obtain ⟨N₀, (hN₀ : _ = d * _)⟩ := a.val.gcd_dvd_right N\n  refine ⟨d, ⟨N₀, hN₀⟩, ?_⟩\n  -- Show `a` is a unit mod `N / d`.\n  have hu₀ : IsUnit (a₀ : ZMod N₀) := by\n    refine (isUnit_iff_coprime _ _).mpr (Nat.isCoprime_iff_coprime.mp ?_)\n    obtain ⟨p, q, hpq⟩ : ∃ (p q : ℤ), d = a.val * p + N * q := ⟨_, _, Nat.gcd_eq_gcd_ab _ _⟩\n    rw [ha₀, hN₀, Nat.cast_mul, Nat.cast_mul, mul_assoc, mul_assoc, ← mul_add, eq_comm,\n      mul_comm _ p, mul_comm _ q] at hpq\n    exact ⟨p, q, Int.eq_one_of_mul_eq_self_right (Nat.cast_ne_zero.mpr hd) hpq⟩\n  -- Lift it arbitrarily to a unit mod `N`.\n  obtain ⟨u, hu⟩ := (unitsMap_surjective (⟨d, mul_comm d N₀ ▸ hN₀⟩ : N₀ ∣ N)) hu₀.unit\n  rw [unitsMap_def, ← Units.val_inj, Units.coe_map, IsUnit.unit_spec, MonoidHom.coe_coe] at hu\n  refine ⟨u.val, u.isUnit, ?_⟩\n  rw [← natCast_zmod_val a, ← natCast_zmod_val u.1, ha₀, ← Nat.cast_mul,\n    natCast_eq_natCast_iff, mul_comm _ d, Nat.ModEq]\n  simp only [hN₀, Nat.mul_mod_mul_left, Nat.mul_right_inj hd]\n  rw [← Nat.ModEq, ← natCast_eq_natCast_iff, ← hu, natCast_val, castHom_apply]\n\ntheorem coe_int_mul_inv_eq_one {n : ℕ} {x : ℤ} (h : IsCoprime x n) :\n    (x : ZMod n) * (x : ZMod n)⁻¹ = 1 := by\n  by_cases hn : n = 0\n  · simp only [hn, Nat.cast_zero, isCoprime_zero_right] at h\n    rcases Int.isUnit_eq_one_or h with h | h <;> simp [h]\n  haveI : NeZero n := ⟨hn⟩\n  rw [← natCast_zmod_val x]\n  apply coe_mul_inv_eq_one\n  rwa [Int.isCoprime_iff_gcd_eq_one, ← Int.gcd_emod, ← val_intCast] at h\n\ntheorem coe_int_inv_mul_eq_one {n : ℕ} {x : ℤ} (h : IsCoprime x n) :\n    (x : ZMod n)⁻¹ * (x : ZMod n) = 1 := by\n  rw [mul_comm, coe_int_mul_inv_eq_one h]\n\nlemma coe_int_mul_val_inv {n : ℕ} [NeZero n] {m : ℤ} (h : IsCoprime m n) :\n    (m * (m⁻¹ : ZMod n).val : ZMod n) = 1 := by\n  rw [natCast_zmod_val, coe_int_mul_inv_eq_one h]\n\nlemma coe_int_val_inv_mul {n : ℕ} [NeZero n] {m : ℤ} (h : IsCoprime m n) :\n    ((m⁻¹ : ZMod n).val : ZMod n) * m = 1 := by\n  rw [mul_comm, coe_int_mul_val_inv h]\n\n/-- The unit of `ZMod m` associated with an integer prime to `n`. -/\ndef unitOfIsCoprime {m : ℕ} (n : ℤ)\n    (h : IsCoprime n (m : ℤ)) : (ZMod m)ˣ where\n  val := n\n  inv := n⁻¹\n  val_inv := coe_int_mul_inv_eq_one h\n  inv_val := coe_int_inv_mul_eq_one h\n\n@[simp]\ntheorem coe_unitOfIsCoprime {m : ℕ} (n : ℤ) (h : IsCoprime n ↑m) :\n    (unitOfIsCoprime n h : ZMod m) = n := rfl\n\nTarget:\ntheorem isUnit_inv {m : ℕ} {n : ℤ} (h : IsUnit (n : ZMod m)) :\n    IsUnit (n : ZMod m)⁻¹ :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_7c3c6fabe63d","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"730c95b19a4e55853480c1cfa4f0282f9ea8ae2205ac7e70ab0404f5ac258b17","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Data/ZMod","family_id":"isunit_inv","file_id":"mathlib/Mathlib/Data/ZMod/Units.lean","sample_id":"7c3c6fabe63da232039bfcfad352f1cc6f001b2ce6fe12a8d6099283072984c1"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7e80ec87feb47441a23ea0489b04ea9d8954472b614c08ce067659d50bc27e69","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a0f03b458ed50734a3025d61af53351276c215c38aa315f74b586dfa99d0d93","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"ec0e8191257eb538c96b568fc34a540eaf43a4e996b7fbafb697f19a5d8137e8","source_sha256":"8edd066203db5b8ec30e5a0f3cf792a4af8c9625cc6277b004728c090fd82f43","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [hasBasis_self, (hasBasis_nhdsSet _).forall_iff (by grind)]\n  intro U ⟨hU, h_KU⟩\n  obtain ⟨L, hL, hKL, hLU⟩ := exists_compact_between hK hU h_KU\n  exact ⟨L, by rwa [← subset_interior_iff_mem_nhdsSet], hL, hLU⟩","hard_negative":false,"metrics":{"chosen_tokens":53,"rejected_tokens":2,"token_jaccard":0.03125,"token_length_ratio":0.037736},"negative_category":"by_exact_placeholder","negative_mode":"by?","pair_id":"fa974d87bd4888292b6b3209e08ac3e2f96c7b2b8f753d1fa553ba2110ed6c5d","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Topology.Compactness.Compact\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov\n-/\n/-!\n# Locally compact spaces\n\nThis file contains basic results about locally compact spaces.\n-/\n\npublic section\n\nopen Set Filter Topology TopologicalSpace\n\nvariable {X : Type*} {Y : Type*} {ι : Type*}\nvariable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}\n\ninstance [WeaklyLocallyCompactSpace X] [WeaklyLocallyCompactSpace Y] :\n    WeaklyLocallyCompactSpace (X × Y) where\n  exists_compact_mem_nhds x :=\n    let ⟨s₁, hc₁, h₁⟩ := exists_compact_mem_nhds x.1\n    let ⟨s₂, hc₂, h₂⟩ := exists_compact_mem_nhds x.2\n    ⟨s₁ ×ˢ s₂, hc₁.prod hc₂, prod_mem_nhds h₁ h₂⟩\n\ninstance {ι : Type*} [Finite ι] {X : ι → Type*} [(i : ι) → TopologicalSpace (X i)]\n    [(i : ι) → WeaklyLocallyCompactSpace (X i)] :\n    WeaklyLocallyCompactSpace ((i : ι) → X i) where\n  exists_compact_mem_nhds f := by\n    choose s hsc hs using fun i ↦ exists_compact_mem_nhds (f i)\n    exact ⟨pi univ s, isCompact_univ_pi hsc, set_pi_mem_nhds univ.toFinite fun i _ ↦ hs i⟩\n\ninstance (priority := 100) [CompactSpace X] : WeaklyLocallyCompactSpace X where\n  exists_compact_mem_nhds _ := ⟨univ, isCompact_univ, univ_mem⟩\n\nprotected theorem Topology.IsClosedEmbedding.weaklyLocallyCompactSpace [WeaklyLocallyCompactSpace Y]\n    {f : X → Y} (hf : IsClosedEmbedding f) : WeaklyLocallyCompactSpace X where\n  exists_compact_mem_nhds x :=\n    let ⟨K, hK, hKx⟩ := exists_compact_mem_nhds (f x)\n    ⟨f ⁻¹' K, hf.isCompact_preimage hK, hf.continuous.continuousAt hKx⟩\n\nprotected theorem IsClosed.weaklyLocallyCompactSpace [WeaklyLocallyCompactSpace X]\n    {s : Set X} (hs : IsClosed s) : WeaklyLocallyCompactSpace s :=\n  hs.isClosedEmbedding_subtypeVal.weaklyLocallyCompactSpace\n\ntheorem IsOpenQuotientMap.weaklyLocallyCompactSpace [WeaklyLocallyCompactSpace X]\n    {f : X → Y} (hf : IsOpenQuotientMap f) : WeaklyLocallyCompactSpace Y where\n  exists_compact_mem_nhds := by\n    refine hf.surjective.forall.2 fun x ↦ ?_\n    rcases exists_compact_mem_nhds x with ⟨K, hKc, hKx⟩\n    exact ⟨f '' K, hKc.image hf.continuous, hf.isOpenMap.image_mem_nhds hKx⟩\n\n/-- In a weakly locally compact space,\nevery compact set is contained in the interior of a compact set. -/\ntheorem exists_compact_superset [WeaklyLocallyCompactSpace X] {K : Set X} (hK : IsCompact K) :\n    ∃ K', IsCompact K' ∧ K ⊆ interior K' := by\n  choose s hc hmem using fun x : X ↦ exists_compact_mem_nhds x\n  rcases hK.elim_nhds_subcover _ fun x _ ↦ interior_mem_nhds.2 (hmem x) with ⟨I, -, hIK⟩\n  refine ⟨⋃ x ∈ I, s x, I.isCompact_biUnion fun _ _ ↦ hc _, hIK.trans ?_⟩\n  exact iUnion₂_subset fun x hx ↦ interior_mono <| subset_iUnion₂ (s := fun x _ ↦ s x) x hx\n\n/-- In a weakly locally compact space,\nthe filters `𝓝 x` and `cocompact X` are disjoint for all `X`. -/\ntheorem disjoint_nhds_cocompact [WeaklyLocallyCompactSpace X] (x : X) :\n    Disjoint (𝓝 x) (cocompact X) :=\n  let ⟨_, hc, hx⟩ := exists_compact_mem_nhds x\n  disjoint_of_disjoint_of_mem disjoint_compl_right hx hc.compl_mem_cocompact\n\ntheorem compact_basis_nhds [LocallyCompactSpace X] (x : X) :\n    (𝓝 x).HasBasis (fun s => s ∈ 𝓝 x ∧ IsCompact s) fun s => s :=\n  hasBasis_self.2 <| by simpa only [and_comm] using LocallyCompactSpace.local_compact_nhds x\n\ntheorem local_compact_nhds [LocallyCompactSpace X] {x : X} {n : Set X} (h : n ∈ 𝓝 x) :\n    ∃ s ∈ 𝓝 x, s ⊆ n ∧ IsCompact s :=\n  LocallyCompactSpace.local_compact_nhds _ _ h\n\ntheorem LocallyCompactSpace.of_hasBasis {ι : X → Type*} {p : ∀ x, ι x → Prop}\n    {s : ∀ x, ι x → Set X} (h : ∀ x, (𝓝 x).HasBasis (p x) (s x))\n    (hc : ∀ x i, p x i → IsCompact (s x i)) : LocallyCompactSpace X :=\n  ⟨fun x _t ht =>\n    let ⟨i, hp, ht⟩ := (h x).mem_iff.1 ht\n    ⟨s x i, (h x).mem_of_mem hp, ht, hc x i hp⟩⟩\n\ninstance Prod.locallyCompactSpace (X : Type*) (Y : Type*) [TopologicalSpace X]\n    [TopologicalSpace Y] [LocallyCompactSpace X] [LocallyCompactSpace Y] :\n    LocallyCompactSpace (X × Y) :=\n  have := fun x : X × Y => (compact_basis_nhds x.1).prod_nhds' (compact_basis_nhds x.2)\n  .of_hasBasis this fun _ _ ⟨⟨_, h₁⟩, _, h₂⟩ => h₁.prod h₂\n\nsection Pi\n\nvariable {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i, LocallyCompactSpace (X i)]\n\n/-- In general it suffices that all but finitely many of the spaces are compact,\n  but that's not straightforward to state and use. -/\ninstance Pi.locallyCompactSpace_of_finite [Finite ι] : LocallyCompactSpace (∀ i, X i) :=\n  ⟨fun t n hn => by\n    rw [nhds_pi, Filter.mem_pi] at hn\n    obtain ⟨s, -, n', hn', hsub⟩ := hn\n    choose n'' hn'' hsub' hc using fun i =>\n      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)\n    refine ⟨(Set.univ : Set ι).pi n'', ?_, subset_trans (fun _ h => ?_) hsub, isCompact_univ_pi hc⟩\n    · exact (set_pi_mem_nhds_iff (@Set.finite_univ ι _) _).mpr fun i _ => hn'' i\n    · exact fun i _ => hsub' i (h i trivial)⟩\n\n/-- For spaces that are not Hausdorff. -/\ninstance Pi.locallyCompactSpace [∀ i, CompactSpace (X i)] : LocallyCompactSpace (∀ i, X i) :=\n  ⟨fun t n hn => by\n    rw [nhds_pi, Filter.mem_pi] at hn\n    obtain ⟨s, hs, n', hn', hsub⟩ := hn\n    choose n'' hn'' hsub' hc using fun i =>\n      LocallyCompactSpace.local_compact_nhds (t i) (n' i) (hn' i)\n    refine ⟨s.pi n'', ?_, subset_trans (fun _ => ?_) hsub, ?_⟩\n    · exact (set_pi_mem_nhds_iff hs _).mpr fun i _ => hn'' i\n    · exact forall₂_imp fun i _ hi' => hsub' i hi'\n    · classical\n      rw [← Set.univ_pi_ite]\n      refine isCompact_univ_pi fun i => ?_\n      by_cases h : i ∈ s\n      · rw [if_pos h]\n        exact hc i\n      · rw [if_neg h]\n        exact CompactSpace.isCompact_univ⟩\n\ninstance Function.locallyCompactSpace_of_finite [Finite ι] [LocallyCompactSpace Y] :\n    LocallyCompactSpace (ι → Y) :=\n  Pi.locallyCompactSpace_of_finite\n\ninstance Function.locallyCompactSpace [LocallyCompactSpace Y] [CompactSpace Y] :\n    LocallyCompactSpace (ι → Y) :=\n  Pi.locallyCompactSpace\n\nend Pi\n\ninstance (priority := 900) [LocallyCompactSpace X] : LocallyCompactPair X Y where\n  exists_mem_nhds_isCompact_mapsTo hf hs :=\n    let ⟨K, hKx, hKs, hKc⟩ := local_compact_nhds (hf.continuousAt hs); ⟨K, hKx, hKc, hKs⟩\n\ninstance (priority := 100) [LocallyCompactSpace X] : WeaklyLocallyCompactSpace X where\n  exists_compact_mem_nhds (x : X) :=\n    let ⟨K, hx, _, hKc⟩ := local_compact_nhds (x := x) univ_mem; ⟨K, hKc, hx⟩\n\n/-- A reformulation of the definition of locally compact space: In a locally compact space,\n  every open set containing `x` has a compact subset containing `x` in its interior. -/\ntheorem exists_compact_subset [LocallyCompactSpace X] {x : X} {U : Set X} (hU : IsOpen U)\n    (hx : x ∈ U) : ∃ K : Set X, IsCompact K ∧ x ∈ interior K ∧ K ⊆ U := by\n  rcases LocallyCompactSpace.local_compact_nhds x U (hU.mem_nhds hx) with ⟨K, h1K, h2K, h3K⟩\n  exact ⟨K, h3K, mem_interior_iff_mem_nhds.2 h1K, h2K⟩\n\n/-- If `f : X → Y` is a continuous map in a locally compact pair of topological spaces,\n`K : set X` is a compact set, and `U` is an open neighbourhood of `f '' K`,\nthen there exists a compact neighbourhood `L` of `K` such that `f` maps `L` to `U`.\n\nThis is a generalization of `exists_mem_nhds_isCompact_mapsTo`. -/\nlemma exists_mem_nhdsSet_isCompact_mapsTo [LocallyCompactPair X Y] {f : X → Y} {K : Set X}\n    {U : Set Y} (hf : Continuous f) (hK : IsCompact K) (hU : IsOpen U) (hKU : MapsTo f K U) :\n    ∃ L ∈ 𝓝ˢ K, IsCompact L ∧ MapsTo f L U := by\n  choose! V hxV hVc hVU using fun x (hx : x ∈ K) ↦\n    exists_mem_nhds_isCompact_mapsTo hf (hU.mem_nhds (hKU hx))\n  rcases hK.elim_nhds_subcover_nhdsSet hxV with ⟨s, hsK, hKs⟩\n  exact ⟨_, hKs, s.isCompact_biUnion fun x hx ↦ hVc x (hsK x hx), mapsTo_iUnion₂.2 fun x hx ↦\n    hVU x (hsK x hx)⟩\n\n/-- In a locally compact space, for every containment `K ⊆ U` of a compact set `K` in an open\n  set `U`, there is a compact neighborhood `L` such that `K ⊆ L ⊆ U`: equivalently, there is a\n  compact `L` such that `K ⊆ interior L` and `L ⊆ U`.\n  See also `exists_compact_closed_between`, in which one guarantees additionally that `L` is closed\n  if the space is regular. -/\ntheorem exists_compact_between [LocallyCompactSpace X] {K U : Set X} (hK : IsCompact K)\n    (hU : IsOpen U) (h_KU : K ⊆ U) : ∃ L, IsCompact L ∧ K ⊆ interior L ∧ L ⊆ U :=\n  let ⟨L, hKL, hL, hLU⟩ := exists_mem_nhdsSet_isCompact_mapsTo continuous_id hK hU h_KU\n  ⟨L, hL, subset_interior_iff_mem_nhdsSet.2 hKL, hLU⟩\n\n/-- In a (possibly non-Hausdorff) locally compact space, for every compact set `K`,\n`𝓝ˢ K` has a basis consisting of compact sets. -/\n\nTarget:\ntheorem IsCompact.nhdsSet_basis_isCompact [LocallyCompactSpace X] {K : Set X} (hK : IsCompact K) :\n    (𝓝ˢ K).HasBasis (fun L ↦ L ∈ 𝓝ˢ K ∧ IsCompact L) id :=\n\nProof body:\n","rejected":"by?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Topology/Compactness","family_id":"iscompact","file_id":"mathlib/Mathlib/Topology/Compactness/LocallyCompact.lean","sample_id":"ec0e8191257eb538c96b568fc34a540eaf43a4e996b7fbafb697f19a5d8137e8"},"split":"train"}
{"binding":{"chosen_proof_sha256":"105da6d71d1fe288f69e471a4b967a3a862bded59f3b72ae1eccbacf108cf793","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"274b4f9cd10909114b550d1e9e2d114135554b5d70ba738af9dee102127bceab","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"316e9edc7144d415a7e46a6a2a53e203a71e0c03a7d0b39706e889dc27b42fb3","source_sha256":"23217497e3239734dae1b907161ba142454fe3246530e95a20731c7a2dcfd18b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  -- Again, it is a \"left inverse equals right inverse\" argument in the convolution monoid.\n  apply left_inv_eq_right_inv\n    (M := Conv A (A ⊗ A))\n    (a := Δ[A])\n  · rw [Conv.mul_eq, Conv.one_eq]\n    simp only [comp_whiskerRight, tensor_whiskerLeft, MonObj.tensorObj.mul_def, Category.assoc,\n      MonObj.tensorObj.one_def]\n    simp only [tensorμ]\n    simp only [Category.assoc, Iso.inv_hom_id_assoc]\n    exact antipode_comul₁ A\n  · rw [Conv.mul_eq, Conv.one_eq]\n    simp only [whiskerLeft_comp, tensor_whiskerLeft, Category.assoc, Iso.inv_hom_id_assoc,\n      MonObj.tensorObj.mul_def, MonObj.tensorObj.one_def]\n    simp only [tensorμ]\n    simp only [Category.assoc, Iso.inv_hom_id_assoc]\n    exact antipode_comul₂ A","hard_negative":true,"metrics":{"chosen_tokens":156,"rejected_tokens":2,"token_jaccard":0.018519,"token_length_ratio":0.012821},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"faefdb0c1a4dfb16e565cd8401ff605d23b532bd779df295657a8ef2735dac88","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Monoidal.Bimon_\npublic import Mathlib.CategoryTheory.Monoidal.Conv\n\nNamespace:\nCategoryTheory.HopfObj\n\nLocal context:\n/-\nCopyright (c) 2024 Lean FRO LLC. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kim Morrison\n-/\n/-!\n# The category of Hopf monoids in a braided monoidal category.\n\n\n## TODO\n\n* Show that in a Cartesian monoidal category Hopf monoids are exactly group objects.\n* Show that `Hopf (ModuleCat R) ≌ HopfAlgCat R`.\n-/\n\n@[expose] public section\n\nnoncomputable section\n\nuniverse v₁ v₂ u₁ u₂ u\n\nopen CategoryTheory MonoidalCategory\n\nnamespace CategoryTheory\nvariable {C : Type u₁} [Category.{v₁} C] [MonoidalCategory.{v₁} C] [BraidedCategory C]\n\nopen scoped MonObj ComonObj\n\n/--\nA Hopf monoid in a braided category `C` is a bimonoid object in `C` equipped with an antipode.\n-/\nclass HopfObj (X : C) extends BimonObj X where\n  /-- The antipode is an endomorphism of the underlying object of the Hopf monoid. -/\n  antipode : X ⟶ X\n  antipode_left (X) : Δ ≫ antipode ▷ X ≫ μ = ε ≫ η := by cat_disch\n  antipode_right (X) : Δ ≫ X ◁ antipode ≫ μ = ε ≫ η := by cat_disch\n\nnamespace HopfObj\n\n@[inherit_doc] scoped notation \"𝒮\" => HopfObj.antipode\n@[inherit_doc] scoped notation \"𝒮[\" M \"]\" => HopfObj.antipode (X := M)\n\nattribute [reassoc (attr := simp)] antipode_left antipode_right\n\n\nend HopfObj\n\nvariable (C)\n\n/--\nA Hopf monoid in a braided category `C` is a bimonoid object in `C` equipped with an antipode.\n-/\nstructure Hopf where\n  /-- The underlying object in the ambient monoidal category -/\n  X : C\n  [hopf : HopfObj X]\n\nattribute [instance] Hopf.hopf\n\nnamespace Hopf\n\nvariable {C}\n\n/-- A Hopf monoid is a bimonoid. -/\ndef toBimon (A : Hopf C) : Bimon C := .mk' A.X\n\n/--\nMorphisms of Hopf monoids are just morphisms of the underlying bimonoids.\nIn fact they automatically intertwine the antipodes, proved below.\n-/\ninstance : Category (Hopf C) :=\n  inferInstanceAs <| Category (InducedCategory (Bimon C) Hopf.toBimon)\n\nend Hopf\n\nnamespace HopfObj\n\nvariable {C}\n\n/-- Morphisms of Hopf monoids intertwine the antipodes. -/\ntheorem hom_antipode {A B : C} [HopfObj A] [HopfObj B] (f : A ⟶ B) [IsBimonHom f] :\n    f ≫ 𝒮 = 𝒮 ≫ f := by\n  -- We show these elements are equal by exhibiting an element in the convolution algebra\n  -- between `A` (as a comonoid) and `B` (as a monoid),\n  -- such that the LHS is a left inverse, and the RHS is a right inverse.\n  apply left_inv_eq_right_inv\n    (M := Conv A B)\n    (a := f)\n  · rw [Conv.mul_eq, Conv.one_eq]\n    simp only [comp_whiskerRight, Category.assoc]\n    slice_lhs 3 4 =>\n      rw [← whisker_exchange]\n    slice_lhs 2 3 =>\n      rw [← tensorHom_def]\n    slice_lhs 1 2 =>\n      rw [← IsComonHom.hom_comul f]\n    slice_lhs 2 4 =>\n      rw [antipode_left]\n    slice_lhs 1 2 =>\n      rw [IsComonHom.hom_counit]\n  · rw [Conv.mul_eq, Conv.one_eq]\n    simp only [whiskerLeft_comp, Category.assoc]\n    slice_lhs 2 3 =>\n      rw [← whisker_exchange]\n    slice_lhs 3 4 =>\n      rw [← tensorHom_def]\n    slice_lhs 3 4 =>\n      rw [← IsMonHom.mul_hom]\n    slice_lhs 1 3 =>\n      rw [antipode_right]\n    slice_lhs 2 3 =>\n      rw [IsMonHom.one_hom]\n\n@[reassoc (attr := simp)]\ntheorem one_antipode (A : C) [HopfObj A] : η[A] ≫ 𝒮[A] = η[A] := by\n  have := (rfl : η[A] ≫ Δ[A] ≫ (𝒮[A] ▷ A) ≫ μ[A] = _)\n  conv at this =>\n    rhs\n    rw [antipode_left]\n  rw [Bimon.one_comul_assoc, tensorHom_def_assoc, unitors_inv_equal,\n    ← rightUnitor_inv_naturality_assoc, whisker_exchange_assoc, ← rightUnitor_inv_naturality_assoc,\n    rightUnitor_inv_naturality_assoc] at this\n  simpa\n\n@[reassoc (attr := simp)]\ntheorem antipode_counit (A : C) [HopfObj A] : 𝒮[A] ≫ ε[A] = ε[A] := by\n  have := (rfl : Δ[A] ≫ (𝒮[A] ▷ A) ≫ μ[A] ≫ ε[A] = _)\n  conv at this =>\n    rhs\n    rw [antipode_left_assoc]\n  rw [Bimon.mul_counit, tensorHom_def', Category.assoc, ← whisker_exchange_assoc] at this\n  simpa [unitors_equal]\n\n/-!\n## The antipode is an antihomomorphism with respect to both the monoid and comonoid structures.\n-/\n\ntheorem antipode_comul₁ (A : C) [HopfObj A] :\n    Δ[A] ≫\n      𝒮[A] ▷ A ≫\n      Δ[A] ▷ A ≫\n      (α_ A A A).hom ≫\n      A ◁ A ◁ Δ[A] ≫\n      A ◁ (α_ A A A).inv ≫\n      A ◁ (β_ A A).hom ▷ A ≫\n      A ◁ (α_ A A A).hom ≫\n      (α_ A A (A ⊗ A)).inv ≫\n      (μ[A] ⊗ₘ μ[A]) =\n    ε[A] ≫ (λ_ (𝟙_ C)).inv ≫ (η[A] ⊗ₘ η[A]) := by\n  slice_lhs 3 5 =>\n    rw [← associator_naturality_right, ← Category.assoc, ← tensorHom_def]\n  slice_lhs 3 9 =>\n    rw [Bimon.compatibility]\n  slice_lhs 1 3 =>\n    rw [antipode_left]\n  simp [MonObj.tensorObj.one_def]\n\n/--\nAuxiliary calculation for `antipode_comul`.\nThis calculation calls for some ASCII art out of This Week's Finds.\n\n```\n   |   |\n   n   n\n  | \\ / |\n  |  /  |\n  | / \\ |\n  | | S S\n  | | \\ /\n  | |  /\n  | | / \\\n  \\ / \\ /\n   v   v\n    \\ /\n     v\n     |\n```\n\nWe move the left antipode up through the crossing,\nthe right antipode down through the crossing,\nthe right multiplication down across the strand,\nreassociate the comultiplications,\nthen use `antipode_right` then `antipode_left` to simplify.\n-/\ntheorem antipode_comul₂ (A : C) [HopfObj A] :\n    Δ[A] ≫\n      Δ[A] ▷ A ≫\n      (α_ A A A).hom ≫\n      A ◁ A ◁ Δ[A] ≫\n      A ◁ A ◁ (β_ A A).hom ≫\n      A ◁ A ◁ (𝒮[A] ⊗ₘ 𝒮[A]) ≫\n      A ◁ (α_ A A A).inv ≫\n      A ◁ (β_ A A).hom ▷ A ≫\n      A ◁ (α_ A A A).hom ≫\n      (α_ A A (A ⊗ A)).inv ≫\n      (μ[A] ⊗ₘ μ[A]) =\n    ε[A] ≫ (λ_ (𝟙_ C)).inv ≫ (η[A] ⊗ₘ η[A]) := by\n  -- We should write a version of `slice_lhs` that zooms through whiskerings.\n  slice_lhs 6 6 =>\n    simp only [tensorHom_def', whiskerLeft_comp]\n  slice_lhs 7 8 =>\n    rw [← whiskerLeft_comp, associator_inv_naturality_middle, whiskerLeft_comp]\n  slice_lhs 8 9 =>\n    rw [← whiskerLeft_comp, ← comp_whiskerRight, BraidedCategory.braiding_naturality_right,\n      comp_whiskerRight, whiskerLeft_comp]\n  slice_lhs 9 10 =>\n    rw [← whiskerLeft_comp, associator_naturality_left, whiskerLeft_comp]\n  slice_lhs 5 6 =>\n    rw [← whiskerLeft_comp, ← whiskerLeft_comp, ← BraidedCategory.braiding_naturality_left,\n      whiskerLeft_comp, whiskerLeft_comp]\n  slice_lhs 11 12 =>\n    rw [tensorHom_def', ← Category.assoc, ← associator_inv_naturality_right]\n  slice_lhs 10 11 =>\n    rw [← whiskerLeft_comp, ← whisker_exchange, whiskerLeft_comp]\n  slice_lhs 6 10 =>\n    simp only [← whiskerLeft_comp]\n    rw [← BraidedCategory.hexagon_reverse_assoc, Iso.inv_hom_id_assoc,\n      ← BraidedCategory.braiding_naturality_left]\n    simp only [whiskerLeft_comp]\n  rw [ComonObj.comul_assoc_flip_assoc, Iso.inv_hom_id_assoc]\n  slice_lhs 2 3 =>\n    simp only [← whiskerLeft_comp]\n    rw [ComonObj.comul_assoc]\n    simp only [whiskerLeft_comp]\n  slice_lhs 3 7 =>\n    simp only [← whiskerLeft_comp]\n    rw [← associator_naturality_middle_assoc, Iso.hom_inv_id_assoc]\n    simp only [← comp_whiskerRight]\n    rw [antipode_right]\n    simp only [comp_whiskerRight]\n    simp only [whiskerLeft_comp]\n  slice_lhs 2 3 =>\n    simp only [← whiskerLeft_comp]\n    rw [ComonObj.counit_comul]\n    simp only [whiskerLeft_comp]\n  slice_lhs 3 4 =>\n    simp only [← whiskerLeft_comp]\n    rw [BraidedCategory.braiding_naturality_left]\n    simp only [whiskerLeft_comp]\n  slice_lhs 4 5 =>\n    simp only [← whiskerLeft_comp]\n    rw [whisker_exchange]\n    simp only [whiskerLeft_comp]\n  slice_lhs 5 7 =>\n    rw [associator_inv_naturality_right_assoc, whisker_exchange]\n  simp only [braiding_tensorUnit_left,\n    whiskerLeft_comp, whiskerLeft_rightUnitor_inv,\n    whiskerRight_id, whiskerLeft_rightUnitor, Category.assoc, Iso.hom_inv_id_assoc,\n    Iso.inv_hom_id_assoc, whiskerLeft_inv_hom_assoc, antipode_right_assoc]\n  rw [rightUnitor_inv_naturality_assoc, tensorHom_def]\n  monoidal\n\nTarget:\ntheorem antipode_comul (A : C) [HopfObj A] :\n    𝒮[A] ≫ Δ[A] = Δ[A] ≫ (β_ _ _).hom ≫ (𝒮[A] ⊗ₘ 𝒮[A]) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_316e9edc7144","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"16903bab18cca94395c17353e82074094cf1cafb49b76cb066438de74df85702","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Monoidal","family_id":"antipode_comul","file_id":"mathlib/Mathlib/CategoryTheory/Monoidal/Hopf_.lean","sample_id":"316e9edc7144d415a7e46a6a2a53e203a71e0c03a7d0b39706e889dc27b42fb3"},"split":"train"}
{"binding":{"chosen_proof_sha256":"9342ddf1a3670b64a91f411236ca705a4f712a410fef1d38cc662b4389a061cb","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7eb57228a163703ead56c3c31e0edfa6a8a9e924d0c95b243604ebfe3b29b339","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"83c5bc19a8475efdecefba11a168fc6bb576240b8897c80b4ded959f3ff7248c","source_sha256":"42f43c83c3b42913128aa30933a73363c485cf2bc69fc613214486004b504ca0","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← image_commutatorSet_closureCommutatorRepresentatives G]\n  exact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _))","hard_negative":true,"metrics":{"chosen_tokens":24,"rejected_tokens":3,"token_jaccard":0.105263,"token_length_ratio":0.125},"negative_category":"circular_dependency","negative_mode":"circular_dependency","pair_id":"fb9fa7aeef1da5dc41f91f04f721b32151067b936f9319513ba7add25384a86b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Group.Subgroup.Finite\npublic import Mathlib.GroupTheory.Commutator.Basic\npublic import Mathlib.GroupTheory.Rank\npublic import Mathlib.GroupTheory.Index\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jordan Brown, Thomas Browning, Patrick Lutz\n-/\n/-!\nThe commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n\npublic section\n\nvariable {G : Type*} [Group G]\n\nnamespace Subgroup\n\n/-- The commutator of a finite direct product is contained in the direct product of the commutators.\n-/\n@[to_additive /-- The commutator of a finite direct product is contained in the direct product of\nthe commutators. -/]\ntheorem commutator_pi_pi_of_finite {η : Type*} [Finite η] {Gs : η → Type*} [∀ i, Group (Gs i)]\n    (H K : ∀ i, Subgroup (Gs i)) : ⁅Subgroup.pi Set.univ H, Subgroup.pi Set.univ K⁆ =\n    Subgroup.pi Set.univ fun i => ⁅H i, K i⁆ := by\n  classical\n    apply le_antisymm (commutator_pi_pi_le H K)\n    rw [pi_le_iff]\n    intro i hi\n    rw [map_commutator]\n    apply commutator_mono <;>\n      · rw [le_pi_iff]\n        intro j _hj\n        rintro _ ⟨x, hx, rfl⟩\n        by_cases h : j = i\n        · subst h\n          simpa using hx\n        · simp [h, one_mem]\n\nvariable [Finite (commutatorSet G)]\n\ninstance : Group.FG (_root_.commutator G) := by\n  rw [commutator_eq_closure]; apply Group.closure_finite_fg\n\nvariable (G) in\nlemma rank_commutator_le_card : Group.rank (_root_.commutator G) ≤ Nat.card (commutatorSet G) := by\n  rw [Subgroup.rank_congr (commutator_eq_closure G)]\n  apply Subgroup.rank_closure_finite_le_nat_card\n\nvariable [Group.FG G]\n\ninstance finiteIndex_center : FiniteIndex (center G) := by\n  obtain ⟨S, -, hS⟩ := Group.rank_spec G\n  exact ⟨mt (Finite.card_eq_zero_of_embedding (quotientCenterEmbedding hS)) Finite.card_pos.ne'⟩\n\nvariable (G) in\nlemma index_center_le_pow : (center G).index ≤ Nat.card (commutatorSet G) ^ Group.rank G := by\n  obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G\n  rw [← hS1, ← Fintype.card_coe, ← Nat.card_eq_fintype_card, ← Finset.coe_sort_coe, ← Nat.card_fun]\n  exact Finite.card_le_of_embedding (quotientCenterEmbedding hS2)\n\nend Subgroup\n\nsection commutatorRepresentatives\n\nopen Subgroup\n\nTarget:\nlemma card_commutatorSet_closureCommutatorRepresentatives :\n    Nat.card (commutatorSet (closureCommutatorRepresentatives G)) = Nat.card (commutatorSet G) :=\n\nProof body:\n","rejected":"by\n  exact card_commutatorSet_closureCommutatorRepresentatives","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"a492ca3d778cae007620ac29501c34872e3f5b01f8fb7718b2f0f59efe97e960","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"GroupTheory/Commutator","family_id":"card_commutatorset_closurecommutatorrepresentatives","file_id":"mathlib/Mathlib/GroupTheory/Commutator/Finite.lean","sample_id":"83c5bc19a8475efdecefba11a168fc6bb576240b8897c80b4ded959f3ff7248c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"1e7bb9669789dabc1d0b4a44d7bd38a7c58136fdfc801571fd7b5e8b4e1c7606","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6a4eb052d860f24edc427522755e82d280520425b9508477ebb40527feb38df0","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4c02a391b7ac277a192dd766770fc91e8893ee8cde8243ce667618594d9c0e76","source_sha256":"81278dc64ee13852107171790942ce238560fcc410268d3130067bd338a7e15e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  conv_lhs => rw [← (mateEquiv adj₁ adj₂).right_inv α]\n  exact (mateEquiv_counit adj₁ adj₂ ((mateEquiv adj₁ adj₂).symm α) d)","hard_negative":true,"metrics":{"chosen_tokens":38,"rejected_tokens":5,"token_jaccard":0.136364,"token_length_ratio":0.131579},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"fc07bc3e3b9a592cff36ec0921c3ff297fc31b4c4fe7b6002c4086fbf020a8c2","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Adjunction.Basic\npublic import Mathlib.CategoryTheory.Functor.TwoSquare\npublic import Mathlib.CategoryTheory.HomCongr\n\nNamespace:\nCategoryTheory\n\nLocal context:\n/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Emily Riehl, Joël Riou\n-/\n/-!\n# Mate of natural transformations\n\nThis file establishes the bijection between the 2-cells\n\n```\n         L₁                  R₁\n      C --→ D             C ←-- D\n    G ↓  ↗  ↓ H         G ↓  ↘  ↓ H\n      E --→ F             E ←-- F\n         L₂                  R₂\n```\n\nwhere `L₁ ⊣ R₁` and `L₂ ⊣ R₂`. The corresponding natural transformations are called mates.\n\nThis bijection includes a number of interesting cases as specializations. For instance, in the\nspecial case where `G,H` are identity functors then the bijection preserves and reflects\nisomorphisms (i.e. we have bijections `(L₂ ⟶ L₁) ≃ (R₁ ⟶ R₂)`, and if either side is an iso then the\nother side is as well). This demonstrates that adjoints to a given functor are unique up to\nisomorphism (since if `L₁ ≅ L₂` then we deduce `R₁ ≅ R₂`).\n\nAnother example arises from considering the square representing that a functor `H` preserves\nproducts, in particular the morphism `H A ⨯ H- ⟶ H (A ⨯ -)`. Then provided `(A ⨯ -)` and `H A ⨯ -`\nhave left adjoints (for instance if the relevant categories are Cartesian closed), the transferred\nnatural transformation is the exponential comparison morphism: `H (A ^ -) ⟶ H A ^ H-`.\nFurthermore if `H` has a left adjoint `L`, this morphism is an isomorphism iff its mate\n`L (H A ⨯ -) ⟶ A ⨯ L-` is an isomorphism, see\nhttps://ncatlab.org/nlab/show/Frobenius+reciprocity#InCategoryTheory.\nThis also relates to Grothendieck's yoga of six operations, though this is not spelled out in\nmathlib: https://ncatlab.org/nlab/show/six+operations.\n-/\n\nset_option backward.defeqAttrib.useBackward true\n\n@[expose] public section\n\nuniverse v₁ v₂ v₃ v₄ v₅ v₆ v₇ v₈ v₉ u₁ u₂ u₃ u₄ u₅ u₆ u₇ u₈ u₉\nnamespace CategoryTheory\n\nopen Category Functor Adjunction NatTrans TwoSquare\n\nsection mateEquiv\n\nvariable {C : Type u₁} {D : Type u₂} {E : Type u₃} {F : Type u₄}\nvariable [Category.{v₁} C] [Category.{v₂} D] [Category.{v₃} E] [Category.{v₄} F]\nvariable {G : C ⥤ E} {H : D ⥤ F} {L₁ : C ⥤ D} {R₁ : D ⥤ C} {L₂ : E ⥤ F} {R₂ : F ⥤ E}\nvariable (adj₁ : L₁ ⊣ R₁) (adj₂ : L₂ ⊣ R₂)\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- Suppose we have a square of functors (where the top and bottom are adjunctions `L₁ ⊣ R₁`\nand `L₂ ⊣ R₂` respectively).\n\n```\n      C ↔ D\n    G ↓   ↓ H\n      E ↔ F\n```\n\nThen we have a bijection between natural transformations `G ⋙ L₂ ⟶ L₁ ⋙ H` and\n`R₁ ⋙ G ⟶ H ⋙ R₂`. This can be seen as a bijection of the 2-cells:\n\n```\n         L₁                  R₁\n      C --→ D             C ←-- D\n    G ↓  ↗  ↓ H         G ↓  ↘  ↓ H\n      E --→ F             E ←-- F\n         L₂                  R₂\n```\n\nNote that if one of the transformations is an iso, it does not imply the other is an iso.\n-/\n@[simps]\ndef mateEquiv : TwoSquare G L₁ L₂ H ≃ TwoSquare R₁ H G R₂ where\n  toFun α := .mk _ _ _ _ <|\n    (rightUnitor _).inv ≫\n    whiskerLeft (R₁ ⋙ G) adj₂.unit ≫\n    (associator _ _ _).hom ≫ whiskerLeft _ (associator _ _ _).inv ≫\n    whiskerLeft R₁ (whiskerRight α.natTrans R₂) ≫\n    whiskerLeft _ (associator _ _ _).hom ≫ (associator _ _ _).inv ≫\n    whiskerRight adj₁.counit (H ⋙ R₂) ≫\n    (leftUnitor _).hom\n  invFun β := .mk _ _ _ _ <|\n    (leftUnitor _).inv ≫\n    whiskerRight adj₁.unit (G ⋙ L₂) ≫\n    (associator _ _ _).inv ≫ whiskerRight (associator _ _ _).hom _ ≫\n    whiskerRight (whiskerLeft L₁ β.natTrans) L₂ ≫\n    whiskerRight (associator _ _ _).inv _ ≫ (associator _ _ _).hom ≫\n    whiskerLeft (L₁ ⋙ H) adj₂.counit ≫\n    (rightUnitor _).hom\n  left_inv α := by\n    ext\n    simp only [comp_obj, whiskerLeft_comp, whiskerLeft_twice, assoc, Iso.hom_inv_id_assoc,\n      whiskerRight_comp, comp_app, id_obj, leftUnitor_inv_app, Functor.whiskerRight_app,\n      Functor.comp_map, associator_inv_app, associator_hom_app, map_id, Functor.whiskerLeft_app,\n      rightUnitor_inv_app, leftUnitor_hom_app, rightUnitor_hom_app, comp_id, id_comp,\n      counit_naturality, counit_naturality_assoc, left_triangle_components_assoc]\n    rw [← assoc, ← Functor.comp_map, α.natTrans.naturality, Functor.comp_map, assoc, ← H.map_comp,\n      left_triangle_components, map_id]\n    simp only [comp_obj, comp_id]\n  right_inv β := by\n    ext\n    simp only [comp_obj, whiskerRight_comp, whiskerRight_twice, assoc, Iso.inv_hom_id_assoc,\n      whiskerLeft_comp, comp_app, id_obj, rightUnitor_inv_app, Functor.whiskerLeft_app,\n      associator_hom_app, associator_inv_app, Functor.whiskerRight_app, leftUnitor_inv_app, map_id,\n      Functor.comp_map, rightUnitor_hom_app, leftUnitor_hom_app, comp_id, id_comp,\n      unit_naturality_assoc, right_triangle_components_assoc]\n    rw [← assoc, ← Functor.comp_map, assoc, ← β.natTrans.naturality, ← assoc, Functor.comp_map,\n      ← G.map_comp, right_triangle_components, map_id, id_comp]\n\nset_option backward.defeqAttrib.useBackward true in\n/-- A component of a transposed version of the mates correspondence. -/\ntheorem mateEquiv_counit (α : TwoSquare G L₁ L₂ H) (d : D) :\n    L₂.map ((mateEquiv adj₁ adj₂ α).app _) ≫ adj₂.counit.app _ =\n      α.app _ ≫ H.map (adj₁.counit.app d) := by simp\n\n/-- A component of a transposed version of the inverse mates correspondence. -/\n\nTarget:\ntheorem mateEquiv_counit_symm (α : TwoSquare R₁ H G R₂) (d : D) :\n    L₂.map (α.app _) ≫ adj₂.counit.app _ =\n      ((mateEquiv adj₁ adj₂).symm α).app _ ≫ H.map (adj₁.counit.app d) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_4c02a391b7ac","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"5a4fab0c29198f803ecee62039477850c92cd9e568d5eb3d98f72e4e757ad36c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Adjunction","family_id":"mateequiv_counit_symm","file_id":"mathlib/Mathlib/CategoryTheory/Adjunction/Mates.lean","sample_id":"4c02a391b7ac277a192dd766770fc91e8893ee8cde8243ce667618594d9c0e76"},"split":"train"}
{"binding":{"chosen_proof_sha256":"19091c655ecaa51563cbf7438e93b5ad161e207efc740ddfb20c17af665fcd74","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"d3ed95d2114a8fd9d1ee96ff835de25fc3a1d3f259eceb409df00724ed57f1ad","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"1d731b6f1a0aa28042e1c006480d5c7f9dc42d3f0a8f8f5b52bbb265d67e141d","source_sha256":"5fbd6e0c9aed4ae0af15e49b0a292eb4e252a42e876e277383b0144f2f430dad","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  intro x y h\n  by_cases hx : x ∈ xs <;> by_cases hy : y ∈ xs\n  · rw [List.mem_iff_getElem?] at hx hy\n    obtain ⟨i, hx⟩ := hx\n    obtain ⟨j, hy⟩ := hy\n    suffices some x = some y by injection this\n    have h₂ := h₁.length_eq\n    rw [List.applyId_zip_eq h₀ h₂ _ _ _ hx] at h\n    rw [← hx, ← hy]; congr\n    apply (List.getElem?_inj _ (h₁.nodup_iff.1 h₀)).mp\n    · symm; rw [h]\n      rw [← List.applyId_zip_eq] <;> assumption\n    · rw [← h₁.length_eq]\n      rw [List.getElem?_eq_some_iff] at hx\n      obtain ⟨hx, hx'⟩ := hx\n      exact hx\n  · rw [← applyId_mem_iff h₀ h₁] at hx hy\n    rw [h] at hx\n    contradiction\n  · rw [← applyId_mem_iff h₀ h₁] at hx hy\n    rw [h] at hx\n    contradiction\n  · rwa [List.applyId_eq_self, List.applyId_eq_self] at h <;> assumption","hard_negative":true,"metrics":{"chosen_tokens":215,"rejected_tokens":5,"token_jaccard":0.047619,"token_length_ratio":0.023256},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"fc4c2b5ab9a129497a5ad09940749ee763eec3aa784deb5d1df78d23c9c5e290","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Batteries.Data.MLList.Basic\npublic import Mathlib.Algebra.Order.Group.Nat\npublic import Mathlib.Algebra.Order.ZeroLEOne\npublic import Mathlib.Data.DFinsupp.Defs\npublic import Mathlib.Data.Finsupp.Defs\npublic import Mathlib.Data.Int.Range\npublic import Mathlib.Data.List.Sigma\npublic import Mathlib.Data.PNat.Notation\npublic import Mathlib.Tactic.Bound.Init\n\nNamespace:\nPlausible.InjectiveFunction\n\nLocal context:\n/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\npublic meta import Mathlib.Data.Finsupp.ToDFinsupp\npublic meta import Mathlib.Data.Int.Range\npublic meta import Mathlib.Data.List.Sigma\npublic meta import Plausible.Functions\n\n/-!\n## `Plausible`: generators for functions\n\nThis file defines `Sampleable` instances for `ℤ → ℤ` injective functions.\n\nInjective functions are generated by creating a list of numbers and\na permutation of that list. The permutation insures that every input\nis mapped to a unique output. When an input is not found in the list\nthe input itself is used as an output.\n\nInjective functions `f : α → α` could be generated easily instead of\n`ℤ → ℤ` by generating a `List α`, removing duplicates and creating a\npermutation. One has to be careful when generating the domain to make\nit vast enough that, when generating arguments to apply `f` to,\nthey argument should be likely to lie in the domain of `f`. This is\nthe reason that injective functions `f : ℤ → ℤ` are generated by\nfixing the domain to the range `[-2*size .. 2*size]`, with `size`\nthe size parameter of the `gen` monad.\n\nMuch of the machinery provided in this file is applicable to generate\ninjective functions of type `α → α` and new instances should be easy\nto define.\n\nOther classes of functions such as monotone functions can generated using\nsimilar techniques. For monotone functions, generating two lists, sorting them\nand matching them should suffice, with appropriate default values.\nSome care must be taken for shrinking such functions to make sure\ntheir defining property is invariant through shrinking. Injective\nfunctions are an example of how complicated it can get.\n-/\n\n@[expose] public meta section\n\nuniverse u v\n\nvariable {α : Type u} {β : Type v}\n\nnamespace Plausible\n\nnamespace TotalFunction\n\nsection Finsupp\n\nvariable [DecidableEq α]\n\n/--\nThis theorem exists because plausible does not have access to dlookup but\nmathlib has all the theory for it and wants to use it. We probably want to\nbring these two together at some point.\n-/\nprivate theorem apply_eq_dlookup (m : List (Σ _ : α, β)) (y : β) (x : α) :\n     (withDefault m y).apply x = (m.dlookup x).getD y := by\n  dsimp only [apply]\n  congr 1\n  induction m with\n  | nil => simp\n  | cons p m ih =>\n    rcases p with ⟨fst, snd⟩\n    by_cases heq : fst = x\n    · simp [heq]\n    · rw [List.dlookup_cons_ne]\n      · simp [heq, ih]\n      · symm\n        simp [heq]\n\nvariable [Zero β] [DecidableEq β]\n\n/-- Map a `TotalFunction` to one whose default value is zero so that it represents a `Finsupp`. -/\n@[simp]\ndef zeroDefault : TotalFunction α β → TotalFunction α β\n  | .withDefault A _ => .withDefault A 0\n\n/-- The support of a zero default `TotalFunction`. -/\ndef zeroDefaultSupp : TotalFunction α β → Finset α\n  | .withDefault A _ =>\n    List.toFinset <| (A.dedupKeys.filter fun ab => Sigma.snd ab ≠ 0).map Sigma.fst\n\n/-- Create a finitely supported function from a total function by taking the default value to\nzero. -/\ndef applyFinsupp (tf : TotalFunction α β) : α →₀ β where\n  support := zeroDefaultSupp tf\n  toFun := tf.zeroDefault.apply\n  mem_support_toFun := by\n    intro a\n    rcases tf with ⟨A, y⟩\n    simp only [zeroDefaultSupp, List.mem_map, List.mem_filter, exists_and_right,\n      List.mem_toFinset, exists_eq_right, Sigma.exists, Ne, zeroDefault]\n    rw [apply_eq_dlookup]\n    constructor\n    · rintro ⟨od, hval, hod⟩\n      have := List.mem_dlookup (List.nodupKeys_dedupKeys A) hval\n      rw [(_ : List.dlookup a A = od)]\n      · simpa using hod\n      · simpa [List.dlookup_dedupKeys]\n    · intro h\n      use (A.dlookup a).getD (0 : β)\n      rw [← List.dlookup_dedupKeys] at h ⊢\n      simp only [h, ← List.mem_dlookup_iff A.nodupKeys_dedupKeys, not_false_iff, Option.mem_def]\n      cases haA : List.dlookup a A.dedupKeys\n      · simp [haA] at h\n      · simp\n\nvariable [SampleableExt α] [SampleableExt β] [Repr α]\n\ninstance Finsupp.sampleableExt : SampleableExt (α →₀ β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\n-- TODO: support a non-constant codomain type\ninstance DFinsupp.sampleableExt : SampleableExt (Π₀ _ : α, β) where\n  proxy := TotalFunction α (SampleableExt.proxy β)\n  interp := fun f => (f.comp SampleableExt.interp).applyFinsupp.toDFinsupp\n  sample := SampleableExt.sample (α := α → β)\n  -- note: no way of shrinking the domain without an inverse to `interp`\n  shrink := { shrink := letI : Shrinkable α := {}; TotalFunction.shrink }\n\nend Finsupp\nend TotalFunction\n\nopen _root_.List\n\n/-- Data structure specifying a total function using a list of pairs\nand a default value returned when the input is not in the domain of\nthe partial function.\n\n`mapToSelf f` encodes `x ↦ f x` when `x ∈ f` and `x ↦ x`,\ni.e. `x` to itself, otherwise.\n\nWe use `Σ` to encode mappings instead of `×` because we\nrely on the association list API defined in `Mathlib/Data/List/Sigma.lean`.\n-/\ninductive InjectiveFunction (α : Type u) : Type u\n  | mapToSelf (xs : List (Σ _ : α, α)) :\n      xs.map Sigma.fst ~ xs.map Sigma.snd → List.Nodup (xs.map Sigma.snd) → InjectiveFunction α\n\ninstance : Inhabited (InjectiveFunction α) :=\n  ⟨⟨[], List.Perm.nil, List.nodup_nil⟩⟩\n\nnamespace InjectiveFunction\n\n/-- Apply a total function to an argument. -/\ndef apply [DecidableEq α] : InjectiveFunction α → α → α\n  | InjectiveFunction.mapToSelf m _ _, x => (m.dlookup x).getD x\n\n/-- Produce a string for a given `InjectiveFunction`.\nThe output is of the form `[x₀ ↦ f x₀, .. xₙ ↦ f xₙ, x ↦ x]`.\nUnlike for `TotalFunction`, the default value is not a constant\nbut the identity function.\n-/\nprotected def repr [Repr α] : InjectiveFunction α → String\n  | InjectiveFunction.mapToSelf m _ _ => s! \"[{TotalFunction.reprAux m}x ↦ x]\"\n\ninstance (α : Type u) [Repr α] : Repr (InjectiveFunction α) where\n  reprPrec f _p := InjectiveFunction.repr f\n\n/-- Interpret a list of pairs as a total function, defaulting to\nthe identity function when no entries are found for a given function -/\ndef List.applyId [DecidableEq α] (xs : List (α × α)) (x : α) : α :=\n  ((xs.map Prod.toSigma).dlookup x).getD x\n\n@[simp]\ntheorem List.applyId_cons [DecidableEq α] (xs : List (α × α)) (x y z : α) :\n    List.applyId ((y, z)::xs) x = if y = x then z else List.applyId xs x := by\n  simp only [List.applyId, List.dlookup, eq_rec_constant, Prod.toSigma, List.map]\n  split_ifs <;> rfl\n\nopen Function\nopen List\n\nopen Nat\n\ntheorem List.applyId_zip_eq [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs)\n    (h₁ : xs.length = ys.length) (x y : α) (i : ℕ) (h₂ : xs[i]? = some x) :\n    List.applyId.{u} (xs.zip ys) x = y ↔ ys[i]? = some y := by\n  induction xs generalizing ys i with\n  | nil => cases h₂\n  | cons x' xs xs_ih =>\n    cases i\n    · simp only [length_cons, lt_add_iff_pos_left, add_pos_iff, Nat.lt_add_one, or_true,\n        getElem?_eq_getElem, getElem_cons_zero, Option.some.injEq] at h₂\n      subst h₂\n      cases ys\n      · cases h₁\n      · simp\n    · cases ys\n      · cases h₁\n      · obtain - | ⟨h₀, h₁⟩ := h₀\n        simp only [getElem?_cons_succ, zip_cons_cons, applyId_cons] at h₂ ⊢\n        rw [if_neg]\n        · apply xs_ih <;> solve_by_elim [Nat.succ.inj]\n        · apply h₀; apply List.mem_of_getElem? h₂\n\ntheorem applyId_mem_iff [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs) (h₁ : xs ~ ys)\n    (x : α) : List.applyId.{u} (xs.zip ys) x ∈ ys ↔ x ∈ xs := by\n  simp only [List.applyId]\n  cases h₃ : List.dlookup x (List.map Prod.toSigma (xs.zip ys)) with\n  | none =>\n    dsimp [Option.getD]\n    rw [h₁.mem_iff]\n  | some val =>\n    have h₂ : ys.Nodup := h₁.nodup_iff.1 h₀\n    replace h₁ : xs.length = ys.length := h₁.length_eq\n    dsimp\n    induction xs generalizing ys with\n    | nil => contradiction\n    | cons x' xs xs_ih =>\n      rcases ys with - | ⟨y, ys⟩\n      · cases h₃\n      dsimp [List.dlookup] at h₃; split_ifs at h₃ with h\n      · rw [Option.some_inj] at h₃\n        subst x'; subst val\n        simp only [List.mem_cons, true_or]\n      · obtain - | ⟨h₀, h₅⟩ := h₀\n        obtain - | ⟨h₂, h₄⟩ := h₂\n        have h₆ := Nat.succ.inj h₁\n        specialize xs_ih h₅ h₃ h₄ h₆\n        simp only [Ne.symm h, xs_ih, List.mem_cons]\n        suffices val ∈ ys by tauto\n        rw [← Option.mem_def, List.mem_dlookup_iff] at h₃\n        · simp only [Prod.toSigma, List.mem_map, Prod.exists] at h₃\n          rcases h₃ with ⟨a, b, h₃, h₄, h₅⟩\n          apply (List.of_mem_zip h₃).2\n        simp only [List.NodupKeys, List.keys, comp_def, Prod.fst_toSigma, List.map_map]\n        rwa [List.map_fst_zip (le_of_eq h₆)]\n\ntheorem List.applyId_eq_self [DecidableEq α] {xs ys : List α} (x : α) :\n    x ∉ xs → List.applyId.{u} (xs.zip ys) x = x := by\n  intro h\n  dsimp [List.applyId]\n  rw [List.dlookup_eq_none.2]\n  · rfl\n  simp only [List.keys, not_exists, Prod.toSigma, exists_and_right, exists_eq_right, List.mem_map,\n    Function.comp_apply, List.map_map, Prod.exists]\n  intro y hy\n  exact h (List.of_mem_zip hy).1\n\nTarget:\ntheorem applyId_injective [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs) (h₁ : xs ~ ys) :\n    Injective.{u + 1, u + 1} (List.applyId (xs.zip ys)) :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_1d731b6f1a0a","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"0d94b5475912bb5aa03ee546cb4afa1233159c376e0c56e4a9e9a37b39be414c","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Testing/Plausible","family_id":"applyid_injective","file_id":"mathlib/Mathlib/Testing/Plausible/Functions.lean","sample_id":"1d731b6f1a0aa28042e1c006480d5c7f9dc42d3f0a8f8f5b52bbb265d67e141d"},"split":"train"}
{"binding":{"chosen_proof_sha256":"e08724fba84126ab1ada747c46865f50f539e3028cb4c27da440863f309c14d1","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"76ec52606c6847d51d5dec2714d2a3c51c8c687ad90026d53e798cf8cb2cf44c","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"5d6d44820ec47fa0236df113ccc6391b9ab3172c1aa71f5359c1564641b4264c","source_sha256":"b397fa221cf5b605f3212baf045d24b21f44907b35b75875e7daf454ffb5e047","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  borelize G\n  rw [modularCharacterFun_eq_haarScalarFactor MeasureTheory.Measure.haar g]\n  exact haarScalarFactor_pos_of_isHaarMeasure _ _","hard_negative":true,"metrics":{"chosen_tokens":17,"rejected_tokens":5,"token_jaccard":0.176471,"token_length_ratio":0.294118},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"fc5b70ce8cac9b3505c98f5a706d3419f21b03b28a57c907f93fd39d4b0110fb","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.MeasureTheory.Function.LocallyIntegrable\npublic import Mathlib.MeasureTheory.Group.Integral\npublic import Mathlib.MeasureTheory.Group.Measure\npublic import Mathlib.Topology.Metrizable.Urysohn\npublic import Mathlib.MeasureTheory.Measure.Haar.Unique\npublic import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic\n\nNamespace:\nMeasureTheory.Measure\n\nLocal context:\n/-\nCopyright (c) 2025 Noam Atar. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Noam Atar\n-/\n/-!\n# Modular character of a locally compact group\n\nOn a locally compact group, there is a natural homomorphism `G → ℝ≥0*`, which for `g : G` gives the\nvalue `μ (· * g⁻¹) / μ`, where `μ` is an (inner regular) Haar measure. This file defines this\nhomomorphism, called the modular character, and shows that it is independent of the chosen Haar\nmeasure.\n\nTODO: Show that the character is continuous.\n\n## Main Declarations\n\n* `modularCharacterFun`: Define the modular character function. If `μ` is a left Haar measure on `G`\n  and `g : G`, the measure `A ↦ μ (A g⁻¹)` is also a left Haar measure, so by uniqueness is of the\n  form `Δ(g) μ`, for `Δ(g) ∈ ℝ≥0`. This `Δ` is the modular character. The result that this does not\n  depend on the measure chosen is `modularCharacterFun_eq_haarScalarFactor`.\n* `modularCharacter`: The homomorphism G →* ℝ≥0 whose toFun is `modularCharacterFun`.\n-/\n\n@[expose] public section\n\nopen MeasureTheory\nopen scoped NNReal\n\nnamespace MeasureTheory\n\nnamespace Measure\n\nvariable {G : Type*} [TopologicalSpace G] [Group G] [IsTopologicalGroup G] [LocallyCompactSpace G]\n\n/-- The modular character as a map is `g ↦ μ (· * g⁻¹) / μ`, where `μ` is a left Haar measure.\n\n  See also `modularCharacter` that defines the map as a homomorphism. -/\n@[to_additive /-- The additive modular character as a map is `g ↦ μ (· - g) / μ`, where `μ` is an\n  left additive Haar measure. -/]\nnoncomputable def modularCharacterFun (g : G) : ℝ≥0 :=\n  letI : MeasurableSpace G := borel G\n  haveI : BorelSpace G := ⟨rfl⟩\n  haarScalarFactor (map (· * g) MeasureTheory.Measure.haar) MeasureTheory.Measure.haar\n\n/-- Independence of modularCharacterFun from the chosen Haar measure. -/\n@[to_additive /-- Independence of addModularCharacterFun from the chosen Haar measure -/]\nlemma modularCharacterFun_eq_haarScalarFactor [MeasurableSpace G] [BorelSpace G] (μ : Measure G)\n    [IsHaarMeasure μ] (g : G) : modularCharacterFun g = haarScalarFactor (map (· * g) μ) μ := by\n  let ν := MeasureTheory.Measure.haar (G := G)\n  obtain ⟨⟨f, f_cont⟩, f_comp, f_nonneg, f_one⟩ :\n    ∃ f : C(G, ℝ), HasCompactSupport f ∧ 0 ≤ f ∧ f 1 ≠ 0 := exists_continuous_nonneg_pos 1\n  have int_f_ne_zero (μ₀ : Measure G) [IsHaarMeasure μ₀] : ∫ x, f x ∂μ₀ ≠ 0 :=\n    ne_of_gt (f_cont.integral_pos_of_hasCompactSupport_nonneg_nonzero f_comp f_nonneg f_one)\n  apply NNReal.coe_injective\n  have t : (∫ x, f (x * g) ∂ν) = (∫ x, f (x * g) ∂(haarScalarFactor ν μ • μ)) := by\n    refine integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport ν μ ?_ ?_\n    · exact Continuous.comp' f_cont (continuous_mul_const g)\n    · have j : (fun x ↦ f (x * g)) = (f ∘ (Homeomorph.mulRight g)) := rfl\n      rw [j]\n      exact HasCompactSupport.comp_homeomorph f_comp _\n  have r : (haarScalarFactor ν μ : ℝ) / (haarScalarFactor ν μ) = 1 := by\n    refine div_self ?_\n    rw [NNReal.coe_ne_zero]\n    apply (ne_of_lt (haarScalarFactor_pos_of_isHaarMeasure _ _)).symm\n  calc\n  ↑(modularCharacterFun g) = ↑(haarScalarFactor (map (· * g) ν) ν) := by borelize G; rfl\n  _ = (∫ x, f x ∂(map (· * g) ν)) / ∫ x, f x ∂ν :=\n    haarScalarFactor_eq_integral_div _ _ f_cont f_comp (int_f_ne_zero ν)\n  _ = (∫ x, f (x * g) ∂ν) / ∫ x, f x ∂ν := by\n    rw [integral_map (AEMeasurable.mul_const aemeasurable_id' _)\n    (Continuous.aestronglyMeasurable f_cont)]\n  _ = (∫ x, f (x * g) ∂(haarScalarFactor ν μ • μ)) / ∫ x, f x ∂ν := by rw [t]\n  _ = (∫ x, f (x * g) ∂(haarScalarFactor ν μ • μ)) / ∫ x, f x ∂(haarScalarFactor ν μ • μ) := by\n    rw [integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport ν μ f_cont f_comp]\n  _ = (haarScalarFactor ν μ • ∫ x, f (x * g) ∂μ) / (haarScalarFactor ν μ • ∫ x, f x ∂μ) := by\n    rw [integral_smul_nnreal_measure, integral_smul_nnreal_measure]\n  _ = (haarScalarFactor ν μ / haarScalarFactor ν μ) * ((∫ x, f (x * g) ∂μ) / ∫ x, f x ∂μ) :=\n    mul_div_mul_comm _ _ _ _\n  _ = 1 * ((∫ x, f (x * g) ∂μ) / ∫ x, f x ∂μ) := by rw [r]\n  _ = (∫ x, f (x * g) ∂μ) / ∫ x, f x ∂μ := by rw [one_mul]\n  _ = (∫ x, f x ∂(map (· * g) μ)) / ∫ x, f x ∂μ := by\n    rw [integral_map (AEMeasurable.mul_const aemeasurable_id' _)\n    (Continuous.aestronglyMeasurable f_cont)]\n  _ = haarScalarFactor (map (· * g) μ) μ :=\n    (haarScalarFactor_eq_integral_div _ _ f_cont f_comp (int_f_ne_zero μ)).symm\n\n@[to_additive]\nlemma map_right_mul_eq_modularCharacterFun_smul [MeasurableSpace G] [BorelSpace G] (μ : Measure G)\n    [IsHaarMeasure μ] [InnerRegular μ] (g : G) : map (· * g) μ = modularCharacterFun g • μ := by\n  rw [modularCharacterFun_eq_haarScalarFactor μ _]\n  exact isMulLeftInvariant_eq_smul_of_innerRegular _ μ\n\n@[to_additive]\n\nTarget:\nlemma modularCharacterFun_pos (g : G) : 0 < modularCharacterFun g :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_5d6d44820ec4","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"6d2e42bdb4e1b04aade046cc6168629faec230f54b8cfd10b70b6b2a6771f3b5","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":true},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"MeasureTheory/Group","family_id":"modularcharacterfun_pos","file_id":"mathlib/Mathlib/MeasureTheory/Group/ModularCharacter.lean","sample_id":"5d6d44820ec47fa0236df113ccc6391b9ab3172c1aa71f5359c1564641b4264c"},"split":"train"}
{"binding":{"chosen_proof_sha256":"d216e83441f49f61f8d591a6008f2029d6a48f6f63ee5482956becb053185e63","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"6235791abd41c06ce11aa0d46e38fd7ac25c86c2a81a2c78dd7083262d6efd02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9c77b43c796b9a66a9282957c2b67d46b27f3beda4c44634fa70becc056cd2ca","source_sha256":"ee7b415cab50ba8cea4663cf633679dc26a3069a4531b1546132f03d2045e5e6","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [commute_iff_eq, hp.le_iff_mul_eq_right hq |>.mp h, hp.le_iff_mul_eq_left hq |>.mp h]","hard_negative":false,"metrics":{"chosen_tokens":25,"rejected_tokens":30,"token_jaccard":0.75,"token_length_ratio":1.2},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"fcf3c2aec84165d9c63bc7475349dfc1ae9077d41f52e556095f10ed35be4064","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order\npublic import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric\nimport Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Projection\n\nNamespace:\nIsStarProjection\n\nLocal context:\n/-\nCopyright (c) 2025 Monica Omar. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Monica Omar, Jireh Loreaux\n-/\n/-!\n\n# Projections in C⋆-algebras\n\nHere we collect results about projections specific to C⋆-algebras.\n\n## Main results\n\n+ `isStarProjection_iff_isIdempotentElem_and_isStarNormal`: star projections are precisely\n  idempotent normal elements.\n+ `IsStarProjection.le_tfae`: for star projections `p` and `q`, the following are equivalent:\n  - `p ≤ q`\n  - `q * p = p`\n  - `p * q = p`\n  - `q - p` is a star projection\n  - `q - p` is an idempotent element\n\n-/\n\npublic section\n\nopen scoped CStarAlgebra\n\nsection NonUnital\nvariable {A : Type*} [TopologicalSpace A] [NonUnitalRing A] [StarRing A]\n\nlemma isStarProjection_iff_quasispectrum_subset_and_isSelfAdjoint [Module ℝ A] [IsScalarTower ℝ A A]\n    [SMulCommClass ℝ A A] [NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint] {p : A} :\n    IsStarProjection p ↔ quasispectrum ℝ p ⊆ {0, 1} ∧ IsSelfAdjoint p :=\n  (isStarProjection_iff p).eq ▸\n    and_congr_left_iff.mpr fun h ↦ isIdempotentElem_iff_quasispectrum_subset ℝ p h\n\nsection Normal\nvariable [Module ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A]\n  [NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal]\n\n/-- An idempotent element in a non-unital C⋆-algebra is self-adjoint iff it is normal. -/\ntheorem IsIdempotentElem.isSelfAdjoint_iff_isStarNormal {p : A} (hp : IsIdempotentElem p) :\n    IsSelfAdjoint p ↔ IsStarNormal p := by\n  simp only [isSelfAdjoint_iff_isStarNormal_and_quasispectrumRestricts,\n    QuasispectrumRestricts.real_iff, and_iff_left_iff_imp]\n  intro h x hx\n  rcases hp.quasispectrum_subset _ hx with (hx | hx) <;> simp [Set.mem_singleton_iff.mp hx]\n\n/-- An element in a non-unital C⋆-algebra is a star projection\nif and only if it is idempotent and normal. -/\ntheorem isStarProjection_iff_isIdempotentElem_and_isStarNormal {p : A} :\n    IsStarProjection p ↔ IsIdempotentElem p ∧ IsStarNormal p :=\n  (isStarProjection_iff p).eq ▸ and_congr_right_iff.eq ▸ fun h => h.isSelfAdjoint_iff_isStarNormal\n\ntheorem isStarProjection_iff_quasispectrum_subset_and_isStarNormal {p : A} :\n    IsStarProjection p ↔ quasispectrum ℂ p ⊆ {0, 1} ∧ IsStarNormal p :=\n  isStarProjection_iff_isIdempotentElem_and_isStarNormal (p := p).eq ▸\n    and_congr_left_iff.mpr fun h ↦ isIdempotentElem_iff_quasispectrum_subset ℂ p h\n\nend Normal\nend NonUnital\n\nsection Unital\nvariable {A : Type*} [TopologicalSpace A] [Ring A] [StarRing A]\n\nlemma isStarProjection_iff_spectrum_subset_and_isSelfAdjoint [Algebra ℝ A]\n    [NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint] {p : A} :\n    IsStarProjection p ↔ spectrum ℝ p ⊆ {0, 1} ∧ IsSelfAdjoint p :=\n  (isStarProjection_iff p).eq ▸\n    and_congr_left_iff.mpr fun h ↦ isIdempotentElem_iff_spectrum_subset ℝ p h\n\ntheorem isStarProjection_iff_spectrum_subset_and_isStarNormal [Algebra ℂ A]\n    [NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal] {p : A} :\n    IsStarProjection p ↔ spectrum ℂ p ⊆ {0, 1} ∧ IsStarNormal p :=\n  isStarProjection_iff_isIdempotentElem_and_isStarNormal (p := p).eq ▸\n    and_congr_left_iff.mpr fun h ↦ isIdempotentElem_iff_spectrum_subset ℂ p h\n\nend Unital\n\nnamespace IsStarProjection\n\nvariable {A : Type*} [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A] {p q : A}\n\nopen CFC in\nlemma le_tfae (hp : IsStarProjection p) (hq : IsStarProjection q) :\n  List.TFAE\n    [p ≤ q,\n    q * p = p,\n    p * q = p,\n    IsStarProjection (q - p),\n    IsIdempotentElem (q - p)] := by\n  tfae_have 1 → 2 := fun h ↦ (hq.mul_right_and_mul_left_of_nonneg_of_le hp.nonneg h).2\n  tfae_have 2 → 3 := fun h ↦ by\n    simpa [hp.isSelfAdjoint.star_eq, hq.isSelfAdjoint.star_eq] using congr(star $h)\n  tfae_have 3 → 4 := hp.sub_of_mul_eq_left hq\n  tfae_have 4 → 1 := fun h ↦ by simpa using h.nonneg\n  tfae_have 4 ↔ 5 := by simp [isStarProjection_iff, hq.isSelfAdjoint.sub hp.isSelfAdjoint]\n  tfae_finish\n\nlemma le_iff_mul_eq_right (hp : IsStarProjection p) (hq : IsStarProjection q) :\n    p ≤ q ↔ q * p = p :=\n  hp.le_tfae hq |>.out 0 1\n\nlemma le_iff_mul_eq_left (hp : IsStarProjection p) (hq : IsStarProjection q) :\n    p ≤ q ↔ p * q = p :=\n  hp.le_tfae hq |>.out 0 2\n\nlemma le_iff_sub (hp : IsStarProjection p) (hq : IsStarProjection q) :\n    p ≤ q ↔ IsStarProjection (q - p) :=\n  hp.le_tfae hq |>.out 0 3\n\nlemma le_iff_idempotent_sub (hp : IsStarProjection p) (hq : IsStarProjection q) :\n    p ≤ q ↔ IsIdempotentElem (q - p) :=\n  hp.le_tfae hq |>.out 0 4\n\nTarget:\nlemma commute_of_le (hp : IsStarProjection p) (hq : IsStarProjection q) (h : p ≤ q) :\n    Commute p q :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  rw [commute_iff_eq, hp.le_iff_mul_eq_right hq |>.mp h, hp.le_iff_mul_eq_left hq |>.mp h]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/CStarAlgebra","family_id":"commute_of_le","file_id":"mathlib/Mathlib/Analysis/CStarAlgebra/Projection.lean","sample_id":"9c77b43c796b9a66a9282957c2b67d46b27f3beda4c44634fa70becc056cd2ca"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b310f14e4f1b5c03181254cbcff43ceec8a366fb3cce3389a0bcba30d3ef880d","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"da9f340528cf03713b6735daff262145f2c76509df440f189b265704baed3570","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"9a2df3c5de7a292cbf62d67b3b00525a929d8c851243d6ce0af4fd45360425d2","source_sha256":"69b176e2c392735eaa07198e23b2563fecf64b20f095a33f543e0985594e4963","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  by_cases h0 : p = 0\n  · simp [h0]\n  by_cases h_top : p = ∞\n  · simp only [h_top, eLpNorm, eLpNormEssSup, ENNReal.top_ne_zero, if_true,\n      if_false]\n    apply essSup_eq\n    exact h.comp (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)\n  simp only [eLpNorm_eq_eLpNorm' h0 h_top, eLpNorm', one_div]\n  congr 1\n  apply lintegral_eq\n  exact h.comp (Measurable.pow_const (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)\n    p.toReal)","hard_negative":false,"metrics":{"chosen_tokens":81,"rejected_tokens":86,"token_jaccard":0.909091,"token_length_ratio":1.061728},"negative_category":"markdown_preamble_trailing","negative_mode":"preamble","pair_id":"fd32fa56b05702cc6d93cd99fa80b39ffb29cc39d57c28cc677cc53f4bb63998","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Probability.HasLaw\npublic import Mathlib.Probability.Moments.Variance\npublic import Mathlib.MeasureTheory.Function.UniformIntegrable\n\nNamespace:\nProbabilityTheory.IdentDistrib\n\nLocal context:\n/-\nCopyright (c) 2022 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n/-!\n# Identically distributed random variables\n\nTwo random variables defined on two (possibly different) probability spaces but taking value in\nthe same space are *identically distributed* if their distributions (i.e., the image probability\nmeasures on the target space) coincide. We define this concept and establish its basic properties\nin this file.\n\n## Main definitions and results\n\n* `IdentDistrib f g μ ν` registers that the image of `μ` under `f` coincides with the image of `ν`\n  under `g` (and that `f` and `g` are almost everywhere measurable, as otherwise the image measures\n  don't make sense). The measures can be kept implicit as in `IdentDistrib f g` if the spaces\n  are registered as measure spaces.\n* `IdentDistrib.comp`: being identically distributed is stable under composition with measurable\n  maps.\n\nThere are two main kinds of lemmas, under the assumption that `f` and `g` are identically\ndistributed: lemmas saying that two quantities computed for `f` and `g` are the same, and lemmas\nsaying that if `f` has some property then `g` also has it. The first kind is registered as\n`IdentDistrib.foo_fst`, the second one as `IdentDistrib.foo_snd` (in the latter case, to deduce\na property of `f` from one of `g`, use `h.symm.foo_snd` where `h : IdentDistrib f g μ ν`). For\ninstance:\n\n* `IdentDistrib.measure_mem_eq`: if `f` and `g` are identically distributed, then the probabilities\n  that they belong to a given measurable set are the same.\n* `IdentDistrib.integral_eq`: if `f` and `g` are identically distributed, then their integrals\n  are the same.\n* `IdentDistrib.variance_eq`: if `f` and `g` are identically distributed, then their variances\n  are the same.\n\n* `IdentDistrib.aestronglyMeasurable_snd`: if `f` and `g` are identically distributed and `f`\n  is almost everywhere strongly measurable, then so is `g`.\n* `IdentDistrib.memLp_snd`: if `f` and `g` are identically distributed and `f`\n  belongs to `ℒp`, then so does `g`.\n\nWe also register several dot notation shortcuts for convenience.\nFor instance, if `h : IdentDistrib f g μ ν`, then `h.sq` states that `f^2` and `g^2` are\nidentically distributed, and `h.norm` states that `‖f‖` and `‖g‖` are identically distributed, and\nso on.\n-/\n\npublic section\n\n\nopen MeasureTheory Filter Finset\n\nnoncomputable section\n\nopen scoped Topology MeasureTheory ENNReal NNReal\n\nvariable {α β γ δ : Type*} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ]\n  [MeasurableSpace δ]\n\nnamespace ProbabilityTheory\n\n/-- Two functions defined on two (possibly different) measure spaces are identically distributed if\ntheir image measures coincide. This only makes sense when the functions are ae measurable\n(as otherwise the image measures are not defined), so we require this as well in the definition. -/\nstructure IdentDistrib (f : α → γ) (g : β → γ)\n    (μ : Measure α := by volume_tac)\n    (ν : Measure β := by volume_tac) : Prop where\n  aemeasurable_fst : AEMeasurable f μ\n  aemeasurable_snd : AEMeasurable g ν\n  map_eq : Measure.map f μ = Measure.map g ν\n\nnamespace IdentDistrib\n\nopen TopologicalSpace\n\nvariable {μ : Measure α} {ν : Measure β} {f : α → γ} {g : β → γ}\n\nprotected theorem refl (hf : AEMeasurable f μ) : IdentDistrib f f μ μ :=\n  { aemeasurable_fst := hf\n    aemeasurable_snd := hf\n    map_eq := rfl }\n\nprotected theorem symm (h : IdentDistrib f g μ ν) : IdentDistrib g f ν μ :=\n  { aemeasurable_fst := h.aemeasurable_snd\n    aemeasurable_snd := h.aemeasurable_fst\n    map_eq := h.map_eq.symm }\n\nprotected theorem trans {ρ : Measure δ} {h : δ → γ} (h₁ : IdentDistrib f g μ ν)\n    (h₂ : IdentDistrib g h ν ρ) : IdentDistrib f h μ ρ :=\n  { aemeasurable_fst := h₁.aemeasurable_fst\n    aemeasurable_snd := h₂.aemeasurable_snd\n    map_eq := h₁.map_eq.trans h₂.map_eq }\n\nprotected theorem comp_of_aemeasurable {u : γ → δ} (h : IdentDistrib f g μ ν)\n    (hu : AEMeasurable u (Measure.map f μ)) : IdentDistrib (u ∘ f) (u ∘ g) μ ν :=\n  { aemeasurable_fst := hu.comp_aemeasurable h.aemeasurable_fst\n    aemeasurable_snd := by rw [h.map_eq] at hu; exact hu.comp_aemeasurable h.aemeasurable_snd\n    map_eq := by\n      rw [← AEMeasurable.map_map_of_aemeasurable hu h.aemeasurable_fst, ←\n        AEMeasurable.map_map_of_aemeasurable _ h.aemeasurable_snd, h.map_eq]\n      rwa [← h.map_eq] }\n\nprotected theorem comp {u : γ → δ} (h : IdentDistrib f g μ ν) (hu : Measurable u) :\n    IdentDistrib (u ∘ f) (u ∘ g) μ ν :=\n  h.comp_of_aemeasurable hu.aemeasurable\n\nprotected theorem of_ae_eq {g : α → γ} (hf : AEMeasurable f μ) (heq : f =ᵐ[μ] g) :\n    IdentDistrib f g μ μ :=\n  { aemeasurable_fst := hf\n    aemeasurable_snd := hf.congr heq\n    map_eq := Measure.map_congr heq }\n\nlemma _root_.MeasureTheory.AEMeasurable.identDistrib_mk\n    (hf : AEMeasurable f μ) : IdentDistrib f (hf.mk f) μ μ :=\n  IdentDistrib.of_ae_eq hf hf.ae_eq_mk\n\nlemma _root_.MeasureTheory.AEStronglyMeasurable.identDistrib_mk\n    [TopologicalSpace γ] [PseudoMetrizableSpace γ] [BorelSpace γ]\n    (hf : AEStronglyMeasurable f μ) : IdentDistrib f (hf.mk f) μ μ :=\n  IdentDistrib.of_ae_eq hf.aemeasurable hf.ae_eq_mk\n\ntheorem measure_mem_eq (h : IdentDistrib f g μ ν) {s : Set γ} (hs : MeasurableSet s) :\n    μ (f ⁻¹' s) = ν (g ⁻¹' s) := by\n  rw [← Measure.map_apply_of_aemeasurable h.aemeasurable_fst hs, ←\n    Measure.map_apply_of_aemeasurable h.aemeasurable_snd hs, h.map_eq]\n\nalias measure_preimage_eq := measure_mem_eq\n\ntheorem ae_snd (h : IdentDistrib f g μ ν) {p : γ → Prop} (pmeas : MeasurableSet {x | p x})\n    (hp : ∀ᵐ x ∂μ, p (f x)) : ∀ᵐ x ∂ν, p (g x) := by\n  apply (ae_map_iff h.aemeasurable_snd pmeas).1\n  rw [← h.map_eq]\n  exact (ae_map_iff h.aemeasurable_fst pmeas).2 hp\n\ntheorem ae_mem_snd (h : IdentDistrib f g μ ν) {t : Set γ} (tmeas : MeasurableSet t)\n    (ht : ∀ᵐ x ∂μ, f x ∈ t) : ∀ᵐ x ∂ν, g x ∈ t :=\n  h.ae_snd tmeas ht\n\ntheorem _root_.ProbabilityTheory.HasLaw.identDistrib {κ : Measure γ} (h₀ : HasLaw f κ μ)\n    (h₁ : HasLaw g κ ν) : IdentDistrib f g μ ν :=\n  ⟨h₀.aemeasurable, h₁.aemeasurable, by simp [h₀.map_eq, h₁.map_eq]⟩\n\ntheorem hasLaw {κ : Measure γ} (h₀ : IdentDistrib f g μ ν) (h₁ : HasLaw f κ μ) : HasLaw g κ ν :=\n  ⟨h₀.aemeasurable_snd, by simp [h₀.map_eq, ← h₁.map_eq]⟩\n\n/-- In a second countable topology, the first function in an identically distributed pair is a.e.\nstrongly measurable. So is the second function, but use `h.symm.aestronglyMeasurable_fst` as\n`h.aestronglyMeasurable_snd` has a different meaning. -/\ntheorem aestronglyMeasurable_fst [TopologicalSpace γ] [PseudoMetrizableSpace γ]\n    [OpensMeasurableSpace γ] [SecondCountableTopology γ] (h : IdentDistrib f g μ ν) :\n    AEStronglyMeasurable f μ :=\n  h.aemeasurable_fst.aestronglyMeasurable\n\n/-- If `f` and `g` are identically distributed and `f` is a.e. strongly measurable, so is `g`. -/\ntheorem aestronglyMeasurable_snd [TopologicalSpace γ] [PseudoMetrizableSpace γ] [BorelSpace γ]\n    (h : IdentDistrib f g μ ν) (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable g ν := by\n  refine aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨h.aemeasurable_snd, ?_⟩\n  rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 hf).2 with ⟨t, t_sep, ht⟩\n  refine ⟨closure t, t_sep.closure, ?_⟩\n  apply h.ae_mem_snd isClosed_closure.measurableSet\n  filter_upwards [ht] with x hx using subset_closure hx\n\ntheorem aestronglyMeasurable_iff [TopologicalSpace γ] [PseudoMetrizableSpace γ] [BorelSpace γ]\n    (h : IdentDistrib f g μ ν) : AEStronglyMeasurable f μ ↔ AEStronglyMeasurable g ν :=\n  ⟨fun hf => h.aestronglyMeasurable_snd hf, fun hg => h.symm.aestronglyMeasurable_snd hg⟩\n\ntheorem essSup_eq [ConditionallyCompleteLinearOrder γ] [TopologicalSpace γ] [OpensMeasurableSpace γ]\n    [OrderClosedTopology γ] (h : IdentDistrib f g μ ν) : essSup f μ = essSup g ν := by\n  have I : ∀ a, μ {x : α | a < f x} = ν {x : β | a < g x} := fun a =>\n    h.measure_mem_eq measurableSet_Ioi\n  simp_rw [essSup_eq_sInf, I]\n\ntheorem lintegral_eq {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (h : IdentDistrib f g μ ν) :\n    ∫⁻ x, f x ∂μ = ∫⁻ x, g x ∂ν := by\n  change ∫⁻ x, id (f x) ∂μ = ∫⁻ x, id (g x) ∂ν\n  rw [← lintegral_map' aemeasurable_id h.aemeasurable_fst, ←\n    lintegral_map' aemeasurable_id h.aemeasurable_snd, h.map_eq]\n\ntheorem integral_eq [NormedAddCommGroup γ] [NormedSpace ℝ γ] [BorelSpace γ]\n    (h : IdentDistrib f g μ ν) : ∫ x, f x ∂μ = ∫ x, g x ∂ν := by\n  by_cases hf : AEStronglyMeasurable f μ\n  · have A : AEStronglyMeasurable id (Measure.map f μ) := by\n      rw [aestronglyMeasurable_iff_aemeasurable_separable]\n      rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 hf).2 with ⟨t, t_sep, ht⟩\n      refine ⟨aemeasurable_id, ⟨closure t, t_sep.closure, ?_⟩⟩\n      rw [ae_map_iff h.aemeasurable_fst]\n      · filter_upwards [ht] with x hx using subset_closure hx\n      · exact isClosed_closure.measurableSet\n    change ∫ x, id (f x) ∂μ = ∫ x, id (g x) ∂ν\n    rw [← integral_map h.aemeasurable_fst A]\n    rw [h.map_eq] at A\n    rw [← integral_map h.aemeasurable_snd A, h.map_eq]\n  · rw [integral_non_aestronglyMeasurable hf]\n    rw [h.aestronglyMeasurable_iff] at hf\n    rw [integral_non_aestronglyMeasurable hf]\n\nTarget:\ntheorem eLpNorm_eq [NormedAddCommGroup γ] [OpensMeasurableSpace γ] (h : IdentDistrib f g μ ν)\n    (p : ℝ≥0∞) : eLpNorm f p μ = eLpNorm g p ν :=\n\nProof body:\n","rejected":"Here is the proof:\nby\n  by_cases h0 : p = 0\n  · simp [h0]\n  by_cases h_top : p = ∞\n  · simp only [h_top, eLpNorm, eLpNormEssSup, ENNReal.top_ne_zero, if_true,\n      if_false]\n    apply essSup_eq\n    exact h.comp (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)\n  simp only [eLpNorm_eq_eLpNorm' h0 h_top, eLpNorm', one_div]\n  congr 1\n  apply lintegral_eq\n  exact h.comp (Measurable.pow_const (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)\n    p.toReal)","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Probability","family_id":"elpnorm_eq","file_id":"mathlib/Mathlib/Probability/IdentDistrib.lean","sample_id":"9a2df3c5de7a292cbf62d67b3b00525a929d8c851243d6ce0af4fd45360425d2"},"split":"train"}
{"binding":{"chosen_proof_sha256":"0e69e1ffdf18797894f9d6a7bd81718318be487366506848fac73b20c0a23f03","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"229982fa3e3692bb975dd7914f9e2dfa6cda96f4ff1e049d7a06c93313e7e6e0","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"e1cca0717031d58aaaf848a8d08234c2dff2f8f85f108568fd9abbcb33e5e463","source_sha256":"71fe80c3d870cb012c2842689740577b7ac73128332425c67807885eb1ca905f","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [internallyProjective_iff_tensor_condition]\n  refine ⟨fun h A B e he S g ↦ ?_, fun h A B e he S g ↦ ?_⟩\n  · specialize h e S ((β_ _ _).hom ≫ g)\n    obtain ⟨S', π, hπ, g', hh⟩ := h\n    refine ⟨S', π, hπ, (β_ _ _).inv ≫ g', ?_⟩\n    simp [← hh]\n  · specialize h e S ((β_ _ _).inv ≫ g)\n    obtain ⟨S', π, hπ, g', hh⟩ := h\n    refine ⟨S', π, hπ, (β_ _ _).hom ≫ g', ?_⟩\n    simp [← hh]","hard_negative":false,"metrics":{"chosen_tokens":151,"rejected_tokens":155,"token_jaccard":0.925,"token_length_ratio":1.02649},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"fd68541312e92487163b18e8a1c37e82393465edc3e32d3541774dc8031b5ef2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Preadditive.Projective.Internal\npublic import Mathlib.Condensed.Light.Epi\npublic import Mathlib.Condensed.Light.Functors\npublic import Mathlib.Condensed.Light.Monoidal\n\nNamespace:\nLightCondensed\n\nLocal context:\n/-\nCopyright (c) 2025 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# Characterization of internal projectivity in light condensed modules\n\nThis file gives an explicit condition on light condensed modules over a ring `R` to be internally\nprojective, namely the following:\n\n`internallyProjective_iff_tensor_condition`: `P : LightCondMod R` is internally projective if and\nonly if, for all `A B : LightCondMod R`, for all epimorphisms `e : A ⟶ B`, for all\n`S : LightProfinite` and all morphisms `g : P ⊗ R[S] ⟶ B`, there exists a `S' : LightProfinite`\nwith a surjection `π : S' ⟶ S` and a morphism `g' : P ⊗ R[S'] ⟶ A`, making the diagram\n```\nP ⊗ R[S'] --> A\n  |           |\n  v           v\nP ⊗ R[S]  --> B\n```\ncommute.\n\nWe also provide the analogous characterization with the tensor product commuted the other way around\n(see `internallyProjective_iff_tensor_condition'`), and the special cases when `P` is the free\ncondensed module on a condensed set (`free_internallyProjective_iff_tensor_condition`,\n`free_internallyProjective_iff_tensor_condition'`) and when `P` is the free condensed module on a\nlight profinite set (`free_lightProfinite_internallyProjective_iff_tensor_condition`/\n`free_lightProfinite_internallyProjective_iff_tensor_condition'`).\n-/\n\n@[expose] public section\n\nuniverse u\n\nopen CategoryTheory Category MonoidalCategory Functor Monoidal LaxMonoidal OplaxMonoidal\n\nvariable (R : Type u) [CommRing R]\n\nnamespace LightCondensed\n\n/--\nThe `S`-valued points of the internal hom `A ⟶[LightCondMod R] B` are in bijection with\nmorphisms `A ⊗ R[S] ⟶ B`.\n-/\nnoncomputable def ihomPoints (A B : LightCondMod.{u} R) (S : LightProfinite) :\n    (A ⟶[LightCondMod R] B).obj.obj ⟨S⟩ ≃ ((A ⊗ ((free R).obj S.toCondensed)) ⟶ B) :=\n  (((freeForgetAdjunction R).homEquiv _ _).trans\n    (coherentTopology _).yonedaEquiv).symm.trans\n      ((ihom.adjunction A).homEquiv _ _).symm\n\nlemma ihomPoints_apply (A B : LightCondMod.{u} R) (S : LightProfinite)\n    (x : (A ⟶[LightCondMod R] B).obj.obj ⟨S⟩) :\n    ihomPoints R A B S x = (MonoidalClosed.uncurry (((freeForgetAdjunction R).homEquiv _ _).symm\n      ((coherentTopology LightProfinite.{u}).yonedaEquiv.symm x))) :=\n  rfl\n\nlemma ihomPoints_symm_apply (A B : LightCondMod.{u} R) (S : LightProfinite)\n    (x : (A ⊗ ((free R).obj S.toCondensed)) ⟶ B) :\n    (ihomPoints R A B S).symm x = (coherentTopology LightProfinite.{u}).yonedaEquiv\n      ((freeForgetAdjunction R).homEquiv _ _ (MonoidalClosed.curry x)) :=\n  rfl\n\nset_option backward.isDefEq.respectTransparency false in\nlemma ihom_map_val_app (A B P : LightCondMod.{u} R) (S : LightProfinite) (e : A ⟶ B)\n    (x : (P ⟶[LightCondMod R] A).obj.obj ⟨S⟩) :\n    (((ihom P).map e).hom.app ⟨S⟩) x = (ihomPoints R P B S).symm (ihomPoints R P A S x ≫ e) := by\n  apply (ihomPoints R P B S).injective\n  simp only [ihomPoints_apply, ← MonoidalClosed.uncurry_natural_right,\n    ← Adjunction.homEquiv_naturality_right_symm, Equiv.apply_symm_apply]\n  congr\n  apply (coherentTopology LightProfinite.{u}).yonedaEquiv.injective\n  simp [dsimp% GrothendieckTopology.yonedaEquiv_comp]\n\nset_option backward.isDefEq.respectTransparency false in\nlemma ihomPoints_symm_comp (B P : LightCondMod.{u} R) (S S' : LightProfinite) (π : S ⟶ S')\n    (f : P ⊗ (free R).obj S'.toCondensed ⟶ B) :\n    (ihomPoints R P B S).symm (P ◁ (free R).map (lightProfiniteToLightCondSet.map π) ≫ f) =\n      ((P ⟶[LightCondMod R] B).obj.map π.op) ((ihomPoints R P B S').symm f) := by\n  simpa [ihomPoints_symm_apply, MonoidalClosed.curry_natural_left, Adjunction.homEquiv_apply] using!\n    (GrothendieckTopology.yonedaEquiv_naturality _ _ _).symm\n\nset_option backward.defeqAttrib.useBackward true in\n/--\n`P : LightCondMod R` is internally projective if and\nonly if, for all `A B : LightCondMod R`, for all epimorphisms `e : A ⟶ B`, for all\n`S : LightProfinite` and all morphisms `g : P ⊗ R[S] ⟶ B`, there exists a `S' : LightProfinite`\nwith a surjection `π : S' ⟶ S` and a morphism `g' : P ⊗ R[S'] ⟶ A`, making the diagram\n```\nP ⊗ R[S'] --> A\n  |           |\n  v           v\nP ⊗ R[S]  --> B\n```\ncommute.\n-/\nlemma internallyProjective_iff_tensor_condition (P : LightCondMod R) : InternallyProjective P ↔\n    ∀ {A B : LightCondMod R} (e : A ⟶ B) [Epi e],\n      (∀ (S : LightProfinite) (g : P ⊗ (free R).obj S.toCondensed ⟶ B), ∃ (S' : LightProfinite)\n        (π : S' ⟶ S) (_ : Function.Surjective π) (g' : P ⊗ (free R).obj S'.toCondensed ⟶ A),\n          (P ◁ ((lightProfiniteToLightCondSet ⋙ free R).map π)) ≫ g = g' ≫ e) := by\n  refine ⟨fun ⟨h⟩ A B e he S g ↦ ?_, fun h ↦ ⟨⟨fun {A B} e he ↦ ?_⟩⟩⟩\n  · have hh := h.1 e\n    rw [LightCondMod.epi_iff_locallySurjective_on_lightProfinite] at hh\n    specialize hh S ((ihomPoints R P B S).symm g)\n    obtain ⟨S', π, hπ, g', hh⟩ := hh\n    refine ⟨S', π, hπ, (ihomPoints _ _ _ _) g', ?_⟩\n    rw [ihom_map_val_app] at hh\n    apply (ihomPoints R P B S').symm.injective\n    rw [hh]\n    exact ihomPoints_symm_comp R B P S' S π g\n  · rw [LightCondMod.epi_iff_locallySurjective_on_lightProfinite]\n    intro S g\n    specialize h e S ((ihomPoints _ _ _ _) g)\n    obtain ⟨S', π, hπ, g', hh⟩ := h\n    refine ⟨S', π, hπ, (ihomPoints _ _ _ _).symm g', ?_⟩\n    rw [ihom_map_val_app]\n    have := ihomPoints_symm_comp R B P S' S π ((ihomPoints R P B S) g)\n    dsimp at hh\n    rw [hh] at this\n    simp [this, Quiver.Hom.op]\n\nset_option backward.defeqAttrib.useBackward true in\n/--\n`P : LightCondMod R` is internally projective if and\nonly if, for all `A B : LightCondMod R`, for all epimorphisms `e : A ⟶ B`, for all\n`S : LightProfinite` and all morphisms `g : R[S] ⊗ P ⟶ B`, there exists a `S' : LightProfinite`\nwith a surjection `π : S' ⟶ S` and a morphism `g' : R[S'] ⊗ P ⟶ A`, making the diagram\n```\nR[S'] ⊗ P --> A\n  |           |\n  v           v\nR[S] ⊗ P  --> B\n```\ncommute.\n-/\n\nTarget:\nlemma internallyProjective_iff_tensor_condition' (P : LightCondMod R) : InternallyProjective P ↔\n    ∀ {A B : LightCondMod R} (e : A ⟶ B) [Epi e],\n      (∀ (S : LightProfinite) (g : (free R).obj S.toCondensed ⊗ P ⟶ B), ∃ (S' : LightProfinite)\n        (π : S' ⟶ S) (_ : Function.Surjective π) (g' : (free R).obj S'.toCondensed ⊗ P ⟶ A),\n          (((lightProfiniteToLightCondSet ⋙ free R).map π) ▷ P) ≫ g = g' ≫ e) :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  rw [internallyProjective_iff_tensor_condition]\n  refine ⟨fun h A B e he S g ↦ ?_, fun h A B e he S g ↦ ?_⟩\n  · specialize h e S ((β_ _ _).hom ≫ g)\n    obtain ⟨S', π, hπ, g', hh⟩ := h\n    refine ⟨S', π, hπ, (β_ _ _).inv ≫ g', ?_⟩\n    simp [← hh]\n  · specialize h e S ((β_ _ _).inv ≫ g)\n    obtain ⟨S', π, hπ, g', hh⟩ := h\n    refine ⟨S', π, hπ, (β_ _ _).hom ≫ g', ?_⟩\n    simp [← hh]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed/Light","family_id":"internallyprojective_iff_tensor_condition","file_id":"mathlib/Mathlib/Condensed/Light/InternallyProjective.lean","sample_id":"e1cca0717031d58aaaf848a8d08234c2dff2f8f85f108568fd9abbcb33e5e463"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c628935d9750505c8adb2f5d44da9ce697c92da40b87a39fbb8e2a552d65d0b4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"e6432d29ef4ba24cfd7ea5c2ca105f906cb2d6c17cea1ff2533b15568a777269","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"8fa0d3b17dbb462bc253c216aedf5600d245ea36ea5fef5dc975ff5afcf7c43e","source_sha256":"c7531e909ba94c250fd36a161457eaacef5f4b4e592debc46a5c88309057a427","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext; exact geometricallyReduced_iff _","hard_negative":false,"metrics":{"chosen_tokens":6,"rejected_tokens":5,"token_jaccard":0.375,"token_length_ratio":0.833333},"negative_category":"sorry_admit_sorryAx_proof_hole","negative_mode":"sorryAx","pair_id":"fd6d204074abcf13128165747e074d7a0749e58f853353eb9dc71697df24d484","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.AlgebraicGeometry.Artinian\npublic import Mathlib.AlgebraicGeometry.Geometrically.Basic\npublic import Mathlib.AlgebraicGeometry.Morphisms.SchemeTheoreticallyDominant\n\nNamespace:\nAlgebraicGeometry\n\nLocal context:\n/-\nCopyright (c) 2026 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\n/-!\n# Geometrically Reduced Schemes\n\n## Main results\n- `AlgebraicGeometry.GeometricallyReduced`:\n  We say that morphism `f : X ⟶ Y` is geometrically reduced if for all `Spec K ⟶ Y` with `K`\n  a field, `X ×[Y] Spec K` is reduced.\n  We also provide the fact that this is stable under base change (by `infer_instance`)\n- `GeometricallyReduced.iff_geometricallyReduced_fiber`:\n  A scheme is geometrically reduced over `S` iff the fibers of all\n  `s : S` are geometrically reduced.\n- `AlgebraicGeometry.GeometricallyReduced.isReduced_of_flat_of_isLocallyNoetherian`:\n  If `X` is geometrically reduced and flat over a reduced and locally noetherian scheme,\n  then `X` is also reduced.\n  In particular, the base change of a geometrically reduced and flat scheme to an\n  reduced and locally noetherian scheme is reduced (by `infer_instance`).\n\n## TODO\nGet rid of the noetherian assumption.\n-/\n\npublic section\n\nopen CategoryTheory MorphismProperty Limits\n\nnamespace AlgebraicGeometry\n\nvariable {X Y Z S : Scheme} (f : X ⟶ S) (g : Y ⟶ S)\n\n/-- We say that morphism `f : X ⟶ Y` is geometrically reduced if for all `Spec K ⟶ Y` with `K`\na field, `X ×[Y] Spec K` is reduced. -/\n@[mk_iff]\nclass GeometricallyReduced (f : X ⟶ Y) : Prop where\n  geometrically_isReduced : geometrically IsReduced f\n\nTarget:\nlemma GeometricallyReduced.eq_geometrically :\n    @GeometricallyReduced = geometrically IsReduced :=\n\nProof body:\n","rejected":"by\n  exact sorryAx _ true","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"AlgebraicGeometry/Geometrically","family_id":"geometricallyreduced","file_id":"mathlib/Mathlib/AlgebraicGeometry/Geometrically/Reduced.lean","sample_id":"8fa0d3b17dbb462bc253c216aedf5600d245ea36ea5fef5dc975ff5afcf7c43e"},"split":"train"}
{"binding":{"chosen_proof_sha256":"b2349a1019905ca8cc28d0ef2d79317628410a89c7793fafcbce2e7a084a9bdd","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"12e7ffd87a87551cb23a2885c902209a7a7da30fe1990c8658984fa1fd83d570","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"16580c7b1066ea68f7db2b4cc23ac07bcdd4aa4217971b759537bed8271b6fb6","source_sha256":"b8e3cefa8cfc385c8eb0b597e4d568bbff6c04129d45f91f63ded9f92406e88b","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [← Ioi_union_left, ← Ioi_union_left]\n  exact\n    (surjOn_Ioi_of_monotone_surjective h_mono h_surj a).union_union\n      (@image_singleton _ _ f a ▸ surjOn_image _ _)","hard_negative":true,"metrics":{"chosen_tokens":30,"rejected_tokens":2,"token_jaccard":0.043478,"token_length_ratio":0.066667},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"fe1f79d77fd6aed889086f3d6d975d142ca2816bfb1e9510f64c8f9cff8c87a7","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Data.Set.Function\npublic import Mathlib.Order.Interval.Set.LinearOrder\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2020 Heather Macbeth. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Heather Macbeth\n-/\n/-!\n# Monotone surjective functions are surjective on intervals\n\nA monotone surjective function sends any interval in the domain onto the interval with corresponding\nendpoints in the range.  This is expressed in this file using `Set.surjOn`, and provided for all\npermutations of interval endpoints.\n-/\n\npublic section\n\n\nvariable {α : Type*} {β : Type*} [LinearOrder α] [PartialOrder β] {f : α → β}\n\nopen Set Function\n\nopen OrderDual (toDual)\n\ntheorem surjOn_Ioo_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    (a b : α) : SurjOn f (Ioo a b) (Ioo (f a) (f b)) := by\n  intro p hp\n  rcases h_surj p with ⟨x, rfl⟩\n  refine ⟨x, mem_Ioo.2 ?_, rfl⟩\n  contrapose! hp\n  exact fun h => h.2.not_ge (h_mono <| hp <| h_mono.reflect_lt h.1)\n\ntheorem surjOn_Ico_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    (a b : α) : SurjOn f (Ico a b) (Ico (f a) (f b)) := by\n  obtain hab | hab := lt_or_ge a b\n  · intro p hp\n    rcases eq_left_or_mem_Ioo_of_mem_Ico hp with (rfl | hp')\n    · exact mem_image_of_mem f (left_mem_Ico.mpr hab)\n    · exact image_mono Ioo_subset_Ico_self <|\n        surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'\n  · rw [Ico_eq_empty (h_mono hab).not_gt]\n    exact surjOn_empty f _\n\ntheorem surjOn_Ioc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    (a b : α) : SurjOn f (Ioc a b) (Ioc (f a) (f b)) := by\n  simpa using! surjOn_Ico_of_monotone_surjective h_mono.dual h_surj (toDual b) (toDual a)\n\n-- to see that the hypothesis `a ≤ b` is necessary, consider a constant function\ntheorem surjOn_Icc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    {a b : α} (hab : a ≤ b) : SurjOn f (Icc a b) (Icc (f a) (f b)) := by\n  intro p hp\n  rcases eq_endpoints_or_mem_Ioo_of_mem_Icc hp with (rfl | rfl | hp')\n  · exact ⟨a, left_mem_Icc.mpr hab, rfl⟩\n  · exact ⟨b, right_mem_Icc.mpr hab, rfl⟩\n  · exact image_mono Ioo_subset_Icc_self <|\n      surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'\n\ntheorem surjOn_Ioi_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    (a : α) : SurjOn f (Ioi a) (Ioi (f a)) := by\n  rw [← compl_Iic, ← compl_compl (Ioi (f a))]\n  refine MapsTo.surjOn_compl ?_ h_surj\n  exact fun x hx => (h_mono hx).not_gt\n\ntheorem surjOn_Iio_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    (a : α) : SurjOn f (Iio a) (Iio (f a)) :=\n  @surjOn_Ioi_of_monotone_surjective _ _ _ _ _ h_mono.dual h_surj a\n\nTarget:\ntheorem surjOn_Ici_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)\n    (a : α) : SurjOn f (Ici a) (Ici (f a)) :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_16580c7b1066","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"b8ad03a9377f68f1de44062eea13597ed3a30f54cd3f9d2ecd60049637c36c7d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Order/Interval","family_id":"surjon_ici_of_monotone_surjective","file_id":"mathlib/Mathlib/Order/Interval/Set/SurjOn.lean","sample_id":"16580c7b1066ea68f7db2b4cc23ac07bcdd4aa4217971b759537bed8271b6fb6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"5668706c687646eb1a543fe99da9ca45de8dce6aa203fed4f6e41809be74067a","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"587a3a6ebe4fb7f9e8b2bca4769899b66e41c67fa88aa76f552baac57f831a85","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"f7c063a4e5099ef9d392d9d1440be2d9799586ee41a1ffac00a85cda728bea8f","source_sha256":"abc84df69b88508535f62fd151f496b3587000aca07e5e3a2f098ce3ffb6b87d","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos\n  have hk₀' : (0 : ℝ) < k := Int.cast_pos.mpr hk₀\n  have hden : ((j / k : ℚ).den : ℤ) ≤ k := by\n    convert! le_of_dvd hk₀ (Rat.den_dvd j k)\n    exact Rat.intCast_div_eq_divInt _ _\n  refine ⟨j / k, ?_, Nat.cast_le.mp (hden.trans hk₁)⟩\n  rw [← div_div, le_div_iff₀ (Nat.cast_pos.mpr <| Rat.pos _ : (0 : ℝ) < _)]\n  refine (mul_le_mul_of_nonneg_left (Int.cast_le.mpr hden : _ ≤ (k : ℝ)) (abs_nonneg _)).trans ?_\n  rwa [← abs_of_pos hk₀', Rat.cast_div, Rat.cast_intCast, Rat.cast_intCast, ← abs_mul, sub_mul,\n    div_mul_cancel₀ _ hk₀'.ne', mul_comm]","hard_negative":false,"metrics":{"chosen_tokens":192,"rejected_tokens":196,"token_jaccard":0.956522,"token_length_ratio":1.020833},"negative_category":"missing_wrong_import","negative_mode":"import","pair_id":"fea5e445f0ecf531c447510cf4a6d9a8a18de16ebf60d8a1e12148fd46792c65","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.NumberTheory.Real.Irrational\npublic import Mathlib.RingTheory.Coprime.Lemmas\npublic import Mathlib.RingTheory.Int.Basic\npublic import Mathlib.Tactic.Basic\n\nNamespace:\nReal\n\nLocal context:\n/-\nCopyright (c) 2022 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Geißer, Michael Stoll\n-/\n/-!\n# Diophantine Approximation\n\nThe first part of this file gives proofs of various versions of\n**Dirichlet's approximation theorem** and its important consequence that when $\\xi$ is an\nirrational real number, then there are infinitely many rationals $x/y$ (in lowest terms)\nsuch that\n$$\\left|\\xi - \\frac{x}{y}\\right| < \\frac{1}{y^2} \\,.$$\nThe proof is based on the pigeonhole principle.\n\nThe second part of the file gives a proof of **Legendre's Theorem** on rational approximation,\nwhich states that if $\\xi$ is a real number and $x/y$ is a rational number such that\n$$\\left|\\xi - \\frac{x}{y}\\right| < \\frac{1}{2y^2} \\,,$$\nthen $x/y$ must be a convergent of the continued fraction expansion of $\\xi$.\n\n## Main statements\n\nThe main results are three variants of Dirichlet's approximation theorem:\n* `Real.exists_int_int_abs_mul_sub_le`, which states that for all real `ξ` and natural `0 < n`,\n  there are integers `j` and `k` with `0 < k ≤ n` and `|k*ξ - j| ≤ 1/(n+1)`,\n* `Real.exists_nat_abs_mul_sub_round_le`, which replaces `j` by `round(k*ξ)` and uses\n  a natural number `k`,\n* `Real.exists_rat_abs_sub_le_and_den_le`, which says that there is a rational number `q`\n  satisfying `|ξ - q| ≤ 1/((n+1)*q.den)` and `q.den ≤ n`,\n\nand\n* `Real.infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational`, which states that\n  for irrational `ξ`, the set `{q : ℚ | |ξ - q| < 1/q.den^2}` is infinite.\n\nWe also show a converse,\n* `Rat.finite_rat_abs_sub_lt_one_div_den_sq`, which states that the set above is finite\n  when `ξ` is a rational number.\n\nBoth statements are combined to give an equivalence,\n`Real.infinite_rat_abs_sub_lt_one_div_den_sq_iff_irrational`.\n\nThere are two versions of Legendre's Theorem. One, `Real.exists_rat_eq_convergent`, uses\n`Real.convergent`, a simple recursive definition of the convergents that is also defined\nin this file, whereas the other, `Real.exists_convs_eq_rat` defined in the file\n`Mathlib/NumberTheory/DiophantineApproximation/ContinuedFractions.lean`, uses\n`GenContFract.convs` of `GenContFract.of ξ`.\n\n## Implementation notes\n\nWe use the namespace `Real` for the results on real numbers and `Rat` for the results\non rational numbers. We introduce a secondary namespace `Real.ContfracLegendre`\nto separate off a definition and some technical auxiliary lemmas used in the proof\nof Legendre's Theorem. For remarks on the proof of Legendre's Theorem, see below.\n\n## References\n\n<https://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem>\n<https://de.wikipedia.org/wiki/Kettenbruch> (The German Wikipedia page on continued\nfractions is much more extensive than the English one.)\n\n## Tags\n\nDiophantine approximation, Dirichlet's approximation theorem, continued fraction\n-/\n\n@[expose] public section\n\n\nnamespace Real\n\nsection Dirichlet\n\n/-!\n### Dirichlet's approximation theorem\n\nWe show that for any real number `ξ` and positive natural `n`, there is a fraction `q`\nsuch that `q.den ≤ n` and `|ξ - q| ≤ 1/((n+1)*q.den)`.\n-/\n\n\nopen Finset Int\n\n/-- *Dirichlet's approximation theorem:*\nFor any real number `ξ` and positive natural `n`, there are integers `j` and `k`,\nwith `0 < k ≤ n` and `|k*ξ - j| ≤ 1/(n+1)`.\n\nSee also `Real.exists_nat_abs_mul_sub_round_le`. -/\ntheorem exists_int_int_abs_mul_sub_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :\n    ∃ j k : ℤ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - j| ≤ 1 / (n + 1) := by\n  let f : ℤ → ℤ := fun m => ⌊fract (ξ * m) * (n + 1)⌋\n  have hn : 0 < (n : ℝ) + 1 := mod_cast Nat.succ_pos _\n  have hfu := fun m : ℤ => mul_lt_of_lt_one_left hn <| fract_lt_one (ξ * ↑m)\n  conv in |_| ≤ _ => rw [mul_comm, le_div_iff₀ hn, ← abs_of_pos hn, ← abs_mul]\n  let D := Icc (0 : ℤ) n\n  by_cases! H : ∃ m ∈ D, f m = n\n  · obtain ⟨m, hm, hf⟩ := H\n    have hf' : ((n : ℤ) : ℝ) ≤ fract (ξ * m) * (n + 1) := hf ▸ floor_le (fract (ξ * m) * (n + 1))\n    have hm₀ : 0 < m := by\n      have hf₀ : f 0 = 0 := by\n        simp only [f, cast_zero, mul_zero, fract_zero, zero_mul, floor_zero]\n      refine Ne.lt_of_le (fun h => n_pos.ne ?_) (mem_Icc.mp hm).1\n      exact mod_cast hf₀.symm.trans (h.symm ▸ hf : f 0 = n)\n    refine ⟨⌊ξ * m⌋ + 1, m, hm₀, (mem_Icc.mp hm).2, ?_⟩\n    rw [cast_add, ← sub_sub, sub_mul, cast_one, one_mul, abs_le]\n    refine\n      ⟨le_sub_iff_add_le.mpr ?_, sub_le_iff_le_add.mpr <| le_of_lt <| (hfu m).trans <| lt_one_add _⟩\n    simpa only [neg_add_cancel_comm_assoc] using! hf'\n  · have hD : #(Ico (0 : ℤ) n) < #D := by rw [card_Icc, card_Ico]; exact lt_add_one n\n    have hfu' : ∀ m, f m ≤ n := fun m => lt_add_one_iff.mp (floor_lt.mpr (mod_cast hfu m))\n    have hwd : ∀ m : ℤ, m ∈ D → f m ∈ Ico (0 : ℤ) n := fun x hx =>\n      mem_Ico.mpr\n        ⟨floor_nonneg.mpr (mul_nonneg (fract_nonneg (ξ * x)) hn.le), Ne.lt_of_le (H x hx) (hfu' x)⟩\n    obtain ⟨x, hx, y, hy, x_lt_y, hxy⟩ : ∃ x ∈ D, ∃ y ∈ D, x < y ∧ f x = f y := by\n      obtain ⟨x, hx, y, hy, x_ne_y, hxy⟩ := exists_ne_map_eq_of_card_lt_of_maps_to hD hwd\n      rcases lt_trichotomy x y with (h | h | h)\n      exacts [⟨x, hx, y, hy, h, hxy⟩, False.elim (x_ne_y h), ⟨y, hy, x, hx, h, hxy.symm⟩]\n    refine\n      ⟨⌊ξ * y⌋ - ⌊ξ * x⌋, y - x, sub_pos_of_lt x_lt_y,\n        sub_le_iff_le_add.mpr <| le_add_of_le_of_nonneg (mem_Icc.mp hy).2 (mem_Icc.mp hx).1, ?_⟩\n    convert_to! |fract (ξ * y) * (n + 1) - fract (ξ * x) * (n + 1)| ≤ 1\n    · congr; push_cast; simp only [fract]; ring\n    exact (abs_sub_lt_one_of_floor_eq_floor hxy.symm).le\n\n/-- *Dirichlet's approximation theorem:*\nFor any real number `ξ` and positive natural `n`, there is a natural number `k`,\nwith `0 < k ≤ n` such that `|k*ξ - round(k*ξ)| ≤ 1/(n+1)`.\n-/\ntheorem exists_nat_abs_mul_sub_round_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :\n    ∃ k : ℕ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - round (↑k * ξ)| ≤ 1 / (n + 1) := by\n  obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos\n  have hk := toNat_of_nonneg hk₀.le\n  rw [← hk] at hk₀ hk₁ h\n  exact ⟨k.toNat, natCast_pos.mp hk₀, Nat.cast_le.mp hk₁, (round_le (↑k.toNat * ξ) j).trans h⟩\n\n/-- *Dirichlet's approximation theorem:*\nFor any real number `ξ` and positive natural `n`, there is a fraction `q`\nsuch that `q.den ≤ n` and `|ξ - q| ≤ 1/((n+1)*q.den)`.\n\nSee also `AddCircle.exists_norm_nsmul_le`. -/\n\nTarget:\ntheorem exists_rat_abs_sub_le_and_den_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :\n    ∃ q : ℚ, |ξ - q| ≤ 1 / ((n + 1) * q.den) ∧ q.den ≤ n :=\n\nProof body:\n","rejected":"import Mathlib.WrongImport\nby\n  obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos\n  have hk₀' : (0 : ℝ) < k := Int.cast_pos.mpr hk₀\n  have hden : ((j / k : ℚ).den : ℤ) ≤ k := by\n    convert! le_of_dvd hk₀ (Rat.den_dvd j k)\n    exact Rat.intCast_div_eq_divInt _ _\n  refine ⟨j / k, ?_, Nat.cast_le.mp (hden.trans hk₁)⟩\n  rw [← div_div, le_div_iff₀ (Nat.cast_pos.mpr <| Rat.pos _ : (0 : ℝ) < _)]\n  refine (mul_le_mul_of_nonneg_left (Int.cast_le.mpr hden : _ ≤ (k : ℝ)) (abs_nonneg _)).trans ?_\n  rwa [← abs_of_pos hk₀', Rat.cast_div, Rat.cast_intCast, Rat.cast_intCast, ← abs_mul, sub_mul,\n    div_mul_cancel₀ _ hk₀'.ne', mul_comm]","rejection":{"accepted":false,"all_failure_types":["not_exact_by_body","top_level_command"],"failure_type":"not_exact_by_body","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"NumberTheory/DiophantineApproximation","family_id":"exists_rat_abs_sub_le_and_den_le","file_id":"mathlib/Mathlib/NumberTheory/DiophantineApproximation/Basic.lean","sample_id":"f7c063a4e5099ef9d392d9d1440be2d9799586ee41a1ffac00a85cda728bea8f"},"split":"train"}
{"binding":{"chosen_proof_sha256":"ff5b125214a3c976ea99dae2474b0a2f292cf08626af894b2e62bc6e985e11b4","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"81c79c52eecec98e2493470604cad60abe03dbfbae1d3ae748117604c4663df8","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"2c8f4c46b465c3b61a9d9f12832774c7822ae516e1e560bc50ecc34f4e2ef998","source_sha256":"d2b257e0c7de516a23c8b73fe5ea40e7a7931660c3e6a7b037f3eda8ab6e5072","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  apply freeYonedaHomEquiv.injective\n  simp only [freeYonedaHomEquiv_comp, Equiv.apply_symm_apply]\n\nvariable (C)","hard_negative":false,"metrics":{"chosen_tokens":18,"rejected_tokens":23,"token_jaccard":0.809524,"token_length_ratio":1.277778},"negative_category":"markdown_preamble_trailing","negative_mode":"trailing","pair_id":"fec7b034d69a1dc3d34be425dc3d2ca348dc1f03cb5763bee58dbcebf9fcded5","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Generator.Basic\npublic import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic\n\nNamespace:\nCategoryTheory.Presheaf\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Generators in the category of presheaves\n\nIn this file, we show that if `A` is a category with zero morphisms that\nhas a separator (and suitable coproducts), then the category of\npresheaves `Cᵒᵖ ⥤ A` also has a separator.\n\n-/\n\n@[expose] public section\n\nuniverse w v' v u' u\n\nnamespace CategoryTheory\n\nopen Limits Opposite\n\nnamespace Presheaf\n\nvariable {C : Type u} [Category.{v} C] {A : Type u'} [Category.{v'} A]\n  [HasCoproducts.{v} A]\n\n/-- Given `X : C` and `M : A`, this is the presheaf `Cᵒᵖ ⥤ A` which sends\n`Y : Cᵒᵖ` to the coproduct of copies of `M` indexed by `Y.unop ⟶ X`. -/\n@[simps]\nnoncomputable def freeYoneda (X : C) (M : A) : Cᵒᵖ ⥤ A where\n  obj Y := ∐ (fun (i : (yoneda.obj X).obj Y) ↦ M)\n  map f := Sigma.map' ((yoneda.obj X).map f) (fun _ ↦ 𝟙 M)\n\nset_option backward.isDefEq.respectTransparency false in\n/-- The bijection `(Presheaf.freeYoneda X M ⟶ F) ≃ (M ⟶ F.obj (op X))`. -/\nnoncomputable def freeYonedaHomEquiv {X : C} {M : A} {F : Cᵒᵖ ⥤ A} :\n    (freeYoneda X M ⟶ F) ≃ (M ⟶ F.obj (op X)) where\n  toFun f := Sigma.ι (fun (i : (yoneda.obj X).obj _) ↦ M) (𝟙 _) ≫ f.app (op X)\n  invFun g :=\n    { app Y := Sigma.desc (fun φ ↦ g ≫ F.map φ.op)\n      naturality _ _ _ := Sigma.hom_ext _ _ (by simp) }\n  left_inv f := by\n    ext Y\n    refine Sigma.hom_ext _ _ (fun φ ↦ ?_)\n    simpa using (Sigma.ι _ (𝟙 _) ≫= f.naturality φ.op).symm\n  right_inv g := by simp\n\nset_option backward.defeqAttrib.useBackward true in\n@[reassoc]\nlemma freeYonedaHomEquiv_comp {X : C} {M : A} {F G : Cᵒᵖ ⥤ A}\n    (α : freeYoneda X M ⟶ F) (f : F ⟶ G) :\n    freeYonedaHomEquiv (α ≫ f) = freeYonedaHomEquiv α ≫ f.app (op X) := by\n  simp [freeYonedaHomEquiv]\n\n@[reassoc]\n\nTarget:\nlemma freeYonedaHomEquiv_symm_comp {X : C} {M : A} {F G : Cᵒᵖ ⥤ A} (α : M ⟶ F.obj (op X))\n    (f : F ⟶ G) :\n    freeYonedaHomEquiv.symm α ≫ f = freeYonedaHomEquiv.symm (α ≫ f.app (op X)) :=\n\nProof body:\n","rejected":"by\n  apply freeYonedaHomEquiv.injective\n  simp only [freeYonedaHomEquiv_comp, Equiv.apply_symm_apply]\n\nvariable (C)\nThis completes the proof.","rejection":{"accepted":false,"all_failure_types":["prose_or_trailing_explanation"],"failure_type":"prose_or_trailing_explanation","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Generator","family_id":"freeyonedahomequiv_symm_comp","file_id":"mathlib/Mathlib/CategoryTheory/Generator/Presheaf.lean","sample_id":"2c8f4c46b465c3b61a9d9f12832774c7822ae516e1e560bc50ecc34f4e2ef998"},"split":"train"}
{"binding":{"chosen_proof_sha256":"156fa55c2e6ebaab51efb61e1b6415d519f8bca6eaddaf4eff8b3327fd1b34c8","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"7023cf0fd56d90184e0f93e416074e4ba1a0d78779cec37f38498aa989407b8b","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"b9d038f0b31926725a67cabc50ad8dfeb3e914d41076f9b7c6d575a332e9b255","source_sha256":"e2806df73ecfc81f91bdaba3be881cdfeef7117f485c371a23cf9b886b2f1661","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n    rcases le_total x 0 with (neg | pos)\n    · exact .inr <| isSquare_of_nonneg (neg_nonneg_of_nonpos neg)\n    · exact .inl <| isSquare_of_nonneg pos\n  exists_isRoot_of_odd_natDegree := exists_isRoot_of_odd_natDegree\n\nvariable [IsRealClosed R]","hard_negative":true,"metrics":{"chosen_tokens":38,"rejected_tokens":3,"token_jaccard":0.074074,"token_length_ratio":0.078947},"negative_category":"unknown_identifier","negative_mode":"unknown_identifier","pair_id":"fece8a02b77f73127ee44b266ecdbee6ca4989dc4e3ff87d1583b4cf58727cc1","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Algebra.Polynomial.Degree.Domain\npublic import Mathlib.Algebra.Polynomial.Eval.Defs\npublic import Mathlib.Algebra.Ring.Semireal.Defs\npublic import Mathlib.Tactic.LinearCombination\n\nNamespace:\nIsRealClosed\n\nLocal context:\n/-\nCopyright (c) 2025 Artie Khovanov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Artie Khovanov\n-/\n/-!\n# Real Closed Field\n\nA field `R` is real closed if all of the following hold:\n1. `R` is real (that is, `-1` is not a sum of squares in `R`).\n2. for every `x` in `R`, one of `x` or `-x` is a square.\n3. every odd-degree polynomial over `R` has a root in `R`.\n\nA real closed field is an algebraic generalisation of the real numbers.\n\nIn this file we define real closed fields and prove some of their properties.\n\nTODO (Artie Khovanov) : equivalent conditions for a real field to be real closed\nTODO (Artie Khovanov) : real numbers, real algebraic numbers, hyperreals form a real closed field\n\n## Main Definitions\n\n- `IsRealClosed R` is the typeclass saying `R` is a real closed field.\n\n## Tags\n\nreal closed, rcf\n\n-/\n\npublic section\n\nopen Polynomial\n\n/--\nA field `R` is real closed if all of the following hold:\n1. `R` is real (that is, `-1` is not a sum of squares in `R`).\n2. for every `x` in `R`, one of `x` or `-x` is a square.\n3. every odd-degree polynomial over `R` has a root in `R`.\n-/\nclass IsRealClosed (R : Type*) [Field R] : Prop extends IsSemireal R where\n  isSquare_or_isSquare_neg (x : R) : IsSquare x ∨ IsSquare (-x)\n  exists_isRoot_of_odd_natDegree {f : R[X]} (hf : Odd f.natDegree) : ∃ x, f.IsRoot x\n\nattribute [aesop 90% forward] IsRealClosed.isSquare_or_isSquare_neg\n\nnamespace IsRealClosed\n\nuniverse u\n\nvariable {R : Type u} [Field R]\n\nTarget:\ntheorem of_linearOrderedField [LinearOrder R] [IsStrictOrderedRing R]\n    (isSquare_of_nonneg : ∀ {x : R}, 0 ≤ x → IsSquare x)\n    (exists_isRoot_of_odd_natDegree : ∀ {f : R[X]}, Odd f.natDegree → ∃ x, f.IsRoot x) :\n    IsRealClosed R where\n  isSquare_or_isSquare_neg {x} :=\n\nProof body:\n","rejected":"by\n  exact __oprover_missing_b9d038f0b319","rejection":{"accepted":false,"failure_type":"lean_elaboration_error","output_sha256":"b0e35e417854898447c4fafe8f7055e9110b898bd9f9b856d255df8dd4875df9","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"FieldTheory/IsRealClosed","family_id":"of_linearorderedfield","file_id":"mathlib/Mathlib/FieldTheory/IsRealClosed/Basic.lean","sample_id":"b9d038f0b31926725a67cabc50ad8dfeb3e914d41076f9b7c6d575a332e9b255"},"split":"train"}
{"binding":{"chosen_proof_sha256":"7d031507a20582e22ca4729c3a20ae04f876f5f252c02c99897df94087378755","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"eefa6d8a0659afef7ba5bbcd19f7c9f76ed0b03180e70dc321558069713aa686","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"623cf51d346b229ca13fed1da60cd6df5ba3c3809b92103ef6f67007cec38024","source_sha256":"b7ad320c59fef5b242b6e2dd8c96baf9488a66a587c50e32c8af235ebfc77c21","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  ext j\n  rw [assoc, assoc, enrichedComp_π, id_comp, tensorHom_def', assoc,\n    ← MonoidalCategory.whiskerLeft_comp_assoc, enrichedId_π,\n    whisker_exchange_assoc, MonoidalCategory.whiskerRight_id, assoc, assoc,\n    Iso.inv_hom_id_assoc]\n  dsimp\n  rw [e_comp_id, comp_id]","hard_negative":true,"metrics":{"chosen_tokens":47,"rejected_tokens":5,"token_jaccard":0.074074,"token_length_ratio":0.106383},"negative_category":"wrong_namespace_qualification","negative_mode":"wrong_namespace_qualification","pair_id":"fedb638c5be7d8eb5573f67caa2aa6e798f500d18a8f84ace0958ef16d5b661c","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.CategoryTheory.Monoidal.FunctorCategory\npublic import Mathlib.CategoryTheory.Enriched.Ordinary.Basic\npublic import Mathlib.CategoryTheory.Functor.Category\npublic import Mathlib.CategoryTheory.Limits.Shapes.End\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2024 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n/-!\n# Functor categories are enriched\n\nIf `C` is a `V`-enriched ordinary category, then `J ⥤ C` is also\nboth a `V`-enriched ordinary category and a `J ⥤ V`-enriched\nordinary category, provided `C` has suitable limits.\n\nWe first define the `V`-enriched structure on `J ⥤ C` by saying\nthat if `F₁` and `F₂` are in `J ⥤ C`, then `enrichedHom V F₁ F₂ : V`\nis a suitable limit involving `F₁.obj j ⟶[V] F₂.obj j` for all `j : C`.\nThe `J ⥤ V` object of morphisms `functorEnrichedHom V F₁ F₂ : J ⥤ V`\nis defined by sending `j : J` to the previously defined `enrichedHom`\nfor the \"restriction\" of `F₁` and `F₂` to the category `Under j`.\nThe definition `isLimitConeFunctorEnrichedHom` shows that\n`enriched V F₁ F₂` is the limit of the functor `functorEnrichedHom V F₁ F₂`.\n\n-/\n\nset_option backward.defeqAttrib.useBackward true\n\n@[expose] public section\n\nuniverse v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄\n\nnamespace CategoryTheory.Enriched.FunctorCategory\n\nopen Category MonoidalCategory Limits Functor\n\nvariable (V : Type u₁) [Category.{v₁} V] [MonoidalCategory V]\n  {C : Type u₂} [Category.{v₂} C] {J : Type u₃} [Category.{v₃} J]\n  {K : Type u₄} [Category.{v₄} K] [EnrichedOrdinaryCategory V C]\n\nvariable (F₁ F₂ F₃ F₄ : J ⥤ C)\n\n/-- Given two functors `F₁` and `F₂` from a category `J` to a `V`-enriched\nordinary category `C`, this is the diagram `Jᵒᵖ ⥤ J ⥤ V` whose end shall be\nthe `V`-morphisms in `J ⥤ V` from `F₁` to `F₂`. -/\n@[simps!]\ndef diagram : Jᵒᵖ ⥤ J ⥤ V := F₁.op ⋙ eHomFunctor V C ⋙ (whiskeringLeft J C V).obj F₂\n\n/-- The condition that the end `diagram V F₁ F₂` exists, see `enrichedHom`. -/\nabbrev HasEnrichedHom := HasEnd (diagram V F₁ F₂)\n\nsection\n\nvariable [HasEnrichedHom V F₁ F₂]\n\n/-- The `V`-enriched hom from `F₁` to `F₂` when `F₁` and `F₂` are functors `J ⥤ C`\nand `C` is a `V`-enriched category. -/\nnoncomputable abbrev enrichedHom : V := end_ (diagram V F₁ F₂)\n\n/-- The projection `enrichedHom V F₁ F₂ ⟶ F₁.obj j ⟶[V] F₂.obj j` in the category `V`\nfor any `j : J` when `F₁` and `F₂` are functors `J ⥤ C` and `C` is a `V`-enriched category. -/\nnoncomputable abbrev enrichedHomπ (j : J) : enrichedHom V F₁ F₂ ⟶ F₁.obj j ⟶[V] F₂.obj j :=\n  end_.π _ j\n\n@[reassoc]\nlemma enrichedHom_condition {i j : J} (f : i ⟶ j) :\n    enrichedHomπ V F₁ F₂ i ≫ eHomWhiskerLeft V (F₁.obj i) (F₂.map f) =\n    enrichedHomπ V F₁ F₂ j ≫ eHomWhiskerRight V (F₁.map f) (F₂.obj j) :=\n  end_.condition (diagram V F₁ F₂) f\n\n@[reassoc]\nlemma enrichedHom_condition' {i j : J} (f : i ⟶ j) :\n    enrichedHomπ V F₁ F₂ i ≫ (ρ_ _).inv ≫\n      _ ◁ (eHomEquiv V) (F₂.map f) ≫ eComp V _ _ _ =\n    enrichedHomπ V F₁ F₂ j ≫ (λ_ _).inv ≫\n      (eHomEquiv V) (F₁.map f) ▷ _ ≫ eComp V _ _ _ :=\n  end_.condition (diagram V F₁ F₂) f\n\nvariable {F₁ F₂}\n\nset_option backward.defeqAttrib.useBackward true in\nset_option backward.isDefEq.respectTransparency false in\n/-- Given functors `F₁` and `F₂` in `J ⥤ C`, where `C` is a `V`-enriched ordinary category,\nthis is the bijection `(F₁ ⟶ F₂) ≃ (𝟙_ V ⟶ enrichedHom V F₁ F₂)`. -/\nnoncomputable def homEquiv : (F₁ ⟶ F₂) ≃ (𝟙_ V ⟶ enrichedHom V F₁ F₂) where\n  toFun τ := end_.lift (fun j ↦ eHomEquiv V (τ.app j)) (fun i j f ↦ by\n    trans eHomEquiv V (τ.app i ≫ F₂.map f)\n    · dsimp\n      simp only [eHomEquiv_comp, tensorHom_def_assoc, MonoidalCategory.whiskerRight_id,\n        ← unitors_equal, assoc, Iso.inv_hom_id_assoc, eHomWhiskerLeft]\n    · dsimp\n      simp only [← NatTrans.naturality, eHomEquiv_comp, tensorHom_def', id_whiskerLeft,\n        assoc, Iso.inv_hom_id_assoc, eHomWhiskerRight])\n  invFun g :=\n    { app := fun j ↦ (eHomEquiv V).symm (g ≫ end_.π _ j)\n      naturality := fun i j f ↦ (eHomEquiv V).injective (by\n        simp only [eHomEquiv_comp, Equiv.apply_symm_apply, Iso.cancel_iso_inv_left]\n        conv_rhs =>\n          rw [tensorHom_def_assoc, MonoidalCategory.whiskerRight_id_assoc, assoc,\n            enrichedHom_condition' V F₁ F₂ f]\n        conv_lhs =>\n          rw [tensorHom_def'_assoc, MonoidalCategory.whiskerLeft_comp_assoc,\n            id_whiskerLeft_assoc, id_whiskerLeft_assoc, Iso.inv_hom_id_assoc, unitors_equal]) }\n  left_inv τ := by aesop\n  right_inv g := by aesop\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma homEquiv_apply_π (τ : F₁ ⟶ F₂) (j : J) :\n    homEquiv V τ ≫ enrichedHomπ V _ _ j = eHomEquiv V (τ.app j) := by\n  simp [homEquiv]\n\nend\n\nsection\n\nvariable [HasEnrichedHom V F₁ F₁]\n\n/-- The identity for the `V`-enrichment of the category `J ⥤ C`. -/\nnoncomputable def enrichedId : 𝟙_ V ⟶ enrichedHom V F₁ F₁ := homEquiv _ (𝟙 F₁)\n\n@[reassoc (attr := simp)]\nlemma enrichedId_π (j : J) : enrichedId V F₁ ≫ end_.π _ j = eId V (F₁.obj j) := by\n  simp [enrichedId]\n\n@[simp]\nlemma homEquiv_id : homEquiv V (𝟙 F₁) = enrichedId V F₁ := rfl\n\nend\n\nsection\n\nvariable [HasEnrichedHom V F₁ F₂] [HasEnrichedHom V F₂ F₃] [HasEnrichedHom V F₁ F₃]\n\n/-- The composition for the `V`-enrichment of the category `J ⥤ C`. -/\nnoncomputable def enrichedComp : enrichedHom V F₁ F₂ ⊗ enrichedHom V F₂ F₃ ⟶ enrichedHom V F₁ F₃ :=\n  end_.lift (fun j ↦ (end_.π _ j ⊗ₘ end_.π _ j) ≫ eComp V _ _ _) (fun i j f ↦ by\n    dsimp\n    trans (end_.π (diagram V F₁ F₂) i ⊗ₘ end_.π (diagram V F₂ F₃) j) ≫\n      (ρ_ _).inv ▷ _ ≫ (_ ◁ (eHomEquiv V (F₂.map f))) ▷ _ ≫ eComp V _ (F₂.obj i) _ ▷ _ ≫\n        eComp V _ (F₂.obj j) _\n    · have := end_.condition (diagram V F₂ F₃) f\n      dsimp [eHomWhiskerLeft, eHomWhiskerRight] at this ⊢\n      conv_lhs => rw [assoc, tensorHom_def_assoc]\n      conv_rhs =>\n        rw [tensorHom_def_assoc, whisker_assoc_assoc, e_assoc,\n          triangle_assoc_comp_right_inv_assoc, ← MonoidalCategory.whiskerLeft_comp_assoc,\n          ← MonoidalCategory.whiskerLeft_comp_assoc, ← MonoidalCategory.whiskerLeft_comp_assoc,\n          assoc, assoc, ← this, MonoidalCategory.whiskerLeft_comp_assoc,\n          MonoidalCategory.whiskerLeft_comp_assoc, MonoidalCategory.whiskerLeft_comp_assoc,\n          ← e_assoc, whiskerLeft_rightUnitor_inv_assoc, associator_inv_naturality_right_assoc,\n          Iso.hom_inv_id_assoc, whisker_exchange_assoc, MonoidalCategory.whiskerRight_id_assoc,\n          Iso.inv_hom_id_assoc]\n    · have := end_.condition (diagram V F₁ F₂) f\n      dsimp [eHomWhiskerLeft, eHomWhiskerRight] at this ⊢\n      conv_lhs =>\n        rw [tensorHom_def'_assoc, ← comp_whiskerRight_assoc,\n          ← comp_whiskerRight_assoc, ← comp_whiskerRight_assoc,\n          assoc, assoc, this, comp_whiskerRight_assoc, comp_whiskerRight_assoc,\n          comp_whiskerRight_assoc, leftUnitor_inv_whiskerRight_assoc,\n          ← associator_inv_naturality_left_assoc, ← e_assoc',\n          Iso.inv_hom_id_assoc, ← whisker_exchange_assoc, id_whiskerLeft_assoc,\n          Iso.inv_hom_id_assoc]\n      conv_rhs => rw [assoc, tensorHom_def'_assoc])\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma enrichedComp_π (j : J) :\n    enrichedComp V F₁ F₂ F₃ ≫ end_.π _ j =\n      (end_.π (diagram V F₁ F₂) j ⊗ₘ end_.π (diagram V F₂ F₃) j) ≫ eComp V _ _ _ := by\n  simp [enrichedComp]\n\nvariable {F₁ F₂ F₃}\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc]\nlemma homEquiv_comp (f : F₁ ⟶ F₂) (g : F₂ ⟶ F₃) :\n    (homEquiv V) (f ≫ g) = (λ_ (𝟙_ V)).inv ≫ ((homEquiv V) f ⊗ₘ (homEquiv V) g) ≫\n    enrichedComp V F₁ F₂ F₃ := by\n  ext j\n  simp only [homEquiv_apply_π, NatTrans.comp_app, eHomEquiv_comp, assoc,\n    enrichedComp_π, Functor.op_obj, tensorHom_comp_tensorHom_assoc]\n\nend\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\nlemma enriched_id_comp [HasEnrichedHom V F₁ F₁] [HasEnrichedHom V F₁ F₂] :\n    (λ_ (enrichedHom V F₁ F₂)).inv ≫ enrichedId V F₁ ▷ enrichedHom V F₁ F₂ ≫\n      enrichedComp V F₁ F₁ F₂ = 𝟙 _ := by\n  ext j\n  rw [assoc, assoc, enrichedComp_π, id_comp, tensorHom_def, assoc,\n    ← comp_whiskerRight_assoc, enrichedId_π, ← whisker_exchange_assoc,\n    id_whiskerLeft, assoc, assoc, Iso.inv_hom_id_assoc]\n  dsimp\n  rw [e_id_comp, comp_id]\n\nset_option backward.isDefEq.respectTransparency false in\n@[reassoc (attr := simp)]\n\nTarget:\nlemma enriched_comp_id [HasEnrichedHom V F₁ F₂] [HasEnrichedHom V F₂ F₂] :\n    (ρ_ (enrichedHom V F₁ F₂)).inv ≫ enrichedHom V F₁ F₂ ◁ enrichedId V F₂ ≫\n      enrichedComp V F₁ F₂ F₂ = 𝟙 _ :=\n\nProof body:\n","rejected":"by\n  exact OProverMissing.missing_623cf51d346b","rejection":{"accepted":false,"failure_type":"unknown_identifier","output_sha256":"d6bf0ee5819e999d5abbba29e9aedcb278bf30ca9caaae706151f9b794ff6e6d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"CategoryTheory/Enriched","family_id":"enriched_comp_id","file_id":"mathlib/Mathlib/CategoryTheory/Enriched/FunctorCategory.lean","sample_id":"623cf51d346b229ca13fed1da60cd6df5ba3c3809b92103ef6f67007cec38024"},"split":"train"}
{"binding":{"chosen_proof_sha256":"c311ad4b581caa0a3b895b5e4dcbcd3cc30063d829252ea41d678e32db406435","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"704efeeee592c3ee6480336bbb69c4f75dd795846a471cd26e230716ffdaed02","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"4edfa6f1dfc9284bddb186d5cd60c15b919148603ac348edfd11fc15fed97087","source_sha256":"f341bd031305e6d14f35c1e928ff55667e741d845f3190520037617945b7b186","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  tfae_have 1 ↔ 2 := Sheaf.isConstant_iff_isIso_counit_app _ _ _\n  tfae_have 1 ↔ 3 := ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩\n  tfae_have 1 ↔ 4 := Sheaf.isConstant_iff_mem_essImage _ LightProfinite.isTerminalPUnit adjunction X\n  tfae_have 1 ↔ 5 :=\n    have : functor.Faithful := inferInstance\n    have : functor.Full := inferInstance\n    -- These `have` statements above shouldn't be needed, but they are.\n    Sheaf.isConstant_iff_isIso_counit_app' _ LightProfinite.isTerminalPUnit adjunction X\n  tfae_have 6 → 4 := fun h ↦\n    mem_locallyConstant_essImage_of_isColimit_mapCocone X (fun S ↦ (h S).some)\n  tfae_have 4 → 6 := fun ⟨Y, ⟨i⟩⟩ S ↦\n    ⟨IsColimit.mapCoconeEquiv ((sheafToPresheaf _ _).mapIso i)\n      (LightCondensed.isColimitLocallyConstantPresheafDiagram Y S)⟩\n  tfae_finish","hard_negative":false,"metrics":{"chosen_tokens":149,"rejected_tokens":3,"token_jaccard":0.016667,"token_length_ratio":0.020134},"negative_category":"by_exact_placeholder","negative_mode":"exact?","pair_id":"ff0c30ecd470d8af41854c5ade7a55215e2ddc6f58bb84cf5a9e2546d4c4f5d2","pair_index":0,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Condensed.Discrete.Colimit\npublic import Mathlib.Condensed.Discrete.Module\n\nNamespace:\nLightCondSet\n\nLocal context:\n/-\nCopyright (c) 2024 Dagur Asgeirsson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Dagur Asgeirsson\n-/\n/-!\n\n# Characterizing discrete condensed sets and `R`-modules.\n\nThis file proves a characterization of discrete condensed sets, discrete condensed `R`-modules over\na ring `R`, discrete light condensed sets, and discrete light condensed `R`-modules over a ring `R`.\nsee `CondensedSet.isDiscrete_tfae`, `CondensedMod.isDiscrete_tfae`, `LightCondSet.isDiscrete_tfae`,\nand `LightCondMod.isDiscrete_tfae`.\n\nInformally, we can say: The following conditions characterize a condensed set `X` as discrete\n(`CondensedSet.isDiscrete_tfae`):\n\n1. There exists a set `X'` and an isomorphism `X ≅ cst X'`, where `cst X'` denotes the constant\n   sheaf on `X'`.\n2. The counit induces an isomorphism `cst X(*) ⟶ X`.\n3. There exists a set `X'` and an isomorphism `X ≅ LocallyConstant · X'`.\n4. The counit induces an isomorphism `LocallyConstant · X(*) ⟶ X`.\n5. For every profinite set `S = limᵢSᵢ`, the canonical map `colimᵢX(Sᵢ) ⟶ X(S)` is an isomorphism.\n\nThe analogues for light condensed sets, condensed `R`-modules over any ring, and light\ncondensed `R`-modules are nearly identical (`CondensedMod.isDiscrete_tfae`,\n`LightCondSet.isDiscrete_tfae`, and `LightCondMod.isDiscrete_tfae`).\n-/\n\npublic section\n\nuniverse u\n\nopen CategoryTheory Limits Functor FintypeCat\n\nnamespace Condensed\n\nvariable {C : Type*} [Category* C] [HasWeakSheafify (coherentTopology CompHaus.{u}) C]\n\n/--\nA condensed object is *discrete* if it is constant as a sheaf, i.e. isomorphic to a constant sheaf.\n-/\nabbrev IsDiscrete (X : Condensed.{u} C) := X.IsConstant (coherentTopology CompHaus)\n\nend Condensed\n\nnamespace CondensedSet\n\nopen CompHausLike.LocallyConstant\n\nlemma mem_locallyConstant_essImage_of_isColimit_mapCocone (X : CondensedSet.{u})\n    (h : ∀ S : Profinite.{u}, IsColimit <|\n      (profiniteToCompHaus.op ⋙ X.obj).mapCocone S.asLimitCone.op) :\n    CondensedSet.LocallyConstant.functor.essImage X := by\n  let e : CondensedSet.{u} ≌ Sheaf (coherentTopology Profinite) _ :=\n    (Condensed.ProfiniteCompHaus.equivalence (Type (u + 1))).symm\n  let i : (e.functor.obj X).obj ≅ (e.functor.obj (LocallyConstant.functor.obj _)).obj :=\n    Condensed.isoLocallyConstantOfIsColimit _ h\n  exact ⟨_, ⟨e.functor.preimageIso ((sheafToPresheaf _ _).preimageIso i.symm)⟩⟩\n\n/--\n`CondensedSet.LocallyConstant.functor` is left adjoint to the forgetful functor from condensed\nsets to sets.\n-/\nnoncomputable abbrev LocallyConstant.adjunction :\n    CondensedSet.LocallyConstant.functor ⊣ Condensed.underlying (Type (u + 1)) :=\n  CompHausLike.LocallyConstant.adjunction _ _\n\nopen Condensed\n\nopen CondensedSet.LocallyConstant List in\ntheorem isDiscrete_tfae (X : CondensedSet.{u}) :\n    TFAE\n    [ X.IsDiscrete\n    , IsIso ((Condensed.discreteUnderlyingAdj _).counit.app X)\n    , (Condensed.discrete _).essImage X\n    , CondensedSet.LocallyConstant.functor.essImage X\n    , IsIso (CondensedSet.LocallyConstant.adjunction.counit.app X)\n    , Sheaf.IsConstant (coherentTopology Profinite)\n        ((Condensed.ProfiniteCompHaus.equivalence _).inverse.obj X)\n    , ∀ S : Profinite.{u}, Nonempty\n        (IsColimit <| (profiniteToCompHaus.op ⋙ X.obj).mapCocone S.asLimitCone.op)\n    ] := by\n  tfae_have 1 ↔ 2 := Sheaf.isConstant_iff_isIso_counit_app _ _ _\n  tfae_have 1 ↔ 3 := ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩\n  tfae_have 1 ↔ 4 := Sheaf.isConstant_iff_mem_essImage _ CompHaus.isTerminalPUnit adjunction _\n  tfae_have 1 ↔ 5 :=\n    have : functor.Faithful := inferInstance\n    have : functor.Full := inferInstance\n    -- These `have` statements above shouldn't be needed, but they are.\n    Sheaf.isConstant_iff_isIso_counit_app' _ CompHaus.isTerminalPUnit adjunction _\n  tfae_have 1 ↔ 6 :=\n    (Sheaf.isConstant_iff_of_equivalence (coherentTopology Profinite)\n      (coherentTopology CompHaus) profiniteToCompHaus Profinite.isTerminalPUnit\n      CompHaus.isTerminalPUnit _).symm\n  tfae_have 7 → 4 := fun h ↦\n    mem_locallyConstant_essImage_of_isColimit_mapCocone X (fun S ↦ (h S).some)\n  tfae_have 4 → 7 := fun ⟨Y, ⟨i⟩⟩ S ↦\n    ⟨IsColimit.mapCoconeEquiv (isoWhiskerLeft profiniteToCompHaus.op\n      ((sheafToPresheaf _ _).mapIso i))\n      (Condensed.isColimitLocallyConstantPresheafDiagram Y S)⟩\n  tfae_finish\n\nend CondensedSet\n\nnamespace CondensedMod\n\nvariable (R : Type (u + 1)) [Ring R]\n\nlemma isDiscrete_iff_isDiscrete_forget (M : CondensedMod R) :\n    M.IsDiscrete ↔ ((Condensed.forget R).obj M).IsDiscrete :=\n  Sheaf.isConstant_iff_forget (coherentTopology CompHaus)\n    (forget (ModuleCat R)) M CompHaus.isTerminalPUnit\n\ninstance : HasLimitsOfSize.{u, u + 1} (ModuleCat.{u + 1} R) :=\n  hasLimitsOfSizeShrink.{u, u + 1, u + 1, u + 1} _\n\nopen CondensedMod.LocallyConstant List in\ntheorem isDiscrete_tfae (M : CondensedMod.{u} R) :\n    TFAE\n    [ M.IsDiscrete\n    , IsIso ((Condensed.discreteUnderlyingAdj _).counit.app M)\n    , (Condensed.discrete _).essImage M\n    , (CondensedMod.LocallyConstant.functor R).essImage M\n    , IsIso ((CondensedMod.LocallyConstant.adjunction R).counit.app M)\n    , Sheaf.IsConstant (coherentTopology Profinite)\n        ((Condensed.ProfiniteCompHaus.equivalence _).inverse.obj M)\n    , ∀ S : Profinite.{u}, Nonempty\n        (IsColimit <| (profiniteToCompHaus.op ⋙ M.obj).mapCocone S.asLimitCone.op)\n    ] := by\n  tfae_have 1 ↔ 2 := Sheaf.isConstant_iff_isIso_counit_app _ _ _\n  tfae_have 1 ↔ 3 := ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩\n  tfae_have 1 ↔ 4 := Sheaf.isConstant_iff_mem_essImage _ CompHaus.isTerminalPUnit (adjunction R) _\n  tfae_have 1 ↔ 5 :=\n    have : (functor R).Faithful := inferInstance\n    have : (functor R).Full := inferInstance\n    -- These `have` statements above shouldn't be needed, but they are.\n    Sheaf.isConstant_iff_isIso_counit_app' _ CompHaus.isTerminalPUnit (adjunction R) _\n  tfae_have 1 ↔ 6 :=\n    (Sheaf.isConstant_iff_of_equivalence (coherentTopology Profinite)\n      (coherentTopology CompHaus) profiniteToCompHaus Profinite.isTerminalPUnit\n      CompHaus.isTerminalPUnit _).symm\n  tfae_have 7 → 1 := by\n    intro h\n    rw [isDiscrete_iff_isDiscrete_forget, ((CondensedSet.isDiscrete_tfae _).out 0 6 :)]\n    intro S\n    letI : PreservesFilteredColimitsOfSize.{u, u} (forget (ModuleCat R)) :=\n      preservesFilteredColimitsOfSize_shrink.{u, u + 1, u, u + 1} _\n    exact ⟨isColimitOfPreserves (forget (ModuleCat R)) (h S).some⟩\n  tfae_have 1 → 7 := by\n    intro h S\n    rw [isDiscrete_iff_isDiscrete_forget, ((CondensedSet.isDiscrete_tfae _).out 0 6 :)] at h\n    letI : ReflectsFilteredColimitsOfSize.{u, u} (forget (ModuleCat R)) :=\n      reflectsFilteredColimitsOfSize_shrink.{u, u + 1, u, u + 1} _\n    exact ⟨isColimitOfReflects (forget (ModuleCat R)) (h S).some⟩\n  tfae_finish\n\nend CondensedMod\n\nnamespace LightCondensed\n\nvariable {C : Type*} [Category* C] [HasWeakSheafify (coherentTopology LightProfinite.{u}) C]\n\n/--\nA light condensed object is *discrete* if it is constant as a sheaf, i.e. isomorphic to a constant\nsheaf.\n-/\nabbrev IsDiscrete (X : LightCondensed.{u} C) := X.IsConstant (coherentTopology LightProfinite)\n\nend LightCondensed\n\nnamespace LightCondSet\n\nlemma mem_locallyConstant_essImage_of_isColimit_mapCocone (X : LightCondSet.{u})\n    (h : ∀ S : LightProfinite.{u}, IsColimit <|\n      X.obj.mapCocone (coconeRightOpOfCone S.asLimitCone)) :\n    LightCondSet.LocallyConstant.functor.essImage X := by\n  let i : X.obj ≅ (LightCondSet.LocallyConstant.functor.obj _).obj :=\n    LightCondensed.isoLocallyConstantOfIsColimit _ h\n  exact ⟨_, ⟨((sheafToPresheaf _ _).preimageIso i.symm)⟩⟩\n\n/--\n`LightCondSet.LocallyConstant.functor` is left adjoint to the forgetful functor from light condensed\nsets to sets.\n-/\nnoncomputable abbrev LocallyConstant.adjunction :\n    LightCondSet.LocallyConstant.functor ⊣ LightCondensed.underlying (Type u) :=\n  CompHausLike.LocallyConstant.adjunction _ _\n\nopen LightCondSet.LocallyConstant List in\n\nTarget:\ntheorem isDiscrete_tfae (X : LightCondSet.{u}) :\n    TFAE\n    [ X.IsDiscrete\n    , IsIso ((LightCondensed.discreteUnderlyingAdj _).counit.app X)\n    , (LightCondensed.discrete _).essImage X\n    , LightCondSet.LocallyConstant.functor.essImage X\n    , IsIso (LightCondSet.LocallyConstant.adjunction.counit.app X)\n    , ∀ S : LightProfinite.{u}, Nonempty\n        (IsColimit <| X.obj.mapCocone (coconeRightOpOfCone S.asLimitCone))\n    ] :=\n\nProof body:\n","rejected":"by\n  exact?","rejection":{"accepted":false,"all_failure_types":["proof_hole_or_search_placeholder"],"failure_type":"proof_hole_or_search_placeholder","rejection_layer":"strict_parser"},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Condensed/Discrete","family_id":"isdiscrete_tfae","file_id":"mathlib/Mathlib/Condensed/Discrete/Characterization.lean","sample_id":"4edfa6f1dfc9284bddb186d5cd60c15b919148603ac348edfd11fc15fed97087"},"split":"train"}
{"binding":{"chosen_proof_sha256":"3f320a505bb6b269ad39068f91895095fa32692c32719fd89a78bd89e6e3c623","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"0d002db9cc8388d0bb048c967b6cc0ba1b001333a19addabc857b0bfa660e31e","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"308f6cf4c900ea2a1076236867cd7cbcb72edd4a9f392f88b684001d707ae5c6","source_sha256":"220aaaefdf1c03061afd6055e7419dc5fe5b656993ba2d93b412d09922822f9e","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  classical\n  rw [LinearMap.trace_eq_matrix_trace 𝕜 b.toBasis T]\n  apply Fintype.sum_congr\n  intro i\n  rw [Matrix.diag_apply, T.toMatrix_apply, b.coe_toBasis, b.coe_toBasis_repr_apply,\n    b.repr_apply_apply]\n\nvariable [FiniteDimensional 𝕜 E]\nvariable {n : ℕ} (hn : Module.finrank 𝕜 E = n)","hard_negative":true,"metrics":{"chosen_tokens":64,"rejected_tokens":8,"token_jaccard":0.046512,"token_length_ratio":0.125},"negative_category":"unsolved_goals","negative_mode":"unsolved_goals","pair_id":"ff36e99145c397b086e6dcc40beee2c688d1383b4ff6365cf6d05c579689caad","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Analysis.InnerProductSpace.PiL2\npublic import Mathlib.Analysis.InnerProductSpace.Spectrum\npublic import Mathlib.LinearAlgebra.Eigenspace.Charpoly\n\nNamespace:\nLinearMap\n\nLocal context:\n/-\nCopyright (c) 2025 Iván Renison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Iván Renison\n-/\n/-!\n# Traces in inner product spaces\n\nThis file contains various results about traces of linear operators in inner product spaces.\n-/\n\npublic section\n\nnamespace LinearMap\n\nvariable {𝕜 E ι : Type*} [RCLike 𝕜] [Fintype ι]\nvariable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]\n\nopen scoped InnerProductSpace\n\nTarget:\nlemma trace_eq_sum_inner (T : E →ₗ[𝕜] E) (b : OrthonormalBasis ι 𝕜 E) :\n    T.trace 𝕜 E = ∑ i, ⟪b i, T (b i)⟫_𝕜 :=\n\nProof body:\n","rejected":"by\n  have __oprover_safe_h : True := by trivial","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"aaa10d9e2e18985a11b384b2b89e78e9eb706e3ec6647e65a0f70c4870acdb74","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"Analysis/InnerProductSpace","family_id":"trace_eq_sum_inner","file_id":"mathlib/Mathlib/Analysis/InnerProductSpace/Trace.lean","sample_id":"308f6cf4c900ea2a1076236867cd7cbcb72edd4a9f392f88b684001d707ae5c6"},"split":"train"}
{"binding":{"chosen_proof_sha256":"956d0d39593774896edc51480ad38b07e1edca7a7a86a5d842d9b626672e0346","mathlib_revision":"360da6fa66c1273b76b6b2d8c5666fd5ac2e3b56","negative_generator_hash":"8285991fca7eded1551939aba257955b2916c5e9006a0b0c6c621296d1e6c2d9","rejected_proof_sha256":"06ed38f6debd03daf214edd7588ee3722cd943639cd891e6a8c134ce91dfab71","source_artifact_hash":"5ebec3f7ed71d0af202924ea9d5ca12a857c5363b89b028dbf74268bf2e32f24","source_sample_id":"a21a121517cc485ec8b0fac3320d37a95ffa767d8d52dbb2b3c8baa9ed0c81a3","source_sha256":"d2377d09b51f8862ecd1e9604b616865bc29923d42ea22ca4ee787791f41d5e7","toolchain":"leanprover/lean4:v4.32.0-rc1"},"chosen":"by\n  rw [isNoetherian_iff', wellFoundedGT_iff_monotone_chain_condition]\n\nvariable [IsNoetherian R M]","hard_negative":true,"metrics":{"chosen_tokens":13,"rejected_tokens":2,"token_jaccard":0.083333,"token_length_ratio":0.153846},"negative_category":"invalid_tactic","negative_mode":"invalid_tactic","pair_id":"ffa3fe7d3fdde395aa3cbd116fee881dfc77f13916abfbdf23ab6e53f7a1044b","pair_index":1,"prompt":"Return exactly one Lean proof body beginning with `by`. Do not emit prose, fences, declarations, sorry, admit, by?, or warnings.\n\nImports:\npublic import Mathlib.Order.Filter.AtTopBot.Basic\npublic import Mathlib.RingTheory.Finiteness.Basic\n\nNamespace:\n(root)\n\nLocal context:\n/-\nCopyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Kevin Buzzard\n-/\n/-!\n# Noetherian rings and modules\n\nThe following are equivalent for a module M over a ring R:\n1. Every increasing chain of submodules M₁ ⊆ M₂ ⊆ M₃ ⊆ ⋯ eventually stabilises.\n2. Every submodule is finitely generated.\n\nA module satisfying these equivalent conditions is said to be a *Noetherian* R-module.\nA ring is a *Noetherian ring* if it is Noetherian as a module over itself.\n\n(Note that we do not assume yet that our rings are commutative,\nso perhaps this should be called \"left-Noetherian\".\nTo avoid cumbersome names once we specialize to the commutative case,\nwe don't make this explicit in the declaration names.)\n\n## Main definitions\n\nLet `R` be a ring and let `M` and `P` be `R`-modules. Let `N` be an `R`-submodule of `M`.\n\n* `IsNoetherian R M` is the proposition that `M` is a Noetherian `R`-module. It is a class,\n  implemented as the predicate that all `R`-submodules of `M` are finitely generated.\n\n## Main statements\n\n* `isNoetherian_iff` is the theorem that an R-module M is Noetherian iff `>` is well-founded on\n  `Submodule R M`.\n\nNote that the Hilbert basis theorem, that if a commutative ring R is Noetherian then so is R[X],\nis proved in `RingTheory.Polynomial`.\n\n## References\n\n* [M. F. Atiyah and I. G. Macdonald, *Introduction to commutative algebra*][atiyah-macdonald]\n* [P. Samuel, *Algebraic Theory of Numbers*][samuel1967]\n\n## Tags\n\nNoetherian, noetherian, Noetherian ring, Noetherian module, noetherian ring, noetherian module\n\n-/\n\n@[expose] public section\n\nassert_not_exists Finsupp.linearCombination Matrix Pi.basis\n\nopen Set Pointwise\n\n/-- `IsNoetherian R M` is the proposition that `M` is a Noetherian `R`-module,\nimplemented as the predicate that all `R`-submodules of `M` are finitely generated.\n-/\n-- TODO: should this be renamed to `Noetherian`?\nclass IsNoetherian (R M) [Semiring R] [AddCommMonoid M] [Module R M] : Prop where\n  noetherian : ∀ s : Submodule R M, s.FG\n\nattribute [inherit_doc IsNoetherian] IsNoetherian.noetherian\n\nsection\n\nvariable {R : Type*} {M : Type*} {P : Type*}\nvariable [Semiring R] [AddCommMonoid M] [AddCommMonoid P]\nvariable [Module R M] [Module R P]\n\nopen IsNoetherian\n\n/-- An R-module is Noetherian iff all its submodules are finitely-generated. -/\ntheorem isNoetherian_def : IsNoetherian R M ↔ ∀ s : Submodule R M, s.FG :=\n  ⟨fun h => h.noetherian, IsNoetherian.mk⟩\n\ntheorem isNoetherian_submodule {N : Submodule R M} :\n    IsNoetherian R N ↔ ∀ s : Submodule R M, s ≤ N → s.FG := by\n  refine ⟨fun ⟨hn⟩ => fun s hs =>\n    have : s ≤ LinearMap.range N.subtype := N.range_subtype.symm ▸ hs\n    Submodule.map_comap_eq_self this ▸ (hn _).map _,\n    fun h => ⟨fun s => ?_⟩⟩\n  specialize h (s.map N.subtype) (Submodule.map_subtype_le N s)\n  exact Submodule.fg_of_fg_map_injective N.subtype Subtype.val_injective h\n\ntheorem isNoetherian_submodule_left {N : Submodule R M} :\n    IsNoetherian R N ↔ ∀ s : Submodule R M, (N ⊓ s).FG :=\n  isNoetherian_submodule.trans ⟨fun H _ => H _ inf_le_left, fun H _ hs => inf_of_le_right hs ▸ H _⟩\n\ntheorem isNoetherian_submodule_right {N : Submodule R M} :\n    IsNoetherian R N ↔ ∀ s : Submodule R M, (s ⊓ N).FG :=\n  isNoetherian_submodule.trans ⟨fun H _ => H _ inf_le_right, fun H _ hs => inf_of_le_left hs ▸ H _⟩\n\ninstance isNoetherian_submodule' [IsNoetherian R M] (N : Submodule R M) : IsNoetherian R N :=\n  isNoetherian_submodule.2 fun _ _ => IsNoetherian.noetherian _\n\ntheorem isNoetherian_of_le {s t : Submodule R M} [ht : IsNoetherian R t] (h : s ≤ t) :\n    IsNoetherian R s :=\n  isNoetherian_submodule.mpr fun _ hs' => isNoetherian_submodule.mp ht _ (le_trans hs' h)\n\nend\n\nopen IsNoetherian Submodule Function\n\nsection\n\nuniverse w\n\nvariable {R M P : Type*} {N : Type w} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N]\n  [Module R N] [AddCommMonoid P] [Module R P]\n\ntheorem isNoetherian_iff' : IsNoetherian R M ↔ WellFoundedGT (Submodule R M) := by\n  refine .trans ?_ ((CompleteLattice.wellFoundedGT_characterisations <| Submodule R M).out 0 3).symm\n  exact\n    ⟨fun ⟨h⟩ => fun k => (fg_iff_compact k).mp (h k), fun h =>\n      ⟨fun k => (fg_iff_compact k).mpr (h k)⟩⟩\n\ntheorem isNoetherian_iff :\n    IsNoetherian R M ↔ WellFounded ((· > ·) : Submodule R M → Submodule R M → Prop) := by\n  rw [isNoetherian_iff', ← isWellFounded_iff]\n\nalias ⟨IsNoetherian.wf, _⟩ := isNoetherian_iff\n\nalias ⟨IsNoetherian.wellFoundedGT, isNoetherian_mk⟩ := isNoetherian_iff'\n\ninstance wellFoundedGT [h : IsNoetherian R M] : WellFoundedGT (Submodule R M) :=\n  h.wellFoundedGT\n\ntheorem isNoetherian_iff_fg_wellFounded :\n    IsNoetherian R M ↔ WellFoundedGT { N : Submodule R M // N.FG } := by\n  let α := { N : Submodule R M // N.FG }\n  constructor\n  · intro H\n    let f : α ↪o Submodule R M := OrderEmbedding.subtype _\n    exact OrderEmbedding.wellFoundedLT f.dual\n  · intro H\n    constructor\n    intro N\n    obtain ⟨⟨N₀, h₁⟩, e : N₀ ≤ N, h₂⟩ :=\n      WellFounded.has_min H.wf { N' : α | N'.1 ≤ N } ⟨⟨⊥, Submodule.fg_bot⟩, @bot_le _ _ _ N⟩\n    convert! h₁\n    refine (e.antisymm ?_).symm\n    by_contra h₃\n    obtain ⟨x, hx₁ : x ∈ N, hx₂ : x ∉ N₀⟩ := Set.not_subset.mp h₃\n    apply hx₂\n    rw [eq_of_le_of_not_lt (le_sup_right : N₀ ≤ _) (h₂\n      ⟨_, Submodule.FG.sup ⟨{x}, by rw [Finset.coe_singleton]⟩ h₁⟩ <|\n      sup_le ((Submodule.span_singleton_le_iff_mem _ _).mpr hx₁) e)]\n    exact (le_sup_left : R ∙ x ≤ _) (Submodule.mem_span_singleton_self _)\n\n/-- A module is Noetherian iff every nonempty set of submodules has a maximal submodule among them.\n-/\ntheorem set_has_maximal_iff_noetherian :\n    (∀ a : Set <| Submodule R M, a.Nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬M' < I) ↔ IsNoetherian R M := by\n  rw [isNoetherian_iff, WellFounded.wellFounded_iff_has_min]\n\n/-- A module is Noetherian iff every increasing chain of submodules stabilizes. -/\n\nTarget:\ntheorem monotone_stabilizes_iff_noetherian :\n    (∀ f : ℕ →o Submodule R M, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsNoetherian R M :=\n\nProof body:\n","rejected":"by\n  __oprover_invalid_tactic_a21a121517cc","rejection":{"accepted":false,"failure_type":"unsolved_goals","output_sha256":"fde5879b0007d8a93a1b9953ea86fc4366956237ea64b4d16897ba203885227d","rejection_layer":"lean","returncode":1,"timed_out":false,"warning":false},"safe_lean_text_only":true,"schema_version":1,"source":{"dependency_family_id":"RingTheory/Noetherian","family_id":"monotone_stabilizes_iff_noetherian","file_id":"mathlib/Mathlib/RingTheory/Noetherian/Defs.lean","sample_id":"a21a121517cc485ec8b0fac3320d37a95ffa767d8d52dbb2b3c8baa9ed0c81a3"},"split":"train"}